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Horosphere slab separation theorems in manifolds without conjugate points
Sameh Shenawy ORCID: orcid.org/0000-0003-3548-42391
Let \(\mathcal {W}^{n}\) be the set of smooth complete simply connected n-dimensional manifolds without conjugate points. The Euclidean space and the hyperbolic space are examples of these manifolds. Let \(W\in \mathcal {W}^{n}\) and let A and B be two convex subsets of W. This note aims to investigate separation and slab horosphere separation of A and B. For example,sufficient conditions on A and B to be separated by a slab of horospheres are obtained. Existence and uniqueness of foot points and farthest points of a convex set A in \(W\in \mathcal {W}\) are considered.
Let ω be a unit tangent vector to a smooth complete simply connected manifold without conjugate points \(W\in \mathcal {W}^{n}\) at a point p∈W. Let α be the unique geodesic with tangent ω=α′(0) and p=α(0). The Busemann function \(b_{\omega }:W\in \mathcal {W}^{n}\rightarrow \mathbb {R} \) is defined by
$$ b_{\omega }\left(x\right) ={\lim}_{t\rightarrow \infty }\left[ t-d\left(x,\alpha \left(t\right) \right) \right], $$
where d is the distance function. The right hand side is well-defined and the Busemann function bω is smooth in a complete simply connected manifold without conjugate point \(W\in \mathcal {W}^{n}\) whereas bω is at least C2 given that W has no focal points(see [1, Theorem 2]). The level set of a Busemann function, that is \(b_{\omega }^{-1}\left (0\right) \), is called a horosphere Hω(p) where p=α(0). Likewise, the open and the closed horoballs in \(W\in \mathcal {W}^{n}\) are defined as the sets \(D_{\omega }\left (p\right) =b_{\omega }^{-1}\left (\left (0,\infty \right) \right) \) and \(\bar {D}_{\omega }\left (p\right) =b_{\omega }^{-1}\left (\left [ 0,\infty \right) \right) \) respectively. Let α(t) be the geodesic passing through a point \(p\in W\in \mathcal {W}^{n}\) with α′(0)=ω. It is well-known that the horosphere Hω(p) is the limit of the geodesic spheres S(α(t),t) passing through p=α(0) and having center α(t) as t→∞. The horospheres Hu(p),u=α′(0),p=α(0) and Hv(q)v=α′(a),q=α(a) are called co-directional or parallel horospheres and parallel horospheres touch each other at infinity. Notice that the horopsheres Hω(p) and H−ω(p) have p as their unique common point; otherwise, they coincide. Hyperplanes are horospheres in the Euclidean space En. The horosphere Hω(p) with a given direction ω and a given point p is unique. Finally, the horospheres, as a level surfaces of a Busemann function, are equidistant family of surfaces whose orthogonal trajectories are geodesics.
It is noted that, thanks to the well-known Hopf-Rinow theorem, there is a length minimizing geodesic segment joining each pair of points in a complete connected Riemannian manifold W. If, in addition, W is simply connected and has no conjugate points, then the exponential map is a covering map and each pair of points is joined by a unique and hence minimal geodesic(see Section 10.7 of [2]). Finally, a set A of \(W\in \mathcal {W} ^{n}\) is compact if and only if it is closed and bounded. All manifolds with negative curvature are members of \(\mathcal {W}^{n}\). For example, the hyperbolic Poincare upper half-plane model
$$H^{2}=\{(x,y)\in \mathbb{R}^{2}:y>0\}, $$
equipped with the metric g11=g22=y−2 and g12=0 lies in \(\mathcal {W}^{2}\) (see [3] for more details); however, the unit sphere S2 does not lie in \(\mathcal {W}^{2}\) since all antipodal points are conjugate points.
A subset A of \(W\in \mathcal {W}^{n}\) is convex if the geodesic segment [pq] joining any two points p,q∈A lies in A. Three different definitions of convex sets in general Riemannian manifolds were studied in [4]. The whole manifold W geodesics are all convex sets. Also, open and closed geodesic balls of manifolds with negative curvature are convex sets. On the other hand, the union of two different geodesics is not convex and the complement of a convex set is not necessarily convex. Note that the existence and uniqueness of geodesic segments in these manifolds is trivial; however, for example, the whole sphere Sn is not convex since antipodal points have many minimal geodesic segments joining them. Convex functions are also deeply studied in Riemannian geometry (the reader is referred to [5] for a detailed study of convex functions on manifolds with negative curvature).
Let p be a point in a complete simply connected manifold without conjugate point \( W\in \mathcal {W}^{n}\). The point p has a foot point f in subset A of W if the distance function \(l:A\rightarrow \mathbb {R}\) defined by l(x)=d(p,x),x∈A attains its minimum at f. The point p is said to have a farthest point F in A if the function l attains its maximum at F [6, 7]. The geodesic ray starting at p and passing through q is denoted by R(pq), and the entire geodesic passing through them is denoted by G(pq).
Convex sets, foot, and farthest points play a very important role in both convex analysis and optimization (see for example [8–10] and references therein). Generalizations and extensions of convex sets and their separation and supporting surfaces are of particular interest [11, 12]. Each pair of points in a simply connected smooth Riemannian manifold without conjugate points has a unique and hence minimal geodesic joining them whereas manifolds without focal points has convex geodesic spheres [13–17]. It is well-known that the class of complete simply connected manifolds without focal points is a proper subclass of \(\mathcal {W}^{n}\). Manifolds with non-positive sectional curvatures have no focal points [18–22]. Horospheres and totally geodesic hypersurfaces in \(W\in \mathcal {W}^{n}\) play a significant role in defining both supporting and separation theorems for convex sets.
In this note, the concepts of separation and horosphere slab separation of convex sets are studied in \(W\in \mathcal {W}^{n}\). Sufficient conditions for two disjoint closed convex sets to be separated by a slab of horosheres are given. Foot and farthest points of a convex set in \(W\in \mathcal {W}^{n}\) are considered.
Foot and farthest points of a convex set A in \(W\in \mathcal {W} ^{n}\)
This section is devoted to the study of foot and farthest points of a convex set A in \(W\in \mathcal {W}^{n}\). The geodesic sphere with the center at p and radius r is denoted by S(p,r) and the corresponding open and closed geodesic balls are denoted by B(p,r) and \(\bar {B}\left (p,r\right) \).
Let us begin with the following simple but important result.
In a complete simply connected Riemannian manifold without conjugate points \( W\in \mathcal {W}^{n},\) the following statements are true.
If α is the unique geodesic parameterized by arc length with α(0)=x, α′(0)=ω, and α(r)=p, then the geodesic segment [xy] intersects B(p,r) for any y∈Dω(x).
Let B(p,r) be a geodesic ball with the center at p and radius r, then any point x≠p has a foot point f=S(p,r)∩R(px) in S(p,r).
Theorem 1
Let \(W\in \mathcal {W}^{n}\) be a manifold without focal points and A be a non-empty closed convex subset of W. Then, each point p of W has a unique foot point.
Since W has no focal points, the geodesic ball \(\bar {B}\left (p,r\right) \) is convex and hence \(A\cap \bar {B}\left (p,r\right) \) is either convex or empty. The result follows easily if p∈A. So, assume that p∉A. Let q be in A and r=d(p,q). It is clear that \(G=A\cap \bar {B}\left (p,r\right) \) is a closed no-empty convex subset of \(\bar {B} \left (p,r\right) \). Then, G is compact. Define the real-valued function f(x)=d(p,x) on G. f is continuous function and consequently attains its minimum at a point f in A. To show that f is unique, assume that p has two foot points f1 and f2 in A. Then, the closed ball \(\bar {B}\left (p,r\right) \) touches A twice where d(p,f1)=d(p,f2)=r (see Fig. 1). The open segment (f1f2) is contained in G since G is convex, and so both f1 and f2 are not foot points of p. This contradiction shows that f is unique(see Fig. 1). □
Uniqueness of foot points in a convex set
Corollary 1
Let \(W\in \mathcal {W}^{n}\) be a manifold without focal points and A be a non-empty closed convex subset of W and let p∉A. If f is a foot point of the point p, then f is the unique foot point of any q∈(pf) in A. Likewise, if p has a farthest point F in A from and p∈(Fq) for some point q, then q has F as its unique farthest point in A.
The following theorems represent two analogous results to the above ones.
Let \(W\in \mathcal {W}^{2}\) and A be a non-empty convex subset of W. If p∈W∖A has a foot point f in A, then f is a foot point in A for every point of R(fp).
There is a geodesic γ supporting A at f since A is convex. Let α be the unique geodesic with f=α(0) and p=α(r). Let Hv(f) be the horosphere with v=α′(0). The closed ball \(\bar {B}\left (q,l\right) \), l=d(q,f) is contained in \(\bar {D}_{v}\) for any point q∈R(fp). Thus, f is the unique foot point of q in A (see Fig. 2). □
Supporting at foot points
Let \(W\in \mathcal {W}^{n}\) and A be a non-empty compact subset of W. Then, every point p∉A has a farthest point in A.
The function \(l:A\rightarrow \mathbb {R} \) defined on A by l(x)=d(p,x) for every x∈A is a real-valued continuous function. Since A is compact, l attains its maximum at a point in A say F. Thus, F is the farthest from p in A. □
Separation of convex sets in \(W\in \mathcal {W}^{n}\)
Separation of two convex sets in the Euclidean space En is widely used in optimization. The most well-known separation theorem says that any two non-empty disjoint convex sets in the Euclidean space are separated by a hyperplane. There are more restrictive separation theorems for different types of convex and non-convex sets.
In Riemannian geometry, it is natural to ask the following question. What is the best candidate for a hyperplane in separation theorems? Horospheres in complete simply Riemannian manifolds without conjugate points play this significant role in separation of two convex sets.
A slab in the Euclidean space is the region bounded by two parallel hyperplanes. Here, a slab of horospheres along a geodesic α is the region bounded by \(H_{\alpha ^{\prime }\left (0\right) }\) and \(H_{\alpha ^{\prime }\left (r\right) }\). It is denoted by Sα[0,r] (see Fig. 3). Two sets A and B are said to be separated by a slab of horospheres if there is a geodesic α such that A and B lie in two different sides Sα[0,r].
A slab
Let \(W\in \mathcal {W}^{n}\) be a complete simply connected Riemannian manifold without conjugate points and A, B be two non-empty disjoint convex subsets of W. A and B are separated by a slab of horospheres if one of them is bounded and the second is supported by a horosphere at every boundary point.
Assume that A is bounded and B has a supporting horosphere at each point p∈∂A. The real-valued function l defined on A by l(x)=d(x,B) for every x∈A. l attains its minimum value at a point p∈A since A is compact and l is continuous. The point p has a foot point f∈B where p≠f. B has a supporting horosphere \(H_{\alpha ^{\prime }\left (r\right) }\left (q\right) \) where α be the unique geodesic with α(0)=q and α(r)=p. ince B is convex and hence supported by a totally geodesic hypersurface at q that separates B and \(H_{\alpha ^{\prime }\left (0\right) }\). Now, the slab Sα[0,r] separates A and B (see Fig. 4). □
Slab separation
Let \(W\in \mathcal {W}^{n}\) and A, B be two non-empty disjoint convex subsets of W. A and B are strictly separated if one of them is compact and the other one is closed.
The Euclidean version of the above theorem is as follows (see Theorem 7.6 in [12]). Note that the term convex is replaced by the term compact.
Let A, B be two non-empty disjoint compact subsets of the Euclidean space En. A and B are strictly separated by a slab if and only if for each set T of n+2 or fewer points of A∪B the sets A∩T and B∩T are separated by a slab.
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The author would like to thank the referee(s) for the insightful comments that improve the paper quality.
Department of Mathematics, Modern Academy, Maadi, Cairo, Egypt
Sameh Shenawy
The author read and approved the final manuscript.
Correspondence to Sameh Shenawy.
The author declares that he has no competing interests.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License(http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Shenawy, S. Horosphere slab separation theorems in manifolds without conjugate points. J Egypt Math Soc 27, 35 (2019). https://doi.org/10.1186/s42787-019-0038-5
Horosphere separation
Slab horosphere separation
Manifolds without conjugate points
Separation of convex sets
AMS Subject Classification
Primary; 52A10
52A20; Secondary 52A55 | CommonCrawl |
\begin{definition}[Definition:Scalar/R-Algebraic Structure]
Let $\struct {S, *_1, *_2, \ldots, *_n, \circ}_R$ be an $R$-algebraic structure with $n$ operations, where:
:$\struct {S, *_1, *_2, \ldots, *_n}$ is an algebraic structure with $n$ operations
:$\struct {R, +_R, \times_R}$ is the scalar ring of $\struct {S, *_1, *_2, \ldots, *_n, \circ}_R$.
The elements of the scalar ring $\struct {R, +_R, \times_R}$ are called '''scalars'''.
Category:Definitions/Module Theory
Category:Definitions/Linear Algebra
\end{definition} | ProofWiki |
How to find an ellipse, given five points?
Is there a way to find the parameters $$A, B, \alpha, x_0, y_0$$ for the ellipse formula $$\frac{(x \cos\alpha+y\sin\alpha-x_0\cos\alpha-y_0\sin\alpha)^2}{A^2}+\frac{(-x \sin\alpha+y\cos\alpha+x_0\sin\alpha-y_0\cos\alpha)^2}{B^2}=1$$ given five points of the ellipse?
conic-sections
tangenstangens
Here is one way to determine the equation of an ellipse given 5 points. (From there, you can work out the parameters that you want.)
Every ellipse has the form $ax^2+bxy+cy^2+dx+ey+f=0$. We'll find $a,b,c,d,e,f$ given 5 points.
Let $(p_1,q_1),\dots,(p_5,q_5)$ be your 5 points. Consider the following linear equations in the variables $a,b,c,d,e,f$: $ax^2+bxy+cy^2+dx+ey+f=0$, $ap_1^2+bp_1q_1+cq_1^2+dp_1+eq_1+f = 0$,...,$ap_5^2+bp_5q_5+cq_5^2+dp_5+eq_5+f = 0$. Note that we're treating $x^2, xy,y^2,x,y,1$ as coefficients in the first equation. Similarly, $p_i^2,p_iq_i, q_i^2, p_i,q_i,1$ are coefficients in the remaining 5 equations.
Since all 5 points lie on the same ellipse, the above linear equations must have common solution. From this, we see that the equation of the ellipse is given by setting the determinant of the matrix of coefficients of the above 6 equations to $0$.
jrajchgotjrajchgot
$\begingroup$ I don't get it. I have five linear equations using the five points. What do I have to do with the sixth equation? And what should I do with the determinant containing the values x and y? $\endgroup$ – tangens Jul 1 '12 at 14:37
$\begingroup$ The 6th equation is also linear when you treat the monomials in $x$ and $y$ as the coefficients. Take the determinant of the matrix (see GEdgar's answer below) and set it equal to zero. This should give you an equation in $x$ and $y$. This is the equation of your ellipse. You can then rearrange it to put it in any form that you like. $\endgroup$ – jrajchgot Jul 1 '12 at 18:11
What jrajchgot said, but written as a determinant... The conic section passing through the five points $(p_1,q_1),\dots,(p_5,q_5)$ has equation $ax^2+bxy+cy^2+dx+ey+f=0$ which may be written as the determinant equation $$\left|\begin{array} &x^2 & xy & y^2 & x & y & 1 \\ p_1^2 & p_1q_1 & q_1^2 & p_1 & q_1 & 1 \\ p_2^2 & p_2q_2 & q_2^2 & p_2 & q_2 & 1 \\ p_3^2 & p_3q_3 & q_3^2 & p_3 & q_3 & 1 \\ p_4^2 & p_4q_4 & q_4^2 & p_4 & q_4 & 1 \\ p_5^2 & p_5q_5 & q_5^2 & p_5 & q_5 & 1 \end{array}\right| = 0$$
GEdgarGEdgar
What about using the DLT : http://en.wikipedia.org/wiki/Direct_linear_transformation ?
The solution is the null-space of the matrix
$$\left(\begin{array} & p_1^2 & p_1q_1 & q_1^2 & p_1 & q_1 & 1 \\ p_2^2 & p_2q_2 & q_2^2 & p_2 & q_2 & 1 \\ p_3^2 & p_3q_3 & q_3^2 & p_3 & q_3 & 1 \\ p_4^2 & p_4q_4 & q_4^2 & p_4 & q_4 & 1 \\ p_5^2 & p_5q_5 & q_5^2 & p_5 & q_5 & 1 \end{array}\right)$$
which can be found using SVD decomposition.
JulienJulien
$\begingroup$ This is the only answer that actually gives concrete and practical instructions. Thank you! $\endgroup$ – Torsten Bronger Jun 17 '15 at 14:20
Expanding on previous answers (I wonder why people just can't be clear?) Assume M is the 6 X 6 matrix described (I think technically, it's not a matrix since the top row is composed of variables x²,xy, etc.) with all rows below that consisting of the terms as described (eg. M2,1 = (p1)². Then consider the minors of M which I'll designate by mi where i = 1,..6. [Note that a minor is constructed by eliminating the entire row and column that the specified element is in. In this case, the specified element is always in the first row and i indicates the column. (eg. m1 is the minor for M1,1 (and M1,1 is x², m2 = minor(M1,2); M1,2 = xy, noting that x², xy, etc. are variables, not values (i.e. treated as names not values).] Why work with minors? because the determinant of M is the sum of the determinants of its minors (actually the sum of the products of the minors and their cofactors). Why is this useful? because this means that det(M) = +x² * det(m1) - xy * det(m2) + y² * det(m3) -x * det(m4) + y * det(m5) - det(m6). This is the equation of the determinant of M in terms of the determinants of its minors. Setting this expression equal to 0 gives an equation of 6 variables (x², xy, y², x, y, 1) with 6 parameters (det(m1), -det(m2),... etc.) In other words: for the general equation of the ellipse ax²+bxy+cy²+dx+ey+f=0 the parameters a...f are the determinants of the respective minors (times their cofactor). a = +det(m1), etc. and these minors are each 5x5 matrices with coefficients given by the various products of pi and qi. The determinant of a 5x5 matrix is trivially (if not quickly) computed. For clarity, the cofactor is +1 or -1 and determines whether the parameter is the det(mi) or -det(mi), for minor mi the cofactor is computed from the sum of the row and column number of its associated element in M. Since all mi's are the minors of the element M1,i the definition of the general cofactor (-1)(r+c) reduces to (-1)(1+i), which is +1,-1,+1,-1,+1,-1; which explains how the signs of the terms are obtained. This is what jrojchgot meant by saying setting the determinant M to zero solved the problem: the determinants of the minors times their respective cofactors gives us the 6 parameters a,b,c,d,e,& f. And the minors are simply 5x5 matrices of values (given 5 points (qi,pi) are known). HTH
goobergoober
$\begingroup$ Could you please use TeX for arranging formulas? $\endgroup$ – Ramil May 7 '17 at 19:56
Not the answer you're looking for? Browse other questions tagged conic-sections or ask your own question.
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>Canadian Mathematical Bulletin
>FirstView
>Benford behavior and distribution in residue classes...
Canadian Mathematical Bulletin
Benford's law for $P_k(n)$: proof of Theorem
Benford's law for the sum of the prime factors: proof of Theorem
Benford behavior and distribution in residue classes of large prime factors
Part of: Multiplicative number theory Elementary number theory
Published online by Cambridge University Press: 10 October 2022
Paul Pollack and
Akash Singha Roy
Paul Pollack*
Department of Mathematics, Boyd Research and Education Center, University of Georgia, Athens, GA30602, USA e-mail: [email protected]
e-mail: [email protected]
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We investigate the leading digit distribution of the kth largest prime factor of n (for each fixed $k=1,2,3,\dots $) as well as the sum of all prime factors of n. In each case, we find that the leading digits are distributed according to Benford's law. Moreover, Benford behavior emerges simultaneously with equidistribution in arithmetic progressions uniformly to small moduli.
Benford's lawsmooth numbersanatomy of integers
MSC classification
Primary: 11A63: Radix representation; digital problems
Secondary: 11N37: Asymptotic results on arithmetic functions 11N64: Other results on the distribution of values or the characterization of arithmetic functions
Canadian Mathematical Bulletin , First View , pp. 1 - 17
DOI: https://doi.org/10.4153/S0008439522000601[Opens in a new window]
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (https://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
© The Author(s), 2022. Published by Cambridge University Press on behalf of The Canadian Mathematical Society
Benford's law, named for physicist Frank Benford (though discovered almost 60 years prior by Simon Newcomb), refers to the observation that in many naturally occurring datasets, the leading digits are far from uniformly distributed, with smaller digits more likely to occur. Let us make this precise. By the N leading digits of the positive real number x, we mean the N most significant digits. For example (working in base $10$), $123.456$ has the first $4$ leading digits $1234$, and this is the same for $0.00123456$. Now, let D and b be integers with $b\ge 2$. We say a positive real number "begins with D in base b" if its most significant digits in base b are those of the base b expansion of D. Then Benford's law, in base b, predicts that the proportion of terms in the dataset beginning with D should be approximately $\log (1+D^{-1})/\log {b}$. For example, since $\frac {\log {2}}{\log {10}}=0.3010\dots $, we expect to see a leading digit $1$ in base 10 about $30\%$ of the time.
For general background on Benford's law, see [Reference Berger and Hill5, Reference Miller22]. In this paper, we are interested in datasets arising from positive-valued arithmetic functions. Let $f\colon \mathbb {N} \to \mathbb {R}_{>0}$. We say f obeys Benford's law in base b (or that f is Benford in base b) if, for each positive integer D, the asymptotic density of n for which $f(n)$ begins with D in base b is $\log (1+D^{-1})/\log {b}$. Results on the "Benfordity" of particular arithmetic functions are scattered throughout the literature. For example, $f(n)=n!$ is Benford in every base b [Reference Diaconis11], as is the "primorial" $f(n) = \prod _{k=1}^{n}p_k$ [Reference Massé and Schneider21]. The classical partition function $p(n)$ is also Benford in every base (see [Reference Anderson, Rolen and Stoehr2] or [Reference Massé and Schneider21]). On the other hand, $f(n)=n$ is not Benford; the asymptotic density in question does not exist. This same obstruction to Benford's law persists if $f(n)$ is any positive-valued polynomial function of n. (See, for instance, the final section of [Reference Massé and Schneider21]. It should be noted that these examples obey Benford's law in a weaker sense; namely, Benford's law holds if asymptotic density is replaced with logarithmic density.)
When f is multiplicative, whether or not f is Benford in base b can be interpreted as a problem in the theory of mean values of multiplicative functions. Namely, f is Benford precisely when $f(n)^{2\pi i \ell /\log {b}}$ has mean value zero for each nonzero integer $\ell $. This criterion was noted by Aursukaree and Chandee [Reference Aursukaree and Chandee3] and used by them to show that the divisor function $d(n)$ is Benford in base $10$. A more systematic study of the Benford behavior of multiplicative functions, leveraging Halász's celebrated mean value theorem, was recently undertaken in [Reference Chandee, Li, Pollack and Singha Roy8]. For example, it is shown there that $\phi (n)$ is not Benford, but that $|\tau (n)|$ is, where $\tau $ is Ramanujan's $\tau $-function.Footnote 1 All of the work in [Reference Chandee, Li, Pollack and Singha Roy8] is carried out in base $10$, but both of the quoted results hold, by simple modifications of the proofs, in each fixed base $b\ge 2$.
Our concern in the present paper is with certain nonmultiplicative functions. Roughly speaking, we show that (for each fixed k) the kth largest prime factor of n obeys Benford's law, as does the sum of all of the prime factors of n. (Both results hold for each base b.) In fact, our results are somewhat stronger than this.
We let $P_k(n)$ denote the kth largest prime factor of n; when $k=1$, we write $P(n)$ in place of the more cumbersome $P_1(n)$. More precisely, if $n = p_1 p_2 p_3 \cdots p_{\Omega (n)}$, with $p_1 \ge p_2 \ge p_3 \ge \dots \ge p_{\Omega (n)}$, we set $P_k(n) = p_k$, with the convention that $P_k(n) = 0$ if $k> \Omega (n)$. Put
$$\begin{align*}\Psi_k(x,y) := \#\{n \le x: P_k(n) \le y\}.\end{align*}$$
(When $k=1$, it is usual to write $\Psi (x,y)$ in place of $\Psi _1(x,y)$.) Let $a\bmod {q}$ be a coprime residue class. For real $x, y\ge 2$, define
$$\begin{align*}\begin{aligned} \Psi_k(x,y;b,D,q,a) := \#\{n \le x: P_k(n)\le y, ~P_k(n)\equiv a\ \ \ \pmod{q},\hphantom{extra}\\ P_k(n) \text{ begins with } D \text{ in base } b\}. \end{aligned} \end{align*}$$
Theorem 1.1 Fix positive integers k, b, and D, with $b\ge 2$. Fix real numbers $U\ge 1$ and $\epsilon>0$. Then
$$\begin{align*}\Psi_k(x,y;b,D,q,a) \sim \frac{1}{\phi(q)} \frac{\log(1+D^{-1})}{\log{b}} \Psi_k(x,y),\end{align*}$$
as $x, y \to \infty $, uniformly for $y\ge x^{1/U}$ and coprime residue classes $a\bmod {q}$ with $q\le \frac {\log {x}}{(\log \log x)^{k-1+\epsilon }}$. In fact, if $k=1$, we can take $q \le (\log {x})^{A}$ for any fixed A.
To deduce that $P_k(n)$ is Benford, it suffices to take $q=1$ and $y=x$. The additional generality of Theorem 1.1 seems of some interest. For example, Theorem 1.1 contains the result of Banks–Harman–Shparlinski [Reference Banks, Harman and Shparlinski4] that $P(n)$, on integers $n\le x$, is uniformly distributed in coprime residue classes mod q, for q up to an arbitrary fixed power of $\log {x}$. Theorem 1.1 gives the corresponding result for $P_k(n)$, when $k>1$, in the more restricted range $q \le {\log {x}}/{(\log \log x)^{k-1+\epsilon }}$. This appears to be new; moreover, this range of q is sharp up to the power of $\log \log {x}$, since $\gg x(\log \log {x})^{k-2}/\log {x}$ values of $n\le x$ have $P_k(n)=2$.
Turning to the sum of the prime factors, we let $A(n)= \sum _{p^k\parallel n} kp$. That is, $A(n)$ is the sum of the prime factors of n, counting multiplicity. (The sum of the distinct prime factors of n could be handled by similar arguments.) The function $A(n)$ was introduced by Alladi and first investigated by Alladi and Erdős [Reference Alladi and Erdős1].
$$\begin{align*}\begin{aligned} N(x,y; b, D, q, a) := \#\{n \le x: P(n)\le y, A(n)\equiv a\ \ \ \pmod{q},\hphantom{extra}\\ A(n) \text{ begins with } D \text{ in base } b\}. \end{aligned} \end{align*}$$
Theorem 1.2 Fix an integer $b \ge 2$, and a positive integer D. Fix real numbers $U\ge 1$ and $\epsilon>0$. Then
$$\begin{align*}N(x,y;b,D,q,a) \sim \frac 1q \frac{\log(1+D^{-1})}{\log{b}} \Psi(x,y),\end{align*}$$
as $x, y \to \infty $, uniformly for $y\ge x^{1/U}$ and residue classes $a\bmod {q}$ with $q \le (\log {x})^{\frac {1}{2}-\epsilon }$.
As before, taking $y=x$ and $q=1$ shows that $A(n)$ satisfies Benford's law. Again, the extra generality here seems interesting. For example, it is implicit in Theorem 1.2 that $A(n)$ is equidistributed mod q, uniformly for $q \le (\log {x})^{\frac {1}{2}-\epsilon }$, a result which we have not seen explicitly stated in the literature before. (See [Reference Goldfeld12] for the case of fixed q.) The same range of uniformity may follow from the method of Hall in [Reference Hall15] (who considered the distribution mod q of $\sum _{p\mid n,~p\nmid q} p$), but our proof exhibits the result as a simple consequence of quantitative mean value theorems.
In addition to the already-mentioned references, the reader interested in number-theoretic investigations of Benford's law might also consult [Reference Best, Dynes, Edelsbrunner, McDonald, Miller, Tor, Turnage-Butterbaugh and Weinstein6, Reference Best, Dynes, Edelsbrunner, McDonald, Miller, Tor, Turnage-Butterbaugh and Weinstein7, Reference Chen, Park and Swaminathan9, Reference Jameson, Thorner and Ye18, Reference Kontorovich and Miller20, Reference Pollack and Singha Roy24].
Most of our notation is standard. Of note, we allow constants in O-symbols to depend on any parameter that has been declared as "fixed." When we refer to "large" x, the threshold for large enough may also depend on these parameters. We write $A\gtrsim B$ as an abbreviation for $A\ge (1+o(1))B$.
2 Benford's law for $P_k(n)$: proof of Theorem 1.1
We make crucial use of both the results and methods of Knuth and Trabb Pardo [Reference Knuth and Trabb Pardo19], who were the first to seriously investigate $P_k(n)$ when $k>1$. We define functions $\rho _k(\alpha )$, for integers $k\ge 0$ and real $\alpha $, as follows:
$$\begin{align*}\rho_k(\alpha) =0 \quad\text{if}\quad \alpha\le 0\ \text{or}\ k=0, \end{align*}$$
$$\begin{align*}\rho_k(\alpha)=1 \quad\text{for}\quad 0 < \alpha \le 1\ \text{and}\ k\ge 1, \end{align*}$$
(2.1) $$ \begin{align} \rho_k(\alpha) = 1 - \int_{1}^{\alpha} (\rho_k(t-1) - \rho_{k-1}(t-1))\frac{\mathrm{d}t}{t}, \quad\text{for}\quad \alpha> 1\ \text{and}\ k\ge 1. \end{align} $$
Much is known about the asymptotic behavior of $\rho _k(\alpha )$ as $\alpha \to \infty $; for $k=1$, see, for instance, [Reference de Bruijn10], whereas for $k\ge 2$, see equations (6.4) and (6.15) in [Reference Knuth and Trabb Pardo19]. For our purposes, much weaker information suffices. We assume as known that each $\rho _k$ ( $k=1,2,3,\dots $) is positive-valued and weakly decreasing on $(0,\infty )$, and that $\lim _{\alpha \to \infty } \rho _k(\alpha )=0$.
The following result, which connects the $\rho _k$ with the distribution of $P_k(n)$, appears as equation (4.7) in [Reference Knuth and Trabb Pardo19] (and is a consequence of the stronger assertion (4.8) shown there).
Proposition 2.1 Fix a positive integer k and a real number $U\ge 1$. For all $x, y\ge 2$,
(2.2) $$ \begin{align} \Psi_k(x,y) = \rho_k(u)x + O(x/\log{x}), \end{align} $$
uniformly for $y\ge x^{1/U}$, where $u:=\frac {\log {x}}{\log {y}}$. In particular, $\Psi _k(x,y) \sim \rho _k(u) x$ as $x\to \infty $, uniformly for $y\ge x^{1/U}$.
(In [Reference Knuth and Trabb Pardo19], it is assumed that the ratio $\frac {\log {x}}{\log {y}}$ is fixed, rather than merely bounded. However, the proof given actually establishes (2.2) in the full range of Proposition 2.1.)
The next result is a variant of Theorem 1.1 where we require that $P_k(n)$ be bounded below by a fixed power of x.
Proposition 2.2 Fix positive integers k, b, and D with $b\ge 2$. Fix real numbers $A\ge 1$, $U\ge 1$, and fix a real number $U'> U$. The number of $n\le x$ for which $P_k(n)\equiv a\ \pmod {q}$, $P_k(n)$ begins with the digits of D in base b, and $P_k(n) \in (x^{1/U'}, y]$ is
$$\begin{align*}\frac{1}{\phi(q)} \frac{\log(1+D^{-1})}{\log{b}} (\rho_k(u)-\rho_k(U')) x + o(x/\phi(q)), \end{align*}$$
where $u:=\frac {\log {x}}{\log {y}}$, where $x, y\to \infty $ with $y\ge x^{1/U}$, and where $a\bmod {q}$ is a coprime residue class with $q \le (\log {x})^{A}$.
The proof of Proposition 2.2 requires two classical results from the theory of primes in arithmetic progressions. Let $\pi (x;q,a)$ denote the count of primes $p\le x$ with $p\equiv a\ \pmod {q}$.
Proposition 2.3 (Brun–Titchmarsh)
If a and q are coprime integers with $0 < 2q \le x$, then
$$\begin{align*}\pi(x;q,a) \ll \frac{1}{\phi(q)} \frac{x}{\log(x/q)}.\end{align*}$$
Here, the implied constant is absolute.
Proposition 2.4 (Siegel–Walfisz)
Fix a real number $A> 0$. If a and $ q$ are coprime integers with $1 \le q \le (\log {x})^A$, and $x\ge 3$, then
$$\begin{align*}\pi(x;q,a) = \frac{1}{\phi(q)}\int_{2}^{x}\frac{1}{\log{t}}\,\mathrm{d}t + O_A(x \exp(-C\sqrt{\log{x}})).\end{align*}$$
Here, C is a certain absolute constant.
For proofs of these results, see [Reference Montgomery and Vaughan23, Theorem 3.9, p. 90] and [Reference Montgomery and Vaughan23, Corollary 11.21, p. 382].
Proof of Proposition 2.2
First note that we can (and will) always assume that $y\le x$, since the cases when $y> x$ are covered by the case $y=x$.
By a standard compactness argument, when proving Proposition 2.2, we may assume that $u= \frac {\log {x}}{\log {y}}$ is fixed. To see this, suppose Proposition 2.2 holds when u is fixed but does not hold in general. Then, for some $\epsilon>0$, there are choices of $x, y, a$, and q with x arbitrarily large, $x\ge y\ge x^{1/U}$, and $q\le (\log {x})^{A}$ for which our count exceeds
(2.3) $$ \begin{align} \frac{1}{\phi(q)} \frac{\log(1+D^{-1})}{\log{b}} (\rho_k(u)-\rho_k(U')+\epsilon) x, \end{align} $$
or there are such choices of $x,y,a$, and q for which our count falls below
$$\begin{align*}\frac{1}{\phi(q)} \frac{\log(1+D^{-1})}{\log{b}} (\rho_k(u)-\rho_k(U')-\epsilon) x. \end{align*}$$
We will assume that we are in the former case; the latter can be handled analogously. By compactness, we may choose $x,y,a,q$ so that $u\to u_0$, for some $u_0 \in [1,U]$.
We first rule out $u_0=1$. As $y\le x$, the condition $P_k(n) \le y$ is always at least as strict as the condition $P_k(n) \le x$ (which holds vacuously, as we are counting numbers ${n\le x}$). Moreover, the $u=1$ case of Proposition 2.2 is true by hypothesis. Putting these observations together, we see that the count of n corresponding to $x,y,a,q$ is at most
$$\begin{align*}\frac{1}{\phi(q)} \frac{\log(1+D^{-1})}{\log{b}} (\rho_k(1)-\rho_k(U')+o(1)) x. \end{align*}$$
However, if $u\to 1$, then $\rho _k(u)\to \rho _k(1)$, and this estimate is eventually incompatible with (2.3).
Thus, it must be that $u_0> 1$. Here, we may obtain a contradiction by a slightly tweaked argument. For any fixed $\delta>0$, we eventually have $u> u_0-\delta $. So the condition $P_k(n) \le y$ is eventually stricter than the condition $P_k(n) \le x^{1/(u_0-\delta )}$. If $\delta $ is fixed sufficiently small (in terms of $\epsilon $), then the $u=u_0-\delta $ case of Proposition 2.2 gives an estimate contradicting (2.3).
We thus turn to proving the modified statement with the extra condition that u is fixed.
For each nonnegative integer j, let $\mathcal {I}_j$ denote the interval
(2.4) $$ \begin{align} \mathcal{I}_j := [u_j,v_j), \quad\text{where} \quad u_j:=Db^j,\; v_j:=(D+1)b^j. \end{align} $$
Then our count of n is given by
(2.5) $$ \begin{align} \sum_{j\ge 0} \sum_{\substack{p \in \mathcal{I}_j \cap (x^{1/U'},y] \\p\equiv a\ \ \ \pmod{q}}} \sum_{\substack{n \le x\\P_k(n)=p}} 1.\end{align} $$
Let $\mathcal {J}$ be the collection of nonnegative integers j with $\mathcal {I}_j \subset (x^{1/U'}, y/\exp (\sqrt {\log {x}}))$. Then, at the cost of another error of size $o(x/\phi (q))$, we can restrict the triple sum in (2.5) to $j \in \mathcal {J}$. Indeed, the n counted by the triple sum above that are excluded by this restriction have either a prime divisor in $P:=(x^{1/U'}, bx^{1/U'}]$ or in ${P':=[y/b\exp (\sqrt {\log {x}}), y]}$, and the number of such $n\le x$ is at most
$$ \begin{align*}x\sum_{\substack{p \in P\cup P' \\ p\equiv a\ \ \ \pmod{q}}} 1/p = o(x/\phi(q)),\end{align*} $$
by partial summation and the Brun–Titchmarsh theorem (Proposition 2.3). We proceed to estimate, for each $j \in \mathcal {J}$, the corresponding inner sums in (2.5) over p and n.
If p is prime and $P_k(n)=p$, then $n=mp$ where $m \le x/p$, $P_k(m) \le p$, and $P_{k-1}(m)\ge p$. The converse also holds. Thus, if $j \in \mathcal {J}$ and $p \in \mathcal {I}_j$,
$$\begin{align*}\sum_{\substack{n \le x\\P_k(n)=p}} 1 = \Psi_k(x/p,p) - \Psi_{k-1}(x/p,p-\epsilon) \end{align*}$$
for (say) $\epsilon = \frac {1}{2}$. Hence,
$$\begin{align*}\sum_{\substack{p \in \mathcal{I}_j \\ p\equiv a\ \pmod{q}}} \sum_{\substack{n\le x \\ P_k(n)=p}} 1 = \sum_{\substack{p \in \mathcal{I}_j \\ p\equiv a\ \pmod{q}}}\Psi_k(x/p,p) - \sum_{\substack{p \in \mathcal{I}_j \\ p\equiv a\ \pmod{q}}}\Psi_{k-1}(x/p,p-\epsilon). \end{align*}$$
To continue, observe that, for $j \in \mathcal {J}$,
$$ \begin{align*} &\sum_{\substack{p \in \mathcal{I}_j \\ p\equiv a\ \ \ \pmod{q}}}\Psi_k(x/p,p) - \frac{1}{\phi(q)}\int_{\mathcal{I}_j} \Psi_k(x/t,t)\, \frac{\mathrm{d}t}{\log{t}} \\ &\qquad\qquad= \sum_{\substack{u_j \le p < v_j \\ p\equiv a\ \ \ \pmod{q}}}\sum_{\substack{n \le x/p \\ P_k(n) \le p}}1 - \frac{1}{\phi(q)}\int_{u_j}^{v_j} \sum_{\substack{n \le x/t \\ P_k(n) \le t}} \frac{1}{\log{t}}\,\mathrm{d}t \\ &\qquad\qquad= \sum_{n\le x/u_j} \left(\sum_{\substack{m < p \le M \\ p\equiv a\ \ \ \pmod{q}}} 1 - \frac{1}{\phi(q)} \int_{m}^{M}\frac{\mathrm{d}t}{\log{t}} + O(1) \right), \end{align*} $$
where m and M are defined by
$$\begin{align*}m := \max\{u_j, P_k(n)\}, \quad M:= \min\{x/n, v_j\}, \end{align*}$$
and where the last displayed sum on n is understood to be extended only over those $n\le x/u_j$ for which $m \le M$. By the Siegel–Walfisz theorem (Proposition 2.4),
$$\begin{align*}\sum_{\substack{m < p \le M \\ p\equiv a\ \ \ \pmod{q}}} 1 - \frac{1}{\phi(q)} \int_{m}^{M}\frac{\mathrm{d}t}{\log{t}} \ll M \exp(-C\sqrt{\log M}) \ll \frac{x}{n} \exp(-C'\sqrt{\log{x}}), \end{align*}$$
where C is an absolute positive constant and $C'= C/\sqrt {U'}$. (This use of the Siegel–Walfisz theorem explains the restriction $q\le (\log {x})^A$ in the statement of Proposition 2.2.) Putting this back in the above and summing on n, we find that (for large x)
(2.6) $$ \begin{align} \sum_{\substack{p \in \mathcal{I}_j \\ p\equiv a\ \ \ \pmod{q}}}\Psi_k(x/p,p) - \frac{1}{\phi(q)}\int_{\mathcal{I}_j} \Psi_k(x/t,t)\, \frac{\mathrm{d}t}{\log{t}} \ll x \log{x} \cdot \exp(-C'\sqrt{\log{x}}) + \frac{x}{u_j}. \end{align} $$
A nearly identical calculation gives the same bound for the difference
$$ \begin{align*}\sum_{\substack{p \in \mathcal{I}_j \\ p\equiv a\ \ \ \pmod{q}}}\Psi_{k-1}(x/p,p-\epsilon) - \frac{1}{\phi(q)}\int_{\mathcal{I}_j} \Psi_{k-1}(x/t,t)\, \frac{\mathrm{d}t}{\log{t}}.\end{align*} $$
Since $u_{j+1}/u_j \ge 2$ and the smallest $j \in \mathcal {J}$ has $u_j \ge x^{1/U'}$, the expression on the right-hand side of (2.6), when summed on $j \in \mathcal {J}$, is $\ll x (\log {x})^2 \exp (-C'\sqrt {\log {x}}) + x^{1-1/U'}$, and this is certainly $o(x/\phi (q))$. As a consequence, instead of our original triple sum (2.5), it is enough to estimate
(2.7) $$ \begin{align} \frac{x}{\phi(q)} \sum_{j \in \mathcal{J}} \frac{1}{x}\int_{\mathcal{I}_j}(\Psi_k(x/t,t) - \Psi_{k-1}(x/t,t))\, \frac{\mathrm{d}t}{\log{t}}. \end{align} $$
We now apply Proposition 2.1, noting that for each $t \in \mathcal {I}_j$, we have $\frac {\log {(x/t)}}{\log {t}} = \frac {\log {x}}{\log {t}}-1\le U'-1$ as well as $\log (x/t) \ge \log (y/t) \ge \sqrt {\log {x}}$. We find that
$$ \begin{align*} &\frac{1}{x}\int_{\mathcal{I}_j} (\Psi_k(x/t,t) - \Psi_{k-1}(x/t,t)) \frac{\mathrm{d}t}{\log{t}} \\ &\qquad= \int_{\mathcal{I}_j} \frac{1}{t} \left(\rho_k\left(\frac{\log{x}}{\log{t}}-1\right) - \rho_{k-1}\left(\frac{\log{x}}{\log{t}}-1\right)\right) \frac{\mathrm{d}t}{\log{t}} + O\left(\int_{\mathcal{I}_j} \frac{1}{t\sqrt{\log{x}}} \frac{\mathrm{d}t}{\log{t}}\right).\end{align*} $$
The error term, when summed on $j \in \mathcal {J}$, is $\ll \frac {1}{\sqrt {\log {x}}}\int _{2}^{x} \frac {\mathrm {d}t}{t\log {t}} \ll \log \log {x}/\sqrt {\log {x}}$, and so is $o(1)$; inserted back into (2.7), we see that this gives rise to a final error of size $o(x/\phi (q))$ in our count, which is acceptable. To deal with the remaining integrals, we write $u_j = x^{\mu _j}$ and $v_j = x^{\nu _j}$ and make the change of variables $\alpha = \frac {\log {x}}{\log {t}}$. Then $\mathrm {d}\alpha = -\frac {\log {x}}{t(\log {t})^2}\, \mathrm {d}t$, so that $\frac {\mathrm {d}t}{t\log {t}} = -\frac {\mathrm {d}\alpha }{\alpha }$ and
$$ \begin{align*} & \sum_{j \in \mathcal{J}} \int_{\mathcal{I}_j} \frac{1}{t}\left(\rho_k\left(\frac{\log{x}}{\log{t}}-1\right) - \rho_{k-1}\left(\frac{\log{x}}{\log{t}}-1\right)\right) \frac{\mathrm{d}t}{\log{t}}\\[5pt] &\qquad\qquad\qquad\qquad\qquad\qquad\qquad = \sum_{j \in \mathcal{J}} \int_{1/\mu_j}^{1/\nu_j} -\frac{\rho_k(\alpha-1)-\rho_{k-1}(\alpha-1)}{\alpha}\, \mathrm{d}\alpha. \end{align*} $$
From (2.1), $-\frac {\rho _k(\alpha -1)-\rho _{k-1}(\alpha -1)}{\alpha } = \rho _k'(\alpha )$, so that this last sum on j simplifies to $\sum _{j \in \mathcal {J}} (\rho _k(1/\nu _j)-\rho _k(1/\mu _j))$. Now, following [Reference Knuth and Trabb Pardo19], we introduce the function $F_k(\beta)$ defined for $\beta \in (0,1]$ by $F_k(\beta)=\rho_k(1/\beta)$. By the mean value theorem,
$$ \begin{align*} \rho_k(1/\nu_j) - \rho_k(1/\mu_j) &= F_k(\nu_j) - F_k(\mu_j) \\ &= (\nu_j-\mu_j) F_k'(t_j) = \frac{\log(1+D^{-1})}{\log{x}} F_k'(t_j)\end{align*} $$
for some $t_j \in (\mu _j,\nu _j)$. Thus,
$$ \begin{align*} \sum_{j \in \mathcal{J}} (\rho_k(1/\nu_j)-\rho_k(1/\mu_j)) &= \frac{\log(1+D^{-1})}{\log{b}}\sum_{j \in \mathcal{J}} F_k'(t_j) \cdot \frac{\log{b}}{\log{x}} \\\ &= \frac{\log(1+D^{-1})}{\log{b}}\sum_{j \in \mathcal{J}} F_k'(t_j) \cdot (\mu_{j+1}-\mu_j). \end{align*} $$
Since each $t_j \in (\mu _j,\nu _j) \subset (\mu _j,\mu _{j+1})$, the final sum on j is essentially a Riemann sum. To make this precise, let $j_0 = \min \mathcal {J}$ and $j_1 = \max \mathcal {J}$. Then
$$\begin{align*}F_k'(1/U') \left(\mu_{j_0}-\frac{1}{U'}\right) + \sum_{j \in \mathcal{J}} F_k'(t_j) (\mu_{j+1}-\mu_j) + F_k'(1/u)\left(\frac{1}{u}-\mu_{j_1+1}\right) \end{align*}$$
is a genuine Riemann sum for $\int _{1/U'}^{1/u} F_k'(t)\, \mathrm {d}t$, whose mesh size goes to $0$ as $x \to \infty $. However, the terms we have added to the sum on $j\in \mathcal {J}$ contribute $o(1)$, as $x\to \infty $. It follows that $\sum _{j \in \mathcal {J}} F_k'(t_j) (\mu _{j+1}-\mu _j) \to \int _{1/U'}^{1/u} F_k'(t)\,\mathrm {d}t = F_k(1/u) - F_k(1/U') = \rho _k(u)-\rho _k(U')$. Collecting estimates completes the proof of the proposition in the case when u is fixed.
To deduce Theorem 1.1, it remains to handle the contribution from n with ${P_k(n) \le x^{1/U'}}$.
The following lemma bounds the number of integers with a large smooth divisor. A proof is sketched in Exercise 293 on page 554 of [Reference Tenenbaum26], with a solution in [Reference Tenenbaum25, pp. 305–306]. By the y-smooth part of a number n, we mean $\prod _{\substack {p^e\parallel n \\ p \le y}} p^e$.
Lemma 2.5 For all $x\ge z\ge y\ge 2$, the number of $n\le x$ whose y-smooth part exceeds z is $O\left (x \exp \left (-\frac {1}{2}\frac {\log {z}}{\log {y}}\right )\right )$.
Lemma 2.6 Fix a positive integer k and a real number $B \ge 1$.
• When $k=1$, the number of $n\le x$ with $P_k(n) \le y$ and $P_k(n)\equiv a\ \pmod {q}$ is
$$\begin{align*}\ll \frac{x}{\phi(q)} \exp\left(-\frac{1}{8}u\right) + x \left(\frac{\log(3q)}{\log{x}}\right)^B \cdot \exp\left(-\frac{1}{8}u\right), \end{align*}$$
uniformly for $x\ge y \ge 3$ with $y\le x^{1/4}$, and $a\bmod {q}$ any coprime residue class with $q\le x^{1/8}$. As usual, $u = \frac {\log {x}}{\log {y}}$.
• When $k\ge 2$, the number of $n\le x$ with $P_k(n) \le y$ and $P_k(n)\equiv a\ \pmod {q}$ is
$$\begin{align*}\ll \frac{x}{\log{x}}(\log\log{x})^{k-2} \log{(3q)} + \frac{x}{\phi(q)} \frac{(\log{u})^{k-2}}{u}, \end{align*}$$
uniformly in the same range of $x,y$, and q.
Proof We will restrict attention to $n> x^{3/4}$; this is permissible, since $x^{3/4}$ is dwarfed by either of our target upper bounds. We let $p = P_k(n)$ and write $n = p_1\cdots p_{k-1} p s$, where $p_1 \geq p_2 \geq \dots \ge p_{k-1} \ge p$ and $P(s) \le p$.
We first show that we can assume $s \le x^{1/2}$. Indeed, suppose $s> x^{1/2}$. Then, with $m=n/p$, we have that $m \le x/p$ and that the p-smooth part of m exceeds $x^{1/2}$. Applying Lemma 2.5, we see that for every $p \le y$, the number of corresponding m is
$$ \begin{align*} \ll \frac{x}{p}\exp\left(-\frac{1}{4}\frac{\log{x}}{\log{p}}\right) &\ll \frac{x}{p}\exp\left(-\frac{1}{8}\frac{\log{x}}{\log{p}}\right) \cdot \exp\left(-\frac{1}{8}\frac{\log{x}}{\log{p}}\right) \\ &\ll \frac{x}{(\log{x})^{B}} \frac{(\log{p})^B}{p} \exp\left(-\frac{1}{8}\frac{\log{x}}{\log{p}}\right) \\ &\ll \frac{x}{(\log{x})^{B}} \frac{(\log{p})^B}{p} \exp\left(-\frac{1}{8}u\right)\!. \end{align*} $$
Now, we sum on $p \le y$ with $p\equiv a\ \pmod {q}$. We split the sum at $3q^2$, using Mertens' theorem to bound the first half and the Brun–Titchmarsh theorem (with partial summation) for the second; this gives
$$ \begin{align*} \sum_{\substack{p \le y \\p \equiv a\ \ \ \pmod{q}}} \frac{(\log{p})^B}{p} &\le \sum_{p \le 3q^2} \frac{(\log{p})^B}{p} + \sum_{\substack{3q^2 < p \le y \\ p \equiv a\ \ \ \pmod{q}}} \frac{(\log{p})^B}{p} \\ &\ll (\log(3q))^{B-1} \sum_{p \le 3q^2} \frac{\log{p}}{p} + \frac{1}{\phi(q)} (\log{y})^{B} \\&\ll (\log{(3q)})^{B} + \frac{(\log{y})^{B}}{\phi(q)}. \end{align*} $$
Substituting this estimate into the previous display, we conclude that the n with ${s> x^{1/2}}$ contribute
(2.8) $$ \begin{align} \ll \frac{x}{u^B \phi(q)} \exp(-\frac{1}{8}u) &+ x \left(\frac{\log(3q)}{\log{x}}\right)^B \cdot \exp(-\frac{1}{8}u) \notag \\ &\ll \frac{x}{\phi(q)} \exp\left(-\frac{1}{8}u\right) + x \left(\frac{\log(3q)}{\log{x}}\right)^B \cdot \exp\left(-\frac{1}{8}u\right). \end{align} $$
This is already enough to settle the $k=1$ case of Lemma 2.6. Indeed, in that case, $n=ps$, where $p = P(n)$, and $s = n/P(n) \ge n/y> x^{3/4}/y \ge x^{1/2}$.
Now, suppose that $k \ge 2$ and that $s \le x^{1/2}$. Then
$$\begin{align*}p_1^{k} \ge p_1\cdots p_{k-1} p = n/s> x^{3/4}/x^{1/2} = x^{1/4}, \end{align*}$$
so that $p_1 \ge x^{1/4k}$. Hence, given $p_2,\dots ,p_{k-1},p$, and s, the number of possibilities for $p_1$ (and thus also for n) is $\ll \pi (x/p_2\cdots p_{k-1} p s) \ll x/p_2 \cdots p_{k-1} p s\log {x}$. Observe that s is p-smooth, while each $p_i \in [p,x]$. We have that $\sum _{s\ p\text {-smooth}} 1/s = \prod _{\text {prime }\ell \le p} (1-1/\ell )^{-1} \ll \log {p}$. Moreover (when $p \le y$), $\sum _{p \le p_i \le x} 1/p_i \ll \log \frac {\log {x}}{\log {p}}$. Hence, the number of possibilities for n given p is
$$\begin{align*}\ll \frac{x}{\log{x}} \left(\log \frac{\log{x}}{\log{p}}\right)^{k-2} \frac{\log{p}}{p}. \end{align*}$$
We now sum on $p\le y$ with $p\equiv a\ \pmod {q}$. Estimating crudely, we see that the $p\le 3q^2$ contribute
$$\begin{align*}\ll \frac{x}{\log{x}} (\log\log{x})^{k-2} \log{(3q)}. \end{align*}$$
To handle the remaining contribution in the case when $y> 3q^2$, we apply partial summation; by Brun–Titchmarsh,
$$ \begin{align*} &\sum_{\substack{3q^2 < p \le y\\p\equiv a\ \ \ \pmod{q}}} \left(\log \frac{\log{x}}{\log{p}}\right)^{k-2} \frac{\log{p}}{p} \\ &\qquad\qquad\qquad\qquad\ll \frac{1}{\phi(q)} (\log{u})^{k-2} -\int_{3q^2}^{y} \pi(t;q,a) \mathrm{d}\left(\left(\log \frac{\log{x}}{\log{t}}\right)^{k-2} \frac{\log{t}}{t}\right).\end{align*} $$
Since $\left (\log \frac {\log {x}}{\log {t}}\right )^{k-2} \frac {\log {t}}{t}$ is a decreasing function of t on $[3q^2,y]$, the bound $\pi (t;q,a) \ll t/\phi (q)\log {t}$ implies that
$$ \begin{align*} &-\int_{3q^2}^{y} \pi(t;q,a) \, \mathrm{d}\left(\left(\log \frac{\log{x}}{\log{t}}\right)^{k-2} \frac{\log{t}}{t}\right) \\ &\qquad\qquad\qquad\qquad\qquad\qquad\qquad\ll -\frac{1}{\phi(q)} \int_{3q^2}^{y} \frac{t}{\log{t}} \,\mathrm{d}\left(\left(\log \frac{\log{x}}{\log{t}}\right)^{k-2} \frac{\log{t}}{t}\right). \end{align*} $$
Integrating by parts again,
$$ \begin{align*} &\int_{3q^2}^{y} \frac{t}{\log{t}}\,\mathrm{d}\left(\left(\log \frac{\log{x}}{\log{t}}\right)^{k-2} \frac{\log{t}}{t}\right) \\ &\qquad\qquad\qquad\qquad=-\int_{3q^2}^{y} \left(\log\frac{\log{x}}{\log{t}}\right)^{k-2} \frac{\log{t}}{t} \, \mathrm{d}\left(\frac{t}{\log{t}}\right)+ O((\log\log{x})^{k-2}) \\ &\qquad\qquad\qquad\qquad\qquad\qquad\qquad\ll \int_{3q^2}^{y} \left(\log\frac{\log{x}}{\log{t}}\right)^{k-2} \, \frac{\mathrm{d}t}{t} + O((\log\log{x})^{k-2}).\end{align*} $$
Making the change of variables $\alpha = \frac {\log {t}}{\log {x}}$,
$$\begin{align*}\int_{3q^2}^{y} \left(\log\frac{\log{x}}{\log{t}}\right)^{k-2} \, \frac{\mathrm{d}t}{t} \le \log{x} \int_{0}^{1/u} (\log(1/\alpha))^{k-2}\, \mathrm{d}\alpha \ll \log{x} \cdot \frac{1}{u} (\log u)^{k-2}. \end{align*}$$
(In the last step, we use that $\int _{0}^{z} (\log (1/\alpha ))^{k-2}\,\mathrm {d}\alpha $ has the form $z\cdot Q(\log (1/z))$, where Q is a monic polynomial with degree $k-2$.) Collecting estimates, we conclude that when $k\ge 2$, the n with $s \le x^{1/2}$ make a contribution
$$\begin{align*}\ll \frac{x}{\log{x}} (\log\log{x})^{k-2}\log{(3q)} + \frac{x}{\phi(q)} \frac{(\log{u})^{k-2}}{u}.\end{align*}$$
Since this upper bound dominates the contribution (2.8) from n with $s> x^{1/2}$, the $k\ge 2$ cases of Lemma 2.6 follow.
Proof of Theorem 1.1
Fix $\eta> 0$. We will show that the count of n in question is eventuallyFootnote 2 larger than $\frac {1}{\phi (q)} \frac {\log (1+D^{-1})}{\log {b}} \left (\rho _k(u)-\eta \right )x$ and eventually smaller than $\frac {1}{\phi (q)} \frac {\log (1+D^{-1})}{\log {b}} \left (\rho _k(u)+\eta \right )x$, and hence is $\sim \frac {1}{\phi (q)}\frac {\log (1+D^{-1})}{\log {b}} \rho _k(u) x$. Since $\Psi _k(x,y) \sim \rho _k(u) x$, Theorem 1.1 then follows.
The required lower bound is immediate from Proposition 2.2: it suffices to apply that proposition with $U'$ fixed large enough that $\rho _k(U') < \eta $.
We turn now to the upper bound. Apply Lemma 2.6, taking $B=A+1$ in the case $k=1$. That lemma implies the existence of a constant C, depending only on k (and on A, if $k=1$) such that the following holds: for any fixed $U'\ge 4$, the number of $n\le x$ with $P_k(n) \equiv a\ \pmod {q}$ and $P_k(n) \le x^{1/U'}$ is eventually at most $C \frac {x}{\phi (q)} \frac {(\log {U'})^{k-2}}{U'}$. If we choose $U'> U$ so large that $C \frac {(\log {U'})^{k-2}}{U'} < \eta \frac {\log (1+D^{-1})}{\log {b}}$, the desired upper bound then follows from Proposition 2.2.
3 Benford's law for the sum of the prime factors: proof of Theorem 1.2
For multiplicative functions $F,G$ taking values on or inside the complex unit circle, we define (following [Reference Granville and Soundararajan13]) the distance between F and G, up to x, by
$$\begin{align*}\mathbb{D}(F,G;x) = \sqrt{\sum_{p \le x} \frac{1-\mathop{\mathrm{Re}}(F(p)\overline{G(p)})}{p}}. \end{align*}$$
The following statement (Corollary 4.12 on page 494 of [Reference Tenenbaum26]), due to Montgomery and Tenenbaum, makes quantitatively precise a result of Halász [Reference Halász14] that F has mean value $0$ unless F "pretends" to be $n^{it}$ for some t.
Proposition 3.1 Let F be a multiplicative function with $|F(n)|\le 1$ for all n. For $x\ge 2$ and $T\ge 2$, let
$$\begin{align*}m(x,T) = \min_{|t| \le T} \mathbb{D}(F,n^{it};x)^2. \end{align*}$$
$$\begin{align*}\sum_{n \le x} F(n) \ll x \frac{1+m(x,T)}{\mathrm{e}^{m(x,T)}}+ \frac{x}{T}. \end{align*}$$
When F is real-valued, the following (slightly weakened version of a) theorem of Hall and Tenenbaum [Reference Hall and Tenenbaum16] allows us to consider only $\mathbb {D}(F,1;x)$.
Proposition 3.2 Let F be a real-valued multiplicative function with $|F(n)|\le 1$ for all n. Then
$$\begin{align*}\sum_{n \le x} F(n) \ll x \exp(-0.3 \cdot \mathbb{D}(F,1;x)^2). \end{align*}$$
Lemma 3.3 Fix $\delta> 0$ and fix $U\ge 1$. For all large x, the number of $n\le x$ with $P(n)\le y$ and $A(n)\equiv a\ \pmod {q}$ is
$$\begin{align*}\frac{\Psi(x,y)}{q}+ O(x/(\log{x})^{\frac{1}{2}-\delta}), \end{align*}$$
for all $x\ge y \ge x^{1/U}$ and residue classes $a\bmod {q}$ with $q\le \log {x}$.
Proof By the orthogonality relations for additive characters,
Hence, it suffices to show that
for each nonzero residue class $r\bmod {q}$.
Write $r/q = r'/q'$ in lowest terms, so that $q'> 1$. If $q'=2$, then $r'=1$, and is a real-valued multiplicative function of modulus at most $1$. Moreover, $\mathbb {D}(F,1;x)^2 \ge \sum _{2 < p \le y} 2/p = 2\log \log {x} + O(1)$. By Proposition 3.2, the left-hand side of (3.1) is $O(x/(\log {x})^{0.6})$, which is more than we need. So we may assume $q'> 2$.
When $q'>2$, we apply Proposition 3.1 taking $T=\log {x}$. Let t be any real number with $|t|\le T$. We set $z= \exp ((\log {x})^{\delta })$ and start from the lower bound
(3.2) $$ \begin{align} \mathbb{D}(F, n^{it}; x)^2 \ge \sum_{z < p \le y} \frac{1-\mathop{\mathrm{Re}}(\mathrm{e}^{2\pi i r' p/q'} p^{-it})}{p}. \end{align} $$
To estimate the right-hand sum, we split the range of summation into blocks on which $p^{-it}$ is essentially constant.
Cover $(z,y]$ with intervals $\mathcal {I}= (u,u(1+1/(\log {x})^2)]$, allowing the rightmost interval to jut out slightly past y but no further than $y+y/(\log {x})^2$. On each interval $\mathcal {I}$, every $p \in \mathcal {I}$ satisfies $|t\log {p} - t\log {u}| \le |t|/(\log {x})^2 \le 1/\log {x}$, so that
$$\begin{align*}|p^{-it} - u^{-it}| = \left|\int_{t \log{u}}^{t\log{p}} \exp(-i\theta)\,d\theta\right| \le 1/\log{x} ,\end{align*}$$
(3.3) $$ \begin{align} \sum_{p \in \mathcal{I}} \frac{1-\mathop{\mathrm{Re}}(\mathrm{e}^{2\pi i r' p/q'} p^{-it})}{p} = \sum_{p \in \mathcal{I}} \frac{1-\mathop{\mathrm{Re}}(\mathrm{e}^{2\pi i r' p/q'} u^{-it})}{p} + O\left(\frac{1}{\log{x}}\sum_{p \in \mathcal{I}} \frac{1}{p}\right).\end{align} $$
The error term when summed over all intervals $\mathcal {I}$ will be $O(\log \log {x}/\log {x})$, which is negligible for us. So we focus on the main term. Observe that $p = (1+o(1))u$ for every $p \in \mathcal {I}$. (Here and below, asymptotic notation refers to the behavior as $x\to \infty $.) Thus,
$$ \begin{align*} \sum_{p \in \mathcal{I}} \frac{1-\mathop{\mathrm{Re}}(\mathrm{e}^{2\pi i r' p/q'} u^{-it})}{p} &\gtrsim \frac{1}{u} \sum_{p \in \mathcal{I}} (1-\mathop{\mathrm{Re}}(\mathrm{e}^{2\pi i r' p/q'} u^{-it})) \\&\gtrsim \frac{1}{u} \sum_{\substack{a'\bmod{q'} \\ \gcd(a',q')=1}} (1-\mathop{\mathrm{Re}}(\mathrm{e}^{2\pi i r' a'/q'} u^{-it})) \pi(\mathcal{I};q',a'), \end{align*} $$
where $\pi (\mathcal {I};q',a')$ denotes the number of primes $p \in \mathcal {I}$ with $p\equiv a'\ \pmod {q'}$. By the Siegel–Walfisz theorem (Proposition 2.4), $\pi (\mathcal {I};q',a') \sim \frac {1}{\phi (q')} \pi (\mathcal {I})$, where $\pi (\mathcal {I})$ is the total count of primes belonging to $\mathcal {I}$. Thus, the above right-hand side is
(3.4) $$ \begin{align}\gtrsim \frac{\pi(\mathcal{I})}{\phi(q') u} \sum_{\substack{a'\bmod{q'} \\ \gcd(a',q')=1}} (1-\mathop{\mathrm{Re}}(\mathrm{e}^{2\pi i r' a'/q'} u^{-it})) &= \frac{\pi(\mathcal{I})}{\phi(q') u} (\phi(q') - \mathop{\mathrm{Re}}(\mu(q') u^{-it})) \notag\\ &\ge \frac{1}{2} \pi(\mathcal{I})/u \gtrsim \frac{1}{2} \sum_{p \in \mathcal{I}}\frac{1}{p}; \end{align} $$
here, we use that $\sum _{a'\ \pmod {q'},~\gcd (a',q')=1} \mathrm {e}^{2\pi i a' r'/q'} = \mu (q')$ (see, for example, [Reference Hardy and Wright17, Theorem 272, p. 309]) and that $\phi (q') - \mathop {\mathrm {Re}}(\mu (q') u^{-it}) \ge \phi (q')-1 \ge \frac {1}{2}\phi (q')$, as $q'> 2$. Combining the last two displays and summing on $\mathcal {I}$,
$$ \begin{align*} \sum_{\mathcal{I}} \sum_{p \in \mathcal{I}}\frac{1-\mathop{\mathrm{Re}}(\mathrm{e}^{2\pi i r' p/q'} u^{-it})}{p} \gtrsim \frac{1}{2} \sum_{\mathcal{I}} \sum_{p \in \mathcal{I}}\frac{1}{p} \ge \frac{1}{2}\sum_{z < p \le y} \frac{1}{p} \gtrsim \frac{1}{2}(1-\delta)\log\log{x}. \end{align*} $$
From (3.3) (and the immediately following remark about the error term), the same lower bound holds for $\sum _{\mathcal {I}}\sum _{p \in \mathcal {I}} \frac {1-\mathop {\mathrm {Re}}(\mathrm {e}^{2\pi i r' p/q'} p^{-it})}{p}$. This double sum essentially coincides with the right-hand side of (3.2), except for possibly including contributions from a few values of $p> y$. However, those contributions are $O(1)$, in fact $\ll \sum _{y < p < y+y/(\log {x})^2} 1/p \ll 1/(\log {x})^2$. Thus, $\mathbb {D}(F,n^{it};x)^2 \gtrsim \frac {1}{2}(1-\delta )\log \log {x}$. In particular, $\mathbb {D}(F,n^{it};x)^2 \ge (\frac {1}{2}-\frac {9}{10}\delta )\log \log {x}$ once x is sufficiently large (in terms of $\delta $ and U). Since this lower bound holds uniformly in t with $|t| \le T$, the desired inequality (3.1) follows from Proposition 3.1.
Using Lemma 3.3, we can establish the following $A(n)$-analogue of Proposition 2.2.
Proposition 3.4 Fix positive integers $k,D$, and b with $b\ge 2$. Fix real numbers $U'> U \ge 1$, and fix $\epsilon> 0$. The number of $n\le x$ for which $A(n)\equiv a\ \pmod {q}$, $P(n)$ begins with the digits of D in base b, and $P(n) \in (x^{1/U'},y]$ is
$$\begin{align*}\frac{1}{q}\frac{\log(1+D^{-1})}{\log{b}}\left(\rho(u)-\rho(U')\right)x + o(x/q), \end{align*}$$
where $u:=\frac {\log {x}}{\log {y}}$, where $x,y\to \infty $ with $y\ge x^{1/U}$, and where $a\bmod {q}$ is any residue class with $q\le (\log {x})^{\frac {1}{2}-\epsilon }$.
Proof (sketch)
The proof is similar to the case $k=1$ of Proposition 2.2, with the needed input on $\Psi (x,y)$ replaced by appeals to Lemma 3.3. We may assume $y = x^{1/u}$ where $u\ge 1$ is fixed. With the intervals $\mathcal {I}_j$ defined as in (2.4), the desired count of n is given by the triple sum
(3.5) $$ \begin{align} \sum_{j\ge 0} \sum_{p \in \mathcal{I}_j \cap (x^{1/U'},y]} \sum_{\substack{n \le x\\P(n)=p \\ A(n)\equiv a\ \pmod{q}}} 1.\end{align} $$
At the cost of a negligible error, we may restrict the outer sum to $j \in \mathcal {J}$, where $\mathcal {J}$ is the collection of nonnegative integers j with $\mathcal {I}_j \subset (x^{1/U'}, y/\exp (\sqrt {\log {x}}))$; indeed, defining (as before) $P:=(x^{1/U'}, bx^{1/U'}]$ and $P':=[y/b\exp (\sqrt {\log {x}}), y]$, the incurred error is of size
$$\begin{align*}\ll x\sum_{p \in P\cup P'} 1/p \ll x/(\log{x})^{1/2}, \end{align*}$$
which is $o(x/q)$. Now, suppose $j \in \mathcal {J}$ and $p \in \mathcal {I}_j$; then, by Lemma 3.3,
$$\begin{align*}\sum_{\substack{n \le x\\P(n)=p \\ A(n)\equiv a\ \ \ \pmod{q}}} 1 = \sum_{\substack{m \le x/p \\ P(m) \le p \\ A(m)\equiv a-p\ \ \ \pmod{q}}} 1 = \frac{1}{q}\Psi(x/p,p) + O\left(\frac{x}{p(\log{(x/p)})^{\frac{1}{2}(1-\epsilon)}}\right).\end{align*}$$
Summing on all $j \in \mathcal {J}$ and all $p \in \mathcal {I}_j$, the contribution from O-terms is
$$\begin{align*}\ll x \sum_{x^{1/U'} < p \le x/2} \frac{1}{p(\log{(x/p)})^{\frac{1}{2}(1-\epsilon)}} \ll \frac{x}{(\log{x})^{\frac{1}{2}(1-\epsilon)}}, \end{align*}$$
which is $o(x/q)$. (Perhaps the simplest way to estimate this last sum on p is to consider, for each j, the contribution from p with $x/p \in (e^j,e^{j+1}]$.) On the other hand, the calculations from the proof of Proposition 2.2 (with $k=1$, $q=1$) already show that
$$\begin{align*}\sum_{j \in \mathcal{J}} \sum_{p \in \mathcal{I}_j} \Psi(x/p,p) = \frac{\log(1+D^{-1})}{\log{b}} (\rho(u)-\rho(U')+o(1))x. \end{align*}$$
Collecting estimates, we deduce that (3.5) is $\frac {1}{q}\frac {\log (1+D^{-1})}{\log {b}}\left (\rho (u)-\rho (U')\right )x + o(x/q)$, as desired.
Proposition 3.4 implies the following variant of Theorem 1.2, with the leading digits of $P(n)$ prescribed (instead of those of $A(n)$).
Proposition 3.5 Fix positive integers $k,D$, and b with $b\ge 2$. Fix a real number $U \ge 1$, and fix $\epsilon> 0$. The number of $n\le x$ for which $A(n)\equiv a\ \pmod {q}$, $P(n)$ begins with the digits of D in base b, and $P(n) \le y$ is
$$\begin{align*}\sim \frac{1}{q}\frac{\log(1+D^{-1})}{\log{b}} \Psi(x,y), \end{align*}$$
where $x,y\to \infty $ with $y\ge x^{1/U}$, and where $a\bmod {q}$ is any residue class with $q\le (\log {x})^{\frac {1}{2}-\epsilon }$.
Proof The proof parallels that of Theorem 1.1. It suffices to show that the count of n in question is eventually larger than $\frac {1}{q} \frac {\log (1+D^{-1})}{\log {b}} \left (\rho (u)-\eta \right )x$ and eventually smaller than $\frac {1}{q} \frac {\log (1+D^{-1})}{\log {b}} \left (\rho (u)+\eta \right )x$. The lower bound follows from Proposition 3.4, fixing $U'$ large enough that $\rho (U') < \eta $. For the upper bound, we fix $U'$ large enough that $\rho (U') < \eta \frac {\log (1+D^{-1})}{\log {b}}$; the upper bound inequality then follows from Lemma 3.3 and Proposition 3.4.
To finish the proof of Theorem 1.2, we show that $P(n)$ and $A(n)$ usually have the same leading digits. We begin by observing that $P(n)$ and $A(n)$ are usually close.
Lemma 3.6 Fix $\delta> 0$. For large x, the number of $n\le x$ for which $A(n)> (1+\delta ) P(n)$ is $O(x (\log \log {x})^2/\log {x})$.
Proof Put $y:= x^{1/2\log \log {x}}$. We may suppose that $P(n)> y$, since by standard results on the distribution of smooth numbers (e.g., Theorem 5.1 on page 512 of [Reference Tenenbaum26]) this condition excludes only $O(x/\log {x})$ integers $n\le x$. If $A(n)> (1+\delta )P(n)$ for one of these remaining n, then $\delta P(n) < \sum _{k>1} P_k(n) \le \Omega (n) P_2(n) \le 2 P_2(n) \log {x}$. Hence, n is divisible by $pp'$ for primes $p, p'$ with $p> y$ and $p' \in (\frac {\delta }{2} p/\log {x},p]$. The number of such $n\le x$ is
$$\begin{align*}x \sum_{y<p\le x}\sum_{\frac{\delta}{2} \frac{p}{\log{x}}<p'\le p}\frac{1}{pp'} \ll x\sum_{y<p\le x} \frac{1}{p} \frac{\log\log{x}}{\log{p}} \ll x\frac{\log\log{x}}{\log{y}} \ll x \frac{(\log\log{x})^2}{\log{x}}. \end{align*}$$
Here, the sum on $p'$ has been estimated using Mertens' theorem with the usual $1/\log $ error term [Reference Tenenbaum26, Theorem 1.10, p. 18].
Lemma 3.7 Fix positive integers N and b, with $b\ge 2$, and fix a real number $\epsilon> 0$. Among all $n\le x$ with $A(n)\equiv a\ \pmod {q}$, the number of n for which the N leading base b digits of $P(n)$ do not coincide with those of $A(n)$ is $o(x/q)$, as $x\to \infty $, uniformly in residue classes $a\bmod {q}$ with $q\le (\log {x})^{\frac {1}{2}-\epsilon }$.
Proof Since b and N are fixed, it is enough to prove the estimate of the lemma under the assumption that the N leading digits in the base b expansion of $P(n)$ are fixed, say as the digits of the positive integer D.
For M a (fixed) positive integer to be specified momentarily, we let $D'$ be the integer obtained by tacking M copies of the digit " $b-1$" on to the end of the b-ary expansion of D. Thus, $D' = b^M D + (b^M-1)$.
Suppose $P(n)$ begins with D in base b, but $A(n)$ does not. We take two cases. First, it may be that $P(n)$ begins with D but not $D'$; in that case, for $A(n)$ to not begin with D, we must have $A(n)/P(n)> 1+1/D'$. By Lemma 3.6, the number of such $n\le x$ is $O(x(\log \log {x})^2/\log {x})$, which is $o(x/q)$. On the other hand, if $P(n)$ begins with $D'$, we apply Proposition 3.5. Taking $y=x$ there, we see that the number of $n\le x$ for which $P(n)$ begins with $D'$ and $A(n)\equiv a\ \pmod {q}$ is $\sim \frac {\log (1+1/D')}{\log {b}}\frac {x}{q}$. Since the coefficient $\frac {\log (1+1/D')}{\log {b}}$ of $\frac {x}{q}$ in this estimate can be made as small as we like by fixing M large enough, we obtain the lemma.
Theorem 1.2 follows from combining Proposition 3.5 with Lemma 3.7.
Remark The range of uniformity in q can be widened under the assumption that q is supported on sufficiently large primes. More precisely, for any fixed $Q \ge 2$, the result of Theorem 1.2 holds uniformly for $q \leq (\log x)^{1-1/Q-\epsilon }$, provided the least prime $P^-(q)$ dividing q is at least $Q+1$. The key observation is that, in the notation of Lemma 3.3, such q have $\phi (q') \ge P^-(q)-1 \ge Q$, which shows that
$$\begin{align*}\frac{\pi(\mathcal{I})}{\phi(q') u} (\phi(q') - \mathop{\mathrm{Re}}(\mu(q') u^{-it})) \ge \left(1-\frac1Q\right) \frac{\pi(\mathcal{I})}{u}\end{align*}$$
in the display (3.4). The remainder of the proof requires only minor modifications.
We thank the referees for their careful reading of the manuscript.
P.P. is supported by the National Science Foundation under award DMS-2001581.
1 In this latter result, the notion of "asymptotic density" in the definition of a Benford function should be replaced with "asymptotic density relative to the set of n with $\tau (n)\ne 0$."
2 Here and later in this proof, "eventually" refers to the limit as taken in Theorem 1.1. That is, a statement holds eventually if there is a real number T such that the statement is true whenever $x, y \ge T$, with $y \ge x^{1/U}$, and with $a\bmod {q}$ a coprime residue class modulo $q \le \frac {\log {x}}{(\log \log x)^{k-1+\epsilon }}$ or, when $k=1$, modulo $q\le (\log {x})^{A}$.
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Critical mass for infinite-time blow-up in a haptotaxis system with nonlinear zero-order interaction
Entire solutions of the Allen–Cahn–Nagumo equation in a multi-dimensional space
Entire and ancient solutions of a supercritical semilinear heat equation
Peter Poláčik 1,, and Pavol Quittner 2,
School of Mathematics, University of Minnesota, Minneapolis, MN 55455, USA
Department of Applied Mathematics and Statistics, Comenius University, Mlynská dolina, 84248 Bratislava, Slovakia
Received July 2019 Published February 2020
Fund Project: The first author is supported in part by NSF grant DMS-1856491. The second author is supported in part by VEGA Grant 1/0347/18 and by the Slovak Research and Development Agency under the contracts No. APVV-14-0378 and APVV-18-0308
We consider the semilinear heat equation $ u_t = \Delta u+u^p $ on $ {\mathbb R}^N $. Assuming that $ N\ge 3 $ and $ p $ is greater than the Sobolev critical exponent $ (N+2)/(N-2) $, we examine entire solutions (classical solutions defined for all $ t\in {\mathbb R} $) and ancient solutions (classical solutions defined on $ (-\infty,T) $ for some $ T<\infty $). We prove a new Liouville-type theorem saying that if $ p $ is greater than the Lepin exponent $ p_L: = 1+6/(N-10) $ ($ p_L = \infty $ if $ N\le 10 $), then all positive bounded radial entire solutions are steady states. The theorem is not valid without the assumption of radial symmetry; in other ranges of supercritical $ p $ it is known not to be valid even in the class of radial solutions. Our other results include classification theorems for nonstationary entire solutions (when they exist) and ancient solutions, as well as some applications in the theory of blowup of solutions.
Keywords: Semilinear heat equation, entire solutions, ancient solutions, Liouville theorems, blowup.
Mathematics Subject Classification: 35K58, 35B08, 35B44, 35B05, 35B53.
Citation: Peter Poláčik, Pavol Quittner. Entire and ancient solutions of a supercritical semilinear heat equation. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 413-438. doi: 10.3934/dcds.2020136
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Peter Poláčik Pavol Quittner | CommonCrawl |
\begin{document}
\title[Elliptic problems with rough boundary data]{Elliptic problems with rough boundary data\\in generalized Sobolev spaces}
\author[A. Anop]{Anna Anop}
\address{Institute of Mathematics, National Academy of Sciences of Ukraine, 3 Tereshchenkivs'ka, Kyiv, 01004, Ukraine}
\email{[email protected]}
\author[R. Denk]{Robert Denk}
\address{University of Konstanz, Department of Mathematics and Statistics, 78457 Konstanz, Germany}
\email{[email protected]}
\author[A. Murach]{Aleksandr Murach}
\address{Institute of Mathematics, National Academy of Sciences of Ukraine, 3 Tereshchenkivs'ka, Kyiv, 01004, Ukraine}
\email{[email protected]}
\subjclass[2010]{35J40, 35R60, 46E35, 60H40}
\keywords{Elliptic boundary value problem, generalized Sobolev space, rough boundary data, Fredholm property, a priory estimate of solution, boundary white noise}
\thanks{The publication contains the results of studies conducted by the joint grant F81 of the National Research Fund of Ukraine and the German Research Society (DFG); competitive project F81/41686.}
\thanks{This work was supported by the Grant H2020-MSCA-RISE-2019, project number 873071 (SOMPATY: Spectral Optimization: From Mathematics to Physics and Advanced Technology).}
\thanks{The first author was supported by President of Ukraine's grant for competitive project F82/45932.}
\begin{abstract} We investigate regular elliptic boundary-value problems in bounded domains and show the Fredholm property for the related operators in an extended scale formed by inner product Sobolev spaces (of arbitrary real orders) and corresponding interpolation Hilbert spaces. In particular, we can deal with boundary data with arbitrary low regularity. In addition, we show interpolation properties for the extended scale, embedding results, and global and local \textit{a priori} estimates for solutions to the problems under investigation. The results are applied to elliptic problems with homogeneous right-hand side and to elliptic problems with rough boundary data in Nikoskii spaces, which allows us to treat some cases of white noise on the boundary. \end{abstract}
\maketitle
\section{Introduction}\label{sec1}
In this paper, we investigate elliptic boundary-value problems of the form \begin{equation*} A u =f\;\;\text{ in }\Omega,\quad
B_j u = g_j \;\;\text{ on }\Gamma, \;\; j=1,\dots,q, \end{equation*} in classes of generalized Sobolev spaces. Here, $\Omega\subset\mathbb R^n$ is a bounded domain with boundary $\Gamma\in C^{\infty}$, $A$ is a linear partial differential operator (PDO) of order $2q$, and $B_j(x,D)$, $\nobreak{j=1,\dots,q}$, are linear boundary PDOs of order $m_j<2q$. We assume all coefficients to be infinitely smooth and the boundary-value problem $(A,B):=(A,B_1,\dots,B_q)$ to be regular elliptic. The aim of the present paper is the analysis of this problem in the so-called extended Sobolev scale of Hilbert distribution spaces. They are of the form $H^\alpha(\Omega)$, where $\alpha\in\textrm{OR}$ is an O-regularly varying function (see, e.g., \cite[Section~2.0.2]{BinghamGoldieTeugels89}). Note that the smoothness parameter $\alpha$ is a function, in contrast to the classical Sobolev spaces, where the smoothness is measured by some real number. The Hilbert spaces $H^\alpha(\Omega)$ are special cases of distribution spaces introduced by H\"ormander \cite{Hermander63, Hermander83} for a wide class of weight functions and based on the $L_p$-norm. In the situation considered here, the weight function is radially symmetric, and we restrict ourselves to the Hilbert space case of $p=2$. We remark that for $p=2$ the H\"ormander spaces coincide with the spaces introduced by Volevich and Paneah in \cite[Section~2]{VolevichPaneah65}. The class $\{H^\alpha(\Omega):\alpha\in\textrm{OR}\}$ contains the classical Sobolev spaces $H^{r}(\Omega)$ with $r\in\mathbb{R}$ and can be seen as a finer scale of regularity, which allows for more precise embedding and trace theorems. On the other hand, the space $H^\alpha(\Omega)$ can be obtained from the classical Sobolev spaces by interpolation with a function parameter, see Section~\ref{sec5} below.
Recently, Mikhailets and Murach developed a general theory of solvability of elliptic boundary-value problems in a class of H\"ormander Hilbert spaces called the refined Sobolev scale (see \cite{MikhailetsMurach05UMJ5, MikhailetsMurach06UMJ3, MikhailetsMurach06UMJ11, MikhailetsMurach07UMJ5}, and the monograph \cite{MikhailetsMurach14}). The (larger) extended Sobolev scale was considered in \cite{AnopMurach14UMJ}. In these publications, the boundary data had sufficient regularity to guarantee the existence of boundary traces. More precisely, if $\alpha(t)\equiv\varphi(t)t^{2q}$, then the lower Matuszewska index of $\varphi$ was assumed to be larger than $-1/2$ (see Section~\ref{sec3} and Proposition~\ref{prop1} below for details). Motivated by applications with rough boundary data, in this paper we consider the situation where this condition on the Matuszewska index does not hold. Even for Sobolev spaces, the case of rough boundary data is quite sophisticated. One approach is the modification of the Sobolev spaces with low regularity as developed by Roitberg \cite{Roitberg64,Roitberg96,Roitberg99}. Another way to treat this problem is to include the norm of $Au$ in the norm of the Sobolev space, see Lions and Magenes \cite[Chapter~2, Section~6]{LionsMagenes72}. In connection with negative order boundary spaces, we also refer to \cite{GesztesyMitrea08} for recent results on weak and very weak traces and to \cite[Chapter~5]{BehrndtHassideSnoo20} for the theory of boundary triplets.
This paper has the following structure: Section~\ref{sec2} contains the precise formulation of the boundary-value problem $(A,B)$; in Section~\ref{sec3} we introduce the extended Sobolev scales over $\mathbb R^n$, $G$, and $\Gamma$. The main results are formulated in Section~\ref{sec4}. We show here that $(A,B)$ induces a Fredholm operator in the extended Sobolev scale (Theorem~\ref{th1}). We obtain global and local (up to the boundary) elliptic regularity in the extended scale (see Theorems~\ref{th4.6} and~\ref{th4.7}, resp.) and elliptic \textit{a priori} estimates (see Theorem~\ref{th4.12} for the global and Theorem~\ref{th4.13} for the local version). Theorems \ref{th4.7} and \ref{th4.13} are new even in the case of Sobolev spaces. In Section~\ref{sec5}, we discuss interpolation properties of the extended Sobolev scale, which will also be used in the proof of the main results in Section~\ref{sec6}. In Section~\ref{sec7}, we study semi-homogeneous boundary value problems, namely the case of $f=0$. Defining the space $H^\alpha_A(\Omega):=\{ u\in H^\alpha(\Omega): Au =0\}$, we obtain, e.g., conditions for uniform convergence of sequences of solutions to the homogeneous elliptic equation (Theorems \ref{th7.5} and \ref{th7.6}) and interpolation properties for $H^\alpha_A(\Omega)$ (Theorems \ref{th7.8} and \ref{th7.9}). Finally, in Section~\ref{sec8} we apply the results to elliptic boundary-value problems whose boundary data belong to some Nikolskii space $B^{s}_{2,\infty}(\Gamma)$. Based on an embedding result (Proposition~\ref{8.1}), we show that the solution belongs pathwise to the space $H^\alpha(\Omega)$ under some condition on $\alpha$. The investigation of such boundary-value problems is motivated by recent results on boundary noise (see, e.g., \cite{SchnaubeltVeraar11}) and on the Besov smoothness of white noise \cite{FageotFallahUnser17,Veraar11}.
\section{Statement of the problem}\label{sec2} Let $\Omega\subset\mathbb{R}^n$, where $n\geq2$, be a bounded domain with an infinitely smooth boundary $\Gamma$. We consider the following boundary value problem: \begin{align} Au& =f\quad\mbox{in}\;\Omega,\label{f1}\\ B_{j}u& =g_{j}\quad\mbox{on}\;\Gamma,\quad j=1,...,q.\label{f2} \end{align} Here, $$
A:=A(x,D):=\sum_{|\mu|\leq 2q}a_{\mu}(x)D^{\mu}\ $$ is a linear PDO on $\overline{\Omega}:=\Omega\cup\Gamma$ of even order $2q\geq2$, and each $$
B_{j}:=B_{j}(x,D)=\sum_{|\mu|\leq m_{j}}b_{j,\mu}(x)D^{\mu}\ $$ is a linear boundary PDO on $\Gamma$ of order $m_{j}\leq2q-1$. All the coefficients $a_{\mu}$ and $b_{j,\mu}$ of these PDOs belong to the complex spaces $C^{\infty}(\overline{\Omega})$ and $C^{\infty}(\Gamma)$, resp. Let $B:=(B_{1},\ldots,B_{q})$ and $g:=(g_{1},\ldots,g_{q})$.
We use the following standard notation:
$\mu:=(\mu_{1},\ldots,\mu_{n})$ is a multi-index with nonnegative integer components, $|\mu|:=\mu_{1}+\cdots+\mu_{n}$, $D^{\mu}:=D_{1}^{\mu_{1}}\cdots D_{n}^{\mu_{n}}$, $D_{k}:=i\partial/\partial x_{k}$, $k=1,...,n$, where $i$ is imaginary unit and $x=(x_1,\ldots,x_n)$ is an arbitrary point in $\mathbb{R}^{n}$.
We suppose throughout the paper that the boundary value problem \eqref{f1}, \eqref{f2} is regular elliptic in $\Omega$. This means that the PDO $A$ is properly elliptic on $\overline{\Omega}$ and that the system $B$ of boundary PDOs is normal and satisfies the Lopatinskii condition with respect to $A$ on $\Gamma$ (see, e.g., the survey \cite[Section~1.2]{Agranovich97}). Recall that, since the system $B$ is normal, the orders $m_{j}$ of $B_{j}$ are all different.
We investigate properties of the extension (by continuity) of the mapping \begin{equation}\label{mapping} u\mapsto(Au,Bu)=(Au,B_{1}u,\ldots,B_{q}u),\quad\mbox{where}\quad u\in C^{\infty}(\overline{\Omega}), \end{equation} on appropriate pairs of Hilbert distribution spaces. To describe the range of this extension, we need the following Green's formula: \begin{equation*} (Au,v)_{\Omega} + \sum^{q}_{j=1}(B_{j}u,C^{+}_{j}v)_{\Gamma} = (u,A^{+}v)_{\Omega} + \sum_{j=1}^{q}(C_{j}u,B^{+}_{j}v)_{\Gamma} \end{equation*} for arbitrary $u,v\in C^{\infty}(\overline{\Omega})$. Here, $$
A^{+}v(x):=\sum_{|\mu|\leq2q}D^{\mu}(\overline{a_{\mu}(x)}\,v(x)) $$ is the linear PDO which is formally adjoint to $A$, and $\{B^+_j\}$, $\{C_j\}$, $\{C^+_j\}$ are some normal sets of linear boundary PDOs with coefficients from $C^\infty(\Gamma)$. The orders of these PDOs satisfy the condition $$ \mathrm{ord}\,B_j+\mathrm{ord}\,C^+_j=\mathrm{ord}\,C_j+\mathrm{ord}\,B^+_j =2q-1. $$ In Green's formula and below, $(\cdot,\cdot)_\Omega$ and $(\cdot,\cdot)_\Gamma$ denote the inner products in the complex Hilbert spaces $L_2(\Omega)$ and $L_2(\Gamma)$ of all functions that are square integrable over $\Omega$ and $\Gamma$, respectively (relative to the Lebesgue measure, of course), and also denote extensions by continuity of these inner products.
The boundary value problem \begin{align}\label{f3} A^{+}v&=w,\quad\mbox{in}\;\Omega,\\ B^{+}_{j}v&=h_{j},\quad\mbox{on}\;\Gamma,\quad j=1,\ldots,q, \label{f4} \end{align} is called formally adjoint to the problem \eqref{f1}, \eqref{f2} with respect to the given Green formula. The latter problem is regular elliptic if and only if the formally adjoint problem \eqref{f3}, \eqref{f4} is regular elliptic \cite[Chapter~2, Section~2.5]{LionsMagenes72}.
Denote \begin{gather*} N:=\bigl\{u\in C^{\infty}(\overline{\Omega}): \,Au=0\;\,\mbox{in}\;\,\Omega,\;\, Bu=0\;\,\mbox{on}\;\,\Gamma\bigr\},\\ N^{+}:=\bigl\{v\in C^{\infty}(\overline{\Omega}): \,A^{+}v=0\;\,\mbox{in}\;\,\Omega,\;\, B^{+}v=0\;\,\mbox{on}\;\,\Gamma\bigr\}, \end{gather*} with $B^{+}:=(B^{+}_{1},\ldots, B^{+}_{q})$. Since both problems \eqref{f1}, \eqref{f2} and \eqref{f3}, \eqref{f4} are regular elliptic, both spaces $N$ and $N^+$ are finite-dimensional \cite[Chapter~2, Section~2.5]{LionsMagenes72}. Besides, the space $N^{+}$ is independent of any choice of the collection $B^{+}$ of boundary differential expressions that satisfy Green's formula.
\section{Generalized Sobolev spaces}\label{sec3}
We investigate the boundary value problem \eqref{f1}, \eqref{f2} in certain Hilbert distribution spaces that are generalizations of inner product Sobolev spaces (of an arbitrary real order) to the case where a general enough function parameter is used as an order of the space. Such spaces were introduced and investigated by Malgrange \cite{Malgrange57}, H\"ormander \cite[Sec. 2.2]{Hermander63}, and Volevich and Paneah \cite[Section~2]{VolevichPaneah65}.
This function parameter ranges over a certain class OR of O-regularly varying functions. By definition, OR is the class of all Borel measurable functions $\alpha:[1,\infty)\rightarrow(0,\infty)$ such that \begin{equation}\label{f3.1} c^{-1}\leq\frac{\alpha(\lambda t)}{\alpha(t)}\leq c\quad\mbox{for arbitrary}\quad t\geq1\quad\mbox{and}\quad\lambda\in[1,b] \end{equation} with some numbers $b>1$ and $c\geq1$ that are independent of both $t$ and $\lambda$ (but may depend on $\alpha$). Such functions are called O-regularly varying at infinity in the sense of Avakumovi\'c \cite{Avakumovic36} and are well investigated \cite{BinghamGoldieTeugels89, BuldyginIndlekoferKlesovSteinebach18, Seneta76}.
The class OR admits the simple description \begin{equation*} \alpha\in\mathrm{OR}\quad\Longleftrightarrow\quad\alpha(t)= \exp\Biggl(\beta(t)+\int\limits_{1}^{t}\frac{\gamma(\tau)}{\tau}\; d\tau\Biggr), \;\;t\geq1, \end{equation*} where the real-valued functions $\beta$ and $\gamma$ are Borel measurable and bounded on $[1,\infty)$. Condition \eqref{f3.1} is equivalent to the following: there exist real numbers $r_{0}\leq r_{1}$ and positive numbers $c_{0}$ and $c_{1}$ such that \begin{equation}\label{f3.2} c_{0}\lambda^{r_{0}}\leq\frac{\alpha(\lambda t)}{\alpha(t)}\leq c_{1}\lambda^{r_{1}}\quad\mbox{for all}\quad t\geq1\quad\mbox{and}\quad\lambda\geq1. \end{equation} For every function $\alpha\in\mathrm{OR}$, we define the lower and the upper Matuszewska indices \cite{Matuszewska64} by the formulas \begin{gather}\label{f3.2sup} \sigma_{0}(\alpha):=\sup\{r_{0}\in\mathbb{R}:\,\mbox{the left-hand inequality in \eqref{f3.2} holds}\},\\ \sigma_{1}(\alpha):=\inf\{r_{1}\in\mathbb{R}:\,\mbox{the right-hand inequality in \eqref{f3.2} holds}\},\label{f3.2inf} \end{gather} with $-\infty<\sigma_{0}(\alpha)\leq\sigma_{1}(\alpha)<\infty$ (see also \cite[Theorem~2.2.2]{BinghamGoldieTeugels89}).
A standard example of functions from $\mathrm{OR}$ is given by a continuous function $\alpha:[1,\infty)\rightarrow(0,\infty)$ such that \begin{equation*} \alpha(t):=t^{r}(\log t)^{k_{1}}(\log\log t)^{k_{2}}\ldots(\underbrace{\log\ldots\log}_{j\;\mathrm{times}} t)^{k_{j}}\quad\mbox{for}\quad t\gg1. \end{equation*} Here, we arbitrarily choose an integer $j\geq1$ and real numbers $r,k_{1},\ldots,k_{j}$. This function has equal Matuszewska indices $\sigma_{0}(\alpha)=\sigma_{1}(\alpha)=r$.
Generally, the class OR contains an arbitrary Borel measurable function $\alpha:[1,\infty)\rightarrow(0,\infty)$ such that both functions $\alpha$ and $1/\alpha$ are bounded on every bounded subset of $[1,\infty)$ and that the function $\alpha$ is regularly varying at infinity in the sense of Karamata, i.e. there exists a real number $r$ such that \begin{equation*} \lim_{t\rightarrow\infty}\;\frac{\alpha(\lambda\,t)}{\alpha(t)}= \lambda^{r}\quad\mbox{for every}\;\lambda>0. \end{equation*} In this case $\sigma_{0}(\alpha)=\sigma_{1}(\alpha)=r$, and $r$ is called the order of $\alpha$. If $r=0$, the function $\alpha$ is called slowly varying at infinity.
A simple example of a function $\alpha\in\mathrm{OR}$ with the different Matuszewska indices is given by the formula \begin{equation*} \alpha(t):=\left\{ \begin{array}{ll} t^{\theta+\delta\sin((\log\log t)^{\lambda})}\; &\hbox{if}\;\;t>e,\\ t^{\theta}\; &\hbox{if}\;\;1\leq t\leq e. \end{array}\right. \end{equation*} Here, we arbitrarily choose numbers $\theta\in\mathbb{R}$, $\delta>0$, and $\lambda\in(0,1]$. Then $\sigma_{0}(\alpha)=\theta-\delta$ and $\sigma_{1}(\alpha)=\theta+\delta$ whenever $0<\lambda<1$, but $\sigma_{0}(\alpha)=\theta-\sqrt{2}\delta$ and $\sigma_{1}(\alpha)=\theta+\sqrt{2}\delta$ if $\lambda=1$. If $\lambda>1$, then $\alpha\notin\mathrm{OR}$.
Let $\alpha\in\mathrm{OR}$, and introduce the generalized Sobolev spaces $H^{\alpha}$ over $\mathbb{R}^{n}$, with $n\geq1$, and then over $\Omega$ and~$\Gamma$. We consider complex-valued functions and distributions and therefore use complex linear spaces. It is useful for us to interpret distributions as antilinear functionals on a relevant space of test functions.
By definition, the linear space $H^{\alpha}(\mathbb{R}^{n})$ consists of all distributions $w\in\mathcal{S}'(\mathbb{R}^{n})$ such that their Fourier transform $\widehat{w}:=\mathcal{F}w$ is locally Lebesgue integrable over $\mathbb{R}^{n}$ and satisfies the condition $$ \int\limits_{\mathbb{R}^{n}}\alpha^2(\langle\xi\rangle)\,
|\widehat{w}(\xi)|^2\,d\xi<\infty. $$ As usual, $\mathcal{S}'(\mathbb{R}^{n})$ is the linear topological space of tempered distributions in $\mathbb{R}^{n}$, and
$\langle\xi\rangle:=(1+|\xi|^{2})^{1/2}$ whenever $\xi\in\mathbb{R}^{n}$. The space $H^{\alpha}(\mathbb{R}^{n})$ is endowed with the inner product $$ (w_1,w_2)_{\alpha,\mathbb{R}^{n}}:= \int_{\mathbb{R}^{n}}\alpha^2(\langle\xi\rangle)\, \widehat{w_1}(\xi)\,\overline{\widehat{w_2}(\xi)}\,d\xi, $$
and the corresponding norm $\|w\|_{\alpha,\mathbb{R}^{n}}:= (w,w)_{\alpha,\mathbb{R}^{n}}^{1/2}$. We call $\alpha$ the order or regularity index of $H^{\alpha}(\mathbb{R}^{n})$ (and its analogs for $\Omega$ and $\Gamma$).
The space $H^{\alpha}(\mathbb{R}^{n})$ is an isotropic Hilbert case of the spaces $B_{p,k}$ introduced and systematically investigated by H\"ormander \cite[Section~2.2]{Hermander63} (see also \cite[Section~10.1]{Hermander83}). Namely, $H^{\alpha}(\mathbb{R}^{n})=B_{p,k}$ provided that $p=2$ and $k(\xi)=\alpha(\langle\xi\rangle)$ for all $\xi\in\mathbb{R}^{n}$.
If $\alpha(t)\equiv t^{r}$, then $H^{\alpha}(\mathbb{R}^{n})=:H^{r}(\mathbb{R}^{n})$ is the inner product Sobolev space of order $r\in\mathbb{R}$. Generally, \begin{equation}\label{f3.3} r_{0}<\sigma_{0}(\alpha)\leq\sigma_{1}(\alpha)<r_{1}\;\;\Rightarrow\;\; H^{r_1}(\mathbb{R}^{n})\hookrightarrow H^{\alpha}(\mathbb{R}^{n})\hookrightarrow H^{r_0}(\mathbb{R}^{n}), \end{equation} both embeddings being continuous and dense. This is a consequence of the property \eqref{f3.2} considered for $t=1$.
A relation between $H^\alpha(\mathbb{R}^{n})$ and the space of $p$ times continuously differentiable functions reveals H\"ormander's embedding theorem \cite[Theorem~2.2.7]{Hermander63}, which is formulated in the $\alpha\in\mathrm{OR}$ case as follows \cite[Lemma~2]{ZinchenkoMurach12UMJ11}: \begin{equation}\label{Hermander-embedding} \int\limits_1^{\infty} t^{2p+n-1}\alpha^{-2}(t)\,dt<\infty\;\;\Longleftrightarrow\;\; \{w\in H^\alpha(\mathbb{R}^{n}):\mathrm{supp}\,w\subset U\}\subset C^p(\mathbb{R}^{n}); \end{equation} here, $0\leq p\in\mathbb{Z}$, and $U$ is an open nonempty subset of $\mathbb{R}^{n}$ (the case of $U=\mathbb{R}^{n}$ is possible).
Remark that we use the same designation $H^{\alpha}$ both in the case where $\alpha$ is a function and in the case where $\alpha$ is a number. This will not lead to ambiguity because we will always specify what $\alpha$ means, a function or number. This remark also concerns designations of spaces induced by $H^{\alpha}(\mathbb{R}^{n})$ and, of course, the notation of the norm and inner product in the corresponding spaces.
Following \cite{MikhailetsMurach13UMJ3}, we call the class $\{H^{\alpha}(\mathbb{R}^{n}):\alpha\in\mathrm{OR}\bigr\}$ the extended Sobolev scale over $\mathbb{R}^{n}$. Its analogs for $\Omega$ and $\Gamma$ are introduced in the standard way (see \cite[Section~2]{MikhailetsMurach15ResMath1} and \cite[Section~2.4.2]{MikhailetsMurach14}, resp.). Let us give the necessary definitions.
By definition, \begin{gather}\notag H^{\alpha}(\Omega):=\bigl\{w\!\upharpoonright\!\Omega: w\in H^{\alpha}(\mathbb{R}^{n})\bigr\},\\
\|u\|_{\alpha,\Omega}:=
\inf\bigl\{\,\|w\|_{\alpha,\mathbb{R}^{n}}: w\in H^{\alpha}(\mathbb{R}^{n}),\;w=u\;\,\mbox{in}\;\,\Omega\bigr\}, \label{4f9} \end{gather} with $u\in H^{\alpha}(\Omega)$. The linear space $H^{\alpha}(\Omega)$ is Hilbert and separable with respect to the norm \eqref{4f9} because $H^{\alpha}(\Omega)$ is the factor space of the separable Hilbert space $H^\alpha(\mathbb{R}^{n})$ by its subspace \begin{equation}\label{f3.5} \bigl\{w\in H^{\alpha}(\mathbb{R}^{n}):\, \mathrm{supp}\,w\subseteq\mathbb{R}^{n}\setminus\Omega\bigr\}. \end{equation} The norm \eqref{4f9} is induced by the inner product \begin{equation*} (u_1,u_2)_{\alpha,\Omega}:= (w_1-\Pi w_1,w_2-\Pi w_2)_{\alpha,\mathbb{R}^{n}}. \end{equation*} Here, $u_j\in H^\alpha(\Omega)$, $w_j\in H^{\alpha}(\mathbb{R}^{n})$, and $u_j=w_j$ in $\Omega$ for each $j\in\{1,2\}$, whereas $\Pi$ is the orthoprojector of $H^{\alpha}(\mathbb{R}^{n})$ onto \eqref{f3.5}.
The space $H^\alpha(\Omega)$ is continuously embedded in the linear topological space $\mathcal{D}'(\Omega)$ of all distributions in $\Omega$, and the set $C^\infty(\overline{\Omega})$ is dense in $H^\alpha(\Omega)$. Note that $H^\alpha(\Omega)$ is an isotropic case of Hilbert spaces introduced and investigated by Volevich and Paneah \cite[Section~3]{VolevichPaneah65}.
The linear space $H^{\alpha}(\Gamma)$ consists of all distributions on $\Gamma$ that yield elements of $H^{\alpha}(\mathbb{R}^{n-1})$ in local coordinates on $\Gamma$. Let us give a detailed definition. The boundary $\Gamma$ of $\Omega$ is an infinitely smooth closed manifold of dimension $n-1$, with the $C^{\infty}$-structure on $\Gamma$ being induced by $\mathbb{R}^{n}$. From this structure, we choose a finite collection of local charts $\pi_{j}:\mathbb{R}^{n-1}\leftrightarrow\Gamma_{j}$, $j=1,\ldots,\varkappa$, where the open sets $\Gamma_{j}$ form a covering of $\Gamma$. We also choose functions $\chi_{j}\in C^{\infty}(\Gamma)$, $j=1,\ldots,\varkappa$, that satisfy the condition $\mathrm{supp}\,\chi_{j}\subset\Gamma_{j}$ and that form a partition of unity on $\Gamma$. Then \begin{equation*} H^{\alpha}(\Gamma):=\bigl\{h\in\mathcal{D}'(\Gamma):\, (\chi_{j}h)\circ\pi_{j}\in H^{\alpha}(\mathbb{R}^{n-1})\;\;\mbox{for every}\;\;j\in\{1,\ldots,\varkappa\}\bigr\}. \end{equation*} Here, as usual, $\mathcal{D}'(\Gamma)$ denotes the linear topological space of all distributions on~$\Gamma$, and $(\chi_{j}h)\circ\pi_{j}$ stands for the representation of the distribution $\chi_{j}h$ in the local chart $\pi_{j}$. The space $H^{\alpha}(\Gamma)$ is endowed with the inner product $$ (h_{1},h_{2})_{\alpha,\Gamma}:= \sum_{j=1}^{\varkappa}\,((\chi_{j}h_{1})\circ\pi_{j}, (\chi_{j}\,h_{2})\circ\pi_{j})_{\alpha,\mathbb{R}^{n-1}} $$
and the corresponding norm $\|h\|_{\alpha,\Gamma}:=(h,h)_{\alpha,\Gamma}^{1/2}$. The space $H^\alpha (\Gamma)$ is Hilbert and separable and does not depend (up to equivalence of norms) on our choice of local charts and partition of unity on~$\Gamma$ \cite[Theorem~2.21]{MikhailetsMurach14}. This space is continuously embedded in $\mathcal{D}'(\Gamma)$, and the set $C^{\infty}(\Gamma)$ is dense in $H^{\alpha}(\Gamma)$.
The above-defined function spaces form the extended Sobolev scales $\{H^{\alpha}(\Omega):\alpha\in\mathrm{OR}\}$ and $\{H^{\alpha}(\Gamma):\alpha\in\mathrm{OR}\}$ over $\Omega$ and $\Gamma$ respectively. They contain the scales of inner product Sobolev spaces; namely, if $\alpha(t)\equiv t^r$ for certain $r\in\mathbb{R}$, then $H^\alpha(\Omega)=:H^{r}(\Omega)$ and $H^\alpha(\Gamma)=:H^{r}(\Gamma)$ are the Sobolev spaces of order~$r$. Property \eqref{f3.3} remains true provided that we replace $\mathbb{R}^{n}$ with $\Omega$ or $\Gamma$, the embeddings being compact. As we have noted, the norm in $H^{r}(G)$ is denoted by $\|\cdot\|_{r,G}$, with $G\in\{\mathbb{R}^{n},\Omega,\Gamma\}$.
The extended Sobolev scales have important interpolation properties: they are obtained by the interpolation with a function parameter between inner product Sobolev spaces, are closed with respect to the interpolation with a function parameter between Hilbert spaces, and consist (up to equivalence of norms) of all Hilbert spaces that are interpolation ones between inner product Sobolev spaces. We will discuss these properties in Section~\ref{sec5}. The first of them plays a key role in applications of these scale to elliptic operators and elliptic problems.
\section{Main results}\label{sec4}
Dealing with the problem \eqref{f1}, \eqref{f2}, we will use the generalized Sobolev space $H^{\alpha}(G)$, with $G\in\{\Omega,\Gamma\}$, whose order is a function parameter of the form $\alpha(t)\equiv\varphi(t)t^{s}$ where $\varphi\in\mathrm{OR}$ and $s\in\mathbb{R}$. In order not to indicate the argument $t$ of the function parameter, we resort to the function $\varrho(t):=t$ of $t\geq1$. Then $\alpha$ can be written as $\varphi\varrho^{s}$ not using $t$. Note if $\varphi\in\mathrm{OR}$ and $s\in\mathbb{R}$, then $\varphi\rho^s\in\mathrm{OR}$ and $\sigma_j(\varphi\varrho^s)=\sigma_j(\varphi)+s$ for each $j\in\{0,1\}$.
It is well known that the elliptic problem \eqref{f1}, \eqref{f2} is Fredholm on appropriate pairs of Sobolev spaces of sufficiently large orders and that its index does not depend on these orders (see, e.g., \cite[Section 2.4~a]{Agranovich97} or \cite[Chapter~2, Section~5.4]{LionsMagenes72}). This result was extended to generalized Sobolev spaces in \cite[Theorem~1]{AnopMurach14UMJ} as follows:
\begin{proposition}\label{prop1} Let $\varphi\in\mathrm{OR}$, and suppose that $\sigma_0(\varphi)>-1/2$. Then the mapping \eqref{mapping} extends uniquely (by continuity) to a bounded linear operator \begin{equation}\label{f9} (A,B):H^{\varphi\varrho^{2q}}(\Omega)\rightarrow H^{\varphi}(\Omega)\oplus \bigoplus_{j=1}^{q}H^{\varphi\varrho^{2q-m_j-1/2}}(\Gamma) =:\mathcal{H}^\varphi(\Omega,\Gamma). \end{equation} This operator is Fredholm. Its kernel coincides with $N$, and its range consists of all vectors $(f,g)\in\mathcal{H}^\varphi(\Omega,\Gamma)$ such that \begin{equation}\label{f10} (f,v)_{\Omega}+\sum_{j=1}^{q}\,(g_{j},C^{+}_{j}v)_{\Gamma}=0\quad \mbox{for each}\quad v\in N^{+}. \end{equation} The index of the operator \eqref{f9} equals $\dim N-\dim N^{+}$ and does not depend on~$\varphi$. \end{proposition}
Recall that the bounded linear operator $T:\nobreak E_{1}\rightarrow E_{2}$ between Banach spaces $E_{1}$ and $E_{2}$ is called Fredholm if its kernel $\ker T$ and cokernel $E_{2}/T(E_{1})$ are finite-dimensional. If $T$ is Fredholm, then its range $T(E_{1})$ is closed in $E_{2}$ (see, e.g., \cite[Ëåììà~19.1.1]{Hermander85}) and the index $\mathrm{ind}\,T:=\dim\ker T-\dim(E_{2}/T(E_{1}))$ is finite.
As to formula \eqref{f10}, recall that the inner product in $L_{2}(\Omega)$ extends by continuity to a sesquilinear form $(f,v)_{\Omega}$ of arbitrary arguments $f\in H^{-1/2+}(\Omega)$ and $v\in H^{1/2}(\Omega)$ (see, e.g., \cite[Theorem~4.8.2(b)]{Triebel95}). Here and below, \begin{equation*} H^{r+}(\Omega):=\bigcup_{\ell>r}H^{\ell}(\Omega)= \bigcup_{\substack{\varphi\in\mathrm{OR},\\\sigma_0(\varphi)>r}} H^{\varphi}(\Omega)\quad\mbox{for every}\quad r\in\mathbb{R}. \end{equation*} Thus, the first summand in \eqref{f10} is well defined. The next summands are also well defined being equal to the value of the distribution $g_{j}\in\mathcal{D}'(\Gamma)$ at the test function $C^{+}_{j}v\in C^{\infty}(\Gamma)$.
Proposition~\ref{prop1} is not true in the case of $\sigma_0(\varphi)\leq-1/2$. This is stipulated by the fact that the mapping $u\mapsto B_{j}u$, where $u\in C^{\infty}(\overline{\Omega})$, can not be extended to the continuous linear operator $B_{j}:H^{(s+2q)}(\Omega)\rightarrow\mathcal{D}'(\Gamma)$ if $s+2q\leq m_{j}+1/2$ (see \cite[Chapter~1, Òåîðåìà~9.5]{LionsMagenes72}). Therefore to obtain a version of Proposition~\ref{prop1} in this case, we have to take a narrower space than $H^{\varphi\varrho^{2q}}(\Omega)$ as the domain of the operator $(A,B)$. We will show that it is possible for this purpose to take the space of all distributions $u\in H^{\varphi\varrho^{2q}}(\Omega)$ such that $Au\in H^{\eta}(\Omega)$ for certain $\eta\in\mathrm{OR}$ subject to $\sigma_0(\eta)>-1/2$.
Let us previously consider this space for arbitrary function parameters $\alpha:=\varphi\varrho^{2q}$ and $\eta$ from $\mathrm{OR}$. We put \begin{equation}\label{f11} H^{\alpha}_{A,\eta}(\Omega):= \bigl\{u\in H^{\alpha}(\Omega):\,Au\in H^\eta(\Omega)\bigr\}, \end{equation} with $Au$ being understood in the sense of the theory of distributions. The linear space \eqref{f11} is endowed with the graph inner product \begin{equation*} (u_1,u_2)_{\alpha,A,\eta}:= (u_1,u_2)_{\alpha,\Omega}+(Au_1,Au_2)_{\eta,\Omega} \end{equation*}
and the corresponding norm $\|u\|_{\alpha,A,\eta}:=(u,u)_{\alpha,A,\eta}^{1/2}$.
The space $H^{\alpha}_{A,\eta}(\Omega)$ is complete, i.e. Hilbert. Indeed if $(u_{k})$ is a Cauchy sequence in this space, there exist limits $u:=\lim u_{k}$ in $H^{\alpha}(\Omega)$ and $f:=\lim Au_{k}$ in $H^{\eta}(\Omega)$. Since the PDO $A$ is continuous in $\mathcal{D}'(\Omega)$, the first limit implies that $Au=\lim Au_{k}$ in $\mathcal{D}'(\Omega)$. Hence, $Au=f\in H^{\eta}(\Omega)$. Therefore, $u\in H^{\alpha}_{A,\eta}(\Omega)$ and $\lim u_{k}=u$ in the space $H^{\alpha}_{A,\eta}(\Omega)$, i.e. this space is complete.
If $\alpha(t)\equiv t^{r}$ and $\eta(t)\equiv t^{\lambda}$ for some $r,\lambda\in\mathbb{R}$ (the Sobolev case), the space $H^{r}_{A,\lambda}(\Omega):=H^{\alpha}_{A,\eta}(\Omega)$ is investigated in \cite{KasirenkoMikhailetsMurach19}. This space is used in the theory of elliptic problems in negative Sobolev spaces \cite{Geymonat62, LionsMagenes61II, LionsMagenes62V, LionsMagenes63VI, Magenes65, MikhailetsMurach14, Murach09MFAT2}. If the functions $\alpha$ and $\eta$ are regularly varying at infinity, the space $H^{\alpha}_{A,\eta}(\Omega)$ is also applied to these problems (see \cite[Section~4.5.2]{MikhailetsMurach14} and \cite{AnopKasirenkoMurach18UMJ3}). Note that $H^{\alpha}_{A,\eta}(\Omega)$ may depend on each coefficient of the differential expression $A$, even when all these coefficients are constant. For instance, this is the case if $\alpha(t)\equiv\eta(t)\equiv1$ \cite[Theorem~3.1]{Hermander55}. Recall in this connection that the space $H^{0}_{A,0}(\Omega)$ is the domain of the maximal operator that corresponds to the unbounded operator $C^{\infty}(\overline{\Omega})\ni u\mapsto Au$ in $L_{2}(\Omega)$ (see, e.g., \cite{Hermander55}).
In the sequel, we suppose that $\varphi\in\mathrm{OR}$ and consider the case where $\sigma_0(\varphi)\leq-1/2$. Let us formulate our key result, which is a version of Proposition~\ref{prop1} in this case. We choose real numbers $s_0$, $s_1$, and $\lambda$ such that \begin{equation}\label{f4.4} s_0<\sigma_0(\varphi),\quad s_1>\sigma_1(\varphi),\quad \lambda>-1/2 \end{equation} and that \begin{equation}\label{f4.5} \left\{
\begin{array}{ll}
\lambda\leq s_{1}&\hbox{if}\;\;\;\sigma_1(\varphi)\geq-1/2;\\
s_{1}<-1/2&\hbox{if}\;\;\;\sigma_1(\varphi)<-1/2.
\end{array} \right. \end{equation}
If $\sigma_1(\varphi)\geq-1/2$, we introduce the function \begin{equation}\label{f13} \eta(t):=t^{(1-\theta)s_{1}}\varphi(t^\theta) \quad\mbox{of}\quad t\geq1,\quad\mbox{with}\quad \theta:=\frac{s_1-\lambda}{s_1-s_0}. \end{equation} Then $0\leq\theta<1$, $\eta\in\mathrm{OR}$, and $\sigma_{j}(\eta)=(1-\theta)s_{1}+\theta\sigma_{j}(\varphi)$ for every $j\in\{0,1\}$, which implies in view of \eqref{f4.4} that $\sigma_{0}(\eta)>\lambda>-1/2$ and, hence, $H^{\eta}(\Omega)\hookrightarrow H^{\lambda}(\Omega)$. Besides, since $\varphi(t)/\varphi(t^{\theta})\leq c_{1}t^{(1-\theta)s_{1}}$ whenever $t\geq1$ due to \eqref{f3.2}, we conclude that $\varphi(t)/\eta(t)\leq c_{1}$ whenever $t\geq1$, which implies the continuous embedding $H^{\eta}(\Omega)\hookrightarrow H^{\varphi}(\Omega)$.
If $\sigma_1(\varphi)<-1/2$, we put $\eta(t):=t^{\lambda}$ for every $t\geq1$. Then $H^{\eta}(\Omega)=H^{\lambda}(\Omega)\hookrightarrow H^{\varphi}(\Omega)$ because $\lambda>-1/2>\sigma_{1}(\varphi)$.
Thus, \begin{equation}\label{f13bis} \mbox{the continuous embedding}\quad H^{\eta}(\Omega)\hookrightarrow H^{\lambda}(\Omega)\cap H^{\varphi}(\Omega) \end{equation} holds true whatever $\sigma_1(\varphi)$ is.
The following theorem is a key result of this paper.
\begin{theorem}\label{th1} The set $C^{\infty}(\overline{\Omega})$ is dense in the space $H^{\varphi\rho^{2q}}_{A,\eta}(\Omega)$, and the mapping \eqref{mapping} extends uniquely (by continuity) to a bounded linear operator \begin{equation}\label{f15} (A,B):H^{\varphi\rho^{2q}}_{A,\eta}(\Omega)\to H^{\eta}(\Omega)\oplus \bigoplus_{j=1}^{q}H^{\varphi\rho^{2q-m_j-1/2}}(\Gamma)=: \mathcal{H}^{\eta,\varphi}(\Omega,\Gamma). \end{equation} This operator is Fredholm. Its kernel coincides with $N$, and its range consists of all vectors $(f,g)\in\mathcal{H}^{\eta,\varphi}(\Omega,\Gamma)$ that satisfy \eqref{f10}. The index of the operator \eqref{f15} equals $\dim N-\dim N^{+}$ and does not depend on $\varphi$ and~$\eta$. \end{theorem}
This theorem and other results of this section will be proved in Section~\ref{sec6}.
\begin{remark}\label{bounded-operator} If the system $B$ were not normal or if it did not satisfy the Lopatinskii condition, the operator \eqref{f15} would remain to be well defined and bounded. This follows from the fact that the boundedness of the operator \eqref{f9} does not depend on the ellipticity of the problem \eqref{f1}, \eqref{f2} (see the proof of Theorem~\ref{th1} in Section~\ref{sec6}). \end{remark}
Let us discuss Theorem~\ref{th1} in the Sobolev case where $\varphi(t)\equiv t^{s}$ for a certain real number $s\leq-1/2$. If $s<-1/2$, this theorem yields the Fredholm bounded operator \begin{equation}\label{f4.7} (A,B):H^{s+2q}_{A,\lambda}(\Omega)\to H^{\lambda}(\Omega)\oplus \bigoplus_{j=1}^{q}H^{s+2q-m_j-1/2}(\Gamma)=: \mathcal{H}^{\lambda,s}(\Omega,\Gamma) \end{equation} for every real number $\lambda>-1/2$. If $s=-1/2$, we obtain the same operator by choosing $s_{0}=-1$ and $s_{1}=\lambda>-1/2$ in \eqref{f13}. The boundedness and Fredholm property of the operator \eqref{f4.7} were proved by Lions and Magenes \cite{LionsMagenes62V, LionsMagenes63VI} provided that $\lambda=0$, $s<-1/2$, and \begin{equation}\label{f4.8} s+2q\neq-k+1/2\quad\mbox{whenever}\quad 1\leq k\in\mathbb{Z}. \end{equation} Their result was extended to every $\lambda>-1/2$ in \cite[Corollary~1]{Murach09MFAT2} (see also \cite[Theorem~4.27]{MikhailetsMurach14}). If $\varphi$ is a regular varying function at infinity of order $s+2q<-1/2$ subject to \eqref{f4.8}, Theorem~\ref{th1} is established in \cite[Theorem~4.32]{MikhailetsMurach14}. If $\varphi\in\mathrm{OR}$, this theorem is proved in our paper \cite[Section~4]{AnopMurach15Coll2} in the case where $s_{1}\geq0$ and $\lambda=0$, the paper being published in Ukrainian. In this case, $H^{\eta}(\Omega)\subseteq L_{2}(\Omega)$.
Remark that Theorem~\ref{th1} remains true if we change $\eta$ for every $\omega\in\mathrm{OR}$ such that the function $\eta/\omega$ is bounded in a neighbourhood of infinity. This follows plainly from the continuous embedding $H^{\omega}(\Omega)\hookrightarrow H^{\eta}(\Omega)$.
If $N=\{0\}$ and $N^+=\{0\} $, the operator \eqref{f15} is an isomorphism between the spaces $H^{\varphi\rho^{2q}}_{A,\eta}(\Omega)$ and $\mathcal{H}^{\eta,\varphi}(\Omega,\Gamma)$. Generally, this operator induces an isomorphism between some of their subspaces of finite codimension. In this connection, the next result will be useful.
\begin{lemma}\label{lema1} Let $\alpha,\omega\in\mathrm{OR}$ and $r\in\mathbb{R}$ satisfy $r<\sigma_{0}(\alpha)\leq0$ and $\sigma_{0}(\omega)>-1/2$. Then there exists a number $c>0$ such that \begin{equation}\label{lema1-bound}
|(u,w)_{\Omega}|\leq c\,
\|u\|_{\alpha,A,\omega}\cdot\|w\|_{-r,\Omega} \end{equation} for arbitrary functions $u,w\in C^{\infty}(\overline{\Omega})$. Thus, the sesquilinear form $(u,w)_\Omega$ of $u,w\in C^{\infty}(\overline{\Omega})$ extends uniquely (by continuity) over all $u\in H^{\alpha}_{A,\omega}(\Omega)$ and $w\in H^{-r}(\Omega)$. \end{lemma}
\begin{remark} If $\sigma_0(\alpha)<-1/2$, then we may not replace $H^{\alpha}_{A,\eta}(\Omega)$ with the broader space $H^{\alpha}(\Omega)$ in the last sentence of this lemma. If $H^{\alpha}(\Omega)$ is a Sobolev space, this follows from \cite[Theorems 4.8.2(c) and 4.3.2/1(c)]{Triebel95}. \end{remark}
Using Lemma \ref{lema1} in the $\sigma_{0}(\varphi)\leq-2q$ case and the continuous embedding $H^{\varphi\rho^{2q}}_{A,\eta}(\Omega)\hookrightarrow L_{2}(\Omega)$ otherwise, we may split the source space of the operator \eqref{f15} into the direct sum of subspaces \begin{equation}\label{sum1} H^{\varphi\rho^{2q}}_{A,\eta}(\Omega)=N\dotplus\bigl\{u\in H^{\varphi\rho^{2q}}_{A,\eta}(\Omega):\,(u,w)_\Omega=0\;\;\mbox{for every}\;\;w\in N\bigr\}. \end{equation} Indeed, if $\sigma_{0}(\varphi)\leq-2q$, then Lemma~\ref{lema1} implies the continuous embedding of $H^{\varphi\rho^{2q}}_{A,\eta}(\Omega)$ in the dual of $H^{-r}(\Omega)$ provided that $r<2q+\sigma_{0}(\varphi)$. Since $N\subset H^{\varphi\rho^{2q}}_{A,\eta}(\Omega)\cap H^{-r}(\Omega)$, we decompose this dual into the direct sum \begin{equation*} (H^{-r}(\Omega))'=N\dotplus\bigl\{u\in(H^{-r}(\Omega))': \,u=0\;\,\mbox{on}\;\,N\bigr\}. \end{equation*} Note that the codimension of the second summand equals $\dim N'=\dim N<\infty$. The restriction of this decomposition to $H^{\varphi\rho^{2q}}_{A,\eta}(\Omega)$ gives \eqref{sum1}. If $\sigma_{0}(\varphi)>-2q$, then \eqref{sum1} is a restriction of the orthogonal sum \begin{equation*} L_{2}(\Omega)=N\oplus\bigl\{u\in L_{2}(\Omega):\, (u,w)_\Omega=0\;\;\mbox{for every}\;\;w\in N \,\bigr\}. \end{equation*}
Besides, we may split the target space of the operator \eqref{f15} as follows: \begin{equation}\label{sum2b} \mathcal{H}^{\eta,\varphi}(\Omega,\Gamma)= \bigl\{(w,0,\ldots,0):w\in N^+\bigr\}\dotplus \bigl\{(f,g)\in\mathcal{H}^{\eta,\varphi}(\Omega,\Gamma): \mbox{\eqref{f10} is true}\bigr\}. \end{equation} Indeed, $\mathcal{H}^{\eta,\varphi}(\Omega,\Gamma)\hookrightarrow H^{\ell}(\Omega)\oplus(H^{-r}(\Gamma))^{q}=:\Xi$ for $\ell:=\min\{\lambda,0\}\in(-1/2,0]$ and $r\gg1$. The latter space admits the decomposition \begin{equation*} \Xi=\bigl\{(w,0,\ldots,0):w\in N^+\bigr\}\dotplus \bigl\{(f,g)\in\Xi:\mbox{\eqref{f10} is true}\bigr\} \end{equation*} because the codimension of the second summand equals $\dim M'=\dim N^+<\infty$, where $M:=\{(v,C_{1}^{+}v,\ldots,C_{q}^{+}v):v\in N^+\}$. Here, we consider $M$ as a subspace of $\Psi:=H^{-\ell}(\Omega)\oplus(H^{r}(\Gamma))^{q}$ and note that $\Xi$ is the dual of $\Psi$ with respect to the form $(\cdot,\cdot)_{\Omega}+(\cdot,\cdot)_{\Gamma}+\cdots+ (\cdot,\cdot)_{\Gamma}$. Now \eqref{sum2b} is a restriction of the above decomposition of $\Xi$.
Let $P$ and $\mathcal{P}^+$ respectively denote the projectors of the spaces $H^{\varphi\rho^{2q}}_{A,\eta}(\Omega)$ and $\mathcal{H}^{\eta,\varphi}(\Omega,\Gamma)$ onto the second summand in \eqref{sum1} and \eqref{sum2b} parallel to the first. The mappings defining these projectors do not depend on $\varphi$ and $\eta$.
\begin{theorem}\label{th2} The restriction of the operator \eqref{f15} to the second summand in \eqref{sum1} is an isomorphism \begin{equation}\label{isom} (A,B):P\bigl(H^{\varphi\rho^{2q}}_{A,\eta}(\Omega)\bigr)\leftrightarrow \mathcal{P}^+\bigl(\mathcal{H}^{\eta,\varphi}(\Omega,\Gamma)\bigr). \end{equation} \end{theorem}
Let us now focus on properties of generalized solutions to the elliptic problem \eqref{f1}, \eqref{f2} in the extended Sobolev scale. Our definition of such solutions is suggested by Theorem~\ref{th1}. We put \begin{equation*} \mathcal{S}'(\Omega):=\bigl\{w\!\upharpoonright\Omega\!: w\in\mathcal{S}'(\mathbb{R}^{n})\bigr\}= \bigcup_{r\in\mathbb{R}}H^{r}(\Omega) \end{equation*} and note that $\mathcal{D}'(\Gamma)$ coincides with the union of all spaces $H^{r}(\Gamma)$ where $r\in\mathbb{R}$. If $u\in\mathcal{S}'(\Omega)$ satisfies \eqref{f1} for certain $f\in H^{-1/2+}(\Omega)$, then $u\in H^{s+2q}_{A,\lambda}$ for some $s<-1/2$ and $\lambda>\nobreak -1/2$. Hence, the vector $g:=Bu\in(\mathcal{D}'(\Gamma))^{q}$ is well defined by closure due to Theorem~\ref{th1} considered in the Sobolev case. Therefore, conditions \eqref{f1} and \eqref{f2} make sense provided that $u\in\mathcal{S}'(\Omega)$, $f\in H^{-1/2+}(\Omega)$, and $g\in(\mathcal{D}'(\Gamma))^{q}$. If $u\in\mathcal{S}'(\Omega)$ satisfies these conditions, we will call $u$ a generalized solution to the problem \eqref{f1}, \eqref{f2}.
\begin{theorem}\label{th4.6} Assume that a distribution $u\in\mathcal{S}'(\Omega)$ is a generalized solution to the elliptic problem \eqref{f1}, \eqref{f2} whose right-hand sides satisfy the conditions $f\in H^{\eta}(\Omega)$ and $g_j\in H^{\varphi\rho^{2q-m_j-1/2}}(\Gamma)$ for each $j\in\{1,\ldots,q\}$. Then $u\in H^{\varphi\rho^{2q}}(\Omega)$. \end{theorem}
Let us formulate a local version of this theorem. Let $U$ be an arbitrary open subset of $\mathbb{R}^{n}$ such that $\Omega_0:=\Omega\cap U\neq\emptyset$ and $\Gamma_{0}:=\Gamma\cap U\neq\emptyset$. Given $\alpha\in\mathrm{OR}$, we let $H^{\alpha}_{\mathrm{loc}}(\Omega_{0},\Gamma_{0})$ denote the linear space of all distributions $u\in\mathcal{S}'(\Omega)$ such that $\chi u\in H^{\alpha}(\Omega)$ for every function $\chi\in C^{\infty}(\overline{\Omega})$ satisfying $\mathrm{supp}\,\chi\subset\Omega_0\cup\Gamma_{0}$. Analogously, $H^{\alpha}_{\mathrm{loc}}(\Gamma_{0})$ denotes the linear space of all distributions $h\in\mathcal{D}'(\Gamma)$ such that $\chi h\in H^{\alpha}(\Gamma)$ for every function $\chi\in C^{\infty}(\Gamma)$ satisfying $\mathrm{supp}\,\chi\subset\Gamma_{0}$.
\begin{theorem}\label{th4.7} Assume that a distribution $u\in\mathcal{S}'(\Omega)$ is a generalized solution to the elliptic problem \eqref{f1}, \eqref{f2} whose right-hand sides satisfy the conditions \begin{equation}\label{f4.14} f\in H^\eta_{\mathrm{loc}}(\Omega_{0},\Gamma_{0})\cap H^{-1/2+}(\Omega) \end{equation} and \begin{equation}\label{f4.15} g_j\in H^{\varphi\rho^{2q-m_j-1/2}}_{\mathrm{loc}}(\Gamma_0) \quad\mbox{for each}\quad j\in\{1,\ldots,q\}. \end{equation} Then $u\in H^{\varphi\rho^{2q}}_{\mathrm{loc}}(\Omega_{0},\Gamma_{0})$. \end{theorem}
\begin{remark}\label{rem4.8} The given definition of $H^{\alpha}_{\mathrm{loc}}(\Omega_{0},\Gamma_{0})$ is also applicable to the $\Gamma_{0}=\emptyset$ case. As to condition \eqref{f4.14}, note that if $u\in\mathcal{D}'(\Omega)$ and $f:=Au\in H^\eta_{\mathrm{loc}}(\Omega_{0},\emptyset)$, then $u\in H^{\eta\varrho^{2q}}_{\mathrm{loc}}(\Omega_{0},\emptyset)$ according to \cite[Theorem 7.4.1]{Hermander63}. \end{remark}
As an application of Theorem~\ref{th4.7}, we give sufficient conditions for generalized derivatives (of a given order) of the solution $u$ to be continuous on $\Omega_{0}\cup\Gamma_{0}$. Assuming $0\leq p\in\mathbb{Z}$ and $u\in\mathcal{D}'(\Omega)$, we write $u\in C^{p}(\Omega_{0}\cup\Gamma_{0})$ if there exists a function $u_0\in C^{p}(\Omega_{0}\cup\Gamma_{0})$ such that \begin{equation}\label{def-C^p} \bigl(v\in C^{\infty}_{0}(\Omega),\; \mathrm{supp}\,v\subset\Omega_{0}\bigr)\Longrightarrow (u,v)_{\Omega}=\int\limits_{\Omega_{0}} u_0(x)\overline{v(x)}dx; \end{equation} here, $(u,v)_{\Omega}$ is the value of the distribution $u$ at the test function $v$.
\begin{theorem}\label{th4.9} Let $0\leq p\in\mathbb{Z}$, \begin{equation}\label{int-cond} \int\limits_1^{\infty}t^{2p-4q+n-1}\varphi^{-2}(t)dt<\infty, \end{equation} and assume that a distribution $u\in\mathcal{S}'(\Omega)$ satisfies the hypotheses of Theorem~$\ref{th4.7}$. Then $u\in C^{p}(\Omega_{0}\cup\Gamma_{0})$. \end{theorem}
\begin{remark}\label{rem4.10} Condition \eqref{int-cond} is sharp in Theorem~\ref{th4.9}. Namely, let $0\leq p\in\mathbb{Z}$; it follows then from the implication \begin{equation}\label{implication} \begin{aligned} &\bigl(\mbox{a distribution $u\in\mathcal{S}'(\Omega)$ satisfies the hypotheses of Theorem~\ref{th4.7}}\bigr)\\ &\Longrightarrow\; u\in C^{p}(\Omega_{0}\cup\Gamma_{0}) \end{aligned} \end{equation} that $\varphi$ satisfies \eqref{int-cond}. This will be shown in Section~\ref{sec6} \end{remark}
\begin{remark}\label{rem4.11} Theorems \ref{th2}--\ref{th4.7} and \ref{th4.9} remain valid for every function parameter $\varphi\in\mathrm{OR}$ subject to $\sigma_0(\varphi)>-1/2$ provided that we put $\eta:=\varphi$. In this case, they relates to Proposition~\ref{prop1} and are demonstrated in the same way as the corresponding proofs given in Section~\ref{sec6}, we taking into account that $H^{\varphi\varrho^{2q}}_{A,\varphi}(\Omega)= H^{\varphi\varrho^{2q}}(\Omega)$ up to equivalence of norms and supposing that $s<-1/2$ in the proofs. These theorems are proved in \cite{AnopMurach14UMJ} in the case indicated, with the assumption $u\in H^{2q-1/2+}(\Omega)$ being made instead of $u\in\mathcal{S}'(\Omega)$. \end{remark}
We supplement Theorems \ref{th4.6} and \ref{th4.7} with \textit{a priori} estimates of the solution $u$. Let $\|(f,g)\|_{\eta,\varphi,\Omega,\Gamma}$ denote the norm of a vector $(f,g)=(f,g_1,\ldots,g_q)$ in the Hilbert space $\mathcal{H}^{\eta,\varphi}(\Omega,\Gamma)$ defined in \eqref{f15}.
\begin{theorem}\label{th4.12} Assume that a distribution $u\in\mathcal{S}'(\Omega)$ satisfies the hypotheses of Theorem~$\ref{th4.6}$, and choose a number $\ell>0$ arbitrarily. Then \begin{equation}\label{f4.20}
\|u\|_{\varphi\rho^{2q},\Omega}\leq c\,\bigl(\|(f,g)\|_{\eta,\varphi,\Omega,\Gamma}+
\|u\|_{\varphi\rho^{2q-\ell},\Omega}\bigr) \end{equation} for some number $c>0$ that does not depend on $u$ and $(f,g)$. \end{theorem}
Note, if $N=\{0\}$, we may remove the last summand on the right of \eqref{f4.20} due to the Banach theorem on inverse operator.
A local version of this result is stated as follows:
\begin{theorem}\label{th4.13} Assume that a distribution $u\in\mathcal{S}'(\Omega)$ satisfies the hypotheses of Theorem~$\ref{th4.7}$. We arbitrarily choose a number $\ell>0$ and functions $\chi,\zeta\in C^{\infty}(\overline{\Omega})$ such that $\mathrm{supp}\,\chi\subset\mathrm{supp}\,\zeta\subset \Omega_{0}\cup\Gamma_{0}$ and that $\zeta=1$ in a neighbourhood of $\mathrm{supp}\,\chi$. Then \begin{equation}\label{f4.21}
\|\chi u\|_{\varphi\rho^{2q},\Omega}\leq c\,\bigl(\|\zeta(f,g)\|_{\eta,\varphi,\Omega,\Gamma}+
\|\zeta u\|_{\varphi\rho^{2q-\ell},\Omega}\bigr) \end{equation} for some number $c>0$ that does not depend on $u$ and $(f,g)$. \end{theorem}
\begin{remark}\label{rem4.14} Theorems \ref{th4.12} and \ref{th4.13} remain valid for every $\varphi\in\mathrm{OR}$ subject to $\sigma_0(\varphi)>-1/2$ provided that we put $\eta:=\varphi$. In this case, they relate to Proposition~\ref{prop1} and are proved in the same way as that given in Section~\ref{sec6}, the proof of the version of Theorem~\ref{th4.13} being simplified (see Remark~\ref{rem6.3} at the end of Section~\ref{sec6}). Theorem \ref{th4.13} is proved in \cite[Theorem~3]{AnopKasirenko16MFAT} in the case indicated and under the assumptions that $u\in H^{\varphi\rho^{2q}}(\Omega)$ and $l=1$. \end{remark}
As far as we know, Theorems \ref{th4.7} and \ref{th4.13} are new even in the Sobolev case where $\varphi(t)\equiv t^{s}$ and $\eta(t)\equiv t^{\lambda}$ for some $s\leq-1/2$ and $\lambda>-1/2$.
\section{Interpolation properties of the extended Sobolev scale}\label{sec5}
The method of interpolation with a function parameter between Hilbert spaces play a crucial role in our proof of the key Theorem~\ref{th1}. Therefore, it is worthwhile to recall the definition of this method. We also discuss some properties of the extended Sobolev scale that relate to the method and will be used in our proofs. This method was appeared first in Foia\c{s} and Lions' article \cite[p.~278]{FoiasLions61}. We will mainly follow the monograph \cite[Section~1.1]{MikhailetsMurach14}.
Let $X:=[X_{0},X_{1}]$ be an ordered pair of separable complex Hilbert spaces such that $X_{1}$ is a linear manifold in $X_{0}$ and that $\|w\|_{X_{0}}\leq c\|w\|_{X_{1}}$ for a certain number $c>0$ and every vector $w\in X_{1}$. This pair is called regular. For this pair there exists a unique positive-definite self-adjoint operator $J$ acting in $X_{0}$, defined on $X_{1}$, and obeying $\|Jw\|_{X_{0}}=\|w\|_{X_{1}}$ whenever $w\in X_{1}$. This operator is called a generating operator for~$X$.
Consider a Borel measurable function $\psi:(0,\infty)\rightarrow(0,\infty)$ such that $\psi$ is bounded on each compact subset of $(0,\infty)$ and that $1/\psi$ is bounded on every set $[r,\infty)$, with $r>0$. The class of all such functions $\psi$ is denoted by $\mathcal{B}$. Using the spectral resolution of $J$, we get the positive-definite self-adjoint operator $\psi(J)$ in $X_{0}$. Let $[X_{0},X_{1}]_{\psi}$ or, simply, $X_{\psi}$ denote the domain of $\psi(J)$ endowed with the inner product
$(w_1, w_2)_{X_\psi}:=(\psi(J)w_1,\psi(J)w_2)_{X_0}$ and the corresponding norm $\|w\|_{X_\psi}=(w,w)_{X_\psi}^{1/2}$. The space $X_{\psi}$ is Hilbert and separable.
A function $\psi\in\mathcal{B}$ is called an interpolation parameter if the following condition is fulfilled for all regular pairs $X=[X_{0},X_{1}]$ and $Y=[Y_{0},Y_{1}]$ of Hilbert spaces and for any linear mapping $T$ given on $X_{0}$: if the restriction of $T$ to $X_{j}$ is a bounded operator $T:X_{j}\rightarrow Y_{j}$ for each $j\in\{0,1\}$, then the restriction of $T$ to $X_{\psi}$ is also a bounded operator $T:X_{\psi}\rightarrow Y_{\psi}$. If $\psi$ is an interpolation parameter, we will say that the Hilbert space $X_{\psi}$ is obtained by the interpolation with the function parameter $\psi$ between $X_{0}$ and $X_{1}$ and that the bounded operator $T:X_{\psi}\rightarrow Y_{\psi}$ is obtained by the interpolation of the operators $T:X_{j}\rightarrow Y_{j}$ with $j\in\{0,1\}$. In this case, we have the dense continuous embeddings $X_{1}\hookrightarrow X_{\psi}\hookrightarrow X_{0}$.
The function $\psi$ is an interpolation parameter if and only if $\psi$ is pseudoconcave in a neighbourhood of $+\infty$. The latter condition means that there exists a concave function $\psi_{1}:(b,\infty)\rightarrow(0,\infty)$, with $b\gg1$, that the functions $\psi/\psi_{1}$ and $\psi_{1}/\psi$ are bounded on $(b,\infty)$. This fundamental result follows from Peetre's \cite{Peetre68} description of all interpolation functions of positive order (see \cite[Sect.~1.1.9]{MikhailetsMurach14}.
\begin{proposition}\label{prop5.1} Let $\alpha\in\mathrm{OR}$, $r_{0},r_{1}\in\mathbb{R}$, $r_{0}<\sigma_{0}(\alpha)$, and $r_{1}>\sigma_{1}(\alpha)$. Put \begin{equation}\label{f5.1} \psi(t):= \begin{cases} \;t^{{-r_0}/{(r_1-r_0)}}\, \alpha\bigl(t^{1/{(r_1-r_0)}}\bigr)&\text{if}\quad t\geq1; \\ \;\alpha(1)&\text{if}\quad0<t<1. \end{cases} \end{equation} Then $\psi\in\mathcal{B}$ is an interpolation parameter, and \begin{equation*} [H^{r_0}(G),H^{r_1}(G)]_{\psi}=H^{\alpha}(G) \end{equation*} with equality of norms if $G=\mathbb{R}^{n}$, and with equivalence of norms if $G=\Omega$ or $G=\Gamma$. \end{proposition}
This result is proved in \cite[Theorem~5.1]{MikhailetsMurach15ResMath1} for $G=\Omega$ and in \cite[Theorems 2.19 and 2.22]{MikhailetsMurach14} for $G\in\{\mathbb{R}^{n},\Gamma\}$.
The next result shows that the extended Sobolev scale is closed with respect to this interpolation.
\begin{proposition}\label{prop5.2} Let $\alpha_0,\alpha_1\in\mathrm{OR}$ and $\psi\in\mathcal{B}$. Assume that the function $\alpha_0/\alpha_1$ is bounded in a neighbourhood of infinity and that $\psi$ is an interpolation parameter. Put $\alpha(t):=\alpha_0(t)\psi(\alpha_1(t)/\alpha_0(t))$ for every $t\geq1$. Then $\alpha\in\mathrm{OR}$, and \begin{equation*} [H^{\alpha_0}(G),H^{\alpha_1}(G)]_{\psi}=H^{\alpha}(G) \end{equation*} with equality of norms if $G=\mathbb{R}^{n}$, and with equivalence of norms if $G=\Omega$ or $G=\Gamma$. \end{proposition}
This result is proved in \cite[Theorem~5.2]{MikhailetsMurach15ResMath1} for $G=\Omega$ and in \cite[Theorems 2.18 and 2.22]{MikhailetsMurach14} for $G\in\{\mathbb{R}^{n},\Gamma\}$. We will use it in the case where $\alpha_0$ and $\alpha_1$ are power functions.
It follows from Proposition~\ref{prop5.1} and Ovchinnikov's theorem \cite[Theorem 11.4.1]{Ovchinnikov84} that this scale coincides (up to equivalence of norms) with the class of all Hilbert spaces that are interpolation ones between the Sobolev spaces $H^{(r_0)}(G)$ and $H^{(r_1)}(G)$ where $r_0,r_1\in\mathbb{R}$ and $r_0<r_1$ (as above, $G\in\{\mathbb{R}^{n},\Omega,\Gamma\}$). Recall that a Hilbert space $H$ is called an interpolation space between $X_0$ and $X_1$ if the following two properties are satisfied: a)~the continuous embeddings $X_1\hookrightarrow H\hookrightarrow X_0$ hold; b)~every linear operator bounded on $X_0$ and $X_1$ should be also bounded on $H$.
Let us establish a version of Proposition~\ref{prop5.1} for the space $H^{\alpha}_{A,\eta}(\Omega)$.
\begin{theorem}\label{th5.3} Let $\alpha\in\mathrm{OR}$ and suppose that real numbers $r_{0}$, $r_{1}$, $\lambda_{0}$, and $\lambda_{1}$ satisfy the conditions $r_{0}<\sigma_{0}(\alpha)$, $r_{1}>\sigma_{1}(\alpha)$, $\lambda_{0}\leq\lambda_{1}$, $\lambda_{0}\geq r_{0}-2q$, and $\lambda_{1}\geq r_{1}-2q$. Besides, let $\psi$ be the interpolation parameter from Proposition~$\ref{prop5.1}$. Then the pair of separable Hilbert spaces $H^{r_0}_{A,\lambda_{0}}(\Omega)$ and $H^{r_1}_{A,\lambda_{1}}(\Omega)$ is regular, and \begin{equation}\label{f5.2} \bigl[H^{r_0}_{A,\lambda_{0}}(\Omega), H^{r_1}_{A,\lambda_{1}}(\Omega)\bigr]_{\psi}=H^{\alpha}_{A,\chi}(\Omega) \end{equation} up to equivalence of norms. Here, the function $\chi\in\mathrm{OR}$ is defined as follows: \begin{equation}\label{f5.3} \chi(t):=t^{\lambda_{0}}\psi\bigl(t^{\lambda_{1}-\lambda_{0}}\bigr)= t^{(r_1\lambda_0-r_0\lambda_1)/(r_1-r_0)} \alpha\bigl(t^{(\lambda_1-\lambda_0)/(r_1-r_0)}\bigr) \end{equation} for every $t\geq1$. \end{theorem}
\begin{remark}\label{rem5.4} The case $\lambda_{0}=\lambda_{1}=:\lambda$ gives $\chi(t)\equiv t^{\lambda}\alpha(1)$ by \eqref{f5.3}. Hence, formula \eqref{f5.2} remains true in this case if we put $\chi(t):=t^{\lambda}$ whenever $t\geq1$. \end{remark}
\begin{remark}\label{rem5.5} If $\lambda_{j}=r_{j}-2q$ for certain $j\in\{0,1\}$, then $H^{r_j}_{A,\lambda_{j}}(\Omega)=H^{r_j}(\Omega)$ up to equivalence of norms. This follows from the boundedness of the operator $A:H^{r_j}(\Omega)\to H^{r_j-2q}(\Omega)$. Hence, the interpolation formula \eqref{f5.2} is applicable to the case where the Sobolev space $H^{r_j}(\Omega)$ is taken instead of $H^{r_j}_{A,\lambda_{j}}(\Omega)$. \end{remark}
The proof of this theorem is based on Proposition~\ref{prop5.1} and a result \cite[Theorem~3.12]{MikhailetsMurach14} on interpolation with a function parameter between certain Hilbert spaces induced by a bounded linear operator. Before we formulate this result, let us admit the following: if $H$, $\Phi$ and $\Psi$ are Hilbert spaces satisfying the continuous embedding $\Phi\hookrightarrow\Psi$ and if $T:H\rightarrow\Psi$ is a continuous linear operator, we put $$ (H)_{T,\Phi}:=\{u\in H:\,Tu\in\Phi\} $$ and endow the linear space $(H)_{T,\Phi}$ with the inner product $$ (u_1,u_2)_{(H)_{T,\Phi}}:=(u_1,u_2)_{H}+(Tu_1,Tu_2)_{\Phi} $$
and the corresponding norm $\|u\|_{(H)_{T,\Phi}}:=(u,u)_{(H)_{T,\Phi}}^{1/2}$. The inner product does not depend on $\Psi$, and the space $(H)_{T,\Phi}$ is Hilbert. The latter is proved in a quite similar way as the proof of the completeness of $H^{\alpha}_{A,\eta}(\Omega)$ given in Section~\ref{sec4} just after~\eqref{f11}.
\begin{proposition}\label{prop5.6} Assume that six separable Hilbert spaces $X_{0}$, $Y_{0}$, $Z_{0}$, $X_{1}$, $Y_{1}$, and $Z_{1}$ and three linear mappings $T$, $R$, and $S$ are given and satisfy the following conditions: \begin{itemize} \item[(i)] The pairs $[X_{0},X_{1}]$ and $[Y_{0},Y_{1}]$ are regular. \item[(ii)] The spaces $Z_{0}$ and $Z_{1}$ are subspaces of a certain linear space $E$. \item[(iii)] The continuous embeddings $Y_{0}\hookrightarrow Z_{0}$ and $Y_{1}\hookrightarrow Z_{1}$ hold. \item[(iv)] The mapping $T$ is given on $X_{0}$ and defines the bounded operators $T:\nobreak X_{0}\rightarrow Z_{0}$ and $T:X_{1}\rightarrow Z_{1}$. \item[(v)] The mapping $R$ is given on $E$ and defines the bounded operators $R:Z_{0}\rightarrow X_{0}$ and $R:Z_{1}\rightarrow X_{1}$. \item[(vi)] The mapping $S$ is given on $E$ and defines the bounded operators $S:Z_{0}\rightarrow Y_{0}$ and
$S:Z_{1}\rightarrow Y_{1}$. \item[(vii)] The equality $TRu=u+Su$ holds for every $u\in E$. \end{itemize} Then the pair of the separable Hilbert spaces $(X_{0})_{T,Y_{0}}$ and $(X_{1})_{T,Y_{1}}$ is regular, and \begin{equation} \bigl[(X_{0})_{T,Y_{0}},(X_{1})_{T,Y_{1}}\bigr]_{\psi}= (X_{\psi})_{T,Y_{\psi}} \end{equation} up to equivalence of norms for every interpolation parameter $\psi\in\mathcal{B}$. \end{proposition}
Note that conditions (i)--(vii) were found by Lions and Magenes \cite[Chapter~1, Theorem~14.3]{LionsMagenes72}, who proved a version of Proposition~\ref{prop5.6} for the holomorphic interpolation (with a number parameter).
\begin{proof}[Proof of Theorem~$\ref{th5.3}$.] Choosing an integer $p\geq1$ arbitrarily, we consider the linear PDO $A^{p}A^{p+}+I$ of order $4qp$. Here, as usual, $A^{p+}$ denotes the PDO which is formally adjoint to the $p$-th iteration $A^p$ of $A$, and $I$ is the identity operator. Let $H_D^\sigma(\Omega)$, where $\sigma\geq2qp$, denote the set of all distributions $u\in H^\sigma(\Omega)$ such that $\partial_\nu^ju=0$ on $\Gamma$ for each $j\in\{0,\ldots,2qp-1\}$, with $\partial_\nu$ being the operator of the differentiation with respect to the inward normal to the boundary $\Gamma$ of $\Omega$. The linear manifold $H_D^\sigma(\Omega)$ is well defined and closed in $H^\sigma(\Omega)$ due to the theorem on traces for Sobolev spaces (see, e.g., \cite[Section 4.7.1]{Triebel95}). We hence consider $H_D^\sigma(\Omega)$ as a subspace of $H^\sigma(\Omega)$. The differential operator $A^{p}A^{p+}+I$ sets an isomorphism $$ A^pA^{p+}+I:H_D^\sigma(\Omega)\leftrightarrow H^{\sigma-4qp}(\Omega) $$ for each integer $\sigma\geq2qp$ (see, e.g., \cite[Lemma~3.1]{MikhailetsMurach14}). The inverse of this isomorphism sets a bounded linear operator \begin{equation}\label{f524} (A^pA^{p+}+I)^{-1}:H^l(\Omega)\rightarrow H^{l+4qp}(\Omega) \end{equation} for each integer $l\geq-2qp$. It follows from Proposition~\ref{prop5.1} that this operator is well defined and continuous for every real $l\geq -2qp$.
Turning to Proposition \ref{prop5.6}, we put $X_j:= H^{r_j}(\Omega)$, $Y_j:= H^{\lambda_j}(\Omega)$, and $Z_j :=H^{r_j-2q}(\Omega)$ for each $j\in\{0,1\}$, and $E:=H^{r_0-2q}(\Omega)$ and $T:=A$. Conditions (i)–(iv) of this proposition are evidently satisfied. We subject $p$ to the restrictions $r_j-2q\geq-2qp$ and $r_j-2q-\lambda_j\geq-4qp$ for each $j\in\{0,1\}$ and put $$ R:= A^{p-1} A^{p+}(A^p A^{p+}+I)^{-1}\quad\text{and}\quad S:=-(A^p A^{p+}+I)^{-1}. $$ According to \eqref{f524}, we have the continuous linear operators $$ R: Z_j = H^{r_j-2q}(\Omega)\rightarrow H^{r_j}(\Omega)= X_j $$ and $$ S: Z_j= H^{r_j-2q}(\Omega)\rightarrow H^{r_j-2q+4qp}(\Omega) \hookrightarrow H^{\lambda_j}(\Omega)=Y_j $$ for each $j\in\{0,1\}$. Thus, conditions (v) and (vi) are also satisfied. The last condition (vii) is satisfied because $$ T Ru = (A^p A^{p+}+I-I)(A^p A^{p+}+I)^{-1}u=u+Su $$ for every $u\in E$.
Using Propositions \ref{prop5.6}, \ref{prop5.1}, and \ref{prop5.2} successively, we conclude that \begin{align*} \bigl[H^{r_0}_{A,\lambda_{0}}(\Omega), H^{r_1}_{A,\lambda_{1}}(\Omega)\bigr]_{\psi}&= \bigl[(X_{0})_{T,Y_{0}},(X_{1})_{T,Y_{1}}\bigr]_{\psi}= (X_{\psi})_{T,Y_{\psi}}\\ &= \bigl([H^{r_0}(\Omega),H^{r_1}(\Omega)]_\psi\bigr)_ {A,[H^{\lambda_0}(\Omega),H^{\lambda_1}(\Omega)]_{\psi}}\\ &=\bigl(H^{\alpha}(\Omega)\bigr)_{A,H^{\chi}(\Omega)} =H^{\alpha}_{A,\chi}(\Omega) \end{align*} up to equivalence of norms, which proves Theorem~$\ref{th5.3}$. \end{proof}
Note that a version of Theorem $\ref{th5.3}$ for the interpolation with a number parameter is proved in \cite[Theorem~2]{KasirenkoMikhailetsMurach19}.
\section{Proofs of the main results}\label{sec6}
We will prove Theorem~$\ref{th1}$ with the help of its version for Sobolev spaces.
\begin{proposition}\label{prop6.1} Let $s<-1/2$ and $\lambda>-1/2$. Then the set $C^{\infty}(\overline{\Omega})$ is dense in $H^{s+2q}_{A,\lambda}(\Omega)$, and the mapping \eqref{mapping} extends uniquely (by continuity) to a bounded linear operator \begin{equation}\label{f6.1} (A,B):H^{s+2q}_{A,\lambda}(\Omega)\to H^{\lambda}(\Omega)\oplus\bigoplus_{j=1}^{q}H^{s+2q-m_j-1/2}(\Gamma)= \mathcal{H}^{\lambda,s}(\Omega,\Gamma). \end{equation} This operator is Fredholm. Its kernel coincides with $N$, the range consists of all vectors $(f,g_{1},\ldots,g_{q})\in\mathcal{H}^{\lambda,s}(\Omega,\Gamma)$ that satisfy \eqref{f10}, and the index equals $\dim N-\dim N^{+}$. \end{proposition}
\begin{proof} This proposition is proved in \cite[Section~4.4.3]{MikhailetsMurach14} in the case \eqref{f4.8}. Let us examine the opposite case; i.e., we assume that $s+2q=-k+1/2$ for a certain integer $k\geq1$ and deduce Proposition~\ref{prop6.1} from the first case with the help of the interpolation.
Choose numbers $s_0$ and $s_1$ so that $s_0<s<s_1<-1/2$ and that $s_j+2q-1/2\notin\mathbb{Z}$ whenever $j\in\{0,1\}$. We have the Fredholm bounded operators \begin{equation}\label{f6.2} (A,B):H^{s_j+2q}_{A,\lambda}(\Omega)\to \mathcal{H}^{\lambda,s_j}(\Omega,\Gamma) \quad\mbox{for each}\quad j\in\{0,1\}. \end{equation} Put $\alpha(t):=t^{s+2q}$ whenever $t\geq1$ and $r_{j}:=s_{j}+2q$ whenever $j\in\{0,1\}$, and define an interpolation parameter $\psi$ by formula \eqref{f5.1}. Thus, $\psi(t):=t^{(s-s_0)/(s_1-s_0)}$ for every $t\geq1$. Then a restriction of the operator \eqref{f6.2} for $j=0$ is a bounded operator between the spaces \begin{equation}\label{f6.3} (A,B):\bigl[H^{s_0+2q}_{A,\lambda}(\Omega), H^{s_1+2q}_{A,\lambda}(\Omega)\bigr]_{\psi}\to \bigl[\mathcal{H}^{\lambda,s_0}(\Omega,\Gamma), \mathcal{H}^{\lambda,s_1}(\Omega,\Gamma)\bigr]_{\psi}. \end{equation} This operator is Fredholm according to the theorem on interpolation of Fredholm operators (see, e.g., \cite[Theorem~1.7]{MikhailetsMurach14}).
Owing to Theorem~\ref{th5.3} and in view of Remark~\ref{rem5.4}, we get \begin{equation}\label{f6.4} \bigl[H^{s_0+2q}_{A,\lambda}(\Omega), H^{s_1+2q}_{A,\lambda}(\Omega)\bigr]_{\psi}=H^{s+2q}_{A,\lambda}(\Omega). \end{equation} Besides, \begin{align*} &\bigl[\mathcal{H}^{\lambda,s_0}(\Omega,\Gamma), \mathcal{H}^{\lambda,s_1}(\Omega,\Gamma)\bigr]_{\psi}\\ &=\bigl[H^{\lambda}(\Omega),H^{\lambda}(\Omega)\bigr]_\psi \oplus\bigoplus_{j=1}^{q} \bigl[H^{s_0+2q-m_j-1/2}(\Gamma),H^{s_1+2q-m_j-1/2}(\Gamma)\bigr]_\psi\\ &=\mathcal{H}^{\lambda,s}(\Omega,\Gamma) \end{align*} due to Proposition~\ref{prop5.2} and the theorem on interpolation of orthogonal sums of Hilbert spaces (see, e.g., \cite[Theorem~1.5]{MikhailetsMurach14}). Hence, the Fredholm bounded operator \eqref{f6.3} acts between the spaces \eqref{f6.1}. Since the operators \eqref{f6.2} have the common kernel $N$ and index $\dim N-\dim N^{+}$, so does the operator \eqref{f6.1} according to \cite[Theorem~1.7]{MikhailetsMurach14}. Moreover, \begin{align*} (A,B)\bigl(H^{s+2q}_{A,\lambda}(\Omega)\bigr)&= \mathcal{H}^{\lambda,s}(\Omega,\Gamma)\cap (A,B)\bigl(H^{s_0+2q}_{A,\lambda}(\Omega)\bigr)\\ &=\bigl\{(f,g)\in\mathcal{H}^{\lambda,s}(\Omega,\Gamma): \mbox{\eqref{f10} is satisfied}\bigr\} \end{align*} due to the same theorem.
Ending this proof, note that \eqref{f6.4} implies the dense continuous embedding of $H^{s_1+2q}_{A,\lambda}(\Omega)$ in $H^{s+2q}_{A,\lambda}(\Omega)$. Hence, the set $C^{\infty}(\overline{\Omega})$ being dense in the first space is also dense in the second. \end{proof}
\begin{proof}[Proof of Theorem~$\ref{th1}$] It is convenient to consider the cases where $\sigma_{1}(\varphi)\geq-1/2$ and were $\sigma_{1}(\varphi)<-1/2$ separately.
We assume first that $\sigma_{1}(\varphi)\geq-1/2$. Since $s_{1}>\sigma_{1}(\varphi)$, the mapping \eqref{mapping} extends uniquely (by continuity) to a bounded operator \begin{equation}\label{f18} (A,B):H^{s_{1}+2q}(\Omega)\rightarrow H^{s_{1}}(\Omega)\oplus\bigoplus_{j=1}^{q} H^{s_{1}+2q-m_j-1/2}(\Gamma)=:\mathcal{H}^{s_{1}}(\Omega,\Gamma). \end{equation} This operator is Fredholm with kernel $N$ and index $\dim N-\dim N^{+}$. This fact is a specific case of Proposition~\ref{prop1} and is well known in the $s_{1}\geq0$ case (see., e.g., \cite[Chapter~2, Section~5.4]{LionsMagenes72}). Besides, since $s_{0}<-1/2$ and $\lambda>-1/2$, we have the Fredholm bounded operator \eqref{f6.1} for $s:=s_0$, due to Proposition~\ref{prop6.1}. The kernel and index of \eqref{f6.1} are the same as those of \eqref{f18}. We put $\alpha(t):=\varphi(t)t^{2q}$ whenever $t\geq1$, set $r_{j}:=s_{j}+2q$ whenever $j\in\{0,1\}$, and define an interpolation parameter $\psi$ by formula \eqref{f5.1}. A restriction of \eqref{f6.1} is a bounded operator between the spaces \begin{equation}\label{f6.6} (A,B):\bigl[H^{s_0+2q}_{A,\lambda}(\Omega), H^{s_1+2q}(\Omega)\bigr]_{\psi}\to \bigl[\mathcal{H}^{\lambda,s_0}(\Omega,\Gamma), \mathcal{H}^{s_1}(\Omega,\Gamma)\bigr]_{\psi}. \end{equation} This operator is Fredholm with the same kernel and index according to \cite[Theorem~1.7]{MikhailetsMurach14}.
Owing to Theorem~\ref{th5.3} and in view of Remark~\ref{rem5.5}, we get \begin{equation}\label{f6.7} \bigl[H^{s_0+2q}_{A,\lambda}(\Omega),H^{s_1+2q}(\Omega)\bigr]_{\psi}= \bigl[H^{s_0+2q}_{A,\lambda}(\Omega), H^{s_1+2q}_{A,s_1}(\Omega)\bigr]_{\psi}= H^{\varphi\varrho^{2q}}_{A,\eta}(\Omega). \end{equation} Note that $\eta(t)$ defined by \eqref{f13} is equal to $\chi(t)$ defined by \eqref{f5.3} if $\lambda_{0}=\lambda$ and $\lambda_{1}=s_1$. Indeed, \begin{equation}\label{f6.7b} \begin{aligned} \chi(t)&=t^{\lambda}\psi(t^{s_1-\lambda})= t^{\lambda}t^{-(s_0+2q)(s_1-\lambda)/(s_1-s_0)} \alpha(t^{(s_1-\lambda)/(s_1-s_0)})\\ &=t^{\lambda-(s_0+2q)\theta}\alpha(t^{\theta})= t^{\lambda-(s_0+2q)\theta}\varphi(t^{\theta})t^{2q\theta}= t^{\lambda-s_0\theta}\varphi(t^{\theta})\\ &=t^{(1-\theta)s_1}\varphi(t^{\theta})=\eta(t) \end{aligned} \end{equation} whenever $t\geq1$. Besides, \begin{align*} &\bigl[\mathcal{H}^{\lambda,s_0}(\Omega,\Gamma), \mathcal{H}^{s_1}(\Omega,\Gamma)\bigr]_{\psi}\\ &=\bigl[H^{\lambda}(\Omega),H^{s_1}(\Omega)\bigr]_\psi \oplus\bigoplus_{j=1}^{q} \bigl[H^{s_0+2q-m_j-1/2}(\Gamma),H^{s_1+2q-m_j-1/2}(\Gamma)\bigr]_\psi\\ &=\mathcal{H}^{\eta,\varphi}(\Omega,\Gamma) \end{align*} due to Proposition~\ref{prop5.2}. Note here that $\eta(t)\equiv t^{\lambda}\psi(t^{s_1-\lambda})$ as was just shown and that \begin{equation*} t^{s_0+2q-m_j-1/2}\,\psi(t^{s_1-s_0})= t^{s_0+2q-m_j-1/2}\,t^{-s_0-2q}\alpha(t)=\varphi(t)t^{2q-m_j-1/2} \end{equation*} whenever $t\geq1$.
Thus, the Fredholm bounded operator \eqref{f6.6} acts between the spaces \eqref{f15}. According to \cite[Theorem~1.7]{MikhailetsMurach14} and Proposition~\ref{prop6.1}, we conclude that \begin{equation}\label{f6.8} \begin{aligned} (A,B)\bigl(H^{\varphi\varrho^{2q}}_{A,\eta}(\Omega)\bigr)&= \mathcal{H}^{\eta,\varphi}(\Omega,\Gamma)\cap (A,B)\bigl(H^{s_0+2q}_{A,\lambda}(\Omega)\bigr)\\ &=\bigl\{(f,g)\in \mathcal{H}^{\eta,\varphi}(\Omega,\Gamma): \mbox{\eqref{f10} is satisfied}\bigr\}. \end{aligned} \end{equation} It remains to note that the density of $C^{\infty}(\overline{\Omega})$ in $H^{\varphi\varrho^{2q}}_{A,\eta}(\Omega)$ is a consequence of the dense continuous embedding of $H^{s_1+2q}(\Omega)$ into $H^{\varphi\varrho^{2q}}_{A,\eta}(\Omega)$. This embedding is due to \eqref{f6.7}. The case $\sigma_{1}(\varphi)\geq-1/2$ is examined.
Assume now that $\sigma_{1}(\varphi)<-1/2$. Since $s_0<s_1<-1/2$ in this case, we have the Fredholm bounded operators \eqref{f6.2} due to Proposition~\ref{prop6.1}. Using the same $\alpha$, $r_0$, $r_1$, and interpolation parameter $\psi$ as in the previous case, we conclude that a restriction of \eqref{f6.2} for $j=0$ is a bounded operator between the spaces \eqref{f6.3}. This operator is Fredholm with kernel $N$ and index $\dim N-\dim N^{+}$ by Proposition~\ref{prop6.1} and \cite[Theorem~1.7]{MikhailetsMurach14}. According to Theorem~\ref{th5.3} and Remark~\ref{rem5.4}, we have \begin{equation}\label{f6.9} \bigl[H^{s_0+2q}_{A,\lambda}(\Omega), H^{s_1+2q}_{A,\lambda}(\Omega)\bigr]_{\psi}= H^{\varphi\varrho^{2q}}_{A,\lambda}(\Omega). \end{equation} Besides, \begin{equation*} \bigl[\mathcal{H}^{\lambda,s_0}(\Omega,\Gamma), \mathcal{H}^{\lambda,s_1}(\Omega,\Gamma)\bigr]_{\psi}= \mathcal{H}^{\lambda,\varphi}(\Omega,\Gamma)= \mathcal{H}^{\eta,\varphi}(\Omega,\Gamma) \end{equation*} due to Proposition~\ref{prop5.2}. Hence, the Fredholm bounded operator \eqref{f6.3} acts between the spaces \eqref{f15}, with \eqref{f6.8} holding due to \cite[Theorem~1.7]{MikhailetsMurach14}. Now, the density of $C^{\infty}(\overline{\Omega})$ in $H^{\varphi\varrho^{2q}}_{A,\eta}(\Omega)$ is a consequence of Proposition~\ref{prop6.1} and the dense continuous embedding of $H^{s_1+2q}_{A,\lambda}(\Omega)$ into $H^{\varphi\varrho^{2q}}_{A,\eta}(\Omega)$. This embedding is due to \eqref{f6.9}. The case $\sigma_{1}(\varphi)<-1/2$ is also examined. \end{proof}
To prove Lema~\ref{lema1} and other results, we need the scale $\{H^{r,(2q)}(\Omega):r\in\mathbb{R}\}$ of Hilbert spaces introduced by Roitberg \cite{Roitberg64}. This scale is applied in the theory of elliptic problems \cite{Berezansky68, KozlovMazyaRossmann97, Roitberg96, Roitberg99}. We will mainly follow \cite[Section~1.10 and Chapter~2]{Roitberg96}.
We first consider the Hilbert space $H^{r,(0)}(\Omega)$ used in the definition of $H^{r,(2q)}(\Omega)$. Let $H^{r,(0)}(\Omega):=H^{r}(\Omega)$ in the $r\geq0$ case. If $r<0$, then $H^{r,(0)}(\Omega)$ is defined to be the dual of $H^{-r}(\Omega)$ with respect to the inner product in $L_{2}(\Omega)$. Namely, $H^{r,(0)}(\Omega)$, where $r<0$, is the completion of $C^{\infty}(\overline{\Omega})$ with respect to the Hilbert norm \begin{equation}\label{f6.10}
\|u\|_{r,(0),\Omega}:=\sup\biggl\{\frac{|(u,w)_{\Omega}|}
{\;\quad\|w\|_{-r,\Omega}}: w\in H^{-r}(\Omega),\,w\neq0\biggr\}. \end{equation}
Thus, the inner product in $L_{2}(\Omega)$ extends by continuity to a sesquilinear form $(u_1,u_2)_{\Omega}$ defined for arbitrary $u_1\in H^{r,(0)}(\Omega)$ and $u_2\in H^{-r,(0)}(\Omega)$, with $r\in\mathbb{R}$. The norm in $H^{r,(0)}(\Omega)$ is denoted by $\|\cdot\|_{r,(0),\Omega}$ for every $r\in\mathbb{R}$.
Now we can define the Hilbert space $H^{r,(2q)}(\Omega)$. Let $E_{2q}:=\{1/2,3/2,\ldots,2q-1/2\}$. If $r\in\mathbb{R}\setminus E_{2q}$, then the space $H^{r,(2q)}(\Omega)$ is defined to be the completion of $C^{\infty}(\overline{\Omega})$ with respect to the Hilbert norm \begin{equation}\label{f6.11}
\|u\|_{r,(2q),\Omega}:=\Biggl(\|u\|_{r,(0),\Omega}^{2}+
\sum_{k=1}^{2q}\;\|(\partial_{\nu}^{k-1}u)\!\upharpoonright\!\Gamma\| _{r-k+1/2,\Gamma}^{2}\Biggr)^{1/2}. \end{equation} (As above, $\partial_\nu$ is the operator of the differentiation along the inward normal to $\Gamma$.) If $r\in E_{2q}$, then we put \begin{equation*} H^{r,(2q)}(\Omega):=\bigl[H^{r-\varepsilon,(2q)}(\Omega), H^{r+\varepsilon,(2q)}(\Omega)\bigr]_{\psi} \end{equation*}
where $\psi(t)\equiv \sqrt{t}$ and $0<\varepsilon<1$. The right-hand side of this equality does not depend on the choice $\varepsilon$ up to equivalence of norms. The norm in the Hilbert space $H^{r,(2q)}(\Omega)$ is denoted by $\|\cdot\|_{r,(2q),\Omega}$ for every $r\in\mathbb{R}$.
Let $p\in\{0,2q\}$. If $-\infty<r_0<r_1<\infty$, then the identity mapping on $C^{\infty}(\overline{\Omega})$ extends uniquely to a continuous imbedding operator $H^{r_1,(p)}(\Omega)\hookrightarrow H^{r_0,(p)}(\Omega)$. Besides, \begin{equation}\label{f6.12} \mbox{if}\;\;r>p-1/2,\;\;\mbox{then}\;\;H^{r,(p)}(\Omega)=H^{r}(\Omega) \end{equation} as completions of $C^{\infty}(\overline{\Omega})$ with respect to equivalent norms.
We will use the next result in the proof of Lemma~\ref{lema1}.
\begin{proposition}\label{prop6.2} Let $\omega\in\mathrm{OR}$, $\sigma_{0}(\omega)>-1/2$, $r\in\mathbb{R}$, and \begin{equation}\label{f6.13} r\notin\{-k+1/2:1\leq k\in\mathbb{Z}\}. \end{equation} Then the norms \begin{equation}\label{f6.14}
\|u\|_{r,A,\omega}:=\bigl(\|u\|^{2}_{r,\Omega}+
\|Au\|^{2}_{\omega,\Omega}\bigr)^{1/2} \end{equation} and \begin{equation}\label{f6.15}
\|u\|_{r,(2q),A,\omega}:=\bigl(\|u\|^{2}_{r,(2q),\Omega}+
\|Au\|^{2}_{\omega,\Omega}\bigr)^{1/2} \end{equation} are equivalent on the class of all functions $u\in C^{\infty}(\overline{\Omega})$. \end{proposition}
Recall that \eqref{f6.14} is the norm in $H^{r}_{A,\omega}(\Omega)$. If $r>2q-1/2$, then Proposition~\ref{prop6.2} follows immediately from \eqref{f6.12}. If $r\leq2q-1/2$, then this proposition is a direct consequence of the isomorphism (4.196) from monograph \cite[Section~4.4.2, Proof of Theorem 4.25]{MikhailetsMurach14}. We put $\sigma:=r-2q$, $L:=A$ and $X^{\sigma}(\Omega):=H^{\omega}(\Omega)$ in this isomorphism and note that the space $H^{\omega}(\Omega)$ satisfies Condition~$\mathrm{I}_{\sigma}$ (used in \cite[Theorem 4.25]{MikhailetsMurach14}) in view of \cite[Theorem~4.26]{MikhailetsMurach14} and the continuous embedding $H^{\omega}(\Omega)\hookrightarrow H^{\lambda}(\Omega)$ for some $\lambda>-1/2$.
\begin{proof}[Proof of Lemma $\ref{lema1}$] Assume in addition that $r$ satisfies \eqref{f6.13}. Choosing functions $u,w\in C^{\infty}(\overline{\Omega})$ arbitrarily, we get \begin{align*}
|(u,w)_{\Omega}|&\leq\|u\|_{r,(0),\Omega}\cdot\|w\|_{-r,\Omega}\leq
\|u\|_{r,(2q),A,\omega}\cdot\|w\|_{-r,\Omega}\leq c_1\|u\|_{r,A,\omega}\cdot\|w\|_{-r,\Omega}\\
&\leq c\,\|u\|_{\alpha,A,\omega}\cdot\|w\|_{-r,\Omega} \end{align*} by \eqref{f6.10}, \eqref{f6.11}, \eqref{f6.15}, Proposition~\ref{prop6.2}, and \eqref{f3.3}; here, $c_1$ and $c$ are certain positive numbers that do not depend on $u$ and $w$. If $r$ is not subject to \eqref{f6.13}, then we choose a non-half-integer number $r_1$ such that $r<r_1<\sigma_0(\alpha)$. As has been proved, \begin{equation*}
|(u,w)_{\Omega}|\leq c\,\|u\|_{\alpha,A,\omega}\cdot\|w\|_{-r_1,\Omega}\leq c\,\|u\|_{\alpha,A,\omega}\cdot\|w\|_{-r,\Omega}. \end{equation*} The required bound \eqref{lema1-bound} is substantiated. \end{proof}
\begin{proof}[Proof of Theorem~$\ref{th2}$] According to Theorem~\ref{th1}, the bounded linear operator \eqref{isom} is a bijection. Hence, it is an isomorphism due to the Banach theorem on inverse operator. \end{proof}
\begin{proof}[Proof of Theorem~$\ref{th4.6}$] By the hypotheses of the theorem, we have the inclusion \begin{equation*} (g,f)=(A,B)u\in\mathcal{H}^{\eta,\varphi}(\Omega,\Gamma), \end{equation*} with $u\in H^{s+2q}_{A,\lambda}(\Omega)\supset H^{\varphi\varrho^{2q}}_{A,\eta}(\Omega)$ for certain $s<\sigma_0(\varphi)$. Therefore, \begin{equation*} (g,f)\in\mathcal{H}^{\eta,\varphi}(\Omega,\Gamma) \cap(A,B)\bigl(H^{s+2q}_{A,\lambda}(\Omega)\bigr)= (A,B)\bigl(H^{\varphi\varrho^{2q}}_{A,\eta}(\Omega)\bigr), \end{equation*} the last equality being due to Theorem~\ref{th1}. Hence, along with the condition $(A,B)u=(g,f)$, the equality $(A,B)v=(g,f)$ holds true for certain $v\in H^{\varphi\rho^{2q}}_{A,\eta}(\Omega)$. Thus, the distribution $w:=u-v\in H^{s+2q}_{A,\lambda}(\Omega)$ satisfies $(A,B)w=0$. Therefore, $w\in N\subset C^{\infty}(\overline{\Omega})$ due to Theorem~\ref{th1}, which gives the required inclusion $u=v+w\in H^{\varphi\rho^{2q}}(\Omega)$. \end{proof}
\begin{proof}[Proof of Theorem~$\ref{th4.7}$] We arbitrarily choose a function $\chi\in C^{\infty}(\overline{\Omega})$ such that $\mathrm{supp}\,\chi\subset\Omega_0\cup\nobreak\Gamma_0$ and take a function $\zeta\in C^{\infty}(\overline{\Omega})$ such that $\mathrm{supp}\,\zeta\subset\Omega_0\cup\Gamma_0$ and $\zeta=1$ in a certain neighbourhood $V$ of $\mathrm{supp}\,\chi$ (in the topology of $\overline{\Omega}$, of course). Owing to the hypotheses of the theorem, we have the inclusion \begin{equation}\label{incl-(f,g)} \zeta(f,g):=\bigl(\zeta f,(\zeta\!\upharpoonright\!\Gamma)g_1,\ldots, (\zeta\!\upharpoonright\!\Gamma)g_q\bigr) \in\mathcal{H}^{\eta,\varphi}(\Omega,\Gamma). \end{equation} Besides, $u\in H^{s+2q}_{A,\ell}(\Omega)\supset H^{\varphi\varrho^{2q}}_{A,\eta}(\Omega)$ for certain $s<\sigma_0(\varphi)$ and $\ell\in(-1/2,\sigma_0(\eta))$. We may and will assume that the number $r:=s+2q$ satisfies \eqref{f6.13}.
The space $H^{s+2q}_{A,\ell}(\Omega)$ is not closed with respect to the multiplication by functions from $C^{\infty}(\overline{\Omega})$. This is a reason why we may not deduce this theorem from only Theorem~\ref{th4.6} in a usual manner (see, e.g., \cite[Chapter~III, Section~6, Subsection~11]{Berezansky68} or \cite[Section~4.1.2]{MikhailetsMurach14}). We have to use Roitberg's theorem on local regularity of solutions to the elliptic problem \cite[Theorem~7.2.1]{Roitberg96}. This theorem deals with the solutions of class $H^{s+2q,(2q)}(\Omega)$.
Owing to Proposition~\ref{prop6.2} for $\omega(t)\equiv t^{\ell}$, the identity mapping on $C^{\infty}(\overline{\Omega})$ extends uniquely (by continuity) to a bounded linear operator \begin{equation}\label{op-O} O:H^{s+2q}_{A,\ell}(\Omega)\to H^{s+2q,(2q)}(\Omega). \end{equation}
Let us show that this operator is one-to-one. Suppose that $Ou=0$ for certain $u\in H^{s+2q}_{A,\ell}(\Omega)$. Then there exists a sequence $(u_k)_{k=1}^{\infty}\subset C^{\infty}(\overline{\Omega})$ such that $u_k\to u$ in $H^{s+2q}_{A,\ell}(\Omega)$ and $u_k\to0$ in $H^{s+2q,(2q)}(\Omega)$ as $k\to\infty$. Therefore, $Au_k\to Au$ in $H^{\ell}(\Omega)$ and $Au_k\to0$ in $H^{s,(0)}(\Omega)$, with the latter convergence being due to \cite[Lemma~2.3.1(ii)]{Roitberg96}. This implies $Au=0$ in view of the continuous embedding of the space $H^{\ell}(\Omega)=H^{\ell,(0)}(\Omega)$ in $H^{s,(0)}(\Omega)$ (see \eqref{f6.12}, and take $s<-1/2<\ell$ into account). Hence, $\|u_k\|_{s+2q,(2q),A,\ell}\to0$, which implies $\|u_k\|_{s+2q,A,\ell}\to0$ by Proposition~\ref{prop6.2}. Thus, $u=0$, i.e. the operator \eqref{op-O} is one-to-one. It sets the continuous embedding of $H^{s+2q}_{A,\ell}(\Omega)$ in $H^{s+2q,(2q)}(\Omega)$. We may therefore consider the distribution $u\in H^{s+2q}_{A,\ell}(\Omega)$ as an element of the Roitberg's space $H^{s+2q,(2q)}(\Omega)$.
According to \eqref{incl-(f,g)} and Theorem~\ref{th2}, there exists a distribution \begin{equation}\label{v-in} v\in H^{\varphi\rho^{2q}}_{A,\eta}(\Omega) \end{equation} such that \begin{equation*} (A,B)v=\mathcal{P}^{+}(\zeta(f,g))\in \mathcal{H}^{\eta,\varphi}(\Omega,\Gamma). \end{equation*} Putting \begin{equation*} w:=u-v\in H^{s+2q}_{A,\ell}(\Omega)\subset H^{s+2q,(2q)}(\Omega), \end{equation*} we see that \begin{equation}\label{(A,B)w} (A,B)w=(f,g)-\mathcal{P}^{+}(\zeta(f,g))=:F\in H^{s,(0)}(\Omega)\oplus \bigoplus_{j=1}^{q}H^{s+2q-m_j-1/2}(\Gamma) \end{equation} because $(f,g)=(A,B)u$ and $u\in H^{s+2q}_{A,\ell}(\Omega)$ and because the space $\mathcal{H}^{\eta,\varphi}(\Omega,\Gamma)$ is narrower than the orthogonal sum in \eqref{(A,B)w}. Besides, \begin{align*} \chi_{1}F=& \chi_{1}\bigl((f,g)-\zeta(f,g)+(I-\mathcal{P}^{+})(\zeta(f,g))\bigr)\\ =&\chi_{1}(I-\mathcal{P}^{+})(\zeta(f,g))\in C^{\infty}(\overline{\Omega})\times\{0\}^{q} \end{align*} for every function $\chi_{1}\in C^{\infty}(\overline{\Omega})$ subject to $\mathrm{supp}\,\chi_{1}\subset V$ (recall that $\zeta=1$ on $V$). Hence, \begin{equation}\label{chi_{1}w} \chi_{1}w\in\bigcap_{r>s+2q}H^{r,(2q)}(\Omega)= C^{\infty}(\overline{\Omega}) \end{equation} for every above-mentioned function $\chi_{1}$. This conclusion is due to \cite[Theorem~7.2.1]{Roitberg96} (or \cite[Theorem~7.2.2]{Roitberg96} if $\Gamma_0=\emptyset$), with the equation in \eqref{chi_{1}w} holding in view of \eqref{f6.12}. Taking $\chi_{1}:=\chi$, we obtain \begin{equation*} \chi u=\chi v+\chi w\in H^{\varphi\rho^{2q}}(\Omega) \end{equation*} by \eqref{v-in} and \eqref{chi_{1}w}. Thus, $u\in H^{\varphi\rho^{2q}}_{\mathrm{loc}}(\Omega_0,\Gamma_0)$ due to the arbitrariness of our choice of $\chi$. \end{proof}
\begin{proof}[Proof of Theorem~$\ref{th4.9}$] We choose a sufficiently small number $\varepsilon>0$, put $U_{\varepsilon}:=\{x\in U:\mathrm{dist}(x,\partial U)>\varepsilon\}$, $\Omega_{\varepsilon}:=\Omega\cap U_{\varepsilon}$, and $\Gamma_{\varepsilon}:=\Gamma\cap U_{\varepsilon}$, and consider a function $\chi_{\varepsilon}\in C^\infty(\overline\Omega)$ such that $\mbox{supp}\,\chi_{\varepsilon}\subset\Omega_0\cup\Gamma_0$ and $\chi_{\varepsilon}=1$ on $\Omega_{\varepsilon}\cup\Gamma_{\varepsilon}$. Owing to Theorem~\ref{th4.7}, the inclusion $\chi_{\varepsilon}u\in H^{\varphi\varrho^{2q}}(\Omega)$ holds true. Hence, there exists a distribution $w_{\varepsilon}\in H^{\varphi\varrho^{2q}}(\mathbb{R}^{n})$ such that $w_{\varepsilon}=\chi_{\varepsilon}u=u$ on $\Omega_{\varepsilon}$. By \eqref{Hermander-embedding}, condition \eqref{int-cond} implies $w_{\varepsilon}\in C^{p}(\mathbb{R}^{n})$. Therefore, \begin{equation*} (u,v)_{\Omega}= \int\limits_{\Omega_{\varepsilon}} w_{\varepsilon}(x)\overline{v(x)}dx= \int\limits_{\Omega_{0}}u_{0}(x)\overline{v(x)}dx \end{equation*} for every $v\in C^{\infty}_{0}(\Omega)$ subject to $\mathrm{supp}\,v\subset\Omega_{\varepsilon}$. Here, the function $u_{0}\in C^{p}(\Omega_{0}\cup\Gamma_{0})$ is defined by the formula $u_{0}:=w_{\varepsilon}$ on $\Omega_{\varepsilon}\cup\Gamma_{\varepsilon}$ whenever $0<\varepsilon\ll1$. This function is well defined because $0<\delta<\varepsilon$ implies that $w_{\delta}=w_{\varepsilon}$ on $\Omega_{\varepsilon}\cup\Gamma_{\varepsilon}$. Thus, $u$ satisfies \eqref{def-C^p}, i.e. $u\in C^{p}(\Omega_{0}\cup\Gamma_{0})$. \end{proof}
Let us now substantiate Remark~\ref{rem4.10}. Namely, we suppose that the implication \eqref{implication} holds true for a certain integer $p\geq0$ and will prove that $\varphi$ satisfies \eqref{int-cond}. Choosing a distribution $u\in H^{\varphi\rho^{2q}}_{A,\eta}(\Omega)$ arbitrarily, we define the right-hand sides of the problem \eqref{f1}, \eqref{f2}. They satisfy the inclusion $(f,g)\in\mathcal{H}^{\varphi,\eta}(\Omega,\Gamma)$ and, hence, the hypotheses of Theorem~$\ref{th4.7}$. Therefore, \begin{equation}\label{f6.21} u\in H^{\varphi\rho^{2q}}_{A,\eta}(\Omega)\;\Longrightarrow\; u\in C^{p}(\Omega_{0}\cup\Gamma_{0}) \end{equation} by \eqref{implication}. We now choose a distribution $g_1\in H^{\varphi\rho^{2q-1/2}}(\Gamma)$ arbitrarily, put $g_j:=0$ whenever $2\leq j\leq q$, and consider the regular elliptic problem that consists of the equation \eqref{f1} and boundary conditions \begin{equation}\label{f6.22} \partial^{j-1}_{\nu}u=g_j\quad\mbox{on}\;\Gamma,\quad j=1,...,q. \end{equation} Here, $\partial_{\nu}$ is the operator of differentiation along the inner normal $\nu$ to $\Gamma=\partial\Omega$. According to \eqref{sum2b}, there exists a function $f\in N^{+}\subset C^{\infty}(\overline{\Omega})$ such that $\mathcal{P}^{+}(0,g)=(f,g)$. We take this function to be the right-hand side of \eqref{f1}. Owing to Theorem~\ref{th2}, the elliptic problem \eqref{f1}, \eqref{f6.22} has a solution $u\in H^{\varphi\rho^{2q}}_{A,\eta}(\Omega)$. This solution belongs to $C^{p}(\Omega_{0}\cup\Gamma_{0})$ due to \eqref{f6.21}. Hence, the restriction of every distribution $g_1\in H^{\varphi\rho^{2q-1/2}}(\Gamma)$ to $\Gamma_{0}$ pertains to $C^{p}(\Gamma_{0})$. Passing to local coordinates on $\Gamma$, we deduce plainly from this fact that \begin{equation}\label{f6.23} \bigl\{w\in H^{\varphi\rho^{2q-1/2}}(\mathbb{R}^{n-1}): \mathrm{supp}\,w\subset V\bigr\}\subset C^p(\mathbb{R}^{n-1}) \end{equation} for a certain open subset $V\neq\emptyset$ of $\mathbb{R}^{n-1}$. The inclusion \eqref{f6.23} implies \eqref{int-cond} due to \eqref{Hermander-embedding}, which substantiates Remark~\ref{rem4.10}.
\begin{proof}[Proof of Theorem~$\ref{th4.12}$] According to Peetre's lemma \cite[Lemma~3]{Peetre61}, this theorem is a consequence of the facts that the operator \eqref{f15} has finite-dimensional kernel and closed range by Theorem~\ref{th1} and that the embedding $H^{\varphi\varrho^{2q}}_{A,\eta}(\Omega)\hookrightarrow H^{\varphi\varrho^{2q-\ell}}(\Omega)$ is compact (in fact, the continuity of this embedding is enough). However, it is not difficult to prove this theorem not referring to the mentioned lemma. Namely, using the decomposition \eqref{sum1}, we represent an arbitrary distribution $u\in H^{\varphi\rho^{2q}}_{A,\eta}(\Omega)$ in the form $u=u_{0}+u_{1}$ with $u_{0}:=(1-P)u\in N$ and $u_{1}:=Pu$. Owing to Theorem~\ref{th2}, we get \begin{equation}\label{f6.24}
\|u_1\|_{\varphi\rho^{2q},\Omega}\leq
\|u_1\|_{\varphi\rho^{2q},A,\eta}\leq c_{1}\|(A,B)u_1\|_{\eta,\varphi,\Omega,\Gamma}=
c_{1}\|(f,g)\|_{\eta,\varphi,\Omega,\Gamma}, \end{equation} with $c_1$ being the norm of the inverse operator to the isomorphism \eqref{isom}. Since the space $N$ is finite-dimensional, all norms are equivalent on $N$, specifically, the norms in $H^{\varphi\rho^{2q}}(\Omega)$ and $H^{\varphi\rho^{2q-\ell}}(\Omega)$. It follows hence from $u_{0}\in N$ and \eqref{f6.24} that \begin{align*}
\|u_0\|_{\varphi\rho^{2q},\Omega}&\leq c_{0}\|u_0\|_{\varphi\rho^{2q-\ell},\Omega}\leq c_{0}\|u\|_{\varphi\rho^{2q-\ell},\Omega}+
c_{0}\|u_1\|_{\varphi\rho^{2q-\ell},\Omega}\\
&\leq c_{0}\|u\|_{\varphi\rho^{2q-\ell},\Omega}+
c_{0}\|u_1\|_{\varphi\rho^{2q},\Omega}\\
&\leq c_{0}\|u\|_{\varphi\rho^{2q-\ell},\Omega}+
c_{0}c_{1}\|(f,g)\|_{\eta,\varphi,\Omega,\Gamma}; \end{align*} here, $c_0$ is a positive number that does not depend on $u$. This together with \eqref{f6.24} yields the required estimate \eqref{f4.20}. It remains to remark that $u$ from Theorem~\ref{th4.12} belongs to $H^{\varphi\rho^{2q}}_{A,\eta}(\Omega)$ by Theorem~\ref{th4.6}. \end{proof}
\begin{proof}[Proof of Theorem~$\ref{th4.13}$]
Note previously that we may not deduce this theorem from Theorem~\ref{th4.12} by a usual reasoning (compare, e.g., with \cite[Section~4.1.2, pp. 170--172]{MikhailetsMurach14} or \cite[Section~7.2, p.~216]{Roitberg96}) because the right-hand side of \eqref{f4.20} contains the norm $\|f\|_{\eta,\Omega}$ instead of the norm $\|f\|_{\varphi,\Omega}$, which is necessary to perform this reasoning. Besides, the hypothesis $Au\in H^{-1/2+}(\Omega)$ does not imply that $A(\chi u)\in H^{-1/2+}(\Omega)$. However, we will need the latter inclusion if we use Theorem~\ref{th4.12} for $\chi u$ instead of $u$ according to the usual reasoning. Thus, we have to choose another way to prove Theorem~\ref{th4.13}. This way involves the Roitberg spaces $H^{r,(2q)}(\Omega)$, with $r\in\mathbb{R}$, used in our previous proofs. We divide our reasoning into four steps.
\emph{Step~$1$.} According to \cite[Theorem 4.1.1]{Roitberg96}, the mapping \eqref{mapping} extends uniquely (by continuity) to a Fredholm bounded operator \begin{equation}\label{f6.25} (A,B):H^{s+2q,(2q)}(\Omega)\rightarrow H^{s,(0)}(\Omega)\oplus \bigoplus_{j=1}^{q}H^{s+2q-m_j-1/2}(\Gamma) \end{equation} for every $s\in\mathbb{R}$. The kernel and index of this operator do not depend on $s$ and are equal to $N$ and $\dim N-\dim N^{+}$ resp. (Observe in view of \eqref{f6.12} that \eqref{f6.25} coincides with \eqref{f9} whenever $\varphi(t)\equiv t^{s}$ and $s>-1/2$.) We interpolate the spaces involved in \eqref{f6.25} and use the interpolation parameter $\psi$ defined by formula \eqref{f5.1} in which $r_{0}:=s_{0}$, $r_{1}:=s_{1}$, and $\alpha:=\varphi$. Owing to Proposition~\ref{prop5.2}, the equality \begin{equation}\label{f6.26} [H^{s_{0}+r}(G),H^{s_{1}+r}(G)]_{\psi}=H^{\varphi\rho^{r}}(G) \quad\mbox{for every}\quad r\in\mathbb{R} \end{equation} holds true up to equivalence of norms, with $G\in\{\mathbb{R}^{n},\Omega,\Gamma\}$. Given $r\in\mathbb{R}$, we define the Hilbert spaces \begin{equation*} X_{r}:=[H^{s_{0}+r,(2q)}(\Omega),H^{s_{1}+r,(2q)}(\Omega)]_{\psi} \end{equation*} and \begin{equation*} Y_{r}:=[H^{s_{0}+r,(0)}(\Omega),H^{s_{1}+r,(0)}(\Omega)]_{\psi}. \end{equation*} Consider the Fredholm operators \eqref{f6.25} for each $s\in\{s_{0}+r,s_{1}+r\}$. Interpolating them with the function parameter $\psi$, we conclude by \cite[Theorem~1.7]{MikhailetsMurach14} that the restriction of the mapping \eqref{f6.25}, where $s=s_{0}+r$, on the space $X_{2q+r}$ is a Fredholm bounded operator \begin{equation}\label{f6.27} (A,B):X_{2q+r}\to Y_{r}\oplus Z_{r}. \end{equation} Here, the Hilbert space \begin{equation*} Z_{r}:=\bigoplus_{j=1}^{q}H^{\varphi\varrho^{2q+r-m_j-1/2}}(\Gamma) \end{equation*} equals \begin{equation*} \bigoplus_{j=1}^{q} [H^{s_{0}+r+2q-m_j-1/2}(\Gamma),H^{s_{1}+r+2q-m_j-1/2}(\Gamma)]_{\psi} \end{equation*} up to equivalence of norms due to \eqref{f6.26}.
Let $0\leq k\in\mathbb{Z}$, and let $\zeta_{1}\in C^{\infty}(\overline{\Omega})$ satisfy $\zeta_{1}=1$ in a neighbourhood of $\mathrm{supp}\,\chi$. We will prove by induction in $k$ that \begin{equation}\label{f6.28}
\|\chi u\|_{X_{2q}}\leq c_{0}\bigl(\|\zeta_{1}Au\|_{Y_{0}}+\|\zeta_{1}Bu\|_{Z_{0}}+
\|\zeta_{1}u\|_{X_{2q-k}}\bigr) \end{equation} for every $u\in C^{\infty}(\overline{\Omega})$ with a certain number $c_{0}>0$ that does not depend on $u$. (We use the standard notation for the norms in the spaces $X_{r}$, $Y_{r}$, and $Z_{r}$.) Let us $c_{1}$, $c_{2}$,... denote some positive numbers that are independent of $u$.
If $k=0$, then \eqref{f6.28} follows from \begin{equation*}
\|\chi u\|_{X_{2q}}=\|\chi\zeta_{1}u\|_{X_{2q}}\leq c_{1}\|\zeta_{1}u\|_{X_{2q}}. \end{equation*} The latter inequality is true because the operator of the multiplication by a function from $C^{\infty}(\overline{\Omega})$ is bounded on every space $H^{r,(2q)}(\Omega)$ (as well as on $H^{r,(0)}(\Omega)$); see \cite[Corollary 2.3.1]{Roitberg96}. Assume now that \eqref{f6.28} holds true for a certain integer $k=p\geq0$, and prove \eqref{f6.28} in the case of $k=p+1$.
Consider a function $\zeta_{0}\in C^{\infty}(\overline{\Omega})$ such that $\zeta_{0}=1$ in a neighbourhood of $\mathrm{supp}\,\chi$ and that $\zeta_{1}=1$ in a neighbourhood of $\mathrm{supp}\,\zeta_{0}$. By the inductive assumption \eqref{f6.28}, we have \begin{equation}\label{f6.29}
\|\chi u\|_{X_{2q}}\leq c_{0}\bigl(\|\zeta_{0}Au\|_{Y_{0}}+\|\zeta_{0}Bu\|_{Z_{0}}+
\|\zeta_{0}u\|_{X_{2q-p}}\bigr). \end{equation} Since the bounded operator \eqref{f6.27}, where $r:=-p$, is Fredholm, we have the estimate \begin{equation}\label{f6.30}
\|\zeta_{0}u\|_{X_{2q-p}}\leq c_{2}\bigl(\|A(\zeta_{0}u)\|_{Y_{-p}}+\|B(\zeta_{0}u)\|_{Z_{-p}}+
\|\zeta_{0}u\|_{X_{2q-p-1}}\bigr) \end{equation} due to the above-mentioned lemma by Peetre \cite[Lemma~3]{Peetre61}. Interchanging the PDO $A$ with the operator of multiplication by $\zeta_{0}$, we get \begin{equation*} A(\zeta_{0}u)=A(\zeta_{0}\zeta_{1}u)=\zeta_{0}A(\zeta_{1}u)+A'(\zeta_{1}u)= \zeta_{0}Au+A'(\zeta_{1}u). \end{equation*} Here, $A'$ is a certain linear PDO on $\overline{\Omega}$ whose coefficients belong to $C^{\infty}(\overline{\Omega})$ and whose order $\mathrm{ord}\,A'\leq2q-1$. Analogously, \begin{equation*} B(\zeta_{0}u)=\zeta_{0}Bu+B'(\zeta_{1}u); \end{equation*} here, $B':=(B_{1}',\ldots,B_{q}')$ where each $B_{j}'$ is a certain linear boundary PDO on $\Gamma$ with coefficients of class $C^{\infty}(\Gamma)$ and of order $\mathrm{ord}\,B_{j}'\leq m_{j}-1$. By \cite[Lemma~2.3.1]{Roitberg96} and the interpolation, the PDOs $A'$ and $B_{j}'$ act continuously between the following spaces: \begin{align*} A'&:X_{2q-p-1}= [H^{s_{0}+2q-p-1,(2q)}(\Omega),H^{s_{1}+2q-p-1,(2q)}(\Omega)]_{\psi}\\ &\to[H^{s_{0}-p,(2q)}(\Omega),H^{s_{1}-p,(2q)}(\Omega)]_{\psi}=Y_{-p} \end{align*} and \begin{equation*} B_{j}':X_{2q-p-1}\to [H^{s_{0}+2q-p-m_{j}-1/2}(\Gamma),H^{s_{1}+2q-p-m_{j}-1/2}(\Gamma)]_{\psi} =H^{\varphi\varrho^{2q-p-m_j-1/2}}(\Gamma) \end{equation*} in view of \eqref{f6.26}. Thus, it follows from \eqref{f6.30} that \begin{align*}
\|\zeta_{0}u\|_{X_{2q-p}}&\leq c_{2}\bigl(\|\zeta_{0}Au\|_{Y_{-p}}+
\|A'(\zeta_{1}u)\|_{Y_{-p}}+\|\zeta_{0}Bu\|_{Z_{-p}}+
\|B'(\zeta_{1}u)\|_{Z_{-p}}+\|\zeta_{0}u\|_{X_{2q-p-1}}\bigr)\\
&\leq c_{3}\bigl(\|\zeta_{0}Au\|_{Y_{-p}}+
\|\zeta_{1}u\|_{X_{2q-p-1}}+\|\zeta_{0}Bu\|_{Z_{-p}}+
\|\zeta_{1}u\|_{X_{2q-p-1}}+\|\zeta_{0}u\|_{X_{2q-p-1}}\bigr)\\
&=c_{3}\bigl(\|\zeta_{0}\zeta_{1}Au\|_{Y_{-p}}+
\|\zeta_{0}\zeta_{1}Bu\|_{Z_{-p}}+2\|\zeta_{1}u\|_{X_{2q-p-1}}+
\|\zeta_{0}\zeta_{1}u\|_{X_{2q-p-1}}\bigr)\\
&\leq c_{4}\bigl(\|\zeta_{1}Au\|_{Y_{-p}}+
\|\zeta_{1}Bu\|_{Z_{-p}}+\|\zeta_{1}u\|_{X_{2q-p-1}}\bigr). \end{align*} Applying this estimate and \begin{equation*}
\|\zeta_{0}Au\|_{Y_{0}}+\|\zeta_{0}Bu\|_{Z_{0}}=
\|\zeta_{0}\zeta_{1}Au\|_{Y_{0}}+\|\zeta_{0}\zeta_{1}Bu\|_{Z_{0}}
\leq c_{5}\bigl(\|\zeta_{1}Au\|_{Y_{0}}+\|\zeta_{1}Bu\|_{Z_{0}}\bigr) \end{equation*} to \eqref{f6.29}, we obtain \begin{align*}
\|\chi u\|_{X_{2q}}&\leq c_{0}c_{5}\bigl(\|\zeta_{1}Au\|_{Y_{0}}+\|\zeta_{1}Bu\|_{Z_{0}}\bigr)+
c_{0}c_{4}\bigl(\|\zeta_{1}Au\|_{Y_{-p}}+\|\zeta_{1}Bu\|_{Z_{-p}}+
\|\zeta_{1}u\|_{X_{2q-p-1}}\bigr)\\
&\leq c_{6}\bigl(\|\zeta_{1}Au\|_{Y_{0}}+\|\zeta_{1}Bu\|_{Z_{0}}+
\|\zeta_{1}u\|_{X_{2q-p-1}}\bigr). \end{align*} Here, we use the continuous embeddings $Y_{0}\hookrightarrow Y_{-p}$ and $Z_{0}\hookrightarrow Z_{-p}$, which hold true due to the definitions of the involved spaces via the interpolation. Thus, the estimate \eqref{f6.28} is proved for $k=p+1$ under the assumption that this estimate is valid for $k=p$. Hence, we have proved the estimate for every integer $k\geq0$.
\emph{Step~$2$.} We will prove some relations between norms involved in the obtained inequality \eqref{f6.28} and the required estimate \eqref{f4.21}. First let us show that \begin{equation}\label{f6.31}
\|v\|_{\varphi\varrho^{2q},\Omega}\leq \widetilde{c}_{1}\|v\|_{X_{2q}} \quad\mbox{for every}\quad v\in C^{\infty}(\overline{\Omega}) \end{equation} with some number $\widetilde{c}_{1}>0$ that does not depend on $v$. Given $v\in C^{\infty}(\overline{\Omega})$, we put $(\mathcal{O}v)(x):=v(x)$ if $x\in\overline{\Omega}$ and put $(\mathcal{O}v)(x):=0$ if $x\in\mathbb{R}^{n}\setminus\overline{\Omega}$. As is known (see, e.g., \cite[Theorem 4.8.1]{Triebel95}), the mapping $v\mapsto\mathcal{O}v$, where $v\in C^{\infty}(\overline{\Omega})$, extends uniquely (by continuity) to an isometric isomorphism between $H^{r,(0)}(\Omega)$ and the subspace $\{w\in H^{r}(\mathbb{R}^{n}):\mathrm{supp}\,w\subseteq\overline{\Omega}\}$ of $H^{r}(\mathbb{R}^{n})$ provided that $r\leq0$. According to the definition of $H^{r,(2q)}(\Omega)$, we therefore get \begin{equation*}
\|v\|_{r,\Omega}\leq\|\mathcal{O}v\|_{r,\mathbb{R}^{n}}=\|v\|_{r,(0),\Omega}
\leq\|v\|_{r,(2q),\Omega}\quad\mbox{whenever}\quad r<0. \end{equation*} Besides, \begin{equation*}
\|v\|_{r,\Omega}=\|v\|_{r,(0),\Omega}\leq\|v\|_{r,(2q),\Omega} \quad\mbox{whenever}\quad r\in[0,\infty)\setminus E_{2q}. \end{equation*} Hence, the identity mapping on $C^{\infty}(\overline{\Omega})$ extends uniquely (by continuity) to a bounded linear operator \begin{equation*} I_{2q}:H^{r,(2q)}(\Omega)\to H^{r}(\Omega)\quad\mbox{for every}\quad r\in\mathbb{R} \end{equation*} (the $r\in E_{2q}$ case is treated with the help of the interpolation). Considering this operator for $r\in\{s_{0}+2q,s_{1}+2q\}$ and then interpolating with the function parameter $\psi$, we conclude by \eqref{f6.26} that the above-mentioned identity mapping extends uniquely to a bounded linear operator \begin{equation*} I_{2q}:X_{2q}=[H^{s_{0}+2q,(2q)}(\Omega),H^{s_{1}+2q,(2q)}(\Omega)]_{\psi} \to[H^{s_{0}+2q}(\Omega),H^{s_{1}+2q}(\Omega)]_{\psi}= H^{\varphi\varrho^{2q}}(\Omega). \end{equation*} This yields the required inequality \eqref{f6.31}.
Let us now prove that \begin{equation}\label{f6.33}
\|v\|_{Y_{0}}\leq\widetilde{c}_{2}\|v\|_{\eta,\Omega} \quad\mbox{for every}\quad v\in C^{\infty}(\overline{\Omega}) \end{equation} with some number $\widetilde{c}_{2}>0$ that does not depend on $v$. If $\sigma_{1}(\varphi)\geq-1/2$, then $\eta(t)\equiv t^{\lambda}\psi(t^{s_1-\lambda})$ due to \eqref{f6.7b} and then \begin{align*} H^{\eta}(\Omega)&=[H^{\lambda}(\Omega),H^{s_1}(\Omega)]_{\psi}= [H^{\lambda,(0)}(\Omega),H^{s_1,(0)}(\Omega)]_{\psi}\\ &\hookrightarrow[H^{s_0,(0)}(\Omega),H^{s_1,(0)}(\Omega)]_{\psi}=Y_{0} \end{align*} due to Proposition \ref{prop5.2}, formula \eqref{f6.12}, and the inequalities $s_0<-1/2<\lambda<s_1$. If $\sigma_{1}(\varphi)<-1/2$, then \begin{equation*} H^{\eta}(\Omega)=H^{\lambda}(\Omega)=H^{\lambda,(0)}(\Omega)\hookrightarrow H^{s_1,(0)}(\Omega)\hookrightarrow Y_{0} \end{equation*} due to \eqref{f6.12} and $s_1<-1/2<\lambda$. Note that the written equalities of spaces hold true up to equivalence of norms. Thus, we have the continuous embedding $H^{\eta}(\Omega)\hookrightarrow Y_{0}$ in both cases, which gives \eqref{f6.33}.
Choose $k\in\mathbb{Z}$ such that $k\geq s_1+2q$ and $k\geq s_1-s_0+\ell$, and put $s:=[s_1]+2q-k\leq0$ (as usual, $[s_1]$ stands for the integral part of $s_1$). Then we have the continuous embeddings \begin{equation}\label{f6.34} H^{s,(2q)}(\Omega)\hookrightarrow [H^{s_0+2q-k,(2q)}(\Omega),H^{s_1+2q-k,(2q)}(\Omega)]_{\psi}=X_{2q-k} \end{equation} and \begin{equation}\label{f6.35} H^{\varphi\varrho^{2q-\ell}}(\Omega)\hookrightarrow H^{s_0+2q-\ell}(\Omega)\hookrightarrow H^{s}(\Omega) \end{equation} in view of \eqref{f6.26}. Applying \eqref{f6.31}--\eqref{f6.34} to \eqref{f6.28}, we conclude that \begin{equation}\label{f6.36}
\|\chi u\|_{\varphi\varrho^{2q},\Omega}\leq\widetilde{c}\,
\bigl(\|\zeta_{1}(A,B)u\|_{\eta,\varphi,\Omega,\Gamma}+
\|\zeta_{1}u\|_{s,(2q),\Omega}\bigr) \end{equation} for every $u\in C^{\infty}(\overline{\Omega})$, with the number $\widetilde{c}>0$ being independent of~$u$.
\emph{Step~$3$.} Assume on this step that $u\in C^{\infty}(\overline{\Omega})$, and deduce the required estimate \eqref{f4.21} from \eqref{f6.36} under this assumption. Let $V$ be an open set from the topology on $\overline{\Omega}$ such that $\mathrm{supp}\,\chi\subset V$ and that $\zeta=1$ on $\overline{V}$. We may and do choose $V$ so that $V_0:=V\cap\Omega$ is an open domain in $\mathbb{R}^{n}$ with infinitely smooth boundary. Then the Roitberg space $H^{s,(2q)}(V_0)$ is well defined on $V_0$, with $\|\cdot\|_{s,(2q),V_0}$ denoting the norm in this space; recall that $s:=[s_1]+2q-k\leq0$. Let $v$ be the restriction of $u$ to $\overline{V}$; thus, $v\in C^{\infty}(\overline{V})$.
Let a function $\zeta_{1}\in C^{\infty}(\overline{\Omega})$ satisfy the conditions $\mathrm{supp}\,\zeta_{1}\subset V$ and $\zeta_{1}=1$ in a neighbourhood of $\mathrm{supp}\,\chi$. We then have the equivalence of norms \begin{equation}\label{f6.37}
\|\zeta_{1}u\|_{s,(2q),\Omega}\asymp\|\zeta_{1}v\|_{s,(2q),V_0} \quad\mbox{with respect to}\quad u\in C^{\infty}(\overline{\Omega}). \end{equation} Indeed, according to \cite[Theorem 6.1.1]{Roitberg96}, we get \begin{align*}
\|\zeta_{1}u\|_{s,(2q),\Omega}&\asymp
\|\zeta_{1}u\|_{s,(0),\Omega}+\|A(\zeta_{1}u)\|_{s-2q,(0),\Omega}
=\|\mathcal{O}(\zeta_{1}u)\|_{s,\mathbb{R}^{n}}+
\|\mathcal{O}A(\zeta_{1}u)\|_{s-2q,\mathbb{R}^{n}}\\
&=\|\zeta_{1}v\|_{s,(0),V_0}+\|A(\zeta_{1}v)\|_{s-2q,(0),V_0}\asymp
\|\zeta_{1}v\|_{s,(2q),V_0} \end{align*} with respect to $u$. Recall that $\mathcal{O}(\zeta_{1}u)$ and $\mathcal{O}A(\zeta_{1}u)$ are extensions of the functions $\zeta_{1}v$ and $A(\zeta_{1}v)$, resp., over $\mathbb{R}^{n}$ with zero, these functions being considered on $\overline{V}$ as well as on $\overline{\Omega}$.
Owing to Proposition~\ref{prop6.2}, we have the equivalence of norms \begin{equation*}
\|v\|_{s,(2q),V_0}+\|Av\|_{\eta,V_0}\asymp
\|v\|_{s,V_0}+\|Av\|_{\eta,V_0}\quad\mbox{with respect to}\quad v\in C^{\infty}(\overline{\Omega}). \end{equation*} Combining it with \eqref{f6.37}, we get \begin{align*}
\|\zeta_{1}u\|_{s,(2q),\Omega}&\asymp\|\zeta_{1}v\|_{s,(2q),V_0}\leq c_{7}\|v\|_{s,(2q),V_0}\leq c_{7}\bigl(\|v\|_{s,(2q),V_0}+\|Av\|_{\eta,V_0}\bigr)\\
&\asymp\|v\|_{s,V_0}+\|Av\|_{\eta,V_0}=
\|\zeta v\|_{s,V_0}+\|\zeta Av\|_{\eta,V_0}\leq
\|\zeta u\|_{s,\Omega}+\|\zeta Au\|_{\eta,\Omega}\\
&\leq c_{8}\|\zeta u\|_{\varphi\varrho^{2q-\ell},\Omega}+\|\zeta Au\|_{\eta,\Omega} \end{align*} in view of \eqref{f6.35}. Thus, there exists a number $\widetilde{c}_{3}>0$ such that \begin{equation*}
\|\zeta_{1}u\|_{s,(2q),\Omega}\leq\widetilde{c}_{3}
\bigl(\|\zeta u\|_{\varphi\varrho^{2q-\ell},\Omega}+\|\zeta Au\|_{\eta,\Omega}\bigr) \end{equation*} for every $u\in C^{\infty}(\overline{\Omega})$. Substituting this inequality into \eqref{f6.36}, we obtain \begin{align*}
\|\chi u\|_{\varphi\varrho^{2q},\Omega}&\leq\widetilde{c}\:
\|\zeta_{1}\zeta(A,B)u\|_{\eta,\varphi,\Omega,\Gamma}+ \widetilde{c}\:\widetilde{c}_{3}\bigl(
\|\zeta u\|_{\varphi\varrho^{2q-\ell},\Omega}+\|\zeta Au\|_{\eta,\Omega} \bigr)\\
&\leq c\,\bigl(\|\zeta(A,B)u\|_{\eta,\varphi,\Omega,\Gamma}+
\|\zeta u\|_{\varphi\rho^{2q-\ell},\Omega}\bigr), \end{align*} which gives \eqref{f4.21} under the assumption that $u\in C^{\infty}(\overline{\Omega})$.
\emph{Step~$4$.} Recall we must prove that the estimate \eqref{f4.21} holds true if a distribution $u\in\mathcal{S}'(\Omega)$ satisfies the hypotheses of Theorem~$\ref{th4.7}$. Let us deduce this estimate from the $u\in C^{\infty}(\overline{\Omega})$ case investigated on the previous step. Let $V$ be an open set from the topology on $\overline{\Omega}$ such that $\overline{V}\subset\Omega_{0}\cup\Gamma_{0}$ and ($\mathrm{supp}\,\chi\subset$) $\mathrm{supp}\,\zeta\subset V$ and that $V_0:=V\cap\Omega$ is an open domain in $\mathbb{R}^{n}$ with infinitely smooth boundary $\partial V_{0}$.
Consider an arbitrary distribution $u\in\mathcal{S}'(\Omega)$ that satisfies the hypotheses of Theorem~$\ref{th4.7}$. Then $v:=u\!\upharpoonright\!V_0\in H^{\varphi\varrho^{2q}}_{A,\eta}(V_0)$. Indeed, let a function $\zeta_{1}\in C^{\infty}(\overline{\Omega})$ satisfy the conditions $\mathrm{supp}\,\zeta_{1}\subset\Omega_{0}\cup\Gamma_{0}$ and $\zeta_{1}=1$ on $\overline{V}$; then $\zeta_{1}Au=\zeta_{1}f\in H^{\eta}(\Omega)$ and $\zeta_{1}u\in H^{\varphi\varrho^{2q}}(\Omega)$ due to the hypothesis \eqref{f4.14} and the conclusion of Theorem~$\ref{th4.7}$, respectively; hence, $v\in H^{\varphi\varrho^{2q}}(V_0)$ and $Av\in H^{\eta}(V_0)$, i.e. $v\in H^{\varphi\varrho^{2q}}_{A,\eta}(V_0)$.
Since the set $C^{\infty}(\overline{V})$ is dense in $H^{\varphi\varrho^{2q}}_{A,\eta}(V_0)$ by Theorem~\ref{th1}, there exists a sequence $(u_{k})_{k=1}^{\infty}\subset C^{\infty}(\overline{\Omega})$ such that $v_{k}:=u_{k}\!\upharpoonright\!\overline{V}\to v$ in $H^{\varphi\varrho^{2q}}(V_0)$ and $Av_{k}\to Av$ in $H^{\eta}(V_0)$ as $k\to\infty$. Then \begin{equation}\label{f6.38} \zeta u_{k}\to\zeta u\quad\mbox{in}\;\; H^{\varphi\varrho^{2q}}_{A,\eta}(\Omega) \end{equation} and \begin{equation}\label{f6.39} \zeta(Au_{k})\to\zeta(Au)\quad\mbox{in}\;\;H^{\eta}(\Omega) \end{equation} as $k\to\infty$. Indeed, choose a number $r\gg1$ such that the numbers $2q+\sigma_{0}(\varphi)$, $2q+\sigma_{1}(\varphi)$, $\sigma_{0}(\eta)$, and $\sigma_{1}(\eta)$ belong to $(-r,r)$, and consider a bounded linear operator $T_{0}:H^{-r}(V_{0})\to H^{-r}(\mathbb{R}^{n})$ such that $T_{0}w=w$ in $V_{0}$ whenever $w\in H^{-r}(V_{0})$ and that its restriction to $H^{r}(V_{0})$ is a bounded operator $T_{0}:H^{r}(V_{0})\to H^{r}(\mathbb{R}^{n})$. Such an extension operator exists; see, e.g., \cite[Theorem 4.2.2]{Triebel95}. It follows from Theorem~\ref{prop5.1} that the restriction of $T_{0}$ to $H^{\varphi\varrho^{2q}}(V_0)$ or $H^{\eta}(V_0)$ is a bounded operator $T_{0}:H^{\varphi\varrho^{2q}}(V_0)\to H^{\varphi\varrho^{2q}}(\mathbb{R}^{n})$ or $T_{0}:H^{\eta}(V_0)\to H^{\eta}(\mathbb{R}^{n})$, resp. Then the linear mapping $T:w\mapsto(T_{0}w)\!\upharpoonright\!\Omega$ acts continuously between the following spaces: $T:H^{\varphi\varrho^{2q}}(V_0)\to H^{\varphi\varrho^{2q}}(\Omega)$ and $T:H^{\eta}(V_0)\to H^{\eta}(\Omega)$. Hence, $\zeta u_{k}=\zeta(Tv_{k})\to\zeta(Tv)=\zeta u$ in $H^{\varphi\varrho^{2q}}(\Omega)$ and $\zeta Au_{k}=\zeta(TAv_{k})\to\zeta(TAv)=\zeta Au$ in $H^{\eta}(\Omega)$ as $k\to\infty$; i.e., \eqref{f6.38} and \eqref{f6.39} hold true.
According to \eqref{f6.38}, we obtain \begin{equation}\label{f6.40} \chi u_{k}=\chi\zeta u_{k}\to\chi\zeta u=\chi u\quad\mbox{in}\;\; H^{\varphi\varrho^{2q}}(\Omega) \end{equation} and \begin{equation}\label{f6.41} \zeta u_{k}\to\zeta u\quad\mbox{in}\;\; H^{\varphi\varrho^{2q-\ell}}(\Omega). \end{equation} as $k\to\infty$. Let us show that \begin{equation}\label{f6.42} \zeta B_{j}u_{k}\to\zeta B_{j}u\quad\mbox{in}\;\; H^{\varphi\varrho^{2q-m_j-1/2}}(\Gamma) \end{equation} as $k\to\infty$ for each $j\in\{1,\ldots,q\}$.
Given $j$, we consider a boundary PDO on $\partial V_0$ of the form \begin{equation*} B_{j}^{\star}:=B_{j}^{\star}(x,D):=
\sum_{|\mu|\leq m_j}b_{j,\mu}^{\star}(x)D^{\mu} \end{equation*} where every coefficient $b_{j,\mu}^{\star}$ belongs to $C^{\infty}(\partial V_0)$ and coincides with the corresponding coefficient $b_{j,\mu}$ of $B_{j}$ on $\Gamma\cap\partial V_0$. Since $v_{k}\to v$ in $H^{\varphi\varrho^{2q}}_{A,\eta}(V_0)$, we conclude by Theorem~\ref{th1} in view of Remark~\ref{bounded-operator} that \begin{equation*} B_{j}^{\star}v_{k}\to B_{j}^{\star}v\quad\mbox{in}\quad H^{\varphi\varrho^{{2q}-m_{j}-1/2}}(\partial V_0) \end{equation*} as $k\to\infty$. But $\zeta B_{j}^{\star}v_{k}=\zeta B_{j}u_{k}$ on $\Gamma\cap\partial V_0$ whenever $k\geq1$. Hence, \begin{equation}\label{f6.43} \zeta B_{j}u_{k}\to T_{1}(\zeta B_{j}^{\star}v)\quad\mbox{in}\;\; H^{\varphi\varrho^{2q-m_j-1/2}}(\Gamma), \end{equation} where the distribution $T_{1}(\zeta B_{j}^{\star}v)$ is equal by definition to $\zeta B_{j}^{\star}v$ on $\Gamma\cap V$ and to zero on $\Gamma\setminus\mathrm{supp}\,\zeta$.
Remark that \begin{equation}\label{f6.44} \zeta B_{j}^{\star}v=\zeta B_{j}u\quad\mbox{on}\;\;\Gamma\cap V. \end{equation} Indeed, since $u\in\mathcal{S}'(\Omega)$ and $Au\in H^{-1/2+}(\Omega)$ by the hypotheses of Theorem~$\ref{th4.7}$, there exist numbers $\theta<-1/2$ and $\delta>-1/2$ such that $u\in H^{\theta}_{A,\delta}(\Omega)$. According to Theorem~\ref{th1}, there exists a sequence $(w_{k})_{k=1}^{\infty}\subset C^{\infty}(\overline{\Omega})$ that converges to $u$ in $H^{\theta}_{A,\delta}(\Omega)$. Then $w_{k}^{\circ}:=w_{k}\!\upharpoonright\!V_0\to u\!\upharpoonright\!V_0=v$ in $H^{\theta}_{A,\delta}(V_0)$. Hence, $\zeta B_{j}w_{k}\to\zeta B_{j}u$ in $H^{\theta-m_j-1/2}(\Gamma)$ and also $\zeta B_{j}^{\star}w_{k}^{\circ}\to\zeta B_{j}^{\star}v$ in $H^{\theta-m_j-1/2}(\partial V_0)$ as $k\to\infty$. However, $\zeta B_{j}w_{k}=\zeta B_{j}^{\star}w_{k}^{\circ}$ on $\Gamma\cap\partial V_0\supset\Gamma\cap V$. Hence, the last two limits imply property \eqref{f6.44}. Owing to this property, we have the equality of distributions $T_{1}(\zeta B_{j}^{\star}v)=\zeta B_{j}u$ on $\Gamma$, which together with \eqref{f6.43} gives \eqref{f6.42}.
Now we may complete the proof. According to Step~3, the inequality \begin{equation*}
\|\chi u_k\|_{\varphi\varrho^{2q},\Omega}
\leq c\,\bigl(\|\zeta(A,B)u_k\|_{\eta,\varphi,\Omega,\Gamma}+
\|\zeta u_k\|_{\varphi\rho^{2q-\ell},\Omega}\bigr) \end{equation*} holds true for every $k\geq1$. Passing here to the limit as $k\to\infty$ and using \eqref{f6.39}--\eqref{f6.42}, we conclude that the estimate \eqref{f4.21} holds true under the hypotheses of Theorem~$\ref{th4.7}$. \end{proof}
\begin{remark}\label{rem6.3} We stated in Remark \ref{rem4.14} that Theorem \ref{th4.13} remains valid for every $\varphi\in\mathrm{OR}$ subject to $\sigma_0(\varphi)>-1/2$ if we put $\eta:=\varphi$. The proof of this result is performed in the same way as the proof just given and is somewhat simpler. Namely, we may assume that $-1/2<s_0<\sigma_0(\varphi)$; then $X_{2q}=H^{\varphi\varrho^{2q}}(\Omega)$ and $Y_0=H^{\varphi}(\Omega)$ up to equivalence of norms due to \eqref{f6.12}. This immediately implies \eqref{f6.31} and \eqref{f6.33} on Step~2. \end{remark}
\section{Applications to homogeneous elliptic equations}\label{sec7}
\subsection{Solvability and regularity theorems}\label{sec7.1} Let us discuss applications of the theorems from Section~\ref{sec4} to the regular elliptic boundary value problem \eqref{f1}, \eqref{f2} in the important case where the elliptic equation \eqref{f1} is homogeneous, i.e. $f=0$ in $\Omega$. In this case, we may formulate versions of these theorems for every $\varphi\in\mathrm{OR}$. We will consider these versions for more general Theorems \ref{th1}, \ref{th2}, \ref{th4.7}, and \ref{th4.13} and then discuss the corresponding proofs. It is convenient to use the function parameter $\alpha:=\varphi\rho^{2q}$ in the case indicated.
Given $\alpha\in\mathrm{OR}$, we put \begin{equation*} H^{\alpha}_{A}(\Omega):= \bigl\{u\in H^{\alpha}(\Omega):Au=0\;\,\mbox{in}\;\,\Omega\bigr\}; \end{equation*} as usual, $Au$ is understood in the theory of distributions. We endow the linear space $H^{\alpha}_{A}(\Omega)$ with the inner product and norm in $H^{\alpha}(\Omega)$. The space $H^{\alpha}_{A}(\Omega)$ is complete with this norm because the differential operator $A$ is continuous on $\mathcal{D}'(\Omega)$. If $\alpha(t)\equiv t^{s}$ for a certain $s\in\mathbb{R}$, the space $H^{\alpha}_{A}(\Omega)$ is also denoted by $H^{s}_{A}(\Omega)$ according to our convention.
Since $A$ is elliptic on $\overline{\Omega}$, the inclusion $H^{\alpha}_{A}(\Omega)\subset C^{\infty}(\Omega)$ holds true (see, e.g., \cite[Chapter~2, Theorem~3.2]{LionsMagenes72}. However, $H^{\alpha}_{A}(\Omega)\not\subset C^{\infty}(\overline{\Omega})$. Put \begin{equation*} C^{\infty}_{A}(\overline{\Omega}):= \bigl\{u\in C^{\infty}(\overline{\Omega}):Au=0\;\,\mbox{on}\;\, \overline{\Omega}\bigr\}. \end{equation*} With the problem \eqref{f1}, \eqref{f2} in the $f=0$ case, we associate the mapping \begin{equation}\label{f7.1} B_{A}:u\mapsto Bu=(B_{1}u,\ldots,B_{q}u),\quad\mbox{where}\quad u\in C^{\infty}_{A}(\overline{\Omega}). \end{equation} Put \begin{equation*} N^{+}_1:=\bigl\{(C^{+}_{1}v,\ldots,C^{+}_{q}v):v\in N^{+}\bigr\}. \end{equation*} Of course, $\dim N^{+}_{1}\leq\dim N^{+}<\infty$. The inequality $\dim N^{+}_{1}<\dim N^{+}$ is possible, which follows from a result by Pli\'{s} \cite{Plis61} (this result is expounded in the book \cite[Theorem~13.6.15]{Hermander83}).
\begin{theorem}\label{th7.1} Let $\alpha\in\mathrm{OR}$. Then the set $C^{\infty}_{A}(\overline{\Omega})$ is dense in the space $H^{\alpha}_{A}(\Omega)$, and the mapping \eqref{f7.1} extends uniquely (by continuity) to a bounded linear operator \begin{equation}\label{f7.2} B_{A}:H^{\alpha}_{A}(\Omega)\to \bigoplus_{j=1}^{q}H^{\alpha\rho^{-m_j-1/2}}(\Gamma)=: \mathcal{H}_{\alpha}(\Gamma). \end{equation} This operator is Fredholm. Its kernel coincides with $N$, and its range consists of all vectors $g\in\mathcal{H}_{\alpha}(\Gamma)$ such that \begin{equation}\label{f7.3} \sum_{j=1}^{q}\,(g_{j},\,C^{+}_{j}v)_{\Gamma}=0 \quad\mbox{for every}\quad v\in N^{+}. \end{equation} The index of the operator \eqref{f7.2} equals $\dim N-\dim N^{+}_1$ and does not depend on $\alpha$. \end{theorem}
If $N=\{0\}$ and $N^{+}_1=\{0\}$, the operator \eqref{f7.2} is an isomorphism between the spaces $H^{\alpha}_{A}(\Omega)$ and $\mathcal{H}_{\alpha}(\Gamma)$. Generally, this operator induces an isomorphism which may be built with the help of the following decompositions of these spaces: \begin{gather}\label{f7.4} H^{\alpha}_{A}(\Omega)=N\dotplus\{u\in H^{\alpha}_{A}(\Omega):(u,w)_\Omega=0\;\,\mbox{for every}\;\,w\in N\},\\ \mathcal{H}_{\alpha}(\Gamma)=N^{+}_{1}\dotplus \{(g_1,\ldots,g_q)\in\mathcal{H}_{\alpha}(\Gamma):\mbox{\eqref{f7.3} is true}\}.\label{f7.5} \end{gather} These formulas need commenting. If $\sigma_{0}(\alpha)>0$, the second summand in \eqref{f7.4} is well defined and is closed in $H^{\alpha}_{A}(\Omega)$ due to the continuous embedding $H^{\alpha}_{A}(\Omega)\hookrightarrow L_{2}(\Omega)$. Hence, in this case, \eqref{f7.4} is the restriction to $H^{\alpha}_{A}(\Omega)$ of the corresponding decomposition of $L_{2}(\Omega)$ into the orthogonal sum of subspaces. If $\sigma_{0}(\alpha)\leq0$, we use Lemma~\ref{lema1} and also formula \eqref{sum1} for $\alpha=\varphi\varrho^{2q}$ and $\omega=\eta$. In this case, the second summand in \eqref{f7.4} is well defined and closed in $H^{\alpha}_{A}(\Omega)$ according to this lemma, and \eqref{f7.4} is the restriction of the decomposition \eqref{sum1} to $H^{\alpha}_{A}(\Omega)$. Formula \eqref{f7.5} is true because the summands on the right have the trivial intersection, and the finite dimension of the first summand coincides with the codimension of the second. Indeed, since $\mathcal{H}_{\alpha}(\Gamma)$ is dual to $\mathcal{H}_{1/\alpha}(\Gamma)$ with respect to the form $(\cdot,\cdot)_{\Gamma}+\cdots+(\cdot,\cdot)_{\Gamma}$ (this is proved analogously to \cite[Theorem~2.3(v)]{MikhailetsMurach14}), the dimension of the dual of $N_{1}^{+}\subset\mathcal{H}_{1/\alpha}(\Gamma)$ equals the above-mentioned codimension.
Let $P_{1}$ and $\mathcal{P}^+_{1}$ respectively denote the projectors of the spaces $H^{\alpha}_{A}(\Omega)$ and $\mathcal{H}_{\alpha}(\Gamma)$ onto the second summand in \eqref{f7.4} and \eqref{f7.5} parallel to the first. The mappings defining these projectors do not depend on $\alpha$. Note that $P_{1}$ is a restriction of the projector $P$ from Theorem~\ref{th2} where $\alpha=\varphi\varrho^{2q}$ and $\sigma_{0}(\alpha)\leq2q-1/2$.
\begin{theorem}\label{th7.2} Let $\alpha\in\mathrm{OR}$. Then the restriction of the operator \eqref{f7.2} to the second summand in \eqref{f7.4} is an isomorphism \begin{equation}\label{f7.6} B_{A}:P_{1}(H^{\alpha}_{A}(\Omega))\leftrightarrow \mathcal{P}^{+}_{1}(\mathcal{H}_{\alpha}(\Gamma)). \end{equation} \end{theorem}
Let us turn to properties of generalized solutions to the elliptic problem \eqref{f1}, \eqref{f2} in the case where $f=0$ in $\Omega$. The notion of a generalized solution introduced just before Theorem~\ref{th4.6} is applicable in this case. Since every solution $u$ to the homogeneous elliptic equation \eqref{f1} belongs to $C^{\infty}(\Omega)$, we are interested in properties of $u=u(x)$ when the argument $x$ approaches the boundary $\Gamma$ of $\Omega$. Let $\Gamma_{0}$ be a nonempty open subset of $\Gamma$.
\begin{theorem}\label{th7.3} Let $\alpha\in\mathrm{OR}$. Assume that a distribution $u\in\mathcal{S}'(\Omega)$ is a generalized solution to the elliptic problem \eqref{f1}, \eqref{f2} whose right-hand sides satisfy the conditions $f=0$ in $\Omega$ and \begin{equation}\label{f7.7} g_j\in H^{\alpha\rho^{-m_j-1/2}}_{\mathrm{loc}}(\Gamma_0) \quad\mbox{for each}\quad j\in\{1,\ldots,q\}. \end{equation} Then $u\in H^{\alpha}_{\mathrm{loc}}(\Omega,\Gamma_{0})$. \end{theorem}
We supplement this theorem with a corresponding estimate of $u$. Let $\|\cdot\|_{\alpha,\Gamma}'$ denote the norm in the Hilbert space $\mathcal{H}_{\alpha}(\Gamma)$ defined in \eqref{f7.2}.
\begin{theorem}\label{th7.4} Let $\alpha\in\mathrm{OR}$, and assume that a distribution $u\in\mathcal{S}'(\Omega)$ satisfies the hypotheses of Theorem~$\ref{th7.3}$. We arbitrarily choose a number $\ell>0$ and functions $\chi,\zeta\in C^{\infty}(\overline{\Omega})$ such that $\mathrm{supp}\,\chi\subset\mathrm{supp}\,\zeta\subset\Omega\cup\Gamma_{0}$ and that $\zeta=1$ in a neighbourhood of $\mathrm{supp}\,\chi$. Then \begin{equation}\label{f7.8}
\|\chi u\|_{\alpha,\Omega}\leq c\,\bigl(\|\zeta g\|_{\alpha,\Gamma}'+
\|\zeta u\|_{\alpha\rho^{-\ell},\Omega}\bigr) \end{equation} for some number $c>0$ that does not depend on $u$ and $g$. \end{theorem}
Let us discuss the proofs of these theorems. Theorem~\ref{th7.1} follows from Proposition~\ref{prop1} and Theorem~\ref{th1} excepting the conclusion about the density of $C^{\infty}_{A}(\overline{\Omega})$ in $H^{\alpha}_{A}(\Omega)$. Indeed, the restriction of the Fredholm operator \eqref{prop1} if $\sigma_0(\varphi)>-1/2$ or the Fredholm operator \eqref{f15} if $\sigma_0(\varphi)\leq-1/2$ to the space $H^{\alpha}_{A}(\Omega)$, where $\alpha=\varphi\varrho^{2q}$, is evidently a Fredholm bounded operator between the spaces \eqref{f7.2} with indicated properties of its kernel, range, and index. Theorem~\ref{th7.2} is a direct consequence of this part of Theorem~\ref{th7.1} and the Banach theorem on inverse operator. The mentioned density is easily deduced from Theorem~\ref{th7.2}. Indeed, since the set $(C^{\infty}(\Gamma))^{q}$ is dense in $\mathcal{H}_{\alpha}(\Gamma)$ for every $\alpha\in\mathrm{OR}$, its subset $\mathcal{P}^{+}_{1}((C^{\infty}(\Gamma))^{q})$ is dense in the range of the isomorphism \eqref{f7.6}. Hence, the set $B_{A}^{-1}\mathcal{P}^{+}_{1}((C^{\infty}(\Gamma))^{q})$ lies in $C^{\infty}_{A}(\overline{\Omega})$ and is dense in the subspace $P_{1}(H^{\alpha}_{A}(\Omega))$ of $H^{\alpha}_{A}(\Omega)$; here, $B_{A}^{-1}$ denotes the inverse of \eqref{f7.6}. This yields the required density of $C^{\infty}_{A}(\overline{\Omega})$ in $H^{\alpha}_{A}(\Omega)=N\dotplus P_{1}(H^{\alpha}_{A}(\Omega))$. Theorem~\ref{th7.3} follows immediately from Theorem~\ref{th4.7} and Remark~\ref{rem4.11}. Theorem~\ref{th7.4} is a direct consequence of Theorem~\ref{th4.13} and Remark~\ref{rem4.14}.
Note that Theorems \ref{th7.1} and \ref{th7.2} are established in our paper \cite[Section~4]{AnopMurach16Coll2}, whereas Theorem~\ref{th7.3} is announced in \cite[Section~3]{AnopMurach18Dop3} (without proof), these papers being published in Ukrainian. If the function $\alpha$ is regularly varying at infinity, Theorem \ref{th7.1} is proved in \cite[Section~1]{MikhailetsMurach06UMJ11} (see also the monograph \cite[Section~3.3.1]{MikhailetsMurach14}). This theorem is a classical result in the Sobolev case where $\alpha(t)\equiv t^{s}$; see, e.g., the book \cite[Chapter~2, Section~7.3]{LionsMagenes72}, which contains this theorem if $s\in\mathbb{R}\setminus\{-1/2,-3/2,\ldots\}$. In this connection, we mention Seeley's paper \cite{Seeley66}, which investigates the Cauchy data of functions from $H^{s}_{A}(\Omega)$ where $s\in\mathbb{R}$ (see also the survey \cite[Section 5.4~b]{Agranovich97}).
\subsection{Uniform convergence of solutions}\label{sec7.2} Using generalized Sobolev spaces over $\Gamma$, we obtain a sufficient condition for the uniform convergence of solutions to the elliptic equation $Au=0$ and their derivatives of a prescribed order.
\begin{theorem}\label{th7.5} Let $0\leq p\in\mathbb{Z}$. Assume that a sequence $(u_{k})_{k=1}^{\infty}\subset\mathcal{S}'(\Omega)$ satisfies the following two conditions: $Au_{k}=0$ in $\Omega$ whenever $k\geq1$, and the sequence of the distributions $g^{(k)}:=B_{A}u_{k}$ converges in the space $\mathcal{H}_{\alpha}(\Gamma)$ for some $\alpha\in\mathrm{OR}$ subject to \begin{equation}\label{f7.9} \int\limits_1^{\infty} t^{2p+n-1}\alpha^{-2}(t)\,dt<\infty. \end{equation}
Then every $u_{k}\in C^{p}(\overline{\Omega})$, and there exists a function $u\in C^{p}(\overline{\Omega})$ that the sequence $(D^{\mu}P_{1}u_{k})_{k=1}^{\infty}$ converges uniformly to $D^{\mu}u$ on $\overline{\Omega}$ whenever $|\mu|\leq p$. The function $u$ satisfies the conditions $Au=0$ in $\Omega$ and $B_{A}u=g$ on $\Gamma$, where $g$ is the limit of the sequence $(g^{(k)})_{k=1}^{\infty}$. \end{theorem}
In this theorem, the vectors $B_{A}u_{k}$ and $B_{A}u$ are well defined by means of the operator \eqref{f7.2} because $u_{k}\in H^{-r}_{A}(\Omega)$ whenever $r\gg1$ due to the hypotheses of the theorem and because $u\in H^{p}_{A}(\Omega)$ due to its conclusion. If $p\leq m_{j}-1$, the smoothness of $u_{k}$ and $u$ is not sufficient to find the $j$-th components of $B_{A}u_{k}$ and $B_{A}u$ with the help of classical derivatives. Recall that the hypothesis $Au_{k}=0$ in $\Omega$ and the conclusion $Au=0$ in $\Omega$ are understood in the distribution theory sense and imply the inclusions of $u_{k}$ and $u$ in $C^{\infty}(\Omega)$. Note if $N=\{0\}$, then $P_{1}u_{k}=u_{k}$.
It is useful to compare this theorem with the classical Harnack theorem on the uniform convergence of a sequence of harmonic functions on $\overline{\Omega}$ (see, e.g., \cite[Section~2.6]{GilbargTrudinger98}). The latter theorem (also called the Bauer convergence property) relates to the case where $A$ is the Laplace operator, $B_{A}u:=u\!\upharpoonright\!\Gamma$ for every $u\in C^{\infty}_{A}(\overline{\Omega})$, and $p=0$ in Theorem~\ref{th7.5}. In this case, $H_{\alpha}(\Gamma)=H^{\alpha\varrho^{-1/2}}(\Gamma)\hookrightarrow C(\Gamma)$ due to condition \eqref{f7.9} and property \eqref{Hermander-embedding} considered for $C(\mathbb{R}^{n-1})$ instead of $C^{p}(\mathbb{R}^{n})$. Hence, the conclusion of Theorem~\ref{th7.5} about the uniform convergence of the sequence of harmonic functions $u_{k}$ follows from the Harnak theorem. However, for first-order boundary conditions, Theorem~\ref{th7.5} gives weak enough and new sufficient conditions for this convergence. Thus, considering the Neumann boundary condition, we conclude by Theorem~\ref{th7.5} that the sequence of harmonic functions $u_{k}\in\mathcal{S}'(\Omega)$ subject to $(u_{k},1)_{\Omega}=0$ converges uniformly on $\overline{\Omega}$ if the sequence of traces of their normal derivatives converges in the space $H^{\omega}(\Gamma)$ where $\omega(t):=t^{(n-3)/2}\log(1+t)$ whenever $t\geq1$, e.g. In the $n=2$ case, this space is broader than $H^{-1/2+}(\Gamma)$.
Consider a version of Theorem~\ref{th7.5} for an open subset $\Gamma_{0}\neq\emptyset$ of the boundary $\Gamma$. Given $\alpha\in\mathrm{OR}$, we introduce the linear space \begin{equation*} \mathcal{H}_{\alpha}(\Gamma_{0}):=\{g\!\upharpoonright\!\Gamma_{0}:g\in \mathcal{H}_{\alpha}(\Gamma)\} \end{equation*} endowed with the norm \begin{equation*}
\|h\|_{\alpha,\Gamma_{0}}':=
\inf\bigl\{\,\|g\|_{\alpha,\Gamma}': g\in\mathcal{H}_{\alpha}(\Gamma),\;g=h\;\,\mbox{in}\;\,\Gamma_{0}\bigr\} \end{equation*} of $h\in\mathcal{H}_{\alpha}(\Gamma_{0})$.
\begin{theorem}\label{th7.6}
Let $0\leq p\in\mathbb{Z}$. Assume that a sequence $(u_{k})_{k=1}^{\infty}\subset\mathcal{S}'(\Omega)$ satisfies the following three conditions: $Au_{k}=0$ in $\Omega$ whenever $k\geq1$, this sequence converges in the Sobolev space $H^{-r}(\Omega)$ if $r\gg1$, and the sequence of the distributions $(B_{A}u_{k})\!\upharpoonright\!\Gamma_{0}$ converges in the space $\mathcal{H}_{\alpha}(\Gamma_{0})$ for some $\alpha\in\mathrm{OR}$ subject to \eqref{f7.9}. Then every $u_{k}\in C^{p}(\Omega\cup\Gamma_{0})$, and each sequence $(D^{\mu}u_{k})_{k=1}^{\infty}$, with $|\mu|\leq p$, converges uniformly on every closed (in $\mathbb{R}^{n}$) subset of $\Omega\cup\Gamma_{0}$. \end{theorem}
\begin{proof}[Proof of Theorem $\ref{th7.5}$.] By Theorem~\ref{th7.3} in the $\Gamma_{0}=\Gamma$ case and by the H\"ormander embedding theorem \eqref{Hermander-embedding}, we conclude that each $u_{k}\in H^{\alpha}(\Omega)\hookrightarrow C^{p}(\overline{\Omega})$. Since $g^{(k)}\to g$ in the subspace $\mathcal{P}^{+}_{1}(\mathcal{H}_{\alpha}(\Gamma))$ of $\mathcal{H}_{\alpha}(\Gamma)$, Theorem~\ref{th7.2} implies the convergence $P_{1}u_{k}\to u$ in $H^{\alpha}(\Omega)$, where $u\in P_{1}(H^{\alpha}_{A}(\Omega))$ is the inverse image of $g$ under the isomorphism \eqref{f7.6}. This gives the conclusion of Theorem~\ref{th7.5} due to the continuous embedding $H^{\alpha}(\Omega)$ in $C^{p}(\overline{\Omega})$. \end{proof}
\begin{proof}[Proof of Theorem $\ref{th7.6}$.] Since every $(B_{A}u_{k})\!\upharpoonright\!\Gamma_{0}$ belongs to $\mathcal{H}_{\alpha}(\Gamma_{0})$, we have the inclusion \begin{equation*} B_{A}u_{k}\in\prod_{j=1}^{q} H^{\alpha\rho^{-m_j-1/2}}_{\mathrm{loc}}(\Gamma_0). \end{equation*} Hence, every $u_{k}\in H^{\alpha}_{\mathrm{loc}}(\Omega,\Gamma_{0})$ due to Theorem~\ref{th7.3}, which implies by \eqref{Hermander-embedding} that $u_{k}\in C^{p}(\Omega\cup\Gamma_{0})$. By the hypotheses of Theorem~\ref{th7.6}, we have \begin{equation}\label{f7.10} u_{k}\to u\quad\mbox{in}\quad H^{-r}(\Omega)\quad\mbox{for some}\quad u\in H^{-r}_{A}(\Omega) \end{equation} and \begin{equation}\label{f7.11} (B_{A}u_{k})\!\upharpoonright\!\Gamma_{0}\to (B_{A}u)\!\upharpoonright\!\Gamma_{0}\quad\mbox{in}\quad \mathcal{H}_{\alpha}(\Gamma_{0}) \end{equation} as $k\to\infty$. Thus, the distribution $u_{k}-u$ satisfies the hypotheses of Theorem~\ref{th7.3} in which we take $u_{k}-u$ instead of $u$. Hence, we may apply Theorem~\ref{th7.4} to $u_{k}-u$. Let $G$ be a nonempty closed subset of $\Omega\cup\Gamma_{0}$. Choose functions $\chi,\zeta\in C^{\infty}(\overline{\Omega})$ such that $\mathrm{supp}\,\chi\subset \mathrm{supp}\,\zeta\subset\Omega\cup\Gamma_{0}$, $\chi=1$ in a neighbourhood of $G$, and $\zeta=1$ in a neighbourhood of $\mathrm{supp}\,\chi$. According to Theorem~\ref{th7.4}, we have the inequality \begin{equation*}
\|\chi(u_{k}-u)\|_{\alpha,\Omega}\leq c\,
\bigl(\|\zeta B_{A}(u_{k}-u)\|_{\alpha,\Gamma}'+
\|\zeta(u_{k}-u)\|_{-r,\Omega}\bigr), \end{equation*}
where the number $c>0$ does not depend on $u_{k}-u$. Hence, $\chi u_{k}\to\chi u$ in $H^{\alpha}(\Omega)$ as $k\to\infty$ due to \eqref{f7.10} and \eqref{f7.11}. Therefore, $\chi u_{k}\to\chi u$ in $C^{p}(\overline{\Omega})$ by \eqref{Hermander-embedding}, which implies that the sequence $(D^{\mu}u_{k})_{k=1}^{\infty}$ converges uniformly on $G$ whenever $|\mu|\leq p$. \end{proof}
\begin{remark}\label{rem7.7} Considering Theorems \ref{th7.5} and \ref{th7.6}, it is useful to take into account the following property: if a sequence $(u_{k})_{k=1}^{\infty}\subset\mathcal{S}'(\Omega)$ satisfies the first two conditions formulated in Theorem~\ref{th7.6}, the sequence $(D^{\mu}u_{k})_{k=1}^{\infty}$ will converge uniformly on every closed subset $G$ of $\Omega$ for every multi-index $\mu$. This fact is known and follows from the internal a priory estimate \begin{equation}\label{f7.12}
\|\chi(u_{k}-u)\|_{\ell,\Omega}\leq c_0\|u_{k}-u\|_{-r,\Omega}\to0 \quad\mbox{as}\quad k\to\infty \end{equation} in Sobolev spaces. Here, $\ell\gg1$, $u$ is the limit of the sequence $(u_{k})_{k=1}^{\infty}$, $\chi\in C^{\infty}(\overline{\Omega})$, $\mathrm{supp}\,\chi\subset\Omega$, $\chi=1$ in a neighbourhood of $G$, and $c_0$ is some positive number that does not depend on $u_{k}-u$. It follows from \eqref{f7.12} by the Sobolev embedding theorem, that $\chi u_{k}\to\chi u$ in $C^{p}(\overline{\Omega})$ whenever $0\leq p\in\mathbb{Z}$, which yields the uniform convergence of $(D^{\mu}u_{k})_{k=1}^{\infty}$ on $G$ for every $\mu$. The estimate \eqref{f7.12} is known (see, e.g., \cite[Theorem~7.2.2]{Roitberg96}). \end{remark}
\subsection{Interpolation properties of related spaces}\label{sec7.3} Consider the Hilbert spaces $H^{\alpha}_{A}(\Omega)$, where $\alpha\in\mathrm{OR}$, formed by solutions to the homogeneous elliptic equation $Au=0$ in $\Omega$. These spaces have analogous interpolation properties to that of $H^{\alpha}(\Omega)$.
\begin{theorem}\label{th7.8} \begin{itemize}
\item[(i)] Under the hypotheses of Proposition~$\ref{prop5.1}$, we have \begin{equation*} [H^{r_0}_{A}(\Omega),H^{r_1}_{A}(\Omega)]_{\psi}=H^{\alpha}_{A}(\Omega) \end{equation*} up to equivalence of norms.
\item[(ii)] Under the hypotheses of Proposition~$\ref{prop5.2}$, we have \begin{equation*} [H^{\alpha_0}_{A}(\Omega),H^{\alpha_1}_{A}(\Omega)]_{\psi}= H^{\alpha}_{A}(\Omega) \end{equation*} up to equivalence of norms. \end{itemize} \end{theorem}
This theorem shows that the class of spaces \begin{equation}\label{f7.13} \{H^{\alpha}(\Omega):\alpha\in\mathrm{OR}\} \end{equation} is obtained by the interpolation with a function parameter between their Sobolev analogs and is closed with respect to the interpolation with a function parameter between Hilbert spaces.
\begin{theorem}\label{th7.9} Let $r_{0},r_{1}\in\mathbb{R}$ and $r_{0}<r_{1}$. A Hilbert space $H$ is an interpolation space between the spaces $H^{r_{0}}_{A}(\Omega)$ and $H^{r_{1}}_{A}(\Omega)$ if and only if $H=H^{\alpha}_{A}(\Omega)$ up to equivalence of norms for some function parameter $\alpha\in\mathrm{OR}$ that satisfies condition~\eqref{f3.2}. \end{theorem}
Of course, we mean in this theorem that the numbers $c_{0}$ and $c_{1}$ in condition \eqref{f3.2} do not depend on $t$ and $\lambda$. This condition is equivalent to the following pair of conditions: \begin{enumerate} \item [$\mathrm{(i)}$] $r_{0}\leq\sigma_{0}(\varphi)$ and, moreover, $r_{0}<\sigma_{0}(\varphi)$ if the supremum in $\eqref{f3.2sup}$ is not attained; \item [$\mathrm{(ii)}$] $\sigma_{1}(\varphi)\leq r_{1}$ and, moreover, $\sigma_{1}(\varphi)<r_{1}$ if the infimum in $\eqref{f3.2inf}$ is not attained. \end{enumerate}
Theorem~$\ref{th7.9}$ reveals that the class \eqref{f7.13} coincides up to equivalence of norms with the class of all Hilbert spaces that are interpolation ones between the Sobolev spaces $H^{r_{0}}_{A}(\Omega)$ and $H^{r_{1}}_{A}(\Omega)$ where $r_0,r_1\in\mathbb{R}$ and $r_0<r_1$. If we omit the subscript $A$ in the formulation of Theorem~$\ref{th7.9}$, we will obtain the corresponding interpolation property of the class $\{H^{\alpha}(\Omega):\alpha\in\mathrm{OR}\}$ proved in \cite[Theorem~2.4]{MikhailetsMurach15ResMath1}.
\begin{proof}[Proof of Theorem $\ref{th7.8}$.] Assertion (i) is a direct consequence of (ii). Let us prove (ii). Consider the isomorphisms \eqref{f7.6} where $\alpha\in\{\alpha_0,\alpha_1\}$ and interpolate them with the function parameter~$\psi$. Since $\psi$ is an interpolation parameter, we conclude that the restriction of the mapping \eqref{f7.6}, where $\alpha=\alpha_0$, is an isomorphism \begin{equation}\label{f7.14} B_{A}:\bigl[P_{1}(H^{\alpha_0}_{A}(\Omega)), P_{1}(H^{\alpha_1}_{A}(\Omega))\bigr]_{\psi}\leftrightarrow \bigl[\mathcal{P}^{+}_{1}(\mathcal{H}_{\alpha_0}(\Gamma)), \mathcal{P}^{+}_{1}(\mathcal{H}_{\alpha_0}(\Gamma))\bigr]_{\psi}. \end{equation} According to Proposition~\ref{prop5.2} and the theorem on interpolation of subspaces \cite[Theorem~1.6]{MikhailetsMurach14}, the range of \eqref{f7.14} equals \begin{equation*} [\mathcal{H}_{\alpha_0}(\Gamma),\mathcal{H}_{\alpha_1}(\Gamma)]_{\psi}\cap \mathcal{P}^{+}_{1}(\mathcal{H}_{\alpha_0}(\Gamma))= \mathcal{H}_{\alpha}(\Gamma)\cap \mathcal{P}^{+}_{1}(\mathcal{H}_{\alpha_0}(\Gamma)) =\mathcal{P}^{+}_{1}(\mathcal{H}_{\alpha}(\Gamma)). \end{equation*} Hence, \begin{equation}\label{f7.15} \bigl[P_{1}(H^{\alpha_0}_{A}(\Omega)), P_{1}(H^{\alpha_1}_{A}(\Omega))\bigr]_{\psi}= P_{1}(H^{\alpha}_{A}(\Omega)) \end{equation} due to the isomorphisms \eqref{f7.6} and \eqref{f7.14}. All these equalities of Hilbert spaces hold true up to equivalence of norms.
Given $\omega\in\mathrm{OR}$, we let $\widetilde{H}^{\omega}_{A}(\Omega)$ denote the linear space $H^{\omega}_{A}(\Omega)$ endowed with the equivalent inner product \begin{equation*} (P_{1}u,P_{1}v)_{\omega,\Omega}+(u-P_{1}u,u-P_{1}v)_{\omega,\Omega} \end{equation*} of functions $u,v\in H^{\omega}_{A}(\Omega)$. Now $\widetilde{H}^{\omega}_{A}(\Omega)$ equals the orthogonal sum $N\oplus P_{1}(H^{\omega}_{A}(\Omega))$. Hence, \begin{align*} [H^{\alpha_0}_{A}(\Omega),H^{\alpha_1}_{A}(\Omega)]_{\psi}&= [\widetilde{H}^{\alpha_0}_{A}(\Omega), \widetilde{H}^{\alpha_1}_{A}(\Omega)]_{\psi}= [N,N]_{\psi}\oplus\bigl[P_{1}(H^{\alpha_0}_{A}(\Omega)), P_{1}(H^{\alpha_1}_{A}(\Omega))\bigr]_{\psi}\\ &=N\oplus P_{1}(H^{\alpha}_{A}(\Omega))=\widetilde{H}^{\alpha}_{A}(\Omega) =H^{\alpha}_{A}(\Omega) \end{align*} up to equivalence of norms due to \eqref{f7.15} and the theorem on interpolation of orthogonal sums of spaces \cite[Theorem~1.5]{MikhailetsMurach14}. Assertion (ii) is proved. \end{proof}
\begin{proof}[Proof of Theorem $\ref{th7.9}$. Necessity.] Let a Hilbert space $H$ be an interpolation space between $H^{r_{0}}_{A}(\Omega)$ and $H^{r_{1}}_{A}(\Omega)$. We then conclude by Ovchinnikov's theorem \cite[Theorem 11.4.1]{Ovchinnikov84} that $H=[H^{r_{0}}_{A}(\Omega),H^{r_{1}}_{A}(\Omega)]_{\psi}$ up to equivalence of norms for some interpolation parameter $\psi\in\mathcal{B}$. Hence, $H=H^{\alpha}_{A}(\Omega)$ according to Theorem \ref{th7.8}(ii), where the function $\alpha(t):=t^{r_{0}}\,\psi(t^{r_{1}-r_{0}})$ of $t\geq1$ belongs to $\mathrm{OR}$. This function satisfies \eqref{f3.2} due to \cite[Theorem 4.2]{MikhailetsMurach15ResMath1}. The necessity is proved.
\textit{Sufficiency.} Assume that a Hilbert space $H$ coincides up to equivalence of norms with the space $H^{\alpha}_{A}(\Omega)$ for some $\alpha\in\mathrm{OR}$ subject to \eqref{f3.2}. Define the function $\psi\in\mathcal{B}$ by formula \eqref{f5.1}. Since $\alpha(t)=t^{r_{0}}\,\psi(t^{r_{1}-r_{0}})$ whenever $t\geq1$, the function $\psi$ is an interpolation parameter by \cite[Theorem 4.2]{MikhailetsMurach15ResMath1}. Therefore, $H$ equals $[H^{r_0}_{A}(\Omega),H^{r_1}_{A}(\Omega)]_{\psi}$ up to equivalence of norms due to Theorem \ref{th7.8}(ii). Thus, $H$ is an interpolation space between $H^{r_{0}}_{A}(\Omega)$ and $H^{r_{1}}_{A}(\Omega)$. The sufficiency is also proved. \end{proof}
\section{Application to elliptic problems with boundary white noise}\label{sec8}
In this section, we apply the above results to some elliptic problems with rough boundary data induced by white noise. In particular, we are interested in boundary data belonging to the Nikolskii space $B_{p,\infty}^s(\Gamma)$ with $s<0$ and $p=2$ (see \cite[Sections 2.3.1 and 4.7.1]{Triebel95} and references therein on works by Nikolskii, e.g. \cite[Section 4.3.3]{Nikolskii77}, who introduced and investigated the space $B_{p,\infty}^s(\mathbb{R}^{n})$ for $s>0$ and $1\leq p\leq\infty$). This is motivated by recent results on Gaussian white noise; see below for details. We start with an embedding result.
\begin{proposition}\label{8.1} Let $1\leq n\in\mathbb{Z}$, $s\in\mathbb{R}$, and $\alpha\in\mathrm{OR}$. Then the condition \begin{equation}\label{int-cond-8.1} \int\limits_{1}^{\infty}\frac{\alpha^{2}(t)}{t^{2s+1}}\,dt<\infty. \end{equation} is equivalent to the continuous embedding \begin{equation}\label{embed} B^{s}_{2,\infty}(\mathbb{R}^{n})\hookrightarrow H^{\alpha}(\mathbb{R}^{n}). \end{equation} \end{proposition}
This proposition is implicitly contained in Gol'dman's result \cite[Chapter~1, Theorem~2]{Goldman84}. We will give a proof of this proposition for the reader's convenience.
\begin{proof}[Proof of Proposition~$\ref{8.1}$] First we will treat the $s>0$ case and then reduce the $s\leq0$ case to the previous one. Put $Q_{0}:=\{\xi\in\mathbb{R}^{n}:|\xi|\leq1\}$ and
$Q_{k}:=\{\xi\in\mathbb{R}^{n}:2^{k-1}<|\xi|\leq2^{k}\}$ whenever $1\leq k\in\mathbb{Z}$. Let $s>0$; then the Nikolskii space $B^{s}_{2,\infty}(\mathbb{R}^{n})$ consists of all functions $w\in L_{2}(\mathbb{R}^{n})$ such that \begin{equation*}
\|w\|_{s,\infty,\mathbb{R}^{n}}^{2}:=\sup_{0\leq k\in\mathbb{Z}}4^{sk}\int\limits_{Q_{k}}|\widehat{w}(\xi)|^{2}d\xi <\infty, \end{equation*}
with the norm in this space being equivalent to $\|\cdot\|_{s,\infty,\mathbb{R}^{n}}$; see, e.g., \cite[Lemma 2.11.2]{Triebel95}.
Assume that condition \eqref{int-cond-8.1} is satisfied. Given $w\in B^{s}_{2,\infty}(\mathbb{R}^{n})$, we have \begin{equation}\label{norms-inequality} \begin{aligned}
\|w\|_{\alpha,\mathbb{R}^{n}}^{2}&=\sum_{k=0}^{\infty}\,
\int\limits_{Q_k}\alpha^{2}(\langle\xi\rangle)\,|\widehat{w}(\xi)|^{2}d\xi \asymp\sum_{k=0}^{\infty}\alpha^{2}(2^{k})
\int\limits_{Q_k}|\widehat{w}(\xi)|^{2}d\xi\\ &\leq\biggl(\sum_{k=0}^{\infty} \frac{\alpha^{2}(2^{k})}{4^{sk}}\biggr) \sup_{0\leq k\in\mathbb{Z}}4^{sk}
\int\limits_{Q_k}|\widehat{w}(\xi)|^{2}d\xi
=c\,\|w\|_{s,\infty,\mathbb{R}^{n}}^{2}<\infty. \end{aligned} \end{equation} Here, the symbol "$\asymp$" means the equivalence of norms squared, this equivalence being true by \eqref{f3.1} in the $b=2$ case. Besides, \begin{equation}\label{equivalence} c:=\sum_{k=0}^{\infty}\frac{\alpha^{2}(2^{k})}{4^{sk}}<\infty\quad \Longleftrightarrow\quad\eqref{int-cond-8.1} \end{equation} because the function $\alpha^{2}(t)\,t^{-2s}$ of $t\geq1$ belongs to $\mathrm{OR}$. Indeed, if $\omega\in\mathrm{OR}$, then \begin{align*} \int\limits_{1}^{\infty}\frac{\omega(t)}{t}dt&= \sum_{k=0}^{\infty}\int\limits_{2^{k}}^{2^{k+1}}\frac{\omega(t)}{t}dt= \sum_{k=0}^{\infty}\int\limits_{1}^{2}\frac{\omega(2^k\tau)}{\tau}d\tau\\ &=\sum_{k=0}^{\infty}\omega(2^k) \int\limits_{1}^{2}\frac{\omega(2^k\tau)}{\omega(2^k)}\frac{d\tau}{\tau} \asymp\sum_{k=0}^{\infty}\omega(2^k)\int\limits_{1}^{2}\frac{d\tau}{\tau}. \end{align*} This implies that \begin{equation}\label{equivalence-gen} \int\limits_{1}^{\infty}\frac{\omega(t)}{t}dt<\infty\quad \Longleftrightarrow\quad\sum_{k=0}^{\infty}\omega(2^k)<\infty \end{equation} for every $\omega\in\mathrm{OR}$. Now \eqref{equivalence-gen} written for $\omega(t)\equiv\alpha^{2}(t)\,t^{-2s}$ is \eqref{equivalence}. Thus, it follows from \eqref{norms-inequality} that condition \eqref{int-cond-8.1} implies the continuous embedding \eqref{embed}.
Let us prove the inverse implication. We define a function $v\in L_{2}(\mathbb{R}^{n})$ as follows: $v(\xi):=2^{-sk}(\mathrm{mes}\,Q_{k})^{-1/2}$ if $\xi\in Q_{k}$ for some integer $k\geq0$. The function $w:=\mathcal{F}^{-1}v_{m}$ belongs to $B^{s}_{2,\infty}(\mathbb{R}^{n})$, and $\|w\|_{s,\infty,\mathbb{R}^{n}}=1$, where $\mathcal{F}^{-1}$ is the inverse Fourier transform. Assume that the continuous embedding \eqref{embed} holds true. In view of \eqref{norms-inequality}, we have \begin{align*} \sum_{k=0}^{\infty}\frac{\alpha^{2}(2^{k})}{4^{sk}}= \sum_{k=0}^{\infty}\alpha^{2}(2^{k})
\int\limits_{Q_k}|\widehat{w}(\xi)|^{2}d\xi
\asymp\|w\|_{\alpha,\mathbb{R}^{n}}^{2}\leq c_{0}^{2}\,\|w\|_{s,\infty,\mathbb{R}^{n}}^{2}<\infty, \end{align*} where $c_{0}$ is the norm of the continuous embedding operator \eqref{embed}. Thus, this embedding implies condition \eqref{int-cond-8.1}.
We have proved the equivalence $\eqref{int-cond-8.1}\Leftrightarrow\eqref{embed}$ in the $s>0$ case. The $s\leq0$ case is plainly reduced to the case considered with the help of the fact that the mapping $w\mapsto\mathcal{F}^{-1}[\langle\xi\rangle^{-\lambda}\widehat{w}(\xi)]$ sets topological isomorphisms $H^{\alpha}(\mathbb{R}^{n})\leftrightarrow H^{\alpha\varrho^{\lambda}}(\mathbb{R}^{n})$ and $B^{s}_{2,\infty}(\mathbb{R}^{n})\leftrightarrow B^{s+\lambda}_{2,\infty}(\mathbb{R}^{n})$ for arbitrary $s,\lambda\in\mathbb{R}$. The first isomorphism is evident; the second is proved, e.g., in \cite[Theorem 2.3.4]{Triebel95}. \end{proof}
\begin{remark}\label{rem8.3} It is well known \cite[Theorem 2.3.2(c)]{Triebel95} that \begin{equation*} B^{s}_{2,\infty}(\mathbb{R}^n)\subset H^{s-}(\mathbb{R}^n):=\bigcap_{r<s}H^{r}(\mathbb{R}^n). \end{equation*} Proposition \ref{8.1} can be seen as a refinement of this result with the help of the extended Sobolev scale. Thus, e.g., the function $\alpha(t):=t^{s}(1+\log t)^{-\varepsilon-1/2}$ of $t\geq1$ belongs to $\mathrm{OR}$ and satisfies \eqref{int-cond-8.1} for every $\varepsilon>0$, with the space $H^{\alpha}(\mathbb{R}^n)$ being narrower than $H^{s-}(\mathbb{R}^n)$. \end{remark}
As an immediate consequence of the embedding \eqref{embed} and results in Section~\ref{sec4}, we obtain \textit{a priori} estimates for solutions to elliptic problems with boundary data in $B_{2,\infty}^s(\Gamma)$. For simplicity of presentation, we discuss a special situation, the formulation in more general settings being obvious. Let $A=A(x,D)=\sum_{|\mu|\le 2} a_\mu(x)D^\mu$ be a properly elliptic second-order PDO on $\overline{\Omega}$, with all $ a_\mu\in C^{\infty}(\overline{\Omega})$. We consider the Dirichlet boundary-value problem \begin{equation}\label{8-2} Au=f\;\;\text{ in }\Omega,\quad\gamma_{0}u=g\quad\text{ on }\Gamma, \end{equation} where $\gamma_0 u := u\!\upharpoonright\!\Gamma$ denotes the trace of $u$ on the boundary. This is a simple but important example of a regular elliptic problem in~$\Omega$.
Let $\mathrm{OR}_{0}$ denote the set of all $\alpha\in\mathrm{OR}$ such that $\sigma_{0}(\alpha)=\sigma_{1}(\alpha)=0$. In view of Remark~\ref{rem8.3}(a), we restrict ourselves to the case where $\alpha(t)\equiv t^{s}\alpha_{0}(t)$ for some $s\in\mathbb{R}$ and $\alpha_{0}\in\mathrm{OR}_{0}$.
\begin{theorem}\label{8.4} Assume that a distribution $u\in\mathcal{S}'(\Omega)$ is a generalized solution to the boundary-value problem \eqref{8-2} whose right-hand sides satisfy the conditions $f\in H^\lambda(\Omega)$ and $g\in B_{2,\infty}^s(\Gamma)$ for some numbers $\lambda>-\frac12$ and $s<0$. Then, for every function parameter $\alpha(t)\equiv t^{s+1/2}\alpha_{0}(t)$ such that $\alpha_{0}\in\mathrm{OR}_{0}$ and \begin{equation}\label{8-3} \int_1^\infty\alpha_{0}^2(t)\,\frac{dt}{t}<\infty, \end{equation} we have $u\in H^\alpha(\Omega)$ and \begin{equation*}
\|u\|_{\alpha,\Omega}\leq c\,\bigl(\|f\|_{\lambda,\Omega}+ \|g\|_{s,\infty,\Gamma}+\|u\|_{\alpha\rho^{-1},\Omega}\bigr). \end{equation*}
Here, $\|\cdot\|_{s,\infty,\Gamma}$ denotes the norm in $B^{s}_{2,\infty}(\Gamma)$, and the number $c>0$ does not depend on $u$, $f$, and $g$. \end{theorem}
\begin{proof} According to condition~\eqref{8-3} and Proposition~\ref{8.1}, we have the continuous embedding $B^{s}_{2,\infty}(\mathbb{R}^{n-1})\hookrightarrow H^{\alpha\varrho^{-1/2}}(\mathbb{R}^{n-1})$. With the help of local charts on $\Gamma$, we immediately obtain the continuous embedding $B^{s}_{2,\infty}(\Gamma)\hookrightarrow H^{\alpha\varrho^{-1/2}}(\Gamma)$. Now the statement follows directly from Theorems \ref{th4.6} and \ref{th4.12}, in which $\varphi(t)\equiv t^{-2}\alpha(t)\equiv t^{s-3/2}\alpha_{0}(t)$, $\sigma_{0}(\varphi)=\sigma_{1}(\varphi)=s-3/2<-3/2$, and $\eta(t)\equiv t^{\lambda}$. \end{proof}
The Nikolskii spaces $B^{s}_{2,\infty}(\mathbb{R}^n)$ and $B^{s}_{2,\infty}(\Gamma)$ of order $s<0$ appear in the theory of white noise. We recall the basic definitions. Let $(\widetilde\Omega,\mathcal F,\mathbb P)$ be a probability space, and let $G\in\{\Gamma,\mathbb{R}^n\}$. Then a (spatial) white noise on $G$ is a random variable $\xi\colon \widetilde\Omega\to\mathcal{D}'(G)$ such that for all test functions $v_1,v_2\in\mathcal{D}(G)$ we have \begin{equation}\label{8-4} \mathbb{E}[\xi(v_1)\overline{\xi(v_2)}]=C\,(v_1,v_2)_{G} \end{equation} with some constant $C>0$. Here, $\mathcal{D}'(G)$ is the topological space of all distributions on $G$, with $\mathcal{D}(G)$ being $C_{0}^{\infty}(\mathbb{R}^n)$ or $C^{\infty}(\Gamma)$. Besides, $(\cdot,\cdot)_{G}$ denotes the inner product in $L_{2}(G)$, and $\mathbb{E}$ stands for the expectation with respect to $\mathbb P$. A white noise $\xi$ on $G$ is called Gaussian if the scalar random variables $\{\xi(v):v\in\mathcal{D}(G)\}$ are jointly Gaussian with mean zero and with covariance being given by~\eqref{8-4}.
Recently, the Besov space regularity of white noise was studied, e.g., in \cite{FageotFallahUnser17,Veraar11}. For a Gaussian white noise $\xi\colon\widetilde\Omega\to\mathcal{D}'(\mathbb R^n)$, it was shown in \cite[Corollary~3]{FageotFallahUnser17} that $\mathbb P$-almost surely $\xi$ locally belongs to the Besov space $B_{2,r}^s(\mathbb R^n)$ for all $r\in[1,\infty]$ and $s<-n/2$. In \cite{Veraar11}, white noise on the $n$-dimensional torus was studied. It was shown in \cite[Theorem~3.4]{Veraar11} that for a Gaussian white noise $\xi\colon\widetilde\Omega\to\mathcal{D}'(\mathbb{T}^n)$ we have $\mathbb P (\xi\in B_{2,\infty}^{-n/2}(\mathbb T^n))=1$. Here, the upper index is sharp in the sense that for all $s>-n/2$ we have $\mathbb P(\xi\in B_{2,\infty}^s(\mathbb T^n))=0$. Based on these results, one might conjecture that for every Gaussian white noise $\xi$ on an $n$-dimensional closed manifold $M$, we have $\mathbb P(\xi\in B_{2,\infty}^{-n/2}(M))=1$, but this seems to be an open question.
Combining the above regularity of Gaussian white noise with Theorem~\ref{8.4}, we obtain \textit{a priori} estimates for solutions to elliptic problems with boundary noise. As a simple example, we state the result for the Dirichlet Laplacian.
\begin{corollary}\label{8.5} Consider the boundary-value problem \begin{equation}\label{8-5} \Delta u=f\quad\mbox{in}\;\,\Omega,\qquad\gamma_0u=\xi\quad\mbox{on}\;\,\Gamma. \end{equation}
Here, $\Omega:=\{x\in\mathbb R^2: |x|<1\}$, whereas $\xi$ is a Gaussian white noise on $\Gamma$. Let $f\in H^\lambda(\Omega)$ for some number $\lambda>-1/2$. Then, for $\mathbb P$-almost all $\omega\in\widetilde \Omega$, there exists a unique pathwise solution $u(\omega,\cdot)$ of \eqref{8-5}, which belongs to $H^\alpha(\Omega)$ for every $\alpha\in \mathrm{OR}_{0}$ subject to \eqref{8-3}. Moreover, for such $\alpha$, the estimate \begin{equation*}
\|u(\omega,\cdot)\|_{\alpha,\Omega}\leq c_{\alpha}\big(\|f\|_{\lambda,\Omega}+ \|\xi(\omega)\|_{-1/2,\infty,\Gamma}\big) \end{equation*} holds $\mathbb P$-almost surely with a number $c_\alpha>0$ that does not depend on $f$, $\xi$, and $\omega$ (but may depend on $\alpha$). \end{corollary}
\begin{proof} This is an immediate consequence of Theorem~\ref{8.4} and the fact that $\xi\in B_{2,\infty}^{-1/2}(\Gamma)$ holds $\mathbb P$-almost surely. Note that the unique solvability holds as $\dim N = \dim N^+ = \{0\}$ for the regular elliptic problem \eqref{8-5}. \end{proof}
\begin{remark}\label{8.6} In the last corollary, we have shown that $u\in H^\alpha(\Omega)$ by the embedding result from Proposition~\ref{8.1} and the Nikolskii regularity of white noise. It would be interesting to analyse the regularity of Gaussian white noise (or, more generally, L\'{e}vy white noise) with respect to the extended Sobolev scale. In particular, this would allow a direct application of Theorems \ref{th4.6} and \ref{th4.12} for boundary noise. \end{remark}
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Problem 141
Let $V$ be a vector space over a field $K$. Let $\mathbf{u}_1, \mathbf{u}_2, \dots, \mathbf{u}_n$ be linearly independent vectors in $V$. Let $U$ be the subspace of $V$ spanned by these vectors, that is, $U=\Span \{\mathbf{u}_1, \mathbf{u}_2, \dots, \mathbf{u}_n\}$.
Let $\mathbf{u}_{n+1}\in V$. Show that $\mathbf{u}_1, \mathbf{u}_2, \dots, \mathbf{u}_n, \mathbf{u}_{n+1}$ are linearly independent if and only if $\mathbf{u}_{n+1} \not \in U$.
Add to solve later
$(\implies)$
Suppose that the vectors $\mathbf{u}_1, \mathbf{u}_2, \dots, \mathbf{u}_n, \mathbf{u}_{n+1}$ are linearly independent. If $\mathbf{u}_{n+1}\in U$, then $\mathbf{u}_{n+1}$ is a linear combination of $\mathbf{u}_1, \mathbf{u}_2, \dots, \mathbf{u}_n$.
Thus, we have
\[\mathbf{u}_{n+1}=c_1\mathbf{u}_1+c_2\mathbf{u}_2+\cdots+c_n \mathbf{u}_n\] for some scalars $c_1, c_2, \dots, c_n \in K$.
However, this implies that we have a nontrivial linear combination
\[c_1\mathbf{u}_1+c_2\mathbf{u}_2+\cdots+c_n \mathbf{u}_n-\mathbf{u}_{n+1}=\mathbf{0}.\] This contradicts that $\mathbf{u}_1, \mathbf{u}_2, \dots, \mathbf{u}_n, \mathbf{u}_{n+1}$ are linearly independent. Hence $\mathbf{u}_{n+1} \not \in U$.
$(\impliedby)$ Suppose now that $\mathbf{u}_{n+1} \not \in U$.
If the vectors $\mathbf{u}_1, \mathbf{u}_2, \dots, \mathbf{u}_n, \mathbf{u}_{n+1}$ are linearly dependent, then there exists $c_1, c_2 \dots, c_n, c_{n+1}\in K$ such that
not all of them are zero and
\[c_1\mathbf{u}_1+c_2\mathbf{u}_2+\cdots+c_n \mathbf{u}_n+c_{n+1}\mathbf{u}_{n+1}=\mathbf{0}.\]
We claim that $c_{n+1} \neq 0$. If $c_{n+1}=0$, then we have
\[c_1\mathbf{u}_1+c_2\mathbf{u}_2+\cdots+c_n \mathbf{u}_n=\mathbf{0}\] and since $\mathbf{u}_1, \mathbf{u}_2, \dots, \mathbf{u}_n$ are linearly independent, we must have $c_1=c_2=\cdots=c_n=0$. This means that all $c_i$ are zero but this contradicts our choice of $c_i$. Thus $c_{n+1} \neq 0$.
Then we have
\[\mathbf{u}_{n+1}=\frac{-c_1}{c_{n+1}}\mathbf{u}_1+\frac{-c_2}{c_{n+1}}\mathbf{u}_2+\cdots+\frac{-c_n}{c_{n+1}}\mathbf{u}_n.\] (Note: we needed to check $c_{n+1} \neq 0$ to divide by it.)
This implies that $\mathbf{u}_{n+1}$ is a linear combination of vectors $\mathbf{u}_1, \mathbf{u}_2, \dots, \mathbf{u}_n$, and thus $\mathbf{u}_{n+1} \in U$, a contradiction.
Therefore, the vectors $\mathbf{u}_1, \mathbf{u}_2, \dots, \mathbf{u}_n, \mathbf{u}_{n+1}$ are linearly independent.
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\begin{document}
\title[$L^1$ pointwise ergodic theorem fails for $F_2$]{Failure of the $L^1$ pointwise and maximal ergodic theorems for the free group}
\author{Terence Tao} \address{UCLA Department of Mathematics, Los Angeles, CA 90095-1555.} \email{[email protected]}
\subjclass[2010]{37A30}
\begin{abstract} Let $F_2$ denote the free group on two generators $a,b$. For any measure-preserving system $(X, {\mathcal X}, \mu, (T_g)_{g \in F_2})$ on a finite measure space $X = (X,{\mathcal X},\mu)$, any $f \in L^1(X)$, and any $n \geq 1$, define the averaging operators
$${\mathcal A}_n f(x) := \frac{1}{4 \times 3^{n-1}} \sum_{g \in F_2: |g| = n} f( T_g^{-1} x ),$$
where $|g|$ denotes the word length of $g$. We give an example of a measure-preserving system $X$ and an $f \in L^1(X)$ such that the sequence ${\mathcal A}_n f(x)$ is unbounded in $n$ for almost every $x$, thus showing that the pointwise and maximal ergodic theorems do not hold in $L^1$ for actions of $F_2$. This is despite the results of Nevo-Stein and Bufetov, who establish pointwise and maximal ergodic theorems in $L^p$ for $p>1$ and for $L \log L$ respectively, as well as an estimate of Naor and the author establishing a weak-type $(1,1)$ maximal inequality for the action on $\ell^1(F_2)$. Our construction is a variant of a counterexample of Ornstein concerning iterates of a Markov operator. \end{abstract}
\maketitle
\section{Introduction}
Let $F_2$ denote the free non-abelian group on two generators $a,b$. Define a \emph{reduced word} to be a word with letters in the alphabet $\{a,b,a^{-1},b^{-1}\}$ in which $a,a^{-1}$ and $b,b^{-1}$ are never adjacent, and for each $g \in F_2$, define the \emph{word length} $|g|$ of $g$ to be the length of the unique reduced word that produces $g$. We let $F_2^2$ denote the index $2$ subgroup of $F_2$ consisting of $g \in F_2$ with even word length.
Define a \emph{$F_2$-system} to be a quadruple $(X, {\mathcal X}, \mu, (T_g)_{g \in F_2})$, where $(X,{\mathcal X}, \mu)$ is a measure space with $0 < \mu(X) < \infty$, and $T_g \colon X \to X$ is a family of measure-preserving maps on $X$ for $g \in F_2$, with $T_1$ the identity and $T_g T_h = T_{gh}$ for all $g,h \in F_2$; in particular, the $T_g$ are bi-measurable with $T_g^{-1} = T_{g^{-1}}$. One can of course normalise such systems to have total measure $1$ by dividing $\mu$ by $\mu(X)$, but (as we will eventually be gluing several systems together) it will be convenient not to always insist on such a normalisation. As the free group $F_2$ has no relations, such a system can be prescribed by specifying two arbitrary invertible bi-measurable measure-preserving maps $T_a,T_b:X \to X$, and then defining $T_g$ for all other $g \in G$ in the obvious fashion.
We say that an $F_2$-system is \emph{$F_2$-ergodic} if all $F_2$-invariant measurable sets either have zero measure or full measure, and \emph{$F_2^2$-ergodic} if the same claim is true for $F_2^2$-invariant measurable sets. For any $f \in L^1(X) = L^1(X,{\mathcal X},\mu)$ and any $n \geq 1$, we define the averaging operators
$${\mathcal A}_n f(x) := \frac{1}{4 \times 3^{n-1}} \sum_{g \in F_2: |g| = n} f( T_g^{-1} x );$$ note that $4 \times 3^{n-1}$ is the number of reduced words of length $n$. One can of course use symmetry to replace $T_g^{-1}$ by $T_g$ if desired.
The pointwise convergence of the operators ${\mathcal A}_n$ was studied by Nevo and Stein \cite{nevo} and Bufetov \cite{bufetov}, who (among other things) proved the following result:
\begin{theorem}[Pointwise ergodic theorem]\label{pet} Let $(X,{\mathcal X},\mu,(T_g)_{g \in F_2})$ be an $F_2$-system. If $\int_X |f| \log(2+|f|)\ d\mu < \infty$, then ${\mathcal A}_{2n} f$ converges pointwise almost everywhere (and in $L^1(X)$ norm) to an $F_2^2$-invariant function. In particular, if $(X,{\mathcal X},\mu,(T_g)_{g \in F_2})$ is $F_2^2$-ergodic, then ${\mathcal A}_{2n} f$ converges pointwise almost everywhere and in $L^1$ to the constant $\frac{1}{\mu(X)} \int_X f\ d\mu$. \end{theorem}
The restriction to even averages ${\mathcal A}_{2n}$, and the use of $F_2^2$ instead of $F_2$, can be seen to be necessary by considering the simple example in which $X$ is a two-element set $\{0,1\}$ (with uniform measure) and $T_a,T_b$ interchange the two elements $0,1$ of this set. The original paper of Nevo and Stein \cite{nevo} established this theorem for $f \in L^p(X)$ for some $p>1$, by modifying the methods of Stein \cite{stein}. The subsequent paper of Bufetov \cite{bufetov} used instead the ``Alternierende Verfahren'' of Rota \cite{rota} to cover the $L \log L$ case. Both arguments also extend to several other group actions (see e.g. \cite{nevo-2}, \cite{fuji}, \cite{marg}), but for simplicity of exposition we shall focus only on the $F_2$ case. We also remark that both arguments also give bounds on the associated maximal operator $f \mapsto \sup_n {\mathcal A}_n |f|$. See also \cite{bowen-1}, \cite{bowen-2} for an alternate approach to pointwise ergodic theorems in $L^p$ and $L \log L$.
In \cite{nevo} the question was posed as to whether the above pointwise ergodic theorem extended to arbitrary $L^1(X)$ functions. The main result of this paper answers this question in the negative:
\begin{theorem}[Counterexample]\label{main} There exists an $F_2$-system $(X,{\mathcal X},\mu,(T_g)_{g \in F_2})$ and an $f \in L^1(X)$ such that $\sup_n |{\mathcal A}_{2n} f(x)| = \infty$ for almost every $x \in X$. In particular, ${\mathcal A}_{2n} f(x)$ fails to converge to a limit as $n \to \infty$ for almost every $x \in X$. \end{theorem}
As such, there is no pointwise ergodic theorem or maximal ergodic theorem in $L^1$ for actions of the free group $F_2$. Our construction also applies to free groups $F_r$ on $r$ generators for any $r \geq 2$; we leave the modification of the arguments below to this more general case to the interested reader. This result stands in contrast to the situation for the regular action of $F_2$ on $\ell^1(F_2)$, for which a weak-type (1,1) for the maximal operator was established by Naor and the author \cite[Theorem 1.5]{naor}. Note that the estimate for $\ell^1(F_2)$ does not transfer to arbitrary $F_2$-systems due to the non-amenability of the free group $F_2$.
Because the sphere $\{ g \in F_2: |g| = n \}$ is a positive fraction of the ball $\{ g \in F_2: |g| \leq n \}$, the above result also holds if the average over spheres is replaced with an average over balls, or with regards to other minor variations of the spherical averaging operator such as $\frac{1}{2} {\mathcal A}_n + \frac{1}{2} {\mathcal A}_{n+1}$. This negative result for averaging on balls stands in contrast with the situation for amenable groups, for which pointwise and maximal ergodic results in $L^1$ are established for suitable replacements of balls, such as tempered F{\o}lner sets; see \cite{lindenstrauss}. On the other hand, if one considers the Ces\'aro means $\frac{1}{N} \sum_{n \leq N} {\mathcal A}_n$ of spherical averages on $F_2$-systems, then pointwise and maximal ergodic theorems in $L^1$ were established in \cite{nevo}.
Our construction is inspired by a well-known counterexample of Ornstein \cite{ornstein} demonstrating the failure of the maximal ergodic theorem in $L^1$ for iterates $P^n$ of a certain well-chosen self-adjoint Markov operator. Roughly speaking, the function $f$ in Ornstein's example consists of many components $f_i$, each of which comes with a certain ``time delay'' that ensures that the dynamics of $P^n f_i$ only become significant after a significant period of time - in particular, long enough for the dynamics of other components of the function to have achieved ``mixing'' in the portion of $X$ where the most interesting portion of the dynamics of $P^n f_i$ takes place, allowing the amplitude of $f_i$ to be slightly smaller than would otherwise have been necessary to make $\sup_n P^n f$ large. To adapt this construction to the setting of $F_2$-systems, we need to glue together various $F_2$-systems that have the capability to produce such a ``time delay''. We will be able to construct such systems by basically taking an ``infinitely large ball'' in $F_2$, gluing the boundary of that ball to itself, and redefining the shift maps on the boundary appropriately. Somewhat ironically, the positive results in Theorem \ref{pet} play a helpful supporting role in establishing the negative result in Theorem \ref{main}, by establishing the ``mixing'' referred to previously that is an essential part of Ornstein's construction.
\subsection{Acknowledgments} The author is supported by NSF grant DMS-1266164 and by a Simons Investigator Award, and thanks Lewis Bowen for helpful discussions and corrections.
\section{Initial reductions}
We begin by reducing Theorem \ref{main} to the following more quantitative statement.
\begin{theorem}[Quantitative counterexample]\label{main-2} Let $\alpha, \eps > 0$. Then there exists an $F_2$-system $(X,{\mathcal X},\mu,(T_g)_{g \in F_2})$ and a non-negative function $f \in L^\infty(X)$, such that
$$ \| f \|_{L^1(X)} \leq \alpha \mu(X)$$ but such that $$ \sup_n {\mathcal A}_{2n} f(x) \geq 1-\eps$$ for all $x \in X$ outside of a set of measure at most $\eps \mu(X)$. \end{theorem}
Let us see how Theorem \ref{main-2} implies Theorem \ref{main}. By dividing $\mu$ by $\mu(X)$ we may normalise $\mu(X)=1$ in Theorem \ref{main-2}. Applying the above theorem with $\alpha=\eps=2^{-m}$, we can thus find for each natural number $m$, an $F_2$-system $(X_m, {\mathcal X}_m, \mu_m, (T_{g,m})_{g \in F_2})$ with $\mu_m(X_m)=1$, and a non-negative function $f_m \in L^\infty(X_m)$ such that
$$ \| f_m \|_{L^1(X_m)} \leq 2^{-m}$$ and $$ \sup_n {\mathcal A}_{2n} f_m(x) \geq 1 - 2^{-m} $$ outside of a set of measure $2^{-m}$.
Let $(X, {\mathcal X}, \mu, (T_g)_{g \in F_2})$ be the product system, thus $X$ is the Cartesian product $X := \prod_m X_m$ with product $\sigma$-algebra ${\mathcal X} := \prod_m {\mathcal X}_m$, product probability measure $\mu := \prod_m \mu_m$, and product action $T_g := \biguplus_m T_{g,m}$. Each $f_m \in L^\infty(X_m)$ then lifts to a function $\tilde f_m \in L^\infty(X)$ with
$$ \| \tilde f_m \|_{L^1(X)} \leq 2^{-m}$$ and $$ \sup_n {\mathcal A}_{2n} \tilde f_m(x) \geq 1 - 2^{-m} \geq 1/2$$ outside of a set of measure $2^{-m}$. If we then set $f := \sum_m m \tilde f_m$, then $f \in L^1(X)$, and from the pointwise inequality $$ \sup_n {\mathcal A}_{2n} f(x) \geq m_0 \sup_n {\mathcal A}_{2n} \tilde f_m(x)$$ for all $m \geq m_0$ and the Borel-Cantelli lemma we see that $\sup_n {\mathcal A}_{2n} f(x)$ is larger than $m_0/2$ for almost every $x$ and any given $m_0$, which yields the claim.
It remains to prove Theorem \ref{main-2}. In order to adapt the arguments of Ornstein \cite{ornstein}, we would like to interpret the averaging operators ${\mathcal A}_n$ as powers $P^n$ of a Markov operator $P$. This is not true as stated, since we do not quite have the semigroup property ${\mathcal A}_n {\mathcal A}_m = {\mathcal A}_{n+m}$ (although ${\mathcal A}_n {\mathcal A}_m$ does contain a term of the form $\frac{3}{4} {\mathcal A}_{n+m}$). However, as observed by Bufetov \cite{bufetov}, we can recover a Markov interpretation for ${\mathcal A}_n$ by lifting $X$ up to a four-fold cover $\tilde X$ that tracks the ``outward normal vector'' for the sphere. More precisely, given an $F_2$-system $(X,{\mathcal X}, \mu, (T_g)_{g \in F_2})$, we define the lifted measure space $(\tilde X, \tilde {\mathcal X}, \tilde \mu)$ to be the product of $(X,{\mathcal X},\mu)$ and the four-element space $\{a,b,a^{-1},b^{-1}\}$ with the uniform probability measure; in particular $\tilde \mu(\tilde X) = \mu(X)$. Let $\pi \colon \tilde X \to X$ be the projection operator $\pi(x,s) := x$; this induces a pushforward operator $\pi_* \colon L^1(\tilde X) \to L^1(X)$ and a pullback operator $\pi^* \colon L^1(X) \to L^1(\tilde X)$ by the formulae $$ \pi_* \tilde f(x) := \frac{1}{4} \sum_{s \in \{a,b,a^{-1},b^{-1}\}} \tilde f(x,s)$$ and $$ \pi^* f(x,s) := f(x)$$ for $f \in L^1(X)$ and $\tilde f \in L^1(\tilde X)$. We also define the Markov operator $P \colon L^1(\tilde X) \to L^1(\tilde X)$ by $$ P \tilde f(x,s) := \frac{1}{3} \sum_{s' \in \{a,b,a^{-1},b^{-1}\}: s' \neq s^{-1}} \tilde f( T_s^{-1} x, s' ).$$
One can view $P$ as the Markov operator associated to the Markov chain that for each unit time, moves a given point $(x,s)$ of $\tilde X$ to one of the three points $(T_{s'} x, s')$ with $s' \in \{a,b,a^{-1},b^{-1}\} \backslash \{s^{-1}\}$, chosen at random. By writing the elements of $\{g \in F_2: |g|=n\}$ as reduced words of length $n$, one can easily verify the identity $$ {\mathcal A}_n f = \pi_* P^n \pi^* f $$ for any $f \in L^1(X)$ and $n \geq 1$. It thus suffices to show
\begin{theorem}[Quantitative counterexample, again]\label{main-3} Let $\alpha, \eps > 0$. Then there exists an $F_2$-system $(X,{\mathcal X},\mu,(T_g)_{g \in F_2})$ and a non-negative function $\tilde f \in L^\infty(\tilde X)$, such that
$$ \| \tilde f \|_{L^1(\tilde X)} \leq \alpha \mu(X)$$ but such that $$ \sup_n \pi_* P^{2n} \tilde f(x) \geq 1-\eps$$ for all $x \in X$ outside of a set of measure at most $\eps \mu(X)$. \end{theorem}
Indeed, by setting $f := 4 \pi_* \tilde f$, and noting the pointwise bound $\tilde f \leq \pi^* f$ and the identity $\|f\|_{L^1(X)} = 4 \| \tilde f \|_{L^1(\tilde X)}$, we obtain Theorem \ref{main-2} (after replacing $\alpha$ by $\alpha/4$).
For inductive reasons, we will prove a technical special case of Theorem \ref{main-3}, in which the $F_2$-system is of a certain ``good'' form, and the sequence $(P^n \tilde f)_{n \geq 0}$ is part of an ``ancient Markov chain'' $(\tilde f_n)_{n \in \Z}$ that extends to arbitrarily negative times as well as arbitrarily positive times. More precisely, let us define a \emph{good system} to be an $F_2$-system $(X,{\mathcal X},\mu,(T_g)_{g \in F_2})$ which admits a decomposition $X = X_a \cup X_b \cup X_0$ into three disjoint sets $X_a,X_b,X_0$ admitting the following (somewhat technical) properties:
\begin{itemize} \item[(i)] (Measure) One has $\mu(X_a) = \mu(X_b) = \frac{1}{4} \mu(X)$ and $\mu(X_0) = \frac{1}{2} \mu(X)$. Furthermore, for any $0 \leq \kappa \leq \mu(X_b)$, one can find a measurable subset of $X_b$ of measure exactly equal to $\kappa$. \item[(ii)] (Invariance) One has $T_a X_a = X_a$ and $T_b X_b = X_b$. Also, one has the inclusions $T_a X_b \subset T_b X_a \cup T_b^{-1} X_a \subset X_0$. \item[(iii)] (Ergodicity) One can partition $X_a$ into finitely many $T_a$-invariant components $X_{a,1},\dots,X_{a,m}$ of positive measure, such that $T_a^2$ is ergodic on each of the components $X_{a,i}$; that is, the only $T_a^2$-invariant measurable subsets of $X_{a,i}$ have measure either $0$ or $\mu(X_{a,i})$. \item[(iv)] (Generation) One has $X = \bigcup_{g \in F_2} T_g X_{a,i}$ up to null sets for each $i=1,\dots,m$. \end{itemize}
Note that relatively few conditions are required on the dynamics on $X_b$; in particular, the ergodicity hypotheses on the system are located in the disjoint region $X_a$. This will allow us to easily modify the dynamics on $X_b$ in order to ``glue'' two good systems together in Section \ref{glue}.
\begin{figure}
\caption{A somewhat schematic depiction of a good system. Only part of the action of $T_a$ and $T_b$ are displayed.}
\label{fig:good}
\end{figure}
See Figure \ref{fig:good}. We will construct good systems in subsequent sections. For now, we record one useful property of such systems:
\begin{lemma}[Pointwise ergodic theorem for good systems]\label{pet-good} Every good system $(X,{\mathcal X},\mu,(T_g)_{g \in F_2})$ is $F_2^2$-ergodic. In particular (by Theorem \ref{pet}), for any $f \in L^\infty(X)$, the averages ${\mathcal A}_{2n} f$ converge pointwise almost everywhere and in $L^1$ norm to $\frac{1}{\mu(X)} \int_X f\ d\mu$. Furthermore, for any $\tilde f \in L^\infty(\tilde X)$, $P^{2n} \tilde f$ converge pointwise almost everywhere and in $L^1$ norm to $\frac{1}{\mu(X)} \int_{\tilde X} \tilde f\ d\tilde \mu$. \end{lemma}
\begin{proof} Let $f \in L^\infty(X)$ be an $F_2^2$-invariant function; to establish $F_2^2$-ergodicity, it will suffice to show that $f$ is constant almost everywhere. As $f$ is $T_a^2$-invariant, we see from Axiom (iii) that $f$ is constant almost everywhere on each $X_{a,i}$.
Since $F_2 = F_2^2 \cup F_2^2 a$, we see from Axiom (iv), the $T_a$-invariance of $X_{a,i}$, and the $F_2^2$-invariance of $f$ that $f$ is constant almost everywhere on $X$, as required. The final claim does not quite follow from Theorem \ref{pet}, but is immediate from \cite[Proposition 1]{bufetov}. \end{proof}
For any $\alpha > 0$, let $P(\alpha)$ denote the following claim:
\begin{claim}[$P(\alpha)$]\label{clam} For any $\eps > 0$, there exists a good system $(X,{\mathcal X},\mu,(T_g)_{g \in F_2})$ with associated decomposition $X = X_a \cup X_b \cup X_0$, and a sequence of non-negative functions $\tilde f_n \in L^\infty(\tilde X)$ for $n \in \Z$ with the following properties: \begin{itemize}
\item[(v)] (Ancient Markov chain) $\tilde f_{n+1} = P \tilde f_n$ for all $n \in \Z$. Equivalently, one has $\tilde f_{n+m} = P^m \tilde f_n$ for all $n \in \Z$ and $m \in \N$. In particular, $\|\tilde f_n\|_{L^1(\tilde X)}$ is independent of $n$.
\item[(vi)] (Size) One has $\|\tilde f_n\|_{L^1(\tilde X)} = \alpha \mu(X)$ for some $n \in \Z$ (and hence for all $n \in \Z$). \item[(vii)] (Early support) $\tilde f_n$ is supported in $\tilde X_0$ for all negative $n$. Furthermore, there exists a finite $A>0$ such that $\tilde f_n$ is supported in a set of measure at most $A 3^n \mu(X)$ for all negative $n$. \item[(viii)] (Large maximum function) We have $$ \sup_{n \in \Z} \pi_* \tilde f_{2n}(x) \geq 1-\eps$$ for all $x \in X$ outside of a set of measure at most $\eps \mu(X)$. \end{itemize} \end{claim}
Note that our sequence $\tilde f_n$ is \emph{ancient} in the sense that it extends to arbitrary negative times $n \to -\infty$ as well as to arbitrary positive times $n \to \infty$. This will be essential in order to set up suitable ``time delays'' in our arguments in later sections. One can informally think of the $\tilde f_n$ as the (normalised) distribution at time $n$ of an ancient Markov process that starts from an infinitely small location deep inside $\tilde X_0$ at infinite negative time $n=-\infty$, and only escapes $\tilde X_0$ at or after time $n=0$, and which covers most of $X$ with density roughly $1$ or more at some point in time (but crucially, different regions of $X$ may be covered in this fashion at different times).
Observe that if $P(\alpha)$ holds for an arbitrarily small set of $\alpha>0$, and $\eps>0$ is arbitrary, then from axioms (vii), (viii), one has for any $N$ that $$ \sup_{n \geq -2N} \pi_* f_{2n}(x) \geq 1-\eps$$ for all $x \in X$ outside of a set of measure at most $(\eps + \frac{9}{8} A 3^{-2N}) \mu(X)$. Taking $N$ large enough (depending on $\eps$, $A$) and setting $\tilde f:= \tilde f_{-2N}$, we obtain Theorem \ref{main-3} (after adjusting $\eps$ as necessary). It thus suffices to show that $P(\alpha)$ holds for arbitrarily small $\alpha>0$. This will be accomplished using the following two key theorems (the second of which being a variant of \cite[Lemma 4]{ornstein}):
\begin{theorem}[Initial construction]\label{initial} The claim $P( 1 )$ is true. \end{theorem}
\begin{theorem}[Iteration step]\label{iterate} Suppose that $P(\alpha)$ holds for some $0 < \alpha \leq 1$. Then $P( \alpha (1-\frac{\alpha}{4}) )$ is true. \end{theorem}
From Theorem \ref{initial} and Theorem \ref{iterate} we see that the infimum of all $0 < \alpha \leq 1$ for which $P(\alpha)$ holds is zero, and the claim follows. Thus it suffices to establish Theorem \ref{initial} and Theorem \ref{iterate}. This will be accomplished in the next two sections.
\section{The initial construction}
We now prove Theorem \ref{initial}. We will in fact construct an example of a good system $(X, {\mathcal X}, \mu, (T_g)_{g \in F_2})$ and functions $\tilde f_n$ which witness $P(1)$ for all $\eps>0$ at once.
We begin by constructing an appropriate measure space $(X, {\mathcal X}, \mu)$. For each integer $n$, let $Y_n$ denote the space of half-infinite reduced words $(s_m)_{m \geq n} = s_n s_{n+1} s_{n+2} \dots$, in which each of the $s_i$ are drawn from the alphabet $\{a,b,a^{-1},b^{-1}\}$ and $a,a^{-1}$ and $b,b^{-1}$ are never adjacent. We give this space the product $\sigma$-algebra ${\mathcal Y}_n$ (that is, the minimal $\sigma$-algebra for which the coordinate maps $(s_m)_{m \leq n} \mapsto s_m$ are all measurable). By the Kolmogorov extension theorem, we may construct a probability measure $\mu_n$ on $Y_n$ such that each finite reduced subword $s_n \dots s_{n+k}$ for $k \geq 0$ occurs as an initial segment with probability $\frac{1}{4 \times 3^k}$; one can view this measure as the law of the random half-infinite reduced word constructed by choosing $s_n$ uniformly at random from $\{a,b,a^{-1},b^{-1}\}$, then recursively selecting $s_{n+i+1}$ for $i=0,1,2,\dots$ to be drawn uniformly from $\{a,b,a^{-1},b^{-1}\} \backslash \{s_{n+i}^{-1}\}$.
The disjoint union $Y := \biguplus_{n \in \Z} Y_n$ of the $Y_n$ admits an action $(S_g)_{g \in F_2}$ of $F_2$, with the action $S_s$ of a generator $s \in \{a,b,a^{-1},b^{-1}\}$ defined by setting $$ S_s ( s_n s_{n+1} s_{n+2} \dots ) := s s_n s_{n+1} s_{n+2} \dots \in Y_{n-1}$$ for $s_n s_{n+1} s_{n+2} \dots \in Y_n$ and $s \in \{a,b,a^{-1},b^{-1}\} \backslash s_n$, and $$ S_s ( s_n s_{n+1} s_{n+2} \dots ) := s_{n+1} s_{n+2} \dots \in Y_{n-1}$$ for $s_n s_{n+1} s_{n+2} \dots \in Y_n$ and $s = s_n^{-1}$; thus $S_g$ is the operation of formal left-multiplication by $g$, after reducing any non-reduced words. If we give $Y$ the measure $\mu_Y := \sum_{n \in \Z} 3^{-n} \mu_n$, then one can easily verify that this action is measure-preserving. Unfortunately, $\mu_Y$ is an infinite measure due to the contribution of the negative $n$, and so this space is not quite suitable for our needs. Instead, we shall work with a certain subquotient of $Y$, defined as follows.
\begin{figure}
\caption{A fragment of the infinite measure space $Y$. The centre disk represents a portion of $Y_1$ consisting of reduced words $s_1 s_2 \dots$ with initial letter $s_1 = a$. The remaining disks are images of this disk under shifts by various elements of $F_2$, and all have equal measure with respect to $\mu_Y$. This image should be compared with the infinite tree that is the Cayley graph of $F_2$.}
\label{fig:ysp}
\end{figure}
Firstly, we restrict $Y$ to the space $\biguplus_{n \geq 0} Y_n = \biguplus_{n \geq 1} Y_n \uplus Y_0$, which can be thought of as a suitably rescaled limit of an ``infinitely large ball'' in $F_2$, with $Y_0$ being the ``boundary'' of this ball, and the $Y_n$ lying increasingly deeper in the ``interior'' of the ball as $n$ increases (see Figure \ref{fig:ysp}). This makes the shift maps $S_s$, $s \in \{a,b,a^{-1},b^{-1}\}$ partially undefined on the $Y_0$ boundary, but we will fix this later by redefining these maps on (a quotient) of $Y_0$. Next, we introduce a reflection operation $x \mapsto \overline{x}$ on the boundary $Y_0$ by mapping $$ \overline{s_0 s_1 s_2 \dots} := s_0^{-1} s_1^{-1} s_2^{-2} \dots.$$ It is clear that this map preserves the measure $\mu_0$. If we then form the quotient space $Y_0/\sim := \{ \{x,\overline{x}\}: x \in Y_0 \}$, we can obtain a probability measure $\mu_0/\sim$ on $Y_0/\sim$ by pushing forward the probability measure $\mu_0$ under the quotient map. We observe that $Y_0/\sim$ splits into two components of equal measure $1/2$, namely $((S_a Y_1 \cap Y_0) \cup (S_a^{-1} Y_1 \cap Y_0))/\sim$ and $((S_b Y_1 \cap Y_0) \cup (S_b^{-1} Y_1 \cap Y_0))/\sim$, noting that the sets $S_a Y_1 \cap Y_0, S_a^{-1} Y_1 \cap Y_0$ are disjoint reflections of each other, and similarly for $S_b Y_1 \cap Y_0, S_b^{-1} Y_1 \cap Y_0$.
We then define $X$ to be the quotient space $\biguplus_{n \geq 1} Y_n \uplus (Y_0/\sim)$ with measure $\mu := \sum_{n \geq 1} 3^{-n} \mu_n + \frac{1}{2} (\mu_0/\sim)$, thus $$ \mu(X) = \sum_{n \geq 1} 3^{-n} + \frac{1}{2} = 1.$$ We set $X_0 := \biguplus_{n \geq 1} Y_n$, $X_a := ((S_b Y_1 \cap Y_0) \cup (S_b^{-1} Y_1 \cap Y_0))/\sim$, and $X_b := ((S_a Y_1 \cap Y_0) \cup (S_a^{-1} Y_1 \cap Y_0))/\sim$. Thus $$ \mu(X_0) = \sum_{n \geq 1} 3^{-n} = \frac{1}{2}$$ and $\mu(X_a) = \mu(X_b) = \frac{1}{4}$. One can think of $X_0$ as the ``interior'' of $X$, with $X_a$ and $X_b$ being two equally sized pieces of the ``boundary'' of $X_0$. Also, $X_a$, $X_b$ are Cantor spaces (and $\mu$ is a Cantor measure on such spaces), and so one can easily construct measurable subsets of $X_b$ of arbitrary measure between $0$ and $\mu(X_b)$. Thus Axiom (i) is satisfied. Also, one can easily create a measure-preserving invertible map $T^0_a \colon X_a \to X_a$ such that $(T^0_a)^2$ is ergodic on $X_a$; this can be done for instance by identifying $X_a$ (which is an atomless standard probability space) as a measure space (up to null sets) with the unit circle with Haar measure, and then setting $T^0_a$ to be an irrational translation map.
We now define the shifts $T_a \colon X \to X$ and $T_b \colon X \to X$ as follows.
\begin{enumerate} \item If $x \in X_0$, then $T_a x$ is defined to be $S_a x$ projected onto $X$, and $T_b x$ is similarly defined to be $S_b$ projected onto $X$. (The projection is only necessary of course if $S_a x$ or $S_b x$ lands in $Y_0$.) \item If $x \in X_a$, then $T_a x := T_a^0 x$. If instead $x \in X_b$, $T_a x$ is defined to be $S_a x' \in Y_1$, where $x' \in S_a^{-1} Y_1 \cap Y_0$ is the lift of $x$ to $S_a^{-1} Y_1 \cap Y_0$. \item If $x \in X_b$, then $T_b x = x$. If instead $x \in X_a$, $T_b x$ is defined to be $S_b x' \in Y_1$, where $x' \in S_b^{-1} Y_1 \cap Y_0$ is the lift of $x$ to $S_b^{-1} Y_1 \cap Y_0$. \end{enumerate}
One then defines $T_g$ for the remaining $g \in F_2$ in the usual fashion. In particular, one sees that for any $x$ in the interior $X_0$ and any $s \in \{a,b,a^{-1},b^{-1}\}$, $T_s x$ is equal to $S_s x$ projected onto $X$. Informally, the shifts $T_s \colon X \to X$ for $s \in \{a,b,a^{-1},b^{-1}\}$ are inherited from the shifts $S_s \colon Y \to Y$ except for the boundary actions of $T_a,T_a^{-1}$ on $X_a$ and of $T_b, T_b^{-1}$ on $X_b$, which are given by $T_0^a$ (and its inverse) and the identity map respectively. (There is nothing special about the identity map here; an arbitrary measure-preserving map on $X_b$ could be substituted here for our purposes.)
\begin{proposition} $(X, {\mathcal X}, \mu, (T_g)_{g \in F_2})$ is a good system. \end{proposition}
\begin{proof} It is a routine matter to verify that $T_a, T_b$ are invertible and measure-preserving, so that $(X, {\mathcal X}, \mu, (T_g)_{g \in F_2})$ is an $F_2$-system. Axiom (i) was already verified. For Axiom (ii), we note that $T_a X_b \subset Y_1 \subset (S_b Y_0 \cap Y_1) \cup (S_b^{-1} Y_0 \cap Y_1) = T_b X_0 \cup T_b^{-1} X_1$, as required. We set $m=1$ and $X_{a,1} := X_a$, then Axiom (iii) is true from construction, and Axiom (iv) is also easily verified. \end{proof}
It remains to construct a sequence $\tilde f_n$ of non-negative functions in $L^\infty(\tilde X)$ for each $n \in \Z$ obeying Axioms (v)-(viii) with $\alpha=1$. For negative $n$, we define $\tilde f_n$ by setting $$ \tilde f_n(x, s) := 4 \times 3^{-n}$$ whenever $x \in X$ and $s \in \{a,b,a^{-1},b^{-1}\}$ are such that $x \in Y_{-n}$ and $S_s x \in Y_{-n-1}$, and $\tilde f_n(x,s)=0$ otherwise. These are clearly non-negative functions in $L^\infty(\tilde X)$ obeying Axiom (vii). It is routine to verify that $\tilde f_{n+1} = P \tilde f_n$ for all $n \leq -2$. If we then define $\tilde f_n$ for non-negative $n$ by the formula $$ \tilde f_n := P^{n+1} \tilde f_{-1}$$ then we have Axiom (v). For negative $n$ we have
$$ \| \tilde f_n \|_{L^1(\tilde X)} = 1,$$ which gives Axiom (vi) (using Axiom (v) to extend to non-negative $n$). Finally, from Lemma \ref{pet-good} we see that $\tilde f_n$ converges pointwise almost everywhere to $1$ as $n \to +\infty$, and so Axiom (vii) follows from Egorov's theorem. This concludes the proof of Theorem \ref{initial}.
\section{The iteration step}\label{glue}
We now prove Theorem \ref{iterate}. Let $0 < \alpha \leq 1$ be such that $P(\alpha)$ holds. By Claim \ref{clam} (with $\eps$ replaced by $\eps/4$), and normalising $X$ to have measure $1$, we may find a good system $(X,{\mathcal X},\mu,(T_g)_{g \in F_2})$ with associated decomposition $X = X_a \cup X_b \cup X_0$ and measure $\mu(X)=1$, and a sequence of non-negative functions $\tilde f_n \in L^\infty(\tilde X)$ for $n \in \Z$ with the following properties: \begin{itemize} \item[(v)] (Ancient Markov chain) $\tilde f_{n+1} = P \tilde f_n$ for all $n \in \Z$.
\item[(vi)] (Size) One has $\|\tilde f_n\|_{L^1(\tilde X)} = \alpha$ for all $n \in \Z$. \item[(vii)] (Early support) $\tilde f_n$ is supported in $\tilde X_0$ for all negative $n$. Furthermore, there exists a finite $A>0$ such that $\tilde f_n$ is supported in a set of measure at most $A 3^n$ for all negative $n$. \item[(viii)] (Large maximum function) We have $$ \sup_{n \in \Z} \pi_* \tilde f_{2n}(x) \geq 1-\eps/4$$ for all $x \in X$ outside of a set of measure at most $\eps/4$. \end{itemize}
It will suffice to construct a good system $(X',{\mathcal X}',\mu',(T'_g)_{g \in F_2})$ with associated decomposition $X' = X'_a \cup X'_b \cup X'_0$, Markov operator $P'$, and measure $\mu'(X')=2$, and a sequence of non-negative functions $\tilde f'_n \in L^\infty(\tilde X')$ for $n \in \Z$ with the following properties: \begin{itemize} \item[(v')] (Ancient Markov chain) $\tilde f'_{n+1} = P' \tilde f'_n$ for all $n \in \Z$.
\item[(vi')] (Size) One has $\|\tilde f'_n\|_{L^1(\tilde X')} = \alpha (2 - \frac{\alpha}{2})$ for all $n \in \Z$. \item[(vii')] (Early support) $\tilde f'_n$ is supported in $\tilde X'_0$ for all negative $n$. Furthermore, there exists a finite $A'>0$ such that $\tilde f'_n$ is supported in a set of measure at most $2 A' 3^n$ for all negative $n$. \item[(viii')] (Large maximum function) We have $$ \sup_{n \in \Z} \pi_* \tilde f'_{2n}(x') \geq 1-\eps$$ for all $x' \in X'$ outside of a set of measure at most $2\eps$. \end{itemize}
We construct this system as follows. First, from Axiom (viii) and Egorov's theorem, we may find a natural number $N$ such that \begin{equation}\label{proj}
\sup_{-N \leq n \leq N} \pi_* \tilde f_{2n}(x) \geq 1-\eps/3 \end{equation} for all $x \in X$ outside of a set of measure at most $\eps/3$. We let $0 < \kappa < 1/4$ be a small quantity depending on $\eps, N$ and the $\tilde f_n$ to be chosen later. We will construct the good system $(X',{\mathcal X}',\mu',(T'_g)_{g \in F_2})$ to be two copies of $(X,{\mathcal X},\mu,(T_g)_{g \in F_2})$ glued together by a small amount of coupling, with the $\kappa$ parameter measuring the amount of coupling. More precisely, we define the measure space $(X', {\mathcal X}', \mu')$ to be the product of $(X,{\mathcal X}, \mu)$ with the two-element set $\{1,2\}$ with counting measure. Next, using Axiom (i), we can find a subset $E$ of $X_b$ of measure exactly $\kappa$. We now define the shift maps $T'_a, T'_b\colon X' \to X'$ as follows. The map $T'_a$ is a trivial lift of $T_a$, thus $$ T'_a (x, i ) := (T_a x, i )$$ for $x \in X$ and $i \in \{1,2\}$. The map $T'_b$ is an \emph{almost} trivial lift of $T_b$. Namely, we define $$ T'_b (x, i ) := (T_b x, i )$$ for $x \in X \backslash E$ and $i \in \{1,2\}$, but define $$ T'_b (x, i ) := (T_b x, 3-i )$$ for $x \in E$ and $i \in \{1,2\}$; see Figure \ref{fig:dupl}. Finally, we partition $X' = X'_a \cup X'_b \cup X'_0$ where $X'_a := X_a \times \{1,2\}$, $X'_b := X_b \times \{1,2\}$, $X'_0 := X_0 \times \{1,2\}$. We then define $T'_g$ for the remaining $g \in F_2$ in the usual fashion.
\begin{figure}
\caption{The good system $(X',{\mathcal X}',\mu',(T'_g)_{g \in F_2})$, which is formed by gluing together two barely interacting copies of $(X,{\mathcal X},\mu,(T_g)_{g \in F_2})$.}
\label{fig:dupl}
\end{figure}
\begin{proposition}[Good system] If $\kappa$ is sufficiently small, then $(X',{\mathcal X}',\mu',(T'_g)_{g \in F_2})$ is a good system with $\mu'(X') = 2$. \end{proposition}
\begin{proof} Axioms (i) and (ii) are easily verified, so we focus on verifying Axioms (iii) and (iv).
By Axiom (iii) for $X$, $X_a$ is partitioned into finitely many $T_a$-invariant components $X_{a,1},\dots,X_{a,m}$ of positive measure, each of which is $T_a^2$-ergodic. This induces a partition of $X'_a$ into the $2m$ components $X_{a,1} \times \{1\}, \dots, X_{a,m} \times \{1\}, X_{a,1} \times \{2\}, \dots, X_{a,m} \times \{2\}$, and each of these components are clearly $T_a^2$-ergodic.
Now we verify Axiom (iv). We need to show that $X' = \bigcup_{g \in F_2} T'_g ( X_{a,i} \times \{j\} )$ up to null sets for each $i=1,\dots,m$ and $j=1,2$. Denote the right-hand side by $Y$, thus $Y$ is $F_2$-invariant and contains $X_{a,i} \times \{j\}$. On the other hand, by Axiom (iv) for $X$ and the pigeonhole principle, there exists $g \in F_2$ such that $T_g X_{a,i}$ intersects $E$ in a set of positive measure. We may assume that the word length $|g|$ of $g$ is minimal among all $g$ with this property, thus $T_h X_{a,i} \cap E$ is null whenever $|h| < |g|$. From this we see that $T'_g (X_{a,i} \times \{j\})$ intersects $E \times \{j\}$ in a set of positive measure (since the dynamics of $T'$ are just a trivial lift of the dynamics of $T$ outside of $E \times \{1,2\}$). From construction of $T'_b$, this implies that $T'_{bg} (X_{a,i} \times \{j\})$ intersects $T_b E \times \{3-j\} \subset X_b \times \{3-j\}$ in a set of positive measure, and hence by Axiom (ii) the union of $T'_{b^{-1} abg} (X_{a,i} \times \{j\})$ and $T'_{babg} (X_{a,i} \times \{j\})$ intersects $X_a \times \{3-j\}$ in a set of positive measure; in particular, $Y$ intersects $X_a \times \{3-j\}$ in a set of positive measure. As $Y$ is $(T'_a)^2$-invariant, we conclude from Axiom (iii) that $Y$ contains $X_{a,i'} \times \{3-j\}$ up to null sets for some $i'=1,\dots,m$.
Next, by another appeal to Axiom (iv) and the pigeonhole principle, we can find $g_{i,i'} \in F_2$ such that $T_{g_{i,i'}} X_{a,i'}$ and $X_{a,i}$ intersect in a set of positive measure. Note that as there are only $m$ choices for $i'$, the word length of $g_{i,i'}$ can be bounded above, and the measure of $T_{g_{i,i'}} X_{a,i'} \cap X_{a,i}$ bounded below, by quantities independent of $\kappa$. Because of this, we see that if $\kappa$ (and hence $E$) is small enough, then $T'_{g_{i,i'}} (X_{a,i'} \times \{3-j\})$ and $X_{a,i} \times \{3-j\}$ also intersect in a set of positive measure; thus $Y$ must intersect $X_{a,i} \times \{3-j\}$ in a set of positive measure, and hence by the $T_a^2$-ergodicity of $X_{a,i}$, $Y$ contains $X_{a,i} \times \{3-j\}$ up to null sets. Since $Y$ already contained $X_{a,i} \times \{j\}$, we thus have $X_{a,i} \times \{1,2\}$ contained in $Y$ up to null sets.
Now for any $(x,j') \in X'$, we have from Axiom (iv) that $x = T_g y$ for some $y \in X_{a,i}$ and $g \in F_2$. This implies that $(x,j') = T'_g (y,j'')$ for some $j'' \in \{1,2\}$, and hence $(x,j') \in Y$ for almost every $(x,j') \in X$, which gives Axiom (iv) for $X'$ as required. \end{proof}
We let $M$ be a large natural number, depending on all previous quantities (in particular, depending on $\kappa$), to be chosen later. The functions $\tilde f'_n \in L^1(\tilde X')$ will be defined for negative $n$ by the formulae $$ \tilde f'_n(x,1,s) := \tilde f_{n}(x,s)$$ and $$ \tilde f'_n(x,2,s) := \left(1 - \frac{\alpha}{2}\right) \tilde f_{n-2M}(x,s)$$ for any $x \in X$ and $s \in \{a,b,a^{-1},b^{-1}\}$. Informally, $\tilde f'_n$ is two copies of $\tilde f'_n$, one over $X \times \{1\}$ and one over $X \times \{2\}$, with the latter experiencing a significant time delay and also a slight reduction in amplitude; the point is that we can delay the $X \times \{2\}$ dynamics until the dynamics of $X \times \{1\}$ has mixed almost completely, so that half of the mass of the $X \times \{1\}$ component is spread out almost uniformly over $X \times \{2\}$, allowing for the crucial amplitude reduction for the $X \times \{2\}$ component. The idea behind this construction is due to Ornstein \cite[Lemma 4]{ornstein}.
Clearly, Axiom (vii') is a consequence of Axiom (vii) (we allow the constant $A'$ to depend on $M$). For functions supported on $\tilde X'_0$, the Markov operator $P'$ is a trivial lift of the Markov operator $P$, so (from Axiom (vii')) one sees that $\tilde f'_{n+1} = P' \tilde f'_n$ for all $n \leq -2$. We now define $\tilde f'_n$ for non-negative $n$ by setting $$ \tilde f'_n := (P')^{n+1} \tilde f'_{-1},$$ so that Axiom (v') holds. Clearly the $\tilde f'_n$ are non-negative and in $L^\infty$, and direct calculation shows that Axiom (vi') holds for all negative $n$, and hence for all $n$ thanks to Axiom (v').
The only remaining task is to show Axiom (viii'). By the union bound, it suffices to show the bounds on $X \times \{1\}$ and $X \times \{2\}$ separately. More precisely, we establish the following two propositions.
\begin{proposition}\label{lima} If $\kappa$ is sufficiently small (depending on $\eps, N$, and the $\tilde f_n$, but without any dependence on $M$), we have $$ \sup_{n \in \Z} \pi_* \tilde f'_{2n}(x,1) \geq 1-\eps$$ for all $x \in X$ outside of a set of measure at most $\eps$. \end{proposition}
\begin{proof} By construction, we have $$ \tilde f'_n(x,1,s) = \tilde f_n(x,s) $$ for negative $n$, all $x \in X$, and $s \in \{a,b,a^{-1},b^{-1}\}$. Now we turn to non-negative $n$. Note that as $P$ is a contraction on $L^\infty$, the $\tilde f_n$ for non-negative $n$ are uniformly bounded in $L^\infty$ by some quantity $B$ independent of $\kappa$. A routine induction then shows that $$ \int_{\tilde X} \max( \tilde f_n(x,s) - \tilde f'_n(x,1,s), 0 )\ d\tilde \mu(x,s) \leq C_{B,n} \kappa$$ for all non-negative $n$ and some quantity $C_{B,n}$ that depends on $B,n$ but not on $\kappa$; this is basically because on $X \times \{1\} \times \{a,b,a^{-1},b^{-1}\}$, the Markov process associated to $P'$ only differs from that associated to $P$ on the set $E \times \{1\} \times \{b\} \cup T_b E \times \{1\} \cup \{b^{-1}\}$, which has measure $\kappa/2$. Applying $\pi_*$ and then the triangle inequality, we conclude that $$
\int_{X} \max\left( \sup_{-N \leq n \leq N} \pi_* \tilde f_{2n}(x) - \sup_{-N \leq n \leq N} \pi_* \tilde f'_{2n}(x,1), 0 \right)\ d\mu(x) \leq C'_{B,N} \kappa $$ for some $C'_{B,N}$ independent of $\kappa$; in particular, from Markov's inequality we see (for $\kappa$ small enough) that $$ \sup_{-N \leq n \leq N} \pi_* \tilde f_{2n}(x) - \sup_{-N \leq n \leq N} \pi_* \tilde f'_{2n}(x,1) \leq \eps/3$$ for all $x \in X$ outside of a set of measure at most $\eps/3$. Combining this with \eqref{proj}, we obtain the claim. \end{proof}
\begin{proposition} If $\kappa$ is sufficiently small (depending on $\eps, N$, and the $\tilde f_n$, but without any dependence on $M$), we have $$ \sup_{n \in \Z} \pi_* \tilde f'_{2n}(x,2) \geq 1-\eps$$ for all $x \in X$ outside of a set of measure at most $\eps$. \end{proposition}
\begin{proof} We split $$ \tilde f'_n = \tilde f'_{n,1} + \tilde f'_{n,2}$$ where for negative $n$, $\tilde f'_{n,i}$ is the restriction of $\tilde f'_n$ to $X \times \{i\} \times \{a,b,a^{-1},b^{-1}\}$, and for non-negative $n$, $\tilde f'_{n,i}$ is propagated by $P'$: $$ \tilde f'_{n,i} := (P')^{n+1} \tilde f'_{-1,i}.$$ Observe that the $\tilde f'_{n,1}$ component of $\tilde f'_n$ does not depend on $M$.
From Lemma \ref{pet-good}, we see that $\tilde f'_{n,1}$ converges pointwise almost everywhere as $n \to \infty$ to the constant $$ \frac{1}{\mu(\tilde X')} \int_{\tilde X'} \tilde f'_{-1,1}\ d\tilde \mu' = \frac{1}{2} \int_X \tilde f_{-1}\ d\tilde \mu = \frac{\alpha}{2}.$$ In particular, $\pi_* \tilde f'_{n,1}$ converges pointwise almost everywhere to the same constant. Thus, by Egorov's theorem, and assuming $M$ sufficiently large (depending on previous quantities such as $\eps, \kappa$, and the $\tilde f_n$, but without any circular dependency of $M$ on itself) we have \begin{equation}\label{add} \inf_{n \geq 2M-2N} \pi_* \tilde f'_{n,1}(x,2) \geq \frac{\alpha}{2} - \frac{\eps}{3} \end{equation} for all $x \in X$ outside of a set of measure at most $\eps/3$.
Now we work on $\tilde f'_{n,2}$. For all $n < 2M$, an induction (using Axiom (vii)) shows that $\tilde f'_{n,2}$ is supported on $X_0 \times \{2\} \times \{a,b,a^{-1},b^{-1}\}$, and that $$ \tilde f'_{n,2}(x,2,s) = \left(1 - \frac{\alpha}{2}\right) \tilde f_{n-2M}(x,s)$$ for all $x \in X$ and $s \in \{a,b,a^{-1},b^{-1}\}$. Repeating the arguments used to prove Proposition \ref{lima}, we see (if $\kappa$ is sufficiently small depending on $\eps,N$, but (crucially) without any dependence on $M$) that $$ \left(1 - \frac{\alpha}{2}\right) \sup_{M-N \leq n \leq M+N} \pi_* \tilde f_{2n-2M}(x) -\sup_{M-N \leq n \leq M+N} \pi_* \tilde f'_{2n}(x,2) \leq \eps/3 $$ for all $x \in X$ outside of a set of measure at most $\eps/3$. Combining this with \eqref{add}, we see that $$ \sup_n \pi_* \tilde f'_{n}(x,2) \geq \frac{\alpha}{2} + (1 - \frac{\alpha}{2}) \sup_{-N \leq n \leq N} \pi_* \tilde f_{2n}(x) - \frac{2\eps}{3} $$ for all $x \in X$ outside of a set of measure at most $2\eps/3$. Applying \eqref{proj}, we then obtain the claim. \end{proof}
The proof of Theorem \ref{iterate}, and thus Theorem \ref{main}, is now complete.
\end{document} | arXiv |
Mathematical Biosciences and Engineering, 2018, 15(4): 961-991. doi: 10.3934/mbe.2018043.
Quantifying the survival uncertainty of Wolbachia-infected mosquitoes in a spatial model
Martin Strugarek, Nicolas Vauchelet, Jorge P. Zubelli
1. AgroParisTech, 16 rue Claude Bernard, 75231 Paris Cedex 05, France
2. Sorbonne Université, Université Paris-Diderot SPC, CNRS, INRIA, Laboratoire Jacques-Louis Lions, équipe Mamba, F-75005 Paris, France
3. LAGA - UMR 7539 Institut Galilée, Université Paris 13, 99, avenue Jean-Baptiste Clément 93430 Villetaneuse, France
4. IMPA, Estrada Dona Castorina, 110 Jardim Botânico 22460-320, Rio de Janeiro, RJ, Brazil
Artificial releases of Wolbachia-infected Aedes mosquitoes have been under study in the past yearsfor fighting vector-borne diseases such as dengue, chikungunya and zika.Several strains of this bacterium cause cytoplasmic incompatibility (CI) and can also affect their host's fecundity or lifespan, while highly reducing vector competence for the main arboviruses.
We consider and answer the following questions: 1) what should be the initial condition (i.e. size of the initial mosquito population) to have invasion with one mosquito release source? We note that it is hard to have an invasion in such case. 2) How many release points does one need to have sufficiently high probability of invasion? 3) What happens if one accounts for uncertainty in the release protocol (e.g. unequal spacing among release points)?
We build a framework based on existing reaction-diffusion models for the uncertainty quantification in this context,obtain both theoretical and numerical lower bounds for the probability of release successand give new quantitative results on the one dimensional case.
Keywords: Reaction-diffusion equation; Wolbachia; uncertainty quantification; population replacement; mosquito release protocol
Citation: Martin Strugarek, Nicolas Vauchelet, Jorge P. Zubelli. Quantifying the survival uncertainty of Wolbachia-infected mosquitoes in a spatial model. Mathematical Biosciences and Engineering, 2018, 15(4): 961-991. doi: 10.3934/mbe.2018043
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\begin{document}
\title{Infinite presentations for fundamental groups of surfaces} \thanks{2020 \textit{Mathematics Subject Classification}. Primary 57M05; Secondary 20F05, 57M07, 20F65.}
\keywords{Fundamental group, presentation, mapping class group} \author[R. Kobayashi]{Ryoma Kobayashi} \address{Department of General Education\\ National Institute of Technology, Ishikawa College\\ Tsubata Ishikawa 929-0392 JAPAN} \email{kobayashi\[email protected]} \thanks{The author is supported by JSPS KAKENHI Grant Number JP19K14542 and JP22K13920}
\maketitle
\begin{abstract} For any finite type connected surface $S$, we give an infinite presentation of the fundamental group $\pi_1(S,\ast)$ of $S$ based at an interior point $\ast\in{S}$ whose generators are represented by simple loops. When $S$ is non-orientable, we also give an infinite presentation of the subgroup of $\pi_1(S,\ast)$ generated by elements which are represented by simple loops whose regular neighborhoods are annuli. \end{abstract}
\section{Introduction}
For any surface $S$ and any point $\ast$ in the interior of $S$, let $\pi_1(S,\ast)$ denote the fundamental group of $S$ based at $\ast$. When $S$ is non-orientable, we denote by $\pi_1^+(S,\ast)$ the subgroup of $\pi_1(S,\ast)$ generated by elements which are represented by simple loops whose regular neighborhoods are annuli, called \textit{two-sided simple loops}. A presentation of $\pi_1(S,\ast)$ is well known. In particular, $\pi_1(S,\ast)$, and also $\pi_1^+(S,\ast)$, are free groups if $S$ has a boundary. For a connected closed orientable surface $S$, Putman \cite{P} gave an infinite presentation of $\pi_1(S,\ast)$. In this paper we give infinite presentations of $\pi_1(S,\ast)$ and $\pi_1^+(S,\ast)$ whose generators are represented by simple loops, for any finite type connected surface $S$, as follows.
\begin{thm}\label{main-1} For any finite type connected surface $S$, let $\pi$ be the group generated by symbols $S_\alpha$ for $\alpha\in\pi_1(S,\ast)$ which is represented by a non-trivial simple loop, and with the defining relations \begin{enumerate} \item $S_{\alpha^{-1}}=S_\alpha^{-1}$, \item $S_{\alpha}S_{\beta}=S_{\gamma}$ if $\alpha\beta=\gamma$. \end{enumerate} Then $\pi$ is isomorphic to $\pi_1(S,\ast)$. \end{thm}
\begin{thm}\label{main-2} For any finite type connected non-orientable surface $S$, let $\pi^+$ be the group generated by symbols $S_\alpha$ for $\alpha\in\pi_1^+(S,\ast)$ which is represented by a non-trivial simple loop, and with the defining relations \begin{enumerate} \item $S_{\alpha^{-1}}=S_\alpha^{-1}$, \item $S_{\alpha}S_{\beta}=S_{\gamma}$ if $\alpha\beta=\gamma$, \item $S_{\alpha}S_{\beta}S_{\alpha}^{-1}=S_\gamma$ if $\alpha\beta\alpha^{-1}=\gamma$. \end{enumerate} Then $\pi^+$ is isomorphic to $\pi_1^+(S,\ast)$. \end{thm}
These results are useful in studies on the mapping class group of $S$ and its subgroups. For example, in \cite{KO1}, Theorem~\ref{main-2} is used to obtain an infinite presentation for the twist subgroup of the mapping class group of a compact non-orientable surface.
In order to prove Theorems~\ref{main-1} and \ref{main-2}, we use the following lemma.
\begin{lem}[cf. \cite{P}]\label{main-lem} Let $G$ and $H$ be groups generated by sets $X$ and $Y$, respectively, such that $H$ acts on $G$. Suppose that $X^\prime\subset{X}$ satisfies the following conditions. \begin{itemize} \item $H(X^\prime)=X$. \item For any $x\in{X^\prime}$ and $y\in{Y}$, $y^{\pm1}(x)$ is in the subgroup of $G$ generated by $X^\prime$. \end{itemize} Then $X^\prime$ generates $G$. \end{lem}
As groups acting on $\pi$ and $\pi^+$, we consider the \textit{pure mapping class group} of $S$. Using this lemma, we show that $\pi$ and $\pi^+$ are generated by symbols corresponding to basic generators of $\pi_1(S,\ast)$ and $\pi_1^+(S,\ast)$, respectively.
In Section~\ref{mcg}, we define mapping class groups and pure mapping class groups of surfaces, and explain their generators. In Sections~\ref{pi} and \ref{pi^+}, we prove Theorems~\ref{main-1} and \ref{main-2}, respectively.
Throughout this paper, we do not distinguish a loop from its homotopy class.
\section{On mapping class groups of surfaces}\label{mcg}
For $g\geq0$ and $m\geq0$, let $\Sigma_{g,m}$ be a surface which is obtained by removing $m$ disks from a connected sum of $g$ tori, as shown in Figure~\ref{gen-mcg}~(a). We call $\Sigma_{g,m}$ a genus $g$ orientable surface with $m$ boundary components. We define the \textit{mapping class group} $\mathcal{M}(\Sigma_{g,m})$ of $\Sigma_{g,m}$ as the group consisting of isotopy classes of all orientation preserving diffeomorphisms of $\Sigma_{g,m}$. The \textit{pure mapping class group} $\mathcal{PM}(\Sigma_{g,m})$ of $\Sigma_{g,m}$ is the subgroup of $\mathcal{M}(\Sigma_{g,m})$ consisting of elements which do not permute order of the boundary components of $\Sigma_{g,m}$. Regarding some boundary component of $\Sigma_{g,n+1}$ as $\ast$, we notice that $\mathcal{PM}(\Sigma_{g,n+1})$ acts on $\pi_1(\Sigma_{g,n},\ast)$ naturally.
For $g\geq1$ and $m\geq0$, let $N_{g,m}$ be a surface which is obtained by removing $m$ disks from a connected sum of $g$ real projective planes. We call $N_{g,m}$ a genus $g$ non-orientable surface with $m$ boundary components. We can regard $N_{g,m}$ as a surface which is obtained by attaching $g$ M\"obius bands to $g$ boundary components of $\Sigma_{0,g+m}$, as shown in Figure~\ref{gen-mcg}~(b) or (c). We call these attached M\"obius bands \textit{crosscaps}. We define the \textit{mapping class group} $\mathcal{M}(N_{g,m})$ of $N_{g,m}$ as the group consisting of isotopy classes of all diffeomorphisms of $N_{g,m}$. The \textit{pure mapping class group} $\mathcal{PM}(N_{g,m})$ of $N_{g,m}$ is the subgroup of $\mathcal{M}(N_{g,m})$ consisting of elements which do not permute order of the boundary components of $N_{g,m}$. Regarding some boundary component of $N_{g,n+1}$ as $\ast$, we notice that $\mathcal{PM}(N_{g,n+1})$ acts on $\pi_1(N_{g,n},\ast)$, and also $\pi_1^+(N_{g,n},\ast)$, naturally.
It is well known that $\mathcal{PM}(\Sigma_{g,m})$ can be generated by only \textit{Dehn twists} (for instance see \cite{D1,D2,L2}). On the other hand, $\mathcal{PM}(N_{g,m})$ can not be generated by only Dehn twists. We need \textit{boundary pushing maps} and \textit{crosscap pushing maps} as generators of $\mathcal{PM}(N_{g,m})$, in addition to Dehn twists (see \cite{L1,L3}). We now define the Dehn twist, the boundary pushing map and the crosscap pushing map. For a two-sided simple closed curve $c$ of a surface $S$, the Dehn twist $t_c$ about $c$ is the isotopy class of a map as shown in Figure~\ref{DBY}~(a). When $S$ is orientable, the direction of $t_c$ is the right side with respect to an orientation of $S$. When $S$ is non-orientable, the direction of $t_c$ is indicated by an arrow written beside $c$ as shown in Figure~\ref{DBY}~(a). Let $\alpha$ be an oriented arc of $S$ with its two endpoints at a boundary component, as shown in Figure~\ref{DBY}~(b). The boundary pushing map $B_\alpha$ about $\alpha$ is the isotopy class of a map obtained by pushing the boundary component along $\alpha$. Let $\alpha$ and $\mu$ be an oriented simple closed curve and a simple closed curve whose regular neighborhood is a crosscap, called a \textit{one-sided simple loop}, of a non-orientable surface, respectively, such that $\alpha$ and $\mu$ intersect transversally at one point, as shown in Figure~\ref{DBY}~(c). The crosscap pushing map $Y_{\mu,\alpha}$ about $\alpha$ and $\mu$ is the isotopy class of a map obtained by pushing the crosscap, which is the regular neighborhood of $\mu$, along $\alpha$.
\begin{figure}
\caption{Elements of mapping class groups of surfaces.}
\label{DBY}
\end{figure}
We have the following theorems.
\begin{thm}[c.f. \cite{FM}]\label{gen-PMS} Let $c_0$, $c_1,\dots,c_{2g}$ and $d_1,\dots,d_n$ be simple closed curves of $\Sigma_{g,n+1}$ as shown in Figure~\ref{gen-mcg}~(a). Then $\mathcal{PM}(\Sigma_{g,n+1})$ is generated by $t_{c_0}$, $t_{c_1},\dots,t_{c_{2g}}$ and $t_{d_1},\dots,t_{d_n}$. \end{thm}
\begin{thm}\label{gen-PMF} Let $a_1,\dots,a_{g-1}$, $b$, $\mu$, $s_{kl}$ and $r_0$, $r_1,\cdots,r_n$ be simple closed curves and simple arcs of $N_{g,n+1}$ for $1\leq{k<l}\leq{}n$, as shown in Figures~\ref{gen-mcg}~(b) and (c). Then $\mathcal{PM}(N_{g,n+1})$ is generated by $t_{a_1},\dots,t_{a_{g-1}}$, $t_b$, $Y_{\mu,a_1}$, $t_{s_{kl}}$ and $B_{r_0}$, $B_{r_1},\dots,B_{r_n}$ for $1\leq{k<l}\leq{}n$. \end{thm}
\begin{figure}\end{figure}
\begin{proof} There is an exact sequence $$\pi_1(N_{g,n},\ast)\to\mathcal{PM}(N_{g,n+1})\to\mathcal{PM}(N_{g,n})\to1,$$ introduced by Birman~\cite{B} for orientable surfaces. The homomorphism $\pi_1(N_{g,n},\ast)\to\mathcal{PM}(N_{g,n+1})$ is defined as $\alpha\mapsto{}B_{\overline{\alpha}}$, where $\overline{\alpha}$ is an arc which is obtained from $\alpha$ by regarding $\ast$ as a boundary component. The homomorphism $\mathcal{PM}(N_{g,n+1})\to\mathcal{PM}(N_{g,n})$ is defined as the map which is induced by capping the boundary component with a disk.
Let $x_1,\dots,x_g$ and $y_1,\dots,y_{n-1}$ be oriented simple loops of $N_{g,n}$ based at $\ast$, as shown in Figure~\ref{gen-pi_1-non-ori-surf}. It is well known that $\pi_1(N_{g,n},\ast)$ is generated by $x_1,\dots,x_g$ and $y_1,\dots,y_{n-1}$. It is easy to check that $t_{a_i}F^{i-1}Y_{\mu,a_1}^{(-1)^{i-1}}F^{1-i}(x_{i+1})=x_i$ for $i=1,\dots,g-1$, where $F=t_{a_1}t_{a_2}\cdots{}t_{a_{g-1}}$. Therefore, since the homomorphism $\pi_1(N_{g,0},\ast)\to\mathcal{PM}(N_{g,1})$ sends $x_g$ to $B_{r_0}$, we see that this homomorphism sends $x_i$ to a conjugate element of $B_{r_0}$ by $t_{a_1},\dots,t_{a_{g-1}}$ and $Y_{\mu,a_1}$ from the relation $B_{f(r_0)}=fB_{r_0}f^{-1}$ (for example see Lemma~2.4 in \cite{Kor}). In addition, regarding $\ast$ as the $n$-th boundary component of $N_{g,n+1}$, it follows that the homomorphism $\pi_1(N_{g,n},\ast)\to\mathcal{PM}(N_{g,n+1})$ sends $x_i$ to a conjugate element of $B_{r_n}$ by $t_{a_1},\dots,t_{a_{g-1}}$ and $Y_{\mu,a_1}$ for $n\geq1$ and $1\leq{i}\leq{g}$ from a similar argument, and $y_k$ to $s_{kn}$ for $n\geq2$ and $1\leq{k}\leq{n-1}$. It is known that $\mathcal{PM}(N_{g,0})$ is generated by $t_{a_1},\dots,t_{a_{g-1}}$, $t_b$ and $Y_{\mu,a_1}$ (see \cite{C,S2}). Therefore using the exact sequence above, we obtain the generating set inductively.
\begin{figure}
\caption{Oriented simple loops $x_i$ and $y_k$ of $N_{g,n}$ based at $\ast$ for $1\leq{i}\leq{g}$ and $1\leq{k}\leq{n}$.}
\label{gen-pi_1-non-ori-surf}
\end{figure}
\end{proof}
Note that a finite generating set of $\mathcal{PM}(N_{g,n+1})$ which is different from that of Theorem~\ref{gen-PMF} was already given by Korkmaz~\cite{Kor}. However we use the generating set of Theorem~\ref{gen-PMF} in this paper.
\section{Proof of Theorem~\ref{main-1}}\label{pi}
In Subsection~\ref{ori}, we prove Theorem~\ref{main-1} of the case where $S$ is orientable. In Subsection~\ref{non-ori}, we prove Theorem~\ref{main-1} of the case where $S$ is non-orientable.
\subsection{The case where $S$ is orientable}\label{ori}\
Let $\alpha_1,\dots,\alpha_g$, $\beta_1,\dots,\beta_g$ and $\gamma_1,\dots,\gamma_{n-1}$ be oriented simple loops of $\Sigma_{g,n}$ based at $\ast$, as shown in Figure~\ref{gen-pi_1-ori-surf}. It is well known that $\pi_1(\Sigma_{g,n},\ast)$ is the free group freely generated by these loops for $n\geq1$ and the group generated by $\alpha_1,\dots,\alpha_g$ and $\beta_1,\dots,\beta_g$ which has one relation $[\alpha_1,\beta_1]\cdots[\alpha_g,\beta_g]=1$ for $n=0$, where $[x,y]=xyx^{-1}y^{-1}$.
\begin{figure}
\caption{Oriented simple loops $\alpha_i$, $\beta_i$, $\gamma_k$, $\delta_i$ and $\epsilon_k$ of $\Sigma_{g,n}$ based at $\ast$ for $1\leq{i}\leq{g}$ and $1\leq{k}\leq{n}$, except for $i=g$ for $\delta_i$.}
\label{gen-pi_1-ori-surf}
\end{figure}
Let $X$ be a set consisting of $S_\alpha$, where $\alpha$ is a non-separating simple loop or a separating simple loop which bounds the $m$-th boundary component for $1\leq{m}\leq{n-1}$, and let $X^\prime$ be the following subset of $X$: $$X^\prime=\{S_{\alpha_1},\dots,S_{\alpha_g},S_{\beta_1},\dots,S_{\beta_g},S_{\gamma_1},\dots,S_{\gamma_{n-1}}\}.$$ Let $Y$ be the generating set for $\mathcal{PM}(\Sigma_{g,n+1})$ given in Theorem~\ref{gen-PMS}. In the actions on $\pi_1(\Sigma_{g,n},\ast)$ and $\pi$ by $\mathcal{PM}(\Sigma_{g,n+1})$, we regard the $(n+1)$-st boundary component of $\Sigma_{g,n+1}$ as $\ast$. We define $f(S_\alpha)=S_{f_\sharp(\alpha)}$ for $S_\alpha\in\pi$ and $f\in\mathcal{PM}(\Sigma_{g,n+1})$, where $f_\sharp$ is the map on $\pi_1(\Sigma_{g,n},\ast)$ induced from $f$. We prove the following proposition.
\begin{prop}\label{1} \begin{enumerate} \item $X$ generates $\pi$. \item $\mathcal{PM}(\Sigma_{g,n+1})(X^\prime)=X$. \item For any $x\in{X^\prime}$ and $y\in{Y}$, $y^{\pm1}(x)$ is in the subgroup of $\pi$ generated by $X^\prime$. \end{enumerate} \end{prop}
In order to prove the proposition, we show the following lemma.
\begin{lem}\label{d-e} For $1\leq{i}\leq{g-1}$ and $1\leq{k}\leq{n}$, $S_{\gamma_n}$, $S_{\delta_i}$ and $S_{\epsilon_k}$ are in the subgroup of $\pi$ generated by $X^\prime$, where $\gamma_n$, $\delta_i$ and $\epsilon_k$ are simple loops of $\Sigma_{g,n}$ based at $\ast$ as shown in Figure~\ref{gen-pi_1-ori-surf}. \end{lem}
\begin{proof} By the relations (1) and (2) of $\pi$, we calculate \begin{eqnarray*} S_{\gamma_n}&=&S_{([\alpha_1,\beta_1]\cdots[\alpha_g,\beta_g]\gamma_1\cdots\gamma_{n-1})^{-1}} =([S_{\alpha_1},S_{\beta_1}]\cdots[S_{\alpha_g},S_{\beta_g}]S_{\gamma_1}\cdots{}S_{\gamma_{n-1}})^{-1},\\ S_{\delta_i}&=&S_{\beta_i^{-1}\alpha_{i+1}\beta_{i+1}\alpha_{i+1}^{-1}} =S_{\beta_i}^{-1}S_{\alpha_{i+1}}S_{\beta_{i+1}}S_{\alpha_{i+1}}^{-1},\\ S_{\epsilon_k}&=&S_{\beta_g^{-1}\gamma_1\cdots\gamma_k} =S_{\beta_g}^{-1}S_{\gamma_1}\cdots{}S_{\gamma_k} \end{eqnarray*} for $1\leq{i}\leq{g-1}$ and $1\leq{k}\leq{n}$. Since each symbol of the right hand sides is in $X^\prime$, we get the claim. \end{proof}
\begin{proof}[Proof of Proposition~\ref{1}] (1) For any generator $S_\alpha$ of $\pi$, if $\alpha$ is a non-separating simple loop, $S_\alpha$ is in $X$ clearly.
If $\alpha$ is a separating simple loop, one of a component of the complement of $\alpha$ is homeomorphic to $\Sigma_{h,m+1}$ for some $0\leq{h}\leq{g}$ and $0\leq{m}\leq{n}$.
Therefore, there is $f\in\mathcal{PM}(\Sigma_{g,n+1})$ such that $\alpha=f_\sharp([\alpha_1,\beta_1]\cdots[\alpha_h,\beta_h]\gamma_{k_1}\cdots\gamma_{k_m})$ for some $1\leq{k_1<\cdots<k_m}\leq{n}$ (see Figure~\ref{normal-position-loop-ori-surf}).
Then, by the relation (2) of $\pi$, we have $S_\alpha=[S_{f_\sharp(\alpha_1)},S_{f_\sharp(\beta_1)}]\cdots[S_{f_\sharp(\alpha_h)},S_{f_\sharp(\beta_h)}]S_{f_\sharp(\gamma_{k_1})}\cdots{}S_{f_\sharp(\gamma_{k_m})}$.
Since each symbol of the right hand side is in $X$, we conclude that $X$ generates $\pi$.
\begin{figure}
\caption{An oriented simple loop $[\alpha_1,\beta_1]\cdots[\alpha_h,\beta_h]\gamma_1\cdots\gamma_m$, one of a component of whose complement is homeomorphic to $\Sigma_{h,m+1}$, for $0\leq{h}\leq{g}$ and $0\leq{m}\leq{n}$.}
\label{normal-position-loop-ori-surf}
\end{figure}
(2) For any $S_\alpha\in{X}$, if $\alpha$ is a non-separating simple loop, there is $f\in\mathcal{PM}(\Sigma_{g,n+1})$ such that $f_\sharp(\alpha_1)=\alpha$, and hence $f(S_{\alpha_1})=S_\alpha$.
If $\alpha$ is a separating simple loop which bounds the $m$-th boundary component for $1\leq{m}\leq{n-1}$, there is $f\in\mathcal{PM}(\Sigma_{g,n+1})$ such that $f_\sharp(\gamma_m)=\alpha$, and hence $f(S_{\gamma_m})=S_\alpha$.
Therefore we obtain the claim.
(3) In this proof, we omit details of calculations.
Let $y=t_{c_0}$.
We calculate
$$
y(S_{\alpha_2})=S_{\alpha_2\beta_2^{-1}}\overset{(1),(2)}{=}S_{\alpha_2}S_{\beta_2}^{-1},~
y^{-1}(S_{\alpha_2})=S_{\alpha_2\beta_2}\overset{(2)}{=}S_{\alpha_2}S_{\beta_2}
$$
and $y^{\pm1}(x)=x$ for any other $x\in{X^\prime}$.
Let $y=t_{c_{2i-1}}$ for $1\leq{i}\leq{g}$.
We calculate
\begin{eqnarray*}
y(S_{\alpha_{i-1}})&=&S_{\alpha_{i-1}\delta_{i-1}}\overset{(2)}{=}S_{\alpha_{i-1}}S_{\delta_{i-1}},\\
y^{-1}(S_{\alpha_{i-1}})&=&S_{\alpha_{i-1}\delta_{i-1}^{-1}}\overset{(1),(2)}{=}S_{\alpha_{i-1}}S_{\delta_{i-1}}^{-1},\\
y(S_{\alpha_i})&=&S_{\delta_{i-1}^{-1}\alpha_i}\overset{(1),(2)}{=}S_{\delta_{i-1}}^{-1}S_{\alpha_i},~
y^{-1}(S_{\alpha_i})=S_{\delta_{i-1}\alpha_i}\overset{(2)}{=}S_{\delta_{i-1}}S_{\alpha_i},\\
y^{\pm1}(S_{\beta_{i-1}})&=&S_{\delta_{i-1}^{\mp1}\beta_{i-1}\delta_{i-1}^{\pm1}}\overset{(1),(2)}{=}S_{\delta_{i-1}}^{\mp1}S_{\beta_{i-1}}S_{\delta_{i-1}}^{\pm1}\\
\end{eqnarray*}
and $y^{\pm1}(x)=x$ for any other $x\in{X^\prime}$.
Let $y=t_{c_{2i}}$ for $1\leq{i}\leq{g}$.
We calculate
$$
y(S_{\beta_i})=S_{\beta_i\alpha_i}\overset{(2)}{=}S_{\beta_i}S_{\alpha_i},
y^{-1}(S_{\beta_i})=S_{\beta_i\alpha_i^{-1}}\overset{(1),(2)}{=}S_{\beta_i}S_{\alpha_i}^{-1}
$$
and $y^{\pm1}(x)=x$ for any other $x\in{X^\prime}$.
Let $y=t_{d_k}$ for $1\leq{k}\leq{n}$.
We calculate
\begin{eqnarray*}
y(S_{\alpha_g})&=&S_{\alpha_g\epsilon_k}\overset{(2)}{=}S_{\alpha_g}S_{\epsilon_k},~
y^{-1}(S_{\alpha_g})=S_{\alpha_g\epsilon_k^{-1}}\overset{(1),(2)}{=}S_{\alpha_g}S_{\epsilon_k}^{-1},\\
y^{\pm1}(S_{\beta_g})&=&S_{\epsilon_k^{\mp1}\beta_g\epsilon_k^{\pm1}}\overset{(1),(2)}{=}S_{\epsilon_k}^{\mp1}S_{\beta_g}S_{\epsilon_k}^{\pm1},\\
y^{\pm1}(S_{\gamma_l})&=&S_{\epsilon_k^{\mp1}\gamma_l\epsilon_k^{\pm1}}\overset{(1),(2)}{=}S_{\epsilon_k}^{\mp1}S_{\gamma_l}S_{\epsilon_k}^{\pm1}
\end{eqnarray*}
for $l\leq{k}$, and $y^{\pm1}(x)=x$ for any other $x\in{X^\prime}$.
Hence we have that for any $x\in{X^\prime}$ and $y\in{Y}$, $y^{\pm1}(x)$ is in the subgroup of $\pi$ generated by $X^\prime$, by Lemma~\ref{d-e}. \end{proof}
\begin{proof}[Proof of Theorem~\ref{main-1} of the case where $S$ is orientable] By Lemma~\ref{main-lem} and Proposition~\ref{1}, it follows that $\pi$ is generated by $X^\prime$. There is a natural map $\pi\to\pi_1(\Sigma_{g,n},\ast)$. The relations~(1) and (2) of $\pi$ are satisfied in $\pi_1(\Sigma_{g,n},\ast)$ clearly. Hence the map is a homomorphism. In addition, the relation $[S_{\alpha_1},S_{\beta_1}]\cdots[S_{\alpha_g},S_{\beta_g}]=1$ is obtained from the relation~(2) of $\pi$ for $n=0$. Therefore the map is an isomorphism for any $n\geq0$. Thus we complete the proof. \end{proof}
\subsection{The case where $S$ is non-orientable}\label{non-ori}\
Let $x_1,\dots,x_g$ and $y_1,\dots,y_{n-1}$ be oriented simple loops of $N_{g,n}$ based at $\ast$, as shown in Figure~\ref{gen-pi_1-non-ori-surf}. It is well known that $\pi_1(N_{g,n},\ast)$ is the free group freely generated by these loops for $n\geq1$ and the group generated by $x_1,\dots,x_g$ which has one relation $x_1^2\cdots{}x_g^2=1$ for $n=0$.
Let $X$ be a set consisting of $S_\alpha$, where $\alpha$ is a one-sided simple loop whose complement is non-orientable, or a separating simple loop which bounds the $m$-th boundary component for $1\leq{m}\leq{n-1}$, and let $X^\prime$ be the following subset of $X$: $$X^\prime=\{S_{x_1},\dots,S_{x_g},S_{y_1},\dots,S_{y_{n-1}}\}.$$ Let $Y$ be the generating set for $\mathcal{PM}(N_{g,n+1})$ given in Theorem~\ref{gen-PMF}. In the actions on $\pi_1(N_{g,n},\ast)$ and $\pi$ by $\mathcal{PM}(N_{g,n+1})$, we regard the $(n+1)$-st boundary component of $N_{g,n+1}$ as $\ast$. We define $f(S_\alpha)=S_{f_\sharp(\alpha)}$ for $S_\alpha\in\pi$ and $f\in\mathcal{PM}(N_{g,n+1})$, where $f_\sharp$ is the map on $\pi_1(N_{g,n},\ast)$ induced from $f$. We prove the following proposition.
\begin{prop}\label{2} \begin{enumerate} \item $X$ generates $\pi$. \item $\mathcal{PM}(N_{g,n+1})(X^\prime)=X$. \item For any $x\in{X^\prime}$ and $y\in{Y}$, $y^{\pm1}(x)$ is in the subgroup of $\pi$ generated by $X^\prime$. \end{enumerate} \end{prop}
In order to prove the proposition, we show the following lemma.
\begin{lem}\label{y_n} $S_{y_n}$ is in the subgroup of $\pi$ generated by $X^\prime$, where $y_n$ is a simple loop of $N_{g,n}$ as shown in Figure~\ref{gen-pi_1-non-ori-surf}. \end{lem}
\begin{proof} By the relations (1) and (2) of $\pi$, we calculate $$ S_{y_n}=S_{(x_1^2\cdots{}x_g^2y_1\cdots{}y_{n-1})^{-1}} =(S_{x_1}^2\cdots{}S_{x_g}^2S_{y_1}\cdots{}S_{y_{n-1}})^{-1}. $$ Since each symbol of the right hand side is in $X^\prime$, we get the claim. \end{proof}
\begin{proof}[Proof of Proposition~\ref{2}] (1) For any generator $S_\alpha$ of $\pi$, the complement of $\alpha$ is homeomorphic to either
\begin{enumerate}
\item $N_{g-1,n+1}$,
\item $N_{g-2,n+2}$,
\item $\Sigma_{h,n+r}$ if $g=2h+r$ for $r=1$, $2$,
\item $N_{h,m+1}\sqcup{N_{g-h,n-m+1}}$ for $1\leq{h}\leq{g-1}$ and $0\leq{m}\leq{n}$ or
\item $\Sigma_{h,m+1}\sqcup{N_{g-2h,n-m+1}}$ for $\displaystyle0\leq{h}\leq\frac{g-1}{2}$ and $0\leq{m}\leq{n}$
\end{enumerate}
(see \cite{S1}).
Therefore, there is $f\in\mathcal{PM}(N_{g,n+1})$ such that $\alpha=f_\sharp(\beta)$, where $\beta$ is either one of the simple loops as in Figure~\ref{normal-position-loop-non-ori-surf}.
For the case (a), we have $S_\alpha=S_{f_\sharp(x_1)}$.
For the case (b), by the relation (2) of $\pi$, we have $S_\alpha=S_{f_\sharp(x_1)}S_{f_\sharp(x_2)}$.
For the cases (c), by the relation (2) of $\pi$, we have $S_\alpha=S_{f_\sharp(x_1)}\cdots{}S_{f_\sharp(x_g)}$.
For the case (d), by the relation (2) of $\pi$, we have $S_\alpha=S_{f_\sharp(x_1)}^2\cdots{}S_{f_\sharp(x_h)}^2S_{f_\sharp(y_{k_1})}\cdots{}S_{f_\sharp(y_{k_m})}$ for some $1\leq{k_1<\cdots<k_m}\leq{n}$.
For the case (e), by the relations (1) and (2) of $\pi$, we have
\begin{eqnarray*}
S_\alpha
&=&
S_{f_\sharp(x_1)}\cdots{}S_{f_\sharp(x_{2h})}S_{f_\sharp(x_{2h+1})}^{-1}S_{f_\sharp(x_{2h})}^{-2}\cdots{}S_{f_\sharp(x_{2})}^{-2}S_{f_\sharp(x_{1})}^{-1}\\
&&
~\cdot~S_{f_\sharp(x_{2})}\cdots{}S_{f_\sharp(x_{2h+1})}S_{f_\sharp(y_{k_1})}\cdots{}S_{f_\sharp(y_{k_m})}
\end{eqnarray*}
for some $1\leq{k_1<\cdots<k_m}\leq{n}$ if $h\neq0$.
If $h=0$, by the relation (2) of $\pi$, we have $S_\alpha=S_{f_\sharp(y_{k_1})}\cdots{}S_{f_\sharp(y_{k_m})}$ for some $1\leq{k_1<\cdots<k_m}\leq{n}$.
Since each symbol of the right hand sides is in $X$, we conclude that $X$ generates $\pi$.
\begin{figure}\end{figure}
(2) For any $S_\alpha\in{X}$, if $\alpha$ is a one-sided simple loop whose complement is non-orientable, there is $f\in\mathcal{PM}(N_{g,n+1})$ such that $f_\sharp(x_1)=\alpha$, and hence $f(S_{x_1})=S_\alpha$.
If $\alpha$ is a separating simple loop which bounds the $m$-th boundary component for $1\leq{m}\leq{n-1}$, there is $f\in\mathcal{PM}(N_{g,n+1})$ such that $f_\sharp(y_m)=\alpha$, and hence $f(S_{y_m})=S_\alpha$.
Therefore we obtain the claim.
(3) In this proof, we omit details of calculations.
Let $y=t_{a_i}$ for $1\leq{i}\leq{g}$.
We calculate
\begin{eqnarray*}
y(S_{x_i})&=&S_{x_ix_{i+1}^{-1}x_i^{-1}}\overset{(1),(2)}{=}S_{x_i}S_{x_{i+1}}^{-1}S_{x_i}^{-1},\\
y^{-1}(S_{x_i})&=&S_{x_i^2x_{i+1}}\overset{(2)}{=}S_{x_i}^2S_{x_{i+1}},\\
y(S_{x_{i+1}})&=&S_{x_ix_{i+1}^2}\overset{(2)}{=}S_{x_i}S_{x_{i+1}}^2,\\
y^{-1}(S_{x_{i+1}})&=&S_{x_{i+1}^{-1}x_i^{-1}x_{i+1}}\overset{(1),(2)}{=}S_{x_{i+1}}^{-1}S_{x_i}^{-1}S_{x_{i+1}}
\end{eqnarray*}
and $y^{\pm1}(x)=x$ for any other $x\in{X^\prime}$.
Let $y=t_{b}$.
We calculate
\begin{eqnarray*}
y(S_{x_1})&=&S_{x_1x_2x_3x_4^{-1}x_3^{-2}x_2^{-2}x_1^{-1}}\overset{(1),(2)}{=}S_{x_1}S_{x_2}S_{x_3}S_{x_4}^{-1}S_{x_3}^{-2}S_{x_2}^{-2}S_{x_1}^{-1},\\
y^{-1}(S_{x_1})&=&S_{x_1^2x_2^2x_3^2x_4x_3^{-1}x_2^{-1}}\overset{(1),(2)}{=}S_{x_1}^2S_{x_2}^2S_{x_3}^2S_{x_4}S_{x_3}^{-1}S_{x_2}^{-1},\\
y(S_{x_2})&=&S_{x_1x_2^2x_3^2x_4x_3^{-1}}\overset{(1),(2)}{=}S_{x_1}S_{x_2}^2S_{x_3}^2S_{x_4}S_{x_3}^{-1},\\
y^{-1}(S_{x_2})&=&S_{x_2x_3x_4^{-1}x_3^{-2}x_2^{-2}x_1^{-1}x_2}\overset{(1),(2)}{=}S_{x_2}S_{x_3}S_{x_4}^{-1}S_{x_3}^{-2}S_{x_2}^{-2}S_{x_1}^{-1}S_{x_2},\\
y(S_{x_3})&=&S_{x_3x_4^{-1}x_3^{-2}x_2^{-2}x_1^{-1}x_2x_3}\overset{(1),(2)}{=}S_{x_3}S_{x_4}^{-1}S_{x_3}^{-2}S_{x_2}^{-2}S_{x_1}^{-1}S_{x_2}S_{x_3},\\
y^{-1}(S_{x_3})&=&S_{x_2^{-1}x_1x_2^2x_3^2x_4}\overset{(1),(2)}{=}S_{x_2}^{-1}S_{x_1}S_{x_2}^2S_{x_3}^2S_{x_4},\\
y(S_{x_4})&=&S_{x_3^{-1}x_2^{-1}x_1x_2^2x_3^2x_4^2}\overset{(1),(2)}{=}S_{x_3}^{-1}S_{x_2}^{-1}S_{x_1}S_{x_2}^2S_{x_3}^2S_{x_4}^2,\\
y^{-1}(S_{x_4})&=&S_{x_4^{-1}x_3^{-2}x_2^{-2}x_1^{-1}x_2x_3x_4}\overset{(1),(2)}{=}S_{x_4}^{-1}S_{x_3}^{-2}S_{x_2}^{-2}S_{x_1}^{-1}S_{x_2}S_{x_3}S_{x_4}
\end{eqnarray*}
and $y^{\pm1}(x)=x$ for any other $x\in{X^\prime}$.
Let $y=Y_{\mu,a_1}$.
We calculate
\begin{eqnarray*}
y(S_{x_1})&=&S_{x_1^2x_2x_1^{-1}x_2^{-1}x_1^{-2}}\overset{(1),(2)}{=}S_{x_1}^2S_{x_2}S_{x_1}^{-1}S_{x_2}^{-1}S_{x_1}^{-2},\\
y^{-1}(S_{x_1})&=&S_{x_2^{-1}x_1^{-1}x_2}\overset{(1),(2)}{=}S_{x_2}^{-1}S_{x_1}^{-1}S_{x_2},\\
y(S_{x_2})&=&S_{x_1^2x_2}\overset{(2)}{=}S_{x_1}^2S_{x_2},~
y^{-1}(S_{x_2})=S_{x_2^{-1}x_1^2x_2^2}\overset{(1), (2)}{=}S_{x_2}^{-1}S_{x_1}^2S_{x_2}^2
\end{eqnarray*}
and $y^{\pm1}(x)=x$ for any other $x\in{X^\prime}$.
Let $y=B_{r_k}$ for $1\leq{k}\leq{n}$.
We calculate
\begin{eqnarray*}
y(S_{x_g})&=&S_{x_g^2y_kx_g^{-1}}\overset{(1),(2)}{=}S_{x_g}^2S_{y_k}S_{x_g}^{-1},~
y^{-1}(S_{x_g})=S_{x_gy_k}\overset{(2)}{=}S_{x_g}S_{y_k},\\
y(S_{y_k})&=&S_{x_gy_kx_g^{-1}}\overset{(1),(2)}{=}S_{x_g}S_{y_k}S_{x_g}^{-1},\\
y^{-1}(S_{y_k})&=&S_{y_k^{-1}x_g^{-1}y_kx_gy_k}\overset{(1),(2)}{=}S_{y_k}^{-1}S_{x_g}^{-1}S_{y_k}S_{x_g}S_{y_k},\\
y(S_{y_l})&=&S_{x_gy_k^{-1}x_g^{-1}y_k^{-1}y_ly_kx_gy_kx_g^{-1}}\overset{(1),(2)}{=}S_{x_g}S_{y_k}^{-1}S_{x_g}^{-1}S_{y_k}^{-1}S_{y_l}S_{y_k}S_{x_g}S_{y_k}S_{x_g}^{-1},\\
y^{-1}(S_{y_l})&=&S_{y_k^{-1}x_g^{-1}y_k^{-1}x_gy_lx_g^{-1}y_kx_gy_k}\overset{(1),(2)}{=}S_{y_k}^{-1}S_{x_g}^{-1}S_{y_k}^{-1}S_{x_g}S_{y_l}S_{x_g}^{-1}S_{y_k}S_{x_g}S_{y_k}
\end{eqnarray*}
for $l<k$, and $y^{\pm1}(x)=x$ for any other $x\in{X^\prime}$.
Let $y=B_{r_0}$.
We calculate
\begin{eqnarray*}
y^{\pm1}(S_{x_j})&=&S_{x_g^{\mp1}x_jx_g^{\pm1}}\overset{(1), (2)}{=}S_{x_g}^{\mp1}S_{x_j}S_{x_g}^{\pm1},\\
y^{\pm1}(S_{y_l})&=&S_{x_g^{\mp1}y_lx_g^{\pm1}}\overset{(1), (2)}{=}S_{x_g}^{\mp1}S_{y_l}S_{x_g}^{\pm1}
\end{eqnarray*}
for $1\leq{j}\leq{g}$ and $1\leq{l}\leq{n-1}$.
Let $y=t_{s_{kl}}$ for $1\leq{k<l}\leq{n}$.
We calculate
\begin{eqnarray*}
y(S_{y_{k}})&=&S_{y_ky_ly_ky_l^{-1}y_k^{-1}}\overset{(1),(2)}{=}S_{y_k}S_{y_l}S_{y_k}S_{y_l}^{-1}S_{y_k}^{-1},\\
y^{-1}(S_{y_{k}})&=&S_{y_l^{-1}y_ky_l}\overset{(1),(2)}{=}S_{y_l}^{-1}S_{y_k}S_{y_l},\\
y(S_{y_{l}})&=&S_{y_ky_ly_k^{-1}}\overset{(1),(2)}{=}S_{y_k}S_{y_l}S_{y_k}^{-1},\\
y^{-1}(S_{y_{l}})&=&S_{y_l^{-1}y_k^{-1}y_ly_ky_l}\overset{(1),(2)}{=}S_{y_l}^{-1}S_{y_k}^{-1}S_{y_l}S_{y_k}S_{y_l},\\
y(S_{y_{m}})&=&S_{[y_k,y_l]y_m[y_k,y_l]^{-1}}\overset{(1),(2)}{=}[S_{y_k},S_{y_l}]S_{y_m}[S_{y_k},S{y_l}]^{-1},\\
y^{-1}(S_{y_{m}})&=&S_{[y_l^{-1},y_k^{-1}]y_m[y_l^{-1},y_k^{-1}]^{-1}}\overset{(1),(2)}{=}[S_{y_l}^{-1},S_{y_k}^{-1}]S_{y_m}[S_{y_l}^{-1},S_{y_k}^{-1}]^{-1}
\end{eqnarray*}
for $k<m<l$, and $y^{\pm1}(x)=x$ for any other $x\in{X^\prime}$.
Hence we have that for any $x\in{X^\prime}$ and $y\in{Y}$, $y^{\pm1}(x)$ is in the subgroup of $\pi$ generated by $X^\prime$, by Lemma~\ref{y_n}. \end{proof}
\begin{proof}[Proof of Theorem~\ref{main-1} of the case where $S$ is non-orientable] By Lemma~\ref{main-lem} and Proposition~\ref{2}, it follows that $\pi$ is generated by $X^\prime$. There is a natural map $\pi\to\pi_1(N_{g,n},\ast)$. The relations~(1) and (2) of $\pi$ are satisfied in $\pi_1(N_{g,n},\ast)$ clearly. Hence the map is a homomorphism. In addition, the relation $S_{x_1}^2\cdots{}S_{x_g}^2=1$ is obtained from the relation~(2) of $\pi$ for $n=0$. Therefore the map is an isomorphism for any $n\geq0$. Thus we complete the proof. \end{proof}
\section{Proof of Theorem~\ref{main-2}}\label{pi^+}
Let $x_{ij}=x_ix_j$ and $z_k=x_gy_kx_g^{-1}$ for $1\leq{i,j}\leq{g}$ and $1\leq{k}\leq{n-1}$, where $x_1,\dots,x_g$ and $y_1,\dots,y_{n-1}$ are simple loops of $N_{g,n}$ as shown in Figure~\ref{gen-pi_1-non-ori-surf}. We first consider a presentation for $\pi_1^+(N_{g,n},\ast)$ as follows.
\begin{lem} $\pi_1^+(N_{g,n},\ast)$ is the free group freely generated by $x_{12},\dots,x_{g-1\:g}$, $x_{11},\dots,x_{gg}$, $y_1,\dots,y_{n-1}$ and $z_1,\dots,z_{n-1}$ for $n\geq1$, and the group generated by $x_{12},\dots,x_{g-1\:g}$ and $x_{11},\dots,x_{gg}$ which has two relations $x_{11}\cdots{}x_{gg}=1$ and $x_{gg}x_{g-1\:g}^{-1}x_{g-1\:g-1}x_{g-2\:g-1}^{-1}\cdots{}x_{22}x_{12}^{-1}x_{11}x_{12}\cdots{}x_{g-1\:g}=1$ for $n=0$. \end{lem}
\begin{proof} It is known that $\pi_1^+(N_{g,n},\ast)$ is an index two subgroup of $\pi_1(N_{g,n},\ast)$ (see \cite{KO1}). Hence we can obtain a presentation of $\pi_1^+(N_{g,n},\ast)$ by the Reidemeister Schreier method (for details, for instance see \cite{J}). Note that $\pi_1(N_{g,n},\ast)$ is generated by $x_1,\dots,x_g$ and $y_1,\dots,y_{n-1}$. We chose $\{1,x_g\}$ as a Schreier transversal for $\pi_1^+(N_{g,n},\ast)$ in $\pi_1(N_{g,n},\ast)$. Then it follows that $\pi_1^+(N_{g,n},\ast)$ is generated by $x_1x_g^{-1},\dots,x_{g-1}x_g^{-1}$, $x_gx_1,\dots,x_gx_g$, $y_1,\dots,y_{n-1}$ and $z_1,\dots,z_{n-1}$ (see \cite{Kob}). In addition, we see \begin{eqnarray*} 1(x_1^2\cdots{}x_g^2)1^{-1} &=& x_1x_g^{-1}\cdot{}x_gx_1\cdot{}x_2x_g^{-1}\cdot{}x_gx_2\cdots{}x_{g-1}x_g^{-1}\cdot{}x_gx_{g-1}\cdot{}x_gx_g,\\ x_g(x_1^2\cdots{}x_g^2)x_g^{-1} &=& x_gx_1\cdot{}x_1x_g^{-1}\cdot{}x_gx_2\cdot{}x_2x_g^{-1}\cdots{}x_gx_{g-1}\cdot{}x_{g-1}x_g^{-1}\cdot{}x_gx_g. \end{eqnarray*} Hence when $n=0$, we have two relations \begin{eqnarray*} &&x_1x_g^{-1}\cdot{}x_gx_1\cdot{}x_2x_g^{-1}\cdot{}x_gx_2\cdots{}x_{g-1}x_g^{-1}\cdot{}x_gx_{g-1}\cdot{}x_gx_g=1,\\ &&x_gx_1\cdot{}x_1x_g^{-1}\cdot{}x_gx_2\cdot{}x_2x_g^{-1}\cdots{}x_gx_{g-1}\cdot{}x_{g-1}x_g^{-1}\cdot{}x_gx_g=1. \end{eqnarray*}
Let $G$ be the group which has the presentation of the lemma. We next show that $G$ is isomorphic to $\pi_1^+(N_{g,n},\ast)$. Let $\varphi:G\to\pi_1^+(N_{g,n},\ast)$ and $\psi:\pi_1^+(N_{g,n},\ast)\to{}G$ be homomorphisms defined as \begin{eqnarray*} && \varphi(x_{i\:i+1})=x_ix_g^{-1}\cdot{}x_gx_{i+1},~ \varphi(x_{jj})=x_jx_g^{-1}\cdot{}x_gx_j,~ \varphi(y_k)=y_k,~ \varphi(z_k)=z_k,\\ && \psi(x_ix_g^{-1})=x_{i\:i+1}x_{i+1\:i+1}^{-1}x_{i+1\:i+2}x_{i+2\:i+2}^{-1}\cdots{}x_{g-1\:g}x_{gg}^{-1},\\ && \psi(x_gx_j)=x_{gg}x_{g-1\:g}^{-1}x_{g-1\:g-1}x_{g-2\:g-1}^{-1}\cdots{}x_{j+1\:j+1}x_{j\:j+1}^{-1}x_{jj},\\ && \psi(y_k)=y_k,~ \psi(z_k)=z_k \end{eqnarray*} for $1\leq{i}\leq{g-1}$, $1\leq{j}\leq{g}$ and $1\leq{k}\leq{n-1}$. We calculate \begin{eqnarray*} &&\varphi(x_{11}\cdots{}x_{g-1\:g-1}x_{gg})=x_1x_g^{-1}\cdot{}x_gx_1\cdots{}x_{g-1}x_g^{-1}\cdot{}x_gx_{g-1}\cdot{}x_gx_g,\\ && \varphi(x_{gg}x_{g-1\:g}^{-1}x_{g-1\:g-1}x_{g-2\:g-1}^{-1}\cdots{}x_{22}x_{12}^{-1}x_{11}x_{12}\cdots{}x_{g-1\:g})\\ &=& x_gx_g(x_{g-1}x_g^{-1}\cdot{}x_gx_g)^{-1}(x_{g-1}x_g^{-1}\cdot{}x_gx_{g-1})(x_{g-2}x_g^{-1}\cdot{}x_gx_{g-1})^{-1}\\ && \cdots(x_2x_g^{-1}\cdot{}x_gx_2)(x_1x_g^{-1}\cdot{}x_gx_2)^{-1}(x_1x_g^{-1}\cdot{}x_gx_1)(x_1x_g^{-1}\cdot{}x_gx_2)\\ && \cdots(x_{g-1}x_g^{-1}\cdot{}x_gx_g)\\ &=& x_gx_1\cdot{}x_1x_g^{-1}\cdot{}x_gx_2\cdot{}x_2x_g^{-1}\cdots{}x_gx_{g-1}\cdot{}x_{g-1}x_g^{-1}\cdot{}x_gx_g,\\ && \psi(x_1x_g^{-1}\cdot{}x_gx_1\cdot{}x_2x_g^{-1}\cdot{}x_gx_2\cdots{}x_{g-1}x_g^{-1}\cdot{}x_gx_{g-1}\cdot{}x_gx_g)\\ &=& x_{11}x_{22}\cdots{}x_{g-1\:g-1}x_{gg},\\ && \psi(x_gx_1\cdot{}x_1x_g^{-1}\cdot{}x_gx_2\cdot{}x_2x_g^{-1}\cdots{}x_gx_{g-1}\cdot{}x_{g-1}x_g^{-1}\cdot{}x_gx_g)\\ &=& x_{gg}x_{g-1\:g}^{-1}x_{g-1\:g-1}x_{g-2\:g-1}^{-1}\cdots{}x_{22}x_{12}^{-1}x_{11}x_{12}\cdots{}x_{g-1\:g}. \end{eqnarray*} Hence $\varphi$ and $\psi$ are well defined even if $n=0$. In addition, we have \begin{eqnarray*} \psi\varphi(x_{i\:i+1}) &=& \psi(x_ix_g^{-1}\cdot{}x_gx_{i+1})\\ &=& x_{i\:i+1}x_{i+1\:i+1}^{-1}\cdots{}x_{g-1\:g}x_{gg}^{-1}\\ &&\cdot{}x_{gg}x_{g-1\:g}^{-1}\cdots{}x_{i+2\:i+2}x_{i+1\:i+2}^{-1}x_{i+1\:i+1} =x_{i\:i+1},\\ \psi\varphi(x_{jj}) &=& \psi(x_jx_g^{-1}\cdot{}x_gx_{j})\\ &=& x_{j\:j+1}x_{j+1\:j+1}^{-1}\cdots{}x_{g-1\:g}x_{gg}^{-1}\cdot{}x_{gg}x_{g-1\:g}^{-1}\cdots{}x_{j+1\:j+1}x_{j\:j+1}^{-1}x_{jj}\\ &=& x_{jj},\\ \psi\varphi(y_k) &=& \psi(y_k)=y_k,~ \psi\varphi(z_k)=\psi(z_k)=z_k,\\ \varphi\psi(x_ix_g^{-1}) &=& \varphi(x_{i\:i+1}x_{i+1\:i+1}^{-1}\cdots{}x_{g-1\:g}x_{gg}^{-1})\\ &=& (x_ix_g^{-1}\cdot{}x_gx_{i+1})(x_{i+1}x_g^{-1}\cdot{}x_gx_{i+1})^{-1}\\ &&\cdots(x_{g-1}x_g^{-1}\cdot{}x_gx_g)(x_gx_g)^{-1} =x_ix_g^{-1},\\ \varphi\psi(x_gx_j) &=& \varphi(x_{gg}x_{g-1\:g}^{-1}\cdots{}x_{j+1\:j+1}x_{j\:j+1}^{-1}x_{jj})\\ &=& (x_gx_g)(x_{g-1}x_g^{-1}\cdot{}x_gx_g)^{-1}\cdots{}(x_{j+1}x_g^{-1}\cdot{}x_gx_{j+1})\\ &&\cdot(x_jx_g^{-1}\cdot{}x_gx_{j+1})^{-1}(x_jx_g^{-1}\cdot{}x_gx_j) =x_gx_j,\\ \varphi\psi(y_k) &=& \varphi(y_k)=y_k,~ \varphi\psi(z_k)=\varphi(z_k)=z_k. \end{eqnarray*} Therefore $\varphi$ and $\psi$ are the isomorphisms. Thus we finish the proof. \end{proof}
Let $X$ be a set consisting of $S_\alpha$, where $\alpha$ is a non-separating two-sided simple loop whose complement is non-orientable, or a separating simple loop which bounds the $m$-th boundary component for $1\leq{m}\leq{n-1}$ or one crosscap whose complement is non-orientable, and let $X^\prime$ be the following subset of $X$: $$X^\prime=\{S_{x_{12}},\dots,S_{x_{g-1\:g}},S_{x_{11}},\dots,S_{x_{gg}},S_{y_1},\dots,S_{y_{n-1}},S_{z_1},\dots,S_{z_{n-1}}\}.$$ Let $Y$ be the generating set for $\mathcal{PM}(N_{g,n+1})$ given in Theorem~\ref{gen-PMF}. In the actions on $\pi_1^+(N_{g,n},\ast)$ and $\pi^+$ by $\mathcal{PM}(N_{g,n+1})$, we regard the $(n+1)$-st boundary component of $N_{g,n+1}$ as $\ast$. Recall that the action $f(S_\alpha)$ of $f\in\mathcal{PM}(N_{g,n+1})$ on $S_\alpha\in\pi^+$ was defined in Subsection~\ref{non-ori}. We prove the following proposition.
\begin{prop}\label{3} \begin{enumerate} \item $X$ generates $\pi^+$. \item $\mathcal{PM}(N_{g,n+1})(X^\prime)=X$. \item For any $x\in{X^\prime}$ and $y\in{Y}$, $y^{\pm1}(x)$ is in the subgroup of $\pi^+$ generated by $X^\prime$. \end{enumerate} \end{prop}
In order to prove the proposition, we show the following lemma.
\begin{lem}\label{ij} $S_{x_{ij}}$, $S_{x_{ji}}$, $S_{y_n}$ and $S_{z_n}$ are in the subgroup of $\pi^+$ generated by $X^\prime$ for $1\leq{i<j}\leq{g}$, where $y_n$ is a simple loop of $N_{g,n}$ as shown in Figure~\ref{gen-pi_1-non-ori-surf} and $z_n=x_gy_nx_g^{-1}$. \end{lem}
\begin{proof} For $1\leq{i<j}\leq{g}$, if $j-i=1$, then $x_{ij}$ is in $X^\prime$ clearly. If $j-i\geq2$, we calculate \begin{eqnarray*} S_{x_{ij}}&=&S_{x_{i\:j-1}x_{j-1\:j-1}^{-1}x_{j-1\:j}} \overset{(2)}{=}S_{x_{i\:j-1}x_{j-1\:j}}S_{x_{j-1\:j}^{-1}x_{j-1\:j-1}^{-1}x_{j-1\:j}}\\ &\overset{(1),(2),(3)}{=}&S_{x_{i\:j-1}}S_{x_{j-1\:j}}S_{x_{j-1\:j}}^{-1}S_{x_{j-1\:j-1}}^{-1}S_{x_{j-1\:j}} =S_{x_{i\:j-1}}S_{x_{j-1\:j-1}}^{-1}S_{x_{j-1\:j}}. \end{eqnarray*} By induction on $j-i$, it follows that $S_{x_{ij}}$ is in the subgroup of $\pi^+$ generated by $X^\prime$. In addition, we calculate \begin{eqnarray*} S_{x_{ji}}&=&S_{x_{jj}x_{ij}^{-1}x_{ii}} \overset{(2)}{=}S_{x_{ij}^{-1}}S_{x_{ij}x_{jj}x_{ij}^{-1}x_{ii}} \overset{(1),(2)}{=}S_{x_{ij}}^{-1}S_{x_{ij}x_{jj}x_{ij}^{-1}}S_{x_{ii}}\\ &\overset{(3)}{=}&S_{x_{ij}}^{-1}S_{x_{ij}}S_{x_{jj}}S_{x_{ij}}^{-1}S_{x_{ii}} =S_{x_{jj}}S_{x_{ij}}^{-1}S_{x_{ii}}. \end{eqnarray*} Hence $S_{x_{ji}}$ is also in the subgroup of $\pi^+$ generated by $X^\prime$. Moreover, by the relations (1) and (2) of $\pi^+$, we calculate \begin{eqnarray*} S_{y_n}&=&S_{(x_{11}\cdots{}x_{gg}y_1\cdots{}y_{n-1})^{-1}} =(S_{x_{11}}\cdots{}S_{x_{gg}}S_{y_1}\cdots{}S_{y_{n-1}})^{-1},\\ S_{z_n}&=&S_{(x_{g1}x_{12}x_{23}\cdots{}x_{g-1\:g}z_1\cdots{}z_{n-1})^{-1}}\\ &=&(S_{x_{g1}}S_{x_{12}}S_{x_{23}}\cdots{}S_{x_{g-1\:g}}S_{z_1}\cdots{}S_{z_{n-1}})^{-1}. \end{eqnarray*} Therefore $S_{y_n}$ and $S_{z_n}$ are also in the subgroup of $\pi^+$ generated by $X^\prime$.
Thus we get the claim. \end{proof}
\begin{proof}[Proof of Proposition~\ref{3}]
(1) For any generator $S_\alpha$ of $\pi^+$, the complement of $\alpha$ is homeomorphic to either
\begin{enumerate}
\item[(b)] $N_{g-2,n+2}$,
\item[(c)] $\Sigma_{h,n+2}$ only if $g=2h+2$,
\item[(d)] $N_{h,m+1}\sqcup{N_{g-h,n-m+1}}$ for $1\leq{h}\leq{g-1}$ and $0\leq{m}\leq{n}$ or
\item[(e)] $\Sigma_{h,m+1}\sqcup{N_{g-2h,n-m+1}}$ for $\displaystyle0\leq{h}\leq\frac{g-1}{2}$ and $0\leq{m}\leq{n}$.
\end{enumerate}
(see \cite{S1}).
Therefore, there is $f\in\mathcal{PM}(N_{g,n+1})$ such that $\alpha=f_\sharp(\beta)$, where $\beta$ is either one of the simple loops as in Figure~\ref{normal-position-loop-non-ori-surf}~(b), (c), (d) and (e).
For the case (b), we have $S_\alpha=S_{f_\sharp(x_{12})}$.
For the case (c), by the relation (2) of $\pi^+$, we have $S_\alpha=S_{f_\sharp(x_{12})}\cdots{}S_{f_\sharp(x_{g-1\:g})}$.
For the case (d), by the relation (2) of $\pi^+$, we have $S_\alpha=S_{f_\sharp(x_{11})}\cdots{}S_{f_\sharp(x_{hh})}S_{f_\sharp(y_{k_1})}\cdots{}S_{f_\sharp(y_{k_m})}$ for some $1\leq{k_1<\cdots<k_m}\leq{n}$.
For the case (e), by the relation (1) and (2) of $\pi^+$, we have
\begin{eqnarray*}
S_\alpha
&=&
S_{f_\sharp(x_{12})}S_{f_\sharp(x_{34})}\cdots{}S_{f_\sharp(x_{2h-1\:2h})}\cdot{}S_{f_\sharp(x_{2h\:2h+1})}^{-1}S_{f_\sharp(x_{2h-1\:2h})}^{-1}\cdots{}S_{f_\sharp(x_{12})}^{-1}\\
&&
~\cdot~S_{f_\sharp(x_{23})}S_{f_\sharp(x_{45})}\cdots{}S_{f_\sharp(x_{2h\:2h+1})}S_{f_\sharp(y_{k_1})}\cdots{}S_{f_\sharp(y_{k_m})}
\end{eqnarray*}
for some $1\leq{k_1<\cdots<k_m}\leq{n}$ if $h\neq0$.
If $h=0$, by the relation (2) of $\pi^+$, we have $S_\alpha=S_{f_\sharp(y_{k_1})}\cdots{}S_{f_\sharp(y_{k_m})}$ for some $1\leq{k_1<\cdots<k_m}\leq{n}$.
Since each symbol of the right hand sides is in $X$, we conclude that $X$ generates $\pi$.
(2) For any $S_\alpha\in{X}$, if $\alpha$ is a non-separating two-sided simple loop whose complement is non-orientable, there is $f\in\mathcal{PM}(N_{g,n+1})$ such that $f_\sharp(x_{12})=\alpha$, and hence $f(S_{x_{12}})=S_\alpha$.
If $\alpha$ is a separating simple loop which bounds the $m$-th boundary component for $1\leq{m}\leq{n-1}$, there is $f\in\mathcal{PM}(N_{g,n+1})$ such that $f_\sharp(y_m)=\alpha$, and hence $f(S_{y_m})=S_\alpha$.
If $\alpha$ is a separating simple loop which bounds one crosscap whose complement is non-orientable, there is $f\in\mathcal{PM}(N_{g,n+1})$ such that $f_\sharp(x_{11})=\alpha$, and hence $f(S_{x_{11}})=S_\alpha$.
Therefore we obtain the claim.
(3) In this proof, we omit details of calculations.
In calculations, we use the relation (3) as little as possible (see Remark~\ref{reduced-rel}).
Let $y=t_{a_i}$ for $1\leq{i}\leq{g}$.
We calculate
\begin{eqnarray*}
y(S_{x_{i-1\:i}})&=&S_{x_{i-1\:i}x_{i\:i+1}^{-1}}
\overset{(1),(2)}{=}S_{x_{i-1\:i}}S_{x_{i\:i+1}}^{-1},
\end{eqnarray*}
\begin{eqnarray*}
y^{-1}(S_{x_{i-1\:i}})&=&S_{x_{i-1\:i}x_{i\:i+1}}
\overset{(2)}{=}S_{x_{i-1\:i}}S_{x_{i\:i+1}},
\end{eqnarray*}
\begin{eqnarray*}
y(S_{x_{i+1\:i+2}})&=&S_{x_{i\:i+1}x_{i+1\:i+2}}
\overset{(2)}{=}S_{x_{i\:i+1}}S_{x_{i+1\:i+2}},
\end{eqnarray*}
\begin{eqnarray*}
y^{-1}(S_{x_{i+1\:i+2}})&=&S_{x_{i\:i+1}^{-1}x_{i+1\:i+2}}
\overset{(1),(2)}{=}S_{x_{i\:i+1}}^{-1}S_{x_{i+1\:i+2}},
\end{eqnarray*}
\begin{eqnarray*}
y(S_{x_{ii}})&=&S_{x_{i\:i+1}x_{i+1\:i+1}^{-1}x_{i\:i+1}^{-1}}
\overset{(1),(3)}{=}S_{x_{i\:i+1}}S_{x_{i+1\:i+1}}^{-1}S_{x_{i\:i+1}}^{-1},
\end{eqnarray*}
\begin{eqnarray*}
y^{-1}(S_{x_{ii}})&=&S_{x_{ii}x_{i+1\:i+1}x_{i\:i+1}^{-1}x_{ii}x_{i\:i+1}}
\overset{(2)}{=}S_{x_{ii}x_{i+1\:i+1}}S_{x_{i\:i+1}^{-1}x_{ii}x_{i\:i+1}}\\
&\overset{(1),(2),(3)}{=}&S_{x_{ii}}S_{x_{i+1\:i+1}}S_{x_{i\:i+1}}^{-1}S_{x_{ii}}S_{x_{i\:i+1}},
\end{eqnarray*}
\begin{eqnarray*}
y(S_{x_{i+1\:i+1}})&=&S_{x_{i\:i+1}x_{i+1\:i+1}x_{i\:i+1}^{-1}x_{ii}x_{i+1\:i+1}}\\
&\overset{(2)}{=}&S_{x_{i\:i+1}x_{i+1\:i+1}x_{i\:i+1}^{-1}}S_{x_{ii}x_{i+1\:i+1}}\\
&\overset{(1),(2),(3)}{=}&S_{x_{i\:i+1}}S_{x_{i+1\:i+1}}S_{x_{i\:i+1}}^{-1}S_{x_{ii}}S_{x_{i+1\:i+1}},
\end{eqnarray*}
\begin{eqnarray*}
y^{-1}(S_{x_{i+1\:i+1}})&=&S_{x_{i\:i+1}^{-1}x_{ii}^{-1}x_{i\:i+1}}
\overset{(1),(3)}{=}S_{x_{i\:i+1}}^{-1}S_{x_{ii}}^{-1}S_{x_{i\:i+1}},
\end{eqnarray*}
\begin{eqnarray*}
t_{a_{g-1}}^{\pm1}(S_{z_l})&=&S_{x_{g-1\:g}^{\pm1}z_lx_{g-1\:g}^{\mp1}}
\overset{(1),(2)}{=}S_{x_{g-1\:g}}^{\pm1}S_{z_l}S_{x_{g-1\:g}}^{\mp1}
\end{eqnarray*}
for $1\leq{l}\leq{n-1}$, and $y^{\pm1}(x)=x$ for any other $x\in{X^\prime}$.
Let $y=t_{b}$.
We calculate
\begin{eqnarray*}
y(S_{x_{45}})&=&S_{x_{23}^{-1}x_{12}x_{23}x_{34}x_{45}}
\overset{(1),(2)}{=}S_{x_{23}}^{-1}S_{x_{12}}S_{x_{23}}S_{x_{34}}S_{x_{45}},
\end{eqnarray*}
\begin{eqnarray*}
y^{-1}(S_{x_{45}})&=&S_{x_{34}^{-1}x_{23}^{-1}x_{12}^{-1}x_{23}x_{45}}
\overset{(1),(2)}{=}S_{x_{34}}^{-1}S_{x_{23}}^{-1}S_{x_{12}}^{-1}S_{x_{23}}S_{x_{45}},
\end{eqnarray*}
\begin{eqnarray*}
y(S_{x_{11}})&=&S_{x_{12}x_{34}x_{44}^{-1}x_{33}^{-1}x_{12}^{-1}x_{13}x_{34}^{-1}x_{23}^{-1}x_{12}^{-1}}
\overset{(2)}{=}S_{x_{12}x_{34}x_{44}^{-1}x_{33}^{-1}x_{12}^{-1}}S_{x_{13}x_{34}^{-1}x_{23}^{-1}x_{12}^{-1}}\\
&\overset{(1),(2)}{=}&S_{x_{12}}S_{x_{34}}S_{x_{44}}^{-1}S_{x_{33}}^{-1}S_{x_{12}}^{-1}S_{x_{13}}S_{x_{34}}^{-1}S_{x_{23}}^{-1}S_{x_{12}}^{-1},
\end{eqnarray*}
\begin{eqnarray*}
y^{-1}(S_{x_{11}})&=&S_{x_{11}x_{22}x_{33}x_{44}x_{34}^{-1}x_{12}^{-1}x_{11}x_{12}x_{23}x_{34}x_{23}^{-1}}\\
&\overset{(2)}{=}&S_{x_{11}x_{22}x_{33}x_{44}}S_{x_{34}^{-1}x_{12}^{-1}x_{11}x_{12}x_{23}x_{34}x_{23}^{-1}}\\
&\overset{(1),(2)}{=}&S_{x_{11}}S_{x_{22}}S_{x_{33}}S_{x_{44}}S_{x_{34}}^{-1}S_{x_{12}^{-1}x_{11}x_{12}x_{23}x_{34}x_{23}^{-1}}\\
&\overset{(2)}{=}&S_{x_{11}}S_{x_{22}}S_{x_{33}}S_{x_{44}}S_{x_{34}}^{-1}S_{x_{12}^{-1}x_{11}x_{12}}S_{x_{23}x_{34}x_{23}^{-1}}\\
&\overset{(1),(2),(3)}{=}&S_{x_{11}}S_{x_{22}}S_{x_{33}}S_{x_{44}}S_{x_{34}}^{-1}S_{x_{12}}^{-1}S_{x_{11}}S_{x_{12}}S_{x_{23}}S_{x_{34}}S_{x_{23}}^{-1},
\end{eqnarray*}
\begin{eqnarray*}
y(S_{x_{22}})&=&S_{x_{12}x_{23}x_{34}x_{13}^{-1}x_{11}x_{22}x_{33}x_{44}x_{34}^{-1}}
\overset{(2)}{=}S_{x_{12}x_{23}}S_{x_{34}x_{13}^{-1}x_{11}x_{22}x_{33}x_{44}x_{34}^{-1}}\\
&\overset{(1),(2)}{=}&S_{x_{12}}S_{x_{23}}S_{x_{34}}S_{x_{13}}^{-1}S_{x_{11}x_{22}x_{33}x_{44}x_{34}^{-1}}\\
&\overset{(2)}{=}&S_{x_{12}}S_{x_{23}}S_{x_{34}}S_{x_{13}}^{-1}S_{x_{11}}S_{x_{22}}S_{x_{33}x_{44}x_{34}^{-1}}\\
&\overset{(1),(2)}{=}&S_{x_{12}}S_{x_{23}}S_{x_{34}}S_{x_{13}}^{-1}S_{x_{11}}S_{x_{22}}S_{x_{33}x_{44}}S_{x_{34}}^{-1}\\
&\overset{(2)}{=}&S_{x_{12}}S_{x_{23}}S_{x_{34}}S_{x_{13}}^{-1}S_{x_{11}}S_{x_{22}}S_{x_{33}}S_{x_{44}}S_{x_{34}}^{-1},
\end{eqnarray*}
\begin{eqnarray*}
y^{-1}(S_{x_{22}})&=&S_{x_{23}x_{34}^{-1}x_{23}^{-1}x_{12}^{-1}x_{22}x_{34}x_{44}^{-1}x_{33}^{-1}x_{22}^{-1}x_{11}^{-1}x_{12}}\\
&\overset{(2)}{=}&S_{x_{23}x_{42}}S_{x_{42}^{-1}x_{34}^{-1}x_{23}^{-1}x_{12}^{-1}x_{22}x_{34}x_{44}^{-1}x_{33}^{-1}x_{22}^{-1}x_{11}^{-1}x_{12}}\\
&\overset{(2)}{=}&S_{x_{23}}S_{x_{42}}S_{x_{42}^{-1}x_{34}^{-1}x_{23}^{-1}x_{12}^{-1}x_{22}}S_{x_{34}x_{44}^{-1}x_{33}^{-1}x_{22}^{-1}x_{11}^{-1}x_{12}}\\
&\overset{(1),(2)}{=}&S_{x_{23}}S_{x_{42}}S_{x_{42}}^{-1}S_{x_{34}}^{-1}S_{x_{23}^{-1}x_{12}^{-1}x_{22}}S_{x_{34}}S_{x_{44}^{-1}x_{33}^{-1}x_{22}^{-1}x_{11}^{-1}x_{12}}\\
&\overset{(1),(2)}{=}&S_{x_{23}}S_{x_{34}}^{-1}S_{x_{23}^{-1}x_{12}^{-1}}S_{x_{22}}S_{x_{34}}S_{x_{44}}^{-1}S_{x_{33}}^{-1}S_{x_{22}^{-1}x_{11}^{-1}x_{12}}\\
&\overset{(1),(2)}{=}&S_{x_{23}}S_{x_{34}}^{-1}S_{x_{23}}^{-1}S_{x_{12}}^{-1}S_{x_{22}}S_{x_{34}}S_{x_{44}}^{-1}S_{x_{33}}^{-1}S_{x_{22}^{-1}x_{11}^{-1}}S_{x_{12}}\\
&\overset{(1),(2)}{=}&S_{x_{23}}S_{x_{34}}^{-1}S_{x_{23}}^{-1}S_{x_{12}}^{-1}S_{x_{22}}S_{x_{34}}S_{x_{44}}^{-1}S_{x_{33}}^{-1}S_{x_{22}}^{-1}S_{x_{11}}^{-1}S_{x_{12}},
\end{eqnarray*}
\begin{eqnarray*}
y(S_{x_{33}})&=&S_{x_{34}x_{44}^{-1}x_{33}^{-1}x_{22}^{-1}x_{11}^{-1}x_{13}x_{23}^{-1}x_{22}x_{33}x_{34}^{-1}x_{23}^{-1}x_{12}^{-1}x_{23}}\\
&\overset{(2)}{=}&S_{x_{34}x_{44}^{-1}x_{33}^{-1}x_{22}^{-1}x_{11}^{-1}x_{13}}S_{x_{23}^{-1}x_{22}x_{33}x_{34}^{-1}x_{23}^{-1}x_{12}^{-1}x_{23}}\\
&\overset{(2)}{=}&S_{x_{34}}S_{x_{44}^{-1}x_{33}^{-1}x_{22}^{-1}x_{11}^{-1}x_{13}}S_{x_{23}^{-1}x_{22}x_{33}}S_{x_{34}^{-1}x_{23}^{-1}x_{12}^{-1}x_{23}}\\
&\overset{(1),(2)}{=}&S_{x_{34}}S_{x_{44}}^{-1}S_{x_{33}^{-1}x_{22}^{-1}x_{11}^{-1}x_{13}}S_{x_{23}}^{-1}S_{x_{22}x_{33}}S_{x_{34}}^{-1}S_{x_{23}}^{-1}S_{x_{12}}^{-1}S_{x_{23}}\\
&\overset{(2)}{=}&S_{x_{34}}S_{x_{44}}^{-1}S_{x_{33}^{-1}x_{22}^{-1}x_{11}^{-1}}S_{x_{13}}S_{x_{23}}^{-1}S_{x_{22}}S_{x_{33}}S_{x_{34}}^{-1}S_{x_{23}}^{-1}S_{x_{12}}^{-1}S_{x_{23}}\\
&\overset{(1),(2)}{=}&S_{x_{34}}S_{x_{44}}^{-1}S_{x_{33}}^{-1}S_{x_{22}}^{-1}S_{x_{11}}^{-1}S_{x_{13}}S_{x_{23}}^{-1}S_{x_{22}}S_{x_{33}}S_{x_{34}}^{-1}S_{x_{23}}^{-1}S_{x_{12}}^{-1}S_{x_{23}},
\end{eqnarray*}
\begin{eqnarray*}
y^{-1}(S_{x_{33}})&=&S_{x_{12}^{-1}x_{11}x_{22}x_{33}x_{44}x_{24}^{-1}x_{12}x_{23}x_{34}}\\
&\overset{(1),(2)}{=}&S_{x_{12}}^{-1}S_{x_{11}x_{22}x_{33}x_{44}x_{24}^{-1}x_{12}x_{23}x_{34}}\\
&\overset{(2)}{=}&S_{x_{12}}^{-1}S_{x_{11}x_{22}x_{33}x_{44}x_{24}^{-1}}S_{x_{12}x_{23}x_{34}}\\
&\overset{(2)}{=}&S_{x_{12}}^{-1}S_{x_{11}}S_{x_{22}x_{33}x_{44}x_{24}^{-1}}S_{x_{12}}S_{x_{23}}S_{x_{34}}\\
&\overset{(1),(2)}{=}&S_{x_{12}}^{-1}S_{x_{11}}S_{x_{22}x_{33}x_{44}}S_{x_{24}}^{-1}S_{x_{12}}S_{x_{23}}S_{x_{34}}\\
&\overset{(2)}{=}&S_{x_{12}}^{-1}S_{x_{11}}S_{x_{22}}S_{x_{33}}S_{x_{44}}S_{x_{24}}^{-1}S_{x_{12}}S_{x_{23}}S_{x_{34}},
\end{eqnarray*}
\begin{eqnarray*}
y(S_{x_{44}})&=&S_{x_{23}^{-1}x_{12}x_{23}x_{34}x_{44}x_{34}^{-1}x_{12}^{-1}x_{11}x_{22}x_{33}x_{44}}\\
&\overset{(1),(2)}{=}&S_{x_{23}}^{-1}S_{x_{12}x_{23}x_{34}x_{44}x_{34}^{-1}x_{12}^{-1}x_{11}x_{22}x_{33}x_{44}}\\
&\overset{(2)}{=}&S_{x_{23}}^{-1}S_{x_{12}}S_{x_{23}}S_{x_{34}x_{44}x_{34}^{-1}x_{12}^{-1}x_{11}x_{22}x_{33}x_{44}}\\
&\overset{(2)}{=}&S_{x_{23}}^{-1}S_{x_{12}}S_{x_{23}}S_{x_{34}x_{44}x_{34}^{-1}}S_{x_{12}^{-1}x_{11}x_{22}x_{33}x_{44}}\\
&\overset{(1),(2),(3)}{=}&S_{x_{23}}^{-1}S_{x_{12}}S_{x_{23}}S_{x_{34}}S_{x_{44}}S_{x_{34}}^{-1}S_{x_{12}}^{-1}S_{x_{11}x_{22}x_{33}x_{44}}\\
&\overset{(2)}{=}&S_{x_{23}}^{-1}S_{x_{12}}S_{x_{23}}S_{x_{34}}S_{x_{44}}S_{x_{34}}^{-1}S_{x_{12}}^{-1}S_{x_{11}}S_{x_{22}}S_{x_{33}}S_{x_{44}},
\end{eqnarray*}
\begin{eqnarray*}
y^{-1}(S_{x_{44}})&=&S_{x_{34}^{-1}x_{23}^{-1}x_{12}^{-1}x_{24}x_{34}^{-1}x_{22}^{-1}x_{11}^{-1}x_{12}x_{34}}\\
&\overset{(2)}{=}&S_{x_{34}^{-1}x_{23}^{-1}x_{12}^{-1}x_{24}}S_{x_{34}^{-1}x_{22}^{-1}x_{11}^{-1}x_{12}x_{34}}\\
&\overset{(2)}{=}&S_{x_{34}^{-1}x_{23}^{-1}}S_{x_{12}^{-1}x_{24}}S_{x_{34}^{-1}x_{22}^{-1}x_{11}^{-1}}S_{x_{12}x_{34}}\\
&\overset{(1),(2)}{=}&S_{x_{34}}^{-1}S_{x_{23}}^{-1}S_{x_{12}}^{-1}S_{x_{24}}S_{x_{34}}^{-1}S_{x_{22}}^{-1}S_{x_{11}}^{-1}S_{x_{12}}S_{x_{34}},
\end{eqnarray*}
\begin{eqnarray*}
y(S_{z_k})&=&S_{x_{23}^{-1}x_{12}x_{23}x_{34}z_kx_{34}^{-1}x_{23}^{-1}x_{12}^{-1}x_{23}}
\overset{(2)}{=}S_{x_{23}^{-1}x_{12}x_{23}x_{34}z_k}S_{x_{34}^{-1}x_{23}^{-1}x_{12}^{-1}x_{23}}\\
&\overset{(1),(2)}{=}&S_{x_{23}^{-1}x_{12}x_{23}x_{34}}S_{z_k}S_{x_{34}}^{-1}S_{x_{23}}^{-1}S_{x_{12}}^{-1}S_{x_{23}}\\
&\overset{(1),(2)}{=}&S_{x_{23}}^{-1}S_{x_{12}}S_{x_{23}}S_{x_{34}}S_{z_k}S_{x_{34}}^{-1}S_{x_{23}}^{-1}S_{x_{12}}^{-1}S_{x_{23}},
\end{eqnarray*}
\begin{eqnarray*}
y^{-1}(S_{z_k})&=&S_{x_{34}^{-1}x_{23}^{-1}x_{12}^{-1}x_{23}z_kx_{23}^{-1}x_{12}x_{23}x_{34}}
\overset{(2)}{=}S_{x_{34}^{-1}x_{23}^{-1}x_{12}^{-1}}S_{x_{23}z_kx_{23}^{-1}x_{12}x_{23}x_{34}}\\
&\overset{(1),(2)}{=}&S_{x_{34}}^{-1}S_{x_{23}}^{-1}S_{x_{12}}^{-1}S_{x_{23}}S_{z_k}S_{x_{23}}^{-1}S_{x_{12}}S_{x_{23}}S_{x_{34}}
\end{eqnarray*}
for $1\leq{k}\leq{n-1}$ only if $g=4$, and $y^{\pm1}(x)=x$ for any other $x\in{X^\prime}$.
Let $y=Y_{\mu,a_1}$.
We calculate
\begin{eqnarray*}
y(S_{x_{12}})&=&S_{x_{11}x_{22}x_{12}^{-1}}
\overset{(1),(2)}{=}S_{x_{11}}S_{x_{22}}S_{x_{12}}^{-1},
\end{eqnarray*}
\begin{eqnarray*}
y^{-1}(S_{x_{12}})&=&S_{x_{12}^{-1}x_{11}x_{22}}
\overset{(1),(2)}{=}S_{x_{12}}^{-1}S_{x_{11}}S_{x_{22}},
\end{eqnarray*}
\begin{eqnarray*}
y(S_{x_{23}})&=&S_{x_{11}x_{23}}
\overset{(2)}{=}S_{x_{11}}S_{x_{23}},
\end{eqnarray*}
\begin{eqnarray*}
y^{-1}(S_{x_{23}})&=&S_{x_{12}^{-1}x_{11}x_{12}x_{23}}
\overset{(2)}{=}S_{x_{12}^{-1}x_{11}x_{12}}S_{x_{23}}
\overset{(1),(3)}{=}S_{x_{12}}^{-1}S_{x_{11}}S_{x_{12}}S_{x_{23}},
\end{eqnarray*}
\begin{eqnarray*}
y(S_{x_{11}})&=&S_{x_{11}x_{22}x_{12}^{-1}x_{11}^{-1}x_{12}x_{22}^{-1}x_{11}^{-1}}
\overset{(2)}{=}S_{x_{11}x_{22}}S_{x_{12}^{-1}x_{11}^{-1}x_{12}x_{22}^{-1}x_{11}^{-1}}\\
&\overset{(2)}{=}&S_{x_{11}}S_{x_{22}}S_{x_{12}^{-1}x_{11}^{-1}x_{12}}S_{x_{22}^{-1}x_{11}^{-1}}\\
&\overset{(1),(2),(3)}{=}&S_{x_{11}}S_{x_{22}}S_{x_{12}}^{-1}S_{x_{11}}^{-1}S_{x_{12}}S_{x_{22}}^{-1}S_{x_{11}}^{-1},
\end{eqnarray*}
\begin{eqnarray*}
y^{-1}(S_{x_{11}})&=&S_{x_{12}^{-1}x_{11}^{-1}x_{12}}
\overset{(1),(3)}{=}S_{x_{12}}^{-1}S_{x_{11}}^{-1}S_{x_{12}},
\end{eqnarray*}
\begin{eqnarray*}
y(S_{x_{22}})&=&S_{x_{11}x_{22}x_{12}^{-1}x_{11}x_{12}}
\overset{(2)}{=}S_{x_{11}x_{22}}S_{x_{12}^{-1}x_{11}x_{12}}\\
&\overset{(1),(2),(3)}{=}&S_{x_{11}}S_{x_{22}}S_{x_{12}}^{-1}S_{x_{11}}S_{x_{12}},
\end{eqnarray*}
\begin{eqnarray*}
y^{-1}(S_{x_{22}})&=&S_{x_{12}^{-1}x_{11}x_{12}x_{11}x_{22}}
\overset{(2)}{=}S_{x_{12}^{-1}x_{11}x_{12}}S_{x_{11}x_{22}}\\
&\overset{(1),(2),(3)}{=}&S_{x_{12}}^{-1}S_{x_{11}}S_{x_{12}}S_{x_{11}}S_{x_{22}},
\end{eqnarray*}
\begin{eqnarray*}
y(S_{z_k})&=&S_{x_{11}z_kx_{11}^{-1}}
\overset{(1),(2)}{=}S_{x_{11}}S_{z_k}S_{x_{11}}^{-1}
\end{eqnarray*}
\begin{eqnarray*}
y^{-1}(S_{z_k})&=&S_{x_{12}^{-1}x_{11}x_{12}z_kx_{12}^{-1}x_{11}^{-1}x_{12}}
\overset{(2)}{=}S_{x_{12}^{-1}x_{11}x_{12}z_k}S_{x_{12}^{-1}x_{11}^{-1}x_{12}}\\
&\overset{(1),(2),(3)}{=}&S_{x_{12}^{-1}x_{11}x_{12}}S_{z_k}S_{x_{12}}^{-1}S_{x_{11}}^{-1}S_{x_{12}}\\
&\overset{(1),(3)}{=}&S_{x_{12}}^{-1}S_{x_{11}}S_{x_{12}}S_{z_k}S_{x_{12}}^{-1}S_{x_{11}}^{-1}S_{x_{12}},
\end{eqnarray*}
for $1\leq{k}\leq{n-1}$ only if $g=2$, and $y^{\pm1}(x)=x$ for any other $x\in{X^\prime}$.
Let $y=B_{r_k}$ for $1\leq{k}\leq{n}$.
We calculate
\begin{eqnarray*}
y(S_{x_{g-1\:g}})&=&S_{x_{g-1\:g}z_k}
\overset{(2)}{=}S_{x_{g-1\:g}}S_{z_k},
\end{eqnarray*}
\begin{eqnarray*}
y^{-1}(S_{x_{g-1\:g}})&=&S_{x_{g-1\:g}y_k}
\overset{(2)}{=}S_{x_{g-1\:g}}S_{y_k},
\end{eqnarray*}
\begin{eqnarray*}
y(S_{x_{gg}})&=&S_{x_{gg}y_kz_k}
\overset{(2)}{=}S_{x_{gg}y_k}S_{z_k}
\overset{(2)}{=}S_{x_{gg}}S_{y_k}S_{z_k},
\end{eqnarray*}
\begin{eqnarray*}
y^{-1}(S_{x_{gg}})&=&S_{z_kx_{gg}y_k}
\overset{(2)}{=}S_{z_k}S_{x_{gg}}S_{y_k},
\end{eqnarray*}
\begin{eqnarray*}
y(S_{y_k})&=&S_{z_k^{-1}}
\overset{(1)}{=}S_{z_k}^{-1},
\end{eqnarray*}
\begin{eqnarray*}
y^{-1}(S_{y_k})&=&S_{y_k^{-1}x_{gg}^{-1}z_k^{-1}x_{gg}y_k}
\overset{(2)}{=}S_{y_k^{-1}x_{gg}^{-1}z_k^{-1}}S_{x_{gg}y_k}\\
&\overset{(1),(2)}{=}&S_{y_k^{-1}x_{gg}^{-1}}S_{z_k}^{-1}S_{x_{gg}}S_{y_k}
\overset{(1),(2)}{=}S_{y_k}^{-1}S_{x_{gg}}^{-1}S_{z_k}^{-1}S_{x_{gg}}S_{y_k},
\end{eqnarray*}
\begin{eqnarray*}
y(S_{y_m})&=&S_{z_k^{-1}y_k^{-1}y_my_kz_k}
\overset{(2)}{=}S_{z_k^{-1}y_k^{-1}y_my_k}S_{z_k}
\overset{(1),(2)}{=}S_{z_k}^{-1}S_{y_k}^{-1}S_{y_m}S_{y_k}S_{z_k},
\end{eqnarray*}
\begin{eqnarray*}
y^{-1}(S_{y_m})&=&S_{y_k^{-1}x_{gg}^{-1}z_k^{-1}x_{gg}y_mx_{gg}^{-1}z_kx_{gg}y_k}
\overset{(2)}{=}S_{y_k^{-1}x_{gg}^{-1}z_k^{-1}}S_{x_{gg}y_mx_{gg}^{-1}z_kx_{gg}y_k}\\
&\overset{(1),(2)}{=}&S_{y_k}^{-1}S_{x_{gg}}^{-1}S_{z_k}^{-1}S_{x_{gg}y_mx_{gg}^{-1}}S_{z_kx_{gg}y_k}\\
&\overset{(1),(2)}{=}&S_{y_k}^{-1}S_{x_{gg}}^{-1}S_{z_k}^{-1}S_{x_{gg}y_m}S_{x_{gg}}^{-1}S_{z_k}S_{x_{gg}}S_{y_k}\\
&\overset{(2)}{=}&S_{y_k}^{-1}S_{x_{gg}}^{-1}S_{z_k}^{-1}S_{x_{gg}}S_{y_m}S_{x_{gg}}^{-1}S_{z_k}S_{x_{gg}}S_{y_k},
\end{eqnarray*}
\begin{eqnarray*}
y(S_{z_k})&=&S_{x_{gg}y_k^{-1}x_{gg}^{-1}}
\overset{(1),(2)}{=}S_{x_{gg}}S_{y_k}^{-1}S_{x_{gg}}^{-1},
\end{eqnarray*}
\begin{eqnarray*}
y^{-1}(S_{z_k})&=&S_{y_k^{-1}}
\overset{(1)}{=}S_{y_k}^{-1},
\end{eqnarray*}
\begin{eqnarray*}
y(S_{z_m})&=&S_{z_k^{-1}z_mz_k}
\overset{(1),(2)}{=}S_{z_k}^{-1}S_{z_m}S_{z_k},
\end{eqnarray*}
\begin{eqnarray*}
y^{-1}(S_{z_m})&=&S_{y_k^{-1}z_my_k}
\overset{(2)}{=}S_{y_k^{-1}z_m}S_{y_k}
\overset{(1),(2)}{=}S_{y_k}^{-1}S_{z_m}S_{y_k},
\end{eqnarray*}
\begin{eqnarray*}
y(S_{z_{m^\prime}})&=&S_{x_{gg}y_kx_{gg}^{-1}z_{m^\prime}x_{gg}y_k^{-1}x_{gg}^{-1}}
\overset{(2)}{=}S_{x_{gg}y_kx_{gg}^{-1}}S_{z_{m^\prime}x_{gg}y_k^{-1}x_{gg}^{-1}}\\
&\overset{(1),(2)}{=}&S_{x_{gg}y_k}S_{x_{gg}}^{-1}S_{z_{m^\prime}}S_{x_{gg}}S_{y_k}^{-1}S_{x_{gg}}^{-1}
\overset{(2)}{=}S_{x_{gg}}S_{y_k}S_{x_{gg}}^{-1}S_{z_{m^\prime}}S_{x_{gg}}S_{y_k}^{-1}S_{x_{gg}}^{-1},
\end{eqnarray*}
\begin{eqnarray*}
y^{-1}(S_{z_{m^\prime}})&=&S_{z_kz_{m^\prime}z_k^{-1}}
\overset{(1),(2)}{=}S_{z_kz_{m^\prime}}S_{z_k}^{-1}
\overset{(1),(2)}{=}S_{z_k}S_{z_{m^\prime}}S_{z_k}^{-1}
\end{eqnarray*}
for $1\leq{m}<k$ and $k<m^\prime\leq{n-1}$, and $y^{\pm1}(x)=x$ for any other $x\in{X^\prime}$.
Let $y=B_{r_0}$.
We calculate
\begin{eqnarray*}
y(S_{x_{i\:i+1}})&=&S_{x_{ig}^{-1}x_{ii}x_{i+1\:g}}
\overset{(1),(2)}{=}S_{x_{ig}}^{-1}S_{x_{ii}}S_{x_{i+1\:g}},
\end{eqnarray*}
\begin{eqnarray*}
y^{-1}(S_{x_{i\:i+1}})&=&S_{x_{gi}x_{i+1\:i+1}x_{g\:i+1}^{-1}}
\overset{(1),(2)}{=}S_{x_{gi}x_{i+1\:i+1}}S_{x_{g\:i+1}}^{-1}\\
&\overset{(2)}{=}&S_{x_{gi}}S_{x_{i+1\:i+1}}S_{x_{g\:i+1}}^{-1},
\end{eqnarray*}
\begin{eqnarray*}
y(S_{x_{g-1\:g}})&=&S_{x_{g-1\:g}^{-1}x_{g-1\:g-1}x_{gg}}
\overset{(1),(2)}{=}S_{x_{g-1\:g}}^{-1}S_{x_{g-1\:g-1}}S_{x_{gg}},
\end{eqnarray*}
\begin{eqnarray*}
y^{-1}(S_{x_{g-1\:g}})&=&S_{x_{g\:g-1}},
\end{eqnarray*}
\begin{eqnarray*}
y(S_{x_{jj}})&=&S_{x_{jg}^{-1}x_{jj}x_{jg}}
\overset{(1),(3)}{=}S_{x_{jg}}^{-1}S_{x_{jj}}S_{x_{jg}},
\end{eqnarray*}
\begin{eqnarray*}
y^{-1}(S_{x_{jj}})&=&S_{x_{gj}x_{jj}x_{gj}^{-1}}
\overset{(1),(3)}{=}S_{x_{gj}}S_{x_{jj}}S_{x_{gj}}^{-1},
\end{eqnarray*}
\begin{eqnarray*}
y^{\pm1}(S_{x_{gg}})&=&S_{x_{gg}},
\end{eqnarray*}
\begin{eqnarray*}
y(S_{y_l})&=&S_{x_{gg}^{-1}z_lx_{gg}}
\overset{(1),(2)}{=}S_{x_{gg}}^{-1}S_{z_l}S_{x_{gg}},
\end{eqnarray*}
\begin{eqnarray*}
y^{-1}(S_{y_l})&=&S_{z_l},
\end{eqnarray*}
\begin{eqnarray*}
y(S_{z_l})&=&S_{y_l},
\end{eqnarray*}
\begin{eqnarray*}
y^{-1}(S_{z_l})&=&S_{x_{gg}y_lx_{gg}^{-1}}
\overset{(1),(2)}{=}S_{x_{gg}y_l}S_{x_{gg}}^{-1}
\overset{(2)}{=}S_{x_{gg}}S_{y_l}S_{x_{gg}}^{-1}
\end{eqnarray*}
for $1\leq{i}\leq{g-2}$, $1\leq{j}\leq{g-1}$ and $1\leq{l}\leq{n-1}$.
Let $y=t_{s_{kl}}$ for $1\leq{k<l}\leq{n}$.
We calculate
\begin{eqnarray*}
y(S_{y_k})&=&S_{y_ky_ly_ky_l^{-1}y_k^{-1}}
\overset{(2)}{=}S_{y_ky_l}S_{y_ky_l^{-1}y_k^{-1}}
\overset{(1),(2)}{=}S_{y_k}S_{y_l}S_{y_k}S_{y_l}^{-1}S_{y_k}^{-1},
\end{eqnarray*}
\begin{eqnarray*}
y^{-1}(S_{y_k})&=&S_{y_l^{-1}y_ky_l}
\overset{(1),(2)}{=}S_{y_l}^{-1}S_{y_k}S_{y_l},
\end{eqnarray*}
\begin{eqnarray*}
y(S_{y_l})&=&S_{y_ky_ly_k^{-1}}
\overset{(1),(2)}{=}S_{y_ky_l}S_{y_k}^{-1}
\overset{(2)}{=}S_{y_k}S_{y_l}S_{y_k}^{-1},
\end{eqnarray*}
\begin{eqnarray*}
y^{-1}(S_{y_l})&=&S_{y_l^{-1}y_k^{-1}y_ly_ky_l}
\overset{(2)}{=}S_{y_l^{-1}y_k^{-1}y_l}S_{y_ky_l}\\
&\overset{(2)}{=}&S_{y_l^{-1}y_k^{-1}}S_{y_l}S_{y_k}S_{y_l}
\overset{(1),(2)}{=}S_{y_l}^{-1}S_{y_k}^{-1}S_{y_l}S_{y_k}S_{y_l},
\end{eqnarray*}
\begin{eqnarray*}
y(S_{y_m})&=&S_{y_ky_ly_k^{-1}y_l^{-1}y_my_ly_ky_l^{-1}y_k^{-1}}
\overset{(2)}{=}S_{y_ky_ly_k^{-1}y_l^{-1}y_my_l}S_{y_ky_l^{-1}y_k^{-1}}\\
&\overset{(1),(2)}{=}&S_{y_ky_ly_k^{-1}}S_{y_l^{-1}y_my_l}S_{y_k}S_{y_l}^{-1}S_{y_k}^{-1}\\
&\overset{(1),(2)}{=}&S_{y_ky_l}S_{y_k}^{-1}S_{y_l}^{-1}S_{y_m}S_{y_l}S_{y_k}S_{y_l}^{-1}S_{y_k}^{-1}\\
&\overset{(2)}{=}&S_{y_k}S_{y_l}S_{y_k}^{-1}S_{y_l}^{-1}S_{y_m}S_{y_l}S_{y_k}S_{y_l}^{-1}S_{y_k}^{-1},
\end{eqnarray*}
\begin{eqnarray*}
y^{-1}(S_{y_m})&=&S_{y_l^{-1}y_k^{-1}y_ly_ky_my_k^{-1}y_l^{-1}y_ky_l}
\overset{(2)}{=}S_{y_l^{-1}y_k^{-1}y_l}S_{y_ky_my_k^{-1}y_l^{-1}y_ky_l}\\
&\overset{(1),(2)}{=}&S_{y_l}^{-1}S_{y_k}^{-1}S_{y_l}S_{y_ky_my_k^{-1}}S_{y_l^{-1}y_ky_l}\\
&\overset{(1),(2)}{=}&S_{y_l}^{-1}S_{y_k}^{-1}S_{y_l}S_{y_ky_m}S_{y_k}^{-1}S_{y_l}^{-1}S_{y_k}S_{y_l}\\
&\overset{(2)}{=}&S_{y_l}^{-1}S_{y_k}^{-1}S_{y_l}S_{y_k}S_{y_m}S_{y_k}^{-1}S_{y_l}^{-1}S_{y_k}S_{y_l},
\end{eqnarray*}
\begin{eqnarray*}
y(S_{z_k})&=&S_{z_kz_lz_kz_l^{-1}z_k^{-1}}
\overset{(2)}{=}S_{z_kz_l}S_{z_kz_l^{-1}z_k^{-1}}
\overset{(1),(2)}{=}S_{z_k}S_{z_l}S_{z_k}S_{z_l}^{-1}S_{z_k}^{-1},
\end{eqnarray*}
\begin{eqnarray*}
y^{-1}(S_{z_k})&=&S_{z_l^{-1}z_kz_l}
\overset{(1),(2)}{=}S_{z_l}^{-1}S_{z_k}S_{z_l},
\end{eqnarray*}
\begin{eqnarray*}
y(S_{z_l})&=&S_{z_kz_lz_k^{-1}}
\overset{(1),(2)}{=}S_{z_kz_l}S_{z_k}^{-1}
\overset{(2)}{=}S_{z_k}S_{z_l}S_{z_k}^{-1},
\end{eqnarray*}
\begin{eqnarray*}
y^{-1}(S_{z_l})&=&S_{z_l^{-1}z_k^{-1}z_lz_kz_l}
\overset{(2)}{=}S_{z_l^{-1}z_k^{-1}z_l}S_{z_kz_l}\\
&\overset{(2)}{=}&S_{z_l^{-1}z_k^{-1}}S_{z_l}S_{z_k}S_{z_l}
\overset{(1),(2)}{=}S_{z_l}^{-1}S_{z_k}^{-1}S_{z_l}S_{z_k}S_{z_l},
\end{eqnarray*}
\begin{eqnarray*}
y(S_{z_m})&=&S_{z_kz_lz_k^{-1}z_l^{-1}z_mz_lz_kz_l^{-1}z_k^{-1}}
\overset{(2)}{=}S_{z_kz_lz_k^{-1}z_l^{-1}z_mz_l}S_{z_kz_l^{-1}z_k^{-1}}\\
&\overset{(1),(2)}{=}&S_{z_kz_lz_k^{-1}}S_{z_l^{-1}z_mz_l}S_{z_k}S_{z_l}^{-1}S_{z_k}^{-1}\\
&\overset{(1),(2)}{=}&S_{z_kz_l}S_{z_k}^{-1}S_{z_l}^{-1}S_{z_m}S_{z_l}S_{z_k}S_{z_l}^{-1}S_{z_k}^{-1}\\
&\overset{(2)}{=}&S_{z_k}S_{z_l}S_{z_k}^{-1}S_{z_l}^{-1}S_{z_m}S_{z_l}S_{z_k}S_{z_l}^{-1}S_{z_k}^{-1},
\end{eqnarray*}
\begin{eqnarray*}
y^{-1}(S_{z_m})&=&S_{z_l^{-1}z_k^{-1}z_lz_kz_mz_k^{-1}z_l^{-1}z_kz_l}
\overset{(2)}{=}S_{z_l^{-1}z_k^{-1}z_l}S_{z_kz_mz_k^{-1}z_l^{-1}z_kz_l}\\
&\overset{(1),(2)}{=}&S_{z_l}^{-1}S_{z_k}^{-1}S_{z_l}S_{z_kz_mz_k^{-1}}S_{z_l^{-1}z_kz_l}\\
&\overset{(1),(2)}{=}&S_{z_l}^{-1}S_{z_k}^{-1}S_{z_l}S_{z_kz_m}S_{z_k}^{-1}S_{z_l}^{-1}S_{z_k}S_{z_l}\\
&\overset{(2)}{=}&S_{z_l}^{-1}S_{z_k}^{-1}S_{z_l}S_{z_k}S_{z_m}S_{z_k}^{-1}S_{z_l}^{-1}S_{z_k}S_{z_l}
\end{eqnarray*}
for $k<m<l$, and $y^{\pm1}(x)=x$ for any other $x\in{X^\prime}$.
Hence we have that for any $x\in{X^\prime}$ and $y\in{Y}^{\pm1}$, $y(x)$ is in the subgroup of $\pi^+$ generated by $X^\prime$, by Lemma~\ref{ij} \end{proof}
\begin{proof}[Proof of Theorem~\ref{main-2}] By Lemma~\ref{main-lem} and Proposition~\ref{3}, it follows that $\pi^+$ is generated by $X^\prime$. There is a natural map $\pi^+\to\pi_1^+(N_{g,n},\ast)$. The relations~(1), (2) and (3) of $\pi^+$ are satisfied in $\pi_1^+(N_{g,n},\ast)$ clearly. Hence the map is a homomorphism. In addition, the relations $S_{x_{11}}\cdots{}S_{x_{gg}}=1$ and $S_{x_{gg}}S_{x_{g-1\:g}}^{-1}S_{x_{g-1\:g-1}}S_{x_{g-2\:g-1}}^{-1}\cdots{}S_{x_{22}}S_{x_{12}}^{-1}S_{x_{11}}S_{x_{12}}\cdots{}S_{x_{g-1\:g}}=1$ are obtained from the relations~(1), (2) and (3) of $\pi^+$ for $n=0$. Therefore the map is an isomorphism for any $n\geq0$. Thus we complete the proof. \end{proof}
\begin{rem}\label{reduced-rel} Simple loops $\alpha$, $\beta$ and $\gamma$ of the relation~(3) in Theorem~\ref{main-2} can be reduced to the form as shown in Figure~\ref{rel-3}. In fact, we used only this reduced relation as the relation~(3), in the proof of Theorem~\ref{main-2}. We do not know whether the relation~(3) in Theorem~\ref{main-2} can be obtained from the relations~(1) and (2) there or not.
\begin{figure}
\caption{The reduced relation $S_{\alpha}S_{\beta}S_{\alpha}^{-1}=S_\gamma$.}
\label{rel-3}
\end{figure}
\end{rem}
\end{document} | arXiv |
A club has 15 members and needs to choose 2 members to be co-presidents. In how many ways can the club choose its co-presidents?
If the co-president positions are unique, there are 15 choices for the first president and 14 choices for the second president. However, since the positions are identical, we must divide by $2$, since $15\cdot 14$ counts each possible pair of co-presidents twice, once for each order in which they are selected. This gives us $\dfrac{15 \times 14}{2} = \boxed{105}$ ways to choose the co-presidents. | Math Dataset |
# Understanding the basics of signals and signal processing
Signals are a fundamental concept in many fields, including engineering, physics, and computer science. In simple terms, a signal is a function that conveys information. It can represent various types of data, such as sound, images, or electrical signals.
Signal processing is the field that deals with analyzing, modifying, and extracting information from signals. It involves techniques to enhance the quality of signals, remove noise, and extract useful features.
In this section, we will explore the basics of signals and signal processing. We will learn about different types of signals, their properties, and the techniques used to process them.
A signal can be represented as a function of time, denoted as $x(t)$, where $t$ is the time variable. The value of the signal at any given time represents the amplitude or intensity of the signal at that time.
Signals can be classified into two main categories: continuous-time signals and discrete-time signals.
Continuous-time signals are defined for all values of time within a given interval. They are represented by continuous functions, such as $x(t) = \sin(t)$ or $x(t) = e^{-t}$.
Discrete-time signals, on the other hand, are defined only at specific time instances. They are represented by sequences of values, such as $x[n] = \{1, 2, 3, 4, 5\}$ or $x[n] = \{0, 1, 0, -1, 0\}$.
An example of a continuous-time signal is a sine wave. The equation $x(t) = \sin(2\pi ft)$ represents a sine wave with frequency $f$.
An example of a discrete-time signal is a sequence of numbers. The sequence $x[n] = \{1, 2, 3, 4, 5\}$ represents a discrete-time signal with five values.
## Exercise
Think of a real-world example of a continuous-time signal and a discrete-time signal. Describe each signal and explain why it falls into the category of continuous-time or discrete-time.
### Solution
A real-world example of a continuous-time signal is the voltage waveform of an audio signal. It is continuous because it can take any value at any given time within a range.
A real-world example of a discrete-time signal is the daily temperature recorded at a specific location. It is discrete because it is measured at specific time instances, such as every hour or every day.
# Introduction to the Fast Fourier Transform algorithm
The Fast Fourier Transform (FFT) algorithm is a widely used technique for efficiently computing the Discrete Fourier Transform (DFT) of a sequence. The DFT is a mathematical operation that transforms a sequence of values in the time domain into a sequence of values in the frequency domain.
The FFT algorithm was developed by Cooley and Tukey in 1965 and revolutionized the field of signal processing. It significantly reduces the computational complexity of computing the DFT, making it practical for real-world applications.
In this section, we will introduce the FFT algorithm and explain how it works. We will also discuss its advantages over the naive approach of computing the DFT.
The DFT of a sequence $x[n]$ of length $N$ is defined as:
$$X[k] = \sum_{n=0}^{N-1} x[n]e^{-j2\pi kn/N}$$
where $X[k]$ represents the frequency components of the signal and $k$ is an integer between 0 and $N-1$.
The naive approach to compute the DFT involves directly evaluating this equation for each value of $k$. However, this approach has a computational complexity of $O(N^2)$, which becomes impractical for large sequences.
The FFT algorithm exploits the symmetry and periodicity properties of the DFT to reduce the number of calculations required. It divides the sequence into smaller subproblems and recursively applies the DFT to each subproblem. This divide-and-conquer strategy reduces the computational complexity to $O(N\log N)$.
Consider a sequence $x[n] = \{1, 2, 3, 4\}$ of length 4. The DFT of this sequence can be computed using the FFT algorithm.
The first step is to divide the sequence into two subproblems: $x_0[n] = \{1, 3\}$ and $x_1[n] = \{2, 4\}$. We then compute the DFT of each subproblem separately.
For $x_0[n]$, the DFT is $X_0[k] = \{4, -2\}$.
For $x_1[n]$, the DFT is $X_1[k] = \{6, -2\}$.
Next, we combine the results of the subproblems to obtain the final DFT of the original sequence. The formula for combining the results is:
$$X[k] = X_0[k] + e^{-j2\pi k/N}X_1[k]$$
Using this formula, we get $X[k] = \{10, -4, -2, -4\}$.
## Exercise
Compute the DFT of the sequence $x[n] = \{1, -1, 1, -1\}$ using the FFT algorithm. Show all the intermediate steps and the final result.
### Solution
The sequence $x[n]$ can be divided into two subproblems: $x_0[n] = \{1, 1\}$ and $x_1[n] = \{-1, -1\}$.
For $x_0[n]$, the DFT is $X_0[k] = \{2, 0\}$.
For $x_1[n]$, the DFT is $X_1[k] = \{0, 0\}$.
Combining the results, we get $X[k] = \{2, 0, 0, 0\}$.
# Using the Numpy library to implement FFT in Python
Python is a popular programming language for scientific computing and signal processing. It provides several libraries that make it easy to work with signals and perform FFT computations.
One such library is Numpy, which stands for Numerical Python. Numpy provides efficient implementations of mathematical functions and data structures for working with arrays and matrices.
In this section, we will learn how to use the Numpy library to implement FFT in Python. We will explore the functions and methods provided by Numpy for FFT computations.
To use the FFT functionality in Numpy, we need to import the `fft` module from the `numpy.fft` package. This module provides several functions for computing the FFT of a sequence.
The most commonly used function is `numpy.fft.fft`, which computes the 1-dimensional FFT of a sequence. It takes a sequence as input and returns the complex-valued FFT coefficients.
Here is an example of how to use the `numpy.fft.fft` function:
```python
import numpy as np
x = np.array([1, 2, 3, 4])
X = np.fft.fft(x)
print(X)
```
The output of this code will be an array of complex numbers representing the FFT coefficients.
Let's consider another example to illustrate the usage of the `numpy.fft.fft` function. Suppose we have a sequence $x[n] = \{1, 0, -1, 0\}$. We can compute its FFT using Numpy as follows:
```python
import numpy as np
x = np.array([1, 0, -1, 0])
X = np.fft.fft(x)
print(X)
```
The output will be an array of complex numbers: `[0.+0.j, 2.+0.j, 0.+0.j, 2.+0.j]`.
## Exercise
Compute the FFT of the sequence $x[n] = \{1, 2, 3, 4, 5\}$ using the `numpy.fft.fft` function. Print the result.
### Solution
```python
import numpy as np
x = np.array([1, 2, 3, 4, 5])
X = np.fft.fft(x)
print(X)
```
The output will be an array of complex numbers representing the FFT coefficients.
# Understanding the concept of frequency analysis
Frequency analysis is a fundamental technique in signal processing. It involves analyzing the frequency components of a signal to gain insights into its characteristics and behavior.
In this section, we will explore the concept of frequency analysis and its importance in signal processing. We will learn about the frequency spectrum, which represents the distribution of frequency components in a signal.
The frequency spectrum of a signal is a plot that shows the amplitude or power of each frequency component in the signal. It provides valuable information about the signal's frequency content and helps identify important features.
The frequency spectrum is obtained by computing the Fourier Transform of the signal. The Fourier Transform is a mathematical operation that decomposes a signal into its frequency components.
The frequency spectrum is typically represented as a graph with frequency on the x-axis and amplitude or power on the y-axis. The x-axis is usually scaled logarithmically to better visualize the distribution of frequency components.
Consider a simple example of a sine wave signal with a frequency of 10 Hz. The frequency spectrum of this signal will have a peak at 10 Hz, indicating the presence of a frequency component at that frequency.
If we add another sine wave with a frequency of 20 Hz to the original signal, the frequency spectrum will show two peaks at 10 Hz and 20 Hz, representing the two frequency components.
## Exercise
Think of a real-world example where frequency analysis is used. Describe the example and explain how frequency analysis helps in understanding the signal.
### Solution
One real-world example where frequency analysis is used is in audio processing. When analyzing an audio signal, frequency analysis helps identify different components of the sound, such as the fundamental frequency, harmonics, and noise. This information is useful for tasks like speech recognition, music analysis, and noise reduction.
# Exploring different types of signals and their frequency components
In signal processing, signals can take various forms and have different frequency components. Understanding the characteristics of different types of signals and their frequency components is crucial for effective signal analysis and processing.
In this section, we will explore different types of signals commonly encountered in signal processing and examine their frequency components.
1. Continuous-time signals: Continuous-time signals are signals that exist over a continuous range of time. These signals can be represented by mathematical functions, such as sine waves, square waves, and triangular waves. The frequency components of continuous-time signals can be determined using Fourier analysis.
2. Discrete-time signals: Discrete-time signals are signals that are defined only at specific time points. These signals are often obtained by sampling continuous-time signals. Discrete-time signals can have frequency components that are multiples of the sampling frequency.
3. Periodic signals: Periodic signals are signals that repeat themselves after a certain period of time. These signals can be represented by Fourier series, which decomposes the signal into a sum of sinusoidal components. The frequency components of periodic signals are harmonically related.
4. Non-periodic signals: Non-periodic signals, also known as aperiodic signals, do not repeat themselves after a certain period of time. These signals can have a wide range of frequency components and do not follow a specific pattern.
Consider a continuous-time signal that represents a musical tone. This signal can be modeled as a sine wave with a specific frequency. The frequency component of this signal corresponds to the pitch of the musical tone.
Now, let's consider a discrete-time signal obtained by sampling the continuous-time signal at regular intervals. The frequency components of this discrete-time signal will be multiples of the sampling frequency. These frequency components can provide information about the harmonics present in the musical tone.
## Exercise
Think of a real-world example for each type of signal mentioned above (continuous-time, discrete-time, periodic, and non-periodic). Describe the example and explain the frequency components associated with each signal type.
### Solution
- Continuous-time signal: An example of a continuous-time signal is an analog audio signal from a microphone. The frequency components of this signal can represent different sounds or voices in the environment.
- Discrete-time signal: An example of a discrete-time signal is a digital audio signal obtained by sampling the analog audio signal. The frequency components of this signal will be multiples of the sampling frequency and can represent the harmonics of the original sound.
- Periodic signal: An example of a periodic signal is a sine wave generated by an electronic oscillator. The frequency components of this signal will be harmonically related and can represent the fundamental frequency and its harmonics.
- Non-periodic signal: An example of a non-periodic signal is a random noise signal. This signal can have a wide range of frequency components and does not follow a specific pattern.
# Applying the FFT algorithm to real-world signals
To apply the FFT algorithm, we first need to obtain a signal that we want to analyze. This signal can be obtained from various sources, such as audio recordings, sensor data, or simulated signals. Once we have the signal, we can use the Numpy library to perform the FFT.
The Numpy library provides the `fft` function, which takes an input signal and returns its frequency spectrum. The frequency spectrum represents the amplitudes of different frequency components present in the signal. By analyzing the frequency spectrum, we can gain insights into the characteristics of the signal.
Let's say we have a recorded audio signal of a musical instrument. We want to analyze the frequency components present in the signal to identify the notes being played.
First, we need to load the audio signal into Python using a library such as `scipy.io.wavfile`. Then, we can use the `fft` function from Numpy to compute the frequency spectrum of the signal.
```python
import numpy as np
from scipy.io import wavfile
# Load the audio signal
sampling_rate, signal = wavfile.read('audio.wav')
# Compute the FFT
fft_signal = np.fft.fft(signal)
# Get the frequency spectrum
frequency_spectrum = np.abs(fft_signal)
# Plot the frequency spectrum
import matplotlib.pyplot as plt
plt.plot(frequency_spectrum)
plt.xlabel('Frequency')
plt.ylabel('Amplitude')
plt.show()
```
By plotting the frequency spectrum, we can visualize the amplitudes of different frequency components in the signal. This can help us identify the notes being played and analyze the characteristics of the audio signal.
## Exercise
Apply the FFT algorithm to a real-world signal of your choice. This can be an audio signal, an image signal, or any other type of signal that you find interesting. Compute the frequency spectrum of the signal and plot it using the Numpy and Matplotlib libraries.
### Solution
```python
import numpy as np
import matplotlib.pyplot as plt
# Load the signal
signal = ...
# Compute the FFT
fft_signal = np.fft.fft(signal)
# Get the frequency spectrum
frequency_spectrum = np.abs(fft_signal)
# Plot the frequency spectrum
plt.plot(frequency_spectrum)
plt.xlabel('Frequency')
plt.ylabel('Amplitude')
plt.show()
```
Replace `signal` with the actual signal you want to analyze.
# Analyzing the results of the FFT and interpreting the frequency spectrum
Once we have computed the frequency spectrum using the FFT algorithm, we can analyze the results to gain insights into the characteristics of the signal.
The frequency spectrum is a plot that shows the amplitudes of different frequency components present in the signal. The x-axis represents the frequency, and the y-axis represents the amplitude. By analyzing the frequency spectrum, we can identify the dominant frequencies in the signal and their corresponding amplitudes.
One important concept to understand when analyzing the frequency spectrum is the Nyquist frequency. The Nyquist frequency is defined as half of the sampling rate, which is the rate at which the signal is sampled. The Nyquist frequency represents the maximum frequency that can be accurately represented in the frequency spectrum. Frequencies above the Nyquist frequency will be aliased, meaning they will appear as lower frequencies in the spectrum.
Let's say we have a signal that was sampled at a rate of 1000 Hz. This means that the Nyquist frequency is 500 Hz. If we compute the frequency spectrum of the signal using the FFT algorithm, any frequencies above 500 Hz will be aliased and appear as lower frequencies in the spectrum.
For example, if there is a frequency component at 600 Hz in the signal, it will appear as a component at 400 Hz in the frequency spectrum. This is because 600 Hz is above the Nyquist frequency of 500 Hz.
## Exercise
Consider a signal that was sampled at a rate of 2000 Hz. Compute the frequency spectrum of the signal using the FFT algorithm and plot it using the Numpy and Matplotlib libraries. Identify any aliased frequencies in the spectrum.
### Solution
```python
import numpy as np
import matplotlib.pyplot as plt
# Load the signal
signal = ...
# Compute the FFT
fft_signal = np.fft.fft(signal)
# Get the frequency spectrum
frequency_spectrum = np.abs(fft_signal)
# Plot the frequency spectrum
plt.plot(frequency_spectrum)
plt.xlabel('Frequency')
plt.ylabel('Amplitude')
plt.show()
```
Replace `signal` with the actual signal you want to analyze.
# Understanding the limitations of FFT and when to use alternative methods
While the FFT algorithm is a powerful tool for analyzing the frequency components of a signal, it does have some limitations. It assumes that the signal is periodic and stationary, meaning that its properties do not change over time. If these assumptions are not met, the FFT may not provide accurate results.
Additionally, the FFT algorithm is most effective when the signal contains a limited number of frequency components. If the signal is highly complex and contains a large number of frequency components, the FFT may not be able to accurately resolve all of them.
In such cases, alternative methods may be more suitable. For example, if the signal is non-periodic or non-stationary, time-frequency analysis techniques such as the short-time Fourier transform (STFT) or wavelet transform can provide more accurate results.
Let's say we have a signal that contains both low-frequency and high-frequency components, and the high-frequency components are transient and occur only for a short duration. If we use the FFT algorithm to analyze this signal, the high-frequency components may not be accurately resolved due to the limited time resolution of the FFT.
In this case, the STFT or wavelet transform can provide a better representation of the signal, as they can capture both the frequency and time information of the signal.
## Exercise
Consider a signal that is non-periodic and contains both low-frequency and high-frequency components. Compare the results of analyzing the signal using the FFT algorithm and the STFT algorithm. Which method provides a more accurate representation of the signal?
### Solution
To compare the results of the FFT and STFT algorithms, you can compute the frequency spectrum using the FFT algorithm and the spectrogram using the STFT algorithm. Plot both the frequency spectrum and the spectrogram and compare the results.
```python
import numpy as np
import matplotlib.pyplot as plt
from scipy.signal import spectrogram
# Load the signal
signal = ...
# Compute the FFT
fft_signal = np.fft.fft(signal)
frequency_spectrum = np.abs(fft_signal)
# Compute the spectrogram
frequencies, times, spectrogram = spectrogram(signal)
# Plot the frequency spectrum
plt.subplot(2, 1, 1)
plt.plot(frequency_spectrum)
plt.xlabel('Frequency')
plt.ylabel('Amplitude')
# Plot the spectrogram
plt.subplot(2, 1, 2)
plt.pcolormesh(times, frequencies, np.log10(spectrogram))
plt.xlabel('Time')
plt.ylabel('Frequency')
plt.colorbar()
plt.show()
```
Replace `signal` with the actual signal you want to analyze.
# Advanced techniques for signal processing using FFT
The FFT algorithm can be used for more than just analyzing the frequency components of a signal. It can also be used for various signal processing tasks, such as filtering, convolution, and correlation.
One common signal processing task is filtering, which involves removing or attenuating certain frequency components of a signal while preserving others. The FFT algorithm can be used to implement various types of filters, such as low-pass filters, high-pass filters, and band-pass filters.
Another signal processing task is convolution, which involves combining two signals to create a third signal that represents the combined effect of the original signals. The FFT algorithm can be used to efficiently compute the convolution of two signals.
Additionally, the FFT algorithm can be used for correlation, which involves measuring the similarity between two signals. The FFT algorithm can compute the cross-correlation and auto-correlation of signals, which can be useful in various applications such as pattern recognition and signal synchronization.
Let's say we have a noisy signal and we want to remove the noise using a low-pass filter. We can use the FFT algorithm to compute the frequency spectrum of the signal, apply a filter to attenuate the high-frequency components, and then compute the inverse FFT to obtain the filtered signal.
Similarly, we can use the FFT algorithm to compute the convolution of two signals, such as an input signal and an impulse response, to obtain the output signal.
## Exercise
Consider a noisy signal and a low-pass filter. Use the FFT algorithm to remove the noise from the signal by applying the filter. Plot the original signal, the filtered signal, and the frequency spectrum of the filtered signal.
### Solution
```python
import numpy as np
import matplotlib.pyplot as plt
# Load the noisy signal
noisy_signal = ...
# Compute the FFT of the noisy signal
fft_noisy_signal = np.fft.fft(noisy_signal)
# Apply a low-pass filter to the frequency spectrum
filtered_spectrum = ...
filtered_signal = np.fft.ifft(filtered_spectrum)
# Plot the original signal
plt.subplot(3, 1, 1)
plt.plot(noisy_signal)
plt.xlabel('Time')
plt.ylabel('Amplitude')
# Plot the filtered signal
plt.subplot(3, 1, 2)
plt.plot(filtered_signal)
plt.xlabel('Time')
plt.ylabel('Amplitude')
# Plot the frequency spectrum of the filtered signal
plt.subplot(3, 1, 3)
plt.plot(np.abs(filtered_spectrum))
plt.xlabel('Frequency')
plt.ylabel('Amplitude')
plt.show()
```
Replace `noisy_signal` with the actual noisy signal you want to filter.
# Combining FFT with other signal processing techniques
The FFT algorithm can be combined with other signal processing techniques to perform more advanced analysis and manipulation of signals.
One common technique is windowing, which involves multiplying the signal by a window function before applying the FFT algorithm. Windowing can help reduce the spectral leakage effect, which occurs when the frequency components of a signal spread out and interfere with each other in the frequency spectrum.
Another technique is zero-padding, which involves adding zeros to the end of the signal before applying the FFT algorithm. Zero-padding can increase the frequency resolution of the FFT, allowing for more accurate analysis of the frequency components.
Additionally, the FFT algorithm can be used in conjunction with other time-frequency analysis techniques, such as the wavelet transform or the short-time Fourier transform (STFT), to obtain a more detailed representation of the signal in both the time and frequency domains.
Let's say we have a signal with a short duration and we want to analyze its frequency components with high resolution. We can use zero-padding to increase the length of the signal, which will result in a higher frequency resolution in the frequency spectrum.
Similarly, we can apply a window function to the signal before applying the FFT algorithm to reduce the spectral leakage effect and obtain a cleaner frequency spectrum.
## Exercise
Consider a signal with a short duration and a high-frequency component. Use zero-padding to increase the length of the signal and analyze its frequency components with high resolution. Plot the original signal, the zero-padded signal, and the frequency spectrum of the zero-padded signal.
### Solution
```python
import numpy as np
import matplotlib.pyplot as plt
# Load the original signal
original_signal = ...
# Zero-pad the signal
zero_padded_signal = ...
# Compute the FFT of the zero-padded signal
fft_zero_padded_signal = np.fft.fft(zero_padded_signal)
# Plot the original signal
plt.subplot(3, 1, 1)
plt.plot(original_signal)
plt.xlabel('Time')
plt.ylabel('Amplitude')
# Plot the zero-padded signal
plt.subplot(3, 1, 2)
plt.plot(zero_padded_signal)
plt.xlabel('Time')
plt.ylabel('Amplitude')
# Plot the frequency spectrum of the zero-padded signal
plt.subplot(3, 1, 3)
plt.plot(np.abs(fft_zero_padded_signal))
plt.xlabel('Frequency')
plt.ylabel('Amplitude')
plt.show()
```
Replace `original_signal` with the actual original signal you want to analyze.
# Optimizing FFT performance for large datasets
The FFT algorithm can be computationally intensive, especially for large datasets. However, there are several techniques that can be used to optimize the performance of the FFT algorithm.
One technique is to use a radix-2 FFT algorithm, which is the most common implementation of the FFT algorithm. The radix-2 FFT algorithm is based on the divide-and-conquer approach, where the input signal is divided into smaller sub-signals and the FFT is computed recursively.
Another technique is to use a multi-threaded or parallel implementation of the FFT algorithm, which can take advantage of multiple processors or cores to compute the FFT faster.
Additionally, the performance of the FFT algorithm can be improved by using optimized libraries or hardware accelerators, such as the Intel Math Kernel Library (MKL) or graphics processing units (GPUs).
Let's say we have a large dataset and we want to compute the FFT of the entire dataset. We can optimize the performance of the FFT algorithm by using a radix-2 FFT implementation, a multi-threaded or parallel implementation, and an optimized library or hardware accelerator.
By using these techniques, we can significantly reduce the computation time of the FFT and analyze large datasets more efficiently.
## Exercise
Consider a large dataset and compute the FFT of the entire dataset using the optimized techniques mentioned above. Compare the computation time of the optimized FFT algorithm with the computation time of the standard FFT algorithm.
### Solution
To compare the computation time of the optimized FFT algorithm with the standard FFT algorithm, you can use the `time` module to measure the execution time of each algorithm.
```python
import numpy as np
import time
# Load the large dataset
large_dataset = ...
# Compute the FFT using the standard algorithm
start_time = time.time()
fft_standard = np.fft.fft(large_dataset)
end_time = time.time()
computation_time_standard = end_time - start_time
# Compute the FFT using the optimized algorithm
start_time = time.time()
fft_optimized = ...
end_time = time.time()
computation_time_optimized = end_time - start_time
# Compare the computation times
print('Computation time (standard):', computation_time_standard)
print('Computation time (optimized):', computation_time_optimized)
```
Replace `large_dataset` with the actual large dataset you want to analyze.
# Practical applications of FFT in various industries
The Fast Fourier Transform (FFT) algorithm has a wide range of practical applications in various industries. It is a powerful tool for analyzing signals and extracting useful information from them.
In the field of telecommunications, the FFT algorithm is used for signal processing and modulation. It is used to analyze and manipulate signals in order to improve the quality of communication and reduce noise and interference.
In the field of audio and music, the FFT algorithm is used for audio signal processing, such as equalization, filtering, and spectral analysis. It is used to analyze the frequency components of audio signals and apply various effects and enhancements.
In the field of image processing, the FFT algorithm is used for image compression, filtering, and enhancement. It is used to analyze the frequency components of images and remove noise, blur, or other unwanted artifacts.
In the field of finance, the FFT algorithm is used for time series analysis and forecasting. It is used to analyze the frequency components of financial data and identify patterns and trends.
In the field of medical imaging, the FFT algorithm is used for image reconstruction and analysis. It is used to analyze the frequency components of medical images and enhance the visibility of certain structures or abnormalities.
In the field of geophysics, the FFT algorithm is used for seismic data analysis and processing. It is used to analyze the frequency components of seismic signals and detect and locate underground structures or resources.
These are just a few examples of the practical applications of the FFT algorithm in various industries. The FFT algorithm is a versatile and powerful tool that can be applied to a wide range of signal processing tasks. | Textbooks |
Large deviations theory
In probability theory, the theory of large deviations concerns the asymptotic behaviour of remote tails of sequences of probability distributions. While some basic ideas of the theory can be traced to Laplace, the formalization started with insurance mathematics, namely ruin theory with Cramér and Lundberg. A unified formalization of large deviation theory was developed in 1966, in a paper by Varadhan.[1] Large deviations theory formalizes the heuristic ideas of concentration of measures and widely generalizes the notion of convergence of probability measures.
Roughly speaking, large deviations theory concerns itself with the exponential decline of the probability measures of certain kinds of extreme or tail events.
Introductory examples
An elementary example
Consider a sequence of independent tosses of a fair coin. The possible outcomes could be heads or tails. Let us denote the possible outcome of the i-th trial by $X_{i}$, where we encode head as 1 and tail as 0. Now let $M_{N}$ denote the mean value after $N$ trials, namely
$M_{N}={\frac {1}{N}}\sum _{i=1}^{N}X_{i}$.
Then $M_{N}$ lies between 0 and 1. From the law of large numbers it follows that as N grows, the distribution of $M_{N}$ converges to $0.5=\operatorname {E} [X]$ (the expected value of a single coin toss).
Moreover, by the central limit theorem, it follows that $M_{N}$ is approximately normally distributed for large $N$. The central limit theorem can provide more detailed information about the behavior of $M_{N}$ than the law of large numbers. For example, we can approximately find a tail probability of $M_{N}$, $P(M_{N}>x)$, that $M_{N}$ is greater than $x$, for a fixed value of $N$. However, the approximation by the central limit theorem may not be accurate if $x$ is far from $\operatorname {E} [X_{i}]$ unless $N$ is sufficiently large. Also, it does not provide information about the convergence of the tail probabilities as $N\to \infty $. However, the large deviation theory can provide answers for such problems.
Let us make this statement more precise. For a given value $0.5<x<1$, let us compute the tail probability $P(M_{N}>x)$. Define
$I(x)=x\ln {x}+(1-x)\ln(1-x)+\ln {2}$.
Note that the function $I(x)$ is a convex, nonnegative function that is zero at $x={\tfrac {1}{2}}$ and increases as $x$ approaches $1$. It is the negative of the Bernoulli entropy with $p={\tfrac {1}{2}}$; that it's appropriate for coin tosses follows from the asymptotic equipartition property applied to a Bernoulli trial. Then by Chernoff's inequality, it can be shown that $P(M_{N}>x)<\exp(-NI(x))$.[2] This bound is rather sharp, in the sense that $I(x)$ cannot be replaced with a larger number which would yield a strict inequality for all positive $N$.[3] (However, the exponential bound can still be reduced by a subexponential factor on the order of $1/{\sqrt {N}}$; this follows from the Stirling approximation applied to the binomial coefficient appearing in the Bernoulli distribution.) Hence, we obtain the following result:
$P(M_{N}>x)\approx \exp(-NI(x))$.
The probability $P(M_{N}>x)$ decays exponentially as $N\to \infty $ at a rate depending on x. This formula approximates any tail probability of the sample mean of i.i.d. variables and gives its convergence as the number of samples increases.
Large deviations for sums of independent random variables
Main article: Cramér's theorem (large deviations)
In the above example of coin-tossing we explicitly assumed that each toss is an independent trial, and the probability of getting head or tail is always the same.
Let $X,X_{1},X_{2},\ldots $ be independent and identically distributed (i.i.d.) random variables whose common distribution satisfies a certain growth condition. Then the following limit exists:
$\lim _{N\to \infty }{\frac {1}{N}}\ln P(M_{N}>x)=-I(x)$.
Here
$M_{N}={\frac {1}{N}}\sum _{i=1}^{N}X_{i}$,
as before.
Function $I(\cdot )$ is called the "rate function" or "Cramér function" or sometimes the "entropy function".
The above-mentioned limit means that for large $N$,
$P(M_{N}>x)\approx \exp[-NI(x)]$,
which is the basic result of large deviations theory.[4][5]
If we know the probability distribution of $X$, an explicit expression for the rate function can be obtained. This is given by a Legendre–Fenchel transformation,[6]
$I(x)=\sup _{\theta >0}[\theta x-\lambda (\theta )]$,
where
$\lambda (\theta )=\ln \operatorname {E} [\exp(\theta X)]$
is called the cumulant generating function (CGF) and $\operatorname {E} $ denotes the mathematical expectation.
If $X$ follows a normal distribution, the rate function becomes a parabola with its apex at the mean of the normal distribution.
If $\{X_{i}\}$ is an irreducible and aperiodic Markov chain, the variant of the basic large deviations result stated above may hold.
Moderate deviations for sums of independent random variables
The previous example controlled the probability of the event $[M_{N}>x]$, that is, the concentration of the law of $M_{N}$ on the compact set $[-x,x]$. It is also possible to control the probability of the event $[M_{N}>xa_{N}]$ for some sequence $a_{N}\to 0$. The following is an example of a moderate deviations principle:[7][8]
Theorem — Let $X_{1},X_{2},\dots $ be a sequence of centered i.i.d variables with finite variance $\sigma ^{2}$ such that $\forall \lambda \in \mathbb {R} ,\ \ln \mathbb {E} [e^{\lambda X_{1}}]<\infty $. Define $M_{N}:={\frac {1}{N}}\sum \limits _{n\leq N}X_{N}$. Then for any sequence $1\ll a_{N}\ll {\sqrt {N}}$:
$\lim \limits _{N\to +\infty }{\frac {a_{N}^{2}}{N}}\ln \mathbb {P} [a_{N}M_{N}\geq x]=-{\frac {x^{2}}{2\sigma ^{2}}}$
In particular, the limit case $a_{N}={\sqrt {N}}$ is the central limit theorem.
Formal definition
Given a Polish space ${\mathcal {X}}$ let $\{\mathbb {P} _{N}\}$ be a sequence of Borel probability measures on ${\mathcal {X}}$, let $\{a_{N}\}$ be a sequence of positive real numbers such that $\lim _{N}a_{N}=\infty $, and finally let $I:{\mathcal {X}}\to [0,\infty ]$ be a lower semicontinuous functional on ${\mathcal {X}}.$ The sequence $\{\mathbb {P} _{N}\}$ is said to satisfy a large deviation principle with speed $\{a_{n}\}$ and rate $I$ if, and only if, for each Borel measurable set $E\subset {\mathcal {X}}$,
$-\inf _{x\in E^{\circ }}I(x)\leq \varliminf _{N}a_{N}^{-1}\log(\mathbb {P} _{N}(E))\leq \varlimsup _{N}a_{N}^{-1}\log(\mathbb {P} _{N}(E))\leq -\inf _{x\in {\overline {E}}}I(x)$,
where ${\overline {E}}$ and $E^{\circ }$ denote respectively the closure and interior of $E$.
Brief history
The first rigorous results concerning large deviations are due to the Swedish mathematician Harald Cramér, who applied them to model the insurance business.[9] From the point of view of an insurance company, the earning is at a constant rate per month (the monthly premium) but the claims come randomly. For the company to be successful over a certain period of time (preferably many months), the total earning should exceed the total claim. Thus to estimate the premium you have to ask the following question: "What should we choose as the premium $q$ such that over $N$ months the total claim $C=\Sigma X_{i}$ should be less than $Nq$?" This is clearly the same question asked by the large deviations theory. Cramér gave a solution to this question for i.i.d. random variables, where the rate function is expressed as a power series.
A very incomplete list of mathematicians who have made important advances would include Petrov,[10] Sanov,[11] S.R.S. Varadhan (who has won the Abel prize for his contribution to the theory), D. Ruelle, O.E. Lanford, Amir Dembo, and Ofer Zeitouni.[12]
Applications
Principles of large deviations may be effectively applied to gather information out of a probabilistic model. Thus, theory of large deviations finds its applications in information theory and risk management. In physics, the best known application of large deviations theory arise in thermodynamics and statistical mechanics (in connection with relating entropy with rate function).
Large deviations and entropy
Main article: asymptotic equipartition property
The rate function is related to the entropy in statistical mechanics. This can be heuristically seen in the following way. In statistical mechanics the entropy of a particular macro-state is related to the number of micro-states which corresponds to this macro-state. In our coin tossing example the mean value $M_{N}$ could designate a particular macro-state. And the particular sequence of heads and tails which gives rise to a particular value of $M_{N}$ constitutes a particular micro-state. Loosely speaking a macro-state having a higher number of micro-states giving rise to it, has higher entropy. And a state with higher entropy has a higher chance of being realised in actual experiments. The macro-state with mean value of 1/2 (as many heads as tails) has the highest number of micro-states giving rise to it and it is indeed the state with the highest entropy. And in most practical situations we shall indeed obtain this macro-state for large numbers of trials. The "rate function" on the other hand measures the probability of appearance of a particular macro-state. The smaller the rate function the higher is the chance of a macro-state appearing. In our coin-tossing the value of the "rate function" for mean value equal to 1/2 is zero. In this way one can see the "rate function" as the negative of the "entropy".
There is a relation between the "rate function" in large deviations theory and the Kullback–Leibler divergence, the connection is established by Sanov's theorem (see Sanov[11] and Novak,[13] ch. 14.5).
In a special case, large deviations are closely related to the concept of Gromov–Hausdorff limits.[14]
See also
• Large deviation principle
• Cramér's large deviation theorem
• Chernoff's inequality
• Sanov's theorem
• Contraction principle (large deviations theory), a result on how large deviations principles "push forward"
• Freidlin–Wentzell theorem, a large deviations principle for Itō diffusions
• Legendre transformation, Ensemble equivalence is based on this transformation.
• Laplace principle, a large deviations principle in Rd
• Laplace's method
• Schilder's theorem, a large deviations principle for Brownian motion
• Varadhan's lemma
• Extreme value theory
• Large deviations of Gaussian random functions
References
1. S.R.S. Varadhan, Asymptotic probability and differential equations, Comm. Pure Appl. Math. 19 (1966),261-286.
2. "Large deviations for performance analysis: queues, communications, and computing", Shwartz, Adam, 1953- TN: 1228486
3. Varadhan, S.R.S.,The Annals of Probability 2008, Vol. 36, No. 2, 397–419,
4. http://math.nyu.edu/faculty/varadhan/Spring2012/Chapters1-2.pdf
5. S.R.S. Varadhan, Large Deviations and Applications (SIAM, Philadelphia, 1984)
6. Touchette, Hugo (1 July 2009). "The large deviation approach to statistical mechanics". Physics Reports. 478 (1–3): 1–69. arXiv:0804.0327. Bibcode:2009PhR...478....1T. doi:10.1016/j.physrep.2009.05.002. S2CID 118416390.
7. Dembo, Amir; Zeitouni, Ofer (3 November 2009). Large Deviations Techniques and Applications. Springer Science & Business Media. p. 109. ISBN 978-3-642-03311-7.
8. Sethuraman, Jayaram; O., Robert (2011), "Moderate Deviations", in Lovric, Miodrag (ed.), International Encyclopedia of Statistical Science, Berlin, Heidelberg: Springer Berlin Heidelberg, pp. 847–849, doi:10.1007/978-3-642-04898-2_374, ISBN 978-3-642-04897-5, retrieved 2 July 2023
9. Cramér, H. (1944). On a new limit theorem of the theory of probability. Uspekhi Matematicheskikh Nauk, (10), 166-178.
10. Petrov V.V. (1954) Generalization of Cramér's limit theorem. Uspehi Matem. Nauk, v. 9, No 4(62), 195--202.(Russian)
11. Sanov I.N. (1957) On the probability of large deviations of random magnitudes. Matem. Sbornik, v. 42 (84), 11--44.
12. Dembo, A., & Zeitouni, O. (2009). Large deviations techniques and applications (Vol. 38). Springer Science & Business Media
13. Novak S.Y. (2011) Extreme value methods with applications to finance. Chapman & Hall/CRC Press. ISBN 978-1-4398-3574-6.
14. Kotani M., Sunada T. Large deviation and the tangent cone at infinity of a crystal lattice, Math. Z. 254, (2006), 837-870.
Bibliography
• Special invited paper: Large deviations by S. R. S. Varadhan The Annals of Probability 2008, Vol. 36, No. 2, 397–419 doi:10.1214/07-AOP348
• A basic introduction to large deviations: Theory, applications, simulations, Hugo Touchette, arXiv:1106.4146.
• Entropy, Large Deviations and Statistical Mechanics by R.S. Ellis, Springer Publication. ISBN 3-540-29059-1
• Large Deviations for Performance Analysis by Alan Weiss and Adam Shwartz. Chapman and Hall ISBN 0-412-06311-5
• Large Deviations Techniques and Applications by Amir Dembo and Ofer Zeitouni. Springer ISBN 0-387-98406-2
• Random Perturbations of Dynamical Systems by M.I. Freidlin and A.D. Wentzell. Springer ISBN 0-387-98362-7
• "Large Deviations for Two Dimensional Navier-Stokes Equation with Multiplicative Noise", S. S. Sritharan and P. Sundar, Stochastic Processes and Their Applications, Vol. 116 (2006) 1636–1659.
• "Large Deviations for the Stochastic Shell Model of Turbulence", U. Manna, S. S. Sritharan and P. Sundar, NoDEA Nonlinear Differential Equations Appl. 16 (2009), no. 4, 493–521.
| Wikipedia |
Alan J. Goldman
Alan J. Goldman (1932–2010) was an American expert in operations research.[1][2]
Career
Goldman was born in 1932 and grew up in Brooklyn, where his parents both worked for the public school system. In 1949, he was a winner of the Westinghouse Science Talent Search. He studied mathematics and physics at Brooklyn College, graduating in 1952.[1][2] He went on to graduate study in mathematics at Princeton University, completing his doctorate in topology in 1957 under the supervision of Ralph Fox.[3] Goldman worked at the National Bureau of Standards from 1956 until 1979, when he became a professor of mathematical sciences at Johns Hopkins University. He retired in 1999.[1][2]
While at Princeton, Goldman came under the influence of Albert W. Tucker, with whom he published three "seminal papers" in Annals of Mathematics Studies on linear programming and convex polytopes. His work at the National Bureau of Standards included work on facility location for the US Postal Service and on transportation planning; he was also a mentor there to Jack Edmonds and George Nemhauser.[1] After moving to Johns Hopkins, his doctoral students included combinatorialist Arthur T. Benjamin.[3]
In 1976, Goldman won the Gold Medal for Excellence in Service of the US Department of Commerce. He was elected to the National Academy of Engineering in 1989.[1][2]
Selected publications
• Goldman, A. J.; Tucker, A. W. (1956), "Polyhedral convex cones", Linear equalities and related systems, Annals of Mathematics Studies, vol. 38, Princeton, N.J.: Princeton University Press, pp. 19–40, MR 0087974.
• Goldman, A. J. (1956), "Resolution and separation theorems for polyhedral convex sets", Linear inequalities and related systems, Annals of Mathematics Studies, vol. 38, Princeton, N.J.: Princeton University Press, pp. 41–51, MR 0089113.
• Goldman, A. J.; Tucker, A. W. (1956), "Theory of linear programming", Linear inequalities and related systems, Annals of Mathematics Studies, vol. 38, Princeton, N.J.: Princeton University Press, pp. 53–97, MR 0101826.
• Goldman, A. J. (1971), "Optimal center location in simple networks", Transportation Science, 5 (2): 212–221, doi:10.1287/trsc.5.2.212, MR 0359738.
References
1. Naiman, Daniel Q.; Witzgall, Christoff (2013), "Alan J. Goldman 1932–2010", Memorial Tributes of the National Academies, 17: 122–126.
2. "Alan J. Goldman, 77, expert in operations research", The JHU Gazette, Johns Hopkins University, March 1, 2010, archived from the original on May 3, 2017.
3. Alan J. Goldman at the Mathematics Genealogy Project
Authority control: Academics
• MathSciNet
• Mathematics Genealogy Project
• zbMATH
| Wikipedia |
Journal of Wood Science
Official Journal of the Japan Wood Research Society
Fuel and material utilization of a waste shiitake (Lentinula edodes) mushroom bed derived from hardwood chips I: characteristics of calorific value in terms of elemental composition and ash content
Noboru Sekino1 &
Zhuoqiu Jiang2
Journal of Wood Science volume 67, Article number: 1 (2021) Cite this article
To understand the fuel characteristics of a waste shiitake mushroom bed derived from hardwood chips, the moisture content at the time of disposal and after 1 month, as well as its calorific value, ash content, and elemental composition, were investigated. The moisture content on a wet basis (MCw) was 78% at the time of disposal and was as high as 63% even 1 month after disposal. It is considered that the slow drying process is caused by the low moisture permeability of the skin of mushroom bed, and therefore, it is preferable to crush the waste mushroom bed before drying. Comparing the gross calorific value on a dry basis of the waste mushroom bed with that of the cultivation bed wood chips, the value inside of the waste mushroom bed was similar, while that of its skin was significantly lower (by 11%). The reason for this lies in the significantly higher ash content and nitrogen content compared to those of wood. When analyzed from the combustion heat of the contained elements, it was found that both the cultivation bed wood chips and the waste mushroom bed had almost no hydrogen contributing to combustion due to their high oxygen content, and they were dependent on the heat generation of carbon. As a result of finding the relationship between the net calorific value that can be used as a boiler fuel and MCw, for example, the value at an MCw of 50% was calculated to be 7.6 MJ/kg, which was almost the same as that of sugi (Cryptomeria Japonica) sapwood and bark. The ash content of the waste mushroom bed was about 7%, which is close to that of bark and about ten times that of the wood used for the cultivation bed. When the waste mushroom bed is used as boiler fuel, appropriate ash treatment is required as in the case of using bark.
Mushroom production in each country is on the rise due to the recent boom in health foods. According to FAO statistics (Food and Agricultural Organization of the United Nations, 2017 results) [1], there are 30 countries that produce more than 10,000 tons per year, with a total of 10.18 million tons. There are 43 countries with less than 10,000 tons, but because their production in total is tens of thousands of tons, the total production of the world is approximately 10.2 million tons. The growth in mushroom production is supported by the conversion from raw wood cultivation to mushroom bed cultivation, and the effective utilization of mushroom beds after harvesting (hereinafter referred to as waste mushroom beds) has become an issue from the view point of a resource recycling society. For example, in China, which has the highest mushroom production in the world, efforts are being made to produce wood pellets from waste mushroom beds and convert the heat source for district heating from coal to wood pellets [2].
According to Japanese government statistics [3], the total production of mushrooms in Japan is 456,000 tons (2018 results). Among several kinds of mushrooms, shiitake mushrooms are the largest on a production value basis, and the number of large-scale shiitake mushroom bed cultivation farms is increasing. Figure 1 shows the production process at the shiitake farm in the Kuji area of the Iwate prefecture, Japan. At this shiitake farm, hardwood logs collected from a nearby area are chipped to form a cultivation bed, which is disposed of after about 11 months through six times of harvesting. Traditionally, the treatment of waste mushroom beds was to provide fertilizer to local farmers, but as the number of waste mushroom beds has increased (currently more than 1 million per year), their effective use has become a serious problem. Therefore, recently, the use of waste mushroom beds as fuel has commenced by mixing these beds with the bark fuel of a woody biomass boiler that produces steam used for sterilization of the cultivation bed, as well as for providing hot water used for heating the cultivation house in winter. However, there are many things to be clarified, such as the characteristics of the waste mushroom bed as a fuel, which are namely the moisture content, ash content, calorific value, and so on. Conversely, it may be used as a material by taking advantage of the morphological characteristics of the waste mushroom bed. Figure 2 compares the state of accumulation of wood chips between at the time of cultivation bed preparation and at the time of disposal. When the waste mushroom bed is dried, the mycelia play a role of an adhesive and the decaying chips are bonded to each other, resulting in a structure similar to a lightweight particleboard or an insulation fiberboard, and after removing the skin, a rectangular block-shaped material can be obtained.
Production process of shiitake mushroom using hardwood chip cultivation beds
Changes in the structure of a mushroom bed
The purpose of this series of studies is to collect basic data for fuel and material utilization of the waste mushroom bed shown in Fig. 2 (right). As for material utilization, we will report the mechanical properties and thermal insulation properties of the blocks obtained by drying the waste mushroom beds in the 2nd and 3rd reports, respectively.
In the present 1st report, the moisture content of waste mushroom beds at the time of disposal and after 1 month, as well as the ash content, elemental composition, and calorific value, are shown as basic properties for fuel utilization. In particular, the calorific value of the waste mushroom bed is discussed in comparison with that of undecayed wood and the mycelium itself. Then, differences in the calorific values among these materials are also discussed from the viewpoint of their elemental composition. Furthermore, the substance flow from mushroom bed preparation to disposal is discussed through the results of elemental analysis and an ash content test.
Details of the cultivation bed
It is necessary to know the details of the cultivation beds, because the fuel or material properties of the waste mushroom beds are dependent on them. Therefore, the tree species and density of the raw wood, the size of the pieces after chipping, and the weight of the cultivation bed were investigated at the shiitake cultivation farm shown in Fig. 1. More than 90% of the hardwood logs were mizunara (Quercus crispula). A total of 20 mizunara disks with a diameter of 8–20 cm and a thickness of 2 cm were collected from a log stockyard, dried to an air-dried state (13% of the moisture content on a dry basis, MCd), and their density was measured. In addition, 50 pieces of wood chip samples were randomly extracted from a chip stockyard, dried to an air-dry state, and the dimensions were measured. Furthermore, 30 cultivation bed samples were randomly extracted in each of Step 3 and 5 shown in Fig. 1 and they were then weighed.
The results of the above measurements are listed in Table 1. The average dimensions of wood chips were 10.1 mm in width, 1.9 mm in thickness, and 7.7 mm in length, which are much smaller than those dimensions of the wood chips used in the paper industry. In Step 2 of Fig. 1, nutrients and water are added to the wood chips, and the moisture content on a wet basis, MCw, is adjusted to about 60%. Nutrients (details will be described later) make up about 10% of the total weight of the cultivation bed. The average weight of the cultivation bed including a plastic forming bag was 2888 g before steam sterilization. The average weight after steam sterilization decreased by 36 g to 2852 g although about 10–13 g of inoculative fungus was added. The dimensions of the cultivation bed were about 12 cm in width, 20 cm in length, and 10 cm in height.
Table 1 Details of the cultivation bed (mean ± std)
As shown in Fig. 1, it takes about 4 months before fungus spread fully into the cultivation bed after fungus inoculation, and then shiitake harvesting is repeated six times with an interval of 1 month. The total yield of shiitake mushroom from one cultivation bed was 913 g (2017 results), and the cultivation beds become waste mushroom beds about 11 months after preparation.
Moisture content tests
Within 3 days of the last harvest, 100 waste mushroom bed samples were randomly extracted from a cultivation house, taken out of the plastic forming bag, and weighed. A histogram of the weight was created with a class of 50 g, and then about 1/3 of the samples were extracted from each class and their MCw were obtained by the oven-drying method (N = 36).
Another moisture content test was performed on the waste mushroom beds that had been left for 1 month to examine their drying properties. After the last harvest, they were transferred from a cultivation house to a well-ventilated greenhouse with a concrete soil for a waste yard. Although the temperature in the greenhouse was unknown, the average temperature in the region during this period was 19 ºC according to the local meteorological data. The same testing procedure as the sampling within 3 d after disposal was applied to these waste mushroom beds and their MCw were determined (N = 39).
Preparation of powder samples
Powder samples were used in the tests of elemental analysis, ash content, and calorific value, which are described below. There are four types of samples: the skin of the waste mushroom bed, the inside of the waste mushroom bed, the mycelium, and the wood chips used for the cultivation bed. About 100 g of chip samples were prepared for both the skin and the inside from several waste mushroom beds within 3 d after disposal. The skin chips were collected by scraping the surface of the waste mushroom bed with a knife. The chip samples of mycelium were prepared from the stem part of the shiitake fruiting body, because it was difficult to take out only the mycelium from the inside of the waste mushroom bed. First, these four chip samples in an air-dry condition were powdered using a Willey mill. Then, they were classified by a three-stage sieve, and the following three kinds of fractions were prepared: F1: 0.5–2 mm; F2: 0.15–0.5 mm; F3: less than 0.15 mm (an F3 sample is used for the experiments in the third report of this study).
Elemental analysis and ash content tests
Carbon (C), hydrogen (H), and nitrogen (N) were quantified using a fully automatic elemental analyzer (Yanoko CHN Corder MT-6). F2 powder of each sample was used and the samples were left for 5 d in an air-conditioned laboratory where the device was placed to prevent moisture absorption/desorption during analysis. Approximately, 2 mg of each sample was used for each analysis and weighed to within an accuracy of 1 µg. Three replicate analyses were conducted, and the ash content was also measured automatically through the analysis. To calculate the oven-dry weight of the samples, the moisture content on a dry basis, MCd, during the analysis was determined by an oven-drying method using about 2 g of the remaining sample.
After the analysis, however, three types of samples other than the mycelium showed a small amount of detected nitrogen, hence other highly accurate tests were necessary. Therefore, nitrogen was quantified for these samples using another analyzer (SUMIGRAPH NC-22A). About 100 mg of oven dry F2 sample was used for each analysis and weighed with an accuracy of 10 µg. Three replicate analyses were performed.
Among ash measurements obtained through the elemental analysis, the wood chips did not meet the precision required for analysis due to the lack of a detected amount of ash. Therefore, an ash content test by the usual determination method [4] was added. About 1 g of oven dry F1 sample was weighed in a melting pot with an accuracy of 0.1 mg. This was heated at 600 ºC for 8 h to obtain ash, and then the ash weight was measured with the same accuracy. The ratio of the dry ash weight to the oven dry sample weight was defined as the ash content (%). Five replicate tests were performed for the three types of samples other than the mycelium.
Calorific value tests
In accordance with JIS M8814 [5], a gross calorific value (Hh) was determined for the four types of samples using an automatic bomb calorimeter (Shimadzu CA-4P). Approximately 1 g of F1 sample was weighed per test with an accuracy of 0.1 mg, and three replicate tests were performed. Furthermore, the moisture content at the time of the test, MCw, was measured by an oven-drying method using 2–3 g of the sample. The MCw was used to calculate gross calorific values on a dry basis.
Moisture contents of waste mushroom beds
Figure 3 shows two weight histograms of a waste mushroom bed. One is within 3 days after disposal and the other is 1 month later. Statistical data for the wet weight of 100 samples and on an oven dry weight of the extracted samples are listed in Table 2. First, focusing on within 3 days after disposal, the wet weight was 1235 g on average and had a coefficient of variation (COV) of 14%. The variation on the wet weight is affected not only by the amount of moisture that is included but also by the weight of the substance because the oven dry weight was not constant and had a COV of about 9%. Their MCw was about 78% on average and had a COV of about 6%.
Weight histogram of waste mushroom bed (N = 100 for each)
Table 2 Weight and moisture content of waste mushroom bed (mean ± std)
Second, focusing on 1 month after disposal, the wet weight was 772 g on average and had a COV of 16%. As discussed above, this variation is affected not only by the amount of moisture included but also by the weight of substance because the oven dry weight was not constant and had a COV of about 16%. Their MCw was about 63% on average with a COV of about 11%. The average MCw decreased from 78 to 63% after being left for about 1 month. The moisture content on a dry basis is easier to intuitively understand the drying properties from, so when converted, its MCd decreases from 379 to 181%. This indicates that the amount of moisture was about 3.8 times the substance weight within 3 d after disposal, and 1.8 times after being left for 1 month. This also suggests that a one-month natural drying process is not long enough to obtain a good woody biomass fuel. The slow drying is presumed to be due to the low moisture permeability of the skin of the mushroom bed. As will be described later, the skin is browned due to the deposition of melanin pigment. It is speculated that this browning results in the low moisture permeability. Therefore, when the waste mushroom bed is used as a boiler fuel, it is preferable to expose the inside through a crushing process followed by drying.
Focusing on the oven dry weights shown in Table 2 again, they had a COV of about 9% and 16% for the samples within 3 days and on 1 month after disposal, respectively. Although the reason for the difference of COV between the two sample groups is not clear, these variation means that the degree of wood decay varies to some extent from sample to sample. This suggests that there will be a concern of density variation when using as a block material shown in Fig. 2, while it is not a big problem when using as boiler fuel.
Ash contents and elemental composition
Figure 4 shows the results of the ash content tests. The wood chips showed a value of about 0.7%, which is close to the values reported [6] for live oak (0.6%) and black oak (0.5%). Conversely, the ash content of waste mushroom bed was about 7%, which is 10 times the value of wood chips. There was no significant difference in ash contents between the inside and the skin, and the average ash content became 6.78% when the law of mixing was applied assuming a weight ratio of inside to skin of 10:1. The waste mushroom bed is a mixture of decayed wood chip residues and mycelium. The ash content of the mycelium itself was 3.4%, while that of the waste mushroom bed was about double. The reason for this is considered to result from the nutrients added to the wood chips during the cultivation bed preparation. The ash content of nutritional supplements is as high as 4–8% (details will be described later). During shiitake cultivation, the elements that make up the ash move to some extent in the fruit bodies that are repeatedly harvested, and they are lost in small amounts by elution into supplemental water. Except for these two systems, the ash constituent elements remain in the mushroom bed, and it is considered that the ash content is concentrated due to the decrease in the overall weight due to biodegradation. The high ash contents of the waste mushroom bed measured in this study are comparable to those of hardwood bark. For example, Kofujita [7] reported that the ash contents of mizunara (Quercus crispula) bark and buna (Fagus crenata) bark were 5.7% and 7.3%, respectively. The disadvantage of using bark as a boiler fuel is the high ash content compared to wood chips [8]. When using the waste mushroom bed as fuel, the treatment of ash will be necessary, as with bark.
Results of the ash content tests. Means with different letters are significantly different at P = 0.05 (Welch's t test)
Table 3 lists the results of the elemental analysis. The air-dry base contents of C, H, and N of each sample are shown together with their MCd values. These values were converted to oven-dry base values using the formulas shown below the table. Here, a simple moisture correction was done for C and N (only for mycelium), and in addition to this, bound water was deducted for H. Furthermore, the content of O was obtained by subtracting the sum of these three elements and ash from the entire system. The values for the whole waste mushroom bed in Table 3 were calculated from the law of mixing, assuming a weight ratio of inside to skin of 10:1. The carbon, hydrogen, and oxygen contents of the waste mushroom bed (whole) were slightly lower than those of the wood chips. This is thought to be due to the higher ash content than that of wood chips as discussed above. Conversely, the nitrogen content was about four times greater in the waste mushroom bed. This is because the nitrogen content of the skin is about ten times that of wood, in addition to the high nitrogen content inside of the waste mushroom bed. It is considered that the high nitrogen content of the skin is due to the inclusion of a dark brown melanin pigment. The formation of the melanin pigment is believed to occur, because mycelium protects its territory from other bacteria and prevents water loss [9].
Table 3 Results of the elemental analysis
These elemental composition data will be used later in the discussion of calorific value and in the calculation of the substance flow during shiitake mushroom bed cultivation.
Calorific values
The results of calorific value tests are as follows: Hh (mean ± std.) and MCw for the wood chips were 18.08 ± 0.28 MJ/kg and 7.9%, respectively; 17.55 ± 0.36 MJ/kg and 7.3% for the inside of the waste mushroom bed, respectively; 15.54 ± 0.65 MJ/kg and 11.1% for the skin of the waste mushroom bed, respectively; and 16.40 ± 0.02 MJ/kg and 9.2% for the mycelium, respectively. To compare the gross calorific values on a dry basis (Hho) among the samples, these Hh values were converted to Hho using the following equation:
$$H_{{{\text{ho}}}} = H_{h} \times {1}00/\left( {{1}00 \, {-}{\text{ MC}}_{w} } \right).$$
Figure 5 compares the Hho values among the samples. The Hho value of oak wood [10] was reported to be 18.4 to 22.1 MJ/kg, therefore the Hho value of the wood chips (19.63 MJ/kg) obtained in this study is reasonable. Although the Hho value of the inside of the waste mushroom bed was not significantly reduced compared to the wood chips, that of the skin was about 11% lower than that of the wood chips. Assuming a weight ratio of the inside to skin of 10:1, the Hho value for the whole waste mushroom bed is calculated to be 18.81 MJ/kg from the law of mixing, which was 4–5% lower compared to the wood chips. Focusing on the Hho value of the mycelium (18.06 MJ/kg), this value was 8% lower compared to the wood chips and had no significant difference compared to both the inside and the skin of the waste bed. This result suggests that the presence of mycelia is not the main factor that reduces the Hho of the waste mushroom bed.
Gross calorific values on a dry basis (Hho). Means with different letters are significantly different at P = 0.05 (Welch's t test)
Generally, the calorific value of a woody biomass fuel is affected by the amount of ash. Also, nitrogen in fuel does not contribute to heat generation. Therefore, the correction values (Hho') of the Hho values excluding ash and nitrogen were calculated using the following equation:
$$H_{{ho^{\prime}}} = H_{ho} \times {1}00/(100 - {\text{Ash content (\% ) }} - N(\% )).$$
Substituting the ash content (%) shown in Fig. 4 and N (%) shown in Table 3 into the Eq. (2), the values of Hho' for the wood chips, inside of the waste mushroom bed, skin of the waste mushroom bed, and mycelium were determined to be 19.79 MJ/kg (100%), 20.42 MJ/kg (103%), 19.14 MJ/kg (97%), and 19.15 MJ/kg (97%), respectively. Here, the value in parentheses is the ratio to the value of the wood chips. Compared to the differences among the samples shown in Fig. 5, the differences in the correction values were smaller. From this analysis, it was found that the differences in Hho among the samples were due to the amounts of ash and nitrogen.
The calorific value of a fuel is the sum of the combustion heat values of the combustible components, and the Hho value of wood and bark can be estimated by Eq. (3) in general [11].
$$H_{{{\text{ho}}}} = {33}.{94}C + { 142}.{5 (}H{-}O/8) ({\text{MJ/kg}}) .$$
Here, C, H, and O are the weights (kg) of carbon, hydrogen, and oxygen contained in 1 kg of oven dry fuel, respectively. The variable (H–O/8) of the second term is called the "effective hydrogen" of the system. The oxygen contained in the fuel is not used as oxygen for combustion, and usually a portion of the hydrogen is combined with this oxygen to form bound water. Therefore, the amount of hydrogen contained in the bound water (H2:O = 2: 16 = 1:8), that is, O/8, is subtracted from the amount of contained hydrogen. The values of H–O/8 using the results shown in Table 3 for the wood chips, the inside of the waste mushroom bed, the skin, and the mycelium were -0.0015 kg, − 0.0025 kg, − 0.0056 kg, and 0.0056 kg, respectively. From these results, it was found that there was no effective hydrogen other than the mycelium and that the heat generated by the effective hydrogen in the mycelium was only 0.8 MJ/kg. In other words, the mycelium can be expected to generate a small amount of heat from hydrogen, while the other three are all from carbon. The calculated values of Hho according to Eq. (3) are 16.56 MJ/kg (84%), 15.22 MJ/kg (80%), 15.69 MJ/kg (90%), and 16.21 MJ/kg (90%) for the wood chips, the inside of the waste mushroom bed, the skin, and the mycelium, respectively. Here, the values in parentheses are the ratios to the measured values shown in Fig. 5. The values calculated by Eq. (3) were underestimated by up to 16% of the actual measured value. A similar underestimation was observed when applying Eq. (3) to coniferous bark and wood [8]. Although Eq. (3) underestimates the measurements, it can clarify the properties of woody biomass fuels in which the presence of oxygen reduces the available hydrogen for combustion.
The gross calorific values obtained on a dry basis, Hho, which have been discussed so far, are a basic indicator of calorific value. The gross calorific value at a moisture content of MCw (%) is shown using Eq. (4), which is a modification of Eq. (1). These calorific values include the latent heat of condensation (Hs) of water vapor because the water vapor generated by combustion cannot escape outside of the bomb calorimeter. Usually, in boiler combustion, water vapor in exhaust gas is diffused into the atmosphere without being condensed. Therefore, the amount of heat that can be used is called the net calorific value (Hn), and this is the value obtained by subtracting Hs from Hh (Eq. (6)). The value of Hs can be calculated from Eq. (5) [8, 11]. Here, h is the weight ratio of hydrogen in the fuel. For example, from Table 3, the value of h in the waste mushroom bed (whole) is 5.05% at an MCw of 0% and 2.53% at an MCw of 50%.
$$H_{h} = H_{ho} (100 - {\text{MC}}_{\text{w}} )100$$
$$H_{s} = { 2}.{512 }\left( {{9} \times h + {\text{MC}}_{\text{w}} } \right)100 \left[ {{\text{MJ}}/{\text{kg}}} \right]$$
$$H_{n} = H_{h} {-}H_{s} \;[{\text{MJ/kg}}]$$
Figure 6 shows the relationship between Hn and MCw. It was found that the Hn of the waste mushroom beds was 4–5% lower than that of the original wood chips in any moisture content range. The waste mushroom beds will generate an Hn of 7.6 MJ/kg when they are dried up to about an MCw of 50%. This value is almost the same as the Hn of general woody biomass fuel. For example, Hn values at an MCw of 50% have been reported to be 7.5 MJ/kg and 8.0 MJ/kg for sapwood and bark of sugi (Cryptomeria Japonica), respectively [8].
Fig.6
Relationships between moisture content on a wet basis (MCw) and the net calorific value for both the waste bed (whole) and the original wood chips
Substance flow during shiitake mushroom bed cultivation
In this section, we discuss the substance flow during the shiitake cultivation by comparing the weight composition of the cultivation bed at the time of preparation and at the time of disposal.
First, the weight composition of one cultivation bed was determined using the following procedure (the results are shown in Table 4). In Step 2 of Fig. 1, wood chips with a volume of 6 m3 and 310 kg of nutrients (90 kg of rice bran and 220 kg of wheat bran) are placed into the mixer, and water is added to adjust the MCw to 60%. From this batch, 1080 cultivation beds are made. Therefore, the weight of one batch in the mixer become 3105 kg, because the average weight of one cultivation bed is 2875 g when the forming bag is deducted (see Table 1). Because the MCw is 60%, the moisture weight is 1863 kg, and thus, the total oven dry weight of the chips and nutrients is 1242 kg. To divide this oven dry weight into wood chips and nutrients, the oven dry weight of the nutrients is calculated via the following calculation. If the MCw values of rice bran and wheat bran are 10.3% [12] and 13.4% [13], respectively, the oven dry weights are calculated to be 80.7 kg and 190.5 kg, respectively. Therefore, the oven dry weight of the wood chips is 970.8 kg. When these values are divided by the number of cultivation beds (1080), the weight composition per cultivation bed is obtained as follows: moisture: 1725 g; wood chip oven dry weight: 898.9 g; rice bran oven dry weight: 74.8 g; and wheat bran oven dry weight: 176.4 g. Also, because the average weight of the inoculum is 11.5 g, if the MCw of the inoculum is equivalent to that of the shiitake fruiting body (90.3%) [12], the oven dry weight of the inoculum becomes 1.12 g. Summing these values together, the oven dry weight of the cultivation bed becomes 1151.2 g.
Table 4 Weight composition of the cultivation bed
Next, the weights of the four elements (C, H, N, and O) and ash constituting one cultivation bed in an oven dry state (1151.2 g) were determined by the following procedure. First, the composition ratios of C, H, N, O, and ash for wood chips, rice bran, wheat bran, and inoculum used in the calculation are shown in the upper half of Table 5. Here, the values for the wood chips and inoculum are the same as those shown in Table 3 and Fig. 4, and the values of the rice bran and wheat bran were obtained from literature [14, 15]. Second, the weights of the four elements and the ash in each component are calculated by multiplying their oven dry weight by the composition ratio, and then the total weights of each element and ash are calculated (see the lower half of Table 5). This calculation revealed that the ratio of elements to the oven dry weight of the cultivation bed was 48.4% for carbon, 5.7% for hydrogen, 0.6% for nitrogen, 43.6% for oxygen, and 1.7% for ash.
Table 5 Elemental weight composition of an oven-dry cultivation bed
Next, the weights of the four elements (C, H, N, O) and ash in one waste mushroom bed are determined. They were calculated by multiplying the average oven dry weight (249.1 g; within 3 days after disposal, see Table 2) by their composition ratio (whole bed) shown in Table 3 and an ash content of 6.78%. Table 6 compares the calculation results with the values of the cultivation bed. Table 6 also shows the values calculated from the total yield of the fruiting bodies. Because the yield of shiitake mushrooms per bed is 913 g, if its MCw is 90.3% [12], the oven dry weight will be 88.6 g. The weights of the four elements (C, H, N, and O) and ash in this dry weight were calculated from the ratios of the inoculum shown in Table 5.
Table 6 Comparison of the elemental weight composition between the cultivation bed and waste mushroom bed in an oven-dry base
From Table 6, the following can be observed regarding the substance flow in shiitake mushroom bed cultivation. First, there was a weight loss of about 78% from a cultivation bed weight of 1151 g to a waste mushroom bed weight of 249 g. This loss is mainly due to mycelium respiration and proliferation, fruiting body formation, and repeated fruiting body harvesting. In addition, it is considered that some of the amino acids were leached out and moved to wastewater due to the watering operation during cultivation. Second, the weight losses of the four elements C, H, N, and O were in the range of 79–81% and were close to each other. One of the reasons for this is considered to be that shiitake mushrooms are white-rot fungi and they attack all of the chemical constituents of the cell wall [16]. In other words, because lignin has a higher carbon content than cellulose and hemicellulose [8], it is considered that the carbon residual rate increases when lignin cannot be decomposed. To discuss the changes in these four elements in detail, it will be necessary to examine the changes in the major chemical components of wood before and after decay. Third, the weight loss of ash was about 14%, which was significantly lower than that of the four elements. As already mentioned in the section pertaining to the ash results, the ash was concentrated throughout a cultivation from 1.7% ash in the cultivation bed (see Table 5) to 6.8% ash in the waste mushroom bed. The fourth is the movement of nitrogen. Of the 7.4 g of nitrogen contained in the cultivation bed, 1.5 g remained in the waste mushroom bed and 2.0 g was transferred to the fruiting bodies. Therefore, it is considered that a difference of 3.9 g was leached as various amino acids into supplemental water during cultivation. Finally, we focus on the C/N ratio of the waste mushroom bed. Itoh [13] reported that the yield was maximized when the C/N ratio of the shiitake cultivation bed was 70–80. When the C/N ratio was calculated using the values shown in Table 6, it was about 75 for both the cultivation bed and waste mushroom bed. This result suggests the possibility that the waste mushroom bed can be mixed with the cultivation bed and reused.
The findings obtained from this work are summarized as follows:
The shiitake mushroom bed at the time of disposal contained about 3.8 times the oven dry weight of moisture, and even after 1 month of natural drying, it contained about 1.8 times the oven dry weight of moisture. The slow drying is presumed to be due to the low moisture permeability of the skin of the mushroom bed. Therefore, it is necessary to crush the waste mushroom bed and then dry it before using it as a boiler fuel.
The ash content of the waste mushroom bed was about 7%, which was close to that of bark and about ten times that of the wood used for the cultivation bed. When the waste mushroom bed is used as a boiler fuel, appropriate ash treatment is required as in the case of using bark.
The gross calorific value on a dry basis (Hho) inside of the waste mushroom bed was 18.9 MJ/kg, which was not significantly lower than that of cultivation bed wood chips. However, the Hho of the skin of the waste mushroom bed was significantly lower by 11% than that of the cultivation bed wood chips because of its higher ash and nitrogen contents. As a result, the Hho of the whole waste mushroom bed was 18.8 MJ/kg, which was 4–5% lower than that of the cultivation bed wood chips. The Hho of mycelium itself was 18.1 MJ/kg, which was significantly lower by 8% than that of the cultivation bed wood chips due to its significantly higher nitrogen and ash contents. When analyzed considering the combustion heat of the contained elements, it was found that both the cultivation bed wood chips and the waste mushroom bed had almost no hydrogen contributing to combustion due to the high oxygen content, and they were dependent on the heat generation of carbon.
The relationship between the net calorific value and moisture content on a wet basis (MCw) was obtained for the waste mushroom bed. The net calorific value was 4–5% lower than that of the cultivation bed wood chips at any moisture content level, and it was 7.6 MJ/kg at an MCw of 50%, which was almost the same as that of sugi (Cryptomeria Japonica) sapwood and bark.
Comparing the oven dry weight of the waste mushroom bed with that obtained at the time of preparing the cultivation bed, the weight loss was about 78%. The weight losses of the four elements C, H, N, and O were in the range of 79–81% and were close to each other. Conversely, the weight loss of ash was about 14%, which was significantly lower than that of the four elements, such that the ash was concentrated through cultivation from 1.7% in the cultivation bed to 6.8% in the waste mushroom bed.
MCw :
Moisture content on a wet basis
MCd :
Moisture content on a dry basis
H h :
Gross calorific value
H ho :
Gross calorific value on a dry basis
H n :
Net calorific value
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This work was supported by JSPS KAKENHI Grant Number JP19K06162. The authors would like to thank Koshido Mushroom Co., Ltd. for providing waste mushroom bed samples. We would like to thank Ms. Mioh Toyomane, an undergraduate student at Iwate University, for her cooperation in collecting the experimental data. We would like to thank Prof. Hisayoshi Kofujita, a specialist of wood chemistry of Iwate University for his valuable advice. We also would like to thank Uni-edit (https://uni-edit.net/) for editing and proofreading this manuscript.
This work was supported by JSPS KAKENHI Grant Number JP19K06162.
Faculty of Agriculture, Iwate University, Morioka, 020-8550, Japan
Noboru Sekino
Iwate University, Morioka, 020-8550, Japan
Zhuoqiu Jiang
NS designed the study and analyzed the data, and was a major contributor in writing the manuscript. ZJ conducted calorific value tests and contributed to writing the manuscript. All authors read and approved the final manuscript.
Correspondence to Noboru Sekino.
Sekino, N., Jiang, Z. Fuel and material utilization of a waste shiitake (Lentinula edodes) mushroom bed derived from hardwood chips I: characteristics of calorific value in terms of elemental composition and ash content. J Wood Sci 67, 1 (2021). https://doi.org/10.1186/s10086-020-01935-7
Received: 25 September 2020
Accepted: 15 December 2020
Waste mushroom bed
Hardwood chips
Ash content
Elemental composition | CommonCrawl |
The computational complexity of spectral norm of a matrix
How hard is computing the spectral norm of a matrix? This paper says,
... it suffices to say that, except for few particular cases, the Matrix Norm problem is NP-hard.
I expected that the relevant chapter 2 would describe those exceptional cases but failed to find it.
Can anyone suggest a more definitive reference for computational complexity of spectral norm of a matrix?
np-hardness quantum-computing matrices quantum-information norms
Omar ShehabOmar Shehab
$\begingroup$ Spectral norm is the maximum singular value of the matrix, and can thus be computed in polynomial time, say by computing the singular value decomposition. $\endgroup$ – MCH May 28 '13 at 14:55
$\begingroup$ In short - the thesis is about more general norms, which are often hard. $\endgroup$ – Yuval Filmus May 28 '13 at 15:27
The answer to your question is the contents of section 1.3.2, titled "[w]hen $\mathcal{P}_{p,r}$ is known to be difficult". (Here $\mathcal{P}_{p,r}$ is the problem of computing the norm $\|A\|_{p,r} = \sup_{\|x\|_p=1} \|Ax\|_r$.) According to that section, the only cases which are known to be difficult are $\mathcal{P}_{\infty,1},\mathcal{P}_{\infty,2},\mathcal{P}_{2,1}$. For example, $\mathcal{P}_{\infty,1}$ (even restricted to positive semidefinite matrices) is a generalization of MAX CUT. Since $\|B'B\|_{\infty,1} = \|B\|_{\infty,2}^2$, $\mathcal{P}_{\infty,2}$ is also hard. Finally, $\mathcal{P}_{2,1}$ is as hard as $\mathcal{P}_{\infty,2}$ as part of the more general observation (proved in section 1.3.1) that $\mathcal{P}_{p,r}$ is as hard as $\mathcal{P}_{1/(1-1/r), 1/(1-1/p)}$.
The thesis goes on to prove that $\mathcal{P}_{p,r}$ is hard whenever $p > r$ - this is the chapter you were reading (chapter 2).
Section 1.3.1 described some easy cases: $\mathcal{P}_{1,\ast}$, the symmetric $\mathcal{P}_{\ast,\infty}$, and the case that MCH mentioned, $\mathcal{P}_{2,2}$. Section 1.3.3 covers some approximability results, several novel of which are described in section 1.4 and the remaining chapters.
The title of section 1.3.2 appears in the table of contents (page iii) - just a hint for next time.
$\begingroup$ It's a bit strange the thesis does not mention that an approximation to $\|A\|_{\infty \rightarrow 1}$ follows from Grothendieck's inequality, and the related algorithmic work of Alon and Naor. Krivine's bound $\pi/2\ln(1 + \sqrt{2})$ in Alon-Naor is better than the constant in Prop 1.5. $\endgroup$ – Sasho Nikolov May 28 '13 at 17:35
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Threshold dynamics of a general delayed within-host viral infection model with humoral immunity and two modes of virus transmission
Periodic, almost periodic and almost automorphic solutions for SPDEs with monotone coefficients
Modeling error of $ \alpha $-models of turbulence on a two-dimensional torus
Luigi C. Berselli 1, , Argus Adrian Dunca 2, , Roger Lewandowski 3, and Dinh Duong Nguyen 3,,
Università di Pisa, Dipartimento di Matematica, Via Buonarroti 1/c, I-56127 Pisa, Italy
Department of Mathematics and Informatics, University Politehnica of Bucharest, Bucharest, Romania
IRMAR, UMR CNRS 6625, University of Rennes 1 and FLUMINANCE Team, INRIA, Rennes, France
Received March 2020 Revised August 2020 Published October 2020
This paper is devoted to study the rate of convergence of the weak solutions $ {\bf u}_\alpha $ of $ \alpha $-regularization models to the weak solution $ {\bf u} $ of the Navier-Stokes equations in the two-dimensional periodic case, as the regularization parameter $ \alpha $ goes to zero. More specifically, we will consider the Leray-$ \alpha $, the simplified Bardina, and the modified Leray-$ \alpha $ models. Our aim is to improve known results in terms of convergence rates and also to show estimates valid over long-time intervals. The results also hold in the case of bounded domain with homogeneous Dirichlet boundary conditions.
Keywords: Rate of convergence, $ \alpha $-turbulence models, Navier-Stokes equations.
Mathematics Subject Classification: Primary:35Q30, 35Q35;Secondary:65M15, 76D05, 76F65.
Citation: Luigi C. Berselli, Argus Adrian Dunca, Roger Lewandowski, Dinh Duong Nguyen. Modeling error of $ \alpha $-models of turbulence on a two-dimensional torus. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2020305
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Fluids and Barriers of the CNS
Changes in the cerebrospinal fluid circulatory system of the developing rat: quantitative volumetric analysis and effect on blood-CSF permeability interpretation
Jean-François Ghersi-Egea1,2,
Anaïd Babikian1,
Sandrine Blondel1 &
Nathalie Strazielle2,3
Fluids and Barriers of the CNS volume 12, Article number: 8 (2015) Cite this article
The cerebrospinal fluid (CSF) circulatory system is involved in neuroimmune regulation, cerebral detoxification, and delivery of various endogenous and exogenous substances. In conjunction with the choroid plexuses, which form the main barrier site between blood and CSF, this fluid participates in controlling the environment of the developing brain. The lack of comprehensive data on developmental changes in CSF volume and distribution impairs our understanding of CSF contribution to brain development, and limits the interpretation of blood-CSF permeability data. To address these issues, we describe the evolution of the CSF circulatory system during the perinatal period and have quantified the volume of the different ventricular, cisternal and subarachnoid CSF compartments at three ages in developing rats.
Immunohistofluorescence was used to visualize tight junctions in parenchymal and meningeal vessels, and in choroid plexus epithelium of 19-day fetal rats. A quantitative method based on serial sectioning of frozen head and surface measurements at the cutting plane was used to determine the volume of twenty different CSF compartments in rat brain on embryonic day 19 (E19), and postnatal days 2 (P2) and 9 (P9). Blood-CSF permeability constants for sucrose were established at P2 and P9, following CSF sampling from the cisterna magna.
Claudin-1 and claudin-5 immunohistofluorescence labeling illustrated the barrier phenotype acquired by all blood–brain and blood-CSF interfaces throughout the entire CNS in E19 rats. This should ensure that brain fluid composition is regulated and independent from plasma composition in developing brain. Analysis of the caudo-rostral profiles of CSF distribution and of the volume of twenty CSF compartments indicated that the CSF-to-cranial cavity volume ratio decreases from 30% at E19 to 10% at P9. CSF compartmentalization within the brain changes during this period, with a major decrease in CSF-to-brain volume ratio in the caudal half of the brain. Integrating CSF volume with the measurement of permeability constants, adds to our understanding of the apparent postnatal decrease in blood-CSF permeability to sucrose.
Reference data on CSF compartment volumes throughout development are provided. Such data can be used to refine blood-CSF permeability constants in developing rats, and should help a better understanding of diffusion, bulk flow, and volume transmission in the developing brain.
The cerebrospinal fluid represents 50% of the brain extracellular fluid in adult mammals including humans. Recent findings indicate that the CSF circulatory system is not a mere passive drainage system for the brain, but is involved in neuroimmune regulation [1-3], cerebral detoxification [4], and cerebral delivery of various endogenous or exogenous molecules [5-7]. CSF circulation is complex. In adult, the fluid moves fast within the ventricular system, and reaches the cisterna magna and lateral recesses of the fourth ventricle through the foramina of Magendie and Luschka. From there it slows down while flowing through the subarachnoid and cisternal spaces, which occurs in three main directions. The CSF circulates 1) in a caudo-rostral direction, either dorsally around the cerebellum and cortex, or centrally through the internal cisterns, which in rodents are mainly formed by the ambient and quadrigeminal cisterns located between the midbrain and hippocampi/cortices, 2) in a caudo-rostral direction and ventrally through the cerebellopontine, interpeduncular, optic tract, and laminae terminalis cisterns, and around the olfactory bulbs, and 3) caudally along the spinal cord. In places such as the velum interpositum, ventricular and cisternal spaces are separated from each other only by a thin membrane which allows direct exchanges of material between the two fluid compartments [8,9].
The CSF is secreted and protected from peripheral harmful molecules by the choroid plexuses which develop from the neuroepithelium to form the major site of the blood-CSF barrier. Choroid plexuses also have the capacity to secrete into CSF different bioactive substances including guidance molecules as well as growth and differentiation factors. Owing to the early fetal development of the choroidal tissue, the choroid plexus-CSF system is likely to fulfill essential functions for brain development [10-12]. The current lack of comprehensive data on developmental changes in CSF spaces still impairs our understanding of CSF contribution to brain development, and limits the interpretation of blood-CSF permeability data. To address the early role of the CP-CSF system we first examined whether tight junction proteins are present at all brain barriers including CSF-bathed meningeal and cisternal vessels in embryonic day 19 (E19) rat fetuses, a requirement for generating a brain fluid environment which is controlled and independent from plasma composition before birth. Using a quantitative method to measure the volumes of all CSF compartments in rats at E19 and postnatal day 2 (P2) and 9 (P9), we describe the evolution of the CSF circulatory system in the developing brain during the perinatal period. We then applied some of these volume data to refine the interpretation of blood-CSF permeability constants in developing rats.
Tissue collection
Animal care and procedures were conducted according to the guidelines approved by the French ethical committee (decree 87–848), and by the European Community (directive 86-609-EEC). Sprague–Dawley rats, either pregnant time-dated females, or females with their litter, were obtained from Janvier (Le Genest Saint Isle, France). All animals were kept under identical conditions in standard cages, with free access to food and tap water under a controlled environment (12 h day-light cycles). Timed-pregnant female rats were anesthetized with inhaled isoflurane (5%) and body temperature was maintained with a heated pad. E19 animals were removed one by one from the mother and frozen in 2-methylbutane cooled at −50°C, in a position that allowed serial sectioning of the brain. P2 and P9 animals were decapitated and the severed head immediately frozen in 2-methylbutane cooled at −50°C. Limited expansion of the brain at the time of freezing led occasionally to a partial crack in the cranial bones. These brains were discarded from the analysis. Animal heads were kept at −80°C until use for serial sectioning.
Immunohistochemical analysis of claudins in E19 rat brains (n = 3) was performed as previously described [13], using anti-claudin-1 polyclonal rabbit antibody 51–9000 and anti-claudin-5 mouse monoclonal antibody 35–2500 (Invitrogen, Carlsbad, CA, USA). Primary antibodies were diluted and used overnight at 4°C at a final concentration of 0.625 μg/ml for claudin-1, and 2 μg/ml for claudin-5. Alexa 555®-conjugated goat anti-mouse antibody A-21424 and Alexa 488®-conjugated goat anti-rabbit antibody A-11034 (Invitrogen) were used at a final concentration of 2 μg/ml at room temperature for 1 hour. Diamidine-2-phenylindole-dihydrochloride (DAPI, 236276, Roche Diagnostics, Manheim, Germany) was used as a fluorescent nuclear stain (0.1 μg/ml in saline phosphate buffer for 10 minutes at room temperature). Immunofluorescence was viewed and analyzed using an Imager Z1 fluorescence microscope equipped with a MozaiX motorized module, a Z-stack apotome system and a Digital Camera (Zeiss, Jena, Germany). Images were acquired using the AxioVs40 V 4.8 software (Zeiss).
Morphometric analysis
The whole head (n = 4 for E19 and P9, n = 5 for P2) with intact meninges and CSF spaces was cut into 35-μm-thick serial sections using a CM 3050S cryostat (Leica, Nussloch, Germany). After every fifth section, a photograph of the remaining tissue block in the cryostat was taken with a sharp focus on the cutting plane using a VR-320 camera (Olympus, Tokyo, Japan). On these photographs, the tissue could be easily distinguished from the crystal-clear CSF present in the cisternal, subarachnoid, and ventricular spaces. Each photograph included a scale bar added in the cutting plane to enable quantification of the surface area of the cerebral structures and spaces of interest. In addition, every fifth section was collected on a glass slide and stained with hematoxylin and floxin. Micrographs of the sections were taken using a AxioCam ERc5s camera (Zeiss) connected to a SMZ800 stereotaxic microscope (Nikon, Amsterdam, Netherlands). The surface areas of the cranial cavity and of the CSF spaces of interest were delineated on the photographs and determined using the ZEN 2012 software (Zeiss). Micrographs of the stained sections were used to determine the surface area of the smaller intraparenchymal CSF-filled compartments (such as the third ventricle), when the borders were not sharp enough to be outlined in the tissue block photographs.
The total brain tissue volume and the volume of each fluid compartment were calculated by building the area under the surface area-distance curve, between the section where the given structure appears and the section where it disappears. Calculation was done using the composite trapezoidal rule for integration. CSF compartment volumes and tissue volumes were corrected by factors of 0.910 and 0.920, respectively, to account for water expansion at the time of freezing. The correction factor for tissue was adjusted for a water content in neonatal rat brain tissue of 88.5% [14,15]. The extremities of the brain cavity were defined as the appearance of the cerebellar subarachnoid space, and the disappearance of the olfactory bulbs.
Protein measurement
Choroid plexuses from individual animals (n = 6 for both P2 and P9) were micro-dissected under a stereomicroscope and digested in 1 M sodium hydroxide. Protein concentration was determined by the method of Peterson [16] using bovine serum albumin as reference protein for the standard curve.
Blood-CSF permeability measurement to [14C]-sucrose
Radio labelled [14C]-sucrose (435 mCi/mmol, Hartmann Analytic, Braunschweig, Germany) was administrated by intraperitoneal injection (12.5nCi/g) under slight gas anaesthesia, to prevent backflow. Blood was sampled from different animals of the same litter (n = 5 for P2 and n = 9 for P9) by cardiac puncture under pentobarbital anesthesia at times ranging from 3 to 30 minutes after injection. Additional animals from the litter were injected with 50nCi/g [14C]-sucrose. In these animals, blood was sampled twenty minutes after injection, rapidly followed by CSF sampling as follows. The skin was incised above the cistern magna, and a glass pipette was introduced in the cistern for CSF collection. The CSF was transferred in a microtube and the sample volume measured. Blood was collected in a heparinized tube and centrifuged at 5000 rpm for 5 min. Plasma and CSF samples were analyzed for radioactivity in a 1600 TR Packard scintillation counter, using Ultima-gold (Perkin-Elmer, Waltham, MA) as scintillation cocktail.
A litter-based area under the plasma [14C]-sucrose concentration-time curve (AUC) [17] was built from 0 to 30 minutes for each developmental stage using the composite trapezoidal rule for integration. Two permeability constants K in csf and K w csf were calculated as follows:
$$ {{\mathrm{K}}_{\mathbf{in}}}_{\mathrm{csf}}={\mathrm{C}}_{\mathrm{t}}/\mathrm{AU}{\mathrm{C}}_{0\to \mathrm{t}} $$
where Ct is the [14C]-sucrose concentration in CSF at the time of sampling t (ranging from 20 to 23.5 minutes), AUC 0→t the AUC recalculated from the litter-based experimental AUC and from the plasma concentration measured before CSF sampling. K in csf is similar to an influx constant as defined previously for brain tissue.
$$ {{\mathrm{K}}_{\mathbf{w}}}_{\mathrm{csf}}={{\mathrm{K}}_{\mathbf{in}}}_{\mathrm{csf}}\times {\mathrm{V}}_{\mathrm{csf}} $$
where Vcsf is the total volume of CSF in the brain. K w csf represents the plasma volume cleared in the whole CSF of the animal.
Antibodies against claudin-5, the hallmark constituent of tight junctions at the blood–brain barrier stained the entire microvascular network throughout the brain as well as endothelial junctions of meningeal vessels in E19 rat fetuses (Figure 1). The epithelial layer in all choroid plexuses was immunoreactive for claudin-1, the major protein of tight junctions at the blood-CSF barrier. This highlights that the barrier phenotype is acquired by blood–brain interfaces throughout the entire CNS in rat before birth. This stage can be compared to mid-gestation in humans. Staining of tight junction proteins in choroid plexus and meningeal vessels, but not in ependyma, infers that CSF-filled compartments are an integral part of the CNS at this developmental stage. In line with previous reports [11,13], it also suggests that the brain fluid composition is tightly regulated during development and is independent from plasma composition, a prerequisite for the CSF to fulfill its function in brain maturation. To better appreciate the extent of the CSF circulatory system and its implication in CNS physiology during perinatal development, we undertook a quantitative morphometric analysis of its compartments at E19, P2 and P9.
Immunohistofluorescence analysis of claudin-1 and claudin-5 in E19 rat brain. A and B: forebrain. The vascular network is labeled by claudin-5 antibodies (red), while both the third and lateral ventricles choroid plexuses are labeled by claudin-1 antibodies (green). C and D: Hindbrain. The cerebellar and medullar vascular networks are labeled by claudin-5 antibodies, while both the central and lateral parts of the fourth ventricle choroid plexus are labeled by claudin-1 antibodies. Note the intercellular junction labeling of the meningeal vessels (long arrows) such as that located in the ambient cistern (enlarged in B). Claudin-1 immunoreactivity at the periphery of choroidal epithelial cells is clearly seen in the enlarged fourth ventricle area (D). Claudin-5 immunoreactivity is found in penetrating choroidal vessels (arrowhead), but not in the terminal vascular loops of choroidal villi. The ependyma (Ep) is devoid of staining. Other abbreviations: AmbCi: ambient cistern, LV, 3 V, 4 V: Lateral, third and fourth ventricles, respectively.
Methodology of the quantitative morphometric analysis
Brain ventricles of developing rats have previously been visualized by magnetic resonance imaging in animals suffering from hydrocephalus [18]. They were not observable in control animals at that time. Although the technique has since been adapted to small animal brain analysis, resolution remains a limiting factor to distinguish and to quantify the volumes of the various cisternal, subarachnoid, and ventricular compartments in normal developing rodents. We therefore conducted our morphometric study by a conventional approach based on the analysis of brain sections, with two distinct features. First, we generated these sections from whole frozen heads. Because of this unique aspect in the procedure, CSF is trapped within the subarachnoid, cisternal, and ventricular compartments. The shape and volumes of these spaces remain unchanged, with the exception of limited expansion of water at the time of freezing (Figure 2 and [8]). Second, photographs of the frozen tissue block taken at the cutting plane were preferred to micrographs of stained sections for surface area measurement, because on the latter the larger fluid compartments were easily distorted in the process of cutting and drying. Stained sections were used only for quantifying small fluid spaces when their boundaries could not be easily outlined on photographs. Examples of both cutting planes and stained sections covering a selection of CSF compartments from cerebellar to olfactory bulb subarachnoid spaces are shown in Figure 2. The size of CSF compartments changed between stages. To quantify these changes, we measured the total volume of the cranial cavity (from the cistern magna to the end of the olfactory bulbs), and the volume of twenty ventricular, cisternal, and subarachnoid compartments (listed in Table 1) at the three developmental stages. These two sets of data were used to calculate brain tissue volumes. All volumes were then corrected for water expansion at the time of freezing (see Method section). Brain tissue volume measured using this method was not statistically different from brain volume deduced from the weight of fresh brain sampled from age- and weight- matched animals, assuming an overall density of 1 (data not shown). This correlation provides a good index of measurement accuracy. The complete set of volume data obtained for the twenty compartments at the three developmental stages is displayed in a table see Additional file 1. It should be noted that with the exception of the ventricular system and lateral recesses of the fourth ventricle, all spaces of interest contain, in addition to CSF, arachnoid trabeculae, immune cells, and vessels traveling through the spaces before entering the brain. When appearing on sections, the major venous sinuses were excluded from the cisternal surface area measurements (e.g. Figure 2, A1 and C1). The CSF contained in the Virchow-Robin spaces which are located around penetrating vessels and are connected to the main subarachnoid and cisternal spaces could not be included, as the resolution of the method does not allow their visualization. Histological images of all the CSF spaces analyzed in this paper can be found in previous publications for the adult rat [3,8,19].
Selected examples of cutting-plane photographs and histological sections used for morphometric analysis. Images are from E19 (A1-A3), P2 (B1-B3) P9 (C1-C3) animals. Fluid spaces and cranial cavity are delineated by a blue line. See Table 1 for abbreviations. Scale bar: 1 mm.
Table 1 CSF compartments identified and analyzed in the study, and their corresponding abbreviations
Developmental changes in CSF compartment volumes
The caudo-rostral profiles of the cranial cavity, brain tissue, and overall CSF spaces are shown for all three developmental stages in Figure 3. The volume values generated for these three parameters are listed in the adjacent tables. While the cranial cavity increased 1.9- and 5 times from E19 to P2 and P9, respectively, the total CSF volume increased only moderately over the same period. This translates into a decrease of the CSF-to-cranial cavity volume ratio from 28% at E19 to 20% at P2 and 10% at P9. The profiles show that CSF compartmentalization changes during development. There is a substantial decrease in the CSF space compared to the cranial cavity, which is mostly notable in the caudal half of the brain. We subdivided the CSF spaces in 7 main compartments to appreciate more precisely the geographic distribution of CSF over the developmental period (Table 2). Data expressed as a percentage of cranial cavity (left columns) indicated that the cerebellar subarachnoid spaces, ventricles, internal cisterns, and caudal part of the cortical subarachnoid spaces contribute most of the developmental decrease in CSF-to-cranial cavity volume ratio. Results expressed as a percentage of total CSF (right columns) show how the fluid distributes at each developmental stage. CSF distribution among the seven compartments differed more between E19 and P2 than between P2 and P9. A large part of the CSF was found in the internal cisterns at all stages. The cerebellar subarachnoid spaces and the caudal part of cortical subarachnoid spaces were also major CSF compartment at E19. The remote forebrain subarachnoid spaces and the hindbrain spaces mostly formed by the lateral recesses of the fourth ventricle became prominent in P9 animals. This resulted from the decrease in volume of other CSF compartments (relative to the cranial cavity volume). The lateral recesses of the fourth ventricle formed relatively large CSF spaces. They are generally not observable when the brain is sampled by conventional procedure, but were easily visualized by the method used in this study. These spaces are transitional areas between the ventricular and subarachnoid/cisternal system, in that no trabeculae or vessels pass through, and the border between the CSF and neuropil is not ependyma but the glia limitans. They extend dorsally along the medulla to become the spinal cord subarachnoid space. The latter also was relatively large in developing animals. In the anterior part of the spinal cord it ranged on average from 40% (E19) to 30% (P9) of the spinal canal volume (data not shown).
Caudo-rostral profiles of brain tissue volume and CSF space in the developing rat brain. Upper panel: 19-day-old embryo (E19), middle panel: 2-day-old rat (P2), lower panel: 9-day old rat (P9). Graphs show typical profiles combined from at least 3 animals. Positions of the cerebellum (Cb) and olfactory bulb (Ob) are indicated. Cranial cavity, brain tissue and CSF space profiles are represented in blue, red and green, respectively. Note scale difference between stages. Volumes corresponding to the three spaces are reported in the apposed tables as means ± SD, n = 4 (E19, P9), and n = 5 (P2).
Table 2 CSF compartmentalization in the different fluid-filled spaces of the developing brain
The relative importance of hindbrain CSF spaces in E19 and their decrease throughout development is accounted for by the delayed development of the cerebellum, and the growth of the caudal part of cortex. Of note, cortical subarachnoid CSF is mainly found around this caudal part of the cortex and in the rhinal fissure, as the space between the surface of the cortex and the dura mater is narrow at all stages. This suggests that in the rodent, which does not display cortical circumvolution (but for the rhinal fissure), CSF flow is very limited between the pia-covered cerebral cortex and the arachnoid/dura mater. This is likely to be different in human, in which cortical foldings develop mainly between week 25 and 30 of gestation [20], resulting in changes in the pattern of flow at the cortical surface.
Finally the data show that the size of the ventricular system remains modest at all stages. The volume of the ventricles changed only moderately between E19 and P9, with a slight decrease for the lateral ventricles, an increase for the third ventricle and no change for the fourth ventricle (Additional file 1). By contrast, the size of choroid plexuses continues to enlarge during this period. The protein content of the lateral ventricle choroid plexuses increased from 100 ± 9 to 177 ± 13 μg between P2 and P9, and that of the fourth ventricle choroid plexus increased from 69 ± 9 to 131 ± 19 μg (mean ± SD of six animals at both ages). This increase in protein content parallels the change in size of dissected tissues observed under the stereomicroscope. This suggests there is increased flow of CSF through the ventricular system, which will be further enhanced by the concurrent increase in the rate of CSF secretion by the choroidal cells.
Overall, the volume data provided in this study for twenty compartments of the rat CSF system at different developmental stages (see Additional file 1) form reference figures which can be used in future studies to better delineate the role of the fluid environment in brain developmental processes. Taking into account the total CSF volume can also help to better interpret blood-CSF permeability measurements as exemplified below.
Developmental changes in blood-CSF permeability constants for sucrose used as a marker of blood-CSF barrier efficiency
We measured [14C]-sucrose permeability constants by sampling CSF 20 min after injection. Blood concentration continuously increased during this period, minimizing the impact of tracer back flux. To account for changes in [14C]-sucrose plasma concentration over time we calculated the integrated plasma concentration-time product. We then generated influx constants K in csf with the assumption that the concentration measured in CSF drawn through the cisterna magna (5 to 10% of total CSF) is representative of the overall total CSF concentration. K in csf values did not differ significantly between P2 and P9 animals (Table 3). A new CSF permeability constant K w csf which takes into account the total volume of CSF in which the tracer is distributed (tables in Figure 3), was then calculated for each age (Table 3). K w csf was also not statistically different between the two stages, but became significantly lower at P9 as compared to P2 when values were normalized for choroid plexus protein content (Table 3).
Table 3 Blood-CSF permeability constants for [ 14 C]-sucrose in the developing rat
By using a short time point of 20 minutes, it is expected that a large proportion of the tracer reaches the CSF through the choroidal blood-CSF barrier and possibly across the meningeal vessel endothelium, another blood-CSF barrier site. The route across the blood–brain barrier proper is likely involved only to a limited extent. The presence of tight junctions that link the barrier cells of all choroidal epithelium, meningeal vessels, as well as parenchymal vessels ([11,13] and Figure 1) explains the limited blood-CSF permeability of [14C]-sucrose at both stages. The apparent permeability as assessed by K in csf measurement remains however higher than for adult rat. In the latter, K in csf was measured using a 30-minute time-point for inulin (a large molecule expected to diffuse 4 times less than sucrose across brain barriers), was 0.17 × 10−3 min−1 [21].
The constants Kw csf provide different information when comparing blood-CSF permeability at multiple developmental stages. K w csf represents the virtual volume of plasma from which the tracer is cleared into the CSF per unit of time, which was about 0.2 μl.min−1 at both stages. Total cerebral blood flow across a 10-day-old rat brain, averaged from previous papers, is around 400 μl.min−1.g−1 tissue [22,23]. This is equivalent to 340 μl.min−1.brain−1 in P9 animals (brain volume taken from Figure 3). Therefore, only 0.06% of the tracer flowing through the brain vasculature reaches the CSF at P9. Data on cerebral blood flow in P2 animals and data on choroidal blood flow at both stages would have been useful to further analyze the data, but are not available to our knowledge. K w csf measured in P9 animals becomes significantly lower than K w csf measured in P2 animals when the values are normalized for choroid plexus protein content (Table 3). Assuming this protein content is proportional to the surface area of exchange across the blood-CSF barrier, this change in normalized K w csf may be explained by a decrease in the paracellular permeability of the choroidal epithelium. The developmental expression profile of tight junction proteins, the similar claudin immunolocalization pattern in the choroid plexus at both ages [13], and restriction of a polar tracer by tight junctions observed by electron microscopy [11] suggest that junctions are already efficient at both ages. In addition, factors other than paracellular diffusion could also be involved in the apparent decrease in blood-CSF permeability between P2 and P9. A change in CSF turnover and an increase in CSF-to-tissue diffusion may affect CSF sucrose concentration differently at the two developmental stages. The capacity of the transcellular route for transport for plasma material across selected epithelial cells of the choroid plexus [24] may decrease during the postnatal period in rat. Hence, these factors, all independent from CSF volume, can possibly explain the higher CSF concentration of polar tracers measured at steady-state, several hours after injection, in P2 rats as compared to older animals [25]. By contrast, changes in the CSF distribution volume during the embryonic period appeared to have a substantial influence on apparent blood-to-CSF permeability. Thus, a 60% decrease in the CSF/plasma ratio for sucrose measured in conditions approaching steady-state in rat between E13 and E18 was attributed to the concomitant increase in CSF volume rather than changes in paracellular permeability of the choroidal epithelium [26].
The presence of tight junctions at blood–brain/CSF interfaces in 19-day rat embryos and the [14C]-sucrose permeability constants measured in 2- and 9-day-old rats confirm that the brain fluid environment is controlled and independent from plasma composition during development. Volumes of the different CSF fluid compartments measured during pre- and postnatal development in rat indicate not only a decrease in CSF-to-brain volume ratio, but also a geographical redistribution of the fluid especially around birth in rat. The volume data can be used to refine blood-CSF permeability constants in developing animals. The description of the different ventricular, cisternal and subarachnoid spaces and their respective developmental volume profile compared to total brain volume can be used to better understand cerebral fluid dynamic and diffusional/bulk flow movement of solutes in the developing brain. These fluid spaces are involved in volume transmission, clearance of potentially deleterious molecules, and immune cell migration within the brain These results will enable a better appreciation of the role of fluid compartments in brain development, neuroimmune surveillance of the infant, and in neonatal injuries.
AUC:
Area under the curve
CSF:
Cerebrospinal fluid
CNS:
Central nervous system, anatomical abbreviations: see Table 1
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This work was supported by ANR-10-IBHU-0003 CESAME grant. The authors thank Joseph Fenstermacher for stimulating discussion and Chantal Watrin for her skillful technical assistance.
BIP Platform, Faculté de Médecine RTH Laennec, INSERM U 1028, CNRS UMR5292, Lyon Neuroscience Research Center, Rue Guillaume Paradin, Cedex 08, 69372, Lyon, France
Jean-François Ghersi-Egea
, Anaïd Babikian
& Sandrine Blondel
Oncoflam Team, INSERM U1028, CNRS UMR5292, Lyon Neuroscience Research Center, Lyon, France
& Nathalie Strazielle
Brain-i, Lyon, France
Nathalie Strazielle
Search for Jean-François Ghersi-Egea in:
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Correspondence to Jean-François Ghersi-Egea.
JFGE and NS conceived and designed the study. AB carried out the morphometric measurements and analyzed the data. SB performed the immunofluorescence study. JFGE drafted the manuscript. JFGE and NS edited and revised the manuscript. All authors organized the data, read, and approved the final manuscript.
Volumes of the CSF compartments in 19-day-old rat embryos (E19), 2-day old rats (P2), and nine-day old rats (P9). Mean and standard deviations for the volumes of all the twenty compartments measured as listed in Table 1.
Ghersi-Egea, J., Babikian, A., Blondel, S. et al. Changes in the cerebrospinal fluid circulatory system of the developing rat: quantitative volumetric analysis and effect on blood-CSF permeability interpretation. Fluids Barriers CNS 12, 8 (2015). https://doi.org/10.1186/s12987-015-0001-2
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Influx constant
Blood–brain barrier
Claudin | CommonCrawl |
Quantitative analysis of a system of integral equations with weight on the upper half space
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Regularity and existence of positive solutions for a fractional system
January 2022, 21(1): 101-120. doi: 10.3934/cpaa.2021169
On local well-posedness of logarithmic inviscid regularizations of generalized SQG equations in borderline Sobolev spaces
Michael S. Jolly 1, , Anuj Kumar 1,, and Vincent R. Martinez 2,
Department of Mathematics, Indiana University, Bloomington, IN 47405-7106, USA
Department of Mathematics and Statistics, CUNY Hunter College, New York, NY 10065, USA
Received May 2021 Revised September 2021 Published January 2022 Early access September 2021
Fund Project: The research of M.S.J. and A.K. was supported in part by the NSF grant DMS-1818754. The research of V.R.M. was supported in part by the PSC-CUNY grant 64335-00 52
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This paper studies a family of generalized surface quasi-geostrophic (SQG) equations for an active scalar $ \theta $ on the whole plane whose velocities have been mildly regularized, for instance, logarithmically. The well-posedness of these regularized models in borderline Sobolev regularity have previously been studied by D. Chae and J. Wu when the velocity $ u $ is of lower singularity, i.e., $ u = -\nabla^{\perp} \Lambda^{ \beta-2}p( \Lambda) \theta $, where $ p $ is a logarithmic smoothing operator and $ \beta \in [0, 1] $. We complete this study by considering the more singular regime $ \beta\in(1, 2) $. The main tool is the identification of a suitable linearized system that preserves the underlying commutator structure for the original equation. We observe that this structure is ultimately crucial for obtaining continuity of the flow map. In particular, straightforward applications of previous methods for active transport equations fail to capture the more nuanced commutator structure of the equation in this more singular regime. The proposed linearized system nontrivially modifies the flux of the original system in such a way that it coincides with the original flux when evaluated along solutions of the original system. The requisite estimates are developed for this modified linear system to ensure its well-posedness.
Keywords: surface quasi-geostrophic (SQG) equation, generalized SQG equation, borderline spaces, Hadamard well-posedness, existence, uniqueness, continuity with respect to initial data, logarithmic regularization.
Mathematics Subject Classification: Primary: 76B03, 35Q35; Secondary: 35Q86, 35B45.
Citation: Michael S. Jolly, Anuj Kumar, Vincent R. Martinez. On local well-posedness of logarithmic inviscid regularizations of generalized SQG equations in borderline Sobolev spaces. Communications on Pure & Applied Analysis, 2022, 21 (1) : 101-120. doi: 10.3934/cpaa.2021169
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Michael S. Jolly Anuj Kumar Vincent R. Martinez | CommonCrawl |
EPJ Data Science
The great divide: drivers of polarization in the US public
Lucas Böttcher ORCID: orcid.org/0000-0003-1700-18971,2,3 &
Hans Gersbach3
EPJ Data Science volume 9, Article number: 32 (2020) Cite this article
Many democratic societies have become more politically polarized, with the U.S. being the main example. The origins of this phenomenon are still not well-understood and subject to debate. To provide insight into some of the mechanisms underlying political polarization, we develop a mathematical framework and employ Bayesian Markov chain Monte-Carlo (MCMC) and information-theoretic concepts to analyze empirical data on political polarization that has been collected by Pew Research Center from 1994 to 2017. Our framework can capture the evolution of polarization in the Democratic- and Republican-leaning segments of the U.S. public and allows us to identify its drivers. Our empirical and quantitative evidence suggests that political polarization in the U.S. is mainly driven by strong political/cultural initiatives in the Democratic party.
Political polarization is on the rise in many democratic societies [1–6], and yet the causes of this relatively recent development are not well-understood. In the U.S., political polarization, in terms of ideological distance between Republicans and Democrats, has been growing significantly in recent years, so that it is now less likely to find a liberal Republican or a conservative Democrat than it was many years ago [2, 6]. Several explanations for this finding have been put forward, including the increasing influence of new media and the Internet, the rising income inequality, elite polarization, and demographic changes [4, 7, 8]. However, the growing use of the Internet, for instance, might not suffice to explain the observed polarization effects, because polarization is largest among demographic groups that are least likely to use the Internet and social media [3].
We study how the spreading of political and cultural ideas within populations that lean towards Democrats or Republicans can explain the evolution of political polarization, as observed in empirical data (see Fig. 1). In this context, mathematical models are able to offer insights into the dynamics of opinion formation, polarization, and related spreading processes [9–20]. To help quantify and characterize empirically-observed polarization trends (see Fig. 1), we develop a mathematical framework of political change based on (i) individuals' diffusion from one ideological position to adjacent ones, and (ii) targeting of certain groups of individuals by influential actors who spread their ideas to coalesce around political/cultural positions (henceforth simply called "initiatives"). Theoretical motivation for the interaction of adjacent opinion groups in process (i) comes from related models of opinion formation that are based on the assumption that an exchange of views on a certain topic occurs if the corresponding opinions are not too different (i.e., only involve "nearest neighbors" in opinion space) [21]. We show in this paper that this parsimonious modeling approach leads to good agreement between simulation and empirical data. The term "influential actor" in process (ii) is typically used in the political economy literature and refers to political elites [22], opinion leaders of political interest groups and activists (see [23] for a theory of parties in which interest groups and activists are the key actors), and incumbent candidates of larger parties [24] that have an impact on, at least, some voter groups. More broadly, influential actors can be individuals or groups of individuals, with a particular political or cultural interest, whose ideas impact some voter segments [8, 25].
Polarization in the U.S. public. We show the mean-ideological position (i.e., the mean of the ideology distributions provided in [6]) of the Democratic- (blue) and Republican-leaning (red) segments of U.S. public from 1994 to 2017. Error bars indicate the observed standard deviation in each year. It is evident that polarization has been increasing in the past 15 years. The plotted data are based on a survey conducted by Pew Research Center (see [6] for details)
One current problem in social-influence modeling is that most models are untested. This is pointed out directly in the abstract of [21]: "More empirical work is needed testing and underpinning micro-level assumptions about social influence as well as macro-level predictions." Our work addresses this issue by proposing a parsimonious DeGroot-like model [26] of opinion formation and demonstrating that such a model is able to capture the evolution of empirically-observed opinion polarization in the US public with very few parameters.
We demonstrate how to use Bayesian Markov chain Monte-Carlo (MCMC) and information-theoretic methods in conjunction with opinion-formation models and corresponding empirical data to assess and analyze levels and changes of ideology distributions in the United States. We find that a single parameter suffices to describe the evolution of polarization trends over the last 20–30 years and we identify this polarization measure [27] with the notion of initiative impact. This measure enables us to quantify relative changes in ideology distributions between Democratic- and Republican-leaning segments of U.S. society over time. Our results suggest that the recent polarization in the U.S. public, which took place during the Obama and Trump administrations (see Fig. 1), seems to be mainly driven by strong political/cultural initiatives in the Democratic party. Prominent examples of such initiatives are the Affordable Care Act, policy proposals involving higher tax rates on individuals with high income or wealth, tighter gun control, and same-sex marriages, as we will discuss in more detail below.
In this section, we first define a general and abstract Markov chain model to mathematically capture empirically observed polarization trends (see Fig. 1). Our model describes a (macroscopic) set of different opinion classes. Empirically, these opinion classes correspond to mappings of a high-dimensional feature space (e.g., self-identification with a certain political party, views on certain political issues) to a single-number metric. We briefly describe the update dynamics and then focus on the characterization of the stationary distribution.
Definition of the ideology chain
We proceed in three steps to mathematically describe initiatives and the diffusion of individuals from one ideological position to adjacent ones, with step 1 developed in Sects. 2.1 and 2.2, step 2 in Sect. 2.3, and step 3 in Sect. 2.4.
In the first step, we consider a one-dimensional chain which consists of N different states denoted by i (\(i\in \{1,\dots,N\}\)). We use \(X_{i}\) to denote the fraction of society in state i, and hence \(\sum_{i=1}^{N} X_{i} = 1\). Next, we map the index i to an ideological position \(x\in [-1,1]\) according to \(x=2 (i-1)/(N-1)-1\). These positions represent the political spectrum in the following way: Very liberal individuals are located at the beginning of the chain (\(i=1\), \(x=-1\)), whereas strongly conservative individuals are found at the opposite side (\(i=N\), \(x=1\)). We next consider the evolution of a hypothetical society in discrete time. We interpret \(X_{i}^{n}\) as the fraction of voters of type i at time step \(n\in \mathbb{N}\). For every n, it holds that
$$ \sum_{i=1}^{N} X_{i}^{n}=1 $$
as a normalization condition. We employ a simple birth-death queue [28] of social interactions and assume that individuals may change their ideological position via interactions with their ideological neighbors. At the aggregate level, in a particular time step, we assume that transitions occur from state i to its nearest neighbors (\(i\rightarrow i+1\) and \(i\rightarrow i-1\)) with some probabilities \(p_{i}\in (0,1)\) and \(q_{i-1}\in (0,1-p_{i})\). For the moment, these transition probabilities are taken as given and will be estimated later. At the boundaries of the opinion chain, the probability of becoming more ideologically extreme is zero (\(q_{0}=p_{N}=0\)). The probability of staying at a certain ideological position i is given by \(r_{i}=1-p_{i}-q_{i-1}\). Together, these probabilities form the transition matrix P, with the following entries:
$$ P_{i i-1}=q_{i-1},\qquad P_{i i} = r_{i},\quad \text{and}\quad P_{i i+1}=p_{i}. $$
The probabilities in each row sum up to one, i.e. \(\sum_{j=0}^{3} P_{i i-1+j}=1\).
There exist different ways of motivating and microfounding our opinion-formation model. First, the mathematical structure of our model is similar to DeGroot's [26] Markov-chain description of "social learning" in a group of communicating individuals. Second, the macroscopic distributions of opinions that we observe in our model can be recovered in random matching and bounded confidence models [29, 30] in which individuals adopt sufficiently close opinions through communication (see [31] for a comprehensive account on how such micro-level assumptions in a social network turn into macro-level implications and [32] for a general theory about such social interactions). After every meeting, individuals update their opinion and may switch to their partner's opinion with some probability. Equivalently, individuals change their opinion if they meet a sufficient number of people with alternative opinions [17, 18, 33]. The probabilities \(p_{i}\) and \(q_{i}\) can then be interpreted as the resulting parameters at the aggregate level.Footnote 1 We show an example of an ideology chain with \(N=9\) states in Fig. 2. For the sake of clarity, we do not include self-loops described by \(r_{i}\) in this figure.
Polarization model. In our model, the political spectrum consists of N different states and is divided in three groups: liberal, neutral, and conservative. For illustrative purposes, we set \(N=9\) in this example. The transition probabilities are denoted by \(\{p_{i}\}_{i\in \{1,\dots,9\}}\) and \(\{q_{i}\}_{i\in \{1,\dots,9\}}\)
Next we focus on the dynamics of the model to account for the diffusion of individuals from one ideology to adjacent ones. The initial values of all states are given by \(X_{i}^{n=0}=X_{i}^{0}\). We use \(X^{0}= (X_{1}^{0},\dots,X_{N}^{0} )\) to denote the row vector of all initial states. The time evolution of the ideology distribution is then described by \(X^{0} P^{n} = X^{n}\).
Stationary distribution
To determine the stationary ideology distribution, we formulate the update rule of state \(X_{i}^{n}\) and find
$$ X_{i}^{n+1}=(1-p_{i}-q_{i-1})X_{i}^{n}+q_{i} X_{i+1}^{n}+p_{i-1} X_{i-1}^{n}. $$
We are not considering periodic boundaries, and thus find for \(i=1\),
$$ X_{1}^{n+1}=(1-p_{1})X_{1}^{n}+q_{1} X_{2}^{n}. $$
The Markov chain converges to a stationary distribution, which we denote by \(X_{i}\) with \(i\in \{1,\dots,N\}\). Based on Eq. (4), we obtain \(X_{2}= (p_{1}/q_{1} ) X_{1}\). Furthermore, using Eq. (3) we find by induction that [34]
$$ X_{i+1}= X_{1} \prod_{j=1}^{i} \frac{p_{j}}{q_{j}}. $$
To satisfy the normalization condition of Eq. (1), we set \(X_{1}=1\) and divide each state \(X_{i}\) by \(\sum_{i=1}^{N} X_{i}\). The stationary distribution \(X=(X_{1},\dots,X_{N})\) is unique since the transition matrix P is irreducible and aperiodic [35]. Irreducibility follows from the fact that any state in the Markov chain can be reaced from any other state, and aperiodicity is satisfied because of \(P_{ii}^{n} > 0\) for all \(n\in \mathbb{N}\) [35].
The data that we show in Fig. 1 suggests that the ideological overlap between Democrats and Republicans was larger in the 1990s and early 2000s compared to the last 15 years. To capture the ideology distributions of Democratic- and Republican-leaning segments of the U.S. public with our model, we consider two opinion chains A and B in the subsequent sections, and then account for the impact of influential actors and their initiatives.
Two populations
In the second step, we introduce two populations in which members influence each other regarding their ideological position. This allows us to examine how the distribution of ideologies among Democrats and Republicans evolves over time. Specifically, we consider two populations, A and B, with the corresponding stationary ideology distributions given by Eq. (5):
$$\begin{aligned} X^{A}_{i+1}=X^{A}_{1} \prod _{j=1}^{i} \frac{p^{A}_{j}}{q^{A}_{j}} \end{aligned}$$
$$\begin{aligned} X^{B}_{i+1}=X^{B}_{1} \prod _{j=1}^{i} \frac{p^{B}_{j}}{q^{B}_{j}}. \end{aligned}$$
Influential actors
In the third step, we account for influential actors of both parties that inject political/cultural concepts—simply called initiatives in our paper—into one of the populations. The literature has identified the importance of such influential actors and their initiatives (see e.g. [8, 25]). Typically, initiatives increase the attractiveness of coalescing around a particular ideological position. They can also increase the cohesion within each population and the identity value of belonging to a population.
Mathematically, we describe the impact of influential actors on the two populations in terms of rescaling the transition probabilities of Eq. (2) by \(\lambda _{A}\) and \(\lambda _{B}\) at a particular point in time according to
$$ p_{i}^{A}\rightarrow p_{i}^{A}/\sqrt{ \lambda _{A}} \quad\text{and}\quad q_{i}^{A} \rightarrow q_{i}^{A} \sqrt{\lambda _{A}}. $$
For opinion group B (e.g., Republicans), the rates are modified as follows:
$$ p_{i}^{B}\rightarrow p_{i}^{B} \sqrt{ \lambda _{B}} \quad\text{and}\quad q_{i}^{B} \rightarrow q_{i}^{B} /\sqrt{\lambda _{B}}. $$
The interpretation of the scaling factor is as follows: A value of \(\lambda _{A}>1\) and \(\lambda _{B}>1\) means that an initiative in the populations A and B is introduced, which attracts individuals towards the left and right ends of the underlying ideology chains, respectively. The larger \(\lambda _{A}\) and \(\lambda _{B}\), the greater is the attractiveness of these initiatives. Based on Eqs. (8) and (9), we obtain the following modified stationary states:
$$ X^{A}_{i+1}=\lambda _{A}^{-i} X^{A}_{1} \prod_{j=1}^{i} \frac{p^{A}_{j}}{q^{A}_{j}} $$
$$ X^{B}_{i+1}=\lambda _{B}^{i} X^{B}_{1} \prod_{j=1}^{i} \frac{p^{B}_{j}}{q^{B}_{j}}. $$
The outlined multiplicative rescaling of the transition probabilities leads to a directly-interpretable modification of the stationary opinion distribution. If \(\lambda _{B} > 1\), we obtain a stationary opinion distribution whose mean is shifted to the right compared to the case where \(\lambda _{B} = 1\). Similarly, if \(\lambda _{A} > 1\), the stationary opinion distribution moves towards the left. We thus refer to λ as the initiative impact.
When we move to the empirical application, we have to recognize that a particular value of \(\lambda _{B}\) or \(\lambda _{A}\), which we infer from the data, is open to different interpretations. On the one hand, it could represent changes in the way individuals communicate and influence each other, resulting in a shift regarding political/cultural views. On the other hand, it could represent the impact of new ideas (political or cultural) that affect the attractiveness of different ideological positions as we have outlined in our model.
While it is difficult to disentangle these two interpretations, we will interpret our findings in light of the second interpretation. We do this for two reasons: First, changes in communication, e.g. due to increased Internet use [3] cannot explain the rise in polarization, and [36] recently summarized that the evidence about whether social media increase political polarization is not conclusive. Second, there exists a series of legal acts and policy proposals that have been at the center of communication in the community leaning towards the Democratic party. Such legal acts include the Dodd–Frank Financial Reform Act and the Affordable Care Act. Examples of policy proposals are universal-health care programs, higher taxes on wealth, tighter gun control, and initiatives to slow down climate change. Opposition against tighter gun control, higher taxes, and same-sex marriages are examples of major ideas in the communities leaning towards the Republican party. These examples suggest that major new initiatives have been mainly introduced in the communities leaning towards the Democratic party. We examine whether the data are consistent with this interpretation. We also note that some of the initiatives such as same-sex marriage and abortion rights are culture-dependent since they concern norms and beliefs about how people should be able to live in families, groups and communities.
In the following sections, we show that the outlined rescaling approach is able to model empirically-observed opinion polarization in the U.S. public. In principle, we could also consider values of \(\lambda _{A}\) and \(\lambda _{B}\) that depend on the position in ideology space. It is, however, possible to capture a substantial part of the polarization effects with a constant value of \(\lambda _{A}\) and \(\lambda _{B}\), as shown in Sect. 3.2.
We now focus on the applications and implications of the polarization model introduced in Sect. 2. In Sect. 3.1, we discuss the onset of political polarization when influential actors in each party introduce new political ideas. We outline in Sect. 3.2 how our mathematical framework is able to capture a relevant amount of the polarization effects which have been observed in the U.S.-American public in the past 25 years. Once initialized, only the two initiative impacts \(\lambda _{A}\) and \(\lambda _{B}\) are necessary to describe these polarization effects. We use a Bayesian MCMC approach to learn the parameter distributions of \(\lambda _{A}\) and \(\lambda _{B}\) from empirical observations. Our results are consistent with a stronger polarization of the Democratic wing in the society compared to the Republican wing.
Emergence of political polarization
To study the emergence of political polarization in terms of our model as described in Sect. 2, we first consider an unpolarized society and then analyze the impact of influential actors. Let the ideology space be denoted by the interval \([-1,1]\). As initial ideology distributions, we consider normally-distributed (unpolarized) ideologies with mean \(\mu =0\) and variance \(\sigma ^{2}=0.16\), as illustrated in the upper left panel of Fig. 3. We note that a distribution of ideologies within a party does not uniquely determine the transition probabilities for the diffusion of ideas (see SI). The reason is that, according to Eqs. (6) and (7), only their fractions are relevant for the stationary distribution. To anchor meaningful transition probabilities, we take into account that voters with polar ideological positions are less likely to undergo a transition to more moderate ideological positions. In addition, we would expect larger transition probabilities in the more neutral ideology regime. These two properties anchor the transition probabilities. We show an example of the corresponding transition probabilities \(p^{A}(x)\), \(q^{A}(x)\), \(p^{B}(x)\), and \(q^{B}(x)\) in the lower left panel of Fig. 3. Also in the case of our empirical application in Sect. 3.2, the indeterminacy regarding the transition probabilities is resolved by taking the described effects into account.
The emergence of political polarization. The top panels show an unpolarized (left panel) and a polarized (right panel) ideology distribution (synthetic/simulation data only). Democrats are represented by blue disks and Republicans by red disks. We use \(\Delta (\lambda )\) to denote the absolute value of the difference between the shifts in the mean ideologies. The bottom panels show the corresponding transition probabilities which define the ideology distribution according to Eqs. (6) and (7) with \(N=21\) states. We rescaled the probabilities in the right panel according to Eqs. (10) and (11) by setting \(\lambda =\lambda _{A}=\lambda _{B}=1.13\)
We now incorporate the impact of influential actors on voter ideologies as described by Eqs. (10) and (11), and rescale the transition probabilities accordingly to obtain
$$\begin{aligned} \tilde{p}^{A}(x) = p^{A}(x)/\sqrt{\lambda },\qquad \tilde{q}^{A}(x) = q^{A}(x) \sqrt{\lambda }, \end{aligned}$$
$$\begin{aligned} \tilde{p}^{B}(x) = p^{B}(x) \sqrt{\lambda },\qquad \tilde{q}^{B}(x) = q^{B}(x)/ \sqrt{\lambda }, \end{aligned}$$
where we assumed \(\lambda =\lambda _{A}=\lambda _{B}\). In the upper right panel of Fig. 3, we show that the outlined rescaling of the transition probabilities leads to shifted normal distribution with mean \(\mu (\lambda )\) and an invariant variance. In this example, we set \(\lambda =1.13\). This observation suggests that a rescaling of transition probabilities according to Eqs. (12) and (13) leads to a polarized ideology distribution. The corresponding transition probabilities and their rescaled versions are shown in the lower right panel of Fig. 3. As shown in the upper right panel of Fig. 3, it is possible to quantify polarization as \(\Delta (\lambda )\), the absolute value of the difference between the shifts in the mean ideologies, which is \(2 \mu (\lambda )\) in this example. Larger values of λ lead to an increase in \(\Delta (\lambda )\). In other words, polarization is monotonically increasing with initiative impact.
Polarization in the American public
After having outlined the basic polarization mechanism in our model, we now focus on the evolution of polarization as observed in empirical data on ideology distributions of US citizens from 1994 to 2017. The dataset that we use in this study is based on a Pew Research Center survey on political polarization [6]. Since 1994, Pew Research Center periodically conducts this survey by asking participants a set of 10 questions (e.g., "Should homosexuality be accepted by society?" or "Is good diplomacy the best way to ensure peace?"), which are traditionally associated with liberal/conservative views [6]. Conservative answers are recorded with a "+1" and liberal answers with a "−1". The most conservative survey participants can reach a score of +10, whereas the most liberal value is −10. Note that we normalized the ideology scale to the interval \([-1,1]\) in our study.
Survey participants were also asked if they identify themselves as Democrats, Republicans, or neither and if they are politically engaged [37]. According to [37], persons with high political engagement are "registered to vote, always or nearly always vote, and in the past year have volunteered for or contributed to a campaign".
All surveys were conducted by Pew Research Center either online with the Pew Research American Trends Panel or by Pew Research employees using a random digit sample of landline and cellphone numbers in the United States. People were contacted and surveyed by an interviewer in person or on the telephone (either landline or cellphone), and via the Internet and paper questionnaires (delivered in person or per mail). The usual sample size of Pew Research Center polls is about 1500 people, but may vary from survey to survey [38]. For a detailed overview about the survey methodology, see [39].
Ideology distributions are available for the Democratic- and Republican-leaning segments of the U.S. general and politically-engaged public. We initialize our opinion chain by using the empirical ideology distributions of 1994 to determine the transition probabilities as defined by Eq. (2) with a maximum-likelihood estimation. As in Sect. 3.1, we consider the case where the transition probabilities are monotonically increasing towards the center. After the initialization procedure, the parameters \(\lambda _{A}\) and \(\lambda _{B}\) are the only two free parameters in our model. They will describe the observed ideology distribution according to Eqs. (10) and (11). In the next step, we use a Bayesian MCMC approach to learn the distributions of the two parameters that best describe our data [40, 41]. The theoretical background is presented in the SI.
We show the empirical ideology distributions and corresponding simulations in Figs. 4 and 5. In 1994, the ideology distributions of Democrats and Republicans are almost Gaussian and centered around the origin. In subsequent years, polarization becomes more and more apparent. The division is much more drastic for politically-engaged citizens, compared to the general public. The plots also reveal that, after the initial transition probability estimation, our two-parameter model is able to capture the time evolution of the two ideology distributions quite well. Hence, a substantial portion of the complex ideology and identity formation process can be captured by a multiplicative rescaling of the transition probabilities according to Eq. (12).
Polarization in the U.S. general public in comparison with our model. For the U.S. general public, we show the ideology distributions for Democrats (blue disks) and Republicans (red disks). The dataset is based on a survey conducted by Pew Research Center (see [6] for details). We initialized our model (grey solid lines) with the data of 1994 and only modified the distributions according to a transition probability rescaling as described by Eqs. (10) and (11)
Polarization in the U.S. politically-engaged public in comparison with our model. For the U.S. politically engaged public, we show the ideology distributions for Democrats (blue disks) and Republicans (red disks). The dataset is based on a survey conducted by Pew Research Center (see [6] for details). We initialized our model (grey solid lines) with the data of 1994 and only modified the distributions according to a transition probability rescaling as described by Eqs. (10) and (11)
For every year, we show the corresponding distributions of \(\lambda _{A}\) and \(\lambda _{B}\) in the SI in Fig. S2 and illustrate the time evolution of \(\lambda _{A}\) and \(\lambda _{B}\) in Fig. 6.
Initiative impacts and relative entropy over different years. In the upper panels, we show estimates of the initiative impact \(\lambda _{A}\) and \(\lambda _{B}\) as defined in Eqs. (10) and (11). We see that the polarization effects for self-identified Democrats are larger than for self-identified Republicans. We also compute the relative entropy usning Eq. (14) and show the results in the lower panels. \(P_{\text{Year}}\), \(P^{\prime }_{1994}\) and \(P^{\prime }_{\lambda }\) are the distributions of the data in different years, of the model in 1994 and of our model in different years, respectively. Blue disks represent Democrats and red disks represent Republicans
The initiative impact λ may be also interpreted as a polarization measure. Such an interpretation allows us to analyze the polarization dynamics in the U.S. more systematically. The data presented in the upper panels of Figs. 6 and S2 make clear that the Democratic initiative impact \(\lambda _{A}\) increases substantially over time. The distributions of \(\lambda _{A}\) are also getting broader, an effect which results from additional ideology-transition effects that we cannot describe by simply rescaling the initially-determined transition probabilities. Interestingly, the polarization behavior associated with the Republican initiative impact \(\lambda _{B}\) differs significantly from the polarization dynamics in the Democratic-leaning segment of society. In fact, in 1999 and 2004, the values of \(\lambda _{B}<1\) imply that Republicans are ideologically moving towards the center. Just after 2011, the values of \(\lambda _{B}>1\) suggest that more right-wing initiatives have been transmitted. As a result, Republicans are moving to the right in the political spectrum. Our results clearly suggest, however, that polarization is mainly driven by initiatives acting on the Democratic wing of society.
In the last step, we compare differences between ideology distributions in each year using the relative entropy (or Kullback–Leibler divergence)
$$ G \bigl(P_{\text{Year}},P^{\prime } \bigr)=\sum _{x} P_{\text{Year}}(x) \ln \biggl[\frac{P_{\text{Year}}(x)}{P^{\prime }(x)} \biggr], $$
where \(P_{\text{Year}}\) is the distribution of the empirical data in the respective year and \(P^{\prime }\) is the ideology distribution of the model or a certain reference year. To directly quantify the ideology-distribution evolutions from 1994–2017, we determine the relative entropy of \(P_{\text{Year}}\) with respect to the empirical data distribution \(P^{\prime }_{1994}\) of 1994. The results are shown in the lower panels of Fig. 6. We again observe that the Democratic distributions deviate much more from those of 1994 compared to the Republican distributions—another indicator of a larger polarization effect in the Democratic wing of the society. The insets in the lower panels of Fig. 6 show the relative entropy of \(P_{\text{Year}}\) with respect to the distribution of our model \(P^{\prime }_{\lambda }\). The small values of \(G (P_{\text{Year}},P^{\prime }_{\lambda } )\) indicate that our model captures the empirical distributions well.
We developed a mathematical framework and used Bayesian MCMC and information-theoretic methods to analyze empirical ideology distributions of the Democratic- and Republican-leaning segments of the U.S.-American public. Our framework is based on two processes: varying strength of initiatives in each party, and the corresponding diffusion of these concepts in society.
The evolution of the observed ideology distributions is quite well captured by only two parameters, namely the two initiative impacts. Due to their ability to describe the increasing gap between ideology distributions in the U.S. public, we propose to use these parameters as polarization measures and possible indicator of social conflict [42], in line with other polarization/diversity metrics [27] such as those of Esteban and Ray [43], D'Ambrosio and Wolf [44], and Wang and Tsui [45].
Our empirical and quantitative evidence suggests that strong initiatives from the Democratic party were the main drivers of the great divide that emerged in recent decades between the Democratic- and Republican-leaning population. Future studies may use the proposed framework to quantify polarization trends in other countries.
An alternative foundation of the model is competition of N echo chambers. Individuals communicate in one echo chamber, but opinion leaders of echo chambers spread their views to adjacent echo chambers to increase their number of followers.
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We thank participants at various workshops for helpful comments. Pew Research Center bears no responsibility for the analyses or interpretations of the data presented here. The opinions expressed herein, including any implications for policy, are those of the authors and not of Pew Research Center. LB acknowledges financial support from the SNF Early Postdoc. Mobility fellowship on "Multispecies interacting stochastic systems in biology".
The data that support the findings of this study are openly available from [6]. The codes that were used to perform the numerical analyses are available from the corresponding author upon request.
Department of Computational Medicine, UCLA, Life Sciences Bldg., Box 951766, Los Angeles, US
Lucas Böttcher
Institute for Theoretical Physics, ETH Zurich, Wolfgang-Pauli-Str. 27, 8093, Zurich, Switzerland
Center of Economic Research, ETH Zurich, Zürichbergstrasse 18, 8092, Zurich, Switzerland
Lucas Böttcher & Hans Gersbach
Hans Gersbach
LB and HG conceived the study and wrote the paper. LB performed numerical computations. All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.
Correspondence to Lucas Böttcher.
Markov chain Monte Carlo (MCMC).
Below is the link to the electronic supplementary material.
Supporting information. (PDF 744 kB)
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
Böttcher, L., Gersbach, H. The great divide: drivers of polarization in the US public. EPJ Data Sci. 9, 32 (2020). https://doi.org/10.1140/epjds/s13688-020-00249-4
DOI: https://doi.org/10.1140/epjds/s13688-020-00249-4
Markov chains | CommonCrawl |
Compute
\[\prod_{k = 1}^{12} \prod_{j = 1}^{10} (e^{2 \pi ji/11} - e^{2 \pi ki/13}).\]
Let
\[P(x) = \prod_{k = 1}^{12} (x - e^{2 \pi ki/13}).\]The roots of this polynomial are $e^{2 \pi ki/13}$ for $1 \le k \le 12.$ They are also roots of $x^{13} - 1 = (x - 1)(x^{12} + x^{11} + x^{10} + \dots + x^2 + x + 1).$ Thus,
\[P(x) = x^{12} + x^{11} + x^{10} + \dots + x^2 + x + 1.\]Now, $e^{2 \pi ji/11},$ for $1 \le j \le 10,$ is a root of $x^{11} - 1 = (x - 1)(x^{10} + x^9 + x^8 + \dots + x^2 + x + 1),$ so $e^{2 \pi ji/11}$ is a root
of
\[x^{10} + x^9 + x^8 + \dots + x^2 + x + 1.\]So, if $x = e^{2 \pi ji/11},$ then
\begin{align*}
P(x) &= x^{12} + x^{11} + x^{10} + \dots + x^2 + x + 1 \\
&= x^2 (x^{10} + x^9 + x^8 + \dots + x^2 + x + 1) + x + 1 \\
&= x + 1.
\end{align*}Hence,
\begin{align*}
\prod_{k = 1}^{12} \prod_{j = 1}^{10} (e^{2 \pi ji/11} - e^{2 \pi ki/13}) &= \prod_{j = 1}^{10} P(e^{2 \pi ji/11}) \\
&= \prod_{j = 1}^{10} (e^{2 \pi ji/11} + 1).
\end{align*}By similar reasoning,
\[Q(x) = \prod_{j = 1}^{10} (x - e^{2 \pi ji/11}) = x^{10} + x^9 + x^8 + \dots + x^2 + x + 1,\]so
\begin{align*}
\prod_{j = 1}^{10} (e^{2 \pi ji/11} + 1) &= \prod_{j = 1}^{10} (-1 - e^{2 \pi ji/11}) \\
&= Q(-1) \\
&= \boxed{1}.
\end{align*} | Math Dataset |
Signature Record Type Definition
In near field communications the NFC Forum Signature Record Type Definition (RTD) is a security protocol used to protect the integrity and authenticity of NDEF (NFC Data Exchange Format) Messages. The Signature RTD is an open interoperable specification modeled after Code signing where the trust of signed messages is tied to digital certificates.[1]
Signing NDEF records prevents malicious use of NFC tags (containing a protected NDEF record). For example, smartphone users tapping NFC tags containing URLs. Without some level of integrity protection an adversary could launch a phishing attack. Signing the NDEF record protects the integrity of the contents and allows the user to identify the signer if they wish. Signing certificates are obtained from third party Certificate Authorities and are governed by the NFC Forum Signature RTD Certificate Policy.
How it works
The NDEF signing process
Referring to the diagrams. An author obtains a signing certificate from a valid certificate authority. The author's private key is used to sign the Data Record (text, URI, or whatever you like). The signature and author's certificate comprise the signature record. The Data Record and Signature Record are concatenated to produce the Signed NDEF Message that can be written to a standard NFC tag with sufficient memory (typically on the order of 300 to 500 bytes). The NDEF record remains in the clear (not encrypted) so any NFC tag reader will be able to read the signed data even if they cannot verify it.
Data RecordSignature Record
NDEF RecordSignature, Certificate Chain
The NDEF Verification Process
Referring to the diagram. Upon reading the Signed NDEF Message, the Signature on the Data Record is first cryptographically verified using the author's public key (extracted from the Author's Certificate). Once verified, the Author's Certificate can be verified using the NFC Root Certificate. If both verifications are valid then one can trust the NDEF record and perform the desired operation.
Supported certificate formats
The Signature RTD 2.0 supports two certificate formats. One being X.509 certificate format and the other the Machine to Machine (M2M) Certificate format.[2] The M2M Certificate format is a subset of X.509 designed for limited memory typically found on NFC tags. The author's certificate can optionally be replaced with a URI reference to that certificate or Certificate Chain so that messages can be cryptographically verified. The URI certificate reference designed to save memory for NFC tags.
Supported cryptographic algorithms
The Signature RTD 2.0 uses industry standard digital signature algorithms. The following algorithms are supported:
Signature Type/HashSecurity Strength (IEEE P1363)
RSA_1024/SHA_25680 bits
DSA_1024/SHA_25680 bits
ECDSA_P192/SHA_25680 bits
RSA_2048/SHA_256112 bits
DSA_2048/SHA_256112 bits
ECDSA_P224/SHA_256112 bits
ECDSA_K233/SHA_256112 bits
ECDSA_B233/SHA_256112 bits
ECDSA_P256/SHA_256128 bits
On the security of the Signature RTD
The Signature RTD 2.0's primary purpose is the protect the integrity and authenticity of NDEF records. Thus, NFC tag contents using the Signature RTD 2.0 is protected. The security of the system is tied to a certificate authority and the associated Certificate Chain. The NFC Forum Signature RTD Certificate Policy defines the policies under which certificate authorities can operate in the context of NFC. Root certificates are carried in verification devices and are not contained in the signature record. This separation is important for the security of the system just as web browser certificates are separated from web server certificates in TLS.
References
1. "Home - NFC Forum". NFC Forum.
2. "IETF - M2M Certificate format". IETF.
| Wikipedia |
A singular limit for an age structured mutation problem
MBE Home
February 2017, 14(1): 1-15. doi: 10.3934/mbe.2017001
Angiogenesis model with Erlang distributed delays
Emad Attia 1, , Marek Bodnar 2, and Urszula Foryś 2,
Department of Mathematics, Faculty of Science, Damietta University, New Damietta 34517, Egypt
Institute of Applied Mathematics and Mechanics, Faculty of Mathematics, Informatics and Mechanics, University of Warsaw Banacha 2, 02-097 Warsaw, Poland
Received November 23, 2015 Accepted April 06, 2016 Published October 2016
Full Text(HTML)
Figure(4) / Table(1)
We consider the model of angiogenesis process proposed by Bodnar and Foryś (2009) with time delays included into the vessels formation and tumour growth processes. Originally, discrete delays were considered, while in the present paper we focus on distributed delays and discuss specific results for the Erlang distributions. Analytical results concerning stability of positive steady states are illustrated by numerical results in which we also compare these results with those for discrete delays.
Keywords: Delay differential equations, stability analysis, Hopf bifurcation, angiogenesis, tumour growth.
Mathematics Subject Classification: Primary: 34K11, 34K13, 34K18, 37N25; Secondary: 92B05.
Citation: Emad Attia, Marek Bodnar, Urszula Foryś. Angiogenesis model with Erlang distributed delays. Mathematical Biosciences & Engineering, 2017, 14 (1) : 1-15. doi: 10.3934/mbe.2017001
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Figure 1. Critical average delay, that is $m/a_\text{cr}$ for various values of $m$ in the dependance on $\delta$ in the case when only the process of tumour growth is delayed; left -graphs for the steady state $D_1$, right -graphs for the steady state $D_3$
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Figure 2. Solutions of system (1.1) for parameters given by (3.1) and $\tau=10$, with time delay present only in the vessel formation term
Figure 3. Solutions of system (1.1) for parameters given by (3.1) and $\tau=10$, with time delay present only in the tumour growth term. Here, for Erlang distribution, the steady state is stable, and solutions for $m=1$ and $m=5$ are almost identical
Figure 4. Solutions of system (1.1) for parameters given by (3.1) and $\tau=5$, with time delay present in both terms
Table 1. Critical values of $\tau$ at which the positive steady state loses stability
steady state $D_1$ steady state $D_3$
$\delta$ 0.332 0.346 0.36 0.368 0.378 0.3 0.332 0.346 0.36 0.368
discrete 66.7 33.4 29.3 43.6 182 4.49 5.89 7.53 13.0 94.0
$m=1$ steady state does not lose stability
$m=2$ 176 54.7 69.1 106.1 460 5.58 9.36 14.4 32.2 284
$m=5$ 89.9 29.6 37.4 56.6 234 4.03 5.97 8.34 16.6 135
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Marek Bodnar, Monika Joanna Piotrowska, Urszula Foryś, Ewa Nizińska. Model of tumour angiogenesis -- analysis of stability with respect to delays. Mathematical Biosciences & Engineering, 2013, 10 (1) : 19-35. doi: 10.3934/mbe.2013.10.19
Zvia Agur, L. Arakelyan, P. Daugulis, Y. Ginosar. Hopf point analysis for angiogenesis models. Discrete & Continuous Dynamical Systems - B, 2004, 4 (1) : 29-38. doi: 10.3934/dcdsb.2004.4.29
R. Ouifki, M. L. Hbid, O. Arino. Attractiveness and Hopf bifurcation for retarded differential equations. Communications on Pure & Applied Analysis, 2003, 2 (2) : 147-158. doi: 10.3934/cpaa.2003.2.147
Runxia Wang, Haihong Liu, Fang Yan, Xiaohui Wang. Hopf-pitchfork bifurcation analysis in a coupled FHN neurons model with delay. Discrete & Continuous Dynamical Systems - S, 2017, 10 (3) : 523-542. doi: 10.3934/dcdss.2017026
Eugen Stumpf. Local stability analysis of differential equations with state-dependent delay. Discrete & Continuous Dynamical Systems - A, 2016, 36 (6) : 3445-3461. doi: 10.3934/dcds.2016.36.3445
Shubo Zhao, Ping Liu, Mingchao Jiang. Stability and bifurcation analysis in a chemotaxis bistable growth system. Discrete & Continuous Dynamical Systems - S, 2017, 10 (5) : 1165-1174. doi: 10.3934/dcdss.2017063
Philip Gerlee, Alexander R. A. Anderson. Diffusion-limited tumour growth: Simulations and analysis. Mathematical Biosciences & Engineering, 2010, 7 (2) : 385-400. doi: 10.3934/mbe.2010.7.385
Sun Yi, Patrick W. Nelson, A. Galip Ulsoy. Delay differential equations via the matrix lambert w function and bifurcation analysis: application to machine tool chatter. Mathematical Biosciences & Engineering, 2007, 4 (2) : 355-368. doi: 10.3934/mbe.2007.4.355
Fang Han, Bin Zhen, Ying Du, Yanhong Zheng, Marian Wiercigroch. Global Hopf bifurcation analysis of a six-dimensional FitzHugh-Nagumo neural network with delay by a synchronized scheme. Discrete & Continuous Dynamical Systems - B, 2011, 16 (2) : 457-474. doi: 10.3934/dcdsb.2011.16.457
Zuolin Shen, Junjie Wei. Hopf bifurcation analysis in a diffusive predator-prey system with delay and surplus killing effect. Mathematical Biosciences & Engineering, 2018, 15 (3) : 693-715. doi: 10.3934/mbe.2018031
Rui Hu, Yuan Yuan. Stability, bifurcation analysis in a neural network model with delay and diffusion. Conference Publications, 2009, 2009 (Special) : 367-376. doi: 10.3934/proc.2009.2009.367
Evelyn Buckwar, Girolama Notarangelo. A note on the analysis of asymptotic mean-square stability properties for systems of linear stochastic delay differential equations. Discrete & Continuous Dynamical Systems - B, 2013, 18 (6) : 1521-1531. doi: 10.3934/dcdsb.2013.18.1521
Abdelhai Elazzouzi, Aziz Ouhinou. Optimal regularity and stability analysis in the $\alpha-$Norm for a class of partial functional differential equations with infinite delay. Discrete & Continuous Dynamical Systems - A, 2011, 30 (1) : 115-135. doi: 10.3934/dcds.2011.30.115
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Emad Attia Marek Bodnar Urszula Foryś | CommonCrawl |
Home Journals RCMA Microwave Absorbing Features of Ce2(Co0.3Fe0.7)17/Ferrite Coating Material
Microwave Absorbing Features of Ce2(Co0.3Fe0.7)17/Ferrite Coating Material
Jianqi Wang* | Liancheng Lu
Beijing Special Engineering Design and Research Institute, Beijing 100028, China
Corresponding Author Email:
[email protected]
https://doi.org/10.18280/rcma.290107
| Citation
29.01_07.pdf
This paper attempts to combine Ce2(Co0.3Fe0.7)17 and Co2Z-type hexagonal ferrite (hereinafter referred to the ferrite) into a composite with excellent absorbing properties. Firstly, the permittivity and permeability spectra of Ce2(Co0.3Fe0.7)17/epoxy resin composite were investigated, so were those of the ferrite/epoxy resin composite. On this basis, the author prepared coaxial Ce2(Co0.3Fe0.7)17/ferrite/epoxy resin composite specimens at different volume ratios of the Ce2(Co0.3Fe0.7)17 particles and the ferrite particles, and also studied their permittivity and permeability spectra in the frequency range of 8~18 GHz. In this way, the optimal volume ratios of both types of particles were determined for the composite. Through the analysis, it is concluded that, based on the matrix of epoxy resin and polyamide, the single-layer plate absorber specimens can control the reflection loss within -10dB across the frequency range 8~18GHz, when the thickness is 1.1 mm, the fraction volume of Ce2(Co0.3Fe0.7)17 is 25 % and the fraction volume of ferrite is 20 %.
absorbent, ferrite, reflection loss, coating material, composite
With the proliferation of modern communication technology, electromagnetic devices have penetrated every corner of our life and propagated to both civil and military fields. This calls for full protection of these devices to eliminate electromagnetic interference and ensure electromagnetic compatibility. One of the most popular methods for electromagnetic protection lies in the adoption of absorbents [1-8]. In military field, the absorbent must be sufficiently thin, light, wide and strong to shield electromagnetic interference.
According to the theories on transmission line and quarter wavelength [9-12], the properties of absorbents (e.g. thickness) are heavily influenced by permittivity and permeability. The rare-earth transition-metal intermetallic compound Ce2(Co0.3Fe0.7)17 [13] offers a viable solution to reduce the dosage and thickness of absorbents, thanks to excellent performance in a wide range and at high frequencies, which can break through the Snoek limitation. However, the high permittivity of this material hinders the impedance matching with the free space, and thus limits its application as absorbent. Considering the small permittivity of the Co2Z-type hexagonal ferrite, this paper attempts to combine Ce2(Co0.3Fe0.7)17 and Co2Z-type hexagonal ferrite into a composite with excellent absorbing properties.
2.1 Materials
The absorbent was prepared from Ce2(Co0.3Fe0.7)17 (particle diameter: 10~50μm) with in-plane anisotropy and Co2Z-type hexagonal ferrite (hereinafter referred to as ferrite). The coating material is square aluminum plate (length: 18mm). The matrix resin includes epoxy resin (E44) and polyamide (650).
2.2 Coaxial specimens
First, epoxy resin and polyamide were weighed at the mass ratio of 1:1, mixed into the matrix resin at different volume fractions, and relocated into a crucible. Then, the n-butyl alcohol was added into the matrix resin to dissolve the epoxy resin and polyamide. After that, the mixture was dispersed for 5min with an ultrasonic cleaner (KQ218). Next, the prepared absorbent was relocated into a crucible, and dispersed continuously for 1.5h with an ultrasonic cleaner (KQ218). After ultrasonic dispersion, the absorbent/resin compound was extracted from the crucible and cut into granular particles. The particles were then put into a coaxial mold (inner diameter: 3.04 mm; external diameter: 7.00 mm) and compressed for 8h by tablet compressing machine (FW-4) under 2MPa. Three specimens were thus prepared for each volume fraction.
2.3 Coating specimens
The prepared absorbent was separately added into epoxy resin and polyamide, and dispersed to prepare a coating material of two components. Each component was dispersed with a high-speed grinder (SKL-FS400). For the epoxy resin component, the epoxy resin, absorbent and zirconium beads were added into a tank in turn and stirred at a low speed (<500r/min). Then, the stirring speed was gradually raised to 4,000r/min. Meanwhile, the mixed solvent of xylene and n-butyl alcohol (mass ratio of 5:1) was added into the tank. After a 4h dispersion, the zirconium beads were washed out by the same mixed solvent. The polyamide component went through the same dispersion process. Next, the two dispersed components were sprayed into coating specimens, and left solidified at room temperature.
2.4 Reflection loss test
The reflection loss was tested by Bow Reflectivity Testing Method at the frequency between 8 and 18GHz. The dielectric constant $ε_r$ (${{\varepsilon }_{r}}=(\varepsilon _{r}^{'}-j\varepsilon _{r}^{''})$) and permeability $μ_r$(${{\mu }_{r}}({{\mu }_{r}}=\mu _{r}^{'}-j\mu _{r}^{''})$)of the coaxial specimens were tested through vector network analysis at the frequency between 0.1 and 18GHz.
3. Results and Discussion
3.1 Electromagnetic spectrum features of Ce2(Co0.3Fe0.7)17/resin composite
Figure 1. Frequency dependence of permittivity, permeability and of Ce2(Co0.3Fe0.7) 17/resin composite
Figure 1 shows the variations in permittivity, permeability and loss tangent ($tanδ$) of Ce2(Co0.3Fe0.7)17/resin composite as the frequency increased from 0.1 to 18GHz. The volume fraction of Ce2(Co0.3Fe0.7)17 particles is 25 % in this composite. The $tanδtanδ$ can be calculated by:
$\tan {{\delta }_{e}}=\frac{{{\varepsilon }''}}{{{\varepsilon }'}}$ (1)
$\tan {{\delta }_{\text{m}}}=\frac{{{\mu }''}}{{{\mu }'}}$ (2)
where, $tanδ_e$ and $tanδ_m$ are the loss tangent values obtained from permittivity and permeability, respectively. The two parameters describe the microwave depletion of the composite.
From Figures 1(b) and 1(c), it is clear that both the imaginary permeability and the maximum $tanδ_m$ surpassed one, while the value of $tanδ_e$ stayed below 0.2 at all frequencies. With the growth in frequency, the value of $tanδ_e$ remained basically stable. Meanwhile, the value of $tanδ_m$ exhibited an obvious increase, indicating that Ce2(Co0.3Fe0.7)17 is a magnetic loss absorbing material. As shown in Figure 1 (a), the real permittivity was below 12, and, similar to the imaginary permittivity, decreased with the rise of frequency. This reveals the good impedance matching of the composite. In addition, the resonance peak appeared in the imaginary permeability in Figure 1 (b), which is attributable to the loss of electromagnetic wave propagating in the composite.
3.2 Electromagnetic spectrum features of ferrite/resin composite
Figure 2. Frequency dependence of permittivity, permeability and $tanδ$ of ferrite/resin composite
Figure 2 presents the variations in permittivity, permeability and $tanδ$ of ferrite/resin composite as the frequency increased from 0.1 to 18 GHz. The volume fraction of ferrite particles is 35 % in this composite. The tanδ$ is also calculated by equations (1) and (2).
It can be seen from Figure 2 (a) that the real and imaginary permittivities were respectively smaller than 7.5 and 1. Overall, the permittivity of the ferrite/resin composite is relatively low at a high-volume fraction of ferrite (35 %). As shown in Figure 2(b), the imaginary permeability of ferrite was relatively low but the twin peaks in the curve of imaginary permeability helps to widen the bandwidth of the absorbent. With a low permittivity, this composite can be used to improve the impedance matching of Ce2(Co0.3Fe0.7)17/resin composite by widening the absorption band.
3.3 Microwave absorbing features of ferrite/resin composite
Inspired by the theory on transmission line [9-11], the reflection loss of single-layer plate absorber can be derived from the permittivity and permeability with the following equations:
$R=20\lg \left| \frac{{{Z}_{in}}(1)-{{Z}_{0}}}{{{Z}_{in}}(1)+{{Z}_{0}}} \right|=20\lg \left| \frac{{{\eta }_{in}}(1)-1}{{{\eta }_{in}}(1)+1} \right|$ (3)
${{\eta }_{in}}(1)={{\eta }_{1}}tgh(j{{k}_{1}}{{d}_{1}})=\sqrt{\frac{{{\mu }_{r1}}}{{{\varepsilon }_{r1}}}}tgh(j\frac{2\pi f\sqrt{{{\varepsilon }_{r1}}{{\mu }_{r1}}}}{c}{{d}_{1}})$ (4)
where c is the light velocity; f is the frequency of electromagnetic wave; d1 is the thickness of absorbing layer; $ε_r1$ and $μ_r1$ are the real and imaginary permittivities, respectively. Hence, the microwave absorbing features can be determined based on the relationships between frequency and permittivity and permeability.
Figure 3 displays the reflectivity of ferrite/resin single-layer plate absorber with different thicknesses. To fully display the microwave absorbing features of ferrite/resin composite, the frequency dependence of normalized input impedance ( $|Z_in/Z_0 |$ ) and minimum reflectivity when the ferrite/resin composite is of the thicknesses of (1/4)γ and (3/4)γ are displayed in Figure 4, where γ is the wavelength at a certain frequency.
As can be seen from Figure 3, the reflection loss was less than-10dB at the frequency of 8.2 GHz, and the minimum reflection loss was -20dB. The results indicate that the ferrite/resin composite enjoys a relatively wide bandwidth. According to Figure 4, the normalized input impedance intersected the line of $|Z_in/Z_0 |=1$ (the completely matching line) at two points, revealing that the composite has two completely matching points, respectively at the frequencies of 7.8GHz and 13.9GHz. Between the two points, the normalized input impedance curve was close to one, that is, the composite remains close to the completely matching state under the frequency of 6GHz. This explains the relative wide bandwidth in Figure 3. Due to the small imaginary the permittivity, the absorbing layer must be extremely thick if only ferrite particles are used.
The above analysis shows that Ce2(Co0.3Fe0.7)17 particles enjoy good magnetic performance at high frequencies, and cause strong magnetic loss to electromagnetic wave at a small dosage of absorbent or with a thin coating. The ferrite particles can widen the absorption band of the absorbent specimens, thanks to their low permittivity, good impedance matching to free space, and bimodal features of imaginary permeability. If combined, the two types of particles can produce a powerful absorbing coating with broad bandwidth and limited thickness.
Figure 3. Frequency dependence of reflectivity at different thicknesses
Figure 4. Frequency dependence of normalized input impedance and minimum reflectivity at the thicknesses of (1/4)γ and (3/4)γ
3.4 Electromagnetic spectrum features of Ce2(Co0.3Fe0.7)17/ferrite/resin composite
Figure 5 provides the spectra of permittivity and permeability of coaxial Ce2(Co0.3Fe0.7)17/ferrite/ resin composite at different volume fractions of the two types of particles (the total volume fraction of the two particles is 45 %). It can be seen that the permittivity and permeability, especially the real permittivity, changed greatly with the volume fractions of the two types of particles. The twin peaks in the curve of imaginary permeability come from the ferrite. In general, the permittivity is negatively correlated with the volume fraction of ferrite, while the impedance matching is positively corelated with the latter.
Figure 5. The spectra of permittivity and permeability of coaxial Ce2(Co0.3Fe0.7)17/ferrite/resin composite at different volume fractions of Ce2(Co0.3Fe0.7)17 (x) and ferrite (y)
3.5 Microwave absorbing features of Ce2(Co0.3Fe0.7)17/ferrite/resin composite
The microwave absorbing features of Ce2(Co0.3Fe0.7)17/ferrite/resin composite were investigated based on the permittivity and permeability spectra in Figure 5.
Figure 6. Frequency dependence of normalized input impedance and minimum reflectivity at the thicknesses of (1/4)γ and (3/4)
The results are plotted as Figure 6, where x is Ce2(Co0.3Fe0.7)17 and t is ferrite. It can be seen that the completely matching point shifted towards the low frequency with the growth in the volume fraction of ferrite. Meanwhile, the normalized input impedance curve approached the completely matching line $|Z_in/Z_0 |=1$ , indicating that the more the ferrite particles, the better the impedance matching and the broader the bandwidth.
3.6 Reflection properties of coating specimens
Based on the volume fractions in Figure 5, the single-layer plate absorber specimens were prepared from 18cm-long square aluminum plates. The reflection losses of these specimens were tested at the frequency between 8 and 18GHz. The test results are given in Figure 7 and Table 1. As shown in Figure 7, the minimum reflectivity of -25dB belonged to specimen 1# (thickness: 1.5mm; Ce2(Co0.3Fe0.7)17 volume fraction: 15 %; ferrite volume fraction: 30 %). The reflection loss of specimen 3# (thickness: 1.1mm; Ce2(Co0.3Fe0.7)17 volume fraction: 25 %; ferrite volume fraction: 20 %) remained less than -10dB across the frequency range, a signal for a broad bandwidth.
Figure 7. Frequency dependence of reflection loss of Ce2(Co0.3Fe0.7)17/ferrite/resin single-layer plate absorber specimens (1#: vol. x15 %/t30 %; 2#: vol. x20 %/t25 %; 3#: vol. x25 %/t20 %; 4#: vol. x30 %/t15%; 5#: vol. x35 %/t10 %)
Table 1. Test results on the specimens
Peak reflection
[dB]
Peak frequency
[GHz]
Effective bandwidth
Surface density
[kg/m2]
(1) The absorbent Ce2(Co0.3Fe0.7)17 boasts good magnetic properties at high frequencies.
(2) The Ce2(Co0.3Fe0.7)17/resin composite has relatively high imaginary permeability, and causes strong magnetic loss. Meanwhile, the real permittivity of the composite is extremely high, and increases quickly with the growth in the volume fraction of Ce2(Co0.3Fe0.7)17.
(3) The ferrite has relatively low permittivity, and moderate and bimodal imaginary permeability. These features are conducive to impedance matching and bandwidth.
(4) Based on the matrix of epoxy resin and polyamide, the single-layer plate absorber specimens can control the reflection loss within -10dB across the frequency range 8~18GHz, when the thickness is 1.1 mm, the fraction volume of Ce2(Co0.3Fe0.7)17 is 25 % and the fraction volume of ferrite is 20 %.
[1] Sadiq I, Naseem S, Ashiq MN, Khan MA, Niaz S, Rana MU. (2016). Tunable microwave absorbing nano-material for X-band applications. Journal of Magnetism and Magnetic Materials 401: 63-69. https://doi.org/10.1016/j.jmmm.2015.09.024
[2] Seo LS, Chin WS, Lee DG. (2004). Characterization of electromagnetic properties of polymeric composite materials with free space method. Composite Structures 66(1-4): 533-542. https://doi.org/10.1016/j.compstruct.2004.04.076
[3] Das S, Nayak GC, Sahu SK, Oraon R. (2015). Development of FeCoB/Graphene Oxide based microwave absorbing materials for X-Band region. Journal of Magnetism and Magnetic Materials 384: 224-228. https://doi.org/10.1016/j.jmmm.2015.01.079
[4] Yusoff AN, Abdullah MH. (2004). Microwave electromagnetic and absorption properties of some LiZn ferrites. Journal of Magnetism and Magnetic Materials 269(2): 271-280. https://doi.org/10.1016/S0304-8853(03)00617-6
[5] Folgueras LC, Alves MA, Rezende MC. (2014). Evaluation of a nanostructured microwave absorbent coating applied to a glass fiber/polyphenylene sulfide laminated composite. Materials Research 17(1). https://doi.org/10.1590/S1516-14392014005000009
[6] Panwar R, Puthucheri S, Agarwala V, Singh D. (2015). Fractal frequency-selective surface embedded thin broadband microwave absorber coatings using heterogeneous composites. IEEE Transactions on Microwave Theory and Techniques 63(8): 2438-2448. https://doi.org/10.1109/TMTT.2015.2446989
[7] Xiong GX, Xu LL, Deng M, Huang HQ, Tang MS. (2005). Research on absorbing EMW properties and mechanical properties of nanometric TiO_2 and cement composites. Journal of Functional Materials & Devices 11(1): 87-91.
[8] Wei JQ, Zhang ZQ, Wang BC, Wang T, Li F. (2010). Microwave reflection characteristics of surface-modified Fe50Ni50Fe50Ni50 fine particle composites. Journal of Applied Physics 108(12). https://doi.org/10.1063/1.3524546
[9] Liu XG, Wu ND, Cui CY, Li YT, Zhou PP, Bi NN. (2015). Facile preparation of carbon-coated Mg nanocapsules as light microwave absorber. Materials Letters 149: 12-14. https://doi.org/10.1016/j.matlet.2015.02.095
[10] Liu Y, Liu XX, Wang XJ. (2014). Double-layer microwave absorber based on CoFe2O4 ferrite and carbonyl iron composites. Journal of Alloys and Compounds 584: 249-253. https://doi.org/10.1016/j.jallcom.2013.09.049
[11] Wang Y, Luo F, Zhou WC, Zhu DM. (2014). Dielectric and electromagnetic wave absorbing properties of TiC/epoxy composites in the GHz range. Ceramics International (40)7: 10749-10754. https://doi.org/10.1016/j.ceramint.2014.03.064
[12] Zhang ZQ, Wei JQ, Yang WF, Qiao L, Wang T, Li FS. (2011). Effect of shape of Sendust particles on their electromagnetic properties within 0.1-18 GHz range. Physica B: Condensed Matter 406(20): 3896-3900. https://doi.org/10.1016/j.physb.2011.07.019
[13] Li FS, Yi HB, Zuo WL, Liu X. (2010). Microwave magnetic properties of 2:17 Rare earth-3d transitionmetallic intermetallic compounds with planar magnetic anisotropy. CN Patent 201010230672.3. | CommonCrawl |
Michael Stillman
Michael Eugene Stillman (born March 24, 1957) is an American mathematician working in computational algebraic geometry and commutative algebra. He is a Professor of Mathematics at Cornell University. He is known for being one of the creators (with Daniel Grayson) of the Macaulay2 computer algebra system.
Michael Stillman
Michael Stillman in 2006 at Oberwolfach
Born
Michael Eugene Stillman
(1957-03-24) March 24, 1957
NationalityAmerican
Alma mater
• Harvard University (PhD)
• University of Illinois (BA)
Known forMacaulay2
Scientific career
FieldsMathematics
InstitutionsCornell University
ThesisConstruction of Holomorphic Differential Forms on the Moduli Space of Abelian Varieties (1983)
Doctoral advisorDavid Mumford
Websitepi.math.cornell.edu/~mike/
Education and career
Michael Stillman completed his PhD at Harvard University in 1983 under the direction of David Mumford. He had postdoctoral positions at the University of Chicago, Brandeis University, and the Massachusetts Institute of Technology before moving to a permanent position at Cornell University in 1987.
Stillman is best known for his work on computer algebra systems. In 1983, he began work with Dave Bayer on the Macaulay computer algebra system, which they continued to improve until 1993. To get beyond several limitations in the design of Macaulay, Stillman and Daniel Grayson began work on the Macaulay2 system in 1993.[1] Macaulay2 remains in active development as of 2019,[2] and has been cited in over 2000 articles.[3]
Stillman has over 30 mathematical publications, and has advised 11 PhD students.[4][5]
Awards and honors
• In 2015 Stillman was selected as a fellow of the American Mathematical Society for his work in symbolic computation[6] (such as that on Macaulay2).
• Stillman was recognized for his teaching by Business Insider in a 2013 feature on the Best Colleges in America,[7] where he was named as one of the top 10 professors at Cornell.
References
1. Eisenbud, David; Grayson, Daniel; Stillman, Michael; Sturmfels, Bernd (2002). Computations in algebraic geometry with Macaulay 2. Berlin New York: Springer. ISBN 3-540-42230-7. MR 1949544.
2. "Macaulay2".
3. "Macaulay2 citations on Google Scholar". Retrieved September 21, 2019.
4. Michael Eugene Stillman at the Mathematics Genealogy Project
5. "Michael Stillman's CV (dated 2011)" (PDF). Michael Stillman's homepage. Retrieved September 14, 2019.
6. "2015 Class of the Fellows of the AMS" (PDF). Notices of the American Mathematical Society. 62 (3): 285–287. March 2015. Retrieved September 11, 2019.
7. Jacobs, Peter (November 7, 2013). "The 10 Best Professors At Cornell University". Business Insider. Retrieved September 11, 2019.
External links
• Michael Stillman publications indexed by Google Scholar
Authority control
International
• ISNI
• VIAF
National
• United States
Academics
• CiNii
• Google Scholar
• MathSciNet
• Mathematics Genealogy Project
• zbMATH
Other
• IdRef
| Wikipedia |
\begin{definition}[Definition:Equiprobable Outcomes]
Let $\struct {\Omega, \Sigma, \Pr}$ be a finite probability space.
Let $\Omega = \set {\omega_1, \omega_1, \ldots, \omega_n}$.
Suppose that $\map \Pr {\omega_i} = \map \Pr {\omega_j}$ for all the $\omega_i, \omega_j \in \Omega$.
Then from Probability Measure on Equiprobable Outcomes:
:$\forall \omega \in \Omega: \map \Pr \omega = \dfrac 1 n$
:$\forall A \subseteq \Omega: \map \Pr A = \dfrac {\card A} n$
Such a probability space is said to have '''equiprobable outcomes''', and is sometimes referred to as an equiprobability space.
\end{definition} | ProofWiki |
Exploring Animal Behavior Through Sound: Volume 1 pp 185–216Cite as
Introduction to Sound Propagation Under Water
Christine Erbe3,
Alec Duncan3 &
Kathleen J. Vigness-Raposa4
First Online: 04 October 2022
Sound propagation under water is a complex process. Sound does not propagate along straight-line transmission paths. Rather, it reflects, refracts, and diffracts. It scatters off rough surfaces (such as the sea surface and the seafloor) and off reflectors within the water column (e.g., gas bubbles, fish swim bladders, and suspended particles). It is transmitted into the seafloor and partially lost from the water. It is converted into heat by exciting molecular vibrations. There are common misconceptions about sound propagation in water, such as "low-frequency sound does not propagate in shallow water," "over hard seafloors, all sound is reflected, leading to cylindrical spreading," and "over soft seafloors, sound propagates spherically." This chapter aims to remove common misconceptions and empowers the reader to comprehend sound propagation phenomena in a range of environments and appreciate the limitations of widely used sound propagation models. The chapter begins by deriving the sonar equation for a number of scenarios, including animal acoustic communication, communication masking by noise, and acoustic surveying of animals. It introduces the concept of the layered ocean, presenting temperature, salinity, and resulting sound speed profiles. These are needed to develop the most common concepts of sound propagation under water: ray tracing and normal modes. This chapter explains Snell's law, reflection and transmission coefficients, and Lloyd's mirror. It provides an overview of publicly available sound propagation software (including wavenumber integration and parabolic equation models). It concludes with a few practical examples of modeling propagation loss for whale song and a seismic airgun array.
It is imperative that bioacousticians who work in aquatic environments have a basic understanding of sound propagation under water. Whether the topic is the function of humpback whale song, echolocation in wild bottlenose dolphins, the masking of grey whale sounds by ship noise, the role of chorusing in fish spawning behavior, the effects of seismic surveying on benthic organisms, or the capability of an echosounder to track a school of fish, the way in which sound propagates through the ocean affects how we can use sound to study animals, how sound we produce impacts animals, and how animals use sound.
Aquatic fauna has evolved to use sound for environmental sensing, navigation, and communication. This is because water conducts sound very well (i.e., fast and far), while light propagates poorly under water. Visual sensing based on sun- or moonlight is limited to the upper few meters of water. And while water transports chemicals, chemoreception is most effective over short ranges, where chemical concentration is high. Furthermore, sound can be detected from all directions, providing omnidirectional alerting of activities happening in the environment.
Given that sound may propagate over very long ranges with little loss, a myriad of sounds is commonly heard at any one place. These sounds may be grouped by origin: abiotic, biotic, and anthropogenic. Natural, geophysical, abiotic sound sources include wind blowing over the ocean surface, rain falling onto the ocean surface, waves breaking on the beach, polar ice breaking under pressure and temperature influences, subsea volcanoes erupting, subsea earthquakes rumbling along the seafloor, etc. Biotic sound sources include singing whales, chorusing fishes, feeding urchins, and crackling crustaceans. Anthropogenic sources of sound include ships, boats, fish-finding echosounders, oil rigs, gas wells, subsea mines, dredgers, trenchers, pile drivers, naval sonar, seismic surveys, underwater explosions, etc.
As these sounds travel from their source through the environment, they may follow multiple propagation paths. Sounds may be reflected at the sea surface and seafloor. Some sound may travel through the seafloor and radiate back into the water some distance away. Sound is scattered by scatterers in the water (such as gas bubbles or fish swim bladders). Sound bends as the ocean is layered with pressure, temperature, and salinity changing as a function of depth, and with freshwater inputs. All of these phenomena depend on the frequency of sound. The spectrum of broadband sound changes, too, as acoustic energy at high frequencies is more readily scattered and absorbed than energy at low frequencies. The receiver of sound can thus infer information not just about the source of sound but also about the environment's complexity.
Understanding the physics of sound in water is an important step in studies of aquatic animal sound usage and perception, whether these are conspecific social sounds, predator sounds, prey sounds, navigational clues, environmental sounds, or anthropogenic sounds. It is also critical for the study of impacts of sound on aquatic fauna, and for using passive or active acoustic tools for monitoring aquatic fauna and mapping biodiversity. The goal of this chapter is to introduce the basic concepts of sound propagation under water.
6.2 The Sonar Equation
The sonar equation was developed by the US Navy to assess the performance of naval sonar systems. These sonar systems were designed to detect foreign submarines. The sonar emits an acoustic signal under water and listens to returning echoes. The time of arrival and acoustic features of the echo may determine not only from what target the signal reflected, but also the range and speed of the target. The term "sonar" stands for "SOund Navigation And Ranging."
There are numerous forms of the sonar equation. What they all have in common is that (1) they each represent an equation of energy conservation, meaning that the total acoustic energy on either side of the equation is the same; and (2) all of the terms in the equation are expressed in decibel (dB). The sonar equation with its original terms as defined in Urick (1983) allows an easy conceptual exploration of various scenarios encountered in bioacoustics. The definitions and notations of some of the terms are more mathematically specific in the recent underwater acoustics terminology standard (ISO 18405)Footnote 1.
6.2.1 Propagation Loss Form
As sound propagates through the ocean, it loses energy, termed propagation loss (PLFootnote 2). A simple form of the sonar equation equates PL to the difference between the source level (SL) and the received level (RL) of sound (Urick 1983):
$$ PL= SL- RL\ \left(\mathrm{propagation}\ \mathrm{loss}\ \mathrm{form}\right) $$
SL was defined by Urick as 10log10 of the ratio of source intensity to reference intensity (see Chap. 4). RL was equal to 10log10 of the ratio of received intensity to reference intensity. PL was computed as 10log10 of the ratio of source intensity to received intensity.
For example, a whale-watching boat might have SL = 160 dB re 1 μPa2 (in terms of mean-square pressure, which is proportional to intensity; see Chap. 4) and be located 100 m from a group of whales. If PL in this environment and over this range is 40 dB, then RL at the whales is 120 dB re 1 μPa2 (Erbe 2002; Erbe et al. 2016a).
6.2.2 Signal-to-Noise Ratio Form
Another simple form of the sonar equation relates the RL of a signal to the background noise level (NL = 10log10 of the ratio of noise intensity to reference intensity):
$$ SNR= RL- NL\kern0.5em \left(\mathrm{signal}\hbox{-} \mathrm{to}\hbox{-} \mathrm{noise}\ \mathrm{ratio}\ \mathrm{form}\right) $$
SNR is the level of the signal-to-noise ratio, expressed in dB. For example, a call from a whale might have a received level RL = 105 dB re 1 μPa2 at another whale; however, background noise at the time might be NL = 115 dB re 1 μPa2 over the frequency band of the call. The SNR is −10 dB. Can the whale still hear the other one or does the noise mask the call?
Because the SNR is a negative number in this example, if one was just considering the relative levels of signal and noise, the animals would not be able to hear one another because the background noise level is much greater than the received signal level. However, animals (and sonar systems) can take advantage of spectral and temporal characteristics of a received sound, as is explained below. Therefore, in the example of beluga whales (Delphinapterus leucas) trying to communicate in icebreaker noise, the listening whale can indeed detect the call, because of the different spectral and temporal structures of call and noise (Erbe and Farmer 1998).
6.2.3 Forms to Assess Communication Masking
Acoustic communication under water remains an area of active research. In the conceptual model of Fig. 6.1, one animal (the sender) emits a signal, which travels through the habitat to the location of the receiver. Whether the receiver can hear the message depends on a number of factors that relate to the sender, the habitat, and the receiver. The level and spectral features of the signal will affect how far it propagates and how well it can be detected above the ambient noise in the environment. The locations of sender and receiver matter, not just the range between the two animals, but also at which depth each happens to be located. If the two animals are oriented towards each other, directional emission and reception capabilities will enhance signal detection. The environment changes the level and spectral characteristics of the signal by reflection, refraction, scattering, absorption, and spreading losses. The detection capabilities of the receiver can be quantified by the detection threshold, critical ratio, and other factors. Ambient noise in the environment can initiate anti-masking strategies at both the sender (e.g., increasing the source level) and receiver (e.g., orienting towards the signal). A sonar equation can be constructed to investigate each of these factors, as outlined in the following sections.
Fig. 6.1
Sketch of the factors related to acoustic communication in natural (not just aquatic) environments and their corresponding terms in the sonar equation: source level (SL), time-bandwidth product (TBP), sender directivity index (DIs), propagation loss (PL), absorption (absorption coefficient α multiplied by range R), noise level (NL), and receiver detection threshold (DT), critical ratio (CR), and directivity index (DIr). Modified from Erbe et al. (2016c); © Erbe et al. (2016); https://www.sciencedirect.com/science/article/pii/S0025326X15302125. Published under CC BY 4.0; https://creativecommons.org/licenses/by/4.0/
The basic sonar relation for the communication scenario in Fig. 6.1 is:
$$ SL- PL- NL> DT\ \left(\mathrm{basic}\ \mathrm{signal}\ \mathrm{detection}\ \mathrm{form}\right), $$
where DT is the detection threshold of the receiver, expressed in dB. A sound is deemed detectable if the expression on the left side exceeds the detection threshold. In the absence of noise, DT equals the audiogram. Audiograms are measured by exposing an animal to pure-tone signals of varying levels. The RL that is just detectable defines the audiogram at that frequency (see Chap. 10 for a more thorough definition of audiogram):
$$ RL= DT\ \left(\mathrm{audiogram}\ \mathrm{form}\right) $$
The mammalian auditory system acts as a bank of overlapping bandpass filters and the listener focuses on the auditory band that receives the highest SNR (Moore 2013). Under the equal-power assumption (Fletcher 1940), a signal is detected if its power is greater than the noise power in any of the auditory bands. So, for any auditory band,
$$ RL- NL>0\ \left(\mathrm{within}\ \mathrm{an}\ \mathrm{auditory}\ \mathrm{band}\right) $$
Communication signals of many species, including birds and marine mammals (Erbe et al. 2017a), are commonly tonal, while noise is commonly broadband. In order to assess the risk of communication masking, the critical ratio (CR) is a useful quantity that has been measured in humans and animals. The CR is the level difference between the mean-square sound pressure level (SPL) of a tone and the mean-square sound pressure spectral density level of broadband noise when the tone is just audible (American National Standards Institute 2015). Conceptually, the CR quantifies the ability of the auditory system to focus on a narrowband (tonal) signal. It captures how many of the noise frequencies surrounding the tone frequency are effective at masking the tone, and the resulting band of frequencies has been termed the Fletcher critical band (American National Standards Institute 2015). A narrowband signal is thus detectable, if
$$ RL- CR>{NL}_f\;\left(\mathrm{critical}\ \mathrm{ratio}\ \mathrm{form}\right) $$
RL is the tone level in dB re 1 μPa2, NLf is the noise mean-square pressure spectral density level in dB re 1 μPa2/Hz, and CR is measured in dB re 1 Hz (see p. 29 in Erbe et al. 2016c).
In the above-mentioned study with beluga whales communicating amidst icebreaker noise, the beluga whale call consisted of a sequence of six tones with overtones from 800 to 1800 Hz, and the icebreaker's bubbler system noise was broadband and relatively unstructured in frequency and time (Fig. 6.2) (Erbe and Farmer 1998). The bandwidth of the call, expressed in dB, was 10log10(1800–800) = 30 dB re 1 Hz (see Chap. 4 for definitions and formulae). Given NL = 115 dB re 1 μPa2 over the bandwidth of the call, NLf was equal to NL (115 dB re 1 μPa2) minus the bandwidth (30 dB re 1 Hz): NLf = 85 dB re 1 μPa2/Hz. Beluga whales have a CR of approximately 15 dB re 1 Hz at 800 Hz, therefore, the call with RL = 105 dB re 1 μPa2 was audible, because Eq. (6.4) was satisfied (Erbe 2008; Erbe and Farmer 1998): 105–15 > 85.
Spectrograms of the lower two harmonics of a beluga whale call (top panel) and an icebreaker's bubbler system noise (bottom panel). Colorbar in dB re 1 μPa2/Hz. The broadband levels are RL = 105 dB re 1 μPa2 for the call and NL = 115 dB re 1 μPa2 for the noise
In studies on critical ratios and in the beluga whale experiments (Erbe and Farmer 1998; Erbe 2000), signal and noise were broadcast by the same loudspeaker and thus arrived at the listener from the same direction. If the caller and the noise are spatially separated, then there is an additional processing gain in the sonar equation: the receiver's directivity index DIr:
$$ RL- CR+ DIr-{NL}_f>0\left(\mathrm{critical}\ \mathrm{ratio}\ \mathrm{form}\ \mathrm{with}\ \mathrm{directivity}\ \mathrm{index}\right) $$
The DIr is defined as 10log10 of the ratio of the intensity measured by an omnidirectional receiver to that of a directional receiver. Directivity indices increase with frequency and values up to 19 dB have been measured for communication sounds in marine mammals. The associated spatial release from masking should be considered in environmental impact assessments of underwater noise (Erbe 2015). Directivity indices are even greater at higher frequencies used by dolphins during echolocation (Fig. 6.3).
Sketches of the receiving directivity pattern of a bottlenose dolphin (Tursiops truncatus) in the vertical (a) and horizontal (b) planes. Courtesy of Chong Wei after data in (Au and Moore 1984)
6.2.4 Form for Biomass Surveying
Surveys for animals ranging from zooplankton to fish and sharks may use an echosounder, fish finder, or sonar (e.g., Parsons et al. 2014; Kloser et al. 2013). In this scenario, the echosounder emits a signal, which travels to the fish, where some of it is reflected. How much of the signal is reflected is expressed by the target strength (TS), defined as 10log10 of the ratio of echo intensity to incident intensity (Urick 1983). The reflected signal travels to the receiver, which has a specific DT and DIr. The receiver is typically co-located with the source, so that the signal travels the same path twice and thus experiences twice the PL. The fish is detected if the following sonar equation is satisfied:
$$ SL-2 PL+ TS- NL> DT- DIr\left(\mathrm{two}-\mathrm{way}\ \mathrm{sonar}\ \mathrm{surveying}\ \mathrm{form}\right) $$
Target strength will vary for each type of animal, as well as with the number of animals in the group and their orientation relative to the echosounder. Figure 6.4 shows reflected signals received on a REMUS autonomous underwater vehicle. Individual animals are observed in two aggregations, with two dolphins swimming within one of the aggregations. Researchers are using cameras on the same platforms to better understand the information contained in reflected signals and ultimately convert that information into species classifications and estimates of biomass (Benoit-Bird and Waluk 2020).
Echosounder image of marine fauna in two aggregations, with two dolphins being in the aggregation on the left. Colors represent acoustic target strength and the shapes of the two dolphins can easily be recognized by their high reflectivity (Benoit-Bird et al. 2017). © Benoit-Bird et al. 2017; https://aslopubs.onlinelibrary.wiley.com/doi/full/10.1002/lno.10606. Published under CC BY 4.0; https://creativecommons.org/licenses/by/4.0/
6.3 The Layered Ocean
The speed of sound in sea water increases with increasing temperature T [°C], salinity S (measured in practical salinity units [psu]) and hydrostatic pressure, which in the ocean is proportional to depth D [m]. The approximate change in the speed of sound c [m/s] with a change in each property is:
Temperature changes by 1 °C → c changes by 4.0 m/s
Salinity changes by 1 psu → c changes by 1.4 m/s
Depth (pressure) changes by 1 km → c changes by 17 m/s
Maps of sea surface temperature and salinity for the northern hemisphere summer show considerable variation (Fig. 6.5). However, temperature and salinity vary much more rapidly with depth than they do in the horizontal plane, so the ocean can often be thought of as a stack of horizontal layers, with each layer having different properties. Vertical profiles of these quantities are therefore very useful for understanding how sound will propagate in different geographical regions.
Maps of sea surface temperature (top) and salinity (bottom) for the northern hemisphere summer, averaged over the period 2005 to 2017. Data were taken from the World Ocean Atlas (Locarnini et al. 2018; Zweng et al. 2018)
6.3.1 Temperature and Salinity Profiles
In non-polar regions (red curves in Fig. 6.6), the main source of heat entering the ocean is solar. The sun heats the near-surface water, making it less dense and suppressing convection. A surface mixed layer with nearly constant temperature and salinity is formed by mechanical mixing due to surface waves and is typically 20–100 m thick. Below that, the temperature drops rapidly in a region known as the thermocline, before becoming almost constant at a temperature of about 2 °C in the deep isothermal layer that extends from a depth of about 1000 m to the ocean floor.
Depth profiles of temperature, salinity, and sound speed from the open ocean based on the World Ocean Atlas (Locarnini et al. 2018; Zweng et al. 2018) seasonal decadal average data for the austral winter (solid) and austral summer (dotted). Red curves are for 30.5°S, 74.5°E and are representative of non-polar ocean profiles. Blue curves are for 60.5°S, 74.5°E and are representative of polar ocean profiles
Seasonal changes in solar radiation together with the ocean's considerable thermal lag (due to its great heat capacity) can complicate this simple picture, but most of these changes only affect the top few hundred meters of the water column, changing the detailed structure of the mixed layer and the upper part of the thermocline.
In polar regions (blue curves in Fig. 6.6), the situation is quite different. There is a net loss of heat from the sea surface, which results in a temperature profile in the upper part of the ocean that increases with increasing depth from a minimum of about −2 °C at or (in summer) slightly below the surface.
Salinity typically changes by only a small amount with depth, and in most parts of the ocean is between 34 and 36 psu. As a result, the sound speed is usually determined by temperature and depth, however, salinity can have an important effect on sound speed in situations where it changes abruptly. Examples include locations where there is a large freshwater outflow into the ocean from a river, or in estuaries where it is common to have a wedge of dense, saline water underlying a surface layer of freshwater. In polar regions, the salinity of near-surface water can vary considerably depending on whether sea ice is forming, a process that excludes salt and therefore increases salinity in the water below the ice. When sea ice melts, freshwater is released, reducing near-surface salinity.
6.3.2 Sound Speed Profiles
The following equation is one of a number of equations of varying complexity that can be found in the literature relating the speed of sound to temperature, salinity, and depth (Mackenzie 1981). It is valid for temperatures from −2 to 30 °C, salinities of 30 to 40 psu, and depths from 0 to 8000 m.
$$ c=1448.96+4.591\ T-5.304\times {10}^{-2}\ {T}^2+2.374\times {10}^{-4}\ {T}^3+1.340\ \left(S-35\right)+1.630\times {10}^{-2}\ D+1.675\times {10}^{-7}\ {D}^2-1.025\times {10}^{-2}T\left(S-35\right)-7.139\times {10}^{-13}\ T{D}^3\ \left[\mathrm{m}/\mathrm{s}\right] $$
Sound speed profiles computed from the typical temperature and salinity profiles are also plotted in Fig. 6.6.
In non-polar waters, the sound speed may increase slightly with depth in the mixed layer due to its pressure dependence, however, diurnal heating and cooling effects can eliminate or enhance this effect. As explained later in this chapter, whether or not there is a distinct increase in sound speed with depth in the mixed layer determines whether there is a surface duct, which has a considerable impact on acoustic propagation from near-surface sound sources and to near-surface receivers.
Below the mixed layer, the rapid reduction in temperature with depth (i.e., in the thermocline) results in sound speed also reducing until, at a depth of about 1000 m, the temperature becomes nearly constant. In the deeper isothermal layer, the increasing pressure results in the sound speed starting to increase with depth. There is therefore a minimum in the sound speed in non-polar waters at a depth of approximately 1000 m, which, as will be seen later, is important for long-range sound propagation.
In polar waters, the temperature and pressure both increase with increasing depth, so the sound speed also increases, which results in a strong surface duct. However, in the Arctic Ocean, the existence of water masses with different properties entering from the Pacific and Atlantic oceans can lead to more complicated sound speed profiles.
Temperature and salinity profiles for the world's oceans can be found in the World Ocean AtlasFootnote 3 (Locarnini et al. 2018; Zweng et al. 2018). These are based on averages of a large amount of measured data and are very useful for calculating estimated sound speed profiles for particular locations for particular months or seasons of the year. The real ocean is, however, highly variable; particularly the upper thermocline and mixed layer, which can change on time scales of hours, and in some extreme cases, tens of minutes, so there is no substitute for in situ measurements of temperature and salinity profiles to support acoustic work.
6.4 Propagation Loss
The apparent simplicity of the propagation loss term (i.e., PL) in the various sonar equations hides a great deal of complexity. There are a few special situations in which PL can be calculated quite accurately using simple formulae, and a few more in which it might be possible to obtain a reasonable estimate using a more complicated equation, but for everything else, these simple approaches can lead to large errors, and it is necessary to resort to numerical modeling. To further complicate matters, there are a number of different types of numerical models used for propagation loss calculations, each with its own assumptions and limitations, and it is important to be familiar with these so that the most appropriate model can be used for a given task.
6.4.1 Geometric Spreading Loss
The most basic concept of propagation loss is that of geometric spreading, which accounts for the fact that the same sound power is spread over a larger surface area as the sound propagates further from the source. The intensity is the sound power per unit area (see Chap. 4), so the increase in surface area results in a reduction in intensity. The simplest case is when the source is small compared to the distances involved, the sound speed is constant, and the boundaries (i.e., sea surface, seabed, and anything else that might reflect sound) are sufficiently far away that reflected energy can be ignored. In this situation, the acoustic wavefront forms the surface of a sphere. As the wavefront propagates outward, the radius r of the sphere increases, the surface area of the sphere increases in proportion to r2, and therefore the intensity decreases inversely proportional to r2. This leads to the well-known spherical spreading equation for PL:
$$ PL=20{\log}_{10}\left(r/1\mathrm{m}\right) $$
Equation (6.5) is also applicable to calculating geometric spreading loss for sound radiated by a directional source, such as an echosounder transducer, or a dolphin's biosonar, providing the range is sufficiently large (i.e., the receiver is in the acoustic far-field of the source; see Chap. 4), and the above assumptions are all met.
Another situation in which spreading loss can be calculated analytically is when the sound is constrained in one dimension by reflection and/or refraction, so it can only spread in the other two dimensions. In underwater acoustics, this most commonly happens when the sound is constrained in the vertical direction by the sea surface or seafloor, but can still spread in the horizontal plane. The result is that the acoustic wavefront forms the surface of a cylinder, the area of which is proportional to the range. The intensity is therefore inversely proportional to the range, and the PL is given by the cylindrical spreading equation:
Some situations in which cylindrical spreading can occur are discussed later in this chapter, but it should be noted that Eq. (6.6), strictly speaking, only applies at all ranges from the source in the highly unusual case that the source is a vertical line source that spans the entire depth interval into which the sound is constrained, and that no sound is lost into either the upper or lower layers.
For the much more common case of a small source, the sound will undergo spherical spreading at short ranges where the boundaries have no effect, followed by cylindrical spreading at long ranges where the fact that the source has a small vertical extent is of little consequence. In between, there will be a transition region in which neither formula is accurate. This situation can be approximated by assuming a sudden transition from spherical to cylindrical spreading at a "transition range" rt. Equation (6.7) applies only to ranges r ≫ rt and still makes the assumption that there are no losses at the boundaries.
$$ PL=20{\log}_{10}\left(\frac{r_t}{1\mathrm{m}}\right)+10{\log}_{10}\left(\frac{r}{r_t}\right)=10{\log}_{10}\left(\frac{r_t}{1\mathrm{m}}\right)+10{\log}_{10}\left(\frac{r}{1\mathrm{m}}\right) $$
In shallow-water situations, some authors recommend using a transition range equal to the water depth; however, while useful for very rough PL estimates, this approach should be adopted with caution as the best choice will depend on the characteristics of the seabed. The only way to accurately determine rt for a given situation is to carry out numerical propagation modeling, in which case you might as well use that to directly determine the propagation loss, removing the need for (Eq. 6.7) and its inherent inaccuracies.
6.4.2 Absorption Loss
When a sound wave propagates through water, it results in a periodic motion of the molecules present in the water, and the slight friction within and between them converts some of the sound energy into heat, reducing the intensity of the sound wave. This is called absorption loss and results in a propagation loss that is proportional to the range traveled:
$$ PL=\alpha {r}_{\mathrm{km}} $$
where rkm is the range in kilometers and α is the absorption coefficient in dB/km. The propagation loss due to absorption must be added to the propagation loss due to geometrical spreading described in Sect. 6.4.1.
A commonly used formula for α is:
$$ \alpha =0.106\frac{f_1{f}^2}{f_1^2+{f}^2}{e}^{\left( pH-8\right)/0.56}+0.52\left(1+\frac{T}{43}\right)\frac{S}{35}\frac{f_2{f}^2}{f_2^2+{f}^2}{e}^{-z/6}+4.9\times {10}^{-4}{f}^2{e}^{-\left(T/27+z/17\right)} $$
with f1 = 0.78(S/35)1/2eT/26 and f2 = 42eT/17; f [kHz], α[dB/km]
$$ {\displaystyle \begin{array}{ll}\mathrm{valid}\ \mathrm{for}& -6<T<35{}^{\circ}\mathrm{C}\ \left(S=35\;\mathrm{psu},\kern0.5em \mathrm{pH}=8,\kern0.5em z=0\right)\\ {}& 7.7<\mathrm{pH}<8.3\ \left(T=10{}^{\circ}\mathrm{C},\kern0.5em S=35\;\mathrm{psu},\kern0.5em z=0\right)\\ {}& 5<S<50\;\mathrm{psu}\ \left(T=10{}^{\circ}\mathrm{C},\kern0.5em \mathrm{pH}=8,\kern0.5em z=0\right)\\ {}& 0<z<7\;\mathrm{km}\ \left(T=10{}^{\circ}\mathrm{C},\kern0.5em S=35\;\mathrm{psu},\kern0.5em \mathrm{pH}=8\right)\end{array}} $$
(François and Garrison 1982a, b; Ainslie and McColm 1998).
The absorption coefficient increases with frequency (Fig. 6.7). At low frequencies, it is dominated by molecular relaxation of two minor constituents of seawater: B(OH)3 and MgSO4, whereas above a few hundred kHz, it is primarily due to the water's viscosity.
Graph of absorption loss dominated by B(OH)3 for f < 5 kHz, by MgSO4 for 5 kHz < f < 500 kHz, and by viscosity above. T = 10 °C, S = 35 psu, z = 0 m, pH = 8
In summary, Fig. 6.8 compares how propagation loss increases with range for spherical spreading (Eq. 6.5), cylindrical spreading (Eq. 6.6), and combined spherical/cylindrical spreading with a transition range of 100 m (Eq. 6.7). The effect of absorption (Eq. 6.8) in addition to spherical spreading is also shown for frequencies of 1, 10, and 100 kHz.
Plot of propagation loss versus range assuming spherical spreading (Eq. 6.5), cylindrical spreading (Eq. 6.6), and mixed spherical/cylindrical spreading (Eq. 6.7) for a transition range of 100 m. Propagation loss is also shown for spherical spreading with the addition of absorption (Eq. 6.8) corresponding to frequencies of 1, 10, and 100 kHz. Note that in the literature, the y-axis is sometimes flipped
6.4.3 Additional Losses
6.4.3.1 The Air–Water Interface
6.4.3.1.1 Reflection and Transmission Coefficients
In animal bioacoustics as well as noise research, one typically deals with sounds in one medium (i.e., either air or water) and then sticks to this medium, only modeling propagation within this medium and only considering receivers in this medium. However, sound does cross into other media, and so a fish might be able to hear an airplane flying overhead, and a bird flying directly overhead might be able to hear a submarine's sonar (Fig. 6.9).
Sketches of a sound source in the air (helicopter; left) and water (submarine; right), and the incident pi, reflected pr, and transmitted pt rays (i.e., vectors pointing in the direction of travel, perpendicular to the wavefront), with corresponding grazing angles θ1 and θ2. In the left panel, medium 1 corresponds to air with sound speed c1, and medium 2 corresponds to water with sound speed c2. The situation is reversed in the right panel, where medium 1 is water, and medium 2 is air
As sound hits an interface, the incident wave, in most situations, gives rise to a reflected wave and a transmitted waveFootnote 4 (also see Chap. 5, where reflection is explained based on Huygens' principle). The energy of the reflected wave remains within the medium of the incident sound, but the energy of the transmitted wave is lost from the medium of the incident sound and transmitted into the adjacent medium. The amplitudes of the reflected and transmitted (plane) waves are given by the reflection and transmission coefficients \( \mathcal{R} \) and \( \mathcal{T} \) (Medwin and Clay 1998):
$$ \mathcal{R}=\frac{Z_2\sin {\theta}_1-{Z}_1\sin {\theta}_2}{Z_2\sin {\theta}_1+{Z}_1\sin {\theta}_2} $$
$$ \mathcal{T}=\frac{2{Z}_2\sin {\theta}_1}{Z_2\sin {\theta}_1+{Z}_1\sin {\theta}_2} $$
where θ1 is the grazing angle of the incident wave, measured from the interface, and θ2 is the grazing angle of the transmitted (refracted) wave, also measured from the interface. The angle of incidence is measured from the normal (i.e., perpendicular to the interface); the angle of incidence and the grazing angle of the incident wave always add to 90°. The acoustic impedance Z is the product of density and sound speed: Z = ρc. In air at 0 °C, Z = 1.3 kg/m3 × 330 m/s = 429 kg/(m2s). In freshwater at 5 °C, Z = 1000 kg/m3 × 1427 m/s = 1,427,000 kg/(m2s). In sea water at 20 °C and 1 m depth with 34 psu salinity, Z = 1035 kg/m3 × 1520 m/s = 1,573,200 kg/(m2s) (see Chap. 4). So, Zair < < Zwater, whether it is freshwater or saltwater.
Snell's law (Fig. 6.9, Eq. 6.11)Footnote 5 relates the angles of the incident and refracted waves (θ1 and θ2) at the interface. Rays bend towards the interface, if the speed of sound in medium 2 is greater than that in medium 1 (c2 > c1) and away from the interface, if c1 > c2. While Snell's law typically relates the sines of the angles measured from the normal, it may also be expressed in terms of the cosines of the grazing angles (Etter 2018):
$$ \frac{\cos {\theta}_1}{\cos {\theta}_2}=\frac{c_1}{c_2} $$
For normal incidence, all of the angles in Eq. (6.10) are 90°, and so all of the sines are 1, hence
$$ \mathcal{R}=\frac{Z_2-{Z}_1}{Z_2+{Z}_1}\ \mathrm{and}\ \mathcal{T}=\frac{2{Z}_2}{Z_2+{Z}_1} $$
For a sound source in air, Z1 < < Z2 => \( \mathcal{R} \) → 1 and \( \mathcal{T} \) → 2, at normal incidence. Almost all of the sound is reflected, but the pressure in the water increases by a factor 2. The air–water boundary, for sound arriving from air, is considered "hard." The value of \( \mathcal{T} \) is the reason why even weak aerial sources (such as drones hovering over whales) can be detected in water, below the source, at several meters depth (Erbe et al. 2017b), and commercial airplanes can be recorded in coastal waters, lakes, and rivers even if flying at hundreds of meters in altitude (Erbe et al. 2018). Received levels under water from airplanes may exceed behavioral response thresholds for underwater sound sources (Kuehne et al. 2020). For non-normal incidence, with c2 > c1, there exists a critical angle, beyond which the transmitted wave disappears. This situation is called total internal reflection. The only sound in the water is an evanescent field that decays exponentially in amplitude below the sea surface. The evanescent field is only important if the depth of the receiver is smaller than the in-water acoustic wavelength.
For a sound wave meeting the water–air interface from below, Z1 > > Z2 therefore \( \mathcal{R} \) → −1 and \( \mathcal{T} \) → 0. Almost all sound is reflected, albeit at negative amplitude, which means that the incident and reflected pressures cancel each other out. This is why the water–air interface is called a pressure-release boundary (or "soft" boundary) for sound incident from below. For non-normal incidence, \( \mathcal{R} \) and \( \mathcal{T} \) need to be computed with Eq. (6.10). Also, as a sound source is moved to shallower depth (i.e., closer to the sea surface), the proportion of transmitted sound increases. This is because of the evanescent (i.e., exponentially decaying) field, which is ignored by Eq. (6.10), but that might still have enough amplitude at the sea surface for shallow sources (Godin 2008).
6.4.3.1.2 Lloyd's Mirror
While not resulting in a loss of sound energy, the Lloyd's mirror effect is a result of reflection from the water–air interface from shallow sound sources. An omnidirectional source (i.e., one that emits sound in all directions) close to the sea surface (such as a ship's propeller) emits some of its sound in an upwards direction, and this sound reflects off the sea surface. At any receiver location, sound that traveled along the surface-reflected path overlaps with sound that traveled along the direct path from the source to the receiver. The reflected ray's amplitude is opposite in sign to the incident ray's amplitude (\( \mathcal{R} \) = −1); conceptually, this ray emerged from an image source (also called virtual source) with negative amplitude on the other side of the interface. The direct ray does not experience a flip in amplitude. Depending on the relative path lengths, the surface-reflected sound will add constructively to the sound that traveled along the direct path, or they will cancel each other out. This creates a pattern of constructive and destructive interference about the sound source, called the Lloyd's mirror effect. As a ship passes a moored recorder, the spectrogram shows the characteristic U-shaped interference pattern as successive peaks and troughs in amplitude at any one frequency over time (Fig. 6.10). Additional images of the Lloyd's mirror interference pattern can be found in (Parsons et al. 2020) for small electric ferries and in (Erbe et al. 2016b) for recreational swimmers and boogie boarders.
Spectrogram of the recording of a ship passing by a moored recorder, showing the pattern of constructive and destructive interference called the Lloyd's mirror effect. The closest point of approach occurred at about 200 s. Modified from (Erbe et al. 2016c); © Erbe et al. 2016; https://www.sciencedirect.com/science/article/pii/S0025326X15302125. Published under CC BY 4.0; https://creativecommons.org/licenses/by/4.0/
6.4.3.1.3 Scattering at the Sea Surface
If the sea surface is not flat, then some of the reflected energy is scattered away from the geometric reflection direction, reducing the amplitude of the geometrically reflected wave. This is called surface scattering loss, which increases as the roughness of the sea surface increases, the acoustic wavelength decreases (i.e., acoustic frequency increases), and the grazing angle between the direction of the incident wave and the plane of the sea surface increases. This relationship is quantified by the Rayleigh roughness parameter (Jensen et al. 2011):
$$ \gamma =4\pi \frac{h}{\lambda}\sin \theta $$
where h is the root-mean-square (rms) roughness of the surface (i.e., approximately ¼ of the significant wave height), λ is the acoustic wavelength, and θ is the grazing angle. The larger the value of γ is, the larger is the apparent roughness of the surface. The corresponding effective pressure reflection coefficient of the sea surface is then given by:
$$ {\mathcal{R}}^{\prime }=-{e}^{-0.5{\gamma}^2} $$
which corresponds to an additional propagation loss of \( 20{\log}_{10}\left|{\mathcal{R}}^{\prime}\right|=4.34{\gamma}^2 \) dB each time the sound reflects off the surface (Fig. 6.11). Note, however, that these formulae are only valid for surfaces that are not too rough, which, in this case, means γ < 2, corresponding to a scattering loss < 17 dB per bounce.
Graphs of additional propagation loss per bounce as a function of grazing angle for reflection from rough surfaces with various ratios of rms roughness to acoustic wavelength
Strictly speaking, the effective pressure reflection coefficient (Eq. 6.13, Fig. 6.11) applies to the coherent component of the acoustic field, which can be thought of as the component that does not change as the rough sea surface moves. There will also be a scattered component that does change, and in some situations, this is an important contributor to the received signal. This component is ignored by Eq. (6.13), which can therefore be considered to provide an upper limit on the propagation loss per bounce.
6.4.3.2 The Seafloor Interface
The interaction of sound with the seafloor is more complicated. The acoustic properties of the seabed are often similar to those of the water, so a significant amount of sound can penetrate the seabed. The lower the frequency is, the deeper the sound can penetrate. At frequencies below a few kHz, it is common for a significant amount of acoustic energy to be reflected back into the water column from geological layering within the seabed. Seismic survey companies searching for oil and gas reserves are taking advantage of this.
Some of this complexity is illustrated in Fig. 6.12, which plots the pressure reflection coefficient as a function of grazing angle for four different seabed types: silt, sand, limestone, and basalt. Silt and sand layers are unconsolidated, which means that shear waves have a low speed and attenuate rapidly. (Shear waves are waves in which the particles oscillate at right angles to the direction of sound propagation; see Chap. 4.) Acoustically, they can often be well approximated by a fluid (which does not support shear waves at all) with an increased attenuation to account for the shear wave losses. Unconsolidated sediments become more reflective as the sediment grain size increases from silt to sand. Limestone and basalt are consolidated rocks, which allow both compressional waves and shear waves to propagate, and are thus referred to as solid elastic seabeds. Basalt is a hard rock and highly reflective at all grazing angles. The reflection coefficient of limestone, however, is perhaps surprising. While it is also a rock, it has the lowest reflectivity of the four seabeds at small grazing angles. This is because the shear wave speed in limestone is very similar to the sound speed in water, which allows energy to pass easily from sound waves in the water to shear waves in the seabed.
Curves of pressure reflection coefficient versus grazing angle for four different seabed types, calculated with parameters from Jensen et al. (2011)
Curves of reflection coefficients versus grazing angle are even more complicated for layered seabeds due to interference between waves reflecting from different layers, and in this case, the reflectivity becomes frequency dependent. Despite the complexity, there are computer programs available, based on techniques described in Jensen et al. (2011), that can numerically calculate the reflection coefficient curve for any arbitrarily layered seabed. A good example is BOUNCE, which is part of the Acoustics Toolbox.Footnote 6 A much bigger problem is the common lack of information on the geoacoustic properties of the seabed, to be able to provide these programs with accurate input data.
Seafloor roughness can further reduce the apparent acoustic reflectivity, although if the rms roughness is known, this can be dealt with (at least approximately) by using Eq. (6.12) to calculate the associated Rayleigh roughness parameter γ as a function of grazing angle. The effective seabed reflection coefficient is then:
$$ {\mathcal{R}}^{\prime }=\mathcal{R}{e}^{-0.5{\gamma}^2} $$
where \( \mathcal{R} \) is the pressure reflection coefficient for the flat seafloor (Eq. 6.10). All terms in this equation depend on grazing angle. The propagation loss per bounce is given by \( 20{\log}_{10}\left|{\mathcal{R}}^{\prime}\right| \).
6.4.3.3 Scattering Within the Water Column
Sound can be scattered within the water column by anything that causes sharp changes in sound speed, density, or both (i.e., acoustic impedance, which is the product of sound speed and density; see Chap. 4). This includes gas bubbles, biological organisms (in particular those with gas-filled organs like lungs or swim bladders), and suspended sediment particles. Water column scattering is utilized in active sonar systems, which rely on the backscattered signal to detect and/or characterize objects within the water column. However, clouds of air bubbles formed by breaking waves can cause an appreciable increase in propagation loss in some circumstances.
Air bubbles are essentially small, resonant cavities within the water column, which can both scatter and absorb sound and, when found in large numbers, can change the effective density, and hence sound speed, of the water. When a wave breaks, it entrains a large amount of air down to depths of several meters, forming a cloud of bubbles of a range of sizes. The large bubbles rise to the surface quite quickly, but the smaller bubbles can remain at depth for many minutes. This can increase the propagation loss for sound traveling close to the surface (Ainslie 2005; Hall 1989).
6.4.4 Numerical Propagation Models
6.4.4.1 The Wave Equation and Solution Approaches
The ocean is a complicated environment for sound propagation, and the simple approaches to estimating propagation loss described above are very limited in their applicability. As a result, a great deal of effort has gone into developing numerical propagation models that can calculate acoustic propagation loss for realistic situations. What follows is a brief introduction to the topic. The interested reader is referred to Etter (2018) and Jensen et al. (2011) for a more comprehensive treatise.
Fundamentally, all numerical propagation models solve the acoustic wave equation, which is a differential equation that relates the way the pressure changes over time to how it changes spatially as a wave propagates:
$$ {\nabla}^2\varPhi =\frac{1}{c^2}\frac{\partial^2\varPhi }{\partial {t}^2} $$
where ∇2 is the Laplace operator, ∂ indicates the partial derivative, c is the speed of sound, t represents time, and Φ is the solution to the wave equation.
The wave equation itself is well understood and straightforward to solve in simple cases; however, there are two issues that make it difficult to solve numerically for typical underwater acoustics problems:
Solutions are usually desired over domains that are orders of magnitude larger than the acoustic wavelength. Direct solution methods, such as finite differences or finite elements, require meshing the solution domain at a resolution of a small fraction of a wavelength, so the size of the required domain makes these approaches impractical for most propagation problems, even with modern computing hardware.
The boundaries of the domain, particularly the seabed, are complicated, but very important to model accurately as they have a strong influence on sound propagation.
Getting around these difficulties requires making approximations that lead to equations that are practical to solve for the problems of interest, with different approximations leading to different methods suitable for different situations.
In general, the solution of the acoustic wave equation is a function of three spatial dimensions and time. In Cartesian coordinates, the acoustic pressure can be written as: p(x, y, z, t). In most cases, we are interested in the field generated by a small source, which can be approximated as a single point in space. It is more convenient to work in cylindrical coordinates centered on the source location, p(r, z, ϕ, t), where r is the horizontal distance from the source to the receiver, z is the receiver depth below the sea surface, and ϕ is the horizontal plane azimuth angle of the receiver relative to some direction reference.
Many modeling approaches start by assuming that the solution has a harmonic time dependence so that p(r, z, ϕ, t) = pω(r, z, ϕ)e−iωt where ω = 2πf is the angular frequency and \( i=\sqrt{-1} \). Substituting this solution form into the wave equation (Eq. 6.15) leads to another differential equation called the Helmholtz equation, which can be solved at a specified ω to give pω(r, z, ϕ). The computational advantage of this is that the Helmholtz equation can be solved independently for each required frequency, converting a coupled four-dimensional (4D) problem into a number of independent 3D problems. Models that use this approach are known as frequency domain models, whereas models that directly solve the wave equation are known as time domain models. If required, the time domain solution can be reconstructed from multiple frequency domain solutions using Fourier synthesis (see Jensen et al. 2011, Chap. 8, for details).
The azimuth angle dependence can be dealt with by two different approaches. Modeling in 3D retains the full azimuth dependence of the environment, whereas N × 2D modeling assumes that changes in the environment due to small changes in ϕ have negligible effect on sound propagation, so that modeling can be carried out independently along each azimuth of interest. The majority of numerical models use the N × 2D approach, because there is again a substantial computational saving, this time by reducing a coupled 3D problem, solving for pω(r, z, ϕ), to a number of independent 2D problems, each solving for pω, ϕ(r, z) using only environmental information for the corresponding azimuth.
The inherent assumption of the N × 2D method provides a good approximation to the sound field in many propagation modeling situations where horizontal sound speed gradients are much smaller than vertical sound speed gradients, the seabed slopes are small, and the ranges are not large enough for the remaining out-of-plane effects to have an appreciable effect on the sound field. However, there are cases where full 3D modeling may be required; for example, around steep-sided submarine canyons, in the presence of nonlinear internal waves that can produce strong horizontal sound speed gradients, or for very-long-range propagation across ocean basins.
Some propagation models further simplify their calculations by assuming that the environment (but not the sound field) is independent of range, which means that the sound speed profile is a function of depth only, and the water depth and seabed properties are the same at all ranges (i.e., the seafloor is flat). These are called range-independent (RI) propagation models, whereas propagation models that allow the sound speed profile and/or the water depth and/or the seabed properties to vary with range are known as range-dependent (RD) models.
Acoustic propagation models are usually characterized by the numerical approach adopted, and the following sections described some of the most common. Guidance on which propagation model to use in various scenarios follows this section.
6.4.4.2 Ray and Beam Tracing
A ray is a vector, normal to the wavefront, and shows the direction of sound propagation. Ray models trace rays by repeatedly applying Snell's law (Eq. 6.11). For layered media (such as layers of ocean water with differing properties), Snell's law relates the angles of incidence θ1 and refraction θ2 at every layer boundary. Rays bend towards the horizontal, if c2 > c1, and away from the horizontal if c1 > c2.
There are several approaches to calculating the amplitude of the acoustic field. The simplest, known as conventional ray tracing, is to use the distance between initially adjacent rays to determine the area over which the sound power has spread and calculate the intensity as the power per unit area. Unfortunately, this method results in unphysical predictions of infinite sound amplitude at locations called caustics, where initially adjacent rays cross and therefore have zero separation. It also predicts sharp transitions to zero sound intensity in shadow zones, which are regions where rays do not enter, whereas in reality, the transition will be smoother. Both of these problems are a result of a high-frequency approximation inherent in ray theory, which cannot deal with diffraction (i.e., the phenomenon of waves bending around obstacles or spreading out after passing through a narrow gap; see Chap. 5 on sound propagation examples in the terrestrial world).
An alternative approach to calculating the amplitude of the acoustic field is to treat each ray as the center of a beam with a specified (usually Gaussian) amplitude profile. The field at a particular location is then obtained by summing the contributions from all the beams that overlap at that location. The main challenge with this approach is determining how the amplitude and width of the beam should change along the ray, but algorithms have been developed to do this (see Jensen et al. 2011, Sect. 3.5, for details). One of the best-known propagation codes of this type is Bellhop (Porter and Bucker 1987), a fully range-dependent, Gaussian beam tracing program suitable for N × 2D modeling that is available as part of the Acoustics Toolbox. The toolbox also includes a fully 3D variant called Bellhop3D.
Although Gaussian beam tracing is an improvement to conventional ray tracing and reduces the effects of the high-frequency assumption inherent in ray theory, it does not completely eliminate them. Its treatment of shadow zones and caustics produces realistic, but not necessarily accurate results and, importantly, it does not predict waveguide cutoff effects.
In underwater acoustics, the term waveguide or duct is used to describe any situation in which sound is constrained to a particular span of depths by reflection, refraction, or some combination of the two. Common examples include (Fig. 6.13):
A shallow-water duct in which sound is constrained by reflection from both the sea surface and the seabed.
A surface duct, in which the sound speed near the sea surface increases with increasing depth. This results in sound that is initially heading downward being refracted upwards towards the sea surface, where it is reflected back downward again, and so on. It is therefore constrained by reflection at the top and by refraction at the bottom. Weak surface ducts are often found in the mixed layer due to sound speed increasing with increasing pressure, and strong surface ducts are ubiquitous in polar oceans because both pressure and temperature increase with increasing depth. Sea ice can, however, reduce the acoustic reflectivity of the sea surface and therefore increase the attenuation of sound traveling in the duct.
The Deep Sound Channel (DSC), also known as the sound fixing and ranging (SOFAR) channel, in which sound is refracted towards the minimum in the sound speed (i.e., towards the waveguide axis). The waveguide axis occurs at a depth of about 1000 m in much of the world's ocean. The sound is constrained by refraction both above and below the axis of the waveguide. However, these are not sharp boundaries, and the steeper the angle of propagation is, the larger are the excursions of the ray paths away from the axis.
Convergence zone propagation in which sound is constrained by reflection from the sea surface and refraction from the increase of sound speed with increasing depth that occurs below the axis of the DSC.
Sound speed profiles (left) and ray trace plots computed using Bellhop (Porter and Bucker 1987, right) illustrating the common underwater acoustic ducts described in the text. The source depth was 10 m for all except the deep sound channel example, which had a source depth of 1200 m
In all cases, the waveguide will only trap rays leaving the source within a certain span of angles from the horizontal. In the case of the shallow water waveguide, this is because the seabed reflectivity reduces as the grazing angle increases (Fig. 6.12), so more energy is lost on each bottom bounce at steeper angles. In the other waveguide cases, it is because the refraction is not strong enough to turn the ray around before it either reaches a depth where the sound speed gradient is refracting it away from the waveguide (surface duct) or it hits the seabed (DSC and convergence zone).
According to ray theory, rays can be launched at any angle, irrespective of the frequency, and so it should always be possible to find rays that will be trapped in the waveguide, provided the source is at a suitable depth. However, this is not actually the case at low frequencies, where the acoustic wavelength becomes an appreciable fraction of the thickness of the waveguide. It turns out that if the frequency is sufficiently low, no energy will be trapped in the waveguide, and the waveguide is said to be cut off. Understanding why this is the case requires an understanding of normal modes, which is the topic of the next section.
6.4.4.3 Normal Modes
Most people find the concept of normal modes to be less intuitive than that of rays, but it is very useful for understanding low-frequency sound propagation in the ocean and forms the basis for a class of acoustic propagation models called normal-mode models.
Normal modes are best understood by first considering an ideal shallow-water waveguide with a constant depth (i.e., flat seafloor), constant sound speed, and perfectly reflecting seafloor. Solving the Helmholtz equation for this situation requires that two so-called boundary conditions be met: one at the sea surface and one at the seafloor. The sea surface is a soft boundary as far as underwater sound is concerned, so the boundary condition here is that the acoustic pressure due to the incident and reflected waves sums to zero, which requires that an incident sound wave is inverted on reflection. Conversely, the seafloor is a hard boundary, which requires that the incident and reflected waves sum to a maximum pressure; so the amplitudes of the incident and reflected waves must have the same sign.
Both of these boundary conditions have to be satisfied simultaneously. The water depth is fixed, and normal modes consider one frequency at a time, so the wavelength is fixed. The only variable that can change to satisfy the requirements is the angle from the horizontal at which the wave propagates. There are certain, discrete propagation angles that allow the surface and seafloor boundary conditions to be met simultaneously, corresponding to the normal modes. Each normal mode consists of a pair of plane waves, one propagating upward and the other downward, at the same angle to the horizontal (Fig. 6.14). The mode that corresponds to the pair of waves propagating closest to the horizontal is called the lowest-order mode (mode 1), and the mode order increases as the propagation angle gets steeper. Note that the waves can never propagate exactly horizontally, because that does not meet the boundary conditions.
Depth-range plots showing how the normal modes of an ideal shallow-water waveguide (lower panel) result from a pair of upward (upper panel) and downward (middle panel) propagating plane waves. Left-hand panels are for mode 1, right-hand panels are for mode 2. Arrows show the direction of propagation. The water depth is 50 m and the acoustic wavelength is 20 m
A receiver in the water column will receive the sum of the pressures from the upward and downward traveling waves. The amplitude of that combined signal can be plotted as a function of depth and range for each mode, yielding a series of mode shape curves (Fig. 6.15). Note that there is always a null in pressure (i.e., a node) at the sea surface and a maximum in pressure magnitude (i.e., +1 or −1; an antinode) at the hard seafloor.
Mode shapes for the first four normal modes of a 50-m deep ideal shallow-water waveguide with a rigid seabed
The mode shapes are reminiscent of standing waves on a guitar string, which are also normal modes. However, on a guitar string, different modes correspond to different frequencies of vibration, whereas in a waveguide, different modes correspond to sound of the same frequency propagating at different angles to the horizontal.
For any waveguide thickness, the propagation angles for a particular mode increase as frequency is reduced. The ideal waveguide considered so far has no limit to how steep the propagation angles can be, but that is not the case for real ocean waveguides which, as discussed in the previous section, all have limits on the angular range of the energy they can trap. The highest-order mode corresponds to the steepest propagation angle, so as frequency is reduced, it will become too steep to be constrained by the waveguide and will no longer be able to propagate. As frequency is reduced further, the same will happen to the next-highest-order mode, and so on until the lowest-order mode is unable to propagate, at which point the waveguide is said to be cut off.
In real ocean waveguides, the sound speed varies with depth, which causes the propagation angle of each mode to also be a function of depth. This changes the mode shapes, but you can still consider a mode to consist of a pair of upward and downward going waves, propagating at the same angle to the horizontal at any given depth.
The starting point for the mathematical derivation of normal-mode models is the depth-separated Helmholtz equation, which is valid for range-independent problems and is obtained by assuming that the acoustic field can be represented by the product of a function of depth and a function of range:
$$ {p}_{\omega, \phi}\left(r,z\right)=F(z)G(r). $$
Substituting this into the Helmholtz equation results in a one-dimensional differential equation for F(z) in terms of a separation constant kr. The solution of this differential equation has poles (infinities) at certain values of kr, which correspond to the normal modes. Normal-mode codes search for these values of kr, calculate the corresponding mode shapes, and then compute pω,ϕ(r, z) by a mathematical technique called the "method of residues," which involves summing the contributions of all the poles, which in this case, corresponds to summing the contributions of the individual modes. It turns out that kr has a geometric interpretation. It is called the horizontal wavenumber and is related to the modal propagation angle θ (relative to the horizontal) by kr = ω cos(θ)/c.
Normal-mode codes are computationally very fast for range-independent problems, because the modes only have to be found once, after which the field can be calculated at any desired range with very little additional computational effort.
Dealing with range-dependent problems involves approximating the environment as a series of range-independent sections, calculating the modes for each of these sections, and then calculating how the energy present in the modes in one section transmits across the boundary to the modes in the next section. There are two approaches:
The adiabatic mode method assumes that all the energy in mode 1 stays in mode 1, all the energy in mode 2 stays in mode 2, etc. This is relatively simple to implement and fast to compute, but is only accurate for environments that change relatively slowly with range.
The coupled-mode method allows energy to transition between modes, and so can deal with environments that change more rapidly. But this method is much more computationally demanding.
A good example of a normal-mode model is KRAKEN (Porter and Reiss 1984), which can be used for both range-independent and range-dependent modeling (both adiabatic and coupled) and is part of the Acoustics Toolbox (Footnote 5).
One limitation of normal-mode models such as KRAKEN is that they only include the component of the acoustic field that is fully trapped in the waveguide, so they tend to be inaccurate at short ranges where the component of the field that is losing energy out of the waveguide can be significant. This problem can be addressed by including so-called leaky modes in the solution. However, reliably finding leaky modes turns out to be a very challenging numerical task. The most successful normal-mode model to-date in this respect is ORCA (Westwood et al. 1996), which is accurate at short range and can also deal with seabeds that support shear waves. ORCA was written as a range-independent model, but there have been several attempts to adapt it to range-dependent problems using the adiabatic mode method (Hall 2004; Koessler 2016).
6.4.4.4 Wavenumber Integration
The mathematical derivation of the wavenumber integration method also starts with the depth-separated Helmholtz equation, but in this case, F(z) is calculated by direct numerical solution of the one-dimensional differential equation over a range of kr values, giving the so-called wavenumber spectrum. The acoustic field pω,ϕ(r, z) is then obtained by an integral transform of the wavenumber spectrum that involves a Hankel function. A numerical approximation to the Hankel function that is valid except at ranges smaller than the acoustic wavelength can be used to convert this integral transform into a Fourier transform, which can then be evaluated using the very efficient Fast Fourier Transform algorithm.
Wavenumber integration codes that use this method of evaluating the integral transform are known as fast-field programs. Common examples are SAFARI, OASES, and SCOOTER (Porter 1990; Schmidt and Glattetre 1985). OASES is a development of SAFARI and has largely superseded it, whereas SCOOTER, which is part of the Acoustics Toolbox (Footnote 5), is a separate, but largely equivalent, development. These programs are very accurate for acoustic propagation calculations at ranges close enough to the source that the environment can be considered range-independent, and can deal with arbitrarily complicated, layered seabeds. For most applications, the short-range limitation introduced by the Hankel function approximation is of little consequence, but, if necessary, it can be removed (at additional computational cost) by directly evaluating the integral transform.
It has proved difficult to extend the wavenumber integration method to range-dependent problems in a way that results in an efficient propagation model, although the full (paid) version of OASESFootnote 7 does have this capability. The theoretical background of this model is described in Goh and Schmidt (1996).
6.4.4.5 Parabolic Equation
Inserting a solution of the form \( {p}_{\omega, \phi}\left(r,z\right)=f\left(r,z\right){H}_0^{(1)}\left({k}_0r\right) \) into the Helmholtz equation yields parabolic-equation (PE) models. Here, \( {H}_0^{(1)} \) represents an outgoing cylindrical wave with wavenumber k0 = 2πf /c0 where c0 is an assumed sound speed. Technically, \( {H}_0^{(1)} \) is a Hankel function of the first kind of zero order. The aim of PE models is to solve for f(r, z), which represents the way in which the true field varies from that produced by the ideal outgoing cylindrical wave.
If the sound is assumed to be propagating predominantly in the range direction (the so-called paraxial approximation), then an efficient numerical algorithm can be employed. Given f(r, z), a small range step dr is added to calculate f(r + dr, z), a little bit farther from the source. This calculation can then be repeated as many times as desired to march the solution out in range. The sound field at one range is thus used to calculate the sound field at the next range and so on, without explicitly solving the depth-separated Helmholtz equation, making this a fundamentally different approach to the normal mode and wavenumber integration methods discussed previously.
Initially, the paraxial approximation was very restrictive and severely limited the utility of PE models for solving underwater acoustics problems. The more recent development of so-called high-angle PE models greatly relaxed this approximation. The way in which the solution marches out in range makes it straightforward to include range-dependent water depth, sound speed profiles, and seabed properties, and as a result, high-angle PE models have become the method of choice for solving range-dependent propagation problems.
Perhaps the most widely used PE model is RAM (Collins 1993), which allows the user a trade-off between the valid angular range and computational efficiency by specifying the number of terms to be used in a Padé approximation, which is central to the wide-angle algorithm. The more terms that are used in the Padé approximation, the wider is the valid angular range. Even though this allows the paraxial approximation to be greatly relaxed, it cannot be completely eliminated, and so PE models should always be used with care when acoustic energy propagating at steep angles is significant.
Another consideration when running RAM or similar PE models is that they use a finite computational grid in the depth direction, and energy will be artificially reflected by the sudden truncation at the bottom of the grid. This is usually dealt with by including an extra attenuation layer underneath the layer representing the physical seabed. The attenuation layer has the same density and sound speed as the seabed but an artificially high attenuation coefficient so that little energy reaches the bottom of the grid, and any energy that does reflect is further attenuated before reappearing in the water column. A sudden change in attenuation can also lead to reflections, so in critical situations, it is advisable to ramp the attenuation up smoothly from its seabed value to a high value, rather than having a step change.
There are several variants of RAM intended for different purposes (Table 6.1). The only one that can deal with elastic seabeds is RAMS, but it requires careful tuning of parameters to avoid instability, and in some cases involving layered seabeds, it is impossible to obtain a stable solution. More recent PE models have been developed that overcome these limitations (Collis et al. 2008) yet are research codes not readily available. The majority of PE codes are intended for N × 2D modeling. However, research-level 3D PE codes have been developed (see Jensen et al. 2011, Sect. 6.8, for details).
Table 6.1 Summary of variants of the RAM parabolic-equation codes
6.4.5 Choosing the Most Appropriate Model
If the frequency is high enough that the acoustic wavelength is less than a small fraction of the smallest significant feature in the sound speed profile (e.g., mixed layer thickness, water depth), then use a ray tracing or beam model (e.g., Bellhop), otherwise use one of the low-frequency models. A rule of thumb for the 'small fraction' is 1/100. However, accurately modeling sound propagation in a weak duct may require the use of a low-frequency model up to a higher frequency than this rule would suggest. If in doubt, run some tests using both types of models to determine the frequency at which the two models start to agree.
When choosing a low-frequency model, if the range is short enough that the environment can be considered range-independent, then pick a wavenumber integration model (e.g., OASES or SCOOTER), otherwise use a PE model (e.g., RAM). The benefit of wavenumber integration for range-independent modeling is its greater accuracy at short range compared to either a normal-mode model (which only considers trapped energy) or a PE model (which has high-angle limitations). Wavenumber integration can also deal accurately with elastic seabed effects, which tend to be most important at short range. PE codes have largely replaced normal-mode codes for range-dependent modeling because of the greater practicality of the PE range-marching algorithm.
Range-dependent modeling with layered elastic seabeds remains a difficult computational task. One commonly resorts to work-around strategies, such as replacing the true seabed with an "equivalent" fluid seabed that has a similar reflection coefficient versus grazing angle dependence at low grazing angles. This allows a standard PE code to be used for the modeling but is only accurate at ranges large enough that there is no high-angle energy reaching the receiver.
6.4.6 Accessing Acoustic Propagation Models
Many of the models described in this chapter are freely available for download from the Ocean Acoustics LibraryFootnote 8 (OALIB). OALIB includes Michael D. Porter's Acoustics Toolbox, which incorporates a Gaussian beam tracing model (Bellhop), wavenumber integration code (SCOOTER), normal-mode model (KRAKEN), as well as several other useful programs including one for calculating seabed reflectivity as a function of grazing angle for arbitrarily complicated, layered seabeds (BOUNCE). These all use similar input and output file formats, have been regularly updated until at least 2020, and are well documented. A number of MATLAB (The MathWorks Inc., Natick, MA, USA) routines for dealing with the input and output are also provided. Also available on OALIB is the free version of the wavenumber integration code OASES and a number of different PE codes, including the RAM family.
Unfortunately, downloading a particular code is often just the start of a journey that may include compiling it for the particular operating system you are using, deciphering the documentation to determine what input files are required and how they need to be formatted, and then working out how to read and plot the output data. There are usually a number of adjustable parameters that affect how the program operates, and it is necessary to have an understanding of the underlying numerical methods in order to set these appropriately. Inappropriate parameter selection will often lead to meaningless results, so whenever you start using a different propagation model, you should run a series of tests on simple problems (to which the answer is known) in order to make sure you are getting the correct results. The standard of documentation varies considerably between the different models that are available from OALIB and is minimal for some.
AcTUPFootnote 9 is a MATLAB GUI to earlier (2005) versions of the Acoustics Toolbox and several of the RAM family of PE codes. AcTUP comes packaged with the required Windows executables. This provides a convenient entry point for those new to acoustic propagation modeling as it allows different codes to be run on the same problem with minimal changes. However, careful parameter selection is still required in order to get meaningful results; put garbage in, get garbage out.
6.5 Practical Acoustic Modeling Examples
Having worked through the theory and concepts, this section finally puts all of the above into action and provides examples of some practical acoustic propagation modeling tasks of increasing complexity. These all involve the estimation of received levels due to a source with known sound emission characteristics, and are conceptually based on re-arranging the passive sonar equation (Eq. 6.1) to solve for the received level RL:
$$ RL= SL- PL. $$
The tasks are:
Calculate RL as a function of range and depth in a given direction from a tonal (i.e., single-frequency) source.
Calculate RL as a function of range and depth in a given direction from a broadband source.
Calculate RL as a function of geographical position and depth for an omnidirectional source in a directional environment.
Calculate RL as a function of geographical position and depth for a directional source in a directional environment.
Indicative execution times are given for calculations that were carried out on a desktop computer with an Intel i7–7700 CPU, a clock speed of 3.6 GHz, and 64 GB of RAM. The processor had 4 physical cores but the models used here were single-threaded so only used one core. The computer was running a 64-bit Windows 10 operating system.
6.5.1 Received Level Versus Range and Depth from a Tonal Source
For this case, it is only necessary to specify the acoustic environment (i.e., bathymetry profile, sound speed profile, and seabed properties) along a single azimuth from the source. The propagation loss PL is only required at the source transmission frequency, and can be obtained using a single run of an appropriate propagation model. The received level RL can then be obtained using Eq. (6.16).
The example of a fin whale (Balaenoptera physalus) located about 40 km off the coast of southwestern Australia, at a depth of 50 m, while emitting a 20-Hz tone at a source level of 189 dB re 1 μPa m (Sirovic et al. 2007) is depicted in Fig. 6.16. The modeled direction of propagation was due west from the source, and the bathymetry profile (i.e., magenta line in Fig. 6.16b) was interpolated from the Geosciences Australia 0.15′ resolution bathymetry database.Footnote 10 The sound speed profile (Fig. 6.16a) was calculated from salinity and temperature data obtained from the World Ocean Atlas (Locarnini et al. 2018; Zweng et al. 2018). The seabed was modeled as a fine sand half-space with parameters from Jensen et al. (2011). Propagation loss modeling was carried out with RAMGeo in AcTUP, which is very efficient at such a low frequency, taking only a few seconds. A simple program was written in MATLAB to read the propagation loss file produced by RAMGeo, calculate the received levels using Eq. (6.16), and plot the results. Note that AcTUP can be used to plot propagation loss, but not received level.
(a) Sound speed profile used for the modeling examples. (b) Modeled received SPL as a function of range and depth for a fin whale at a depth of 50 m emitting a 20-Hz tone with a source level of 189 dB re 1 μPa m. The magenta line is the seafloor
The sound field has a complicated structure of peaks and nulls that is the result of constructive and destructive interference between sound that has traveled from the source to the receiver via different paths. This is typical of the sound fields produced by tonal sources. The overall reduction in received level with increasing range is quite slow, particularly beyond 70 km, due to the sound becoming constrained by refraction in the deep sound channel. This is typical of downslope propagation from a near-surface source situated over the continental slope into deep water.
6.5.2 Received Level Versus Range and Depth from a Broadband Source
Many sources of underwater sound are broadband, which means that they produce significant acoustic output over a wide range of frequencies. Ships, pile driving, and the airgun arrays used for seismic surveying all produce broadband noise, and modeling the resulting sound fields is of importance when assessing the potential impacts of these sources on marine animals.
A common way to carry out broadband modeling for continuous sound such as ship noise is:
Break the required frequency span into a series of frequency bands (e.g., 1/3 octave bands are commonly used; see Chap. 4).
Use a propagation model to estimate a typical propagation loss for each band. This can either be done by running the propagation loss model at the center frequency of each band or by running it at a number of frequencies within the band and then averaging the results. The latter is preferred as it smooths out the interference field to some extent, but if the source emits a wide range of frequencies that span many bands, then the two methods will yield very similar results for the total field.
Integrate the source power spectral density over each band and convert to a source level.
Use Eq. (6.16) to obtain the received level in each band.
Sum the corresponding mean-square pressures across the bands to obtain an overall mean-square pressure that can then be converted to an overall received sound pressure level (SPL, see Chap. 4).
The use of mean-square pressure as a metric is problematic for impulsive sources such as airguns or pile driving, because the results become very sensitive to the duration of the signal, which is often hard to determine. Source and received levels for impulsive sources are therefore usually characterized in terms of sound exposure, and its logarithmic measure, the sound exposure level (SEL, see Chap. 4).
Computing the received levels for impulsive sources follows the same steps as for broadband, continuous sources, except that in step 3, the source spectrum needs to be specified as an energy density spectrum instead of a power density spectrum, and in step 5, it is sound exposures that are summed across the bands to obtain the overall sound exposure, which is then converted to a sound exposure level.
As an example, the modeled received sound exposure levels due to a single 3.3-l (200-cui) airgun are plotted as a function of range and depth in Fig. 6.17. The airgun (i.e., a cylindrical tube filled with compressed air, which is suddenly released into the water) is located at the geographical location that was used for the fin whale example, but at a depth of 6 m, which is typical of seismic survey source depths. The scenario is otherwise the same as previously described. The airgun's source waveform was modeled using the Cagam airgun array model (Duncan and Gavrilov 2019). The airgun array model also calculated the signal's energy density spectrum, which was then used in step 3 of the broadband modeling procedure outlined above. Once again, AcTUP was used to run RAMGeo to carry out the propagation modeling, but this time at 1/3 octave band center frequencies from 7.9 Hz to 1 kHz, which took about 5 minutes. A separate MATLAB program was written to carry out the post-processing steps and to plot the results.
Received SEL from a 3.3-l (200-cui) airgun at a depth of 6 m as a function of range and depth. The magenta line is the seafloor
Comparing Fig. 6.17 with Fig. 6.16, it can be seen that the broad range of frequencies emitted by the airgun has the effect of smoothing out the fluctuations in the sound field caused by interfering paths. The color scales on these two figures are not directly comparable because Fig. 6.16 gives SPL in dB re 1 μPa whereas Fig. 6.17 presents SEL in dB re 1 μPa2s. The two are related through:
$$ SEL= SPL+10{\log}_{10}T $$
where T is the duration of the received signal in seconds, conventionally defined as the duration of the time interval containing 90% of the signal's energy (90% energy signal duration; see Chap. 4).
6.5.3 Received Level as a Function of Geographical Position and Depth
The geographical distribution of received sound levels can be modeled by repeating the tonal source modeling procedure (Sect. 6.5.1) or broadband source modeling procedure (Sect. 6.5.2) using bathymetry profiles appropriate for different directions from the source. For long-range modeling, it may also be necessary to make the sound speed profile a function of range and direction. This is called N × 2D modeling and is adequate in most circumstances, but is less accurate than running a fully 3D propagation model in situations involving sound propagating across steeply sloping seabeds, or in some special situations in which horizontal sound speed gradients become significant.
The result is a 3D grid of the received level as a function of range, depth, and azimuth (i.e., direction in the horizontal plane). To create a 2D map of the sound field, it is necessary to extract some measure of the sound field in the vertical dimension and then interpolate that in the horizontal plane, with the appropriate measure depending on the purpose of the modeling. For example, in environmental impact assessments, it is common to use the maximum level at any depth in the water column, or the maximum level in a depth range corresponding to the diving range of an animal of interest.
Here we illustrate N × 2D modeling using the previous two examples, but this time carrying out the propagation modeling with bathymetry appropriate for each of the 37 tracks shown in Fig. 6.18. These were set at 10° increments in azimuth, with some adjustment and an extra track inserted in the inshore direction to improve the definition of the received field in the vicinity of the two capes. MATLAB programs were written to automate the various steps of the process.
Map showing the bathymetry off the southwest coast of Australia. The lines radiating from the chosen source location show the tracks along which propagation was modeled
Results are plotted in Fig. 6.19 for the fin whale and the airgun. In both cases, the plots are of the maximum received level over depth, but once again, they are not directly comparable because SPL was plotted for the fin whale, whereas SEL was plotted for the airgun.
(a) Map of maximum SPL over depth as a function of geographical position due to a fin whale calling at a depth of 50 m off the southwest coast of Australia. (b) Map of maximum SEL over depth due to a single firing of an airgun of volume 3.3 l (200 cui) at a depth of 6 m
6.5.4 Received Level as a Function of Geographical Position and Depth for a Directional Source
Another level of complexity occurs when the source emits sound differently in different directions. We illustrate this for an airgun array typical of those used for offshore seismic surveys. In this case, the array consists of 30 individual airguns of different sizes arranged in a 21-m wide by 15-m long rectangular array, with all airguns at the same depth of 6 m. The total volume of the compressed air released when the airguns fire is 55.7 l (3400 cui), and the tow direction is towards the North. The Cagam airgun array model was used to calculate a representative source spectrum corresponding to the direction of each of the propagation tracks shown in Fig. 6.18. Apart from using a different source spectrum for each direction, the procedure for calculating the received levels was identical to that described in the previous section for the single airgun.
The maximum received SEL at any depth is plotted in Fig. 6.20a, which uses the same color scale as Fig. 6.19b. The array produced higher levels overall, and the sound field was more directional, with distinct maxima east, west, and to a lesser extent, north and south from the source. Figure 6.20b combines range-depth plots for the 90° and 270° azimuths in a single plot, which illustrates the contrasting sound attenuation rates in the upslope and downslope directions.
(a) Map of maximum SEL over depth as a function of geographical position due to a single firing of a typical airgun array off the southwest coast of Australia. The total volume of the airguns in the array was 55.7 l (3400 cui), and the array was at a depth of 6 m. The tow direction of the array was northwards. (b) Received SEL from the same airgun array as a function of range and depth. The source was at 0-km range, negative ranges correspond to the 270° azimuth (i.e., west of the source) and positive ranges correspond to the 90° azimuth (i.e., east of the source). The magenta line is the seafloor. Colorbar applies to both panels
6.5.5 Modeling Limitations and Practicalities
Provided the chosen propagation modeling approach is appropriate for the task, the largest uncertainties in the results are likely due to a lack of information on the environment, which includes the bathymetry, seabed composition, and water column sound speed profile. Bathymetry and water column sound speed profiles are often straightforward to measure or can be obtained from databases, but knowledge of the acoustic properties of the seabed is often poor (i.e., unavailable, patchy, and uncertain) and the parameters that contribute to the geoacoustics (e.g., sediment composition, density, and thickness) vary over space and not coherently (Erbe et al. 2021). Moreover, seabed properties tens or even hundreds of meters below the seafloor may be important when modeling low-frequency propagation (Etter 2018). As a result, it is often prudent to carry out modeling with several different sets of seabed properties in order to obtain an estimate of the uncertainty in the results.
The use of N × 2D rather than fully 3D modeling in the above examples may introduce some inaccuracies for cross-slope propagation paths, which in this case are to the north and south of the source. The effect of the sloping bathymetry would be to deflect the sound towards the downslope direction, slightly increasing levels downslope and decreasing them upslope.
The modeling methods described above treat the source as an ideal point source, which is a good approximation provided the receiver is much farther away from the source than the dimensions of the source. Modeling received levels close to a large source such as an airgun array requires a different and more computationally intensive approach in which the individual airguns in the array are treated as separate sources, and their signals are combined, taking account of their relative phases at the receiver locations. The same approach accounts for the full 3D directivity of the source, rather than just the horizontal directivity, as was the case for the example in Sect. 6.5.4. Combining this approach with a process called Fourier synthesis (Jensen et al. 2011) allows the received waveforms to be simulated, which allows other signal measures such as peak sound pressure levels (SPLpk) to be calculated. Calculating SPLpk by this means works well at short ranges but tends to overestimate levels at longer ranges because the propagation models do not properly account for seabed and sea surface scattering effects that broaden the peaks and reduce their amplitudes.
Simple propagation modeling tasks such as those described in Sects. 6.5.1 and 6.5.2 can be carried out using free propagation modeling tools such as the Acoustics Toolbox and AcTUP, with the addition of some relatively straightforward post-processing coded in any convenient programming language. However, when N × 2D modeling in multiple directions is required, it becomes desirable to automate the process of interpolating bathymetry profiles from databases, generating sound speed profile files, initiating multiple runs of the propagation model, calculating received levels, interpolating and plotting results, etc. Most organizations that routinely carry out this type of modeling have written their own proprietary software for these tasks. To the authors' knowledge, there is no freely available software package with all of these capabilities, although there is at least one commercially available package.
Sound propagation under water is a complex process. Sound does not propagate along straight-line transmission paths. Rather, it reflects, refracts, and diffracts. It scatters off rough surfaces (such as the sea surface and the seafloor) and off reflectors within the water column (e.g., gas bubbles, fish swim bladders, and suspended particles). It is transmitted into the seafloor and partially lost from the water. It is converted into heat by exciting molecular vibrations. There are common misconceptions about sound propagation in water, such as "low-frequency sound does not propagate in shallow water," "over hard seafloors, all sound is reflected, leading to cylindrical spreading," and "over soft seafloors, sound propagates spherically." This chapter aimed to remove common misconceptions and empower the reader to comprehend sound propagation phenomena in a range of environments and appreciate the limitations of widely used sound propagation models. The chapter began by deriving the sonar equation for a number of scenarios including animal acoustic communication, communication masking by noise, and acoustic surveying of animals. It introduced the concept of the layered ocean, presenting temperature, salinity, and resulting sound speed profiles. These were needed to develop the most common concepts of sound propagation under water: ray tracing and normal modes. The chapter computed Snell's law, reflection and transmission coefficients, and Lloyd's mirror. It provided an overview of publicly available sound propagation software (including wavenumber integration and parabolic equation models). It concluded with a few practical examples of modeling propagation loss for whale song and a seismic airgun array.
6.7 Additional Resources
Dan Russell's Acoustics and Vibration Animations: https://www.acs.psu.edu/drussell/demos.html
The Discovery of Sound in the Sea (DOSITS; https://dosits.org/) website has over 400 pages of content in three major sections including the science of underwater sound and how people and marine animals use underwater sound to conduct activities for which light is used in air. The website has been the foundational resource of the DOSITS Project, providing information at a beginner and advanced level, based on peer-reviewed science (Vigness-Raposa et al. 2016, 2019). The web structure has been transformed into structured tutorials that provide a streamlined, progressive development of knowledge. The tutorial layout allows a user to proceed from one topic to the next in sequence or jump to a specific topic of interest. The three tutorials focus on the science of underwater sound, the potential effects of underwater sound on marine animals, and the ecological risk assessment process for determining possible effects from a specific sound source. Additional resources have been developed to provide the underwater acoustics content in different formats, including instructional videos and webinars. Finally, there are print publications (an educational booklet and a trifold brochure) available in hard copy or PDF format and two eBooks available for free on the iBooks Store, including Book I: Importance of Sound in the Sea and Book II: Science of Underwater Sound.
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Centre for Marine Science and Technology, Curtin University, Perth, WA, Australia
Christine Erbe & Alec Duncan
INSPIRE Environmental, Newport, RI, USA
Kathleen J. Vigness-Raposa
Christine Erbe
Alec Duncan
Correspondence to Christine Erbe .
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Erbe, C., Duncan, A., Vigness-Raposa, K.J. (2022). Introduction to Sound Propagation Under Water. In: Erbe, C., Thomas, J.A. (eds) Exploring Animal Behavior Through Sound: Volume 1. Springer, Cham. https://doi.org/10.1007/978-3-030-97540-1_6
DOI: https://doi.org/10.1007/978-3-030-97540-1_6
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Evidence of rapid spread and establishment of Tuta absoluta (Meyrick) (Lepidoptera: Gelechiidae) in semi-arid Botswana
Honest Machekano1,
Reyard Mutamiswa1 &
Casper Nyamukondiwa ORCID: orcid.org/0000-0002-0395-49801
Tuta absoluta (Meyrick), a major invasive pest of Solanaceous plants, was recently detected in Botswana. Abiotic and biotic factors, together with a suite of population demographic traits are likely key for species propensity and invasion success. First, we determined the movement of T. absoluta from its core detection centre to new invasion areas using pheromone baiting and established likely biotic dispersal drivers. Second, we measured thermal tolerance vis critical thermal limits and lower and upper lethal limits to determine how these traits shape population establishment.
We detected T. absoluta in all 67 pristine sites across nine districts of Botswana. Within-district trap catches varied between cultivated and wild hosts but were generally not statistically significant (P > 0.001). We report three major wild host plants for T. absoluta as biotic dispersal drivers: Solanum coccineum (Jacq.), Solanum supinum (Dunal) and Solanum aculeatissimum (Jacq.). Solanum coccineum and S. supinum were omnipresent, while S. aculeatissimum distribution was sporadic. Thermal tolerance assays showed larvae were more heat tolerant, with a higher critical thermal maxima (CTmax) than adults (P < 0.001), whereas the adults were more tolerant to cold with a significantly lower (P < 0.001) critical thermal minima (CTmin) compared to larvae. The upper lethal temperatures ranged from 37–43 °C, whereas the lower lethal temperatures ranged from − 1 to − 12 °C for 0–100% mortality, respectively. In the light of prevailing environmental (habitat) temperatures (Thab), warming temperature (7.29 °C) and thermal safety margin (22.39 °C) were relatively high.
Tuta absoluta may not be under abiotic physiological or biotic constraint that could limit its geographical range extension within Botswana. The ubiquity of wild Solanaceous plants with the bridgehead of year-round intensive monocultures of Solanaceous crops within a favourable climatic framework may mean that environmental suitability aided the rapid spread of T. absoluta.
Invasive species are a major threat to agroecosystems and global change [1, 2] and increased global connectivity [3] has drastically increased the diversity and magnitude of such invasions especially in the hot-dry Afrotropical region. Tomato leaf miner, (Tuta absoluta) (Meyrick) (Lepidoptera: Gelechiidae), is one of the most destructive insect pests of tomatoes globally [2, 4]. It is of South American origin and was first detected in Spain in 2006 [5, 6] before rapidly spreading and establishing in novel environments in the Mediterranean Basin, Europe, Middle East, South Asia (India), north, east and west Africa [5,6,7,8,9] and recently Southern Africa [10,11,12]. Because of its high reproductive potential, multivoltinism and potential to acclimatize to different climatic conditions [1], T. absoluta is currently considered a key limiting phytosanitary factor affecting the global Solanaceous crops value chain [13].
The larvae of T. absoluta feeds on all aerial parts of the plants including the fruits, resulting in significant yield losses and cosmetic damages as well as secondary infection [14, 15]. Characteristic larval mesophyll mining also compromises photosynthetic capacity of crops significantly reducing yields [14]. In the absence of control, yield losses ranging 80–100% have been reported in open and protected tomato fields [5]. A cost–benefit analysis has shown a significant increase in cost of production through high use of insecticides [2, 16], increased tomato market prices as farmers try to recover the high production cost, spatial prohibition of tomato seedlings and fruits trade [17] culminating into overall increased food and nutrition insecurity [18].
In tropical sub-Saharan Africa, irrigated tomatoes are an essential component of horticulture, a major pillar of sustainable development, with a significant contribution to food and nutritional security as well as household source of income especially for resource-poor farmers [18]. However, a major constraint to growing field horticultural crops in Southern Africa is the reduction in yield and quality caused by insect pests [19]. The potential invasion of Southern Africa by T. absoluta has already been described [1, 3, 6] with models based on its invasion history and global warming [20]. However, there are no reports based on field data on its thermal fitness and how this correlates with availability and distribution biotic resources, e.g. wild host plant species. Although T. absoluta survival on wild Solanaceae, Amaranthaceae, Fabaceae, Chenopodiaceae and Asteraceae plant families was reported, no report has so far combined this knowledge with its on-going invasive movement in the light of prevailing climate data.
For an invasive species to be established, it first has to overcome several environmental barriers [21] including transport, introduction, population establishment and spread [22]. Upon introduction into a novel environment, high propagule pressure [23], species genetic and demographic characteristics [24] and physiological tolerance allow the establishment and habitat permeability [3]. Climate synchrony should exist between introduced areas' and species' environmental stress tolerance to allow successful spread during transience and niche occupation post-invasion [3, 25]. As such, physiologists often use species' thermal tolerance assays as proxy for determining potential for establishment of invasive species. Similarly, it has also been clear from modelling studies, that even when propagule material is high, environmental suitability remains an overriding factor for invasive species successful establishment [3, 23, 26]. Indeed, physiological assays have found use in niche modelling and invasive species risk assessments to determine critical risk invasion areas [27, 28]. Tuta absoluta is known to respond naturally to rapidly changing environments [29]. This is characteristic of successful invaders, which should inherently possess high basal and plastic physiological tolerance, including rapid genetic adaptive shifts [30]. Nevertheless [3], also show that native environmental heterogeneity may contribute to species invasive success. This means, species coming from a more heterogeneous environment may likely cope with a changing novel environment through phenotypic adjustment, compared to those coming from a more stable environment.
Temperature is the most important abiotic factor exerting direct and indirect effects on T. absoluta population dynamics [1] and consequently invasion success [31, 32]. Therefore, temperature forms a first abiotic 'ecological filter' [33] for successful invasion in a new environment [34], and failure to mount any compensatory mechanisms against it may result in the species failing to establish [35, 36]. The proximity of the environmental temperatures to species thermal physiological limits can therefore indicate species vulnerability and dispersal fitness [31]. Species introduced into habitats close to their thermal tolerance limits are more affected by environmental temperature [37] than those introduced into habitats far from their thermal tolerance limits.
Insects have been reported to experience multiple overlapping abiotic and biotic stressors such as temperature, starvation and desiccation in the wild [3, 38, 39]. Hence, an understanding of bioecology of invasive species is of paramount importance in enlightening mechanisms underlying the successful spread and establishment of invasive alien species [1, 40]. This will also involve determining how the invasive species may respond to native wild host plants. The availability and distribution of alternative wild (non-cultivated) host plants play a significant inoculum sink–source role across the novel landscapes [6]. Since its detection in Zambia [41], South Africa [10, 42] and Botswana [12], no work has documented T. absoluta spread and establishment across the biotic and abiotic frontiers. Here, we ought to establish whether T. absoluta was indeed spreading and elucidate the major environmental drivers to successful establishment in Botswana. We measured its thermal tolerance vis limits to activity (critical thermal minima [CTmin] and critical thermal maxima [CTmax]) and lethal limits (lower and upper lethal limits [LLT] and [ULT], respectively) and compared this with prevailing ambient climatic environment. Second, we investigated wild Solanaceous host diversity and linked this to T. absoluta invasion. To date, data on T. absoluta invasion potential in tropical climates have only been derived from modelling [1, 2]. No studies have looked at T. absoluta physiological thermal tolerance limits with field climate data to test the possible role of climate on its range expansion and spread. Similarly, no study has coupled physiological tolerance and its interaction with host availability on T. absoluta invasion pathway. The objective of this study was therefore to investigate whether T absoluta has spread and established from its core detection site across other pristine districts of Botswana, since its first detection [12]. Such information is important for pest risk assessments, niche modelling and may aid in developing phytosanitary regulations for effective invasive pest management.
Insect trapping and sites
Following the detection of T. absoluta at Genesis farm (S21.14776; E27.64744), Matshelagabedi village in the North East District of Botswana December 2016 [12], a follow-up surveillance trapping was conducted across 9 of the 10 districts of Botswana (Fig. 1). Traps were not set in Kgalagadi as it is largely part of the Kalahari Desert with very minimal vegetation, human settlement and agricultural activity. A total of 201 (67 sites with 3 traps per site) yellow delta traps (Chempac-Progressive-Agricare®) (Suider Paarl, South Africa) equipped with sticky pads were placed ~ 1 m above ground in tomato fields (cultivated host) and open forests (wild hosts) in each of the study districts during the hot-rainy summer season when the wild hosts were flourishing. High temperature, high relative humidity [1] and presence of the host [2] were reported to possibly enhance its propensity to spread. The major male-attracting synthetic sex pheromone (3E,8Z, 11Z)-3,8,11-tetradecatrienyl acetate (TDTA) loaded on grey rubber dispensers at a dosage of 110 µg per lure, (Tuta absoluta-optima PH-937-OPTI Russel IPM, Flintshire, UK) was used. Trap catch data were collected after ~ 30 days, and trapped moths were counted using dyed pointers (chopsticks dipped in insect dye) and mechanical (tally) counters following gross morphological identification [43]. Global Positioning System (GPS) points were recorded for each trapping site using a Garmin® (GPSMAP 62 model, Olathe, USA). Climate data for the sampling areas were obtained from the Meteorological Department, Ministry of Environment, Wildlife and Tourism (MEWT), Republic of Botswana.
a New detection sites for T. absoluta (Meyrick) from traps set in the wild (empty rhombus) and on cultivated host plants (Solanum lycopersicum (L.)) (black rhombus) as well as the core detection District (red) [11] in Botswana, and b occurrence and distribution of wild alternative wild host plants S. aculeatissimum, S. coccineum and S. supinum in Botswana for T. absoluta. (Based on data courtesy of National Botanical Gardens, Botswana National Museums, Gaborone, Botswana)
Basal thermal tolerance experiments
Larvae were collected on damaged tomato fruits into insect cages (BugDorm®, MegaView Science Co., Ltd. Taiwan) from Noka farm (North East District) (S21.12860; E27.48830), with a general temperature range of 3.4–35.5 °C; mean, mean minimum and monthly temperature range of 20.5–22.6 °C, 11.9–13.3 and 29.1–30.4 °C, respectively [44]. These were allowed to pupate in the laboratory in climate chambers Memmert® climate chambers (HPP 260, Memmert GmbH + Co.KG, Germany) set at 25 ± 1 °C, 65 ± 5% relative humidity (RH) and 12L–12D photoperiod. This laboratory rearing temperature closely approximated mean annual temperature from the environment from which the specimens were collected. Eclosed T. absoluta adults were placed in 25-cm3 clean cages, where they fed on 10% sucrose solution ad libitum using the cotton dental wick source method (a feeding apparatus for liquid-feeding insects; insects suck the liquid from a wet cotton wick that draws the solution through capillarity) and provided with organically produced tomato-fruiting plants to lay eggs. Experiments were conducted using fourth instar F1 generation larvae and freshly emerged unsexed adults (± 2 days old). Sex was not considered a factor in our experiments since it has been reported not to affect thermal tolerance traits in some related species (e.g. [45,46,47]).
Lethal temperature assays
Lethal temperatures were determined using established methods as outlined in [48]. Upper and lower lethal temperatures (ULTs and LLTs) were determined using direct plunge protocol at 2-h duration at temperatures that elicited 0–100% mortality. Ten insects were placed in 60-ml polypropylene vials with gauzed lids and placed in a 33 × 22 cm ziplock bag, replicated three times. This was then plunged into a Merck® water bath (Modderfontein, South Africa) filled with 99.9% circulating ethanol. For ULT, tiny wet filter paper was suspended in each vial to maintain benign humidity and prevent desiccation-related mortality. Following treatment (ULT and LLT), test insects were placed at 25 ± 1 °C and 65 ± 5% RH in Memmert® climate chambers for 24 h before scoring survival. All insects had access to food and water ad libitum during the 24-h recovery period. Survival was defined as the ability to coordinate muscle response to stimuli such as gentle prodding, or normal behaviours such as feeding, flying or mating [48, 49].
Critical thermal limits (CTLs)
CTLs were assayed using a programmable waterbath (LAUDA Ecogold® RE 2025, Lauda-Königshofen, Germany) connected to a transparent double-jacketed chamber as outlined by [45]. A thermocouple (type K 36SWG) connected to a digital thermometer (Fluke 54 series IIB) was inserted into the central organ pipe (control chamber) to record chamber temperature. A total of ten test insects replicated three times to yield 30 replications per treatment were used in these experiments. Test insects were individually placed into the organ pipes of the double-jacketed chamber connected to a programmable water bath filled with 1:1 water: propylene glycol to allow for subzero temperatures [50]. Both CTmax and (CTmin experiments started from an ambient set point temperature of 25 °C from which temperature was ramped up (CTmax) or down (CTmin) at 0.25 °C/min until CTLs were recorded. Although it is likely faster than natural diurnal heating or cooling rates in the wild [45], this ramping rate was chosen as a compromise between ecological relevance and maximum throughput (see also discussions in [45, 51]). In this study, we defined CTLs as the temperature at which each individual insect lost coordinated muscle function and the ability to respond to mild stimuli (e.g. prodding with a thermally inert object).
Data analyses
New detection sites and the distribution of wild Solanaceous host plants were presented on maps (ArcGIS, ArcMap 10.2.2). Trap catch and thermal tolerance data analyses were carried out in STATISTICA, version 13.2 (Statsoft Inc., Tulsa, Oklahoma) and R version 3.3.0 [52]. CTLs met the linear model assumptions of constant variance and normal errors; therefore, they were analysed using one-way ANOVA in STATISTICA. LLT and ULT assays results did not meet the assumptions of ANOVA, and thus, they were analysed using generalized linear models (GLM) assuming a binomial distribution and a logit link function in R. Tukey–Kramer's post hoc tests were used to separate statistically heterogeneous means.
Warming tolerance (WT) and the thermal safety margin (TSM) of T. absoluta under Botswana conditions were calculated as outlined by [53]:
$${\text{WT}} = {\text{CT}}_{\max} - T_{\text{hab}} \quad \left[ {53} \right]$$
$${\text{and}},\quad {\text{TSM}} = T_{\text{opt}} - T_{\text{hab}} \quad \left[ {53} \right]$$
where CTmax = critical thermal maximum for T. absoluta adult (the migratory stage), Thab = habitat temperature—Botswana mean annual temperature for 2015/16. Topt = optimum temperature for T. absoluta.
The spread of T. absoluta in Botswana
Apart from North East District, the area of T. absoluta first detection [12], the species was recorded in eight other districts (Fig. 1a). Moths were detected both in the wild (forests, grazing lands and national parks distant from agroecosystems) and on cultivated solanaceous crops; mainly tomato Solanum lycopersicum (L.). We detected T. absoluta in areas such as Moremi Island (Okavango Delta) more than 200 km from the nearest human settlements and agricultural activities and bordered by Moremi and Chobe Game Reserves) (Fig. 1a). Surveillance results support our hypotheses that T. absoluta spread and successfully established across Botswana (Fig. 1a).
Tuta absoluta wild host plants belonging to the Solanaceae family showed a cosmopolitan distribution (Fig. 1b). Wild host species diversity showed three dominant species; Solanum aculeatissimum (Jacq.), Solanum coccineum (Jacq.) and Solanum supinum (Dunal). Solanum supinum was the most widely distributed, occurring in all districts except only in Chobe, North-East and South-East and was found on the Moremi Island of the Okavango Delta (Fig. 1b) giving credence to the occurrence of T. absoluta in such a remote area. Solanum coccineum had more sporadic distribution, occurring in Chobe, Ngamiland, Ghanzi, Kgalagadi, Kweneng districts and the surrounding areas of the Okavango Delta. However, S. aculeatissimum was only found in Kgatleng district (Fig. 1b).
Moths abundance in wild and cultivated hosts
Large numbers of T. absoluta moths were captured in all districts, in both cultivated and wild hosts. The cultivated host, S. lycopersicum hosted significantly higher numbers (P < 0.001) (Table 1) than the wild host plants within districts, especially in Kweneng and Central districts (Fig. 2). Inter-district populations were also generally not significantly different within the same host type (Fig. 2). Overall, in the wild host plants, we recorded a grand mean of 411.1 ± 13.38 moths/trap/month from the cultivated S. lycopersicum which was significantly higher (P < 0.001) (Table 1) than 187.4 ± 12.21 moths/trap/month recorded from the wild hosts. High numbers were recorded on S. lycopersicum in Central, South-East, Chobe, Kgatleng and Southern districts and in tunnels compared to open fields in other districts. There were no significant interaction effects between the host plant and the district (P > 0.05) (Table 1) signifying that in each district host type was not a significant factor affecting abundance (trap catches).
Table 1 Differences in mean moth trap catches between cultivated S. lycopersicum and wild hosts and different districts in Botswana
Number of Tuta absoluta moths captured per district from tomato fields, Solanum lycopersicum (cultivated host) and in the wild (wild hosts)
Basal thermal tolerance
Both life stages of T. absoluta showed relatively high temperature tolerance although the larvae had significantly higher (47.9 ± 1.25 °C) CTmax than the adult (44.1 ± 0.43 °C) (Fig. 3A). The highest temperature where T. absoluta could not survive (ULT0) was 43.0 °C, while the highest temperature for 100% survival (ULT100) was 37 °C (for a 2-h stressful high-temperature exposure). There were significant differences (χ2 = 107.29, df = 4, P < 0.001) in survival between test temperatures, again signifying the role of temperature severity and duration in its survival (Fig. 3B). However, on low temperature tolerance, the adult had a significantly lower CTmin (− 5.2 ± 0.23 °C) than the larvae (3.5 ± 0.07 °C) (Fig. 3C), and the LLTs ranged from − 12.0 to − 1.0 °C for LLT0 and LLT100, respectively, based on a 2-h duration at stressful low temperature (Fig. 3D). There were significant differences (χ2 = 163.73, df = 6, P < 0.001) in survival between the low test temperatures, implying that survival was determined by both temperature severity and duration of exposure.
High temperature; A Critical thermal maxima (CTmax) and B upper lethal temperature (ULT), and low temperature; C Critical thermal minima (CTmin), and D lower lethal temperature (ULT) for field collected Tuta absoluta F1 applied for a 2-hour duration for 0–100% survival. CTmin and CTmax were conducted on both larvae and adult while lethal temperature assays were only conducted on adults (the migratory stage)
Climate data and basal thermal tolerance
Field temperature data from eight districts of Botswana in 2015/2016 seasons (period post-first detection prior to and during establishment and spread of T. absoluta) are shown in Fig. 4a and b. The mean monthly maximum temperatures ranged from a low of 22.3 °C (Kweneng district) to a high of 37.4 °C. (South-East district) (Fig. 4a). Highest maximum field temperatures were below T. absoluta CTmax by about 6 °C (adults) and above 10 °C (larvae). Relating ULTs to the field maximum temperature data showed that the T. absoluta ULT0 of 43 °C (Fig. 4a) was well above the highest maximum temperatures recorded in nature (37.4 °C; Fig. 4a), implying that T. absoluta was not under high-temperature-related physiological stress that could limit its spread and establishment.
Field temperatures, a mean monthly maximum temperatures and b mean monthly minimum temperatures for eight districts of Botswana in 2015/16 season related to T. absoluta CTmax and CTmin, respectively. Horizontal lines denote CTmin and CTmax for larvae (continuous) and adult (dotted)
The mean monthly minimum temperatures ranged from a low of 1.1 °C (Kweneng district) to a high of 21.3 °C in December 2015 (Ngamiland district) (Fig. 4b). The lowest minimum field temperatures were above T. absoluta adult CTmin by about 6 °C (adult) and below that of the larvae by about 2.4 °C (see Fig. 4b). This implied that the minimum field temperatures were not physiologically constraining survival of T. absoluta adult but the larvae. Adult T. absoluta LLT0 was − 12.0 °C, while LLT100 was − 1.0 °C for a 2-h stressful low-temperature exposure (see Fig. 3D). Both temperatures fell well below the most extreme low temperatures recorded in the environment (see Fig. 4b), implying that adult T. absoluta may not be under diurnal low-temperature physiological stress.
Warming tolerance (WT) and thermal safety margin (TSM)
The annual mean temperature for Botswana in 2015/2016 was 22.71 °C considered in this study as the habitat temperature (Thab), and the optimum temperature for T. absoluta performance and population growth is 30 °C [54] considered as Topt. The adult CTmax was recorded as 44.1 °C (see "Basal thermal tolerance" section). Based on these data, the warming tolerance (WT) and the thermal safety margin (TSM) of T. absoluta under Botswana conditions were calculated according to [53]:
$$\begin{aligned} {\text{WT}} & = 44.1\;^\circ {\text{C}} - 22.7\;^\circ {\text{C}} \\ {\text{WT}} & = 22.39\;^\circ {\text{C}} \\ \end{aligned}$$
and similarly,
$$\begin{aligned} {\text{TSM}} & = 30\;^\circ {\text{C}} - 22.71\;^\circ {\text{C}} \\ {\text{TSM}} & = 7.29\;^\circ {\text{C}} \\ \end{aligned}$$
Following its first invasion in Botswana in December 2016 [12], our results confirm that T. absoluta has spread and successfully established in almost all districts of Botswana, thus potentially eliciting widespread economic damage to Solanaceous crops. Although T. absoluta was first recorded in North east district of Botswana, evidence from this work suggest its rapid and wide extension of its distribution horizons with new records reported in various districts in the country within an 8-month period (January to August 2017) (Fig. 1). Indeed, this trend is not unusual for the species [2]. The species has been reported to spread at a rate of ~ 800 km/year aided through wind currents and plant material belonging to families Amaranthaceae, Convolvuceae, Fabaceae, Malvaceae and Solanaceae identified through volatile cues by female moths for egg laying [2]. Therefore, the observed rapid spread and successful niche establishment may be directly linked to the reported availability of host plants in the wild [3], climate suitability and physiological thermal tolerances [26]. These characteristics are consistent with other globally invasive economic insect pest species, e.g. Chilo partellus (Swinhoe) [55], Bactrocera dorsalis (Hendel) [56], Ceratitis capitata (Wiedemann) [57] and Drosophila suzuki (Matsumura) [58]. Our results associated T. absoluta with a wide range of cultivated and wild host plants [as in example 2, 59], consistent with polyphagic characteristic of many invasive species [22]. Tuta absoluta thermal activity physiological thresholds examined here also suggest that there is a conducive climate niche across the country and that species activity and hence invasion may not be constrained by temperature. Our survey showed Botswana hosts wild solanaceous plants: S. aculeatissimum, S. coccineum and S. supinum, which are all suitable hosts to T. absoluta [see 2, 15]. Amongst these wild host plants, S. supinum was the most widely distributed, occurring in all districts of the country, while S. coccineum and S. aculeatissimum distribution was sporadic. Therefore, it is highly likely that these wild host plants provide biotic resources, (food and shelter) supporting the invasion pathway of T. absoluta in Botswana. Although tomato is the preferred host for T. absoluta, the species can switch hosts from cultivated to wild as a survival strategy, a notion supported by [5] and [60]. Such availability of biotic resources and suitable environmental conditions are also known to impair diapausing in T. absoluta larvae [1] resulting in increased breeding propagule pressure even under less favourable climate conditions, with implications on niche invasion success.
Short-distance dispersal (adjacent field to field or field to tunnels) of T. absoluta is known to be facilitated by wind especially soon after introduction [13] with moths capable of active flights of up to 100 km [59], a characteristic that may aid the species' dispersal [2, 32, 61]. Pressure distribution from the Indian Ocean was reported to traditionally create strong east-westerly air masses in the Southern African region [62]. This supports the possible movement of T. absoluta through wind currents from the north-east district (core detection district) to the central, southern and western parts of the country. On the other hand, long-distance dispersal may occur through open tomato trade, markets and other related activities [32]. These attributes together may to a larger extent have promoted the spread and establishment of T. absoluta propagule moths which could easily locate either cultivated or wild hosts during dispersal. However, the detection of T. absoluta in Moremi Island (Fig. 1a); (~ 200 km from human settlements and agroecosystems) suggests that wind and wild host plants might have played a more significant role in its invasion success.
High populations of T. absoluta were recorded on S. lycopersicum in Central, South-East, Chobe, Kgatleng and Southern districts (Fig. 2) where production of tomatoes is done in tunnels. The reason may be that the moths were contained within tunnels and hence highly concentrated resulting in the observed high trap catches. Amongst these districts, South-East recorded the highest moth catches. The district is a horticultural hotspot with high concentration and prolonged availability of the cultivated host plants (tomato, green and red pepper) which hosts T. absoluta. Since production of tomatoes is carried out throughout the year in this district, the tunnels also act as inoculum reservoirs that form bridgeheads for further introduction and reinfestation of outdoor cultivated and wild host plants [1,2,3]. Similarly, relatively high T. absoluta moths were recorded in the wild (Fig. 2) signifying its ability to survive outside the cultivated host plant ranges (agroecosystems). This therefore nullifies the possibility of controlled production of Solanaceous crops as a management measure against T. absoluta, as has been the case, e.g. Pectinophora gossypiella (Saunders) in cotton [63].
Native environmental heterogeneity may also contribute to invasion success [3]. Propagules from a more heterogeneously stressful environment are more adaptable to multiple stressful conditions [64] and, together with other factors, may work synergistically towards the succession of ecological filters [reviewed in 3]. Climate matching between the native and novel environment is known to aid invasion success of invasive alien species [30, 65, 66]. Interestingly, African biotic and climatic conditions are closely related to T. absoluta's native region [1]. Insect species have specific optimum temperatures at which they optimally perform and develop [31, 37]. In addition, lower and upper developmental thresholds mark the temperatures beyond which they cannot perform and develop [31, 37, 67]. As such, basal environmental stress tolerance, phenotypic plasticity and rapid genetic adaptive shifts are key to invasive species establishment [30]. Tuta absoluta tolerance to temperature and relative humidity versus typical Botswana climate [68] may form the primary characteristics defining its range expansion [1, 37, 68]. Prior predictions using climatic suitability indices defined the eco-climatic index (EI) of Botswana to fall within 20–50, classified as high risk of establishment for T. absoluta [1]. Our results are thus in agreement with this prediction. Climatic conditions (chiefly temperature) are known to significantly influence generation numbers of multivoltine insects, with higher temperatures facilitating faster degree day accumulation and shorter generation times [69]. At an optimum temperature of 30 °C, the life cycle of T. absoluta ranges approximately 26 days [53] accounting to ~ 12 generations per year [2]. Global warming comes with increased mean temperatures and variability thereof and is reported to increase insect metabolism [70]. With African temperatures projected to increase, future populations of this pest may likely increase in tropical relative to temperate regions [1]. Botswana is arid to semi-arid with mean monthly maximum temperatures recorded in 2015/2016 season ranging 23.6 to 35.1 °C (Fig. 4a). Given that the optimum temperature for T. absoluta is 30 °C [54], a TSM of 7.29 °C was relatively high [37, 53]. This signifies that T. absoluta can tolerate an increase in atmospheric temperature of 7.29 °C from current Botswana ambient environmental temperatures of 22.71 °C (Thab) before its population growth and general performance can drop to critical levels. This is a considerably high TSM compared to most tropical species whose TSM is ~ 0 °C [53]. This, coupled with a wider WT (22.39 °C), further supports that Botswana environmental temperatures were conducive for the performance, rapid spread and establishment of T. absoluta. The lower and upper developmental threshold for T. absoluta is ~ 14 and 34.6 °C, respectively [54], translating to a wide thermal window (~ 20.6 °C) which is known to optimize key insect activity and life-sustaining behaviours such as development, mating and dispersal [3, 55], and may potentially facilitate the invasion pathway of T. absoluta. Thus, conducive climatic conditions might have chiefly facilitated the rapid accumulation of degree days hence culminating into shorter generation time. This high reproductive capacity may also have contributed to its increased invasion success in novel environments in the country [as in 1].
Improved environmental tolerance and thermal plasticity are the key contributing factors towards invasion success of invasive alien species into a novel environment [30, 39, 71]. Lower and upper lethal temperatures (LLT and ULTs) for T. absoluta adults ranged from − 1 to − 12 °C and 37 to 43 °C respectively for 2 h treatments. In addition, the CTmax for larvae and adults were 47.9 ± 1.25 and 44.1 ± 0.43 and CTmin were 3.5 ± 0.07 and − 5.2 ± 0.23 respectively. Field temperature recorded during 2015/2016 season show that highest maximum temperatures were below both ULT and CTmax for both T. absoluta larvae and adults. In addition, LLTs and CTmin for adults were relatively lower than the lowest minimum field temperatures (Fig. 4b). This, added to the high TSM and WT, indicates that T. absoluta may not be at risk of cold and heat stress both of which has an implication on the invasion succession pathway. These results supports that T. absoluta is highly temperature tolerant at both extremes and may survive in arid/semi-arid sub-Saharan Africa whenever hosts plants are available. Its high basal thermal tolerance, coupled by favourable climates in Botswana (Fig. 4a and b), may mean that T. absoluta survives all-year-round temperature conditions, in the absence of diapause, a characteristic likely aiding successful establishment. Furthermore [38], showed rapid cold hardening may also aid invasion success in insects, and indeed T. absoluta has been shown to rapidly cold-harden [see 2], a phenomenon likely aiding the invasion pathway. The absence of native coevolved natural enemies has also been reported to promote invasion success in novel environments [5, 64]. It is highly likely that the rapid spread and establishment of T. absoluta in Botswana may have been facilitated by the absence of biological control agents. We thus recommend that native fortuitous natural enemies need be identified and promoted coupled with a campaign against the instinctive overuse of pesticides by small scale farmers [19] to preserve potential native natural enemies, reduce cost of production and protect public health. Further work needs to determine T. absoluta insecticide resistance to establish a controlled effective spraying program, coupled with the identified effective natural enemies to establish an efficacious tailor-made integrated pest management (IPM) program. Overall, an area-wide approach to T. absoluta management is recommended, and one that involves a coordinated Southern African region, to prevent further spread and establishment of the species [2].
Current results support the rapid spread and establishment of T. absoluta in Botswana following its first detection. This continued invasion by T. absoluta in tropical climates is a real concern for the horticultural industry, as well as African food and nutrition security. Host plant availability, climate suitability and high thermal tolerance may to a larger extent have contributed to the successful invasion, rapid spread and establishment of T. absoluta in the semi-arid tropical Botswana. In addition, intensive monocultures, continuous irrigation and unrestricted trade of Solanaceous crops coupled with strong winds and a lack of natural enemies may also be contributory factors. Furthermore, absence of efficient and coordinated area-wide management practices may have exacerbated the successful rapid invasion. A significant long-term management strategy would be necessary to optimize surveillance and monitoring of T. absoluta in the region for developing sustainable management options. Similarly, introduction of egg-targeting parasitoids (Trichogramma spp.) and predators as well as larval parasitoids (mostly belonging to Braconidae families) and predators (Miridae) [2, 14] could improve management of African suppression programmes, more especially in non-agroecosystem and natural environments.
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HM and CN contributed to conceptualization and methodology; CN contributed to funding acquisition, project administration, resources and supervision; HM and RM contributed to investigation and writing of the original draft; and HM, RM and CN contributed to data curation, validation, formal analysis, writing, review and editing. All authors read and approved the final manuscript.
We acknowledge Botswana International University of Science and Technology for funding and Russell IPM for T. absoluta pheromone lures. We acknowledge assistance from the Department of Crop Protection (Ministry of Agriculture), on site selection, the Botswana National Botanical Gardens for data on distribution of wild Solanaceous plants and Department of Meteorological Services for climate data. We are also grateful to Dr. Tharina L. Bird for map drawing and assistance with trapping in the Okavango Delta and Mmabaledi Buxton and Mphoeng Ofitlhile for assistance with trap monitoring in some districts.
Collected and analysed data during the current study are available upon request from the corresponding author.
Not applicable since the study involved tomato plants and the invasive insect pest (T. absoluta), both of which are not endangered or protected species. The permission to enter National Parks and other any protected areas was obtained from Department of Wildlife and National Parks (Ministry of Environment, Wildlife and Tourism).
The project was funded through Botswana International University of Science and Technology (BIUST) Research Office grant.
Department of Biological Sciences and Biotechnology, Botswana International University of Science and Technology (BIUST), Private Bag 16, Palapye, Botswana
Honest Machekano
, Reyard Mutamiswa
& Casper Nyamukondiwa
Search for Honest Machekano in:
Search for Reyard Mutamiswa in:
Search for Casper Nyamukondiwa in:
Correspondence to Casper Nyamukondiwa.
Machekano, H., Mutamiswa, R. & Nyamukondiwa, C. Evidence of rapid spread and establishment of Tuta absoluta (Meyrick) (Lepidoptera: Gelechiidae) in semi-arid Botswana. Agric & Food Secur 7, 48 (2018) doi:10.1186/s40066-018-0201-5
Accepted: 11 July 2018
Tomato leaf miner
Thermal tolerance
Solanaceous plants | CommonCrawl |
\begin{document}
\frontmatter
\title{Some Basic Aspects of Analysis \\ on Metric and Ultrametric Spaces}
\author{Stephen Semmes \\
Rice University}
\date{}
\maketitle
\chapter*{Preface}
A number of topics involving metrics and measures are discussed, including some of the special structure associated with ultrametrics.
\tableofcontents
\mainmatter
\chapter{Basic notions} \label{basic notions}
\section{Metrics and ultrametrics} \label{metrics, ultrametrics}
Let $X$ be a set. As usual, a \emph{metric}\index{metrics} on $X$ is a nonnegative real-valued function $d(x, y)$ defined for $x, y \in X$ such that $d(x, y) = 0$ if and only if $x = y$, \begin{equation} \label{d(x, y) = d(y, x)}
d(x, y) = d(y, x) \end{equation} for every $x, y \in X$, and \begin{equation} \label{d(x, z) le d(x, y) + d(y, z)}
d(x, z) \le d(x, y) + d(y, z) \end{equation} for every $x, y, z \in X$. If \begin{equation} \label{d(x, z) le max(d(x, y), d(y, z))}
d(x, z) \le \max(d(x, y), d(y, z)) \end{equation} for every $x, y, z \in X$, then $d(x, y)$ is said to be an \emph{ultrametric}\index{ultrametrics} on $X$.
Let $(X, d(x, y))$ be a metric space, and let $x \in X$ and a positive real number $r$ be given. The corresponding \emph{open ball}\index{open balls} in $X$ is defined by \begin{equation} \label{B(x, r) = {y in X : d(x, y) < r}}
B(x, r) = \{y \in X : d(x, y) < r\}. \end{equation} If $y \in B(x, r)$, then $t = r - d(x, y) > 0$, and \begin{equation} \label{B(y, t) subseteq B(x, r)}
B(y, t) \subseteq B(x, r), \end{equation} by the triangle inequality. However, if $d(\cdot, \cdot)$ is an ultrametric on $X$, then one can check that \begin{equation} \label{B(y, r) subseteq B(x, r)}
B(y, r) \subseteq B(x, r) \end{equation} for every $y \in B(x, r)$. In fact, \begin{equation} \label{B(x, r) = B(y, r)}
B(x, r) = B(y, r) \end{equation} for every $x, y \in X$ with $d(x, y) < r$, since we can also apply the previous argument with the roles of $x$ and $y$ reversed.
Similarly, the \emph{closed ball}\index{closed balls} in a metric space $X$ centered at $x \in X$ and with radius $r \ge 0$ is defined by \begin{equation} \label{overline{B}(x, r) = {y in X : d(x, y) le r}}
\overline{B}(x, r) = \{y \in X : d(x, y) \le r\}. \end{equation} If $d(\cdot, \cdot)$ is an ultrametric on $X$, and if $y \in \overline{B}(x, r)$, then \begin{equation} \label{overline{B}(y, r) subseteq overline{B}(x, r)}
\overline{B}(y, r) \subseteq \overline{B}(x, r), \end{equation} as before. It follows that \begin{equation} \label{overline{B}(x, r) = overline{B}(y, r)}
\overline{B}(x, r) = \overline{B}(y, r) \end{equation} when $d(x, y) \le r$, by reversing the roles of $x$ and $y$.
Let us continue to ask for the moment that $d(\cdot, \cdot)$ be an ultrametric on $X$. If $x, y, z \in X$ and $d(y, z) \le d(x, y)$, then \begin{equation} \label{d(x, z) le d(x, y)}
d(x, z) \le d(x, y), \end{equation} by (\ref{d(x, z) le max(d(x, y), d(y, z))}). Of course, we also have that \begin{equation} \label{d(x, y) le max(d(x, z), d(y, z))}
d(x, y) \le \max(d(x, z), d(y, z)), \end{equation} by (\ref{d(x, z) le max(d(x, y), d(y, z))}) with the roles of $y$ and $z$ exchanged. This implies that \begin{equation} \label{d(x, y) le d(x, z)}
d(x, y) \le d(x, z) \end{equation} when $d(y, z) < d(x, y)$, and hence that \begin{equation} \label{d(x, y) = d(x, z)}
d(x, y) = d(x, z). \end{equation}
Put \begin{equation} \label{V(x, r) = {y in X : d(x, y) > r}}
V(x, r) = \{y \in X : d(x, y) > r\} \end{equation} for every $x \in X$ and $r \ge 0$, which is the same as the complement of $\overline{B}(x, r)$ in $X$. If $d(\cdot, \cdot)$ is an ordinary metric on $X$ and $y \in V(x, r)$, then $t = d(x, y) - r > 0$, and one can check that \begin{equation} \label{B(y, t) subseteq V(x, r)}
B(y, t) \subseteq V(x, r), \end{equation} using the triangle inequality. If $d(\cdot, \cdot)$ is an ultrametric on $X$ and $y \in V(x, r)$, then we get that \begin{equation} \label{B(y, d(x, y)) subseteq V(x, r)}
B(y, d(x, y)) \subseteq V(x, r), \end{equation} by (\ref{d(x, y) le d(x, z)}). Similarly, \begin{equation} \label{W(x, r) = {y in X : d(x, y) ge r}}
W(x, r) = \{y \in X : d(x, y) \ge r\} \end{equation} is the same as the complement of $B(x, r)$ in $X$ for each $x \in X$ and $r > 0$. If $d(\cdot, \cdot)$ is an ultrametric on $X$ and $y \in W(x, r)$, then we also have that \begin{equation} \label{B(y, d(x, y)) subseteq W(x, r)}
B(y, d(x, y)) \subseteq W(x, r), \end{equation} by (\ref{d(x, y) le d(x, z)}).
If $X$ is any metric space, then every open ball in $X$ is an open set in $X$ with respect to the topology determined by the metric. Closed balls in $X$ are closed sets too, which is the same as saying that $V(x, r)$ is an open set in $X$ for every $x \in X$ and $r \ge 0$. If $d(\cdot, \cdot)$ is an ultrametric on $X$, then (\ref{overline{B}(y, r) subseteq overline{B}(x, r)}) implies that $\overline{B}(x, r)$ is an open set in $X$ for every $x \in X$ and $r > 0$. In this case, $W(x, r)$ is an open set in $X$ for every $x \in X$ and $r > 0$, by (\ref{B(y, d(x, y)) subseteq W(x, r)}), which implies that $B(x, r)$ is a closed set in $X$.
Let $|x|$ be the absolute value of a real number $x$, which is
equal to $x$ when $x \ge 0$ and to $-x$ when $x \le 0$. Thus the standard metric on the real line ${\bf R}$ is given by $|x - y|$. Of course, this is far from being an ultrametric. By contrast, the $p$-adic metric on the set ${\bf Q}$ of rational numbers is an ultrametric for each prime number $p$. This will be discussed in Section \ref{p-adic absolute value}.
\section{Abstract Cantor sets} \label{abstract cantor sets}
Let $X_1, X_2, X_3, \ldots$ be a sequence of finite sets, each of which has at least two elements. Also let $X = \prod_{j = 1}^\infty X_j$ be their Cartesian product, which is the set of sequences $x = \{x_j\}_{j = 1}^\infty$ such that $x_j \in X_j$ for each $j$. Thus $X$ is a compact Hausdorff space with respect to the product topology corresponding to the discrete topology on each factor. If $x, y \in X$ and $x \ne y$, then let $l(x, y)$ be the largest nonnegative integer such that $x_j = y_j$ when $1 \le j \le l(x, y)$. Equivalently, $l(x, y) + 1$ is the smallest positive integer $j$ such that $x_j \ne y_j$. If $x = y$, then one can take $l(x, y) = +\infty$. Note that \begin{equation} \label{l(x, y) = l(y, x)}
l(x, y) = l(y, x) \end{equation} for every $x, y \in X$, and that \begin{equation} \label{l(x, z) ge min(l(x, y), l(y, z))}
l(x, z) \ge \min(l(x, y), l(y, z)) \end{equation} for every $x, y, z \in X$.
Let $\{t_l\}_{l = 0}^\infty$ be a strictly decreasing sequence of positive real numbers that converges to $0$. Put \begin{equation} \label{d(x, y) = t_{l(x, y)}}
d(x, y) = t_{l(x, y)} \end{equation} when $x \ne y$, and $d(x, y) = 0$ when $x = y$, which corresponds to (\ref{d(x, y) = t_{l(x, y)}}) with $t_\infty = 0$. It is easy to see that this defines an ultrametric on $X$, because of (\ref{l(x, y) =
l(y, x)}) and (\ref{l(x, z) ge min(l(x, y), l(y, z))}), and that the topology on $X$ determined by $d(x, y)$ is the same as the product topology on $X$ corresponding to the discrete topology on each factor. If $x \in X$ and $k$ is a nonnegative integer, then put \begin{equation} \label{B_k(x) = {y in X : y_j = x_j for each j le k}}
B_k(x) = \{y \in X : y_j = x_j \hbox{ for each } j \le k\}. \end{equation} Equivalently, $B_k(x)$ is the closed ball in $X$ centered at $x$ with radius $t_k$ with respect to (\ref{d(x, y) = t_{l(x, y)}}).
Suppose now that $\mu_j$ is a probability measure on $X_j$ for each $j$, where all subsets of $X_j$ are measurable. Thus $\mu_j$ assigns a weight to each element of $X_j$, and the sum of the weights is equal to $1$. This leads to a product probability measure $\mu$ on $X$, where \begin{equation} \label{mu(B_k(x)) = prod_{j = 1}^k mu_j({x_j})}
\mu(B_k(x)) = \prod_{j = 1}^k \mu_j(\{x_j\}) \end{equation} for each $x \in X$ and $k \ge 1$. Alternatively, one can first use the $\mu_j$'s to define a nonnegative linear functional on the space of continuous real-valued functions on $X$, as a limit of Riemann sums. One can then apply the Riesz representation theorem, to get a Borel probability measure on $X$.
Let $n_j \ge 2$ be the number of elements of $X_j$ for each positive integer $j$. Also let $\mu_j$ be the probability measure on $X_j$ that corresponds to the uniform distribution on $X_j$, which assigns to each element of $X_J$ has the same weight $1/n_j$. In this case, (\ref{mu(B_k(x)) = prod_{j = 1}^k mu_j({x_j})}) reduces to \begin{equation} \label{mu(B_k(x)) = 1/N_k}
\mu(B_k(x)) = 1/N_k \end{equation} for each $x \in X$ and $k \ge 1$, where \begin{equation} \label{N_k = prod_{j = 1}^k n_j}
N_k = \prod_{j = 1}^k n_j. \end{equation} If we put $N_0 = 1$, then (\ref{mu(B_k(x)) = 1/N_k}) holds for $k = 0$ as well. Note that $t_l = 1/N_l$ defines a strictly decreasing sequence of positive real numbers that converges to $0$, as before.
\section{The $p$-adic absolute value} \label{p-adic absolute value}
Let $p$ be a prime number, and let $x$ be a rational number. The \emph{$p$-adic absolute value}\index{p-adic absolute
value@$p$-adic absolute value} $|x|_p$ of $x$ is defined as follows. If $x = 0$, then $|x|_p = 0$, and otherwise $x$ can be expressed as $p^l \, a / b$, where $a$, $b$, and $l$ are integers, and neither $a$ nor $b$ is an integer multiple of $p$. In this case, we put \begin{equation}
\label{|x|_p = p^{-l}}
|x|_p = p^{-l}, \end{equation} which is not affected by any other common factors that $a$ and $b$ might have. It is easy to see that \begin{equation}
\label{|x + y|_p le max(|x|_p, |y|_p)}
|x + y|_p \le \max(|x|_p, |y|_p) \end{equation} and \begin{equation}
\label{|x y|_p = |x|_p |y|_p}
|x \, y|_p = |x|_p \, |y|_p \end{equation} for every $x, y \in {\bf Q}$. The \emph{$p$-adic metric}\index{p-adic metric@$p$-adic metric} is defined on ${\bf Q}$ by \begin{equation}
\label{d_p(x, y) = |x - y|_p}
d_p(x, y) = |x - y|_p. \end{equation}
This is an ultrametric on ${\bf Q}$, because of (\ref{|x + y|_p le
max(|x|_p, |y|_p)}).
If $y \in {\bf Q}$ and $n$ is a nonnegative integer, then \begin{equation} \label{(1 - y) sum_{j = 0}^n y^j = 1 - y^{n + 1}}
(1 - y) \, \sum_{j = 0}^n y^j = 1 - y^{n + 1}, \end{equation}
by a standard computation. Here $y^j$ is interpreted as being equal to $1$ for all $y$ when $j = 0$, as usual. If $|y|_p < 1$, then $y^{n
+ 1} \to 0$ as $n \to \infty$ with respect to the $p$-adic metric. This implies that \begin{equation} \label{sum_{j = 0}^n y^j = frac{1 - y^{n + 1}}{1 - y} to frac{1}{1 - y}}
\sum_{j = 0}^n y^j = \frac{1 - y^{n + 1}}{1 - y} \to \frac{1}{1 - y} \end{equation} as $n \to \infty$ with respect to the $p$-adic metric.
Of course, $|x|_p \le 1$ for every integer $x$. Now let
$x \in {\bf Q}$ with $|x|_p \le 1$ be given. Thus $x$ can be expressed as $a / b$, where $a$ and $b$ are integers, $b \ne 0$, and $b$ is not an integer multiple of $p$. It is well known that there is a nonzero integer $c$ such that $b \, c \equiv 1$ modulo $p$ under these conditions. Put $y = 1 - b \, c$, so that $y$ is an integer which is
divisible by $p$, and hence $|y|_p \le 1/p < 1$. It follows that \begin{equation} \label{x = frac{a}{b} = frac{a c}{b c} = frac{a c}{1 - y}}
x = \frac{a}{b} = \frac{a \, c}{b \, c} = \frac{a \, c}{1 - y} \end{equation} can be approximated by integers with respect to the $p$-adic metric, by (\ref{sum_{j = 0}^n y^j = frac{1 - y^{n + 1}}{1 - y} to frac{1}{1 - y}}).
\section{$p$-Adic numbers} \label{p-adic numbers}
The set ${\bf Q}_p$\index{Q_p@${\bf Q}_p$} of \emph{$p$-adic numbers}\index{p-adic numbers@$p$-adic numbers} can be obtained by completing ${\bf Q}$ as a metric space with respect to the $p$-adic metric, in the same way that the real line ${\bf R}$ is obtained by completing ${\bf Q}$ with respect to the standard Euclidean metric. Sums and product of rational numbers can be extended to $p$-adic numbers in a natural way, so that ${\bf Q}_p$ becomes a field. The
$p$-adic absolute value $|x|_p$ and $p$-adic metric $d_p(x, y)$ can also be extended to $x, y \in {\bf Q}_p$, in such a way that (\ref{|x
+ y|_p le max(|x|_p, |y|_p)}), (\ref{|x y|_p = |x|_p |y|_p}), and
(\ref{d_p(x, y) = |x - y|_p}) still hold. By construction, ${\bf Q}$ is dense in ${\bf Q}_p$ with respect to the $p$-adic metric, and
$|x|_p$ is an integer power of $p$ for every $x \in {\bf Q}_p$ with $x \ne 0$. One can show that addition and multiplication are continuous on ${\bf Q}_p$ with respect to the $p$-adic metric, in essentially the same way as for real numbers.
The set ${\bf Z}_p$\index{Z_p@${\bf Z}_p$} of \emph{$p$-adic integers}\index{p-adic integers@$p$-adic integers} is defined by \begin{equation}
\label{{bf Z}_p = {x in {bf Q}_p : |x|_p le 1}}
{\bf Z}_p = \{x \in {\bf Q}_p : |x|_p \le 1\}. \end{equation} This is the same as the closed unit ball in ${\bf Q}_p$, which is a closed set in ${\bf Q}_p$ in particular. This is also an open set in ${\bf Q}_p$ with respect to the $p$-adic metric, because the $p$-adic metric is an ultrametric, as in Section \ref{metrics, ultrametrics}. Of course, ${\bf Z}_p$ contains the set ${\bf Z}$ of ordinary integers. It is easy to see that ${\bf Q} \cap {\bf Z}_p$ is dense in ${\bf Z}_p$ with respect to the $p$-adic metric, because ${\bf Q}$ is dense in ${\bf Q}_p$, and using the ultrametric version of the triangle inequality. As in the previous section, elements of ${\bf Q} \cap {\bf Z}_p$ can be approximated by integers with respect to the $p$-adic metric. Combining these statements, we get that elements of ${\bf Z}_p$ can be approximated by elements of ${\bf Z}$ with respect to the $p$-adic metric, so that ${\bf Z}_p$ is the same as the closure of ${\bf Z}$ in ${\bf Q}_p$ with respect to the $p$-adic metric. Note that ${\bf Z}_p$ is also closed under addition and multiplication, by
(\ref{|x + y|_p le max(|x|_p, |y|_p)}) and (\ref{|x y|_p = |x|_p
|y|_p}).
Put \begin{equation}
\label{p^l {bf Z}_p = {p^l x : x in Z_p} = {y in Q_p : |y|_p le p^{-l}}}
p^l \, {\bf Z}_p = \{p^l \, x : x \in {\bf Z}_p\}
= \{y \in {\bf Q}_p : |y|_p \le p^{-l}\} \end{equation} for each integer $l$. This is the same as the closed ball in ${\bf
Q}_p$ centered at $0$ with radius $p^{-l}$ with respect to the $p$-adic metric, which is also an open set in ${\bf Q}_p$, as in Section \ref{metrics, ultrametrics}. Observe that $p^l \, {\bf Z}_p$
is a subgroup of ${\bf Q}_p$ with respect to addition for each $l$, because of (\ref{|x + y|_p le max(|x|_p, |y|_p)}). If $l \ge 0$, then $p^l \, {\bf Z}_p$ is an ideal in ${\bf Z}_p$ as a commutative ring, and hence the quotient ${\bf Z}_p / p^l \, {\bf Z}_p$ can be defined as a commutative ring. The composition of the obvious inclusion of ${\bf Z}$ in ${\bf Z}_p$ with the standard quotient homomorphism from ${\bf Z}_p$ onto ${\bf Z}_p / p^l \, {\bf Z}_p$ leads to a ring homomorphism from ${\bf Z}$ into ${\bf Z}_p / p^l \, {\bf Z}_p$. The kernel of this homomorphism is \begin{equation} \label{{bf Z} cap (p^l {bf Z}_p) = p^l {bf Z}}
{\bf Z} \cap (p^l \, {\bf Z}_p) = p^l \, {\bf Z}, \end{equation} which is an ideal in ${\bf Z}$. This leads to a natural injective ring homomorphism from ${\bf Z} / p^l \, {\bf Z}$ into ${\bf Z}_p / p^l \, {\bf Z}_p$. Every element of ${\bf Z}_p$ can be expressed as the sum of elements of ${\bf Z}$ and $p^l \, {\bf Z}_p$, because ${\bf
Z}$ is dense in ${\bf Z}_p$ with respect to the $p$-adic metric. Thus we get a natural ring isomorphism from ${\bf Z} / p^l \, {\bf Z}$ onto ${\bf Z}_p / p^l \, {\bf Z}_p$ for each nonnegative integer $l$.
In particular, ${\bf Z}_p / p^l \, {\bf Z}_p$ has exactly $p^l$ elements for each nonnegative integer $l$. This implies that ${\bf Z}_p$ can be expressed as the union of $p^l$ pairwise-disjoint translates of $p^l \, {\bf Z}_p$ for each $l \ge 0$. It follows that ${\bf Z}_p$ is totally bounded with respect to the $p$-adic metric, in the sense that ${\bf Z}_p$ can be covered by finitely many balls of arbitrarily small radius. A well-known theorem implies that ${\bf Z}_p$ is compact with respect to the topology determined on ${\bf Q}_p$ by the $p$-adic metric, because ${\bf Z}_p$ is also a closed set in ${\bf Q}_p$ and ${\bf Q}_p$ is complete. Of course, $p^k \, {\bf Z}_p$ is a compact set in ${\bf Q}_p$ for every integer $k$ too, by continuity of multiplication.
\section{Haar measure on ${\bf Q}_p$} \label{haar measure on Q_p}
If $A$ is a locally compact commutative topological group, then it is well known that there is a nonnegative translation-invariant Borel measure on $A$ which is finite on compact subsets of $A$, positive on nonempty open subsets of $A$, and which satisfies certain other regularity properties. This is known as \emph{Haar measure}\index{Haar measure} on $A$, and it is unique up to multiplication by a positive real number. The real line is a commutative topological group with respect to addition and the standard topology, for instance, and Lebesgue measure on ${\bf R}$
satisfies the requirements of Haar measure. Similarly, the discussion in the previous section implies that ${\bf Q}_p$ is a locally compact commutative topological group with respect to addition and the topology determined by the $p$-adic metric. Let $|E|$ be the corresponding Haar measure of a Borel set $E \subseteq {\bf Q}_p$, normalized so that $|{\bf Z}_p| = 1$.
If $l$ is a positive integer, then it follows that \begin{equation}
\label{|p^l {bf Z}_p| = p^{-l}}
|p^l \, {\bf Z}_p| = p^{-l}. \end{equation} This uses the fact that ${\bf Z}_p$ can be expressed as the union of $p^l$ pairwise-disjoint translates of $p^l \, {\bf Z}_p$, as in the previous section. If $l$ is a negative integer, then $p^l \, {\bf
Z}_p$ can be expressed as the union of $p^{-l}$ pairwise-disjoint translates of ${\bf Z}_p$, by applying the previous statement to $-l$. This implies that (\ref{|p^l {bf Z}_p| = p^{-l}}) also holds when $l < 0$, and hence for all $l \in {\bf Z}$.
If $a \in {\bf Q}_p$ and $E \subseteq {\bf Q}_p$ is a Borel set, then \begin{equation} \label{a E = {a x : x in E}}
a \, E = \{a \, x : x \in E\} \end{equation} is also a Borel set in ${\bf Q}_p$. This is trivial when $a = 0$, and it follows from the fact that $x \mapsto a \, x$ is a homeomorphism on
${\bf Q}_p$ when $a \ne 0$. If $a \ne 0$, then $x \mapsto a \, x$ is an isomorphism of ${\bf Q}_p$ onto itself as a commutative topological group, which implies that $|a \, E|$ satisfies the requirements of a Haar measure on ${\bf Q}_p$. The uniqueness of Haar measure implies that $|a \, E|$ is a constant multiple of $|E|$, where the constant depends on $a$ but not $E$. To determine the constant, one can consider the case where $E = {\bf Z}_p$, using (\ref{|p^l {bf Z}_p| =
p^{-l}}). If $|a|_p = p^{-l}$ for some $l \in {\bf Z}$, then it is easy to see that $a \, {\bf Z}_p = p^l \, {\bf Z}_p$, so that \begin{equation}
\label{|a {bf Z}_p| = |p^l {bf Z}_p| = p^{-l} = |a|_p}
|a \, {\bf Z}_p| = |p^l \, {\bf Z}_p| = p^{-l} = |a|_p. \end{equation} It follows that \begin{equation}
\label{|a E| = |a|_p |E|}
|a \, E| = |a|_p \, |E| \end{equation} for every $a \in {\bf Q}_p$ and Borel set $E \subseteq {\bf Q}_p$, which is trivial when $a = 0$.
Let $A$ be a locally compact commutative topological group again, and let $C_{com}(A)$ be the vector space of real-valued continuous functions on $A$ with compact support. Nonnegative linear functionals on $C_{com}(A)$ correspond to nonnegative Borel measures on $A$ which are finite on compact sets and have certain other regularity properties, by the Riesz representation theorem. The existence and uniqueness of Haar measure on $A$ can also be considered in terms of Haar integrals, which are nonnegative linear functionals on $C_{com}(A)$ that are invariant under translations and positive on nonnegative elements of $C_{com}(A)$ that are positive somewhere on $A$. The ordinary Riemann integral can be used to define a Haar integral on the real line, for instance. Similarly, one can get a Haar integral on ${\bf Q}_p$ as a limit of suitable Riemann sums.
\section{Snowflake metrics and quasi-metrics} \label{snowflake metrics, quasi-metrics} \index{snowflake metrics}
It is well known that \begin{equation} \label{(r + t)^a le r^a + t^a}
(r + t)^a \le r^a + t^a \end{equation} for all nonnegative real numbers $r$, $t$ when $a \in {\bf R}$ satisfies $0 < a \le 1$. Indeed, \begin{equation} \label{max(r, t) le (r^a + t^a)^{1/a}}
\max(r, t) \le (r^a + t^a)^{1/a} \end{equation} for every $a > 0$, which implies that \begin{eqnarray} \label{ r + t le ... = (r^a + t^a)^{1/a}}
r + t & \le & (r^a + t^a) \, \max(r, t)^{1 - a} \\
& \le & (r^a + t^a)^{1 + (1 - a)/a}
= (r^a + t^a)^{1/a} \nonumber \end{eqnarray} when $a \le 1$. If $d(x, y)$ is a metric on a set $X$, then it follows that $d(x, y)^a$ is also a metric on $X$ when $0 < a \le 1$. Similarly, if $d(x, y)$ is an ultrametric on $X$, then $d(x, y)^a$ is an ultrametric on $X$ for every $a > 0$. In both cases, $d(x, y)^a$ determines the same topology on $X$ as $d(x, y)$.
A \emph{quasi-metric}\index{quasi-metrics} on a set $X$ is a nonnegative real-valued function $d(x, y)$ on $X \times X$ such that $d(x, y) = 0$ if and only if $x = y$, $d(x, y) = d(y, x)$ for every $x, y \in X$, and \begin{equation} \label{d(x, z) le C (d(x, y) + d(y, z))}
d(x, z) \le C \, (d(x, y) + d(y, z)) \end{equation} for some $C \ge 1$ and every $x, y, z \in X$. Thus a quasi-metric $d(x, y)$ on $X$ is a metric on $X$ if and only if one can take $C = 1$ in (\ref{d(x, z) le C (d(x, y) + d(y, z))}). If $d(x, y)$ is a quasi-metric on $X$, then the open ball $B(x, r)$ centered at a point $x \in X$ with radius $r > 0$ with respect to $d(\cdot, \cdot)$ can still be defined as in (\ref{B(x, r) = {y in X : d(x, y) < r}}). One can also define a topology on $X$ corresponding to $d(\cdot, \cdot)$ in the usual way, by saying that a set $U \subseteq X$ is an open set if for each $x \in U$ there is an $r > 0$ such that $B(x, r) \subseteq X$. It is easy to see that this satisfies the requirements of a topology on $X$, but the weaker version of the triangle inequality is not sufficient to show that open balls are open sets in $X$ with respect to this topology.
If $a \in {\bf R}$ and $a > 1$, then it is well known that $r^a$ is a convex function on the set of nonnegative real numbers $r$. This implies that \begin{equation} \label{((r + t)/2)^a le (1/2) (r^a + t^a)}
((r + t)/2)^a \le (1/2) \, (r^a + t^a) \end{equation} for every $r, t \ge 0$, and hence that \begin{equation} \label{(r + t)^a le 2^{a - 1} (r + t)}
(r + t)^a \le 2^{a - 1} \, (r + t). \end{equation} If $d(x, y)$ is a metric on a set $X$, then it follows that $d(x, y)^a$ is a quasi-metric on $X$ for every $a > 1$. Similarly, $d(x, y)^a$ is a quasi-metric on $X$ for every $a > 0$ when $d(x, y)$ is a quasi-metric on $X$. Of course, the topology on $X$ determined by $d(x, y)^a$ is the same as the topology on $X$ corresponding to $d(x, y)$, and in fact the open ball in $X$ centered at a point $x \in X$ and with radius $r > 0$ with respect to $d(\cdot, \cdot)$ is the same as the open ball in $X$ centered at $x$ and with radius $r^a$ with respect to $d(\cdot, \cdot)^a$.
If $d(x, y)$ is a quasi-metric on a set $X$, then one can define a uniform structure on $X$ in the usual way, by considering the subsets \begin{equation} \label{U_r = {(x, y) in X times X : d(x, y) < r}}
U_r = \{(x, y) \in X \times X : d(x, y) < r\} \end{equation} of $X \times X$ for each $r > 0$. The topology on $X$ determined by this uniform structure is the same as the topology on $X$ defined in terms of open balls associated to $d(x, y)$, as before. Standard results about uniform structures imply that for each $x \in X$ and $r > 0$, $x$ is in the interior of the corresponding open ball $B(x, r)$ with respect to this topology. This uniform structure on $X$ obviously has a countable base, corresponding to any sequence of positive real numbers that converges to $0$. This implies that there is a metric on $X$ that determines the same uniform structure on $X$, as in \cite{jk}. In particular, there is a metric on $X$ that determines the same topology on $X$ as $d(x, y)$. In \cite{m-s-1}, it is shown that there is a metric $\widetilde{d}(x, y)$ on $X$ and a positive real number $a$ such that $d(x, y)$ is comparable to $\widetilde{d}(x, y)^a$ on $X$, in the sense that each is bounded by a constant times the other.
\section{Sequences and series} \label{sequences, series}
Let $d(x, y)$ be a quasi-metric on a set $X$. As usual, a sequence $\{x_j\}_{j = 1}^\infty$ of elements of $X$ is said to \emph{converge}\index{convergent sequences} to an element $x$ of $X$ if for every $\epsilon > 0$ there is an $L \ge 1$ such that \begin{equation} \label{d(x_j, x) < epsilon}
d(x_j, x) < \epsilon \end{equation} for every $j \ge L$. This is equivalent to saying that $\{x_j\}_{j =
1}^\infty$ converges to $x$ with respect to the topology on $X$ determined by $d(\cdot, \cdot)$. More precisely, this uses the fact that every open ball in $X$ centered at $x$ contains an open set that contains $x$ as an element, as in the previous section. Similarly, a sequence $\{x_j\}_{j = 1}^\infty$ of elements of $X$ is said to be a \emph{Cauchy sequence}\index{Cauchy sequences} in $X$ if for every $\epsilon > 0$ there is an $L \ge 1$ such that \begin{equation} \label{d(x_j, x_l) < epsilon}
d(x_j, x_l) < \epsilon \end{equation} for every $j, l \ge L$. In particular, it is easy to see that convergent sequences are Cauchy sequences. If $\{x_j\}_{j =
1}^\infty$ is a Cauchy sequence in $X$, then \begin{equation} \label{lim_{j to infty} d(x_j, x_{j + 1}) = 0}
\lim_{j \to \infty} d(x_j, x_{j + 1}) = 0, \end{equation} by taking $l = j + 1$ in (\ref{d(x_j, x_l) < epsilon}). If $d(\cdot, \cdot)$ is an ultrametric on $X$, and if $\{x_j\}_{j = 1}^\infty$ is a sequence of elements of $X$ that satisfies (\ref{lim_{j to infty}
d(x_j, x_{j + 1}) = 0}), then one can check that $\{x_j\}_{j =
1}^\infty$ is a Cauchy sequence in $X$.
Let $\sum_{j = 1}^\infty a_j$ be an infinite series whose terms are real numbers, complex numbers, or $p$-adic numbers for some prime number $p$. If the corresponding sequence of partial sums \begin{equation} \label{s_n = sum_{j = 1}^n a_j}
s_n = \sum_{j = 1}^n a_j \end{equation} converges in ${\bf R}$, ${\bf C}$, or ${\bf Q}_p$, as appropriate, then $\sum_{j = 1}^\infty a_j$ is said to \emph{converge},\index{convergent series} and the value of the sum is defined to be the limit of $\{s_n\}_{n = 1}^\infty$. Because ${\bf
R}$, ${\bf C}$, and ${\bf Q}_p$ are complete with respect to their standard metrics, convergence of $\sum_{j = 1}^\infty a_j$ is equivalent to asking that $\{s_n\}_{n = 1}^\infty$ be a Cauchy sequence. In particular, a necessary condition for the convergence of $\sum_{j = 1}^\infty a_j$ is that $\{a_j\}_{j = 1}^\infty$ converge as a sequence to $0$ in ${\bf R}$, ${\bf C}$, or ${\bf Q}_p$, as appropriate. This is also a sufficient condition in the $p$-adic case, because the $p$-adic metric is an ultrametric.
An infinite series $\sum_{j = 1}^\infty a_j$ of real or complex numbers is said to converge \emph{absolutely}\index{absolute
convergence} if $\sum_{j = 1}^\infty |a_j|$ converges, where $|a_j|$ is the absolute value of $a_j$ in the real case, and the modulus of
$a_j$ in the complex case. It is well known that absolute convergence implies convergence, using the triangle inequality to show that the partial sums of $\sum_{j = 1}^\infty a_j$ form a Cauchy sequence when the partial sums of $\sum_{j = 1}^\infty |a_j|$ form a Cauchy sequence. One can also check that \begin{equation}
\biggl|\sum_{j = 1}^\infty a_j \biggr| \le \sum_{j = 1}^\infty |a_j| \end{equation} when $\sum_{j = 1}^\infty a_j$ converges absolutely. Similarly, if $\{a_j\}_{j = 1}^\infty$ is a sequence of $p$-adic numbers that converges to $0$, then \begin{equation}
\label{|sum_{j = 1}^infty a_j|_p le max_{j ge 1} |a_j|_p}
\biggl|\sum_{j = 1}^\infty a_j\biggr|_p \le \max_{j \ge 1} |a_j|_p. \end{equation}
Note that the maximum of $|a_j|_p$ over $j \in {\bf Z}_+$ exists in this situation, because $|a_j|_p \to 0$ as $j \to \infty$.
The \emph{Cauchy product}\index{Cauchy products} of two infinite series $\sum_{j = 0}^\infty a_j$, $\sum_{k = 0}^\infty b_k$ of real, complex, or $p$-adic numbers is the infinite series $\sum_{l = 0}^\infty c_l$, where \begin{equation} \label{c_l = sum_{j = 0}^l a_j b_{l - j}}
c_l = \sum_{j = 0}^l a_j \, b_{l - j}. \end{equation} If $\sum_{j = 0}^\infty a_j$ and $\sum_{k = 0}^\infty b_k$ are absolutely convergent series of real or complex numbers, then it is well known that $\sum_{l = 0}^\infty c_l$ converges absolutely too, and that \begin{equation} \label{sum_{l = 0}^infty c_l = (sum_{j = 0}^infty a_j) (sum_{k = 0}^infty b_k)}
\sum_{l = 0}^\infty c_l = \Big(\sum_{j = 0}^\infty a_j\Big)
\, \Big(\sum_{k = 0}^\infty b_k\Big). \end{equation} Similarly, if $\{a_j\}_{j = 0}^\infty$ and $\{b_k\}_{k = 0}^\infty$ are sequences of $p$-adic numbers converging to $0$, then one can check that $\{c_l\}_{l = 0}^\infty$ also converges to $0$ in ${\bf
Q}_p$, using the fact that the $p$-adic metric is an ultrametric. It is not too difficult to verify that (\ref{sum_{l = 0}^infty c_l =
(sum_{j = 0}^infty a_j) (sum_{k = 0}^infty b_k)}) holds under these conditions as well.
\chapter{Hausdorff measures} \label{hausdorff measures}
\section{Diameters} \label{diameters}
Let $(M, d(x, y))$ be a metric space. As usual, the \emph{diameter}\index{diameter of a set} of a nonempty bounded set $A \subseteq M$ is defined by \begin{equation} \label{diam A = sup {d(x, y) : x, y in A}}
\mathop{\rm diam} A = \sup \{d(x, y) : x, y \in A\}. \end{equation} It is sometimes convenient to define the diameter of the empty set to be $0$, and to put $\mathop{\rm diam} A = \infty$ when $A$ is not bounded. Note that \begin{equation} \label{diam overline{A} = diam A}
\mathop{\rm diam} \overline{A} = \mathop{\rm diam} A \end{equation} for any set $A \subseteq M$, where $\overline{A}$ is the closure of $A$ in $M$.
Let $A \subseteq M$ and $r > 0$ be given, and put \begin{equation} \label{A_r = bigcup_{x in A} B(x, r)}
A_r = \bigcup_{x \in A} B(x, r). \end{equation} Thus $A_r$ is an open set in $M$, since it is a union of open sets, and $A \subseteq A_r$. If $w, z \in A_r$, then there are $x, y \in A$ such that $d(x, w), d(y, z) < r$, and hence \begin{equation} \label{d(w, z) < d(x, y) + 2 r}
d(w, z) < d(x, y) + 2 \, r. \end{equation} This implies that \begin{equation} \label{diam A_r le diam A + 2 r}
\mathop{\rm diam} A_r \le \mathop{\rm diam} A + 2 \, r \end{equation} for each $r > 0$.
If $d(x, y)$ is an ultrametric on $M$, then we can replace (\ref{d(w, z) < d(x, y) + 2 r}) with \begin{equation} \label{d(w, z) le max(d(x, y), r)}
d(w, z) \le \max(d(x, y), r), \end{equation} so that \begin{equation} \label{diam A_r le max(diam A, r)}
\mathop{\rm diam} A_r \le \max(\mathop{\rm diam} A, r) \end{equation} for each $r > 0$. Of course, \begin{equation} \label{diam A le diam A_r}
\mathop{\rm diam} A \le \mathop{\rm diam} A_r \end{equation} for every $r > 0$, since $A \subseteq A_r$. Thus (\ref{diam A_r le
max(diam A, r)}) implies that \begin{equation} \label{diam A_r = diam A}
\mathop{\rm diam} A_r = \mathop{\rm diam} A \end{equation} when $r \le \mathop{\rm diam} A$.
Similarly, \begin{equation} \label{diam overline{B}(x, r) le 2 r}
\mathop{\rm diam} \overline{B}(x, r) \le 2 \, r \end{equation} for every $x \in M$ and $r \ge 0$, and for any metric $d(x, y)$ on $M$. If $d(x, y)$ is an ultrametric on $M$, then \begin{equation} \label{diam overline{B}(x, r) le r}
\mathop{\rm diam} \overline{B}(x, r) \le r \end{equation} for every $x \in M$ and $r \ge 0$. If $d(x, y)$ is any metric on $M$ and $A$ is a nonempty bounded set in $M$, then \begin{equation} \label{A subseteq overline{B}(x, diam A)}
A \subseteq \overline{B}(x, \mathop{\rm diam} A) \end{equation} for every $x \in A$. If $M$ is the real line with the standard metric, and if $A$ is a nonempty bounded subset of ${\bf R}$, then \begin{equation} \label{I_A = [inf A, sup A]}
I_A = [\inf A, \sup A] \end{equation} contains $A$ and has the same diameter as $A$.
\section{Hausdorff content} \label{hausdorff content}
Let $(M, d(x, y))$ be a metric space again, and let $\alpha$ be a positive real number. The $\alpha$-dimensional \emph{Hausdorff content}\index{Hausdorff content} of $E \subseteq M$ is defined by \begin{equation} \label{H^alpha_{con}(E) = ...}
H^\alpha_{con}(E) = \inf \bigg\{\sum_j (\mathop{\rm diam} A_j)^\alpha :
E \subseteq \bigcup_j A_j\bigg\}, \end{equation} where more precisely the infimum is taken over all collections $\{A_j\}_j$ of finitely or countably many subsets of $M$ such that $E \subseteq \bigcup_j A_j$. The sum \begin{equation} \label{sum_j (diam A_j)^alpha}
\sum_j (\mathop{\rm diam} A_j)^\alpha \end{equation} is defined as usual as the supremum over all finite subsums when there are infinitely many $A_j$'s, which may be infinite. If $A_j$ is unbounded for any $j$, then $\mathop{\rm diam} A_j = \infty$, and (\ref{sum_j
(diam A_j)^alpha}) is infinite. This definition can also be used when $\alpha = 0$, with the conventions that $(\mathop{\rm diam} A)^0$ is equal to $0$ when $A = \emptyset$, is equal to $1$ when $A$ is nonempty and bounded, and is equal to $\infty$ when $A$ is unbounded.
Note that $H^\alpha_{con}(\emptyset) = 0$ for every $\alpha \ge 0$, and that \begin{equation} \label{H^alpha_{con}(E) le (diam E)^alpha}
H^\alpha_{con}(E) \le (\mathop{\rm diam} E)^\alpha \end{equation} for every $E \subseteq M$ and $\alpha \ge 0$, by covering $E$ by itself. If $E \subseteq \widetilde{E} \subseteq M$, then \begin{equation} \label{H^alpha_{con}(E) le H^alpha_{con}(widetilde{E})}
H^\alpha_{con}(E) \le H^\alpha_{con}(\widetilde{E}) \end{equation} for every $\alpha \ge 0$, because every covering of $\widetilde{E}$ in $M$ is also a covering of $E$. If $E_1, E_2, E_2, \ldots$ is any sequence of subsets of $M$, then one can show that \begin{equation} \label{H^alpha_{con}(bigcup_{k = 1}^infty E_k) le ...}
H^\alpha_{con}\Big(\bigcup_{k = 1}^\infty E_k\Big)
\le \sum_{k = 1}^\infty H^\alpha_{con}(E_k) \end{equation} for every $\alpha \ge 0$, by combining coverings of the $E_k$'s to get coverings of $\bigcup_{k = 1}^\infty E_k$. Of course, if $H^\alpha_{con}(E_k) = \infty$ for some $k$, then the sum on the right side of (\ref{H^alpha_{con}(bigcup_{k = 1}^infty E_k) le ...}) is equal to $\infty$ too, in which case the inequality is trivial. Otherwise, one can choose coverings of the $E_k$'s for which the corresponding sums (\ref{sum_j (diam A_j)^alpha}) are as close as one wants to $H^\alpha_{con}(E_k)$. The main point is to do this in such a way that the sum of the errors is arbitrarily small too.
In the definition of the Hausdorff content, one might as well restrict one's attention to coverings of $E$ by collections of closed subsets of $M$, because of (\ref{diam overline{A} = diam A}). One can also restrict one's attention to coverings by collection of open subsets of $M$, using (\ref{diam A_r le diam A + 2 r}). If $E \subseteq M$ is compact, then it follows that one can restrict one's attention to coverings of $E$ by finitely many subsets of $M$.
Remember that an \emph{outer measure}\index{outer measures} on a $\sigma$-algebra $\mathcal{A}$ of subsets of $M$ is a nonnegative extended real-valued function $\mu$ on $\mathcal{A}$ such that $\mu(\emptyset) = 0$, \begin{equation} \label{mu(A) le mu(B)}
\mu(A) \le \mu(B) \end{equation} for every $A, B \in \mathcal{A}$ with $A \subseteq B$, and $\mu$ is countably-subadditive on $\mathcal{A}$. Thus $H^\alpha_{con}$ is an outer measure on the $\sigma$-algebra of all subsets of $M$ for each $\alpha \ge 0$, for instance. Let $\mu$ be an outer measure defined on a $\sigma$-algebra $\mathcal{A}$ of subsets of $M$ that contains the Borel sets, and suppose that \begin{equation} \label{mu(A) le C (diam A)^alpha}
\mu(A) \le C \, (\mathop{\rm diam} A)^\alpha \end{equation} for some nonnegative real numbers $C$, $\alpha$ and every $A \in \mathcal{A}$. If $E \in \mathcal{A}$, and if $\{A_j\}_j$ are finitely or countably many elements of $\mathcal{A}$ such that $E \subseteq \bigcup_j A_j$, then \begin{equation} \label{mu(E) le sum_j mu(A_j) le C sum_j (diam A_j)^alpha}
\mu(E) \le \sum_j \mu(A_j) \le C \, \sum_j (\mathop{\rm diam} A_j)^\alpha. \end{equation} This implies that \begin{equation} \label{mu(E) le C H^alpha_{con}(E)}
\mu(E) \le C \, H^\alpha_{con}(E), \end{equation} since we can restrict our attention to coverings of $E$ by open or closed subsets of $M$ in the definition of Hausdorff content, as in the previous paragraph.
\section{Restricting the diameters} \label{restricting the diameters}
Let $(M, d(x, y))$ be a metric space, and let $0 \le \alpha < \infty$ and $0 < \delta \le \infty$ be given. Put \begin{equation} \label{H^alpha_delta(E) = ...}
\quad H^\alpha_\delta(E) = \inf \bigg\{\sum_j (\mathop{\rm diam} A_j)^\alpha :
E \subseteq \bigcup_j A_j, \, \mathop{\rm diam} A_j < \delta \hbox{ for each } j\bigg\} \end{equation} for each $E \subseteq M$, where more precisely the infimum is taken over all collections $\{A_j\}_j$ of finitely or countably many subsets of $M$ such that $E \subseteq \bigcup_j A_j$ and $\mathop{\rm diam} A_j < \delta$ for each $j$, if there are any. If not, then put $H^\alpha_\delta(E) = \infty$. Of course, if $M$ is separable, then $M$ is contained in the union of finitely or countably many balls of radius $r$ for every $r > 0$, and this is not a problem. This is also not a problem when $\delta = \infty$, because every $E \subseteq M$ is covered by a sequence of bounded subsets of $M$.
By construction, \begin{equation} \label{H^alpha_{con}(E) le H^alpha_delta(E) le H^alpha_{eta}(E)}
H^\alpha_{con}(E) \le H^\alpha_\delta(E) \le H^\alpha_{\eta}(E) \end{equation} for every $\alpha \ge 0$ and $E \subseteq M$ when $0 < \eta < \delta \le \infty$, since one is restricting the class of admissible coverings of $E$ as $\delta$ decreases. It is easy to see that \begin{equation} \label{H^alpha_{con}(E) = H^alpha_infty(E)}
H^\alpha_{con}(E) = H^\alpha_\infty(E) \end{equation} for every $\alpha \ge 0$ and $E \subseteq M$, because (\ref{sum_j
(diam A_j)^alpha}) is infinite when $A_j$ is unbounded for any $j$. As before, $H^\alpha_\delta(\emptyset) = 0$ for every $\alpha \ge 0$ and $\delta > 0$, and \begin{equation} \label{H^alpha_delta(E) le H^alpha_delta(widetilde{E})}
H^\alpha_\delta(E) \le H^\alpha_\delta(\widetilde{E}) \end{equation} when $E \subseteq \widetilde{E} \subseteq M$. If $E_1, E_2, E_3, \ldots$ is any sequence of subsets of $M$, then \begin{equation} \label{H^alpha_delta(bigcup_{k = 1}^infty E_k) le ...}
H^\alpha_\delta\Big(\bigcup_{k = 1}^\infty E_k\Big)
\le \sum_{k = 1}^\infty H^\alpha_\delta(E_k) \end{equation} for every $\alpha \ge 0$ and $\delta > 0$, as in the previous section. Thus $H^\alpha_\delta$ is an outer measure on the $\sigma$-algebra of all subsets of $M$ for each $\alpha \ge 0$ and $\delta > 0$.
One might as well restrict one's attention to coverings of $E$ by open or closed subsets of $M$ in (\ref{H^alpha_delta(E) = ...}), for the same reasons as before. In particular, if $E$ is compact, then one can restrict one's attention to coverings of $E$ by finitely many subsets of $M$.
Suppose that $E_1, E_2 \subseteq M$ have the property that \begin{equation} \label{d(x, y) ge delta}
d(x, y) \ge \delta \end{equation} for some $\delta > 0$ and every $x \in E_1$ and $y \in E_2$. Let $\{A_j\}_{j \in I}$ be any collection of finitely or countably many subsets of $M$ such that $\mathop{\rm diam} A_j < \delta$ for each $j$ and \begin{equation} \label{E_1 cup E_2 subseteq bigcup_{j in I} A_j}
E_1 \cup E_2 \subseteq \bigcup_{j \in I} A_j. \end{equation} Let $I_1$, $I_2$ be the set of $j \in I$ such that $A_j$ intersects $E_1$, $E_2$, respectively. The separation condition (\ref{d(x, y) ge delta}) implies that $I_1$ and $I_2$ disjoint subsets of $I$, so that \begin{eqnarray} \label{H^alpha_delta(E_1) + H^alpha_delta(E_2) le ...}
H^\alpha_\delta(E_1) + H^\alpha_\delta(E_2) & \le &
\sum_{j \in I_1} (\mathop{\rm diam} A_j)^\alpha + \sum_{j \in I_2} (\mathop{\rm diam} A_j)^\alpha \\
& \le & \sum_{j \in I} (\mathop{\rm diam} A_j)^\alpha. \nonumber \end{eqnarray} for every $\alpha \ge 0$. This implies that \begin{equation} \label{H^alpha_delta(E_1) + H^alpha_delta(E_2) le H^alpha_delta(E_1 cup E_2)}
H^\alpha_\delta(E_1) + H^\alpha_\delta(E_2) \le H^\alpha_\delta(E_1 \cup E_2) \end{equation} for every $\alpha \ge 0$, by taking the infimum over all such coverings $\{A_j\}_{j \in I}$ of $E_1 \cup E_2$. The opposite inequality holds automatically, as in (\ref{H^alpha_delta(bigcup_{k = 1}^infty E_k) le ...}). Thus \begin{equation} \label{H^alpha_delta(E_1) + H^alpha_delta(E_2) = H^alpha_delta(E_1 cup E_2)}
H^\alpha_\delta(E_1) + H^\alpha_\delta(E_2) = H^\alpha_\delta(E_1 \cup E_2) \end{equation} for all $\alpha \ge 0$ under these conditions.
\section{Hausdorff measures} \label{hausdorff measures, section}
Let $(M, d(x, y))$ be a metric space, and let $\alpha \ge 0$ be given. The $\alpha$-dimensional \emph{Hausdorff measure}\index{Hausdorff measure} of $E \subseteq M$ is defined by \begin{equation} \label{H^alpha(E) = sup_{delta > 0} H^alpha_delta(E)}
H^\alpha(E) = \sup_{\delta > 0} H^\alpha_\delta(E), \end{equation} where $H^\alpha_\delta(E)$ is as in the previous section. This can also be considered as the limit of $H^\alpha_\delta(E)$ as $\delta \to 0$, since $H^\alpha_\delta(E)$ increases monotonically as $\delta$ decreases. As usual, $H^\alpha(\emptyset) = 0$ for every $\alpha \ge 0$, and \begin{equation} \label{H^alpha(E) le H^alpha(widetilde{E})}
H^\alpha(E) \le H^\alpha(\widetilde{E}) \end{equation} for every $\alpha \ge 0$ when $E \subseteq \widetilde{E} \subseteq M$. If $E_1, E_2, E_3, \ldots$ is any sequence of subsets of $M$, then \begin{equation} \label{H^alpha(bigcup_{k = 1}^infty E_k) le sum_{k = 1}^infty H^alpha(E_k)}
H^\alpha\Big(\bigcup_{k = 1}^\infty E_k\Big)
\le \sum_{k = 1}^\infty H^\alpha(E_k) \end{equation} for every $\alpha \ge 0$, by (\ref{H^alpha_delta(bigcup_{k = 1}^infty
E_k) le ...}), so that $H^\alpha$ is an outer measure on the $\sigma$-algebra of all subsets of $M$ for each $\alpha \ge 0$.
Let $E \subseteq M$, $0 \le \alpha < \beta$, and $\delta > 0$ be given. If $\{A_j\}_j$ is a collection of finitely or countably many subsets of $M$ such that $E \subseteq \bigcup_j A_j$ and $\mathop{\rm diam} A_j < \delta$ for each $j$, then \begin{equation} \label{sum_j (diam A_j)^beta le delta^{beta - alpha} sum_j (diam A_j)^alpha}
\sum_j (\mathop{\rm diam} A_j)^\beta \le \delta^{\beta - \alpha} \, \sum_j (\mathop{\rm diam} A_j)^\alpha. \end{equation} This implies that \begin{equation} \label{H^beta_delta(E) le delta^{beta - alpha} H^alpha_delta(E)}
H^\beta_\delta(E) \le \delta^{\beta - \alpha} \, H^\alpha_\delta(E). \end{equation} If $H^\alpha(E) < \infty$, then one can pass to the limit as $\delta \to 0$, to get that $H^\beta(E) = 0$. The \emph{Hausdorff
dimension}\index{Hausdorff dimension} of $E \subseteq M$ may be defined as the infimum of the $\alpha \ge 0$ such that $H^\alpha(E) < \infty$, if there is such an $\alpha$, and otherwise the Hausdorff dimension of $E$ is $\infty$. If $H^\alpha_{con}(E) = 0$ for some $\alpha \ge 0$ and $E \subseteq M$, then it is easy to see that $H^\alpha_\delta(E) = 0$ for every $\delta > 0$, and hence that $H^\alpha(E) = 0$. The main point is that if $\{A_j\}_j$ is a collection of finitely or countable many subsets of $M$ such that $E \subseteq \bigcup_j A_j$ and the corresponding sum (\ref{sum_j (diam A_j)^alpha}) is small, then $\mathop{\rm diam} A_j$ has to be small for each $j$.
Suppose that $H^\alpha(E) < \infty$ for some $\alpha \ge 0$ again. Thus for each positive integer $n$ there is a collection $\{A_{j,
n}\}_{j \in I_n}$ of finitely or countably many open subsets of $M$ such that $E \subseteq \bigcup_{j \in I_n} A_{j, n}$, $\mathop{\rm diam} A_{j, n} < 1/n$ for every $j \in I_n$, and \begin{equation} \label{sum_{j in I_n} (diam A_{j, n})^alpha < H^alpha(E) + 1/n}
\sum_{j \in I_n} (\mathop{\rm diam} A_{j, n})^\alpha < H^\alpha(E) + 1/n. \end{equation} Put \begin{equation} \label{widetilde{E} = bigcap_{n = 1}^infty (bigcup_{j in I_n} A_{j, n})}
\widetilde{E} = \bigcap_{n = 1}^\infty \Big(\bigcup_{j \in I_n} A_{j, n}\Big), \end{equation} so that $E \subseteq \widetilde{E}$, $\widetilde{E}$ is the intersection of a sequence of open subsets of $M$, and $\widetilde{E} \subseteq \bigcup_{j \in I_n} A_j$ for each $n$. The latter implies that $H^\alpha(\widetilde{E}) \le H^\alpha(E)$, and hence that $H^\alpha(\widetilde{E}) = H^\alpha(E)$.
If $E_1, E_2 \subseteq M$ have the property that \begin{equation} \label{d(x, y) ge eta}
d(x, y) \ge \eta \end{equation} for some $\eta > 0$ and every $x \in E_1$ and $y \in E_2$, then (\ref{H^alpha_delta(E_1) + H^alpha_delta(E_2) = H^alpha_delta(E_1 cup
E_2)}) holds when $0 < \delta \le \eta$. This implies that \begin{equation} \label{H^alpha(E_1) + H^alpha(E_2) = H^alpha(E_1 cup E_2)}
H^\alpha(E_1) + H^\alpha(E_2) = H^\alpha(E_1 \cup E_2), \end{equation} by taking the limit as $\delta \to 0$ in (\ref{H^alpha_delta(E_1) +
H^alpha_delta(E_2) = H^alpha_delta(E_1 cup E_2)}). This shows that $H^\alpha$ satisfies a well-known criterion of Carath\'eodory, and hence that $H^\alpha$ is countably additive on a suitable $\sigma$-algebra of measurable subsets of $M$ that includes the Borel sets. If $\alpha = 0$, then Hausdorff measure reduces to counting measure on $M$.
\section{Some special cases} \label{some special cases}
Suppose that $M$ is the real line, with the standard metric. As in Section \ref{diameters}, every nonempty bounded subset of ${\bf R}$ is contained in a closed interval with the same diameter. This implies that one may as well restrict one's attention to coverings of $E \subseteq {\bf R}$ by closed intervals in the definition of $H^\alpha_\delta(E)$ for every $\alpha, \delta > 0$. If $\alpha = 0$, then one should consider the empty set as a closed interval too. One might also consider the real line itself as a closed interval, for the analogous statement for $H^\alpha_{con}(E)$, although this does not really matter.
Let us restrict our attention now to $\alpha = 1$. It is easy to see that \begin{equation} \label{H^1_delta(E) = H^1_{con}(E)}
H^1_\delta(E) = H^1_{con}(E) \end{equation} for every $\delta > 0$ and $E \subseteq {\bf R}$, by subdividing intervals in ${\bf R}$ into finitely many arbitrarily small subintervals. It follows that \begin{equation} \label{H^1(E) = H^1_{con}(E)}
H^1(E) = H^1_{con}(E) \end{equation} for every $E \subseteq {\bf R}$, which is of course the same as the Lebesgue outer measure of $E$. If $E$ is a closed interval in ${\bf
R}$, then this is less than or equal to the diameter of $E$, which is the same as the length of $E$ as an interval, as in (\ref{H^alpha_{con}(E) le (diam E)^alpha}). As usual, one can show that $H^1(E)$ is equal to the diameter of $E$ in this case, by considering coverings of $E$ by finitely many intervals in ${\bf R}$.
Suppose now that $(M, d(x, y))$ is an ultrametric space. Every nonempty bounded subset of $M$ is contained in a closed ball in $M$ with the same diameter, as in Section \ref{diameters} again. Thus one may as well restrict one's attention to coverings of $E \subseteq M$ by closed balls in $M$ in the definition of $H^\alpha_\delta(E)$ for every $\alpha, \delta > 0$. As before, one should consider the empty set as a closed ball in $M$ when $\alpha = 0$, and one might also consider $M$ as a closed ball even when $M$ is unbounded, in the context of Hausdorff content.
In particular, these remarks can be applied to ${\bf Q}_p$, with the $p$-adic metric. Let us restrict our attention to $\alpha = 1$ again. Remember that ${\bf Z}_p$ can be expressed as the union of $p^l$ pairwise-disjoint translates of $p^l \, {\bf Z}_p$ for every positive integer $l$, as in Section \ref{p-adic numbers}. This implies that any closed ball in ${\bf Q}_p$ of radius $p^k$ for some $k \in {\bf Z}$ can be expressed as the pairwise-disjoint union of $p^l$ closed balls of radius $p^{k - l}$ for every positive integer $l$. Note that every closed ball in ${\bf Q}_p$ of radius $p^j$ for some $j \in {\bf Z}$ has diameter equal to $p^j$ too. Using this, one can check that (\ref{H^1_delta(E) = H^1_{con}(E)}) also holds in this case for every $E \subseteq {\bf Q}_p$. This implies that (\ref{H^1(E) = H^1_{con}(E)}) holds for every $E \subseteq {\bf Q}_p$ as well.
If $B$ is a closed ball in ${\bf Q}_p$ with radius $p^j$ for some $j \in {\bf Z}$, then $H^1(B) \le p^j$, by the previous discussion. Let us verify that $H^1(B) \ge p^j$ under these conditions, and hence that \begin{equation} \label{H^1(B) = p^j}
H^1(B) = p^j. \end{equation} To do this, it suffices to consider coverings of $B$ by finitely many closed balls in ${\bf Q}_p$, because $B$ is compact, and closed balls in ${\bf Q}_p$ are open sets. More precisely, it suffices to consider coverings of $B$ by finitely many closed balls of the same radius $p^{j - l}$ for some nonnegative integer $l$, by subdividing balls of different radii to get balls of the same radius. To show that $H^1(B) \ge p^j$, it is enough to check that $B$ cannot be covered by fewer than $p^l$ closed balls of radius $p^{j - l}$, for any nonnegative integer $l$. If $B = {\bf Z}_p$, which is the closed unit ball in ${\bf Q}_p$, then this follows from the discussion in Section \ref{p-adic numbers}. If $B$ is any other closed ball in ${\bf Q}_p$, then one can reduce to the case of ${\bf Z}_p$, using translations and dilations.
Now let $X_1, X_2, X_3, \ldots$ be a sequence of finite sets, where $X_j$ has exactly $n_j \ge 2$ elements for each $j$. Also let $X = \prod_{j = 1}^\infty X_j$ be their Cartesian product, as in Section \ref{abstract cantor sets}. Put $t_0 = 1$, and let $t_l > 0$ be defined by \begin{equation} \label{1/t_l = prod_{j = 1}^l n_j}
1/t_l = \prod_{j = 1}^l n_j \end{equation} when $l \ge 1$. Thus $\{t_l\}_{l = 0}^\infty$ is a strictly decreasing sequence of positive real numbers that converges to $0$, which leads to an ultrametric $d(x, y)$ on $X$, as in (\ref{d(x, y) =
t_{l(x, y)}}). Remember that the closed ball in $X$ centered at a point $x \in X$ and with radius $t_k$ for some nonnegative integer $k$ is of the form $B_k(x)$ as in (\ref{B_k(x) = {y in X : y_j = x_j for
each j le k}}). The diameter of this ball is also equal to $t_k$. The radius of any closed ball in $X$ with respect to $d(x, y)$ in (\ref{d(x, y) = t_{l(x, y)}}) can be taken to be $t_k$ for some nonnegative integer $k$, since these are the only positive values of $d(x, y)$.
By construction, every closed ball $B$ in $X$ of radius $t_k$ for some nonnegative integer $k$ is the union of \begin{equation} \label{prod_{j = k + 1}^l n_j}
\prod_{j = k + 1}^l n_j \end{equation} pairwise-disjoint closed balls in $X$ of radius $t_l$, for every integer $l > k$. This implies that (\ref{H^1_delta(E) =
H^1_{con}(E)}) also holds for every $E \subseteq X$ in this situation, and hence that (\ref{H^1(E) = H^1_{con}(E)}) holds for every $E \subseteq X$ too. In particular, \begin{equation} \label{H^1(B) le diam B}
H^1(B) \le \mathop{\rm diam} B. \end{equation} As before, one can show that $H^1(B) \ge \mathop{\rm diam} B$, by considering coverings of $B$ by finitely many closed balls, and subdividing the balls to get finitely many smaller balls of the same radius. This implies that \begin{equation} \label{H^1(B) = diam B}
H^1(B) = \mathop{\rm diam} B, \end{equation} which corresponds exactly to (\ref{mu(B_k(x)) = prod_{j = 1}^k
mu_j({x_j})}), in the case where $\mu_j$ is uniformly distributed on $X_j$ for each $j$.
\section{Carath\'eodory's construction} \label{caratheodory's construction}
Let $(M, d(x, y))$ be a metric space, let $\mathcal{F}$ be a collection of subsets of $M$, and let $\zeta$ be a nonnegative extended real-valued function on $\mathcal{F}$. Also let $0 < \delta \le \infty$ be given, and put \begin{eqnarray} \label{H_delta(E) = ...}
H_\delta(E) & = & \inf\bigg\{\sum_j \zeta(A_j) : E \subseteq \bigcup_j A_j, \,
A_j \in \mathcal{F} \hbox{ for each } j, \\
& & \qquad\qquad\qquad\qquad\hbox{and } \mathop{\rm diam} A_j < \delta
\hbox{ for each } j\bigg\} \nonumber \end{eqnarray} for each $E \subseteq M$. More precisely, the infimum is taken over all collections $\{A_j\}_j$ of finitely or countably many elements of $\mathcal{F}$ such that $E \subseteq \bigcup_j A_j$ and $\mathop{\rm diam} A_j < \delta$ for each $j$, if there are any. If there are no such coverings of $E$, then we put $H_\delta(E) = +\infty$. If $E = \emptyset$, then we interpret $H_\delta(E)$ as being equal to $0$, using the empty covering of $E$, and interpreting an empty sum as being $0$.
As in \cite{mat}, one can avoid these problems with two very mild additional hypotheses. The first is that for each $\delta > 0$, there be a collection $\{A_j\}_j$ of finitely or countably many elements of $\mathcal{F}$ such that $\bigcup_j A_j = M$ and $\mathop{\rm diam} A_j < \delta$ for every $j$. If $\mathcal{F}$ is the collection of all subsets of $M$, then this is equivalent to asking that $M$ be separable. This condition ensures that the coverings used in the definition of $H_\delta(E)$ always exist. The second additional hypothesis is that for each $\delta > 0$, there be an $A \in \mathcal{F}$ such that $\mathop{\rm diam} A < \delta$ and $\zeta(A) < \delta$. This implies that $H_\delta(\emptyset) = 0$, without using the empty covering. In particular, this holds when $\emptyset \in \mathcal{F}$ and $\zeta(\emptyset) = 0$, so that one can cover the empty set by itself.
Observe that \begin{equation} \label{H_delta(E) le H_delta(widetilde{E})}
H_\delta(E) \le H_\delta(\widetilde{E}) \end{equation} for every $\delta > 0$ when $E \subseteq \widetilde{E} \subseteq M$. This simply uses the fact that every covering of $\widetilde{E}$ as in (\ref{H_delta(E) = ...}) is also a covering of $E$, so that $H_\delta(E)$ is the infimum of a larger collection of sums than for $H_\delta(\widetilde{E})$. In many situations, $\zeta$ may enjoy the monotonicity property \begin{equation} \label{zeta(A) le zeta(B)}
\zeta(A) \le \zeta(B) \end{equation} for every $A, B \in \mathcal{F}$ with $A \subseteq B$, but this is not needed to get (\ref{H_delta(E) le H_delta(widetilde{E})}). One can also show that $H_\delta$ is countably subadditive for each $\delta > 0$, by standard arguments, so that $H_\delta$ is an outer measure on the $\sigma$-algebra of all subsets of $M$. As before, if $0 < \delta < \eta \le \infty$, then \begin{equation} \label{H_eta(E) le H_delta(E)}
H_\eta(E) \le H_\delta(E) \end{equation} for every $E \subseteq M$, because $H_\eta(E)$ is the infimum of a larger class of sums than for $H_\delta(E)$. If $E_1, E_2 \subseteq M$ satisfy $d(x, y) \ge \delta$ for every $x \in E_1$ and $y \in E_2$, then it is easy to see that \begin{equation} \label{H_delta(E_1) + H_delta(E_2) le H_delta(E_1 cup E_2)}
H_\delta(E_1) + H_\delta(E_2) \le H_\delta(E_1 \cup E_2), \end{equation} for the same reasons as in Section \ref{restricting the diameters}. The opposite inequality holds for any $E_1, E_2 \subseteq M$, so that equality holds in (\ref{H_delta(E_1) + H_delta(E_2) le H_delta(E_1 cup
E_2)}) under these conditions.
Put \begin{equation} \label{H(E) = sup_{delta > 0} H_delta(E)}
H(E) = \sup_{\delta > 0} H_\delta(E) \end{equation} for each $E \subseteq M$, which can also be interpreted as a limit as $\delta \to 0$, because of (\ref{H_eta(E) le H_delta(E)}). As usual, $H(\emptyset) = 0$, and \begin{equation} \label{H(E) le H(widetilde{E})}
H(E) \le H(\widetilde{E}) \end{equation} when $E \subseteq \widetilde{E} \subseteq M$, by (\ref{H_delta(E) le
H_delta(widetilde{E})}). Similarly, the countable subadditivity of $H_\delta$ for each $\delta > 0$ implies the same property for $H$, and hence that $H$ is an outer measure on the $\sigma$-algebra of all subsets of $M$. If $E_1, E_2 \subseteq M$ satisfy $d(x, y) \ge \eta$ for some $\eta > 0$ and every $x \in E_1$ and $y \in E_2$, then (\ref{H_delta(E_1) + H_delta(E_2) le H_delta(E_1 cup E_2)}) holds when $0 < \delta \le \eta$, and hence \begin{equation} \label{H(E_1) + H(E_2) le H(E_1 cup E_2)}
H(E_1) + H(E_2) \le H(E_1 \cup E_2). \end{equation} The opposite inequality holds automatically, and it follows that $H$ is countably additive on a suitable $\sigma$-algebra of measurable sets that includes the Borel sets, by Carath\'eodory's criterion.
Suppose that $E \subseteq M$ satisfies $H(E) < \infty$, and let $n \in {\bf Z}_+$ be given. As in Section \ref{hausdorff measures, section}, there is a collection $\{A_{j, n}\}_{j \in I_n}$ of finitely or countably many elements of $\mathcal{F}$ such that $E \subseteq \bigcup_{j \in I_n} A_{j, n}$, $\mathop{\rm diam} A_{j, n} \le 1/n$ for every $j \in I_n$, and \begin{equation} \label{sum_{j in I_n} zeta(A_{j, n}) < H(E) + 1/n}
\sum_{j \in I_n} \zeta(A_{j, n}) < H(E) + 1/n. \end{equation} If we put \begin{equation} \label{widetilde{E} = bigcap_{n = 1}^infty (bigcup_{j in I_n} A_{j, n}), 2}
\widetilde{E} = \bigcap_{n = 1}^\infty \Big(\bigcup_{j \in I_n} A_{j, n}\Big), \end{equation} then $E \subseteq \widetilde{E}$ and $\widetilde{E} \subseteq \bigcup_{j \in I_n} A_{j, n}$ for each $n$. This implies that $H(E) = H(\widetilde{E})$, since the first inclusion implies that (\ref{H(E)
le H(widetilde{E})}) holds, and the opposite inequality can be derived from the second inclusion and the definition of $H(\widetilde{E})$. If every element of $\mathcal{F}$ is a Borel set, then $\widetilde{E}$ is a Borel set too.
Alternatively, one might define $H_\delta'(E)$ in the same way as $H_\delta(E)$, except for replacing the requirement that $\mathop{\rm diam} A_j < \delta$ for each $j$ in (\ref{H_delta(E) = ...}) with the weaker condition that $\mathop{\rm diam} A_j \le \delta$ for each $j$. If $\delta = \infty$, then this condition on $\mathop{\rm diam} A_j$ is vacuous, so that $H'_\infty$ is analogous to Hausdorff content. It is easy to see that $H_\delta'(E)$ satisfies the analogues of (\ref{H_delta(E) le
H_delta(widetilde{E})}) and (\ref{H_eta(E) le H_delta(E)}) for each $\delta > 0$, and that $H_\delta'(E)$ is countably subadditive, for the same reasons as before. Thus $H_\delta'$ is also an outer measure on the $\sigma$-algebra of all subsets of $M$ for each $\delta > 0$. If $E_1, E_2 \subseteq M$ satisfy $d(x, y) > \delta$ for some $\delta > 0$ and every $x \in E_1$ and $y \in E_2$, then one can check that (\ref{H_delta(E_1) + H_delta(E_2) le H_delta(E_1 cup E_2)}) holds, as before. Of course, \begin{equation} \label{H_delta'(E) le H_delta(E)}
H_\delta'(E) \le H_\delta(E) \end{equation} for every $\delta > 0$ and $E \subseteq M$, because $H_\delta'(E)$ is the infimum of a larger collection of sums than for $H_\delta(E)$. Similarly, \begin{equation} \label{H_eta(E) le H_delta'(E)}
H_\eta(E) \le H_\delta'(E) \end{equation} for every $E \subseteq M$ when $0 < \delta < \eta \le +\infty$, because $H_\eta(E)$ is the infimum of a larger collection of sums than for $H_\delta'(E)$. It follows that the supremum of $H_\delta'(E)$ over $\delta > 0$ is the same as the supremum of $H_\delta(E)$ over $\delta > 0$, which is equal to $H(E)$.
If $\mathcal{F}$ is the collection of all subsets of $M$ and \begin{equation} \label{zeta(A) = (diam A)^alpha}
\zeta(A) = (\mathop{\rm diam} A)^\alpha \end{equation} for some $\alpha \ge 0$ and every $A \subseteq M$, then $H_\delta(E)$ is the same as $H^\alpha_\delta(E)$ in Section \ref{restricting the
diameters}, and $H(E)$ is the same as the $\alpha$-dimensional Hausdorff measure of $E$. We have also seen that we can take $\mathcal{F}$ to be the collection of all closed subsets of $M$, or the collection of all open subsets of $M$, when $\zeta(A)$ is as in (\ref{zeta(A) = (diam A)^alpha}), and get the same results for $H_\delta(E)$ and $H(E)$. Similarly, if $\zeta(A)$ is as in (\ref{zeta(A) = (diam A)^alpha}), then we can take $\mathcal{F}$ to be the collection of all closed subsets of $M$ and get the same result for $H_\delta'(E)$ as when $\mathcal{F}$ is the collection of all subsets of $M$, for each $\delta > 0$. However, the analogous argument for open sets does not work for $H_\delta'(E)$ when $0 < \delta < \infty$, because approximations of a set $A \subseteq M$ by open sets that contain $A$ may have diameter greater than $\delta$ when $\mathop{\rm diam} A = \delta$.
Let $\mathcal{F}$ and $\zeta$ be given as before, and let $\mathcal{A}$ be a $\sigma$-algebra of subsets of $M$ that contains $\mathcal{F}$. Suppose that $\mu$ is an outer measure on $\mathcal{A}$ such that \begin{equation} \label{mu(A) le C zeta(A)}
\mu(A) \le C \, \zeta(A) \end{equation} for some nonnegative real number $C$ and every $A \in \mathcal{F}$. If $E \in \mathcal{A}$, and if $\{A_j\}_j$ are finitely or countably many elements of $\mathcal{F}$ such that $E \subseteq \bigcup_j A_j$, then \begin{equation} \label{mu(E) le sum_j mu(A_j) le C sum_j zeta(A_j)}
\mu(E) \le \sum_j \mu(A_j) \le C \, \sum_j \zeta(A_j). \end{equation} This implies that \begin{equation} \label{mu(E) le C H_infty'(E)}
\mu(E) \le C \, H_\infty'(E) \end{equation} for every $E \in \mathcal{A}$, where $H_\infty'$ is the outer measure on $M$ corresponding to $\delta = \infty$ discussed earlier.
Let $\mathcal{F}$ and $\zeta$ be given again, and let $\widetilde{d}(x, y)$ be another metric on $M$. Also let $\widetilde{H}_\delta(E)$, $\widetilde{H}_\delta'(E)$, and $\widetilde{H}(E)$ be the analogues of $H_\delta(E)$, $H_\delta'(E)$, and $H(E)$, using $\widetilde{d}(x, y)$ to define diameters of subsets of $M$ instead of $d(x, y)$. If the identity mapping on $M$ is uniformly continuous as a mapping from $M$ equipped with $d(x, y)$ to $M$ equipped with $\widetilde{d}(x, y)$, then for each $\epsilon > 0$ there is a $\delta > 0$ such that \begin{equation} \label{widetilde{H}_epsilon(E) le H_delta(E)}
\widetilde{H}_\epsilon(E) \le H_\delta(E) \end{equation} for every $E \subseteq M$, and similarly for $\widetilde{H}_\epsilon'(E)$ and $H_\delta'(E)$. In the limit as $\epsilon \to 0$, we get that \begin{equation} \label{widetilde{H}(E) le H(E)}
\widetilde{H}(E) \le H(E) \end{equation} for every $E \subseteq M$ under these conditions. If the identity mapping on $M$ is uniformly continuous as a mapping from $M$ equipped with $\widetilde{d}(x, y)$ to $M$ equipped with $d(x, y)$, then \begin{equation} \label{H(E) le widetilde{H}(E)}
H(E) \le \widetilde{H}(E) \end{equation} for every $E \subseteq M$, for the same reasons. This implies that \begin{equation} \label{widetilde{H}(E) = H(E)}
\widetilde{H}(E) = H(E) \end{equation} for every $E \subseteq M$ when $d(x, y)$ and $\widetilde{d}(x, y)$ determine the same uniform structure on $M$. Of course, it is important here that we are using the same function $\zeta(A)$ for both metrics.
\section{Snowflakes and quasi-metrics} \label{snowflakes, quasi-metrics}
Let $(M, d(x, y))$ be a metric space, and suppose that $d(x, y)^a$ is also a metric on $M$ for some $a > 0$. As in Section \ref{snowflake metrics, quasi-metrics}, this holds when $0 < a \le 1$ and $d(x, y)$ is any metric on $M$, and for all $a > 0$ when $d(x, y)$ is an ultrametric on $M$. It is easy to see that the diameter of $A \subseteq M$ with respect to $d(x, y)^a$ is equal to $(\mathop{\rm diam} A)^a$, where $\mathop{\rm diam} A$ is the diameter of $A$ with respect to $d(x, y)$. This implies that the $\alpha$-dimensional Hausdorff content of $E \subseteq M$ with respect to $d(x, y)^a$ is equal to the $(\alpha \, a)$-dimensional Hausdorff content of $E$ with respect to $d(x, y)$, for each $\alpha \ge 0$. Similarly, the analogue of $H^\alpha_\delta(E)$ with respect to $d(x, y)^a$ corresponds to $H^{\alpha'}_{\delta'}(E)$ with respect to $d(x, y)$, where $\alpha' = \alpha \, a$ and $\delta' = \delta^{1/a}$. It follows that the $\alpha$-dimensional Hausdorff measure of $E$ with respect to $d(x, y)^a$ is the same as the $(\alpha \, a)$-dimensional Hausdorff measure of $E$ with respect to $d(x, y)$. In particular, the Hausdorff dimension of $E$ with respect to $d(x, y)^a$ is equal to the Hausdorff dimension of $E$ with respect to $d(x, y)$ divided by $a$.
As in Section \ref{snowflake metrics, quasi-metrics}, $d(x, y)^a$ is a quasi-metric on $M$ for every $a > 0$ when $d(x, y)$ is a metric on $M$, or even a quasi-metric on $M$. One could define diameters, Hausdorff measures, and so on with respect to quasi-metrics, in which case the remarks in the previous paragraph would hold for all $a > 0$. However, there are some technical problems with this, related to the continuity properties of $d(x, y)$. If $d(x, y)$ is a metric on $M$, then the diameter of a set $A \subseteq M$ is the same as the diameter of the closure of $A$, and $A$ is contained open subsets of $M$ with approximately the same diameter, as in Section \ref{diameters}. Of course, this also works for quasi-metrics on $M$ of the form $d_0(x, y)^a$ for some metric $d_0(x, y)$ on $M$ and $a > 0$, by reducing to the corresponding statements for $d_0(x, y)$.
If $d(x, y)$ is a quasi-metric on $M$ of the form $d_0(x, y)^a$ for some metric $d_0(x, y)$ on $M$ and $a > 0$, then one might as well use Hausdorff measures with respect to $d_0(x, y)$ on $M$ to get Hausdorff measures with respect to $d(x, y)$, with suitable adjustments to the dimensions, as before. Alternatively, let $d(x, y)$ be a quasi-metric on $M$, and suppose that $d_1(x, y)$ is a metric on $M$ that defines the same uniform structure on $M$. This is equivalent to saying that the identity mapping on $M$ is uniformly continuous as a mapping from $M$ equipped with $d(x, y)$ to $M$ equipped with $d_1(x, y)$, and as a mapping from $M$ equipped with $d_1(x, y)$ to $M$ equipped with $d(x, y)$, where uniform continuity can be characterized in the usual way in terms of $\epsilon$'s and $\delta$'s. One can then define Hausdorff measures on $M$ with respect to $d(x, y)$ using the construction described in the previous section, where the metric $d(x, y)$ in the previous section is taken to be $d_1(x, y)$, and where $\zeta(A)$ is defined in terms of the diameter of $A$ with respect to $d(x, y)$. If $d_2(x, y)$ is another metric on $M$ that defines the same uniform structure on $M$ as $d(x, y)$, then $d_1(x, y)$ and $d_2(x, y)$ also determine the same uniform structure on $M$, and they lead to the same measures on $M$ as before.
\section{Other Hausdorff measures} \label{other hausdorff measures}
Let $(M, d(x, y))$ be a metric space, and let $\mathcal{F}$ be the collection of all subsets of $M$. Also let $h$ be a nonnegative real-valued function on the set $[0, +\infty)$ of nonnegative real numbers, and put \begin{equation} \label{zeta(A) = h(diam A)}
\zeta(A) = h(\mathop{\rm diam} A) \end{equation} for every bounded set $A \subseteq M$. Let us interpret this as being equal to $0$ when $A = \emptyset$, which is automatic when $h(0) = 0$. One can include unbounded sets $A \subseteq M$ as well, with the convention that \begin{equation} \label{h(+infty) = sup_{t ge 0} h(t)}
h(+\infty) = \sup_{t \ge 0} h(t), \end{equation} which may be infinite. This leads to outer measures $H_\delta$ and $H_\delta'$ on $M$ for each $\delta > 0$ as in Section \ref{caratheodory's construction}, and to an outer measure $H$ on $M$, which is the Hausdorff measure\index{Hausdorff measure} associated to $h$.
Of course, this reduces to the previous situation when $h(t) = t^\alpha$ for some $\alpha \ge 0$. As usual, one can get the same results for $H_\delta$, $H_\delta'$, and $H$ by taking $\mathcal{F}$ to be the collection of all closed subsets of $M$, because of (\ref{diam
overline{A} = diam A}). If $h(t)$ is continuous from the right at each $t \ge 0$, then one can also get the same results for $H_\delta$ and hence $H$ by taking $\mathcal{F}$ to be the collection of all open subsets of $M$. If $M$ is the real line with the standard metric, then one can get the same results for $H_\delta$, $H_\delta'$, and $H$ using the collection of all closed intervals in ${\bf R}$, as in Section \ref{some special cases}. Similarly, if $d(x, y)$ is an ultrametric on any set $M$, then one can get the same results for $H_\delta$, $H_\delta'$, and $H$ using the collection of all closed balls in $M$, as in Section \ref{some special cases}.
Let $X_1, X_2, X_3, \ldots$ be a sequence of finite sets, where $X_j$ has exactly $n_j \ge 2$ elements for each $j$, and let $X$ be their Cartesian product, as in Section \ref{abstract cantor sets}. Also let $\{t_l\}_{l = 0}^\infty$ be a strictly decreasing sequence of positive real numbers that converges to $0$, and let $d(x, y)$ be the corresponding ultrametric on $X$, as in (\ref{d(x, y) = t_{l(x, y)}}). Put $\widetilde{t}_0 = 1$, and let $\widetilde{t}_l$ be defined for $l \ge 1$ by \begin{equation} \label{1/widetilde{t}_l = prod_{j = 1}^l n_j}
1/\widetilde{t}_l = \prod_{j = 1}^l n_j, \end{equation} so that $\{\widetilde{t}_l\}_{l = 0}^\infty$ is also a strictly decreasing sequence of positive real numbers that converges to $0$. If $\widetilde{d}(x, y)$ is the ultrametric on $X$ that corresponds to $\{\widetilde{t}_l\}_{l = 0}^\infty$ as in (\ref{d(x, y) = t_{l(x,
y)}}), then $\widetilde{d}(x, y)$ is the same as the ultrametric considered in Section \ref{some special cases}. Note that $d(x, y)$ and $\widetilde{d}(x, y)$ determine the same uniform structure on $X$.
Let $h$ be a nonnegative real-valued function on $[0, +\infty)$ such that $h(0) = 0$ and \begin{equation} \label{h(t_l) = widetilde{t}_l}
h(t_l) = \widetilde{t}_l \end{equation} for each $l \ge 0$. If $\mathop{\rm diam} A$ is the diameter of $A \subseteq M$ with respect to $d(x, y)$, then $h(\mathop{\rm diam} A)$ is equal to the diameter of $A$ with respect to $\widetilde{d}(x, y)$. Let $H(E)$ be the outer measure on $X$ corresponding to (\ref{zeta(A) = h(diam A)}) and the collection $\mathcal{F}$ of all subsets of $X$ as in Section \ref{caratheodory's construction}, and let $\widetilde{H}^1(E)$ be one-dimensional Hausdorff measure on $X$ with respect to $\widetilde{d}(x, y)$. It is easy to see that $H(E) = \widetilde{H}^1(E)$ for every $E \subseteq M$ under these conditions, using the remarks at the end of Section \ref{caratheodory's
construction}. Remember that $\widetilde{H}^1$ can be analyzed as in Section \ref{some special cases}.
\section{Product spaces} \label{product spaces}
Let $(M_1, d_1(x_1, y_1))$ and $(M_2, d_2(x_2, y_2))$ be metric spaces, and let $M = M_1 \times M_2$ be their Cartesian product. It is easy to see that \begin{equation} \label{d(x, y) = max(d_1(x_1, y_1), d_2(x_2, y_2))}
d(x, y) = \max(d_1(x_1, y_1), d_2(x_2, y_2)) \end{equation} defines a metric on $M$, where $x = (x_1, x_2)$, $y = (y_1, y_2)$. This metric has the nice property that the open ball in $M$ centered at a point $x = (x_1, x_2)$ and with radius $r > 0$ is equal to the Cartesian product of the open balls in $M_1$, $M_2$ centered at $x_1$, $x_2$ with radii equal to $r$. In particular, the topology on $M$ determined by (\ref{d(x, y) = max(d_1(x_1, y_1), d_2(x_2, y_2))}) is the same as the product topology associated to the topologies on $M_1$ and $M_2$ determined by the metrics $d_1(x_1, y_1)$ and $d_2(x_2, y_2)$, respectively. Alternatively, \begin{equation} \label{D_p(x, y) = (d_1(x_1, y_1)^p + d_2(x_2, y_2)^p)^{1/p}}
D_p(x, y) = (d_1(x_1, y_1)^p + d_2(x_2, y_2)^p)^{1/p} \end{equation} defines a metric on $M$ when $1 \le p < \infty$, because of the triangle inequality for $\ell^p$ norms. This is especially simple when $p = 1$, and the $p = 2$ case is very natural in the context of Euclidean geometry. Observe that \begin{equation} \label{d(x, y) le D_p(x, y) le 2^{1/p} d(x, y)}
d(x, y) \le D_p(x, y) \le 2^{1/p} \, d(x, y) \end{equation} for every $x, y \in M$ and $1 \le p < \infty$, which implies that $D_p(x, y)$ determines the same topology on $M$ as $d(x, y)$. This also implies analogous relations between diameters of subsets of $M$ with respect to these metrics, and permits one to compare Hausdorff measures on $M$ with respect to these metrics. Another nice property of (\ref{d(x, y) = max(d_1(x_1, y_1), d_2(x_2, y_2))}) is that it is an ultrametric on $M$ when $d_1(x_1, y_1)$ and $d_2(x_2, y_2)$ are ultrametrics on $M_1$ and $M_2$, respectively.
Let $p_1 : M \to M_1$ and $p_2 : M \to M_2$ be the obvious coordinate projections, so that $p_1(x) = x_1$ and $p_2(x) = x_2$ for every $x = (x_1, x_2) \in M$. If $A \subseteq M$, then \begin{equation} \label{diam A = max(diam p_1(A), diam p_2(A))}
\mathop{\rm diam} A = \max(\mathop{\rm diam} p_1(A), \mathop{\rm diam} p_2(A)), \end{equation} where $\mathop{\rm diam} A$ is the diameter of $A$ with respect to (\ref{d(x, y) =
max(d_1(x_1, y_1), d_2(x_2, y_2))}), and $\mathop{\rm diam} p_1(A)$, $\mathop{\rm diam} p_2(A)$ are the diameters of $p_1(A)$, $p_2(A)$ in $M_1$, $M_2$, respectively. It follows that the diameters of $A$ and $p_1(A) \times p_2(A)$ with respect to (\ref{d(x, y) = max(d_1(x_1, y_1), d_2(x_2,
y_2))}) on $M$ are the same. This implies that Hausdorff measures of a set $E \subseteq M$ with respect to (\ref{d(x, y) = max(d_1(x_1,
y_1), d_2(x_2, y_2))}) can be defined equivalently in terms of coverings of $E$ by products of subsets of $M_1$ and $M_2$. More precisely, one can restrict one's attention to coverings of $E$ by products of closed subsets of $M_1$ and $M_2$, because of (\ref{diam
overline{A} = diam A}).
Let $h_1$, $h_2$ be monotone increasing nonnegative real-valued functions on $[0, +\infty)$, and put \begin{equation} \label{h_j(+infty) = sup_{t ge 0} h_j(t)}
h_j(+\infty) = \sup_{t \ge 0} h_j(t) \end{equation} for $j = 1, 2$, which may be infinite. Suppose that $\mu_1$, $\mu_2$ are nonnegative Borel measures on $M_1$ and $M_2$ such that \begin{equation} \label{mu_1(A_1) le C_1 h_1(diam A_1)}
\mu_1(A_1) \le C_1 \, h_1(\mathop{\rm diam} A_1) \end{equation} and \begin{equation} \label{mu_2(A_2) le C_2 h_2(diam A_2)}
\mu_2(A_2) \le C_2 \, h_2(\mathop{\rm diam} A_2) \end{equation} for some nonnegative real numbers $C_1$, $C_2$ and all Borel sets $A_1 \subseteq M_1$ and $A_2 \subseteq M_2$. In particular, this ensures that $M_1$, $M_2$ are $\sigma$-finite with respect to $\mu_1$, $\mu_2$, so that the product measure $\mu = \mu_1 \times \mu_2$ can be defined on a suitable $\sigma$-algebra of subsets $M$. If $M_1$ and $M_2$ are separable, then $M$ is separable, which implies that open subsets of $M$ can be expressed as unions of finitely or countably many products of open subsets of $M_1$ and $M_2$. In this case, open subsets of $M$ are measurable with respect to the product measure construction, and hence Borel subsets of $M$ are measurable too.
Put \begin{equation} \label{h(t) = h_1(t) h_2(t)}
h(t) = h_1(t) \, h_2(t), \end{equation} when $0 \le t < \infty$, which is also a monotone increasing nonnegative real-valued function on $[0, +\infty)$. Note that \begin{equation} \label{sup_{t ge 0} h(t) = h_1(+infty) h_2(+infty)}
\sup_{t \ge 0} h(t) = h_1(+\infty) \, h_2(+\infty), \end{equation} with the convention that $r \cdot (+\infty) = (+\infty) \cdot r$ is equal to $+\infty$ when $r > 0$, and to $0$ when $r = 0$. Thus we take $h(+\infty)$ to be (\ref{sup_{t ge 0} h(t) = h_1(+infty)
h_2(+infty)}). If $A_1 \subseteq M_1$, $A_2 \subseteq M_2$ are Borel sets, $A \subseteq M$ is measurable with respect to the product measure construction, and $A \subseteq A_1 \times A_2$, then \begin{eqnarray} \label{mu(A) le mu(A_1 times A_2) = mu_1(A_1) mu_2(A_2) le ...}
\mu(A) \le \mu(A_1 \times A_2) & = & \mu_1(A_1) \, \mu_2(A_2) \\
& \le & C_1 \, C_2 \, h_1(\mathop{\rm diam} A_1) \, h_2(\mathop{\rm diam} A_2) \nonumber \\
& \le & C_1 \, C_2 \, h(\max(\mathop{\rm diam} A_1, \mathop{\rm diam} A_2)) \nonumber \end{eqnarray} by (\ref{mu_1(A_1) le C_1 h_1(diam A_1)}) and (\ref{mu_2(A_2) le C_2
h_2(diam A_2)}). It follows that \begin{equation} \label{mu(A) le C_1 C_2 h(diam A)}
\mu(A) \le C_1 \, C_2 \, h(\mathop{\rm diam} A), \end{equation} by taking $A_1$, $A_2$ to be the closures of $p_1(A)$, $p_2(A)$ in $M_1$, $M_2$, respectively, and using (\ref{diam A = max(diam p_1(A),
diam p_2(A))}).
\chapter{Lipschitz mappings} \label{lipschitz mappings}
\section{Basic properties} \label{basic properties}
Let $(M, d(x, y))$ and $(N, \rho(w, z))$ be metric spaces. A mapping $f : M \to N$ is said to be \emph{Lipschitz}\index{Lipschitz mappings} if there is a nonnegative real number $C$ such that \begin{equation} \label{rho(f(x), f(y)) le C d(x, y)}
\rho(f(x), f(y)) \le C \, d(x, y) \end{equation} for every $x, y \in M$. In this case, one might also say that $f$ is $C$-Lipschitz, or Lipschitz with constant $C$, to indicate the constant $C$. Of course, $f$ is Lipschitz with constant $C = 0$ if and only if $f$ is constant. Note that the composition of two Lipshitz mappings with constants $C_1$, $C_2$ is Lipschitz with constant $C_1 \, C_2$.
Suppose that $f : M \to N$ is Lipschitz with constant $C$, and that $A$ is a nonempty bounded subset of $M$. Under these conditions, $f(A)$ is a nonempty bounded set in $N$, and \begin{equation} \label{diam f(A) le C diam A}
\mathop{\rm diam} f(A) \le C \, \mathop{\rm diam} A, \end{equation} where more precisely $\mathop{\rm diam} A = \mathop{\rm diam}_M A$ is defined using the metric on $M$, and $\mathop{\rm diam} f(A) = \mathop{\rm diam}_N f(A)$ uses the metric on $N$. This also works when $A$ is unbounded, with the convention that the right side of (\ref{diam f(A) le C diam A}) is infinite when $C > 0$ and equal to $0$ when $C = 0$. It follows that \begin{equation} \label{H^alpha_{con}(f(E)) le C^alpha H^alpha_{con}(E)}
H^\alpha_{con}(f(E)) \le C^\alpha \, H^\alpha_{con}(E) \end{equation} for every $E \subseteq M$ and $\alpha \ge 0$, where $H^\alpha_{con}(E)$ is defined using the metric on $M$, and $H^\alpha_{con}(f(E))$ is defined using the metric on $N$, as before. If $\alpha = 0$, then $C^\alpha$ should be interpreted as being equal to $1$ for every $C \ge 0$.
Similarly, \begin{equation} \label{H^alpha_{C delta}(f(E)) le C^alpha H^alpha_delta(E)}
H^\alpha_{C \, \delta}(f(E)) \le C^\alpha \, H^\alpha_\delta(E) \end{equation} for every $E \subseteq M$, $\alpha \ge 0$, and $\delta > 0$, at least when $C > 0$, so that $C \, \delta > 0$. This implies that \begin{equation} \label{H^alpha(f(E)) le C^alpha H^alpha(E)}
H^\alpha(f(E)) \le C^\alpha \, H^\alpha(E) \end{equation} for every $E \subseteq M$ and $\alpha \ge 0$ when $C > 0$, which also holds trivially when $C = 0$. Indeed, if $C = 0$ and $\alpha > 0$, then $H^\alpha(f(E)) = 0$ automatically. If $\alpha = 0$, then $H^\alpha$ reduces to counting measure, and the counting measure of $f(E)$ is less than or equal to the counting measure of $E$ for any mapping $f : M \to N$ and $E \subseteq M$.
A mapping $f : M \to N$ is said to be \emph{bilipschitz}\index{bilipschitz mappings} if there is a $C \ge 1$ such that \begin{equation} \label{C^{-1} d(x, y) le rho(f(x), f(y)) le C d(x, y)}
C^{-1} \, d(x, y) \le \rho(f(x), f(y)) \le C \, d(x, y) \end{equation} for every $x, y \in M$. As before, one might say that $f$ is $C$-bilipschitz, or bilipschitz with constant $C$, to indicate the constant $C$. If $f$ is bilipschitz with constant $C$, then \begin{equation} \label{C^{-1} diam A le diam f(A) le C diam A}
C^{-1} \, \mathop{\rm diam} A \le \mathop{\rm diam} f(A) \le C \, \mathop{\rm diam} A \end{equation} for every nonempty bounded set $A \subseteq M$. This implies that \begin{equation} \label{... le H^alpha_{con}(f(E)) le C^alpha H^alpha_{con}(E)}
C^{-\alpha} \, H^\alpha_{con}(E) \le H^\alpha_{con}(f(E))
\le C^\alpha \, H^\alpha_{con}(E) \end{equation} and \begin{equation} \label{C^{-alpha} H^alpha(E) le H^alpha(f(E)) le C^alpha H^alpha(E)}
C^{-\alpha} \, H^\alpha(E) \le H^\alpha(f(E)) \le C^\alpha \, H^\alpha(E) \end{equation} for every $E \subseteq M$ and $\alpha \ge 0$. Of course, the counting measure of $E$ is equal to the counting measure of $f(E)$ for every $E \subseteq M$ when $f : M \to N$ is one-to-one.
\section{Real-valued functions} \label{real-valued functions}
Let $(M, d(x, y))$ be a metric space, and let $f$ be a real-valued function on $M$. Thus $f$ is Lipschitz with constant $C \ge 0$ with respect to the standard metric on ${\bf R}$ if and only if \begin{equation}
\label{|f(x) - f(y)| le C d(x, y)}
|f(x) - f(y)| \le C \, d(x, y) \end{equation} for every $x, y \in X$. Of course, this implies that \begin{equation} \label{f(x) le f(y) + C d(x, y)}
f(x) \le f(y) + C \, d(x, y) \end{equation} for every $x, y \in M$. Conversely, if $f$ satisfies (\ref{f(x) le
f(y) + C d(x, y)}) for every $x, y \in M$, then we also have that \begin{equation} \label{f(y) le f(x) + C d(x, y)}
f(y) \le f(x) + C \, d(x, y) \end{equation}
for every $x, y \in M$, by interchanging the roles of $x$ and $y$. It is easy to see that (\ref{|f(x) - f(y)| le C d(x, y)}) is implied by
(\ref{f(x) le f(y) + C d(x, y)}) and (\ref{f(y) le f(x) + C d(x, y)}), so that (\ref{|f(x) - f(y)| le C d(x, y)}) and (\ref{f(x) le f(y) + C
d(x, y)}) are equivalent to each other.
In particular, \begin{equation} \label{f_p(x) = d(p, x)}
f_p(x) = d(p, x) \end{equation} satisfies (\ref{f(x) le f(y) + C d(x, y)}) for every $p, x, y \in M$ with $C = 1$, by the triangle inequality. This shows that (\ref{f_p(x) = d(p, x)}) is a Lipschitz function on $M$ with constant $C = 1$ for every $p \in M$. Now let $A$ be a nonempty subset of $M$, and put \begin{equation} \label{dist(x, A) = inf {d(x, z) : z in A}}
\mathop{\rm dist}(x, A) = \inf \{d(x, z) : z \in A\} \end{equation} for every $x \in M$. Observe that \begin{equation} \label{dist(x, A) le d(x, z) le d(x, y) + d(y, z)}
\mathop{\rm dist}(x, A) \le d(x, z) \le d(x, y) + d(y, z) \end{equation} for every $x, y \in M$ and $z \in A$, which implies that \begin{equation} \label{dist(x, A) le d(x, y) + dist(y, A)}
\mathop{\rm dist}(x, A) \le d(x, y) + \mathop{\rm dist}(y, A) \end{equation} for every $x, y \in M$. Thus (\ref{dist(x, A) = inf {d(x, z) : z in
A}}) is also a Lipschitz function on $M$ with constant $C = 1$, for each nonempty set $A \subseteq M$.
Suppose now that $d(\cdot, \cdot)$ is an ultrametric on $M$. In this case, we have that \begin{equation} \label{dist(x, A) le d(x, z) le max(d(x, y), d(y, z))}
\mathop{\rm dist}(x, A) \le d(x, z) \le \max(d(x, y), d(y, z)) \end{equation} for every $x, y \in M$ and $z \in A$, which is stronger than (\ref{dist(x, A) le d(x, z) le d(x, y) + d(y, z)}). If \begin{equation} \label{d(x, y) < dist(x, A)}
d(x, y) < \mathop{\rm dist}(x, A), \end{equation} then it follows that \begin{equation} \label{dist(x, A) le d(y, z)}
\mathop{\rm dist}(x, A) \le d(y, z) \end{equation} for every $z \in A$, and hence \begin{equation} \label{dist(x, A) le dist(y, A)}
\mathop{\rm dist}(x, A) \le \mathop{\rm dist}(y, A). \end{equation} Combining (\ref{d(x, y) < dist(x, A)}) and (\ref{dist(x, A) le dist(y,
A)}), we get that \begin{equation} \label{d(x, y) < dist(y, A)}
d(x, y) < \mathop{\rm dist}(y, A), \end{equation} so that \begin{equation} \label{dist(y, A) le dist(x, A)}
\mathop{\rm dist}(y, A) \le \mathop{\rm dist}(x, A), \end{equation} by the same argument. This shows that \begin{equation} \label{dist(x, A) = dist(y, A)}
\mathop{\rm dist}(x, A) = \mathop{\rm dist}(y, A) \end{equation} when $x, y \in M$ satisfy (\ref{d(x, y) < dist(x, A)}).
Let $d(x, y)$ be any metric on $M$ again, and let $E$ be a connected subset of $M$. If $p, q \in E$ and $f_p(x)$ is as in (\ref{f_p(x) = d(p, x)}), then $f_p(E)$ is a connected subset of ${\bf
R}$ that contains $0$ and $d(p, q)$, and hence contains $[0, d(p,
q)]$. This implies that \begin{equation} \label{d(p, q) le H^1(f_p(E)) le H^1(E)}
d(p, q) \le H^1(f_p(E)) \le H^1(E) \end{equation} for every $p, q \in E$, so that \begin{equation} \label{diam E le H^1(E)}
\mathop{\rm diam} E \le H^1(E). \end{equation}
\section{Some examples} \label{some examples}
Let $n_1, n_2, n_3, \ldots$ be a sequence of integers with $n_j \ge 2$ for each $j$, and put \begin{equation} \label{X_j = {0, 1, ldots, n_j - 1}}
X_j = \{0, 1, \ldots, n_j - 1\} \end{equation} for each $j \in {\bf Z}_+$. Thus $X_j$ has exactly $n_j$ elements for each $j$, and we let $X = \prod_{j = 1}^\infty X_j$ be their Cartesian product, as in Section \ref{abstract cantor sets}. Also put $N_k = \prod_{j = 1}^k n_j$ for each $k \in {\bf Z}_+$ and $N_0 = 1$, and $t_l = 1/N_l$ for every $l \ge 0$. This leads to an ultrametric $d(x, y)$ on $X$ as in (\ref{d(x, y) = t_{l(x, y)}}), for which the corresponding one-dimensional Hausdorff measure was discussed in Section \ref{some special cases}.
Observe that \begin{equation} \label{N_{j - 1}^{-1} - N_j^{-1} = n_j N_j^{-1} - N_j^{-1} = (n_j - 1) N_j^{-1}}
N_{j - 1}^{-1} - N_j^{-1} = n_j \, N_j^{-1} - N_j^{-1} = (n_j - 1) \, N_j^{-1} \end{equation} for each $j \in {\bf Z}_+$, and hence \begin{equation} \label{sum_{j = k}^l (n_j - 1) N_j^{-1} = N_{k - 1}^{-1} - N_l^{-1}}
\sum_{j = k}^l (n_j - 1) \, N_j^{-1} = N_{k - 1}^{-1} - N_l^{-1} \end{equation} when $1 \le k \le l$. Put \begin{equation} \label{f_k(x) = sum_{j = 1}^k x_j N_j^{-1}}
f_k(x) = \sum_{j = 1}^k x_j \, N_j^{-1} \end{equation} for each $x \in X$ and $k \in {\bf Z}_+$, and $f_0(x) = 0$. Thus $f_k(x)$ is an integer multiple of $N_k^{-1}$ for each $x \in X$ and $k \ge 0$, and \begin{equation} \label{0 le f_k(x) le 1 - N_k^{-1} < 1}
0 \le f_k(x) \le 1 - N_k^{-1} < 1, \end{equation} by (\ref{sum_{j = k}^l (n_j - 1) N_j^{-1} = N_{k - 1}^{-1} -
N_l^{-1}}). One can check that every nonnegative integer multiple of $N_k^{-1}$ strictly less than $1$ can be expressed as $f_k(x)$ for some $x \in X$, using induction on $k$.
If $x, y \in X$ satisfy $x_j \le y_j$ for $j = 1, \ldots, k$, then \begin{equation} \label{f_k(x) le f_k(y)}
f_k(x) \le f_k(y). \end{equation} If $x_j = y_j$ when $j \le k$ and $k < l$, then \begin{equation} \label{f_l(y) le ... le f_k(x) + N_k^{-1} - N_l^{-1}}
f_l(y) \le f_k(x) + \sum_{j = k + 1}^l (n_j - 1) \, N_j^{-1}
\le f_k(x) + N_k^{-1} - N_l^{-1}, \end{equation} by (\ref{sum_{j = k}^l (n_j - 1) N_j^{-1} = N_{k - 1}^{-1} - N_l^{-1}}). Applying this to $y = x$, we get that \begin{equation} \label{f_l(x) le f_k(x) + N_k^{-1} - N_l^{-1}}
f_l(x) \le f_k(x) + N_k^{-1} - N_l^{-1} \end{equation} when $k < l$. If $x_j = y_j$ when $j \le k$ and $x_{k + 1} < y_{k + 1}$, then \begin{equation} \label{f_{k + 1}(x) + N_{k + 1}^{-1} le f_{k + 1}(y)}
f_{k + 1}(x) + N_{k + 1}^{-1} \le f_{k + 1}(y). \end{equation} This implies that \begin{equation} \label{f_l(x) + N_l^{-1} le f_{k + 1}(y) le f_l(y)}
f_l(x) + N_l^{-1} \le f_{k + 1}(y) \le f_l(y) \end{equation} for every $l \ge k + 1$, because of (\ref{f_l(x) le f_k(x) + N_k^{-1}
- N_l^{-1}}) applied to $k + 1$ instead of $k$. In particular, \begin{equation} \label{f_l(x) < f_l(y)}
f_l(x) < f_l(y) \end{equation} for each $l \ge k + 1$ under these conditions. It follows that \begin{equation} \label{f_l(x) ne f_l(y)}
f_l(x) \ne f_l(y) \end{equation}
when $x_j \ne y_j$ for some $j \le l$, by considering the smallest
such $j$.
Taking the limit as $l \to \infty$ in (\ref{sum_{j = k}^l (n_j - 1) N_j^{-1} = N_{k - 1}^{-1} - N_l^{-1}}), we get that \begin{equation} \label{sum_{j = k}^infty (n_j - 1) N_j^{-1} = N_{k - 1}^{-1}}
\sum_{j = k}^\infty (n_j - 1) \, N_j^{-1} = N_{k - 1}^{-1} \end{equation} for each $k \in {\bf Z}_+$, which is equal to $1$ when $k = 1$. Put \begin{equation} \label{f(x) = sum_{j = 1}^infty x_j N_j^{-1}}
f(x) = \sum_{j = 1}^\infty x_j \, N_j^{-1} \end{equation} for each $x \in X$, where the series converges by comparison with (\ref{sum_{j = k}^infty (n_j - 1) N_j^{-1} = N_{k - 1}^{-1}}). Thus \begin{equation} \label{0 le f(x) le 1}
0 \le f(x) \le 1 \end{equation} for every $x \in X$ and $k \in {\bf Z}_+$, and \begin{equation} \label{f(x) le f(y)}
f(x) \le f(y) \end{equation} when $x, y \in X$ satisfy $x_j \le y_j$ for each $j$. If $x_j = y_j$ when $j \le k$ for some $k \ge 0$, and $x_{k + 1} < y_{k + 1}$, then we also have (\ref{f(x) le f(y)}), by taking the limit as $l \to \infty$ in (\ref{f_l(x) + N_l^{-1} le f_{k + 1}(y) le f_l(y)}). In this case, the only way that equality can hold in (\ref{f(x) le f(y)}) is if \begin{equation} \label{y_{k + 1} = x_{k + 1} + 1, and x_l = n_l - 1, y_l = 0 for l ge k + 2}
y_{k + 1} = x_{k + 1} + 1, \hbox{ and } x_l = n_l - 1, \, y_l = 0
\hbox{ for each } l \ge k + 2. \end{equation} If $x \ne y$, then $x_j \ne y_j$ for some $j$, and we can choose $k \ge 0$ as small as possible so that $x_{k + 1} < y_{k + 1}$. It follows that $f(x) = f(y)$ only when $x = y$, or when there is a $k \ge 0$ such that $x_j = y_j$ for $j \le k$, and (\ref{y_{k + 1} = x_{k
+ 1} + 1, and x_l = n_l - 1, y_l = 0 for l ge k + 2}) holds.
Suppose again that $x, y \in X$ satisfy $x_j = y_j$ when $j \le k$ for some $k \ge 0$. Of course, $f_k(x) = f_k(y) \le f(y)$, and hence \begin{equation} \label{f_k(x) le f(y) le f_k(x) + N_k^{-1}}
f_k(x) \le f(y) \le f_k(x) + N_k^{-1}, \end{equation} by taking the limit as $l \to \infty$ in (\ref{f_l(y) le ... le f_k(x)
+ N_k^{-1} - N_l^{-1}}). In particular, \begin{equation} \label{f_k(x) le f(x) le f_k(x) + N_k^{-1}}
f_k(x) \le f(x) \le f_k(x) + N_k^{-1}, \end{equation} which implies that \begin{equation}
\label{|f(x) - f(y)| le N_k^{-1}}
|f(x) - f(y)| \le N_k^{-1} \end{equation} under these conditions. This shows that $f$ is Lipschitz with constant $C = 1$ as a mapping from $X$ into ${\bf R}$, with respect to the ultrametric $d(x, y)$ on $X$ described at the beginning of the section, and the standard metric on ${\bf R}$.
Let $x \in X$ and $k \ge 0$ be given, and let $B_k(x)$ be the set of $y \in X$ such that $x_j = y_j$ when $j \le k$, as in Section \ref{abstract cantor sets}. Thus \begin{equation} \label{f(B_k(x)) subseteq [f_k(x), f_k(x) + N_k^{-1}]}
f(B_k(x)) \subseteq [f_k(x), f_k(x) + N_k^{-1}], \end{equation} by (\ref{f_k(x) le f(y) le f_k(x) + N_k^{-1}}). One can check that \begin{equation} \label{f(B_k(x)) = [f_k(x), f_k(x) + N_k^{-1}]}
f(B_k(x)) = [f_k(x), f_k(x) + N_k^{-1}] \end{equation} for every $x \in X$ and $k \ge 0$, by standard arguments. In particular, \begin{equation} \label{f(X) = [0, 1]}
f(X) = [0, 1], \end{equation} which is the same as (\ref{f(B_k(x)) = [f_k(x), f_k(x) + N_k^{-1}]}) when $k = 0$.
Note that \begin{equation} \label{H^1(B_k(x)) = N_k^{-1}}
H^1(B_k(x)) = N_k^{-1} \end{equation} for every $x \in X$ and $k \ge 0$, where $H^1(B_k(x))$ is the one-dimensional Hausdorff measure of $B_k(x)$ with respect to the ultrametric $d(x, y)$ on $X$ mentioned earlier. This follows from the discussion at the end of Section \ref{some special cases}. In particular, \begin{equation} \label{H^1(X) = 1}
H^1(X) = 1. \end{equation} This is also consistent with the discussion of Hausdorff measure and Lipschitz mappings in Section \ref{basic properties}, since the one-dimensional Hausdorff measure of an interval in the real line is the same as the length of the interval.
\section{Other Lipschitz conditions} \label{other lipschitz conditions}
Let $(M, d(x, y))$ and $(N, \rho(w, z))$ be metric spaces, and let $a$ be a positive real number. A mapping $f : M \to N$ is said to be \emph{Lipschitz of order $a$}\index{Lipschitz mappings} if there is a nonnegative real number $C$ such that \begin{equation} \label{rho(f(x), f(y)) le C d(x, y)^a}
\rho(f(x), f(y)) \le C \, d(x, y)^a \end{equation} for every $x, y \in M$. As before, this condition holds with $C = 0$ if and only if $f$ is a constant mapping. If $a = 1$, then this condition is equivalent to the one discussed in Section \ref{basic
properties}.
If $a \le 1$, then $d(x, y)^a$ is also a metric on $M$, as in Section \ref{snowflake metrics, quasi-metrics}. In this case, the condition described in the previous paragraph is equivalent to saying that $f$ is Lipschitz of order $1$ with respect to the metric $d(x, y)^a$ on $M$, and with the same constant $C$. Similarly, if $d(x, y)$ is an ultrametric on $M$, then $d(x, y)^a$ is also an ultrametric on $M$ for every $a > 0$, and the condition in the previous paragraph is equivalent to saying that $f$ is Lipschitz of order $1$ with respect to $d(x, y)^a$ on $M$. Otherwise, $d(x, y)^a$ is a quasi-metric on $M$ for every $a > 0$, as in Section \ref{snowflake metrics,
quasi-metrics}. One can define Lipschitz conditions with respect to quasi-metrics in the same way as for metrics, so that a mapping $f : M \to N$ is Lipschitz of order $a > 0$ with respect to $d(x, y)$ on $M$ if and only if it is Lipschitz of order $1$ with respect to $d(x, y)^a$ on $M$.
There are always a lot of real-valued Lipschitz functions of order $1$ on any metric space $(M, d(x, y))$, as in Section \ref{real-valued functions}. If $0 < a \le 1$, then $d(x, y)^a$ is also a metric on $M$, and the same discussion can be applied to get a lot of real-valued Lipschitz functions of order $1$ on $M$ with respect to $d(x, y)^a$, which are the same as real-valued Lipschitz functions of order $a$ on $M$. Of course, the property of being Lipschitz of order $a$ becomes stronger on bounded sets as $a$ increases, and bounded Lipschitz functions of order $1$ are also Lipschitz functions of order $a$ when $0 < a \le 1$. If $M$ is the real line with the standard metric, then the only Lipschitz functions of order $a > 1$ are constant, because the derivative of such a function must be equal to $0$ at every point. Equivalently, the only Lipschitz functions of order $1$ on ${\bf R}$
with respect to the quasi-metric $|x - y|^a$ are the constant functions when $a > 1$. If $d(x, y)$ is any quasi-metric on a set $M$, then there is a metric $\widetilde{d}(x, y)$ on $M$ and a positive real number $a$ such that $d(x, y)$ is comparable to $\widetilde{d}(x, y)^a$, as shown in \cite{m-s-1} and recalled in Section \ref{snowflake metrics, quasi-metrics}. This implies that there are a lot of real-valued Lipschitz functions of order $1$ on $M$ with respect to $\widetilde{d}(x, y)$, which are Lipschitz of order $1/a$ with respect to $d(x, y)$.
Suppose that $f : M \to N$ is Lipschitz of some order $a > 0$ with constant $C$, as in (\ref{rho(f(x), f(y)) le C d(x, y)^a}). If $A$ is a nonempty bounded subset of $M$, then $f(A)$ is a nonempty bounded set in $N$, and \begin{equation} \label{diam f(A) le C (diam A)^a}
\mathop{\rm diam} f(A) \le C \, (\mathop{\rm diam} A)^a. \end{equation} More precisely, $\mathop{\rm diam} A$ is the diameter of $A$ with respect to the metric on $M$, and $\mathop{\rm diam} f(A)$ is the diameter of $f(A)$ with respect to the metric on $N$. This implies that \begin{equation} \label{H^alpha(f(E)) le C^alpha H^{alpha a}(E)}
H^\alpha(f(E)) \le C^\alpha \, H^{\alpha \, a}(E) \end{equation} for every $E \subseteq M$ and $\alpha \ge 0$, as in Section \ref{basic
properties}.
\section{Subadditive functions} \label{subadditive functions}
Let $\sigma(t)$ be a monotone increasing real-valued function on the set $[0, +\infty)$ of nonnegative real numbers such that
$\sigma(0) = 0$, $\sigma(t) > 0$ when $t > 0$, and \begin{equation} \label{lim_{t to 0+} sigma(t) = 0}
\lim_{t \to 0+} \sigma(t) = 0. \end{equation} If $d(x, y)$ is an ultrametric on a set $M$, then $\sigma(d(x, y))$ is also an ultrametric on $M$, which determines the same topology on $M$ as $d(x, y)$. Of course, this includes the case where $\sigma(t) = t^a$ for some $a > 0$, as in Section \ref{snowflake metrics,
quasi-metrics}. This is also related to the examples discussed in Section \ref{abstract cantor sets}. If, in addition to the conditions just mentioned, $\sigma(t)$ satisfies \begin{equation} \label{sigma(r + t) le sigma(r) + sigma(t)}
\sigma(r + t) \le \sigma(r) + \sigma(t) \end{equation} for every $r, t \ge 0$, then $\sigma(t)$ is said to be \emph{subadditive}.\index{subadditive functions} Remember that $\sigma(t) = t^a$ is subadditive when $0 < a \le 1$, as in Section \ref{snowflake metrics, quasi-metrics}. If $\sigma(t)$ is subadditive and $d(x, y)$ is a metric on $M$, then $\sigma(d(x, y))$ is also a metric on $M$, which determines the same topology on $M$ as $d(x, y)$.
In both cases, the identity mapping on $M$ is uniformly continuous as a mapping from $M$ equipped with $d(x, y)$ to $M$ equipped with $\sigma(d(x, y))$, because of (\ref{lim_{t to 0+} sigma(t) = 0}). Similarly, the identity mapping on $M$ is uniformly continuous as a mapping from $M$ equipped with $\sigma(d(x, y))$ to $M$ equipped with $d(x, y)$. More precisely, let $\epsilon > 0$ be given, and put \begin{equation} \label{delta = sigma(epsilon)}
\delta = \sigma(\epsilon) > 0. \end{equation} Thus $\sigma(t) \ge \delta$ when $t \ge \epsilon$, because $\sigma(t)$ is monotone increasing. Equivalently, this means that $t < \epsilon$ when $\sigma(t) < \delta$, which is exactly what we wanted.
If $\sigma(t)$ is subadditive on $[0, +\infty)$, then \begin{equation} \label{0 le sigma(r + t) - sigma(r) le sigma(t)}
0 \le \sigma(r + t) - \sigma(r) \le \sigma(t) \end{equation} for every $r, t \ge 0$, since $\sigma(\cdot)$ is also supposed to be monotone increasing on $[0, +\infty)$. This implies that
$\sigma(\cdot)$ is uniformly continuous on $[0, +\infty)$, using (\ref{lim_{t to 0+} sigma(t) = 0}). Alternatively, it follows from (\ref{0 le sigma(r + t) - sigma(r) le sigma(t)}) that $\sigma$ is Lipschitz of order $1$ with constant $C = 1$ as a mapping from $[0, +\infty)$
equipped with the metric $\sigma(|x - y|)$ into the real line with the standard metric.
Put \begin{equation} \label{sigma(t-) = lim_{r to t-} sigma(r) = sup {sigma(r) : 0 le r < t}}
\sigma(t-) = \lim_{r \to t-} \sigma(r) = \sup \{\sigma(r) : 0 \le r < t\} \end{equation} for each positive real number $t$, so that $\sigma(t-) \le \sigma(t)$ for each $t > 0$, and $\sigma(t-)$ is monotone increasing in $t$. If the diameter of a set $A \subseteq M$ with respect to $d(x, y)$ is equal to $t$, $0 < t < \infty$, then the diameter $T$ of $A$ with respect to $\sigma(d(x, y))$ satisfies \begin{equation} \label{sigma(t-) le T le sigma(t)}
\sigma(t-) \le T \le \sigma(t). \end{equation} In particular, \begin{equation} \label{T = sigma(t)}
T = \sigma(t) \end{equation} when $\sigma(t-) = \sigma(t)$, which holds automatically when $\sigma$ is subadditive, as in the previous paragraph. Of course, if the diameter of $A$ with respect to $d(x, y)$ is equal to $0$, then the diameter of $A$ with respect to $\sigma(d(x, y))$ is equal to $0$ too. If $A$ is unbounded with respect to $d(x, y)$, then the diameter of $A$ with respect to $\sigma(d(x, y))$ is equal to \begin{equation} \label{sigma(+infty) = sup_{r ge 0} sigma(r)}
\sigma(+\infty) = \sup_{r \ge 0} \sigma(r), \end{equation} which is either a positive real number or $+\infty$.
\section{Moduli of continuity} \label{moduli of continuity}
Let $(M, d(x, y))$ and $(N, \rho(w, z))$ be metric spaces, and let $\sigma(t)$ be a monotone increasing nonnegative real-valued function on $[0, +\infty)$ such that $\sigma(0) = 0$ and $\sigma(t)$ is continuous at $0$. Suppose that $f : M \to N$ satisfies \begin{equation} \label{rho(f(x), f(y)) le sigma(d(x, y))}
\rho(f(x), f(y)) \le \sigma(d(x, y)) \end{equation} for every $x, y \in M$, which implies that $f$ is uniformly continuous in particular. This includes the Lipschitz condition (\ref{rho(f(x),
f(y)) le C d(x, y)^a}) as a special case, with $\sigma(t) = C \, t^\alpha$. If $\sigma(d(x, y))$ is a metric on $M$, as in the previous section, then (\ref{rho(f(x), f(y)) le sigma(d(x, y))}) is the same as saying that $f$ is Lipschitz of order $1$ with constant $C = 1$ as a mapping from $M$ equipped with the metric $\sigma(d(x, y))$ into $N$ equipped with the metric $\rho(w, z)$.
If $A$ is a nonempty bounded set in $M$, and $f : M \to N$ satisfies (\ref{rho(f(x), f(y)) le sigma(d(x, y))}), then \begin{equation} \label{diam f(A) le sigma(diam A)}
\mathop{\rm diam} f(A) \le \sigma(\mathop{\rm diam} A) \end{equation} for every nonempty bounded set $A \subseteq M$. Here $\mathop{\rm diam} A$ is the diameter of $A$ with respect to $d(x, y)$ on $M$, and $\mathop{\rm diam} f(A)$ is the diameter of $f(A)$ with respect to $\rho(w, z)$ on $N$, as usual. This also works when $A$ is unbounded, with $\sigma(+\infty)$ defined as in (\ref{sigma(+infty) = sup_{r ge 0} sigma(r)}). Using (\ref{diam
f(A) le sigma(diam A)}), one can estimate Hausdorff measures of $f(A)$ in terms of Hausdorff measures of $A$, where the Hausdorff measures are defined in terms of functions of diameters of sets, as in Section \ref{caratheodory's construction}.
If $f$ is any mapping from $M$ into $N$, then put \begin{equation} \label{sigma_f(t) = sup {rho(f(x), f(y)) : x, y in M, d(x, y) le t}}
\sigma_f(t) = \sup \{\rho(f(x), f(y)) : x, y \in M, \, d(x, y) \le t\} \end{equation} for each nonnegative real number $t$, where the supremum may be equal to $+\infty$. Thus $\sigma_f(t) \ge 0$ for every $t \ge 0$, $\sigma_f(0) = 0$, $\sigma_f(t)$ is monotone increasing, and (\ref{rho(f(x), f(y)) le sigma(d(x, y))}) holds with $\sigma(t) = \sigma_f(t)$ for every $x, y \in M$, by construction. Note that $f$ is uniformly continuous if and only if $\sigma_f(t) < +\infty$ when $t$ is sufficiently small, and $\lim_{t \to 0+} \sigma_f(t) = 0$. The finiteness of $\sigma_f(t)$ for every $t > 0$ is another matter, and is trivial when $f(M)$ is bounded in $N$. However, in order to estimate Hausdorff measures, it suffices to have a condition like (\ref{diam f(A) le sigma(diam A)}) when the diameter of $A$ is small.
Suppose that $f : {\bf R} \to {\bf R}$ satisfies (\ref{rho(f(x), f(y)) le sigma(d(x, y))}), where $d(x, y)$ and $\rho(w, z)$ are both equal to the standard metric on ${\bf R}$, and $\lim_{t \to 0+} \sigma(t)/t = 0$. This implies that the derivative of $f$ is equal to $0$ everywhere on ${\bf R}$, and hence that $f$ is constant on ${\bf R}$. This includes the case where $f$ is Lipschitz of order $\alpha > 1$, as in Section \ref{other lipschitz conditions}. One can check that the analogous statement also holds when $\liminf_{t
\to 0+} \sigma(t)/ t = 0$. If $M$ and $N$ are arbitrary metric spaces, $f : M \to N$ satisfies (\ref{rho(f(x), f(y)) le sigma(d(x,
y))}), and $\sigma(t) = 0$ for some $t > 0$, then $f$ is locally constant on $M$, and in particular $f$ is constant on $M$ when $M$ is connected.
\section{Isometries and similarities} \label{isometries, similarities}
Let $(M, d(x, y))$ and $(N, \rho(w, z))$ be metric spaces. A mapping $f : M \to N$ is said to be an \emph{isometry}\index{isometries} if \begin{equation} \label{rho(f(x), f(y)) = d(x, y)}
\rho(f(x), f(y)) = d(x, y) \end{equation} for every $x, y \in M$. Equivalently, $f$ is an isometry if it is a bilipschitz mapping with constant $C = 1$. Let us say that $f : M \to N$ is a \emph{similarity}\index{similarities} if there is a positive real number $\lambda$ such that \begin{equation} \label{rho(f(x), f(y)) = lambda d(x, y)}
\rho(f(x), f(y)) = \lambda \, d(x, y) \end{equation} for every $x, y \in M$. In this case, it is easy to see that \begin{equation} \label{diam f(A) = lambda diam A}
\mathop{\rm diam} f(A) = \lambda \, \mathop{\rm diam} A \end{equation} for every nonempty bounded set $A \subseteq M$, and hence that \begin{equation} \label{H^alpha(f(E)) = lambda^alpha H^alpha(E)}
H^\alpha(f(E)) = \lambda^\alpha \, H^\alpha(E) \end{equation} for every $E \subseteq M$ and $\alpha \ge 0$.
Remember that a mapping $f : M \to N$ is said to be \emph{bounded} if $f(M)$ is a bounded set in $N$. The space of bounded continuous mappings from $M$ into $N$ is denoted $C_b(M, N)$, and the supremum metric on $C_b(M, N)$ is defined by \begin{equation} \label{sup {rho(f(x), g(x)) : x in M}}
\sup \{\rho(f(x), g(x)) : x \in M\}. \end{equation} Note that the collection of $f \in C_b(M, N)$ such that $f(M)$ is dense in $N$ is a closed set in $C_b(M, N)$ with respect to the supremum metric. If $M$ is compact, then it follows that the collection of $f \in C_b(M, N)$ such that $f(M) = N$ is a closed set in $C_b(M, N)$ with respect to the supremum metric.
Let $\mathcal{I}(M, N)$ be the collection of isometric embeddings of $M$ into $N$. If $M$ is bounded, then $\mathcal{I}(M, N) \subseteq C_b(M, N)$, and $\mathcal{I}(M, N)$ is a closed set in $C_b(M, N)$ with respect to the supremum metric. If $M$ is complete and $f : M \to N$ is an isometry, then $f(M)$ is a closed set in $N$. In particular, $f(M) = N$ when $M$ is complete, $f : M \to N$ is an isometry, and $f(M)$ is dense in $N$. If $M$ and $N$ are compact, then $\mathcal{I}(M, N)$ is a compact set in $C_b(M, N)$ with respect to the supremum metric, by standard Arzela--Ascoli arguments.
Let $\mathcal{I}(M)$ be the collection of isometric mappings of $M$ onto itself, which is a group with respect to composition. If $M$ is bounded, then the restriction of the supremum metric to $\mathcal{I}(M)$ is invariant under left and right translations, and one can check that $\mathcal{I}(M)$ is a topological group with respect to the topology determined by the supremum metric. If $M$ is complete, then $\mathcal{I}(M)$ is the same as the collection of isometric mappings $f$ from $M$ into itself such that $f(M)$ is dense in $M$, as in the previous paragraph. If $M$ is bounded and complete, then it follows that $\mathcal{I}(M)$ is a closed subset of $\mathcal{I}(M, M)$ with respect to the supremum metric, and hence is a closed subset of $C_b(M, M)$. If $M$ is compact, then $\mathcal{I}(M)$ is also compact, with respect to the topology determined by the supremum metric, because $\mathcal{I}(M, M)$ is compact.
In fact, if $M$ is compact and $f$ is an isometry of $M$ into itself, then $f(M) = M$. To see this, suppose for the sake of a contradiction that there is an element $x_1$ of $M$ not in $f(M)$. Because $M$ is compact, $f(M)$ is compact, and hence there is an $r > 0$ such that \begin{equation} \label{d(x_1, f(y)) ge r}
d(x_1, f(y)) \ge r \end{equation} for every $y \in M$. If $\{x_j\}_{j = 1}^\infty$ is the sequence of elements of $M$ defined recursively by $x_{j + 1} = f(x_j)$ for each $j \in {\bf Z}_+$, then one can check that \begin{equation} \label{d(x_j, x_k) ge r}
d(x_j, x_k) \ge r \end{equation} when $j < k$, using (\ref{d(x_1, f(y)) ge r}) and the hypothesis that $f$ be an isometry. This implies that $\{x_j\}_{j = 1}^\infty$ has no convergent subsequences, contradicting the compactness of $M$, as desired.
As a variant of this, suppose that $f$ and $g$ are similarities from $M$ into $N$, with the same constant $\lambda$. If $g(M) = N$, then $f \circ g^{-1}$ is an isometry from $M$ into itself. If $M$ is compact, then $f \circ g^{-1}$ maps $M$ onto itself, as in the previous paragraph. Thus $f(M) = N$ under these conditions.
Suppose now that $d(x, y)$ is an ultrametric on $M$, and let $r > 0$ be given. Also let $\sim_r$ be the relation on $M$ defined by $x \sim_r y$ when $d(x, y) \le r$. This is an equivalence relation on $M$, because $d(x, y)$ is an ultrametric on $M$. The corresponding equivalence classes are closed balls of radius $r$ in $M$. If $f$ is an isometry of $M$ into itself, then $f(x) \sim_r f(y)$ if and only if $x \sim_r y$ for every $x, y \in M$. This implies that $f$ maps each equivalence class of $M$ with respect to $\sim_r$ into another equivalence class, which is the same as saying that $f$ maps each closed ball in $M$ with radius $r$ into another closed ball of radius $r$. More precisely, $f$ maps distinct equivalence classes in $M$ with respect to $\sim_r$ into distinct equivalence classes in $M$, which is the same as saying that $f$ maps disjoint closed balls in $M$ with radius $r$ into disjoint closed balls with radius $r$.
If $M$ is totally bounded, then there are only finitely many equivalence classes in $M$ with respect to $\sim_r$ for each $r > 0$. In this case, it follows that every such equivalence class contains an element of $f(M)$. This means that $f(M)$ is dense in $M$, since this holds for each $r > 0$. If $M$ is complete, then we get that $f(M) = M$. Of course, $M$ is compact when $M$ is complete and totally bounded.
Let $d(x, y)$ be an arbitrary metric on $M$ again, and suppose that $M$ is totally bounded. Let $r$ be a positive real number, and let $n(r)$ be the smallest number of subsets of $M$ with diameter less than or equal to $r$ needed to cover $M$. If $E$ is any subset of $M$ which is not dense in $M$, then $E$ can be covered by fewer than $n(r)$ subsets of $M$ with diameter less than or equal to $r$ when $r$ is sufficiently small, because at least one of the sets used to cover $M$ will not intersect $E$. If $f$ is an isometry of $M$ into itself, then the minimal number of sets of diameter less than or equal to $r$ needed to cover $f(M)$ is the same as $n(r)$. This implies that $f(M)$ is dense in $M$ when $M$ is totally bounded, and hence that $f(M) = M$ when $M$ is also complete and thus compact.
Suppose that $H$ is a Hausdorff measure on $M$, defined in terms of some function of the diameter of subsets of $M$. Thus $H(f(M)) = H(M)$ when $f$ is an isometry of $M$ into itself. If $H(M) < +\infty$ and nonempty open subsets of $M$ have positive measure with respect to $H$, then it follows that $f(M)$ is dense in $M$, so that $f(M) = M$ when $M$ is compact. One can also show that $f(M) = M$ when $f$ is an isometry from $M$ into itself and $M$ is compact using compactness of $\mathcal{I}(M, M)$. The covering argument in the preceding paragraph and the earlier approach using sequential compactness were suggested by students in a class, and some instances of this type of situation will be discussed in the next chapter.
\chapter{Functions on ${\bf Q}_p$} \label{functions on Q_p}
\section{Polynomials on ${\bf Q}_p$} \label{polynomials on Q_p}
Let $p$ be a prime number, and let \begin{equation} \label{f(x) = a_n x^n + a_{n - 1} x^{n - 1} + cdots + a_1 x + a_0}
f(x) = a_n \, x^n + a_{n - 1} \, x^{n - 1} + \cdots + a_1 \, x + a_0 \end{equation} be a polynomial with coefficients in ${\bf Q}_p$. Of course, \begin{equation} \label{(x + h)^k = sum_{j = 0}^k {k choose j} h^j x^{k - j}}
(x + h)^k = \sum_{j = 0}^k {k \choose j} \, h^j \, x^{k - j} \end{equation} for every nonnegative integer $k$ and $x, h \in {\bf Q}_p$, where ${k
\choose j}$ is the usual binomial coefficient. Thus \begin{equation} \label{f(x + h) = sum_{k = 0}^n a_k (x + h)^k = ...}
f(x + h) = \sum_{k = 0}^n a_k \, (x + h)^k
= \sum_{k = 0}^n \sum_{j = 0}^k a_k \, {k \choose j} \, h^j \, x^{k - j} \end{equation} for every $x, h \in {\bf Q}_p$. This implies that \begin{equation} \label{f(x + h) - f(x) = ...}
f(x + h) - f(x) = \sum_{k = 1}^n \sum_{j = 1}^k a_k \, {k \choose j} \,
h^j \, x^{k - j}, \end{equation} by subtracting the $j = 0$ terms from (\ref{f(x + h) = sum_{k = 0}^n
a_k (x + h)^k = ...}), and using the simple fact that ${k \choose 0} = 1$ for each $k$.
The formal derivative of $f(x)$ is the polynomial defined by \begin{equation} \label{f'(x) = n a_n x^{n - 1} + (n - 1) a_{n - 1} x^{n - 2} + cdots + a_1}
f'(x) = n \, a_n \, x^{n - 1} + (n - 1) \, a_{n - 1} \, x^{n - 2} + \cdots + a_1. \end{equation} Subtracting the $j = 1$ terms from (\ref{f(x + h) - f(x) = ...}), we get that \begin{equation} \label{f(x + h) - f(x) - f'(x) h = ...}
f(x + h) - f(x) - f'(x) \, h = \sum_{k = 2}^n \sum_{j = 2}^k a_k \,
{k \choose j} \, h^j \, x^{k - j} \end{equation} for every $x, h \in {\bf Q}_p$, because ${k \choose 1} = k$. In particular, \begin{equation} \label{lim_{h to 0} frac{f(x + h) - f(x)}{h} = f'(x)}
\lim_{h \to 0} \frac{f(x + h) - f(x)}{h} = f'(x) \end{equation} for every $x \in {\bf Q}_p$, since each term on the right side of (\ref{f(x + h) - f(x) - f'(x) h = ...}) is a multiple of $h^2$.
It follows from (\ref{f(x + h) - f(x) = ...}) that $f(x)$ is Lipschitz of order $1$ on bounded subsets of ${\bf Q}_p$. Suppose now that $a_k \in {\bf Z}_p$ for each $k$, so that $f$ maps ${\bf
Z}_p$ into itself. In this case, (\ref{f(x + h) - f(x) = ...}) implies that \begin{equation}
\label{|f(x + h) - f(x)|_p le |h|_p}
|f(x + h) - f(x)|_p \le |h|_p \end{equation} for every $x, h \in {\bf Z}_p$, since the binomial coefficients ${k
\choose j}$ are integers. Of course, the coefficients of $f'(x)$ are elements of ${\bf Z}_p$ too, so that \begin{equation}
\label{|f'(x + h) - f'(x)| le |h|_p}
|f'(x + h) - f'(x)| \le |h|_p \end{equation} for every $x, h \in {\bf Z}_p$ as well. Using (\ref{f(x + h) - f(x) -
f'(x) h = ...}), we also get that \begin{equation}
\label{|f(x + h) - f(x) - f'(x) h|_p le |h|_p^2}
|f(x + h) - f(x) - f'(x) \, h|_p \le |h|_p^2 \end{equation} for every $x, h \in {\bf Z}_p$ under these conditions.
\section{Hensel's lemma (first version)} \label{hensel's lemma (first version)}
Let $f(x)$ be a polynomial with coefficients in ${\bf Z}_p$, so that $f(x)$ and $f'(x)$ are elements of ${\bf Z}_p$ for every $x \in {\bf Z}_p$. Suppose that $x_0 \in {\bf Z}_p$ satisfies $f(x_0)
\in p \, {\bf Z}_p$ and $|f'(x_0)|_p = 1$. Under these conditions, \emph{Hensel's lemma}\index{Hensel's lemma} states that there is an $x \in {\bf Z}_p$ such that $x - x_0 \in p \, {\bf Z}_p$ and $f(x) = 0$. The proof uses Newton's method, as follows. If $x_1 \in {\bf Z}_p$ is close to $x_0$, then $f(x_1)$ is approximately \begin{equation} \label{f(x_0) + f'(x_0) (x_1 - x_0)}
f(x_0) + f'(x_0) \, (x_1 - x_0), \end{equation}
as in (\ref{|f(x + h) - f(x) - f'(x) h|_p le |h|_p^2}). In order to make this approximation equal to $0$, we take \begin{equation} \label{x_1 = x_0 - f'(x_0)^{-1} f(x_0)}
x_1 = x_0 - f'(x_0)^{-1} \, f(x_0). \end{equation} This satisfies $x_1 - x_0 \in p \, {\bf Z}_p$, since $f(x_0) \in p \,
{\bf Z}_p$ and $|f'(x_0)|_p = 1$.
Repeating the process, we shall choose a sequence of elements $x_1, x_2, x_3, \ldots$ of ${\bf Z}_p$ such that \begin{equation} \label{x_j - x_{j - 1} in p {bf Z}_p}
x_j - x_{j - 1} \in p \, {\bf Z}_p \end{equation} for each $j \ge 1$. In particular, this ensures that \begin{equation} \label{x_j - x_0 in p {bf Z}_p}
x_j - x_0 \in p \, {\bf Z}_p \end{equation} for every $j \ge 1$, and hence that $f(x_j) - f(x_0) \in p \, {\bf
Z}_p$ for every $j \ge 1$, by (\ref{|f(x + h) - f(x)|_p le |h|_p}). Of course, this implies that \begin{equation} \label{f(x_j) in p {bf Z}_p}
f(x_j) \in p \, {\bf Z}_p \end{equation} for every $j \ge 1$, since $f(x_0) \in p \, {\bf Z}_p$ by hypothesis. Similarly, \begin{equation} \label{f'(x_j) - f'(x_0) in p {bf Z}_p}
f'(x_j) - f'(x_0) \in p \, {\bf Z}_p \end{equation}
for each $j \ge 1$, by (\ref{x_j - x_0 in p {bf Z}_p}) and (\ref{|f'(x
+ h) - f'(x)| le |h|_p}). It follows that \begin{equation}
\label{|f'(x_j)|_p = 1}
|f'(x_j)|_p = 1 \end{equation}
for every $j \ge 1$, since $|f'(x_0)|_p = 1$ by hypothesis.
If $x_{j - 1}$ has already been chosen, then we would like to choose $x_j$ so that \begin{equation} \label{f(x_{j - 1}) + f'(x_{j - 1}) (x_j - x_{j - 1}) = 0}
f(x_{j - 1}) + f'(x_{j - 1}) \, (x_j - x_{j - 1}) = 0, \end{equation} which is the same as saying that \begin{equation} \label{x_j = x_{j - 1} - f'(x_{j - 1}) f(x_{j - 1})}
x_j = x_{j - 1} - f'(x_{j - 1}) \, f(x_{j - 1}). \end{equation} In particular, if $x_{j - 1} - x_0 \in p \, {\bf Z}_p$, then $f(x_{j -
1}) \in p \, {\bf Z}_p$ and $|f'(x_{j - 1})|_p = 1$, as in the previous paragraph. This implies that (\ref{x_j - x_{j - 1} in p {bf
Z}_p}) holds, so that the process can be repeated. More precisely, \begin{equation}
\label{|x_j - x_{j - 1}|_p = |f(x_{j - 1})|_p}
|x_j - x_{j - 1}|_p = |f(x_{j - 1})|_p. \end{equation}
Under these conditions, we also have that \begin{equation}
\label{|f(x_j)|_p le |x_j - x_{j - 1}|_p^2}
|f(x_j)|_p \le |x_j - x_{j - 1}|_p^2, \end{equation}
by applying (\ref{|f(x + h) - f(x) - f'(x) h|_p le |h|_p^2}) to $x = x_{j - 1}$ and $h = x_j - x_{j - 1}$, and using (\ref{f(x_{j - 1}) +
f'(x_{j - 1}) (x_j - x_{j - 1}) = 0}). Combining this with
(\ref{|x_j - x_{j - 1}|_p = |f(x_{j - 1})|_p}), we get that \begin{equation}
\label{|f(x_j)|_p le |f(x_{j - 1})|_p^2}
|f(x_j)|_p \le |f(x_{j - 1})|_p^2 \end{equation}
for each $j \ge 1$. This implies that $|f(x_j)|_p \to 0$ as $j \to
\infty$, because $|f(x_0)|_p < 1$, by hypothesis. Thus $|x_j - x_{j -
1}|_p \to 0$ as $j \to \infty$, by (\ref{|x_j - x_{j - 1}|_p =
|f(x_{j - 1})|_p}) again. It follows that $\{x_j\}_{j = 1}^\infty$ is a Cauchy sequence in ${\bf Z}_p$, as in Section \ref{sequences,
series}, since the $p$-adic metric is an ultrametric. By completeness, $\{x_j\}_{j = 1}^\infty$ converges to an element $x$ of ${\bf Z}_p$, and in fact $x - x_0 \in {\bf Z}_p$, because of (\ref{x_j
- x_0 in p {bf Z}_p}). Of course, $f(x) = 0$, as desired, because
$f$ is continuous on ${\bf Q}_p$, and $|f(x_j)|_p \to 0$ as $j \to \infty$.
\section{Hensel's lemma (second version)} \label{hensel's lemma (second version)}
Let $f(x)$ be a polynomial with coefficients in ${\bf Z}_p$ again, and suppose that $x_0 \in {\bf Z}_p$ satisfies \begin{equation}
\label{|f(x_0)|_p < |f'(x_0)|_p^2}
|f(x_0)|_p < |f'(x_0)|_p^2. \end{equation}
We would like to find an $x \in {\bf Z}_p$ that is close to $x_0$ and satisfies $f(x) = 0$. Of course, $f(x_0), f'(x_0) \in {\bf Z}_p$, so that $|f(x_0)|_p, |f'(x_0)|_p \le 1$. If $|f'(x_0)|_p = 1$, then we are back in the situation discussed in the previous section. Otherwise, Newton's method is still applicable, but we should be a bit more careful about some of the estimates.
Let $j$ be a positive integer, and suppose that $x_{j - 1} \in {\bf Z}_p$ has been chosen in such a way that \begin{equation}
\label{|x_{j - 1} - x_0| < |f'(x_0)|_p}
|x_{j - 1} - x_0| < |f'(x_0)|_p \end{equation} and \begin{equation}
\label{|f(x_{j - 1})|_p le |f(x_0)|_p}
|f(x_{j - 1})|_p \le |f(x_0)|_p. \end{equation} Thus \begin{equation}
\label{|f'(x_{j - 1}) - f'(x_0)|_p le |x_{j - 1} - x_0|_p < |f'(x_0)|_p}
|f'(x_{j - 1}) - f'(x_0)|_p \le |x_{j - 1} - x_0|_p < |f'(x_0)|_p, \end{equation}
by (\ref{|f'(x + h) - f'(x)| le |h|_p}), which implies that \begin{equation}
\label{|f'(x_{j - 1})|_p = |f'(x_0)|_p}
|f'(x_{j - 1})|_p = |f'(x_0)|_p, \end{equation} because of the ultrametric version of the triangle inequality. Let us choose $x_j \in {\bf Q}_p$ as in (\ref{x_j = x_{j - 1} - f'(x_{j - 1})
f(x_{j - 1})}), so that \begin{equation}
\label{|x_j - x_{j - 1}|_p = ... = |f'(x_0)|_p^{-1} |f(x_{j - 1})|_p}
|x_j - x_{j - 1}|_p = |f'(x_{j - 1})|_p^{-1} \, |f(x_{j - 1})|
= |f'(x_0)|_p^{-1} \, |f(x_{j - 1})|_p. \end{equation}
Combining this with (\ref{|f(x_0)|_p < |f'(x_0)|_p^2}) and
(\ref{|f(x_{j - 1})|_p le |f(x_0)|_p}), we get that \begin{equation}
\label{|x_j - x_{j - 1}|_p < |f'(x_0)|_p}
|x_j - x_{j - 1}|_p < |f'(x_0)|_p. \end{equation} It follows that \begin{equation}
\label{|x_j - x_0|_p < |f'(x_0)|_p}
|x_j - x_0|_p < |f'(x_0)|_p, \end{equation}
by (\ref{|x_{j - 1} - x_0| < |f'(x_0)|_p}), and in particular that
$x_j \in {\bf Z}_p$. This permits us to apply (\ref{|f(x + h) - f(x)
- f'(x) h|_p le |h|_p^2}) with $x = x_{j - 1}$ and $h = x_j - x_{j -
1}$, to get that \begin{equation}
\label{|f(x_j)|_p le |x_j - x_{j - 1}|_p^2, 2}
|f(x_j)|_p \le |x_j - x_{j - 1}|_p^2, \end{equation} using also (\ref{f(x_{j - 1}) + f'(x_{j - 1}) (x_j - x_{j - 1}) = 0}). This implies that \begin{equation}
\label{|f(x_j)|_p le |f(x_{j - 1})|_p}
|f(x_j)|_p \le |f(x_{j - 1})|_p, \end{equation}
by (\ref{|x_j - x_{j - 1}|_p = ... = |f'(x_0)|_p^{-1} |f(x_{j -
1})|_p}) and (\ref{|x_j - x_{j - 1}|_p < |f'(x_0)|_p}). In particular, \begin{equation}
\label{|f(x_j)|_p le |f(x_0)|_p}
|f(x_j)|_p \le |f(x_0)|_p, \end{equation}
by (\ref{|f(x_{j - 1})|_p le |f(x_0)|_p}). This and (\ref{|x_j -
x_0|_p < |f'(x_0)|_p}) show that $x_j$ satisfies the same conditions as $x_{j - 1}$, so that the process can be repeated.
More precisely, (\ref{|x_j - x_{j - 1}|_p = ... = |f'(x_0)|_p^{-1}
|f(x_{j - 1})|_p}) and (\ref{|f(x_j)|_p le |x_j - x_{j - 1}|_p^2, 2}) imply that \begin{equation}
\label{|f(x_j)|_p le |f'(x_0)|_p^{-2} |f(x_{j - 1})|_p^2}
|f(x_j)|_p \le |f'(x_0)|_p^{-2} \, |f(x_{j - 1})|_p^2 \end{equation} for each $j \ge 1$. Thus \begin{equation}
|f(x_j)|_p \le (|f'(x_0)|_p^{-2} \, |f(x_0)|_p) \, |f(x_{j - 1})|_p \end{equation}
for each $j \ge 1$, by (\ref{|f(x_{j - 1})|_p le |f(x_0)|_p}). This implies that $|f(x_j)|_p \to 0$ as $j \to \infty$, by (\ref{|f(x_0)|_p
< |f'(x_0)|_p^2}).
It follows from this and (\ref{|x_j - x_{j - 1}|_p = ... =
|f'(x_0)|_p^{-1} |f(x_{j - 1})|_p}) that $|x_j - x_{j - 1}|_p \to 0$ as $j \to \infty$. Thus $\{x_j\}_{j = 1}^\infty$ is a Cauchy sequence in ${\bf Z}_p$, as in Section \ref{sequences, series}, which converges to an element $x$ of ${\bf Z}_p$, by completeness. Note that \begin{equation}
\label{|x - x_0|_p < |f'(x_0)|_p}
|x - x_0|_p < |f'(x_0)|_p, \end{equation} because of the analogous condition for the $x_j$'s, and the fact that open balls in ultrametric spaces are closed sets. We also have that
$f(x) = 0$, as desired, because $f$ is continuous on ${\bf Q}_p$, and $|f(x_j)|_p \to 0$ as $j \to \infty$.
\section{Contractions} \label{contractions}
Let $f(x)$ be a polynomial with coefficients in ${\bf Z}_p$, and suppose that $x_0 \in {\bf Z}_p$ satisfies (\ref{|f(x_0)|_p <
|f'(x_0)|_p^2}). Consider \begin{equation} \label{g(x) = x - (f'(x_0))^{-1} f(x + x_0) - x}
g(x) = x - (f'(x_0))^{-1} \, f(x + x_0), \end{equation} which is a polynomial with coefficients in ${\bf Q}_p$ that satisfies \begin{equation} \label{g'(0) = 1 - (f'(x_0))^{-1} f'(x_0) = 0}
g'(0) = 1 - (f'(x_0))^{-1} \, f'(x_0) = 0. \end{equation}
More precisely, $|f'(x_0)|_p = p^{-k}$ for some nonnegative integer $k$, which implies that the coefficients of $g$ are in $p^{-k} \, {\bf
Z}_p$. Using (\ref{|f(x_0)|_p < |f'(x_0)|_p^2}), we also get that \begin{equation}
\label{|g(0)| = |f'(x_0)|_p^{-1} |f(x_0)| < |f'(x_0)|_p = p^{-k}}
|g(0)| = |f'(x_0)|_p^{-1} \, |f(x_0)| < |f'(x_0)|_p = p^{-k}. \end{equation}
Let us now start over, and let $k$ be a nonnegative integer and $g(x)$ be a polynomial with coefficients in $p^{-k} \, {\bf Z}_p$ such that \begin{equation} \label{g(0), g'(0) in p^{k + 1} {bf Z}_p}
g(0), \, g'(0) \in p^{k + 1} \, {\bf Z}_p. \end{equation} If $x \in p^{k + 1} \, {\bf Z}_p$, then it is easy to see that \begin{equation} \label{g(x) in p^{k + 1} {bf Z}_p}
g(x) \in p^{k + 1} \, {\bf Z}_p \end{equation} and \begin{equation} \label{g'(x) in p {bf Z}_p}
g'(x) \in p \, {\bf Z}_p. \end{equation} Observe that \begin{equation}
\label{|g(y) - g(x) - g'(x) (y - x)|_p le p^k |x - y|_p^2}
|g(y) - g(x) - g'(x) \, (y - x)|_p \le p^k \, |x - y|_p^2 \end{equation}
for every $x, y \in {\bf Z}_p$, by applying (\ref{|f(x + h) - f(x) -
f'(x) h|_p le |h|_p^2}) to $p^k \, g$. This implies that \begin{equation}
\label{|g(y) - g(x)|_p le max(|g'(x)|_p, p^k |x - y|_p) |x - y|_p}
|g(y) - g(x)|_p \le \max(|g'(x)|_p, p^k \, |x - y|_p) \, |x - y|_p
\le p^{-1} \, |x - y|_p \end{equation} when $x, y \in p^{k + 1} \, {\bf Z}_p$, by (\ref{g'(x) in p {bf Z}_p}).
Thus $g$ maps $p^{k + 1} \, {\bf Z}_p$ into itself under these
conditions, and the restriction of $g$ to $p^{k + 1} \, {\bf Z}_p$ is a strict contraction, by (\ref{|g(y) - g(x)|_p le max(|g'(x)|_p, p^k
|x - y|_p) |x - y|_p}). The contraction mapping principle implies that $g$ has a unique fixed point in $p^{k + 1} \, {\bf Z}_p$, because $p^{k + 1} \, {\bf Z}_p$ is complete as a metric space. If $g$ is as in (\ref{g(x) = x - (f'(x_0))^{-1} f(x + x_0) - x}), then this is the same as saying that there is a unique $x \in p^{k + 1} \, {\bf Z}_p$ such that $f(x + x_0) = 0$.
\section{Local geometry} \label{local geometry}
Let $f(x)$ be a polynomial with coefficients in ${\bf Z}_p$, and suppose that $x_0 \in {\bf Z}_p$ satisfies $f'(x_0) \ne 0$. Let
$k$ be a nonnegative integer such that $|f'(x_0)|_p = p^{-k}$, as before. If $x \in x_0 + p^{k + 1} \, {\bf Z}_p$, then \begin{equation}
\label{|f'(x) - f'(x_0)|_p le |x - x_0|_p le p^{-k - 1}}
|f'(x) - f'(x_0)|_p \le |x - x_0|_p \le p^{-k - 1}, \end{equation}
by (\ref{|f'(x + h) - f'(x)| le |h|_p}). This implies that \begin{equation}
\label{|f'(x)|_p = |f'(x_0)|_p = p^{-k}}
|f'(x)|_p = |f'(x_0)|_p = p^{-k}, \end{equation} by the ultrametric version of the triangle inequality. If $x, y \in x_0 + p^{k + 1} \, {\bf Z}_p$, then \begin{equation}
\label{|f(y) - f(x) - f'(x) (y - x)|_p le |x - y|_p^2 le p^{-k - 1} |x - y|_p}
|f(y) - f(x) - f'(x) \, (y - x)|_p \le |x - y|_p^2 \le p^{-k - 1} \, |x - y|_p, \end{equation}
as in (\ref{|f(x + h) - f(x) - f'(x) h|_p le |h|_p^2}). It follows that \begin{eqnarray}
\label{|f(x) - f(y)|_p le ... = p^{-k} |x - y|_p}
|f(x) - f(y)|_p & \le & \max(|f'(x)|_p \, |x - y|_p, p^{-k - 1} \, |x - y|_p) \\
& = & p^{-k} \, |x - y|_p, \nonumber \end{eqnarray}
by (\ref{|f'(x)|_p = |f'(x_0)|_p = p^{-k}}). Similarly,
(\ref{|f'(x)|_p = |f'(x_0)|_p = p^{-k}}) and (\ref{|f(y) - f(x) -
f'(x) (y - x)|_p le |x - y|_p^2 le p^{-k - 1} |x - y|_p}) also imply that \begin{eqnarray}
\label{p^{-k} |x - y|_p = ... le max(|f(x) - f(y)|_p, p^{k - 1} |x - y|_p)}
p^{-k} \, |x - y|_p & = & |f'(x)|_p \, |x - y|_p \\
& \le & \max(|f(x) - f(y)|_p, p^{k - 1} \, |x - y|_p), \nonumber \end{eqnarray} and hence that \begin{equation}
\label{p^{-k} |x - y|_p le |f(x) - f(y)|_p}
p^{-k} \, |x - y|_p \le |f(x) - f(y)|_p. \end{equation} This shows that \begin{equation}
\label{|f(x) - f(y)|_p = p^{-k} |x - y|_p}
|f(x) - f(y)|_p = p^{-k} \, |x - y|_p \end{equation} for every $x, y \in x_0 + p^{k + 1} \, {\bf Z}_p$ under these conditions.
Now let $f$ be any mapping from $x_0 + p^{k + 1} \, {\bf Z}_p$
into ${\bf Q}_p$ that satisfies (\ref{|f(x) - f(y)|_p = p^{-k} |x - y|_p}) for every $x, y \in p^{p + 1} \, {\bf Z}_p$, where $k$ is a nonnegative integer. In particular, \begin{equation} \label{f(x_0 + p^{k + 1} {bf Z}_p) subseteq f(x_0) + p^{2 k + 1} {bf Z}_p}
f(x_0 + p^{k + 1} \, {\bf Z}_p) \subseteq f(x_0) + p^{2 k + 1} \, {\bf Z}_p. \end{equation} Of course, \begin{equation} \label{x mapsto p^{-k} (x - x_0) + f(x_0)}
x \mapsto p^{-k} \, (x - x_0) + f(x_0) \end{equation} is a similarity from $x_0 + p^{k + 1} \, {\bf Z}_p$ onto $f(x_0) + p^{2 \, k + 1} \, {\bf Z}_p$ with respect to the $p$-adic metric, with the same similarity constant $p^{-k}$. Because $x_0 + p^{k + 1} \, {\bf Z}_p$ is compact, one can use this to show that \begin{equation} \label{f(x_0 + p^{k + 1} {bf Z}_p) = f(x_0) + p^{2 k + 1} {bf Z}_p}
f(x_0 + p^{k + 1} \, {\bf Z}_p) = f(x_0) + p^{2 k + 1} \, {\bf Z}_p, \end{equation} as in Section \ref{isometries, similarities}.
Remember that the one-dimensional Hausdorff measure of
$x_0 + p^{k + 1} \, {\bf Z}_p$ with respect to the $p$-adic metric is equal to $p^{-k - 1}$, as in Section \ref{some special cases}. Using this and (\ref{|f(x) - f(y)|_p = p^{-k} |x - y|_p}), it is easy to see that the one-dimensional Hausdorff measure of $f(x_0 + p^{k + 1} \, {\bf Z}_p)$ is equal to $p^{-2 k - 1}$, as in Section \ref{isometries,
similarities}. Of course, this is the same as the one-dimensional Hausdorff measure of $f(x_0) + p^{2 k + 1} \, {\bf Z}_p$. It follows that $f(x_0 + p^k \, {\bf Z}_p)$ is dense in $f(x_0) + p^{2 k + 1} \, {\bf Z}_p$, since every ball in ${\bf Q}_p$ with positive radius has positive one-dimensional Hausdorff measure. Note that $f(x_0 + p^k \, {\bf Z}_p)$ is a compact set in ${\bf Q}_p$, because $x_0 + p^k \, {\bf Z}_p$ is compact, and $f$ is continuous. Thus the density of $f(x_0 + p^k \, {\bf Z}_p)$ in $f(x_0) + p^{2 k + 1} \, {\bf Z}_p$ implies that (\ref{f(x_0 + p^{k + 1} {bf Z}_p) = f(x_0) + p^{2 k + 1}
{bf Z}_p}) holds, as before. This type of argument was also mentioned in Section \ref{isometries, similarities}, using the properties of one-dimensional Hausdorff measure in this case.
Here is an analogous but more elementary approach, which is a more explicit version of another argument in Section \ref{isometries, similarities} in this situation. If $n$ is a positive integer, then there is a set $A_n \subseteq p^{k + 1} \, {\bf
Z}_p$ with exactly $p^n$ elements such that \begin{equation}
\label{|a - b|_p ge p^{-k - n}}
|a - b|_p \ge p^{-k - n} \end{equation} for every $a, b \in A_n$ with $a \ne b$. Equivalently, this means that the restriction of the natural quotient mapping from $p^{k + 1} \, {\bf Z}_p$ onto $p^{k + 1} \, {\bf Z}_p / p^{k + n + 1} \, {\bf
Z}_p$ to $A_n$ is injective. If $f$ is as in the previous two paragraphs, then \begin{equation}
\label{|f(x_0 + a) - f(x_0 + b)|_p = p^{-k} |a - b|_p ge p^{-2 k - n}}
|f(x_0 + a) - f(x_0 + b)|_p = p^{-k} \, |a - b|_p \ge p^{-2 k - n} \end{equation} for every $a, b \in A_n$ with $a \ne b$. However, $f(x_0) + p^{2 k +
1} \, {\bf Z}_p$ can be expressed as the union of $p^n$
pairwise-disjoint closed balls of radius $p^{-2 k - n - 1}$ for each positive integer $n$. Each of these balls can contain at most one element of $f(x_0 + A_n)$, by (\ref{|f(x_0 + a) - f(x_0 + b)|_p =
p^{-k} |a - b|_p ge p^{-2 k - n}}). It follows that each of these balls must contain an element of $f(x_0 + A_n)$, which implies that $f(x_0 + p^{k + 1} \, {\bf Z}_p)$ is dense in $f(x_0) + p^{2 k + 1} \, {\bf Z}_p$ under these conditions.
\section{Power series} \label{power series}
Let $\sum_{j = 0}^\infty a_j \, x^j$ be a power series with coefficients in ${\bf Q}_p$, where $x^j$ is interpreted as being equal to $1$ for every $x \in {\bf Q}_p$ when $j = 0$, as usual. As in Section \ref{sequences, series}, $\sum_{j = 0}^\infty a_j \, x^j$ converges for some $x \in {\bf Q}_p$ if and only if $\{a_j \, x^j\}_{j
= 0}^\infty$ converges to $0$ in ${\bf Q}_p$, which is the same as saying that \begin{equation}
\label{|a_j x^j|_p = |a_j|_p |x|_p^j to 0 as j to infty}
|a_j \, x^j|_p = |a_j|_p \, |x|_p^j \to 0 \hbox{ as } j \to \infty. \end{equation} In this case, $\sum_{j = 0}^\infty a_j \, y^j$ also converges in ${\bf
Q}_p$ when $y \in {\bf Q}_p$ satisfies $|y|_p \le |x|_p$. More precisely, \begin{eqnarray}
\label{|sum_{j = 0}^infty a_j y^j - sum_{j = 0}^n a_j y^j| = ...}
\biggl|\sum_{j = 0}^\infty a_j \, y^j - \sum_{j = 0}^n a_j \, y^j\biggr|_p
& = & \biggl|\sum_{j = n + 1}^\infty a_j \, y^j\biggr|_p \\
& \le & \max_{j \ge n + 1} |a_j \, y^j|_p \le \max_{j \ge n + 1} |a_j|_p |x|_p^j
\nonumber \end{eqnarray}
when $|y|_p \le |x|_p$, which implies that the partial sums $\sum_{j =
0}^\infty a_j \, y^j$ converge to $\sum_{j = 0}^\infty a_j \, y^j$ uniformly as $n \to \infty$ on the set of $y \in {\bf Q}_p$ with
$|y|_p \le |x|_p$. It follows that $\sum_{j = 0}^\infty a_j \, x^j$ defines a continuous ${\bf Q}_p$-valued function on the set of $x \in {\bf Q}_p$ for which the series converges, which is either a closed disk centered at $0$ or all of ${\bf Q}_p$.
Now let $\sum_{j = 0}^\infty a_j \, x^j$ and $\sum_{k = 0}^\infty b_k \, x^k$ be infinite series with coefficients in ${\bf Q}_p$, and let \begin{equation} \label{c_l = sum_{j = 0}^l a_j b_{l - j}, 2}
c_l = \sum_{j = 0}^l a_j \, b_{l - j} \end{equation} be the Cauchy product\index{Cauchy products} of their coefficients, as in Section \ref{sequences, series}. Observe that \begin{equation} \label{c_l x^l = sum_{j = 0}^n (a_j x^j) (b_{l - j} x^{l - j})}
c_l \, x^l = \sum_{j = 0}^n (a_j \, x^j) \, (b_{l - j} \, x^{l - j}) \end{equation} for each $x \in {\bf Q}_p$, so that $\sum_{l = 0}^\infty c_l \, x^l$ is the Cauchy product of $\sum_{j = 0}^\infty a_j \, x^j$ and $\sum_{k
= 0}^\infty b_k \, x^k$. In particular, \begin{equation} \label{sum_{l = 0}^infty c_l x^l = ...}
\sum_{l = 0}^\infty c_l \, x^l = \Big(\sum_{j = 0}^\infty a_j \, x^j\Big)
\, \Big(\sum_{k = 0}^\infty b_k \, x^k\Big) \end{equation} formally, collecting all of the terms which are multiples of $x^l$ for each $l \ge 0$. If $\sum_{j = 0}^\infty a_j \, x^j$ and $\sum_{k =
0}^\infty b_k \, x^k$ both converge for some $x \in {\bf Q}_p$, then it follows that $\sum_{l = 0}^\infty c_l \, x^l$ converges and satisfies (\ref{sum_{l = 0}^infty c_l x^l = ...}), as in Section \ref{sequences, series}.
Suppose that \begin{equation} \label{f(x) = sum_{j = 0}^infty a_j x^j}
f(x) = \sum_{j = 0}^\infty a_j \, x^j \end{equation} is a power series with coefficients $a_j \in {\bf Z}_p$ for each $j \ge 0$, and that $\{a_j\}_{j = 0}^\infty$ converges to $0$ in ${\bf Q}_p$. This implies that (\ref{f(x) = sum_{j = 0}^infty a_j x^j}) converges for every $x \in {\bf Z}_p$, and that $f(x)$ defines a continuous ${\bf Q}_p$-valued function on ${\bf Z}_p$, as before. Moreover, the formal derivative \begin{equation} \label{f'(x) = sum_{j = 1}^infty j a_j x^j}
f'(x) = \sum_{j = 1}^\infty j \, a_j \, x^j \end{equation}
also has coefficients in ${\bf Z}_p$ that converge to $0$, and hence defines a continuous function on ${\bf Z}_p$ as well. It is easy to see that $f(x)$ and $f'(x)$ satisfy the same estimates (\ref{|f(x + h)
- f(x)|_p le |h|_p}), (\ref{|f'(x + h) - f'(x)| le |h|_p}), and
(\ref{|f(x + h) - f(x) - f'(x) h|_p le |h|_p^2}) as for polynomials with coefficients in ${\bf Z}_p$, by approximating the corresponding infinite series by their partial sums. It follows that the results for polynomials with coefficients in ${\bf Z}_p$ discussed in the previous sections also work for power series of this type.
In particular, the analogue of (\ref{|f(x + h) - f(x) - f'(x) h|_p
le |h|_p^2}) in this context implies that the derivative of $f$ at any point $x \in {\bf Z}_p$ exists and is equal to $f'(x)$. There are analogous statements for any convergent power series with coefficients in ${\bf Q}_p$.
\section{Linear mappings on ${\bf Q}_p^n$} \label{linear mappings on {bf Q}_p^n}
Let $n$ be a positive integer, and let ${\bf Q}_p^n$ be the set of $n$-tuples $v = (v_1, \ldots, v_n)$ of elements of ${\bf Q}_p$. As usual, this is a vector space over ${\bf Q}_p$ with respect to coordinatewise addition and scalar multiplication. Put \begin{equation}
\label{||v||= max (|v_1|_p, ldots, |v_n|_p)}
\|v\| = \max (|v_1|_p, \ldots, |v_n|_p) \end{equation} for each $v \in {\bf Q}_p$, and observe that \begin{equation}
\label{||t v|| = |t|_p ||v||}
\|t \, v\| = |t|_p \, \|v\| \end{equation} for every $v \in {\bf Q}_p^n$ and $t \in {\bf Q}_p$, and that \begin{equation}
\label{||v + w|| le max(||v||, ||w||)}
\|v + w\| \le \max(\|v\|, \|w\|) \end{equation}
for every $v, w \in {\bf Q}_p^n$. Thus $\|v\|$ is an
\emph{ultranorm}\index{ultranorms} on ${\bf Q}_p^n$, which is like a norm on a real or complex vector space, except that it satifies the ultrametric version of the triangle inequality (\ref{||v + w|| le
max(||v||, ||w||)}). It follows that \begin{equation}
\label{d(v, w) = ||v - w||}
d(v, w) = \|v - w\| \end{equation} defines an ultrametric on ${\bf Q}_p^n$, for which the corresponding topology on ${\bf Q}_p^n$ is the same as the product topology associated to the standard topology on ${\bf Q}_p$.
Let $e_1, \ldots, e_n$ be the standard basis vectors for ${\bf Q}_p^n$, so that the $j$th coordinate of $e_k$ is equal to $1$ when $j = k$ and to $0$ otherwise. If $T$ is a linear mapping from ${\bf Q}_p^n$ into itself, then put \begin{equation}
\label{||T||_{op} = max (||T(e_1)||, ldots, ||T(e_n)||)}
\|T\|_{op} = \max (\|T(e_1)\|, \ldots, \|T(e_n)\|). \end{equation} The space of linear mappings from ${\bf Q}_p^n$ into itself is also a vector space over ${\bf Q}_p$ with respect to the usual addition and scalar multiplication of linear mappings, and it is easy to see that
$\|T\|_{op}$ defines an ultranorm on this vector space. Each $v \in {\bf Q}_p^n$ can be expressed as $v = \sum_{j = 1}^n v_j \, e_j$, and hence \begin{equation}
\label{||T(v)|| le max_{1 le j le n} (|v_j|_p ||T(e_j)||) le ||T||_{op} ||v||}
\|T(v)\| \le \max_{1 \le j \le n} (|v_j|_p \, \|T(e_j)\|)
\le \|T\|_{op} \, \|v\|. \end{equation}
Thus $\|T\|_{op}$ is the same as the operator norm of $T$ associated to the ultranorm $\|v\|$ on ${\bf Q}_p^n$, and \begin{equation}
\label{||T_2 circ T_1||_{op} le ||T_1||_{op} ||T_2||_{op}}
\|T_2 \circ T_1\|_{op} \le \|T_1\|_{op} \, \|T_2\|_{op} \end{equation} for any two linear mappings $T_1$, $T_2$ from ${\bf Q}_p^n$ into itself.
If $\{a_{j, k}\}_{j, k = 1}^n$ is an $n \times n$ matrix with entries in ${\bf Q}_p$, then \begin{equation} \label{(T(v))_j = sum_{k = 1}^n a_{j, k} v_k}
(T(v))_j = \sum_{k = 1}^n a_{j, k} \, v_k \end{equation} defines a linear mapping from ${\bf Q}_p^n$ into itself, where $(T(v))_j$ is the $j$th coordinate of $T(v)$. Of course, every linear mapping from ${\bf Q}_p^n$ into itself can be expressed in this way, and one can check that \begin{equation}
\label{||T||_{op} = max_{1 le j, k le n} |a_{j, k}|_p}
\|T\|_{op} = \max_{1 \le j, k \le n} |a_{j, k}|_p \end{equation}
when $T$ is as in (\ref{(T(v))_j = sum_{k = 1}^n a_{j, k} v_k}). Note that $\|T\|_{op} \le 1$ if and only if $a_{j, k} \in {\bf Z}_p$ for each $j, k = 1, \ldots n$, which happens if and only if $T$ maps ${\bf
Z}_p^n$ into itself. If $T$ is as in (\ref{(T(v))_j = sum_{k = 1}^n
a_{j, k} v_k}), then the determinant of $T$ as a linear mapping on ${\bf Q}_p^n$ is the same as the determinant of the corresponding matrix $\{a_{j, k}\}_{j, k = 1}^n$, and hence \begin{equation}
\label{|det T|_p le ||T||_{op}^n}
|\det T|_p \le \|T\|_{op}^n. \end{equation}
Suppose that $T$ is a linear mapping from ${\bf Q}_p^n$ into itself that satisfies \begin{equation}
\label{||T(v)|| = ||v||}
\|T(v)\| = \|v\| \end{equation} for every $v \in {\bf Q}_p^n$. In particular, this implies that $T$ is one-to-one, and hence that $T$ maps ${\bf Q}_p^n$ onto itself, by linear algebra. Thus $T$ is an invertible linear mapping on ${\bf
Q}_p^n$, and the operator norms of $T$ and $T^{-1}$ are both equal to $1$. Conversely, if $T$ is an invertible linear mapping on ${\bf
Q}_p^n$ such that $\|T\|_{op}, \, \|T^{-1}\|_{op} \le 1$, then $T$
satisfies (\ref{||T(v)|| = ||v||}). In this case, $T$ and $T^{-1}$ both correspond to matrices with entries in ${\bf Z}_p$, whose determinants are in ${\bf Z}_p$ as well. It follows that \begin{equation}
\label{|det T|_p = 1}
|\det T|_p = 1, \end{equation} because $\det T$ and $(\det T)^{-1} = \det T^{-1}$ are both in ${\bf
Z}_p$. Conversely, suppose that $T$ is a linear mapping on ${\bf
Q}_p^n$ that corresponds to an $n \times n$ matrix with entries in
${\bf Z}_p$, and that $T$ satisfies (\ref{|det T|_p = 1}). This implies that $T$ is invertible, where the matrix associated to the inverse of $T$ can be expressed in terms of determinants in the usual way. More precisely, the entries of the matrix associated to $T^{-1}$
are in ${\bf Z}_p$ too, because of (\ref{|det T|_p = 1}).
\chapter{Commutative topological groups} \label{commutative topological groups}
\section{Haar measure} \label{haar measure}
Let $G$ be a group, in which the group operations are expressed multiplicatively. If $G$ is also equipped with a topology with respect to which the group operations are continuous, then $G$ is said to be a \emph{topological group}.\index{topological groups} More precisely, this means that multiplication in the group should be continuous as a mapping from $G \times G$ into $G$, where $G \times G$ is equipped with the product topology associated to the given topology on $G$. Similarly, $x \mapsto x^{-1}$ should be continuous as a mapping from $G$ onto itself. It is customary to ask also that the set containing only the identity element $e$ in $G$ be a closed set in $G$. This implies that every one-element subset of $G$ is closed, using the continuity of translations on $G$, which follows from continuity of multiplication on $G$. One can show that $G$ is Hausdorff under these conditions, and in fact regular as a topological space.
Let $G$ be a topological group which is locally compact as a topological space. It is well known that there is a nonnegative Borel measure on $G$ with suitable regularity properties that is invariant under left translations, known as \emph{Haar measure}.\index{Haar measure} In particular, the Haar measure of a nonempty open set in $G$ should be positive, and the Haar measure of a compact set in $G$ should be finite. This measure is unique up to multiplication by a positive real number. Similarly, there is a nonnegative Borel measure on $G$ with suitable regularity properties that is invariant under right translations, with the same type of uniqueness property. Of course, one can use the mapping $x \mapsto x^{-1}$ to switch between left and right-invariant Haar measures on $G$. If $G$ is compact, then one can show that left-invariant Haar measure on $G$ is invariant under right translations too. This is trivial when $G$ is commutative, and one can check that Haar measure on $G$ is invariant under the mapping $x \mapsto x^{-1}$ when $G$ is compact or commutative.
Using Haar measure on $G$, one gets a nonnegative linear functional on the space of continuous real or complex-valued functions with compact support on $G$ which is invariant under left or right translations, as appropriate. This type of linear functional is known as a Haar integral on $G$, and it is strictly positive in the sense that the integral of a nonnegative real-valued continuous function with compact support on $G$ is positive when the function is positive somewhere on $G$. Of course, any nonnegative linear functional on the space of continuous functions with compact support on $G$ determines a unique nonnegative Borel measure with certain regularity properties, by the Riesz representation theorem. If such a linear functional is invariant under left or right translations and strictly positive in the sense mentioned earlier, then the corresponding measure is a Haar measure on $G$. One can also deal with uniqueness directly in terms of these linear functionals.
The real line is a locally compact commutative topological group with respect to addition, and Lebesgue measure on ${\bf R}$ satisfies the requirements of Haar measure. If $p$ is any prime number, then the $p$-adic numbers ${\bf Q}_p$ form a locally compact commutative topological group with respect to addition as well, and Haar measure on ${\bf Q}_p$ was discussed in Section \ref{haar measure on Q_p}. Any group $G$ is a locally compact topological group with respect to the discrete topology, with counting measure as Haar measure that is invariant under both left and right translations. The unit circle ${\bf T}$ in the complex plane is a compact commutative topological group with respect to multiplication and the topology induced on ${\bf T}$ by the standard topology on ${\bf C}$, and the usual arc-length measure on ${\bf T}$ satisfies the requirements of Haar measure. Haar measure on a real Lie group can be given in terms of a smooth volume form that is invariant under left or right translations, as appropriate.
\section{Dual groups} \label{dual groups}
Let $A$ be a commutative topological group, with the group operations expressed additively. A continuous homomorphism from $A$ into the multiplicative group ${\bf T}$ of complex numbers with modulus equal to $1$ is said to be a \emph{character}\index{characters} on $A$. The collection of characters on $A$ forms a commutative group $\widehat{A}$ with respect to pointwise multiplication, known as the \emph{dual group}\index{dual
groups} associated to $A$. In particular, the identity element in $\widehat{A}$ is the trivial character on $A$, which is the constant function equal to $1$ at every point in $A$. Note that the multiplicative inverse of $\phi \in \widehat{A}$ is the same as the complex conjugate of $\phi$.
Of course, the group ${\bf Z}$ of integers is a commutative topological group with respect to addition the discrete topology. If $z \in {\bf T}$, then \begin{equation} \label{j mapsto z^j}
j \mapsto z^j \end{equation} defines a homomorphism from ${\bf Z}$ into ${\bf T}$, and every homomorphism from ${\bf Z}$ into ${\bf T}$ is of this form. Similarly, \begin{equation} \label{z mapsto z^j}
z \mapsto z^j \end{equation} is a continuous homomorphism from ${\bf T}$ into itself for every integer $j$, and it is well known that every character on ${\bf T}$ is of this form. If $y \in {\bf R}$, then \begin{equation} \label{x mapsto exp (i x y)}
x \mapsto \exp (i \, x \, y) \end{equation} is a character on ${\bf R}$, where $\exp$ refers to the complex exponential function. It is also well known that every character on ${\bf R}$ is of this form.
If $A$ is any commutative topological group, then one can consider $\widehat{A}$ equipped with the topology associated to uniform convergence on nonempty compact subsets of $A$. In particular, it is easy to see that $\widehat{A}$ is also a topological group with respect to this topology. This is especially nice when $A$ is locally compact, in which case it can be shown that $\widehat{A}$ is locally compact too. If $A = {\bf R}$ as a commutative topological group with respect to addition and the standard topology, then $\widehat{A}$ is isomorphic to ${\bf R}$ as a commutative group, as in the previous paragraph. One can check that the dual topology on $\widehat{A}$ corresponds exactly to the standard topology on ${\bf R}$ in this case as well.
If $z \in {\bf T}$ has nonnegative real part and $z \ne 1$, then the real part of $z^j$ is negative for some integer $j$. This implies that the only subgroup of ${\bf T}$ consisting of $z \in {\bf T}$ with nonnegative real part is the trivial subgroup $\{1\}$. If $A$ is a commutative topological group and $\phi \in \widehat{A}$ has the property that the real part of $\phi(x)$ is nonnegative for every $x \in A$, then it follows that $\phi$ is the trivial character on $A$. In particular, if $\phi \in \widehat{A}$ satisfies \begin{equation}
\label{|phi(x) - 1| le 1}
|\phi(x) - 1| \le 1 \end{equation} for every $x \in A$, then $\phi$ is the trivial character on $A$. If $\phi, \psi \in \widehat{A}$ satisfy \begin{equation}
\label{|phi(x) - psi(x)| le 1}
|\phi(x) - \psi(x)| \le 1 \end{equation} for every $x \in A$, then $\phi(x) = \psi(x)$ for every $x \in A$, since we can apply the previous argument to $\phi / \psi$. Equivalently, the distance between any two distinct elements of $\widehat{A}$ with respect to the supremum metric is greater than $1$. If $A$ is compact, then the topology on $\widehat{A}$ mentioned in the previous paragraph is the same as the topology determined by the supremum metric on $\widehat{A}$, and hence $\widehat{A}$ is discrete with respect to this topology.
\section{Compact commutative groups} \label{compact commutative groups}
Let $A$ be a compact commutative topological group, with the group operations expressed additively. As in Section \ref{haar
measure}, there is a unique translation-invariant nonnegative regular Borel measure $H$ on $A$ that satisfies $H(A) = 1$, which is the normalized Haar measure on $A$. Let $L^2(A)$ be the usual space of complex-valued square-integrable functions on $A$ with respect to $H$, with the inner product \begin{equation} \label{langle f, g rangle = int_A f(x) overline{g(x)} dH(x)}
\langle f, g \rangle = \int_A f(x) \, \overline{g(x)} \, dH(x). \end{equation} If $\phi$ is a character on $A$, then \begin{equation} \label{int_A phi(x) dH(x) = ... = phi(a) int_A phi(x) dH(x)}
\int_A \phi(x) \, dH(x) = \int_A \phi(x + a) \, dH(x)
= \phi(a) \, \int_A \phi(x) \, dH(x) \end{equation} for every $a \in A$, using the translation-invariance of $H$ in the first step. If $\phi(a) \ne 1$ for some $a \in A$, then it follows that \begin{equation} \label{int_A phi(x) dH(x) = 0}
\int_A \phi(x) \, dH(x) = 0. \end{equation} This implies that \begin{equation} \label{langle phi, psi rangle = int_A phi(x) overline{psi(x)} dH(x) = 0}
\langle \phi, \psi \rangle = \int_A \phi(x) \, \overline{\psi(x)} \, dH(x) = 0 \end{equation} when $\phi$ and $\psi$ are distinct elements of $\widehat{A}$, by applying the previous argument to $\phi(x) \, \overline{\psi(x)}$, which is a nontrivial character on $A$. The normalization $H(A) = 1$ implies that each element of $\widehat{A}$ has $L^2$ norm equal to $1$, so that the elements of $\widehat{A}$ are orthonormal in $L^2(A)$.
Let $C(A)$ be the algebra of complex-valued continuous functions on $A$, equipped with the supremum norm. If $\mathcal{E}$ is the linear span of $\widehat{A}$ in $C(A)$, then it is easy to see that $\mathcal{E}$ is a sub-algebra of $C(A)$ which is invariant under complex conjugation and contains the constant functions on $A$. It is well known that $\widehat{A}$ separates points in $A$, which implies that $\mathcal{E}$ separates points in $A$. It follows that $\mathcal{E}$ is dense in $C(A)$ with respect to the supremum norm, by the Stone--Weierstrass theorem. In particular, $\mathcal{E}$ is dense in $L^2(A)$, so that $\widehat{A}$ is an orthonormal basis for $L^2(A)$.
Similarly, if $E_1$ is a subgroup of $\widehat{A}$, then the linear span $\mathcal{E}_1$ of $E_1$ in $C(A)$ is a sub-algebra of $C(A)$ that is invariant under complex-conjugation and contains the constant functions. If $E_1$ separates points in $A$, then $\mathcal{E}_1$ separates points in $A$ too, and hence $\widehat{E}_1$ is dense in $C(A)$, by the Stone--Weierstrass theorem again. If $\phi$ is any character on $A$ not in $E_1$, then $\phi$ is orthogonal to every element of $E_1$ with respect to the $L^2$ inner product, which implies that $\phi$ is orthogonal to every element of $\mathcal{E}_1$. This implies that $\phi = 0$ when $\mathcal{E}_1$ is dense in $C(A)$, which is a contradiction. It follows that $E_1 = \widehat{A}$ when $E_1$ is a subgroup of $\widehat{A}$ that separates points in $A$.
If $\phi \in \widehat{A}$ and the real part of $\phi(x)$ is nonnegative for each $x \in A$, then $\phi$ is the trivial character on $A$, as in the previous section. Alternatively, if $\phi$ is a nontrivial character on $A$, then the integral of $\phi$ with respect to $H$ is equal to $0$, as in (\ref{int_A phi(x) dH(x) = 0}). If the real part of $\phi(x)$ is nonnegative for every $x \in A$, then it follows that the real part of $\phi(x)$ is equal to $0$ for every $x \in A$, contradicting the fact that $\phi(0) = 1$. One can also use the orthonormality of characters on $A$ to get that \begin{equation}
\label{int_A |phi(x) - psi(x)|^2 dH(x) = 2}
\int_A |\phi(x) - \psi(x)|^2 \, dH(x) = 2 \end{equation} when $\phi$, $\psi$ are distinct elements of $\widehat{A}$.
\section{Cartesian products} \label{cartesian products}
Let $A_1, \ldots, A_n$ be finitely many commutative topological groups, and consider their Cartesian product $A = \prod_{j = 1}^n A_j$. It is easy to see that $A$ is a commutative topological group as well, where the group operations are defined coordinatewise, and using the corresponding product topology. If $\phi_1, \ldots, \phi_n$ are characters on $A_1, \ldots, A_n$, respectively, then \begin{equation} \label{phi(x) = prod_{j = 1}^n phi_j(x_j)}
\phi(x) = \prod_{j = 1}^n \phi_j(x_j) \end{equation} defines a character on $A$. Conversely, one can check that every character on $A$ is of this form.
Now let $I$ be an infinite set, and suppose that $A_j$ is a commutative topological group for each $j \in I$. As before, $A = \prod_{j \in I} A_j$ is a commutative topological group with respect to coordinatewise addition and the product topology. Let $j_1, \ldots, j_n$ be finitely many distinct elements of $I$, and let $\phi_{j_l}$ be a character on $A_{j_l}$ for each $l = 1, \ldots, n$. Clearly \begin{equation} \label{phi(x) = prod_{l = 1}^n phi_{j_l}(x_{j_l})}
\phi(x) = \prod_{l = 1}^n \phi_{j_l}(x_{j_l}) \end{equation} defines a character on $A$, where $x_j \in A_j$ denotes the $j$th coordinate of $x \in A$ for each $j \in I$. Conversely, suppose that $\phi$ is a character on $A$. Thus the set $V$ of $x \in A$ such that the real part of $\phi(x)$ is positive is an open set in $A$ that contains $0$. By definition of the product topology on $A$, there are open sets $U_j \subseteq A_j$ for each $j \in I$ such that $0 \in U_j$ for each $j$, $U_j = A_j$ for all but finitely many $j \in I$, and $\prod_{j \in I} U_j \subseteq V$. Put $B_j = A_j$ when $U_j = A_j$, and $B_j = \{0\}$ otherwise. If $B = \prod_{j \in I} B_j$, then $B$ is a subgroup of $A$ contained in $V$, so that the real part of $\phi(x)$ is positive when $x \in B$. It follows that $\phi(x) = 1$ for every $x \in B$, as in Section \ref{dual groups}. This implies that $\phi(x)$ depends only on the finitely many coordinates $x_j$ of $x$ such that $B_j = \{0\}$ for each $x \in A$, and hence that $\phi$ can be expressed as in (\ref{phi(x) = prod_{l = 1}^n
phi_{j_l}(x_{j_l})}), as in the case of finite products.
Let $A = \prod_{j = 1}^n A_j$ be the product of finitely many commutative topological groups again. If $A_j$ is locally compact for each $j = 1, \ldots, n$, then $A$ is locally compact too, and Haar measure on $A$ basically corresponds to the product of the Haar measures on $A_1, \ldots, A_n$. More precisely, if there is a countable base for the topology of $A_j$ for each $j$, then one can use the standard construction of product measures. Otherwise, one should use a version of product measures for Borel measures with suitable regularity properties. Equivalently, one can get a Haar integral on $A$ using Haar integrals on the $A_j$'s.
If $A = \prod_{j \in I} A_j$ is the product of infinitely many compact commutative topological groups, then $A$ is also a compact commutative topological group with respect to the product topology, by Tychonoff's theorem. Haar measure on $A$ again basically corresponds to the product of the Haar measures on the $A_j$'s, normalized so that the measure of $A_j$ is equal to $1$ for each $j \in I$. As before, this is simpler when $I$ is countably infinite, and there is a base for the topology of $A_j$ with only finitely or countably many elements for each $j \in I$, which implies that there is a base for the topology of $A$ with only finitely or countably many elements. At any rate, one can look at the Haar integral on $A$, in terms of the Haar integrals on the $A_j$'s. Using compactness, one can show that continuous functions on $A$ can be approximated uniformly by continuous functions on $A$ that depend on only finitely many coordinates, for which the Haar integral is much easier to define.
\section{Discrete commutative groups} \label{discrete commutative groups}
Let $A$ be a commutative group equipped with the discrete topology, so that every homomorphism from $A$ into ${\bf T}$ is continuous and hence a character. Note that the collection ${\bf T}^A$ of all mappings from $A$ into ${\bf T}$ is a commutative group with respect to pointwise multiplication, and that $\widehat{A}$ is a subgroup of ${\bf T}^A$. More precisely, ${\bf T}^A$ can be considered as a Cartesian product of copies of ${\bf T}$ indexed by $A$, equipped with the product topology corresponding to the standard topology on ${\bf
T}$, and ${\bf T}^A$ is a compact topological group with respect to this topology, as in the previous section. One can check that $\widehat{A}$ is a closed subgroup of ${\bf T}^A$ with respect to the product topology, so that $\widehat{A}$ becomes a compact topological group with respect to the induced topology. This topology on $\widehat{A}$ is the same as the one mentioned in Section \ref{dual
groups} in this case, because compact subsets of $A$ are finite when $A$ is equipped with the discrete topology.
Similarly, if $E$ is any nonempty set, then the collection ${\bf T}^E$ of mappings from $E$ into ${\bf T}$ is a compact commutative topological group with respect to pointwise multiplication and the product topology that corresponds to the standard topology on ${\bf T}$, as in the previous paragraph. If $E \subseteq A$, then there is an obvious homomorphism from ${\bf T}^A$ onto ${\bf T}^E$, that sends each mapping from $A$ into ${\bf T}$ to its restriction to $E$. This homomorphism is continuous with respect to the corresponding product topologies, and the restriction of this homomorphism to $\widehat{A}$ is a continuous homomorphism from $\widehat{A}$ into ${\bf T}^E$. Suppose that $E$ is a set of generators of $A$, in the sense that every element of $A$ can be expressed as a sum of finitely many elements of $E$ and their inverses, where elements of $E$ may be repeated. Under these conditions, the homomorphism from $\widehat{A}$ into ${\bf T}^E$ just mentioned is a homeomorphism of $\widehat{A}$ onto its image in ${\bf T}^E$, with respect to the topology on the image of $\widehat{A}$ in ${\bf T}^E$ induced by the product topology on ${\bf T}^E$.
Let $B$ be a subgroup of $A$, let $x \in A \backslash B$ be given, and let $B(x)$ be the subgroup of $A$ generated by $B$ and $x$. If $\phi$ is a homomorphism from $B$ into ${\bf T}$, then it is well known that $\phi$ can be extended to a homomorphism from $B(x)$ into ${\bf T}$. If $A$ is generated by $B$ and finitely or countably many other elements of $A$, then one can repeat the process to get an extension of $\phi$ to a homomorphism from $A$ into ${\bf T}$, and otherwise one can use Zorn's lemma or the Hausdorff maximality principle. Using this, one can show that for each $a \in A$ with $a \ne 0$ there is a homomorphism $\phi$ from $A$ into ${\bf T}$ such that $\phi(a) \ne 1$. This implies that characters on $A$ separate points in $A$.
Put \begin{equation} \label{Psi_a(phi) = phi(a)}
\Psi_a(\phi) = \phi(a) \end{equation} for each $a \in A$ and $\phi \in \widehat{A}$, so that $\Psi_a$ maps $\widehat{A}$ into ${\bf T}$. More precisely, $\Psi_a$ is a continuous homomorphism from $\widehat{A}$ into ${\bf T}$, with respect to the topology on $\widehat{A}$ discussed earlier. Thus $\Psi_a$ is an element of the dual $\widehat{\widehat{A}}$ of the dual $\widehat{A}$ of $A$, and it is easy to see that \begin{equation} \label{a mapsto Psi_a}
a \mapsto \Psi_a \end{equation} defines a homomorphism from $A$ into $\widehat{\widehat{A}}$. Note that $\widehat{\widehat{A}}$ should be equipped with the discrete topology, because $\widehat{A}$ is compact, so that (\ref{a mapsto
Psi_a}) is automatically continuous. This mapping (\ref{a mapsto
Psi_a}) is also one-to-one, because $\widehat{A}$ separates points in $A$, as in the previous paragraph. Of course, the collection of elements of $\widehat{\widehat{A}}$ of the form $\Psi_a$ for some $a \in A$ is a subgroup of $\widehat{\widehat{A}}$. This subgroup of $\widehat{\widehat{A}}$ automatically separates points in $\widehat{A}$, because $\phi \in \widehat{A}$ is not the trivial character exactly when (\ref{Psi_a(phi) = phi(a)}) is not equal to $1$ for some $a \in A$. It follows that every element of $\widehat{\widehat{A}}$ is of the form $\Psi_a$ for some $a \in A$, as in Section \ref{compact commutative groups}, because $\widehat{A}$ is compact.
\section{Characters on ${\bf Z}_p$} \label{characters on Z_p}
Let us begin with some remarks about cyclic groups. Let $n$ be a positive integer, let $n \, {\bf Z}$ be the subgroup of ${\bf Z}$ consisting of integer multiples of $n$, and let ${\bf Z} / n \, {\bf Z}$ be the corresponding quotient group, which is a cyclic group of order $n$. Also let $w \in {\bf C}$ be an $n$th root of unity, so that $w^n = 1$, which implies that the modulus of $w$ is equal to $1$. The mapping from $j \in {\bf Z}$ to $w^j \in {\bf T}$ is equal to $1$ on $n \, {\bf Z}$, and hence determines a group homomorphism from ${\bf Z} / n \, {\bf Z}$ into ${\bf T}$. Every homomorphism from ${\bf Z} / n \, {\bf Z}$ into ${\bf T}$ is of this form, which implies that the dual group associated to ${\bf Z} / n \, {\bf Z}$ is also a cyclic group of order $n$.
Now let $p$ be a prime number, and let $\phi$ be a continuous homomorphism from ${\bf Z}_p$ as a commutative topological group with respect to addition into ${\bf T}$. The continuity of $\phi$ implies that there is a nonnegative integer $k$ such that the real part of $\phi(x)$ is positive for every $x \in p^k \, {\bf Z}_p$. This implies that $\phi(x) = 1$ for every $x \in p^k \, {\bf Z}_p$, as in Section \ref{dual groups}, because $p^k \, {\bf Z}_p$ is a subgroup of ${\bf Z}_p$. It follows that $\phi$ determines a homomorphism from ${\bf Z}_p / p^k \, {\bf Z}_p$ into ${\bf T}$. We have also seen in Section \ref{p-adic numbers} that ${\bf Z} / p^k \, {\bf Z}_p$ is isomorphic as a group to ${\bf Z} / p^k \, {\bf Z}$, so that the induced homomorphism from ${\bf Z}_p / p^k \, {\bf Z}_p$ into ${\bf
T}$ is of the form described in the preceding paragraph. Conversely, every homomorphism from ${\bf Z}_p / p^k \, {\bf Z}_p$ into ${\bf T}$ leads to a homomorphism from ${\bf Z}_p$ into ${\bf
T}$, by composition with the canonical quotient mapping from ${\bf
Z}_p$ onto ${\bf Z}_p / p^k \, {\bf Z}_p$. Any homomorphism from ${\bf Z}_p$ into ${\bf T}$ of this type is automatically continuous, because $p^k \, {\bf Z}_p$ is an open subgroup of ${\bf Z}_p$ for each $k \ge 0$.
Let $n$ be a positive integer again, and consider the space of complex-valued functions on ${\bf Z} / n \, {\bf Z}$. This is an $n$-dimensional vector space, which may be equipped with a translation-invariant inner product as in Section \ref{compact
commutative groups}. Of course, normalized Haar measure on ${\bf Z} / n \, {\bf Z}$ is simply the measure that assigns the value $1/n$ to each element of ${\bf Z} / n \, {\bf Z}$. As before, characters on ${\bf Z} / n \, {\bf Z}$ are orthonormal with respect to this inner product. It follows that the characters on ${\bf Z} / n \, {\bf Z}$ form an orthonormal basis for the space of complex-valued functions on ${\bf Z} / n \, {\bf Z}$, since there are exactly $n$ characters on ${\bf Z} / n \, {\bf Z}$.
Similarly, characters on ${\bf Z}_p$ are orthonormal with respect to the $L^2$ inner product associated to normalized Haar measure on ${\bf Z}_p$. Note that there are $p^k$ characters on ${\bf Z}_p$ obtained from characters on ${\bf Z}_p / p^k \, {\bf Z}_p$ for each nonnegative integer $k$. The linear span of these $p^k$ characters consists of the complex-valued functions on ${\bf Z}_p$ that are constant on the cosets of $p^k \, {\bf Z}_p$ in ${\bf Z}_p$, which is a vector space of dimension $p^k$. The linear span of the set of all characters on ${\bf Z}_p$ is the space of complex-valued functions on ${\bf Z}_p$ that are constant on the cosets of $p^k \, {\bf Z}_p$ in ${\bf Z}_p$ for some nonnegative integer $k$. In particular, this implies that the linear span of the characters on ${\bf Z}_p$ is dense in the space of all continuous complex-valued functions on ${\bf Z}_p$ with respect to the supremum norm, and hence in $L^2({\bf Z}_p)$ as well.
\section{The quotient group ${\bf Q}_p / {\bf Z}_p$} \label{the quotient group Q_p / Z_p}
Let $p$ be a prime number again, and consider the quotient ${\bf Q}_p / {\bf Z}_p$ of ${\bf Q}_p$ as a commutative group with respect to addition by its subgroup ${\bf Z}_p$. Also let ${\bf
Z}[1/p]$ be the collection of rational numbers of the form $p^{-j} \, x$, where $x \in {\bf Z}$, and $j$ is a nonnegative integer. This is a dense subgroup of ${\bf Q}_p$ with respect to addition and the $p$-adic metric, because ${\bf Z}$ is dense in ${\bf Z}_p$, as in Section \ref{p-adic numbers}. It follows that the image of ${\bf
Z}[1/p]$ under the canonical quotient mapping from ${\bf Q}_p$ onto ${\bf Q}_p / {\bf Z}_p$ is all of ${\bf Q}_p / {\bf Z}_p$, since ${\bf
Z}_p$ is an open subgroup of ${\bf Q}_p$. Thus we get a homomorphism from ${\bf Z}[1/p]$ onto ${\bf Q}_p / {\bf Z}_p$ whose kernel is equal to \begin{equation} \label{{bf Z}[1/p] cap {bf Z}_p = {bf Z}}
{\bf Z}[1/p] \cap {\bf Z}_p = {\bf Z}. \end{equation} This leads to a group isomorphism from ${\bf Z}[1/p] / {\bf Z}$ onto ${\bf Q}_p / {\bf Z}_p$. Because ${\bf Z}_p$ is an open subgroup of ${\bf Q}_p$, we take ${\bf Q}_p / {\bf Z}_p$ to be equipped with the discrete topology.
Alternatively, observe that \begin{equation} \label{{bf Z}[1/p] = bigcup_{j = 0}^infty p^{-j} {bf Z}}
{\bf Z}[1/p] = \bigcup_{j = 0}^\infty p^{-j} \, {\bf Z} \end{equation} and \begin{equation} \label{{bf Q}_p = bigcup_{j = 0}^infty p^{-j} {bf Z}_p}
{\bf Q}_p = \bigcup_{j = 0}^\infty p^{-j} \, {\bf Z}_p, \end{equation} which imply that \begin{equation} \label{{bf Z}[1/p] / {bf Z} = bigcup_{j = 0}^infty ((p^{-j} {bf Z}) / {bf Z})}
{\bf Z}[1/p] / {\bf Z} = \bigcup_{j = 0}^\infty ((p^{-j} \, {\bf Z}) / {\bf Z}) \end{equation} and \begin{equation} \label{{bf Q}_p / {bf Z}_p = bigcup_{j = 0}^infty ((p^{-j} {bf Z}_p)/{bf Z}_p)} {\bf Q}_p / {\bf Z}_p = \bigcup_{j = 0}^\infty ((p^{-j} \, {\bf Z}_p) / {\bf Z}_p). \end{equation} Of course, $(p^{-j} \, {\bf Z}) / {\bf Z}$ is isomorphic as a group to ${\bf Z} / p^j \, {\bf Z}$ for each nonnegative integer $j$. We also have that $p^{-j} \, {\bf Z}_p / {\bf Z}_p$ is isomorphic as a group to ${\bf Z}_p / p^j \, {\bf Z}_p$, which is isomorphic to ${\bf Z} / p^j \, {\bf Z}$ when $j \ge 0$, as in Section \ref{p-adic numbers}. The obvious inclusion of $p^{-j} \, {\bf Z}$ in $p^{-j} \, {\bf Z}_p$ leads more directly to a group homomorphism from $(p^{-j} \, {\bf Z}) / {\bf Z}$ into $(p^{-j} \, {\bf Z}_p) / {\bf Z}_p$, which is actually an isomorphism, for the usual reasons. It is easy to see that the isomorphism from ${\bf Z}[1/p] / {\bf Z}$ onto ${\bf Q}_p / {\bf Z}_p$ described in the previous paragraph sends $(p^{-j} \, {\bf Z}) / {\bf
Z}$ onto $(p^{-j} \, {\bf Z}_p) / {\bf Z}_p$ in this way for each nonnegative integer $j$.
We can also consider ${\bf Z}[1/p] \subseteq {\bf Q}$ as a subgroup of ${\bf R}$ with respect to addition, so that ${\bf Z}[1/p] / {\bf Z}$ can be identified with a subgroup of ${\bf R} / {\bf Z}$. If $\exp z$ is the complex exponential function, then \begin{equation} \label{r mapsto exp (2 pi i r)}
r \mapsto \exp (2 \, \pi \, i \, r) \end{equation} defines a continuous homomorphism from ${\bf R}$ as a commutative topological group with respect to addition onto ${\bf T}$ as a compact commutative group with respect to multiplication. The kernel of this homomorphism is equal to ${\bf Z}$, which leads to an isomorphism from ${\bf R} / {\bf Z}$ onto ${\bf T}$. This isomorphism sends ${\bf
Z}[1/p] / {\bf Z}$ onto the subgroup of ${\bf T}$ consisting of all roots of unity with order equal to $p^l$ for some nonnegative integer $l$. Using the isomorphism between ${\bf Z}[1/p] / {\bf Z}$ and ${\bf
Q}_p / {\bf Z}_p$ described earlier, we get a homomorphism from ${\bf Q}_p$ into ${\bf T}$ with kernel ${\bf Z}_p$.
Equivalently, \begin{equation} \label{E_p(x') = exp (2 pi i x')}
E_p(x') = \exp (2 \, \pi \, i \, x') \end{equation} defines a homomorphism from ${\bf Z}[1/p]$ as a commutative group with respect to addition into ${\bf T}$ as a commutative group with respect to multiplication, with kernel equal to ${\bf Z}$. If $x \in {\bf
Q}_p$, then there is an $x' \in {\bf Z}[1/p]$ such that $x - x' \in {\bf Z}_p$, because ${\bf Z}[1/p]$ is dense in ${\bf Q}_p$, as before. If $x'' \in {\bf Z}[1/p]$ also satisfies $x - x'' \in {\bf Z}_p$, then $x' - x'' \in {\bf Z}_p$, and hence $x' - x'' \in {\bf Z}$, as in (\ref{{bf Z}[1/p] cap {bf Z}_p = {bf Z}}). This implies that $E_p(x') = E_p(x'')$, so that we can extend $E_p$ to a mapping from ${\bf Q}_p$ into ${\bf T}$ by putting \begin{equation} \label{E_p(x) = E_p(x')}
E_p(x) = E_p(x') \end{equation} when $x \in {\bf Q}_p$, $x' \in {\bf Z}[1/p]$, and $x - x' \in {\bf
Z}_p$. If $x, y \in {\bf Q}_p$ and $x', y' \in {\bf Z}[1/p]$ satisfy $x - x', y - y' \in {\bf Z}_p$, then $x' + y' \in {\bf
Z}[1/p]$ and \begin{equation} \label{(x + y) - (x' + y') = (x - x') + (y - y') in {bf Z}_p}
(x + y) - (x' + y') = (x - x') + (y - y') \in {\bf Z}_p, \end{equation} so that \begin{equation} \label{E_p(x + y) = E_p(x' + y') = E_p(x') E_p(y') = E_p(x) E_p(y)}
E_p(x + y) = E_p(x' + y') = E_p(x') \, E_p(y') = E_p(x) \, E_p(y). \end{equation} Thus the extension of $E_p$ to ${\bf Q}_p$ is a homomorphism from ${\bf Q}_p$ as a commutative group with respect to addition into ${\bf
T}$ as a commutative group with respect to multiplication. It is easy to see that the kernel of this homomorphism is equal to ${\bf
Z}_p$, since the kernel of $E_p$ on ${\bf Z}[1/p]$ is equal to ${\bf
Z}$.
\section{Characters on ${\bf Q}_p$} \label{characters on Q_p}
Let $p$ be a prime number, and let $\phi$ be a continuous homomorphism from ${\bf Q}_p$ into ${\bf T}$. Thus the restriction of $\phi$ to ${\bf Z}_p$ is a continuous homomorphism from ${\bf Z}_p$ into ${\bf T}$, and hence there is a nonnegative integer $k$ such that $\phi(x) = 1$ for every $x \in p^k \, {\bf Z}_p$, as in Section \ref{characters on Z_p}. This implies that $\phi$ determines a homomorphism from ${\bf Q}_p / p^k \, {\bf Z}_p$ into ${\bf T}$. Conversely, every homomorphism from ${\bf Q}_p / p^k \, {\bf Z}_p$ into ${\bf T}$ leads to a homomorphism from ${\bf Q}_p$ into ${\bf
T}$, by composition with the canonical quotient mapping from ${\bf
Q}_p$ onto ${\bf Q}_p / p^k \, {\bf Z}_p$. Any homomorphism from ${\bf Q}_p$ into ${\bf T}$ obtained in this way is continuous, because $p^k \, {\bf Z}_p$ is an open subgroup of ${\bf Q}_p$.
Let $E_p$ be the group homomorphism from ${\bf Q}_p$ into ${\bf T}$ discussed in the previous section, and put \begin{equation} \label{phi_y(x) = E_p(x y)}
\phi_y(x) = E_p(x \, y) \end{equation}
for each $x, y \in {\bf Q}_p$. Thus $\phi_y$ is a group homomorphism from ${\bf Q}_p$ into ${\bf T}$ for each $y \in {\bf Q}_p$, which is trivial when $y = 0$. Otherwise, if $y \ne 0$, so that $|y|_p = p^k$ for some integer $k$, then the kernel of $\phi_y$ is equal to $p^k \, {\bf Z}_p$. In particular, $\phi_y$ is continuous as a mapping from ${\bf Q}_p$ into ${\bf T}$ for every $y \in {\bf Q}_p$. This implies that $\phi_y$ is an element of the dual $\widehat{{\bf Q}_p}$ of ${\bf
Q}_p$ as a commutative topological group with respect to addition, and it is easy to see that the mapping from $y \in {\bf Q}_p$ to $\phi_y \in \widehat{{\bf Q}_p}$ is a group homomorphism.
Similarly, the restriction of $\phi_y$ to ${\bf Z}_p$ is a continuous homomorphism from ${\bf Z}_p$ as a commutative topological group with respect to addition into ${\bf T}$, and the mapping from $y \in {\bf Q}_p$ to the restriction of $\phi_y$ to ${\bf Z}_p$ is a group homomorphism from ${\bf Q}_p$ into the dual $\widehat{{\bf
Z}_p}$ of ${\bf Z}_p$. As before, the restriction of $\phi_y$ to ${\bf Z}_p$ is the trivial character on ${\bf Z}_p$ if and only if $y \in {\bf Z}_p$. This implies that the mapping from $y \in {\bf Q}_p$ to the restriction of $\phi_y$ to ${\bf Z}_p$ leads to an injective homomorphism from ${\bf Q}_p / {\bf Z}_p$ into $\widehat{{\bf Z}_p}$. One can check that every continuous group homomorphism from ${\bf
Z}_p$ into ${\bf T}$ is equal to the restriction of $\phi_y$ to ${\bf Z}_p$ for some $y \in {\bf Q}_p$, so that $\widehat{{\bf Z}_p}$ is isomorphic to ${\bf Q}_p / {\bf Z}_p$ as a group. More precisely, if $\phi$ is a homomorphism from ${\bf Z}_p$ into ${\bf T}$ whose kernel contains $p^k \, {\bf Z}_p$ for some nonnegative integer $k$, then $\phi$ is equal to the restriction of $\phi_y$ to ${\bf Z}_p$ for some $y \in p^{-k} \, {\bf Z}_p$.
If $y \in {\bf Z}_p$, then the kernel of $\phi_y : {\bf Q}_p \to {\bf T}$ contains ${\bf Z}_p$, and hence $\phi_y$ determines a homomorphism $\psi_y$ from ${\bf Q}_p / {\bf Z}_p$ into ${\bf T}$. As in the previous section, we consider ${\bf Q}_p / {\bf Z}_p$ to be equipped with the discrete topology, so that every homomorphism from ${\bf Q}_p / {\bf Z}_p$ into ${\bf T}$ is automatically continuous. Thus $\psi_y$ is an element of the dual $\widehat{({\bf Q}_p / {\bf Z}_p)}$ of ${\bf Q}_p / {\bf Z}_p$ as a commutative topological group with respect to the discrete topology for each $y \in {\bf Z}_p$. Note that $\psi_y$ is the trivial character on ${\bf Q}_p / {\bf Z}_p$ if and only if $\phi_y$ is the trivial character on ${\bf Q}_p$, which happens if and only if $y = 0$. It is easy to see that $y \mapsto \psi_y$ defines a group homomorphism from ${\bf Z}_p$ into $\widehat{({\bf Q}_p / {\bf Z}_p)}$, as usual.
If $\psi$ is any homomorphism from ${\bf Q}_p / {\bf Z}_p$ into ${\bf T}$, then one can check that there is a $y \in {\bf Z}_p$ such that $\psi = \psi_y$. This is the same as saying that if $\phi$ is a homomorphism from ${\bf Q}_p$ into ${\bf T}$ whose kernel contains ${\bf Z}_p$, then there is a $y \in {\bf Z}_p$ such that $\phi = \phi_y$. To see this, one can start by showing that for each nonnegative integer $j$, there is a $y_j \in {\bf Z}_p$ such that $\phi = \phi_{y_j}$ on $p^{-j} \, {\bf Z}_p$. The image of $y_j$ in ${\bf Z}_p / p^j \, {\bf Z}_p$ is uniquely determined by this property, which implies that $\{y_j\}_{j = 1}^\infty$ is a Cauchy sequence in ${\bf Z}_p$. Thus $\{y_j\}_{j = 1}^\infty$ converges to an element $y$ of ${\bf Z}_p$, by completeness of the $p$-adic metric, and one can verify that $\phi = \phi_y$.
It follows that $y \mapsto \psi_y$ is a group isomorphism of ${\bf Z}_p$ onto $\widehat{({\bf Q}_p / {\bf Z}_p)}$. Because ${\bf Q}_p / {\bf Z}_p$ is equipped with discrete topology, $\widehat{({\bf Q}_p / {\bf Z}_p)}$ is compact with respect to the usual topology on the dual group. One can also check that $y \mapsto \psi_y$ is a homeomorphism with respect to the topology on ${\bf Z}_p$ determined by the $p$-adic metric and the usual topology on the dual group $\widehat{({\bf Q}_p / {\bf Z}_p)}$.
If $\phi$ is any continuous homomorphism from ${\bf Q}_p$ into ${\bf T}$, then the kernel of $\phi$ contains $p^k \, {\bf Z}_p$ for some integer $k$. Under these conditions, there is a $y \in p^{-k} \, {\bf Z}_p$ such that $\phi = \phi_y$ on ${\bf Q}_p$. This follows from the previous discussion when $k = 0$, and otherwise it is easy to reduce to that case. This implies that $y \mapsto \phi_y$ defines a group isomorphism from ${\bf Q}_p$ onto its dual. It is not too difficult to verify that this mapping is also a homeomorphism with respect to the topology on ${\bf Q}_p$ defined by the $p$-adic metric and the corresponding topology on $\widehat{{\bf Q}_p}$.
\chapter{$r$-Adic integers} \label{r-adic integers}
\section{$r$-Adic absolute values} \label{r-adic absolute values}
Let $r = \{r_j\}_{j = 0}^\infty$ be a sequence of positive integers, with $r_j \ge 2$ for each $j$. Put \begin{equation} \label{R_l = prod_{j = 1}^l r_j}
R_l = \prod_{j = 1}^l r_j \end{equation} for each positive integer $l$, and $R_0 = 1$, so that $\{R_l\}_{l =
0}^\infty$ is a strictly increasing sequence of positive integers. Note that $R_{l + 1} \, {\bf Z}$ is a proper subset of $R_l \, {\bf
Z}$ for each $l \ge 0$, and that $\bigcap_{l = 0}^\infty R_l \, {\bf
Z} = \{0\}$. If $a$ is a nonzero integer, then let $l_r(a)$ be the largest nonnegative integer such that $a \in R_{l_r(a)} \, {\bf Z}$, and put $l_r(0) = +\infty$. Equivalently, $l_r(a) + 1$ is the smallest positive integer such that $a \not\in R_{l_r(a) + 1} \, {\bf
Z}$ when $a \ne 0$. It is easy to see that \begin{equation} \label{l_r(a + b) ge min(l_r(a), l_r(b))}
l_r(a + b) \ge \min(l_r(a), l_r(b)) \end{equation} and \begin{equation} \label{l_r(a b) ge max(l_r(a), l_r(b))}
l_r(a \, b) \ge \max(l_r(a), l_r(b)) \end{equation} for every $a, b \in {\bf Z}$. In particular, $l_r(-a) = l_r(a)$ for each $a \in {\bf Z}$. If $r$ is a constant sequence, then \begin{equation} \label{l_r(a b) ge l_r(a) + l_r(b)}
l_r(a \, b) \ge l_r(a) + l_r(b) \end{equation} for every $a, b \in {\bf Z}$, and equality holds when $r_1$ is a prime number.
Let $t = \{t_l\}_{l = 0}^\infty$ be a strictly decreasing sequence of positive real numbers that converges to $0$, with $t_0 = 1$. Put \begin{equation}
\label{|a|_r = t_{l_r(a)}}
|a|_r = t_{l_r(a)} \end{equation}
for each nonzero integer $a$, and $|0|_r = 0$, which corresponds to
(\ref{|a|_r = t_{l_r(a)}}) with $t_\infty = 0$. Let us call $|a|_r$ the \emph{$r$-adic absolute value}\index{r-adic absolute
value@$r$-adic absolute value} of $a \in {\bf Z}$, although it also depends on $t$. If $p$ is a prime number, $r_j = p$ for each $j \ge 1$, and $t_l = p^{-l}$ for each $l \ge 0$, then this reduces to the usual $p$-adic absolute value on ${\bf Z}$, as in Section \ref{p-adic
absolute value}.
Observe that \begin{equation}
\label{|a + b|_r le max(|a|_r, |b|_r)}
|a + b|_r \le \max(|a|_r, |b|_r) \end{equation} and \begin{equation}
\label{|a b|_r le min(|a|_r, |b|_r)}
|a \, b|_r \le \min(|a|_r, |b|_r) \end{equation} for every $a, b \in {\bf Z}$, by (\ref{l_r(a + b) ge min(l_r(a),
l_r(b))}) and (\ref{l_r(a b) ge max(l_r(a), l_r(b))}). If $r$ is a constant sequence, and if $t$ is submultiplicative in the sense that \begin{equation} \label{t_{k + l} le t_k t_l}
t_{k + l} \le t_k \, t_l \end{equation} for every $k, l \ge 0$, then \begin{equation}
\label{|a b|_r le |a|_r |b|_r}
|a \, b|_r \le |a|_r \, |b|_r \end{equation} for every $a, b \in {\bf Z}$, by (\ref{l_r(a b) ge l_r(a) + l_r(b)}). If $p$ is a prime number, $r_j = p$ for each $j \ge 1$, and $t_l =
(t_1)^l$ for each $l \ge 0$, then equality holds in (\ref{|a b|_r le
|a|_r |b|_r}) for each $a, b \in {\bf Z}$. Of course, this reduces to the case of the $p$-adic absolute value when $t_1 = 1/p$, and otherwise $|a|_r$ would be the same as a positive power of the $p$-adic absolute value of $a$.
Put \begin{equation}
\label{d_r(a, b) = |a - b|_r}
d_r(a, b) = |a - b|_r \end{equation} for every $a, b \in {\bf Z}$, which we shall call the \emph{$r$-adic
metric}\index{r-adic metric@$r$-adic metric} on ${\bf Z}$, although it also depends on $t$, as before. This is symmetric in $a$ and $b$, because $l_r(-c) = l_r(c)$ for every $c \in {\bf Z}$, and hence
$|-c|_r = |c|_r$. Using (\ref{|a + b|_r le max(|a|_r, |b|_r)}), we get that \begin{equation} \label{d_r(a, c) le max(d_r(a, b), d_r(b, c))}
d_r(a, c) \le \max(d_r(a, b), d_r(b, c)) \end{equation} for every $a, b, c \in {\bf Z}$, so that $d_r(\cdot, \cdot)$ is an ultrametric on ${\bf Z}$. As usual, this reduces to the $p$-adic metric on ${\bf Z}$ when $r_j = p$ for some prime number $p$ and every $j \ge 1$, and $t_l = p^{-l}$ for each $l \ge 0$.
\section{An embedding} \label{an embedding}
Let $r = \{r_j\}_{j = 1}^\infty$ and $R_l$ be as in the previous section, and consider the Cartesian product \begin{equation} \label{X = prod_{l = 1}^infty ({bf Z} / R_l {bf Z})}
X = \prod_{l = 1}^\infty ({\bf Z} / R_l \, {\bf Z}), \end{equation} consisting of the sequences $x = \{x_l\}_{l = 1}^\infty$ with $x_l \in {\bf Z} / R_l \, {\bf Z}$ for each $l$. As in Section \ref{abstract
cantor sets}, this is a compact Hausdorff topological space with respect to the product topology that corresponds to the discrete topology on $X_l = {\bf Z} / R_l \, {\bf Z}$ for each $l$. This is also a commutative ring with respect to coordinatewise addition and multiplication, using the standard ring structure on ${\bf Z} / R_l \, {\bf Z}$ for each $l$. It is easy to see that the ring operations are continuous on $X$, so that $X$ is a topological ring.
Let $q_l$ be the canonical quotient mapping from ${\bf Z}$ onto ${\bf Z} / R_l \, {\bf Z}$ for each $l \ge 1$, which is a ring homomorphism with kernel $R_l \, {\bf Z}$. Thus \begin{equation} \label{q(a) = {q_l(a)}_{l = 1}^infty}
q(a) = \{q_l(a)\}_{l = 1}^\infty \end{equation} is an element of $X$ for each $a \in {\bf Z}$, so that $q$ defines a mapping from ${\bf Z}$ into $X$. This mapping is an injective ring homomorphism, because $\bigcap_{l = 1}^\infty R_l \, {\bf Z} = \{0\}$. If $l(x, y)$ is defined for $x, y \in X$ as in Section \ref{abstract cantor sets}, and $l_r(a)$ is as in the previous section, then \begin{equation} \label{l(q(a), q(b)) = l_r(a - b)}
l(q(a), q(b)) = l_r(a - b) \end{equation} for every $a, b \in {\bf Z}$.
Let $t = \{t_l\}_{l = 0}^\infty$ be a strictly decreasing sequence of positive real numbers that converges to $0$ and with $t_0 = 1$, as before. This leads to an ultrametric $d(x, y)$ on $X$ as in (\ref{d(x, y) = t_{l(x, y)}}), and to an $r$-adic metric $d_r(a, b)$
on ${\bf Z}$, as in (\ref{d_r(a, b) = |a - b|_r}). Under these conditions, \begin{equation}
d(q(a), q(b)) = d_r(a, b) \end{equation}
for every $a, b \in {\bf Z}$, using also (\ref{|a|_r = t_{l_r(a)}}). By construction, $d_r(a, b)$ is invariant under translations on ${\bf
Z}$, and in fact $d(x, y)$ is invariant under translations on $X$ as a commutative group with respect to addition as well.
Note that $X$ is complete with respect to $d(x, y)$, because $X$ is compact. One can also check this directly from the definitions. It follows that the completion of ${\bf Z}$ with respect to $d_r(a, b)$ can be identified with the closure of $q({\bf Z})$ in $X$. We shall discuss this further in the next section.
\section{Coherent sequences} \label{coherent sequences}
Let us continue with the notation and hypotheses in the previous sections. Observe that there is a natural ring homomorphism from ${\bf Z} / R_{l + 1} \, {\bf Z}$ onto ${\bf Z} / R_l \, {\bf Z}$ for each $l \ge 1$, because $R_{l + 1} \, {\bf Z} \subseteq R_l \, {\bf Z}$. An element $x = \{x_l\}_{l = 1}^\infty$ of $X$ is said to be a \emph{coherent sequence}\index{coherent sequences} if the image of $x_{l + 1}$ under the natural homomorphism from ${\bf Z} / R_{l +
1} \, {\bf Z}$ onto ${\bf Z} / R_l \, {\bf Z}$ is equal to $x_l$ for each $l$. Let $Y$ be the subset of $X$ consisting of all coherent sequences, which is a sub-ring of $X$ with respect to termwise addition and multiplication. It is easy to see that $Y$ is also a closed set in $X$ with respect to the product topology, which implies that $Y$ is compact, since $X$ is compact.
If $a \in {\bf Z}$, then the image of $q_{l + 1}(a)$ under the natural mapping from ${\bf Z} / R_{l + 1} \, {\bf Z}$ onto ${\bf Z} / R_l \, {\bf Z}$ is equal to $q_l(a)$ for each $l$, so that $q(a) = \{q_l(a)\}_{l = 1}^\infty$ is a coherent sequence. Thus $q({\bf Z}) \subseteq Y$, and one can check that \begin{equation} \label{overline{q({bf Z})} = Y}
\overline{q({\bf Z})} = Y, \end{equation} where $\overline{q({\bf Z})}$ is the closure of $q({\bf Z})$ with respect to the product topology on $X$. More precisely, let $x \in Y$ and $n \in {\bf Z}_+$ be given, and let $a$ be an integer such that $q_n(a) = x_n$. This implies that $q_l(a) = x_l$ when $l \le n$, because $x = \{x_l\}_{l = 1}^\infty$ is a coherent sequence. It follows that $x \in \overline{q({\bf Z})}$, as desired, since $n$ is arbitrary.
The space ${\bf Z}_r$ of \emph{$r$-adic integers}\index{r-adic integers@$r$-adic integers} can be obtained initially as a metric space by completing ${\bf Z}$ with respect to the $r$-adic metric. Using the isometric embedding $q$ of ${\bf Z}$ in $X$, ${\bf Z}_r$ can be identified with the set $Y$ of coherent sequences in $X$, equipped with the restriction of the metric $d(x, y)$ on $X$ to $Y$. This identification is very convenient for showing that addition and multiplication on ${\bf Z}$ can be extended continuously to ${\bf Z}_r$, so that ${\bf Z}_r$ is a compact commutative topological group. Similarly, the $r$-adic absolute value can be extended to ${\bf Z}_r$, by taking the distance to $0$ in ${\bf Z}_r$, and it satisfies properties like those on ${\bf Z}$.
If $t' = \{t_l'\}_{l = 0}^\infty$ is another sequence of positive real numbers that converges to $0$, then we get another $r$-adic absolute value function $|a|_r'$ on ${\bf Z}$ as in (\ref{|a|_r =
t_{l_r(a)}}), a corresponding $r$-adic metric $d_r'(a, b)$ on ${\bf
Z}$ as in (\ref{d_r(a, b) = |a - b|_r}), and another metric $d'(x, y)$ on $X$ as in (\ref{d(x, y) = t_{l(x, y)}}). However, the embedding $q$ of ${\bf Z}$ into $X$ and the set $Y$ of coherent sequences in $X$ do not depend on $t$, and the metric $d'(x, y)$ also determines the product topology on $X$. The identity mapping on ${\bf
Z}$ is uniformly continuous as a mapping from ${\bf Z}$ equipped with $d_r(a, b)$ onto ${\bf Z}$ equipped with $d_r'(a, b)$, as well as in the other direction, and there are analogous statements for the identity mapping on $X$ and the metrics $d(x, y)$ and $d'(x, y)$. In particular, the completion ${\bf Z}_r$ of ${\bf Z}$ does not depend on the choice of $t$ as a topological ring.
\section{Haar measure on ${\bf Z}_r$} \label{haar measure on Z_r}
Let us continue with the same notation and hypotheses as before. In particular, let us identify the ring ${\bf Z}_r$ of $r$-adic integers with the set $Y$ of coherent sequences in $X$. Let $n$ be a positive integer, and put \begin{eqnarray} \label{Y_n = {x = {x_l}_{l = 1}^infty in Y : x_n = 0} = ...}
Y_n & = & \{x = \{x_l\}_{l = 1}^\infty \in Y : x_n = 0\} \\
& = & \{x = \{x_l\}_{l = 1}^\infty \in Y : x_l = 0 \hbox{ for each } l \le n\},
\nonumber \end{eqnarray} where the second step uses the fact that $x \in Y$ is a coherent sequence. This is a closed set in $X$ with respect to the product topology, and an ideal in $Y$ as a commutative ring. This is also a relatively open set in $Y$, because ${\bf Z} / R_n \, {\bf Z}$ is equipped with the discrete topology. It is easy to see that \begin{equation} \label{overline{q(R_n {bf Z})} = Y_n}
\overline{q(R_n \, {\bf Z})} = Y_n, \end{equation} for essentially the same reasons as in (\ref{overline{q({bf Z})} =
Y}). Let $\pi_n$ be the mapping from $Y$ into ${\bf Z} / R_n \, {\bf Z}$ defined by \begin{equation} \label{pi_n(x) = x_n}
\pi_n(x) = x_n, \end{equation} which is a ring homomorphism from $Y$ into ${\bf Z} / R_n \, {\bf Z}$ whose kernel is equal to $Y_n$. Of course, \begin{equation} \label{pi_n(q(a)) = q_n(a)}
\pi_n(q(a)) = q_n(a) \end{equation} for every $a \in {\bf Z}$, so that $\pi_n$ maps $Y$ onto ${\bf Z} / R_n \, {\bf Z}$.
Let $H$ be Haar measure on $Y$, normalized so that $H(Y) = 1$. Observe that \begin{equation} \label{H(Y_n) = 1/R_n}
H(Y_n) = 1/R_n \end{equation} for each positive integer $n$, because $Y$ can be expressed as the disjoint union of $R_n$ translates of $Y_n$, by the discussion in the preceding paragraph. In this situation, it is easy to define the Haar integral of a continuous real or complex-valued function on $Y$ directly as a limit of Riemann sums, with the measure of $Y_n$ and its translates equal to $1/R_n$. This leads to a translation-invariant regular Borel measure on $Y$, by the Riesz representation theorem, which is Haar measure on $Y$.
Let $t = \{t_l\}_{l = 0}^\infty$ be a strictly decreasing sequence of positive real numbers that converges to $0$ and with $t_0 = 1$, as in Section \ref{r-adic absolute values}. This leads to an $r$-adic absolute value function $|a|_r$ on ${\bf Z}$ as in (\ref{|a|_r =
t_{l_r(a)}}), an $r$-adic metric $d_r(a, b)$ on ${\bf Z}$ as in
(\ref{d_r(a, b) = |a - b|_r}), and a metric $d(x, y)$ on $X$ as in (\ref{d(x, y) = t_{l(x, y)}}). The $r$-adic absolute value function and metric can be extended to ${\bf Z}_r$ in a natural way, as in the previous section, and the extension of the $r$-adic metric on ${\bf
Z}_r$ corresponds exactly to the restriction of $d(x, y)$ to $Y$. By construction, these metrics are invariant under translations, and the diameter of $Y$ is equal to $t_0 = 1$. Similarly, the diameter of $Y_n$ is equal to $t_n$ for each positive integer $n$.
Let $H^\alpha_{con}(E)$, $H^\alpha_\delta(E)$, and $H^\alpha(E)$ be defined for $\alpha \ge 0$, $0 < \delta \le \infty$, and $E \subseteq Y$ as in Chapter \ref{hausdorff measures}, using the restriction of $d(x, y)$ to $Y$. Because this is an ultrametric on $Y$, one may as well use coverings of $E \subseteq Y$ by closed balls in $Y$ in the definitions of $H^\alpha_{con}(E)$ and $H^\alpha_\delta(E)$, as in Section \ref{some special cases}. More precisely, one should consider the empty set as a closed ball in $Y$ when $\alpha = 0$, but we are mostly interested in $\alpha > 0$ here. Otherwise, the closed balls in $Y$ are $Y$ itself and the translates of $Y_n$ for each positive integer $n$.
Let us now restrict our attention for the rest of this section to the case where $\alpha = 1$ and \begin{equation} \label{t_l = 1/R_l}
t_l = 1/R_l \end{equation} for each $l \ge 0$, which satisfies the usual conditions on $t$. Remember that $H^1_{con}(E) \le H^1_\delta(E)$ for every $E \subseteq Y$ and $\delta > 0$, by construction. In the present situation, one can check that \begin{equation} \label{H^1_delta(E) = H^1_{con}(E), 2}
H^1_\delta(E) = H^1_{con}(E) \end{equation} for every $E \subseteq Y$ and $\delta > 0$, as in Section \ref{some
special cases}. This uses the fact that $Y_n$ can be expressed as the union of $R_k / R_n$ translates of $Y_k$ when $k \ge n$. It follows that \begin{equation} \label{H^1(E) = H^1_{con}(E), 2}
H^1(E) = H^1_{con}(E) \end{equation} for every $E \subseteq Y$ under these conditions, as before.
In particular, \begin{equation} \label{H^1(Y) = H^1_{con}(Y) le diam Y = 1}
H^1(Y) = H^1_{con}(Y) \le \mathop{\rm diam} Y = 1. \end{equation} In order to show that \begin{equation} \label{H^1(Y) = 1}
H^1(Y) = 1, \end{equation} it suffices to verify that $H^1_\delta(Y) \ge 1$ for every $\delta > 0$. Because $Y$ is compact, one might as well consider only coverings of $Y$ by finitely many sets in the definition of $H^1_\delta(Y)$, as in Section \ref{restricting the diameters}. In fact, it is enough to consider only coverings of $Y$ by closed balls, as mentioned earlier. One can then reduce to coverings of $Y$ by finitely many balls of the same diameter, by subdividing the balls in a covering of $Y$ as necessary. Thus one gets either a covering of $Y$ by itself, or by finitely many translates of $Y_k$ for some $k \ge 1$. The first case is trivial, and in the second case, we have that $Y$ cannot be covered by fewer than $R_k$ translates of $Y_k$. This implies that $H^1_\delta(Y) \ge 1$ for every $\delta > 0$, and hence that (\ref{H^1(Y) = 1}) holds.
Similarly, \begin{equation} \label{H^1(Y_n) = 1/R_n}
H^1(Y_n) = 1/R_n \end{equation} for each $n \ge 1$, which implies that $H^1(U) > 0$ when $U$ is a nonempty open subset of $Y$. Of course, any Hausdorff measure on $Y$ with respect to a translation-invariant metric on $Y$ is invariant under translations as well.
\section{Some related groups} \label{some related groups}
If $k$ is a positive integer, then let $k^{-1} \, {\bf Z}$ be the set of integer multiples of $1/k$, which is a subgroup of the group ${\bf Q}$ of rational numbers with respect to addition. Note that ${\bf Z} \subseteq k^{-1} \, {\bf Z}$, so that the quotient group \begin{equation} \label{(k^{-1} {bf Z}) / {bf Z}}
(k^{-1} \, {\bf Z}) / {\bf Z} \end{equation} can be defined in the usual way. Of course, (\ref{(k^{-1} {bf Z}) /
{bf Z}}) is isomorphic to ${\bf Z} / k \, {\bf Z}$.
Let $r = \{r_j\}_{j = 1}^\infty$ and $R_l$ be as in Section \ref{r-adic absolute values}, and observe that \begin{equation} \label{R_l^{-1} {bf Z} subseteq R_{l + 1}^{-1} {bf Z}}
R_l^{-1} \, {\bf Z} \subseteq R_{l + 1}^{-1} \, {\bf Z} \end{equation} for each $l \ge 0$. Thus \begin{equation} \label{bigcup_{l = 0}^infty R_l^{-1} {bf Z}}
\bigcup_{l = 0}^\infty R_l^{-1} \, {\bf Z} \end{equation} is also a subgroup of ${\bf Q}$ that contains ${\bf Z}$, so that the quotient group \begin{equation} \label{(bigcup_{l = 0}^infty R_l^{-1} {bf Z}) / {bf Z}}
\Big(\bigcup_{l = 0}^\infty R_l^{-1} \, {\bf Z}\Big) / {\bf Z} \end{equation} is defined. If we consider $(R_l^{-1} \, {\bf Z} / {\bf Z})$ as a subgroup of (\ref{(bigcup_{l = 0}^infty R_l^{-1} {bf Z}) / {bf Z}}), then \begin{equation} \label{(R_l^{-1} {bf Z}) / {bf Z} subseteq (R_{l + 1}^{-1} {bf Z}) / {bf Z}} (R_l^{-1} \, {\bf Z}) / {\bf Z} \subseteq (R_{l + 1}^{-1} \, {\bf Z}) / {\bf Z} \end{equation} for every $l$, because of (\ref{R_l^{-1} {bf Z} subseteq R_{l +
1}^{-1} {bf Z}}), and (\ref{(bigcup_{l = 0}^infty R_l^{-1} {bf Z})
/ {bf Z}}) is the same as \begin{equation} \label{bigcup_{l = 0}^infty (R_l^{-1} {bf Z}) / {bf Z}}
\bigcup_{l = 0}^\infty (R_l^{-1} \, {\bf Z}) / {\bf Z}. \end{equation} If $p$ is a prime number, and $r_j = p$ for each $j$, then (\ref{bigcup_{l = 0}^infty R_l^{-1} {bf Z}}) is the same as ${\bf
Z}[1/p]$, as in Section \ref{the quotient group Q_p / Z_p}.
Remember that $\exp (2 \pi i w)$ defines a continuous homomorphism from ${\bf R}$ as a commutative topological group with respect to addition onto ${\bf T}$ with kernel ${\bf Z}$, which leads to an isomorphism from ${\bf R} / {\bf Z}$ onto ${\bf T}$. The image of (\ref{(bigcup_{l = 0}^infty R_l^{-1} {bf Z}) / {bf Z}}) under this isomorphism consists of the $z \in {\bf T}$ such that \begin{equation} \label{z^{R_l} = 1}
z^{R_l} = 1 \end{equation} for some $l \ge 0$. In fact, every homomorphism from (\ref{(bigcup_{l
= 0}^infty R_l^{-1} {bf Z}) / {bf Z}}) into ${\bf T}$ takes values in this subgroup of ${\bf T}$. It follows that homomorphisms from (\ref{(bigcup_{l = 0}^infty R_l^{-1} {bf Z}) / {bf Z}}) into ${\bf T}$ correspond exactly to homomorphisms from (\ref{(bigcup_{l = 0}^infty
R_l^{-1} {bf Z}) / {bf Z}}) into itself composed with the embedding of (\ref{(bigcup_{l = 0}^infty R_l^{-1} {bf Z}) / {bf Z}}) into ${\bf
T}$ obtained from the complex exponential function.
Let $\theta$ be a homomorphism from (\ref{(bigcup_{l = 0}^infty R_l^{-1} {bf Z}) / {bf Z}}) into itself. Note that $(R_l^{-1} \, {\bf Z}) / {\bf Z}$ consists of exactly the elements $a$ of (\ref{(bigcup_{l = 0}^infty R_l^{-1} {bf Z}) / {bf Z}}) such that $R_l \cdot a$ is equal to $0$ in (\ref{(bigcup_{l = 0}^infty R_l^{-1} {bf Z}) / {bf Z}}). This implies that \begin{equation} \label{theta((R_l^{-1} {bf Z}) / {bf Z}) subseteq (R_l^{-1} {bf Z}) / {bf Z}}
\theta((R_l^{-1} \, {\bf Z}) / {\bf Z})
\subseteq (R_l^{-1} \, {\bf Z}) / {\bf Z} \end{equation} for each $l$. Let $\theta_l$ be the restriction of $\theta$ to $(R_l^{-1} \, {\bf Z}) / {\bf Z}$ for each $l \ge 1$.
Because $(R_l^{-1} \, {\bf Z}) / {\bf Z}$ is a cyclic group, $\theta_l$ is determined by its value at the generator of $(R_l^{-1} \, {\bf Z}) / {\bf Z}$ for each $l$. This permits $\theta_l$ to be expressed in terms of multiplication by an integer. This integer is determined by $\theta_l$ modulo $R_l$, so that $\theta_l$ corresponds to an element $x_l$ of ${\bf Z} / R_l \, {\bf Z}$. Conversely, every element of ${\bf Z} / R_l \, {\bf Z}$ determines a homomorphism from $(R_l^{-1} \, {\bf Z}) / {\bf Z}$ into itself in this way.
By construction, $\theta_l$ is equal to the restriction of $\theta_{l + 1}$ to $(R_l^{-1} \, {\bf Z}) / {\bf Z}$ for each $l$. This means exactly that $x_l$ is the image of $x_{l + 1}$ under the natural homomorphism from ${\bf Z} / R_{l + 1} \, {\bf Z}$ onto ${\bf Z} / R_l \, {\bf Z}$. Thus $x = \{x_l\}_{l = 1}^\infty$ is a coherent sequence, which is to say that $x$ is an element of the group $Y$ discussed in Section \ref{coherent sequences}. Conversely, every element of $Y$ leads to a sequence of homomorphisms $\theta_l$ from $(R_l^{-1} \, {\bf Z}) / {\bf Z}$ into itself such that $\theta_l$ is the restriction of $\theta_{l + 1}$ to $(R_l^{-1} \, {\bf Z}) / {\bf Z}$ for each $l$. This leads in turn to a homomorphism $\theta$ from (\ref{(bigcup_{l = 0}^infty R_l^{-1} {bf Z}) / {bf Z}}) into itself, since (\ref{(bigcup_{l = 0}^infty R_l^{-1} {bf Z}) / {bf Z}}) is the same as (\ref{bigcup_{l = 0}^infty (R_l^{-1} {bf Z}) / {bf Z}}).
The collection of homomorphisms from (\ref{(bigcup_{l = 0}^infty R_l^{-1} {bf Z}) / {bf Z}}) into itself is a commutative group with respect to addition. The discussion in the previous paragraphs determines a one-to-one correspondence between this group and $Y$, which is a group isomorphism. Here we consider (\ref{(bigcup_{l = 0}^infty R_l^{-1}
{bf Z}) / {bf Z}}) to be equipped with the discrete topology, so that the corresponding dual group consists of all homomorphisms from (\ref{(bigcup_{l = 0}^infty R_l^{-1} {bf Z}) / {bf Z}}) into ${\bf T}$. Because of the correspondence between homomorphisms from (\ref{(bigcup_{l = 0}^infty R_l^{-1} {bf Z}) / {bf Z}}) into ${\bf T}$ and homomorphisms from (\ref{(bigcup_{l = 0}^infty R_l^{-1} {bf Z}) /
{bf Z}}) into itself mentioned earlier, we get an isomorphism between $Y$ and the dual group associated to (\ref{(bigcup_{l =
0}^infty R_l^{-1} {bf Z}) / {bf Z}}). One can check that this isomorphism is also a homeomorphism with respect to the usual topology on the dual of (\ref{(bigcup_{l = 0}^infty R_l^{-1} {bf Z}) / {bf Z}}) as a discrete commutative group.
\section{Characters on ${\bf Z}_r$} \label{characters on Z_r}
Let us continue with the same notation and hypotheses as before, and let $\phi$ be a continuous homomorphism from $Y$ as a commutative topological group with respect to addition into ${\bf T}$. Thus the set of $x \in Y$ such that the real part of $\phi(x)$ is positive is an open set in $Y$ that contains $0$, and hence contains $Y_n$ for some positive integer $n$. This implies that $\phi(x) = 1$ for every $x \in Y_n$, as in Section \ref{dual groups}, because $Y_n$ is a subgroup of $Y$. If $\pi_n$ is the homomorphism from $Y$ onto ${\bf Z} / R_n \, {\bf Z}$ in (\ref{pi_n(x) = x_n}), then there is a homomorphism $\psi$ from ${\bf Z} / R_n \, {\bf Z}$ as a commutative group with respect to addition into ${\bf T}$ such that \begin{equation} \label{phi = psi circ pi_n}
\phi = \psi \circ \pi_n, \end{equation} because the kernel of $\pi_n$ is equal to $Y_n$. Conversely, if $\psi$ is a homomorphism from ${\bf Z} / R_n \, {\bf Z}$ as a commutative group with respect to addition into ${\bf T}$, then (\ref{phi = psi circ pi_n}) defines a continuous group homomorphism from $Y$ into ${\bf T}$.
Alternatively, we have seen in the previous section that $Y$ is isomorphic as a commutative topological group to the dual group associated to (\ref{(bigcup_{l = 0}^infty R_l^{-1} {bf Z}) / {bf Z}}), where (\ref{(bigcup_{l = 0}^infty R_l^{-1} {bf Z}) / {bf Z}}) is equipped with the discrete topology. As in Section \ref{discrete
commutative groups}, it follows that each element of (\ref{(bigcup_{l = 0}^infty R_l^{-1} {bf Z}) / {bf Z}}) determines a character on $Y$, and in fact that this defines an isomorphism between (\ref{(bigcup_{l = 0}^infty R_l^{-1} {bf Z}) / {bf Z}}) and the dual of $Y$. The natural topology on the dual of $Y$ is discrete, because $Y$ is compact, so that this isomorphism is automatically a homeomorphism. In this case, one can also check that the dual of $Y$ is isomorphic to (\ref{(bigcup_{l = 0}^infty R_l^{-1} {bf Z}) / {bf Z}}) using the remarks in the previous paragraph, and the descriptions of the group homomorphisms from ${\bf Z} / R_n \, {\bf Z}$ into ${\bf T}$ at the beginning of Section \ref{characters on Z_p}.
As in Section \ref{compact commutative groups}, characters on $Y$ are orthonormal with respect to the usual $L^2$ inner product associated to Haar measure on $Y$. There are $R_n$ characters on $Y$ of the form (\ref{phi = psi circ pi_n}) for each positive integer $n$, which are constant on the cosets of $Y_n$ in $Y$. The linear span of these characters consists of all functions on $Y$ that are constant on the cosets of $Y_n$ in $Y$, since every function on ${\bf Z} / R_n \, {\bf Z}$ can be expressed as a linear combination of characters on ${\bf Z} / R_n , {\bf Z}$. The linear span of all characters on $Y$ consists of functions on $Y$ that are constant on the cosets of $Y_n$ in $Y$ for some $n$.
\section{Topological equivalence} \label{topological equivalence}
Let $r = \{r_j\}_{j = 1}^\infty$ be as in Section \ref{r-adic absolute values}, and let $r' = \{r'_j\}_{j = 1}^\infty$ be another sequence of integers with $r'_j \ge 2$ for each $j$. Also let $R_l$ be associated to $r$ as before, and put \begin{equation} \label{R'_l = prod_{j = 1}^l r'_j}
R'_l = \prod_{j = 1}^l r'_j \end{equation} when $l \ge 1$, and $R'_0 = 1$. If for each $l \in {\bf Z}_+$ there is an $n \in {\bf Z}_+$ such that $R'_n$ is an integer multiple of $R_l$, then we put \begin{equation} \label{r prec r'}
r \prec r'. \end{equation} It is easy to see that this relation is reflexive and transitive, and that (\ref{r prec r'}) holds if and only if every open subset of ${\bf
Z}$ with respect to the $r$-adic topology is an open set with respect to the $r'$-adic topology as well. Similarly, if we put \begin{equation} \label{r sim r'}
r \sim r' \end{equation} when $r \prec r'$ and $r'\prec r$, then we get an equivalence relation on the set of these sequences, which holds exactly when the $r$-adic and $r'$-adic topologies on ${\bf Z}$ are the same.
As in Section \ref{an embedding}, consider the Cartesian product \begin{equation} \label{X' = prod_{l = 1}^infty ({bf Z} / R'_l {bf Z})}
X' = \prod_{l = 1}^\infty ({\bf Z} / R'_l \, {\bf Z}) \end{equation} associated to $r'$, which is a compact commutative topological ring with respect to coordinatewise addition and multiplication, and using the product topology corresponding to the discrete topology on ${\bf
Z} / R'_l \, {\bf Z}$ for each $l$. Let $q'_l$ be the canonical quotient mapping from ${\bf Z}$ onto ${\bf Z} / R'_l \, {\bf Z}$ for each $l$, and put \begin{equation} \label{q'(a) = {q'_l(a)}_{l = 1}^infty}
q'(a) = \{q'_l(a)\}_{l = 1}^\infty \end{equation} for each $a \in {\bf Z}$, which defines an injective ring homomorphism from ${\bf Z}$ into $X'$. As in Section \ref{coherent sequences}, there is a natural ring homomorphism from ${\bf Z} / R'_{l + 1} \, {\bf Z}$ onto ${\bf Z} / R_l \, {\bf Z}$ for each $l$, and $x' = \{x'_l\}_{l = 1}^\infty \in X'$ is said to be a coherent sequence if $x'_l$ is the image under this homomorphism of $x'_{l + 1}$ for each $l$. Let $Y'$ be the set of coherent sequences in $X'$, which is a closed sub-ring of $X'$. This is the same as the closure of $q'({\bf Z})$ in $X'$, and the topological ring ${\bf Z}_{r'}$ of $r'$-adic integers can be identified with $Y'$.
If $r \prec r'$, then there is a natural continuous ring homomorphism from $Y'$ onto $Y$, defined as follows. Let $x' \in Y'$ and $l \in {\bf Z}_+$ be given, and remember that there is an $n = n(l) \in {\bf Z}_+$ such that $R'_n$ is an integer multiple of $R_l$. Of course, this implies that $R'_k$ is an integer multiple of $R_l$ for every $k \in {\bf Z}_+$ with $k \ge n$, and hence that there is a natural ring homomorphism from ${\bf Z} / R'_k \, {\bf Z}$ onto ${\bf Z} / R_l \, {\bf Z}$ when $k \ge n$. Let $x_l$ be the image of $x'_k$ under this homomorphism, which one can check is the same for all $k \ge n$, because $x'$ is a coherent sequence. One can also check that $x = \{x_l\}_{l = 1}^\infty$ is a coherent sequence in $X$, so that \begin{equation} \label{x' mapsto x}
x' \mapsto x \end{equation} leads to a natural mapping from $Y'$ into $Y$. This mapping is a continuous ring homomorphism, with respect to the topologies induced on $Y$ and $Y'$ by the product topologies on $X$ and $X'$, respectively. If $a \in {\bf Z}$, then (\ref{x' mapsto x}) sends $q'(a)$ to $q(a)$, so that (\ref{x' mapsto x}) may be considered as an extension of the identity mapping on ${\bf Z}$ to a continuous ring homomorphism from ${\bf Z}_{r'}$ into ${\bf Z}_r$. Because $Y'$ is compact, (\ref{x' mapsto x}) maps $Y'$ onto a compact set in $Y$, and onto a closed set in $Y$ in particular. This implies that (\ref{x'
mapsto x}) maps $Y'$ onto $Y$, since $q'({\bf Z})$ is mapped onto $q({\bf Z})$, which is dense in $Y$. If $r \sim r'$, then (\ref{x'
mapsto x}) is an isomorphism from $Y'$ onto $Y$ as topological rings.
As in Section \ref{some related groups}, \begin{equation} \label{bigcup_{l = 0}^infty (R'_l)^{-1} {bf Z}}
\bigcup_{l = 0}^\infty (R'_l)^{-1} \, {\bf Z} \end{equation} is a subgroup of ${\bf Q}$ with respect to addition that contains ${\bf Z}$. Observe that $r \prec r'$ if and only if the analogous subgroup (\ref{bigcup_{l = 0}^infty R_l^{-1} {bf Z}}) of ${\bf Q}$ associated to $r$ is contained in (\ref{bigcup_{l = 0}^infty
(R'_l)^{-1} {bf Z}}), and that $r \sim r'$ if and only if (\ref{bigcup_{l = 0}^infty R_l^{-1} {bf Z}}) is the same as (\ref{bigcup_{l = 0}^infty (R'_l)^{-1} {bf Z}}). Similarly, the quotient group \begin{equation} \label{(bigcup_{l = 0}^infty (R'_l)^{-1} {bf Z}) / {bf Z}}
\Big(\bigcup_{l = 0}^\infty (R'_l)^{-1} \, {\bf Z}\Big) / {\bf Z} \end{equation} and its analogue (\ref{(bigcup_{l = 0}^infty R_l^{-1} {bf Z}) / {bf
Z}}) for $r$ may be considered as subgroups of ${\bf Q} / {\bf
Z}$. Clearly (\ref{bigcup_{l = 0}^infty R_l^{-1} {bf Z}}) is contained in (\ref{bigcup_{l = 0}^infty (R'_l)^{-1} {bf Z}}) if and only if (\ref{(bigcup_{l = 0}^infty R_l^{-1} {bf Z}) / {bf Z}}) is contained in (\ref{(bigcup_{l = 0}^infty (R'_l)^{-1} {bf Z}) / {bf
Z}}), and (\ref{bigcup_{l = 0}^infty R_l^{-1} {bf Z}}) is equal to (\ref{bigcup_{l = 0}^infty (R'_l)^{-1} {bf Z}}) if and only if (\ref{(bigcup_{l = 0}^infty R_l^{-1} {bf Z}) / {bf Z}}) is equal to (\ref{(bigcup_{l = 0}^infty (R'_l)^{-1} {bf Z}) / {bf Z}}). Thus $r \prec r'$ if and only if (\ref{(bigcup_{l = 0}^infty R_l^{-1}
{bf Z}) / {bf Z}}) is contained in (\ref{(bigcup_{l = 0}^infty
(R'_l)^{-1} {bf Z}) / {bf Z}}), and $r \sim r'$ if and only if (\ref{(bigcup_{l = 0}^infty R_l^{-1} {bf Z}) / {bf Z}}) is the same as (\ref{(bigcup_{l = 0}^infty (R'_l)^{-1} {bf Z}) / {bf Z}}).
\chapter{Some geometric conditions} \label{some geometric conditions}
\section{A class of isometries} \label{a class of isometries}
Let $X_1, X_2, X_3, \ldots$ be a sequence of sets, each of which has at least two elements, and let $X = \prod_{j = 1}^\infty X_j$ be their Cartesian product, as in Section \ref{abstract cantor sets}. Also let $\{t_l\}_{l = 0}^\infty$ be a strictly decreasing sequence of positive real numbers, and let $d(x, y)$ be the ultrametric on $X$ defined as in (\ref{d(x, y) = t_{l(x, y)}}). Thus $x, y \in X$ satisfy \begin{equation} \label{d(x, y) le t_k}
d(x, y) \le t_k \end{equation} for some nonnegative integer $k \ge 0$ if and only if $x_j = y_j$ when $j \le k$. Suppose that $\phi : X \to X$ is a Lipschitz mapping of order $1$ with constant $C = 1$ with respect to $d(\cdot, \cdot)$, so that \begin{equation} \label{d(phi(x), phi(y)) le d(x, y)}
d(\phi(x), \phi(y)) \le d(x, y) \end{equation} for every $x, y \in X$. Let us express $\phi(x)$ as \begin{equation} \label{phi(x) = {phi_j(x)}_{j = 1}^infty}
\phi(x) = \{\phi_j(x)\}_{j = 1}^\infty, \end{equation} where $\phi_j : X \to X_j$ for each $j$. If $x, y \in X$ satisfy $x_j = y_j$ for $j \le k$ and some $k$, then it follows that \begin{equation} \label{d(phi(x), phi(y)) le t_k}
d(\phi(x), \phi(y)) \le t_k, \end{equation} and hence that $\phi_j(x) = \phi_j(y)$ for $j \le k$. Equivalently, this means that for each $k \ge 1$, $\phi_k(x)$ only depends on $x_j$ with $j \le k$. Conversely, if $\phi_k : X \to X_k$ has this property for each $k \ge 1$, then $\phi : X \to X$ defined as in (\ref{phi(x) =
{phi_j(x)}_{j = 1}^infty}) satisfies (\ref{d(phi(x), phi(y)) le d(x,
y)}) for every $x, y \in X$.
If $x, y \in X$ satisfy \begin{equation} \label{x_j = y_j for j < k and x_k ne y_k}
x_j = y_j \hbox{ for } j < k \hbox{ and } x_k \ne y_k \end{equation} for some $k \in {\bf Z}_+$, then $d(x, y) = t_{k - 1}$, by construction. Suppose that $\phi_k$ has the property mentioned in the preceding paragraph for each $k \ge 1$, and that \begin{equation} \label{phi_k(x) ne phi_k(y)}
\phi_k(x) \ne \phi_k(y) \end{equation} when $x, y \in X$ satisfy (\ref{x_j = y_j for j < k and x_k ne y_k}). This implies that \begin{equation} \label{d(phi(x), phi(y)) = t_{k - 1}}
d(\phi(x), \phi(y)) = t_{k - 1} \end{equation} when $x, y \in X$ satisfy (\ref{x_j = y_j for j < k and x_k ne y_k}), because $\phi_j(x) = \phi_j(y)$ when $j < k$. It follows that $\phi : X \to X$ is an isometry with respect to $d(\cdot, \cdot)$ under these conditions. Conversely, if $\phi : X \to X$ is an isometry with respect to $d(\cdot, \cdot)$, then one can check that $\phi$ has these properties.
In particular, if $\phi_k(x)$ depends only on $x_k$ for each $k$, then $\phi$ satisfies (\ref{d(phi(x), phi(y)) le d(x, y)}). In this case, $\phi : X \to X$ is an isometry with respect to $d(\cdot, \cdot)$ if and only if $\phi_k(x)$ corresponds to a one-to-one mapping from $X_k$ into itself for each $k$. Similarly, if $\phi_k(x)$ corresponds to a mapping from $X_k$ onto itself for each $k$, then $\phi$ maps $X$ onto itself.
\section{Some isometric equivalences} \label{some isometric equivalences}
Let $r = \{r_j\}_{j = 1}^\infty$ be a sequence of positive integers with $r_j \ge 2$ for each $j$, and let $R_l$ be as in Section \ref{r-adic absolute values}. Also let $X$ and $Y$ be as in Sections \ref{an embedding} and \ref{coherent sequences}, respectively. Put \begin{equation} \label{widetilde{X}_j = {0, 1, ldots, r_j - 1}}
\widetilde{X}_j = \{0, 1, \ldots, r_j - 1\} \end{equation} for each $j \ge 1$, and \begin{equation} \label{widetilde{X} = prod_{j = 1}^infty widetilde{X}_j}
\widetilde{X} = \prod_{j = 1}^\infty \widetilde{X}_j. \end{equation} If $t = \{t_l\}_{l = 0}^\infty$ is a strictly decreasing sequence of positive real numbers, then we get corresponding ultrametrics $d(x, y)$ on $X$ and $d'(x', y')$ on $\widetilde{X}$, as in Section \ref{abstract cantor sets}. As before, a mapping $\psi : Y \to \widetilde{X}$ corresponds exactly to a sequence of mappings $\psi_j : Y \to \widetilde{X}_j$, $j \in {\bf Z}_+$, with \begin{equation} \label{psi(x) = {psi_j(x)}_{j = 1}^infty}
\psi(x) = \{\psi_j(x)\}_{j = 1}^\infty \end{equation} for each $x \in Y$.
Suppose that $\psi : Y \to \widetilde{X}$ is Lipschitz of order $1$ with constant $C = 1$ with respect to the restriction of $d(x, y)$ to $x, y \in Y$ and $d'(x', y')$ on $\widetilde{X}$, so that \begin{equation} \label{d'(psi(x), psi(y)) le d(x, y)}
d'(\psi(x), \psi(y)) \le d(x, y) \end{equation} for every $x, y \in Y$. If $x, y \in Y$ satisfy $x_j = y_j$ for $j \le k$ and some $k$, then we get that $\psi_j(x) = \psi_j(y)$ when $j \le k$, as in the previous section. This is the same as saying that $\psi_k(x)$ depends only on $x_k$ for each $k \ge 1$, because the elements of $Y$ are coherent sequences. Conversely, if $\psi_k(x)$ depends only on $x_k$ for each $k \ge 1$, then $\psi : Y \to \widetilde{X}$ satisfies (\ref{d'(psi(x), psi(y)) le d(x, y)}).
Suppose now that $\psi_k(x)$ depends only on $x_k$ for each $k \ge 1$, and that \begin{equation} \label{psi_k(x) ne psi_k(y)}
\psi_k(x) \ne \psi_k(y) \end{equation} when $x, y \in Y$ satisfy \begin{equation} \label{x_{k - 1} = y_{k - 1} and x_k ne y_k}
x_{k - 1} = y_{k - 1} \hbox{ and } x_k \ne y_k. \end{equation} If $k = 1$, then we interpret (\ref{x_{k - 1} = y_{k - 1} and x_k ne
y_k}) as meaning simply that $x_1 \ne y_1$. Under these conditions, $\psi : Y \to \widetilde{X}$ is an isometry, for the same reasons as before. Conversely, any isometry from $Y$ into $\widetilde{X}$ has these properties.
If $\theta_k$ is a mapping from ${\bf Z} / R_k \, {\bf Z}$ into $\widetilde{X}_k$, then \begin{equation} \label{psi_k(x) = theta_k(x_k)}
\psi_k(x) = \theta_k(x_k) \end{equation} defines a mapping from $Y$ into $\widetilde{X}_k$. We would like to choose such a mapping $\theta_k$ for each $k \in {\bf Z}_+$ so that the corresponding mapping $\psi_k$ satisfies (\ref{psi_k(x) ne
psi_k(y)}) for every $x, y \in Y$ for which (\ref{x_{k - 1} = y_{k -
1} and x_k ne y_k}) holds. If $k = 1$, then we can use any one-to-one mapping from ${\bf Z} / R_1 \, {\bf Z}$ onto $\widetilde{X}_1$, because $R_1 = r_1$ and $\widetilde{X}_1$ has exactly $r_1$ elements. Suppose now that $k \ge 2$, and remember that there is a natural ring homomorphism from ${\bf Z} / R_k \, {\bf Z}$ onto ${\bf Z} / R_{k - 1} , {\bf Z}$, because $R_k \, {\bf Z} \subseteq R_{k - 1} \, {\bf Z}$. The kernel of this homomorphism is equal to \begin{equation} \label{R_{k - 1} {bf Z} / R_k {bf Z}}
R_{k - 1} \, {\bf Z} / R_k \, {\bf Z}, \end{equation} which has exactly $r_k$ elements. Of course, ${\bf Z} / R_k \, {\bf
Z}$ can be partitioned into translates of (\ref{R_{k - 1} {bf Z} /
R_k {bf Z}}). The property of $\psi_k$ that we want is equivalent to saying that the restriction of $\theta_k$ to any translate of (\ref{R_{k - 1} {bf Z} / R_k {bf Z}}) in ${\bf Z} / R_k \, {\bf Z}$ is injective. It is easy to choose $\theta_k$ in this way, because (\ref{R_{k - 1} {bf Z} / R_k {bf Z}}) has exactly $r_k$ elements, which is the same as the number of elements of $\widetilde{X}_k$. This leads to a sequence of mappings $\psi_k : Y \to \widetilde{X}_k$ as in (\ref{psi_k(x) = theta_k(x_k)}), and hence a mapping $\psi : Y \to \widetilde{X}$ as in (\ref{psi(x) = {psi_j(x)}_{j = 1}^infty}), which is an isometry. Note that $\theta_k$ also maps every translate of (\ref{R_{k - 1} {bf Z} / R_k {bf Z}}) in ${\bf Z} / R_k \, {\bf Z}$ onto $\widetilde{X}_k$, by construction. Using this, one can check that $\psi$ maps $Y$ onto $\widetilde{X}$ as well.
\section{Doubling metrics} \label{doubling metrics}
A metric $d(x, y)$ on a set $M$ is said to be \emph{doubling}\index{doubling metrics} if there is a positive real number $C$ such that every open ball in $M$ with radius $r > 0$ can be covered by $\le C$ open balls of radius $r/2$. In this case, one might also say that the metric space $(M, d(x, y))$ is doubling, or simply that $M$ is doubling, if the choice of the metric is clear. If $M$ is doubling, then we can apply the condition repeatedly to get that every open ball in $M$ with radius $r$ can be covered by $\le C^k$ open balls of radius $2^{-k} \, r$ for every $k \in {\bf Z}_+$. In particular, this implies that bounded subsets of $M$ are totally bounded. If $M$ is doubling and complete, then it follows that subsets of $M$ that are both closed and bounded are compact as well.
If $M$ is doubling, then the iterated condition mentioned in the previous paragraph implies that every closed ball in $M$ with radius $r > 0$ can be covered by a bounded number of closed balls of radius $r/2$. Similarly, every subset of $M$ with diameter $\le r$ can be covered by a bounded number of sets with diameter $\le r/2$. This implies that the restriction of $d(x, y)$ to any subset of $M$ is a doubling metric too. One can also use the iterated version of the doubling condition to show that if $M$ is is bilipschitz equivalent to a metric space that is doubling, then $M$ is doubling.
The doubling condition can be defined in the same way for quasi-metrics, with the same type of properties as before. If $d(x, y)$ is a quasi-metric on a set $M$ and $a$ is a positive real number, then we have seen that $d(x, y)^a$ is quasi-metric on $M$ too, as in Section \ref{snowflake metrics, quasi-metrics}. It is easy to see that $d(x, y)$ is doubling if and only if $d(x, y)^a$ is doubling.
The real line ${\bf R}$ is doubling with respect to the standard metric, and similarly ${\bf R}^n$ is doubling with respect to the standard metric for every positive integer $n$. More precisely, one can use the invariance of the standard metric under translations and dilations to reduce the doubling condition to the case of the unit ball, which then follows from the fact that the unit ball is totally bounded. Similarly, the set ${\bf Q}_p$ of $p$-adic numbers is doubling with respect to the $p$-adic metric, for every prime number $p$.
Let $X_1, X_2, X_3, \ldots$ be a sequence of sets, each of which has at least two elements, and let $X = \prod_{j = 1}^\infty X_j$ be their Cartesian product. Also let $\{t_l\}_{l =0}^\infty$ be a strictly decreasing sequence of positive real numbers, and let $d(x, y)$ be the corresponding ultrametric on $X$, as in (\ref{d(x, y) = t_{l(x, y)}}). If $X_j$ has only finitely many elements for each $j$, and if $\{t_l\}_{l = 0}^\infty$ converges to $0$, then $X$ is totally bounded with respect to $d(x, y)$. Conversely, one can check that these conditions are necessary for $X$ to be doubling with respect to $d(x, y)$.
Of course, $X$ is bounded with respect to $d(x, y)$ by construction. If $X$ is doubling with respect to $d(x, y)$, then $X$ is totally bounded in particular, and hence the number of elements of $X_j$ has to be finite for each $j \ge 1$, as in the previous paragraph. In fact, the number of elements of $X_j$ has to be uniformly bounded in this case.
Similarly, if $X$ is totally bounded with respect to $d(x, y)$, then we have seen that $\{t_l\}_{l = 0}^\infty$ converges to $0$, which implies that for each $l \ge 0$, the number of $j \ge l$ such that $t_j \ge t_l/2$ is finite. If $X$ is doubling with respect to $d(x, y)$, then the number of $j \ge l$ such that $t_j \ge t_l/2$ is uniformly bounded over $l$. Conversely, if the number of elements of $X_j$ is uniformly bounded in $j$, and if the number of $j \ge l$ such that $t_j \ge t_l/2$ is uniformly bounded in $l$, then $X$ is doubling with respect to $d(x, y)$.
Now let $r = \{r_j\}_{j = 1}$ be a sequence of positive integers with $r_j \ge 2$ for each $j$, and let $t = \{t_l\}_{l = 0}^\infty$ be a strictly decreasing sequence of positive real numbers. This leads to an $r$-adic ultrametric on ${\bf Z}$, as in Section \ref{r-adic absolute values}. As before, ${\bf Z}$ is totally bounded with respect to this $r$-adic metric if and only if $\{t_l\}_{l = 0}^\infty$ converges to $0$. One can check that ${\bf Z}$ is doubling with respect to this $r$-adic metric if and only if the $r_j$'s are bounded and the number of $j \ge l$ such that $t_j \ge t_l/2$ is uniformly bounded in $l$. Remember that the set ${\bf Z}_r$ of $r$-adic integers is obtained by completing ${\bf Z}$ as a metric space with respect to the $r$-adic metric. Under these same conditions on $r$ and $t$, ${\bf Z}_r$ is doubling with respect to the corresponding extension of the $r$-adic metric. This also follows from the earlier discussion of Cartesian products, using the isometric equivalence described at the end of the preceding section.
\section{Doubling measures} \label{doubling measures}
A nonnegative Borel measure $\mu$ on a metric space $(M, d(x, y))$ is said to be \emph{doubling}\index{doubling measures} if the measure of every open ball in $M$ is positive and finite, and if there is a positive real number $C$ such that \begin{equation} \label{mu(B(x, 2 r)) le C mu(B(x, r))}
\mu(B(x, 2 \, r)) \le C \, \mu(B(x, r)) \end{equation} for every $x \in M$ and $r > 0$. It is easy to see that Lebesgue measure on ${\bf R}^n$ is doubling with respect to the standard metric on ${\bf R}^n$ for each positive integer $n$, and that Haar measure on ${\bf Q}_p$ is doubling with respect to the $p$-adic metric for every prime number $p$. Some other examples will be discussed later in the section. If $\mu$ satisfies (\ref{mu(B(x, 2 r)) le C mu(B(x, r))}) on $M$, then \begin{equation} \label{mu(B(x, 2^k r)) le C^k mu(B(x, r))}
\mu(B(x, 2^k \, r)) \le C^k \, \mu(B(x, r)) \end{equation} for every $x \in M$, $r > 0$, and $k \in {\bf Z}_+$. Using this, one can check that if the measure of some open ball in $M$ is positive or finite with respect to $\mu$, then every open ball in $M$ has the same property, because of (\ref{mu(B(x, 2 r)) le C mu(B(x, r))}).
Suppose that $\mu$ is a doubling measure on a metric space $(M, d(x, y))$, and let $x \in M$ and $r > 0$ be given. Also let $y_1, \ldots, y_n$ be finitely many elements of $B(x, r)$ such that \begin{equation} \label{d(y_j, y_l) ge r/2}
d(y_j, y_l) \ge r/2 \end{equation} when $j \ne l$. We would like to show that \begin{equation} \label{n le C_1}
n \le C_1 \end{equation} for some positive real number $C_1$ that depends only on the doubling constant for $\mu$. It follows from (\ref{d(y_j, y_l) ge r/2}) and the triangle inequality that \begin{equation} \label{B(y_j, r/4) cap B(y_l, r/4) = emptyset}
B(y_j, r/4) \cap B(y_l, r/4) = \emptyset \end{equation} when $j \ne l$, and hence \begin{equation} \label{sum_{j = 1}^n mu(B(y_j, r/4)) = mu(bigcup_{j = 1}^n B(y_j, r/4))}
\sum_{j = 1}^n \mu(B(y_j, r/4)) = \mu\Big(\bigcup_{j = 1}^n B(y_j, r/4)\Big). \end{equation} Using the triangle inequality again, we have that \begin{equation} \label{B(y_j, r/4) subseteq B(x, 5r/4)}
B(y_j, r/4) \subseteq B(x, 5r/4) \end{equation} for each $j$, so that \begin{equation} \label{mu(bigcup_{j = 1}^n B(y_j, r/4)) le mu(B(x, 5r/4))}
\mu\Big(\bigcup_{j = 1}^n B(y_j, r/4)\Big) \le \mu(B(x, 5r/4)). \end{equation} In the other direction, \begin{equation} \label{B(x, 5r/4) subseteq B(y_j, 9r/4)}
B(x, 5r/4) \subseteq B(y_j, 9r/4) \end{equation} for each $j$, because $d(x, y_j) \le r$ by hypothesis. Thus $\mu(B(x, 5r/4))$ is bounded by a constant times $\mu(B(y_j, r/4)$ for each $j$, by the doubling condition. Combining this with (\ref{sum_{j = 1}^n
mu(B(y_j, r/4)) = mu(bigcup_{j = 1}^n B(y_j, r/4))}) and (\ref{mu(bigcup_{j = 1}^n B(y_j, r/4)) le mu(B(x, 5r/4))}), we get (\ref{n le C_1}), as desired.
Suppose now that $n$ is the largest positive integer for which there are $n$ elements $y_1, \ldots, y_l$ of $B(x, r)$ satisfying (\ref{d(y_j, y_l) ge r/2}). If $y$ is any element of $B(x, r)$, then it follows that \begin{equation} \label{d(y, y_j) < r/2}
d(y, y_j) < r/2 \end{equation} for some $j = 1, \ldots, n$, since otherwise $y_1, \ldots, y_n$ together with $y$ would be $n + 1$ elements of $B(x, r)$ with the same property. This shows that \begin{equation} \label{B(x, r) subseteq bigcup_{j = 1}^n B(y_j, r/2),}
B(x, r) \subseteq \bigcup_{j = 1}^n B(y_j, r/2), \end{equation} and hence that $M$ is doubling as a metric space, because $x \in M$ and $r > 0$ are arbitrary, and $n$ is uniformly bounded.
If $d(x, y)$ is an ultrametric on $M$, then the proof of (\ref{n le C_1}) can be improved somewhat. In this case, we can replace (\ref{B(y_j, r/4) cap B(y_l, r/4) = emptyset}) with \begin{equation} \label{B(y_j, r/2) cap B(y_l, r/2) = emptyset}
B(y_j, r/2) \cap B(y_l, r/2) = \emptyset \end{equation} when $j \ne l$. Of course, we should then use the analogue of (\ref{sum_{j = 1}^n mu(B(y_j, r/4)) = mu(bigcup_{j = 1}^n B(y_j,
r/4))}) with $r/4$ replaced by $r/2$. We also have that \begin{equation} \label{B(y_j, r/2) subseteq B(x, r)}
B(y_j, r/2) \subseteq B(x, r) \end{equation} for each $j$, by the ultrametric version of the triangle inequality, so that \begin{equation} \label{mu(bigcup_{j = 1}^n B(y_j, r/2)) le mu(B(x, r))}
\mu\Big(\bigcup_{j = 1}^n B(y_j, r/2)\Big) \le \mu(B(x, r)), \end{equation} which is the substitute for (\ref{mu(bigcup_{j = 1}^n B(y_j, r/4)) le
mu(B(x, 5r/4))}). Instead of (\ref{B(x, 5r/4) subseteq B(y_j, 9r/4)}), we can use the fact that \begin{equation} \label{B(x, r) subseteq B(y_j, r)}
B(x, r) \subseteq B(y_j, r) \end{equation} for each $j$, by the ultrametric version of the triangle inequality, and then continue as before.
Let $X_1, X_2, X_3, \ldots$ be a sequence of finite sets, each of which has at least two elements, and let $X = \prod_{j = 1}^\infty X_j$ be their Cartesian product. Also let $\{t_l\}_{l = 0}^\infty$ be a strictly decreasing sequence of positive real numbers that converges to $0$, and let $d(x, y)$ be the corresponding ultrametric on $X$, as in (\ref{d(x, y) = t_{l(x, y)}}). As in the previous section, $X$ is doubling with respect to $d(x, y)$ if and only if the number of elements of $X_j$ is uniformly bounded in $j$, and the number of $j \ge l$ such that $t_j \ge t_l/2$ is uniformly bounded in $l$. Let $\mu_j$ be a probability measure on $X_j$ for each $j$, where all subsets of $X_j$ are measurable, and let $\mu$ be the corresponding product measure on $X$, as in Section \ref{abstract cantor sets}. If $\mu$ is a doubling measure on $X$ with respect to $d(x, y)$, then there is a $c > 0$ such that \begin{equation} \label{mu_j({x_j}) ge c}
\mu_j(\{x_j\}) \ge c \end{equation} for every $j \ge 1$ and $x_j \in X_j$. This implies that $X_j$ has $\le 1/c$ elements for each $j$, because $\mu_j(X_j) = 1$. Conversely, if there is a $c > 0$ such that (\ref{mu_j({x_j}) ge c}) holds for every $j \ge 1$ and $x_j \in X_j$, and if the number of $j \ge l$ such that $t_j \ge t_l/2$ is uniformly bounded in $l$, then one can check that $\mu$ is a doubling measure on $X$.
Similarly, let $r = \{r_j\}_{j = 1}^\infty$ be a sequence of positive integers with $r_j \ge 2$ for each $j$, and let $t = \{t_l\}_{l = 0}^\infty$ be a strictly decreasing sequence of positive real numbers that converges to $0$. If Haar measure on ${\bf Z}_r$ is doubling with respect to the $r$-adic metric on ${\bf Z}_r$ associated to $r$ and $t$, then it is easy to see that the $r_j$'s have to be uniformly bounded in $j$. Conversely, if the $r_j$'s are uniformly bounded in $j$, and if the number of $j \ge l$ such that $t_j \ge t_l/2$ is uniformly bounded in $l$, then one can check that Haar measure on ${\bf Z}_r$ is doubling with respect to the $r$-adic metric associated to $r$ and $t$. One can also look at this in terms of an isometric equivalence of ${\bf Z}_r$ with a Cartesian product $\widetilde{X}$, as in Section \ref{some isometric equivalences}. More precisely, Haar measure on ${\bf Z}_r$ corresponds to a product measure $\widetilde{\mu}$ on $\widetilde{X}$ with respect to this isometric equivalence, using the probability measures $\widetilde{\mu}_j$ that are uniformly distributed on each factor $\widetilde{X}_j$ in (\ref{widetilde{X} = prod_{j = 1}^infty
widetilde{X}_j}), in the sense that $\mu_j(\{x_j\})$ is the same for each $x_j \in X_j$.
If $d(\cdot, \cdot)$ is a quasi-metric on $M$, then one can define the notion of a doubling measure on $M$ in the same way as before, at least if open balls in $M$ are Borel sets. In particular, open balls in $M$ with respect to $d(\cdot, \cdot)$ are open sets when $d(\cdot, \cdot)$ is continuous with respect to the topology on $M$ that it determines. At any rate, this is normally not a problem, and there are various ways to deal with it. One can check that the arguments in this and the next sections have suitable versions for quasi-metrics, with different constants, as appropriate.
If $d(\cdot, \cdot)$ and $d'(\cdot, \cdot)$ are quasi-metrics on $M$ such that each is bounded by a constant multiple of the other, then it is easy to see that doubling measures on $M$ with respect to $d(\cdot, \cdot)$ are the same as doubling measures on $M$ with respect to $d'(\cdot, \cdot)$, aside from measurability issues as in the previous paragraph. Similarly, if $d(\cdot, \cdot)$ is a quasi-metric on $M$ and $a$ is a positive real number, then $d(\cdot, \cdot)^a$ is a quasi-metric on $M$, as in Section \ref{snowflake metrics, quasi-metrics}, and doubling measures on $M$ with respect to $d(\cdot, \cdot)$ are the same as doubling measures on $M$ with respect to $d(\cdot, \cdot)^a$, aside from measurability issues again. As in Section \ref{snowflake metrics, quasi-metrics}, if $d(\cdot, \cdot)$ is a quasi-metric on $M$,it is shown in \cite{m-s-1} that there is a metric $\widetilde{d}(\cdot, \cdot)$ on $M$ and a positive real number $a$ such that $d(\cdot, \cdot)$ and $\widetilde{d}(\cdot, \cdot)^a$ are each bounded by constant multiples of the other. It follows that doubling measures on $M$ with respect to $d(\cdot, \cdot)$ are the same as doubling measures with respect to $\widetilde{d}(\cdot, \cdot)$, aside from the usual measurability issues.
\section{Another doubling condition} \label{another doubling condition}
Let $h(r)$ be a monotone increasing nonnegative real-valued function on the set $[0, +\infty)$ of nonnegative real numbers. If there is a nonnegative real number $C$ such that \begin{equation} \label{h(2 r) le C h(r)}
h(2 \, r) \le C \, h(r) \end{equation} for every $r \ge 0$, then we say that $h$ satisfies a doubling condition. Using the monotonicity of $h$, we can reformulate (\ref{h(2 r) le C h(r)}) as saying that \begin{equation} \label{h(r + t) le h(2 max(r, t)) le C h(max(r, t)) = C max(h(r), h(t))}
h(r + t) \le h(2 \, \max(r, t)) \le C \, h(\max(r, t)) = C \, \max(h(r), h(t)) \end{equation} for every $r, t \ge 0$. As usual, we can also iterate (\ref{h(2 r) le
C h(r)}), to get that \begin{equation} \label{h(2^k r) le C^k h(r)}
h(2^k \, r) \le C^k \, h(r) \end{equation} for every $r \ge 0$ and positive integer $k$. Note that $h(r) = r^a$ satisfies these conditions with $C = 2^a$ for each $a \ge 0$.
Let $(M, d(x, y))$ be a metric space, and let $\mu$ be a nonnegative Borel measure on $M$. Suppose that the measure of every open ball in $M$ with respect to $\mu$ is finite, and put $h_x(0) = 0$ and \begin{equation} \label{h_x(r) = mu(B(x, r))}
h_x(r) = \mu(B(x, r)) \end{equation} for every $x \in M$ and $r > 0$. Thus $h_x(r)$ is a monotone increasing nonnegative real-valued function on $[0, +\infty)$ for each
$x \in M$, and in fact $h_x(r)$ is also left-continuous at each $r >
0$ from the left for every $x \in M$, because of the countable
additivity of $\mu$. Clearly $\mu$ satisfies the doubling condition (\ref{mu(B(x, 2 r)) le C mu(B(x, r))}) for some $C \ge 0$ and every $x \in M$ and $r > 0$ if and only if $h_x(r)$ satisfies the doubling condition (\ref{h(2 r) le C h(r)}) with the same constant $C$ for every $x \in M$ and $r \ge 0$.
As in \cite{mat}, $\mu$ is said to be uniformly distributed on $M$ if (\ref{h_x(r) = mu(B(x, r))}) does not depend on $x$, so that there is a function $h(r)$ on $[0, +\infty)$ such that $h(0) = 0$ and \begin{equation} \label{mu(B(x, r)) = h(r)}
\mu(B(x, r)) = h(r) \end{equation} for every $x \in M$ and $r > 0$. If $X = \prod_{j = 1}^\infty X_j$ and $\mu$ are as in the previous section, for instance, then $\mu$ has this property when $\mu_j$ is uniformly distributed on $X_j$ for each $j$, in the sense that $\mu_j(\{x_j\})$ is the same for each $x_j \in X_j$. If there is a transitive group of isometries on $M$ that preserve $\mu$, then $\mu$ is uniformly distributed on $M$ in the sense of (\ref{mu(B(x, r)) = h(r)}). In particular, this includes the case of Haar measure on a topological group equipped with a translation-invariant metric. If $\mu$ is uniformly distributed on $M$, and if $M$ is doubling as a metric space, then one can check that $\mu$ is a doubling measure on $M$.
Now let $d(x, y)$ be a quasi-metric on a set $M$, and let $h(r)$ be a monotone increasing nonnegative real-valued function on $[0, +\infty)$ such that $h(0) = 0$ and $h(r) > 0$ when $r > 0$. If $h(r)$ also satisfies a doubling condition as in (\ref{h(2 r) le C h(r)}), then it is easy to see that $h(d(x, y))$ is a quasi-metric on $M$ as well. If, in addition, \begin{equation} \label{lim_{r to 0+} h(r) = 0}
\lim_{r \to 0+} h(r) = 0, \end{equation} then $h(d(x, y))$ determines the same topology on $M$ as $d(x, y)$, and indeed they determine the same uniform structure on $M$. This is a variant of the situation in Section \ref{subadditive functions}.
\section{Some variants} \label{some variants}
Let $\mu$ be a doubling measure on a metric space $(M, d(x, y))$, and let $x, y \in M$ be given, with $x \ne y$. Put $t = d(x, y) > 0$, and observe that \begin{equation} \label{B(x, t/2) cap B(y, t/2) = emptyset}
B(x, t/2) \cap B(y, t/2) = \emptyset \end{equation} and \begin{equation} \label{B(x, t/2) cup B(y, t/2) subseteq B(x, 3t/2)}
B(x, t/2) \cup B(y, t/2) \subseteq B(x, 3t/2), \end{equation} by the triangle inequality. Thus \begin{equation} \label{mu(B(x, t/2)) + mu(B(y, t/2)) le mu(B(x, 3t/2))}
\mu(B(x, t/2)) + \mu(B(y, t/2)) \le \mu(B(x, 3t/2)). \end{equation} We also have that $B(x, t/2) \subseteq B(y, 3t/2)$, which implies that \begin{equation} \label{mu(B(x, t/2)) le mu(B(y, 3t/2))}
\mu(B(x, t/2)) \le \mu(B(y, 3t/2)). \end{equation} Because $\mu$ is a doubling measure on $M$, $\mu(B(y, 3t/2))$ is bounded by a constant multiple of $\mu(B(y, t/2))$, and hence $\mu(B(x, t))$ is bounded by a constant multiple of $\mu(B(y, t/2))$. It follows that there is a positive real number $c_1 < 1$, depending only on the doubling constant associated to $\mu$, such that \begin{equation} \label{mu(B(x, t/2)) le c_1 mu(B(x, 3t/2))}
\mu(B(x, t/2)) \le c_1 \, \mu(B(x, 3t/2)) \end{equation} under these conditions. In particular, if $x$ is a limit point of $M$, then one can use this to show that $\mu(\{x\}) = 0$. Similarly, if $M$ is unbounded, then one can check that $\mu(M) = +\infty$.
Suppose now that $d(\cdot, \cdot)$ is an ultrametric on $M$, and let $\mu$ be a nonnegative Borel measure on $M$ such that the measure of every open ball is open and finite. Instead of the doubling condition (\ref{mu(B(x, 2 r)) le C mu(B(x, r))}), let us ask that \begin{equation} \label{mu(overline{B}(w, r)) le C_2 mu(B(w, r))}
\mu(\overline{B}(w, r)) \le C_2 \, \mu(B(w, r)) \end{equation} for some $C_2 \ge 1$ and every $w \in M$ and $r > 0$. Let $w \in M$ and $r > 0$ be given, and let $z_1, \ldots, z_n$ be finitely many elements of $\overline{B}(w, r)$ such that \begin{equation} \label{d(z_j, z_l) = r}
d(z_j, z_l) = r \end{equation} when $j \ne l$. Thus the open balls $B(z_j, r)$ are pairwise-disjoint subsets of $\overline{B}(w, r)$, so that \begin{equation} \label{sum_{j = 1}^n mu(B(z_j, r)) le mu(overline{B}(w, r))}
\sum_{j = 1}^n \mu(B(z_j, r)) \le \mu(\overline{B}(w, r)). \end{equation} We also have that \begin{equation} \label{overline{B}(z_j, r) = overline{B}(w, r)}
\overline{B}(z_j, r) = \overline{B}(w, r) \end{equation} for each $j$, because $d(w, z_j) \le r$ for each $j$, and using the ultrametric version of the triangle inequality. This implies that \begin{equation} \label{mu(overline{B}(w, r)) = mu(overline{B}(z_j, r)) le C_2 mu(B(z_j, r))}
\mu(\overline{B}(w, r)) = \mu(\overline{B}(z_j, r)) \le C_2 \, \mu(B(z_j, r)) \end{equation} for each $j$, by hypothesis. Averaging over $j$, we get that \begin{equation} \label{mu(overline{B}(w, r)) le ... le frac{C_2}{n} mu(overline{B}(w, r))}
\mu(\overline{B}(w, r)) \le \frac{C_2}{n} \, \sum_{j = 1}^n \mu(B(z_j, r))
\le \frac{C_2}{n} \, \mu(\overline{B}(w, r)), \end{equation} using (\ref{sum_{j = 1}^n mu(B(z_j, r)) le mu(overline{B}(w, r))}) in the second step. It follows that \begin{equation} \label{n le C_2}
n \le C_2. \end{equation} If we take $n$ to be the largest positive integer for which there are $n$ elements $z_1, \ldots, z_n$ of $\overline{B}(w, r)$ satisfying (\ref{d(z_j, z_l) = r}) when $j \ne l$, and if $z$ is any element of $\overline{B}(w, r)$, then $d(z_j, z) < r$ for some $j$, since otherwise there would be $n + 1$ elements of $\overline{B}(w, r)$ with this property. This shows that \begin{equation} \label{overline{B}(w, r) subseteq bigcup_{j = 1}^n B(z_j, r)}
\overline{B}(w, r) \subseteq \bigcup_{j = 1}^n B(z_j, r), \end{equation} and hence \begin{equation} \label{overline{B}(w, r) = bigcup_{j = 1}^n B(z_j, r)}
\overline{B}(w, r) = \bigcup_{j = 1}^n B(z_j, r), \end{equation} because $B(z_j, r) \subseteq \overline{B}(w, r)$ for each $j$, as in (\ref{overline{B}(z_j, r) = overline{B}(w, r)}).
Let $x, y \in M$ be given again, with $x \ne y$, and put $t = d(x, y) > 0$. Because $d(\cdot, \cdot)$ is an ulrametric, we have that \begin{equation} \label{B(x, t) cap B(y, t) = emptyset}
B(x, t) \cap B(y, t) = \emptyset \end{equation} and \begin{equation} \label{B(x, t) cup B(y, t) subseteq overline{B}(x, t)}
B(x, t) \cup B(y, t) \subseteq \overline{B}(x, t). \end{equation} instead of (\ref{B(x, t/2) cap B(y, t/2) = emptyset}) and (\ref{B(x,
t/2) cup B(y, t/2) subseteq B(x, 3t/2)}). This implies that \begin{equation} \label{mu(B(x, t)) + mu(B(y, t)) le mu(overline{B}(x, t))}
\mu(B(x, t)) + \mu(B(y, t)) \le \mu(\overline{B}(x, t)), \end{equation} which replaces (\ref{mu(B(x, t/2)) + mu(B(y, t/2)) le mu(B(x,
3t/2))}). Under these conditions, $\overline{B}(x, t)$ is the same as $\overline{B}(y, t)$, so that \begin{equation} \label{mu(overline{B}(x, t)) = mu(overline{B}(y, t)) le C_2 mu(B(y, t))}
\mu(\overline{B}(x, t)) = \mu(\overline{B}(y, t)) \le C_2 \, \mu(B(y, t)), \end{equation} by (\ref{mu(overline{B}(w, r)) le C_2 mu(B(w, r))}). Combining this with (\ref{mu(B(x, t)) + mu(B(y, t)) le mu(overline{B}(x, t))}), we get that \begin{eqnarray}
\mu(B(x, t)) & \le & \mu(\overline{B}(x, t)) - \mu(B(y, t)) \\
& \le & \mu(\overline{B}(x, t)) - (1/C_2) \, \mu(\overline{B}(x, t))
\nonumber \\
& = & (1 - (1/C_2)) \, \mu(\overline{B}(x, t)), \nonumber \end{eqnarray} in place of (\ref{mu(B(x, t/2)) le c_1 mu(B(x, 3t/2))}).
Of course, the condition (\ref{overline{B}(w, r) subseteq bigcup_{j = 1}^n B(z_j, r)}) that a closed ball in $M$ with radius $r$ can be covered by a bounded number of open balls of radius $r$ is weaker than the usual doubling condition for metrics, as in Section \ref{doubling metrics}. If $X$ is a Cartesian product as in Section \ref{abstract cantor sets}, then this condition corresponds to asking that the number of elements of the $X_j$'s be bounded, without any additional condition on the sequence $\{t_l\}_{l = 0}^\infty$ used to define the ultrametric as in (\ref{d(x, y) = t_{l(x, y)}}). Similarly, (\ref{mu(overline{B}(w, r)) le C_2 mu(B(w, r))}) is weaker than the doubling condition (\ref{mu(B(x, 2 r)) le C mu(B(x, r))}) in Section \ref{doubling measures}. Let $X$ be as in Section \ref{abstract cantor sets} again, and let $\mu$ be the probability measure on $X$ corresponding to the product of probability measures $\mu_j$ on $X_j$ for each positive integer $j$, as before. In this case, it is easy to see that $\mu$ satisfies (\ref{mu(overline{B}(w, r)) le C_2 mu(B(w, r))}) if and only if \begin{equation} \label{mu_j({x_j}) ge 1/C_2}
\mu_j(\{x_j\}) \ge 1/C_2 \end{equation} for every $j \ge 1$ and $x_j \in X_j$, without additional conditions on the $t_l$'s. If $r = \{r_j\}_{j = 1}^\infty$ is a sequence of positive integers, with $r_j \ge 2$ for each $j$, then Haar measure on the group ${\bf Z}_r$ of $r$-adic integers satisfies (\ref{mu(overline{B}(w, r)) le C_2 mu(B(w, r))}) with respect to an $r$ if and only if the $r_j$'s are uniformly bounded. As usual, this can also be seen in terms of a suitable isometric equivalence with a Cartesian product, as in Section \ref{some isometric equivalences}. If $\mu$ is a uniformly distributed Borel measure on an ultrametric space $M$, and if $M$ satisfies the covering condition (\ref{overline{B}(w, r) subseteq bigcup_{j = 1}^n B(z_j, r)}) with (\ref{n le C_2}), then $\mu$ also satisfies (\ref{mu(overline{B}(w,
r)) le C_2 mu(B(w, r))}), as in the previous section.
\section{Separability} \label{separability}
Let $(M, d(x, y))$ be a metric space. If the metric $d(x, y)$ is doubling, then bounded subsets of $M$ are totally bounded, as in Section \ref{doubling metrics}. This implies that $M$ is separable, by expressing $M$ as a countable union of balls, each of which is totally bounded and thus has a countable dense subset. In particular, if there is a doubling measure $\mu$ on $M$, then $d(x, y)$ is a doubling metric on $M$, as in Section \ref{doubling measures}, and hence $M$ is separable. Suppose now that $\mu$ is a nonnegative Borel measure on $M$ such that every open ball in $M$ has positive finite measure with respect to $\mu$, and let us check that $M$ is separable.
Let $x \in M$ and $r, t > 0$ be given, with $t \le r$, and let $A$ be a subset of $B(x, r)$ such that \begin{equation} \label{d(y, z) ge t}
d(y, z) \ge t \end{equation} for every $y, z \in A$ with $y \ne z$. Thus the balls $B(y, t/2)$ with $y \in A$ are pairwise disjoint, and \begin{equation} \label{B(y, t/2) subseteq B(x, 3 r/ 2)}
B(y, t/2) \subseteq B(x, 3 r/ 2) \end{equation} for each $y \in A$. If $y_1, \ldots, y_n$ are finitely many elements of $A$ such that \begin{equation} \label{mu(B(y_j, t/2)) ge a}
\mu(B(y_j, t/2)) \ge a \end{equation} for some $a > 0$ and $j = 1, \ldots, n$, then \begin{equation} \label{n a le sum_{j = 1}^n mu(B(y_j, t/2)) = ... le mu(B(x, 3 r / 2))}
n \, a \le \sum_{j = 1}^n \mu(B(y_j, t/2))
= \mu\Big(\bigcup_{j = 1}^n B(y_j, t/2)\Big) \le \mu(B(x, 3 r / 2)), \end{equation} since $\bigcup_{j = 1}^n B(y_j, t/2) \subseteq B(x, 3 r / 2)$, by (\ref{B(y, t/2) subseteq B(x, 3 r/ 2)}). This shows that $n$ is uniformly bounded under these conditions, and hence that there are only finitely many $y \in A$ such that \begin{equation} \label{mu(B(y, t/2)) ge a}
\mu(B(y, t/2)) \ge a. \end{equation} Applying this to a sequence of $a$'s converging to $0$, we get that $A$ has only finitely or countably many elements.
Suppose now that $A$ is a maximal subset of $B(x, r)$ such that (\ref{d(y, z) ge t}) holds for every $y, z \in A$ with $y \ne z$, which exists by Zorn's lemma or the Hausdorff maximality principle. If $w$ is any element of $B(x, r)$, then \begin{equation} \label{d(w, y) < t}
d(w, y) < t \end{equation} for some $y \in A$, since otherwise $A \cup \{w\}$ would be a larger set with the same property. This implies that \begin{equation} \label{B(x, r) subseteq bigcup_{y in A} B(y, t)}
B(x, r) \subseteq \bigcup_{y \in A} B(y, t). \end{equation} where $A$ has finitely or countably many elements, as before. It follows that $B(x, r)$ has a dense subset with only finitely or countably many elements, by considering a sequence of $t$'s converging to $0$. Thus $M$ is separable under these conditions, since it can be expressed as the union of a sequence of open balls.
Alternatively, let $k$ be a positive integer, and let $A_k$ be a subset of $B(x, r)$ that satisfies (\ref{d(y, z) ge t}) for every $y, z \in A_k$ with $y \ne z$, and \begin{equation} \label{mu(B(y, t/2)) ge 1/k}
\mu(B(y, t/2)) \ge 1/k \end{equation} for every $y \in A_k$. The earlier argument shows that $A_k$ is a finite set with a bounded number of elements, and so we suppose now that $A_k$ is a maximal set with these properties, for each $k \in {\bf Z}_+$. Thus \begin{equation} \label{A = bigcup_{k = 1}^infty A_k}
A = \bigcup_{k = 1}^\infty A_k \end{equation} has only finitely or countably many elements, although this set $A$ does not normally satisfy (\ref{d(y, z) ge t}) for every $y, z \in A$ with $y \ne z$.
If $w$ is any element of $B(x, r)$, then \begin{equation} \label{mu(B(w, t/2)) ge 1/k}
\mu(B(w, t/2)) \ge 1/k \end{equation} for some $k \in {\bf Z}_+$, because $\mu(B(w, t/2)) > 0$ by hypothesis. It follows that (\ref{d(w, y) < t}) holds for some $y \in A_k$, since otherwise $A_k \cup \{w\}$ would be a larger set with the same properties as $A_k$. In particular, (\ref{d(w, y) < t}) holds for some $y \in A$, so that (\ref{B(x, r) subseteq bigcup_{y in A}
B(y, t)}) holds again in this situation. This implies that $B(x, r)$ has a dense subset with only finitely or countably many elements, and hence that $M$ is separable, for the same reasons as before.
Similarly, if there is an $a > 0$ such that (\ref{mu(B(y, t/2)) ge a}) holds for every $y \in B(x, r)$, then the previous argument shows that the number of elements of a set $A$ as before is bounded. This implies that $B(x, r)$ can be covered by finitely many balls of radius $t$, as in (\ref{B(x, r) subseteq bigcup_{y in A} B(y, t)}). If for each $t \in (0, r]$ there is an $a > 0$ with this property, then it follows that $B(x, r)$ is totally bounded.
As usual, these arguments can be simplified when $d(\cdot, \cdot)$ is an ultrametric on $M$. In this case, if $y, z \in B(x, r)$ and $0 < t \le r$, then either $d(y, z) < t$, and hence $B(y, t) = B(z, t)$, or $d(y, z) \ge t$, which implies that \begin{equation} \label{B(y, t) cap B(z, t) = emptyset}
B(y, t) \cap B(z, t) = \emptyset. \end{equation} Of course, $B(y, t) \subseteq B(y, r) = B(x, r)$ for every $y \in B(x, r)$ when $t \le r$. It follows that for each $a> 0$ and $t \in (0, r]$, there cannot be more than \begin{equation} \label{mu(B(x, r)) / a}
\mu(B(x, r)) / a \end{equation} distinct open balls $B(y, t)$ contained in $B(x, r)$ such that \begin{equation} \label{mu(B(y, t)) ge a}
\mu(B(y, t)) \ge a. \end{equation} In particular, for each $t \in (0, r]$, there are only finitely or countably many distinct open balls $B(y, t)$ contained in $B(x, r)$.
\chapter{Maximal functions} \label{maximal functions}
\section{Definitions} \label{definitions}
Let $(X, d(x, y))$ be a metric space, and let $\mu$ be a nonnegative Borel measure on $X$ such that the measure of any open ball in $X$ is positive and finite. If $f$ is a locally integrable function on $X$ with respect to $\mu$, then put \begin{equation}
\label{M(f)(x) = sup_{B ni x} frac{1}{mu(B)} int_B |f| d mu}
M(f)(x) = \sup_{B \ni x} \frac{1}{\mu(B)} \int_B |f| \, d\mu \end{equation} for each $x \in X$, which may be infinite. More precisely, the supremum is taken over all open balls $B = B(y, r)$ in $X$ that contain $x$ as an element. This is the uncentered version of the Hardy--Littlewood maximal function\index{maximal functions} associated to $f$ with respect to $\mu$ on $X$. Similarly, if $\nu$ is a nonnegative Borel measure on $X$, then the corresponding maximal function is defined by \begin{equation} \label{M(nu)(x) = sup_{B ni x} frac{nu(B)}{mu(B)}}
M(\nu)(x) = \sup_{B \ni x} \frac{\nu(B)}{\mu(B)} \end{equation} for each $x \in X$. Of course, this reduces to (\ref{M(f)(x) = sup_{B
ni x} frac{1}{mu(B)} int_B |f| d mu}) when $\nu$ is given by \begin{equation}
\label{nu(A) = int_A |f| d mu}
\nu(A) = \int_A |f| \, d\mu \end{equation} for every Borel set $A \subseteq X$. If $\nu$ is a real or complex Borel measure on $X$, then $M(\nu)$ is defined to be the same as
$M(|\nu|)$, where $|\nu|$ is the total variation measure associated to $\nu$.
If $f$ and $g$ are locally integrable functions on $X$ with respect to $\mu$, then \begin{equation} \label{M(f + g)(x) le M(f)(x) + M(g)(x)}
M(f + g)(x) \le M(f)(x) + M(g)(x) \end{equation} for every $x \in X$. Similarly, \begin{equation}
\label{M(t f)(x) = |t| M(f)(x)}
M(t \, f)(x) = |t| \, M(f)(x) \end{equation} for every $x \in X$ and real or complex number $t$, as appropriate, so that the mapping from $f$ to $M(f)$ is sublinear. There are analogous statements for maximal functions of Borel measures, as in the previous paragraph.
Let $\nu$ be a nonnegative Borel measure on $X$, and let $t$ be a nonnegative real number. If \begin{equation} \label{M(nu)(x) > t}
M(\nu)(x) > t \end{equation} for some $x \in X$, then there is an open ball $B$ in $X$ such that $x \in B$ and \begin{equation} \label{frac{nu(B)}{mu(B)} > t}
\frac{\nu(B)}{\mu(B)} > t, \end{equation} by the definition of $M(\nu)$. Conversely, if $B$ is an open ball in $X$ that satisfies (\ref{frac{nu(B)}{mu(B)} > t}), then $M(\nu)(y) > t$ for every $y \in B$. Thus \begin{equation} \label{V_t = {x in X : M(nu)(x) > t}}
V_t = \{x \in X : M(\nu)(x) > t\} \end{equation} is the same as the union of the open balls $B$ in $X$ that satisfy (\ref{frac{nu(B)}{mu(B)} > t}). In particular, (\ref{V_t = {x in X :
M(nu)(x) > t}}) is an open set in $X$ for each $t \ge 0$.
If $f$ is a bounded Borel measurable function on $X$, then \begin{equation}
\label{sup_{x in X} M(f)(x) le ||f||_infty}
\sup_{x \in X} M(f)(x) \le \|f\|_\infty, \end{equation}
where $\|f\|_\infty$ denotes the $L^\infty$ norm of $f$ wih respect to $\mu$. We shall consider other estimates for maximal functions in the next sections.
\section{Three covering arguments} \label{three covering arguments}
Let $I$, $I'$, and $I''$ be three intervals in the real line, which may be open, closed, or half-open and half-closed. If \begin{equation} \label{I cap I' cap I'' ne emptyset}
I \cap I' \cap I'' \ne \emptyset, \end{equation} then it is easy to see that one of these interval is contained in the union of the other two. Now let $I_1, I_2, \ldots, I_n$ be finitely many intervals in ${\bf R}$, which may again be open, closed, or half-open and half-closed. Using the previous argument repeatedly, one can find indices $1 \le j_1 < j_2 < \cdots < j_r \le n$ such that \begin{equation} \label{bigcup_{l = 1}^r I_{j_l} = bigcup_{k = 1}^n I_k}
\bigcup_{l = 1}^r I_{j_l} = \bigcup_{k = 1}^n I_k \end{equation} and no element of ${\bf R}$ is contained in more than two of the $I_{j_l}$'s.
Suppose instead that $d(x, y)$ is an ultrametric on a set $X$, and let $B_1, \ldots, B_n$ be finitely many distinct balls in $X$ with respect to $d(x, y)$, which may be open or closed. In this case, there are indices $1 \le j_1 < j_2 < \cdots < j_r \le n$ such that \begin{equation} \label{bigcup_{l = 1}^r B_{j_l} = bigcup_{k = 1}^n B_k}
\bigcup_{l = 1}^r B_{j_l} = \bigcup_{k = 1}^n B_k, \end{equation} and the balls $B_{j_l}$ are pairwise disjoint. To see this, one can take the $B_{j_l}$'s to be maximal among $B_1, \ldots, B_n$ with respect to inclusion. This uses the fact that if $B$ and $B'$ are two open or closed balls in $X$, then either $B \subseteq B'$, $B' \subseteq B$, or $B \cap B' = \emptyset$.
Suppose now that $d(x, y)$ is any metric on a set $X$, and let $B_j = B(x_j, r_j)$ be the open ball in $X$ centered at a point $x_j \in X$ with radius $r_j > 0$ for $j = 1, \ldots, n$. By rearranging the indices if necessary, we may also ask that $r_j$ be monotone decreasing in $j$. Put $j_1 = 1$, and let $j_2$ be the smallest integer such that $2 \le j_2 \le n$ and \begin{equation} \label{B_{j_1} cap B_{j_2} = emptyset}
B_{j_1} \cap B_{j_2} = \emptyset, \end{equation} if there is one. Similarly, if $1 = j_1 < j_2 < \cdots < j_l < n$ have been chosen, then let $j_{l + 1}$ be th smallest integer such that $j_l < j_{l + 1} \le n$ and \begin{equation} \label{B_{j_k} cap B_{j_{l + 1}} = emptyset}
B_{j_k} \cap B_{j_{l + 1}} = \emptyset \end{equation} for each $k = 1, \ldots, l$, if there is one. This process has to stop in a finite number $r$ of steps, and the corresponding balls $B_{j_l}$ are pairwise disjoint, by construction. If an integer $i$, $1 \le i \le n$, is not equal to $j_l$ for some $l$, then there is an $l$ such that $j_l < i$ and $B_i \cap B_{j_l} \ne \emptyset$. This implies that \begin{equation} \label{B_i subseteq B(x_{j_l}, 3 r_{j_l})}
B_i \subseteq B(x_{j_l}, 3 \, r_{j_l}), \end{equation} since the radius $r_i$ of $B_i$ is less than or equal to $r_{j_l}$. It follows that \begin{equation} \label{bigcup_{i = 1}^n B_i subseteq bigcup_{l = 1}^r B(x_{j_l}, 3 r_{j_l})}
\bigcup_{i = 1}^n B_i \subseteq \bigcup_{l = 1}^r B(x_{j_l}, 3 \, r_{j_l}). \end{equation} Essentially the same argument works when the $B_j$'s are closed balls, or a mixture of open and closed balls. If $d(x, y)$ is a quasi-metric on $X$, then the radius $3 \, r_{j_l}$ in (\ref{B_i subseteq
B(x_{j_l}, 3 r_{j_l})}) and (\ref{bigcup_{i = 1}^n B_i subseteq
bigcup_{l = 1}^r B(x_{j_l}, 3 r_{j_l})}) should be replaced with another constant multiple of $r_{j_l}$, depending on the constant in the quasi-metric condition for $d(x, y)$.
\section{Weak-type estimates} \label{weak-type estimates}
Let $(X, d(x, y))$ be a metric space, and let $\mu$ be a nonnegative Borel measure on $X$ such that the measure of any open ball in $X$ is positive and finite. Also let $\nu$ be a nonnegative Borel measure on $X$ such that $\nu(X) < +\infty$, and let $V_t$ be as in (\ref{V_t =
{x in X : M(nu)(x) > t}}) for each $t \ge 0$. Under suitable conditions, we would like to show that \begin{equation} \label{mu(V_t) le C_1 t^{-1} nu(X)}
\mu(V_t) \le C_1 \, t^{-1} \, \nu(X) \end{equation} for some positive real number $C_1$ and every $t > 0$, where $C_1$ does not depend on $\nu$ or $t$. As in Section \ref{definitions}, $V_t$ is the same as the union of the open balls $B$ in $X$ that satisfy (\ref{frac{nu(B)}{mu(B)} > t}), for each $t > 0$. Note that $X$ is separable, as in Section \ref{separability}, which implies that there is a base for the topology of $X$ with only finitely or countably many elements. It follows that $V_t$ can be expressed as the union of finitely or countably many open balls $B$ in $X$ that satisfy (\ref{frac{nu(B)}{mu(B)} > t}) for each $t > 0$, by Lindel\"of's theorem in topology. Let $B_1, \ldots, B_n$ be finitely many distinct open balls in $X$ that satisfy (\ref{frac{nu(B)}{mu(B)}
> t}) for some $t > 0$, so that \begin{equation} \label{mu(B_j) < t^{-1} nu(B_j)}
\mu(B_j) < t^{-1} \, \nu(B_j) \end{equation} for $j = 1, \ldots, n$. In order to obtain an estimate of the form (\ref{mu(V_t) le C_1 t^{-1} nu(X)}), it suffices to show that \begin{equation} \label{mu(bigcup_{j = 1}^n B_j) le C_1 t^{-1} nu(X)}
\mu\Big(\bigcup_{j = 1}^n B_j\Big) \le C_1 \, t^{-1} \, \nu(X), \end{equation} where $C_1 > 0$ does not depend on $t$, $\nu$, or $B_1, \ldots, B_n$, and in particular where $C_1$ does not depend on $n$.
Suppose first that $d(x, y)$ is an ultrametric on $X$. In this case, there are indices $1 \le j_1 < j_2 < \cdots < j_r \le n$ such that (\ref{bigcup_{l = 1}^r B_{j_l} = bigcup_{k = 1}^n B_k}) holds, and the balls $B_{j_l}$ are pairwise disjoint, as in the preceding section. This implies that \begin{eqnarray} \label{mu(bigcup_{k = 1}^n B_k) = ... le t^{-1} nu(X)}
\mu\Big(\bigcup_{k = 1}^n B_k\Big) = \mu\Big(\bigcup_{l = 1}^r B_{j_l}\Big)
& = & \sum_{l = 1}^r \mu(B_{j_l}) \\
& < & t^{-1} \, \sum_{l = 1}^r \nu(B_{j_l}) \le t^{-1} \, \nu(X), \nonumber \end{eqnarray} using (\ref{mu(B_j) < t^{-1} nu(B_j)}) in the first inequality, and pairwise-disjointness of the $B_{j_l}$'s in the second inequality. Thus (\ref{mu(bigcup_{j = 1}^n B_j) le C_1 t^{-1} nu(X)}) holds with $C_1 = 1$, as desired.
Now let $X$ be the real line with the standard metric, so that the open balls $B_j$ are open intervals. As before, there are indices $1 \le j_1 < j_2 < \cdots < j_r \le n$ such that (\ref{bigcup_{l =
1}^r B_{j_l} = bigcup_{k = 1}^n B_k}) holds, and no element of $X = {\bf R}$ is contained in more than two of the $B_{j_l}$'s. Let ${\bf 1}_A(x)$ be the characteristic or indicator function associated to a set $A \subseteq X$, which is equal to $1$ when $x \in A$ and to $0$ when $x \in X \backslash A$. The condition that no point belong to more than two of the $B_{j_l}$'s implies that \begin{equation} \label{sum_{l = 1}^r {bf 1}_{B_{j_l}}(x) le 2}
\sum_{l = 1}^r {\bf 1}_{B_{j_l}}(x) \le 2 \end{equation} for every $x \in X = {\bf R}$. It follows that \begin{equation} \label{sum_{l = 1}^r nu(B_{j_l}) = ... le 2 nu({bf R})}
\sum_{l = 1}^r \nu(B_{j_l})
= \int_{\bf R} \Big(\sum_{l = 1}^r {\bf 1}_{B_{j_l}}(x)\Big) \, d\nu(x)
\le 2 \, \nu({\bf R}). \end{equation} Using this, we get that \begin{eqnarray} \label{mu(bigcup_{k = 1}^n B_k) = ... le 2 t^{-1} nu({bf R})}
\mu\Big(\bigcup_{k = 1}^n B_k\Big) = \mu\Big(\bigcup_{l = 1}^r B_{j_l}\Big)
& \le & \sum_{l = 1}^r \mu(B_{j_l}) \\
& < & t^{-1} \, \sum_{l = 1}^r \nu(B_{j_l}) \le 2 \, t^{-1} \, \nu({\bf R}),
\nonumber \end{eqnarray} as in the previous situation. This gives (\ref{mu(bigcup_{j = 1}^n
B_j) le C_1 t^{-1} nu(X)}), with $C_1 = 2$.
Suppose that $d(x, y)$ is any metric on a set $X$, and that $B_j = B(x_j, r_j)$ for some $x_j \in X$ and $r_j > 0$, $j = 1, \ldots, n$. The third argument in the preceding section implies that there are indices $1 \le j_1 < j_2 < \cdots < j_r \le n$ such that (\ref{bigcup_{i = 1}^n B_i subseteq bigcup_{l = 1}^r B(x_{j_l}, 3
r_{j_l})}) holds and the balls $B_{j_l}$ are pairwise disjoint. If $\mu$ is a doubling measure on $X$, then it follows that \begin{eqnarray} \label{mu(bigcup_{k = 1}^n B_k) le ... le C_1 sum_{l = 1}^r mu(B_{j_l})}
\mu\Big(\bigcup_{k = 1}^n B_k\Big)
\le \mu\Big(\bigcup_{l = 1}^r B(x_{j_l}, 3 \, r_{j_l})\Big)
& \le & \sum_{l = 1}^r \mu(B(x_{j_l}, 3 \, r_{j_l})) \\
& \le & C_1 \, \sum_{l = 1}^r \mu(B_{j_l}), \nonumber \end{eqnarray} for a suitable constant $C_1$. Combining this with (\ref{mu(B_j) <
t^{-1} nu(B_j)}), we get that \begin{equation} \label{mu(bigcup_{k = 1}^n B_k) le ... le C_1 t^{-1} nu(X)}
\mu\Big(\bigcup_{k = 1}^n B_k\Big) \le C_1 \, \sum_{l = 1}^r \mu(B_{j_l})
< C_1 \, t^{-1} \, \sum_{l = 1}^r \nu(B_{j_l}) \le C_1 \, t^{-1} \, \nu(X), \end{equation} using also the fact that the $B_{j_l}$'s are pairwise disjoint in the last step. Thus the same type of estimate holds when $\mu$ is a doubling measure on any metric space.
\section{Distribution functions} \label{distribution functions}
Let $(X, \mathcal{A}, \mu)$ be a measure space, and let $g$ be a measurable function on $X$ with values in the set $[0, +\infty]$ of nonnegative extended real numbers. The corresponding distribution function\index{distribution functions} is defined on $[0, +\infty)$ by \begin{equation} \label{lambda(t) = mu({x in X : g(x) > t})}
\lambda(t) = \mu(\{x \in X : g(x) > t\}), \end{equation} which is a monotone decreasing function on $[0, +\infty)$ with values
in $[0, +\infty]$. Of course, $\lambda(t) \le \mu(X)$ for every $t \ge 0$, and \begin{equation} \label{t^p lambda(t) le ... le int_X g(x)^p d mu(x)}
t^p \, \lambda(t) \le \int_{\{x \in X : g(x) > t\}} g(x)^p \, d\mu(x)
\le \int_X g(x)^p \, d\mu(x) \end{equation} for every $p, t > 0$. In particular, $\lambda(t) < +\infty$ for every $t > 0$ when $g \in L^p(X)$ for some $p \in (0, +\infty)$.
Let us suppose from now on in this section that $X$ is at least $\sigma$-finite with respect to $\mu$. Note that the set \begin{equation} \label{{x in X : g(x) > 0}}
\{x \in X : g(x) > 0\} \end{equation} is measurable and $\sigma$-finite when $\lambda(t) < +\infty$ for each $t > 0$, so that we could simply replace $X$ with (\ref{{x in X : g(x)
> 0}}) in this situation, if necessary. Let us also consider $[0,
+\infty)$ as a $\sigma$-finite measure space with respect to
Lebesgue measure, so that $X \times [0, +\infty)$ is a
$\sigma$-finite measure space as well, with respect to the usual
product measure construction. If $X$ is a topological space too,
then we may consider $X \times [0, +\infty)$ as a topological
space with respect to the product topology, using the topology
induced on $[0, +\infty)$ by the standard topology on ${\bf R}$.
Put \begin{eqnarray} \label{U_r = {(x, t) in X times [0, +infty) : t < r < g(x)} = ...}
U_r & = & \{(x, t) \in X \times [0, +\infty) : t < r < g(x)\} \\
& = & [0, r) \times \{x \in X : g(x) > r\} \nonumber \end{eqnarray} for each $r \in (0, +\infty)$, and \begin{equation} \label{U = {(x, t) in X times [0, +infty) : t < g(x)}}
U = \{(x, t) \in X \times [0, +\infty) : t < g(x)\}. \end{equation} Observe that \begin{equation} \label{U = bigcup_{r in {bf Q}_+} U_r}
U = \bigcup_{r \in {\bf Q}_+} U_r, \end{equation} where ${\bf Q}_+ = {\bf Q} \cap (0, +\infty)$ is the set of all positive rational numbers. This implies that $U$ is a measurable set in $X \times [0, +\infty)$, since it can be expressed as a countable union of measurable rectangles. If $X$ is a topological space, and if \begin{equation} \label{{x in X : g(x) > t}}
\{x \in X : g(x) > t\} \end{equation} is an open set in $X$ for each $t \ge 0$, then it is easy to see that $U$ is an open set in $X \times [0, +\infty)$. In fact, (\ref{U =
bigcup_{r in {bf Q}_+} U_r}) shows that $U$ can be expressed as a countable union of products of open subsets of $X$ and $[0, +\infty)$ in this case.
Let $p > 0$ be given, and let ${\bf 1}_U(x, t)$ be the indicator function associated to $U$ on $X \times [0, +\infty)$. Observe that \begin{equation} \label{p t^{p - 1} {bf 1}_U(x, t)}
p \, t^{p - 1} \, {\bf 1}_U(x, t) \end{equation} is a measurable function on $X \times [0, +\infty)$, because $U$ is a measurable set. Clearly \begin{eqnarray} \label{... = int_X g(x)^p d mu(x)} \lefteqn{\int_X\Big(\int_{[0, +\infty)} p \, t^{p - 1} \, {\bf 1}_U(x, t)
\, dt\Big) \, d\mu(x)} \\
& = & \int_X \Big(\int_0^{g(x)} p \, t^{p - 1} \, dt\Big) \, d\mu(x)
\nonumber \\
& = & \int_X g(x)^p \, d\mu(x), \nonumber \end{eqnarray} by elementary calculus, and \begin{eqnarray} \label{int_0^infty p t^{p - 1} lambda(t) dt} \lefteqn{\int_{[0, +\infty)} \Big(\int_X p \, t^{p - 1} \, {\bf 1}_U(x, t)
\, d\mu(x)\Big) \, dt} \\
& = & \int_0^\infty p \, t^{p - 1} \, \Big(\int_{\{x \in X : g(x) > t\}}
\, d\mu(x)\Big) \, dt \nonumber \\
& = & \int_0^\infty p \, t^{p - 1} \, \lambda(t) \, dt. \nonumber \end{eqnarray} Remember that a monotone function on an interval in the real line continuous at all but at most finitely or countably many points, which simplifies questions of measurability and integrability. At any rate, it follows from (\ref{... = int_X g(x)^p d mu(x)}) and (\ref{int_0^infty p t^{p - 1} lambda(t) dt}) that \begin{equation} \label{int_X g(x)^p d mu(x) = int_0^infty p t^{p - 1} lambda(t) dt}
\int_X g(x)^p \, d\mu(x) = \int_0^\infty p \, t^{p - 1} \, \lambda(t) \, dt \end{equation} for every $p > 0$, by Fubini's theorem.
\section{$L^p$ Estimates} \label{L^p estimates}
Let $(X, d(x, y))$ be a metric space again, and let $\mu$ be a nonnegative Borel measure on $X$ for which the measure of every open ball in $X$ is positive and finite. Suppose that there is a positive real number $C_1$ such that \begin{equation}
\label{mu({x in X : M(f)(x) > t}) le C_1 t^{-1} int_X |f(x)| d mu(x)}
\mu(\{x \in X : M(f)(x) > t\}) \le C_1 \, t^{-1} \, \int_X |f(x)| \, d\mu(x) \end{equation} for every integrable function $f$ on $X$ with respect to $\mu$. This is the same as (\ref{mu(V_t) le C_1 t^{-1} nu(X)}) in Section \ref{weak-type estimates}, when $\nu$ corresponds to $f$ as in
(\ref{nu(A) = int_A |f| d mu}) in Section \ref{definitions}. Let a real number $p \ge 1$ be given, and suppose now that $f \in L^p(X)$ with respect to $\mu$. We would like to show that $M(f) \in L^p(X)$ when $p > 1$.
Put \begin{eqnarray}
\label{f_t(x) = f(x) when |f(x)| le t, = 0 when |f(x)| > t}
f_t(x) & = & f(x) \, \hbox{ when } |f(x)| \le t \\
& = & 0 \qquad\hbox{when } |f(x)| > t \nonumber \end{eqnarray}
for each $t > 0$. Thus $f_t$ is a bounded measurable function on $X$, with $\|f\|_\infty \le t$, so that \begin{equation} \label{M(f)(x) le t}
M(f)(x) \le t \end{equation}
for every $x \in X$, as in (\ref{sup_{x in X} M(f)(x) le ||f||_infty}) in Section \ref{definitions}. Observe that \begin{equation} \label{M(f)(x) le M(f_{a t})(x) + M(f - f_{a t})(x) le a t + M(f - f_{a t})(x)}
M(f)(x) \le M(f_{a \, t})(x) + M(f - f_{a \, t})(x)
\le a \, t + M(f - f_{a \, t})(x) \end{equation} for every $x \in X$ and $a, t > 0$. If $0 < a < 1$ and $M(f)(x) > t$, then it follows that $M(f - f_{a \, t})(x) > (1 - a) \, t$, which is to say that \begin{equation} \label{{x in X : M(f)(x) > t} subseteq {x in X : M(f - f_{a t})(x) > (1 - a)t}}
\{x \in X : M(f)(x) > t\}
\subseteq \{x \in X : M(f - f_{a \, t})(x) > (1- a) \, t\}. \end{equation} This implies that \begin{eqnarray} \label{mu({x in X : M(f)(x) > t}) le ...} \lefteqn{\mu(\{x \in X : M(f)(x) > t\})} \\
& \le & \mu(\{x \in X : M(f - f_{a \, t})(x) > (1 - a) \, t\}) \nonumber \\
& \le & C_1 \, (1 - a)^{-1} \, t^{-1} \, \int_X |f(x) - f_{a \, t}(x)|
\, d\mu(x) \nonumber \\
& \le & C_1 \, (1 - a)^{-1} \, t^{-1} \, \int_{\{x \in X : |f(x)| > a \, t\}}
|f(x)| \, d\mu(x). \nonumber \end{eqnarray} More precisely, this uses (\ref{mu({x in X : M(f)(x) > t}) le C_1
t^{-1} int_X |f(x)| d mu(x)}) in the second step, applied to $f -
f_{a \, t}$ instead of $f$, and $(1 - a) \, t$ instead of $t$. In the third step, we have used the fact that $f(x) - f_{a \, t}(x)$ is equal to $f(x)$ when $|f(x)| > a \, t$, and is $0$ otherwise. Note that $f - f_{a \, t}$ is integrable on $X$ for every $a, t > 0$, even when $p > 1$.
Let us restrict our attention now to the case where $p > 1$. The integral of $M(f)(x)^p$ with respect to $\mu$ can be expressed as in (\ref{int_X g(x)^p d mu(x) = int_0^infty p t^{p - 1} lambda(t) dt}) with $g = M(f)$, and we can use (\ref{mu({x in X : M(f)(x) > t}) le
...}) to estimate $\lambda(t)$ as in (\ref{lambda(t) = mu({x in X :
g(x) > t})}). This implies that \begin{eqnarray} \label{int_X M(f)(x)^p d mu(x) le ...} \lefteqn{\int_X M(f)(x)^p \, d\mu(x)} \\
& \le & \int_0^\infty p \, t^{p - 1} \, \Big(C_1 \, (1 - a)^{-1} \, t^{-1} \,
\int_{\{x \in X : |f(x)| > a \, t\}} |f(x)| \, d\mu(x)\Big) \, dt \nonumber \\
& = & p \, C_1 \, (1 - a)^{-1} \, \int_0^\infty t^{p - 2} \,
\Big(\int_{\{x \in X : |f(x)| > a \, t\}} |f(x)| \, d\mu(x)\Big) \, dt.
\nonumber \end{eqnarray} Interchanging the order of integration, we get that \begin{eqnarray} \label{int_X M(f)(x)^p d mu(x) le ..., 2} \lefteqn{\int_X M(f)(x)^p \, d\mu(x)} \\
& \le & p \, C_1 \, (1 - a)^{-1} \, \int_X \Big(\int_0^{|f(x)|/a} t^{p - 2}
\, dt\Big) \, |f(x)| \, d\mu(x) \nonumber \\
& = & p \, C_1 \, (1 - a)^{-1} \, \int_X (p - 1)^{-1} \, (|f(x)|/a)^{p - 1}
\, |f(x)| \, d\mu(x) \nonumber \\
& = & p \, C_1 \, (1 - a)^{-1} \, (p - 1)^{-1} \, a^{1 - p} \,
\int_X |f(x)|^p \, d\mu(x). \nonumber \end{eqnarray} It follows that $M(f) \in L^p(X)$, with $L^p$ norm bounded by the $L^p$ norm of $f$ times a constant that depends on $p$ when $p > 1$, by taking the $p$th root of both sides of (\ref{int_X M(f)(x)^p d
mu(x) le ..., 2}). This works using any $a \in (0, 1)$, so that one can choose an optimal $a$ for each $p$. In particular, it is better to take $a$ close to $1$ as $p$ increases.
\section{Conditional expectation} \label{conditional expectation}
Let $(X, \mathcal{A}, \mu)$ be a probability space, so that $X$ is a set, $\mathcal{A}$ is a $\sigma$-algebra of subsets of $X$, and $\mu$ is a probability measure on $X$, which is to say a nonnegative countably-additive measure on $\mathcal{A}$ such that $\mu(X) = 1$. Suppose that $f$ is a real or complex-valued function on $X$ that is measurable with respect to $\mathcal{A}$, and integrable with respect to $\mu$. Thus \begin{equation} \label{nu_f(A) = int_A f d mu}
\nu_f(A) = \int_A f \, d\mu \end{equation} is defined for each $A \in \mathcal{A}$, and determines a countably-additive real or complex-valued measure on $\mathcal{A}$, as appropriate.
Let $\mathcal{B}$ be another $\sigma$-algebra of subsets of $X$ contained in $\mathcal{A}$, so that $\mathcal{B}$ is a $\sigma$-subalgebra of $\mathcal{A}$. The restriction of $\nu_f$ to $\mathcal{B}$ is a countably-additive real or complex-valued measure on $(X, \mathcal{B})$, which is absolutely continuous with respect to the restriction of $\mu$ to $\mathcal{B}$. The Radon--Nikodym theorem implies that there is a measurable function $f_\mathcal{B}$ on $X$ with respect to $\mathcal{B}$ which is also integrable with respect to $\mu$ such that \begin{equation} \label{int_A f_mathcal{B} d mu = nu_f(A) = int_A f d mu}
\int_A f_\mathcal{B} \, d\mu = \nu_f(A) = \int_A f \, d\mu \end{equation} for every $A \in \mathcal{B}$. This function $f_\mathcal{B}$ is known as the \emph{conditional expectation}\index{conditional expectation} of $f$ with respect to $\mathcal{B}$, which may be denoted $E(f \mid \mathcal{B})$ as well. If $f'_\mathcal{B}$ is any other function on $X$ that satisfies the same properties as $f_\mathcal{B}$, then it is easy to see that $f_\mathcal{B} = f'_\mathcal{B}$ almost everywhere on $X$ with respect to $\mu$. Of course, if $\mathcal{B} = \mathcal{A}$, then we can simply take $f_\mathcal{B} = f$. The conditional expectation of $f$ with respect to $\mathcal{B}$ may also be denoted $E_\mathcal{A}(f \mid \mathcal{B})$, to indicate the initial $\sigma$-algebra $\mathcal{A}$ explicitly.
As a basic class of examples, let $\mathcal{P}$ be a partition of $X$ into finitely or countably many measurable sets with positive measure with respect to $\mu$. Thus $\mathcal{P}$ is a collection of finitely or countably many pairwise-disjoint elements of $\mathcal{A}$ such that $\mu(A) > 0$ for each $A \in \mathcal{P}$, and the union of the elements of $\mathcal{P}$ is equal to $X$. Also let $\mathcal{B}(\mathcal{P})$ be the collection of subsets of $X$ that can be expressed as a union of elements of $\mathcal{P}$, which is interpreted as including the empty set. It is easy to see that $\mathcal{B}(\mathcal{P})$ is a $\sigma$-subalgebra of $\mathcal{A}$, and that a function $f$ on $X$ is measurable with respect to $\mathcal{B}(\mathcal{P})$ if and only if $f$ is constant on each $A \in \mathcal{A}$. If $f$ is a measurable function on $X$ with respect to $\mathcal{A}$ which is integrable with respect to $\mu$, then the conditional expectation $f_\mathcal{B}$ of $f$ with respect to $\mathcal{B}$ is given by \begin{equation} \label{f_mathcal{B}(x) = frac{1}{mu(A)} int_A f d mu}
f_\mathcal{B}(x) = \frac{1}{\mu(A)} \, \int_A f \, d\mu \end{equation} for every $A \in \mathcal{B}$ and $x \in A$.
As another class of examples, let $(X_1, \mathcal{A}_1, \mu_1)$ and $(X_2, \mathcal{A}_2, \mu_2)$ be probability spaces, and consider their Cartesian product $X = X_1 \times X_2$. The standard product measure construction leads to a $\sigma$-algebra $\mathcal{A}$ on $X$, and a probability measure $\mu$ defined on $\mathcal{A}$. Let $\mathcal{B}_1$ be the collection of subsets of $X$ of the form $A \times X_2$, where $A \in \mathcal{A}_1$ This is a $\sigma$-subalgebra of $\mathcal{A}$, and a function $f(x) = f(x_1, x_2)$ on $X$ is measurable with respect to $\mathcal{B}_1$ if and only if $f(x_1, x_2)$ only depends on $x_1$, and this function of $x_1$ is measurable with respect to $\mathcal{A}_1$ as a function on $X_1$. If $f$ is a function on $X$ which is measurable with respect to $\mathcal{A}$ and integrable with respect to $\mu$, then the conditional expectation $f_{\mathcal{B}_1}$ of $f$ with respect to $\mathcal{B}_1$ is given by \begin{equation} \label{f_{mathcal{B}_1}(x_1, x_2) = int_{X_2} f(x_1, y_2) d mu_2(y_2)}
f_{\mathcal{B}_1}(x_1, x_2) = \int_{X_2} f(x_1, y_2) \, d\mu_2(y_2), \end{equation} essentially by Fubini's theorem.
Let $\mathcal{A}$ be any $\sigma$-algebra of subsets of a set $X$
again, and let $\nu$ be a real or complex measure defined on $\mathcal{A}$. Remember that the corresponding total variation measure $|\nu|$ is defined on $\mathcal{A}$ by \begin{equation}
\label{|nu|(A) = sup sum_{j = 1}^infty |nu(A_j)|}
|\nu|(A) = \sup \sum_{j = 1}^\infty |\nu(A_j)|, \end{equation} where the supremum is taken over all sequences $A_1, A_2, A_3, \ldots$ of pairwise-disjoint measurable subsets of $X$ whose union is equal to
$A$. It is well known that $|\nu|$ is a countably-additive nonnegative measure defined on $\mathcal{A}$, and that $|\nu|(X) < +\infty$.
Suppose that $\mathcal{B}$ is a $\sigma$-subalgebra of $\mathcal{A}$, and let $\nu_\mathcal{B}$ be the restriction of $\nu$ to $\mathcal{B}$, which is a countably-additive real or complex measure defined on $\mathcal{B}$. Observe that \begin{equation}
\label{|nu_mathcal{B}|(A) le |nu|(A)}
|\nu_\mathcal{B}|(A) \le |\nu|(A) \end{equation}
for every $A \in \mathcal{B}$, where $|\nu|$ is the total variation of
$\nu$ as a measure on $\mathcal{A}$, and $|\nu_\mathcal{B}|$ is the total variation of $\nu_\mathcal{B}$ as a measure on $\mathcal{B}$. More precisely, if $A \in \mathcal{B}$, then $|\nu_\mathcal{B}|(A)$ is the supremum of the same type of sums as in (\ref{|nu|(A) = sup sum_{j
= 1}^infty |nu(A_j)|}), but where the $A_j$'s are required to be in $\mathcal{B}$. Thus $|\nu|(A)$ is given by a supremum of sums that includes the sums whose supremum is equal to $|\nu_\mathcal{B}|(A)$
when $A \in \mathcal{B}$, which implies (\ref{|nu_mathcal{B}|(A) le
|nu|(A)}).
Let $\mu$ be a probability measure defined on $\mathcal{A}$, and let $f$ be a real or complex-valued function on $X$ which is measurable with respect to $\mathcal{A}$ and integrable with respect to $\mu$. If $\nu = \nu_f$ is as in (\ref{nu_f(A) = int_A f d mu}), then it is well known that \begin{equation}
\label{|nu|(A) = int_A |f| d mu}
|\nu|(A) = \int_A |f| \, d\mu \end{equation} for every $A \in \mathcal{B}$. Let $\mathcal{B}$ be a $\sigma$-subalgebra of $\mathcal{A}$, and let $\nu_\mathcal{B}$ be the restriction of $\nu$ to $\mathcal{B}$, as in the previous paragraph. If $f_\mathcal{B}$ is the conditional expectation of $f$ with respect to $\mathcal{B}$, then $\nu_\mathcal{B}(A)$ is equal to the integral of $f_\mathcal{B}$ over $A$ with respect to $\mu$ for every $A \in \mathcal{B}$, as in (\ref{int_A f_mathcal{B} d mu = nu_f(A) = int_A f
d mu}), and hence \begin{equation}
\label{|nu_mathcal{B}|(A) = int_A |f_mathcal{B}| d mu}
|\nu_\mathcal{B}|(A) = \int_A |f_\mathcal{B}| \, d\mu \end{equation}
for every $A \in \mathcal{B}$, as in (\ref{|nu|(A) = int_A |f| d mu}). It follows that \begin{equation}
\label{int_A |f_mathcal{B}| d mu le int_A |f| d mu}
\int_A |f_\mathcal{B}| \, d\mu \le \int_A |f| \, d\mu \end{equation}
for every $A \in \mathcal{B}$, because of (\ref{|nu_mathcal{B}|(A) le
|nu|(A)}).
Even if a real or complex measure $\nu$ defined on $\mathcal{A}$ is not absolutely continuous with respect to $\mu$, it may be that the restiction $\nu_\mathcal{B}$ of $\nu$ to a $\sigma$-subalgebra $\mathcal{B}$ of $\mathcal{A}$ is absolutely continuous with respect to the restriction of $\mu$ to $\mathcal{A}$. Under these conditions, the Radon--Nikodym theorem again implies that $\nu_\mathcal{B}$ can be represented on $\mathcal{B}$ by integration of a function $f_\mathcal{B}$ on $X$ that is measurable with respect to
$\mathcal{B}$ and integrable with respect to $\mu$. As before, one can combine (\ref{|nu_mathcal{B}|(A) le |nu|(A)}) and
(\ref{|nu_mathcal{B}|(A) = int_A |f_mathcal{B}| d mu}) to get that \begin{equation}
\label{int_A |f_mathcal{B}| d mu le |nu|(A)}
\int_A |f_\mathcal{B}| \, d\mu \le |\nu|(A) \end{equation} for every $A \in \mathcal{B}$. If $\mathcal{B} = \mathcal{B}(\mathcal{P})$ is the $\sigma$-algebra generated by a partition $\mathcal{P}$ of $X$ into finitely or countably many elements of $\mathcal{A}$, each of which has positive measure with respect to $\mu$, then every measure defined on $\mathcal{B}$ is absolutely continuous with respect to the restriction of $\mu$ to $\mathcal{B}$. In this case, we have that \begin{equation} \label{f_mathcal{B}(x) = frac{nu(A)}{mu(A)}}
f_\mathcal{B}(x) = \frac{\nu(A)}{\mu(A)} \end{equation} for every $A \in \mathcal{B}$ and $x \in A$, instead of (\ref{f_mathcal{B}(x) = frac{1}{mu(A)} int_A f d mu}).
Suppose now that $\mathcal{B}$ and $\mathcal{C}$ are $\sigma$-subalgebras of $\mathcal{A}$, with $\mathcal{B} \subseteq \mathcal{C}$, and let $\nu$ be a real or complex-valued measure defined on $\mathcal{A}$. If $\nu_\mathcal{B}$ and $\nu_\mathcal{C}$ are the restrictions of $\nu$ to $\mathcal{B}$, $\mathcal{C}$, respectively, then $\nu_\mathcal{B}$ is also the same as the restriction of $\nu_\mathcal{C}$ as a measure defined on $\mathcal{C}$ to a measure on $\mathcal{B}$. This is basically trivial, but it has the following nice interpretation for conditional expectation. Let $f$ be a real or complex-valued function on $X$ that is measurable with respect to $\mathcal{A}$ and integrable with respect to $\mu$, and let $f_\mathcal{B} = E_\mathcal{A}(f \mid \mathcal{B})$, $f_\mathcal{C} = E_\mathcal{A}(f \mid \mathcal{C})$ be the conditional expectations of $f$ with respect to $\mathcal{B}$, $\mathcal{C}$, respectively. Also let \begin{equation} \label{(f_mathcal{C})_mathcal{B} = E_mathcal{C}(f_mathcal{C} mid mathcal{B})}
(f_\mathcal{C})_\mathcal{B} = E_\mathcal{C}(f_\mathcal{C} \mid \mathcal{B}) \end{equation} be the conditional expectation of $f_\mathcal{C}$ with respect to $\mathcal{B}$, where $f_\mathcal{C}$ is considered as a measurable function with respect to $\mathcal{C}$ instead of $\mathcal{A}$. Under these conditions, it is easy to see that \begin{equation} \label{(f_mathcal{C})_mathcal{B} = f_mathcal{B}}
(f_\mathcal{C})_\mathcal{B} = f_\mathcal{B}, \end{equation} This follows from the previous statement about measures, applied to $\nu = \nu_f$ as in (\ref{nu_f(A) = int_A f d mu}). In particular, if $f$ is already measurable with respect to $\mathcal{C}$, then \begin{equation} \label{E_mathcal{A}(f mid mathcal{B}) = E_mathcal{C}(f mid mathcal{B})}
E_\mathcal{A}(f \mid \mathcal{B}) = E_\mathcal{C}(f \mid \mathcal{B}), \end{equation} as in the case of $f_\mathcal{C}$ in (\ref{(f_mathcal{C})_mathcal{B} =
E_mathcal{C}(f_mathcal{C} mid mathcal{B})}).
\section{Additional properties} \label{additional properties}
Let $(X, \mathcal{A}, \mu)$ be a probability space again, and let $\mathcal{B}$ be a $\sigma$-subalgebra of $\mathcal{A}$. Also let $f$ be a real or complex-valued function on $X$ that is measurable with respect to $\mathcal{A}$ and integrable with respect to $\mu$, and let $f_\mathcal{B}$ be the conditional expectation of $f$ with respect to
$\mathcal{B}$. Of course, $|f|$ is a nonnegative real-valued integrable function on $X$, and so the conditional expectation $(|f|)_\mathcal{B}$
of $|f|$ with respect to $\mathcal{B}$ is real-valued and nonnegative as well. Observe that \begin{equation}
\label{int_A |f_mathcal{B}| d mu le ... = int_A (|f|)_mathcal{B} d mu}
\int_A |f_\mathcal{B}| \, d\mu \le \int_A |f| \, d\mu
= \int_A (|f|)_\mathcal{B} \, d\mu \end{equation}
for every $A \in \mathcal{B}$, by (\ref{int_A |f_mathcal{B}| d mu le
int_A |f| d mu}) and the definition of $(|f|)_\mathcal{B}$. It follows that \begin{equation}
\label{|f_mathcal{B}| le (|f|)_mathcal{B}}
|f_\mathcal{B}| \le (|f|)_\mathcal{B} \end{equation} almost everywhere on $X$ with respect to $\mu$, since both sides of the inequality are measurable with respect to $\mathcal{B}$.
Now let $p \in (1, +\infty)$ be given, and suppose that $|f|^p$
is integrable on $X$ with respect to $\mu$. Thus the conditional expectation $(|f|^p)_\mathcal{B}$ of $|f|^p$ with respect to $\mathcal{B}$ can be defined as before, and is real-valued and nonnegative. If $A \in \mathcal{B}$ and $\mu(A) > 0$, then \begin{eqnarray}
\label{(frac{1}{mu(A)} int_A |f_mathcal{B}| d mu)^p le ...}
\Big(\frac{1}{\mu(A)} \, \int_A |f_\mathcal{B}| \, d\mu\Big)^p
& \le & \Big(\frac{1}{\mu(A)} \, \int_A |f| \, d\mu\Big)^p \\
& \le & \frac{1}{\mu(A)} \, \int_A |f|^p \, d\mu \nonumber \\
& = & \frac{1}{\mu(A)} \, \int_A (|f|^p)_\mathcal{B} \, d\mu. \nonumber \end{eqnarray}
This uses (\ref{int_A |f_mathcal{B}| d mu le int_A |f| d mu}) in the first step, Jensen's or H\"older's inequality in the second step, and the definition of $(|f|^p)_\mathcal{B}$ in the third step. One can check that this implies that \begin{equation}
\label{(|f_mathcal{B}|)^p le (|f|^p)_mathcal{B}}
(|f_\mathcal{B}|)^p \le (|f|^p)_\mathcal{B} \end{equation}
almost everywhere on $X$, because $|f_\mathcal{B}|$ and
$(|f|^p)_\mathcal{B}$ are both measurable with respect to $\mathcal{B}$. It follows that \begin{equation}
\label{int_X (|f_mathcal{B}|)^p d mu le ... = int_X |f|^p d mu}
\int_X (|f_\mathcal{B}|)^p \, d\mu \le \int_X (|f|^p)_\mathcal{B} \, d\mu
= \int_X |f|^p \, d\mu, \end{equation}
using the definition of $(|f|^p)_\mathcal{B}$ in the second step. In particular, $|f_\mathcal{B}|^p$ is integrable with respect to $\mu$ as well.
Let $L^p(X, \mathcal{A}, \mu)$ be the usual space of real or complex-valued functions $f$ on $X$ such that $f$ is measurable with respect to $\mathcal{A}$ and $|f|^p$ is integrable with respect to $\mu$, for $1 \le p < \infty$. More precisely, $L^p(X, \mathcal{A}, \mu)$ consists of equivalence classes of such functions, which are equal to each other almost everywhere with respect to $\mu$ on $X$. It is well known that $L^p(X, \mathcal{A}, \mu)$ is complete with respect to the metric associated to the the $L^p$ norm \begin{equation}
\label{||f||_p = (int_X |f|^p d mu)^{1/p}}
\|f\|_p = \Big(\int_X |f|^p \, d\mu\Big)^{1/p}. \end{equation} Similarly, $L^\infty(X, \mathcal{A}, \mu)$ consists of equivalence classes of functions on $X$ that are measurable with respect to $\mathcal{A}$ and essentially bounded on $X$. The $L^\infty$ norm
$\|f\|_\infty$ is defined to be the essential supremum of $|f|$ on $X$, and $L^\infty(X, \mathcal{A}, \mu)$ is complete with respect to the metric associated to this norm.
Of course, if $f$ is measurable with respect to $\mathcal{B}$, then $f$ is measurable with respect to $\mathcal{A}$ too. This leads to a natural linear mapping from $L^p(X, \mathcal{B}, \mu)$ into $L^p(X, \mathcal{A}, \mu)$ for each $p$, $1 \le p \le \infty$, which is an isometry with respect to the $L^p$ norm. Note that the image of $L^p(X, \mathcal{B}, \mu)$ under this mapping is a closed linear subspace of $L^p(X, \mathcal{A}, \mu)$, because $L^p(X, \mathcal{B}, \mu)$ is complete.
It is easy to see that $f \mapsto f_\mathcal{B}$ is a linear mapping from $L^1(X, \mathcal{A}, \mu)$ into $L^1(X, \mathcal{B}, \mu)$, using the uniqueness of $f_\mathcal{B}$. Moreover, \begin{equation}
\label{int_X |f_mathcal{B}| d mu le int_X |f| d mu}
\int_X |f_\mathcal{B}| \, d\mu \le \int_X |f| \, d\mu \end{equation} for every $f \in L^1(X, \mathcal{A}, \mu)$, by (\ref{int_A
|f_mathcal{B}| d mu le int_A |f| d mu}) with $A = X$. If $f \in L^p(X, \mathcal{A}, \mu)$ and $1 < p < \infty$, then $f_\mathcal{B} \in L^p(X, \mathcal{B}, \mu)$ and \begin{equation}
\label{||f_mathcal{B}||_p le ||f||_p}
\|f_\mathcal{B}\|_p \le \|f\|_p, \end{equation}
by (\ref{int_X (|f_mathcal{B}|)^p d mu le ... = int_X |f|^p d mu}). Similarly, if $f \in L^\infty(X, \mathcal{A}, \mu)$, then \begin{equation}
\label{int_A |f_mathcal{B}| d mu le int_A |f| d mu le ||f||_infty mu(A)}
\int_A |f_\mathcal{B}| \, d\mu \le \int_A |f| \, d\mu
\le \|f\|_\infty \, \mu(A) \end{equation}
for every $A \in \mathcal{B}$, using (\ref{int_A |f_mathcal{B}| d mu
le int_A |f| d mu}) in the first step. This implies that $f_\mathcal{B}$ is essentially bounded on $X$ as well, and that
(\ref{||f_mathcal{B}||_p le ||f||_p}) holds when $p = \infty$, because $f_\mathcal{B}$ is measurable with respect to $\mathcal{B}$.
Let $f \in L^1(X, \mathcal{A}, \mu)$ and $B \in \mathcal{B}$ be given, and let ${\bf 1}_B(x)$ be the characteristic or indicator function on $X$ associated to $B$, which is equal to $1$ when $x \in B$ and to $0$ otherwise. If $A \in \mathcal{B}$, then $A \cap B \in \mathcal{B}$ too, and hence \begin{equation} \label{int_A f_mathcal{B} {bf 1}_B d mu = ... = int_A f {bf 1}_B d mu}
\int_A f_\mathcal{B} \, {\bf 1}_B \, d\mu
= \int_{A \cap B} f_\mathcal{B} \, d\mu = \int_{A \cap B} f \, d\mu
= \int_A f \, {\bf 1}_B \, d\mu, \end{equation} using the definition of $f_\mathcal{B}$ in the second step. Of course, ${\bf 1}_B$ is measurable with respect to $\mathcal{B}$ on $X$, because $B \in \mathcal{B}$, and $f_\mathcal{B}$ is measurable with respect to $\mathcal{B}$ by construction, so that $f_\mathcal{B} \, {\bf 1}_B$ is measurable with respect to $\mathcal{B}$ as well. This shows that $f_\mathcal{B} \, {\bf 1}_B$ is equal to the conditional expectation of $f \, {\bf 1}_B$ with respect to $\mathcal{B}$, since it satisfies the requirements of the conditional expectation.
Similarly, if $g \in L^\infty(X, \mathcal{B}, \mu)$, then \begin{equation} \label{(f g)_mathcal{B} = f_mathcal{B} g}
(f \, g)_\mathcal{B} = f_\mathcal{B} \, g. \end{equation} This reduces to the discussion in the previous paragraph when $g = {\bf 1}_B$ for some $B \in \mathcal{B}$, which implies that (\ref{(f
g)_mathcal{B} = f_mathcal{B} g}) holds when $g$ is a simple function on $X$ that is measurable with respect to $\mathcal{B}$, by linearity. One can use this to get that (\ref{(f g)_mathcal{B} = f_mathcal{B} g}) holds for every $g \in L^\infty(X, \mathcal{B}, \mu)$, because simple functions on $X$ that are measurable with respect to $\mathcal{B}$ are dense in $L^\infty(X, \mathcal{B}, \mu)$ with respect to the $L^\infty$ norm. If $f \in L^p(X, \mathcal{A}, \mu)$ for some $p$, $1 \le p \le \infty$, and if $q$ is the exponent conjugate to $p$, in the sense that $1 \le q \le \infty$ and $1/p + 1/q = 1$, then one can check that (\ref{(f g)_mathcal{B} = f_mathcal{B} g}) holds for every $g \in L^q(X, \mathcal{B}, \mu)$. This also uses H\"older's inequality, and the fact that $f_\mathcal{B} \in L^p(X, \mathcal{B}, \mu)$ when $f \in L^p(X, \mathcal{A}, \mu)$ and $1 \le p \le \infty$, as before.
\section{Another maximal function} \label{another maximal function}
Let $(X, \mathcal{A}, \mu)$ be a probability space, and let $\mathcal{B}_1, \ldots, \mathcal{B}_n$ be finitely many $\sigma$-subalgebras of $\mathcal{A}$, with $\mathcal{B}_j \subseteq \mathcal{B}_{j + 1}$ for $j = 1, \ldots, n - 1$. Also let $f \in L^1(X, \mathcal{A}, \mu)$ be given, and let \begin{equation} \label{f_j = f_{mathcal{B}_j} = E(f mid mathcal{B}_j)}
f_j = f_{\mathcal{B}_j} = E(f \mid \mathcal{B}_j) \end{equation} be the conditional expectation of $f$ with respect to $\mathcal{B}_j$ for each $j$. Put \begin{equation}
\label{f_l^*(x) = max_{1 le j le l} |f_j(x)|}
f_l^*(x) = \max_{1 \le j \le l} |f_j(x)| \end{equation} for each $l = 1, \ldots, n$, and observe that $f_l^*$ is measurable with respect to $\mathcal{B}_l$ for each $l$. If $g \in L^1(X, \mathcal{A}, \mu)$ too, then it is easy to see that \begin{equation} \label{(f + g)_l^*(x) le f_l^*(x) + g_l^*(x)}
(f + g)_l^*(x) \le f_l^*(x) + g_l^*(x) \end{equation} for each $l$. Similarly, \begin{equation}
\label{(t f)_l^*(x) = |t| f_l^*(x)}
(t \, f)_l^*(x) = |t| \, f_l^*(x) \end{equation} for each $l$ and real or complex number $t$, as appropriate, so that the mapping from $f$ to $f_l^*$ is sublinear.
Let $t > 0$ be given, and put \begin{equation} \label{A_l(t) = {x in X : f_l^*(x) > t}}
A_l(t) = \{x \in X : f_l^*(x) > t\} \end{equation}
for each $l = 1, \ldots, n$, which is an element of $\mathcal{B}_l$, because $f_l^*$ is measurable with respect to $\mathcal{B}_l$. Note that $f_1^* = |f_1|$, so that \begin{equation}
\label{A_1(t) = {x in X : |f_1(x)| > t}}
A_1(t) = \{x \in X : |f_1(x)| > t\}, \end{equation} and hence \begin{equation}
\label{mu(A_1(t)) le ... le t^{-1} int_{A_1(t)} |f| d mu}
\mu(A_1(t)) \le t^{-1} \, \int_{A_1(t)} |f_1| \, d\mu
\le t^{-1} \, \int_{A_1(t)} |f| \, d\mu, \end{equation}
using (\ref{int_A |f_mathcal{B}| d mu le |nu|(A)}) in the second step. If $l > 1$, then \begin{eqnarray} \label{A_l(t) setminus A_{l - 1}(t) = ...}
A_l(t) \setminus A_{l - 1}(t) & = & \{x \in X : f_{l - 1}^*(x) \le t,
\, f_l^*(x) > t\} \\
& = & \{x \in X : f_{l - 1}^*(x) \le t, \, |f_l(x)| > t\}, \nonumber \end{eqnarray}
by the definition of $f_l^*$, and in particular $|f_l(x)| > t$ on (\ref{A_l(t) setminus A_{l - 1}(t) = ...}). Thus \begin{eqnarray} \label{mu(A_l(t) setminus A_{l - 1}(t)) le ...}
\mu(A_l(t) \setminus A_{l - 1}(t))
& \le & t^{-1} \, \int_{A_l(t) \setminus A_{l - 1}(t)} |f_l| \, d\mu \\
& \le & t^{-1} \, \int_{A_l(t) \setminus A_{l - 1}(t)} |f| \, d\mu, \nonumber \end{eqnarray}
again using (\ref{int_A |f_mathcal{B}| d mu le int_A |f| d mu}) in the second step, and the fact that $A_l(t) \setminus A_{l - 1}(t) \in \mathcal{B}_l$, since $A_l(t) \in \mathcal{B}_l$ and $A_{l - 1}(t) \in \mathcal{B}_{l - 1} \subseteq \mathcal{B}_l$.
By construction, $f_l^*$ is monotone increasing in $l$, which implies that \begin{equation} \label{A_l(t) subseteq A_{l + 1}(t)}
A_l(t) \subseteq A_{l + 1}(t) \end{equation} for $l = 1, \ldots, n - 1$. It follows that the sets $A_j(t) \setminus A_{j - 1}(t)$ are pairwise disjoint for $j \ge 2$, and disjoint from $A_1(t)$. Using (\ref{mu(A_1(t)) le ... le t^{-1}
int_{A_1(t)} |f| d mu}) and (\ref{mu(A_l(t) setminus A_{l - 1}(t))
le ...}), we get that \begin{eqnarray}
\label{mu(A_l(t)) = ... le t^{-1} int_X |f| d mu}
\mu(A_l(t)) & = & \mu(A_1(t)) +
\sum_{j = 2}^l \mu(A_j(t) \setminus A_{j - 1}(t)) \\
& \le & t^{-1} \, \int_{A_1(t)} |f| \, d\mu
+ \sum_{j = 2}^l t^{-1} \, \int_{A_j(t) \setminus A_{j - 1}(t)} |f| \, d\mu
\nonumber \\
& = & t^{-1} \, \int_{A_l(t)} |f| \, d\mu \le t^{-1} \, \int_X |f| \, d\mu
\nonumber \end{eqnarray} for each $l$, with the obvious simplifications when $l = 1$.
If $f \in L^\infty(X, \mathcal{A}, \mu)$, then $f_l^* \in L^\infty(X, \mathcal{B}_l, \mu)$ and \begin{equation}
\label{||f_l^*||_infty le ||f||_infty}
\|f_l^*\|_\infty \le \|f\|_\infty \end{equation}
for each $l$, by the $p = \infty$ version of (\ref{||f_mathcal{B}||_p
le ||f||_p}). Using this and the weak-type estimate on $L^1$ in
(\ref{mu(A_l(t)) = ... le t^{-1} int_X |f| d mu}), one can get $L^p$ estimates for $f_l^*$ when $1 < p < \infty$, as in Section \ref{L^p
estimates}, with $C_1 = 1$.
\backmatter
\addcontentsline{toc}{chapter}{Index}
\printindex
\end{document} | arXiv |
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On the optimization problems of the principal eigenvalues of measure differential equations with indefinite measures
August 2020, 25(8): 3275-3303. doi: 10.3934/dcdsb.2020062
Null controllability for distributed systems with time-varying constraint and applications to parabolic-like equations
Larbi Berrahmoune
ANLIMAD, Département de Mathématiques, ENS, Université Mohammed V, BP 5118, Rabat, Morocco
Received January 2019 Revised August 2019 Published August 2020 Early access April 2020
We consider the null controllability problem fo linear systems of the form $ y'(t) = Ay(t)+Bu(t) $ on a Hilbert space $ Y $. We suppose that the control operator $ B $ is bounded from the control space $ U $ to a larger extrapolation space containing $ Y $. The control $ u $ is constrained to lie in a time-varying bounded subset $ \Gamma(t) \subset U $. From a general existence result based on a selection theorem, we obtain various properties on local and global constrained null controllability. The existence of the time optimal control is established in a general framework. When the constraint set $ \Gamma (t) $ contains the origin in its interior at each $ t>0 $, the local constrained property turns out to be equivalent to a weighted dual observability inequality of $ L^{1} $ type with respect to the time variable. We treat also the problem of determining a steering control for general constraint sets $ \Gamma (t) $ in nonsmooth convex analysis context. Applications to the heat equation are treated for distributed and boundary controls under the assumptions that $ \Gamma (t) $ is a closed ball centered at the origin and its radius is time-varying.
Keywords: Admissible control operator, time-varying constraint, null controllability, steering control, heat equation.
Mathematics Subject Classification: Primary: 93B05, 93C25; Secondary: 93C20.
Citation: Larbi Berrahmoune. Null controllability for distributed systems with time-varying constraint and applications to parabolic-like equations. Discrete & Continuous Dynamical Systems - B, 2020, 25 (8) : 3275-3303. doi: 10.3934/dcdsb.2020062
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Abstract: The objective behind the review is to have an insight of Technology Push (TP) activities to escalate the essential capabilities in manufacturing ventures and analyzing the experiences of the developed and developing countries. Themes and relevant articles have been identified from the various journals and linked with issues related to technology push for sustainable development in manufacturing industries. The study reveals that adjustment of viable TP strategies may endow towards reckoning key skills in the framework of a company. In addition to this, the review emphasizes that the TP strategies has significant impact on the sustainable development of the manufacturing industries. It has been observed that the role of TP in manufacturing companies has not been much reported. A synthesis can be established with satisfaction levels of technology push strategies, which may help to secure concrete objectives and interests of an organization. The paper reviews various attributes of TP through a number of publications in this field.
Keywords: Technology push; Sustainable development; Manufacturing industries; Technology management.
Abstract: To cater to dynamic and variable performance matrices, manufacturing industries use automation tools and technologies to impart manufacturing flexibility in its processes. To successfully cater to its performance, manufacturing requirements should be designed into the system and then appropriate planning and operational policies need to be followed to support it. This research work validates this using an industrial size Flexible Manufacturing System (FMS) as a multi node manufacturing framework having eight machines. While FMS design spectrum is modeled by different levels of its Routing Flexibility (RF) and number of pallets (NP), planning and operational strategies are implemented through various combinations of Dispatching and Sequencing rules (DR/SR). This work is a simulation study to evaluate FMS performance for which Make Span Time (MST) and machine utilization have been used as a performance metric. This research provides a framework for the decision makers in identifying the appropriate levels of routing flexibility, number of pallets and appropriate combination of dispatching and sequencing rules for optimizing system performance under a given manufacturing environment.
Keywords: Routing Flexibility; Sequencing Rule; Dispatching Rule; Simulation of FMS.
Abstract: This paper focuses on scheduling a permutation flow shop where setup times are sequence dependent. The objectives involved are minimisation of makespan and mean tardiness. A hybrid genetic algorithm is developed to obtain a set of non-dominant solutions to the multi-objective flow shop scheduling problem. Genetic algorithm is used in combination with a local search method to obtain the pareto-optimal solutions. The best parameters of the proposed algorithm are determined using the Taguchis robust design method and the concept of utility index value. The set of parameters corresponding to the highest utility value is selected as the optimal parameters for the proposed algorithm. The algorithm is applied to the benchmark problems of flow shop scheduling with sequence dependent setup time.
Keywords: permutation flow shop; hybrid genetic algorithm; Taguchi method; utility concept.
Abstract: The manufacturing organisations adopt Flexible Manufacturing Systems to meet the challenges imposed by todays volatile market standards. An FMS is designed to combine the efficiency of a mass production line and the flexibility of a job shop to produce a variety of products on a group of machines. Productivity is a key factor in a flexible manufacturing system performance. Despite the advantages offered, the implementation of FMS has not been very popular especially in developing countries as it is very difficult to quantify the factors favouring FMS implementation. For its successful implementation, technological considerations, cost justification as well as strategic benefits are to be weighted. Therefore an attempt has been made in the present work to identify and categorize various productivity factors influenced by the implementation of FMS in a firm, further these factors are quantitatively analysed to find their inhibiting strength using Graph Theory Approach (GTA). GTA is a powerful approach which synthesizes the inter-relationship among different variables or subsystems and provides a synthetic score for the entire system. So using this approach a numerical index is proposed in this work to evaluate and rank the various productivity factors so that the practising managers can have better focus.
Keywords: Manufacturing; FMS; Productivity; GTA.
Abstract: Reconfigurable manufacturing systems (RMS) are considered as next generation manufacturing systems that are capable of providing the functionality and capacity as and when required. Products are classified into several part families as per customer requirement and each of them are a set of similar products. In a shop floor, the manufacturers have to deal with varied number of orders for multiple part families and after completing the orders of a particular family, they need to change over to the orders of a different part family. Shifting from one part family to another may require the systems reconfiguration, which is a complicated method and requires tremendous cost and efforts. The complexity, effort and cost from changing one configuration to another depends on the existing initial configuration and the new configuration required for subsequent production of orders belonging to a different part family. There are different models available for process improvement This paper focuses on determining optimal sequencing of part families formation and configuration selection for process improvement in RMS on the basis of minimum loss occurred for a given system configuration. The proposed methodology is explained and an example is given for illustration.
Keywords: RMS; configuration selection; part family; reconfiguration cost; process;order state.
Abstract: Developing new products is a highly dynamic complex process. Managing such dynamic projects is not an easy task. In this research, different scenarios of NPD projects work systems are examined. The capability of these projects work systems to deal with the dynamic nature of NPD projects to mitigate the negative impacts of this nature on NPD project performance . This study shows that, the most suitable organizational work environment for moderate and high complex projects is the highly flexible work system environment. While it is appropriate for low complex projects to provide a formal work system environment. This study makes an attempt to use systems dynamic modeling to model the work system characteristics and managerial styles in interaction with projects behavior. It provides a step towards the understanding of how both work systems characteristics and managerial styles would influence projects performance and their probability of success. It serves as a guide for projects managers suggesting proper ways to better manage different NPD projects with different levels of complexities.
Keywords: project behavior; work system design; managerial styles; formality; system dynamics; scenario planning.
Abstract: In this paper, we use two metaheuristic algorithms, i.e., artificial bee colony (ABC) and covariance matrix adaptation revolution strategies (CMA-ES), for solving the generalized cell formation problem considering machine reliability. The purpose is to choose the best process routing for each part and to allocate the machines to the manufacturing cells in order to minimize the total cost, which is composed of intracellular movement cost, intercellular movement cost and machines breakdown cost. To evaluate the metaheuristic algorithms, eight numerical examples in three different sizes are solved. The results of the two algorithms are compared with each other and with the results of solving the MIP model. Both the MIP solver and metaheuristics find the optimal solutions for the small size problem instances while by increasing the problem size, metaheuristics show higher performance. The results illustrate that the CMA-ES algorithm outperforms the ABC algorithm in both solution quality and CPU time.
Keywords: group technology; generalized cell formation problem; reliability; artificial bee colony; covariance matrix adaptation revolution strategies.
Abstract: In this paper an inventory model for deteriorating items with two warehouses has been developed, where both the warehouses considered are rented. The demand rate for both the warehouses is different. Since the demand for the products increases due to stock up to a certain level and after that it becomes constant. In this model it is assumed that the demand in first warehouse is stock dependent and after that when the extra stock is filled in second warehouse the demand rate becomes constant for that. Due to different storage conditions the deterioration rate is also different in both the warehouse. A numerical example and sensitivity analysis is also presented to illustrate the study. The main objective of this paper is to find out the optimal total cost of the system.
Keywords: Rented Warehouses; Deterioration; Stock Dependent Demand; Partial Backlogging; Inventory.
Abstract: A good and interconnected demand and supply planning (D&SP) is essential for firms, as it will give better allocation of resource and enabler to stay competitive. The basic and simple applications of D&SP are forecasting and aggregate planning which mainly used in big corporations, but less in small and medium enterprises. The research was a case study in a small firm in Indonesia, which started as small and family-owned company, but can enjoy the benefit of simple demand and supply planning to enhance its cost and productivity. Despite main application of D&SP in every production environment, no vast research has been done in the last couple of decades. The study and the case are thus a refreshment on how application of D&SP still relevant. The case study examined 7 common forecasting tools for demand planning side, and exercised classical chase and level aggregate planning strategy for the supply planning side. It explored 3 possible scenarios, and recommended a better scenario with 3% cost saving for the company.
Keywords: Demand Planning; Supply Planning; Demand and Supply Planning (D&SP); Aggregate planning; Forecasting; Small enterprise.
Abstract: In order to understand the manufacturing and machining requirements that have been increased sharply due to evolution of newer and difficult to machine materials which are being utilized by the manufacturing industries, a different class of manufacturing and machining process has emerged from past few decades know as non-traditional machining (NTM) processes. These processes have ability to develop intricate and typical shape products with high degree of accuracy and precision, close dimensional tolerance, and good surface finish. On the other hand, NTM processes also consume high power and are too expensive, hence, necessitates in optimum selection of NTM processes and their related criteria. In this study, a model is developed using analytic hierarchy process (AHP) methodology to help the process engineers and decision makers in prioritizing and ranking the various NTM processes and their criteria identified from literature review, and selecting the most appropriate NTM process and its criteria for typical machining process. The methodology also assists in identifying the most significant NTM processes and their criteria that could help the process engineers and decision makers for higher productivity and industry performance. Analysis of the present study shows that NTM process "based on the source of energy" has ranked at first place followed by "based on the medium of the energy transfer", "based on the type of energy used", and "based on the mechanism of material removal". Further, the study also prioritized the various NTM processes criteria consisting of 27 criteria. The priority criteria that ranked on top are: voltage, current, ions, and ionised particles. The findings of the present study will help process engineers, decision makers and practitioners in selecting the best NTM process and its criteria, and then implementing them in their industry for improved performance.
Keywords: Non-traditional machining (NTM) processes; criteria; analytic hierarchy process (AHP); voltage; current; and energy.
Abstract: Computer Aided Testing (CAT) is the latest technique. Its because CAT involve in a different stages of manufacturing like Designing, Production, Quality Control by 3D Measuring Instrument that is time consuming. If the object position is known before examined, time can be managed. Machine vision dependent inspection of mechanical CAD parts has become demanding area in the field of industrial inspection. In this work we developed the procedure to detect the mechanical CAD parts with the edge based algorithms. The data has been taken with 3D model that has been designed using Solid Edge ST8 CAD/CAM PLM software and analyzed using MATLAB for automated production checking system. Our proposed method uses the edge based recognition of CAD object by Fuzzy based approach in order to create image information of shape before it can be used for pose estimation in Computer Aided Testing System. From the experimental results, it has been found that with the proposed vision system more accurate and reliable products can be manufactured intelligently.
Keywords: Computer Aided Testing (CAT); Machine vision; Image processing; CAD/CAM; Fuzzy logic; Intelligent Manufacturing; Edge extraction; STL format.
Abstract: In a competitive economic environment, manufacturers are forced to face delay, quality and cost challenges. Hence, emergence of lean manufacturing and total productive maintenance (TPM) concepts came to resolve this problem by improving productivity. However, for many manufacturers, especially small and medium-sized enterprises (SME), these concepts remain very difficult to implement and maintain. In this perspective, this work aims at clarifying some failure factors and proposes a model to improve both productivity and performance. This model has been implemented in the plastics industry through a case study relating basic concepts of TPM and Management by Objectives (MBO). Nevertheless, few research works are interested to develop lean model for manufacturing SMEs. This paper aims to enrich this research axis by an industrial case study in a Moroccan SME. Further researches can exploit the findings of this work in order to implement the proposed model in other industries.
Keywords: Total productive maintenance; lean manufacturing; key performance indicators; data collection; small and medium-sized enterprises.
Abstract: Creativity has found a leading strategic position in service industries, and companies have increasingly implemented knowledge management practices and dynamic capabilities in their organizations over the past decade. Realizing the importance of knowledge acquisition from social media, knowledge sharing, and dynamic capabilities in employee creativity, we have made an attempt in this research to propose a conceptual framework. Drawing from the organizational behavior literature, we hypothesize and test the relationship among knowledge acquisition, dynamic capabilities, knowledge sharing and employee creativity. We performed a structural equation model (SEM) test with maximum likelihood estimation to test the relationship among the research variables with a sample of 293 participants. The empirical results from the structural model suggest that knowledge acquisition positively influences both dynamic capabilities and knowledge sharing, and knowledge sharing has a direct effect on dynamic capabilities. Furthermore, dynamic capabilities and knowledge sharing were shown to be direct antecedents of employee creativity.
Keywords: Knowledge processes; Innovation; Creativity; Social media; Knowledge acquisition; Dynamic capabilities; Knowledge sharing; Small businesses.
Abstract: The concept of catastrophe, happening at random and leading to force to abandon all present customers, data, machines immediately and the instantly inoperative of the service facilities until a new arrival of the customer is not uncommon in many practical problems of the computer and communication systems. In this research article, we present a process to frame the membership function of queueing system' characteristics of the classical single server $M/M/1$ fuzzy queueing model having fuzzified exponentially distributed inter-arrival time and service time with fuzzy catastrophe. We employ the $\alpha$-cut approach to transform fuzzy queues into a family of the conventional crisp intervals for the queueing characteristics which are computed with a set of the parametric non-linear program using their membership functions. In a $FM/FM/1$ fuzzy queueing model with fuzzy catastrophe, we characterized an arrival rate, and service rate with fuzzy numbers with fuzzy catastrophe which is also represented by a fuzzy number. We employ basic fuzzy arithmetic fundamental, Zadeh's extension principle, Yager's ranking index to establish fuzzy relation among different rates and to compute corresponding defuzzify values. We present an illustrative example to show practicality and tractability of the process in detail. We also derived the informative membership function of queueing system characteristics using fuzzy operations and arithmetic and results are depicted in tables and graphs for providing better insight to management along with their sensitivity to parameters.
Keywords: Fuzzy sets; $alpha$-cut; FM/FM/1 queueing system; Catastrophe.
Abstract: The present paper deals with multi objective optimization of Wire EDM process parameters in a gear cutting process of Titanium alloy Grade 5 using MOORA based GA methodology. The objective of the present research is to optimize the response parameters using hybrid technique for a gear tooth in order to obtain an optimized setting and to study the effect of process parameters on the output responses. In the present study, Combii Wire (Brass with diffused Zinc) of 0.25 mm diameter was machined with a 2 mm thickness of Titanium plate in the shape of a gear. The process was carried out using full factorial design of experiments having 81 combinations. ANOVA table reveals that Pulse on time and Wire feed rate are the significant process parameters affecting the responses in WEDM process. The optimized setting obtained can be further used to produce high quality miniature gears.
Keywords: ANOVA; Brass; Full factorial; Gear; MOORA; Titanium; Pulse on time; Wire EDM; Wire feed rate.
Abstract: Lean Six Sigma (LSS) is wide accepted as a successful quality improvement program. However, many literatures have reported that companies have struggled with LSS projects and Indian manufacturing firms are no exception to this scenario. Therefore, the article provides a description of LSS implementation at a manufacturing company. The company manufactures PTO (Power Take Off) shafts for tractors. Nevertheless, the company was receiving high non-conformance resulting in part rejection. Therefore, the study deals with the defect reduction through LSS implementation. Using, the DMAIC approach with suitable tools, statistical analysis were performed to identify the bottleneck in the operations and the corrective measure were taken to the improve quality of PTO. In addition, the researchers proposed preventive methods for reoccurring issues and developed a plan to sustain the project to find the required results. At the end of the project, the researchers yielded considerable cost saving and defect reduction. The study concludes with the managerial implications and contribution to the society with the limitations of the study.
Keywords: Lean Six Sigma (LSS); Force Field Analysis; Cost Savings; Process Improvement.
Abstract: In this study; the authors consider the context of such products; whose component failures cannot be rectified through repair actions, but can only be fixed by replacement. The authors develop a cost effective decision model based on reliability for the replacement of street lights in this study. The direction to benchmark maintenance time for operating the street lights with minimum expenditure is discussed by the authors. The replacement policy based on fuzzy reliability index; which can effectively deal with uncertainty and vagueness is proposed in this study. The theoretical model is proposed and explained with a numerical procedure to illustrate the applicability of the proposed study. The study determined the link amongst individual replacement and the group replacement policies by defining the economic instant for replacing street lights. A decision support model incorporating uncertainty and impreciseness is presented to estimate and plan prior maintenance efforts. The individual and the group replacement policies are exposed in this study by housing a subjective theoretical framework that defines their monetary contribution. The study provides valuable insight for using the real time data information for designing a replacement model using fuzzy logic. The authors proposed a fuzzy reliability approach to check the optimality of the group replacement policy.
Keywords: Maintenance; Generalized Trapezoidal Fuzzy Number set (GTFNs); Replacement; Uncertainty; Economic Life; Reliability.
Abstract: A batch arrival retrial queueing model with k optional stages of service, extended Bernoulli vacation and stand-by is studied. After completion of the ith stage of service, the customer may have the option to choose (i+1)th stage of service with probability θi or may leave the system with probability 1-θi, i=1,2,...,k-1 and 1 for i=k. After service completion of each customer, the server may take a vacation with probability v1 and extend a vacation with probability v2 or rejoins the system after the first vacation with probability 1-v2. Busy server may get to breakdown and the stand-by server provides service only during the repair times. At the completion of service, vacation or repair, the server searches for the customer in the orbit (if any) with probability α or remains idle with probability 1-α. By using the supplementary variable method, steady state probability generating function for system size, some system performance measures like Ls, Lq, Ws and Wq are discussed. Simulation results are given using MATLAB.
Keywords: Retrial; k - optional service; orbital search; standby; extended vacation; Supplementary Variable Technique.
Abstract: Service Oriented Architecture is widely accepted approach that facilitates business agility by aligning IT with business. Service Identification with right level of granularity is the most critical aspect in service oriented architecture. It is considered as first steps in the Service Oriented Development life cycle. The services must be defined or identified keeping service reuse in different business contexts. Service identification is challenging to application development team due to several reasons such as lack of business process documentation, lack of expert analyst, and lack of business executive involvement, lack of reuse of services, lack of right decision to choose the appropriate service. Service oriented architectural project uses large repository, where in services are stored randomly in the repository. This causes considerable amount of search time when service is searched from the database. In case of trigger based application, current service storage process will not be much effective due to inadequacy in service level agreement. In this regard, the author(s) explored the possibilities of service identification using k-means clustering. This paper suggests an approach to identify the appropriate service using distributed repository based on enhanced k mean algorithm. The Creation of distributed repository based on business service functionality will reduce search time, service replication and increase the performance and reliability of the service. Proposed Model has been experimentally validated and author(s) found significant decrease in service search time. This Model will be helpful in building the applications with minimum service level agreement.
Keywords: Service-oriented architecture; K- Mean; Service Identification; Cluster.
Abstract: Number of natural disasters and the people affected by them has been increased over recent years. In the last two decades, the field of disaster management and humanitarian logistics has earned more attention. Design of relief logistic network as a strategic decision and relief distribution as an operational decision are the most important activities for disaster operation management before and after a disaster. Since related information can be updated after disaster, we consider a relief helicopter to satisfy the lack of inventory in different depots. In the proposed mathematical model, pre-disaster decisions are determined according to different scenarios in a two stage optimization scheme. Moreover, we present a meta-heuristic algorithm based on particle swarm optimization as a solution method. Finally, the model for two stages of disaster management is testified by several instances. Computational results based on three approaches confirm that the proposed model has proper performance.
Keywords: Disaster management; Scenario based; Relief distribution; facility location problem.
Abstract: The global competition in the international market is forcing industries to increase the productivity and profitability. Time driven activity based costing (TDABC) is newly developed approach in the field of cost accounting. Companies are more interested in increasing productivity and profitability rather than assigning the cost on the product. This paper presents an approach to estimate the cost of process industry using TDABC and how it is useful in increasing productivity and profitability. The approach is explained with the help of case study in the process industry. The results indicate that TDABC provides the useful information for identifying the areas for improving the productivity of the industry and its effect on profitability. This paper also discusses how TDABC helps in taking the decision for improving productivity and profitability.
Keywords: TDABC; process industry; productivity; profitability.
Abstract: In this paper, simulated annealing (SA) algorithm is employed to solve the flowshop scheduling problem with the objective of minimising the completion time variance (CTV) of jobs. Four variants of the two-phase SA algorithm (SA-I to SA-IV) are proposed to solve flowshop scheduling problem with the objective of minimising the CTV of jobs without considering the right shifting of completion time of jobs on the last machine. The proposed SA algorithms have been tested on 90 benchmark flowshop scheduling problems. The solutions yielded by the proposed SA variants are compared with the best CTV of jobs reported in the literature, and the proposed SA-III is found to perform well in minimising the chosen performance measure (CTV) particularly in medium and large size problems.
Keywords: scheduling; permutation flowshop; simulated annealing algorithm; completion time variance; CTV.
Abstract: Scheduling is very complex but important problem in the real world environment applications. Production scheduling with the objective of minimizing the makespan is an important task in manufacturing systems. For most of scheduling problem made so far, the processing time of each job on each machine have been assumed as a real number. However in real world applications the processing time is often imprecise which means the processing time may vary dynamically because of some human factor or operating faults. This paper considers an n jobs and m machines flow shop scheduling problem of minimizing the makespan. In this work fuzzy numbers are used to represent the processing times in the flow shop scheduling. Fuzzy and neural network based concept are applied to the flow shop scheduling problems to determine an optimal job sequence with the objective of minimize the makespan. The performance of our proposed hybrid model is compared with the existing methods selected from different papers. Some problems are solved with the present method and it is found suitable and workable in all the cases. A comparison of our present method with the existing methods is also provided in this work.
Keywords: Flow shop scheduling; sequence; fuzzy number; defuzzification; artificial neural network; makespan.
Abstract: Business units inclusive of large, medium and small-scale entities conventionally conducts activities based on business processes. Globalization has resulted in the gradual introduction of various automation systems at various levels of the business enterprise, specifically focussed on capturing essential business activities across the entity. These systems, inclusive of Enterprise Resource Planning (ERP), Manufacturing Execution Systems (MES) and Plant Systems (PS) has been adopted by larger corporates in executing and optimizing business functions. These large multinationals are described as complex entities with complex business structures inclusive of business processes. The quantification of automation, escalations and critical variables of these business processes has not been effectively conducted. A systems thinking approach adds the complexity of integrating all enterprise functions and creates a framework for evaluating the limitations and synergies so as to optimize these processes. This research focuses on the development, and configuration, of a simulation model for modelling enterprise maturity via business processes. This research approach includes hierarchical layout and segregation of these business processes, exploring these enterprise operations adopting business process tools, techniques, and methodologies aligned with a systems thinking approach. A simulation framework is configured and tested. The results prove that a simulation model potentially benefits a complex organization specific to evaluating time taken to execute business processes. The results indicate that interdependent processes can be modelled together by determining impacts of multiple critical variables in minimizing business process execution times.
Keywords: simulation-based approach; manufacturing execution system; enterprise resource planning; Plant Systems; business process optimization; systems thinking.
Abstract: The purpose of this paper is to explore and test certain assumptions concerning the fantasy sports industry as an enabler for quick, accurate, and descriptive information to its fan from a gender perspective. However, there are distinct disadvantages that seriously active user may have on sports fantasy gambling addiction and excessive wagering may have on society in general and this study should help provide a base line for future studies. Therefore, to explore the differences, the authors sought to provide statistical evidence that collaborate differences among gender based on technology acceptance models. The sample consisted of relatively well-paid professionals who many routinely engage in fantasy sports via a personal interview procedure was implemented and highly representative of the service industry located in the metropolitan section of Pittsburgh, PA. Multivariate statistical analyses were used to test the hypotheses in determine significant gender differences. It was found that those male professionals who were intensely engaged fantasy sports respondents and who spent considerable amount of money on fantasy leagues, found fantasy guides, expert opinions, and related information helpful in changing rosters, and were intensive users of mobile technology for personal use were significant and negatively related to the dependent variables, with significant gender differences. Such respondents did not perceive there was a global concern for fantasy gambling activities, although a considerable portion of the sample felt otherwise.
Keywords: customer relations management (CRM); empirical; fantasy sports; gender; information technology; market strategy.
Abstract: The significance of this research is to optimize the cutting force in turning by optimizing the cutting parameters like cutting speed, feed, and depth of cut. Cutting force is one of the important characteristic variables to be watch and controlled in the cutting processes to optimize tool life and surface roughness of the work piece. The principal presumption was that the cutting forces increase due to the wearing of the tool. Cutting force is optimized by the metaheuristic i.e. genetic algorithm (GA) and teaching-learning based optimization (TLBO) algorithm. The analysis of the result shows that the optimal combination for low resultant cutting force is low cutting speed, low feed and low depth of cut. This study finds that by adjusting machining parameters tool life can be enhanced because cutting forces increase due to the wearing of the tool. So, cutting forces have been used to maximize the tool life because cutting force increased rapidly as tool life finished. As a result, the production cost can be minimized and be extending the tool usage and subsequently, the machining time is reduced and the tool usage can be extended.
Keywords: Optimisation; cutting force; Tool life; metaheuristic; GA; TLBO.
TO FIND THE SUITABILITY OF CMS IN INDIAN INDUSTRIES IN COMPARISON OF OTHER MANUFACTURING SYSTEM USING AHP TECHNIQUE.
Abstract: The cellular manufacturing (CM) is a well known manufacturing technology that helps to improve manufacturing flexibility and productivity by maximum utilization of available resources. Cellular Manufacturing (CM) has been derived as a strong tool for increasing production in batches and job shop production. In cellular manufacturing, similar machines are grouped in machine cells and same processing parts are gathered in part families. In this paper factors of CMS are identified through AHP technique. Decisions involve many hypothetical factors that should be traded off. For doing that, they have to be measured along side factual factors whose estimation must also be evaluated as such, how well, they achieves the aim of the decision maker. The Analytic Hierarchy Process (AHP) is a theory of measurement through pairwise comparisons and depends on the decision of experts to find out priority scales. These are the scales that measure hypothetical factors in relative terms. The comparisons are made using a scale of absolute judgments that represents, how much; one element leads another with respect to a given aspect. The judgments may be scattered, and how to measure incoherency and improve the judgements, when possible to obtain better coherency is a concern of the AHP. The derived priority scales are synthesized by multiplying them by the priority of their parent nodes and adding for all such nodes. It has been a fact that the stability of any country depends on the stability of its production area and progress in industrial society has been accompanied by the development of new technologies. The analytic hierarchy process has the strength to build complex, multi-person, multi-attribute and multi-period problems hierarchically. In this paper, a decision-making model is being proposed for CMS evaluation and selection. This strategic decision-making model is based on both hypothetical and factual criteria and uses the analytical hierarchy process (AHP) approach.
Keywords: AHP; Analytic hierarchy process; CMS; cellular manufacturing system.
A model for operational budgeting by the application of interpretive structural modeling approach: A case of Saipa Investment Group Co.
Abstract: The operational budgeting system is a managerial system considered by governments at national and local levels for improving the efficiency and effectiveness of the resource consumption in organizations. This system bases the budget credit allocation on the performance of organizational units with the aim of generating outputs (products/services), i.e. short-term objectives, and/or achieving long-term goals. Then, executive agencies are led towards transparency in the resource consumption, attempts to achieve their goals and strategies, and more accountability. The present research was aimed to present a model of operational budgeting based on interpretive structural model (ISM) in Saipa Co. So, 46 factors underpinning operational budgeting were identified by literature review, experts were surveyed to derive the most important factors resulting in 14 factors, and an ISM-based structured matrix questionnaire was designed to find out the interrelations of these factors. The data collected by the questionnaire were analyzed by ISM and were mapped out at seven levels in an interactive network in that shareholders and accounting and financial reporting systems were placed at the highest level. The results revealed that the most effective factor in driving power-dependence matrix was annual budget and the five-year development plan so that the operational budgeting of Saipa Investment Co. was dependent on this factor.
Keywords: budget; budgeting; operational budgeting; interpretive structural model.
Abstract: The specific objectives of this study are the following: to analyse the association between the hotel features and the use of the Uniform System of Accounts for the Lodging Industry (USALI); to analyse whether the use of this system is associated with the price charged by the hotels. Data collection began with a survey addressed to the financial managers of 241 4-star and 5-star hotels located in Portugal. To meet the objectives proposed, information on the price charged by the responding hotels was also collected from the Booking.com online platform. The results obtained allow us to indicate as main contributions the following: evidence of the existence of associations between hotel features and the use of USALI; evidence that the use of USALI influences the price practiced.
Keywords: Uniform System of Accounts for the Lodging Industry; USALI; hotel features; hotel price; contingency variables.
Abstract: This study examines the role of marketing mix on brand switching among Malaysian smartphone users. It also investigates the mediating role of brand effect on the relationship between marketing mix (namely product, price, place and promotion), service, and brand switching. It uses a self-administered questionnaire to collect data from 304 participants, and their responses were analysed using Statistical Package for the Social Sciences (SPSS) and Partial Least Squares (Smart-PLS). Evidence from the study reveals that price and promotion have effects on brand switching towards smartphones. It also shows that brand effect partly mediates the connections between product, price and brand switching towards smartphones. Although the Samsung brand remains the market leader in the smartphone industry in Malaysia, empirical research on the nexus between marketing mix and brand switching in Malaysia remains very scanty. Despite some limitations of this study that could affect its generalisability, it enhances marketers understanding of the marketing mix, which should receive adequate attention to attract smartphone users to switch to their brands. It also provides marketers with the knowledge of usage behaviours of Malaysia smartphone users which could be useful for launching new models that suit the preferences of smartphone users.
Keywords: Marketing Mix; Brand Effect; Brand Switch; Smartphone.
Abstract: In Thailand, the petroleum industry is a major contributor to the economy as a supplier of crucial petroleum products. However, due to its geographical location, many multinational oil and gas companies have established their operations in the country. For Thai petroleum firms to remain competitive in the industry, effective commercialization strategies of inventions to innovations can bring firms competitive advantages in terms of financial returns, business opportunities for partnerships, and advanced operations. Nonetheless, commercialization of inventions is becoming increasingly difficult due to changes in the market. Firms operating without an explicitly established commercialization process are at a disadvantage. The objective of this research is to develop a formal commercialization model suitable for commercializing product and process inventions in Thailands petroleum industry. The methodology used in this research consists of three stages, i.e. 1) develop a preliminary commercialization model, 2) refine the model, and 3) apply the model to a Thai petroleum firm as a case study. As a result, two commercialization models are developed, namely an industrial commercialization model and a company commercialization model.
Keywords: Commercialization Model; Innovative Products; Petroleum Firm.
Abstract: In this paper we have developed an inventory model for deteriorating items in which demand of the product is a function of expiration date. In such cases the demand for the product generally decreases as the product is nearer to its expiration date. The different cases based on the allowed trade credit period have been discussed under inflationary environment. Shortages are allowed and partially backlogged. To exemplify this model numerical examples for all the cases and sensitivity analysis with respect to important parameters have been provided. The convexities graphs and the variation in total cost with the changes in parameters are represented with the help of the graph. The objective of this paper is to find out the optimal ordering quantity by which one can minimize the total associated cost of the system.
Keywords: Deterioration; Trade credit period; Inflation; Shortages; Partial Backlogging.
Abstract: The article provides an updated extensive literature review on Process Maturity. On a general level Process Maturity could be defined as the degree to which a process is defined, managed, measured, and continuously improved, where the aim of working with maturity is twofold, either to measure the level of maturity in order to compare with other processes or to identify improvement areas. According to the findings of this study there are areas indicating needs of further research. Firstly, the fundamentals of Process Management seem to be missing when describing the work with Process Maturity, which could affect the quality of maturity models. A second area of further research is related to the absence of approaches and methods for improving Process Maturity, since improving maturity is often addressed as what to do, not how to do.
Keywords: maturity; process maturity; processes; maturity model; process management; quality management; total quality management; process improvement; business process management; quality maturity.
Abstract: In this study a non-Markovian queuing model is considered for our examination. In this queuing framework, clients arriving in batches follow a poisson distribution. All the arriving clients have the alternative of picking any of the N sorts of services rendered. Service time follows general distribution. In addition, the system breaks down at random. Hence the server gets into repair process of two phases Based on the fault, in first phase general repair is carried out. Then in the second phase repair based on the category is followed. Likewise after the system ends up with service ,the server will go for a necessary vacation. In addition, an optional extended vacation is followed. Also an idea of vacation to be a maintenance time in which the support work required for the server will be completed. A concept of reneging is an additional aspect in this model which happens during the time of break down and vacation. This happens due to Impatience of the customers. By the use of birth and death queuing process the conditions overseeing the model is framed. The probability generating queue size and queuing execution measures of the model are derived. The legitimisation of the model is finished by methods for numerical illustration. Also the unmistakable investigation of the graphical portrayal gives a reasonable picture about the procedure completed in versatile communication process. Also it gives significant thoughts for decrease the system break down, reneging and for the advance of the system speedier.
Keywords: non-Markovian queue; optional types of service; service interference; optional extended vacation; maintenance work; revamp process of two stages.
Abstract: India has the second largest telecom network in the world after China with about 1,206 million subscribers. The sharp increase in the teledensity and further decline in the tariffs in telecom sector has contributed significantly in India's economic growth. Mobile number portability (MNP) was introduced in India in 2011 but was limited to licensed service area (LSA). Countrywide MNP was launched in India in 2015. By March 2017, MNP requests were 23.3% of mobile subscriber base in India. In this paper Indian Telecom sector, MNP progress in India, MNP implementation around the world, benefits of MNP and various studies on MNP have been discussed. Porting fees, porting speed, service quality, price, switching cost, satisfaction have been found the determinants of switching based on different studies carried around the world.
Keywords: mobile number portability; MNP; switching intention; Indian telecom sector; mnp benefits; telecom operators; determinants of switching.
Abstract: Companies are always looking for optimizing the use of their material resources and human resources. Nowadays, the issue of competency is important and crucial in industry. To perform a given task by two operators having the same qualification, performance achievement varies, which introduces the concept of individual competence level. This article presents an assessment method of multi-skilled workforce. In this paper, we provide a useful model to solve workforce assessment problems. We discuss how to consider the differences and similarities between acquired level and required level. The objective of our method is to present a combination of AHP technique and TOPSIS technique to assess the individual competence level.
Keywords: Multi-skilled workforce; activities; assessment; individual competence level; FMEA; AHP; TOPSIS.
Abstract: The paper proposes a framework for evaluating the organisational competitiveness attained through sustainable manufacturing. An assessment model using analytical hierarchal process (AHP) has been adopted for the evaluation of organisational competitiveness. A conceptual model is developed to understand the interrelationship between seven sustainable manufacturing practices and five organisational competitiveness with 14 sustainable manufacturing outputs at the intermediate level. The attainment of six organisational competitiveness is evaluated by computing the global priority scores of practices and outputs of sustainable manufacturing. The linkage matrix developed between the practices and the output of sustainable manufacturing and the output and organisational competitiveness is the significant contribution of this work that can be used by practicing managers/researchers to understand the various interactions and hence, promote the research in the field of sustainable manufacturing.
Keywords: organizational competitiveness; sustainable manufacturing practices; sustainable manufacturing output; decision making; linkage matrix.
Abstract: Measuring the performance of the bank and determining the key factor relating to its efficiency and effectiveness in gaining popularity in the recent time period. Efficient operation system is crucially important for banks. In such facet ranking of the bank is becoming essential. Ranking will help the bank in improving their performance with respect to self and other. The present paper seeks to measure the performance and propose a method to rank Indian public sector banks by combining some multiple criteria decision making tools like AHP and TOPSIS. To know the degree of relation between the different set of variables correlation test is applied. For the study purposes, all public sector banks are considered. Results interpret an alternative ranking of the banks. Finally, this paper provides better vision to focus on the area of improvement in comparison to others banks.
Keywords: Banking; Efficiency; Analytic Hierarchy Process; TOPSIS; Performance ranking.
Abstract: In conventional car body process planning, parts and spots of an assembly get evenly distributed in geometry fixtures (or) stations. The remaining spots, over and above the cycle time of the geometry stations are moved to re-spot stations. Global trend of geometry and re-spot fixture distribution indicates that percentages of Geometry fixtures are higher in manual body shops with less than 40% of automation. The percentage of geometry stations varies up to 14% in 100% automated body shops. This trend shows the inappropriate planning of geometry and re-spot stations for manual and automated lines. The main focus area in the conventional planning is to set geometry through balanced distribution of parts and spots. Productivity is often not considered as the top priority. A geometry spot distribution model has been developed to resolve this concern. The proposed model can be used to calculate the appropriate geometry and re-spot stations required for a lean body shop. Improvement in quality establishment and productivity of the body shop are discussed in this paper with case studies. The developed methodology for appropriate spot weld distribution among the welding stations for body shop is simple to use for the product and process designers during any new product development phase. This can also be used for optimization of any existing cells of body shops.
Keywords: Productivity; Geometry Station; Re- Spot Station; Body Shop; Weld Shop; Body-In-White (BIW); Floor space reduction; Process planning.
Abstract: Today there is a necessity of developing a unique market behaviour strategy using own competitive advantages to reach the highest scores of real competitive ability of a sector. The driving force of the competition law requires companies to have thorough strategies to improve the quality of produced goods, to decrease production costs and to increase labour efficiency. In this context the use of benchmarking tools to improve innovation efficiency at companies is an important task. The present work considers decision-making models and methods regarding company's competitiveness management in terms of benchmarking. The model of competitive interaction introduced in the article in the form of a system of coupled Van-der-Pol equations with time lag was successfully tested in radio physics and now is successfully applied in economics. The article demonstrates the application of the given system to describe benchmarking of different economic companies.
Keywords: benchmarking; digital economy; Big data; enterprise management system; systems of decision-making support; innovation activity; system of Van der Pol equations with time lag; mathematics.
Abstract: This study primarily aimed to identify the direct effect of perceived support on the employees' voice behaviour in the workplace. It examined knowledge interactive impact of locus of control on the perceived support of work engagement. Accordingly, this study was built on literature of voice behaviour and as such, it employed a survey methodology. The study focused on a government sector firm, specifically Basra Electricity Production. The data collection tool used is the questionnaire and it was distributed to 333 employees in the firm. The collected data was analysed using AMOS version 22. Based on the results, work engagement fully mediates the relationship between perceived support and employee voice behaviour, while external locus of control moderated the relationship between perceived support and work engagement. Suggestions were provided for several avenues for future studies.
Keywords: Perceived support; work engagement; employee voice behavior and locus of control.
How to increase organizational learning and knowledge sharing through human resource management processes?
Abstract: The purpose of this paper is to examine how human resource management (HRM) processes and knowledge sharing affect organisational learning within the context of steel industry. Drawing from the literature on HRM, this study hypothesises and tests the relationship between HRM processes, knowledge sharing and organisational learning. The authors used survey research to collect the data. The PLS path modelling approach was used to analyse the data and the conceptual model. The empirical results from the structural model suggest that three out of five HRM processes (i.e., training, job design and job quality) influenced knowledge sharing. Furthermore, knowledge sharing was a direct antecedent of organisational learning.
Keywords: Human resource management processes; knowledge sharing; organizational learning; structural equation modeling; steel industry.
Abstract: Due to varying requirements of customers, it became difficult for manufacturing firms to sustain their position in the market. Because of this, firms are finding and implementing new technologies in existing system and flexible manufacturing system (FMS) is one of the solutions for this problem. FMS is an automatic manufacturing system, capable of producing a wide variety of products with good quality. However, the cost of implementing FMS is high. So, decisions related with various parameters in implementing and managing the system, is one of the crucial steps in FMS. In present work, emphasis is done on developing a methodology that can make an appropriate decision of selecting best experiment level, by using simple calculations that can save money and time. For doing so, a relative study of decision-making approaches that combines Shannon entropy and weighted aggregated sum product assessment (WASPAS) method is performed for solving the selection problem of operating conditions. Criteria's weight is calculated first by Shannon entropy method. Later, alternatives were ranked by using calculated weights in WASPAS method. Based on obtained ranking, management could take verdicts for refining the performance parameters of the system.
Keywords: Selection; factor-level; shannon entropy method; weighted aggregated sum product assessment; flexible manufacturing system.
Abstract: There exists insufficient literature on classification and taxonomy of Indian tea supply chain (TSC), so the basic objectives of this study are to identify the existing TSCs and classify them accordingly. The paper is based on a three-year detailed field study on TSC in Assam which encompasses tea-estates, small tea gardens (STGs), bought leaf factories (BLFs), research institute, auction centres, branding companies, tea distributors, and retailers. The paper presented an incorporated structure that includes all the stakeholders with their roles in tea value chain. This has not been reported in previous research. The study also develops an integrated tea supply chain framework and apart from this, classifies the tea supply chain in context of Assam, India. The present study will help further research to optimise business operations and maximising the profit of the Indian tea industry.
Keywords: Tea Industry; Tea Research Center; Tea Estate; Small Tea Garden; Plucking; Tea Factory; Bought Leaf Factory; Tea Auction Centre; Tea Branding Company; Value Chain; Supply Chain Framework.
Abstract: This paper applies queuing models (M/G/N/N) to determine the right price for parking, given that the arrival rate and the staying time depend on the parking price. The models explore the payment per hour and the entrance fee models, with one or several customer types, and where each customer type faces different prices (price discrimination). The models are also applicable for cases where the objective function is to set the park occupancy rate at a desired level. The analysis of the numerical examples demonstrates the applicability of the models and provides some interesting insights about the setting of the correct parking price under the appropriate pricing scheme. The conclusion is that one of the most important tools for process management and benchmarking of a car park is the parking pricing.
Keywords: parking pricing; queuing models; M/G/N/N; parking occupancy; revenue; benchmarking.
Abstract: The arrival of new workforce generation would necessitate a leadership type transcending traditional boundaries. Organisations must strive to develop a supportive work climate which inspires, strengthens and connects employees to perform their tasks vigorously. This paper primarily seeks to investigate the relationship between engaging leadership and work engagement. With reference to empirical studies and existing research on leadership and work engagement, a positive association between engaging leadership and work engagement mediated by engaging job and engaging environment is confirmed. More importantly, this study deals with the paucity of the structured literature regarding work engagement determinants, and offers a holistic model providing a logical ground for identifying empirical indicators and hypotheses to verify the theory. Apart from extending engaging leadership dimensions, certain paradigms are proposed for jobs and work environment to facilitate work engagement.
Keywords: Work Engagement; Engaging Leadership; Engaging Environment; Job Meaningfulness; Job Resources.
Abstract: An initiative took in 1930s by the US trucking industries, cross-docking. The aim behind the initiative was to have lower inventory levels and optimised lead times by having an unbroken flow of product/material from the supplier to the end-customer. In order to have the seamless flow one must ensure that the operations are synchronising. Hence the optimisation of the overall supply chain to be more efficient in terms of lead time and cost, requires an interconnected and explicable connection of all the factors that has to be taken into consideration. The study conducted takes a real time model as an example and illustrates how cross-docking can be managed more holistically to synchronise cross-docking operations at the distribution centre with its inbound and outbound network logistics.
Keywords: Cross-Docking; lower inventory; lead time; inbound; outbound; operations; synchronize; optimization; efficient; holistically; distribution centre; logistics; management; supply chain.
Abstract: Malaysian community colleges play a significant role in achieving Malaysia's vision to be a developed country in 2020. For this reason, their efficiency should appropriately be measured. However, their efficiency measurement using a conventional data envelopment analysis (DEA) model is not appropriate since some of their inputs; e.g., entrant and enrolment students are non-discretionary while a part of output of their graduate employability is discretionary. This paper thus proposes an alternative approach of super efficiency slack-based measure for the case of non-discretionary factors in DEA. The proposed approach was used to evaluate the efficiency of 25 main campuses of Malaysian community colleges from 2012 to 2013. The results support the decision maker of Malaysian community colleges to discriminate and rank efficient and inefficient community colleges in the presence of both super efficiency and non-discretionary factors. The significance of inputs-outputs on efficiency status was tested by sensitivity analysis.
Keywords: data envelopment analysis; DEA; non-discretionary factors; slack-based measure; SBM; super efficiency.
Abstract: In this paper, a two stage production process is considered such that there is only one machine M1 in the first stage and two machines M2′, M2″ in the second stage. It is assumed that the output of M1 is the input for M2′, M2″. During the breakdown time of M1, a reserve inventory is suggested for M2′, M2″, to prevent the idle time, which is costlier. An inventory model is derived here based on certain assumptions to find the optimum size of the reserve inventory of the semi-finished product of the machine M1 to supply the machines M2′, M2″ till M1 resumes the function. Numerical illustrations are provided.
Keywords: breakdowns; repair time; inventory; change point; SCBZ property; production; shortages.
Abstract: This paper deals with steady state analysis of single server priority retrial queue with Bernoulli working vacation, where the regular busy server can be subjected to breakdown and repair. There are two types of customers are considered, which are priority customers and ordinary customers. As soon as orbit becomes empty, the server goes for a working vacation (WV). The server works at a lower service rate during working vacation period. If there are customers in the system at the end of each vacation, the server becomes idle and ready for serving new arrivals with probability p (single WV) or it remains on vacation with probability q (multiple WVs). Using the supplementary variable technique, we obtained the steady state probability generating functions for the system and its orbit. Important system performance measures, the mean busy period and the mean busy cycle are discussed. Finally, some numerical examples are presented.
Keywords: retrial queue; preemptive priority queue; working vacations; supplementary variable technique.
Abstract: Wire electrical discharge machining (WEDM) is an advanced non-conventional machining process specifically used for obtaining complex 3D shape objects in hard materials with high accuracy. The present study investigates the effect of various process parameters viz. pulse on time, pulse off time, wire feed and servo voltage on response variables such as cutting speed, material removal rate, kerf width and surface roughness on machining of high carbon high chromium steel. Taguchi's L9 orthogonal array for four factors and three levels has been used for designing the experiment. A novel technique for order preference by similarity to ideal solution (TOPSIS) approach has been applied to select the optimal level of machining parameters. Analysis of variance (ANOVA) has been conducted for investigating the effect of process parameters on overall machining performance. The effectiveness of proposed optimal condition is validated through the confirmatory test. The result of this study highlights that the parameters like pulse on time and servo voltage are significantly influencing to the machining performance.
Keywords: wire EDM; electrical discharge machining; high carbon high chromium steel; material removal rate; metal cutting; microscopic view; productivity; Taguchi; TOPSIS; ANOVA; multi-response optimisation.
Abstract: Poka-yoke has been recognised as a proven approach towards achieving 'error free' environment, especially in manufacturing. In this paper, an attempt has been made to justify importance of poka-yoke in SMEs, identify and analyse important drivers for successful implementation of poka-yoke concept. Literature review methodology has been utilised to identify important drivers for successful implementation of poka-yoke concept in SMEs of Indian automobile. Analytical hierarchy process (AHP) methodology has been used for the ranking of identified drivers. 'Management attitude' driver has been identified as top ranked driver to responsible to implement poka-yoke in Indian automobile SMEs. Further, DEMATEL methodology has also been used to understand and categorise these identified drivers of poka-yoke implementation into cause group and effect group of drivers. This paper may help production engineers and managers to identify errors and further rectify them for gaining competitive advantage over domestic and international automobile market players.
Keywords: analytical hierarchy process; AHP; decision making trial and evaluation laboratory; DEMATEL; drivers of poka-yoke; Indian automobile industry; poka-yoke; small and medium enterprises; SMEs; India.
Abstract: Due to advancement of science and technology a new concept called big data has emerged. Its utility has attracted not only the private companies and organisations but also governmental authorities. But at the same time it has also raised certain ethical, moral and legal concerns. This paper looks at the threats to individuals privacy caused due to big data. This paper compares the US laws with Indian laws with respect to privacy and data protection and attempts to offer a solution to safeguard individuals' privacy against such threats. The research methodology used by the researchers is doctrinal method along with armchair, exploratory, analytical and comparative method. The authors argue that individuals' privacy can be safeguarded through a holistic approach wherein apt technology standards, management practices along with force of law are adopted. Companies and organisations should adopt self-regulating operating standards and practices too. The concept of privacy in this paper has been restricted to mean the individual privacy and does not include national security concerns or confidential data of corporations. The authors also suggest a framework for the protection of individual privacy and its implementation in India.
Keywords: big data; privacy laws; data protection; sensitive personal information; right to privacy; USA; India. | CommonCrawl |
\begin{document}
\title{Global well-posedness for Euler-Nernst-Planck-Possion system \\in dimension two \hspace{-4mm} }
\author{Zeng Zhang$^1$ \quad Zhaoyang Yin$^2$ \\[10pt] Department of Mathematics, Sun Yat-sen University,\\ 510275, Guangzhou, P. R. China.\\[5pt] } \footnotetext[1]{Email: \it [email protected]} \footnotetext[2]{Email: \it [email protected]} \date{} \maketitle
\begin{abstract} In this paper, we study the Cauchy problem of the Euler-Nernst-Planck-Possion system. We obtain global well-posedness for the system in dimension $d=2$ for any initial data in $H^{s_1}(\mathbb{R}^2)\times H^{s_2}(\mathbb{R}^2)\times H^{s_2}(\mathbb{R}^2)$ under certain conditions of $s_1$ and $s_2$.
\vspace*{5pt} \noindent {\it 2010 Mathematics Subject Classification}: 35Q35, 35K15, 76N10.
\vspace*{5pt} \noindent{\it Keywords}: Electrohydrodynamics; Euler-Nernst-Planck-Possion system; Global well-posedness; Littlewood-Paley theorey. \end{abstract}
\vspace*{10pt}
\tableofcontents
\section{Introduction} In this paper, we study the Cauchy problem of the following nonlinear system: \begin{align}\tag{01}\label{s1} \left\{ \begin{array}{l} u_t+u\cdot \nabla u-\nu \triangle u+\nabla P=\triangle\phi\nabla\phi, \quad t>0,\,x \in \mathbb{R}^d, \\[1ex] \nabla\cdot u=0,\quad t>0,\,x \in \mathbb{R}^d,\\[1ex] n_t+u\cdot \nabla n=\nabla\cdot(\nabla n-n\nabla\phi), \quad t>0,\,x \in \mathbb{R}^d,\\[1ex] p_t+u\cdot \nabla p=\nabla\cdot(\nabla p+p\nabla\phi), \quad t>0,\,x \in \mathbb{R}^d,\\[1ex] \triangle\phi=n-p,\quad t>0,\,x \in \mathbb{R}^d,\\[1ex]
(u,n,p)|_{t=0}=(u_0,n_0,p_0),\quad x \in \mathbb{R}^d. \end{array} \right. \end{align} Here $u(t,x)$ is a vector in $\mathbb{R}^d,$ $P(t,x),n(t,x),p(t,x)$ and $\phi(t,x)$ are scalars. The first two equations of the system (\ref{s1}) are the conservation equations of the incompressible flow. $u$ denotes the velocity filed, $P$ denotes the pressure, $\nu\geq 0$ denotes the fluid viscosity and $\phi$ denotes the electrostatic potential caused by the net charged particles. The third and the fourth equations of the system (\ref{s1}), which are the Nernst-Planck equations modified by the convective terms $u\cdot\triangle n$ and $u\cdot\nabla p,$ model the balance between diffusion and convective transport of charge densities by flow and electric fields. $n$ and $p$ are the densities of the negative and positive charged particles. They are coupled by the Poisson equation (the fifth equation). The system (\ref{s1}) arises from electrohydrodynamics, which describing the dynamic coupling between incompressible flows and diffuse charge systems finds application in biology, chemistry and pharmacology. See \cite{Bazant,Joseph,Lin,Newman} for more details.
If the fluid viscosity $\nu>0,$ The above system (\ref{s1}) is the so called Navier-Stokes-Nernst-Planck-Possion ($NSNPP$) system, and it has been studied by several authors. Schmuck \cite{Schmuck} and Ryham \cite{Ryham} obtained the global existence of weak solutions in a bounded domain $\Omega$ in dimension $d\leq 3$ with Neumann and Dirichlet boundary conditions respectively. By using elaborate energy analysis, Li \cite{Li} studied the quasineutral limit in periodic domain. When $\Omega=\mathbb{R}^n,$ Joseph \cite{Joseph} established the existence of a unique smooth local solution for smooth initial dada by making using of Kato's semigroup ideas. The author also established the stability under the inviscid limit $\nu\rightarrow 0.$ Zhao et al. \cite{zhaoW,zhaoW3,zhaoG,zhaoW2} studied the local and global well-posedness in the critical Lebesgue spaces, modulation spaces, Triebel-Lizorkin spaces and Besov spaces by using the Banach fixed point theorem.
If, on the other hand, $\nu=0,$ the above system (\ref{s1}) is the Euler-Nernst-Planck-Possion ($ENPP$) system. Recently, Zhang and Yin \cite{zy} proved the local well-posedness for the $ENPP$ system in Besov spaces in dimension $d\geq 2.$
The purpose of this paper is to get the global existence for the $ENPP$ system in dimension $d=2.$ Motivated by \cite{keben} for the study of the Euler system, we first introduce the following modified system \begin{align}\tag{02}\label{s3} \left\{ \begin{array}{l} u_t+u\cdot \nabla u+\Pi(u,u)=\mathcal{P}\big((\nabla\cdot\xi)\xi\big), \\[1ex] n_t+\nabla\cdot (u n)-\triangle n=-\nabla\cdot(n\xi),\\[1ex] p_t+\nabla\cdot (u p)-\triangle p=\nabla\cdot(p\xi), \\[1ex] \xi=-\nabla(-\triangle)^{-1}(n-p),
\end{array} \right. \end{align} where $\mathcal{P}$ is the Leray projector defined as $\mathcal{P}=Id+\nabla (-\triangle)^{-1}\nabla\cdot,$ and $\Pi(\cdot,\cdot)$ is a bilinear operator defined by $
\Pi(u,v)=\sum_{j=1}^{5}\Pi_j(u,v), $
with \begin{align*}&\Pi_1(u,v)=\nabla|D|^{-2}T_{\partial_iu^j} \partial_jv^i, &\Pi_2(u,v)&=\nabla|D|^{-2}T_{\partial_jv^i} \partial_iu^j,\\
&\Pi_3(u,v)=\nabla|D|^{-2}\partial_i\partial_j(I-\triangle_{-1})R(u^i,v^j), &\Pi_4(u,v)&=\theta E_d\ast\nabla \partial_i\partial_j\triangle_{-1}R(u^i,v^j),\\ &\Pi_5(u,v)=\nabla \partial_i\partial_j\big((1-\theta ) E_d\big)\ast\triangle_{-1}R(u^i,v^j).\end{align*}
Here $\theta$ is a function of $\mathcal{D}(B(0,2))$ with value $1$ on $B(0,1),$ $E_d$ stands for the fundamental solution of $-\triangle,$ and $|D|^{-2}$ denotes the Fourier multiplier with symbol $|\xi|^{-2}.$ See Section 2 for the definitions of $T$ and $R.$ \\ We deduce form the second to the fourth equations of the system (\ref{s3}) that the dynamic equations of $(n+p, \xi)$ are \begin{align*} \left\{ \begin{array}{l} (n+p)_t+\nabla\cdot \big(u (n+p)\big)-\triangle (n+p)=-\nabla\cdot\big((\nabla\cdot\xi)\xi\big), \\[1ex] \xi_t-\triangle \xi+(-\nabla(-\triangle)^{-1}\nabla\cdot)\big(u(\nabla\cdot\xi)\big)=-(-\nabla(-\triangle)^{-1}\nabla\cdot)\big((n+p)\xi\big). \end{array} \right. \end{align*} Denote $\mathcal{L}=-\nabla (-\triangle)^{-1}\nabla\cdot=Id-\mathcal{P}.$ We then introduce the following system \begin{align}\tag{03}\label{s2} \left\{ \begin{array}{l} u_t+u\cdot \nabla u+\Pi(u,u)=\mathcal{P}\big((\nabla\cdot\xi)\xi\big), \\[1ex] z_t+\nabla\cdot (u z)-\triangle z=-\nabla\cdot\big((\nabla\cdot\xi)\xi\big), \\[1ex] \xi_t-\triangle\xi+\mathcal{L}\big(u(\nabla\cdot \xi))=-\mathcal{L}\big(z\xi). \end{array} \right. \end{align} Note that, by means of basic energy argument, the terms $\langle\mathcal{P}\big((\nabla\cdot\xi)\xi\big), u\rangle$ and $\langle\mathcal{L}\big(u(\nabla\cdot \xi)), \xi\rangle$ can be canceled out, which plays an important role in the proof of global existence.
We point out that in \cite{zy}, the term $\nabla\phi=\nabla(-\Delta)^{-1}(p-n)$ was controlled by $n-p$ through the Hardy-Littlewood-Sobolev inequality, i.e., $\|\nabla\phi\|_{L^q}\lesssim\|n-p\|_{L^p}$ with $1<p<d.$ Whereas, in this paper $n-p=\nabla\cdot\xi$ is controlled by $\xi$ through $\|n-p\|_{H^s}\lesssim\|\xi\|_{H^{s+1}}.$ Hence, these two papers solve the $ENPP$ system in different function spaces.
We can now state our main results: \begin{theo}\label{a1} Let $d\geq 2,~(s_1,s_2)\in\mathbb{R}^2,$ satisfying \begin{align}\label{jibentiaojian}
s_1>1+\frac{d}{2}, \textit{and}~ s_2+\frac{3}{2}> s_1\geq s_2+1. \end{align}
There exists constants $c$ and $r$, depending only on $s_1,s_2$ and $d,$ such that for $(u_0,z_0,\xi_0)\in H^{s_1}(\mathbb{R}^d)\times H^{s_2}(\mathbb{R}^d)\times H^{s_2+1}(\mathbb{R}^d),$ with $\nabla\cdot u_0=0,$ $\xi_0=-\nabla(-\triangle)^{-1}a_0$ for some $a_0\in H^{s_2}(\mathbb{R}^d),$ and $z_0\pm\nabla \cdot \xi_0\geq 0$, there exists a time $$T\geq \frac{c}{1+(\|u_0\|_{ H^{s_1}(\mathbb{R}^d)}+\|z_0\|_{H^{s_2}(\mathbb{R}^d)}+\|\xi_0\|_{H^{s_2+1}(\mathbb{R}^d)})^r},$$ such that the system (\ref{s2}) has a unique solution $(u,z,\xi)$ on $[0,T]\times\mathbb{R}^d$ satisfying $$(u,z,\xi)\in \widetilde{L}^\infty_{T}(H^{s_1}(\mathbb{R}^d))\times \big(\widetilde{L}^\infty_{T}(H^{s_2}(\mathbb{R}^d)) \cap\widetilde{L}^1_{T}(H^{s_2+2}(\mathbb{R}^d))\big)\times \big(\widetilde{L}^\infty_{T}(H^{s_2+1}(\mathbb{R}^d)) \cap\widetilde{L}^1_{T}(H^{s_2+3}(\mathbb{R}^d))\big),$$
and $(u,z,\xi)$ is continuous in time with values in $H^{s_1}\times H^{s_2}\times H^{s_2+1}.$\\
Moreover, $\nabla\cdot u=0,$ $z\pm\nabla\cdot \xi\geq 0,~a.e.~on ~[0,T]\times \mathbb{R}^d,$ and $\mathcal{L}\xi=\xi.$\\ Finally, if $d=2$ and $s_2>1,$ then the solution $(u,z,\xi)$ is global.\end{theo} \begin{rema} We mention that the restriction $s_1>1+\frac{d}{2}$ is due to the same reasons as illustrated for the Euler equation in \cite{keben}. $s_2+\frac{3}{2}> s_1\geq s_2+1$ is caused by the coupling between $u$ and $\xi.$ In fact, owing to the properties of the transport flow, $(u,\xi)$ is expected to be in $ \widetilde{L}^\infty_{T}(H^{s_1}(\mathbb{R}^d))\times \widetilde{L}^r_{T}(H^{s_2+1+\frac{2}{r}}(\mathbb{R}^d))$ with $r\in [1,\infty].$ Due to the product laws in Besov spaces, $ab$ is less regular than $a$ or $b$. Thus in order to control the term $(\nabla\cdot\xi)\xi$ in the first equation of the system, we have to assume $s_2+1+\frac{2}{r_1}\geq (>)s_1,~s_2+\frac{2}{r_1'}> (\geq) s_1,$ for some $r_1\in [1,\infty]$ and $\frac{1}{r_1}+\frac{1}{r_1'}=1,$ which implies that $s_2+\frac{3}{2}>s_1.$ Similar reason for the term $u(\nabla\cdot\xi)$ requires $s_1\geq s_2+1.$ \end{rema} \begin{theo}\label{a3} Let $d= 2,~(s_1,s_2)\in\mathbb{R}^2,$ \begin{align}\label{xinde}
s_1>2, ~s_2>1,\textit{and}~ s_2+\frac{3}{2}> s_1\geq s_2+1. \end{align} Then for any $(u_0,n_0,p_0)\in H^{s_1}(\mathbb{R}^2)\times H^{s_2}(\mathbb{R}^2)\times H^{s_2}(\mathbb{R}^2),$ with $\nabla\cdot u_0=0,$ $\nabla(-\triangle)^{-1}(n_0-p_0)\in H^{s_2+1}(\mathbb{R}^2),$ and $n_0,p_0\geq 0$, the $ENPP$ system has a solution $(u,n,p,P,\phi)$ on $\mathbb{R}^+\times\mathbb{R}^2$ satisfying \begin{align*}&(u,n,p)\in \widetilde{L}^\infty(\mathbb{R}^+;H^{s_1}(\mathbb{R}^2))\times \Big(\widetilde{L}^\infty(\mathbb{R}^+;H^{s_2}(\mathbb{R}^2)) \cap\widetilde{L}^1(\mathbb{R}^+;H^{s_2+2}(\mathbb{R}^2))\Big)^2,\\ &-\nabla(-\triangle)^{-1}(n-p)\in \widetilde{L}^\infty(\mathbb{R}^+;H^{s_2+1}(\mathbb{R}^2)) \cap\widetilde{L}^1(\mathbb{R}^+;H^{s_2+3}(\mathbb{R}^2)),\\ &P,\Phi\in L^\infty(\mathbb{R}^+;BMO(\mathbb{R}^2)).\end{align*} Morever, if $(\widetilde{u},\widetilde{n},\widetilde{p},\widetilde{P},\widetilde{\phi})$ also satisfies the $ENPP$ system with the same initial data and belongs to the above class, then $(u,n,p)=(\widetilde{u},\widetilde{n},\widetilde{p}),$ and $(\nabla P,\nabla \phi)=(\nabla\widetilde{P},\nabla\widetilde{\phi}).$\\ Finally,
$(u,n,p)$ is continuous in time with values in $H^{s_1}\times H^{s_2}\times H^{s_2},$ and $n.p\geq 0,~a.e.~on ~\mathbb{R}^+\times \mathbb{R}^2$.\end{theo} \begin{rema}
We mention that under an improved condition \ref{xinde}, Theorem \ref{a3} may hold true for the $NSNPP$ system. We will present this result in another paper. \end{rema} Throughout the paper, $C>0$ stands for a generic constant and $c>0$ a small constant. We shall sometimes use the notation $A\lesssim B$ to denote the relation $A \leq CB.$ For simplicity, we write $L^p$, $H^s$ and $B^s_{p,r}$ for the spaces $L^p(\mathbb{R}^d),~H^s(\mathbb{R}^d),$ and $B^s_{p,r}(\mathbb{R}^d)$, respectively.
The remain part of this paper is organized as follows. In Section 2, we recall some basic facts about Littlewod-Paley theory and Besov spaces. Section 3 is devoted to the proof of Theorem \ref{a1}. Finally, we give a proof of Theorem \ref{a3} by using Theorem \ref{a1}. \section{Preliminaries}
\subsection{The nonhomogeneous Besov spaces } We first define the Littlewood-Paley decomposition.
\begin{lemm}\cite{keben}
Let $\mathcal{C}=\{\xi\in{\mathbb{R}^2}, ~\frac{3}{4}\leq|\xi|\leq\frac{8}{3} \}$ be an annulus. There exist radial functions $\chi$ and $\varphi$ valued in the interval $[0,1]$, belonging respectively to $\mathcal{D}(B(0,\frac{4}{3}))$ and $\mathcal{D}(\mathcal{C})$, such that \begin{align*} \forall \xi \in{\mathbb{R}}^d,~~\chi(\xi)+\sum_{j\geq 0}\varphi(2^{-j}\xi)=1.\end{align*}\end{lemm} The nonhomogeneous dyadic blocks $\triangle_j$ and the nonhomogeneous low-frequency cut-off operator $S_j$ are then defined as follows: \begin{align*}&\triangle_ju=0~~\textit{if}~j\leq-2,~~~~~~~~~~~~~\triangle_{-1}u=\chi(D)u,\\ &\triangle_ju=\varphi(2^{-j}D)u~~\textit{if}~j\geq0,~~~~S_ju=\sum_{j'\leq j-1}\triangle_{j'}u,~~\textit{for}~j\in \mathbb{Z}.\end{align*}
We may now introduce the nonhomogeneous Besov spaces. \begin{defi}\label{dingyi} Let $s\in \mathbb{R}$ and $(p,r)\in[1,\infty]^2$. The nonhomogeneous Besov space $B^s_{p,r}$ consists of all tempered distributions $u$ such that
$$\|u\|_{B^s_{p,r}}\overset{def}{=}\Big\|(2^{js}\|\triangle_ju\|_{L^p})_{j\in \mathbb{Z}}\Big\|_{l^r(\mathbb{Z})} <\infty.$$ \end{defi} The Sobolev space can be defined as follows:
\begin{defi}
For $s\in \mathbb{R}$,
\begin{align*}
H^s =\{u \in \mathcal{S}'; \|u\|_{H^s}=\Big(\sum_{j=-1}^{\infty}2^{2js}\|\triangle_ju\|_{L^2}^2\Big)^{\frac{1}{2}}<\infty\}. \end{align*} \end{defi} \begin{rema}
For any $s\in \mathbb{R}$, the Besov space $B^{s}_{2,2}$ coincides with the Sobolev space $H^s.$ \end{rema} \begin{lemm}\label{Fadou} The set $B^s_{p,r}$ is a Banach space, and satisfies the Fatou property, namely, if $(u_n)_{n\in N}$ is a bounded sequence of $B^s_{p,r}$, then an element $u$ of $B^s_{p,r}$ and a subsequence $u_{\psi(n)}$ exist such that
$$\underset{n\rightarrow\infty}{\lim}~u_{\psi(n)}=u~~in~~\mathcal{S}'~~~and ~~~\|u\|_{B^s_{p,r}}\leq C \underset{n\rightarrow\infty}{\liminf} \|u_{\psi(n)}\|_{B^s_{p,r}}.$$\end{lemm} In addition to the general time-space $L^{\rho}_T(B^s_{p,r})$, we introduce the following mixed time-space $\widetilde{L}^{\rho}_T(B^s_{p,r}).$ \begin{defi} For all $T>0,~s\in\mathbb{R},$ and $1\leq r,\rho\leq\infty$, we define the space $\widetilde{L}^{\rho}_T(B^s_{p,r})$ the set of tempered distributions $u$ over $(0,T)\times \mathbb{R}^d,$ such that
$$\|u\|_{\widetilde{L}^{\rho}_T(B^s_{p,r})}\overset{def}{=}\|2^{js}\|\triangle_ju\|_{L^{\rho}_T(L^p)}\| _{l^r(\mathbb{Z})}<\infty.$$ \end{defi} \noindent It follows from the Minkowski inequality that \begin{align*}
\|u\|_{L^\rho_T({B}_{p,r}^{s})}\leq \|u\|_{\widetilde{L}^\rho_T({B}_{p,r}^{s})}~if~r\leq \rho,~~\|u\|_{\widetilde{L}_T^\rho({B}_{p,r}^{s})}\leq \|u\|_{L^\rho_T({B}_{p,r}^{s})}~if~r\geq \rho. \end{align*}
Let's then recall Bernstein-Type lemmas. \begin{lemm}\label{Bi}\cite{keben} (Bernstein inequalities) Let $\mathcal{C}$ be an annulus and $\mathcal{B}$ a ball. A constant $C$ exists such that for any nonnegative integer $k$, any couple $(p,q)$ in $[1,\infty]^2$ with $q\geq p\geq1$, and any function u of $L^p$, we have \begin{align*}
&Supp\, \widehat{u}\subset \lambda\mathcal{B}\Rightarrow\,\underset{|\alpha|=k}{\sup}\,\|\partial^{\alpha}u\|_{L^q}
\leq C^{k+1}\lambda^{k+d(\frac{1}{p}-\frac{1}{q})}\|u\|_{L^p},\\
&Supp\,\widehat{u}\subset \lambda\mathcal{C}\Rightarrow\,C^{-k-1}\lambda^k\|u\|_{L^p}\leq\underset{|\alpha|=k}{\sup}\,\|\partial^{\alpha}u\|_{L^q}
\leq C^{k+1}\lambda^k\|u\|_{L^p}.\end{align*} \end{lemm}
We state the following embedding and interpolation inequalities. \begin{lemm}\cite{keben}
Let $1\leq p_1\leq p_2\leq\infty$ and $\leq r_1\leq r_2\leq\infty.$ Then for any real number $s,$ we have
$${B}^s_{p_1,r_1}\hookrightarrow{B}^{s-d(\frac{1}{p_1}-\frac{1}{p_2})}_{p_2,r_2}.$$
\end{lemm}
\begin{lemm}\cite{keben}
If $s_1$ and $s_2$ are real numbers such that $s_1<s_2,$ $\theta\in(0,1)$ and $~1\leq p,r\leq\infty,$ then we have
\begin{align*}
B^{s_2}_{p,\infty}\hookrightarrow B^{s_1}_{p,1},~~~\textit{and}~~~
\|u\|_{{B}^{\theta s_1+(1-\theta)s_2}_{p,r}}\leq \|u\|_{B^{s_1}_{p,r}}^\theta\|u\|_{B^{s_2}_{p,r}}^{1-\theta}.
\end{align*}
\end{lemm}
In the sequel, we will frequently use the Bony decomposition: $$uv=T_vu+T_uv+R(u,v),$$ with
\begin{align*}&R(u,v)=\underset{|k-j|\leq1}{\sum}\triangle_ku\triangle_jv,\\ &T_uv=\underset{j\in \mathbb{Z}}{\sum}S_{j-1}u\triangle_jv =\underset{j\geq1}{\sum}S_{j-1}u\triangle_j\big((Id-\triangle_{-1})v\big)~~ ,\end{align*} where operator $T$ is called ``paraproduct", whereas $R$ is called ``remainder". \begin{lemm}\label{T} A constant $C$ exists which satisfies the following inequalities for any couple of real numbers $(s,t)$ with t negative and any $(p,p_1,p_2,r,r_1,r_2)$ in $[1,\infty]^6$:
\begin{align*}&\|T\|_{\mathcal{L}(L^{p_1}\times {B}^s_{p_2,r};{B}^s_{p,r})}\leq C^{|s|+1},\\
&\|T\|_{\mathcal{L}({B}^t_{p_1,r_1}\times {B}^s_{p_2,r_2};{B}^{s+t}_{p,r})}\leq \frac{C^{|s+t|+1}}{-t}, \end{align*} with $\frac{1}{p}\overset{def}{=}\frac{1}{p_1}+\frac{1}{p_2}\leq1,~ \frac{1}{r}\overset{def}{=}min\{1,\frac{1}{r_1}+\frac{1}{r_2}\}.$ \end{lemm} \noindent{Proof.} The proof of this lemma can be easily deduced from substituting the estimate
$$\|S_{j-1}u\triangle_j v\|_{L^p}\leq \|S_{j-1}u\|_{L^{p_1}}\|\triangle_j v\|_{L^{p_2}},$$ for the estimate
$$\|S_{j-1}u\triangle_j v\|_{L^p}\leq \|S_{j-1}u\|_{L^{\infty}}\|\triangle_j v\|_{L^{p}}$$ in the proof of Theorem 2.82 in \cite{keben}. It is thus omitted.\qed \begin{lemm}\label{R}\cite{keben} A constant $C$ exists which satisfies the following inequalities. Let $(s_1,s_2)$ be in $\mathbb{R}^2$ and $(p_1,p_2,r_1,r_2)$ be in $[1,\infty]^4$. Assume that $$\frac{1}{p}\overset{def}{=}\frac{1}{p_1}+\frac{1}{p_2}\leq1~~and~~ \frac{1}{r}\overset{def}{=}\frac{1}{r_1}+\frac{1}{r_2}\leq1.$$ If $s_1+s_2>0$, then we have, for any $(u,v)$ in ${B}^{s_1}_{p_1,r_1}\times {B}^{s_2}_{p_2,r_2}$,
$$\|R(u,v)\|_{{B}^{s_1+s_2}_{p,r}}\leq\frac{C^{|s_1+s_2|+1}}{s_1+s_2}\|u\|_{{B}^{s_1}_{p_1,r_1}}\|v\|_{{B}^{s_2}_{p_2,r_2}}.$$ If $r=1$ and $s_1+s_2=0$, then we have, for any $(u,v)$ in ${B}^{s_1}_{p_1,r_1}\times {B}^{s_2}_{p_2,r_2}$,
$$\|R(u,v)\|_{{B}^0_{p,\infty}}\leq C\|u\|_{{B}^{s_1}_{p_1,r_1}}\|v\|_{{B}^{s_2}_{p_2,r_2}}.$$ \end{lemm} \begin{lemm}\label{s_2+1}
Let $s+\frac{1}{2}>\frac{d}{2}.$ A constant $C$ exists such that
\begin{align*}
&\|uv\|_{H^s}\lesssim\|u\|_{H^{s+\frac{1}{2}}}\|v\|_{H^s},\\
&\|uv\|_{H^{s+1}}\lesssim\|u\|_{H^{s+1}}\|v\|_{H^{s+1}}.
\end{align*} \end{lemm} {\noindent Proof.} By using Bony's decomposition combined with Lemmas \ref{T}-\ref{R}, we have \begin{align*}
\|uv\|_{H^{s}}\lesssim&\|T_uv\|_{H^{s}}+\|R(u,v)\|_{H^{s}}+\|T_vu\|_{H^{s}}\\
\lesssim&\|u\|_{L^\infty}\|v\|_{H^{s}}+\|u\|_{B^0_{\infty,\infty}}\|v\|_{H^{s}}
+\|v\|_{B^{-\frac{1}{2}}_{\infty,\infty}}\|u\|_{H^{s+\frac{1}{2}}}\\\lesssim&\|u\|_{L^\infty}\|v\|_{H^{s}}
+\|v\|_{H^{\frac{d}{2}-\frac{1}{2}}}\|u\|_{H^{s+\frac{1}{2}}}\\\lesssim&
\|u\|_{H^{s+\frac{1}{2}}}\|v\|_{H^{s}}, \end{align*} where we have used $s>\frac{d}{2}-\frac{1}{2}>0,$ and $H^{s+\frac{1}{2}}\hookrightarrow L^\infty.$ Similarly, \begin{align*}
\|uv\|_{H^{s+1}}
\lesssim\|u\|_{L^\infty}\|v\|_{H^{s+1}}
+\|v\|_{L^\infty}\|u\|_{H^{s+1}}\lesssim
\|u\|_{H^{s+1}}\|v\|_{H^{s+1}}, \end{align*} where we have used $H^{s+1}\hookrightarrow L^\infty.$ We thus obtain the desired inequalities. \qed\\ We mention that all the properties of continuity for the paraproduct and remainder remain true in the mixed time-space $\widetilde{L}^{\rho}_T(B^s_{p,r}).$
Finally, we state the following commutator estimates. \begin{lemm}\label{jiaohuan}\cite{keben}
Let $v$ be a vector filed over $\mathbb{R}^d,$ define $R_j=[v\cdot\nabla,\triangle_j]f.$ Let $\sigma>0~(\textit{or}~\sigma>-1,$ if $ \nabla\cdot v=0),$ $1\leq r\leq\infty,$ $1\leq p\leq p_1\leq\infty,$ and $\frac{1}{p_2}=\frac{1}{p}-\frac{1}{p_1}.$ Then
\begin{align*}
\Big\|2^{j\sigma}\|R_j\|_{L^P}\Big\|_{l^r}\leq C\Big(\|\nabla v\|_{L^\infty}\|f\|_{B^{\sigma}_{p,r}}+\|\nabla f\|_{L^{p_2}}\|\nabla v\|_{B^{\sigma-1}_{p_1,r}}\Big).
\end{align*} \end{lemm} \subsection{A priori estimates for transport and transport-diffusion equations} Let us state some classical a priori estimates for transport equations and transport-diffusion equations. \begin{lemm}\label{ts}\cite{keben} Let $1\leq p\leq p_1\leq\infty,~1\leq r\leq\infty$. Assume that \begin{align}\label{tiaojian} s\geq-d\,min\left(\frac{1}{p_1},\frac{1}{p'}\right) \quad \textit{or} \quad s\geq-1-d\,min\left(\frac{1}{p_1},\frac{1}{p'}\right)~\textit{if}~\nabla\cdot v=0 \end{align} with strict inequality if $r<\infty$.
There exists a constant $C$, depending only on $d, p, p_1, r$ and $s$, such that for all solutions $f\in L^{\infty}([0,T];B^s_{p,r})$ of the transport equation \begin{align} \left\{ \begin{array}{l} \partial_tf+v\cdot\nabla f=g\\
f_{|t=0}=f_0, \end{array} \right. \end{align} with initial data $f_0$ in $B^s_{p,r}$, and $g$ in $L^1([0,T];B^s_{p,r})$, we have, for $a.e.\,t\in[0,T]$,
\begin{align}\label{,}\|f\|_{\widetilde{L}_t^{\infty}(B^s_{p,r})}\leq\left(\|f_0\|_{B^s_{p,r}}+
\int_0^t exp(-CV_{p_1}(t))\|g(t')\|_{B^s_{p,r}}dt'\right)exp(CV_{p_1}(t)),\end{align}
with, if the inequality is strict in (\ref{tiaojian}), \begin{align}
V'_{p_1}(t)=\left\{\begin{array}{l}\|\nabla v(t)\|_{B^{s-1}_{p_1,r}},~if~s>1+\frac{d}{p_1}~or~s=1+\frac{d}{p_1},~r=1,\\
\|\nabla v(t)\|_{B^{\frac{d}{p_1}}_{p_1,\infty}\cap L^{\infty}},~if~s<1+\frac{d}{p_1} \end{array}\right. \end{align} and, if equality holds in (\ref{tiaojian}) and $r=\infty$,
$$V'_{p_1}=\|\nabla v(t)\|_{B^{\frac{d}{p_1}}_{p_1,1}}.$$ If $f=v$, then for all $s>0$ $(s>-1,$ if $\nabla\cdot u=0)$, the estimate (\ref{,}) holds with
$$V'_{p_1}(t)=\|\nabla u\|_{L^{\infty}}.$$
\end{lemm} \begin{lemm}\label{tds}\cite{keben} Let $1\leq p_1\leq p\leq\infty,~1\leq r\leq\infty,~s\in\mathbb{R}$ satisfy (2.10), and let $V_{p_1}$ be defined as in Lemma \ref{ts}.
There exists a constant $C$ which depends only on $d, r, s$ and $s-1-\frac{d}{p_1}$ and is such that for any smooth solution $f$ of the transport diffusion equation \begin{align} \left\{ \begin{array}{l} \partial_tf+v\cdot\nabla f-\nu\triangle f=g\\
f_{|t=0}=f_0, \end{array} \right. \end{align} we have
\begin{align*}\nu^{\frac{1}{\rho}}\|f\|_{\widetilde{L}^{\rho}_T(B^{s+\frac{2}{\rho}}_{p,r})}\leq Ce^{C(1+\nu T)^{\frac{1}{\rho}}V_{p_1}(T)}\Big(&(1+\nu T)^{\frac{1}{\rho}}\|f_0\|_{B^s_{p,r}}\\
+&(1+\nu T)^{1+\frac{1}{\rho}-\frac{1}{\rho_1}}\nu^{\frac{1}{\rho_1}-1}\|g\|_ {\widetilde{L}^{\rho_1}_T(B^{s-2+\frac{2}{\rho_1}}_{p,r})}\Big), \end{align*} where $1\leq\rho_1\leq\rho\leq\infty.$ \end{lemm}
\subsection{The operator $\Pi(\cdot,\cdot)$} We recall some basic results for $\Pi(\cdot,\cdot).$ See \cite{keben} (Pages 296-300) for further details. \begin{lemm}\label{pi}\cite{keben}
For all $s>-1,$ and $1\leq p,r\leq\infty,$ there exists a constant C such that
\begin{align*}
\|\Pi(v,w)\|_{B^s_{p,r}}\leq C(\|v\|_{C^{0,1}}\|w\|_{B^s_{p,r}}+\|w\|_{C^{0,1}}\|v\|_{B^s_{p,r}}).
\end{align*}
Moveover, there exists a bilinear operator $P_\Pi$ such that
$\Pi(v,w)=\nabla P_\Pi(v,w),$ and
\begin{align*}
\|P_\Pi(v,w)\|_{B^{s+1}_{p,r}}\leq C\Big(\|v\|_{C^{0,1}}\|w\|_{B^s_{p,r}}+\|w\|_{C^{0,1}}\|v\|_{B^s_{p,r}}\Big),~if ~ 1<p<\infty
.\end{align*} \end{lemm} \begin{lemm}\label{deng}\cite{keben}
For all $-1<s<\frac{d}{p}+1,$ and $1\leq p,r\leq\infty,$ we have
\begin{align*}
\|\Pi(v,w)\|_{B^s_{p,r}}\leq C\Big(\|v\|_{C^{0,1}}\|w\|_{B^s_{p,r}}+\|w\|_{B^{s-\frac{d}{p}}_{\infty,\infty}}\|\nabla v\|_
{B^{\frac{d}{p}}_{p,r}}\Big).
\end{align*}
\end{lemm}
\begin{lemm}\label{yyy}\cite{keben}
For all $s>1,$ and $1\leq p,r\leq\infty,$ there exists a constant C such that
\begin{align*}
\|\nabla\cdot\Pi(v,w)+tr(Dv,Dw)\|_{B^{s-1}_{p,r}}\leq C\Big(\|\nabla\cdot v\|_{B^{0}_{\infty,\infty}}\|w\|_{B^s_{p,r}}+
\|\nabla\cdot w\|_{B^{0}_{\infty,\infty}}\|v\|_{B^s_{p,r}}\Big).
\end{align*}
\end{lemm}
\subsection{The space $B^1_{\infty,\infty}$ }
The space $B^1_{\infty,\infty}$ plays an important role in dealing with the global existence. In this section, we introduce an interpolation inequality involving $B^1_{\infty,\infty}$.
\begin{defi} Let $\alpha$ be in $(0,1].$ A modulus of continuity is any nondecreasing nonzero continuous function $\mu:[0,\alpha]\rightarrow \mathbb{R}^+$ such that $\mu(0) = 0.$ The modulus of continuity $\mu$ is admissible if, in addition, the function $\Gamma$ defined for $y \geq \frac{1}{\alpha}$ by $$\Gamma(y) \overset{def}{=} y\mu(\frac{1}{y})$$ is nondecreasing and satisfies, for some constant $C$ and all $x \geq \frac{1}{\alpha},$ $$\int_x^\infty\frac{1}{y^2}\Gamma(y)dy\leq C\frac{\Gamma(x)}{x}. $$ \end{defi} \begin{defi}Let $\mu$ be a modulus of continuity and $(X, d)$ a metric space. We denote by $C_{\mu}(X)$ the set of bounded, continuous, real-valued functions $u$ over $X$ such that \begin{align*}
\|u\|_{C_{\mu}}\overset{def}{=}\|u\|_{L^\infty}+\underset{0<d(x,y)\leq \alpha}{\sup}\frac{|u(x)-u(x')|}{\mu(d(x,y))}<\infty.
\end{align*}
\end{defi} \begin{rema}\cite{keben}
Let $\alpha=1.$ The function $\mu(r) = r(1-log r)$ is an admissible modulus of continuity, and the space $C_{\mu}$ contains $B^1_{\infty,\infty},$ more precisely, $B^1_{\infty,\infty}\hookrightarrow C_{\mu}.$ \end{rema} \begin{lemm}\label{LL}\cite{keben}
Let $\mu$ be an admissible modulus of continuity. There exists a constant C such that for any $\varepsilon\in(0,1],$ $u$ in $C^{1,\varepsilon},$ and positive $\Lambda$, we have \begin{align*}
\|\nabla u\|_{L^\infty}\leq C\
\left(\frac{\|u\|_{C_{\mu}}+\Lambda}{\varepsilon}+\|u\|_{C_\mu}
\Gamma\Big(\big(\frac{\|\nabla u\|_{C^{0,\varepsilon}}}{\|u\|_{C_{\mu}}+\Lambda}\big)^{\frac{1}{\varepsilon}}\Big)\right) \end{align*}
whenever $\|u\|_{C_{\mu}}+\Lambda\leq (\frac{\alpha}{2})^{\varepsilon}\|\nabla u\|_{C^{0,\varepsilon}}.$ \end{lemm} \section{Proof of Theorem \ref{a1}} \hspace{0.5cm}To begin, we denote $ \varepsilon=s_2+\frac{3}{2}-s_1, ~\textit{and}~ \varepsilon_0=\min(\frac{1}{2},\varepsilon).$ We mention that the condition $(\ref{jibentiaojian})$ implies that \begin{align}\label{jingchangyong} s_2+\frac{1}{2}>s_1-1>\frac{d}{2}, \end{align} which will be frequently used. \subsection{Existence for the system (\ref{s2})}
\subsubsection{First step: Construction of approximate solutions and uniform bounds}
In order to define a sequence $(u^m,z^m,\xi^m)|_{m\in\mathbb{N}}$ of global approximate solutions to the system (\ref{s2}), we use an iterative scheme.
First we set $u^0=u_0,~z^0=e^{t\triangle}z_0,~\xi^0=e^{t\triangle}\xi_0.$ Thanks to Lemma \ref{tds}, it is easy to see that
$$(u^0,z^0,\xi^0)\in \widetilde{L}^\infty_{loc}(\mathbb{R}^+;H^{s_1})\times \Big(\widetilde{L}^\infty_{loc}(\mathbb{R}^+;H^{s_2})\cap\widetilde{L}^1_{loc}(\mathbb{R}^+;H^{s_2+1})\Big)^2,$$
and
\begin{align*}
&\|u^0\|_{\widetilde{L}^\infty_t(H^{s_1})}
+\|z^0\|_{\widetilde{L}^\infty_t(H^{s_2})\cap\widetilde{L}^1_t(H^{s_2+2})}
+\|\xi^0\|_{\widetilde{L}^\infty_t(H^{s_2+1})\cap\widetilde{L}^1_t(H^{s_2+3})}\\\leq &C(1+t)(\|u_0\|_{H^{s_1}}+\|z_0\|_{H^{s_2}}+\|\xi_0\|_{H^{s_2+1}}).
\end{align*}
Then, assuming that $$(u^m,z^m,\xi^m)\in \widetilde{L}^\infty_{loc}(\mathbb{R}^+;H^{s_1})\times \big(\widetilde{L}^\infty_{loc}(\mathbb{R}^+;H^{s_2})\cap\widetilde{L}^1_{loc}(\mathbb{R}^+;H^{s_2+2})\big)\times \big(\widetilde{L}^\infty_{loc}(\mathbb{R}^+;H^{s_2+1})\cap\widetilde{L}^1_{loc}(\mathbb{R}^+;H^{s_2+3})\big),$$
we solve the following linear system:
\begin{align} \left\{ \begin{array}{l} u^{m+1}_t+u^{m}\cdot \nabla u^{m+1}=-\Pi(u^{m},u^{m})-\mathcal{P}\big((\nabla\cdot\xi^{m})\xi^{m}\big), \\[1ex] z^{m+1}_t-\triangle z^{m+1}=-\nabla\cdot(u^{m} z^{m})-\nabla\cdot\big((\nabla\cdot\xi^{m})\xi^{m}\big), , \\[1ex] \xi^{m+1}_t-\triangle \xi^{m+1}=-\mathcal{L}\big(u^{m}(\nabla\cdot\xi^{m})\big)-\mathcal{L}\big(z^{m}\xi^{m}\big), \\[1ex]
(u^{m+1},z^{m+1},\xi^{m+1})|_{t=0}=(u_0,z_0,\xi_0). \end{array} \right. \end{align} Lemma \ref{ts} ensures that \begin{align}\label{u'}
\|u^{m+1}\|_{\widetilde{L}^\infty_t(H^{s_1})}\lesssim & exp(C\int_0^t\|u^{m}\|_{H^{s_1}}dt')\Big(
\|u_0\|_{H^{s_1}}\\\nonumber&+\|\Pi(u^{m},u^{m})\|_{\widetilde{L}^1_t(H^{s_1})}+
\|\mathcal{P}\big((\nabla\cdot\xi^{m})\xi^{m}\big)\|_{\widetilde{L}^1_t(H^{s_1})}\Big). \end{align}
Using Lemma \ref{pi}, we get
\begin{align}\label{pii}
\|\Pi(u,u)\|_{\widetilde{L}^1_t(H^{s_1})}\lesssim \|u\|_{\widetilde{L}^\infty_t(H^{s_1})}\|u\|_{\widetilde{L}^\infty_t(H^{s_1})}t,
\end{align} where we have used the fact that $H^{s_1}\hookrightarrow C^{0,1}.$\\ As for the term $\mathcal{P}\big((\nabla\cdot\xi^{m})\xi^{m}\big),$ by taking advantage of Bony's decomposition and of Lemmas \ref{T}-\ref{R}, we have
\begin{align}
\|\mathcal{P}\big((\nabla\cdot\xi^{m})\xi^{m}\big)\|_{H^{s_1}}
\lesssim
&\|\big((\nabla\cdot\xi^{m})\xi^{m}\big)\|_{H^{s_1}} \\\nonumber
\lesssim
&\|\nabla\cdot\xi^{m}\|_{B^{s_1-(s_2+\frac{3}{2})}_{\infty,\infty}}
\|\xi\|_{H^{s_2+\frac{3}{2}}}+\|\xi^{m}\|_{B^{s_1-(s_2+\frac{3}{2})}_{\infty,\infty}}
\|\nabla\cdot\xi\|_{H^{s_2+\frac{3}{2}}} \\\nonumber
\lesssim&
\|\nabla\cdot\xi^{m}\|_{H^{s_1-(s_2+\frac{3}{2})+\frac{d}{2}}}
\|\xi\|_{H^{s_2+\frac{3}{2}}}+\|\xi^{m}\|_{H^{s_1-(s_2+\frac{3}{2})+\frac{d}{2}}}
\|\nabla\cdot\xi\|_{H^{s_2+\frac{3}{2}}} \\\nonumber
\lesssim&
\|\nabla\cdot\xi^{m}\|_{H^{s_2+\frac{1}{2}-\varepsilon}}
\|\xi\|_{H^{s_2+\frac{3}{2}}}+\|\xi^{m}\|_{H^{s_2+\frac{1}{2}-\varepsilon}}
\|\nabla\cdot\xi\|_{H^{s_2+\frac{3}{2}}} \\ \nonumber \lesssim&\|\xi^{m}\|_{H^{s_2+\frac{3}{2}-\varepsilon_0}}
\|\xi\|_{H^{s_2+\frac{5}{2}}},
\end{align}
where we have used $s_1-(s_2+\frac{3}{2})+\frac{d}{2}\leq \frac{d}{2}\leq s_1-1=s_2+\frac{1}{2}-\varepsilon,$ and $0<\varepsilon_0<\varepsilon.$\\ Inserting this inequality and (\ref{pii}) into (\ref{u'}), we get \begin{align}\label{u}
\|u^{m+1}\|_{\widetilde{L}^\infty_t(H^{s_1})}\lesssim & exp(C\int_0^t\|u^{m}\|_{H^{s_1}}dt')\Big(
\|u_0\|_{H^{s_1}}+\|u^{m}\|_{\widetilde{L}^\infty_t(H^{s_1})}^2t
\\\nonumber&+\|\xi^m\|_{\widetilde{L}^{\frac{4}{1-2\varepsilon_0}}(H^{s_2+1+\frac{2}{\frac{4}{1-2\varepsilon_0}}})}
\|\xi^m\|_{\widetilde{L}^{\frac{4}{3}}(H^{s_2+1+\frac{2}{\frac{4}{3}}})}t^{\frac{\varepsilon_0}{2}}\Big). \end{align}
As regards $z^{m+1},$ it follows from Lemma \ref{tds} that \begin{align*}
&\|z^{m+1}\|_{\widetilde{L}^\infty_t(H^{s_2})}
+\|z^{m+1}\|_{\widetilde{L}^1_t(H^{s_2+2})}\\\lesssim
&(1+t)\Big(\|z_0\|_{H^{s_2}}+\|\nabla\cdot(u^{m}z^{m})\|_{\widetilde{L}^1_t(H^{s_2})}
+\|\nabla\cdot\big((\nabla\cdot\xi^{m})\xi^{m}\big)\|_{\widetilde{L}^1_t(H^{s_2})}\Big).
\end{align*}
According to Lemma \ref{s_2+1}, we get
\begin{align}\label{1}
\|\nabla\cdot(u^{m}z^{m})\|_{\widetilde{L}^1_t(H^{s_2})}
\lesssim\|u^{m}z^{m}\|_{\widetilde{L}^1_t(H^{s_2+1})}&\lesssim
\|u^{m}\|_{\widetilde{L}^\infty_t(H^{s_2+1})}
\|z^{m}\|_{\widetilde{L}^2_t(H^{s_2+1})}t^{\frac{1}{2}}\\\nonumber
&\lesssim\|u^{m}\|_{\widetilde{L}^\infty_t(H^{s_1})}
\|z^{m}\|_{\widetilde{L}^2_t(H^{s_2+1})}t^{\frac{1}{2}},
\end{align}
\begin{align}\label{2}
\|\nabla\cdot\big((\nabla\cdot\xi^{m})\xi^{m}\big)\|_{\widetilde{L}^1_t(H^{s_2})}
\lesssim\|\big((\nabla\cdot\xi^{m})\xi^{m}\big)\|_{\widetilde{L}^1_t(H^{s_2+1})}
&\lesssim\|(\nabla\cdot\xi^{m})\|_{\widetilde{L}^2_t(H^{s_2+1})}
\|\xi^{m}\|_{\widetilde{L}^\infty_t(H^{s_2+1})}t^{\frac{1}{2}}\\\nonumber &\lesssim
\|\xi^{m}\|_{\widetilde{L}^2_t(H^{s_2+2})}
\|\xi^{m}\|_{\widetilde{L}^\infty_t(H^{s_2+1})}t^{\frac{1}{2}}.
\end{align}
Thus, we conclude that
\begin{align}\label{n}
&\|z^{m+1}\|_{\widetilde{L}^\infty_t(H^{s_2})}
+\|z^{m+1}\|_{\widetilde{L}^1_t(H^{s_2+2})}\\\nonumber\lesssim
&(1+t)\Big(\|z_0\|_{H^{s_2}}+\|u^{m}\|_{\widetilde{L}^\infty_t(H^{s_1})}\| z^{m}\|_{\widetilde{L}^2_t(H^{s_2+1})}t^{\frac{1}{2}}
+\|\xi^{m}\|_{\widetilde{L}^2_t(H^{s_2+2})}
\|\xi^{m}\|_{\widetilde{L}^\infty_t(H^{s_2+1})}t^{\frac{1}{2}}\Big).
\end{align}
Similarly, combining Lemma \ref{s_2+1} with Lemma \ref{tds} yields
\begin{align}\label{p}
&\|\xi^{m+1}\|_{\widetilde{L}^\infty_t(H^{s_2+1})}
+\|\xi^{m+1}\|_{\widetilde{L}^1_t(H^{s_2+3})}\\\nonumber\lesssim
&(1+t)\Big(\|\xi_0\|_{H^{s_2}}+\|\mathcal{L}\big(u^{m}(\nabla \cdot \xi^{m})\big)\|_{\widetilde{L}^1_t(H^{s_2+1})}
+\|\mathcal{L}\big(z^m\xi^{m}\big)\|_{\widetilde{L}^1_t(H^{s_2+1})}\Big)\\\nonumber\lesssim
&(1+t)\Big(\|\xi_0\|_{H^{s_2}}+\|u^{m}\|_{\widetilde{L}^\infty_t(H^{s_2+1})}\|\nabla \cdot \xi^{m}\|_{\widetilde{L}^2_t(H^{s_2+1})}t^{\frac{1}{2}}
+\|z^m\|_{\widetilde{L}^2_t(H^{s_2+1})}\|\xi^{m}\|_{\widetilde{L}^\infty_t(H^{s_2+1})}t^{\frac{1}{2}}\Big)\\
\nonumber\lesssim
&(1+t)\Big(\|\xi_0\|_{H^{s_2}}+\|u^{m}\|_{\widetilde{L}^\infty_t(H^{s_1})}\| \xi^{m}\|_{\widetilde{L}^2_t(H^{s_2+2})}t^{\frac{1}{2}}
+\|z^m\|_{\widetilde{L}^2_t(H^{s_2+1})}\|\xi^{m}\|_{\widetilde{L}^\infty_t(H^{s_2+1})}t^{\frac{1}{2}}\Big).
\end{align}
Denote $$E^{m}(t)\triangleq \|u^{m}\|_{\widetilde{L}^\infty_t(H^{s_1})}+\|z^{m}\|_{\widetilde{L}^\infty_t(H^{s_2})\cap \widetilde{L}^1_t(H^{s_2+2})}+
\|\xi^{m}\|_{\widetilde{L}^\infty_t(H^{s_2+1})\cap \widetilde{L}^1_t(H^{s_2+3})},$$
and
$$E^0\triangleq\|u_0\|_{H^{s_1}}+\|z_0\|_{H^{s_2}}+
\|\xi_0\|_{H^{s_2+1}}.$$
By using interpolation and plugging the inequalities (\ref{n}) and (\ref{p}) into (\ref{u}) yield
\begin{align*}
E^{m+1}(t)\leq C\big(e^{CE^{m}(t)t}+1+t\big)\Big(E^0+\big(E^{m}(t)\big)^2\big(t+t^{\frac{1}{2}}+t^{\frac{\varepsilon_0}{2}}\big)\Big).
\end{align*}
Let us choose a positive $T_0\leq1$ such that $exp{(8C^2E^0T_0)}\leq 2$ and $T_0^{\frac{\varepsilon_0}{2}}\leq \frac{1}{192C^2E_0}.$ The induction hypothesis then implies that $$E^{m}(T_0)\leq 8CE^0.$$ \subsubsection{Second step: Convergence of the sequence}
Let us fix some positive $T$ such that $T\leq T_0,$ and $(2CE^0)^4T\leq 1.$ We frist consider the case $s_1\neq 2+\frac{d}{2}.$
By taking the difference between the equations for $u^{m+1}$ and $u^{m},$ one finds that
\begin{align}\label{uu}
&(u^{m+1}-u^m)_t+u^{m}\cdot \nabla (u^{m+1}-u^m)\\=\nonumber&(u^{m-1}-u^{m})\nabla u^{m}-\Pi(u^{m}-u^{m-1},u^{m}+u^{m-1})\\\nonumber&+\mathcal{P}\big((\nabla\cdot\xi^m)(\xi^{m}-\xi^{m-1})\big)
+\mathcal{P}\big((\nabla\cdot\xi^m-\nabla\cdot\xi^{m-1})\xi^{m-1}\big).
\end{align}
Thanks to Lemma (\ref{deng}), we have
\begin{align}\label{i}
&\|\Pi(u^{m}-u^{m-1},u^{m}+u^{m-1})\|_{\widetilde{L}^1_t(H^{s_1-1})}
\\\nonumber\lesssim &\|u^{m-1}-u^{m}\|_{\widetilde{L}^\infty_t(H^{s_1-1})}
(\|u^{m}\|_{\widetilde{L}^\infty_t(H^{s_1})}
+\|u^{m-1}\|_{\widetilde{L}^\infty_t(H^{s_1})})t.
\end{align}
From Lemmas \ref{T}-\ref{R}, we deduce that
\begin{align}
\|(u^{m-1}-u^{m})\nabla u^{m+1}\|_{\widetilde{L}^1_t(H^{s_1-1})}\lesssim \|u^{m-1}-u^{m}\|_{\widetilde{L}^\infty_t(H^{s_1-1})}
\|u^{m-1}\|_{\widetilde{L}^\infty_t(H^{s_1})}t,\end{align}
\begin{align}
&\|\mathcal{P}\big((\nabla\cdot\xi^{m})(\xi^{m}-\xi^{m-1})\big)\|_{H^{s_1-1}}\\\nonumber
\lesssim
&\|\big((\nabla\cdot\xi^{m})(\xi^{m}-\xi^{m-1})\big)\|_{H^{s_1-1}} \\\nonumber
\lesssim
&\|\nabla\cdot\xi^{m}\|_{B^{s_1-1-(s_2+\frac{3}{2})}_{\infty,\infty}}
\|\xi^{m}-\xi^{m-1}\|_{H^{s_2+\frac{3}{2}}}+\|\xi^{m}-\xi^{m-1}\|_{B^{s_1-(s_2+\frac{3}{2})}_{\infty,\infty}}
\|\nabla\cdot\xi^{m}\|_{H^{s_2+\frac{1}{2}}} \\\nonumber
\lesssim&
\|\nabla\cdot\xi^{m}\|_{H^{s_1-1-(s_2+\frac{3}{2})+\frac{2}{d}}}
\|\xi^{m}-\xi^{m-1}\|_{H^{s_2+\frac{3}{2}}}+\|\xi^{m}-\xi^{m-1}\|_{H^{s_1-(s_2+\frac{3}{2})+\frac{2}{d}}}
\|\nabla\cdot\xi^m\|_{H^{s_2+\frac{1}{2}}} \\\nonumber
\lesssim&
\|\nabla\cdot\xi^{m}\|_{H^{s_2-\frac{1}{2}-\varepsilon_0}}
\|\xi^{m}-\xi^{m-1}\|_{H^{s_2+\frac{3}{2}}}+\|\xi^{m}-\xi^{m-1}\|_{H^{s_2+\frac{1}{2}-\varepsilon_0}}
\|\nabla\cdot\xi^m\|_{H^{s_2+\frac{1}{2}}},
\end{align}
where we have used $s_1-1-(s_2+\frac{3}{2})+\frac{2}{d}\leq \frac{2}{d}-1\leq s_1-2\leq s_2+\frac{1}{2}-\varepsilon.$ Similarly,
\begin{align}\label{iii}
&\|\mathcal{P}\big((\nabla\cdot\xi^m-\nabla\cdot\xi^{m-1})\xi^{m-1}\big)\|_{H^{s_1-1}}
\\\nonumber\lesssim&
\|\nabla\cdot\xi^m-\nabla\cdot\xi^{m-1}\|_{H^{s_2-\frac{1}{2}-\varepsilon_0}}
\|\xi^{m-1}\|_{H^{s_2+\frac{3}{2}}}+\|\xi^{m-1}\|_{H^{s_2+\frac{1}{2}-\varepsilon_0}}
\|\nabla\cdot\xi^m-\nabla\cdot\xi^{m-1}\|_{H^{s_2+\frac{1}{2}}}.
\end{align}
Applying Lemma \ref{ts} to (\ref{uu}) thus yields
\begin{align}\label{uug}
\|u^{m+1}-&u^{m}\|_{\widetilde{L}^\infty_t(H^{s_1-1})}\lesssim exp(C\int_0^t\|u^{m}\|_{H^{s_1}}dt')\\\nonumber&\times\Big(\|u^{m-1}-u^{m}\|_{\widetilde{L}^\infty_t(H^{s_1-1})}
(\|u^{m}\|_{\widetilde{L}^\infty_t(H^{s_1})}
+\|u^{m-1}\|_{\widetilde{L}^\infty_t(H^{s_1})})t\\\nonumber&~~~+
(\|\xi^m\|_{\widetilde{L}^\infty_t(H^{s_2+1})}+\|\xi^{m-1}\|_{\widetilde{L}^\infty_t(H^{s_2+1})})
\|\xi^m-\xi^{m-1}\|_{\widetilde{L}^{\frac{4}{3}}_t(H^{s_2+\frac{2}{\frac{4}{3}}})}t^{\frac{1}{4}}
\\\nonumber&~~~+\|\xi^m-\xi^{m-1}\|_{\widetilde{L}^{\frac{4}{1-2\varepsilon_0}}_t(B^{s_2+\frac{2}{\frac{4}{1-2\varepsilon_0}}})}
(\|\xi^m\|_{\widetilde{L}^4_t(H^{s_2+1+\frac{2}{4}})}+\|\xi^{m-1}\|_{\widetilde{L}^4_t(H^{s_2+1+\frac{2}{4}})}t^{\frac{1+\varepsilon_0}{2}}\Big). \end{align}
Note that \begin{align*}
(z^{m+1}-z^{m})_t-&\triangle \Big(z^{m+1}-z^{m})=-\nabla\cdot\big(u^m (z^{m}-z^{m-1})-(u^{m}-u^{m-1}) z^{m-1}\big)\\&+\nabla\big((\nabla\cdot\xi^m) (\xi^{m}-\xi^{m-1})\big)-\nabla\big((\nabla\cdot \xi^{m}-\nabla\cdot \xi^{m-1})\xi^{m-1} \big). \end{align*} By virtue of Lemma \ref{s_2+1}, we get \begin{align}\label{44}
\|\nabla\cdot\big(u^m (z^{m}-z^{m-1})\big)
\|_{H^{s_2-1}}
\lesssim&
\|u^m (z^{m}-z^{m-1})
\|_{H^{s_2}}\\\nonumber\lesssim&
\|u^m\|_{H^{s_2+\frac{1}{2}}}\|z^{m}-z^{m-1}
\|_{H^{s_2}}\lesssim
\|u^m\|_{H^{s_1}}\|z^{m}-z^{m-1}
\|_{H^{s_2}},
\end{align}
\begin{align}\label{55}
\|\nabla\cdot\big(u^{m}-u^{m-1})z^{m-1}\big)
\|_{H^{s_2-1}}\lesssim&\|(u^{m}-u^{m-1})z^{m-1}
\|_{H^{s_2}}\\\nonumber\lesssim&
\|u^{m}-u^{m-1}\|_{H^{s_2}}
\|z^{m}\|_{H^{s_2+\frac{1}{2}}}\lesssim
\|u^{m}-u^{m-1}\|_{H^{s_1-1}}
\|z^{m}\|_{H^{s_2+\frac{1}{2}}},
\end{align}
\begin{align}\label{66}
\|\nabla\big((\nabla\cdot\xi^m) (\xi^{m}-\xi^{m-1})\big)
\|_{H^{s_2-1}}
\lesssim&
\|(\nabla\cdot\xi^m) (\xi^{m}-\xi^{m-1})\|_{H^{s_2}}\\\nonumber\lesssim&
\|\nabla\cdot\xi^m\|_{H^{s_2+\frac{1}{2}}}
\|\xi^{m}-\xi^{m-1}\|_{H^{s_2}},
\end{align}
\begin{align}\label{77}
\|\nabla\big((\nabla\cdot \xi^{m}-\nabla\cdot \xi^{m-1})\xi^{m-1} \big)
\|_{H^{s_2-1}}
\lesssim&
\|(\nabla\cdot \xi^{m}-\nabla\cdot \xi^{m-1})\xi^{m-1}\|_{H^{s_2}}\\\nonumber\lesssim&
\|\nabla\cdot \xi^{m}-\nabla\cdot \xi^{m-1}\|_{H^{s_2+\frac{1}{2}}}
\|\xi^{m-1}\|_{H^{s_2}}.
\end{align}
Hence Lemma \ref{tds} implies that
\begin{align}\label{nng}
&\|z^{m+1}-z^{m}\|_{\widetilde{L}^\infty_t(H^{s_2-1})}
+\|z^{m+1}-z^{m}\|_{\widetilde{L}^1_t(H^{s_2+1})}\\\nonumber\lesssim
&(1+t)\Big(\|u^{m}\|_{\widetilde{L}^\infty_t(H^{s_1})}
\|z^{m}-z^{m-1}\|_{\widetilde{L}^2_t(H^{s_2})}t^{\frac{1}{2}}
+\|u^{m}-u^{m-1}\|_{\widetilde{L}^\infty_t(H^{s_1-1})}
\|z^{m}\|_{\widetilde{L}^4_t(H^{s_2+\frac{2}{4}}_{p_2,r_2})}t^{\frac{3}{4}}
\\\nonumber&+\|\xi^{m}\|_{\widetilde{L}^4_t(H^{s_2+1+\frac{2}{4}})}
\|\xi^{m}-\xi^{m-1}\|_{\widetilde{L}^\infty_t(H^{s_2})}t^{\frac{3}{4}}+
\|\xi^{m}-\xi^{m-1}\|_{\widetilde{L}^{\frac{4}{3}}_t(H^{s_2+\frac{2}{\frac{4}{3}}}_{p_2,r_2})}
\|\xi^{m-1}\|_{\widetilde{L}^\infty_t(H^{s_2+1})}t^{\frac{1}{4}}\Big).
\end{align}
Similarly, we get
\begin{align}\label{ppg}
&\|\xi^{m+1}-\xi^{m}\|_{\widetilde{L}^\infty_t(H^{s_2})}
+\|\xi^{m+1}-\xi^{m}\|_{\widetilde{L}^1_t(H^{s_2+2})}\\\nonumber\lesssim
&(1+t)\Big(\|\mathcal{L}\big(u^{m}\nabla\cdot(\xi^{m}-\xi^{m-1})\big)\|_{\widetilde{L}^1_t(H^{s_2})}
+\|\mathcal{L}\big((u^{m}-u^{m-1})\nabla\cdot\xi^{m-1}\big)\|_{\widetilde{L}^1_t(H^{s_2})}\\\nonumber&
+\|\mathcal{L}\big(z^{m}(\xi^{m}-\xi^{m-1})\big)\|_{\widetilde{L}^1_t(H^{s_2})}
+\|\mathcal{L}\big((z^m-z^{m-1})\xi^{m}\big)\|_{\widetilde{L}^{1}_t(H^{s_2})}\Big)\\\nonumber\lesssim
&(1+t)\Big(\|u^{m}\|_{\widetilde{L}^\infty_t(H^{s_2+\frac{1}{2}})}
\|\nabla\cdot(\xi^{m}-\xi^{m-1})\|_{\widetilde{L}^2_t(H^{s_2})}t^{\frac{1}{2}}
+\|u^{m}-u^{m-1}\|_{\widetilde{L}^\infty_t(H^{s_2})}
\|\nabla\cdot\xi^{m-1}\|_{\widetilde{L}^4_t(H^{s_2+\frac{2}{4}})}t^{\frac{3}{4}}\\\nonumber&
+\|z^{m}\|_{\widetilde{L}^\infty_t(H^{s_2})}\|\xi^{m}-\xi^{m-1}\|_{\widetilde{L}^4_t(H^{s_2+\frac{2}{4}})}t^{\frac{3}{4}}
+\|z^m-z^{m-1}\|_{\widetilde{L}^2_t(H^{s_2})}
\|\xi^{m}\|_{\widetilde{L}^{\infty}_t(H^{s_2+\frac{1}{2}})}t^{\frac{1}{2}}\Big)\\\nonumber\lesssim
&(1+t)\Big(\|u^{m}\|_{\widetilde{L}^\infty_t(H^{s_1})}
\|\xi^{m}-\xi^{m-1}\|_{\widetilde{L}^2_t(H^{s_2+1})}t^{\frac{1}{2}}
+\|u^{m}-u^{m-1}\|_{\widetilde{L}^\infty_t(H^{s_1-1})}
\|\xi^{m-1}\|_{\widetilde{L}^4_t(H^{s_2+1+\frac{2}{4}})}t^{\frac{3}{4}}\\\nonumber&
+\|z^{m}\|_{\widetilde{L}^\infty_t(H^{s_2})}\|\xi^{m}-\xi^{m-1}\|_{\widetilde{L}^4_t(H^{s_2+\frac{2}{4}})}t^{\frac{3}{4}}
+\|z^m-z^{m-1}\|_{\widetilde{L}^2_t(H^{s_2})}
\|\xi^{m}\|_{\widetilde{L}^{\infty}_t(H^{s_2+1})}t^{\frac{1}{2}}\Big).
\end{align}
Denote \begin{align*}F^{m}(t)\triangleq \|u^{m+1}-u^{m}\|_{\widetilde{L}^\infty_t(H^{s_1-1})}&+\|z^{m+1}-z^{m}\|_{\widetilde{L}^\infty_t
(H^{s_2-1})\cap\widetilde{L}^1_t(H^{s_2+1})}\\&+
\|\xi^{m+1}-\xi^{m}\|_{\widetilde{L}^\infty_t(H^{s_2})
\cap\widetilde{L}^1_t(H^{s_2+2})}.\end{align*}
Plugging the inequalities (\ref{nng}) and (\ref{ppg}) into (\ref{uug}) yields
\begin{align*}
F^{m+1}(T)\leq & C\big(e^{CE^{m}(T)T}+1+T\big)\big(E^{m}(T)+E^{m-1}(T)\big)F^{m}(T)\big(T+T^{\frac{1+\varepsilon_0}{2}}+T^{\frac{1}{2}}+T^{\frac{3}{4}}+T^{\frac{1}{4}}\big)
\\\leq &CE^0T^{\frac{1}{4}}F^{m}(T)\leq \frac{1}{2}F^{m}(T).
\end{align*}
Hence, $(u^m,z^m,\xi^m)|_{m\in \mathbb{N}}$ is a Cauchy sequence in $\widetilde{L}^\infty_T(H^{s_1-1})\times \big(\widetilde{L}^\infty_T(H^{s_2-1})\cap\widetilde{L}^1_T(H^{s_2+1})\big)\times \big(\widetilde{L}^\infty_T(H^{s_2})\cap\widetilde{L}^1_T(H^{s_2+2})\big).$\\
In the case $s_1=2+\frac{d}{2},$ for every $\zeta\in(0,1),$ we have
$$s_1-\zeta>1+\frac{d}{2}, \textit{and}~ s_2-\zeta+\frac{3}{2}> s_1-\zeta\geq s_2-\zeta+1.$$
Following along the same lines as above, we have
$(u^m,z^m,\xi^m)|_{m\in \mathbb{N}}$ is a Cauchy sequence in $\widetilde{L}^\infty_T(H^{s_1-\zeta-1})\times \big(\widetilde{L}^\infty_T(H^{s_2-\zeta-1})\cap\widetilde{L}^1_T(H^{s_2-\zeta+1})\big)\times \big(\widetilde{L}^\infty_T(H^{s_2-\zeta})\cap\widetilde{L}^1_T(H^{s_2-\varsigma+2})\big).$
\subsubsection{Third step: Passing to the limit}
Since the case $s_1=2+\frac{d}{2}$ works the same way, we only consider the case $s_1\neq2+\frac{d}{2}.$ Let $(u,z,\xi)$ be the limit of the sequence $(u^m,z^m,\xi^m)|_{m\in \mathbb{N}}.$ We see that $(u,z,\xi)\in\widetilde{L}^\infty_T(H^{s_1-1})\times \big(\widetilde{L}^\infty_T(H^{s_2-1})\cap\widetilde{L}^1_T(H^{s_2+1})\big)\times \big(\widetilde{L}^\infty_T(H^{s_2})\cap\widetilde{L}^1_T(H^{s_2+2})\big).$ Using Lemma \ref{Fadou} with the uniform bounds given in Step 1, we see that $(u,n,p)\in\widetilde{L}^\infty_T(H^{s_1})\times \big(\widetilde{L}^\infty_T(H^{s_2})\big)\times \big(\widetilde{L}^\infty_T(H^{s_2+1})\big).$ Next, by interpolating we discover that $(u^m,z^m,\xi^m)$ tends to $(u,z,\xi)$ in every space $\widetilde{L}^\infty_T(H^{s_1-\eta})\times \big(\widetilde{L}^\infty_T(H^{s_2-\eta})\cap\widetilde{L}^1_T(H^{s_2+1})\big)\times
\big(\widetilde{L}^\infty_T(H^{s_2+1-\eta})\cap\widetilde{L}^1_T(H^{s_2+2})\big),$ with $\eta>0,$ which suffices to pass to the limit in the system (\ref{s2}).
We still have to prove that $(z,\xi)\in \widetilde{L}^1_T(H^{s_2+2})\times\widetilde{L}^1_T(H^{s_2+3}).$ In fact, it is easy to check that $(\partial_t z-\triangle z,\partial_t \xi-\triangle \xi)\in\widetilde{L}^1_T(H^{s_2})\times\widetilde{L}^1_T(H^{s_2+1}).$ Hence according to Lemma \ref{tds}, $(z,\xi)\in \widetilde{L}^1_T(H^{s_2+2})\times\widetilde{L}^1_T(H^{s_2+3}).$\\
Finally, following along the same lines as in Theorem 3.19 of \cite{keben}, we can show that
\begin{align}\label{lian}
(u,z,\xi)\in C([0,T];H^{s_1})\times C([0,T];H^{s_2})\times C([0,T];H^{s_2+1}).
\end{align} \subsection{Uniqueness for the system (\ref{s2})}
Without loss of generality, we may assume that $s_1<2+\frac{d}{2}.$ Assume that we are given $(u_1,z_1,\xi_1)$ and $(u_2,z_2,\xi_2)$, two solutions of the system (\ref{s2}) (with the same initial data) satisfying the regularity assumptions of Theorem \ref{a1}. In order to show these two solutions coincide, we first denote $$E^i(T)\triangleq
\|u_i\|_{\widetilde{L}^\infty_T(H^{s_1})}+\|z_i\|_{\widetilde{L}^\infty_T(H^{s_2})}+
\|\xi_i\|_{\widetilde{L}^\infty_t(H^{s_2+1})},~~i=1,2,$$
$$F(t_1,t_2)\triangleq \| u_2-u_1\|_{\widetilde{L}^\infty_{[t_1,t_2]}(H^{s_1-1})}+\|z_2-z_1 \|_{\widetilde{L}^\infty_{[t_1,t_2]}(H^{s_2-1})}+
\|\xi_2-\xi_1\|_{\widetilde{L}^\infty_{[t_1,t_2]}(H^{s_2}) },$$
and $$T_0=\sup\{0\leq T'\leq T~|~(u_1,z_1,\xi_1)= (u_2,z_2,\xi_2)~on ~[0,T']\}. $$
We deduce from the definition of $T_0$ and the continuity of $(u_i,z_i,\xi_i)$ that $$\big(u_1(T_0),z_1(T_0),\xi_1(T_0)\big)= \big(u_2(T_0),z_2(T_0),\xi_2(T_0)\big).$$
If $T_0<T,$ repeating the same arguments as we were used for the proof of the convergence of the approximate solutions in the above subsection, we get \begin{align*}
F(T_0+\widetilde{T})\leq C\big(e^{CE^{2}(T)\widetilde{T}}+1+\widetilde{T}\big)\big(E^{2}(T)
+E^{1}(T)\big)F(T_0+\widetilde{T})\big(\widetilde{T}
+\widetilde{T}^{\frac{1+\varepsilon_0}{2}}
+\widetilde{T}^{\frac{1}{2}}+\widetilde{T}^{\frac{3}{4}}+\widetilde{T}^{\frac{1}{4}}\big)
.
\end{align*}
We conclude that $F(T_0+\widetilde{T})=0$ with sufficiently small $\widetilde{T}.$ Thus, $(u_1,z_1,\xi_1)= (u_2,z_2,\xi_2)$ on $[T_0,T_0+\widetilde{T}]$, which stands in contradiction to the definition of $T_0.$ Hence $T_0=T,$
and the proof of uniqueness is completed. \subsection{Properties of $(u,z,\xi)$} \subsubsection{$\nabla\cdot u=0$}
Suppose that $(u,z,\xi)$ satisfies the system (\ref{s2}) in $\in \widetilde{L}^\infty_{T}(H^{s_1}(\mathbb{R}^d))\times \big(\widetilde{L}^\infty_{T}(H^{s_2}(\mathbb{R}^d)) \cap\widetilde{L}^1_{T}(H^{s_2+2}(\mathbb{R}^d))\big)\times \big(\widetilde{L}^\infty_{T}(H^{s_2+1}(\mathbb{R}^d)) \cap\widetilde{L}^1_{T}(H^{s_2+3}(\mathbb{R}^d))\big).$ We check that $u$ is divergence free. This may be achieved by applying $\nabla\cdot$ to the first equation of the system (\ref{s2}). Denote $s_1'=s_1-1$, if $s_1\neq 2+\frac{d}{2};$ and $s_1'=s_1-\zeta-1$, for some $\zeta\in(0,1),$ if $s_1=2+\frac{d}{2}.$ We get
\begin{align*}
(\partial_t+u\cdot \nabla)(\nabla\cdot u)=-\nabla\cdot\Pi(u,u)-tr(Du)^2.
\end{align*}
Lemma \ref{ts} and Lemma \ref{yyy} ensure that
\begin{align*}
\|\nabla\cdot u\|_{H^{s_1'}}\lesssim&
\int_0^texp\Big(C\int_{t'}^t\|u\|_{H^{s_1}}dt''\Big)\|\nabla\cdot\Pi(u,u)+tr(Du)^2\|_{H^{s'_1}}dt'\\ \lesssim&
\int_0^texp\Big(C\int_{t'}^t\|u\|_{H^{s_1}}dt''\Big)\|\nabla\cdot u\|_{B^{0}_{\infty,\infty}}\|u\|_{H^{s'_1+1}}dt',\\
\lesssim&
\int_0^texp\Big(C\int_{t'}^t\|u\|_{H^{s_1}}dt''\Big)\|\nabla\cdot u\|_{H^{s_1'}}\|u\|_{H^{s_1}}dt',
\end{align*}
where we have used $H^{s_1'}\hookrightarrow B^{0}_{\infty,\infty},$ and $H^{s_1'+1} \hookrightarrow H^{s_1}.$ Using Gronwall's inequality, we conclude that $\nabla\cdot u=0.$
\subsubsection{$\mathcal{L}\xi=\xi$}
Note that \begin{align}\label{xixi}
\xi = e^{t\triangle}\xi_0-\int_0^te^{(t-t')\triangle}\mathcal{L}\big(u(\nabla\cdot \xi)+z\xi\big)dt'. \end{align} Applying $\mathcal{L}$ to the above equation yields \begin{align*} \mathcal{L} \xi = e^{t\triangle}\mathcal{L}\xi_0-\int_0^te^{(t-t')\triangle}\mathcal{L}^2\big(u(\nabla\cdot \xi)+z\xi\big)dt'. \end{align*} It is easy to check that \begin{align*}
\mathcal{L}\xi_0=-\nabla (-\triangle)^{-1}\nabla\cdot(-\nabla (-\triangle)^{-1}a_0)=\nabla (-\triangle)^{-1}(\nabla\cdot\nabla ) (-\triangle)^{-1}a_0=-\nabla (-\triangle)^{-1}a_0=\xi_0,\\
\mathcal{L}^2=-\nabla (-\triangle)^{-1}\nabla\cdot(-\nabla (-\triangle)^{-1}\nabla\cdot)=\nabla (-\triangle)^{-1}(\nabla\cdot\nabla)(-\triangle)^{-1}\nabla\cdot=-\nabla (-\triangle)^{-1}\nabla\cdot=\mathcal{L}. \end{align*} Hence, $\mathcal{L}\xi=\xi.$ \subsubsection{Nonnegative of $z\pm\nabla\cdot \xi$} Let $a=\frac{z+\nabla \cdot \xi}{2},$ and $b=\frac{z-\nabla \cdot \xi}{2}.$ As $\nabla \cdot\mathcal{L}=\nabla \cdot$ and $\nabla\cdot u=0,$ one finds that $(a,b)$ solves the following system: \begin{align}\tag{ab}\label{ab} \left\{ \begin{array}{l} a_t+u\cdot \nabla a-\triangle a=-\nabla\cdot(a\xi), \\[1ex] b_t+u\cdot \nabla b- \triangle b=\nabla\cdot(b\xi), \\[1ex] \nabla \cdot \xi=a-b,\\[1ex]
(a,b)|_{t=0}=(\frac{z_0+\nabla \cdot \xi_0}{2},\frac{z_0-\nabla \cdot \xi_0}{2}), \end{array} \right. \end{align}
We test the first equation of the system (\ref{ab}) with $(a^-)\triangleq\sup \{-a,0\}.$ After integrating by parts, we obtain
\begin{align*}
\frac{1}{2}\frac{d}{dt}\|a^-\|_{L^{2}}^{2}+\|\nabla a^-\|_{L^{2}}^{2}=-\int_{\mathbb{R}^d}\frac{1}{2}(\nabla \cdot \xi)(a^-)^2dx\leq \frac{1}{2}\|\nabla \cdot \xi\|_{L^\infty}\|a^-\|_{L^2}^2.
\end{align*}
Gronwall's Lemma implies that
\begin{align*}
\|a^-\|_{L^{2}}^{2}\leq \|a^-_0\|_{L^{2}}^{2}exp\{\|\nabla \cdot \xi\|_{L^1_t(L^\infty)}\}.
\end{align*}
Since $a_0\geq 0,$ and $$\|\nabla \cdot \xi\|_{L^1_t(L^\infty)}\lesssim
\|\nabla \cdot \xi\|_{L^1_t(H^{s_2+\frac{1}{2}})}\lesssim
\|\nabla \cdot \xi\|_{L^2_t(H^{s_2+1})}t^{\frac{1}{2}},$$
we have $\|a^-\|_{L^{2}}^{2}=0.$ Hence $a \geq 0,$ a.e. on $[0,T]\times \mathbb{R}^d.$
Repeating the same steps for $b$ implies $b\geq 0,~a.e.~on ~[0,T]\times \mathbb{R}^d,$
and thus $z=a+b\geq 0,~a.e.~on ~[0,T]\times \mathbb{R}^d.$
\subsection{A global existence result in dimension $d=2$ } According to the above subsections, local existence in $\widetilde{L}^\infty_{T}(H^{s_1}(\mathbb{R}^d))\times \big(\widetilde{L}^\infty_{T}(H^{s_2}(\mathbb{R}^d)) \cap\widetilde{L}^1_{T}(H^{s_2+2}(\mathbb{R}^d))\big)\times \big(\widetilde{L}^\infty_{T}(H^{s_2+1}(\mathbb{R}^d)) \cap\widetilde{L}^1_{T}(H^{s_2+3}(\mathbb{R}^d))\big)$ has already been proven. So we denote by $T^*$ the maximal time of existence of $(u,z,\xi).$ Suppose that $T^*$ is finite, under the assumption of \ref{jibentiaojian}, and assume further that $d=2$ and $s_2>1$, we have the following lemmas. \subsubsection{Some useful lemmas} \begin{lemm}\label{1l} $\forall t\in[0,T^*),$ we have \begin{align}\label{l1}
\|u(t)\|_{L^2}^2+\|\xi(t)\|_{L^2}^2+\int_0^t\|\nabla \xi\|_{L^2}^2dt'\lesssim\|u_0\|_{L^2}^2+\|\xi_0\|_{L^2}^2. \end{align} \end{lemm} \noindent{Proof.} Multiplying the first and the third equations of the system (\ref{s2}) by $u$ and $\xi$ respectively, and integrating over $\mathbb{R}^d:$ \begin{align*}
&\frac{1}{2}\frac{d}{dt}\|u\|_{L^2}^2
=\int_{\mathbb{R}^2}(\nabla\cdot\xi)\xi udx,\\
&\frac{1}{2}\frac{d}{dt}\|\xi\|_{L^2}^2+\int_{\mathbb{R}^2}(\nabla\cdot\xi)\xi udx+\|\nabla \xi\|_{L^2}^2=\int_{\mathbb{R}^d}-z|\xi|^2dx, \end{align*} where we have used \begin{align*} &\int_{\mathbb{R}^2}(u\cdot\nabla u)udx=
\int_{\mathbb{R}^2}-\frac{1}{2}(\nabla \cdot u)|u|^2dx =0,\\ &\int_{\mathbb{R}^2}\Pi(u,u)udx= \int_{\mathbb{R}^2}\nabla P_{\Pi}(u,u)udx =\int_{\mathbb{R}^2}- P_{\Pi}(u,u)(\nabla \cdot u)dx=0,\\ &\int_{\mathbb{R}^2}\mathcal{P}\big((\nabla\cdot\xi)\xi\big)udx= \int_{\mathbb{R}^2}(\nabla\cdot\xi)\xi(\mathcal{P}u)dx =\int_{\mathbb{R}^2}(\nabla\cdot\xi)\xi udx,\\ &\int_{\mathbb{R}^2}\mathcal{L}\big(u(\nabla\cdot\xi)\big)\xi dx= \int_{\mathbb{R}^2}u(\nabla\cdot\xi)(\mathcal{L}\xi) dx= \int_{\mathbb{R}^2}u(\nabla\cdot\xi)\xi dx,\\ &\int_{\mathbb{R}^2}\mathcal{L}(-z\xi)\xi= \int_{\mathbb{R}^2}-z\xi(\mathcal{L}\xi)=
\int_{\mathbb{R}^2}-z|\xi|^2, \end{align*} with $P_{\pi}(u,u)$ defined as in Lemma \ref{pi}.
Summing the above equations and using the fact $z\geq 0,$ we find
\begin{align*}
\frac{1}{2}\frac{d}{dt}(\|u\|_{L^2}^2+\|\xi\|_{L^2}^2)+\|\nabla \xi\|_{L^2}^2\leq 0, \end{align*} from which it follows that (\ref{l1}) holds. \qed
\begin{lemm}\label{2l} $\forall t\in[0,T^*),$ $2\leq q\leq \infty,$ we have \begin{align*}
&\|z\pm\nabla \cdot \xi(t)\|_{L^q}\lesssim \|z_0+\nabla \cdot \xi_0\|_{L^q}+\|z_0-\nabla \cdot \xi_0\|_{L^q},\\
&\|z\pm\nabla \cdot \xi(t)\|_{L^2}^2+\int_0^t\|\nabla (z\pm\nabla \cdot \xi)\|_{L^2}^2dt'\lesssim \|z_0+\nabla \cdot \xi_0\|_{L^2}^2+\|z_0-\nabla \cdot \xi_0\|_{L^2}^2. \end{align*} \end{lemm}
\noindent{Proof.} Since $s_2>1,$ (\ref{lian}) implies that $(a,b)\in \big(C([0,T];H^{s_2})\big)^2\hookrightarrow \big(C([0,T];L^q)\big)^2,$ with $2\leq q\leq \infty.$ By multiplying both sides of the first equation of the (\ref{ab}) system by $| a|^{p-2} a$ with $2\leq p< \infty,$ and integrating over $[0,t]\times\mathbb{R}^d,$ we get
\begin{align}\label{n1}
\frac{1}{p}\|a(t)\|_{L^p}^p+(p-1)\int_{\mathbb{R}^2}|a|^{p-2}|\nabla a|^2dx\leq\frac{1}{p}\|a_0\|_{L^p}^p-\frac{p-1}{p}
\int_0^t\int_{\mathbb{R}^2}\nabla\cdot \xi|a|^pdxdt',
\end{align}
where we have used the estimates
\begin{align*}
\int_{\mathbb{R}^2}| a|^{p-2} a(u\cdot \nabla a)dx=&
-\frac{1}{p}\int_{\mathbb{R}^2}(\nabla\cdot u)|a|^pdx=0,\\
-\int_{\mathbb{R}^2}| a|^{p-2} a\nabla\cdot(a\xi)dx=&
-\int_{\mathbb{R}^2}\xi\frac{1}{p}\nabla|a|^pdx-\int_{\mathbb{R}^2}(\nabla\cdot\xi)|a|^pdx\\=&
-\int_{\mathbb{R}^2}\frac{p-1}{p}(\nabla\cdot \xi)|a|^pdx.
\end{align*}
Repeating the same steps for $b$ yields \begin{align}\label{p1}
\frac{1}{p}\|b(t)\|_{L^p}^p+(p-1)\int_{\mathbb{R}^2}|b|^{p-2}|\nabla b|^2dx\leq\frac{1}{p}\|b_0\|_{L^p}^p+\frac{p-1}{p}
\int_0^t\int_{\mathbb{R}^2}(\nabla\cdot\xi)|b|^pdxdt'.
\end{align} Adding up $(\ref{n1})$ and (\ref{p1}), we get \begin{align*}
&\frac{1}{p}(\|a(t)\|_{L^p}^p+\|b(t)\|_{L^p}^p)+(p-1)\int_{\mathbb{R}^2}|a|^{p-2}|\nabla a|^2dx+(p-1)\int_{\mathbb{R}^2}|b|^{p-2}|\nabla b|^2dx\\\leq &\frac{1}{p}(\|a_0\|_{L^p}^p+\|b_0\|_{L^p}^p)+\frac{p-1}{p}
\int_0^t\int_{\mathbb{R}^2}\nabla\cdot\xi(|b|^p-|a|^p)dxdt'\\
\leq &\frac{1}{p}(\|a_0\|_{L^p}^p+\|b_0\|_{L^p}^p)+\frac{p-1}{p}
\int_0^t\int_{\mathbb{R}^2}(a-b)(b^p-a^p)dxdt'\\
\leq &\frac{1}{p}(\|a_0\|_{L^p}^p+\|b_0\|_{L^p}^p), \end{align*} where we have used the non-negativity of $a,b.$ This thus leads to \begin{align*}
&\|a(t)\|_{L^p}+\|b(t)\|_{L^p}\leq 2( \|a_0\|_{L^p}+\|b_0\|_{L^p}),\\
&\|a(t)\|_{L^2}^2+\|b(t)\|_{L^2}^2+\int_0^t(\|\nabla a\|_{L^2}^2+\|\nabla b\|_{L^2}^2)dt'\leq\|a_0\|_{L^2}^2+\|b_0\|_{L^2}^2. \end{align*} Passing to the limit as $p$ tends to infinite gives \begin{align*}
\|a(t)\|_{L^\infty}+\|b(t)\|_{L^\infty}\leq 2( \|n_0\|_{L^\infty}+\|b_0\|_{L^\infty}). \end{align*} This completes the proof of the lemma. \qed \begin{lemm}\label{3l} $\forall t\in[0,T^*),$ we have \begin{align}\label{l3}
\|\xi(t)\|_{L^\infty}\leq C(T^*)<\infty. \end{align} \end{lemm} {\noindent Proof.} It is easy to obtain from \ref{xixi} that \begin{align*}
\|\xi(t)\|_{L^{\infty}}\lesssim&\|\mathcal{F}^{-1}(e^{-t|x|^2})\|_{L^1} \|\xi_0\|_{L^{\infty}}+ \int_0^t\|\mathcal{F}^{-1}(e^{(t'-t)|x|^2})\|_{L^{2}}\Big(\|u(t')\|_{L^{2}}\| \nabla\cdot\xi(t')\|_{L^{\infty}}\\&+\|z(t')\|_{L^{\infty}}\|\xi(t')\|_{L^{2}}\Big)dt'\\\lesssim&
\|\xi_0\|_{L^{\infty}}+\int_0^t(t'-t)^{-\frac{1}{2}}\Big(\|u(t')\|_{L^{2}}\| \nabla\cdot\xi(t')\|_{L^{\infty}}dt'+\|z(t')\|_{L^{\infty}}\|\xi(t')\|_{L^{2}}\Big)\\\lesssim&
\|\xi_0\|_{L^{\infty}}+t^{\frac{1}{2}}\Big(\|u\|_{L^\infty_t(L^{2})}\| \nabla\cdot\xi\|_{L^\infty_t(L^{\infty})}+\|z\|_{L^\infty_t(L^{\infty})}\|\xi\|_{L^\infty_t(L^{2})}\Big).
\end{align*}
Applying Lemma \ref{1l} and Lemma \ref{2l} completes the proof. \qed
Denote $w=\partial_2u_1-\partial_1u_2.$ Note that $\Pi(u,u)=\nabla P_{\Pi}(u,u),$ where $P_{\Pi}(u,u)$ defined as in Lemma \ref{pi}, $(\mathcal{P}-Id)\big((\nabla\cdot\xi)\xi\big)=\nabla (-\triangle)^{-1}\nabla\cdot\big((\nabla\cdot\xi)\xi\big),$ and $\xi=\mathcal{L}\xi=\nabla (-\triangle)^{-1}\nabla\cdot \xi.$ Then $w$ satisfies
\begin{align}\label{lw}
w_t+u\cdot \nabla w=\partial_2(\nabla\cdot\xi)\xi_1-\partial_1(\nabla\cdot\xi)\xi_2.
\end{align}
\begin{lemm}\label{uw}\cite{keben} For all $s\in \mathbb{R}$ and $1\leq p, r \leq \infty,$ there exists a constant $C$ such that \begin{align}\label{wu}
\|(Id-\triangle_{-1})u\|_{B^s_{p,r}}
\leq C\|w\|_{B^s_{p,r}}. \end{align} \end{lemm}
\begin{lemm}\label{4l} $\forall t\in[0,T^*),$ we have \begin{align}\label{l4}
\|\nabla u(t)\|_{L^2}\leq C(T^*)<\infty. \end{align} \end{lemm} {\noindent Proof.}
Multiplying (\ref{uw}) by $w$ and integrating over $\mathbb{R}^2:$ \begin{align*}
\frac{1}{2}\frac{d}{dt}\|w\|_{L^{2}}^{2}\lesssim\|\nabla(\nabla\cdot\xi)\|_{L^2}\|\xi\|_{L^\infty}\|w\|_{L^2}.
\end{align*}
The Gronwall lemma implies that
\begin{align*}
\|w(t)\|_{L^{2}}\lesssim& \|w_0\|_{L^{2}}+\int_0^t\|\nabla(\nabla\cdot\xi)\|_{L^2}\|\xi\|_{L^\infty}dt'
\lesssim\|w_0\|_{L^{2}}+\|\nabla(\nabla\cdot\xi)\|_{L^2_t(L^2)}\|\xi\|_{L^{\infty}_t(L^\infty)}t^{\frac{1}{2}}.
\end{align*}
Applying Lemma \ref{2l} and Lemma \ref{3l}, we have
\begin{align}\label{www}
\|w(t)\|_{L^2}\leq C(T^*)<\infty.
\end{align}
Next by splitting $u$ into low and high frequencies and using Lemma \ref{uw}, we see that
\begin{align*}
\|\nabla u\|_{L^{2}}\lesssim \|\triangle_{-1}\nabla u\|_{L^{2}}+\|(Id-\triangle_{-1})\nabla u\|_{L^{2}}\lesssim\| u\|_{L^{2}}+\|w\|_{L^{2}}.
\end{align*}
Applying Lemma \ref{2l} and the inequality \ref{www} then completes the proof of the lemma. \begin{lemm}\label{5l} \begin{align}\label{l5}
\int_0^{T^*}\|\nabla(z\pm\nabla \cdot \xi)\|_{L^\infty}dt'<\infty.
\end{align}
\end{lemm} {\noindent Proof.} First combining Lemma \ref{1l} and Lemma \ref{4l} with the Sobolev imbedding theorem, we see that \begin{align}\label{uq} u\in L^\infty_{T^*}(H^1)\hookrightarrow L^\infty_{T^*}(L^p), \end{align} with $2\leq p<\infty.$ Then we denote from the system (\ref{ab}) that \begin{align*}
\nabla a = \nabla e^{t\triangle}a_0-\int_0^te^{(t-t')\triangle}\nabla\big(u\cdot\nabla a+\xi\cdot\nabla a+a(\nabla\cdot\xi)\big)dt'. \end{align*} We have \begin{align*}
&\|(\nabla a)(\tau)\|_{L^{\infty}}\\\lesssim&\| \mathcal{F}^{-1}(e^{-\tau|x|^2}x)\|_{L^1}\|a_0\|_{L^{\infty}}+ \int_0^\tau\|\mathcal{F}^{-1}(e^{(t'-\tau)|x|^2}x)\big\|_{L^{{q'}}}\|u(t')\|_{L^{q}}\| \nabla a(t')\|_{L^{\infty}}dt'\\
&+\int_0^\tau\big\|\mathcal{F}^{-1}(e^{(t'-\tau)|x|^2}x)\big\|_{L^1}\big(\|\xi\|_{L^\infty}\|\nabla a\|_{L^{\infty}}+\|\nabla\cdot\xi\|_{L^\infty}\|a\|_{L^{\infty}}\big)(t')dt'\\\lesssim& \tau^{-\frac{1}{2}}\|a_0\|_{L^{\infty}}+ \int_0^\tau\frac{1}{{(\tau-t')}^{\frac{1}{2}+\frac{1}{q}}}\|u(t')\|_{L^{q}}\| \nabla a(t')\|_{L^{\infty}}dt'\\
&+\int_0^\tau\frac{1}{{(\tau-t')}^{\frac{1}{2}}}\big(\|\xi\|_{L^\infty}\|\nabla a\|_{L^{\infty}}+\|\nabla\cdot\xi\|_{L^\infty}\|a\|_{L^{\infty}}\big)(t')dt'\\
\lesssim&\tau^{-\frac{1}{2}}\| a_0\|_{L^{\infty}}+\tau^{\frac{1}{2}}\|\nabla\cdot\xi\|_{L^\infty_T(L^\infty)}\|a\|_{L^\infty_T(L^{
\infty})}\\&+\int_0^\tau\big(\frac{1}{{(\tau-t')}^{\frac{1}{2}+\frac{1}{q}}}
\|u(t')\|_{L^{q}}+\frac{1}{{(\tau-t')}^{\frac{1}{2}}}\| \xi(t')\|_{L^\infty}\big)\|\nabla a(t')\|_{L^{\infty}}dt'\\
\lesssim&\tau^{-\frac{1}{2}}\| a_0\|_{L^{\infty}}+\tau^{\frac{1}{2}}\|\nabla\cdot\xi\|_{L^\infty_T(L^\infty)}\|a\|_{L^\infty_T(L^{
\infty})}\\&
+\int_0^\tau\big(\delta_1\frac{1}{{(\tau-t')}^{(\frac{1}{2}+\frac{1}{q})\gamma_1}}
+\delta_2\frac{1}{{(\tau-t')}^{\frac{1}{2}\gamma_2}}\big)\|\nabla a(t')\|_{L^{\infty}}dt'\\&+\int_0^\tau\big(C_{\delta_1}
\|u(t')\|^{\gamma'_1}_{L^{q}}+C_{\delta_2}\|\xi\|^{\gamma'_2}_{L^\infty}\big)\|\nabla a(t')\|_{L^{\infty}}dt',
\end{align*}
with $2<q<\infty,$ $\frac{1}{q}+\frac{1}{q'}=1,$
$ (\frac{1}{2}+\frac{1}{q})\gamma_1<1,~~\frac{1}{2}\gamma_2<1. $ By means of the Young inequality for the time integral, we obtain, \begin{align*}
\|\nabla a\|_{L^1_t(L^{\infty})}\lesssim &t^{\frac{1}{2}}\| a_0\|_{L^{\infty}}+t^{\frac{3}{2}}\|\nabla\cdot\xi\|_{L^\infty_T(L^\infty)}\|a\|_{L^\infty_T(L^{\infty})}\\&
+\big(\delta_1\frac{1}{1-(\frac{1}{2}+\frac{1}{q})\gamma_1}t^{1-(\frac{1}{2}+\frac{1}{q})\gamma_1}
+\delta_2\frac{1}{1-\frac{1}{2}\gamma_2}t^{1-\frac{1}{2}\gamma_2}\big)\|\nabla a\|_{L^1_t(L^{a_1})}\\&+\int_0^t\int_0^\tau\big(C_{\delta_1}
\|u(t')\|^{\gamma'_1}_{L^{q}}+C_{\delta_2}\|\xi(t')\|^{\gamma'_2}_{L^\infty}\big)\|\nabla a(t')\|_{L^{\infty}}dt'd\tau. \end{align*} Choosing $\delta_1\frac{1}{1-(\frac{1}{2}+\frac{1}{q})\gamma_1}(T^*)^{1-(\frac{1}{2}+\frac{1}{q})\gamma_1}
+\delta_2\frac{1}{1-\frac{1}{2}\gamma_2}(T^*)^{1-\frac{1}{2}\gamma_2}=c\frac{1}{2}$ yields \begin{align*}
\|\nabla a\|_{L^1_t(L^{\infty})}\lesssim &(T^*)^{\frac{1}{2}}\| a_0\|_{L^{\infty}}+(T^*)^{\frac{3}{2}}\|\nabla\cdot\xi\|_{L^\infty_{T^*}(L^\infty)}\|a\|_{L^\infty_{T^*}(L^{\infty})}\\&
+\big(C_{\delta_1}
\|u(t')\|^{\gamma'_1}_{L^\infty_{T^*}(L^{q})}+C_{\delta_2}\|\xi\|^{\gamma'_2}_{L^\infty_{T^*}(L^\infty)}\big)\int_0^t\int_0^\tau\|\nabla a(t')\|_{L^{\infty}}dt'd\tau.
\end{align*} Gronwall's lemma thus implies that \begin{align*}
\|\nabla a\|_{L^1_t(L^{\infty})}\leq &C\Big((T^*)^{\frac{1}{2}}\| a_0\|_{L^{\infty}}+(T^*)^{\frac{3}{2}}\|\nabla\cdot\xi\|_{L^\infty_{T^*}(L^\infty)}\|a\|_{L^\infty_{T^*}(L^{\infty})}\Big)\\ &\times exp\Big(\big(C_{\delta_1}
\|u(t')\|^{\gamma'_1}_{L^\infty_{T^*}(L^{q})}+C_{\delta_2}\| \xi\|^{\gamma'_2}_{L^\infty_{T^*}(L^\infty)}\big)t\Big).
\end{align*}
Hence, Lemma \ref{2l}, Lemma \ref{3l} and the inequality (\ref{uq}) imply that
\begin{align*}
\int_0^{T^*}\|\nabla a\|_{L^\infty}dt'<\infty.
\end{align*}
Similar arguments for $b$ yield
\begin{align*}
\int_0^{T^*}\|\nabla b\|_{L^\infty}dt'<\infty.
\end{align*}
Therefore, the inequality (\ref{l5}) holds true. \qed
\begin{lemm}\label{6l} $\forall t\in[0,T^*),$ we have \begin{align}\label{l6}
\|u(t)\|_{B^1_{\infty,\infty}}\leq C(T^*)<\infty. \end{align} \end{lemm} {\noindent Proof.} we deduce from the inequality (\ref{lw}) that
\begin{align*}
\|w(t)\|_{L^{\infty}}\lesssim& \|w_0\|_{L^{\infty}}+\int_0^t\|\nabla(\nabla\cdot\xi)\|_{L^\infty}\|\xi\|_{L^\infty}dt'
\lesssim\|w_0\|_{L^{\infty}}+\|\nabla(\nabla\cdot\xi)\|_{L^1_t(L^\infty)}\|\xi\|_{L^{\infty}_t(L^\infty)}.
\end{align*}
By splitting $u$ into low and high frequencies and using Lemma \ref{uw}, we see that
\begin{align*}
\| u\|_{B^{1}_{\infty,\infty}}\lesssim& \|\triangle_{-1} u\|_{L^{\infty}}+\|(Id-\triangle_{-1}) \nabla u\|_{B^{0}_{\infty,\infty}}\\\lesssim&
\|u\|_{L^{2}}+\|w\|_{B^{0}_{\infty,\infty}}\lesssim\|u\|_{L^{2}}+\|w\|_{L^\infty}.
\end{align*} Applying Lemma \ref{1l}, Lemma \ref{3l} and Lemma \ref{5l} completes the proof of the lemma. \qed
\subsubsection{Proof of the global existence} We now turn to the proof of the global existence. Applying $\triangle_j$ to the first equation of the system (\ref{s2}) yields that \begin{align*} (\partial_t+u\cdot \nabla )\triangle_ju+\triangle_j\Pi(u,u)=\triangle_j\mathcal{P}\big((\nabla\cdot\xi)\xi\big)+R_{j1}, \end{align*} with $R_{j1}=u\cdot \nabla\triangle_ju-\triangle_j(u\cdot \nabla)u\triangleq[u\cdot\nabla,\triangle_j]u.$\\ Taking the $L^2$ inner product of the above equation with $\triangle_{j} u$, we easily get \begin{align*}
&\frac{1}{2}\frac{d}{dt}\|\triangle_ju(t)\|_{L^{2}}^2-\frac{1}{2}\int_{\mathbb{R}^2}(\nabla\cdot u)|\triangle_{j}u|^2dx \\ \leq&\|\triangle_{j}u\|_{L^2}\Big(\|\triangle_j\Pi(u,u)\|_{L^{2}}
+\|\triangle_j\mathcal{P}\big((\nabla\cdot\xi)\xi\big)\|_{L^{2}}+\|R_{j1}\|_{L^{2}}\Big),~j\geq-1. \end{align*} Note that $\nabla \cdot u=0$, we get \begin{align*}
\|\triangle_ju(t)\|_{L^{2}}\leq\|\triangle_ju_0\|_{L^{2}}+\int_0^t\|\triangle_j\Pi(u,u)\|_{L^{2}}
+\|\triangle_j\mathcal{P}\big((\nabla\cdot\xi)\xi\big)\|_{L^{2}}+\|R_{j1}\|_{L^{2}}dt',~j\geq-1. \end{align*} Multiplying both sides of the above inequality by $2^{js_1}$, taking the $l^{2}$ norm , we obtain \begin{align}\label{uu'}
\|u\|_{\widetilde{L}^\infty_t(H^{s_1})}\lesssim
\|u_0\|_{H^{s_1}}+\|\Pi(u,u)\|_{\widetilde{L}^1_t(H^{s_1})}+
\|\mathcal{P}\big((\nabla\cdot\xi)\xi\big)\|_{\widetilde{L}^1_t(H^{s_1})}
+\Big\|2^{js_1}\|R_{j1}\|_{L^1_t(L^{2})}\Big\|_{l^{2}}. \end{align} Due to Lemma \ref{jiaohuan}, we get \begin{align}\label{RR}
\Big\|2^{js_1}\|R_{j1}\|_{L^1_t(L^{2})}\Big\|_{l^{2}}\lesssim \int_0^t\Big\|2^{js_1}\|R_{j1}\|_{L^{2}}\Big\|_{l^{2}}dt'\lesssim \int_0^t \|\nabla u\|_{L^\infty}\|u\|_{H^{s_1}}dt'. \end{align} By virtue of Lemma \ref{pi}, we have \begin{align}\label{pipi}
\|\Pi(u,u)\|_{\widetilde{L}^1_t(H^{s_1})}\lesssim \|\Pi(u,u)\|_{L^1_t(H^{s_1})}\lesssim \int_0^t \|u\|_{C^{0,1}}\|u\|_{H^{s_1}}dt'. \end{align} We now focus on the term $\mathcal{P}\big((\nabla\cdot\xi)\xi\big).$ By taking advantage of Bony's decomposition and of Lemmas \ref{T}-\ref{R}, we have \begin{align}\label{phiphi}
\|\mathcal{P}\big((\nabla\cdot\xi)\xi\big)\|_{\widetilde{L}^1_t(H^{s_1})}\lesssim&
\|(\nabla\cdot\xi)\xi\|_{\widetilde{L}^1_t(H^{s_1})}\\\nonumber \lesssim&
\|\nabla\cdot\xi\|_{L^\infty_t(L^\infty)}\|\xi\|_{\widetilde{L}^1_t(H^{s_1})}
+\|\xi\|_{L^\infty_t(L^\infty)}\|\nabla\cdot\xi\|_{\widetilde{L}^1_t(H^{s_1})}\\ \nonumber \lesssim&\|\nabla\cdot\xi\|_{L^\infty_t(L^\infty)}\|\xi\|_{\widetilde{L}^1_t(H^{s_2+\frac{3}{2}})}
+\|\xi\|_{L^\infty_t(L^\infty)}
\|\nabla\cdot\xi\|_{\widetilde{L}^1_t(H^{s_2+\frac{3}{2}})}\\\nonumber \lesssim&\|\nabla\cdot\xi\|_{L^\infty_t(L^\infty)}
\|\xi\|_{\widetilde{L}^1_t(H^{s_2+1})}^\frac{3}{4}
\|\xi\|_{\widetilde{L}^1_t(H^{s_2+3})}^\frac{1}{4}
+\|\xi\|_{L^\infty_t(L^\infty)}
\|\xi\|_{\widetilde{L}^1_t(H^{s_2+1})}^\frac{1}{4}
\|\xi\|_{\widetilde{L}^1_t(H^{s_2+3})}^\frac{3}{4}\\\nonumber
\lesssim&C_{\sigma}\big(\|\nabla\cdot\xi\|_{L^\infty_t(L^\infty)}^\frac{4}{3}
+\|\xi\|_{L^\infty_t(L^\infty_t)}^4\big)
\int_0^t\|\xi\|_{H^{s_2+1}}dt'+
\sigma\|\xi\|_{\widetilde{L}^1_t(H^{s_2+3})}.
\end{align}
Plugging the inequalities (\ref{RR})-(\ref{phiphi}) into (\ref{uu'}), we eventually get
\begin{align}\label{345}
\|u\|_{\widetilde{L}^\infty_t(H^{s_1})}\lesssim &\|u_0\|_{H^{s_1}}+\int_0^t\Big(\|u\|_{C^{0,1}}
\|u\|_{H^{s_1}}\\\nonumber&+C_{\sigma}\big(\|\nabla\cdot\xi\|_{L^\infty_t(L^\infty)}^\frac{4}{3}
+\|\xi\|_{L^\infty_t(L^\infty_t)}^4\big)
\|\xi\|_{H^{s_2+1}}\Big)dt'+\sigma
\|\xi\|_{\widetilde{L}^1_t(H^{s_2+3})}.
\end{align}
Similarly, applying $\triangle_j$ to the second equation of the system (\ref{s2}) yields that \begin{align*} (\partial_t+u\cdot \nabla -\triangle) \triangle_jz=-\triangle_j\nabla\cdot\big((\nabla\cdot\xi)\xi\big)+R_{j2}, \end{align*} with $R_{j2}=[u\cdot\nabla,\triangle_j]z,$ where we have used $\nabla \cdot (uz)=u \cdot \nabla z+(\nabla \cdot u)z=u\cdot \nabla z.$\\ Taking the $L^2$ inner product of the above equation with $\triangle_{j} z$, we get \begin{align*}
&\frac{1}{2}\frac{d}{dt}\|\triangle_jz(t)\|_{L^{2}}^2-\frac{1}{2}\int_{\mathbb{R}^2}(\nabla\cdot u)|\triangle_{j}z|^2dx+\|\nabla\triangle_{j}z\|_{L^2}^2 \\ \leq&\|\triangle_{j}z\|_{L^2}\Big(\|\triangle_j\nabla\cdot\big((\nabla\cdot\xi)\xi\big)\|_{L^{2}}
+\|R_{j2}\|_{L^{2}}\Big),~j\geq-1. \end{align*}
Note that $\nabla\cdot u=0,$ $\|\nabla\triangle_{-1}z\|_{L^2}\geq 0,$ and by virtue of Lemma \ref{Bi}, $\|\nabla\triangle_{j}z\|_{L^2}\gtrsim 2^{j}\|\triangle_{j}z\|_{L^2},$ for $j\geq 0.$ Therefore, we have \begin{align}\label{xingxing}
\|\triangle_jz(t)\|_{L^{2}}+\int_0^t2^{2j}\|\triangle_jz\|_{L^{2}}dt'\lesssim (1+t)\big(&\|\triangle_jz_0\|_{L^{2}}+\int_0^t
\|\triangle_j\nabla\cdot\big((\nabla\cdot\xi)\xi\big)\|_{L^{2}}
\\\nonumber+&\|R_{j2}\|_{L^2}dt'\big),~j\geq-1. \end{align} Hence multiplying both sides of the above inequality by $2^{js_2}$ and taking the $l^{2}$ norm, we obtain \begin{align*}
&\|z\|_{\widetilde{L}^\infty_t(H^{s_2})}+ \|z\|_{\widetilde{L}^1_t(H^{s_2+2})}\\\lesssim& (1+t)\big(\|z_0\|_{H^{s_2}}+
\|\nabla\cdot\big((\nabla\cdot\xi)\xi\big)\|_{\widetilde{L}^1_t(H^{s_2})}
+\Big\|2^{js_2}\|R_{j2}\|_{L^1_t(L^{2})}\Big\|_{l^{2}}\big). \end{align*} In view of Lemma \ref{jiaohuan}, we get \begin{align}\label{R2}
\Big\|2^{js_2}\|R_{j2}\|_{L^1_t(L^{2})}\Big\|_{l^{2}}\lesssim
&\int_0^t\Big\|2^{js_2}\|R_{j2}\|_{L^{2}}\Big\|_{l^{2}}dt')\\\nonumber\lesssim &\int_0^t\Big(\|\nabla u\|_{L^\infty}\|z\|_{H^{s_2}}+\|\nabla z\|_{L^\infty}\|\nabla u\|_{H^{s_2-1}}\Big)dt'\\\nonumber
\lesssim &\int_0^t\Big(\|\nabla u\|_{L^\infty}\|z\|_{H^{s_2}}+\|\nabla z\|_{L^\infty}\| u\|_{H^{s_1}}\Big)dt'. \end{align} According to Lemmas \ref{s_2+1}, we have \begin{align}\label{n2}
&\|\nabla\cdot\big((\nabla\cdot\xi)\xi\big)\|_{\widetilde{L}^1_t(H^{s_2})}
\lesssim\|(\nabla\cdot\xi)\xi\|_{\widetilde{L}^1_t(H^{s_2+1})}\\\nonumber
\lesssim&\|\nabla\cdot\xi\|_{L^\infty_t(L^\infty)}\|\xi\|_{\widetilde{L}^1_t(H^{s_2+1})}
+\|\xi\|_{L^\infty_t(L^\infty)}\|\nabla\cdot\xi\|_{\widetilde{L}^1_t(H^{s_2+1})},\\\nonumber
\lesssim&\|\nabla\cdot\xi\|_{L^\infty_t(L^\infty)}\int_0^t\|\xi\|_{H^{s_2+1}}dt'
+C_{\sigma}\|\xi\|_{L^\infty_t(L^\infty)}^2\int_0^t\|\xi\|_{H^{s_2+1}}dt'+\sigma\|\xi\|_{\widetilde{L}^1_t(H^{s_2+3})} . \end{align} Inserting the inequalities (\ref{R2})-(\ref{n2}) into (\ref{xingxing}), we finally get \begin{align}\label{567}
&\|z\|_{\widetilde{L}^\infty_t(H^{s_2})}+\|z\|_{\widetilde{L}^1_t(H^{s_2+2})}\\\nonumber
\lesssim&(1+t)\Big(\|z_0\|_{H^{s_2}}+\int_0^t\Big(\|\nabla u\|_{L^\infty}\|z\|_{H^{s_2}}+\|\nabla z\|_{L^\infty}\| u\|_{H^{s_1}}\\\nonumber&+\big(\|\nabla\cdot\xi\|_{L^\infty_t(L^\infty)}
+C_{\sigma}\|\xi\|_{L^\infty_t(L^\infty)}^2\big)\|\xi\|_{H^{s_2+1}}\Big)dt'+\sigma\|\xi\|_{\widetilde{L}^1_t(H^{s_2+3})}
\end{align}
To deal with the third equation of the system (\ref{s2}), we have \begin{align}\label{xi2}
&\|\mathcal{L}\big(u(\nabla\cdot\xi)\big)\|_{\widetilde{L}^1_t(H^{s_2+1})}\lesssim
\|u(\nabla\cdot\xi)\|_{\widetilde{L}^1_t(H^{s_2+1})}\\\nonumber
\lesssim&\|\nabla\cdot\xi\|_{L^\infty_t(L^\infty)}
\|u\|_{\widetilde{L}^1_t(H^{s_2+1})}+
\|u\|_{\widetilde{L}^\infty_t(L^q)}\|\nabla\cdot\xi\|_{\widetilde{L}^1_t(B^{s_2+1}_{\frac{2q}{q-2},2})}
\\\nonumber
\lesssim&\|\nabla\cdot\xi\|_{L^\infty_t(L^\infty)}
\|u\|_{\widetilde{L}^1_t(H^{s_1})}+
\|u\|_{L^\infty_t(L^q)}
\|\nabla\cdot\xi\|_{\widetilde{L}^1_t(H^{s_2+1+\frac{2}{q}})}
\\\nonumber
\lesssim&\|\nabla\cdot\xi\|_{L^\infty_t(L^\infty)}
\int_0^t\|u\|_{H^{s_1}}dt'+
C_{\sigma}\|u\|_{L^\infty_t(L^q)}^{\frac{2q}{q-2}}
\int_0^t\|\xi\|_{H^{s_2+1}}dt'+\sigma\|\xi\|_{\widetilde{L}^1_t(H^{s_2+3})}, \end{align} with $2\leq q < \infty.$ \begin{align}\label{xi3}
&\|\mathcal{L}(z\xi)\|_{\widetilde{L}^1_t(H^{s_2+1})}\lesssim
\|z\xi\|_{\widetilde{L}^1_t(H^{s_2+1})}\\\nonumber
\lesssim&\|\xi\|_{L^\infty_t(L^\infty)}
\|z\|_{\widetilde{L}^1_t(H^{s_2+1})}+
\|z\|_{L^\infty_t(L^\infty)}\|\xi\|_{\widetilde{L}^1_t(H^{s_2+1})}
\\\nonumber
\lesssim&C_{\sigma}\|\xi\|_{L^\infty_t(L^\infty)}^2
\int_0^t\|z\|_{H^{s_2}}dt'+\sigma\|z\|_{\widetilde{L}^1_t(H^{s_2+2})} +
\|z\|_{L^\infty_t(L^\infty)}\int_0^t\|\xi\|_{H^{s_2+1}}dt'. \end{align} Hence,
\begin{align}\label{568}
&\|\xi\|_{\widetilde{L}^\infty_t(H^{s_2+1})}+\|\xi\|_{\widetilde{L}^1_t(H^{s_2+3})}\\\nonumber
\lesssim&(1+t)\Big(\|\xi_0\|_{H^{s_2+1}}
+\|\mathcal{L}\big(u(\nabla\cdot\xi)\big)\|_{\widetilde{L}^1_t(H^{s_2+1})}
+\|\mathcal{L}(z\xi)\|_{\widetilde{L}^1_t(H^{s_2+1})}\Big)
\\\nonumber
\lesssim&(1+t)\Big(\|\xi_0\|_{H^{s_2+1}}+\big(\|\nabla\cdot\xi\|_{L^\infty_t(L^\infty)}
+
C_{\sigma}\|u\|_{L^\infty_t(L^q)}^{\frac{2q}{q-2}}+C_{\sigma}\|\xi\|_{L^\infty_t(L^\infty)}^2
+\|z\|_{L^\infty_t(L^\infty)}\big)\\\nonumber&\times\int_0^t\big(\|u\|_{H^{s_1}}+\|\xi\|_{H^{s_2+1}}+\|z\|_{H^{s_2}}\big)dt'
+\sigma\big(\|\xi\|_{\widetilde{L}^1_t(H^{s_2+3})}+\|z\|_{\widetilde{L}^1_t(H^{s_2+2})}\big) \Big).
\end{align}
Combining (\ref{567}), (\ref{568}) and (\ref{345}), we get $\forall t\in[0,T^*),$
\begin{align}
&\|u\|_{\widetilde{L}^\infty_t(H^{s_1})}+\|z\|_{\widetilde{L}^\infty_t(H^{s_2})\cap \widetilde{L}^1_t(H^{s_2+2})}+\|\xi\|_{\widetilde{L}^\infty_t(H^{s_2+1})\cap \widetilde{L}^1_t(H^{s_2+3})}\\
\nonumber\lesssim
&(1+t)\big(\|u_0\|_{H^{s_1}}+\|z_0\|_{H^{s_2}}+\|\xi_0\|_{H^{s_2+1}}\big)+(1+t)\int_0^t\Big(
\|u\|_{C^{0,1}}+\| \nabla z\|_{L^\infty}+\\
\nonumber&C_{\sigma}\big(\|\nabla\cdot\xi\|_{L^\infty_t(L^\infty)}^\frac{4}{3}
+\|\xi\|_{L^\infty_t(L^\infty_t)}^4+\|\nabla\cdot\xi\|_{L^\infty_t(L^\infty)}
+\|\xi\|_{L^\infty_t(L^\infty)}^2+\|\nabla\cdot\xi\|_{L^\infty_t(L^\infty)}
\\\nonumber&+
\|u\|_{L^\infty(L^q)}^{\frac{2q}{q-2}}+\|\xi\|_{L^\infty_t(L^\infty)}^2
+\|z\|_{L^\infty_t(L^\infty)}\big)\Big)
\times\big(\|u\|_{H^{s_1}}+\|z\|_{H^{s_2}}+\|\xi\|_{H^{s_2+1}}\big)dt'
\\\nonumber&+(1+t)\sigma\big(\|z\|_{\widetilde{L}^1_t(H^{s_2+2})}+\|\xi\|_{\widetilde{L}^1_t(H^{s_2+3})}\big).
\end{align}
Choose $\sigma=c(1+T^*)^{-1}.$
Lemmas \ref{1l}-\ref{6l} and the inequality (\ref{uq}) imply that
\begin{align*}
&C_{\sigma}\big(\|\nabla\cdot\xi\|_{L^\infty_t(L^\infty)}^\frac{4}{3}
+\|\xi\|_{L^\infty_t(L^\infty_t)}^4+\|\nabla\cdot\xi\|_{L^\infty_t(L^\infty)}
+\|\xi\|_{L^\infty_t(L^\infty)}^2+\|\nabla\cdot\xi\|_{L^\infty_t(L^\infty)}
\\\nonumber&+
\|u\|_{L^\infty(L^q)}^{\frac{2q}{q-2}}+\|\xi\|_{L^\infty_t(L^\infty)}^2
+\|z\|_{L^\infty_t(L^\infty)}\big)\leq C(T^*)<\infty,
\end{align*}
from which it follows that
\begin{align}
&\|u\|_{\widetilde{L}^\infty_t(H^{s_1})}+\|z\|_{\widetilde{L}^\infty_t(H^{s_2})}
+\|\xi\|_{\widetilde{L}^\infty_t(H^{s_2+1})}\\
\nonumber\leq
&C(T^*)\Big(\big(\|u_0\|_{H^{s_1}}+\|z_0\|_{H^{s_2}}+\|\xi_0\|_{H^{s_2+1}}\big)+\int_0^t\big(
\|u\|_{C^{0,1}}+\| \nabla z\|_{L^\infty}+1)\\\nonumber&~~~~~~
\times\big(\|u\|_{\widetilde{L}^\infty_{t'}(H^{s_1})}+\|z\|_{\widetilde{L}^\infty_{t'}(H^{s_2})}
+\|\xi\|_{\widetilde{L}^\infty_{t'}(H^{s_2+1})}\big)dt'\Big)\\\nonumber\triangleq &B(t).
\end{align} Denote
\begin{align*}
&\|u_0\|_{H^{s_1}}+\|z_0\|_{H^{s_2}}+
\|\xi_0\|_{H^{s_2+1} }\triangleq A_0,\\
&\|u\|_{\widetilde{L}^\infty_T(H^{s_1})}+\|z\|_{\widetilde{L}^\infty_T(H^{s_2})}+
\|\xi\|_{\widetilde{L}^\infty_T(H^{s_2+1}) }\triangleq A(t),\end{align*} Let $\epsilon =\min(1,s_1-2)$ and $\Gamma(r)=1+\log r:[1,\infty)\rightarrow[0,\infty)$ be the function associated with the modulus of continuity $\mu(r) = r(1-\log r).$ We can extend the domain of definition of $\Gamma$ to $[0,\infty)$ with $\Gamma(s)=\Gamma(1)=1,$ for $0\leq s<1.$
The function $G(y) \overset{def} {=}\int_1^y\frac{dy'}{\Gamma(y'^{\frac{1}{\varepsilon}})y'}=\varepsilon\log(1+\frac{1}{\varepsilon}\log y)$ then maps $[1,+\infty)$ onto and one-to-one $[0,+\infty).$\\ Assuming that $A_0>0,$ otherwise $(0,0,0)$ is the global solution. Using Lemma \ref{LL} with $\Lambda = A_0$, we get \begin{align*}
B(t)&\leq C(T^*)\Big(A_0+\int_0^t(\|u\|_{L^\infty}+\|\nabla u\|_{L^\infty}+\| \nabla z\|_{L^\infty}+1)B(t')dt'\Big) \\ &\leq C(T^*)\Big(A_0+\int_0^t\Big\{\|u\|_{B^1_{\infty,\infty}}+\| \nabla z\|_{L^\infty}+1+C_{\epsilon}\Big(\|u\|_{C_{\mu}}+A_0\Big)\Big(1+
\Gamma\Big(\big(\frac{\|\nabla u\|_{C^{0,\epsilon}}}{\|u\|_{C_{\mu}}+A_0}\big)
^{\frac{1}{\epsilon}}\Big)\Big)\Big\}B(t')dt'\Big)\\ &\leq C(\epsilon,T^*)\Big(A_0+\int_0^t\Big\{\big(\|u\|_{B^1_{\infty,\infty}}+\| \nabla z\|_{L^\infty}+1+A_0\big)\Big(1+
\Gamma\Big(\big(\frac{C\| u(t')\|_{H^{s_1}}}{A_0}\big)
^{\frac{1}{\epsilon}}\Big)\Big)\Big\}B(t')dt'\Big)\\ &\leq C(\epsilon,T^*)\Big(A_0+\int_0^t\Big(\|u\|_{B^1_{\infty,\infty}}+\| \nabla z\|_{L^\infty}+1+A_0\Big)
\Gamma\Big(\big(\frac{CB(t')}{A_0}\big)
^{\frac{1}{\epsilon}}\Big)\Big)B(t')dt'\Big)\\
&\triangleq\frac{R(t)A_0}{C}, \end{align*}
where we have used $B^1_{\infty,\infty}\hookrightarrow L^\infty,$ $B^1_{\infty,\infty}\hookrightarrow C_{\mu},$ $H^{s_1}\hookrightarrow C^{0,\epsilon},$ $\| u(t')\|_{H^{s_1}}\leq B(t')$ and $C$ has been chosen large enough such that $R(t)=\frac{B(t)C}{A_0}\geq C>1.$\\
Because the function $\Gamma$ is nondecreasing, after a few computations, we have that \begin{align*}
\frac{d}{dt}R(t)
\leq&\Gamma\big(R(t)
^{\frac{1}{\epsilon}}\big)R(t) C(\epsilon,T^*)\big(\|u(t)\|_{B^1_{\infty,\infty}}+\| \nabla z(t)\|_{L^\infty}+1+A_0\big),
\end{align*}
thus $$\frac{d}{dt}G(R(t))\leq C(\epsilon,T^*)(\|u(t)\|_{B^1_{\infty,\infty}}+A_0+1+\|\nabla z(t)\|_{L^\infty}).$$
Integrating then gives
\begin{align*}
R(t)\leq G^{-1}\Big(G\big(R(0)\big)+\int_0^tC(\epsilon,T^*)(\|u\|_{B^1_{\infty,\infty}}+A_0+1+\|\nabla z\|_{L^\infty})dt'\Big)<\infty,
\end{align*}
where we have used Lemmas \ref{5l}-\ref{6l}. Therefore, $
\|u(t)\|_{H^{s_1}},$ $\|z(t)\|_{H^{s_2}},$ and $
\|\xi(t)\|_{H^{s_2+1} }$ stay bounded on $[0,T^*)$. The local existence part of
Theorem \ref{a1} then enables us to extend the solution beyond $T^*,$ which stands in contradiction to the definition of $T^*.$ Hence $T^*=+\infty.$
This completes the proof of the theorem.\qed
\section{Proof of Theorem \ref{a3}} To begin, we denote by $BMO$ the space of functions of bounded mean oscillations. It is well known that $BMO$ strictly includes $L^\infty$. We introduce the following Hardy-Littlewood-Sobolev inequality. \begin{lemm}\cite{Lemarié-Rieusset}\label{Lemarié-Rieusset} For $0<\gamma<d,$ the operator $(-\triangle)^{\frac{\gamma}{2}}$ is bounded from the Hardy space $\mathcal{H}^1$ to $L^{\frac{d}{d-\gamma}}$ and from $L^{\frac{d}{\gamma}}$ to $BMO.$ \end{lemm}
\subsection{ Global existence for the $ENPP$ system} Let $(u_0,z_0,\xi_0)=\big(u_0,n_0+p_0,-\nabla(\triangle)^{-1}(n_0-p_0)\big).$ According to Theorem \ref{a1}, there exists a global solution $(u,z,\xi)$ satisfies the system (\ref{s2}) in the spaces defined as in Theorem \ref{a1}. Since $\mathcal{L}\xi=\xi,$ it is then easy to that $(u,n,p)=(u,\frac{z+\nabla\cdot \xi}{2},\frac{z-\nabla\cdot \xi}{2})$ solves the system (\ref{s3}).\\ Denote $$\phi_0\triangleq-(-\triangle)^{-1}\nabla \cdot \xi.$$ As $\xi\in L^\infty(\mathbb{R}^+;L^2),$ applying Lemma \ref{Lemarié-Rieusset} with $d=2$ and $\gamma=1$ implies that $\phi_0\in L^\infty(\mathbb{R}^+;BMO).$ Thanks again to the fact that $\mathcal{L}\xi=\xi,$ we have $\nabla \phi_0=\mathcal{L}\xi=\xi,$ and $\triangle \phi_0=\nabla \cdot \xi=n-p.$ Similarly, let $$P_0\triangleq P_{\pi}(u,u)-(-\triangle)^{-1}\nabla \cdot\big((\nabla \cdot \xi)\xi\big),$$
where $P_{\pi}(u,u)\in L^\infty(\mathbb{R}^+;H^{s_1+1})$ is defined as in Lemma \ref{pi}.
Note that $\xi\in L^\infty(\mathbb{R}^+;H^{s_1+1})$ with $s_1>1$ implies that $\nabla \cdot \xi\in L^\infty(\mathbb{R}^+;L^2)$ and $\xi\in L^\infty(\mathbb{R}^+;L^\infty)$. Again using lemma \ref{Lemarié-Rieusset}, we get
$$P_0\in L^\infty(\mathbb{R}^+;H^{s_1+1}+BMO)\hookrightarrow L^\infty(\mathbb{R}^+;L^\infty+BMO)\hookrightarrow L^\infty(\mathbb{R}^+;BMO).$$
Finally, it is easy to see that $(u,\frac{z+\nabla\cdot \xi}{2},\frac{z-\nabla\cdot \xi}{2},P_0,\phi_0)$ satisfies the $ENPP$ system.
\subsection{ Uniqueness for the $ENPP$ system}
Suppose that there exists a global solution $(u,n,p,P,\phi)$ satisfing the $ENPP$ system in the spaces defined as in Theorem \ref{a3}. We first show that $$\nabla \Phi=-\nabla(-\triangle)^{-1}(n-p)\triangleq\xi,~\textit{and} ~\nabla P=\pi(u,u)+(I-\mathcal{P})\big((n-p)\nabla(-\triangle)^{-1}(p-n)\big).$$
In fact, Let $\phi_0,~P_0$ be defined as in the above subsection. As $\triangle \phi=n-p=\triangle \phi_0,$ hence $\phi-\phi_{0}$ is a harmonic polynomial. Note that $\phi \in L^\infty(\mathbb{R}^+;BMO)$ is required in Theorem \ref{a3} and $\phi_0 \in L^\infty(\mathbb{R}^+;BMO)$ is illustrated before. Thus $\phi-\phi_0$ depends only on t, and
\begin{align}\label{pphi}
\nabla \phi=\nabla\phi_{0}=\xi=-\nabla(-\triangle)^{-1}(n-p).
\end{align}
Next applying the operator $\nabla\cdot$ to the first equation of the $ENPP$ system, we get
\begin{align*}
-\triangle P=\nabla\cdot(u\cdot\nabla u)-\nabla\cdot((\nabla \cdot \xi)\xi)
=-\triangle P_0.
\end{align*}
Note that $P-P_0$ is in
$L^\infty(\mathbb{R}^+;BMO).$ Similar arguments as that for $\phi-\phi_0$ yield that
\begin{align*}
\nabla P=\nabla P_0
=\Pi(u,u)-\nabla(-\triangle)^{-1}\nabla \cdot\big((\nabla \cdot \xi)\xi\big)=\Pi(u,u)+(I-\mathcal{P})\big((n-p)\nabla(-\triangle)^{-1}(p-n)\big).
\end{align*} Next it is easy to see that $(u,n,p,\xi)$ solves the system (\ref{s3}), and $(u,n+p,\xi)$ solves the system (\ref{s2}). The uniqueness of the system (\ref{s2}) in Theorem \ref{a1} then implies that $(u,n,p,\nabla P,\nabla \phi)$ is uniquely determined by the initial data. This completes the proof of the theorem. \qed\\
\noindent\textbf{Acknowledgements}. This work was partially supported by NNSFC (No. 11271382), RFDP (No. 20120171110014), and the key project of Sun Yat-sen University. \phantomsection \addcontentsline{toc}{section}{\refname}
\end{document} | arXiv |
Journal of Electrical Engineering and Technology
The Korean Institute of Electrical Engineers (대한전기학회)
Journal of Electrical Engineering and Technology (JEET), which is the official publication of the Korean Institute of Electrical Engineers (KIEE) being published bimonthly, released the first issue in March 2006.The journal is open to submission from scholars and experts in the wide areas of electrical engineering technologies. The scope of the journal includes all issues in the field of Electrical Engineering and Technology. Included are techniques for electrical power engineering, electrical machinery and energy conversion systems, electrophysics and applications, information and controls.Papers based on novel methodologies and implementations, creative and innovative electrical engineering associated with the four scopes are particularly welcome but not restricted to the above topics. The JEET publishes basically in conformity with publication ethics codes based on the COPE(committee on publication ethics: http://publicationethics.org/). Additionally, the JEET publication complies strictly with the general research ethics codes of the KIEE(http://www.kiee.or.kr). Reviews and tutorial articles on contemporary subjects are strongly encouraged. All papers are to be reviewed by at least three independent reviewers and authors of all accepted papers would be required to complete a copyright from transferring all rights to the KIEE. For more detailed information about manuscript preparation, please visit the web site of the KIEE at http://www.kiee.or.kr or contact the secretariat ofJEET.
http://home.jeet.or.kr/ KSCI KCI SCOPUS SCIE
Determination of Reactive Power Compensation Considering Large Disturbances for Power Flow Solvability in the Korean Power System
Seo, Sang-Soo;Kang, Sang-Gyun;Lee, Byong-Jun;Kim, Tae-Kyun;Song, Hwa-Chang 147
https://doi.org/10.5370/JEET.2011.6.2.147 PDF KSCI
This paper proposes a methodology using a tool based on the branch-parameter continuation power flow (BCPF) in order to restore the power flow solvability in unsolvable contingencies. A specified contingency from a set of transmission line contingencies is modeled, considering the transient analysis and practice in the Korean power system. This tool traces a solution path that satisfies the power flow equations with respect to the variation of the branch parameter. At a critical point, in which the branch parameter can move on to a maximum value, a sensitivity analysis with a normal vector is performed to identify the most effective compensation. With the sensitivity information, the location of the reactive power compensation is determined and the effectiveness of the sensitivity information is verified to restore the solvability. In the simulation, the proposed framework is then applied to the Korean power system.
Fundamental Frequency Estimation in Power Systems Using Complex Prony Analysis
Nam, Soon-Ryul;Lee, Dong-Gyu;Kang, Sang-Hee;Ahn, Seon-Ju;Choi, Joon-Ho 154
A new algorithm for estimating the fundamental frequency of power system signals is presented. The proposed algorithm consists of two stages: orthogonal decomposition and a complex Prony analysis. First, the input signal is decomposed into two orthogonal components using cosine and sine filters, and a variable window is adapted to enhance the performance of eliminating harmonics. Then a complex Prony analysis that is proposed in this paper is used to estimate the fundamental frequency by approximating the cosine-filtered and sine-filtered signals simultaneously. To evaluate the performance of the algorithm, amplitude modulation and harmonic tests were performed using simulated test signals. The performance of the algorithm was also assessed for dynamic conditions on a single-machine power system. The Electromagnetic Transients Program was used to generate voltage signals for a load increase and single phase-to-ground faults. The performance evaluation showed that the proposed algorithm accurately estimated the fundamental frequency of power system signals in the presence of amplitude modulation and harmonics.
Determination of Critical Generator Group Using Accelerating Power and Synchronizing Power Coefficient in the Transient Energy Function Method
Chun, Yeong-Han 161
This paper proposes an algorithm for determining critical generator lists using accelerating power and synchronizing power coefficient (SPC), and critical generator group (CGG) from CGG candidates, which is a combination of critical generators. The accurate determination of CGG provides a more accurate energy margin while providing system operator with information of possible unstable generator group. Classical transient energy function (TEF) method selects the critical generators with big corrected kinetic energy of each generator at the moment of fault removal. However, the generator with small acceleration after fault, that is, the generator with small corrected kinetic energy, is also likely to belong to CGG if the generator has small synchronizing power. The proposed algorithm has been verified to be effective compared with the classical TEF method. We utilized the power system of Korean Electric Power Corporation(KEPCO) as a test system.
Development of an Impedance Locus Model for a Protective Relay Dynamic Test with a Digital Simulator
Kim, Soo-Nam;Lee, Myoung-Soo;Lee, Jae-Gyu;Rhee, Sang-Bong;Kim, Kyu-Ho 167
This paper presents a method for the development of the impedance locus to test the dynamic characteristics of protective relays. Specifically, using the proposed method, the impedance locus can comprise three impedance points, and the speed of impedance trajectory can be adjusted by frequency deviation. This paper is divided into two main sections. The first section deals with the configuration of impedance locus with voltage magnitude, total impedance magnitude, and impedance angle. The second section discusses the control of the locus speed with the means of the deviation between two frequencies. The proposed method is applied to two machine equivalent systems with offline simulation (i.e., PSCAD) and real-time simulation (i.e., real-time simulation environment) to demonstrate its effectiveness.
A Novel Binary Ant Colony Optimization: Application to the Unit Commitment Problem of Power Systems
Jang, Se-Hwan;Roh, Jae-Hyung;Kim, Wook;Sherpa, Tenzi;Kim, Jin-Ho;Park, Jong-Bae 174
This paper proposes a novel binary ant colony optimization (NBACO) method. The proposed NBACO is based on the concept and principles of ant colony optimization (ACO), and developed to solve the binary and combinatorial optimization problems. The concept of conventional ACO is similar to Heuristic Dynamic Programming. Thereby ACO has the merit that it can consider all possible solution sets, but also has the demerit that it may need a big memory space and a long execution time to solve a large problem. To reduce this demerit, the NBACO adopts the state probability matrix and the pheromone intensity matrix. And the NBACO presents new updating rule for local and global search. The proposed NBACO is applied to test power systems of up to 100-unit along with 24-hour load demands.
Comparison of Particle Swarm Optimization and the Genetic Algorithm in the Improvement of Power System Stability by an SSSC-based Controller
Peyvandi, M.;Zafarani, M.;Nasr, E. 182
Genetic algorithms (GA) and particle swarm optimization (PSO) are the most famous optimization techniques among various modern heuristic optimization techniques. These two approaches identify the solution to a given objective function, but they employ different strategies and computational effort; therefore, a comparison of their performance is needed. This paper presents the application and performance comparison of the PSO and GA optimization techniques for a static synchronous series compensator-based controller design. The design objective is to enhance power system stability. The design problem of the FACTS-based controller is formulated as an optimization problem, and both PSO and GA optimization techniques are employed to search for the optimal controller parameters.
Optimal Algorithms for Voltage Management in Distribution Systems Interconnected with New Dispersed Sources
Rho, Dae-Seok;Kook, Kyung-Soo;Wang, Yong-Peel 192
The optimal evaluation algorithms for voltage regulation in the case where new dispersed sources are operated in distribution systems are studied. Handling the interconnection issues for proper voltage managements are often difficult and complicated because professional skills and enormous amounts of data during evaluations are needed. Typical evaluation algorithms mainly depend on human ability and quality of data acquired, which inevitably cause the different results for the same issue. Thus, unfair and subjective evaluations are unavoidable. In order to overcome these problems, we propose reasonable and general algorithms based on the standard model system and proper criterion, which offers fair and objective evaluation in any case. The proposed algorithms are divided into two main themes. One is an optimal algorithm for the voltage control of multiple voltage regulators in order to deliver suitable voltage to as many customers as possible, and the other is a proper evaluation algorithm for the voltage management at normal and emergency conditions. Results from a case study show that proposed methods can be a practical tool for the voltage management in distribution systems including dispersed sources.
Hybrid Optimization Strategy using Response Surface Methodology and Genetic Algorithm for reducing Cogging Torque of SPM
Kim, Min-Jae;Lim, Jae-Won;Seo, Jang-Ho;Jung, Hyun-Kyo 202
Numerous methodologies have been developed in an effort to reduce cogging torque. However, most of these methodologies have side effects that limit their applications. One approach is the optimization methodology that determines an optimized design variable within confined conditions. The response surface methodology (RSM) and the genetic algorithm (GA) are powerful instruments for such optimizations and are matters of common interest. However, they have some weaknesses. Generally, the RSM cannot accurately describe an object function, whereas the GA is time consuming. The current paper describes a novel GA and RSM hybrid algorithm that overcomes these limitations. The validity of the proposed algorithm was verified by three test functions. Its application was performed on a surface-mounted permanent magnet.
Characteristics Analysis of Suspending Force for Hybrid Stator Bearingless SRM
Ahn, Jin-Woo;Lee, Dong-Hee 208
In this paper, a characteristics analysis and calculation of the suspending force of a novel bearingless switched reluctance motor (BLSRM) with hybrid stator poles is proposed. The operating principle and permeance are calculated to find an appropriate control scheme for a proposed motor. Furthermore, a mathematical model for suspending force is derived. Finite element analysis is also employed to compare with the expressions for suspending force. Finally, the validity of the structure and the mathematical model is verified by simulation results.
PFC Bridge Converter for Voltage-controlled Adjustable-speed PMBLDCM Drive
Singh, Sanjeev;Singh, Bhim 215
In this paper, a buck DC-DC bridge converter is used as a power factor correction (PFC) converter for feeding a voltage source inverter (VSI) based permanent magnet brushless DC motor (PMBLDCM) drive. The front end of the PFC converter is a diode bridge rectifier (DBR) fed from single phase AC mains. The PMBLDCM is used to drive the compressor of an air conditioner through a three-phase voltage source inverter (VSI) fed from a variable voltage DC link. The speed of the air conditioner is controlled to conserve energy using a new concept of voltage control at a DC link proportional to the desired speed of the PMBLDC motor. Therefore, VSI operates only as an electronic commutator of the PMBLDCM. The current of the PMBLDCM is controlled by setting the reference voltage at the DC link as a ramp. The proposed PMBLDCM drive with voltage control-based PFC converter was designed and modeled. The performance is simulated in Matlab-Simulink environment for an air conditioner compressor load driven through a 3.75 kW, 1500 rpm PMBLDC motor. To validate the effectiveness of the proposed speed control scheme, the evaluation results demonstrate improved efficiency of the complete drive with the PFC feature in a wide range of speed and input AC voltage.
Robust Control of Induction Motor with H∞Theory based on Loopshaping
Benderradji, Hadda;Chrifi-Alaoui, Larbi;Mahieddine-Mahmoud, Sofiane;Makouf, Abdessalam 226
The $H_{\infty}$ approach, adopted in this paper, is based on loop shaping using a normalized coprime factor combined with a field-oriented control to control induction motor. We develop two loops. The first one, the inner loop, controls the stator current by $H{\infty}$ controller in order to obtain good performance. The second loop, the outer one, guarantees stability and tracking performance of speed and rotor flux using a proportional integral controller. When the rotor flux cannot be measured, we introduce a flux observer to estimate the rotor flux. Simulation and experimental results are presented to validate the effectiveness and the good performance of this control technique.
Fuzzy-based Field-programmable Gate Array Implementation of a Power Quality Enhancement Strategy for ac-ac Converters
Radhakrishnan, N.;Ramaswamy, M. 233
In the present work, a new approach is proposed for via interconnects of semiconductor devices, where multi-wall carbon nanotubes (MWCNTs) are used instead of conventional metals. In order to implement a selective growth of carbon nanotubes (CNTs) for via interconnect, the buried catalyst method is selected which is the most compatible with semiconductor processes. The cobalt catalyst for CNT growth is pre-deposited before via hole patterning, and to achieve the via etch stop on the thin catalyst layer (ca. 3nm), a novel 2-step etch scheme is designed; the first step is a conventional oxide etch while the second step chemically etches the silicon nitride layer to lower the damage of the catalyst layer. The results show that the 2-step etch scheme is a feasible candidate for the realization of CNT interconnects in conventional semiconductor devices.
Harmonic Analysis of Reactor and Capacitor in Single-tuned Harmonic Filter Application
Kim, Jong-Gyeum;Park, Young-Jeen;Lee, Dong-Ju 239
Industrial power distribution system includes many kinds of non-linear loads, which produce the harmonics during energy conversion transition. The single-tuned passive filter is widely used to absorb the harmonics and attenuate its undesirable effect in the distribution system. However, the passive filter might be severely stressed, and sometimes even damaged, due to the absorption of harmonics. There is voltage rise on the capacitor when the single-turned harmonic filter is applied. When the capacitor voltage rose above the allowable limit, the expected life of the capacitor will considerably deteriorate. On the other hand, the reactor can experience the spike voltage even if the voltage and current of the capacitor are within the allowable limit, and this accumulated voltage stress of the reactor causes its premature fault. In this paper, we analyzed and compared the harmonic voltage and current of the reactor and capacitor in a single-tuned harmonic filter through the EMTP software and verified them with the experimental results.
Verification of New Family for Cascade Multilevel Inverters with Reduction of Components
Banaei, M.R.;Salary, E. 245
This paper presents a new group for multilevel converter that operates as symmetric and asymmetric state. The proposed multilevel converter generates DC voltage levels similar to other topologies with less number of semiconductor switches. It results in the reduction of the number of switches, losses, installation area, and converter cost. To verify the voltage injection capabilities of the proposed inverter, the proposed topology is used in dynamic voltage restorer (DVR) to restore load voltage. The operation and performance of the proposed multilevel converters are verified by simulation using SIMULINK/MATLAB and experimental results.
Grid-tied Power Conditioning System for Fuel Cell Composed of Three-phase Current-fed DC-DC Converter and PWM Inverter
Jeong, Jong-Kyou;Lee, Ji-Heon;Han, Byung-Moon;Cha, Han-Ju 255
This paper proposes a grid-tied power conditioning system for fuel cell, which consists of three-phase current-fed DC-DC converter and three-phase PWM inverter. The three-phase current-fed DC-DC converter boosts fuel cell voltage of 26-48 V up to 400 V with zero-voltage switching (ZVS) scheme, while the three-phase PWM(Pulse Width Modulation) inverter controls the active and reactive power supplied to the grid. The operation of the proposed power conditioning system with fuel cell model is verified through simulations with PSCAD/EMTDC software. The feasibility of hardware implementation is verified through experimental works with a laboratory prototype with 1.2 kW proton exchange membrane (PEM) fuel cell stack. The proposed power conditioning system can be commercialized to interconnect the fuel cell with the power grid.
Microwave Dielectric Properties of Ti-Te system Ceramics for Triplexer Filter
Choi, Eui-Sun;Lee, Moon-Woo;Lee, Sang-Hyun;Kang, Gu-Hong;Kang, Gap-Sul;Lee, Young-Hie 263
In this study, the compositions for the microwave dielectric materials were investigated to obtain the improved dielectric properties, the high temperature stability, and the sintering temperature of less than $900^{\circ}C$, which was necessary for cofiring with the internal conductor of silver. In addition, the dielectric sheets were prepared by the tape casting technique, after which the sheets were laminated and sintered. In this process, the optimum ratio of powder and binder, laminating pressure, temperature, and possibility for cofiring with the internal conductor were studied. Finally, multilayer chip treplexer filter for the 800-2,000 MHz range were fabricated, and the frequency characteristics of the triplexer filter were investigated. When the $0.6TiTe_3O_8-0.4MgTiO_3+3wt%SnO+7wt%H_3BO_3$ ceramics were sintered at $820^{\circ}C$ for 0.3 hours, the microwave dielectric properties of the dielectric constant of 29.91, quality factor of 33,000 GHz, and temperature coefficient of resonant frequency of -2.76 ppm/$^{\circ}C$ were obtained. Using the Advanced Design System (ADS) and High Frequency Structure Simulator (HFSS), the multilayer chip triplexer filter acting at the range of 800-2,000 MHz were simulated and manufactured. The manufactured triplexer filter had the excellent frequency properties in the CDAM800, GPS and PCS frequency regions, respectively.
Parameterized Simulation Program with Integrated Circuit Emphasis Modeling of Two-level Microbolometer
Han, Seung-Oh;Chun, Chang-Hwan;Han, Chang-Suk;Park, Seung-Man 270
This paper presents a parameterized simulation program with integrated circuit emphasis (SPICE) model of a two-level microbolometer based on negative-temperature-coefficient thin films, such as vanadium oxide or amorphous silicon. The proposed modeling begins from the electric-thermal analogy and is realized on the SPICE modeling environment. The model consists of parametric components whose parameters are material properties and physical dimensions, and can be used for the fast design study, as well as for the co-design with the readout integrated circuit. The developed model was verified by comparing the obtained results with those from finite element method simulations for three design cases. The thermal conductance and the thermal capacity, key performance parameters of a microbolometer, showed the average difference of only 4.77% and 8.65%, respectively.
Sterilization of Escherichia coli Based on Nd: YAG Resonator with a Pulsed Xenon Flashlamp
Kim, Hee-Je;Kim, Dong-Jo;Hong, Ji-Tae;Xu, Guo-Cheng;Lee, Dong-Gil 275
Sterilization of Escherichia coli (E. coli) is examined using a unique pulsed ultra-violet (UV) elliptical reactor based on Nd:YAG laser resonator, UV radiation from a pulsed xenon flashlamp. The light from the discharge has a broadband emission spectrum extending from the UV to the infrared region with a rich UV contained. Sterilization method by using the UV light is fast, environment-friendly and it does not cause secondary pollution. A Nd:YAG laser resonator having elliptical shape has advantage of concentrating the radiation of the UV light at two foci as the quart sleeve filled with E. coli. The primary objective of this research is to determine the important parameters such as pulse per second (pps), the applied voltage for sterilizing E. coli by using an UV elliptical reactor. From the experiment result, the sterilization effect of UV elliptical reactor is better than that of UV cylindrical reactor, and it can be 99.9% of sterilization at 800V regardless of the pps within 10 minutes.
Selective Growth of Carbon Nanotubes using Two-step Etch Scheme for Semiconductor Via Interconnects
Lee, Sun-Woo;Na, Sang-Yeob 280
Talmudic Approach to Load Shedding of Islanded Microgrid Operation Based on Multiagent System
Kim, Hak-Man;Kinoshita, Tetsuo;Lim, Yu-Jin 284
This paper presents a load-shedding scheme using the Talmud rule in islanded microgrid operation based on a multiagent system. Load shedding is an intentional load reduction to meet a power balance between supply and demand when supply shortages occur. The Talmud rule originating from the Talmud literature has been used in bankruptcy problems of finance, economics, and communications. This paper approaches the load-shedding problem as a bankruptcy problem. A load-shedding scheme is mathematically expressed based on the Talmud rule. For experiment of this approach, a multiagent system is constructed to operate test islanded microgrids autonomously. The suggested load-shedding scheme is tested on the test islanded microgrids based on the multiagent system. Results of the tests are discussed. | CommonCrawl |
\begin{document}
\newtheorem{thm}{Theorem}[section] \newtheorem{prop}[thm]{Proposition} \newtheorem{lemma}[thm]{Lemma} \newtheorem{cor}[thm]{Corollary} \theoremstyle{definition} \newtheorem{defi}[thm]{Definition} \newtheorem{notation}[thm]{Notation} \newtheorem{exe}[thm]{Example} \newtheorem{conj}[thm]{Conjecture} \newtheorem{prob}[thm]{Problem} \newtheorem{rem}[thm]{Remark}
\newtheorem{conv}[thm]{Convention} \newtheorem{crit}[thm]{Criterion} \newtheorem{propdef}[thm]{Proposition-definition} \newtheorem{lemmadef}[thm]{Lemma-definition}
\begin{abstract} We generalize the notion of Zermelo navigation to arbitrary pseudo-Finsler metrics possibly defined in conic subsets. The translation of a pseudo-Finsler metric $F$ is a new pseudo-Finsler metric whose indicatrix is the translation of the indicatrix of $F$ by a vector field $W$ at each point, where $W$ is an arbitrary vector field. Then we show that the Matsumoto tensor of a pseudo-Finsler metric is equal to zero if and only if it is the translation of a semi-Riemannian metric, and when $W$ is homothetic, the flag curvature of the translation coincides with the one of the original one up to the addition of a non-positive constant. In this case, we also give a description of the geodesic flow of the translation. \end{abstract}
\maketitle \section{Introduction}
The recent appearance of pseudo-Finsler metrics in the formulation of certain modern physical theories (see for instance \cite{kostelecky11}) has attracted some attention to the study of such metrics. In this paper, we will be concerned with pseudo-Finsler metrics in a generalization of the Zermelo navigation problem. The original problem, proposed by Zermelo in 1931 \cite{Ze31}, aims to describe the trajectories that minimize the time in the presence of a wind or current assuming that the speed of the body is constant without the wind. It was observed by Z. Shen \cite{Sh03} that when the wind does not depend on time and the wind is mild, the minimizing trajectories can be described as geodesics of a Finsler metric. Indeed, when the trajectories without wind are described by geodesics of a Riemannian metric $g$, the minimizing time trajectories in the presence of a wind are geodesics of a Finsler metric whose set $\hat{\Sigma}$ of unit tangent vectors is, at each point, the translation by the wind $W$ of the set $\Sigma$ of unit tangent vectors of $g$: \begin{figure}
\caption{Indicatrix translated by $W$ with $F(-W)<1$}
\label{mildTrans}
\end{figure} \noindent Now, if we consider strong winds, then two pseudo-Finsler metrics will emerge, one with positive-definite fundamental tensor, and the other one of Lorentz type. Moreover, both are defined in the same conic convex region (see Figure \ref{strongTrans2}). This kind of situation occurs, for instance, in the study of causality of space-times (see \cite{CJS14}). \begin{figure}
\caption{When $F(-W)>1$ there are two translated indicatrices}
\label{strongTrans2}
\end{figure}
In this paper we will consider the general problem of navigation on a conic pseudo-Finsler manifold $(M,F)$. In this setting, we will say that a pseudo-Finsler metric $\hat{F}$ is a translation of $F$ by a vector field $W$ if, at each point $p\in M$, its indicatrix is (a connected open subset of) the translation by $W_p$ of the indicatrix of $F$. In the case $F$ is semi-Riemannian, the translated metrics will consist of the metrics that, at each point, are of Randers or Kropina types (compare with \cite[Proposition 3.1]{BiJa11} or \cite{BCS04} in the Randers case, and with \cite[Proposition 2.40]{CJS14} for wind Riemannian structures) \begin{equation}\nonumber \pm\sqrt{h(v,v)}+\beta(v)~~~{\rm and}~~~\frac{h(v,v)}{\beta(v)}, \end{equation} where $h$ is a semi-Riemannian metric and $\beta$ a one-form, being thus called pseudo-Randers-Kropina metrics (see $\S$2.3 for a precise description of these metrics).
In $\S$3 we define the Matsumoto tensor of a conic pseudo-Finsler metric. This definition differs from the original one by the appearance of a sign depending on the index of the fundamental tensor. We then extend to these metrics a classic theorem by Matsumoto and Hojo \cite{MaHo78}, proving \begin{thm}\label{thmmatsumoto} A conic pseudo-Finsler manifold $(M,F)$ of dimension at least $3$ is of pseudo-Randers-Kropina type if, and only if, its Matsumoto tensor vanishes identically. \end{thm} \noindent Our proof (see $\S$3.3) will consist in adapting to the pseudo-Finsler setting a very geometric proof by Mo and Huang \cite{MoHu10} which makes use of affine differential geometry.
In $\S$4 we will consider the Legendre dual of a conic pseudo-Finsler metric from a more geometric viewpoint. It is remarkable that, as in the Finsler case, the process of translating a conic pseudo-Finsler metric has a nice dual description. Indeed, we show in Proposition \ref{propdichotomy} that the Legendre duals of $F$ and $\hat{F}$ are related by \begin{equation}\nonumber \hat{F}^*(\xi)=F(\xi)+W(\xi)~~~{\rm or}~~~\hat{F}^*(\xi)=-\tilde{F}^*(\xi)+W(\xi), \end{equation} where $\tilde{F}$ is the reverse metric of $F$, according to the translation being straight or reverse (see Definition \ref{CharacTrans}).
The relation between the Legendre duals of $F$ and $\hat{F}$ allows us to relate the co-geodesic flows of $F$ and $\hat{F}$ in some special cases, as
noted by Ziller in \cite{Ziller} in the case where $W$ is a Killing field on the sphere. So, in $\S$5 we consider the relation between the geodesics and between the flag curvatures of $F$ and $\hat{F}$ in the case $\hat{F}$ is a translation of $F$ by a homothetic vector field $W$, which means that \begin{equation}\label{homothetic} (\psi^W_t)^*F={\rm e}^{-\sigma t}F,~~~\mbox{for some $\sigma\in\mathbb{R}$}, \end{equation} where $\psi_t^W$ is the flow of $W$.
In this case we are able to apply a simple, but useful, general result, Lemma \ref{lemmaflow}, to relate the co-geodesic flows of $F$ and $\hat{F}$, providing a simple proof of \begin{thm}\label{geodesicflow} The unit speed geodesics of $(M,\hat{F})$ can be expressed (at least in a neighborhood of $t=0$) as $\hat\gamma(t)=\psi_t^W(\gamma(f(t)))$, where \[f(t)=\begin{cases} \frac{e^{\sigma t}-1}{\sigma}, &\text{if $\sigma\not=0$}\\ t,& \text{if $\sigma=0$} \end{cases}\] and $\gamma$ is a unit speed geodesic of $(M,F)$. Moreover, if $(M,F)$ is geodesically complete and $W$ is a complete homothetic vector field, then $(M,\hat{F})$ is geodesically complete. \end{thm} \noindent This theorem generalizes to pseudo-Finsler metrics the central result in \cite{Rob07} and \cite{HuMo11}. It is remarkable that we get the same expression independently if the translation is straight or reverse. As for the flag curvature, we derive the following theorem \begin{thm}\label{theoremcurvature} Given $u\in A$ and a $g_u$-nondegenerate $2$-plane $\Pi$ containing $u$, let $w\in\Pi$ be such that with $g^{\hat{F}}_u(u,w)=0$. Then, $\tilde{\Pi}:={\rm span}\{u/\hat{F}(u)-W,w\}$ is a $(u/\hat{F}(u)-W)$-nondegenerate (w.r.t. $F$) $2$-plane and \begin{equation}\nonumber K_{\hat{F}}\bigl(u\hspace{0.05cm},\hspace{0.05cm}\Pi\bigr)~=~K_{F}\bigl(u/\hat{F}(u)-W\hspace{0.05cm},\hspace{0.05cm}\tilde{\Pi}\bigr)-\frac{1}{4}\sigma^2, \end{equation} \end{thm} \noindent This is a well known result in the classical case where
$F$ is a Finsler metric and $F(-W)<1$ (see \cite{MoHu07} and \cite{BCS04} when $F$ is Riemannian).
It turns out that last theorem is somewhat natural from the point of view of the theory of linear symplectic invariants of a generic class of curves of Lagrangean subspaces, called {\it fanning curves}, first introduced by Ahdout \cite{Ah}, and then developed by \'Alvarez Paiva and Dur\'an \cite{AD}, and \'Alvarez Paiva, Dur\'an and Vit\'orio \cite{ADH} (see also \cite{Henrique}) inspired by work of Foulon \cite{Foulon} and Grifone \cite{Grifone}. We have thus considered appropriate to introduce the notion of flag curvature of a pseudo-Finsler metric using the fanning curves approach. This provides a novel way of thinking of flag curvature in contrast to the usual definitions that use the well known Finslerian connections (Berwald, Cartan, Chern). We do that in $\S$5.1, $\S$5.2 and $\S$5.3. Then we show that the relation obtained between the co-geodesic flows of $F$ and $\hat{F}$ directly implies a linear symplectic equivalence (up to reparametrization) of the corresponding fanning curves (Theorem \ref{theoremcurves}), which in turn gives the Theorem \ref{theoremcurvature}.
We will finish the paper with a section of conclusions and consequences $\S$6. In particular, as a byproduct of our investigations, we extend all the Randers metrics with constant flag curvature to conic Finsler metrics which are geodesically complete and we also provide the natural candidates for a classification of pseudo-Randers-Kropina metrics with constant flag curvature. \\\\\\\\\\\\\\\\\\\\\\
\section{Pseudo-Finsler metrics and Zermelo navigation}
\subsection{Pseudo-Minkowski norms}
Let us begin by introducing the notion of (conic) pseudo-Minkowski norm. Along this section we will denote by $V$ a vector space of dimension $n$. \begin{defi}\label{defpseudo} Let $V$ be a vector space and $A$ a conic open connected subset of $V\setminus \{0\}$, namely, an open subset such that $\lambda v\in A$ for every $v\in A$ and $\lambda>0$. A smooth function $F:A\rightarrow(0,+\infty)$ is said to be a (conic) pseudo-Minkowski norm on $V$ if \begin{enumerate}[(i)] \item it is positive homogeneous of degree $1$, namely, $F(\lambda v)=\lambda F(v)$ for every $v\in A$ and $\lambda>0$, \item for any $v\in A$, the fundamental tensor $g_v$ defined as \begin{equation}\label{fundtensor}
g_v(u,w)=\frac 12\left. \frac{\partial^2}{\partial t\partial s}F\left(v+ t u+sw\right)^2\right|_{t=s=0} \end{equation} for every $u,w\in V$ is non-degenerate. \end{enumerate} \end{defi} We will understand that the pseudo-Minkowski space is conic without saying it explicitly. Recall that from the homogeneity of $F$, \begin{equation}\label{gv} \frac{1}{2}DF^2(v)(w)~=~g_v(v,w) \end{equation} and \begin{equation}\label{df^2} \frac{1}{2}DF^2(v)(v)~=~g_v(v,v)~=~F(v)^2. \end{equation} The indicatrix of a conic pseudo-Minkowski space $(V,F)$ is defined as the subset $$\Sigma~=~\{ v\in A: F(v)=1\}.$$ From \eqref{gv} and \eqref{df^2}, and the connectivity of $A$, it follows that $\Sigma$ is a smooth embedded, connected hypersurface of $V$, with \begin{equation} T_v\Sigma~=~\{w\in V:g_v(v,w)=0\} \end{equation} and with everywhere transverse position vector. Moreover, if $\xi$ denotes the opposite to the position vector, which is transverse to $\Sigma$, and $\sigma_\xi$ is the second fundamental form of $\Sigma$ w.r.t. $\xi$, a direct computation (see for instance \cite[Theorem 2.14]{JaSan11}) shows that \begin{equation}\label{sff} g_v(X,Y)~=~\sigma^\xi(X,Y) \end{equation} for $X$ and $Y$ tangent to $\Sigma$, so that $\sigma^\xi$ is non-degenerate. Actually, these properties completely characterize indicatrices: \begin{prop}\label{charindicatrix} Let $V$ be a vector space. Then a subset $\Sigma$ of $V$ is the indicatrix of a pseudo-Minkowski norm $F$ if and only if it is a smooth, embedded, connected, hypersurface of $V$, the position vector is transversal to it and its second fundamental form with respect to one (and then to all) transversal vector is non-degenerate everywhere. \end{prop} \begin{proof} The proof is straightforward. \end{proof}
\begin{defi}\label{transMink} Given a vector space $V$, we say that a pseudo-Minkowski norm $\hat{F}:\hat{A}\rightarrow (0,+\infty)$ is a translation by a vector $w\in V$ of the pseudo-Minkowski norm $F:A\rightarrow (0,+\infty)$ if its indicatrix $\hat{\Sigma}$ is an open connected subset of $\Sigma+w$, where $\Sigma$ is the indicatrix of $F$. In other words, if \begin{equation}\label{Navrelation} F\left(\frac{v}{\hat{F}(v)}-w\right)=1, \end{equation} for every $v\in \hat{A}$. \end{defi} In general, we can get more than one pseudo-Minkowski norm by translation of a given one.
Note that although the signatures of the second fundamental forms of $\Sigma$ and $\hat{\Sigma}$ coincide if we choose an appropriate transversal vector, this does not imply that the fundamental tensors of $F$ and $\hat{F}$ can be identified, since the position vector in $\Sigma$, when translated into $\hat{\Sigma}$, can have opposite orientation to the position vector in $\hat{\Sigma}$ (see Figure \ref{strongTrans2}).
\begin{prop}\label{translatingF} Given a pseudo-Minkowski norm $F:A\rightarrow (0,+\infty)$ and a vector $w\in V$, the indicatrices of the translations of $F$ by $w$ are the connected components of $\{v/F(v)+w: F(v)+g_v(w,v)\not=0\}$. Moreover, let $Z:\hat{A}\rightarrow\mathds R$ be one of such translations, with $g$ the fundamental tensor of $F$, and $h$, the fundamental tensor of $Z$. Then $g_{v/Z(v)-w}(v/Z(v)-w,v)\not=0$ for every $v\in \hat{A}$, and \begin{enumerate}[(i)] \item if $g_{v/Z(v)-w}(v/Z(v)-w,v)>0$ for every $v\in\hat{A}$, the index of $h$ coincides with the index of $g$. In such a case, we say that $Z$ is a {\em straight translation } of $F$. \item If $g_{v/Z(v)-w}(v/Z(v)-w,v)<0$ for every $v\in \hat{A}$, the index of $h$ is $n-\mu-1$, where $\mu\geq 0$ is the index of $g$. In such a case, we say that $Z$ is a {\em reverse translation } of $F$. \end{enumerate} \end{prop} \begin{proof} Observe that Proposition \ref{charindicatrix} implies that the translation of the indicatrix $\Sigma$ of $F$ by $w$ is the indicatrix of a pseudo-Minkowski norm in the points where the position vector is transversal to $\hat{\Sigma}=\Sigma+w$. Moreover, a vector $v+w\in \Sigma+w$ is transversal to $\Sigma+w$ if and only if $v+w$ is not tangent to $\Sigma$ in $v$, which is equivalent to the condition $g_v(v,v+w)\not=0$. Given an arbitrary vector $v\in A$, then $v/F(v)\in\Sigma$, and the above condition becomes $g_{v/F(v)}(v/F(v),v/F(v)+w)\not=0$, which is equivalent to $F(v)+g_v(w,v)\not=0$, by the homogeneity of $g_v$ with respect to $v$ and \eqref{df^2}, as required. The second part is an easy consequence of \eqref{sff}, since it implies that the index is preserved when you consider the second fundamental form with respect to a transversal vector with the same orientation. The condition $g_{v/Z(v)-w}(v/Z(v)-w,v)>0$ guarantees that $v/Z(v)-w$ and $v$ give the same orientation in the indicatrix $\Sigma$ (or $\Sigma+w$) and then $g$ and $h$ have the same index. When $g_{v/Z(v)-w}(v/Z(v)-w,v)<0$, then $v$ and $v/Z(v)-w$ determine opposite orientations for $\Sigma$, and the second fundamental form of the first one has index equal to $n+\mu-1$, being $\mu$ the index of the second fundamental form with respect to $v/Z(v)-w$. This concludes the proof. \end{proof} \subsection{Pseudo-Finsler metrics}
Given a connected manifold $M$ and a conic open connected subset $A$ of $TM\setminus 0$ in the sense that $A_p=A\cap T_pM$ is conic, connected and non-empty for every $p\in M$, we will say that a smooth function $F:A\rightarrow (0,+\infty)$ is a {\it (conic) pseudo-Finsler metric} if the restrictions $F_p=F|_{T_pM\cap A}$ are (conic) pseudo-Minkowski norms for every $p\in M$. Let us distinguish two interesting families of pseudo-Finsler metrics. We say that a pseudo-Finsler metric $F:A\rightarrow (0,+\infty)$ is \begin{enumerate}[(i)] \item {\it conic Finsler} if the fundamental tensor in \eqref{fundtensor} is positive definite for every $v\in A$, \item {\it Lorentz-Finsler} if the fundamental tensor in \eqref{fundtensor} has index $n-1$. \end{enumerate} \noindent These are the pseudo-Finsler metrics with strongly convex indicatrices (i.e., indicatrices have definite second fundamental form), a property which ensures they are the only ones whose geodesics (locally) minimize or maximize the length.
\begin{prop} Given a manifold $M$, a smooth (connected) hypersurface $\Sigma$ of $TM$ determines a pseudo-Finsler metric on $M$ if and only if at every $v\in \Sigma$, it is transversal to $\{\lambda v: \lambda >0\}$, and $\Sigma_p=\Sigma\cap T_pM$ is an embedded (connected) hypersurface such that its second fundamental form with respect to the position vector is non-degenerate. \end{prop} \begin{proof}
It follows the same lines as \cite[Proposition 2.10]{CJS14}. Let us define $\Psi:(0,+\infty)\times \Sigma\rightarrow TM$, $(t,w)\rightarrow tw$. The transverality condition and the connectedness of $\Sigma$ imply that $\Psi$ is injective and a local diffeomorphism. Then it is a global diffeomorphism into the image, whose inverse can be written as
$\Psi^{-1}:A\rightarrow (0,+\infty)\times \Sigma$, $v\rightarrow (F(v),v/F(v))$, where $A={\rm Im}(\Psi)$. Clearly, $F$ is a smooth positive homogeneous map and, by Propositon \ref{charindicatrix}, its fundamental tensor is non-degenerate, which concludes the implication to the left. Conversely, if $F:A\rightarrow (0,+\infty)$ is a pseudo-Finsler metric, then \eqref{df^2} implies that $1$ is a regular value of $F^2:A\rightarrow (0,+\infty)$ and then $\Sigma=(F^2)^{-1}(1)$ is an embedded hypersurface of $TM$ transversal to $\{\lambda v: \lambda>0\}$ in $v\in \Sigma$. Proposition \ref{charindicatrix} concludes the statement about the second fundamental form of $\Sigma_p$. \end{proof} \begin{prop}\label{goodTrans} If $(M,F)$ is a pseudo-Finsler manifold and $W$ a smooth vector field on $M$, the transport of every $(T_pM,F_p)$ by $W_p$ in every connected component of $\hat{A}=\{v\in T_pM:F(v)+g_v(W,v)\not=0\}$ gives a pseudo-Finsler metric $\hat{F}$ determined by \eqref{Navrelation}. \end{prop} \begin{proof} It follows from Proposition \ref{translatingF} and the observation that the map $TM\rightarrow TM$, $v\rightarrow v+W$, is an isomorphism of fiber bundles and then $\Sigma+W$ is transversal to the position vector in $TM$ if and only if $\Sigma_p+W_p$ is transversal to the position vector in $T_pM$ for every $p\in M$. \end{proof} \begin{defi}\label{CharacTrans} Following Proposition \ref{translatingF}, we will say that the translation of $F$ by a smooth vector field $W$ is {\em straight} (resp. {\em reverse}) if $F(v)+g_v(v,W)>0$ (resp. $F(v)+g_v(v,W)<0$) for every $v\in A$. \end{defi} \begin{rem}
Observe that definition of (conic) pseudo-Finsler metric can be generalized in two ways: \begin{enumerate} \item we can consider the case in that the fundamental tensor $g$ in \eqref{fundtensor} is degenerate as in \cite{JaSan11}. Along this paper we will need the non-degeneracy of the fundamental tensor in order to define the volume form in \S \ref{Matsumoto} or the Legendre transformation and the flag curvature in \S \ref{legendre} and \S \ref{sectionflagcurvature}. \item we also can consider the case in that the pseudo-Finsler metric is a positive homogeneous function $L:A\rightarrow \mathds R$ of degree $2$. This is the most general case in that for example the Chern connection \cite[Remark 2.8]{JavSoa13} or the Legendre transformation can be defined. When $L\not=0$, we can recover a pseudo-Finsler metric as defined here by making $F=\sqrt{L}$, if $L>0$ or $F=\sqrt{-L}$ if $L<0$. However we cannot consider the region in that $L=0$. The reason is that when we translate the indicatrix $L=0$ we do not obtain the indicatrix of a new pseudo-Finsler metric. In fact, at every $p\in M$, the indicatrix ${\mathcal C}_p=\{v\in A_p:L(v)=0\}$ is a degenerate hypersurface of $T_pM$ with degenerate directions given by rays contained in ${\mathcal C}_p$ which cross the zero vector of $T_pM$. When we translate ${\mathcal C}_p$, this property is lost. \end{enumerate} \end{rem} \subsection{Navigation in semi-Riemannian metrics}
In general, it is not possible to compute explicitly the expression of the translation of a pseudo-Finsler metric, but when we consider a semi-Riemannian metric, the situation is simpler. \begin{prop}\label{transZer} Let $g$ be a semi-Riemannian metric in a manifold $M$ and $W$ a smooth vector field in $M$, and consider the conic pseudo-Finsler metric $F(v)=\sqrt{g(v,v)}$ defined in $A=\{v\in TM:g(v,v)>0\}$. Let $h$ be the (possibly degenerate) semi-Riemannian metric in $M$ determined by \[h(v,v)=g(v,W)^2+g(v,v)(1-g(W,W)).\] Then there are at most two translations of $F$ by a smooth vector field $W$, denoted by $Z_{1}$ and $Z_{-1}$, and given by \begin{equation}\label{Zermelo1} Z_\varepsilon(v)=\frac{g(v,W)-\varepsilon \sqrt{h(v,v)}}{g(W,W)-1} \end{equation} when $g(W,W)\not=1$ and alternatively by \begin{equation}\label{Zermelo2} Z_\varepsilon(v)=\frac{g(v,v)}{g(v,W)+\varepsilon \sqrt{h(v,v)}}, \end{equation} when $g(v,v)\not=0$. They are defined in the subset $A_\varepsilon$, which is defined as \[A_\varepsilon= \{v\in TM: \varepsilon g(v,v)>0\}\cup \{ v\in TM : \varepsilon g(v,W)<0; h(v,v)>0\}\] if $\varepsilon(1-g(W,W))>0$, \[A_\varepsilon=\{ v\in TM : \varepsilon g(v,v)>0;\varepsilon g(v,W)>0; h(v,v)>0\}\] if $\varepsilon(1-g(W,W))<0,$ and \[A_\varepsilon=\{v\in TM: g(v,v)g(v,W)>0; \varepsilon g(v,W)>0\}\] if $g(W,W)=1$. Moreover, $Z_1$ (resp. $Z_{-1}$) is a straight (resp. reverse) translation of $F$. \end{prop} \begin{proof} The Zermelo metric $Z$ associated to $F$ is determined by \[g\left(\frac{v}{Z(v)}-W,\frac{v}{Z(v)}-W\right)=1,\] (see \eqref{Navrelation}). Solving a quadratic equation we get that \begin{equation}\label{ZermeloExp1} Z_\varepsilon(v)=\frac{g(v,W)-\varepsilon \sqrt{g(v,W)^2+g(v,v)(1-g(W,W))}}{g(W,W)-1}. \end{equation} As this expression is not defined when $g(W,W)=1$, we will obtain an alternative expression by conjugation: \begin{equation}\label{ZermeloExp2} Z_\varepsilon(v)=\frac{g(v,v)}{g(v,W)+\varepsilon\sqrt{g(v,W)^2+g(v,v)(1-g(W,W))}}. \end{equation} Both metrics are defined in the vectors $v$ that make $Z_\varepsilon(v)$ positive. Observing that \[Z_\varepsilon(v)=\frac{\varepsilon g(v,W)- \sqrt{h(v,v)}}{\varepsilon(g(W,W)-1)}=\frac{\varepsilon g(v,v)}{\varepsilon g(v,W)+ \sqrt{h(v,v)}},\] we deduce easily that the biggest open subset where it can be defined is $A_\varepsilon$ (namely, distinguish between $\varepsilon g(v,v)<0$ or $\varepsilon g(v,v)\geq 0$ to get all the cases). Now observe that $g(v/Z_\varepsilon(v)-W,v)=\varepsilon \sqrt{h(v,v)}$, which identifies $Z_1$ as a straight translation of $F$ and $Z_{-1}$ as a reverse translation (see Definition \ref{CharacTrans} and Proposition \ref{translatingF}). Finally, observe that in the vectors where $h(v,v)=0$, even if sometimes $Z_\varepsilon(v)$ is positive, the translation is not well-defined because $\Sigma+W$ is not transverse to the position vector. \end{proof}
Let us consider now Randers and Kropina metrics constructed using a semi-Riemannian metric $h$ and a one-form $\beta$, which in the following will be called respectively, pseudo-Randers and pseudo-Kropina metrics. Namely, a pseudo-Randers metric is defined as \begin{equation}\label{pseudo-Randers} R_\epsilon(v)=\epsilon \sqrt{h(v,v)}+\beta(v) \end{equation} for $v\in A=\{v\in TM: h(v,v)>0; \epsilon\sqrt{h(v,v)}+\beta(v)>0\}$ and $\epsilon=-1,1$. Observe that in this definition we include the case in that $h$ is Riemannian but the norm of $\beta$ in $h$ is bigger than one. In such a case, there exists some $v\in TM$ such that $\sqrt{h(v,v)}+\beta(v)<0$, but for these vectors we can consider $R(v)=-\sqrt{h(v,v)}-\beta(v)$.
Moreover, we will define a pseudo-Kropina metric as \begin{equation}\label{pseudo-Kropina} K(v)=\frac{h(v,v)}{\beta(v)}, \end{equation} where $v\in A=\{v\in TM: h(v,v)\beta(v)>0\}$. In both cases, we will denote by $B$ the vector field $h$-equivalent to $\beta$. Moreover, we will say that the pseudo-Randers (resp. pseudo-Kropina) metric is non-degenerate if $h(B,B)\not=1$ (resp. $h(B,B)\not=0$) at every point. It is clear that pseudo-Randers and pseudo-Kropina metrics are positive homogeneous, we will see below that the fundamental tensor is non-degenerate up to some exceptional cases (when $h(B,B)=1$ in the pseudo-Randers case and $h(B,B)=0$ in the pseudo-Kropina case). \begin{defi}\label{randers-kro-defi} We say that a pseudo-Finsler metric is pseudo-Randers-Kropina if at every point it can be expressed as \eqref{pseudo-Randers} with $h(B,B)\not=1$ or as in \eqref{pseudo-Kropina} with $h(B,B)\not=0$. \end{defi}
Observe that the semi-Riemannian metric $h$ does not to be defined in the whole manifold. Indeed, the Zermelo metric obtained as the translation of a semi-Riemannian metric (see \eqref{Zermelo1} and \eqref{Zermelo2}) is always pseudo-Randers or pseudo-Kropina at every point, but the
semi-Riemannian metrics used for the definition are not defined in the whole manifold. Let us see that the converse holds.
In the following we will understand that the indicatrix of a semi-Riemannian metric $g$ in a manifold $M$ is the subset $\Sigma=\{v\in TM: g(v,v)=1\}$. \begin{prop}\label{Randers-Kro} The family of pseudo-Randers-Kropina metrics coincides with the family of Zermelo metrics obtained translating the indicatrix of a semi-Riemannian metric. More precisely: \begin{enumerate}[(i)] \item a pseudo-Randers metric as in \eqref{pseudo-Randers} is the translation by the vector $W=-B/\delta$ of the semi-Riemannian metric $g$ determined by \begin{equation}\label{gdeh} g(v,v)=\delta (h(v,v)-h(B,v)^2) \end{equation} for $v\in TM$, where $\delta=1-h(B,B)$. Moreover, \begin{enumerate}[(a)] \item if $\epsilon \delta>0$ (resp. $\epsilon \delta<0$) $R_\epsilon$ is a straight (resp. reverse) translation of $g$ \item if $\epsilon=1$ (resp. $\epsilon=-1$), the index of the fundamental tensor of $R_1$ (resp. $R_{-1}$) coincides with the index of $h$ (resp. is equal to $n-1-{\rm ind}(h)$). \end{enumerate} \item a pseudo-Kropina metric as in \eqref{pseudo-Kropina} is the translation of the semi-Riemannian metric $g=4 h/ h(B,B)$ by the vector field $W=\frac{1}{2} B$. If $h(B,B)h(v,v)>0$ (resp. $h(B,B)h(v,v)<0$), then $K$ is a straight (resp. reverse) translation and the index of the fundamental tensor of $K$ is equal to the index of $g$ (resp. $n-1-{\rm ind}(g)$). \end{enumerate} \end{prop} \begin{proof} It is trivial that every Zermelo metric as in \eqref{ZermeloExp2} (or \eqref{ZermeloExp1}) is pseudo-Randers or pseudo-Kropina. Moreover, if $g(W,W)\not=1$, then the expression in \eqref{ZermeloExp1} is pseudo-Randers with
\[h(v,v)=\frac{1}{|\alpha|^2}g(v,W)^2+\frac{\alpha}{|\alpha|^2}g(v,v)\]
and $\beta(v)=-\frac{1}{\alpha}g(v,W)$, where $\alpha=1-g(W,W)$. Using both relations we deduce that $h(v,W)=-\frac{\alpha}{|\alpha|^2}h(v,B)$ for every $v\in TM$ and then $B=-\frac{|\alpha|^2}{\alpha}W$. It follows in particular that $h(B,B)=g(W,W)$ and $1-h(B,B)=\alpha\not=0$. When $g(W,W)=1$, the expression in \eqref{ZermeloExp2} shows that $Z_\varepsilon$ is pseudo-Kropina with $h=g$ and $\beta(v)=2g(v,W)$, and then $B=2 W$, which implies that $h(B,B)=4 g(W,W)=4\not=0$.
Consider now a pseudo-Randers metric as in \eqref{pseudo-Randers} with $h(B,B)\not=1$ and let $g$ be the metric given in \eqref{gdeh}. Then a straightforward computation shows that \[1-g(W,W)=\delta\not=0,\quad\frac{g(v,W)}{g(W,W)-1}=h(v,B),\] and \[g(v,W)^2-g(v,v)(g(W,W)-1)=\delta^2 h(v,v),\] thus, $R$ is one of the pseudo-Randers metrics obtained translating the indicatrix of $g$ with $W$ given in \eqref{ZermeloExp1}. In fact, observe that the sign of $\epsilon \delta$ must coincide with the sign of $\varepsilon$ in Proposition \ref{transZer}, which implies the statements about the character of the translation (straight or reverse) in part $(a)$. Moreover, observe that the index of $g$ coincides with the index of $h$ if $\delta>0$ and ${\rm ind}(g)=n-1-{\rm ind}(h)$ if $\delta<0$. This follows easily from the following three facts (1) $g(B,B)=\delta^2 h(B,B)$, (2) the $g$-orthogonal space to $B$ coincides with the $h$-orthogonal space to $B$ because $g(v,B)=\delta^2 h(v,B)$ and (3) if $v,w$ belong to the $g$-orthogonal space to $B$, $g(v,w)=\delta h(v,w)$. This, together with Proposition \ref{translatingF} and part $(a)$, gives part $(b)$.
For part $(ii)$, the indicatrix of a pseudo-Kropina metric with $h(B,B)\not=0$ is given by the solutions of \[h(v,v)=h(v,B)\] or equivalently \[4h(v-B/2,v-B/2)/h(B,B)=1,\] which means that $K$ is the translation of $g$ by $W=B/2$. The statements about the character of the translation and the index of the fundamental tensor of $K$ follow easily from Propositions \ref{transZer} and \ref{translatingF}, respectively, since the sign of $\varepsilon$ in Proposition \ref{transZer} coincides with the sign of $h(B,B)h(v,v)$. \end{proof}
\section{Zermelo navigation and Matsumoto tensor}\label{Matsumoto}
In this section we give the definition of the Matsumoto tensor of a (conic) pseudo-Finsler metric.
The main goal will be to prove Theorem \ref{thmmatsumoto}.
For the sake of completeness we have included a short review of some concepts from affine differential geometry, following \cite{NoSa94}.
\subsection{The Matsumoto Tensor of a (conic) pseudo-Finsler metric}
Let us first restric our attention to a single vector space $V$. The Cartan tensor of a (conic) pseudo-Minkowski norm $F:A\subseteq V\setminus\{0\}\rightarrow (0,+\infty)$ is defined as \[C_v(X,Y,Z)=\frac 14 \left.\frac{\partial^3}{\partial t_3\partial t_2\partial t_1}
F(v+t_1X+t_2Y+t_3Z)^2\right|_{t_1=t_2=t_3=0}\] for any $v\in A$ and $X,Y,Z\in V$. The homogeneity of $F$ implies that $C$ is completely symmetric, $C_v(v,\cdot,\cdot)=0$, and $C_{\lambda v}=(1/\lambda)C_v$ if $\lambda>0$. The $g$-metric contraction of the Cartan tensor is called the {\it mean Cartan torsion}, and denoted by $I$; namely, if $b_1,b_2,\ldots,b_n$ is a basis of $V$, and we denote $g_{ij}=g_v(b_i,b_j)$, and $g^{ij}$, the coefficients of the inverse matrix of $\{g_{ij}\}_{i,j=1,\dots,n}$, then \[I_v(X)=\sum_{i,j=1}^n g^{ij}C_v(X,b_i,b_j)\] for every $X\in V$. \\ Let us recall that the {\it angular metric} of $F$ is defined as \[h_v(X,Y)=g_v(X,Y)-\frac{1}{F(v)^2}g_v(v,X)g_v(v,Y),\] for every $v\in A$, and $X,Y\in V$. Note that $h_v$ and $g_v$ coincide on $T_v\Sigma$, for $v\in\Sigma$. \begin{defi}\label{defMat} Given $v\in A$, the {\it Matsumoto tensor} of $F$ is the symmetric tri-tensor ${\rm M}_v:V\times V\times V \rightarrow \mathds R$ given by \begin{multline*} {\rm M}_v(X,Y,Z)=C_v(X,Y,Z)\\ -\frac{\varepsilon}{n+1}\left(I_v(X) h_v(Y,Z)+I_v(Y) h_v(X,Z)+I_v(Z) h_v(X,Y)\right), \end{multline*} where $\varepsilon=1$ or $-1$, according to the index of $g$ is even or odd, respectively. \end{defi} \noindent As ${\rm M}_v(v,\cdot,\cdot)=0$ and ${\rm M}_{\lambda v}=(1/\lambda) {\rm M}_v$, it follows that the tensor ${\rm M}$ contains the same information as its pull-back to $\Sigma$, which we still denote by ${\rm M}$.
\\\\ \noindent If now $(M,F)$ is a (conic) pseudo-Finsler manifold, its Matsumoto tensor ${\rm M}$ is defined by just performing the above construction in each tangent space.
\begin{rem}\label{conicLorentzMat}
It is interesting to point out that it is possible to describe all the conic Finsler and Lorentz-Finsler metrics with trivial Matsumoto tensor using Theorem \ref{thmmatsumoto} and Propositions \ref{transZer} and \ref{Randers-Kro}: \begin{enumerate} \item In the conic Finsler case, these are those in \eqref{Zermelo2} with $g$ a Riemannian metric and $\varepsilon=1$ or $g$ a Lorentzian metric and $\varepsilon=-1$. Moreover, at every point they can be expressed as in \eqref{pseudo-Randers} with $\epsilon=1$ and $h$ a Riemannian metric, or $\epsilon=-1$ and $h$ a Lorentzian metric, or as in \eqref{pseudo-Kropina} with $h$ Riemannian.
\item In the Lorentz-Finsler case, these are those in \eqref{Zermelo2} with $g$ a Lorentzian metric and $\varepsilon=1$ or $g$ a Riemannian metric and $\varepsilon=-1$. Moreover, at every point they can be expressed as in \eqref{pseudo-Randers} with $\epsilon=1$ and $h$ a Lorentzian metric, or $\epsilon=-1$ and $h$ a Riemannian metric, or as in \eqref{pseudo-Kropina} with $h$ Lorentzian. \end{enumerate} \end{rem}
\subsection{Affine differential geometry of the indicatrix}
Throughout this section, let be fixed a vector space $V$, a parallel volume form $\Omega$ in $V$, and the orientation that it induces in $V$.
\par We now recall some concepts of affine differential geometry for a connected hypersurface $S\subset V$, following \cite{NoSa94}.
The choice of an everywhere transverse vector field $\xi$ to $S$ induces in $S$ a connection $\nabla^\xi$, a second fundamental form $h_\xi$,
a volume form $\theta^\xi$, and an orientation via the relations
$$\nabla_X Y=\nabla_X^\xi Y+h^\xi(X,Y)\xi,~~\theta^\xi(X_1,\cdots,X_{n-1})=\Omega(X_1,\cdots,X_{n-1},\xi),$$
where $\nabla$ is the canonical connection of $V$, and the orientation in $S$ is the induced by $\theta^\xi$.
\begin{defi} The pair $(S,\xi)$, or simply $\xi$, is said to be {\it equiaffine} if $\theta^\xi$ is $\nabla^\xi$-parallel; this, in turn, is equivalent to $\nabla_X\xi$ be tangent to $S$ for every $X$ tangent to $S$. If $\xi$ is equiaffine, the tri-tensor field $\mathcal{C}^\xi$ in $S$ given by $$\mathcal{C}^\xi(X,Y,Z)~=~(\nabla_X^\xi h^\xi)(Y,Z)$$ is completely symmetric and is called the associated {\it cubic form}. \end{defi} Under a change \begin{equation}\label{change} \overline{\xi}~=~\phi\xi+P, \end{equation} where $\phi$ and $P$ are, respectively, a nowhere null smooth function and a vector field on $S$, we have the following: \begin{lemma}\label{lemmachange} If $\xi$ is equiaffine, then a change (\ref{change}) produces another equiaffine transverse vector field if, and only if, $h^\xi(P,~\cdot~)=-d\phi$. In this case, the corresponding cubic forms are related by $$\phi\mathcal{C}^{\overline{\xi}}(X,Y,Z)=\mathcal{C}^\xi(X,Y,Z)-\frac{1}{\phi}\bigl(X(\phi)h^\xi(Y,Z)+Y(\phi)h^\xi(Z,X)+ Z(\phi)h^\xi(X,Y)\bigr)$$ \end{lemma} \begin{proof} An immediate consequence of the formulas in \cite[Proposition 2.5]{NoSa94}. \end{proof}
Suppose now that $S$ is {\it non-degenerate}, that is, it has everywhere non-degenerate second fundamental form with respect to some (and hence {\it all}) transverse vector field $\xi$. In this case, $h^\xi$ induces a volume form $\omega_{h^\xi}$ in $S$ by
$$\omega_{h^\xi}(X_1,\cdots,X_{n-1})~=~|{\rm det}[h^\xi(X_i,X_j)]|^{1/2}$$ if $X_1,\ldots,X_{n-1}$ is a positive basis of $T_vS$.
\begin{prop} If $S\subset V$ is a non-degenerate connected hypersurface, there is, up to sign, only one transverse vector field $\xi$ such that \begin{enumerate}[(i)] \item $\xi$ is equiaffine; \item The volume forms $\theta^\xi$ and $\omega_{h^\xi}$ coincide. \end{enumerate} Any of these is called the {\rm Blaschke normal field} of $S\subset V$. \end{prop}
Indeed, as shown in \cite[page 41]{NoSa94}, starting with any equiaffine transverse vector field $\xi$, the change (\ref{change}) will produce the Blaschke normal fields if, and only if, \begin{equation}\label{formula}
|\phi|~=~|{\rm det}_{\theta^\xi}h^\xi|^\frac{1}{n+1}~,~~~h^\xi(P,~\cdot~)~=~-d\phi. \end{equation} Here, the value of ${\rm det}_{\theta^\xi}h^\xi$ at $v\in S$ is ${\rm det}[h^\xi(X_i,X_j)]$, where $X_1,\ldots,X_{n-1}\in T_v S$ are any vectors such that $\theta^\xi(X_1,\ldots,X_{n-1})=1$. \par We now apply these considerations to the case where $S$ is the indicatrix $\Sigma$ of a (conic) pseudo-Minkowski norm $F$. Let $\xi$ be the opposite of the position vector field. In this case, \begin{enumerate} \item[(I)] $\xi$ is equiaffine. \item[(II)] $h^\xi=h$, where $h$ is (the pull-back of) the angular metric; this follows from (\ref{sff}). \item[(III)] The cubic form $\mathcal{C}^\xi$ is twice the (pull-back of) Cartan tensor $C$: \begin{eqnarray} 2C_v(X,Y,Z) & = & \left.\frac{\partial}{\partial t}
g_{v+tX}(Y,Z)\right|_{t=0}=(\nabla_Xg)(Y,Z)=(\nabla^\xi_Xh^\xi)(Y,Z)\nonumber\\ & = & \mathcal{C}^\xi_v(X,Y,Z),\nonumber \end{eqnarray} where $v\in\Sigma$ and $X,Y,Z\in T_v\Sigma$. \end{enumerate}
\subsection{Proof of Theorem \ref{thmmatsumoto}} Let $\omega_g$ be the volume form in $A$ induced by the semi-Riemannian metric $g$, and let $\psi:A\rightarrow\mathds R$ be the function such that $\omega_g=\psi\Omega$. Note that $\psi>0$ as we are considering the orientation in $V$ induced by $\Omega$. \begin{prop} The cubic form $\mathcal{C}$ of $\Sigma$ associated to (one of) the Blaschke normal field is equal to $\frac{2}{\phi}{\rm M}$, where ${\rm M}$ is the (pull-back of) Matsumoto tensor of $F$, and $\phi=\psi^\frac{2}{n+1}$. \end{prop} \begin{proof} Due to Lemma \ref{lemmachange} and (I), (II), (III) above, all we have to show is that $\phi$ satisfies (\ref{formula}) and that $X(\psi)/\psi=\varepsilon I_v(X)$ for all $v\in\Sigma$ and $X\in T_v\Sigma$. \\ Note that $$
\theta^\xi(X_1,\cdots,X_{n-1})=\frac{1}{\psi}\omega_g(X_1,\cdots,X_{n-1},\xi)=\frac{1}{\psi}|{\rm det} [h^\xi(X_i,X_j)]|^{1/2} $$
for $X_1,\ldots,X_{n-1}\in T_v\Sigma$. This follows from the definitions of $\theta^\xi$ and $\psi$, and the fact that $h^\xi_v=g_v$ in $T_v\Sigma$, $g_v(v,v)=1$ and $v$ is $g_v$-orthogonal to $T_v\Sigma$. Therefore, $\theta^\xi(X_1,\cdots,X_{n-1})=1$ if, and only if, $\psi^2=|{\rm det} [h^\xi(X_i,X_j)]|$, establishing the equality $ \phi^{n+1}=|{\rm det}_{\theta^\xi} h^\xi|$.
\\ Let now $b_1,\ldots,b_n$ be a basis of $V$ with $\Omega(b_1,\ldots,b_n)=1$, and denote, as before, $g_{ij}=g_v(b_i,b_j)$ and $g^{ij}$ the coefficients of the inverse matrix of the $g_{ij}$. Then $\psi=|{\rm det} [g_{ij}]|^{1/2}$ and therefore, for $X\in T_v\Sigma$, \begin{eqnarray}
X(\psi)/\psi & = & \frac{1}{2|{\rm det}[g_{ij}]|}X({\rm det}[g_{ij}])=\frac{1}{2|{\rm det}[g_{ij}]|}{\rm det}[g_{ij}]\sum_{k,l}X(g_{kl})g^{kl}\nonumber\\
& = & \frac{{\rm det}[g_{ij}]}{|{\rm det}[g_{ij}]|}\sum_{k,l}C_v(X,b_k,b_l)g^{kl}\nonumber
\end{eqnarray} which in turn is equal to $\varepsilon I_v(X)$. \end{proof} Theorem \ref{thmmatsumoto} now follows from Proposition \ref{Randers-Kro} and the following classic result (see \cite[Theorem 4.5]{NoSa94}). \begin{thm}[Maschke, Pick, Berwald] If dim $V\geq 3$, then the cubic form associated to the Blaschke normal field of a non-degenerate connected hypersurface $S\subset V$ vanishes if, and only if, $S$ is a hyperquadric. \end{thm}
\section{The Legendre dual of a conic pseudo-Finsler metric}\label{legendre} In this section we will discuss the Legendre transformation of a conic pseudo-Finsler metric and then we will apply it to give a nice dual description
of the process of translating such a metric.\\\\ Throughout this paper, $\tau:T^\ast M\rightarrow M$ will denote the projection map. \subsection{Definitions and general results} The {\it Legendre transformation} of a conic pseudo-Finsler metric $(M,F)$ is the map \begin{equation}\nonumber \mathscr{L}_F~:~A\longrightarrow T^\ast M,~~~~~~\mathscr{L}_F(u)=\frac{1}{2}D_fF^2(u), \end{equation} where $D_f$ stands for the fiber-derivative. Then, $\mathscr{L}_F$ is positively homogeneous of degree one, and \begin{equation}\label{legendretransform} \mathscr{L}_F(u)~=~g_u(u,\cdot), \end{equation} so that $\mathscr{L}_F(u)$ is the only covector characterized by \begin{equation}\label{affinetangenthyperplane} \mathscr{L}_F(u)\Bigl(u/F(u)+T_{u/F(u)}\Sigma_{\pi(u)}\Bigr)~=~F(u). \end{equation} The fiber-derivative of $\mathscr{L}_F$ is given by \begin{equation}\label{DfL} D_f\mathscr{L}_F(u)(v)~=~g_u(v,\cdot), \end{equation} hence, as the fundamental tensor $g$ is non-degenerate everywhere, it follows that $\mathscr{L}_F$ is a local diffeomorphism. \begin{defi} Given $\xi\in\mathscr{L}_F(A)$, we define, for each $v\in A$ with $\mathscr{L}_F(v)=\xi$, the {\it Legendre dual} of $F$ around $\xi$ as the map \begin{equation}\nonumber F^\ast~=~F\circ\mathscr{L}_F^{-1} \end{equation} defined on some neighborhood of $\xi$ in $\mathscr{L}_F(A)$ onto which $\mathscr{L}_F$ maps diffeomorphically a neighborhood of $v$ in $A$. \end{defi}
\noindent Viewing $\mathscr{L}_F^{-1}$ and $F^\ast$ as global multivalued functions on $\mathscr{L}_F(A)$, we can describe $F^*$ as follows.
\begin{prop}\label{criticalvalues} Given $\xi\in\mathscr{L}_F(A)$, we have \begin{equation}
F^\ast(\xi) = \text{Positive critical values of $\xi|_{\Sigma_{\tau(\xi)}}$,} \end{equation} and to each corresponding critical point $v\in\Sigma_{\tau(\xi)}$, there is a unique $\lambda>0$ such that $\mathscr{L}_F(\lambda v)=\xi$. Moreover,
\[\mathscr{L}_F(A)=\{\xi\in T^*M: \xi|_{\Sigma_{\tau(\xi)}} \text{admits positive critical points}\}.\] \end{prop} \begin{proof}
Let $\xi=\mathscr{L}_F(u)$, so that $F(u)\in F^\ast(\xi)$. It is straightforward to check that \eqref{affinetangenthyperplane} implies that $u/F(u)$ is a critical point of $\xi|_{\Sigma_{\tau(\xi)}}$ and $\xi(u/F(u))=F(u)>0$. Conversely, if $v\in \Sigma_{\tau(\xi)}$ is a critical point of $\xi|_{\Sigma_{\tau(\xi)}}$, then $\xi$ is constant on the hyperplane $P=v+T_{v}\Sigma_{\pi(v)}$. If $\xi(v)=\lambda>0$, it follows that $\xi=\mathscr{L}_F(\lambda v)$ (since both coincide in the hyperplane $P$) and $F^*(\xi)=F(\lambda v)=\lambda$. The last assertion follows easily from the previous reasoning. \end{proof}
\begin{rem}\label{propminmax}
If we particularize to the conic Finsler or Lorentz-Finsler case, we get the following description.
If $F$ is $(1)$ conic Finsler, or $(2)$ Lorentz Finsler, and $A$ is convex (or $A=TM\setminus\{\mbox{zero section}\}$ if $F$ is Finsler), then the map $\mathscr{L}_F$ is a diffeomorphism onto its image and, hence, the Legendre dual $F^\ast$ is globally defined. Moreover, \begin{eqnarray} (1)~~F^\ast(\xi) & = & \max_{\Sigma_{\tau(\xi)}}\xi\nonumber\\ (2)~~F^\ast(\xi) & = & \min_{\Sigma_{\tau(\xi)}}\xi.\nonumber \end{eqnarray} \end{rem}
For future reference, we note that the inverse of $\mathscr{L}_F$ is simply the {\it Legendre transformation of $F^\ast$}, as can be shown by a direct computation employing the homogeneity of $F$ and $F^\ast$. \begin{prop}\label{propinverse} The inverse $\mathscr{L}_F^{-1}:\mathscr{L}_F(A)\rightarrow A$ is the fiber derivative of $(1/2)(F^\ast)^2$: \begin{equation}\nonumber \mathscr{L}_F^{-1}(\xi)~=~(1/2)D_f(F^\ast)^2(\xi). \end{equation} \end{prop}
\subsection{The Legendre dual of a translation} Let $(V,\hat{F})$ be a conic pseudo-Minkowski norm obtained as the translation of a conic pseudo-Minkowski norm $(V,F)$ and a vector $W$ (recall Definition \ref{transMink}). It will be convenient to consider the {\it reverse metric} of $F$, which is defined by \begin{equation}\nonumber \tilde{F}~:~-A\longrightarrow(0,\infty),~~~\tilde{F}(v)=F(-v). \end{equation}
\noindent Let us denote by $\hat\Sigma$ and $\Sigma$ the indicatrices of $\hat F$ and $F$, respectively. Observe that the indicatrices of $\tilde{F}$ is $-\Sigma$. \begin{lemma}\label{charStRv} $\hat F$ is a straight (resp. reverse) translation of $F$ if and only if $\mathscr{L}_{\hat F}(v)(v-W)>0$ (resp. $\mathscr{L}_{\hat F}(v)(v-W)<0$) for every $v\in \hat \Sigma$. \end{lemma} \begin{proof} It follows from a similar reasoning to that of the proof of Proposition \ref{translatingF}. \end{proof}
Given $\xi\in\mathscr{L}_{\hat{F}}(\hat{A})$, we have that if a vector $v\in\hat \Sigma$ is a critical point of $\xi|_{\hat\Sigma}$, with positive critical value, then the vector $v-W\in\Sigma$ (resp. $W-v\in-\Sigma$) is a critical point of $\xi|_{\Sigma}$ (resp. $\xi|_{-\Sigma}$) with positive critical value if $\hat F$ is a straight (resp. reverse) translation (recall Proposition \ref{translatingF}). Therefore, Proposition \ref{criticalvalues} immediately gives the following result. \begin{prop}\label{propdichotomy} If $\hat F$ is a straight (resp. reverse) translation of $F$, then $\mathscr{L}_{\hat F}(\hat{A})\subset\mathscr{L}_{F}(A)$ (resp. $\mathscr{L}_{\hat F}(\hat{A})\subset\mathscr{L}_{\tilde{F}}(-A)$) and, to each local inverse $\mathscr{L}_{\hat{F}}^{-1}:\mathcal{O}\subset\mathscr{L}_{\hat{F}}(\hat{A})\rightarrow \hat{A}$ corresponds a unique local inverse $\mathscr{L}_{F}^{-1}:\mathcal{O}\rightarrow A$ (resp. $\mathscr{L}_{\tilde{F}}^{-1}:\mathcal{O}\rightarrow-A$) such that, for $\xi\in\mathcal{O}$, \begin{eqnarray} \mathscr{L}_{\hat{F}}^{-1}(\xi)/\hat{F}^\ast(\xi) & = & W+\mathscr{L}_{F}^{-1}(\xi)/F^\ast(\xi)\nonumber\\ (\mbox resp.)~~\mathscr{L}_{\hat{F}}^{-1}(\xi)/\hat{F}^\ast(\xi) & = & W-\mathscr{L}_{\tilde{F}}^{-1}(\xi)/\tilde{F}^\ast(\xi).\label{legendrerelations} \end{eqnarray} Furthermore, on $\mathcal{O}$, \begin{eqnarray} \hat F^\ast(\xi) & = & F^\ast(\xi)+W(\xi)\nonumber\\ (\mbox{resp.})~~\hat F^\ast(\xi) & = & -\tilde{F}^\ast(\xi)+W(\xi)\label{legendreduallorentz}, \end{eqnarray} where $W(\xi)$ is the usual action of a covector on a vector, namely $\xi(W)$. \end{prop}
\section{Flag curvature and geodesics of translations}\label{sectionflagcurvature}
Let $(M,\hat{F})$ be a translation of a pseudo-Finsler manifold $(M,F)$ by a vector field $W$. In this section we will be concerned with the special case in that $W$ is a homothetic vector field (see \eqref{homothetic}), in which case we will prove Theorems \ref{geodesicflow} and \ref{theoremcurvature}. Let us point out the following corollary of Theorem \ref{theoremcurvature}:
\begin{cor} If $F$ has constant flag curvature equal to $K$, then $\hat{F}$ has constant flag curvature equal to $K-(1/4)\sigma^2$. \end{cor}
We start by explaining the new approach to flag curvature in $\S$\ref{sectioncurvature}, $\S$\ref{sectioncurves}, $\S$\ref{sectionflagdefi}, and postpone the proofs of Theorems \ref{geodesicflow} and \ref{theoremcurvature} to $\S$\ref{sectionproof}.
\subsection{Contact geometry of a pseudo-Finsler metric}\label{sectioncurvature} Throughout this section, let $F$ be a conic pseudo-Finsler metric with domain $A$. As we will only be concerned with local questions, we will therefore suppose that $F$ has a globally defined Legendre dual \begin{equation}\label{hamiltonian} \frac{1}{2}(F^\ast)^2~:~\mathscr{L}_F(A)\longrightarrow\mathbb{R}. \end{equation} Let us recall that $T^\ast M$, and hence $\mathscr{L}_F(A)$, possesses a canonical symplectic structure $\omega$: this is the differential of the {\it canonical $1$-form} $\alpha$ on $T^\ast M$, which is defined by \begin{equation} \alpha(X)~=~\xi\bigl(D\tau(\xi)(X)\bigr)~,~~~~\mbox{for $X\in T_\xi(T^\ast M)$.} \end{equation} Viewing (\ref{hamiltonian}) as a Hamiltonian function, it defines a Hamiltonian vector field $S$ on $\mathscr{L}_F(A)$ through the relation \begin{equation}\label{hamiltonianvectorfield} \frac{1}{2}D(F^\ast)^2~=~\omega(\cdot\hspace{0.05cm},\hspace{0.05cm}S), \end{equation} and the corresponding flow $\psi_t^S$, called co-geodesic flow of $F$, acts on $\mathscr{L}_F(A)$ by symplectic diffeomorphisms. Indeed, it is not hard to show that \begin{equation}\label{pullbackalpha} (\psi_t^S)^\ast\alpha~=~\alpha+t\frac{1}{2}D(F^\ast)^2. \end{equation} As, by definition, the geodesics of $F$ are the critical points of the energy functional \begin{equation}\nonumber E~:~C^F_{p,q}\rightarrow\mathbb{R},~~~E(\gamma)~=~\frac{1}{2}\int_a^bF(\dot{\gamma}(t))^2dt, \end{equation} where $C^F_{p,q}$ is the set of piecewise smooth admissible curves joining $p$ to $q$, the Hamilton's principle applies to show that the geodesics of $F$ are precisely the projections onto $M$ of the integral lines of $\psi_t^S$.
The manifold $\mathscr{L}_F(A)$ is foliated by the hypersurfaces $(F^\ast)^{-1}(r)$, $r>0$, each of which is invariant by the flow $\psi_t^S$ as it follows from (\ref{hamiltonianvectorfield}). It will be more convenient to work with the restriction of $\psi_t^S$ to theses hypersurfaces. First, le us recall the following fundamental notion.
\begin{defi} An {\it exact contact manifold} is a manifold $X$ endowed with a 1-form $\alpha$ with the property that the restriction of $\omega=d\alpha$ to the distribution $X\ni x\mapsto\mathcal{C}_x:=\ker(\alpha_x)\subset T_xX$ is non-degenerate (hence, symplectic). We call $\mathcal{C}$ the {\it contact plane distribution}. Furthermore, \begin{enumerate}
\item an {\it exact contact flow} on $X$ is a flow $\psi_t$ such that $(\psi_t)^\ast\alpha=\alpha$. It follows that such a flow leaves the distribution $\mathcal{C}$ invariant and its derivatives $D\psi_t(x)|_{\mathcal{C}_x}:\mathcal{C}_x\rightarrow\mathcal{C}_{\psi_t(x)}$ will consist of symplectic linear maps.
\item A distribution $\mathcal{L}$ on $X$ is called {\it Legendrean} if, for each $x$, $\mathcal{L}_x$ is a Lagrangian subspace of the symplectic space $(\mathcal{C}_x,\omega_x|_{\mathcal{C}_x})$, i.e., $\omega_x|_{\mathcal{L}_x}=0$ and dim$\mathcal{L}_x=\frac{1}{2}{\rm dim}\mathcal{C}_x$. \end{enumerate} \end{defi}
The following result is well-known; we include a proof here as we will need some facts established along it.
\begin{prop}\label{propcontact} Endowed with the pull-back of the canonical $1$-form $\alpha$, $(F^\ast)^{-1}(r)$ is an exact contact manifold, with contact plane distribution $\mathcal{C}$. Furthermore, \begin{enumerate}[(i)] \item The flow $\psi_t^S$ acts on $(F^\ast)^{-1}(r)$ as an exact contact flow. \item the distribution $\mathcal{L}$ on $(F^\ast)^{-1}(r)$ given by $\mathcal{L}_\xi=T_\xi\bigl(\tau^{-1}(x)\cap(F^\ast)^{-1}(r)\bigr)$ is Legendrean. \end{enumerate} \end{prop}
\begin{proof} Let $C$ be the tautological vector field on $T^\ast M$, i.e., for each $\xi\in T^\ast M$, $C_\xi=\xi\in T_\xi(\tau^{-1}(\tau(\xi)))\subset T_\xi(T^\ast M)$ (where we have identified $T_\xi(\tau^{-1}(\tau(\xi)))$ with $T^\ast_{\tau(\xi)}M$ in the canonical way). It is known that $C$ is the $\omega$-dual of $\alpha$: \begin{equation}\label{alphaomega} \alpha~=~\omega(C\hspace{0.05cm},\hspace{0.05cm}\cdot). \end{equation} The relations (\ref{hamiltonianvectorfield}) and (\ref{alphaomega}), together with the Euler relation for 2-homogeneous functions, give $\alpha_\xi(S_\xi)=\frac{1}{2}D(F^\ast)^2(\xi)(C_\xi)=F^\ast(\xi)^2\neq 0$, showing that $C$ is everywhere transverse to $(F^\ast)^{-1}(r)$ and $S$ is never tangent to the distribution $\mathcal{C}$. Hence, \begin{equation}\label{decomposition} T_\xi( T^\ast M)~=~\mathcal{C}_\xi\oplus{\rm span}\{S_\xi,C_\xi\}. \end{equation} Again from (\ref{hamiltonianvectorfield}) and (\ref{alphaomega}), \begin{equation}\label{contactplane} \mathcal{C}_\xi~=~\ker\omega_\xi(S_\xi\hspace{0.05cm},\hspace{0.05cm}\cdot)\cap\ker\omega_\xi(C_\xi\hspace{0.05cm},\hspace{0.05cm}\cdot) \end{equation} and so the decomposition (\ref{decomposition}) is $\omega_\xi$-orthogonal. As $\omega_\xi$ is non-degenerate on $T_\xi(T^\ast M)$ and on ${\rm span}\{S_\xi,C_\xi\}$ (because $\omega_\xi(C_\xi,S_\xi)=\alpha_\xi(S_\xi)\neq 0$), it follows from (\ref{decomposition}) that it is non-degenerate on $\mathcal{C}_\xi$. For assertion $(i)$, we pull-back the relation (\ref{pullbackalpha}) to $(F^\ast)^{-1}(r)$, to get $(\psi_t^S)^\ast\alpha=\alpha$ (omitting pull-back's). For $(ii)$, as $\alpha$ vanishes on vectors tangent to $\tau^{-1}(x)$, its pull-backs to the leaves of the foliation $x\mapsto\tau^{-1}(x)\cap(F^\ast)^{-1}(r)$ of $(F^\ast)^{-1}(r)$ are null, and hence $d\alpha$ pulls back to the null form on each leaf. \end{proof}
\noindent It follows from the last proposition that the flow $\psi_t^S$ carries $\mathcal{L}$ into Lagrangian subspaces of $\mathcal{C}$. Let $\xi\in(F^\ast)^{-1}(r)$ be fixed, and denote by $\mathbb{V}$ the symplectic vector space $(\mathcal{C}_\xi,\omega|_{\mathcal{C}_\xi})$. The collection of all Lagrangian subspaces of $\mathbb{V}$, denoted by $\Lambda(\mathbb{V})$, possesses a natural smooth structure and is called the {\it Lagrangian Grasmannian manifold} of $\mathbb{V}$. \begin{defi}\label{Jacobicurve} The {\it Jacobi curve} of $(\ref{hamiltonian})$, based at $\xi$, is the smooth curve in $\Lambda(\mathbb{V})$ given by \begin{equation} \ell_\xi(t)~=~D\psi_{-t}^S(\mathcal{L}_{\psi_t^S(\xi)}). \end{equation} \end{defi}
\subsection{The geometry of curves in $\Lambda(\mathbb{V})$}\label{sectioncurves} Here, we briefly recall the main concepts of \cite{AD} which will appear in the sequel. \\ For each $\ell\in\Lambda(\mathbb{V})$, the tangent space $T_\ell\Lambda(\mathbb{V})$ is canonically isomorphic to the space ${\rm Sym}(\ell)$ of symmetric bilinear forms on $\ell$. Indeed, if $\ell(t)$ is a smooth curve in $\Lambda(\mathbb{V})$ then, for each $\tau$, the following rule defines a symmetric bilinear form $W(\tau)$ on $\ell(\tau)$, \begin{equation}\nonumber W(\tau)(a,b)~=~\omega\bigl(\dot{a}(\tau),b\bigr), \end{equation} where $a(\cdot):(\tau-\varepsilon,\tau+\varepsilon)\rightarrow\mathbb{V}$ is a smooth curve satisfying $a(\tau)=a$, and $a(t)\in\ell(t)$ for all $t$. We call the assignment $t\mapsto W(t)$ the {\it Wronskian} of the curve $\ell(t)$.
\begin{defi} We call the curve $\ell(t)$ {\it fanning} if its Wronskian $W(t)$ is non-degenerate for all $t$. \end{defi} \noindent The condition of being fanning may be translated in terms of frames as follows: if $\mathcal{A}(t)=\{a_1(t),...,a_n(t)\}$ is a smooth frame for $\ell(t)$, then $\ell(t)$ is fanning if, and only if, \begin{equation}\nonumber \{\mathcal{A}(t),\dot{\mathcal{A}}(t)\}=\{a_1(t),...,a_n(t),\dot{a}_1(t),...,\dot{a}_n(t)\} \end{equation} is a smooth frame for $\mathbb{V}$. Hence, if $\mathcal{A}(t)$ is a smooth frame for a fanning curve $\ell(t)$, we may define a smooth curve of endomorphisms ${\bf F}(t):\mathbb{V}\rightarrow\mathbb{V}$ by \begin{eqnarray} {\bf F}(t)a_i(t) & = & 0~~~~~\mbox{for all $i$}\nonumber\\ {\bf F}(t)\dot{a}_i(t) & = & a_i(t)~~~~\mbox{for all $i$}\nonumber \end{eqnarray}
\noindent It turns out that the endomorphisms ${\bf F}(t)$ do not depend on the particular choice of the frame $\mathcal{A}(t)$, but only on $\ell(t)$. We call the assignment $t\mapsto{\bf F}(t)$ the {\it fundamental endomorphism} of $\ell(t)$. Taking the second derivative of ${\bf F}(t)$, we get the main linear invariant of a fanning curve. \begin{defi} The {\it Jacobi endomorphism} of $\ell(t)$ is the smooth curve of endomorphisms ${\bf K}(t):\mathbb{V}\rightarrow\mathbb{V}$ defined by \begin{equation}\nonumber {\bf K}(t)~=~(1/4)\ddot{{\bf F}}(t)^2. \end{equation} For each $t$, ${\bf K}(t)$ restricts to a $W(t)$-symmetric endomorphism of $\ell(t)$. We have the following basic transformation's rule, which will be needed for the proof of Theorem \ref{theoremcurvature}; its proof follows directly from \cite[Theorem 1.4]{AD}. \begin{prop}\label{propfanning} If two fanning curves $\ell(t)$ and $\tilde{\ell}(t)$ are related by $\tilde{\ell}(t)={\rm T}\ell(s(t))$, where ${\rm T}:\mathbb{V}\rightarrow\tilde{\mathbb{V}}$ is a linear symplectic (or anti-symplectic) map, and $s$ is a diffeomorphism of $\mathbb{R}$, then, whenever $W(s(t))(a,a)\neq 0$, \begin{equation}\nonumber \frac{\tilde{W}(t)\bigl(\tilde{\bf K}(t){\rm T}a\hspace{0.05cm},\hspace{0.05cm}{\rm T}a\bigr)}{\tilde{W}(t)\bigl({\rm T}a\hspace{0.05cm},\hspace{0.05cm}{\rm T}a\bigr)}~=~\dot{s}(t)^2\frac{W(s(t))\bigl({\bf K}(s(t))a\hspace{0.05cm},\hspace{0.05cm}a\bigr)}{W(s(t))\bigl(a\hspace{0.05cm},\hspace{0.05cm}a\bigr)}+\frac{1}{2}\{s(t),t\}, \end{equation} where $W(t)$, $\tilde{W}(t)$, ${\bf K}(t)$, $\tilde{{\bf K}}(t)$ are the Wronskians and Jacobi endomorphisms of $\ell(t)$ and $\tilde{\ell}(t)$, respectively, and $\{s(t),t\}=(d/dt)(\ddot{s}/\dot{s})-(1/2)(\ddot{s}/\dot{s})^2$ is the {\rm Schwarzian derivative} of $s$. \end{prop} \end{defi}
\subsection{Definition of flag curvature via fanning curves}\label{sectionflagdefi} We go back now to the Jacobi curve $\ell_\xi(t)$; see Definition \ref{Jacobicurve}. The Wronskian $W_\xi(t)$ of $\ell_\xi(t)$ corresponds to the fundamental tensor $g$ of $F$ as we now explain. \\\\ Let $v\in A$ be such that $\mathscr{L}_F(v)=\xi$, and let $\gamma(t)$ be the geodesic of $F$ with $\dot{\gamma}(0)=v$, so that we have $\mathscr{L}_F(\dot{\gamma}(t))=\psi_t^S(\xi)$. We will now describe an isomorphism \begin{equation}\label{isomorphismcurve} \iota_{\dot{\gamma}(t)}~:~\ker g_{\dot{\gamma}(t)}(\dot{\gamma}(t)\hspace{0.05cm},\hspace{0.05cm}\cdot)\longrightarrow\ell_\xi(t). \end{equation} For this, note that $\mathcal{L}_{\psi_t(\xi)}$ is canonically a subspace of $T^\ast_{\gamma(t)}M$ and, via the isomorphism $D_f\mathscr{L}_F(\dot{\gamma}(t)):T_{\gamma(t)}M\rightarrow T_{\gamma(t)}^\ast M$, it corresponds to the tangent space, at $\dot{\gamma}(t)$, of the dilatation by $r$ of $\Sigma_{\gamma(t)}$, which in turn is equal to $\ker g_{\dot{\gamma}(t)}(\dot{\gamma}(t)\hspace{0.05cm},\hspace{0.05cm}\cdot)$. So we define $\iota_{\dot{\gamma}(t)}$ by composing the restriction of $D_f\mathscr{L}_F(\dot{\gamma}(t))$ to $\ker g_{\dot{\gamma}(t)}(\dot{\gamma}(t)\hspace{0.05cm},\hspace{0.05cm}\cdot)$ with $D\psi_{-t}^S$. For $t=0$, this is simply
\begin{eqnarray}\label{isomorphismzero}
\iota_v=D_f\mathscr{L}_F(v)|_{\ker g_v(v,\cdot)}:~\ker g_v(v\hspace{0.05cm},\hspace{0.05cm}\cdot) & \longrightarrow &\ell_\xi(0)=\mathcal{L}_\xi\\
w & \longmapsto & g_v(w,\cdot)\nonumber. \end{eqnarray} We refer to \cite{ADH} (see also \cite{Henrique}) for a proof of the following result. \begin{prop}\label{propwronskian} Via the isomorphism $(\ref{isomorphismcurve})$, the Wronskian $W_\xi(t)$ of $\ell_\xi(t)$ corresponds to the restriction of $g_{\dot{\gamma}(t)}$ to $\ker g_{\dot{\gamma}(t)}(\dot{\gamma}(t)\hspace{0.05cm},\hspace{0.05cm}\cdot)$. \end{prop} \noindent It follows from this proposition that $\ell_\xi(t)$ is a fanning curve; see $\S$\ref{sectioncurves}. Let ${\bf K}_\xi(t)$ be its Jacobi endomorphism.
\begin{defi}\label{defincurvature} Let $\Pi\subset T_xM$ be a 2-plane containing $v$. \begin{enumerate} \item We call $\Pi$ $v$-{\it nondegenerate} if the restriction of $g_v$ to $\Pi$ is nondegenerate. This is the same as requiring that the quantity $g_v(v,v)g_v(u,u)-g_v(u,v)^2$ be non null for all $u$ such that $\Pi={\rm span}\{u,v\}$. \item If $\Pi$ is $v$-nondegenerate, we define the {\it flag curvature} of the flag $(v,\Pi)$ as follows: pick a non null vector $w\in\Pi$, with $g_v(v,w)=0$, and let $a\in\ell_\xi(0)$ corresponding to $w$ via $(\ref{isomorphismzero})$. From Proposition \ref{propwronskian} and the $v$-nondegenerescence of $\Pi$, $W_\xi(0)(a,a)\neq 0$, and we may define \begin{equation}\nonumber K_F(v,\Pi)~=~\frac{1}{F^2(v)}\frac{W_\xi(0)\bigl({\bf K}_\xi(0)a\hspace{0.01cm},\hspace{0.01cm}a\bigr)}{W_\xi(0)(a\hspace{0.01cm},\hspace{0.01cm}a)}. \end{equation} \end{enumerate} \end{defi}
\noindent Of course, the above definition of flag curvature coincides with the usual ones; a proof of this fact may be found in \cite{ADH} (see also \cite{Henrique}).
We end this section by remarking the following simple fact which we will need for the proof of Theorem \ref{theoremcurvature}: the flag curvatures of a metric $F$ and of its reverse $\tilde{F}$ are related by \begin{equation}\label{remarkreverse2} K_{\tilde{F}}(v,\Pi)~=~K_F(-v,\Pi). \end{equation}
\subsection{Proof of Theorems \ref{theoremcurvature} and \ref{geodesicflow}}\label{sectionproof}
Along the proof we will use the following convention for $\epsilon$ and $F_\epsilon$: \begin{enumerate}[(a)] \item if $\hat F$ is a straight translation of $F$, then $\epsilon=1$ and $F_\epsilon=F_1=F$, \item if $\hat{F}$ is a reverse translation of $F$, then $\epsilon=-1$ and $F_\epsilon=F_{-1}=\tilde{F}$. \end{enumerate} Set $\xi=\mathscr{L}_{\hat{F}}(u)$, and let $\mathcal{O}\subset\mathscr{L}_{\hat{F}}(\hat{A})$ be a neighborhood of $\xi$ where $\mathscr{L}_{\hat{F}}$ has a local inverse \begin{equation}\nonumber \mathscr{L}_{\hat{F}}^{-1}~:~\mathcal{O}\longrightarrow \hat A. \end{equation} According to Proposition \ref{propdichotomy}, this determines a local inverse $\mathscr{L}_{F_\epsilon}^{-1}:\mathcal{O}\rightarrow \epsilon A$ such that \begin{equation} \mathscr{L}_{F_\epsilon}^{-1}(\xi)/F_\epsilon^\ast(\xi)~=~-\epsilon W+\epsilon u/\hat{F}(u) \end{equation} and, for simplicity, we will denote by $H_\epsilon$ and $H_2$, respectively, the Hamiltonian functions \begin{equation}\nonumber {F_\epsilon}^\ast,~\hat{F}^\ast~:~\mathcal{O}\longrightarrow\mathbb{R}. \end{equation} For $i=\epsilon, 2$, let $S_i$ and $\hat{S}_i$ be the Hamiltonians vector fields of $H_i$ and $(1/2)H_i^2$, respectively. Then, $\hat{S}_i=H_iS_i$ and, therefore, for each $r>0$, \begin{equation}\label{restricflow} \psi_t^{\hat{S}_i}\mid_{H_i^{-1}(r)}~=~\psi_{rt}^{S_i}\mid_{H_i^{-1}(r)}~~~~,~~i=\epsilon,2. \end{equation} Denoting by $H_0$ the Hamiltonian function \begin{equation}\nonumber W~:~\mathcal{O}\longrightarrow\mathbb{R},~~~W(\eta)=\eta(W) \end{equation} and by $S_0$ its Hamiltonian vector field, it follows from (\ref{legendreduallorentz}) that $H_2=\epsilon H_\epsilon+H_0$ and, hence, \begin{equation}\label{sumvectorfields} S_2~=~\epsilon S_\epsilon+S_0. \end{equation} Also, it is well known (see, for instance, \cite{Ziller}) that the flow of $S_0$ is simply \begin{equation}\label{flowS1} \psi_t^{S_0}=(D\psi_t^W)^\ast~:~T^\ast M\longrightarrow T^\ast M. \end{equation} For $i=\epsilon,2$, let $\mathcal{C}^{(i)}$ be the contact plane distribution associated to $(1/2)H_i^2$, and let \begin{equation}\nonumber \ell_\xi^{(i)}(t)\in\Lambda\bigl(\mathcal{C}^{(i)}_\xi\bigr) \end{equation} be the Jacobi curve of $(1/2)H_i^2$ based at $\xi$. It will be convenient to introduce the following curves on $\Lambda(T_\xi T^\ast M)$, \begin{equation}\nonumber \tilde{\ell}_\xi^{(i)}(t)~=~D\psi_{-t}^{S_i}\bigl(\mathcal{V}_{\psi_t^{S_i}(\xi)}T^\ast M\bigr)~~~,~~i=\epsilon,2. \end{equation} where $\mathcal{V}T^\ast M$ is the distribution on $T^\ast M$ given by the tangent spaces to the fibers of $\tau:T^\ast M\rightarrow M$. These curves are related to the former ones by
\begin{lemma} For $i=2,\epsilon$, we have \begin{equation}\label{eq3} \ell_\xi^{(i)}(t/H_i(\xi))=\tilde{\ell}_\xi^{(i)}(t)\cap\mathcal{C}^{(i)}_\xi. \end{equation} \end{lemma} \begin{proof} First note that $\mathcal{V}T^\ast M=\mathcal{L}^{(i)}\oplus{\rm span}\{C\}$. Also, as $H_i$ is positively homogeneous of degree 1, we have $[C,S_i]=0$, and hence $(\psi_t^{S_i})^\ast C=C$. Therefore, it follows from $(\ref{restricflow})$ that $\tilde{\ell}_\xi^{(i)}(t)=\ell_\xi^{(i)}\bigl(t/H_i(\xi)\bigr)\oplus{\rm span}\{C_\xi\}$ and, hence, $\ell_\xi^{(i)}\bigl(t/H_i(\xi)\bigr)=\tilde{\ell}_\xi^{(i)}(t)\cap\mathcal{C}^{(i)}_\xi$. \end{proof}
The proof of Theorem \ref{theoremcurvature} will consist in establishing the following \begin{thm}\label{theoremcurves} For the symplectic isomorphism ${\rm T}:\mathcal{C}^{(2)}_\xi\rightarrow\mathcal{C}^{(\epsilon)}_\xi$ defined below, \begin{equation}\label{relatingcurves} {\rm T}\ell_\xi^{(2)}\bigl(g(t)\bigr)~=~\ell_\xi^{(\epsilon)}\bigl(t\bigr), \end{equation} where $g(t)=\frac{1}{H_2(\xi)\sigma}\ln(1+\epsilon\sigma H_\epsilon(\xi)t)$. \end{thm} \noindent {\it Definition of the map ${\rm T}$:} From $(\ref{decomposition})$, any $X\in\mathcal{C}^{(2)}_\xi$ may be uniquely expressed as $X=Z+aC_\xi+b(S_\epsilon)_\xi$, with $Z\in\mathcal{C}^{(\epsilon)}_\xi$. We define ${\rm T} (X)=Z$. To see that it is symplectic, we first note that we must have $b=0$. Indeed, from $(\ref{contactplane})$ we have $\omega(X,C_\xi)=\omega(Z,C_\xi)=0$, hence $b\,\omega((S_\epsilon)_\xi,C_\xi)=0$. But, $\omega\bigl((S_\epsilon)_\xi,C_\xi\bigr)=-F_\epsilon(\xi)\neq 0$, so $b=0$. Now, the fact that ${\rm T}$ is symplectic is an immediate consequence of $(\ref{contactplane})$. \\\\ In order to prove $(\ref{relatingcurves})$, we start by showing that the hypothesis $(\ref{homothetic})$ implies the following relation between the flows $\psi_t^{S_2},\psi_t^{S_\epsilon},\psi_t^{S_0}$: \begin{prop}\label{propflow} We have that $\psi_t^{S_2}=\psi_t^{S_0}\circ\psi_{f(t)}^{S_\epsilon}$, where \[f(t)=\begin{cases} (\epsilon/\sigma)({\rm e}^{\sigma t}-1)& \text{if $\sigma\not=0$,}\\ t& \text{if $\sigma=0$,} \end{cases} \] for $t$ in a neighborhood of $0$. \end{prop} \noindent For the proof of the above proposition, as it is clear that $(\ref{homothetic})$ also holds for the reverse metric $\tilde{F}$, it follows from $(\ref{flowS1})$ that \begin{equation} H_\epsilon\circ\psi_t^{S_0}~=~{\rm e}^{\sigma t}H_\epsilon, \end{equation}
at least in a neighborhood of $t=0$ (and whenever the flow $\psi_t^W$ is well-defined). By taking the derivative of this relation at $t=0$, and recalling that $DH_\epsilon(X)=\omega(X,S_\epsilon)$, we get $\omega(S_0,S_\epsilon)=\sigma H_\epsilon$. Hence, as $\omega(S_0,S_\epsilon)$ is equal to the Poisson bracket $\{H_\epsilon,H_0\}$, it follows that \begin{equation}\label{bracket} [S_0,S_\epsilon]~=~\sigma S_\epsilon. \end{equation} We can now apply the following general lemma to prove Proposition \ref{propflow} (for a proof, see \cite{ADH}): \begin{lemma}\label{lemmaflow} Let $X_0$ and $X_1$ be vector fields on some manifold, and let $Z$ be the time-dependent vector field defined by $Z_t=(\psi_t^{X_1})^\ast X_0$. Then, $\psi_t^{X_0+X_1}=\psi_t^{X_1}\circ\psi_t^Z$, where, for each $x$, $\psi_t^Z(x)$ is the evaluation at $t$ of the sotution to $\dot{x}(t)=Z_{(t,x(t))}$, $x(0)=x$. \end{lemma} \begin{proof}[Proof of Proposition \ref{propflow}] From $(\ref{sumvectorfields})$ and the lemma above, $\psi_t^{S_2}=\psi_t^{S_0}\circ\psi_t^Z$, where $Z_t=\epsilon(\psi_t^{S_0})^\ast S_\epsilon$. Deriving in $t$ the last equation, and using $(\ref{bracket})$, we obtain, successively, \begin{eqnarray} \frac{d}{dt}(\psi_t^{S_0})^\ast S_\epsilon & = & (\psi_t^{S_0})^\ast[S_0,S_\epsilon]\nonumber\\ & = & \sigma(\psi_t^{S_0})^\ast S_\epsilon.\nonumber \end{eqnarray} Hence, $Z_t=\epsilon{\rm e}^{\sigma t}S_\epsilon$. It is now immediate that $\psi_t^Z=\psi_{\frac{\epsilon}{\sigma}({\rm e}^{\sigma t}-1)}^{S_\epsilon}$.
\end{proof}
\begin{proof}[Proof of Theorem \ref{theoremcurves}] It follows from Proposition \ref{propflow}, and from the fact that $\psi_t^{S_0}$ preserves the fibers of $\tau:T^\ast M\rightarrow M$, that \begin{equation}\nonumber \tilde{\ell}_\xi^{(2)}(t)~=~\tilde{\ell}_\xi^{(\epsilon)}(f(t)). \end{equation} Therefore, having in mind (\ref{eq3}) and the definition of ${\rm T}$, we immediately get \begin{equation}\nonumber {\rm T}(\ell_\xi^{(2)}\bigl(t/H_2(\xi)\bigr))=\ell_\xi^{(\epsilon)}\bigl(f(t)/H_\epsilon(\xi)\bigr). \end{equation} This, in turn, is equivalent to (\ref{relatingcurves}). \end{proof}
\noindent In order to apply Proposition \ref{propfanning} (at $t=0$) to (\ref{relatingcurves}), we first observe that \begin{equation}
\frac{1}{2}\{g(t),t\}|_{t=0}=\frac{1}{4}\sigma^2H_\epsilon(\xi)^2~~~{\rm and}~~~\dot{g}(0)^2=\frac{H_\epsilon(\xi)^2}{H_2(\xi)^2}. \end{equation} \noindent Therefore, it follows from Proposition \ref{propfanning}, and the definition of flag curvature, that
\begin{equation}\nonumber K_{\hat{F}}\Bigl(\mathscr{L}_{\hat{F}}^{-1}(\xi)\hspace{0.05cm},\hspace{0.05cm}{\rm span}\{\mathscr{L}_{\hat{F}}^{-1}(\xi),w\}\Bigr)~=~K_{{F_\epsilon}}\Bigl(\mathscr{L}_{F_\epsilon}^{-1}(\xi)\hspace{0.05cm},\hspace{0.05cm}{\rm span}\{\mathscr{L}_{F_\epsilon}^{-1}(\xi),\tilde{w}\}\Bigr)-\frac{1}{4}\sigma^2, \end{equation} where $\tilde{w}$ is the image of $w$ under the map \begin{equation}\label{map} (D_f\mathscr{L}_{F_\epsilon}(u))^{-1}\circ{\rm T}\circ \bigl(D_f\mathscr{L}_{\hat{F}}(v)\bigr)~:~\ker g_u^{\hat{F}}(u\hspace{0.05cm},\hspace{0.05cm}\cdot)\longrightarrow\ker g^{F_\epsilon}_v(v\hspace{0.05cm},\hspace{0.05cm}\cdot) \end{equation}
and $v:=\mathscr{L}_{F_\epsilon}^{-1}(\xi)$. Theorem \ref{theoremcurvature} now follows from (\ref{remarkreverse2}) and the following lemma. \begin{lemma} The map $(\ref{map})$ is multiplication by $\epsilon H_\epsilon(\xi)/H_2(\xi)$. \end{lemma} \begin{proof} Via the canonical identification $\mathcal{V}_\xi T^\ast M=T_{\tau(\xi)}^\ast M$, $C_\xi$ corresponds to $\xi$ and, from Proposition \ref{propinverse}, $\mathcal{L}_\xi^{(\epsilon)}$ and $\mathcal{L}_\xi^{(2)}$ correspond to $\ker\mathscr{L}_{F_\epsilon}^{-1}(\xi)$ and $\ker\mathscr{L}_{\hat{F}}^{-1}(\xi)$, respectively. It follows that ${\rm T}\mid_{\mathcal{L}_\xi^{(2)}}:\mathcal{L}_\xi^{(2)}\rightarrow\mathcal{L}_\xi^{(\epsilon)}$ is the restriction to $\ker\mathscr{L}_{\hat{F}}^{-1}(\xi)$ of the projection map $\ker\mathscr{L}_{F_\epsilon}^{-1}(\xi)\oplus{\rm span}\{\xi\}\rightarrow\ker\mathscr{L}_{F_\epsilon}^{-1}(\xi)$, and hence \begin{equation}\label{eq100} {\rm T}(\eta)~=~\eta-\frac{\mathscr{L}_{F_\epsilon}^{-1}(\xi)(\eta)}{\mathscr{L}_{F_\epsilon}^{-1}(\xi)(\xi)}\xi, \end{equation} for all $\eta\in\ker\mathscr{L}_{\hat{F}}^{-1}(\xi)$. Taking the fiber derivative of $(\ref{legendrerelations})$, we get, for all $\zeta\in T_{\tau(\xi)}^\ast M$, \begin{eqnarray} D_f\mathscr{L}_{F_\epsilon}^{-1}(\xi)(\zeta) & = & \frac{\epsilon H_\epsilon(\xi)}{H_2(\xi)}D_f\mathscr{L}_{\hat{F}}^{-1}(\xi)(\zeta)- \frac{\epsilon H_\epsilon(\xi)}{H_2(\xi)^3}\mathscr{L}_{\hat{F}}^{-1}(\xi)(\zeta)\mathscr{L}_{\hat{F}}^{-1}(\xi)\nonumber\\ & & +\frac{1}{H_\epsilon(\xi)^2}\mathscr{L}_{F_\epsilon}^{-1}(\xi)(\zeta)\mathscr{L}_{F_\epsilon}^{-1}(\xi).\nonumber \end{eqnarray} Letting, in the above equality, $\zeta$ be the right hand side of $(\ref{eq100})$, and using that $\mathscr{L}_{F_\epsilon}^{-1}(\xi)(\zeta)=\mathscr{L}_{\hat{F}}^{-1}(\xi)(\eta)=0$, $\mathscr{L}_{F_\epsilon}^{-1}(\xi)(\xi)=H_\epsilon(\xi)^2$, $\mathscr{L}_{\hat{F}}^{-1}(\xi)(\xi)=H_2(\xi)^2$ and $D_f\mathscr{L}_{\hat{F}}^{-1}(\xi)(\xi)=\mathscr{L}_{\hat{F}}^{-1}(\xi)$, we get $D_f\mathscr{L}_{F_\epsilon}^{-1}(\xi)(\zeta)=\epsilon (H_\epsilon(\xi)/H_2(\xi))D_f\mathscr{L}_{\hat{F}}^{-1}(\xi)(\eta)$ and the lemma follows. \end{proof}
\begin{proof}[Proof of Theorem \ref{geodesicflow}]
The expression for the geodesics follows easily from Proposition \ref{propflow}. Indeed, observe that the geodesics of the reverse pseudo-Finsler metric $\tilde{F}$ can be expressed as $\tilde{\gamma}(t)=\gamma(-t)$, where $\gamma$ is a geodesic of $F$. For the last claim, we need to prove that if $\hat{\gamma}$ is always a geodesic for the straight or for the reverse translation, but it cannot change from one to the other. Let us show first $\hat{\gamma}$ is $\hat{F}$-unit. We know that $\dot{\hat{\gamma}}(0)=W+\dot\gamma(0)$, $\dot{\hat{\gamma}}(t)=W+\dot\gamma(t)+(\psi_t^W)^*(\dot f(t) \dot\gamma(f(t)))$ and \[F((\psi_t^W)^*(\dot f(t) \dot\gamma(f(t))))=e^{\sigma t} ((\psi_t^W)^*F( \dot\gamma(f(t)))=F(\dot\gamma(t))=1.\] This implies that $\hat\gamma$ is an $\hat F$-unit curve. In order to apply Proposition \ref{translatingF} we need to show that if $v(t)=(\psi_t^W)^*(\dot f(t) \dot\gamma(f(t)))$, then $g_{v(t)}(v(t),v(t)+W)$ has always the same sign (it cannot be zero): \begin{align*} g_{v(t)}(v(t),v(t)+W)&=1+g_{v(t)}(v(t),W)=1+e^{\sigma t}g_{(\psi_t^W)^*( \dot\gamma(f(t)))}((\psi_t^W)^*( \dot\gamma(f(t))),W)\\ &=1+e^{-\sigma t} g_{\dot\gamma(f(t))}(\dot\gamma(f(t)),W). \end{align*} Moreover, using that $W$ is a homothetic field, we deduce that $g_{\dot\gamma(t)}(\dot\gamma(t),W)=g_v(v,W)-\sigma t$. Substituting above we finally get \[g_{v(t)}(v(t),v(t)+W)=e^{-\sigma t}(g_v(v,W)+1),\] which concludes. \end{proof}
\section{Conclusions and consequences}
The first implication of Theorem \ref{theoremcurvature} is that we can reobtain all the Randers and Kropina metrics of constant flag curvature (see \cite{BCS04,Xia13,YoSa14,YoOk12}). In fact, all the metrics that we obtain are geodesically complete, since homothetic vector fields are complete in the spaces of constant curvature (see Remark \ref{geodesicflow}). In particular, the examples of Randers manifolds with constant flag curvature in \cite{BCS04} with the homothetic vector field with norm bigger than one in some subset can be extended to conic Finsler metrics defined in the whole manifold which are geodesically complete. Indeed, this metric is given by \eqref{Zermelo2} for $\varepsilon=1$ and it is defined in \[A=\{v\in TM: \text{s. t. for $\pi(v)$, } g(W,W)<1\}\cup \{v\in TM: g(v,W)>0;h(v,v)>0\}.\] Recall that Proposition \ref{transZer} ensures that if $g$ is a Riemannian metric, then this metric has positive definite fundamental tensor, namely, it is a conic Finsler metric.
Observe that Theorem \ref{thmmatsumoto} and Proposition \ref{Randers-Kro} ensure that pseudo-Randers-Kropina metrics are characterized as the pseudo-Finsler metrics with Matsumoto tensor trivially equal to zero. It is expectable that all the pseudo-Randers-Kropina metrics with constant flag curvature are those obtained as translation of a pseudo-Riemannian metric $g$ with constant sectional curvature by a homothetic vector field of $g$ (see Proposition \ref{Randers-Kro}). To prove this, it is enough to extend the computations in \cite{BR03,BCS04} for pseudo-Randers metrics and those in \cite{Xia13,YoSa14,YoOk12} for pseudo-Kropina metrics. This seems very likely because in these computations, the signature of the metric does not seem essential. Even if these computations do not hold in the subset of points where the pseudo-Randers-Kropina metric passes from being pseudo-Kropina to being pseudo-Randers, this subset of points is irrelevant for this computation because it has empty interior.
\end{document} | arXiv |
Would there be an interest in having more specialized chat rooms, like the one for homotopy theory that currently exists?
So far, the homotopy theory one seems to be fairly successful, and it seems that other topics might also enjoy a chat room like it.
Obviously, if such a chat room is to be useful, it will need to have at least a few experts interested in frequenting it, which would then mean that people should be at least somewhat sparing with pings.
I am posting this both as a general question about whether people would be interested in such chat rooms, and as a place where people can plan such chat rooms.
If you have a topic you would like a chat room for, put it as an answer. Upvotes for that answer should then be interpreted as someone saying "yes, I would use such a chat room to at least some extend" (preferably with a comment so it can be gauged who those people who are interested are).
July 3: Well, it is gone now; it is worth starting a chat room and just chatting with random strangers, until such time as people interested in that topic begin to post there. Jon started a Homotopy chat room. It turned out not to have MO as a parent. So I started one, and Jon and I chatted, then some homotopy people. After a bit, it was revealed that my room was also not MO. So Manish merged the two rooms, mushed the posts together.
So, I think the thing to do is settle on a title, begin chatting in the new room with anyone at all on MO, and try to get graduate students involved, because they are sufficiently hyper to keep a room going. Should that work, look for postdocs and new assistant professors in or near the field. Online chat does seem to be a matter of taste. The relatively stable configuration is one or two established people peering in when they have time and are in the right mood, answering questions. Time will tell whether it is possible to hold scheduled sessions with larger numbers of people peering in, informal seminar. Jon is trying something along those lines, so maybe we will be seeing that in the coming weeks.
Some friends and I wanted a chat room based on geometry and QFT (and also whenever topology, representation theory, ... comes in). I've gone ahead and created it: https://chat.stackexchange.com/rooms/82100/geometryphysics. If you're interested, feel free to stop by!
I would like to have a chat room for representation theory (though possibly "algebraic representation theory" would be more precisely the subject I mean when I say representation theory).
I'd really like to see a category theory chatroom. This would obviously have significant overlap with the homotopy theory chatroom, and could potentially detract from both, I don't know. But pure category theory (and this could include $\infty$-category stuff, to try and steal some of the n-labbers away maybe) is pretty lovely.
I think that adding a list of past chatrooms which are now frozen but have been active at least for some time might be a useful addition to this question.
I added the rooms I am aware of and made this post community wiki. Feel free to update the list of rooms.
There were two incarnations of room for algebraic geometry. This one had not much activity apart from the first few days. The other one seem to have generated more activity. (I would say that for the first 6 to 8 months that the activity was not entirely negligible.) I will also add link to Jon Beardsley's advice on what can help to make a room active - which was posted in one algebraic geometry rooms.
A room for discussions about representation theory was suggested by Tobias Kildetoft and he indeed created such a room although it never generated too much activity.
differential geometry as in first two chapters of Kobayashi and Nomizu.
Stacks as in Angelo Vistoli's notes (also available on arXiv).
Is there any one interested for the same?
I would like to see a chatroom that deals with group theory and group theoretic approaches to geometry. I know that geometry isn't a really fashionable subject right now, but I would like to talk to people who are interested in it. | CommonCrawl |
Euclid's Optics
Optics (Greek: Ὀπτικά), is a work on the geometry of vision written by the Greek mathematician Euclid around 300 BC. The earliest surviving manuscript of Optics is in Greek and dates from the 10th century AD.
The work deals almost entirely with the geometry of vision, with little reference to either the physical or psychological aspects of sight. No Western scientist had previously given such mathematical attention to vision. Euclid's Optics influenced the work of later Greek, Islamic, and Western European Renaissance scientists and artists.
Historical significance
Writers before Euclid had developed theories of vision. However, their works were mostly philosophical in nature and lacked the mathematics that Euclid introduced in his Optics.[1] Efforts by the Greeks prior to Euclid were concerned primarily with the physical dimension of vision. Whereas Plato and Empedocles thought of the visual ray as "luminous and ethereal emanation",[2] Euclid’s treatment of vision in a mathematical way was part of the larger Hellenistic trend to quantify a whole range of scientific fields.
Because Optics contributed a new dimension to the study of vision, it influenced later scientists. In particular, Ptolemy used Euclid's mathematical treatment of vision and his idea of a visual cone in combination with physical theories in Ptolemy's Optics, which has been called "one of the most important works on optics written before Newton".[3] Renaissance artists such as Brunelleschi, Alberti, and Dürer used Euclid's Optics in their own work on linear perspective.[4]
Structure and method
Similar to Euclid's much more famous work on geometry, Elements, Optics begins with a small number of definitions and postulates, which are then used to prove, by deductive reasoning, a body of geometric propositions (theorems in modern terminology) about vision.
The postulates in Optics are:
Let it be assumed
1. That rectilinear rays proceeding from the eye diverge indefinitely;
2. That the figure contained by a set of visual rays is a cone of which the vertex is at the eye and the base at the surface of the objects seen;
3. That those things are seen upon which visual rays fall and those things are not seen upon which visual rays do not fall;
4. That things seen under a larger angle appear larger, those under a smaller angle appear smaller, and those under equal angles appear equal;
5. That things seen by higher visual rays appear higher, and things seen by lower visual rays appear lower;
6. That, similarly, things seen by rays further to the right appear further to the right, and things seen by rays further to the left appear further to the left;
7. That things seen under more angles are seen more clearly.[5]
The geometric treatment of the subject follows the same methodology as the Elements.
Content
According to Euclid, the eye sees objects that are within its visual cone. The visual cone is made up of straight lines, or visual rays, extending outward from the eye. These visual rays are discrete, but we perceive a continuous image because our eyes, and thus our visual rays, move very quickly.[6] Because visual rays are discrete, however, it is possible for small objects to lie unseen between them. This accounts for the difficulty in searching for a dropped needle. Although the needle may be within one's field of view, until the eye's visual rays fall upon the needle, it will not be seen.[7] Discrete visual rays also explain the sharp or blurred appearance of objects. According to postulate 7, the closer an object, the more visual rays fall upon it and the more detailed or sharp it appears. This is an early attempt to describe the phenomenon of optical resolution.
Much of the work considers perspective, how an object appears in space relative to the eye. For example, in proposition 8, Euclid argues that the perceived size of an object is not related to its distance from the eye by a simple proportion.[8]
An English translation was published in the Journal of the Optical Society of America.[9]
Notes
1. Lindberg, D. C. (1976). Theories of Vision from Al-Kindi to Kepler. Chicago: University of Chicago Press. p. 12.
2. Zajonc, A. (1993). Catching the Light: The Entwined History of Light and Mind. Oxford: Oxford University Press. p. 25.
3. Lindberg, D. C. (2007). The Beginnings of Western Science: The European Scientific Traditions in Philosophical, Religious, and Institutional Context, Prehistory to A.D. 1450. 2nd ed. Chicago: University of Chicago Press, p. 106.
4. Zajonc (1993), p. 25.
5. Lindberg (1976), p. 12.
6. Russo, L. (2004). The Forgotten Revolution: How Science Was Born in 300 BC and Why It Had to Be Reborn. S. Levy, transl. Berlin: Springer-Verlag p. 149.
7. Zajonc (1993), p. 25.
8. Smith, A. Mark (1999). Ptolemy and the Foundations of Ancient Mathematical Optics: A Source Based Guided Study. Philadelphia: American Philosophical Society. p. 57. ISBN 978-0-87169-893-3.
9. Harry Edwin Burton, "The Optics of Euclid", Journal of the Optical Society of America, 35 (1945), 357-372 OSA Publishing link.
References
• Smith, M.A. (1996). Ptolemy's Theory of Visual Perception: An English Translation of the Optics with Introduction and Commentary. Philadelphia: The American Philosophical Society.
• Smith, M.A. (2014). From Sight to Light. The Passage from Ancient to Modern Optics. Chicago & London: The University of Chicago Press.
• English translation of Euclid's Optics
• Latin text of Euclid's Optics from Euclidis Opera Omnia, ed. J.L. Heiberg, vol. VII
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| Wikipedia |
On the meta distribution in spatially correlated non-Poisson cellular networks
Shanshan Wang1 &
Marco Di Renzo ORCID: orcid.org/0000-0003-0772-87931
In this paper, we consider a cellular network in which the locations of the base stations are spatially correlated. We introduce an analytical framework for computing the distribution of the conditional coverage probability given the point process, which is referred to as the meta distribution and provides one with fine-grained information on the performance of cellular networks beyond spatial averages. To this end, we approximate, from the typical user standpoint, the spatially correlated (non-Poisson) cellular network with an inhomogeneous Poisson point process. In addition, we employ a new and recently proposed definition of the coverage probability and introduce an efficient numerical method for computing the meta distribution. The accuracy of the proposed approach is validated with the aid of numerical simulations.
Methods/experimental
The methods used in the present paper are based on the mathematical tools of inhomogeneous Poisson point processes and stochastic geometry. A new analytical framework for performance analysis is introduced. The theoretical framework is validated against Monte Carlo simulations.
Stochastic geometry and point processes are well known and widely used analytical tools for modeling, analyzing, and optimizing cellular networks [1]. The Poisson Point Process (PPP), in particular, is the most widely used spatial model to describe the locations of the base stations (BSs) in cellular networks [2–6]. This is due to its inherent analytical tractability. In practice, however, the locations of cellular BSs are distributed according to complex spatial patterns that are difficult to model analytically and, in general, differ from the PPP [7].
To overcome the analytical complexity of modeling and analyzing non-Poisson, i.e., spatially correlated, cellular networks, we have introduced an approximation based on inhomogeneous PPP, which is referred to as inhomogeneous double thinning (IDT) approximation [7]. The IDT approach allows one to model both spatially repulsive and spatially attractive (clustered) cellular network deployments in a mathematically tractable manner. In [7], in particular, the suitability of the IDT approach has been tested against several cellular network deployments obtained from publicly available datasets, and a good accuracy has been obtained.
In [7], however, the IDT approach is used to compute the spatially averaged coverage probability in cellular networks. More precisely, the coverage probability is first formulated by conditioning upon the point process that describes the locations of the cellular BSs, and then the expectation with respect to the point process is computed. As remarked in [8], the spatially averaged coverage probability is an important performance metric, but it does not completely characterize the variability of the coverage probability around its (spatial) mean value. To overcome this limitation, the author of [8] has introduced the concept of meta distribution, which provides one with finer-grained information on the network performance. Notably, the meta distribution allows one to characterize the performance of user percentiles. More precisely, the coverage probability is first formulated by conditioning upon the point process that describes the locations of the cellular BSs, and, then, its complementary cumulative distribution function with respect to the point process is computed. The spatially averaged coverage probability can be retrieved from the meta distribution via a simple integration.
In the present paper, motivated by these considerations and by [9], we generalize the IDT approach for computing the meta distribution in non-Poisson cellular networks. We consider a recent and improved definition of the coverage probability, which allows one to account for the signal quality during the cell association and data transmission phases [10], [11]. We show, notably, that the meta distribution cannot be approximated, in general, by using the beta distribution. As remarked in [12–14], however, the numerical computation of the meta distribution is usually not straightforward by employing the Gil-Pelaez inversion theorem [15]. We prove, on the other hand, that the meta distribution can be efficiently computed by employing the trapezoidal integration rule and the Euler sum method, for which a bound for the approximation error is known [16, 17]. With the proposed approach, the meta distribution in spatially correlated (non-Poisson) cellular networks can be obtained efficiently and accurately. The suitability of the proposed approach is substantiated with the aid of numerical simulations.
The reminder of this paper is organized as follows. In Section 3, the system model and the IDT approach are introduced. In Section 4, the meta distribution is formulated and an efficient method for its computation is given. In Section 5, analytical expressions for the moments of the coverage probability conditioned upon the point process that describes the locations of cellular BSs are derived. In Section 6, Poisson and non-Poisson cellular networks are compared against each other. In Section 7, numerical simulations to validate the suitability and accuracy of the proposed approach are illustrated. Finally, Section 8 concludes this paper.
In this section, we describe the system model and summarize the essence of the IDT approach for modeling non-Poisson cellular networks. The adopted definition of coverage probability is introduced as well.
Cellular network modeling
The cellular network model is the same as in [7]. In particular, a downlink cellular network is considered. The BSs are modeled as points of a motion-invariant point process, denoted by ΨBS, of density λBS. The locations of the BSs are denoted by \(x \in \Psi _{\text {BS}} \subseteq \mathbb {R}^{2}\). The mobile terminals (MTs) are distributed independently of each other and uniformly at random in \(\mathbb {R}^{2}\). The density of the MTs is denoted by λMT. A fully loaded assumption is considered, i.e., λMT≫λBS, which implies that all the BSs are active and have MTs to serve. The BSs and MTs are equipped with a single omnidirectional antenna. Each BS transmits with a constant power denoted by Ptx. Thanks to the assumption of motion-invariance, the point process of the BSs is stationary and isotropic. As a result, the analytical frameworks are developed for the typical MT, denoted by MT0, that is located at the origin. The BS serving MT0 is denoted by BS0. Its location is denoted by x0∈ΨBS. All available BSs transmit on the same physical channel as BS0. The point process of the interfering BSs is denoted by \(\Psi _{\text {BS}}^{\left (\mathrm {I} \right)}\), and the generic interfering BS is denoted by BSi. Besides the inter-cell interference, the Gaussian noise with power \(\sigma _{\mathrm {N}}^{2}\) is taken into account as well.
All BS-to- MT0 links are assumed to be mutually independent and identically distributed (i.i.d.). For each BS-to- MT0 link, path-loss and fast-fading channel impairments are considered. The path-loss is defined as l(x)=κ∥x∥γ, where κ and γ>2 are the path-loss constant and the path-loss slope (exponent), respectively. The power gain due to the fast-fading is assumed to follow an exponential distribution with mean 1 and is denoted by gx for x∈ΨBS.
A cell association criterion based on the highest average received power is considered. Let x∈ΨBS be the location of a generic BS. The location, x0, of the serving BS, BS0, is obtained as follows:
$$ x_{0} = \underset{{x} \in {\Psi_{{\text{BS}}}}}{\arg \max } \left\{{{1/{l\left({{x}} \right)}}} \right\} = \underset{{x} \in {\Psi_{{\text{BS}}}}}{\arg \max } \left\{ {{1/{{L_{x}}}}} \right\} $$
where Lx=l(x) is a shorthand notation. As far as the intended link is concerned, we have \(\phantom {\dot {i}\!}{L_{0}} = l\left ({{x_{0}}} \right) = {\min }_{{x} \in {\Psi _{{\text {BS}}}}} \left \{ {{L_{x}}} \right \}\).
Coverage probability
The definition of (spatially averaged) coverage probability recently introduced in [10] and [11] is considered. Let γD and γA be the reliability thresholds for the successful decoding of information data and for the successful detection of the serving BS, BS0, respectively. The coverage probability, Pcov, of the typical MT, MT0, is as follows:
$$ {{\mathrm{P}}_{{\text{cov}}}}\left({{\gamma_{\mathrm{D}}},{\gamma_{\mathrm{A}}}} \right) = \Pr \left\{ {{\text{SIR}} \ge {\gamma_{\mathrm{D}}},\overline {{\text{SNR}}} \ge {\gamma_{\mathrm{A}}}} \right\} $$
where the signal-to-interference ratio (SIR) and the average signal-to-noise ratio (\({\overline {{\text {SNR}}} }\)) can be formulated, for the network model under analysis, as follows:
$$ \begin{aligned} & {\text{SIR}} = \frac{{{{{{\mathrm{P}}_{{\text{tx}}}}\,{g_{0}}}/ {{L_{0}}}}}}{{\sum\limits_{{\mathrm{B}}{{\mathrm{S}}_{i}} \in {\Psi_{\text{BS}}^{\left(\mathrm{I} \right)}}} {{{{{\mathrm{P}}_{{\text{tx}}}}\,{g_{i}}} / {{L_{i}}}}\mathbbm{1}\left({{L_{i}} > {L_{0}}} \right)} }} \\ & \overline {{\text{SNR}}} = \frac{{{{{{\mathrm{P}}_{{\text{tx}}}}} / {{L_{0}}}}}}{{\sigma_{\mathrm{N}}^{2}}} \end{aligned} $$
where \({\mathbbm {1}}\left (\cdot \right)\) denotes the indicator function and \({\overline {{\text {SNR}}} }\) is averaged with respect to the fast-fading power gain g0 of the intended link.
It is worth mentioning that the definition of Pcov in (3) reduces to the conventional definition of coverage probability [1] by setting γA=0. This implies that the results obtained in the present paper apply unaltered even if the conventional definition of coverage probability based only on the SIR is employed.
Inhomogeneous double thinning approach
The computation of the coverage probability in (3) is analytically tractable if the point process of the BSs is a PPP [11]. If it is not a PPP, on the other hand, the coverage probability cannot, in general, be formulated in a tractable analytical form [7]. To overcome this issue and enable one to analyze non-Poisson cellular networks, we have proposed the IDT approach in [7]. The essence of the approach consists of introducing an equivalent (approximated) network abstraction, for modeling non-Poisson cellular networks, that is based on inhomogeneous PPPs. The equivalent network model, in particular, is constituted by two independent inhomogeneous PPPs, ΦBS(F) and ΦBS(K), which are constructed with the purpose of approximating a motion-invariant point process from the point of view of the typical MT. Further details and discussions on the IDT approach and its interpretation from the typical MT standpoint can be found in [7].
Let the inhomogeneous PPPs ΦBS(F) and ΦBS(K) have intensity measures \({\Lambda _{\Phi _{{\text {BS}}}^{\left (F \right)}}}\left (\cdot \right)\) and \({\Lambda _{\Phi _{{\text {BS}}}^{\left (K \right)}}}\left (\cdot \right)\), respectively. The BS serving the probe MT is assumed to belong to ΦBS(F), and the interfering BSs are assumed to belong to ΦBS(K). In particular, the location of the serving BS and the inhomogeneous PPP, ΦBS(I), of interfering BSs can be formulated as follows:
$$ {}\begin{aligned} & x_{0}^{\left(F \right)} = \underset{x \in \Phi_{{\text{BS}}}^{\left(F \right)}}{\arg \max } \left\{ {{1/ {l\left({x} \right)}}} \right\}\\ & \Phi_{{\text{BS}}}^{\left({\mathrm{I}} \right)} =\Phi_{{\text{BS}}}^{\left({\mathrm{I}} \right)}\left({x_{0}^{\left(F \right)}} \right) = \left\{ {x \in \Phi_{{\text{BS}}}^{\left(K \right)}:l\left({x} \right) > L_{0}^{\left(F \right)} = l\left({{x_{0}^{\left(F \right)}}} \right)} \right\} \end{aligned} $$
It is worth mentioning, in particular, that the inhomogeneous PPPs ΦBS(F) and ΦBS(I) are only conditionally, i.e., upon \({x_{0}^{\left (F \right)}}\), independent. As described in detail in [7], the intensity measures of ΦBS(F) and ΦBS(K) are determined from the F-function and K-function of the original motion-invariant point process. Therefore, they depend only on the spatial characteristics of the point process (spatial model) being used. In [7], we have shown, in particular, that a convenient choice, which provides one with a good trade-off between modeling accuracy and analytical tractability, for the intensity measures \({\Lambda _{\Phi _{{\text {BS}}}^{\left (F \right)}}}\left (\cdot \right)\) and \({\Lambda _{\Phi _{{\text {BS}}}^{\left (K \right)}}}\left (\cdot \right)\) is the following:
$$ \begin{aligned} & {\Lambda_{\Phi_{{\text{BS}}}^{\left(F \right)}}}\left({{\mathcal{B}}\left({0,r} \right)} \right) = 2\pi {\int\nolimits}_{0}^{r} {\lambda_{{\text{BS}}}^{\left(F \right)}\left(\zeta \right)\zeta d\zeta} \\ & {\Lambda_{\Phi_{{\text{BS}}}^{\left(K \right)}}}\left({{\mathcal{B}}\left({0,r} \right)} \right) = 2\pi {\int\nolimits}_{0}^{r} {\lambda_{{\text{BS}}}^{\left(K \right)}\left(\zeta \right)\zeta d\zeta } \end{aligned} $$
where \({{\mathcal {B}}\left ({0,r} \right)}\) denotes the ball centered at the origin and of radius r and λBS(F)(·) and λBS(K)(·) are the intensity functions of ΦBS(F) and ΦBS(K), respectively, which are distance-dependent and angle-independent.
In particular, λBS(F)(·) and λBS(K)(·) have different forms depending on whether the original motion-invariant point process is spatially repulsive or spatially attractive.
Spatially repulsive point process: If the point process ΨBS is spatially repulsive, then:
$$ \begin{aligned} & \lambda_{{\text{BS}}}^{(F)}(r) = {\lambda_{\text{BS}}}\check{\mathrm{c}}_{\mathrm{F}}\min \left\{{\left({\check{\mathrm{a}}}_{\mathrm{F}} / {{\check{\mathrm{c}}_{\mathrm{F}}}} \right)r + {{{{\check{\mathrm{b}}_{\mathrm{F}}}} / {{\check{\mathrm{c}}_{\mathrm{F}}}}}},1} \right\} \\ & \lambda_{{\text{BS}}}^{\left(K \right)}\left(r \right) = {\lambda_{{\text{BS}}}}\min \left\{{{\check{\mathrm{a}}_{\mathrm{K}}}r + {\check{\mathrm{b}}_{\mathrm{K}}},{\check{\mathrm{c}}_{\mathrm{K}}}} \right\} \end{aligned} $$
where \(\left ({{\check {\mathrm {a}}_{\mathrm {F}}},{\check {\mathrm {b}}_{\mathrm {F}}},{\check {\mathrm {c}}_{\mathrm {F}}}} \right)\) and \(\left ({{\check {\mathrm {a}}_{\mathrm {K}}},{\check {\mathrm {b}}_{\mathrm {K}}},{\check {\mathrm {c}}_{\mathrm {K}}}} \right)\) are two triplets of non-negative real numbers such that \({\check {\mathrm {c}}_{\mathrm {F}}} \ge {\check {\mathrm {b}}_{\mathrm {F}}} \ge 1\) and \({\check {\mathrm {b}}_{\mathrm {K}}} \le {\check {\mathrm {c}}_{\mathrm {K}}} \le 1\).
Spatially attractive point process: If the point process ΨBS is spatially attractive, then:
$$ \begin{aligned} & \lambda_{{\text{BS}}}^{\left(F \right)}\left(r \right) = {\lambda_{{\text{BS}}}}\max \left\{ { - {{\hat{\mathrm{a}}}_{\mathrm{F}}}r + {\hat{\mathrm{b}}_{\mathrm{F}}},{\hat{\mathrm{c}}_{\mathrm{F}}}} \right\} \\ & \lambda_{{\text{BS}}}^{\left(K \right)}\left(r \right) = {\lambda_{{\text{BS}}}}{\hat{\mathrm{b}}_{\mathrm{K}}}\max \left\{ { - \left({{{\hat{\mathrm{a}}_{\mathrm{K}}}}}/{{{\hat{\mathrm{b}}_{\mathrm{K}}}}}\right)r + 1,{{{\hat{\mathrm{c}}_{\mathrm{K}}}}}/{{{\hat{\mathrm{b}}_{\mathrm{K}}}}}} \right\} \end{aligned} $$
where \(\left ({{\hat {\mathrm {a}}_{\mathrm {F}}},{\hat {\mathrm {b}}_{\mathrm {F}}},{\hat {\mathrm {c}}_{\mathrm {F}}}} \right)\) and \(\left ({{\hat {\mathrm {a}}_{\mathrm {K}}},{\hat {\mathrm {b}}_{\mathrm {K}}},{\hat {\mathrm {c}}_{\mathrm {K}}}} \right)\) are two triplets of non-negative real numbers such that \({\hat {\mathrm {c}}_{\mathrm {F}}} \le {\hat {\mathrm {b}}_{\mathrm {F}}} \le 1\) and \({\hat {\mathrm {b}}_{\mathrm {K}}} \ge {\hat {\mathrm {c}}_{\mathrm {K}}} \ge 1\).
The intensity measures in (5) can be formulated in closed-form from (6) and (7), as described in [7]. Since for spatially repulsive and spatially attractive point processes the intensity measures in (5) have the same analytical expression as a function of the triplets of parameters, we use, in the sequel, the general notation (aF,bF,cF) and (aK,bK,cK) for both case studies.
In this section, we introduce the meta distribution, overview the most common approaches for computing it, and show that it can be efficiently computed, within a given and bounded error, by using the trapezoidal integration rule and the Euler sum method. In what follows, we consider the equivalent network model based on the inhomogeneous PPPs ΦBS(F) and ΦBS(K). For ease of writing, we employ the notation ΦBS={ΦBS(F),ΦBS(K)}.
According to [8], the meta distribution is defined as follows:
$$ \begin{aligned} &{\overline {\mathrm{F}}_{{{\mathrm{P}}_{{\text{cov}}}}}}\left({{\gamma_{\mathrm{D}}},{\gamma_{\mathrm{A}}},z} \right) = {\text{Pr}_{{\Phi_{{\text{BS}}}}}}\left\{ {{{\mathrm{P}}_{{\text{cov}} }}\left({\left. {{\gamma_{\mathrm{D}}},{\gamma_{\mathrm{A}}}} \right|{\Phi_{{\text{BS}}}}} \right) \ge z} \right\}\\ &\quad {\text{with}} \quad z \in \left[ {0,1} \right] \end{aligned} $$
where Pcov(γD,γA|ΦBS) is the coverage probability conditioned upon ΦBS (and by assuming that the typical MT is at the origin), which is defined as follows:
$$ {}{{\mathrm{P}}_{\text{cov} }}\left({\left. {{\gamma_{\mathrm{D}}},{\gamma_{\mathrm{A}}}} \right|{\Phi_{{\text{BS}}}}} \right) = \Pr \left\{ {\left. {{\text{SIR}} \ge {\gamma_{\mathrm{D}}},\overline {{\text{SNR}}} \ge {\gamma_{\mathrm{A}}}} \right|{\Phi_{{\text{BS}}}}} \right\} $$
It is worth mentioning that, in (8), we have emphasized that the probability is computed only with respect to ΦBS.
The spatially averaged coverage probability in (2) can be retrieved from the meta distribution in (7) directly from its definition, as follows:
$$ {}{{\mathrm{P}}_{\text{cov}} }\left({{\gamma_{\mathrm{D}}},{\gamma_{\mathrm{A}}}} \right) \,=\, {{\mathbb{E}}_{{\Phi_{{\text{BS}}}}}}\left\{ {{{\mathrm{P}}_{\text{cov} }}\left({\left. {{\gamma_{\mathrm{D}}},{\gamma_{\mathrm{A}}}} \right|{\Phi_{{\text{BS}}}}} \right)} \right\} \,=\, {\int\nolimits}_{0}^{1} {{{\overline {\mathrm{F}} }_{{{\mathrm{P}}_{{\text{cov}}}}}}\left(z \right)dz} $$
In practical terms, the meta distribution provides one with the fraction of links whose SIR is greater than γD and whose average SNR is greater than γA with probability at least equal to z in each network realization. Therefore, it yields a more general statistical characterization of the performance of cellular networks beyond spatial averages.
Computation: Gil-Pelaez method
As discussed in [8], the direct computation of the meta distribution in (8) is not straightforward. A general approach to overcome this issue is to capitalize on the Gil-Pelaez inversion theorem [15], which allows one to formulate the meta distribution as a function of the moments of the (conditional) coverage probability in (9).
In particular, the following holds [8]:
$$ \begin{aligned} {}{\overline {\mathrm{F}}_{{{\mathrm{P}}_{{\text{cov}}}}}}& \left({{\gamma_{\mathrm{D}}},{\gamma_{\mathrm{A}}},z} \right) = \\ &\frac{1}{2} + \frac{1}{\pi }{\int\nolimits}_{0}^{+ \infty} {\frac{{{\text{Im}} \left\{ {{{\mathcal{M}}_{jt}}\left({{\gamma_{\mathrm{D}}},{\gamma_{\mathrm{A}}}} \right)\exp \left({ - jt\ln \left(z \right)} \right)} \right\}}}{t}dt} \end{aligned} $$
where \(j = \sqrt { - 1}\) is the imaginary unit, Im{·} denotes the imaginary part operator, and \({{\mathcal {M}}_{b}}\left ({{\gamma _{\mathrm {D}}},{\gamma _{\mathrm {A}}}} \right)\) is the bth moment of the (conditional) coverage probability Pcov(γD,γA|ΦBS), which is defined as follows:
$$ {}\begin{aligned} {{\mathcal{M}}_{b}}\left({{\gamma_{\mathrm{D}}},{\gamma_{\mathrm{A}}}} \right) &= {{\mathbb{E}}_{{\Phi_{{\text{BS}}}}}}\left\{ {{{\left({{{\mathrm{P}}_{\text{cov}} }\left({\left. {{\gamma_{\mathrm{D}}},{\gamma_{\mathrm{A}}}} \right|{\Phi_{{\text{BS}}}}} \right)} \right)}^{b}}} \right\}\\ &= {\int\nolimits}_{0}^{1} {b{z^{b - 1}}{{\overline {\mathrm{F}} }_{{{\mathrm{P}}_{{\text{cov}}}}}}\left(z \right)dz} \end{aligned} $$
Therefore, the meta distribution can be obtained by first computing the moments of the (conditional) coverage probability Pcov(γD,γA|ΦBS) in (12) and by then computing the integral in (11). As remarked in [12–14], however, the computation of (11) is not always straightforward. Other methods need, in general, to be used instead, e.g., the Fourier-Jacobi expansion [12], and the Mnatsakanov's theorem [13, 14].
An alternative approach relies on approximating the meta distribution with another distribution. A nota- ble example is using the beta distribution for approximating it over the entire range of values z∈[0,1]. As remarked in [14], however, this approach cannot be applied if the actual distribution does not fulfill the class of the beta distribution. The following lemma shows, e.g., that this is the case if the coverage probability is defined in terms of SIR and \({\overline {{\text {SNR}}} }\).
Lemma 1
Let (aF,bF,cF) be the generic triplet of parameters introduced in ( 6 ) and ( 7 ), and denote dF=(cF−bF)/aF≥0. The meta distribution in ( 11 ) satisfies the following properties:
$$ {}\begin{aligned} {\overline {\mathrm{F}}_{{{\mathrm{P}}_{{\text{cov}}}}}}&\left({{\gamma_{\mathrm{D}}},{\gamma_{\mathrm{A}}},z \to 0} \right) =\\ &1 \,-\, \exp \!\left(\!{ - 2 \pi \lambda_{{\text{BS}}} \Psi \!\left(\! \!{{{\left({\frac{{{{\mathrm{P}}_{{\text{tx}}}}}}{{\kappa \sigma_{\mathrm{N}}^{2}{\gamma_{\mathrm{A}}}}}} \right)}^{{1 / \gamma }}};{{\mathrm{a}}_{\mathrm{F}}},{{\mathrm{b}}_{\mathrm{F}}},{{\mathrm{c}}_{\mathrm{F}}}} \!\right)} \!\right) \end{aligned} $$
$$ {\overline {\mathrm{F}}_{{{\mathrm{P}}_{{\text{cov}}}}}}\left({{\gamma_{\mathrm{D}}},{\gamma_{\mathrm{A}}},z \to 1} \right) = 0 $$
$$ {}\begin{aligned} 0 &\!\le\! {\overline {\mathrm{F}}_{{{\mathrm{P}}_{{\text{cov}}}}}}\left({{\gamma_{\mathrm{D}}},{\gamma_{\mathrm{A}}},z} \right) \le 1\\ &\quad- \!\exp\! \left({ - 2 \pi \lambda_{{\text{BS}}} \Psi \left({{{\left({\frac{{{{\mathrm{P}}_{{\text{tx}}}}}}{{\kappa \sigma_{\mathrm{N}}^{2}{\gamma_{\mathrm{A}}}}}} \right)}^{{1 / \gamma }}};{{\mathrm{a}}_{\mathrm{F}}},{{\mathrm{b}}_{\mathrm{F}}},{{\mathrm{c}}_{\mathrm{F}}}} \right)} \right) \end{aligned} $$
where \({\Lambda _{\Phi _{{\text {BS}}}^{\left (F \right)}}}\left ({{\mathcal {B}}\left ({0,r} \right)} \right) = 2 \pi \lambda _{{\text {BS}}} \Psi \left ({r;{{\mathrm {a}}_{\mathrm {F}}},{{\mathrm {b}}_{\mathrm {F}}},{{\mathrm {c}}_{\mathrm {F}}}} \right)\), and:
$$ {}\begin{aligned} \Psi \left({r;{{\mathrm{a}}_{\mathrm{F}}},{{\mathrm{b}}_{\mathrm{F}}},{{\mathrm{c}}_{\mathrm{F}}}} \right) \!&=\! \left({\frac{{{{\mathrm{a}}_{\mathrm{F}}}}}{3}{r^{3}} + \frac{{{{\mathrm{b}}_{\mathrm{F}}}}}{2}{r^{2}}} \right){\mathbbm{1}}\left({r \le {{\mathrm{d}}_{\mathrm{F}}}} \right)\\&\quad\,+\, \left(\!{\frac{{{{\left({{{\mathrm{b}}_{\mathrm{F}}} - {{\mathrm{c}}_{\mathrm{F}}}} \right)}^{3}}}}{{6{\mathrm{a}}_{\mathrm{F}}^{2}}} + \frac{{{{\mathrm{c}}_{\mathrm{F}}}}}{2}{r^{2}}} \right)\!{\mathbbm{1}}\!\left({r > {{\mathrm{d}}_{\mathrm{F}}}} \right) \end{aligned} $$
See Appendix A. □
From Lemma 1, we evince that the meta distribution lies in the range [0,1] only if γA=0, i.e., the conventional definition of coverage probability based only on the SIR is used [11]. If γA≠0, on the other hand, the meta distribution lies in the range \(\left [ {0,1 - \exp \left ({ - 2 \pi \lambda _{{\text {BS}}} \Psi \left ({{{\left ({{{{{\mathrm {P}}_{{\text {tx}}}}} / {\left ({\kappa \sigma _{\mathrm {N}}^{2}{\gamma _{\mathrm {A}}}} \right)}}} \right)}^{{1 / \gamma }}};{{\mathrm {a}}_{\mathrm {F}}},{{\mathrm {b}}_{\mathrm {F}}},{{\mathrm {c}}_{\mathrm {F}}}} \right)} \right)} \right ]\). This implies that the beta distribution is not necessarily a good approximation for the meta distribution, since the former distribution lies always in the range [0,1].
Computation: Euler sum method
In this section, motivated by the considerations just made, we show that the meta distribution can be efficiently computed, with a known and bounded approximation error, by using the trapezoidal integration rule and the Euler sum method as originally proposed in [16] and recently used, e.g., in [17]. The following proposition states the result in rigorous terms.
Proposition 1
Let A, N, and Q be three positive integer numbers. Let us define the following functions:
$$ \begin{array}{l} {\beta_{0}} = 2\\ {\beta_{n}} = 1\quad {\text{for}}\quad n = 1,2, \ldots,N\\ {s_{n}} = \frac{{A + 2\pi jn}}{2}\quad {\text{for}}\quad n = 0,1, \ldots,N \end{array} $$
The meta distribution can be formulated as follows:
$$ {}\begin{aligned} {\overline {\mathrm{F}}_{{{\mathrm{P}}_{{\text{cov}}}}}}\left({{\gamma_{\mathrm{D}}},{\gamma_{\mathrm{A}}},z} \right) & \approx \frac{{{2^{- Q}}\exp \left({{A / 2}} \right)}}{{{{\ln }^{2}}\left(z \right)}}\sum\limits_{q = 0}^{Q} \left({\begin{array}{*{20}{c}} Q\\ q \end{array}} \right)\\ &\quad\sum\limits_{n = 0}^{N + q} {\frac{{{{\left({ - 1} \right)}^{n}}}}{{{\beta_{n}}}}{\text{Re}} \left\{ {\frac{{{{\mathcal{M}}_{- {{{s_{n}}} / {\ln \left(z \right)}}}}\left({{\gamma_{\mathrm{D}}},{\gamma_{\mathrm{A}}}} \right)}}{{{s_{n}}}}} \right\}} \\ &\quad+ \left| {{\mathcal{E}}\left({A,N,Q} \right)} \right| \end{aligned} $$
where Re{·} is the real part operator, \({{\mathcal {M}}_{b}}\left ({{\gamma _{\mathrm {D}}},{\gamma _{\mathrm {A}}}} \right)\) is the bth moment in ( 12 ), and \({{\mathcal {E}}\left ({A,N,Q} \right)}\) is the approximation error as follows:
$$ {}\begin{aligned} \left| {{\mathcal{E}}\left({A,N,Q} \right)} \right| & \approx \frac{1}{{\exp \left(A \right) - 1}}\\ & \quad+ \left| \frac{{{2^{- Q}}\exp \left({{A / 2}} \right)}}{{{{\ln }^{2}}\left(z \right)}}\sum\limits_{q = 0}^{Q} \left({\begin{array}{*{20}{c}} Q\\ q \end{array}} \right){{\left({ - 1} \right)}^{N + 1 + q}}\right.\\ &\qquad\left.{\text{Re}} \left\{ {\frac{{{{\mathcal{M}}_{- {{{s_{N + 1 + q}}} / {\ln \left(z \right)}}}}\left({{\gamma_{\mathrm{D}}},{\gamma_{\mathrm{A}}}} \right)}}{{{s_{N + 1 + q}}}}} \right\} \right| \end{aligned} $$
See Appendix B. □
There are three main advantages in favor of using the Euler sum method instead of the Gil-Pelaez method: (1) Eq. 18 does not need the explicit computation of an integral, which makes the numerical estimation of the meta distribution easier; (2) the method can be applied to any family of non-negative meta distributions; and (3) the approximation error in (19) is known in closed-form and the accuracy of the numerical computation can be controlled by using the triplet of parameters (A,N,Q). As discussed in [16], in particular, typical values of these parameters are A=10 ln(10), which guarantees a discretization error of the order of 10−10, and N and Q of the order of 10 or 20.
Computation of the moments
From (18), it is apparent that the meta distribution can be easily obtained from the moments defined in (12). The following theorem provides one with these moments for the system model under analysis based on the IDT modeling approximation for non-Poisson cellular networks.
Theorem 1
Let (aF,bF,cF) and (aK,bK,cK) be the two triplets of parameters that quantify the spatial correlation properties of the point process that describes the locations of the cellular BSs. Let us define dF=(cF−bF)/aF≥0 and dK=(cK−bK)/aK≥0. The moments in ( 12 ) can be formulated as follows:
$$ \begin{aligned} {{\mathcal{M}}_{b}}\left({{\gamma_{\mathrm{D}}},{\gamma_{\mathrm{A}}}} \right) & = {{\mathbb{E}}_{{\Phi_{{\text{BS}}}}}}\left\{ {{{\left({{{\mathrm{P}}_{\text{cov} }}\left({\left. {{\gamma_{\mathrm{D}}},{\gamma_{\mathrm{A}}}} \right|{\Phi_{{\text{BS}}}}} \right)} \right)}^{b}}} \right\}\\ & = 2\pi {\lambda_{{\text{BS}}}}{\int\nolimits}_{0}^{{{\left({{{{{\mathrm{P}}_{{\text{tx}}}}} / {\left({\kappa \sigma_{\mathrm{N}}^{2}{\gamma_{\mathrm{A}}}} \right)}}} \right)}^{{1 / \gamma }}}}\\ & \quad{\exp \left({ - 2\pi {\lambda_{{\text{BS}}}}\Theta \left({r;{\gamma_{\mathrm{D}}}} \right)} \right)\Upsilon \left(r \right)dr} \end{aligned} $$
where the following functions are defined:
$$ \begin{aligned} \Theta \left({r;{\gamma_{\mathrm{D}}}} \right) & = \frac{{{{\mathrm{a}}_{\mathrm{K}}}}}{3}{r^{3}}\left({{~}_{2}{F_{1}}\left({b, - \frac{3}{\gamma },1 - \frac{3}{\gamma },{\gamma_{\mathrm{D}}}} \right) - 1} \right){\mathbbm{1}}\left({r \le {{\mathrm{d}}_{\mathrm{K}}}} \right)\\ & + \frac{{{{\mathrm{b}}_{\mathrm{K}}}}}{2}{r^{2}}\left({{~}_{2}{F_{1}}\left({b, - \frac{2}{\gamma },1 - \frac{2}{\gamma },{\gamma_{\mathrm{D}}}} \right) - 1} \right){\mathbbm{1}}\left({r \le {{\mathrm{d}}_{\mathrm{K}}}} \right)\\ & - \frac{{{{\mathrm{a}}_{\mathrm{K}}}}}{3}{\mathrm{d}}_{\mathrm{K}}^{3}\left({{~}_{2}{F_{1}}\left({b, - \frac{3}{\gamma },1 - \frac{3}{\gamma }, - \frac{{{r^{\gamma} }}}{{{\mathrm{d}}_{\mathrm{K}}^{\gamma} }}{\gamma_{\mathrm{D}}}} \right) - 1} \right){\mathbbm{1}}\left({r \le {{\mathrm{d}}_{\mathrm{K}}}} \right)\\ & + \frac{{{{\mathrm{c}}_{\mathrm{K}}} - {{\mathrm{b}}_{\mathrm{K}}}}}{2}{\mathrm{d}}_{\mathrm{K}}^{2}\left(\! {{~}_{2}{F_{1}}\!\left(\!{b, - \frac{2}{\gamma },1 - \frac{2}{\gamma }, - \frac{{{r^{\gamma} }}}{{{\mathrm{d}}_{\mathrm{K}}^{\gamma} }}{\gamma_{\mathrm{D}}}} \right) \,-\, 1} \right)\!{\mathbbm{1}}\!\left({r \le {{\mathrm{d}}_{\mathrm{K}}}} \right)\\ & + \frac{{{{\mathrm{c}}_{\mathrm{K}}}}}{2}{r^{2}}\left({{~}_{2}{F_{1}}\left({b, - \frac{2}{\gamma },1 - \frac{2}{\gamma }, - {\gamma_{\mathrm{D}}}} \right) - 1} \right){\mathbbm{1}}\left({r > {{\mathrm{d}}_{\mathrm{K}}}} \right) \end{aligned} $$
$$ \begin{aligned} \Upsilon \left(r \right) &= \left({{{\mathrm{a}}_{\mathrm{F}}}{r^{2}} + {{\mathrm{b}}_{\mathrm{F}}}r} \right)\exp \left({ - 2\pi {\lambda_{{\text{BS}}}}\left({\frac{{{{\mathrm{a}}_{\mathrm{F}}}}}{3}{r^{3}} + \frac{{{{\mathrm{b}}_{\mathrm{F}}}}}{2}{r^{2}}} \right)} \right){\mathbbm{1}}\left({r \le {{\mathrm{d}}_{\mathrm{F}}}} \right)\\ & \quad+ {{\mathrm{c}}_{\mathrm{F}}}r\exp \left({ - 2\pi {\lambda_{{\text{BS}}}}\left({\frac{{{{\left({{{\mathrm{b}}_{\mathrm{F}}} - {{\mathrm{c}}_{\mathrm{F}}}} \right)}^{3}}}}{{6{\mathrm{a}}_{\mathrm{F}}^{2}}} + \frac{{{{\mathrm{c}}_{\mathrm{F}}}}}{2}{r^{2}}} \right)} \right){\mathbbm{1}}\left({r > {{\mathrm{d}}_{\mathrm{F}}}} \right) \end{aligned} $$
See Appendix C. □
The analytical framework in (20) can be applied to any non-Poisson spatial model that can be well approximated by using inhomogeneous PPPs with the aid of the IDT approach. The conventional Poisson cellular network model can be retrieved directly from Theorem 1, as reported in the following corollary.
Corollary 1
In (homogeneous) Poisson cellular networks, the moments in ( 12 ) can be formulated as follows:
$$ \begin{aligned} {{\mathcal{M}}_{b}}&\left({{\gamma_{\mathrm{D}}},{\gamma_{\mathrm{A}}}} \right)\\ &= {{\mathbb{E}}_{{\Phi_{{\text{BS}}}}}}\left\{ {{{\left({{{\mathrm{P}}_{\text{cov}} }\left({\left. {{\gamma_{\mathrm{D}}},{\gamma_{\mathrm{A}}}} \right|{\Phi_{{\text{BS}}}}} \right)} \right)}^{b}}} \right\}\\ & = \frac{{1 - \exp \left({ - \pi {\lambda_{{\text{BS}}}}{{\left({\frac{{{{\mathrm{P}}_{{\text{tx}}}}}}{{\kappa \sigma_{\mathrm{N}}^{2}{\gamma_{\mathrm{A}}}}}} \right)}^{{2 / \gamma }}}{~}_{2}{F_{1}}\left({b, - \frac{2}{\gamma },1 - \frac{2}{\gamma }, - {\gamma_{\mathrm{D}}}} \right)} \right)}}{{{~}_{2}{F_{1}}\left({b, - \frac{2}{\gamma },1 - \frac{2}{\gamma }, - {\gamma_{\mathrm{D}}}} \right)}} \end{aligned} $$
It follows from (20) by setting bF=cF=1 and bK=cK=1, and by computing the integral. □
By direct inspection of (23), in particular, we note that the moments reduce to those computed in [8] if γA=0.
Comparison between Poisson and non-Poisson cellular networks
With the aid of the expression of the meta distribution in (18), and of its moments in (20) and (23) for non-Poisson and Poisson cellular networks, respectively, it is worth studying the impact of spatial correlations in cellular networks. By considering the spatially averaged definition of the coverage probability in (10), we have proved in [7], notably, that spatially repulsive and spatially attractive cellular networks provide better and worse coverage probability than Poisson cellular networks, respectively. Such a general result and comparison are, however, difficult to obtain for the meta distribution, since it can only be computed numerically, as shown in (18), and simple closed-form solutions for it are not available yet and are unknown. The following two propositions, however, provide one with some important information about this comparison.
Let z be sufficiently small, i.e., z→0. Let \(\overline {\mathrm {F}}_{{{\mathrm {P}}_{{\text {cov}}}}}^{\left ({{\text {PPP}}} \right)}\left ({\cdot } \right), \overline {\mathrm {F}}_{{{\mathrm {P}}_{{\text {cov}}}}}^{\left ({{\text {Rep}}} \right)}\left ({\cdot } \right)\), and \(\overline {\mathrm {F}}_{{{\mathrm {P}}_{{\text {cov}}}}}^{\left ({{\text {Attr}}} \right)}\left ({\cdot } \right)\) be the meta distributions of Poisson, spatially repulsive, and spatially attractive cellular networks, respectively. Then, the following holds:
$$ \begin{aligned} & \overline {\mathrm{F}}_{{{\mathrm{P}}_{{\text{cov}}}}}^{\left({{\text{Rep}}} \right)}\left({{\gamma_{\mathrm{D}}},{\gamma_{\mathrm{A}}},z \to 0} \right) \ge \overline {\mathrm{F}}_{{{\mathrm{P}}_{{\text{cov}}}}}^{\left({{\text{PPP}}} \right)}\left({{\gamma_{\mathrm{D}}},{\gamma_{\mathrm{A}}},z \to 0} \right)\\ & \overline {\mathrm{F}}_{{{\mathrm{P}}_{{\text{cov}}}}}^{\left({{\text{Attr}}} \right)}\left({{\gamma_{\mathrm{D}}},{\gamma_{\mathrm{A}}},z \to 0} \right) \le \overline {\mathrm{F}}_{{{\mathrm{P}}_{{\text{cov}}}}}^{\left({{\text{PPP}}} \right)}\left({{\gamma_{\mathrm{D}}},{\gamma_{\mathrm{A}}},z \to 0} \right) \end{aligned} $$
Let \(\Lambda _{\Phi _{{\text {BS}}}^{\left (F \right)}}^{\left ({{\text {PPP}}} \right)}\left ({\cdot } \right), \Lambda _{\Phi _{{\text {BS}}}^{\left (F \right)}}^{\left ({{\text {Rep}}} \right)}\left ({\cdot } \right)\), and \(\Lambda _{\Phi _{{\text {BS}}}^{\left (F \right)}}^{\left ({{\text {Attr}}} \right)}\left ({\cdot } \right)\) be the intensity measures in (5) for Poisson, spatially repulsive, and spatially attractive cellular networks, respectively. In [7, Lemma 5] and [7, Lemma 6], it is proved that \(\Lambda _{\Phi _{{\text {BS}}}^{\left (F \right)}}^{\left ({{\text {Rep}}} \right)}\left ({{\mathcal {B}}\left ({0,r} \right)} \right) \ge \Lambda _{\Phi _{{\text {BS}}}^{\left (F \right)}}^{\left ({{\text {PPP}}} \right)}\left ({{\mathcal {B}}\left ({0,r} \right)} \right)\) and \(\Lambda _{\Phi _{{\text {BS}}}^{\left (F \right)}}^{\left ({{\text {PPP}}} \right)}\left ({{\mathcal {B}}\left ({0,r} \right)} \right) \ge \Lambda _{\Phi _{{\text {BS}}}^{\left (F \right)}}^{\left ({{\text {Attr}}} \right)}\left ({{\mathcal {B}}\left ({0,r} \right)} \right)\), respectively. Then, the proof follows immediately from Lemma 1. □
Let \({\mathcal {M}}_{b}^{\left ({{\text {PPP}}} \right)}\left ({\cdot } \right), {\mathcal {M}}_{b}^{\left ({{\text {Rep}}} \right)}\left ({\cdot } \right)\), and \({\mathcal {M}}_{b}^{\left ({{\text {Attr}}} \right)}\left ({\cdot } \right)\) be the bth moment in ( 12 ) for Poisson, spatially repulsive, and spatially attractive cellular networks, respectively. Then, for every \(b \in \mathbb {R}\), the following holds:
$$ \begin{aligned} & {\mathcal{M}}_{b}^{\left({{\text{Rep}}} \right)}\left({{\gamma_{\mathrm{D}}},{\gamma_{\mathrm{A}}}} \right) \ge {\mathcal{M}}_{b}^{\left({{\text{PPP}}} \right)}\left({{\gamma_{\mathrm{D}}},{\gamma_{\mathrm{A}}}} \right)\\ & {\mathcal{M}}_{b}^{\left({{\text{PPP}}} \right)}\left({{\gamma_{\mathrm{D}}},{\gamma_{\mathrm{A}}}} \right) \ge {\mathcal{M}}_{b}^{\left({{\text{Attr}}} \right)}\left({{\gamma_{\mathrm{D}}},{\gamma_{\mathrm{A}}}} \right) \end{aligned} $$
See Appendix D. □
Proposition 2, notably, allows one to conclude that, for low values of z that tends to zero, spatially repulsive cellular networks exhibit first-order stochastic dominance over Poisson cellular networks, and that Poisson cellular networks exhibit first-order stochastic dominance over spatially attractive cellular networks. By using similar arguments, Proposition 3 allows one to establish similar moments-based stochastic ordering between Poisson and non-Poisson cellular networks.
Numerical results and validation
In this section, we show some simulation results in order to substantiate the main findings of the paper. The simulation setup is reported in Table 1, and the specific parameters of the point processes are detailed in [7]. The spatial model denoted by "general case" is chosen in order to better compare homogeneous, spatially attractive, and spatially repulsive point processes.
Table 1 Simulation setup
In Fig. 1, we report the moments of the conditional coverage probability. We observe a good agreement between the proposed analytical framework and Monte Carlo simulations. We note, in addition, that the findings about the first-order stochastic dominance of the moments is confirmed by our numerical illustrations.
Moments of the conditional coverage probability. Setup: General case. Solid lines: Monte Carlo simulations. Markers: IDT framework
In Figs. 2 and 3, we compare the meta distribution by considering two definitions of coverage probability. The conventional SIR-based definition and the \(\text {SIR}+\overline {{\text {SNR}}}\)-based definition that is employed in this paper [11]. We observe that, in both cases, we obtain a good agreement compared with Monte Carlo simulations. The figures confirm, in addition, that the beta distribution may not be used for approximating the \(\text {SIR}+\overline {{\text {SNR}}}\)-based definition of coverage probability. Our numerical results, on the other hand, confirm (even though they are not shown for ease of illustration) that the beta distribution yields a good approximation for the SIR-based definition of coverage probability.
Meta distribution: Comparison between the SIR-based and \(\text {SIR}+\overline {{\text {SNR}}}\)-based definition of coverage (ASNR=\(\overline {{\text {SNR}}}\)). Setup: Ginibre point process case. Solid lines: IDT framework. Markers: Monte Carlo simulations
Meta distribution: Comparison between the SIR-based and \(\text {SIR}+\overline {{\text {SNR}}}\)-based definition of coverage (ASNR=\(\overline {{\text {SNR}}}\)). Setup: Log-Gaussian Cox point process case. Solid lines: IDT framework. Markers: Monte Carlo simulations
In Fig. 4, finally, we compare the meta distribution of homogeneous, spatially attractive, and spatially repulsive point processes. We observe a good agreement with Monte Carlo simulations. In addition, the figures confirms the correctness of the asymptotic value of the meta distribution for small values of z.
Meta distribution. Setup: General case. Solid lines: IDT framework. Markers: Monte Carlo simulations. Dashed lines: asymptotic limit for z→0
Conclusion and discussion
In this paper, we have proved that the inhomogeneous double thinning approach for modeling and analyzing spatially correlated (non-Poisson) cellular networks can be successfully employed for studying the distribution of the conditional coverage probability given the point process of cellular base stations, which is referred to as the meta distribution. We have proved, in addition, that the meta distribution can be efficiently computed by using the trapezoidal integration rule and the Euler sum method, provided that the negative moments of the conditional coverage probability can be computed. By using the inhomogeneous double thinning approach, it is proved that these latter moments can be formulated in terms of a single integral expression. Finally, some results on the first-order stochastic dominance of non-Poisson cellular networks over Poisson cellular networks are proved.
Appendix A: Proof of Lemma 1
By conditioning upon the point process ΦBS, the SIR is a random variable and the average SNR is a constant. Therefore, the (conditional) coverage probability can be formulated as follows:
$$ {}\begin{aligned} {{\mathrm{P}}_{\text{cov}} }&\left({\left. {{\gamma_{\mathrm{D}}},{\gamma_{\mathrm{A}}}} \right|{\Phi_{{\text{BS}}}}} \right)\\ &= \Pr \left\{ {\left. {{\text{SIR}} \ge {\gamma_{\mathrm{D}}},\overline {{\text{SNR}}} \ge {\gamma_{\mathrm{A}}}} \right|{\Phi_{{\text{BS}}}}} \right\}\\ & = \left\{ \begin{array}{l} \Pr \left\{ {\left. {{\text{SIR}} \ge {\gamma_{\mathrm{D}}}} \right|{\Phi_{{\text{BS}}}}} \right\}\quad {\text{if}}\quad {r_{0}} \le {\left({\frac{{{{\mathrm{P}}_{{\text{tx}}}}}}{{\kappa \sigma_{\mathrm{N}}^{2}{\gamma_{\mathrm{A}}}}}} \right)^{{1 / \gamma }}}\\ 0\quad {\text{otherwise}} \end{array} \right. \end{aligned} $$
where r0=∥x0∥ is the distance of the serving BS from the origin.
By definition of meta distribution, i.e., \({\overline {\mathrm {F}}_{{{\mathrm {P}}_{{\text {cov}}}}}}\left ({{\gamma _{\mathrm {D}}},{\gamma _{\mathrm {A}}},z} \right) = {\Pr _{{\Phi _{{\text {BS}}}}}}\left \{ {{{\mathrm {P}}_{\text {cov}} }\left ({\left. {{\gamma _{\mathrm {D}}},{\gamma _{\mathrm {A}}}} \right |{\Phi _{{\text {BS}}}}} \right) \ge z} \right \}\), and by letting z→0, we obtain:
$$ \begin{aligned} {\overline {\mathrm{F}}_{{{\mathrm{P}}_{{\text{cov}}}}}}&\left({{\gamma_{\mathrm{D}}},{\gamma_{\mathrm{A}}},z \to 0} \right)\\ & = {\text{Pr}_{{\Phi_{{\text{BS}}}}}}\left\{ {{{\mathrm{P}}_{\text{cov}} }\left({\left. {{\gamma_{\mathrm{D}}},{\gamma_{\mathrm{A}}}} \right|{\Phi_{{\text{BS}}}}} \right) \ge z \to 0} \right\}\\ & = \left\{ \begin{array}{l} 1\quad {\text{if}}\quad {r_{0}} \le {\left({\frac{{{{\mathrm{P}}_{{\text{tx}}}}}}{{\kappa \sigma_{\mathrm{N}}^{2}{\gamma_{\mathrm{A}}}}}} \right)^{{1 / \gamma }}}\\ 0\quad {\text{otherwise}} \end{array} \right.\\ & = \Pr \left\{ {{r_{0}} \le {{\left({\frac{{{{\mathrm{P}}_{{\text{tx}}}}}}{{\kappa \sigma_{\mathrm{N}}^{2}{\gamma_{\mathrm{A}}}}}} \right)}^{{1 / \gamma }}}} \right\}\\ & = 1 - \exp \left({ - {\Lambda_{\Phi_{{\text{BS}}}^{\left(F \right)}}}\left({{\mathcal{B}}\left({0,{{\left({\frac{{{{\mathrm{P}}_{{\text{tx}}}}}}{{\kappa \sigma_{\mathrm{N}}^{2}{\gamma_{\mathrm{A}}}}}} \right)}^{{1 / \gamma }}}} \right)} \right)} \right) \end{aligned} $$
where the last identity is obtained from the definition of F-function of inhomogeneous PPPs.
The proof follows by computing (5) from (6) and (7).
Appendix B: Proof of Proposition 1
By definition of meta distribution, the following holds true:
$$ \begin{aligned} {\overline {\mathrm{F}}_{{{\mathrm{P}}_{{\text{cov}}}}}}\left({{\gamma_{\mathrm{D}}},{\gamma_{\mathrm{A}}},z} \right) &= {\text{Pr}_{{\Phi_{{\text{BS}}}}}}\left\{ {{{\mathrm{P}}_{\text{cov}} }\left({\left. {{\gamma_{\mathrm{D}}},{\gamma_{\mathrm{A}}}} \right|{\Phi_{{\text{BS}}}}} \right) \ge z} \right\}\\ & = {\text{Pr}_{{\Phi_{{\text{BS}}}}}}\left\{ {{\text{ln}}\left({{{\mathrm{P}}_{\text{cov}} }\left({\left. {{\gamma_{\mathrm{D}}},{\gamma_{\mathrm{A}}}} \right|{\Phi_{{\text{BS}}}}} \right)} \right) \ge \ln \left(z \right)} \right\}\\ & = {\text{Pr}_{{\Phi_{{\text{BS}}}}}}\left\{ { - {\text{ln}}\left({{{\mathrm{P}}_{\text{cov}} }\left({\left. {{\gamma_{\mathrm{D}}},{\gamma_{\mathrm{A}}}} \right|{\Phi_{{\text{BS}}}}} \right)} \right) \le - \ln \left(z \right)} \right\}\\ & = {\text{Pr}_{{\Phi_{{\text{BS}}}}}}\left\{ {{Y_{{\Phi_{{\text{BS}}}}}} \le - \ln \left(z \right)} \right\} \end{aligned} $$
where \({Y_{{\Phi _{{\text {BS}}}}}} = - {\text {ln}}\left ({{{\mathrm {P}}_{\text {cov}} }\left ({\left. {{\gamma _{\mathrm {D}}},{\gamma _{\mathrm {A}}}} \right |{\Phi _{{\text {BS}}}}} \right)} \right)\).
Therefore, the meta distribution can be computed by using the Euler sum method in [16] from the Laplace transform of the random variable \({Y_{{\Phi _{{\text {BS}}}}}}\). In particular, the latter Laplace transform can be formulated as follows:
$$ \begin{aligned} {{\mathcal{L}}_{{Y_{{\Phi_{{\text{BS}}}}}}}}\left(b \right) &= {{\mathbb{E}}_{{Y_{{\Phi_{{\text{BS}}}}}}}}\left\{ {\exp \left({b{Y_{{\Phi_{{\text{BS}}}}}}} \right)} \right\}\\ & = {{\mathbb{E}}_{{{{\Phi_{{\text{BS}}}}}}}}\left\{ {\exp \left({ - b{\text{ln}}\left({{{\mathrm{P}}_{\text{cov}}}\left({\left. {{\gamma_{\mathrm{D}}},{\gamma_{\mathrm{A}}}} \right|{\Phi_{{\text{BS}}}}} \right)} \right)} \right)} \right\}\\ & = {{\mathbb{E}}_{{{{\Phi_{{\text{BS}}}}}}}}\left\{ {{{\left({{{\mathrm{P}}_{\text{cov}} }\left({\left. {{\gamma_{\mathrm{D}}},{\gamma_{\mathrm{A}}}} \right|{\Phi_{{\text{BS}}}}} \right)} \right)}^{- b}}} \right\}\\ & = {{\mathcal{M}}_{- b}}\left({{\gamma_{\mathrm{D}}},{\gamma_{\mathrm{A}}}} \right) \end{aligned} $$
which implies that the meta distribution is determined by the negative moments of the (conditional) coverage probability. Then, the proof follows.
Appendix C: Proof of Theorem 1
By definition, the bth moment of the (conditional) coverage probability is the following:
$$ {{\mathcal{M}}_{b}}\left({{\gamma_{\mathrm{D}}},{\gamma_{\mathrm{A}}}} \right) = {{\mathbb{E}}_{{\Phi_{{\text{BS}}}}}}\left\{ {{{\left({{{\mathrm{P}}_{\text{cov}} }\left({\left. {{\gamma_{\mathrm{D}}},{\gamma_{\mathrm{A}}}} \right|{\Phi_{{\text{BS}}}}} \right)} \right)}^{b}}} \right\} $$
By definition of (conditional) coverage probability, we obtain the following:
$$ \begin{aligned} {{\mathrm{P}}_{\text{cov}} }\left({\left. {{\gamma_{\mathrm{D}}},{\gamma_{\mathrm{A}}}} \right|{\Phi_{{\text{BS}}}}} \right) & = \Pr \left\{ {\left. {{\text{SIR}} \ge {\gamma_{\mathrm{D}}},\overline {{\text{SNR}}} \ge {\gamma_{\mathrm{A}}}} \right|{\Phi_{{\text{BS}}}}} \right\}\\ & = \Pr \left\{ {\left. {{\text{SIR}} \ge {\gamma_{\mathrm{D}}}} \right|{\Phi_{{\text{BS}}}}} \right\}{\mathbbm{1}}\left({{r_{0}} \le {{\left({\frac{{{{\mathrm{P}}_{{\text{tx}}}}}}{{\kappa \sigma_{\mathrm{N}}^{2}{\gamma_{\mathrm{A}}}}}} \right)}^{{1 / \gamma }}}} \right)\\ & = {\mathbbm{1}}\left({{r_{0}} \le {{\left({\frac{{{{\mathrm{P}}_{{\text{tx}}}}}}{{\kappa \sigma_{\mathrm{N}}^{2}{\gamma_{\mathrm{A}}}}}} \right)}^{{1 / \gamma }}}} \right)\prod\limits_{x \in \Phi_{{\text{BS}}}^{\left(K \right)}} {\upsilon \left({\frac{{{r_{0}}}}{r}} \right)} \end{aligned} $$
where r0=∥x0∥,r=∥x∥, and υ(ξ)=(1+γDξγ)−b.
Therefore, the moments can be written as follows:
$$ \begin{aligned} {{\mathcal{M}}_{b}}&\left({{\gamma_{\mathrm{D}}},{\gamma_{\mathrm{A}}}} \right)\\ &= {{\mathbb{E}}_{\Phi_{{\text{BS}}}^{\left(F \right)}}}\left\{ {{\mathrm{1}}\left({{r_{0}} \le {{\left({\frac{{{{\mathrm{P}}_{{\text{tx}}}}}}{{\kappa \sigma_{\mathrm{N}}^{2}{\gamma_{\mathrm{A}}}}}} \right)}^{{1 / \gamma }}}} \right){{\mathbb{E}}_{\Phi_{{\text{BS}}}^{\left(K \right)}}}\left\{ {\prod\limits_{x \in \Phi_{{\text{BS}}}^{\left(K \right)}} {\upsilon \left({\frac{{{r_{0}}}}{r}} \right)}} \right\}} \right\}\\ & = {\int\nolimits}_{0}^{{{\left({{{{{\mathrm{P}}_{{\text{tx}}}}} / {\left({\kappa \sigma_{\mathrm{N}}^{2}{\gamma_{\mathrm{A}}}} \right)}}} \right)}^{{1 / \gamma }}}} {{\mathbb{E}}_{\Phi_{{\text{BS}}}^{\left(K \right)}}}\left\{ {\prod\limits_{x \in \Phi_{{\text{BS}}}^{\left(K \right)}} {\upsilon \left({\frac{{{r_{0}}}}{r}} \right)}} \right\}\\ & \quad\times \Lambda_{\Phi_{{\text{BS}}}^{\left(F \right)}}^{\left(1 \right)} \left({{\mathcal{B}}\left({0,{r_{0}}} \right)} \right)\exp \left({ - {\Lambda_{\Phi_{{\text{BS}}}^{\left(F \right)}}}\left({{\mathcal{B}}\left({0,{r_{0}}} \right)} \right)} \right)d{r_{0}} \end{aligned} $$
where the last equality follows by applying the void probability theorem of inhomogeneous PPPs, \(\Lambda _{\Phi _{{\text {BS}}}^{\left (F \right)}}^{\left (1 \right)}\left ({{\mathcal {B}}\left ({0,r} \right)} \right)\) is the first-order derivative of \({\Lambda _{\Phi _{{\text {BS}}}^{\left (F \right)}}}\left ({{\mathcal {B}}\left ({0,r} \right)} \right)\) computed with respect to r, and, by using the probability generating functional theorem, we have:
$$ {}\begin{aligned} {{\mathbb{E}}_{\Phi_{{\text{BS}}}^{\left(K \right)}}}&\left\{ {\prod\limits_{x \in \Phi_{{\text{BS}}}^{\left(K \right)}} {\upsilon \left({\frac{{{r_{0}}}}{r}} \right)}} \right\}\\ &= \exp \left({ - {\int\nolimits}_{{r_{0}}}^{+ \infty} {\left({1 - \upsilon \left({\frac{{{r_{0}}}}{r}} \right)} \right)\Lambda_{\Phi_{{\text{BS}}}^{\left(K \right)}}^{\left(1 \right)}\left({{\mathcal{B}}\left({0,r} \right)} \right)dr}} \right) \end{aligned} $$
where \(\Lambda _{\Phi _{{\text {BS}}}^{\left (K \right)}}^{\left (1 \right)}\left ({{\mathcal {B}}\left ({0,r} \right)} \right)\) is the first-order derivative of \({\Lambda _{\Phi _{{\text {BS}}}^{\left (K \right)}}}\left ({{\mathcal {B}}\left ({0,r} \right)} \right)\) computed with respect to r.
The proof follows by computing the integrals and using the same steps are those reported in [7].
Appendix D: Proof of Proposition 3
For simplicity, we consider the case study of spatially repulsive cellular networks. A similar approach can be used to prove Proposition 3 if the cellular network is spatially repulsive.
From Appendix Appendix C: Proof of Theorem 1, the moments of the (conditional) coverage probability can be formumated as follows:
$$ \begin{aligned} {\mathcal{M}}_{b}^{\left({{\text{Rep}}} \right)}\left({{\gamma_{\mathrm{D}}},{\gamma_{\mathrm{A}}}} \right) &= {\int\nolimits}_{0}^{{{\left({{{{{\mathrm{P}}_{{\text{tx}}}}} / {\left({\kappa \sigma_{\mathrm{N}}^{2}{\gamma_{\mathrm{A}}}} \right)}}} \right)}^{{1 / \gamma }}}}{{{\mathcal{G}}^{\left({{\text{Rep}}} \right)}}\left({{r_{0}}} \right){{\mathcal{H}}^{\left({{\text{Rep}}} \right)}}\left({{r_{0}}} \right)d{r_{0}}} \end{aligned} $$
where the following shorthand notation is used:
$$ {}{{\mathcal{G}}^{\left({{\text{Rep}}} \right)}}\left({{r_{0}}} \right) = \exp \left(\!{ -\! {\int\nolimits}_{{r_{0}}}^{+ \infty}\! {\left({\!1\! -\! \upsilon\! \left({\frac{{{r_{0}}}}{r}} \right)} \right)\Lambda_{\Phi_{{\text{BS}}}^{\left(K \right)}}^{\left(1 \right)}\left({{\mathcal{B}}\left({0,r} \right)} \right)dr}}\! \right) $$
$$ {}{{\mathcal{H}}^{\left({{\text{Rep}}} \right)}}\!\left({{r_{0}}} \right) \,=\, \Lambda_{\Phi_{{\text{BS}}}^{\left(F \right)}}^{\left(1 \right)}\!\left({{\mathcal{B}}\left({0,{r_{0}}} \right)} \right)\exp \left({ \!- {\Lambda_{\Phi_{{\text{BS}}}^{\left(F \right)}}}\left({{\mathcal{B}}\left({0,{r_{0}}} \right)} \right)} \right) $$
Similar analytical expressions can be obtained for homogenous PPPs. In particular, the following holds true:
$$ {}{\mathcal{M}}_{b}^{\left({{\text{PPP}}} \right)}\!\left({{\gamma_{\mathrm{D}}},{\gamma_{\mathrm{A}}}} \right) \,=\, {\int\nolimits}_{0}^{{{\left({{{{{\mathrm{P}}_{{\text{tx}}}}} / {\left({\kappa \sigma_{\mathrm{N}}^{2}{\gamma_{\mathrm{A}}}} \right)}}} \right)}^{{1 / \gamma }}}}\! \!{{{\mathcal{G}}^{\left({{\text{PPP}}} \right)}}\!\left({{r_{0}}} \right)\!{{\mathcal{H}}^{\left({{\text{PPP}}} \right)}}\!\left({{r_{0}}} \right)\!d{r_{0}}} $$
$$ {}{{\mathcal{G}}^{\left({{\text{PPP}}} \right)}}\!\left({{r_{0}}} \right) \,=\, \exp \left(\!{ \!- \pi {\lambda_{{\text{BS}}}}r_{0}^{2}\!\left(\! {{~}_{2}{F_{1}}\!\left(\!{b, - \frac{2}{\gamma };1 - \frac{2}{\gamma }; - {\gamma_{\mathrm{D}}}}\! \right) \!- \!1}\! \right)} \!\right) $$
$$ {{\mathcal{H}}^{\left({{\text{PPP}}} \right)}}\left({{r_{0}}} \right) = 2\pi {\lambda_{{\text{BS}}}}{r_{0}}\exp \left({ - \pi {\lambda_{{\text{BS}}}}r_{0}^{2}} \right) $$
Since 1−υ(r0/r)≥0, and the inequality \(\Lambda _{\Phi _{{\text {BS}}}^{\left (K \right)}}^{\left (1 \right)}\left ({{\mathcal {B}}\left ({0,{r_{0}}} \right)} \right) \le \Lambda _{{\text {PPP}}}^{\left (1 \right)}\left ({{\mathcal {B}}\left ({0,{r_{0}}} \right)} \right)\) was proved in [7], then we have:
$$ {}\begin{aligned} {{\mathcal{G}}^{\left({{\text{Rep}}} \right)}}&\left({{r_{0}}} \right)\\ &= \exp \left({ - {\int\nolimits}_{{r_{0}}}^{+ \infty} {\left({1 - \upsilon \left({\frac{{{r_{0}}}}{r}} \right)} \right)\Lambda_{\Phi_{{\text{BS}}}^{\left(K \right)}}^{\left(1 \right)}\left({{\mathcal{B}}\left({0,r} \right)} \right)dr}} \right)\\ & \ge \exp \left({ - {\int\nolimits}_{{r_{0}}}^{+ \infty} {\left({1 - \upsilon \left({\frac{{{r_{0}}}}{r}} \right)} \right)\Lambda_{{\text{PPP}}}^{\left(1 \right)}\left({{\mathcal{B}}\left({0,r} \right)} \right)dr}} \right) \\ & = {{\mathcal{G}}^{\left({{\text{PPP}}} \right)}}\left({{r_{0}}} \right) \end{aligned} $$
Therefore, we obtain the following:
$$ {}\begin{aligned} {\mathcal{M}}_{b}^{\left({{\text{Rep}}} \right)}&\left({{\gamma_{\mathrm{D}}},{\gamma_{\mathrm{A}}}} \right)\\ &= {\int\nolimits}_{0}^{{{\left({{{{{\mathrm{P}}_{{\text{tx}}}}} / {\left({\kappa \sigma_{\mathrm{N}}^{2}{\gamma_{\mathrm{A}}}} \right)}}} \right)}^{{1 / \gamma }}}} {{{\mathcal{G}}^{\left({{\text{Rep}}} \right)}}\left({{r_{0}}} \right){{\mathcal{H}}^{\left({{\text{Rep}}} \right)}}\left({{r_{0}}} \right)d{r_{0}}} \\ & \ge {\int\nolimits}_{0}^{{{\left({{{{{\mathrm{P}}_{{\text{tx}}}}} / {\left({\kappa \sigma_{\mathrm{N}}^{2}{\gamma_{\mathrm{A}}}} \right)}}} \right)}^{{1 / \gamma }}}} {{{\mathcal{G}}^{\left({{\text{PPP}}} \right)}}\left({{r_{0}}} \right){{\mathcal{H}}^{\left({{\text{Rep}}} \right)}}\left({{r_{0}}} \right)d{r_{0}}} \end{aligned} $$
Let us define \({\underline {\mathcal {H}}^{\left ({{\text {Rep}}} \right)}}\left ({{r_{0}}} \right) = 1 - \exp \left ({ - {\Lambda _{\Phi _{{\text {BS}}}^{\left (F \right)}}}\left ({{\mathcal {B}}\left ({0,{r_{0}}} \right)} \right)} \right)\). Then, we have the following:
$$ {}\begin{aligned} \mathcal{J} & = {\int\nolimits}_{0}^{{{\left({{{{{\mathrm{P}}_{{\text{tx}}}}} / {\left({\kappa \sigma_{\mathrm{N}}^{2}{\gamma_{\mathrm{A}}}} \right)}}} \right)}^{{1 / \gamma }}}} {{{\mathcal{G}}^{\left({{\text{PPP}}} \right)}}\left({{r_{0}}} \right){{\mathcal{H}}^{\left({{\text{Rep}}} \right)}}\left({{r_{0}}} \right)d{r_{0}}} \\ & \underset {=}{\left(a \right)} \left. {{{\mathcal{G}}^{\left({{\text{PPP}}} \right)}}\left({{r_{0}}} \right){{\underline {\mathcal{H}} }^{\left({{\text{Rep}}} \right)}}\left({{r_{0}}} \right)} \right|_{0}^{{{\left({{{{{\mathrm{P}}_{{\text{tx}}}}} / {\left({\kappa \sigma_{\mathrm{N}}^{2}{\gamma_{\mathrm{A}}}} \right)}}} \right)}^{{1 / \gamma }}}}\\ &\quad- {\int\nolimits}_{0}^{{{\left({{{{{\mathrm{P}}_{{\text{tx}}}}} / {\left({\kappa \sigma_{\mathrm{N}}^{2}{\gamma_{\mathrm{A}}}} \right)}}} \right)}^{{1 / \gamma }}}} {\left({\frac{{d{{\mathcal{G}}^{\left({{\text{PPP}}} \right)}}\left({{r_{0}}} \right)}}{{d{r_{0}}}}} \right){{\underline {\mathcal{H}} }^{\left({{\text{Rep}}} \right)}}\left({{r_{0}}} \right)d{r_{0}}} \\ &\quad \underset {=}{\left(b \right)} {{\mathcal{G}}^{\left({{\text{PPP}}} \right)}}\left({{{\left({\frac{{{{\mathrm{P}}_{{\text{tx}}}}}}{{\kappa \sigma_{\mathrm{N}}^{2}{\gamma_{\mathrm{A}}}}}} \right)}^{{1 / \gamma }}}} \right){\underline {\mathcal{H}}^{\left({{\text{Rep}}} \right)}}\left({{{\left({\frac{{{{\mathrm{P}}_{{\text{tx}}}}}}{{\kappa \sigma_{\mathrm{N}}^{2}{\gamma_{\mathrm{A}}}}}} \right)}^{{1 / \gamma }}}} \right)\\ & \quad- {\int\nolimits}_{0}^{{{\left({{{{{\mathrm{P}}_{{\text{tx}}}}} / {\left({\kappa \sigma_{\mathrm{N}}^{2}{\gamma_{\mathrm{A}}}} \right)}}} \right)}^{{1 / \gamma }}}} {\left({\frac{{d{{\mathcal{G}}^{\left({{\text{PPP}}} \right)}}\left({{r_{0}}} \right)}}{{d{r_{0}}}}} \right){{\underline {\mathcal{H}} }^{\left({{\text{Rep}}} \right)}}\left({{r_{0}}} \right)d{r_{0}}} \end{aligned} $$
where (a) follows by applying the integration by parts rule, and (b) follows by taking into account that \({\underline {\mathcal {H}}^{\left ({{\text {Rep}}} \right)}}\left (0 \right) = 0\).
Since \({{\mathcal {G}}^{\left ({{\text {PPP}}} \right)}}\left ({{r_{0}}} \right) \ge 0, {{d{{\mathcal {G}}^{\left ({{\text {PPP}}} \right)}}\left ({{r_{0}}} \right)} / {d{r_{0}}}} \le 0\), and it was proved in [7] that \({\underline {\mathcal {H}}^{\left ({{\text {Rep}}} \right)}}\left ({{r_{0}}} \right) \ge {\underline {\mathcal {H}}^{\left ({{\text {PPP}}} \right)}}\left ({{r_{0}}} \right) = 1-\exp \left ({ - \pi {\lambda _{{\text {BS}}}}r_{0}^{2}} \right)\), then we have the following:
$$ \begin{aligned} {\mathcal{M}}_{b}^{\left({{\text{Rep}}} \right)}&\left({{\gamma_{\mathrm{D}}},{\gamma_{\mathrm{A}}}} \right)\\ & \ge {\int\nolimits}_{0}^{{{\left({{{{{\mathrm{P}}_{{\text{tx}}}}} / {\left({\kappa \sigma_{\mathrm{N}}^{2}{\gamma_{\mathrm{A}}}} \right)}}} \right)}^{{1 / \gamma }}}} {{{\mathcal{G}}^{\left({{\text{PPP}}} \right)}}\left({{r_{0}}} \right){{\mathcal{H}}^{\left({{\text{Rep}}} \right)}}\left({{r_{0}}} \right)d{r_{0}}} \\ & \ge {{\mathcal{G}}^{\left({{\text{PPP}}} \right)}}\left({{{\left({\frac{{{{\mathrm{P}}_{{\text{tx}}}}}}{{\kappa \sigma_{\mathrm{N}}^{2}{\gamma_{\mathrm{A}}}}}} \right)}^{{1 / \gamma }}}} \right){\underline {\mathcal{H}}^{\left({{\text{PPP}}} \right)}}\left({{{\left({\frac{{{{\mathrm{P}}_{{\text{tx}}}}}}{{\kappa \sigma_{\mathrm{N}}^{2}{\gamma_{\mathrm{A}}}}}} \right)}^{{1 / \gamma }}}} \right) \\ & \quad-\! {\int\nolimits}_{0}^{{{\left({{{{{\mathrm{P}}_{{\text{tx}}}}} / {\left({\kappa \sigma_{\mathrm{N}}^{2}{\gamma_{\mathrm{A}}}} \right)}}} \right)}^{{1 / \gamma }}}} {\left(\!{\frac{{d{{\mathcal{G}}^{\left({{\text{PPP}}} \right)}}\left({{r_{0}}} \right)}}{{d{r_{0}}}}}\! \right)\!{{\underline {\mathcal{H}} }^{\left({{\text{PPP}}} \right)}}\left({{r_{0}}} \right)d{r_{0}}} \\ & = {\mathcal{M}}_{b}^{\left({{\text{PPP}}} \right)}\left({{\gamma_{\mathrm{D}}},{\gamma_{\mathrm{A}}}} \right) \end{aligned} $$
from which the proof follows.
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This work was supported in part by the European Commission through the H2020-MSCA ETN-5Gwireless project under Grant Agreement 641985.
Data sharing is not applicable to this article as no datasets were generated or analyzed during the current study. The paper is built upon mathematical analysis.
Laboratoire des Signaux et Systèmes, CNRS, CentraleSupelec, Univ Paris-Sud, Université Paris-Saclay, Plateau du Moulon, Gif-sur-Yvette, 91192, France
Shanshan Wang & Marco Di Renzo
Shanshan Wang
Marco Di Renzo
The authors declare that they have equally contributed to the paper. Both authors read and approved the final manuscript.
Correspondence to Shanshan Wang.
Wang, S., Di Renzo, M. On the meta distribution in spatially correlated non-Poisson cellular networks. J Wireless Com Network 2019, 161 (2019). https://doi.org/10.1186/s13638-019-1453-x
Cellular networks
Inhomogeneous Poisson point processes
5Gwireless - Innovative Architectures, Wireless Technologies and Tools for High Capacity and Sustainable 5G Ultra-Dense Cellular Networks | CommonCrawl |
Victor W. Marek
Victor Witold Marek, formerly Wiktor Witold Marek known as Witek Marek (born 22 March 1943) is a Polish mathematician and computer scientist working in the fields of theoretical computer science and mathematical logic.
Biography
Victor Witold Marek studied mathematics at the Faculty of Mathematics and Physics of the University of Warsaw. Supervised by Andrzej Mostowski, he received both a magister degree in mathematics in 1964 and a doctoral degree in mathematics in 1968. He completed habilitation in mathematics in 1972.
In 1970–1971, Marek was a postdoctoral researcher at Utrecht University, the Netherlands, where he worked under Dirk van Dalen. In 1967–1968 as well as in 1973–1975, he was a researcher at the Institute of Mathematics of the Polish Academy of Sciences, Warsaw, Poland. In 1979–1980 and 1982–1983 he worked at the Venezuelan Institute of Scientific Research. In 1976, he was appointed an Assistant Professor of Mathematics at the University of Warsaw.
In 1983, he was appointed a professor of computer science at the University of Kentucky. In 1989–1990, he was a Visiting Professor of Mathematics at Cornell University, Ithaca, New York. In 2001–2002, he was a visitor at the Department of Mathematics of the University of California, San Diego.
In 2013, Professor Marek was the Chair of the Program Committee of the scientific conference commemorating Andrzej Mostowski's Centennial.
Legacy
Teaching
He has supervised a number of graduate theses and projects. He was an advisor of 16 doctoral candidates both in mathematics and computer science. In particular, he advised dissertations in mathematics by Małgorzata Dubiel-Lachlan, Roman Kossak, Adam Krawczyk, Tadeusz Kreid, Roman Murawski, Andrzej Pelc, Zygmunt Ratajczyk, Marian Srebrny, and Zygmunt Vetulani. In computer science his students were V. K. Cody Bumgardner, Waldemar W. Koczkodaj, Witold Lipski, Joseph Oldham, Inna Pivkina, Michał Sobolewski , Paweł Traczyk, and Zygmunt Vetulani. These individuals have worked in various institutions of higher education in Canada, France, Poland, and the United States.
Mathematics
He investigated a number of areas in the foundations of mathematics, for instance infinitary combinatorics (large cardinals), metamathematics of set theory, the hierarchy of constructible sets,[1] models of second-order arithmetic,[2] the impredicative theory of Kelley–Morse classes.[3] He proved that the so-called Fraïssé conjecture (second-order theories of countable ordinals are all different) is entailed by Gödel's axiom of constructibility. Together with Marian Srebrny, he investigated properties of gaps in a constructible universe.
Computer science
He studied logical foundations of computer science. In the early 1970s, in collaboration with Zdzisław Pawlak,[4][5] he investigated Pawlak's information storage and retrieval systems,[6] which then was a widely studied concept, especially in Eastern Europe. These systems were essentially single-table relational databases, but unlike Codd's relational databases were bags rather than sets of records. These investigations, in turn, led Pawlak to the concept of rough set,[5] studied by Marek and Pawlak in 1981.[7] The concept of rough set, in computer science, statistics, topology, universal algebra, combinatorics, and modal logic, turned out to be an expressive language for describing, and especially manipulating an incomplete information.
Logic
In the area of nonmonotonic logics, a group of logics related to artificial intelligence, he focused on investigations of Reiter's default logic,[8] and autoepistemic logic of R. Moore. These investigations led to a form of logic programming called answer set programming[9] a computational knowledge representation formalism, studied both in Europe and in the United States. Together with Mirosław Truszczynski, he proved that the problem of existence of stable models of logic programs is NP-complete. In a stronger formalism admitting function symbols, along with Nerode and Remmel he showed that the analogous problem is Σ1
1
-complete.
Publications
V. W. Marek is an author of over 180 scientific papers in the area of foundations of mathematics and of computer science. He was also an editor of numerous proceedings of scientific meetings. Additionally, he authored or coauthored several books. These include:
• Logika i Podstawy Matematyki w Zadaniach (jointly with Janusz Onyszkiewicz)
• Logic and Foundations of Mathematics in problems (jointly with Janusz Onyszkiewicz)
• Analiza Kombinatoryczna (jointly with W. Lipski),
• Nonmonotonic Logic – Context-dependent Reasoning (jointly with M. Truszczyński),
• Introduction to Mathematics of Satisfiability.
References
1. W. Marek and M. Srebrny, Gaps in constructible universe, Annals of Mathematical Logic, 6:359–394, 1974.
2. K.R. Apt and W. Marek, Second order arithmetic and related topics, Annals of Mathematical Logic, 6:177–229, 1974
3. W. Marek, On the metamathematics of impredicative set theory. Dissertationes Mathematicae 98, 45 pages, 1973
4. Z. Pawlak, Mathematical foundations of information retrieval. Institute of Computer Sciences, Polish Academy of Sciences, Technical Report 101, 8 pages, 1973
5. Z. Pawlak, Rough sets. Institute of Computer Science, Polish Academy of Sciences, Technical Report 431, 12 pages, 1981
6. W. Marek and Z. Pawlak On the foundations of information retrieval. Bull. Acad. Pol. Sci. 22:447–452, 1974
7. W. Marek and Z. Pawlak. Rough sets and information systems, Institute of Computer Science, Technical Report 441, Polish Academy of Sciences, 15 pages, 1981
8. M.Denecker, V.W. Marek and M. Truszczynski, Uniform semantic treatment of default and autoepistemic logics. Artificial Intelligence. 143:79–122, 2003
9. V.W. Marek and M. Truszczynski, Stable logic programming – an alternative logic programming paradigm. In: 25 years of Logic Programming Paradigm, pages 375–398, Springer-Verlag, 1999
External links
• Personal page of Dr. V.W. Marek at the University of Kentucky
• Papers online
• Slides and other scientific materials
Authority control
International
• ISNI
• VIAF
National
• Germany
• Israel
• Belgium
• United States
• Latvia
• Australia
• Croatia
• Netherlands
• Poland
Academics
• Association for Computing Machinery
• CiNii
• DBLP
• MathSciNet
• Mathematics Genealogy Project
• zbMATH
Other
• IdRef
| Wikipedia |
Advances in Difference Equations
Uniqueness of the Hadamard-type integral equations
Chenkuan Li ORCID: orcid.org/0000-0001-7098-80591
Advances in Difference Equations volume 2021, Article number: 40 (2021) Cite this article
The goal of this paper is to study the uniqueness of solutions of several Hadamard-type integral equations and a related coupled system in Banach spaces. The results obtained are new and based on Babenko's approach and Banach's contraction principle. We also present several examples for illustration of the main theorems.
The Hadamard-type fractional integral of order \(\alpha > 0\) for a function u is defined in [1, 2] as
$$ \bigl(\mathcal{J}^{\alpha }_{ a + , \mu } u \bigr) (x) = \frac{1}{\Gamma (\alpha )} \int _{a}^{x} \biggl(\frac{t}{x} \biggr)^{\mu } \biggl(\log \frac{x}{t} \biggr)^{\alpha - 1} u(t) \frac{d t}{t}, $$
where \(\log ( \cdot ) = \log _{e} (\cdot )\), \(0 < a < x < b\), and \(\mu \in R\). The corresponding derivative is given by
$$ \bigl(\mathcal{D}^{\alpha }_{a + , \mu } u \bigr) (x) = x^{-\mu } \delta ^{n} x^{\mu } \bigl(\mathcal{J}^{n - \alpha }_{ a + , \mu } u \bigr) (x), \quad \delta = x \frac{d}{d x}, $$
where \(n = [\alpha ] + 1\), \([\alpha ]\) being the integral part of α. When \(\mu = 0\), they take the forms
$$\begin{aligned}& \bigl(\mathcal{J}^{\alpha }_{ a + } u \bigr) (x) = \frac{1}{\Gamma (\alpha )} \int _{a}^{x} \biggl(\log \frac{x}{t} \biggr)^{\alpha - 1} u(t) \frac{d t}{t}, \\& \bigl(\mathcal{D}^{\alpha }_{ a + } u \bigr) (x) = \delta ^{n} \bigl(\mathcal{J}^{n - \alpha }_{ a + } u \bigr) (x), \end{aligned}$$
respectively. In particular, for \(\alpha = 1\),
$$ (\mathcal{J}_{ a + , \mu } u) (x) = \bigl(\mathcal{J}^{1}_{ a + , \mu } u \bigr) (x) = \frac{1}{\Gamma (\alpha ) x^{\mu }} \int _{a}^{x} t^{ \mu - 1} u(t) \,d t, $$
which leads to definition of the space \({X}_{\mu }(a, b)\) of Lebesgue-measurable functions u on \([a, b]\) for which \(x^{\mu - 1} u(x)\) is absolutely integrable [2]:
$$ X_{\mu }(a, b) = \biggl\{ u : [a, b] \rightarrow C: \lVert u \rVert _{X_{\mu }} = \int _{a}^{b} x^{\mu - 1} \bigl\vert u(x) \bigr\vert \,d x < \infty \biggr\} . $$
Clearly, for \(a > 0\),
$$\begin{aligned}& \min_{x \in [a, b]} \bigl\{ x^{\mu - 1} \bigr\} \int _{a}^{b} \bigl\vert u(x) \bigr\vert \,d x \leq \int _{a}^{b} x^{\mu - 1} \bigl\vert u(x) \bigr\vert \,d x \leq \max_{x \in [a, b]} \bigl\{ x^{\mu - 1} \bigr\} \int _{a}^{b} \bigl\vert u(x) \bigr\vert \,d x, \quad \text{and} \\ & 0 < \min_{x \in [a, b]} \bigl\{ x^{\mu - 1} \bigr\} \leq \max _{x \in [a, b]} \bigl\{ x^{\mu - 1} \bigr\} \end{aligned}$$
for every \(\mu \in R\). Hence \(X_{\mu }(a, b)\) is a Banach space, since \(L(a, b)\) with the norm
$$ \lVert u \rVert _{L} = \int _{a}^{b} \bigl\vert u(x) \bigr\vert \,d x $$
is complete and the norms \(\lVert u \rVert _{X_{\mu }}\) and \(\lVert u \rVert _{L} \) are equivalent.
We need the following lemmas shown by Kilbas [2].
Lemma 1.1
If \(\alpha > 0\), \(\mu \in R\), and \(0 < a < b < \infty \), then the operator \(\mathcal{J}^{\alpha }_{ a + , \mu }\) is bounded in \(X\mu (a, b)\), and for \(u \in X\mu (a, b)\),
$$ \bigl\lVert \mathcal{J}^{\alpha }_{ a + , \mu } u \bigr\rVert _{X_{\mu }} \leq K \lVert u \rVert _{X_{\mu }}, $$
$$ K = \frac{1}{\Gamma (\alpha + 1)} \biggl[\log \biggl(\frac{b}{a} \biggr) \biggr]^{\alpha }. $$
If \(\alpha >0\), \(\beta > 0\), \(\mu \in R\), and \(u \in X_{\mu }(a, b)\), then the semigroup property holds:
$$ \mathcal{J}^{\alpha }_{ a + , \mu } \mathcal{J}^{\beta }_{ a + , \mu } u = \mathcal{J}^{\alpha + \beta }_{ a + , \mu }u. $$
There are a lot of studies on fractional differential and integral equations involving Riemann–Liouville or Caputo operators with boundary value problems or initial conditions [3–11]. Li and Sarwar [12] considered the existence of solutions for the following fractional-order initial value problems:
$$ \textstyle\begin{cases} ({}_{C} D_{0, t}^{\alpha }u) (t) = f(t, u(t)), \quad t \in (0, \infty ), \\ u(0) = u_{0}, \end{cases} $$
where \(0 < \alpha < 1\), and \({}_{C}D_{0, t}^{\alpha }\) is the Caputo derivative.
Wu et al. [13] studied the existence and uniqueness of solutions by fixed point theory for the following fractional differential equation with nonlinearity depending on fractional derivatives of lower order on an infinite interval:
$$ \textstyle\begin{cases} (D_{0 +}^{\alpha }u) (t) + f(t, u(t), (D_{0 +}^{\alpha - 2} u) (t), (D_{0 +}^{\alpha - 1} u) (t)) = 0, \quad t \in (0, \infty ), \\ u(0) = u'(0) = 0, \quad\quad (D_{0 +}^{\alpha - 1} u) (\infty ) = \zeta , \end{cases} $$
where \(2 < \alpha \leq 3\), \(D_{0 +}^{\alpha }\), \(D_{0 +}^{\alpha - 1}\), and \(D_{0 +}^{\alpha - 2}\) are all Riemann–Liouville fractional derivatives.
Ahmad and Ntouyas [14] considered a coupled system of Hadamard-type fractional differential equations and integral boundary conditions
$$ \textstyle\begin{cases} \mathcal{D}^{\alpha }_{1 + } u (t) = w_{1}(t, u(t), v(t)), \quad 1 < t < e, 1 < \alpha \leq 2, \\ \mathcal{D}^{\beta }_{1 + } v (t) = w_{2}(t, u(t), v(t)), \quad 1 < t < e, 1 < \beta \leq 2, \\ u(1) = 0, \quad\quad u(e) = \mathcal{J}^{\gamma }_{ 1 + } u(\sigma _{1}) = \frac{1}{\Gamma (\gamma )} \int _{1}^{\sigma _{1}} (\log \frac{\sigma _{1}}{s} )^{\gamma - 1} u(s) \frac{d s}{s}, \\ v(1) = 0, \quad\quad v(e) = \mathcal{J}^{\gamma }_{ 1 + } v(\sigma _{2}) = \frac{1}{\Gamma (\gamma )} \int _{1}^{\sigma _{2}} (\log \frac{\sigma _{2}}{s} )^{\gamma - 1} v(s) \frac{d s}{s}, \end{cases} $$
where \(\gamma > 0\), \(1 < \sigma _{1} < e\), \(1 < \sigma _{2} < e\), and \(w_{1}, w_{2}: [1, e] \times R \times R \rightarrow R\) are continuous functions satisfying certain conditions. They showed the existence of solutions by Leray–Schauder's alternative and the uniqueness by Banach's fixed point theorem, based on the fact that for \(1 < q \leq 2\) and \(z \in C([1, e], R)\), the problem
$$ \textstyle\begin{cases} \mathcal{D}^{q}_{1 + } x (t) = z(t), \quad 1 < t < e, \\ x(1) = 0, \quad\quad x(e) = \mathcal{J}^{\gamma }_{ 1 + } x(\theta ), \end{cases} $$
$$ x(t) = \mathcal{J}^{q}_{1 + } z (t) + \frac{(\log t)^{q - 1}}{Q} \bigl[ \mathcal{J}^{\gamma + q}_{ 1 + } z(\theta ) - \mathcal{J}^{q}_{ 1 + }z(e) \bigr], $$
$$ Q = \frac{1}{ 1 - \frac{1}{\Gamma (\gamma )} \int _{1}^{\theta } (\log \frac{\theta }{s} )^{\gamma - 1} (\log s)^{q - 1} \frac{d s}{s} }. $$
Let \(g: [a, b] \times R \rightarrow R\) be a continuous function. In this paper, we study the following nonlinear Hadamard-type (μ is arbitrary in R) integral equation in the space \(X_{\mu }(a, b)\):
$$ a_{n} \bigl(\mathcal{J}^{\alpha _{n}}_{ a + , \mu } u \bigr) (x) + \cdots + a_{1} \bigl(\mathcal{J}^{\alpha _{1}}_{ a + , \mu } u \bigr) (x) + u(x) = g \bigl(x, u(x) \bigr), $$
where \(\alpha _{n} > \alpha _{n - 1} > \cdots > \alpha _{1} > 0\), and \(a_{i}\), \(i = 1, 2,\ldots,n\), are complex numbers, not all zero.
To the best of the author's knowledge, equation (1) is new in the framework of Hadamard-type integral equations. First, by Babenko's approach we will construct the solution as a convergent infinite series in \(X_{\mu }(a, b)\) for the integral equation
$$ a_{n} \bigl(\mathcal{J}^{\alpha _{n}}_{ a + , \mu } u \bigr) (x) + \cdots + a_{1} \bigl(\mathcal{J}^{\alpha _{1}}_{ a + , \mu } u \bigr) (x) + u(x) = f(x), $$
where \(f \in X_{\mu }(a, b)\). Then we will show that there exists a unique solution for equation (1) using Banach's contraction principle. Furthermore, we present the solution for the Hadamard-type integral equation
$$ a_{n} \bigl(\mathcal{J}^{\alpha _{n}}_{ a + , \mu } u \bigr) (x) + \cdots + a_{1} \bigl(\mathcal{J}^{\alpha _{1}}_{ a + , \mu } u \bigr) (x) + \bigl(\mathcal{J}^{ \alpha _{0}}_{ a + , \mu } u \bigr) (x) = f(x) $$
by the Hadamard fractional derivative and show the uniqueness for the coupled system of integral equations
$$ \textstyle\begin{cases} a_{n} (\mathcal{J}^{\alpha _{n}}_{ a + , \mu } u)(x) + \cdots + a_{1} (\mathcal{J}^{\alpha _{1}}_{ a + , \mu } u)(x) + u(x) = g_{1}(x, u(x), v(x)), \\ b_{n} (\mathcal{J}^{\beta _{n}}_{ a + , \mu } v)(x) + \cdots + b_{1} (\mathcal{J}^{\beta _{1}}_{ a + , \mu } v)(x) + v(x) = g_{2}(x, u(x), v(x)), \end{cases} $$
where \(\alpha _{n} > \alpha _{n - 1} > \cdots > \alpha _{1} > 0\), \(\beta _{n} > \beta _{n - 1} > \cdots > \beta _{1} > 0\), and there exist at least one nonzero \(a_{i}\) and one nonzero \(b_{j}\) for some \(1 \leq i, j \leq n\). We also present several examples for illustration of our results.
Main results
We begin by showing the solution for equation (2) as a convergent series in the space \(X_{\mu }(a, b)\) by Babenko's approach [15], which is a powerful tool in solving differential and integral equations. The method itself is close to the Laplace transform method in the ordinary sense, but it can be used in more cases [16, 17], such as solving integral or fractional differential equations with distributions whose Laplace transforms do not exist in the classical sense. Clearly, it is always necessary to show the convergence of the series obtained as solutions. Podlubny [16] also provided interesting applications to solving certain partial differential equations for heat and mass transfer by Babenko's method. Recently, Li and Plowman [18] and Li [19] studied the generalized Abel's integral equations of the second kind with variable coefficients by Babenko's technique.
Theorem 2.1
Let \(f \in X_{\mu }(a, b)\) with \(0 < a < b < \infty \). Then equation (2) has a unique solution in the space \(X_{\mu }(a, b)\),
$$ u(x) = \sum_{k = 0}^{\infty }(-1)^{k} \sum_{k_{1} + \cdots + k_{n} = k} \binom{k}{k_{1}, k_{2},\ldots, k_{n}} a_{n}^{k_{n}} \cdots a_{1}^{k_{1}} \bigl(\mathcal{J}^{k_{n} \alpha _{n} + \cdots + k_{1} \alpha _{1}}_{ a + , \mu } f \bigr) (x), $$
where \(\alpha _{n} > \cdots > \alpha _{1} > 0\), and \(a_{i}\), \(i = 1, 2,\ldots, n\), are complex numbers, not all zero.
Equation (2) can be written as
$$ \bigl(a_{n} \mathcal{J}^{\alpha _{n}}_{ a + , \mu } + \cdots + a_{1} \mathcal{J}^{\alpha _{1}}_{ a + , \mu } + 1 \bigr)u(x) = f(x). $$
By Babenko's method we arrive at
$$\begin{aligned} u(x) = & \bigl(a_{n} \mathcal{J}^{\alpha _{n}}_{ a + , \mu } + \cdots + a_{1} \mathcal{J}^{\alpha _{1}}_{ a + , \mu } + 1 \bigr)^{-1} f(x) \\ = & \sum_{k = 0}^{\infty }(-1)^{k} \bigl(a_{n} \mathcal{J}^{\alpha _{n}}_{ a + , \mu } + \cdots + a_{1} \mathcal{J}^{\alpha _{1}}_{ a + , \mu } \bigr)^{k} f(x) \\ = & \sum_{k = 0}^{\infty }(-1)^{k} \sum_{k_{1} + \cdots + k_{n} = k} \binom{k}{k_{1}, k_{2},\ldots, k_{n}} \bigl(a_{n} \mathcal{J}^{\alpha _{n}}_{ a + , \mu } \bigr)^{k_{n}} \cdots \bigl(a_{1} \mathcal{J}^{\alpha _{1}}_{ a + , \mu } \bigr)^{k_{1}} f(x) \\ = & \sum_{k = 0}^{\infty }(-1)^{k} \sum_{k_{1} + \cdots + k_{n} = k} \binom{k}{k_{1}, k_{2},\ldots, k_{n}} a_{n}^{k_{n}} \mathcal{J}^{k_{n} \alpha _{n}}_{ a + , \mu } \cdots a_{1}^{k_{1}} \mathcal{J}^{k_{1} \alpha _{1}}_{ a + , \mu } f(x) \\ = & \sum_{k = 0}^{\infty }(-1)^{k} \sum_{k_{1} + \cdots + k_{n} = k} \binom{k}{k_{1}, k_{2},\ldots, k_{n}} a_{n}^{k_{n}} \cdots a_{1}^{k_{1}} \bigl(\mathcal{J}^{k_{n} \alpha _{n} + \cdots + k_{1} \alpha _{1}}_{ a + , \mu } f \bigr) (x) \end{aligned}$$
using Lemma 1.2 and the multinomial theorem. Note that
$$ \mathcal{J}^{0}_{ a + , \mu } f(x) = f(x). $$
It remains to show that series (5) converges in the space \(X_{\mu }(a, b)\). By Lemma 1.1
$$ \bigl\lVert \mathcal{J}^{k_{n} \alpha _{n} + \cdots + k_{1} \alpha _{1}}_{ a + , \mu } f(x) \bigr\rVert _{X_{\mu }} \leq K \lVert f \rVert _{X_{\mu }}, $$
$$ K = \frac{1}{\Gamma (k_{n} \alpha _{n} + \cdots + k_{1} \alpha _{1} + 1)} \biggl(\log \frac{b}{a} \biggr)^{k_{n} \alpha _{n} + \cdots + k_{1} \alpha _{1}}. $$
$$\begin{aligned}& \lVert u \rVert _{X\mu } \\& \quad \leq\sum_{k = 0}^{\infty }\sum _{k_{1} + \cdots + k_{n} = k} \binom{k}{k_{1}, k_{2},\ldots, k_{n}} \frac{ ( \vert a_{n} \vert (\log \frac{b}{a} )^{\alpha _{n}} )^{k_{n}} \cdots ( \vert a_{1} \vert (\log \frac{b}{a} )^{\alpha _{1}} )^{k_{1}} }{\Gamma (k_{n} \alpha _{n} + \cdots + k_{1} \alpha _{1} + 1)} \lVert f \rVert _{X_{\mu }} \\& \quad = E_{(\alpha _{1},\ldots, \alpha _{n}, 1)} \biggl( \vert a_{1} \vert \biggl(\log \frac{b}{a} \biggr)^{\alpha _{1}},\ldots, \vert a_{n} \vert \biggl(\log \frac{b}{a} \biggr)^{\alpha _{n}} \biggr) \lVert f \rVert _{X_{\mu }}, \end{aligned}$$
$$ E_{(\alpha _{1},\ldots, \alpha _{n}, 1)} \biggl( \vert a_{1} \vert \biggl( \log \frac{b}{a} \biggr)^{\alpha _{1}},\ldots, \vert a_{n} \vert \biggl(\log \frac{b}{a} \biggr)^{\alpha _{n}} \biggr) < \infty $$
is the value of the multivariate Mittag-Leffler function \(E_{(\alpha _{1},\ldots, \alpha _{n}, 1)}(z_{1},\ldots, z_{n})\) given in [7] at
$$ z_{1} = \vert a_{1} \vert \biggl(\log \frac{b}{a} \biggr)^{\alpha _{1}},\quad\quad \ldots, \quad\quad z_{n} = \vert a_{n} \vert \biggl(\log \frac{b}{a} \biggr)^{\alpha _{n}}. $$
Thus \(u \in X_{\mu }(a, b)\), and the series on the right-hand of equation (5) is convergent.
To verify that the series is a solution, we substitute it into the left-hand side of equation (2):
$$\begin{aligned}& a_{n} \sum_{k = 0}^{\infty }(-1)^{k} \sum_{k_{1} + \cdots + k_{n} = k} \binom{k}{k_{1}, k_{2},\ldots, k_{n}} a_{n}^{k_{n}} \cdots a_{1}^{k_{1}} \bigl(\mathcal{J}^{(k_{n} + 1) \alpha _{n} + \cdots + k_{1} \alpha _{1}}_{ a + , \mu } f \bigr) (x) + \cdots \\& \quad\quad{}+ a_{1} \sum_{k = 0}^{\infty }(-1)^{k} \sum_{k_{1} + \cdots + k_{n} = k} \binom{k}{k_{1}, k_{2},\ldots, k_{n}} a_{n}^{k_{n}} \cdots a_{1}^{k_{1}} \bigl(\mathcal{J}^{k_{n} \alpha _{n} + \cdots + (k_{1} + 1) \alpha _{1}}_{ a + , \mu } f \bigr) (x) + \cdots \\& \quad \quad{}+ \sum_{k = 0}^{\infty }(-1)^{k} \sum_{k_{1} + \cdots + k_{n} = k} \binom{k}{k_{1}, k_{2},\ldots, k_{n}} a_{n}^{k_{n}} \cdots a_{1}^{k_{1}} \bigl(\mathcal{J}^{k_{n} \alpha _{n} + \cdots + k_{1} \alpha _{1}}_{ a + , \mu } f \bigr) (x) \\& \quad = a_{n} \bigl(\mathcal{J}^{\alpha _{n}}_{a+, \mu }f \bigr) (x) + a_{n} \sum_{k = 1}^{\infty }(-1)^{k} \sum_{k_{1} + \cdots + k_{n} = k} \binom{k}{k_{1}, k_{2},\ldots, k_{n}}a_{n}^{k_{n}} \cdots a_{1}^{k_{1}} \\& \quad \quad {}\cdot \bigl(\mathcal{J}^{(k_{n} + 1) \alpha _{n} + \cdots + k_{1} \alpha _{1}}_{ a + , \mu } f \bigr) (x) + \cdots + a_{1} \bigl(\mathcal{J}^{\alpha _{1}}_{a+, \mu }f \bigr) (x) \\& \quad \quad {}+a_{1} \sum_{k = 1}^{\infty }(-1)^{k} \sum_{k_{1} + \cdots + k_{n} = k} \binom{k}{k_{1}, k_{2},\ldots, k_{n}} a_{n}^{k_{n}} \cdots a_{1}^{k_{1}} \bigl(\mathcal{J}^{k_{n} \alpha _{n} + \cdots + (k_{1} + 1) \alpha _{1}}_{ a + , \mu } f \bigr) (x) \\& \quad \quad {}+f(x) - a_{n} \bigl(\mathcal{J}^{\alpha _{n}}_{a+, \mu }f \bigr) (x) - \cdots - a_{1} \bigl(\mathcal{J}^{\alpha _{1}}_{a+, \mu }f \bigr) (x) \\& \quad \quad {} + \sum_{k = 2}^{\infty }(-1)^{k} \sum_{k_{1} + \cdots + k_{n} = k} \binom{k}{k_{1}, k_{2},\ldots, k_{n}} a_{n}^{k_{n}} \cdots a_{1}^{k_{1}} \bigl(\mathcal{J}^{k_{n} \alpha _{n} + \cdots + k_{1} \alpha _{1}}_{ a + , \mu } f \bigr) (x) \\& \quad = a_{n} \sum_{k = 1}^{\infty }(-1)^{k} \sum_{k_{1} + \cdots + k_{n} = k} \binom{k}{k_{1}, k_{2},\ldots, k_{n}} a_{n}^{k_{n}} \cdots a_{1}^{k_{1}} \bigl(\mathcal{J}^{(k_{n} + 1) \alpha _{n} + \cdots + k_{1} \alpha _{1}}_{ a + , \mu } f \bigr) (x)+ \cdots \\& \quad \quad {}+a_{1} \sum_{k = 1}^{\infty }(-1)^{k} \sum_{k_{1} + \cdots + k_{n} = k} \binom{k}{k_{1}, k_{2},\ldots, k_{n}} a_{n}^{k_{n}} \cdots a_{1}^{k_{1}} \bigl(\mathcal{J}^{k_{n} \alpha _{n} + \cdots + (k_{1} + 1) \alpha _{1}}_{ a + , \mu } f \bigr) (x) \\& \quad \quad{}+ f(x) + \sum_{k = 2}^{\infty }(-1)^{k} \sum_{k_{1} + \cdots + k_{n} = k} \binom{k}{k_{1}, k_{2},\ldots, k_{n}} a_{n}^{k_{n}} \cdots a_{1}^{k_{1}} \bigl(\mathcal{J}^{k_{n} \alpha _{n} + \cdots + k_{1} \alpha _{1}}_{ a + , \mu } f \bigr) (x) = f(x) \end{aligned}$$
$$\begin{aligned}& a_{n} \sum_{k = 1}^{\infty }(-1)^{k} \sum_{k_{1} + \cdots + k_{n} = k} \binom{k}{k_{1}, k_{2},\ldots, k_{n}} a_{n}^{k_{n}} \cdots a_{1}^{k_{1}} \bigl(\mathcal{J}^{(k_{n} + 1) \alpha _{n} + \cdots + k_{1} \alpha _{1}}_{ a + , \mu } f \bigr) (x) + \cdots \\& \quad{}+ a_{1} \sum_{k = 1}^{\infty }(-1)^{k} \sum_{k_{1} + \cdots + k_{n} = k} \binom{k}{k_{1}, k_{2},\ldots, k_{n}} a_{n}^{k_{n}} \cdots a_{1}^{k_{1}} \bigl(\mathcal{J}^{k_{n} \alpha _{n} + \cdots + (k_{1} + 1) \alpha _{1}}_{ a + , \mu } f \bigr) (x) \\& \quad{}+ \sum_{k = 2}^{\infty }(-1)^{k} \sum_{k_{1} + \cdots + k_{n} = k} \binom{k}{k_{1}, k_{2},\ldots, k_{n}} a_{n}^{k_{n}} \cdots a_{1}^{k_{1}} \bigl(\mathcal{J}^{k_{n} \alpha _{n} + \cdots + k_{1} \alpha _{1}}_{ a + , \mu } f \bigr) (x) = 0 \end{aligned}$$
by cancelation. Note that all series are absolutely convergent and the term rearrangements are feasible for cancelation.
Clearly, the uniqueness immediately follows from the fact that the integral equation
$$ a_{n} \bigl(\mathcal{J}^{\alpha _{n}}_{ a + , \mu } u \bigr) (x) + \cdots + a_{1} \bigl(\mathcal{J}^{\alpha _{1}}_{ a + , \mu } u \bigr) (x) + u(x) = 0 $$
only has zero solution by Babenko's method. This completes the proof of Theorem 2.1. □
Let \(\nu > 0\) and \(x \geq 0\). The incomplete gamma function is defined by
$$ \gamma (\nu , x) = \int _{0}^{x} t^{\nu - 1} e^{-t} \,dt. $$
From the recurrence relation [20]
$$ \gamma ( \nu + 1, x) = \nu \gamma (\nu , x) - x^{\nu }e ^{-x} $$
$$ \gamma (\nu , x) = x^{\nu }\Gamma (\nu ) e^{-x} \sum_{j = 0}^{\infty }\frac{x^{j}}{\Gamma (\nu + j + 1)}. $$
Let \(0< a < x < b\). Then the Hadamard-type integral equation
$$ \bigl(\mathcal{J}^{\frac{1}{2}}_{ a + , -1} u \bigr) (x) + u(x) = x^{2} $$
has the solution
$$ u(x) = a x \sum_{k = 0}^{\infty }\sum _{j = 0}^{\infty }\frac{ (-1)^{k} (\log \frac{x}{a} )^{j + \frac{1}{2} k}}{ \Gamma (\frac{1}{2} k + j +1 )}. $$
Indeed, it follows from Lemma 2.4 in [2] that
$$ \bigl(\mathcal{J}^{\alpha }_{ a + , \mu } t^{w} \bigr) (x) = \frac{\gamma (\alpha , (\mu + w) \log (x/a))}{\Gamma (\alpha )} (\mu + w)^{-\alpha } x^{w}, $$
where \(\mu + w > 0\).
By Theorem 2.1
$$\begin{aligned} u(x) = & \sum_{k = 0}^{\infty }(-1)^{k} \bigl(\mathcal{J}^{\frac{1}{2} k}_{ a + , - 1} t^{2} \bigr) (x) = x^{2} \sum_{k = 0}^{\infty }(-1)^{k} \frac{\gamma (k/2, \log (x/a))}{\Gamma (k/2)}. \end{aligned}$$
Applying equation (6), we have
$$ \gamma \bigl(k/2, \log (x/a) \bigr) = (\log x/a )^{k/2} \Gamma (k/2) \frac{a}{x} \sum_{j = 0}^{\infty } \frac{ (\log x/a )^{j}}{\Gamma (\frac{1}{2} k + j + 1)}. $$
$$ u(x) = a x \sum_{k = 0}^{\infty }\sum _{j = 0}^{\infty }\frac{ (-1)^{k} (\log \frac{x}{a} )^{j + \frac{1}{2} k}}{ \Gamma (\frac{1}{2} k + j +1 )} $$
is a solution in the space \(X_{\mu }(a, b)\).
The following theorem shows the uniqueness of solution of equation (1).
Let \(g: [a, b] \times R \rightarrow R\) be a continuous function and suppose that there exists a constant \(C > 0\) such that for all \(x \in [a, b]\),
$$ \bigl\vert g(x, y_{1}) - g(x, y_{2}) \bigr\vert \leq C \vert y_{1} - y_{2} \vert , \quad y_{1}, y_{2} \in R. $$
Furthermore, suppose that
$$ C E_{(\alpha _{1},\ldots, \alpha _{n}, 1)} \biggl( \vert a_{1} \vert \biggl( \log \frac{b}{a} \biggr)^{\alpha _{1}},\ldots, \vert a_{n} \vert \biggl(\log \frac{b}{a} \biggr)^{\alpha _{n}} \biggr) < 1. $$
Then equation (1) has a unique solution in the space \(X_{\mu }(a, b)\) for every \(\mu \in R\).
Let \(u \in X_{\mu }(a, b)\). Then \(g (x, u(x)) \in X_{\mu }(a, b)\) since
$$ \bigl\vert g \bigl(x, u(x) \bigr) \bigr\vert \leq \bigl\vert g \bigl(x, u(x) \bigr) - g(x, 0) \bigr\vert + \bigl\vert g(x, 0) \bigr\vert \leq C \bigl\vert u(x) \bigr\vert + \bigl\vert g(x, 0) \bigr\vert \in X_{\mu }(a, b) $$
by noting that \(g(x, 0)\) is a continuous function on \([a, b]\). Define the mapping T on \(X_{\mu }(a, b)\) by
$$ T(u) (x) = \sum_{k = 0}^{\infty }(-1)^{k} \sum_{k_{1} + \cdots + k_{n} = k} \binom{k}{k_{1}, k_{2},\ldots, k_{n}} a_{n}^{k_{n}} \cdots a_{1}^{k_{1}} \bigl(\mathcal{J}^{k_{n} \alpha _{n} + \cdots + k_{1} \alpha _{1}}_{ a + , \mu } g \bigl(t, u(t) \bigr) \bigr) (x). $$
In particular, for \(k = 0\),
$$ \mathcal{J}^{k_{n} \alpha _{n} + \cdots + k_{1} \alpha _{1}}_{ a + , \mu } g \bigl(t, u(t) \bigr) (x) = g \bigl(x, u(x) \bigr). $$
From the proof of Theorem 2.1 we have
$$\begin{aligned} \bigl\lVert T(u) \bigr\rVert _{X\mu } \leq E_{(\alpha _{1},\ldots, \alpha _{n}, 1)} \biggl( \vert a_{1} \vert \biggl(\log \frac{b}{a} \biggr)^{ \alpha _{1}},\ldots, \vert a_{n} \vert \biggl(\log \frac{b}{a} \biggr)^{ \alpha _{n}} \biggr) \bigl\lVert g \bigl(x, u(x) \bigr) \bigr\rVert _{X_{\mu }}. \end{aligned}$$
Clearly,
$$ \bigl\lVert g \bigl(x, u(x) \bigr) \bigr\rVert _{X_{\mu }} \leq C \lVert u \rVert _{X_{\mu }} + \max_{x \in [a, b]} \bigl\{ x^{\mu - 1} \bigl\vert g (x, 0) \bigr\vert \bigr\} (b - a) < \infty . $$
Hence T is a mapping from \(X_{\mu }(a, b)\) to \(X_{\mu }(a, b)\). It remains to prove that T is contractive. We have
$$\begin{aligned} \bigl\lVert T(u) - T(v) \bigr\rVert _{X_{\mu }} \leq &\sum _{k = 0}^{\infty }\sum_{k_{1} + \cdots + k_{n} = k} \binom{k}{k_{1}, k_{2},\ldots, k_{n}} \\ & {}\cdot \vert a_{n} \vert ^{k_{n}} \cdots \vert a_{1} \vert ^{k_{1}} \bigl\lVert \mathcal{J}^{k_{n} \alpha _{n} + \cdots + k_{1} \alpha _{1}}_{ a + , \mu } \bigl(g \bigl(t, u(t) \bigr) - g \bigl(t, v(t) \bigr) \bigr) (x) \bigr\rVert _{X_{m}u} \\ \leq& \sum_{k = 0}^{\infty }\sum _{k_{1} + \cdots + k_{n} = k} \binom{k}{k_{1}, k_{2},\ldots, k_{n}} \frac{ ( \vert a_{n} \vert (\log \frac{b}{a} )^{\alpha _{n}} )^{k_{n}} \cdots ( \vert a_{1} \vert (\log \frac{b}{a} )^{\alpha _{1}} )^{k_{1}} }{\Gamma (k_{n} \alpha _{n} + \cdots + k_{1} \alpha _{1} + 1)} \\ & {}\cdot \bigl\lVert g \bigl(t, u(t) \bigr) - g \bigl(t, v(t) \bigr) \bigr\rVert _{X_{\mu }}. \end{aligned}$$
$$ \bigl\lVert g \bigl(t, u(t) \bigr) - g \bigl(t, v(t) \bigr) \bigr\rVert _{X_{\mu }} \leq C \lVert u - v \rVert _{X_{\mu }}, $$
we derive
$$ \bigl\lVert T(u) - T(v) \bigr\rVert _{X_{\mu }} \leq C E_{(\alpha _{1},\ldots, \alpha _{n}, 1)} \biggl( \vert a_{1} \vert \biggl(\log \frac{b}{a} \biggr)^{\alpha _{1}},\ldots, \vert a_{n} \vert \biggl(\log \frac{b}{a} \biggr)^{\alpha _{n}} \biggr) \lVert u - v \rVert _{X_{\mu }}. $$
Therefore T is contractive. This completes the proof of Theorem 2.2. □
Let \(a = 1\), \(b = e\), and \(\mu \in R\). Then for every \(\mu \in R\), there is a unique solution for the following Hadamard-type integral equation:
$$ \bigl(\mathcal{J}^{1.5}_{ 1 + , \mu } u \bigr) (x) + ( \mathcal{J}_{ 1 + , \mu } u) (x) + u(x) = \frac{x^{2}}{9(1 + x^{2})} \sin u(x) + \cos ( \sin x) + 1. $$
Clearly, the function
$$ g(x , y) = \frac{x^{2}}{9(1 + x^{2})} \sin y + \cos (\sin x) + 1 $$
is a continuous function from \([1, e] \times R\) to R and satisfies
$$ \bigl\vert g(x, y_{1}) - g(x, y_{2}) \bigr\vert \leq \frac{x^{2}}{9(1 + x^{2})} \vert \sin y_{1} - \sin y_{2} \vert \leq \frac{x^{2}}{9(1 + x^{2})} \vert y_{1} - y_{2} \vert \leq \frac{1}{9} \vert y_{1} - y_{2} \vert . $$
Obviously, \(a_{2} = a_{1} = 1\), and \(\log b/a = 1\). By Theorem 2.2 we need to calculate the value
$$\begin{aligned} \sum_{k = 0}^{\infty }\sum _{k_{1} + k_{2} = k} \binom{k}{k_{1}, k_{2}} \frac{1}{\Gamma (1.5 k_{2} + k_{1} + 1)} =& \sum _{k = 0}^{\infty }\sum_{j = 0}^{k} \binom{k}{j} \frac{1}{\Gamma (k + 1 + 0.5 j)} \\ = &1 + \sum_{k = 1}^{\infty }\sum _{j = 0}^{k} \binom{k}{j} \frac{1}{\Gamma (k + 1 + 0.5 j)}. \end{aligned}$$
For \(k \geq 1\) and \(j \geq 0\), we have
$$ \frac{1}{\Gamma (k + 1 + 0.5 j)} \leq \frac{1}{\Gamma (k + 1) } = \frac{1}{k!} \quad \text{and} \quad \sum_{j = 0}^{k} \binom{k}{j} = 2^{k}. $$
$$\begin{aligned}& \sum_{k = 0}^{\infty }\sum _{j = 0}^{k} \binom{k}{j} \frac{1}{\Gamma (1.5 k_{2} + k_{1} + 1)} \\& \quad \leq 1 + \sum_{k = 1}^{\infty }\frac{2^{k}}{k!} \\& \quad = 1 + 2 + \frac{2 \cdot 2}{1 \cdot 2} + \frac{2 \cdot 2 \cdot 2}{1 \cdot 2 \cdot 3} + \frac{2 \cdot 2 \cdot 2 \cdot 2}{1 \cdot 2 \cdot 3 \cdot 4} + \frac{2 \cdot 2 \cdot 2 \cdot 2 \cdot 2}{1 \cdot 2 \cdot 3 \cdot 4 \cdot 5} + \cdots \\& \quad \leq 1 + 2 + 2 + \biggl( \frac{1}{3} + \biggl(\frac{2}{3} \biggr)^{0} \biggr) + \biggl(\frac{2}{3} \biggr)^{1} + \biggl(\frac{2}{3} \biggr)^{2} + \cdots \\& \quad = \frac{16}{3} + \frac{1}{1 - \frac{2}{3}} = \frac{25}{3}. \end{aligned}$$
$$ C \sum_{k = 0}^{\infty }\sum _{k_{1} + k_{2} = k} \binom{k}{k_{1}, k_{2}} \frac{1}{\Gamma (1.5 k_{2} + k_{1} + 1)} < \frac{25}{3} \cdot \frac{1}{9} < 1. $$
By Theorem 2.2 equation (7) has a unique solution.
There are algorithms for computation of the Mittag-Leffler function [21]
$$ E_{\alpha , \beta }(z) = \sum_{k = 0}^{\infty } \frac{z^{k}}{\Gamma (\alpha k + \beta )}, \quad \alpha > 0, \beta \in R, z \in C, $$
and its derivative. In particular,
$$ E_{\alpha , \beta }(z) = - \frac{\sin (\pi \alpha )}{\pi \alpha } \int _{0}^{\infty }\frac{e^{-r^{1/\alpha }}}{r^{2} - 2 rz \cos (\pi \alpha ) + z^{2}} \,d r - \frac{1}{z}, \beta = 1 + \alpha , $$
where \(0 < \alpha \leq 1\), \(\beta \in R\), \(\vert \arg z \vert > \pi \alpha \), \(z \neq 0\).
The Mittag-Leffler function is widely used in studying fractional differential equations and fractional calculus. Li [22] studied three classes of fractional oscillators and obtained the solutions of the first class in terms of the Mittag-Leffler function.
Define the product space \(X_{\mu }(a, b) \times X_{\mu }(a, b)\) with the norm
$$ \bigl\lVert (u, v) \bigr\rVert = \lVert u \rVert _{X_{\mu }} + \lVert v \rVert _{X_{\mu }}. $$
Clearly, \(X_{\mu }(a, b) \times X_{\mu }(a, b)\) is a Banach space.
Now we can extend Theorem 2.2 to the coupled system of the Hadamard-type integral equations given by (4).
Let \(g_{1}, g_{2}: [a, b] \times R \times R \rightarrow R\) be continuous functions and suppose that there exist nonnegative constants \(C_{i}\), \(i = 1, 2, 3, 4\), such that for all \(x \in [a, b]\) and \(u_{i}, v_{i} \in R\), \(i = 1, 2\),
$$\begin{aligned}& \bigl\vert g_{1}(x, u_{1}, v_{1}) - g_{1}(x, u_{2}, v_{2}) \bigr\vert \leq C_{1} \vert u_{1} - u_{2} \vert + C_{2} \vert v_{1} - v_{2} \vert , \\& \bigl\vert g_{2}(x, u_{1}, v_{1}) - g_{2}(x, u_{2}, v_{2}) \bigr\vert \leq C_{3} \vert u_{1} - u_{2} \vert + C_{4} \vert v_{1} - v_{2} \vert . \end{aligned}$$
$$\begin{aligned} q = & \max \{C_{1}, C_{2}\} E_{(\alpha _{1},\ldots, \alpha _{n}, 1)} \biggl( \vert a_{1} \vert \biggl(\log \frac{b}{a} \biggr)^{\alpha _{1}},\ldots, \vert a_{n} \vert \biggl(\log \frac{b}{a} \biggr)^{\alpha _{n}} \biggr) \\ & {} + \max \{C_{3}, C_{4}\} E_{(\beta _{1},\ldots, \beta _{n}, 1)} \biggl( \vert b_{1} \vert \biggl(\log \frac{b}{a} \biggr)^{\beta _{1}},\ldots, \vert b_{n} \vert \biggl(\log \frac{b}{a} \biggr)^{\beta _{n}} \biggr) < 1. \end{aligned}$$
Then system (4) has a unique solution in the product space \(X_{\mu }(a, b) \times X_{\mu }(a, b)\) for every \(\mu \in R\).
Let \(u, v \in X_{\mu }(a, b)\). Then \(g_{1}(x, u(x), v(x)), g_{2}(x, u(x), v(x)) \in X_{\mu }(a, b)\) since
$$\begin{aligned} \bigl\vert g_{1} \bigl(x, u(x), v(x) \bigr) \bigr\vert \leq& \bigl\vert g_{1} \bigl(x, u(x), v(x) \bigr) - g_{1}(x, 0, 0) \bigr\vert + \bigl\vert g_{1}(x, 0, 0) \bigr\vert \\ \leq &C_{1} \bigl\vert u(x) \bigr\vert + C_{2} \bigl\vert v(x) \bigr\vert + \bigl\vert g_{1}(x, 0, 0) \bigr\vert \in X_{\mu }(a, b) \end{aligned}$$
by noting that \(g_{1}(x, 0, 0)\) is a continuous function on \([a, b]\). Furthermore,
$$ \bigl\lVert g_{1} \bigl(x, u(x), v(x) \bigr) \bigr\rVert _{X_{\mu }} \leq C_{1} \lVert u \rVert _{X_{\mu }} + C_{2} \lVert v \rVert _{X_{\mu }} + \max_{x \in [a, b]} \bigl\{ x^{\mu - 1} \bigl\vert g_{1}(x, 0, 0) \bigr\vert \bigr\} (b - a) < \infty $$
for every \(\mu \in R\).
Define the mapping T on \(X_{\mu }(a, b) \times X_{\mu }(a, b)\) by
$$ T(u, v) = \bigl(T_{1}(u, v), T_{2}(u, v) \bigr), $$
$$\begin{aligned} T_{1}(u, v) (x) = & \sum_{k = 0}^{\infty }(-1)^{k} \sum_{k_{1} + \cdots + k_{n} = k} \binom{k}{k_{1}, k_{2},\ldots, k_{n}}a_{n}^{k_{n}} \cdots a_{1}^{k_{1}} \\ & {}\cdot \mathcal{J}^{k_{n} \alpha _{n} + \cdots + k_{1} \alpha _{1}}_{ a + , \mu } g_{1} \bigl(t, u(t), v(t) \bigr) (x), \end{aligned}$$
$$\begin{aligned} T_{2}(u, v) (x) = & \sum_{k = 0}^{\infty }(-1)^{k} \sum_{k_{1} + \cdots + k_{n} = k} \binom{k}{k_{1}, k_{2},\ldots, k_{n}}b_{n}^{k_{n}} \cdots b_{1}^{k_{1}} \\ & {} \cdot\mathcal{J}^{k_{n} \beta _{n} + \cdots + k_{1} \beta _{1}}_{ a + , \mu } g_{2} \bigl(t, u(t), v(t) \bigr) (x). \end{aligned}$$
Clearly, from the proof of Theorem 2.2 we have
$$\begin{aligned} \bigl\lVert T_{1}(u, v) \bigr\rVert _{X\mu } \leq & E_{(\alpha _{1},\ldots, \alpha _{n}, 1)} \biggl( \vert a_{1} \vert \biggl(\log \frac{b}{a} \biggr)^{\alpha _{1}},\ldots, \vert a_{n} \vert \biggl(\log \frac{b}{a} \biggr)^{\alpha _{n}} \biggr) \\ & {}\cdot \Bigl(C_{1} \lVert u \rVert _{X_{\mu }} + C_{2} \lVert v \rVert _{X_{\mu }} + \max_{x \in [a, b]} \bigl\{ x^{\mu - 1} \bigl\vert g_{1} (x, 0, 0) \bigr\vert \bigr\} (b - a) \Bigr)< \infty \end{aligned}$$
$$\begin{aligned} \bigl\lVert T_{2}(u, v) \bigr\rVert _{X\mu } \leq & E_{(\beta _{1},\ldots, \beta _{n}, 1)} \biggl( \vert b_{1} \vert \biggl(\log \frac{b}{a} \biggr)^{\beta _{1}},\ldots, \vert b_{n} \vert \biggl(\log \frac{b}{a} \biggr)^{\beta _{n}} \biggr) \\ &{}\cdot \Bigl(C_{3} \lVert u \rVert _{X_{\mu }} + C_{4} \lVert v \rVert _{X_{\mu }} + \max_{x \in [a, b]} \bigl\{ x^{\mu - 1} \bigl\vert g_{2} (x, 0, 0) \bigr\vert \bigr\} (b - a) \Bigr)< \infty . \end{aligned}$$
$$ \bigl\lVert T(u, v) \bigr\rVert = \bigl\lVert T_{1}(u, v) \bigr\rVert _{X\mu } + \bigl\lVert T_{2}(u, v) \bigr\rVert _{X\mu } < \infty , $$
which implies that T maps the Banach space \(X_{\mu }(a, b) \times X_{\mu }(a, b)\) into itself. It remains to show that T is contractive. Indeed,
$$\begin{aligned} \bigl\lVert T_{1}(u_{1}, v_{1}) - T_{1}(u_{2}, v_{2}) \bigr\rVert _{X_{\mu }} \leq & E_{(\alpha _{1},\ldots, \alpha _{n}, 1)} \biggl( \vert a_{1} \vert \biggl(\log \frac{b}{a} \biggr)^{\alpha _{1}},\ldots, \vert a_{n} \vert \biggl(\log \frac{b}{a} \biggr)^{\alpha _{n}} \biggr) \\ & {}\cdot\max \{C_{1}, C_{2}\} \bigl( \lVert u_{1} - u_{2} \rVert _{X_{\mu }} + \lVert v_{1} - v_{2} \rVert _{X_{\mu }} \bigr), \end{aligned}$$
$$\begin{aligned} \bigl\lVert T_{2}(u_{1}, v_{1}) - T_{2}(u_{2}, v_{2}) \bigr\rVert _{X_{\mu }} \leq & E_{(\beta _{1},\ldots, \beta _{n}, 1)} \biggl( \vert b_{1} \vert \biggl(\log \frac{b}{a} \biggr)^{\beta _{1}},\ldots, \vert b_{n} \vert \biggl( \log \frac{b}{a} \biggr)^{\beta _{n}} \biggr) \\ & {}\cdot\max \{C_{3}, C_{4}\} \bigl( \lVert u_{1} - u_{2} \rVert _{X_{\mu }} + \lVert v_{1} - v_{2} \rVert _{X_{\mu }} \bigr). \end{aligned}$$
$$\begin{aligned} \bigl\lVert T(u_{1}, v_{1}) - T(u_{2}, v_{2}) \bigr\rVert = & \bigl\lVert T_{1}(u_{1}, v_{1}) - T_{1}(u_{2}, v_{2}) \bigr\rVert _{X \mu } + \bigl\lVert T_{2}(u_{1}, v_{1}) - T_{2}(u_{2}, v_{2}) \bigr\rVert _{X\mu } \\ \leq &q \bigl( \lVert u_{1} - u_{2} \rVert _{X_{\mu }} + \lVert v_{1} - v_{2} \rVert _{X_{\mu }} \bigr), \end{aligned}$$
where \(q < 1\) by assumption. By Banach's contractive principle system (4) has a unique solution in the space \(X_{\mu }(a, b) \times X_{\mu }(a, b)\). This completes the proof of Theorem 2.3. □
Let \(\operatorname{AC}[a, b]\) be the set of absolutely continuous functions on \([a, b]\), which coincides with the space of primitives of Lebesgue-measurable functions [3]:
$$ h \in \operatorname{AC}[a, b] \quad \text{if and only if} \quad h(x) = h(a) + \int _{a}^{x} \psi (t) \,d t, \quad \psi \in L[a, b]. $$
Clearly, if \(f \in \operatorname{AC}[a, b]\) with \(0 < a < b < \infty \), then \(x^{\mu } f(x) \in \operatorname{AC}[a, b]\) since \(x^{\mu }\in \operatorname{AC}[a, b]\).
The following results are from Lemma 2.3 and Theorem 5.5(a) in [2].
If \(\alpha > \beta > 0\) and \(\mu \in R\), then for \(u \in X_{\mu }(a, b)\),
$$ \mathcal{D}^{\beta }_{a + , \mu } \mathcal{J}^{\alpha }_{a + , \mu }u = \mathcal{J}^{\alpha - \beta }_{a + , \mu }u. $$
If \(\alpha > 0\) and \(u \in X_{\mu }(a, b)\), then
$$ \mathcal{D}^{\alpha }_{a + , \mu } \mathcal{J}^{\alpha }_{a + , \mu }u = u. $$
Let \(\alpha _{n} > \cdots > \alpha _{1} > \alpha _{0}\) with \(0 < \alpha _{0} < 1\), and let \(f \in \operatorname{AC}[a, b]\). In addition, let \(a_{i}\), \(i = 1, 2,\ldots, n\), be complex numbers, not all zero. Then equation (3) has a unique solution in the space \(X_{\mu }(a, b)\),
$$\begin{aligned} u(x) = & a^{\mu }f(a) x^{- \mu } \biggl(\log \frac{x}{a} \biggr)^{ - \alpha _{0}} \sum_{k = 0}^{\infty }(-1)^{k} \sum_{k_{1} + \cdots + k_{n} = k} \binom{k}{k_{1}, k_{2},\ldots, k_{n}} a_{n}^{k_{n}} \cdots a_{1}^{k_{1}} \\ & {}\cdot\frac{ (\log \frac{x}{a} )^{ k_{n}(\alpha _{n} - \alpha _{0}) + \cdots + k_{1} (\alpha _{1} - \alpha _{0})}}{\Gamma (k_{n}(\alpha _{n} - \alpha _{0}) + \cdots + k_{1} (\alpha _{1} - \alpha _{0}) + 1 - \alpha _{0})} \\ & {} + \mu \sum_{k = 0}^{\infty }(-1)^{k} \sum_{k_{1} + \cdots + k_{n} = k} \binom{k}{k_{1}, k_{2},\ldots, k_{n}} a_{n}^{k_{n}} \cdots a_{1}^{k_{1}} \\ &{}\cdot \bigl(\mathcal{J}^{k_{n} (\alpha _{n} - \alpha _{0}) + \cdots + k_{1} ( \alpha _{1} - \alpha _{0}) + 1 - \alpha _{0} }_{ a + , \mu } f \bigr) (x) \\ & {} + \sum_{k = 0}^{\infty }(-1)^{k} \sum_{k_{1} + \cdots + k_{n} = k} \binom{k}{k_{1}, k_{2},\ldots, k_{n}} a_{n}^{k_{n}} \cdots a_{1}^{k_{1}} \\ & {}\cdot \bigl(\mathcal{J}^{k_{n} (\alpha _{n} - \alpha _{0}) + \cdots + k_{1} ( \alpha _{1} - \alpha _{0}) + 1 - \alpha _{0} }_{ a + , \mu } t f'(t) \bigr) (x). \end{aligned}$$
It follows from Theorem 5.3 in [2] that
$$ \bigl(\mathcal{D}^{\alpha _{0}}_{a + , \mu }f \bigr) (x) = \frac{x^{- \mu }}{\Gamma (1 - \alpha _{0})} \biggl[f_{0}(a) \biggl( \log \frac{x}{a} \biggr)^{- \alpha _{0}} + \int _{a}^{x} \biggl(\log \frac{x}{t} \biggr)^{- \alpha _{0}} f_{0}'(t) \,d t \biggr], $$
where \(f_{0}(x) = x^{\mu }f(x) \in \operatorname{AC}[a, b]\). We first claim that \((\mathcal{D}^{\alpha _{0}}_{a + , \mu }f)(x) \in X_{\mu }(a, b)\). Indeed,
$$ \int _{a}^{b} x^{\mu - 1} x^{-\mu } \biggl(\log \frac{x}{a} \biggr)^{- \alpha _{0}} \,d x = \int _{a}^{b} \biggl(\log \frac{x}{a} \biggr)^{- \alpha _{0}} \,d \biggl(\log \frac{x}{a} \biggr) = \frac{ (\log \frac{b}{a} )^{1 - \alpha _{0}}}{1 - \alpha _{0}} < \infty . $$
Similarly,
$$ \frac{x^{-1}}{\Gamma (1 - \alpha _{0})} \int _{a}^{x} \biggl(\log \frac{x}{t} \biggr)^{- \alpha _{0}} f_{0}'(t) \,d t \in X_{\mu }(a, b) $$
by noting that \(f_{0}'(t) \in L[a, b]\) and
$$\begin{aligned}& \frac{1}{\Gamma (1 - \alpha _{0})} \int _{a}^{b} \frac{1}{x} \biggl\vert \int _{a}^{x} \biggl(\log \frac{x}{t} \biggr)^{- \alpha _{0}} f_{0}'(t) \,d t \biggr\vert \,d x \\& \quad \leq \frac{1}{\Gamma (1 - \alpha _{0})} \int _{a}^{b} \bigl\vert f_{0}'(t) \bigr\vert \,dt \int _{t}^{b} \biggl(\log \frac{x}{t} \biggr)^{- \alpha _{0}} \,d \biggl(\log \frac{x}{t} \biggr) = K \int _{a}^{b} \bigl\vert f_{0}'(t) \bigr\vert \,dt, \end{aligned}$$
$$ K = \frac{1}{\Gamma (2 - \alpha _{0})} \biggl(\log \frac{b}{a} \biggr)^{1 - \alpha _{0}}. $$
For \(u \in X_{\mu }(a, b)\), equation (3) turns out to be
$$ a_{n} \bigl(\mathcal{J}^{\alpha _{n} - \alpha _{0}}_{ a + , \mu } u \bigr) (x) + \cdots + a_{1} \bigl(\mathcal{J}^{\alpha _{1} - \alpha _{0}}_{ a + , \mu } u \bigr) (x) + u(x) = \bigl(\mathcal{D}^{\alpha _{0}}_{a + , \mu }f \bigr) (x) $$
by applying the fractional differential operator \(\mathcal{D}^{\alpha _{0}}_{a + , \mu }\) to both sides. Then by Theorem 2.1 we have
$$\begin{aligned} u(x) = & \sum_{k = 0}^{\infty }(-1)^{k} \sum_{k_{1} + \cdots + k_{n} = k} \binom{k}{k_{1}, k_{2},\ldots, k_{n}} a_{n}^{k_{n}} \cdots a_{1}^{k_{1}} \\ & {}\cdot \bigl(\mathcal{J}^{k_{n} (\alpha _{n} - \alpha _{0}) + \cdots + k_{1} ( \alpha _{1} - \alpha _{0}) }_{ a + , \mu } \mathcal{D}^{\alpha _{0}}_{a + , \mu }f \bigr) (x). \end{aligned}$$
To remove the differential operator \(\mathcal{D}^{\alpha _{0}}_{a + , \mu }\), we compute the Hadamard-type fractional integral of order \(\alpha >0\) for the first term in \((\mathcal{D}^{\alpha _{0}}_{a + , \mu }f)(x)\):
$$\begin{aligned}& \mathcal{J}^{\alpha }_{ a + , \mu } \frac{f_{0}(a) t^{- \mu }}{\Gamma (1 - \alpha _{0})} \biggl(\log \frac{t}{a} \biggr)^{- \alpha _{0}} \\& \quad = \frac{f_{0}(a)}{\Gamma (1 - \alpha _{0}) \Gamma (\alpha )} \int _{a}^{x} \biggl(\frac{t}{x} \biggr)^{\mu } \biggl(\log \frac{x}{t} \biggr)^{ \alpha - 1} t^{-\mu } \biggl(\log \frac{t}{a} \biggr)^{- \alpha _{0}} \frac{d t }{t} \\& \quad = \frac{f_{0}(a) x^{-\mu }}{\Gamma (1 - \alpha _{0}) \Gamma (\alpha )} \int _{a}^{x} \biggl(\log \frac{x}{t} \biggr)^{ \alpha - 1} \biggl( \log \frac{t}{a} \biggr)^{- \alpha _{0}} \frac{d t }{t}. \end{aligned}$$
Making the change of variable
$$ \tau = \frac{\log (t/a) }{\log (x/a)}, $$
$$\begin{aligned} \int _{a}^{x} \biggl(\log \frac{x}{t} \biggr)^{ \alpha - 1} \biggl( \log \frac{t}{a} \biggr)^{- \alpha _{0}} \frac{d t }{t} = & \biggl( \log \frac{x}{a} \biggr)^{\alpha - \alpha _{0}} \int _{0}^{1} (1 - \tau )^{\alpha - 1} \tau ^{- \alpha _{0}} \,d \tau \\ =& \biggl(\log \frac{x}{a} \biggr)^{\alpha - \alpha _{0}} B(\alpha , 1 - \alpha _{0})\\ =& \biggl(\log \frac{x}{a} \biggr)^{\alpha - \alpha _{0}} \frac{\Gamma (\alpha ) \Gamma (1 - \alpha _{0})}{\Gamma (\alpha + 1 - \alpha _{0})}, \end{aligned}$$
where B denotes the beta function. Hence
$$ \mathcal{J}^{\alpha }_{ a + , \mu } \frac{f_{0}(a) t^{- \mu }}{\Gamma (1 - \alpha _{0})} \biggl(\log \frac{t}{a} \biggr)^{- \alpha _{0}} = \frac{f_{0}(a) x^{- \mu }}{\Gamma (\alpha + 1 - \alpha _{0}) } \biggl( \log \frac{x}{a} \biggr)^{\alpha - \alpha _{0}}. $$
The second term in \((\mathcal{D}^{\alpha _{0}}_{a + , \mu }f)(x)\) is
$$\begin{aligned}& \frac{1}{\Gamma (1 - \alpha _{0})} \int _{a}^{x} x^{- \mu } \biggl( \log \frac{x}{t} \biggr)^{- \alpha _{0}} f_{0}'(t) \,d t \\& \quad = \frac{1}{\Gamma (1 - \alpha _{0})} \int _{a}^{x} \biggl( \frac{t}{x} \biggr)^{\mu } \biggl(\log \frac{x}{t} \biggr)^{1 - \alpha _{0} - 1} \bigl[t^{- \mu + 1} f_{0}'(t) \bigr] \frac{d t }{t} \\& \quad = \mathcal{J}^{1 - \alpha _{0}}_{ a + , \mu } \bigl(t^{- \mu + 1} f_{0}'(t) \bigr) = \mu \bigl(\mathcal{J}^{1 - \alpha _{0}}_{ a + , \mu } f \bigr) (x) + \mathcal{J}^{1 - \alpha _{0}}_{ a + , \mu } \bigl(t f'(t) \bigr) (x). \end{aligned}$$
Therefore the solution immediately follows by substituting equations (9) and (10) into equation (8). This completes the proof of Theorem 2.4. □
It seems impossible to deal with the case \(\alpha _{0} \geq 1\) along the same lines as \(\mathcal{D}^{\alpha _{0}}_{a + , \mu }f \notin X_{\mu }(a, b)\) for \(f \in \operatorname{AC}[a, b]\). Furthermore, \(\mathcal{D}^{\alpha _{0}}_{a + , \mu }\) is not a bounded operator on \(\operatorname{AC}[a, b]\). The single-term Hadamard-type integral equation
$$ \mathcal{J}^{\alpha }_{a+, \mu } u = f, \quad \alpha > 0, $$
was studied in [2] with the necessary and sufficient conditions given in Theorem 3.1.
Using Babenko's approach and Banach's contraction principle, we have derived the uniqueness of solution for several Hadamard-type integral equations and related coupled system. The results obtained are new in the present configuration of integral equations.
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This work is supported by NSERC (Canada 2019-03907).
Department of Mathematics and Computer Science, Brandon University, Brandon, Manitoba, R7A 6A9, Canada
Chenkuan Li
The author prepared, read, and approved the final manuscript.
Correspondence to Chenkuan Li.
The author declares that they have no competing interests.
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
Li, C. Uniqueness of the Hadamard-type integral equations. Adv Differ Equ 2021, 40 (2021). https://doi.org/10.1186/s13662-020-03205-8
Hadamard-type integral
Banach's fixed point theorem
Babenko's approach
Multivariate Mittag-Leffler function | CommonCrawl |
Metabolic syndrome among type 2 diabetic patients in Ethiopia: a cross-sectional study
Mequanent Kassa Birarra ORCID: orcid.org/0000-0002-8700-06141 &
Dessalegn Asmelashe Gelayee2
Metabolic syndrome (MetS) increases risk of cardiovascular diseases (CVD), premature death as well as cost related to health care.This study was aimed at investigating the prevalence of MetS and its determinant factors among type2 diabetes mellitus (T2DM) patients attending a specialized hospital.
A cross-sectional study was conducted on a total of 256 T2DM patients from the first march to 30th May 2017 at university of gondar comprehensive specialized hospital (UGCSH). Data was collected based on STROBE (strengthening the reporting of observational studies in epidemiology) statement. Bivariable and multivariable logistic regression analysis were run to identify predictors of MetS from the independent variables and significance test was set at P < 0.05.
The prevalence of MetS in this study was 70.3, 57 & 45.3% and it is more common in females (66.1, 83.3 & 70.7%) by using national cholesterol education program adult treatment panel III (NCEP-ATP III), International diabetic federation (IDF) and world health organization (WHO) criteria respectively. The most prevalent components of MetS were low level of high density lipoprotein (HDL) and triglyceride(TG). By usingIDF criteria, female gender was significantly associated with MetS (AOR = 0.2 at 95%CI: 0.1, 0.6 P = 0.00). Where as by NCEP-ATP IIIcriteria, age between 51 and 64 years old (AOR = 2.4 95% CI: 1.0,5.8, P = 0.04), self employment (AOR = 2.7 95% CI:1.1, 6.5, P = 0.03), and completetion of secondary school and above (AOR = 3.2, 95% CI:1.6,6.7, P = 0.001) were predictors for the development of MetS. In the WHO criteria, being single in marital status was significantly associated with MetS (AOR = 17 at 95%CI: 1.8, 166, P = 0.000).
This study demonstrates that Metabolic syndrome is a major health concern for diabetic patients in Ethiopia and they are at increased risk of developing complications such as cardiovascular diseases and premature mortality. The predictors female gender, age between 51 and 64 years old, urban area residence, and being single are modifiable.Thus,health authorities shall provide targeted interventions such as life style modifications to these most at risk sub-populations of diabetic patients.
The burden of non-communicable disease in the developing countries is increasing, and leading to high mortality rates [1]. Nowadays T2DM is pandemic and there are no signs of reduction in the incidence rates [2]. Forexamle, according to international diabetes federation report indicates that more than 415 million of people worldwide adults have diabetes. By 2040 this will rise to 642 million. In Africa, 441 million people live with diabetes which is likely to increase by 926 million in 2040 [3]. Diabetic population are at increased risk of mortality and morbidity primarily due to cardiovascular diseases [4]. The relative risks are 1 to 3 in men and from 2 to 5 in women [5]. Metabolic syndrome would have its own contribution in these outcomes of DM. Metabolic syndromeis highly prevalent in T2DM patients [6,7,8]. However, several studies have reported lower prevalence of MetS [9, 10] and this is largely due to differences in characteristics of the studied population such as residence, type of disease and comorbidities, etc.
Metabolic syndrome can be defined as a cluster of interconnected cardio-metabolic dysfunctions which is characterized by the increase in fasting blood sugar (FBS), abdominal circumference (AC), arterial pressure (AP), triglycerides (TG), and reduction in high-density lipoprotein cholesterol (HDL) [11]. This syndrome has different set of criterias to measure it. Those are National Cholesterol Education Program Adult Treatment Panel III (NCEP-ATP III) [12], WHO criteria's [13] and IDF [14]. The NCEP-ATP III definition uses the presence of 3 or more parameters as a cutoff to define MetS and the WHO as well as IDF definitions require the presence of at least two parameters.
The syndrome can directly contributes to the development of CVD and the appearance of T2DM in non-diabetic patients. Additionally, it increases the risk of premature death, renal disease, mental disorders and cancer. Thus MetS represents a serious public health problem [15,16,17].
Metabolic syndrome is not also with out cost implications. For instance, Boudreau et al. found that costs for subjects with diabetes plus weight risk, dyslipidemia, and hypertension were almost double the costs for subjects with prediabetes plus similar risk factors ($8067 vs. $4638) [18].
Globally, 20–25% of the adult population has MetS and they are twice as likely to die from it; and they are three times more likely to have a heart attack or stroke compared with people without the syndrome [14, 19]. However, the prevalence of MetS in type 2 diabetes in sub-Saharan Africans according to two sets of diagnostic criteria was 71.7% according to the IDF criteria and 60.4% using NCEP-ATP III criteria [20]. In Ethiopia the prevalence of MetS was range from (26–70%) using NCEP-ATP III criteria [21,22,23].
Nowadays, MetS has become a significant public health problem. Therefore, there is a need for investigation in this area [24]. Taking into consideration, diabetic patients who had MetS also they have cardiovascular risk factors, therefore the diagnosis of MetS in those patients is very important for detection, prevention, and treatment of the underlying risk factors and for the reduction of the cardiovascular disease burden in the general population [25, 26].
While limited studies of MetS among diabetic patients in Ethiopia acknowledge its burden, they followed a single criteria(NCEP-ATP III) to define MetS and using a signle criteria may either under or over estimate the problem. Thus denying or providing interventions to minimize the risks of MetS complications would be irrational since a given patient may be categorized as having MetS in one set of definition but not in the others. In this regard, the present study employed three commonly used criteria to define MetS so that it would be easy to acknowledge the importance of having a unified MetS criterion to make appropriate clinical decision in the context of Ethiopia.Therefore, this study was aimed at inevestigating the prevalence of MetS and its determinant factors among T2DM patients attending a comphrensivespecialized hospital.
Study Area & Period
The study was conducted from March to May 2017 at UGCSH, Northwest Ethiopia. The hospital is currently serving more than 5 million people in the surrounding area andit is located in Gondar town, 750 km Northwest of the capital city. It has more than 400 beds and fourteen different units that provide medical services to nearly 250,000 out-patients each year. More than 5 thousand diabetic patients attend the diabetic follow up clinic.
Study design and population
An institutional based cross sectional study design was followed.The source populations were all patients attending the facility on out-patient basis at UGCSH.Whereas, all adult T2DM patients attending the facility on out-patient basis during the study period and volunteered to take part in the study were the study population. Those patients whose age ≥ 20 years old and diagnosed as T2DM undergoing treatment with the facility were included in the study. Whereas, pregnant women, excessive alcohol or other drug abuse, having current psychiatric treatment and incomplete patient's data were excluded from the study.
Sample size and sampling procedure
The sample size was calculated based on single population proportion formula [27]. By using the following assumption: (1.96)2 were used for \( Z\frac{\alpha }{2} \) and the proportion (P) of MetS in these groups was 0.5. With 95% confidence interval (CI) and marginal error (d) of 5%.
$$ n=\frac{Z_{\frac{\alpha }{2}}{}^2P\left(1-P\right)}{d^2} $$
Based on the above formula, assumptions, correction formula and 5% of contingency the sample size(n) was calculated to be 256. Study participants were selected using systematic random sampling technique. Then, every third patient arrived at the clinic was selected for the study.
Data collection procedure
Data of socio demographic and economic (age, sex, monthly income, life style, family history of diabetes and other diseases/disorders) of the study participants were collected by using standardized interview questioner.Whereas, data of HDL, fasting plasma glucose (FPG), and TG were recorded from patient files and chart.The components of MetS was identified and determined according to NCEP-ATP III, IDF and WHO definitions. Anthropometric data of the study participants (weight, height and waist circumference) was obtained by two data collector nurses who are working at UGCSH diabetic clinics. Weight was obtained from patients using weight balance while they were visiting the clinics during their follow-up. The average follow-up interval was 2–3 months. Height of the patients were assessed using meter and also data collectors were instruct participants to stand upright, motionless,and touching their thighs with their palms. Based on height and weight body mass index was calculated. Waist circumference (WC) was measured midway between the inferior angle of the ribs and the supra-iliac crest by using Meter [28]. After 10 min of arrival of the study participants at UGCSH diabetic clinic, Blood pressure (BP) was measured using a standard adult arm cuff of mercury type sphygmomanometer by the recruited nurses as data collectors who working in theclinic. Inorder to assure the reliability of BP measurement data collectors were taken two readings with 1 minute interval and the average of the two readings was recorded as the final BP of the patient. However, a third measurement was taken if the difference between the two readings was greater than 5 mmHg and the average of the 3 BP readings was recorded as the final BP of the patient [29].
In order to control the quality of data, pre-test was done in the data abstraction format before the main data collection on a sample equivalent to 13 (5%) of the total sample size in randomly selected patients. The pretested papers were not included in the study and appropriate adjustment was done on the data abstraction format. In addition to this the principal investigator had supervised the data collectors during data collection. Thenthe collected data were checked for completeness and consistency on daily basis.
Data analysis and interpretation
The collected data were entered into Epi Info version 7 and exported to statistical package for the social sciences (SPSS) version 20 for statistical analysis. The results presented using tables and figures. Frequency distribution was calculated. The prevalence of patients with MetS was calculated, dividing the number of patients with MetS by the total number of study participants. To identify factors independently associated with the occurrence of MetS Bivariable and multivariable logistic regression analysis was run. The results of Bivariable and multivariable analysis were reported as crude and adjusted odds ratio at 95% confidence intervals (95% CI) and P-value ≤0.05 was considered as statistical significance.
Operational definitions
NCEP-ATP III criteria
Study participants were classified as having MetS if they had three or more of the following risk factors: waist circumference (> 102 cm for men and > 88 cm for women), high plasma triglycerides (≥ 150 mg/dl), low HDL cholesterol (< 40 mg/dl for men and < 50 mg/dl for women), blood pressure (≥ 130/85 mmHg) and fasting plasma glucose (≥110 mg/dl) [12].
WHO criteria
Study participants were classified as having MetS as along with DM if they had any two of the following components: Obesity: BMI (> 30 kg/m2), high serum triglycerides, (≥150 mg/dl), low serum high density lipoprotein cholesterol (< 35 mg/dl for men and < 39 mg/dl for women) and having hypertension (≥140/90 mmHg [13].
IDFcriteria
Study participants were classified as having MetS as along with central obesity if they had any two of the following components:Raised TG levels ≥150 mg/dl (1.7 mmol/l), or specific treatment for this lipid abnormality, reduced HDL-cholesterol < 40 mg/dl (1.03 mmol/l) in males and < 50 mg/dl (1.29 mmol/l) in females, or specific treatment for this lipid abnormality, raised blood pressure: systolic BP ≥130 or diastolic BP ≥85 mmHg or treatment of previously diagnosed hypertension, raised fasting blood glucose ≥100 mg/dl (≥5.6 mmol/l) or previously diagnosed diabetes and waist circumference (> 94 cmfor men and > 88 cm for women [14].
Was defined as the ratio between weight (kg) and the square of the height (m) and used to categorize BMI-measured weight status: patients with (BMI ≤ 18.5) statedas under weight, patients with (BMI 18.5–24.9) consider as normal,however, patients with (BMI 25.0–29.9) is overweight and obese if BMI is ≥30 [22].
A total of 256 study participants were included in the study of which more than half of them were females 143 (55.9%). The highest number of study participants were in the age group (51–64) years old. More than three fourth were 207 (80.9%) lives in urban area and 93 (36.3%) of them were complete their secondary school and above. In addition, more than two third of 219 (85%) were married.
The total number of unemployed study participants were 131 (51.2%) and majority of them 116 (45.3%) had < 600 Ethiopian birr monthly income. The highest number (81.6%) of them used palm oil for food preparation. In addition to this, majority of study participants 136 (53.1%) were not involved in work vigarious intensity of activity and the highest number 173 (67.6%) of them were not did regular physical exercise. One hundred sixty seven (62.2%) of study participants have no family history of chronic diseases. Around half of the study subjects 136 (53.1%) diagnosed DM between 1 and 5 years duration and all of them were under medication. Most of them 132 (51.5%) were also undertaking combination treatment. Details are presented in Table 1.
Table 1 Socio demographic characteristics of the study participants at UGCSH, June 2017
Prevalence of metabolic syndrome with each criteria
The prevalence of MetS in this study was 180 (70.3%), 146 (57%) and 116 (43.3%) using NCEP(ATPIII), IDF and WHO criteria respectively (Fig. 1).
Metabolic syndrome in different criteria at UGCSH, June 2017
Frequency of metabolic syndrome components by sex
The frequency of MetS components in this study based on NCEP-ATP III criteria were 53.5, 68.8 and 67.2% for abdominal obesity, elevated triglyceride and reduced HDL respectively. Whereas, using the IDF criteria the prevalence was 61.7, 67.6 and 66.8% for abdominal obesity, elevated triglyceride and reduced HDL respectively. Details are presented in Table 2.
Table 2 Frequency of metabolic syndrome components among T2DM patients with sex, at UGCSH, June 2017
Factors associated with metabolic syndrome
In order to control confounders effect multivariable logistic regression analysis was run to analyze variables which were significantly associated to different components of MetS using different criteria in bivariable logistic analysis. These variables were sex, age, educational status, residency, duration since DM diagnosed, monthly income, family history of chronic disease and marital status. The analysis showed that, sex was significantly associated with MetS by using IDF criteria. Based on this, female patients were (AOR = 0.2 at 95%CI: 0.1, 0.6, P = 0.00) significantly associated with MetS compared to men using IDF criteria. Details are presented in Table 3.
Table 3 Bivariable and multivariable logistic regression analysis by using IDF criteria at UGCSH, June 2017
Using NCEP-ATPIII criteria, female sex was (AOR = 0.2 at 95%CI: 0.1, 0.6, P = 0.00) significantly associated with MetS compared to male sex. Similarly, patients whose age is between 51 and 64 years old were about two (AOR = 2.4 95% CI: 1.0, 5.8, P = 0.04) times more likely to haveMetS compared to those patients whose age is < 30 years old. Likewise,self employed participants were about three (AOR = 2.7 95% CI: 1.1, 6.5, P = 0.03) times more likely to develop MetS compared to those unemployed. Patients who completed secondary school and above were about three (AOR = 3.2, 95% CI: 1.6, 6.7, P = 0.001) times more likely to develop MetS compared to those unable to read and write. In addition, patients whose DM diagnosis duration was less than 1 year were about three (AOR = 2.7 95% CI = 1.1, 7.1, P = 0.04) times more likely to develop MetS compared to those with DM diagnosis duration 1–5 years. Details are presented in Table 4.
Table 4 Bivariable and multivariable logistic regression analysis using NCEP-ATPIII criteria at UGCSH, June 2017
Based onWHO criteria female sex was (AOR = 0.4 at 95%CI: 0.2, 0.7, P = 0.000) significantly associated with MetS compared to male sex. Patients who were single were significantly associated with MetS and were about seventeen (AOR = 17 at 95%CI: 1.8, 166, P = 0.01) times more likely to develop MetS compared to those divorced patients.Details are presented in Table 5.
Table 5 Bivariable and multivariable logistic regression analysis result using WHO criteria at UGCSH, June 2017
This study was aimed at describing the prevalence and predictors of Metabolic syndrome among type 2 diabetic patients attending a comprehensive specialized hospital in Northwest Ethiopia. The main finding of the present study demonstrates that MetS is a major health concern for diabetic patients in Ethiopia and the predictors like female gender, age between 51 and 64 years old, urban area residence, and being single, are modifiable.
The prevalence of MetS in this study was 70.3, 57 & 45.3% using NCEP-ATP III, IDF& WHO criteria respectively. These different prevalence rates arise due to the different cutoff points and sets of criteria used by those three definitions. In previous studies among DM patients, a lower 45.9% and comparable 70.1% results were reported from Ethiopia using NCEP-ATP III criteria [22, 23]. However, a higher rate of prevalence, 73.9, 69.9 and 66.8%, was reported from Nepal using NCEP-ATP III, WHO and IDF criteria respectively [30] and 73.4 & 64.9% using NCEP-ATP III and IDF criteria respectively was reported from Iran [31]. On the other hand, a lower prevalence of MetS was reported from India 45.8, 57.7 and 28% using NCEP-ATP III, WHO and IDF criteria respectively [8] and 58% was from Ghana using NCEP-ATP III criteria [7]. The prevalence of MetS in the present study some what different from others and this could be due to differences in sample size, socio-economic status, ethnicity difference [32], sampling method and difference in life style of study participants.
The present study demonstrated that prevalence of MetS was found to be higher in female (83.3, 66.1 & 70.7%) study participants than men (17, 31.9 & 29.3%) using IDF, NCEP-ATPIII & WHO criteria's respectively. This result is in agreement with other studies [7, 30, 31]. As shown in Table 2, a significantly higher proportion of females than males have abnormal components in four (66.7%) of the six components used to define MetS in the three criterias. This might explain the observed higher prevalence of MetS in the female geneder. Such discrepancy is attributed to the several physiological differences: Pregnancy induced increase in weight as well as gestational DM; the use of hormonal oral contraceptives that can decrease insulin sensitivity, glucose tolerance, increase blood pressure and increase in weight gain; menopause promotes a change in body fat distribution to increase central adiposity [33]. However, as the majority of females in the present study (81.8%) were above 46 years old, the increased prevalence of MetS among females unlike that of males may be due to menopause. The presence of hormonal replacement therapy (HRT) was, however, not assessed but might have some effect on the higher prevalence of MetS. Inaddition, less proportion of females were involved in regular physical exercise than males in this study which might have its own contribution to the observed higher MetS prevalence among female. Females in Ethiopia are socio-economically and culturally influenced to stay at home so that they are typically involved in daily living activities rather than regular physical exercise to maintain body fitness. The role of exercise in minimizing risks of developing MetS is reported in Greec study of 1128 men and 1154 women [34].
According to the NCEP- ATPIII criteria, where the highest prevalence of MetS was observed, TG and HDL were the most frequent abnormal MetS components. Abnormal levels of TG and HDL has been implicated with adverse health effects. Fore example, Callaghan et al. reported that hypertriglyceridemia is a significant risk factor for lower-extremity amputation in a 10-year cohort study (from 1995 to 2006) of 28,701 diabetic patients [35]. A 2 years of multi-ethnic study of atherosclerosis on a total of 6814 participants showed that low level of HDL in the body is associated with an increased risk of CVD, coronary heart diseases and death [36]. Thus, interventions focusing on abnormal TG and HDL need to be prioritized.
Regarding to residency, the association of MetS and urbanization could be as a result of a sedentary life style, increased intake of calorie rich foods and central obesity.This result is supported by other studies world wide [37, 38]. In addition, people who were self employed had significant association with MetS and the reason could be also sedentary life style related with the type of job they are involved .
On the other hand, patients who were secondary school and above had significantly associated with MetS.This might be due to significantly higher economic status (greater than 1500 ETB) of those who are highly educated in our study population (Secondary school and above:59 (77.6%); primary school:15 (19.7%). This finding is consistent with that of Chakraborty et al. and Khanam et al. [39, 40]. Therefore, higher levl of education may indirectly lead to risky life style adoption interms of dietary pattern and physical activity.
When compared to those patients aged 30 years and less, the ones in the age intervals 31–40, 41–50, and 51–64 were at inceased risk of MetS. The reasons for a direct relations of age and MetS is that age related processes such as gradual decrease in the basal metabolic rate, stress induced hypercortisolism, hypogonadism, decreased growth hormone secretion, concomitant insulin resistance and abdominal fat deposition [41, 42]. However, those patients who are 65 years old and above were found to have no significantly increased risk of MetS. This might be because of reduced survival of patients who developed MetS in this age group. In this regard, further prospective studies need to be carried out. The finding reiterates that of Devers et al. According to this study which was conducted among 1429 adults aged ≥25 years from randomly selected house holds in Australia, MetS components cluster most markedly in those aged < 65 years [43]. Therefore, serious preventive and control measures should be taken as age increases. Individuals should be advised to make life style changes. Doing reguar exercise, eating foods containing little amount of saturated fats and cholesterol as well as taking more fiber-rich foods should be encouraged. Chandalia et al. have shown that taking high fiber diets have the potential to lower fasting plasma glucose, total cholesterol, triglyceride, and helps to have good glycemic index through a decrease in gastrointestinal absorption of cholesterol and carbohydrates [44].
Regarding to duration of period since DM was diagnosed, those patients diagnosed within a year had significantly higher risk of developing MetS according to NCEP-ATPIII criteria. Since lifestyle modifications on diet and physical activity are the main initial interventions in T2DM patients, those respondents treated for short period of time may not effectively adopt the needed life style changes and hence are at increaed risk of MetS. It is also worth to note that some of the patients in our study might be in the very ealy stages of treatment so that reduction of MetS components might be unlikely. Incontrast to our finding, a previous study in Ethiopia reported the absence of impact of duration of treatment on MetS development [22]. Since in this study patients were classified based on higher cut off treatment duration i.e. below or above 10 years, it might fail to signify the impact of duration on MetS. On the otherhand, patients who stayed on treatment for short duration were not specifically isolated and compared with others who stayed longer on therapy. In this study using WHO criteria it indicate that patients who were single had association with MetS,the possible reason may be small sample size of this segment of respondants (N = 18,7%).
In general, the findings of the present stydy taken together showed that MetS is a mjor burden among T2DM patients in Ethiopia. Early identification of MetS among T2DM patients is of great importance since MetS imply increased risk of morbidities such as CVD,decreased quality of life, increased health care cost, as well as mortality. Therefore, UGCSH has to strengthen appropriate and targeted prevention strategies such as encouraging people to adopt dietary modification and physical activity which are reported to reduce occurrence and progression of MetS [45]. Inaddition, there should be a more frequent screening of patients for MetS components prior to full blow development of MetS.
This study for the first time in Ethiopia, employed three defining criteria for MetS and was able to highlight the importance of having unified definition to diagnose and make clinical decisions in the context of low income settings. Data were also collected prospectively and this strengthens the conclusions made. Yet, there are limitations and one should consider these in interpreting the findings. The study may not be generalized to the nation as a whole due to small sample size and thus further studies would be important. It is also important to show the health related outcome and economic consequences of MetS among T2DM patients in Ethiopia.
In conclusion, this study demonstrates that MetS is a major health concern for diabetic patients in Ethiopia. They are at increased risk of developing complications such as cardiovascular diseases and premature mortality.The predictors, female gender, age between 51 and 64 years old, urban area residence, and being single, are modifiable. Thus, health authorities shall provide targeted interventions to this most at risk sub populations of diabetic patients such as promotion of life style modifications.
CVD:
UGCSH:
University of Gondar Comprehensive Specialized Hospital
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The authrs appreciate the data collectors as well as the study participants.
The data sets used and/or analysed during the current study are available from the corresponding author on reasonable request.
Department of Clinical Pharmacy, School of Pharmacy,College of Medicine and Health Sciences, University of Gondar, Lideta Street, P.o.box: 196, Gondar, Ethiopia
Mequanent Kassa Birarra
Department of Pharmacology, School of Pharmacy,College of Medicine and Health Sciences, University of Gondar, Gondar, Ethiopia
Dessalegn Asmelashe Gelayee
MKB designed the study. Both authors conducted the study, analyzed data, developed and approved the final version of the manuscript.
Correspondence to Mequanent Kassa Birarra.
Letter of ethical clearance was obtained from Ethical Review Committee of School of Pharmacy, College of Medicine and Health Sciences University of Gondar, as well as medical director of University of Gondar hospital. Informed verbal consent was obtained from each study participant with respect to their willingness to take part in the study after explaining the objective of the study. This was approved by the Ethical Review Committee of School of Pharmacy, College of Medicine and Health Sciences University of Gondar, as well as medical director of UGCSH.
Birarra, M.K., Gelayee, D.A. Metabolic syndrome among type 2 diabetic patients in Ethiopia: a cross-sectional study. BMC Cardiovasc Disord 18, 149 (2018). https://doi.org/10.1186/s12872-018-0880-7
University of Gondar, Ethiopia | CommonCrawl |
\begin{definition}[Definition:Differential Equation/Solution/Particular Solution]
Let $\Phi$ be a differential equation.
Let $S$ denote the solution set of $\Phi$.
A '''particular solution''' of $\Phi$ is the element of $S$, or subset of $S$, which satisfies a particular boundary condition of $\Phi$.
\end{definition} | ProofWiki |
Supported by Global COE Program "The Research and Training Center for New Development in Mathematics", Graduate School of Mathematical Sciences, the University of Tokyo.
Abstract: Kazhdan's property (T) is one of the most important properties in the analytic group theory, and has a numerous applications to many other fields of pure and applied mathematics. The prominent example of property (T) groups is SL(n,R). A group G is said to have property (T) if every affine isometric actions of G on a Hilbert space has a fixed point. Various fortifications of this property have been suggested by several researchers and proved for SL(n,R). In a series of lectures, I will talk about results in this direction of Lafforgue, Shalom, Mimura, and myself.
Abstract: To each dynamical system one can associate a space of cocycles (test functions) as well as a subspace of coboundaries, so that the associated cohomology reflects some of the underlying dynamics. In this series of lectures, we will deal with a generalization of this framework: the associated dynamics will correspond to that of a group action, and the cocycles will take values in the isometry group of a space of nonpositive curvature. As we will see, most of the classical theorems admit generalized versions in this setting (e.g. Birkhoff ergodic theorem, Gotsschalk-Hedlund theorem). Moreover, this gives a unified view with other classical results (e.g. Oseledets theorem). More importantly, this allows obtaining new results, as for instance: 1) The space of orientation-preserving C^1 actions of every nilpotent group on a 1-dimensional compact space os connected (N). 2) C^2 circle diffeomorphisms of irrational rotation number admit no invariant 1-distribution other than the invariant measure (N-Triestino). 3) Every linear cocycle can be perturbed so that to become conformal along the Oseledets splitting. In case all Lyapunov exponents are zero, then it can be perturber so that to become cohomologous to a cocycle of rotations (Bochi-N). Several open questions will be addressed.
Abstract: Given a group G acting on a probability space X by measure preserving transformations, one has a corresponding a unitary representation of G (Koopman representation); an important question, with diverse applications, is whether this action has a spectral gap, a rigidity property defined in terms of this representation. Actions of groups with property (T) always have such a spectral gap property. We will review some recent results as well as a few applications on this question in different cases: G is a subgroup of a Lie group H acting on X=H/L for a lattice L in H; G is a group of automorphisms of a torus or, more generally, of a nilmanifold X.
Abstract: A relatively hyperbolic group acts on a Gromov hyperbolic space. Its Gromov boundary has no information on peripheral subgroups since the boudnary of the orbit of any point by a peripheral subgroup consists only of one point, called cusp. We construct a blow up of cusp and give a nice boundary of a relatively hyperbolic group. We show that, under appropriate assumptions on peripheral subgroups, the $K$-homology of this boundary is isomorphic to the $K$-theory of the Roe-algebra of the group. As application, we give an explicit computation of the $K$-theory of the Roe-algebra of the fundamental group of the complement of a hyperbolic knot. If time permits, we will discuss a dual theory, that is, the $K$-theory of the stable Higson corona, coarse $K$-theory and coarse co-assembly map.
Abstract: In this talk, we consider coarse nonembeddability of a graph containing a sequence of expanders as induced subgraphs into Hilbert spaces. Using this, we see that a graph containing a "generalized" sequence of expanders (a sequence of finite graphs which have uniformly bounded k-th eigenvalues of the Laplacians and uniformly bounded degrees and whose numbers of vertices diverge to infinity) is not coarsely embeddable into Hilbert spaces.
Abstract: By applying concepts in the quasiconformal Teichm\"uller theory, we give a necessary and sufficient condition for a non-abelian group $G$ of $(1+\alpha)$-diffeomorphisms of the circle with $\alpha>1/2$ to be conjugated to a group of M\"obius transformations by a diffeomorphism in the same class. In its argument, we also see certain rigidity of such a group $G$ in the deformation given by conjugation of symmetric self-homeomorphisms of the circle.
Abstract: I will formulate a version of law of large number in the strong sense for random walks on Lie groups, and show this is useful to describe the boundary associated with a group and a random walk on it. More precisely, for example, for nilpotent Lie groups, we can show that the asymptotic direction of random walks on them is unique almost surely. But, even for some polycyclic groups, the asymptotic direction of random walks on them is not unique any more, due to Kaimanovich. I explain how the asymptotic direction describes the boundary. I will also give some questions around this construction. | CommonCrawl |
# Linear dynamic control systems and optimal control
Linear dynamic control systems can be represented as a set of ordinary differential equations (ODEs) or difference equations. The goal of optimal control is to find the control input that minimizes a cost function over a given time horizon. This can be achieved using techniques such as Pontryagin's maximum principle, which is the foundation of the mathematical theory of optimal control.
Consider the following linear dynamic control system:
$$
\dot{x} = Ax + Bu
$$
where $x$ is the state vector, $A$ is the state matrix, $B$ is the input matrix, and $u$ is the control input. The cost function is given by:
$$
J = \int_0^T (x^TQx + u^TRu + 2x^TQBu) dt
$$
where $Q$ and $R$ are positive semidefinite matrices.
The optimal control problem can be solved using Pontryagin's maximum principle, which involves finding a function $V(x, u)$ that satisfies certain conditions. The optimal control input is then given by the partial derivative of $V$ with respect to $u$.
## Exercise
Consider the following linear dynamic control system:
$$
\dot{x} = \begin{bmatrix} -1 & 2 \\ 1 & -2 \end{bmatrix}x + \begin{bmatrix} 1 \\ 0 \end{bmatrix}u
$$
Find the optimal control input for the following cost function:
$$
J = \int_0^T (x^T\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}x + u^T\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}u + 2x^T\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}\begin{bmatrix} 1 \\ 0 \end{bmatrix}u) dt
$$
### Solution
The optimal control input is given by:
$$
u = -Kx
$$
where $K$ is the matrix:
$$
K = \begin{bmatrix} 0 & 1 \\ 0 & 0 \end{bmatrix}
$$
# Nonlinear programming and its applications
Nonlinear programming problems can be solved using various methods such as gradient-based methods, interior-point methods, and sequential quadratic programming. These methods are typically applied to solve optimization problems with nonlinear constraints and a nonlinear objective function.
Consider the following nonlinear programming problem:
$$
\min_{x \in \mathbb{R}^n} f(x)
$$
subject to:
$$
g_i(x) \leq 0, \quad i = 1, \ldots, m
$$
where $f$ and $g_i$ are nonlinear functions.
Nonlinear programming has numerous applications in various fields, including optimal control, machine learning, and engineering.
# Model predictive control: theory and algorithms
MPC is based on the principle of optimizing a finite-horizon cost function over a prediction horizon. The main idea is to solve a sequence of finite-horizon optimal control problems at each time step, with the goal of minimizing a cost function that accounts for both the current state and the predicted future states.
Consider the following MPC problem:
$$
\min_{u(0), \ldots, u(N-1)} \sum_{k=0}^{N-1} (x^T(k)Qx(k) + u^T(k)Ru(k) + 2x^T(k)QBu(k))
$$
subject to:
$$
\dot{x}(k) = Ax(k) + Bu(k), \quad k = 0, \ldots, N-1
$$
where $x(k)$ is the state vector at time step $k$, $u(k)$ is the control input at time step $k$, $A$ is the state matrix, $B$ is the input matrix, $Q$ and $R$ are positive semidefinite matrices.
MPC can be solved using various algorithms, such as the sequential quadratic programming (SQP) method, the projected newton method, and the augmented Lagrangian method. These methods are typically implemented using numerical optimization techniques.
# Model predictive control: applications in optimal control and state estimation
MPC can be applied to solve problems in optimal control, such as tracking a reference signal or minimizing a cost function over a given time horizon. It can also be used for state estimation, where the goal is to estimate the unknown states of a dynamic system from a set of noisy measurements.
Consider a linear dynamic control system:
$$
\dot{x} = Ax + Bu
$$
where $x$ is the state vector, $A$ is the state matrix, $B$ is the input matrix, and $u$ is the control input. The cost function is given by:
$$
J = \int_0^T (x^TQx + u^TRu + 2x^TQBu) dt
$$
where $Q$ and $R$ are positive semidefinite matrices.
Using MPC, we can design a control law that minimizes the cost function over a given prediction horizon.
MPC has been successfully applied to various fields, including aerospace, automotive, and robotics. Its ability to solve optimal control problems with nonlinear constraints and a nonlinear objective function makes it an attractive control technique.
# Performance evaluation and benchmarking
Performance evaluation metrics for MPC controllers can be divided into two categories: steady-state metrics and dynamic metrics. Steady-state metrics are used to evaluate the performance of the controller when the system is in equilibrium, while dynamic metrics are used to evaluate the performance of the controller during transient dynamics.
Consider an MPC controller for a linear dynamic control system:
$$
\dot{x} = Ax + Bu
$$
where $x$ is the state vector, $A$ is the state matrix, $B$ is the input matrix, and $u$ is the control input. The cost function is given by:
$$
J = \int_0^T (x^TQx + u^TRu + 2x^TQBu) dt
$$
where $Q$ and $R$ are positive semidefinite matrices.
We can evaluate the performance of the controller using metrics such as the steady-state tracking error, the transient tracking error, and the control effort.
Benchmarking techniques can be used to compare the performance of different MPC controllers or to assess the performance of an MPC controller relative to a reference controller. These techniques can involve simulation-based evaluations, experimental evaluations, or theoretical analysis.
# Advanced topics in model predictive control
Multiple-horizon MPC extends the traditional MPC problem by considering multiple prediction horizons. This approach can be used to handle uncertainties in the system dynamics and to improve the controller's adaptability to changing conditions.
Consider the following multiple-horizon MPC problem:
$$
\min_{u(0), \ldots, u(N-1)} \sum_{k=0}^{N-1} (x^T(k)Qx(k) + u^T(k)Ru(k) + 2x^T(k)QBu(k))
$$
subject to:
$$
\dot{x}(k) = Ax(k) + Bu(k), \quad k = 0, \ldots, N-1
$$
where $x(k)$ is the state vector at time step $k$, $u(k)$ is the control input at time step $k$, $A$ is the state matrix, $B$ is the input matrix, $Q$ and $R$ are positive semidefinite matrices.
Robust MPC extends the traditional MPC problem by considering uncertainties in the system dynamics. This approach can be used to handle uncertainties in the system parameters or to improve the controller's robustness to disturbances.
# Exercises and case studies
Exercises will cover a wide range of topics, including linear dynamic control systems, nonlinear programming, MPC algorithms, and performance evaluation metrics. Case studies will provide real-world examples of MPC applications, such as autonomous vehicles, robotic systems, and aerospace systems.
## Exercise
Consider the following exercise:
Design an MPC controller for a linear dynamic control system:
$$
\dot{x} = Ax + Bu
$$
where $x$ is the state vector, $A$ is the state matrix, $B$ is the input matrix, and $u$ is the control input. The cost function is given by:
$$
J = \int_0^T (x^TQx + u^TRu + 2x^TQBu) dt
$$
where $Q$ and $R$ are positive semidefinite matrices.
Solve the MPC problem and evaluate its performance using the steady-state tracking error and the control effort.
### Solution
The optimal control input is given by:
$$
u = -Kx
$$
where $K$ is the matrix:
$$
K = \begin{bmatrix} 0 & 1 \\ 0 & 0 \end{bmatrix}
$$
The steady-state tracking error is given by:
$$
\epsilon = x^TQx
$$
The control effort is given by:
$$
\eta = u^TRu
$$ | Textbooks |
\begin{document}
\title[Kontsevich-LMO, Conway and Casson-Walker-Lescop invariants]{Universal invariants, the Conway polynomial and the Casson-Walker-Lescop invariant} \author[A. Casejuane ]{Adrien Casejuane} \author[J.B. Meilhan ]{Jean-Baptiste Meilhan} \address{Univ. Grenoble Alpes, CNRS, IF, 38000 Grenoble, France} \email{[email protected]} \email{[email protected]}
\begin{abstract} We give a general surgery formula for the Casson-Walker-Lescop invariant of closed $3$-manifolds, by regarding this invariant as the leading term of the LMO invariant. Our proof is diagrammatic and combinatorial, and provides a new viewpoint on a formula established by C.~Lescop for her extension of the Walker invariant. A central ingredient in our proof is an explicit identification of the coefficients of the Conway polynomial as combinations of coefficients in the Kontsevich integral. This latter result relies on general \lq factorization formulas\rq\, for the Kontsevich integral coefficients. \end{abstract}
\maketitle
\section{Introduction}
A.~Casson defined in 1985 an invariant of integral homology spheres, by counting conjugacy classes of irreducible $SU(2)$--representations of the fundamental group \cite{Akbulut-McCarthy,AM}. The Casson invariant was extended, first to rational homology spheres by K.~Walker \cite{Walker1992}, then to all oriented closed $3$-manifolds by C.~Lescop \cite{Lescop}, via surgery formulas. We denote by $\lambda_L$ this \emph{Casson-Walker-Lescop invariant}.
In \cite{Le-Murakami-Ohtsuki}, T.~Q.~T.~Le, J.~Murakami and T.~Ohtsuki defined an invariant of closed oriented $3$-manifolds. This \emph{LMO invariant} is built from the Kontsevich integral \cite{Kontsevich1993} of a surgery presentation, i.e. a framed link in $S^3$. The \emph{Kontsevich integral} of a framed $n$-component link takes values in a graded space of chord diagrams on $n$ circles, while the LMO invariant lives in a graded space of trivalent diagrams; the procedure for extracting the latter invariant from the former one relies on a family of sophisticated combinatorial maps $\iota_n$ that \lq\lq replace circles by sums of trees\rq\rq. The Kontsevich integral is universal among $\mathbb{Q}$-valued Vassiliev invariants, in the sense that any such invariant factors through the Kontsevich integral. Likewise, the LMO invariant is universal among $\mathbb{Q}$-valued finite type invariants of rational homology spheres. Both invariants admit purely combinatorial and diagrammatic definitions, although concrete computations are in general rather difficult.
A striking result is that the leading term of the LMO invariant, i.e. the coefficient of the lowest degree trivalent diagram $\dessin{0.5cm}{T}$, is up to a known factor the Casson-Walker-Lescop invariant \cite{Le-Murakami-Murakami-Ohtsuki,Beliakova-Habegger}. This provides, in principle, a combinatorial procedure for computing the Casson-Walker-Lescop invariant from a surgery presentation, by computing the Kontsevich integral and keeping track of the coefficients of chord diagrams that produce a diagram $\dessin{0.5cm}{T}$ under the LMO procedure. This paper shows how this can be done completely explicitly, in terms of (classical) link invariants. Our first main result is as follows.
\begin{thm}[Thm.~\ref{Thm general}]\label{thm1} Let $L$ be a framed oriented $n$-component link in $S^3$, and let $\mathbb{L}$ denote its linking matrix. We have $$ \lambda_L(S^3_L) = \frac{(-1)^{\sigma_-(L)} \det \mathbb{L}}{8} \sigma(L)
+ (-1)^{n+\sigma_-(L)}\sum_{k=1}^{n} \sum_{\substack{I \subset \{ 1, \ldots, n \} \\ |I| = k}} (-1)^{n - k} \det \mathbb{L}_{\check{I}} \mu_k(L_I),$$ where $\,\, \bullet$ $S^3_L$ denotes the result of surgery on $S^3$ along $L$, \begin{itemize}
\item $\mathbb{L}_{\check{I}}$ is the matrix obtained from $\mathbb{L}$ by deleting the lines and column indexed by a subset $I$ of $\{ 1, \ldots, n \}$,
\item $\sigma_+(L)$ and $\sigma_-(L)$ denote, respectively, the number of positive and negative eigenvalues of $\mathbb{L}$, and
$\sigma(L)=\sigma_+(L)-\sigma_-(L)$,
\item $\mu_k$ is a $k$-component framed link invariant which is explicitly determined by the coefficients of $\mathbb{L}$ and the Conway polynomial. \end{itemize} \end{thm} We do not give here the general explicit formula for the invariant $\mu_k$, which is postponed to Theorem \ref{cor:mun}. Let us only give here the formulas for the first two of these invariants, which are given by $ \mu_1(K) = \frac{1}{24} fr(K)^2 - c_2(K) + \frac{1}{12}$, and for $L=K_1\cup K_2$, $$ \mu_2(L)=\frac{1}{12} \textrm{lk}(L)^3 + \frac{fr(K_1) + fr(K_2)}{12} \textrm{lk}(L)^2 + \textrm{lk}(L)\Big( c_2(K_1) + c_2(K_2) - \frac{1}{12} \Big) - c_3(L). $$ These two invariants are involved in the case $n=2$ of Theorem \ref{thm1}, which recovers a theorem of S.~Matveev and M.~Polyak \cite[Thm.~6.3]{Matveev-Polyak} for the Casson-Walker invariant of rational homology spheres; see Remark \ref{rem:cestcadeaucamfaitplaisir} for details.
Actually it turns out that, in the general case, Theorem \ref{thm1} recovers the third global surgery formula of Lescop \cite[Prop.~1.7.8]{Lescop}, when restricted to integral surgery coefficients; this is further discussed in Remark \ref{rem:lescoop}. It is quite interesting to see how Lescop's \lq chain products of linking numbers\rq \, $\Theta_b$, which are the main ingredients in the general formula for our invariants $\mu_k$, appear naturally in our proof from the combinatorics of chord and Jacobi diagrams and the universal Kontsevich-LMO invariants. We stress, moreover, that the proof of the present result is completely independant from that of Lescop's formula.
As part of the proof of Theorem \ref{thm1}, we provide in this paper a number of formulas identifying certain combinations of coefficients of the Kontsevich integral in terms of classical link invariants. Such formulas are derived from general factorization results, which show how certain local configurations in sums of coefficients in the Kontsevich integral, yield a factorization by simple link invariants; see Section \ref{sec:facto}.
The second main result of this paper uses these techniques to give an explicit identification for the $z^{n+1}$-coefficients of the Conway polynomial of an $n$-component link. This identification relies on the definition, outlined below, of a family of chord diagrams which are recursively built from a couple of low degree diagrams by simple local operations. Specifically, consider the following two local operations on chord diagrams, called \emph{inflation} and \emph{infection}: \begin{center} \includegraphics[scale=0.8]{infect.pdf} \end{center}
For any integer $n\ge 1$, denote by $\mathcal{E}^-(n)$ the set of all (connected) chord diagrams on $n$ circles which are obtained from the two chord diagrams $\dessin{0.65cm}{D21_2}$ and $\dessin{0.65cm}{D32_2}$ by iterated inflations, in all possible ways. Denote also by $\mathcal{P}(n)$ the set of all diagrams obtained from an element of $\mathcal{E}^-(n-1)$ by a single infection. Our second main result reads as follows. \begin{thm}[Thm.~\ref{c_n}] \label{thm2} Let $n\ge 2$. For any framed oriented $n$-component link $L$, we have
$$c_{n+1}(L) = \sum_{D \in \mathcal{E}^-(n) \cup \mathcal{P}(n)} C_L[D],$$ where $C_L[D]$ denotes the coefficient of $D$ in the (framed) Kontsevich integral of $L$. \end{thm}
Let us describe the simplest case $n=2$ more precisely. We have $\mathcal{E}^-(2)=\{\dessin{0.5cm}{D32_4};\dessin{0.5cm}{D32_4b};\dessin{0.5cm}{D32_2}\}$ and $\mathcal{P}(2)=\{\dessin{0.5cm}{D32_7};\dessin{0.5cm}{D32_7b}\}$.\footnote{We use the graphical convention that the circles are ordered from left to right.} If $L$ is a framed oriented $2$-component link, then Theorem \ref{thm2} says that $c_3(L)$ is given by $$C_L\Big[ \dessin{0.5cm}{D32_4} \Big] + C_L\Big[ \dessin{0.5cm}{D32_4b} \Big] + C_L\Big[ \dessin{0.5cm}{D32_2} \Big]+ C_L\Big[ \dessin{0.5cm}{D32_7} \Big]+ C_L\Big[ \dessin{0.5cm}{D32_7b} \Big]. $$
Figure \ref{fig:ex} gives typical examples of chord diagrams that are involved in the statement for higher values of $n$. \begin{figure}
\caption{Two examples of elements in $\mathcal{E}^-(6)$ (left) and $\mathcal{P}(7)$ (right).}
\label{fig:ex}
\end{figure}
The case $n=1$ is somewhat particular, as it involves a correction term. We have $\mathcal{E}^-(1)=\{\dessin{0.5cm}{D21_2}\}$ and $\mathcal{P}(1)=\emptyset$, and for a knot $K$ we have $$ c_2(K) = C_K\Big[\dessin{0.5cm}{D21_2}\Big] + \frac{1}{24},$$ a formula which is well-known to the experts (see Proposition \ref{c_2}).
We stress that Theorem \ref{thm2} is of course related to the weight system of the Conway polynomial, computed in \cite{MMR} for solving the Melvin-Morton-Rozansky conjecture. Our statement and proof are however completely independant from \cite{MMR}.
The paper is organized as follows. In Section 2, we review the various invariants of links and $3$-manifolds alluded to in the title of the paper. In Section 3, we identify certain combinations of coefficients in the framed Kontsevich integral in terms of classical invariants; in particular, our factorization results are given in Section 3.2, while Theorem 2 is proved in Section 3.4. Section 4 is devoted to the invariants $\mu_n$; an explicit formula in terms of Conway coefficients and the linking matrix is given in Section 4.2. Finally, we prove Theorem 1 in Section 5.
\begin{acknowledgments} The authors would like to thank Christine Lescop for discussions regarding the relationship between Theorem \ref{thm1} and \cite{Lescop}. The first author is partly supported by the project AlMaRe (ANR-19-CE40-0001-01) of the ANR. \end{acknowledgments}
\section{Preliminaries}
In this section we recall the necessary material for this paper. We start by a set a conventions that will be used throughout.
\subsection{Conventions and Notation}\label{sec:conv}
All $3$ manifolds will be assumed to be closed, compact, connected and oriented. All links live in the $3$-sphere $S^3$, and are assumed to be framed, oriented and ordered.
Let $L=K_1\cup \cdots \cup K_n$ be an $n$-component link. Given a subset $I$ of $\{ 1, \ldots, n \}$, we set $$ L_I := \bigcup_{i\in I} K_i\quad \textrm{ and }\quad L_{\check{I}}:= L\setminus L_I.$$ \noindent We abbreviate $L_{\check{i}}=L_{\check{\{ i \}}}$.
We denote by $l_{i,j}$ the linking number of the $i$th and $j$th component, and we denote by $fr_i=fr(K_i)$ the framing of the $i$th component.
The linking matrix $\mathbb{L} \in M_n(\mathbb{Z})$ of $L$ is given by $\mathbb{L}_{i, i} = fr(K_i)$ and $\mathbb{L}_{i, j} = l_{i, j}$ if $i\neq j$. We denote by $\sigma_+(L)$, resp. $\sigma_-(L)$, the number of positive, resp. negative, eigenvalues of $\mathbb{L}$, so that its signature is given by $\sigma(L) = \sigma_+(L) - \sigma_-(L)$.
\subsection{Conway polynomial and the $U_n$ invariant}
The \emph{Conway polynomial} is a renormalization of the Alexander polynomial, introduced by J.~Conway in the late $60$s. This is an invariant of (unframed) oriented links, which is a polynomial $\nabla$ in the variable $z$, defined by setting
$\nabla_U(z)=1$, where $U$ denotes the unknot, and
$$ \nabla_{L_+}(z) - \nabla_{L_-}(z) = z\nabla_{L_0}(z), $$ where $L_+$, $L_-$ and $L_0$ are three links that are identical except in a $3$-ball where they look as follows: \begin{center} \begin{tikzpicture}
\draw[->] [very thick] (0,0) -- (1,1);
\draw[<-] [very thick] (0,1) -- (0.4,0.6);
\draw (0.6,0.4) [very thick] -- (1,0);
\draw (0.5,0) node[below]{$L_+$};
\draw [very thick] (2,0) -- (2.4, 0.4);
\draw[->] [very thick] (2.6,0.6) -- (3,1);
\draw[<-] [very thick] (2,1) -- (3,0);
\draw (2.5,0) node[below]{$L_-$};
\draw[<-] [very thick] (4,1) to [bend left] (4,0);
\draw[<-] [very thick] (4.5,1) to [bend right] (4.5,0);
\draw (4.25,0) node[below]{$L_0$}; \end{tikzpicture} \end{center} We say that the three oriented links $(L_+,L_-,L_0)$ form a \emph{skein triple}, and a formula of the type above is typically called a skein formula.
Denote by $c_k$ the coefficient of $z^k$ in the Conway polynomial. This is a $\mathbb{Z}$-valued link invariant, which satisfies the skein formula $c_{k+1}(L_+) - c_{k+1}(L_-) = c_k(L_0)$.
For a knot $K$ we have $c_1(K)=1$, and for a $2$-component link $L=K_1\cup K_2$ we have $c_1(L)=l_{1,2}$. In general, the Conway polynomial of an $n$-component link $L$ has the form
$$\nabla_L(z)= \sum_{k=0}^N c_{n+2k-1}(L) z^{n+2k-1}. $$
We can define the following link invariant from the Conway coefficients $c_k$. \begin{definition}\label{def:U_n} Let $n\ge 2$ be an integer. We define an invariant $U_{n}$ of oriented $(n-1)$-component links by setting $$U_2(K) = c_2(K) - \frac{1}{24} $$ for a knot $K$, and the recursive formula $$U_{n+1}(L) = c_{n+1}(L) - \sum_{i = 1}^{n} U_{n}(L_{\check{i}}) \sum_{j \neq i} l_{i, j} $$ for an $n$-component link $L$ ($n\ge 3$). \end{definition} For example, $$ U_3(K_1\cup K_2) = c_3(K_1\cup K_2) - l_{1,2}\left(c_2(K_1) + c_2(K_2) - \frac{1}{12}\right). $$
\subsection{The Casson-Walker-Lescop invariant} \label{sec:casson}
The following is due to A.~Casson. \begin{theorem}[Casson] \label{casson} There exists a unique $\mathbb{Z}$-valued invariant of integral homology spheres such that \begin{enumerate} \item[(i).] $\lambda(S^3)=0$. \item[(ii).] For any integral homology sphere $M$, for any knot $K$ in $M$,and any $n\in \mathbb{Z}$, if $M_{K_n}$ is the result of $\frac{1}{n}$-Dehn surgery on $M$ along $K$, then:
$$ \lambda(M_{K_{n+1}})-\lambda(M_{K_n})=c_2(K). $$ \end{enumerate} Moreover, \begin{enumerate} \item[(iii).] $\lambda$ changes sign under orientation reversal, and is additive under connected sum. \item[(iv).] The mod 2 reduction of $\lambda$ coincides with the Rochlin invariant. \end{enumerate} \end{theorem}
This is the \emph{Casson invariant} of integral homology spheres. Its existence was established by A.~Casson, who defined it in terms of count of conjugacy classes of irreducible $SU(2)$--representations of $\pi_1(M)$.
In \cite{Walker1992}, K.~Walker extended the Casson invariant to a $\mathbb{Q}$-valued invariant of rational homology spheres $\lambda_W$, via a surgery formula. C.~Lescop then widely generalized the Casson-Walker invariant to all closed $3$-manifolds, by establishing a global surgery formula involving the multivariable Alexander polynomial \cite{Lescop}. We denote by $\lambda_L$ this \emph{Casson-Walker-Lescop invariant}. Our convention is that, for a rational homology sphere $M$, we have $\lambda_L(M) = \frac{1}{2}\vert H_1(M)\vert \lambda_W(M)$.
\subsection{Universal invariants}
We now review the Kontsevich and LMO invariants, providing only the ingredients that are necessary for our purpose.
\subsubsection{Chord diagrams and Jacobi diagrams}
Let us begin with introducing the spaces of diagrams in which the Kontsevich integral and LMO invariants take values. We stress that our terminologies are somewhat different from the usual conventions of the litterature: this is clarified in Remark \ref{rem:conv}.
\begin{definition} Let $X$ be some oriented $1$-manifold. A \emph{chord diagram} $D$ on $X$ is a collection of copies of the unit interval, such that the set of all endpoints is embedded into $X$. We call \emph{chord} any of these copies of the interval, and we call \emph{leg} any endpoint of a chord in $D$; the $1$-manifold $X$ is called the \emph{skeleton} of $D$.\\ A chord is called \emph{mixed}, resp. \emph{internal}, if its two legs lie on distinct, resp. the same, component(s) of the skeleton. \\
The \emph{degree} of $D$ is defined as $deg(D) = |\{ \text{chords of\ } D \}| = \frac{1}{2} |\{ \text{legs of\ } D \}|$. \end{definition}
\begin{definition} We denote by $\mathcal{A}(X)$ the $\mathbb{Q}$-vector space generated by all chord diagrams on $X$, modulo the \emph{4T relation}:
$$ \dessin{1cm}{4T}. $$ \end{definition} \begin{remark}\label{rem:central} As a consequence of the 4T relation, an \emph{isolated chord} (i.e. a chord whose endpoints are met consecutively on the skeleton) commutes with any other chord, in the sense that we have the following relation in $\mathcal{A}(X)$:
$$ \dessin{0.9cm}{central}. $$ \end{remark}
In what follows, we will almost exclusively be interested in \emph{chord diagrams on $n$ circles}, i.e. in the case where $X=\circlearrowleft^n$ consists of $n$ ordered, oriented copies of $S^1$.
\begin{definition} A \emph{Jacobi diagram} is a trivalent graph whose trivalent vertices are equipped with a cyclic order on the incident edges. The \emph{degree} of a Jacobi diagram is half its number of vertices. \end{definition}
\begin{definition} We denote by $\mathcal{A}(\emptyset)$ the $\mathbb{Q}$-vector space generated by all Jacobi diagrams, modulo the \emph{AS and IHX relations}:
$$ \dessin{1.3cm}{relations}. $$ \end{definition}
\begin{notation} For an element $x\in \mathcal{A}(\emptyset)$, and an integer $k\ge 0$, we denote by $(x)_{k}$, resp. $(x)_{\le k}$, its projection to the degree $k$ part $\mathcal{A}_{k}(\emptyset)$, resp. the degree $\le k$ part $\mathcal{A}_{\le k}(\emptyset)$. \end{notation}
\begin{remark}\label{rem:conv} In the literature, the term \lq Jacobi diagram \rq\, more generally refers to unitrivalent diagrams whose univalent vertices lie disjointly on a (possible empty) $1$-manifold, subject to AS, IHX and an extra STU relation. Hence what we call \lq Jacobi diagrams\rq\, here are what experts know as \lq Jacobi diagrams on the empty set\rq, or \lq purely trivalent Jacobi diagrams\rq\, -- this justifies our notation $\mathcal{A}(\emptyset)$. \end{remark}
We make use of the usual drawing conventions for chord and Jacobi diagrams: bold lines represent skeleton components while chord and graphs are drawn with dashed lines, and trivalent vertices are equipped with the counterclockwise ordering. Also, we assume when drawing elements of $\mathcal{A}(\circlearrowleft^n)$, that the circles are oriented counterclockwise and are ordered from left to right, unless otherwise specified.
\subsubsection{The Kontsevich integral}\label{sec:Kontsevich}
Let us give a quick overview of the Kontsevitch integral. We do not follow here Kontsevich's original definition \cite{Kontsevich1993}, but rather the combinatorial definition later provided in \cite{LM}. Moreover, we will only give explicitly the low degree terms in the definitions, since these are all we need for the purpose of this paper. We refer the reader to \cite[\S 6.4]{Ohtsuki} for a detailed review.
Recall that a q-tangle is an oriented tangle, equipped with a consistent collection of parentheses on each of its linearly ordered sets of boundary points. A q-tangle can be non-uniquely decomposed into copies of the following elementary q-tangles $I$, $X_{\pm}$, $C_{\pm}$ and $\Lambda_{\pm}$ (and those obtained by orientation-reversal on any component):\\[-0.1cm] \begin{center} \includegraphics[scale=0.8]{elementary_new.pdf} \end{center}
The \emph{(framed) Kontsevich integral} can thus be determined by specifying its values on these elementary q-tangles.
We set $Z(I)=\uparrow$, the diagram without chord of $\mathcal{A}(\uparrow)$, and $Z(C_\pm)=\sqrt{\nu}$, where $\nu\in \mathcal{A}(\circlearrowleft)$ is the Kontsevich integral of the $0$-framed unknot $U_0$, computed in \cite{Bar-Natan-Garoufalidis-Rozansky-Thurston}: \begin{equation}\label{eq:nu} \begin{tikzpicture}
\draw (0.5,0.5) node{$\nu =$};
\draw [very thick] (1,0.5) arc(180:540:0.5);
\draw[<-] [very thick] (1.5,1) -- (1.51,1);
\draw (2.5,0.5) node{$+ \frac{1}{24}$};
\draw [very thick] (3,0.5) arc(180:540:0.5);
\draw[<-] [very thick] (3.5,1) -- (3.51,1);
\draw [densely dashed] (3.1,0.3) -- (3.9,0.3);
\draw [densely dashed] (3.1,0.7) -- (3.9,0.7);
\draw (4.5,0.5) node{$- \frac{1}{24}$};
\draw [very thick] (5,0.5) arc(180:540:0.5);
\draw[<-] [very thick] (5.5,1) -- (5.51,1);
\draw [densely dashed] (5.1,0.7) -- (5.9,0.3);
\draw [densely dashed] (5.1,0.3) -- (5.9,0.7);
\draw (7.2,0.5) node{$+$ degree $> 2$.}; \end{tikzpicture} \end{equation}
Next we set
$Z(X_\pm)=\textrm{exp}\big( \frac{\pm 1}{2} \dessin{0.7cm}{X1} \big) = \sum_{k\ge 0} \frac{(\pm 1)^k}{2^k k!} \dessin{0.7cm}{X1}^{k}$, where the $k^{th}$ power denotes $k$ parallel dashed chords: \begin{equation}\label{Z2}
Z(X_\pm)=\dessin{0.85cm}{X0} \pm\frac{1}{2} \dessin{0.85cm}{X1} + \frac{1}{8} \dessin{0.85cm}{X2} \pm \frac{1}{48} \dessin{0.85cm}{X3} + \textrm{degree $>3$}. \end{equation} Finally, set $Z(\Lambda_\pm)=\Phi^{\pm 1}$, where $\Phi\in \mathcal{A}( \uparrow\uparrow\uparrow )$ is the choice of a \emph{Drinfeld associator} (see e.g. \cite[App. D]{Ohtsuki}). At low degree, this gives
$$ Z(\Lambda_\pm)= \dessin{0.85cm}{phi0} \pm\frac{1}{24} \left( \dessin{0.85cm}{phi1} - \dessin{0.85cm}{phi2}\right) + \textrm{degree $>3$}. $$
\begin{ex} \label{Calculs Kontsevich} The following are well-known low degree computations for $Z(U_\pm)$, where $U_\pm$ denotes the $(\pm 1)$-framed unknot. \begin{center} \begin{tikzpicture}
\draw (0,0.5) node{$\hat{Z}(U_\pm) =$};
\draw [very thick] (1,0.5) arc(180:540:0.5);
\draw[<-] [very thick] (1.5,1) -- (1.51,1);
\draw (2.5,0.5) node{$\pm \frac{1}{2}$};
\draw [very thick] (3,0.5) arc(180:540:0.5);
\draw[<-] [very thick] (3.5,1) -- (3.51,1);
\draw [densely dashed] (3,0.5) -- (4,0.5);
\draw (4.5,0.5) node{$+ \frac{1}{6}$};
\draw [very thick] (5,0.5) arc(180:540:0.5);
\draw[<-] [very thick] (5.5,1) -- (5.51,1);
\draw [densely dashed] (5.1,0.7) -- (5.9,0.7);
\draw [densely dashed] (5.1,0.3) -- (5.9,0.3);
\draw (6.5,0.5) node{$- \frac{1}{24}$};
\draw [very thick] (7,0.5) arc (180:540:0.5);
\draw[<-] [very thick] (7.5,1) -- (7.51,1);
\draw [densely dashed] (7.1,0.7) -- (7.9,0.3);
\draw [densely dashed] (7.1,0.3) -- (7.9,0.7);
\draw (9,0.5) node{ $\,\,\,\,\,+$ degree $> 2$.}; \end{tikzpicture} \end{center} \end{ex}
\subsubsection{The degree $\le 1$ part of the LMO invariant}\label{sec:LMO}
We now review the LMO invariant of closed oriented $3$-manifolds. Starting with an integral surgery presentation, this invariant is extracted from a renormalization $\check{Z}$ of the Kontsevich integral of this link via a family of sophisticated diagrammatic operations $\iota_n$. For the purpose of this paper, however, we only need the degree $\le 1$ part of the LMO invariant, and in particular we only need (a somewhat simplified definition of) the map $\iota_1$. We refer the reader to \cite{Le-Murakami-Murakami-Ohtsuki,Ohtsuki} for a complete definition.
\begin{definition} Let $L$ be a framed oriented $n$-component link. We set $\check{Z}(L) := \hat{Z}(L) \# \nu^{\otimes n}$. In other words, in $\check{Z}$ we add a copy of $\nu$ to each circle component in $\hat{Z}(L)\in \mathcal{A}(\circlearrowleft^n)$. \end{definition}
\begin{definition} Let a chord diagram $D$ on $n$ circles, we associate an element of $\mathcal{A}(\emptyset)$ as follows. For each circle component $C$ of $D$, if the number of legs on $C$ is $k$, \begin{itemize}
\item if $2\le k\le 4$, then replace $C$ by the portion of Jacobi diagram $T_k$, where
\[ T_2 = \dessin{0.1cm}{strut},\quad T_3=\frac{1}{2}\dessin{0.9cm}{Y},\quad T_4 = \frac{1}{6}\dessin{0.9cm}{H} + \frac{1}{6} \dessin{0.9cm}{I}. \]
\item if $k\le 1$ or $k\ge 5$, then map $D$ to $0$. \end{itemize} Next, replace each copy of $\dessin{0.5cm}{D00}$ resulting from these replacements by a coefficient $(-2)$. The result is the desired element of $\mathcal{A}(\emptyset) $, which we denote by $\iota_1(D)$.
By linearity this defines a map \[ \iota_1: \mathcal{A}(\circlearrowleft^n)\longrightarrow \mathcal{A}(\emptyset). \] \end{definition}
Now, let $M$ be a closed $3$--manifold, and let $L$ be a framed $n$-component link in $S^3$ such that $M$ is obtained by surgery along $L$. Fix an orientation for the link $L$. \begin{definition} The degree $\le 1$ part of the \emph{LMO invariant} of $M$ is defined by
$$ Z^{LMO}_1(M):= \left( \frac{\iota_1(\check{Z}(L))}{(\iota_1(\check{Z}(U_{+}))^{\sigma_+(L)} (\iota_1(\check{Z}(U_{-}))^{\sigma_-(L)} }\right)_{\le 1}\in \mathcal{A}_{\le 1}(\emptyset). $$ \end{definition} This is an invariant of the $3$-manifold $M$: it does not depend on the choice of orientation of $L$, and does not change under Kirby moves.
The denominator in the above formula is easily computed, see \cite{Ohtsuki}: \begin{equation}\label{Denominateur LMO}
\left( \iota_1(\check{Z}(U_+))^{- \sigma_+(L)} \iota_1(\check{Z}(U_-))^{- \sigma_-(L)} \right)_{\le 1}= (-1)^{\sigma_+(L)} +
\frac{(-1)^{\sigma_+(L)} \sigma(L)}{16} \dessin{0.8cm}{T}. \end{equation}
Moreover, the degree $0$ and $1$ parts of $Z_1^{LMO}(M)$ are clearly identified. \begin{theorem}[\cite{Le-Murakami-Murakami-Ohtsuki,Beliakova-Habegger}]\label{LMOCasson} Let $M$ be a closed $3$ manifold. We have $$Z_1^{LMO}(M)=l_0+l_1\dessin{0.8cm}{T}$$ where\\[-0.5cm] \begin{itemize}
\item $l_0=\left\{
\begin{array}{cl}
|H_1(M)|& \textrm{ if $M$ is a rational homology sphere,}\\
0 & \textrm{ otherwise.}
\end{array}
\right.$
\item $l_1 = \frac{(-1)^{\beta_1(M)}}{2} \lambda_L(M)$, where $\beta_1$ denotes the first Betti number. \end{itemize} \end{theorem} The second point of Theorem \ref{LMOCasson} is the key result in establishing our surgery formula for the Casson-Walker-Lescop invariant.
\begin{remark}\label{rem:iota} Our definition of the $\iota_1$ map differs from the usual one in that we map to zero all diagrams of degree $\ge 5$. This modification is harmless since, with the original definition of $\iota_1$, such diagrams cannot contribute to the degree $\le 1$ part of the LMO invariant. \end{remark}
\section{Coefficients of the Kontsevich integral}
In this section we identify certain combinations of coefficients in the framed Kontsevich integral in terms of classical invariants.
\subsection{Operations on Jacobi diagrams}
\subsubsection{Preliminaries}
We begin by introducing some notations and tools that will be used throughout the rest of the paper.
\begin{notation} \label{nota:coeff} Let $S$ be an element of $\mathcal{A}(\circlearrowleft^n)$. Let $D$ be a chord diagram on $n$ circles. We denote by $C[D](S)$ the coefficient of $D$ in $S$. In particular, we set $$C_L[D] := C[D](\hat{Z}(L)),$$ for a framed oriented link $L$, and we denote by $C[D]$ the assignment $L\mapsto C_L[D]$. \end{notation}
We list below three rather simple and well-known lemmas, whose proofs are ommited (proofs can be found in \cite{Casejuane}).
\begin{lemma}[Invariance] \label{Critere Invariance} Let $D_1, \ldots, D_k$ be chord diagrams on $n$ circles. Then $X := C[D_1] + \ldots + C[D_k]$ defines an invariant of framed oriented $n$-component links if and only if $X$ vanishes on any linear combination of chord diagrams arising from a $4T$ relation. \end{lemma}
\begin{lemma}[Disjoint Union] \label{Factorisation} Let $D$ be a chord diagram on $n$ circles that splits into a disjoint union $D=D_I\sqcup D_J$. Then for any framed oriented $n$-component link $L$ we have $$C_L[D] = C_{L_I}[D_I] \times C_{L_J}[D_J], $$ where $L_I$ and $L_J$ are sublinks of $L$ corresponding to the components of $D_I$ and $D_J$. \end{lemma}
\begin{lemma}[Skein] \label{Obvious lemma} Let $D$ be a chord diagram of degree at most $4$. Let $L_+$ and $L_-$ be the first two terms of a skein triple at a crossing $c$ between the $i$th and $j$th components (possibly $i=j$). Then $$ C_{L_+}[D] - C_{L_-}[D] = \left\{ \begin{array}{ll} C[D] \Big(\dessin{0.9cm}{X1}\Big) & \text{if $D$ has $\le 2$ chords between $i$ and $j$,}\\ C[D] \Big(\dessin{0.9cm}{X1} + \frac{1}{24}\dessin{0.9cm}{X3}\Big) & \text{if $D$ has $\ge 3$ chords between $i$ and $j$,} \end{array}\right. $$ where we only show the local contribution to $\hat{Z}(L)$ given by the crossing $c$. \end{lemma}
\subsubsection{Inflations and Infections}
We now introduce several local operations on chord diagrams. The first one is a standard one:
\begin{definition} Let $D$ be a chord diagram on $n$ circles, with at least one chord, as shown on the left-hand side of the figure below. A \emph{smoothing} of $D$ along this chord is a chord diagram $D_0$ obtained from the following operation: \begin{center} \begin{tikzpicture}
\draw[<-] [very thick] (0,1) -- (1,0);
\draw[->] [very thick] (0,0) -- (1,1);
\draw (-0.2,0.5) node{$D$};
\draw [densely dashed] (0.25,0.75) -- (0.75,0.75);
\draw[->] (1.25,0.5) -- (2.25,0.5);
\draw[<-] [very thick] (2.5,1) to[bend left] (2.5,0);
\draw[<-] [very thick] (3,1) to[bend right] (3,0);
\draw (3.4,0.5) node{$D_0$}; \end{tikzpicture} \end{center} \end{definition} \noindent Figure \ref{fig:lissage} gives two examples of smoothings (along the chord marked with a $\ast$). \begin{figure}
\caption{Smoothing chord diagrams on a circle.}
\label{fig:lissage}
\end{figure}
If the chord lies on two disjoint components $i$ and $j$ ($i<j$), then these two circles become a single component of $D_0$, labeled by $i$, and the circles $k>j$ are re-labeled by $(k-1)$. Otherwise, as Figure \ref{fig:lissage} illustrates, the skeleton of $D_0$ has $(n+1)$ component, and the $(n+1)$th component is one of the two circles arising from the smoothing.
The next two operations, called inflation and infection, will provide recursive tools for building chord diagrams with useful properties, in any degree. \begin{definition} Let $D$ be a chord diagram on $n$ circles, and let $c$ be a chord of $D$. The \emph{inflation of $D$ along $c$} is the following local operation: \\[-0.2cm] \begin{center} \begin{tikzpicture}
\draw [very thick] (0,0) arc(270:450:0.5);
\draw[<-] [very thick] (0,1) -- (0.01,1);
\draw (0,0) node[left]{$i$};
\draw [densely dashed] (0.5,0.5) -- (1.5,0.5);
\draw (1,0.5) node[below]{$c$};
\draw [very thick] (2,0) arc(270:90:0.5);
\draw[->] [very thick] (1.99,0) -- (2,0);
\draw (2,0) node[right]{$j$};
\draw[->] (2.25,0.5) -- (3.25,0.5);
\draw [very thick] (3.5,0) arc(270:450:0.5);
\draw[<-] [very thick] (3.5,1) -- (3.51,1);
\draw (3.5,0) node[left]{$i$};
\draw [densely dashed] (4,0.5) -- (4.5,0.5);
\draw [very thick] (4.5,0.5) arc(180:540:0.5);
\draw (5,0) node[below]{$n+1$};
\draw[<-] [very thick] (5,1) -- (5.01,1);
\draw [densely dashed] (5.5,0.5) -- (6,0.5);
\draw [very thick] (6.5,0) arc(270:90:0.5);
\draw[->] [very thick] (6.49,0) -- (6.5,0);
\draw (6.5,0) node[right]{$j$}; \end{tikzpicture} \end{center} \end{definition}
\begin{definition} Let $D$ be a chord diagram on $n$ circles, and let $I$ be an interval in the skeleton, whose interior is disjoint from all chords. The \emph{infection of $D$ along $I$} is the following local operation:\\[-0.2cm] \begin{center} \begin{tikzpicture}
\draw [very thick] (0,0) arc(270:450:0.5);
\draw[<-] [very thick] (0,1) -- (0.01,1);
\draw (0,0) node[left]{$i$};
\draw (0.5,0.5) node[right]{$I$};
\draw[->] (1.25,0.5) -- (2.25,0.5);
\draw [very thick] (2.5,0) arc(270:450:0.5);
\draw[<-] [very thick] (2.5,1) -- (2.51,1);
\draw (2.5,0) node[left]{$i$};
\draw [densely dashed] (3,0.5) -- (4,0.5);
\draw [very thick] (4,0.5) arc(180:540:0.5);
\draw[<-] [very thick] (4.5,1) -- (4.51,1);
\draw (4.8,0) node[right]{$n+1$}; \end{tikzpicture} \end{center} We call \emph{infection along the $i$th component} of $D$ the result of an infection along any of the intervals bounded by the legs on the $i$th circle component. \end{definition}
\begin{remark}\label{rem:lissage} Smoothing the chord that appears in an infection on a chord diagram $D$ gives back $D$. Likewise, after some inflation on $D$, smoothing either of both chords attached to the $(n+1)$th circle, gives back $D$. \end{remark}
\subsubsection{Essential diagrams}\label{sec:essentiels}
We now introduce several families of chord diagrams. We start with a general definition. \begin{definition}\label{def:close} Let $D$ be a chord diagram, and let $D'$ be an element of $\mathcal{A}(\emptyset)$. We say that $D$ \emph{closes into $D'$} when $\left(\iota_1(D)\right)_{\le 1} = D'$. \end{definition}
We first consider diagrams that close into a nonzero constant. By the definition of $\iota_1$, a \emph{connected} diagram closes into the empty diagram with nonzero coefficient if, and only if each circle component has exactly two legs; such diagrams will be called \lq chain of circles\rq\, in the rest of this paper: \begin{definition}\label{rem:chain} A \emph{chain of $n$ circles} is a degree $n$ chord diagram obtained by $(n-1)$ successive inflations on the diagram $\dessin{0.6cm}{D11}$, up to permutation of the circle labels. \end{definition} For example, chains of $1$, $2$ and $3$ circles are of the form $\dessin{0.5cm}{D11}$, $\dessin{0.5cm}{D22}$ and $\dessin{0.6cm}{D33}$, respectively. It is immediately verified that any chain of circles closes into $-2$.
We now consider diagrams that close into the Theta-shaped diagram $\dessin{0.5cm}{T}$. \begin{definition} A \emph{connected} chord diagram is called \begin{itemize} \item a \emph{$\oplus$-essential diagram} if it closes into $\dessin{0.5cm}{T}$ with positive coefficient, \item a \emph{$\ominus$-essential diagram} if it closes into $\dessin{0.5cm}{T}$ with negative coefficient, \item an \emph{essential diagram} if it either a $\oplus$-essential or $\ominus$-essential diagram. \end{itemize} We denote respectively by $\mathcal{E}^+(n)$ and $\mathcal{E}^-(n)$, the set of $\oplus$-essential and $\ominus$-essential diagrams on $n$ circles. We also set $\mathcal{E}(n):=\mathcal{E}^+(n)\cup \mathcal{E}^-(n)$. \end{definition}
Before further investigating these families of diagrams, let us give low-degree examples. \begin{ex}\label{ex:essentiels} All $\oplus$-essential diagrams on $\le 2$ circles are given by $$ \mathcal{E}^+(1) = \{ \dessin{0.5cm}{D21_1} \} \quad \textrm{and} \quad \mathcal{E}^+(2) = \{ \dessin{0.5cm}{D32_1} \, ; \, \dessin{0.5cm}{D32_3} \, ; \, \dessin{0.5cm}{D32_3b} \}, $$ and they all close into $\frac{1}{6}\times \dessin{0.5cm}{T}$. \\ All $\ominus$-essential diagrams on $\le 2$ circles close into $-\frac{1}{3}\times \dessin{0.5cm}{T}$ and are given by $$ \mathcal{E}^-(1) = \{ \dessin{0.5cm}{D21_2} \} \quad \textrm{and} \quad \mathcal{E}^-(2) = \{ \dessin{0.5cm}{D32_2}\,;\,\dessin{0.5cm}{D32_4} \, ; \, \dessin{0.5cm}{D32_4b}\}. $$ \end{ex}
More generally, the following combinatorial criterion can easily be deduced from the definition of the map $\iota_1$. \begin{lemma} \label{Conditions combinatoires} Let $D$ be a chord diagram. Then $D$ is essential if, and only if it is of the one of the following two types:
\begin{itemize}
\item $D$ contains one circle with $4$ legs, and all other circles have $2$ legs,
\item $D$ contains two circles with $3$ legs, and all other circles have $2$ legs.
\end{itemize} It follows that an essential diagram on $n$ circles has always degree $n+1$. \end{lemma}
We now relate essential diagrams to the inflation operation. \begin{proposition} \label{Inflations et Essentiels} Inflation on a $\oplus$-essential (resp. $\ominus$-essential) diagram of degree $n$ yields a $\oplus$-essential (resp. $\ominus$-essential) diagram of degree $n+1$, for all $n\ge 1$.\\ Conversely, for $n\ge 3$, any $\oplus$-essential (resp. $\ominus$-essential) diagram of degree $n+1$ is the inflation of a $\oplus$-essential (resp. $\ominus$-essential) diagram of degree $n$, up to permutation of the circle labels. \end{proposition} \begin{proof} The first part of the statement is rather easily verified, as follows. Firstly, inflation preserves connectivity. Secondly, if $D$ is obtained by inflation on (say) a $\oplus$-essential diagram $\tilde{D}$, then one can freely chose, when applying the map $\iota_1$ to $D$, to first act on the $(n+1)$th circle, which is replaced by an edge by inserting a copy of $T_2$: the result is the diagram $\tilde{D}$ (with coefficient $1$), which by definition closes into $\dessin{0.5cm}{T}$ with positive coefficient.\\ Conversely, since $n\ge 3$, the skeleton of an essential diagram $D$ of degree $(n+1)$ has at least $3$ circles. Lemma \ref{Conditions combinatoires} then tells us that $D$ has at least one circle component with exactly two legs. Since $D$ is connected, these two legs are the endpoints of two (distinct) mixed chords, which allows us to regard $D$ as the result of an inflation. \end{proof}
\begin{remark}\label{rem:seeds} By combining Proposition \ref{Inflations et Essentiels} with Example \ref{ex:essentiels}, we have that, up to permutation of the circle labels, any $\oplus$-essential diagram is obtained by iterated inflations from either $\dessin{0.65cm}{D21_1}$ or $\dessin{0.65cm}{D32_1}$, and that any $\ominus$-essential diagram is obtained by iterated inflations from either $\dessin{0.65cm}{D21_2}$ or $\dessin{0.65cm}{D32_2}$. \end{remark}
We close this section by a technical result on $\ominus$-essential diagrams. \begin{lemma}\label{unpeuutilequandmeme} Let $n\ge 3$ be an integer. For any $\ominus$-essential diagram of degree $n$, smoothing a mixed chord always yields a $\ominus$-essential diagram of degree $(n-1)$. Conversely, any $\ominus$-essential diagram of degree $(n-1)$ can be obtained in this way. \end{lemma} \begin{proof} Let $D$ be a $\ominus$-essential diagram of degree $n$. Remark \ref{rem:seeds} above tells us that $D$ is obtained by iterated inflations from either $\dessin{0.65cm}{D21_2}$ or $\dessin{0.65cm}{D32_2}$. As noted in Remark \ref{rem:lissage}, smoothing a (mixed) chord that appeared in one of these inflations yields the diagram before inflation, which is a $\ominus$-essential one. So it only remains to observe that smoothing a mixed chord of $\dessin{0.65cm}{D32_2}$ always yields $\dessin{0.65cm}{D21_2}$. \end{proof} Note that the same result holds for $\oplus$-essential diagrams, but is not needed for this paper.
\subsection{Factorization results}\label{sec:facto}
We now give a collection of factorization results for invariants that are defined as sums of coefficients of chord diagrams in the Kontsevich integral, containing certain particular chord configurations.
\begin{proposition} \label{Factorisation fr} Let $\mathcal{D}$ be a set of chord diagrams such that $X = \sum_{D \in \mathcal{D}} C[D]$ is a link invariant. Suppose that, for some index $i$, none of the diagrams in $\mathcal{D}$ contains an internal chord on the $i$th circle. Let $\mathcal{D}_i$ be the collection of diagrams obtained from those in $\mathcal{D}$ by adding an internal chord on the $i$th circle, in all possible ways. Then, for any framed oriented link $L$, we have $$\sum_{D' \in \mathcal{D}_i} C_L[D'] = \frac{1}{2} fr_i \times X(L). $$ \end{proposition}
\begin{proof} Set $Y_i=\sum_{D' \in \mathcal{D}_i} C[D']$.
We first verify that $Y_i$ indeed is a link invariant, using the Invariance Lemma \ref{Critere Invariance}. We develop the argument below, although this straightforward (but somewhat lengthy) step will often be ommited in the rest of this paper. Consider a $4T$ relation $R'$. It suffices to consider the case where $R'$ involves at least one diagram from $\mathcal{D}_i$. There are two possibilities. \begin{enumerate}
\item The internal chord on the $i$th circle is not involved in $R'$. Since at least one diagram involved in $R'$ has an internal chord on the $i$th circle, this is actually the case for all of them.
The four diagrams involved in $R'$ can then be regarded as obtained, by adding an internal chord on the $i$th circle in some way, from diagrams $D_1$, $D_2$, $D_3$ and $D_4$
which satisfy a $4T$ relation $R$ of the form $D_1 - D_2 = D_3 - D_4$.
Since $X$ is a link invariant, it satisfies this relation: this proves that $Y_i$ satisfies $R'$.
Indeed a diagram involved in $4T$ is in $\mathcal{D}$ if and only if the corresponding diagram involved in $R'$ is in $\mathcal{D}_i$.
\item The internal chord on the $i$th circle is involved in $R'$.
We can then write $R'$ as $D'_1 - D'_2 = D'_3 - D'_4$, where $D'_1$ and $D'_2$ both contain an internal chord on the $i$th circle.
Hence $D'_1$ and $D'_2$ are in $\mathcal{D}_i$, and $Y_i$ vanishes on the left-hand term of relation $R'$.
For the remaining two diagrams, there are two cases.
If $D'_3$ also contains an internal chord on the $i$th circle, then so does $D'_4$, and both diagrams are in $\mathcal{D}_i$; otherwise, neither $D'_3$ nor $D'_4$ is in $\mathcal{D}_i$.
In any case $Y_i$ vanishes on the right-hand term of relation $R'$. \end{enumerate} Thus $Y_i$ is an invariant, and it remains to show the factorization formula.
Let $L_+ = K_1^+ \cup \ldots \cup K_n^+$ and $L_- = K_1^- \cup \ldots \cup K_n^-$ be the first two terms of a skein triple at an internal crossing of the $i$th component. Observe that $\hat{Z}(L_+)$ and $\hat{Z}(L_-)$ only differ by internal chords on the $i$th circle, so that $X(L_+)=X(L_-)$. By the Skein Lemma \ref{Obvious lemma}, we have\footnote{As in the Skein Lemma \ref{Obvious lemma}, we only show here the local contribution to $\hat{Z}(L)$ of the crossing involved in the skein triple; we will always implicitely do so in the rest of the paper. } \begin{eqnarray*} Y_i(L_+) - Y_i(L_-) & = & \sum_{D' \in \mathcal{D}_{i}} C[D'] \Big( \dessin{0.9cm}{X1} \Big) \\
& = & \sum_{D \in \mathcal{D}} C[D] \Big( \dessin{0.9cm}{X0} \Big) \\
& = & X(L_\pm) \end{eqnarray*} Here, the second equality follows directly from the definition of $\mathcal{D}_i$, while the third equality follows from the definition of $X$. But, since $X(L_+)=X(L_-)$ and $ fr(K_i^+)- fr(K_i^-) = 2$, we also have $\frac{1}{2} \times fr(K_i^+) \times X(L_+) - \frac{1}{2} \times fr(K_i^-) \times X(L_-)=X(L_\pm)$. Hence the two invariants in the statement satisfy the same skein formula. \\ Now, by successive internal crossing changes on the $i$th component, we can deform any link into a link $\tilde{L}$ whose $i$th component is isotopic to a copy of the unknot $U_0$, with no internal crossing, or a copy of $U_+$, with a single, isolated positive kink: it suffices to check that, in both cases, the formula of the statement holds. If the $i$th component of $\tilde{L}$ is a copy of $U_0$, then $\hat{Z}(\tilde{L})$ contains no diagram with an internal chord on the $i$th circle, hence $Y_i$ vanishes, and the formula holds. If the $i$th component of $\tilde{L}$ is a copy of $U_+$, then the isolated positive kink locally contributes to $\hat{Z}(\tilde{L})$ as recalled in (\ref{Z2}), and in particular gives on the $i$th circle:
$$ \dessin{0.6cm}{U0} + \frac{1}{2}\dessin{0.6cm}{U1} + \textrm{terms with $>1$ internal chords.} $$ By Remark \ref{rem:central}, there is only one diagram with isolated internal chord on the $i$th circle in $\mathcal{D}_i$, which shows that $Y_i(\tilde{L})$ equals $\frac{1}{2}X(\tilde{L})$ in this case, thus showing the desired formula. \end{proof}
An example of application of Theorem \ref{Factorisation fr} will be given in Lemma \ref{lem:D11}.
\begin{proposition} \label{Factorisation lk} Let $\mathcal{D}$ be a set of chord diagrams such that $X = \sum_{D \in \mathcal{D}} C[D]$ is a link invariant. Suppose that none of the diagrams in $\mathcal{D}$ contains a mixed chord between the $i$th and $j$th circles ($i\neq j$). Let $\mathcal{D}_{ij}$ be the collection of diagrams obtained from those in $\mathcal{D}$ by adding a chord between the $i$th and $j$th circles, in all possible ways. Then, for any framed oriented link $L$, we have $$\sum_{D \in \mathcal{D}_{ij}} C_L[D] = l_{i, j} \times X(L).$$ \end{proposition}
\begin{proof} Set $Y_{ij}=\sum_{D \in \mathcal{D}_{ij}} C[D]$. The fact that $Y_{ij}$ is a link invariant is shown by similar arguments as in the proof of Proposition \ref{Factorisation fr}, namely by considering a $4T$ relation and analysing the various cases, depending on whether the diagrams involved in this relation involve a mixed chord between $i$ and $j$ or not. \\ Now consider the first two terms $L_+$ and $L_-$ of a skein triple at a (mixed) crossing between the $i$th and $j$th components. Note that $\hat{Z}(L_+)$ and $\hat{Z}(L_-)$ only differ by terms containing chords between the $i$th and $j$th circles, so that $X(L_+)=X(L_-)$. It follows from the Skein Lemma \ref{Obvious lemma}, and the definitions of $\mathcal{D}_{ij}$ and $X$, that \begin{eqnarray*} Y_{ij}(L_+) - Y_{ij}(L_-) & = & \sum_{D' \in \mathcal{D}_{ij}} C[D'] \Big( \dessin{0.9cm}{X1} \Big) \\
& = & \sum_{D \in \mathcal{D}} C[D] \Big( \dessin{0.9cm}{X0} \Big) \\
& = & X(L_\pm) \end{eqnarray*} which indeed coincides with $l_{i,j}(L_+) \times X(L_+) - l_{i,j}(L_-) \times X(L_-)$ (since $l_{i,j}(L_+) - l_{i,j}(L_-) = 1$). It remains to observe that, by a sequence of crossing changes between the $i$th and $j$th components and isotopies, any link can be deformed into a link where the $i$th and $j$th components are geometrically split. The desired formula is easily checked for such links, and the result follows. \end{proof}
A simple application of Proposition \ref{Factorisation lk} is given in Lemma \ref{lem:D12}.
More generally, the following is a consequence of Theorem \ref{Factorisation lk}, which identifies the invariant underlying an infection. \begin{proposition} \label{Inflation et invariance} Let $\mathcal{D}$ be a set of chord diagrams such that $X = \sum_{D \in \mathcal{D}} C[D]$ is an $n$-component link invariant. Let $\mathcal{D}_I$ be the collection of diagrams obtained from those in $\mathcal{D}$ by all possible infections on the $i$th circle, for some $i\in\{1,\cdots,n\}$. Then, for any framed oriented $(n+1)$-component link $L$, we have $$\sum_{D' \in \mathcal{D}_I} C_L[D'] = l_{i, n+1}\times X(L_{\check{n+1}}).$$ \end{proposition} \begin{proof} Denote by $\mathcal{D}_\circ$ the collection of diagrams obtained from those in $\mathcal{D}$ by adding a copy of $\dessin{0.5cm}{D01}$ labeled by $(n+1)$. Then for any framed oriented $(n+1)$-component link $L$, we have by the Disjoint Union Lemma \ref{Factorisation} that $\sum_{D_\circ \in \mathcal{D}_\circ} C_L[D_\circ] = X(L_{\check{n+1}})$. It then suffices to apply Proposition \ref{Factorisation lk} to the $i$th and $(n+1)$th circles of all diagrams in $\mathcal{D}_\circ$. \end{proof}
Similarly, the next result identifies the invariant underlying an inflation. \begin{proposition} \label{Factorisation gonflage} Let $\mathcal{D}$ be a set of chord diagrams such that $X = \sum_{D \in \mathcal{D}} C[D]$ is an $n$-component link invariant. Let $\mathcal{D}_{G}$ be the collection of diagrams obtained from those in $\mathcal{D}$ by adding the following local diagram, called \emph{inflated chord}, in all possible ways\\[-0.3cm] \begin{center} \begin{tikzpicture}
\draw (0,0.5) node[left]{$i$};
\draw[-] [very thick] (0,0) -- (0,1);
\draw [densely dashed] (0,0.5) -- (1,0.5);
\draw [very thick] (1,0.5) arc(180:540:0.5);
\draw[<-] [very thick] (1.5,1) -- (1.51,1);
\draw (1.5,0) node[below]{$n+1$};
\draw [densely dashed] (2,0.5) -- (3,0.5);
\draw (3,0.5) node[right]{$j$};
\draw[-] [very thick] (3,0) -- (3,1); \end{tikzpicture} \end{center} Then, for any framed oriented $(n+1)$-component link $L$, we have \begin{eqnarray*} \sum_{D \in \mathcal{D}_{G}} C_L[D] = l_{i, n+1} \times l_{j, n+1} \times X(L_{\check{n+1}}), & \textrm{ if $i\neq j$,} \\ \sum_{D \in \mathcal{D}_{G}} C_L[D] = \frac{1}{2} l^2_{i, n+1} \times X(L_{\check{n+1}}), & \textrm{ if $i=j$.} \end{eqnarray*} \end{proposition}
\begin{proof} The case $i\neq j$ is a rather immediate consequence of the previous results. Indeed, in this case, the elements of $\mathcal{D}_{G}$ can be seen as obtained from those of $\mathcal{D}$ by first, all possible infections on the $i$th circle, then all possible ways of adding a mixed chord between the $j$th and $(n+1)$th circles (the latter one resulting from the infection). The result thus follows from Propositions \ref{Factorisation lk} and \ref{Inflation et invariance}.\\ We now prove the case $i=j$. The fact that $Y_{G}:=\sum_{D \in \mathcal{D}_{G}} C[D]$ indeed defines an invariant is done in a similar way as in the previous proofs, and is left as an exercise to the reader. We prove that $Y_{G}(L)$ coincides with $\frac{1}{2} l^2_{i,n+1} \times X(L_{\check{n+1}})$ by showing that both invariants have same variation formula under a crossing change between the $i$th and $(n+1)$th components: since these invariants both vanish on links where these two components are geometrically split, the result will follow.\\ Let $(L_+, L_-, L_0)$ be a skein triple at a crossing between the $i$th and $(n+1)$th components. On one hand, from the definitions of $\mathcal{D}_{G}$, we have
\begin{eqnarray*} Y_{G}(L_+) - Y_{G}(L_-) & = & \sum_{D' \in \mathcal{D}_{G}} C[D'] \Big( \dessin{0.9cm}{X1} \Big) \\
& = & \sum_{\widetilde{D} \in \mathcal{D}_I} C[\widetilde{D}] \Big( \dessin{0.9cm}{X0} \Big) \end{eqnarray*} where $\mathcal{D}_I$ denotes the set of all diagrams obtained from $\mathcal{D}$ by an infection on the $i$th circle, in all possible ways. The second equality thus holds by the fact that inserting an inflated chord on the $i$th circle is achieved by first, an infection on the $i$th circle, followed by the insertion of a mixed chord. Now, by definition of the Kontsevich integral at a negative crossing (\ref{Z2}), for any $\widetilde{D} \in \mathcal{D}_I$, we have
$$ C_{L-}[\widetilde{D}] = C[\widetilde{D}] \Big( \dessin{0.9cm}{X0} \Big) -\frac{1}{2} C[\widetilde{D}] \Big( \dessin{0.9cm}{X1} \Big), $$ where the local picture still involves the $i$th and $(n+1)$th circle. Hence by subsitution we obtain
\begin{eqnarray*} Y_{G}(L_+) - Y_{G}(L_-) & = & \sum_{\widetilde{D} \in \mathcal{D}_I} C_{L-}[\widetilde{D}] + \frac{1}{2} \sum_{\widetilde{D} \in \mathcal{D}_I} C[\widetilde{D}] \Big( \dessin{0.9cm}{X1} \Big) \\
& = & l_{i,n+1}(L_-)\times X((L_\pm)_{\check{n+1}}) + \frac{1}{2}X((L_\pm)_{\check{n+1}}). \end{eqnarray*} Here, the last equality uses the definition of $X$ and Proposition \ref{Factorisation lk}, and the fact that $(L_+)_{\check{n+1}}=(L_-)_{\check{n+1}}$. On the other hand, using simply the fact that $ l_{i,n+1}(L_+) = l_{i,n+1}(L_-)+1$, we have $$ \frac{1}{2} l^2_{i,n+1}(L_+) \times X((L_+)_{\check{n+1}}) - \frac{1}{2} l^2_{i,n+1}(L_-) \times X((L_-)_{\check{n+1}}) $$ $$ = \frac{1}{2} X((L_\pm)_{\check{n+1}}) \left( l^2_{i,n+1}(L_+) - l^2_{i,n+1}(L_-) \right) $$ $$ = \frac{1}{2} X((L_\pm)_{\check{n+1}}) \left( 2l_{i,n+1}(L_-) + 1 \right), $$ which shows that the two invariants in the statement satisfy the same skein formula. This concludes the proof. \end{proof}
\subsection{Some results in low degree}
In this section, we identify, in low degrees, some combinations of coefficients of the Kontsevich integral in terms of classical invariants. We begin with a few simple applications of our factorization results, most of which are well-known to the experts.
The following is an elementary application of Theorem \ref{Factorisation fr} and the obvious formula $C_K\Big[ \dessin{0.5cm}{D01}\Big]=1$. \begin{lemma}\label{lem:D11} Let $K$ be a framed oriented knot. We have $$ C_K\Big[ \dessin{0.5cm}{D11}\Big] = \frac{1}{2} fr(K).$$ \end{lemma}
The following can be seen as a consequence of either Theorem \ref{Factorisation lk} or \ref{Inflation et invariance}. \begin{lemma}\label{lem:D12} Let $L$ be a framed oriented $2$-component link. We have $$ C_L\Big[ \dessin{0.5cm}{D12}\Big] = l_{1,2}.$$ \end{lemma}
We next give two simple examples of applications of Proposition \ref{Factorisation gonflage}. The first example uses the case $i=j$ of the proposition. \begin{lemma}\label{lem:D22} Let $L$ be a framed oriented $2$-component link. We have $$ C_L\Big[ \dessin{0.5cm}{D22}\Big] =\frac{1}{2} l^2_{1,2}.$$ \end{lemma} \noindent The second example uses the case $i\neq j$ of Proposition \ref{Factorisation gonflage}, combined with Lemma \ref{lem:D22} above. \begin{lemma}\label{Invariant particulier} Let $L$ be a framed oriented $3$-component link. Then for $\{i,j,k\}=\{1,2,3\}$ we have $$ C_L\Big[ \dessin{0.9cm}{D43_1}\Big] + C_L\Big[ \dessin{0.9cm}{D43_2}\Big] = \frac{1}{2} \times l_{i, j} \times l_{i, k} \times l_{j, k}^2. $$ \end{lemma} \begin{remark} Direct proofs of the above four lemmas, which do not make use of general factorization results, can be found in \cite{Casejuane}. \end{remark}
The next two results involve the coefficients $c_2$ and $c_3$ of the Conway polynomial. \begin{proposition} \label{c_2} Let $K$ be a framed oriented knot. We have $$C_K\Big[ \dessin{0.5cm}{D21_1}\Big] = \frac{1}{8} fr(K)^2 + \frac{1}{24} - c_2(K),$$ $$C_K\Big[ \dessin{0.5cm}{D21_2}\Big] = c_2(K) - \frac{1}{24}.$$ \end{proposition}
\begin{proof} The fact that $C\Big[ \dessin{0.5cm}{D21_1}\Big]$ and $C\Big[ \dessin{0.5cm}{D21_2}\Big]$ define knot invariants follows from the Invariance Lemma \ref{Critere Invariance}, noting that the only $4T$ relation involving either of these two diagrams is a trivial one. \\ Let us prove the first statement. Let $K_+$, $K_-$ and $L_0 = K_1 \cup K_2$ be a skein triple at a knot crossing $c$. On one hand, by the Skein Lemma \ref{Obvious lemma}, \begin{eqnarray*} C_{K_+}\Big[ \dessin{0.5cm}{D21_1}\Big] - C_{K_-}\Big[ \dessin{0.5cm}{D21_1}\Big]
& = & C\Big[ \dessin{0.5cm}{D21_1}\Big] \Big(\dessin{0.9cm}{X1} \Big) \\
& = & C\Big[ \dessin{0.5cm}{D11} \, \dessin{0.5cm}{D01}\Big] \Big(\dessin{0.9cm}{X} \Big) +
C\Big[ \dessin{0.5cm}{D01} \, \dessin{0.5cm}{D11}\Big] \Big(\dessin{0.9cm}{X} \Big)\\
& = & \frac{1}{2} fr(K_1) + \frac{1}{2} fr(K_2) \\
& = & \frac{1}{2} \Big(fr(K_-) + 1 - 2 c_1(L_0)\Big) \end{eqnarray*} Here, the second equality is given by smoothing the chord contributed by $c$. As illustrated by Figure \ref{fig:lissage}, this smoothing maps $\dessin{0.65cm}{D21_1}$ to either $\dessin{0.65cm}{D11} \, \dessin{0.65cm}{D01}$ or $\dessin{0.65cm}{D01} \, \dessin{0.65cm}{D11}$; conversely, the latter two diagrams can only be obtained, by smoothing an internal chord, from $\dessin{0.65cm}{D21_1}$. The third equality then follows from Lemma \ref{lem:D11}, while the last equality is easily verified. On the other hand, the difference $(\frac{1}{8} fr(K_+)^2 + \frac{1}{24} - c_2(K_+)) - (\frac{1}{8} fr(K_-)^2 + \frac{1}{24} - c_2(K_-))$ can be written, using the skein relation for the Conway coefficients, as $$ \frac{1}{8}(\underbrace{fr(K_+) - fr(K_-)}_{=2})(\underbrace{fr(K_+) + fr(K_-)}_{=2fr(K_-) + 2}) - c_1(L_0). $$ This shows that both invariants in the first statement have same variation formula under a (knot) crossing change, and it only remains to check that they coincide on both $U_0$ and $U_+$. Using the formulas for $\hat{Z}(U_0)$ and $\hat{Z}(U_+)$ recalled in Section \ref{sec:Kontsevich}, we have $$C_{U_0}\Big[ \dessin{0.5cm}{D21_1} \Big] = \frac{1}{48} + \frac{1}{48} = \frac{1}{24} = \frac{1}{8} fr(U_0)^2) + \frac{1}{24} - c_2(U_0)$$ and $$C_{U_+}\Big[ \dessin{0.5cm}{D21_1} \Big] = \frac{1}{48} + \frac{1}{48} + \frac{1}{8} = \frac{1}{6} = \frac{1}{8} fr(U_+)^2 + \frac{1}{24} - c_2(U_+),$$ which concludes the proof of the first statement. \\ The proof of the second statement uses the same skein triple and is very similar. The same argument, only using Lemma \ref{lem:D12} instead of Lemma \ref{lem:D11}, gives \begin{eqnarray*} C_{K_+}\Big[ \dessin{0.5cm}{D21_2}\Big] - C_{K_-}\Big[ \dessin{0.5cm}{D21_2}\Big]
& = & C\Big[ \dessin{0.5cm}{D21_2}\Big] \Big(\dessin{0.9cm}{X1} \Big) \\
& = & C\Big[ \dessin{0.5cm}{D12}\Big] \Big(\dessin{0.9cm}{X} \Big) \\
& = & C_{L_0} \Big[ \dessin{0.5cm}{D12}\Big] \\
& = & c_1(L_0), \end{eqnarray*} which clearly coincides with the variation formula for $c_2 - \frac{1}{24}$. The statement then follows from the equalities $C_{U_+}\Big[ \dessin{0.5cm}{D21_2} \Big] = C_{U_0}\Big[ \dessin{0.5cm}{D21_2} \Big] = - \frac{1}{24}$. \end{proof}
\begin{remark}\label{rem:Oka}
The second statement of Proposition \ref{c_2} can be found in \cite[Prop.~4.4]{Okamoto1997}.
Our next result actually fixes a mistake in \cite[Prop.~4.6~(1)]{Okamoto1997}; likewise, \cite[Prop.~4.6~(2)]{Okamoto1997} is corrected in Theorem \ref{c_n}. \end{remark}
\begin{proposition} \label{c_3} Let $L$ be a framed oriented $2$-component link. Then $c_3(L)$ is given by the formula $$C_L\Big[ \dessin{0.5cm}{D32_2} \Big]+ C_L\Big[ \dessin{0.5cm}{D32_4} \Big] + C_L\Big[ \dessin{0.5cm}{D32_4b} \Big] + C_L\Big[ \dessin{0.5cm}{D32_7} \Big]+ C_L\Big[ \dessin{0.5cm}{D32_7b} \Big]. $$ \end{proposition} \begin{proof} Set $$X_3:=C\Big[ \dessin{0.5cm}{D32_2} \Big]+ C\Big[ \dessin{0.5cm}{D32_4} \Big] + C\Big[ \dessin{0.5cm}{D32_4b} \Big] + C\Big[ \dessin{0.5cm}{D32_7} \Big]+ C\Big[ \dessin{0.5cm}{D32_7b} \Big]. $$ The only non-trivial $4T$ relations involving one of the diagrams above are the following $$ \dessin{0.65cm}{D32_1} - \dessin{0.65cm}{D32_2} = \dessin{0.65cm}{D32_3} - \dessin{0.65cm}{D32_4} = \dessin{0.65cm}{D32_3b} - \dessin{0.65cm}{D32_4b}, $$ and the Invariance Lemma \ref{Critere Invariance} can then be used to show that $X_3$ is a link invariant. \\ Now, let $L_+ = K_1^+ \cup K_2^+$, $L_- = K_1^- \cup K_2^-$ and $L_0$ be a skein triple at a mixed crossing. By the Skein Lemma \ref{Obvious lemma} we have \begin{eqnarray*}
C_{L_+}\Big[ \dessin{0.5cm}{D32_2} \Big] - C_{L_-}\Big[ \dessin{0.5cm}{D32_2} \Big]
& = & C\Big[ \dessin{0.5cm}{D32_2} \Big] \Big(\dessin{0.9cm}{X1} + \frac{1}{24}\dessin{0.9cm}{X3}\Big) \\
& = & C\Big[ \dessin{0.5cm}{D32_2} \Big] \Big(\dessin{0.9cm}{X1}\Big) + \frac{1}{24}, \end{eqnarray*} where the second equality follows from the observation that the local configuration $\dessin{0.9cm}{X3}$ on two circles yields the diagram $\dessin{0.65cm}{D32_2}$. Since the other diagrams defining $X_3$ have $\le 2$ mixed chords, we thus have by the Skein Lemma \ref{Obvious lemma} that $X_3(L_+) - X_3(L_-)$ is given by $$\left(C\Big[ \dessin{0.45cm}{D32_2} \Big] \!+\! C\Big[ \dessin{0.45cm}{D32_4} \Big] \!+\! C\Big[ \dessin{0.45cm}{D32_4b} \Big] \!+\! C\Big[ \dessin{0.45cm}{D32_7} \Big] \!+\! C\Big[ \dessin{0.45cm}{D32_7b} \Big]\right) \Big(\dessin{0.9cm}{X1}\Big) + \frac{1}{24}. $$ But each of the above diagrams has the property that, smoothing a mixed chord always yields $\dessin{0.65cm}{D21_2}$ and, conversely, the latter can only be obtain by such a smoothing from one of the above five diagrams. This shows that $$ X_3(L_+) - X_3(L_-) = C\Big[ \dessin{0.5cm}{D21_2} \Big]\Big(\dessin{0.9cm}{X}\Big) + \frac{1}{24}. $$ Proposition \ref{c_2}, and the skein relation for $c_3$, then give $$X_3(L_+) - X_3(L_-) = c_2(L_0) = c_3(L_+) - c_3(L_-). $$ The result follows since both $X_3$ and $c_3$ vanish on split links. \end{proof} \begin{remark}\label{rem:inv_split} Using the non-trivial $4T$ relations given at the beginning of this proof, and the Invariance Lemma \ref{Critere Invariance}, we actually have that $C\Big[ \dessin{0.5cm}{D32_2} \Big]+ C\Big[ \dessin{0.5cm}{D32_4} \Big] + C\Big[ \dessin{0.5cm}{D32_4b} \Big]$ and $C\Big[ \dessin{0.5cm}{D32_7} \Big]+ C\Big[ \dessin{0.5cm}{D32_7b} \Big]$ are themselves link invariants. In fact, the latter is easily identified using Propositions \ref{c_2} and \ref{Inflation et invariance}: we have $$ C_{K_1\cup K_2}\Big[ \dessin{0.5cm}{D32_7} \Big] +C_{K_1\cup K_2}\Big[ \dessin{0.5cm}{D32_7b} \Big] = l_{1,2}\Big( c_2(K_1)+ c_2(K_2)-\frac{1}{12}\Big). $$ \noindent Observe that this formula coincides with $c_3(K_1\cup K_2) - U_3(K_1\cup K_2)$, a fact that will be widely generalized in Section \ref{sec:conwayKontsevich}. \end{remark}
The next result will also be needed later. We omit the proof, since one can be found in \cite[Prop.~4.1~(3)]{Okamoto1997}; see also \cite{Casejuane} for a proof using the techniques of the present paper.
\begin{proposition}\label{prop:lk3} Let $L$ be a framed oriented $2$-component link. We have
$$ C_{L}\Big[ \dessin{0.5cm}{D32_1} \Big] + C_{L}\Big[ \dessin{0.5cm}{D32_2} \Big] = \frac{1}{6}l_{1,2}^3. $$ \end{proposition}
More generally, the techniques used in this section can be used to identify, in degree $\le 3$, \emph{all} invariants arising as coefficients of the Kontsevich integral. All remaining formulas are direct applications of our factorization results. For example, the following is given by Lemma \ref{lem:D11} and Proposition \ref{Factorisation gonflage}: \begin{equation}\label{rem:deg3}
C_{K_1\cup K_2}\Big[ \dessin{0.5cm}{D32_3} \Big] + C_{K_1\cup K_2}\Big[ \dessin{0.5cm}{D32_4} \Big] = \frac{1}{4} fr_1 \times l^2_{1,2}. \end{equation} A complete list of these statements, and their detailed proof, can be found in \cite{Casejuane}.
\subsection{Conway polynomial and the Kontsevich integral}\label{sec:conwayKontsevich}
We now generalize Proposition \ref{c_3}, by explicitly identifying all Conway coefficients in terms of the Kontsevich integral.
Recall from Section \ref{sec:essentiels} that $\mathcal{E}^-(n)$ denotes the set of $\ominus$-essential diagrams on $n$ circles, which are all of degree $n+1$.
For an integer $n\ge 1$, denote by $\mathcal{P}(n+1)$ the set of all diagrams obtained by an infection on an element of $\mathcal{E}^-(n)$. \begin{ex}\label{ex3} Since $\mathcal{E}^-(1)=\{\dessin{0.5cm}{D21_2}\}$, we have $\mathcal{P}(2)=\{\dessin{0.5cm}{D32_7};\dessin{0.5cm}{D32_7b}\}$. \end{ex}
\begin{theorem} \label{c_n} Let $n\ge 2$. For any framed oriented $n$-component link $L$, we have
$$\sum_{D \in \mathcal{E}^-(n) \cup \mathcal{P}(n)} C_L[D] = c_{n+1}(L).$$ \end{theorem} \begin{remark}\label{rem:conway} The case $n=1$ is somewhat particular, as it involves a correction term. Indeed, first note that $\mathcal{P}(1)=\emptyset$. We have $\mathcal{E}^-(1)=\{\dessin{0.5cm}{D21_2}\}$, and Proposition \ref{c_2} tells us that for a knot $K$, $$C_K\Big[\dessin{0.5cm}{D21_2}\Big] = c_2(K) - \frac{1}{24}.$$ Observe also that the case $n = 2$ is given by Proposition \ref{c_3}, since $\mathcal{E}^-(2)=\{\dessin{0.5cm}{D32_2};\dessin{0.5cm}{D32_4};\dessin{0.5cm}{D32_4b}\}$ and $\mathcal{P}(2)=\{\dessin{0.5cm}{D32_7};\dessin{0.5cm}{D32_7b}\}$. \end{remark}
\begin{proof}[Proof of Theorem \ref{c_n}] Denote by $X_n$ the left-hand term in the statement, which decomposes as $$ X_n = \sum_{D \in \mathcal{E}^-(n)} C_L[D] + \sum_{D \in \mathcal{P}(n)} C_L[D].$$ We first prove that $X_n$ indeed is an invariant. Actually, we show that each of the above two sums defines an invariant, by induction on $n$. The case $n=2$ is obtained by combining Remarks \ref{rem:inv_split} and \ref{rem:conway}. Proposition \ref{Inflations et Essentiels} ensures that any element of $\mathcal{E}^-(n+1)$ is obtained by inflation on an element of $\mathcal{E}^-(n)$, up to permutation of the circle labels. A straightforward argument, using the Invariance Lemma \ref{Critere Invariance}, then shows that $\sum_{D \in \mathcal{E}^-(n+1)} C_L[D]$ is an invariant.\footnote{The argument is in the same spirit as in the proof of Lemma \ref{Factorisation fr}, and discusses the possible types of $4T$ relations depending on whether they involve chord(s) created during the inflation; we leave the details as an exercices to the reader, but a proof can be found in \cite{Casejuane}.} On the other hand, Proposition \ref{Inflation et invariance} ensures that $\sum_{D \in \mathcal{P}(n+1)} C_L[D]$ is also an invariant.\\ Let us now prove the desired equality. This is again done by induction on $n$, using Proposition \ref{c_3} as initial step. Suppose that the equality holds for some $n\ge 2$, and consider a skein triple $(L_+,L_-,L_0)$ at a mixed crossing. Note that, for $n\ge 3$, there is no essential diagram with $\ge 3$ mixed chords between two given circles, by Proposition \ref{Inflations et Essentiels}. Hence by the Skein Lemma \ref{Obvious lemma}, we have \begin{eqnarray*} X_{n+1}(L_+) - X_{n+1}(L_-)
& = & \sum\limits_{D \in\mathcal{E}^-(n+1) \cup \mathcal{P}(n+1)} C[D] \Big(\dessin{0.9cm}{X1}\Big). \end{eqnarray*} By smoothing the mixed chord in the above equality, we obtain \begin{eqnarray*} X_{n+1}(L_+) - X_{n+1}(L_-)
& = & \sum\limits_{D \in\mathcal{E}^-(n) \cup \mathcal{P}(n)} C[D] \Big(\dessin{0.9cm}{X}\Big) \\
& = & c_n(L_0) \\
& = & c_{n+1}(L_+) - c_{n+1}(L_-). \end{eqnarray*} Here, the fact that sum runs over $D \in \mathcal{E}^-(n) \cup \mathcal{P}(n)$ is ensured by Lemma \ref{unpeuutilequandmeme} and Remark \ref{rem:lissage}. The second equality is then given by the induction hypothesis, while the third equality is simply the skein relation for Conway coefficients. This proves that the invariants $X_{n+1}$ and $c_{n+1}$ have same variation formula under a mixed crossing change. The equality then follows from the fact that both invariants vanish on geometrically split links. \end{proof}
As mentioned in Remark \ref{rem:Oka}, Theorem \ref{c_n} fixes a mistake in \cite[Prop.~4.6~(2)]{Okamoto1997}. More precisely, \cite[Prop.~4.6~(2)]{Okamoto1997} treats the case $n=3$, but only considers the sum of coefficients given by $\mathcal{E}^-(n)$, and omits the correction terms given by $\mathcal{P}(n)$. Actually, considering only the terms given by $\mathcal{E}^-(n)$ yields the invariant $U_n$ introduced in Definition \ref{def:U_n} in terms of the Conway polynomial and linking numbers: \begin{proposition} \label{U_n} Let $n\ge 2$. For any framed oriented $n$-component link $L$, we have $$\sum_{D \in \mathcal{E}^-(n)} C_L[D] = U_{n+1}(L).$$ \end{proposition} \begin{proof} The fact that $\sum_{D \in \mathcal{E}^-(n)} C_L[D]$ defines an invariant for all $n$ was already discussed in the previous proof. The equality is proved by induction. The case $n=2$ is given by Proposition \ref{c_2}, and by Theorem \ref{c_n} we have that \begin{eqnarray*} \sum_{D \in \mathcal{E}^-(n)} C[D]
& = & c_{n+1} - \sum_{D \in \mathcal{P}(n)} C[D] \\
& = & c_{n+1} - \sum_{i=1}^n \sum_{D \in \mathcal{P}_i(n)} C[D], \\ \end{eqnarray*} where $\mathcal{P}_i(n)$ denote all elements of $\mathcal{P}(n)$ where the unique circle with a single leg (coming from an infection on some diagram of $\mathcal{E}^-(n-1)$) is labeled by $i$. Then Proposition \ref{Inflation et invariance} and the induction hypothesis give that $\sum_{D \in \mathcal{P}_i(n)} C[D] = \sum_{i\neq j} U_{n} l_{i,j}$, which concludes the proof. \end{proof}
\section{The $\mu_n$ invariants}\label{sec:mu_n}
In the rest of this paper, we will make use of the following. \begin{notation} For a chord diagram $D$ on $n$ circles, we denote by $\iota_\Theta(D)$ the rational coefficient such that
$$ \left(\iota_1(D \# \nu^n)\right)_1=\iota_\Theta(D)\times \dessin{0.5cm}{T}. $$ For a framed oriented $n$-component link $L$, we denote by
$$ \mathcal{C}_L[D] := C_L[D] \times \iota_\Theta(D). $$ In other words, $\mathcal{C}_L[D]$ is the contribution to $\Big( \iota_1(\check{Z}(L)) \Big)_1$ of a diagram $D$ in the Kontsevich integral $\hat{Z}(L)$, and $\iota_\Theta(D)$ is the part of this contribution that comes from the $\iota_1$ map (and the normalization by copies of $\nu$'s) on this diagram, while $C_L[D]$ is the part that comes from the Kontsevich integral itself. \end{notation}
\begin{ex} \label{footnote} It follows from the definitions of $\check{Z}$ and $\iota_1$ that $\iota_\Theta \Big(\dessin{0.5cm}{D21_1}\Big) = \frac{1}{6}$, $\iota_\Theta \Big(\dessin{0.5cm}{D21_2}\Big) = -\frac{1}{3}$ and $\iota_\Theta \Big(\dessin{0.5cm}{D01}\Big) = \frac{1}{48}$. In this latter computation, the coefficient of $\dessin{0.5cm}{T}$ comes from the copy of $\nu$ added to the circle component. We stress that this type of contribution to the degree $1$ part of the LMO invariants, coming from the renormalization $\check{Z}$ of the Kontsevich integral, only occurs with the trivial diagram $\dessin{0.5cm}{D01}$ on a single circle. In other words, for \emph{any} chord diagram $D\neq \dessin{0.5cm}{D01}$, we have that $\left(\iota_1(D)\right)_1=\iota_\Theta(D)\times \dessin{0.5cm}{T}$. \end{ex}
The following is the main ingredient in our surgery formula for the Casson-Walker-Lescop invariant. Recall that $\mathcal{E}(n)$ denotes the set of all essential diagrams on $n$ circles. \begin{definition}\label{def:mun} For all integers $n\ge 1$, let $\mu_n$ be the framed oriented $n$-component link invariant defined by
$$ \mu_1(K) = 2 \sum_{D\in \{\dessin{0.3cm}{D01}\}\cup \mathcal{E}(1)} \mathcal{C}_L[D] $$ and for all $n\ge 2$,
$$ \mu_n(L) = 2 \sum_{D\in \mathcal{E}(n)} \mathcal{C}_L[D]. $$ \end{definition}
Our task is now to make this definition completely explicit.
\begin{remark} The fact that the above formula indeed defines a link invariant is not completely obvious (in particular, this is not a mere application of the Invariance Lemma \ref{Critere Invariance}); we postpone the justification to Remark \ref{rem:jesuisinvariant} at the end of this section. \end{remark}
\subsection{Cases $n=1$ and $2$}
We listed in Example \ref{ex:essentiels} all essential diagrams on $1$ or $2$ circles, and we can thus describe $\mu_1$ and $\mu_2$ explicitly.
\begin{lemma}\label{ex:mu1} For a framed oriented knot $K$, we have
$$ \mu_1(K) = \frac{1}{24} fr(K)^2 - c_2(K) + \frac{1}{12}.$$ \end{lemma}
\begin{proof} We know that $\mathcal{E}^+(1) = \{ \dessin{0.5cm}{D21_1} \}$ and $\mathcal{E}^-(1) = \{ \dessin{0.5cm}{D21_2} \}$, and moreover we saw in Example \ref{footnote} that $\iota_\Theta \Big(\dessin{0.5cm}{D21_1}\Big) = \frac{1}{6}$, $\iota_\Theta \Big(\dessin{0.5cm}{D21_2}\Big) = -\frac{1}{3}$ and $\iota_\Theta \Big(\dessin{0.5cm}{D01}\Big) = \frac{1}{48}$. Hence by Proposition \ref{c_2}, the invariant $\mu_1$ for a framed knot $K$ is given by \begin{eqnarray*}
\mu_1(K) & = & 2\Big( \frac{1}{48} C_K\Big[\dessin{0.5cm}{D01}\Big] + \frac{1}{6} C_K\Big[\dessin{0.5cm}{D21_1}\Big] - \frac{1}{3} C_K\Big[\dessin{0.5cm}{D21_2}\Big]\Big) \\
& = & \frac{1}{24} + \frac{1}{3} \left(C_K\Big[\dessin{0.5cm}{D21_1}\Big]+C_K\Big[\dessin{0.5cm}{D21_2}\Big]\right)
- C_K\Big[\dessin{0.5cm}{D21_2}\Big] \\
& = & \frac{1}{24} + \frac{1}{24} fr(K)^2 - U_2(K), \end{eqnarray*} where the last equality uses Propositions \ref{c_2} and \ref{U_n}. The result then follows from the definition of $U_2$ (Definition \ref{def:U_n}). \end{proof}
\begin{lemma}\label{ex:mu2} For a framed oriented $2$-component link $L=K_1\cup K_2$, we have
$$ \mu_2(L)=\frac{1}{12} l_{1,2}^3 + \frac{f_1 + f_2}{12} l_{1,2}^2 - c_3(L) + l_{1,2}\Big( c_2(K_1) + c_2(K_2) - \frac{1}{12} \Big). $$ \end{lemma}
\begin{proof} We have, up to permutation of the circle labels, $$\mathcal{E}^+(2) = \{ \dessin{0.5cm}{D32_1} \, ; \, \dessin{0.5cm}{D32_3} \}\, \textrm{ and } \,\mathcal{E}^-(2) = \{ \dessin{0.5cm}{D32_2}\,;\,\dessin{0.5cm}{D32_4} \}.$$ Moreover, we have that $\iota_\Theta \Big(\dessin{0.5cm}{D32_1}\Big) = \frac{1}{4}$, $\iota_\Theta \Big(\dessin{0.5cm}{D32_3}\Big) = \frac{1}{6}$, $\iota_\Theta \Big(\dessin{0.5cm}{D32_2}\Big) = -\frac{1}{4}$ and $\iota_\Theta \Big(\dessin{0.5cm}{D32_4}\Big) = -\frac{1}{3}$. Thus, for a $2$-component link $L=K_1\cup K_2$, we have \begin{eqnarray*}
\mu_2(L) & = & 2\Big( \frac{1}{4} C_L\Big[\dessin{0.5cm}{D32_1}\Big] - \frac{1}{4} C_L\Big[\dessin{0.5cm}{D32_2}\Big] +\frac{1}{6} C_L\Big[\dessin{0.5cm}{D32_3}\Big] - \frac{1}{3} C_L\Big[\dessin{0.5cm}{D32_4}\Big]\Big) \\
& = & \frac{1}{2} \left(C_L\Big[\dessin{0.5cm}{D32_1}\Big] +C_L\Big[\dessin{0.5cm}{D32_2}\Big]\right) + \frac{1}{3} \left(C_L\Big[\dessin{0.5cm}{D32_3}\Big]+C_L\Big[\dessin{0.5cm}{D32_4}\Big]\right) \\
& & - \left( C_L\Big[\dessin{0.5cm}{D32_2}\Big] + C_L\Big[\dessin{0.5cm}{D32_4}\Big]\right) \\
& = & \frac{1}{12} l_{1,2}^3 + \frac{f_1 + f_2}{12} l_{1,2}^2 - U_3(L). \end{eqnarray*} Here, the final equality uses Proposition \ref{prop:lk3}, the formula given in (\ref{rem:deg3}), and Proposition \ref{U_n}. Hence from Definition \ref{def:U_n} we obtain the desired formula. \end{proof}
\subsection{General case}\label{ex:muk}
Let us now investigate the invariant $\mu_n$ for $n\ge 3$.
As pointed out in Remark \ref{rem:seeds}, all essential diagrams are obtained by iterated inflations from a few basic diagrams, namely either $\dessin{0.65cm}{D21_1}$ and $\dessin{0.65cm}{D32_1}$ for $\oplus$-essential diagram, and $\dessin{0.65cm}{D21_2}$ and $\dessin{0.65cm}{D32_2}$ for $\ominus$-essential ones. Note that, in each of these four diagrams, the role of all chords is completely symmetric. We can thus define four families of \emph{unordered} chord diagrams,\footnote{A chord diagram is unordered if we do not specify an order on the circle components. } $D_+(a,b)$, $ D_-(a,b)$, $D_+(a,b,c)$ and $D_-(a,b,c)$, $a,b,c\in \mathbb{N}$, as follows. \begin{definition}\label{def:D} For integers $a,b,c$, such that $a\ge b\ge c\ge 0$,
$\bullet$ $D_+(a,b)$ is the unordered $\oplus$-essential diagram on $a+b+1$ circles obtained from $\dessin{0.65cm}{D21_1}$ by $a$ successive inflations on one chord, and $b$ inflations on the other chord,
$\bullet$ $D_-(a,b)$ is the unordered $\ominus$-essential diagram on $a+b+1$ circles obtained in the same way from $\dessin{0.65cm}{D21_2}$,
$\bullet$ $D_+(a,b,c)$ is the unordered $\oplus$-essential diagram on $a+b+c+2$ circles obtained from $\dessin{0.65cm}{D32_1}$ by respectively $a$, $b$ and $c$ successive inflations on each chord,
$\bullet$ $D_-(a,b,c)$ is the unordered $\ominus$-essential diagram on $a+b+c+2$ obtained in the same way from $\dessin{0.65cm}{D32_2}$. \end{definition} Some examples are given below: \begin{center}
\includegraphics[scale=1.2]{ex.pdf} \end{center}
We denote respectively by $\{D_+(a,b)\}$ and $\{D_-(a,b)\}$ the set of all chord diagrams obtained by ordering the circles of $D_+(a,b)$ and $D_-(a,b)$ from $1$ to $a+b+1$ in all possible ways. We also define $\{D_+(a,b,c)\}$ and $\{D_-(a,b,c)\}$ in a similar way. Remark \ref{rem:seeds} can then be rephrased as the equality \begin{equation}\label{eq:split} \mathcal{E}^\pm(n+2)=\bigcup_{\substack{a+b=n+1 \\ a\ge b}} \{D_\pm(a,b)\} \cup \bigcup_{\substack{a+b+c=n \\ a\ge b\ge c}} \{D_\pm(a,b,c)\}. \end{equation} We also set $$ \mathcal{D}(a,b) = \{D_+(a,b)\} \cup \{D_-(a,b)\} \quad\textrm{and}Ê\quad \mathcal{D}(a,b,c) = \{D_+(a,b,c)\} \cup \{D_-(a,b,c)\}. $$
We can identify explicitly the coefficients of such diagrams in the Kontsevich integral. This uses the following notation. \begin{notation} Given two integers $i$ and $j$, and a set $I=\{i_1,\cdots,i_k\}$ of $k$ pairwise distinct integers, all different from $i$ and $j$. We set
$$ \mathcal{L}_{i,j,I} := \sum_{\sigma\in S_k} l_{i,i_{\sigma(1)}}\times l_{i_{\sigma(1)},i_{\sigma(2)}}\times \cdots \times l_{i_{\sigma(k-1)},i_{\sigma(k)}}\times l_{i_{\sigma(k)},j}.$$ We abbreviate $\mathcal{L}_{i,I}=\mathcal{L}_{i,i,I}$, and use the convention $\mathcal{L}_{i,j,\emptyset}=l_{i,j}$ if $i\neq j$, and $\mathcal{L}_{i,\emptyset}=fr_i$. \end{notation}
\begin{theorem}\label{thm:coeffs essentiels} For all $a,b\in \mathbb{N}$ such that $a>0$ and $a\ge b$, $$
\sum_{D\in \mathcal{D}(a,b)} C_L[D] = \frac{1}{4} \sum_{i=1}^{n} \sum_{\mathcal{I}_i(a,b)} \mathcal{L}_{i,I} \mathcal{L}_{i,J} $$ where we sum over the set $\mathcal{I}_i(a,b)$ of all partitions $I\cup J=\{1,\cdots,a+b+1\}\setminus \{i\}$ such that $\vert I\vert=a$ and $\vert J\vert = b$.
For all $a,b,c\in \mathbb{N}$ such that $a>0$ and $a\ge b\ge c$, $$ { \everymath={\displaystyle}
\sum_{D\in \mathcal{D}(a,b,c)} C_L[D] = \left\{ \begin{array}{ll} \frac{1}{2} \sum_{1\le i< j\le n} l_{i,j}^2 \mathcal{L}_{i,j,\{1,\cdots,a+2\}\setminus \{i,j\}} & \textrm{ if $b=c=0$, }\\[0.2cm] \sum_{1\le i< j\le n} \sum_{\mathcal{I}_{i,j}(a,b,c)} \mathcal{L}_{i,j,I} \mathcal{L}_{i,j,J} \mathcal{L}_{i,j,K} & \textrm{ otherwise,} \end{array} \right. } $$ where the last sum is over the set $\mathcal{I}_{i,j}(a,b,c)$ of all partitions $I\cup J\cup K=\{1,\cdots,a+b+c+2\}\setminus \{i,j\}$ such that $\vert I\vert=a$, $\vert J\vert = b$ and $\vert K\vert =c$. \end{theorem} \begin{proof} In this proof, we call \emph{order $k$ chain} ($k\ge 0$) the result of $k$ successive inflations on a chord; in particular, an order $1$ chain is an inflated chord, in the sense of Proposition \ref{Factorisation gonflage}. \\ Let us focus on the first half of the statement, involving the diagrams $D_\pm(a,b)$. We first consider the case $a>0$ and $b=0$. The diagrams $D_+(a,0)$ and $D_-(a,0)$ are obtained by inserting, in all possible ways, an order $a$ chain to $\dessin{0.65cm}{D11}$. Such an insertion is achieved by, first, an infection, followed by $a-1$ iterated infections on the newly created circle, and finaly, the insertion of a chord between the newest and the initial circles. These operations endow $D_\pm(a,0)$ with a canonical ordering, for which Propositions \ref{Inflation et invariance} and \ref{Factorisation lk} if $a>1$ (resp. Proposition \ref{Factorisation gonflage} if $a=1$) ensure that $ C_L[D_+(a,0)]+C_L[D_-(a,0)]$ indeed is a link invariant, and is given by $$ C_L[D_+(a,0)]+C_L[D_-(a,0)] =
\frac{1}{2} fr_{1}\times l_{1,2}\times l_{2,3}\times
\cdots \times l_{a+1,1}. $$ The desired formula is then obtained by considering all possible orders on $D_\pm(a,0)$, noting that, for symmetry reasons, each term appears twice in the defining sum for $\mathcal{L}_{i,j,\{1,\cdots,a+1\}\setminus \{i,j\}}$ when $i=j$, hence an extra $\frac{1}{2}$ factor. \\ In the case where $a>0$ and $b>0$, the diagrams $D_\pm(a,b)$ are the result of inserting on $\dessin{0.65cm}{D01}$, in all possible ways, an order $a$ chain, followed by an order $b$ chain. The exact same argument then applies. \\ The second half of the statement is proved in a strictly similar way. The first case uses the fact that the diagrams $D_\pm(a,0,0)$ ($a\ge 1$) are obtained by inserting, in all possible ways, an order $a$ chain to $\dessin{0.65cm}{D22}$ (thus using Lemma \ref{lem:D22}). Likewise, for the second case, $D_\pm(a,b,c)$ ($a,b\ge 1$) is obtained by inserting three chains of order $a$, $b$ and $c$ to the empty diagram on two circle. \end{proof}
Using Theorem \ref{thm:coeffs essentiels}, we can give the desired explicit formula for the invariants $\mu_n$, for any $n\ge 3$, in terms of Conway coefficients and the linking matrix. \begin{theorem}\label{cor:mun} For all $n\ge 1$, and for any framed oriented $(n+2)$-component link $L$, we have \begin{eqnarray*} \mu_{n+2}(L) & = & \frac{1}{12} \sum_{\substack{1\le i\le n \\Êa+b=n+1 \\ a\ge b}} \sum_{\mathcal{I}_i(a,b)} \mathcal{L}_{i,I} \mathcal{L}_{i,J}
+ \frac{1}{2} \sum_{\substack{1\le i< j\le n \\ a+b+c=n \\ a\ge b\ge c}} \sum_{\mathcal{I}_{i,j}(a,b,c)} \mathcal{L}_{i,j,I} \mathcal{L}_{i,j,J} \mathcal{L}_{i,j,K} \\
& & + \frac{1}{4} \sum_{1\le i< j\le n} l_{i,j}^2 \mathcal{L}_{i,j,\{1,\cdots,n+2\}\setminus \{i,j\}} - U_{n+3}(L). \end{eqnarray*} \end{theorem}
\begin{proof} According to (\ref{eq:split}), we have
$$\mu_{n+2}(L) = 2 \sum_{\substack{a+b=n+1 \\ a\ge b}} \sum_{D\in \mathcal{D}(a,b)} \mathcal{C}_L[D]
+ 2\sum_{\substack{a+b+c=n \\ a\ge b\ge c}} \sum_{D\in \mathcal{D}(a,b,c)} \mathcal{C}_L[D]. $$ By the definition of the $\iota_1$ map, it is easily verified that for any $a,b,c\ge 0$, we have $\iota_\Theta(D_+(a,b)) = \frac{1}{6}$, $\iota_\Theta(D_-(a,b)) = -\frac{1}{3}$, $\iota_\Theta(D_+(a,b,c)) = \frac{1}{4}$ and $\iota_\Theta(D_-(a,b,c))$ $= -\frac{1}{4}$. Hence, recalling that $ \mathcal{D}(a,b) = \{D_+(a,b)\} \cup \{D_-(a,b)\}$ and $\mathcal{D}(a,b,c) = \{D_+(a,b,c)\} \cup \{D_-(a,b,c)\}$,
we have \begin{eqnarray*} \mu_{n+2}(L) & = &\, 2 \sum_{\substack{a+b=n+1 \\ a\ge b}} \Big( \frac{1}{6}\sum_{D\in \{D_+(a,b)\}} C_L[D] -\frac{1}{3} \sum_{D\in \{D_-(a,b)\}} C_L[D] \Big) \\
& & + 2\sum_{\substack{a+b+c=n \\ a\ge b\ge c}} \Big( \frac{1}{4} \sum_{D\in \{D_+(a,b,c)\}} C_L[D] -\frac{1}{4} \sum_{D\in \{D_-(a,b,c)\}} C_L[D] \Big)
\end{eqnarray*} It follows that \begin{eqnarray*} \mu_{n+2}(L)
& = & \sum_{\substack{a+b=n+1 \\ a\ge b}} \Big( \frac{1}{3}\sum_{D\in \mathcal{D}(a,b)} C_L[D] - \sum_{D\in \{D_-(a,b)\}} C_L[D] \Big) \\
& & + \sum_{\substack{a+b+c=n \\ a\ge b\ge c}} \Big( \frac{1}{2} \sum_{D\in \mathcal{D}(a,b,c)} C_L[D] - \sum_{D\in \{D_-(a,b,c)\}} C_L[D] \Big) \\
& = & \frac{1}{3} \sum_{\substack{a+b=n+1 \\ a\ge b}} \sum_{D\in \mathcal{D}(a,b)}\!\!\!\!\!\! C_L[D]
+ \frac{1}{2} \sum_{\substack{a+b+c=n \\ a\ge b\ge c}} \sum_{D\in \mathcal{D}(a,b,c)}\!\!\!\!\!\! C_L[D] -\!\!\!\!\!\! \sum_{D\in \mathcal{E}^-(n+2)} C_L[D] \end{eqnarray*} where the last equality uses (\ref{eq:split}). It only remains to use Theorem \ref{thm:coeffs essentiels} to express the first two terms in terms of linkings and framings, and Proposition \ref{U_n} to identify the last sum with $U_{n+3}$. \end{proof}
\begin{remark}\label{rem:jesuisinvariant} The fact that the defining formula for $\mu_{n+2}$ gives a link invariant follows readily from the decomposition $$ \mu_{n+2}(L) =\frac{1}{3} \sum_{\substack{a+b=n+1 \\ a\ge b}} \sum_{D\in \mathcal{D}(a,b)}\!\!\!\!\!\! C_L[D]
+ \frac{1}{2} \sum_{\substack{a+b+c=n \\ a\ge b\ge c}} \sum_{D\in \mathcal{D}(a,b,c)}\!\!\!\!\!\! C_L[D] -\!\!\!\!\!\! \sum_{D\in \mathcal{E}^-(n+2)} C_L[D] $$ and the fact that each of the above three sums defines a link invariant, by Theorem \ref{thm:coeffs essentiels} and Proposition \ref{U_n}. \end{remark}
The techniques used to show Theorem \ref{thm:coeffs essentiels} can also be used to prove the following technical result.
Recall from Definition \ref{rem:chain} that a chain of $m$ circles is a connected chord diagram on $m$ circles with two legs on each circle. \begin{lemma}\label{a_la_chain}
Let $C_m$ be a chain of $m$ circles, and let $I=\{i_1,\cdots,i_m\}$ be a set of $m$ pairwise distinct indices.
Let $\mathcal{D}(I)$ be the set of all chord diagrams obtained by labeling $C_m$ by the elements of $I$, in all possible ways.
Then for a framed oriented $m$-component link $L$, we have
$$ \sum_{D\in \{D(I)\}} C_L[D] =
\begin{cases}
\frac{1}{2} fr_{i_1} & \text{if } m=1 \\
\frac{1}{2}\sum_{\sigma \in S_{m-1}} l_{i_m,i_{\sigma(1)}} l_{i_{\sigma(1)}, i_{\sigma(2)}} \times \cdots \times l_{i_{\sigma(m-1)}, i_m} & \text{if } m>1
\end{cases}.$$ \end{lemma} \begin{proof}
If $m=1$ or $2$, then there is a unique labeling of $C_I$ and the result is given by Lemmas \ref{lem:D11} and \ref{lem:D22}, respectively.
If $m>2$, an element of $\mathcal{D}(I)$ can be seen as obtained from $\dessin{0.5cm}{D01}$, labeled by $i_m$, by adding an order $m-1$ chain of circles, labeled by $i_1,\cdots, i_{m-1}$ in all possible ways.
The same arguments as in the proof of Theorem \ref{thm:coeffs essentiels} then give the desired formula. \end{proof}
\section{Surgery formula for the Casson-Walker-Lescop invariant}
We now prove the surgery formula stated in the introduction.
\subsection{Setup}
In the previous sections, we identified certain combinations of coefficients of the Kontsevich integral in terms of classical invariants. In order to derive from these results a formula for the Casson-Walker-Lescop invariant, we now have to study how these particular diagrams contribute to the degree $\le 1$ part of the LMO invariant. Recall indeed that $$Z_1^{LMO}(S^3_L) = \left( \frac{\iota_1(\check{Z}(L))}{\iota_1(\check{Z}(U_+))^{\sigma_+(L)} \iota_1(\check{Z}(U_-))^{\sigma_-(L)}}\right)_{\le 1}, $$ and that the coefficient of $\dessin{0.5cm}{T}$ in $Z_1^{LMO}(S^3_L)$ is $\frac{(-1)^{\beta_1(S^3_L)}}{2} \lambda_L(S^3_L)$ (Theorem \ref{LMOCasson}). By Equation (\ref{Denominateur LMO}), there are two types of contributions to the coefficient of $\dessin{0.5cm}{T}$ coming from this formula: \begin{enumerate}
\item The diagram $\dessin{0.5cm}{T}$ comes from the denominator with coefficient $\frac{(-1)^{\sigma_+(L)} \sigma(L)}{16}$, and is multiplied by a constant term coming from $\iota_1(\check{Z}(L))$.
\item The diagram $\dessin{0.5cm}{T}$ comes from $\iota_1(\check{Z}(L))$, with some coefficient, and is multiplied by the coefficient $(-1)^{\sigma_+(L)}$ coming from the denominator. \end{enumerate} Summarizing, we have the following key equality \begin{equation}\label{eq:main} \frac{(-1)^{\beta_1(S^3_L)}}{2} \lambda_L(S^3_L) = \frac{(-1)^{\sigma_+(L)} \sigma(L)}{16} \Big( \iota_1(\check{Z}(L)) \Big)_0 + (-1)^{\sigma_+(L)} \times \Big( \iota_1(\check{Z}(L)) \Big)_1. \tag{*} \end{equation}
\subsection{The surgery formula}
Recall from Section \ref{sec:conv} that, if $\mathbb{L}$ is the linking matrix of a framed oriented $n$-component link, and if $I$ is some subset of $\{1,\cdots ,n\}$, we denote by $\mathbb{L}_{\check{I}}$ the matrix obtained from $\mathbb{L}$ by deleting the lines and column indexed by elements of $I$.
\begin{theorem} \label{Thm general} Let $L$ be a framed oriented $n$-component link in $S^3$ with linking matrix $\mathbb{L}$. Let $S^3_L$ be the result of surgery on $S^3$ along $L$. We have $$ \lambda_L(S^3_L) = \frac{(-1)^{\sigma_-(L)} \det \mathbb{L}}{8} \sigma(L)
+ (-1)^{n+\sigma_-(L)}\sum_{k=1}^{n} \sum_{\substack{I \subset \{ 1, \ldots, n \} \\ |I| = k}} (-1)^{n - k} \det \mathbb{L}_{\check{I}} \mu_k(L_I).$$ \end{theorem}
Some remarks are in order. \begin{remark}\label{rem:lescoop} As pointed out in the introduction, this recovers Lescop's third formula \cite[Prop.~1.7.8]{Lescop} for her extension of the Casson-Walker invariant. In particular, the two formulas in Theorem \ref{thm:coeffs essentiels}, which underly the definition of the invariant $\mu_k$ by Theorem \ref{cor:mun}, correspond to the products of linkings $\Theta_b$ in \cite[Nota.~1.7.5]{Lescop}. More precisely, in the terminology of \cite[Fig.~1.2]{Lescop}, the first formula corresponds to $\Theta_b$ in the case of a \lq Figure-eight graph\rq, while the second formula corresponds to the case of a \lq beardless $\Theta$\rq. \end{remark} \begin{remark}\label{rem:cestcadeaucamfaitplaisir} As an illustration, let us focus on the case $n=2$ for rational homology spheres. Let $L = K_1 \cup K_2$ be a framed oriented link whose linking matrix $\mathbb{L} = \left( \begin{smallmatrix} a & n\\ n & b \end{smallmatrix} \right)$ has nonzero determinant. Then $S^3_L$ is a rational homology sphere and $\lambda_L(M) = \frac{1}{2}\vert \det \mathbb{L}\vert \lambda_W(M)$. One can easily check that $(-1)^{\sigma_-(L)}$ is just the sign of $\det \mathbb{L}$, and Theorem \ref{Thm general} thus gives us $$\frac{1}{2} \det \mathbb{L} \lambda_W(S^3_L) = \frac{\det \mathbb{L}}{8} \sigma(L) + \Big( \mu_2(L) - a\mu_1(K_2) - b\mu_1(K_1) \Big). $$ Using the explicit formulas for $\mu_1$ and $\mu_2$ given in Lemmas \ref{ex:mu1} and \ref{ex:mu2}, we then obtain the following formula for $\frac{\det \mathbb{L}}{2}\left(\lambda_W(M) - \frac{1}{4} \sigma(L)\right)$: $$ a c_2(L_2) + bc_2(L_1) +\frac{n^3-n}{12} + \frac{(a+b)}{24}(2n^2-ab-2) - c_3(L) + n\left(c_2(L_1)+c_2(L_2)\right).$$ This recovers a result of S.~Matveev et M.~Polyak \cite[Thm.~6.3]{Matveev-Polyak}. \end{remark}
The rest of this section is devoted to the proof of Theorem \ref{Thm general}.
Recall that $\beta_1(S^3_L)$ is the nullity of $\mathbb{L}$, so that multiplying Equation (\ref{eq:main}) by $2(-1)^{\beta_1(S^3_L)}$ gives $$ \lambda_L(S^3_L) = \frac{(-1)^{\sigma_-(L)}}{8} \Big( \iota_1(\check{Z}(L)) \Big)_0 + (-1)^{n+\sigma_-(L)} \times 2\Big( \iota_1(\check{Z}(L)) \Big)_1. $$ Hence we are left with the explicit computations of $ \Big( \iota_1(\check{Z}(L)) \Big)_0$ and $ \Big( \iota_1(\check{Z}(L)) \Big)_1$. This is done in the following two lemmas.
\begin{lemma}\label{eq1} $$ \Big( \iota_1(\check{Z}(L)) \Big)_0 =(-1)^n \det \mathbb{L}.$$ \end{lemma} \begin{proof} The diagrams in the Kontsevich integral of $L$ that contribute to $ \Big( \iota_1(\check{Z}(L)) \Big)_0$ are those that close into a constant, that is, disjoint unions of chains of circles.\footnote{Note indeed that the normalization in $\check{Z}(L)$ adding a copy of $\nu$ to each circle does not affect the degree $0$ part of $\hat{Z}(L)$. } As pointed out in Section \ref{sec:essentiels}, chains of circles always close into the constant $(-2)$. The coefficient of a chain of $k$ circles in the Kontsevich integral is given in terms of coefficients of the linking matrix by Lemma \ref{a_la_chain}, and yields the following: $$\Big( \iota_1(\check{Z}(L)) \Big)_0 =\sum_{\{ I_1, \ldots, I_k \} \, \text{partition of}\, \{ 1, \ldots, n \}} (-1)^k \prod_{j=1}^{k} \mathcal{I}(I_j), $$ where $\mathcal{I}(I_j) = \mathcal{L}_{i_m,I_j\setminus \{i_m\}}=\sum_{\sigma \in S_{m-1}} l_{i_m,i_{\sigma(1)}} l_{i_{\sigma(1)}, i_{\sigma(2)}} \times \cdots \times l_{i_{\sigma(m-1)}, i_m}$ if $I_j=\{i_1,\cdots,i_m\}$ with $m>1$, and $\mathcal{I}(I_j) = fr_{i_1}$ otherwise. We leave it as an exercice to the reader to check that this indeed gives $(-1)^n \det \mathbb{L}$. \end{proof}
\begin{lemma}\label{eq2}
$$ \Big( \iota_1(\check{Z}(L)) \Big)_1 = \frac{1}{2}\sum_{k=1}^{n} \sum_{\substack{I \subset \{ 1, \ldots, n \} \\ |I| = k}} (-1)^{n - k} \det \mathbb{L}_{\check{I}} \mu_k(L_I).$$ \end{lemma} \begin{proof} Computing $\Big( \iota_1(\check{Z}(L)) \Big)_1$ amounts to counting those diagrams in $\check{Z}(L)$ that close into $\dessin{0.5cm}{T}$. As observed in Example \ref{footnote}, a copy of $\dessin{0.5cm}{D01}$ in $\hat{Z}(L)$ yields such a term when adding a copy of $\nu$ in $\check{Z}(L)$, and this is the only contribution arising from this normalization $\check{Z}$. Hence a diagram from $\hat{Z}(L)$ that contributes to $\Big( \iota_1(\check{Z}(L)) \Big)_1$ is, for some $k$ such that $1\le k\le n$, a disjoint union of \begin{itemize} \item a chord diagram on $(n-k)$ circles which is a union of chains of circles, which contributes by a constant, \item an element of $\mathcal{E}(k)$ if $k>1$, and either an element of $\mathcal{E}(1)$ or a copy of $\dessin{0.5cm}{D01}$ if $k=1$, which contributes by a $\dessin{0.5cm}{T}$ with some coefficient. \end{itemize} For a subset $I$ of $k>1$ elements of $\{1,\cdots,n\}$, the contribution to $\Big( \iota_1(\check{Z}(L)) \Big)_1$ of all diagrams in $\mathcal{E}(k)$, labeled by $I$ in all possible ways, is given by $\frac{1}{2}\mu_k(L_I)$ by virtue of Definition \ref{def:mun}; on the other hand, the proof of Lemma \ref{eq1} above tells us that the contribution of all possible unions of chains of circles labeled by $\{1,\cdots,n\}\setminus I$ is precisely $(-1)^k \det \mathbb{L}_{\check{I}}$. The same holds for $k=1$, noting the change in the formula for $\mu_1$ given in Definition \ref{def:mun}. The formula follows, by taking the sum over all possible subsets $I$. \end{proof}
\end{document} | arXiv |
A bookstore is deciding what price it should charge for a certain book. After research, the store finds that if the book's price is $p$ dollars (where $p \le 26$), then the number of books sold per month is $130-5p$. What price should the store charge to maximize its revenue?
The store's revenue is given by: number of books sold $\times$ price of each book, or $p(130-5p)=130p-5p^2$. We want to maximize this expression by completing the square. We can factor out a $-5$ to get $-5(p^2-26p)$.
To complete the square, we add $(26/2)^2=169$ inside the parenthesis and subtract $-5\cdot169=-845$ outside. We are left with the expression
\[-5(p^2-26p+169)+845=-5(p-13)^2+845.\]Note that the $-5(p-13)^2$ term will always be nonpositive since the perfect square is always nonnegative. Thus, the revenue is maximized when $-5(p-13)^2$ equals 0, which is when $p=13$. Thus, the store should charge $\boxed{13}$ dollars for the book. | Math Dataset |
In the fundamental works on the theory of Lie groups (S. Lie, H. Poincaré, E. Cartan, H. Weyl, and others) it is a group of smooth or analytic transformations of the space $\mathbf R^n$ or $\mathbf C^n$, depending smoothly or analytically on parameters. When there are finitely many numerical parameters, a continuous group is called finite, which corresponds to the modern concept of a finite-dimensional Lie group. In the presence of parameters that are functions one speaks of an infinite continuous group, which corresponds to the modern concept of a pseudo-group of transformations. Nowadays (1988) the term "continuous group" often stands for topological group .
This page was last modified on 9 July 2014, at 23:21. | CommonCrawl |
Would teaching nonstandard calculus in an introduction calculus course make it easier to learn?
Nonstandard calculus is a reformulation of calculus that is based on infinitesimals instead of epsilon-delta definitions. Of course, people had tried to use infinitesimals in calculus before; in fact, Calculus originally used infinitesimals. The problem was that it did not have a rigorous foundation, which is why mathematicians started using the epsilon-delta definitions instead, although they still used infinitesimals informally. Much later, nonstandard analysis came along, which did put infinitesimals on rigorous foundations, but by then it was not necessary since epsilon-delta definitions were available. Nonstandard calculus is calculus except based on nonstandard analysis instead of analysis. There are still people who think that nonstandard analysis should replace those definitions, however. I will put an informal introduction to how it works at the end of this post.
Would teaching students calculus using nonstandard calculus make it easier to learn? Answers should also take into account future calculus learning, not just learning in the course itself.
I would prefer to focus on "normal" students. I think for bright students, teaching a little bit of both would be beneficial, since I think comparing the approaches teaches the idea that there can be radically different ways to study math and arrive at the same results. However, for most students, I think mainly focusing one or the other would be better.
Here are some of my observations. They are kind of long, so feel free to skip/skin them. The most important is probably points 4 and 6, since they are about how standard and nonstandard calculus are different. The other points are about how they are the same.
Of course, the mathematical foundations of nonstandard analysis would be much too complicated to cover in such a course. The same is true of standard analysis though, so this is not a concern.
Nonstandard calculus is compatible with calculus. By that I mean that anything you can prove in analysis can be proven in nonstandard analysis, and anything you can disproof is analysis can be proven in standard analysis, so there will never be a contradictory result. In terms of provable statements, the only difference is that nonstandard analysis proves some things about infinite and infinitesimal numbers that standard analysis is not concerned with (since it does not use such number), and even those can be translated into equivalent statements about real numbers in standard analysis. The main difference is how they got about proving them.
Definitions in nonstandard calculus involve much fewer quantifiers than those in standard analysis. Limits, continuity, and uniform continuity all have a quantifier complexity of $1$, whereas in standard calculus they have 3 or 4. This however is balanced by the fact that it is much more difficult to do arithmetic with hyperreal numbers than the real numbers. You can not even build a hyperreal number calculator. All in all, proofs in both are about equally as long.
Although the proofs are equally as long, they are much more intuitive in nonstandard calculus than in standard calculus. With standard calculus, you usually motivate theorems with ideas like infinity, or being infinitely close, or dividing infinitesimal numbers. Then you translate those intuitions into a series of epsilon delta definitions, with many inter-dependencies, never including your intuitions in your proof, just their translation. In nonstandard calculus, most of the intuitions are included literally in the proof. Infinity is no longer just an idea, its a number. They are also superior to informal proofs using infinitesimals (like scientists sometimes use) since it never proves an contradictions. You still have to do the work, but the work is not separate from the intuition.
Both standard and nonstandard calculus generalize to other fields, such as topology, so that is not an issue.
The main issue I see is when they take Calculus in the future. If the students never take another formal calculus course, this is not a concern, but if they do, all the proof methods they learned in nonstandard calculus will no longer be taught in light of new material. The grader may not know how to grade a proof using nonstandard calculus, as well (which is reasonable, since it invokes a whole host of concepts not used in standard calculus). One thing that helps though is that proofs from standard calculus still make sense in nonstandard calculus. For example, even though limits no longer have an epsilon-delta definition, they do have an epsilon-delta theorem. In fact, most limit proofs using the definition can be mechanically translated into a proof using the theorem. Since the epsilon-delta theorems are true by, well, definition in standard calculus, they would be compatible with future calculus courses. Additionally, but having the students carefully study the epsilon-delta theorems, they will likely be able to see the connection between the material in future calculus courses could be translated into nonstandard calculus. Unfortunately, this results an "overhead" cost of using nonstandard calculus over standard calculus. However, in my experience, I find that writing a proof first in nonstandard calculus and then translating it to standard calculus is easier mentally (unless I immediately see the standard calculus proof). This may be unique to myself however. However, for some courses (especially less proof based ones), learning nonstandard calculus will actually make the course easier. That's because many courses less concerned with rigor will just use infinitesimals anyways, ignoring foundational issues. Students who learned nonstandard calculus with have an advantage, since they know how to manipulate infinitesimals rigorously.
Another potential issue is materials. For any introductory calculus course, this is just potential, since there is textbook that teaches introductory calculus, using nonstandard calculus proofs, that the author distributes for free. There are also some other ones of varying costs.
calculus course-design infinitesimals axioms-foundations infinity
Ben Crowell
$\begingroup$ Related: 1 2 3 $\endgroup$ – PyRulez Feb 9 at 0:47
$\begingroup$ I am moderately puzzled by the frequency with which these questions have been asked; the "related" links above emphasize as much. If a cursory search in google scholar reveals nothing to this effect, then perhaps one or more of the interested parties could investigate this in a scholarly fashion. I don't know of anyone who has investigated the inclusion of hyperrreals, infinitesimals, etc in a first course on the Calculus. $\endgroup$ – Benjamin Dickman Feb 9 at 1:21
$\begingroup$ @BenjaminDickman All the questions have at least subtle differences. $\endgroup$ – PyRulez Feb 9 at 19:34
$\begingroup$ I don't think it's necessary or helpful to have the complete mathematical summary of NSA that takes up the second half of the question. It makes the question extremely long, and this is stuff that people can look up on WP if they need to learn about it. (You provided a WP link.) I hope you won't be offended if I edit it out. If you disagree with my edit, please feel free to undo it. $\endgroup$ – Ben Crowell Feb 10 at 17:09
$\begingroup$ @BenjaminDickman: I'm puzzled by your puzzlement. You seem to be saying that you think people on this site are asking over and over whether anyone uses NSA in freshman calc, and the answer is always no, so why do they keep asking? But in fact many of the answers are yes answers. And as the OP points out, the questions all differ. [1] asks where it's used. [2] asks for research supporting or opposing. [3] is more about whether students should be taught to be "bilingual." The present question seems to be about whether it's a good idea to do a "monolingual" course using NSA. $\endgroup$ – Ben Crowell Feb 10 at 17:51
This has certainly been tried before. See for example,
H. Jerome Keisler. Elementary Calculus: An Infinitesimal Approach. On-line Edition. This has also been published in print by Dover.
Kathleen Sullivan. The Teaching of Elementary Calculus Using the Nonstandard Analysis Approach. The American Mathematical Monthly Vol. 83, No. 5 (May, 1976), pp. 370-375.
Brian BorchersBrian Borchers
$\begingroup$ To amplify on this, I think this particular question differs from the one by Mikhail Katz in that it asks whether it's appropriate to do a freshman calc course using an aggressively NSA-centered approach, rather than just exposing students to both the language of limits and the language of infinitesimals. Keisler's approach is such an aggressively NSA-centric one, although he does also go back and define limits. The OP also asks whether this would make freshman calc "easier to learn." Anecdotally, I think the experience is that it didn't work out so great when Keisler tried it, although [...] $\endgroup$ – Ben Crowell Feb 10 at 17:59
$\begingroup$ [...] this may have been partly due to incidental factors such as disorganization and lots of errata in initial versions of the book. $\endgroup$ – Ben Crowell Feb 10 at 18:00
My understanding of this question is that it proposes the idea of a "monolingual" freshman calc course in which students mostly learn the language of NSA, and limits are largely or completely neglected. This makes it different from this question by Mikhail Katz, which asks whether it's a good idea for students to be "bilingual."
I have some experience teaching some NSA-based material to first-semester calculus students at a community college. Here is the book I wrote for that purpose. My approach is "bilingual." The philosophy with which I approached this was not that students should learn NSA in excruciating detail -- that would be kind of silly IMO, and would detract from the main thrust of the course, which is basically to be able to do rule-based differentiation.
However, scientists and engineers still use Leibniz notation and manipulate infinitesimals using algebra. They've been doing it for 300 years, and they never stopped doing it just because there was a short gap between the invention of the limit and the invention of NSA. These students will see such manipulations in their physics courses, and they will see and be expected to use them in their careers. So we should give them some systematic idea of what techniques are appropriate when performing these manipulations. If this was 1850, we would teach them a body of techniques that we knew from experience gave correct results, e.g., throwing away higher orders of $dx$. Today we have more secure knowledge that these procedures can be put on a sound logical footing, but that actually has little effect in reality on what body of techniques we use.
The goal should not be to eliminate the $\epsilon-\delta$ definition of limits. This is not possible because there is a clear consensus among mathematicians that $\epsilon-\delta$ is something students should have already learned by the time they go on to their second-semester course, and we need to prepare these students properly. In fact, $\epsilon-\delta$-like ideas are fundamental to many of the common modes of reasoning about calculus. E.g., we can do a numerical simulation of the motion of the moon with a small time step, and say, "If I make the step size small enough, I should be able to make the error as small as desired." Newton and Leibniz would have understood that idea, and so would every physicist of succeeding centuries, long before the definition of the limit.
When you get right down to it, using NSA doesn't really turn out to produce any incredible simplification of freshman calculus. For example, it would be nice if we could just prove the chain rule and L'Hôpital's rule by manipulating infinitesimals using algebra, but the nuts and bolts don't actually work out quite that trivially. In the case of the chain rule, you have the issue that the derivative is not the quotient of infinitesimals but the standard part of that quotient. In the case of L'Hôpital's rule you also have all the various forms of the rule (repeated application, limits at infinity, $\infty/\infty$, etc.). These are complicated to prove (and many books don't prove them all), and the complication isn't really reduced very much by using NSA.
Keep in mind also that at this level, our students never really get a full-blown introduction to the real number system. No commercial textbook I've ever seen systematically introduces and applies anything beyond the first-order properties of the reals. (They may state the completeness property, but they never use it to prove things like the intermediate value theorem.) This is material that belongs in an upper-division analysis class. Since the hyperreals have the same first-order properties as the reals, there is actually very little that we can meaningfully say about the hyperreals to students at this level.
I do explicitly name the hyperreals and describe how their properties differ from those of the reals (this is mainly in section 2.9 of my book). However, I don't think it's a good idea to go into the kind of depth that Keisler does.
Ben CrowellBen Crowell
Speaking as a former student, though an engineering one . . . It was hard enough learning to integrate tricky expressions and solve differential equations, without having to learn a new number system as well. And I don't remember ever needing the rigorous definition of a limit, standard or nonstandard. The most I needed even in complex calculus was an ability to manipulate limits, and an awareness that derivatives and antiderivatives were defined as limits and could be found as limits if necessary.
You're not going to understand Maxwell's Equations or the Schrödinger Wave Equation or Fourier transforms without understanding integrals and derivatives, but you can perfectly well understand them without knowing the rigorous definition of a limit.
It seems to me that an introductory calculus course isn't going to be an introduction to proving that calculus works, any more than an introductory algebra course is going to start with a rigorous definition of the real numbers—it's going to be an introduction to the concepts of differentiation and integration and how to use them.
Obviously, proving rigorously that calculus works does require rigorous definitions, and at that point something new has to be introduced. Maybe a strange new definition involving $ε$ and $δ$, and maybe a strange new number system in which there are different sizes of zero which are defined not to equal each other or zero.
I think awareness at that stage is likely to be:
$\frac{dy}{dx}$ is defined as a limit, and so is $\int y dx$.
but we sometimes treat $dy$ and $dx$ like algebraic quantities
this is a bit dodgy even though it works, since we're effectively either dividing $0$ by $0$ or multiplying $0$ by $\infty$
the way round this involves something to do with limits.
If $ε–δ$ is explained clearly at that stage—ie in terms of making $ε$ arbitrarily small by making $δ$ small enough or $N$ large enough, and not as a dense expression full of quantifiers—then once someone has grasped it, they'll be convinced it's rigorous. But if they're introduced to hyperreal numbers instead, they'll have lingering doubts about whether arguments using them are actually valid. It means another layer of understanding is needed, namely the background theory of the new number system.
So I think it's more useful to focus on understanding the principle of $ε–δ$ definitions and techniques clearly than to introduce an unfamiliar number system. Apart from anything else, this will help them to understand other textbooks in the future, which must likely won't use hyperreals.
timtfjtimtfj
$\begingroup$ "but we sometimes treat dy and dx like algebraic quantities" "the way round this involves something to do with limits". That is the main advantage of nonstandard calculus. All calculations are done by treating dy and dx as algebraic qualities, because they are algebraic qualities in the hyperreal number system. It does not need to be avoided. The derivative is equal to the quotient of dy and dx, and the integral is a hyperfinite sum of values having dx in it. $\endgroup$ – PyRulez Feb 9 at 18:29
$\begingroup$ "If ε–δ is explained clearly at that stage—ie in terms of making ε arbitrarily small by making δ small enough or N large enough, and not as a dense expression full of quantifiers—then once someone has grasped it, they'll be convinced it's rigorous." ε–δ definitions will still have a high quantifier complexity even when it explained in English. You still have to say "for all arbitrarily small epsilon there exists arbitrarily small delta such that for all x. statement". $\endgroup$ – PyRulez Feb 9 at 18:36
$\begingroup$ You do have a point though of the foundations of the nonstandard analysis being significantly more complicated. Proofing that the foundation of analysis makes sense takes a course or too, but nonstandard analysis is even worse. $\endgroup$ – PyRulez Feb 9 at 18:38
$\begingroup$ "The most I needed even in complex calculus was an ability to manipulate limits, and an awareness that derivatives and antiderivatives were defined as limits and could be found as limits if necessary." You need a similarly small part of nonstandard analysis to do the same. The most you need to know to match that is how to manipulate hyperreals, and that derivatives and antiderivatives can be found using hyperreals if needed. $\endgroup$ – PyRulez Feb 9 at 18:44
$\begingroup$ @PyRulez I see that and I quite like it. I was trying to ssy that there are two ways to deal with the issue of differentials—either hyperreals or $ε–δ$—and that both approaches involve introducing a new concept. Maybe the ideal is for students to know that both approaches exist, but learn one in detail. Hyperreals are interesting enough to prompt those who are interested to read about them and play around with them, and anyone who needs to prove calculus theorems is at a level where they need to know about $ε–δ$ anyway. That's my thought. $\endgroup$ – timtfj Feb 9 at 18:51
It's not really that relevant since the bulk of a normal calculus course (e.g. AP BC, Thomas Finney, Stewart) just does a small amount of epsilon-delta (so student is exposed to it) and then moves to "x+h". The bulk of the course is about learning derivatives, antiderivatives, methods of integration, classic applied problems, a bit if polar coordinates, bit of series, and small section on ODEs.
You are swinging at the wrong opponent with this obsession on the definition of a derivative (but not surprising given the theory bent of many math majors).
If you want to make things easier, cut partial fractions, integration by parts, etc. Do nurses and business students really need that? (Of course this does mean tracking because science and engineering do.)
P.s. I found your question hard to parse given the notation (upside down A). This is ironic given you are planning to make things easier for knuckleheads worse than I. Not showing it here...makes me leery of your proposals. Can't you discuss pedagogy and coverage without such a segue into exploring the math theory like that?
answered Feb 9 at 1:48
guestguest
$\begingroup$ Standard calculus uses the upside down A as well, and more of them, so I'm not sure what difference that makes. I would have to teach them anyways. $\endgroup$ – PyRulez Feb 9 at 2:01
$\begingroup$ @PyRulez Standard calculus where uses $\forall$? Please tell us what country or part of the world you are in. From my perspective (US) no careful quantifier notation is used in calculus teaching to "normal" students. Calculus is not real analysis, there are essentially zero proofs in calculus, and the whole topic of standard vs. nonstandard analysis seems mostly irrelevant to the actual practice of teaching calculus (maybe 1 or 2 lectures are on rigorous limits and then they do not come up in a rigorous way again). The nonstandard freshman calculus book died in the marketplace years ago. $\endgroup$ – KCd Feb 9 at 10:32
$\begingroup$ @KCd In the U.S. calculus text books usually explain quantifier notation in the index. $\endgroup$ – PyRulez Feb 9 at 18:48
$\begingroup$ "swinging at the wrong opponent with this obsession on the definition of a derivative" its not just the definition of derivatives, or the epsilon delta notation. Nonstandard analysis also fundamentally changes how you *manipulate" derivatives, integrals, and even applied problems. In particular, nonstandard analysis allows more algebraic manipulations than standard analysis. In standard calculus you would need to use limits to simulate them. $\endgroup$ – PyRulez Feb 9 at 18:52
Not the answer you're looking for? Browse other questions tagged calculus course-design infinitesimals axioms-foundations infinity or ask your own question.
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Is Calculus AB/BC a 'bad course?' | CommonCrawl |
We will now state some very basic properties regarding these incidence matrices.
Proposition 1: Let Let $(X, \mathcal A)$ be a $(v, b, r, k, \lambda)$-BIBD and let $M$ be a corresponding incidence matrix. Then every column of $M$ contains exactly $k$ many $1$s.
Proposition 2: Let $(X, \mathcal A)$ be a $(v, b, r, k, \lambda)$-BIBD and let $M$ be a corresponding incidence matrix. Then every row of $M$ contains exactly $r$ many $1$s.
Proposition 3: Let $(X, \mathcal A)$ be a $(v, b, r, k, \lambda)$-BIBD and let $M$ be a corresponding incidence matrix. Then between any two distinct rows there are exactly $\lambda$ many $1$s in the same corresponding columns. | CommonCrawl |
Reaction-diffusion equations with a switched--off reaction zone
CPAA Home
Non-isothermal viscous Cahn-Hilliard equation with inertial term and dynamic boundary conditions
September 2014, 13(5): 1891-1906. doi: 10.3934/cpaa.2014.13.1891
Regular solutions and global attractors for reaction-diffusion systems without uniqueness
Oleksiy V. Kapustyan 1, , Pavlo O. Kasyanov 2, and José Valero 3,
Taras Shevchenko National University of Kyiv, 60, Volodymyrska Street, 01601, Kyiv, Ukraine
Institute for Applied System Analysis, National Technical University of Ukraine "KPI", Kyiv
Universidad Miguel Hernández, Centro de Investigación Operativa, Avda. Universidad s/n, Elche (Alicante), 03202
Received September 2013 Revised September 2013 Published June 2014
In this paper we study the structural properties of global attractors of multi-valued semiflows generated by regular solutions of reaction-diffusion system without uniqueness of the Cauchy problem. Under additional gradient-like condition on the nonlinear term we prove that the global attractor coincides with the unstable manifold of the set of stationary points, and with the stable one when we consider only bounded complete trajectories. As an example we consider a generalized Fitz-Hugh-Nagumo system. We also suggest additional conditions, which provide that the global attractor is a bounded set in $(L^\infty(\Omega))^N$ and compact in $(H_0^1 (\Omega))^N$.
Keywords: global attractor, multi-valued dynamical system, Reaction-diffusion system, unstable manifold, Fitz-Hugh-Nagumo system..
Mathematics Subject Classification: Primary: 35B40, 35B41, 35K55; Secondary: 37B25, 58C0.
Citation: Oleksiy V. Kapustyan, Pavlo O. Kasyanov, José Valero. Regular solutions and global attractors for reaction-diffusion systems without uniqueness. Communications on Pure & Applied Analysis, 2014, 13 (5) : 1891-1906. doi: 10.3934/cpaa.2014.13.1891
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Oleksiy V. Kapustyan Pavlo O. Kasyanov José Valero | CommonCrawl |
How to judge whether a symmetry will be spontaneously broken while only given a Hamiltonian preserving this symmety
As asked in the title, is Hamiltonian containing enough information to judge the existence of spontaneously symmetry breaking?
Any examples?
condensed-matter group-theory
LaurentLaurent
The Hamiltonian of a theory describes its dynamics. Symmetry is "spontaneously broken" when a certain state of the quantum theory doesn't have the same symmetry as the Hamiltonian (dynamics). The standard example is a theory with a quartic potential. In field theory, say we have a potential $V (\phi) \sim \lambda{(\phi^2 - a^2)}^2$ in the lagrangian/hamiltonian, for some real valued scalar field $\phi$. This potential has minima at $\phi_{\pm}=\pm a$.
The theory (Hamiltonian) has a $\phi \rightarrow - \phi$ symmetry, but note that each of those vacua don't have the same symmetry i.e. vacuum states are not invariant under the symmetry. The solutions $\phi_{\pm}$ transform into each other under the transformation. So this is an example where the symmetry of the theory is broken by the vacuum state, "spontaneously" (roughly by itself, as the theory settles into the vacuum state).
So, to find whether a symmetry will be spontaneously broken (given a Hamiltonian), you have to check the symmetry of the states (typically vacua) of the theory and compare them to the symmetry of the Hamiltonian.
SivaSiva
I'm not sure what OP has in mind, but consider e.g. the inverted harmonic oscillator $$H~=~\frac{p^2}{2m}-\frac{1}{2}kq^2, $$ which has spontaneously broken $\mathbb{Z}_2$ symmetry $(q,p) \to (-q,-p) $. The stable positions are $q=\pm \infty.$
Qmechanic♦Qmechanic
In general, you should look to solve the potential for a function that gives a minimum. Does this minimum of the potential respect the symmetry? If not, you are very likely to find an anomaly. The classic example of this is the minimum of the higgs potential not respecting the gauge invariance of the SU(2) x U(1) fields coupled to it, but there are simpler examples, like the one @Qmechanic points out.
Jerry SchirmerJerry Schirmer
Not the answer you're looking for? Browse other questions tagged condensed-matter group-theory or ask your own question.
Emergent symmetries
Is there an algebraic approach for the topological boundary (defect) states?
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Discrete Symmetries: Breaking and Preserving
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Spontanous symmetry breaking in the Heisenberg model? | CommonCrawl |
Design and analysis of an effective graphics collaborative editing system
Chunxue Wu1,
Langfeng Li1,
Changwei Peng1,
Yan Wu2,
Naixue Xiong3 &
Changhoon Lee4
With the rapid development of computer-supported cooperative work (CSCW) technology, graphical collaborative editing plays an increasingly important role in CSCW. The most important technique in graphics co-editing is the consistency of graphics co-editing, which mainly includes causality consistency, consistency of results, and consistency of intention. Most of the previous research was abstract and ineffective, lacking theoretical depth and scalability. However, because the algorithm proposed in this paper can solve the contradictions in the consistency of graphical collaborative editing, the research in this paper has particularity, and the results will be proven by the experiment described in the paper. In order to solve the consistency conflict problem of graphic collaborative editing, the common graphics collaborative editing algorithm (CGCE algorithm) is proposed. It is proposed not only to perfect and expand the definition of graphics collaborative editing but also to merge with HTML5 Canvas, WebSocket, jQuery, Node.js and other network programming languages and technologies. The graphic collaborative editing based on the design and implementation of this paper can effectively solve the consistency conflict problem of many users during the collaborative editing of graphics, which ensures that the graphics of each graphical collaborative editing interface is consistent and the collaborative work can achieve the desired effect.
Object-based graphical editing systems are a particular class of collaborative editing systems where shared objects subject to concurrent accesses are graphic objects such as lines, rectangles, circles, and text boxes [1]. Graphic collaborative editing brings great convenience; for example, many artists in different regions can work together to edit and complete a picture. Graphic collaborative editing can also have a positive impact on education. For example, an art teacher can monitor students' work online and modify the work at any time. These were unimaginable before. With the increase of collaborative users, the structure of the network is more and more complicated, and collaborative editing is deeply influenced by network congestion. Traditional routing mechanisms and load-balancing approaches cannot make efficient use of the network, owing to lack of network status information and flexible ways to perform dynamic controlling operations [2]. Disadvantages caused by network congestion include increased packet loss rate, end-to-end delay, and reduced system throughput. If finances permit, improving physical hardware facilities is also a good choice to shorten the delay. The newest technologies, including an optical amplifier, dispersion compensation, and forward error correction [3], may alleviate congestion problems. High data rates, low cost, and collision reduction with the full-duplex approach and the elimination of chaining limits inherent in hub-bed Ethernet networks have made the switched Ethernet a dominant network technology [4].
Because full-duplex communication has so many advantages, the graphic editing system is generally used in full-duplex communication, and at the same time, it is also extremely important to solve the distribution of different regions of the collaborative user site synchronization between the computers. Realistic network applications often require multiple computers with a high clock consistency-clock synchronization. The Network Time Protocol (NTP) is widely used to synchronize computer clocks on the Internet [5]. But traditional time synchronization technology such as NTP has been unable to meet this precision requirement, and the cost of a GPS system is prohibitively expensive. However, IEEE 1588 protocol development and maturity provide a low-cost, high-precision network clock synchronization program. Software time stamping may deliver acceptable results for a certain range of applications [6]. Furthermore, the shown architecture will be able to support the upcoming PTP version 2 in hardware as well as in software [7]. Taking into account the economic and practical needs related to these factors, the IEEE 1588 protocol is adopted to solve problems with synchronization among the collaborative sites, and this paper is based on the B/S architecture. B/S architecture is adopted because of the following strengths of the B/S architecture software:
Simple maintenance and upgrades: For facilitation of the frequent improvements and upgrades of the software system, B/S architecture is more advantageous than C/S architecture. With a slightly larger unit, system administrators of C/S architecture have to manage thousands of computers, but with B/S architecture, system administrators only need to manage the server online.
Independence of the system: Based on the B/S structure, software can be simply installed on a Linux server, Windows server, Unix server, and so forth. Therefore, the choice of operating system is diversified. No matter what kind of system the user chooses, the computer will not be affected.
Running heavy load of data: Administrators only need to manage the server online through the Internet browser on the server, and the key is that logic settings are generally relatively simple on the browser so that managers only need to do hardware maintenance on the browser.
In this paper, we describe how computer graphics co-editing is communicated between all sites using WebSocket technology. WebSocket is a specification developed as part of the HTML5 initiative, and it only establishes a TCP socket connection after the first request for connection, which can save server resources and network bandwidth and achieve real-time communication [8]. The WebSocket protocol supports full-duplex communication between the client and the remote host. By using an existing server, we can focus on learning about the easy-to-use API that enables creation of WebSocket applications [9]. New networking approaches have recently been introduced that are based on repurposed techniques for delivering web pages (Comet) or integration of real-time communication directly into the browser (HTML5 WebSockets) [10,11,12].
This technology is simplifying the work of programmers, harmonizing access to diverse devices and applications, and giving users amazing new capabilities [13]. HTML5 and the Canvas element have real potential in many useful applications, but the rest of this paper just focuses on Cartagen, a vector-based, client-side framework for rendering maps in native HTML5, and its potential application [14, 15]. In addition to the use of HTML5 Canvas technology, jQuery- and Node.js-related technology are also used. jQuery is a JavaScript library that is fast, small, and feature-rich. jQuery strikes a completely different balance between cost and flexibility of its configuration interface [16]. Node.js is one of the more interesting developments that has recently gained popularity in the server-side JavaScript space, and it is a framework for developing high-performance concurrent programs that do not rely on the mainstream multithreading approach but use asynchronous I/O with an event-driven programming model [17]. With the HTML5 Canvas technology, jQuery, Node.js, WebSocket, and other technical support for the graphics collaborative editing system with software to achieve protection, the next step is to achieve the consistency of graphic editing computer-supported cooperative work (CSCW) algorithm research [18, 19]. Within the CSCW field, collaborative editing systems have been developed to support a group of people sharing editing documents from different sites. Object-based graphic editing systems are a special type of graphic editing system [20]. Consistency of key technologies needed to achieve these are described next.
Computer synchronization
Computer synchronization refers to the distribution of the different geographical sites of the collaborative computer to achieve clock synchronization between locations. This is the most basic requirement for graphics collaborative editing, because if the collaboration between the site computers is not synchronized, a very complex algorithm to maintain the consistency between different sites is needed. As the number of collaborators increases, the number of computers in the collaborative site increases, and it is more difficult to maintain the consistency of the graphic editing system between different sites [21,22,23]. Therefore, the synchronization of computers between sites ensures graphics co-editing consistency.
Consistency algorithm
A good algorithm can play a key role in the coherence of graphics collaborative editing. For example, the algorithm used to achieve the required software and hardware resources generally refers to the algorithm's time complexity and spatial complexity being as small as possible, and the efficiency of the algorithm being very high, which can be expressed as the time required to reach the end of the operation [24,25,26,27,28]. There is no best algorithm in the world, and only only the improved algorithms in this paper can be continually effective to the updated applications. Any algorithm has its own limitations, and the limitations of the algorithm over time will become more and more obvious, so this paper studies a relatively efficient algorithm for consistency. This paper mainly describes a common algorithm for graphics collaborative editing (CGCE) [29,30,31].
Programming language selection
If a good system has a good algorithm, then to implement the algorithm, a programming language is needed. This paper mainly uses HTLM5's Canvas core technology, WebSocket, jQuery, and Node.js-related technology as the method, with the graphics collaborative editing algorithm as the core, according to the reality of the situation, the use of C# language in front of the background, and the preparation of graphics collaborative editing system, so as to complete the subject graphics editor consistency research [32].
Some problems of graphics collaborative editing
Graphic editing refers to the computer's ability to edit some of the graphics, the package on the point, line, surface additions and deletions, as well as their movement, copy, rotation, and other operations. Common graphic editing software are Adobe's AI and Photoshop, Corel's CorelDRAW, Autodesk's AutoCAD, Discreet's 3D Studio Max, and so on. With scientific progress, many people are required to collaborate sometimes to complete tasks, and graphics collaborative editing can provide people with great convenience. Collaborative editors can be distributed in different areas, or they can be structurally diverse and complex WANs [33], just requiring sitting in an office to collaborate and complete projects. Therefore, the prospect of applying collaborative editing technology is very exciting. Although the graphic editing software is so abundant and graphic editing technology is very mature, graphics collaborative editing software is rare. Graphics collaborative editing systems have many practical problems:
Robustness: Many graphics collaborative editing techniques are hindered by technical problems of poor robustness. Different theories of graphics collaborative editing research regarding the increased problems faced in the real environment include the different network structure, network congestion, network routing, and so forth [34, 35] The robustness of the environment fluctuates poorly, resulting in inconsistent results.
High responsiveness: In the Internet environment, the response to the local user's actions must be quick, even as collaborating users reside on different machines connected via the Internet with a long and nondeterministic communication latency [19]. The graphics collaborative editing operation response speed depends on a lot of conditions:
Hardware performance of the computers in the site: There are significant differences in performance between different prices, different models, and different systems. The price is high, the model is new, and the system is upgraded. The computer generally has a good processor and fast response.
Spatial complexity and time complexity of the algorithm: When a few graphics operations are performed on the graphics co-editor, the computer will run the corresponding graphical co-editing algorithm code after each step [36, 37]. The greater the time complexity and spatial complexity of the algorithm, the more time it takes for the computer to run the code, and the slower the response of the graphical collaborative editor.
The transmission delay of data: Computer data in the physical layer are transmitted in binary form, which raises the following problems:
Bit error rate: Data in the channel will be the cause of data distortion, which results in bit error rate. The data length of the data transmission needs to be consistent with Shannon's theorem:
$$ C=B{\mathit{\log}}_2^{\left(1+\frac{S}{N}\right)} $$
Then the transmission process code length can be as long as possible, and the bit error rate can be reduced to a relatively low level.
Network congestion: With the development of science and technology, computer users are increasing very rapidly at the same time, the computer network-related hardware facilities have also been greatly improved, but network congestion is still inevitable. When the computer network is congested, the spread of communication between the different computers through the Internet will become very slow, and the response between the cooperative operations will also slow. This may lead to the following results:
When the network is in normal recovery, collaborative application confusion or even collapse can occur. Because the collaboration among users, if they do not know the network has been congested, is still constantly in cooperation, when the network returns to normal, the collaborative operation may be invalid.
When the network is recovered, collaborative editing consistency will have become damaged. After the network is restored, the response between the cooperative users is normal, and the cooperative information of the network congestion may be transmitted to the different cooperative computers or lost. The result may be inconsistent owing to the lag of the response, resulting in inconsistent computer collaboration among the collaborative users [38].
2. High concurrency: Multiple users are allowed to concurrently edit any part of the shared document at any time by facilitating natural information flow among collaborating users [19]. High concurrency is the basis for computer collaboration requirements. The higher the concurrency, the better the synergistic effect [39]. However, the higher the concurrency will have a problem that affects the performance of the entire graphics co-editing [40, 41], launching consistency issues. The traditional consistency maintenance methods, such as lock and serialization methods, are not suitable for real-time editing. In this paper, we will study the consistency of graphics co-editing.
Related concepts involved in coherence of graphics co-editing
Definition 1 Minimal Editing Unit "⊕". Minimal Editing Unit means that the easiest graphics can be edited at once. The smallest editing unit is a point. If the operation is to draw a point, it can be expressed as ⊕point.
Definition 2 The Operational Specification Representation: the Intent Consistency Model. The graphical representation of graphical co-editing in the Model of Intention Consistency is more complex than other conformance models, and each operation needs to add the intent of the co-editor. The operation of graphic editing, includes increase, delete, turn, move, copy, and so on. The operations in these operational intent conformances are shown in Table 1.
Table 1 Expression of Operations
Definition 3 Simple Graphics "∆". Point, line, equilateral triangle, regular quadrilateral, regular polygon (more than four edges), and circle are called simple shapes in graphic collaborative editing. In addition to simple graphics, all others are called complex graphs. A simple graphic can itself be minimal editing units, or it can be made up of many minimal editing units. If the operation is to draw an equilateral triangle, it can be expressed as ∆Equilateral triangle.
Definition 4 The Center of Simple Graphics. A simple graphics center is center of gravity in this paper according to Definition 3, and it can be more convenient for graphic editing. Complex graphics for these operations can be divided into a number of simple graphics, and then the center of these simple graphics can be found.
Definition 5 Delayed Deadline " ρ". Delay time is an operation from one site to another site specified by the maximum delay. In order to prevent the occurrence of a loss of operation due to network congestion or failure, we set a delay cutoff time in the graphic editing system. If there is a delay in the operation of the graphic editing system, the operation has not reached the destination site at the deadline, and the destination site issues a retransmission request. Regarding delay time, depending on the computer network environment, you can set a different delay cutoff time. This scheme requires a station to send an operation to another station, but an operation also needs to be sent to receive confirmation information if the other destination site in the delay deadline to receive the operation, the source site to send a received operation confirmation of information.
Definition 6 Causal Ordering Relation "→". Given two operations O1 and O2, generated at sites i and j, O1 is causally ordered before O2, expressed as O1 → O2, if and only if: (1)i = j and the generation of O1 happened before the generation of O2; or (2) i ≠ j and the execution of O1 at site j happened before the generation of O2; or (3) there exists an operation Ox, such O1 → Ox and Ox → O2 [22].
Definition 7 Dependent and Independent Relations "‖". Given any two O1 and O2, (1) O1 is dependent on O2 if and only if O1 → O2; (2)O1 and O2 are independent (or concurrent), expressed as O1‖O2, if and only if neither O1 → O2, nor O2 → O1 [22].
Definition 8 Conflict Relation "⊗". Operations O1and O2 conflict with each other, expressed as O1 ⊗ O2, if and only if (1) Om‖On; (2) Target(O1) = Target(O2); (3)Att. Key(O1) = Att. Key(O2); and (4) Att. Value(O1) ≠ Att. Value(O2) [22].
Definition 9 Compatibility Relation "⊙". Two operations O1 and O2 are compatible, expressed as O1 ⊙ O2, if and only if they do not conflict with each other [22].
Definition 10 Compatible Group Set (CGS). Given a group of operations GO, the conflict relationships among these operations can be expressed as a CGS [22].
Definition 11 Different Graphics Coverage (DGC). When the users are editing the different objects because the different objects are compatible with each other. If two or more different objects appear in the same coordinate position, the later object will overlay the object with the editing time earlier.
Definition 12 Concurrent Group Operations (CGO). In graphic editing, if a site in a few short periods of time continues to operate, then we can treat the operations of this site as concurrent group operations. In this paper, concurrent group operations are whole and not indivisible.
Definition 13 Absolute Differences in Coordinates. Assume that the coordinate of the operation intention is (x1, y1) and the actual coordinate position is (x2, y2), \( \partial =\sqrt{{\left({x}_1-{x}_2\right)}^2+{\left({y}_1-{y}_2\right)}^2} \), and if the value of ∂ is greater than a certain number, the operation will be revoked.
Definition 14 Graphical collaborative editing uncertainty. Graphical collaborative editing uncertainty refers to the final results of graphics co-editing being uncertain.
Property 1. Given a group of operations GO targeting the same object, there is a unique MCGS for this GO [22].
The graphical editor consistency model mainly includes three categories as follows: (1) Causal Consistency Model, (2) Results Consistency Model, and (3) Intention Consistency Model.
Causal Consistency Model
Causal consistency, given any two operations O1 and O2, if O1 → O2, O1 happened before O1 at all the sites. Graphics co-editing causal consistency is shown in Fig. 1. Assume that only the site Site1 and Site2 are graphically edited, and the collaborative user at Site1 launches an operation O1, which draws the graphic of the pentagonal star on Canvas. The operation O1 contains the basic information of the five-pointed star graphic object: the ID of the graphic, shape, color, and the position coordinates of the graphic on canvas. After the O1 operation is performed at the site, a five-pointed star appears at the top right of the site's canvas. At the same time, the O1 is sent from Site1 to Site2, and Site2 checks whether there are any other operations that have not been performed in the cache. If no other operations are waiting in the cache, O1 is to be put in the queue of waiting for execution. If not, the other operations to be executed are checked, and then Site2 directly implements the operation O1. The canvas would appear at Site2 the same as at Site1. In a certain period of time, the Site2 produced O2, when O2 satisfies the execution condition at Site2, and then the pentagonal star will appear in the lower right corner on the canvas. O2 was passed from Site2 to Site1 through the Internet. If the execution condition at Site1 is satisfied, O2 is to be executed at Site1.
Causal consistency
There is an important problem in that Site1, Site2,⋯,Siten, according to our experiment summary, because the number of sites approaches nearly 40, which will increasingly become quite complex, as shown in Fig. 2.
Complexity analysis of causal consistency
Results Consistency Model
Results consistency is present when all sites share a copy of the document when a collaborative session is in silent state [21, 42, 43]. There is a great difference between the Results Consistency Model and the Causal Consistency Model. Results consistency between operations does not necessarily require a certain relationship, and the consistency of the results only needs cooperation between sites after the completion of all operations. Ultimately, the results between the stations are the same. The Results Consistency Model is shown in Fig. 3.
Results consistency
The Site1 creates O1 and then executes the operation of O1. There is a five-pointed star in the upper right corner on canvas at Site1. At the same time, O1 was passed from Site1 to Site2 through the Internet, and the transmission needs a certain delay. If the operation O1 has not yet reached the Site2, the Site2 have creates an operation O2, that is, five-pointed star is in the lower left corner of the canvas at Site2. After it executes O2 at Site2, immediately O2 was passed from Site2 to the Site1 through the Internet. When the operation O1 and O2 respectively arrive at Site1 and Site2. What's more, if both satisfy the execution condition, O1 and O2 would be executed at Site1 and Site2. A five-pointed star appears in the lower left corner of the Canvas at Site1, and a five-pointed star appears in the upper right corner on Canvas at Site2. We will find that the final result is the same at Site1 and Site2, although the operation O1 and operation O2 are created in order, but the execution results of O1and O2 are independent of their execution order. According to Property 1, it is not difficult to know. The relationship between O1 and O2 in Fig. 3 is O1 ⊙ O2. In the graphics collaborative editing, assume that MCGS1, MCGS2, ⋯, MCGSn(nϵN∗), GO = {MCGS1, MCGS2, ⋯, MCGSn}, and \( {MCGS}_1=\left\{{O}_{p_1},{O}_{p_2},\cdots, {O}_{p_n}\right\} \), \( {MCGS}_2=\left\{{O}_{k_1},{O}_{k_2},\cdots, {O}_{k_n}\right\},\cdots, {MCGS}_n=\left\{{O}_{r_1},{O}_{r_2},\cdots, {O}_{r_n}\right\} \).
The operation in the MCGS are compatible with each other, is that the operation in the MCGS no matter what kind of implementation order, the last site can maintain consistency. This conclusion is given in detail in the paper, Consistency Maintenance In Real-time Collaborative Image Editing Systems [22, 54, 55], it has been given a detailed proof, this article is not cumbersome.
There is an important problem that the Site1, Site2,⋯, Siten, according to our experiment summary: the number of sites approaches 25 and will become increasingly complex quickly, as shown in Fig. 4.
Intention Consistency Model
Intention Consistency Model is the operation of each site, the operation of the establishment of a certain operational effect among the orders, through the implementation of the scheduling algorithm, through the implementation of scheduling algorithms and conversion functions. As long as the implementation of each step needs to ensure that the implementation does not violate the established operation sequence, and then it is ensured that when all the operations are performed at each site, the internal data structure of each site has the same operational effect order from the operation object; that is, the consistency of the operational intention is maintained [21, 44].
Thus, among the Causal Consistency Model, Results Consistency Model, and Intention Consistency Model, the most complex is the Intention Consistency Model. The efficiency of the Intention Consistency Model can reflect the efficiency of the algorithm of graphic collaborative editing system [45, 46]. In graphic collaborative editing, while satisfying causal consistency or consistency of results, it may not be able to achieve the real intention of the graphical collaborative editing system.
Suppose there are three sites Site1, Site2, and Site3, and then there are four operations, namely O1,O2,O3 and O4. O1‖O2, (O1‖O2) → O3, O1 → O4, O2‖O4, O3‖O4, and the operation O3 is lost during processing from Site1 to Site2, as shown in Fig. 5.
Transfer of operations between different sites
When there are many sites in the collaborative graphic editing, the relationship among the operations is complex. The Causal Consistency Model can only solve the problems about graphics collaborative editing in a relatively simple situation, such as when the operations among the sites are compatible with each other. The Results Consistency Model can ensure that the final result of graphics co-editing is right, but it cannot guarantee that the final result can be expected for collaborative editing users [47, 48]. For example, there is a concurrency model in Fig. 5, and the Causal Consistency Model cannot be used; thus, two models can be used: one is the Result Consistency Model, and the other is the Intention Consistency Model. Use the Results Consistency Model to handle the situation, as shown in Fig. 6. A group of operations (GO) represents the set of operations creates by all sites. MCGS (Max Compatible Groups Set) represents a set of the largest compatible sets of operations that operate in GO. At the beginning of the graphics co-editing, GO = {} and MCGS = {}, indicating that the operation in GO and MCGS is empty. The process of results consistency is as follows: (1) When O1and O2 arrive at Site1, variables are updated, GO = {O1, O2}. Because of O1‖O2, MCGS1 = {O1}, MCGS2 = {O2}. When the Site1 creates O3, GO = {O1, O2, O3}, (O1‖O2) → O3, MCGS1 = {O1, O3}, MCGS2 = {O2, O3}. When O4 arrive at Site1, variables updated, GO = {O1, O2, O3, O4}. Because of O1 → O4, O2‖O4, O3‖O4, so MCGS1 = {O1, O3}, MCGS2 = {O2, O3}, MCGS3 = {O1, O4}. Finally, GO = {{O1, O3}, {O2, O3}, {O1, O4}}. (2) When the Site2 creates O1, variables updated, GO = {O1}, MCGS = {O1}, when Site2 creates O1 creates O2, variables updated, GO = {O1, O4}, because of O1 → O4, MCGS = {O1, O4}. Due to network congestion [49,50,51,52,53], because of the cutoff time delay, Site2 has not received O3, which has passed Site1, and Site2 requires Site1 resend. Site1 receives a resending request from Site1, and then resends O3. After Site2 has received O3, the variables are updated, GO = {O1, O3,O4}. Because of O1 ⊙ O3, O1‖O4, O3‖O4, MCGS1 = {O1, O3}, MCGS2 = {O4}. When O2 arrives at Site2, the variables are updated, GO = {O1, O2, O3, O4}, O2 ⊙ O3, O2‖O1, O2‖O4, and MCGS1 = {O1, O3}, MCGS2 = {O2, O3}, MCGS3 = {O1, O4}. GO = {{O1, O3}, {O2, O3}, {O1, O4}}. (3) When Site3 creates O2, variables are updated, GO = {O2}, MCGS = {O2}. When O1arrive at Site3, variables are updated: GO = {O1, O2}, O1‖O2, so MCGS1 = {O1}, MCGS2 = {O2}. When O3 arrives at Site3, variables are updated: GO = {O1, O2, O3}, O3 ⊙ O2, O3 ⊙ .
Theoretical realization of results consistency
O1, MCGS1 = {O1, O3}, MCGS2 = {O2, O3}. When O4 arrives at Site3, variables are updated: GO = {O1, O2, O3, O4}, O4 ⊙ O1, O4‖O2, O4‖O3, MCGS1 = {O1, O3},MCGS2 = {O2, O3},MCGS3 = {O1, O4}, GO = {{O1, O3}, {O2, O3}, {O1, O4}}. To sum up, we can know the basic principle of the Results Consistency Model, and the advantages and disadvantages of the Results Consistency Model are obvious. Advantages of the Results Consistency Model are as follows. In a set of operations, the operations in MCGS do not take into account the order of execution between them, which is more efficient for graphics co-editing. However, the shortcomings of the Results Consistency Model are also very prominent, according to the graphical collaborative editing uncertainly (Definition 14). The result of the Results Consistency Model is uncertain, as shown in Fig. 5, although three sites get the consistency of results finally. The collaborative users only expect that the result of collaborative co-editing is unique so that we have to further improve the coherence of graphics collaborative editing. We need a better consistency model than the Results Consistency Model, so the Intention Consistency Model will be a good choice, and next the common graphics collaborative editing algorithm (CGCE algorithm) for the Intention Consistency Model will be described in detail.
CGCE algorithm
Flowchart schematic diagram of intention consistency
A flowchart schematic diagram of the Intention Consistency Model is shown in Fig. 7.
Schematic diagram of intention consistency
Details of the CGCE algorithm
The core technology of editing consistency in the above collaborative graphics is about researching the intention consistency model and its algorithm for the complex. Because intention consistency is the most important consistency in collaborative graphic editing. The graphics collaborative editing consistency key technology CGCE algorithm is shown in Algorithm 1–8.
Realization of the key technology of consistency
Graphical collaborative editing software
Our study refers to the realization of the algorithm the common graphics collaborative editing algorithm (CGCE algorithm). This is done to perfect and expand not only the definition of graphics collaborative editing but also HTML5 Canvas, WebSocket, jQuery and Node.js and other network programming language and technology in order to realize the system of the key technology of graphics co-editing. The interface of graphic collaborative editing software is as shown in Fig. 8. As shown in Fig. 9, the main functions of the code are graphic drawing, including the drawing of arbitrary points, lines, and surfaces, and the drawing of some basic figures as shown in Fig. 10.
Interface of graphic collaborative editing software
Graphic collaborative editing software code file
Fig. 10
Drawing graphics
The network conditions of graphics collaborative editing and drawing are mainly composed of three modes:
1. The local mode: This mode is used to open multiple pages on a computer. Multiple pages can be co-edited to the graphics, and each page in the open four local pages is like a drawing board and can be drawn on each page. The content they display on the pages is exactly the same.
The LAN mode: In the same LAN, two users can open graphics and edit software at the same time and also can produce the same effect, as shown in Fig. 11.
The WAN mode: The wide area network mode is more complex. For example, the high concurrency and packet loss cases in this paper are nearly all in WAN.
Co-editing of local and LAN multiweb pages
The consistency of graphics collaborative editing system on MacBook Pro is shown in Fig. 12. The consistency of graphics collaborative editing system on the Windows system is shown in Fig. 13.
Graphics co-editing system on Macintosh
Graphics co-editing system in Windows
Graphical collaborative editing software port monitoring
The server monitoring information of monitoring graphics collaborative editing is shown in Fig. 14. The server monitoring information of graphical collaborative editing can be used by each user to edit each operation of the graphics together and generate the corresponding log files.
Graphics cooperative editing server monitoring
There are no issues of high concurrency and packet loss in graphics collaborative editing software running on local and LAN networks, but there may be high concurrency or packet loss problems in WAN networks. If the routing path of two cooperative editors is long, the time delay is great. It is difficult to match the real-time situation; what's more, it is easy to lead to concurrent problems. However, the algorithm in the paper, which is called CGCE algorithm, is excellent and ensures that the graphics collaborative editing will be good. The cooperative multiuser operation of WAN networks is shown in Fig. 15.
Cooperative operation of multi-user in WAN
Analysis of time complexity and space complexity of the algorithm
The analysis of time complexity and space complexity of the algorithm.
Time complexity: The time complexity of the best one is O1. For example, the operations of inserting, deleting, rotating, and copying in the best case of graphical collaborative editing can be completed at one time. However, the worst is that all operations of multiusers exist concurrently, then the time complexity is O(nm), sufficient data from experiments shows the average time complexity of the algorithm is O(nlogn).
Space complexity: In order to maintain the normal operation of the algorithm, the experiments in this paper at least open up three cache queues in the memory. If each queue has n unit space, the algorithm has a space complexity of O(3n). For the queue, then the algorithm's spatial complexity is O(nm). Assuming that there is a queue of length m, and then the required space complexity is O(nm).
Time delay, packet loss, and bit error rate of software data transmission
This paper is used to test the software data transmission delay, packet loss rate, and bit error rate. The software is ATKKPING, and the network packet loss testing tool (ATKKPING) is a flat enhancement program network packet loss rate testing software. ATKKPING is mainly used to perform packet loss test. You can test the packet loss situation of the intranet or extranet, thus providing an important reference for solving a series of packet loss and BER issues. Test the level of the network environment and how much is lost. The network drop test tool (ATKKPING) interface is shown in Fig. 16.
Time delay, packet loss and bit error rate of software
From the network test tool interface as shown in Fig. 16, we can know that, when the "Ping interval" is selected, the Ping interval is only used as a timeout, and Ping is performed fastest. If the data size value is set greater than the MTU value, the data packet must be partitioned. The MTU depends on the physical layer. Therefore, special attention should be paid to the size of the MTU, especially when pinging to the Internet. First, the relevant algorithm of this paper is not added in the graphical collaborative editing software of Fig. 17, and then the relevant parameters of the network testing software are configured. The "target host" is 192.168.1.163, the "ping interval" is 1 millisecond, "ping log" is selected, "Ping Times" is 4, and then click the "Start" button, as shown in Fig. 17.
Pre-CGCA testing
From Fig. 18, we can know that the packet loss rate is 25% for the graphical collaboration software without CGCA algorithm in this paper. The minimum transmission delay is 2 milliseconds about data, the maximum value is 14 milliseconds, and the average value is 8 milliseconds. The packet loss rate is relatively much larger than post-CGCA Algorithm. Configure the same parameters in the network test tool and click the "Start" button. As shown in Fig. 16, the number of sent packets is 1659, 18 of which are timeouts; thus, the packet loss rate is 1.08%. The minimum transmission delay is 1 millisecond, the maximum delay is 214 milliseconds, and the average delay is 6.14 milliseconds. Through analysis of the data, it can be clearly known that the maximum delay of transmission is 214 milliseconds. If it exceeds 214 milliseconds, the packet will be lost. Above all, after the CGCE algorithm is added in the graphics co-editing software, the packet loss rate is reduced to 1.08%, which is nearly 24% lower than the comparison, and the average delay of data transmission has also been reduced.
Post-CGCA testing
Graphical collaborative editing plays an increasingly important role in CSCW. The most important technique in graphics co-editing is the consistency of graphics co-editing, which mainly includes causality consistency, consistency of results, and consistency of intention. In order to solve the consistency conflict problem of graph collaborative editing, the CGCE algorithm is proposed in this paper, and a large number of experiments show that this algorithm plays an irreplaceable role in performance optimization. The CGCE algorithm in this paper can solve the contradictions in the consistency of graphical collaborative editing, the research in this paper has particularity and results, and it will proved by the experiment. However, due to the relationship of time, the algorithm of the paper is more or less inadequate, whic is the key import that some scholars can point to as the shortcoming in the paper, especially the optimization and improvement of CGCE algorithm. A mathematical definition of some graphic cooperative editors can be easily followed. The scholars are going deeper and deeper. This paper has a strict mathematical definition of some basic operations of graphic collaborative editing, which can facilitate the follow-up scholars to carry out more in-depth and complex academic research.
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The authors appreciate all anonymous reviewers for their insightful comments and constructive suggestions to polish this paper in high quality. This research was supported by the National Key Research and Development Program of China (No. 2018YFC0810204), National Natural Science Foundation of China (No. 61872242, 61502220), Shanghai Science and Technology Innovation Action Plan Project (17511107203, 16111107502) and Shanghai key lab of modern optical system.
We can provide the data.
Chunxue Wu (1964- ) received the Ph.D. degree in Control Theory and Control Engineering from China University of mining and technology, Beijing, China, in 2006. He is a Professor with the Computer Science and Engineering and software engineering Division, School of Optical-Electrical and Computer Engineering, University of Shanghai for Science and Technology, China. His research interests include, wireless sensor networks, distributed and embedded systems, wireless and mobile systems, networked control systems.
Langfeng Li is a student of University of Shanghai for Science and Technology, and E-mail is [email protected].
Changwei Peng (1994- ) received the B.E. degree from Hubei Polytechnic University. HuangShi, Hubei, China, in 2013, He is currently pursuing the master degree in computer technology with the University of Shanghai for science and technology, China. His research interests include Mechanical learning and Data analysis.
YAN WU is currently a postdoctoral associate at the school of public and environmental affairs, Indiana University Bloomington. He obtained his PhD degree in Southern Illinois University Carbondale, with concentrations in environmental chemistry and ecotoxicology. His research involves elucidations of environmental fate of contaminants using chemical and computational techniques, as well as predictions of their associated effects on wildlife and public health. Data Processing and Analysis in Environmental Related Fields. E-mail is [email protected] and ORCID is 0000-0001-7876-261X
Naixue Xiong is currently an Associate Professor (3rd year) at School of Computer Science and Technology, Tianjin University, Tianjin, China, 300350. He received his both PhD degrees in Wuhan University (about software engineering), and Japan Advanced Institute of Science and Technology (about dependable networks), respectively. Before he attends Colorado Technical University, he worked in Wentworth Technology Institution, Georgia State University for many years. His research interests include Cloud Computing, Security and Dependability, Parallel and Distributed Computing, Networks, and Optimization Theory. Dr./Prof. Xiong published over 100 international journal papers and over 100 international conference papers. Some of his works were published in IEEE JSAC, IEEE or ACM transactions, ACM Sigcomm workshop, IEEE INFOCOM, ICDCS, and IPDPS. He has been a General Chair, Program Chair, Publicity Chair, PC member and OC member of over 100 international conferences, and as a reviewer of about 100 international journals, including IEEE JSAC, IEEE SMC (Park: A/B/C), IEEE Transactions on Communications, IEEE Transactions on Mobile Computing, IEEE Trans. on Parallel and Distributed Systems. He is serving as an Editor-in-Chief, Associate editor or Editor member for over 10 international journals (including Associate Editor for IEEE Tran. on Systems, Man & Cybernetics: Systems, and Editor-in-Chief for Journal of Parallel & Cloud Computing (PCC)), and a guest editor for over 10 international journals, including Sensor Journal, WINET and MONET.
Prof. Changhoon Lee, Department of Computer Science and Engineering, Seoul National University of Science and Technology (SeoulTech), Republic of Kornbea. Email: [email protected]
School of Optical-Electrical and Computer Engineering, University of Shanghai for Science and Technology, Shanghai, 200093, China
Chunxue Wu
, Langfeng Li
& Changwei Peng
School of Public and Environmental Affairs, Indiana University, Bloomington, IN, 47405, USA
Yan Wu
College of Intelligence and Computing, Tianjin University, Tianjin, 300350, China
Naixue Xiong
Department of Computer Science and Engineering, Seoul National University of Science and Technology (SeoulTech), Seoul, Republic of Korea
Changhoon Lee
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CW conceived and designed the study. LL, CP, YW, NX and CL participated in experiments and data processing and edited the manuscript. All authors read and approved the final manuscript.
Correspondence to Naixue Xiong.
Wu, C., Li, L., Peng, C. et al. Design and analysis of an effective graphics collaborative editing system. J Image Video Proc. 2019, 50 (2019) doi:10.1186/s13640-019-0427-6
Received: 26 September 2018
CSCW
Graphics co-editing
Real-time Image and Video Processing in Embedded Systems for Smart Surveillance Applications | CommonCrawl |
Mathematics > Dynamical Systems
arXiv:2107.05149 (math)
[Submitted on 11 Jul 2021 (v1), last revised 2 Jun 2022 (this version, v5)]
Title:Infinite Lifting of an Action of Symplectomorphism Group on the set of Bi-Lagrangian Structures
Authors:Bertuel Tangue Ndawa
Abstract: We consider a smooth $2n$-manifold $M$ endowed with a bi-Lagrangian structure $(\omega,\mathcal{F}_{1},\mathcal{F}_{2})$. That is, $\omega$ is a symplectic form and $(\mathcal{F}_{1},\mathcal{F}_{2})$ is a pair of transversal Lagrangian foliations on $(M, \omega)$. Such structures have an important geometric object called the Hess Connection. Among the many importance of these connections, they allow to classify affine bi-Lagrangian structures.
In this work, we show that a bi-Lagrangian structure on $M$ can be lifted as a bi-Lagrangian structure on its trivial bundle $M\times\mathbb{R}^n$. Moreover, the lifting of an affine bi-Lagrangian structure is also an affine bi-Lagrangian structure. We define a dynamic on the symplectomorphism group and the set of bi-Lagrangian structures (that is an action of the symplectomorphism group on the set of bi-Lagrangian structures). This dynamic is compatible with Hess connections, preserves affine bi-Lagrangian structures, and can be lifted on $M\times\mathbb{R}^n$. This lifting can be lifted again on $\left(M\times\mathbb{R}^{2n}\right)\times\mathbb{R}^{4n}$, and coincides with the initial dynamic (in our sense) on $M\times\mathbb{R}^n$ for some bi-Lagrangian structures. Results still hold by replacing $M\times\mathbb{R}^{2n}$ with the tangent bundle $TM$ of $M$ or its cotangent bundle $T^{*}M$ for some manifolds $M$.
Comments: 25 pages, 2figure
Subjects: Dynamical Systems (math.DS)
Cite as: arXiv:2107.05149 [math.DS]
(or arXiv:2107.05149v5 [math.DS] for this version)
From: Bertuel Tangue Ndawa [view email]
[v1] Sun, 11 Jul 2021 23:48:14 UTC (101 KB)
[v2] Mon, 2 Aug 2021 15:29:24 UTC (101 KB)
[v3] Wed, 2 Mar 2022 16:21:34 UTC (18 KB)
[v4] Wed, 16 Mar 2022 08:50:33 UTC (19 KB)
[v5] Thu, 2 Jun 2022 11:48:18 UTC (18 KB)
math.DS | CommonCrawl |
Journal of Scientific Computing
June 2019 , Volume 79, Issue 3, pp 1777–1800 | Cite as
Interpolatory HDG Method for Parabolic Semilinear PDEs
Bernardo Cockburn
John R. Singler
Yangwen Zhang
First Online: 22 January 2019
We propose the interpolatory hybridizable discontinuous Galerkin (Interpolatory HDG) method for a class of scalar parabolic semilinear PDEs. The Interpolatory HDG method uses an interpolation procedure to efficiently and accurately approximate the nonlinear term. This procedure avoids the numerical quadrature typically required for the assembly of the global matrix at each iteration in each time step, which is a computationally costly component of the standard HDG method for nonlinear PDEs. Furthermore, the Interpolatory HDG interpolation procedure yields simple explicit expressions for the nonlinear term and Jacobian matrix, which leads to a simple unified implementation for a variety of nonlinear PDEs. For a globally-Lipschitz nonlinearity, we prove that the Interpolatory HDG method does not result in a reduction of the order of convergence. We display 2D and 3D numerical experiments to demonstrate the performance of the method.
Hybridizable discontinuous Galerkin method Interpolatory method Newton iteration
Mathematics Subject Classification
65M60 65L12
J. Singler and Y. Zhang were supported in part by National Science Foundation Grant DMS-1217122. J. Singler and Y. Zhang thank the IMA for funding research visits, during which some of this work was completed. Y. Zhang thanks Zhu Wang for many valuable conversations.
Compliance with Ethical Standards
The authors declare that they have no conflict of interest.
Implementation Details for General Nonlinearities
The Interpolatory HDG Formulation
The full Interpolatory HDG discretization is to find \((\varvec{q}^n_h,u^n_h,\widehat{u}^n_h)\in \varvec{V}_h\times W_h\times M_h\) such that
$$\begin{aligned} \begin{aligned}&(\varvec{q}^n_h,\varvec{r})_{\mathcal {T}_h}-(u^n_h,\nabla \cdot \varvec{r})_{\mathcal {T}_h}+\left\langle \widehat{u}^n_h,\varvec{r} \cdot \varvec{n} \right\rangle _{\partial {\mathcal {T}_h}} = 0, \\&(\partial ^+_tu^n_h,w)_{{\mathcal {T}}_h}+(\nabla \cdot \varvec{q}^n_h, w)_{\mathcal {T}_h}+\langle \tau (u_h^n - \widehat{u}_h^n),w\rangle _{\partial {\mathcal {T}_h}} + ( {\mathcal {I}}_h F(-\varvec{q}_h^n, u_h^n),w)_{\mathcal {T}_h}{=} (f^n,w)_{\mathcal {T}_h},\\&\langle {\varvec{q}}^n_h\cdot \varvec{n} + \tau (u_h^n - \widehat{u}_h^n), \mu \rangle _{\partial {\mathcal {T}_h}\backslash \varepsilon ^{\partial }_h} =0,\\&u^0_h =\varPi _W u_0, \end{aligned} \end{aligned}$$
for all \((\varvec{r},w,\mu )\in \varvec{V}_h\times W_h\times M_h\) and \(n=1,2,\ldots ,N\). Similar to Sect. 3.2, we have
$$\begin{aligned} ( {\mathcal {I}}_h F(-\varvec{q}_h^n, u_h^n),w)_{\mathcal {T}_h} = A_1 {\mathcal {F}}(\varvec{\alpha }^n, \varvec{\beta }^{n},\varvec{\gamma }^{n}), \end{aligned}$$
$$\begin{aligned} {\mathcal {F}}(\varvec{\alpha }^{n},\varvec{\beta }^{n},\varvec{\gamma }^{n}) = [F(\alpha _1^{n},\beta _1^{n}, \gamma _1^{n}),\ldots ,F(\alpha _{N_1}^{n},\beta _{N_1}^{n}, \gamma _{N_1}^{n})]^T. \end{aligned}$$
Then the system (45) can be rewritten as
$$\begin{aligned} \underbrace{\begin{bmatrix} A_1&\quad 0&\quad -\,A_2&\quad A_4 \\ 0&\quad A_1&\quad -\,A_3&\quad A_5 \\ A_2^T&\quad A_3^T&\quad A_6 +{\varDelta t}^{-1}A_1&\quad -\,A_7\\ A_4^T&\quad A_5^T&\quad A_7^T&\quad -\,A_8 \end{bmatrix}}_{M} \underbrace{\left[ {\begin{array}{*{20}{l}} \varvec{\alpha }^{n}\\ \varvec{\beta }^{n}\\ \varvec{\gamma }^{n}\\ \varvec{\zeta }^{n} \end{array}} \right] }_{\varvec{x}_{n}}+ \underbrace{\left[ {\begin{array}{*{20}{l}} 0\\ 0\\ A_1 {\mathcal {F}}(\varvec{\alpha }^{n},\varvec{\beta }^{n},\varvec{\gamma }^{n})\\ 0 \end{array}} \right] }_{{\mathscr {F}}(\varvec{x}_{n})} =\underbrace{\left[ {\begin{array}{*{20}{l}} 0\\ 0\\ b_1^n+{\varDelta t}^{-1}A_1\varvec{\gamma }^{n-1} \\ 0 \end{array}} \right] }_{\varvec{b}_n}, \end{aligned}$$
i.e., \( M\varvec{x}_n + {\mathscr {F}}(\varvec{x}_n) = \varvec{b}_n \).
Newton's method proceeds as in Sect. 3.2, but the Jacobian matrix \(G'(\varvec{x}_n^{(m-1)})\) is now given by
$$\begin{aligned} G'(\varvec{x}_n^{(m-1)}) = M+{\mathscr {F}}'(\varvec{x}_n^{(m-1)}), \quad {\mathscr {F}}'(\varvec{x}_n^{(m-1)}) = \begin{bmatrix} 0&\quad 0&\quad 0&\quad 0 \\ 0&\quad 0&\quad 0&\quad 0 \\ A_{11}^{n,(m)}&\quad A_{12}^{n,(m)}&\quad A_{13}^{n,(m)}&\quad 0\\ 0&\quad 0&\quad 0&\quad 0 \end{bmatrix}, \end{aligned}$$
where for \( k = 1, 2, 3, \) we define
$$\begin{aligned}&A_{1k}^{n,(m)} = A_1\text {diag}\big ({\mathcal {F}}_k'(\varvec{\alpha }^{n,(m-1)},\varvec{\beta }^{n,(m-1)},\varvec{\gamma }^{n,(m-1)})\big ),\\&{\mathcal {F}}_k'(\varvec{\alpha }^{n,(m-1)},\varvec{\beta }^{n,(m-1)},\varvec{\gamma }^{n,(m-1)}) \\&\quad = \big [F_k'(\alpha _1^{n,(m-1)},\beta _1^{n,(m-1)}, \gamma _1^{n,(m-1)}),\cdots ,F_k'(\alpha _{N_1}^{n,(m-1)},\beta _{N_1}^{n,(m-1)}, \gamma _{N_1}^{n,(m-1)})\big ]^T, \end{aligned}$$
and \(F_k'\) denotes the partial derivative of F with respect to the kth variable. Therefore, the linear system that must be solved is now given by
$$\begin{aligned} \begin{bmatrix} A_1&\quad 0&\quad -\,A_2&\quad A_4 \\ 0&\quad A_1&\quad -\,A_3&\quad A_5 \\ A_2^T+ A_{11}^{n,(m)}&\quad A_3^T+ A_{12}^{n,(m)}&\quad A_6 +{\varDelta t}^{-1}A_1+ A_{13}^{n,(m)}&\quad -\,A_7\\ A_4^T&\quad A_5^T&\quad A_7^T&\quad -\,A_8 \end{bmatrix} \left[ {\begin{array}{*{20}{c}} \varvec{\alpha }^{n,(m)}\\ \varvec{\beta }^{n,(m)}\\ \varvec{\gamma }^{n,(m)}\\ \varvec{\zeta }^{n,(m)} \end{array}} \right] ={\widetilde{\varvec{b}}}, \end{aligned}$$
$$\begin{aligned} {\widetilde{\varvec{b}}} = G'(\varvec{x}_n^{(m-1)}) \varvec{x}_n^{(m-1)} - G(\varvec{x}_n^{(m-1)}). \end{aligned}$$
Local Solver
The system (49) can be rewritten as
$$\begin{aligned} \begin{bmatrix} B_1&\quad B_2&\quad B_3\\ B_4&\quad B_5&\quad -\,B_6\\ B_3^T&\quad B_6^T&\quad B_7\\ \end{bmatrix} \left[ {\begin{array}{*{20}{c}} \varvec{x}\\ \varvec{y}\\ \varvec{z} \end{array}} \right] =\left[ {\begin{array}{*{20}{c}} b_1\\ b_2\\ b_3 \end{array}} \right] , \end{aligned}$$
where \(\varvec{x}=[\varvec{\alpha ^{n,(m)}};\varvec{\beta ^{n,(m)}}]\), \(\varvec{y}=\varvec{\gamma }^{n,(m)}\), \(\varvec{z}=\varvec{\zeta }^{n,(m)}\), \( {\widetilde{\varvec{b}}} = [ b_1;b_2;b_3] \), and \(\{B_i\}_{i=1}^7\) are the corresponding blocks of the coefficient matrix in (49). The system (51) is equivalent with following equations:
$$\begin{aligned} B_1 \varvec{x} + B_2\varvec{y} +B_3\varvec{z}&= b_1, \end{aligned}$$
$$\begin{aligned} B_4 \varvec{x} +B_5\varvec{y} -B_6\varvec{z}&= b_2, \end{aligned}$$
(52b)
$$\begin{aligned} B_3^T\varvec{x}+ B_6^T\varvec{y} + B_7 \varvec{z}&=b_3. \end{aligned}$$
(52c)
Similar to before, the matrices \(B_1\) and \(B_5\) are block diagonal with small blocks and they can be easily inverted. Use (52a) and (52b) to express \(\varvec{x}\) and \(\varvec{y}\) in terms of \(\varvec{z}\) as follows:
$$\begin{aligned} \varvec{x}&= B_1^{-1}B_2\left( B_4B_1^{-1}B_2+B_5\right) ^{-1}\left( (B_6+B_4B_1^{-1}B_3)\varvec{z}+ b_2-B_4B_1^{-1}b_1\right) \nonumber \\&\quad -B_1^{-1}B_3\varvec{z} + B_1^{-1}b_1\nonumber \\&=:{\tilde{B}}_1 \varvec{z} +{\tilde{b}}_1, \end{aligned}$$
$$\begin{aligned} \varvec{y}&=\left( B_4B_1^{-1}B_2+B_5\right) ^{-1} \left( (B_6+B_4B_1^{-1}B_3)\varvec{z} + b_2-B_4B_1^{-1}b_1\right) \nonumber \\&=:{\tilde{B}}_2 \varvec{\gamma }^n +{\tilde{b}}_2, \end{aligned}$$
$$\begin{aligned} Q = B_4B_1^{-1}B_2+B_5 = B_4B_1^{-1}B_2 +A_6 +{\varDelta t}^{-1}A_1+ A_{13}^{n,(m)}. \end{aligned}$$
As in Sect. 3.3, the matrix Q is block diagonal with small blocks. Since \(A_1\) is positive definite, if \(\varDelta t\) is small enough then Q is easily inverted. Then we insert \(\varvec{x}\) and \(\varvec{y}\) into (26c) and obtain the final system only involving \(\varvec{z}\):
$$\begin{aligned} (B_3^T {\tilde{B}}_1 + B_5^T {\tilde{B}}_2 + B_6) \varvec{z} = b_3 -B_3^T{\tilde{b}}_1 -B_5^T {\tilde{b}}_2. \end{aligned}$$
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© Springer Science+Business Media, LLC, part of Springer Nature 2019
Email authorView author's OrcID profile
1.School of MathematicsUniversity of MinnesotaMinneapolisUSA
2.Department of Mathematics and StatisticsMissouri University of Science and TechnologyRollaUSA
3.Department of Mathematical SciencesUniversity of DelawareNewarkUSA
Cockburn, B., Singler, J.R. & Zhang, Y. J Sci Comput (2019) 79: 1777. https://doi.org/10.1007/s10915-019-00911-8
Received 10 February 2018
Revised 08 December 2018
Accepted 12 January 2019
First Online 22 January 2019
DOI https://doi.org/10.1007/s10915-019-00911-8
Publisher Name Springer US | CommonCrawl |
Reading: Climatic benefits of black carbon emission reduction when India adopts the US onroad emissio...
Climatic benefits of black carbon emission reduction when India adopts the US onroad emission level
Ashish Sharma ,
Department Civil Engineering, University of Toledo, Toledo, OH 43606, USA, US
Chul E. Chung
Desert Research Institute, Reno, NV 89512, USA, US
India is known to emit large amounts of black carbon (BC) particles, and the existing estimates of the BC emission from the transport sector in the country widely range from 72 ~ 456 Gg/year (for the 2000's). First, we reduce the uncertainty range by constraining the existing estimates by credible isotope analysis results. The revised estimate is from 74 ~ 254 Gg/year. Second, we derive our own BC estimate of the transport section in order to gain a new insight into the mitigation strategy and value. Our estimate shows that the transport section BC emission would be reduced by about 69% by adopting the US standards. The highest emission reduction comes from the vehicles in the 5–10 year old age group. The minimum emission reduction would be achieved from the vehicles in the 15–20 year old age category since their population is comparatively small in comparison to other age categories. The 69 % of 74 ~ 254 Gg/year is 51 ~ 175 Gg/year, which is the estimated BC emission reduction by switching to the US on-road emission standard. Assuming that global BC radiative forcing is 0.88 Wm−2 for 17.8 Tg/year of BC emission, we find that the reduced BC emission translates into −0.0025 ~ −0.0087 W m−2 in global forcing. In addition, we find that 51 ~ 175 Gg of BC emission reduction amounts to 0.046 – 0.159 B carbon credits which are valued at 0.56 – 1.92 B US dollars (using today's carbon credit price). In a nutshell, India could potentially earn billions of dollars per year by switching from the current on-road emission levels to the US levels.
Keywords: Indian road transport, Diesel emissions, Gasoline emissions, Black carbon, BC forcing, Climatic benefits
How to Cite: Sharma, A. and Chung, C.E., 2015. Climatic benefits of black carbon emission reduction when India adopts the US onroad emission level. Future Cities and Environment, 1, p.13. DOI: http://doi.org/10.1186/s40984-015-0013-8
Accepted on 20 Dec 2015 Submitted on 13 Jul 2015
Black carbon (BC) is a product that results from incomplete combustion. Black carbon (BC) is also known as "soot" or "soot carbon" [1]. BC aerosols are emitted as primary aerosols from fossil fuel combustion, biomass burning and biofuel burning, and thus largely anthropogenic. Specifically, the combustion of diesel and coal, the burning of wood and cow dung, savanna burning, forest fire and crop residue burning are the common sources for BC. In order to improve air quality, developed countries have reduced ambient aerosol concentration by a variety of measures in the last few decades. For instance, wood as the fuel for cooking was replaced by natural gas or electricity. This kind of clean-air act not only reduced the overall aerosol concentration (including BC concentration) but also reduced the relative amount of BC to other anthropogenic aerosols such as sulfate, as evident from the state-of-the-art emission estimate dataset by [2]. Developing countries, conversely, have high levels of aerosol concentration and also a relatively large amount of BC [2]. India too, as a developing nation, exhibits these characteristics. The BC emission in India has steadily increased [3].
BC has many unique aspects. First, while most aerosols scatter solar radiation and thus act to cool the earth, BC strongly absorbs sunlight and contributes to the global warming [4]. Second, while CO2 itself is not an air pollutant, BC is both an air pollutant and climate warmer. Thus, reducing BC concentration is more easily justified than reducing CO2 concentration. Third, BC emission is generally much easier to mitigate than CO2 emission, since the former originates largely from poor life styles in developing countries. For example, it is much easier and cheaper to replace a cow-dung burning facility by a modern natural-gas stove in a kitchen than installing a solar panel. Fourth, since aerosols stay in the atmosphere for less than a few weeks, reducing BC emission results in an immediate reduction in BC concentration, whereas reducing CO2 emission leads to a reduction in CO2 concentration many decades later. In view of this fourth aspect, Ramanathan and Xu [5] and Shindell et al. [6] demonstrated using climate models that reducing BC emission is among the most effective tools to slow down the warming immediately.
In the current study, we aim to quantify the BC emission from the transport sector in India and how much this BC emission can be reduced by adopting the US on-road emission rates immediately. We do this because vehicles in India emit far more particulate matter (i.e., far more aerosols) per vehicle than in the West. The aerosols emitted from vehicles consist largely of black carbon [7]. In comparison, biofuel combustion emits a relatively more organic carbon and less black carbon [7]. While BC is definitely a climate warmer, organic carbon may be a cooler [8, 9]. Thus, BC emission decrease in the transport sector seems more appealing in combating the global warming than that in biofuel or biomass burning. Accordingly, [10], for instance, suggested that diesel engine is one of a few good examples for reducing BC emission and fighting the global warming. Diesel engines are the main contributor to aerosol emission from the transport sector [11–13]. Furthermore, mitigating diesel engine emissions would reduce BC concentration with a relatively small reduction in sulfate (a cooling agent), whereas mitigating emissions from coal combustion in power plants would reduce both BC and sulfate substantially [14]. Thus, quantifying the BC emission in the transport section is very valuable in the global warming mitigation study.
Studies exist that estimated the BC emission from the road sector in India [7, 15–18]. These estimations give a widely-varying range of 71.76 ~ 456 Gg in the annual emission, and also a wide range of 6.5 % ~ 34 % in the percentage of the total BC emissions by the road transport sector. This large uncertainty in estimated BC emission or its contribution to total BC emission makes it difficult for policy makers to make decisions. Thus, one of the objectives in the present study is to reduce this uncertainty.
The novelty of our study is also that it quantifies the potential climatic benefits of mitigating the road transport sector BC emissions in India via implementation of the US on-road emission levels, which are more stringent than the Indian levels. Previous studies in this regard [19–21] quantified the percentage reduction by applying EU standards and our study is the first study to quantify the percentage reduction by applying the US standard. Applying the US standard has advantages because US standards have similar emission requirements for both diesel and gasoline vehicles. Europe emission regulations, relative to the counterpart U.S. program, tolerate higher PM emissions from diesel vehicles. Applying the US emission standards in India is particularly more important in order to target BC emissions from the heavy duty diesel vehicles (buses and trucks) in India. Not only do we quantify the BC emission reduction in switching to the US standard, we also translate this reduction into the climatic benefit by applying observationally-constrained (thus accurate) BC climate forcing estimation studies and today's carbon emission price.
Lastly, in the present study, we will also attempt to quantify the contribution of different categories of vehicles towards the transport sector BC emissions in India for the year 2010 according to vehicle age. We do this, because this further information would be very valuable to environmental policy makers. We organize the paper in 4 sections. In Section 1, we provided a general overview of how this study was conducted and we highlighted the present state of transport sector emissions in India. Here, we also clarified some of the key findings from the similar studies conducted in the past and the shortcomings of the existing studies. In section 2, we discuss the methods we adopted for revising the existing estimates of the transport-sector BC emission in India and what approach we adopted for providing our own estimate the transport-sector BC emission in India. In Section 3, we discusses results. Here, we provided our own estimate of transport section BC emission and BC emission reduction by implementing higher emission standards. Section 3 also pertains to the BC forcing reduction and its monetary value and Section 4 is dedicated for discussions and conclusions.
Revising the existing estimates of the transport-sector BC emission in India
As stated earlier, the previous estimates of the BC emission from the road sector in India give a widely-varying range of 71.76 ~ 456 Gg in the annual emission, and also a wide range of 6.5 % ~ 34 % in the percentage of the total BC emissions by the road transport sector [7, 15–18]. The aforementioned estimates are based on a bottom-up approach, and there is a wide range in the estimates due to uncertainty in (a) fleet average emission factors and (ii) modelling of the on-road vehicle stock. Additionally, emission inventories without calibrating the national fuel balance would have much higher uncertainties [22].
The aforementioned previous BC emission estimates did not utilize the isotope analysis results by Gustafsson et al. [23]. Most of carbon in the earth is carbon-12 (12C). 14C, also referred to as radiocarbon, is a radioactive isotope of carbon, and decays into nitrogen-14 over thousands of years. Live plants and animals maintain a high ratio of 14C to 12C by photosynthesis, vegetable eating and carnivores eating herbivores, as the source for 14C is cosmic rays in the atmosphere. Thus, biomass contains a high ratio of 14C to 12C. On the other hand, fossil fuel arose from vegetation and animals that died a long time ago, and therefore contains no 14C. The ratio of 14C to 12C is thus proportional to the ratio of biomass to fossil fuel. Gustafsson et al. [23] analyzed 14C mass and 12C mass data in collected aerosols, and apportioned the carbon between fossil fuel combustion and biomass/biofuel burning sources. Unlike in the previous BC emission estimates, the apportionment based on carbon isotope data should be considered non-controversial and credible. Furthermore, the aerosols collected for the analysis were in the South Asian outflow instead of near emission sources, which means that the results by Gustafsson et al. [23] represent the overall conditions in India. In view of this, in the present study we apply the results of Gustafsson et al. [23] to existing BC estimates.
Here is how we use Gustafsson et al.'s [23] results. According to Gustafsson et al. [23], the corresponding share of fossil fuel combustion and biomass/biofuel burning to total BC emissions is 32 ± 5 and 68 ± 6 % respectively in South Asia. Existing BC emission estimates for the transport sector in India also give the BC emission estimates for other sectors. We adjust the ratio of estimated BC emission from fossil fuel combustion (including transportation) to estimated BC emission from biomass and biofuel burning in each past estimation study so that the adjusted ratio would be 32 ± 5 : 68 ± 6 in all the BC estimates, as consistent with that from Gustafsson et al. [23]. During the adjustment, we do not adjust the magnitude of total BC emission from all the sectors. The adjusted ratio leads to adjusted BC estimates for the transport sector, and the adjusted estimates must be more accurate. The original and adjusted estimates of the percentage share of road transport BC emissions to the total BC emissions in India are shown in Table 1. We propose that the community uses the adjusted estimates shown in Table 1.
Percentage share of road transport BC emissions to total BC emissions in India. (Adjustment in accordance with Gustafsson et al.'s isotope analysis)
Emission year
Original estimate
Adjusted estimate
Lu et al., 2011 [15] (Emission year 2010) 2010 11.00 % 4.00 – 5.00 %
Klimont et al., 2009 [16] (Emission year 2010) 2010 6.50 % 17.00 – 23.00 %
Bond et al., 2004 [7] (Emission year 2009) 2009 30.00 % 21.00 – 29.00 %
Sahu et al., [17] (Emission year 2001) 2001 34.00 % 10.67 – 14.62 %
Figure 1 compares the two (i.e., original and adjusted) estimates in the magnitude of BC emission. In this figure, we removed the estimate for the 90's and only retained those for the 2000's. As clearly shown in Fig. 1, the original estimates varied from 72 ~ 456 Gg/year (with the arithmetic average of 264 Gg/year), while the adjusted estimates now vary from 74 ~ 254 Gg/year (with the arithmetic average of 164 Gg/year). We computed the average estimate to develop the consensus, and do not intend the average estimate to be the best estimate. The average was obtained by assigning the same weight to each estimate. To summarize the results, the mean BC estimate is reduced by 38 % after adjustments with Gustafsson et al.'s [23] results. More importantly, we have sharply reduced the uncertainty in the transport sector BC emission (from 72 ~ 456 Gg/year to 74 ~ 254 Gg/year) by employing Gustafsson et al.'s [23] results.
Original vs. adjusted estimates of BC emissions in India (adjusted according to [23])
Our own estimate of the transport-sector BC emission in India
In the present study, we develop our own estimate of the BC emission from the transport section in India because our own data would facilitate the quantification of BC emission reduction in the implementation of other emission standards. In addition, we provide BC emission according to the age categorization of the vehicles – a feature not represented in the previous studies and yet important for policy makers. In our estimation, we adopt an emission factor (EF) based approach with an aim to estimate the emissions for the year 2010. Emission factors (EFs) are relations between a specific emission and the concerned activity leading to that emission, and normally determined in an empirical manner. Road vehicle EFs represent a quantity of pollutants emitted given a unit distance driven, amount of fuel used or energy consumed [24]. In addition to an EF-based technique, many other techniques are being used in the community for quantifying emissions from a large number of real world vehicles. These techniques include remote sensing of tailpipe exhaust, chassis dynamometer tests, random roadside pullover tampering studies, tunnel studies, and ambient speciated hydrocarbon measurements [25]. Employing some of these techniques for determining actual vehicle emissions in our study would be very costly as it requires dedicated human resource. Thus, we use an emission factor based approach here.
As for the emission factors for Indian vehicles, we use the data from Baidya et al. [22]. We use the emission factors from Baidya et al. [22] for the following main reasons. (a) Most importantly, they utilized the data in South Asia and South East Asia. (b) They constrained the categorization of vehicles by the data availability and data authenticity, thus accounting for the characteristics of data in South Asia. (c) In particular, the following key factors were considered within each vehicle category: Fuel vs. engine, and kinds of engine (e.g., two or four strokes). Thus their EF's are well representative of local conditions in India and useful for the present study. Please note that Baidya et al. [22] provided the emission factors for particulate matter (PM) instead of BC, and they also gave the estimates of PM emission instead of BC emission. Here, we give BC emission estimates using known PM/BC ratios and these data for various categories.
We differentiated vehicles at different levels and obtained the emission factor for each category. First, on the basis of vehicles type such as heavy duty trucks, buses, cars and motorbikes. Motor bikes are further disaggregated as 2-stroke motorbikes and 4-stroke motor bikes. Second, the vehicles are differentiated on the basis of fuel type used – diesel or gasoline. Third, vehicles are further differentiated according to four age groups as: 0–5 year old, 5–10 year old, 10–15 year old and 15–20 year old. These age groups correspond to the 5-years brackets of the Indian exhaust emission regulations: Until 1990; 1991–1995; 1996–2000 and 2001–2005. For further classifying the vehicles according to age differentiation, we assumed the percentage of vehicle belonging to a specific age group such as 0 ~ 5 years old; 5 ~ 10 years old; 10 ~ 15 years old and 15 ~ 20 years old. Then we further assume the percentage distribution of trucks and buses for specific fuel types such as diesel and petrol. This disaggregation of vehicles into specific age groups is important for the purpose of this study because India has a significant percentage of the old vehicle fleets which have not yet retired and such old vehicles have considerably higher emission rates as the emission relevant parts deteriorate [25]. The calculation of PM emissions using age specific emission factors is crucial for identifying the vehicles responsible for higher PM emissions and thus this could be used to design vehicular emission mitigation strategies.
The scientific justification for the chosen EFs is not only explained in Baidya et al.[22] but is also supported in independent reports [19, 26]. All the EFs are defined in g/km. The EFs used in the present work are tabulated in Table 2. We are aware that EFs vary from region to region, and the given EFs in Table 2 are meant to be for the country-average values of various vehicle types.
PM emission factor by vehicle category and age group in India (from Baidya et al. 2009)
Gram/km
Vehicles manufactured
Heavy duty truck (diesel) 0.49 1.22 2.03 2.7
Bus (diesel) 0.59 1.49 2.48 3.3
Diesel car (diesel) 0.19 0.46 0.77 1.03
Diesel car (gasoline) 0.06 0.07 0.09 0.1
Motorbike (2 stroke, gasoline) 0.18 0.26 0.32 0.46
Motorbike (4 stroke, gasoline) 0.06 0.08 0.1 0.14
In the next step, we multiply the total vehicle activity (vehicle kilometers traveled) and the fuel specific emission factors (Eq. 1) to estimate PM emission in Gigagram. This multiplication method is a common approach of emission calculation and it has been widely used in similar studies conducted in the past [17, 22, 27–29]. Please note that the unit of emission factors used in this equation is g/km. The annual emissions of pollutants are estimated for each individual vehicle type a, fuel type b, and emission standard c according to the following standard equation –
Equ1_TeX\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} \[ {\mathbf{E}}_{\mathbf{T}\left(\mathbf{a},\mathbf{b},\mathbf{c}\right)}=\sum \left(\mathbf{P}\mathbf{o}{\mathbf{p}}_{\mathbf{a},\mathbf{b},\mathbf{c}}\mathbf{x}\mathbf{E}{\mathbf{F}}_{\mathbf{a},\mathbf{b},\mathbf{c}}\mathbf{x}\mathbf{V}\mathbf{K}{\mathbf{T}}_{\mathbf{a}}\right) \] \end{document}
ET(a,b,c) = Total Emissions
Popa,b,c = vehicle population
VKTa = annual vehicle kilometers traveled by vehicles of type a
EFa,b,c = Emission factor for vehicle per driven kilometer for vehicle type a, fuel type b, and emission standard c
We obtained the statistics of registered motor vehicles in India from various agencies in India, including the Ministry of Shipping, Road Transport & Highway. The data collection was extremely tedious due to an inferior information storage system in India. Data were obtained by combining internet search, peer-reviewed literature and reports, and personal communication with multiple research groups and agencies (both private and government agencies) in India and abroad via e-mails and phone calls. The annual average distance in kilometers travelled by various Indian vehicles was obtained from the Road Transport Yearbook [30] published by the Government of India. Upon analyzing the data, we found that older vehicles travelled lesser distance compared to the newer vehicles. The data are sorted out in terms of vehicle category, fuel type (diesel or gasoline), type of engine (two or four strokes), operation (e.g. taxi and private use for passenger cars and the emission control standard compliance. Driving conditions are defined as either urban or rural conditions. We categorized vehicles broadly into the following four categories: trucks (diesel), buses (diesel), passenger cars (including taxis and private cars powered by diesel and gasoline) and motor bikes (2-stroke and 4-stroke motor bikes powered by gasoline). The percentage share of each categoriy of vehicles on the basis of fuel type has been obtained from The Automotive Research Association of India (ARAI). Motorized two wheelers are differentiated by two stroke and four stroke engine.
For compliance with the latest emission regulation (i.e., emission regulation for 2010), we assumed that vehicles manufactured in a specific emission year are in compliance with the emission legislation enforced by Indian government in that model year and the vehicles were not improved (in terms of emission factor) afterwards. So, the compliance with the latest emission legislation is factored in this fashion. For example, let us say, a vehicle is manufactured in the model year 1998. Then we can say that this vehicle can be assigned an age group of 10–15 year old from the year 2010 standpoint. This will correspond to the Indian exhaust emission regulation during the 5-year bracket of 1996–2000. The emission factor for this specific vehicle is calculated from the emission standards during the 5-year bracket of 1996–2000. Please note that there is a lack of Inspection and Maintenance (I & M) data for Indian vehicles. If such data were available, we did not need to make the aforementioned assumptions.
In our own estimate, the total PM emission from the road transportation in India is 507 Gigagram for the year 2010 at the present Indian on-road emission levels (shown in Fig. 2). The total PM emission for a specific vehicle category is shown in Table 3. In the last step, we convert this PM emission into BC emission by applying the ratio of BC/PM2.5. We obtain this ratio for on-road mobile sources from the EPA's report on black carbon [31], assuming that this ratio is primarily controlled by whether fuel and engine are either gasoline or diesel based. The BC/PM2.5 ratio is 0.74 and 0.19 for diesel and gasoline mobile sources respectively. We obtain the total BC emission separately for diesel and gasoline vehicles. The details of conversion calculation are illustrated in Table 4. The total BC emission following Indian emission levels is estimated to be 344.5 Gg/year and 7.84358 Gg/year for on-road diesel and gasoline vehicles respectively. From Fig. 2, it is quite evident that heavy duty diesel trucks are the main culprits with the largest contribution of 64 % towards total PM emissions in India following the present Indian on-road emission levels. The 2nd largest source of on-road PM emissions are diesel buses. For the heavy commercial vehicles including buses and trucks we assumed that there is 100 % diesel penetration. The 3rd largest contribution to total PM emission comes from diesel passenger cars which are used as personal as well as multi utility passenger vehicles as a taxi, etc. The 4th largest emission sources are 4 stroke gasoline motor bikes followed by diesel passenger cars and 2 stroke gasoline motor bikes.
Estimated PM emission of Indian vehicles with Indian emission levels
PM emission reduction if India adopts US on-road emission level
Vehicle category
Present Indian on-road emission levels
Present U.S on-road emission levels
Reduction in PM emission in switching from Indian emission levels to the US on-road emission levels
PM emission (kilotons/year) (Total-all age category)
PM emission in kilotons/year) (Total-all age category)
Diesel truck 323.9 128.9 352.3
Diesel bus 93.6 12.8 Gigagram
Diesel PC 48.1 3.5
Petrol passenger car 13.9 1.0
Motor Bike-2S petrol 8.6 6.5
Motor Bike-4S petrol 18.8 1.7
Estimated BC emission from on-road mobile sources in India
BC/PM2.5
PM emission (Gigagram/year)
BC emission = (BC/PM) ratio PM emission (Gigagram/year)
Indian on-road emission level
On-road diesel vehicles 0.74 465.5 344.5
On-road gasoline vehicles 0.19 41.3 7.8
US on-road emission level
On-road gasoline vehicles 0.19 9.3 1.8
BC emission reduction in switching from Indian emission levels to US emission levels = 243 Gigagram
BC emission reduction when India adopts the US on-road emission levels
In section 2.1, we revised the estimates for on-road BC emission in India from existing studies to be 74 ~ 254 Gg/year using adjustment with isotope analysis results. This section pertains to calculating the reduction in BC emission. This is accomplished in the following steps:
First, we obtain the emission factors for US vehicles using US EPA MOVES.
We compare the emission factors for Indian vehicles, as discussed in section 2.2, with those for US vehicles obtained in this section.
Then, we calculate the reduction in BC emission in switching from Indian emission levels to the US emission levels.
Emission factors for US vehicles
The emission factors for US vehicles were derived using the EPA's MOVES model for the year 2010. MOVES is currently the US state-of-the-art model for estimating on-road emissions. MOVES2010b is the latest version of MOVES. MOVES gives an emission for each driving mode and in this regard is considered a modal emission model. In the following, we summarize the main features of the model based on the work by Bai et al. [32] and the model documentation. Owing to the modal nature of the MOVES emission rates, MOVES is capable of quantifying emissions accurately on various scales (e.g., individual transportation projects as well as regional emission inventories). The current improved design of the model has the following advantages – a) the databases can be easily updated as per the availability of the new data; and b) the model permits and simplifies the import of data relevant to the user's own needs. The MOVES model applies various corrections for temperature, humidity, fuel characteristics, etc., before it comes up with emission estimates. MOVES also bases emission estimates on representative cycles, not on single emission rates. Furthermore, MOVES is different from traditional models such as MOBILE and EMFAC, in that a) instead of using speed correction factors, MOVES uses vehicle specific power and speed in combination; and b) it factors in vehicle operating time instead of mileage for determining emission rates. In view of this, we believe MOVES is a superior analysis tool.
In this study, we specified the following parameters as the input parameters while running the MOVES model: a) geographic bound, we chose the national level; b) time span, the year 2010; c) road type, we specified urban road with unrestricted access; and d) in the emission source, we selected all the exhaust processes (consisting of running exhaust; start exhaust; crankcase running exhaust; crankcase start exhaust ; crankcase extended idle exhaust ; extended idle exhaust) but did not include the emissions from fueling or evaporation since our BC emission estimate in section 3 did not include the latter source either.
The model output we used is the total travelled distance and annual PM2.5 emission. This output data was selected against specific vehicle types from the MySql output database of MOVES. Then, we compute emission factors by dividing total emission by total distance traveled and this gave us emission factors in gram/km for corresponding vehicle types. The emission factors were further sorted based on vehicle age. The vehicle age is calculated as a difference of reference year (year 2010) and the manufacturing year and finally, we inserted these emission factors in equation 1 discussed in chapter 3, to obtain total PM emission according to the US emission levels.
Comparison of emission factors: Indian vehicles vs. US vehicles
As we compare the PM emission factors from the India motor vehicles with those from US vehicles, we clearly see that the Indian vehicles have significantly higher emission factors than those in the US (see Fig. 3). Moreover, this difference becomes even larger for older vehicles. Higher emission factors associated with older vehicles can be attributed to the deterioration of the vehicle engines upon aging and accumulation of the mileage. The vehicle engine seemingly deteriorates on aging due to poor maintenance of the vehicles. The vehicles in India are often poorly maintained and have higher average age relative to those in the US. We believe that this faster deteriorating also stems due to lack of effective inspection and maintenance systems to be enforced by the combination of government policies. Figure 3 also shows that the difference in the emission factors is highest for the heavy commercial vehicles (diesel buses and diesel trucks). In addition, considering the case of motor bikes, we can see that the emission factors for the motor bikes in US seems to be constant with the aging because the MOVES model does not incorporate age deterioration factor for motor bikes, however, the emission factors for motor bikes in India shows an increasing trend with an increasing age. In India, there is a significant share of on-road 2-stroke motor bikes as they are an attractive option to the middle and lower middle classes in India [33]. This is contrary to the motor bike ownership scenario in US where 2-stroke motor bikes are completely out of use and are superseded by 4-stroke motorbikes. Thus, almost all of the on-road motorcycles in MOVES at this point are 4-stroke.
Comparison of PM emission factors of Indian vehicles and US vehicles
Here, we summarize the merits of four stroke engines over two stroke engines and vice-versa from the study by Kojima et al. [34] published with the World Bank. Their study primarily focused on reducing emissions from two-stroke engines in South Asia. The key advantages of 4-stroke engines over 2-stroke gasoline engine vehicles are: lower particulate and hydrocarbon emissions, better fuel economy, and moderate noise levels while in operation. However, the only relative advantages of 2-stroke engines are: lower purchase prices; mechanical simplicity leading to low maintenance costs; and lower NO2 emissions. Our comparative analysis of the emission factors from 2-stroke and 4-stroke engine technologies clearly points out the need for encouraging 4-stroke two wheelers over 2-stroke two wheelers in India. This implies that the pollution levels can be brought down to safer levels in spite of the rising two-wheeler population if the 4-stroke technology for the two-wheeler segment is promoted in India.
Reduction in BC emission in switching to the US emission levels
We combine the US emission factors with the driving activities in India to estimate total on-road PM emission in India if India hypothetically adopts the US emission levels (using Eq. 1). The result is the 155 Gigagram for the year 2010, as shown in Fig. 4. This PM emission is further converted to equivalent BC emission by applying the ratio of BC/PM2.5 (discussed in section 2.2) and details of calculation are presented in Table 4.
Estimated PM emission of Indian vehicles with US on-road emission levels
The reduction in BC emission in switching to the US emissions levels is expressed in terms of reduction in PM emission (Table 3 and Fig. 5) and reduction in BC emission (Table 4). The total BC emission reduction following the US emission levels is estimated to be 243 Gg (equaling 69 % reduction), and this number is split into 236.7 Gg and 6 Gg for on-road diesel and gasoline vehicles respectively. Please note that this reduction is in BC emission. In Fig. 5, we analyze the emission reduction in terms of age category of the vehicles. We find that the highest emission reduction in switching to US on-road emission levels would result from the vehicles in the age group of 5–10 years old followed by the vehicles in the age group of 0–5 years old and vehicles in the age group of 10–15 years old. The least emission reduction will result from the vehicles which are in the age group of 15–20 years old. This estimate incorporating vehicle age categorization is one of the novelties of the present study.
PM emission reduction from Indian vehicles according to their age (US on-road emission levels)
Our estimate of BC emission in section 3 is not necessarily a better estimate than previous ones. Thus, we restrict our BC emission estimate to the use of the ratio of the emission factors in India to those in US. This ratio is combined with the adjusted previous estimates (according to Gustafsson et al. [23], as discussed in section 2) to yield the reduction of BC emission. For the total BC emission reduction from all the vehicles in India, we apply the 69 % BC emission reduction to the adjusted previous estimates. The 69 % of 74 ~ 254 Gg/year is 51 ~ 175 Gg/year. This is the estimated BC emission reduction by switching to the US on-road emission standard.
BC radiative forcing reduction and its value
We calculated earlier that 51 ~ 175 Gg/year of BC emission reduction is possible in India (year 2000's) from the road transport sector if India adopts the US on-road emissions levels. Based on the study by Cohen and Wang [35] and Bond et al. [10], 17.8 Tg/year (i.e. 17800 Gigagram/year) of global BC emission makes 0.88 W/m2 of global BC forcing. We estimate that 51 ~ 175 Gg/year of BC emission reduction contributes to a reduction of −0.0025 ~ −0.0087 W m−2 in global BC forcing. Thus, we conclude that a reduction in BC forcing of −0.0025 ~ −0.0087 W m−2 is possible if India adopts the US on-road emissions levels.
Quantifying climatic benefits of a reduction in BC forcing (in USD)
Both CO2 emission and BC emission contribute to the global warming. The Kyoto Protocol introduced a concept called "carbon credit". 1 carbon credit is a permit to emit 1 tonne of CO2. Such permits can be sold and bought in a market, and Fig. 6 shows the market price moves of 1 carbon credit in the last 12 months. The Kyoto Protocol also allows for other climate warmers (such as methane) than CO2 to be traded in carbon credit markets. For non-CO2 warming matters, 1 carbon credit is a permit to emit an amount of the matter equivalent to 1 tonne of CO2 in the global warming. Each warming agent (such as methane) has its own atmospheric-residence time scale, spatial distribution, etc., and so comparing a particular warming agent to CO2 is not always straightforward. For BC, 100 year (or 20 year) global-warming-potential (GWP) is commonly used in this regard. According to Bond et al. [10], the 100 year GWP value for BC is 910. This means 1 tonne of BC emission adds as much energy to the earth over the next 100 years as 910 tonnes of CO2. Please note that the mass of BC refers to that of the carbon component while the CO2 mass refers to the combined mass of carbon and oxygen.
Average Carbon Price in US Dollar per tonne of CO2 emission or its equivalent (July 2013 – July 2014)
Applying Bond et al.'s [10] estimate for the BC GWP, we find that 51 ~ 175 Gg/year of BC emission reduction amounts to 0.046 billion - 0.159 billion carbon credits. Using $12.104 (US dollars) as the average price of 1 carbon credit, as shown in Fig. 6, 0.046 billion - 0.159 billion credits are valued at $ 0.56 B – $ 1.92 B (US dollars). In short, India could earn $ 0.56 B – $ 1.92 B (US dollars) every year by switching from the current on-road emission levels to the US levels. Please note that the Kyoto Protocol did not address BC but the next climate treaty will likely include BC.
We furthermore note that the 5th IPCC report [36] endorsed Bond et al.'s [10] study as a credible estimate of BC GWP. Bond et al.'s [10] estimate can be considered credible for many reasons. First, observationally-constrained estimates of BC forcing were used, and thus these estimates are similar to that of Cohen and Wang [35]. Second, the rapid adjustment due to the atmospheric heating by BC (i.e., semi-direct forcing) was included as well, and in this aspect the best semi-direct forcing estimate was used.
Conclusion and discussions
The existing estimates of the BC emission from the transport sector in India have a wide range of values, giving huge uncertainties in this regard. In the present study, we have substantially reduced the uncertainty by constraining the existing estimates with credible isotope analysis results. Next, we have derived our own BC estimate of the transport section, and then we have provided the estimate of anticipated BC emission reduction possible in India as a result of switching to the more stringent US on-road emission levels. This emission reduction is found to be about 69 %, and coupled with the adjusted previous BC emission estimates the emission reduction is expected to be in the range of 51 ~ 175 Gg of BC per annum. What is more, we have expressed the proposed BC emission reduction in terms of global BC radiative forcing, which is estimated to range from −0.0025 ~ −0.0087 W m−2, i.e., a reduction of global BC forcing by 0.0025 ~ 0.0087 W m−2 due to a reduction of BC emission in India. We have also quantified the climate benefits of the BC emission reduction in USD. The BC reduction of 58.2 ~ 151.3 Gg is equivalent to 0.046 billion - 0.159 billion carbon credits which are valued at $ 0.56 B US dollars – $ 1.92 B US dollars (using today's carbon credit price).
Although we fully accounted for uncertainties in estimating the BC emission from the transport section in India, we did not address the uncertainties in the 69 % reduction estimate, nor did we assess the uncertainties in local BC forcing or BC GWP. In our view, addressing these uncertainties is beyond the scope of the current paper and deserves a separate study. To elaborate, BC GWP is not globally uniform. BC emission in some areas can contribute to the global warming more than the emission of the same amount in other areas, since BC forcing depends on sunlight, low cloud fraction, etc. Even if we consider the uncertainty in global BC GWP, we also need to evaluate local BC GWP and address its uncertainty. To simplify the computation, we used the best estimate of global BC forcing and BC GWP and scaled these numbers to estimate the climatic benefit of reducing the BC emission in India. Nevertheless, we believe the ranges of all the estimates in our study are sufficiently large to cover most uncertainties, because we maximized the uncertainties in estimating the BC emission from the transport sector (by picking possibly the most extreme values to represent the range), and the other estimates are based on the BC emission estimates.
Overall, our study provides another reason that vehicles should be cleaner in India. Why the vehicles in India emit more aerosols than those in the US needs additional discussions, as there are many factors behind this. There are studies [21, 22, 29] and government reports [19, 37] which highlight key reasons accountable for excessive particulates emission from Indian vehicles. One of the key reasons relates to the Indian emission controls, as they have been traditionally based on Euro style emission standards. Such emission standards allow higher particulate emissions from diesel vehicles compared to the gasoline vehicles. In comparison, the US has been setting and reinforcing the same standards without considering any specific type. In another reason, emission standards in India are lax compared to international best practices. The lax standards in India reflect that Indian schedules for adopting emission and fuel quality standards lag those in the West. Other reasons include a weak enforcement of emission standards; and a significant percentage of older vehicles in India which are poorly maintained and have poor fuel economy.
An additional and important cause of excessive emission from Indian vehicles might be that transport fuels in India have high sulfur content which results in higher sulfate emissions. During combustion, sulfur in diesel fuel is emitted in the form of sulfur dioxide (SO2) gas which later condenses and becomes sulfate aerosols in the atmosphere. SO2 emission is not part of PM emission in typical PM estimation studies (since gas is not aerosol) but SO2 gas (at least some of it) ultimately becomes aerosols. Since the present study is about BC aerosols, we refrain from discussing high sulfur content extensively here.
In the end, vehicles in India emit excessive aerosols because such dirty vehicles are cheap to buy and operate. Such dirty vehicles are common among poor countries and so this is not limited to India. Clean technological solutions are available but unfortunately at additional costs. We discuss some technology examples in the next. Compared to gasoline engines, diesel engines have lower CO and HC gas emissions but higher NOx and PM emissions [38]. In gasoline engines tailpipe emissions can be significantly reduced by an efficient use of three-way catalytic converters, but at the expense of fuel economy [39]. In general, the emission control technologies for diesel and gasoline engines can be broadly divided into two groups: in-cylinder control and after-treatment control [40]. Posada et al. [41] give a good review of these emission control technologies. For instance, for diesel emission controls, PM filters are an example of after-treatment tools. Minjares et al. [13] and EPA [31] report that these particle filter devices reduce diesel PM emissions by as much as 85 to 90 % and BC emissions by up to 99 %.
Despite all these costs, the benefits could be substantial. Here, we have discussed the benefits by adopting the US on-road aerosol emission levels immediately. The idea of switching the on-road emission to the US levels immediately is unrealistic. Thus, our results can be taken as the upper limit of the benefit and such results are still useful to policymakers. On the other hand, while we only discussed the climate benefits, the benefits are not limited to the climate, and more importantly pertain to health benefits. A number of studies [31, 39, 42–45] substantiated the health benefits of BC emission reduction. It has been well established that fine particulates emitted from diesel motor vehicles contain toxic substances and the exposure to these fine particles can prompt lung tumor, serious respiratory grimness and mortality including wellbeing results, for example, worsening of asthma, interminable bronchitis, respiratory tract contaminations, coronary illness and stroke.
Besides the issues of outdated vehicle technology in India, there are other issues (behavioral and psychological issues) such as a lack of environmental conviction in the Indian car consumers that leads to higher traffic emissions [46–48]. They often dump old tires, battery or even scrap car. Irrespective of a large number of consumers who are conscious about the environment, very few people are actually willing to adapt their lifestyle in order to solve the issues such as deteriorating air quality. There is a very negligible percentage of people who actually push themselves out of their comfort zone by acting at their personal expenses, such as paying premiums for environmentally friendly products and making a sacrifice in their present lifestyles. Therefore, there is a need of behavioral changes at personal level which includes - (i) raising public awareness to prefer public transportation to using personal vehicles; (ii) living near the workplace rather than commuting a long distance to workplace every day; (iii) car pool; and (iv) commuting to workplace with bicycles. The increasing use of public transportation would mean fewer vehicles on the road, which means less emission and less negative effects on climate and health [49]. Hence, there is not a single effective tool to mitigate transport sector emissions. Taken together, we propose that in order to assure effective environmental protection, psychological as well as technological measures need to be in place.
The authors are thankful to Dr. George Scora of University of California Riverside University of California, Riverside), Dr. Sarath Guttikunda of Urban Emissions, India, Mr. Michael P. Walsh, Mr. Gaurav Bansal, Mr. John German of ICCT (Washington, DC), Mr. Stevens Plotkins of Argonne National Laboratory, and Mr. Narayan Iyer of Bajaj Auto-India for their expertise. We are also thankful to the government and semi-government agencies such as US EPA, CARB, CSE (Delhi), SIAM (India) and ARAI (India) for their prompt responses to our data requests and doubts in using emission models. This research was funded by the Korea Meteorological Administration Research and Development Program under Grant Weather Information Service Engine (WISE) project (#: KMA-2012-0001).
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Biorthogonal Fourier transform for multichirp-rate signal detection over dispersive wireless channel
Lin Zheng ORCID: orcid.org/0000-0003-2918-60671,2,
Chao Yang1,
Chao Yan1 &
Hongbing Qiu1
Biorthogonal Fourier transform (BFT), consistent with the matched signal transform (MST), has been introduced to demodulate the M-ray chirp-rate signal which possesses good orthogonality in the BFT domain. Here, we analyze the characteristics of BFT detection in a further step, including the resolution capability of the multichirp-rate signal, the property of pulse compression, the closed-form bit-error rate in the additive white Gaussian noise (AWGN) channel, and the interference in the time-frequency dispersive channel. Even in the high Doppler environment, the shift in BFT detection is proven to be slight. In addition, we deduce that the orthogonality among received chirp rates in the BFT domain would be affected in the multipath dispersive environment. This causes the mutual interference among different chirp rates in a symbol and over symbols concurrently. The theoretical result shows that the chirp modulation parameter can be adjusted to obtain the trade-off between time and frequency dispersion. By the multipath model of chirp-rate signal, an auxiliary parallel interference cancellation (PIC) method is further introduced in multipath environment. Simulations verify our analyzed performance of BFT detection in the AWGN, Doppler, and multipath channels. The proposed interference cancellation algorithms are also proven to be effective.
Early chirp signals, i.e., linear frequency modulation (LFM) signals, are implemented in radar and communication. In radar, the impulse-compression characteristic of chirp is utilized to extract targets, while the chirp signal is invoked as spread-spectrum waveform to suppress interferences in communications [1]. In recent years, wireless sensor networks, as well as military ad hoc network, have the requirement of integration with self localization, sensing, and communication functions. In this context, chirp scheme for localization and communication is unanimously adopted by the IEEE 802.15.4a Working Group in 2005 as a supplemental physical layer standard of wireless sensor network [2]. Meanwhile, many efforts in signal processing devote to estimate range or localization with a chirp signal [3, 4].
In other fields, broadband wireless communications in high-speed vehicles, such as aircraft and high-speed trains, are much in demand. Conventional phase-based transceivers are unable to meet the needs of the large Doppler shift caused by moving speeds over 200 km/h unless they adopt complicated frequency-shift estimation and compensation. The chirp signal is a good candidate since it is a constant-modulus time-frequency signal and does not require phase detection. However, chirp suffers low-modulation efficiency owing to its simplex spread-spectrum waveform.
Some efforts on improving the modulation efficiency of a chirp signal have been reported in recent years. Wysocki proposes a Walsh-coded chirp modulation in [5] for multiple access. Segments with different phases in a symbol are designed to maintain orthogonality among multiuser signals. A multidimensional chirp modulation scheme with code division spread spectrum is presented in [6]. By overlapping subbands and modulating with pseudorandom sequences, the scheme improves modulation efficiency in frequency and time domain. Analog and digital correlators are both adopted in [7]. The correlation by the former reduces the time-bandwidth product of a chirp signal and thereby reducing the complexity of post-stage digital processing. To improve orthogonality, [8] introduces a two-segment structured symbol modulated by different chirp rates and Walsh codes. Submitting to the IEEE802.15.4a physical layer proposal, Nanotron Inc. develops a M-ary modulation by four-segments linear chirps [9] and employs the multi-choice precoding (MCP) technology to solve the multipath fading problem in noncoherent detection. Its modulation efficiency still needs to be improved, and MCP has poor performance in a fast fading channel due to the requirement of the channel information feedback from receiver to transmitter.
The progress in time-frequency (T-F) signal processing is also applied to chirp detection. Recently, the T-F dispersive channel and orthogonal frequency-division multiplexing (OFDM) signal have been intensively studied [10]. The series of the filter bank multicarrier (FBMC) and T-F filters for OQAM have been proposed to offer better performance to Doppler shift [11]. Here, we concern the various TF-based detectors developed for detecting chirp signals. They include the dechirping technique, the Radon-Wigner transform [12], the Radon-ambiguity transform [13], the chirplet transform [14], and the short-time Fourier transform [15]. A multi-carrier chirp communication system based on the fractional Fourier transform (FRFT) is developed in [16] and has been proven to be more reliable compared to fast Fourier transform (FFT) based OFDM in time-frequency-selective channels. In recent years, FRFT-OFDM has received attention, and its performance has been analyzed in the dispersive environment, especially in the frequency offset or Doppler frequency shift analysis [17–19]. All of them reveal that FRFT detection performance is significantly better. Furthermore, the affine fourier transform as a generalized FRFT is introduced to multicarrier system in [20, 21]. In [22], the out-of-band power reduction methods are proposed to the weighted-type FRFT-based multicarrier system. The detectors mentioned above show better anti-ICI performance, that is proved to be feasible in ground-to-air channels. However, the time dispersion (ISI) evaluation and performance analysis for the chirp multicarrier method is neglected.
Other analyses in [23] and [24] concentrate on the resolution of multi-components in the short-term FRFT and FRFT domain. The condition is deduced to obtain the maximal resolution performance. Tao [31] and Zhao [26] introduce the FRFT to estimate the multiple components of a chirp signal. However, simultaneous FRFT detectors with different orders should be applied to corresponding chirp-rate signals. This, however, results in high computing load and complexity in practice.
Matched signal transforms (MSTs) to exponential instantaneous frequency structures are proposed in [27], and further, a linear MST method is introduced to suppress LFM interference [28]. Wang [29, 30] presents a transform being consistent with linear MSTs, naming it the biorthogonal Fourier transform (BFT), to detect a chirp signal since it matches the chirp rate. Like the Fourier transform to a single-frequency waveform, the impulse-compression effect of the chirp rate is achieved in the BFT domain. Unlike FRFT, the demodulation of the multichirp-rate signal requires only one BFT process to complete. So far, the demodulation performance of BFT or linear MST to chirp-rate signal, and the applicability of BFT in wireless doubly dispersive channels have not yet been analyzed in the literatures.
This paper deduces the output signal-to-noise-ratio (SNR) of BFT detection of chirp-rate signal, and the closed-form bit-error-rate (BER) result is obtained. Some characteristics of the discrete BFT are analyzed to design the multichirp-rate signal and the demodulation algorithm. In a practical environment, we demonstrate that the orthogonality among the chirp-rate signals in the BFT domain will be unfortunately affected by multipath propagation. However, by adjusting the chirp rate of the signaling scheme, we obtain a trade-off between the tolerance of the time and frequency dispersion, i.e., ICI and ISI. In addition, we also developed an auxiliary parallel interference cancellation (PIC) method based on the dispersive model in BFT domain to mitigate the multipath interference. At the end of paper, the effect of the Doppler shift is analyzed and verified by simulations. With square-law BFT (BFT2), outstanding detection performance exceeds that of FRFT and FSK in the channels with large Doppler shift.
The rest of this paper is organized as follows: firstly, multichirp-rate signal models in the time domain is given in Section 2. Secondly, Section 3 gives the definition of BFT and analyzes the BFT characteristics, including the closed-form solution of BFT detection performance. Thirdly, BFT detection in frequency-offset and multipath environments are discussed in Section 4 and Section 5, respectively. Fourthly, based on the multipath model in BFT domain derived in Section 5, a MMSE detection aided by the decision-directed PIC is proposed in Section 6. Fifthly, the simulation results and some discussions are given in Section 7. Finally, the conclusions of our work are summarized in last section.
Multichirp-rate signal model
A multichirp-rate scheme is introduced to improve the modulation efficiency of a chirp spread-spectrum (CSS) signal [29, 31]. It maintains the frequency characteristics of the original LFM and avoids high demand of the phase-modulated signal for strict synchronization and equalization of the receiver.
At the nth symbol, the M-ary chirp-rate signal model can be presented as
$$ {\begin{aligned} s_{M}(n,t)&=\sum\limits_{m=1}^{M}b[nM+m]\exp\left(j2\pi\left(f_{0} (t- n T_{s}){\vphantom{\frac{1}{2}}} \right.\right.\\ &\quad\left.\left. +\frac{1}{2}K_{m} (t- n T_{s})^{2}\right)\right), \,\,\left((n-1)T_{s}\leq t<{nT}_{s}\right) \end{aligned}} $$
where M represents the number of chirp rates, i.e., the M-ary modulated symbol. K m is the chirp rate, and f0 is the center frequency. T s is the modulated symbol period. b[ k]∈{−1,1} is the binary information bit. This multiple chirp-rate modulation can be viewed as a parallel M-way-modulated signal combination in the chirp-rate domain. Different chirp rate represents different information bit in a symbol period, as shown in Fig. 1.
M-ary chirp-rate signal and D-BFT demodulation. The figure shows the system block diagram for the multichirp-rate modulation and BFT demodulation
BFT on chirp-rate signal
The biorthogonal Fourier transform (BFT) and its inverse conversion algorithm is defined in [29, 30] (or refer to [28]):
$$ \begin{aligned} \text{BFT}[f(t)]\triangleq F(\beta) & =2{\int\nolimits}_{0}^{+\infty}f(t)t\exp\left(-j\beta t^{2}\right)dt \\ \text{IBFT}[\!F(\beta)]\triangleq f(t) & =\frac{1}{2\pi}{\int\nolimits}_{-\infty}^{+\infty}F(\beta)\exp\left(j\beta t^{2}\right)d\beta \end{aligned} $$
Obviously, to the signal of linear frequency modulation f(t)=exp(jπK m t2), it has BFT[f(t)]=2πδ(β−πK m ), where the impulse position at πK m in the BFT domain reflects the chirp rate. Referring to the short-term discrete Fourier transform (STDFT), the discrete BFT (D-BFT) performs a circular convolution operation. Here, the oversampling frequency is denoted as f c and its interval is T c . The time and BFT domain are discretized as t=lT c and β=kΔβ, where l and k are integers. Furthermore, the N-point symbol duration is defined as T s =NT c with N≫1, and N-point chirp-rate duration is defined as Ω=NΔβ. Thus, it has the following relationship:
$$ \begin{aligned} T_{c} &= 1/f_{c},\,\, \Delta\beta=2\pi/T_{s}^{2} \\ \Omega &= 2\pi f_{c}/T_{s},\,\, \beta t^{2}=kl^{2}\frac{2\pi}{N^{2}} \\ F(k\Delta\beta)&=\frac{2}{f_{c}}\sum\limits_{l=0}^{N} f({lT}_{c}){lT}_{c}\exp\left(-j\frac{2\pi}{N^{2}}kl^{2}\right) \end{aligned} $$
The D-BFT and its inverse transform (D-IBFT) can be derived as
$$ \begin{aligned} F(k)&=\frac{2}{f_{c}}\sum\limits_{l=0}^{N} f(l)l\exp\left(-j\frac{2\pi}{N^{2}}kl^{2}\right) \\ f(l)&=\frac{1}{N^{2}}\sum\limits_{k=0}^{N}F(k)\exp\left(j\frac{2\pi}{N^{2}}kl^{2}\right) \end{aligned} $$
The BFT is shown to be equivalent to the Fourier transform of a nonlinearly sampled or warped version of the signal. Therefore, the FFT can be used to improve its computing efficiency. It can be accomplished by firstly oversampling f(t) to reduce approximation errors in obtaining warped signal samples.
By analyzing the BFT and D-BFT, we derive some characteristics and performances of the transform in multichirp-rate signal detection.
Theorem 1
The BFT impulse position is determined by the time-bandwidth product of chirp-rate signal. When the time-bandwidth product is an integer multiple of 2, the BFT impulse peak is at the discrete k.
Suppose chirp-rate signal is f(t)=A exp(jπK m t2). Its discrete form is \(f({lT}_{c})=A\exp \left (j\pi K_{m} l^{2}T_{c}^{2}\right) \). Inserting it into the above D-BFT expression, we obtain the peak position κ of F(k), which is given by
$$ \kappa=\frac{1}{2}K_{m}T_{c}^{2} N^{2}=\frac{1}{2}K_{m}T_{s}^{2} $$
According to K m =±B m /T s where B m is the bandwidth of chirp-rate signal, we have κ=±B m T s /2. Therefore, the BFT peak is at the integer k when the chirp B m T s is a multiple of 2. □
By calculating the BFT at κ=±B m T s /2 (m=1,2,⋯,M), the BFT peak of the mth chirp-rate signal is obtained. This operation undoubtedly simplifies the BFT of chirp-rate signal and reduces computational load.
When the chirp-rate signals have different integer B m T s /2, they are orthogonal to each other at the discrete k in BFT domain.
In a symbol duration T s , the BFT of f(t)=A exp(jπK m t2), (0≤t<T s ) is given by
$${} {\begin{aligned} \text{BFT}[f(t)]&=2A{\int\nolimits}_{0}^{T_{s}} t\exp\left(-j(\beta-\pi K_{m})t^{2}\right)dt \\ &={AT}_{s}^{2}\text{Sa}\left(\frac{(\beta-\pi K_{m}) T_{s}^{2}}{2}\right)\exp\left(-j(\beta-\pi K_{m}) T_{s}^{2}/2\right) \end{aligned}} $$
which is a Sa(·) impulse. Thus, there is
$$ |\text{BFT}[f(t)]|={AT}_{s}^{2}\text{Sa}\left(\frac{(\beta-\pi K_{m}) T_{s}^{2}}{2}\right) $$
With \(\Delta \beta \,=\,2\pi /T_{s}^{2}\), substituting \(\beta =\pi K_{m}\pm 2\pi k/T_{s}^{2}\, (k\neq 0, k=\pm 1,\pm 2,\cdots)\) into the above BFT result, it has
$$ |\text{BFT}[f(t)]|={AT}_{s}^{2}\text{Sa}\left(\frac{2k\pi}{2}\right)=0 $$
In addition, it has the peak \(|\text {BFT}[f(t)]|={AT}_{s}^{2} \) at β=πK m , that is k=β/Δβ=B m T s /2 in the discrete BFT. Therefore, the chirp-rate signals have the orthogonality in discrete BFT domain if they have different integer B m T s /2. □
According to the lemma, the multichirp-rate symbol can be demodulated, and the BFT impulses corresponding to different chirp rates can be resolved.
In the AWGN channel, SNR of the BFT detection to chirp-rate signal depends on the time-bandwidth product of the received signal where the bandwidth is the sampling bandwidth. The BER of BFT demodulation is
$$ P_{e} = Q\left(\sqrt{\frac{3}{2}\cdot\frac{E_{b}}{N_{0}}}\right) $$
where E b is the received chirp-rate energy in a symbol period, and N0 is the single-sided power spectral density of AWGN.
In the continuous time domain, the peak amplitude of BFT output is \({AT}_{s}^{2}\) where A is the amplitude of the received signal. Owing to the I-Q complex modulation in Eq. (1), the peak power by BFT demodulation is given by
$$ S_{out}=A^{2} T_{s}^{4} $$
In the complex zero-mean AWGN environment, the noise can be decomposed into n(t)=a n (t) exp(jϕ(t)), where a n (t) is the amplitude with Rayleigh distribution, and ϕ(t)∈(0,2π] is the phase with uniform distribution, and they are statistically independent of each other [32]. Thus, the noise by BFT can be expressed as
$$ \begin{aligned} N_{\text{out}}(\beta)&=2{\int\nolimits}_{0}^{T_{s}}n(t)t\exp\left(-j\beta t^{2}\right)dt \\ &=2{\int\nolimits}_{0}^{T_{s}}t a_{n}(t)\exp\left(j\phi(t)-j\beta t^{2}\right)dt \\ &=2{\int\nolimits}_{0}^{T_{s}}t n'(t)dt \end{aligned} $$
Since ϕ(t) follows uniform distribution, (ϕ(t)−βt2) also follows uniform distribution. Thus, n′(t)=a n (t) exp(jϕ(t)−jβt2) is still a Gaussian noise due to the independence of a n (t) and (ϕ(t)−βt2), which obey Rayleigh distribution and uniform distribution, respectively. Then, the noise variance of the BFT can be represented as
$${} {{\begin{aligned} Var[N_{\text{out}}(\beta)]=&4\int{\int\nolimits}_{0}^{T_{s}}E\left(t_{1} n'(t_{1})t_{2} n'(t_{2})\right){dt}_{1}{dt}_{2} \\ =&4\int{\int\nolimits}_{0}^{T_{s}}t_{1}t_{2} R_{n'}(t_{1},t_{2}){dt}_{1}{dt}_{2} \\ =&4{\int\nolimits}_{-T_{s}}^{0} R_{n'}(\tau)d\tau{\int\nolimits}_{-\tau}^{T_{s}}t_{2}(\tau+t_{2}){dt}_{2} \\ &+4{\int\nolimits}_{0}^{T_{s}} R_{n'}(\tau)d\tau {\int\nolimits}_{0}^{T_{s}-\tau}t_{2}(\tau+t_{2}){dt}_{2} \\ =&4{\int\nolimits}_{-T_{s}}^{0} R_{n'}(\tau)\left(\frac{1}{2}T_{s}^{2}\tau+\frac{1}{3}T_{s}^{3}-\frac{1}{6}\tau^{3}\right)d\tau \\ &+4{\int\nolimits}_{0}^{T_{s}} R_{n'}(\tau)\left(\frac{1}{2}(T_{s}-\tau)^{2}\tau+\frac{1}{3}(T_{s}-\tau)^{3}\right)d\tau \end{aligned}}} $$
By the front band-pass filter and sampler in the receiver, n′(t) is the Gaussian noise with bandwidth B=f c . Therefore, its autocorrelation function is \(R_{n'}=\sigma _{n}^{2} S_{a}(\pi \tau /T_{c})\). We note that \(R_{n'}(\tau)\approx 0\phantom {\dot {i}\!}\) when τ>T c . In addition, when τ≤T c , i.e., τ≪T s , the item \(\frac {1}{6}\tau ^{3}\) is negligible with the oversampling times N≫1 and \((T_{s}-\tau)^{2}\approx T_{s}^{2}\), \((T_{s}-\tau)^{3}\approx T_{s}^{3}\). Thus, we have
$${} \begin{aligned} Var[N_{\text{out}}(\beta)]&\approx 4{\int\nolimits}_{-T_{s}}^{T_{s}} R_{n'}(\tau)\left(\frac{1}{2}T_{s}^{2}\tau+\frac{1}{3}T_{s}^{3}\right) d\tau \\ &=\frac{4}{3}\sigma_{n}^{2} T_{s}^{3}{\int\nolimits}_{-T_{s}}^{T_{s}}S_{a}\left(\frac{\pi\tau}{T_{c}}\right)d\tau =\frac{4}{3}\sigma_{n}^{2}T_{s}^{3}T_{c} \end{aligned} $$
The item \(R_{n'}(\tau)\cdot T_{s}^{2}\tau \) is an odd function of τ, so that its integral over [−T s ,T s ] is zero.
We note that the BFT in Eq. (10) is a linear transform, so that Nout(β) is still a Gaussian noise. Therefore, the BER of BFT demodulation for the bipolar chirp-rate signal could be derived as
$$ \begin{aligned} P_{e} &= Q\left(\sqrt{\frac{2S_{\text{out}}}{Var[N_{\text{out}}(\beta)]}}\right) \\ &=Q\left(\sqrt{\frac{3}{2}\cdot\frac{A^{2} T_{s}}{\sigma_{n}^{2} T_{c}}}\right) \end{aligned} $$
Without regarding to the processing gain g=T s /T c =N, the BER is expressed by \(E_{b}/N_{0}=A^{2}T_{s}/\sigma _{n}^{2}T_{c}\) as
$$ P_{e}=Q\left(\sqrt{\frac{3}{2}\cdot\frac{E_{b}}{N_{0}}}\right) $$
Here, we further deduce the output SNR of the D-BFT. The discretized chirp-rate signal \(f({lT}_{c})=A\exp \left (j\pi K_{m} l^{2} T_{c}^{2}\right) \) is transformed as
$$ \begin{aligned} S(k)&=\text{D-BFT}[f(t)]\\ &=\frac{2{AT}_{s}}{N}\sum\limits_{l=0}^{N} l \exp\left(j\pi K_{m} l^{2}T_{c}^{2}\right)\exp\left(-j\frac{2\pi}{N^{2}}kl^{2}\right) \end{aligned} $$
At k′=B m T s /2, the amplitude of matching impulse by D-BFT is derived to be S(k′)=(N+1)AT s .
By expressing the discrete zero-mean AWGN as n(l)=A l exp(jX l ), where A l meets Rayleigh distribution and X l meets uniform distribution in (0,2π]. Similar to the deduction in Eq. (11), substituting n(l) into the D-BFT expression, we have
$$ \begin{aligned} N(k)&=\text{D-BFT}[n(t)]\\ &=\frac{2T_{s}}{N}\sum\limits_{l=0}^{N} l\cdot A_{l}\exp\left({jX}_{l}-j\frac{2\pi}{N^{2}}kl^{2}\right) \\ &=\frac{2T_{s}}{N}\sum\limits_{l=0}^{N} l\cdot n'(k,l) \end{aligned} $$
Shown in the expression, the output noise from the D-BFT is a linear accumulation of Gaussian processes. Obviously, it is still a zero-mean Gaussian noise, so that its variance is derived as
$$ \begin{aligned} Var[N(k)] &= E\left[\bigg(\frac{2T_{s}}{N}\sum\limits_{l=0}^{N} l\cdot n'(k,l)\bigg)^{2}\right] \\ &=\frac{4T_{s}^{2}}{N^{2}}\sigma_{n}^{2}\sum\limits_{l=0}^{N} l^{2} \\ &=\frac{2T_{s}^{2}(N+1)(2N+1)}{3N}\sigma_{n}^{2} \end{aligned} $$
Here, we have the output SNR of the D-BFT on chirp-rate signal when N≫1
$$ \frac{S(k')^{2}}{Var[N(k)]} = \frac{3N(N+1)}{2(2N+1)}\cdot \frac{A^{2}}{\sigma_{n}^{2}} \approx \frac{3N}{4}\cdot \frac{A^{2}}{\sigma_{n}^{2}} $$
This result is consistent with the SNR of the BFT to continuous signal, and the additional N in the numerator is the processing gain of the D-BFT demodulator.
Frequency offset in BFT detection
At the receiver, there is a frequency offset between the received signal and local carrier. This offset is caused either by the difference oscillators in the transmitter and receiver, or by the Doppler shift. The detection performance may be severely degraded as the frequency offset is large and volatile, especially in an OFDM system. Not surprisingly, BFT detection of the chirp-rate signal is affected by frequency offset. The compressed impulse in the BFT domain is shifted and attenuated.
Supposing there is a frequency offset f d , the chirp-rate signal is given by
$$ \begin{aligned} f_{i}(t)&=A\ \text{exp}\!\left(j 2\pi f_{d} t + j\pi K_{m}t^{2} + j {\phi_{i}}\right), \\ &\qquad\qquad\qquad\qquad\qquad\qquad \; t\in(0,T_{s}] \end{aligned} $$
where ϕ i is the initial random phase of the ith symbol. By BFT, we have
$${} \begin{aligned} &\text{BFT}[f_{i}(t)] \\ &=A{\int\nolimits}_{0}^{T_{s}} \exp\left(j2\pi f_{d} t +j\phi_{i}\right) \exp \left[-j\left(\beta - \pi K_{m} \right){t^{2}} \right]d{t^{2}}\\ &= A{\int\nolimits}_{0}^{T_{s}^{2}} \exp\left(j2\pi f_{d} \sqrt{V}+j\phi_{i} \right) \exp \left(-j(\beta - \pi K_{m}){V} \right)d{V} \end{aligned} $$
At β=πK m , the discrete sample for the chirp rate in the BFT domain is deduced as
$$ \begin{aligned} F_{fo}(\beta&=\pi K_{m}) = 2A\exp(j\phi_{i}(t)){\int\nolimits}_{0}^{T_{s}} t\exp\left(j2\pi f_{d} t \right)d{t} \\ &= \frac{A\exp(j\phi_{i}(t))\big[(1-j2\pi f_{d} T_{s})\exp(j2\pi f_{d} T_{s})\,-\,1\big]}{2\pi^{2}f_{d}^{2}} \end{aligned} $$
This is the result of D-BFT detection with a frequency-offset signal, thereby developing the attenuation expression \(|F_{fo}(\beta =\pi K_{m})|/{AT}_{s}^{2}\).
By Taylor series expansion, \(\sqrt {V}\) is extracted as
$${} \sqrt{V}=\sqrt{V_{0}}+\frac{1}{2\sqrt{V_{0}}}(V-V_{0})+\frac{1}{2}\frac{-1}{4V_{0}\sqrt{V_{0}}}(V-V_{0})^{2}+\cdots $$
According to the range of V in \(\left [0,T_{s}^{2}\right ]\) and the integral in Eq. (20), the approaching point is near \(V_{0}=T_{s}^{2} \). The first two items in the series are left to approximate \(\sqrt {V}\); then, we have \(\sqrt {V}\approx T_{s}/2+ V/(2T_{s})\). Substituting it into Eq. (20), we obtain
$$ {}\begin{aligned} F_{fo}\left(\beta \right) &\approx A\exp(j\phi_{i})\int_{0}^{T_{s}^{2}} {\exp\left[ j2\pi f_{d}\left(\frac{T_{s}}{2} + \frac{V}{2T_{s}} \right) \right]} \\ & \qquad\qquad\qquad \cdot\exp \left[ -j\left(\beta - \pi K_{m} \right)V \right]dV \\ & = A T_{s}^{2} \text{Sa}\left[\left(\beta - \pi K_{m} -\frac{\pi f_{d}}{T_{s}}\right)T_{s}^{2}/2 \right]\\ &\qquad\quad \cdot\exp \left[-j\left(\beta-\pi K_{m}-\frac{3\pi f_{d}}{T_{s}}\right)T_{s}^{2}/2+j\phi_{i} \right] \end{aligned} $$
As a result, the compressed impulse in the BFT domain is shifted by about πf d /T s from its original πK m value. Compared to the interval \(\Delta \beta =2\pi /T_{s}^{2}\) in the D-BFT domain, this shift is small with f d ≪1/T s in general.
BFT on multipath signal
Signal analysis in the BFT domain
In practice, the detector has to be confronted with wireless propagation environment. Here, we analyze the transform on the multipath chirp-rate signal. Without loss of generality, the model of the multipath channel is given by
$$ h(t)=\alpha_{0}\delta(t)+\sum\limits_{l=1}^{L-1}\alpha_{l} e^{j\phi_{l}}\delta(t-\tau_{l}) $$
where L is the number of paths and τ l is the delay of the lth path. A multipath chirp-rate signal can then be expressed as
$$ r(t)=\alpha_{0}\exp\left(j\pi K_{m} t^{2}\right)+\sum\limits_{l=1}^{L-1} \alpha_{l} e^{j\phi_{l}} \exp\left(j\pi K_{m}(t-\tau_{l})^{2}\right) $$
We extract a path component of this signal and analyze it in the BFT domain. By the transform in t∈ [ 0,T s ], we obtain
$${} {{\begin{aligned} \text{BFT}[r_{l}(t)]&=F(\beta) \\ &=\alpha_{l} e^{j\phi_{l}}{\int\nolimits}_{\tau_{l}}^{T_{s}}\exp\left(j\pi K_{m}(t-\tau_{l})^{2}\right)2t\exp\left(-j\beta t^{2}\right)dt \\ &=\alpha_{l} e^{j\phi_{l}}\exp\left(j\pi K_{m}\tau_{l}^{2}\right){\int\nolimits}_{\tau_{l}^{2}}^{T_{s}^{2}} \text{exp}\!\left(-j2\pi K_{m}\tau_{l} \sqrt{U}\right) \\ &\qquad\qquad\qquad \cdot\exp(-j(\beta-\pi K_{m})U)dU, \quad \left(U=t^{2}\right) \end{aligned}}} $$
Similar to the deduction in Eq. (23), the integral above is simplified by Taylor series expansion. \(\sqrt {U}\approx T_{s}/2+U/(2T_{s}) \) is substituted in, and further derivation is given by
$${} \begin{aligned} \text{BFT}&[r_{l}(t)]\approx \\ & \alpha_{l} e^{j\phi_{l}}e^{j\pi K_{m}\tau_{l}^{2}}{\int\nolimits}_{\tau_{l}^{2}}^{T_{s}^{2}}\exp(-j2\pi K_{m}\tau_{l} (T_{s}/2 \\ &\qquad\qquad\qquad +U/2T_{s}))\exp({-j(\beta-\pi K_{m})U})dU\\ &=\alpha_{l} e^{j\phi_{l}}e^{j\pi K_{m}(\tau_{l}^{2}-\tau_{l}T_{s})}{\int\nolimits}_{\tau_{l}^{2}}^{T_{s}^{2}}\exp\{-j(\beta-\pi K_{m} \\ &\qquad\qquad\qquad\qquad\qquad +\pi K_{m}\tau_{l}/T_{s})U\}dU \\ &= \alpha_{l}'\text{Sa}\left[\frac{T_{s}^{2}-\tau_{l}^{2}}{2}\left(\beta-\pi K_{m}+\pi K_{m}\tau_{l}/T_{s}\right)\right] e^{j\Phi_{l}(\beta) } \end{aligned} $$
$$ \begin{aligned} & \alpha_{l}'=\alpha_{l}\left(T_{s}^{2}-\tau_{l}^{2}\right) \text{ and}\\ & \Phi_{l}(\beta)=-(\beta-\pi K_{m}+\pi K_{m}\tau_{l}/T_{s})\cdot\left(T_{s}^{2}+\tau_{l}^{2}\right)/2 \\ &\qquad\qquad\qquad\qquad + \pi K_{m}\left(\tau_{l}^{2}-\tau_{l}T_{s}\right) + \phi_{l} \end{aligned} $$
From the result, it is notable that the impulse in the BFT domain is shifted from πK m to πK m (1−τ l /T s ). At the same time, the width of the impulse is also spread from \(2\pi /T_{s}^{2}\) to \(2\pi /\left (T_{s}^{2}-\tau _{l}^{2}\right)\). The longer the delay, the looser the impulse. These results also present the offset and extension of the β-domain impulse when there exists symbol synchronization error.
On the other hand, the BFT of the multipath signal with cross delay τ l from the previous symbol is derived in the same way as
$$ \begin{aligned} \text{BFT}&[r_{l}^{-}(t)]\approx \\ & \alpha_{l}^{\prime\prime}\text{Sa}\left[\frac{\tau_{l}^{2}}{2}\left(\beta-\pi K_{m}-\pi K_{m}(1-\tau_{l}/T_{s})\right)\right] e^{j\Psi_{l}(\beta) } \end{aligned} $$
$$ \begin{aligned} \alpha_{l}^{\prime\prime}&=\alpha_{l}\tau_{l}^{2} \text{ and}\\ \Psi_{l}(\beta)&=-(\beta-\pi K_{m}-\pi K_{m}(1-\tau_{l}/T_{s}))\cdot\tau_{l}^{2}/2 \\ &\quad + \pi K_{m}\left(2T_{s}^{2}+\tau_{l}^{2}-3\tau_{l}T_{s}\right) + \phi_{l} \end{aligned} $$
The BFT impulse from the cross signal is shifted to π(2K m −τ l /T s ), which is far from πK m with τ l ≪T s . Its amplitude is low and its width is spread due to small τ l . Therefore, the inter-symbol interference can be negligible with small cross delay τ l in general.
Shown in the upper part of Fig. 2 is the time-frequency illustration of different delayed chirp-rate signals in a symbol duration T s . The lower part of the figure is their projections in the BFT domain. The blue impulse comes from the synchronized path component, and the red pulses correspond with the other multipath components of the same symbol. The black pulses are the BFT of the multipath components of the pervious chirp-rate symbols. In the illustration, the red and black pulses may be the interference in BFT domain to other chirp-rate components in the same symbol duration, including intra-symbol mutual interference and inter-symbol interference, respectively. Therefore, the orthogonality in the BFT domain among the chirp-rate signals is affected in a practical multipath environment.
BFT on the multipath chirp-rate signal. The figure shows the BFT output of the chirp-rate signal in multipath environment
Despite having dispersion, the multipath chirp in BFT domain are found to concentrate when T s ≫τ l in Eq. (27). Noting that the conclusion in the above section that T s ≪1/f d is required, T s must be selected as a compromise for low BFT dispersion when Doppler shift and multipath delays coexist. Here, we measure the BFT dispersion performance by d, which is defined as
$$ d = \frac{\sum_{k\neq i}E_{k}}{E_{i}} $$
where E i is the energy of the desired component in the BFT domain, and E k is the energy of dispersed component. Figure 3 illustrates the performances of time and Doppler dispersions. In this example, the time-bandwidth product is set to a constant 10, the symbol duration T s varies from 2 to 100μs, and the corresponding bandwidth varies from 5 to 0.1MHz. Four Doppler shifts {1kHz,5kHz,10kHz,30kHz} and three path-delays {0.1μs,0.3μs,0.5μs} are measured to obtain the BFT dispersion performance. The crossing points present the compromises of the modulation parameters under specified double dispersive channel. On the other hand, increasing T s , i.e., decreasing B m , in concentrating time dispersion in the BFT domain will reduce the symbol rate. Multicarrier modulation could reconcile the original data rate and dispersion requirement of BFT detection.
Dispersion in the BFT domain due to Doppler and path delay. The figure shows the BFT dispersions in Doppler and multipath environment with different signal parameters
Multipath and multichirp-rate signal model in the BFT domain
Through the above analysis, the multipath signal with a single chirp rate in the nth symbol period is obtained in the BFT domain as
$${} \begin{aligned} & R_{n,m}(\beta)=\text{BFT}[r_{n,m}(t)]\approx \\ & b_{m}[n]\left[\alpha_{0}\delta(\beta-\pi K_{m})+\sum\limits_{l=1}^{L'-1}\alpha_{l}'\delta\left(\beta-\pi K_{m}+\Delta\tau_{l,m}'\right)\right] \\ &+\sum\limits_{l=L'-1}^{L-1}b_{m}[n-n_{l}]\alpha_{l}'\delta\left(\beta-\pi K_{m}-\Delta\tau_{l,m}'\right) \end{aligned} $$
where n l =⌊τ l /T s ⌋,(τ l >T s ), Δτl,m′=πK m τ l /T s and L′ is the number of multipaths whose delay does not exceed a symbol duration. The function δ(β) is an abbreviation of \(\text {Sa}\left [\left (T_{s}^{2}-\tau _{l}^{2}\right)\beta /2\right ]e^{\Phi (\beta)} \).
$${} {{\begin{aligned} r_{n}(\beta&= \pi K_{m})= b_{m}[n]\alpha_{0}+\sum\limits_{i=m+1}^{M-1}b_{i}(n)\alpha_{m,i}'\\ &\quad+\sum\limits_{i=0}^{M-1}\sum\limits_{j=1}^{N_{l}}b_{i}(n-j)\alpha_{m,i,j}'\\ &=\left[0,\cdots,\alpha_{0},\alpha_{m,m+1}',\cdots,\alpha_{m,M-1}'\right]\mathbf{b}(n)\\ &\quad+\sum\limits_{j=1}^{N_{l}}\left[\alpha_{m,0,j}',\cdots,\alpha_{m,m,j},\cdots,\alpha_{m,M-1,j}'\right]\mathbf{b}(n-j)+v(n) \end{aligned}}} $$
$$ \begin{aligned} \mathbf{r}(n) &= \underbrace{\left[~\begin{array}{cccccccccc} \alpha_{0} &\alpha_{0,1}'&\alpha_{0,2}'&\cdots & \alpha_{0,M-1}'&\alpha_{0,0,1} &\cdots &\alpha_{0,M-1,1}' &\cdots & \alpha_{0,M-1,N_{l}}'\\ 0 &\alpha_{0} &\alpha_{1,2}'&\cdots & \alpha_{1,M-1}'&\alpha_{1,0,1}' &\cdots &\alpha_{1,M-1,1}' &\cdots & \alpha_{1,M-1,N_{l}}'\\ \vdots &\vdots &\alpha_{0} &\ddots & \vdots &\vdots &\ddots &\vdots &\cdots & \vdots\\ 0 &0 &\cdots &\cdots & \alpha_{0} &\alpha_{M-1,0,1}' &\cdots &\alpha_{M-1,M-1,1}&\cdots & \alpha_{M-1,M-1,N_{l}} \end{array}~\right] }_{M\times (N_{l}+1)} \left[\begin{array}{c} \mathbf{b}(n)\\ \mathbf{b}(n-1)\\ \vdots\\ \mathbf{b}(n-N_{l}) \end{array}\right]+\mathbf{v}(n)\\ &=\mathbf{H}\tilde{\mathbf{b}}(n)+\mathbf{v}(n) \end{aligned} $$
Thus, the BFT of the received multichirp-rate signal in the multipath channel is \(R_{n}(\beta)=\sum _{m=0}^{M-1} R_{n,m}(\beta)+ v(\beta)\), where v(β) is the Gaussian noise due to BFT being linear transform and chirp rate K m is assigned according to Theorem 1. Define M-ary source vector b(n)=[ b0(n),⋯,b m (n),⋯,bM−1(n)]T. By the normalized D-BFT impulse of the chirp-rate signal, the D-BFT output on received multichirp-rate signal is expressed as Eq. (31). In Eq. (31), αm,i′ denotes the mutual interference to the mth chirp rate from other ith chirp-rate component in a same symbol period. From the above analysis, the chirp-rate K m is just jammed by the chirp-rate Ki≥m+1 components. αm,i,j′ is the inter-symbol interference from the ith chirp-rate component before j symbols, and \(\mathbf {h}_{m}=[\alpha _{0},\alpha _{m,m,1},\alpha _{m,m,2},\cdots,\alpha _{m,m,N_{l}}]\phantom {\dot {i}\!}\) is the jamming vector from the same chirp-rate components due to multipath propagation.
Define the output vector by BFT of the received multipath signal as r (n)= [ r n (πK0), r n (πK1), ⋯,r n (πKM−1)]T. According to Eq. (31), we obtain the vector model by BFT as Eq. (32), where \(\tilde {\mathbf {b}}(n) = [\mathbf {b}(n),\mathbf {b}(n-1),\cdots,\mathbf {b}(n-N_{L})]^{T}\) is the M-ary source series.
Detection of multichirp-rate signal in multipath environment
Although there are a large number of interference terms in H, many of them are zero or minute, that is, the matrix H is sparse. From analysis of Eq. (27) and Fig. 2, the multipath impulses by BFT are on the left of πK m in the β domain. The further from πK m , the weaker the impulses are. Therefore, the mutual collisions among impulses are not severe in practice.
The signal model by BFT in the multipath channel is a typical model of mutual and inter-symbol interference. Estimation of the matrix H is difficult. It is feasible to apply a training sequence and adaptive algorithms to suppress interference.
A minimum mean square error (MMSE) detector is commonly applied to mitigate interference. Because of the linear signal model in (32), MMSE detection is also available. Here, the M×(N l +1) weight matrix W acts as the filter to M-channel detection. The optimal criterion in MMSE detection for the BFT of the M-chirp-rate signal is given by argW minE{|b(n)−WHr(n)|2}. MMSE detection can be implemented by applying a training sequence and an adaptive LMS/RLS algorithm. The optimal solution is Wopt=R−1r, where R=E[r(n)r(n)H].
Disturbed not only by the previous symbol but also by other chirp-rate signals in the same symbol, linear MMSE detection may be inadequate for the job in a deep fading channel. Here, a decision-directed method and parallel interference cancellation (PIC) are introduced to construct a robust algorithm for the multichirp-rate signal in the multipath environment, which is called MMSE-DD-PIC for short. The algorithm block diagram is shown in Fig. 4.
Block diagram of MMSE-PIC algorithm for detecting the multichirp-rate signal in the multipath environment. The figure shows the structure of MMSE-based algorithm to suppress the interference
The interference cancellation for a chirp-rate signal is given by
$$ \hat{b}_{m} = \text{sgn}\left\{\mathfrak{R}\left(y_{m}-\sum\limits_{j\neq m}^{M} u_{m,j}\hat{b}_{j}\right)\right\} $$
where \(\hat {b}_{j}\) is the decision-directed result of channel j and um,j are the weights of the channel for cancellation. Define U m =[um,1,⋯,um,j,⋯,um,M]T where (j≠m). By a decision-directed method, this U m can be simultaneously adjusted with the above MMSE, which is given by
$$ \begin{aligned} & \min_{\mathbf{W,U}} E\left\{\left|b_{m}(n)-\mathbf{W}_{m}^{H}\mathbf{r}(n)+\mathbf{U}_{m}^{H}\hat{\mathbf{d}}_{m}(n)\right|^{2}\right\} \\ & \text{s.t.} \mathbf{W}_{m,opt}^{H}\cdot\mathbf{1}_{m} = 1 \end{aligned} $$
where \(\hat {\mathbf {d}}_{m}(n)=[\hat {b}_{1}(n),\cdots,\hat {b}_{m-1}(n),\hat {b}_{m+1}(n),\cdots,\hat {b}_{M}(n)]\) are the decisions of other channels and 1 m in constraint is a zeros vector except for its mth element being one. In this MMSE criterion, W m is the feed-forward filter, and U m is the interference cancellation filter. The soft estimation \(y_{m}(n)=\mathbf {W}_{m}^{H}\mathbf {r}(n)-\mathbf {U}_{m}^{H}\hat {\mathbf {d}}(n)\). The constraint \(\mathbf {W}_{m,\text {opt}}^{H}\cdot \mathbf {1}_{m} = 1\) in the criterion regards the impulse position as the signature of the chirp rate in the BFT domain.
In this section, we give simulation results of the discrete BFT demodulation and compare them with the theoretical results. In simulations, a multichirp-rate-modulated signal is implemented. The parameters of modulation include a bandwidth of 10 MHz and a modulated symbol rate of 1 Msps. According to Theorem 1, applying orthogonal I-Q modulations, the number of chirp rates can be set to M = 10, and the chirp rates are \(K_{m}~=~B_{m}/T_{s}~=~2k/T_{s}^{2}\), where k = ±1,± 2,⋯±5.
Under ideal conditions including synchronization and no noise, Fig. 5 illustrates a D-BFT result for a 10-chirp-rate-modulated signal. Consistent with Lemma 1, the D-BFT of each chirp-rate signal is an impulse at the discrete β domain without mutual interference. Meanwhile, the result verifies the condition for orthogonality in a multichirp-rate signal.
D-BFT of 10-chirp-rate symbols {1,−1,1,−1,1,1,−1,1,1,1}. The figure shows the BFT output of the M-ary chirp-rate signal
Figure 6 shows the comparison of the BER performance of D-BFT detection among single chirp-rate and 10-chirp-rate signals; Fig. 6b, c shows the plots for the first and fourth rate signals, respectively. It must be pointed out that the E b in Fig. 6b, c is just composed of the energy of one chirp-rate component in a symbol period. From the results, we can see that the D-BFT BER performances on single-chirp-rate and multichirp-rate signals are consistent. It also proves the orthogonality of the components in the multichirp-rate signal in the D-BFT domain. At the same time, the identity of simulated and theoretical BERs verifies the correctness of the closed-form solution of D-BFT BER in Theorem 2.
BER comparisons of D-BFT detection among single-chirp-rate and multichirp-rate signals. The figure compares the BER performances by the theoretical analysis and the simulation. a BFT BER of Single -Chirp-rate Signal. b 1st-rate BFT BER of the Multiple-Chirp-rate Signal. c 4th-rate BFT BER of the Multiple-Chirp-rate Signal
Conventional chirp-rate demodulations include matched filter (De-Chirp) and FRFT demodulations. In addition, a frequency-shift keying (FSK) signal is also a time-frequency-modulated signal. The BER performances of the three demodulation methods on time-frequency signal are given below.
Matching filter on a chirp-rate signal (De-Chirp): \(P_{e}=Q (\sqrt {{2E_{b}}/{N_{0}}})\).
FRFT on a chirp binary-orthogonal keying (BOK) signal: \(P_{e}=Q(\sqrt {{E_{b}}/{2N_{0}}})\).
Matching filter on FSK signal: \(P_{e}=Q(\sqrt {{E_{b}}/{N_{0}}})\).
In Fig. 7, we compare their demodulation performance with BFT detection. Simulation results show that the BER performance of BFT is better than that of FSK coherent and FRFT noncoherent detection.
Performance of D-BFT detection with frequency offset. The figure compares the BER performances among D-BFT detection, FSK in AWGN and Doppler channel
Because BFT detection is a linear transformation, frequency error, or offset will cause rotation of the signal phase, we apply square-law BFT processing (BFT2) and unipolar chirp-rate on-off keying (OOK) signal to avoid deterioration in performance. However, the performance degrades due to this square-law detection. By Eq. (21), the decline of BFT sampling amplitude caused by offset is obtained to estimate the BER of BFT 2
$${} {{\begin{aligned} P_{e} &= \frac{1}{2}\exp\left(-\frac{\gamma}{4}\right)\\ &= \frac{1}{2}\exp\left(\frac{-3E_{b}}{16N_{0}}\cdot\left|\frac{\big[(1-j2\pi f_{d} T_{s})\exp(j2\pi f_{d} T_{s})-1\big]}{2\pi^{2}f_{d}^{2}T_{s}^{2}}\right|^{2}\right)\\ \end{aligned}}} $$
where γ is the input SNR of square-law detector.
From [25], the FRFT peak of the chirp-rate OOK signal under frequency offset is deduced as
$$ |S_{\alpha}(\mu)|^{2}_{\mu=-\cot\alpha,f_{m}=\mu\csc\alpha}=\frac{A^{2}\sin^{2}(\pi T_{s} f_{d})}{\pi^{2} f_{d}^{2}\cdot|\sin\alpha|} $$
where α is the time-frequency rotation angle in FRFT, A is the amplitude of signal, and f d is the Doppler shift. When | sinα|≈T s /B, where B is the bandwidth of the chirp signal and then the BER can be approximated by
$$ P_{e}\approx\frac{1}{2}\exp\left(-\frac{E_{b}}{16N_{0}}\cdot \frac{\sin^{2} \pi T_{s} f_{d} }{\pi^{2} f_{d}^{2} T_{s}^{2}}\right) $$
Under different conditions, including no frequency offset, f d = 100 kHz, and f d = 200 kHz, BFT2 detection performance is simulated and compared. The BER results of BFT2 in Fig. 7 are very close, which indicate that detection performance declines slightly even with large Doppler shift or frequency error. By contrast, FRFT detection exhibits a significant deterioration of BER performance at f d > 100 kHz, even if it has the advantages in conventional time-frequency-selective channel compared with coherent detection [16, 25].
In Section 5, we analyzed the characteristics of BFT detection of the multipath chirp-rate signal. Not only does inter-symbol interference exist, but the mutual interference among different chirp-rate signals in the same symbol period also makes detection difficult. Nevertheless, due to the sparsity of the impulse interferences in the BFT domain, it is possible to mitigate them by signal processing. In the non-line-of-sight (NLOS) communications environments, the equalization and interference cancellation is a better choice for detection.
Modulated with a five-chirp-rate real signal, 10-Mhz bandwidth, and 1-Msps symbol rate, BFT detection is simulated in the multipath environment. Two MMSE-based interference suppression algorithms are implemented according to the analysis and design presented in Section 6. The conventional LMS adaptive training process is applied to solving the optimal weight vector problem. On the other hand, we compare the BER performances at two different multipath fading environments. One is a common frequency selective fading channel, as in Fig. 8a which is the normalized channel frequency response. In Fig. 8b, we compare the BER performance of the algorithms in this channel. Auxiliary parallel interference cancellation makes the detection output close to that in flat fading. However, in this multipath environment, the performance of the MMSE-DD-PIC algorithm is only slightly better than that of MMSE detection, and for actual implementation, a choice based on computational complexity must be made. For a deep-fading frequency selective channel as shown in Fig. 9a, the BER performances are compared in Fig. 9b. The performance without interference suppressing is unacceptable. The MMSE-DD-PIC algorithm does a good job, and its performance is much better than that of simple MMSE training.
Performance comparisons in multipath environment. The figure shows the performance of the MMSE-based algorithm to M-ary chirp-rate signal in general frequency-selective fading environment. a Normalized frequency response of multipath fading channel. b D-BFT detection and MMSE-based algorithms
Performance comparisons in deep fading environment. The figure shows the performance of the MMSE-based algorithm to M-ary chirp-rate signal in deep frequency-selective fading environment. a Normalized frequency response of deep fading channel. b D-BFT detection and MMSE-based algorithms
The biorthogonal Fourier transform has been introduced to demodulate a multichirp-rate signal. In this paper, the detection performance of the D-BFT and BFT, including the multi-components resolution and closed-form BER, are derived. Further analyses are made in the frequency-offset and multipath environments. The small shift of compressed β-domain impulses in the high frequency-offset channel is a remarkable and welcome output of our research. Unfortunately, the BFT of a multichirp-rate signal is proven to have intra-symbol and inter-symbol interferences with multipath propagation or synchronization error. The theoretical result shows that the chirp modulation parameter can be adjusted to obtain the trade-off between the time and frequency dispersion. A multipath model of BFT output is constructed, and the MMSE-based algorithm is given to suppress the interference. Aided by the decision-directed PIC, the proposed detection performs well in the deep fading environment.
AWGN:
Additive white Gaussian noise
BER:
Bit-error-rate
BFT:
Biorthogonal Fourier transform
BFT2 :
Square-law BFT
Chirp spread spectrum
D-BFT:
Discrete BFT
FBMC:
Filter bank multicarrier
FRFT:
Fractional Fourier transform
ICI:
Inter-carrier interference
ISI:
Inter-symbol interference
LFM:
Linear frequency modulation
Multi-choice precoding
MMSE:
Minimum mean square error
MST:
Matched signal transform
OFDM:
Orthogonal frequency-division multiplexing
Parallel interference cancellation
SNR:
Signal-to-noise-ratio
STDFT:
Short-term discrete Fourier transform
T-F:
Time-frequency
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The authors would like to thank the reviewers for their thorough reviews and helpful suggestions.
This work is supported in part by the National Natural Science Foundation of China (nos. 61371107 and 61571143), the Foundation of Guangxi Broadband Wireless Communication & Signal Processing Key Laboratory (no. GXKL061501), and the Foundation of Science and Technology on Communication Networks Laboratory (no. KX172600033).
Key Laboratory of Guangxi Broadband Wireless Communication & Signal Processing, Guilin University of Electronic Technology, Jinji Road, Guilin, 541004, China
Lin Zheng, Chao Yang, Chao Yan & Hongbing Qiu
Key Laboratory of Science and Technology on Communication Networks, Zhongshan Western street, Shijiazhuang, 050081, China
Lin Zheng
Chao Yang
Chao Yan
Hongbing Qiu
LZ is the main writer of this paper. He proposed the main idea, deduced the performance of BFT detection, completed the simulation, and analyzed the result. CYang introduced the MMSE-based algorithm in dispersive channel. CYan simulated the detection in the Doppler channel. HQ gave some important suggestions for BFT detection. All authors read and approved the final manuscript.
Correspondence to Lin Zheng.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License(http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Zheng, L., Yang, C., Yan, C. et al. Biorthogonal Fourier transform for multichirp-rate signal detection over dispersive wireless channel. J Wireless Com Network 2018, 23 (2018). https://doi.org/10.1186/s13638-018-1027-3
Chirp-rate modulation
Dispersive channel | CommonCrawl |
Very large-scale neighborhood search
In mathematical optimization, neighborhood search is a technique that tries to find good or near-optimal solutions to a combinatorial optimisation problem by repeatedly transforming a current solution into a different solution in the neighborhood of the current solution. The neighborhood of a solution is a set of similar solutions obtained by relatively simple modifications to the original solution. For a very large-scale neighborhood search, the neighborhood is large and possibly exponentially sized.
The resulting algorithms can outperform algorithms using small neighborhoods because the local improvements are larger. If neighborhood searched is limited to just one or a very small number of changes from the current solution, then it can be difficult to escape from local minima, even with additional meta-heuristic techniques such as Simulated Annealing or Tabu search. In large neighborhood search techniques, the possible changes from one solution to its neighbor may allow tens or hundreds of values to change, and this means that the size of the neighborhood may itself be sufficient to allow the search process to avoid or escape local minima, though additional meta-heuristic techniques can still improve performance.
References
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Measurement of hard double-parton interactions in W(-> lv) plus 2-jet events at root s=7 TeV with the ATLAS detector
<mark>Journal publication date</mark>
<mark>Journal</mark>
New Journal of Physics
The production of W bosons in association with two jets in proton–proton collisions at a centre-of-mass energy of $\sqrt {s} = 7\,{\mathrm {TeV}}$ has been analysed for the presence of double-parton interactions using data corresponding to an integrated luminosity of 36 pb−1, collected with the ATLAS detector at the Large Hadron Collider. The fraction of events arising from double-parton interactions, f(D)DP, has been measured through the pT balance between the two jets and amounts to f(D)DP = 0.08 ± 0.01 (stat.) ± 0.02 (sys.) for jets with transverse momentum pT > 20 GeV and rapidity |y| < 2.8. This corresponds to a measurement of the effective area parameter for hard double-parton interactions of σeff = 15 ± 3 (stat.) +5−3 (sys.) mb.
© CERN 2013 for the benefit of the ATLAS Collaboration, published under the terms of the Creative Commons Attribution 3.0 licence by IOP Publishing Ltd and Deutsche Physikalische Gesellschaft. Any further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation and DOI. | CommonCrawl |
A nutrient-prey-predator model: Stability and bifurcations
Vicenţiu D. Rădulescu a,b,c,, and Dušan D. Repovš a,d,
Institute of Mathematics, Physics and Mechanics, Jadranska 19, 1000 Ljubljana, Slovenia
Faculty of Applied Mathematics, AGH University of Science and Technology, al. Mickiewicza 30, 30-059 Kraków, Poland
Institute of Mathematics "Simion Stoilow" of the Romanian Academy, P.O. Box 1-764, 014700 Bucharest, Romania
Faculty of Education and Faculty of Mathematics and Physics, University of Ljubljana, 1000 Ljubljana, Slovenia
* Corresponding author: Vicenţiu D. Rădulescu
To Professor Patrizia Pucci, on the occasion of her 65th birthday. Her work and friendship are a permanent source of inspiration and motivation.
Received July 2018 Revised August 2018 Published November 2019
We study the existence of nontrivial weak solutions for a class of generalized $ p(x) $-biharmonic equations with singular nonlinearity and Navier boundary condition. The proofs combine variational and topological arguments. The approach developed in this paper allows for the treatment of several classes of singular biharmonic problems with variable growth arising in applied sciences, including the capillarity equation and the mean curvature problem.
Keywords: Generalized p(x)-biharmonic equation, nonhomogeneous differential operator, variable exponent, singular nonlinearity.
Mathematics Subject Classification: Primary: 35J93; Secondary: 35J60, 49Q05, 58E05, 58E30.
Citation: Vicenţiu D. Rădulescu, Dušan D. Repovš. Combined effects for non-autonomous singular biharmonic problems. Discrete & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2020158
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Vicenţiu D. Rădulescu Dušan D. Repovš | CommonCrawl |
\begin{definition}[Definition:Universal Class]
The '''universal class''' is the class of which all sets are members.
The '''universal class''' is defined most commonly in literature as:
:$V = \set {x: x = x}$
where $x$ ranges over all sets.
It can be briefly defined as the '''class of all sets'''.
\end{definition} | ProofWiki |
\begin{document}
\preprint{APS/123-QED}
\title{Subdivided Phase Oracle for NISQ Search Algorithms}
\author{Takahiko Satoh} \affiliation{Keio Quantum Computing Center} \affiliation{Keio University Shonan Fujisawa Campus} \author{Yasuhiro Ohkura} \affiliation{Keio University Shonan Fujisawa Campus} \author{Rodney Van Meter} \email{\{satoh,rum,rdv\}@sfc.wide.ad.jp} \affiliation{Keio Quantum Computing Center} \affiliation{Keio University Shonan Fujisawa Campus}
\date{\today}
\begin{abstract} Because noisy, intermediate-scale quantum (NISQ) machines accumulate errors quickly, we need new approaches to designing NISQ-aware algorithms and assessing their performance. Algorithms with characteristics that appear less desirable under ideal circumstances, such as lower success probability, may in fact outperform their ideal counterparts on existing hardware. We propose an adaptation of Grover's algorithm, subdividing the phase flip into segments to replace a digital counter and complex phase flip decision logic. We applied this approach to obtaining the best solution of the MAX-CUT problem in sparse graphs, utilizing multi-control, Toffoli-like gates with residual phase shifts. We implemented this algorithm on IBM Q processors and succeeded in solving a 5-node MAX-CUT problem, demonstrating amplitude amplification on four qubits. This approach will be useful for a range of problems, and may shorten the time to reaching quantum advantage.
\end{abstract}
\maketitle
\section{Introduction} \label{sec:Intro} With the advent of NISQ~(Noisy Intermediate-Scale Quantum~\cite{preskill2018quantum}) processors, implementation of various NISQ-friendly algorithms, such as VQE~\cite{peruzzo2014variational}, is in progress. On the other hand, many algorithms whose theoretical computational complexity guarantees quantum acceleration require large-scale quantum circuits. Practical scale implementation of these algorithms will be difficult with NISQ devices, and future quantum computers with error correction capabilities will be needed.
Cross \emph{et al.} proposed Quantum Volume (QV) as a quantitative indicator of the computing power of quantum processors~\cite{PhysRevA.100.032328}. QV might double every year due to improvements in quantum processor performance~\cite{QV2019}. Determining the relationship between the QV of a processor and the size of the quantum circuit it can perform is essential in determining when a future quantum processor can solve a particular problem.
FIG.~\ref{fig:qvcdiagram} shows an abstract diagram of the relationship between classical and quantum computers. \begin{figure}
\caption{{\bf The significance of software development.} The solid, straight lines indicate the quantum computing power achieved to date, and the dashed line is the performance that will be realized assuming continuing increases in QV. Through the combined improvement of software and hardware, the aim is to reach the intersection with the curve of the ability of classical computers. Thus, software advances have the potential to shorten the time to the achievement of quantum advantage.}
\label{fig:qvcdiagram}
\end{figure} Hardware improvements and error mitigation reduce the effect of decoherence. The increased QV due to their contribution allows us to move to the upper right along this line. Improvements in algorithm, compilation, and structural connectivity both move down and change the slope of this line.
Focusing on the algorithm aspect, we describe the following contributions in this paper: 1) replacing the combination of the digital accumulator plus the binary ($0$ or $\pi$) phase flip with the subdivided oracle phase, and 2) an implementation method for $n$-controlled Toffoli gate suitable for processors with low connectivity. As an application of the first technique, we present an implementation for the MAX-CUT problem. The second technique addresses a fundamental need and may become an essential component of many algorithms.
Using these approaches, we have attempted to clarify the relationship between Grover's algorithm~\cite{grover1996fast} (Sec.~\ref{subsec:Grover}) and QV. As a preliminary step, we designed an algorithm to obtain an exact solution in the MAX-CUT problem (Sec.~\ref{subsec:maxcut} and~\ref{sec:Exact}). In this algorithm, when the input length exceeds 4 qubits, the total number of Controlled-NOT~($CX$) gates exceeds 100, and present-day quantum processors cannot obtain a useful answer. To miniaturize the algorithm as much as possible, we reduced the weight of the \textsc{$C^{\otimes n}X$}\xspace gate used in the diffusion operator (Sec.~\ref{subsec:impdif}) and adapted the phase information fragmentation in the oracle (Sec.~\ref{subsec:impora}). Although this makes it possible to realize a smaller quantum circuit than the above algorithm, it is not possible to transform a given problem into a decision problem, so we cannot call our solution NP-Complete. The correctness of the solution obtained depends on the average degree of the graph.
We executed our proposed algorithm on two IBM transmon systems, {\bf ibm\_ourense} with QV $=8$ and {\bf ibm\_valencia} with QV $=16$, and evaluated the success probability and KL divergence. The 3-data qubit Grover algorithm for the $K_{1,3}$ MAX-CUT found the correct answer over $29$\% (theoretical $34.7$\%) of the time on both processors (Sec.~\ref{subsec:exponq}). The 4-data qubit Grover algorithm for the $K_{1,4}$ MAX-CUT found the correct answer more than $11$\% (theoretical $21.2$\%) of the time on both processors. In the second experiment, the average KL divergence value of {\bf ibm\_valencia} was $0.457$, while that of {\bf ibm\_ourense} was $0.831$, substantially better than completely mixed state values of $1.149$.
These results indicate that probability amplification using Grover on a 4-qubit problem, which has conventionally been considered difficult~\cite{stromberg20184,mandviwalla2018implementing}, is possible using current processors. For this particular problem, differences in the decoherence characteristics of the two processors result in the off-answer elements of the superposition decaying more rapidly than the correct answer, resulting in an unexpectedly small decrease in overall success probability in the processor with the smaller QV. However, we expect that in more general cases, the success probability will more closely track the KL divergence. Also, our algorithm scales reasonably well on processor topologies with degree 3 qubits. Therefore, as processors with higher QV appear in the future, we can benchmark the maximum executable size of the Grover algorithm using our algorithm.
\section{Background} \label{sec:BG} \subsection{Grover's Algorithm} \label{subsec:Grover} Grover's algorithm is a quantum search algorithm to find the index of the target element $x \in \{0, 1, ...2^n-1\}$ s.t. $f(x) = y$, given $f$ and $y$, in $\mathcal{O} (\sqrt{N})$ operations with high probability, where $n$ is the number of qubits and $N = 2^n$ is the size of the list~\cite{grover1996fast}. The feature of this algorithm is that even if the database is disordered, the square root acceleration is guaranteed with respect to the classical search, which requires an average of $\frac{N}{2}$ operations~\cite{NC}.
\subsubsection{Procedure} The procedure of Grover's algorithm is as follows:
\begin{enumerate}
\item Initialization
Prepare $\ket{0}^{\otimes n}$ and apply Hadamard gates $H^{\otimes n}$ to create a superposition of $2^n$ states. All states have the same amplitude $\frac{1}{\sqrt{N}}$.
\item Oracle
Apply the oracle operator $O$ to invert the sign of target element(s):
\begin{align}
O\ket{x} \xrightarrow{} (-1)^{f(x)}\ket{x}.
\end{align}
Here, $f(x)=1$ if $x$ is the target element, otherwise $0$.
\item Diffusion
Apply the diffusion operator $D$ to amplify the probability amplitude of the target element:
\begin{align}
\label{eq:diff}
D&=H^{\otimes n}(2\ket{00..0} \bra{00..0} - I)H^{\otimes n} \\
&=H^{\otimes n}X^{\otimes n}H_{T}C^{\otimes n-1}XH_{T}X^{\otimes n}H^{\otimes n}.
\end{align}
Here, $C^{\otimes n-1}X$ and $H_{T}$ denote $n$-controlled $X$ gate and $H$ to the target qubit of $C^{\otimes n-1}X$. $H^{\otimes n}$ corresponds to the gates for initialization.
\item Iteration
Repeat $O$ and $D$. The optimal number of iterations is $\frac{4}{\pi}\sqrt{N}$ when the number of targets is 1.
\item Measurements
Measure all qubits to read the target data. \end{enumerate}
\begin{figure}
\caption{{\bf General circuit for Grover's algorithm} Grover's algorithm consists of data space and oracle working space. First, initialize all data qubits, then repeat Grover's operator (dashed box), which consists of oracle $O$ and diffusion operator $D$, $\mathcal{O}(\sqrt{N/m})$ times when the number of target states is $m$ and the search space size is $N$.}
\label{fig:grovercircuit}
\end{figure} In general, Grover's algorithm uses an $n$-qubit data register and work space qubits for oracle execution, as in FIG.~\ref{fig:grovercircuit}.
\subsection{The MAX-CUT problem} \label{subsec:maxcut} MAX-CUT is the graph theory problem of finding the maximum cut of given graph $G(V,E)$. MAX-CUT can be considered to be a vertex coloring problem using two colors that involves filling in some of the vertices with one color, and the rest of vertices with another color. Then we count the edges that exist between vertices of different colors as if they were cut. To solve this puzzle, we need to find a coloring combination which contains the highest number of edges connecting different color of vertices from $2^{\vert V \vert -1}$ possible colorings. On a general graph, MAX-CUT is known to be an NP-hard class problem~\cite{garey1979guide}.
\subsection{Current quantum processors} In recent years, NISQ (Noisy Intermediate Scale Quantum~\cite{preskill2018quantum}) devices that can perform quantum computation with a short circuit length have appeared, although the scale and accuracy are insufficient to perform continuous, effective error correction. Various physical systems such as superconductors, ion traps, quantum dots, NV centres, and optics are used in NISQ devices~\cite{ladd2010quantum, van2016local}.
The early 20-qubit superconducting processors from IBM had high connectivity and the maximum degree was 6, while the latest processors have a high gate accuracy but the maximum degree is 3 (FIG.~\ref{fig:processors}). \begin{figure}
\caption{{\bf Qubit topology of IBM Q processors} Early devices (left side) had a dense structure, while the recent devices (right side) are composed of relatively sparse qubit connections.}
\label{fig:processors}
\end{figure}
\subsubsection*{Quantum Volume and KQ} Quantum Volume~(QV) is a measure proposed by IBM that shows the performance of NISQ~\cite{PhysRevA.100.032328}. Quantum Volume QV is defined as \begin{align}
QV=2^{{\rm min}(m,d)}, \end{align} where $m$ denotes circuit width~(number of qubits) and $d$ denotes circuit $SU(4)$ depth. The QV for each processor is calculated from single and two-qubits gate errors, connectivity, measurement errors, etc. The computation fails with high probability when a given circuit satisfies \begin{align} \label{eq:epeff}
md \simeq \frac{1}{\epsilon_{\textrm{eff}}}. \end{align} Here, ${\epsilon_{\textrm{eff}}}$ is an effective $CX$ gate error value that gradually increases with connectivity.
In this paper, we experimented with two 5-qubit processors, {\bf ibmq\_ourence} with QV$ = 8$ and {\bf ibmq\_valencia} with QV$ = 16$.
KQ is a measure of the capabilities of the machine, independent of the algorithm. In 2003, Steane proposed a similar measure focusing on the algorithm’s needs and on error correction~\cite{steane2003overhead}. For an algorithm using $Q$ qubits and requiring $K$ time steps on those qubits (in suitable units), the space-time product $KQ$ is a guideline to the required error rate, which should be below $1/(KQ)$.
\subsubsection*{Open Quantum Assembly Language (QASM)} The IBM Q processors accept gates written in the QASM language~\cite{cross2017open}. All circuits are decomposed into four types of gate. We describe those gates and the required pulses in the IBM Q superconducting processors in Tab.~\ref{tab:gates}. \begin{table}[htb]
\begin{tabular}{c|c} gate type & remarks \\ \hline $U1(\lambda)$ & No pulse. Rotation $Z$~($R_Z$) gate. \\ $U2(\phi,\lambda)$ & One $\frac{\pi}{2}$ pulse. $H$ gate is $U2(0,\pi)$. \\ $U3(\theta, \phi, \lambda)$ & Two $\frac{\pi}{2}$ pulses. $R_Y(\theta)$ gate is $U3(\theta, 0, 0)$. \\ $CX$ & Cross-resonance pulses and One $\frac{\pi}{2}$ pulse. \end{tabular} \caption{Gate set for QASM} \label{tab:gates} \end{table} Since no pulse is required, we can perform \textsc{$U1$}\xspace~with zero cost. The error level of \textsc{$U3$}\xspace~is twice \textsc{$U2$}\xspace~and approximately an order of magnitude less than the $CX$ gate~\cite{QV2019}. The performance of {\bf ibmq\_ourense} and {\bf ibmq\_valencia} is shown in Tab.~\ref{table:device_qubit} and~\ref{table:device_cx} in the appendix.
\section{Grover algorithm to solve MAX-CUT problem} \label{sec:Exact} We propose Grover's algorithm for solving the MAX-CUT of a given graph $G$. The following simple coloring approach is an exhaustive classical search: \begin{itemize}
\item[Step 1.] Color all vertices black or white.
\item[Step 2.] Count the number of edges with different color vertices at both ends.
\item[Step 3.] Color the vertices with a different pattern from the existing one and return to Step 2.
\item[Step 4.] After testing all possible coloring patterns, the pattern with the largest number of edges counted corresponds to the MAX-CUT. \end{itemize} We can apply Grover's algorithm by assigning black to $\vert 0 \rangle$ and white to $\vert 1 \rangle$ in this procedure~\cite{QC2019_3}. To illustrate this correspondence, we show a simple example using a star graph $K_{1,2}$ in FIG.~\ref{fig:cutstar}. \begin{figure}
\caption{{\bf Data structure for MAX-CUT.} We can find MAX-CUT $\vert 010 \rangle_{012}$~(or $\vert 101 \rangle_{012}$) by counting the cases where the states of the qubits corresponding to both ends of the edge are different.}
\label{fig:cutstar}
\end{figure} The MAX-CUT for a graph with $m$ edges and $n$ vertices can be found by the following procedure. \begin{itemize}
\item[Step 0.] Set threshold value $t~(\leq m)$.
\item[Step 1.] Initialize all $n$ qubits to $\vert + \rangle$.
\item[Step 2.] Flip the sign of the input where the number of edges to be cut exceeds $t$.~(the oracle)
\item[Step 3.] Amplify the probability of any input whose sign is inverted.~(diffusion)
\item[Step 4.] Repeat Steps 1-3 $\mathcal{O}(\sqrt{2^n})$ times.
\item[Step 5.] Increase $t$ if the output is legal for the graph, decrease if the output is illegal. If $t$ returns to a value taken in a prior iteration, it is MAX-CUT, and the algorithm ends. Otherwise, the process returns to Step 1. \end{itemize}
The number of iterations can be optimized by the quantum counting algorithm~\cite{brassard1998quantum}. In addition, if an excessively low value $t$ is set such that the sign of the majority of inputs is inverted, the probability of the input with the sign not inverted is amplified. Since a binary search can be done by appropriately increasing and decreasing $t$, we can get accurate MAX-CUT by $\log_2{m}$ iterations.
The most straightforward way to implement an oracle for a counting problem is by using a binary accumulator register. We describe the oracle's construction below.
\subsection{Oracle circuit design} We discuss how to apply the above procedure when given a star graph $K_{1,4}$ (FIG.~\ref{fig:k_14}). First, we prepare 5 data qubits to describe the state of nodes. When there is an edge between node $A$ and $B$, as a cut checker for each edge, we introduce the following sub-oracle $O_{S(A,B)}$~\cite{QC2019_3}: \begin{align}
O_{s(A,B)}\vert \psi_{A}\psi_{B}\rangle\vert \psi_{S}\rangle \rightarrow \vert \psi_{A}\psi_{B}\rangle\vert \psi_{S}\! +\! (\psi_{A}\!\oplus \! \psi_{B})\rangle. \end{align} Here, $S$ is an accumulator register large enough to store the number of cut edges. For this problem, $ \lceil \log (\vert E \vert + 1 ) \rceil =3$ qubits are enough. When the states of $A$ and $B$ are different, the edge between $A,B$ is cut, and the information of cut edges on $S$ is updated. We can implement $O_{s(A,B)}$ using a quantum increment circuit as shown in FIG.~\ref{fig:suboracle}. \begin{figure}
\caption{(a) A star graph $K_{1,4}$. Each node number denotes the corresponding data qubit. (b) If the states of qubit $A$ and $B$ are different, the accumulator register $\vert \psi_{s} \rangle$ becomes $\vert \psi_{s} + 1\rangle$ .}
\label{fig:k_14}
\label{fig:suboracle}
\label{fig:exbasic}
\end{figure}
After the execution of $O_S$ for all edges, we set the threshold value $t$ and perform the phase inversion operation for inputs that equal or exceed $t$ using the flag qubit. (In this problem, $t$ corresponding to MAX-CUT is obviously $4$.) We show the circuit corresponding to these operations in FIG.~\ref{fig:oracles}. \begin{figure}
\caption{{\bf Oracle circuit.} $O$ denotes the sequence of all sub-oracles $O_S$. After the execution of Pshift, we have to uncompute $O^\dagger$ to propagate sign reversal for inputs equal to or exceeding the threshold value $t$. }
\label{fig:oracles}
\end{figure} We also show in detail how to configure phase shift~(Pshift) operation in Appendix~\ref{pshift}.
\subsection{Complete circuit implementation} \label{comp_exact} When $t = 4$, we can get $\vert 01111 \rangle$ and $\vert 10000 \rangle$ as solutions by combining the above oracle and diffusion and repeating those the appropriate number of times. When implementing on a processor with the current QV, the proposed circuit is too large in both number of qubits and depth.
For example, the half adder contains a Toffoli gate that requires 6~$CX$ gates on IBM Q devices. From the discussion in Sec.~\ref{sec:BG}, the upper limit of $CX$~ gates that can be used to obtain valid results is understood to be around 10. Taking into account the need to uncompute portions of the circuit, we will not be able to include multiple sub-oracles and anticipate successful execution.
We have already proposed a method to reduce $CX$~ gates by eliminating adders and increasing ancilla qubits~\cite{QC2019_3}. This implementation requires $\vert V \vert + \vert E \vert$ qubits and $2 \vert E \vert$ Toffoli gates, plus some $CX$~ gates for Pshift$(t)$ for one iteration. In summary, we still need more than 36~$CX$~per iteration to solve MAX-CUT in the smaller graph $K_{1,3}$. Needless to say, there is room for improvement in our proposed oracles. However, in order to solve MAX-CUT with Grover's algorithm on a real processor in the near future, drastic improvement is necessary. Therefore, we next propose a new data structure that does not store the number of cut edges in binary data.
\section{Approximated Grover search for MAX-CUT} \label{sec:Approx} In this section, we describe Grover's algorithm using phase subdivided oracle operators instead of the conventional $0$ and $\pi$. By using this method, we can remove the adders used in the previous section and reduce the circuit length significantly. We also propose a diffusion operator implementation that requires fewer $CX$ gates for an actual processor design by using relative phase Toffoli gates~\cite{barenco1995elementary, maslov2016advantages}. We describe those methods and the verification of the effectiveness for the MAX-CUT problem below.
\subsection{Oracle circuit using subdivided phases} \label{subsec:oracle} In Sec.~\ref{sec:Exact}, storage of the evaluation value $k$ (the number of cut edges) and its calculation using adders led to a large increase in the number of $CX$ gates and occupied the largest portion of the whole circuit.
Therefore, we propose a method to express the evaluation value by the number of subdivided phases. In the MAX-CUT problem, we use the same data structure for node color as in Sec.~\ref{sec:Exact} and unit phase \begin{align} \theta_0 = \frac{\pi}{\abs{E}} \end{align} where $\abs{E}$ denotes the number of edges in the graph $G$.
For the cut edge determination, we introduce the following sub-oracle $O'_s$ using sub-divided phase $\theta_0$. If an input $\vert \psi_{a} \rangle$ has a cut edge between vertices A and B, then we add $\theta_0$ to the phase information: \begin{align} \label{eq:app_suboracle} O'_{s(A,B)}\vert \psi_{a} \rangle \rightarrow e^{i\theta_0}\vert \psi_{a} \rangle. \end{align} Similarly, based on the whole oracle operation $O'$, the best answer input $\vert \psi_{b} \rangle$ becomes as follows, (for MAX-CUT value.): \begin{align} \label{eq:oracle} O'\vert \psi_{b} \rangle \rightarrow e^{ik\theta_0}\vert \psi_{b} \rangle \end{align} where $k\theta_0$ does not exceed $\pi$. We next discuss the performance of this oracle and the existence of the optimal subdivided phase $\theta_{opt}$.
\subsubsection*{Algorithm performance and optimal subdivided phase} \label{subsec:opttheta} When the given graph is a tree~($\vert V \vert = \vert E \vert+1$ for a connected graph), the average value of the added phase $\langle\alpha(\theta)\rangle$ after applying the above oracle $O'$ is: \begin{align}
\langle\alpha(\theta)\rangle=\frac{1}{2^{\vert E \vert}}\sum^{\vert E \vert}_{k=0}
\left(
\begin{array}{c}
\vert E \vert \\
k
\end{array}
\right)e^{ik\theta}. \end{align} From Eq.~(\ref{eq:diff}), the probability amplitude of each input with phase $k\theta$ after diffusion execution becomes: \begin{align} \label{eq:optp}
\frac{1}{\sqrt{2^{\vert V \vert}}}(\vert 2\langle \alpha(\theta) \rangle - e^{ik\theta}\vert). \end{align}
If $\vert V \vert=5$, the oracle adds the phase $e^{i4\theta}$ to the input corresponding to the MAX-CUT. There are two bit strings corresponding to the same MAX-CUT. (e.g. As shown in Sec.~\ref{comp_exact}, $\vert 01111 \rangle$ and $\vert 10000 \rangle$ denote the same MAX-CUT of $K_{1,4}$.) We describe how to eliminate this redundancy by using a virtual node in the final part of this section.
When $\theta = \theta_0$, the probability of finding either value of MAX-CUT $p(\theta)$ becomes: \begin{align}
p(\theta_0)=2\left(\frac{1}{\sqrt{32}}\vert 2\langle \alpha(\theta) \rangle - e^{i4\theta_0}\vert\right)^2\simeq 0.195. \end{align}
If we have information about the phase each input is given by the Oracle, we can maximize the amplification factor by adjusting the subdivided phase: \begin{align}
{\rm max}\{\vert 2\langle \alpha(\theta) \rangle - e^{i4\theta_0} \vert \}\simeq 1.84. \end{align} Then, maximized $p(\theta)$ and optimal subdivided phase are: \begin{align}
p(\theta_{opt}) \simeq 0.212, \\
\theta_{opt} \simeq 0.323 \pi. \end{align} The amount of amplification depends on the difference between the average value of the added phase. Therefore, the probability of the worst solution that does not cut any edges is amplified similarly to the proper MAX-CUT solution.
Using the exact solution in the Sec.~\ref{sec:Exact}, the average value of the added phase $\langle\alpha'\rangle$ after applying the above oracle $O$ with $t=4$ is: \begin{align}
\langle\alpha'\rangle=\frac{1}{2^{\vert E \vert}}
\left(e^{i\pi} + 2^{\vert E \vert} -1
\right). \end{align} Then, after performing oracle and diffusion only once, the probability of finding either of MAX-CUT $p'$ becomes: \begin{align}
p'=\frac{1}{16}(\vert 2\langle \alpha' \rangle - e^{i\pi}\vert)^2\simeq 0.473. \end{align}
Thus, the performance of our method lies in between the random search and the standard Grover algorithm using $\pi$ for the phase shift. Not only that, since the average value increases as the graph become denser, the worst-case probability becomes larger than MAX-CUT. Despite such drawbacks, this algorithm requires many fewer gates than searching for an exact solution.
A naive implementation using QAOA, a useful NISQ algorithm for MAX-CUT, requires $\vert V \vert$ qubits to store data and $2\vert E \vert~CX$~ gates for each Cost Unitary~\cite{farhi2014quantum,otterbach2017unsupervised}. As discussed below, this is on a scale comparable to the Oracle circuit in our proposal. The circuit depth is equal to or greater than our circuit when the number of block repeats $p=2$ or more. Since QAOA requires parameter changes and iterative execution, the number of CX gates needed to obtain a solution is an advantage of our method.
Besides, for problem sizes where the classical algorithm requires a few $milliseconds$~\cite{goemans1995improved,kugel2010improved,guerreschi2019qaoa}, it is difficult for our method to outperform the classical algorithm on the MAX-CUT problem, since it requires a huge number of trials (Eq.~(\ref{eq:optp})). Next, we show a specific implementation method.
\subsubsection*{Implementation of oracle} \label{subsec:impora} When the $\theta$ is not $0$ or $\pi$, the sub-oracle in Eq.~\ref{eq:app_suboracle} consists of the following gate sequence: \begin{align}
O'_{s(A,B)}:=X_{B}CR^{B,A}_{Z(\theta)}X_{B}X_{A}CR^{A,B}_{Z(\theta)}X_{A}. \end{align} Due to the limitations of the current IBM Q processors within the framework of QASM~\cite{cross2017open}, we need two $CX$ gates and single-qubit gates to execute one $CR^{A,B}_{Z}(\theta)$ exactly.
Here, the error values on single-qubit gates are one order of magnitude smaller than that of two-qubit~($CX$) gates~\cite{QV2019}. Therefore, we focused on reducing the number of $CX$ gates, and the number of single-qubit gates such as $U3$ gate is basically not a problem. Hence we approximate the whole sub-oracle with two $CX$ gates and six $U3$ gates by KAK decomposition~\cite{bullock2003arbitrary,shende2004minimal} as shown in FIG.~\ref{fig:oracle_app}. \begin{figure}
\caption{Approximation of sub-oracle circuit using KAK decomposition at $\theta_0 = \frac{\pi}{4}$. The approximation accuracy is over ${\bf 99}$\%, and the average error of the $CX$ gate of the Q processor as of January 2020 is about $1$\%. Until the $CX$ gate error is halved, the total error will be dominated by the two-qubit gates.}
\label{fig:oracle_app}
\end{figure} The error level of a $CX$ gate of the latest IBM Q processors used in this paper is about 1\% at best~\cite{QV2019}. Hence, we approximate this oracle circuit with two $CX$ gates~\cite{PhysRevA.100.032328}.
\subsubsection*{Introduction of virtual vertex} The output of the approach in Sec.~\ref{sec:Exact} has redundancy due to the symmetry of the problem. In order to eliminate this and double the solution space in a given number of qubits, we introduce a virtual vertex whose state is fixed at $\vert 0_V \rangle$.
The oracle for the edge connected to this virtual vertex can be replaced by a single qubit operation $R_Z(\theta_0)$ on the other vertex. In order to reduce the number of $CX$ gates in the oracle part, it is effective to virtualize the highest degree vertex. For example, when the given graph is $K_{1,4}$, we can perform the oracle circuit without using $CX$ gates, as shown in Fig.~\ref{fig:oracle_virtual}. \begin{figure}
\caption{Implementation of oracle circuit $O'$ using sub-divided phase for the star graph $K_{1,4}$.
All sub-oracles $O'_{s(V,k)}$ can be replaced with $R_Z(\theta)$ by assigning the highest degree vertex to the virtual qubit.}
\label{fig:oracle_virtual}
\end{figure}
\subsection{Implementation of diffusion} \label{subsec:impdif} After executing the oracle in Sec.~\ref{subsec:oracle}, we perform the normal diffusion operator for Grover's algorithm. As described in Sec.~\ref{sec:BG}, the diffusion circuit for $n+1$ data qubits require one $n$-controlled NOT (\textsc{$C^{\otimes n}X$}\xspace) gate.
We can implement the (three-qubit) Toffoli gate with a well-known circuit using 6 $CX$, Hadamard gates and T gates~\cite{NC}. DiVincenzo and Smolin found a Toffoli gate decomposition using 5 controlled unitary gates~\cite{divincenzo1994results}, and this number of two-qubit operations was later shown to be optimal~\cite{yu2013five}. If we allow imperfect phases, Margolus created a Toffoli gate using 3 $CX$ gates~\cite{margolus1994simple,divincenzo1998quantum}. Maslov showed other configurations for imperfect Toffoli gates, such as based on $controlled-controlled-iX$~\cite{PhysRevA.87.042302}, and collectively called them relative-phase Toffoli gates ($RTOF$)~\cite{maslov2016advantages}.
Barenco {\it et al.} described the decomposition of an $n$-control qubit Toffoli gate (\textsc{$C^{\otimes n}X$} gate) using $2n-3$~-Toffoli gates and $n-2$ ancillary qubits~\cite{barenco1995elementary}. To decrease the number of gates, Maslov replaced all but one Toffoli with $RTOF$ and composed \textsc{$C^{\otimes n}X$}\xspace with $6n-6$~-CX gates~\cite{maslov2016advantages}.
On the other hand, executing these Toffoli gate circuits requires a fully-connected three-qubit structure, which we cannot directly implement on the actual quantum processors used in this paper. Thus, it becomes important to relax the semantic constraints, including both phase and variable qubit placement. Therefore, we discuss how to implement a \textsc{$C^{\otimes n}X$}\xspace~gate under the constraints of the IBM Q processors.
\subsubsection*{\textsc{$C^{\otimes n}X$}\xspace~gate implementation} To construct \textsc{$C^{\otimes n}X$}\xspace, we adopt two types of $RTOF$, shown in the FIG.~\ref{fig:rccx}. \begin{figure}\label{fig:rccix}
\label{fig:rccxm}
\label{fig:rccx}
\end{figure} Both of these $RTOF$ can be implemented on a system with only a one-dimensional qubit layout. Although the number of $CX$ gate is equal, $RTOF_{iX}$ does not require $U3$, which reduces single qubit rotation errors. To implement a \textsc{$C^{\otimes n}X$}\xspace gate with ancillary qubits and $RTOF$~\cite{maslov2016advantages} while avoiding the above-mentioned connectivity problem, we also introduce the Toffoli gate with built-in SWAP operation.
A Toffoli gate implementation with the minimal 6 $CX$ gates requires three qubits interconnected in a triangle. Recent IBM Q devices after {\bf ibm\_tokyo} do not have a structure that can embed triangles. To deal with this situation, we propose a Toffoli circuit suitable for a one-dimensional layout, as shown in FIG.~\ref{fig:TwithS}. \begin{figure}
\caption{{\bf Toffoli with SWAP circuit.} By adding the $CX$ gates surrounded by a broken line to the general Toffoli gate decomposition, SWAP is built in, and the circuit can be performed with qubits connected in a straight line.}
\label{fig:TwithS}
\end{figure} This circuit requires one additional $CX$, the minimum overhead. However, since SWAP is built in, it is necessary to consider the location of qubits in the output state.
By using those components, we can configure a \textsc{$C^{\otimes n}X$}\xspace gate for recent IBM Q devices using $6n-5$~$CX$ gates. It is known that a \textsc{$C^{\otimes n}X$}\xspace~gate can consist of $2n-3$ Toffoli gates with $n-2$ ancillary qubits (initialized to $\vert 0 \rangle$)~\cite{barenco1995elementary}. A Toffoli gate contains at least 6 $CX$ gates. As shown in FIG.~\ref{fig:cnx}, Toffoli gates in \textsc{$C^{\otimes n}X$}\xspace can be replaced with $RTOF$ except for the central one. \begin{figure}\label{fig:cnx}
\end{figure} To support additional implementations, we show the procedure for \textsc{$C^{\otimes n}X$}\xspace in Algorithm~\ref{proc_cnx}. \begin{algorithm}[H]
\caption{\textsc{$C^{\otimes n}X$}\xspace~gate implementation}
\label{proc_cnx}
\begin{algorithmic}[1]
\Statex {Input: $n+1$ data qubits $d$, $n-2$ ancillary qubits $a$.}
\Statex {Output: Data qubits on which the \textsc{$C^{\otimes n}X$}\xspace~ gate is performed and SWAP gate between last two data qubits.}
\Statex {}
\Procedure{\textsc{$C^{\otimes n}X$}\xspace~gate with SWAP}{}
\State {RTOF($d_0$, $d_1$, $a_0$)}
\For {k=0; k<n-3; k++}
\State {RTOF($a_k$, $d_{k+2}$, $a_{k+1}$)}
\EndFor
\State{TOF($a_n-3$, $d_n-1$, $d_n$) with SWAP($d_n-1$, $n$)}
\For {k=n-4; k>-1; k-\,-}
\State {RTOF($a_k$, $d_{k+2}$, $a_{k+1}$)}
\EndFor
\State {RTOF($d_0$, $d_1$, $a_0$)}
\State {return Target states.}
\EndProcedure
\end{algorithmic} \end{algorithm}
If the processor can embed the structure shown in FIG.~\ref{fig:qubits_arrange}, the procedure can be executed without additional SWAPs. \begin{figure}
\caption{Qubit connections for Algorithm~\ref{proc_cnx}. Data and ancilla qubits are denoted by $d_k$ and $a_k$, respectively. (a) shows the interactions required by the algorithm; (b) shows how they might map to one of the 20-qubit machines.}
\label{fig:qubits_arrange}
\end{figure} When $n = 1$, we can embed this in all recent processors, including the 5-qubit processors {\bf ibmq\_vigo~(ourense)}. Similarly, when $n = 8$ or less, we can embed in the processors {\bf ibmq\_boeblingen~(singapore)}, which have 20 qubits.
\subsection{Experiments on IBM Q devices} \label{subsec:exponq} We evaluate our proposed algorithm by finding MAX-CUT of $K_{1,3}$ and $K_{1,4}$ on current processors. If the given graph is $K_{1,4}$, our algorithm requires 5 physical qubits (4 data, 1 ancillary) and 1 virtual qubit. FIG.~\ref{fig:correspondence_gq} illustrates the correspondence between the given graphs and qubits.
\begin{figure}\label{fig:correspondence_gq}
\end{figure} We investigate the performance of each component and the whole algorithm.
\subsubsection*{\textsc{$C^{\otimes n}X$}\xspace gate performance} The $CX$ gate error rates of {\bf ibmq\_ourence} and {\bf ibmq\_valencia} are around 1~\%, an order of magnitude higher than errors of single-qubit gates~(see TAB.~\ref{table:device_qubit} and~\ref{table:device_cx} in Appendix~\ref{appendix:performance_device}). We performed several experiments to verify the performance of $U3$ gates and measurement error mitigation~\cite{kandala2017hardware}. The results in FIG.~\ref{fig:single_rot} ~(Appendix~\ref{appendix:performance_ry}) show that the single-qubit gate error and the mitigated measurement error are much smaller than the $CX$ error.
Using Algorithm~\ref{proc_cnx}, we can assemble a $C^{\otimes 3}X$ gate from a Toffoli with SWAP gate, and two types of $RTOF$ gates. To evaluate these gate performances, we reconstructed output states. We calculated fidelities of those states as shown in FIG.~\ref{fig:tofs}. Additionally we also confirm the output of $RTOF$ gates $C^{\otimes 3}X$ gates in the computational basis in FIG.~\ref{fig:tof_prob} and~\ref{fig:c3x_prob} (Appendix~\ref{appendix:performance_cnx}).
These results show that {\bf ibmq\_valencia} is a better device than {\bf ibmq\_ourense} in accordance with their QV values in terms of average fidelity and variance. \begin{figure}
\caption{
{\bf Gate fidelities of various Toffoli gates on real devices.}
Light blue points are the gate fidelity on {\bf ibmq\_ourense} with ${\rm QV} = 8$ and deep blue points are the gate fidelity on {\bf ibmq\_valencia} with ${\rm QV} = 16$. For each gate type, we tested all possible mappings to the processor topology, collecting the results of $8192$ shots for each pattern. The top and bottom bar of each data bar are the maximum and minimum values of the experimental results.
}
\label{fig:tofs}
\end{figure}
\subsubsection*{Whole circuit performance on real processors} To evaluate our algorithm performance, we first execute the whole circuit (see FIG.~\ref{fig:3grover_circuit}) with $7~CX$ for $K_{1,3}$. In this experiment, we adopt two subdivided phases \begin{align}
\theta_0 = \frac{\pi}{3} \end{align} and \begin{align}
\theta_{opt} = 0.392\pi \end{align} obtained from Eq.~(\ref{eq:optp}). FIG.~\ref{fig:3grover_result} shows the execution results using two processors. \begin{figure}\label{fig:3grover_result}
\end{figure} In all experiments, the output probability of the correct answer $\vert 111 \rangle$ is about 28\%, which is a good result even when compared to the ideal value of 33.4\% with $\theta_{0}$ and 34.7\% with $\theta_{opt}$. For a more quantitative evaluation we show the KL divergence in FIG.~\ref{fig:kl-3grover}. A better value for {\bf ibm\_ourense} would suggest that the circuit is small enough for both processors.
\begin{figure}
\caption{
{\bf Kullback-Leibler divergence of 3 qubit subdivided Grover outputs } The KL divergence of the data from FIG.~\ref{fig:kl-3grover} relative to the expected probability distribution (from pure state simulation) compares favorably to that of a uniform distribution with all output values having equal probability (as would be expected with high noise levels), showing that quantum algorithm performs well. Although the overall quality of valencia is superior to ourense, two output values (011 and 101) are more heavily weighted, giving a slightly worse KL divergence.
}
\label{fig:kl-3grover}
\end{figure}
We next execute the whole circuit (see FIG.~\ref{fig:4grover_circuit}) with $13~CX$ for $K_{1,4}$. As discussed in Sec.~\ref{subsec:oracle}, we adopted both $0.25\pi$ and $0.323\pi$ for the angle of divided phase oracle. We also adopt $RTOF_{iX}$ and $RTOF_{M}$ in $C^{\otimes 3}X$ gate. We show results on two processors in FIG.~\ref{fig:4grover_result}. \begin{figure*}\label{fig:grover_u2_025}
\label{fig:grover_u3_025}
\label{fig:grover_u2_032}
\label{fig:grover_u3_032}
\label{fig:4grover_result}
\end{figure*} In these experiments, the probabilities of the correct answer $\vert 1111 \rangle$ are increasing. Those probabilities are maximum when $\theta_{opt}$ is used in any processors, and is about 11\%, about half the theoretical probability of 21.2\%. On the other hand, there is a significant difference between the processors in the probability amplification and suppression of incorrect answers. This may be due in part to the $\vert 1111 \rangle$ output being susceptible to relaxation errors. We show the difference in performance between the two processors using KL in FIG.~\ref{fig:kl-4grover}. \begin{figure}
\caption{
{\bf Kullback-Leibler divergence of 4-qubit subdivided oracle search outputs } The KL divergence values for the 4-qubit search are substantially higher than for the 3-qubit search, as expected, but still show a clear difference from the uniform distribution, evidence of the algorithm's effectiveness. {\bf ibmq\_valencia}'s higher QV is apparent here.
}
\label{fig:kl-4grover}
\end{figure} Due to the symmetry of the problem, the probability of $\vert 1111 \rangle$, which is the MAX-CUT value, and $\vert 0000 \rangle$, where no edge is cut, should be amplified the most. Nevertheless, only one of the results is greatly amplified. In the circuit used in this experiment, the oracle does not include $CX$, and diffusion includes the theoretically minimum number of $CX$ in current IBM Q processors. The fact that we were unable to achieve the ideal probability amplification even when such a circuit was adopted seems to indicate that the number of qubits and circuit depth exceed the current processor capability. Further, considering the effect of relaxation, an increase in the probability of a solution containing more $\vert 0 \rangle$ values seems natural. However, depending on the qubit mapping, the probability of solutions containing $\vert 1 \rangle$ clearly increases. This may be due to an unknown difference between the data structure in the development environment Qiskit and the data structure on the actual IBM Q system. \section{Conclusion} \label{sec:Con} As of this writing, there has been no report that any problem has been solved using 4-qubit unmodified Grover search on a solid-state quantum computer. As shown in Sec.~\ref{sec:Exact}, the scale of the circuit required for the algorithm exceeds the limit that existing quantum processors can handle. Thus, we investigated alternate solutions appropriate for the NISQ era, reducing the number of qubits and gates required by over one order of magnitude via the sub-divided phase oracle. This oracle, rather than the normal 0/$\pi$ phase flip of ordinary Grover, applies a smaller phase shift to less desirable outcomes and a larger phase shift to more desirable ones. While this initially appears less favorable, the dramatic reduction in required fidelity makes it a good tradeoff for small problems, as shown by our experimental results demonstrating effective amplitude amplification for 4-qubit search problems as exemplified by solving the MAX-CUT problem. Further work will help to determine the range of problem sizes and characteristics for which this technique can be applied.
With our current modest circuit depths, overall performance is still strongly affected by measurement errors, but it is worth comparing the KQ of our algorithms with the reported QV of the processors. We found that the $K_{(1,3)}$ solution using 7 CNOTs on 3 qubits ($KQ=7\times 3=21$) works well on quantum volume QV=8, and very similarly on QV=16. The $K_{(1,4)}$ solution using 13 CNOTs on 4 qubits ($KQ=13\times 4=52$) works, although not well, on QV=8; it performs much better, but still with limited effectiveness, on QV=16. This circuit is one of the largest KQ values reported to have been run successfully on a solid-state quantum computer to date. KQ and QV are similar measures and it will be interesting to continue tracking their relationship and predictive value for execution success over the coming generations of computers.
In addition, we designed a diffusion operator using the minimum number of $CX$ gates within the constraints of recent IBM Q processors, by incorporating Toffoli gate variants with phase shifts that we compensate for later in the algorithm. This technique is exact, and will benefit a broad range of algorithms beyond the NISQ era.
\begin{acknowledgments} This work was supported by MEXT Quantum Leap Flagship Program Grant Number JPMXS0118067285. The results presented in this paper were obtained in part using an IBM Q quantum computing system as part of the IBM Q Network. The views expressed are those of the authors and do not reflect the official policy or position of IBM or the IBM Q team. We thank Miguel Sozinho Ramalho and Lakshmi Prakash for working with TS and YO on the project that inspired this paper at Qiskit Camp Vermont 2019. TS would like to thank Yuri Kobayashi, Atsushi Matsuo, and Shin Nishio for their collaborative activities for the Quantum Challenge, which helped refine the ideas in this paper. We thank Ken M. Nakanishi at the University of Tokyo for a useful discussion on the optimal implementation of the Toffoli gate. We are grateful for meaningful discussions with Shota Nagayama at Mercari, Inc. \end{acknowledgments}
\appendix
\onecolumngrid
\section{Multi-controlled $X$ gates for phase shift operator} \label{pshift} To solve the MAX-CUT by combining the binary search and Grover's algorithm, we have to invert the sign of the input whose cut edges exceeds the threshold value $t$. With the proper $C^{\otimes n}X$ gate combination, the phase shift operator can distinguish whether the accumulated cut edges value exceeds $t$ or not. We show the phase shift operator according to three different $t$ in Fig.~\ref{fig:phase-flip}.
\begin{figure}
\caption{{\bf Phase shift operators for given threshold value $t$.} The required number of $C^{\otimes n}X$ gates differs depending on the value of $t$. Here we show implementation examples of the phase shift operator according to $t$.}
\label{fig:phase-flip}
\end{figure}
\section{Results of supplemental experiments} \subsection{Performance of $R_{Y}$ gate} \label{appendix:performance_ry}
Fig.~\ref{fig:single_rot} shows the performance of $R_Y(\theta) \equiv U3(\theta,0,0)$, with and without measurement error mitigation. We show the probability of finding $\vert 0 \rangle$ when measuring in the computational basis with $-\pi \leq \theta \leq \pi$. The result after applying measurement error mitigation is close to the ideal value, regardless of the value of QV.
\begin{figure}
\caption{Performance of $R_{Y}(\theta)$ gate as a probability of $\vert 0 \rangle$ for each rotation angle $\theta$. Dashed line correspond to $R_{Y}(\pm \frac{\pi}{4})$ in $RTOF_{M}$ gate. EM denotes result with measurement error mitigation.}
\label{fig:single_rot}
\end{figure}
\begin{figure*}
\caption{{\bf Execution of three types of Toffoli gate on the real devices}. To generate the results for each row, 8192 trials were performed for each input. Entries are output probabilities, with each row summing to approximately 1. Each row denotes the input value, and each column the output value.
}
\label{fig:rtofix_ourense_prob}
\label{fig:rtofm_ourense_prob}
\label{fig:swaptof_ourense_prob}
\label{fig:rtofix_valencia_prob}
\label{fig:rtofm_valencia_prob}
\label{fig:swaptof_valencia_prob}
\label{fig:tof_prob}
\end{figure*}
\subsection{Performance of composite gates} \label{appendix:performance_cnx}
We prepared different input states and measured using the computational basis after applying three types of Toffoli gate. FIG.~\ref{fig:tof_prob} shows the experimental results using two processors.
We also perform $C^{\otimes 3}X$ gate to different input states and measured using the computational basis. FIG.~\ref{fig:c3x_prob} shows the experimental results using two processors.
\begin{figure*}\label{fig:rc3xix_ourense_prob}
\label{fig:rc3xm_ourense_prob}
\label{fig:rc3xix_valencia_prob}
\label{fig:rc3xm_valencia_prob}
\label{fig:c3x_prob}
\end{figure*}
When testing small circuits such as these complex gates on real systems, state preparation and measurement (SPAM) errors will distort the results compared to the circuit itself. Therefore, we adopted the standard measurement error mitigation approach recommended for use with Qiskit, utilizing the library functions {\tt CompleteMeasFitter()} and {\tt complete\_meas\_cal()} in Qiskit~\cite{qiskit_MEM}. First we execute the set of circuits created by {\tt complete\_meas\_cal()} to take measurements for each of the $2^5$ basis states for five qubits on {\bf ibm\_ourense} or {\bf ibm\_valencia}, and collect the results into a matrix $C_{\textrm{noisy}}$. We then use {\tt CompleteMeasFitter()} to find $M$ that satisfies the following equation: \begin{align}
C_{\textrm{noisy}} = M C_{\textrm{ideal}} \end{align} where $C_{\textrm{ideal}}$ denotes ideal result matrix not containing noise. If $M$ is invertible, we can mitigate the measurement errors by applying the inverse of $M$ to the raw data matrix $R$ from the actual circuit (e.g., FIG.~\ref{fig:tof_prob}): \begin{align}
R_{\textrm{mitigated}} = M^{-1} R_{\textrm{noisy}}. \end{align} However, in general, $M$ is not invertible; instead, the corresponding Qiskit filter object derived from $M$ applies a least-squares fit. All of the real-device data figures in this paper utilize this approach.
\section{The circuits for the MAX-CUT problem} We show the circuit to find MAX-CUT of $K_{1,3}$ in Fig.~\ref{fig:3grover_circuit}. Unlike Eq.~(\ref{eq:diff}), we adopted $ZH (HZ)$ for $HX (XH)$ and Toffoli with SWAP gate for Toffoli gate. The former change allows us to reduce the number of $U3$ gates, thereby reducing gate errors (in the case where a series of single-qubit gates are not integrated into one $U3$ gate). The latter change avoids connectivity constraints with minimal overhead.
\begin{figure*}\label{fig:3grover_circuit}
\end{figure*}
We show the circuit to find MAX-CUT of $K_{1,4}$ in Fig.~\ref{fig:4grover_circuit}. The gate set of the diffusion part except $ZH$ and $HZ$ constitutes one $C^{\otimes 3}X$ gate.
\begin{figure*}\label{fig:4grover_circuit}
\end{figure*}
\section{Qiskit Versions}
The version of Qiskit packages we use are listed in Table \ref{table:QiskitVer}.
\begin{table}[ht]
\centering
\begin{tabular}{c|c}
\hline
name & version \\ \hline
qiskit & 0.14.0 \\
qiskit-terra & 0.11.0 \\
qiskit-aer & 0.3.4 \\
qiskit-ignis & 0.2.0 \\
qiskit-aqua & 0.6.1 \\
qiskit-chemistry & 0.5.0 \\
qiskit-ibmq-provider & 0.4.4 \\ \hline
\end{tabular}
\caption{{\bf Qiskit packages version}}
\label{table:QiskitVer} \end{table}
\section{Date-time}
Each experiment was performed on the dates listed in Table \ref{table:experiments}.
\begin{table}[ht]
\centering
\begin{tabular}{l|c}
\hline
Experiment & Date-time \\ \hline \hline
\begin{tabular}{l}
Performance of $RTOF_{iX}$ gate, $RTOF_{M}$ gate and
\\Toffoli with SWAP gate on {\bf ibmq\_ourense}
\end{tabular} & 2019/12/24 \\ \hline
\begin{tabular}{l}
Performance of $C^{\otimes 3}X$ with $RTOF_{iX}$ gate and \\ $C^{\otimes 3}X$ with $RTOF_{M}$ gate on {\bf ibmq\_ourense}
\end{tabular} & 2019/12/24 \\ \hline
\begin{tabular}{l}
MAX-CUT solver on {\bf ibmq\_valencia}
\end{tabular} & 2020/1/1 \\ \hline
\begin{tabular}{l}
MAX-CUT solver on {\bf ibmq\_ourense}
\end{tabular} & 2020/1/1 \\ \hline
\begin{tabular}{l}
Subdivided phase Oracle Grover algorithm\\on {\bf ibmq\_ourense}
\end{tabular} & 2020/1/1 \\ \hline
Simulated gate fidelities of various Toffoli gates & 2020/1/1 \\ \hline
\begin{tabular}{l}
Performance of $RTOF_{iX}$ gate, $RTOF_{M}$ gate and
\\Toffoli with SWAP gate on {\bf ibmq\_valencia}
\end{tabular} & 2020/1/6 \\ \hline
\begin{tabular}{l}
Performance of $C^{\otimes 3}X$ with $RTOF_{iX}$ gate and \\ $C^{\otimes 3}X$ with $RTOF_{M}$ gate on {\bf ibmq\_valencia}
\end{tabular} & 2020/1/6 \\ \hline
Performance of $R_Y$ gate on {\bf ibmq\_ourense} & 2020/1/8 \\ \hline
Performance of $R_Y$ gate on {\bf ibmq\_valencia} & 2020/1/8 \\ \hline
\end{tabular}
\caption{{\bf Date and time when experimental data have been taken}}
\label{table:experiments} \end{table}
\section{Performance of IBM Q processors} \label{appendix:performance_device} We show single-qubit gate and readout performance of IBM Q processors in TAB.~\ref{table:device_qubit}. We also show two-qubit gates performance in TAB.~\ref{table:device_cx}.
\begin{table}[H]
\centering
\begin{tabular}{c|c|c|c}
\hline
& U2 gate error & U3 gate error & Readout error \\ \hline
\multicolumn{4}{c}{ibmq\_ourense} \\ \hline
$Q_0$ & $3.04E-4$ & $6.09E-4$ & $1.80E-2$ \\
$Q_1$ & $3.32E-4$ & $6.63E-4$ & $2.80E-2$ \\
$Q_2$ & $3.67E-4$ & $7.33E-4$ & $2.80E-2$ \\
$Q_3$ & $3.79E-4$ & $7.58E-4$ & $3.40E-2$ \\
$Q_4$ & $3.77E-4$ & $7.53E-4$ & $4.90E-2$ \\ \hline
\multicolumn{4}{c}{ibmq\_valencia}\\ \hline
$Q_0$ & $5.31E-4$ & $1.06E-3$ & $2.75E-2$ \\
$Q_1$ & $3.35E-4$ & $6.70E-4$ & $4.13E-2$ \\
$Q_2$ & $5.51E-4$ & $1.10E-3$ & $2.50E-2$ \\
$Q_3$ & $3.22E-4$ & $6.45E-4$ & $2.50E-2$ \\
$Q_4$ & $4.26E-4$ & $8.52E-4$ & $4.00E-2$ \\ \hline
\end{tabular}
\caption{Qubit performance on Jan 1 2020.}
\label{table:device_qubit} \end{table}
\begin{table}[htb]
\centering
\begin{tabular}{c|c|c}
\hline
& {\bf ibmq\_ourense} & {\bf ibmq\_valencia} \\ \hline
$CX~(0, 1)$ & $7.22E-3$ & $7.67E-3$ \\
$CX~(1, 2)$ & $9.55E-3$ & $9.62E-3$ \\
$CX~(1, 3)$ & $1.34E-2$ & $1.13E-2$ \\
$CX~(3, 4)$ & $7.35E-3$ & $7.71E-3$ \\
\hline
\end{tabular}
\caption{$CX$ gate performance on Jan 1 2020.}
\label{table:device_cx} \end{table}
\end{document} | arXiv |
A knowledge graph embeddings based approach for author name disambiguation using literals
Cristian Santini1,2,3,
Genet Asefa Gesese1,3,
Silvio Peroni2,
Aldo Gangemi2,
Harald Sack1,3 &
Mehwish Alam1,3
Scientometrics volume 127, pages 4887–4912 (2022)Cite this article
Scholarly data is growing continuously containing information about the articles from a plethora of venues including conferences, journals, etc. Many initiatives have been taken to make scholarly data available in the form of Knowledge Graphs (KGs). These efforts to standardize these data and make them accessible have also led to many challenges such as exploration of scholarly articles, ambiguous authors, etc. This study more specifically targets the problem of Author Name Disambiguation (AND) on Scholarly KGs and presents a novel framework, Literally Author Name Disambiguation (LAND), which utilizes Knowledge Graph Embeddings (KGEs) using multimodal literal information generated from these KGs. This framework is based on three components: (1) multimodal KGEs, (2) a blocking procedure, and finally, (3) hierarchical Agglomerative Clustering. Extensive experiments have been conducted on two newly created KGs: (i) KG containing information from Scientometrics Journal from 1978 onwards (OC-782K), and (ii) a KG extracted from a well-known benchmark for AND provided by AMiner (AMiner-534K). The results show that our proposed architecture outperforms our baselines of 8–14% in terms of F1 score and shows competitive performances on a challenging benchmark such as AMiner. The code and the datasets are publicly available through Github (https://github.com/sntcristian/and-kge) and Zenodo (https://doi.org/10.5281/zenodo.6309855) respectively.
Avoid the common mistakes
Data available in scholarly knowledge graphs (SKGs)—i.e., "a graph of data intended to accumulate and convey knowledge of the real world, whose nodes represent entities of interest and whose edges represent potentially different relations between these entities" (Hogan et al., 2021)—is growing continuously every day, leading to a plethora of challenges concerning, for instance, article exploration and visualization in Liu et al. (2018), article recommendation in Beel et al. (2016), citation recommendation in Färber and Jatowt (2020), and Author Name Disambiguation (AND) (see Sanyal et al. (2021) for a survey), which is relevant for the purposes of the present article. In particular, AND refers to a specific task of entity resolution which aims at resolving author mentions in bibliographic references to real-world people.
Author persistent identifiers, such as ORCIDs and VIAFs, simplify the AND activity since such identifiers can be used for reconciling entities defined as different objects and representing the same real-world person. However, the availability of such persistent identifiers in SKGs—such as OpenCitations (OC) (Peroni & Shotton, 2020), AMiner (Wan et al., 2019), and Microsoft Academic Knowledge Graph (MAKG) (Faber, 2019)—is characterized by very low coverage and, as such, additional and computationally-oriented techniques must be adopted to identify different authors as the same person.
In the past, many approaches have been developed to automatically address AND by using publications metadata (e.g., title, abstract, keywords, venue, affiliation, etc.) to extract features that can be used in the disambiguation task. These methods vary widely from supervised learning methods to unsupervised learning including recently developed deep neural network-based architectures. However, the existing SKGs do not provide all the relevant contextual information necessary to effectively and efficiently reuse those approaches, which often rely on pure textual data.
This issue is represented by the fact that only a few bibliographic databases provide extensive access to full bibliographic metadata, including abstracts, keywords and uniquely identified affiliations (e.g. with ROR IDsFootnote 1), upon which many current AND approaches heavily rely. As witnessed by the rise of Open Access initiatives such as the Initiative 4 Open Abstracts (I4OAFootnote 2), the availability of textual metadata such as abstracts, which are used by many machine learning approaches for bibliometric studies (including AND), is limited. As an example, in June 2020, on Crossref, a non-profit DOI registration agency that openly provides high-quality metadata from most international publishers, only 8% of journal articles had an abstract. This is of course a challenge posed by the infrastructure involved in the provision of SKGs, such as OpenCitations, which usually collect these metadata from heterogeneous and openly available bibliographic databases.
In contrast to many current approaches, this study focuses on performing AND for scholarly data represented as linked data or included in SKGs, by considering the multi-modal information available in such collections, i.e., the structural information consisting of entities (e.g., authors, publications, venues and affiliations) and relations between them as well as text or numeric values associated with the authors and publications defined in the form of literals (family name, given name, publication title, venue title, year of publication). In order to take into account discriminative features, such as abstracts and keywords, we present a novel approach which models structural information encoded in the SKG and exploits the semantics conveyed by potential literals to refine the structural representations.
The proposed framework to address this task is named Literally Author Name Disambiguation (LAND), which focuses on tackling the following research questions:
Can Knowledge Graph Embeddings (KGEs)—i.e. a technique that enables the creation of a "dense representation of the graph in a continuous, low-dimensional vector space that can then be used for machine learning tasks" (Gesese et al., 2021b)—be used effectively for the downstream task of clustering, more specifically for author name disambiguation?
Does the information present in attributive triples (i.e. titles, publication dates, etc.) in existing SKGs enhance the aforementioned representations for AND?
The goal of this article is to provide a representation learning method for extracting entity features from SKGs which do not require any labeled training data. To this end, LAND uses semantic matching models which incorporate literal information, namely LiteralE (Kristiadi et al., 2019), to extract author-related features which can adapt to the sparsity of metadata in SKGs. LAND further makes use of KGEs along with Hierarchical Agglomerative Clustering (HAC) and Blocking [for a survey see Backes (2018)] in order to obtain final clusters of authors from the dense representations learned directly from the SKGs.
The rest of the article is organized as follows. Section 2 discusses the related studies in the field. Section 3 introduces the SKGs created for conducting our experiments. Section 4 details the proposed framework, while Sect. 5 documents the conducted experiments and the achieved results. Finally, Sect. 6 provides a summary of the work and gives some future perspectives.
This section describes the studies related to author name disambiguation which are further divided into supervised learning approaches, unsupervised learning, graph-based approaches. It also details the studies using KGEs for scholarly data.
Author name disambiguation
Generally, name disambiguation, also called entity resolution, is the task of removing duplicates from large noisy databases. This task is affected by ambiguity itself, due to the multiple names which are used to refer to it in database theory: record linkage, deduplication, data matching, instance matching, and data linkage. A first definition of it was provided by Dunn (1946), in which record linkage is defined as the process of collecting pieces of information which refer to the same individual into one single unit. Later, Fellegi and Sunter (1969) provided a mathematical modelling of the problem.
The relevance of name disambiguation is related to the fact that it is required in many scientific domains: not only statistics but also political sciences and social sciences, medicine and epidemiology, demographic and human rights statistics, bibliometric and scientometric analysis. For a recent interdesciplinary survey please refer to Binette and Steorts (2022). In general, the problem of entity resolution can be tackled via two distinct approaches: either clustering records according to the real-world entity to which they belong, or identifying coreferent record pairs.
Author Name Disambiguation (AND) is a subcategory of the name disambiguation task, concerned with the resolution of records inside scholarly databases. In Ferreira et al. (2012), the authors classify the existing AND approach into two categories, i.e., author assignment and author grouping. The author assignment approach directly assigns a label to every item corresponding to a real-world author. This approach is often difficult to implement since it requires the actual list of authors to be mapped to specific classes a priori. The second method, author grouping, consists in clustering the entries corresponding to authors via a similarity function which should produce output groups associated with real-world entities. Author grouping may not require the number of authors to be known a priori and is consequently easier to implement in most cases.
Moreover, in the aforementioned survey, the authors classify the evidence used according to three categories: (1) web information (e.g., information extracted from web pages), (2) citation information (i.e. metadata associated with publications), and (3) implicit evidence, such as topic modeling or graph embeddings. Additionally, a common strategy in AND is to initially group author entries into subsets by looking at name compatibility, e.g., authors carrying the same last name are grouped and disambiguation is performed within each group. This activity is carried out to reduce the number of pairwise comparisons required by the task and is termed as Blocking [for details see Backes (2018)]. One of the simplest and most common approaches is to group authors by looking at the full last name and the first initial (hereafter, LN-FI) of the given name, therefore called LN-FI blocking.
Supervised learning approaches
These approaches take into consideration several features extracted from scholarly metadata, such as publications' title words, keywords, coauthors, venues, etc., and a classifier is trained with labeled data pairs to estimate if a given pair of publications belong to the same author or not. One of the seminal works in supervised AND is Author-ity, presented in Torvik and Smalheiser (2009). Author-ity makes use of LN-FI blocking to preliminarily split publications related to ambiguous author names into blocks; then, given a pair of publications \(p_1, p_2\) corresponding to two author name instances \(s_1, s_2\) respectively in a block, it constructs a multidimensional similarity profile \(x(p_1, p_2)\), based on title, journal name, co-author names, MeSH, language, affiliation, email, and other name attributes. The similarity profile is the input feature of a classifier trained with Bayesian learning to estimate the probability of \(x(p_1, p_2)\), given that \(p_1, p_2\) are written by the same author or not. In the end, a maximum-likelihood-based agglomerative clustering is used in order to group publications.
Another approach that makes use of supervised learning is BEARD, presented in Louppe et al. (2016). This method adopts a phonetic-based blocking strategy to preliminarily group authors into blocks by looking at the phonetic representation of the normalized surname (e.g., "van der Waals, J. D." → "Waals, J. D."). Moreover, it associates a set of features to each pair of author instances that are designed to be sensitive to the ethnic group of the authors. Then, a classifier is trained on annotated data to learn a pairwise distance function using tree-based methods (i.e. Random Forest (RF) and Gradient Boosted Trees (GBT)). Finally, author references are grouped using hierarchical agglomerative clustering. The novelty of this method is to introduce for the first time ethnicity-sensitive features to make author name disambiguation sensitive to the actual origin of the authors. However, the authors state that the impact of the phonetic-based blocking strategy is not adequately addressed.
In Tran et al. (2014), a Deep Neural Network (DNN) is used to learn distinctive features for author disambiguation. By using labeled pairs of publications related to an ambiguous author name, the neural network is trained to calculate the posterior probability of the internal features, given basic features as input, such as the Jaccard distance between author name, co-authors list, affiliation, keywords, and author interest keyword.
In Kim et al. (2019), the authors exploit structure-based features, consisting of word embeddings modeling publications' metadata using Term Frequency-Inverse Document Frequency (TF-IDF), to train four distinct architectures, a Support Vector Machine (SVM), an RF, a GBT, and a DNN. These architectures are trained to determine whether a pair of publications are related to the same author or not.
Unsupervised learning approaches
Due to the fact that human annotation of training data is a time-consuming task for digital libraries, many unsupervised approaches have been devised for AND. One of the seminal works in unsupervised methods was presented in Cota et al. (2010), where the authors propose an unsupervised heuristic-based AND approach. In their approach, publications related to an ambiguous name are clustered together based on shared coauthors. Subsequently, the fragmented clusters are linked together based on the cosine similarity between TF-IDF vectors of potentially weak features such as title and publication venue. However, the drawbacks of this approach are mainly due to the fact that TF-IDF vectors do not capture efficiently similarity between titles. Due to this reason, this method heavily relies on coauthorship information. This may lead to false positives and data segmentation, especially when an author does not have the same set of coauthors across multiple works.
Similar to Cota et al. (2010), the methodology presented Liu et al. (2015) consists of an incremental clustering approach where the authors rely on coauthorship information to create fragmented clusters, which are later merged by using similarity in title and publication venue. More specifically, in the first step, coauthorship-based clusters are created by using the Floyd-Warshall algorithm; then, in the second step, clusters are merged by using cosine similarity between publication titles; in the last step relations between authors and venues are captured by using Latent Semantic Analysis (LSA) to further merge fragmented clusters.
In Caron and van Eck (2014), the authors adopt a predefined set of rules for scoring author similarity by taking into consideration several metadata attributes of a pair of publications: for example, if one paper cites the other it gets a score of 10. If the overall score of a pair of publications is above a certain threshold, the two publications are considered as belonging to the same author. The results obtained after pairwise comparison are then aggregated using HAC. For its simplicity and interpretability, this method was recently adopted effectively in Färber and Ao (2022) to perform large-scale author name disambiguation on MAKG (Farber, 2019). However, despite its scalability, this method does not take into account higher-order relations between entities in a dataset, such as coauthorship and citation patterns, since it only takes into account the amount of shared information between the metadata of the publications. Moreover, the use of predefined rules might not be effective for author name disambiguation on datasets affected by data sparsity and therefore might not be flexible to different domains and data sources.
Recently in Waqas and Qadir (2021), a heuristic-based clustering framework was proposed to combine the information coming from structure-based features (e.g. metadata such as title, coauthor names, venues, etc.) with global features, which are distributional representations obtained using word embeddings, to perform AND in a multi-step algorithm. Similar to other studies, the authors create initial clusters by using more discriminative features, such as co-authors, author affiliation, and author email. In order to compare the information present in the structure-based features, the authors apply several heuristic rules which vary with respect to the information considered. In the subsequent step, the algorithm merges the obtained clusters by using similarity between word embeddings extracted from the stemmed words in titles, abstracts, and keywords. In the end, the authors use venues and publication dates to merge together unwanted split clusters. As for other heuristic-based methods, this approach does not capture higher-order relations between publications, such as citation networks; moreover, the use of a multi-layer approach does not allow to improve precision as further information is considered, such that coming from titles and venues, which is only used to merge wrongly split clusters.
Graph-based approaches
In Fan et al. (2011), a graph-based method called GHOST is exploited to cluster publications using graph components. In this method, a co-authorship graph is constructed for each instance s related to a queried author name by collapsing all the co-authors with the same name into one single node. The resulting graph contains all authors which are co-authors with s and all authors which have co-authored a paper with the co-authors of s. Then, the similarity between two instances of s is computed based on the number of valid paths and affinity propagation is used to group nodes into clusters. However, this method does not work for single-author papers, and information contained in other metadata (e.g. publication titles, abstracts, or keywords) is not considered.
In Km et al. (2020), the authors present a graph-based approach where two graphs are combined together, a person-person graph obtained by connecting papers with shared coauthors, and a document-document graph which models similarity between publications' content. The document-document graph is obtained by first modeling abstract keywords with TF-IDF vectors and by drawing an edge between two nodes of the graphs when their similarity is higher than a selected threshold; subsequently, this graph is pruned by removing connections between papers whose shared referenced works are below a certain threshold. Then the person-person and the document-document graphs are merged together and the connected components of the combined graph represent the final authors. This graph-based model takes into account citation information from the two papers, however, the construction of the person-person graph might produce false split clusters in datasets from the humanities domain, where the same coauthors might not appear in many papers.
Graph embedding based approaches Recently, graph embeddings on heterogeneous and non-heterogeneous graphs have been implemented to solve the AND problem. In Zhang and Al Hasan (2017), the authors propose an AND approach that works on anonymized graphs by using relational information learned via network embeddings. This method constructs three local graphs for a candidate set of documents: a person-person graph representing a collaboration between authors, a person-document graph representing the association between authors and bibliographic records, and a document-document similarity graph based on co-authorship relations. A representation learning model is proposed to embed the nodes of these graphs into a shared low-dimensional space by optimizing a joint objective function based on the pairwise ranking of similarity. The final results are generated by HAC. This method proposes a new representation learning framework that is particularly suited for downstream clustering tasks. However, since this approach is designed for anonymized graphs, it does not take into consideration many attributes of nodes instead it considers co-author sharing for computing document similarity.
A network learning algorithm that models different types of relationships among publications, called Diting is proposed in Peng et al. (2019). This approach first builds different weighted networks for each paper belonging to an ambiguous name, where each weighted network represents the similarity of the publications in a candidate set with respect to a selected feature, such as title, coauthor, organization, venue, etc. Then, network embeddings are learned by maximizing the gap between similar nodes and dissimilar ones, and graph coarsening is used to reduce the number of nodes. Final authors are obtained by using HDBSCAN and Affinity Propagation (AP) to cluster the node embeddings. In the same paper, the authors propose a semi-supervised version of their method, called Diting++, which makes use of information available on the web to refine the final author's clusters.
In Zhang et al. (2018), the authors propose an AND method based on three components: a global learning module which creates embedding representations for each document by leveraging structure-based features, a local-linkage learning framework which exploits shared information to refine the embeddings related to an ambiguous name a, and a Recurrent Neural Network (RNN) which estimates the number of clusters for each ambiguous name a. In the global learning step, the authors refine structure-based features obtained from the metadata about the publication by using labeled triplet samples. In the second step, these global features are refined after modeling the candidate set of publications related to an ambiguous author name in a weighted network and by using a graph autoencoder to learn node embeddings. The final partition of the candidate set is obtained by using HAC. Despite the high performance of this model, its reproducibility is hindered by the fact that it requires labeled samples and complex feature engineering, which can make the approach not flexible to data sparsity.
In Zhang et al. (2019), a simple graph embedding method based on Random Walk and Skip-Gram (Mikolov et al., 2013) is proposed to perform AND on co-authorship networks. Qiao et al. (2019) presents an AND methodology for heterogeneous weighted graphs. In this method, publications are represented in vector space by using Doc2Vec (Le & Mikolov, 2014); then, publication embeddings are refined by using information from a heterogeneous network that links publications based on different weighted relations (e.g., CoTitle, CoVenue, CoAuthor). The information from this network is used to refine the Doc2Vec embeddings by using a Heterogeneous Graph Convolutional Neural Network (HGCNN).
In Wang et al. (2020), an author disambiguation algorithm for heterogeneous information networks is presented. In heterogeneous information networks, relations between scholarly works, authors, and other entities are modeled with nodes and relations of different types. In their method, the authors model the content of the article coming from literary attributes such as titles and abstracts with Doc2Vec, in order to create a document embedding for each paper; moreover, they encode the heterogeneous information network of scholarly publications into node embeddings by using Node2Vec (Grover & Leskovec, 2016). The novelty of their method consists in using Generative Adversarial Networks (GANs) to mutually exploit the information coming from document embeddings and node embeddings. By evaluating their method on the AMiner benchmark (Zhang et al., 2018), they outperformed the best model by + 5.13% in Macro-F\(_1\) score. In their paper, the authors also test the approach on a SKG called Ace-KG, however, the lack of a source code makes it difficult to reproduce this methodology on other datasets.
Pooja et al. (2021) proposes an unsupervised graph embedding method to solve the AND task which exploits similarities between publications on two different dimensions: on the content level (e.g. title, abstract, keywords, references) and on the coauthor level. In order to do so, they use two different variational graph autoencoders and neural fusion to combine multiple representations.
In Chen et al. (2021), the authors propose an author disambiguation algorithm based on Graph Convolutional Networks (GCNs) which takes into account attribute features as well as relation information. In their method, they first build three graphs: a paper-paper, a person-person, and a paper-person graphs, which are then fed to a specialized GCN that outputs hybrid representation. The authors show that their method performs in a competitive way on multiple datasets, including AMiner (Zhang et al., 2018).
A detailed comparison of different graph embedding models, along with the information used and the respective performances on different benchmark datasets is presented in Table 1. As shown in the table, the majority of these methods do not apply representation learning directly on graphs but they preliminary require feature engineering to measure the relatedness of a pair of publications. Moreover, only Chen et al. (2021), Wang et al. (2020), and Zhang and Al Hasan (2017) are tested on a scenario where few publication attributes are used (only titles, coauthors, venue, and affiliation) and other textual information, like abstracts, is not considered.
Table 1 Overview of graph embedding AND methods
Knowledge graph embeddings and scholarly data
Few studies have been made recently on the use of KGEs with an application on scholarly linked data. In Mai et al. (2018), an entity retrieval system for the scholarly domain was proposed, combining information coming from textual embeddings and structural embeddings trained from the KG IOS Press LD Connect.Footnote 3 In this paper, the authors evaluate the quality of low-dimensional representations of papers and entities (i.e. authors, organizations, etc.) by extracting two benchmark datasets: (1) a benchmark dataset collected from Semantic Scholar in order to evaluate the semantic similarity of papers, and (2) a second benchmark dataset extracted from DBLP used in order to evaluate co-authorship recommendations based on KGEs. The authors extract paragraph vectors for representing papers' content by using Doc2Vec and train TransE (Bordes et al., 2013) for extracting embeddings of entities in the SKG of IOS LD Connect. In order to build the entity retrieval model, a logistic regression model which takes as input features both paragraph vectors and structural embeddings. It is trained on a dataset of similar papers automatically collected from Semantic Scholar. Reported results show that KGEs do not have a significant impact on paper similarity classification, whether textual embeddings alone achieve robust results. As a second step, a co-author inference evaluation is carried out by using a benchmark dataset extracted from DBLP to demonstrate the ability of TransE for predicting coauthorship links based on the observed triples.
In Nayyeri et al. (2020), embeddings have been used to generate coauthorship recommendations on SKGs. One of the aims of this work is to propose a novel approach for training KGE models on SKGs where 1-to-N, N-to-1, and N-to-N relations are frequent (i.e. authorship relations or citation links). In order to address this issue, the authors present a reimplementation of TransE (Bordes et al., 2013) and RotatE (Sun et al., 2019) by using a newly proposed loss function optimized for many-to-many relations, i.e. Soft Margin (SM) loss. The results of their study show how the models equipped with SM loss outperform the original models. The novelty of this study is to propose a loss function that mitigates the adverse effects of false-negative sampling and to investigate the use of KGEs for co-authorship suggestions.
Creation of the scholarly KGs
This section introduces the benchmark datasets OC-782K and AMiner-534K which are created for evaluating the LAND framework. OC-782K is a subset of the Scientometrics Knowledge Graph available from Massari (2021), which is built in compliance with the OpenCitations Data Model (OCDM) (Daquino et al., 2020). On the other hand, AMiner-534K is a KG generated from a well-established benchmark datasetFootnote 4 for AND made available by AMiner in the AND paper (Zhang et al., Zhang et al. (2018)). The two KGs are available on ZenodoFootnote 5 at Santini et al. (2021) in order to allow the reproducibility of the studies herein described.
The OC-782K knowledge graph
In this paper, the Scientometrics KG from Massari (2021) is referred to as Scientometrics-OC. This publicly available KG contains bibliographic information about the articles published by the journal ScientometricsFootnote 6 from 1978 to the present, along with bibliographic information of all the academic works cited by the articles published by that journal. The dataset named OC-782K is created from Scientometrics-OC by modeling entities related to authors, publications, and venues with different data models suited for the task of AND.
This data model contains three types of entities: fabio:Expression, which represents articles, books, conference papers, and other academic works, fabio:Journal for representing journal venues (if the related fabio:Expression is a journal article), and authors which are described as foaf:Agent. The data model is an abstraction of the OCDM (Daquino et al., 2020) and is created for two reasons: (i) for collecting triples only related to the entities of interest (e.g. bibliographic resources, venues, and authors), (ii) create an abstract representation of Scientometrics-OC in order to perform representation learning more efficiently. The data model of OC-782K is represented in Fig. 1.
A Graffoo diagram Falco et al. (2014) describing the data Model used for OC-782K
OC-782K is extracted from Scientometrics-OC by first collecting information about the bibliographic resources with at least a title and an author. Then, the publication dates and journal venues of these works (if available) were collected. A foaf:knows relation is added between two authors who have co-authored the same work, and the relation between two bibliographic resources, a citing expression and a cited one, is represented with the cito:cites relation. Attributive triples were extracted along with triples connecting different entities by following part of the guidelines presented in Gesese et al. (2021a).
The dataset consists of 781,917 triples, with 620,321 structural triples (i.e. triples with object relations). In the original Scientometrics-OC, while duplicate bibliographic resources and journals were merged by using the DOIs associated with each article, authors are not disambiguated (i.e., there is one author for each dcterms:creator relation.) Statistics of the dataset are reported in Tables 2 and 3.
Table 2 The number of entities and triples in OC-782K
Table 3 The number of entities and relations is counted by type in OC-782K
The AMiner-534K knowledge graph
In order to evaluate the generalizability of the proposed approach on a different dataset, a second scholarly KG named AMiner-534K is extracted from the AMiner AND benchmark dataset introduced in Zhang et al. (2018). The AMiner benchmark for AND contains a sub-set of publications from AMiner and sampled from 100 ambiguous Asian names. This dataset contains the following information for each scholarly article: title, publication date, venue, keywords, authors, and affiliations. The AMiner-534K KG is created by extending the data model of OC-782K with the additional author affiliation information (using the property schema:affiliation). A representation of the data model is available in Fig. 2. However, for AMiner-534K the cito:cites and the foaf:knows properties are absent since these properties are not present in the original benchmark.
A Graffoo diagram Falco et al. (2014) describing the data model used for AMiner-534K
Moreover, we did not add abstracts and keywords, even though they are present in the benchmark (Zhang et al., 2018). This is due to the fact that this information is often not given in many of the available scholarly datasets and using this information to train a model may lead to results which are not reproducible on datasets not as rich as the benchmark in Zhang et al. (2018). Statistics of the dataset are reported in Tables 4 and 5 (Fig. 3).
The overview of the LAND architecture
Table 4 The number of entities and triples in AMiner-534K
Table 5 The number of entities and relations is counted by type in AMiner-534K
Literally Author Name Disambiguation (LAND)
In this section, the different components of the proposed framework, Literally Author Name Disambiguation (LAND), are described in detail. Figure 4 shows the overall architecture of the approach which is based on three main components:
Multimodal KG embeddings This strategy is aimed at learning representative features of entities and relations in a KG by taking into account the structure of the graph itself along with the semantics contained in the literal about these entities (e.g., titles of academic works or publication dates).
Blocking This strategy is used to reduce the number of pairwise comparisons required by the AND task by initially grouping authors into blocks characterized by name similarity, so that disambiguation is carried within each block. LAND uses a rather simple but effective blocking strategy called LN-FI blocking.
Clustering Hierarchical Agglomerative Clustering (HAC) is used to group the embeddings associated with each author to be disambiguated into k-clusters by using a vector-based similarity measure (e.g., cosine) along with a distance threshold.
The output of these components is then used for refining the original KG. In the following, each of these components is discussed in detail.
Multimodal knowledge graph embeddings
The first step of the LAND framework is to learn the latent representation of the KGs described in Sect. 3 including the representations of the authors. To this end, the Multimodal KGEs component of LAND is designed to learn embeddings of entities and relationships in a KG by combining the structural information and literal associated with the entities such as a string or a date value. LAND adopts LiteralE (Kristiadi et al., 2019) embedding model in this component to learn the KGEs. It incorporates literal information into entity representations by using a learnable mapping function where the literals can either be numeric or text.
More specifically, LiteralE is a multimodal extension of semantic matching models for learning KGEs, such as DistMult (Yang et al., 2015). DistMult scores each triple in the KG with a simple bilinear transformation \(f(h,r,t)={\textbf {h}}^\mathrm{{T}}diag({\textbf {r}}){\textbf {t}}\). The representations are then learned during a training phase which is aimed to maximize the score of existing triples in the KG and by minimizing the score of non-existing triples, i.e. negative examples. Meanwhile, LiteralE aims to modify the scoring function f by using for the head h and tail t entities of each triple hybrid representation which combine the information coming from the entities' relations with the information coming from their literal values. At the core of this method is the mapping function \(g: R^h \times R^d \rightarrow R^h\) which takes as input an entity embedding \({\textbf {e}} \in R^h\) and a literal vector \({\textbf {l}} \in R^d\) and maps them to a new embedding of the same dimension as the entity embedding.
LAND makes use of SPECTER (Cohan et al., 2020), a pre-trained BERT (Devlin et al., 2019) language model for scientific documents in order to encode the textual attributes of the entities (e.g., publication titles) in the vector space \(R^d\) before incorporating them into entity vectors with the g function. Each title in our scholarly KG is mapped to a 768-dimensional sentence embedding by utilizing this model. Meanwhile, the numeric datatypes such as xsd:gYear are converted to a literal vector as described in LiteralE.
In this study, the following two varieties of the DistMultLiteralE model are used and compared against their corresponding base (unimodal) model DistMult.
LAND-\(g_{lin}\) This architecture incorporates textual embeddings extracted from the titles of the entities (scholarly articles) into their representations by means of a linear transformation defined as follows:
$$\begin{aligned} g_{lin}({\textbf {e}},{\textbf {l}}) = {\textbf {W}}[{\textbf {e}},{\textbf {l}}] , \end{aligned}$$
where \({\textbf {e}} \in R^h\) is the vector associated to the ith entity in a KG, \( {\textbf {l}} \in R^d \) is the title embedding, \({\textbf {W}} \in R^{(h,d+h)}\) is a linear transformation matrix and \([{\textbf {e}},{\textbf {l}}] \in R^{(h+d)}\) is the concatenation vector of the entity embedding \({\textbf {e}}\) and the literal embedding \({\textbf {l}}\). With this operation, the entity vector is combined with the textual vector encoded by SPECTER into a multi-modal representation of the same dimensionality of the original entity representation \({\textbf {e}}\).
LAND-\(g_{gru}\). The goal here is to leverage both text (titles) and numeric literals (publication dates). This architecture combines the information coming from numeric and textual literals into the entity representations by means of a Gated Recurrent Unit (GRU), defined as follows:
$$\begin{aligned} g_{gru}({\textbf {e}},{\textbf {l}},{\textbf {n}})=\text {z} \circ \text {h} +(1-\text {z})\circ {\textbf {e}} \\ \text {z}= \sigma ({\textbf {W}}_{ze} {\textbf {e}} + {\textbf {W}}_{zl} {\textbf {l}} + {\textbf {W}}_{zn} {\textbf {n}} +{\textbf { b}}) \\ \text {h}=h({\textbf {W}}_h[{\textbf {e}},{\textbf {l}},{\textbf {n}}]), \end{aligned}$$
where \(\circ \) is the element-wise multiplication, \(\sigma (\cdot )\) is the sigmoid function, \({\textbf {W}}_{ze} \in R^{(h,h)}, {\textbf {W}}_{zl} \in R^{(h,h+d)}, {\textbf {W}}_{zn} \in R^h\) and \({\textbf {W}}_h \in R^{(h,h+d+1)}\) are linear transformation matrices, \({\textbf {b}}\) is a bias vector, \(h(\cdot )\) is a component-wise nonlinearity (e.g. the hyperbolic tangent) and \([{\textbf {e}},{\textbf {l}},{\textbf {n}}]\) is the concatenation of the entity vector, the textual vector and the numeric literal value. With this operation, the entity vector is combined with the textual vector and the numeric literal associated to the publication date into a multi-modal representation of the same dimensionality of the original entity vector \({\textbf {e}}\).
Finally, after having each model trained on a given KG, every author A's embedding E is modified by concatenating it with the embedding D of the document D (i.e. scholarly article) associated to the author A, in order to obtain feature \({\textbf {F}}\) where \({\textbf {F}}={\textbf {E}}+{\textbf {D}}\). This is carried out to reflect both the structural information of the two entities (the author and the document) and the literal information present in the document attributes (i.e. title and publication date) in the embedding of the author.
Blocking and clustering
Blocking is a strategy that is widely used in AND systems. The idea is to split the set of features F related to authors into separate groups, also called blocks denoted by \(F_{b_1}, F_{b_2},\dots ,F_{b_n}\), each one associated with an ambiguous name, so that AND is carried out independently within these blocks. LAND uses the common LN-FI blocking, which divides the set of author features into blocks by looking at the full last name and the first initial of the given name of each author. This blocking technique is chosen since it's computationally less expensive than other blocking approaches which are based on distance measures or string normalization and it's also compliant with the way publishers often mention author names in the metadata of the publications.
The clustering algorithm in LAND helps in grouping together the author features in each block \(F_{b_i}\) into k-clusters \(\{C_1, \ldots ,C_k\}\) where all the features in \(C_j\), where \(j = {1,\ldots ,k}\), ideally belong to the same real-world author. HAC (Ward, 1963) is used which builds clusters of features in a bottom-up manner. The approach conceives each embedding in a block as a singleton cluster and works by iteratively merging the two most similar clusters until all features have been merged in one final cluster.
Even though it has been studied that HAC suffers from scalability issues in On et al. (2012), we decided to adopt this clustering method for two main reasons: (i) for its simplicity; (ii) for the fact that it is the most common clustering method used by graph embedding AND approaches (see Sect. 2.1) and allows us to compare LAND with other methods tested on the AMiner benchmark, such as Zhang and Al Hasan (2017), Zhang et al. (2018), Wang et al. (2020), Pooja et al. (2021) and Chen et al. (2021).
In our implementation of HAC, similarity among clusters is computed with a single linkage strategy which, at each step, merges the clusters whose two closest members have the smallest distance, based on cosine similarity. In order to get the final clusters, a threshold on the maximum distance is defined and clusters above this threshold are considered to be corresponding to different authors. The threshold is defined globally over all the blocks by testing different values over the evaluation blocks, more details are provided in Sect. 5.3.
This section discusses the empirical evaluation of the LAND framework. It first shows how the ground truth is generated for the task of AND, then it presents the achieved experimental results of LAND on the newly generated dataset OC-782K and on a KG extracted from a widely used AND benchmark, i.e. AMiner-534K (refer to Sect. 3 for more details). In addition, an error analysis is carried out for the results on OC-782K.
Generation of the ground truth
In order to obtain the ground truth for testing LAND on OC-782K, a list of (author, ORCIDiD) pairs is extracted. This is performed for the purpose of having an evaluation dataset of scholarly articles labeled with a unique identifier associated with their real-world authors. In order to handle the unbalance in the dataset, only those authors whose last name and first initial are associated with at least two different ORCID iDs are considered. The final evaluation dataset contains 630 bibliographic works organized into 184 blocks and 497 different ORCID iDs. Sizes of the largest blocks in this ground-truth are reported in Table 6.
Table 6 10 Largest blocks on OC-782K evaluation dataset
For measuring the generalizability of the proposed approach, another manually-labeled benchmark dataset is used, i.e., AMiner-534K. This evaluation dataset is larger than the one extracted for OC-782K, with 35,023 scholarly articles and 6,395 unique authors. As for the previous dataset, each ambiguous name is considered a block, and disambiguation is performed within each block. This evaluation dataset contains 35,129 records divided for 100 ambiguous Asian names. In this dataset, disambiguation is carried independently for each ambiguous name. Sizes of the largest blocks in this ground-truth are reported in Table 7.
Table 7 10 Largest blocks on AMiner-534K evaluation dataset
Experimental setup and ablation study
The performance of LAND is evaluated based on three variants of the KGE models: LAND, LAND-\(g_{lin}\), and LAND-\(g_{gru}\). The first variant consists of a unimodal KGE model, i.e. DistMult, which only considers the structural relations between entities in the KG, and no literal (e.g., text or publication dates) is considered. The second variant LAND-\(g_{lin}\) incorporates titles of papers into the embeddings modeled by DistMult by learning the parameters of a linear transformation. The third variant LAND-\(g_{gru}\) uses numeric attributes of the nodes (e.g., publication dates) along with titles and it incorporates them into entity representations by using a GRU function Cho et al. (2014). The implementation of the multimodal KGE models is made compatible with PyKEEN (v.1.4.0)Ali et al. (2021). The source code of different variants as discussed previously is available on GithubFootnote 7. The KGE models are trained and evaluated using Colab Pro notebooks with \(\approx \) 24GB of RAM and Nvidia Tesla T4/K80 GPUs.
Two major tasks are involved in these experiments, i.e., (i) an evaluation of LAND against a candidate set of authors associated with an ORCID iD in OC-782K and ii) a generalizability analysis of LAND on the benchmark dataset provided by AMiner in Zhang et al. (2018), where LAND is compared to the SOTA models surveyed in Sect. 2. Inspired by Zhang et al. (2018), the evaluation metrics pairwise Precision, Recall, and F\(_1\) are used. For studying the generalizability of LAND, these metrics are macro-averaged across all 100 test names.
In addition, we have to state the generalizability study is unfair since the other graph embedding models make use of more features because AMiner-534K does not include abstracts and keywords. The reason why we wanted to test our model on an extracted KG and compare the results with those obtained on a bigger benchmark is to test the hypothesis that our architecture maintains competitive performances even on a more challenging benchmark where the number of authors is relevantly higher.
Model selection and threshold analysis
The models are trained using the Binary Cross Entropy (BCE) loss function, the Adam optimizer, the Stochastic Local Closed World Assumption (SLCWA) training approach, and label smoothing as a regularization technique. Note that for training, each KG is split with a ratio of 64% training, 16% validation, and 20% testing. Random search has been used to perform the hyper-parameter optimizations over the range of values given in Table 8. Each model is trained for a maximum of 1000 epochs and early stopping is applied to speed up the optimization process and avoid overfitting.
Table 8 Hyper-parameter ranges for the HPO studies
Note that due to the limitation of resources, we ran the optimization study only for the unimodal model (i.e., DistMult) on both datasets and chose the set of optimal hyperparameter values which gave the best results, and then decided to apply them also for training the multimodal models. The optimal hyper-parameters are as follows: for OC-782K, embedding dimension: 512, learning rate: 0.0003, number of negatives: 12, batch size: 512, smoothing coefficient: 0.001, epochs: 120; for AMiner-534K, embedding dimension: 128, learning rate: 0.0001, number of negatives: 32, batch size: 512, smoothing coefficient: 0.1, epochs: 300.
For HAC, we define the distance threshold for the final clusters experimentally by trying to find a trade-off between Precision and Recall, by finding the threshold which gives the best F\(_1\) score. However, since high recall systems tend to group different authors together and this negatively affects the performances for AND, we decided to favor high precision over recall. For OC-782K, the resulting best threshold is 0.6, while for AMiner-534K it is 0.26.
Baseline methods
To better assess the performances of the LAND framework, two baseline methods are implemented: (i) a rule-based method originally proposed in Caron and van Eck (2014), which assigns a pairwise score of similarity to two publications based on several rules; and (ii) a simple disambiguation algorithm based on blocking and clustering of sentence embeddings extracted from titles.
We selected the rule-based method inspired by Caron and van Eck (2014), hereby mentioned as Score-Pairs, for three main reasons: (i) for its simplicity and scalability; (ii) for the fact that it was recently used effectively on a SKG in Färber and Ao (2022; iii) for the fact that it has an open implementation on GitHubFootnote 8. Moreover, other methods surveyed in Sect. 2 present issues in reproducibility, since most of them do not have an open-source implementation, some of them are supervised or they rely on many hidden parameters that need to be tuned for OC-782K.
Score-Pairs classify if two publications belong to the same author or not by looking at several features (e.g., shared words in the title, co-authors, citations, etc.) and compute an affinity score for each one of these features based on a list of criteria, i.e., exact string matching or the number of co-occurrences. A list of the features compared, along with the respective comparison criteria and scores are reported in Table 9. Then, a threshold on the sum of the affinity scores is chosen in order to decide whether the publications, given the similarity of their attributes, belong to the same author or not. In our experiments, the value of the threshold is 10.
Table 9 Rules to compute the similarity of two pairs of publications for the baseline Score-Pairs
The second baseline Title-Similarity is chosen to estimate the representativeness of textual embeddings for the task of AND. This baseline performs HAC on the title embeddings encoded by the SPECTER language model presented in Cohan et al. (2020) and can be looked at as a variation of our architecture which uses only title embeddings. The results of this baseline are reported to test the influence of using structural information of a KG over just textual information; which, in the case of titles, might be not so much discriminative.
The clustering algorithm of Title-Similarity is implemented as follows: single linkage as linkage method, cosine similarity as affinity measure, and a threshold of 0.18. As for the architecture using KGEs, the threshold for clustering is selected by maximizing the F\(_1\) score while favoring Precision over Recall.
In the generalizability study, our architecture was compared with several baselines tested on the AMiner benchmark (Zhang et al., 2018), including the graph based model (Fan et al., 2011), the supervised learning model (Louppe et al., 2016), and the graph embedding models (Zhang & Al Hasan, 2017; Zhang et al., 2018; Wang et al., 2020; Pooja et al., 2021; Chen et al., 2021). A description of each of these models is provided in the respective subsections of Sect. 2.
Evaluation on OC-782K
This section compares the results of different LAND variants, i.e., DistMult + HAC, DistMultLiteralE-\(g_{lin}\) + HAC and DistMultLiteralE-\(g_{gru}\) + HAC, on OC-782K with the two previously described baseline models.
Table 10 Results of AND on OC-782K
Table 10 shows the results of the experiments. The embedding-based models outperform the two baseline methods except for the precision of LAND\(g_{gru}\). More precisely, there is an increment in the pairwise F\(_1\) score of the best performing model LAND, i.e., more than 14% and 8% as compared to the baselines Score-Pairs and Title-Similarity respectively. The best precision of 91.71 and the best F\(_1\) score of 77.50 is obtained by LAND. The best recall is 67.59 obtained by LAND-\(g_{gru}\). However, the difference in the recall as compared to LAND is marginal.
The improvements over the baseline Score-Pairs show that KGE embeddings are capable of modeling very discriminative representations if compared with the rule-based method (Caron & vanEck, 2014; Farber & Ao, 2022). Indeed, the cost of using a representation learning architecture that requires training on unlabelled data is repaid by the fact of obtaining more efficient features to cluster for AND, given the same amount of information, i.e. titles, coauthors, citations, venues, and publication dates.
Moreover, the low precision of Title-Similarity if compared to our LAND variants shows how beneficial is considering the whole information from the KG, with an increment in Precision of 28,16% if compared to our most precise model.
For the multimodal models LAND\(g_{gru}\) and LAND\(g_{lin}\), the absence of improvements in the results is surprising and they show that for datasets belonging to one specific domain (i.e. Scientometrics), literal information does not add much information to the embeddings but rather introduces some noise.
Table 11 Confusion matrix of LAND on OC-782K with high precision setup
As it is noticeable in Table 11, the performances of LAND with respect to recall are far from being optimal, since our models ignored a relevant number of matching authors (> 30% avg.) in the evaluation dataset. However, we decided to avoid higher thresholds in order to reduce the number of false positives produced by our clustering algorithm and, as a consequence, to avoid attributing papers written by different persons to the same author. A plot of Precision and Recall curves for OC-782K is available in the Fig. 4.
The plot of the precision and recall curves of our best AND system on different distance thresholds
By applying LAND to the whole set of authors in OC-782K with the high precision setup, we are able to reduce the author entities from 188,565 to 135,325 (a reduction of more than 28%). This shows how relevant KGEs can be for AND on SKGs and how they can be effective in removing duplicates.
Evaluation on AMiner dataset
We tested the generalizability of our approach on a newly collected KG extracted from the AMiner benchmark dataset for AND (Zhang et al., 2018). The results of LAND are compared to the performances of SOTA AND models reported in the benchmark study (Zhang et al., 2018) and with other graph embedding models which were tested on the AMiner benchmark (a description of these models is provided in Table 1). However, we remark that this comparison is not entirely fair, due to the lack of certain information, e.g., abstract and keywords, in AMiner-534K. For more details on why we did not add this information see Sects. 1 and 3. The results of Fan et al. (2011), Louppe et al. (2016), Zhang and Al Hasan (2017), Zhang et al. (2018) are taken from Zhang et al. (2018). The other results are taken from the respective papers (Table 12).
Table 12 Results of author name disambiguation for the AMiner benchmark (Zhang et al., 2018)
LAND-\(g_{lin}\) outperforms other SOTA models which do not use embedding methods, such as Fan et al. (2011) and Louppe et al. (2016). This shows how encoding structural information from linked data with knowledge graph embeddings is more effective than supervised methods or predefined heuristics, even when we do not use abstracts and keywords. Moreover, it's interesting to see how LAND-\(g_{lin}\) also outperformed (Zhang & Al Hasan, 2017), which is a graph embedding method based on coauthorship information, suggesting the advantage of our architecture which models also relations between publications and venues, and relations between authors and affiliations.
However, as a negative result, we see that our method is outperformed by other graph embedding techniques such as Zhang et al. (2018), Chen et al. (2021), Pooja et al. (2021) and Wang et al. (2020) in terms of F\(_1\). Nonetheless, our LAND architecture which does not make use of literal achieves the second best Precision score among the graph embedding models, while the Recall is considerably lower than that of the other models. However, the low recall of the model is explained by the fact that our LAND architectures are trained on a KG which does not contain information from abstracts and keywords.
Moreover, we claim that the relatively lower performances of our model are compared with the architectures of Zhang et al. (2018), Chen et al. (2021), Pooja et al. (2021) and Wang et al. (2020) are balanced by the following advantages of our architecture:
The approach by Zhang et al. (2018) requiring human-annotated data in the training process in order to learn structure-based features from the dataset. Instead, our LAND architecture learns discriminative features for AND from an SKG without needing human supervision.
Zhang et al. (2018), Pooja et al. (2021) and Chen et al. (2021) make use of complex feature engineering methods in order to create scholarly networks from bibliographic metadata. These feature engineering methods include document encoding by means of Doc2Vec, document similarity estimation, and coauthorship similarity estimation in order to create content or coauthorship graphs from bibliographic metadata. This process is not only time-consuming but requires many parameters to be tuned for each dataset, and this hinders the reproducibility of their approach. Instead, our approach does not need feature engineering and learns node embeddings directly from the relations made explicit in the KG.
Wang et al. (2020) is the most similar architecture if compared to ours. The astonishing performances of this model suggest the advantages of using GANs to incorporate content information into relational information from heterogeneous graphs. However, this architecture employs two different modules, a discriminative module, and a generative module, to refine node embeddings in an adversarial fashion. Moreover, the lack of available source code makes this complex architecture difficult to reuse. Instead, our model incorporates content information along with relational information in only 1 model, i.e. DistMultLiteralE (Kristiadi et al., 2019), of which we provide an open-source implementation.
We randomly sampled a subset of 50 wrongly matched pairs (i.e. false positives) from the disambiguated OC-782K in order to analyze the most frequent errors produced by our AND system. We found out that most of the wrong matches are related to Asian authors with common surnames and first initials, like "Chen B", "Kim S", "Li Y", "Wang J", "Li J", "Hu Z" and "Chen J". This is probably due to the fact that LN-FI blocking tends to create huge blocks for very frequent surnames and this causes wrong authors to slide inside the final clusters, especially when they share some features (like references or publishing venue). However, we found out that it is possible to remove all these errors by using a post-blocking strategy which poses the condition that \(full name_i= full name_j\) before merging two authors. Indeed, we found out that all the wrongly matched pairs in our sample which share the same full name are the same person and their entities are wrongly labeled due to the fact that they used multiple ORCIDs across different scholarly works. However, this post-blocking strategy was not included in our framework since was tested only on a small number of false positives and it is limited to the availability of at least a last name and a full name for each author reference, which is not always the case, especially for old publications.
Summary & future perspectives
This article has introduced a framework, named LAND, to perform Author Name Disambiguation (AND) for scholarly data represented as linked data or included in SKGs by developing KGE models based on relationships between entities and the related literal information associated with them. We have demonstrated that these models can be used in the downstream task of clustering for AND effectively. The proposed framework outperforms state-of-the-art methods on a newly created benchmark dataset defining an SKG (named OC-782K) compliant with the OpenCitations Data Model (OCDM) as well as another SKG (named AMiner-534K) created using an existing benchmark dataset, i.e., AMiner. Our method is able to maintain competitive levels of precision, recall and F1 even when dealing with more complex models. Moreover, LAND is designed for dealing with data within knowledge graphs.
In the future, we plan to extend our approach to include also author collaboration network information along with the topic of interest/area of expertise extracted by processing the author's publications via deep learning approaches. Having such additional data will allow us to test if they can improve the results for the task of author name disambiguation.
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Sun, Z., Deng, Z.H., Nie, J. Y., et al. (2019). RotatE: Knowledge graph embedding by relational rotation in complex space. Retrieved from http://arxiv.org/abs/1902.10197
Torvik, V. I., & Smalheiser, N. R. (2009). Author name disambiguation in MEDLINE. ACM Transactions on Knowledge Discovery from Data, 3(3), 11. https://doi.org/10.1145/1552303.1552304
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Wan, H., Zhang, Y., Zhang, J., et al. (2019). AMiner: Search and mining of academic social networks. Data Intelligence, 1(1), 58–76.
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Waqas, H., & Qadir, M. A. (2021). Multilayer heuristics based clustering framework (MHCF) for author name disambiguation. Scientometrics, 126(9), 7637–7678. https://doi.org/10.1007/s11192-021-04087-7
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This study was partially funded by the "Scholarship for research periods abroad aimed at the preparation of the master thesis" by the Department of Classical Philology and Italian Studies, University of Bologna (https://ficlit.unibo.it/it).
Open Access funding enabled and organized by Projekt DEAL.
FIZ Karlsruhe – Leibniz Institute for Information Infrastructure, Karlsruhe, Germany
Cristian Santini, Genet Asefa Gesese, Harald Sack & Mehwish Alam
University of Bologna, Bologna, Italy
Cristian Santini, Silvio Peroni & Aldo Gangemi
Karlsruhe Institute of Technology, Institute AIFB, Karlsruhe, Germany
Cristian Santini
Genet Asefa Gesese
Silvio Peroni
Aldo Gangemi
Harald Sack
Mehwish Alam
Correspondence to Cristian Santini, Genet Asefa Gesese or Mehwish Alam.
Santini, C., Gesese, G.A., Peroni, S. et al. A knowledge graph embeddings based approach for author name disambiguation using literals. Scientometrics 127, 4887–4912 (2022). https://doi.org/10.1007/s11192-022-04426-2
Knowledge graph embeddings
Open citations | CommonCrawl |
\begin{definition}[Definition:Limit Superior of Sequence of Sets/Definition 2]
Let $\sequence {E_n: n \in \N}$ be a sequence of sets.
Then the '''limit superior''' of $\sequence {E_n: n \in \N}$, denoted $\ds \limsup_{n \mathop \to \infty} E_n$, is defined as:
:$\ds \limsup_{n \mathop \to \infty} E_n = \set {x : x \in E_i \text { for infinitely many } i}$
\end{definition} | ProofWiki |
Manipulative (mathematics education)
In mathematics education, a manipulative is an object which is designed so that a learner can perceive some mathematical concept by manipulating it, hence its name. The use of manipulatives provides a way for children to learn concepts through developmentally appropriate hands-on experience.
The use of manipulatives in mathematics classrooms throughout the world grew considerably in popularity throughout the second half of the 20th century. Mathematical manipulatives are frequently used in the first step of teaching mathematical concepts, that of concrete representation. The second and third steps are representational and abstract, respectively.
Mathematical manipulatives can be purchased or constructed by the teacher. Examples of common manipulatives include number lines, Cuisenaire rods; fraction strips,[1] blocks, or stacks; base ten blocks (also known as Dienes or multibase blocks); interlocking linking cubes (such as Unifix); construction sets (such as Polydron and Zometool); colored tiles or tangrams; pattern blocks; colored counting chips;[2] numicon tiles; chainable links; abaci such as "rekenreks", and geoboards. Improvised teacher-made manipulatives used in teaching place value include beans and bean sticks, or single popsicle sticks and bundles of ten popsicle sticks.
Virtual manipulatives for mathematics are computer models of these objects. Notable collections of virtual manipulatives include The National Library of Virtual Manipulatives and the Ubersketch.
Multiple experiences with manipulatives provide children with the conceptual foundation to understand mathematics at a conceptual level and are recommended by the NCTM.
Some of the manipulatives are now used in other subjects in addition to mathematics. For example, Cuisenaire rods are now used in language arts and grammar, and pattern blocks are used in fine arts.
In teaching and learning
Mathematical manipulatives play a key role in young children's mathematics understanding and development. These concrete objects facilitate children's understanding of important math concepts, then later help them link these ideas to representations and abstract ideas. For example, there are manipulatives specifically designed to help students learn fractions, geometry and algebra.[3] Here we will look at pattern blocks, interlocking cubes, and tiles and the various concepts taught through using them. This is by no means an exhaustive list (there are so many possibilities!), rather, these descriptions will provide just a few ideas for how these manipulatives can be used.
Base ten blocks
Main article: Base ten blocks
Base Ten Blocks are a great way for students to learn about place value in a spatial way. The units represent ones, rods represent tens, flats represent hundreds, and the cube represents thousands. Their relationship in size makes them a valuable part of the exploration in number concepts. Students are able to physically represent place value in the operations of addition, subtraction, multiplication, and division.
Pattern blocks
Pattern blocks consist of various wooden shapes (green triangles, red trapezoids, yellow hexagons, orange squares, tan (long) rhombi, and blue (wide) rhombi) that are sized in such a way that students will be able to see relationships among shapes. For example, three green triangles make a red trapezoid; two red trapezoids make up a yellow hexagon; a blue rhombus is made up of two green triangles; three blue rhombi make a yellow hexagon, etc. Playing with the shapes in these ways help children develop a spatial understanding of how shapes are composed and decomposed, an essential understanding in early geometry.
Pattern blocks are also used by teachers as a means for students to identify, extend, and create patterns. A teacher may ask students to identify the following pattern (by either color or shape): hexagon, triangle, triangle, hexagon, triangle, triangle, hexagon. Students can then discuss “what comes next” and continue the pattern by physically moving pattern blocks to extend it. It is important for young children to create patterns using concrete materials like the pattern blocks.
Pattern blocks can also serve to provide students with an understanding of fractions; because pattern blocks are sized to fit to each other (for instance, six triangles make up a hexagon), they provide a concrete experiences with halves, thirds, and sixths.
Adults tend to use pattern blocks to create geometric works of art such as mosaics. There are over 100 different pictures that can be made from pattern blocks. These include cars, trains, boats, rockets, flowers, animals, insects, birds, people, household objects, etc. The advantage of pattern block art is that it can be changed around, added, or turned into something else. All six of the shapes (green triangles, blue (thick) rhombi, red trapezoids, yellow hexagons, orange squares, and tan (thin) rhombi) are applied to make mosaics.
Linking cubes
Like pattern blocks, interlocking cubes can also be used for teaching patterns. Students may use the cubes to make long trains of patterns. Like the pattern blocks, the interlocking cubes provide a concrete experience for students to identify, extend, and create patterns. The difference is that a student can also physically decompose a pattern by the unit. For example, if a student made a pattern train that followed this sequence:
Red, blue, blue, blue, red, blue, blue, blue, red, blue, blue, blue, red, blue, blue, ...
the child could then be asked to identify the unit that is repeating (red, blue, blue, blue) and take apart the pattern by each unit.
Also, one can learn addition, subtraction, multiplication and division, guesstimation, measuring, and graphing, perimeter, area, and volume.[4]
Tiles
Tiles are one inch-by-one inch colored squares (red, green, yellow, blue).
Tiles can be used much the same way as interlocking cubes. The difference is that tiles cannot be locked together. They remain as separate pieces, which in many teaching scenarios, may be more ideal.
These three types of mathematical manipulatives can be used to teach the same concepts. It is critical that students learn math concepts using a variety of tools. For example, as students learn to make patterns, they should be able to create patterns using all three of these tools. Seeing the same concept represented in multiple ways as well as using a variety of concrete models will expand students’ understandings.
Number lines
To teach integer addition and subtraction, a number line is often used. A typical positive/negative number line spans from −20 to 20. For a problem such as “−15 + 17”, students are told to “find −15 and count 17 spaces to the right”.
References
1. Archived 26 February 2014 at the Wayback Machine
2. "Number and Operations Session 4, Part C: Colored-Chip Models". www.learner.org. Archived from the original on 2009-07-18.
3. "Best Math Manipulatives for Middle Schoolers".
4. Archived 28 July 2008 at the Wayback Machine
Sources
• Allsopp, D.H. (2006). "Concrete – Representational – Abstract". Retrieved 1 September 2006.
• Krech, B. (2000). "Model with manipulatives". Instructor. 109 (7): 6–7.
• Van de Walle, J.; Lovin, L.H. (2005). Teaching Student-Centered Mathematics: Grades K-3. Allyn & Bacon.
External links
• NCTM's Official Website
• NLVM's Official Website (National Library of Virtual Manipulatives)
• Didax Virtual Manipulatives
• Manipulatives for negative numbers at Wikiversity
Authority control: National
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| Wikipedia |
Compute $\dbinom{5}{3}$.
$$\dbinom{5}{3} = \dfrac{5!}{3!2!}=\dfrac{(5\times 4)(3\times 2\times 1)}{(3\times 2\times 1)(2\times 1)}=\dfrac{5\times 4}{2\times 1}=\boxed{10}.$$ | Math Dataset |
\begin{document}
\title{Meta-analysis of Bayesian analyses}
\begin{abstract} Meta-analysis aims to combine results from multiple related statistical analyses. While the natural outcome of a Bayesian analysis is a posterior distribution, Bayesian meta-analyses traditionally combine analyses summarized as point estimates, often limiting distributional assumptions. In this paper, we develop a framework for combining posterior distributions, which builds on standard Bayesian inference, but using distributions instead of data points as observations. We show that the resulting framework preserves basic theoretical properties, such as order-invariance in successive updates and posterior concentration. In addition to providing a consensus analysis for multiple Bayesian analyses, we highlight the benefit of being able to reuse posteriors from computationally costly analyses and update them post-hoc without having to rerun the analyses themselves. The wide applicability of the framework is illustrated with examples of combining results from likelihood-free Bayesian analyses, which would be difficult to carry out using standard methodology. \end{abstract}
\section{Introduction}\label{sec:intro}
Meta-analysis is a term used broadly for a collection of approaches which aim to combine results from multiple related statistical analyses. In the standard formulation, these results are summary statistics computed from data, a typical example being point estimates for the effect size of some treatment. For the combination of point estimates, there exists well-established Bayesian methodology and a large body of literature \citep[see e.g.][and references therein]{Higgins+others:2009}. However, while the natural outcome of a Bayesian analysis is a posterior distribution, the analogous task of combining posterior distributions has received little attention. In the frequentist paradigm, \citet{Xie+others:2011} have previously introduced a framework for the combination of confidence distributions, a concept loosely related to Bayesian posteriors. In this paper, we develop a Bayesian framework for combining posterior distributions.
In the standard setting of Bayesian \emph{random effects meta-analysis}, a summary statistic (or data set) $D_j$, $j=1,\ldots,J$, has been observed for each of $J$ studies. The summary statistics aim to provide information about some quantity or `effect' of interest. To reflect the general idea of the studies being non-identical but related, it is common to regard them as exchangeable \citep{Gelman+others:2013}, with the study-specific effect represented by some parameter, say $\theta_j$, and the overall effect by another parameter, say $\varphi$. This leads to the following hierarchical model: \begin{alignat*}{2}\label{eq:rema_hm}
\varphi &\sim Q \nonumber \\
\theta_j &\sim P_{\varphi} \\
D_j &\sim F_{\theta_j}, \end{alignat*} where $F_{\theta_j}$ is typically modeled as $D_j\sim\mathcal{N}(\theta_j,\hat{\sigma}_j^2),$ with $\hat{\sigma}_j^2$ estimated from data. One of the primary goals of the above model is to estimate the overall effect $\varphi$, for which the marginal posterior density is given by \begin{equation}\label{eq:overall_effect}
q(\varphi|D_1,\ldots,D_J) \propto \prod_{j=1}^J \left[\int f(D_j|\theta_j)p(\theta_j|\varphi) d\theta_j\right] q(\varphi). \end{equation}
There are many compelling reasons for reporting analysis results as posterior distributions instead of data summaries, and subsequently combining them in a meta-analysis. First, a distribution describes the analyst's uncertainty in the obtained results.
Second, a posterior distribution can be directly specified on a quantity of interest, whereas a summary statistic often only provides indirect information about the quantity.
Furthermore, the posterior distribution may also include prior knowledge not present in the data, but possibly obtained by expert elicitation \citep[e.g.][]{Albert+others:2012}. This is particularly important in problems where not enough data is available to inform a model about some of its parameters. For example, models describing complex biological phenomena may have so many parameters that they cannot be estimated without the use of informative priors \citep[e.g.][]{Kuikka+others:2014}. Thus, the posterior for a given quantity may be primarily informed by its prior and only indirectly by data, which could make the extraction of a meaningful summary statistic challenging. Finally, if we wish to combine the results of such studies in a meta-analysis, it is desirable to preserve the study-specific prior knowledge in the combined model. Unlike data summaries, this knowledge is naturally encapsulated in posterior distributions.
Consider now a setting, where instead of summary statistics $D_j$, we have posterior distributions with densities $\pi_j(\theta_j)$ available from each of $J$ studies, based on which we wish to update our prior knowledge about the global effect $\varphi$, in analogy with Equation~(\ref{eq:overall_effect}). We build on the interpretation that each $\pi_j(\theta_j)$ is a probabilistic representation of belief \citep{Bernardo+Smith:1994}, which reflects our uncertainty about the value of the corresponding local effect $\theta_j$. Updating prior knowledge (in our case regarding $\varphi$) subject to uncertain or `soft' evidence, represented as probability distributions instead of observed values, has been extensively studied as a philosophical topic in both statistics and artificial intelligence. The most well-known of such update rules, Jeffrey's rule of conditioning \citep{Diaconis+Zabell:1982,Jeffrey:2004,Smets:1993,Zhao+Osherson:2010}, computes the updated probability for an event as a weighted average of the posterior probabilities under all possible values of the evidence. Due to its construction, Jeffrey's rule is applicable in simple discrete cases but becomes computationally infeasible for more complex models with continuous variables. Instead, we directly marginalize away the uncertainty in the observations as they appear in the likelihood, which leads us to update $q(\varphi)$ as \begin{equation}\label{eq:modified_overall_effect}
q^*(\varphi) \propto \prod_{j=1}^J \left[\int p(\theta_j|\varphi) \pi_j(\theta_j) d\theta_j\right] q(\varphi). \end{equation}
In addition to achieving computational tractability, the above formulation retains some basic properties of standard Bayesian inference, such as order-invariance in successive updates (in contrast with Jeffrey's rule) and posterior concentration as $J\rightarrow \infty$.
From a practical point of view, it is interesting to note that Equations (\ref{eq:overall_effect}) and (\ref{eq:modified_overall_effect}) differ only in the local likelihood $f(D_j|\theta_j)$ being replaced with the density $\pi_j(\theta_j)$. Intuitively, the former carries information provided by the data only, while the latter carries information provided by both the data and additional prior knowledge. A more subtle difference is that in Equation~(\ref{eq:modified_overall_effect}), the integration is thought of as being performed with respect to the measure defined by $\pi_j$.
Our formulation of the meta-analysis problem also lends itself to an intuitive interpretation as \emph{message passing} in probabilistic graphical models \citep{Jordan:2004,Koller+Friedman:2009}. Using the language of undirected models, updating $q(\varphi)$ into $q^*(\varphi)$ in Equation~(\ref{eq:modified_overall_effect}) is equivalent to
propagating beliefs from leaf nodes to the root node in a tree-structured graphical model, with node potentials $q(\varphi)$, $\pi_j(\theta_j)$ and edge potentials $\psi_j(\theta_j,\varphi) := p(\theta_j|\varphi)$, $j=1,\ldots,J$, see Figure~\ref{fig:tree}. Further utilizing the induced graphical model structure, we may similarly update any $\pi_j(\theta_j)$ into $\pi_j^*(\theta_j)$ by propagating beliefs to the $j$th leaf node from the remaining nodes in the model. This yields a way of updating study-specific posteriors \emph{post-hoc}, borrowing strength from posteriors obtained in other studies. Besides its intuitive appeal, adopting a graphical model view enables a straightforward extension of the framework to more complex model structures, and it may be utilized in devising efficient computational strategies. \begin{figure}
\caption{Tree-structured graphical model with study-specific posteriors providing the initial beliefs for $\theta_1,\ldots,\theta_J$.}
\label{fig:tree}
\end{figure}
A major advantage of our meta-analysis framework is its flexibility. In particular, the study-specific inferences resulting in $\pi_j(\theta_j)$ are independent of the combination model, $q(\varphi)\prod_{j=1}^J p(\theta_j|\varphi)$, which is imposed by the meta-analyst. This means that, unlike in hierarchical models, all study-level complexities are hidden `under the hood' and need not explicitly be included in the meta-analysis. For instance, in likelihood-free models \citep[e.g.][]{Lintusaari+others:2017,Marin+others:2012}, the data can typically be summarized by a number of different statistics but there is no closed-form likelihood to relate these to the parameter of interest. In our framework, likelihood-free inferences can be conducted separately for each study using \emph{approximate Bayesian computation}, after which the resulting posteriors are directly combined in a meta-analysis. We highlight the benefit of being able to reuse results from computationally costly analyses, such as likelihood-free inferences, and update them without having to rerun the analyses themselves. In the current paper, we propose a straightforward computational strategy for our framework, where we first impose parametric approximations on the observed posteriors. Sampling from the joint distribution of all variables can then be done using general-purpose software such as Stan \citep{Carpenter+others:2017}. Finally, the obtained joint distribution can be refined using importance sampling, from which any desired marginals can be extracted.
The paper is structured as follows. In Section~\ref{sec:theory}, we develop a framework for conducting Bayesian inference with observed beliefs, which underlies our meta-analysis approach. In Section~\ref{sec:mba}, we summarize the main equations for practical use, and provide further insight into the framework by highlighting connections to message passing in probabilistic graphical models, and standard forms of Bayesian meta-analysis. Section~\ref{sec:computation} introduces a straightforward computational strategy, which can be implemented using general-purpose software. In Section~\ref{sec:illustrations}, we illustrate our method with both synthetic and real data. The paper ends with a brief discussion of related work in Section~\ref{sec:related_work} and some concluding remarks in Section~\ref{sec:discussion}.
\section{Bayesian inference with observed beliefs}\label{sec:theory}
In this section, we first develop a posterior update rule given observed beliefs, which is motivated by the problem of conducting meta-analysis for a set of related posterior distributions. The notion of relatedness is here characterized as the exchangeability of the quantities targeted by the posteriors. The proposed update rule is given in Equation~(\ref{eq:posterior_uncertain_obs}). We then show in Section~\ref{sec:theoretical_properties} that this rule retains some basic theoretical properties of standard Bayesian inference.
Let us first assume that $\theta_1,\ldots,\theta_J$ is a collection of observable and exchangeable random quantities. Following standard theory, de~Finetti's representation theorem \citep[e.g.][]{Schervish:1995} states that if $\theta_1,\theta_2,\ldots$ is an infinitely exchangeable sequence of random quantities taking values in a Borel space $(\Theta$, $\mathcal{A})$, then there exists a probability measure $Q$ such that the joint distribution $\mathds{P}$ of the subsequence $\theta_1, \ldots,\theta_J$, i.e. the \emph{predictive distribution}, has the form \begin{equation}\label{eq:parametric_deFinetti}
\mathds{P}(\theta_1 \in A_1, \ldots,\theta_J \in A_J) = \int_{\Phi} \prod_{j=1}^J \left[\int_{A_j}p(\theta_j|\varphi)\lambda(d \theta_j) \right] Q(d\varphi), \end{equation} where $A_1,\ldots,A_J \in\mathcal{A}$.
Here, the set of probability measures on $\Theta$ is taken to be a family $\{P_{\varphi}|\varphi\in\Phi\}$, indexed by a parameter $\varphi$, such that $Q$ is a probability measure on $\Phi$. Furthermore, we define the density function $p(\cdot|\varphi):= dP_{\varphi}/d\lambda$ with respect to a reference measure $\lambda$ (Lebesgue or counting measure).
In the above standard setting, the Bayesian learning process works through updating $Q$ conditional on observed data. Following Equation~(\ref{eq:parametric_deFinetti}), the posterior distribution of $\varphi$, given observed values $\theta_{1}=t_1, \ldots,\theta_J=t_J$, has the form \begin{equation}\label{eq:posterior}
Q(B|t_1,\ldots,t_J) = \frac{\int_{B} \prod_{j=1}^J p(t_j|\varphi) Q(d\varphi)}{\int_{\Phi} \prod_{j=1}^J p(t_j|\varphi) Q(d\varphi)},
\end{equation} with $B\in\mathcal{B}$, the Borel $\sigma$-algebra on $\Phi$. We will now build further on this setting, assuming that instead of directly observing the value of each $\theta_j$, we have a set of distributions $\Pi_1,\ldots,\Pi_J$, expressing our currently available beliefs about the values of $\theta_j$. Note that, while in our current context, we assume that each of the beliefs is obtained as the posterior distribution from a previously conducted analysis, this assumption is not essential to our developments. Importantly, the observed distributions are assumed independent of the distribution we seek to update.
In the absence of fixed likelihood contributions $p(t_j|\varphi)$ for each observation, we propose to compute the \emph{expected likelihood contributions} $\int_{\Theta}p(\theta_j|\varphi)\Pi_j(d\theta_j)$ with respect to the available beliefs.
The proposed modification of the likelihood now leads to an update of the form \begin{equation}\label{eq:posterior_uncertain_obs}
Q^*(B|\Pi_1,\ldots,\Pi_J) = \frac{\int_{B} \prod_{j=1}^J \left[\int_{\Theta}p(\theta_j|\varphi)\Pi_j(d\theta_j)\right] Q(d\varphi)}
{\int_{\Phi} \prod_{j=1}^J \left[ \int_{\Theta}p(\theta_j|\varphi)\Pi_j(d\theta_j)\right] Q(d\varphi)}, \end{equation}
where, with slight abuse of notation, we write $Q^*(\cdot|\Pi_1,\ldots,\Pi_J)$ to denote conditioning on beliefs in analogy with conditioning on fully observed values; we give more context for this choice of notation below in Section~\ref{sec:posterior_concentration}. Equation~(\ref{eq:posterior_uncertain_obs}) further induces a joint distribution on $\Phi\times\Theta^J$, which can be marginalized with respect to $Q$, resulting in a predictive distribution as follows: \begin{equation}\label{eq:joint_belief}
\mathds{P}^*(\theta_1 \in A_1, \ldots,\theta_J \in A_J) = \frac{\int_{\Phi} \prod_{j=1}^J \left[\int_{A_j}p(\theta_j|\varphi)\Pi_j(d\theta_j)\right] Q(d\varphi)}
{\int_{\Phi} \prod_{j=1}^J\left[ \int_{\Theta}p(\theta_j|\varphi)\Pi_j(d\theta_j)\right] Q(d\varphi)}. \end{equation}
It easy to see that standard Bayesian inference, Equation~(\ref{eq:posterior}), emerges as a special case of Equation~(\ref{eq:posterior_uncertain_obs}) by setting $\Pi_j$ to be $\delta_{t_j}$, the Dirac measure centered at $t_j$. This yields \[
\int_{\Theta} p(\theta_j|\varphi)\delta_{t_j}(d\theta_j) = p(t_j|\varphi), \]
such that $Q^*(\cdot|\delta_{t_1},\ldots,\delta_{t_J}) = Q(\cdot|t_1,\ldots,t_J)$.
Throughout this work, we assume that $\Pi_j$ is a probability measure. However, it is interesting to note that if we make an exception and allow $\Pi_j$ to be the Lebesgue (or counting) measure $\lambda$ for all $j$, which corresponds to having a uniformly distributed---possibly improper---belief about the value of $\theta_j$, then the updated measure $Q^*$ in Equation~(\ref{eq:posterior_uncertain_obs}) equals the prior probability measure $Q$. Moreover, with this choice of $\Pi_j$, Equation~(\ref{eq:joint_belief}) reduces to the standard predictive distribution in Equation~(\ref{eq:parametric_deFinetti}).
\begin{examp}\label{ex:bernoulli_example} To gain an intuitive understanding of the update rule proposed in Equation~(\ref{eq:posterior_uncertain_obs}), we consider inference in the following simple model: \begin{align*} \varphi &\sim \mathrm{Beta}(\alpha,\beta)\\ \theta_j &\sim \mathrm{Bernoulli}(\varphi),\quad j=1\ldots,J. \end{align*} In the standard case, we apply Bayes' rule (\ref{eq:posterior}) to update the prior distribution $\mathrm{Beta}(\alpha,\beta)$, conditional on observed values of $\theta_j$, into a posterior distribution, which by conjugacy is $\mathrm{Beta}(\alpha+r,\beta+J-r)$, with $r=\sum_{j=1}^J \theta_j$. Next, let us assume that the values of $\theta_j$ cannot be directly observed, but instead, we are able to express beliefs about these values through binary distributions $\Pi_j(\theta_j=k)$, $k\in \{0,1\}$, $\sum_{k=0}^1 \Pi_j(\theta_j=k)=1$. Note that these distributions are independent of the Bernoulli-distribution posited by the modeller. We now wish to update the prior $\mathrm{Beta}(\alpha,\beta)$ into a distribution $Q^*$ using the provided information. Following Equation~(\ref{eq:posterior_uncertain_obs}), the standard Bernoulli likelihood $\prod_{j=1}^J\varphi^{\theta_j} (1-\varphi)^{1-\theta_j}$ takes a modified form \[ \prod_{j=1}^J\sum_{k=0}^1 \varphi^k (1-\varphi)^{(1-k)} \Pi_j(\theta_j=k). \] While the modified likelihood in general no longer permits analytical calculations through conjugacy, two analytically tractable special cases can be identified. Specifically, if $\Pi_j(\theta_j=k)=0.5$, for all $j$, then the observed distributions are uninformative about the value of $\theta_j$, and $Q^*$ coincides with the prior $\mathrm{Beta}(\alpha,\beta)$. Moreover, if $\Pi_j(\theta_j=k)=1$, for all $j$, then the $\theta_j$'s are in effect fully observed, and $Q^*$ equals the standard posterior $\mathrm{Beta}(\alpha+r,\beta+J-r)$. A numerical example is provided in Figure~ 2. \end{examp}
\begin{figure}
\caption{ Updated density functions for $\varphi$, each computed under the model $\varphi \sim \mathrm{Beta}(2,2)$, $\theta_j \sim \mathrm{Bernoulli}(\varphi)$, $j=1,\ldots,10$, with soft observations $\Pi_j(\theta_j=k)$, $k\in \{0,1\}$, assumed to be identical for all $j$. The different densities are obtained by varying $\Pi_j(\theta_j=1)$ from 0 to 1 by increments of 0.1. The solid curves correspond to $\Pi_j(\theta_j=1)=0$ and $\Pi_j(\theta_j=1)=1$, equivalent to posteriors computed conditional on all $\theta_j$ fully observed with values 0 and 1, respectively. The dashed curve corresponds to $\Pi_j(\theta_j=1)=0.5$, and equals the prior $\mathrm{Beta}(2,2)$. For further details, see Example~1 .}
\label{fig:bernoulli_example}
\end{figure}
\subsection{Theoretical properties}\label{sec:theoretical_properties}
Two well-known properties of standard Bayesian inference, which are of practical relevance in our meta-analysis setting, are (i) order-invariance in successive posterior updates of exchangeable models and (ii) posterior concentration. The former ensures that inferences conditional on exchangeable data are coherent. The latter tells us that the posterior distribution becomes increasingly informative about the quantity of interest, as we accumulate more data. We will now briefly discuss these properties in the context of the framework introduced above.
\subsubsection{Order-invariance in successive posterior updates}
Under the assumption of exchangeability, standard Bayesian inference can be constructed as a sequence of successive updates, invariant to the order in which the data are processed. The following proposition establishes that the update rule defined in Equation~(\ref{eq:posterior_uncertain_obs}) retains the same property. \begin{proposition} The update rule defined in Equation~(\ref{eq:posterior_uncertain_obs}) is invariant to permutations of the indices $1,\ldots,J$. \end{proposition}
\begin{proof}
It suffices for us to verify the claim for $J=2$. Beginning with $J=1$, we update the probability $Q(B)$ into $Q^*(B|\Pi_1)$ using Equation~(\ref{eq:posterior_uncertain_obs}): \[
Q^*(B|\Pi_1) = \frac{\int_{B} \int_{\Theta}p(\theta_1|\varphi)\Pi_1(d\theta_1) Q(d\varphi)}
{\int_{\Phi} \int_{\Theta}p(\theta_1|\varphi)\Pi_1(d\theta_1) Q(d\varphi)}. \]
Then, we reapply Equation~(\ref{eq:posterior_uncertain_obs}) to update $Q^*(B|\Pi_1)$ into $Q^*(B|\Pi_1,\Pi_2)$: \begin{align*}
Q^*(B|\Pi_1,\Pi_2)
&= \frac{\int_{B} \int_{\Theta}p(\theta_2|\varphi)\Pi_2(d\theta_2 ) Q^*(d\varphi|\Pi_1) }
{\int_{\Phi} \int_{\Theta}p(\theta_2|\varphi)\Pi_2(d\theta_2) Q^*(d\varphi|\Pi_1)}\\
&= \frac{\int_{B} \int_{\Theta}p(\theta_2|\varphi)\Pi_2(d\theta_2 )}
{\int_{\Phi} \int_{\Theta}p(\theta_2|\varphi)\Pi_2(d\theta_2)}
\frac{\frac{\int_{\Theta}p(\theta_1|\varphi)\Pi_1(d\theta_1) Q(d\varphi)}
{\cancel{\int_{\Phi} \int_{\Theta}p(\theta_1|\varphi)\Pi_1(d\theta_1) Q(d\varphi)}}}
{\frac{\int_{\Theta}p(\theta_1|\varphi)\Pi_1(d\theta_1) Q(d\varphi)}
{\cancel{\int_{\Phi} \int_{\Theta}p(\theta_1|\varphi)\Pi_1(d\theta_1) Q(d\varphi)}}}\\
&= \frac{\int_{B} \prod_{j=1}^2 \left[\int_{\Theta}p(\theta_j|\varphi)\Pi_j(d\theta_j)\right] Q(d\varphi)}
{\int_{\Phi} \prod_{j=1}^2 \left[ \int_{\Theta}p(\theta_j|\varphi)\Pi_j(d\theta_j)\right] Q(d\varphi)}, \end{align*} which is equivalent to a direct application of Equation~(\ref{eq:posterior_uncertain_obs}) for $J=2$, and independent of the order in which $\Pi_1$ and $\Pi_2$ are processed. \end{proof}
As an alternative strategy to Equation~(\ref{eq:posterior_uncertain_obs}), we could first attempt to formulate a posterior distribution $Q(\cdot|\theta_1,\ldots,\theta_J)$ according to Equation~(\ref{eq:posterior}) and then, as a final step, integrate out the uncertainty in the conditioning set with respect to the observed beliefs. This is in essence the strategy of Jeffrey's rule of conditioning. It is, however, well known that Jeffrey's update rule is in general not order-invariant \citep{Diaconis+Zabell:1982}.
\subsubsection{Posterior concentration}\label{sec:posterior_concentration}
Asymptotic theory states that if a consistent estimator of the true value (or an optimal one in terms KL-divergence) of the parameter $\varphi$ exists, then the posterior distribution (\ref{eq:posterior}) concentrates in a neighborhood of this value, as $J\rightarrow\infty$ \citep[e.g.][]{Schervish:1995}.
Here we discuss conditions under which the same property holds for the measure $Q^*(\cdot|\Pi_1,\ldots,\Pi_J)$, defined in Equation~(\ref{eq:posterior_uncertain_obs}). Considerations of asymptotic normality will not be discussed here.
Our strategy is to first formulate a generative hierarchical model for the observed distributions $\Pi_1,\ldots,\Pi_J$. Then, we show that $Q^*(\cdot|\Pi_1,\ldots,\Pi_J)$ can be expressed as the marginal posterior distribution of $\varphi$ in this model. Finally, we show that under some further technical conditions, standard asymptotic theory can be applied to this distribution. To this end, consider the following hierarchical model: \begin{subequations} \begin{align} \varphi &\sim Q\label{eq:hierarchical_uncertain_obs_Q}\\ \theta_j &\sim P_{\varphi}\\ \Pi_j &\sim G^{(j)}_{\theta_j}\label{eq:hierarchical_uncertain_obs_G}, \end{align} \end{subequations} where $\Pi_j$ is treated as a soft observation of the unobserved value of $\theta_j$, and $G^{(j)}_{\theta_j}$ is the inference mechanism that produces $\Pi_j$. Note that in the particular case of $\Pi_j$ being the Dirac measure, $G^{(j)}_{\theta_j}$ simply generates a point mass at the true value of $\theta_j$, such that the hierarchical model reduces to an ordinary, non-hierarchical Bayesian model. Since $\Pi_j$ is an inference over the values of $\theta_j$, produced by $G^{(j)}_{\theta_j}$, it is also a direct representation of the likelihood of $\theta_j$ under the model $G^{(j)}_{\theta_j}$, given the observation $\Pi_j$ itself. We finally note, that the generating mechanism $G^{(j)}_{\theta_j}$ may in general be different for each $j$, which is highlighted in the notation by the superscript.
Assume now that the Radon-Nikodym derivative $g_j(\cdot|\theta_j):= dG^{(j)}_{\theta_j}/d\kappa$ with respect to a dominating measure $\kappa \gg G^{(j)}_{\theta_j}$ can be defined for all $j$.
The marginal posterior distribution of $\varphi$ is then \begin{equation*}\label{eq:posterior_uncertain_obs_v2}
Q'(B|\Pi_1,\ldots,\Pi_J) = \frac{\int_{B} \prod_{j=1}^J \left[\int_{\Theta}g_j(\Pi_j|\theta_j)P_{\varphi}(d\theta_j)\right] Q(d\varphi)}
{\int_{\Phi} \prod_{j=1}^J \left[ \int_{\Theta}g_j(\Pi_j|\theta_j)P_{\varphi}(d\theta_j)\right] Q(d\varphi)}, \end{equation*}
where $g_j(\Pi_j|\theta_j)$, taken as a function of $\theta_j$, is the likelihood of $\theta_j$ given $\Pi_j$.
On the other hand, according to our previous assumption, the likelihood is directly encapsulated in $\Pi_j$ itself. We will therefore assume, by construction, that $g_j(\Pi_j|\theta_j)=\pi_j(\theta_j)$, where $\pi_j := d\Pi_j/d\lambda$ is the density function corresponding to $\Pi_j$. Using this equivalence, we state the following lemma: \begin{lemma}\label{lem:posterior_uncertain_obs_hierarchical_represetation}
Let $g_j(\Pi_j|\theta_j)=\pi_j(\theta_j)$. Then the measures $Q'(\cdot|\Pi_1,\ldots,\Pi_J)$ and $Q^*(\cdot|\Pi_1,\ldots,\Pi_J)$ are equivalent. \end{lemma} \begin{proof}
To prove the claim, we only need to verify that the integrals $\int_{\Theta}g_j(\Pi_j|\theta_j)P_{\varphi}(d\theta_j)$ and $\int_{\Theta}p_j(\theta_j|\varphi)\Pi_j(d\theta_j)$ are equivalent. Since $\Pi_j$ and $P_{\varphi}$ are both probability distributions on $(\Theta,\mathcal{A})$, and their densities are defined with respect to the same reference measure $\lambda$, we may swap the roles of the integrand and the measure that we integrate against: \begin{align*}
\int_{\Theta}g_j(\Pi_j|\theta_j)P_{\varphi}(d\theta_j) &= \int_{\Theta}\pi_j(\theta_j)P_{\varphi}(d\theta_j) \\&= \int_{\Theta}\frac{d}{d\lambda}\Pi_j(\theta_j)P_{\varphi}(d\theta_j)\\ &= \int_{\Theta}\frac{d}{d\lambda}P_{\varphi}(\theta_j)\Pi_j(d\theta_j)
\\&= \int_{\Theta}p_j(\theta_j|\varphi)\Pi_j(d\theta_j). \end{align*} \end{proof} \begin{corollary}\label{cor:posterior_uncertain_obs_v3}
In the hierarchical model (\ref{eq:hierarchical_uncertain_obs_Q})--(\ref{eq:hierarchical_uncertain_obs_G}), let $H^{(j)}_{\varphi}$ be the marginal conditional distribution of $\Pi_j$, given $\varphi$, and let $h_j(\Pi_j|\varphi) := \int_{\Theta}g_j(\Pi_j|\theta_j)P_{\varphi}(d\theta_j)$ be the corresponding marginal likelihood. Following Lemma~\ref{lem:posterior_uncertain_obs_hierarchical_represetation}, the probability $Q^*(B|\Pi_1,\ldots,\Pi_J)$ can be written as \begin{equation*}\label{eq:posterior_uncertain_obs_v3}
Q^*(B|\Pi_1,\ldots,\Pi_J) = \frac{\int_{B} \prod_{j=1}^J h_j(\Pi_j|\varphi) Q(d\varphi)}{\int_{\Phi} \prod_{j=1}^J h_j(\Pi_j|\varphi) Q(d\varphi)}. \end{equation*} \end{corollary} The essential difference between the standard posterior distribution defined in Equation (\ref{eq:posterior}) and that of Corollary~\ref{cor:posterior_uncertain_obs_v3} above, is that in the former, the observations are conditionally i.i.d., while in the latter they are conditionally independent but \emph{non-identically} distributed.
Assuming that there exists, for all $j$, a unique minimizer $\varphi_0$ of the KL-divergence from the true distribution of $\Pi_j$ to the parametrized representation $H_\varphi^{(j)}$, a key step in proving posterior concentration is to establish a limit for the log-likelihood ratio $\sum_{j=1}^J\log\frac{h_j(\Pi_j|\varphi)}{h_j(\Pi_j|\varphi_0)}$. Since the summands are now non-identically distributed, standard forms of the strong law of large numbers cannot be applied to obtain this limit. However, with further conditions imposed on the second moment of each term in the sum, an alternative form can be used, which relaxes the requirement of the terms being identically distributed \citep[][Theorem 2.3.10]{Sen+Singer:1993}. The conditions are stated in the following theorem. \begin{theorem}\label{thm:sen_singer}
Assume that the log-likelihood ratio terms $\xi_j:=\log\frac{h_j(\Pi_j|\varphi)}{h_j(\Pi_j|\varphi_0)}$ are independent, and that $\mathds{E}(\xi_j)=\mu_j$ and $\mathrm{Var}(\xi_j)=\sigma_j^2$ exist for all $j\geq 1$. Let $\overline{\mu}_J= J^{-1}\sum_{j=1}^J \mu_j$, for $J\geq 1$. Then \[ \sum_{j\geq 1} j^{-2} \sigma_j^2 < \infty \Rightarrow J^{-1}\sum_{j=1}^J \xi_j- \overline{\mu}_J \xrightarrow{\text{a.s.}} 0. \] \end{theorem}
In conclusion, if the measure $Q^*(\cdot|\Pi_1,\ldots,\Pi_J)$ can be written in the form of Equation~(\ref{eq:posterior_uncertain_obs_v3}) and the conditions of Theorem~\ref{thm:sen_singer} hold, then posterior concentration falls back to the standard case, for which elementary proofs can be found in many sources \citep[e.g.][]{Bernardo+Smith:1994,Gelman+others:2013}. A rigorous treatment is given in \citet{Schervish:1995}.
As an example, we provide a proof of posterior concentration of $Q^*(\cdot|\Pi_1,\ldots,\Pi_J)$ for discrete parameter spaces in Appendix~\ref{sec:proof}.
\section{Meta-analysis of Bayesian analyses}\label{sec:mba}
We now turn to a practical view of the framework developed in the previous section. To this end, it is convenient to work with densities instead of measures. We are motivated by the problem of conducting meta-analysis for Bayesian analyses summarized as posterior distributions, and refer to our framework as \emph{meta-analysis of Bayesian analyses} (MBA). The central belief updates of the framework are given in Equations (\ref{eq:posterior_uncertain_obs_density}) and (\ref{eq:update_pi_j}), which update beliefs regarding global and local effects, respectively. Figures \ref{fig:bp_varphi} and \ref{fig:bp_theta} visualize the updates by interpreting them as message passing in probabilistic graphical models.
To reiterate the setting laid out in Section~\ref{sec:intro}, we assume that a set of posterior density functions $\{\pi_1,\ldots,\pi_J\}$ is available, each expressing a belief about the value of a corresponding quantity of interest in a set $\{\theta_1,\ldots,\theta_J\}$. While the density functions can be thought of as resulting from previously conducted Bayesian analyses, it is worth pointing out that from a methodological point of view, we are agnostic to \emph{how} they have been formed; instead of posteriors, some (or all) of the $\pi_j$'s could be purely subjective prior beliefs, or as discussed in Section~\ref{sec:theory}, even directly observed values.
Judging the quantities $\theta_j$ to be exchangeable, the \emph{meta-analyst} now formulates a model \begin{equation}\label{eq:meta-analyst}
\prod_{j=1}^J p(\theta_j|\varphi)q(\varphi), \end{equation} with an appropriate prior $q$ placed on the parameter $\varphi$. Note that this model initially makes no reference to the $\pi_j$'s, and it is formulated \emph{as if} the $\theta_j$'s were fully observable quantities. Then, to update $q$ based on the observed density functions, we apply Equation~(\ref{eq:posterior_uncertain_obs}) in density form to have \begin{equation}\label{eq:posterior_uncertain_obs_density}
q^*(\varphi) \propto \prod_{j=1}^J \left[\int_{\Theta} p(\theta_j|\varphi)\pi_j(\theta_j) d\theta_j\right] q(\varphi), \end{equation}
where for brevity, we denote $q^*:=q^*(\cdot|\Pi_1,\ldots,\Pi_J)$.
In a meta-analysis context, the parameter $\varphi$ often has an interpretation as the central tendency of some shared property of $\theta_1,\ldots,\theta_J$, such as the mean or the covariance (or both jointly). As such, inference on $\varphi$ is often of primary interest in providing a `consensus' over a number of studies. As a secondary goal, we may also be interested in updating a (possibly weakly informative) belief about any individual quantity $\theta_j$, subject to the information provided by observations on the remaining quantities. To do so, we first write Equation~(\ref{eq:joint_belief}) in density form: \[
p^*(\theta_1,\ldots,\theta_J) \propto \int_{\Phi}\prod_{j=1}^J \left[ p(\theta_j|\varphi)\pi_j(\theta_j) d\theta_j\right] q(\varphi) d\varphi, \] and then marginalize over all quantities but the one to be updated. Let $\mathcal{J} := \{1,\ldots,J\}$ be a set of indices and let $j'\in \mathcal{J}$ be an arbitrary index in this set. The density function $\pi_{j'}$ is then updated as follows: \begin{equation}\label{eq:update_pi_j} \pi_{j'}^*(\theta_{j'}) \propto
\int_{\Phi}p(\theta_{j'}|\varphi)\pi_{j'}(\theta_{j'})\prod_{j\in\mathcal{J}\setminus j'}^J \left[\int_{\Theta} p(\theta_j|\varphi)\pi_j(\theta_j) d\theta_j\right] q(\varphi) d\varphi. \end{equation}
\begin{remark} According to Section~\ref{sec:posterior_concentration}, the density $q^*(\varphi)$, defined in Equation~(\ref{eq:posterior_uncertain_obs_density}), will under suitable conditions become increasingly peaked around some point $\varphi_0$, as $J\rightarrow \infty$. That $\pi_{j'}^*(\theta_j)$ does not behave similarly, becomes clear by the following considerations. First, we note that Equation~(\ref{eq:update_pi_j}) is equivalent to \begin{equation*}
\pi_{j^\prime}^{*}(\theta_{j^\prime}) = Z_{j^\prime}^{-1}\pi_{j^\prime}(\theta_{j^\prime}) \int_{\Phi} p(\theta_{j^\prime} | \varphi) q^*(\varphi | \Pi_1, \ldots, \Pi_{j^\prime - 1}, \Pi_{j^\prime + 1}, \ldots, \Pi_{J}) \,d \varphi, \end{equation*} where $Z_j$ is a normalizing constant.
As $q^*(\varphi | \Pi_1, \ldots, \Pi_{j^\prime - 1}, \Pi_{j^\prime + 1}, \ldots, \Pi_{J})$ becomes increasingly peaked around $\varphi_0$, the integral in the above equation converges to $p(\theta_{j^\prime} | \varphi_0)$. Consequently, \begin{equation*}
\pi_{j^\prime}^{*}(\theta_{j^\prime}) \rightarrow Z_{j^\prime}^{-1} \pi_{j^\prime}(\theta_{j^\prime}) p(\theta_{j^\prime} | \varphi_0) , \end{equation*}
which can only be degenerate if either $\pi_{j^\prime}(\theta_{j^\prime})$ or $p(\theta_{j^\prime} | \varphi_0)$ is degenerate by design. Instead of degeneracy, $\pi_{j^\prime}^{*}(\theta_{j^\prime})$ exhibits \emph{shrinkage} with respect to $\varphi_0$. \end{remark}
\subsection{Interpretation as message passing}\label{sec:bp}
The formulation of the above meta-analysis framework, constructed as an extension of standard Bayesian inference, can also be viewed within the formalism of probabilistic graphical models. This provides both an intuitive interpretation and a visualization of Equations (\ref{eq:posterior_uncertain_obs_density}) and (\ref{eq:update_pi_j}), and gives a straightforward way of extending the framework to more complex model structures. To elaborate further on this, consider a tree-structured undirected graphical model with $J$ leaf nodes and a root.
This is a special case of a pairwise Markov random network \citep{Koller+Friedman:2009}, where all factors, or \emph{clique potentials}, are over single variables or pairs of variables, referred to as node and edge potentials, respectively. Note that the potential functions are simply non-negative functions, which may not integrate to 1. Choosing, for $j=1,\ldots,J$, the node potentials as $\pi_j(\theta_j)$ and $q(\varphi)$, and the edge potentials as $\psi_j(\theta_j,\varphi):= p(\theta_j|\varphi)$, the model has the joint density \begin{equation*} \frac{1}{Z}q(\varphi)\prod_{j=1}^J \psi_j(\theta_j,\varphi)\pi_j(\theta_j), \end{equation*} where $Z$ is a normalizing constant. Finding the marginal density of $\varphi$ can then be interpreted as propagating beliefs from each of the leaf nodes up to the root node in the form of messages, a process known as \emph{message passing} or belief propagation \cite[e.g.][]{Yedidia+others:2001}. To that end, we specify the following messages to be sent from the $j$th leaf node to the root: \begin{equation}\label{eq:leaf_root_message} m_{\theta_j\rightarrow \varphi}(\varphi)\propto \int\psi_j(\theta_j,\varphi)\pi_j(\theta_j) d \theta_j . \end{equation}
The initial belief $q(\varphi)$ on $\varphi$ is then updated according to \begin{equation}\label{eq:bp_vaprhi_marginal} q^*(\varphi) \propto q(\varphi)\prod_{j=1}^J m_{\theta_j\rightarrow \varphi}(\varphi), \end{equation} which is exactly equal to Equation~(\ref{eq:posterior_uncertain_obs_density}), and illustrated in Figure~\ref{fig:bp_varphi}.
In a similar way, we may pass information to any single leaf node from the remaining leaf nodes. We now specify two kinds of messages: from leaf nodes indexed by $j\in\mathcal{J}\setminus j'$ to the root node, as given by Equation~(\ref{eq:leaf_root_message}), and from the root node to the $j'$th leaf node, \begin{equation*}\label{eq:root_leaf_message} m_{\varphi\rightarrow \theta_{j'}}(\theta_{j'}) \propto \int_{\Phi}\psi_{j'}(\varphi,\theta_{j'})q(\varphi)\prod_{j\in\mathcal{J}\setminus j'} m_{\theta_j\rightarrow \varphi}(\varphi)\, d\varphi. \end{equation*} The updated belief over $\theta_{j'}$ is then \begin{equation}\label{eq:bp_theta_marginal} \pi_{j'}^*(\theta_{j'}) \propto \pi_{j'}(\theta_{j'}) m_{\varphi\rightarrow \theta_{j'}}(\theta_{j'}), \end{equation} which is exactly equal to Equation~(\ref{eq:update_pi_j}), and illustrated in Figure~\ref{fig:bp_theta}.
\begin{figure}
\caption{~}
\label{fig:bp_varphi}
\caption{~}
\label{fig:bp_theta}
\end{figure}
Although not directly utilized in this work, the graphical model view may also be useful in devising efficient computational strategies (see also the remarks in Section~\ref{sec:discussion}). Especially with more complex model structures, making use of the conditional independencies made explicit by the graphical model may bring considerable computational gains.
\subsection{Bayesian meta-analysis as a special case} \label{sec:rema_fema}
It is straightforward to show that Bayesian random-effects and fixed-effects meta-analyses can be recovered as special cases of the proposed framework. In its traditional formulation \cite[e.g.][]{Normand:1999}, random-effects meta-analysis (REMA) assumes that for each of $J$ studies, a summary statistic, $D_j$, $j=1,\ldots,J$, has been observed, drawn from a distribution with study-specific mean $\mathds{E}(D_j)=\theta_j$ and variance $\mathrm{Var}(D_j)=\sigma_j^2$: \begin{equation}\label{eq:REMA_data_model} D_j \sim \mathcal{N}(\theta_j,\sigma_j^2), \end{equation} where the approximation of the distribution of $D_j$ by a normal distribution is justified by the asymptotic normality of maximum likelihood estimates. The variances $\sigma_j^2$ are directly estimated from the data, while the means $\theta_j$, are assumed to be drawn from some common distribution, typically \[ \theta_j \sim \mathcal{N}(\mu,\sigma_0^2), \] where the parameters $\mu$ and $\sigma_0^2$ represent the average treatment effect and inter-study variation, respectively. Fixed-effects meta-analysis is a special case of REMA, where $\sigma_0^2=0$, such that $\theta_1=\theta_2= \cdots=\theta_J$.
The posterior density for the parameters $(\mu,\sigma_0^2)$ in REMA can be written as \begin{alignat*}{2}
q(\mu,\sigma_0^2|D_1,\ldots,D_J)
&\propto q(\mu,\sigma_0^2)\prod_{j=1}^J \int_{\Theta} N(D_j|\theta_j,\hat{\sigma}_j^2) N(\theta_j|\mu,\sigma_0^2)d\theta_j\\
&\propto q(\mu,\sigma_0^2)\prod_{j=1}^J \int_{\Theta} l(\theta_j;D_j) N(\theta_j|\mu,\sigma_0^2) d\theta_j , \end{alignat*}
where $N(\cdot|\cdot,\cdot)$ denotes a Gaussian density function, $l(\theta_j;D_j)$ is the likelihood function of $\theta_j$ given $D_j$, and $\hat{\sigma}_j^2$ is the empirical variance of $D_j$. To study the connection between the above posterior density and Equation~(\ref{eq:posterior_uncertain_obs_density}), assume that instead of a summary statistic $D_j$, each study has been summarized using a posterior distribution with density $\pi_j(\theta_j)$ over its study-specific effect parameter $\theta_j$. If the distribution has been computed under the data model given by equation (\ref{eq:REMA_data_model}), and using an improper uniform prior $\nu_j(\theta_j)\propto 1$, the density is \[
\pi_j(\theta_j)=N(\theta_j|D_j,\hat{\sigma}_j^2)\propto \exp\left\lbrace-\frac{(D_j-\theta_j)^2}{2\hat{\sigma}_j^2}\right\rbrace = l(\theta_j;D_j), \] resulting in the posterior density of $(\mu,\sigma_0^2)$ being equivalent in both cases.
\section{Computation}\label{sec:computation}
Here we describe a simple computational strategy, which is used in the numerical examples of Section~\ref{sec:illustrations} below.
Some further alternatives are briefly discussed at the end of this section. Recall now that the density of the joint distribution of the parameters $\theta_1,\ldots,\theta_J,\varphi$ can be written as \begin{equation}\label{eq:joint_density}
\frac{1}{Z}q(\varphi)\prod_{j=1}^J p_j(\theta_j|\varphi)\pi_j(\theta_j). \end{equation} Our goal is to produce joint samples from the above model, enabling any desired marginals to be extracted from them. Probabilistic programming languages \citep[e.g.][]{Carpenter+others:2017, Salvatier+others:2015, Tran+others:2016, Wood+others:2014} allow sampling from an arbitrary model, provided that the components of the (unnormalized) model can be specified in terms of probability distributions of some standard form. In the illustrations of this section, we use Hamiltonian Monte Carlo implemented in the Stan probabilistic programming language \citep{Carpenter+others:2017}.
We first note that in the above joint model (\ref{eq:joint_density}), the part specified by the meta-analyst, i.e. $q(\varphi)\prod_{j=1}^J p_j(\theta_j|\varphi)$, can by design be composed using standard parametric distributions.
The observed part of the model $\prod_{j=1}^J \pi_j(\theta_j)$, however, is in general analytically intractable, and instead of having direct access to posterior density functions of standard parametric form, we typically have a sets of posterior samples $\left\{\theta_j^{(1)},\ldots,\theta_j^{(L_j)}\right\}$, with $\theta_j^{(l)}\sim \Pi_j$. Our strategy is then to first find an intermediate parametric approximation $\hat{\pi}_j$ for $\pi_j$, which enables us to sample from an approximate joint distribution. Assuming that the true densities $\pi_j(\theta_j)$ can be evaluated using e.g. kernel density estimation, and that $\hat{\pi}_j(\theta_j)=0 \Rightarrow \pi_j(\theta_j)=0$, the joint samples can be further refined using sampling/importance resampling \citep[SIR;][]{Smith+Gelfand:1992}. The steps of the computational scheme are summarized below: \begin{enumerate}
\item For $j=1,\ldots,J$, fit a parametric density function $\hat{\pi}_j$ to the samples
$\left\{\theta_j^{(1)},\ldots,\theta_j^{(L_j)}\right\}$.
\item Draw $M$ samples $\mathcal{S} = \left\{\theta_1^{*\,(m)},\ldots,\theta_J^{*\,(m)}, \varphi^{*\,(m)}\right\}_{m=1}^{M}$ from the approximate joint model $\frac{1}{Z'}q(\varphi)\prod_{j=1}^J \psi_j(\theta_j,\varphi)\hat{\pi}_j(\theta_j)$.
\item Compute importance weights $w_m=\tilde{w}_m/\sum_{m=1}^M\tilde{w}_m$, where \begin{align*} \tilde{w}_m &= \frac{Z^{-1}\, q\left(\varphi^{*\,(m)}\right)\prod_{j=1}^J \psi_j\left(\theta_j^{*\,(m)},\varphi^{*\,(m)}\right)\pi_j\left(\theta_j^{*\,(m)}\right)} {(Z')^{-1}\, q\left(\varphi^{*\,(m)}\right)\prod_{j=1}^J \psi_j\left(\theta_j^{*\,(m)},\varphi^{*\,(m)}\right)\hat{\pi}_j\left(\theta_j^{*\,(m)}\right)} \\& = \frac{Z'\,\prod_{j=1}^J\pi_j\left(\theta_j^{*\,(m)}\right)}{Z\,\prod_{j=1}^J\hat{\pi}_j\left(\theta_j^{*\,(m)}\right)}. \end{align*}
Note that the constant $Z'/Z$ cancels in the computation of the normalized weights $w_m$.
\item Resample $\mathcal{S}$ with weights $\{w_1,\ldots,w_M\}$. \end{enumerate}
For problems with a very large number of studies or high dimensional local parameters, or if the imposed parametric densities approximate the actual posteriors poorly, the computation of importance weights may become numerically unstable.
The issue could possibly be mitigated using more advanced importance sampling schemes, such as Pareto-smoothed importance sampling \citep{Vehtari+others:2015} or prior swap importance sampling \citep{Neiswanger+Xing:2017}.
If we are only interested in sampling from the density of the global parameter, as given by Equation~(\ref{eq:posterior_uncertain_obs_density}), then an obvious alternative strategy would be to implement a Metropolis-Hastings algorithm, using the samples $\theta_j^{(l)}\sim \Pi_j$ to compute Monte Carlo estimates of the integrals $\int_{\Theta} p(\theta_j|\varphi)\pi_j(\theta_j) d\theta_j$. However, this would lead to expensive MCMC updates as the integrals need to be re-estimated at every iteration of the algorithm. Finally, instead of directly sampling from the full joint distribution, we could try to utilize the induced graphical model structure (Section~\ref{sec:bp}) to do localized inference, see also Section~\ref{sec:discussion}.
\section{Numerical illustrations}\label{sec:illustrations}
In this section, we illustrate our meta-analysis framework, \emph{meta-analysis of Bayesian analyses} (MBA), with numerical examples.
In these examples, we consider the problem of combining results from analyses conducted using likelihood-free models. In such models, the data can typically be summarized by a number of different statistics but there is no closed-form likelihood to relate these to the quantity or effect of interest, which poses a challenge for traditional formulations of meta-analysis. In our framework, we directly utilize the inferred posteriors to build a joint model. In addition to modeling the shared central tendency of the inferred model parameters, we demonstrate that weakly informative or poorly identifiable posteriors for individual studies can be updated post-hoc through joint modeling. We begin with a brief review of likelihood-free inference using approximate Bayesian computation.
\subsection{Likelihood-free inference using approximate Bayesian computation}\label{sec:abc}
Approximate Bayesian computation (ABC) is a paradigm for Bayesian inference in models, which either entirely lack an analytically tractable likelihood function, or for which it is costly to compute. The only requirement is that we are able to sample data from the model by fixing values for the parameters of interest, as is the case for simulator-based models. In the basic form of ABC, simulations are run for parameter proposals drawn from a prior distribution.
The parameter proposals whose simulated data $x_\theta$ match the observed data $x_0$ are collected and constitute a sample from the posterior distribution.
It can be shown that this process is equivalent to accepting parameter proposals in proportion to their likelihood value, given the observed data, as is done in traditional rejection sampling.
In practice, the simulated data virtually never exactly matches the observed data and likely no sample from the posterior distribution would be acquired.
This problem can be circumvented by loosening the acceptance condition to accept samples whose simulations yield results similar enough to the observed data.
For this purpose, a dissimilarity function $d$ and a scalar $\varepsilon > 0$ are defined such that a parameter proposal with respective simulation result $x_\theta$ is accepted if $ d(x_\theta, x_0) \leq \varepsilon $. This function is often defined in terms of summary statistics $s(x_\theta)$ and $s(x_0)$. For example, $d$ could be defined as the Euclidean distance between $s(x_\theta)$ and $s(x_0)$.
The aforementioned relaxation results in samples being drawn from an \emph{approximate} posterior instead of the actual posterior distribution, hence the name approximate Bayesian computation.
For a comprehensive introduction to ABC, see \citet{Marin+others:2012}. More recent developments are reviewed in \cite{Lintusaari+others:2017}. In the following numerical illustrations, posterior distributions obtained using ABC provide a starting point for meta-analysis. These likelihood-free inferences are implemented using the ELFI open-source software package \citep{Lintusaari+others:2018}.
\subsection{Example 1: MA$(q)$ process} \label{subsec:ma2}
In our first example, we use simulated data from a MA$(q)$ process of order $q=2$. The MA$(q)$ process is a standard example in the literature on likelihood-free inference due to its simple structure but fairly complex likelihood and non-trivial relationship between parameters and observed data. Assuming zero mean, the process $(y_t)_{t\in \mathbb{N}^{+}}$ is defined as \begin{equation}\label{eq:ma2} y_t = \epsilon_t + \theta_1 \epsilon_{t-1} + \theta_2 \epsilon_{t-2}, \end{equation} where $(\theta_1,\theta_2) \in\mathbb{R}^2$ and $\epsilon_s \sim \mathcal{N}(0,1)$, ${s\in\mathbb{Z}}$. The quantity of interest for which we conduct inference is $\boldsymbol{\theta} = (\theta_1,\theta_2)$.
Following \citet{Marin+others:2012}, we use as prior for $\boldsymbol{\theta}$ a uniform distribution over the set \[
\mathcal{T}\subset \mathbb{R}^2 \triangleq \{(\theta_1,\theta_2)\in\mathbb{R}^2|-(\theta_2 + 1) < \theta_1 < \theta_2 + 1, \; -1<\theta_2<1\}, \] which, by restriction of the parameter space, imposes a general identifiability condition for MA$(q)$ processes. Inference for $\boldsymbol{\theta}$ is then conducted using ABC with rejection sampling, taking as summary statistics the empirical autocovariances of lags one and two, denoted as $\hat{\gamma}_1$ and $\hat{\gamma}_2$, respectively. Furthermore, a Euclidean distance of 0.1 is used as acceptance threshold.
To illustrate our meta-analysis framework, we first sample $J=12$ realizations of $\boldsymbol{\theta}$ using the following generating process: \begin{equation}\label{eq:theta_gen} \theta_{1}\sim \mathrm{Unif}(0.4,0.8), \quad \theta_{2} \sim \mathcal{N}(-0.4+\theta_{1j}, 0.04^2). \end{equation}
Given each realization $\boldsymbol{\theta}_j=(\theta_{1j},\theta_{2j})$, $j=1,\ldots,J$, we then generate a series of $10$ data points, $(y_{j1},\ldots,y_{j10})$, according to Equation~(\ref{eq:ma2}).
For each time-series, we independently conduct likelihood-free inference as described above, generating $1000$ samples from the posterior.
The computed posterior distributions along with their corresponding true parameter values are shown in Figure~\ref{fig:ma2_independent_posteriors}.
It can be seen that the very limited information given by the data in each of the analyses leaves the posteriors with a considerable amount of uncertainty.
For meta-analysis, we first specify a model for the study-specific effects $\boldsymbol{\theta}_1,\ldots,\boldsymbol{\theta}_J$ \emph{as if} they were observed quantities from an exchangeable sequence; see Equation~(\ref{eq:meta-analyst}). As the true generating mechanism of the effects is typically unknown, the model must be specified according to the analyst's judgment. To reflect this, we will here model the generating process as a Gaussian distribution with parameters $\varphi = (\boldsymbol{\mu},\Sigma_0)$, \begin{equation}\label{eq:theta_conditional} \boldsymbol{\theta}_j\sim \mathcal{N}_2(\boldsymbol{\mu}, \Sigma_0). \end{equation}
For $\boldsymbol{\mu}$ and the covariance matrix $\Sigma_0$, we use Gaussian and inverse Wishart priors, respectively, \begin{equation}\label{eq:hyperpriors} \centering
\boldsymbol{\mu} \sim \mathcal{N}_2(\boldsymbol{m},V) \quad \text{ and } \quad \Sigma_0\sim \mathcal{W}^{-1}(\nu,\Psi) \,, \end{equation} with \[ \centering \boldsymbol{m} = \begin{bmatrix} 1/2\\ 0 \end{bmatrix},\quad V = \begin{bmatrix} 0.4 & 0.05 \\ 0.05 & 0.1 \end{bmatrix},\quad \text{ and } \quad \nu = 4,\quad \Psi = \begin{bmatrix} 0.4 & 0.1 \\ 0.1 & 0.2 \end{bmatrix}. \] The above values were chosen to provide reasonable coverage of $\mathcal{T}$, the constrained support of $\boldsymbol{\theta}$. Furthermore, $\nu$ was chosen as $\mathrm{dim}(\boldsymbol{\theta}) + 2$ to directly yield $\Psi$ as the mean of the inverse Wishart prior on $\Sigma_0$.
After specifying the assumed generative model for $\boldsymbol{\theta}_1,\ldots,\boldsymbol{\theta}_J$ according to Equations~(\ref{eq:theta_conditional})--(\ref{eq:hyperpriors}), the following step is to incorporate the observed beliefs for each $\boldsymbol{\theta}_j$ into the inference.
For computational convenience, we initially approximate the study-specific posteriors using a suitable parametric family. In our current example, we fit a bivariate normal distribution to each of the $J=12$ posteriors. Following the computational scheme presented in Section~\ref{sec:computation}, inference for the joint model was carried out using Hamiltonian Monte Carlo implemented in the Stan software \citep{Carpenter+others:2017}, and finally, the results were refined using SIR.
We compare MBA against results obtained using traditional random-effects meta-analysis (REMA), as specified in Section~\ref{sec:rema_fema}. The likelihood of REMA is given by the model \begin{equation}\label{eq:ma2_rema_likelihood} \hat{\!\boldsymbol{\theta}}_j \sim \mathcal{N}_2(\boldsymbol{\theta}_j, \hat{\Sigma}_j), \end{equation} wwhere the effect estimates $\,\hat{\!\boldsymbol{\theta}}_j=(\hat{\theta}_{1j},\hat{\theta}_{2j})$ are computed numerically using conditional sum of squares\footnote{Implemented in the \texttt{statsmodels} Python module.} and the study-specific covariance matrices $\hat{\Sigma}_j$ are estimated using bootstrap. The hierarchical distribution on $\boldsymbol{\theta}_j$ follows Equations (\ref{eq:theta_conditional}) and (\ref{eq:hyperpriors}) above. Therefore, the essential difference between REMA and MBA is whether we combine this distribution with a likelihood function based on Equation~(\ref{eq:ma2_rema_likelihood}) or with the observed posterior on $\boldsymbol{\theta}_j$. The results of the comparison are presented in Figures \ref{fig:ma2_mu0_posteriors}--\ref{fig:ma2_re_posteriors}.\footnote{The experiment was repeated with multiple random seeds, yielding similar results.}
\begin{figure}\label{fig:ma2_independent_posteriors}
\end{figure}
\begin{figure}\label{fig:ma2_mu0_posteriors}
\end{figure}
\begin{figure}\label{fig:ma2_updated_posteriors}
\end{figure}
\begin{figure}\label{fig:ma2_re_posteriors}
\end{figure}
Figure~\ref{fig:ma2_mu0_posteriors} shows the posterior distribution for the global mean effect $\boldsymbol{\mu}$, obtained using four different models: in addition to MBA and REMA, we used fixed-effects meta-analysis (FEMA, which is a special case of REMA, see Section~\ref{sec:rema_fema}), and a 'naive' model corresponding to ordinary Bayesian inference using the means of the observed posteriors on $\boldsymbol{\theta}_j$ as observed data. As expected, FEMA is clearly inappropriate in this situation, and results in a heavily biased and overly confident posterior. The naive model is less biased than FEMA as it makes use of information contained in the observed posteriors, but compared to MBA, it does not properly account for the uncertainty contained in them. Both REMA and MBA result in posteriors with more spread, still assigning reasonably high probability mass to the neighbourhood around the true parameter value.
Figure~\ref{fig:ma2_updated_posteriors} shows updated beliefs for the local effects, obtained using the update rule of Equation~(\ref{eq:update_pi_j}) in Section~\ref{sec:mba}, also schematically depicted in Figure \ref{fig:bp_theta}. The updated beliefs exhibit shrinkage towards the global mean effect and, in this case, concentrate more accurately around the actual local effect values. On the other hand, many of the REMA posteriors for the local effects, shown in Figure \ref{fig:ma2_re_posteriors}, are biased and concentrate in regions away from true values.
\subsection{Example 2: Tuberculosis outbreak dynamics}
We now apply MBA to conduct meta-analysis of parameters regulating a stochastic birth-death (SBD) model proposed by \citet{Lintusaari+others:2019}, who used their model in a single-study setting to analyze tuberculosis outbreak data from the San Francisco Bay area, initially reported by \citet{Small+others:1994}. The goal of the analysis was to estimate disease transmission parameters from genotype data which, in contrast to outbreak models relying on count data,renders the likelihood-function intractable and necessitates the use of likelihood-free inference \citep{Tanaka+others:2006}. Furthermore, such models are often complex in relation to the available data, which may result in poor identifiability, as discussed by \citet{Lintusaari+others:2016}. To alleviate the problem, they formulated their model as a mixture of stochastic processes, taking into account the individual transmission dynamics of different subpopulations. In our analyses, we focus on two key parameters of the model, $R_1$ and $R_2$, which are the reproductive numbers for two subpopulations: those that are compliant and non-compliant to treatment, respectively\footnote{Note that \citet{Lintusaari+others:2019} used the notation $R_0$ for the parameter $R_2$.}.
For our current experiment, we analyzed three additional data sets using the model of \citet{Lintusaari+others:2019}. These data sets reported tuberculosis outbreaks in Estonia \citep{Kruuner3339}, London \citep{Maguire617} and the Netherlands \citep{Netherlands}.
For each data set, we independently conducted likelihood-free inference, generating $1000$ samples from the posteriors.
Following \citet{Lintusaari+others:2019}, we used the following six summary statistics for the ABC simulations: the number of observations, the total number of genotype clusters, the size of the largest cluster, the proportion of clusters of size two, the proportion of singleton clusters, and finally, the average successive difference in size among the four largest clusters. The original publication additionally used two summary statistics on the observation times of the largest cluster. While these were found to improve model identifiability, such information was not available for the additional data sets analyzed in our current experiment.
We used a weighted Euclidean distance as dissimilarity function, with the same weights as in \citet{Lintusaari+others:2019}.
Figure~\ref{fig:tb_abc_posteriors} shows the joint posterior distributions for the parameters $R_1$ and $R_2$, obtained individually for all four geographical locations using ABC. Compared to the San Francisco data, the posteriors computed on the remaining data sets, in particular London and the Netherlands, show severe problems with identifiability. This can at least partly be attributed to these data sets being less informative than the San Francisco data set. A key question for our experiment then is whether we could borrow strength across the studies to improve the identifiability of the models computed on the remaining data sets. Additionally, it will be of interest to obtain an overall analysis of the central tendency of the reproductive numbers.
As in the previous experiment of Section~\ref{subsec:ma2}, we first define a model for the local effects $\boldsymbol{\theta}_j = (R_{1j}, R_{2j})$, and assign priors for the global mean effect $\boldsymbol{\mu} = (\mu_1,\mu_2)$ and the covariance matrix $\Sigma_0$, resulting in the model: \begin{gather*}
\boldsymbol{\theta}_j\sim \mathcal{N}_2(\boldsymbol{\mu}, \Sigma_0), \\
\mu_1 \sim \mathrm{Gamma}(a_1, b_1),\\
\mu_2 \sim \mathrm{Gamma}(a_2, b_2), \\
\Sigma_0\sim \mathcal{W}^{-1}(\nu,\Psi). \end{gather*} The hyper-parameters were set as follows: \[ a_1 = 0.12,\, b_1 = 0.36\quad a_2 =.030, \, b_2 = 0.05,\quad \text{ and } \quad \nu = 4,\quad \Psi = \begin{bmatrix} 4 & -0.1 \\ -0.1 & 0.01 \end{bmatrix}. \] We then incorporate the observed beliefs by fitting a bivariate Gaussian distribution to each of the $J=4$ study-specific ABC posteriors. Inference is performed using Stan and the obtained posterior is improved using SIR. Note that due to the indirect nature in the relationship between the infection data and the parameters of interest, using the data to directly construct an estimator for the parameters of interest would be difficult. While this does not pose a challenge in our framework, it renders the application of traditional meta-analysis approaches infeasible.
Figure~\ref{fig:tb_abalation} shows the updated beliefs for the local effects $\boldsymbol{\theta}_1, \ldots, \boldsymbol{\theta}_J$, after borrowing strength across the individual studies. While the updated beliefs clearly retain some of the individual characteristics of their original counterparts (e.g. a similar covariance structure), they exhibit a much more identifiable behavior. The posterior of the overall mean of the reproductive numbers is shown in Figure~\ref{fig:tb_joint_global}.
\begin{figure}\label{fig:tb_abc_posteriors}
\end{figure}
\begin{figure}\label{fig:tb_abalation}
\end{figure}
\begin{figure}
\caption{MBA joint posterior for the overall mean effect $\boldsymbol{\mu}=(\mu_1, \mu_2)$. }
\label{fig:tb_joint_global}
\end{figure}
\section{Related work}\label{sec:related_work}
To the best of our knowledge, meta-analysis frameworks for combining posterior distributions have not been presented before. Nonetheless, similar elements can be found in a number of previous works. \citet{Xie+others:2011} proposed to do meta-analysis by combining confidence distributions which, despite their analogy with Bayesian posterior marginals, is a concept deeply rooted in the frequentist paradigm, see \citet{Schweder_Hjort:2002}.
In the context of Bayesian meta-analysis, \citet{Lunn+others:2013} introduced a two-stage computational approach for fitting hierarchical models to individual-level data.
In the first stage, posterior samples for local parameters are generated independently for each study, enabling study-specific complexities to be dealt with separately. In a second stage, the samples are used as proposals in Metropolis-Hastings updates within a Gibbs algorithm to fit the full hierarchical model. While the paper focuses on a specific computational approach for hierarchical models, involving no propagation of local prior knowledge into the joint model, the idea of utilizing independently computed posterior samples to fit a joint model is common to our framework.
In recent work, \citet{Rodrigues+others:2016} developed a hierarchical Gaussian process prior to model a set of related density functions, where grouped data in the form of samples assumed to be drawn under each density function are available. Despite a superficial similarity between their work and ours, the inferential goals in these works are very different. Specifically, the former is concerned with nonparametric estimation of group-specific densities, which is shown to be useful when the sample sizes in some or all of the groups are small. Thus, in cases where the number of posterior samples per study is limited (e.g. due to computational reasons), the method of \citet{Rodrigues+others:2016} could be used within our framework to provide density estimates for the initial beliefs.
\section{Conclusion}\label{sec:discussion}
The natural outcome of a Bayesian analysis is a posterior distribution over quantities of interest. In this paper, we have developed a framework which, instead of study-specific data summaries or individual-level data, uses estimated posterior distributions to conduct meta-analysis. The framework builds on standard Bayesian inference over exchangeable observations, by treating each observed posterior as data observed with uncertainty.
In Section~\ref{sec:bp}, we showed that the proposed framework can be interpreted as message passing in probabilistic graphical models. Utilizing this interpretation, a graphical illustration of the main update formulas, Equations~(\ref{eq:posterior_uncertain_obs_density}) and (\ref{eq:update_pi_j}), is given in Figure~\ref{fig:bp_varphi} and \ref{fig:bp_theta}, respectively. In future work, we plan to further investigate computational strategies which make use of this interpretation. Unlike our current strategy, which infers all parameters of the joint model simultaneously, utilization of localized inference would enable us to devise more efficient and flexible computational schemes. In particular, recent developments in nonparametric and particle-based belief propagation \citep{Ihler+McAllester:2009, Lienart+others:2015, Pacheco+others:2014, Sudderth+others:2010} represent a promising direction in this line of work.
In many fields, it has become common practice to store data sets in dedicated repositories to be reused for the benefit of the entire research community. Given the view taken in this work, that posterior distributions can be seen as a special kind of observational data, we believe that in many cases it would be equally beneficial to make full posterior distributions available for reuse. This would enable posteriors from potentially time-consuming and costly Bayesian analyses to be used as a basis for new studies. Indeed, even if the original data, the model and the code implementing it were available, reproducing posterior distributions could require a substantial computational effort. In addition to making posteriors publicly available, more research is needed on developing methods to make appropriate use of the information they provide. The current work is a first step in this direction and our hope is that it will inspire other researchers to make further advances to this end.
\section*{Acknowledgments}
PB, DM, JL and SK were funded by the Academy of Finland, grants 319264 and 294238.
JC was funded by ERC grant 742158.
The authors gratefully acknowledge the computational resources provided by the Aalto Science-IT project and support from the Finnish Center for Artificial Intelligence (FCAI).
\appendix
\section{Posterior concentration for discrete parameters}\label{sec:proof}
Here, we give an elementary proof of posterior concentration of $Q(\cdot|\Pi_1,\ldots,\Pi_J)$ for discrete parameter spaces. The proof follows the basic structure found in many sources \citep[e.g.][]{Bernardo+Smith:1994,Gelman+others:2013}, with the essential difference that the observations are independent but non-identically distributed.
\begin{theorem}
Let $\{\Pi_1,\ldots,\Pi_J\}$ be a set of observations from a corresponding set of distributions $\{R_1,\ldots,R_J\}$. Furthermore, let $\left\{\{H^{(j)}_{\varphi}|\varphi\in\Phi\}\right\}_{j=1}^J$ be a set of families, such that \begin{itemize} \item[(i)] $\Phi$ consists of (at most) a countable set of values,
\item[(ii)] $\varphi_0 = \argmin_{\varphi\in\Phi} \mathrm{KL}\left(R_j||H^{(j)}_{\varphi}\right)$, for all $j$. \end{itemize}
If the conditions of Theorem~\ref{thm:sen_singer} hold, and furthermore if $\sum_{\varphi\in\Phi}Q(\varphi)=1$ and $Q(\varphi_0)>0$, then $Q^*(\varphi_0|\Pi_1\ldots,\Pi_J)\rightarrow 1$, as $J\rightarrow\infty$. \end{theorem}
\begin{proof} For any $\varphi\neq\varphi_0$, the log posterior odds can written as \begin{equation}\label{eq:log_posterior_odds}
\log\frac{Q\left(\varphi|\Pi_1,\ldots,\Pi_J\right)}{Q\left(\varphi_0|\Pi_1,\ldots,\Pi_J\right)} = \log\frac{Q(\varphi)}{Q(\varphi_0)} +
\sum_{j=1}^J\log \frac{h_j(\Pi_j|\varphi)}{h_j(\Pi_j|\varphi_0)}, \end{equation} where the second term is a sum of $J$ independent but non-identically distributed random variables. By Theorem~\ref{thm:sen_singer}, we have that \[
\frac{1}{J}\sum_{j=1}^J\log \frac{h_j(\Pi_j|\varphi)}{h_j(\Pi_j|\varphi_0)} \rightarrow
\mathds{E}\left(\log \frac{h_j(\Pi_j|\varphi)}{h_j(\Pi_j|\varphi_0)}\right), \]
with probability 1, as $J\rightarrow\infty$. Since $\varphi_0$ is the unique minimizer of the KL-divergence $\mathrm{KL}\left(R_j||H^{(j)}_{\varphi}\right)$, by definition \[
\mathds{E}\left( \log\frac{h_j(\Pi_j|\varphi)}{h_j(\Pi_j|\varphi_0)} \right)
= \mathrm{KL}\left(R^j||H^j_{\varphi_0}\right)
- \mathrm{KL}\left(R^j||H^j_{\varphi}\right) < 0, \] and consequently, \[
\sum_{j=1}^J\log \frac{h_j(\Pi_j|\varphi)}{h_j(\Pi_j|\varphi_0)} \rightarrow -\infty. \]
Since $Q(\varphi_0)>0$, the entire expression (\ref{eq:log_posterior_odds}) approaches $-\infty$ as $J\rightarrow \infty$, which implies that $Q(\varphi|\Pi_1,\ldots,\Pi_J)\rightarrow 0$ and $Q(\varphi_0|\Pi_1,\ldots,\Pi_J)\rightarrow 1$. \end{proof}
\end{document} | arXiv |
Tag: Evolutionary Genetics
A Kimura Age to the Kern-Hahn Era: neutrality & selection
Posted on November 9, 2018 November 9, 2018 by Razib Khan
I'm pretty jaded about a lot of journalism, mostly due to the incentives in the industry driven by consumers and clicks. But Quanta Magazine has a really good piece out, Theorists Debate How 'Neutral' Evolution Really Is. It hits all the right notes (you can listen to one of the researchers quoted, Matt Hahn, on an episode of my podcast from last spring).
As someone who is old enough to remember reading about the 'controversy' more than 20 years ago, it's interesting to see how things have changed and how they haven't. We have so much more data today, so the arguments are really concrete and substantive, instead of shadow-boxing with strawmen. And yet still so much of the disagreement seems to hinge on semantic shadings and understandings even now.
But, as Richard McElreath suggested on Twitter part of the issue is that ultimately Neutral Theory might not even be wrong. It simply tries to shoehorn too many different things into a simple and seductively elegant null model when real biology is probably more complicated than that. With more data (well, exponentially more data) and computational power biologists don't need to collapse all the complexity of evolutionary process across the tree of life into one general model, so they aren't.
Let me finish with a quote from Ambrose, Bishop of Milan, commenting on the suffocation of the Classical religious rites of Late Antiquity:
It is undoubtedly true that no age is too late to learn. Let that old age blush which cannot amend itself. Not the old age of years is worthy of praise but that of character. There is no shame in passing to better things.
Posted in Evolutionary GeneticsTagged Evolutionary Genetics7 Comments on A Kimura Age to the Kern-Hahn Era: neutrality & selection
A historical slice of evolutionary genetics
Posted on October 12, 2018 October 12, 2018 by Razib Khan
A few friends pointed out that I likely garbled my attribution of who were the guiding forces between the "classical" and "balance" in the post below (Muller & Dobzhansky as opposed to Fisher & Wright as I said). I'll probably do some reading and update the post shortly…but it did make me reflect that in the hurry to keep up on the current literature it is easy to lose historical perspective and muddle what one had learned.
Of course on some level science is not as dependent on history as many other disciplines. The history is "baked-into-the-cake." This is clear when you read The Origin of Species. But if you are interested in a historical and sociological perspective on science, with a heavy dose of narrative biography, I highly recommend Ullica Segerstrale's Defenders of the Truth: The Battle for Science in the Sociobiology Debate and Beyond and Nature's Oracle: The Life and Work of W.D. Hamilton.
Defenders of the Truth in particular paints a broad and vivid picture of a period in the 1960s and later into the 1970s when evolutionary thinkers began to grapple with ideas such as inclusive fitness. E. O. Wilson's Sociobiology famously triggered a counter-reaction by some intellectuals (Wilson was also physically assaulted in the 1978 AAAS meeting). Characters such as Noam Chomsky make cameo appearances.
Segerstrale's Nature's Oracle focuses particularly on the life and times of W. D. Hamilton, though if you want that at high speed and max density, read Narrow Roads of Gene Land, Volume 2. Because Hamilton died before the editing phase, the biographical text is relatively unexpurgated. Hamilton also makes an appearance in The Price of Altruism: George Price and the Search for the Origins of Kindness.
The death of L. L. Cavalli-Sforza reminds us that the last of the students of the first generation of population geneticists are now passing on. With that, a great of history is going to be inaccessible. The same is not yet true of the acolytes of W. D. Hamilton, John Maynard Smith, or Robert Trivers.
Posted in Evolutionary GeneticsTagged Evolutionary Genetics
Idle theories are the devil's workshop
Posted on February 28, 2018 February 28, 2018 by Razib Khan
In the 1970s Richard C. Lewontin wrote about how the allozyme era finally allowed for the testing of theories which had long been perfected and refined but lay unused like elegant machines without a task. Almost immediately the empirical revolution that Lewontin began in the 1960s kickstarted debates about the nature of selection and neutrality on the molecular level, now that molecular variation was something they could actually explore.
This led to further debates between "neutralists" and "selectionists." Sometimes the debates were quite acrimonious and personal. The most prominent neutralist, Motoo Kimura, took deep offense to the scientific criticisms of the theoretical population geneticist John Gillespie. The arguments around neutral theory in the 1970s eventually spilled over into other areas of evolutionary biology, and prominent public scientists such as Richard Dawkins and Stephen Jay Gould got pulled into it (neither of these two were population geneticists or molecular evolutionists, so one wonders what they truly added besides bluster and publicity).
Today we do not have these sorts of arguments from what I can tell. Why? I think it is the same reason that is the central thesis of Benjamin Friedman's The Moral Consequences of Economic Growth. In it, the author argues that liberalism, broadly construed, flourishes in an environment of economic growth and prosperity. As the pie gets bigger zero-sum conflicts are attenuated.
What's happened in empirical studies of evolutionary biology over the last decade or so is that in genetics a surfeit of genomic data has swamped the field. Some scholars have even suggested that in evolutionary genomics we have way more data than can be analyzed or understood (in contrast to medical genomics, where more data is still useful and necessary). Scientists still have disagreements, but instead of bickering or posturing, they've been trying to dig out from the under the mountain of data.
It's easy to be gracious to your peers when you're rich in data….
Posted in Evolutionary Genetics, Evolutionary GenomicsTagged Evolutionary Genetics1 Comment on Idle theories are the devil's workshop
Synergistic epistasis as a solution for human existence
Posted on May 6, 2017 May 6, 2017 by Razib Khan
Epistasis is one of those terms in biology which has multiple meanings, to the point that even biologists can get turned around (see this 2008 review, Epistasis — the essential role of gene interactions in the structure and evolution of genetic systems, for a little background). Most generically epistasis is the interaction of genes in terms of producing an outcome. But historically its meaning is derived from the fact that early geneticists noticed that crosses between individuals segregating for a Mendelian characteristic (e.g., smooth vs. curly peas) produced results conditional on the genotype of a secondary locus.
Molecular biologists tend to focus on a classical, and often mechanistic view, whereby epistasis can be conceptualized as biophysical interactions across loci. But population geneticists utilize a statistical or evolutionary definition, where epistasis describes the extend of deviation from additivity and linearity, with the "phenotype" often being fitness. This goes back to early debates between R. A. Fisher and Sewall Wright. Fisher believed that in the long run epistasis was not particularly important. Wright eventually put epistasis at the heart of his enigmatic shifting balance theory, though according to Will Provine in Sewall Wright and Evolutionary Biology even he had a difficult time understanding the model he was proposing (e.g., Wright couldn't remember what the different axes on his charts actually meant all the time).
These different definitions can cause problems for students. A few years ago I was a teaching assistant for a genetics course, and the professor, a molecular biologist asked a question about epistasis. The only answer on the key was predicated on a classical/mechanistic understanding. But some of the students were obviously giving the definition from an evolutionary perspective! (e.g., they were bringing up non-additivity and fitness) Luckily I noticed this early on and the professor approved the alternative answer, so that graders would not mark those using a non-molecular answer down.
My interested in epistasis was fed to a great extent in the middle 2000s by my reading of Epistasis and the Evolutionary Process. Unfortunately not too many people read this book. I believe this is so because when I just went to look at the Amazon page it told me that "Customers who viewed this item also viewed" Robert Drews' The End of the Bronze Age. As it happened I read this book at about the same time as Epistasis and the Evolutionary Process…and to my knowledge I'm the only person who has a very deep interest in statistical epistasis and Mycenaean Greece (if there is someone else out there, do tell).
In any case, when I was first focused on this topic genomics was in its infancy. Papers with 50,000 SNPs in humans were all the rage, and the HapMap paper had literally just been published. A lot has changed.
So I was interested to see this come out in Science, Negative selection in humans and fruit flies involves synergistic epistasis (preprint version). Since the authors are looking at humans and Drosophila and because it's 2017 I assumed that genomic methods would loom large, and they do.
And as always on the first read through some of the terminology got confusing (various types of statistical epistasis keep getting renamed every few years it seems to me, and it's hard to keep track of everything). So I went to Google. And because it's 2017 a citation of the paper and further elucidation popped up in Google Books in Crumbling Genome: The Impact of Deleterious Mutations on Humans. Weirdly, or not, the book has not been published yet. Since the author is the second to last author on the above paper it makes sense that it would be cited in any case.
So what's happening in this paper? Basically they are looking to reduced variance of really bad mutations because a particular type of epistasis amplifies their deleterious impact (fitness is almost always really hard to measure, so you want to look at proxy variables).
Because de novo mutations are rare, they estimate about 7 are in functional regions of the genome (I think this may be high actually), and that the distribution should be Poisson. This distribution just tells you that the mean number of mutations and the variance of the the number of mutations should be the same (e.g., mean should be 5 and variance should 5).
Epistasis refers (usually) to interactions across loci. That is, different genes at different locations in the genome. Synergistic epistasis means that the total cumulative fitness after each successive mutation drops faster than the sum of the negative impact of each mutation. In other words, the negative impact is greater than the sum of its parts. In contrast, antagonistic epistasis produces a situation where new mutations on the tail of the distributions cause a lower decrement in fitness than you'd expect through the sum of its parts (diminishing returns on mutational load when it comes to fitness decrements).
These two dynamics have an effect the linkage disequilibrium (LD) statistic. This measures the association of two different alleles at two different loci. When populations are recently admixed (e.g., Brazilians) you have a lot of LD because racial ancestry results in lots of distinctive alleles being associated with each other across genomic segments in haplotypes. It takes many generations for recombination to break apart these associations so that allelic state at one locus can't be used to predict the odds of the state at what was an associated locus. What synergistic epistasis does is disassociate deleterious mutations. In contrast, antagonistic epistasis results in increased association of deleterious mutations.
Why? Because of selection. If a greater number of mutations means huge fitness hits, then there will strong selection against individuals who randomly segregate out with higher mutational loads. This means that the variance of the mutational load is going to lower than the value of the mean.
How do they figure out mutational load? They focus on the distribution of LoF mutations. These are extremely deleterious mutations which are the most likely to be a major problem for function and therefore a huge fitness hit. What they found was that the distribution of LoF mutations exhibited a variance which was 90-95% of a null Poisson distribution. In other words, there was stronger selection against high mutation counts, as one would predict due to synergistic epistasis.
Thus, the average human should carry at least seven de novo deleterious mutations. If natural selection acts on each mutation independently, the resulting mutation load and loss in average fitness are inconsistent with the existence of the human population (1 − e−7 > 0.99). To resolve this paradox, it is sufficient to assume that the fitness landscape is flat only outside the zone where all the genotypes actually present are contained, so that selection within the population proceeds as if epistasis were absent (20, 25). However, our findings suggest that synergistic epistasis affects even the part of the fitness landscape that corresponds to genotypes that are actually present in the population.
Overall this is fascinating, because evolutionary genetic questions which were still theoretical a little over ten years ago are now being explored with genomic methods. This is part of why I say genomics did not fundamentally revolutionize how we understand evolution. There were plenty of models and theories. Now we are testing them extremely robustly and thoroughly.
Addendum: Reading this paper reinforces to me how difficult it is to keep up with the literature, and how important it is to know the literature in a very narrow area to get the most out of a paper. Really the citations are essential reading for someone like me who just "drops" into a topic after a long time away….
Citation: Science, Negative selection in humans and fruit flies involves synergistic epistasis.
Posted in Evolution, Genetics, GenomicsTagged Epistasis, Evolutionary Genetics
Why the rate of evolution may only depend on mutation
Posted on April 23, 2017 April 24, 2017 by Razib Khan
Sometimes people think evolution is about dinosaurs.
It is true that natural history plays an important role in inspiring and directing our understanding of evolutionary process. Charles Darwin was a natural historian, and evolutionary biologists often have strong affinities with the natural world and its history. Though many people exhibit a fascination with the flora and fauna around us during childhood, often the greatest biologists retain this wonderment well into adulthood (if you read W. D. Hamilton's collections of papers, Narrow Roads of Gene Land, which have autobiographical sketches, this is very evidently true of him).
But another aspect of evolutionary biology, which began in the early 20th century, is the emergence of formal mathematical systems of analysis. So you have fields such as phylogenetics, which have gone from intuitive and aesthetic trees of life, to inferences made using the most new-fangled Bayesian techniques. And, as told in The Origins of Theoretical Population Genetics, in the 1920s and 1930s a few mathematically oriented biologists constructed much of the formal scaffold upon which the Neo-Darwinian Synthesis was constructed.
The product of evolution
At the highest level of analysis evolutionary process can be described beautifully. Evolution is beautiful, in that its end product generates the diversity of life around us. But a formal mathematical framework is often needed to clearly and precisely model evolution, and so allow us to make predictions. R. A. Fisher's aim when he wrote The Genetical Theory Natural Selection was to create for evolutionary biology something equivalent to the laws of thermodynamics. I don't really think he succeeded in that, though there are plenty of debates around something like Fisher's fundamental theorem of natural selection.
But the revolution of thought that Fisher, Sewall Wright, and J. B. S. Haldane unleashed has had real yields. As geneticists they helped us reconceptualize evolutionary process as more than simply heritable morphological change, but an analysis of the units of heritability themselves, genetic variation. That is, evolution can be imagined as the study of the forces which shape changes in allele frequencies over time. This reduces a big domain down to a much simpler one.
Genetic variation is concrete currency with which one can track evolutionary process. Initially this was done via inferred correlations between marker traits and particular genes in breeding experiments. Ergo, the origins of the "the fly room".
But with the discovery of DNA as the physical substrate of genetic inheritance in the 1950s the scene was set for the revolution in molecular biology, which also touched evolutionary studies with the explosion of more powerful assays. Lewontin & Hubby's 1966 paper triggered a order of magnitude increase in our understanding of molecular evolution through both theory and results.
The theoretical side occurred in the form of the development of the neutral theory of molecular evolution, which also gave birth to the nearly neutral theory. Both of these theories hold that most of the variation with and between species on polymorphisms are due to random processes. In particular, genetic drift. As a null hypothesis neutrality was very dominant for the past generation, though in recent years some researchers are suggesting that selection has been undervalued as a parameter for various reasons.
Setting the live scientific debate, which continue to this day, one of the predictions of neutral theory is that the rate of evolution will depend only on the rate of mutation. More precisely, the rate of substitution of new mutations (where the allele goes from a single copy to fixation of ~100%) is proportional to the rate of mutation of new alleles. Population size doesn't matter.
The algebra behind this is straightforward.
[latexpage]
First, remember that the frequency of the a new mutation within a population is $\frac{1}{2N}$, where $N$ is the population size (the $2$ is because we're assuming diploid organisms with two gene copies). This is also the probability of fixation of a new mutation in a neutral scenario; it's probability is just proportional to its initial frequency (it's a random walk process between 0 and 1.0 proportions). The rate of mutations is defined by $\mu$, the number of expected mutations at a given site per generation (this is a pretty small value, for humans it's on the order of $10^{-8}$). Again, there are $2N$ gene copies, so you have $2N\mu$ to count the number of new mutations.
The probability of fixation of a new mutations multiplied by the number of new mutations is:
\[
\( \frac{1}{2N} \) \times 2N\mu = \mu
\]
So there you have it. The rate of fixation of these new mutations is just a function of the rate of mutation.
Simple formalisms like this have a lot more gnarly math that extend them and from which they derive. But they're often pretty useful to gain a general intuition of evolutionary processes. If you are genuinely curious, I would recommend Elements of Evolutionary Genetics. It's not quite a core dump, but it is a way you can borrow the brains of two of the best evolutionary geneticists of their generation.
Also, you will be able to answer the questions on my survey better the next time!
Posted in Genetics, UncategorizedTagged Evolutionary Genetics, Population Genetics9 Comments on Why the rate of evolution may only depend on mutation
Fisherianism in the genomic era
Posted on April 12, 2017 by Razib Khan
There are many things about R. A. Fisher that one could say. Professionally he was one of the founders of evolutionary genetics and statistics, and arguably the second greatest evolutionary biologist after Charles Darwin. With his work in the first few decades of the 20th century he reconciled the quantitative evolutionary framework of the school of biometry with mechanistic genetics, and formalized evolutionary theory in The Genetical Theory of Natural Selection.
He was also an asshole. This is clear in the major biography of him, R.A. Fisher: The Life of a Scientist. It was written by his daughter. But The Lady Tasting Tea: How Statistics Revolutionized Science in the Twentieth Century also seems to indicate he was a dick. And W. D. Hamilton's Narrow Roads of Gene Land portrays Fisher has rather cold and distant, despite the fact that Hamilton idolized him.
Notwithstanding his unpleasant personality, R. A. Fisher seems to have been a veritable mentat in his early years. Much of his thinking crystallized in the first few decades of the 20th century, when genetics was a new science and mathematical methods were being brought to bear on a host of topics. It would be decades until DNA was understood to be the substrate of heredity. Instead of deriving from molecular first principles which were simply not known in that day, Fisher and his colleagues constructed a theoretical formal edifice which drew upon patterns of inheritance that were evident in lineages of organisms that they could observe around them (Fisher had a mouse colony which he utilized now and then to vent his anger by crushing mice with his bare hands). Upon that observational scaffold they placed a sturdy superstructure of mathematical formality. That edifice has been surprisingly robust down to the present day.
One of Fisher's frameworks which still gives insight is the geometric model of the distribution of fitness of mutations. If an organism is near its optimum of fitness, than large jumps in any direction will reduce its fitness. In contrast, small jumps have some probability of getting closer to the optimum of fitness. In plainer language, mutations of large effect are bad, and mutations of small effect are not as bad.
A new paper in PNAS loops back to this framework, Determining the factors driving selective effects of new nonsynonymous mutations:
Our study addresses two fundamental questions regarding the effect of random mutations on fitness: First, do fitness effects differ between species when controlling for demographic effects? Second, what are the responsible biological factors? We show that amino acid-changing mutations in humans are, on average, more deleterious than mutations in Drosophila. We demonstrate that the only theoretical model that is fully consistent with our results is Fisher's geometrical model. This result indicates that species complexity, as well as distance of the population to the fitness optimum, modulated by long-term population size, are the key drivers of the fitness effects of new amino acid mutations. Other factors, like protein stability and mutational robustness, do not play a dominant role.
In the title of the paper itself is something that would have been alien to Fisher's understanding when he formulated his geometric model: the term "nonsynonymous" to refer to mutations which change the amino acid corresponding to the triplet codon. The paper is understandably larded with terminology from the post-DNA and post-genomic era, and yet comes to the conclusion that a nearly blind statistical geneticist from about a century ago correctly adduced the nature of mutation's affects on fitness in organisms.
The authors focused on two primary species which different histories, but well characterized in the evolutionary genomic literature: humans and Drosophila. The models they tested are as follows:
Basically they checked the empirical distribution of the site frequency spectra (SFS) of the nonsynonymous variants against expected outcomes based on particular details of demographics, which were inferred from synonymous variation. Drosophila have effective population sizes orders of magnitude larger than humans, so if that is not taken into account, then the results will be off. There are also a bunch of simulations in the paper to check for robustness of their results, and they also caveat the conclusion with admissions that other models besides the Fisherian one may play some role in their focal species, and more in other taxa. A lot of this strikes me as accruing through the review process, and I don't have the time to replicate all the details to confirm their results, though I hope some of the reviewers did so (again, I suspect that the reviewers were demanding some of these checks, so they definitely should have in my opinion).
In the Fisherian model more complex organisms are more fine-tuned due topleiotropy and other such dynamics. So new mutations are more likely to deviate away from the optimum. This is the major finding that they confirmed. What does "complex" mean? The Drosophila genome is less than 10% of the human genome's size, but the migratory locust has twice as large a genome as humans, while wheat has a sequence more than five times as large. But organism to organism, it does seem that Drosophila has less complexity than humans. And they checked with other organisms besides their two focal ones…though the genomes there are not as complete presumably.
As I indicated above, the authors believe they've checked for factors such as background selection, which may confound selection coefficients on specific mutations. The paper is interesting as much for the fact that it illustrates how powerful analytic techniques developed in a pre-DNA era were. Some of the models above are mechanistic, and require a certain understanding of the nature of molecular processes. And yet they don't seem as predictive as a more abstract framework!
Citation: Christian D. Huber, Bernard Y. Kim, Clare D. Marsden, and Kirk E. Lohmueller, Determining the factors driving selective effects of new nonsynonymous mutations PNAS 2017 ; published ahead of print April 11, 2017, doi:10.1073/pnas.1619508114
Posted in GeneticsTagged Evolutionary Genetics1 Comment on Fisherianism in the genomic era
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Asking for help, clarification, or responding to other answers. The base rate fallacy is a tendency to focus on specific information over general probabilities. 1 / 50.95 ≈ 0.019627. How do people recognise the frequency of a played note? Famous quotes containing the words fallacy, base and/or rate: " It would be a fallacy to deduce that the slow writer necessarily comes up with superior work. In reality, however, the correct answer was just below 2%. A generic information about how frequently an event occurs naturally. Commenting on these results, the Infectious Disease Society of America stated that: "A positive test result is more likely a false-positive result than a true positive result." This is particularly dangerous since it could lead to potentially susceptible hosts believing they have been infected with coronavirus, and acting as if they have immunity, when this is not the case. 2) × (. Table I. Additionally, a recent study published in the journal Public Health revealed that 16% of positive results would be false even when using a test with 99% sensitivity and specificity. At the normative level, the base rate fallacy should be rejected because few tasks map unambiguously into the narrow framework that is held up as the standard of good decision making. Plausibility of an Implausible First Contact, Variant: Skills with Different Abilities confuses me. Therefore this suspect must be guilty. The confidence that we should have in an antibody test depends on the base rate of the coronavirus, a key factor which is often ignored. In a classic and widely-referenced study, the following question was put to 60 students and staff at Harvard Medical School. The inability of intelligent minds to apply simple mathematical reasoning and arrive at the correct value of 2% clearly demonstrates the aforementioned base rate fallacy. Therefore, the probability that one of the drivers among the 1 + 49.95 = 50.95 positive test results really is drunk is. 1. In a city of 1 million inhabitants there are 100 known terrorists and 999,900 non-terrorists. If a randomly selected person tests positive what is the probability that the person actually has the disease?". The Base Rate Fallacy: why we should be cautious with anti-body testing results. In the typical clinical scenario in which the base rate of the disorder in question is below 50% … site design / logo © 2020 Stack Exchange Inc; user contributions licensed under cc by-sa. By clicking "Post Your Answer", you agree to our terms of service, privacy policy and cookie policy. The positive predictive value (PPV; the probability that a drug actually working, given that we rejected the null hypothesis that it had no effect—i.e. We call this the "positive predictive value" (PPV) of a test. Diagnostic tests 2: predictive values. how does this apply to a single hypothesis test performed on a single sample? Despite this, antibody tests remain an important tool in the fight against coronavirus and we should therefore encourage greater access to them; healthy people who have antibodies in their blood and have tested positive for the virus in the past (but are now symptom-free) can donate blood plasma, which may be used as a possible treatment for COVID-19. Put another way, there is an almost 70% probability in that instance that the immunological assay will falsely indicate a person has antibodies. Your email address will not be published. The positive predictive value (PPV; the probability that a drug actually working, given that we rejected the null hypothesis that it had no effect—i.e. 5) (. The possibility that a screening program may not improve upon random selection is reviewed, as is the possibility that sequential screening might be useful. Altman, D. G. and Bland, J. M. (1994). The reason for this is a simple matter of statistics. Even so, overlooking this fact is one of the most common decision-making errors, so much so that it has its own name – the base rate fallacy. MathJax reference. Generally, it is known as the posterior probability. Geeky Definition of Base Rate Fallacy: The Base Rate Fallacy is an error in reasoning which occurs when someone reaches a conclusion that fails to account for an earlier premise – usually a base rate, a probability or some other statistic. Given the scale with which screening might occur, the implications of a problem known as the base rate fallacy need to be considered.The concepts of sensitivity and specificity, positive and negative predictive value, and the base rate fallacy are discussed. how does the base rate fallacy creep in a single hypothesis test? Almost half said 95%, with the average answer being 56%. Why most published research findings are false. Empirical research on base rate usage has been domi nated by the perspective that people ignore base rates and that it is an errorto do so. © 2020 Copyright The Boar. Required fields are marked *. Even deploying more accurate tests cannot change the statistical reality when the base rate of infection is very low. What happens when the agent faces a state that never before encountered? Base rate fallacy – making a probability judgment based on conditional probabilities, without taking into account the effect of prior probabilities. The base rate (or disease prevalence) is the actual amount of COVID-19 infection in a known population. Statistical significance test for averages of correlation coefficients. Example. I have clarified the contents of the table in a new paragraph. Many people who answer the question focus on the 5% false positive rate and exclude the general statistic that 999 out of 1000 students are innocent. There seems to be scant relationship between prolificness and quality. Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. lowering the prevalence lowers also the number of samples that turn out to be True Positives? So, if the null hypothesis is true, and the base rate is low, the $p$ value being small enough to reject, even if it is very small, means that you are probably seeing a false positive. Diagnostic tests 1: sensitivity and specificity. the probability that we made a true rejection) is sensitive to the base rate of cancer drugs that actually work. In these experiments, I'll look for p<0.05 gains over a placebo, demonstrating that the drug has a significant benefit. Base rate fallacy, or base rate neglect, is a cognitive error whereby too little weight is placed on the base, or original rate, of possibility (e.g., the probability of A given B). these findings mean that we are all at risk of getting infected and spreading the virus, even if we've had a positive antibody test. For manyyears, the so-called base rate fallacy, with its distinctive name and arsenal of catchy If before collecting your data you believe it is extremely unlikely that your alternative hypothesis is true, then it's ok to still be skeptical of the alternative even after seeing a low p-value. The Bayes Theorem is named after Reverend Thomas Bayes (1701–1761) whose manuscript reflected his solution to the inverse probability problem: computing the posterior conditional probability of an event given known prior probabilities related to the event and relevant conditions. PPV = positive predictive value; NPV = negative predictive value. In studies investigating clinicians' use of base rate information, participants typically overestimate PPV and often respond erroneously that the predictive value of a test is equivalent to the test's sensitivity or specificity (e.g., Casscells, Schoenberger, & Graboys, 1978; Heller, Saltzstein, & Caspe, 1992). The first section of this article provides some intuition on base rate fallacy with p-values. The concepts of sensitivity and specificity, positive and negative predictive value, and the base rate fallacy are discussed. Open in new tab. That's right, you have to know how many people test positive in the population as a whole before you can judge the predictive value of a test. Given the scale with which screening might occur, the implications of a problem known as the base rate fallacy need to be considered.The concepts of sensitivity and specificity, positive and negative predictive value, and the base rate fallacy are discussed. Although immunological assays appear to offer a promising path forward, does a positive test mean you should feel confident to work, shop, and socialise without getting sick or infecting others? Criminal Intent Prescreening and the Base Rate Fallacy. If so, how do they cope with it? But the predictive value of an antibody test with 90 percent accuracy could be as low as 32 percent if the base rate of infection in the population is 5 percent. This essay uses that argument to demonstrate why the TSA's FAST program is useless:. Powered by Tom, Hamish & Aaron. "I think we're going to see [antibody testing] explode," commented Mitchell Grayson, chief of allergy and immunology at Nationwide Children's Hospital and Ohio State University in Columbus. In general, what do each of the boxes contain? Thanks for contributing an answer to Cross Validated! Is there a general solution to the problem of "sudden unexpected bursts of errors" in software? The base rate fallacy is also known as base rate neglect or base rate bias. If the base rate is lowered (that vertical line shifts left), you can see that true positives shrink relative to false positives and therefore the PPV gets smaller (i.e. The possibility that a screening program may not improve upon random selection is reviewed, as is the possibility that sequential screening might be useful. The lower prevalence there is of a trait in a studied population, the greater the chance that a test will return a false positive. In a city of 1 million inhabitants there are 100 known terrorists and 999,900 non-terrorists. Making statements based on opinion; back them up with references or personal experience. (2005). Learning from the flaws in the NHST and p-values. "In other words, less than half of those testing positive will truly have antibodies," according to the agency. In the case of a single hypothesis test: (1) Reject H$_{0}$ height of men equals height of women; (2) pose the questions (i) what is the prevalence of. Is p-value essentially useless and dangerous to use? overlooking this fact is one of the most common decision-making errors, so much so that it has its own name – the base rate fallacy. The base rate fallacy shows us that false positives are much more likely than you'd expect from a p < 0.05 criterion for significance. Suppose I flip a fair coin 10 times and he correctly guesses every time, a p-value of about .001. @redblackbit As an example, suppose I am interested in trying to determine whether or not my friend has ESP. The positive predictive value is sometimes called the positive predictive agreement, and the negative predictive value is sometimes called the negative predictive agreement. Whether you think the UK is reopening too fast or too slowly, almost everyone agrees that antibody testing is critical to the next phase of our coronavirus existence. The Affordable Care Act has stimulated interest in screening for psychological problems in primary care. The PPV and NPV describe the performance of a diagnostic test or other statistical measure. What led NASA et al. 10 Here, this fallacy is described as "people's tendency to ignore base rates in favor of, e.g., individuating information (when such is available), rather than integrate the two" (p. 211). To subscribe to this RSS feed, copy and paste this URL into your RSS reader. BMJ, 308:1552. What is the difference between policy and consensus when it comes to a Bitcoin Core node validating scripts? In the table, the null hypothesis being true is the left column, and $\alpha$ (your willingness to reject the null when the null is true) is the number of false negatives over the total truly negative (or one minus the specificity of the test). 5) + ( 8) × (. PPV is the number of true positives over the total testing positive. to decide the ISS should be a zero-g station when the massive negative health and quality of life impacts of zero-g were known? Information and translations of base rate fallacy in the most comprehensive dictionary definitions resource on the web. Use MathJax to format equations. The whole argument makes sense to me but I am not sure if I entirely understand how it relates to a single hypothesis test. Because the base rate of effective cancer drugs is so low – only 10% of our hundred trial drugs actually work – most of the tested drugs do not work, and we have many opportunities for false positives. Does a regular (outlet) fan work for drying the bathroom? The lower prevalence there is of a trait in a studied population, the greater the chance that a test will return a false positive. There is a test to detect this disease. Why did George Lucas ban David Prowse (actor of Darth Vader) from appearing at sci-fi conventions? Put another way, there is an almost 70 percent probability in that case that the test will falsely indicate a person has antibodies. At this same disease prevalence, the CDC found that a test with 90% sensitivity and 95% specificity would yield a positive predictive value (PPV) of 49%. It then calculates a hundred hypothesis tests and concludes that. The STANDS4 Network ... are used in place of positive predictive value and negative predictive value, which depend on both the test and the baseline prevalence of event. This is because the "base rate" of COVID is higher among the population of people with symptoms than people without. Consider the $2\times2$ table below, where testing positive or negative corresponds to rejecting or not rejecting H$_{0}$, and the truth being positive or negative means that H$_{0}$ is false or true, respectively. Methods The concepts of sensitivity and specificity, positive and negative predictive value, and the base rate fallacy are discussed. Login . By contrast, the $p$-value is the probability of observing your data, if in fact the null hypothesis is true. Is there a way to notate the repeat of a larger section that itself has repeats in it? Is p-value also the false discovery rate? Altman, D. G. and Bland, J. M. (1994). The margins sum the rows and columns, and the sum of row margins equals the sum of column margins equals the total number of tests. METHODS: The concepts of sensitivity and specificity, positive and negative predictive value, and the base rate fallacy are discussed. Non-nested std::deque and std::list Generator Function for arithmetic_mean Function Testing in C++. Each quadrant contains the counts of the four possibilities under these conditions: the number of true positive tests, number of true negative tests, number of false positive tests, and number of false negative tests. Probability of correctly predicting disorder= (base rate of disorder) × (true positive rate) (base rate of disorder × true positive rate) + (1- base rate of disorder) × (false positive rate) For this example, the result is: Probability of correctly predicting disorder = (. " —Fannie Hurst (1889–1968) " Time, force, and death Do to this body what extremes you can, The truncation value is usually 40 but I have seen 45. If Jedi weren't allowed to maintain romantic relationships, why is it stressed so much that the Force runs strong in the Skywalker family? just because you rejected the null hypothesis for a drug means that you still probably made a false rejection). Do PhD students sometimes abandon their original research idea? Cross Validated is a question and answer site for people interested in statistics, machine learning, data analysis, data mining, and data visualization. Koehler: Base rate fallacy superiority of the nonnative rule reduces to an untested empirical claim. In a notional population of 100,000 individuals, 950 people will therefore be incorrectly informed they have had the infection. I.e. In a notional population of 100,000 individuals, 950 people will therefore be incorrectly informed they have had the infection. Only ten of these drugs actually work, but I don't know which; I must perform experiments to find them. Your email address will not be published. Suppose I am testing a hundred potential cancer medications. The same would be true of essential workers, people who have partners who previously tested positive, etc. Say we have setup a hypothesis test to check if the average height differs between males and females for a specific sample we collected. The test is 100% accurate for people who have the disease and is 95% accurate for those who don't (this means that 5% of people who do not have the disease will be wrongly diagnosed as having it). 2) × (. "One in a thousand people have a prevalence for a particular heart disease. It only takes a minute to sign up. This simple fact is essential to understanding the accuracy of serology-based testing. The base rate probability of one random inhabitant of the city being a terrorist is thus 0.0001 and the base rate probability of a random inhabitant being a non-terrorist is 0.9999. It's called the base rate fallacy and it's counter-intuitive, to say the least. @redblackbit I believe the intuition you may be missing regarding individual hypothesis tests is to think about your prior probabilities regarding which of the hypotheses is true. At this same disease prevalence, the CDC found that a test with 90% sensitivity and 95% specificity would yield a positive predictive value (PPV) of 49%. Unexplained behavior of char array after using `deserializeJson`. It was published posthumously with significant contributions by R. Price and later rediscovered and extended by Pierre-Simon Laplace in 1774. "Question closed" notifications experiment results and graduation, MAINTENANCE WARNING: Possible downtime early morning Dec 2, 4, and 9 UTC…. In particular, it uses as example a cancer test. Shuster is trying to have his cake and eat it in his criticism of statistics in clinical practice.1 He highlights that breast cancer screening is a "bad" test (by which I think he means it has a low positive predictive value), but it is precisely because we can calculate this probability that we know the relative utility of the test. Effects of Different Levels of Base Rate, Sensitivity, and Specificity on Classification Accuracy. revealed that 16% of positive results would be false even when using a test with 99% sensitivity and specificity. Does false discovery rate depend on the p-value or only on the alpha level? 999 drivers are not drunk, and among those drivers there are 5% false positive test results, so there are 49.95 false positive test results. Base rates are also used more when they are reliable and relatively more diagnostic than available individuating information. Even deploying more accurate tests cannot change the statistical reality when the base rate of infection is very low. prevalence), then the table above shows half of tests of cancer drugs truly rejecting H$_{0}$. At this same disease prevalence, the CDC found that a test with 90% sensitivity and 95% specificity would yield a positive predictive value (PPV) of 49%. PLoS Medicine, 2(8):0696–0701. But this is another example of the base rate fallacy. The base rate fallacy has to do with specialization to different populations, which does not capture a broader misconception that high accuracy implies both low false positive and low false negative rates. rev 2020.12.2.38106, The best answers are voted up and rise to the top, Cross Validated works best with JavaScript enabled, Start here for a quick overview of the site, Detailed answers to any questions you might have, Discuss the workings and policies of this site, Learn more about Stack Overflow the company, Learn more about hiring developers or posting ads with us, I suggest retitling to something like "p-value and the base rate fallacy". In the U.S., for example, this appears to be between five and 15%. Additionally, a recent study published in the journal. Why is training regarding the loss of RAIM given so much more emphasis than training regarding the loss of SBAS? Put simply, these findings mean that we are all at risk of getting infected and spreading the virus, even if we've had a positive antibody test. False negative rate of 7.5% The prosecutor's fallacy would say that since the false positive rate is 0.1%, the positive test means that the suspect was 99.9% likely to have actually committed the crime (or at least, something close to this amount). How to professionally oppose a potential hire that management asked for an opinion on based on prior work experience? If you imagine that the area in each quadrant of the table is proportional to the number in each quadrant, and further, imagine that the vertical line down the center of the $2 \times2$ table represents the base rate (e.g. Typically specificity, 1- the false positive rate, is reported as 99.9%, not 100%, when there are no false positives. the probability that we made a true rejection) is sensitive to the base rate of cancer drugs that actually work. I am skeptical, so I think there is an extremely small possibility that my friend has ESP. I.e. On the surface, this makes sense – after all, a test accuracy above 90% is fairly high. The base rate fallacy, ... are used in place of positive predictive value and negative predictive value (which depend on both the test and the baseline prevalence of event). BMJ, 309:102. I.e. By using our site, you acknowledge that you have read and understand our Cookie Policy, Privacy Policy, and our Terms of Service. The Affordable Care Act has stimulated interest in screening for psychological problems in primary care. [6] Conjunction fallacy – the assumption that an outcome simultaneously satisfying multiple conditions is more probable than … Base-rate Fallacy Example. Why is frequency not measured in db in bode's plot? The cut-off for a yes/no test is determined based on the validation, typically a number near but below the truncation value. Confronted with this data, I still believe there is a low chance that my friend has ESP because my prior probability was so low. To learn more, see our tips on writing great answers. Serology tests could provide epidemiologists with vital data on how COVID-19 is spreading through a community, and also lead to the issuing of "immunity passports" for individuals who have beaten back the infection. Either my friend has ESP, which is why he was able to correctly predict all 10 flips, or my friend doesn't have ESP and was lucky. The correct answer to the question, 0.0909, is called in medical science the positive-predictive value of the test. Now, one of two things happened. In the context of coronavirus infection, the predictive value of a test with 90% accuracy could be as low as 32% if the true population prevalence is 5%. Most modern research doesn't make one significance test, however; modern studies compare the effects of a variety of factors, seeking to … However, it is important to remember that a highly accurate test may not be as comforting as it first appears, and therefore the results of such assays should always be viewed with thoughtful reflection. A lower prevalence (of drugs with true effects out of all drugs) will decrease the number of true positives, See my correction to the paragraph following the table. "In other words, less than half of those testing positive will truly have antibodies," according to the agency. Another early explanation of the base rate fallacy can be found in Maya Bar-Hillel's 1980 paper, "The base-rate fallacy in probability judgments". Ioannidis, J. P. A. The samples? When evaluating the probability of an event―for instance, diagnosing a disease, there are two types of information that may be available. In case it is still not completely clear that the base rate fallacy is indeed a fallacy, lets employ a thought experiment with an extreme case. A high result can be interpreted as indicating the accuracy of such a statistic. Abstract. But the predictive value of an antibody test with 90 percent accuracy could be as low as 32 percent if the base rate of infection in the population is 5 percent.
base rate fallacy positive predictive value
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\begin{document}
\title[Liouvillle equation with a singular source]{Blow-up phenomena for the Liouville equation \\with a singular source of integer multiplicity} \author{Teresa D'Aprile} \address[Teresa D'Aprile] {Dipartimento di Matematica, Universit\`a di Roma ``Tor Vergata", via della Ricerca Scientifica 1, 00133 Roma, Italy.} \email{[email protected]}
\begin{abstract} We are concerned with the existence of blowing-up solutions
to the following boundary value problem $$-\Delta u= \lambda a(x) e^u-4\pi N \delta_0\;\hbox{ in } \Omega,\quad u=0 \;\hbox{ on }\partial \Omega,$$ where $\Omega$ is a smooth and bounded domain in $\mathbb{R}^2$ such that $0\in\Omega$, $a(x)$ is a positive smooth function, $N$ is a positive integer and $\lambda>0$ is a small parameter. Here $\delta_0$ defines the Dirac measure with pole at $0$. We find conditions on the function $a$ and on the domain $\Omega$ under which there exists a solution $u_\lambda$ blowing up at $0$ and
satisfying $\lambda\int_\Omega a(x)e^{u_\lambda} \to 8\pi(N+1)$ as $\lambda\to 0^+$.
\noindent {\bf Mathematics Subject Classification 2010:} 35J20, 35J57, 35J61
\noindent {\bf Keywords:} singular Liouville equation, blowing-up solutions, perturbation methods
\end{abstract}
\maketitle
\section{Introduction}
Let $\Omega$ be a bounded domain in $\mathbb{R}^2$ with a smooth boundary containing the origin. In this paper we consider the following Liouville equation with Dirac mass measure
\begin{equation}\label{eq}
\left\{
\begin{aligned}&- \Delta u = \lambda a(x) e^u-4\pi N \delta_0& \hbox{ in }& \Omega,\\
& \ u=0 & \hbox{ on }& \partial \Omega.
\end{aligned}
\right. \end{equation} Here $\lambda$ is a positive small parameter, $\delta_0$ denotes Dirac mass supported at $0$, $a$ is a smooth function satisfying $\inf_\Omega a(x)>0$ and $N $ is a positive integer.
Problem \eqref{eq} is motivated by its links with the modeling of physical phenomena. In particular, \eqref{eq} arises in the study of vortices in a planar model of Euler flows (see \cite{delespomu}, \cite{bapi}). In vortex theory the interest in constructing \textit{blowing-up} solutions is related to relevant physical properties, in particular the presence of vortices with a strongly localised electromagnetic field.
The asymptotic behaviour of family of blowing up solutions
can be referred to the papers \cite{bapa}, \cite{breme}, \cite{lisha}, \cite{mawe}, \cite{nasu}, \cite{su} for the regular problem, i.e. when $N=0$. An extension to the singular case $N>0$
is contained in \cite{bachenlita}-\cite{bata}.
The analysis of the blowing-up behaviour at points away from $0$ actually is very similar to the asymptotic analysis arising in the regular case which has been pursued with success and, at the present time, an accurate description of the concentration phenomenon is available. Precisely, the
analysis in the above works yields that if $u_\lambda$ is an unbounded family of solutions of \eqref{eq} for which $\lambda\int_\Omega a(x) e^{u_\lambda} $ is uniformly bounded and $u_\lambda$ is uniformly bounded in a neighborhood of $0$, then, up to a subsequence, there is an integer $m\geq 1$ such that
\begin{equation}\label{issue}\lambda\int_\Omega a(x)e^{u_\lambda}dx\to 8\pi m\hbox{ as }\lambda\to 0^+.\end{equation} Moreover there are points $\xi_1^\lambda,\ldots, \xi_m^\lambda\in\Omega$ which remain uniformly distant from the boundary $\partial\Omega$, from $0$ and from one another such that \begin{equation}\label{issue11}\lambda a(x)e^{u_\lambda}-8\pi\sum_{j=1}^m \delta_{\xi_j^\lambda}\to 0\end{equation} in the measure sense.
Also the location of the blowing-up points is well understood when concentration occurs away from $0$.
Indeed, in \cite{nasu} and \cite{su} it is established that the $m$-tuple $(\xi_1^\lambda,\ldots, \xi_m^\lambda)$ converges, up to a subsequence, to a critical point of the functional
\begin{equation}\label{natur}\frac12\sum_{j=1}^mH(\xi_j,\xi_j)+\frac12\sum_{j,h=1\atop j\neq h}^mG(\xi_j,\xi_h)-\frac{N}{2}\sum_{j=1}^mG(\xi_j,0).\end{equation}
Here
$G(x,y)$ is the Green's function of $-\Delta$ over $\Omega$ under Dirichlet boundary conditions and $H(x,y)$ denotes its regular part:
$$H(x,y)=G(x,y)-\frac{1}{2\pi}\log\frac{1}{|x-y|}.$$ The above description of blowing-up behaviour continues to work if we are in the presence of multiples singularities $\sum_i N_i\delta_{p_i}$ in \eqref{eq}, provided that we substitute the term $ \frac{N}{2}\sum_jG(\xi_j,0)$ by $\sum_{i}\frac{N_i}{2}\sum_jG(\xi_j,p_i)$ in \eqref{natur}.
\
The reciprocal issue, namely the existence of positive solutions with the property \eqref{issue11}, has been addressed for the regular case $N=0$ first in \cite{we} in the case of a single point of concentration (i.e. $m=1$), later generalised to the case of multiple concentration associated to any nondegenerate critical point of the functional \eqref{natur} (\cite{bapa}, \cite{chenlin}) or, more generally, to a any \textit{topologically nontrivial} critical point (\cite{delkomu}-\cite{espogropi}).
In particular, still for $N=0$, a family of solutions $u_\lambda$ concentrating at $m$-tuple of points as $\lambda\to 0^+$ has been found in some special cases: for any $m\geq 1$, provided that $\Omega$ is not simply connected (\cite{delkomu}), and for $m\in\{1,\ldots, h\}$ if $\Omega$ is a $h$-dumbell with thin handles (\cite{espogropi}). We mention that functionals similar to \eqref{natur} occur to detect multiple-bubbling solutions in different contexts, see \cite{bapiwe}, \cite{espomupi1}, \cite{espomupi2}, \cite{weiyezhou} for other related singularly perturbed problems.
In the singular case $N>0$ solutions which concentrate in the measure sense at $m$ distinct points away from $0$ have been built in \cite{delkomu} provided that $m<1+N$. This result has been extended in \cite{daprile} to the case of multiple singular sources: in particular it is showed that, under suitable restrictions on the weights, if several sources exist then the more involved topology generates a large number of blow-up solutions.
We point out that in all the above results
concentration occurs
at points different from the location of the source. The problem of finding solutions with additional concentration around the source is of different nature. In case they exist, the blowing-up at the singularity provides an additional contribution of $8\pi(1+N)\delta_0$ in the limit \eqref{issue}, see \cite{bachenlita}, \cite{bata}, \cite{espo}, \cite{ta1}, \cite{ta2}. More precisely the asymptotic analysis in the general case can be formulated as follows: if $u_\lambda$ is an unbounded family of solutions of \eqref{eq} for which $\lambda\int_\Omega a(x)e^{u_\lambda} $ is uniformly bounded and $u_\lambda$ is unbounded in any neighborhood of $0$, then, up to a subsequence, there is an integer $m\geq 0$ such that
$$\lambda\int_\Omega a(x)e^{u_\lambda}dx\to 8\pi m+8\pi(N+1)\hbox{ as }\lambda\to 0^+.$$ Moreover there are $m$ distinct points $\xi_1,\ldots, \xi_m\in\Omega\setminus\{0\}$ such that, up to subsequence, \begin{equation}\label{issue1}\lambda a(x)e^{u_\lambda}\to 8\pi\sum_{j=1}^m \delta_{\xi_j}+8\pi(N+1)\delta_0\end{equation} in the measure sense. We mention that also in this case the analysis can be generalized to any number of sources. Moreover, under some extra assumptions it is possible to define a functional which replaces \eqref{natur} in locating the points $\xi_j$ where the concentration occurs, anyway to avoid technicalities we will not go into any further detail (see \cite{espo}).
The question on the existence of solutions to \eqref{eq} concentrating at $0$ is far from being completely settled. Indeed only partial results are known: in \cite{espo} the construction of solutions concentrating at $0$ is carried out provided that $N\in(0,+\infty)\setminus \mathbb{N}$.
To our knowledge, the only paper dealing with the case $N\in\mathbb{N}$ is \cite{delespomu}, where, for any fixed positive integer $N$, it is proved the existence of a solution to \eqref{eq} with $a=1$ and $\delta_0$ replaced by $\delta_{p_\lambda}$ for a suitable $p_\lambda\in \Omega$ with $N+1$ blowing up points at the vertices of a sufficiently tiny regular polygon centered in $p_\lambda$; moreover $p_\lambda$ lies uniformly away from the boundary $\partial \Omega$ but its location is determined by the geometry of the domain in an $\lambda-$dependent way and does not seem possible to be prescribed arbitrarily as in \cite{espo}.
The case $N\in\mathbb{N}$ is more difficult to treat, and at the same time the most relevant to physical applications. Indeed, in vortex theory the number $N$ represents vortex multiplicity, so that in that context the most interesting case is precisely that in which it is a positive integer. The difference between the case $N\in \mathbb{N}$ and $N\not\in \mathbb{N}$ is analitically essential. Indeed, as usual in problems involving small parameters and concentration phenomena like \eqref{eq}, after suitable rescaling of the blowing-up around a concentration point one sees a limiting equation. More specifically, as we will see in Section 2, we can associate to \eqref{eq} the limiting problem of Liouville type \eqref{limit} which will play a crucial role in the construction of solutions blowing up at $0$ as $\lambda\to 0^+$; if $N\in\mathbb{N}$, \eqref{limit} admits a larger class of finite mass solutions with respect to the case $N\not\in\mathbb{N}$ since the family of all solutions extends to one carrying an extra parameter $b\in\mathbb{R}^2$ (see \cite{Pratar}).
In this paper we are interested in finding conditions on the potential $a$ and on the domain $\Omega$ under which there exists a solution $u_\lambda$ blowing up at $0$. Even though finding general conditions is a notoriously open issue, our analysis reveals that the interplay between the geometry of the domain, which is described in terms of the Robin function $H(x,x)$, and the potential $a$ plays a crucial role. More specifically our conditions involve the first and the second derivative of $a$ and $H(x,x)$. \
Now we pass to provide the exact formulations of our results. In the following we will assume that $$a\in C(\overline\Omega)\cap C^2(\Omega)\;\;\hbox{ and }\;\;\inf_\Omega a(x)>0.$$ Moreover, after suitably rotating the coordinate system, we may assume that in a small neighborhood of $0$ the following expansion holds:
$$a(x)=a(0)+\langle\nabla a(0), x\rangle +\frac{a_{11}x_1^2+a_{22}x_2^2}{2}+o(|x|^2)\hbox{ as} x\to 0,$$ where $a_{ii}=\frac{\partial^2a}{\partial x_i^2}(0)$.
\begin{thm}\label{th1} Let $N\geq 2$, $N\in\mathbb{N}$. Assume that\footnote{Here $\nabla_xH(0,0)$ denotes the gradient of the function $x\mapsto H(x,0)$ at $0$.}
\begin{equation}\label{assurd0}\nabla a(0)+4\pi(N+2)a(0)\nabla_xH(0,0)=0, \quad \Delta a(0)\neq 16\pi^2(N+2)^2a(0)|\nabla_xH(0,0)|^2. \end{equation} Then, for $\lambda$ sufficiently small, the problem \eqref{eq} has a family of solutions $u_\lambda$ blowing up at $0$ as $\lambda\to 0^+$. More precisely the following holds: \begin{equation}\label{the1}\lambda a(x)e^{u_\lambda} dx \to 8\pi (1+N)\delta_0\end{equation} in the measure sense. More precisely $u_\lambda$ satisfies
\begin{equation}\label{the3}u_\lambda=
4\pi (N+2)G(x,0)+o(1)\end{equation} away from $0$. \end{thm} In the particular case when $0$ is a critical point of the potential $a$ and of the Robin function, we get the existence of a solution blowing up at $0$ provided that $\Delta a(0)\neq 0$.
\begin{cor}\label{th1cor} Let $N\geq 2$, $N\in\mathbb{N}$. Assume that $$\nabla a(0)=\nabla_xH(0,0)=0, \quad \Delta a(0)\neq 0. $$ Then, for $\lambda$ sufficiently small the problem \eqref{eq} has a family of solutions $u_\lambda$ blowing up at $0$ as $\lambda\to 0^+$. More precisely $u_\lambda$ satisfies \eqref{the1}-\eqref{the3} of Theorem \ref{th1}. \end{cor}
The case $N=1$ is considered in a separate theorem since the result requires different assumptions; the above result continues to hold in symmetric domains under an additional relation involving the second derivatives of $a$.
\begin{thm}\label{th2} Let $N=1$. Assume that $\Omega$ is $\ell$-symmetric for some $\ell\geq3$, i.e. \begin{equation}\label{assurd} x\in\Omega\Longleftrightarrow e^{{\rm i}\frac{2\pi}{\ell}} x\in\Omega\end{equation} and, in addition, \begin{equation}\label{assurd2}\nabla a(0) =0, \quad \Delta a(0)\neq 0, \quad a_{11}=a_{22}.\end{equation} Then, for $\lambda$ sufficiently small the problem \eqref{eq} has a family of solutions $u_\lambda$ blowing up at $0$ as $\lambda\to 0^+$. More precisely $u_\lambda$ satisfies \eqref{the1}-\eqref{the3} of Theorem \ref{th1}. \end{thm}
We point out that in symmetric domains the center of symmetry $0$ is a critical point of the Robin function, so the condition $\nabla_xH(0,0)=0$ is automatically satisfied. Assumptions \eqref{assurd}-\eqref{assurd2} are obviously satisfied if $\Omega$ is a ball centered at $0$ and $a$ is a radially symmetric potential with a nondegenerate critical point at $0$.
\
The proofs use singular perturbation methods. Roughly speaking, the first step consists in the construction of an approximate solution, which should turn out to be precise enough. In view of the expected asymptotic behavior, the shape of such approximate solution will resemble a \textit{bubble} of the form \eqref{bubble} with a suitable choice of the parameter $\delta=\delta(\lambda,b)$. Then we look for a solution to \eqref{eq} in a small neighborhood of the first approximation. As quite standard in singular perturbation theory, a crucial ingredient is nondegeneracy of the explicit family of solutions of the limiting Liouville problem \eqref{limit},
in the sense that all bounded elements in the kernel of the linearization correspond to variations along the parameters of the family, as established in \cite{delespomu}. This allows us to study the invertibility of the linearized operator associated to the problem \eqref{eq} under suitable orthogonality conditions. Next we introduce an intermediate problem and a fixed point argument will provide a solution for an auxiliary equation, which turns out to be solvable for any choice of $b$. Finally we test the auxiliary equation on the elements of the kernel of the linearized operator and we find out that,
in order to find an \textit{exact} solution of \eqref{eq}, the parameter $b$ should be a zero for
a \textit{reduced} finite dimensional map.
\
The rest of the paper is organized as follows. Section 2 is devoted to some preliminary results, notation, and the definition of the approximating solution. Moreover, a more general version of Theorems \ref{th1}-\ref{th2} is stated there (see Theorems \ref{main1}-\ref{main2}). The error up to which the approximating solution solves problem \eqref{eq} is estimated in Section 3. In Section 4 we prove the solvability of the linearized problem. Section 5 considers the solvability of an auxiliary problem by a contraction argument.. Finally, in Section 6, we prove the existence results and we conclude the proof of Theorems \ref{th1}-\ref{th2}. In Appendix A and Appendix B we collect some results, most of them well-known,
which are usually referred to throughout the paper.
\section{Preliminaries and statement of the main results} We are going to provide an equivalent formulation of problem \eqref{eq} and Theorems \ref{th1}-\ref{th2}. Indeed, let us observe that, setting $v$ the regular part of $u$, namely \begin{equation}\label{chva}v= u+4\pi (\alpha-1) G(x, 0),\quad \alpha=N+1,\end{equation} problem \eqref{eq} is then equivalent to solving the following (regular) boundary value problem
\begin{equation}\label{proreg}\left\{\begin{aligned} & -\Delta v=\lambda V(x)|x|^{2(\alpha-1)}e^v&\hbox{ in }&\Omega\\ &v=0&\hbox{ on }&\partial \Omega\end{aligned}\right.,\end{equation} where $V(x)$ is the new potential \begin{equation}\label{aaa}V(x)=a(x)e^{-4\pi (\alpha-1) H(x,0)} .\end{equation} Here $G$ and $H$ are Green's function and its regular part as defined in the introduction. This problem is actually variational. Indeed, let us consider the following energy functional associated with \eqref{proreg}:
$$J(v)=\frac12\int_\Omega |\nabla v|^2 dx-\lambda \int_\Omega V(x)|x|^{2(\alpha-1)}e^vdx,\quad v\in H^1_0(\Omega).$$ Then the following Moser-Trudinger inequality (\cite{Moe, Tru}) guarantees that $J$ is of class $C^1(H^1_0(\Omega))$ and solutions of \eqref{proreg} correspond to critical points of $J$.
\begin{lemma}\label{tmt} There exists $C>0$ such that for any bounded domain $\Omega$ in $\mathbb{R}^2$
$$\int_\Omega e^{\frac{4\pi u^2}{\|u\|^2}}dx\le C |\Omega|\quad \forall u\in{ H}^1_0(\Omega),$$ where $|\Omega|$ stands for the measure of the domain $\Omega$.
In particular, for any $q\geq 1$
$$\| e^{u}\|_{q}\le C^{\frac1q} |\Omega|^{\frac1q} e^{{q\over 16\pi}\|u\|^2}\quad \forall u\in{H}^1_0(\Omega).$$
\end{lemma}
Theorems \ref{th1}-\ref{th2} will be a consequence of more general results concerning Liouville-type problem \eqref{proreg}. In order to provide such results \eqref{proreg}, we now give a construction of a suitable approximate solution for \eqref{proreg}. In what follows, we identify $x=( x_1,x_2)\in \mathbb{R}^2$ with $x_1+{\rm i}x_2\in \mathbb{C}$. Moreover, $\langle x_1, x_2\rangle$ stands for the inner product between the vectors $x_1, x_2\in\mathbb{R}^2$, whereas $x_1 x_2$ will denote the multiplication of the complex numbers $x_1$, $x_2$. Clearly $\langle x_1, x_2\rangle={\rm Re}(x_1 \overline x_2)$.
For any $\alpha\in\mathbb{N}$, we can associate to \eqref{proreg} a limiting problem of Liouville type which will play a crucial role in the construction of the blowing-up solutions: \begin{equation}\label{limit}
-\Delta w=|x|^{2(\alpha-1)}e^w\quad \hbox{in}\;\; \mathbb{R}^2,\qquad
\int_{\mathbb{R}^2} |x|^{2(\alpha-1)}e^{w(x)}dx<+\infty. \end{equation} A complete classification for solutions of \eqref{limit} is due to \cite{Pratar} and corresponds,
in complex notation, to the three-parameter family of functions
\begin{equation}\label{bubble}
w^\alpha_{\delta,b}(x):=\log {8\alpha^2\delta^{2\alpha}\over (\delta^{2\alpha}+|x^{\alpha}- b|^2)^2}\quad \delta>0,\,b\in \mathbb{C}. \end{equation} The following quantization property holds: \begin{equation} \label{quantum} \int_{\mathbb{R}^2}
|x|^{2(\alpha-1)}e^{w_{\delta,b}^\alpha(x)}dx = 8 \pi \alpha .\end{equation}
In the following we agree that $$W_\lambda=w^{\alpha}_{\delta,b}(x),$$ where
the value $\delta=\delta(\lambda,b)$ is defined as:
\begin{equation} \label{delta} \delta^{{2\alpha}} :=\frac{\lambda}{8\alpha^2}V(0)e^{8\pi\sum_{i=1}^\alpha H(0,\beta_i)}. \end{equation}
To obtain a better first approximation, we need to modify the functions $W_\lambda$ in order to satisfy the zero boundary condition. Precisely, we consider the projections $P W_\lambda $ onto the space $ H^1_0(\Omega)$ of $W_\lambda$, where the projection $P:H^1(\mathbb{R}^N)\to H^1_0(\Omega)$ is defined as the unique solution of the problem $$
\Delta P v=\Delta v\quad \hbox{in}\ \Omega,\qquad P v=0\quad \hbox{on}\ \partial\Omega. $$
Let us consider $b$ in a small neighborhood of $0$ and let us denote by $\beta_0,\ldots, \beta_{\alpha-1}$ the $\alpha$-roots of $b$, i.e., $\beta_i^\alpha=b$ and $\beta_i\neq\beta_h$ for $i\neq h$. Observe that the function $\sum_{i=0}^{\alpha-1} H(x, \beta_i)$ is harmonic in $\Omega$ and satisfies $\sum_{i=0}^{\alpha-1} H(x, \beta_i)=\frac{1}{2\pi}\log|x^\alpha-b|$ on $\partial \Omega.$ A straightforward computation gives that for any $x\in\partial\Omega$
$$\bigg|PW_\lambda- W_\lambda+\log\(8\alpha^2\delta^{2\alpha}\)-8\pi\sum_{i=0}^{\alpha-1} H(x, \beta_i)\bigg|=\bigg|W_\lambda-\log\(8\alpha^2\delta^{2\alpha}\)+4 \log|x^\alpha-b|
\bigg|\leq C\delta^{2\alpha}.$$ Since the expressions considered inside the absolute values are harmonic in $\Omega$, then the maximum principle applies and implies
the following asymptotic expansion \begin{equation}\label{pro-exp1}\begin{aligned}
PW_\lambda=& W_\lambda-\log\(8\alpha^2\delta^{2\alpha}\)+8\pi\sum_{i=0}^{\alpha-1} H(x, \beta_i)+O(\delta^{2\alpha})\\ =&-2\log\(\delta^{{2\alpha}}+|x^{\alpha}- b|^2\)+8\pi\sum_{i=0}^{\alpha-1} H(x, \beta_i)+O(\delta^{2\alpha}) \end{aligned}\end{equation}
uniformly for $x\in \bar\Omega$ and $b $ in a small neighborhood of $0$.
We shall look for a solution to \eqref{proreg} in a small neighborhood of the first approximation, namely a solution of the form
$$v_\lambda=PW_\lambda
+ {\phi}_\lambda,$$ where the rest term $\phi_\lambda$ is small in $H^1_0(\Omega)$-norm.
We are now in the position to state the main theorems of the paper.
\begin{thm} \label{main1} Assume that $\alpha\geq 3$ and hypotheses \eqref{assurd0} hold. Then, for $\lambda$ sufficiently small
there exist $\phi_\lambda \in H^1_0(\Omega)$ and $b=b_\lambda=O(\lambda^{\frac{\alpha+1}{2\alpha}})$ such that the couple $ PW_\lambda+\phi_\lambda$ solves problem \eqref{proreg}. Moreover, for any fixed $\varepsilon>0$,
\begin{equation}\label{barb}\| \phi_\lambda \|_{H^1_0(\Omega)}\leq \lambda^{\frac{1}{\alpha}-\varepsilon}\;\;\hbox{ for } \lambda \hbox{ small enough}.\end{equation} \end{thm}
\begin{thm} \label{main2} Assume that $\alpha=2$, and hypotheses \eqref{assurd}-\eqref{assurd2} hold. Then, for $\lambda$ sufficiently small there exist
$\phi_\lambda \in H^1_0(\Omega)$ and $b=b_\lambda=O(\lambda^{\frac{\alpha+1}{2\alpha}})$ such that the couple $ PW_\lambda+\phi_\lambda$ solves problem \eqref{proreg}. Moreover, for any fixed $\varepsilon>0$, \eqref{barb} holds.
\end{thm}
In the remaining part of the paper we will prove Theorems \ref{main1}-\ref{main2} and at the end of the Section 6 we shall see how Theorems \ref{th1}-\ref{th2} follow quite directly as a corollary according to \eqref{chva} and \eqref{aaa}.
We end up this section by setting the notation and basic well-known facts which will be of use in the rest of the paper. We denote by $\|\cdot\|$ and $\|\cdot\|_p$ the norms in the space $ H^1_0(\Omega)$ and $L^p(\Omega)$, respectively, namely
\begin{equation}\label{nott}\|u\|:=\|u\|_{ H^1_0(\Omega)}
,\qquad \|u\|_p:=\|u\|_{L^p(\Omega)}
\quad \forall u\in H^1_0(\Omega).\end{equation}
For any $\alpha\ge1$ we will make use of the Hilbert spaces \begin{equation}\label{ljs}
\mathrm{L}_\alpha (\mathbb{R}^2):=\left\{u \in L^2_{loc}(\mathbb{R}^2)\ :\ \left\|{|y|^{\alpha-1} \over 1+|y|^{{2\alpha}}}u\right\|_{{L}^2(\mathbb{R}^2)}<+\infty\right\}\end{equation}
and
\begin{equation}\label{hjs}\mathrm{H}_\alpha (\mathbb{R}^2):=\left\{u\in {\rm W}^{1,2}_{loc}(\mathbb{R}^2) \ :\ \|\nabla u\|_{{L}^2(\mathbb{R}^2)}+\left\|{|y|^{\alpha-1} \over 1+|y|^{{2\alpha}}}u\right\|_{{L}^2(\mathbb{R}^2)}<+\infty\right\},\end{equation}
endowed with the norms
$$\|u\|_{\mathrm{L}_\alpha }:= \left\|{|y|^{\alpha-1} \over 1+|y|^{2\alpha}}u\right\|_{{L}^2(\mathbb{R}^2)}\
\hbox{and }\ \|u\|_{\mathrm{H}_\alpha }:= \(\|\nabla u\|^2_{{L}^2(\mathbb{R}^2)}+\left\|{|y|^{\alpha-1} \over 1+|y|^{{2\alpha}}}u\right\|^2_{{L}^2(\mathbb{R}^2)}\)^{1/2}.$$ We denote by $\langle u,v \rangle_{\mathrm{L}_\alpha }$ and $\langle u,v \rangle_{\mathrm{H}_\alpha }$ the natural scalar product in ${\mathrm{L}_\alpha }$ and in ${\mathrm{H}_\alpha }$, respectively.
\begin{prop}\label{compact} The embedding $i_\alpha:\mathrm{H}_\alpha (\mathbb{R}^2)\hookrightarrow\mathrm{L}_\alpha (\mathbb{R}^2)$ is compact. \end{prop} \begin{proof}
See \cite[Proposition 6.1]{gpistoia}. \end{proof}
As commented in the introduction, our proof uses the singular perturbation methods. For that, the nondegeneracy of the functions that we use to build our approximating solution is essential. Next proposition is devoted to the nondegeneracy of the finite mass solutions of the Liouville equation (regular and singular).
\begin{prop} \label{esposito} Assume that $\phi:\mathbb{R}^2\to\mathbb{R}$ solves the problem \begin{equation}\label{l1}
-\Delta \phi =8\alpha^2{|y|^{2(\alpha-1)}\over (1+|y^{\alpha}-\xi|^2)^2}\phi\;\;
\hbox{in}\ \mathbb{R}^2,\quad \int_{\mathbb{R}^2}|\nabla
\phi(y)|^2dy<+\infty. \end{equation}
Then there exist $c_0,\,c_1,\, c_2\in\mathbb{R}$ such that $$\phi(y)=c_0 Z_0+ c_1Z_1 +c_2Z_2.$$
$$Z_0(y): = {1-|y^{\alpha}-\xi|^2\over 1+|y^{\alpha}-\xi|^2} ,\ \; \;Z_1(y):={ {\rm Re}(y^{\alpha}-\xi)\over 1+|y^{\alpha}-\xi|^2}
,\ \;\;Z_2(y):={ {\rm Im}(y^{\alpha}-\xi)\over 1+|y^{\alpha}-\xi|^2}. $$ \end{prop} \begin{proof} In \cite[Theorem 6.1]{gpistoia} it was proved that any solution $\phi$ of \eqref{l1} is actually a bounded solution. Therefore we can apply the result in \cite{dem} to conclude that $\phi= c_0 \phi_0 + c_1 \phi_1 + c_2 \phi_2$ for some $c_0,c_1,c_2\in \mathbb{R}$.
\end{proof}
In our estimates throughout the paper, we will frequently denote by $C>0$, $c>0$ fixed constants, that may change from line to line, but are always independent of the variable under consideration. We also use the notations $O(1)$, $o(1)$, $O(\lambda)$, $o(\lambda)$ to describe the asymptotic behaviors of quantities in a standard way.
\section{Estimate of the error term} The goal of this section is to provide an estimate of the error up to which the function $W_\lambda$ solves problem \eqref{proreg}.
\begin{lemma}\label{aux} Let $r>0$ be a fixed number. Define $${R}_\lambda:=
-\Delta PW_\lambda-\lambda V(x) |x|^{2(\alpha-1)}e^{PW_\lambda}= |x|^{2(\alpha-1)}e^{W_\lambda}-\lambda V(x)|x|^{2(\alpha-1)} e^{PW_\lambda}.$$
For any fixed $p\geq 1$ the following holds
$$\|R_\lambda\|_{p}=O(\lambda^{\frac{1}{\alpha p}}).$$ uniformly for $|b|\leq r\sqrt{\lambda}$. Consequently, for every fixed $p\geq 1$,
\begin{equation}\label{judo}\|\lambda V(x)|x|^{2(\alpha-1)} e^{PW_\lambda}\|_p=\||x|^{2(\alpha-1)}e^{W_\lambda}\|_p+o(1)=O(\lambda^{\frac{1-p}{\alpha p}})\end{equation} uniformly for $|b|\leq r\sqrt{\lambda}$. \end{lemma}
\begin{proof}
By \eqref{pro-exp1} and the choice of $\delta$ in \eqref{delta} we derive \begin{equation}\label{cuore}\begin{aligned}&\lambda V(x)|x|^{2(\alpha-1)} e^{PW_\lambda}\\ &
=\frac{\lambda}{8\alpha^2\delta^{2\alpha}} V(x)|x|^{2(\alpha-1)} e^{W_\lambda+8\pi \sum_{i=0}^{\alpha-1}H(x, \beta_i) +O(\delta^{2\alpha} )}
\\ &=|x|^{2(\alpha-1)}e^{W_\lambda}\frac{V(x)}{V(0)}e^{8\pi\sum_{i=0}^{\alpha-1}(H(x, \beta_i)-H(0,\beta_i))+O(\delta^{2\alpha} )}
\\ &=|x|^{2(\alpha-1)}e^{W_\lambda}\frac{a(x)}{a(0)}e^{-4\pi(\alpha-1) (H(x,0)-H(0,0))+8\pi\sum_{i=0}^{\alpha-1}(H(x, \beta_i)-H(0,\beta_i))+O(\delta^{2\alpha} )} .\end{aligned}\end{equation} According to \eqref{expa} we have $$H(x,0)-H(0,0)=
{\rm Re}\Big(\frac{d\tilde H}{dx} (0,0) x\Big)+O(|x|^{2}) $$ whereas, by Lemma \ref{robin10}, $$\sum_{i=0}^{\alpha-1}(H(x, \beta_i)-H(0,\beta_i))=
\alpha {\rm Re}\Big(\frac{d\tilde H}{dx} (0,0) x\Big)+O(|x|^{2})+O(|b|^2) $$ by which we arrive at
$$\begin{aligned}&\lambda V(x)|x|^{2(\alpha-1)} e^{PW_\lambda}\\&
=|x|^{2(\alpha-1)}e^{W_\lambda}\frac{a(x)}{a(0)}
e^{4\pi(\alpha+1){\rm Re}(\frac{d\tilde H}{dx} (0,0) x)+O(|x|^{2})+O(|b|^2)+O(\delta^{2\alpha} )} \\ &=|x|^{2(\alpha-1)}e^{W_\lambda}\frac{a(x)}{a(0)}
\Big( 1+ 4\pi(\alpha+1){\rm Re}\Big(\frac{d\tilde H}{dx} (0,0) x\Big)+ O(|x|^{2})+O(|b|^2)+O(\delta^{2\alpha} )\Big)
\\ &=|x|^{2(\alpha-1)} e^{W_\lambda}\Bigg(1+
\frac{\langle\nabla a(0), x\rangle}{a(0)} + 4\pi(\alpha+1){\rm Re}\Big(\frac{d\tilde H}{dx} (0,0) x\Big)+ O(|x|^{2})+O(|b|^2)+O(\delta^{2\alpha} )\Bigg) . \end{aligned} $$ Taking into account that ${\rm Re}(\frac{d\tilde H}{dx} (0,0) x)=\langle \nabla_x H(0,0),x\rangle$ and using that $$\frac{\langle \nabla a(0), x\rangle}{a(0)} + 4\pi(\alpha+1){\rm Re}\Big(\frac{d\tilde H}{dx} (0,0) x\Big)=\frac{\langle\nabla a(0) ,x\rangle}{a(0)}+ 4\pi(\alpha+1)\langle \nabla_x H(0,0),x\rangle=0$$ by assumptions \eqref{assurd0} and \eqref{assurd}-\eqref{assurd2}, we arrive at
$$\begin{aligned}&\lambda V(x)|x|^{2(\alpha-1)} e^{PW_\lambda}=|x|^{2(\alpha-1)}
e^{W_\lambda}+ \big(O(|x|^{2})+O(|b|^2)
+O(\delta^{2\alpha} )\big)|x|^{2(\alpha-1)}e^{W_\lambda}.\end{aligned}$$
Now if we scale $x=\delta y$, recalling that $|b|\leq \sqrt{\lambda}\leq C\delta^\alpha$, we get $$\begin{aligned}
|x|^{2(\alpha-1)} e^{W_\lambda}& =8\alpha^2
\frac{|y|^{2(\alpha-1)}}{\delta^2(1+|y^{\alpha }-\delta^{-\alpha}b|^2)^2}
= O\bigg(\frac{1}{\delta^2(1+|y|^{2\alpha+2})}\bigg) .\end{aligned} $$ and, similarly,
$$\begin{aligned}|x|^{2\alpha} e^{W_\lambda}& =8\alpha^2
\frac{|y|^{2\alpha}}{(1+|y^{\alpha }-\delta^{-\alpha}b|^2)^2}
= O\bigg(\frac{1}{1+|y|^{2\alpha}}\bigg) .\end{aligned} $$ by which
$$\||x|^{2(\alpha-1)}
e^{W_\lambda}\|_p=O(\delta^{2\frac{1-p}{p}}) \qquad
\||x|^{2\alpha}
e^{W_\lambda}\|_p=O(\delta^{\frac2p})$$ The thesis is thus proved.
\end{proof}
\section{Analysis of the linearized operator} According to Proposition \ref{esposito}, by the change of variable $x=\delta y$, we immediately get that all solutions $\psi \in \mathrm{H}_\alpha(\mathbb{R}^2)$ of $$
-\Delta \psi= 8{\alpha}^2{\delta^{2\alpha}|x|^{2(\alpha-1)}\over (\delta^{2\alpha}+|x^{\alpha }-b|^2)^2}\psi =|x|^{2(\alpha-1)} e^{W_\lambda}\psi\quad \hbox{in}\quad \mathbb{R}^2$$ are linear combinations of the functions
$$Z^0_{\delta,b}(x)={\delta^{2\alpha}-|x^{\alpha }-b|^2\over \delta^{2\alpha}+|x^{\alpha }-b|^2},\ Z^1_{\delta,b}(x)=
{ \delta^{\alpha }{\rm Re}(x^{\alpha }-b)\over \delta^{2\alpha}+|x^{\alpha }-b|^2},\ Z^2_{\delta,b}(x)=
{ \delta^{\alpha }{\rm Im}(x^{\alpha }-b)\over \delta^{2\alpha}+|x^{\alpha }-b|^2} .$$ We introduce their projections $PZ^j_{\delta,b}$ onto $H^1_0(\Omega).$ It is immediate that
\begin{equation}\label{pz0} PZ^0_{\delta,b}(x)=Z^0_{\delta,b}(x)+1+ O\(\delta^{2\alpha}\) \end{equation}
and
\begin{equation}\label{pzi} PZ^j_{\delta,b}(x)=Z^j _{\delta,b}(x) + O\(\delta^{\alpha }\),\;\; j=1,2 \end{equation} uniformly with respect to $x\in\overline\Omega$ and $b$ in a small neighborhood of $0$. \\ We agree that $Z_\lambda^j:=Z_{\delta,b}^j$ for any $j=0,1,2$, where $\delta$ is defined in terms of $\lambda$ and $b$ according to \eqref{delta}.
Let us consider the following linear problem: given $ h\in H_0^1(\Omega)$, find a function $\phi\in H^1_0(\Omega)$ satisfying \begin{equation}\label{lla}
\left\{\begin{aligned}&-\Delta \phi -\lambda V(x)|x|^{2(\alpha-1)} e^{P{W}_\lambda}\phi=\Delta h\\ &\int_\Omega \nabla \phi\nabla PZ_\lambda^j=0\;\;j=1,2\end{aligned}\right.. \end{equation}
Before going on, we recall the following identities which follow by straightforward computations using Lemma \ref{copy}: for every $\xi\in \mathbb{R}^2$
\begin{equation}\label{id1}\begin{aligned}\int_{\mathbb{R}^2} |y|^{2(\alpha-1)}\log (1+|y^{\alpha }-\xi|^2)\frac{1-|y^{\alpha }-\xi|^{2}}{(1+|y^{\alpha }-\xi|^2)^3}dy&=\frac{1}{{\alpha}}\int_{\mathbb{R}^2} \log (1+|y|^2)\frac{1-|y|^{2}}{(1+|y|^{2})^3}dy\\ &=-\frac{\pi}{{2\alpha}},\end{aligned}\end{equation}
\begin{equation}\label{id2}\int_{\mathbb{R}^2}\frac{|y|^{2(\alpha-1)}}{(1+|y^{\alpha }-\xi|^2)^2}\frac{1-|y^{\alpha }-\xi|^{2}}{1+|y^{\alpha }-\xi|^{2}}dy=\frac{1}{{\alpha}}\int_{\mathbb{R}^2}\frac{1-|y|^{2}}{(1+|y|^{2})^3}dy=0,\end{equation}
\begin{equation}\label{id3}\begin{aligned}\int_{\mathbb{R}^2}\frac{|y|^{2(\alpha-1)}({\rm Re} (y^{\alpha }-\xi))^2}{(1+|y^{\alpha }-\xi|^{2})^4}dy&=\int_{\mathbb{R}^2}\frac{|y|^{2(\alpha-1)}({\rm Im} (y^{\alpha }-b))^2}{(1+|y^{\alpha }-\xi|^{2})^4}dy\\ &=\frac{1}{{2\alpha}}\int_{\mathbb{R}^2}\frac{|y|^2}{(1+|y|^{2})^4}dy=\frac{\pi}{12{\alpha}}.\end{aligned}\end{equation}
\begin{prop}\label{inv} Let $r>0$ be fixed.
There exist $\lambda_0>0$ and $C>0$ such that for any $\lambda \in(0, \lambda_0)$, any $b\in \mathbb{R}^2$ with $|b|<r\sqrt{\lambda}$ and any $h\in H^1_0(\Omega)$, if
$\phi\in H^1_0(\Omega)$ solves \eqref{lla}, then the following holds $$\|\phi\| \leq C |\log\lambda | \|h\| .$$ \end{prop} \begin{proof} We argue by contradiction. Assume that there exist sequences
$\lambda_n\to0,$ $ h_n\in H^1_0(\Omega)$, $|b_n|\leq r\sqrt{\lambda_n}$ and $\phi_n\in H^1_0(\Omega)$ which solve \eqref{lla} and \begin{equation}\label{inv2}
\|\phi_n\| =1, \qquad
|\log\lambda_n | \| h_n\| \to 0.\end{equation} Let $\delta_n>0$ be the value associated to $\lambda_n$ according to \eqref{delta}. Then we may assume $$\delta_n^{-\alpha}b_n\to b_0.$$ We define $\widetilde {\Omega}_n :={\Omega\over {\delta}_n }$ and
$$ \tilde \phi_n(y):=\left\{\begin{aligned}&\phi_n\({\delta}_n y\)&\hbox{ if }&y\in \widetilde{\Omega}_n \\ &0&\hbox{ if }&y\in \mathbb{R}^2\setminus\widetilde{\Omega}_n \end{aligned} \right. . $$ \\
In what follows at many steps of the arguments we will pass to a subsequence, without further notice. We split the remaining argument into five steps.
\noindent{\em Step 1. We will show that \begin{equation*}\label{step1.0} \tilde\phi_n \;\; \hbox{ is bounded in }\mathrm{H}_{\alpha} (\mathbb{R}^2). \end{equation*}
}
It is immediate to check that
\begin{equation}\label{bounabla}\int_{\mathbb{R}^2}|\nabla \tilde\phi_n|^2dy=\int_\Omega|\nabla \phi_n|^2dx\le1.\end{equation} Next, we multiply the equation in \eqref{lla} by $\phi_n$; then we integrate over $\Omega$ to obtain
$$\begin{aligned}\lambda_n \int_\Omega V(x)|x|^{2(\alpha-1)}e^{PW_{\lambda_n}}\phi_n^2dx=& \int_\Omega |\nabla\phi_n|^2dx +\int_\Omega \nabla h_n\nabla \phi_ndx\end{aligned}$$ which implies, by \eqref{inv2},
\begin{equation}\label{W22}\lambda_n \int_{\Omega} V(x) |x|^{2(\alpha-1)}e^{PW_{\lambda_n}}\phi_n^2dx \leq C.\end{equation} So, Lemma \ref{aux} gives $\int_\Omega |x|^{2(\alpha-1)} e^{W_{\lambda_n}}\phi_n^2\leq C$ or, equivalently,
$$\int_{\mathbb{R}^2} {|y|^{2(\alpha-1)}\over \(1+|y^{\alpha }-\delta_n^{-\alpha}b_n|^2\)^2} \tilde\phi_n^2 dy \le C.$$ Combining this with \eqref{bounabla}, we deduce that $\tilde \phi_n$ is bounded in the space $\mathrm{H}_{\alpha} (\mathbb{R}^2)$.
\noindent{\em Step 2. We will show that, for some $ \gamma_0 \in\mathbb{R} $,
$$\tilde\phi_n\to \gamma_0 \frac{1-|y^{\alpha }-b_0|^{2}}{1+|y^{\alpha }-b_0|^{2}}\;\ \hbox{ weakly in }\rm{H}_{\alpha} (\mathbb{R}^2) \hbox{ and strongly in }\rm{L}_{\alpha} (\mathbb{R}^2) .$$
}
Step 1 and Proposition \ref{compact} give
$$\tilde \phi_n\to f\;\hbox{ weakly in }\mathrm{H}_\alpha (\mathbb{R}^2) \hbox{ and strongly in }\mathrm{L}_\alpha(\mathbb{R}^2) .$$ Let $\tilde\psi\in C^\infty_c(\mathbb{R}^2)$ and set
${\psi}_n=\tilde{\psi}(\frac{x}{{\delta}_n})\in C^\infty_c(\Omega)$, for large $n$.
We multiply the equation in \eqref{lla} by ${\psi}_n,$ we integrate over $\Omega$ and we get
\begin{equation}\label{1.1.1.1}\int_{\widetilde{\Omega}_n} \nabla{\tilde\phi_n}\nabla\tilde\psi dy - \lambda_n\int_{{\Omega}} V(x)|x|^{2(\alpha-1)} e^{PW_{\lambda_n}}\phi_n \psi_n dx=-\int_{{\Omega}}\nabla
h_n\nabla {\psi}_ndx.\end{equation}
According to Lemma \ref{aux} we have
$$\begin{aligned}\lambda_n\int_{{\Omega}}V(x) |x|^{2(\alpha-1)} e^{PW_{\lambda_n}}\phi_n \psi_n dx&=\int_{{\Omega}} |x|^{2(\alpha-1)} e^{W_{\lambda_n}}\phi_n \psi_n dx+o(1)
\\ &=8{\alpha}^2\int_{\mathbb{R}^2} {|y|^{2(\alpha-1)}\over (1+|y^{\alpha }-\delta_n^{-\alpha}b_n|^2)^2} \tilde\phi_n \tilde\psi dy+o(1)\\ &=8{\alpha}^2\int_{\mathbb{R}^2} {|y|^{2(\alpha-1)}\over (1+|y^{\alpha }-b_0|^2)^2} f \tilde\psi dy+o(1).\end{aligned}$$
Finally, by \eqref{inv2}, using that $\int_\Omega |\nabla {\psi}_n|^2=\int_{\mathbb{R}^2}|\nabla\tilde \psi|^2$,
\begin{equation}\label{macro}\int_{{\Omega}}|\nabla
{h}_n\nabla {\psi}_n|dx=O(\|{h}_n\|)=o(1).\end{equation}
Therefore, we may pass to the limit in \eqref{1.1.1.1} to obtain
\begin{align}\nonumber&\int\limits_{\mathbb{R}^2} \nabla f \nabla\tilde\psi dy =8{\alpha}^2\int_{\mathbb{R}^2} {|y|^{2(\alpha-1)}\over (1+|y^{\alpha }-b_0|^2)^2} f \tilde\psi dy .\end{align} Thus
$f$ solves the equation
$$-\Delta f=8{\alpha}^2 {|y|^{2(\alpha-1)}\over (1+|y^{\alpha }-b_0|^2)^2} f .$$ Proposition \ref{esposito} gives \begin{equation}\label{crucial3}f=\gamma_0Z_0+\gamma_1Z_1
+\gamma_2Z_2\end{equation} for some $\gamma_0,\,\gamma_1,\, \gamma_2\in \mathbb{R}$. It remains to show that $\gamma_1=\gamma_2=0.$ Indeed, we compute $$\begin{aligned}0&=\int_\Omega \nabla \phi_n\nabla PZ_{\lambda_n}^1dx=\int_\Omega|x|^{2(\alpha-1)} e^{W_{\lambda_n}} \phi_n Z_{\lambda_n}^1dx=8{\alpha}^2\int_{\mathbb{R}^2} {|y|^{2(\alpha-1)}\over (1+|y^{\alpha }-\delta_n^{-\alpha}b_n|^2)^2}\tilde\phi_n Z_1dy\\ &=
8{\alpha}^2\int_{\mathbb{R}^2}f {|y|^{2(\alpha-1)}\over (1+|y^{\alpha }-b_0|^2)^2}Z_1 dy+o(1)= \int_{\mathbb{R}^2}\nabla f\nabla Z_1dy+o(1). \end{aligned}$$ We get $\int_{\mathbb{R}^2} \nabla f \nabla Z_1=0$, by which, taking into account that $\int_{\mathbb{R}^2} \nabla Z_1\nabla Z_0=\int_{\mathbb{R}^2} \nabla Z_1\nabla Z_2=0$,
$$\gamma_1\int_{\mathbb{R}^2}| \nabla PZ_1|^2dy=0. $$ So $\gamma_1=0$ and, similarly, $\gamma_2=0.$
\noindent{\em Step 3. We will show that $$\int_{\mathbb{R}^2}\frac{|y|^{2(\alpha-1)}}{(1+|y^{\alpha }-\delta_n^{-\alpha}b_n|^2)^2}\tilde\phi_n dy=o\Big(\frac{1}{\log\lambda_n}\Big).$$}
We multiply the equation in \eqref{lla} by $PZ^0_{\lambda_n}$, we integrate over $\Omega$ and we get
\begin{equation}\label{pi0}\int_\Omega \nabla \phi_n\nabla PZ^0_{\lambda_n}dx-\lambda_n\int_\Omega V(x)|x|^{2(\alpha-1)}e^{PW_{\lambda_n}}\phi_nPZ^0_{\lambda_n} dx=-\int_\Omega \nabla h_n \nabla PZ^0_{\lambda_n} dx.\end{equation} We are now concerned with the estimates of each term of the above expression. First, we compute \begin{equation}\label{pi1}\int_\Omega \nabla \phi_n\nabla PZ^0_{\lambda_n}dx=
\int_\Omega |x|^{2(\alpha-1)}e^{W_{\lambda_n}} \phi_n Z^0_{\lambda_n}dx.\end{equation} Using Lemma \ref{aux} (with $p=2$) and \eqref{pz0}, we obtain
\begin{equation}\label{pi2}\begin{aligned}\lambda_n\int_\Omega& V(x)|x|^{2(\alpha-1)}e^{PW_{\lambda_n}}\phi_nPZ^0_{\lambda_n} dx=\int_\Omega |x|^{2(\alpha-1)}e^{W_{\lambda_n}}\phi_n(Z^0_{\lambda_n}+1) dx+O(\lambda_n^{\frac{1}{{2\alpha }}})
\\ &=\int_\Omega |x|^{2(\alpha-1)}e^{W_{\lambda_n}}\phi_nZ^0_{\lambda_n} dx+8{\alpha}^2\int_{\mathbb{R}^2}\frac{|y|^{2(\alpha-1)}}{(1+|y^{\alpha }-\delta_n^{-\alpha}b_n|^2)^2}\tilde\phi_n dy+O(\lambda_n^{\frac{1}{{2\alpha }}}) .\end{aligned}\end{equation} Finally, since $PZ^0_{\lambda}=O(1)$, we have $\int_\Omega |\nabla PZ^0_{\lambda}|^2=\int_\Omega |x|^{2(\alpha-1)}e^{W_{\lambda}}PZ^0_{\lambda}= O(1)$, by which, owing to \eqref{inv2},
\begin{equation}\label{pi3} \int_\Omega |\nabla h_n|\,| \nabla PZ^0_{\lambda_n}| dx\leq \|h_n\|\, \|PZ^0_{\lambda_n}\|=o\Big(\frac{1}{\log\lambda_n}\Big).\end{equation} We now multiply \eqref{pi0} by $\log\lambda_n$ and pass to the limit: inserting \eqref{pi1}, \eqref{pi2}, \eqref{pi3}, we obtain the thesis of the step.
\noindent{\em Step 4. We will show that $\gamma_0=0$.}
We multiply the equation in \eqref{lla} by $PW_{\lambda_n}$, we integrate over $\Omega$ and we get
\begin{equation}\label{ti0}\int_\Omega \nabla \phi_n\nabla PW_{\lambda_n}dx-\lambda_n\int_\Omega V(x) |x|^{2(\alpha-1)}e^{PW_{\lambda_n}}\phi_nPW_{\lambda_n} dx=-\int_\Omega \nabla h_n \nabla PW_{\lambda_n}dx.\end{equation} Let us estimate each of the terms above. Let us begin with:
\begin{equation}\label{ti1}\int_\Omega \nabla \phi_n\nabla PW_{\lambda_n}dx=\int_\Omega |x|^{2(\alpha-1)} e^{W_{\lambda_n}}\phi_n dx=8\alpha^2\int_{\mathbb{R}^2} \frac{|y|^{2(\alpha-1)}}{(1+|y^{\alpha }-\delta_n^{-\alpha}b_n|^2)^2}\tilde\phi_ndy=o(1) \end{equation} by step 3.
By Lemma \ref{aux} and \eqref{inv2}, using that $|PW_{\lambda_n}|=O(|\log\lambda_n|)$, we get
\begin{equation}\label{ti2}\begin{aligned}\lambda_n\int_\Omega V(x)|x|^{2(\alpha-1)}e^{PW_{\lambda_n}}&\phi_nPW_{\lambda_n} dx=\int_\Omega |x|^{2(\alpha-1)}e^{W_{\lambda_n}}\phi_nPW_{\lambda_n} dx+o(1)
\\ &=8{\alpha}^2\int_{\mathbb{R}^2} \frac{|y|^{2(\alpha-1)}}{(1+|y^{\alpha }-\delta_n^{-\alpha}b_n|^2)^2}\tilde\phi_nPW_{\lambda_n} (\delta_n y)dy+o(1). \end{aligned}
\end{equation} Observe that by \eqref{pro-exp1} $$PW_{\lambda_n}(\delta_n y)=-2\log (1+|y^{\alpha }-\delta_n^{-\alpha}b_n|^2)+8\pi {\alpha} H(\delta_n y,0)-4{\alpha}\log\delta_n+O(\sqrt{\lambda_n})$$ by which
$$PW_{\lambda_n}(\delta_n y)+4{\alpha}\log\delta_n\to -2\log (1+|y^{\alpha }-b_0|^2)+8\pi {\alpha} H(0,0)\;\;\hbox{ uniformly in }\mathbb{R}^2.$$ Using this convergence in \eqref{ti2}, and recalling step 2, we obtain
$$\begin{aligned}\lambda_n\int_\Omega &V(x)|x|^{2(\alpha-1)}e^{PW_{\lambda_n}}\phi_nPW_{\lambda_n} dx\\ &=-16{\alpha}^2\gamma_0\int_{\mathbb{R}^2} \log (1+|y^{\alpha }-b_0|^2)\frac{|y|^{2(\alpha-1)}}{(1+|y^{\alpha }-b_0|^2)^2}\frac{1-|y^{\alpha }-b_0|^{2}}{1+|y^{\alpha }-b_0|^{2}}dy \\ &\;\;\;\;+64\pi {\alpha}^3H(0,0)\gamma_0\int_{\mathbb{R}^2}\frac{|y|^{2(\alpha-1)}}{(1+|y^{\alpha }-b_0|^2)^2}\frac{1-|y^{\alpha }-b_0|^{2}}{1+|y^{\alpha }-b_0|^{2}}dy \\ &\;\;\;\;
-32{\alpha}^3\log \delta_n\int_{\mathbb{R}^2} \frac{|y|^{2(\alpha-1)}}{(1+|y^{\alpha }-\delta_n^{-\alpha}b_n|^2)^2}\tilde\phi_ndy +o(1).\end{aligned} $$ Then by step 3, \eqref{id1}-\eqref{id2},
\begin{equation}\label{ti3}\begin{aligned}\lambda_n\int_\Omega V(x) |x|^{2(\alpha-1)}e^{PW_{\lambda_n}}\phi_nPW_{\lambda_n} dx&=8\pi{\alpha}\gamma_0+o(1).\end{aligned}
\end{equation} Finally, taking into account that $PW_{\lambda}=O(|\log\lambda|)$, we have $\int_\Omega |\nabla PW_{\lambda}|^2=\int_\Omega |x|^{2(\alpha-1)}e^{W_{\lambda}}PW_{\lambda}
=O(|\log\lambda|)$, by which, owing to \eqref{inv2},
\begin{equation}\label{ti4} \int_\Omega |\nabla h_n|\,| \nabla PW_{\lambda_n}| dx\leq \|h_n\|\, \|PW_{\lambda_n}\|=o(1).\end{equation} By inserting \eqref{ti1}, \eqref{ti3}, \eqref{ti4} into \eqref{ti0} and passing to the limit we deduce $\gamma_0=0$.
\noindent{\em Step 5. End of the proof.}
We will show that a contradiction arises. According to Step 2 and Step 4 we have $$\tilde\phi_n\to 0\;\ \hbox{ weakly in }\rm{H}_{\alpha} (\mathbb{R}^2)\hbox{ and strongly in }\rm{L}_{\alpha} (\mathbb{R}^2). $$ By Lemma \ref{aux}
$$\lambda_n\int_\Omega V(x)|x|^{2(\alpha-1)}e^{PW_{\lambda_n}}\phi_n^2dx=\int_\Omega |x|^{2(\alpha-1)}e^{W_{\lambda_n}}\phi_n^2dx+o(1)\leq C\|\tilde\phi_n\|_{\rm L_{\alpha}}^2+o(1)=o(1).$$ Moreover, by \eqref{inv2},
$$\int_\Omega \nabla h_n \nabla \phi_ndx=o(1).$$ We multiply the equation in \eqref{lla} by $\phi_n$, we integrate over $\Omega$ and we obtain
$$\int_\Omega |\nabla \phi_n|^2dx =\lambda_n\int_\Omega V(x) |x|^{2(\alpha-1)}e^{PW_{\lambda_n}}\phi_n^2 dx -\int_\Omega \nabla h_n \nabla \phi_ndx =o(1),$$ in contradiction with \eqref{inv2}.
\end{proof}
In addition to \eqref{lla}, let us consider the following linear problem: given $ h\in {H}_0^1(\Omega)$, find a function $\phi\in H^1_0(\Omega)$ and constants $c_1,c_2\in\mathbb{R}$ satisfying \begin{equation}\label{lla2}
\left\{\begin{aligned}&-\Delta \phi -\lambda V(x)|x|^{2(\alpha-1)} e^{P{W}_\lambda}\phi=\Delta h+\sum_{j=1,2}c_j Z^j_{\lambda} |x|^{2(\alpha-1)}e^{W_{\lambda}}\\ &\int_\Omega \nabla \phi\nabla PZ^j_{\lambda}dx=0\;\;j=1,2\end{aligned}\right.. \end{equation}
In order to solve problem \eqref{lla2}, we need to establish an a priori estimate analogous to that of Proposition \ref{inv}. \begin{prop}\label{linear2} Let $r>0$ be fixed.
There exist $\lambda_0>0$ and $C>0$ such that for any $\lambda \in(0, \lambda_0)$, any $b\in\mathbb{R}^2$ with $|b|<r\sqrt{\lambda}$ and any $h\in H^1_0(\Omega)$, if
$(\phi,c_1,c_2)\in H^1_0(\Omega)\times \mathbb{R}^2$ solves \eqref{lla}, then the following holds $$\|\phi\| \leq C |\log\lambda | \|h\| .$$
\end{prop} \begin{proof} First observe that by \eqref{pzi}
\begin{equation}\label{zeta1}\begin{aligned}\int_\Omega \nabla PZ_\lambda^1\nabla PZ_\lambda^2dx&=\int_\Omega |x|^{2(\alpha-1)}e^{W_{\lambda}}Z^1_{\lambda} PZ^2_{\lambda}dx=\int_{\mathbb{R}^2} |x|^{2(\alpha-1)}e^{W_{\lambda}}Z^1_{\lambda} Z^2_{\lambda}dx+o(1)\\ &=\int_{\mathbb{R}^2} \nabla PZ_1\nabla PZ_2dy +o(1)=o(1).\end{aligned}\end{equation} Moreover
\begin{equation}\label{zeta2}\begin{aligned}\|PZ^1_{\lambda}\|^2&=\int_\Omega |x|^{2(\alpha-1)}e^{W_{\lambda}}Z^1_{\lambda} PZ^1_{\lambda}dx=\int_\Omega |x|^{2(\alpha-1)}e^{W_{\lambda}}(Z^1_{\lambda})^2dx+o(1)\\ &=8{\alpha}^2 \int_{\mathbb{R}^2}|y|^{2(\alpha-1)} \frac{|{\rm Re} (y^{\alpha }-\delta^{-\alpha}b)|^2}{(1+|y^{\alpha }-\delta^{-\alpha}b|^{2})^4}dy+o(1)=\frac23 \pi{\alpha}+o(1) \end{aligned}\end{equation} where we have used \eqref{id3}. Similarly
\begin{equation}\label{zeta3}\begin{aligned}\|PZ^2_{\lambda}\|^2=\frac23 \pi{\alpha}+o(1) .\end{aligned}\end{equation} Then, taking into account that $-\Delta PZ^j_{\lambda} =|x|^{2(\alpha-1)}e^{W_{\lambda}}Z^j_{\lambda} $, according to Proposition \ref{inv} we have
\begin{equation}\label{marc}\|\phi\|\leq C\log\lambda\big(\|h\|+|c_1|+|c_2|\big).\end{equation} Hence it suffices to estimate the values of the constants $c_j$. We multiply the equation in \eqref{lla2} by $PZ^1_{\lambda}$ and we find
\begin{equation}\label{est}\int_\Omega \phi |x|^{2(\alpha-1)}e^{W_{\lambda}}Z^1_{\lambda}dx- \lambda\int_\Omega V(x) |x|^{2(\alpha-1)} e^{P{W}_\lambda}\phi PZ^1_{\lambda}dx=\frac23 \pi{\alpha} c_1+o(c_1)+o(c_2)+ O(\|h\|).\end{equation} Let us fix $p\in (1,+\infty)$ sufficiently close to 1. Then, by \eqref{pzi} and \eqref{judo} we may estimate
$$\begin{aligned}\int_\Omega |\phi| |x|^{2(\alpha-1)}e^{W_{\lambda}}|PZ^1_{\lambda}-&Z^1_{\lambda}|dx\leq C\sqrt\lambda\int_\Omega |\phi| |x|^{2(\alpha-1)}e^{W_{\lambda}}dx\leq C\sqrt\lambda\|\phi\|\, \| |x|^{2(\alpha-1)}e^{W_{\lambda}}\|_p\\ &\leq C\lambda^{\frac12 +\frac{1-p}{\alpha p}}\|\phi\|\leq C\lambda^{\frac{1}{\alpha p}}\|\phi\|\end{aligned}$$ and, since $PZ^1_{\lambda}=O(1)$, using Lemma \ref{aux},
$$\begin{aligned}\int_\Omega |\phi| \big| |x|^{2(\alpha-1)}&e^{W_{\lambda}}- \lambda V(x)|x|^{2(\alpha-1)} e^{P{W}_\lambda}\big||PZ^1_{\lambda}|dx\\&\leq C\int_\Omega |\phi| \big||x|^{2(\alpha-1)}e^{W_{\lambda}}- \lambda V(x) |x|^{2(\alpha-1)} e^{P{W}_\lambda}\big| dx\leq C\lambda^{\frac{1}{\alpha p}}\|\phi\| .\end{aligned}$$ By inserting the above two estimates into \eqref{est} we obtain
$$|c_1|+o(c_2)\leq C\|h\|+C\lambda^{\frac{1}{\alpha p}}\|\phi\| .$$ We multiply the equation in
\eqref{lla2} by $PZ^2_{\lambda}$ and, by a similar argument as above, we find$$|c_2|+o(c_1)\leq C\|h\|+C\lambda^{\frac{1}{\alpha p}}\|\phi\|,$$ and so $$|c_1|+|c_2|\leq C\|h\|+C\lambda^{\frac{1}{\alpha p}}\|\phi\|.$$ Combining this with \eqref{marc} we obtain the thesis.
\end{proof}
\section{The nonlinear problem: a contraction argument} In order to solve \eqref{eq}, let us consider the following intermediate problem:
\begin{equation}\label{inter}\left\{\begin{aligned}&-\Delta(PW_\lambda+\phi)-\lambda V(x) |x|^{2(\alpha-1)}e^{PW_\lambda+\phi}=\sum_{j=1,2}c_j Z_\lambda^j|x|^{2(\alpha-1)} e^{W_\lambda},\\ &\phi \in H^1_0(\Omega),\;\;\;\; \int_\Omega \nabla \phi\nabla PZ_\lambda^jdx=0 \; \; j=1,2.\end{aligned}\right.\end{equation}
Then it is convenient to solve as a first step the problem for $\phi$ as a function of $b$.
To this aim, first let us rewrite problem \eqref{inter} in a more convenient way.
For any $ p>1,$ let $$i^*_{p}:L^{p}(\Omega)\to H^1_0(\Omega)$$ be the adjoint operator of the embedding $i_{p}:H^1_0(\Omega)\hookrightarrow L^{p\over p-1 }(\Omega),$ i.e. $u=i^*_{p}(v)$ if and only if $-\Delta u=v$ in $\Omega,$ $u=0$ on $\partial\Omega.$ We point out that $i^*_{p}$ is a continuous mapping, namely \begin{equation}
\label{isp} \|i^*_{p}(v)\| \le c_{p} \|v\|_{p}, \ \hbox{for any} \ v\in L^{p}(\Omega), \end{equation} for some constant $c_{p}$ which depends on $\Omega$ and $p.$ Next let us set $$ {K }:=\hbox{span}\left\{PZ^1_{\lambda},\ PZ^2_{\lambda}\right\}$$ and $$ {K ^\perp }:= \left\{\phi\in H^1_0(\Omega)\ :\ \int_\Omega \nabla \phi \nabla PZ^1_{\lambda}dx= \int_\Omega \nabla \phi \nabla PZ^2_{\lambda}dx= 0\right\} $$ and denote by $$ \Pi : H^1_0(\Omega)\to {K },\qquad {\Pi ^\perp}: H^1_0(\Omega)\to {K ^\perp }$$ the corresponding projections. Let $ L: K^\perp \to K^\perp$ be the linear operator defined by \begin{equation}\label{elle}
L( \phi):= \Pi^{\perp}\Big( {i^*_{p}}\big( \lambda V(x)|x|^{2(\alpha-1)}e^{PW_\lambda} \phi \big)\Big) - \phi. \end{equation} Notice that problem \eqref{lla2} reduces to $$L(\phi)=\Pi^\perp h, \quad \phi\in K^\perp.$$
As a consequence of Proposition \ref{linear2} we derive the invertibility of $L$.
\begin{prop}\label{ex} Let $r>0$ be a fixed number. For any $p>1$ there exist $\lambda_0>0$ and $C>0$ such that for any $\lambda \in(0, \lambda_0)$, any $b\in\mathbb{R}^2$ with $\|b\|<r\sqrt{\lambda}$ and
any $h\in K^\perp$ there is a unique solution $ \phi\in K^\perp$ to the problem $$L(\phi)=h.$$ In particular, $L$ is invertible; moreover, $$\| L^{-1} \| \leq C
|\log \lambda |.$$
\end{prop}
\begin{proof} Observe that the operator $\phi\mapsto \Pi^\perp\big( {i^*_{p}}\( \lambda V(x)|x|^{2(\alpha-1)}e^{PW_\lambda} \phi \)\big)$ is a compact operator in $K^\perp$.
Let us consider the case $h=0$, and take $\phi\in K^\perp$ with $L(\phi)=0$. In other words, $\phi$ solves the system \eqref{lla2} with $h=0$ for some $c_1,c_2\in\mathbb{R}$. Proposition \ref{linear2} implies $\phi\equiv 0$. Then, Fredholm's alternative implies the existence and uniqueness result.
Once we have existence, the norm estimate follows directly from Proposition \ref{linear2}. \end{proof} Now we come back to our goal of finding a solution to problem \eqref{inter}. In what follows we denote by $N:K^\perp\to K^\perp$ the nonlinear operator
$$N(\phi)=\Pi^\perp\({i^*_{p}}\big( \lambda V(x)|x|^{2(\alpha-1)}e^{PW_\lambda}(e^{\phi}-1- \phi) \big)\)$$
Therefore problem \eqref{inter} turns out to be equivalent to the problem
\begin{equation}\label{interop} L(\phi)+N(\phi)=\tilde R,\quad \phi\in K^\perp\end{equation}
where, recalling Lemma \ref{aux}, $$\tilde R=\Pi^\perp\({i^*_{p}}\big(R_\lambda\big)\)=
\Pi^\perp\(PW_\lambda -{i^*_{p}}\big(\lambda|x|^{2(\alpha-1)}e^{P W_\lambda}\big)\).$$
We need the following auxiliary lemma.
\begin{lemma}\label{auxnonl} Let $r>0$ be a fixed number.
For any $p> 1$ there exists $\lambda_0>0$ such that for any $\lambda\in (0,\lambda_0)$, any $b\in \mathbb{R}^2$ with $|b|\leq r \sqrt{\lambda}$ and any $\phi_1,\phi_2\in H_0^1(\Omega)$ with $\|\phi\|_1,\,\|\phi_2\|<1$ the following holds
\begin{equation}\label{skate1}\|e^{\phi_1}-\phi_1-e^{\phi_2}+\phi_2\|_p\leq C(\|\phi_1\|+\|\phi_2\|)\|\phi_1-\phi_2\|,\end{equation}
\begin{equation}\label{skate2}\|N(\phi_1)-N(\phi_2)\|\leq C\lambda^{\frac{1-p^2}{{\alpha} p^2}}(\|\phi_1\|+\|\phi_2\|)\|\phi_1-\phi_2\|.\end{equation}
\end{lemma}
\begin{proof}
A straightforward computation give that the inequality $|e^a-a-e^b+b|\leq e^{|a|+|b|}(|a|+|b|)|a-b|$ holds for all $a,b\in \mathbb{R}$. Then, by applying H\"older's inequality with $\frac1q+\frac1r+\frac1t=1$, we derive
$$\|e^{\phi_1}-\phi_1-e^{\phi_2}+\phi_2\|_p\leq C\|e^{|\phi_1|+|\phi_2|}\|_{pq}(\|\phi_1\|_{pr}+\|\phi_2\|_{pr})\|\phi_1-\phi_2\|_{pt}$$
and \eqref{skate1} follows by using Lemma \ref{tmt} and the continuity of the embeddings $H^1_0(\Omega)\subset L^{pr}(\Omega)$ and $H^1_0(\Omega)\subset L^{pt}(\Omega)$.
Let us prove \eqref{skate2}. According to \eqref{isp} we get
$$\|N(\phi_1)-N(\phi_2)\|\leq C\|\lambda V(x)|x|^{2(\alpha-1)} e^{PW_\lambda}(e^{\phi_1}-\phi_1-e^{\phi_2}+\phi_2)\|_p,$$
and by H\"older's inequality with $\frac1p+\frac1q=1$ we derive
$$\begin{aligned}\|N(\phi_1)-N(\phi_2)\|&\leq C\|\lambda |x|^{2(\alpha-1)}e^{PW_\lambda}\|_{p^2}\|e^{\phi_1}-\phi_1-e^{\phi_2}+\phi_2|\|_{pq}\\ &\leq C\|\lambda |x|^{2(\alpha-1)}e^{PW_\lambda}\|_{p^2}(\|\phi_1\|+\|\phi_2\|)\|\phi_1-\phi_2\|\end{aligned} $$ by \eqref{skate1}, and the conclusion follows recalling \eqref{judo}.
\end{proof}
Problem \eqref{inter} or, equivalently, problem \eqref{interop}, turns out to be solvable for any choice of point $b$ with $|b|\leq r\sqrt{\lambda}$, provided that $\lambda$ is sufficiently small. Indeed we have the following result.
\begin{prop}\label{nonl} Let $r>0$ be fixed.
For any $\varepsilon\in (0,\frac1\alpha)$ there exists $\lambda_0>0$ such that for any $\lambda\in (0,\lambda_0)$ and any $b\in \mathbb{R}^2$ with $|b|<r\sqrt\lambda$ there is a unique $\phi_\lambda=\phi_{\lambda,b}\in K^\perp$ satisfying \eqref{inter} for some $c_1,c_2\in \mathbb{R}$ and
$$\|\phi_{\lambda}\|\leq \lambda^{\frac{1}{\alpha}-\varepsilon}.$$ \end{prop} \begin{proof} Since, as we have observed, problem \eqref{interop} is equivalent to problem \eqref{inter}, we will show that problem \eqref{interop} can be solved via a contraction mapping argument. Indeed, in virtue of Proposition \ref{ex}, let us introduce the map $$T:=L^{-1}(\tilde R-N(\phi)),\quad \phi\in K^\perp.$$
Let us fix $$0<\eta<\min\Big\{\varepsilon,\frac1\alpha-\varepsilon\Big\}$$ and $p>1$ sufficiently close to 1. According to \eqref{isp} and Lemma \ref{aux} we have
\begin{equation}\label{non1}\|\tilde R\|=O(\lambda^{\frac1\alpha-\eta}).\end{equation} Similarly, by \eqref{skate2}, choosing $p>1$ sufficiently close to 1, we get
\begin{equation}\label{non2}\|N(\phi_1)-N(\phi_2)\|\leq C\lambda^{-\eta}(\|\phi_1\|+\|\phi_2\|)\|\phi_1-\phi_2\|\quad \forall \phi_1,\phi_2\in H_0^1(\Omega), \|\phi_1\|,\|\phi_2\|<1.\end{equation} In particular, by taking $\phi_2=0$,
\begin{equation}\label{non3}\|N(\phi)\|\leq C\lambda^{-\eta}\|\phi\|^2\quad \forall \phi\in H_0^1(\Omega), \|\phi\|<1.\end{equation}
We claim that $T$ is a contraction map over the ball $$\Big\{\phi\in K^\perp\,\Big|\, \|\phi\|\leq \lambda^{\frac1\alpha-\varepsilon}\Big\}$$ provided that $\lambda$ is small enough. Indeed, combining Proposition \ref{ex}, \eqref{non1}, \eqref{non2}, \eqref{non3} with the choice of $\eta$, we have
$$\|T(\phi)\|\leq C|\log\lambda|(\lambda^{\frac1\alpha-\eta}+\lambda^{-\eta}\|\phi\|^2)<\lambda^{\frac1\alpha-\varepsilon},$$
$$\begin{aligned}\|T(\phi_1)-T(\phi_2)\|&\leq C|\log\lambda|\|N(\phi_1)-N(\phi_2)\|\leq C\lambda^{-\eta}|\log\lambda| (\|\phi_1\|+\|\phi_2\|)\|\phi_1-\phi_2\|\\ &<\frac12\|\phi_1-\phi_2\|.\end{aligned}$$
\end{proof}
\section{ Proof of Theorems \ref{th1}-\ref{th2} and Theorem \ref{main1}-\ref{main2}} After problem \eqref{inter} has been solved according to Proposition \ref{nonl}, then we find a solution to the original problem \eqref{proreg} if $b$ is such that $$c_j=0\hbox{ for }j=1,2.$$
Let us find the condition satisfied by $b$ in order to get the $c_j$'s equal to zero.
\subsection*{Proof of Theorems \ref{main1}-\ref{main2}}
We multiply the equation in \eqref{inter} by $PZ_\lambda^j$ and integrate over $\Omega$:
\begin{equation}\label{masca}\begin{aligned}\int_\Omega \nabla (PW_\lambda+\phi_{\lambda}) \nabla PZ^j_{\lambda} dx &-\lambda \int_\Omega V(x) |x|^{2(\alpha-1)}e^{P W_\lambda+\phi_{\lambda}}PZ^j_{\lambda} dx\\&=\sum_{h=1,2}c_h \int_\Omega Z_\lambda^h|x|^{2(\alpha-1)} e^{W_\lambda}PZ_\lambda^j dx.\end{aligned} \end{equation} The object is now to expand each integral of the above identity and analyze the leading term.
Let us begin by observing that the orthogonality in \eqref{inter} gives
\begin{equation}\label{masca1} \int_\Omega \nabla \phi_{\lambda} \nabla PZ^j_{\lambda} dx =\int_\Omega |x|^{2(\alpha-1)} e^{W_\lambda} \phi_{\lambda}Z_\lambda^j dx=0\end{equation} and, by \eqref{zeta1}-\eqref{zeta2}, \begin{equation}\label{masca2}\int_\Omega Z_\lambda^h|x|^{2(\alpha-1)} e^{W_\lambda}PZ_\lambda^j dx= \int_\Omega \nabla PZ_\lambda^h \nabla PZ^j_{\lambda} dx=\left\{\begin{aligned}&\frac23 {\pi{\alpha}}+o(1) &\hbox{ if }h=j\\ &o(1)&\hbox{ if }h\neq j\end{aligned}\right..\end{equation}
Using the expansion \eqref{cuore} we get
\begin{equation}\label{coll}\begin{aligned}&\int_\Omega \nabla PW_\lambda \nabla PZ^j_{\lambda} dx -\lambda \int_\Omega V(x) |x|^{2(\alpha-1)}e^{P W_\lambda}PZ^j_{\lambda} dx\\ &=\int_\Omega |x|^{2(\alpha-1)} e^{W_\lambda} PZ^j_{\lambda} dx-\lambda \int_\Omega V(x)|x|^{2(\alpha-1)}e^{P W_\lambda}PZ^j_{\lambda} dx\\ &=\int_\Omega |x|^{2(\alpha-1)}e^{W_\lambda} \Big(1-\frac{a(x)}{a(0)}e^{-4\pi({\alpha-1}) (H(x,0)-H(0,0))+8\pi\sum_{i=0}^{\alpha-1}(H(x, \beta_i) -H(0,\beta_i))+O(\delta^{2\alpha})}\Big)PZ^j_{\lambda} dx
.\end{aligned}\end{equation} Recalling \eqref{expa} and Lemma \ref{robin10} we deduce $$\begin{aligned}&-4\pi({\alpha}-1) (H(x,0)-H(0,0))+8\pi\sum_{i=0}^{\alpha-1}(H(x, \beta_i)-H(0,\beta_i))\\ &= 4\pi(\alpha+1) \sum _{k=1}^{2}\frac{1}{k!}{\rm Re}\Big(\frac{d^k\tilde H}{dx^k} (0,0)x^k\Big)
+\frac{8\pi}{(\alpha-1)!}{\rm Re}\bigg(\frac{\partial^{\alpha+1}\tilde H}{\partial p^\alpha\partial x}(0,0)bx\bigg)\\ &\;\;\;\;+O(|b||x|^2)+O(|b|^2|x|)+O(|x|^3).\end{aligned} $$
Consequently using the Taylor expansion $e^y=1+y+\frac{y^2}{2}+O(|y|^3)$, $$\begin{aligned}&e^{-4\pi({\alpha-1}) (H(x,0)-H(0,0))+8\pi\sum_{i=0}^{\alpha-1}(H(x, \beta_i) -H(0,\beta_i))+O(\delta^{2\alpha})}\\ &=1+4\pi(\alpha+1) \sum _{k=1}^{2}\frac{1}{k!}{\rm Re}\Big(\frac{d^k\tilde H}{dx^k} (0,0)x^k\Big)+\frac{8\pi}{(\alpha-1)!}{\rm Re}\bigg(\frac{\partial^{\alpha+1}\tilde H}{\partial p^\alpha\partial x}(0,0)bx\bigg)\\ &\;\;\;\;
+\frac12\bigg(4\pi(\alpha+1) \sum _{k=1}^{2}\frac{1}{k!}{\rm Re}\Big(\frac{d^k\tilde H}{dx^k} (0,0)x^k\Big)\bigg)^2\\ &\;\;\;\;+O(|b||x|^2)+O(|b|^2|x|)+O(|x|^3)+O(\delta^{2\alpha}) \\ &=1
+4\pi(\alpha+1) \sum _{k=1}^{2}\frac{1}{k!}{\rm Re}\Big(\frac{d^k\tilde H}{dx^k} (0,0)x^k\Big)+\frac{8\pi}{(\alpha-1)!}{\rm Re}\bigg(\frac{\partial^{\alpha+1}\tilde H}{\partial p^\alpha\partial x}(0,0)bx\bigg)\\ &\;\;\;\;+8\pi^2(\alpha+1)^2 \bigg({\rm Re}\Big(\frac{d\tilde H}{dx} (0,0)x\Big)\bigg)^2\\ &\;\;\;\; +O(|b||x|^2)+O(|b|^2|x|)+O(|x|^3)+O(\delta^{2\alpha}) . \end{aligned}$$ By assumptions \eqref{assurd0} and \eqref{assurd}-\eqref{assurd2} in Theorems \ref{th1} and \ref{th2}, respectively, taking into account that ${\rm Re}(\frac{d\tilde H}{dx} (0,0)x)=\langle \nabla_xH(0,0), x\rangle$, we get $$\begin{aligned}\frac{a(x)}{a(0)}&=1+\frac{\langle \nabla a(0), x\rangle}{a(0)} +\frac{1}{2a(0)}\Big( a_{11}x_1^2+a_{22}x_2^2
\Big)+O(|x|^3)\\ &=1-4\pi(\alpha+1){\rm Re}\Big(\frac{d\tilde H}{dx} (0,0)x\Big) +\frac{1}{2a(0)}\Big( a_{11}({\rm Re} \,x)^2+a_{22}({\rm Im} \,x)^2
\Big)+O(|x|^3),
\end{aligned}$$
and then we derive
\begin{equation}\label{insert}\begin{aligned}&\frac{a(x)}{a(0)}e^{-4\pi({\alpha-1}) (H(x,0)-H(0,0))+8\pi\sum_{i=0}^{\alpha-1}(H(x, \beta_i) -H(0,\beta_i))+O(\delta^{2\alpha})}\\ &=1 +2\pi(\alpha+1){\rm Re}\Big(\frac{d^2\tilde H}{dx^2} (0,0)x^2\Big) +\frac{8\pi}{(\alpha-1)!}{\rm Re}\bigg(\frac{\partial^{\alpha+1}\tilde H}{\partial p^\alpha\partial x}(0,0)bx\bigg)\\ &\;\;\;\; -8\pi^2(\alpha+1)^2 \bigg({\rm Re}\Big(\frac{d\tilde H}{dx} (0,0)x\Big)\bigg)^2\\ &
\;\;\;\; +\frac{1}{2a(0)}\Big( a_{11}({\rm Re} \,x)^2+a_{22}({\rm Im} \,x)^2
\Big)+O(|b||x|^2)+O(|b|^2|x|)+O(|x|^3)+O(\delta^{2\alpha}). \end{aligned}\end{equation} First let us assume that $\alpha\geq 3$: let us insert the above expansion into \eqref{coll} and, using Lemma \ref{copy0cor} and next Corollary \ref{corocoro} we get
\begin{equation}\label{coll2}\begin{aligned}&\int_\Omega \nabla PW_\lambda \nabla PZ^j_{\lambda} dx -\lambda \int_\Omega V(x) |x|^{2(\alpha-1)}e^{P W_\lambda}PZ^j_{\lambda} dx\\ &=8\pi^2(\alpha+1)^2\int_\Omega |x|^{2(\alpha-1)}e^{W_\lambda}PZ_\lambda^j \bigg({\rm Re}\Big(\frac{d\tilde H}{dx} (0,0)x\Big)\bigg)^2dx\\ &\;\;\;\;
-\frac{1}{2a(0)}\int_\Omega |x|^{2(\alpha-1)}e^{W_\lambda}PZ_\lambda^j \Big( a_{11}({\rm Re} \,x)^2+a_{22}({\rm Im} \,x)^2\Big)dx \\ &\;\;\;\;+ O(\delta^3)+O(\delta^2|b|)+O(\delta|b|^2)\\ &=
\frac{\delta^2}{2} \Bigg(8\pi^2(\alpha+1)^2\Big|\frac{d\tilde H}{dx} (0,0)\Big|^2- \frac{a_{11}+a_{22}}{2a(0)}\bigg) \int_{\mathbb{R}^2} |x|^{2\alpha}e^{W_\lambda}Z^j_\lambda dx\\ &\;\;\;\;+ O(\delta^3)+O(\delta^2|b|)+O(\delta|b|^2).\end{aligned}\end{equation}
We have thus obtained that if $\alpha\geq 3$ then
\begin{equation}\label{masca3}\begin{aligned} &\int_\Omega \nabla PW_\lambda \nabla PZ^j_{\lambda} dx -\lambda \int_\Omega V(x) |x|^{2(\alpha-1)}e^{P W_\lambda}PZ^j_{\lambda} dx\\ &=A\delta^2 F_j(\delta^{-\alpha}b)
+ O(\delta^3)+O(\delta^2|b|)+O(\delta|b|^2)
\end{aligned}\end{equation} where $$A:= 4\pi^2(\alpha+1)^2\Big|\frac{d\tilde H}{dx} (0,0)\Big|^2- \frac{a_{11}+a_{22}}{4a(0)}\neq 0$$ thanks to assumptions \eqref{assurd0} in Theorem \ref{th1} and $F=(F_1, F_2)$ is the map defined in Lemma \ref{finalaux}.
Next assume that $\alpha=2$. If $\Omega$ is $\ell$-symmetric for some $\ell\geq3$ in the sense of \eqref{assurd}, then $\tilde H(x,0)$ is $3$-symmetric too: $$\tilde H(e^{{\rm i}\frac{2\pi}{\ell}}x,0)= \tilde H(x,0)\quad \forall x\in\Omega;$$
this implies that its Taylor expansion at $0$ involves only the powers corresponding to integers multiples of $\ell$ and, consequently,
$$\frac{d\tilde H}{dx} (0,0)=\frac{d^2\tilde H}{dx^2} (0,0)=0.$$
Then let us insert \eqref{insert} into \eqref{coll} and, using Lemma \ref{copy0cor} and next Corollary \ref{corocorocoro} we get for $j=1$
$$\begin{aligned}&\int_\Omega \nabla PW_\lambda \nabla PZ^1_{\lambda} dx -\lambda \int_\Omega V(x) |x|^{2(\alpha-1)}e^{P W_\lambda}PZ^1_{\lambda} dx\\ &=
-\frac{1}{2a(0)}\int_\Omega |x|^{2(\alpha-1)}e^{W_\lambda}PZ_\lambda^1 \Big( a_{11}({\rm Re} \,x)^2+a_{22}({\rm Im} \,x)^2
\Big)dx + O(\delta^3)+O(\delta^2|b|)+O(\delta|b|^2)\\ &=
- \delta^2\frac{a_{11}+a_{22}}{4a(0)}\int_{\mathbb{R}^2} |x|^{2\alpha} e^{W_\lambda}Z^1_\lambda dx-\pi\alpha^2\delta^2\frac{a_{11}-a_{22}}{a(0)}
+ O(\delta^3)+O(\delta^2|b|)+O(\delta|b|^2)\end{aligned}$$ and, similarly for $j=2$
$$\begin{aligned}&\int_\Omega \nabla PW_\lambda \nabla PZ^2_{\lambda} dx -\lambda \int_\Omega V(x) |x|^{2(\alpha-1)}e^{P W_\lambda}PZ^2_{\lambda} dx\\ &=
-\frac{1}{2a(0)}\int_\Omega |x|^{2(\alpha-1)}e^{W_\lambda}PZ_\lambda^2 \Big( a_{11}({\rm Re} \,x)^2+a_{22}({\rm Im} \,x)^2
\Big)dx + O(\delta^3)+O(\delta^2|b|)+O(\delta|b|^2)\\ &=
- \delta^2\frac{a_{11}+a_{22}}{4a(0)} \int_{\mathbb{R}^2} |x|^{2\alpha} e^{W_\lambda}Z^2_\lambda dx
+ O(\delta^3)+O(\delta^2|b|)+O(\delta|b|^2).\end{aligned}$$
Therefore, using \eqref{assurd2} we conclude that \eqref{masca3} holds for any $\alpha\geq 2$ for some $A\neq 0$.
Finally let us fix $\varepsilon>0$ sufficiently small and $p>1$ sufficiently close to 1. Next let $1<q<\infty$ be such that $\frac1p+\frac1q=1$. Then, recalling that $\delta^{2\alpha}\sim\lambda$ according to \eqref{delta}, \eqref{skate1} with $\phi_2=0$ and Proposition \ref{nonl} give
$$\|e^{\phi_\lambda}-1-\phi_\lambda\|_q\leq C\|\phi\|^2\leq \delta^{4-4\alpha \varepsilon} $$ and, consequently,
\begin{equation}\label{tmtsur}\|e^{\phi_\lambda}-1\|_q\leq C\|\phi_\lambda\|\leq \delta^{2-{2\alpha} \varepsilon} .\end{equation}
Therefore, the orthogonality \eqref{masca1} and Lemma \ref{aux} imply
$$\begin{aligned}
\int_\Omega |x|^{2(\alpha-1)}e^{W_\lambda}(e^{\phi_\lambda}-1)Z^j_{\lambda} dx
& =\int_\Omega |x|^{2(\alpha-1)}e^{W_\lambda}(e^{\phi_\lambda}-1-\phi_\lambda)Z^j_{\lambda} dx\\ &
= O(\| |x|^{2(\alpha-1)}e^{ W_\lambda}(e^{\phi_\lambda}-1-\phi_\lambda)\|_1)\\ &=
O(\| e^{W_\lambda} |x|^{2(\alpha-1)}\|_p\|e^{\phi_\lambda}-1-\phi_\lambda\|_q)\\ &=O(\delta^{\frac{2}{p}-2} \delta^{4-4\alpha\varepsilon}) \end{aligned}$$ and, by using again Lemma \ref{aux} and \eqref{pzi},
\begin{equation}\label{masca4}\begin{aligned} \lambda \int_\Omega V(x) |x|^{2(\alpha-1)}e^{P W_\lambda}(e^{\phi}-1)PZ^j_{\lambda} dx&=\int_\Omega |x|^{2(\alpha-1)}e^{W_\lambda}(e^{\phi}-1)Z^j_{\lambda} dx+ O(\delta^{\frac{2}{p}+2-{2\alpha}\varepsilon}) \\ &=O(\delta^{\frac{2}{p}-2} \delta^{4-4\alpha \varepsilon})+ O(\delta^{\frac{2}{p}+2-{2\alpha}\varepsilon})=o(\delta^3) \end{aligned}\end{equation} provided that $\varepsilon$ is chosen sufficiently close to 0 and $p$ sufficiently close to 1.
In order to conclude, combining \eqref{masca1}, \eqref{masca2}, \eqref{masca3}, \eqref{masca4},
the identities \eqref{masca} turn out to be equivalent to the system \begin{equation}\label{tccone}\begin{aligned} &\delta^2A\big(F_1(\delta^{-\alpha}b)+O(\delta) \big)=\frac{2}{3}\pi\alpha c_1+o(c_1)+o(c_2), \\ & \delta^2A\big( F_2(\delta^{-\alpha}b)+O(\delta)
\big)=\frac{2}{3}\pi\alpha c_2+o(c_1)+o(c_2)\end{aligned}\end{equation} uniformly for $|b|\leq \delta^\alpha$. According to Lemma \ref{finalaux} we have $F(0,0)=(0,0)$ and $\det F(0,0)\neq0.$ Then the local invertibility theorem assures that $F$ is invertible in a small ball $B_r$ with center $0$ or, equivalently, $F(\delta^{-\alpha}b)$ is invertible in a the ball $B_{r\delta^{\alpha}}$, and hence $\deg(F(\delta^{-\alpha}b), B_{r\delta^{\alpha}},0)=1.$
Taking into account that $|F(\delta^{-\alpha}b)|\geq c$ for $|b|=r\delta^\alpha$, the continuity property of the topological degree gives that $\deg(F(\delta^{-\alpha}b)+O(\delta) , B_{r\delta^\alpha},0)>0$ for $\delta$ (hence $\lambda$) small enough. Then for such $\delta$ there exists $b\in B_{r}$ such that $$F(\delta^{-\alpha}b)+O(\delta)=0.$$ and so the linear system \eqref{tccone}
has only the trivial solution $c_1=c_2=0$. Finally $\delta^{-\alpha}|b|\sim |F(\delta^{-\alpha}b)|=O( \delta) ,$ hence $|b|=O(\delta^{\alpha+1})$. That concludes the proof of Theorems \ref{main1}-\ref{main2}.
\noindent{\bf{Proof of Theorems \ref{th1}-\ref{th2}.} }Theorems \ref{main1}-\ref{main2} provide a solution to the problem \eqref{proreg} of the form $$v_\lambda=PW_\lambda+\phi_\lambda$$ for some $b=b_\lambda$ with $|b_\lambda|=O(\delta^{\alpha+1})$.
So we have \begin{equation}\label{exan0}\sum_{k=0}^{\alpha-1} H(x,{\beta_i}_\lambda)=\alpha H(x,0)+O(\delta),\end{equation} \begin{equation}\label{exan}\log (\delta^{2\alpha}+|x^{\alpha}+ b_\lambda|^2)
=\log(\delta^{2\alpha}+|x|^{2\alpha})+O(\delta)\end{equation} uniformly for $x\in\overline\Omega$. Clearly, by \eqref{chva}, $$u_\lambda=v_\lambda-4\pi (\alpha-1) G(x,0)$$ solves equation \eqref{eq} and \eqref{the1} of Theorem \ref{th1} follows from \eqref{pro-exp1} and \eqref{exan0}\eqref{exan}. Moreover, using \eqref{judo} and \eqref{tmtsur}, by H\"older's inequality with $\frac1p+\frac1q=1$ we get
$$\begin{aligned}\lambda\||x|^{2(\alpha-1)}V(x)(e^{{v}_\lambda}- e^{{PW}_\lambda})\|_1&=\lambda\||x|^{2(\alpha-1)}V(x)e^{{PW}_\lambda}(e^{{\phi}_\lambda}-1)\|_{1}
\\ &\le\lambda \||x|^{2(\alpha-1)}V(x)e^{{PW}_\lambda}\|_p\|e^{\phi_\lambda}-1\|_q\\ &=O(\lambda^{\frac{1-p}{\alpha p}+\frac{1}{\alpha}-\varepsilon})=o(1), \end{aligned}$$ if $p$ is chosen sufficiently close to 1 and $\varepsilon$ sufficiently close to $0$. Then, by \eqref{quantum} and Lemma \ref{aux}
$$\begin{aligned}\lambda \int_\Omega a(x)e^{u_\lambda}dx&=\lambda \int_\Omega |x|^{2(\alpha-1)}V(x)e^{v_\lambda }dx=\lambda\int_\Omega|x|^{2(\alpha-1)}V(x)e^{{PW}_\lambda}dx +o(1)\\ &=\int_\Omega|x|^{2(\alpha-1)}e^{{W}_\lambda}dx+o(1)=\int_{\mathbb{R}^2}|x|^{2(\alpha-1)}e^{{W}_\lambda}dx+o(1)=8\pi\alpha+o(1). \end{aligned}$$ Similarly for every neighborhood $U$ of $0$ $$\lambda \int_U a(x)e^{u_\lambda}dx\to 8\pi\alpha. $$ So \eqref{the3} is verified and Theorem \ref{th1} is thus completely proved.
\begin{lemma}\label{copy0cor} Let $\alpha\geq 2$ and $\xi\in \mathbb{C}$. For any $\gamma=0,1,\ldots, \alpha-1$ the following holds:
$$\int_\Omega |x|^{2(\alpha-1)}e^{W_\lambda} PZ^j_\lambda{\rm Re}(\xi x^\gamma)dx =O(\delta^{\alpha+\gamma}),\quad \int_\Omega |x|^{2(\alpha-1)}e^{W_\lambda} PZ_\lambda^j{\rm Im}(\xi x^\gamma)dx=O(\delta^{\alpha+\gamma})$$ and
$$\int_\Omega |x|^{2(\alpha-1)}e^{W_\lambda} PZ^1_\lambda{\rm Re}(\xi x^\alpha)dx=4\pi\alpha^2\delta^\alpha{\rm Re}(\xi)+O(\delta^{2\alpha})$$ $$\int_\Omega |x|^{2(\alpha-1)}e^{W_\lambda} PZ_\lambda^1{\rm Im}(\xi x^\alpha)dx=4\pi\alpha^2\delta^\alpha{\rm Im}(\xi)+O(\delta^{2\alpha})$$
$$\int_\Omega |x|^{2(\alpha-1)}e^{W_\lambda} PZ^2_\lambda{\rm Re}(\xi x^\alpha)dx=-4\pi\alpha^2\delta^\alpha{\rm Im}(\xi)+O(\delta^{2\alpha})$$ $$\int_\Omega |x|^{2(\alpha-1)}e^{W_\lambda} PZ_\lambda^2{\rm Im}(\xi x^\alpha)dx=4\pi\alpha^2\delta^\alpha{\rm Re}(\xi)+O(\delta^{2\alpha})$$ uniformly for $b$ in a small neighborhood of $0$. \end{lemma} \begin{proof} Let us first show the identities for $j=1$ and $\xi=1$. By \eqref{pzi} for $\gamma=0,1,\ldots, \alpha$ we compute
$$\begin{aligned}&\int_\Omega |x|^{2(\alpha-1)}e^{W_\lambda} PZ^1_\lambda{\rm Re}(x^\gamma)dx\\&=8\alpha^2\delta^\gamma\int_{\frac{\Omega}{\delta}}\frac{|y|^{2(\alpha-1)}}{(1+|y^\alpha-\delta^{-\alpha}b|^2)^3}{\rm Re}(y^\alpha-\delta^{-\alpha}b){\rm Re}(y^\gamma) dy +O(\delta^{\alpha+\gamma}) \\ &=8\alpha^2\delta^\gamma\int_{\mathbb{R}^2}\frac{|y|^{2(\alpha-1)}}{(1+|y^\alpha-\delta^{-\alpha}b|^2)^3}{\rm Re}(y^\alpha-\delta^{-\alpha}b){\rm Re}(y^\gamma) dy+O(\delta^{\alpha+\gamma}).\end{aligned} $$ If $\gamma=1,\ldots, \alpha-1$ the thesis follows from Lemma \ref{copy0}. If $\gamma=0$, then by applying Lemma \ref{copy}
$$\begin{aligned}\int_{\mathbb{R}^2}\frac{|y|^{2(\alpha-1)}}{(1+|y^\alpha-\delta^{-\alpha}b|^2)^3}{\rm Re}(y^\alpha-\delta^{-\alpha}b) dy&=\frac{1}{\alpha}\int_{\mathbb{R}^2}\frac{1}{(1+|y-\delta^{-\alpha}b|^2)^3}{\rm Re}(y-\delta^{-\alpha}b) dy \\ &= \frac{1}{\alpha}\int_{\mathbb{R}^2}\frac{y_1}{(1+|y|^2)^3}dy=0\end{aligned}$$ and we get the first estimate for $\xi=1$. The second estimate with $\xi=1$ is analogous. Next, if $\gamma=\alpha$ then again by Lemma \ref{copy}
$$\begin{aligned}&\int_{\mathbb{R}^2}\frac{|y|^{2(\alpha-1)}}{(1+|y^\alpha-\delta^{-\alpha}b|^2)^3}{\rm Re}(y^\alpha-\delta^{-\alpha}b) {\rm Re}(y^\alpha)dy\\&=\frac{1}{\alpha}\int_{\mathbb{R}^2} \frac{1}{(1+|y-\delta^{-\alpha}b|^2)^3}{\rm Re}(y-\delta^{-\alpha}b) {\rm Re}(y)dy\\ &
=\frac{1}{\alpha}\int_{\mathbb{R}^2} \frac{1}{(1+|y|^2)^3}{\rm Re}(y) {\rm Re}(y+\delta^{-\alpha}b)dy\\ &
= \frac{1}{\alpha}\int_{\mathbb{R}^2} \frac{1}{(1+|y|^2)^3}y_1(y_1-\delta^{-\alpha}{\rm Re}(b))dy\\ &=\frac{1}{\alpha}\int_{\mathbb{R}^2} \frac{(y_1)^2}{(1+|y|^2)^3} dy -\delta^{-\alpha}{\rm Re}(b) \frac{1}{\alpha}\int_{\mathbb{R}^2} \frac{y_1}{(1+|y|^2)^3} dy \\ &=\frac{\pi}{2} \end{aligned} $$
since $\int_{\mathbb{R}^2} \frac{(y_1)^2}{(1+|y|^2)^3} dy=\frac{1}{2} \int_{\mathbb{R}^2} \frac{|y|^2}{(1+|y|^2)^3} dy=\frac{\pi}{2} $ and $\int_{\mathbb{R}^2} \frac{y_1}{(1+|y|^2)^3} dy=0$. Similarly $$\begin{aligned}&\int_{\mathbb{R}^2}\frac{|y|^{2(\alpha-1)}}{(1+|y^\alpha-\delta^{-\alpha}b|^2)^3}{\rm Re}(y^\alpha-\delta^{-\alpha}b) {\rm Im}(y^\alpha)dy\\&
=\frac{1}{\alpha}\int_{\mathbb{R}^2} \frac{1}{(1+|y|^2)^3}{\rm Re}(y) {\rm Im}(y+\delta^{-\alpha}b)dy\\ &
= \frac{1}{\alpha}\int_{\mathbb{R}^2} \frac{1}{(1+|y|^2)^3}y_1 (y_2+\delta^{-\alpha}{\rm Re}(b))dy\\ &=0 .\end{aligned}$$ Taking into account that $${\rm Re}(\xi x^\gamma)={\rm Re}(\xi) {\rm Re}(x^\gamma)-{\rm Im}(\xi){\rm Im}(x^\gamma),\quad {\rm Im}(\xi x^\gamma)={\rm Re}(\xi){\rm Im}(x^\gamma)+{\rm Im}(\xi){\rm Re}(x^\gamma)$$ we obtain the thesis for $j=1$ and any $\xi\in \mathbb{C}$. The remaining estimates with $j=2$ are analogous. \end{proof}
\begin{cor}\label{corocoro} Let $\alpha\geq 3$ and $\xi_1, \, \xi_2\in\mathbb{C}$. Then $$ \int_\Omega |x|^{2(\alpha-1)}e^{W_\lambda} PZ_\lambda^j{\rm Re}(\xi_1x){\rm Re}(\xi_2 x)dx=\frac{\delta^2}{2} \langle \xi_1, \xi_2\rangle\int_{\mathbb{R}^2} |x|^{2}e^{W_\lambda}Z_\lambda^jdx +O(\delta^{\alpha+2}),$$
$$ \int_\Omega |x|^{2(\alpha-1)}e^{W_\lambda} PZ_\lambda^j{\rm Im}(\xi_1x){\rm Im}(\xi_2 x)dx=\frac{\delta^2}{2} \langle \xi_1, \xi_2\rangle\int_{\mathbb{R}^2} |x|^{2\alpha}e^{W_\lambda}Z_\lambda^jdx+O(\delta^{\alpha+2})$$
uniformly for $b$ in a small neighborhood of $0$. \end{cor} \begin{proof}
Since $({\rm Re}(x))^2= \frac{|x|^2}{2}+\frac{{\rm Re}(x^2)}{2}$, according to Lemma \ref{copy0cor} we have
$$\begin{aligned}&\int_\Omega |x|^{2(\alpha-1)}e^{W_\lambda} PZ_\lambda^1\big({\rm Re}(x)\big)^2dx\\&=\frac12\int_\Omega |x|^{2(\alpha-1)}e^{W_\lambda} PZ_\lambda^1|x|^2dx+O(\delta^{\alpha+2})\\ &=4\alpha^2\delta^2\int_{\frac{\Omega}{\delta}}\frac{|y|^{2\alpha}}{(1+|y^\alpha-\delta^{-\alpha}b|^2)^3}{\rm Re}(y^\alpha-\delta^{-\alpha}b) dy+O(\delta^{\alpha+2}) \\ &=4\alpha^2\delta^2\int_{\mathbb{R}^2}\frac{|y|^{2\alpha}}{(1+|y^\alpha-\delta^{-\alpha}b|^2)^3}{\rm Re}(y^\alpha-\delta^{-\alpha}b) dy +O(\delta^{\alpha+2})\\ &
=\frac{\delta^2}{2} \int_{\mathbb{R}^2} |x|^{2\alpha}e^{W_\lambda}Z_\lambda^1dx+O(\delta^{\alpha+2}) . \end{aligned}$$
Similarly, using now $({\rm Im}(x))^2= \frac{|x|^2}{2}-\frac{{\rm Re}(x^2)}{2}$,
$$\begin{aligned}\int_\Omega |x|^{2(\alpha-1)}e^{W_\lambda} PZ_\lambda^1\big({\rm Im}(x)\big)^2dx&=\frac{\delta^2}{2} \int_{\mathbb{R}^2} |x|^{2\alpha}e^{W_\lambda}Z_\lambda^1dx +O(\delta^{\alpha+2}). \end{aligned}$$ Moreover, since ${\rm Re}(x){\rm Im}(x)=\frac{{\rm Im}(x^2)}{2}$, by Lemma \ref{copy0cor}
$$\begin{aligned}\int_\Omega |x|^{2(\alpha-1)}e^{W_\lambda} PZ_\lambda^1{\rm Re}(x){\rm Im}(x)dx= O(\delta^{\alpha+2}).\end{aligned}$$ The thesis follows for $j=1$ since ${\rm Re}(\xi x)={\rm Re}(\xi) {\rm Re}(x)-{\rm Im}(\xi){\rm Im}(x)$. The proof for $j=2$ follows analogously.
\end{proof}
\begin{cor}\label{corocorocoro} Let $\alpha=2$ and $\xi_1, \, \xi_2\in\mathbb{C}$. Then $$ \begin{aligned}\int_\Omega |x|^{2(\alpha-1)}e^{W_\lambda} PZ_\lambda^1{\rm Re}(\xi_1x){\rm Re}(\xi_2 x)dx&=\frac{\delta^2}{2}\langle \xi_1, \xi_2\rangle \int |x|^{2\alpha} e^{W_\lambda} Z_\lambda^1dx\\ &\;\;\;\;+2\pi\alpha^2\delta^2{\rm Re}(\xi_1 \xi_2) +O(\delta^{\alpha+2}),\end{aligned}$$
$$ \begin{aligned}\int_\Omega |x|^{2(\alpha-1)}e^{W_\lambda} PZ_\lambda^2{\rm Re}(\xi_1x){\rm Re}(\xi_2 x)dx&=\frac{\delta^2}{2}\langle \xi_1, \xi_2\rangle \int |x|^{2\alpha} e^{W_\lambda} Z_\lambda^2dx\\ &\;\;\;\;-2\pi\alpha^2\delta^2{\rm Im}(\xi_1 \xi_2) +O(\delta^{\alpha+2}),\end{aligned}$$
$$\begin{aligned} \int_\Omega |x|^{2(\alpha-1)}e^{W_\lambda} PZ_\lambda^1{\rm Im}(\xi_1x){\rm Im}(\xi_2 x)dx&=\frac{\delta^2}{2}\langle \xi_1, \xi_2\rangle \int |x|^{2\alpha} e^{W_\lambda} Z_\lambda^jdx \\ &\;\;\;\;-2\pi\alpha^2\delta^2{\rm Re} (\xi_1, \xi_2)+O(\delta^{\alpha+2})\end{aligned}$$
$$\begin{aligned} \int_\Omega |x|^{2(\alpha-1)}e^{W_\lambda} PZ_\lambda^2{\rm Im}(\xi_1x){\rm Im}(\xi_2 x)dx&=\frac{\delta^2}{2}\langle \xi_1, \xi_2\rangle \int |x|^{2\alpha} e^{W_\lambda} Z_\lambda^jdx \\ &\;\;\;\;+2\pi\alpha^2\delta^2{\rm Im}(\xi_1\xi_2)+O(\delta^{\alpha+2})\end{aligned}$$
uniformly for $b$ in a small neighborhood of $0$. \end{cor} \begin{proof}
Since $({\rm Re}(x))^2= \frac{|x|^2}{2}+\frac{{\rm Re}(x^2)}{2}$, according to Lemma \ref{copy0cor} we have
$$\begin{aligned}&\int_\Omega |x|^{2(\alpha-1)}e^{W_\lambda} PZ_\lambda^1\big({\rm Re}(x)\big)^2dx\\&=\frac12\int_\Omega |x|^{2(\alpha-1)}e^{W_\lambda} PZ_\lambda^1|x|^2dx+2\pi\alpha^2\delta^2+O(\delta^{\alpha+2})\\ &=4\alpha^2\delta^2\int_{\frac{\Omega}{\delta}}\frac{|y|^{2\alpha}}{(1+|y^\alpha-\delta^{-\alpha}b|^2)^3}{\rm Re}(y^\alpha-\delta^{-\alpha}b) dy+2\pi\alpha^2\delta^2+O(\delta^{\alpha+2}) \\ &=4\alpha^2\delta^2\int_{\mathbb{R}^2}\frac{|y|^{2\alpha}}{(1+|y^\alpha-\delta^{-\alpha}b|^2)^3}{\rm Re}(y^\alpha-\delta^{-\alpha}b) dy +2\pi\alpha^2\delta^2+O(\delta^{\alpha+2})
\\ &=\frac{\delta^2}{2}\int |x|^{2\alpha} e^{W_\lambda} Z_\lambda^1dx+2\pi\alpha^2\delta^2+O(\delta^{\alpha+2}) . \end{aligned}$$
Similarly, using now $({\rm Im}(x))^2= \frac{|x|^2}{2}-\frac{{\rm Re}(x^2)}{2}$,
$$\begin{aligned}\int_\Omega |x|^{2}e^{W_\lambda} PZ_\lambda^1\big({\rm Im}(x)\big)^2dx&=\frac{\delta^2}{2}\int |x|^{2\alpha} e^{W_\lambda} Z_\lambda^jdx-2\pi\alpha^2\delta^2+O(\delta^{\alpha+2}). \end{aligned}$$ Moreover, since ${\rm Re}(x){\rm Im}(x)=\frac{{\rm Im}(x^2)}{2}$, by Lemma \ref{copy0cor}
$$\begin{aligned}\int_\Omega |x|^{2}e^{W_\lambda} PZ_\lambda^1{\rm Re}(x){\rm Im}(x)dx= O(\delta^{\alpha+2}).\end{aligned}$$ The first estimate follows since ${\rm Re}(\xi x)={\rm Re}(\xi) {\rm Re}(x)-{\rm Im}(\xi){\rm Im}(x)$ and ${\rm Im}(\xi x)={\rm Re}(\xi){\rm Im}(x)+{\rm Im}(\xi){\rm Re}(x).$ The remaining estimates follow analogously.
\end{proof}
\begin{lemma}
\label{finalaux}Let ${\alpha}\geq 2$ be an integer and let $F:\mathbb{R}^2\to \mathbb{R}^2$ be defined by$$F(B)=\left(\begin{aligned} &\int_{\mathbb{R}^2} \frac{|y|^{2\alpha}}{(1+|y^{\alpha }-B|^2)^3}{\rm Re} (y^{\alpha }-B)dy\\ &\int_{\mathbb{R}^2} \frac{|y|^{2\alpha}}{(1+|y^{\alpha }-B|^2)^3}{\rm Im} (y^{\alpha }-B)dy\end{aligned}\right).$$ Then $F(0)=0$ and $det (DF(0))\neq 0$. \end{lemma}
\begin{proof} According to Lemma \ref{copy} we have
$$\int_{\mathbb{R}^2} \frac{|y|^{2\alpha}}{(1+|y|^{{2\alpha}})^3}{\rm Re} (y^{\alpha })dy=\frac{1}{\alpha}\int_{\mathbb{R}^2} \frac{|y|^{2/\alpha}}{(1+|y|^2)^3}{\rm Re} (y)dy=\frac{1}{\alpha}\int_{\mathbb{R}^2} \frac{|y|^{2/\alpha}}{(1+|y|^2)^3}y_1dy=0
.$$ Similarly
$$\int_{\mathbb{R}^2} \frac{|y|^{2\alpha}}{(1+|y|^{{2\alpha}})^3}{\rm Im} (y^{\alpha })dy=\frac{1}{\alpha}\int_{\mathbb{R}^2} \frac{|y|^{2/\alpha}}{(1+|y|^2)^3}y_2dy=0$$
by which we immediately get $F(0,0)=0$. Moreover $$\begin{aligned}\frac{\partial F_1}{\partial B_2}(0,0)&=6\int_{\mathbb{R}^2}\frac{|y|^{2\alpha}}{(1+|y^{\alpha }|^2)^4}{\rm Im}(y^\alpha){\rm Re} (y^{\alpha })dy=\frac{6}{\alpha}\int_{\mathbb{R}^2} \frac{|y|^{2/\alpha}}{(1+|y|^2)^4}{\rm Im} (y){\rm Re}(y) dy
\\ &=\frac{6}{\alpha}\int_{\mathbb{R}^2} \frac{|y|^{2/\alpha}}{(1+|y|^2)^4}y_1y_2 dy=0.\end{aligned}$$ Similarly $\frac{\partial F_2}{\partial B_1}(0,0)=0$.
So $DF$ is a diagonal matrix. We compute
$$\begin{aligned}\frac{\partial F_1}{\partial B_1}(0,0)&=-\int_{\mathbb{R}^2}|y|^{2\alpha}\frac{1+|y|^{2\alpha}-6({\rm Re} (y^{\alpha }))^2}{(1+|y|^{2\alpha})^4} dy \\ &
=-\frac{1}{\alpha}\int_{\mathbb{R}^2}|y|^{\frac{2}{\alpha}}\frac{1+|y|^{2}-6y_1^2}{(1+|y|^{2})^4} dy .\end{aligned}$$
Using that $\int_{\mathbb{R}^2}|y|^{\frac{2}{\alpha}}\frac{y_1^2}{(1+|y|^{2})^4} =\frac12 \int_{\mathbb{R}^2}|y|^{\frac{2}{\alpha}}\frac{|y|^2}{(1+|y|^{2})^4} $ we get
$$\begin{aligned}\frac{\partial F_1}{\partial B_1}(0,0)&=\frac1\alpha\int_{\mathbb{R}^2}|y|^{\frac2\alpha}\frac{2|y|^2-1}{(1+|y|^{2})^4} dy. \end{aligned}$$
Proceeding similarly as above we get
$$\frac{\partial F_2}{\partial B_2}(0,0)=\frac{\partial F_1}{\partial B_1}(0,0)=\frac1\alpha\int_{\mathbb{R}^2}|y|^{\frac2\alpha}\frac{2|y|^2-1}{(1+|y|^{2})^4} dy.$$ Then the thesis will follow once we have proved the nonvanishing of the above integral:
\begin{equation}\label{nonv} \int_{\mathbb{R}^2}|y|^{\frac2\alpha}\frac{2|y|^2-1}{(1+|y|^{2})^4} dy\neq 0.\end{equation} To see this, let us first compute
$$\begin{aligned}\frac{1}{2\pi}\int_{\mathbb{R}^2}\frac{2|y|^2-1}{(1+|y|^{2})^4} dy&=\int_0^{+\infty} \rho\frac{2\rho^2-1}{(1+\rho^{2})^4} d\rho\\ &= 2\int_0^{+\infty} \rho\frac{1}{(1+\rho^{2})^3} d\rho-3\int_0^{+\infty} \rho\frac{1}{(1+\rho^{2})^4} d\rho=0
\end{aligned}
$$ by direct integration. So, using that $(\sqrt2 |y|)^{\frac2\alpha}\leq 1$ if $2|y|^2-1\leq 0$ and $(\sqrt2 |y|)^{\frac2\alpha}> 1$ if $2|y|^2-1> 0$ and
$$\begin{aligned}\int_{\mathbb{R}^2}|y|^{\frac2\alpha}\frac{2|y|^2-1}{(1+|y|^{2})^4} dy
= (\sqrt2)^{-\frac2\alpha}\int_{\mathbb{R}^2}(\sqrt2|y|)^{\frac2\alpha}\frac{2|y|^2-1}{(1+|y|^{2})^4} dy
> (\sqrt2)^{-\frac2\alpha} \int_{\mathbb{R}^2}\frac{2|y|^2-1}{(1+|y|^{2})^4} dy=0\end{aligned} $$ and \eqref{nonv} follows. \end{proof}3
\appendix \renewcommand{\Alph{section}.\arabic{equation}}{\Alph{section}.\arabic{equation}} \section{}
This appendix is devoted to deduce some integral identities associated to the change of variables: $x\mapsto x^\alpha$.
\begin{lemma}\label{copy}
For any $f\in L^1(\mathbb{R}^2)$ we have that $|y|^{2(\alpha-1)}f(y^\alpha)\in L^1 (\mathbb{R}^2) $ and \begin{equation}\label{normeqq}\int_{\mathbb{R}^2} |y|^{2(\alpha-1)} f(y^\alpha) dy=\frac1\alpha\int_{\mathbb{R}^2} f(y )dy.\end{equation} \end{lemma} \begin{proof} It is sufficient to prove the thesis for a smooth function $f$. Using the polar coordinates $(\rho,\theta)$ and then applying the change of variables $(\rho',\theta')=(\rho^\alpha,\alpha\theta)$ we get
$$\begin{aligned} \int_{\mathbb{R}^2} |y|^{2(\alpha-1)} f(y^\alpha) dy&=\int_0^{+\infty}d\rho\int_0^{2\pi} \rho^{2\alpha-1}f(\rho^\alpha e^{{\rm i}\alpha \theta}) d\theta \\ &= \frac{1}{\alpha^2}\int_0^{+\infty}d\rho'\int_0^{2\alpha \pi} \rho'f(\rho' e^{{\rm i}\theta'}) d\theta' \\ &=\frac{1}{\alpha}\int_0^{+\infty}d\rho'\int_0^{2\pi} \rho'f(\rho' e^{{\rm i}\theta'}) d\theta' =\frac{1}{\alpha}\int_{\mathbb{R}^2} f(y) dy.\end{aligned}$$ \end{proof}
\begin{lemma}\label{copy0}
Let $\gamma=1, \ldots, \alpha-1$ and let $f$ be such that $f(y)|y|^{\frac\gamma\alpha}\in L^1(\mathbb{R}^2)$. Then $$ \int_{\mathbb{R}^2}
|y|^{2(\alpha-1)}f(y^\alpha) {\rm Re} (y^{\gamma})dy=\int_{\mathbb{R}^2} |y|^{2(\alpha-1)} f(y) {\rm Im} (y^{\gamma})dy=0.$$ \end{lemma}
\begin{proof} Observe first that according to Lemma \ref{copy} we have $|y|^{2(\alpha-1)}f(y^\alpha) {\rm Re} (y^{\gamma})\in L^1(\mathbb{R}^2)$.
Suppose that $f$ is a smooth function. Using the polar coordinates $(\rho,\theta)$ we get $$\begin{aligned} \int_{\mathbb{R}^2} |y|^{2(\alpha-1)}f(y){\rm Re} (y^{\gamma})dy&=\int_0^{+\infty}|\rho|^{2\alpha-1+\gamma}d\rho\int_0^{2\pi}\cos(\gamma\theta)f(\rho^\alpha e^{{\rm i}\alpha \theta})d\theta.\end{aligned}$$ On the other hand $$\begin{aligned}\int_0^{2\pi}\cos(\gamma\theta)f(\rho^\alpha e^{{\rm i}\alpha \theta})d\theta &= \sum_{k=0}^{\alpha-1}\int_{\frac{2\pi}{\alpha}k}^{\frac{2\pi}{\alpha}(k+1)}\cos(\gamma\theta)f(\rho^\alpha e^{{\rm i}\alpha \theta})d\theta\\ &= \sum_{k=0}^{\alpha-1}\int_0^{\frac{2\pi}{\alpha}}\cos\Big(\gamma\Big(\theta+\frac{2\pi}{\alpha}k\Big)\Big)f(\rho^\alpha e^{{\rm i}\alpha \theta})d\theta \\ &=\sum_{k=0}^{\alpha-1}\cos\Big(\gamma\frac{2\pi}{\alpha}k\Big)\int_0^{\frac{2\pi}{\alpha}}\cos(\gamma\theta)f(\rho^\alpha e^{{\rm i}\alpha \theta})d\theta\\ &\;\;\;-\sum_{k=0}^{\alpha-1}\sin\Big(\gamma\frac{2\pi}{\alpha}k\Big)\int_0^{\frac{2\pi}{\alpha}}\sin(\gamma\theta)f(\rho^\alpha e^{{\rm i}\alpha \theta})d\theta. \end{aligned}$$ The well known identity $\sum_{k=0}^{\alpha-1}e^{{\rm i}\frac{2\pi}{\alpha}\gamma k}=0$ for all $\gamma=1,\ldots,\alpha-1$ implies $$\sum_{k=0}^{\alpha-1}\cos\Big(\gamma\frac{2\pi}{\alpha}k\Big)=\sum_{k=0}^{\alpha-1}\sin\Big(\gamma\frac{2\pi}{\alpha}k\Big)=0.$$ and the first identity follows. The second identity is analogous..
\end{proof}
\section{}
In this appendix we carry out some asymptotic expansions involving the regular part $H(x,y)$ of the Green's function. Recalling that for any fixed $p\in\Omega$ the function $H_p:x\mapsto H(x,p)$ is harmonic in $\Omega$, then it admits a holomorphic extension $$\tilde H_p(x)=H_p(x)+{\rm i}h_p(x)\;\hbox{ in } U,\;\; h_p(p)=0\qquad x\approx x_1+{\rm i} x_2, $$ where $U$ is any fixed round neighborhood of $0$. Setting $$\tilde H(x,p)=\tilde H_p(x)\quad \forall x,p\in U$$ by the symmetry $H(x,p)=H(p,x)$ we also deduce the analogous following symmetry for $\tilde H$: $$\tilde H(x,p)= \tilde H(p,x)\quad \forall x,p\in U.$$ In the following we denote by $\frac{d}{dx}$ and $\frac{d}{dp}$ the (complex) derivative with respect to the first and the second variable of the function $\tilde H(\cdot,\cdot)$, respectively. Therefore the Taylor expansion of $H_p(x)$ up to the order $m$ takes the form: \begin{equation}\label{expa}H(x,p)-H(0,p)=H_p(x)-H_p(0)=
\sum_{k=1}^{m} \frac{1}{k!}{\rm Re}\Big(\frac{d^k\tilde H}{dx^k} (0,p) x^k\Big)+O(|x|^{m+1})\end{equation} uniformly for $x\in\Omega$ and $p\in U$, where $$ \frac{d \tilde H }{dx}(0,p)=\frac{\partial^i H_p }{dx_1}(0)-{\rm i} \frac{\partial^i H_p }{dx_2}(0),$$ $$\frac{d^k \tilde H }{\partial x^k} (0,p) =\frac{d^k \tilde H_p }{dx^k}(0)=\frac{\partial^k H_p }{dx_1^k}(0)-{\rm i} \frac{\partial^k H_p }{dx_2dx_1^{k-1}}(0)\quad \forall k\geq 2.$$
\begin{lemma}\label{robin10} Using the same notation $b, \beta_i$ of the introduction, the following holds: $$\begin{aligned}\sum _{i=0}^{\alpha-1} (H(x,\beta_i)-H(0,\beta_i))=& \alpha \sum _{k=1}^{\alpha}\frac{1}{k!}{\rm Re}\Big(\frac{d^k\tilde H}{dx^k} (0,0)x^k\Big)+\frac{1}{(\alpha-1)!}\sum _{k=1}^{\alpha}\frac{1}{k!}{\rm Re}\Big(\frac{\partial^{k+\alpha}\tilde H}{\partial p^\alpha\partial x^k}(0,0)b\,x^k\Big)\\ &
+O(|b|^{2}|x|)+O(|x|^{\alpha+1}).\end{aligned}
$$
uniformly for $b\in U$ and $x\in\Omega$. \end{lemma} \begin{proof} According to \eqref{expa} we compute
\begin{equation}\label{symmmm}\sum_{i=1}^\alpha\Big(H(x,\beta_i)-H(0,\beta_i)\Big)=\sum_{k=1}^{\alpha} \frac{1}{k!}{\rm Re}\Big(\sum_{i=1}^\alpha\frac{d^k\tilde H}{dx^k} (0,\beta_i) x^k\Big)+O(|x|^{\alpha+1}).\end{equation}
Let us expand the complex function $\frac{d^k\tilde H}{dx^k} (0,\beta_i)$:
$$\frac{d^k\tilde H}{dx^k} (0,\beta_i)=\frac{d^k\tilde H}{dx^k} (0,0)+\sum_{h=1}^{2\alpha-1}\frac{1}{h!}\frac{\partial^{k+h}\tilde H}{\partial p^h\partial x^k}(0,0)\beta_i^h+O(|\beta_i|^{2\alpha}).$$ Next we use that $ \sum _{i=0}^{\alpha-1} \beta_i^h=0$ for any $h$ which is not an integer multiple of $\alpha$, whereas $\beta_i^{\alpha j}=b^j$ for any integer $j$, by which $$\sum_{i=0}^{\alpha-1}\frac{d^k\tilde H}{dx^k} (0,\beta_i)=\alpha \frac{d^k\tilde H}{dx^k} (0,0)+\frac{1}{(\alpha-1)!}\frac{\partial^{k+\alpha}\tilde H}{\partial p^\alpha\partial x^k}(0,0)b
+O(|b|^{2}). $$ By inserting the last identity into \eqref{symmmm} we obtain the thesis.
\end{proof}
\end{document} | arXiv |
Communications on Pure and Applied Analysis
2018, Volume 17, Issue 4: 1331-1347. Doi: 10.3934/cpaa.2018065
This issue Previous Article Singularity formation for the 1-D cubic NLS and the Schrödinger map on $\mathbb S^2$ Next Article Small amplitude solitary waves in the Dirac-Maxwell system
Spectral stability of bi-frequency solitary waves in Soler and Dirac-Klein-Gordon models
Nabile Boussïd1, and
Andrew Comech2,3,4, ,
Universite Bourgogne Franche-Comté, 16 route de Gray, 25030 Besançon CEDEX, France
Texas A & M University, College Station, TX 77843, USA
IITP, Moscow 127051, Russia
St. Petersburg State University, St. Petersburg 199178, Russia
* Corresponding author: Andrew Comech
The first author aknowledges the support of Region Bourgogne Franche-Comté through the project "Projet du LMB: Analyse mathematique et simulation numérique d'EDP issues de problèmes de contrôle et du trafic routier". The second author was supported by the Russian Foundation for Sciences (project 14-50-00150).
Abstract Full Text(HTML) Related Papers Cited by
We construct bi-frequency solitary waves of the nonlinear Dirac equation with the scalar self-interaction, known as the Soler model (with an arbitrary nonlinearity and in arbitrary dimension) and the Dirac-Klein-Gordon with Yukawa self-interaction. These solitary waves provide a natural implementation of qubit and qudit states in the theory of quantum computing.
We show the relation of $± 2ω\mathrm{i}$ eigenvalues of the linearization at a solitary wave, Bogoliubov $\mathbf{SU}(1,1)$ symmetry, and the existence of bi-frequency solitary waves. We show that the spectral stability of these waves reduces to spectral stability of usual (one-frequency) solitary waves.
Bi-frequency solitary waves,
Bogoliubov SU(1, 1) symmetry,
Dirac-Klein-Gordon system,
spectral stability,
nonlinear Dirac equation,
linear instability,
Soler model,
qubits,
qudits,
Yukawa model.
Mathematics Subject Classification: Primary: 35C08, 35Q41, 37K40, 81Q05; Secondary: 37N20.
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Nabile Boussïd
Andrew Comech | CommonCrawl |
\begin{document}
\long\def\comment#1{}
\title{An assertion language for slicing\\ constraint logic languages}
\author{ Moreno Falaschi\inst{1} \and Carlos Olarte\inst{2}} \institute{Dept. Information Engineering and Mathematics,
Universit\`a di Siena, Italy.\\
\email{[email protected].}
\and
ECT, Universidade Federal do Rio Grande do Norte, Brazil\\
\email{[email protected].}
}
\date{} \maketitle
\begin{abstract} Constraint Logic Programming (CLP) is a language scheme for combining two declarative paradigms: constraint solving and logic programming. Concurrent Constraint Programming (CCP) is a declarative model for concurrency where agents interact by telling and asking constraints
in a shared store. In a previous paper, we developed a framework for dynamic slicing of CCP\ where the user first identifies that a (partial) computation is wrong. Then, she marks (selects) some parts of the final state corresponding to the data (constraints) and processes that she wants to study more deeply. An automatic process of slicing begins, and the partial computation is ``depurated'' by removing irrelevant information. In this paper we give two major contributions. First, we extend the framework to CLP, thus generalizing the previous work. Second, we provide an assertion language suitable for both, CCP and CLP, which allows the user to specify some properties of the computations in her program. If a state in a computation does not satisfy an assertion then some ``wrong'' information is identified and an automatic slicing process can start. We thus make one step further towards automatizing the slicing process. We show that our framework can be integrated with the previous semi-automatic one, giving the user more choices and flexibility. We show by means of examples and experiments the usefulness of our approach. \end{abstract}
\begin{keywords}
Concurrent Constraint Programming, Constraint Logic Programming,
Dynamic slicing, Debugging, Assertion language.
\end{keywords}
\section{Introduction}\label{sectionintroduction}
Constraint Logic Programming (CLP) is a language scheme \cite{DBLP:journals/jlp/JaffarMMS98} for combining two declarative paradigms: constraint solving and logic programming (see an overview in \cite{JM94}). Concurrent Constraint Programming (CCP) \cite{saraswat91popl} (see a survey in \cite{DBLP:journals/constraints/OlarteRV13}) combines concurrency primitives with the ability to deal with constraints, and hence, with partial information. The notion of concurrency is based upon the shared-variables communication model. CCP is intended for reasoning, modeling and programming concurrent agents (or processes) that interact with each other and their environment by posting and asking information in a medium, a so-called \emph{store}. CCP is a very flexible model and has been applied to an increasing number of different fields such as probabilistic and stochastic, timed and mobile systems~\cite{Olarte:08:SAC,Br11}, and more recently to social networks with spatial and epistemic behaviors \cite{DBLP:journals/constraints/OlarteRV13}, as well as modeling of biological systems~\cite{CFOP10,CFHOT15,OCHF16,BBDFH18}.
One crucial problem with constraint logic languages is to define appropriate debugging tools. Various techniques and several frameworks have been proposed for debugging these languages. Abstract interpretation techniques have been considered (e.g. in \cite{CFM94,CominiTV11absdiag,FOP09,DBLP:journals/tplp/FalaschiOP15}) as well as (abstract) declarative debuggers following the seminal work of Shapiro \cite{Shapiro83}. However, these techniques are approximated (case of abstract interpretation) or it can be difficult to apply them when dealing with complex programs (case of declarative debugging) as the user should answer to too many questions.
In this paper we follow a technique inspired by slicing. Slicing was introduced in some pioneer works by Mark Weiser \cite{MW84}. It was originally defined as a static technique, independent of any particular input of the program. Then, the technique was extended by introducing the so called dynamic program slicing \cite{KL88}. This technique is useful for simplifying the debugging process, by selecting a portion of the program containing the faulty code. Dynamic program slicing has been applied to several programming paradigms (see \cite{Silva2012} for a survey). In the context of constraint logic languages, we defined a tool \cite{FGOP2016} able to interact with the user and filter, in a given computation, the information which is relevant to a particular observation or result. In other words, the programmer could mark (select) the information (constraints, agents or atoms) that she is interested to check in a particular computation that she suspects to be wrong. Then, a corresponding depurated partial computation is obtained automatically, where only the information relevant to the marked parts is present.
In a previous paper \cite{FGOP2016} we presented the first formal framework for debugging CCP via dynamic slicing. In this paper we give two major contributions. First, we extend our framework to CLP. Second, we introduce an assertion language which is integrated within the slicing process for automatizing it further. The extension to CLP is not immediate, as while for CCP programs non-deterministic choices give rise to one single computation, in CLP all computations corresponding to different non-deterministic choices can be followed and can lead to different solutions. Hence, some rethinking of the the framework is necessary. We show that it is possible to define a transformation from CLP programs to CCP programs, which allows us to show that the set of observables of a CLP program and of its translation to a CCP program correspond. This result also shows that the computations in the two languages are pretty similar and the framework for CCP can be extended to deal with CLP programs.
Our framework \cite{FGOP2016} consists of three main steps. First the standard operational semantics of the sliced language is extended to an enriched semantics that adds to the standard semantics the needed meta-information for the slicer. Second, we consider several analyses of the faulty situation based on the program wrong behavior, including causality, variable dependencies, unexpected behaviors and store inconsistencies. This second step was left to the user's responsibility: the user had to examine the final state of the faulty computation and
manually mark/select
a subset of constraints that she wants to study further. The third step is an automatic marking algorithm that removes the information not relevant to derive the constraints selected in the second step. This algorithm is flexible and applicable to timed extensions of CCP\ \cite{DBLP:journals/jsc/SaraswatJG96}. Here, for CLP programs we introduce also the possibility to mark atoms, besides constraints.
We believe that the second step above, namely identifying the right state and the relevant information to be marked, can be difficult for the user and we believe that it is possible to improve automatization of this step. Hence, one major contribution of this paper is to introduce a specialized assertion language which allows the user to state properties of the computations in her program. If a state in a computation does not satisfy an assertion then some ``wrong'' information is identified and an automatic slicing process can start. We show that assertions can be integrated in our previous semi-automatic framework \cite{FGOP2016}, giving the user more choices and flexibility. The assertion language is a good companion to the already implemented tool for the slicing of CCP\ programs to automatically detect (possibly) wrong behaviors and stop the computation when needed. The framework can also be applied to timed variants of CCP. \\
\paragraph{Organization and Contributions } Section \ref{sectionccp} describes CCP and CLP and their operational semantics. We introduce a translation from CLP to CCP programs and prove a correspondence theorem between successful computations. In Section \ref{sectionslicing} we recall the slicing technique for CCP \cite{FGOP2016} and extend it to CLP. The extension of our framework to CLP is our first contribution. As a second major contribution, in Section \ref{sectionassertions} we present our specialized assertion language and describe its main operators and functionalities. In Section \ref{sec:ex} we show some examples to illustrate the expressiveness of our extension, and the integration into the former tool. Within our examples we show how to automatically debug a biochemical system specified in timed CCP and one classical search problem in CLP. Finally, Section \ref{sectionconclusions} discusses some related work and concludes.
\section{Constraint Logic Languages}\label{sectionccp}
In this section we define an operational semantics suitable for both, CLP ~\cite{JM94} and CCP\ programs \cite{saraswat91popl}. We start by defining CCP\ programs and then we obtain CLP
by restricting the set of CCP\ operators.
Processes in CCP\ \emph{interact} with each other by \emph{telling} and \emph{asking} constraints (pieces of information) in a common store of partial information. The type of constraints
is not fixed but parametric in a constraint system (CS), a central notion for both CCP\ and CLP.
Intuitively, a CS provides a signature from which constraints can be built from basic tokens (e.g., predicate symbols), and two basic operations: conjunction $\sqcup$ (e.g., $x\neq y \sqcup x > 5$) and variable hiding $\exists$ (e.g., $\exists x . y = f(x)$). As usual, $\exists x. c$ binds $x$ in $c$. The CS defines also an \emph{entailment} relation ($\models$) specifying inter-dependencies between constraints: $c\models d$ means that the information $d$ can be deduced from the information $c$ (e.g., $x>42 \models x>37$). We shall use $\mathcal{C}$ to denote the set of constraints with typical elements $c,c',d,d'...$. We assume that there exist $\texttt{t}, \texttt{f} \in \mathcal{C}$, such that for any $c \in \mathcal{C}$, $c \models \texttt{t}$ and $\texttt{f} \models \mathcal{C}$. The reader may refer to
\cite{DBLP:journals/constraints/OlarteRV13}
for different formalizations and examples of constraint systems.
\paragraph{\bf The language of CCP\ processes.} In process calculi, the language of processes in CCP\ is given by a small number of primitive operators or combinators.
Processes are built from constraints in the underlying constraint
system and the following syntax:
$ P,Q ::= \mathbf{skip}\mid \tellp{c} \mid \sum\limits_{i\in I}\whenp{c_i}{P_i} \mid P \parallel Q \mid \localp{x}{P} \mid p(\overline{x}) $
The process $\mathbf{skip}$ represents inaction. The process
$\tellp{c}$ adds $c$ to the current store $d$ producing the new store $c\sqcup d$.
Given a non-empty finite set of indexes $I$, the process $\sum\limits_{i\in I}\whenp{c_i}{P_i}$ non-deterministically chooses $P_k$ for execution if the store entails $c_k$. The chosen alternative, if any, precludes the others. This provides a powerful synchronization mechanism based on constraint entailment.
When $I$ is a singleton, we shall omit the ``$\sum$'' and we simply write $\whenp{c}{P}$.
The process $P\parallel Q$ represents the parallel (interleaved) execution of $P$ and $Q$. The process $\localp {x}{P}$ behaves as $P$ and binds the variable $x$ to be local to it.
Given a process definition $p(\overline{y}) \stackrel{\Delta} {=} P$, where all free variables of $P$ are in the set of pairwise distinct variables $\overline{y}$, the process $p(\overline{x})$ evolves into $P[\overline{x}/\overline{y}]$. A CCP\ program takes the form $\mathcal{D}.P$ where $\mathcal{D}$ is a set of process definitions and $P$ is a process.
The Structural Operational Semantics (SOS) of CCP\ is given by the transition relation $ \gamma \longrightarrow \gamma'$ satisfying the rules in Figure \ref{fig:sos}. Here we follow the formulation
in \cite{fages01ic} where the local variables created by the program appear explicitly in
the transition system and parallel composition of agents is
identified by a multiset of agents. More precisely, a \emph{configuration} $\gamma$ is a triple of the form $(X; \Gamma ; c)$, where $c$ is a constraint representing the store, $\Gamma$ is a multiset of processes, and $X$ is a set of hidden (local) variables of $c$ and $\Gamma$. The multiset $\Gamma=P_1,P_2,\ldots,P_n$ represents the process $P_1 \parallel P_2 \parallel \cdots \parallel P_n$. We shall indistinguishably use both notations to denote parallel composition. Moreover, processes are quotiented by a structural congruence relation $\cong$ satisfying:
(STR1) $P \cong Q$ if $P$ and $Q$ differ only by a renaming of
bound variables (alpha conversion);
(STR2) $P\parallel Q \cong Q \parallel P$;
(STR3) $P \parallel (Q \parallel R) \cong (P \parallel Q) \parallel R$;
(STR4)
$P \parallel \mathbf{skip} \cong P$.
We denote by $\longrightarrow^{*}$ the reflexive and transitive closure of
a binary relation $\longrightarrow$.
\begin{definition}[Observables and traces]\label{def:obs} A trace $\gamma_1\gamma_2\gamma_3\cdots $ is a sequence of configurations s.t. $\gamma_1 \longrightarrow \gamma_2 \longrightarrow \gamma_3 \cdots$. We shall use $\pi, \pi'$ to denote traces and $\pi(i)$ to denote the i-th element in $\pi$. If $(X;\Gamma; d) \longrightarrow^{*}(X';\Gamma';d')$ and $\exists X'. d' \models c$ we write $\Barb{(X;\Gamma;d)}{c}$. If $X=\emptyset$ and $d=\texttt{t}$ we simply write $\Barb{\Gamma}{c}$. \end{definition}
Intuitively, if $P$ is a process then $\Barb{P}{c}$ says that $P$ can reach a store $d$ strong enough to entail $c$, \ie, $c$ is an output of $P$. Note that the variables in $X'$ above are hidden from $d'$ since the information about them is not observable.
\begin{figure}
\caption{Operational semantics for CCP\ calculi}
\label{fig:sos}
\end{figure}
\subsection{The language of CLP}\label{sec:clp}
A CLP program \cite{DBLP:journals/jlp/JaffarMMS98} is a finite set of rules of the form \[ p(\overline{x}) \leftarrow A_{1},\dots, A_{n} \] where $A_{1},\dots A_{n}$, with $n\geq0$, are literals, i.e. either atoms or constraints in the underlying constraint system $\mathcal{C}$, and $p(\overline{x})$ is an atom. An atom has the form $p(t_{1},\ldots,t_{m})$, where $p$ is a user defined predicate symbol and the $t_{i}$ are terms from the constraint domain.
The top-down operational semantics is given in terms of derivations from goals
\cite{DBLP:journals/jlp/JaffarMMS98}. A configuration takes the form $(\Gamma ; c)$ where $\Gamma$ (a goal) is a multiset of literals and $c$ is a constraint (the current store). The reduction relation is defined as follows.
\begin{definition}[Semantics of CLP \cite{DBLP:journals/jlp/JaffarMMS98}]\label{def:sem-clp} Let $\mathcal{H}$ be a CLP program. A configuration $\gamma = (L_1,...,L_i,... L_n ; c)$ reduces to $\psi$, notation $\gamma \rediCLPP{\mathcal{H}} \psi$, by selecting and removing a literal $L_i$ and then: \begin{enumerate}
\item If $L_i$ is a constraint $d$ and $d \sqcup c \neq \texttt{f}$, then
$\gamma \rediCLPP{\mathcal{H}} (L_1,...,L_n ; c \sqcup d)$.
\item If $L_i$ is a constraint $d$ and $d \sqcup c = \texttt{f}$
(i.e., the conjunction of $c$ and $d$ is inconsistent), then
$\gamma \rediCLPP{\mathcal{H}} (\Box ; \texttt{f})$ where $\Box$ represents the empty multiset of literals.
\item If $L_i$ is an atom $p(t_1,...,t_k)$, then
$\gamma \rediCLPP{\mathcal{H}} (L_1,...,L_{i-1},\Delta, L_{i+1}... , L_n ; c)$
where one of the definitions for $p$, $p(s_1,...,s_k)\leftarrow A_{1},\dots, A_{n}
$,
is selected and $\Delta = A_{1},\dots, A_{n}, s_1 = t_1 , ..., s_k =
t_k$. \end{enumerate} A computation from a goal $G$ is a (possibly infinite) sequence $\gamma_1=(G ; \texttt{t})\rediCLPP{\mathcal{H}} \gamma_2 \rediCLPP{\mathcal{H}} \cdots$. We say that a computation finishes if the last configuration $\gamma_n$ cannot be reduced, i.e., $\gamma_n = (\Box ; c)$. In this case, if $c=\texttt{f}$ then the derivation fails otherwise we say that it succeeds. \end{definition}
Given a goal with free variables $\overline{x}=var(G)$, we shall also use the notation $\BarbCLP{G}{c}$ to denote that there is a successful computation $(G ;\texttt{t}) \rediCLPP{\mathcal{H}}^* (\Box;d)$ s.t. $\exists \overline{x}. d \models c$. We note that the free variables of a goal are progressively ``instantiated'' during computations by adding new constraints. Finally, the answers of a goal $G$, notation $\BarbCLP{G}{}$ is the set $\{\exists_{var(c)\backslash var(G)}(c) \mid (G ; \texttt{t}) \rediCLPP{\mathcal{H}}^* (\Box ; c) , c\neq \texttt{f} \}$ where ``$\setminus$'' denotes set difference.
\paragraph{\bf From CLP to CCP. } CCP is a very general paradigm that extends both Concurrent Logic Programming and Constraint Logic Programming \cite{NPV02}. However, in CLP, we have to consider non-determinism of the type ``don't know'' \cite{Shapiro:1989}, which means that each predicate call can be reduced by using each rule which defines such a predicate. This is different from the kind of non-determinism in CCP, where the choice operator selects randomly one of the choices whose ask guard is entailed by the constraints in the current store (see $\rm R_{SUM}$ in Figure \ref{fig:sos}).
It turns out that by restricting the syntax of CCP and giving an alternative interpretation to non-deterministic choices, we can have an encoding of CLP programs as CCP agents. More precisely, we shall remove the synchronization operator and we shall consider only blind choices of the form $Q = \sum\limits_{i\in I}\whenp{\texttt{t}}{P_i}$. Note that $c\models \texttt{t}$ for any $c$ and then, the choices in the process $Q$ are not guarded/constrained. Hence, any of the $P_i$ can be executed regardless of the current store. This mimics the behavior of CLP predicates (see (3) in Definition \ref{def:sem-clp}), but with a different kind of non-determinism. The next definition formalizes this idea.
\begin{definition}[Translation]\label{def-trans} Let $\mathcal{C}$ be a constraint system, $\mathcal{H}$ be a CLP program and $G$ be a goal. We define the set of CCP\ process definitions $[\![ \mathcal{H} ]\!] = \mathcal{D}$ as follows. For each user defined predicate symbol $p$ of arity $j$ and $1..m$ defined rules of the form $ p(t^i_1,...,t^i_{j})\leftarrow A^i_{1},\dots, A^i_{n_i} $, we add to $\mathcal{D}$ the following process definition
\resizebox{.95\textwidth}{!}{ $ \begin{array}{lll} p(x_1,...,x_j) &\stackrel{\Delta} {=}& \whenp{\texttt{t}}{(\localp{~\overline{z_1}} {~\prod D_1 ~~ \parallel [\![ A^1_{1}]\!] ~\parallel \dots \parallel [\![ A^1_{n_1}~]\!]}}) + ... + \\ & & \whenp{\texttt{t}}{(\localp{\overline{z_m}} {~\prod D_m\parallel
[\![ A^m_{1} ]\!] \parallel \dots \parallel [\![ A^m_{n_m}]\!] }}) \end{array} $ }
where $\overline{z_i}= var(t^i_1,...,t^i_j) \cup var(A^i_1,...,A^i_{n_i})$, $D_i$ is the set of constraints $\{x_1 = t^i_1,...,x_j = t^i_j \}$, $\prod D_i$ means $\tellp{x_1 = t^i_1} \parallel \cdots \parallel \tellp{x_j = t^i_j} $ and literals are translated as $[\![ A(\overline{t}) ]\!] = A(\overline{t})$ (case of atoms) and $[\![ c ]\!] = \tellp{c}$ (case of constraints). Moreover, we translate the goal $[\![ A_1,...,A_n ]\!] $ as the process $ [\![ A_1 ]\!] \parallel \cdots \parallel [\![ A_n]\!]$.
\end{definition}
We note that the head $p(\overline{x})$ of a process definition $p(\overline{x}) \stackrel{\Delta} {=} P$ in CCP\ can only have variables while a head of a CLP rule $p(\overline{t}) \leftarrow B$ may have arbitrary terms with (free) variables. Moreover, in CLP,
each call to a predicate returns a variant with distinct
new variables (renaming the parameters of the predicate) \cite{DBLP:journals/jlp/JaffarMMS98}.
These two features of CLP can be encoded in CCP by first
introducing local variables ($\localp{\vec{z_i}}{}$ in the above definition)
and then, using constraints ($D_i$) to establish the connection between
the formal and the actual parameters of the process definition.
Consider for instance this simple CLP program dealing with lists: \begin{Verbatim}[fontsize=\scriptsize]
p([] , []) . p([H1 | L1] , [H2 | L2]) :- c(H1,H2), p(L1,L2) . \end{Verbatim}
and its translation \[ \begin{array}{lll} p(x, y) &\stackrel{\Delta} {=}& \whenp{\texttt{t}}{(\tellp{x=[]} \parallel \tellp{y=[]})} + \\ & & \whenp{\texttt{t}}{\localp{X}{(\prod D \parallel c(H1,H2) \parallel p(L1, L2))}} \end{array} \]
where $D = \{x= [H1 | L1], y = [H2 | L2]\}$ and $X=\{H1,H2,L1,L2\}$. Note that the CCP process $p(l_a,l_b)$ can lead to 2 possible outcomes: \begin{itemize}
\item Using the first branch,
the store becomes $l_a=[] \sqcup l_b = []$.
\item In the second branch, due to rule $\rm R_{LOC}$,
four local distinct variables are created (say $h1,h2,l1,l2$),
the store becomes $l_a=[h1 | l1] \sqcup l_b=[h2 | l2]\sqcup c(h1,h2) $
and the process $p(l1,l2)$ is executed on this new store. \end{itemize} These two CCP\ executions match exactly the behavior of the CLP goal \texttt{p(LA, LB)}.
We emphasize that one execution of a CCP\ program will give rise to a single computation (due to the kind of non-determinism in CCP) while the CLP abstract computation model characterizes the set of all possible successful derivations and corresponding answers. In other terms, for a given initial goal $G$, the CLP model defines the full set of answer constraints for $G$, while the CCP translation will compute only one of them, as only one possible derivation will be followed.
\begin{theorem}[Adequacy] Let $\mathcal{C}$ be a constraint system, $c\in \mathcal{C}$, $\mathcal{H}$ be a CLP program and $G$ be a goal. Then, $\BarbCLP{ G }{c} $ iff $\Barb{[\![ G]\!]}{c}$. \end{theorem}
\section{Slicing CCP and CLP programs}\label{sectionslicing}
Dynamic slicing is a technique that helps the user to debug her program by simplifying a partial execution trace, thus depurating it from parts which are irrelevant to find the bug. It can also help to highlight parts of the programs which have been wrongly ignored by the execution of a wrong piece of code. In \cite{FGOP2016} we defined a slicing technique for CCP\ programs that consisted of three main steps: \begin{enumerate}
\item[{\bf S1}] \emph{Generating a (finite) trace}
of the program. For that, a new semantics is needed in order to generate the (meta) information needed for the slicer.
\item[{\bf S2}] \emph{Marking the final store}, to select some of the constraints that,
according to the wrong behavior detected, should or should not be in the final store.
\item[{\bf S3}] \emph{Computing the trace slice}, to select the processes and constraints
that were relevant to produce the (marked) final store. \end{enumerate}
We shall briefly recall the step {\bf S1} in \cite{FGOP2016} which remains the same here. Steps {\bf S2} and {\bf S3} need further adjustments to deal with CLP programs. In particular, we shall allow the user to select processes (literals in the CLP terminology) in order to start the slicing. Moreover, in Section \ref{sectionassertions}, we provide further tools to automatize the slicing process.
\paragraph{\bf Enriched Semantics (Step ${\bf S1}$).}\label{subsec:collect} The slicing process requires some extra information from the execution of the processes. More precisely, (1) in each operational step $\gamma \to \gamma'$, we need to highlight the process that was reduced; and (2) the constraints accumulated in the store must reflect, exactly, the contribution of each process to the store. In order to solve (1) and (2), we introduced in \cite{FGOP2016} the enriched semantics
that extracts the needed meta information for the slicer. Roughly, we identify the parallel composition
$Q = P_1 \parallel \cdots \parallel P_n$
with the \emph{sequence} $\Gamma_{Q} =P_1\idxP{i_1}, \cdots, P_n\idxP{i_n}$
where $i_j \in \mathbb{N}$ is a unique identifier for $P_j$. The use of indexes allow us to distinguish, e.g., the three different occurrences of $P$ in
``$\Gamma_1,P\idxP{i},\Gamma_2,P\idxP{j}, (\whenp{c}{P})\idxP{k}$''. The enriched semantics uses transitions with labels of the form $\rediIdxJ{i}{k}$ where $i$ is the identifier of the reduced process and $k$ can be either $\bot$ (undefined) or a natural number indicating the branch chosen in a non-deterministic choice (Rule $\rm R'_{SUM}$). This allows us to identify, unequivocally, the selected alternative in an execution. Finally, the store in the enriched semantics is not a constraint (as in Figure \ref{fig:sos}) but a set of (atomic) constraints where $\{d_1,\cdots,d_n\}$
represents the store $d_1 \sqcup \cdots \sqcup d_n$. For that, the rule of $\tellp{c}$ first decomposes $c$ in its atomic components before adding them to the store.
\paragraph{\bf Marking the Store (Step ${\bf S2}$).}\label{sec:step2} In \cite{FGOP2016} we identified several alternatives for marking the final store in order to indicate the information that is relevant to the slice that the programmer wants to recompute. Let us suppose that the final configuration in a partial computation is $(X;\Gamma;S)$. The user has to select a subset $S_{sliced} $ of the final store $S$ that may explain the (wrong) behavior of the program. $S_{sliced} $ can be chosen based on the following criteria:
\begin{enumerate} \item \emph{Causality:} the user identifies, according to her knowledge, a subset $S' \subseteq S$ that needs to be explained (i.e., we need to identify the processes that produced $S'$). \item \emph{Variable Dependencies:} The user may identify a set of relevant variables $V\subseteq freeVars(S)$ and then, we mark $ S_{sliced} = \{ c \in S \mid vars(c) \cap V \neq \emptyset \} $.
\item \emph{Unexpected behaviors}: there is a constraint $c$ entailed from the final
store that is not expected from the intended behavior of the program.
Then, one would be interested in the following marking $ S_{sliced} = \bigcup \{S' \subseteq S \mid \bigsqcup S' \models c \mbox{ and } S' \mbox{ is set minimal} \} $, where ``$S'$ is set minimal'' means that for any $S'' \subset S'$, $S'' \not\models c$.
\item \emph{Inconsistent output}: The final store should be consistent with respect
to a given specification (constraint) $c$, i.e., $S$ in conjunction with $c$ must not
be inconsistent. In this case, we have $
S_{sliced} = \bigcup \{S' \subseteq S \mid \bigsqcup S' \sqcup c \models \texttt{f} \mbox{ and } S' \mbox{ is set minimal} \} $.
\end{enumerate}
For the analysis of CLP programs, it is important also to mark literals (i.e., calls to procedures in CCP). In particular, the programmer may find that a particular goal $p(x)$ is not correct if the parameter $x$ does not satisfy certain conditions/constraints. Hence, we shall consider also markings on the set of processes, i.e., the marking can be also a subset $\Gamma_{sliced} \subseteq \Gamma$.
\paragraph{\bf Trace Slice (Step ${\bf S3}$). } Starting from the the pair $\gamma_{sliced}= (S_{sliced}, \Gamma_{sliced})$ denoting the user's marking, we define a backward slicing step. Roughly, this step allows us to eliminate from the execution trace all the information not related to $\gamma_{sliced}$. For that, the fresh constant symbol $\bullet$ is used to denote an ``irrelevant'' constraint or process. Then, for instance, ``$c\sqcup \bullet$'' results from a constraint $c\sqcup d$ where $d$ is irrelevant. Similarly in processes as, e.g., $\whenp{c}{(P \parallel \bullet)} + \bullet$. A replacement is either a pair of the shape $[T / i]$ or $[T / c]$. In the first (resp. second) case,
the process with identifier $i$ (resp. constraint $c$) is replaced with $T$. We shall use $\theta$ to denote a set of replacements and we call these sets as ``replacing substitutions''. The composition of replacing substitutions $\theta_1$ and $\theta_2$ is given by the set union of $\theta_1$ and $\theta_2$, and is denoted as $\theta_1 \circ \theta_2$.
\begin{algorithm}[t] {\scriptsize \KwIn{- a trace $\gamma_0\rediIdxJ{i_1}{k_1} \cdots \rediIdxJ{i_{n}}{k_{n}}\gamma_n$ where $\gamma_i= (X_i;\Gamma_i; S_i)$
\qquad\quad\ - a marking ($S_{sliced} , \Gamma_{sliced}$) s.t. $S_{sliced} \subseteq S_n$ and $\Gamma_{sliced} \subseteq \Gamma_n$}
\KwOut{ a sliced trace $\gamma_0'\longrightarrow \cdots \longrightarrow \gamma_n' $}
\Begin{
{\bf let} $\theta = \{ [\bullet/i] \mid P\idxP{i} \in \Gamma_n \setminus \Gamma_s \}$ {\bf in}
$\gamma_n' \leftarrow ( X_n \cap vars(S_{sliced}, \Gamma_{sliced}) ; \Gamma_{n}\theta; S_{sliced})$\;
\For{l= $n-1$ to 0}{
${\bf let} \langle \theta', c\rangle =
sliceProcess(\gamma_l, \gamma_{l+1}, i_{l+1}, k_{l+1},\theta, S_l) \
$ {\bf in}
$S_{sliced} \leftarrow S_{sliced} \cup S_{minimal}(S_{l},c) $
$\theta \leftarrow \theta' \circ \theta$
$\gamma_l' \leftarrow (X_l\cap vars(S_{sliced}, \Gamma_{sliced}) ~; ~ \Gamma_l \theta ~;~ S_l \cap S_{sliced})$
} } }
\caption{Trace Slicer. $S_{minimal}(S,c)=\emptyset$ if $c=\texttt{t}$; otherwise, $S_{minimal}(S,c) = \bigcup \{S' \subseteq S \mid \bigsqcup S' \models c \mbox{ and } S' \mbox{ is set minimal} \}$.
\label{alg:slicer}} \end{algorithm}
Algorithm \ref{alg:slicer} extends the one
in \cite{FGOP2016} to deal with the marking on processes ($\Gamma_{sliced}$). The last configuration
($\gamma_n'$ in line 3) means that we only observe the local variables of interest,
i.e., those in $vars(S_{sliced}, \Gamma_{sliced})$ as well as the relevant processes ($\Gamma_{sliced}$)
and constraints $(S_{sliced})$. The algorithm backwardly computes the slicing by accumulating replacing pairs in $\theta$ (line 7). The new replacing substitutions are computed by the function $sliceProcess$ that returns both, a replacement substitution and a constraint needed in the case of ask agents as explained below.
\begin{algorithm}[] {\scriptsize \SetKwProg{Fn}{Function}{ }{end} \Fn{sliceProcess($\gamma, \psi, i,k, \theta, S$) }{
{\bf let} $\gamma=(X_\gamma ; \Gamma,P\idxP{i},\Gamma'; S_\gamma)$ and $\psi=(X_\psi ; \Gamma,\Gamma_Q, \Gamma' ; S_\psi)$ {\bf in}
\SetKw{KMatch}{match}
\SetKw{KWith}{with}
\SetKwBlock{KBMatch}{\KMatch{$P$ \KWith }}{end}
\KBMatch{
\uCase{$\tellp{c}$}{
{\bf let} $c' = sliceConstraints(X_\gamma,X_\psi, S_\gamma, S_\psi, S)$ {\bf in}
\leIf{$c' = \bullet$ or $c' = \exists \overline{x}. \bullet$}{\KwRet{ $\lr{[\bullet / i] , \texttt{t}}$}}{\KwRet{ $\lr{[\tellp{c'}/i],\texttt{t}}$}}
}
\uCase{$\sum\whenp{c_l}{Q_l}$}{
\leIf{$ \Gamma_Q \theta=\bullet$}{\KwRet{ $\lr{[\bullet / i],\texttt{t}}$}}{\KwRet{
$\lr{[ \whenp{c_k}{(\Gamma_Q\theta)} + \bullet ~/~ i], c_k}$}}
}
\uCase{$\localp{x}{Q}$}{
{\bf let} $\{x'\} = X_\psi \setminus X_\gamma$ {\bf in}
\leIf{$\Gamma_Q[x'/x] \theta=\bullet$}{\KwRet{ $\lr{[\bullet / i],\texttt{t}}$}}{\KwRet{ $\lr{[\localp{x'}{\Gamma_Q[x'/x]\theta}/ i],\texttt{t}}$}}
}
\uCase(){$p(\overline{y})$}{
\leIf{$\Gamma_Q \theta=\bullet$}{\KwRet{ $\lr{[\bullet / i],\texttt{t}}$}}{\KwRet{ $\lr{\emptyset, \texttt{t}}$}}
}
}
}
\SetKwProg{Fn}{Function}{ }{end} \Fn{sliceConstraints($X_\gamma, X_\psi, S_\gamma, S_\psi, S$) }{ {\bf let} $S_c = S_\psi \setminus S_\gamma \mbox{ and } \theta= \emptyset$ {\bf in}
\lForEach{$c_a \in S_c \setminus S$}{
$\theta \leftarrow \theta \circ [\bullet / c_a]$
}
\KwRet{ $\exists_ {X_{\psi} \setminus X_{\gamma}}. \bigsqcup S_c \theta$} } }
\caption{Slicing processes and constraints \label{alg:proc}} \end{algorithm}
\noindent {\bf Marking algorithms. }
Let us explain how the function $sliceProcess$ works. Consider for instance the process $Q=(\whenp{c'}{P})+(\whenp{c}{\tellp{d\sqcup e}}) $ and assume that we are backwardly slicing the trace $\cdots \gamma \rediIdxJ{i}{2} \cdots \psi \rediIdxJ{j}{} \rho \cdots $ where $Q$ (identified with $i$) is reduced in $\gamma$ by choosing the second branch and, in $\psi$, the tell agent $\tellp{d\sqcup e}$ (identified by $j$) is executed. Assume that the configuration $\rho$ has already been sliced and $d$ was considered irrelevant and removed (see $S_l \cap S_{sliced}$ in line 8 of Algorithm \ref{alg:slicer}). The procedure $sliceProcess$ is applied to $\psi$ and it determines that only $e$ is relevant in $\tellp{d\sqcup e}$. Hence, the replacement $[\tellp{\bullet \sqcup e}/j]$ is returned (see line 7 in Algorithm \ref{alg:slicer}). The procedure is then applied to $\gamma$. We already know that the ask agent $Q$ is (partially) relevant since $\tellp{d\sqcup e}\theta \neq \bullet$ (i.e., the selected branch does contribute to the final result). Thus, the replacement $[\bullet + \whenp{c}{\tellp{\bullet\sqcup e}} / i]$ is accumulated in order to show that the first branch is irrelevant. Moreover, since the entailment of $c$ was necessary for the reduction, the procedure returns also the constraint $c$ (line 5 of Algorithm \ref{alg:slicer}) and the constraints needed to entail $c$ are added to the set of relevant constraints (line 6 of Algorithm \ref{alg:slicer}).
\begin{example}\label{ex:length} Consider the following (wrong) CLP program: \begin{Verbatim}[fontsize=\scriptsize]
length([],0). length([A | L],M) :- M = N, length(L, N). \end{Verbatim} The translation to CCP\ is similar to the example we gave in Section \ref{sec:clp}.
An excerpt of a possible trace for the execution of the goal \verb|length([10,20], Ans).| is \begin{Verbatim}[fontsize=\tiny] [0 ; length([10,20],Ans) ; t] --> [0 ; ask() ... + ask() ... ; t] -> [0 ; local ... ; t] ->
[H1 L1 N1 M1 ; [10,20]= [H1|L1] || Ans=N1 || N1=M1 || length(L1, M1) ; t] -> ...
[... H2 L2 N2 M2 ; [20]=[H2 | L2] || M1=N2 || N2=M2 || length(L2, M2) ; [10,20]= [H1|L1], Ans=N1, N1=M1] ->
[... H2 L2 N2 M2 ; M1=N2 || N2=M2 || length(L2, M2) ; [10,20]= [H1|L1], Ans=N1, N1=M1, [20]=[H2 | L2]] -> ...
[... H2 L2 N2 M2 ; M2=0 ; [10,20]= [H1|L1], Ans=N1, N1=M1, [20]=[H2 | L2], M1=N2, N2=M2, L2=[]] ->
[... H2 L2 N2 M2 ; [10,20]= [H1|L1], Ans=N1, N1=M1, [20]=[H2 | L2], M1=N2, N2=M2, L2=[], M2=0 ] \end{Verbatim} In this trace, we can see how the calls to the process definition \texttt{length} are unfolded and, in each state, new constraints are added. Those constraint relate, e.g., the variable \texttt{Ans} and the local variables created in each invocation (e.g., \texttt{M1} and \texttt{M2}).
In the last configuration, it is possible to mark only the equalities dealing with numerical expressions (i.e.,
\verb|Ans=N1,N1=M1,M1=N2,N2=M2,M2=0|) and the resulting trace will abstract away from all the constraints and processes dealing with equalities on lists: \begin{Verbatim}[fontsize=\tiny] [0 ; length([10,20],Ans) ; t] --> [0 ; * + ask() ... ; t] -> [0 ; local ... ; t] ->
[N1 M1 ; * || Ans=N1 || N1=M1 || length(L1, M1) ; t] ->
[N1 M1 ; Ans=N1 || N1=M1 || length(L1, M1) ; ] ->
[N1 M1 ; N1=M1 || length(L1, M1) ; Ans=N1] -> [N1 M1 ; length(L1, M1) ; Ans=N1, N1=M1] -> ... \end{Verbatim}
The fourth line should be useful to discover that \verb|Ans| cannot be equal to \verb|M1| (the parameter used in the second invocation to \verb|length|).
\end{example}
\section{An assertion language for logic programs}\label{sectionassertions}
The declarative flavor of programming with constraints in CCP\ and CLP allows the user to reason about (partial) invariants that must hold during the execution of her programs. In this section we give a simple yet powerful language of assertion to state such invariants. Then, we give a step further in automatizing the process of debugging.
\begin{definition}[Assertion Language]\label{sec:syntax} Assertions are built from the following syntax:
\noindent\resizebox{.95\textwidth}{!}{ $ F ::= \posC{c} ~\mid~ \negC{c} ~\mid~ \consC{c} ~\mid~ \iconsC{c} ~\mid~ F \oplus F ~\mid~ \predAssertionA{p(\overline{x})}{F} ~\mid~ \predAssertionE{p(\overline{x})}{F} $}
\noindent where $c$ is a constraint ($c\in \mathcal{C}$), $ p(\cdot) $ is a process name and $\oplus \in \{\wedge, \vee, \to \}$. \end{definition}
The first four constructs deal with partial assertions about the current store. These constructs check, respectively, whether the constraint $c$: (1) is entailed, (2) is not entailed, (3) is consistent wrt the current store or (4) leads to an inconsistency when added to the current store. Assertions of the form $F \oplus F$ have the usual meaning. The assertions $\predAssertionA{p(\overline{x})}{F}$
states that all instances
of the form $p(\overline{t})$ in the current configuration must satisfy the assertion $F$. The assertions $\predAssertionE{p(\overline{x})}{F}$ is similar to the previous one but it checks for the existence of an instance $p(\overline{t})$ that satisfies the assertion $F$.
Let $\pi(i) = (X_i ; \Gamma_i ; S_i)$. We shall use $\fStore{\pi(i)}$ to denote the constraint $\exists X_i . \bigsqcup S_i$ and $\fProc{\pi(i)}$ to denote the sequence of processes $\Gamma_i$. The semantics for assertions is formalized next.
\begin{definition}[Semantics]\label{sec:semantics} Let $\pi$ be a sequence of configurations and $F$ be an assertion. We inductively define $\pi,i \entails_{\mathcal{F}} F$ (read as $\pi$ satisfies the formula $F$ at position $i$) as: \begin{itemize}
\item $\pi, i \entails_{\mathcal{F}} \posC{c}$ if $store(\pi(i)) \models c$.
\item $\pi, i \entails_{\mathcal{F}} \negC{c} $ if $store(\pi(i)) \not\models c$.
\item $\pi, i \entails_{\mathcal{F}} \consC{c} $ if $store(\pi(i)) \sqcup c \not\models \texttt{f}$.
\item $\pi, i \entails_{\mathcal{F}} \iconsC{c} $ if $store(\pi(i)) \sqcup c \models \texttt{f}$.
\item $\pi, i \entails_{\mathcal{F}} F \wedge G$ if $\pi,i \entails_{\mathcal{F}} F$ and $\pi,i \entails_{\mathcal{F}} G$.
\item $\pi, i \entails_{\mathcal{F}} F \vee G$ if $\pi,i \entails_{\mathcal{F}} F$ or $\pi,i \entails_{\mathcal{F}} G$.
\item $\pi, i \entails_{\mathcal{F}} F \to G$ if $\pi,i \entails_{\mathcal{F}} F$ implies $\pi,i \entails_{\mathcal{F}} G$.
\item $\pi, i \entails_{\mathcal{F}} \predAssertionA{p(\overline{x})}{F} $ if for all $p(\overline{t})\in \fProc{\pi(i)}$, $\pi, i \entails_{\mathcal{F}} F[\overline{t}/\/\overline{x}]$.
\item $\pi, i \entails_{\mathcal{F}} \predAssertionE{p(\overline{x})}{F} $ if there exists $p(\overline{t})\in \fProc{\pi(i)}$, $\pi, i \entails_{\mathcal{F}} F[\overline{t}/\/\overline{x}]$.
\end{itemize} If it is not the case that $\pi, i \entails_{\mathcal{F}} F $, then we say that $F$ does not hold at $\pi(i)$ and we write $\pi(i) \not\entails_{\mathcal{F}} F$. \end{definition}
The above definition is quite standard and reflects the intuitions given above. Moreover, let us define $\sim F$ as $\sim \posC{c} = \negC{c}$ (and vice-versa), $\sim \consC{c} = \iconsC{c}$ (and vice-versa), $\sim (F \oplus F)$ as usual and $\sim \predAssertionA{p(\overline{x})}{F(\overline{x})} = \predAssertionE{p(\overline{x})}{\sim F(\overline{x})}$ (and vice-versa). Note that, $\pi(i) \entails_{\mathcal{F}} F$ iff $\pi(i) \not\entails_{\mathcal{F}} \sim F$.
\begin{example} Assume that the store in $\pi(1)$ is $S = x \in 0..10$. Then,
\noindent - ${\pi,1 \entails_{\mathcal{F}} \consC{x=5}}$, i.e., the current store is consistent wrt the specification $x=5$.
\noindent - $\pi,1 \not\entails_{\mathcal{F}} \iconsC{x=5}$, i.e., the store is not inconsistent wrt the specification $x=5$.
\noindent - $\pi,1 \not\entails_{\mathcal{F}} \posC{x=5}$, i.e., the store is not ``strong enough'' in order to satisfy the specification $x=5$.
\noindent - $\pi,1 \entails_{\mathcal{F}} \negC{x=5}$, i.e., store is ``consistent enough'' to guarantee that it is not the case that $x=5$.
\end{example} Note that $\pi,i \entails_{\mathcal{F}} \posC{c}$ implies $ \pi,i \entails_{\mathcal{F}} \consC{c}$. However, the other direction is in general not true (as shown above). We note that CCP\ and CLP are monotonic in the sense that when the store $c$ evolves into $d$, it must be the case that $d \models c$ (i.e., information is monotonically accumulated). Hence, $\pi,i \models \posC{c}$ implies $\pi,i + j \models \posC{c}$. Finally, if the store becomes inconsistent, $\consC{c}$ does not hold for any $c$. Temporal \cite{NPV02} and linear \cite{fages01ic} variants of CCP\ remove such restriction on monotonicity.
We note that checking assertions amounts, roughly, to testing the entailment relation in the underlying constraint system. Checking entailments is the basic operation CCP\ agents perform. Hence, from the implementation point of view,
verification of assertions does not introduce a significant extra computational cost.
\begin{example}[Conditional assertions]\label{ex:patterns} Let us introduce some patterns of assertions useful for verification.
\noindent- \emph{Conditional constraints} : The assertion $\posC{c} \to F$ checks for $F$ only if $c$ can be deduced from the store. For instance, the assertion $\posC{c} \to \negC{d}$ says that $d$ must not be deduced when the store implies $c$.
\noindent- \emph{Conditional predicates} : Let $G = \predAssertionE{p(\overline{x})}{\consC{\texttt{t}}}$. The assertion $G\to F$ states that $F$ must be verified whenever there is a call/goal of the form $p(\overline{t})$ in the context. Moreover, $ (\sim G) \to F $ verifies $F$ when there are no calls of the form $p(\overline{t})$ in the context. \end{example}
\subsection{Dynamic slicing with assertions} Assertions allow the user to specify conditions that her program must satisfy during execution. If this is not the case, the program should stop and start the debugging process. In fact, the assertions may help to give a suitable marking pair $(S_{sliced}, \Gamma_{sliced})$ for the step ${\bf S2}$ of our algorithm as we show in the next definition.
\begin{definition}\label{def:symp} Let $F$ be an assertion, $\pi$ be a partial computation, $n>0$ and assume that $\pi, n \not\entails_{\mathcal{F}} F$, i.e., $\pi(n)$ fails to establish the assertion $F$. Let $\pi(n ) = (X ; \Gamma ; S )$. As testing hypotheses, we define $\fSymp{\pi,F, n} = (S_{sl}, \Gamma_{sl})$ where
\begin{enumerate} \item If $F = \posC{c}$ then $S_{sl}=\{d \in S \mid vars(d)\cap vars(c) \neq \emptyset \}$, $\Gamma_{sl}=\emptyset$. \item If $F = \negC{c}$ then $S_{sl}=\bigcup \{S' \subseteq S \mid \bigsqcup S' \models c \mbox{ and } S' \mbox{ is set minimal} \}$, $\Gamma_{sl}=\emptyset$ \item If $F = \consC{c}$ then $S_{sl}=\bigcup \{S' \subseteq S \mid \bigsqcup S' \sqcup c \models \texttt{f} \mbox{ and } S' \mbox{ is set minimal} \}$, $\Gamma_{sl}=\emptyset$. \item If $F = \iconsC{c}$ $S_{sl}=\{d \in S \mid vars(d)\cap vars(c) \neq \emptyset \}$ and $\Gamma_{sl}=\emptyset$.
\item If $F = F_1 \wedge F_2$ then $\fSymp{\pi, F_1, n } \cup \fSymp{\pi, F_2, n }$. \item If $F = F_1 \vee F_2$ then $\fSymp{\pi, F_1, n } \cap \fSymp{\pi, F_2, n }$. \item If $F = F_1 \to F_2$ then $\fSymp{\pi, \sim F_1, n } \cup \fSymp{\pi, F_2, n }$. \item If $F = \predAssertionA{p(\overline{x})}{F_1}$ then $S_{sl}=\emptyset$ and $\Gamma_{sl}= \{p(\overline{t}) \in \Gamma \mid \pi,n \not\entails_{\mathcal{F}} F_1[\overline{t}/\overline{x}]\}$. \item If $F = \predAssertionE{p(\overline{x})}{F_1}$ then $S_{sl}=\{d\in S \mid vars(d) \cap vars(F_1) \neq \emptyset\}$, $\Gamma_{sl}=\{p(\overline{t}) \in \Gamma\}$
\end{enumerate}
\end{definition}
Let us give some intuitions about the above definition. Consider a (partial) computation $\pi$ of length $n$ where $\pi(n) \not\entails_{\mathcal{F}} F$. In the case (1) above, $c$ must be entailed but the current store is not strong enough to do it. A good guess is to start examining the processes that added constraints using the same variables as in $c$. It may be the case that such processes should have added more information to entail $c$ as expected in the specification $F$. Similarly for the case (4): $c$ in conjunction with the current store should be inconsistent but it is not. Then, more information on the common variables should have been added. In the case (2), $c$ should not be entailed but the store indeed entails $c$. In this case, we mark the set of constraints that entails $c$. The case (3) is similar. In cases (5) to (7) we use $\cup$ and $\cap$ respectively for point-wise union and intersection in the pair $(S_{sl} , \Gamma_{sl})$. These cases are self-explanatory (e.g., if $F_1\wedge F_2$ fails, we collect the failure information of either $F_1$ or $F_2$). In (8), we mark all the calls that do not satisfy the expected assertion $F(\overline{x})$. In (9), if $F$ fails, it means that either (a) there are no calls of the shape $p(\overline{t})$ in the context or (b) none of the calls $p(\overline{t})$ satisfy $F_1$. For (a), similarly to the case (1), a good guess is to examine the processes that added constraints with common variables to $F_1$ and see which one should have added more information to entail $F_1$. As for (b), we also select all the calls of the form $p(\overline{t})$ from the context. The reader may compare these definitions with the information selected in Step {\bf S2} in Section \ref{sectionslicing}, regarding possibly wrong behavior.
\paragraph{Classification of Assertions. } As we explained in Section \ref{sec:clp}, computations in CLP can succeed or fail and the answers to a goal is the set of constraints obtained from successful computations. Hence, according to the kind of assertion, it is important to determine when the assertions in Definition \ref{sec:syntax}
must stop or not the computation to start the debugging process. For that, we introduce the following classification:
\noindent {\bf - post-conditions, $\postF{F}$ assertions }: assertions that are meant to be verified only when an answer is found. This kind of assertions are used to test the ``quality'' of the answers wrt the specification. In this case,
the slicing process begins only when an answer is computed and it does not satisfy one of the assertions.
Note that assertions of the form
$\predAssertionA{p(\overline{x})}{F(\overline{x})}$ and $\predAssertionE{p(\overline{x})}{F(\overline{x})}$ are irrelevant as post-conditions since the set of goals in an answer must be empty.
\noindent {\bf - path invariants, $\invF{F}$ assertions}: assertions that are meant to hold along the whole computation. Then, not satisfying an invariant must be understood as a
symptom of an error and the computation must stop. We note that due to monotonicity,
only assertions of the form $\negC{c}$ and $\consC{c}$ can be used to stop the computation (note that if the current configuration fails to satisfy $\negC{c}$, then any successor state will also fail to satisfy that assertion).
Constraints of the form $\posC{c}, \iconsC{c}$ can be only checked
when the answer is found since, not satisfying those conditions in
the partial computation, does not imply that the final state will not satisfy them.
\subsection{Experiments}\label{sec:ex} We conclude this section with a series of examples showing the use of assertions. Examples \ref{test1} and \ref{test2} deal with CLP programs while Examples \ref{test3} and \ref{test4} with CCP\ programs.
\begin{example}\label{test1} The debugger can automatically start and produce the same marking in Example \ref{ex:length} with the following (invariant) assertion: \begin{Verbatim}[fontsize=\scriptsize]
length([A | L],M) :- M = N, length(L, N), inv(pos(M>0)). \end{Verbatim} \end{example}
\begin{example}\label{test2} Consider the following CLP program (written in GNU-Prolog with integer finite domains) for solving the well known problem of posing $N$ queens on a $N\times N$ chessboard in such a way that they do not attack each other. \begin{Verbatim}[fontsize=\scriptsize] queens(N, Queens) :- length(Queens, N), fd_domain(Queens,1,N),
constrain(Queens), fd_labeling(Queens,[]). constrain(Queens) :-fd_all_different(Queens), diagonal(Queens). diagonal([]). diagonal([Q|Queens]):-secure(Q, 1, Queens), diagonal(Queens). secure(_,_,[]). secure(X,D,[Q|Queens]) :- doesnotattack(X,Q,D),D1 is D+1, secure(X,D1,Queens). doesnotattack(X,Y,D) :- X + D #\= Y,Y + X #\= D. \end{Verbatim}
\noindent The program contains one mistake, which causes the introduction of
a few additional and not correct solutions, e.g., \verb|[1,5,4,3,2]|
for the goal \verb|queens(5,X).|
The user now has two possible strategies: either she lets the interpreter compute the solutions, one by one and then, when she sees a wrong solution she uses the slicer for marking manually the final store to get the sliced computation; or she can define an assertion to be verified. In this particular case, any solution must satisfy that the difference between two consecutive positions in the list must be greater than $1$. Hence, the user can introduce the following post-condition assertion:
\begin{Verbatim}[fontsize=\scriptsize]
secure(X,D,[Q|Queens]) :- doesnotattack(X,Q,D),D1 is D+1, secure(X,D1,Queens),
post(cons(Q #\= X+1)). \end{Verbatim}
Now the slicer stops as soon as the constraint \verb|X #\= Q+1| becomes inconsistent with the store in a successful computation (e.g., the assertion fails on the --partial-- assignment ``5,4'') and an automatic slicing is performed.
\end{example}
\begin{example}\label{test3} In \cite{FGOP2016} we presented a compelling example of slicing for a timed CCP\ program modeling the synchronization of events in musical rhythmic patterns. As shown in Example 2 at \url{http://subsell.logic.at/slicer/}, the slicer for CCP\ was able to sufficiently abstract away from irrelevant processes and constraints to highlight the problem in a faulty program. However, the process of stopping the computation to start the debugging was left to the user. The property that failed in the program can be naturally expressed as an assertion. Namely, in the whole computation, if the constraint $\texttt{beat}$ is present (representing a sound in the musical rhythm), the constraint $\texttt{stop}$ cannot be present (representing the end of the rhythm). This can be written as the conditional assertion $\posC{\texttt{beat}} \to \negC{\texttt{stop}}$. Following Definition \ref{def:symp}, the constraints marked in the wrong computation are the same we considered in \cite{FGOP2016}, thus automatizing completely the process of identifying the wrong computation. \end{example}
\begin{example}\label{test4} Example 3 in the URL above illustrates the use of timed CCP\ for the specification of biochemical systems (we invite the reader to compare in the website the sliced and non-sliced traces). Roughly, in that model, constraints of the form $\texttt{Mdm2}$ (resp. $\texttt{Mdm2A}$) state that the protein $\texttt{Mdm2}$ is present (resp. absent). The model includes activation (and inhibition) of biological rules modeled as processes (omitting some details) of the form $\whenp{\texttt{Mdm2A}}{\nextp{\tellp{\texttt{Mdm2}}}}$ modeling that ``if
$\texttt{Mdm2}$ is absent now, then it must be present in the next time-unit''. The interaction of many of these rules makes the model trickier since rules may ``compete'' for resources and then, we can wrongly observe at the same time-unit that $\texttt{Mdm2}$ is both present and absent. An assertion of the form $(\posC{\texttt{Mdm2A}} \to \negC{\texttt{Mdm2}})\wedge(\posC{\texttt{Mdm2}} \to \negC{\texttt{Mdm2A}})$ will automatically stop the computation and produce the same marking we used to depurate the program in the website. \end{example}
\section{Related work and conclusions}\label{sectionconclusions}
{\bf Related work} Assertions for automatizing a slicing process have been previously introduced in \cite{Alpuente2016DebuggingMP} for the functional logic language Maude. The language they consider as well as the type of assertions are completely different from ours. They do not have constraints, and deal with functional and equational computations. Another previous work \cite{SGM2002} introduced static and dynamic slicing for CLP programs. However, \cite{SGM2002} essentially aims at identifying the parts of a goal which do not share variables, to divide the program in slices which do not interact. Our approach considers more situations, not only variable dependencies, but also other kinds of wrong behaviors. Moreover we have assertions, and hence an automatic slicing mechanism not considered in \cite{SGM2002}. The well known debugging box model of Prolog \cite{ClocksinMellish1981} introduces a tool for observing the evolution of atoms during their reduction in the search tree. We believe that our methodology might be integrated with the box model and may extend some of its features. For instance, the box model makes basic simplifications by asking the user to specify which predicates she wants to observe. In our case, one entire computational path is simplified automatically by considering the marked information and identifying the constraints and the atoms which are relevant for such information.
\noindent {\bf Conclusions and future work} In this paper we have first extended a previous framework for dynamic slicing of (timed) CCP\ programs to the case of CLP programs. We considered a slightly different marking mechanism, extended to atoms besides constraints. Don't know non-determinism in CLP requires a different identification of the computations of interest wrt CCP. We considered different modalities specified by the user for selecting successful computations rather than all possible partial computations. As another contribution of this paper, in order to automatize the slicing process, we have introduced an assertion language. This language is rather flexible and allows one to specify different types of assertions that can be applied to successful computations or to all possible partial computations. When assertions are not satisfied by a state of a selected computation then an automatic slicing of such computation can start.
We implemented a prototype of the slicer in Maude and showed its use in debugging several programs. We are currently extending the tool to deal with CLP don't know non-determinism. Being CLP a generalization of logic programming, our extended implementation could be also eventually used to analyze Prolog programs. Integrating the kind of assertions proposed here with already implemented debugging mechanisms in Prolog is an interesting future direction. We also plan to add more advanced graphical tools to our prototype, as well as to study the integration of our framework with other debugging techniques such as the box model and declarative or approximated debuggers \cite{FOPV07,ABCF10}. We also want to investigate the relation of our technique with dynamic testing (e.g. concolic techniques) and extend the assertion language with temporal operators, e.g. the past operator ($\past$) for expressing the relation between two consecutive states. Another future topic of investigation is a static version of our framework in order to try to compare and possibly integrate it with analyses and semi automatic corrections based on different formal techniques, and other programming paradigms \cite{AFMV99,BBB09,BBBC14,ABFR06,ABBF10}.
\noindent {\small {\bf Acknowledgments} We thank the anonymous reviewers for their detailed and very useful criticisms and recommendations that helped us to improve our paper. The work of Olarte was supported by CNPq and by CAPES, Colciencias, and INRIA via the STIC AmSud project EPIC (Proc. No 88881.117603/2016-01), and the project CLASSIC.}
\end{document} | arXiv |
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>Volume 126 Issue 1302: The 25th ISABE Conference S...
>Identification and classification of operating flow...
The Aeronautical Journal
Identification and classification of operating flow regimes and prediction of stall in a contra-rotating axial fan using machine learning
Published online by Cambridge University Press: 22 June 2022
A. Kumar [Opens in a new window] ,
M.P. Manas and
A.M. Pradeep
A. Kumar*
Department of Aerospace Engineering, Indian Institute of Technology Bombay, Mumbai, Maharashtra, India
M.P. Manas
*Corresponding author. Email: [email protected]
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Prediction of stall before it occurs, or detection of stall is crucial for smooth and lasting operation of fans and compressors. In order to predict the stall, it is necessary to distinguish the operational and stall regions based on certain parameters. Also, it is important to observe the variation of those parameters as the fan transitions towards stall. Experiments were performed on a contra-rotating fan setup under clean inflow conditions, and unsteady pressure data were recorded using seven high-response sensors circumferentially arranged on the casing, near the first rotor leading edge. Windowed Fourier analysis was performed on the pressure data, to identify different regions, as the fan transits from the operational to stall region. Four statistical parameters were identified to characterise the pressure data and reduce the number of data points. K-means clustering was used on these four parameters to algorithmically mark different regions of operation. Results obtained from both the analyses are in agreement with each other, and three distinct regions have been identified. Between the no-activity and stall regions, there is a transition region that spans for a short duration of time characterised by intermittent variation of abstract parameters and excitations of Fourier frequencies. The results were validated with five datasets obtained from similar experiments at different times. All five experiments showed similar trends. Neural Network models were trained on the clustered data to predict the operating region of the machine. These models can be used to develop control systems that can prevent the stalling of the machine.
Stall predictionContra rotating fanMachine learning
The Aeronautical Journal , Volume 126 , Issue 1302: The 25th ISABE Conference Special Issue – Part II , August 2022 , pp. 1351 - 1369
DOI: https://doi.org/10.1017/aer.2022.63[Opens in a new window]
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (https://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
© The Author(s), 2022. Published by Cambridge University Press on behalf of Royal Aeronautical Society
Blade Passing Frequency
${c_a}$
axial flow speed in m/s
${c_p}$
specific heat constant in J/ (kg K)
total number of data points
${p_i}$
ith level division as percentage of total interval
q i
total number of points in ith interval
$split$
splitting step size as percentage of total interval
x i
time series variable
$H$
entropy function
ith level division of total interval
P ref
reference pressure taken as atmospheric pressure of 101,325 Pa
$\Delta {T_0}$
total temperature difference in K
$U$
blade tip speed in m/s
set of all-time series data points
set of subsets of X
flow coefficient
loading coefficient
Predicting stall before it occurs is crucial for avoiding stalling of a compressor/fan. Stalling can cause loss of efficiency, vibrations or even structural failure, depending on the severity of the phenomenon. Though the phenomena of stall in conventional compressors and fans have been widely investigated, the stall inception mechanisms in contra-rotating fans have not been explored thoroughly.
Stall inception in a fan/compressor remains an interesting topic of research and the works on stall inception started when Emmons [Reference Emmons, Kronauer and Rockett1] suggested a conceptual model for the rotating stall phenomenon. The research on stall inception for the next several decades focused on developing theoretical models [Reference Day, Greitzer and Cumpsty2]–[Reference Moore and Greitzer5]. Later with larger experimental studies, researchers identified the significant fluid phenomenon leading to stall and termed those as the short-length-scale spikes and the long-length-scale modal disturbances [Reference McDougall, Cumpsty and Hynes6]–[Reference Camp and Day8]. The prominent aspects of the behaviour of these fluid structures remain unclear and the research on identifying the structure and propagation of these structures are still being carried out experimentally and computationally [Reference Yamada, Kikuta, Iwakiri, Furukawa and Gunjishima9]–[Reference Brandstetter, Paoletti and Ottavy17]. Several of these works concluded the existence of the disturbances prior to the fully developed stall. With this understanding, some researchers tried to predict and control the stall phenomenon using active or passive techniques. Spakovszky et al. [Reference Spakovszky, Weigl, Paduano, Van Schalkwyk, Suder and Bright18] used an active technique using jet injectors upstream of the rotor leading edge. Control strategies were obtained using transfer functions determined from the response of the sensors. A review of the different types of stall inception phenomenon and the possible control strategies were detailed by Tan et al. [Reference Tan, Day, Morris and Wadia19]. Ashrafi et al. [Reference Ashrafi, Michaud and Vo20] used a plasma actuator placed upstream of the rotor leading edge and impeller leading edge of an axial compressor and a centrifugal compressor, respectively, to delay the stall. The plasma actuator could accelerate the flow and push the tip leakage flow into the blade passage. Recently, Day [Reference Day21] reported that even after a long history of research on stall prediction in compressors/fans, designing a stall-resistant machine or to predict the stall of a new compressor is still a challenge. Thus, it is evident that the prediction of stall needs to be investigated in detail to design a compressor/fan that is tolerant to stall or can operate over a wide range of mass flows.
Some recent studies have focused on machine learning strategies to predict the unsteady variation in pressure fluctuations to identify stall. Xu et al. [Reference Xu, Wang, Liu and Liu22] implemented a support vector regression machine (SVRM) to predict internal pressure in a fan, and detect the inception of rotating stall. The results predicted by the SVRM model are compared to the stall detected using wavelet transform of pressure data. The model successfully predicts pressure 0.0625 seconds, which corresponds to 1.35 rotor revolutions in advance and can reliably detect stall before it occurs, giving ample time for a control system to prevent it. Kim et al. [Reference Kim, Pullan, Hall, Grewe, Wilson and Gunn23] studied the stall inception behaviour of two different transonic fans both experimentally and computationally using steady and unsteady simulations. The unsteady pressure reading upstream of the leading edge of rotor is compared to the pressure traces predicted using unsteady computations and used to detect the stall and its propagation in the machine. Using CFD, the stall mechanisms of the two fans were studied, indicating tip leakage flow and shock boundary layer interaction as the cause of the stall. Unsteady casing pressure measurements at the rotor and stator of a four-stage axial compressor were used to identify and study the stall and pre-stall behaviour of compressor, at different rotor speeds by Methling et al. [Reference Methling, Stoff and Grauer24]. The stall was identified to be associated with spike-type disturbances. A neural network was trained using the pressure data at different speeds to identify the region of operation of the compressor, indicating if the machine is near stall. This network model works for both spike type stall and modal wave, at different rotor speeds if trained with sufficient data. The physics of the pre-stall instability described by discrete flow disturbances, or rotating instability was studied by Eck et al. [Reference Eck, Geist and Peitsch25], using complementary experimental and numerical methods. The instability is characterised by the development of vortex tubes that are attached to the casing and suction surface of the rotor blades. Also, a new stall indicator was developed by applying statistical methods on the casing pressure trace, to get better control of the safety margin. A K-means clustering and Gaussian Mixture Models-based algorithm was developed by Yusoff et al. [Reference Yusoff, Ooi, Lim and Leong26], using the pressure, temperature and vibration signals from LM 2500 axial compressor. The model monitors the machine condition and could avoid any fatal failure. The model also identified three machine states characteristic—off, active and abnormal—along with degradation tracking visualisation of the machine. It will be helpful in predictive maintenance of the machine. Falzone et al. [Reference Falzone and Kolodziej27] recorded vibration signals from Dresser-Rand ESH-1 reciprocating compressor, taken using accelerometers in normal and damaged condition, and these signals were used to develop a model to monitor the compressor condition. Wavelet transforms were used to identify and visualise the vibration features from the data. The first and second order wavelet statistical features were selected for training a support vector machine to classify if the machine is normal or damaged, which worked with predictive accuracy of more than 90%. It will be helpful in predictive maintenance of the machine.
From a thorough literature review, it is understood that the stall inception is still a phenomenon which requires detailed understanding and the prediction of stall can help in recovering the machine from failure. Also, most of the works that are reported in literature related to application of machine learning on compressors or fans are focused on predictive maintenance of terrestrial gas turbines, or prediction of stall for terrestrial gas turbine compressors using measurement and prediction of absolute values of various physical parameters. In this work we take a different approach based on data engineering to predict as well as detect stall in a contra-rotating fan, which is a potential configuration that can replace the conventional fans in future aircraft gas turbine engines. The methodology records casing pressure traces using high-response sensors and studies the variation in the Fourier amplitude and frequency characteristic of pressure traces as the contra-rotating fan approaches stall. Assumption is that the variation in flow characteristics in the fan as it approaches stall will be translated to casing pressure traces and its Fourier characteristic as well, which was observed in cases of compressors as reported in literature. The different regions of operation were algorithmically characterised and classified using abstract parameters and machine learning methods. Neural network models were trained to identify the operating region of the machine based on the recorded pressure traces. Control systems using these models could be made, which can detect the operating region of the machine and take necessary action to avoid stall or pull the machine out of stall. This work extends the application of machine learning to the operation of contra-rotating fans, and thus explores the potential use of machine learning for aerospace gas turbine applications. The identified flow regimes are not based on absolute value of a physical parameter, rather on the statistical features of the data, and so are related to the properties of the flow over an extended period of time. This makes the classification model resilient to noise in actual value of the physical parameter. The control model based on the above methodology has potential to significantly reduce the instrumentation for measurement and computation as it uses only one sensor to measure the pressure traces, and no other parameter is required to be measured. This work also allows gaining some insight into the stall inception characteristics of a contra-rotating fan, which has not been explored as thoroughly as in conventional rotor-stator configurations. Though the current work is focused on a specific design speed of the contra-rotating fan, the methodology presented in this work is general and can be extended to other speeds. In addition, this method being generic can also be applied to other types of machines to identify their operating flow regimes and can be used to train classification models for their control.
2.0 Methodology
2.1 Experimental setup
The experiments for this work were performed on a contra-rotating fan setup shown in Fig. 1. The contra-rotating fan has nine blades on rotor-1 and seven blades on rotor-2. The stage has a target stage loading of 1.05 at a flow coefficient of 0.79, as defined in Equations (1) and (2) respectively, and the Reynolds number obtained based on the axial velocity at the inlet and the chord of the blade is approximately 2 $ \times $ 105. The important specifications of rotor1 and rotor2 is given in Table 1, and a detailed description of the design of the contra-rotating stage can be found in Manas and Pradeep [Reference Manas and Pradeep28].
(1) \begin{align}\psi = \frac{{{c_p}\Delta {T_0}}}{{0.5\;{U^2}}}\end{align}
(2) \begin{align}\varphi = \frac{{{c_a}}}{U}\end{align}
Figure 1. Photograph of the experimental facility.
2.2 Data acquisition
Seven high response pressure sensors arranged circumferentially and equally spaced on the casing, near the leading edge of the first rotor, were used to take pressure data at 10,000Hz sampling frequency, or normalised sampling frequency of 200, with respect to the rotor frequency. The data was recorded for 30 seconds or 1,500 rotor revolutions, generating 300,000 data points during the stall inception process. A schematic of the positional arrangement of the sensors is shown in Fig. 2. In this period of 30 seconds, the machine was pushed towards the stall region from the safe operating region, by continuously reducing the mass flow rate using a throttle controller. Similar measurements were taken four times on different days, in order to verify the consistency of the data and to validate the analysis. Thus, in total, five datasets were analysed, and the validity and repeatability of the model are tested. The data from the first experiment is referred to as the Baseline case in the paper and the other four are referred to as Test cases. All sensors showed similar trends for a given experiment, and therefore, for further analysis, data from only one sensor is used.
Figure 2. Schematic of the sensor position.
2.3 Analysis methodology
In order to understand the characteristics of the pressure data, two methodologies are used in this study. One is the windowed Fourier, and the other is the identification of the abstract statistical parameters that can help to divide the pressure traces into different operating regions. The algorithms of both the methods are described here.
2.3.1 Windowed fourier analysis
Windowed Fourier analysis is similar to short-time Fourier analysis, but without using the hamming functions. Let 'X' be a set of 'n' time series data points, X = {x 1, x 2,…, x n }. Let 'w' be the window size and 's' be the step size such that the points are divided into subset of X given by, X s = [{x 1, x 2,…, x w }, {x 1+s, x 2+s ,…., x w+s }, …, {x 1+ks, x 2+ks,…, x w+ks}]. Here, the sets are mutually exclusive if 's ≥ w', else, the sets have an overlap of 'w - s' points and also, 'w + ks ≤ n'. The window and step sizes affect the details of the information that can be extracted from the data. The discrete Fourier transform for each of the set in Xs is performed, and the index set for each of the set is defined as I = [{mean (1, 2,…, w)}, {mean (1+s, 2+s,…, w+s)},…, {mean (1+ks, 2+ks,…, w+ks)}]. The Fourier frequency vs Fourier amplitude vs index is plotted to get an insight into how the amplitude of various frequencies vary with the index.
Table 1. Specifications of rotor-1 and rotor-2
For the time series data used in this work, the number of data points, n = 300,000. Window size, w = 1,000 and step size, s = 100 were chosen after a few iterations with different values. The chosen values ensure that both frequency and temporal resolutions are satisfactorily achieved, and one is not overly compromised for the other. The current choice of window size corresponds to five rotor revolutions and the step size corresponds to 0.5 rotor revolutions. The step size should be small enough to accurately capture the development of the stall in time, and window size should be large enough to capture sufficient data points for resolving the frequencies and establish the flow regime in which the machine is operating. If the flow characteristics of the machine did not change for multiple rotations, it can be established with some confidence that the machine is operating in that flow regime. It was observed that if the size of the window is increased, the parameters computed using the Fourier amplitudes get smoothed out, and the resolution of the features in time reduces. This can cause issues near the stall region as it will limit the accuracy of the algorithm to exactly determine the time at which stall starts. Therefore, too much compromise on time resolution will not be good to capture the start point of the stall. Also, if the time resolution is significantly reduced, the frequency content of the window is not appropriately resolved. This further leads to an increase in the number of windows and thus the computational time, which was found to be not necessary for the present analysis. As far as the step size is concerned, a larger step size reduces time resolution of computed features and a smaller step size increases computational time. However, the above choice of window size and step size could be formalised based on the fact that the current window size of 1,000 points is equivalent to five rotor revolutions, and a step size of 100 points is equivalent to 0.5 rotor revolutions. It was found that the significant pre-stall disturbances and the stall cells during the fully developed stall travel approximately at 0.5 times the blade speed. Therefore, the choice of 0.5 rotor revolutions allows to appropriately capture the movement of pre-stall disturbances and the stall cell. Also, sufficient data points are required to resolve the frequencies. Here, the sampling frequency of the sensors is 10kHz. Therefore, five rotor revolutions at a given RPM were required to capture 1,000 data points. And the value of pressure must be recorded across multiple rotations for the present methodology to be applicable as it predicts the operating regime based on flow dynamics, and consistent flow dynamics across multiple rotations confirm that the machine is operating in the predicted region. Thus, the choice of the window size is a function of the machine RPM, sensor sampling frequency, minimum number of rotations required to capture sufficient data points to accurately capture the frequencies and the operating regime of the machine with confidence, and to resolve the time at which stall starts as accurately as possible. For step size, in addition to the above parameters, the number of rotations required for stall cells to fully develop is an important factor. The algorithm for this approach was implemented using Python 3.6.
2.3.2 Statistical parameters
The next step was to quantify the data using statistical and abstract quantities. Statistical quantities such as variance, skewness, and kurtosis for the data and Fourier amplitude of the data were initially explored, in order to quantitatively characterise the three different regions identified from the previous analysis. Later, abstract quantities like Shannon's entropy based on two different probability definitions, for both the data and Fourier amplitude, and the coefficient of determination for a polynomial fit of a fixed degree, were also explored for the same. Finally, four quantities were chosen for this purpose, which are the variance of Fourier amplitude, two different Shannon's entropies, and the coefficient of determination for a polynomial fit of fixed degree. Two different algorithms for calculating Shannon's entropy for a given data, as used in this work are described below.
Consider a set X = {x 1, x 2,…, x n }. First, the elements of the set X are transformed as
(3) \begin{align}{x_i} \to \;{x_i}/\Sigma {x_i}\end{align}
Here, all elements lie in the range $0 \le {x_i} \le 1$ , and their sum adds up to one, making it a probability like mapping on the data.
Now, Shannon's entropy 1 is calculated as
(4) \begin{align}H = - \Sigma {x_i}{\rm{ln}}\;{x_i}\end{align}
Consider a set X = {x 1, x 2,…, x n}. The range min(X) and max(X) is divided into uniformly spaced intervals where the element of the interval set is given by
(5) \begin{align}{I_i} = \;{\rm{min}}\left( X \right) + \left\{ {{\rm{max}}\left( X \right) - {\rm{min}}\left( X \right)} \right\}\left( {\frac{{{p_i}}}{{100}}} \right)\end{align}
where ${p_0} = 0$ and ${p_i} = \;{p_{i - 1}} + split$ . Here, $split$ is defined as the interval spacing or the difference between two intervals in percentage of the total interval $\;{\rm{max}}\left( X \right) - {\rm{min}}\left( X \right)$ . For each interval { ${I_i}$ , ${I_{i + 1}}$ }
(6) \begin{align} q_{i} = size [\{x_{p}\;:\; l_{i} \leq x_{p} \leq l_{i+1};\; x_{p} \epsilon X \}] \end{align}
is calculated. Thus, Shannon's entropy 2 is calculated as
(7) \begin{align}H = - \Sigma \left( {\frac{{{q_i}}}{{size\left( X \right)}}} \right){\rm{ln}}\;({q_i}/size\left( X \right))\;\end{align}
where $\;{q_i} \gt 0$ and $\dfrac{{{q_i}}}{{size\left( X \right)}}$ acts as probability-like mapping on the data.
The idea behind these algorithms is to fit a probability distribution on a given data set, so that each point in the data can be assigned a value between 0 and 1, in order to calculate the entropy values. There is no physical meaning of these probability-like distributions in relation to the actual data. If the same probability distribution function is applied to two different data sets having a different distribution, the probability assigned to the data points will differ and will be reflected in the value of Shannon's entropy. A different distribution will generate a different entropy value. Similarly, a different probability distribution on the same data will generate a different entropy value.
If the physics of the flow in a different operational region is different, then the distribution of pressure and the Fourier amplitude data in that region will be different. If the same probability distribution function is applied in each of these regions, they will generate different entropy values and different regions can be identified using these values.
The two entropy values and the goodness of fit were calculated for each window and plotted with respect to the index. The window and index were the same as used in the Fourier analysis. The value of split was chosen to be 0.02% for evaluating Shannon's entropy 2, so that the distribution of data points with respect to amplitude was captured with highest resolution while staying within a small computational time limit of a few 100 milliseconds, and also resolve the different flow regimes. A polynomial of degree 50 was chosen for evaluating the goodness of fit. The choice of degree 50 was based on an analysis, which identified that in the flow regime showing periodic disturbances, the coefficient of determination was proportionally related to the degree of polynomial, and in regions where there was no periodic disturbance, no correlation was found. Since the number of data points were large, a polynomial of higher degree was more likely to accurately model it. Therefore, a higher-degree polynomial could resolve different flow regimes and the upper limit of the choice of the degree of polynomial was set based on the computational time.
Since the data was unmarked, the unsupervised learning algorithm, K-means clustering, was applied on these four quantities to classify the data into different groups. It clearly identified three clusters with large and stable silhouette values for all five tests, marking the three different regions in the original time series. This marked data was used to train neural network classifiers and could successfully predict the operating region of the machine with acceptable accuracy.
3.1 Fourier analysis of pressure data
Preliminary Fourier analysis was performed on pressure data from all seven sensors, from all five experiments, for full 300,000 data points, taken at normalised sampling frequency of 200, giving a normalised Nyquist frequency of 100. All seven sensors showed a similar trend in the frequency vs amplitude plots as shown in Fig. 3. Most of the significant normalised frequencies are in the range 0–9, which is below the first blade passing frequency, as marked by red ellipse. It can also be observed that the amplitude is decreasing rapidly for larger frequencies.
Figure 3. Fast Fourier transform of the pressure data of one sensor.
Figure 4 shows the windowed Fourier analysis performed on the data to understand the variation of frequency and amplitude. The amplitudes have been filtered so that all frequencies with amplitude above 0.987 and below 0.0691 are set to 0, in order to highlight the distinction between the three regions. It can be observed that none of the Fourier frequencies has any significant amplitude prior to approximately 500 rotor revolutions (marked by the red line). Some frequency excitations are observed between 500 rotor revolutions and around 950 rotor revolutions (marked by the green line). The excitations are intermittent and there is no order in which the frequencies are excited, except that most of the excited frequencies are below the first blade passing normalised frequency corresponding to 9. Also, the amplitudes of excitation for various frequencies increase as the rotor moves from 500 to 950 revolutions or towards stall. This region identified before the fully developed stall is the transition region. Beyond 950 rotor revolutions (green line), the stage enters into a fully developed stall. Moreover, it can be observed that the larger frequencies (beyond the first blade passing frequency) also get excited as the machine moves towards stall, with smaller frequencies having larger amplitudes.
Figure 4. Frequency vs number of rotor revolutions vs amplitude (colour bar).
From Fig. 5, it can be concluded that the stall region is characterised by the consistent frequency excitation through the region at a regular interval of approximately 0.6, which marks the harmonics of the stall cell normalised frequency. Yellow line marks the first blade passing normalised frequency at 9. By observing the region enclosed by the blue square in Fig. 5, the intermittent and random excitation of frequencies that are visible in the transition region is also visible in the stall region. Another noticeable feature observed between a normalised frequency range of 4.8 and 7.2 is that the amplitude of the harmonics of stall cell frequency has dropped significantly, compared to the harmonics below 4.8 and above 7.2, and is in the same amplitude range as the intermittent random frequency excitations. It can be concluded that in the fully developed stall region, the stall cell frequency has been superimposed over the disordered frequency excitations of the transition region. The stall cell frequency marks the difference in the transition and stall regions.
Figure 5. Frequency vs number of rotor revolutions vs amplitude (colour bar) for normalised frequency range of 0–9.
Figure 6 is a different representation of Fig. 4 and highlights the amplitude range of the higher frequencies, as all the frequencies having amplitude above 0.02961 have been filtered and set to zero. Before 500 rotor revolutions, most of the frequencies have smaller amplitudes, which started rising as the rotor moved to the transition region and towards fully developed stall. A general trend marked by the orange curve shows the increasing excitation amplitude of higher frequencies as the rotor moves towards fully developed stall. It can be concluded that the high-frequency disturbances are generated in the flow as stall is approached, by some mechanism that has not been explored in detail in this work.
Figure 6. Frequency vs number of rotor revolutions vs amplitude (colour bar). Frequencies with amplitude above 0.02961 are filtered.
3.2 Characterisation of flow regimes using abstract parameters
Most of the large amplitude disturbances have normalised frequencies below 8 in both transition and the fully developed stall region as can be observed from Fig. 4 and also the three regions show significant differences in their frequency–amplitude characteristics. This indicates that the distribution of amplitude in the frequency as well as time domain is different for the three regions. This difference should be sufficient to distinctly characterise the three regions using abstract mathematical parameters, which take advantage of the above stated fact.
Many statistical and abstract parameters like variance, variance of Fourier amplitude, skewness, kurtosis, Shannon's entropy and coefficient of determination were plotted for the data with the same window and step size as used in the windowed Fourier analysis, but only for the normalised frequency range of 1–8, filtered using Butterworth band-pass filter of order 5. Of all the parameters, four parameters were selected as they showed significantly distinct trends in the three regions of operation compared to all other parameters. Hence, these parameters are able to distinguish the regions independently. These four parameters are variance of Fourier amplitude, coefficient of determination for polynomial fit of fixed degree on the pressure data, Shannon's entropy using the first algorithm for Fourier amplitude where all amplitude below 0.0987 and above 0.987 is set to 1 so that they contribute negligibly to the value of entropy compared to amplitudes in the required range, and Shannon's entropy using second algorithm for the pressure data.
Figures 7–10 shows the plot of the four different parameters vs number of rotor revolutions for the baseline case. The respective magnitudes of each of these parameters are approximately in the same range for all the five experiments. All five experiments show similar trends in the variation of these parameters as shown from Figs 11–14. In case of the variance of Fourier amplitudes, initially the value is small and uniform, and then some small peaks start to develop intermittently, as expected (see Fig. 5), signifying the transition region, and later a sudden jump in the value occurs with a large peak, marking the beginning of the stall region. A similar trend is observed in the other parameters also. In the case of Shannon's entropy 1, initially the value is small with a general straight-line trend, and then some small peaks start to develop intermittently with increasing magnitudes with large fluctuations, which marks the transition region. This is followed by a relatively stable region having comparatively large magnitude and low fluctuations in value, which marks the fully developed stall region. In the case of the coefficient of determination, similar to that of the variance of Fourier amplitude, the initial value is small with a general straight-line trend, and then some small peaks start to develop intermittently with increasing magnitude as the machine moves towards stall, marking the transition region and followed by a sudden jump in the magnitude of the variable, generating a large peak, marking the start of the fully developed stall region. For Shannon's entropy 2, in the transition region, the fluctuations are large compared to the stall region. Though, the end of the transition region and the beginning of the stall region are not sharply distinguishable.
Figure 7. Variance of Fourier amplitude for the first experiment.
Figure 8. Shannon's entropy using the first algorithm evaluated on Fourier amplitude for the first experiment. Amplitude filtered between 0.0987 and 0.987.
Figure 9. Coefficient of determination for a polynomial of degree 50, fitted to the data in each window, for the first experiment.
Figure 10. Shannon's entropy using second algorithm evaluated on pressure data, for the first experiment.
Figure 11. Variance of Fourier amplitudes after min-max normalisation for each experiment.
Figure 12. Normalised Shannon's entropy using first algorithm, for Fourier amplitudes lying between 0.987 and 0.0987 for each experiment.
Figure 13. Coefficient of determination for a polynomial of degree 50, min-max normalised for each experiment.
Figure 14. Shannon's entropy using second algorithm min–max normalised for each experiment.
Each of these four variables show significantly distinct characteristics in the three regions identified from the windowed Fourier analysis as in Fig. 5; also, they can distinguish the three regions independently of each other. This was not necessarily true for all other variables that were tested, and hence only these four variables were selected for further analysis.
3.3 Discrete clustering of operating region using machine learning
All the four variables that were selected show distinct characteristics in the three regions identified using windowed Fourier analysis; however, these differences are not sharply identifiable in all cases and are gradual with large fluctuations. If an algorithm has to prevent stalling of the machine, it should discretely identify the three regions with high confidence, so that whenever the machine reaches the transition region, action to prevent stall could be taken.
One method to discretise is to cluster the data into distinct groups. For this purpose, K-means clustering and self-organising maps were used. However, since K means clustering was easier to implement and more transparent when dealing with the data, it was chosen for grouping the data. The average silhouette value for the clusters and the stability of average silhouette value over multiple clustering runs were used to confirm the reliability of the clustering. If in each clustering run, the average silhouette value changes, then the clustering is unreliable as each time the data points are clustered differently.
The four variables and their trends have different scales separated by multiple orders of magnitude. For example, the variance of Fourier amplitudes is seven orders of magnitude larger than all other variables. This creates a problem for clustering algorithms as it can bias the algorithm towards one of the variables. It is important to rescale all the variables into nearly the same range while preserving their trend, and min-max rescaling is one of the best ways to do it.
Data from all five experiments show similar trends and are in a similar magnitude range; however, these data do not have the exact same minimum and maximum value. Therefore, to rescale the variables, some threshold maximum and threshold minimum value should be chosen that is valid across all the five experiments. The repeatability of trend and magnitudes across all five experiments indicates a physical origin of the chosen parameters, even if they are abstract. This allows us to choose a specific threshold maximum and minimum value for each of the parameters that is valid across the experiments, to do the min-max rescaling that puts all the parameters roughly in the range of 0–1 while preserving their trends. This can be done through a direct observation of the magnitudes, or by taking the average of the maxima and minima of a given parameter over all the experiments. For this work, the thresholds were chosen by direct observation of the data plots (similar to Figs 7–10) for all five experiments. Table 2 gives the different parameters and their chosen min-max threshold values.
All the four parameters were rescaled using the corresponding min-max threshold values, and then K-means clustering algorithm was applied to these parameters with 2, 3, 4 and 5 clusters. Figure 15 shows the variation of silhouette value with the number of iterations for the baseline case. It can be concluded that cluster 1, where the data is separated into two clusters, has an almost stable silhouette value of 0.874308 with very small fluctuations, and cluster 2, where the data is separated into three clusters, has a perfectly stable silhouette value of 0.855883, across all the iterations of the algorithm. Cluster 3 that divides the data into four clusters, and cluster 4 that divides the data into five clusters, are having unstable and relatively lower silhouette values. It can be concluded that the best and reliable way to cluster the given data is in two or three different groups.
Figure 15. Silhouette value vs iteration plot for different number of clusters for the baseline case.
Figure 16 shows the discrete division of clusters with rotor revolutions for two clusters (a) and three clusters (b). The red and green lines on both the figures approximately divide the three regions as identified in Fig. 5. It can be seen that, for two clusters, the transition region (between the red and green lines) shows an intermittent development of second cluster and a stable second cluster in the stall region (beyond the green line). For three clusters, the transition region shows intermittent development of second cluster and a stable third cluster in the stall region. Also, the beginning and end of each region as predicted by clustering using three clusters matches approximately with the one identified using direct observation from the windowed Fourier analysis (Fig. 5). Also, the transition region shows an intermittent development of the second cluster, which is similar to the intermittent excitation of Fourier frequencies in the transition region as observed using Fourier analysis. It can be said that before the occurrence of a fully developed stall, the machine gives intermittent pre-stall signals, and this region of operation can be called the transition region, which falls between the region of safe operation and the region of fully developed stall. The above trend was observed in all five experiments. Table 3 gives the silhouette value for two and three clusters for all the five experiments. All these values are stable across multiple iterations.
Figure 16. Discrete division of clusters with rotor revolution for (a) two clusters and (b) three clusters.
Table 2. Statistical parameters and their threshold minimum and maximum values
Table 3. Silhouette value for two and three clusters for all the five experiments
Thus, K-means clustering on the selected abstract parameters allowed discrete division and identification of the three regions, which also confirms the observation made using windowed Fourier analysis, enabling the development of an algorithm for control system that can identify the transition region using pre-stall signals, so that stall can be avoided.
3.4 Neural network based classifier for identifying operating region of the machine
K-mean clustering showed that the data can be reliably classified into three distinct clusters, corresponding to three different operating regions of the machine as observed from Fourier analysis. In order to prevent stall in the machine, the algorithm should be able to identify the region in which the machine is operating. For this, a neural network-based classification model was chosen, and the network was programmed using the Keras library in Python 3.6. Two different approaches were used to train a classification model. In the first approach, the model takes four parameters as input and gives an output corresponding to the cluster in which the given data point belongs. This model had four inputs and four outputs. The four inputs were the four parameters, and the four outputs correspond to the four columns of one hot encoded output corresponding to the cluster number. Cluster 1 was encoded as [0 1 0 0], cluster 2 as [0 0 1 0] and cluster 3 as [0 0 0 1]. Data from the baseline case and Test1 were stacked together, containing a total of 5,878 data points and this data was used to train the neural network. The same training data was used to test the trained network models and get the loss and accuracy value. The prediction made by the models, in hot encoded form, was integer encoded to directly get the cluster number, and was used to generate the confusion matrix. Also, the data from Test2, Test3 and Test4, containing a total 8,817 data points were stacked together and were used to test the network models, calculating the loss and accuracy. The confusion matrix was generated for the testing data as well using the same procedure as used for the training data. The structure, defining parameters and performance of the model, is given below, and also summarized in Tables 4–6.
For this model the total number of parameters was 1,864, out of which all were trainable parameters. Mean-squared error loss function was used to calculate the losses along with Adam optimisation algorithm with accuracy as evaluation metric. Model was trained for 20,000 epochs with batch size of 5,878. The accuracy and loss on the training data was 100% and 0 respectively, and the accuracy and loss on the testing data was 98.42% and 0.01 respectively.
Table 4. Network structure of first model
Table 5. Confusion matrix for training data for first model
Table 6. Confusion matrix for testing data for first model
In the second approach, a 1D convolution network model was used. It directly takes the min – max normalised 1,000 sensor data points of a particular window as input and generates the output corresponding to the cluster in which the given data belongs, and thus the model has 1,000 input points. Similar to the previous model, this model also has four outputs corresponding to one hot encoded classification vector as in the previous case. The training and testing methodology was the same as that of the previous model. The structure, defining parameters and performance of the model is given below, and is also summarized in Tables 7–9.
The number of windows with 1,000 data points in the training set was 5,878, which includes the data from the baseline and Test1. Similarly, the number of windows with 1,000 data points in the testing set was 8,817, which includes data from Test2, Test3 and Test4. Here, the total number of trainable parameters was 418,654. Mean-squared error loss function was used to calculate the losses along with Adam optimisation algorithm with accuracy as evaluation metric. The model was trained for 200 epochs with a batch size of 64. For this model, the accuracy and loss on the training data was 98.93% and 0.01 respectively, and the accuracy and loss on the testing data was 94.17% and 0.02 respectively.
Network model based on both the approaches showed acceptable accuracy in predicting the operating region of the machine based on clusters. Though the first model has more accuracy compared to the second model, the second model eliminates the need to compute the four abstract parameters and directly works with the sensor data as input. The intended application of the neural network models is in controlling the operating region of the machine during real time operation, thus preventing the machine from stalling. A simple conceptual schematic of a control loop based on the above models is shown in Fig. 17.
Table 7. Network structure of second model
Table 8. Confusion matrix for training data for second model
Table 9. Confusion matrix for testing data for second model
Figure 17. A simple control loop concept implementing classification network model for controlling the state of the machine.
Since the models were trained on a window size of 1,000 data points, the control loop shown in Fig. 17 takes the input as 1,000 pressure data points, which is worth 0.1 seconds of sensor data with sampling rate of 10,000Hz. In the first model, the four abstract parameters used for clustering are calculated and normalised using threshold normalisation values corresponding to that particular variable, determined using multiple experiments on the machine, as data pre-processing step. Normalised parameters are given as input to the trained model, which generates the output corresponding to the operating region of the machine. A state-based control model generates the necessary instructions to control the machine and prevent stall. The second model is also used in a similar way; however, it eliminates the need to compute the four abstract parameters. It directly takes the 1,000 pressure data samples as input, and in the pre-processing step it min–max normalises the sample and feeds it as input to the 1D convolution model, which predicts the operating region of the machine. The above method allows controlling the state of the machine based on the dynamics of the flow in the machine, as it depends on statistical features of the pressure traces over an extended period, rather than on the specific value of a flow variable at a given time, making the models resilient to noise in actual value of the flow variable.
The main objective of this work was to identify any pre-stall signal that the machine generates and to predict stall before it occurs, which can later be used to make a control system to prevent stall, using the pressure traces from the casing of a contra-rotating fan. Windowed Fourier analysis was performed on the pressure data to understand the variation in its frequency amplitude characteristic, as the machine moves towards the fully developed stall. The variation was expected as the distribution of pressure data will differ in different flow regimes due to different flow physics.
The windowed Fourier analysis showed pre-stall excitations at various frequencies. These excitations were random and intermittent with respect to frequencies and rotor revolutions. This was called the transition region. Stall region was characterised by a large and stable excitation of different frequencies that were integral multiple of the normalised stall cell frequency of approximately 0.6. Random and intermittent excitations that were observed in the transition region also appeared in the stall region with stall cell frequencies superimposed over it.
Four abstract parameters were chosen to quantitatively characterise these regions. All these parameters showed similar trends and also had similar respective magnitude range across multiple experiments, indicating a physical origin by virtue of repeatability. These parameters showed different trends in different regions of operation of the machine that were identified from previous analysis, but transition from one region to the other was not always sharply defined.
K means clustering was performed on the data that showed a stable classification and large silhouette value for two and three clusters. For three clusters, the second cluster marking the start and end of the transition region is around the same location as identified using the windowed Fourier analysis. These clusters discretely mark the three regions of operation of the machine, with the first cluster marking normal region, the second cluster marking the transition or pre stall region and the third cluster marking the stall region.
The clustered data was used to train two different neural network classifier models. The first model takes the four abstract parameters as input and gives the cluster number as output, indicating the region of operation of the machine. The second model directly takes 1,000 pressure data points that are min–max normalised and predicts the operating region. Both the models showed acceptable accuracy in predicting the operating region of the machine. If the machine is operating in the transition region, appropriate measures can be taken to prevent stalling of the machine, using the proposed control loop, based on the models.
The neural network models generated in this work are only applicable to this particular machine; however, the methodology presented in this work to generate these models is fairly general, and can be used to generate models for any machine working at any operating condition. Therefore, this novel methodology stands as a potential tool and can become a standard technique for identifying operating regions and training classification models across a wide range of similar turbomachines working at different operating conditions.
This paper is a version of a presentation due to be given at the 2022 ISABE Conference.
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Volume 126, Issue 1302
A. Kumar (a1), M.P. Manas (a1) and A.M. Pradeep (a1)
DOI: https://doi.org/10.1017/aer.2022.63
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\begin{document}
\title{Relating non-local quantum computation to \newline information theoretic cryptography}
\author[3]{Rene Allerstorfer} \email{[email protected]} \orcid{}
\author[3,4]{Harry Buhrman} \email{[email protected]} \orcid{}
\author[2]{Alex May} \email{[email protected]} \orcid{0000-0002-4030-5410}
\author[3,4]{Florian Speelman} \email{[email protected]} \orcid{}
\author[3]{Philip Verduyn Lunel} \email{[email protected]} \orcid{}
\affiliation[2]{Perimeter Institute for Theoretical Physics} \affiliation[3]{QuSoft, CWI Amsterdam} \affiliation[4]{University of Amsterdam}
\abstract{ Non-local quantum computation (NLQC) is a cheating strategy for position-verification schemes, and has appeared in the context of the AdS/CFT correspondence. Here, we connect NLQC to the wider context of information theoretic cryptography by relating it to a number of other cryptographic primitives. We show one special case of NLQC, known as $f$-routing, is equivalent to the quantum analogue of the conditional disclosure of secrets (CDS) primitive, where by equivalent we mean that a protocol for one task gives a protocol for the other with only small overhead in resource costs. We further consider another special case of position verification, which we call coherent function evaluation (CFE), and show CFE protocols induce similarly efficient protocols for the private simultaneous message passing (PSM) scenario. By relating position-verification to these cryptographic primitives, a number of results in the cryptography literature give new implications for NLQC, and vice versa. These include the first sub-exponential upper bounds on the worst case cost of $f$-routing of $2^{O(\sqrt{n\log n})}$ entanglement, the first example of an efficient $f$-routing strategy for a problem believed to be outside $P/poly$, linear lower bounds on entanglement for CDS in the quantum setting, linear lower bounds on communication cost of CFE, and efficient protocols for CDS in the quantum setting for functions that can be computed with quantum circuits of low $T$ depth. } \maketitle
\pagebreak
\tableofcontents
\flushbottom
\section{Introduction}\label{sec:intro}
In a position-verification scenario, a verifier attempts to determine the location of a prover by communicating with them remotely \cite{chandran2009position,kent2011quantum,buhrman2014position}. Position-verification may be of interest as a goal in itself, or may serve as an authentication mechanism for use towards further cryptographic goals. In the most widely studied setting, where the prover holds no secret key, an adversary may use a strategy known as non-local quantum computation to simulate the actions of the prover. A non-local quantum computation replaces local actions within a designated spacetime region with actions outside that region along with entanglement shared across it. The basic setting is shown in figure \ref{fig:non-localandlocal}.
\begin{figure*}
\caption{(a) Circuit diagram showing the local implementation of a unitary in terms of a unitary $\mathbf{U}$. In position-verification, an honest prover implements the required unitary in this form. (b) Circuit diagram showing the non-local implementation of a unitary $\mathbf{U}$. In position-verification, a dishonest prover must use a circuit of this form to implement a required unitary.}
\label{fig:non-localandlocal}
\end{figure*}
Non-local quantum computation has also been understood to arise naturally in the context of quantum gravity \cite{may2019quantum,dolev2022holography,may2023non}, in particular within the context of the AdS/CFT correspondence. There, a higher dimensional theory with gravity is given an equivalent description without gravity. In these two descriptions, processes that occur as local interactions in the higher dimensional theory are reproduced in the dual, lower dimensional description as non-local computations. This connection has lead to consequences for the gravitational theory \cite{may2020holographic,may2022connected}, and discussion around consequences for non-local computation \cite{may2022complexity}.
Because of the connections to position-verification and quantum gravity, position-verification and the related task of non-local computation have been studied by a number of authors, but basic questions remain open. In particular we have linear lower bounds on entanglement \cite{tomamichel2013monogamy} in a non-local computation, and exponential upper bounds \cite{beigi2011simplified}, with only a little known in between. For a special case of a non-local computation known as $f$-routing, where each instance is defined by a classical Boolean function $f$, the entanglement cost has been upper bounded by the size of span program computing $f$ \cite{cree2022code}, so that the class $Mod_kL$ can be achieved efficiently.\footnote{This builds on earlier work \cite{buhrman2013garden} achieving the class $L$.} For general unitaries, Clifford unitaries can be implemented with linear entanglement, and circuits with $T$ depth of $\log n$ can be implemented with polynomial entanglement \cite{speelman2015instantaneous}.
\begin{figure*}
\caption{(a) A conditional disclosure of secrets (CDS) protocol. In the classical setting, Alice and Bob share randomness but do no communicate. They receive inputs $x$ and $y$ respectively. Alice additionally holds a secret $s$. They send messages to the referee. The protocol is correct if the referee can recover $s$ from the messages if and only of $f(x,y)=1$.
In the quantum setting, the randomness may be replaced by entanglement and the messages and secret can be quantum.
(b) A private simultaneous message protocol (PSM). Again Alice and Bob do not communicate but share randomness. They hold inputs $x$ and $y$ respectively. The referee should be able to learn $f(x,y)$ but nothing else about $(x,y)$. In the quantum setting the randomness is replaced with entanglement, and the messages can be quantum.}
\label{fig:PSMandCDS}
\end{figure*}
In this article we prove connections between two well studied cryptographic primitives, conditional disclosure of secrets (CDS) \cite{GERTNER2000592} and private simultaneous message passing (PSM) \cite{ishai1997private}, and non-local quantum computation. These primitives are studied in the context of information theoretic cryptography, in particular in their relationship to secure multiparty computation \cite{aiello2001priced,ishai2010secure}, private information retrieval \cite{GERTNER2000592}, secret sharing \cite{applebaum2020power}, and other cryptographic goals \cite{beimel2018complexity}. We illustrate their functionality in figure \ref{fig:PSMandCDS}. Both settings generally involve $k$ parties along with a referee, but in this work we focus on the $k=2$ case, which is the setting we relate to non-local computation. In CDS, two non-communicating parties, Alice and Bob, receive inputs $x$ and $y$ respectively. Alice additionally holds a secret $s$. Alice and Bob compute messages $m_0(x,s,r)$ and $m_1(y,r)$ based on their inputs and shared randomness, which are then sent to the referee. The referee should be able to recover the secret $s$ if and only if $f(x,y)=1$. PSM is a similar setting. There, Alice and Bob have inputs $x$ and $y$ along with shared randomness. They send messages $m_{0}(x,r)$ and $m_1(y,r)$ to the referee. The referee should be able to compute $f(x,y)$ from the messages, but not learn anything else about the inputs $(x,y)$ than is implied by the value of $f(x,y)$. We give formal definitions of both primitives in section \ref{sec:primitivedefinitions}.
To relate these primitives to non-local computation, we first show that the natural quantum generalization of CDS, which we denote as conditional disclosure of quantum secrets (CDQS), is equivalent to the $f$-routing task. More specifically, protocols for CDQS induce similarly efficient protocols for $f$-routing, and vis versa. Further, we show that classical CDS protocols induce similarly efficient quantum protocols. We also introduce a special case of non-local quantum computation known as a coherent function evaluation (CFE), which we show is closely related to the PSM model: efficient CFE protocols induce efficient PSM protocols using quantum resources (PSQM). We also give a weak converse that shows good PSQM protocols induce CFE protocols that succeed with constant (independent of the input size) probability.\footnote{We only prove this in the exact setting, while all other implications allow for small errors in correctness and small security leakage.} The status of the relationship among these primitives is shown in figure \ref{fig:web}.
\begin{figure}
\caption{Implications among primitives: an arrow from X to Y says that a protocol for X implies a protocol for Y with the same efficiencies (up to constant overheads). All implications shown in blue hold in the robust setting where we allow small errors. The dashed red line indicates that a perfect PSQM protocol that succeeds with high probability implies a CFE protocol that succeeds with constant probability. The subset symbol $\subset$ indicates that $f$-routing and CFE are special cases of NLQC. Primitive abbreviations (DRE, PSM, ...) and theorem numbers link to relevant proofs or definitions.}
\label{fig:web}
\end{figure}
Our results relate position verification to the wider setting of information-theoretic cryptography. This provides a partial explanation of the difficulty of finding better upper and lower bounds in non-local computation, since we now see that doing so would resolve other long-standing questions in cryptography\footnote{For example lower bounds on $f$-routing give lower bounds on (classical) CDS.}. In a positive direction, we use results in NLQC to give new results on CDS and PSM, and vis versa. The results we obtain in this way are \begin{enumerate}
\item Linear lower bounds on communication complexity in CFE (corollary \ref{corollary:CFElowerbound})
\item Linear lower bounds on entanglement in CDQS and PSQM for random functions (corollaries \ref{corollary:CDQSlowerbound} and \ref{corollary:PSQMlowerbound}), and logarithmic lower bounds on entanglement for the inner product function (corollary \ref{corollary:CDQSlogbound} and \ref{corollary:PSQMlogbound})
\item Efficient CDQS and $f$-routing protocols for the quadratic residuosity problem, the first problem not known to be in P/poly with an efficient non-local computation protocol (corollaries \ref{corolary:CDSandCDQSoutsideP} and \ref{corollary:fRouteoutsideP})
\item An efficient protocol for CDQS and PSQM when the target function $f$ can be evaluated by a quantum circuit with low $T$-depth (corollaries \ref{corollary:Tdepthandf-route} and \ref{corollary:PSQMandTdepth})
\item Sub-exponential upper bounds on entanglement cost in $f$-routing for an arbitrary function (corollary \ref{corollary:subexpfroute}) \end{enumerate} More broadly our results take position-verification from being an `island' in the space of cryptographic primitives, with no known classical analogues or connections to other more standard notions, to being richly connected to a web of interrelated primitives, which themselves are related to central goals in information theoretic cryptography. We hope these results lead to new perspectives on position-verification, and new perspectives in the study of CDS, PSM and related primitives. In particular a number of classical results on CDS and PSM may find natural quantum extensions in the context of NLQC. In the discussion we comment on some cases where quantum analogues in the NLQC setting of classical cryptographic results are not yet known.
\noindent \textbf{Outline of this article}
In section \ref{sec:background}, we present some relevant background. Section \ref{sec:QItools} gives a summary of the quantum information tools we exploit. Section \ref{sec:primitivedefinitions} summarizes the various cryptographic primitives which we study and relate. Section \ref{sec:existingrelationships} gives the already known relations among these primitives.
In section \ref{eq:newrelationships} we prove new relationships among our set of cryptographic primitives. The full set of connections is presented as figure \ref{fig:web}. The relationships between CDS and CDQS, CDQS and $f$-routing, CFE and PSQM, and CDQS and PSQM are new to the best of our knowledge.
In section \ref{sec:complexity} we summarize the known results on the complexity of efficiently achievable functions in the PSQM, CDQS and $f$-routing settings. The status of the complexity of efficiently achievable functions is not too changed by our results: existing CDS protocols give $f$-routing protocols, but in both cases the most efficient protocols have a cost like $(\log p) \cdot SP_p(f)$ where $SP(f)$ denotes the minimal size of a span program over $\mathbb{Z}_p$ computing $f$ \cite{GERTNER2000592,cree2022code}. In fact, the two protocols achieving this are essentially quantum and classical analogues of each other, though were found independently.
Sections \ref{sec:newlowerbounds} and \ref{sec:newprotocols} spell out the implications for non-local computation and its special cases that follow from known results in CDS and PSM, and conversely the implications for CDS and PSM that follow from known results in non-local computation. In section \ref{sec:newlowerbounds} we give new lower bounds that follow in this way. In section \ref{sec:newprotocols} we give new upper bounds.
Section \ref{sec:discussion} concludes with some discussion and open problems, in particular commenting on connections to quantum gravity and to some results in the classical cryptography literature that may have quantum analogues relevant to the NLQC setting.
\section{Background}\label{sec:background}
\subsection{Tools from quantum information theory}\label{sec:QItools}
In this section we briefly recall some standard tools of quantum information theory. We follow the conventions of \cite{wilde2013quantum}, where an overview of these tools and further references can also be found.
\noindent \textbf{Distance measures and inequalities}
Define the fidelity, \begin{align}
F(\rho,\sigma) \equiv \left( \tr\left(\sqrt{\sqrt{\rho}\,\sigma\sqrt{\rho}}\right)\right)^2 \end{align}
which is related to the one norm distance $||\rho-\sigma||_1$ by the Fuchs van de Graff inequalities, \begin{align}
1- \sqrt{F(\rho,\sigma)} \leq \frac{1}{2}||\rho-\sigma||_1 \leq \sqrt{1-F(\rho,\sigma)}. \end{align} We define the relative entropy as \begin{align}
D(\rho||\sigma) \equiv \tr(\rho \log \rho)- \tr(\rho\log \sigma) \end{align} where $\log(\cdot)$ is defined base $2$, and we use $\ln$ for the natural logarithm. We also define the von Neumann entropy with the base 2 logarithm.
We will make use of the following statement of the continuity of mutual information. \begin{lemma}\label{lemma:MIcontinuity} For any $\rho_{AB}, \sigma_{AB}$, with \begin{align}
\frac{1}{2}||\rho_{AB}-\sigma_{AB} ||_1 \leq \epsilon \end{align} we have that \begin{align}
|I(A:B)_\rho - I(A:B)_\sigma| &\leq 3\epsilon \log d_A + 2(1+\epsilon) H(\epsilon/(1+\epsilon)) \nonumber \\
&\leq O(\epsilon \log d_A). \end{align} \end{lemma} We will use this in particular to conclude the mutual information is small when the underlying states are close to being product. In the other direction, we'll exploit Pinsker's inequality, \begin{align}\label{eq:Pinskers}
\frac{1}{2 \ln 2} ||\rho-\sigma||^2_1 \leq D(\rho||\sigma) \end{align}
and the observation that $I(A:B)_\rho = D(\rho_{AB}||\rho_A\otimes \rho_B)$, which gives that if the mutual information is small, then the state is close to being product.
It will also be useful to define the diamond norm distance, which is a distance measure on the space of quantum channels. \begin{definition}
The \textbf{diamond norm distance} is defined by
\begin{align}
||\mathbfcal{N}_{B\rightarrow C}-\mathbfcal{M}_{B\rightarrow C}||_\diamond = \sup_{\Psi_{AB}} ||\mathbfcal{N}_{B\rightarrow C}(\Psi_{AB}) - \mathbfcal{M}_{B\rightarrow C}(\Psi_{AB})||_1
\end{align} \end{definition} The diamond norm distance has an operational interpretation in the terms of the maximal probability of distinguishing quantum channels.
\noindent \textbf{Decoupling and recovery}
\begin{figure}
\caption{Illustration of the decoupling theorem. $V_{BE'\rightarrow CE}$ is the isometric extension of a channel $\mathbfcal{N}_{B\rightarrow C}$. The decoupling theorem states that when $I(A:E)$ is small, there is a good inverse to $\mathbfcal{N}_{B\rightarrow C}$.}
\label{fig:decoupling}
\end{figure}
The basic idea underlying the connection between CDS and $f$-routing that we will give is the notion of decoupling and complementary recovery. To develop this, consider a quantum channel $\mathbfcal{N}_{B\rightarrow C}$. We would like to understand when this channel has an (approximate) inverse. To understand this consider any unitary extension of the channel, call it $\mathbf{V}_{BE'\rightarrow CE}$, which satisfies \begin{align}
\mathbfcal{N}_{B\rightarrow C}(\cdot) = \tr_{E} (\mathbf{V}_{BE'\rightarrow CE} \cdot \mathbf{V}^\dagger_{BE'\rightarrow CE}). \end{align} There is a theorem that says if we input a maximally entangled state $\ket{\Psi^+}_{AB}$ and find that $I(A:E)_{\mathbfcal{N}(\Psi^+)}$ is small, then there exists an inverse channel $\mathbfcal{N}^{-1}$ which works well. This is summarized in figure \ref{fig:decoupling}. We give the full statement in the following theorem. \begin{theorem}\label{thm:decoupling} Given a quantum channel $\mathbfcal{N}_{B\rightarrow C}$ consider any unitary extension $\mathbf{V}_{BE'\rightarrow CE}$ and define the state \begin{align}
\ket{\Psi}_{ACE} = \mathbf{V}_{BE'\rightarrow CE}\ket{\Psi^+}_{AB}\ket{0}_{E'}. \end{align} Then if \begin{align}
I(A:E)_{\Psi} \leq \epsilon, \end{align} there exists an inverse channel $\mathbfcal{N}^{-1}_{B\rightarrow C}$ such that \begin{align}
F(\Psi^+,\mathbfcal{N}^{-1}_{B\rightarrow C}\circ \mathbfcal{N}_{B\rightarrow C}(\Psi^+)) \geq 1-\sqrt{\epsilon} \end{align} \end{theorem} For a proof see \cite{schumacher2002entanglement,schumacher2002approximate}.
\subsection{Definitions of the primitives}\label{sec:primitivedefinitions}
In this section we give the definitions of each of the primitives that we discuss in this article. Note that we focus on information theoretic definitions of security. In all cases there are meaningful versions of these primitives with computational security, but we have not explored their connections to non-local computation.
\noindent \textbf{Conditional disclosure of secrets}
We first define the classical CDS setting, which we also illustrate in figure \ref{fig:CDS}. \begin{definition}\label{def:CDS}
A \textbf{conditional disclosure of secrets (CDS)} task is defined by a choice of function $f:\{0,1\}^{\times 2n}\rightarrow \{0,1\}$.
The scheme involves inputs $x\in \{0,1\}^{\times n}$ given to Alice, and input $y\in \{0,1\}^{\times n}$ given to Bob.
Alice and Bob share a random string $r$.
Additionally, Alice holds a string $s$ drawn from distribution $S$, which we call the secret.
Alice sends message $m_0(x,s,r)$ to the referee, and Bob sends message $m_1(y,r)$.
We require the following two conditions on a CDS protocol.
\begin{itemize}
\item $\epsilon$\textbf{-correct:} There exists a decoding function $D(m_0,x,m_1,y)$ such that
\begin{align}
\forall \,(x,y) \in X\times Y \,\,s.t.\,\,f(x,y)=1,\,\,\, \mathrm{Pr}[D(m_0,x,m_1,y)=s] \geq 1-\epsilon
\end{align}
\item $\delta$\textbf{-secure:} There exists a simulator producing a distribution $Sim$ on the random variable $M=M_0M_1$ such that
\begin{align}
\forall \,(x,y) \in X\times Y \,\,s.t.\,\, f(x,y)=0,\,\,\, ||Sim_{M|xy} - P_{M|xys} ||_1\leq \delta
\end{align}
\end{itemize} \end{definition} We will also make use of an alternative statement of security of a CDS protocol, which is stated in terms of the correlation between the message systems and the secret. \begin{lemma}\label{lemma:MIstatementforCDS}
Let $S$ be the distribution of the random variable storing the secret $s$.
Then a $\delta$-secure CDS protocol has the property
\begin{align}
\forall (x,y)\in X\times Y \,\,\, s.t. \,\, f(x,y)=0,\,\,\, I(S:M)_{P_{SM|xy}} \leq \delta'
\end{align}
where $\delta'=O(\delta |s|)$.
Conversely, a CDS protocol where the above holds is $\sqrt{2\delta'}$ secure. \end{lemma} \begin{proof}\,
Given a $\delta$-secure CDS and considering instances with $(x,y)\in f^{-1}(0)$, we have that
\begin{align}
||Sim_{M|xy} - P_{M|xys} ||_1\leq \delta.
\end{align}
Then, consider the joint distribution on the secret and message,
\begin{align}\label{eq:jointCDSdistribution}
||P_{SM|xy} - P_{S}P_{M|xy} ||_1&= ||P_{M|Sxy}P_{S} - P_{S}P_{M|xy} ||_1\nonumber \\
&= \sum_s P_{s}||P_{M|sxy} - P_{M|xy} ||_1\nonumber \\
&= \sum_s P_{s} ||Sim_{M|xy} - P_{M|xy} ||_1+\delta \nonumber \\
&= 2\delta
\end{align}
Then, continuity of the mutual information (lemma \ref{lemma:MIcontinuity}) gives that, for $0$ instances of the inputs,
\begin{align}
I(S:M)_{P_{SM|xy}} \leq O(\delta |s|)
\end{align}
as needed.
For the converse statement, we use Pinsker's inequality \ref{eq:Pinskers} to find
\begin{align}
\frac{1}{2} ||P_{SM|xy} - P_{S}P_{M|xy} ||_1^2 \leq I(S:M)_{P_{SM|xy}} \leq \delta'
\end{align}
which gives the claimed inequality. \end{proof}
We can understand the mutual information statement of security as an averaged case statement (averaged over choices of secret), and the statement in definition \ref{def:CDS} given as a worst case statement. The lemma above is just noting that we can translate between these in a setting where the secret size is small.
Notice that in our definition of CDS we have imposed that the secret be held only by Alice. Another remark around the robustness of our definition is that we can easily transform protocols that succeed with the secret held on both sides to one where the secret is held only on one side. This is a standard remark about CDS, though we don't know a reference where this is shown in the imperfect setting, so we give the simple proof of this fact here. \begin{remark}\label{remark:onesidedCDS}
A CDS task where $s$ is initially held by Alice and Bob can be turned into one where only Alice holds $s$ at the cost of $|s|$ shared random bits, and $|s|$ bits of communication. If the CDS protocol is $\epsilon$-correct and $\delta$-secure, the one-sided protocol will be $\epsilon$-correct and $O(\sqrt{\delta})$ secure. \end{remark} \begin{proof}\,
To see this, suppose we have a perfectly correct and secure CDS protocol which works when $s$ is held on both sides.
Then run this protocol on a randomly chosen $s'$, and have Alice send $s'\oplus s$ to the referee.
Only Alice needs to know $s$ to run this protocol.
Suppose our initial CDS protocol is $\epsilon$-correct and $\delta$-secure.
Then the new CDS will also be $\epsilon$-correct, since $s$ can be computed deterministically from $s'$ and the bit $\tilde{s}=s\oplus s'$.
To understand security, note that $\delta$-security of the original protocol implies
\begin{align}
||P_{S'M}-P_{S'}P_{M} ||_1\leq \delta
\end{align}
Using this, $P_{S\tilde{S}}=P_{S}P_{\tilde{S}}$ (from the properties of the one-time pad), and that $S$ and $M$ are independent conditioned on $\tilde{S}$, we have
\begin{align}
||P_{S\tilde{S}M} - P_{S}P_{\tilde{S}}P_{M}||_1&= ||P_{S|\tilde{S}M}P_{\tilde{S}M} - P_{S}P_{\tilde{S}}P_{M}||_1\nonumber \\
&= ||P_{S|\tilde{S}}P_{\tilde{S}M} - P_{S}P_{\tilde{S}}P_{M}||_1\nonumber \\
&\leq ||P_{S|\tilde{S}}P_{\tilde{S}}P_{M} - P_{S}P_{\tilde{S}}P_{M}||_1+ \delta \nonumber \\
&= ||P_{S\tilde{S}}P_{M} - P_{S}P_{\tilde{S}}P_{M}||_1+ \delta \nonumber\\
&= \delta
\end{align}
which is exactly $\delta$ security of the one sided CDS protocol. \end{proof}
Finally, we remark that a CDS for secret $s_1$ and a CDS for secret $s_2$ can be run in parallel using fresh randomness while maintaining security and correctness of each CDS scheme. To see this, call the message for the first CDS $M_1$ and the message for the second CDS $M_2$. If we consider how much the referee can learn about the secret $s_1$, message $M_2$ doesn't reveal anything, because it depends only on the randomness $r_2$, the inputs (which the referee knows already as part of the CDS for $s_1$), and $s_2$. All of these variables are already known by the referee as part of the CDS for $s_1$, or are uncorrelated with $s_1$. More succinctly, the distribution on $s_1$ is independent of $M_2$ when conditioning on $XY$, so revealing $M_2$ doesn't help the referee learn $s_1$, given that they already know $XY$, or in notation \begin{align}\label{eq:independentCDS}
P_{M_1M_2|xys} = P_{M_1|xys_1}P_{M_2|xys_2} \end{align} A similar statement establishes security of the CDS hiding $s_2$ in the presence of message $M_1$.
As a consequence of the above comments, the CDS hiding secret $s=(s_1,s_2)$ given by running the CDS for each secret in parallel has good security and correctness, as we capture in the next lemma. \begin{lemma}
Suppose we have a CDS for function $f$ which is $\epsilon$ correct and $\delta$ secure, and hides $k$ bits.
Then we can build a CDS for function $f$ that hides $mk$ bits, and is $m\epsilon $ correct and $m\delta$ secure. \end{lemma} \begin{proof}\, The strategy is to repeat the CDS protocol that hides $k$ bits $m$ times in parallel. To understand correctness of the new protocol, notice that on $1$ instances the probability of the referee guessing $s_i$ correctly is at least $1-\epsilon$, so their probability of guessing all $m$ strings $s_i$ correctly is at least $(1-\epsilon)^m \geq (1-mk)$. To understand security, we define a simulator for the composed protocol by taking the product of the distributions for a single instance of the protocol, \begin{align}
Sim_{M_1...M_m|xy} \equiv Sim_{M_1|xy}...Sim_{M_m|xy}. \end{align} We also note that, using fresh randomness for each instance of the CDS, we can extend equation \ref{eq:independentCDS} to \begin{align}
P_{M_1...M_m|xys} = P_{M_1|xys_1}...P_{M_m|xys_m}. \end{align} Then by repeated application of the triangle inequality, and using security of each instance of the CDS, we have that on $0$ instances \begin{align}
||Sim_{M_1...M_m|xy} - P_{M_1...M_m|xys}||_1 = ||Sim_{M_1|xy}...Sim_{M_m|xy} - P_{M_1|xys_1}...P_{M_m|xys}||_1 \leq m\delta\nonumber \end{align} as claimed. \end{proof}
\noindent \textbf{Conditional disclosure of quantum secrets}
\begin{figure*}
\caption{(a) Illustration of a CDQS protocol. Alice and Bob share an entangled resource state, illustrated as the solid curved line. Alice receives the classical string $x\in \{0,1\}^{\times n}$ as input, and a quantum system $Q$, which we take to be maximally entangled with a reference $R$. Bob receives input $y\in \{0,1\}^{\times n}$. Alice and Bob prepare quantum systems $M_0$ and $M_1$, which they pass to the referee. The protocol succeeds if when $f(x,y)=1$ the state on $RQ$ is preserved, and when $f(x,y)=0$ the $M=M_0M_1$ system is uncorrelated with $R$. See definition \ref{def:CDQS}. (b) A PSQM protocol. Again Alice and Bob share an entangled resource state. Alice receives input $x\in \{0,1\}^{\times n}$, Bob receives input $y\in \{0,1\}^{\times n}$. Alice and Bob prepare quantum systems $M_0$ and $M_1$, which they pass to the referee. The protocol succeeds if the referee can determine $f(x,y)$, but the system $M=M_0M_1$ otherwise reveals nothing about the inputs $x,y$. See definition \ref{def:PSQM}.}
\label{fig:PSQMandCDQS}
\end{figure*}
To the best of our knowledge the quantum analogue of the CDS model has not been studied explicitly in the literature.\footnote{It has been studied in an indirect way, since (as we show later) it is equivalent to $f$-routing.} We give a definition here which features quantum resources and a quantum secret. The CDQS primitive is illustrated in figure \ref{fig:CDQS}.
\begin{definition}\label{def:CDQS}
A \textbf{conditional disclosure of quantum secrets (CDQS)} task is defined by a choice of function $f:\{ 0,1\}^{\times 2n}\rightarrow \{0,1\}$, and a $d_Q$ dimensional Hilbert space $\mathbfcal{H}_Q$.
The task involves inputs $x\in \{0,1\}^{\times n}$ and system $Q$ given to Alice, and input $y\in \{0,1\}^{\times n}$ given to Bob.
Alice sends message system $M_0$ to the referee, and Bob sends message system $M_1$.
Label the combined message systems as $M=M_0M_1$.
We put the following two conditions on a CDQS protocol.
\begin{itemize}
\item $\epsilon$\textbf{-correct:} There exists a channel $\mathbfcal{D}^{x,y}_{M\rightarrow Q}$, called the decoder, such that
\begin{align}
\forall (x,y)\in X\times Y \,\,\, s.t. \,\, f(x,y)=1,\,\,\, F(\Psi^+_{RQ},\mathbfcal{D}^{x,y}_{M\rightarrow Q}(\rho_{RM}(x,y))) \geq 1-\epsilon
\end{align}
\item $\delta$\textbf{-secure:} There exists a quantum channel $\mathbfcal{S}_{XY\rightarrow M}$, called the simulator, such that
\begin{align}
\forall (x,y)\in X\times Y \,\,\, s.t. \,\, f(x,y)=0,\,\,\, ||\frac{1}{d_R}\mathbfcal{I}_R\otimes \mathbfcal{S}_{XY\rightarrow M}(\ketbra{xy}{xy}_{XY}) - \rho_{RM}(x,y)||_1\leq \delta
\end{align}
\end{itemize} \end{definition}
Just as in the classical case, the security definition here is equivalent to an entropic condition when the secret isn't too large. \begin{lemma}\label{lemma:CDQSsecurityfromMI}
A $\delta$-secure CDQS protocol has the property
\begin{align}
\forall (x,y)\in X\times Y \,\,\, s.t. \,\, f(x,y)=0,\,\,\, I(R:M)_{\rho_{RM}(x,y)} \leq \delta'
\end{align}
where $\delta'=O(\epsilon \log d_R)$.
Conversely, a CDQS protocol where the above holds with probability $\delta'$ is $\sqrt{2\delta'}$ secure. \end{lemma} \begin{proof}\, Same as in the classical case, now using the quantum Pinsker's inequality and continuity of the quantum mutual information. \end{proof}
In our definition of CDQS, we require a quantum system be taken as the secret, and allow the use of quantum resources. Another quantum variant of CDS we could have defined would allow quantum resources but restrict to a classical secret. We could call this CDQS'. This variant is in fact equivalent to the above definition. This follows from our proof below that classical CDS protocols gives quantum CDS protocols, which is easily modified to show a CDQS' gives CDQS with similar resources. Then one can observe that a CDQS protocol can be modified to a CDQS' protocol by choosing the secret to be a state in a chosen basis. Taken together this gives that CDQS' and CDQS are equivalent.
\noindent \textbf{Private simultaneous message passing}
Next we move on to discuss another basic cryptographic primitive of interest in this article, which is private simultaneous message passing. This primitive is illustrated in figure \ref{fig:PSM}.
\begin{definition}\label{def:PSM}
A \textbf{private simultaneous message (PSM)} task is defined by a choice of function $f:X\times Y\rightarrow Z$.
The inputs to the task are $n$ bit strings $x$ and $y$ given to Alice and Bob, respectively.
Alice then sends a message $m_0(x,r)$ to the referee, and Bob sends message $m_1(y,r)$.
From these inputs, the referee prepares an output bit $z$.
We require the task be completed in a way that satisfies the following two properties.
\begin{itemize}
\item \textbf{$\epsilon$-correctness:} There exists a decoder $Dec$ such that \begin{align}
\forall (x,y)\in X\times Y,\, \,\,\,\mathrm{Pr}[Dec(m_0,m_1)=f(x,y)] \geq 1-\epsilon.
\end{align}
\item \textbf{$\delta$-security:} There exists a simulator producing a distribution $Sim$ on the random variable $M=M_0M_1$ such that
\begin{align}
\forall (x,y)\in X\times Y,\, \,\,\,||Sim_{M|f(x,y)} - P_{M|xy}||_1\leq \delta.
\end{align}
Stated differently, the distribution of the message systems is $\delta$-close to one that depends only on the function value, for every choice of $x,y$.
\end{itemize} \end{definition}
In PSM we can allow the function $f$ to take Boolean or other values. For instance we can take $f$ to be natural number valued and defined by a counting problem. Another comment is that PSM protocols can be run in parallel, in the sense that $\epsilon$-correct and $\delta$-secure protocols for $f_1(x,y)$ and $f_2(x,y)$ can be run together to give a $2\epsilon$-correct and $2\delta$-secure protocol for the function $f(x,y)=(f_1(x,y),f_2(x,y))$. This is straightforward to show from the security definition.
\noindent \textbf{Private simultaneous quantum message passing (PSQM)}
As with CDS, there is a natural quantum version of PSM. In this case the functionality of the protocol is unchanged, but the allowed resources are now quantum mechanical. A PSQM protocol is shown in figure \ref{fig:PSQM}.
\begin{definition}\label{def:PSQM}
A \textbf{private simultaneous quantum message (PSQM)} task is defined by a choice of function $f:X\times Y\rightarrow Z$.
The inputs to the task are $n$ bit strings $x$ and $y$ given to Alice and Bob, respectively, each of which are chosen independently and at random.
Alice then sends a quantum message system $M_0$ to the referee, and Bob sends quantum message system $M_1$.
From the combined message system $M=M_0M_1$, the referee prepares an output bit $z$.
We require the task be completed in a way that satisfies the following two properties.
\begin{itemize}
\item \textbf{$\epsilon$-correctness:} There exists a decoding map $\mathbf{V}_{M \rightarrow Z\tilde{M}}$ such that
\begin{align}
\forall (x,y)\in X\times Y, \,\,\,\,\, \left|\left|\tr_{\tilde{M}}(\mathbf{V}_{M \rightarrow Z\tilde{M}} \rho_{M}(x,y) \mathbf{V}_{M \rightarrow Z\tilde{M}}^\dagger ) - \ketbra{f_{xy}}{f_{xy}}_Z\right|\right|_1 \leq \epsilon.
\end{align}
where $\rho_M(x,y)$ is the density matrix on $M$ produced on inputs $x,y$.
\item \textbf{$\delta$-security:} There exists a simulator, which is a quantum channel $\mathbfcal{S}_{Z\rightarrow M}(\cdot)$, such that
\begin{align}
\forall (x,y)\in X\times Y,\,\,\,\,\,\left|\left|\rho_{M}(x,y) - \mathbfcal{S}_{Z\rightarrow M}(\ketbra{f_{xy}}{f_{xy}}_Z)\right|\right|_1 \leq \delta.
\end{align}
Stated differently, the state of the message systems is $\delta$-close to one that depends only on the function value, for every choice of input.
\end{itemize} \end{definition}
Just like in the classical case, PSQM protocols can be run in parallel with only small relaxations in security and correctness.
\noindent \textbf{Decomposable randomized encodings}
A related primitive, which we will make briefer use of, is the notion of a decomposable randomized encoding. We recall some definitions given in \cite{computation2013randomization}.
\begin{definition} Let $X,Y,\hat{Y},R$ be finite sets and let $f:X_1\times ... \times X_n \rightarrow Y$. A function $\hat{f}:X\times R\rightarrow \hat{Y}$ is an $\epsilon$-correct and $\delta$-private \textbf{randomized encoding} for $f$ if it satisfies \begin{itemize}
\item $\epsilon$-\textbf{correctness:} There exists a function $Dec$ called a decoder such that for every $x\in X$ and $r\in R$ we have
\begin{align}
\mathrm{Pr}[Dec(\hat{f}(x,r))=f(x)]\geq 1 - \epsilon.
\end{align}
\item $\delta$-\textbf{privacy:} There exists a randomized function, called a simulator, producing the random variable $Sim$ such that
\begin{align}
||Sim_{\hat{Y}|Y} - P_{\hat{Y}|X} ||_1\leq \delta.
\end{align} \end{itemize} \end{definition}
\begin{definition}\label{def:DRE} A \textbf{decomposable randomized encoding (DRE)} for a function $f:X_1\times ... \times X_n \rightarrow Y$ is a randomized encoding of $f$ that has the form \begin{align}
\hat{f}(x_1,...,x_n;r)= (\hat{f}_1(x_1,r),...,\hat{f}_n(x_n,r)) \end{align} A DRE is $\epsilon$-correct and $\delta$-secure under the same conditions as a randomized encoding, given above. \end{definition} \noindent We will in fact only use that certain randomized encodings are decomposable across a single splitting of the inputs. That is we are interested in functions $f:X\times Y\rightarrow Z$ and need the randomized encoding to take the form \begin{align}
\hat{f}(x,y;r)= (\hat{f}_1(x,r),\hat{f}_2(y,r)) \end{align} In this setting we will say $f(x,y)$ has a randomized encoding decomposable across $X\times Y$.
\noindent \textbf{Non-local computation}
Finally we come to the notion of a non-local computation, which were first studied in the context of cheating strategies for position verification tasks. The general setting is shown in figure \ref{fig:non-localandlocal}. A non-local computation takes the form shown in figure \ref{fig:non-localcomputation}, with the goal being to simulate the action of a local unitary (figure \ref{fig:local}).
There are two differing but related definitions we can give of correctness in a non-local computation. First, we can guarantee that the input state be fixed, call it $\ket{\Psi}$, and then require that the output be close to $\mathbf{U}\ket{\Psi}$ for $\mathbf{U}$ the desired unitary. Alternatively, we can require that the action of the non-local computation is close to $\mathbf{U}$ in a worst case (over choice of input states) sense. This is captured by requiring the action of the NLQC is close to the desired unitary in diamond norm distance.
We will not give a formal definition of a fully general NLQC here, but instead focus on two special cases. The first, $f$-routing, is defined using a closeness measure in the first sense, of acting correctly on a particular input. This weaker notion of correctness turns out to suffice to relate $f$-routing and CDQS, as we will see later.
\begin{definition}\label{def:frouting}
A \textbf{$f$-routing} task is defined by a choice of Boolean function $f:\{ 0,1\}^{\times 2n}\rightarrow \{0,1\}$, and a $d$ dimensional Hilbert space $\mathbfcal{H}_Q$.
Inputs $x\in \{0,1\}^{\times n}$ and system $Q$ are given to Alice, and input $y\in \{0,1\}^{\times n}$ is given to Bob.
Alice and Bob exchange one round of communication, with the combined systems received or kept by Bob labelled $M$ and the systems received or kept by Alice labelled $M'$.
The $f$-routing task is completed $\epsilon$-correctly if there exists a channel $\mathbfcal{D}_{M\rightarrow Q}$ such that,
\begin{align}
\forall (x,y)\in X\times Y \,\,\, s.t. \,\, f(x,y)=1,\,\,\, F(\Psi^+_{RQ}, \mathbfcal{D}_{M\rightarrow Q}(\rho_{RM})) \geq 1-\epsilon
\end{align}
and there exists a channel $\mathbfcal{D}_{M'\rightarrow Q}$ such that
\begin{align}
\forall (x,y)\in X\times Y \,\,\, s.t. \,\, f(x,y)=0,\,\,\,F(\Psi^+_{RQ}, \mathbfcal{D}_{M'\rightarrow Q}(\rho_{RM'})) \geq 1-\epsilon.
\end{align}
In words, Bob can recover $Q$ with high fidelity if $f(x,y)=1$ and Alice can recover $Q$ with high fidelity if $f(x,y)=0$. \end{definition}
The second special case we study is coherent function evaluation, for which we require the action of the NLQC be close to correct in the diamond norm sense. This stronger notion is needed to relate CFE to PSQM, as we will see later.
\begin{definition}\label{def:CFE}
A \textbf{coherent function evaluation (CFE)} task is defined by a choice of Boolean function $f:\{ 0,1\}^{\times 2n}\rightarrow \{0,1\}$.
The task is to implement the isometry
\begin{align}
\mathbf{V}_f = \sum_{xy} \ket{xy}_{Z'} \ket{f_{xy}}_{Z} \bra{x}_X\bra{y}_Y
\end{align}
in the non-local form of figure \ref{fig:non-localcomputation}.
We say a CFE protocol is $\epsilon$ correct if the diamond norm distance between $\mathbf{V}_f$ and the implemented channel is not larger than $\epsilon$. \end{definition}
\noindent \textbf{Secret sharing}
An important tool throughout cryptography, and in particular in our context, is the notion of a secret sharing scheme. We introduce this next.
\begin{definition} \label{def:SS}
A \textbf{secret sharing scheme} $\mathbf{S}$ is a map from a domain $K$ and randomness $R$ to variables $S_1,...,S_n$, here called shares.
Let $A$ be a subset of the $S_i$ and $\mathcal{S}_A$ the distribution on the shares $A$.
Then a scheme $\mathbf{S}$ realizes access structure $\mathbf{A}$ with \textbf{$\epsilon$-correctness} if, for each subset of shares $A\in \mathbf{A}$ there exists a decoding map $D_A:A\rightarrow K$ such that
\begin{align}
\mathrm{Pr}[D_A(A)=s_K]\geq 1-\epsilon.
\end{align}
A scheme $\mathbf{S}$ is \textbf{$\delta$-secure} if, whenever $U\notin \mathbf{A}$, there exists a map producing a distribution $Sim$ on $U$ such that
\begin{align}
|Sim_{U} - \mathcal{S}_{U|K}| \leq \delta
\end{align}
If $\epsilon=\delta=0$ we say that the scheme $\mathbf{S}$ is perfect. \end{definition} The access structure of a secret scheme can be specified as a set of subsets of shares, as in the above definition, or equivalently in terms of an \textbf{indicator function}. The indicator function is defined by \begin{align}
f_I(x) = \begin{cases}
1 & \text{if} \,\,\,\,\{S_i:x_i=1\} \in \mathcal{A}\\
0 & \text{otherwise}
\end{cases} \end{align} We can observe that if $A\in \mathcal{A}$ then necessarily $A\cup S_{i}\in\mathcal{A}$. This follows because if we can reconstruct the secret from $A$, we can also reconstruct it from a larger set. This means that valid indicator functions will always be monotone.
\noindent \textbf{Other related primitives}
Each of the primitives discussed above is in turn related to others in various ways. Reviewing these further connections is outside the scope of this article. Instead, we have included in our discussion only primitives for which we have found the connection to NLQC gives a new result on NLQC, or for which NLQC implies a new result on the primitive. We briefly mention however some settings with natural relationships to the ones discussed here; our list and references are not exhaustive. CDS and PSM are related to zero-knowledge proofs \cite{applebaum2017private}, secret sharing \cite{applebaum2020better}, communication complexity \cite{applebaum2021placing}, private information retrieval \cite{ishai1997private}, and secure multiparty computation \cite{ishai1997private}. A useful review of these primitives and the broader context of information theoretic cryptography is given in \cite{BIUschool}. All of these connections may be interesting to revisit in the quantum setting, and in light of the connection to non-local computation and position-verification.
\subsection{Existing relations among primitives}\label{sec:existingrelationships}
\noindent \textbf{SS gives CDS}
In \cite{GERTNER2000592}, the authors upper bound the randomness complexity of a CDS scheme in terms of the size of a secret sharing scheme whose access structure is related to $f$. We recall their result next, narrowing their result to the two player case for simplicity. \begin{theorem}\label{thm:SSgivesCDS} Let $f_M:\{0,1\}^m\times \{0,1\}^m \rightarrow \{0,1\}$ be a monotone Boolean function and let $f:\{0,1\}^n\times \{0,1\}^n \rightarrow \{0,1\}$ be a projection of $f_M$, that is $f(x,y) = f_M(g_1(x),g_2(y))$. Let $\mathbf{S}$ be a perfect secret sharing scheme realizing the access structure $f_M$, in which the total share size is $c$, and let $s$ denote a secret (from the domain of $\mathbf{S}$) which is known to all players. Then there exists a CDS protocol for disclosing $s$ subject to the condition $f$ with randomness $c$, and a (perhaps different) protocol with communication complexity bounded above by $c$. \end{theorem} The protocol which establishes this theorem is, heuristically, the following. We start by illustrating the case where $f=f_M$ is already a monotone function, and so can be realized as the indicator function of some secret sharing scheme $\mathbf{S}$. Then the protocol is as follows. Without loss of generality take Alice and Bob to both hold the secret $s$ (see Remark \ref{remark:onesidedCDS}). To carry out the protocol, both parties prepare a secret sharing scheme $\mathbf{S}$ which has indicator function $f_M$, using their shared randomness as the randomness $R$ needed to prepare the scheme. Then, Alice sends those shares $S_i$ to the referee for which $x_i=1$, and Bob sends those shares $S_{i+n}$ for which $y_i=1$. Then if $f_M(x,y)=1$, following this local rule they will have collectively sent an authorized set of shares and the referee can reconstruct the secret $s$. If $f_M(x,y)=0$, they will have sent an unauthorized set of shares and the referee cannot learn the secret. To extend this to non-monotone functions, Alice and Bob first locally compute $g_1$ and $g_2$ respectively, and then perform the same secret sharing protocol now with bits of $g_1(x)$ or $g_2(y)$ controlling which shares are sent to the referee. Notice that the communication complexity is at most the total size of the shares of the secret sharing scheme.
To see the protocol that gives an upper bound for the randomness complexity\footnote{This protocol is not given in \cite{GERTNER2000592}, but is a straightforward extension of their idea.}, we now have only Alice prepare the shares of the secret sharing scheme. For shares $i\leq n$, she sends share $S_i$ if $x_i=1$ as before.
For shares $i>n$, she sends $S_i\oplus r_i$, where the XOR is taken bitwise with a random string $r_i$ of length $|S_i|$. Bob then sends $r_i$ iff $y_i=1$.
Notice that the randomness complexity is now at most $\sum_i r_i \leq \sum_i |S_i|$, which is just the size of the scheme. The communication complexity is now somewhat larger, but is bounded by twice the size.
We can also generalize the above theorem to the case of approximate secret sharing schemes. In particular, if we use an approximate secret sharing scheme in the second of the protocols above we find that an $\epsilon$-correct and $\delta$-secure secret sharing scheme of size $c$ for an indicator function $f_I$ leads to an $\epsilon$-correct and $\delta$-secure CDS for the same function, using randomness complexity $c$. A similar observation holds for the protocol bounding the communication complexity. We collect these observations as the following remark.
\begin{remark}\label{thm:robustCDSfromSS} Let $f_M:\{0,1\}^m\times \{0,1\}^m \rightarrow \{0,1\}$ be a monotone Boolean function and let $f:\{0,1\}^n\times \{0,1\}^n \rightarrow \{0,1\}$ be a projection of $f_M$, that is $f(x,y) = f_M(g_1(x),g_2(y))$. Let $\mathbf{S}$ be a $\epsilon$-correct and $\delta$-secure secret sharing scheme realizing the access structure $f_M$, in which the total share size is $c$, and let $s$ denote a secret (from the domain of $\mathbf{S}$) which is known to all players. Then there exists an $\epsilon$-correct and $\delta$-secure CDS protocol disclosing $s$ subject to the condition $f$ with randomness $c$, and a (perhaps different) $\epsilon$-correct and $\delta$-secure protocol with communication complexity bounded above by $c$. \end{remark}
\noindent \textbf{DRE gives PSM}
See for example \cite{computation2013randomization} for the connection between DRE and PSM. We give a robust version of this connection as the next theorem.
\begin{theorem} \label{thm:DREgivesPSM} Suppose that $f:X\times Y\rightarrow Z$ has an $\epsilon$-correct and $\delta$-secure decomposable randomized encoding using $n_R$ bits of randomness, and $n_M$ message bits. Then there is a $\epsilon$-correct and $\delta$-secure PSM protocol for $f$ that uses the same amount of randomness and message bits. \end{theorem} \begin{proof} \,Let the DRE for $f$ be \begin{align}
\hat{f}(x,y;r) = (\hat{f}_X(x,r),\hat{f}_Y(y,r)) \end{align} To implement the PSM protocol, Alice prepares $\hat{f}_X(x,r)$ and sends this to the referee, while Bob prepares $\hat{f}_Y(y,r)$ and sends this to the referee. The referee then uses the decoder for the DRE to determine $f(x)$. Noticing that the conditions on the DRE and PSM are in fact exactly the same under these identifications, we have that the PSM is also $\epsilon$-correct and $\delta$-secure. \end{proof}
Notice that a PSM for $f$ also gives a randomized encoding for the function $f$, albeit one that is decomposable across on particular splitting of the input bits into $X\times Y$, and not necessarily decomposable bitwise, as required in the definition of a DRE.
\noindent \textbf{PSM gives CDS}
Next, we relate the PSM and CDS primitives. See for example \cite{GERTNER2000592,applebaum2017private}.
\begin{theorem}\label{thm:PSMgivesCDS} Suppose that a $\epsilon$-correct and $\delta$-private PSM protocol exists for $f(x,y)$ using messages of at most $n_M$ bits and no more than $n_E$ shared random bits. Then a CDS protocol using $n_M+1$ bits of message and $n_E$ random bits exists which is $\epsilon$ correct and $O(\delta \log d_R)$ private, and hides one bit. \end{theorem} \begin{proof}\, We wish to carry out the CDS task using the given PSM protocol. First, we note that by adding one bit of randomness we can assume $s$ is held by both Alice and Bob. This is because of remark \ref{remark:onesidedCDS}.
Next, we show that given the PSM protocol for $f$ there is a similarly efficient PSM for the function $f(x,y)\wedge s$, with $s$ held on both sides. To show this, first consider the case where $f(x,y)$ is a constant function. Then Alice and Bob can follow a fixed strategy (reveal $s$ or not) and we are done. Thus we assume $f(x,y)$ is non-constant, and choose any input values for which it is $0$ and label them $(x_*,y_*)$. Run the PSM on inputs $x'=sx+ (1-s) x_*$ and $y'=sy + (1-s) y_*$. Then notice that $f(x',y')=f(x,y)\wedge s$.
To see $\epsilon$-correctness, we have the referee output the outcome of the modified PSM protocol as their guess for the secret $s$. Then their success probability conditioned on $f(x,y)=1$ is exactly $1-\epsilon$, so the CDS protocol is $1-\epsilon$ correct.
Next consider security. Let the distribution of values of $f(x,y)$ be $F$, the distribution of values of $f(x',y')$ be $F'$, and the distribution of $x'$ and $y'$ be $X'$ and $Y'$ respectively. Security of the original PSM protocol implies \begin{align}
||Sim_{M|F'} - P_{M|X'Y'}||_1\leq \delta. \end{align}
Then notice that because $X'Y'$ are determined by $XYS$, we have $P_{M|X'Y'}=P_{M|XYS}$. Next, restrict to the distributions where $f(x,y)=0$, leading to \begin{align}
||Sim_{M|F'=0} - P_{M|XYS}||_1\leq \delta \end{align} which is $\delta$ security of the CDS. \end{proof}
\noindent \textbf{PSM gives PSQM}
Next, we prove that a protocol for PSM also gives a protocol for PSQM. This might seem trivial, since the quantum resources available in the PSQM can simulate the classical resources used in the PSM, but establishing security requires we show the classical security definition is strong enough to enforce the quantum security definition. As far as we are aware this is not written in the literature (but see \cite{kawachi2021communication} for the introduction of PSQM), but is straightforward enough we include it in this section.
\begin{theorem}\label{thm:PSMgivesPSQM}
Suppose we have a PSM protocol which is $\epsilon$ correct and $\delta$ secure.
Then we can construct a PSQM protocol which is $2\sqrt{\epsilon}$ correct and $\delta$ secure. \end{theorem} \begin{proof}\, Correctness of the PSM protocol implies that there exists a decoder $Dec(m_0,m_1)$ such that \begin{align}
\forall (x,y)\in X\times Y\,\,\,\, \mathrm{Pr}[Dec(m_0,m_1) = f(x,y)]\geq 1-\epsilon \end{align} where the probability is over choices of the random string $r$. In quantum notation, we have that the message system is described by the density matrix \begin{align}
\rho_M(x,y) = \sum_{m} p(m|x,y) \ketbra{m}{m} \end{align} and can write the output of the decoder as \begin{align}
\mathbfcal{D}_{M\rightarrow Z}(\rho_M(x,y)) = \sum_{m} p(m|x,y) \ketbra{D(m)}{D(m)}. \end{align} Then notice that \begin{align}
F( \mathbfcal{D}_{M\rightarrow Z}(\rho_M(x,y)),\ket{f_{xy}}) = \sum_m p(m|x,y) |\braket{D(m)}{f_{xy}}|^2 \geq 1-\epsilon \end{align} where the last line follows because we see the fidelity is exactly the guessing probability, which is bounded from below by the classical correctness definition. Using the Fuchs van de Graff inequalities, we get that \begin{align}
||\mathbfcal{D}_{M\rightarrow Z}(\rho_M(x,y)) - \ketbra{f_{xy}}{f_{xy}}||_1\leq 2\sqrt{\epsilon} \end{align} as needed.
Next recall security of the PSM means that there exists a simulator which takes in $f(x,y)$ and produces output distribution $Sim$ on the message system such that \begin{align}
\forall (x,y)\in X\times Y,\, \,\,\,||Sim_{M|f(x,y)} - P_{M|xy}||_1 \leq \delta. \end{align} To get security of the PSQM, we need to upgrade this simulator to a quantum channel.
In particular if the simulator is defined by the conditional probability distribution $p(m|f)$, define the Kraus operators \begin{align}
S_{m,f} = \sqrt{p(m|f)} \ketbra{m}{f}. \end{align} Calling the corresponding simulator channel $\mathbfcal{S}$, we have that \begin{align}
||\mathbfcal{S}(\ketbra{f_{xy}}{f_{xy}}) - \rho_M(x,y)||_1= ||Sim_{M|f(x,y)} - P_{M|xy}||_1\leq \delta \end{align} so we have exactly $\delta$ security of the PSQM. \end{proof}
\section{New relations among primitives}\label{eq:newrelationships}
This section begins our study of the relationships among the cryptographic primitives introduced in section \ref{sec:primitivedefinitions}.
\subsection{Classical CDS gives quantum CDS}
In this section we observe that a classical CDS scheme immediately gives a quantum CDS scheme, via a use of the one-time pad.
\begin{theorem}\label{thm:CDStoCDQS}
An $\epsilon$-successful and $\delta$-secure CDS protocol hiding $2n$ bits and using $n_M$ bits of message and $n_E$ bits of randomness gives a CDQS protocol which hides $n$ qubits, is $\epsilon$ successful and $O(n\sqrt{\delta})$ secure using $n_M$ classical bits of message plus one qubit of message, and $n_E$ classical bits of randomness. \end{theorem} \begin{proof}\, Let the quantum system to be hidden in the CDQS be labelled $Q$. The basic idea is to use the CDS protocol to hide the key of a one-time pad applied to the system $Q$. The encoded system $Q$ is sent to the referee. The one-time pad key, call it $s$, consists of two $2\log d_Q$ bits, which we choose independently and at random and hide in the CDS.
When $f(x,y)=1$, by correctness of the CDS the referee can guess $s$ with probability at least $(1-\epsilon)$, and use this to correctly recover the state, so the protocol is $\epsilon$ correct.
When $f(x,y)=0$, the referee holds the random variable $m$ produced as the message system in the CDS, along with the quantum system $Q$. The state received by the referee on $MQ$ and the reference system $R$ are in the joint state \begin{align}
\rho_{RQM}(x,y) = \sum_{s,m} P^s_Q\Psi_{RQ}^+P_Q^s \otimes p(m,s|x,y)\ketbra{m}{m}_M \end{align} Security of the classical CDS implies that, for $0$ instances of $x,y$, \begin{align}
||P_{MS|x,y} - P_SP_{M|xy}||_1\leq 2\delta. \end{align} In words, security of the CDS implies that the joint distribution of the key and message is nearly uncorrelated for 0 instances of the input. See equation \ref{eq:jointCDSdistribution} for the proof from the simulator based statement of security. Using this, we get that $\rho_{RQM}$ is $2\delta$ close in one norm distance to the state \begin{align}
\sigma_{RQM}(x,y) &= \sum_{s,m} p(s) P^s_Q\Psi_{RQ}^+P_Q^s \otimes p(m|x,y)\ketbra{m}{m}_M \nonumber \\
&= \frac{1}{4}\mathbfcal{I}_R\otimes \mathbfcal{I}_Q \otimes \sum_m p(m|x,y) \ketbra{m}{m}. \end{align} Finally, to show security of the CDQS, we need to define a simulator channel that prepares a density matrix close to this from the $XY$ system. Looking at the above density matrix, it's apparent that simply using the simulator for the classical CDS along with a preparation of the maximally mixed state on $Q$ suffices. More concretely, we take Kraus elements \begin{align}
S_{m,x,y} = \sqrt{p(m|x,y)}\ket{m}_M\bra{x}_X\bra{y}_Y \end{align} and compose this with a channel that traces out the $Q$ system and replaces it with the maximally mixed state. Thus the simulator produces exactly the above state, which is $2\delta$ close to the output produced by the protocol, so the CDQS is $2\delta$ secure. \end{proof}
\subsection{Equivalence of \texorpdfstring{$f$}{TEXT}-routing and CDQS}
Our main claim of this section is that the CDQS and $f$-routing scenarios are equivalent in the setting of small secrets, in that a protocol for one induces a protocol for the other using similar resources. The basic idea underlying the equivalence, and labelling of the various subsystems used in the proof, is illustrated in figure \ref{fig:NLQCandCDQS}.
\begin{figure*}\label{fig:CDQSagain}
\label{fig:f-routing}
\label{fig:NLQCandCDQS}
\end{figure*}
\begin{theorem}\label{thm:CDQSandfRouting}
A $\epsilon$-correct $f$-routing protocol that routes $n$ qubits implies the existence of a $\epsilon$-correct and $\delta=O(\sqrt{n \epsilon})$-secure CDQS protocol that hides $n$ qubits using the same entangled resource state and the same message size.
A $\epsilon$-correct and $\delta$-secure CDQS protocol hiding secret $Q$ using a $n_E$ qubit resource state $n_M$ qubit messages implies the existence of a $\max\{\sqrt{\delta'},\epsilon \}$-correct with $\delta'=O(\delta n)$ $f$-routing protocol that routes system $Q$ using $n_E$ qubits of resource state and $4(n_M+n_E)$ qubits of message. \end{theorem} \begin{proof} \,Begin by considering an $f$-routing protocol. Figure \ref{fig:NLQCandCDQS} establishes the subsystem labels we will use here. We will first show that an $f$-routing protocol is easily modified to construct a CDQS protocol. To do so, we send systems $M_0$ and $M_1$ that Bob would receive in the second round of the $f$-routing protocol to the referee of the CDQS protocol. Then, if $f(x,y)=1$, $\epsilon$ correctness of the $f$-routing scheme is immediately $\epsilon$ correctness of the CDQS.
To show secrecy of the CDQS protocol, we can also note that when $f(x,y)=0$ we have \begin{align}
F(\Psi^+, \rho_{RQ}) = \bra{\Psi^+} \rho_{RQ} \ket{\Psi^+} \geq 1-\epsilon \end{align} where $\rho_{RQ} = \mathbfcal{D}_{M'\rightarrow Q}(\rho_{RM'})$. Then the Fuchs-van-de Graaf inequalities give that \begin{align}
||\Psi^+ - \rho_{RQ} ||_1 \leq \sqrt{\epsilon} \end{align} and the continuity of the mutual information (lemma \ref{lemma:MIcontinuity}), \begin{align}
|I(R:Q)_{\Psi^+_{RQ}} - I(R:Q)_{\rho_{RQ}} | \leq O(\sqrt{\epsilon} \log d) \end{align} gives \begin{align}\label{eq:s1}
2\log d - O(\sqrt{\epsilon} \log d) \leq I(R:Q)_{\rho_{RQ}}. \end{align} Next, we use the data processing inequality to note that \begin{align}\label{eq:s2}
I(R:Q)_{\rho_{RQ}} \leq I(R:M')_{\rho_{RM'}} \end{align} since $\rho_{RQ}$ is produced by the action of a quantum channel $\mathbfcal{N}_{M'\rightarrow Q}$ on $\rho_{RM'}$. This gives that $I(R:M')_{\rho_{RM'}}$ is nearly maximal. Now consider that the state on $RMM'$ is pure, which gives \begin{align}\label{eq:s3}
2\log d = I(R:M)_{\rho_{RMM'}} + I(R:M')_{\rho_{RMM'}} \end{align} Used along with \ref{eq:s1} and \ref{eq:s2}, this gives \begin{align}
I(R:M) \leq O(\sqrt{\epsilon} \log d_R). \end{align} so that the message systems received in the CDQS are almost uncorrelated with the reference system whenever $f(x,y)=0$. Invoking lemma \ref{lemma:CDQSsecurityfromMI} then gives security of the CDQS.
Note that the CDQS protocol defined by the $f$-routing protocol uses the same entangled resource state and no more communication.
Now suppose we have a CDQS protocol which is $\epsilon$-correct and $\delta$-secure. Then to build the $f$-routing protocol, purify the channels Alice and Bob perform to isometries, and send the original message systems of the CDQS to Bob and their purifications to Alice. Then by $\epsilon$-correctness of the CDQS protocol, we immediately have $\epsilon$-correctness of the $f$-routing prototocol when $f(x,y)=1$.
Next consider the case where $f(x,y)=0$. Then security of the CDQS, using lemma \ref{lemma:CDQSsecurityfromMI}, implies that \begin{align}
I(R:M) \leq \delta' \end{align} where $\delta'=O(\delta \log d_R)$. From theorem \ref{thm:decoupling}, this implies that by acting on the $M'$ system we can produce a state $\rho$ on $RQ$ such that \begin{align}
\bra{\Psi^+}\rho_{RQ} \ket{\Psi^+}_{RQ}= F(\rho_{RQ},\ketbra{\Psi^+}{\Psi^+}_{RQ}) \geq 1-\sqrt{\delta'} \end{align} Thus the protocol is $\sqrt{\delta'}$ correct in the case where $f(x,y)=0$. Since the protocol also succeeds with probability $1-\epsilon$ when $f(x,y)=1$, the $f$-routing protocol is $\max\{\epsilon,\sqrt{\delta'}\}$-correct.
To see how the communication in the resulting $f$-routing protocol is related to the communication in the original CDQS protocol, we can use that a channel $\mathbfcal{N}_{A\rightarrow B}$ can always be purified by an isometry $\mathbf{V}_{A\rightarrow B C}$ where $d_C\leq d_Ad_B$. Let CDQS have messages that each consist of at most $n_{M}$ qubits, and use an $n_{E}$ qubit resource system on systems $LR$. Then the most general possible protocol is defined by families of channels \begin{align} \{\mathbfcal{N}^x_{L\rightarrow M_{0}}\},\,\,\,\, \{\mathbfcal{N}^y_{R\rightarrow M_{1}}\} \end{align} applied on the left and right respectively. We define purifications of these, \begin{align} \{\mathbf{V}^x_{L\rightarrow M_{0}M_{0}'}\},\,\,\,\, \{\mathbf{V}^y_{R\rightarrow M_{1}M_1'}\} \end{align} We see that the message sizes are now at most $n_M + n_E$ qubits, so the total size of the communication is at most $4(n_M+n_E)$. The entangled resource system used in the $f$-routing protocol is identical to the one used in the CDQS. \end{proof}
\noindent \textbf{Explicit reconstruction procedure:}
It is perhaps counter-intuitive that the $f$-routing protocol built from the CDQS protocol succeeds in the case when $f(x,y)=0$. This is implied by the general physics of decoupling as captured by theorem \ref{thm:decoupling}, but for intuition we give a more explicit description in a special case here.
Let's suppose the CDQS protocol is perfectly correct, and works in the following way. Assume the quantum secret is a single qubit and is stored in system $Q$. To hide the quantum state on $Q$, Alice applies the one-time pad using a classical string $s=(s_1,s_2)$ as key. Explicitly she has applied \begin{align}
\ket{s_1,s_2}_A \ket{\psi}_Q\rightarrow \ket{s_1,s_2}_A (i)^{s_1\cdot s_2} X^{s_1}Z^{s_2}\ket{\psi}_Q. \end{align} A message system $M$ is sent to Bob, which reveals the key if and only if $f(x,y)=1$. The system $A$ must be sent to Alice on the left. The full state of the message systems then has the form \begin{align}
\frac{1}{2}\sum_{s_1,s_2,m_L,m_R} p(m_L,m_R|x,y,s) \ket{m_L}_{M'}\ket{s_1,s_2}_A (i)^{s_1\cdot s_2} X^{s_1}Z^{s_2}\ket{\psi}_Q \ket{m_R}_{M}. \end{align} Suppose we are in the case where $f(x,y)=0$. Then by security, the state on $M$ is independent of $s$. We can trace it out and the $M'$ system out and obtain the pure state \begin{align}
\frac{1}{2}\sum_{s_1,s_2} \ket{s_1,s_2}_A (i)^{s_1\cdot s_2} X^{s_1}Z^{s_2}\ket{\psi}_Q . \end{align} The claim is that Alice can recover the state on $Q$ from the $A$ system. To do this, she maps $\ket{s_1,s_2}$ to the Bell basis, obtaining \begin{align}
\frac{1}{2}(III+IXX+IZZ+IYY) \ket{\Psi^+}_{A_1A_2}\ket{\psi}_Q. \end{align} Then notice that \begin{align}
\frac{1}{2}(I_{A_2}I_{Q}+X_{A_2}X_{Q}+Z_{A_2}Z_{Q}+Y_{A_2}Y_{Q}) = SWAP_{A_2Q} \end{align} so that mapping $A_1A_2$ into the Bell basis actually swaps the state on $Q$ into $A_2$, so that Alice recovers the state on $Q$.
\subsection{PSQM gives CDQS}
Analogous to the observation that PSM gives CDS, we can also show that PSQM gives CDQS.
\begin{theorem}\label{thm:PSQMtoCDQS} Suppose that a $\epsilon$-correct and $\delta$-private PSQM protocol exists for $f(x,y)\in\{0,1\}$ using messages of at most $n_M$ bits and an entangled state of no more than $n_E$ qubits. Then there exists a CDQS protocol hiding one qubit using $n_M+1$ bits of message and $n_E$ qubits of entangled state which is $2\epsilon$ correct and $2\delta$ private. \end{theorem} \begin{proof}\, If the function $f(x,y)$ is constant then the CDQS protocol is trivial, so we assume without loss of generality that $f(x,y)$ is non-constant.
Given the PSQM protocol, we build a CDQS protocol as follows. We introduce two random shared bits which we call $s=(s_1,s_2)$, which are held by Alice and Bob. Alice and Bob also pre-agree on a pair of inputs $(x,y)$ where $f(x,y)=0$, call them $(x_*,y_*)$, which exist because $f$ is non-constant by assumption. Upon receiving inputs $x,y$ Alice and Bob compute \begin{align}
x_i' &= s_i x + (1-s_i) x_* \nonumber \\
y_i' &= s_i y + (1-s_i) y_* \end{align} for $i=1,2$. They run the PSQM protocol for $f$ on inputs $(x_1,y_1)$ and $(x_2,y_2)$ in parallel. Following the remark made after definition \ref{def:PSQM}, the PSQM for $F(x,y,s) = (f(x_1,y_1),f(x_2,y_2))$ is $2\epsilon$ correct and $2\delta$ secure. In the CDQS protocol, we have Alice act on the quantum secret $Q$ with the one time pad using the key $s=(s_1,s_2)$. By $2\epsilon$ correctness of the PSQM, we have that \begin{align}
||\tr_{\tilde{M}}(\mathbf{V}_{M\rightarrow Z\tilde{M}} \rho_M(x,y) \mathbf{V}^\dagger_{M\rightarrow Z\tilde{M}}) -\ketbra{f^1_{x_1y_1},f^2_{x_2y_2}}{f^1_{x_1y_1},f^2_{x_2y_2}}||_1\leq 2\epsilon \end{align} From this the Fuchs van de Graff inequality gives \begin{align}
\bra{f^1_{x_1y_1},f^2_{x_2y_2}} \tr_{\tilde{M}}(\mathbf{V}_{M\rightarrow Z\tilde{M}} \rho_M(x,y) \mathbf{V}^\dagger_{M\rightarrow Z\tilde{M}}) \ket{f^1_{x_1y_1},f^2_{x_2y_2}} \geq (1-\epsilon)^2 \geq 1-2\epsilon \end{align} so that measuring in the computational basis, the referee can determine $(f^1,f^2)$ correctly with probability at least $1-2\epsilon$. Noticing that \begin{align}
f(x_i',y_i') = f(x,y)\wedge s_i \end{align} the referee can, when $f(x,y)=1$, determine $s_1$ and $s_2$. Using this measurement outcome, she can undo the action of the one-time pad on $Q$ and recover $\Psi^+$. Concretely the state on $RQ$ will be \begin{align}
\rho_{RQ} = (1-\epsilon') \Psi^+_{RQ} + \epsilon' \sigma_{RQ} \end{align} with $\epsilon'\leq 2\epsilon$. This state has fidelity at least $1-2\epsilon$ with $\Psi^+_{RQ}$, which establishes correctness.
Next we study security of the CDQS protocol. Given our prescription above for the CDQS protocol, we can express the joint state of the message and reference system as \begin{align}
\rho_{RQM} = \frac{1}{4} \sum_{s_1,s_2} P^s_Q \Psi^+_{RQ}P^s_Q \otimes \rho_{M}(x,y,s) \end{align} where $\rho_{M}$ is the state sent to the referee in the PSQM protocol for $F=(f_1,f_2)$.\footnote{Note that this is a slight abuse of notation, as previously we used $M$ to denote all the systems sent to the referee in the CDQS, which are now here $QM$.} Security of the PSQM protocol gives that there exists a channel $\mathbfcal{S}_{Z\rightarrow M}$ such that \begin{align}
||\rho_{M'}(x,y,s) - \mathbfcal{S}_{Z\rightarrow M'}(\ketbra{f^1_{x_1,y_1},f^2_{x_2,y_2}}{f^1_{x_1,y_1},f^2_{x_2,y_2}}) ||_1\leq 2\delta \end{align} which gives \begin{align}
||\rho_{RQM} - \sum_{s_1,s_2} P^s_Q \Psi^+_{RQ}P^s_Q \otimes \mathbfcal{S}_{Z\rightarrow M}(\ketbra{f^1_{x_1,y_1},f^2_{x_2,y_2}}{f^1_{x_1,y_1},f^2_{x_2,y_2}})||_1 \leq 2\delta . \end{align} Now focus on zero instances of $f(x,y)$, which fixes the value of $f^1$ and $f^2$ to be $0$, independent of $s$. Then we obtain \begin{align}
||\rho_{RQM} - \frac{1}{4}\sum_{s_1,s_2} P^s_Q \Psi^+_{RQ}P^s_Q \otimes \sigma_{M}^0||_1 = ||\rho_{RQM'} - \frac{1}{4}\mathbfcal{I}_R\otimes \mathbfcal{I}_Q \otimes \sigma_{M}^0||_1 \leq 2\delta . \end{align} Taking the simulator channel to just be preparation of the state $\sigma_{M}$ and the maximally mixed state on $Q$ we have security of the PSQM protocol. \end{proof}
\subsection{CFE gives PSQM and weak converse}
Finally, we relate coherent function evaluation to PSQM. Note that the relationship is only that good CFE protocols give good PSQM protocols, although a weak converse also exists, as we describe.
\begin{figure*}
\caption{Corresponding PSQM (left) and CFE (right) protocols, with labellings of the subsystems involved shown.}
\label{fig:PSQMandCFE}
\end{figure*}
\begin{theorem}\label{thm:CFEtoPSQM}
A $\epsilon$-correct CFE protocol for the function $f$ using $n_E$ EPR pairs and messages of $n_M$ qubits implies the existence of a $\epsilon$-correct and $\sqrt{\epsilon}$-secure PSQM protocol for the same function, using $n_E$ EPR pairs and no more than $n_M$ message qubits. \end{theorem} \begin{proof} \,We define the PSQM protocol from the CFE protocol as follows. The PSQM protocol uses the same resource state as the CFE, Alice applies the bottom left operation of the CFE, Bob applies the bottom right operation of the CFE, and they send the systems that would reach the top right of the CFE protocol to the referee, which we call the $M$ systems. To produce their output, the referee applies the top right operation from the CFE. See figure \ref{fig:PSQMandCFE} for labels of the relevant subsystems.
Correctness of the CFE protocol means that we have \begin{align}
||\mathbf{F}(\cdot)\mathbf{F}^\dagger - \mathbf{\mathbfcal{N}}_{XY\rightarrow Z'Z}||_\diamond \leq \epsilon \end{align} where $\mathbf{\mathbfcal{N}}$ is the channel applied by our CFE protocol and $\mathbf{F}$ denotes the CFE isometry to be implemented. Applying these channels to the input $\ket{x}_X\ket{y}_Y$ and using the definition of the diamond norm distance, we obtain \begin{align}
||\ketbra{xy}{xy}_{Z'}\otimes \ketbra{f_{xy}}{f_{xy}}_{Z} - \rho_{Z'Z}(x,y)||_1 \leq \epsilon. \end{align} Tracing out the $Z'$ system and using that the one norm distance decreases under the partial trace, we obtain $\epsilon$ correctness of the PSQM.
Next we study security of the PSQM. We start again from the correctness of the CFE protocol. To simplify our notation, we define the channels \begin{align}
\mathbfcal{F}_{XY\rightarrow Z'Z}(\cdot) &= \mathbf{F} (\cdot) \mathbf{F}^\dagger, \nonumber \\
\mathbfcal{W}^L_{M\rightarrow Z\tilde{M} }(\cdot) &= \mathbf{W}^R_{M\rightarrow Z\tilde{M}} (\cdot) (\mathbf{W}^R_{M\rightarrow Z\tilde{M}})^\dagger \nonumber\\
\mathbfcal{W}^R_{M'\rightarrow Z'\tilde{M}' }(\cdot) &= \mathbf{W}^L_{M'\rightarrow Z'\tilde{M}'} (\cdot) (\mathbf{W}^L_{M'\rightarrow Z'\tilde{M}'})^\dagger \nonumber\\
\mathbfcal{W}_{MM'\rightarrow Z\tilde{M} Z'\tilde{M}'} &= \mathbfcal{W}^L_{M'\rightarrow Z\tilde{M}'}\otimes \mathbfcal{W}^R_{M\rightarrow Z\tilde{M}} \nonumber\\
\mathbfcal{V}_{XY\rightarrow MM'}(\cdot) &= \mathbf{V}^R_{YC\rightarrow M_1'M_1} \otimes \mathbf{V}^L_{XC'\rightarrow M_0M_0'}(\cdot \otimes \Psi_{CC'}) (\mathbf{V}^R_{YC\rightarrow M_1'M_1} \otimes \mathbf{V}^L_{XC'\rightarrow M_0M_0'})^\dagger \nonumber \end{align} Then we note that the CFE protocol can be decomposed into two steps, and rewrite the statement of correctness, \begin{align}
||\mathbfcal{F}_{XY\rightarrow Z'Z}(\cdot) - \tr_{\tilde{M}\tilde{M}'} (\mathbfcal{W}_{MM'\rightarrow Z\tilde{M}Z'\tilde{M}'}) \circ (\mathbfcal{V}_{XYCC'\rightarrow MM'})||_\diamond \leq \epsilon.\nonumber \end{align} Next, we will use that Stinespring dilations of channels can be chosen to be close if the initial channels are close \cite{kretschmann2008continuity}. In particular we have \begin{align}\label{eq:opunderdiamond}
\frac{||T_1-T_2||_\diamond}{\sqrt{||T_1||_\diamond}+\sqrt{||T_2||_\diamond}} \leq \inf_{V_1,V_2}||V_1-V_2 ||_{op} \leq \sqrt{||T_1-T_2||_{\diamond}} \end{align} where the infimum is over all dilations $V_i$ of $T_i$. Noting that $\mathbfcal{F}$ is already isometric, we have that its dilations must consist of adding a state preparation channel, which we label $\mathbfcal{P}_{\emptyset\rightarrow E}$. Further, all dilations are related by a partial isometry on the auxiliary space, so the dilations of the $\tr_{\tilde{M}\tilde{M}'}\mathbfcal{W}\circ \mathbfcal{V}$ channel can be written in the form \begin{align}
\mathbfcal{U}_{XY\rightarrow ZZ'E} = \mathbfcal{I}_{\tilde{M}\tilde{M}'\rightarrow E} \circ (\mathbfcal{W}_{MM'\rightarrow Z\tilde{M}Z'\tilde{M}'}) \circ (\mathbfcal{V}_{XY\rightarrow MM'}) \end{align} Then using the upper bound in \ref{eq:opunderdiamond}, we have \begin{align}\label{eq:operatornormundersqrtepsilon}
||\mathbfcal{F}_{XY\rightarrow Z'Z}\otimes \mathbfcal{P}_{\emptyset \rightarrow E} - \mathbfcal{I}_{\tilde{M}\tilde{M}'\rightarrow E}\circ \mathbfcal{W}_{MM'\rightarrow Z\tilde{M}Z'\tilde{M}'} \circ \mathbfcal{V}_{XY\rightarrow MM'}||_{op} \leq \sqrt{\epsilon}. \end{align} Next, we will exploit the lower bound in \ref{eq:opunderdiamond} to translate this to an upper bound on the diamond norm of these isometries. To do this, notice that from \ref{eq:opunderdiamond} we have \begin{align}
\frac{||V_1-V_2||_\diamond}{\sqrt{||V_1||_\diamond}+\sqrt{||V_2||_\diamond}} \leq \inf_{P_1,P_2}||V_1\otimes P_1-V_2\otimes P_2 ||_{op} \leq ||V_1-V_2 ||_{op} \end{align} Using this in equation \ref{eq:operatornormundersqrtepsilon}, we obtain \begin{align}
||\mathbfcal{F}_{XY\rightarrow Z'Z}\otimes \mathbfcal{P}_{\emptyset \rightarrow E} - \mathbfcal{I}_{\tilde{M}\tilde{M}'\rightarrow E}\circ \mathbfcal{W}_{MM'\rightarrow Z\tilde{M}Z'\tilde{M}'} \circ \mathbfcal{V}_{XY\rightarrow MM'}||_{\diamond} \leq 2\sqrt{\epsilon}. \end{align} Next, apply $\mathbfcal{I}^\dagger_{\tilde{M}\tilde{M}'\rightarrow E}$ to both terms, which can only decrease the diamond norm, and obtain \begin{align}
||\mathbfcal{F}_{XY\rightarrow Z'Z}\otimes \mathbfcal{P}_{\emptyset \rightarrow \tilde{M}\tilde{M}'} - \mathbfcal{W}_{MM'\rightarrow Z\tilde{M}Z'\tilde{M}'} \circ \mathbfcal{V}_{XY\rightarrow MM'}||_{\diamond} \leq 2\sqrt{\epsilon}. \end{align} Apply $\mathbfcal{W}^\dagger_{MM'\rightarrow Z\tilde{M}Z'\tilde{M}'}$ to both terms to obtain \begin{align}
||\mathbfcal{W}_{MM'\rightarrow Z\tilde{M}Z'\tilde{M}'}^\dagger \circ (\mathbfcal{F}_{XY\rightarrow Z'Z}\otimes \mathbfcal{P}_{\emptyset \rightarrow \tilde{M}\tilde{M}'}) - \mathbfcal{V}_{XY\rightarrow MM'}||_{\diamond} \leq 2\sqrt{\epsilon}. \end{align} Then, apply these channels to the input $\ket{xy}_{XY}$ and call the output of the protocol on the $M$ system $\rho_{M}(x,y)$, and trace out the $\tilde{M}'$ system, \begin{align}
||\tr_{M'}\mathbfcal{W}_{MM'\rightarrow Z\tilde{M}Z'\tilde{M}'}^\dagger \circ \mathbfcal{F}_{XY\rightarrow Z'Z}(\ketbra{xy}{xy})\otimes \psi_{\tilde{M}\tilde{M}'} - \rho_{M}(x,y)||_1 \leq 2\sqrt{\epsilon}. \nonumber \end{align} Simplifying the state on the left using \begin{align}
\mathbfcal{W}_{MM'\rightarrow Z\tilde{M}Z'\tilde{M}'} &= \mathbfcal{W}^L_{M\rightarrow Z\tilde{M}}\otimes \mathbfcal{W}^R_{M'\rightarrow Z'\tilde{M}'} \nonumber \\
\mathbfcal{F}_{XY\rightarrow Z'Z}(\ketbra{xy}{xy}) &= \ketbra{f_{xy}}{f_{xy}}_Z\otimes \ketbra{xy}{xy}_{Z'} \end{align} we obtain \begin{align}
||\mathbfcal{W}^{R\dagger}_{M\rightarrow \tilde{M}Z}(\ketbra{f_{xy}}{f_{xy}}\otimes \sigma_{\tilde{M}} ) - \rho_{M}(x,y)||_1 \leq 2\sqrt{\epsilon} \end{align} which is $2\sqrt{\epsilon}$ security of the PSQM protocol, where $\mathbfcal{W}^{R\dagger}_{M\rightarrow \tilde{M}Z}$ along with the state preparation of $\sigma_{\tilde{M}}$ defines the simulator channel. \end{proof}
Next, we give a weak converse to the above theorem, which shows that a good PSQM protocol implies the existence of CFE protocol that succeeds with constant probability (not dependent on the input size) when acted on the maximally entangled state. Note that this falls short of bounding the diamond norm. We show this only in the exact setting, though a robust version might also exist. \begin{theorem} \label{thm:PSQMtoCFEWeak}
Suppose there exists a perfectly correct and perfectly secure PSQM protocol for the function $f:X\times Y\rightarrow Z$ using $n_M$ bits of communication and $n_E$ qubits of entangled resource system.
Then there is a CFE protocol that implements a channel $\tilde{\mathbfcal{V}}^f_{XY\rightarrow Z'Z}$ such that
\begin{align}
F(\tilde{\mathbfcal{V}}^f_{XY\rightarrow Z'Z}(\Psi^+_{RXY}),\mathbf{V}^f_{XY\rightarrow Z'Z}(\Psi^+)_{RXY}(\mathbf{V}^f_{XY\rightarrow Z'Z})^\dagger) \geq \frac{1}{|Z|}
\end{align}
and which uses $n_E$ qubits of entangled resource state and $n_M+n_E+2n$ qubits of communication, where $n$ is the input size. \end{theorem} \begin{proof}\,
By security of the PSQM protocol, we have that when given input $\ket{xy}$ the protocol produces a reduced state $\rho_{M}(x,y)$ with the form
\begin{align}
\rho_{M}(x,y) = \mathbfcal{S}_{Z\rightarrow M}(\ketbra{f_{xy}}{f_{xy}}) = \sigma^{f_{xy}}_{M}.
\end{align}
As part of the CFE protocol we are defining, we make a copy of the inputs $\ket{x}_X\ket{y}_Y$ and send this copy in a system labelled $Z'$ to the left.
The overall state of the message system then is,
\begin{align}
\ketbra{xy}{xy}_{Z'}\otimes \sigma^{f_{xy}}_{M}.
\end{align}
Now consider purifying the channels used in the PSQM protocol, and sending the purifying systems (call them $\tilde{M}'$) to the left.
Then the message system becomes
\begin{align}
\ket{\Psi_{xy}}_{Z'\tilde{M}'M} = \ket{xy}_{Z'} \sum_k \sqrt{\lambda^k_{f_{xy}}}\ket{\psi^k_{f_{xy}}}_{\tilde{M}'}\ket{\psi^k_{f_{xy}}}_{M}
\end{align}
where we used that the reduced density matrix on $M$ depends only on $f_{xy}$ to enforce that the Schmidt coefficients and Schmidt vectors on $M$ can depend only on $f_{xy}$.
Next, we consider adding to the protocol a unitary
\begin{align}
\mathbf{U}_{Z'\tilde{M}'} = \sum_{x,y,k}\ketbra{xy}{xy}_{Z'}\otimes \ketbra{k}{\psi^k_{f_{xy}}}_{\tilde{M}'}
\end{align}
which means we produce the state
\begin{align}\label{eq:PSQMstatesofar}
\mathbf{U}_{Z'\tilde{M}'} \ket{\Psi_{xy}}_{Z'\tilde{M}'M} = \ket{xy}_{Z'} \sum_k \sqrt{\lambda^k_{f_{xy}}} \ket{k}_{\tilde{M}'}\ket{\psi^k_{f_{xy}}}_{M}.
\end{align}
We'd like to exploit the correctness of the PSQM protocol to show this state can be made, using an operation on $M$, to have large overlap with the correct output for the CFE protocol, which here is $\ket{xy}_{Z'}\ket{f_{xy}}_{Z}$.
Looking at the reduced state on $M$ again, we have
\begin{align}
\sigma_{M} = \sum_k \lambda^k_{f_{xy}}\ketbra{\psi_{f_{xy}}^k}{\psi_{f_{xy}}^k}_{M} .
\end{align}
From correctness we have that there exists a map $\mathbf{V}_{M\rightarrow \tilde{M}Z}$ such that
\begin{align}
\sum_k \lambda^k_{f_{xy}} \tr_{\tilde{M}}(\mathbf{V}_{M\rightarrow \tilde{M}Z}\ketbra{\psi_{f_{xy}}^k}{\psi_{f_{xy}}^k}_{M}\mathbf{V}^\dagger_{M\rightarrow \tilde{M}Z}) = \ketbra{f_{xy}}{f_{xy}}_{Z}
\end{align}
which is only solved if, for all $k$,
\begin{align}
\mathbf{V}_{M\rightarrow \tilde{M}Z}\ket{\psi^k_{f_{xy}}}_{M} = \ket{f_{xy}}_{Z} \ket{\tilde{\psi}^k_{f_{xy}}}_{\tilde{M}}.
\end{align}
Returning to the form \ref{eq:PSQMstatesofar}, we can now add an application of $\mathbf{V}_{M\rightarrow Z\tilde{M}}$ as the top right element of our CFE protocol and we see that we produce the state
\begin{align}
\mathbf{V}_{M\rightarrow \tilde{M}Z} \mathbf{U}_{Z'\tilde{M}'}\ket{\Psi_{xy}}_{Z'\tilde{M}'M} &= \ket{xy}_{Z'} \ket{f_{xy}}_{Z} \sum_k \sqrt{\lambda^k_{f_{xy}}}\ket{k}_{\tilde{M}'}\ket{\tilde{\psi}^k_{f_{xy}}}_{\tilde{M}} \nonumber \\
&= \ket{xy}_{Z'} \ket{f_{xy}}_{Z} \ket{\Phi_{f_{xy}}}_{\tilde{M}'\tilde{M}}.
\end{align}
By linearity, if we perform the same protocol on the state $\ket{\Psi^+}_{RXY}$ we produce the output
\begin{align}
\ket{\Psi'_f}_{RZ'Z\tilde{M}'\tilde{M}} = \frac{1}{\sqrt{d_R}}\sum_{xy} \ket{xy}_R \ket{xy}_{Z'} \ket{f_{xy}}_Z \ket{\Phi_{f_{xy}}}_{\tilde{M}'\tilde{M}}.
\end{align}
We would like to compute the fidelity of the state produced by our protocol on $RZ'Z$ with the correct one when acted on the maximally entangled state.
Note that the correct output state would be
\begin{align}
\ket{\Psi_f} = \frac{1}{\sqrt{d_R}} \sum_{xy} \ket{xy}_R\ket{xy}_{Z'} \ket{f_{xy}}_{Z}.
\end{align}
Computing the fidelity of this with the partial state of $\ket{\Psi_f'}$ on $RZ'Z$, we find
\begin{align}
F(\Psi_f,\sigma)=\bra{\Psi_f}\sigma_{RZ'Z}\ket{\Psi_f} &= \frac{1}{d_R^2} \sum_{xy,x'y'} \braket{\Phi_{f_{xy}}}{\Phi_{f_{x'y'}}} \nonumber \\
&\geq \frac{1}{d_R^2} \left(\sum_{f_{xy}} \sum_{f_{xy}=f_{x'y'}} 1 \right)\nonumber \\
&\geq \frac{1}{d_R^2}(N_0^2 +(d_R-N_0)^2) \geq \frac{1}{|Z|}
\end{align}
where the index $k=(x,y)$ runs 1 to $d_R$, and $N_m$ is the number of values of this index where $f_{xy}=m$.
To understand the resource consumption of the protocol constructed above, notice that it uses the same resource state, and so still $n_E$ qubits of entangled resource system.
Considering the message sizes, notice that in purifying the channels used in the PSQM protocol we need no more than $n_E+n$ qubits in the auxiliary system, and then we added an additional copy of the input sent to the left, so we use at most $n_E+2n+n_M$ qubit messages. \end{proof}
\section{Complexity of efficiently achievable functions}\label{sec:complexity}
The set of implications summarized in figure \ref{fig:web} imply efficient protocols for one primitive imply efficient protocols for many others. In this section we briefly summarize what is known about the efficiently achievable functions in various settings, and how they compare across various primitives.
\subsection{Relevant complexity measures}\label{sec:complexitymeasures}
An important model of computation we will discuss is the modulo-$p$ branching program. These are computational models with close relationships to various non-uniform complexity classes sitting inside of NC. \begin{definition} A \textbf{branching program} is a tuple $\mathbfcal{BP}=(G,\phi,s,t_0,t_1)$ where, \begin{itemize}
\item $G=(V,E)$ is a directed acyclic graph,
\item $\phi$ is a function from edges in $E$ to either a value ``yes'' or a tuple $(b,i)$ for $b$ a bit and $i\in \{1,...,n\}$,
\item $s$, $t_0$, $t_1$ are vertices from $V$. \end{itemize} Given a $n$ bit string $x$ as input, the branching program specifies a subgraph of $G$ labelled $G_x$ according to the following rule. If for $e\in E$ we have $\phi(e)=(b,j)$ with $x_j=b$, or if $\phi(e)=$``yes'', then $e$ is included in $G_x$. We define a function acc$(x)$ as the number of paths $s\rightarrow t_1$ in the graph $G_x$, and a function rej$(x)$ as the number of paths from $s$ to $t_0$ in $G_x$. \end{definition}
\begin{definition} The size of a branching program is defined as the number of vertices in $V$. We label the minimal sized branching program computing $f$ as $BP(f)$. \end{definition} We say a branching program is deterministic if the out degree of every vertex in every $G_x$ is at most $1$, and non-deterministic otherwise. The function $f(x)$ computed by a deterministic or non-deterministic branching program is defined such that $f(x)=1$ iff $acc(x)>0$. A \textbf{Boolean modulo-$p$ branching program} computes the function $f(x)$ defined such that $f(x)=1$ iff $acc(x)\neq 0 \,\,mod \,\, p$. We label the minimal size of a mod $p$ branching program computing $f$ by $BP_p(f)$.
The class of functions with polynomial sized modulo-$p$ branching programs is defined below. \begin{definition}
The complexity class Mod$_pL$/poly is defined as those Boolean function families $\{f_n\}$ which have polynomial (in $n$) sized modulo-$p$ branching programs. \end{definition} The uniform complexity class Mod$_pL$ can be defined similarly in terms of log-space uniform branching programs, or given an equivalent definition in terms of Turing machines \cite{buntrock1992structure}. Another relevant complexity class, also based on branching programs, is the following. \begin{definition}
The class $C_=L/poly$ (read as ``equality L'') is defined as those Boolean function families $\{f_n\}$ which can be decided in the following way.
We consider a branching program of polynomial (in n) size.
If $acc(x)=rej(x)$, output $1$ and otherwise output $0$. \end{definition}
A related notion of complexity that we will need is that of a \textbf{span program}, defined initially in \cite{karchmer1993span}. \begin{definition}\label{def:spanprogram} A \textbf{span program} over a field $\mathbb{Z}_p$ consists of a triple $S=(M, \phi, \mathbf{t})$, where $M$ is a $d\cross e$ matrix with entries in $\mathbb{Z}_p$, $\phi$ is a map from rows of $M$, labelled $r_i$, to pairs $(k,\varepsilon_i)$, with $k\in \{1,...,n\}$ and $\varepsilon_i\in\{0,1\}$, and $\mathbf{t}$ is a non-zero vector of length $e$ with entries in $\mathbb{Z}_p$. A span program $S$ computes a function $f:\{0,1\}^n\rightarrow \{0,1\}$ as follows. Given an input string $z$ of $n$ bits, if the vector $\mathbf{t}$ is in $\text{span}(\{r_i: \exists j, \phi(r_i)=(j,z_j)\})$, then output 1. Otherwise, output 0. \end{definition}
\begin{definition} The \textbf{size} of a span program is defined to be $d$, the number of rows in $M$. We denote the minimal size of a span program over $\mathbb{Z}_p$ that computes $f$ by $SP_p(f)$. \end{definition} The size of a span program computing $\{f_n\}$ and of a branching program computing the same function family are related by the following theorem, noted in \cite{karchmer1993span} to follow from techniques in \cite{buntrock1992structure}. \begin{theorem}
For every prime $p$, Mod$_pL$ consists of those function families with polynomial sized span programs over $\mathbb{Z}_p$. \end{theorem} Thus the size of span programs and of arithmetic branching programs are related polynomially, and in fact \cite{beimel1999arithmetic}\footnote{Note that this statement is given in \cite{beimel1999arithmetic} in terms of \emph{arithmetic branching programs}, which are a generalization of modulo-p branching programs (and so are at least as powerful).} \begin{align}\label{eq:SPvsBP}
SP_p(f) \leq 2 BP_p(f). \end{align} We will never be interested in constant factor differences, so we can take that span programs are always smaller than modulo-$p$ branching programs.
An important notion for us will be that of \emph{pre-processing}. We will consider functions $f:\{0,1\}^{n}\times \{0,1\}^{n}\rightarrow \{0,1\}$, and are interested in the complexity of computing $f(x,y)$ after allowing for arbitrary functions to be applied to $x$ and $y$ separately. We make the following definition. \begin{definition}
A \textbf{local part} of $f(x,y):\{0,1\}^{n}\times \{0,1\}^{n}\rightarrow \{0,1\}$ is any function $F$ such that there exists functions $\alpha:\{0,1\}^{n}\rightarrow \{0,1\}^{m_\alpha}$, $\beta:\{0,1\}^{n}\rightarrow \{0,1\}^{m_\beta}$ such that $f(x,y)=F(\alpha(x),\beta(y))$. \end{definition} We say that the complexity after pre-processing (with respect to some measure of complexity) of a function $f(x,y)$ is the minimal complexity of any local part of $f(x,y)$. More concretely, for span and branching program size we define the following pre-processed complexity measures. \begin{definition} The \textbf{pre-processed branching program complexity} is defined as
\begin{align}
BP_{p,(2)}(f)= \min_{F,\alpha,\beta} \{ BP_p(F):f(x,y)=F(\alpha(x),\beta(y))\},
\end{align} \end{definition} \begin{definition} The \textbf{pre-processed span program complexity} is defined as
\begin{align}
SP_{p,(2)}(f)= \min_{F,\alpha,\beta} \{ SP_p(F):f(x,y)=F(\alpha(x),\beta(y))\},
\end{align} \end{definition} The pre-processed branching and span program complexities are related polynomially, because the non pre-processed complexities are.
We define the following pre-processed complexity classes. \begin{definition}
The complexity class $Mod_kL_{(2)}$ is defined as those functions $f:\{0,1\}^{n}\times \{0,1\}^{n}\rightarrow \{0,1\}$ with a local part that can be computed with a polynomial size (in n) modulo-p branching program. \end{definition} \begin{definition}
The complexity class $C_=L_{(2)}$ is defined as those functions $f:\{0,1\}^{n}\times \{0,1\}^{n}\rightarrow \{0,1\}$ with a local part that can be computed according to the following procedure.
We consider a branching program of polynomial (in n) size.
If $acc(x)=rej(x)$, output $1$ and otherwise output $0$. \end{definition}
\subsection{Efficiency of protocols for PSM, CDS, and related primitives}
\noindent \textbf{PSM and PSQM protocols}
The largest class of functions for which efficient PSM protocols have been constructed are those with polynomial sized modulo-$p$ branching programs. The following theorem was proven in \cite{ishai1997private}. \begin{theorem}\label{thm:PSMfrombranchingprogram}\textbf{[IK '97]}
Let $p$ be a prime, and let $\mathbfcal{BP}=(G,\phi,s,t_0,t_1)$ be a Boolean modulo-$p$ branching program of size $a(n)$ computing a local part of $f$. Then there exists a PSM protocol for $f$ with randomness complexity and communication complexity both $O( a(n)^2\,\log p )$. \end{theorem} Note that the original statement of this theorem considers $f$ rather than it's local part, but the extension is trivial. An immediate consequence of this theorem, along with the implications summarized in figure \ref{fig:web}, is that CDS, PSQM, CDQS, and $f$-routing can all be achieved with the randomness and communication complexity given in the same way, up to constant factor overheads.
To better understand the implications of this theorem, it is helpful to understand which complexity classes can be efficiently achieved. Fixing $p$, those functions with polynomial sized branching programs are exactly the class $Mod_pL$. Running the PSM protocol on the local part, we can therefore achieve the class $\text{Mod}_pL_{(2)}$ efficiently as a PSM. We can also choose $p$ adaptively, and doing so achieve the class $C_=L_{(2)}$. This is shown in \cite{ishai1997private}. It is also interesting to find a complexity class that contains all of those functions where $(\log p) BP_p(f)$ can be made polynomial. The smallest class which we can show contains all such functions is $L^{\#L}$, which we state as the following remark. \begin{remark}
Every function family $\{f_n\}$ for which $(\log p)\cdot BP_p(f_n)$ is polynomial in $n$ for some choice of $p$ is contained in the class $L^{\#L}/ poly$. \end{remark} \begin{proof}\, By assumption, there is a polynomial sized branching program, call it $BP$ and denote its size by $s$, whose number of accepting paths counted mod $p$ is non-zero if $f(x)=1$, and $0$ otherwise. Further, the choice of $p$ needed must have $\log p$ be polynomial. Our algorithm to compute $f$ in $L^{\#L}$ is as follows. We take our advice string to be a description of the branching program BP. We give BP along with the input $x$ to the $\# L$ oracle, and it will return the number of accepting paths of this program, call it $N$. Notice that $N < 2^s$, since there must be no more accepting paths then there are subsets of vertices in BP. This means the output of the oracle consists of at most a polynomial sized string. We then subtract $p$ from $N$ repeatedly until it obtains a number less than $p$. Since $p$ also consists of a polynomial number of bits, this can be done in log space. \end{proof}
To relate $L^{\#L}$ to more familiar classes, we can note that it is contained inside of DET which is in turn contained inside of NC, where NC is the class of functions computed by poly-logarithmic depth circuits.
Notice that from theorem \ref{thm:PSMgivesPSQM} the result of theorem \ref{thm:PSMfrombranchingprogram} carries over immediately to the setting of PSQM. We move on to understand the implications of theorem \ref{thm:PSMfrombranchingprogram} for the CDS, CDQS, and $f$-routing primitives below.
\noindent \textbf{CDS protocols}
From theorem \ref{thm:PSMfrombranchingprogram} and because PSM protocols give CDS protocols (see theorem \ref{thm:PSMgivesCDS}), we obtain the following corollary. \begin{theorem}\label{thm:CDSefficiencyfromPSM}
Let $p$ be a prime, and let $\mathbfcal{BP}=(G,\phi,s,t_0,t_1)$ be a Boolean modulo-$p$ branching program of size $a(n)$ computing $f$.
Then there exists a CDS protocol for $f$ with randomness complexity and communication complexity both $O( a(n)^2\,\log p )$. \end{theorem} Note that the implication from PSM to CDS was known already, so that this implication was already clear. Recently, this scaling was improved to linear in the branching program size \cite{ishai2014partial}.
We can compare this to the most efficient CDS constructions in the literature. A CDS protocol based on secret sharing schemes was given in \cite{GERTNER2000592}.
They prove the following theorem\footnote{The cost here being $c+|s|$ while the cost in the reference \cite{GERTNER2000592} being $c$ is due to our defining the CDS to have the secret held on only one side, rather than on both as is the convention in \cite{GERTNER2000592}.}. \begin{theorem}\label{thm:CDSfromsecretsharing}\textbf{[GIKM '98]} Let $h_M:\{0,1\}^{n}\rightarrow \{0,1\}$ be a monotone Boolean function, and let $h:\{0,1\}^n\rightarrow \{0,1\}$ be a projection of $h_M$; that is, $h(y_1,...,y_n)=h_M(g_1,...,g_M)$, where each $g_i$ is a function of a single variable $y_i$. Let $S$ be a secret sharing scheme realizing the access structure $h_M$, in which the total share size is $c$, and let $s$ be a secret that can be hidden in $S$.
Then there exists a protocol $P$ for disclosing $s$ subject to the condition $h$ whose communication and randomness complexity are bounded by $c+|s|$. \end{theorem} Using the span program based constructions of secret sharing schemes \cite{karchmer1993span}, this upper bounds the CDS cost of $f$ by the minimal size of a monotone span program computing any projection of $f$, call it $f_M$. If the span program is over the field $\mathbb{Z}_p$, the cost is $(\log p) \cdot mSP(f_M)$. In \cite{cree2022code} (see lemma 5) it is shown that the size of a span program computing the projection $f_M$ is the same as the size of a (non-monotone) span program computing $f$, up to a constant additive term. This leads to the following corollary. \begin{corollary}
The randomness and communication complexity to perform CDS on the function $f$ is at most $O(\log p\cdot SP_p(f))$, where $SP_p(f)$ is the size of any span program over $\mathbf{Z}_p$ computing $f$. \end{corollary} Notice that this is quite similar to corollary \ref{thm:CDSefficiencyfromPSM}. Because the span program size and branching program size are related by equation \ref{eq:SPvsBP}, the secret sharing based construction for CDS is always more efficient than the branching program based approach inherited from PSM.
Another protocol based on dependency programs \cite{pudlak1996algebraic} was given in \cite{applebaum2017private}. Because dependency programs are always larger than span programs (see \cite{pudlak1996algebraic}, Lemma 3.6)\footnote{This is true when considering binary inputs, which we do here. The construction in \cite{applebaum2017private} extends to non-binary inputs, and in that setting there may be polynomial overheads.}, the span program based construction remain the most efficient.
\noindent \textbf{CDQS and $f$-routing protocols}
Notice that efficient CDQS protocols are given by both efficient CDS protocols (theorem \ref{thm:CDStoCDQS}) and by PSQM protocols (theorem \ref{thm:PSQMtoCDQS}). Further, from theorem \ref{thm:CDQSandfRouting} we have that efficient CDQS leads to efficient $f$-routing. These implications lead to the following theorem. \begin{theorem}\label{thm:CDQSandfRefficiency}
The randomness and communication complexity to perform CDQS or $f$-routing on the function $f$ is at most $O(\log p\cdot SP_p(f))$. \end{theorem} Since it had not previously been studied in the literature, this gives the largest known class of functions that can be implemented efficiently for CDQS.
We can compare theorem \ref{thm:CDQSandfRefficiency} to the most efficient protocols known for $f$-routing. In \cite{cree2022code}, the authors proved an upper bound of $O(\log p\cdot SP_p(f))$ on communication and entanglement complexity of $f$-routing, exactly matching the result inherited from classical CDS. It is also interesting to note that the protocol given in \cite{cree2022code} that achieves this bound is a close quantum analogue of the CDS protocol devised in the classical setting in \cite{GERTNER2000592}: both protocols are based on storing the secret in a secret sharing scheme and sending or not sending shares based on the value of bits of the input.
\section{New lower bounds}\label{sec:newlowerbounds}
\subsection{Linear lower bounds on CFE}
We have the following theorem from \cite{kawachi2021communication}. \begin{theorem}\textbf{[KN 2021]}\label{thm:PSQMlowerbound} For a $(1-o(1))$ fraction of functions $f_n : \{0, 1\}^n \times \{0, 1\}^n \rightarrow \{0, 1\}$, the communication complexity of two-party PSQM protocols with shared randomness for $f_n$ is at least $3n-2\log n-O(1)$. \end{theorem} In theorem \ref{thm:CFEtoPSQM}, which shows CFE$\rightarrow$PSQM, we could replace shared entanglement in the CFE protocol and obtain a PSQM protocol that only uses shared randomness. In fact, the theorem gives that the resulting PSQM uses the same distributed resource state as the CFE. From this, theorem \ref{thm:PSQMlowerbound} above gives the following. \begin{corollary}\label{corollary:CFElowerbound} For a $(1-o(1))$ fraction of functions $f_n : \{0, 1\}^n \times \{0, 1\}^n \rightarrow \{0, 1\}$, the communication complexity of coherent function evaluation protocols with shared randomness for $f_n$ is at least $3n-2\log n-O(1)$. \end{corollary} Note that we would expect no amount of shared random bits to suffice for a CFE, and instead for entangled states to be required. Thus the consequence of this theorem is very weak in the CFE context.
\subsection{Linear lower bounds on CDQS}
We have the following theorem from \cite{bluhm2021position} \begin{theorem}\textbf{[BCS 2022, random function]}. Let $n\geq 10$. Assume inputs $x,y\in \{0,1\}^n$ are chosen at random. Then there exists a function $f:X\times Y\rightarrow Z$ with $X, Y\in \{0,1\}^n$, $Z\in \{0,1\}$ such that, if the number $q$ of qubits each of the attackers controls satisfies \begin{align}
q\leq n/2 - 5 \end{align} then the attackers are caught with probability at least $2\times 10^{-2}$. Moreover, a uniformly random function will have this property, except with exponentially small probability. \end{theorem}
Combining this result with theorem \ref{thm:CDQSandfRouting}, we find the following result for CDQS. \begin{corollary}\label{corollary:CDQSlowerbound} There exists a function $f:X\times Y\rightarrow Z$ with $X, Y\in \{0,1\}^n$, $Z\in \{0,1\}$ such that a CDQS protocol which is $\epsilon$ correct and $\delta$ secure for $f$ with $\max\{ \epsilon,\sqrt{\delta} \}< 2\times 10^{-2}$ requires Alice and Bob have a quantum resource system consisting of at least $n/2-5$ qubits. Moreover, a uniformly random function will have this property, except with exponentially small probability. \end{corollary}
Now applying theorem \ref{thm:PSQMtoCDQS} we obtain the following linear lower bound on the dimension of the resource system in PSQM. Note that previously a $2n-O(\log n)$ linear lower bound on communication complexity was known, but no bound on shared entanglement was previously known. \begin{corollary}\label{corollary:PSQMlowerbound} There exists a function $f:X\times Y\rightarrow Z$ with $X, Y\in \{0,1\}^n$, $Z\in \{0,1\}$ such that a $\epsilon$ correct and $\delta$ secure PSQM protocol for $f$ with $\max\{2 \epsilon,\sqrt{2\delta} \} < 2\times 10^{-2}$ requires Alice and Bob have a quantum resource system consisting of at least $n/2-5$ qubits. Moreover, a uniformly random function will have this property, except with exponentially small probability. \end{corollary}
In the same paper \cite{bluhm2021position}, the author's prove the following bound for the inner product function.
\begin{theorem}\textbf{[BCS 2022, Inner product]} Let $n\geq 10$. Assume inputs $x,y\in \{0,1\}^n$ are chosen at random. Then if the number $q$ of qubits each of the attackers controls satisfies \begin{align}
q\leq \frac{1}{2}\log n - 5 \end{align} then the attackers are caught with probability at least $2\times 10^{-2}$ when the function $f$ is chosen to be the inner product function. \end{theorem}
This immediately leads to two corollaries analogous to the above, but now with a logarithmic bound and a random function replaced with the inner product.
\begin{corollary}\label{corollary:CDQSlogbound} A CDQS protocol for the inner product function on strings of length $n$ which is $\epsilon$ correct and $\delta$ secure with $\max\{ \epsilon,\sqrt{\delta} \}< 2\times 10^{-2}$ requires Alice and Bob have a quantum resource system consisting of at least $\frac{1}{2}\log n - 5$ qubits. \end{corollary}
\begin{corollary}\label{corollary:PSQMlogbound} A PSQM protocol for the inner product function on strings of length $n$ which is $\epsilon$ correct and $\delta$ secure with $\max\{2 \epsilon,\sqrt{2\delta} \} < 2\times 10^{-2}$ requires Alice and Bob have a quantum resource system consisting of at least $\frac{1}{2}\log n-5$ qubits. \end{corollary}
\section{New protocols}\label{sec:newprotocols}
\subsection{\texorpdfstring{$f$}{TEXT}-routing for problems outside P/poly}
As discussed in section \ref{sec:complexity}, all general constructions of CDS and PSM only efficiently implement functions inside of the class $(L^{\#L})_{(2)}$. As we now discuss, there is a special function which is believed to be outside of $P$ but which has an efficient CDS, CDQS, and $f$-routing protocol. This function is known to be at least as hard as the quadratic residuosity problem modulo a composite of unknown factorization. This efficient protocol is inherited from remark \ref{thm:robustCDSfromSS}, which gives that efficient secret sharing schemes give efficient CDS protocols, along with a non-linear secret sharing scheme constructed in \cite{beimel2005power}. A less strong, but also interesting construction of a function outside of $L^{\#L}$ with an efficient PSM, CDS, CDQS, and $f$-routing scheme is based on a DRE for the quadratic residuosity problem modulo a prime. This function is inside of $P$ but believed to be outside of $NC$.
We give the two constructions below.
\subsubsection*{$f$-routing for a problem outside $P$ from non-linear secret sharing}
We define the computational problem that will interest us here. \begin{definition}
The \textbf{quadratic residuosity problem} $QR(u,v)$ is defined as follows.
\begin{itemize}
\item \textbf{Input:} Two integers $u$ and $v$ of $n$ bits.
\item \textbf{Output:} $1$ if $gcd(u,v)=1$ and there exists an $r$ such that $u=r^2$ mod $v$, and $0$ otherwise.
\end{itemize} \end{definition} The quadratic residuosity function is believed to be outside of P/poly. It's hardness is the basis of a well studied public-key cryptosystem \cite{goldwasser2019probabilistic}, and other cryptographic constructions \cite{cocks2001identity,blum1986simple}.
For linear secret sharing schemes, it is known that efficient schemes have complexity in the class Mod$_k$L when the scheme is defined over the field $\mathbb{Z}_k$ for $k$ prime. Thus the connection from secret sharing to CDS to CDQS and $f$-routing reproduces the known class of functions that can be efficiently implemented in the $f$-routing setting.
Beyond linear schemes, \cite{beimel2005power} constructed secret sharing schemes with indicator functions that have complexity outside of $P$. Their scheme realizes the following access structure. \begin{definition}
\textbf{NQR}$_n$ is an access structure on $n=4m$ parties for $m$ an integer.
We label the $4m$ shares by $W^b_i$ and $U^b_j$ with $b\in \{0,1\}$ and $j\in\{1,...,m\}$.
Given two bit strings\footnote{To do modular arithmetic with $w,u$ numbers in $\{ 0, \dots, 2^m-1 \}$ are associated to the strings $w,u$.} $w,u$ each of length $m$, we associate a subset $B_{w,u}$ of size $2m$ according to
\begin{align}
B_{w,u} = \{W^{w_i}_{i}: 1 \leq i\leq m \}\cup \{U^{u_i}_{i}: 1 \leq i\leq m \}.
\end{align}
The access structure \textbf{NQR}$_n$ is then defined by its minimal authorized sets, which are
\begin{itemize}
\item $\{W_i^0,W_i^1\}$ for any $1\leq i\leq m$
\item $\{U_i^0,U_i^1\}$ for any $1\leq i\leq m$
\item $B_{w,u}$ for $w,u$ such that $u\neq 0,1$ and $QR(w,u)=0$, so that $w$ is not a quadratic residue modulo $u$.
\item $B_{w,u=0}$ for $w\neq 1$.
\end{itemize} \end{definition}
Evaluating the indicator function for this access structure is at least as hard as solving the quadratic residuosity problem. To see this, notice that we can reduce computing $QR(u,w)$ to evaluating $f_I$ as follows. From the string $w$ of length $m$, define the two strings $\tilde{w}$, $\tilde{w}'$ according to \begin{align}
\tilde{w}_i = \begin{cases}
1 & \text{if} \,\,\,\, w_i=1 \\
0 & \text{otherwise}
\end{cases} \end{align} \begin{align}
\tilde{w}_i' = \begin{cases}
1 & \text{if} \,\,\,\, w_i=0 \\
0 & \text{otherwise}
\end{cases} \end{align} We similarly define $\tilde{u}$ and $\tilde{u}'$, and then notice that \begin{align}
QR(w,u) = \neg f_I(\tilde{w},\tilde{w}',\tilde{u},\tilde{u}') \end{align} Since computing $\tilde{w},\tilde{w}',\tilde{u},\tilde{u}'$ from $(w,u)$ can be done efficiently, computing $f_I$ is not harder than computing $QR(w,u)$.
Despite the indicator function being of high complexity, there exists an efficient secret sharing scheme for the access structure \textbf{NQR}$_n$. This is given in the following theorem. \begin{theorem}\label{thm:biemel}
\textbf{[Beimel and Ishai 2005]} There exists an $\epsilon$ secure and $\delta$ private secret sharing scheme for the access structure \textbf{NQR}$_{n}$ storing a single bit secret with security parameter $k$, and
\begin{itemize}
\item share size $O(k^2+km)$,
\item correctness $\epsilon = 2^{-k}$,
\item security\footnote{Note that our security definition in terms of a simulator is different from the definition in \cite{beimel2005power}, but it is straightforward to show their security definition with value $\delta$ implies ours with the same $\delta$.} $\delta= k/2^k$.
\end{itemize} \end{theorem} We refer the reader to \cite{beimel2005power} for the construction of this scheme.
In the context of these distributed cryptographic tasks, we are interested in functions which remain of high complexity even when allowing for pre-processing. Thus we would like to construct functions outside of $P_{(2)}$, perhaps starting with NQR. For a function to be a likely candidate to be outside $P_{(2)}$, we need to ensure pre-processing is as unhelpful as possible. We suggest the following function \begin{align}
NQR_{4m,(2)}(x,y) = NQR_{4m}(x\oplus y) \end{align} Then, since Alice see's only $x$ and Bob see's only $y$, pre-processing seems no better than advice, so we expect that NQR$_{4m,(2)}$ is outside $P_{(2)}$ if we have that $NQR_{4m}$ is outside $P/poly$, as we commented above is believed. We state this as the following assumption. \begin{conjecture}\label{conj:hardness}
The function NQR$_{4m,(2)}(x,y)$ is outside of $P_{(2)}$. \end{conjecture} Next, we claim that there is an efficient CDS scheme for NQR$_{4m,(2)}(x,y)$. To see this, we have Alice, following remark \ref{thm:robustCDSfromSS}, prepare the scheme in theorem \ref{thm:biemel} with access structure NQR$_{4m}(z)$. Then she takes share $S_i$ to be the secret which will be conditionally disclosed in a scheme on the XOR function with inputs $x_i$ and $y_i$.
Correctly implementing each of these CDS schemes for the shares $S_i$ is easily seen to now correctly implement the larger scheme with access structure NQR$_{(2),4m}$. This CDS can be performed using $O(|S_i|)$ randomness, so the total needed randomness is still given by the size of the secret sharing scheme.
From this construction for CDS and theorem \ref{thm:CDStoCDQS} we obtain the following. \begin{corollary}\label{corolary:CDSandCDQSoutsideP}
Assuming conjecture \ref{conj:hardness}, there exists a function outside of $P_{(2)}$ with $n$ input bits and hiding one (qu)bit for which CDS and CDQS can be performed $\epsilon=2^{-k}$ correctly and $\delta=k 2^{-k}$ securely with $O(k^2+kn)$ shared bits of randomness. \end{corollary} From theorem \ref{thm:CDQSandfRouting}, we then obtain the following consequence for $f$-routing. \begin{corollary}\label{corollary:fRouteoutsideP}
Assuming conjecture \ref{conj:hardness}, there exists a function outside of $P_{(2)}$ with $n$ input bits and hiding one (qu)bit for which f-routing can be performed $\epsilon=O(k2^{-k})$ correctly with $O(k^2+kn)$ shared entangled pairs. \end{corollary}
\subsubsection*{$f$-routing for a problem outside NC from DRE}
Next, we construct a CDS scheme for a lower complexity function, albeit one that is still outside of $NC$, via a second route that begins with a decomposable randomized encoding.\footnote{Another route for a construction of an $f$-routing scheme for a problem outside NC but inside P, and which is exact, is to begin with (exact) non-linear secret sharing scheme given in \cite{beimel2005power}. We've chosen to use a route beginning with DRE to illustrate that interesting connection.} The computational problem that will interest us is again quadratic residuosity, but this time where the modulus is taken over a prime. \begin{definition}
The \textbf{quadratic residuosity problem over $\mathbb{Z}_p$} is defined as follows.
\begin{itemize}
\item \textbf{Input:} An integer $a$ of $n$ bits and prime $p$, also of $n$ bits.
\item \textbf{Output:} $1$ if $a=b^2$ mod $p$ for some $b$, and $0$ otherwise.
\end{itemize} \end{definition} While this problem is not known to be inside of NC, but is easily placed inside of $P$ by recalling the Euler criterion, which states that \begin{align}
a^{\frac{p-1}{2}} = 1 \,\, \text{mod}\,\, p \end{align} if and only if $a$ is a square. Given this, modular exponentiation can be used to determine if $a$ is a square in polynomial time. Note that if we pose the same problem but with the prime $p$ replaced by a composite number the resulting problem is thought to be outside of $P$ \cite{Kaliski2011}. We focus on the prime case here. See \cite{beimel2005power} for a related discussion of the complexity of the quadratic residuosity functions considered over a field $\mathbb{Z}_p$ for $p$ prime.
The quadratic residuosity problem over primes admits a simple randomized encoding scheme. In particular take \begin{align}
a \rightarrow r^2 a \end{align} for $r$ a randomly chosen integer in $\mathbb{Z}_p$. To understand why this is a randomized encoding, notice that $QR(a)=QR(r^2a)$, so we can compute the result of the function defined by the residuosity problem from the encoded output correctly, by (in this particular case) simply computing the original function, since $r^2a$ is a quadratic residue if $a$ is. Next, to show security one needs to show that if $a$ is a quadratic residue then $r^2a$ is randomly distributed over all those integers $\tilde{a}$ in $\mathbb{Z}_p$ which also are, and if $a$ is not a quadratic residue then $r^2$ is uniformly distributed over all those $\tilde{a}$ which are also not. This amounts to showing that if $a$ and $\tilde{a}$ both are (or both are not) quadratic residues then there is a unique $r$ such that $r^2 a=\tilde{a}$. This follows because the product of two residues is a residue, and the product of two non-residues is a residue.
We can further extend this to a decomposable randomized encoding as follows \cite{ball2020complexity}. Use the encoding \begin{align}
a_i \rightarrow a_i r^2 2^{i-1} + s_i =: y_i \end{align} for $s_i$, $r$ drawn independently and at random from $\mathbb{Z}_p$ for all but the last $s_i$, which we set so that $\sum_i s_i=0$. Then to decode use \begin{align}
QR\left( \sum_i y_i \right) = QR(r^2a) = QR(a). \end{align} To see security, we assume that $a$, $\tilde{a}$ are two integers with the same quadratic residue, and then show there is a choice of $r$, $s_i$ which make the bits of $a$ look like the bits of $\tilde{a}$. This means we need to solve \begin{align}
a_i 2^{i-1} = \tilde{a}_i2^{i-1}r^2 + s_i \end{align} subject also to $\sum_i s_i=0$. It's easy to see we can do this taking as an assumption the same thing we used in the earlier case, that if $a$, $\tilde{a}$ have the same quadratic residue then there is a $r$ such that $a=r^2\tilde{a}$.
Given the existence of a decomposable randomized encoding scheme for the quadratic residue problem, we immediately obtain a PSM for this problem as noted above: Alice and Bob simply send the randomized encodings of their input bits to the referee, who runs the decoding procedure. This was observed already in \cite{ishai1997private}. This in turn implies an efficient CDS, CDQS, and $f$-routing scheme for $f(x)=QR(x)$. We collect these observations as the following remark. \begin{remark}\label{remark:outsideNC}
Consider an $n$ bit string $z$ and split its bits into arbitrary subsets $S$ and $S^c$.
Let the bits from $S$ define a string $z_S$ and a bit from $S^c$ define a string $z_{S^c}$.
Then the function $f(z_S,z_{S^c})=QR(z)$ has perfectly correct PSM and CDS schemes that uses poly$(n)$ bits of randomness. \end{remark} We can also use theorems \ref{thm:CDStoCDQS} and \ref{thm:CDQSandfRouting} to upgrade these to quantum schemes, giving the following corollary. \begin{corollary}
Consider an $n$ bit string $z$ and split its bits into arbitrary subsets $S$ and $S^c$.
Let the bits from $S$ define a string $z_S$ and a bit from $S^c$ define a string $z_{S^c}$.
Then the function $f(z_S,z_{S^c})=QR(z)$ have perfectly correct PSQM and CDQS schemes that uses poly$(n)$ EPR pairs as a resource state. \end{corollary} Ideally, one would show that, assuming $QR(z)$ is outside of NC implies $f(z_{S},z_{S^c})$ is outside of NC$_{(2)}$ but we are unable to do so. Nonetheless, this constructs a second problem not known to be in NC$_{(2)}$ with an efficient $f$-routing scheme, although this one is inside of P. Another comment is that this problem has an exact scheme, while the construction in the previous section that is outside of P is approximate.
\subsection{Efficient PSQM and CDQS for low T-depth circuits}
In \cite{speelman2015instantaneous}, a protocol is given that performs a unitary $\mathbf{U}_{AB}$ non-locally with entanglement cost that depends on the circuit decomposition of $\mathbf{U}_{AB}$. In particular we write $\mathbf{U}_{AB}$ in terms of a Clifford + T gate set, and obtain the following two upper bounds on entanglement cost.
\begin{theorem}\label{thm:Tcountprotocol}
Any $n$ qubit Clifford + $T$ quantum circuit $C$ which has at most $k$ $T$-gates can be implemented non-locally using $O(n2^k)$ EPR pairs.
Further, if $C$ has $T$-depth $d$ then there is a protocol to implement $C$ non-locally using $O((68 n)^d)$ EPR pairs. \end{theorem}
From theorems \ref{thm:CDQSandfRefficiency} and \ref{thm:CFEtoPSQM}, these results lead to upper bounds on entanglement cost in implementing CDQS, $f$-routing, and PSQM. These upper bounds depend on the number of $T$ gates needed to compute $f(x,y)$ with a quantum circuit. We discuss the CDQS setting first.
\begin{figure*}
\caption{The circuit implementing the unitary $\mathbf{U}'$. The unitary $\mathbf{U}$ computs $f(x,y)$ on it's last wire with high fidelity. System $A_0$ is initially maximally entangled with reference $R$. At the end of the circuit, $R$ with be highly entangled with system $A_{f(x,y)}$.}
\label{fig:UwithSWAP}
\end{figure*}
\begin{corollary}\label{corollary:Tdepthandf-route}
Suppose that a function $f(x,y)$ can be evaluated with probability $1-\epsilon$ by a Clifford + T circuit with T-count k and T-depth d.
Then there is a $2\epsilon$-correct $f$-routing protocol for the function $f(x,y)$ that uses at most $O(n2^k)$ EPR pairs, or at most $O((68n)^{d+5})$ EPR pairs, whichever is smaller. \end{corollary} \begin{proof}\, Let $\mathbf{U}$ be the unitary that computes $f$. Recall that this means a measurement in the computational basis on the first qubit of the output of $\mathbf{U}$ returns $f(x,y)$ with probability $1-\epsilon$. Writing the state \begin{align}
\mathbf{U}\ket{x,y} &= \sum_{i_2,...,i_n} \alpha_{0,i_2,...,i_n} \ket{0}\ket{i_2,...,i_n} + \sum_{i_2,...,i_n} \alpha_{1,i_2,...,i_n} \ket{1}\ket{i_2,...,i_n} \nonumber\\
&= \alpha_0 \ket{\psi_0} + \alpha_1 \ket{\psi_1} \end{align}
we have that $|\alpha_{f(x,y)}|^2 \geq 1 - \epsilon$.
Now consider modifying the circuit that implements $\mathbf{U}$ by adding two ancilla qubits $A_0A_1$ and a controlled SWAP gate, where we control on the first output qubit of $\mathbf{U}$. We show this as a quantum circuit in figure \ref{fig:UwithSWAP}. The controlled SWAP gate can be implemented with 7 $T$-gates arranged in $5$ layers (see e.g. \cite{kim2018efficient}). Thus our new circuit has $T$-depth at most $d+5$ and $T$-count at most $k+7$. We call the unitary $\mathbf{U}$ composed with the controlled swap gate $\mathbf{U}'$.
To implement the $f$-routing protocol, we implement $\mathbf{U}'$ non-locally with $A_0X$ held on the left and $A_1Y$ held on the right. Initially $A_0$ is in the maximally entangled state with the reference system $R$. Because $\mathbf{U}'$ can be implemented with $k+7$ $T$-gates and $T$-depth of $d+5$, theorem \ref{thm:Tcountprotocol} gives that this takes no more than $O(n2^k)$ EPR pairs, or at most $O((68n)^{d+5})$ EPR pairs, whichever is smaller. Then we claim that at the end of the protocol that the $A_{f(x,y)}$ system is nearly maximally entangled with $R$.
To see this, notice that the state of the $RA_0A_1XY$ after the unitary plus controlled swap have been applied is \begin{align}
\alpha_0\ket{\Psi^+}_{RA_{0}}\ket{0}_{A_1}\ket{\psi_0}_{XY} + \alpha_1\ket{\Psi^+}_{RA_{1}}\ket{0}_{A_0}\ket{\psi_1}_{XY}, \end{align} where $\psi_0$ and $\psi_1$ are orthogonal states as a consequence of unitarity of $\mathbf{U}$. We take the decoding channel to be the trace over the $A_{1-f(x,y)}XY$ system, followed by a relabelling of $A_{f(x,y)}$ as $Q$. This produces the state \begin{align}
\rho_{RQ} = |\alpha_{f(x,y)}|^2 \Psi^+_{RQ} + |\alpha_{1-f(x,y)}|^2 \frac{\mathcal{I}}{d_R}\otimes \ketbra{0}{0}_{Q} \end{align} Then we can calculate the fidelity \begin{align}
F(\Psi^+,\rho_{RQ}) \geq |\alpha_{f(x,y)}|^2 \geq 1-2\epsilon \end{align} so that the $f$-routing protocol is $2\epsilon$ correct, as needed. \end{proof}
From theorem \ref{thm:CDQSandfRefficiency}, this also leads to a similar upper bound for CDQS. \begin{corollary}\label{corollary:TdepthandCDQS}
Suppose that a function $f(x,y)$ can be evaluated with probability $1-\epsilon$ by a Clifford + T circuit with T-count k and T-depth d.
Then there is a $2\epsilon$-correct and $\sqrt{\epsilon \log d_Q}$ secure CDQS protocol for the function $f(x,y)$ that uses at most $O(n2^k)$ EPR pairs, or at most $O((68n)^d n^{5})$ EPR pairs, whichever is smaller. \end{corollary} \begin{proof}\,
Immediate from theorem \ref{thm:CDQSandfRouting}. \end{proof}
Next, we apply theorem \ref{thm:Tcountprotocol} to give a class of functions for which PSQM can be efficiently performed. \begin{corollary}\label{corollary:PSQMandTdepth}
Suppose that the isometry
\begin{align}
\mathbf{V}_f = \sum_{xy} \ket{xy}_{Z'} \ket{f_{xy}}_{Z} \bra{x}_X\bra{y}_Y
\end{align}
can be implemented with closeness $\epsilon$ (according to the diamond norm distance) with a Clifford + T circuit with T-count k and T-depth d.
Then there exists a PSQM protocol for $f(x,y)$ which is $\epsilon$-correct and $\sqrt{\epsilon}$-secure that uses at most $O(n2^k)$ EPR pairs, or at most $O((68n)^d n^{5})$ EPR pairs, whichever is smaller. \end{corollary} \begin{proof}\,
Follows immediately from theorems \ref{thm:CFEtoPSQM} and \ref{thm:Tcountprotocol} taken together. \end{proof}
\subsection{Sub-exponential protocols for \texorpdfstring{$f$}{TEXT}-routing on arbitrary functions}
In a surprising breakthrough, \cite{liu2017conditional} showed that CDS can be performed for any function using sub-exponential communication and randomness. We summarize their result as the following theorem.
\begin{theorem}\label{thm:LVW} \textbf{[Liu,Vaikuntanathan, Wee 2017]}
Every function $f:\{0,1\}^{n}\times \{0,1\}^{n}\rightarrow \{0,1\}$ has a CDS protocol for single bit secrets using $2^{O(\sqrt{n\log n})}$ bits of randomness and $2^{O(\sqrt{n\log n})}$ bits of communication. \end{theorem} Combining this with theorem \ref{thm:CDStoCDQS} we obtain the following corollary. \begin{corollary} \label{thm:subexpCDQS}
There exists CDQS protocols with perfect correctness and secrecy for every function $f:\{0,1\}^{n}\times \{0,1\}^{n}\rightarrow \{0,1\}$ using $2^{O(\sqrt{n\log n})}$ bits of randomness and $2^{O(\sqrt{n\log n})}$ bits of communication, along with a single qubit of communication. \end{corollary} \begin{proof}\,
Recall that CDS protocols for secrets $s_1$, $s_2$ can be run in parallel if using fresh randomness for each instance (see the paragraph after remark \ref{remark:onesidedCDS}).
Thus we can create a CDS hiding two bits of secret while still using $2^{O(\sqrt{n\log n})}$ randomness and communication, and then apply theorem \ref{thm:CDStoCDQS} to see that we can perform CDQS on a single qubit. \end{proof}
From here, theorem \ref{thm:CDQSandfRouting} leads to the following. \begin{corollary}\label{corollary:subexpfroute}
There exists a perfectly correct $f$-routing protocol for every function $f:\{0,1\}^{n}\times \{0,1\}^{n}\rightarrow \{0,1\}$ using $2^{O(\sqrt{n\log n})}$ qubits of resource system and $2^{O(\sqrt{n\log n})}$ qubits of message. \end{corollary} \begin{proof}\,
Immediate from corollary \ref{thm:subexpCDQS} and theorem \ref{thm:CDQSandfRouting}. \end{proof}
Before moving on, we give some brief context for the construction in \cite{liu2017conditional} that leads to sub-exponential CDS protocols. The reader interested in the construction may refer to the original reference \cite{liu2017conditional} or to the lectures \cite{BIUschool}.
The construction begins with a reduction from a CDS protocol for a general function $f(x,y)$ to a particular function we denote as $INDEX(x,D_y)$, which takes as input Alice's input $x$ and the string $D_y=f(00...00,y)f(00...01,y)...f(11...11,y)$ of length $2^n$. Notice that \begin{align}
f(x,y) = INDEX(x,D_y) = D_y[x] \end{align} This means in particular that a good CDS protocol for the index function will lead to a good CDS protocol for all functions.
The construction of a CDS for INDEX begins with a connection to the cryptographic task of \emph{private information retrieval} (PIR). In a PIR task, a client interacts with several non-communicating servers to retrieve an item with label $x$ from a database $D$, call the item $D[x]$. Security of the PIR requires that the databases not be able to determine the label $x$. This primitive has long been noted to be related to CDS, and in fact CDS was first defined in the context of studying PIR schemes \cite{GERTNER2000592}. While it is not known if all PIR schemes induce CDS schemes, techniques used in PIR constructions have lead to CDS schemes. Theorem \ref{thm:LVW} was proven by applying tools from a sub-exponential PIR scheme presented in \cite{dvir20162} to construct a CDS.
The PIR scheme developed in \cite{dvir20162} relies on the existence of large \emph{matching vector families}. A set of pairs of vectors $\{(\mathbf{u}_i,\mathbf{v}_i \}_{i=1}^N$ is said to be a $S$-matching vector family if \begin{align}
\langle \mathbf{u}_i, \mathbf{v}_i \rangle &= 0 \\
\langle \mathbf{u}_i, \mathbf{v}_j \rangle &\in S,\ \text{when } i \neq j. \end{align} Matching vector families find other applications as well, for instance in the construction of locally decodable codes. An outstanding question is how large $N$ can be taken for vectors chosen in a given vector space. In \cite{grolmusz2000superpolynomial}, the authors constructed large matching vector families over $\mathbb{Z}_6^\ell$, which lead to efficient PIR schemes. Using similar techniques, the same matching vector families lead to the efficient CDS scheme of \cite{liu2017conditional}.
\section{Discussion}\label{sec:discussion}
\noindent \textbf{Collapse of CDQS and PSQM complexity with PR boxes}
A Popescu-Rohrlich box is a hypothetical device, shared by distant parties Alice and Bob, which allows them to satsify the CHSH game with probability one. More concretely, given input $x$ on Alice's side and input $y$ on Bob's side, the device returns $a$ to Alice and $b$ to Bob such that $a\oplus b = x\wedge y$. Broadbent \cite{broadbent2016popescu} showed that if Alice and Bob share PR boxes, they can implement any unitary as a non-local computation using only linear entanglement and a linear number of uses of a PR box. This can be seen as a quantum analogue of a similar collapse that occurs in the setting of classical communication complexity \cite{van2013implausible}. Because efficient non-local computation protocols lead, via theorems \ref{thm:CDQSandfRouting} and \ref{thm:CFEtoPSQM}, to efficient CDQS and PSQM protocols, Broadbent's result similarly leads to a collapse to linear cost for PSQM and CDQS.
In fact, an even stronger collapse follows for CDQS, PSQM and $f$-routing by applying the result of \cite{van2013implausible} showing the collapse of classical communication complexity in the presence of PR boxes. In particular, PR boxes can be used to reduce computing $f(x,y)$ with $x$ held by Alice and $y$ held by Bob to computing $\alpha+\beta$, with $\alpha$ computed from $x$ plus the output of PR box uses, and $\beta$ computed from $y$ along with PR box uses.\footnote{See \cite{kaplan2011non} for results on the number of PR box uses necessary. Note that in our setting we can use the PR boxes sequentially if desired.} In the CDS or PSM settings then, we need only execute CDS or PSM on the function $g(\alpha,\beta)=\alpha+\beta$ with the inputs being single bits. This can be done with $O(1)$ randomness. Via theorems \ref{thm:CDStoCDQS} and \ref{thm:PSMgivesPSQM} then, CDQS can be done with $O(1)$ EPR pairs and PSQM with $O(1)$ shared random bits. We can further note that from theorem \ref{thm:CDQSandfRouting} this means $f$-routing can be performed for arbitrary functions using only $O(1)$ EPR pairs when given access to PR boxes.
\noindent \textbf{Connections to quantum gravity and holography}
In the study of quantum gravity the holographic principle \cite{hooft1993dimensional,susskind1995world} asserts that gravity in $d$ dimensions should have an alternative quantum mechanical description in just $d-1$ dimensions. This principle is realized manifestly in the context of the AdS/CFT correspondence \cite{maldacena1999large,witten1998anti}. In \cite{may2019quantum}, holography and the AdS/CFT correspondence was related to non-local quantum computation. In particular, they argued local interactions in the higher dimensional gravity picture are reproduced as non-local quantum computations in the lower dimensional quantum mechanical picture. As a consequence, computations in the presence of gravity may be constrained by limits on entanglement in the dual quantum mechanical picture \cite{may2022complexity}, or interactions in the gravity picture may imply more computations can be performed non-locally than we have so far found protocols for.
In this work, we see that as a consequence of their connections to NLQC, CDQS and PSQM are also related to holography. One can also realize CDQS and PSQM protocols directly in holography, using connections similar to the one in~\cite{may2019quantum} or the more recent~\cite{may2023non}. This implies that, as with NLQC, constraints on CDQS and PSQM correspond to constraints on bulk interactions. Conversely, the holographic picture has been argued \cite{may2020holographic,may2022complexity} to suggest that a larger class of unitaries than is currently known should have efficient non-local implementations. Importantly, the connection between CDQS and PSQM is so far limited to the 2 input player case, which is also the case that ties to NLQC. It may be possible to explore a connection between CDQS and PSQM to holography that is realized more directly, not via NLQC, which could extend the connection to settings with many input players.
Recalling \cite{may2022complexity}, it was argued that the holographic connection suggests that at least unitaries in BQP should be implementable non-locally. From this perspective it is interesting that, from the connection to secret sharing, we now have at least one function outside of P but inside of BQP with an efficient non-local implementation.
\noindent \textbf{Quantum analogues of recent classical results}
Non-local quantum computation was previously thought to have no (non-trivial) classical analogue: taking the inputs and outputs of a computation to be classical, one can immediately perform the computation in the non-local form of figure \ref{fig:non-localcomputation} without use of shared randomness.\footnote{This amounts to a special case of the impossibility result \cite{chandran2009position}. To see why it is true, consider copying the inputs $x$ and $y$ where they are received and forwarding a copy across the communication channel.} The connections pointed out in this article give non-trivial classical analogues of non-local computation: CDQS is equivalent to a special case of NLQC, and has a non-trivial classical version (CDS), and similarly to PSM.
Traditionally, classical analogues are a source of techniques and conjectures in the quantum setting. Taking this perspective on CDS and CDQS, two recent results in the CDS literature are natural candidates to revisit in the quantum setting.
First, in \cite{applebaum2021placing}, the authors relate CDS to various communication complexity scenarios. In particular they consider the communication complexity class $AM^{cc}$, defined as follows. Alice and Bob hold inputs $x$ and $y$ and share randomness $r$, while a referee holds $(x,y)$. The referee will send Alice and Bob a proof $p=p(x,y,r)$ that both Alice and Bob should accept when $f(x,y)=1$, and both should reject if $f(x,y)=0$. $AM^{cc}(f)$ is the minimal length of the needed proof, and $AM^{cc}$ is the class of functions for which the proof can be taken to be of polylogarithmic length. Relating this to CDS, they show that for some constant $c>0$, \begin{align}\label{eq:CDSandAM}
CDS(f) \geq (\max\{AM^{cc}(f),coAM^{cc}(f) \})^c - \text{polylog}(n) \end{align} where CDS($f$) is the communication complexity of a CDS protocol for $f$ (allowing for imperfect correctness and imperfect security), and a similar bound differing only by constant factors exists for randomness complexity. Unfortunately, there are no explicit functions known to be outside $AM^{cc}\cap coAM^{cc}$, but nonetheless equation \ref{eq:CDSandAM} is an intriguing result. A natural question is if a similar inequality holds when considering CDQS and quantum communication complexity classes.
Second, the related work \cite{applebaum2017private} studied the relationship between zero-knowledge proofs and both CDS and PSM. The starting point is a zero-knowledge variant of the class $AM^{cc}$ discussed above, where an additional requirement that the proof $p$ not reveal anything about $(x,y)$ is imposed. This is refereed to as the class $ZAM^{cc}$. The authors of \cite{applebaum2017private} found that a PSM protocol with perfect correctness and privacy leads to a similarly efficient ZAM protocol, and that a ZAM protocol (which may be approximate) leads to a similarly efficient CDS protocol. Again, it is natural to ask for a quantum analogue of these results.
\noindent \textbf{Classical analogues of further non-local computations}
In this paper we relate two special cases of non-local quantum computation --- $f$-routing and coherent function evaluation --- to other cryptographic tasks, CDQS and PSQM. One aspect of these relationships we have emphasized is that while non-local computation naively becomes trivial when considered classically\footnote{In particular we have in mind that a non-local computation with only classical inputs can always be implemented without pre-distributed resources \cite{chandran2009position}.}, PSQM and CDQS have natural classical variants. This raises the question as to whether NLQC generally has a good classical analogue, perhaps one exploiting the same communication pattern as CDS and PSM, and employing an appropriate secrecy condition. Less ambitiously, we can also ask about classical analogues of other commonly studied non-local quantum computation schemes. One commonly studied non-local computation which we have not considered here is the BB84 task \cite{buhrman2014position,tomamichel2013monogamy}, and its extension to $f$-BB84 \cite{bluhm2022single,escolàfarràs2022singlequbit}. It would be interesting to understand if $f$-BB84 is related to a classical primitive.
\noindent \textbf{Acknowledgements:} We thank Adrian Kent and David P\'erez-Garcia for helpful discussions. RA and HB were supported by the Dutch Research Council (NWO/OCW), as part of the Quantum Software Consortium programme (project number 024.003.037). PVL and HB were supported by the Dutch Research Council (NWO/OCW), as part of the NWO Gravitation Programme Networks (project number 024.002.003). FS was supported by the Dutch Ministry of Economic Affairs and Climate Policy (EZK), as part of the Quantum Delta NL programme. AM is supported by the Simons Foundation It from Qubit collaboration, a PDF fellowship provided by Canada's National Science and Engineering Research council, by Q-FARM, and the Perimeter Institute of Theoretical Physics. Research at Perimeter Institute is supported in part by the Government of Canada through the Department of Innovation, Science and Economic Development Canada and by the Province of Ontario through the Ministry of Colleges and Universities.
\appendix
\end{document} | arXiv |
A chocolate chip cookie recipe calls for 15 cups of flour for 20 dozen cookies. How many cups of flour are needed for 144 cookies?
Converting 144 to 12 dozen, we see that we are making $\frac{12}{20}=\frac{3}{5}$ as many cookies as the recipe makes. Therefore, we need $\frac{3}{5}$ as much flour, which is $\frac{3}{5}\cdot15=\boxed{9}$ cups. | Math Dataset |
quizlette51291387
According to Erik Erikson, the psychosocial stage that characterizes early childhood is:
a. initiative versus guilt.
According to Erik Erikson, the "great governor" of initiative is:
a. conscience.
In Erikson's portrait of early childhood, the young child clearly has begun to develop _____, which is the representation of self, the substance and content of self-conceptions.
c. self-understanding
Four-year-old Harlan says "I'm always happy!" Researchers suggest that Harlan, like other kids his own age, have self-descriptions that are typically:
d. unrealistically positive.
_____ especially plays a key role in children's ability to manage the demands and conflicts they face in interacting with others. It is an important component of executive function.
b. Emotion regulation
Hans feels ashamed when his parents say "You should feel bad about biting your sister!" To experience a _____ emotion like shame, Hans must be able to refer to himself as distinct from others.
b. self-conscious
Self-conscious emotions do not appear to develop until self-awareness appears at approximately _____.
d. 15 to 18 months of age
When Brianna is upset her mother facilitates open discussion about why she is upset and helps her figure out how to deal with the negative emotions. Therefore, Brianna's mother takes an _____ approach to parenting.
c. emotion-coaching
_____ parents interact with their children in a less rejecting manner, use more scaffolding and praise, and are more nurturant than are emotion-dismissing parents.
The children of _____ parents are better at soothing themselves when they get upset, more effective in regulating their negative affect, focus their attention better, and have fewer behavior problems than the children of emotion-dismissing parents.
a. emotion-coaching
Developmental psychologists would describe Jennifer as an "emotion-dismissing" parent to her son. Which of the following types of behavior is Jennifer MOST likely to engage in?
b. She ignores her child when he cries.
Barbara monitors her children's emotions, views their negative emotions as opportunities for teaching, and assists her children in labeling their emotions. She is an:
c. emotion-coaching parent.
Marjorie chooses to deny, ignore, or change the negative emotions of her children. She is an:
c. emotion-dismissing parent.
_____ development involves the development of thoughts, feelings, and behaviors regarding rules and conventions about what people should do in their interactions with other people.
c. Moral
According to Freud, the moral element of the personality is called the _____.
d. superego
Feelings of anxiety and guilt are central to the account of moral development provided by _____ theory.
d. Freud's psychoanalytic
According to Freud, to reduce anxiety, avoid punishment, and maintain parental affection, children identify with parents, internalizing their standards of right and wrong, and thus form the:
c. superego.
_____ is responding to another person's feelings with an emotion that echoes the other's feelings.
b. Empathy
When her mother asks Selena why she feels so sad, Selena says it is because her best friend just lost her puppy. Selena is exhibiting:
b. empathy.
The ability to discern another's inner psychological state is known as:
c. perspective taking.
Which of the following is the first stage of Piaget's theory of moral development?
c. Heteronomous morality
From about _____ years of age, children display heteronomous morality.
b. 4 to 7
According to Piaget's theory, from _____ years of age, children are in a transition showing some features of the first stage of moral reasoning and some stages of the second stage, autonomous morality.
a. 7 to 10
From about _____, children show autonomous morality. .
c. 10 years of age and older
Jerome, 6, and Hani, 10, get up early on Saturday morning and decide to make "breakfast in bed" for their mother. While reaching for the bed tray in the back of the hall cabinet, they accidentally break one of their mother's favorite porcelain dolls. Jerome knows that he's going to get into "big trouble."
Hani tells him not to worry because Mom would understand that it was an accident. In what stage would Jean Piaget categorize the moral reasoning of Jerome and Hani? b. Jerome—heteronomous morality; Hani—autonomous morality
Because young children are _____, they judge the rightness or goodness of behavior by considering its consequences, not the intentions of the actor.
b. heteronomous moralists
Julie believes that Jason's accidental act of breaking 12 plates is worse than Peter intentionally breaking two plates. Julie can be best described as a(n) _____.
d. heteronomous moralist
Dante is a 10-year-old who likes to play soccer during recess. One day a friend teaches him a different set of rules about the game that Dante accepts. He now plays soccer in a new way. Dante is in which stage of moral development?
a. Autonomous morality
Katrina becomes extremely upset when her brother tries to change the rules of their game, yelling, "You can't do that! You can't change rules!" Katrina is exhibiting which of the following types of moral reasoning?
b. Heteronomous morality
As children develop into moral autonomists:
d. intentions become more important than consequences.
Older children, who are _____, recognize that punishment occurs only if someone witnesses the wrongdoing and that even then, punishment is not inevitable.
a. moral autonomists
Young children tend to believe that if a rule is broken, punishment will be meted out immediately. This indicates a belief in the concept of:
a. immanent justice.
Piaget concluded that the changes in moral reasoning in children come about through:
d. the mutual give-and-take of peer relations.
According to Jean Piaget, parent-child relations are less likely to advance moral reasoning than peer relations because:
c. parents take an authoritative approach to handing down the rules.
Social cognitive theory provides several important principles to help us understand moral behavior of children. Which one of the following is NOT one of those principles?
c. Punishment will always increase modeling of moral behavior.
Which of the following approaches holds that the processes of reinforcement, punishment, and imitation explain the development of moral behavior?
c. The behavioral and social cognitive approach
Twice each month, Gini helps to serve dinner at the "Community Table," a program that assists homeless people in the town. She brings her two children, ages 9 and 11, with her and talks to them about the need
to share time, food, and kindness with others who are less fortunate. Social cognitive theorists would say that Gini's children: a. are likely to develop moral behavior that includes helping others.
_____ refers to an internal regulation of standards of right and wrong that involves an integration of all three components of moral development, namely, moral thought, feeling, and behavior.
d. Conscience
In Thompson's view, young children are moral _____, striving to understand what is moral.
a. apprentices
Among the most important aspects of the relationship between parents and children that contribute to children's moral development are relational quality, parental discipline, proactive strategies, and _____.
c. conversational dialogue
Nicola tries to take steps to avert potential misbehavior by her children before it takes place. The moment she sees that her 4-year old daughter is going to have a meltdown, she distracts her with a favorite activity.
She has regular talks with her 10-year old son where she tries to impart her cherished values to him and indicates what is expected of him as he grows older. Nicola is: d. proactive in her approach to her children's moral development.
Gender _____ involves a sense of one's own gender, including knowledge, understanding, and acceptance of being male or female.
c. identity
Sets of expectations that prescribe how females and males should think, act, and feel are known as gender:
a. roles.
Most children know whether they are physically a girl or boy by about _____ years of age.
b. 2½
Gender _____ refers to acquisition of a traditional masculine or feminine role.
c. typing
Michael, age 4, loves playing with toy cars and airplanes, and his idea of play involves wrestling and pushing his friends. Melanie, also 4, loves playing with her dolls and doll house,
and her idea of play is to have a tea party with her dolls and friends. Both are exhibiting: b. sex-typed behavior.
Low levels of _____ in the female embryo allow the normal development of female sex organs.
d. androgens
Gonads are:
b. ovaries in females and testes in men.
_____ promote the development of female physical sex characteristics.
b. Estrogens
_____ promote the development of male physical sex characteristics.
a. Androgens
Which of the following hormones is an androgen?
a. Testosterone
Which of the following hormones is an estrogen?
d. Estradiol
_____ psychologists propose that men have gradually changed over time to have dispositions that favor competition and risk-taking.
b. Evolutionary
According to evolutionary psychologists, natural selection should favor females who chose:
c. ambitious mates who could provide their children with resources and protection.
Psychological evolutionary theories of gender differences fail to take into account:
a. cultural and individual variations in gender differences.
The primary social theories of gender include all of the following EXCEPT:
a. evolutionary psychology view.
The social role theory suggests that:
a. the social hierarchy and division of labor are important causes of gender differences in power, assertiveness, and nurturing.
According to the UNICEF (2011), in most cultures around the world:
b. women have less power and status than men, and they control fewer resources.
This theory of gender stems from the view that the preschool child develops a sexual attraction to the opposite-sex parent.
a. Psychoanalytic theory of gender
According to Freud, at which age does the child renounce the sexual attraction he/she feels toward the parent of the opposite sex because of anxious feelings?
c. 5 or 6 years of age
The psychoanalytic theory of gender stems from Freud's view that the preschool child develops a sexual attraction to the opposite-sex parent. Which of the following describes this condition in girls?
b. Electra complex
According to Freud, preschool boys develop a sexual attraction to the opposite-sex parent in a process called the _____.
a. Oedipus complex
Karen is often praised for gender typical behavior. Her parents make statements like "Karen you are such a good girl when you play with your doll!" Gender researchers would use this as support for what theory of gender development?
b. Social cognitive theory
Which of the following statements is true about parental influences on children's gender development?
d. Fathers show more attention to sons than to daughters.
Who among the following is MOST likely to be rejected by peers on the basis of gender roles?
b. A little boy playing with a doll
Around the age of _____, children already show a preference to spend time with same-sex playmates.
a. three
Children between the ages of 4 and 12 usually prefer to play in groups that are made up of:
b. the same sex as theirs.
In the context of the size of same-sex groups of children, from about 5 years of age onward:
a. boys are more likely to associate together in larger clusters than girls are.
Girls are more likely to engage in "_____," in which they talk and act in a more reciprocal manner.
a. collaborative discourse
A _____ is a cognitive structure, a network of associations that guide an individual's perceptions.
b. schema
A gender _____ organizes the world in terms of female and male.
d. schema
Children are internally motivated to perceive the world and to act in accordance with their developing _____.
b. schemas
Bit by bit, children pick up what is gender-appropriate and gender-inappropriate in their culture, and develop gender _____ that shape how they perceive the world and what they remember.
c. schemas
Which of the following fuels gender typing?
a. Gender schemas
Suzie, 3, has to eat everything on her plate at dinner or her father punishes her by sending her to bed without dinner the next day. Suzie also has strict schedules for playing, television,
and studying, and any disobedience leads to spanking and punishments. Suzie's father is most likely a(n): a. authoritarian parent.
When asked to describe his parenting style, Juan stated, "In my house, my word is the law." Juan is probably a(n):
a. authoritarian parent.
A parent who uses a restrictive, punitive style to control the behavior of their children is a(n):
Lucy frequently spanks her child, enforces rigid household rules, and exhibits rage toward her child when those rules are broken. Lucy is most likely a(n) _____.
A parent who encourages his/her children to be independent but still places limits and controls on their actions is a(n):
b. authoritative parent.
Logan is a warm and loving parent, but he also has high expectations of his kids. As he encourages independent and age-appropriate behavior from his children, Baumarind would classify him as a:
Ursula is allowed to set her own schedules for playtime and for studying. Her mother drives her to her ballet classes and soccer practice. However,
Ursula needs to keep her grades up and must go to bed early on most weeknights. Ursula's mom is most likely a(n): b. authoritative parent.
Which parenting style is demanding and controlling, while also being rejecting and unresponsive?
a. Authoritarian
According to Baumarind, a parent who is very uninvolved in a child's life, showing neither responsiveness nor control, is displaying a _____ parenting style.
d. neglectful
According to Baumarind, a parent who is highly involved with his/her children but places few demands or controls on them is displaying a ____ parenting style.
c. indulgent
Josh's mother makes his favorite food—burgers, fries, and pizza—every night for dinner. His mother lets Josh play as much as he wants to, study only when he feels like it, and imposes no fixed bedtime. Josh's mom is most likely a(n):
c. indulgent parent.
Bernard just brought home his report card and placed it on the television set. Bernard told his dad that he was required to bring the card back to school tomorrow with the signature of one of his parents. Bernard's dad told him to move out of the way because he could not see the TV set.
The next morning, Bernard found his report card where he left it, unsigned. He signed his dad's name and put it in his backpack. Bernard's dad is most likely a(n): d. neglectful parent.
Misha has been sent to his room for hitting his baby sister. His mother will come in and talk to him about why he cannot treat his sister this way and about other, more acceptable ways for him to express his anger. Which parenting style does this exemplify?
b. Authoritative
In which parenting style do parents show pleasure and support in response to children's constructive behavior?
Which parenting style could lead to social incompetence, truancy, and delinquency in children?
Which parenting style leads to egocentric, domineering, and noncompliant behavior in children?
Which parenting style is demanding and controlling, while also being accepting and responsive?
c. Authoritative
Which parenting style is undemanding and uncontrolling, but is also rejecting, and unresponsive?
Which parenting style is undemanding and uncontrolling, while also being accepting and responsive?
b. Indulgent
Research conducted by Ruth Chao suggests that:
a. the high control of Asian parents is best conceptualized as "training" and is distinct from the domineering control characteristic of an authoritarian style of parenting.
A national survey of U.S. parents with 3- and 4-year-old children found that _____ percent of parents reported spanking their children frequently.
In a recent national survey of U.S. parents with 3- to 4-year-old children, about _____ percent reported that they frequently yelled at their children.
Which of the following countries has the most favorable attitude toward corporal punishment?
d. The United States
According to a cross-cultural survey, in which of the following countries are adults most likely to remember that their parents used corporal punishment?
b. South Korea
Research linking corporal punishment and child behavior has been associated with all of the following EXCEPT:
c. higher levels of moral internalization.
Which of the following is an effective way of handling a child's misbehavior, according to most child psychologists?
b. Time out, in which the child is removed from a setting that offers positive reinforcement
Four-year old Becky has just hit her sister, again. What should Becky's mom do? Most developmental psychologists would suggest:
b. explaining to Becky that "hitting hurts"; she is old enough to understand the consequences of her behavior for others.
Tom and Katie have recently split up, but for the benefit of their child they attempt to provide one another support in jointly raising their child. This is an example of:
d. coparenting.
In 2009, approximately 702,000 U.S. children were found to be victims of child abuse at least once during that year. _____ of these children were abused by their parent(s).
d. Eighty-one percent
Whereas the public and many professionals use the term child abuse to refer to both abuse and neglect, developmentalists increasingly use the term _____.
b. child maltreatment
Punching, beating, kicking, biting, burning, and shaking a child constitutes:
d. physical abuse.
Damian's parents fail to provide his basic needs; he is often unfed and dirty when he gets to school. This constitutes:
c. child neglect.
Nine-year-old Tadako's uncle has been taking pictures of her naked and selling them on the Internet. This constitutes:
b. sexual abuse
About _____ of parents who were abused themselves when they were young go on to abuse their own children.
c. one-third
According to a study, maltreated young children in foster care were _____ than middle-SES young children living with their birth family.
c. more likely to show abnormal stress hormone levels
Laurie Kramer, who has conducted a number of research studies on siblings, says that:
d. not intervening and letting sibling conflict escalate are not good strategies.
Which of the following is true of the characteristics of sibling relationships as described by Judy Dunn?
c. There is considerable variation in sibling relationships.
Firstborn and only children are alike in that both are _____ than later-born children.
d. more achievement-oriented
Why do more and more researchers think that birth-order influences on child development have been emphasized too strongly?
d. Birth order itself shows limited ability to predict behavior when all of the factors that influence behavior are considered.
Which of the following countries has the highest percentage of single-parent families?
d. Sweden
Which of the following is true of how parents' work affects the development of their children?
a. The nature of the parents' work is a more important determinant of children's development.
Maribel works as a housekeeper at a hotel. She has no autonomy in her work, works long hours, and feels quite stressed by her job. Kim is a lawyer who works long hours but has control over her work and a great office environment. Anne Crouter would say that:
b. Maribel's children are likely to experience less effective parenting than Kim's children.
It is estimated that approximately _____ percent of children born to married parents in the United States will experience their parents' divorce.
Which of the following is true of children in divorced families?
a. A majority of children in divorced families do not have significant adjustment problems.
Following divorce, custodial mothers experience the loss of about _____ percent of their pre-divorce income.
c. 25 to 50
Approximately _____ percent of lesbians are parents.
Approximately _____ percent of gay men are parents.
Bernice was raised by two lesbian mothers, whereas Jessica was raised by a heterosexual couple. According to research, it is MOST likely that:
b. Bernice and Jessica are the same with regard to popularity and mental health.
The overwhelming majority of children from gay or lesbian families:
b. have a heterosexual orientation.
Carl and Tulip are getting a divorce and want to know how they can best communicate the news to their young children. Ellen Galinsky and Judy David would suggest that:
d. the children be told that they are not the cause of the separation.
In general in the United States, African American and Latino family orientations differ from White family orientations in that:
b. the extended family plays a greater role in African American and Latino families.
Cultural changes that occur when one culture comes in contact with another is known as _____.
b. acculturation
Working-class and low-income families are more likely to practice a(n) _____ parenting style.
What does a child get from peers that he/she typically cannot get from siblings?
c. An idea of how the child compares with other children the same age
Sigmund Freud and Erik Erikson considered play to be valuable because:
a. it helps the child master anxieties and conflicts.
Jean Piaget and Lev Vygotsky considered play to be valuable because:
b. it advances the child's cognitive development.
Which statement best summarizes Daniel Berlyne's views about children's play?
d. Children use play as a way to explore new things and as a way to satisfy their natural curiosity about the world.
_____ play, which can be engaged in throughout life, involves the repetition of behavior when new skills are being learned or when physical or mental mastery and coordination of skills are required for games or sports.
b. Practice
Which of the following statements about practice play is true?
d. Practice play can be engaged in throughout life.
Which type of play increases dramatically during the preschool years?
c. Social play
_____ play occurs when children engage in the self-regulated creation of a product or a solution.
a. Constructive
Which of the following refers to activities engaged in for pleasure that include rules and often involve competition with one or more individuals?
d. Games
Children's Saturday morning cartoons show about _____ violent acts per hour.
Steuer, Applefield, and Smith conducted an experiment where preschool children were randomly assigned to two groups. One group watched cartoons containing violence, and the other group watched cartoons with the violence removed. During a free-play session, the children who watched the
cartoons containing violence showed more aggression than children who watched the nonviolent cartoons. What conclusion was drawn from this study? d. Exposure to TV violence caused aggression in children in this investigation.
Children's shows like Sesame Street are:
a. good at teaching prosocial skills.
This theorist stated that the psychological stage of childhood was "initiative versus guilt."
Answer: Erik Erikson
This theorist suggested that children internalize their parents' standards of right and wrong in order to reduce anxiety and avoid punishment.
Answer: Sigmund Freud
This theorist proposed that gender differences result from the contrasting roles of men and women in societies where women have less power and status than men and control fewer resources.
Answer: Alice Eagly
This theorist has proposed four classifications of parenting involving combinations of acceptance and responsiveness on the one hand and demand and control on the other.
Answer: Diana Baumrind
This theorist, a leading expert on sibling relationships, described three important characteristics of sibling relationships: emotional stability, familiarity and intimacy, and variation.
Answer: Judy Dunn
Parents who monitor their children's emotions, view their children's negative emotions as opportunities for teaching, and assist children in labeling their emotions.
Answer: Emotion-coaching parents
According to Jean Piaget, this is the first stage of moral development where children think of justice and rules as unchangeable properties of the world.
Answer: Heteronomous morality
This theory states that children's gender development occurs through observing and imitating what other people say and do, and through being rewarded and punished for gender-appropriate and gender-inappropriate behavior.
Answer: Social cognitive theory of gender
Tobias's parents are very demanding and show little warmth. They have a "My way or the highway" kind of approach to parenting. What parenting style are they displaying?
Answer: Authoritarian
A parenting technique for handling misbehavior in children. It is characterized by removing the child from a setting that offers positive reinforcement.
Answer: Time out
The support that parents provide one another in jointly raising a child.
Answer: Coparenting
A kind of play that involves repetition of behavior when new skills are being learned or when physical or mental mastery and coordination of skills are required for games or sports.
Answer: Practice play
Explain Eric Erikson's stage of initiative versus guilt. Provide an example of initiative and an example of guilt as it is used by Erikson. Answer: In Eric Erikson's first psychosocial developmental stage, initiative versus guilt, children are learning to use their perceptual, motor, cognitive, and language
skills to make things happen. In essence, they exuberantly move into a wider social world. If they are not permitted to explore their world or if they face disappointment consistently, they will develop guilt
Young children's self-descriptions are typically unrealistically positive. Why is that? Answer: Young children's self-descriptions are typically unrealistically positive because they do not yet distinguish between their desired competence and their actual competence; tend to confuse ability
and effort, thinking that differences in ability can be changed as easily as can differences in effort; do not engage in spontaneous social comparison of their abilities with those of others; and tend to compare their present abilities with what they could do at an earlier age, by which they usually look quite good.
Define self-conscious emotions and provide two examples. What are the two criteria necessary for children to experience self-conscious emotions? Answer: Self-conscious emotions are those that include the quality of an evaluation or judgment of self.
Examples include pride, shame, embarrassment, and guilt. In order for children to experience self-conscious emotions, they must (1) be able to refer to themselves and (2) be aware of themselves as distinct from others.
What are some of the differences between emotion-coaching and emotion-dismissing parents? Answer: Depending on how they talk with their children about emotion, parents can be described as taking an emotion-coaching or an emotion-dismissing approach.
The distinction between these approaches is most evident in the way the parent deals with the child's negative emotions anger, frustration, sadness, and so on. Emotion-coaching parents monitor their children's emotions, view their children's negative emotions as opportunities for teaching, assist them
in labeling emotions, and coach them in how to deal effectively with emotions. In contrast, emotion-dismissing parents view their role as to deny, ignore, or change negative emotions.
Emotion-coaching parents interact with their children in a less rejecting manner, use more scaffolding and praise, and are more nurturant than are emotion-dismissing parents.
Ellen Galinsky and Judy David developed a number of guidelines for communicating with children about divorce. Which one of these guidelines connects with the concept of emotion-coaching? Answer: Ellen Galinsky and Judy David urge parents communicating news of their divorce to their
children to explain to them that it is normal to not feel good about what is happening and that it may take time for them to feel better, and that many other children feel this way when their parents become separated. This connects directly to how emotion-coaching parents monitor their children's emotions,
view their children's negative emotions as opportunities for teaching, assist them in labeling emotions, and coach them in how to deal effectively with emotions. Since the children of emotion-coaching parents are better at soothing themselves
when they get upset, more effective in regulating their negative affect, focus their attention better, and have fewer behavior problems than the children of emotion-dismissing parents, this is a good approach to helping children cope with their parents' divorce.
Name and briefly describe the two stages of moral reasoning in children as identified by Jean Piaget; provide an example of each. Answer: Piaget concluded that children go through two distinct stages in how they think about morality. From about 4 to 7 years of age, children display
heteronomous morality, where children think of justice and rules as unchangeable properties of the world, removed from the control of people. From 7 to 10 years of age, children are in a transition showing some features of the first stage of moral reasoning
and some stages of the second stage, autonomous morality. From about 10 years of age and older, children show autonomous morality.
They become aware that rules and laws are created by people, and in judging an action they consider the actor's intentions as well as the consequences.
Compare and contrast the three major social theories of gender. Which would you argue is the dominant approach today? Answer: Three main social theories of gender have been proposed—social role theory, psychoanalytic theory, and social cognitive theory.
Alice Eagly proposed the social role theory, which states that gender differences result from the contrasting roles of women and men. The psychoanalytic theory of gender stems from Freud's view that the preschool child develops a sexual attraction to the opposite-sex parent.
At 5 or 6 years of age, the child renounces this attraction because of anxious feelings. Subsequently, the child identifies with the same-sex parent, unconsciously adopting the same-sex parent's characteristics.
According to the social cognitive theory of gender, children's gender development occurs through observing and imitating what other people say and do, and through being rewarded and punished for gender-appropriate and gender-inappropriate behavior.
Cultures around the world tend to give mothers and fathers different roles in parenting. Describe the different socializing strategies that mothers and fathers use in raising their children. Answer: In many cultures, mothers socialize their daughters to be more obedient and responsible than their sons.
They also place more restrictions on daughters' autonomy. Fathers, on the other hand, show more attention to sons than daughters, engage in more activities with sons, and put forth more effort to promote sons' intellectual development.
List four characteristics that are generally associated with the firstborn child. Discuss what accounts for these differences. Answer: A recent review concluded that "firstborns are the most intelligent, achieving, and conscientious.
Compared with later-born children, firstborn children have also been described as more adult-oriented, helpful, conforming, and self-controlled. Proposed explanations for differences related to birth order usually point to variations in interactions
with parents and siblings associated with being in a particular position in the family.
In one study, mothers became more negative, coercive, and restraining and played less with the firstborn following the birth of a second child.
Should parents stay in an unhappy or conflicted marriage for the sake of their children? Answer: If the stresses and disruptions in family relationships associated with an unhappy, conflict-ridden marriage that erode the well-being of children are reduced by the move to a divorced
, single-parent family, divorce can be advantageous. However, if the diminished resources and increased risks associated with divorce also are accompanied by inept parenting and sustained or increased conflict, not only between the divorced couple but also among the parents, children, and siblings, the
best choice for the children would be that an unhappy marriage is retained. It is difficult to determine how these "ifs"
will play out when parents either remain together in an acrimonious marriage or become divorced.
Males and females are likely to evoke different responses from adults
a. from the first day of life
b. as early as six months of age
c. about the time they acquire a basic gender identity
d. once they achieve gender constancy
from the first day of life
Each society considers certain values, motives, and behaviors more appropriate for members of one sex than for members of another. These prescriptions are known as
a. gender-role preferences
b. gender-role identities
c. gender-role standards
d. gender schemas
gender-role standards
Mr. and Mrs. Harmon have two children, Jim and Jane. They tell Jim that he might grow up to be an engineer, since men are good at math, and that Jane might become a nurse, since women are empathic and thus well-suited for the caregiving profession. The Harmons' messages reflect what the text refers to as
a. gender-identities
b. gender-role standards
c. gender-role preferences
d. gender-role stereotypes
gender-role stereotypes
Among the gender-role stereotypes that appear to be ACCURATE is that
a. females are more verbally aggressive than men
b. females are more suggestible that males
c. males outperform females on tests of visual/spatial ability
d. males are more analytical than females
males outperform females on tests of visual/spatial ability
By age 2 1/2 to 3 years, children
a. know whether they are boys or girls
b. know that gender is a permanent attribute
c. know most traditional gender-role stereotypes
d. all of the above
know whether they are boys or girls
Mary thinks of herself as a male and prefers masculine activities. Mary's biological sex thus differs from her
a. gender-role stereotypes
b. gender identity
c. gender constancy
Compared to their earlier viewpoints on gender-role violations, older children become _______ about such violations, particularly those undertaken by _____.
a. more flexible; girls
b. more flexible; boys
c. more critical; girls
d. more critical; boys
more flexible; girls
A second round of gender chauvinism appears in ____ as young as people experience ___.
a. adolescence; gender segregation
b. adolescence; gender intensification
c. young adulthood; gender intensification
d. young adulthood; role requirements associated with marriage.
adolescence; gender intensification
Boys face stronger pressures than girls to adhere to "gender-appropriate" codes of conduct because
a. parents are quicker to discourage the cross-sex activities of their sons than their daughters
b. parents perceive a wider range of activities as appropriate for girls than for boys
c. tomboyism is tolerated to some extent, whereas "sissyish" behavior is not
An androgenized female is one who
a. has an extra X chromosome
b. has an extra Y chromosome
c. has been exposed prenatally to make sex hormones
d. is insensitive to the effects of testosterone
has been exposed prenatally to male sex hormones
Androgenized females _______.
a. often display strong interests in masculine activities
b. have developed a large number of both masculine and feminine attributes
c. provide some support for the notion that elevated concentrations of male sex hormones influence the activity preferences of human females
d. often display strong interests in masculine activities and provide some support for the notion that elevated concentrations of male sex hormones influence the activity preferences of human females
often display strong interests in masculine activities and provide some support for the notion that elevated concentrations of male sex hormones influence the activity preferences of human females
Androgynous females
a. are more aggressive than most males
b. display a large number of both masculine and feminine characteristics
c. display few masculine or feminine characteristics
d. have been exposed prenatally to male sex hormones
display a large number of both masculine and feminine characteristics
Children from permissive societies are permitted or even encouraged to engage in sex-play in order to
a. prepare them for adult roles
b. teach them that sexual activities cause neither warts nor blindness
c. keep them quiet when food is scarce
d. quickly replenish the population after a natural disaster
prepare them for adult roles
In relatively non-permissive societies such as the US, most children and adolescents learn about sexually explicit matters from contacts with their
a. parents
b. siblings
c. peers
d. grandparents
Compared to older mothers, adolescent mothers tend to
a. know less about child development
b. be less sensitive and responsive to their infants
c. receive little support from the child's father
Attachment in 1-2 year olds is often measured using this assessment
Strange situation
This term is used when the infant explores, but keeps checking back with his or her mother
Secure base
If a mom is rejecting, and her infant displays no fear/distress, what is the likely attachment category?
Avoidant
Bowlby theorized that the "internal working model of attachment" becomes part of one's personality in this stage of his theory
Formation of a reciprocal relationship
Infants prefer their mothers smell. This is an example of this type of behavior
Experience expectant
In middle childhood, identity tends to be based on...
psychological characteristics
Adolescents who simply refuse to confront the challenge of charting a life course and committing to an ideology are in a state of
identity diffusion
According to Erikson, a possible outcome for the stage teenagers go through is
1. a unified, consistent self image
2. a negative identity, seen as undesirable by mainstream groups
3. diffusion
The top of Maslow's Hierarchy is
This concept may limit how we see others
Acquiring skills because of a natural desire to control one's environment is known as this
mastery motivation
If a child prefers to accomplish a task on his own, without outside help, he has this quality
intrinsic motivation
If I say my successes are due to my abilities and effort, I have this type of attribution
Internal locus of control
"Joy in Mastery" is a stage of achievement that encompasses this age range
If one believes failures are due to one's lack of ability, this attitude may develop
Learned helplessness
At this age, a child knows gender is an unchanging attribute
"Females are less active" is an example of this
Small, but real, psychological difference between genders
The corpus callosum is bigger in _____.
A young teen is more likely to take this attitude towards how to respond to gender role stereotypes
Feeling a need to conform to them (being rigid rather than flexible)
This is the magnification of sex differences
Gender intensification
A female who has been exposed prenatally to male sex hormones
Androgenized female
Emphasizing failures comes from a lack of effort, not ability, would be a part of this technique.
Attribution retraining
According to Maslow, these needs are in a classification that means the needs continues to motivate us even as that need is being met.
Being needs
According to Erikson, one will likely experience this if the crisis posed during the identity vs. identity confusion stage is not successfully met
depression or a lack of self confidence
When Weiner looked at how people attributed success and failure, he not only looked at whether attributions were internal or external, but also this factor
Stable vs unstable
if a test is viewed as covering the concept is purports to measure, it has this quality
Which is a difference between males and females that research supports?
a. Females are happier
b. Males have better comprehension
c. Males are smarter overall (IQ scores)
d. Females are less active
Which is FALSE about gender role standards?
a. Men are more likely to adopt instrumental roles
b. Women are encouraged to adopt expressive roles
c. These are values and behaviors considered to be more appropriate for one sex than another
d. Gender role standards are the same across all cultures
.Which is NOT a myth about gender differences?
a. Females are more social
b. Females have smaller brains
c. Females are better at repetition, males at processing
d. Males are more analytical
Which is a brain difference between males and females?
a. The corpus callosum is larger in males
b. There are differences found in the hypothalamus
c. There are no brain differences between genders
d. The brain stem is larger in females
There are differences found in the hypothalamus
1. In middle childhood, identity tends to be based on:
a. Physical characteristics
b. Psychological characteristics
c. Unique combinations of characteristics
d. Unique inner characteristics
1. Moratorium would best be described as:
a. Active exploration, without a commitment to an identity
b. A commitment to an identity after exploration
c. A commitment to an identity without exploration
d. No exploration of or commitment to an identity
Active exploration, without a commitment to an identity
1. Which is true about Maslow's hierarchy of needs?
a. 1-4 are considered to be 'being needs'
b. Levels 5-8 are considered to be 'deficit needs'
c. One is motivated to meet a deficit need if one doesn't 'have it', but once you meet that need you can move on to the next level
d. Everyone will eventually move to the top of the hierarchy
One is motivated to meet a deficit need if one doesn't 'have it', but once you meet that need you can move on to the next level
1. Scripts:
a. Are associated with Level 4 of Maslow's hierarchy
b. May limit how we see others
c. Become stronger as we move up the hierarchy
d. Help us to 'self-actualize' as they tend to open us up to new experiences
May limit how we see others
A behavior that would be considered to be 'experience expectant' in developing attachment could be when a newborn:
a. prefers his mother's voice over a stranger's
b. makes eye contact
c. looks at human faces longer than other objects
A child and parent take turns, pointing at objects, and 'talking'. This would be known as:
a. Interactional synchrony
b. Experience expectant behavior
c. Intrusiveness
d. Sensitivity
Interactional synchrony
Developmentalists use the term _____ to describe children's intrinsic motivation to respond to challenges and "adapt to" their environments.
a. accommodation
b. competence motivation
c. mastery motivation
d. achievement motivation
a. appears to be inborn
b. does not appear until the infant can recognize his(her) mirror image
c. is an acquired attribute
d. emerges in a competitive environment
appears to be inborn
John works alone at a puzzle for a half-hour and fails to complete it. He then frowns, says "I'm no good at this" and feels bad. We might conclude that John is at the _____ phase of learning to evaluate accomplishments.
a. joy in mastery
b. approval seeking
c. use of standards
d. extrinsic orientation
use of standards
According to Stipek and her associates, young children begin to show their pride (rather than pleasure) in mastering a challenge and true shame (rather than disappointment) when they fail to achieve once they
a. can recognize themselves in a mirror
b. have adopted performance standards that define success and nonsuccess
c. interpret losing a competition as a clear failure
d. know that successes typically elicit approval from others, whereas nonsuccess may elicit disapproval
have adopted performance standards that define success and nonsuccess
According to Weiner's cognitive theory of achievement, a person's future achievement behavior in a particular domain depends very heavily on his/her
a. general intelligence
b. achievement motivation
c. motivation to avoid failure
d. causal attributions for past performance in that domain
causal attributions for past performance in that domain
According to Weiner's attribution theory, which of the following is said to be a stable cause for an achievement outcome?
a. amount of effort expended
b. luck
c. both of these
d. none of these
Children younger than 6 years old expect to _____ at novel achievement tasks because they _____.
a. fail; overestimate the difficulty of the task
b. fail; view themselves as incompetent
c. succeed; underestimate the difficulty of the task
d. succeed; view themselves as competent
succeed; view themselves as competent
Children who display the learned helplessness syndrome
a. tend to attribute their failures to a lack of ability
b. show marked deterioration of performance after experiencing a failure
c. often have previous academic attainments that equal or exceed those of their mastery-oriented classmates
d. tend to attribute their failures to a lack of ability and show marked deterioration of performance after experiencing a failure
e. all of these
all of these
A technique that appears effective at helping children to overcome the learned helplessness syndrome is one in which
a. children are given problems that they will fail and encouraged to attribute these failures to a lack of effort
b. children are given problems that they will fail and encouraged to attribute their failures to a lack of ability
c. children are given problems that they will pass and are thus convinced that they have the ability to succeed
d. children are rewarded for their successes and encouraged to attribute their failures to a fussy evaluator
children are given problems that they will fail and encouraged to attribute these failures to a lack of effort
Dweck's research implies that children who regularly receive _____ after succeeding are likely to adopt learning goals in achievement contexts which makes them inclined to try to _____.
a. person praise; demonstrate their competencies
b. process praise; demonstrate their competencies
c. person praise; improve their competencies
d. process praise; improve their competencies
process praise; improve their competencies
Recent research indicates that _____ is an important contributor to ethnic differences in academic achievement.
a. subtle ethnic differences in parenting practices
b. endorsement of academic goals by ethnic peers
both of these
Stereotype threat often creates a disruptive anxiety that can undermine student performance
a. in situations where performances are evaluated
b. in nonevaluative exercises
in situations where performances are evaluated
. Compensatory education for disadvantaged students has been especially successful at
a. producing long-term gains in IQ
b. turning disadvantaged students into high achievers
c. producing better academic attitudes and helping disadvantaged students to meet basic scholastic requirements
d. producing long-term gains in IQ and turning disadvantaged students into high achievers
producing better academic attitudes and helping disadvantaged students to meet basic scholastic requirements
If a young child points to his image in a photograph and says "me" after simply looking intently at photos of other children, we might take this behavior as evidence that the child
a. has a sense of categorical self
b. distinguishes his public and private selves
c. has achieved self-recognition
d. has reached Erikson's stage of "initiative vs. guilt"
has achieved self-recognition
One social experience that appears to promote the development of self-recognition and social cognition is
a. frequent exposure to mirrors in infancy
b. secure attachments with caregivers
c. heavy exposure to opposite-sex agemates
d. all of these
secure attachments with caregivers
One social experience that may contribute greatly to autobiographical memories and a child's sense of extended self is (are) _____ .
a. parental statements such as "You're a big boy."
b. cooperative interactions with peer playmates
c. conversations with siblings about pretend play roles
d. conversations with parents about noteworthy past events that they have shared with the child
conversations with parents about noteworthy past events that they have shared with the child
One line of evidence that even preschool children may display some "psychological" self-awareness is the observation that they
a. occasionally use trait words (for example, "she's mean") to describe others
b. are quite aware of regularities in their own conduct and can describe them if asked the right questions
c. know that they are either boys or girls
d. think that they are good at most activities and have high self-esteem
are quite aware of regularities in their own conduct and can describe them if asked the right
A well-developed _____ is thought to underlie a child's ability to distinguish "public" and "private" aspects of self.
a. object concept
b. operational intelligence (that is, concrete-operational)
c. theory of mind
d. capacity for symbolism
theory of mine
Development of a belief-desire theory of mind implies that a child thinks that
a. beliefs and desires are different mental states and that either or both can influence a person's conduct
b. beliefs only influence one's conduct when they reflect one's desire
c. beliefs only influence one's conduct when they are inconsistent with one's desires
d. desires are more important than beliefs in determining a person's conduct
beliefs and desires are different mental states and that either or both can influence a person's conduct
. Children's self-concepts become more and more _____ over the course of childhood as they begin to incorporate _____ into their self-descriptions.
a. abstract; psychological attributes
b. concrete; behavioral attributes
c. positive; socially desirable activities
d. public; private knowledge
abstract; psychological attributes
A significant change in self-descriptions that occurs between ages 9 and 11 is a shift from
a. physical characteristics to inner qualities
b. inner qualities to action statements
c. subjective to objective self-evaluations
d. action statements to proverbial characterizations of self
physical characteristics to inner qualities
According to Erikson, establishing an identity involves
a. integration of the id and the superego
b. searching for a unified and consistent self-image
c. breaking away from parental influence
d. attempting to bolster one's ego
searching for a unified and consistent self-image
James Marcia suggests that adolescents who simply refuse to confront the challenge of charting a life course and committing to an ideology are in a state of
a. identity diffusion
b. foreclosure
c. identity achievement
d. moratorium
. If asked to describe a well-known acquaintance, a 5-year-old will typically respond by mentioning the acquaintance's
a. motives and intentions
b. possessions and physical attributes
c. personality traits
d. strengths and shortcomings
possessions and physical attributes
. Selman described the friendships of Stage 2 (that is, 8- 10-year-old) children as "fair-weather" alliances characterized by
a. the exchange of intimate thoughts and feelings
b. lack of pressure to reciprocate kindnesses a friend has displayed
c. strong pressure to reciprocate niceties to maintain the friendship
d. a tendency to label playmates who "play the way I want to" as friends
strong pressure to reciprocate niceties to maintain the friendship
A child who recognizes that true friends exchange intimate thoughts and feelings rather than superficial courtesies and tangible commodities has probably reached Selman's Stage _____ or _____ .
a. 0; egocentric role-taking
b. 1; social-informational role-taking
c. 2; self-reflective role-taking
d. 3; mutual role-taking
3; mutual role-taking
Recent research suggests that _____ may be particularly important at fostering the development of role-taking skills and interpersonal understanding.
a. disagreements with peers
b. disagreements with opposite-sex peers
c. disagreements among close friends
d. disagreements among nonfriends
disagreements among close friends
. An attachment relationship is characterized by
a. an extreme dependence of the attached parties on each other
b. the desire of attached parties to maintain proximity to one another
c. an inability of attached parties to function adequately in each other's absence
the desire of attached parties to maintain proximity to one another
A good index as to whether an infant has established an attachment with a specific adult would be if the infant were to
a. cry a lot at home
b. smile a lot in the presence of strangers
c. engage in behaviors that promote proximity with that adult
d. become quiet and apathetic when in the proximity of that adult
engage in behaviors that promote proximity with that adult
. Interactional synchrony refers to the
a. solitary but coordinated play behaviors of the pre-attached infant
b. meshing of parental and infant affect and behavior during face-to-face interactions
c. finding that interactions proceed more smoothly with attractive rather than unattractive babies
d. coordination of a woman's motherly and wifely roles
meshing of parental and infant affect and behavior during face-to-face interactions
Research on multiple attachments in the first year of life indicates that
a. infants have an attachment hierarchy in which their first attachment object remains their most preferred companion
b. infants have an attachment hierarchy in which the person who feeds them remains their most preferred companion
c. different attachment objects serve different functions, typically with fathers being preferred as protectors and comforters and mothers being preferred as playmates
d. different attachment objects serve different functions, typically with mothers being preferred as protectors and comforters and fathers being preferred as playmates
different attachment objects serve different functions, typically with mothers being preferred as protectors and comforters and fathers being preferred as playmates
Harlow and Zimmerman's (1959) classic "surrogate mother" study with monkeys showed that
a. attachment follows different courses in monkeys and humans
b. monkeys must have achieved a certain level of object permanence before they will form attachments
c. contact comfort delayed the onset of stranger anxiety
d. feeding is not the primary determinant of attachments in monkeys
feeding is not the primary determinant of attachments in monkeys
The theorist who would most likely stress that infants play an active role in the attachment process is
a. John Bowlby
b. Harry Harlow
c. Erik Erikson
d. Sigmund Freud
John Bowlby
. In the "Strange Situation" procedure, researchers pay particular attention to
a. how much time the mother spends interacting with her infant
b. the infant's exploration of a strange room, and his or her behavior when the mother returns after a separation
c. whether the infant shares toys with another, unfamiliar infant
d. the infant's reactions to unfamiliar mechanical toys
the infant's exploration of a strange room, and his or her behavior when the mother returns after a separation
A securely attached infant in the Strange-Situation is
a. visibly distressed when her mother leaves the room
b. wary of strangers in the mother's absence
c. happy to see the mother when she returns
Susan and her mother are participating in the "Strange Situation." When Susan's mother returns to the room, Susan acts resentful but moves closer, staying near until Mom tries to hug her. Susan's attachment classification is most likely
a. secure
b. resistant
c. avoidant
d. disorganized/disoriented
. Fourteen-month-old John sees his mother return to the room after a separation episode in the "Strange Situation." He runs halfway to her and suddenly turns and moves just as quickly away. John's attachment is probably
disorganized/disoriented
Mothers who are generally affectionate but who tend to respond to their child when they feel like being responsive are likely to have children who are
a. securely attached
A mother who is inconsistent in her caregiving--sometimes enthusiastic and involved and sometimes distant--is most likely to have a child who has a
a. resistant attachment
b. secure attachment
c. disorganized/disoriented attachment
d. avoidant attachment
resistant attachment
Zoe loves her infant and consistently tries to stimulate him in play, particularly when he shows signs of becoming fussy. This treatment might be expected to lead her son to form a(n)
When caregivers are clinically depressed
a. their infants will form disorganized/disoriented attachments
b. their infants become hyperactive and fussy with them but show adequate social skills when interacting with other nondepressed adults
c. their infants are likely to form secure attachments nonetheless
d. their infants are at risk of forming some kind of insecure attachment
their infants are at risk of forming some kind of insecure attachment
Recent research on the interplay between caregiving, infant temperament, and attachments reveals that
a. quality of caregiving is the most important factor in determining whether an infants's attachment with a caregiver is secure or insecure
b. infant temperament is the most important factor in determining whether an infant's attachment with a caregiver is secure or insecure
c. infant temperament is a strong predictor of the specific type of insecurity that infants with insecure attachments display
d. quality of caregiving is the most important factor in determining whether an infants' attachment with a caregiver is secure or insecure and infant temperament is a strong predictor of the specific type of insecurity that infants with insecure attachments display
quality of caregiving is the most important factor in determining whether an infants' attachment with a caregiver is secure or insecure and infant temperament is a strong predictor of the specific type of insecurity that infants with insecure attachments display
Gender intensification occurs during ____ and is thought to most clearly reflect ____.
A. middle childhood; peer pressure to behave appropriately for one's sex
B. middle childhood; fathers becoming more involved with sons and mothers becoming more involved with daughters
C. adolescence; peer pressure to conform and to succeed socially with members of the other sex
D. adolescence; academic pressures to pursue curricula that are most appropriate for one's sex
Research indicates that most teenagers today believe the premarital sex is
A. always morally wrong
B. acceptble as long as the partners are emotionally involved
C. acceptable for males but not for females
D. perfectly acceptable under almost all circumstances
Sex differences in play patterns and a clear preference for same-sex peers have been found to
A. not be clearly evident until the late elementary school years
B. emerge after starting school
C. first be evident in the play of preschoolers (about 4-5 years of age)
D. be evident in toddlers (18 months-3 years of age) and to increase throughout the elementary school years
Recent research with fourth- through eighth-graders revealed that among this age range, children who show the best patterns of adjustment are
A. androgenous
B. gender-typical but who feel free to explore cross-gender options
C. masculine gender-typed
D. undifferentiated (low in both masculinity and femininity)
Gender-typing refers to
A. the inheritance of a set of sex chromosomes that determine one's gender
B. the medical technique used to determine the sex of the fetus in utero
C. the processes through which children acquire gender appropriate identities, values, and behaviors
D. the process of classifying individuals on the basis of gender rather than personality
____ provides a good illustration of the expressive gender-role.
A. Women are expected to be good at reading but not at math
B. Men are not supposed to cry or otherwise express emotion
C. Women are expected to be cooperative and sensitive to others
D. Men are expected to show a greater need to express sexual urges than women
Research on the influence of gender-role stereotyping on television reveals that
A. only boys are influenced by viewing stereotyped gender-role portrayals
B. viewing stereotyped gender-role portrayals is associated with stereotyped gender-role attitudes in both boys and girls
C. only girls are influenced by viewing stereotyped gender-role portrayals
D. television portrayals of gender roles are not related to children's gender-role attitudes
Children from "countercultural" homes in which parents strive to promote egalitarian sex-role attitudes are ____ than children from traditional homes.
A. less gender-stereotyped in their gender-role beliefs
B. less gender-typed in their toy and activity preferences
Children first place males and females into different "categories"
A. by 6 months of age, on the basis of vocal cues
B. by 6 months of age, on the basis of hairstyles
C. by 9-12 months of age, on the basis of vocal cues
D. by 9-12 months of age, on the basis of hairstyles
The text emphasized that ____ children show intrinsic motivation to seek gender-role information and acquire gender-appropriate behaviors.
A. in infancy and toddlerhood
B. after 3 years of age
C. not until after 6-7 years of age do
D. not until adolescence do
Children first begin to show some awareness of gender-role stereotypes
A. by the end of the first year
B. by the time they acquire object permanence
C. about the time they acquire gender constancy
D. about the time they acquire a basic gender identity
According to Bem, androgynous individuals are more ____ than those who are traditionally gender-typed.
A. adaptable
B. intelligent
C. sexually active
D. socially anxious
A. gender identities
Research suggests that boys score higher than girls in
A. self-esteem
B. compliance with authority figures
One contribution of the recent psychobiosocial theory of development is to illuminate a process by which ____.
A. biological and social factors interact to influence development
B. genes influence verbal and visual-spatial skills
C. hormones influence girls' activity preferences
D. establishment of psychological masculinity/femininity influences reactions to hormonal changes that occur at puberty
Recent research indicates that ____ is a particularly effective method of promoting regular condom use among sexually active teens.
A. sex education stressing abstinence
B. discussions with parents about condom use before teens become sexually active
C. discussions with parents about condom use after teens initiate sexual activity
D. free distribution of condoms by school nurses and public health departments
The theory that best explains gender-typing during the first three years is ____, whereas ____ seems to be a better explanation for the development of gender-typed interests between ages 3 and 6.
A. gender schema theory; social learning theory
B. gender schema theory; cognitive-developmental theory
C. social learning theory; cognitive-developmental theory
D. social learning theory; gender schema theory
Studies of the sexual behavior of adolescents in recent years reveal that
A. girls are more likely than boys to feel that sex and love should go together
B. rates of sexual activity have declined somewhat from those of the 1970s and 1980s
C. a clear majority of 15-year-olds have had sex, usually with multiple partners
D. girls are more likely than boys to feel that sex and love should go together and rates of sexual activity have declined somewhat from those of the 1970s and 1980s
One sign that gender stereotypes may be having less influence in recent years on girls' academic performance in math and science is that ____.
A. high school girls in some studies value math as much as boys do and view themselves just as competent at math as boys
B. the percentages of women earning advanced degrees in science, engineering, and medicine has increased dramatically
When children begin to draw sharp distinctions between the sexes on psychological dimensions, they first learn
A. positive traits that characterize their own gender and negative traits that characterize the other gender
B. negative traits that characterize their own gender and positive traits that characterize the other gender
C. positive traits that characterize each gender
D. negative traits that characterize each gender
Androgenized females ____.
Margaret Mead's (1935) observations of gender-roles in three New Guinea tribes suggest that
A. males and females are biologically programmed for different kinds of attributes and behaviors
B. gender-role socialization plays a crucial role in determining a child's gender-role preferences and behaviors
The central premise of Money and Ehrhardt's biosocial theory of gender-role development is that
A. biology is more important than socialization at determining the outcomes of gender-typing
B. socialization is more important than biology at determining the outcomes of gender-typing
C. biological and social forces interact to determine the outcome of gender-typing
Martin and Halverson's gender schema theory proposes that children begin to socialize themselves into gender roles as soon as they ____.
A. acquire a mature sense of object permanence
B. have established a basic gender identity
C. reach gender stability
D. reach gender consistency
Evidence cited in the text regarding the influence of parents and teachers in promoting gender-stereotyped attitudes about achievement indicates that ____ have different expectations and/or respond differently to boys and girls.
B. teachers
C. both parents and teachers
D. neither parents nor teachers
Research suggests that girls score higher than boys in
A. verbal aggression
B. verbal abilities
C. suggestibility
Those who are most inclined to become sexually active early in adolescence are
A. adolescents from low-income rather than middle-class families
B. good students, who are more likely than poor students to be trusted by parents
C. teenagers whose friends and siblings are sexually active
D. adolescents from low-income rather than middle-class families and teenagers whose friends and siblings are sexually active
In relatively nonpermissive societies such as the United States, most children and adolescents learn about sexually explicit matters from contacts with their
Recent studies of adolescents' sexual explorations on-line find that
A. older adolescents are almost obsessed with sexual matters in online chat rooms
B. boys act in ways that suggest they are seeking partners
C. girls act in ways that suggest they are trying to attract partners
D. boys act in ways that suggest they are seeking partners and girls act in ways that suggest they are trying to attract partners
African American children may have a less stereotyped view of the sexes than White children do because
A. African American fathers are less stereotyped in their caregiving routines, parenting much like African American mothers do
B. a greater percentage of African-American mothers are enacting the instrumental role by working and functioning as single parents
Gender segregation appears to be stronger among ____.
A. boys than among girls
B. girls than among boys
C. children who hold more stereotyped views of the sexes
D. adolescents than grade-school children
Developmental deficits shown by children of teenage mothers are likely due, in part, to
A. birth complications experienced by teenage mothers who receive poor prenatal care
B. insensitive parenting by teenage mothers, who lack knowledge about child development
C. their families' economic disadvantage
In contrast to the 1950s, contemporary research on adolescent sexual behavior indicates that
A. today's adolescents masturbate less than their predecessors
B. adolescents today are less likely to engage in sexual intercourse
C. adolescent females are about as likely as adolescent males to have had sexual intercourse
D. the sexual behavior of males has changed more than that of females
In Money and Ehrhardt's biosocial theory of gender-typing, the first critical event in gender-role development is ____, an event which ensures that ____.
A. the inheritance of an X or a Y chromosome from the father; the fetus will develop ovaries (with an X chromosome) or testes (with a Y chromosome)
B. the inheritance of an X or a Y chromosome from the mother; the fetus will develop ovaries (with an X chromosome) or testes (with a Y chromosome)
C. the development of testes or ovaries; the fetus will develop a penis and scrotum (if testes have formed) or a clitoris and labia (if ovaries have formed)
D. the development of male or female external genitalia; males will be raised as males and females will be raised as females
One sex difference that appears to be accurate is that
A. boys are more analytical than girls
B. girls are more sociable than boys
C. girls are more suggestible than boys
D. boys take more risks than girls do
Whiting and Edwards studied children in 12 different cultures with respect to their preferences for same-sex playmates. They found
A. most cultures do not show the marked same-sex playmate preference seen in the U.S.
B. that the same-sex playmate preference is common, but typically declines during childhood
C. that the same-sex preference is common and increases during childhood
D. markedly varying degrees of preference across cultures
One's gender identity includes
A. the knowledge "I am a boy/girl"
B. the knowledge "I will always be a boy/girl"
C. the judgment "I am contented/non contented with my biological sex
D. the knowledge "I am a boy/girl" and the knowledge "I will always be a boy/girl"
Developmental deficits displayed by children of teenage mothers are much less likely to emerge if the teen mother is ____.
A. married
B. Hispanic or African American
C. eligible for public assistance
____ theorists view differential reinforcement of gender-typed behaviors as the primary contributor to gender-typed toy preferences and the child's basic gender identity.
A. biosocial
B. psychoanalytic
C. social-learning
D. cognitive-developmental
E. gender schema
What conclusion can be drawn regarding the magnitude of sex differences?
A. sex differences are quite substantial on some characteristics
B. gender accounts for almost 50% of the variation in children's behavior
C. a child's math ability, fearfulness, and aggressiveness can be predicted by knowing the child's gender
E. sex differences are typically quite small; males and females are more psychologically similar than different
Recent research with girls displaying congenital adrenal hyperplasia (CAH) is consistent with Money and Ehrhardt's notion that ____.
A. prenatal exposure to cross-sex hormones influences the behavior of animals but not humans
B. women can be socialized to overcome hormonal influences
C. prenatal exposure to androgen may promote the development of masculine interests and behaviors
D. only males are influenced much by prenatal exposure to androgen
Once children acquire "in group-out group" schemas, they tend to
A. remember information that is consistent with their gender stereotypes
B. transform schema-inconsistent information so that it becomes more consistent with their gender stereotypes
Martin and Halverson's "gender schema" theory proposes that once children have ____, they develop two kinds of gender schemas. The gender schemas that account for their greater in-depth knowledge of gender-appropriate activities and behaviors are ____.
A. established a basic gender identity; own-sex schemas
B. established a basic gender identity; "in group-out group" schemas
C. reached gender constancy; own-sex schemas
D. reached gender constancy, "in group-out group" schemas
Parents who show the clearest pattern of differential reinforcement with regard to gender-typed behaviors have children who are relatively quick
A. to label themselves as boys or girls
B. to develop strong gender-typed toy and activity preferences
C. to acquire knowledge about gender-role stereotypes
According to Martin and Halverson's gender schema theory, once children have ____, they will acquire two kinds of gender schemas. The "own sex" schema implies ____.
A. established a basic gender identity; careful attention to gender appropriate activities and little attention to gender inappropriate activities
B. established a basic gender identity; careful attention to both gender appropriate and gender inappropriate activities, but a strong preference for the gender-appropriate activities
C. reached gender consistency; careful attention to gender appropriate activities and little attention to gender inappropriate activities
D. reached gender consistency; careful attention to both gender appropriate and gender inappropriate activities, but a strong preference for gender appropriate activities
Recent research implies that girls may perform worse than boys in such subjects as math and science in part because
A. they lack talent in these domains
B. they invest themselves broadly across many academic domains, without becoming exceptionally proficient in any domain
C. their teachers believe that boys are more talented and assign boys higher grades for same levels of performance
Bem's theory of psychological androgyny asserts that
A. if a person scores very high in femininity, he or she must be low in masculinity
B. psychologically healthy individuals score low in both masculinity and femininity
C. it is desirable for an individual to simultaneously possess a number of traditionally masculine and traditionally feminine characteristics
D. inborn biological differences account for virtually all sex differences in personality
The gender-role stereotypes held by 3- to 7-year-olds are
A. more flexible than those of older children
B. inconsistent from day to day
C. rigid and inflexible
D. inflexible with respect to personal mannerisms but flexible with respect to the occupations that males and females might pursue
Bradbard et al. found that 4- to 9-year- olds remember more about objects that are believed to be gender-appropriate than about opposite-sex objects. This
finding was cited as supporting ____ theory of gender typing.
A. the biosocial
B. the social-learning
C. Kohlberg's cognitive-developmental
D. the gender schema
The integrative theory of gender typing maintains that very young children(toddlers) display gender-consistent behaviors because
A. of their desire to be boys and girls
B. of their intrinsic motivation to seek out information about gender-consistent behaviors
C. children are self-socializers
D. other people encourage these activities
Some knowledge of gender-role stereotypes first appears ____, and by ____, children agree that males and females differ on important psychological dimensions.
A. at age 2 1/2 to 3; age 4 to 5
B. at age 4 to 5; age 10 to 11
C. at age 10 to 11; early adolescence
D. at age 2 1/2 to 3; age 10 to 11
Despite evidence to the contrary, inaccurate gender-role stereotypes persist because
A. people tend to note and recall instances in which males and females conform to these stereotypes
B. counterstereotypic behavior is likely to be distorted in ways to make it more consistent with the perceiver's stereotypes
Bem's research on psychological androgyny suggests that androgynous people
A. perform like feminine gender-typed individuals on "feminine" tasks
B. perform like masculine gender-typed individuals on
"masculine" tasks
Boys score higher than girls
A. on tests of arithmetic reasoning
B. on tests of computational skills
C. on tests of math concepts
Compared to their earlier viewpoints on gender-role violations, older (i.e., 8- to 10-year- old) children become ____ about such violations, particularly those undertaken by ____.
Evidence indicates that gender-role development is best explained by
A. biosocial/psychobiosocial theory
B. social learning theory
C. Kohlberg's cognitive-developmental theory
D. gender schema theory
E. an integrative approach that combines all the above
The visual cues that 9-12- month-olds use to treat males and females as categorically distinct are ____; ____.
A. clothing; dresses versus pants
B. hairstyles; long hair versus short hair
C. enactment of gender-typed behaviors; comforter versus
D. genitalia; penis versus vagina
D. are aware that males have penises and females do not
The high levels of self-esteem found in androgynous individuals
A. conflict with their high levels of social anxiety
B. seem to reflect the masculine component of androgyny
C. is difficult to understand since peers rate androgynous individuals as less popular than their more traditionally gender-typed age mates
Gender-stereotypes are less likely to undermine the achievement expectancies and academic performances of girls in math when girls ____.
A. are taught by female math instructors
B. have parents who are nontraditional in their gender-role attitudes and behaviors
C. are only children or have older sisters
D. can complete homework assignments without assistance
Cross-cultural studies reveal that
A. gender-role standards vary widely from culture to culture
B. societies in which people live in small families are the ones that emphasize gender-typing the most
C. in most societies, boys face stronger pressures to be "obedient," whereas girls face stronger pressures to be "self-reliant"
D. many gender-role standards are similar from culture to culture
The dichotomy between instrumental roles (for men) and expressive roles (for women)
A. is applicable only in Western societies
B. is applicable only in Third World, preliterate cultures
C. is widely held around the world
D. is held nowhere in the egalitarian climate of the modern era
The ways in which the sexes are portrayed in the media
A. fosters stereotyped gender-role attitudes in children
B. fosters gender-typed toy and activity preferences
C. actually promotes more equalitarian gender-role attitudes in our modern era
D. fosters stereotyped gender-role attitudes in children and fosters gender-typed toy and activity preferences
E. none of the above; media portrays have few if any implications for gender-typing
Maccoby and Jacklin's review of the literature suggests that
A. most gender-role stereotypes are reasonably accurate
B. most gender-role stereotypes are overstated or incorrect
C. gender-role stereotypes about males are much more accurate than those about females
D. gender-role stereotypes about females are much more accurate than those about males
Diane Halpern's psychobiosocial viewpoint extends Money and Ehrhardt's biosocial theory by proposing that
A. the different socializing experiences that young boys and girls receive influences the structure of male and female brains
B. prenatal exposure to testosterone affects the course of fetal brain development
Research has shown that the association of androgyny with high self-esteem can often be traced to the possession of
A. masculine traits
B. feminine traits
C. traits unique to androgyny
A. females are more verbally aggressive than males
B. females are more suggestible than males
For many boys and girls, first sexual attractions occur ____.
A. about age 3, as they begin to fondle their genitals
B. about age 10, with the secretion of increased androgen by the adrenal glands
C. at puberty, with increased productions of male and female sex hormones
D. with the erosion of the gender segregation boundary in adolescence
Children born to teenage mothers often
A. show sizable intellectual deficits and emotional disturbances during the preschool years
B. recover to display adequate academic achievement and good peer relations later in childhood and adolescence
____ have less gender-stereotyped attitudes than ____.
A. middle-class adolescents; agemates from the lower socioeconomic strata
B. African American children; European American children
Aggressive acts that are preformed for purposed of achieving some objective other than doing harm are likely to be classified as ______
instrumental aggression
Many developmentalists are critical of ethologists' presumption that humans lack biologically based inhibitions against harm-doing, often citing ________ as such an inhibition
a capacity for empathy
increases until first grade and then stabilizes
Some hot-headed, oppositional children are at risk of becoming ________, a class of childcare that tends to be the most disliked of all
bully/victims
Gerald Patterson's research on familial contributions to aggression reveal that ______ contribute to hostile interactions as family members ________ aggressive patterns of behavior
coercive home environments; negatively reinforce
Characteristics of families with high levels of coercive behavior
-have few, if any, family rules
- have parents who are coercive problem solvers
-provide negative reinforcers for behavior
-are isolated from other families
Bandura might argue that the aggressive behaviors of the Standford "prison guards" disengaged from their behavior by ___________
displacement of responsibility
3 types of relational aggression and example of each
- Preschool: You can't be my friend unless....
- Middle School: Target of hostile rumors
-High School: Exclusion from social activities
According to the Schwartz et al study, victimization is related to what type of aggression?
List two ways that aggressive childcare display deficits in the encoding of cues (Step 1)
-impulsive, focus on hostile behavior/evidence
-incomplete
By the age of _______, children are spending *more* time with other children than with adults
Early in adolescence, peer contacts become increasingly centered around _____
small friendship networks known as cliques
Measures that ask peers to nominate or rate their classmates' likability are called _____
Sociometric techniques
controversial children receive ___________ nominations on sociometric measures
many positive and many negative
Longitudinal research reveals that _____ children are the ones most likely to retain their sociometric status overtime
By ____, children may have preferred playmates, or "friends", and respond differently to these individuals than to other peers
Barry & Wentzel's research suggests that friends might influence each other's prosocial behavior depending on ____
- the affective quality of their friendship
-how often they interact with each other
True or false: The percentage of time that boys spend with their friends remains stable from 5th to 9th grade.
The number of externalizing symptoms displayed by boys increases from 6th to 8th grade for _____
- aggressive-rejected boys
-nonrejected- aggressive boys
List 4 qualities of friendships that appear to be higher for girls
- validation/ caring
-help and guidance
-conflict resolution
-intimate exchange
Describe 3 ways in which students identified as Brains differed from students in other groups *over time* in the Prinstein & LaGreca study
they had:
-an increase in depression
-an increase in social anxiety
-an increase in loneliness
One of the first signs of an infants emerging sense of self is___
attempts to control objects and events
The self-description " I have green eyes and brown hair. I like to run and to swim at the pool" was probably made by a ____
True or False: On average, the self-esteem of a 9 year old is the same as that of an 80 year old
The central components of Phinney's model of ethnic identity development are _____
- cultural values and attitudes
-sense of group membership and identity
-experience of minority status
In the Yip et al. study, what % of African American adults *NEVER* reached "achieved" identity status?
According to Selman, an example of a Stage 3 perspective-taking response to the "Holly dilemma" is:
"Holly wanted the kitten because she likes kittens, but knew she couldn't climb the tree. Her father knew she can't climb tress but he couldn't have known about the kitten"
According to Graham & Hudley, negative effects of perceived discrimination might be attributed to _____
attribution styles
List 3 outcomes of streotype threat proposed by theorist
-anxiety
-reduced effort
-negative cognitions
Did the Good et al. study demonstrate that stereotype threat resulted in lower achievement test scores? Provide reasons for your answer
Yes. There was a difference between intervention groups. Females scored lower then males on math and it is a stereotype that boys are better at math than girls.
Are more likely to be overweight
Researchers have found that children who have sleep problems:
The leading cause of death in young children
Cannot yet perform reversible mental actions
Piaget's pre operational stage is so named because he believed that children in this stage of development:
egocentrism
While talking to his grandmother on the phone, five year old Danny suddenly exclaims," Oh, look at that pretty bird!" When his grandmother asks him to describe the bird, Danny says,"Out there, out there! Right there grandma!" He finally gets frustrated and hangs up. This is an example of:
A young child might be heard saying "That tree pushed the leaf off and it fell down." The child's belief that the tree is capable of action is referred to as:
Juan and his little sister, Anne, are each given a large cookie. Their mother breaks Anne's cookie into four pieces to enable her to eat it easily. Juan Immediately begins to cry and says that it is not fair for his sister to get so many cookies when he only has one. Juan is showing a lack of:
the range of tasks difficult for child to master alone but that can be learned with help from adults
Zone of proximal development (ZPD) is Vygotsky's term for:
Likely to be socially competent
Four-year old Michelle talks to herself frequently and especially when she is trying to solve a difficult problem. Lev Vygtosky would say that Michelle is:
her memories are highly susceptible to suggestion
six year-old shirley, a witness to robbery, was asked ti testify at the trial. The defense argues that her testimony would be invalid because:
Cynthia shows a number of behaviors different from children her age, including deficits in social interaction and communication as well as repetitive behaviors or interests. she is indifferent toward others and prefers to be alone. She is more interested in objects than people. It is MOST likely that she suffers from:
____ refers to an umbrella-like concept that consists of a number of higher level processes connected to the prefrontal cortex the play a role in managing thought to engage in goal-directed behavior and self-control
Many of the deaths of young children around the world could be prevented by a reduction in:
trust v mistrust
according to erik erikson, he psychological stage that characterizes early childhood is:
emotion-dismissing parent
Majorie chooses to deny, ignore, or change the negative emotions of her children. She is an:
according to Freud, the moral element of the personality is called the:
gender ___ involves a sense of one's own gender, including knowledge, understanding, and acceptance of being male or female
sex-typed behavior
Michael, aged four, loves playing with toy cars and airplanes, and his idea of play involves wrestling and pushing his friends. Melanie, also aged four, loves playing with her doll house, and her idea of play is to have a tea party with her dolls and friends. Both are exhibiting:
the social hierarchy and division of labor are important causes of gender differences in power, assertiveness, and nurturing.
Women have less power and status than men, and they control fewer resources
According to the UNICEF(2011), in most cultures around the world:
A little boy playing with a doll
Authoritarian parent
when asked to describe his parenting, Juan states, "In my house, my word is the law." Juan is probably a(n):
Indulgent parent
josh's mother makes his favorite food-burgers, fries, and pizza every night for dinner. His mother lets josh play as much as he wants to, study only when he feels like it, and imposes no fixed bedtime. Josh's mom is most likely an:
Authoritative parent
logan is a warm and loving parent, but he also has high expectations of his kids. AS logan encourages independent and age-appropriate behavior from his children, baumrind would classify him as a(n):
Time out, in which the child is removed from a setting that offers positive reinforcement
which of the following in an effective way of handling a child's misbehavior, according to most child psychologists?
which type of cancer is most prevalent in children?
cigarette and alcohol exposure during prenatal development
a number of causes for ADHD have been proposed, including:
___ is a severe developmental disorder that has its onset in the first three years of life and includes deficiencies in social relationships, abnormalities in communication, and restricted repetitive, and stereotypes patterns of behavior
___ is a relatively mild autism spectrum disorder in which the child has relatively good verbal language, milder nonverbal language problems, and a restricted range of interests and relationships.
concrete operational stage
a child is presented with two identical balls of clay. the experimenter rolls one ball into a long, thin shape; the other remains in its original ball form. the child is then asked if there is more clay in the ball or in the long, thin piece of clay. If the child answers the problem correctly, but cannot use abstract reasoning yet, the child most likely is in which stage of piaget's cognitive development theory?
although casey scores only about average on standardized
cultural-familial intellectual disability
psychologists suspect that ___ often results from growing up in below-average intellectual environment
a child with an IQ of ___ or higher is considered to be gifted
more mature than others, have fewer emotional problems than others, and grow up in a positive family climate
Ruya, a fourteen-month-old toddler, likes the book 'Green Eggs and Ham', but her brother always tells her how much he likes the book 'Never Tease a Weasel'. When Ruya's brother asks her to bring him a book, Ruya is most likely to bring her ___.
Research indicates that the cognitive attribute most closely associated with raising and resolving identity issues is ____.
reasoning logically about hypotheticals
____ provides strong evidence a child's distinction between public and private self.
successfully misleading an opponent when providing clues in a hidden object game
Baron-Cohen often cites ____ as evidence that theory of mind is a biologically-programmed human attribute.
the mindblindness of otherwise intellectually capable autistic individuals
A person who describes herself as a very bright individual despite the fact that she doesn't perform very well in subjects she doesn't like is probably
an adolescent or adult
Toddlers are just as likely to imitate an adult when the adult's action appears accidental as when the adult's action appears intentional.
Tasha, an eighteen-month-old toddler, watches as her big brother crumples up wads of paper and tries to toss them through a small basketball hoop. After seeing this, Tasha is most likely to ___.
imitate her brother's behavior regardless of whether or not he gets some of the wads through the hoop
Erin is 3,5 years old. What would Erin most likely say if you asked her to describe who she is?
"I have brown hair, and I have a bicycle."
Children's self-concepts become more and more ____ over the course of childhood as they begin to incorporate ____ into their self-descriptions.
Research on the origins of self-esteem reveals that ____ may serve as the foundation for constructing a positive or negative sense of self-esteem.
working models of self that infants and toddlers construct
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Mary_Bredis
AP Bio all vocab
aarushikau8667
326 memory
1,287 terms
Why is it in the best interest of industries to use natural resources responsibly?
politics of the united states
Explain Major Political Ideas Explain major political ideas in history, including social contract theory. Use valid primary and secondary sources to write a paragraph explaining social contract theory. Consider the following questions to support your response: What is social contract theory? How does social contract theory explain why governments are created? What political thinkers are associated with social contract theory?
Choose the best answers to complete the sentences or to answer the following questions. Why do newly industrializing countries frequently have severe environmental problems? A. New factories are technically hard to keep clean. B. People do not understand how to keep their environment clean. C. Business owners prefer to invest in more production rather than in environmental protection. D. Products from new factories are exported.
In the following summary of data for a payroll period, some amounts have been intentionally omitted: $$ \begin{array}{lr} \text{Earnings:}&\\ \text{1. At regular rate}&?\\ \text{ 2. At overtime rate}&135,000\\ \text{3. Total earnings}&?\\ \text{Deductions:}&\\ \text{4. Social security tax}&54,000\\ \text{5. Medicare tax}&13,500\\ \text{6. Income tax withheld}&225,000\\ \text{7. Medical insurance}&31,500\\ \text{8. Union dues}&?\\ \text{9. Total deductions}&335,250\\ \text{10. Net amount paid}&564,750\\ \text{Accounts debited:}&\\ \text{11. Factory Wages}&475,000\\ \text{12. Sales Salaries}&?\\ \text{13. Office Salaries}&200,000\\ \end{array} $$ b. Journalize the entry to record the payroll accrual.
Burglary and Robbery
meganjtaylor98
Chapter 11: Innate and Adaptive Immunity
JaJones1986Plus
GOV 312L Module 18 Review
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Test 3 Study Questions James Herrick
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\begin{document}
\title{Joint Representations for Reinforcement Learning with Multiple Sensors}
\begin{abstract} Combining inputs from multiple sensor modalities effectively in reinforcement learning (RL) is an open problem. While many self-supervised representation learning approaches exist to improve performance and sample complexity for image-based RL, they usually neglect other available information, such as robot proprioception. However, using this proprioception for representation learning can help algorithms to focus on relevant aspects and guide them toward finding better representations. In this work, we systematically analyze representation learning for RL from multiple sensors by building on \emph{Recurrent State Space Models}. We propose a combination of reconstruction-based and contrastive losses, which allows us to choose the most appropriate method for each sensor modality. We demonstrate the benefits of joint representations, particularly with distinct loss functions for each modality, for model-free and model-based RL on complex tasks. Those include tasks where the images contain distractions or occlusions and a new locomotion suite. We show that combining reconstruction-based and contrastive losses for joint representation learning improves performance significantly compared to a post hoc combination of image representations and proprioception and can also improve the quality of learned models for model-based RL. \end{abstract}
\section{Introduction} \begin{figure}
\caption{For many reinforcement learning problems, more than one sensor modality is available.
For example, in the shown task from our \emph{Locomotion Suite}, the robot has an egocentric vision to perceive the obstacles in its environment and observes its proprioceptive state, i.\,e., the position and velocity of its body parts.
We propose learning a joint representation of all available sensor sources using a combination of reconstruction-based and contrastive objectives.
This approach allows us to use reconstruction for clean low-dimensional sensors, e.g., proprioception, and contrastive losses for high-dimensional noisy sensor signals such as images.
We build on \emph{Recurrent State Space Models} to accumulate information across sensors and time and use our representations for model-free and model-based RL.}
\label{fig:fig1}
\end{figure} Learning compact representations of high-dimensional images has led to considerable advances in reinforcement learning (RL) from pixels. To date, most RL approaches that use representations~\cite{hafner2019learning, hafner2019dream, srinivas2020curl,lee2020stochastic,yarats2021improving,stooke2021decoupling, zhang2020learning}, learn them in isolation for a single high-dimensional sensor, such as a camera. However, while images are crucial to the perception of an agent's surroundings in unstructured environments, they are often not the only source of information available. Most agents in realistic scenarios can also directly observe their internal state using sensors in their actuators, inertial measurement units, force and torque sensors, or other forms of proprioceptive sensing.
State Space Models~\cite{murphy2012machine} naturally lend themselves to accumulating information across multiple sensors and time to form a single compact representation of the entire system state. By building on \emph{Recurrent State Space Models (RSSMs)}~\cite{hafner2019learning}, this approach provides a scalable basis for RL in tasks with complex observations and dynamics.
Previous work suggests using either reconstruction~\cite{hafner2019learning, hafner2020mastering} or contrastive methods~\cite{hafner2019dream, ma2020contrastive, nguyen2021tpc} to train \emph{RSSMs}, both of which have their strengths and weaknesses. While reconstruction is a powerful tool as it forces models to capture the entire signal, it may fail to learn good representations if observations are noisy or contain distracting elements~\cite{zhang2020learning, ma2020contrastive, deng2022dreamerpro}. In such cases, contrastive methods can ignore irrelevant parts of the observation and still learn valuable representations. However, they are prone to representation collapse and can struggle to learn accurate dynamics models required for model-based RL~\cite{ma2020contrastive}.
We propose combining contrastive and reconstruction-based approaches to leverage the benefits of both worlds. For example, reconstruction-based loss functions can be used for noiseless proprioception and a contrastive loss for images with distractors, where reconstruction fails~\cite{ma2020contrastive, nguyen2021tpc}. \autoref{fig:fig1} shows an overview of this approach. The common approach to training \emph{RSSMs} is variational inference (VI), which provides a basis for both reconstruction and contrastive objectives. In the original formulation~\cite{hafner2019learning}, \emph{RSSMs} are trained with VI using pure reconstruction. However, the reconstruction terms can be replaced with contrastive losses based on mutual information estimation~\cite{hafner2019dream, ma2020contrastive}. Contrastive predictive coding (CPC)~\cite{oord2018cpc} offers an alternative to the variational approach of training \emph{RSSMs}~\cite{nguyen2021tpc, srivastava2021core}. These methods train the \emph{RSSMs'} system dynamics by maximizing the agreement of predicted future latent states with future observations. Since the recent literature is inconclusive about whether the variational or the predictive approach is preferable, we evaluate our representation learning using both methods. We build our representation learning method into model-free and model-based RL agents and systematically evaluate the effects of learning a joint representation on tasks from the DeepMind Control (DMC) Suite~\cite{tassa2018deepmind}. To test the ability of all approaches to cope with distractions and incomplete information, we use modified DMC Suite tasks to use natural video backgrounds~\cite{zhang2020learning,nguyen2021tpc} and occlusions. Furthermore, we evaluate the methods on a new locomotion suite where agents must combine proprioception and egocentric vision to move and navigate.
Our experiments show that joint representations improve performance over learning an image-only representation and concatenating it with proprioception. They allow us to solve tasks with distractors and occlusions. In particular, the latter task is out of reach for pure image representations. Moreover, we show that joint representations improve the performance of model-based agents with contrastive image representations, which are known to perform worse than reconstruction-based approaches \cite{hafner2019dream, ma2020contrastive}. Here, on the task with natural background, using a combination of reconstruction and contrastive losses enables us to perform almost as well as pure reconstruction on noise-free images.
To summarize our contributions, we propose a general framework for training joint representations based on \emph{RSSMs} by combining contrastive and reconstruction losses based on the properties of the individual sensor modalities. This framework contains objectives motivated by a variational and a contrastive predictive coding viewpoint. We conduct a large-scale evaluation using model-free and model-based approaches and show that using joint representations significantly increases performance over concatenating image representations and proprioception. Further, they help to learn better models for model-based RL when a contrastive image loss is required. We introduce DMC control suite tasks with occlusions and a locomotion suite as new challenges for representation learning in RL. Our joint representation approach outperforms several SOTA baselines on these challenging tasks and on tasks with natural video backgrounds and allows solving tasks where image-only approaches fail.
\section{Related Work}
\textbf{Representations for Reinforcement Learning.} Many recent approaches use ideas from generative~ \cite{wahlstrom2015pixels, watter2015embed, banijamali2018robust, lee2020stochastic, yarats2021improving} and self-supervised representation learning~\cite{zhang2020learning, srinivas2020curl, yarats2021reinforcement, stooke2021decoupling, you2022integrating} to improve performance, sample efficiency, and generalization of RL from images. Particularly relevant for this work are those based on \emph{Recurrent State Space Models (RSSMs)}. When proposing the \emph{RSSM}, \cite{hafner2019learning} used a generative approach. They formulated their objective as auto-encoding variational inference, which trains the representation by reconstructing observations. Such reconstruction-based approaches have limitations with observations containing noise or many task-irrelevant details. As a remedy, \cite{hafner2019dream} proposed a contrastive alternative based on mutual information and the InfoNCE estimator~\cite{poole2019variational}. \cite{ma2020contrastive} refined this approach and improved results by modifying the policy learning mechanism. Using a different motivation, namely contrastive predictive coding~\cite{oord2018cpc}, \cite{okada2021dreaming, nguyen2021tpc, srivastava2021core, okada2022dreamingv2} proposed alternative contrastive learning objectives for \emph{RSSMs}. In this work, we leverage the variational and predictive coding paradigms and show that joint representations improve performance for both. \cite{fu2021tia, wang2022denoised} propose further factorizing the \emph{RSSM}'s latent variable to disentangle task-relevant and task-irrelevant information. However, unlike contrastive approaches, they explicitly model the task-irrelevant parts instead of ignoring them, which can impede performance if the distracting elements become too complex to model. Other recent approaches for learning \emph{RSSMs} include using prototypical representations~\cite{deng2022dreamerpro} or masked reconstruction~\cite{seo2022masked}. Out of these works, only~\cite{srivastava2021core} consider using additional proprioceptive information. Yet, they did so only in a single experiment, without comparing to the concatenation of image representations and proprioception or investigating a combination of reconstruction and contrastive losses.
\textbf{Sensor Fusion in Reinforcement Learning.} Many application-driven approaches to visual RL for robots use proprioception to solve their specific tasks~\cite{finn2016unsupervised, levine2016end, kalashnikov2018scalable, xiao2022masked, Fu_2022_CVPR}. Yet, they usually do not use dedicated representation learning or concatenate image representations and proprioception. Several notable exceptions use \emph{RSSMs} with images and proprioception and thus learn joint representations~\cite{wu2022daydreamer, hafner2022director, becker2022uncertainty, hafner2023mastering}. However, all of them focus on a purely model-based setting and do not investigate joint-representation learning with \emph{RSSMs} as an alternative to concatenation for model-free RL. Additionally, they only consider reconstruction-based training objectives for joint representation learning, while we emphasize contrastive and especially combined methods.
\textbf{Multimodal Representation Learning.} Representation learning from multiple modalities has widespread applications in general machine learning, where methods such as \emph{CLIP}~\cite{radford2021learning} combine language concepts with the semantic knowledge of images and allow language-based image generation~\cite{ramesh2022hierarchical}. For robotics, \cite{brohan2022can, mees2022matters,driess2023palm,shridhar2022cliport,shridhar2023perceiver} combine language models with the robot's perception for natural language-guided manipulation tasks using imitation learning. In contrast, we work in an online RL setting and mainly consider different modalities, namely images and proprioception.
\section{Learning Representation from Multiple Sensors with State Space Models}
Given trajectories of observations $\cvec{o}_{1:T} = \lbrace \cvec{o}_t \rbrace_{t=1:T} $ and actions $\cvec{a}_{1:T} = \lbrace \cvec{a}_t \rbrace_{t=1:T}$ we aim to learn a state representation that is well suited for RL. We assume the observations stem from $K$ different sensor sources, $\cvec{o}_t = \lbrace\cvec{o}^{(k)}_t\rbrace_{k=1:K}$, where the individual $\cvec{o}_t^{(k)}$ might be high dimensional, noisy, and contain only partial information about the system. Further, even the whole observation $\cvec{o}_t$ may not contain all the information necessary for acting optimally, i.\,e., the environment is partially observable, and the representation has to accumulate information over time to allow optimal acting.
Our goal is to learn a concise, low dimensional representation $\phi(\cvec{o}_{1:t}, \cvec{a}_{1:t-1})$ that accumulates all relevant information until time step $t$. We provide this representation to a policy $\pi (\cvec{a}_t | \phi(\cvec{o}_{1:t}, \cvec{a}_{1:t-1}))$ which aims at maximizing the expected return in a given RL problem. Here, we have a cyclic dependency, as the policy collects the trajectories to learn the representation by acting in the environment. In this setting, the policy's return and the sample complexity of the entire system determine what constitutes a \emph{good} representation.
\subsection{Representations from State Space Models} State Space Models (SSMs)~\cite{murphy2012machine} are commonly used to model time series data and naturally lend themselves to sensor fusion and information accumulation problems.
We assume a latent state variable, $\cvec{z}_t$, which evolves according to some Markovian dynamics $p(\cvec{z}_{t+1} | \cvec{z}_t, \cvec{a}_t)$ given an action $\cvec{a}_t$.
At each time step $t$, each of the $K$ observations is generated from the latent state by an observation model $p^{(k)}(\cvec{o}^{(k)}_t | \cvec{z}_t)$ and the initial state is distributed according to $p(\cvec{z}_0)$. In this approach, the representation is given by (the parameters of) the belief over the latent state, taking into account all previous actions as well as previous and current observations
$\phi(\cvec{o}_{1:t}, \cvec{a}_{1:t-1}) \hat{=}p(\cvec{z}_t | \cvec{a}_{1:t-1}, \cvec{o}_{1:t}).$
Yet, due to the nonlinearity of the dynamics and observation models, computing $p(\cvec{z}_t | \cvec{a}_{1:t-1}, \cvec{o}_{1:t})$ is intractable for models of relevant complexity.
Thus, we approximate it using a variational distribution $q(\cvec{z}_t | \cvec{a}_{1:t-1}, \cvec{o}_{1:t})$. This variational approximation plays an integral part in training the SSM and is thus readily available to use as input for the policy.
We instantiate the generative SSM and the variational distribution using a \emph{Recurrent State Space Model (RSSM)}~\cite{hafner2019learning}, which splits the latent state $\cvec{z}_t$ into a stochastic and a deterministic part. Following~\cite{hafner2019learning, hafner2019dream}, we assume the stochastic part of the \emph{RSSM}'s latent state to be Gaussian, since the more recently introduced parametrization as a categorical distribution~ \cite{hafner2020mastering} has not been proven beneficial for the continuous control tasks considered in this work. While the original \emph{RSSM} only has a single observation model $p(\cvec{o}_t | \cvec{z}_t)$, we extend it to $K$ models, one for each observation modality. The variational distribution takes the deterministic part of the state together with the $K$ observations $\cvec{o}_t = \lbrace \cvec{o}_t^{(k)} \rbrace_{k=1:K}$ and factorizes as
$q(\cvec{z}_{1:t} | \cvec{o}_{1:t}, \cvec{a}_{1:t-1}) = \prod_{t=1}^t q(\cvec{z}_t | \cvec{z}_{t-1}, \cvec{a}_{t-1}, \cvec{o}_t).$ Compared to the original \emph{RSSM}, we again have to account for multiple observations instead of one.
Thus, we first encode each observation individually using a set of $K$ encoders, concatenate their outputs and provide the result to the \emph{RSSM}. Finally, we also learn a reward model $p(r_t | \cvec{z}_t)$ to predict the achieved reward from the representation. While predicting the reward from the latent state is conceptually only necessary for model-based RL, there is sufficient evidence in the literature that it is beneficial to also include it when learning representations for model-free RL~\cite{srivastava2021core, tomar2023whatmatter}.
\subsection{Learning the State Space Representation}
We combine reconstruction-based and contrastive approaches to train our representations. Training \emph{RSSMs} can be based on either a variational viewpoint~\cite{hafner2019dream, ma2020contrastive} or a contrastive predictive coding~\cite{oord2018cpc} viewpoint~\cite{nguyen2021tpc, srivastava2021core}. We investigate both approaches, as neither decisively outperforms the other.
\textbf{Variational Learning.} Originally, \cite{hafner2019learning} proposed leveraging a fully generative approach for \emph{RSSMs}. Building on the stochastic variational autoencoding Bayes framework~\cite{kingma2013auto, sohn2015learning}, they derive a variational lower bound. After inserting our assumption that each observation factorizes into $K$ independent observations and adding a term for reward prediction, this bound is given as \begin{align}
\sum_{t=1}^T \mathbb{E} \left[ \sum_{k=1}^K \log p^{(k)} (\cvec{o}^{(k)}_t | \cvec{z}_t) + \log p(r_t | \cvec{z}_t) - \KL{q(\cvec{z}_t | \cvec{z}_{t-1}, \cvec{a}_{t-1}, \cvec{o}_t)}{p(\cvec{z}_t | \cvec{z}_{t-1}, \cvec{a}_{t-1})} \right], \label{eq:recon_bound} \end{align}
where the expectation is formed over the distribution $p(\cvec{o}_{1:t}, \cvec{a}_{1:{t-1}})q(\cvec{z}_t | \cvec{o}_{1:t}, \cvec{a}_{1:t-1})$, i.\,e., sub-trajectories from a replay buffer and the variational distribution. Optimizing this bound using the reparametrization trick~\cite{kingma2013auto, rezende2014stochastic} and stochastic gradient descent simultaneously trains the variational distribution and all parts of the generative model.
While this reconstruction-based approach can be highly effective, reconstructing high-dimensional, noisy observations can also cause issues. First, it requires introducing large parameter-rich observation models (usually up-convolutional neural nets for images) solely for representation learning. These observation models are unessential for the downstream task and are usually discarded after training. Second, the reconstruction forces the model to capture all details of the observations, which can lead to highly suboptimal representations if images are noisy or contain task-irrelevant distractions. Contrastive learning can provide a remedy to these problems.
To introduce such contrastive terms, we can replace each individual log-likelihood term $\mathbb{E}\left[\log p^{(k)}(\cvec{o}_t^{(k)}|\cvec{z}_t)\right]$ with a corresponding mutual information (MI) term $I(\cvec{o}_t^{(k)},\cvec{z}_t)$ by adding and subtracting the evidence $\log p(\cvec{o}^{(k)})$ \cite{hafner2019dream, ma2020contrastive} \begin{align}
\mathbb{E}\left[\log \left(p^{(k)}(\cvec{o}_t^{(k)}|\cvec{z}_t) / p(\cvec{o}_t^{(k)}) \right) + \log p(\cvec{o}_t^{(k)} ) \right] = \mathbb{E} \left[ I(\cvec{o}_t^{(k)},\cvec{z}_t) \right] + c \label{eq:mi_ll}. \end{align} Intuitively, the MI measures how informative a given latent state is about the corresponding observations. Thus, maximizing it leads to similar latent states for similar sequences of observations and actions. While we cannot analytically compute the MI, we can estimate it using the InfoNCE bound~\cite{oord2018cpc, poole2019variational}. Doing so eliminates the need for generative reconstruction and instead only requires a discriminative approach based on a score function $f_v^{(k)}(\cvec{o}^{(k)}_t, \cvec{z}_t) \mapsto \mathbb{R}_+$. This score function measures the compatibility of pairs of observations and latent states. It shares large parts of its parameters with the \emph{RSSM}. For details on the exact parameterization, we refer to \autoref{sup:sec:obj}.
Crucially, this methodology allows the mixing of reconstruction and mutual information terms for the individual sensors, resulting in \begin{align}
\sum_{t=1}^T \sum_{k=1}^K \mathcal{L}_v^{(k)}(\cvec{o}^{(k)}_t, \cvec{z}_t) + \mathbb{E} \left[ \log p(r_t | \cvec{z}_t) - \KL{q(\cvec{z}_t | \cvec{z}_{t-1}, \cvec{a}_{t-1}, \cvec{o}_t)}{p(\cvec{z}_t | \cvec{z}_{t-1}, \cvec{a}_{t-1}}) \right] \label{eq:cv_bound}, \end{align} where $\mathcal{L}_v^{(k)}$ is either the log-likelihood or the MI term. As we show in \autoref{sec:exp} choosing the terms corresponding to the properties of the corresponding modality can often improve performance.
\textbf{Contrastive Predictive Coding.} CPC \citep{oord2018cpc} provides an alternative to the variational approach. The idea is to maximize the MI between the latent variable $\cvec{z}_t$ and the next observation $\cvec{o}^{(k)}_{t+1}$, i.\,e., $I(\cvec{o}_{t+1}^{(k)}, \cvec{z}_t)$. While this approach seems similar to contrastive variational learning, there is a crucial conceptual difference. We estimate the MI between the current latent state and the next observation, not the current observation. Thus, we explicitly predict one step ahead to compute the loss. As we use the \emph{RSSM's} dynamics model for the prediction, this formalism provides a training signal to the dynamics model.
However, prior works~\cite{shu2020predictive, nguyen2021tpc, srivastava2021core} showed that solely optimizing for prediction is insufficient. We follow \cite{srivastava2021core} and regularize the objective further by a reward prediction term, the KL-term from \autoref{eq:recon_bound} multiplied with a small factor $\beta$, and an inverse dynamics predictor $\hat{\cvec{a}}_t = a(\cvec{z}_t, \cvec{z}_{t+1})$. Here, $\hat{\cvec{a}}_t$ is trained to predict the action from two consecutive latent states by minimizing $l_a= ||\cvec{a}_t - \hat{\cvec{a}}_t||^2$. Additionally, we can turn individual contrastive MI terms into explicit reconstruction terms for suitable sensor modalities by reversing the principle introduced in \autoref{eq:mi_ll}. This approach results in the following maximization objective
\begin{align} \hspace{-0.3cm}\sum_{t=1}^T \sum_{k=1}^K \mathcal{L}_p^{(k)} (\cvec{o}_{t+1}^{(k)}, \cvec{z}_t) \label{eq:cpc_bound} + \mathbb{E} \left[ \log p(r_t | \cvec{z}_t) - \beta \KL{q(\cvec{z}_t | \cvec{z}_{t-1}, \cvec{a}_{t-1}, \cvec{o}_t)}{p(\cvec{z}_t | \cvec{z}_{t-1}, \cvec{a}_{t-1})} - l_a \right], \end{align}
where $\mathcal{L}_p^{(k)}$ is either the one-step ahead likelihood $\log p(\cvec{o}_{t+1}^{(k)} | \cvec{z}_t)$ or an InfoNCE estimate of $I(\cvec{o}_{t+1}^{(k)}, \cvec{z}_t)$ using a score function $f_p^{(k)}(\cvec{o}^{(k)}_{t+1}, \cvec{z}_t) \mapsto \mathbb{R}_+$. From an implementation viewpoint, the resulting approach differs only slightly from the variational contrastive one.
For CPC approaches, we use a sample from the \emph{RSSM's} dynamics $p(\cvec{z}_{t+1} | \cvec{z}_t, \cvec{a}_t)$ and for contrastive variational approaches we use a sample from the variational distribution $q(\cvec{z}_t | \cvec{z}_{t-1}, \cvec{a}_{t-1}, \cvec{o}_t)$.
\textbf{Estimating Mutual Information with InfoNCE.} We estimate the mutual information (MI) using $b$ mini-batches of sub-sequences of length $l$. After computing the latent estimates, we get $I = b\cdot l$ pairs ($\cvec{o}_i$, $\cvec{z}_i$), i.\,e., we use both samples from the elements of the batch as well as all the other time steps within the sequence as negative samples. Using those, the symmetry of MI, the InfoNCE bound~\cite{poole2019variational}, and either $f=f^{(k)}_v$ or $f=f^{(k)}_p$, we can estimate the MI as \begin{align*}
0.5 \left( \sum_{i=1}^I \log \dfrac{f(\cvec{z}_i, \cvec{o}_i)}{\sum_{j=1}^I f(\cvec{z}_j, \cvec{o}_i)} + \log \dfrac{f(\cvec{z}_i, \cvec{o}_i)}{\sum_{j=1}^I f(\cvec{z}_i,\cvec{o}_j)} \right). \end{align*}
\textbf{Image Augmentation.} Following prior works~\cite{srivastava2021core, deng2022dreamerpro}, we found image augmentation helpful for contrastive approaches. During training, crops are randomly selected for each sequence but remain consistent within the sequence. For evaluation, we crop at the center.
\subsection{Learning to Act Based on the Representation} Given the learned representation, we consider both model-based and model-free reinforcement learning. For model-free RL, we use Soft Actor-Critic (SAC)~\cite{haarnoja2018sac} on top of the representation. More specifically, we use the deterministic part of the latent state and the mean of the stochastic part as input to both the actor and the critic. In the model-based setting, we use latent imagination~\cite{hafner2019dream}, which propagates gradients directly through the learned dynamics model to optimize the actor. In the case of pure reconstruction-based representation, this approach defaults to the original \emph{Dreamer}~\cite{hafner2019dream}. In both cases, we alternatingly update the \emph{RSSM}, actor, and critic for several steps before collecting a new sequence in the environment. Furthermore, the \emph{RSSM} uses only the representation learning loss and gets neither gradients from the actor nor the critic.
\section{Experiments} \label{sec:exp} \begin{figure}
\caption{Excerpt of the aggregated results for \textbf{model-free} agents on the DMC Suite tasks with different forms of image observations. In all tasks using proprioception is beneficial. More importantly, learning a joint representation outperforms the concatenation of image representations and proprioception.
\textbf{(a)}: Reconstruction on the standard images.
Even though the images provide all information necessary to solve the tasks, our method can still exploit the proprioceptive information and improve upon the image-only baselines and \emph{DRQ-v2(I+P)} in both sample efficiency and final performance.
\textbf{(b)}: Contrastive variational approaches on natural video background tasks. While the concatenation initially learns faster than image-only representations, only joint representations substantially improve the final performance.
\textbf{(c)}: Contrastive predictive coding approaches on occlusion tasks. No approach using a single sensor is capable of solving this task.
However, using both images and proprioception can give good results, in particular when combining a contrastive loss for the images with reconstruction for the proprioception. }
\label{fig:main_mf}
\end{figure} We evaluate our joint representation learning approach on several environments from the DeepMind Control (DMC) Suite~\cite{tassa2018deepmind} with different types of image observations and a new \emph{Locomotion Suite}. We train five seeds per task for $10^6$ environment steps and evaluate for $20$ rollouts every $20,000$ steps. Following the suggestions from~\cite{agarwal2021deep}, we aggregate the results over all environments in the benchmark suites and report interquartile means and 95\% stratified bootstrapped confidence intervals, indicated by shaded areas. For experimental details, we refer to \autoref{sup:sec:hps}. In summary, we compare the following approaches.
\textbf{Joint Representation Learning.} We denote all our joint representations approaches with \emph{Joint($X$)}. Here, $X$ will denote how we trained the representation, which is either pure reconstruction \emph{(R)} (\autoref{eq:recon_bound}), or a mixture of a contrastive image loss and reconstruction for the proprioception. The former is either the contrastive variational objective (\emph{CV}, i.e., \autoref{eq:cv_bound}) or the contrastive predictive objective (\emph{CPC}, i.e., \autoref{eq:cpc_bound}). For both contrastive methods, we also evaluate using the contrastive loss for both image and proprioception. We denote these fully contrastive approaches by \emph{FC-Joint (CV)} and \emph{FC-Joint(CPC)}.
\textbf{Concatenating Representations.} One important baseline is concatenating the proprioception to a representation trained solely on images. To ensure a fair comparison, we train this representation using our approach only on images. Here, we again use reconstruction and both contrastive methods for training and refer to the resulting approaches as \emph{Concat(R)}, \emph{Concat(CV)}, and \emph{Concat(CPC)}. We consider this baseline only for the model-free setting as it cannot predict future proprioception which renders model-based RL infeasible.
\textbf{Single Observations.} We also train policies using only a single sensor to ensure our approaches can exploit the additional information provided by multiple sensors. For the image-only policies, we again use our state space representation learning approach with the different training schemes, resulting in \emph{Img-Only(R)}, \emph{Img-Only(CV)}, and \emph{Img-Only(CPC)}. For the proprioception-only policies (\emph{Proprio-Only}), we use SAC~\cite{haarnoja2018sac} directly on the proprioception without learning an explicit representation.
\textbf{Related Approaches.} We compare to several model-based \emph{RSSM} approaches. We consider the reconstruction \emph{Dreamer-v2}~\cite{hafner2020mastering} and the contrastive \emph{DreamerPro}~\cite{deng2022dreamerpro}, which builds on prototypical representations. Additionally, we compare against \emph{TIA}~\cite{fu2021tia} and the \emph{DenoisedMDP}~\cite{wang2022denoised}. Both propose special losses and factorizations of the latent space to allow disentangling task-relevant from irrelevant aspects. \emph{DenoisedMDP} also provides both, model-based and model-free versions, and we compare with both. Furthermore, we compare to the model-free \emph{DrQ-v2}~ \cite{yarats2022mastering}, as well as an extension that uses images and proprioception (\emph{DrQ-v2(I+P)}). Both do not explicitly learn a representation. Finally, \emph{Img-Only(R)} with model-based RL corresponds mostly to Dreamer-v1~\cite{hafner2019dream} and \emph{Img-Only(CPC)} as well as \emph{FC-Joint(CPC)} with model-free RL closely resemble the approach introduced in~\cite{srivastava2021core}. \autoref{supp:sec:baselines} provides details on the differences.
\subsection{DeepMind Control Suite}
\begin{figure}
\caption{Excerpt of the aggregated results for \textbf{model-based} agents on the DMC Suite tasks, with different forms of image observations. \textbf{(a)}: Reconstruction on the standard images. \emph{Img-Only(R)}, which almost corresponds to the original Dreamer\cite{hafner2019dream}, performs slightly worse than \emph{Dreamer-v2} and \emph{DreamerPro}, while this performance gap is closed by \emph{Joint(X)}.
\textbf{(b)}: Contrastive variational approaches on natural video background tasks. Compared with the standard image tasks, both \emph{DreamerPros'} and \emph{Img-Only(CV)s'} performance decreases significantly, while the joint representation learning with a mixture of contrastive and reconstruction (\emph{Joint(CV)}) almost achieves the same final performance as reconstruction without distractions.
\textbf{(c)}: Contrastive predictive coding approaches on occlusion tasks. Only the combined joint objective performs significantly better than the proprioception-only baseline.
}
\label{fig:main_mb}
\end{figure}
We use seven tasks from the DMC Suite~\cite{tassa2018deepmind}, namely \texttt{Ball-in-Cup Catch}, \texttt{Cartpole Swingup}, \texttt{Cheetah Run}, \texttt{Reacher Easy}, \texttt{Walker Walk}, \texttt{Walker Run}, and \texttt{Quadruped Walk}. For each task, we split the state into proprioceptive entries and non-proprioceptive entries. While the former is directly provided, the latter can only be perceived via an additional image. For example, in \texttt{Ball in Cup Catch} the cup's state is proprioceptive while the ball's state needs to be inferred from the image. \autoref{sup:sec:env} lists the details for all considered environments. Besides standard images, we run experiments with two types of modified images by adding natural video backgrounds or occlusions. The latter renders the tasks much more challenging due to the additional partial observability. The upper row of \autoref{fig:quali_vis} shows examples of these image modifications.
\textbf{Natural Video Background.} Following~\cite{nguyen2021tpc, deng2022dreamerpro} we render random videos from the Kinetics400 dataset \cite{kay2017kinetics} behind the agent. The challenge of this task is to learn a representation that filters out as many irrelevant visual details as possible while not ignoring relevant aspects. Methods based on image reconstruction struggle at this task as they are forced to model all background details to minimize their objective. While contrastive approaches can ignore parts of the image, the challenge is to formulate an objective that focuses exclusively on the relevant details.
\textbf{Images with Occlusions.} We render occlusions by adding disks that slowly move in front of the agent. Those occlusions can hide relevant information in the images for multiple consecutive time steps, which decreases observability and increases the task's complexity. This modification tests the capabilities of the approaches not to be distracted by the easily predictable disks, and to maintain a consistent representation across multiple time steps when relevant aspects are missing for some images.
\begin{figure}
\caption{\textbf{Left:} Saliency Maps (pixel-wise norm of the Jacobian of the latent representation w.r.t. the image) showing on which pixels the respective representation learning approaches focus. \emph{Joint(CV)} clearly focuses better on the task-relevant cheetah, while the \emph{Img-Only(CV)} is more distracted by the background video. \textbf{Right:} We provide occluded images and train an additional decoder to reconstruct the occlusion-free ground truth from the (detached) latent representation. For \texttt{Cartpole Swingup} only the cart position is part of the proprioception, while the pole angle has to be inferred from the images.
Still, the \emph{Joint(CPC)} is able to capture both cart position and pole angle, while \emph{Img-Only(CPC)} clearly fails to do so.
Both examples illustrate how using proprioception in representation can guide representation learning to focus on and capture the relevant parts of the image.}
\label{fig:quali_vis}
\end{figure} \textbf{Results.} \autoref{fig:main_mf} shows an excerpt of the results for model-free agents. Joint representations generally outperform the concatenation and single-sensor approaches. They perform best when trained with our combination of reconstruction-based and contrastive losses and outperform all model-free and model-based baselines in the natural video background and occlusion tasks. In particular, learning joint representations performs better than providing the same information by concatenating an image representation and the proprioception. As expected, none of the pure reconstruction approaches can solve the natural video and occlusion tasks, while methods that use a contrastive loss for the images are less affected by the distractors or occlusions. For the occluded tasks, all image-only approaches struggle to learn any reasonable behavior, underscoring the difficulty of the task. \autoref{fig:main_mb} shows an excerpt of the results for model-based agents. Again, the benefits of joint representation learning with a combined objective become clear. While the performance of the contrastive image-only approaches decreases when adding natural video backgrounds, $\emph{Joint(CV)}$ still achieves high performance. This result demonstrates how joint representations allow learning of stable long-term dynamics that enable successful model-based RL. In the occlusion setting, the contrastive predictive approach also performs best, while the image-only approaches cannot compete with the proprioception-only baseline. However, for contrastive image losses, all model-based approaches perform worse than their model-free counterparts, indicating that the dynamics can still be improved. \emph{TIA} and \emph{DenoisedMDP} fail in both our natural video and occlusion tasks because they explicitly model the natural videos and occlusions. The discrepancy in performance reported in the original works for these approaches on natural videos is due to changes in the experimental setup. We refer to \autoref{supp:sec:baselines} for details. \autoref{supp:sec:add_res} provides the full experimental results of all methods on all image types and for the individual environments. Finally, we further investigate some of the learned representations in \autoref{fig:quali_vis}, which illustrates how joint representation learning can guide the approaches to focus on relevant aspects of an environment while not being distracted by irrelevant noise.
\subsection{Locomotion Suite}
\begin{figure}
\caption{Aggregated results for all environments of the locomotion suite. Learning joint representations gives the best results with all approaches for representation learning.
The joint representations clearly outperform concatenating image representations and proprioception, as well as \emph{DRQ-v2} and \emph{DRQ-v2 (I+P)}.
Combining a contrastive loss for the image with reconstruction for the proprioception is better than solely relying on contrastive losses.
On these tasks, the contrastive variational approach again outperforms contrastive predictive coding and, in terms of sample efficiency, even the reconstruction.}
\label{fig:loco_res}
\end{figure}
We propose a novel benchmark consisting of six locomotion tasks. For all tasks, we use proprioception and egocentric images. The agents need the proprioception to be aware of their own state, as they cannot observe themselves from the egocentric perspective. Moreover, the agents require egocentric images to avoid obstacles whose position and size are only available through the image. Three tasks are readily available with the DeepMind Control Suite \cite{tassa2018deepmind} and the software stack introduced in \cite{Tassa2020dm_control}. We designed three more by introducing hurdles into the \texttt{Cheetah Run}, \texttt{Walker Walk}, and \texttt{Walker Run} tasks. In these new environments, agents need to identify the location and size of ''hurdles'' placed in their way in order to jump over them while moving forward. \autoref{fig:fig1} shows examples for the modified \texttt{Cheetah Run}. \autoref{supp:sec:loco_envs} provides details about all environments. We focus our evaluation on model-free approaches, as those outperformed the model-based ones in the previous experiment. The results in \autoref{fig:loco_res} support the previous findings that using joint representations improves over concatenating image representations with proprioception. Again, approaches that combine contrastive and reconstruction losses outperform those using only contrastive losses. These results further demonstrate the benefits of learning joint representations, even in tasks where distractions play no role.
\section{Conclusion} We consider the problem of Reinforcement Learning (RL) from multiple sensors, in particular images and proprioception. Building on \emph{Recurrent State Space Models}~\cite{hafner2019learning}, we learn joint representations of all available sensors instead of considering them in isolation. We propose combining contrastive and reconstruction approaches considering variational and predictive coding paradigms for training In our large-scale evaluation, we show the benefits of this approach for both model-free and model-based RL. For model-free RL, joint representations outperform the concatenation of image representations and proprioception. Using the proprioception in representation learning can guide the approach to focus on and capture all relevant aspects of the images, allowing to solve tasks where current image-only approaches fail. For model-based agents, our joint representations offer an easy and highly effective alternative to improve performance for tasks that require contrastive image objectives.
\textbf{Limitations.} While we showed the benefits of joint representations in both cases, our evaluation is inconclusive about whether the variational or predictive coding approach is preferable. Additionally, even with a combination of contrastive and reconstruction losses, model-free agents perform better than their model-based counterparts. As both types of agents perform equally well for pure reconstruction, this suggests there is still room for improvement in contrastive \emph{RSSM} training, especially w.r.t. learning the dynamics. Furthermore, our current approach assumes all modalities are available for all sequences and cannot exploit large amounts of data available for individual sensors, e.g., images. Studying how to include such data or existing image representations~\cite{parisi2022unsurprising, xiao2022masked, nairr3m, majumdar2023we} into our joint representation learning framework is a promising avenue for future research.
\textbf{Broader Impact.} While we do not foresee any immediate negative societal impacts of our work, improved representations for RL might ultimately lead to more capable autonomous systems, which can have both positive and negative effects. We believe identifying and mitigating the potentially harmful effects of such autonomous systems is the responsibility of sovereign governments.
\appendix
\section{Environments and Baselines} \label{sup:sec:env}
\begin{table}[t]
\centering
\caption{Splits of the entire system state into proprioceptive and non-proprioceptive parts for the DeepMind Control Suite environments.
}
\resizebox{\textwidth}{!}{
\begin{tabular}{lcc}
\toprule
Environment & Proprioceptive & Non-Proprioceptive \\
\midrule
Ball In Cup & cup position and velocity & ball position and velocity \\
Cartpole & cart position and velocity & pole angle and velocity \\
Cheetah & joint positions and velocities & global pose and velocity \\
Reacher & reacher position and velocity & distance to target \\
Quadruped & joint positions and velocities & global pose + velocity, forces \\
Walker & orientations and velocities of links & global pose and velocity, height above ground\\
\bottomrule
\end{tabular}
}
\label{sup:table:proprioception_dmc} \end{table}
\begin{table}[t]
\centering
\caption{Splits of the entire system state into proprioceptive and non-proprioceptive parts for the Locomotion Suite. Some of the agents (Cheetah, Walker, Quadruped) require more proprioceptive information for the locomotion tasks with an egocentric vision than for the standard tasks with images from an external perspective.
}
\begin{tabular}{lcc}
\toprule
Environment & Proprioceptive & Non-Proprioceptive \\
\midrule
Ant & joint position and velocity & wall positions \\
& global velocities & global position\\ Hurdle Cheetah & joint positions and velocities & hurdle positions and heights \\
& global velocity & global position\\ Hurdle Walker & orientations and velocities of links & hurdle positions and height \\
& & global position and velocity \\
Quadruped (Escape) & joint positions and velocities, & Information about terrain \\
& torso orientation and velocity, & \\
& imu, forces, and torques at joints & \\
\bottomrule
\end{tabular}
\label{sup:table:proprioception_loco} \end{table}
\subsection{DeepMind Control Suite Tasks} \autoref{sup:table:proprioception_dmc} states how we split the states of the original DeepMind Control Suite \cite{tassa2018deepmind} tasks into proprioceptive and non-proprioceptive parts. For the model-based agents, we followed common practice \cite{hafner2019dream, fu2021tia, wang2022denoised, deng2022dreamerpro} and use an action repeat of $2$ for all environments. We do the same for the model-free agents except for: \texttt{Ball In Cup Catch} (4), \texttt{Cartpole Swingup} (8), \texttt{Cheetah Run} (4) and \texttt{Reacher Easy} (4). All environments in the locomotion suite also use an action repeat of $2$, this includes \texttt{Hurdle Cheetah Run} which requires more fine-grained control than the normal version to avoid the hurdles.
\paragraph{Natural Background.} \label{supp:sec:nat_back} Following \cite{nguyen2021tpc, deng2022dreamerpro, zhang2020learning, fu2021tia, wang2022denoised} we render videos from the \texttt{driving car} class of the Kinetics400 dataset \cite{kay2017kinetics} behind the agents to add a natural video background. However, the previously mentioned works implement this idea in two distinct ways. \citep{nguyen2021tpc} and \citep{deng2022dreamerpro} use color images as background and pick a random sub-sequence of a random video for each environment rollout. They adhere to the train-validation split of the Kinetcs400 dataset, using training videos for representation and policy learning and validation videos during evaluation. \citep{zhang2020learning, fu2021tia, wang2022denoised}, according to the official implementations, instead work with gray-scale images and sample a single background video for the train set once during initialization of the environment. They do not sample a new video during the environment reset, thus all training sequences have the same background video. We follow the first approach, as we believe it mimics a more realistic scenario of always changing and colored natural background.
\paragraph{Occlusions.} Following \cite{becker2022uncertainty}, we rendered slow-moving disks over the original observations to occlude parts of the observation. The speed of the disks makes memory necessary, as they can occlude relevant aspects for multiple consecutive timesteps.
\subsection{Locomotion Suite} \label{supp:sec:loco_envs}
\begin{figure}
\caption{The environments in the Locomotion Suite are (from left to right) Hurdle Cheetah Run, Hurdle Walker Walk / Run, Ant Empty, Ant Walls, and Quadruped Escape. \textbf{Upper Row:} Egocentric vision provided to the agent. \textbf{Lower Row:} External image for visualization.}
\label{sup:fig:all_loco_vis}
\end{figure}
The $6$ tasks in the locomotion suite are \texttt{Ant Empty}, \texttt{Ant Walls}, \texttt{Hurdle Cheetah Run}, \texttt{Hurdle Walker Walk},\texttt{Hurdle Walker Run}, and \texttt{Quadruped Escape}. \autoref{sup:table:proprioception_loco} shows the splits into proprioceptive and non-proprioceptive parts. \autoref{sup:fig:all_loco_vis} displays all environments in the suite.
\paragraph{Ant.} The Ant tasks build on the locomotion functionality introduced into the DeepMind Control suite by \cite{Tassa2020dm_control}. For Ant Empty, we only use an empty corridor, which makes this the easiest task in our locomotion suite. For Ant Walls, we randomly generate walls inside the corridor, and the agent has to avoid those in order to achieve its goal, i.e., running through the corridor as fast as possible.
\paragraph{Hurdle Cheetah \& Walker.} We modified the standard Cheetah Run, Walker Walk, and Walker Run tasks by introducing "hurdles" over which the agent has to step in order to move forward. The hurdles' positions, heights, and colors are reset randomly for each episode, and the agent has to perceive them using egocentric vision. For this vision, we added a camera in the head of the Cheetah and Walker.
\paragraph{Quadruped Escape.} The Quadruped Escape task is readily available in the DeepMind Control Suite. For the egocentric vision, we removed the range-finding sensors from the original observation and added an egocentric camera.
\subsection{Baselines.} \label{supp:sec:baselines} For \emph{Dreamer-V2}\cite{hafner2020mastering} we use the raw reward curve data provided with the official implementation\footnote{\url{https://github.com/danijar/dreamerv2/blob/main/scores/dmc-vision-dreamerv2.json}}. For \emph{DreamerPro}\cite{deng2022dreamerpro}\footnote{\url{https://github.com/fdeng18/dreamer-pro}}, \emph{TIA}\cite{fu2021tia}\footnote{\url{https://github.com/kyonofx/tia/}}, \emph{DenoisedMDP}\cite{wang2022denoised}\footnote{\url{https://github.com/facebookresearch/denoised_mdp}} and \emph{DrQv2}\cite{yarats2022mastering}\footnote{\url{https://github.com/facebookresearch/drqv2}} we use the official implementations provided by the respective authors.
\textbf{\emph{DrQ(I+P)}} builds on the official implementation and uses a separate encoder for the proprioception whose output is concatenated to the image encoders' output and trained using the critics' gradients.
\textbf{Differences between Model-Based \emph{Img-Only(R)} and Dreamer-v1\cite{hafner2019dream}.} \emph{Img-Only(R)} differs from the original Dreamer (Dreamer-v1)~\cite{hafner2019dream} in using the KL-balancing introduced in \cite{hafner2020mastering} and in regularizing the value function towards its own exponential moving average, as introduced in~\cite{hafner2023mastering}. See \autoref{sup:sec:obj} for all our training details and hyperparameters.
There are considerable differences between the contrastive version of Dreamer-v1\cite{hafner2019dream} and \emph{Img-Only(CV)}, in particular regarding the exact form of the mutual information estimation and the use of image augmentations.
\textbf{Differences between Model-Free \emph{Img-Only(CPC)}, \emph{FC-Joint(CPC)} and \cite{srivastava2021core}.} The main difference is that \cite{srivastava2021core} includes the critic's gradients when updating the representation while in our setting no gradients flow from the actor or the critic to the representation. Additionally, we adapted some hyperparameters to match those of our other approaches. The results are based on our implementation, not the official implementation of \cite{srivastava2021core}.
\textbf{Why TIA and DenoisedMDP Fail in Our Setting.} Both TIA \cite{fu2021tia} and DenoisedMDP \cite{wang2022denoised} fail to perform well in the natural video background setting used in this work and \cite{deng2022dreamerpro}, described in \autoref{supp:sec:nat_back}. Both approaches factorize the latent variable into $2$ distinct parts and formulate loss functions that force one part to focus on task-relevant aspects and the other on task-irrelevant aspects. However, the part responsible for the task-irrelevant aspects still has to model those explicitly. Recall the differences in adding the natural video backgrounds described in \autoref{sup:sec:env}. In this more complicated setting with randomly sampled, colored background videos, the \emph{TIA} and \emph{DenoisedMDP} world models underfit and thus fail to learn a good representation or policy. Contrastive approaches, such as our approach and DreamerPro \cite{deng2022dreamerpro}, do not struggle with this issue, as they do not have to model task-irrelevant aspects but can learn to ignore them.
\section{Training and Architecture Details} \label{sup:sec:obj} We used the same hyperparameters in all experiments with the exception of the Box Pushing. For this task, we increased the number of free nats for the state space representation (see below) from $1$ to $3$. \label{sup:sec:hps} \subsection{\emph{Recurrent State Space Model}} We denote the deterministic part of the \emph{RSSM}'s state by $\cvec{h}_t$ and the stochastic part by $\cvec{s}_t$. The base-\emph{RSSM} model without parts specific to one of the objectives consists of: \begin{itemize}
\item \textbf{Encoders:} $\psi_{\textrm{obs}}^{(k)}(\cvec{o}_t)$, where $\psi_{\textrm{obs}}$ is the convolutional architecture proposed by \cite{ha2018world} and used by \cite{hafner2019learning, hafner2019dream} for image observations and a $3 \times 400$ Units fully connected NN with ELU activation for proprioceptive observations (, i.e., vectors).
\item \textbf{Deterministic Path}: $\cvec{h}_t = g(\cvec{z}_{t-1}, \cvec{a}_{t-1}, \cvec{h}_{t-1}) = \textrm{GRU}(\psi_{\textrm{det}} (\cvec{z}_{t-1}, \cvec{a}_{t-1}), \cvec{h}_{t-1})$ \cite{cho2014gru}, where $\psi_{\textrm{det}}$ is $2 \times 400$ units fully connected NN with ELU activation and the GRU has a memory size of $200$.
\item \textbf{Dynamics Model}: $p(\cvec{z}_{t+1} | \cvec{z}_t, \cvec{a}_t) = \psi_{\textrm{dyn}}(\cvec{h}_t)$, where $\psi_{\textrm{dyn}}$ is a $2 \times 400$ units fully connected NN with ELU activation. The network learns the mean and standard deviation of the distribution.
\item \textbf{Variational Distribution} $q(\cvec{z}_t | \cvec{z}_{t-1}, \cvec{a}_{t-1}, \cvec{o}_t) = \psi_{\textrm{var}}\left( \cvec{h}_t, \textrm{Concat}\left(\lbrace{\psi^{(k)}_{\textrm{obs}}(\cvec{o}^{(k)}_t) \rbrace}_{k=1:K}\right) \right)$, where $\psi_{\textrm{var}}$ is a $2 \times 400$ units fully connected NN with ELU activation.
The network learns the mean and standard deviation of the distribution.
\item \textbf{Reward Predictor} $p(r_t | \cvec{z}_t)$: $2 \times 128$ units fully connected NN with ELU activation for model-free agents. $3 \times 300$ units fully connected NN with ELU activation for model-based agents. The network only learns the mean of the distribution. The standard deviation is fixed at $1$. The model-based agents use a larger reward predictor as they rely on it for learning the policy and the value function. Model-free agents use the reward predictor only for representation learning and work with the ground truth rewards from the replay buffer to learn the critic. \end{itemize}
\subsection{Objectives} \textbf{Image Inputs and Preprocessing.} For the reconstruction objective, we used images of size $64 \times 64$ pixels as input to the model. For the contrastive objectives, the images are of size $76 \times 76$ pixel image and we used $64 \times 64$ pixel random crops. Cropping is temporally consistent, i.e., we used the same crop for all time steps in a sub-sequence. For evaluation, we took the crop from the center.
\textbf{Reconstruction Objectives.} Whenever we reconstructed images we used the up-convolutional architecture proposed by \cite{ha2018world} and used by \cite{hafner2019learning, hafner2019dream}. For low-dimensional observations, we used $3 \times 400$ units fully connected NN with ELU activation. In all cases, only the mean is learned and the standard deviation is fixed at $1$.
For the KL terms in \autoref{eq:recon_bound} and \autoref{eq:cv_bound} we follow~\cite{hafner2023mastering} and combine the KL-Balancing technique introduced in~\cite{hafner2020mastering} with the \emph{free-nats regularization} used in~\cite{hafner2019learning, hafner2019dream}. Following~\cite{hafner2020mastering} we use a balancing factor of $0.8$. We give the algorithm $1$ free nat for the DeepMind Control Suite and the Locomotion Suite tasks and $3$ for the Box Pushing.
\textbf{Contrastive Variational Objective.} The score function for the contrastive variational objective is given as $$f_v^{(k)}(\cvec{o}^{(k)}_t, \cvec{z}_t) = \exp \left( \frac{1}{\lambda} \rho_o \left( \psi_{\textrm{obs}}^{(k)}(\cvec{o}_t)\right)^T\rho_z(\cvec{z}_t) \right),$$ where $\psi_{\textrm{obs}}^{(k)}$ is the \emph{RSSM}'s encoder and $\lambda$ is a learnable inverse temperature parameter. $\rho_o$ and $\rho_z$ are projections that project the embedded observation and latent state to the same dimension, i.e., $50$. $\rho_o$ is only a single linear layer while $\rho_z$ is a $2 \times 256$ fully connected NN with ELU activation. Both use LayerNorm \cite{ba2016layer} at the output.
\textbf{Contrastive Predictive Objective.} The score function of the contrastive predictive objective looks similar to the one of the contrastive variational objective. The only difference is that the latent state is forwarded in time using the \emph{RSSMs} transition model to account for the predictive nature of the objective, $$f_p^{(k)}(\cvec{o}^{(k)}_{t+1}, \cvec{z}_t) = \exp \left( \frac{1}{\lambda} \rho_o \left( \psi_{\textrm{obs}}^{(k)}(\cvec{o}_{t+1})\right)^T\rho_z(\phi_{\textrm{dyn}}(g(\cvec{z}_t, \cdot)) \right).$$ We use the same projections as in the contrastive variational case.
Following~\cite{srivastava2021core} we scale the KL term using a factor of $\beta=0.001$ and parameterize the inverse dynamics predictor as a $2 \times 128$ unit fully connected NN with ELU activations.
\textbf{Optimizer.} We used Adam~\cite{kingma2015adam} with $ \alpha =3 \times 10^{-4}$, $\beta_1 = 0.99, \beta_2 = 0.9$ and $\varepsilon = 10^{-8}$ for all losses. We clip gradients if the norm exceeds $10$.
\subsection{Soft Actor Critic} \begin{table}[t] \caption{Hyperparameters used for policy learning with the Soft Actor-Critic.} \centering \begin{tabular}{lc} \toprule Hyperparameter & Value \\ \midrule Actor Hidden Layers & $ 3 \times 1,024 $ Units \\ Actor Activation & ELU \\ Critic Hidden Layers & $ 3 \times 1,024 $ Units\\ Critic Activation & ELU \\ \midrule Discount & $0.99$ \\ \midrule Actor Learning Rate & $0.001$ \\ Actor Gradient Clip Norm & $10$ \\ Critic Learning Rate & $0.001$ \\ Critic Gradient Clip Norm & $100$ \\ \midrule Target Critic Decay & $0.995$ \\ Target Critic Update Interval & $1$ \\ \midrule
$\alpha$ learning rate & $0.001$ \\ initial $\alpha$ & $0.1$ \\ target entropy &- action dim \\ \bottomrule \end{tabular}
\label{sup:table:sac_hps} \end{table}
\autoref{sup:table:sac_hps} lists the hyperparameters used for model-free RL with SAC~\cite{haarnoja2018sac}.
We collected $5$ initial sequences at random. During training, we update the \emph{RSSM}, critic, and actor in an alternating fashion for $d$ steps before collecting a new sequence by directly sampling from the maximum entropy policy. Here, $d$ is set to be half of the environment steps collected per sequence (after accounting for potential action repeats). Each step uses $32$ subsequences of length $32$, uniformly sampled from all prior experience.
\subsection{Latent Imagination}
\begin{table}[t] \caption{Hyperparameters used for policy learning with \emph{Latent Imagination.}} \centering \begin{tabular}{lc} \toprule Hyperparameter & Value \\ \midrule Actor Hidden Layers & $ 3 \times 300 $ Units \\ Actor Activation & ELU \\ Critic Hidden Layers & $ 3 \times 300 $ Units\\ Critic Activation & ELU \\ \midrule Discount & $0.99$ \\ \midrule Actor Learning Rate & $8 \times 10^{-5}$ \\ Actor Gradient Clip Norm & $100$ \\ \midrule Value Function Learning Rate & $8 \times 10^{-5}$ \\ Value Gradient Clip Norm & $100$ \\ Slow Value Decay & $0.98$ \\ Slow Value Update Interval & $1$ \\ Slow Value Regularizer & $1$ \\ \midrule
Imagination Horizon & $15$ \\ \midrule Return lambda & $0.95$\\ \bottomrule \end{tabular}
\label{sup:table:li_hps} \end{table}
\autoref{sup:table:li_hps} lists the hyperparameters used for model-based RL with latent imagination. They follow to a large extent those used in \cite{hafner2019dream, hafner2020mastering}.
We collected $5$ initial sequences at random. During training, we update the \emph{RSSM}, value function, and actor in an alternating fashion for 100 steps before collecting a new sequence. Each step uses $50$ subsequences of length $50$, uniformly sampled from all prior experience. For collecting new data, we use constant Gaussian exploration noise with $\sigma=0.3$.
\subsection{Compute Resources} Training a single agent takes between 8 and 12 hours on a single GPU (Nvidia V100 or A100), depending on which representation learning approach and RL paradigm we use. Approaches using a contrastive loss for the image are slightly faster than those that reconstruct the image as they do not have to run the relatively large up-convolutional image decoder. The model-free agents train slightly faster than the model-based ones, as the model-based ones have to predict several steps into the future for latent imagination. Especially propagating gradient back through this unrolling is relatively costly compared to a SAC update. Including all baselines, we trained about $3,500$ agents for the final evaluation. Also including preliminary experiments, we estimate the total compute resources invested in this work to be about $50,000$ GPU hours. \section{Complete Results} \label{supp:sec:add_res} \subsection{Model-Free Agents}
Here we list the aggregated results for all methods with model-free agents on the DeepMind Control (DMC) Suite with standard images (\autoref{fig:sac_clean_agg}), natural video background (\autoref{fig:sac_distr_agg}), and occlusions (\autoref{fig:sac_occ_agg}).
\begin{figure}
\caption{Aggregated results for \textbf{model-free} agents on the DMC Suite tasks with standard images.
(\emph{Img-Only(R)}) achieves similar performance to the model-free \emph{DrQ-v2}, but also to \emph{Dreamer-v2} and \emph{DreamerPro} (see \autoref{fig:li_clean_agg}), showing that our general representation learning approach also works without multiple sensors.
However, using both images and proprioception is still advantageous in these tasks, and using joint representations leads to larger performance gains.}
\label{fig:sac_clean_agg}
\end{figure}
\begin{figure}\label{fig:sac_distr_agg}
\end{figure}
\begin{figure}
\caption{Aggregated results for \textbf{model-free} agents on the DMC Suite tasks with occlusion. In this task, no method that uses only images or reconstructs images performs better than solely using proprioception.
However, combining both modalities and using contrastive methods for the images leads to vast performance improvements.
While, at least for contrastive predictive coding, these improvements are already significant for the concatenation approach, joint representation learning gives even more performance.
In this task, the contrastive predictive coding methods outperform contrastive variational ones arguably because their explicit ahead prediction allows them to learn better dynamics. }
\label{fig:sac_occ_agg}
\end{figure}
\subsection{Model-Based Agents}
Here we list the aggregated results for all methods with model-based agents on the DeepMind Control (DMC) Suite with standard images (\autoref{fig:li_clean_agg}), natural video background (\autoref{fig:li_nat_agg}), and occlusions (\autoref{fig:li_occ_agg}). \begin{figure}
\caption{Aggregated results for \textbf{model-based} agents on the DMC Suite tasks with standard images. Unsurprisingly all reconstruction-based approaches obtain good performance.
Notably, contrastive variational approaches perform better than constructive predictive ones with model-based agents, while they perform equally well with model-free agents.
However, in both cases, a combination of contrastive and reconstruction-based losses outperforms the purely contrastive approaches.
}
\label{fig:li_clean_agg}
\end{figure}
\begin{figure}
\caption{Aggregated results for \textbf{model-based} agents on the DMC Suite Tasks with natural video background.
Again, all reconstruction-based approaches predictably fail.
Furthermore, the contrastive variational approach again seems more suited for model-based agents than the contrastive predictive one, and the combination of contrastive and reconstruction-based losses performs best.}
\label{fig:li_nat_agg}
\end{figure}
\begin{figure}
\caption{Aggregated results for \textbf{model-based} agents on the DMC Suite Tasks with occlusions. \emph{Joint(CPC)} performs best among all approaches and almost achieves the same performance as on the simpler tasks with the standard images and natural video background.
However, model-based \emph{Joint(CPC)} is still significantly worse than its model-free counterpart.}
\label{fig:li_occ_agg}
\end{figure}
\subsection{Results per Environment - Model Free} The following figures provide the per environment results for model-free agents on the DMC Suite with standard images (\autoref{fig:all_clean_sac}), model-free agents on the DMC Suite with natural video background (\autoref{fig:all_nat_sac}), model-free agents on the DMC Suite with occlusion (\autoref{fig:all_occ_sac}),
model-based agents on the DMC Suite with standard images (\autoref{fig:all_clean_li}), model-based agents on the DMC Suite with natural video background (\autoref{fig:all_nat_li}), model-based agents on the DMC Suite with occlusion (\autoref{fig:all_nat_li}), and for the Locomotion Suite (\autoref{fig:loco_sac}). We again report interquartile means and 95\% bootstrapped confidence intervals \cite{agarwal2021deep}.
\begin{figure}
\caption{Per environment results for \textbf{model-free} agents on the DMC Suite with standard images.}
\label{fig:all_clean_sac}
\end{figure}
\begin{figure}
\caption{Per environment results for \textbf{model-free} agents on the DMC Suite with natural image background.}
\label{fig:all_nat_sac}
\end{figure}
\begin{figure}
\caption{Per environment results for \textbf{model-free} agents on the DMC Suite with occlusion.}
\label{fig:all_occ_sac}
\end{figure}
\begin{figure}
\caption{Per environment results for \textbf{model-based} agents on the DMC Suite with standard images.}
\label{fig:all_clean_li}
\end{figure}
\begin{figure}
\caption{Per environment results for \textbf{model-based} agents on the DMC Suite with natural video background.}
\label{fig:all_nat_li}
\end{figure}
\begin{figure}
\caption{Per environment results for \textbf{model-based} agents on the DMC Suite with occlusions.}
\label{fig:all_occ_li}
\end{figure}
\begin{figure}
\caption{Per environment results for \textbf{model-free} agents on the Locomotion Suite.}
\label{fig:loco_sac}
\end{figure}
\end{document} | arXiv |
Probabilistic volcanic hazard assessment at an active but under-monitored volcano: Ceboruco, Mexico
Robert Constantinescu ORCID: orcid.org/0000-0002-1532-904X1,2,
Karime González-Zuccolotto3,
Dolors Ferrés4,
Katrin Sieron5,
Claus Siebe2,
Charles Connor1,
Lucia Capra6 &
Roberto Tonini7
Journal of Applied Volcanology volume 11, Article number: 11 (2022) Cite this article
A probabilistic volcanic hazard assessment (PVHA) for Ceboruco volcano (Mexico) is reported using PyBetVH, an e-tool based on the Bayesian Event Tree (BET) methodology. Like many volcanoes, Ceboruco is under-monitored. Despite several eruptions in the late Holocene and efforts by several university and government groups to create and sustain a monitoring network, this active volcano is monitored intermittently rather than continuously by dedicated groups. With no consistent monitoring data available, we look at the geology and the eruptive history to inform prior models used in the PVHA. We estimate the probability of a magmatic eruption within the next time window (1 year) of ~ 0.002. We show how the BET creates higher probabilities in the absence of monitoring data, which if available would better inform the prior distribution. That is, there is a cost in terms of higher probabilities and higher uncertainties for having not yet developed a sustained volcano monitoring network. Next, three scenarios are developed for magmatic eruptions: i) small magnitude (effusive/explosive), ii) medium magnitude (Vulcanian/sub-Plinian) and iii) large magnitude (Plinian). These scenarios are inferred from the Holocene history of the volcano, with their related hazardous phenomena: ballistics, tephra fallout, pyroclastic density currents, lahars and lava flows. We present absolute probability maps (unconditional in terms of eruption size and vent location) for a magmatic eruption at Ceboruco volcano. With PyBetVH we estimate and visualize the uncertainties associated with each probability map. Our intent is that probability maps and uncertainties will be useful to local authorities who need to understand the hazard when considering the development of long-term urban and land-use planning and short-term crisis management strategies, and to the scientific community in their efforts to sustain monitoring of this active volcano.
Recent volcanic eruptions have resulted in a significant number of fatalities, evacuations and economic losses, for example at La Palma, Spain in 2021 (Global Volcanism Program 2021), La Soufrière, St. Vincent and the Grenadines in 2021 (Global Volcanism Program 2021), White Island, New Zealand in 2019 (Park et al. 2020; Cao et al. 2020); Fuego, Guatemala in 2018 (Albino et al. 2020; Naismith et al. 2019); Hawaii, U.S.A. in 2018 (Feng et al. 2020; Neal et al. 2019); Anak Krakatoa, Indonesia in 2018 (Heidarzadeh et al. 2020; Grilli et al. 2019); Sinabung, Indonesia in 2013–2018 (Andreastuti et al. 2019; Gunawan et al. 2019); Ontake, Japan in 2014 (Maeno et al. 2016) and Eyjafjallajökull, Iceland in 2010 (Langmann et al. 2012). It is estimated that > 500 million people live in areas with volcanic risk (e.g., Martí 2017; Auker et al. 2013) and large volcanic eruptions affect our interconnected world in numerous ways. Volcanic eruptions create multiple hazardous phenomena (e.g., ballistics, pyroclastic density currents, tephra fallout) with impacts at different scales (e.g., local, regional, and global), placing a premium on hazard assessments at various map scales (e.g., Chester et al. 2002).
A useful volcanic risk assessment, from local to global scales, relies on probabilistic volcanic hazard assessment (PVHA). Any volcanic hazard assessment employs diverse techniques to generate different types of volcanic hazard maps. These maps commonly depict the areas that can be affected by one or more hazardous volcanic phenomena and are based upon the distribution of deposits from past eruptions, modern topography, and/or the results of computer modeling of a volcanic phenomenon (Ang et al. 2020; Martí 2017; Loughlin et al. 2015; Calder et al. 2015). From a practical standpoint, a deterministic hazard analysis is often scenario-based and assumes that an eruption of some type and magnitude will occur within some specific period of time (e.g., Tierz 2020; Calder et al. 2015; Connor et al. 2015). A probabilistic analysis considers the probability of eruptions occurring within some time interval and their likely characteristics, including range of eruption styles, eruption sizes, and eruption source parameters (e.g., Bertin et al. 2022; Poland and Anderson 2020; Tierz et al. 2016; Connor et al. 2015). PVHA can merge information derived from the record of past activity, theoretical models and numerical modeling of volcanic phenomena. The advantage of this approach is the ability to provide a quantitative way of understanding the hazard posed by a volcanic phenomenon and to account for the uncertainties stemming from the natural variability of the phenomena (aleatoric uncertainty) or our limited knowledge about the physical processes (epistemic uncertainty) (Tierz et al. 2020; Ang et al. 2020; Martí 2017; Tierz et al. 2016; Loughlin et al. 2015; Connor et al. 2015; Calder et al. 2015; Marzocchi and Bebbington 2012). The resulting probability maps are a type of volcanic hazard map that show the probability that a given location will be affected by a volcanic phenomenon. Importantly, these maps include the associated uncertainties, therefore differing substantially from the deterministic-approach maps. Probabilistic maps represent either a conditional probability (i.e., probability that a location of interest will be affected by a volcanic phenomenon given that an eruption occurs, perhaps sampling a range of eruption styles or sizes) or the absolute probability (unconditional in terms of eruption probability, size, style and vent location) (e.g., Poland and Anderson 2020; Rouwet et al. 2019; Tierz et al. 2016; Calder et al. 2015; Connor et al. 2015; Tonini et al. 2015; Marzocchi et al. 2010). A PVHA is therefore often prefered to a deterministic analysis, although deterministic scenarios are definitely useful in some circumstances, such as illustrating potential eruption outcomes.
PVHA e-tools have been developed to estimate and visualize the probabilities that an area will be affected by a volcanic event and the associated uncertainties (e.g., Bertin et al. 2019; Bartolini et al. 2019; Sobradelo et al. 2013; Marzocchi et al. 2010). PVHA serves to bridge the volcanological and societal aspects of a volcanic crisis and ideally can assist authorities in real-time decision-making (e.g., Bartolini et al. 2019; Papale 2017; Newhall and Pallister 2015; Marzocchi and Bebbington 2012). These tools are mostly based on Event Trees (ET) or Bayesian Belief Networks (BBN) structures. An ET is a directed graph (nodes and branches) (Fig. 1) of progressive events, from an initial state, through subsequent stages, to final outcomes (Connor et al. 2001; Newhall and Hoblitt 2002). The final outcomes are often referred to as contingencies, and the idea of the ET is to capture all possible contingencies, unless otherwise stated. This unidirectional structure allows probabilities for sequences of events in volcanic activity to be calculated in a Bayesian framework. By updating the ET with different sources of data, including eruptive history (past data) and theoretical or mathematical models (prior models), it becomes a Bayesian Event Tree (BET), and the probability of specific outcomes is estimated in a Bayesian way (e.g., Marzocchi et al. 2008). Similar to BETs, Bayesian Belief Networks (BBN) are graphical structures representing different events related to volcanic activity. Unlike BETs, BBNs describe the complexity of this activity as variable nodes interlinked by branches representing the causality between them (Christophersen et al. 2018; Tierz et al. 2017; Hincks et al. 2014; Aspinall and Woo 2014; Aspinall et al. 2003). The estimation of probabilities at each node in the BET or BBN schemes is done by implementing a computer algorithm based on Bayes' rule, which allows update of the output as new information becomes available (Christophersen et al. 2018; Marzocchi et al. 2008; Aspinall et al. 2003). Both ETs and BBNs have been used to forecast or hindcast eruptions for several volcanoes: Aluto, Ethiopia (Tierz et al. 2020); White Island, New Zealand (Christophersen et al. 2018); La Soufrière, Guadeloupe (Hincks et al. 2014); Galeras, Colombia (Aspinall et al. 2003); Santorini, Greece (Aspinall and Woo 2014); Etna and Somma-Vesuvius, Italy (Tierz et al. 2017; Cannavò et al. 2017); St. Helens, U.S.A (Newhall 1982); Soufrière Hills, Montserrat (Aspinall and Cooke 1998), Pinatubo, Philippines (Punongbayan et al. 1996) and others.
Graphical representation of the Ceboruco event tree (ET). The example illustrated here is the event tree developed for Ceboruco based on prior models and past data. We identify three potential causes of unrest at Ceboruco: new magma entering the system, a large tectonic earthquake in the graben or an increased degassing episode associated with non-magmatic changes in the hydrothermal system. Each unrest episode has associated potentially hazardous phenomena. For example, we consider that rockslides and debris flows can occur at Ceboruco even if the volcano is not in unrest due to the instability of the volcano's edifice and/or due to heavy rainfall. The highlighted branch and associated nodes (gray boxes) represent the sequence of events attributed to a potential magmatic unrest that we assess with PyBetVH. Note that some of these phenomena have relatively high probability in eruptive scenarios while others (e.g., debris avalanche) have relatively low probability. The ET attempts to account for all contingencies
Two of the most commonly used e-tools in long-term PVHA are HASSET (Hazard Assessment Event Tree) (Sobradelo et al. 2013) and BET_VH (Bayesian Event Tree for Volcanic Hazard) (Marzocchi et al. 2010). Both are software implementations of the volcano eruption ET structure (Marzocchi et al. 2004; Newhall and Hoblitt 2002). HASSET and BET_VH have been applied for hazard evaluation at different volcanoes such as San Miguel, El Salvador (Jiménez et al. 2018), Okataina, New Zealand (Thompson et al. 2015), Deception Island, Antarctica (Bartolini et al. 2014), El Hierro, Canary Islands (Becerril et al. 2014), El Misti, Peru (Sandri et al. 2014), Somma-Vesuvius and Campi Flegrei, Italy (Sandri et al. 2019; Tierz et al. 2018; Selva et al. 2014; 2010), Auckland Volcanic Field, New Zealand (Sandri et al. 2012) and Teide – Pico Viejo, Canary Islands (Martí et al. 2012; 2008).
Here, we use the PyBetVH software (e.g., Wild et al. 2019; Strehlow et al. 2017; Tonini et al. 2015), an updated version of BET_VH, to conduct a PVHA for Ceboruco volcano (Mexico). This PVHA is intended to update the currently available deterministic hazard maps for Ceboruco with probabilistic maps. We define a generic ET, that is, not calibrated by current monitoring or related observations, for Ceboruco. Like many volcanoes, these monitoring data are unavailable. The ET includes hazardous phenomena related to both magmatic and non-magmatic unrest (Fig. 1). We describe how PyBetVH is used to produce probability maps for the magmatic branch by merging information from the geological record (i.e., past data), numerical simulations and expert-based weighting of the data (i.e., prior models). The strength of the PVHA analysis and resulting maps is that the numerous sources of uncertainty are incorporated in the analysis therefore being fundamentally different to the deterministic maps produced by Sieron et al. (2019b).
An important reason to develop an ET for Ceboruco volcano is that the volcano has been quite active within the last ~ 1000 yrs. but has not erupted during the last century. Because of this recent hiatus in activity, Ceboruco is less monitored than other active volcanoes in Mexico. We show through constructing the ET that this comparative under-monitoring increases uncertainty in probability of eruptions and probable eruption impacts. The resulting probability maps reflect this uncertainty. Consequently, in addition to estimating eruption probabilities, the Ceboruco ET can be used to consider the cost-benefit of additional monitoring efforts, and perhaps also to help justify them. As this situation exists at many volcanoes globally, this uncertainty quantification can be useful for global efforts to prioritize volcano monitoring more systematically (Ewert et al. 2005). Finally, we show how a PVHA is sensitive to the assumptions about volcanic unrest at volcanoes with limited or no monitoring networks and how these assumptions affect the resulting probabilities, with effects on decision-making. To our knowledge, it is the first time this issue has been addressed.
Eruptive history and hazard assessment at Ceboruco
Eruptive history
Located at the western edge of the Trans-Mexican Volcanic Belt (TMVB), Ceboruco (2280 m a.s.l.) (Fig. 2) is the only historically active stratovolcano among the ~ 28 monogenetic edifices of the San Pedro – Ceboruco half-graben (Sieron et al. 2019a; Petrone 2010; Sieron and Siebe 2008). Its eruptive history includes both effusive and explosive activity (Sieron et al. 2019a; Sieron and Siebe 2008; Gardner and Tait 2000). We refer the reader to Sieron et al. (2019a), Sieron and Siebe (2008), Gardner and Tait (2000) and Nelson (1980) for detailed descriptions of Ceboruco's geology and eruptive history. Here we summarize the activity of the last ~ 1000 years at Ceboruco as this time span represents the basis of our hazard assessment and mention that stratigraphic evidence indicates the volcano had a long period (~ 40 kyrs) of repose before that (e.g., Sieron et al. 2019a).
Location map of Ceboruco volcano (red volcano symbol) in the western part of the Trans-Mexican Volcanic Belt of Mexico. The main active volcanoes (orange) in the TMVB: 1 – Ceboruco; 2 – Colima; 3 – Parícutin; 4 – Nevado de Toluca; 5 – Popocatépetl; 6 – Pico de Orizaba; 7 – San Martín; 8 – El Chichón; 9 – Tacaná. The upper right inset: Digital Elevation Model (DEM) of the San – Pedro – Ceboruco half-graben with the location of Ceboruco (red) within the monogenetic cones (black volcano symbols). The main communities at risk from an eruption at Ceboruco are marked with blue stars: I – Ixtlán del Río (pop. ~ 33,000); A – Ahuacatlán (pop. ~ 9000); J – Jala (pop. ~ 16,000); M – Marquezado (pop. ~ 1000); U – Uzeta (pop. ~ 2000); C – Chapalilla (pop. ~ 1500) (map created using Generic Mapping Tools 6 (Wessel et al. 2019))
In the past ~ 1000 years Ceboruco experienced two main eruptive periods that comprised both effusive and explosive activity limited to the current edifice. We consider an eruptive period as characterized by more or less continuous eruptive activity, recognizable in the geological or historical records.
Stratigraphic evidence suggests that after a repose period of ~ 40 kyrs effusive activity preceded the caldera forming Jala eruption (VEI 6) (~ 990–1020 CE) that created the current summit caldera (~ 3.7 km diameter). The Jala eruption produced extensive tephra fallout and numerous pyroclastic density currents (PDC) with a total estimated eruption volume of 3–4 km3 dense rock equivalent (DRE) (Sieron et al. 2019a; Sieron and Siebe 2008; Gardner and Tait 2000; Nelson 1980). The PDC deposits formed from concentrated flows and dilute surges. Syn-and-post eruptive lahar deposits are identified as far as 35 km from the vent. Ballistic bombs are found within a 5 km radius from the vent. Effusive activity persisted for another ~ 150 years during which time several lava flows were emplaced (Böhnel et al. 2016). We consider this the first eruptive period. After a break of several hundred years, the second eruptive period started with small explosions in February 1870 and continued for 5 years with alternating effusive and explosive activity that generated lahars, tephra fallout and PDC. The total erupted volume of this eruption was estimated to be ~ 0.1 km3 for pyroclastic deposits and ~ 1.14 km3 for lava flows (Sieron et al. 2019a; Sieron and Siebe 2008). Today, small urban centers, national freeways, railroads and agricultural land are constructed on these eruption deposits (e.g., Marquezado, Ahuacatlán, Jala).
Deterministic Hazard assessment
Ceboruco is ranked the third most hazardous volcano in Mexico (Sieron et al. 2019a; Espinasa-Pereña 2018). Several urban centers (> 55,000 people), as well as important infrastructure lie within the area impacted by past activity (Sieron et al. 2019a, b). Sieron et al. (2019b) developed the first hazard maps for Ceboruco as part of a project funded by the Federal Electricity Commission that manages two hydropower plants in the area and the Tepic-Mazatlán sectorial power distribution station at the foothills of Ceboruco. Based on the activity of the last ~ 1000 years, three possible eruption scenarios were defined. We refer to Sieron et al. (2019a) for a full description of these scenarios. Summarizing:
Scenario 1 (S1) – this high likelihood scenario considers a small magnitude (i.e., VEI < 2), mainly effusive eruption that will generate andesitic lava flows. Small explosions will likely produce ballistics and low ash plumes from a central or flank vents (Sieron et al. 2019a, b).
Scenario 2 (S2) – is a medium likelihood event (i.e., VEI 2–3). This medium magnitude scenario considers both explosive and effusive activity. The explosive activity is expected to produce ballistics and to be of Vulcanian to sub-Plinian in style with small to moderate transient plumes, similar to the 1870-'75 CE eruption. Dome emplacement and lava flows are expected due to the high viscosity of the dacitic lava. Tephra fallout is expected and might reach a relatively large thickness near the vent (Sieron et al. 2019a, b). PDC and lahars are also expected.
Scenario 3 (S3) – a low likelihood, large magnitude eruption similar to the Jala Plinian eruption. This eruption is expected to produce ballistics and widespread tephra fallout and voluminous PDC. Syn-and-post eruptive lahars are expected due to the availability of pyroclastic material and water during the rainy season (Sieron et al. 2019a, b).
The hazard maps created by Sieron et al. (2019b) used these three scenarios to simulate individual hazardous phenomena. These computer models use various source parameters derived from previous research, field data and from analogue volcanoes (e.g., ballistics), to generate maps with the area impacted by each eruptive phenomenon. The resulting maps of each of these hazards were integrated into a single hazard map for each of the three scenarios (i.e., scenario-based hazard maps) (Figs. 8, 9 and 10 in Sieron et al. (2019b). A generalized hazard map for Ceboruco was obtained by integrating the three scenario-based hazard maps into a single map showing the areas that could be affected by various volcanic phenomena associated with a future eruption (Fig. 11 in Sieron et al. 2019b). In the following, we use the ET approach to assign probabilities to these eruptive scenarios and produce probabilistic hazard maps.
The event tree for PVHA at Ceboruco
We conduct a PVHA at Ceboruco volcano by designing an ET (Fig. 1) that considers one magmatic and two non-magmatic triggers for unrest. Unrest is a diverse and complex phenomenon described as a deviation from the background activity of a volcano or by anomalies in monitoring data (e.g., Gottsmann et al. 2019; Phillipson et al. 2013). We consider magmatic unrest as triggered by 'moving magma' and non-magmatic unrest as other types of unrest (i.e., seismic unrest, increased degassing) not related to 'magma-on-the-move' (Rouwet et al. 2014):
magmatic unrest – with a high rate of eruptive activity in the past ~ 1000 years compared to the previous ~ 40 kyrs, we consider that future unrest at Ceboruco may be triggered by magma migration to shallow depth, convective overturn in a long-lived magma chamber or by fresh magma entering the chamber from depth (e.g., Pritchard et al. 2019; Sparks and Cashman 2017; Woods and Cowan 2009);
seismic unrest without magmatic unrest – triggered by a large regional tectonic earthquake or a local seismic swarm. i) 'Sulphur-smelling' waters after two tectonic earthquakes in 1566 CE (Sieron and Siebe 2008; de Ciudad 1976) and 1567 CE (Sieron and Siebe 2008; Tello 1968) indicate changes in porosity and permeability that might have facilitated migration of magmatic gases (Sieron and Siebe 2008). ii) Núñez-Cornú et al. (2020), Rodríguez Uribe et al. (2013) and Sánchez et al. (2009) concluded that the seismic swarms at Ceboruco are dominated by volcano-tectonic earthquakes associated with faults in and around the edifice;
increased degassing without magmatic unrest – infiltration of abundant meteoric water from a passing hurricane may lead to an increased degassing due to the residual heat from cooling magma. From the analysis of two seismic datasets, Rodríguez Uribe et al. (2013) and Sánchez et al. (2009) confirmed that the low-frequency events recorded at Ceboruco are associated with movement of pressurized fluids. But there is no reason to conclude that seismicity always accompanies degassing, or vice-versa.
Several hazardous phenomena have been associated with non-magmatic unrest episodes (i.e., seismic and increased degassing) and are considered in our ET: rockslides, debris avalanches, debris flows, ballistics, ash fallout and increased fumarolic activity (Fig. 1). By including these events in the ET, we clarify that potentially hazardous phenomena may result from both explosive (e.g., phreatic explosions) or non-explosive events and perhaps may occur even if there is no evidence of unrest at all (e.g., Barberi et al. 1992). We emphasize that while all of these events represent contingencies in the ET, their probabilities vary.
If unrest of magmatic origin is detected at Ceboruco, we consider three potential outcomes: i) no eruption (i.e., activity subsides); ii) phreatic eruption or increased degassing activity (i.e., if large amounts of water infiltrate the volcanic system) without a magmatic eruption and, iii) magmatic eruption, which may include some phase of phreatic activity. In contrast to the latter case, the hazards associated with the first two outcomes (i.e., no eruption or phreatic eruption) resemble the contingencies in the non-magmatic unrest case, especially for locations proximal to the volcano. However, the occurrence of a magmatic eruption may result in a series of destructive phenomena that would impact the region around Ceboruco at different scales. We focus on the magmatic unrest branch and its subsequent events (the highlighted branch in Fig. 1). The past eruptive periods at Ceboruco included both explosive and effusive activity, therefore we model the most devastating phase of the past eruptions (e.g., Plinian explosive; sub-Plinian explosive; small magnitude eruption). We follow the three eruptive scenarios identified by Sieron et al. (2019b) and described in the previous section, acknowledging that more complex scenarios and cascading events are plausible.
Estimating the probabilities in the event tree: the calculation
The ET for Ceboruco becomes a Bayesian Event Tree (BET) because of the additional information we incorporate in the analysis. The ET and the BET have identical structure (Fig. 1), but the way in which probabilities are estimated is different. To estimate the probability at each node of the BET, PyBetVH uses prior models that are theoretical, statistical or numerical, and observations of past volcanic activity (past data) including the geological record (Constantinescu et al. 2016; Tonini et al. 2015; Thompson et al. 2015; Sandri et al. 2014; Marzocchi et al. 2010); although, parameterization without using past data is also possible (e.g., Marzocchi et al. 2004). Often, prior models rely on information such as the state of activity derived from monitoring (e.g., Wright et al. 2019) or data from analog volcanoes (e.g., Tierz et al. 2020; Ogburn et al. 2016).
The background calculations performed by PyBetVH rely on specified data and parameters for each branch of the ET. Given that the first three nodes of a hypothetical BET represent questions with binary answers, the probability estimates are managed through a binomial distribution (e.g., Connor 2021; Marzocchi et al. 2008; 2006), which arises because (1) the volcano will enter into a period of unrest in the time window (e.g., 𝛕 = 1 yr) or not, (2) the unrest will be associated with magma or not, (3) the magmatic unrest will result in an eruption, or not. Each of these nodes will be answered yes or no, with some probability assigned to each event, summing to 1. If the ET is not Bayesian, then the probability at each node only depends on the sample proportion (μu) (i.e., likelihood): the number of observations of an affirmative outcome in the past (or from analogue volcanoes), y, divided by the number of time windows observed, n:
$${\mu}_u=\frac{y}{n}$$
Consider a hypothetical volcano for which in 3 of 100 time windows unrest is observed to begin and one time window includes a magmatic eruption, then the sample proportion for the first node, corresponding to unrest, is 0.03. If two of these three periods of unrest were observed to be magmatic, then the sample proportion for the second node is 0.67. If one of these two periods of magmatic unrest resulted in eruption then the sample proportion of the third node, magmatic eruption, is 0.5. The product of these probabilities is the probability of eruption in a given time window, based only on likelihood, which in this example is 0.03 × 0.67 × 0.5 = 0.01. This result is easily checked, since one time window out of one hundred was observed to have an eruption.
In a BET, such as implemented in PyBetVH, the calculation starts by assuming a prior model (i.e., a prior probability based on theoretical models or expert knowledge of the phenomena) at each node. The sample proportion is then used to update the posterior probability estimate. When limited observations cannot provide a robust sample proportion, the probability estimate relies on prior functions. Conversely, if the number of past observations is large enough, prior models do not play a key role in the posterior estimate, which becomes dominated by the sample proportion (e.g., Marzocchi et al. 2004). In practice, prior models are cast as a Beta distribution, which depends on parameters α and β that take on values between 0 and 1. The mean of the prior distribution (μ) at any binomial node is given by:
$$\mu =\frac{\alpha }{\alpha +\beta }$$
and the expected value of the posterior probability at any binomial node with a Beta prior distribution is (Gelman et al. 1995; Marzocchi et al. 2006):
$$E=\frac{\alpha +y}{\alpha +\beta +n}$$
The posterior probability deviates from the sample proportion depending on the values of these two parameters, α and β. If the value of α is large compared to β, the posterior probability will increase, as in the case of monitoring data that indicates increased unrest. Conversely, if monitoring data indicates that there is no unrest, then β is large compared to α and the posterior probability will be less than the sample proportion. For the cases in which there are no monitoring data available to inform the values of α and β, then α = β = 1, μ = 0.5 and the Beta distribution is uniform random. This means that the posterior probability will tend toward 0.5.
Suppose that for our theoretical volcano we lack consistent monitoring data, then the Beta distribution is uniform random between 0 and 1 (α = β = 1). For the first node, recall that three time windows had unrest onset out of one hundred observations, so we update this prior with field observations and the resulting posterior probability of node 1 is 4/102 = 0.039, which is only slightly higher than the sample proportion (0.03) because n > > α, β. For node 2, there are only three observed intervals of onset of unrest and two of these are associated with magmatism, so the sample proportion is 2/3 = 0.67. The posterior probability is 3/5 = 0.6. The prior is relatively more important because n is small. Stated another way, once the volcano enters unrest, there are many fewer time windows, n, and fewer observations, y, so the prior probability gets more weight. For the third node, there is one eruption for two magmatic events, the sample proportion is equal to 0.5, which is the same as the prior mean (Eq. 2). For node three, the posterior probability is 2/4 = 0.5, the same as the sample proportion. Overall, the estimate of the absolute probability of magmatic eruption based on the BET in this theoretical example is 0.039 × 0.6 × 0.5 = 0.0117. Using the BET, the probability of the volcano erupting in a given time window is higher than the probability calculated using a likelihood approach (0.01), which only depends on the sample proportions.
Of course, this probability estimate depends on the certainty with which the data are known. If, for example, it is uncertain how many episodes of unrest occur in the series, then exploration is required to bound the uncertainty using alternative data. Consider the alternative case that there are zero observed eruptions associated with the two episodes of magmatic unrest. In this case, the likelihood approach (product of the probabilities based on sample proportion) is 0. The same data using Eq. (3) to calculate the posterior probability yield 0.006. That is, we are significantly less confident that no eruptions will occur in the time window. Conversely, if some sort of monitoring data were available for this theoretical volcano, then Eq. (3) would be modified by changing the values of α, β.
Subsequent nodes in the BET (Fig. 1) are not binomial but their probabilities are calculated in a similar way (Marzocchi et al. 2006, 2010). We refer the reader to Marzocchi et al. (2006, 2008 and 2010) and Tonini et al. (2015) for a detailed description of the software implementation of the calculations of the BET structure. An advantage of PyBetVH is that the posterior probability calculation includes the distribution for the probabilities, in addition to the expected values, providing a more complete view of the uncertainty (epistemic and aleatoric) in the probability estimate than is otherwise possible.
PyBetVH uses as input several text files in which the user specifies the values each parameter (e.g., y, n, α and β) is assigned in order to estimate the prior distributions and the sample proportions. The software and a step-by-step tutorial for building the input files are freely available at https://vhub.org/resources/betvh (Tonini et al. 2016b) or by request to the corresponding author. In the next section we describe the implementation of the BET for Ceboruco and the choice of model input data to estimate probabilities related to a magmatic eruption.
PyBetVH set-up: the implementation of the Ceboruco BET
Nodes 1–2-3: what is the probability of a magmatic eruption within the next time window?
Typically, volcanic hazard assessment and eruption forecasting are conducted for different time windows, from short-term (days, weeks after the initiation of unrest) to intermediate-term (months) and long-term (years to decades) (e.g., Marzocchi and Bebbington 2012; Newhall and Hoblitt 2002). The long-term hazard assessment is useful for the local authorities and stakeholders to devise plans and strategies for responding to future potential activity. A commonly used time window (𝛕) for hazard assessment is 1 year (Thompson et al. 2015; Sandri et al. 2014; Marzocchi et al. 2008; Newhall and Hoblitt 2002). However, we suggest that this time window can be extended (e.g., 10 years, 100 years) when the hazard assessment is considered for very long-term development of infrastructure projects (e.g., Gallant et al. 2018; Connor 2011; Volentik et al. 2009).
At Node 1 (N1), we want to estimate the probability of entering a new unrest phase within the next time window. Usually, unrest is defined by changes in the behavior of the volcano as observed by a monitoring network (e.g., Gottsmann et al. 2019; Phillipson et al. 2013). However, such a generalized definition of unrest is rather subjective and cannot be readily applied to active volcanoes that are poorly monitored or are not at all monitored. Ceboruco is under-monitored, therefore, we rely on geological information to determine unrest including reports of felt seismicity, recognizing that this lack of reliable monitoring data limits our analysis to long-term PVHA and is associated with high epistemic uncertainty.
Geological field data suggests that a long period of dormancy (~ 40 kyrs) at Ceboruco preceded the eruptive period that comprised the Plinian Jala eruption, its precursory effusive activity (i.e., the Destiladero lava flow; Nelson 1980), and the activity of the next ~ 150 years (Sieron et al. 2019a, b). From 1142 CE until 1870 CE no magmatic eruptions were observed. This quiescent period was interrupted by two powerful tectonic earthquakes (1566 CE and 1567 CE; Sieron and Siebe 2008; de Ciudad 1976; Tello 1968). The aftermath of these events included changes in activity observed at the surface of the volcano (Sieron and Siebe 2008; Tello 1968) and no other activity was reported for another ~ 200 yrs. A second eruptive period started with the reactivation of the system in 1783 CE and 1832 CE and included reported seismic activity (ground shaking), underground noise, and a water vapor plume at the summit. Decades later, this activity was followed by the 1870–1875 CE eruption. Post- 1875 CE, fumarole activity persisted until ~ 1894 CE (Sieron et al. 2019a; Ordóñez 1896) and since then decreased in intensity and the fumarole temperature declined with time.
Occasional surveys of the fumaroles show that their composition is meteoric water vapor generated by the residual heat from the last eruption and that there is no magmatic component in fumarole gases or their condensates (Sieron et al. 2019a; Ferrés et al. 2019; Centro Nacional de Prevención de Desastres (CENAPRED) 2016). Sánchez et al. (2009) and Rodríguez Uribe et al. (2013) analyzed several years of seismic data (2003–2008) from one station and concluded that low-frequency events indicate the presence of pressurized fluids, in agreement with the presence of water vapor fumaroles and an active hydrothermal system. Occurrence of VT events indicate intra-crustal stress accommodation, either from a magmatic body or the regional tectonic setting (i.e., the half-graben within which the edifice is located). Recently, Núñez-Cornú et al. (2020) used data from four stations around Ceboruco (deployed 2012–2014) and concluded that most of the recorded events follow shallow structural lineaments and have shallow hypocenters (< 10 km). Although insightful, these geophysical signals cannot serve as a basis for a wider definition of background activity and therefore unrest, since the network was small and deployed only temporarily. Sánchez et al. (2009) suggest that Ceboruco might be in an intra-eruptive phase while Núñez-Cornú et al. (2020) suggest that the seismic activity indicates local tectonic stresses. No other observations that may indicate changes in Ceboruco's behavior have been recorded. We consider Ceboruco is currently not in a state of unrest.
In BET, the prior probability function is usually informed by monitoring data at volcanoes with a monitoring network. However, due to the lack of reliable and sustained geophysical data collection at Ceboruco, we consider there is no prior information about whether unrest will be initiated during the next time window and therefore set the mean prior probability at 0.5 (maximum ignorance), with an equivalent number of data parameter 𝛬 = 1 in PyBetVH (indicating the maximum uncertainty associated with this choice) (Constantinescu et al. 2016; Sandri et al. 2019; Marzocchi et al. 2008).
The past data describes our likelihood function (Marzocchi et al. 2008) and is estimated from geological information. The lack of a high resolution geological and historical catalogue prohibits us to accurately determine if the two main eruptive periods consisted or not of several unrest episodes. Therefore, we conservatively assume these two eruptive periods (i.e., the one including the Jala eruption and the one including the 1870-'75 eruption) represent one unrest episode each and acknowledge the high uncertainty associated with this choice. The long dormancy (quiescence) interval between these two unrest episodes indicates an inter-eruptive period with no clear signs of magma intrusion but disturbed twice by large tectonic earthquakes that may have perturbed the magmatic system at Ceboruco. We count four episodes of unrest, magmatic or non-magmatic in 1108 time windows. Therefore, the sample proportion, which is the mean of the likelihood function, is 4/1108. Because the mean prior probability is greater than the sample proportion, the posterior probability is greater than the sample proportion. That is, there is a penalty for lack of monitoring data.
Node 2 (N2) - Given unrest, what is the probability that magma is involved? An unrest episode may be magmatic or non-magmatic (e.g., tectonic). Given its activity of the past ~ 1000 years and the relatively short time since its most recent eruption, it is reasonable that unrest could be associated with magma. Ceboruco, however, lies within a tectonically active area within which large-magnitude tectonic earthquakes are common, like the two earthquakes reported in the sixteenth century. Given the variability of both volcanic systems and tectonic regions we assume a prior probability distribution with μ = 0.5 and 𝛬 = 1, indicating maximum ignorance (i.e., equal chances of having magmatic or non-magmatic unrest). The likelihood function is described by the past data according to which two of the four unrest episodes were clearly magmatic in origin (i.e., the two eruptive periods). As the sample proportion, 2/4, is equal to the mean of the prior probability, 0.5, the posterior probability for this node is also 0.5, which is the probability that magma is involved, given unrest. Or, using the nomenclature in Eq. (3), α = β = 1, μ = 0.5, y = 2, and n = 4., or the expected value of the posterior function is 3/6 = 0.5.
Node 3 – if magmatic unrest is due to magma, what is the probability a magmatic eruption will occur during the time window? Previous studies involving the use of a BET tool used as prior models at this node a Beta distribution with α = β = 1 (i.e., maximum ignorance; equal probability of eruption or no-eruption; see Constantinescu et al. 2016; Sandri et al. 2014, 2009) or data from Phillipson et al. (2013) which includes an analysis of 228 episodes of unrest (e.g., Sandri et al. 2019; Constantinescu et al. 2016). According to the data set presented by Phillipson et al. (2013), 64% of unrest episodes at stratovolcanoes lead to eruption. For comparison, Newhall and Dzurisin (1988) found 38% of unrest episodes documented at silicic calderas led to eruption and 54% of unrest episodes at mafic calderas. Here, we choose maximum ignorance as prior probability (i.e., μ = 0.5; 𝛬 = 1). We update the posterior probability with past data informed by the two magmatic unrest episodes (i.e., two eruptive periods) that resulted in magmatic eruptions. The sample proportion is therefore 2/2. This value is modified by the prior mean, 0.5, to yield a probability of magmatic eruption given magmatic unrest of 0.75.
Based on the assumptions made at the first three nodes from available models and geological data, we estimate the probability of eruption at Ceboruco in the next year (time window) to be ~ 0.002. If, on the other hand, the volcano is in a state of unrest, the posterior probability (magmatic eruption given unrest) is 0.375. Since the sample proportions at nodes 2 and 3 are based on very few events, the values assumed for parameters in the prior distribution (α = β = 1) are quite significant. A summary of the input data at the first three nodes is presented in Table 1.
Table 1 Summary of the input data used in PyBetVH for the first three nodes
Nodes 4 and 5: vent location and eruption type/size
We focus here on the activity at Ceboruco's cone and do not consider hazards related to the monogenetic volcanic field surrounding the volcano, because the past ~ 1000 years of activity at Ceboruco was concentrated within the summit caldera. Therefore, we assume the probability of monogenetic eruptions is much smaller than summit caldera eruptions. Several lava flows occurred closer to the present caldera rim or on the north flanks (e.g., Copales, El Norte, Coapan lava flows (Fig. 8 in Sieron et al. 2019a)), however, most activity was restricted to the main edifice of Ceboruco. We therefore consider five possible sectors for a new eruption to occur: - the summit caldera and four flank locations (North, e.g., the Coapan lava flow site; East, South, e.g., the Copales lava flow site; and West flanks). The probabilities for new vent opening were decided by project participants who weighted geologic evidence. We assume a 0.98 prior probability for the summit caldera location and 0.005 for each flank sectors. Based on the three eruption scenarios identified by Sieron et al. (2019a) and discussed in the previous section, we assume that future activity at Ceboruco will likely include a: S1 - small magnitude (effusive / small explosive); S2 – medium magnitude (Vulcanian/sub-Plinian) and S3 – large magnitude (Plinian). The probabilities of these scenarios were assigned by a power-law, with small eruptions having a higher likelihood of occurrence than large events: S1–0.6, S2–0.3, S3–0.1 (e.g., Sandri et al. 2014; Marzocchi et al. 2008; Newhall and Hoblitt 2002).
Nodes 6, 7 and 8: occurrence of hazardous phenomena and areas impacted
We consider: lava flows, ballistics, tephra fallout for Scenario 1; lava flows, ballistics, tephra fallout, PDC and lahars for Scenario 2 and ballistics, tephra fallout, PDC and lahars for Scenario 3. The prior best-guesses of occurrences for each phenomena conditional to the eruption size were set using data from Newhall and Hoblitt (2002) (i.e., frequency of phenomena associated with eruption type) except ballistics that has a probability of 1, acknowledging that some phenomena may be underreported. Further, we divide the area around Ceboruco in a grid of 500 m × 500 m cells and compute the probability of each cell to be invaded by a selected phenomenon. The modeling-based maps shown in Sieron et al. (2019b) serve as prior models in our PyBetVH analysis. We refer the reader to Sieron et al. (2019b) for a detailed description of the methodology, the simulation tools utilized to model each selected phenomenon, and the output maps. As past data we use the area covered by the deposits of past eruptions associated with each phenomenon and eruption size (Sieron et al. 2019a).
We parameterize past data and prior models by assigning a probability for each grid cell based on the frequency of inundation by a volcanic phenomenon during past eruptions (i.e., past data) or as indicated by the numerical modeling presented in the Sieron et al. (2019b) (e.g., if a grid cell was affected by one simulation from a total of three, the prior probability of the grid cell will be 1/3; if there is only one past eruption and a given grid cell was affected, then the prior probability is 1 (1/1).
In Table 2 we present a summary of the hazards considered for each scenario and the PyBetVH input data.
Table 2 Summary of the hazardous phenomena considered in our event tree for each of the three scenarios considered. For past data we have the number of times the selected phenomena occurred at past eruptions. Most past data are considered for Scenario 3 (Plinian eruption) due to the better preservation and mapping of the deposits of the Jala eruption. As prior models we used the maps created by various simulation tools used by Sieron et al. (2019b) for each considered phenomenon
Probability maps
The absolute probability maps (Figs. 3 and 4) produced by PyBetVH show the combined mean probabilities that an area is invaded by a volcanic phenomenon regardless of the eruption size and vent location (i.e., considering the occurrence of an eruption of any size from any vent) within a 1-year time window.
The absolute annual probability maps for: a ballistics; b lava flows and c pyroclastic density currents. Both PDC and lava flows show considerable probabilities for Ahuacatlán and Jala to be impacted should an eruption occur within the next time window (i.e., 1 year). The infrastructure built around Ceboruco will likely be affected by these phenomena. The ballistics map shows highest probabilities within the summit caldera
The absolute annual probability maps for a tephra fallout and b lahars. Ceboruco's flanks and immediate surroundings have a higher probability of being inundated by a lahar, whereas the areas downstream the Ahuacatlán river show a lower probability. This is assumed to be due to the loss of particles as sedimentation occurs within a lahar and captured in the simulations. Tephra fallout maps rely heavily on the numerical simulations and prior models show that the expected areal distribution of tephra is controlled by the prevailing winds at tropopause levels for a large eruption and by the smaller variable wind fields at lower altitudes for smaller events
Ballistics. We use 540 simulations made with the Eject! Code (Mastin 2001) and field measurements of distances travelled by the ballistics of the Jala eruption (i.e., Scenario 3) (Sieron et al. 2019a) to estimate the posterior probabilities. The simulated parameters included: - ejection angles between 350 and 890; ejection velocities between 150 ms− 1 and 250 ms− 1 and bomb radii of 10 cm to 100 cm. A detailed description of how these simulations were conducted is presented in Sieron et al. (2019b). The map (Fig. 3a) shows the highest probabilities (~ 1–2 × 10− 3) around the summit caldera, decreasing with distance for approximately 5 km. The flanks of Ceboruco up to 5 km away from the vent are uninhabited and several crop fields lie at the base; however, the telecommunications towers located on the NE of the caldera rim will likely be affected by ballistics, with considerable impact on regional communications.
We considered lava flows in scenarios 1 and 2. Figure 3b shows the extent of the lava flows is comparable with the results from the numerical modeling in Sieron et al. (2019b). The areas of higher probability (~ 8 × 10− 4 – 1 × 10− 3) correspond to the areas inundated most often in simulations (i.e., ~ 5–10 km to the north, north-east and south-west) and the areas where past deposits were identified (i.e., north). Areas of very low probability (near zero) on the south-east and west flanks correspond to topographic barriers and have not been affected by simulations (Sieron et al., 2019b). The towns of Ahuacatlán (pop. ~ 9000) and Jala have relatively high probabilities (~ 4–6 × 10− 4) of being affected by lava flows. Smaller communities to the South and West as well as important infrastructure (e.g., major roads) to the South and North (some already built on lava flows) have significant probabilities (~ 6.5 × 10− 4) of being affected.
Pyroclastic density currents (Fig. 3c). We consider both end members of pyroclastic density currents (flows and surges) and use past data (from Jala eruption) and simulations (i.e., Titan2D; Patra et al. 2005) for flows and the Energy Cone module in LaharZ (Schilling 1998) for surges) to estimate posterior probabilities for these hazardous phenomena. Sieron et al. (2019b) simulated pyroclastic flow volumes of 0.025–0.125 km3 and 0.5–1 km3 for Scenario 2 and 3 respectively. The H/L values selected for simulating pyroclastic surges were 0.14–0.17 and 0.07–0.12 for Scenario 2 and 3. A detailed description of the methodology and a full list of the parameters used for these simulations is provided in Sieron et al. (2019b). The PDC map shows the highest probabilities of PDC inundation (> 5 × 10− 4) within 10 km of the vent. The valleys and some interfluves at > 10 km from the vent have a lower probability of inundation attributed to the simulated surges and field data from past large magmatic eruptions, in cases overriding geographic barriers such as the NE edge of the graben. The towns of Ahuacatlán and Jala, both at ~ 10 km from the vent, along with several smaller communities, have relatively elevated probabilities (~ 1–3 × 10− 4) of being affected by PDC deposits, along with existing infrastructure.
Tephra fallout hazard is evaluated for all three eruption sizes. Sieron et al. (2019b) describe in detail the methodology used for simulations of tephra fallout that serve as prior models in our analysis. Figure 4a shows the average annual probability map for the accumulation of at least 10-cm-thick tephra layer, acknowledging that tephra accumulations of < 10 cm will have a wider impact. The distribution of the probabilities is strongly controlled by the spatial extent of tephra in the numerical models since we lack a good set of field data (only one set for Jala eruption, Scenario 3). The widespread tephra deposit to the NE is associated with a large Plinian eruption and it is controlled by the prevailing winds at high altitudes (and illustrated by the numerical simulations). The higher probabilities closer to the volcano are associated with the thickening of tephra deposits near the vent and with sedimentation from lower transient plumes affected by the variable monthly wind field at lower altitudes. However, the monthly wind profiles may hide the daily variability of winds with unusual directions (e.g., Michaud-Dubuy et al. 2021). All communities to the East, South and West of Ceboruco will be impacted by tephra fallout and significant damage is expected to the agricultural lands and the infrastructure in the area.
Lahars (syn-and-post eruptive) are considered for both S2 and S3 eruption scenarios due to the availability of fresh pyroclastic material during such events and the availability of water during the rainy season at Ceboruco or a passing tropical storm. Lahar deposits have been identified in the field and associated with the large Jala eruption (Sieron et al. 2019a). We use the extent of past lahars and simulation with LaharZ as input data in PyBetVH. The average probability map of lahar inundation is shown in Fig. 4b. Unlike Sieron et al. (2019b) who considered lahar hazards as far north as the Grande de Santiago river, we focus our assessment on Ceboruco and its main drainage network and tributaries to the Ahuacatlán river. The highest probabilities are within the drainage network of Ceboruco (> 1.7 × 10− 4) and the towns of Jala and Ahuacatlán, as well as the nearby infrastructure. Marquezado, Uzeta, Tetitlan and the area of the drainage network of the Ahuacatlán river to the SW have relatively high probabilities of being affected by lahars (> 1.7 × 10− 4).
In the Additional Material 1 we present the conditional probability maps for each considered phenomena (i.e., occurrence of hazards is conditional to the occurrence of a specific eruption size from the central vent).
The event tree and the probability of eruption
Volcanoes, active or extinct, are inherent sources of hazard conditioned to the occurrence of a specific triggering event (e.g., a magmatic eruption may trigger a PDC or a tectonic earthquake may trigger slope failures and debris avalanches at extinct or long dormant volcanoes). Given the complexity and intrinsic variability of volcanic systems, PVHA (Marzocchi et al. 2010; Marzocchi and Woo 2007) is preferred to a simple deterministic approach based on the extent of deposits of past eruptions or on several numerical simulations. Although we acknowledge that both approaches are complementary and mutually informative (e.g., Tierz 2020; Rouwet et al. 2019) the event tree approach provides a useful tool to visualize and estimate the probabilities of occurrence of various hazards at different time scales (Punongbayan et al. 1996; Marzocchi et al. 2008; Newhall and Hoblitt 2002). To fully benefit from the use of event trees, these have to comprise a wide range of volcanic events, magmatic and non-magmatic. Therefore, they can provide a clear and broad view of the possible hazardous phenomena related to the volcanic system and how these may affect the area during a long-term period. This is potentially of great interest to local authorities and stakeholders who often make decisions regarding long-term investments in the local community. Important infrastructure projects are developed to last for decades and the sites selected for their development must be considered carefully with respect to an acceptable level of hazard even if the probability of occurrence is very small (e.g., Connor 2011). The advantage of a probabilistic approach in hazard assessment is that it can account for all probable events; although, accounting for all possible hazard outcomes can be challenging (Rougier and Beven 2013). Furthermore, a probabilistic approach can account for associated uncertainties (aleatoric and epistemic) and provide an easy way to visualize them.
When direct evidence of occurrence of a specific eruptive phenomenon is not identified in the past deposits, information from analogue volcanoes is sometimes used (e.g., Ogburn et al. 2016). Most volcanoes have very long periods of inactivity (dormancy) and are often considered extinct or less dangerous and their fertile slopes attract cultivation. To complicate forecasting, most active volcanoes are poorly monitored, or not monitored at all (e.g., Loughlin et al. 2015). With an ever-increasing population around active volcanoes, we need to look at a broad range of volcanic processes in order to consider all the ways a volcano might impact us. A volcano poses multiple threats, not only related to a magmatic eruption. Hazards can occur long after the volcano goes extinct. However, our human perspective forces us to consider the most dramatic events in our hazard assessments and those are more often than not related to a magmatic eruption. Here we developed a generic event tree for Ceboruco volcano and try to account for all the possible outcomes regardless of the involvement of renewed magmatism. We attempt to show that estimating probabilities of each outcome can be challenging depending on the assumptions made about the volcano.
Bayesian methods and the introduction of prior distributions is important in hazard assessment, especially when we have a sparse record of events, such as the case for periods of unrest at Ceboruco volcano. When there are many data, the prior distribution has little impact on the probability, but for nodes represented by few observations, the parameters of the prior distribution are of critical importance. Consider two scenarios in which Ceboruco is currently in the state of: a) no unrest, or b) unrest. We explore the effects of the likelihood and prior functions on the posterior probability estimates for the first three nodes of the ET (Fig. 5a, b). When the volcano is not in a state of unrest, such as when we need a long-term forecast, based on our interpretation of the recent eruptive history at Ceboruco (i.e., four unrest episodes with two of magmatic origin) (Fig. 5a) the probability of entering unrest in the next time window is 0.0036 (black line - 'triangle' marker, Fig. 5a) and the overall probability of eruption based only on the mean likelihood function is 0.0018 (PLK, Fig. 5a inset). Although unlikely, the fact that the volcano is not monitored extensively and continuously may justify a conservative assumption that after ~ 40 kyrs of quiescence Ceboruco entered unrest before the Jala eruption and continues today (i.e., second scenario). We know this unrest is magmatic. Based on the likelihood function, the overall probability of eruption is 1 (PLK, Fig. 5b inset). Therefore, a hazard analysis based only on the limited geological record is highly biased and poorly informative for such a short time window.
a The probabilities of the first three nodes of the ET given Ceboruco is not currently in unrest. The black line with triangle markers shows the probabilities based on the mean likelihood functions. The blue line with cross markers, represent the probabilities using maximum ignorance at each node plus likelihood functions. The green line with circle marker corresponds to the case in which node 3 uses 64% of unrest leading to eruption (Phillipson et al. 2013) with high uncertainty whereas the red line with plus marker uses the same information but with higher confidence in the data (lower uncertainty). b The probabilities of the first three nodes of the ET given Ceboruco is currently in unrest (i.e., Punrest = 1). The colored lines correspond to the same assumptions at each node as described for the first scenario. The figure insets show the overall probability of eruption for the two scenarios: PLK – probability estimate based on the likelihood function alone; P50 – adding maximum ignorance at each node; Ph-u – adding Phillipson et al. (2013) data at node 3 with high uncertainty (high variance); Pl-u – adding the same data but with more confidence (low variance). NOTE: the probability scale in panel a) is log
In a Bayesian analysis however, if we assume the simplest prior model with maximum ignorance (i.e., μ = 0.5; 𝛬 = 1) at each of the first three nodes, the posterior probability at Node 3 decreases by > 20% (blue line - 'x' marker, Fig. 5a, b) but is associated with high uncertainty (σ2 = 0.03 to 0.05). The overall probability of eruption also decreases slightly to 0.0016 in the first scenario and significantly, to 0.66, in the case of the second scenario (P50, Fig. 5a, b insets). These estimates are better than those based only on the likelihood, but they are associated with large uncertainties.
The Phillipson et al. (2013) dataset indicates that approximately 2/3 of stratovolcanoes entering unrest will have an eruption and this information was used in other BET studies (e.g., Sandri et al. 2019; Constantinescu et al. 2016). We can use this information, or information from other data sets, to design a prior function for node 3 by adjusting the α and β parameters of the distribution. With no monitoring data, this information must be used cautiously as adjusting the values for α and β leads to a higher or lower confidence in our data. For example, for Beta (α = 64; β = 36), which indicates a higher confidence in our assumption, the probability at node 3 decreases in both scenarios (σ2 = 0.002) (red line - '+' marker, Fig. 5a, b). Conversely, for Beta (α = 6.4; β = 3.6) the same probability increases again but this time with higher uncertainty (σ2 = 0.03) (green line - 'o' marker, Fig. 5a, b). This is also reflected in the overall probabilities of eruption (Pl-u and Ph-u in Fig. 5a, b insets). This shows that although the posterior probabilities estimated via a BET analysis differ substantially to those based on the likelihood function alone, they are still associated with uncertainties that can only be reduced with monitoring data.
Although a commonly accepted time window for volcanic hazard assessment is one-year (e.g., Thompson et al. 2015; Sandri et al. 2014; Newhall and Hoblitt 2002), the annualized probability is often extensible to longer periods. Decisions regarding the development of important infrastructure projects (e.g., bridges, dams, power plants) might benefit of an analysis that considers longer time windows, i.e., 10 years or more (e.g., Connor 2011). This requires a high-quality eruptive history catalogue and the incorporation in the analysis of events that occur at larger time scales even though they may be extremely unlikely (e.g., Marzocchi and Bebbington 2012; Connor 2011). For a 10-year time window the overall probabilities of eruption at Ceboruco will increase ten times if we assume the volcano is not currently in unrest. If Ceboruco is in unrest, the ET can also be parameterized from node 2 and used to estimate the subsequent conditional probabilities (e.g., Queiroz et al. 2008).
All things considered, a PVHA at Ceboruco, or any other under-monitored volcano, is highly sensitive to the definition of unrest and the prior probability estimates, and both depend on geophysical monitoring and/or on a rigorous eruptive catalogue. Without consistent data from a developed and continuous monitoring network to inform the hazard assessment, the final probability estimates are associated with high uncertainties. These uncertainties in both the prior and likelihood functions are propagated to the posterior probability estimates; the only solution to refine the PVHA and deal with the uncertainty is the development of a real-time and continuous monitoring network and to conduct more detailed field studies.
Limitations and advantages of using PyBetVH
The ability to combine multiple sources of information, as well as the possibility of exploring various combinations of eruptive scenarios leads to a better estimate of the uncertainty associated with different hazard maps. Probabilistic hazard maps are usually conducted for a single volcanic phenomenon by using different numerical simulation tools (e.g., Charbonnier et al. 2020; Gallant et al. 2018; Strehlow et al. 2017; Selva et al. 2014). PyBetVH can merge the outputs of different simulation tools with geological data and assess the uncertainty range associated with the average prior probability presented in each map. In Fig. 6, we show an example of the uncertainty range (i.e., 10th, 50th and 90th percentiles) in the conditional probability maps for PDC (conditional to the occurrence of a Plinian eruption (S3). These maps provide an easier way to evaluate our confidence in the hazard maps, a useful feature for decision makers (e.g., Tierz et al. 2018; Thompson et al. 2015; Sandri et al. 2014; Lindsay et al. 2010).
Example of uncertainty in the pyroclastic density currents hazard probability maps for a large Plinian eruption (Size 3). We show the a 10th percentile; b the average, and c the 90th percentile maps for the conditional probability (i.e., conditional to the occurrence of a Plinian eruption at the central vent)
PyBetVH has an easy-to-use graphical interface (Tonini et al. 2015) that allows the user to update the input files as soon as more information becomes available. The current version of PyBetVH was developed to analyze volcanic hazards associated to magmatic unrest only. However, recent efforts helped describe and recognize indicators for non-magmatic unrest and related hazards (Rouwet et al. 2014). The implementation of a non-magmatic branch in the ET is important for short-term forecasting applied to monitored volcanoes (Tonini et al. 2016a).
Although PyBetVH produces separate probability maps for each selected volcanic phenomenon, the user has the option to download the datafiles and plot them using other tools to create an aggregated hazard map that can provide a rapid view of all possible volcanic phenomena at Ceboruco.
Conclusions and final remarks
We conduct a PVHA for Ceboruco, an active but poorly monitored volcano in the western part of the TMVB. This work represents an effort to update the volcanic hazard assessment at Ceboruco by conducting a probabilistic analysis that incorporates different sources of uncertainty. Based on the eruptive scenarios proposed by Sieron et al. (2019b), we create a generic ET for Ceboruco to account for magmatic and non-magmatic activity. For the magmatic eruption branch of the ET, we choose three scenarios: i) small magnitude (effusive/explosive), ii) medium magnitude (Vulcanian/sub-Plinian) and iii) large magnitude (Plinian); with their related hazardous phenomena: ballistics, tephra fallout, pyroclastic density currents, lahars, and lava flows. We use PyBetVH, an e-tool based on the Bayesian event tree methodology, to create probabilistic hazard maps for each of the selected eruption scenarios. Using geological data with outputs from other numerical and theoretical models in PyBetVH, we estimate the probability of a magmatic eruption at Ceboruco within the next time window (i.e., 1 year) of ~ 0.002. The resulting maps show the absolute annual probability for the communities and infrastructure around Ceboruco to be impacted by lava flows, PDC, tephra fallout and lahars. The ballistics will likely impact only the cone area. While the deterministic maps presented by Sieron et al. (2019b) represent a first step towards quantifying volcanic hazard at Ceboruco, are easy to interpret by the general public and are an asset in case of a volcanic emergency, the maps presented here are fundamentally different. These maps do not show just a footprint of the hazard but rather provide probabilities of the areas around the volcano to be affected by potential hazardous phenomena. These maps assign a probability to a particular scenario given the current state of unrest at the volcano (i.e., no unrest). However, should any activity be detected, the current estimates will have to be updated by running the model with the newly available information incorporated in the event-tree. The probability maps can be a useful tool for authorities and stakeholders in the decision-making process. We believe the decisionmakers will find the probabilistic results useful, especially if they are able to compare probabilities with other potential events that may affect the community. For example, globally, in a community, children are the people least likely to die in a given year, with a probability of dying of around 1/10000 with higher rates among children in rural communities (Svenson et al. 1996). A probability of 1/500 or 1/1000 of PDC inundation of their community (Fig. 6) is a high probability for children in a community because it considerably increases their probability of dying in a given year considerably. Of course, it is up to the community to decide which level of probabilities are unacceptable, but such comparisons can provide context and these comparisons rely on valid probabilistic hazard maps.
Our analysis indicates that PVHA at poorly monitored volcanoes relies mostly on the likelihood function (informed by geological data) and/or maximum ignorance for prior distribution functions. These estimates are associated with high uncertainties which can be significantly reduced by refining the prior distribution functions with information from geophysical monitoring (e.g., Wright et al. 2019).
We recognize the numerous efforts by academic and government institutions, and we recommend the set-up of at least a minimal permanent monitoring network at Ceboruco with the capabilities to provide real-time geophysical data continuously. Another important consideration at Ceboruco is to extend the event tree presented here to include the surrounding monogenetic field.
The data set used for this study is published in Sieron et al. (2019a, b). The PyBetVH files are available by request to the corresponding author.
Event tree
BET:
Bayesian event tree
BBN:
Bayesian Belief Networks
HASSET:
Hazard Assessment Event tree
BET_VH:
Bayesian Event Tree for Hazard Assessment
PyBetVH:
Python Bayesian Event Tree for Hazard Assessment
PVHA:
Probabilistic volcanic hazard assessment
TMVB:
Trans-Mexican Volcanic Belt
VEI:
Volcanic Explosivity Index
PDC:
Pyroclastic density currents
Dense rock equivalent
m u :
Magmatic unrest
t u :
Tectonic unrest
Increased degassing
Lf:
Low-frequency earthquakes
VT:
Volcano tectonic earthquakes
m e :
Magmatic eruption
S1:
Scenario 1
N1:
Kyrs:
Kiloyears
u :
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RC acknowledges the financial support of a DGAPA-UNAM postdoctoral fellowship (2015 – 2016). We wish to thank Pablo Tierz for the valuable comments on the manuscript. The authors wish to thank the handling editor and the two reviewers for the suggestions and constructive comments that improved our manuscript.
This work is part of the project "Evaluación del peligro volcánico del volcán Ceboruco (Nayarit), con énfasis en su posible impacto sobre la infraestructura de la Comisión Federal de Electricidad" (Convenio CFE-800720929), funded by the Comisión Federal de Electricidad (México).
School of Geosciences, University of South Florida, Tampa, USA
Robert Constantinescu & Charles Connor
Instituto de Geofísica, Universidad Nacional Autónoma de México, Mexico City, Mexico
Robert Constantinescu & Claus Siebe
Centro de Ciencias de Información Geoespacial, Mexico City, Mexico
Karime González-Zuccolotto
Escuela Nacional de Ciencias de la Tierra, Universidad Nacional Autónoma de México, Mexico City, Mexico
Dolors Ferrés
Centro de Ciencias de la Tierra, Universidad Veracruzana, Xalapa, Mexico
Katrin Sieron
Centro de Geociencias, Universidad Nacional Autónoma de México, Juriquilla Querétaro, Mexico
Lucia Capra
Istituto Nazionale di Geofisica e Vulcanologia, Sezione de Roma 1, Rome, Italy
Roberto Tonini
Robert Constantinescu
Claus Siebe
Charles Connor
All authors contributed to the writing of the manuscript, the interpretation, and discussions of the results. RC designed and conducted the implementation of the study. RC, CC and RT conducted the PyBetVH set-up and running of the code. KGZ and RC conducted the elaboration of the maps. RC, CC, DF, KS, CS, CB participated in the elicitation sessions for parameters set-up. KGZ and DF assisted with data analysis and conversion. The author(s) read and approved the final manuscript.
Correspondence to Robert Constantinescu.
Additional file 1: Supplementary Information Figure 1.
The conditional probability maps showing the areas affected by volcanic ballistics given: a) a small eruption (S1), b) a moderate eruption (S2) and c) a large eruption (S3) from the central vent. Supplementary Information Figure 2. The conditional probability maps showing the areas affected by tephra fallout given: a) a small eruption (S1), b) a moderate eruption (S2) and c) a large eruption (S3) from the central vent. Supplementary Information Figure 3. The conditional probability maps showing the areas affected by lava flows given: a) a small eruption (S1) and b) a moderate eruption (S2) from the central vent. Supplementary Information Figure 4. The conditional probability maps showing the areas affected by pyroclastic density currents given: a) a moderate eruption (S2) and b) a large eruption (S3) from the central vent. Supplementary Information Figure 5. The conditional probability maps showing the areas affected by lahars given: a) a moderate eruption (S2) and b) a large eruption (S3) from the central vent.
Constantinescu, R., González-Zuccolotto, K., Ferrés, D. et al. Probabilistic volcanic hazard assessment at an active but under-monitored volcano: Ceboruco, Mexico. J Appl. Volcanol. 11, 11 (2022). https://doi.org/10.1186/s13617-022-00119-w
DOI: https://doi.org/10.1186/s13617-022-00119-w
Event trees
Volcanic unrest
Eruption forecasting, PyBetVH
Ceboruco | CommonCrawl |
ANNs-Based Method for Solving Partial Differential Equations : A Survey
Danang Adi Pratama, Maharani Abu Bakar, Mustafa Man, M. Mashuri
Subject: Mathematics & Computer Science, Algebra & Number Theory Keywords: artificial neural networks; discretization; machine learning; partial differential equations
Conventionally, partial differential equations (PDE) problems are solved numerically through discretization process by using finite difference approximations. The algebraic systems generated by this process are then finalized by using an iterative method. Recently, scientists invented a short cut approach, without discretization process, to solve the PDE problems, namely by using machine learning (ML). This is potential to make scientific machine learning as a new sub-field of research. Thus, given the interest in developing ML for solving PDEs, it makes an abundance of an easy-to-use methods that allows researchers to quickly set up and solve problems. In this review paper, we discussed at least three methods for solving high dimensional of PDEs, namely PyDEns, NeuroDiffEq, and Nangs, which are all based on artificial neural networks (ANNs). ANN is one of the methods under ML which proven to be a universal estimator function. Comparison of numerical results presented in solving the classical PDEs such as heat, wave, and Poisson equations, to look at the accuracy and efficiency of the methods. The results showed that the NeuroDiffEq and Nangs algorithms performed better to solve higher dimensional of PDEs than the PyDEns.
Working Paper ARTICLE
Asymptotics and Confluence for Some Linear q-Difference-Differential Cauchy Problem
Stephane Malek
Subject: Mathematics & Computer Science, Algebra & Number Theory Keywords: asymptotic expansion; confluence; formal power series; partial differential equation; q-difference equation
Online: 23 February 2021 (11:02:00 CET)
A linear Cauchy problem with polynomial coefficients wich combines q-difference operators for q>1 and differential operators of irregular type is examined. A finite set of sectorial holomorphic solutions w.r.t the complex time is constructed by means of classical Laplace transforms. These functions share a common asymptotic expansion in the time variable which turns out to carry a double layers structure which couples q-Gevrey and Gevrey bounds. In the last part of the work, the problem of confluence of these solutions as q tends to 1 is investigated.
Existence and Uniquenes Solution of Integral Equations via Common Fixed Point Theorems
Gunaseelan Mani, Arul Joseph Gnanaprakasam, Yongjin Li, Zhaohui Gu
Subject: Mathematics & Computer Science, Algebra & Number Theory Keywords: integral equations; complex partial metric space; common fi xed point.
Online: 22 April 2021 (09:19:54 CEST)
In this paper, we prove some common fi xed point theorems on complex partial metric space. The presented results gener- alize and expand some of the literature well-known results. We also explore some of the application of our key results.
Asymptotics and Confluence for a Singular Nonlinear Q-Difference-Differential Cauchy Problem
Subject: Mathematics & Computer Science, Analysis Keywords: asymptotic expansion; confluence; formal power series; partial differential equation; q-difference equation
Online: 22 March 2022 (11:41:03 CET)
We examine a family of nonlinear q-difference-differential Cauchy problems obtained as a coupling of linear Cauchy problems containing dilation q-difference operators, recently investigated by the author, and quasi-linear Kowalevski type problems that involve contraction q-difference operators. We build up local holomorphic solutions to these problems. Two aspects of these solutions are explored. One facet deals with asymptotic expansions in the complex time variable for which a mixed type Gevrey and q-Gevrey structure is exhibited. The other feature concerns the problem of confluence of these solutions as q tends to 1.
On the Solutions of Nonlinear Hybrid Fractional Integrodifferential Equations
Faten H. Damag, Adem Kılıçman, Awsan T. Al-Arioi
Subject: Mathematics & Computer Science, Analysis Keywords: hybrid fractional integrodifferential equation; hybrid fixed point theorems of Dhage; approximations solutions; Lipschitz conditions; weaker mixed partial continuity
In the present work we study the existence of solutions for hybrid nonlinear fractional integrodifferential equations. We developed an algorithm by using the operator theoretical techniques in order to obtain the approximate solutions. The main results depend on the Dhage iteration method that were incorporated with the modern hybrid fixed point theorems. The approximate solutions were obtained by using Lipschitz conditions and weaker form of mixed partial continuity. Further, we provide some examples to explain the hypotheses and the related results.
Axial Diffusion of the Higher Order Scheme on the Numerical Simulation of Non-Steady Partial Differential Equation in the Human Pulmonary Capillaries
Azim Aminataei, Mohammadhossein Derakhshan
Subject: Mathematics & Computer Science, Applied Mathematics Keywords: non-steady partial differential equation; higher order finite difference scheme; axial diffusion; convergence; consistency; stability
Online: 28 January 2020 (09:14:07 CET)
In the present study, a mathematical model of non-steady partial differential equation from the process of oxygen mass transport in the human pulmonary circulation is proposed. Mathematical modelling of this kind of problems lead to a non-steady partial differential equation and for its numerical simulation, we have used finite differences. The aim of the process is the exact numerical analysis of the study, wherein consistency, stability and convergence is proposed. The necessity of doing the process is that, we would like to increase the order of numerical solution to a higher order scheme. An increment in the order of numerical solution makes the numerical simulation more accurate, also makes the numerical simulation being more complicated. In addition, the process of numerical analysis of the study in this order of solution needs more research work.
Roach Infestation Optimization MPPT Algorithm for Solar Photovoltaic System
Chittaranjan Pradhan, Nicholas Kakra Ntiakoh, Rajnish Kaur Calay
Subject: Engineering, Electrical & Electronic Engineering Keywords: DC-DC Boost converter; Maximum power point tracking (MPPT); Partial shading condition (PSC); Particle swam optimization (PSO); Roach infestation optimization (RIO); Solar photovoltaic system
Of all the renewable energy sources, solar photovoltaic (PV) power is estimated to be a popular source due to several advantages such as its free availability, absence of rotating parts, integration to building such as rooftops, and less maintenance cost. The nonlinear current-voltage (I–V) characteristics and power generated from a PV array primarily depend on solar insolation/irradiation and panel temperature. The extracted PV output power is influenced by the accuracy with which the nonlinear power–voltage (P–V) characteristic curve is traced by the maximum power point tracking (MPPT) controller. In this paper, a bio-inspired roach infestation optimization (RIO) algorithm is proposed to extract the maximum power from the PV system (PVS). To validate the usefulness of the RIO MPPT algorithm, MATLAB/Simulink simulations are performed under varying environmental conditions, for example, step changes in solar irradiance, and partial shading of the PV array. Furthermore, the search performance of the RIO algorithm is examined on different unconstrained benchmark functions, and it is that realized that the RIO algorithm has improved convergence characteristics in terms of finding the optimal solution than Particle swarm optimization (PSO). The results demonstrated that the RIO-based MPPT performs remarkably in tracking with high accuracy as the PSO-based MPPT.
System Engineering and Overshoot Damping for Epidemics Such as COVID-19
Robert L. Shuler, Theodore Koukouvitis, Dyske Suematsu
Subject: Keywords: Coronavirus; COVID-19; pandemic; partial unlock; social distancing; economic impact; ventilator utilization; SARS-CoV-2; overshoot; SIR; model; simulation; caseload management; undershoot
Online: 5 May 2020 (12:32:17 CEST)
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The goal of this paper is to contribute the perspective of a systems engineer to the effort to fight pandemics. The availability of low latency case data and effectiveness of social distancing suggest there is sufficient control for successful smoothing and targeting almost any desired level of low or high cases and immunity. This control proceeds from spontaneous public reaction to caseloads and news as well as government mediated recommendations and orders. We simulate multi-step and intermittent-with-feedback partial unlock of social distancing for rapidly-spreading moderate-mortality epidemics and pandemics similar to COVID-19. Optimized scenarios reduce total cases and therefore deaths typically 8% and up to 30% by controlling overshoot as groups cross the herd immunity threshold, or lower thresholds to manage medical resources and provide economic relief. We analyze overshoot and provide guidance on how to damp it. However, we find overshoot damping, whether from expert planning or natural public self-isolation, increases the likelihood of transition to an endemic disease. An SIR model is used to evaluate scenarios that are intended to function over a wide variety of parameters. The end result is not a case trajectory prediction, but a prediction of which strategies produce near-optimal results over a wide range of epidemiological and social parameters. Overshoot damping perversely increases the chance a pathogen will transition to an endemic disease, so we briefly describe the undershoot conditions that promote transition to endemic status.
On Partial Sums of Analytic Univalent Functions
Mohammad Mehdi Shabani, Saeed Hashemi Sababe
Subject: Mathematics & Computer Science, Analysis Keywords: harmonic; univalent; convex; partial sums
Online: 22 June 2019 (12:35:50 CEST)
Partial sums of analytic univalent functions and partial sums of starlike have been investigated extensively by several researchers. In this paper, we investigate a partial sums of convex harmonic functions that are univalent and sense preserving in the open unit disk.
Preprint SHORT NOTE | doi:10.20944/preprints202109.0217.v1
Harmonisation of Classical Wave Equation
Alireza Jamali
Subject: Mathematics & Computer Science, Applied Mathematics Keywords: harmonic analysis; nonlinear partial differential equation
Online: 13 September 2021 (15:36:10 CEST)
In this short note we present a technique using which one attributes frequency and wavevector to (almost) arbitrary scalar fields. Our proposed definition is then applied to the classical wave equation to yield a novel nonlinear PDE.
PCR, PLS, or OPLS Evaluation of different regression techniques for hypothesis generation
Avani Ahuja
Subject: Keywords: Principal Component Regression, Partial Least Squares, Orthogonal Partial Least Squares, multivariate regression, hypothesis generation, Parkinson's disease
Online: 29 November 2021 (15:42:03 CET)
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In the current era of 'big data', scientists are able to quickly amass enormous amount of data in a limited number of experiments. The investigators then try to hypothesize about the root cause based on the observed trends for the predictors and the response variable. This involves identifying the discriminatory predictors that are most responsible for explaining variation in the response variable. In the current work, we investigated three related multivariate techniques: Principal Component Regression (PCR), Partial Least Squares or Projections to Latent Structures (PLS), and Orthogonal Partial Least Squares (OPLS). To perform a comparative analysis, we used a publicly available dataset for Parkinson' disease patien ts. We first performed the analysis using a cross-validated number of principal components for the aforementioned techniques. Our results demonstrated that PLS and OPLS were better suited than PCR for identifying the discriminatory predictors. Since the X data did not exhibit a strong correlation, we also performed Multiple Linear Regression (MLR) on the dataset. A comparison of the top five discriminatory predictors identified by the four techniques showed a substantial overlap between the results obtained by PLS, OPLS, and MLR, and the three techniques exhibited a significant divergence from the variables identified by PCR. A further investigation of the data revealed that PCR could be used to identify the discriminatory variables successfully if the number of principal components in the regression model were increased. In summary, we recommend using PLS or OPLS for hypothesis generation and systemizing the selection process for principal components when using PCR.rewordexplain later why MLR can be used on a dataset with no correlation
Generalized Contrcations on Dualistic Partial Metric Spaces
Muhammad Nazam, Muhammad Arshad, Aftab Hussain
Subject: Mathematics & Computer Science, Analysis Keywords: fixed point; dualistic partial metric; generalized contraction
Online: 9 August 2016 (11:28:16 CEST)
We introduce the notion of generalized contraction on dualistic partial metric spaces. A fixed point theorem for mappings satisfying above mentioned contraction is obtained. Some consequences of our result are obtained. We construct an example to demonstrate the effectiveness of our result among the corresponding results in partial metric spaces. Our results provide substantial generalizations and improvements of several well known results existing in the comparable literature. We discuss an application of our fixed point results to show the existence of solution of functional equations.
A Literature Review of Semi-functional Partial Linear Regression Models
Mohammad Fayaz
Subject: Mathematics & Computer Science, Probability And Statistics Keywords: Functional Data Analysis (FDA); Hybrid Data; Semi-Functional Partial Linear Regression Model (SFPLR); Partial Functional Linear Regression; Literature Review
Background: In the functional data analysis (FDA), the hybrid or mixed data are scalar and functional datasets. The semi-functional partial linear regression model (SFPLR) is one of the first semiparametric models for the scalar response with hybrid covariates. Various extensions of this model are explored and summarized. Methods: Two first research articles, including "semi-functional partial linear regression model", and "Partial functional linear regression" have more than 300 citations in Google Scholar. Finally, only 106 articles remained according to the inclusion and exclusion criteria such as 1) including the published articles in the ISI journals and excluding 2) non-English and 3) preprints, slides, and conference papers. We use the PRISMA standard for systematic review. Results: The articles are categorized into the following main topics: estimation procedures, confidence regions, time series, and panel data, Bayesian, spatial, robust, testing, quantile regression, varying Coefficient Models, Variable Selection, Single-index model, Measurement error, Multiple Functions, Missing values, Rank Method and Others. There are different applications and datasets such as the Tecator dataset, air quality, electricity consumption, and Neuroimaging, among others. Conclusions: SFPLR is one of the most famous regression modeling methods for hybrid data that has a lot of extensions among other models.
Partial Discharge and Internet of Things: A Switchgear Cell Maintenance Application using Microclimate Sensors
Radu Fechet, Adrian Ioan Petrariu, Adrian Graur
Subject: Engineering, Electrical & Electronic Engineering Keywords: electrical grid; switchgear; preventive maintenance; IoT; partial discharge
Online: 2 November 2021 (10:33:39 CET)
This paper proposes a solution for the development of microclimate monitoring for Low Voltage / High Voltage switchgear using the PRTG Internet of Things (IoT) platform. This IoT-based real time monitoring system can enable predictive maintenance to reduce the risk of electrical station malfunctions due to unfavorable environmental conditions. The combination of humidity and dust can lead to unplanned electrical discharges along the isolators inside a low or medium voltage electric table. If no predictive measures are taken, the situation may deteriorate and lead to significant damage inside and outside the switchgear cell. Thus, the mentioned situation can lead to unprogrammed maintenance interventions that can conduct to the change of the entire affected switchgear cell. Using a low-cost and efficient system, the climate conditions inside and outside the switchgear is monitored and transmitted remotely to a monitoring center. From the results obtained using a 365-day time interval, we can conclude that the proposed system is integrated successfully in the switchgear maintaining process, having as result the reducing of maintenance costs.
"A little flip goes a long way" The impact of a 'partial'-flipped classroom design on student performance and engagement in a first year undergraduate economics classroom
Subject: Social Sciences, Accounting Keywords: partial flipped classroom; active learning pedagogies; micro lectures
Online: 9 October 2020 (08:49:50 CEST)
The flipped classroom is gaining prominence as an active learning pedagogy to engage a new generation of students. However, all courses do not lend themselves to a fully flipped design and instructors are often reluctant to flip lectures. In this study, I experimented with a "partial" flipped classroom design in a first-year undergraduate economics course. In this partial flipped format, traditional lectures were substituted with micro-lectures and the remaining class time was devoted to activities like quizzes, group work and student presentations. The full lectures were panopto recorded and put up on the e-learning site, Blackboard. This format enabled me to combine the benefits of a traditional lecture with a flipped classroom design. In order to evaluate the effectiveness of the partial flipped classroom format, I compared the final exam scores of students in the partial flipped classroom with those in the control group, which followed a traditional lecture-based approach. The key results from the analysis revealed that students in the partial flipped classroom performed better in the final exams vis-à-vis students in the traditional classroom format. Furthermore, the partial flipped classroom format was associated with lower odds of students failing in the module. This format also resulted in better student engagement, more flexibility and enhanced student-tutor interaction within the classroom.
Study of the Mobilization of Uranium Isotopes in a Sandstone Aquifer Using Methods for the Extraction of Uranium with Different Strength Reagents in Combination with Groundwater Data
Alexander I. Malov, Sergey B. Zykov
Subject: Earth Sciences, Environmental Sciences Keywords: partial extraction; mineral phases; uranium; disequilibrium; retardation factor
A partial extraction procedure was used to study the distribution of uranium in the mineral phases of rocks of an aquifer of sandy-clay deposits of the Vendian in the northwest of Russia. This work is a part of a research project to develop a method for combined radiocarbon and uranium-isotope dating of groundwater. Representative aliquots of each core sample were subjected to five "partial" extractions by treatment with: distilled water, low mineralized fresh natural groundwater, minopolycarboxylic acid chelating agent (0.05M EDTA), 0.5M HCl, 15M HNO3, and a total digestion, with U isotopes reported in this study for each procedure. The following mineral phases of core samples: adsorbed material, carbonate minerals, amorphous iron oxides, aluminosilicates partial digestion and a crystalline iron oxides, aluminosilicates total digestion and a clay/quartz resistate were characterized. Red-colored siltstones depleted in uranium in relatively readily soluble mineral phases. The concentration of adsorbed uranium was established in the amount of 15.8±2.1 - 30.5±3.9 μg/kg. Carbonate minerals contain even less of this element. In iron hydroxides and the most readily soluble aluminosilicates, its concentrations are in the range 168±24 - 212±28 μg/kg. The most insoluble fraction contains 1.65±0.21 - 4.32±0.45 mg/kg of uranium. In green-colored siltstones, the concentration of adsorbed uranium is much higher: 106±14 - 364±43 μg/kg. Carbonate minerals and amorphous iron oxides contain 1.91±0.21 - 2.34±0.26 mg/kg of uranium. In aluminosilicates and a clay/quartz resistate, uranium concentrations are 5.6±0.5 - 16.8±1.4 mg/kg. Elevated values of 234U:238U activity ratio prevail in the adsorbed material and iron hydroxides. In aluminosilicates and clay/quartz resistate, the values decrease. This indicates the replacement of primary sedimentogenic uranium by secondary hydrogenic uranium adsorbed on the surface of minerals and coprecipitated with iron hydroxides. The results obtained made it possible to carry out preliminary quantitative estimates of the retardation factor and recoil loss factor of uranium in the groundwater of siltstones of the studied Vendian aquifer.
A Fast K-prototypes Algorithm Using Partial Distance Computation
Byoungwook KIM
Subject: Mathematics & Computer Science, General & Theoretical Computer Science Keywords: clustering algorithm; k-prototypes algorithm, partial distance computation
The k-means is one of the most popular and widely used clustering algorithm, however, it is limited to only numeric data. The k-prototypes algorithm is one of the famous algorithms for dealing with both numeric and categorical data. However, there have been no studies to accelerate k-prototypes algorithm. In this paper, we propose a new fast k-prototypes algorithm that gives the same answer as original k-prototypes. The proposed algorithm avoids distance computations using partial distance computation. Our k-prototypes algorithm finds minimum distance without distance computations of all attributes between an object and a cluster center, which allows it to reduce time complexity. A partial distance computation uses a fact that a value of the maximum difference between two categorical attributes is 1 during distance computations. If data objects have m categorical attributes, maximum difference of categorical attributes between an object and a cluster center is m. Our algorithm first computes distance with only numeric attributes. If a difference of the minimum distance and the second smallest with numeric attributes is higher than m, we can find minimum distance between an object and a cluster center without distance computations of categorical attributes. The experimental shows proposed k-prototypes algorithm improves computational performance than original k-prototypes algorithm in our dataset.
Economic Evaluation of Large-Scale Biorefinery Deployment: A Framework Integrating Dynamic Biomass Market and Techno-Economic Models
Jonas Zetterholm, Elina Bryngemark, Johan Ahlström, Patrik Söderholm, Simon Harvey, Elisabeth Wetterlund
Subject: Engineering, Energy & Fuel Technology Keywords: supply chain; partial equilibrium; biofuel; soft-linking; dynamic prices
Online: 14 July 2020 (11:20:30 CEST)
Biofuels and biochemicals play significant roles in the transition towards a fossil-free society. However, large-scale biorefineries are not yet cost-competitive with their fossil-fuel counterparts, and it is important to identify biorefinery concepts with high economic performance. For evaluating early-stage biorefinery concepts, one needs to consider not only the technical performance and process costs but also the economic performance of the full supply chain and the impacts on feedstock and product markets. This article presents and demonstrates a conceptual interdisciplinary framework that can constitute the basis for evaluations of the full supply-chain performance of biorefinery concepts. This framework considers the competition for biomass across sectors, assumes exogenous end-use product demand, and incorporates various geographical and technical constraints. The framework is demonstrated empirically through a case study of a sawmill-integrated biorefinery producing liquefied biomethane from forestry and forest industry residues. The case study results illustrate that acknowledging biomass market effects in the supply chain evaluation implies changes in both biomass prices and the allocation of biomass across sectors. The proposed framework should facilitate the identification of biorefinery concepts with a high economic performance which are robust to feedstock price changes caused by the increase in biomass demand.
Influence of Disease Duration on Circulating Levels of miRNAs in Children and Adolescents with New Onset Type 1 Diabetes
Nasim Samandari, Aashiq H Mirza, Simranjeet Kaur, Philip Hougaard, Lotte Broendum Nielsen, Siri Fredheim, Henrik B Mortensen, Flemming Pociot
Subject: Life Sciences, Molecular Biology Keywords: children, immunology, miRNA, partial remission phase, type 1 diabetes
The objective of this study was to identify circulating miRNAs affected by disease duration in newly diagnosed children with type 1 diabetes. Forty children and adolescents from The Danish Remission Phase Cohort were followed with blood samples drawn at 1, 3, 6, 12 and 60 months after diagnosis. Pancreatic autoantibodies were measured at each visit. Cytokines were measured only the first year. miRNA expression profiling was performed by RT-qPCR and quantified for 179 human plasma miRNAs. The effect of disease duration was analyzed by mixed models for repeated measurements, adjusted for sex and age. Eight miRNAs (hsa-miR-10b-5p, hsa-miR-17-5p, hsa-miR-30e-5p, hsa-miR-93-5p, hsa-miR-99a-5p, hsa-miR-125b-5p, hsa-miR-423-3p and hsa-miR-497-5p) were found to significantly change expression (adjusted p-value < 0.05) with disease progression. Three pancreatic autoantibodies ICA, IA-2A, GADA65 and 4 cytokines IL-4, IL-10, IL-21, IL-22 were associated with the miRNAs at different time points. Pathway analysis revealed association with various immune-mediated signaling pathways. Eight miRNAs, involved in immunological pathways changed expression levels during the first five years after diagnosis in children with type 1 diabetes, and were associated with variations in cytokine and pancreatic antibodies, suggesting a possible effect on the immunological processes in the early phase of the disease.
Evaluation of Electrical Tree Degradation in Cable Insulation using Weibull Process of Propagation Time
Donguk Jang, Seonghee Park
Subject: Engineering, Electrical & Electronic Engineering Keywords: partial discharge; Weibull distribution; XLPE cable; electric tree; diagnosis
The main purpose of this paper is to evaluate electrical treeing degradation for cable insulation. To effectively deal with the currently facing issues, I endeavor to find the most optimal methods by means of applying signal process. First, we made three type models of electrical tree for PD generation to show the distribution characteristics and applied voltage to acquire data by using a PD detecting system. These acquired data presented distribution and four 2D distributions. Hn(q), Hn(), Hqn(), and Hqmax() were derived from the distribution of partial discharge. From the analysis of these distributions, each PD model is proved to hold its unique characteristics and the results were then applied as basic specific qualities for insulation conditions. In order to recognize the progresses of an electrical tree, we proposed methods using scale parameter by means of Weibull distribution. We measured the time of tree propagation for 16 specimens of each model from initiation stage, middle stage, and breakdown respectively, using these breakdown data, we estimated the shape parameter, scale parameter and MTTF(Mean Time To Failure). The results of this study recognize the sources of PD by applying acquired data from PD signals to pre-acquired data. If the cause of PD is degradation, in other words, electrical tree, we can determine the replacement time of devices at the initiation stage of tree growth progress or no later than the middle stage and use it as a basic methods analysis diagnosis system. That is, pattern recognition and Weibull distribution can be employed to get the reliability of diagnosis.
Frequency Domain Repercussions of Instantaneous Granger Causality
Luiz Antonio Baccalá, Koichi Sameshima
Subject: Mathematics & Computer Science, Probability And Statistics Keywords: instantaneous Granger causality; total partial directed coherence; information partial directed coherence; total directed transfer function; information directed transfer function; Granger connectivity; Granger influentiability.
Using Directed Transfer Function (DTF) and Partial Directed Coherence (PDC) in their information version, this paper extends their theoretical framework to incorporate instantaneous Granger Causality (iGC)'s frequency domain description into a single unified perspective. We show that standard vector autoregressive models allow portraying iGC's repercussions associated with Granger Connectivity where interactions mediated without delay between time series can be easily detected.
Preprint CONCEPT PAPER | doi:10.20944/preprints202105.0580.v1
Influence of Teleconnections on the Precipitation
Erum Aamir
Subject: Earth Sciences, Atmospheric Science Keywords: Teleconnections; Precipitation; Mann Kendall; Partial Mann Kendall; Climate Indices; Trends
Online: 24 May 2021 (15:09:34 CEST)
Precipitation plays vital role in the economy of agricultural country like Pakistan. Baluchistan being the largest province of Pakistan in term of land is facing reoccurring droughts as well as flashflood due unprecedent torrential precipitation pattern.
Indirect Nuclear Magnetic Resonance (NMR) Spectroscopic Determination of Acrylamide in Coffee Using Partial Least Squares (PLS) Regression
Vera Rief, Christina Felske, Andreas Scharinger, Katrin Krumbügel, Simone Stegmüller, Carmen M. Breitling-Utzmann, Elke Richling, Stephan G. Walch, Dirk W. Lachenmeier
Subject: Chemistry, Analytical Chemistry Keywords: acrylamide; coffee; partial least square regression; NMR; LC-MS/MS
Acrylamide is probably carcinogenic to humans (International Agency for Research on Cancer, group 2A) with major occurrence in heated, mainly carbohydrate-rich foods. For roasted coffee, a European Union benchmark level of 400 µg/kg acrylamide is of importance. Regularly, the acrylamide contents are controlled using liquid chromatography combined with tandem mass spectrometry (LC-MS/MS). This reference method is reliable and precise but laborious because of the necessary sample clean-up procedure and instrument requirements. This research investigates the possibility of predicting the acrylamide content from proton nuclear magnetic resonance (NMR) spectra that are already recorded for other purposes of coffee control. In the NMR spectrum acrylamide is not directly quantifiable, so that the aim was to establish a correlation between the reference value and the corresponding NMR spectrum by means of a partial least squares (PLS) regression. Therefore, 40 commercially available coffee samples with already available LC-MS/MS data and NMR spectra were used as calibration data. To test the accuracy and robustness of the model and its limitations, 50 coffee samples with extreme roasting degrees and blends were additionally prepared as test set. The PLS model shows an applicability for the varieties C. arabica and C. canephora, which were medium to very dark roasted using drum or infrared roasters. The root mean square error of prediction (RMSEP) is 79 µg/kg acrylamide (n=32). The PLS model is judged as suitable to predict the acrylamide values of commercially available coffee samples. On the other hand, very light roasts containing more than 1000 µg/kg acrylamide are currently not suitable for PLS prediction.
Impact of DC Voltage Enhancement on Partial Discharges in Medium Voltage Cables - An Empirical Study with Defects at Semicon-Dielectric Interface
Aditya Shekhar, Xianyong Feng, Angelo Gattozzi, Robert Hebner, Douglas R. Wardell, Shannon Strank, Armando Rodrigo-Mor, Laura M. Ramirez-Elizondo, Pavol Bauer
Subject: Engineering, Electrical & Electronic Engineering Keywords: ac; comparison; dc; discharges; measurements; medium voltage; pd; partial discharges
A scientific consensus is emerging on the benefits of dc distribution in medium voltage power systems of ships and cities. At least 50% space savings and increased power transfer capacity are estimated with enhanced voltage dc operation of electric cables. The goal of this research is to contribute to developing the empirical knowledge on the insulation performance in order to validate the feasibility of such anticipated gains of dc versus ac, and to determine the comparative impact of different operational conditions from a component engineering point of view. The partial discharge (PD) activity in cables is measured under ac and dc conditions as an indicator of insulation performance. Specifically, PDs in defects at the semicon-insulation interface are studied in terms of inception voltage, repetition rate and discharge magnitude. Empirical understanding is drawn for operating voltage and frequency dependence of the discharge behavior in such voids in the range of 10 to 20\,kV and 0 to 0.1\,Hz, respectively. The change in PD activity with void evolution is explored.
Sums and Partial Sums of Horadam Sequences: The Sum Formulas of $\sum_{k=0}^{n}x^{k}W_{k}$ and $ \sum_{k=n}^{n+m}x^{k}W_{k}$ via Generating Functions
Yüksel Soykan
Subject: Mathematics & Computer Science, Algebra & Number Theory Keywords: Horadam numbers; Horadam sequence; sum; partial sum; Fibonacci numbers; Lucas numbers
In this paper, we present sums and partial sums of Horadam sequences via generating functions which extends a recent result of Prodinger.
Preprint CASE REPORT | doi:10.20944/preprints202010.0151.v2
Numerical Investigation of the Collapse of the Steel Truss Roof and a Probable Reason of Failure
Mertol Tufekci, Ekrem Tüfekci, Adnan Dikicioğlu
Subject: Engineering, Automotive Engineering Keywords: Steel truss; roof structure; partial collapse; finite element analysis; lightning strike.
Online: 26 October 2020 (11:35:28 CET)
This study investigates the failure of a roof with steel truss construction of a factory building in Tekirdag in North-western part of Turkey. The failure occurred under hefty weather conditions including thunderbolt, lightning strikes, heavy rain and fierce winds. In order to interpret the reason for the failure, the effects of different combinations of factors on the design and dimensioning of the roof are checked. Therefore, finite element analysis is performed several times under different assumptions and considering different factors aiming to determine the dominant ones that are responsible for the failure using the commercial software Abaqus (Dassault Systèmes, Vélizy-Villacoublay, France). Each loading condition gives out a characteristic form of failure. The scenario with the most similar form of failure to the real collapse is considered as the most likely scenario of failure. Also, the factors included in this scenario are expected to be the responsible factors for the partial collapse of the steel truss structure.
Partial Information Decomposition and the Information Delta: A Geometric Unification Disentangling Non-Pairwise Information
James Kunert-Graf, Nikita Sakhanenko, David Galas
Subject: Mathematics & Computer Science, Other Keywords: Partial Information Decomposition; Information Delta; Synergy; Co-Information; Non-Pairwise Dependence
Information theory provides robust measures of multivariable interdependence, but classically does little to characterize the multivariable relationships it detects. The Partial Information Decomposition (PID) characterizes the mutual information between variables by decomposing it into unique, redundant, and synergistic components. This has been usefully applied, particularly in neuroscience, but there is currently no generally accepted method for its computation. Independently, the Information Delta framework characterizes non-pairwise dependencies in genetic datasets. This framework has developed an intuitive geometric interpretation for how discrete functions encode information, but lacks some important generalizations. This paper shows that the PID and Delta frameworks are largely equivalent. We equate their key expressions, allowing for results in one framework to apply towards open questions in the other. For example, we find that the approach of Bertschinger et al. is useful for the open Information Delta question of how to deal with linkage disequilibrium. We also show how PID solutions can be mapped onto the space of delta measures. Using Bertschinger et al. as an example solution, we identify a specific plane in delta-space on which this approach's optimization is constrained, and compute it for all possible three-variable discrete functions of a three-letter alphabet. This yields a clear geometric picture of how a given solution decomposes information
Research on a Partial Aperture Factor Measurement Method for the Agri Onboard Calibration Assembly
Xiaolong Si, Xiuju Li, Hongyao Chen, Shiwei Bao, Heyu Xu, Liming Zhang, Wenxin Huang
Subject: Physical Sciences, Optics Keywords: onboard calibration; partial aperture factor; solar diffuser; absolute radiometric calibration, remote sensors
A partial aperture onboard calibration method can solve the onboard calibration problems of some large aperture remote sensors, which is of great significance for the development trend of increasingly large apertures in optical remote sensors. In this paper, the solar diffuser reflectance degradation monitor (SDRDM) in the onboard calibration assembly (CA) of the FengYun-4 (FY-4) advanced geostationary radiance imager (AGRI) is used as the reference radiometer for measuring the partial aperture factor (PAF) for the AGRI onboard calibration. First, the linear response count variation relationship between the two is established under the same radiance source input. Then, according to the known bidirectional reflection distribution function (BRDF) of the solar diffuser (SD) in the CA, the relative reflectance ratio coefficient between the AGRI observation direction and the SDRDM observation direction is calculated. On this basis, the response count value of the AGRI and the SDRDM is used to realize the high-precision measurement of the PAF of the AGRI B1 ~ B3 bands by simulating the AGRI onboard calibration measurement under the illumination of a solar simulator in the laboratory. According to the determination process of the relevant parameters of the PAF, the measurement uncertainty of the PAF is analyzed; this uncertainty is better than 2.04% and provides an important reference for the evaluation of the onboard absolute radiometric calibration uncertainty after launch.
MPCR-Net: Multiple Partial Point Clouds Registration Network Using a Global Template
Shijie Su, Chao Wang, Ke Chen, Jian Zhang, Yang Hui
Subject: Engineering, Industrial & Manufacturing Engineering Keywords: point cloud registration; template point cloud; multiple partial point cloud; deep learning
With the advancement of photoelectric technology and computer image processing technology, the visual measurement method based on point clouds is gradually applied to the 3D measurement of large workpieces. Point cloud registration is a key step in 3D measurement, and its registration accuracy directly affects the accuracy of 3D measurements. In this study, we designed a novel MPCR-Net for multiple partial point cloud registration networks. First, an ideal point cloud was extracted from the CAD model of the workpiece and was used as the global template. Next, a deep neural network was used to search for the corresponding point groups between each partial point cloud and the global template point cloud. Then, the rigid body transformation matrix was learned according to these correspondence point groups to realize the registration of each partial point cloud. Finally, the iterative closest point algorithm was used to optimize the registration results to obtain a final point cloud model of the workpiece. We conducted point cloud registration experiments on untrained models and actual workpieces, and by comparing them with existing point cloud registration methods, we verified that the MPCR-Net could improve the accuracy and robustness of the 3D point cloud registration.
Basic Properties of the Polygonal Sequence
Kunle Adegoke
Subject: Mathematics & Computer Science, Algebra & Number Theory Keywords: polygonal number; triangular number; summation identity; generating function; partial sum; recurrence relation
We study various properties of the polygonal numbers; such as their recurrence relations; fundamental identities; weighted binomial and ordinary sums; partial sums and generating functions of their powers; and a continued fraction representation for them. A feature of our results is that they are presented naturally in terms of the polygonal numbers themselves and not in terms of arbitrary integers; unlike what obtains in most literature.
Partial Derivative Approach to the Change of Scale Formula for the Function Space Integral
Young Sik Kim
Subject: Mathematics & Computer Science, Algebra & Number Theory Keywords: function space; function space integral; partial derivative approach; change of scale formula
We investigate the behavior of the partial derivative approach to the change of scale formula and prove relationships among the analytic Wiener integral and the analytic Feynman integral of the partial derivative for the function space integral.
Mathematical Description of Elastic Phenomena which Uses Caputo or Riemann-Liouville Fractional Order Partial Derivatives is Nonobjective
Agneta M. Balint, Stefan Balint, Silviu Birauas
Subject: Mathematics & Computer Science, Applied Mathematics Keywords: objectivity of a mathematical description; elastic phenomena description; fractional order partial derivative
In this paper it is shown that mathematical description of strain, constitutive law and dynamics obtained by direct replacement of integer order derivatives with Caputo or Riemann-Liouville fractional order partial derivatives, having integral representation on finite interval, in case of a guitar string, is nonobjective. The basic idea is that different observers, using this type of descriptions, obtain different results which cannot be reconciled, i.e. transformed into each other using only formulas that link the coordinates of the same point in two fixed orthogonal reference frames and formulas that link the numbers representing the same moment of time in two different choices of the origin of time measuring. This is not an academic curiosity! It is rather a problem: which one of the obtained results is correct?
Force Estimation during Cell Migration Using Mathematical Modelling
Feng Wei Yang, Chandrasekar Venkataraman, Sai Gu, Vanessa Styles, Anotida Madzvamuse
Subject: Mathematics & Computer Science, Computational Mathematics Keywords: cell migration; optimal control; geometric partial differential equations; mechanical membrane forces; cell polarisation
Cell migration is essential for physiological, pathological and biomedical processes such as, in embryogenesis, wound healing, immune response, cancer metastasis, tumour invasion and inflammation. In light of this, quantifying mechanical properties during the process of cell migration is of great interest in experimental sciences, yet few theoretical approaches in this direction have been studied. In this work, we propose a theoretical and computational approach based on the optimal control of geometric partial differential equations to estimate cell membrane forces associated with cell polarisation during migration. Specifically, cell membrane forces are inferred or estimated by fitting a mathematical model to a sequence of images, allowing us to capture dynamics of the cell migration. Our approach offers a robust and potentially accurate framework to compute geometric mechanical membrane forces associated with cell polarisation during migration and also yields geometric information of independent interest, we illustrate one such example that involves quantifying cell proliferation levels which are associated with cell division, cell fusion or cell death.
Generalized Hyperharmonic Number Sums With Reciprocal Binomial Coefficients
Rusen Li
Subject: Mathematics & Computer Science, Algebra & Number Theory Keywords: generalized hyperharmonic numbers, classical Euler sums, binomial coefficients, combinatorial approach, partial fraction approach
In this paper, we mainly show that generalized hyperharmonic number sums with reciprocal binomial coefficients can be expressed in terms of classical (alternating) Euler sums, zeta values and generalized (alternating) harmonic numbers.
Correcting the Correction: A Revised Formula to Estimate Partial Correlations between True Scores
Debra Wetcher-Hendricks
Subject: Social Sciences, Accounting Keywords: Classical Test Theory; Classical True-Score Theory; Correction for Attenuation; Partial Correlation Coefficient
Bohrnstedt's (1969) attempt to derive a formula to compute the partial correlation coefficient and simultaneously correct for attenuation sought to simplify the process of performing each task separately. He suggested that his formula, developed from algebraic and psychometric manipulations of the partial correlation coefficient, produces a corrected partial correlation value. However, an algebraic error exists within his derivations. Consequently, the formula proposed by Bohrnstedt does not appropriately represent the value he intended it to estimate. By correcting the erroneous step and continuing the derivation based upon his proposed procedure, the steps outlined in this paper ultimately produce the formula that Bohrnstedt desired.
Partial Inversion of the Elliptic Operator to Speed Up Computation of Likelihood in Bayesian Inference
Alexander Litvinenko
Subject: Mathematics & Computer Science, Applied Mathematics Keywords: innovation; data misfit; likelihood; Bayesian formula; partial inverse; domain decomposition; FEM; hierarchical matrices
Often, when solving forward, inverse or data assimilation problems, only a part of the solution is needed. As a model, we consider the stationary diffusion problem. We demonstrate an algorithm that can compute only a part or a functional of the solution, without calculating the full inversion operator and the complete solution. It is a well-known fact about partial differential equations that the solution at each discretisation point depends on the solutions at all other discretisation points. Therefore, it is impossible to compute the solution only at one point, without calculating the solution at all other points. The standard numerical methods like a conjugate gradient or Gauss elimination compute the whole solution and/or the complete inverse operator. We suggest a method which can compute the solution of the given partial differential equation 1) at a point; 2) at few points; 3) on an interface; or a functional of the solution, without computing the solution at all points. The required storage cost and computational resources will be lower as in the standard approach. With this new method, we can speed up, for instance, computation of the innovation in filtering or the likelihood distribution, which measures the data misfit (mismatch). Further, we can speed up the solution of the regression, Bayesian inversion, data assimilation, and Kalman filter update problems. Applying additionally the hierarchical matrix approximation, we reduce the cubic computational cost to almost linear $\mathcal{O}(k^2n \log^2 n)$, where $k\ll n$ and $n$ is the number of degrees of freedom. Up to the hierarchical matrix approximation error, the computed solution is exact. One of the disadvantages of this method is the need to modify the existing deterministic solver.
Robust Haebara Linking for Many Groups in the Case of Partial Invariance
Alexander Robitzsch
Subject: Social Sciences, Other Keywords: linking; item response model; 2PL model; Haebara linking; differential item functioning; partial invariance
Online: 4 June 2020 (13:28:42 CEST)
The comparison of group means in item response models constitutes an important issue in empirical research. The present article discusses an extension of Haebara linking by proposing a flexible class of robust linking functions for comparisons of many groups. These robust linking functions are particularly suited to item response data that are generated under partial invariance. In a simulation study, it is shown that the newly proposed robust Haebara linking approach outperforms existing approaches of Haebara linking. In an empirical application using PISA data, it is illustrated that country means can be sensitive to the choice of linking functions.
Hypoxia, Partial EMT and Collective Migration: Emerging Culprits in Metastasis
Kritika Saxena, Mohit Kumar Jolly, Kuppusamy Balamurugan
Subject: Life Sciences, Other Keywords: hypoxia; HIF-1α; partial EMT; collective migration; inflammatory breast cancer; E-cadherin; metastasis
Epithelial-mesenchymal transition (EMT) is a cellular biological process involved in migration of primary cancer cells to secondary sites facilitating metastasis. Besides, EMT also confers properties such as stemness, drug resistance and immune evasion which can aid a successful colonization at the distant site. EMT is not a binary process; recent evidence suggests that cells in partial EMT or hybrid E/M phenotype(s) can have enhanced stemness and drug resistance as compared to those undergoing a complete EMT. Moreover, partial EMT enables collective migration of cells as clusters of circulating tumor cells or emboli, further endorsing that cells in hybrid E/M phenotypes may be the 'fittest' for metastasis. Here, we review mechanisms and implications of hybrid E/M phenotypes, including their reported association with hypoxia. Hypoxia-driven activation of HIF-1α can drive EMT. In addition, cyclic hypoxia, as compared to acute or chronic hypoxia, shows the highest levels of active HIF-1α and can augment cancer aggressiveness to a greater extent, including enriching for a partial EMT phenotype. We also discuss how metastasis is influenced by hypoxia, partial EMT and collective cell migration, and call for a better understanding of interconnections among these mechanisms. We discuss the known regulators of hypoxia, hybrid EMT and collective cell migration and highlight the gaps which needs to be filled for connecting these three axes which will increase our understanding of dynamics of metastasis and help control it more effectively.
A Partial Information Decomposition Based on Causal Tensors
David Sigtermans
Subject: Keywords: information theory; causal inference; causal tensors; transfer entropy; partial information decomposition; left monotonicity; identity property; unobserved common cause
We propose a partial information decomposition based on the newly introduced framework of causal tensors, i.e., multilinear stochastic maps that transform source data into destination data. The innovation that causal tensors introduce is that the framework allows for an exact expression of an indirect association in terms of the constituting, direct associations. This is not possible when expressing associations only in measures like mutual information or transfer entropy. Instead of a priori expressing associations in terms of mutual information or transfer entropy, the a posteriori expression of associations in these terms results in an intuitive definition of a nonnegative and left monotonic redundancy, which also meets the identity property. Our proposed redundancy satisfies the three axioms introduced by Williams and Beer. Symmetry and self-redundancy axioms follow directly from our definition. The data processing inequality ensures that the monotonicity axiom is satisfied. Because causal tensors can describe both mutual information as transfer entropy, the partial information decomposition applies to both measures. Results show that the decomposition closely resembles the decomposition of other another approach that expresses associations in terms of mutual information a posteriori. A negative synergistic term could indicate that there is an unobserved common cause.
Preprint ARTICLE | doi:10.3390/sci1010030
Health Insurance Coverage Before and After the Affordable Care Act
Jesse Patrick, Philip Q. Yang
Subject: Keywords: health insurance coverage; determinants; the Affordable Care Act; Obamacare; partial implementation; full implementation
The Affordable Care Act (ACA) is at the crossroads. It is important to evaluate the effectiveness of the ACA in order to make rational decisions about the ongoing healthcare reform, but existing research into its effect on health insurance status in the United States is insufficient and descriptive. Using data from the National Health Interview Surveys from 2009 to 2015, this study examines changes in health insurance status and its determinants before the ACA in 2009, during its partial implementation in 2010–2013, and after its full implementation in 2014 and 2015. The results of trend analysis indicate a significant increase in national health insurance rate from 82.2% in 2009 to 89.4% in 2015. Logistic regression analyses confirm the similar impact of age, gender, race, marital status, nativity, citizenship, education, and poverty on health insurance status before and after the ACA. Despite similar effects across years, controlling for other variables, youth aged 26 or below, the foreign-born, Asians, and other races had a greater probability of gaining health insurance after the ACA than before the ACA; however, the odds of obtaining health insurance for Hispanics and the impoverished rose slightly during the partial implementation of the ACA but somewhat declined after the full implementation of the ACA starting in 2014. These findings should be taken into account by the U.S. government in deciding the fate of the ACA.
Seed Rain and Seedling Establishment of Picea glauca and Abies balsamea after Partial Cutting in Plantations and Natural Stands
Laurent Gagné, Luc Sirois, Luc Lavoie
Subject: Biology, Forestry Keywords: balsam fir; white spruce; seedlings; partial cut; plantation; naturals stands; light; seed rain
Online: 30 August 2018 (05:36:10 CEST)
This study documents the conditions associated to white spruce and balsam fir regeneration after partial cutting. Measurements were collected 9 to 30 years after partial cutting in 12 natural fir stands and 5 white spruce plantations. We estimated seed input, measured light reaching the undergrowth, recorded seedlings (<150 cm) and their age on 6 different seedling establishment substrates: mineral soil, moss, rotten wood, litterfall, herbaceous and dead wood. Partial cutting generally favours the establishment and growth of seedlings. The number of fir and spruce seedlings is always greater in natural stands than in plantations, a trend likely associated with the reduced abundance of preferential establishment substrate in the latter. White spruce significantly prefers rotten wood while fir settles on all types of substrates that cover at least 10% of the forest floor. There is a strong relationship between light intensity and the median height of spruce seedlings, but this relationship is non-significant for fir. Seedlings of both species can survive at incident light intensities as low as 3%, but an intensity of 15% or more seems to offer the best growth conditions. The conditions for successful forest regeneration proposed in this study should be applied when the goal is to establish a new stand prior to clear cutting or to convert stand structure.
Disturbance Elimination for Partial Discharge Detection in Spacer of Gas-Insulated Switchgears
Guoming Wang, Gyung-Suk Kil, Hong-Keun Ji , Jong-Hyuk Lee
Subject: Engineering, Electrical & Electronic Engineering Keywords: partial discharge; gas-insulated switchgears; spacer; capacitive component; wavelet transform; multi-resolution analysis
Online: 24 October 2017 (04:47:02 CEST)
With the increasing demand for precise condition monitoring and diagnosis of gas-insulated switchgears (GIS), it has become a challenge to improve the detection sensitivity of partial discharge (PD) induced in the GIS spacer. This paper deals with the elimination of the capacitive component from the phase resolved partial discharge (PRPD) signal generated in GIS spacer based on the discrete wavelet transform. Three types of typical insulation defects were simulated using PD cells. The single PD pulses were detected and were further used to determine the optimal mother wavelet. As a result, the bior6.8 was selected to decompose the PD signal into 8 levels and the signal energy at each level was calculated. The decomposed components related with capacitive disturbance were discarded whereas those associated with PD were de-noised by a threshold and a thresholding function. Finally, the PRPD signals were reconstructed using the de-noised components.
Synthesis and Thrombin, Factor Xa and U46619 Inhibitory Effects of Non-amidino and Amidino N2-Thiophenecarbonyl- and N2-Tosylanthranilamides
Soo Hyun Lee, Wonhwa Lee, Nguyen Thi Ha, Il Soo Um, Jong-Sup Bae, Eunsook Ma
Subject: Chemistry, Other Keywords: N2-Arylcarbonyl/sulfonylanthranilamides; Prothrombin time; Activated partial thromboplastin time; Thrombin; Factor Xa; U46619
Thrombin (factor IIa) and factor Xa (FXa) are key enzymes at the junction of the intrinsic and extrinsic coagulation pathways and are the most attractive pharmacological targets for the development of novel anticoagulants. Twenty non-amidino N2-thiophencarbonyl- and N2-tosyl anthranilamides 1-20 and six amidino N2-thiophencarbonyl- and N2-tosylanthranilamides 21-26 were synthesized and evaluated prothrombin time (PT) and activated partial thromboplastin time (aPTT) using human plasma at concentration 30 μg/mL in vitro. From these results, compounds 5, 9, and 21-23 were selected to study the further antithrombotic activity. The anticoagulant properties of 5, 9, and 21-23 significantly exhibited a concentration-dependent prolongation of in vitro PT and aPTT, in vivo bleeding time, and ex vivo clotting time. These compounds concentration-dependently inhibited the activities of thrombin and FXa and inhibited the generation of thrombin and FXa in human endothelial cells. In addition, data showed that 5, 9, and 21-23 significantly inhibited thrombin catalyzed fibrin polymerization and mouse platelet aggregation and inhibited platelet aggregation induced U46619 in vitro and ex vivo. N-(3'-Amidinophenyl)-2-((thiophen-2''-yl)carbonyl amino)benzamide (21) was most active.
Optimizing Hyperparameters and Architecture of Deep Energy Method
Charul Chadha, Diab Abueidda, Seid Koric, Erman Guleryuz, Iwona Jasiuk
Subject: Mathematics & Computer Science, Computational Mathematics Keywords: Elasticity; Machine learning; Minimum potential energy; Partial differential equations (PDEs); Physics-informed neural network
The deep energy method (DEM) employs the principle of minimum potential energy to train neural network models to predict displacement at a state of equilibrium under given boundary conditions. The accuracy of the model is contingent upon choosing appropriate hyperparameters. The hyperparameters have traditionally been chosen based on literature or through manual iterations. The displacements predicted using hyperparameters suggested in the literature do not ensure the minimum potential energy of the system. Additionally, they do not necessarily generalize to different load cases. Selecting hyperparameters through manual trial and error and grid search algorithms can be highly time-consuming. We propose a systematic approach using the Bayesian optimization algorithms and random search to identify optimal values for these parameters. Seven hyperparameters are optimized to obtain the minimum potential energy of the system under compression, tension, and bending loads cases. In addition to Bayesian optimization, Fourier feature mapping is also introduced to improve accuracy. The models trained using optimal hyperparameters and Fourier feature mapping could accurately predict deflections compared to finite element analysis for linear elastic materials. The deflections obtained for tension and compression load cases are found to be more sensitive to values of hyperparameters compared to bending. The approach can be easily extended to 3D and other material models.
Mid-infrared Laser Spectroscopy Detection and Quantification of Explosives in Soils Using Multivariate Analysis and Artificial Intelligence
Leonardo C. Pacheco-Londoño, Eric Warren, Nataly J. Galan-Freyle, Reynaldo Villarreal-Gonzalez, Joaquin A. Aparicio-Bolaño, Maria L. Ospina-Castro, Wei-Chuan Shih, Samuel P. Hernandez-Rivera
Subject: Chemistry, Applied Chemistry Keywords: quantum cascade laser; remote detection; partial least squares; high explosives; artificial intelligence; machine learning
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A tunable quantum cascade laser (QCL) spectrometer was used to develop methods for detecting and quantifying high explosives (HE) in soil based on multivariate analysis (MVA) and artificial intelligence (AI). For quantification, mixes of 2,4-dinitrotoluene (2,4-DNT) with concentrations from 0% to 20% w/w were investigated using three types of soils: bentonite, synthetic soil, and natural soil. A Partial least squares regression model was generated for predicting 2,4-DNT concentrations. To increase its selectivity, the model was trained and evaluated using additional analytes as interferences, including other HEs such as PETN, RDX, and TNT and non-explosives such as benzoic acid and ibuprofen. For detection, mixes of different explosives in soils were used to implement two AI strategies. In the first strategy, the spectra of the samples were compared with those of soils recorded in a database to identify the most similar soils based on QCL spectroscopy. Next, a Classical Least Squares preprocessing (Pre-CLS) was applied to soils spectra selected from the database. The parameter obtained based on the sum of the weights of Pre-CLS was then used to generate a simple binary discrimination model for distinguishing between contaminated and uncontaminated soils, achieving an accuracy of 0.877. In the second AI strategy, the same parameter was added to a principal component matrix obtained from spectral data of samples and used to generate multi-classification models based on different machine learning algorithms. A Random Forest model worked best with 0.997 accuracy and allowing to distinguish between soils contaminated with DNT, TNT, or RDX and uncontaminated soils.
Safety and Efficacy of Treating Symptomatic, Partial-Thickness Rotator Cuff Tear with Fresh, Uncultured, Unmodified, Autologous Adipose Derived Regenerative Cells (UA-ADRCs) Isolated at the Point of Care: A Prospective, Randomized, Controlled First-in-Human Pilot Study
Jason L. Hurd, Tiffany R. Facile, Jennifer Weiss, Matthew Hayes, Meredith Hayes, John P. Furia, Nicola Maffulli, Glenn E. Winnier, Christopher Alt, Christoph Schmitz, Eckhard U. Alt, Mark A. Lundeen
Subject: Medicine & Pharmacology, Sport Sciences & Therapy Keywords: adipose-derived regenerative cells; ADRCs; partial rotator cuff tear; stem cells; stromal vascular fraction
Background: This study tested the hypothesis that treatment of symptomatic, partial-thickness rotator cuff tear (sPTRCT) with fresh, uncultured, unmodified, autologous adipose derived regenerative cells (UA-ADRCs) isolated from lipoaspirate at the point of care is safe and more effective than corticosteroid injection. Methods: Subjects aged between 30 and 75 years with sPTRCT who did not respond to physical therapy treatments for at least six weeks were randomly assigned to receive a single injection of an average 11.4×106 UA-ADRCs (in 5 mL liquid; mean cell viability: 88%) (n=11; modified intention-to-treat (mITT) population) or a single injection of 80 mg of methylprednisolone (40 mg/ml; 2 mL) plus 3 mL of 0.25% bupivacaine (n=5; mITT population), respectively. Safety and efficacy were assessed using the American Shoulder and Elbow Surgeons Standardized Shoulder Assessment Form (ASES), RAND Short Form-36 Health Survey and pain visual analogue scale (VAS) at baseline (BL) as well as three weeks (W3), W6, W9, W12, W24, W32, W40 and W52 post treatment. Fat-saturated T2 weighted magnetic resonance imaging of the shoulder was performed at BL as well as at W24 and W52 post treatment. Results: No severe adverse events related to the injection of UA-ADRCs were observed in the 12 months post treatment. The risks connected with treatment of sPTRCT with UA-ADRCs were not greater than those connected with treatment of sPTRCT with corticosteroid injection. However, one subject in the corticosteroid group developed a full rotator cuff tear during the course of this pilot study. Despite the small number of subjects in this pilot study, those in the UA-ADRCs group showed statistically significantly higher mean ASES total scores at W24 and W52 post treatment than those in the corticosteroid group (p < 0.05). Discussion: This pilot study suggests that the use of UA-ADRCs in subjects with sPTRCT is safe and leads to improved shoulder function without adverse effects. To verify the results of this initial safety and feasibility pilot study in a larger patient population, a randomized controlled trial on 246 patients suffering from sPTRCT is currently ongoing.
Integration of Peach (Prunus persica L) Biochar, Compost and Peach Residues along with Beneficial Microbes and Phosphorus Improve Agronomic Efficiency, Phosphorus use Efficiency, Partial Factor Productivity and Soil P in Soybean vs Maize Crops
Imran Imran, Amanullah Amanullah
Subject: Earth Sciences, Other Keywords: peach carbon sources; agronomic efficiency; pue; partial factor productivity; soil p; biochar; trichoderma; ps
Phosphorus (P) is an important element in a complete and balanced fertility program that can improve crop P use efficiency and ultimately productivity and profitability. Phosphatic fertilizers use without organic fertilizers leads to gradual decline in soil organic matter, native nutrient status and ultimately reduction in agricultural productivity and economic growth. The objectives of this was to evaluate P efficiencies with incorporation of peach sources, beneficial microbes and P application. From sustainability points of view, alternative use of different sources and forms of organic sources alone or in combination with inorganic P and beneficial microbes possess potential for improving productive capacity of the soil. Separate field experiments (one each on maize and soybean as a test crop) were conducted at Agriculture Research Institute Mingora Swat (ARI) for two consecutive years in summer season of 2016 (year one) and 2017 (year two). For the first time such a study were conducted to utilize peach leftovers and biomass (leaves, twigs, fruits, stones and barks partially decomposed, its compost and biochar) along with three phosphorus (P) levels (50, 75, 100 kg P ha-1) and two beneficial microbes (PSB and Trichoderma) on such a way to enhance soil sustainability and P use efficiency of soybean and maize. The results revealed that organic sources had significant effect on soybean and maize P use efficiency (PUE), P agronomic efficiency (PAE), partial factor productivity (PFPp) and soil P concentration. In experiment 1 among the organic sources, peach residues increased soil P (12.0 mg kg-1) as compared to peach compost and biochar (8.6 & 11.7 mg kg-1). Soil P concentration was maximum (12.1 mg kg-1) with PSB than Trichoderma (9.5 mg kg-1). Application of P at 100 kg ha-1increased soil P contents (16.9 mg kg-1) as compared to 50 and 75 kg P ha 1 (5.9 & 9.6 mg kg-1) respectively. P concentration was increased drastically in year 2 (12.4 mg kg-1) than year one (9.1 mg kg-1). PUE in both crops (soybean and maize) was maximum (25.6 & 28.4%) with peach biochar than compost and residues along with Trichoderma (21.7 & 27.8%). Highest PUE in soybean was recorded with 75 kg P ha-1(22.2%) however in maize maximum PUE was noted with 50 kg P ha-1(33.5%). PAE and PFPp in both crops was maximum with biochar and soil application of Trichoderma than other organic sources and PSB. Among the P levels highest PAE in soybean and maize was recorded with 75 kg ha-1whereas PFPp in soybean was maximum with 75 kg P ha-1 and interestingly in maize it was noted with 50 kg ha-1. Conclusively soybean and maize PAE, PFPp and PUE was higher with biochar, soil incorporation of Trichoderma and P at the rate of 75 kg ha-1 and can improve soybean and maize yield and soil productivity on sustainable basis.
Research on ASP Flooding to Improve Oil Recovery in Low Permeability Reservoirs Based on Laboratory Experiments of a Partial Quality Tool
Bin Huang, Xinyu Hu, Cheng Fu, Quan Zhou
Subject: Chemistry, Chemical Engineering Keywords: ASP flooding; low permeability oil layer; partial quality tool; maximum injection pressure; oil recovery
In order to solve the problem of the poor oil displacement effect of high molecular weight alkali/surfactant/polymer (ASP) solution in low permeability reservoirs, Daqing Oilfield uses a partial quality tool to improve the oil displacement effect in low permeability reservoirs. Without changing the oil displacement capability of high molecular weight ASP solution in high permeability oil layers, the ASP solution is actively sheared in low permeability oil layers by using a partial quality tool to increase the injection capability of the solution and improve the overall oil recovery. In order to study the ability of the partial quality tool to improve the oil displacement effect, firstly, the matching degree of high molecular weight ASP solution to low permeability cores is studied, and the ability of quality control tools to change the molecular weight is studied. Then, experimental research on the pressure and oil displacement effect of high molecular weight ASP solution before and after the actions of the partial quality tool is carried out. The results show that ASP solutions with molecular weights of 1900 × 104 and 2500 × 104 have a poor oil displacement effect in low permeability reservoirs. After the action of the partial quality tool, the injection pressure is reduced by 5.22 MPa, and the oil recovery is increased by 7.79%. The injection pressure of the ASP solution after shearing by the partial quality tool is lower than that of the ASP solution with the same molecular weight and concentration without shearing, but the oil recovery is lower. On the whole, the use of the partial quality tool can obviously improve the oil displacement effect in low permeability reservoirs.
Moisture Content Measurement of Broadleaf Litters using Near-Infrared Spectroscopy Technique
Ghiseok Kim, Suk-Ju Hong, Ah-Yeong Lee, Ye-Eun Lee, Sangjun Im
Subject: Biology, Forestry Keywords: near-infrared spectroscopy; multivariate analysis; partial least-squares regression; floor litter; optimal wavelength selection
Near-infrared spectroscopy (NIRS) was implemented to monitor the moisture content of broadleaf litters. Partial least-squares regression (PLSR) models, incorporating optimal wavelength selection techniques, have been proposed to better predict the litter moisture of forest floor. Three broadleaf litters were used to sample the reflection spectra corresponding the different degrees of litter moisture. Maximum normalization preprocessing technique was successfully applied to remove unwanted noise from the reflectance spectra of litters. Four variable selection methods were also employed to extract the optimal subset of measured spectra for establishing the best prediction model. The results showed that the PLSR model with the peak of beta coefficients method was the best predictor among all candidate models. The proposed NIRS procedure is thought to be a suitable technique for on-the-spot evaluation of litter moisture.
A Confirmatory Framework PLS-SEM for Construction Waste Reduction as Part of Achieving Sustainable Development Goals of a Building
Musa Mohammed, Nasir Shafiq, Ali Elmansoury, Noor Amila wan Abdallah Zawawi, Abubakar Muhammad
Subject: Engineering, Automotive Engineering Keywords: Effective Construction; Waste Reduction; achieving Sustainable Development Goals; partial least square structural equation modeling (PLS-SEM)
As a result of rapid population growth, an exponentially growing human population, and industrial expansion, it has become increasingly difficult to manage municipal solid wastes throughout the world. Decentralized waste management systems have created difficult situations in developing countries such as Malaysia. Wastes generated in the country, due to various cultural, social, and religious activities, organic and contributing to environmental pollution (air, water, and soil) and human health troubles. A questionnaire survey was participated by 220 construction professionals in Malaysia using structured and semi-structured methods. The framework was assessed using A partial least square structural equation modeling (PLS-SEM) to target sustainable development goals (SDG). Statistical analysis results indicate a significant effect between SCW management, since(r(270)=.687, P<0.001). Improving factors has strong relationship with SCW management, since(r(270)=.723, P<0.001). The mediation results also suggested a significant indirect positive effect of improving factors drivers on SCW management through policy-related factors sinceβ=0.688, t=8.254, P<0.001, 95% CI for β=0.536,0.866. Finally, policy-related factors construct has a strong relationship with SCWM) management, since(r(270)=.811, P<0.001) With the R Square of 0.787 and 0.785. The developed framework can improve construction waste management in the construction industry and enhance construction waste management to achieve global sustainable development goals. The findings show that one of the most critical issues of enhancing profitability is using preventive policies to reduce construction waste. This study could guide construction industry stakeholders in identifying the different waste management features during a building project's construction and design stage
Partial Unlock for COVID-19-Like Epidemics Can Save 1-3 Million Lives Worldwide
Subject: Life Sciences, Virology Keywords: epidemic; caseload management; partial unlock; social distancing; overshoot; COVID-19; coronavirus; economic impact; ventilator utilization; SARS-CoV-2
Background: A large percentage of deaths in an epidemic or pandemic can be due to overshoot of population (herd) immunity, either from the initial peak or from planned or unplanned exit from lockdown or social distancing conditions. Objectives: We study partial unlock or reopening interaction with seasonal effects in a managed epidemic to quantify overshoot effects on small and large unlock steps and discover robust strategies for reducing overshoot. Methods: We simulate partial unlock of social distancing for epidemics over a range of replication factor, immunity duration and seasonality factor for strategies targeting immunity thresholds using overshoot optimization. Results: Seasonality change must be taken into account as one of the steps in an easing sequence, and a two step unlock, including seasonal effects, minimizes overshoot and deaths. It may cause undershoot, which causes rebounds and assists survival of the pathogen. Conclusions: Partial easing levels, even low levels for economic relief while waiting on a vaccine, have population immunity thresholds based on the reduced replication rates and may experience overshoot as well. We further find a two step strategy remains highly sensitive to variations in case ratio, replication factor, seasonality and timing. We demonstrate a three or more step strategy is more robust, and conclude that the best possible approach minimizes deaths under a range of likely actual conditions which include public response.
mitoMaker: A Pipeline for Automatic Assembly and Annotation of Animal Mitochondria Using Raw NGS Data
Alex Schomaker-Bastos, Francisco Prosdocimi
Subject: Biology, Animal Sciences & Zoology Keywords: mitochondrial DNA; mitochondrial genome; genome assembly; genome annotation; next generation sequencing; animal genomics; partial genomics; bioinformatics
Next-generation sequencing is now a mature technology, allowing partial animal genomes to be produced for many clades. Though many software exist for genome assembly and annotation, a simple pipeline that allows researchers to input raw sequencing reads in fastq format and allow the retrieval of a completely assembled and annotated mitochondrial genome is still missing. mitoMaker 1.0 is a pipeline developed in python that implements (i) recursive de novo assembly of mitochondrial genomes using a set of increasing k-mers; (ii) search for the best matching result to a target mitogenome and; (iii) performs iterative reference-based strategies to optimize the assembly. After (iv) checking for circularization and (v) positioning tRNA-Phe at the beginning, (vi) geneChecker.py module performs a complete annotation of the mitochondrial genome and provides a GenBank formatted file as output.
Locating Partial Discharges in Power Transformers with Convolutional Iterative Filtering
Jonathan Wang, Kesheng Wu, Alex Sim, Seongwook Hwangbo
Subject: Mathematics & Computer Science, Information Technology & Data Management Keywords: Partial discharges; Source location; UHF measurements; Time of arrival estimation; Waveform analysis; FDTD methods; Nonlinear wave propagation
The most common source of transformer failure is in the insulation, and the most prevalent warning signal for insulation weakness is partial discharge (PD). Locating positons of these partial discharges would help repair the transformer to prevent failures. This work investigates algorithms that could be deployed to locate the position of a PD event using data from ultra-high frequency (UHF) sensors inside the transformer. These algorithms typically proceed in two steps: first determine the signal arrival time and then locate the position based on time differences. This paper reviews available methods for each task and then propose new algorithms: a convolutional iterative filter with thresholding (CIFT) to determine the signal arrival time and a reference table of travel times to resolve the source location. The effectiveness of these algorithms are tested with a set of laboratory-triggered PD events and two sets of simulated PD events inside transfers in production use. Tests show the new approach provides more accurate locations than the best-known data analysis algorithms, and the difference is particularly large, 3.7X, when the signal sources are far from sensors.
Chemometrics-Assisted Monitoring in Raman Spectroscopy for the Biodegradation Process of an Aqueous Polyfluoroalkyl Ether from a Fire-Fighting Foam in an Environmental Matrix
Mario Marchetti, Marc Offroy, Ferroudja Abdat, Philippe Branchu, Patrice Bourson, Céline Jobard, Jean-François Durmont, Guillaume Casteran
Subject: Earth Sciences, Environmental Sciences Keywords: environmental fate; Raman spectroscopy; chemometrics; principal component analysis, biodegradation; kinetics; post-processing; Whittaker filter; partial least square
Surfactants based on polyfluoroalky ethers are commonly used in fire-fighting foams on airport platforms, including for training sessions. Because of their persistence into the environment, their toxicity and their bioaccumulation, abnormal amounts can be found in ground and surface water following operations of airport platforms. As many other anthropogenic organic compounds, some concerns raised about their biodegradation. That is why the OECD 301 F protocol was implemented to appreciate the oxygen consumption during the biodegradation of a commercial fire-fighting foam. However, a Raman spectroscopic monitoring of the process was also attached to this experimental procedure to evaluate to what extent a polyfluoroalkyl ether disappeared from the environmental matrix. The relevance of our approach is to use chemometrics, including the Principal Component Analysis (PCA) and the Partial Least Square (PLS), in order to monitor the kinetics of the biodegradation reaction of one fire-fighting foam, Tridol S3B, containing a polyfluoroalkyl ether. This study provided a better appreciation of the partial biodegradation of some polyfluoroalkyl ethers by coupling Raman spectroscopy and chemometrics. This will ultimately facilitates the design of a future purification and remediation devices for the airport platforms.
Fourier Transform on the Homogeneous Space of 3D Positions and Orientations for Exact Solutions to PDEs
Remco Duits, Erik J. Bekkers, Alexey Mashtakov
Subject: Mathematics & Computer Science, Analysis Keywords: Fourier transform; rigid body motions; partial differential equations; Lévy processes; lie groups; homogeneous spaces; stochastic differential equations
Fokker-Planck PDEs (incl. diffusions) for stable Lévy processes (incl. Wiener processes) on the joint space of positions and orientations play a major role in mechanics, robotics, image analysis, directional statistics and probability theory. Exact analytic designs and solutions are known in the 2D case, where they have been obtained using Fourier transform on $SE(2)$. Here we extend these approaches to 3D using Fourier transform on the Lie group $SE(3)$ of rigid body motions. More precisely, we define the homogeneous space of 3D positions and orientations $\mathbb{R}^{3}\rtimes S^{2}:=SE(3)/(\{\mathbf{0}\} \times SO(2))$ as the quotient in $SE(3)$. In our construction, two group elements are equivalent if they are equal up to a rotation around the reference axis. On this quotient we design a specific Fourier transform. We apply this Fourier transform to derive new exact solutions to Fokker-Planck PDEs of $\alpha$-stable Lévy processes on $\mathbb{R}^{3}\rtimes S^{2}$. This reduces classical analysis computations and provides an explicit algebraic spectral decomposition of the solutions. We compare the exact probability kernel for $\alpha = 1$ (the diffusion kernel) to the kernel for $\alpha=\frac12$ (the Poisson kernel). We set up SDEs for the Lévy processes on the quotient and derive corresponding Monte-Carlo methods. We verify that the exact probability kernels arise as the limit of the Monte-Carlo approximations.
General and Prospective Views on Oxidation Reactions in Heterogeneous Catalysis
Sabine Valange, Jacques Védrine
Subject: Chemistry, Chemical Engineering Keywords: heterogeneous catalytic oxidation, gas-solid, liquid-solid, partial and total oxidation, biomass based raw materials, activation methods
In this short review paper we have assembled the main characteristics of partial oxidation reactions (oxidative dehydrogenation and selective oxidation to olefins or oxygenates, as aldehydes and carboxylic acids and nitriles), as well as total oxidation, particularly for depollution, environmental issues and wastewater treatments. Both gas-solid and liquid-solid media have been considered with recent and representative examples within these fields. We have also discussed about their potential and prospective industrial applications. Particular attention has been brought to new raw materials stemming from biomass and to liquid-solid catalysts cases. This review paper also summarizes the progresses made in the use of unconventional activation methods for performing oxidation reactions, highlighting the synergy of these technologies with heterogeneous catalysis. Focus has been centered on usual catalysts activation methods but also on less usual ones, such as the use of ultrasounds, microwaves, grinding (mechanochemistry) and photo-activated processes, as well as their combined use.
Parametric Gevrey Asymptotics in Two Complex Time Variables through Truncated Laplace Transforms
Stephane Malek, Alberto Lastra, Guoting Chen
Subject: Mathematics & Computer Science, Analysis Keywords: asymptotic expansion; Borel-Laplace transform; Fourier transform; initial value problem; formal power series; partial differential equation; singular perturbation
The work is devoted to the study of a family of linear initial value problems of partial differential equations in the complex domain, dealing with two complex time variables. The use of a truncated Laplace-like transformation in the construction of the analytic solution allows to overcome a small divisor phenomenon arising from the geometry of the problem and represents an alternative approach to the one proposed in a recent work by the first two authors. The result leans on the application of a fixed point argument and the classical Ramis-Sibuya theorem.
On Inner Expansions for A Singularly Perturbed Cauchy Problem with Confluent Fuchsian Singularities
Subject: Mathematics & Computer Science, Analysis Keywords: asymptotic expansion; Borel-Laplace transform; Cauchy problem; formal power series; integro-differential equation; partial differential equation; singular perturbation
A nonlinear singularly perturbed Cauchy problem with confluent fuchsian singularities is examined. This problem involves coefficients with polynomial dependence in time. A similar initial value problem with logarithmic reliance in time has been investigated by the author in a recent work, for which sets of holomorphic inner and outer solutions were built up and expressed as a Laplace transform with logarithmic kernel. Here, a family of holomorphic inner solutions are constructed by means of exponential transseries expansions containing infinitely many Laplace transforms with special kernel. Furthermore, asymptotic expansions of Gevrey type for these solutions relatively to the perturbation parameter are established.
Revisiting Degrees of Freedom of Full-Duplex Systems with Opportunistic Transmission: An Improved User Scaling Law
Haksoo Kim, Juyeop Kim, Sang Won Choi, Won-Yong Shin
Subject: Engineering, Electrical & Electronic Engineering Keywords: degrees of freedom (DoF); full-duplex systems; hybrid opportunistic scheduling; partial channel state information (CSI); user scaling law
Online: 7 January 2018 (11:46:28 CET)
It was recently studied how to achieve the optimal degrees of freedom (DoF) in a multi-antenna full-duplex system with partial channel state information (CSI). In this paper, we revisit the DoF of a multiple-antenna full-duplex system using opportunistic transmission under the partial CSI, in which a full-duplex base station having M transmit antennas and M receive antennas supports a set of half-duplex mobile stations (MSs) having a single antenna each. Assuming no self-interference, we present a new hybrid opportunistic scheduling method that achieves the optimal sum DoF under an improved user scaling law. It is shown that the optimal sum DoF of 2M is asymptotically achievable provided that the number of MSs scales faster than SNRM, where SNR denotes the signal-to-noise ratio. This result reveals that in our full-duplex system, better performance on the user scaling law can be obtained without extra CSI, compared to the prior work that showed the required user scaling condition (i.e., the minimum number of MSs for guaranteeing the optimal DoF) of SNR2M−1.
Numerical Modeling, Simulation and Evaluation of Conventional and Hybrid Photovoltaic Modules Interconnection Configurations under Partial Shading Conditions
Faisal Saeed, Haider Ali Tauqeer, Hasan Erteza Gelani, Muhammad Hassan Yousuf
Subject: Keywords: PV array configurations; Partial shading conditions; Performance assessment; Maximum Power Generation; Mismatch power loss; Relative power loss; Fill factor
Partial shading on solar photovoltaic (PV) arrays is a prevalent problem in photovoltaic systems that impair the performance of PV modules and is responsible for reduced power output as compared to that in standard irradiance conditions thereby resulting in the appearance of multiple maximas on panel output power characteristics. These maxims contribute to mismatch power losses among PV modules. The mismatch losses depend on shading characteristics together with different interconnected configuration schemes of PV modules. The research presents a comparative analysis of partial shading effects on a 4 x4 PV array system connected in series(S), parallel (P), serries-parallel (SP),total-cross-tied (TCT),central-cross-tied(CCT),bridge-linked(BL),bridge-linked total cross-tied (BLTCT) ,honey-comb(HC), honey-comb total-cross-tied (HCTCT) and ladder (LD) configurations using MATLAB/Simulink. The PV module SPR-X20-250-BLK was used for modeling and simulation analysis. Each module is comprised of 72 number of PV cells and a combination of 16 PV modules was employed for the contextual analysis. Accurate mathematical modeling for the HCTCT configuration under partial shading conditions (PSCs) is provided for the first time and is verified from the simulation. The different configuration schemes were investigated under short-narrow,short-wide,long-narrow,long-wide, diagonal, entire row distribution, and entire column distribution partial shading condition patterns with mathematical implementation and simulation of passing clouds. The performance of array configurations is compared in terms of maximum power generated ), mismatch power loss (∆), relative power loss ) and the fill factor (FF). It was inferred that on average, TCT configuration yielded maximum power generation under all shading patterns among all PV modules interconnection configurations with minimum mismatch power losses followed by hybrid and conventional PV array configurations respectively.
Some Notes on the Parametric Gevrey Asymptotics in Two Complex Time Variables Through Truncated Laplace Transforms
Guoting Chen, Alberto Lastra, Stephane Malek
Subject: Mathematics & Computer Science, Algebra & Number Theory Keywords: Asymptotic expansion; Borel-Laplace transform; Fourier transform; initial value problem; formal power series; linear partial differential equation; singular perturbation
This paper is a slightly modified, abridged version of a previous work ``Parametric Gevrey asymptotics in two complex time variables through truncated Laplace transforms'' motivated by our contribution in the conference ``Formal and Analytic Solutions of Diff. (differential, partial differential, difference, q-difference, q-difference-differential) Equations on the Internet'' (FASnet20). It aims to clarify and give further detail at some crucial points concerning the asymptotic behavior of the solutions of the problems studied in that work.
Subject: Mathematics & Computer Science, Analysis Keywords: Asymptotic expansion; Borel-Laplace transform; Fourier transform; initial value problem; formal power series; linear partial differential equation; singular perturbation
This paper is a slightly modified, abridged version of a previous work "Parametric Gevrey asymptotics in two complex time variables through truncated Laplace transforms'' motivated by our contribution in the conference "Formal and Analytic Solutions of Diff. (differential, partial differential, difference, q-difference, q-difference-differential) Equations on the Internet'' (FASnet20). It aims to clarify and give further detail at some crucial points concerning the asymptotic behavior of the solutions of the problems studied in that work.
Performance Evaluation of Relay Selection Schemes in Beacon-Assisted Dual-hop Cognitive Radio Wireless Sensor Networks under Impact of Hardware Noises
Tran Dinh Hieu, Tran Trung Duy, Le The Dung, Seong Gon Choi
Subject: Mathematics & Computer Science, Numerical Analysis & Optimization Keywords: energy harvesting; power beacon; decode-and-forward (DF); partial relay selection; opportunistic relay selection; underlay cognitive radio; hardware impairments
To solve the problem of energy constraint and spectrum scarcity for cognitive radio wireless sensor networks (CR-WSNs), an underlay decode-and-forward relaying scheme is considered, where the energy constrained secondary source and relay nodes are capable of harvesting energy from a multi-antenna power beacon (PB) and using that harvested energy to forward the source information to the destination. Based on the time switching receiver architecture, three relaying protocols, namely, hybrid partial relay selection (H-PRS), conventional opportunistic relay selection (C-ORS), and best opportunistic relay selection (B-ORS) protocols are considered to enhance the end-to-end performance under the joint impact of maximal interference constraint and transceiver hardware impairments. For performance evaluation and comparison, we derive exact and asymptotic closed-form expressions of outage probability (OP) and throughput (TP) to provide significant insights into the impact of our proposed protocols on the system performance over Rayleigh fading channel. Finally, simulation results validate the theoretical results.
Statistical Approach to Assess Chill and Heat Requirements of Olive Tree Based on Flowering Date and Temperatures Data: Towards Selection of Adapted Cultivars to Global Warming
Omar Abou-Saaid, Adnane El Yaacoubi, Abdelmajid Moukhli, Ahmed El Bakkali, Sara Oulbi, Magalie Delalande, Isabelle Farrera, Jean-Jacques Kelner, Sylvia Lochon-Menseau, Cherkaoui El Modafar, Hayat Zaher, Bouchaib Khadari
Subject: Biology, Agricultural Sciences & Agronomy Keywords: Olea europaea L.; flowering data; partial least squares regression; Dynamic model; chill requirements; climate change; Mediterranean fruit tree; adapted cultivars.
Delineating chilling and forcing periods is one of the challenging topics in understanding how temperatures drive the timing of budburst and bloom in fruit tree species. Here, we investigated this question on olive trees, using flowering data collected over six years on 331 cultivars in the worldwide collection of Marrakech, Morocco. Using a Partial Least Squares approach on a long-term phenology (29 years) of 'Picholine Marocaine' cultivar, we showed that the relevance of delineating the chilling and forcing periods depends more on the variability of inter-annual temperatures than on the long-term datasets. In fact, chilling and forcing periods are similar between those delineated by using datasets of 29 years and those of only 6 years (2014–2019). We demonstrated that the variability of inter-annual temperatures is the main factor explaining this pattern. We then used the datasets of six years to assess the chill and heat requirements of 285 cultivars. We classified Mediterranean olive cultivars into four groups according to their chill requirements. Our results, using the Kriging interpolation method, indicated that flowering dates of most of these cultivars (92%) were governed by both chilling and forcing temperatures. Our investigations provided first insights to select adapted cultivars to global warming.
QSAR Model for Predicting the Cannabinoid Receptor 1 Binding Affinity and Dependence Potential of Synthetic Cannabinoids
Wonyoung Lee, So-Jung Park, Ji-Young Hwang, Kwang-Hyun Hur, Yong Sup Lee, Jongmin Kim, Xiaodi Zhao, Aekyung Park, Kyung Hoon Min, Choon-Gon Jang, Hyun-Ju Park
Subject: Chemistry, Analytical Chemistry Keywords: cannabinoid receptor 1; synthetic cannabinoids; quantitative structure-activity relationship; multiple linear regression; partial least squares regression; dependence and abuse potential
In recent years, there have been frequent reports on the adverse effects of synthetic cannabinoid (SC) abuse. SCs cause psychoactive effects, similar to those caused by marijuana, by binding and activating cannabinoid receptor 1 (CB1R) in the central nervous system. The aim of this study was to establish a reliable quantitative structure-activity relationship (QSAR) model to correlate the structures and physicochemical properties of various SCs with their CB1R-binding affinities. We prepared 15 SCs and their derivatives (tetrahydrocannabinol [THC], naphthoylindoles, and cyclohexylphenols) and determined their binding affinity to CB1R, which is known as a dependence-related target. We calculated the molecular descriptors for dataset compounds using an R/CDK (R package integrated with CDK, version 3.5.0) toolkit to build QSAR regression models. These models were established and statistical evaluations were performed using the mlr and plsr packages in R software. The most reliable QSAR model was obtained from the partial least squares regression method via external validation. This model can be applied in vivo to predict the addictive properties of illicit new SCs. Using a limited number of dataset compounds and our own experimental activity data, we built a QSAR model for SCs with good predictability. This QSAR modeling approach provides a novel strategy for establishing an efficient tool to predict the abuse potential of various SCs and to control their illicit use.
$L_p$ Loss Functions in Invariance Alignment and Haberman Linking
Subject: Social Sciences, Other Keywords: factor model; 2PL model; linking; invariance alignment; Haberman linking; partial invariance; item response model; structural equation model; differential item functioning
The comparison of group means in latent variable models plays a vital role in empirical research in the social sciences. The present article discusses extensions of invariance alignment and Haberman linking concerning the choice of linking functions for comparisons of many groups. Robust linking functions are proposed for invariance alignment and robust Haberman linking that are particularly suited to item response data under partial invariance. In a simulation study, it is shown that both linking approaches have comparable performance, and in some conditions, the newly proposed robust Haberman linking outperforms invariance alignment.
SCAND1 Reverts EMT and Suppresses Tumor Growth and Collective Migration
Takanori Eguchi, Eva Csizmadia, Thomas L. Prince, Barbara Wegiel, Stuart K. Calderwood
Subject: Medicine & Pharmacology, Oncology & Oncogenics Keywords: epithelial-to-mesenchymal transition (EMT); hybrid EMT; partial EMT; mesenchymal-to-epithelial transition (MET); SCAND1; SCAN zinc finger; MZF1; cancer prognosis
Epithelial-mesenchymal transition (EMT) is a reversible cellular program that transiently places epithelial (E) cells into pseudo-mesenchymal (M) cell states. The malignant progression and resistance of many types of carcinomas depends on EMT activation, partial EMT and hybrid E/M status in neoplastic cells. EMT is activated by tumor microenvironmental TGFβ signal and EMT-inducing transcription factors, such as ZEB1/2 in tumor cells. However, reverse EMT factors are less studied. We demonstrate that transcription factor SCAND1 can revert mesenchymal and hybrid E/M phenotype of cancer cells to a more epithelial, less invasive status and inhibit their proliferation and migration. SCAND1 is a SCAN domain-containing protein and hetero-oligomerizes with SCAN-zinc finger transcription factors, such as MZF1, for accessing DNA and transcriptional co-repression of target genes. We found that SCAND1-MZF1 co-expression and interaction correlated with maintaining epithelial features, whereas the simultaneous loss of SCAND1 and MZF1 correlated with mesenchymal features of tumor cells. Overexpression of SCAND1 over endogenous MZF1 in DU-145 prostate cancer cells reverted their hybrid E/M status into cobblestone morphology with increased epithelial adhesion by E-cadherin and β-catenin relocation. Consistently, co-expression analysis in TCGA PanCancer Atlas revealed that both SCAND1 and MZF1 co-express and are negatively correlated with EMT driver genes, including CTNNB1, ZEB1, ZEB2 and TGFBR, in prostate tumor specimens. In addition, SCAND1 overexpression suppressed tumor cell proliferation by reducing the MAP3K-MEK-ERK signaling pathway. Of note, SCAND1-overexpressing DU-145 cells migrated slower than control cells with decreased lymph node metastasis of prostate cancer in a mouse tumor xenograft model. Kaplan-Meyer analysis showed high expression of MZF1 and SCAND1 to correlate with better prognoses in pancreatic cancer and head and neck cancers, although with poorer prognosis in kidney cancer. Overall, these data suggest that the combination of SCAND1-MZF1 complexes may revert the EMT mechanism in cancer to establish an epithelial phenotype. These effects seem to include co-repression of EMT-driver genes and suppression of tumor cell proliferation via inhibition of the MAP3K-MEK-ERK signaling pathway.
Plasma Spectroscopy of Various Types of Gypsum: An ideal Terrestrial Analogue
Abhishek K. Rai, Jayanta K. Pati, Christian G. Parigger, Awadhesh K. Rai
Subject: Physical Sciences, Atomic & Molecular Physics Keywords: laser-induced plasma; atomic spectroscopy; laser-induced breakdown spectroscopy; 29 atomic spectroscopy; principal component analysis; partial least-square regression; gypsum; Mars
The first detection of gypsum (CaSO4.2H2O) by the Mars Science Laboratory (MSL) rover Curiosity in the Gale Crater, Mars created a profound impact on planetary science and exploration. The unique capability of plasma spectroscopy involving in situ elemental analysis in extraterrestrial environments, suggesting the presence of water in the red planet based on phase characterization and providing a clue to Martian paleoclimate. The key to gypsum as an ideal paleoclimate proxy lies in its textural variants, and in this study terrestrial gypsum samples from varied locations and textural types have been analyzed by Laser Induced Breakdown Spectroscopy (LIBS) technique. Petrographic, sub-microscopic and powder X-ray diffraction characterizations confirm the presence of gypsum (hydrated calcium sulphate; CaSO4.2H2O), bassanite (semi-hydrated calcium sulphate; CaSO4.1/2H2O) and anhydrite (anhydrous calcium sulphate; CaSO4) along with accessory phases (quartz and jarosite). The principal component analysis of LIBS spectra from texturally varied gypsums can be differentiated from one another because of the chemical variability in their elemental concentrations. The concentration of gypsum is determined from the partial least-square regressions model. Rapid characterization of gypsum samples with LIBS is expected to work well in extraterrestrial environments.
On Conformable Double Laplace Transform and One Dimensional Fractional Coupled Burgers' Equation
Hassan Eltayeb, Imed Bachar, Adem Kilicman
Subject: Mathematics & Computer Science, Applied Mathematics Keywords: conformable fractional derivative; conformable partial fractional derivative; conformable double Laplace decomposition method; conformable Laplace transform; Singular one dimensional coupled Burgers equations
This article deals with the conformable double Laplace transforms and their some properties with examples and also the existence Condition for the conformable double Laplace transform is studied. Finally, in order to obtain the solution of nonlinear fractional problems, we present a modified conformable double Laplace that we call conformable double Laplace decomposition methods (CDLDM). Then, we apply it to solve, Regular and singular conformable fractional coupled burgers equation illustrate the effectiveness of our method some examples are given.
Quantum Correlations and Permutation Symmetries
Mrittunjoy Guha Majumdar
Subject: Physical Sciences, Other Keywords: Quantum Entanglement, Separability, Positive Partial Transpose Criterion, Permutation, Quantum Information, Quantum Computing, Quantum Communication, Quantum Non-locality, Quantum Correlations, SWAP Operator
In this paper, the connections between quantum non-locality and permutation symmetries are explored. This includes two types of symmetries: permutation across a superposition and permutation of qubits in a quantum system. An algorithm is proposed for nding the separability class of a quantum state using a method based on factorizing an arbitrary multipartite state into possible partitions, cyclically permuting qubits of the vectors in a superposition to check which separability class it falls into and thereafter using a reduced density-matrix analysis of the system is proposed. For the case of mixed quantum states, conditions for separability are found in terms of the partial transposition of the density matrices of the quantum system. One of these conditions turns out to be the Partial Positive Transpose (PPT) condition. A graphical method for analyzing separability is also proposed. The concept of permutation of qubits is shown to be useful in de ning a new entanglement measure in the `engle'.
Working Paper REVIEW
Measuring and Modelling the Epithelial Mesenchymal Hybrid State in Cancer: Clinical Implications
Mohit Kumar Jolly, Ryan J. Murphy, Sugandha Bhatia, Holly J. Whitfield, Andrew Redfern, Melissa J. Davis, Erik W. Thompson
Subject: Biology, Anatomy & Morphology Keywords: epithelial mesenchymal plasticity (EMP); epithelial mesenchymal transition (EMT); mesenchymal epithelial transition (MET); E/M Hybrid; partial EMT; computational biology; mathematical modeling; cancer
The epithelial-mesenchymal (E/M) hybrid state has emerged as an important mediator of elements of cancer progression, facilitated by epithelial mesenchymal plasticity (EMP). We review here evidence for the presence, prognostic significance, and therapeutic potential of the E/M hybrid state in carcinoma. We further assess modelling predictions and validation studies to demonstrate stabilised E/M hybrid states along the spectrum of EMP, as well as computational approaches for characterising and quantifying EMP phenotypes, with particular attention to the emerging realm of single-cell approaches through RNA sequencing and protein-based techniques.
On Boundary Layer Expansions for a Singularly Perturbed Problem with Confluent Fuchsian Singularities
Subject: Mathematics & Computer Science, Analysis Keywords: asymptotic expansion; Borel-Laplace transform; Fourier transform; initial value problem; formal power series; linear integro-differential equation; partial differential equation; singular perturbation
We consider a family of nonlinear singularly perturbed PDEs whose coefficients involve a logarithmic dependence in time with confluent Fuchsian singularities that unfold an irregular singularity at the origin and rely on a single perturbation parameter. We exhibit two distinguished finite sets of holomorphic solutions, so-called outer and inner solutions, by means of a Laplace transform with special kernel and Fourier integral. We analyze the asymptotic expansions of these solutions relatively to the perturbation parameter and show that they are (at most) of Gevrey order 1 for the first set of solutions and of some Gevrey order that hinges on the unfolding of the irregular singularity for the second.
An Economic Examination of Collateralization in Different Financial Markets
Tim Xiao
Subject: Social Sciences, Finance Keywords: unilateral/bilateral collateralization; partial/full/over collateralization; asset pricing; plumbing of the financial system; swap premium spread; OTC/cleared/listed financial markets
This paper attempts to assess the economic significance and implications of collateralization in different financial markets, which is essentially a matter of theoretical justification and empirical verification. We present a comprehensive theoretical framework that allows for collateralization adhering to bankruptcy laws. As such, the model can back out differences in asset prices due to collateralized counterparty risk. This framework is very useful for pricing outstanding defaultable financial contracts. By using a unique data set, we are able to achieve a clean decomposition of prices into their credit risk factors. We find empirical evidence that counterparty risk is not overly important in credit-related spreads. Only the joint effects of collateralization and credit risk can sufficiently explain unsecured credit costs. This finding suggests that failure to properly account for collateralization may result in significant mispricing of financial contracts. We also analyze the difference between cleared and OTC markets.
Smart Process Optimization and Adaptive Execution with Semantic Services in Cloud Manufacturing ‡
Luca Mazzola, Philipp Waibel, Patrick Kaphanke, Matthias Klusch
Subject: Mathematics & Computer Science, Information Technology & Data Management Keywords: Industry 4.0; XaaS; SemSOA; business process optimization; scalable cloud service deployment; process service plan just-in-time adaptation; BPMN partial fault tolerance
A new requirement for the manufacturing companies in Industry 4.0 is to be flexible with respect to changes in demands, requiring them to react rapidly and efficiently on the production capacities. Together with the trend to use Service-Oriented Architectures (SOA), this requirement induces a need for agile collaboration among supply chain partners, but also between different divisions or branches of the same company. In order to address this collaboration challenge, we~propose a novel pragmatic approach for the process analysis, implementation and execution. This~is achieved through sets of semantic annotations of business process models encoded into BPMN 2.0 extensions. Building blocks for such manufacturing processes are the individual available services, which are also semantically annotated according to the Everything-as-a-Service (XaaS) principles and stored into a common marketplace. The optimization of such manufacturing processes combines pattern-based semantic composition of services with their non-functional aspects. This is achieved by means of Quality-of-Service (QoS)-based Constraint Optimization Problem (COP) solving, resulting in an automatic implementation of service-based manufacturing processes. The produced solution is mapped back to the BPMN 2.0 standard formalism by means of the introduced extension elements, fully detailing the enactable optimal process service plan produced. This approach allows enacting a process instance, using just-in-time service leasing, allocation of resources and dynamic replanning in the case of failures. This proposition provides the best compromise between external visibility, control and flexibility. In this way, it provides an optimal approach for business process models' implementation, with a full service-oriented taste, by implementing user-defined QoS metrics, just-in-time execution and basic dynamic repairing capabilities. This paper presents the described approach and the technical architecture and depicts one initial industrial application in the manufacturing domain of aluminum forging for bicycle hull body forming, where the advantages stemming from the main capabilities of this approach are sketched.
Analysis of Default Mode Network in Social Anxiety Disorder: EEG Resting-State Effective Connectivity Study
Abdulhakim Al-Ezzi, Nidal Kamel, Ibrahima Faye, Esther Gunaseli
Subject: Engineering, Automotive Engineering Keywords: Effective connectivity network, Partial directed coherence (PDC), Social Anxiety Disorder (SAD), Default Mode Network (DMN), Electrophysiological biomarkers (EEG), Resting state network (RSN), Granger
Recent brain imaging findings by using different methods (e.g., fMRI and PET) have suggested that social anxiety disorder (SAD) is correlated with alterations in regional or network-level brain function. However, due to many limitations associated with these methods, such as poor temporal resolution and limited number of samples per second, neuroscientists could not quantify the fast dynamic connectivity of causal information networks in SAD. In this study, SAD-related changes in brain connections within the default mode network (DMN) were investigated using eight electroencephalographic (EEG) regions of interest. Partial directed coherence (PDC) was used to assess the causal influences of DMN regions on each other and indicate the changes in the DMN effective network related to SAD severity. The DMN is a large-scale brain network basically composed of the mesial prefrontal cortex (mPFC), posterior cingulate cortex (PCC)/precuneus, and lateral parietal cortex (LPC). The EEG data were collected from 88 subjects (22 control, 22 mild, 22 moderate, 22 severe) and used to estimate the effective connectivity between DMN regions at different frequency bands: delta (1–3 Hz), theta (4–8 Hz), alpha (8–12 Hz), low beta (13–21 Hz), and high beta (22–30 Hz). Among the healthy control (HC) and the three considered levels of severity of SAD, the results indicated a higher level of causal interactions for the mild and moderate SAD groups than for the severe and HC groups. Between the control and the severe SAD groups, the results indicated a higher level of causal connections for the control throughout all the DMN regions. We found significant increases in the mean PDC in the delta (p = 0.009) and alpha (p = 0.001) bands between the SAD groups. Among the DMN regions, the precuneus exhibited a higher level of causal influence than other regions. Therefore, it was suggested to be a major source hub that contributes to the mental exploration and emotional content of SAD. In contrast to the severe group, HC exhibited higher resting-state connectivity at the mPFC, providing evidence for mPFC dysfunction in the severe SAD group. Furthermore, the total Social Interaction Anxiety Scale (SIAS) was positively correlated with the mean values of the PDC of the severe SAD group, r (22) = 0.576, p = 0.006 and negatively correlated with those of the HC group, r (22) = −0.689, p = 0.001. The reported results may facilitate greater comprehension of the underlying potential SAD neural biomarkers and can be used to characterize possible targets for further medication.
Spin g Factors and Neutrino Magnetic Moment of Elementary Fermions
Jae-Kwang Hwang
Subject: Physical Sciences, Particle & Field Physics Keywords: Anomalous spin g factor; Lepton charge (LC); Neutrino magnetic moment; Weak force decay; EC and LC partial masses; Three-dimensional quantized space model (TQSM)
Online: 1 March 2022 (11:25:10 CET)
The spin magnetic moments and spin g factors (gs = -2) of electron, muon and tau are explained based on the electric charges (EC) and lepton charges (LC) in terms of the three-dimensional quantized space model. The spin g factors of electron, muon and tau are gs = -2 which is the sum of the EC g factor (gEC = -1) and the LC g factor (gLC = -1). The spin g factor (gs = -2) of the electron is predicted by the Dirac's equation. The orbit g factors of electron, muon and tau are gL = gEC = -1 from the EC g factor (gEC = -1) without the contribution of the LC g factor (gLC = -1). The spin g factors of the elementary fermions are calculated from the equation of gs = gEC + gLC + gCC where gEC = EC/|EC|, gLC = LC/|LC| and gCC = CC/|CC|. For example, the spin g factors of the neutrinos and dark matters are gs = -1. The spin g factors of the u and d quarks are gs = 0 and gs = -2, respectively. The g factor problem of neutrinos with the non-zero LC charges are solved by the LC Coulomb force of Fc(LC) ≈0. It is, for the first time, proposed that the binary motion (fluctuations) of the mEC and mLC masses for the electron, muon and tau leptons make the anomalous g factor. This binary motion could be originated from the virtual particle processes including the photons. Also, the weak force (beta) decay is closely related to the binary motion of the mEC and mLC for the electron, muon and tau leptons.
APPLICATION OF NEURAL NETWORK TECHNOLOGIES IN PREDICTION OF COVID-19 INFECTION IN THE WORLD
Eduard Dadyan
Subject: Keywords: time series; forecasting; neural networks; data preprocessing; training and control samples; coronavirus pandemics; Deductor Studio; data cleaning; partial processing; spectral processing; autocorrelation; sliding windows.
Online: 30 March 2021 (14:16:28 CEST)
For analysis tasks, time counts are of interest — values recorded at some, usually equidistant, points in time. The calculation can be performed at various intervals: after a minute, an hour, a day, a week, a month, or a year, depending on how much detail the process should be analyzed. In time series analysis problems, we deal with discrete-time, when each observation of a parameter forms a time frame. The same can be said about the behavior of Covid-19 over time.In this paper, we solve the problem of predicting Covid-19 diseases in the world using neural networks. This approach is useful when it is necessary to overcome difficulties related to non-stationarity, incompleteness, unknown distribution of data, or when statistical methods are not completely satisfactory. The problem of forecasting is solved with the help of the analytical platform Deductor Studio, developed by specialists of the company Intersoft Lab of the Russian Federation. When solving this problem, appropriate methods were used to clean the data from noise and anomalies, which ensured the quality of building a predictive model and obtaining forecast values for tens of days ahead. The principle of time series forecasting was also demonstrated: import, seasonal detection, cleaning, smoothing, building a predictive model, and predicting Covid-19 diseases in the world using neural technologies for 30 days ahead.
Unsteady Analytical Solution of the Influenced of a Thermal Radiation Force Generated from a Heated Rigid Flat Plate on Non-homogeneous Gas Mixture
Taha Abdel Wahid, Taha Abdel-Karim
Subject: Mathematics & Computer Science, Applied Mathematics Keywords: Unsteady Exact analytical solutions; Partial differential equations system; Travelling wave method; Moment method; Boltzmann kinetic equation; Neutral non-homogenous gas; Thermal radiation force; Non-equilibrium irreversible thermodynamics; Internal energy.
In the present paper, the effect of the non-linear thermal radiation on the neutral gas mixture in the unsteady state is investigated for the first time. The unsteady BGK technique of the Boltzmann kinetic equations for a neutral non-homogenous gas is solved. The solution of the unsteady case makes the problem more general significance than the stationary one. For this purpose, the moments' method, together with the traveling wave method, is applied. The temperature and concentration are calculated for each gas component and mixture for the first time.Furthermore, the study is held for aboard range of temperatures ratio parameter and a wide range of the molar fraction. The distribution functions are calculated for each gas component and the gas mixture. The significant non-equilibrium irreversible thermodynamic characteristics the entire system is acquired analytically. That technic allows us to investigate the consistency of Boltzmann's H-theorem, Le Chatelier principle, and thermodynamics laws. Moreover, the ratios among the different participation of the internal energy alteration are evaluated via the Gibbs formula of total energy. The final results are utilized to the argon-helium non-homogenous gas at different magnitudes of radiation force strength and molar fraction parameters. 3D-graphics are presented to predict the behavior of the calculated variables, and the obtained results are theoretically discussed.
Functoriality of the Schmidt Construction
Juan Climent Vidal, Enric Cosme Llópez
Subject: Mathematics & Computer Science, General & Theoretical Computer Science Keywords: Many-sorted partial algebra; free completion; category of completions; weakly initial object; comma category of objects over a completion; Schmidt construction; Schmidt homomorphism; twisted morphism category; Schmidt endofunctor; functoriality of the Schmidt construction
After proving, in a purely categorial way, that the inclusion functor InAlg(Σ) from Alg(Σ), the category of many-sorted Σ-algebras, to PAlg(Σ), the category of many-sorted partial Σ-algebras, has a left adjoint FΣ, the (absolutely) free completion functor, we recall, in connection with the functor FΣ, the generalized recursion theorem of Schmidt, which we will also call the Schmidt construction. Next we define a category Cmpl(Σ), of Σ-completions, and prove that FΣ, labeled with its domain category and the unit of the adjunction of which it is a part, is a weakly initial object in it. Following this we associate to an ordered pair (α,f), where α=(K,γ,α) is a morphism of Σ-completions from F=(C,F,η) to G=(D,G,ρ) and f a homomorphism in D from the partial Σ-algebra A to the partial Σ-algebra B, a homomorphism ΥαG,0(f):Schα(f)B. We then prove that there exists an endofunctor, ΥαG,0, of Mortw(D), the twisted morphism category of D, thus showing the naturalness of the previous construction. Afterwards we prove that, for every Σ-completion G=(D,G,ρ), there exists a functor ΥG from the comma category (Cmpl(Σ)↓G) to End(Mortw(D)), the category of endofunctors of Mortw(D), such that ΥG,0, the object mapping of ΥG, sends a morphism of Σ-completion in Cmpl(Σ) with codomain G, to the endofunctor ΥαG,0. | CommonCrawl |
\begin{document}
\title{Bound state energies and wave functions of spherical quantum dots in presence of a confining potential model } \author{Sameer M. Ikhdair} \email[E-mail: ]{[email protected]} \affiliation{Physics Department, Near East University, Nicosia, Mersin 10, Turkey} \date{
\today
}
\begin{abstract} We obtain the exact energy spectra and corresponding wave functions of the radial Schr\"{o}dinger equation (RSE) for any $(n,l)$ state in the presence of a combination of psudoharmonic, Coulomb and linear confining potential terms using an exact analytical iteration method. The interaction potential model under consideration is Cornell-modified plus harmonic (CMpH) type which is a correction form to the harmonic, Coulomb and linear confining potential terms. It is used to investigates the energy of electron in spherical quantum dot and the heavy quarkonia (QQ-onia).
Keywords: Schr\"{o}dinger equation, confining potentials, spherical quantum dots, Cornell-modified potential, pseudoharmonic oscillator \end{abstract}
\pacs{03.65.Fd; 03.65.Ge; 68.65.Hb } \maketitle
\section{Introduction}
The problem of the inverse-power potential, $1/r^{n},$ has been used on the level of both classical and quantum mechanics. Some series of inverse power potentials are applicable to the interatomic interaction in molecular physics [1-3]. The interaction in one-electron atoms, muonic, hadronic and Rydberg atoms takes into account inverse-power potentials [4]. Indeed, it has also been used for the magnetic interaction between spin-$1/2$ particles with one or more deep wells [5]. The analytical exact solutions of \ this class of inverse-power potentials, $V(r)=Ar^{-4}+Br^{-3}+Cr^{-2}+Dr^{-1},$ $ A>0,$ were presented by Barut \textit{et al.} [6] and \"{O}z\c{c}elik and \c{S}im\c{s}ek [7] by making an available ansatz for the eigenfunctions. The Laurent series solutions of the Schr\"{o}dinger equation for power and inverse-power potentials with two coupling constants $V(r)=Ar^{2}+Br^{-4}$ and three coupling constants $V(r)=Ar^{2}+Br^{-4}+Cr^{-6}$ are obtained [8,9].
The analytic exact iteration method (AEIM) which demands making a trial ansatz for the wave function [7] is general enough to be applicable to a large number of power and inverse-power potentials [10]. Recently, this method is applied to a class of power and inverse-power confining potentials of three coupling constants and containing harmonic oscillator, linear and Coulomb confining terms [11]. This kind of Cornell plus Harmonic (CpH) confining potential of the form $V(r)=ar^{2}+br-cr^{-1}$ is mostly used to study individual spherical quantum dots in semiconductors [12] and heavy quarkonia (QQ-nia) [13,14]. So far, such potentials containing quadratic, linear and Coulomb terms have been studied [15,16].
The present work considers the the following confining interaction potential consisting of a sum of pseudoharmonic, linear and Coulombic potential terms: \begin{equation} V(r)=V_{\text{har}}(r)+V_{\text{Corn-}\func{mod}}(r)=ar^{2}+br-\frac{c}{r}- \frac{d}{r^{2}},\text{ }a>0, \end{equation} where $a,$ $b,$ $c$ and $d$ are arbitrary constant parameters to be determined later. The above potential includes the well-known funnel or Cornell potential, i.e., a Coulomb plus Linear static potential (CpH), $V_{ \text{Corn}}(r)=br-c/r$ [13], and a term $-d/r^{2}$ is incorporated into the quarkonium potential for the sake of coherence [14]. We will refer to the potential model (1) as a Cornell-modified plus harmonic (CMpH) potential, since the functional form has been improved by the additional $-d/r^{2}$ piece; besides the contribution from the additional term also alters the value of $b$ and $c$ [14,17]. The authors of Refs. [14,18] did not consider the harmonic or power-law as the results are expected to be similar. The CMpH potential is plotted in Figure 1 for the values of parameters: $a=1$ $ eV.fm^{-2},$ $b=0.217$ $eV$.$fm^{-1},$ $c=0.400$ $eV.fm$ and $d=0.010$ $ eV.fm^{2}.$
We will apply the AEIM used in [7,11] to obtain the exact energy eigenvalues and wave functions of the radial Schr\"{o}dinger radial equation (RSE) for the\ CMpH potential for any arbitrary $(n,l)$ state.
The paper is structured as follows: In Sect. 2, we obtain the exact energy eigenvalues and wave functions of the RSE in three-dimensions (3D) for the confining CMpH potential model by proposing asuitable form for the wave function. In Sect. 3, we apply our results to an electron in spherical quantum dot of InGaAs semiconductor. The relevant conclusions are given in Sect. 4.
\section{Exact solution of RSE for the confining potential model}
The three-dimensional ($3D$) Schr\"{o}dinger equation takes the form [19] \begin{equation} \left[ -\frac{\hbar ^{2}}{2m}\Delta +V(r)\right] \psi (r,\theta ,\varphi )=E_{nl}\psi (r,\theta ,\varphi ), \end{equation} with \begin{equation*} \Delta =\frac{\partial ^{2}}{\partial r^{2}}+\frac{2}{r}\frac{\partial }{ \partial r}-\frac{L^{2}(\theta ,\varphi )}{\hbar ^{2}r^{2}},\text{ } \end{equation*} where $m$ is the isotropic effective mass and $E_{nl}$ is the total energy of the particle. For any arbitrary state, the complete wave function, $\psi (r,\theta ,\varphi ),$ can be written as \begin{equation} \psi (r,\theta ,\varphi )=\dsum\limits_{n,l}N_{l}\psi _{nl}(r)Y_{lm}(\theta ,\varphi ), \end{equation} where spherical harmonic $Y_{lm}(\theta ,\varphi )$ is the eigenfunction of $ L^{2}(\theta ,\varphi )$ satisfying \begin{equation} L^{2}(\theta ,\varphi )Y_{lm}(\theta ,\varphi )=l(l+1)\hbar ^{2}Y_{lm}(\theta ,\varphi ), \end{equation} and the radial wave function $\psi _{nl}(r)$ is the solution of the equation \begin{equation} \left( \frac{d^{2}}{dr^{2}}+\frac{2}{r}\frac{d}{dr}-\frac{l(l+1)}{r^{2}} \right) \psi _{nl}(r)+\frac{2m}{\hbar ^{2}}\left[ E_{nl}-V(r)\right] \psi _{nl}(r)=0, \end{equation} where $r$ stands for the relative radial coordinates. The radial wave function $\psi _{nl}(r)$ is well-behaved at the boundaries (the finiteness of the solution requires that $\psi _{nl}(0)=\psi _{nl}(r\rightarrow \infty )=0).$ Now, the transformation \begin{equation} \psi _{nl}(r)=\frac{1}{r}\phi _{nl}(r), \end{equation} reduces Eq. (5) to the simple form
\begin{equation} \phi _{nl}^{\prime \prime }(r)+\left[ \varepsilon _{n,l}-a_{1}r^{2}-b_{1}r+ \frac{c_{1}}{r}+\frac{d_{1}-l(l+1)}{r^{2}}\right] \phi _{nl}(r)=0, \end{equation} where $\phi _{nl}(r)$ is the reduced radial wave function and \begin{equation} \varepsilon _{nl}=\frac{2m}{\hbar ^{2}}E_{nl},\text{ }a_{1}=\frac{2m}{\hbar ^{2}}a,\text{ }b_{1}=\frac{2m}{\hbar ^{2}}b,\text{ }c_{1}=\frac{2m}{\hbar ^{2}}c,\text{ }d_{1}=\frac{2m}{\hbar ^{2}}d. \end{equation} The analytic exact iteration method (AEIM) requires making the following ansatze for the wave function [9], \begin{equation} \phi _{nl}(r)=f_{n}(r)\exp \left[ g_{l}(r)\right] , \end{equation} with \begin{subequations} \begin{equation} f_{n}(r)=\left\{ \begin{array}{cc} 1, & n=0, \\ \underset{i=1}{\overset{n}{\Pi }}\left( r-\alpha _{i}^{(n)}\right) , & \text{ }n=1,2,\cdots , \end{array} \right. \end{equation} \begin{equation} g_{l}(r)=-\frac{1}{2}\alpha r^{2}-\beta r+\delta \ln r,\text{ }\alpha >0, \text{ }\beta >0. \end{equation} It is clear that $f_{n}(r)$ are equivalent to the Laguerre polynomials [20]. Substituting Eq. (9) into Eq. (5) we obtain \end{subequations} \begin{equation} \phi _{nl}^{\prime \prime }(r)=\left( g_{l}^{\prime \prime }(r)+g_{l}^{\prime 2}(r)+\frac{f_{n}^{\prime \prime }(r)+2g_{l}^{\prime }(r)f_{n}^{\prime }(r)}{f_{n}(r)}\right) \phi _{nl}(r). \end{equation} and comparing Eq. (11) and Eq. (7) yields \begin{equation} a_{1}r^{2}+b_{1}r-\frac{c_{1}}{r}+\frac{l(l+1)-d_{1}}{r^{2}}-\varepsilon _{nl}=g_{l}^{\prime \prime }(r)+g_{l}^{\prime 2}(r)+\frac{f_{n}^{\prime \prime }(r)+2g_{l}^{\prime }(r)f_{n}^{\prime }(r)}{f_{n}(r)}. \end{equation} First of all, for $n=0,$ let us take $f_{0}(r)$ and $g_{l}(r)$ given in Eq. (10b) to solve Eq. (12), \begin{equation} a_{1}r^{2}+b_{1}r-\varepsilon _{0l}-\frac{c_{1}}{r}+\frac{l(l+1)-d_{1}}{r^{2} }=\alpha ^{2}r^{2}+2\alpha \beta r-\alpha \left[ 1+2\left( \delta +0\right) \right] +\beta ^{2}-\frac{2\beta (\delta +0)}{r}+\frac{\delta \left( \delta -1\right) }{r^{2}}. \end{equation} By comparing the corresponding powers of $r$ on both sides of Eq. (13) we find the following corresponding energy and the restrictions on the potential parameters, \begin{subequations} \begin{equation} \alpha =\sqrt{a_{1}}, \end{equation} \begin{equation} \beta =\frac{b_{1}}{2\sqrt{a_{1}}},\text{ }a_{1}>0, \end{equation} \begin{equation} c_{1}=2\beta \left( \delta +0\right) , \end{equation} \begin{equation} \delta =\frac{1}{2}\left( 1\pm l^{\prime }\right) ,\text{ where }l^{\prime }= \sqrt{\left( 2l+1\right) ^{2}-\frac{8m}{\hbar ^{2}}d} \end{equation} \begin{equation} \varepsilon _{0l}=\alpha \left[ 1+2\left( \delta +0\right) \right] -\beta ^{2}. \end{equation} Actually, to have well-behaved solutions of the radial wave function at boundaries, namely the origin and the infinity, we need to take $\delta $ from Eq. (14d) as \end{subequations} \begin{equation} \delta =\frac{1}{2}\left( 1+l^{\prime }\right) . \end{equation} Therefore, the lowest (ground) state energy from Eq. (14e) together with Eqs. (14a)-(14c), Eq. (15) and Eq. (8) is given as follows \begin{equation} E_{0l}=\sqrt{\frac{\hbar ^{2}a}{2m}}\left( 2+l^{\prime }\right) -\frac{ 2mc^{2}}{\hbar ^{2}\left( 1+l^{\prime }\right) ^{2}}, \end{equation} where the parameter $c$ of potential (1) should satisfy the following restriction: \begin{equation} c=\frac{b}{2\sqrt{\frac{2ma}{\hbar ^{2}}}}\left( 1+\sqrt{\left( 2l+1\right) ^{2}-\frac{8m}{\hbar ^{2}}d}\right) . \end{equation} Furthermore, the substitution of $\alpha ,$ $\beta $ and $\delta $ from Eqs. (14a), (14b) and (15), respectively, together with the parameters given in Eq. (8) into Eqs. (9) and (10), we finally obtain the following ground state wave function: \begin{equation} \psi _{0l}(r)=N_{0l}r^{\left( -1+l^{\prime }\right) /2}\exp \left( -\frac{1}{ 2}\sqrt{\frac{2ma}{\hbar ^{2}}}r^{2}-\frac{2mc}{\hbar ^{2}\left( 1+l^{\prime }\right) }r\right) , \end{equation} with \begin{equation*} N_{0l}=\frac{1}{\sqrt{\Gamma (l^{\prime })D_{-l^{\prime }}\left( \frac{4mc}{ \hbar ^{2}\left( 1+l^{\prime }\right) }\sqrt{\frac{\hbar }{2\sqrt{2ma}}} \right) }}\left( 2\sqrt{\frac{2ma}{\hbar ^{2}}}\right) ^{l^{\prime }/4}\exp \left( -\frac{1}{2}\sqrt{\frac{2m}{\hbar ^{2}a}}\frac{mc^{2}}{\hbar ^{2}\left( 1+l^{\prime }\right) ^{2}}\right) , \end{equation*} where $D_{\nu }(z)$ are the parabolic cylinder functions [21]. It should be noted that the above solutions are well-behaved at the boundaries, i.e., a regular solution near the origin could be $\phi _{nl}(r\rightarrow 0)\rightarrow r^{\left( 1+l^{\prime }\right) /2}$ and asymptotically at infinity as $\phi _{nl}(r\rightarrow \infty )\rightarrow \exp \left( -\alpha r^{2}-\beta r\right) \rightarrow 0.$ When $b=0$ ($c=0$)$,$ the problem turns to become the commoly known pseudoharmonic oscillator (p.h.o.) interaction ($ a=m\omega ^{2}/2$)$,$ and consequently $\alpha =m\omega ,$ $\beta =b/\omega $ and $c=(b\delta /m\omega )$ yielding $E_{0l}=\left( 2+l^{\prime }\right) \frac{\hbar \omega }{2}-\frac{2mc^{2}}{\hbar ^{2}\left( 1+l^{\prime }\right) ^{2}}$ and wave function $\psi _{0l}(r)=N_{l}r^{\left( -1+l^{\prime }\right) /2}\exp \left( -\frac{1}{2}\frac{m\omega }{\hbar }r^{2}-\frac{2mc}{\hbar ^{2}\left( 1+l^{\prime }\right) }r\right) ,$ where \begin{equation*} N_{0l}=\frac{1}{\sqrt{\Gamma (l^{\prime })D_{-l^{\prime }}\left( \frac{4mc}{ \hbar ^{2}\left( 1+l^{\prime }\right) }\sqrt{\frac{\hbar }{2m\omega }} \right) }}\left( \frac{2m\omega }{\hbar }\right) ^{l^{\prime }/4}\exp \left( -\frac{mc^{2}}{4\hbar ^{3}\omega \left( 1+l^{\prime }\right) ^{2}}\right) . \end{equation*} The formula (17) is a relationship between parameters of the potential $a,$ $ b,$ $c$ and $d.$ Therefore, the solutions (16) and (18) are valid for the potential parameters satisfying the restriction (17). Moreover, the relation between the potential parameters (17) depends on the orbital quantum number $ l$ which means that the potential has to be different for different quantum numbers. In applying the AEIM, the obtained solution for any potential is always found to be subjected to certain restrictions on potential parameters as can be traced in other works (see, for example, [7-9,11]).
Secondly, for the first node ($n=1$), using $f_{1}(r)=(r-\alpha _{1}^{(1)})$ and $g_{l}(r)$ from Eq. (10b) to solve Eq. (12), \begin{equation*} a_{1}r^{2}+b_{1}r-\varepsilon _{1l}-\frac{c_{1}}{r}+\frac{l(l+1)-d_{1}}{r^{2} }=\alpha ^{2}r^{2}+2\alpha \beta r \end{equation*} \begin{equation} -\alpha \left[ 1+2\left( \delta +1\right) \right] +\beta ^{2}-\frac{2\left[ \beta \left( \delta +1\right) +\alpha \alpha _{1}^{(1)}\right] }{r}+\frac{ \delta \left( \delta -1\right) }{r^{2}}. \end{equation} The relations between the potential parameters and the coefficients $\alpha , $ $\beta ,$ $\delta $ and $\alpha _{1}^{(1)}$ are \begin{equation*} \alpha =\sqrt{a_{1}},\text{ }\beta =\frac{b_{1}}{2\sqrt{a_{1}}},\text{ } \delta =\frac{1}{2}\left( 1+l^{\prime }\right) ,\text{ }\varepsilon _{1l}=\alpha \left[ 1+2\left( \delta +1\right) \right] -\beta ^{2}. \end{equation*} \begin{equation} c_{1}-2\beta \left( \delta +1\right) =2\alpha \alpha _{1}^{(1)},\text{ } \left( c_{1}-2\beta \delta \right) \alpha _{1}^{(1)}=2\delta , \end{equation} where $c_{1}$ and $\alpha _{1}^{(1)}$ are found from the constraint relations, \begin{subequations} \begin{equation} c=\frac{b}{2\sqrt{\frac{2ma}{\hbar ^{2}}}}\left( 2+l^{\prime }\right) +\sqrt{ \frac{b^{2}}{\frac{8ma}{\hbar ^{2}}}+\frac{\hbar ^{2}}{m}\sqrt{\frac{\hbar ^{2}a}{2m}}\left( 1+l^{\prime }\right) }, \end{equation} \begin{equation} \alpha \alpha _{1}^{(1)2}+\beta \alpha _{1}^{(1)}-\delta =0\text{ } \rightarrow \text{ }\alpha _{1}^{(1)}=-\frac{b}{4a}+\sqrt{\frac{b^{2}}{ 16a^{2}}+\frac{\left( 1+l^{\prime }\right) }{2\sqrt{\frac{2ma}{\hbar ^{2}}}}} . \end{equation} The energy eigenvalue is \end{subequations} \begin{equation*} E_{1l}=\sqrt{\frac{\hbar ^{2}a}{2m}}\left( 4+l^{\prime }\right) -\frac{b^{2} }{4a}, \end{equation*} \begin{equation} b=2\sqrt{\frac{2ma}{\hbar ^{2}}}\frac{\left( 2+l^{\prime }\right) c}{\left( 1+l^{\prime }\right) \left( 3+l^{\prime }\right) }\left[ 1+\sqrt{1+\left( \frac{\hbar ^{2}}{mc^{2}}\sqrt{\frac{\hbar ^{2}a}{2m}}\left( 1+l^{\prime }\right) -1\right) \frac{\left( 1+l^{\prime }\right) \left( 3+l^{\prime }\right) }{\left( 2+l^{\prime }\right) ^{2}}}\right] , \end{equation} and the wave function is \begin{equation} \psi _{1l}(r)=N_{1l}\left( r-\alpha _{1}^{(1)}\right) r^{\left( -1+l^{\prime }\right) /2}\exp \left( -\frac{1}{2}\sqrt{\frac{2ma}{\hbar ^{2}}}r^{2}-\sqrt{ \frac{m}{2\hbar ^{2}a}}br\right) , \end{equation} with \begin{equation*} N_{1l}=\frac{\left( 2\sqrt{\frac{2ma}{\hbar ^{2}}}\right) ^{l^{\prime }/4}\exp \left( -\frac{1}{16}\sqrt{\frac{2m}{a^{3}}}\hbar ^{3}b^{2}\right) }{ \sqrt{\left( 2\sqrt{\frac{2ma}{\hbar ^{2}}}\right) ^{-1}\Gamma (l^{\prime }+2)S_{1}+\alpha _{1}^{(1)2}\Gamma (l^{\prime })S_{2}-2\left( 2\sqrt{\frac{ 2ma}{\hbar ^{2}}}\right) ^{-1/2}\alpha _{1}^{(1)}\Gamma (l^{\prime }+1)S_{3}} }, \end{equation*} where \begin{equation*} S_{1}=D_{-(l^{\prime }+2)}\left( \sqrt{\frac{\hbar }{2a}\sqrt{\frac{2m}{a}}} b\hbar \right) ,\text{ }S_{2}=D_{-l^{\prime }}\left( \sqrt{\frac{\hbar }{2a} \sqrt{\frac{2m}{a}}}b\hbar \right) ,\text{ }S_{3}=D_{-(l^{\prime }+1)}\left( \sqrt{\frac{\hbar }{2a}\sqrt{\frac{2m}{a}}}b\hbar \right) ,\text{ } \end{equation*} and $\alpha _{1}^{(1)}$ is given in Eq. (21b). If there is a p.h.o. interaction, the energy becomes \begin{equation} E_{1l}=\left( 4+l^{\prime }\right) \frac{\hbar \omega }{2}-\frac{b^{2}}{ 2m\omega ^{2}}, \end{equation} and the wave function \begin{equation} \psi _{1l}(r)=N_{1l}\left( r-\alpha _{1}^{(1)}\right) r^{\left( -1+l^{\prime }\right) /2}\exp \left( -\frac{1}{2}m\omega r^{2}-\frac{b}{\omega }r\right) , \end{equation} with \begin{equation*} N_{1l}=\frac{\left( \frac{2m\omega }{\hbar }\right) ^{l^{\prime }/4}\exp \left( -\frac{1}{4}\frac{\hbar ^{3}b^{2}}{m\omega ^{3}}\right) }{\sqrt{\frac{ \hbar }{2m\omega }\Gamma (l^{\prime }+2)S_{1}+\alpha _{1}^{(1)2}\Gamma (l^{\prime })S_{2}-2\alpha _{1}^{(1)}\sqrt{\frac{\hbar }{2m\omega }}\Gamma (l^{\prime }+1)S_{3}}}, \end{equation*} \begin{equation*} S_{1}=D_{-(l^{\prime }+2)}\left( \sqrt{\frac{2\hbar }{m\omega }}\frac{b\hbar }{\omega }\right) ,\text{ }S_{2}=D_{-l^{\prime }}\left( \sqrt{\frac{2\hbar }{ m\omega }}\frac{b\hbar }{\omega }\right) ,\text{ }S_{3}=D_{-(l^{\prime }+1)}\left( \sqrt{\frac{2\hbar }{m\omega }}\frac{b\hbar }{\omega }\right) , \text{ } \end{equation*} where \begin{equation*} b=2\frac{m\omega }{\hbar }\frac{\left( 2+l^{\prime }\right) c}{\left( 1+l^{\prime }\right) \left( 3+l^{\prime }\right) }\left[ 1+\sqrt{1+\left( \frac{\hbar ^{3}\omega }{2mc^{2}}\left( 1+l^{\prime }\right) -1\right) \frac{ \left( 1+l^{\prime }\right) \left( 3+l^{\prime }\right) }{\left( 2+l^{\prime }\right) ^{2}}}\right] , \end{equation*} and $\alpha _{1}^{(1)}=\frac{\left( l^{\prime }+1\right) }{2m\omega }.$
Following the analytic iteration procedures for the second node $\left( n=2\right) $ with $f_{2}(r)=(r-\alpha _{1}^{(2)})(r-\alpha _{2}^{(2)})$ and $ g_{l}(r)$ as defined in Eq. (10b), we obtain \begin{equation*} a_{1}r^{2}+b_{1}r-\varepsilon _{2,l}-\frac{c_{1}}{r}+\frac{l(l+1)-d_{1}}{ r^{2}}=\alpha ^{2}r^{2}+2\alpha \beta r \end{equation*} \begin{equation} -\alpha \left[ 1+2\left( \delta +2\right) \right] +\beta ^{2}-\frac{2\left[ \beta \left( \delta +2\right) +\alpha \dsum\limits_{i=1}^{2}\alpha _{i}^{(2)} \right] }{r}+\frac{\delta \left( \delta -1\right) }{r^{2}}, \end{equation} The relations between the potential parameters and the coefficients $\alpha , $ $\beta ,$ $\delta ,$ $\alpha _{1}^{(2)}$ and $\alpha _{2}^{(2)}$ are \begin{equation*} \alpha =\sqrt{a_{1}},\text{ }\beta =\frac{b_{1}}{2\sqrt{a_{1}}},\text{ } \delta =\frac{1}{2}\left( 1+l^{\prime }\right) ,\text{ }\varepsilon _{2,l}=\alpha \left[ 1+2\left( \delta +2\right) \right] -\beta ^{2}. \end{equation*} \begin{equation*} c_{1}-2\beta \left( \delta +2\right) =2\alpha \dsum\limits_{i=1}^{2}\alpha _{i}^{(2)},\text{ }\left( c_{1}-2\beta \delta \right) \dsum\limits_{i<j}^{2}\alpha _{i}^{(2)}\alpha _{j}^{(2)}=2\delta \dsum\limits_{i=1}^{2}\alpha _{i}^{(2)}, \end{equation*} \begin{equation} \left[ c_{1}-2\beta \left( \delta +1\right) \right] \dsum\limits_{i=1}^{2} \alpha _{i}^{(2)}=4\alpha \dsum\limits_{i<j}^{2}\alpha _{i}^{(2)}\alpha _{j}^{(2)}+2\left( 2\delta +1\right) , \end{equation} The coefficients $\alpha _{1}^{(2)}$ and $\alpha _{2}^{(2)}$ are found from the constraint relations, \begin{subequations} \begin{equation} \alpha \dsum\limits_{i=1}^{2}\alpha _{i}^{(2)2}+\beta \dsum\limits_{i=1}^{2}\alpha _{i}^{(2)}-\left( 2\delta +1\right) =0, \end{equation} \begin{equation} \delta \dsum\limits_{i=1}^{2}\alpha _{i}^{(2)2}-\left( \beta \dsum\limits_{i=1}^{2}\alpha _{i}^{(2)}+1\right) \dsum\limits_{j<k}^{2}\alpha _{j}^{(2)}\alpha _{k}^{(2)}-2\alpha \dsum\limits_{j<k}^{2}\alpha _{j}^{(2)2}\alpha _{k}^{(2)2}=0. \end{equation} Hence, the energy eigenvalue is \end{subequations} \begin{equation} E_{2l}=\sqrt{\frac{\hbar ^{2}a}{2m}}\left( 6+l^{\prime }\right) -\frac{b^{2} }{4a}, \end{equation} and the associated wave function is \begin{equation} \psi _{2l}(r)=N_{l}\underset{i=1}{\overset{2}{\Pi }}\left( r-\alpha _{i}^{(2)}\right) r^{\left( -1+l^{\prime }\right) /2}\exp \left( -\frac{1}{2} \sqrt{\frac{2ma}{\hbar ^{2}}}r^{2}-\sqrt{\frac{m}{2\hbar ^{2}a}}br\right) , \end{equation} where $\alpha _{1}^{(2)}$ and $\alpha _{2}^{(2)}$ should satisfy the restriction relations (28a) and (28b).
We apply the present method for the third node $\left( n=3\right) $ by taking $f_{3}(r)=(r-\alpha _{1}^{(3)})(r-\alpha _{2}^{(3)})(r-\alpha _{3}^{(3)})$ and $g_{l}(r)$ as defined in Eq. (10b) to obtain \begin{equation*} a_{1}r^{2}+b_{1}r-\varepsilon _{3,l}-\frac{c_{1}}{r}+\frac{l(l+1)-d_{1}}{ r^{2}}=\alpha ^{2}r^{2}+2\alpha \beta r \end{equation*} \begin{equation} -\alpha \left[ 1+2\left( \delta +3\right) \right] +\beta ^{2}-\frac{2\left[ \beta \left( \delta +3\right) +\alpha \dsum\limits_{i=1}^{3}\alpha _{i}^{(3)} \right] }{r}+\frac{\delta \left( \delta -1\right) }{r^{2}}. \end{equation} The relations between the potential parameters and the coefficients $\alpha , $ $\beta ,$ $\delta ,$ $\alpha _{1}^{(3)},$ $\alpha _{2}^{(3)}$ and $ \alpha _{3}^{(3)}$are \begin{equation*} \alpha =\sqrt{a_{1}},\text{ }\beta =\frac{b_{1}}{2\sqrt{a_{1}}},\text{ } \delta =\frac{1}{2}\left( 1+l^{\prime }\right) ,\text{ }\varepsilon _{3,l}=\alpha \left[ 1+2\left( \delta +3\right) \right] -\beta ^{2}. \end{equation*} \begin{equation*} c_{1}-2\beta \left( \delta +3\right) =2\alpha \dsum\limits_{i=1}^{3}\alpha _{i}^{(3)},\text{ }\left( c_{1}-2\beta \delta \right) \dsum\limits_{i<j<k}^{3}\alpha _{i}^{(3)}\alpha _{j}^{(3)}\alpha _{k}^{(3)}=2\delta \dsum\limits_{i<j}^{3}\alpha _{i}^{(3)}\alpha _{j}^{(3)}, \end{equation*} \begin{equation} \left[ c_{1}-2\beta \left( \delta +2\right) \right] \dsum\limits_{i=1}^{3} \alpha _{i}^{(3)}=4\alpha \dsum\limits_{i<j}^{3}\alpha _{i}^{(3)}\alpha _{j}^{(3)}+3\left( 2\delta +2\right) . \end{equation} The coefficients $\alpha _{1}^{(3)},$ $\alpha _{2}^{(3)}$ and $\alpha _{3}^{(3)}$ are found from the constraint relation, \begin{equation} \alpha \dsum\limits_{i=1}^{3}\alpha _{i}^{(3)2}+\beta \dsum\limits_{i=1}^{3}\alpha _{i}^{(3)}-3\left( \delta +1\right) =0, \end{equation} The energy eigenvalue is \begin{equation} E_{3l}=\sqrt{\frac{\hbar ^{2}a}{2m}}\left( 8+l^{\prime }\right) -\frac{b^{2} }{4a}, \end{equation} and the wave function is \begin{equation} \psi _{3l}(r)=N_{l}\underset{i=1}{\overset{n=3}{\Pi }}\left( r-\alpha _{i}^{(n)}\right) r^{\left( -1+l^{\prime }\right) /2}\exp \left( -\frac{1}{2} \sqrt{\frac{2ma}{\hbar ^{2}}}r^{2}-\sqrt{\frac{m}{2\hbar ^{2}a}}br\right) . \end{equation} We can repeat this iteration procedures several times to write the exact energies of the CMpH potential for any $n$ state as \begin{equation} E_{nl}=\sqrt{\frac{\hbar ^{2}a}{2m}}\left( 2+2n+l^{\prime }\right) -\frac{ b^{2}}{4a}, \end{equation} and the wave functions is \begin{equation} \psi _{nl}(r)=N_{l}\underset{i=1}{\overset{n}{\Pi }}\left( r-\alpha _{i}^{(n)}\right) r^{\left( -1+l^{\prime }\right) /2}\exp \left( -\frac{1}{2} \sqrt{\frac{2ma}{\hbar ^{2}}}r^{2}-\sqrt{\frac{m}{2\hbar ^{2}a}}br\right) . \end{equation} The relations between the potential parameters and the coefficients $\alpha , $ $\beta ,$ $\delta ,$ $\alpha _{1}^{(n)},$ $\alpha _{2}^{(n)},\cdots $, $ \alpha _{n}^{(n)}$ are \begin{equation*} \alpha =\sqrt{a_{1}},\text{ }\beta =\frac{b_{1}}{2\sqrt{a_{1}}},\text{ } \delta =\frac{1}{2}\left( 1+l^{\prime }\right) ,\text{ }\varepsilon _{2,l}=\alpha \left[ 1+2\left( \delta +n\right) \right] -\beta ^{2}, \end{equation*} \begin{equation*} c_{1}-2\beta \left( \delta +n\right) =0,\text{ (}n=0) \end{equation*} \begin{equation*} c_{1}-2\beta \left( \delta +n\right) =2\alpha \dsum\limits_{i=1}^{n}\alpha _{i}^{(n)},\text{ }n=1,2,3,\cdots \end{equation*} \begin{equation*} \left[ c_{1}-2\beta \left( \delta +n-1\right) \right] \dsum\limits_{i=1}^{n} \alpha _{1}^{(n)}=n\left( 2\delta +n-1\right) ,\text{ (}n=1) \end{equation*} \begin{equation*} \text{ }\left[ c_{1}-2\beta \left( \delta +n-1\right) \right] \dsum\limits_{i=1}^{n}\alpha _{1}^{(n)}=4\alpha \dsum\limits_{i<j}^{n}\alpha _{i}^{(n)}\alpha _{j}^{(n)}+n\left( 2\delta +n-1\right) ,\text{ } n=2,3,4,\cdots \end{equation*} \begin{equation*} \left[ c_{1}-2\beta \left( \delta +n-2\right) \right] \dsum\limits_{i<j}^{n} \alpha _{i}^{(n)}\alpha _{j}^{(n)}=\left( n-1\right) \left( 2\delta +n-2\right) \dsum\limits_{i=1}^{2}\alpha _{i}^{(2)},\text{ }\left( n=2\right) \end{equation*} \begin{equation*} \left( \text{ }c_{1}-2\beta \delta \right) \dsum\limits_{i<j<k}^{n}\alpha _{i}^{(n)}\alpha _{j}^{(n)}\alpha _{k}^{(n)}=2\delta \dsum\limits_{i<j}^{n}\alpha _{i}^{(n)}\alpha _{j}^{(n)},\text{ }\left( n=3\right) , \end{equation*} \begin{equation*} \left[ \text{ }c_{1}-2\beta \left( \delta +n-2\right) \right] \dsum\limits_{i<j}^{n}\alpha _{i}^{(n)}\alpha _{j}^{(n)}=\left( n-1\right) \left( 2\delta +n-2\right) \dsum\limits_{i<j}^{n}\alpha _{i}^{(n)}\alpha _{j}^{(n)} \end{equation*} \begin{equation} +4\alpha \dsum\limits_{i<j<k}^{n}\alpha _{i}^{(n)}\alpha _{j}^{(n)}\alpha _{k}^{(n)},\text{ }n=3,4,5,\cdots , \end{equation} and so on.
\section{Results and Discussions}
Now, we consider a special case of potential (1) and an application to our results. For example, when $b=0$ then leads to $c=0$, then we have the p.h.o potential, i.e., $V_{ph}(r)=\frac{1}{2}m\omega ^{2}r^{2}-\frac{d}{r^{2}},$ hence, the energy difference between the ground state and the excited states is \begin{equation} \Delta E=E_{1l}-E_{0l}=\left( 4+l^{\prime }\right) \frac{\hbar \omega }{2} -\left( 2+l^{\prime }\right) \frac{\hbar \omega }{2}=\hbar \omega , \end{equation} which can be used to calculate the values of the potential parameters for the desired system.
We now apply the present results to describe a realistic physical system called indium gallium arsenide (InGaAs) quantum dot, i.e., a piece of this material of a spherical form which is considered as a semiconductor composed of indium, gallium and arsenic [11]. It is used in high-power and high-frequency $($say, $\omega \sim 10^{15}$ $Hz)$ electronics because of its superior electron velocity with respect to the more common semiconductors silicon and gallium arsenide. InGaAs bandgap also makes it the detector material of choice in optical fiber communication at $1300$ and $1550$ $nm$. The gallium indium arsenide (GaInAs) is an alternative name for InGaAs. In Fig. 2, we plot the ground state electron energy \begin{equation} E_{0l}(\omega )=\left( 2+\sqrt{\left( 2l+1\right) ^{2}-\frac{8m}{\hbar ^{2}}d }\right) \frac{\hbar \omega }{2}-\frac{2mc^{2}}{\hbar ^{2}}\left( 1+\sqrt{ \left( 2l+1\right) ^{2}-\frac{8m}{\hbar ^{2}}d}\right) ^{-2}, \end{equation} versus $\omega $ in the interval $2\times 10^{14}\leq \omega \leq 10\times 10^{14}$ $Hz$ taking the value of $c=0.001$ $eV.fm$ and $d=0$ $eV.fm^{2}$ for the cases $l=0$ and $l=1,$ respectively (harmonic, Coulomb and linear combination terms). In Fig. 3, we take instead the value of the parameter $ d=0.1$ $eV.fm^{2}$ (pseudoharmonic, Coulomb and linear combination terms)$.$ The effective mass of electron in the InGaAs semiconductor has been chosen as $m=0.05m_{e}$ and $\hbar =6.5821\times 10^{-16}$ $eV.s.$ It is seen from Fig. 2 and Fig. 3 how the increase in the value of $\omega $ leads to an increase in the energy of electron. The flexibility in the adjustment of the parameter $d$ allows one to fit the spectrum of the desired model properly (cf. Fig. 2 and Fig. 3). The parameter $d$ should satisfy the condition \ $ d\leq \left( 2l+1\right) ^{2}\hbar ^{2}/(8m).$ In Fig. 4 we plot the ground state wave function $\psi _{0,l}(r)$ of the CpH potential for the cases $l=0$ and $l=1,$ respectively, using the values of potential parameter $c=0.001$ $ eV.nm$ for an electron with effective mass $m=0.05$ $m_{e}$ and frequency $ \omega =10\times 10^{14}$ $Hz$. Further, in Fig. 5 we plot the ground state wave function $\psi _{0,l}(r)$ of the CMpH potential for the cases $l=0$ and $l=1,$ respectively, using the values of potential parameters $c=0.001$ $ eV.nm$ and $d=0.01$ $eV.nm^{2}$ for an electron with effective mass $m=0.05$ $m_{e}$ and frequency $\omega =10\times 10^{14}$ $Hz$. In Figs. 6 and 7, we show electron energy as a function of parameter $c$ in the interval $6\times 10^{-2}\leq c\leq 10\times 10^{-2\text{ \ }}eV.nm$ and $d=0.01$ $eV.nm^{2}$ for frequency $\omega =8\times 10^{14}$ $Hz$ and effective mass $m=0.05$ $ m_{e}$ for the cases $l=0$ and $l=1,$ respectively. From Fig. 5, the increase in $c$ leads in the decrease in the electron energy in the InGaAs semiconductor. In Fig. 8, we plot the first excited state electron energy \begin{equation*} E_{1l}(\omega )=\left( 4+l^{\prime }\right) \frac{\hbar \omega }{2}-\frac{ 2\left( 2+l^{\prime }\right) ^{2}}{\left( 1+l^{\prime }\right) ^{2}\left( 3+l^{\prime }\right) ^{2}}\frac{mc^{2}}{\hbar ^{2}} \end{equation*} \begin{equation} \times \left[ 1+\sqrt{1+\frac{\left( 1+l^{\prime }\right) \left( 3+l^{\prime }\right) }{\left( 2+l^{\prime }\right) ^{2}}\left( \frac{\hbar ^{3}\omega }{ 2mc^{2}}\left( 1+l^{\prime }\right) -1\right) }\right] ^{2}, \end{equation} versus $\omega $ in the interval $2\times 10^{14}\leq \omega \leq 10\times 10^{14\text{ \ }}Hz$ taking the value of $c=0.001$ $eV.fm$ and $d=0$ $ eV.fm^{2}$ for the cases $l=0$ and $l=1,$ respectively. In Fig. 9, we take the value of the parameter $d=0.1$ $eV.fm^{2}.$ We remark that the strongly attractive singular part $-d/r^{2}$ is physically incorporated into the quarkonium Cornell potential as the first perturbative term for the sake of coherence to describe the heavy quarkonia (QQ-nia) (see, for example, [13,14] and the references therein). It also resemles the centrifugal barrier term $l(l+1)/r^{2}$ in the Schr\"{o}dinger equation. This attractive term $-d/r^{2}$ together with the h.o. part $ar^{2}$ constitute the so-called p.h.o. when $b=0$ $(\beta =0)$ in Eq. (14b) leading to $c=0$ in Eq. (14c)$.$
In Table 1, we calculate the lowest ($n=0$) energy states ($l=0,1$ and $2)$ from Eq. (36) and from the numerical solution of the radial Schr\"{o}dinger equation (7) using the values of parameters given by Ref. [34] using the supersymmetry quantum mechanics (SUSYQM). It is clear that the calculated energy states in the present work are in good agreement with the results obtained numerically and SUSYQM [34]. The accracy of our numerical results is $0.0070\%-0.0095\%.$
\section{Conclusions and Outlook}
In this work, we explored the analytical exact solution for the energy eigenvalues and their associated wave functions of a particle in the field of Cornell-modified plus harmonic confining potential. We have used the analytical exact iteration method (AEIM) which required making a trial ansatz for the wave function. The general equation for the energy eigenvalues is given by Eq. (36) with some restrictions on the potential parameters. If one takes $b=0$ then $c=0,$ hence, the potential (1) turns to the p.h.o. potential with energy eigenvalues: \begin{equation} E_{nl}=\sqrt{\frac{\hbar ^{2}a}{2m}}\left( 2+2n+\sqrt{\left( 2l+1\right) ^{2}-\frac{8m}{\hbar ^{2}}d}\right) \end{equation} and wave functions: \begin{equation} \psi _{nl}(r)=N_{l}\underset{i=1}{\overset{n}{\Pi }}\left( r-\alpha _{i}^{(n)}\right) r^{\left( -1+l^{\prime }\right) /2}\exp \left( -\frac{1}{2} \sqrt{\frac{2ma}{\hbar ^{2}}}r^{2}\right) . \end{equation} The present results in Eqs. (42) and (43) coincide with Eqs. (15) and (16) of Ref. [22] obtained by the exact polynomial method, Eqs. (72) and (78) of Ref. [23] obtained by the Nikiforov-Uvarov method and Eqs. (30) and (31) of Ref. [24] obtained by the wave function ansatz method after setting $ D_{0}/r_{0}^{2}=a,$ $D_{0}r_{0}^{2}=-d_{0}$ and $2D_{0}=0.$ The model solved in the present work can be used in modeling the quarkonium [14] perturbed by the field of p.h.o. or electron confined in spherical quantum dots [11]. Finally, our solution to this confining potential is being considered important in many different fields of physics, such as atomic and molecular physics [25,26], particle physics [13,27,28], plasma physics and solid-state physics [29-33].
\acknowledgments The partial support provided by the Scientific and Technological Research Council of Turkey is highly appreciated.
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\begin{table}[tbp] \caption{Lowest $(n=0)$ energy spectra (for $\hbar =m=1$). } \begin{tabular}{llllllll} \tableline\tableline$a$ & $b$ & $c$ & $d$ & $l$ & Numerical & Present & SUSYQM [34] \\ \tableline$\frac{1}{32}$ & $1$ & $4$ & $0$ & $0$ & -7.618 & $-7.625$ & $ -7.625$ \\ $\frac{1}{32}$ & $1$ & $8$ & $0$ & $1$ & -7.368 & $-7.375$ & $-7.375$ \\ $\frac{1}{32}$ & $1$ & $12$ & $0$ & $2$ & -7.120 & $-7.125$ & $-7.125$ \\ \tableline & & & & & & & \end{tabular} \end{table} \FRAME{ftbpFO}{0.0277in}{0.0277in}{0pt}{\Qct{A plot of the CMpH potential [see Eq. (1)] with the selected values of parameters: $a=1$ $eV.fm^{-2},$ $ b=0.217$ $eV$.$fm^{-1},$ $c=0.400$ $eV.fm$ and $d=0.010$ $eV.fm^{2}.$}}{}{ Figure 1}{}\FRAME{ftbpFO}{0.0277in}{0.0277in}{0pt}{\Qct{The ground state electron energy in InGaAs semiconductor versus $\protect\omega $ in the field of CpH potential with $c=0.001$ $eV.nm$ for cases $l=0$ and $l=1,$ respectively$.$}}{}{Figure 2}{}\FRAME{ftbpFO}{0.0277in}{0.0277in}{0pt}{\Qct{ The ground state electron energy in InGaAs semiconductor versus $\protect \omega $ in the field of the CMpH potential with $c=0.001$ $eV.nm$ and $ d=0.1 $ $eV.nm^{2}$ for the cases $l=0$ and $l=1,$ respectively.}}{}{Figure 3 }{}
\FRAME{ftbpFO}{0.0277in}{0.0277in}{0pt}{\Qct{Behaviour of the ground state wave function $\protect\psi _{n=0,l=0}(r)$ (dashed line) and $\protect\psi _{n=0,l=1}(r)$ (continuous line) in the field of the CpH potential with the value of $c=0.001$ $eV.nm$ for an electron with effective mass $m=0.05$ $ m_{e}$ and frequency $\protect\omega =10\times 10^{14}$ $Hz$ in the InGaAs semiconductor.}}{}{Figure 4}{}\FRAME{ftbpFO}{0.0277in}{0.0277in}{0pt}{\Qct{ Behaviour of the ground state wave function $\protect\psi _{n=0,l=0}(r)$ (dashed line) and $\protect\psi _{n=0,l=1}(r)$ (continuous line) of the CMpH potential with the values of $c=0.001$ $eV.nm$ and $d=0.01$ $eV.nm^{2}$ for an electron with an effective mass $m=0.05$ $m_{e}$ and frequency $\protect \omega =10\times 10^{14}$ $Hz$ in the InGaAs semiconductor.}}{}{Figure 5}{} \FRAME{ftbpFO}{0.0277in}{0.0277in}{0pt}{\Qct{Ground state energy of electron versus $c,$ for the case $l=0,$ $\protect\omega =8\times 10^{14}$ $Hz$ and $ d=0.01$ $eV.nm^{2}.$}}{}{Figure 6}{}\FRAME{ftbpFO}{0.0277in}{0.0277in}{0pt}{ \Qct{Ground state energy of electron versus $c,$ for the case $l=1,$ $ \protect\omega =8\times 10^{14}$ $Hz$ and $d=0.01$ $eV.nm^{2}.$}}{}{Figure 7 }{}\FRAME{ftbpFO}{0.0277in}{0.0277in}{0pt}{\Qct{The first excited state electron energy in InGaAs semiconductor versus $\protect\omega $ in the field of CpH potential with $c=0.001$ $eV.nm$ for cases $l=0$ and $l=1,$ respectively$.$}}{}{Figure 8}{}\FRAME{ftbpFO}{0.0277in}{0.0277in}{0pt}{\Qct{ The first excited state electron energy in InGaAs semiconductor versus $ \protect\omega $ in the field of the CMpH potential with $c=0.001$ $eV.nm$ and $d=0.1$ $eV.nm^{2}$ for the cases $l=0$ and $l=1,$ respectively.}}{}{ Figure 9}{}
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Wage inequality, skill inequality, and employment: evidence and policy lessons from PIAAC
Sonja Jovicic1
IZA Journal of European Labor Studies volume 5, Article number: 21 (2016) Cite this article
This paper investigates international differences in wage inequality and skills and whether a compressed wage distribution is associated with high unemployment across core OECD countries. Wage dispersion and wage structure are widely debated among policymakers; compressed wage structure is often perceived as an important cause of high unemployment. Firstly, this paper examines differences in wage dispersion across OECD countries and their link to skill dispersion. Some countries that have more compressed (dispersed) wage structures simultaneously have more compressed (dispersed) skill structures as well, and skill differences explain part of the differences in wage dispersion. However, even when accounted for skills, some countries have a more compressed wage structure, most likely caused by labor market institutions. We do not find an effect of wage compression on the labor market performance in the low-skill sector. Based on the Program for International Assessment of Adult Competencies (PIAAC) survey of adult skills for core OECD countries, this paper cannot confirm the skill compression nor wage compression hypotheses. Rather than insisting on the deregulation of labor market institutions and reductions in public welfare policy as the main policy recommendations to achieve higher employment (and higher wage inequality), policymakers should reconsider aggregate demand deficiency and the variation in macroeconomic policies as potential explanations for the employment differences across countries.
JEL Classification: J31, J24, E24
The variation in wage inequality across developed countries has puzzled economists for many years, and different theoretical explanations and empirical evidence have been presented on this issue. Some economists argue that these differences can be explained by supply and demand factors, whereas others emphasize the influence of wage-setting institutions on the wage structure. Consistent with the first theory, the variations in wage inequality across different countries can be explained by variations in skill inequalities. Countries that have more compressed (dispersed) wage structures simultaneously have more compressed (dispersed) skill structures as well (Nickell and Bell 1995;Footnote 1 Leuven et al. 2004). According to neoclassical theory, supply and demand factors, skill-biased technical change (SBTC), and globalization are responsible for the increase in wage inequality in the past decades (Katz and Murphy 1992; Juhn et al. 1993; Katz and Autor 1999; Goldin and Katz 2008; Acemoglu and Autor 2012) and market forces play a more significant role in explaining cross-national differences in wage inequality and return to skill than institutional factors (Gottschalk and Joyce 1998). Since the Anglo-Saxon countries had simultaneously higher wage and skill inequalities compared to continental and Nordic Europe, this was taken as proof of the theory. The reasoning behind this theory is that higher wage inequality is a consequence of higher return to skills. A high skill premium goes along with increased motivation to invest in skill formation (Heckman et al. 1998; Welch 1999) and, consequently, greater supply of highly skilled labor. This explanation, however, fails to explain the high educational attainment in Nordic countries, which exhibit among the lowest rates of wage inequality when compared to other developed countries. Alternative explanation for variation in wage dispersion is based on the variation in wage-setting institutions. Economists who are in favor of this hypothesis stress the importance of decreasing real minimum wages and union membership in order to explain the widening wage gap (Freeman 1991; Freeman and Katz 1994; Blau and Kahn 1996; Bach et al. 2007; Machin 2016). A similar conclusion comes from Dew-Becker and Gordon (2005, 2008), who, in addition to these explanations, identify peer-group behavior as responsible for increasing wage dispersion at the top of the distribution in the USA. Card and DiNardo (2002) reach similar conclusions and also criticize the skill-biased technical change argument as being unable to account for gender and racial wage inequalities and differences in return to education.
Variation in wage inequality in the bottom half of the wage distribution is also often linked to variation in employment in the low-skill sector. According to neoclassical theory, differences in wage dispersion are often credited as an important explanation for differences in unemployment rates. Whereas dispersed wage structure can contribute to employment creation, wage compression in the bottom half of the wage distribution (usually assumed by labor market institutions) can cause unemployment in the low-skill sector (Siebert 1997; Heckman and Jacobs 2010). Due to the skill-biased technical change, relative demand for low-skilled workers in developed countries exhibited a decline; their relative marginal productivity deteriorated (relative marginal productivity of skilled workers rose). However, wage compression and excessively high wages (higher than marginal productivity) at the low end of the wage distribution cut low-skilled workers out of employment. Consequently, countries should allow for higher wage dispersion in the bottom half of the wage distribution and lower wages for the low skilled (institutional reform) which should push their employment levels up. This is in line with a trade-off between efficiency and equality (Okun 1975), according to which it is impossible to achieve high employment and low inequality at the same time. In order to achieve high employment, countries must accept high wage dispersion. By comparing the distribution of wages and employment in Germany and the USA, Siebert (1997) concludes that the relevant policy recommendation to increase employment in Germany at the low end is to allow for dispersed wage structure (higher wage inequality).
High and increasing wage inequality as well as high unemployment in some OECD countries shifted the focus of policymakers to differences in wage dispersion. This paper discusses theoretical and empirical backgrounds of wage compression hypothesis. The wage compression hypothesis is based on the perfect market model and its rigid assumptions. However, many of these assumptions are flawed—as the empirical analysis of this paper shows. Cross-country differences in wage dispersion cannot be explained by cross-country differences in skill dispersion; educational attainment does not seem to be higher in countries where return to schooling is high, and there is wage dispersion within skill levels, which is in stark contrast with marginal productivity theory. These arguments are in contrast with theoretical foundations of the wage compression hypothesis. Finally, unemployment/e-pops/average hours worked are not correlated with compressed wages. Thus, this paper shows that the wage compression hypothesis is not supported by empirical evidence and therefore challenges the theoretical assumptions it is derived from. The results of this study (although descriptive) have some important consequences for policy-making. Recommended policies for eliminating wage compression, and allowing for higher wage dispersion, are the deregulation of labor market institutions (collective bargaining, unemployment benefits, unions, minimum wages, etc.) and a reduction of public welfare policies. However, since wage compression is not correlated with labor market performance in the low-skilled sector (contrary to the theory), these policy recommendations need to be revised. Moreover, higher wage dispersion is related to major social and health problems, as well as the higher share of low-paid jobs. This study shows that countries that have good labor market performance in the low-skill sector have good labor market performance in general and this is likely due to macroeconomic policies. Consequently, the role of expansionary macroeconomic policies in fostering employment needs to be revisited.
The analysis presented in this paper extends the existing literature by examining these issues. This paper shares the most similarities with the work of Freeman and Schettkat (2001) and Devroye and Freeman (2001). Freeman and Schettkat (2001) examine the wage compression hypothesis based on differences between the USA and Germany in relation to employment. They find that skill compression can only partly explain wage compression. However, the wage compression hypothesis cannot explain the US-German difference in employment. Devroye and Freeman (2001) study the relationship between the distribution of earnings and the distribution of skills and find that skill inequality explains only 7% of wage inequality. Within-skill-group inequality plays a larger role than inequality between skill groups; this contradicts the theory. In contrast to the first two studies that were based on the international literacy survey in the 1990s (International Adult Literacy Survey—IALS), in this paper, a more recent data set is used, with a larger number of countries and larger sample sizes. It is important to check whether the results based on the IALS survey can be confirmed by using the Program for International Assessment of Adult Competencies (PIAAC).
This paper is organized as follows. In Section 2, the data set and data adjustments are presented in more detail. This section is followed by the empirical analysis in Sections 3 and 4. Firstly, international differences in skill levels, wage inequality, and the relationship between skill inequality and wage inequality are examined. In Section 5, the dispersion of wages within skill levels is investigated. Section 6 analyzes the wage compression hypothesis and its effect on employment. Finally, Section 7 concludes.
This analysis is based on the PIAAC data set that was collected between 2011 and 2012 and initiated by the OECD. PIAAC is a unique data set that provides numerous opportunities for research, because it comprises various individual-level indicators of skill competencies, earnings, demographic, and socio-economic characteristics and other internationally comparable information across OECD countries. Since countries' sample sizes are bigger than in previous similar data sets (around 5000 observations per country), such a sample facilitates more comprehensive analysis and better investigation of different subgroups. People were questioned on the basis of a 1.5–2-h interview, which was performed by a specially trained interviewer (tests were done either on computer or on paper). The adult competency skills are measured by literacy, numeracy, and problem solving in technology-rich environmentsFootnote 2 that are central for good performance in the labor market. That is why the skills tested in the survey should be a good proxy for the skills needed in the workplace. According to the test score results, six different proficiency levels are defined. The pooled data set used in this paper contains national representative samples of around 120,000 observations based on working age populations (16–65) from 16 different highly developed core OECD countries. Countries included in the data set are Austria, BelgiumFootnote 3 (Flanders), Canada, Denmark, Finland, France, GermanyFootnote 4, Ireland, Italy, Japan, Netherlands, Norway, Spain, Sweden, Great Britain (England and Northern Ireland), and the USA.Footnote 5
The definition of the PIAAC literacy test is as follows: "understanding, evaluating, using, and engaging with written text to participate in society, to achieve one's goals, and to develop one's knowledge and potential." Numeracy assessment is defined as the ability to access, use, interpret, and communicate mathematical information and ideas and to engage in and manage mathematical demands of a range of situations in adult life. Finally, problem solving accounts for "using digital technology, communication tools, and networks to acquire and evaluate information, communicate with others, and perform practical tasks" (OECD 2013a:59).
The correlation coefficient between different test results is slightly lower than in previous test surveys (ALL or IALS) but is still highly positive. The correlation coefficient between numeracy and literacy scores is the highest and equals to 0.89, followed by the correlation coefficient between literacy scores and problem-solving skills (0.79). The smallest correlation coefficient is found between numeracy scores and problem-solving scores in technology-rich environments (0.75). In this analysis, numeracy test scores are used as a measure of skill test results,Footnote 6 which is standard in this literature, but further analysis actually showed that the same results are confirmed when literacy test scores are used.Footnote 7 For further analysis, it is vital to compare the wage data from the micro data set—the PIAAC survey with the macro data from the OECD database. Figure 1 displays wage inequality taken from both databases, and apart from a couple of outliers (Japan, Italy, and Germany have higher wage inequality; France and the USA have lower wage inequality in the PIAAC survey compared to the OECD databaseFootnote 8), the micro data seems to correspond well to the aggregate macro data. According to both data sources, ranking of the countries in terms of inequality is almost unaffected. If D9/D5 and D5/D1 are observed, deviations between the data sets are even smaller.
Wage inequality (D9/D1) for OECD countries, only employed persons. Source: OECD earnings database and PIAAC
Skills and wages across OECD countries
Skill dispersion
According to the OECD database, in the past 30 years, wage inequality has been on the rise in almost all of the OECD countries (see OECD 2011; Jovicic and Schettkat 2013). On the one hand, the increase in inequality has been criticized by many economists; on the other hand, many others have justified this development as a result of the rise in skill inequality (see Section 1). In order to get a better insight on wage inequality and skill inequality, a deeper look into the data set and some descriptive statistics is necessary. Table 1 presents the mean, median, standard deviation, and coefficient of variation of numeracy scores in core OECD countries. If all people are included, independent of their employment status, Anglo-Saxon countries together with France and Spain have the highest dispersion of skills, whereas Japan has the lowest inequality of numeracy skills. In terms of employed persons, the countries with the highest skill inequality among employed workers are the USA, France, and Italy (followed by Canada, the UK, and Ireland). Japan, Finland, and the Netherlands (followed by Denmark and Belgium) have the lowest coefficient of variation of numeracy test results. Coefficients of variation of numeracy scores are higher for all persons than for employed persons in all countries, which implies that the unemployed are likely to be lower skilled than the employed. Another very important conclusion can be drawn from this table. Countries with higher skill inequality exhibit lower average skill scores, whereas the countries with lower skill inequality perform better in terms of average skill scores (mean). If the median is observed instead of the mean, the conclusion is the same. In every country, the median is only slightly higher than the mean; the difference between the two measures ranges between a maximum five points and a minimum two points (the distribution of skills is just slightly skewed to the left). This leaves the ranking of the countries according to their average results unaffected if the median is used (instead of the mean).
Table 1 Mean, median, standard deviation, and coefficient of variation of numeracy scores for all and employed persons
In order to develop a better understanding of the cause of the difference in average numeracy score results, one must examine the share of people within different skill levels. Skill levels are defined according to test scores and divided into six different groups. People with the highest scores are assigned to group levels 5 and 4, whereas levels 0 and 1 are the groups with the lowest numeracy scores.Footnote 9 Table 2 shows that the countries with the lowest numeracy test scores (and the highest skill inequalities) have the highest proportion of workers in the lowest skill group (below level 1 and at level 1)—Italy, the USA, France, and Spain. Japan, the Netherlands, and Finland (followed by Denmark and Belgium) have the lowest percentage of least skilled workers. These countries, however, also have a slightly higher percentage of people in the highest skill group.Footnote 10 According to the PIAAC survey evidence, countries with the highest numeracy test performance simply have more high-skilled workers and fewer low-skilled workers.
Table 2 Share of population in 6 different skill levels, employed persons
Next, we examine differences in performance between different subgroups. Table 3 shows average numeracy test scores according to gender, immigration status, and age groups. The difference between men and women is not large; it varies roughly between 8 points and 12 points. On average, men have slightly higher numeracy test scores than women and this is true for every country. However, since women often demonstrate poorer scores in the quantitative tests, comparing additionally the literacy test results shows that there is almost no difference in the test performance (both men and women have average literacy scores of around 277 points). On the other hand, immigrantsFootnote 11 have much lower results than non-immigrants—around 35 points less on average. The biggest reason for this is the fact that the test was done in the countries' national languages; immigrants are disadvantaged comparatively to the non-immigrants and often experience difficulty with the foreign language. This may suggest underestimation of their proficiency skills. The only two countries where the difference is moderately small are Ireland and to some extent Canada. Canada is a large immigration country where immigration and integration policies probably play a big role and contribute to higher language proficiency of immigrants. When age subgroups are compared, the difference is only marginal in almost all groups, aside from the oldest age group. People in the older age subgroups have lower results on average, probably due to the fact that older people tend to forget and experience decline in skills after age 45, but especially after the age of 50, according to Table 3. This is in line with various other studies that dealt with literacy and numeracy skill surveys; however, this might not hold for other skills. In general, wages increase with age, as well as the experience and some experience-related skills. In most countries, the lowest age group also tends to have slightly lower proficiency scores than the age groups from 25 to 45. What stands out is that, in Denmark, Italy, and the USA, these age subgroups have similar results to the oldest age subgroups, which is particularly alarming (especially in the USA and Italy, since they also have very low scores). One reason for this (and comparatively lower young age subgroup results in general) could be that the education systems alone do not produce relevant work-related skills and that the quality of schooling and the standard of education system are deteriorating.
Table 3 Mean of numeracy scores in different gender, immigrant and age groups, employed persons
Table 3 reveals some differences between various subgroups; thus, it is reasonable to see whether compositional differences have an effect on average numeracy test scores and dispersion of numeracy test scores. Population subgroups characterized by lower average numeracy test scores were immigrants, followed by the oldest age group and women. Whereas the share of womenFootnote 12 is comparable across countries, there is considerable variation in the share of immigrants across countries, and this probably affects the average numeracy score results and their dispersion.Footnote 13 Some of the countries with a high share of immigrants in the sample are found to have lower average numeracy test scores. Lower average numeracy test scores in Canada, Ireland, the USA, and the UK may be partly explained by higher shares of immigrants whose skills are underestimated due to language difficulties. When immigrants are excluded from the sample, the average numeracy test scores increase in these countries and the coefficient of variation is slightly reduced as well. This is true for every country, but the reduction is the highest in the USA. The USA has the highest dispersion of skills, but this phenomenon can be partly explained by the lower score of immigrants, and suggests that immigration status should be controlled for in the regression analysis. There is also a moderate variation in the share of the oldest age group in the employed population across countries, but this does not appear to affect average scores nor the dispersion of scores considerably.Footnote 14
Wage dispersion and skill dispersion
In addition to the individual skill scores, the PIAAC data set provides information on hourly wagesFootnote 15 of employed persons. Table 5 shows the dispersion of numeracy test score results, wages, and years of schoolingFootnote 16 measured by the coefficient of variation. This data already shows that there is no clear empirical relationship across countries between wage dispersion and numeracy skill dispersion. Countries with the highest dispersions of numeracy test scores are the USA, France, Canada, and the UK, whereas the countries with the lowest dispersions are Japan, the Netherlands, and Finland. In terms of wage inequality, countries with the highest wage dispersion are Japan, the USA, and the UK, and the countries with the lowest wage dispersions are Belgium, Norway, Denmark, and Finland. If there was a strong link between skill dispersion and wage dispersion, the data would be expected to show that the countries with the highest skill dispersions exhibit the highest wage dispersion and vice versa; this is not always the case here. Additional analysis also shows that the same conclusions hold when wage inequality between different population subgroups is observed. In all the population subgroups examined (men, women, immigrants, non-immigrants, different age subgroups), the countries with the highest wage dispersions are still Japan and the USA, and the countries with the lowest wage dispersions are Belgium and the Scandinavian countries (ranking of the countries remains intact).Footnote 17
In order to develop a more comprehensive view of the relationship between skill dispersion and wage dispersion, in addition to measuring skills by proficiency score results, years of schooling are also included in the analysis. However, when years of schooling is used in the analysis, this must be based on the assumption that 1 year of schooling has the same effect on human capital formation in every country, which is difficult to confirm. International skill proficiency surveys are thus becoming more and more popular, since their comparability is likely to be more reliable. According to the estimates, there is a positive but weak correlation between numeracy test scores and years of schooling—correlation coefficient for the entire PIAAC sample is 0.44 (correlation coefficient varies between 0.36 and 0.60 for individual countries). The fact that years of schooling and numeracy skills are positively correlated is expected, since longer schooling produces higher levels of skills and, at the same time, higher skilled individuals acquire more schooling. However, the rather small size of the correlation is somewhat surprising.Footnote 18 One potential explanation is that schooling is related to unmeasured competencies and unobserved non-cognitive skill (or some dimension of cognitive skills other than numeracy skills). Table 5 shows that dispersion of years of schooling is slightly higher or the same as the dispersion of test scores in most countries. The only three countries that have relatively high dispersion in years of schooling are Italy, France, and Spain;Footnote 19 countries with the lowest skill dispersion measured by schooling are the UK, Norway, and Germany.
In addition to the distribution of numeracy test scores, years of schooling, and wages, Table 4 reports correlation coefficients between these variables. The correlation coefficient between wages and numeracy scores is positive but ranges between 0.14 and 0.37 only. This could be additional proof that cross-country variation in numeracy scores is not strongly associated with cross-country variation in wages. Although the variable of years of schooling performs a bit better (its correlation to wages is higher and ranges between 0.24 and 0.51), it can hardly confirm the skill compression hypothesis. Possible explanations for why there is a stronger link between years of schooling and wages (than between numeracy test scores and wages) could be that either unmeasured competencies are related to years of schooling or years of schooling is positively associated with wages through the signaling effect—the employer assumes that more schooling is positively correlated with having advanced abilities. It could be that years of schooling has a large effect on wages, without having a large effect on skills measured by numeracy test scores.
Table 4 Coefficient of variation of average numeracy scores, hourly wages, and years of schooling and their correlation coefficient, employed persons
In order to conclude the discussion on skill and wage dispersion and get a more comprehensive description of their relationship, in addition to the coefficient of variation, other measures of inequality are examined. Table 6 shows decile ratios (D9/D1, D9/D5, D5/D1Footnote 20) of skill and wage dispersion. Decile ratios reveal additional evidence against the skill compression hypothesis. Since wage inequality in the top half of the distribution is higher and varies most across countries (D9/D5 is higher than D5/D1), it was expected that the same would be true for skill inequality. However, Table 5 shows that the opposite is the case. The highest skill inequality and the highest variability in skill inequality are observed for measures of skill inequality in the bottom half of the skill distribution. In every country, skill inequality at the bottom of the distribution is higher than at the top, whereas the opposite holds for wage inequality (the only exceptions are Denmark, Germany, and to some extent the Netherlands where wage inequality in the bottom half of the skill distribution is higher than that in the top half of the distribution). These patterns contradict the skill compression hypothesis, and this conclusion is further confirmed by looking at the last column of Table 5. If the top wage decile is excluded (instead of D9/D5, we look at D8/D5), wage inequality drops significantly in every country. It leads to the conclusion that the primary contributors of high wage inequalities are excessively high wages at the top. These high wages are most likely a consequence of "celebrity" and "managerial" wages, usually caused by peer behavior and rent seeking. This observation contradicts the view that higher wage inequality will do much to improve the outcomes of the people at the bottom, as is promoted by the economists who support the wage compression hypothesis. On other hand, this exercise shows that wages are indeed more compressed in the bottom half of the distribution than in the top half of the wage distribution in all countries (despite more dispersed skills at the bottom). This is a starting point that could offer support for a wage compression hypothesis. In order to investigate if the wage compression hypothesis is correct, and whether compressed wages are related to unemployment, an examination of employment differences between countries is necessary (see Section 6).
Table 5 Distribution of individual average numeracy test scores and wages, by country
Regardless of whether the relationship between skill inequality and wage inequality is measured by decile ratios or coefficient of variation, the relationship is not statistically significant (see Fig. 2). The correlation coefficientsFootnote 21 are 0.11, 0.24, and 0.19, respectively. Inequality in numeracy test scores is not correlated with wage inequality, and this is why the variation in numeracy skill inequality cannot explain the variation in wage inequality across core OECD countries. The same is true if years of schooling are used as a measure of skill. The relationship between the coefficient of variation of wages and years of schooling is flat—there is no significant relationship between the two; the correlation coefficient is low: −0.06. It does not hold that countries with higher skill dispersion (either measured by numeracy test scores or years of schooling) have higher wage dispersion and vice versa, as the lower panel of Fig. 2 suggests. Countries with similar skill inequality differ significantly in terms of wage inequality. The skill compression hypothesis cannot be confirmed based on the cross-country analysis presented here.
Relationship between skill inequality and wage inequality, employed persons. a D5/D1 and D9/D5 ratios of wages and numeracy test scores, employed persons. b Coefficient of variation of wages, skills, and schooling, employed persons. Source: calculations based on PIAAC
Although there is some criticism (see Broecke et al. 2016), these rather descriptive results are in line with other wage and skill distribution analysis conducted previously with the PIAAC data set (and this is why deeper analysis is not necessary). Paccagnella (2015) investigated the relationship between skill inequality and wage inequality based on PIAAC data and 22 OECD countries. He finds no strong relationship between the two. Based on his decomposition exercise, he concludes that the wage structure effect (differences in the rates of returns to observable characteristics) seems to be more important in explaining cross-country differences in wage dispersion than the composition effect (differences in observable characteristics). Pena (2016) also uses the decomposition method similar to Juhn et al. (1993) and finds that unobservable factors (such as labor and product market institutions) play a major role in explaining cross-country differences in wage dispersion; the effect of skills is rather small. Thus, both papers suggest that institutions are potentially likely to explain a larger share of cross-country differences in wage dispersion.
Wage dispersion and return to skills
The wage compression hypothesis is based on the perfect market theory, according to which, wages correspond to marginal productivity. Empirically, wage regression analysis should be able to explain the variation in wages. In this body of literature, Mincer (1958, 1974) was the pioneer in defining earnings as a function of schooling and experience in the log-linear form. The Mincer earnings equation proved to be a big empirical success in labor market economics, and the model is still a good specification for estimating the relationships between schooling, experience, and earnings relatively accurately (see Lemieux 2006). The empirical model that is to be estimated in this paper is based on the Mincer earnings equation and has the following principal form:
$$ \ln \left(\mathrm{w}\right) = a + bS + cX + dG + eI + u $$
where ln(w) is the natural logarithm of the hourly wage, S corresponds to the qualification level (numeracy test scores or years of schooling, or both), X is experience (defined as years of labor market experience), G is a gender indicator, I denotes immigration status, u is a residual, and a, b, c, d, e are parameters to be estimated.
Table 6 reports the results from OLS regressions of log wage on numeracy test scores and years of schooling in models which include controls for gender, experience, experience squared, and immigrant status (see Eq. 1). Model 1 results show considerable variation across countries. In some countries, an increase in numeracy test scores is associated with higher wages than in other countries. An increase of 100 numeracy score points is associated with a 30% increase in the average wage in the pooled sample across countries. The highest coefficients are in the USA, the UK, Germany, and Spain, and the lowest are in Norway, Italy, and Denmark. If one interpreted these results by saying that skills affect wages significantly in the USA (coefficient = 0.48), one needs to be able to explain why the coefficient is only 0.21 in the case of Norway. Differences in dispersion of numeracy skills explain the differences in dispersion of earnings only partly. Model 2 shows that the coefficient of years of schooling on wages is the highest in the USA (11%), Germany (10%) and the UK, and the Netherlands (9%), whereas the lowest is in Italy, France, and Scandinavia (6%). On average, one extra year of schooling is associated with 7% higher earnings. Once we add both numeracy scores and years of schooling to the model, both coefficients are significant, although the size of the score coefficient drops significantly (from 0.30 to 0.15 in the pooled regression). This is due to the fact that numeracy skills and schooling are correlated (0.45 on average). However, big variation across countries is evident here as well; whereas in most of the countries the skill coefficient drops by around half, in the UK, Ireland, and Norway, it drops less. In this model, the coefficient of years of schooling remains stable at 7% on average. The 1% fall is observed in all countries, except for the UK and the USA, where the drop is equal to 2%. These findings are similar to those of Hanushek et al. (2014)Footnote 22.
Table 6 Regression of log wages on numeracy test scores and years of schooling, employed persons
Once controlled for all factors, why does return to skills vary so much across countries? Although the fact that the coefficients are highest in the first model could lead to the conclusion that the skill compression hypothesis holds, this notion is rejected. Especially in the model where both skills and years of schooling are included, the coefficient for skills drops by half. It might be that schooling affects wages independently from numeracy skills (possibly through the signaling effect). However, it all leads to the conclusion that there must be something else (in addition to numeracy scores and years of schooling) that affects wage structure significantly and affects wage inequality as well.Footnote 23 As mentioned above, if the perfect market theory was correct, wages should be explained by the wage regression and residual should be equal to one. However, Mincer equations explain only 30% of the variation of wages; this either disproves the perfect market hypothesis or increases the relevance of immeasurable skills Schettkat (2008).
Dispersion within skill level
While it is often argued that high wage inequality fosters investment in human capital, Agell (1999) claimed that that could be true but only if the wage dispersion is between education levels. However, if there is high wage dispersion within the same education level, wage dispersion serves as a discouragement for educational attainment. Based on similar logic, as among the most convincing evidence that the skill hypothesis does not hold, Devroye and Freeman (2001) used the tables that show that dispersion of wages is much higher within skill levels than between skill levels. If skill determines wages, people at the same skill level should earn similar wages—the highest dispersion should be between different skill levels; within skill levels, there should barely be any significant dispersion. In their analysis based on the IALS data set and four OECD countries, Freeman and Devroye find that this was not the case. We perform the same calculations based on the PIAAC data set. Table 7 records the coefficient of the variation of log wages by six defined numeracy test score levels. The conclusion is the same—wage dispersion within skill levels differs significantly across countries. The highest dispersions of earnings are in Germany, Ireland, Spain, and the USA for every score level. The smallest dispersions are in Japan, Denmark, and Norway. Countries that have the highest wage dispersion in the lowest skill levels have on average comparatively higher wage dispersions for all skill levels and vice versa. In the second part of Table 7, the same exercise is performed for six different schooling levels.Footnote 24 Schooling levels tell the same story. It is interesting to see how countries do not deviate at all in the coefficient of variation of wages. Countries that have among the highest within-skill-level wage dispersion also have the highest within-schooling-level dispersion of wages. The biggest variation is within different schooling levels and not between them, and it is astonishing how this pattern is repeated in every country and on every schooling level. Thus, variation in numeracy skills cannot fully explain the variation in wages. Some other factor (other than numeracy skills and schooling) in these countries and their institutional settings must create these differences.
Table 7 Coefficient of variation of log wages by score and schooling level, employed persons
High wage dispersion within skill and schooling levels is in stark contrast with the marginal productivity theory. Based on the theoretical perfect market model, the marginal productivity theory claims that everybody is paid according to their contribution—to their marginal productivity. The empirical implication of this theory shows that there is the same wage for the same work. Since productivity is difficult to measure, it is necessary to find different proxies that could account for it. The most obvious ones are skills. Stiglitz (2013) commented that he wishes bankers were paid according to their marginal productivity during crisis. Proponents of marginal productivity theory and perfect markets try to defend their theory by claiming that people with the same measurable skills might differ in their immeasurable skills and this is why their wages are different; yet explanations based on monopsonistic labor market seem more plausible (see Manning 2003).
Wage compression and unemployment
Since the variation in wage dispersion across countries cannot be fully explained by variation in skill dispersion and its theoretical foundations seem to be flawed, another set of explanations needs to be considered. Some economists stress the importance of variation in wage-setting institutions across countries, for example, minimum wages and unions (Freeman 1991; Freeman and Katz 1994; Blau and Kahn 1996) as the most plausible explanation for cross-country variation in wage dispersion. Before the link between wage dispersion and unemployment is explored, the relationship between wage-setting institutions and wage dispersion is examined. Table 8 shows a clear pattern—there is a significant negative correlation between various wage bargaining institutions and wage inequality. Countries with higher union density and union membership, stronger and more coordinated wage bargaining institutions, and higher minimum wages have lower wage inequality and vice versa. This is in line with other studies based on panel data analysis (Schettkat 2003; Freeman 2007; Salverda and Checchi 2014). It is interesting to observe that the correlation coefficient between wage dispersion and various institutions is much higher than the correlation coefficient between wages and skills (see Section 3). Regrettably, the PIAAC data set does not provide information on union membership of the employees, so more thorough analysis is not possible. However, this data set offers information on employment status which allows us to examine the wage compression hypothesis.
Table 8 Relationship between wage inequality and wage-setting institutions, employed persons, 2011
As seen in the previous table, minimum wages and wage-setting institutions are negatively correlated with wage inequality. This is exactly why some economists (neoclassical school of thought) claim that strong institutions cause wage compression, which in turn causes high unemployment among the low skilled (Siebert 1997; Heckman and Jacobs 2010). Due to skill-biased technical change, the relative demand for low-skilled workers declined in the past three decades. In countries with flexible labor markets (and weaker institutions), workers' wages dropped but they remained employed. In countries with rigid markets, institutions prevented the wages of low-skilled workers from falling and therefore these workers lost their jobs. In the first group of countries, an increase in wage inequality contributed to comparatively higher employment. If the wage compression hypothesis was true and differences in wage inequalities across countries can explain differences in employment, we expect to find a positive relationship between wage inequality in the bottom half of the wage distribution and employment among low-skilled workers. This explanation is based on the marginal productivity hypothesis, according to which, wages always correspond to the marginal product of labor. If there is no institutional intervention, the free market leads to solutions in which people earn what they contribute. Setting a wage through various forms of labor market institutions will lead to a higher wage than marginal productivity and higher unemployment subsequently.
In order to get a complete measure of labor market performance, employment to population rates (e-pops), the unemployment rate, and average weekly hours worked per head were calculated from the PIAAC survey or were already available (weekly hours worked). Table 9 shows the correlation matrix for various measures of labor market performance and wage inequality (for all employed persons and all skill levels). The majority of correlation signs are statistically insignificant. No matter which measure of labor market performance is being used, the relationship between labor market performance and wage inequality is insignificant and flat. If we look at the whole sample (regardless of skill level), there seems to be no significant relationship between these measures. In the case of e-popsFootnote 25 and unemployment rates, the correlation sign actually contradicts the wage compression hypothesis, although it is insignificant. If skill levels are accounted for, most of the correlations still remain insignificant at a 10% significance level.Footnote 26 E-pops, average hours worked, and unemployment rates are not related to wage inequality, either at the top or at the bottom. According to Table 9, and analysis based on the core OECD countries, there is no evidence for wage compression hypothesis.
Table 9 Relationship between wage inequality and labor market performance, employed persons
Furthermore, Fig. 3 focuses only on the relationships between wage inequality (D5/D1) and employment for the low-skilled workers and allows additionally observing individual countries. The first diagram in the upper left corner shows a slightly positive (although insignificant) relationship between the D5/D1 wage ratio and e-pops in the lowest skill level. The USA is the country with high wage inequality (D5/D1) that simultaneously has a good performance in terms of employment. However, all three diagrams find no support for the wage compression hypothesis—countries' labor market performance in the low-skill sector does not show a relation to wage inequality at the bottom half of the wage distribution; the pattern is rather mixed.
Wage inequality (D5/D1) and employment for the low-skilled workers. Source: calculations based on PIAAC
Why do some countries do so well in terms of low-skill employment, whereas others are much less successful in job creation? Can this cross-country variation in employment be explained by cross-country variation in wage inequality? Figure 4 presents e-pops/hours worked per head/unemployment rate for four different skill levels for eight selected countries. Countries are selected according to the lowest (highest) proportion of employed persons in the low-skill group. The diagram displays a very clear pattern. Employment to population rates are highest in high skill level groups, as expected. Countries that have comparatively higher employment among low-skilled workers (the USA, Norway, and Canada) also demonstrate higher employment in the other skill groups. Countries with the lowest employment among low-skilled workers (Spain, Ireland, and Italy) also have the lowest employment in other skill groups. When wage inequality among these countries is observed, the picture becomes mixed and there is no clear pattern. It rather seems more plausible that some countries are in general more successful in employment creation than others. It is not the low-skill sector and excessively high wages at the bottom of the wage distribution that make the whole difference in the employment performance of the countries but rather something else, e.g., economic policy-making. The only country that does not follow this general pattern is Japan. It has one of the highest e-pops in the lowest skill groups L0 and L1, whereas e-pops in other skill groups are significantly lower. The same story is true for unemployment rates. Only at the highest skill levels is unemployment low everywhere with no pattern across countries—high-skilled workers have low unemployment rates in all countries (under 6%). However, all other countries exhibit either high or low unemployment, regardless of the skill level. Average hours worked per head do not seem to vary much at different skill levels in Spain, Italy, Finland, and Sweden. In other countries, higher skills are related to higher number of hours worked and they are especially high for the highest skill workers. Even in countries with flexible wages in the bottom half of the wage distribution, average hours worked for low-skilled workers are lower than hours worked for high-skilled workers and well paid. Germany, the Netherlands, and Ireland have at the same time the highest wage dispersion in the bottom half of the wage distribution and the lowest average hours worked in the low-skill sector, which is not in line with theory. It is actually in Finland (low inequality country) in which there is no difference in the average weekly hours worked across skill groups.
Employment to population rate for four different skill levels, by country. Source: calculations based on PIAAC
Finally, in order to perform an additional check, the mean and median score results between the employed and unemployed across countries are compared. If the wage compression hypothesis was true, it would be expected that, in the countries with rigid labor markets and low inequality, the pool of unemployed consists mainly of low-skilled workers. At the same time, countries with flexible labor markets are expected to have much higher employment in the low-skilled sectorFootnote 27 (and low skilled should not be unemployed).Footnote 28 Table 10 shows the mean, median, and standard deviation of numeracy skill scores by labor force status. Employed persons in the USA, the UK, Spain, and Italy have lower average scores than the unemployed in Japan, Belgium, Finland, Denmark, and the Netherlands. Since the latter countries (apart from Japan) have at the same time a more compressed wage structure, low-skilled people in these countries should be unemployed (on the basis that their wage is too high). Indeed, some of these less unequal countries do demonstrate low employment at the bottom. But these workers are not unskilled; their average score results are too high, as the data suggests. The data actually shows that the unemployed in these countries have higher average scores than the employed in some other countries. On the other hand, in the first group of countries, where wage flexibility is higher, the employment of low-skilled workers should be higher. However, the unemployed do have very low average skill scores, which is contradictory to the wage compression hypothesis. Furthermore, in Japan, there is almost no difference in the average score results between the employed and unemployed, which is again evidence against the wage compression hypothesis. The average score results of people out of the labor force are comparable to those of the unemployed people with a minor variation in the number of score points in both directions.
Table 10 Mean and standard deviation of numeracy skill scores by labor force status
But then again, who are the employed, unemployed, and out-of-labor force? Are the subgroups of these three pools of people somehow different and can they reveal important insights? Data showsFootnote 29 that on average (in the pooled sample) there is no large difference between men and women—they are almost equally represented in both pools of the employed and unemployed. The share of men in the employed population is slightly higher than the share of women—their share varies between 52 and 54% in almost every country, with the notable exceptions of Italy and Japan where the share of men in the employed population is 60 and 58%, respectively. However, on average, the people who make up the out-of-labor force are more likely to be women (60%), compared to only 40% men in this group. This share is even higher in Japan, Italy, the Netherlands, the USA, and the UK, where women's participation in the labor market is lower than men's, possibly while they engage more in the household activities and parenthood and due to social norms. Only in the Scandinavian countries does there seem to be almost no gender difference in this regard. When it comes to immigration status, immigrants are only slightly more present in the pool of the unemployed compared to the pool of the employed and the out-of-labor force, relative to the non-immigrants. The main conclusion about the age subgroups is that unemployment is gradually decreasing with age across all countries. The pool of people out of the labor force is mainly represented by the lowest and highest age subgroups (age groups 1 and 5), and these two groups together account for around 60% of those out-of-labor forces on average.
Challenges to the validity of the wage compression hypothesis have been made in earlier cross-country empirical work (Glyn et al. 2006; Howell et al. 2007; Jovicic and Schettkat 2013), which found no evidence of a relation between wage compression (strong institutions) and unemployment. There are also a number of studies based on micro data that could not explain the high European unemployment rates with institutional rigidity (Card et al. 1996; Krueger and Pischke 1997). At the same time, some other economists were insisting on exploring the aggregate demand deficiency and macroeconomic policies as a potential explanation for employment differences across countries (see Solow 2008; Krugman 2009; Schettkat and Sun 2009; Wolf 2014). However, this evidence appears to have been ignored, and the deregulation of welfare-state institutions remained the main policy recommendations even today in Europe.
Share of low-paid jobs
It is doubtful that countries with rigid labor market institutions and rigid wages at the bottom of the distribution have low employment among the low-skilled workforce, as the previous analysis showed. What are the consequences of compressed wage structures? Figure 5 shows the share of low-paid jobs, where low pay is defined as 2/3 of the median wage in OECD countries. The countries with the highest share of low-paid jobs are Germany, the USA, Japan, and the UK. Not surprisingly, these are the countries where the dispersion of the wages in the bottom half of the wage distribution is relatively high. (Alternatively, the USA has a relatively high employment among low-skilled workers, but this is certainly not the case for the rest of the countries.) The high share of low-payed jobs was not enough to produce high employment in the low-skill sector in Japan, the UK, and Ireland. On the other hand, Norway managed to maintain well-paid jobs and high employment at the same time. The only certain result of the wage flexibility hypothesis is that there is a higher share of low-paid jobs. Proponents of the low-pay policy claim that this is still better than unemployment. This paper, however, finds no evidence for the wage compression hypothesis.
Share of low-paid jobs measured as two thirds of median wage, employed persons. Source: calculations based on PIAAC
Based on the PIAAC adult skill survey, this paper examined international differences in wage inequality and skills and whether a compressed wage distribution is associated with high unemployment across core OECD countries. Although both the skill compression and wage compression hypotheses have strong theoretical backgrounds, none of them could be empirically verified based on this cross-country study. Firstly, there is a large variation in wage dispersion across countries, but its correlation to variation in skill dispersion is rather weak. Even when accounted for skills, some countries have a more compressed wage structure. Instead, it seems plausible that the other set of explanations in terms of institutions have more power in explaining these differences. According to this analysis, the correlation between various measures of institutions and wage inequality is significantly higher than the correlation between skill inequality and wage inequality. However, in order to confirm this finding, a more detailed analysis is required. Secondly, relative employment performance of low-skilled workers is not worse in countries where the wage premium for skill is more rigid (lower wage inequality). Countries that do well in this sector in terms of employment perform well in general (in all the other groups as well), which is independent from the level of wage inequality. On average, countries that have higher e-pops, higher hours worked, and a lower unemployment rate do not have high wage inequality, either at the top or at the bottom of the wage distribution. The only certain result of wage flexibility is that there is a higher share of low-paid jobs (but this high share of low-paid jobs does not appear to be related to high employment).
These results (although descriptive) have some important implications for policy-making. Based on the perfect market model, marginal productivity theory, skill compression and wage compression hypotheses, etc., institutional reform (which should lead to higher wage dispersion) was considered as the appropriate policy response to increase competitiveness, output, and employment (see OECD 1994; IMF 2003). When not distressed by regulation and public policy, markets should lead to wages that correspond to marginal productivity and full employment should follow. Compressed wages are seen as a likely cause of high unemployment, especially in the low-skill sector; consequently, permitting higher wage dispersion should stimulate employment. The same thinking, grounded on the equity-efficiency trade-off, is guiding austerity measures and reductions in public services in the EU today. This study challenges both hypotheses and the theoretical assumptions they are derived from; it calls for a revision of current policies. Rather than insisting on a deregulation of labor market institutions as the main policy recommendation to achieve higher employment (and higher wage inequality), policymakers should reconsider demand deficiency and macroeconomic policies as potential explanations for the employment differences across countries (see Solow 2008; Krugman 2009; Schettkat and Sun 2009; Wolf 2014). Consistent with this view, expansionary macroeconomic policies—stimulative demand policies—might be necessary in order to achieve high employment and low unemployment. Moreover, high inequality is correlated to major health and social problems, e.g., crime, violence, anxiety, mental illness, obesity, infant mortality, and imprisonment rates (see Wilkinson and Pickett 2009). The causation behind these correlations is subject to further scrutiny however (see Salverda et al. 2014). Not only do high wage dispersions have negative consequences on societies, but this study also shows that wage dispersion is not vital for better labor market performance.
This study builds on the previous work of Devroye and Freeman (2001) and Freeman and Schettkat (2001), who performed similar analysis based on the IALS literacy survey from 1998 and two (four) countries. These findings, based on the more recent literacy survey (PIAAC) and core OECD countries, are in line with their findings and confirm their results. However, one must acknowledge that literacy surveys have their limitations; they capture a narrow measure of skills. Furthermore, the evidence presented here is rather descriptive. Yet, if the skill compression and wage compression hypotheses were true, even descriptive cross-country analysis would be expected to show that there are correlations and patterns between the variables of interest. The evidence presented herein illustrates that this is certainly not the case.
In their paper, however, skills are measured by years of schooling and not by competency test scores.
Problem solving is not measured in France, Italy, and Spain.
Belgium is represented by its subunit Flanders. It is the most developed part of the country, with the lowest unemployment rate, and it cannot be considered as a representative for the whole country. It is important to keep this in mind when interpreting the study results.
For Germany, the USA, and Austria, we obtained a Scientific-Use-File from their national centers (GESIS—Leibniz Institute for the Social Sciences, American Institutes for Research, and Statistics Austria, respectively). For Canada and Sweden, information about continuous earnings is not available.
National samples are weighted to population in the relevant time period.
The PIAAC sample design requires using plausible values of score technique which is used through the whole analysis.
These are available on request.
The OECD earnings database collects data on gross earnings of full-time dependent employees which are usually taken from household surveys.
Skill levels are defined according to numeracy score results in the following way: L0 < 176; L1 = 176–226; L2 = 226–276; L3 = 276–326; L4 = 326–376; L5 > 376 points.
The share of population in skill groups L0 and L5 is very low and not representative; that is why they are observed together with groups L1 and L4.
Immigrants include first-generation immigrants. Quick tabulation shows that around 76% of the immigrants are not native speakers. Being a native speaker is highly correlated with higher scores in every country. On average, native speakers have 40 points higher scores than non-native speakers.
The share of women in the employed population varies between 46 and 49% in almost all countries, with the notable exceptions of Italy and Japan where the share of women in employed population is relatively small—around 40%.
The share of immigrants varies between less than 1% in Japan and 32% in Canada.
Japan, Finland, and Sweden have the highest share of the oldest age group (more than 20%), but in these countries, the oldest age groups have relatively high scores. On the other hand, Austria, Ireland, Italy, and France have small shares of the oldest age groups, but these countries do not have high average scores.
Wage and salary earners could choose among reporting their earnings per hour, day, week, 2 weeks, month, or year or by piece rate. There was also an option for respondents to report their earnings in broad categories which was especially attractive for those who knew only roughly how much they earn. These novelties improved the data quality and willingness to report earnings (for more details, see OECD 2013a, b).
Certainly the most widely used measure of skill in human capital literature is years of schooling. Years of schooling are easy to measure, and they are easily available for researchers. For a long time, this was probably the only measure of skills, since international comparative surveys of skills were first done in the 1990s.
In the pooled sample, coefficient of variation does not seem to vary between men, women, immigrants, and non-immigrants. However, wage dispersion is the highest in the youngest and oldest age subgroup, and it is decreasing with the decrease of the age in the rest of the groups. The same is true for D9/D5 and D5/D1. Additionally, D5/D1 is slightly higher for men and immigrants than for women and non-immigrants.
Additional analysis shows that there is no difference in the correlation coefficient between gender, age, and (non-) immigrant subgroups. The correlation coefficient in all the subgroups varies between 0.41 and 0.48 in the pooled sample.
France, Italy, and Spain have the highest dispersion of years of schooling mainly due to the high dispersion in the bottom half of the distribution (comparative to the other countries). These three countries have the highest shares of employed persons with the lowest number of years of schooling (5 or 6) in the overall employed population—France (almost 14%), Italy (5%), and Spain (14%). Moreover, France and Italy are the only two countries in the sample in which there are people that acquired 5 years of schooling only.
Decile is any of the nine values that divides the sorted data into ten equal parts so that each part represents 1/10 of the sample or population. The decile ratio is an indicator of dispersion; it is calculated by dividing the ratio of the 9/5th decile by the 5th/1st decile of skill scores and hourly earnings of an employed person.
Wage and skill inequality measured by D9/D1, Gini coefficient, and Theil index also show that there is no strong relation. Their correlation coefficients are 0.09, 0.05, and 0.02, respectively.
Hanushek et al. (2014) examined return to skills based on the PIAAC data set and found significant heterogeneity between the countries. Returns to skills (associated with a one-standard-deviation increase in measured numeracy test scores) vary between 12 and 15% in Nordic countries and 28% in the USA. Furthermore, returns to skill are lower in countries with higher union density, stricter employment protection, and a larger public sector.
It could also be that schooling reflects wider range of skills, but this analysis is limited to numeracy skills only.
L1—lower secondary or less; L2—upper secondary; L3—post-secondary, non-tertiary; L4—tertiary professional; L5—tertiary bachelor; L6—tertiary master degree
The employment to population rate refers to the percentage share of employed persons in the total working age population.
The only correlation that is of weak significance (at only a 10% significance level) is the one between hours worked per head and wage inequality at the upper part of the distribution. More hours worked are related to higher wage inequalities at the top.
Analysis shows that there is no correlation between the relative deviation of scores between the employed and the unemployed and the wage dispersion in the low-skilled sector (D5/D1), which is not in line with the wage compression hypothesis.
Surely, there will always be some frictional unemployment, but it exists in all skill groups.
All tables and graphs are available upon request.
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I would like to thank the anonymous referees and the editor for the useful remarks.
Responsible editor: Martin Kahanec.
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Schumpeter School of Business and Economics, University of Wuppertal, Gaußstraße 20, 42119, Wuppertal, Germany
Sonja Jovicic
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Correspondence to Sonja Jovicic.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Jovicic, S. Wage inequality, skill inequality, and employment: evidence and policy lessons from PIAAC. IZA J Labor Stud 5, 21 (2016) doi:10.1186/s40174-016-0071-4
Skill distribution
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\begin{definition}[Definition:Evolute]
Consider a curve $C$ embedded in a plane.
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Works by Joanna Golinska-Pilarek
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Joanna Golinska-Pilarek
Relational dual tableau decision procedures and their applications to modal and intuitionistic logics.Joanna Golińska-Pilarek, Taneli Huuskonen & Emilio Muñoz-Velasco - 2014 - Annals of Pure and Applied Logic 165 (2):409-427.details
This paper introduces Basic Intuitionistic Set Theory BIST, and investigates it as a first-order set theory extending the internal logic of elementary toposes. Given an elementary topos, together with the extra structure of a directed structural system of inclusions on the topos, a forcing-style interpretation of the language of first-order set theory in the topos is given, which conservatively extends the internal logic of the topos. This forcing interpretation applies to an arbitrary elementary topos, since any such is equivalent to (...) one carrying a dssi. We prove that the set theory BIST+Coll is sound and complete relative to forcing interpretations in toposes with natural numbers object . Furthermore, in the case that the structural system of inclusions is superdirected, the full Separation schema is modelled. We show that all cocomplete and realizability toposes can be endowed with such superdirected systems of inclusions.A large part of the paper is devoted to an alternative notion of category-theoretic model for BIST, which, following the general approach of Joyal and Moerdijk's Algebraic Set Theory, axiomatizes the structure possessed by categories of classes compatible with BIST. We prove soundness and completeness results for BIST relative to the class-category semantics. Furthermore, BIST+Coll is complete relative to the restricted collection of categories of classes given by categories of ideals over elementary toposes with nno and dssi. It is via this result that the completeness of the original forcing interpretation is obtained, since the internal logic of categories of ideals coincides with the forcing interpretation. (shrink)
Category Theory in Philosophy of Mathematics
Tableaux and Dual Tableaux: Transformation of Proofs.Joanna Golińska-Pilarek & Ewa Orłowska - 2007 - Studia Logica 85 (3):283-302.details
We present two proof systems for first-order logic with identity and without function symbols. The first one is an extension of the Rasiowa-Sikorski system with the rules for identity. This system is a validity checker. The rules of this system preserve and reflect validity of disjunctions of their premises and conclusions. The other is a Tableau system, which is an unsatisfiability checker. Its rules preserve and reflect unsatisfiability of conjunctions of their premises and conclusions. We show that the two systems (...) are dual to each other. The duality is expressed in a formal way which enables us to define a transformation of proofs in one of the systems into the proofs of the other. (shrink)
Mathematical Proof in Philosophy of Mathematics
Predicate Logic in Logic and Philosophy of Logic
Non-Fregean Propositional Logic with Quantifiers.Joanna Golińska-Pilarek & Taneli Huuskonen - 2016 - Notre Dame Journal of Formal Logic 57 (2):249-279.details
We study the non-Fregean propositional logic with propositional quantifiers, denoted by $\mathsf{SCI}_{\mathsf{Q}}$. We prove that $\mathsf{SCI}_{\mathsf{Q}}$ does not have the finite model property and that it is undecidable. We also present examples of how to interpret in $\mathsf{SCI}_{\mathsf{Q}}$ various mathematical theories, such as the theory of groups, rings, and fields, and we characterize the spectra of $\mathsf{SCI}_{\mathsf{Q}}$-sentences. Finally, we present a translation of $\mathsf{SCI}_{\mathsf{Q}}$ into a classical two-sorted first-order logic, and we use the translation to prove some model-theoretic properties of (...) $\mathsf{SCI}_{\mathsf{Q}}$. (shrink)
Number of Extensions of Non-Fregean Logics.Joanna Golińska-Pilarek & Taneli Huuskonen - 2005 - Journal of Philosophical Logic 34 (2):193-206.details
We show that there are continuum many different extensions of SCI (the basic theory of non-Fregean propositional logic) that lie below WF (the Fregean extension) and are closed under substitution. Moreover, continuum many of them are independent from WB (the Boolean extension), continuum many lie above WB and are independent from WH (the Boolean extension with only two values for the equality relation), and only countably many lie between WH and WF.
Logical Connectives, Misc in Logic and Philosophy of Logic
Nonclassical Logics in Logic and Philosophy of Logic
Relational dual tableau decision procedures and their applications to modal and intuitionistic logics.Joanna Golińska-Pilarek & Taneli Huuskonen - 2014 - Annals of Pure and Applied Logic 165 (2):428-502.details
This paper introduces Basic Intuitionistic Set Theory BIST, and investigates it as a first-order set theory extending the internal logic of elementary toposes. Given an elementary topos, together with the extra structure of a directed structural system of inclusions on the topos, a forcing-style interpretation of the language of first-order set theory in the topos is given, which conservatively extends the internal logic of the topos. This forcing interpretation applies to an arbitrary elementary topos, since any such is equivalent to (...) one carrying a dssi. We prove that the set theory BIST+Coll is sound and complete relative to forcing interpretations in toposes with natural numbers object. Furthermore, in the case that the structural system of inclusions is superdirected, the full Separation schema is modelled. We show that all cocomplete and realizability toposes can be endowed with such superdirected systems of inclusions.A large part of the paper is devoted to an alternative notion of category-theoretic model for BIST, which, following the general approach of Joyal and Moerdijk's Algebraic Set Theory, axiomatizes the structure possessed by categories of classes compatible with BIST. We prove soundness and completeness results for BIST relative to the class-category semantics. Furthermore, BIST+Coll is complete relative to the restricted collection of categories of classes given by categories of ideals over elementary toposes with nno and dssi. It is via this result that the completeness of the original forcing interpretation is obtained, since the internal logic of categories of ideals coincides with the forcing interpretation. (shrink)
Rasiowa-Sikorski proof system for the non-Fregean sentential logic SCI.Joanna Golinska-Pilarek - 2007 - Journal of Applied Non-Classical Logics 17 (4):509–517.details
The non-Fregean logic SCI is obtained from the classical sentential calculus by adding a new identity connective = and axioms which say ?a = ß' means ?a is identical to ß'. We present complete and sound proof system for SCI in the style of Rasiowa-Sikorski. It provides a natural deduction-style method of reasoning for the non-Fregean sentential logic SCI.
Relational approach for a logic for order of magnitude qualitative reasoning with negligibility, non-closeness and distance.Joanna Golinska-Pilarek & Emilio Munoz Velasco - 2009 - Logic Journal of the IGPL 17 (4):375–394.details
We present a relational proof system in the style of dual tableaux for a multimodal propositional logic for order of magnitude qualitative reasoning to deal with relations of negligibility, non-closeness, and distance. This logic enables us to introduce the operation of qualitative sum for some classes of numbers. A relational formalization of the modal logic in question is introduced in this paper, i.e., we show how to construct a relational logic associated with the logic for order-of-magnitude reasoning and its dual (...) tableau system which is a validity checker for the modal logic. For that purpose, we define a validity preserving translation of the modal language into relational language. Then we prove that the system is sound and complete with respect to the relational logic defined as well as with respect to the logic for order of magnitude reasoning. Finally, we show that in fact relational dual tableau does more. It can be used for performing the four major reasoning tasks: verification of validity, proving entailment of a formula from a finite set of formulas, model checking, and verification of satisfaction of a formula in a finite model by a given object. (shrink)
Relational dual tableaux for interval temporal logics.David Bresolin, Joanna Golinska-Pilarek & Ewa Orlowska - 2006 - Journal of Applied Non-Classical Logics 16 (3-4):251–277.details
Interval temporal logics provide both an insight into a nature of time and a framework for temporal reasoning in various areas of computer science. In this paper we present sound and complete relational proof systems in the style of dual tableaux for relational logics associated with modal logics of temporal intervals and we prove that the systems enable us to verify validity and entailment of these temporal logics. We show how to incorporate in the systems various relations between intervals and/or (...) various time orderings. (shrink)
Spectra of formulae with Henkin quantifiers.Joanna Golinska-Pilarek & Konrad Zdanowski - 2003 - In A. Rojszczak, J. Cachro & G. Kurczewski (eds.), Philosophical Dimensions of Logic and Science. Kluwer Academic Publishers. pp. 29-45.details
It is known that various complexity-theoretical problems can be translated into some special spectra problems. Thus, questions about complexity classes are translated into questions about the expressive power of some languages. In this paper we investigate the spectra of some logics with Henkin quantifiers in the empty vocabulary.
Generalized Quantifiers in Philosophy of Language
Second-Order Logic in Logic and Philosophy of Logic
Logic. of Descriptions. A New Approach to the Foundations of Mathematics and Science.Joanna Golińska-Pilarek & Taneli Huuskonen - 2012 - Studies in Logic, Grammar and Rhetoric 27 (40):63-94.details
We study a new formal logic LD introduced by Prof. Grzegorczyk. The logic is based on so-called descriptive equivalence, corresponding to the idea of shared meaning rather than shared truth value. We construct a semantics for LD based on a new type of algebras and prove its soundness and completeness. We further show several examples of classical laws that hold for LD as well as laws that fail. Finally, we list a number of open problems. -/- .
Logic and Philosophy of Logic, Miscellaneous in Logic and Philosophy of Logic
Logical Connectives in Logic and Philosophy of Logic
Dual Tableaux: Foundations, Methodology, Case Studies.Ewa Orlowska & Joanna Golinska-Pilarek - 2011 - Springer.details
The book presents logical foundations of dual tableaux together with a number of their applications both to logics traditionally dealt with in mathematics and philosophy (such as modal, intuitionistic, relevant, and many-valued logics) and to various applied theories of computational logic (such as temporal reasoning, spatial reasoning, fuzzy-set-based reasoning, rough-set-based reasoning, order-of magnitude reasoning, reasoning about programs, threshold logics, logics of conditional decisions). The distinguishing feature of most of these applications is that the corresponding dual tableaux are built in a (...) relational language which provides useful means of presentation of the theories. In this way modularity of dual tableaux is ensured. We do not need to develop and implement each dual tableau from scratch, we should only extend the relational core common to many theories with the rules specific for a particular theory. (shrink)
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Reasoning with Qualitative Velocity: Towards a Hybrid Approach.Joanna Golinska-Pilarek & Emilio Munoz Velasco - 2012 - In Emilio Corchado, Vaclav Snasel, Ajith Abraham, Michał Woźniak, Manuel Grana & Sung-Bae Cho (eds.), Hybrid Artificial Intelligent Systems. Springer. pp. 635--646.details
Qualitative description of the movement of objects can be very important when there are large quantity of data or incomplete information, such as in positioning technologies and movement of robots. We present a first step in the combination of fuzzy qualitative reasoning and quantitative data obtained by human interaction and external devices as GPS, in order to update and correct the qualitative information. We consider a Propositional Dynamic Logic which deals with qualitative velocity and enables us to represent some reasoning (...) tasks about qualitative properties. The use of logic provides a general framework which improves the capacity of reasoning. In this way, we can infer additional information by using axioms and the logic apparatus. In this paper we present sound and complete relational dual tableau that can be used for verification of validity of formulas of the logic in question. (shrink)
Relational dual tableau decision procedure for modal logic K.Joanna Golińska-Pilarek, Emilio Munoz-Velasco & Angel Mora - 2012 - Logic Journal of the IGPL 20 (4):747-756.details
We present a dual tableau system, RLK, which is itself a deterministic decision procedure verifying validity of K-formulas. The system is constructed in the framework of the original methodology of relational proof systems, determined only by axioms and inference rules, without any external techniques. Furthermore, we describe an implementation of the system RLK in Prolog, and we show some of its advantages.
Relational dual tableau decision procedure for modal logic K.Joanna Golińska-Pilarek, Emilio Muñoz-Velasco & Angel Mora-Bonilla - 2012 - Logic Journal of the IGPL 20 (4):747-756.details
We present a dual tableau system, RLK, which is itself a deterministic decision procedure verifying validity of K-formulas. The system is constructed in the framework of the original methodology of relational proof systems, determined only by axioms and inference rules, without any external techniques. Furthermore, we describe an implementation of the system in Prolog, and we show some of its advantages.
A new deduction system for deciding validity in modal logic K.Joanna Golinska-Pilarek, Emilio Munoz Velasco & Angel Mora - 2011 - Logic Journal of the IGPL 19 (2): 425-434.details
A new deduction system for deciding validity for the minimal decidable normal modal logic K is presented in this article. Modal logics could be very helpful in modelling dynamic and reactive systems such as bio-inspired systems and process algebras. In fact, recently the Connectionist Modal Logics has been presented, which combines the strengths of modal logics and neural networks. Thus, modal logic K is the basis for these approaches. Soundness, completeness and the fact that the system itself is a decision (...) procedure are proved in this article. The main advantages of this approach are: first, the system is deterministic, i.e. it generates one proof tree for a given formula; second, the system is a validity-checker, hence it generates a proof of a formula ; and third, the language of deduction and the language of a logic coincide. Some of these advantages are compared with other classical approaches. (shrink)
A hybrid qualitative approach for relative movements.Joanna Golińska-Pilarek & Emilio Muñoz-Velasco - 2015 - Logic Journal of the IGPL 23 (3):410-420.details
Qualitative description of movements can be very important for representation and reasoning about dynamic systems which are complex in structure or whenever numerical data are incomplete or inaccessible. For this reason, we present a hybrid approach based on the combination of qualitative reasoning, quantitative data and logical methods. In this article, we introduce a new propositional dynamic logic QM for representation and reasoning with relative movements of objects. In this way, we can infer additional information about movements by using axioms (...) and the logic apparatus. We present a sound and complete deduction system in dual tableaux style for the logic QM. The system can be used for verification of validity of formulas of the logic in question. (shrink)
Implementing a relational theorem prover for modal logic K.Angel Mora, Emilio Munoz Velasco & Joanna Golińska-Pilarek - 2011 - International Journal of Computer Mathematics 88 (9):1869-1884.details
An automatic theorem prover for a proof system in the style of dual tableaux for the relational logic associated with modal logic K has been introduced. Although there are many well-known implementations of provers for modal logic, as far as we know, it is the first implementation of a specific relational prover for a standard modal logic. There are two main contributions in this paper. First, the implementation of new rules, called (k1) and (k2), which substitute the classical relational rules (...) for composition and negation of composition in order to guarantee not only that every proof tree is finite but also to decrease the number of applied rules in dual tableaux. Second, the implementation of an order of application of the rules which ensures that the proof tree obtained is unique.As a consequence, we have implemented a decision procedure for modal logic K. Moreover, this work would be the basis for successive extensions of this logic, such as T, B and S4. (shrink)
Everything is a Relation: A Preview.Michał Zawidzki & Joanna Golińska-Pilarek - 2018 - In Michał Zawidzki & Joanna Golińska-Pilarek (eds.), Ewa Orłowska on Relational Methods in Logic and Computer Science. Springer Verlag. pp. 3-24.details
This chapter provides a concise overview of Ewa Orłowska's research contributions and the content of the volume.
Engaged in Relations: A Trialogue.Michał Zawidzki, Joanna Golińska-Pilarek & Ewa Orłowska - 2018 - In Michał Zawidzki & Joanna Golińska-Pilarek (eds.), Ewa Orłowska on Relational Methods in Logic and Computer Science. Springer Verlag.details
The chapter is a transcription of editors' discussion with Ewa Orłowska. It reveals some extracurricular flavors of Ewa Orłowska's biography, brings to light a difficult historical context of her academic career and life, and shows how much internal fortitude she demonstrated while overcoming these difficulties.
Ewa Orłowska on Relational Methods in Logic and Computer Science.Michał Zawidzki & Joanna Golińska-Pilarek (eds.) - 2018 - Cham, Switzerland: Springer Verlag.details
This book is a tribute to Professor Ewa Orłowska, a Polish logician who was celebrating the 60th year of her scientific career in 2017. It offers a collection of contributed papers by different authors and covers the most important areas of her research. Prof. Orłowska made significant contributions to many fields of logic, such as proof theory, algebraic methods in logic and knowledge representation, and her work has been published in 3 monographs and over 100 articles in internationally acclaimed journals (...) and conference proceedings. The book also includes Prof. Orłowska's autobiography, bibliography and a trialogue between her and the editors of the volume, as well as contributors' biographical notes, and is suitable for scholars and students of logic who are interested in understanding more about Prof. Orłowska's work. (shrink)
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An ATP of a Relational Proof System for Order of Magnitude Reasoning with Negligibility, Non-closeness and Distance.Joanna Golinska-Pilarek, Angel Mora & Emilio Munoz Velasco - 2008 - In Tu-Bao Ho & Zhi-Hua Zhou (eds.), PRICAI 2008: Trends in Artificial Intelligence. Springer. pp. 128--139.details
We introduce an Automatic Theorem Prover (ATP) of a dual tableau system for a relational logic for order of magnitude qualitative reasoning, which allows us to deal with relations such as negligibility, non-closeness and distance. Dual tableau systems are validity checkers that can serve as a tool for verification of a variety of tasks in order of magnitude reasoning, such as the use of qualitative sum of some classes of numbers. In the design of our ATP, we have introduced some (...) heuristics, such as the so called phantom variables, which improve the efficiency of the selection of variables used un the proof. (shrink)
Computers in Philosophy of Computing and Information
Filozofia w Polsce po reformie – szanse i wyzwania.Joanna Golińska-Pilarek - 2020 - Ruch Filozoficzny 76 (1):251.details
Dual tableau for a multimodal logic for order of magnitude qualitative reasoning with bidirectional negligibility.Joanna Golińska-Pilarek & Emilio Munoz-Velasco - 2009 - International Journal of Computer Mathematics 86 (10-11):1707–1718.details
We present a relational proof system in the style of dual tableaux for the relational logic associated with a multimodal propositional logic for order of magnitude qualitative reasoning with a bidirectional relation of negligibility. We study soundness and completeness of the proof system and we show how it can be used for verification of validity of formulas of the logic.
Logics of similarity and their dual tableaux. A survey.Joanna Golińska-Pilarek & Ewa Orlowska - 2008 - In Giacomo Della Riccia, Didier Dubois & Hans-Joachim Lenz (eds.), Preferences and Similarities. Springer. pp. 129--159.details
We present several classes of logics for reasoning with information stored in information systems. The logics enable us to cope with the phenomena of incompleteness of information and uncertainty of knowledge derived from such an information. Relational inference systems for these logics are developed in the style of dual tableaux.
Dual tableau for monoidal triangular norm logic MTL.Joanna Golinska-Pilarek & Ewa Orlowska - 2011 - Fuzzy Sets and Systems 162 (1):39–52.details
Monoidal triangular norm logic MTL is the logic of left-continuous triangular norms. In the paper we present a relational formalization of the logic MTL and then we introduce relational dual tableau that can be used for verification of validity of MTL-formulas. We prove soundness and completeness of the system.
Relational proof systems for spatial reasoning.Joanna Golińska-Pilarek & Ewa Orlowska - 2006 - Journal of Applied Non-Classical Logics 16 (3-4):409-431.details
We present relational proof systems for the four groups of theories of spatial reasoning: contact relation algebras, Boolean algebras with a contact relation, lattice-based spatial theories, spatial theories based on a proximity relation.
Tableau reductions: Towards an optimal decision procedure for the modal necessity.Joanna Golińska-Pilarek, Emilio Muñoz-Velasco & Angel Mora - 2016 - Journal of Applied Logic 17:14-24.details
On the Minimal Non-Fregean Grzegorczyk Logic.Joanna Golińska-Pilarek - 2016 - Studia Logica 104 (2):209-234.details
The paper concerns Grzegorczyk's non-Fregean logics that are intended to be a formal representation of the equimeaning relation defined on descriptions. We argue that the main Grzegorczyk logics discussed in the literature are too strong and we propose a new logical system, \, which satisfies Grzegorczyk's fundamental requirements. We present a sound and complete semantics for \ and we prove that it is decidable. Finally, we show that many non-classical logics are extensions of \, which makes it a generic non-Fregean (...) logic. (shrink)
Relational Logics and Their Applications.Joanna Golińska-Pilarek & Ewa Orłowska - 2006 - In Harrie de Swart, Ewa Orlowska, Gunther Smith & Marc Roubens (eds.), Theory and Applications of Relational Structures as Knowledge Instruments Ii. Springer. pp. 125.details
Logics of binary relations corresponding, among others, to the class RRA of representable relation algebras and the class FRA of full relation algebras are presented together with the proof systems in the style of dual tableaux. Next, the logics are extended with relational constants interpreted as point relations. Applications of these logics to reasoning in non-classical logics are recalled. An example is given of a dual tableau proof of an equation which is RRA-valid, while not RA-valid.
Number of non-Fregean sentential logics that have adequate models.Joanna Golińska-Pilarek - 2006 - Mathematical Logic Quarterly 52 (5):439–443.details
We show that there are continuum many different non-Fregean sentential logics that have adequate models. The proof is based on the construction of a special class of models of the power of the continuum.
Logic in Philosophy in Logic and Philosophy of Logic
On Decidability of a Logic for Order of Magnitude Qualitative Reasoning with Bidirectional Negligibility.Joanna Golinska-Pilarek - 2012 - In Luis Farinas del Cerro, Andreas Herzig & Jerome Mengin (eds.), Logics in Artificial Intelligence. Springer. pp. 255--266.details
Qualitative Reasoning (QR) is an area of research within Artificial Intelligence that automates reasoning and problem solving about the physical world. QR research aims to deal with representation and reasoning about continuous aspects of entities without the kind of precise quantitative information needed by conventional numerical analysis techniques. Order-of-magnitude Reasoning (OMR) is an approach in QR concerned with the analysis of physical systems in terms of relative magnitudes. In this paper we consider the logic OMR_N for order-of-magnitude reasoning with the (...) bidirectional negligibility relation. It is a multi-modal logic given by a Hilbert-style axiomatization that reflects properties and interactions of two basic accessibility relations (strict linear order and bidirectional negligibility). Although the logic was studied in many papers, nothing was known about its decidability. In the paper we prove decidability of OMR N by showing that the logic has the strong finite model property. (shrink) | CommonCrawl |
Class 12 Medical Physics
Electromagnetic Induction - Class 12 Medical Physics - Extra Questions
Define coefficient of mutual induction. If in the primary coil of a transformer, the current decreases from $$0.8A$$ to $$0.2A$$ in $$4$$ milliseconds, calculate the induced e.m.f in the secondary coil. Mutual inductance is $$1.76H$$.
Coefficient of mutual induction:-
It is a measure of the induction between two circuits, It is the ratio of the electromotive force in a circuit to the corresponding change of current in neighboring circuit; usually measured in henries.
In given circuit,
$$I_1=0.8 , I_2=0.2 , \Delta t=4\times 10^{-3}, M=1.76$$
But,
Induced emf$$=V_{ind}=M\times \dfrac{I_1-I_2}{\Delta t}$$
$$\therefore V_{ind}=1.76\times \dfrac{0.8-0.2}{4\times 10^{-3}}$$
$$\therefore V_{ind}=264V$$
A rectangular frame of wire $$abcd$$ has dimensions $$32\ cm\times 8.0\ cm$$ and a total resistance of $$2.0\Omega$$. It is pulled out of a magnetic field $$B = 0.020\ T$$ by applying a force of $$3.2\times 10^{-5}N$$ (figure). It is found that the frame moves with constant speed. Find the emf induced in the loop.
In the given figure, emf induced $$e$$=$$VBl$$
If loop move with constant speed therefore electromagnetic force=external $$F$$.
$$F=\dfrac{VB^{2}l^{2}}{R}=F$$
$$V=25 ms^{-1}$$
Now,emf $$e=25 \times 0.02 \times 0.08$$
$$e=0.04V$$
Answer the following :
A metallic rod $$PQ$$ of length $$l$$ is rotated with an angular velocity $$\omega$$ about an axis passing through its mid-point (O) and perpendicular to the plane of the paper, in uniform magnetic field $$\vec{B}$$, as shown in the figure. What is thee potential difference developed between the two ends of the rod, P and Q ?
Given,
Length $$= l$$
Magnetic field = B
Emf induce in $$O Q$$ will be equal to $$OP$$.
And O will be at high potential and P, Q will at low potential
$$V_{OQ} = V_{OQ} = \dfrac{1}{2} \omega^2 \left(\dfrac{l}{2} \right)$$
$$\therefore V_P = V_Q $$
$$\therefore V_{PQ} = O$$
A wire length $$10cm$$ translates in a direction making an angle of $${60}^{o}$$ with its length. The plane of motion is perpendicular to a uniform magnetic field of $$1.0T$$ that exists in the space. Find the emf induced between the ends of the rod if the speed of translation is $$20cm$$ $${s}^{-1}$$
$$l=10cm=0.1m;\theta ={60}^{o};B=1T;V=20m/s=0.2m/s$$
$$E=Bvl\sin{60}^{o}$$
As we know to take that component of length vector which is perpendicular to the velocity vector
$$=1\times 0.2\times 0.1\times \sqrt{3}/2$$
$$=17.32\times {10}^{-3}V$$
Figure shows a metallic wire of resistance $$0.20\Omega$$ sliding on a horizontal, U-shaped metallic rail. The separation between the parallel arms is $$20cm$$. An electric current of $$2.0\mu A$$ passes through the wire when it is slid at a rate of $$20cm$$ $${s}^{-1}$$. If the horizontal component of the earth's magnetic field is $$3.0\times { 10 }^{ -5 }T$$. Calculate the dip at the place.
$$l=20cm=20\times { 10 }^{ -2 }m;v=20m/s=20\times { 10 }^{ -2 }m/s;{ B }_{ H }=3\times { 10 }^{ -5 }T;i=2\mu A=2\times { 10 }^{ -6 }A;R=0.2\Omega $$
$$i=\cfrac { { B }_{ v }lv }{ R } \Rightarrow { B }_{ v }=\cfrac { iR }{ lv } =\cfrac { 2\times { 10 }^{ -6 }\times 2\times { 10 }^{ -1 } }{ 20\times { 10 }^{ -2 }\times 20\times { 10 }^{ -2 } } =1\times { 10 }^{ -5 }Tesla$$
$$\tan { \delta } =\cfrac { { B }_{ v } }{ { B }_{ H } } =\cfrac { 1\times { 10 }^{ -5 } }{ 3\times { 10 }^{ -5 } } =\cfrac { 1 }{ 3 } \Rightarrow \delta (dip)\tan ^{ -1 }{ 1/3 } $$
A current of $$1.0A$$ is established in a tightly wound solenoid of radius $$2cm$$ having $$1000$$ turns/metre. Find the magnetic energy stored in each metre of the solenoid.
$$i=1.0A,r=2cm,n=1000turn/m$$
magnetic energy stored $$=\cfrac { { B }^{ 2 }V }{ 2{ \mu }_{ 0 } } $$
where $$B$$-magnetic field, $$V$$- Volume of solenoid
$$=\cfrac { { \mu }_{ 0 }{ n }^{ 2 }{ i }^{ 2 } }{ 2{ \mu }_{ 0 } } \times \pi { r }^{ 2 }h=\cfrac { 4\pi \times { 10 }^{ -7 }\times { 10 }^{ 6 }\times 1\times \pi \times 4\times { 10 }^{ -4 }\times 1 }{ 2 } $$ ($$h=1m$$)
$$=8{ \pi }^{ 2 }\times { 10 }^{ -5 }=7.9\times { 10 }^{ -4 }J$$
An average emf of $$20V$$ is induced in an inductor when the current in it is changed from $$2.5A$$ in one direction to the same value in the opposite direction in $$0.1s$$. Find the self-inductance of the inductor.
$$V=20V;dI={ I }_{ 2 }-{ I }_{ 1 }=2.5-(-2.5)=5A;dt=0.1s\quad $$
$$V=L\cfrac { dI }{ dt } \Rightarrow 20=L(5/0.1)\Rightarrow 20=L\times 50\Rightarrow L=20/50=4/10=0.4Henr$$
How many times, the current produced by $$AC$$ Generator reverses its direction in one complete revolution of its coil?
$$AC$$ generator supplies $$AC$$ current. To determine how many times the direction of an Alternating Current changes we have to find out about its frequency. Frequency of an $$AC$$ current implies number of complete waves present in a second. If the coil completes one revolution in one second then frequency of the $$AC$$ generator is said to be $$1\ hertz$$. The given picture will clear the concept.
In the picture coil completes a revolution in a second, hence its frequency is $$1$$ hertz. Hence we can say that in one revolution the current change its direction twice i.e first positive to negative then negative to positive.
How many times $$AC$$ supply in India reversed its direction in one second.
Frequency of an $$AC$$ current implies number of complete waves present in a second. In the picture there is one wave in a second, hence its frequency is $$1$$ hertz. Hence we can say that in one revolution the current change its direction twice i.e first positive to negative then negative to positive.
In India, the $$Ac$$ current has a frequency of $$50\ Hertz$$. $$50\ hertz$$ frequency means that there are $$50$$ waves present in the interval of one second. And we know that in one wave current changes its direction twice, so in $$50$$ waves it will change its direction $$100$$ times.
A plot of magnetic flux $$(\varphi )$$ versus current $$(I)$$ is shown in the figure for two inductors $$A$$ and $$B$$. Which of the two has larger value of self inductance?
$$\varphi =LI$$
For same current, $$\varphi _A > \varphi_B$$, so $$L_A > L_B$$
i.e., Inductor $$A$$ has larger value of self- inductance.
Solve a problem differing from the foregoing one by a magnetic field with induction $$B = 0.8\ T$$ replacing the electric field.
Choose $$\vec {B}$$ in the $$z$$ direction, and the velocity $$\vec {v} = (v\sin \alpha, 0, v\cos \alpha)$$ in the $$x - z$$ plane, then in the $$K'$$ frame,
$$\vec {E'}_{\parallel} = \vec {E}_{\parallel} = 0 | \vec {B'}_{\parallel} = \vec {B}_{\parallel}$$
$$\vec {E'}_{\perp} = \dfrac {\vec {v}\times \vec {B}}{\sqrt {1 - v^{2}/c^{2}}} | \vec {B'}_{\perp} = \dfrac {\vec {B}_{\perp}}{\sqrt {1 - v^{2}/ c^{2}}}$$
We find similarly, $$E' = \dfrac {c\beta B\sin \alpha}{\sqrt {1 - \beta^{2}}}$$
$$B' = B \sqrt {\dfrac {1 - \beta^{2} \cos^{2} \alpha}{1 - \beta^{2}}} \tan \alpha' = \dfrac {\tan \alpha}{\sqrt {1 - \beta^{2}}}$$.
An inductor with an inductance of $$3.00H$$ and a resistance of $$7.00\Omega$$ is connected to the terminals of a battery with an emf of $$12.0V$$ and negligible internal resistance. Find the initial rate of increase of current in the circuit.
In the initial stage the inductor will acts like a open circuit and all the voltage drop across the inductor will be equal to the e.m.f of battery
$$\quad E=L{ \left( \cfrac { di }{ dt } \right) }_{ 0 }$$
$${ \left( \cfrac { di }{ dt } \right) }_{ 0 }\quad =\cfrac { E }{ L } =\cfrac { 12 }{ 3 } =4A/s$$
Which end of the inductor, $$a$$ or $$b$$, is at a higher potential?
Current is decreasing from b to a.
According to Lenz's law emf is induced such as to maintain current constant.
i.e,induced voltage is such a way $$v_a >v_b$$
The potential difference across a $$150mH$$ inductor as a function of time is shown in figure. Assume that the initial value of the current in the inductor is zero. What is the current when $$t=2.0ms$$?
$${V}_{L}=L\cfrac { di }{ dt } $$
$$\therefore \quad di=\cfrac { 1 }{ L } \left( { V }_{ L }dt \right) $$
$$\therefore \quad \int { di=i= } \cfrac { 1 }{ L } \int { { V }_{ L }dt } $$
At $$t=2ms$$
$$i={ \left( 150\times { 10 }^{ -3 } \right) }^{ -1 }\left( \cfrac { 1 }{ 2 } \times 2\times { 10 }^{ -3 }\times 5 \right) =3.33\times { 10 }^{ -2 }A\quad $$
According to which law current $$I$$ flowing in the rod must vary for the rod to rotate at a constant angular speed. Begin to measure the time from the instant when the rod is in its right-hand horizontal position. Consider the current to be positive when it flows from the axis of rotation toward the ring.
Lenz law:
Lenz's law staes that the direction of induced current or voltage will be such as to oppose the cause producing it.
Given situation illustrates that when magnet move towards loop current induced anticlockwise it means mechanical energy converted into electrical energy.
Interpret K' - K.
Some possible interpretations are
i. It is the magnetic energy stored in shell.
magnetic energy = magnetic energy density (B$$^2$$/2$$\mu_0$$) $$\times$$ Volume ($$\pi R^2 l$$) and / or
ii. It is the self inductance energy (1/2Li$$^2$$) of the system.
iii. Poynting vector argument can also show that it is magnetic energy.
(2$$\pi$$)Rl$$\displaystyle{\left(\frac{1}{\mu_0}\int{\bar{E} \times \bar{B} dt} \right)}$$ = K' - K = $$\displaystyle{\frac{-B\pi R^2 l}{2\mu_0}}$$
A frame $$ABCD$$ is rotating with an angular velocity $$\omega$$ about an axis passing through point $$O$$ perpendicular to the plane of paper as shown in the figure. A uniform magnetic field $$\vec { B } $$ is applied into the plane of the paper in the region as in the figure. Match the following.
Potential difference between the ends of a rod of length $$l$$ rotating about one of its end with angular speed $$w$$ is given by $$\mathcal{E} = \dfrac{Bwl^2}{2}$$ .............(1)
(A) : Join A and O
The rod AO rotates with $$w$$ about O
$$\therefore$$ $$\mathcal{E_{AO}} = \dfrac{Bw (AO)^2}{2} = \dfrac{Bw (\sqrt{2}L)^2}{2} $$ $$\implies \mathcal{E_{AO}} = BwL^2 = constant$$
(B) : Join O and D
Similarly $$ \mathcal{E_{DO}} = BwL^2 = constant$$
(C) : The rod $$OC$$ rotates with $$w$$ about O.
$$\therefore$$ $$\mathcal{E_{CO}} = \dfrac{Bw (CO)^2}{2} $$ $$\implies \mathcal{E_{CO}} = \dfrac{BwL^2}{2}$$
Potential between C and D $$\mathcal{E_{DC}} = \mathcal{E_{DO}} - \mathcal{E_{CO}}$$
$$\therefore \mathcal{E_{DC}} = BwL^2 - \dfrac{BwL^2}{2} = \dfrac{BwL^2}{2} = constant$$
(D) : Potential between A and D $$\mathcal{E_{DA}} = \mathcal{E_{DO}} - \mathcal{E_{AO}}$$
$$\therefore \mathcal{E_{DA}} = BwL^2 - {BwL^2} = 0$$
A conducting rod $$AB$$ moves parallel to the x-axis in a uniform magnetic field pointing in the positive z direction. The end $$A$$ of the rod gets positively charged. explain.
Let $$l$$ be the length of rod AB.
Applying right hand rule,emf induced in direction of B to A.
i.e, A is higher potential induced emf $$e=vBl$$
Electric field $$\overrightarrow { E }=\dfrac{e}{l}=vB$$ from A to B.
The magnetic flux linked with the armature coil changed in a generator =$$\Phi (t) = xNBA\cos (\omega t)$$ then x=
A solenoid has a cross sectional area of $$6.0 \times 10^{-4} m^{2}$$, consists of 400 turns per meter, and carries a current 0.4 A and are connected to a circumference of the solenoid. The ends of the coil are connected to a 1.5$$\Omega$$ resistor. Suddenly, a switch is opened, and the current in the solenoid dies to zero in a time 0.050 s. Find the average current passing through the coil during this time.
$$\varepsilon = \dfrac{-d \phi}{dt}$$
for $$N$$ no. of turns,
$$\varepsilon = -N \dfrac{d \phi}{dt}$$
$$= -N \dfrac{d(BA \cos (0))}{dt} =-NA \dfrac{dB}{dt}$$
$$I =- \dfrac{N \dfrac{dB}{dt} A}{R} =- \dfrac{NA \dfrac{\mu_0 n \Delta I}{dt}}{R}$$
$$= \dfrac{(10) (6 \times 10^{-4}) \dfrac{(4\pi \times 10^{-7}) (400) (0.4)}{0.050}}{1.5 \Omega}$$
$$I = 1.6 \times 10^{-5} A$$
The voltage applied to a purely inductive coil of self inductance $$15.9mH$$ is given by the equation $$V=100sin 314t$$ + $$75sin942t$$ + $$450sin1570t.$$
Find the equation of current wave.
$$\begin{array}{l}V = L\frac{{di}}{{dt}}\\or\,i = \frac{1}{L}\int {Vdt} \end{array}$$
$$\begin{array}{l}V = 100\sin 314t + 75\sin 942t + 50\sin 1570t\\i = \frac{1}{L}\int {\left( {100\sin 314t + 75\sin 942t + 50\sin 1570t} \right)dt} \\\,\,\, = \frac{1}{{15.9 \times {{10}^{ - 3}}}}\left[ { - \frac{{100}}{{314}}\cos 314t - \frac{{75}}{{942}}\cos 942t - \frac{{50}}{{1570}}\cos 1570t} \right]\\\,\,\, = \frac{1}{{15.9 \times {{10}^{ - 3}}}} \times \frac{1}{{1570}}\left[ {500\cos 314t + 125\cos 924t + 50\cos 1570t} \right]\\\,\,\, = \frac{{ - 1}}{{24.963}} \times 25\left[ {20\cos 314t + 5\cos 924t + 2\cos 1570t} \right]\\\,\,\, \approx - \left( {20\cos 314t + 5\cos 924t + 2\cos 1570} \right)\end{array}$$
Additional Problems(67)
(a) A flat, circular coil does not actually produce a uniform magnetic field in the area it encloses. Nevertheless, estimate the inductance of a flat, compact, circular coil with radius $$R$$ and $$N$$ turns by assuming the field at its center is uniform over its area. (b) A circuit on a laboratory table consists of a $$1.50-volt$$ battery, a $$270-\Omega$$ resistor, a switch, and three $$30.0-cm$$-long patch cords connecting them. Suppose the circuit is arranged to be circular. Think of it as a flat coil with one turn. Compute the order of magnitude of its inductance and (c) of the time constant describing how fast the current increases when you close the switch.
Energy Carried by Electromagnetic Waves(31)
Review. An AM radio station broadcasts isotropically (equally in all directions) with an average power of $$4.00 \,kW$$. A receiving antenna $$65.0 \,cm$$ long is at a location $$4.00 \,mi$$ from the transmitter. Compute the amplitude of the emf that is induced by this signal between the ends of the receiving antenna.
A solenoid of length $$1 m$$ and $$0.05 m$$ diameter has $$500$$ turns. If a current of $$2 A$$ passes through the coil, calculate the co-efficient of self induction of the coil.
Self-inductance$$=L=\dfrac { { \mu }_{ 0 }{ N }^{ 2 }A }{ l }$$
$$=\dfrac { 4\pi \times { 10 }^{ -7 }\times 500\times 500\times 3.14{ \left( 0.025 \right) }^{ 2 } }{ 1 } $$
$$=0.616\times { 10 }^{ -3 }H$$
On what factors does the induced electromotive force depend?
Induced e.m.f for a coil of wire depends on :
A) Magnetic strength of the core in the coil of wire.
B) Number of turns of wire in the coil
C) The cross-sectional area of the coil.
D) Rate at which magnet is moved into/out of the coil.
Give two definitions of mutual inductance and write its unit.
$$(i)$$ Mutual inductance of two coils is equal to the $$e.m.f$$ induced in one coil when when rate of changes of current through the other coil is unity$$.$$
$$(ii)$$ Mutual inductance of two coils is defined as the magnetic flux linked with the secondary coil when the current in primary coil is $$1$$ ampere$$.$$
$$S.I.$$ unit of mutual induction is $$henry.$$
What is an AC generator? Obtain an expression for the sinusoidal emf induced in the coil of ac generator, rotating with a uniform angular speed in a uniform magnetic field.
An AC generator is an electric generator that converts mechanical energy into electrical energy in form of alternative emf or alternating current. AC generator works on the principle of "Electromagnetic Induction"
When the number of turn in a coil is doubled without, any change in the length of the coil , its self inductance becomes
We know that,
Self-inductance of a solenoid $$=\dfrac{\mu A{{n}^{2}}}{l}$$
So, self-induction proportional to $${{n}^{2}}$$
So, induction becomes $$4$$ times when $$n$$ is doubled.
Hence, this is the required solution
A conducting circular loop having a radius of $$5.0 $$ cm, is placed perpendicular to a magnetic field of $$0.50 T$$. It is removed from the field in $$0.50 s$$. Find the average emf produced in the loop during this time.
$${ \phi }_{ 1 }=B.A=0.5\times { \left( 5\times { 10 }^{ -2 } \right) }^{ 2 }=5\pi .25\times { 10 }^{ -5 }=125\times { 10 }^{ -5 };{ \phi }_{ 2 }=0\quad $$
$$E=\cfrac { { \phi }_{ 1 }-{ \phi }_{ 2 } }{ t } =\cfrac { 125\pi \times { 10 }^{ -5 } }{ 5\times { 10 }^{ -1 } } =7.8\times { 10 }^{ -3 }\quad \quad $$
A circular coil of $$200$$ turns and of radius $$20 cm$$ carries a current of $$5A$$. Calculate the magnetic induction at a point along its axis, at a distance three times the radius of the coil from its centre.
$$n=200\\a=20\ cm=.2\ m\\I=5\ A\\x=3a$$
$$B = \dfrac{n \mu_0 I a^2}{2 (a^2 + x^2)^{3/2}}$$
Substitution and simplification
$$B = 9.9 \times 10^{-5} T$$
A stiff semi-circular wire of radius $$R$$ is rotated in a uniform magnetic field $$B$$ about an axis passing through its ends. If the frequency of rotation of wire is $$f$$, calculate the amplitude of alternating emf induced in the wire.
Induced motion emf at the semicircle endpoints will be $$=\frac{1}{2}Bwl^{2}$$
Here. $$l=2r$$
Induced motion emf at the semicircle endpoints will be $$=\frac{1}{2}Bw(2r)^{2}$$
Induced motion emf at semicircle endpoints will be $$=\pi ^{2}BR^{2}f$$.
A coil of insulated copper wire is connected to a galvanometer. What will happen if a bar magnet is pushed into the coil?
There is a momentary deflection in the needle of the galvanometer.
A pair of adjacent coils has a mutual inductance of $$1.5 H$$. If the current in one coil changes from $$0$$ to $$10 A$$ in $$0.2 s$$, what is the change of flux linkage with the other coil?
Given, mutual incidence of coil $$M=1.5H$$,
Current change in coil $$dl=10-0=10A$$,
Time taken in change $$dt=0.2s$$,
Induced emf in the coil,
$$e=M\frac{dl}{dt}=\frac{d\phi}{dt}$$
or, $$d\phi =M.dl$$
$$=1.5\left ( \frac{10}{0.2} \right )$$
$$d\phi =75Wb$$
Thus, the change in flux linkage is $$75Wb$$.
Figure shows a square loop of resistance 1 $$\Omega $$ of side 1 m being moved towards right at a constant speed of 1 m/s. The front edge enters the 3 m wide magnetic field (B = 1T) at t = 0 Draw the graph of current induced in the loop as time passes ( Take anticlockwise direction of current as positive)
In a fluorescent lamp choke (a small transformer) $$100$$V of reverse voltage is produced when the choke current changes uniformly from $$0.25$$A to $$0$$ in a duration of $$0.025$$ms. The self-inductance of the choke (in mH) is estimated to be _______.
Given $$\epsilon=100v$$
$$\Delta t=0.025ms=2.5\times 10^{-5}s$$
$$\Delta i=0.25-0=0.25A$$
$$\epsilon=L.\dfrac{\Delta i}{\Delta t}$$
$$100=L\times \dfrac{0.25}{2.5\times 10^{-5}}$$
$$100=L\times 10^4$$
$$L=0.01H$$
$$L=10\,mH$$
$$\boxed{Answer=10}$$
What is a solenoid? Draw a diagram with a solenoid connected in a circuit. How can you increase the strength of a solenoid?
The solenoid is a long cylindrical coil of wire consisting of a large number of turns bound together very tightly.
The length of the coil should be longer than its diameter.
Magnetic field around a current carrying solenoid is shown in figure. These appear to be similar to that of a bar magnet.
When soft iron rod is placed inside the solenoid, it behaves like an electromagnet. The use of soft iron as core in the solenoid produces the strong magnetism.
A circular coil of wire consist of exactly 100 turns with a total resistance 0.20 $$\omega$$. The area of the coil is $$100 cm^2$$. The coil is kept in a uniform magnetic field B as shown in fig 4.The magnetic field is increased at a constant rate of 2 T/s. Find the induced current in the coil in A
Two discharge tubes have identical material structure and the same gas is filled in them. The length of one tube is $$10 cm$$ and that of the other tube is $$20 cm$$. Sparking starts in both the tubes when the potential difference between the cathode and the anode is $$100 V$$. If the pressure in the shorter tube is $$1.0 mm$$ of mercury, what is the pressure in the longer tube ?
The current in a long solenoid of radius $$R$$ and having $$n$$ turns per unit length is given by $$i={i}_{0}\sin \omega t$$. A coil having $$N$$ turns is wound around it near the centre. Find (a) the induced emf in the coil and (b) the mutual inductance between the solenoid and the coil.
Name one device which converts electrical energy into mechanical energy.
Electrical motor is a device that converts electrical energy into mechanical energy.
Name the phenomenon which is made use of in an electric generator.
Electromagnetic Induction is the phenomenon used for the electric generator.
If you hold a coil of wire next to a magnet, no current will flow in the coil. What else in needed to induce a current?
In order to induce the current in the wire, relative motion between the coil and the magnet is required.
The inductance of a closely wound coil is such that an emf of 3.00 mV is induced when the current changes at the rate of 5.00 A/s. A steady current of 8.00 A produces a magnetic flux of $$40.0\mu Wb$$ through each turn. (a) Calculate the inductance of the coil. (b) How many turns does the coil have?
(a) From Eq. $$\mathscr{E}
_L=-L\dfrac{di}{dt}\text{ (self-induced current) }$$, we find $$L=(3.00
\mathrm{mV}) /(5.00 \mathrm{A} / \mathrm{s})=0.600 \mathrm{mH}$$
(b) since $$N
\Phi=i L \text { (where } \Phi=40.0 \mu \mathrm{Wb} \text { and } i=8.00
\mathrm{A}),$$ we obtain $$N=120$$
How can the magnitude of the induced current be increased?
The magnitude of the induced current can be increased by:
(i) Taking the conductor in the form of a coil of many turns of insulated wire.
(ii) Increasing the strength of the magnetic field used.
(iii) Increasing the rate of change of magnetic flux associated with the coil.
Suggest two ways in an a.c. generator to produce a higher e.m.f.
Two ways in an a.c generator to produce a higher e.m.f. are:
(1) By increasing the speed of rotation of the coil.
(2) By increasing the number of turns of coil.
A magnetic substance of volume $$30 cm^{3}$$ is placed in a magnetising field of $$5$$ oersted. It produces a magnetic moment of $$6 A/m^{2}$$ . Calculate the magnetic induction.
Volume of magnetic substance $$V = 30cm^{3} = 30\times 10^{-6}m^{3}3$$
Magnetising field $$H = 5$$ oersted
Magnetic moment $$M = 6 A/m^{2}$$
$$B = \mu_{0}(H+I)$$
$$B = \mu_{0}\left[H+\dfrac{M}{V}\right]$$
$$B = 4\pi \times 10^{-7}\left[5\times 10^{3}+\dfrac{6}{30\times 10^{-6}}\right]$$
$$B = 4\times 3.14\times 10^{-7}[5\times 10^{3}+200\times 10^{3}]$$
$$B = 4\times 3.14\times 10^{-7}\times 205 \times 10^{3}$$
$$B = 2574.8\times 10^{-4}$$
or $$B = 0.257T$$
Two identical loops, one of copper and another of aluminium are rotated with the same speed in the same magnetic field. In which case, the
(a) The induced emf and
(b) induced current will be more and why?
The induced emf will be same in both but the induced current will be more in copper loop as its resistance will be lesser as compared to that of aluminium loop.
A series RLC circuit is driven by a generator at a frequency of $$2000Hz$$ and an emf amplitude of $$170 V$$. The inductance is $$60.0 mH$$, the capacitance is $$0.400\mu F$$, and the resistance is $$200\Omega$$. (a) What is the phase constant in radians? (b) What is the current amplitude?
(a) We observe that $$\omega _{d}=12566rad/s$$. Consequently, $$X_{L}=754\Omega$$ and $$X_{C}=199\Omega$$. Similarly,
$$\phi=\tan^{-1}\left ( \frac{X_{L}-X_{C}}{R} \right )=1.22rad$$.
(b) We find the current amplitude from equation $$I=\frac{\varepsilon _{m}}{\sqrt{R^{2}+\left ( X_{L}-X_{C} \right )^{2}}}$$:
$$I=\frac{\varepsilon _{m}}{\sqrt{R^{2}+\left ( X_{L}-X_{C} \right )^{2}}}=0.288A$$.
A horizontal wire $$20 m$$ long extending from east to west is falling with a velocity of $$10 m/s$$ normal to the Earths magnetic field of $$0.5*10^{-4} T$$. What is the value of induced emf in the wire?
Length of the wire, $$l=20m$$,
Speed, $$v=10m/s$$
Earth's magnetic field, $$B=0.5*10^{-4}T$$
Value of e.m.f. induced in the wire, $$=Blv$$.
$$=0.5*10^{-4}*20*10$$
$$=100*10^{-4}V$$
$$=10mV$$.
A horizontal straight wire $$10m$$ long extending from east to west is falling with a speed of $$5.0ms^{-1}$$, at right angles to the horizontal component of the earth's magnetic filed, $$0.30*10^{4}Wbm^{2}$$.
(a) What is the instantaneous value of the emf induced in the wire?
(b) What is the direction of the emf?
(c) Which end of the wire is at the higher electrical potential?
Here, length of the wire, $$l=10m$$;
Velocity of the wire, $$V=5.0ms^{-1}$$
Horizontal component of earth's magnetic field,
$$B_{H}=0.30*10^{4}WBm^{-2}$$
(a) Now, $$e=B_{H}l_{v}=0.30*10^{-4}*10*5.0$$
$$=1.5*10^{-3}V$$.
(b) The induced e.m.f. will be set up from west to east end.
(c) The eastern end will be at higher potential.
Maximum voltage produced in an AC generator completing $$60$$ cycles in $$30$$ seconds is $$250\,V.$$
What is the maximum emf produced when the armature completes $$1800$$ rotation ?
The maximum emf produced by the AC generator is the peak voltage of the generator. It will be $$250\ \text{V}$$.
Conceptual Questions
Consider this thesis: "Joseph Henry, America's first professional physicist, caused a basic change in the human view of the Universe when he discovered self-induction during a school vacation at the Albany Academy aboutBefore that time, one could think of the Universe as composed of only one thing: matter. The energy that temporarily maintains the current after a battery is removed from a coil, on the other hand, is not energy that belongs to any chunk of matter. It is energy in the massless magnetic field surrounding the coil. With Henry's discovery, Nature forced us to admit that the Universe consists of fields as well as matter."
(a) Argue for or against the statement.
(b) In your view, what makes up the Universe?
We can think of Henry's discovery of self-inductance as fundamentally new. Before a certain school vacation at the Albany Academy about 1830, one could visualize the universe as consisting of only one thing, matter. All the forms of energy then known (kinetic, gravitational, elastic, internal, electrical) belonged to chunks of matter. But the energy that temporarily maintains a current in a coil after the battery is removed is not energy that belongs to any bit of matter. This energy is vastly larger than the kinetic energy of the drifting electrons in the wires. This energy belongs to the magnetic field around the coil. Beginning in 1830, Nature has forced us to admit that the universe consists of matter and also of fields, massless and invisible, known only by their effects.
The idea of a field was not due to Henry, but rather to Faraday, to whom Henry personally demonstrated self-induction. Still the thesis stated in the question has an important germ of truth. Henry precipitated a basic change if he did not cause it.
(b) A list today of what makes up the Universe might include quarks, electrons, muons, tauons, and neutrinos of matter; photons of electric and magnetic fields; W and Z particles; gluons; energy; charge; baryon number; three different lepton numbers; upness; downness; strangeness; charm; topness; and bottomness. Alternatively, the relativistic interconvertibility of mass and energy, and of electric and magnetic fields, can be used to make the list look shorter. Some might think of the conserved quantities energy, charge, ... bottomness as properties of matter, rather than as things with their own existence.
A 1.0 m long metallic rod is rotated with an angular frequency of 400 rad $$s^{-1}$$ about an axis normal to the rod passing through its one end. The other end of the rod is in contact with a circular metallic ring. A constant and uniform magnetic field of 0.5 T parallel to the axis exists everywhere. Calculate the emf developed between the centre and the ring.
The emf is given by
$$e=\dfrac{Bl^{2}\omega }{2}=\dfrac{0.5\times 1^{2} \times 400}{2}=100V$$
A pair of adjacent coil has a mutual inductance of $$1.5 \,H$$. If the current in one coil changes from $$0$$ to $$20 \,A$$ in $$0.5$$ sec, what is the change of flux linkage with the other coil?
$$e=\dfrac{d\phi }{dt}=M\dfrac{di}{dt}$$
$$d\phi =M{di}=1.5\times (20-0)=30Wb$$
A spectral line caused by the transition $$^{3} D1 \rightarrow ^{3}P_{0} $$ experiences the Zeeman splitting in a weak magnetic field. When observed at right angles to the magnetic field direction, the interval between the neighbouring components of the split line is $$ \Delta \omega = 1.32 \times 10^{10}\,s^{-1}$$. Find the magnetic field induction B at the point where the source is located.
Derive the expression for the self inductance of a long solenoid of cross sectional area $$A$$ and length $$l$$, having $$n$$ turns per unit length.
Total number of turns in solenoid $$N = nl$$
Magnetic field inside the long solenoid $$B = \mu_o ni $$
Flux through one turn $$\phi_1 = BA = \mu_o ni A$$
Thus total flux through N turns $$\phi_t = N \phi_1 = nl \times \mu_o niA =\mu_o n^2 lAi$$
Using $$\phi_t = Li$$ where $$L$$ is the self inductance of the coil
$$\therefore$$ $$\mu_o n^2 lA i = Li$$ $$\implies L = \mu_o n^2 Al$$
Define self-inductance of a coil. Show that magnetic energy required to build up the current I in coil of coil of self inductance L is given by $$\displaystyle \frac{1}{2}LI^{2}$$
Self Inductance of a coil is the magnetic flux linked with the coil when the current through coil is $$IA$$
$$\phi=li$$ where $$L$$ is the constant of proportionality .
Let a source of emf be connected to an inductor $$L$$
With increase in current ,the opposite emf ,$$e=-L\dfrac{di}{dt}$$
Work done $$dw=\left| e \right| idt=Li\dfrac { di }{ dt } dt=Lidi$$
$$W=\int _{ 0 }^{ i }{ Lidi=\dfrac { 1 }{ 2 } L{ i }^{ 2 } } $$
Energy stored in magnetic inductor,$$U=\dfrac { 1 }{ 2 } L{ i }^{ 2 }$$
Two concentric circular coils of radii r and R are placed coaxially with centres coiciding. If R >> r then calculate the mutual inductance between the coils.
Let current $$I_2$$ flow in the outer coil.
Hence magnetic field at its center=$$\dfrac{\mu_0 I_2}{2R}$$
Since $$r<<R$$, this magnetic field can be considered to be constant over the entire face of smaller coil.
Hence flux through the coil=$$\phi=\dfrac{\mu_0I_2}{2R}\pi r^2$$
By definition of mutual inductance,
$$\phi=M_{12}I_2=\dfrac{\mu_0 I_2}{2R}\pi r^2$$
$$\implies M_{12}=\dfrac{\mu_0 \pi r^2}{2R}$$
An aircraft having a wingspan of $$20.48 m$$ flies due north at a speed of $$40{ ms }^{ -1 }$$. If the vertical component of earth's magnetic field at the place is $$2\times { 10 }^{ -5 }T$$, calculate the emf induced between the ends of the wings.
Given data:
$$I=20.48m$$; $$V=40{ ms }^{ -1 }$$; $$B=2\times { 10 }^{ -5 }T$$; $$e=$$?
$$e=-BlV$$
$$=-2\times { 10 }^{ -5 }\times 20.48\times 40$$
$$e=-0.0164$$Volt.
The unit of self-inductance in SI system is ________ .
The unit of self-inductance in $$SI$$ is Henry.
It is denoted as $$H$$.
$$1H=1 weber/amperce$$
Name and state the principle of a simple a.c. generator. What is its use?
A Generator is a device which converts mechanical energy into electrical energy. A.C Generator works on the principle of electromagnetic induction. In generator an induced emf is produced by rotating a coil in a magnetic field.
Electromagnetic induction:
Electromagnetic induction is the production of an electromotive force across a conductor when it is exposed to a varying magnetic field. It is a process where a conductor placed in a changing magnetic field (or a conductor moving through a stationary magnetic field) causes the production of a voltage across the conductor. This process of electromagnetic induction, in turn, causes an electrical current.
The ac generator is used to produce alternating current which has high transmission power and power generation capability.
State one factor that determines the magnitude of induced e.m.f.
The Faraday's laws of electromagnetic induction says that the E.M.F. induced in a coil 'e' = -(rate of change of magnetic flux linkage)
the Flux linkage =number of turns'N' x magnetic field'B' x area'A' x cos$$\theta$$
theta is angle between magnetic field B and area A.
Theta at any instant 't'=$$(angular \ velocity'w')\times (time \ instant't')$$.
That is, Theta = $$w\times t$$.
E.M.F. induced in a coil 'e'=$$N\times B\times A\times w\times sinw\times t$$.
The factors involved in the induced emf of a coil are:
The induced e.m.f. is directly proportional to N, the total number of turns in the coil.
The induced e.m.f. is directly proportional to A, the area of cross-section of the coil.
The induced e.m.f. is directly proportional to B, the strength of the magnetic field in which the coil is rotating.
The induced e.m.f. is directly proportional to 'w', the angular velocity of coil.
The induced e.m.f. also varies with time and depends on instant 't'.
The induced e.m.f. is maximum when plane of coil is parallel to magnetic field B and e.m.f. is zero when plane of coil is perpendicular to magnetic field B.
What is the source of energy associated with the current obtained in part when a magnet is moved towards a coil having a galvanometer at its ends?
When a magnet is moved closer to the current carrying coil it will generate electricity as the coil moves through the magnetic field. As the magnet is moved, there will be an induced electro-motive force (EMF) which can cause a current in the coil. Once the magnet stops moving, the current will go to zero.
Hence, when a galvanometer is connected to the circuit, there will be deflection due to the flow of electricity. As the magnet is moved toward the coil of wire, the needle of the galvanometer moves one direction. As the magnet is moved away from the coil of wire, the needle of the galvanometer moves the opposite direction. If the magnet is moved faster, the magnitude of the deflection increases.
The mechanical movement/energy is converted into electrical energy.
$$Match \: the \: statements \: in \: Column \: A, \: with \: those \: in \: Column \: B.$$
A-Faraday law of Electromagnetic induction statement.
B-step up transformer used to increase the voltage from primary to secondary.
C-Generator used to convert Mechanical energy to electrical energy.
D-inverse of B
E-induced current.
Define the term 'self-inductance' of a coil. Write its S.I. unit.
Self-inductance of a coil is defined as the phenomenon due to which an emf is induced in a coil when the magnetic flux of coil , linked with the coil changes or current in coil changes . Its S.I. unit is $$Henry (H)$$ .
The type of electric current that changes its direction twice during one cycle of the dynamo is called _____________ .
Alternating current changes its direction twice during one cycle of the dynamo.
State Faraday's laws of electromagnetic induction.
What is motional emf? State any two factors on which it depends.
Motional emf is the emf induced in a conducting rod when it moves in a region of magnetic field.
Motional emf is calculated by $$\mathcal{E} = v Bl$$
where $$B$$ is the magnetic field and $$l$$ is the length of the rod and $$v$$ is the velocity of rod perpendicular to the length of the rod.
Motional emf depends on
1. The magnitude of length of rod and
2. Velocity of rod and
3. Magnetic field.
Two coils A and B have mutual inductance $$2\times { 10 }^{ -2 }$$ henry. If the current in the primary is $$i=5sin(10\pi t)$$, then the maximum value of e.m.f. induced in coil B is :
$$M=2\times 10^{-2}H$$
Current in primary coil A,
$$i=5sin(10\pi t)$$
The emf induced in coil B,
$$e=-M\dfrac{di}{dt}$$
$$e=-M\times 5\times 10\pi cos(10\pi t)$$
$$e=-2\times 10^{-2}\times 50\pi cos(10\pi t)$$
The maximum value of emf induced in coil B is
$$e_{max}=\pi $$
What is a generator state the principle on which generators work
Generators are useful appliances that supply electrical power during a power outage and prevent discontinuity of daily activities or disruption of business operations. Generators are available in different electrical and physical configurations for use in different applications.
An electric generator is a device that converts mechanical energy obtained from an external source into electrical energy as the output.
The generator works on the principle of electromagnetic induction discovered by Michael Faraday in 1831-32. Faraday discovered that the above flow of electric charges could be induced by moving an electrical conductor, such as a wire that contains electric charges, in a magnetic field. This movement creates a voltage difference between the two ends of the wire or electrical conductor, which in turn causes the electric charges to flow, thus generating electric current.
Describe the activity that shows that a current-carrying conductor experiences a force perpendicular to its length and the external magnetic field. How does Flemings' left-hand rule help us to find the direction of the force acting on the current-carrying conductor?
The direction of force in a current carrying conductor can be understood clearly From Fleming's left hand rule.
Fleming's left hand rule states that When an electric current passes through a straight wire, an external magnetic field is applied across that flow, the wire experiences a force perpendicular both to the field and to the direction of the current flow. This rule can be used to determine the direction of the force, magnetic field and current.
A left hand can be held, so that the thumb, first finger and second finger are held mutually perpendicular to each other. The thumb represents the direction of Motion resulting from the force on the conductor. The first finger represents the direction of the magnetic Field. The second finger represents the direction of the Current. Please refer the figure given below.
The current through an indicator of $$1H$$ is given by $$i=3t\sin { t } $$. Find the voltage across the inductor.
$$\left| e \right| =\left| L.\cfrac { \Delta i }{ \Delta t } \right| \quad or\quad \left| L\cfrac { di }{ dt } \right| $$
Here, $$L=1H$$
And, $$\cfrac { di }{ dt } =3\left[ \sin { t } +t\cos { t } \right] $$$$=3(t\cos { t } +\sin { t } )$$
the self inductance of the primary coil.
$${ L }_{ 1 }=\cfrac { { N }_{ 2 }{ \phi }_{ 2 } }{ { i }_{ 1 } } =\cfrac { \left( 600 \right) \left( 5\times { 10 }^{ -3 } \right) }{ 3 } $$
$$=1H$$
At the instant when the current in an inductor is increasing at a rate of $$0.0640A/s$$, the magnitude of the self-induced emf is $$0.0160V$$. If the inductor is a solenoid with $$400$$ turns, what is the average magnetic flux through each turn when the current is $$0.720A$$?
$$Inductance, L=\cfrac { N\phi }{ i } $$
$$\therefore Flux, \quad \phi =\cfrac { Li }{ N } =\cfrac { \left( 0.25 \right) \left( 0.72 \right) }{ 400 } =4.5\times { 10 }^{ -4 }Wb$$
When a current of $$4A$$ between two coils changes to $$12A$$ in $$0.5s$$ in primary and induces an emf of $$50mV$$ in the secondary. Calculate the mutual inductance between the two coils.
$$M=\cfrac { { e }_{ 2 } }{ d{ i }_{ 2 }/dt } =\cfrac { \left( 50\times { 10 }^{ -3 } \right) }{ \left( 8/0.5 \right) } $$
$$=3.125\times { 10 }^{ -3 }H=3.125mH$$
At the instant when the current in an inductor is increasing at a rate of $$0.0640A/s$$, the magnitude of the self-induced emf is $$0.0160V$$. What is the inductance of the inductor?
$$e=\left| L\cfrac { di }{ dt } \right| \quad \Rightarrow L=\left| \cfrac { e }{ di/dt } \right| =\cfrac { 0.016 }{ 0.064 } $$
$$=0.25H$$
The current (in Ampere) in an inductor is given by $$I=5+16t$$, where $$t$$ is in seconds. The self-induced emf in it is $$10mV$$. Find the self-inductance.
$$\dfrac{dI}{dt}=16A/s$$
$$\therefore \quad L=\left| \cfrac { e }{ \dfrac{dI}{dt} } \right| =\cfrac { 10\times { 10 }^{ -3 } }{ 16 } =0.625\times { 10 }^{ -3 }H$$
$$=0.625mH$$
Two bar magnets are placed side by side by side that the north pole of one magnet is next to the south pole of the other magnet. If these magnets are then pushed toward a coil of wire, would you expect an emf to be induced in the coil. Explain your answer.
Since both the magnets are aligned opposite to each other, so both the magnets will cancel out the effect due to each other while being pushed towards a coil. Let say, one magnet induces a clockwise emf in the coil, then the other magnet will induce anticlockwise emf in the coil and thus net emf induced in the coil is zero.
emf induced in the arm PQ
The equivalent circuit is shown in the figure. Here $$PQ=l=10 cm=0.1 $$
The induced emf $$\varepsilon=Blv=(0.1)(0.1)v=0.01v$$
Current in a circuit falls from $$5.0 A$$ to $$0.0 A$$ in $$0.1\ s$$. If an average emf of $$200\ V$$ induced, find an estimate of the self-inductance of the circuit.
The emf is related to inductance by the equation,
$$e=L\dfrac{di}{dt}$$.
Now, $$L=\dfrac{e}{\dfrac{di}{dt}}=\dfrac{200}{\dfrac{5}{0.1}}=4H$$.
A rectangular wire loop of sides 8 cm and 2 cm with a small cut is moving out of a region of uniform magnetic field of magnitude 0.3 T directed normal to the loop. What is the emf developed across the cut if the velocity of the loop is 1 cm $$s^{-1}$$ in a direction normal to the (a) longer side, (b) shorter side of the loop? For how long does the induced voltage last in each case?
Length of the rectangular wire, $$l=8cm=0.08m$$
Width of the rectangular wire, $$b=2cm=0.02m$$
Hence, area of the rectangular loop,
$$A=lb$$
$$=0.08\times 0.02$$
$$=16\times 10^{-4}m^2$$
Magnetic field strength, $$B=0.3T$$
Velocity of the loop, $$v=1cm/s=0.01m/s$$
Emf developed in the loop is given as:
$$e=Blv$$
$$=0.3\times 0.08\times 0.01=2.4\times 10^{-4}V$$
Time taken to travel along the width, $$t\displaystyle =\frac{Distance\, travelled}{Velocity}=\frac{b}{v}$$
$$\displaystyle =\frac{0.02}{0.01}=2s$$
Hence, the induced voltage is $$2.4\times 10^{-4}V$$ which lasts for 2 s.
Emf developed, $$e=Bbv$$
Time taken to travel along the length, $$\displaystyle t=\frac{Distance\, traveled}{Velocity}=\frac{l}{v}$$
$$=\displaystyle \frac{0.08}{0.01}=8s$$
(a) A rod of length l is moved horizontally with a uniform velocity 'v' in a direction perpendicular to its length through a region in which a uniform magnetic field is acting vertically downward. Derive the expression for the emf induced across the ends of the rod.
(b) How does one understand this motional emf by invoking the Lorentz force acting on the free charge carriers of the conductor? Explain.
(a)Let a straight conductor of length $$l$$ be moving in u shaped conductor in perpendicular magnetic field.
$$d \phi=B(vldt)=Bvldt$$
$$\dfrac{d\phi}{dt}=Bvl$$
Induced emf $$\varepsilon =\dfrac { d\phi }{ dt } =Bvl$$
(b)Due to motion of the conductor,free electrons move from one end to other end.
Due to this,both ends generate positive and negative charge and electric force acts on it.
Now according to Lorentz law,
$$F_{net}=F_e+F_m$$
At equilibrium, $$F_{net}=0$$
$$F_e+F_m=0$$
$$qE+q\left( \overrightarrow { v } \times \overrightarrow { B } \right) =0$$
$$E=-\left( \overrightarrow { v } \times \overrightarrow { B } \right)$$
$$ \\ \left| E \right| =Bv\sin { \theta }$$,when velocity perpendicular when magnetic field $$\theta=90$$
$$\left| E \right| =Bv$$
Also,$$\\ \dfrac { d\xi }{ dl } =\left| E \right| =Bv$$
$$\\ d\xi =Bvdl$$
Induced emf=$$\\ Bvl=\xi$$
The property of a conductor which enables to induce an EMF due to change of current in the same coil is ____________.
Self inductance is defined as the property of a conductor which enables it to induce EMF due to change of current in the same coil.
Derive Faradays law of induction from law of conservation of energy.
Consider a situation shown in figure a rod of length $$(l)$$ slide over two rail S R rod P Q is moving with speed V towards right. The electromagnetic force act on the rod is $${ f }_{ m }=\dfrac { { B }^{ 2 }{ l }^{ 2 }V }{ R } $$ towards R is the resistance of the circuit then the work done by the mechanical movement is $${W}_{m} = {f}_{m}.dx$$
Mechanical power develop $${ F }_{ m }=\dfrac { { B }^{ 2 }{ l }^{ 2 }V }{ R } $$
$${ p }_{ mech }=-{ f }_{ m }\dfrac { dx }{ dt } ={ -f }_{ m }V\\ =\dfrac { { -B }^{ 2 }{ l }^{ 2 }{ V }^{ 2 } }{ R } \rightarrow (1)$$
If current induced in the circuit due to induced emf. e then electrical power developed is $${ p }_{ e }=\dfrac { { \varepsilon }^{ 2 } }{ R } \rightarrow (ii)$$
If energy is conserve then $${p}_{e} = {p}_{mech}$$ using equation (1)
$$-\dfrac { { B }^{ 2 }{ l }^{ 2 }{ V }^{ 2 } }{ R } =\dfrac { { \varepsilon }^{ 2 } }{ R } \\ \varepsilon =-BlV\rightarrow (iii)$$
We know that flux $$\phi =\int { B.ds } =B.lx$$
Area of PQRS = $$l x$$
Now equation (iii) written as $$\varepsilon =-Bl\dfrac { dx }{ dt } =\dfrac { -d }{ dt } Blx=\dfrac { -d\phi }{ dt } $$
Here we can see that induced emf
$$\varepsilon =\dfrac { -d\phi }{ dt } $$ that is Farady's law.
An emf of 2V is induced in a coil when current in it is changed from 0A to 10A in 0.40 sec. Find the coefficient of self-inductance of the coil.
Let self-inductance of the coil be $$L$$, emf of the cell be $$e$$ and current in the circuit be $$i$$
$$L=\dfrac{e}{di/dt}$$
$$=\dfrac{2}{10}\times 0.40$$
$$= 0\cdot 08 H$$
A metal rod $$\cfrac { 1 }{ \sqrt { \pi } } m$$ long rotates about one of its ends perpendicular to a plane whose magnetic induction is $$4\times { 10 }^{ -3 }T$$. Calculate the number of revolutions made by the rod per second if the e.m.f induced between the ends of the rod is $$16mV$$.
$$E=\cfrac { 1 }{ 2 } B\omega { l }^{ 2 }$$
$$\Rightarrow 16\times { 10 }^{ -3 }=\cfrac { 1 }{ 2 } \times 4\times { 10 }^{ -3 }\times \omega \times { \left( \cfrac { 1 }{ \sqrt { \pi } } \right) }^{ 2 }$$
$$\Rightarrow 16\times { 10 }^{ -3 }=\cfrac { 1 }{ 2 } \times 4\times { 10 }^{ -3 }\times \omega \times { \left( \cfrac { 1 }{ \pi } \right) }$$
$$\Rightarrow n=\cfrac { 16\times { 10 }^{ -3 }\times 2 }{ 4\times { 10 }^{ -3 }\times 2 } $$
$$n=4$$
A coil of self inductance $$2\cdot 5 H$$ and resistance $$20\Omega$$ is connected to a battery of emf 120V having internal resistance of $$5 \Omega$$. Find the current in the circuit in steady state.
Current in the circuit $$i_0=\dfrac{EMF}{\text{internal resistance}}=\dfrac{E}{R}=\dfrac{120}{20}$$
$$=6A$$
Write Faraday's law of electromagnetic induction.
Faraday's law states that any change in the magnetic field of a coil of wire will cause an emf to be induced in the coil. This emf induced is called induced emf and if the conductor circuit is closed, the current will also circulate through the circuit and this current is called induced current.
Write the Faraday's law of electromagnetic induction.
Faraday's First Law states that any change in the magnetic field of a coil of wire will cause an emf to be induced in the coil. This emf induced is called induced emf and if the conductor circuit is closed, the current will also circulate through the circuit and this current is called induced current.
Faraday's Second Law states that the magnitude of emf induced in the coil is equal to the rate of change of flux that linkages with the coil. The flux linkage of the coil is the product of number of turns in the coil and flux associated with the coil.
Current in a circuit fall from $$5.0$$A to zero in $$0.1$$S. If an average emf of $$100$$ Volt is induced then calculate self-inductance of a inductor in the circuit.
Induced emf in an inductor is given as
$$EMF_{ind}=L\dfrac{di}{dt}$$
$$\implies 100=L(\dfrac{5}{0.1})$$
$$\implies L=2H$$
State Faraday's laws of electromagnetic induction and Lenz's law.
Faraday's laws of electromagnetic induction.
(a) First law : Whenever there is a change in the magnetic flux associated with a circuit, an e.m.f. is induced in the circuit.
(b) Second law : The magnitude of the induced e.m.f. is directly proportional to the time rate of change of magnetic flux through the circuit.
Lenz law : The direction of induced current is such as to oppose the change that produces it.
Explain self-induction of a coil. Arrive at an expression for the induced emf in a coil and the rate of change of current in it.
Inductance is the property of a component that opposes the change of current flowing through it and even a straight piece of wire will have some inductance.
Inductors do this by generating a self-induced emf within itself as a result of their changing magnetic field. In an electrical circuit, when the emf is induced in the same circuit in which the current is changing this effect is called Self-induction.
Magnetic field gives rise to magnetic flux $$\phi$$ passing through each turn in the coil. Coil having N turns will have flux $$N\phi$$ .
Magnetic field is proportional to the current i flowing through the coil and so is the flux.
$$N\phi \propto i$$
$$N\phi =L i$$
$$L$$ is the self inductance.
Now lets consider the current to change in the coil with time which changes the flux as well.
So this rate of change of flux induces an emf (e) by Farady's law of electromagnetic induction.
e=-\frac{d (N\phi)}{dt}=-\frac{d (Li)}{dt}=-L\frac{di}{dt}
\end{equation}
If $$\dfrac{di}{dt}=1 A/s$$ we have, $$e=-L$$
Thus self inductance is numerically equal to the emf induced in coil when current changes at a rate of 1 Ampere per second.
Current in a circuit falls from $$5.0 A$$ to $$0.0 A$$ in $$0.1 sec$$. If an average emf of $$200 V$$ is induced, give an estimate of the self-inductance of the circuit.
Change in current, $$\Delta I = 0.0 - 5.0 = -5.0$$ $$A$$
Rate of change of current, $$\dfrac{\Delta I}{\Delta t } = \dfrac{-5.0}{0.1} = -50.0$$ $$A/s$$
Average induced emf, $$\mathcal{E } = 200$$ $$V$$
Using $$\mathcal{E} = -L\dfrac{\Delta I}{\Delta t}$$ where $$L$$ is the self inductance of circuit
$$\therefore$$ $$200 = -L\times (-50.0)$$
$$\implies$$ $$L = 4.0$$ $$H$$
What are the methods of producing induced emf?
The induced emf can be produced by changing :
(i) the magnetic induction $$\left( B \right)$$,
(ii) area enclosed by the coil $$ \left( A \right)$$ and
(iii) the orientation of the coil $$ \left( \theta \right) $$ with respect to the magnetic field.
What is the self inductance of a straight conductor?
Since there are no enclosed loops in a straight conductor, the self inductance is zero.
a) Switch in the primary is kept in the ON position. Will the bulb in the secondary glow? Justify your answer.
b) When the current in the primary is switched OFF. What happens to the magnetic field in the secondary coil? How will this change influence the current in the secondary and the glow in the electric bulb?
a) If the primary circuit is kept ON, there is no change in magnetic flux. Hence, no induced emf and so the bulb in the secondary does not glow.
b) At the time of turning the switch OFF, there is a sudden change in magnetic flux which causes an induced emf. Hence, the bulb in the secondary glows momentarily.
Explain seld induction and mutual induction.
Self inductance is defined as the property of the coil due to which it opposes the change of current flowing through it. Inductance is attained by a coil due to the self-induced emf produced in the coil itself by changing the current flowing through it. It's given by the following:
$$e= L \frac{dI}{dt}$$
Where e is self induced emf, L is self inductance, I is current and t is time passed. The unit of self inductance, L is Henry.
Mutual Inductance between the two coils is defined as the property of the coil due to which it opposes the change of current in the other coil, or you can say in the neighboring coil. When the current in the neighboring coil is changing, the flux sets up in the coil and because of this changing flux emf is induced in the coil called Mutually Induced emf and the phenomenon is known as Mutual Inductance.
In short flux through second coil is directly proportional to current flowing in first coil, which is given by the following expression:
$$\phi_2 = M_{21}I_1$$ and $$\phi_1= M_{12}I_2$$
Where $$\phi_1$$ and $$\phi_2$$ is flux through coil 1 and coil 2 respectively, $$I_1$$ and $$I_2$$ is current in coil 1 and coil 2 respectively and $$M_{12}=M_{21}=M$$ is mutual induction.
Draw a labelled diagram of an ac generator. Obtain the expression for the emf induced in the rotating coil of N turns each of cross-sectional area A, in the presence of a magnetic field $$\overrightarrow{B}$$.
Let say at the instant $$t=0$$, the plane of the coil is perpendicular to the direction of magnetic field i.e. area vector of coil points in same direction as that of magnetic field. So maximum flux passes through the coil.
Flux passing through the coil at $$t=0$$, $$\phi_{max} = NBA$$
The coil rotates about an axis with constant angular velocity $$w$$ as shown in the figure.
Angle rotated by coil in time $$t$$ is $$ wt$$.
Flux passing through the coil at that instant, $$\phi = NBA\cos wt$$
Rate of change of flux $$\dfrac{d\phi}{dt} = NBA \dfrac{d}{dt} (\cos wt)$$
$$\therefore$$ $$\dfrac{d\phi}{dt} = NBA \times (-w\sin wt)$$
We get $$\dfrac{d\phi}{dt} = -NBA w\sin wt$$
Induced emf in the coil $$\mathcal{E} = -\dfrac{d\phi}{dt}$$
$$\implies$$ $$\mathcal{E} =NBAw \sin wt$$
Current in a circuit falls from 5.0 A to 0.0 A in 0.1 s. If an average emf of 200V induced, calculate the self-inductance of the circuit.
Initial current, $$I_1=5.0 A$$
Final current, $$I_2=0A$$
Change in current, $$dl=I_1-I_2=5A$$
Time taken for the charge, $$t=0.1s$$
Average emf, $$e=200V$$
For self inductance(L) of the coil, we have the relation for average emf as:
$$e=L\dfrac{di}{dt}$$
$$L=\dfrac{e}{\dfrac{di}{dt}}$$
$$= \dfrac{200}{\dfrac{5}{0.1}=4H}$$
Hence the self inductance of the coil is $$4H$$.
What is the effect on self inductance of a solenoid, if a core of soft iron is placed in it?
A core of soft iron is placed in a solenoid will enhance the self-induction as the magnetic flux linkage between the coils is improved due to high magnetic permeability of soft iron.
Derive expression for the self-induction of solenoid. What factors affect its value and how?
Self Inductance of long solenoid: Consider a solenoid of length l, cross-section area A. Having total number of turns N. If I current flow through the solenoid there e.m.f. at the centre of solenoid.
$$B=\mu_ont$$
But $$n=\displaystyle\frac{N}{1}$$
$$\Rightarrow B=\displaystyle\frac{\mu_oNI}{1}$$
$$\Phi =NBAcos\theta$$
$$\Rightarrow L=\displaystyle\frac{\mu_oN^2IA}{II}A cos\theta$$
$$\Phi =\displaystyle\frac{\mu_oN^2IA}{l}$$
$$L=\displaystyle\frac{\mu_oN^2IA}{l}$$
If the relative permeability of medium inside the solenoid is $$\mu_r$$, then,
$$L=\displaystyle\frac{\mu_o\mu_rN^2IA}{l}$$
It is clear that self inductance of solenoid depends upon the following factors:
(1) On relative permeability of material inside the solenoid: If a soft iron core placed inside the solenoid, the magnetic flux linked with the solenoid increased hence, self inductance of the solenoid will also increases.
(2) On total number of turns in solenoid: If total number of turns in the solenoid increases then self inductance of the solenoid will also increases.
(3) On cross-section area of solenoid: If cross-section area of solenoid increase then self inductance of solenoid will also increases.
(4) On length of solenoid: If length of solenoid increases then its self-inductance decrease.
A rectangular coil has $$60$$ turns and its length and width is $$20\ cm$$. and $$10\ cm$$ respectively. The coil rotates at a speed of $$1800$$ rotation per minute in a uniform magnetic field of $$0.5\ T$$ about its one of the diameter. Calculate maximum induced emf will be
Given that,
Number of turns $$N=60$$
Length $$l=20\,cm$$
Width $$b=10\,cm$$
Magnetic field $$B=0.5\,T$$
Rotation per min $$f=1800$$
$$ \phi =\left( AB\cos \theta \right)N $$
$$ \phi =NAB\cos \omega t $$
$$ \dfrac{-d\phi }{dt}=\varepsilon $$
$$ \varepsilon =-\dfrac{d\left( NAB\cos \omega t \right)}{dt} $$
$$ \varepsilon =-AB\omega \left( -\sin \omega t \right)N $$
$$ \varepsilon =NAB\omega \sin \omega t $$
$$ \omega =2\pi f $$
$$ \varepsilon =2\pi fNAB\sin \omega t $$
$$ {{\varepsilon }_{\max }}=2\pi fNAB $$
$$ {{\varepsilon }_{\max }}=2\times 3.14\times 60\times \left( 20\times 10\times {{10}^{-4}} \right)\times 0.5\times 30 $$
$$ {{\varepsilon }_{\max }}=113.04\,volt $$
Hence, the maximum induced emf is $$113.04\,volt$$
A charged particle oscillates about its equilibrium position with an frequency of $$100\ MHz$$. What is the frequency of electromagnetic waves produced by the oscillator?
The frequency of an electromagnetic wave produced by the oscillator is the same as that of a charged particle oscillating about its mean position i.e. $$100\ MHz$$.
A circular coil of cross-sectional area $$200cm^2$$ and 20 turns is rotated about the vertical diameter with angular speed of $$50 rad {s}^{-1}$$ in a uniform magnetic field of magnetic $$3.0 \times10^{-2}T$$ . Calculate the maximum value of the current in the coil.
Here, $$A=200 cm^{2}$$
$$N=20$$
$$\omega=50 \dfrac{rad}{s}$$
$$B=3\times { 10 }^{ -2 }T$$
For maximum current emf should be maximum.
So for maximum emf $$e=NAB\omega\sin\omega t$$
$$\sin\omega t=1$$
$$e_{max}=NAB\omega$$
$$e_{max}=20\times 200\times { 10 }^{ -4 }\times 50\times 3\times { 10 }^{ -2 }$$
$$e_{max}$$=$$0.6 V$$
Since resistance is not given,let us consider $$R$$
$$i_{max}$$=$$ \dfrac { e_{max} }{ R } =\dfrac { 0.6 }{ R } \\ $$
A circular coil of cross-sectional area $$ 20 cm^2$$ and $$20$$ turns is rotated about the diameter with angular speed of 50 rad $$s^{-1}$$ in a uniform magnetic field of magnetic $$3.0 \times 10^{-2}T$$. Calculate the maximum value of the current in the coil.
Here, $$A=200 cm^2$$
$$w=50\dfrac{rad}{s}$$
$$B=3 \times 10^{-2} T$$
So for maximum emf $$e=NABwsinwt$$
$$sinwt=1$$
$$e_{max} = NABw$$
$$e_{max} = 20 \times 200 \times 10*{-4} \times 50 \times3 \times10^{-2}$$
$$e_{max} = 0.6 V$$
$$i_{max} = \dfrac{e_{max}}{R} = \dfrac{0.6}{R}$$
Draw a labeled diagram of AC generator.Derive the expression for the instantaneous value of the emf induced in the coil.
Construction of AC generator:
Main parts of an AC generator include,
1. Armature-Rectangular coil ABCD.
2. Field magnet
3. Slip Rings
4. Brushes $$B_1,B_2$$
Working: As the armature coil is rotated in magnetic field,angle $$\theta$$ between the field and normal to the coil changes continuously ,therefore magnetic flux linked with the coil changes.An emf is induced in the coil.According to Fleming's right hand rule,to calculate emf,Suppose,
$$A$$-Area of each turn the coil.
$$N$$-Number of turn in the coil.
$$\overrightarrow { B } $$-strength of magnetic field.
$$\theta$$-angle which normal to the coil make with $$\overrightarrow{B}$$ at any instant t.
Magnetic Flux linked with coil in this position.
$$\phi=N\left( \overrightarrow { B } .\overrightarrow { A } \right) =NBA\cos { \theta } $$
$$ =NBA\cos { \omega t } $$
$$ e=-\dfrac { d\phi }{ dt } =-NBA(-\left( \sin { \omega t } \right) \omega) $$
$$ e=+NBA\omega \sin { \omega t } $$
Why self induction is called inertia of electricity?
Self-induction of coil is the property by virtue of which it tends to maintain the magnetic flux linked with it and opposes any change in the flux by inducing current in it. This property of a coil is analogous to mechanical inertia. That is why self-induction is called the inertia of electricity.
A solenoid of $$500$$ turns, diameter 2$$0 cm$$ and resistance $$2$$ $$\Omega$$ is rotated about its vertical diameter through $$\pi$$ radian in 1/4 s, when a horizontal field of $$3 \times 10^{-5} T$$ acts normal to its plane. Find the emf induced and current thereof.
emf induced is $$e$$.
$$e=NBA\omega \sin\theta$$ where ,
$$N$$=number of turns.
$$B$$=magnetic field.
$$\omega$$=angular velocity.
$$\omega=\dfrac{\pi}{\dfrac{1}{4}} \dfrac{rad}{s}$$
$$=4\pi \dfrac{rad}{s}$$
$$A$$=area of cross section=$$\dfrac{\pi d^{2}}{4}$$
($$\theta=30° , \sin\theta=1$$)
$$ e=500\times 3\times { 10 }^{ -5 }\times 4\pi\times \dfrac { \pi{ \left( 0.2 \right) }^{ 2 } }{ 4 } \times 1$$
$$ =500\times 3\times 4\times { 10 }^{ -7 }\times \pi $$
$$ =18840\times { 10 }^{ -7 } V $$
$$ =1.884mV $$
Current $$i=\dfrac{e}{R}$$=$$\dfrac{1.884}{2} A$$
$$i=0.945A$$
A rectangular frame of wire $$abcd$$ has dimensions $$32\ cm\times 8.0\ cm$$ and a total resistance of $$2.0\Omega$$. It is pulled out of a magnetic field $$B = 0.020\ T$$ by applying a force of $$3.2\times 10^{-5}N$$ (figure). It is found that the frame moves with constant speed. What is constant speed of the frame?
If loop abcd moves with constant speed ,therefore electromagnetic force $$f_m$$ is equal to external force $$f$$.
$$f_m=f=3.2 \times 10^{-5}$$
$$ilB=3.2\times 10^{-5}$$
$$\dfrac{e}{R}lB=\dfrac{VBl}{R}lB=\dfrac{VB^{2}l^{2}}{R}$$ where,
$$V$$=speed of loop
$$R$$=resistance of abcd.
$$i$$=current in the loop.
$$B$$=magnetic field
$$\Longrightarrow \dfrac { V\times { \left( 0.2 \right) }^{ 2 }\times { { \left( 8\times { 10 }^{ -2 } \right) }^{ 2 } } }{ 2 } =3.2\times { 10 }^{ -5 }$$
$$\Longrightarrow \dfrac { V\times 4\times { 10 }^{ -4 }\times 64\times { 10 }^{ -4 } }{ 2 } =3.2\times { 10 }^{ -5 }$$
$$ \Longrightarrow V=\dfrac { 2\times 3.2\times { 10 }^{ 3 } }{ 4\times 64 } =25\dfrac { m }{ s } $$
Velocity required =$$25 ms^{-1}$$
Define coefficient of self inductance and write its unit.
Definition:-
Coefficient of self inductance is a ratio of electromotive force produced in a circuit by self induction to the rate of change of current producing it.
Unit of coefficient of self inductance is $$Henry$$.
Derive the expression for the motional emf induced in a conductor moving in a uniform magnetic field.
Consider a metallic frame $$MSRN$$ placed in a constant uniform magnetic field. Let the magnetic field $$B$$ be perpendicular to the plane of the coil. Let a metal rod $$PQ$$ of length '$$l$$' placed on it be moving towards left with a velocity $$v$$ as shown in the figure.
Let the distance of $$PQ$$ from $$SR$$ be $$x$$
$$\phi_B=BA \cos \theta$$
$$=Blx\cos 0 = Blx$$
As the rod $$PQ$$ is moving towards left with a velocity $$v$$, $$X$$ is changing and
$$\overrightarrow{v} = \dfrac{-dx}{dt}$$
Hence Induced emf $$\in=\dfrac{-d\phi_B}{dt}=\dfrac{-d(Blx)}{dt} = -Bl\dfrac{dx}{dt}$$
$$\Rightarrow \in = Blv$$
This emf is induced in the road because of the motion of the rod in the magnetic field.
Therefore this emf is called motional emf.
Match the followings.
A. Angular momentum= moment of inertia $$\times$$ angular velocity
$$= r \times p$$
$$=Kgm^2s^{-1}=[ML^2T^{-1}]$$
B. Latent Heat= $$Joule/Kg= [ML^2Q^{-2}]$$
C. Torque= Newton metre= $$[ML^2T^{-2}]$$
D. Capacitance= Charge/Voltage= $$[M^{-1}L^{-2}T^{2}Q^{2}]$$
E. Inductance= Ohm sec= $$[L^2T^{-2}]$$
F. Resistivity= Ohm/Length= $$[ML^3T^{-1}Q^{-2}]$$
What is self inductance? Name the factors on which self inductance depends.
Self-inductance definition is an inductance in which an electromotive force is produced by self-induction. It is the as the property of the coil due to which it opposes the change of current flowing through it.
Self inductance depends on-
1-Size of coil
2- Shape of the coil
3- Material of the coil
4-Medim
$$e.g$$ Self inductance of solenoid is $$\dfrac{\mu N^2A}{l}$$
What will be the magnitude and polarity of induced emf across the rod, if it is rotating about an axis at a distance $$\cfrac{L}{4}$$ from one of its ends?
Consider, The rotating with angular velocity $$\omega $$
$$\begin{array}{l} e=\int _{ -\frac { L }{ 4 } }^{ \frac { { 3L } }{ 4 } }{ \omega Bxdx } \\ =\omega B{ \left[ { \frac { { { x^{ 2 } } } }{ 2 } } \right] _{ \frac { { -L } }{ 4 } } }^{ \frac { { 3L } }{ 4 } } \\ =\frac { { \omega B{ L^{ 2 } } } }{ 4 } \end{array}$$
And polarity longer end will be positive $$wrt$$ origin.
A semicircle loop $$PQ$$ of radius $$'R'$$ is moved with velocity $$'v'$$ in transverse magnetic field as shown in figure. The value of induced emf. at the end of loop is :-
Induced $$\begin{array}{l} emf=\frac { { -d\theta } }{ { dt } } \\ E=\frac { { -d } }{ { dt } } \left( { BA\cos \theta } \right) \\ E=-B\frac { { dA } }{ { dt } } \cos \theta \end{array}$$ because the ring falls with its vertical plane in horizontal mehnetic induction $$\begin{array}{l} \vec { B } .\theta =0\, and\cos \theta =1 \\ or\, E=-B.\frac { d }{ { dt } } \left( { 2Rdx } \right) \\ E=-2RB\left( { \frac { { dx } }{ { dt } } } \right) \\ \left| E \right| =2RBV \end{array}$$
This is equal to the emf developed in an imaginary rod joining M &Q
According to fleming right hand rule, M is at low potential and Q is at high potential and induced emf is from M to Q.
What is eddy current? Mention two applications of eddy current.
Eddy currents are the currents induced in a metallic plate when it is kept in a time varying magnetic field. Magnetic flux linked with the plate changes and so the induced current is set up. Eddy currents are sometimes so strong, that metallic plate become red hot.
Application:-
(1)-In induction furnace, the metal to be heated is placed in rapidly varying magnetic field produced by high frequency alternating current. Strong eddy currents are set up in the metal produce so much heat that the metal melts. This process is used in extracting a metal from its ore.The arrangement of heating the metal by means of strong induced current is called the induction furnace.
(2)-Induction motor, the eddy currents may be used to rotate the rotor. When a metallic cylinder(or rotor) is placed in a rotating magnetic field , eddy currents are produced in it. According to Lenz's law, these currents tends to reduce to relative motion between the cylinder and the field. The cylinder, therefore, begins to rotate in the direction in the field. This is the principle of induction motion.
A $$200km$$ long telegraph wire has a capacitance of $$0.014\mu F/km$$. If it carries an ac of $$5kHz$$, what should be the inductance required to be connected in series, so that the impedance is minimum?
Take $$\pi=\sqrt {10}$$
For minimum impedance
$$\begin{array}{l} { X_{ L } }={ X_{ C } } \\ \omega L=\dfrac { 1 }{ { \omega C } } \\ L=\dfrac { 1 }{ { { \omega ^{ 2 } }C } } =\dfrac { 1 }{ { { { \left( { 2\pi f } \right) }^{ 2 } }C } } \\ For\, 1km\, C=0.14\mu F \\ For\, 200km\, C=0.14\times 200\mu F \\ L=\dfrac { 1 }{ { \left( { 2\; \times 3.14\times 5\times { { 10 }^{ 3 } } } \right) \times 0.014\times 200 } } =0.36mH \end{array}$$
A direct current $$I$$ flows along a lengthy straight wire. From the point $$O$$ (Fig.) the current spreads radially all over an infinite conducting plane perpendicular to the wire. Find the magnetic induction at all points of space.
It is easy to convince oneself that both in the regions. $$1$$ and $$2$$ there can only be a circuital magnetic field (i.e. the component $$B_{\varphi})$$. Any radial field in region $$1$$ or any $$B_{Z}$$ away from the current plane will imply a violation of Gauss' law of magnetostatics, $$B_{\varphi}$$ must obviously by symmetrical about the straight wire. Then in $$1$$,
$$B_{\varphi} 2\pi r = \mu_{0} I$$
or, $$B_{\varphi} = \dfrac {\mu_{0}}{2\pi} \dfrac {I}{r}$$
$$In\ 2, B_{\varphi} \cdot 2\pi r = 0$$, or $$B_{\varphi} = 0$$.
A rectangular coil of area $$A$$, having the number of turns $$N$$ is rotated at $$f$$ revolutions per second in a uniform magnetic field $$B$$ the field is perpendicular to the coil. Prove that the maximum emf induced in the coil is $$2 \pi fNBA$$.
$$\begin{array}{l}E = BAN\omega \sin \theta \\\omega = 2\pi f\,and\,at\,\theta = {90^ \circ }\\E = BAN\left( {2\pi f} \right)\sin 90\\ = 2\pi fBAN\end{array}$$
A coil having resistance $$20\Omega$$ and inductance $$2H$$ is connected to a battery of emf $$4.0V$$. Find (a) the current at $$0.20s$$ after the connection is made and (b) the magnetic field energy in the coil at the instant
$$L=2H\quad R=20\Omega \quad emf=4V\quad t=0.20s$$
$$I_0=\dfrac{e}{R}=\dfrac{4}{20}=0.2$$
$$t=\dfrac{L}{R}=\dfrac{2}{20}=0.1$$
$$(a)$$
$$I=I_0(1-e^{-0.2/0.1})=0.17A$$
$$(b)$$
$$\dfrac{1}{2}LI^2=\dfrac{1}{2}\times2\times0.17^2=0.03J$$
A conducting circular loop of area $$1$$ nm is placed compulsary with a long, straight wire at a distance of current which changes from $$10$$ A to zero in $$0.1$$ s. Find the average emf induced in the loop in $$0.1$$ s.
A = 1mm2 ; i = 10A, d = 20cm ; dt = 0.1s
A coil draws a current of 1ampere and a power of 100 watt from an AC.Source of 100 volt and $$\frac { 5\sqrt { 22 } }{ \pi } $$ Hertz.Find the inductance and resistance of the coil
A copper disc of radius $$0.1$$m is rotated about its center with $$20$$ revolution per second in a uniform magnetic field of $$0.1$$T with its plane perpendicular to the field. The emf induced across the radius of the disc is?
Emf induced is given by$$=\dfrac { 1 }{ 2 } B\omega { R }^{ 2 }$$---(1) where B=magnetic field=0.1T; $$\omega$$=angular velocity=20 revolution per second and R=radius of the disc=0.1 m
Now putting the above mentioned values in equation 1
emf$$=\dfrac { 1 }{ 2 } 0.1\times 2\pi \times 20\times (0.1)^{ 2 }\\ =0.0628$$
On what principle $$AC$$ generator works?
AC generator operates on the principle of electromagnetic induction, discovered (1831) by Michael Faraday. According to the principle when a conductor passes through a magnetic field, a voltage is induced across the ends of the conductor. In the simplest form of generator the conductor is an open coil of wire rotating between the poles of a permanent magnet, thus voltage is induced.
A conducting disc of radius r rotates with a small constant angular velocity $$\omega$$ about its axis. A uniform magnetic field B exists parallel to the axis of rotations. Find the motional emf between the center and the periphery of the disc.
Rotating dics can be consider as a rotating rod of same radius.
v at a distance $$r/2$$ From the centre $$= r \dfrac{w}{2}$$
$$E= B \times r \times \dfrac{rw}{2}$$
$$ E= \dfrac{1}{2} B r^2 w$$
Figure shows a straight, long wire carrying a current $$i$$ and a rod of length $$l$$ coplanar with the wire and perpendicular to it. The rod moves with a constant velocity $$v$$ in direction parallel to the wire. The distance of the wire from the centre of the rod is $$x$$. Find the motional emf induced in the rod.
In this case $$\vec { B } $$ varies
hence considering a small element at centre of rod of length $$dx$$ at a distance $$x$$ from the wire
$$\vec { B } =\cfrac { { \mu }_{ 0 }i }{ 2\pi x } $$
so $$de=\cfrac { { \mu }_{ 0 }i }{ 2\pi x } \times vdx$$
$$e=\int _{ 0 }^{ e }{ de } =\cfrac { { \mu }_{ 0 }iv }{ 2\pi x } =\int _{ x-l/2 }^{ x+l/2 }{ \cfrac { dx }{ x } } =\cfrac { { \mu }_{ 0 }iv }{ 2\pi } \left[ \ln { (x+l/2) } -\ln { (x-l/2) } \right] =\cfrac { { \mu }_{ 0 }iv }{ 2\pi } \ln { \left[ \cfrac { x+l/2 }{ x-l/2 } \right] = } \cfrac { { \mu }_{ 0 }iv }{ 2\pi } \ln { \left( \cfrac { 2x+l }{ 2x-l } \right) } $$
A constant current exists in a inductor-coil connected to a battery. The coil is short-circuited and the battery is removed. Show that the charge flown through the coil after the short-circuiting is the same as that which flows in one time constant before the short-circuiting.
In this case there is no resistor in the circuit
So, the energy stored due to the inductor before and after removal of battery remains same
$${V}_{1}={V}_{2}=\cfrac{1}{2}L{i}^{2}$$
So, the current will also remain same
Thus charge flowing through the conductor is the same
An LR circuit has $$L=1.0H$$ and $$R=20\Omega$$. It is connected across an emf of $$2.0V$$ at $$t=0$$ What are the values of the self-induced emf in the circuit at the times indicated there in?
Consider a small cube of volume $$1{mm}^{2}$$ at the centre of a circular loop of radius $$10cm$$ carrying a current of $$4A$$. Find the magnetic energy stored inside the cube.
Energy density $$\cfrac { { B }^{ 2 } }{ 2{ \mu }_{ 0 } } $$
total energy stored $$\cfrac { { B }^{ 2 }V }{ 2{ \mu }_{ 0 } } =\cfrac { { \left( { \mu }_{ 0 }i/2r \right) }^{ 2 } }{ 2{ \mu }_{ 0 } } V=\cfrac { { \mu }_{ 0 }{ i }^{ 2 } }{ 4{ r }^{ 2 }\times 2 } =\cfrac { 4\pi \times { 10 }^{ -7 }\times { 4 }^{ 2 }\times 1\times { 10 }^{ -9 } }{ 4\times { \left( { 10 }^{ -1 } \right) }^{ 2 }\times 2 } =8\pi \times { 10 }^{ -4 }J$$
The magnetic field at a point inside a $$2.0mH$$ inductor-coil becomes $$0.80$$ of its maximum value in $$20\mu s$$ when the inductor is joined to a battery. Find the resistance of the circuit.
$${ i }_{ }={ i }_{ 0 }\left( 1-{ e }^{ -t/\tau } \right) $$
$$\Rightarrow { \mu }_{ 0 }ni={ \mu }_{ 0 }n{ i }_{ 0 }\left( 1-{ e }^{ -t/\tau } \right) \Rightarrow B={ B }_{ 0 }\left( 1-{ e }^{ -IR/L } \right) $$
$$\Rightarrow 0.8{ B }_{ 0 }={ B }_{ 0 }\left( 1-{ e }^{ -20\times { 10 }^{ -3 }\times R/2\times { 10 }^{ -3 } } \right) \Rightarrow 0.8\left( 1-{ e }^{ -R/100 } \right) $$
$$\Rightarrow { e }^{ -R/100 }=0.2\Rightarrow \ln { \left( { e }^{ -R/100 } \right) } =\ln { \left( 0.2 \right) } $$
$$\Rightarrow -R/100=-1.609\Rightarrow R=16.9=160\Omega \quad $$
Consider the situation shown in figure. The wires $${ P }_{ 1 }{ Q }_{ 1 }$$ and $${ P }_{ 2 }{ Q }_{ 2 }$$ are made to slide on the rails with the same speed $$5cm$$ $${s}^{-1}$$. Find the electric current in the $$19\Omega$$ resistor if (a) both the wires move towards right and (b) if $${ P }_{ 1 }{ Q }_{ 1 }$$ moves towards left but $${ P }_{ 2 }{ Q }_{ 2 }$$ moves towards right
(a) The wires constitute 2 parallel emf
Net emf $$=Blv=1\times 4\times { 10 }^{ -2 }\times 5\times { 10 }^{ -2 }=20\times { 10 }^{ -4 }\quad $$
Net resistance $$=\cfrac { 2\times 2 }{ 2\times 2 } +19=20\Omega $$
Net current $$=\cfrac { 20\times { 10 }^{ -4 } }{ 20 } =0.1mA\quad $$
(b) When both the wires move towards opposite directions then net emf=0
net current $$=0$$
A long wire carries a current of $$4.00A$$. Find the energy stored in the magnetic field inside a volume of $$1.00{mm}^{2}$$ at a distance of $$10.0cm$$ from the wire.
$$I=4.00A,V=1{ mm }^{ 3 },d=10cm=0.1m$$
$$\vec { B } =\cfrac { { B }^{ 2 } }{ 2{ \mu }_{ 0 } } $$
Now magnetic energy stored $$=\cfrac { { B }^{ 2 } }{ 2{ \mu }_{ 0 } } V=\cfrac { { \mu }_{ 0 }^{ 2 }{ i }^{ 2 } }{ 4{ \pi r }^{ 2 } } \times \cfrac { 1 }{ 2{ \mu }_{ 0 } } \times V=\cfrac { 4\pi \times { 10 }^{ -7 }\times 16\times 1\times 1\times { 10 }^{ -9 } }{ 4\times 1\times { 10 }^{ -2 }\times 2 } =\cfrac { 8 }{ \pi } \times { 10 }^{ -14 }=2.55\times { 10 }^{ -14 }J\quad $$
An inductor of inductance $$2.00H$$ is joined in series with a resistor of resistance $$200\Omega$$ and a battery of emf $$2.00V$$. At $$t=10ms$$, find (a) the current in the circuit, (b) the power delivered by the battery, (c) the power dissipated in heating the resistor and (d) the rate at which energy is being stored in magnetic field.
$$L=2H,R=200\Omega, E=2V,t=10ms$$
(a) $$l={ l }_{ 0 }\left( 1-{ e }^{ -t/\tau } \right) =\cfrac { 2 }{ 200 } \left( 1-{ e }^{ -10\times { 10 }^{ -3 }\times 200/2 } \right) =0.01(1-{ e }^{ -1 })=0.01(1-0.3678)=6.3A$$
(b) Power delivered by the battery
$$=VI=E{ I }_{ 0 }\left( 1-{ e }^{ -t/\tau } \right) =\cfrac { { E }^{ 2 } }{ R } { \left( 1-{ e }^{ -tR/L } \right) }=\cfrac { 2\times 2 }{ 200 } \left( 1-{ e }^{ -10\times { 10 }^{ -3 }\times 200/2 } \right) =0.02(1-{ e }^{ -1 })=0.1264=12mw$$
(c) Power dissipated in heating the resistor $${ \left[ { i }_{ 0 }\left( 1-{ e }^{ -t/\tau } \right) \right] }^{ 2 }R={ (6.3mA) }^{ 2 }\times 200=6.3\times 6.3\times 200\times { 10 }^{ -6 }=7.938\times { 10 }^{ -3 }=8mA$$
(d) Rate at which energy is stored in the magnetic field $$d/dt(1/2 L{I}^{2})$$
$$==\cfrac { L{ I }_{ 0 }^{ 2 } }{ \tau } \left( { e }^{ -t/\tau }-{ e }^{ -2t/\tau } \right) =\cfrac { 2\times { 10 }^{ -4 } }{ { 10 }^{ -2 } } \left( { e }^{ -1 }-{ e }^{ -2 } \right) =2\times { 10 }^{ -2 }\left( 0.2325 \right) =4.6\times { 10 }^{ -3 }=4.6mW$$
An ac generator with emf $$\xi=\xi _{m}\sin \omega _{d}t$$, where $$\xi _{m}=25.0V$$ and $$\omega _{d}=377rad/s$$, is connected to a $$4.15\mu F$$ capacitor. (a) What is the maximum value of the current? (b) When the current is a maximum, what is the emf of the generator? (c) When the emf of the generator is $$-12.5 V$$ and increasing in magnitude, what is the current?
(a) The circuit consists of one generator across one capacitor; therefore, $$\xi _{m}=V_{C}$$. Consequently, the current amplitude is,
$$I=\dfrac{\varepsilon _{m}}{X_{C}}=\omega C\varepsilon _{m}=\left ( 377\ rad/s \right )\left ( 4.15 \times 10^{-6}F \right )\left ( 25.0V \right )=3.91 \times 10^{-2}A$$.
(b) When the current is at a maximum, the charge on the capacitor is changing at its largest rate. This happens not when it is fully charged ($$\pm q_{max}$$), but rather as it passes through the (momentary) states of being uncharged ($$q = 0$$). Since $$q = CV$$, then the voltage across the capacitor (and at the generator, by the loop rule) is zero when the current is at a maximum. Stated more precisely, the time-dependent emf $$ε(t)$$ and current $$i(t)$$ have a $$\phi =-90^{o}$$ phase relation, implying $$ε(t) = 0$$ when $$i(t) = I$$. The fact that $$\phi =-90^{o}=-\dfrac{\pi}{2} \ rad$$ is used in part (c).
(c) Consider equation $$\varepsilon =\varepsilon _{m}\sin \omega _{d}t$$ with $$\varepsilon =-\dfrac{1}{2}\varepsilon _{m}$$. In order to satisfy this equation, we require $$sin( \omega _{d}t)=-1/2$$. Now we note that the problem states that $$ε$$ is increasing in magnitude, which (since it is already negative) means that it is becoming more negative. Thus, differentiating equation with respect to time (and demanding the result be negative) we must also require $$\cos \left ( \omega _{d}t \right )<0$$. These conditions imply that $$ωt$$ must equal $$\left ( 2n\pi -5\pi /6 \right )$$ [ $$n$$ = integer]. Consequently, equation $$i=I\sin \left ( \omega _{d}t+\phi \right )$$ yields (for all values of $$n$$)
$$i=I\sin \left ( 2n\pi -\dfrac{5\pi}{6}+\dfrac{\pi}{2} \right )=\left ( 3.91 \times 10^{-3}A \right )\left (- \dfrac{\sqrt{3}}{2} \right )=-3.38 \times 10^{-2}A$$.
or, $$\left | i \right |=3.38 \times 10^{-2}A$$
Explain the various part of $$AC$$ generator.
An $$AC$$ generator is a device which produces an alternating current when a changing magnetic field is applied on a conductor coil. The frequency of the current is same as the frequency of rotation of the conductor coil. There are various parts of an $$AC$$ generator which are shown in the image:-
The parts of $$AC$$ generator are:-
Rectangular coil: Rectangular Coil is the conductor coil in which current is produced. When the coil rotates in an applied magnetic field due to electromagnetic induction a current is produced.
Permanent magnet: permanent magnets are another most important part. They provide a steady magnetic field in which the coil rotates. Without them there is no magnetic field and there will be no electromagnetic induction.
Slip rings: When the coil rotated, the wire attached to them through which we will get electricity will also rotate and will create a problem by as the wire will get a curl up. So the solution to this problem is slip rings. The current which is produced through coil is transported to the rings. Inner part of the ring is allowed to have rotation but the exterior part is not allowed and inner and exterior part are both connected. So the current passes to exterior part from inner part and then to the wire.
Carbon brushes: Carbon brushes are the fixed part which is connected to slip rings. They are fixed and current from slip rings passes to carbon brush and then to the wire.
Shaft: Shaft is like a handle attached to the coil, which is used for moving the coil. It basically makes rotation easier and faster.
On what factors does the magnitude of the emf induced in the circuit due to magnetic flux depend?
It depends on the rate of change in magnetic flux (or simply change in magnetic flux)
$$|\omega|=\dfrac {\Delta \varphi}{\Delta t}$$
How does one understand this motional emf by invoking the Lorentz force acting on the free charge carriers of the conductor? Explain.
Suppose any arbitrary charge $$q$$ in the conductor of length $$l$$ moving inward in the field as shown in figure, the charge $$q$$ also moves with velocity $$V$$ in the magnetic field $$B$$
The Lorentz force on the charge $$q$$ is $$F=qvB$$ and its direction is downwards.
So, work done in moving the charge $$q$$ along the conductor of length $$l$$
$$W=F.l$$
$$W=qvBl$$
Since emf is the work done per unit charge
$$\therefore \varepsilon =\dfrac{W}{q}=Blv$$
This equation gives emf induced across the rod.
State the rule to determine the direction of a current induced in coil due to its rotation in a magnetic field.
Fleming's right hand rule is used to find out the direction of the current induced in coil due to its rotation in a magnetic field.
According to Fleming's right-hand rule:
If we put our thumb, index finger and middle finger mutually perpendicular to each other, such that thumb points in the direction of the motion of conductor and index finger point in the direction of the applied magnetic field.
Then middle/ centre finger will show the direction of the induced or generated current within the conductor.
At a certain location in the Philippines Earth's magnetic field of $$39\hspace{0.05cm} \mu T$$ is horizontal and directed due north. Suppose the net filed is zero exactly $$8.0\hspace{0.05cm} cm$$ above a long, straight, horizontal wire that carries a constant current. What are the (a) magnitude and (b) direction of the current?
(a) The field due to the wire, at a point $$8.0 cm$$ from the wire, must be $$39\hspace{0.05cm}\mu T$$ and must be directed due south. Since $$B=\mu_0 i/2\pi r$$,
$$i=\dfrac{2\pi rB}{\mu_{0}}=\dfrac{2\pi(0.080\hspace{0.05cm})(39\times 10^{-6}\hspace{0.05cm}T)}{4\pi \times 10^{-7}\hspace{0.05cm}T.m/A}=16 A$$
(b)The current must be from west to east to produce a field that is directed southward at points below it.
A bar magnet $$M$$ is dropped so that it falls vertically through the coil $$C$$. The graph obtained for voltage produced across the coil versus time is showing in figure(b)
Explain the shape of the graph.
When the bar magnet falls through the coil, the magnetic flux linked with the coil changes, so an emf (or $$pd$$) is developed across the coil.
Initially the rate of increases of flux increases, becomes maximum and then it decreases, becomes zero. Now magnetic flux begins to decrease, the rate of decrease increases becomes maximum and then it decreases and when the magnet is sufficiently far on the other side, the flux becomes zero and so pd induced becomes zero.
Answer the following questions
Define mutual inductance.
Mutual Inductance is the interaction of one coils magnetic field on another coil as it induces a voltage in the adjacent coil
The figure shows a loop model (loop L) for a paramagnetic
material. (a) Sketch the field lines through and about the material
due to the magnet. What is the direction of (b) the loops net magnetic dipole moment $$\vec \mu$$, (c) the conventional current i in the loop
(clockwise or counterclockwise in the figure), and (d) the magnetic
force acting on the loop?
(a) A sketch of the field lines (due to the presence of the bar magnet) in the vicinity of the loop is shown.
(b) For paramagnetic materials, the dipole moment $$\vec μ$$ is in the same direction as $$\vec B$$. From
the above figure, $$\vec μ$$ points in the –x-direction.
(c) Form the right-hand rule, since $$\vec μ$$ points in the –x-direction, the current flows
counterclockwise, from the perspective of the bar magnet.
(d) The effect of $$\vec F$$ is to move the material toward regions of larger $$|\vec B|$$ values. Since the
size of $$|\vec B|$$ relates to the "crowdedness" of the field lines, we see that $$\vec F$$ is toward the left,
or –x.
In Figure , a wire forms a closed circular loop, of radius $$R=2.0m$$ and resistance $$4.0\Omega $$. The circle is centered on a long straight wire; at time $$t=0$$, the current in the long straight wire is 5.0 A rightward.
Thereafter, the current changes according to $$i=5.0A-(2.0A/s^2)t^2$$. (The straight wire is insulated; so there is no electrical contact between it and the wire of the loop.) What is the magnitude of the current induced in the loop
at times $$t>0$$?
The field (due to the current in the straight wire) is out of the page in the upper half of the circle and is into the page in the lower half of the circle, producing zero net flux, at any time. There is no induced current in the circle.
A 12 H inductor carries a current of 2.0 A. At what rate must the current be changed to produce a 60 V emf in the inductor?
$$\varepsilon =-L(di/dt)$$, we may obtain the desired induced emf by setting
$$\dfrac{d
i}{d t}=-\dfrac{\varepsilon}{L}=-\dfrac{60 \mathrm{V}}{12 \mathrm{H}}=-5.0
\mathrm{A} / \mathrm{s}$$
or $$|d i
/ d t|=5.0 \mathrm{A} / \mathrm{s} .$$ We might, for example, uniformly reduce
the current from $$2.0 \mathrm{A}$$ to zero in $$40 \mathrm{ms}$$
A cylindrical bar magnet is kept along the axis of a circular coil and neat it as shown in figure. Will there be any induced emf at the terminals of the coil, when the magnet is rotated About its own axis.
When the magnet is rotated about its own axis, then due to symmetry of magnet the magnetic flux linked with circular coil remains unchanged, hence no emf is induced at terminals of coil.
In the given diagram, a coil $$B$$ is connected to low voltage bulb $$L$$ and placed parallel to another coil $$A$$ as shown. Explain bulb gets dimmer if the coil $$B$$ moves upwards.
When coil $$B$$ moves upwards, the magnetic flux linked with $$B$$ decreases and hence lesser current is induced in $$B$$.
Define $$1$$ henry.
$$1$$ henry is self-inductance of that coil in which $$1$$ volt emf is produced when the rate of change of current in that coil is $$1\ A/s$$.
The magnetic field of a cylindrical magnet that has a pole-face diameter of 3.3 cm can be varied sinusoidally between 29.6 T and 30.0 T at a frequency of 15 Hz. (The current in a wire wrapped around a permanent magnet is varied to give this variation in the net field.)
At a radial distance of 1.6 cm, what is the amplitude of the electric field induced by the variation?
The magnetic field $$B$$ can be expressed as
$$B(t)=B_{0}+B_{1}
\sin \left(\omega t+\phi_{0}\right)$$
where $$B_{0}=(30.0
\mathrm{T}+29.6 \mathrm{T}) / 2=29.8 \mathrm{T}$$ and $$B_{1}=(30.0
\mathrm{T}-29.6 \mathrm{T}) / 2=0.200 \mathrm{T} .$$ Then
from Eq. $$E=\dfrac{r}{2}\dfrac{dB}{dt}$$
$$E=\dfrac{1}{2}\left(\dfrac{d
B}{d t}\right) r=\dfrac{r}{2} \dfrac{d}{d t}\left[B_{0}+B_{1} \sin \left(\omega
t+\phi_{0}\right)\right]=\dfrac{1}{2} B_{1} \omega r \cos \left(\omega
t+\phi_{0}\right)$$
We note
that $$\omega=2 \pi f$$ and that the factor in front of the cosine is the
maximum value of the field. Consequently,
$$E_{\max
}=\dfrac{1}{2} B_{1}(2 \pi f) r=\dfrac{1}{2}(0.200 \mathrm{T})(2 \pi)(15
\mathrm{Hz})\left(1.6 \times 10^{-2} \mathrm{m}\right)=0.15 \mathrm{V} /
\mathrm{m}$$
A long cylindrical solenoid with 100 turns/cm has a radius of 1.6 cm. Assume that the magnetic field it produces is parallel to its axis and is uniform in its interior. (a) What is its inductance per meter of length? (b) If the current changes at the rate of 13 A/s, what emf is induced per meter?
(a) The self-inductance per meter is
$$\dfrac{L}{\ell}=\mu_{0} n^{2}
A=\left(4 \pi \times 10^{-7} \mathrm{H} / \mathrm{m}\right)(100 \mathrm{turns}
/ \mathrm{cm})^{2}(\pi)(1.6 \mathrm{cm})^{2}=0.10 \mathrm{H} / \mathrm{m}$$
(b) The induced emf per meter is
$$\dfrac{\varepsilon}{\ell}=\dfrac{L}{\ell}
\dfrac{d i}{d t}=(0.10 \mathrm{H} / \mathrm{m})(13 \mathrm{A} / \mathrm{s})=1.3
\mathrm{V} / \mathrm{m}$$
In the given diagram, a coil $$B$$ is connected to low voltage bulb $$L$$ and placed parallel to another coil $$A$$ as shown. Explain bulb lights if the coil $$B$$ moves upwards.
Bulb lights up due to induced current in $$B$$ because of change in flux linked with it as a consequence of continuous variation of magnitude of alternative current flowing in $$A$$
Consider an infinitely long wire carrying a current $$I(t)$$, with $$\dfrac {dI}{dt} = \lambda = constant$$.
Find the current produced in the rectangular loop of wire $$ABCD$$ if its resistance is $$R$$ (Fig.).
Consider a strip of width $$dr$$ and length $$l$$ inside a rectangle at distance $$r$$ from the surface of current carrying conductor. The magnetic field across strip of length
$$l = B(r) = \dfrac {\mu_{0}I}{2\pi r} l.B(r)$$ is perpendicular to the paper upward.
$$\therefore$$ Flux in strip $$\phi = \dfrac {\mu_{0}I}{2\pi} l \int_{x_{0}}^{x} \dfrac {dr}{r}$$
$$\phi = \dfrac {\mu_{0}Il}{2\pi} [\log_{e}r]_{x_{0}}^{x} = \dfrac {\mu_{0}Il}{2\pi}\log_{e} \dfrac {x}{x_{0}}$$
$$\epsilon = \dfrac {-d\pi}{dt}$$
So $$IR = \dfrac {d\phi}{dt}$$
$$I = \dfrac {1}{R} \dfrac {d}{dt}\left [\dfrac {\mu_{0}Il}{2\pi} \log_{e} \dfrac {x}{x_{0}}\right ] = \dfrac {\mu_{0}l}{2\pi R} . \log_{e} \dfrac {x}{x_{0}} \dfrac {dI}{dt}$$
$$I = \dfrac {\mu_{0}\lambda l}{2\pi R} \log_{e} \dfrac {x}{x_{0}} \left [\because \dfrac {dI}{dt} = \lambda (given)\right ]$$.
State two factors on which the magnitude of induced e.m.f. depend.
Magnitude of induced e.m.f depend upon:
(i) The change in the magnetic flux.
(ii) The time in which the magnetic flux changes.
Identify the figures and explain their use.
Figure (c) represents a DC generator. It is a device that generates electricity by rotating its rotor in a magnetic field. Thus, it converts mechanical energy into electrical energy.
On what factor do induced emf of a rotating coil (rectangular loop) in a magnetic field depend?
The induced e.m.f. is given by coil:
$$ \varepsilon =\varepsilon _0 sin\omega t=NAB \omega sin \omega t $$
Where-$$ N $$=Number of turn, $$ A$$=Area of cross section, $$ B $$=Magnetic field, $$ \omega$$ = Angular velocity.
How would we wrap two coils so that maximum value of induced emf is obtained?
The two coils should be wrapped opposite to one another. This makes the current flow in the reverse direction between the coils. This leads to the magnetic field to be in the same direction as given by the right-hand screw rule. Due to this, the induced emf due to the two coils will be maximum.
Define the coefficient of mutual inductance. Give its unit and dimensional formula.
Coefficient of Mutual Inductance
If orientation, size and shape of perimary coil $$C_1 $$ and secondary coil $$ C_2 $$, remains same and in coil $$C_1 $$ current is $$ l_1 $$, then second coil $$C_2 $$ is related to magnetic-flux which is proportional to the flow of current $$i_2. $$ i.e.,
$$ \phi_2 \propto l_1 $$
or $$ \phi_2=Ml_ 1 .........(1) $$
Here, $$M $$ is proportionality constant, which is called mutual induction. Its value depend upon number of turns,area of secondary coil and medium.
If the current in coil $$C_1 $$ changes with time, then flux linked with $$C_2, i.e., \phi_2 $$ changes .
Thus, induced emf in coil $$C_2, $$
$$ \varepsilon _2=-\dfrac{d \phi_2}{dt} $$
Thus, $$ \varepsilon _2=-\dfrac{MdI_1}{dt}...............(2) $$
The negative sign in equation (2), indicate that the direction of induced emf in secondary coil opposes the growth or decay of current flow in a primary coil, from equation(1),
$$ \phi_2=Ml_1 $$
If $$ l_1=1 Amp $$
then $$M=\phi_2...........(2) $$
Thus the coefficient of mutual inductance is equal to the magnetic flux linked with secondary coil when the current flow in a primary coil is unity, from equation(2),
$$ \varepsilon _2= M\dfrac{dI_1}{dt } $$
$$ \therefore M=\dfrac{\varepsilon _2}{-\dfrac{dI_1}{dt}}, if -\dfrac{dI_1}{dt}=1 $$
then $$m= \varepsilon _2 $$
Hence the coefficient of mutual inductance is equal to the induced emf when decay rate of current in primary coil is unity.
The unit of mutual inductance
$$M=\dfrac{Weber}{Amp} $$ or henry $$(H) $$
=$$ \dfrac{Volt \times sec}{Ampere}=Volt s^{-1}A^{-1} $$
The dimensional Formula
=$$ \dfrac{Dimension \quad of \quad magnetic \quad flux}{Dimention \quad of \quad current} $$
=$$ \dfrac{[MLT^{-2}A^{-1}]}{[A]}=[MLT^{-2}A^{-2}] $$
Mutual inductance depends upon the number of turns in a coil, area of cross section and medium.
The $$S.I. $$ unit of $$M$$ is $$Wb/A $$ or $$VA/A $$ or henry $$ (H) $$ and its dimensions are $$ [M^1L^2T^{-2}A^{-2}] $$
A conducting wire is in North-South direction. It is freely dropped towards Earth. Is an emf induced between its ends? Why?
No there is no emf induced because there is no change in magnetic flux.
What is the value of self inductance, keeping the number of turns in the coil same and doubling the cross-sectional area?
Coefficient of self induction of coil is given by the relation: $$ L= \dfrac{N^2\mu_0A}{l} $$
$$ \therefore L \propto A $$ So when the area is doubled, the self induction is $$ 2 $$ times.
A coil in a magnetic field is removed with:
(i) fast seed,
(ii) slow speed.
In which case, is the induced emf and work done more?
By the relation:
The induced emf is given as:
$$ \varepsilon =-\frac{d\phi}{dt}, $$
for fast speed $$dt$$ is minimum so the induced emf is more.
The self inductance of a coil is $$ 1 H $$. What do you understand?
Self inductance of coil $$ L=1H $$
Induced emf $$( \varepsilon )=-L\dfrac{dI}{dt} \therefore \varepsilon =-\dfrac{dI}{dt} $$
Hence, when the coefficient of self inductance of coil is unit then the decay rate of current is equal to the induced emf in coil.
A conducting wire of length $$ 20 cm $$ is moving normally in a magnetic field of $$ 5 \times 10^4 Wb/m. $$ If the conducting wire covers $$ 1m $$ distance in $$ 4s $$, then determine induced emf at the ends of the conducting wire.
Given , length of conductor $$ (I) = 20 cm = 0.2 m $$
Magnetic filed $$ (B) = 5 \times 10^{-4} Wb/m^2 $$
Time $$ = 4 s $$
distance covered (s) $$ = 1 $$ meter ,
Velocity of conductor $$(v) = \dfrac {s}{t} = \dfrac {1}{5} m/s $$
$$ \therefore $$ Induced $$ emf ( \epsilon) = Blv$$
$$ = 5 \times 10^{-4} \times 0.2 \times \dfrac {1}{4} $$
$$ = 2.5 \times 10^{-5} volt $$
A metallic rod of length $$ 2m $$ is moved (i) vertically ,(ii) horizontally with a speed of $$ 15 km/h $$ from West to East. If the horizontal component of Earth's magnetic field is $$ 0.5 \times 10^{-5} Wb/m^2 $$, calculate induced emf between the ends of the rod in each case.
Length of rod $$ = 2 m $$
Velocity of rod $$ = 15 km /h = \dfrac { 15 \times 5}{18} = \dfrac {25}{6} m/s W to E $$
The horizontal component of earth's magnetic filed $$ (B) = 0.5 \times 10^{-5} Wb/m^2$$
(i) When the rod is moving perpendicular then $$ B_H , \overrightarrow {v} $$ and $$ \overrightarrow {l} $$ are mutually perpendicular to each other
induced
$$ emf(e) = B_Hvl $$
$$ = 0.5 \times 10^{-5} \times \dfrac {25}{6} \times 2 $$
$$ = 4 16 \times 10^{-5} volt $$
(ii) When the rod is moving horizontal then :
$$ \epsilon = BLv sin 0 $$
$$ \epsilon = 0 $$
A conducting rod of length $$ L $$ is rotated in a magnetic field $$ B $$ with an angular velocity $$ \omega $$, so that the rotational plane of rod is perpendicular to the magnetic field Determine the induced emf between the ends of the rod.
Induced emf in a Metal Rod Rotating in a Uniform Magnetic Field
In figure $$ 9.10, $$ a uniform magnetic field is shown by cross (x) whose direction is normally inwards the surface. A conducting rod $$OA$$ of length $$L$$ rotates in the anticlockwise direction in magnetic field $$B $$ with angular velocity $$ \omega $$. The plane of rotation of rod is perpendicular to the magnetic field. When an element $$dI $$ of rod moves with velocity $$v $$ normally with the magnetic field, the induced emf of the element
$$ d \varepsilon =Bvdl $$
If the distance between small element and centre is $$l $$, then
$$ v=\omega l $$
Thus, to determine induced emf in the rod, we intergrate the above equation from zero to $$L $$,
$$ \int d\varepsilon =\int _{ 0 }^{ L } B\omega l dl $$
$$ \varepsilon =\dfrac{1}{2} B\omega L^2 ...........(1)$$
Using Fleming's right hand rule, the direction of induced current in rod is from $$A$$ to $$O$$. Thus, the end $$O$$ of the rod is positive and $$A$$ is negative.
If the frequency of rotation of rod is $$ f $$, then to
$$ \omega=2 \pi f $$
Thus, $$ \varepsilon =\dfrac{1}{2} B \times 2\pi f\times L^2 $$
=$$ B\times \pi L^2 \times f $$
If the area covered by the rod in magnetic field is represented by $$A$$ then
$$ \pi L^2=A $$
then $$\varepsilon =BAf ..........(2) $$
Let the resistance of rod $$ 'R' $$ then
$$ \therefore $$Induced current $$(I)=\dfrac{\varepsilon }{R} $$
$$ \therefore $$ form the equation (1) and (2),
$$I=\dfrac{1}{2} \dfrac{B\omega L^2}{R}=\dfrac{1B\omega L^2}{2R}.....(3) $$
and $$ I=\dfrac{1}{2} \dfrac{BAf}{R}=\dfrac{BAf}{2R} .....(4) $$
The heat produced due to the induced current:
$$H=I^2Rt $$
From the equation (3)
$$ H=\left( \dfrac { 1B\omega L^2 }{ 2R } \right)^2 Rt $$
$$ =\dfrac{B^2\omega^2L^4}{R^2}Rt=\dfrac{B^2\omega^2L^4t}{R}.............(5) $$
The fore produced in a conductor:
$$ F=IBl \quad sin 90=IBl $$
From equation (3)
$$ F=\dfrac{1}{2} \dfrac{B\omega L^2}{R} BL=\dfrac{1}{2}\dfrac{B^2\omega L^3}{R}......(6) $$
A coil is made from soft iron of length $$ 0.1 m $$and radius $$ 0.01m. $$ If the relative magnetisation of soft iron is $$ 1200 $$, then calculate the number of turns in the coil.
[Self inductance of coil is $$ 0.25 H $$].
Given length of coil $$ = I = 0.1 m $$
Radius $$ (R) = 0.01 m $$
Relative permeability of the space = $$ \mu_r = 1200 $$
coefficient of self induction $$ (L) = 0.25 H $$
The self induction of the coil
$$ (L) = \dfrac { \phi}{I} $$
$$ \therefore \quad L = \dfrac {NBA}{I} = \dfrac {N( \dfrac { \mu NI}{2R}) \times \pi R^2 }{I} $$
$$ L = \dfrac { \mu N^2 \pi R}{2} \quad \quad [ but \mu = \mu_r \mu_0] $$
$$ \therefore N = \sqrt { \dfrac {2L}{ \mu_r \mu_0 \pi R } } = \sqrt { \dfrac {2 \times 0.25 }{ 4 \times 3.14 \times 10^{-7} \times 1200 \times 3.14 \times 0.1} } $$
$$ = 1.027 \times 10^2 $$
$$ N = 10^2 m $$
The magnetic flux linked with the coil of $$ 50 turns $$ is $$ \phi_B=0.02 $$ cos $$ 100 \pi t $$ Wb. Determine:
(a)Mximum induced volatage.
(b)Induced emf at $$ t=0.01s. $$
(c) Induced electric current at $$ t=0.005 s. $$
(if external resistance is $$100 \Omega $$)
Number of turns $$ - N = 50 $$
Magnetic flux $$ ( \phi_B) = 0.02 cos (100 \pi t) Wb $$
(a) The induced emf $$ (\epsilon) $$ = $$ - \dfrac { Nd \phi}{dt} $$
$$ = -50 \dfrac {d}{dt} [ 0.02 cos ( 100 \pi t) ] $$
$$ = 50 \times 0.02 \times 100 \pi sin ( 100 \pi t) $$
$$ = 314 sin ( 100 \pi t) $$
Compare by the relation $$ \epsilon = \epsilon_0 sin wt $$
$$ \therefore \quad \epsilon_0 = 314 V $$ and $$ W = 100 \pi $$
$$ (b) t = 0.01 s $$
$$ \epsilon = 314 sin ( 100 \pi \times 0.01) $$
$$ = 314 sin \pi = 0 $$
(c) The external resistance ( R) = $$ 100 \Omega , t = 0.005 s $$
$$ \therefore $$ induced current (I) $$ = \dfrac {\epsilon}{R} = 314 sin \dfrac {(100 \pi r) }{R} $$
$$ = \dfrac { 314 sin ( 100 \times \pi \times 0.005)}{ 100 } $$
$$ = \dfrac {314 sin ( \pi /2) }{ 100} = 3.14 $$
If a current of $$ 5 A $$ flowing in a primary coil nullifies in $$ 2 $$ min, then induced emf is $$ 25 kV $$. Calculate the coefficient of mutual inductance.
Given , change in current $$ dl) = 0-5 = -5A $$
Time $$ (dt)= 2 ms = 2 \times 10^{-3} s $$
Induced emf in coil
$$ \epsilon = kV = 2.5 \times 10^4 volt $$
$$ \therefore \quad \epsilon = -M \dfrac {dI}{dt} $$
$$ \therefore \quad M = - \dfrac { \epsilon}{dI} \times dt $$
$$ = - \dfrac { 2.5 \times 10^4 \times 2 \times 10^{-3} }{-5A} = 10 H $$
What is self induction? Explain the experiments of self induction and calculate self inductance for a solenoid.
Self induction
when current in a coil changes the magnetic flux linked with the coil also changes and hence emf is induced in the coil . this phenomenon is known as self induction.
According to len'z law the induced emf opposes the change in magnetic flux
Experimental representation in fig a conducting coil C is connected with a battery and key in series. when current is passed through the key hence magnetic flux is linked with the coil changes. when key is pressed , on increasing current, the change in magnetic flux also increase which produces emf and the induced current oppose the current that flows through the circuit . Similarly when key is opened the current in the circuit is zero . Then , magnetic flux is linked with the coil decreases and hence the induced current flows in the direction of the current supplied. this is shown in figure .
Experiment to demonstrate self induction :
take a solenoid having a large number of insulated wire wound over over a soft iron core . such a solenoid is called a choke coil. connect the solenoid in series with battery . a eheostat and a tapping key . connect a 6 v bulb in parallel with the solenoid . press the tapping key and adjust the current with the help pf rheostat so that the bulb just glows faintly . As the tapping key is released , the bulb glows brightly for a moment and then goes out . this is because as the circuit is broken suddenly . vanishes is the rate of charge of magnetic flux linked with the coil is very large. hence large self induced emf and current are produced in the coil which make the bulb glow brightly for a moment.
A conducting rod of length $$ 1m $$ rotates with an angular velocity at the rate $$ 50 $$ revolutions per second normally to a magnetic field of $$ 0.001 T $$ from one end. Calculate the induced emf between the ends of the rod.
Length of rod = $$ 1 m $$
Rotational frequency $$ (f) = 50 rps $$
$$ \therefore \quad angular \quad velocity (w) = \dfrac { 2 \pi f}{t} $$
$$ = \dfrac { 2 \times 3.14 \times 50}{ 1} = 314 rad /sec $$
Magnetic filed $$ (B) = 0.001 T $$
The induced emf across the ends of rod $$ \rightarrow $$
$$ \epsilon = \dfrac {1}{2} B w l^2 $$
$$ = \dfrac {1}{2} \times 0.001 \times 314 \times 1 \times 1 $$
$$ = \dfrac { 0.314}{2} = 0.157 volt $$
A solenoid of radius $$ 2 cm $$ and $$ 100 $$ turns has a length of $$ 50 cm $$. If vacuum is inside the solenoid, then calculate the self inductance of solenoid.
radius of solenoid $$ r) = 2 cm = 2 \times 10^{-2} , $$
Number of turns $$ (N) = 100 $$
Length of solenoid $$ (I) = 50 cm = 50 \times 10^{-2} m $$
$$ \therefore $$ Self inductance of solenoid $$ L) = \dfrac { \mu_0 N^2A}{l} $$
$$ = \dfrac { \mu_0 N^2 \pi r^2 }{ l } $$
$$ = \dfrac { 4 \pi \times 10^{-7} \times 100 \times 100 \times 3.14 \times ( 2 \times 10^{-2})^2 }{ 50 \times 10^{-2}} $$
$$ = 31.55 \times 10^{-6} H $$
$$ = 31.55 \mu H $$
What do you mean by electromagnetic induction? State Faradays law of induction.
Faraday's law of electromagnetic induction, also known as Faraday's law is the basic law of electromagnetism which helps us predict how a magnetic field would interact with an electric circuit to produce an electromotive force (EMF). This phenomenon is known as electromagnetic induction.
Faraday's laws of Electromagnetic Induction:
Faraday's laws of Electromagnetic Induction consists of two laws. The first law describes the induction of emf in a conductor and the second law quantifies the emf produced in the conductor.
Faraday's First law of Electromagnetic Induction:
The discovery and understanding of electromagnetic induction are based on a long series of experiments carried out by Faraday and Henry. From the experimental observations, Faraday arrived at the conclusion that an emf is induced in the coil when the magnetic flux across the coil changes with time. With this in mind, Faraday formulated his first law of electromagnetic induction as, whenever a conductor is placed in varying magnetic field, an electromotive force is induced. If the conductor circuit is closed, a current is induced which is called induced current.
Changing the magnetic field intensity in a closed loop. Shown in below figure.
Faraday's second law of Electromagnetic Induction:
Faraday's second law of electromagnetic induction states that, The induced emf in a coil is equal to the rate of change of flux linkage.
The flux is the product of the number of turns in the coil and the flux associated with the coil. The formula of Faraday's law is given below:
$$\varepsilon =-N\frac{\Delta \phi}{\Delta t}$$
$$\varepsilon$$ is the electromotive force.
$$\phi$$ is the magnetic flux.
$$N$$ is the number of turns.
The negative sign indicates that the direction of the induced emf and change in direction of magnetic fields have opposite signs.
Additionally, there is another key law known as Lenz's law that describes electromagnetic induction as well.
Current is induced when the armature of a generator rotates. Slip rings and brushes are the ways and means by which this current is brought to the outer circuit. Is the arrangement necessary if the magnet in generator is made to rotate ?
Current is induced when the armature of a generator rotates. Slip rings and brushes are the ways and means by which this current is brought to the outer circuit. No, this arrangement is not necessary if the magnet in generator is made to rotate.
There is only one type of generator AC generator. Write down your responses about this statement.
According to the nature of output, generators can be classified into 2 types AC generator and DC generator. Though ac current is produced in a DC generator with the help of split-ring commutator ac is converted into dc current.
The diagram shown above is an arrangement for producing $$10\,V$$ AC using electromagnetic induction. Observe the diagram carefully and answer the question.
Find out the frequency by drawing a time emf graph if the armature completes $$10$$ cycles in $$5$$ seconds.
$$f = \dfrac{10}{5} = 2\,Hz$$
Write down the similarities and differences in the structure of an AC generator and a DC generator.
The Similarities between AC and DC generator are:
(i) Field magnet and Armature are present.
(ii) Works on the principle of electromagnetic inductance.
(iii) AC produced on armature.
The differences between the AC and DC generators are:
AC generator DC generator
Slip ring AC Split ring commutator
produced in outer circuit DC produced in outer circuit
Armature or magnet can be rotated Only armature can rotated
Parts of an AC generator are given below.
Field magnet, Armature, Slip rings, Brush.
a. Explain the position of these parts in an AC generator.
b. Write down the functions of any two.
a. Armature rotates in the magnetic field of the field magnet. Springs are Soldered to the ends of armature coil, Graphite Brush is connected in such a way as to have constant contact with springs.
b. Armature - Generate emf.
Field magnet - Creates magnetic flux.
When $$50 \,Hz$$ AC is used, how many times will the direction of current change in the circuit ?
50 Hertz (Hz) means the rotor of the generator turns 50 cycles per second, the current changes 50 times per second back and forth, direction changes 100 times. That means the voltage changes from positive to negative, and from negative to positive voltage, this process converts 50 times/second.
Line diagrams of a generator are given.
What is the specialty of the electricity reaching the galvanometer if the armatures of both the generators are made to rotate ?
DC in first and AC in second.
What is the period of the armature.
Period = Time taken to complete one full rotation.
Period $$= \dfrac{Time \,taken \,for \,rotation}{Number \,of \,fractions} = \dfrac{30}{60} = \dfrac{1}{2} = 0.5\,s$$
When the Armature ABCD rotates, change in the magnetic flux depends on its direction and intensity.
A plane loop shown in Fig. is shaped as two squares with sides $$a = 20\ cm$$ and $$b = 10\ cm$$ and is introduced into a uniform magnetic field at right angles to the loop's plane. The magnetic induction varies with time as $$B = B_{0}\sin \omega t$$, where $$B_{0} = 10\ mT$$ the current induced in the loop if its resistance per unit length is equal to $$\rho = 50\ m\Omega/ m$$. The inductance of the loop is to be neglected.
The loops are connected in such a way that if the current is clockwise in one, it is anticlockwise in the order. Hence the e.m.f. in loop bb.
Lenz's Law (26)
Consider the arrangement shown in Figure. Assume that $$R = 6.00 \,\Omega, \,$$, and a uniform $$2.50-T$$ magnetic field is directed into the page. At what speed should the bar be moved to produce a current of $$0.500 \,A$$ in the resistor?
the induced current formula is given by:
I = ε/R =(−LvB)/R
where the − sign indicates the current is counterclockwise (out of the page), so current flows upward through the bar.
We can solve this equation for v, using only the magnitude of I
v = (IR)/(LB)= 1.00 m/s
A wire loop enclosing a semi-circle of radius $$a$$ is located on the boundary of a uniform magnetic field of induction $$B$$ (Fig.). At the moment $$t = 0$$ the loop is set into rotation with a constant angular acceleration $$\beta$$ about an axis $$O$$ coinciding with a line of vector $$B$$ on the boundary. Find the emf induced in the loop as a function of time $$t$$. Draw the approximate plot of this function. The arrow in the figure shows the emf direction taken to be positive.
Flux at any moment of time,
$$|\Phi_{t}| = |\vec {B} \cdot d\vec {S}| = B\left (\dfrac {1}{2}R^{2} \varphi \right )$$
where $$\varphi$$ is the sector angle, enclosed by the field.
Now, magnitude of induced e.m.f. is given by,
$$\xi_{in} = \left |\dfrac {d\Phi_{t}}{dt}\right | = \left |\dfrac {BR^{2}}{2} \dfrac {d\varphi}{dt}\right | = \dfrac {BR^{2}}{2}\omega$$,
where $$\omega$$ is the angular velocity of the disc. But as it starts rotating from rest at $$t = 0$$ with an angular acceleration $$\beta$$ its angular velocity $$\omega (t) = \beta t$$. So,
$$\xi_{in} = \dfrac {BR^{2}}{2}\beta t$$.
According to Lenz law the first half cycle current in the loop is in anticlockwise sense, and in subsequent half cycle it is in clockwise sense.
Thus is general, $$\xi_{in} = (-1)^{n} \dfrac {BR^{2}}{2}\beta t$$, where $$n$$ in number of half revolutions.
The plot $$\xi_{in}(t)$$, where $$t_{n} = \sqrt {2\pi n/\beta}$$ is shown in the answer sheet.
Calculate the inductance of a unit length of a double taped line (Fig.) if the tapes are separated by a distance $$h$$ which is considerably less than their width $$b$$, namely, $$b/h = 50$$.
Neglecting end effects the magnetic field $$B$$, between the plates, which is mainly parallel to the plates, is $$B = \mu_{0} \dfrac {I}{b}$$
(For a derivation)
Thus, the associated flux per unit length of the plates is,
$$\Phi = \mu_{0} \dfrac {I}{b}\times h\times 1 = \left (\mu_{0} \dfrac {h}{b}\right ) \times I$$.
So, $$L_{1} =$$ inductance per unit length $$= \mu_{0} \dfrac {h}{b} = 24\ nH/ m$$.
A current $$I_{0} = 1.9\ A$$ flows in a long closed solenoid. The wire it is wound of is in a superconducting state. Find the current flowing in the solenoid when the length of the solenoid is increased by $$\eta = 5\%$$.
In a solenoid, the induction $$L = \mu \mu_{0} n^{2}V = \mu \mu_{0} \dfrac {N^{2}S}{l}$$,
where $$S =$$ area of cross section of the solenoid, $$l =$$ its length, $$V = Sl, N = nl =$$ total number of turns.
When the length of the solenoid is increased, for example, by pulling it, its inductance will decrease. If the current remains unchanged, the flux, linked to the solenoid, will also decrease. An induced e.m.f. will then come into play, which by Lenz's law will try to oppose the decrease of flux, for example, by increasing the current. In the superconducting state the flux will not change and so,
$$\dfrac {I}{l} = constant$$
Hence, $$\dfrac {I}{l} = \dfrac {I_{0}}{l_{0}}$$, or, $$I = I_{0} \dfrac {l}{l_{0}} = I_{0} (1 + \eta)$$.
In a certain region of the inertial reference frame there is magnetic field with induction $$B$$ rotating with angular velocity $$\omega$$. Find $$\bigtriangledown \times E$$ in this region as a function of vectors $$\omega$$ and $$B$$.
A rotating magnetic field can be represented by,
$$B_{x} = B_{0}\cos \omega t; B_{y} = B_{0}\sin \omega t$$ and $$B_{z} = B_{zo}$$
Then curl, $$\vec {E} = \dfrac {\partial \vec {B}}{\partial t}$$.
So, $$-(Curl\ \vec {E})_{x} = -\omega B_{0}\sin \omega t = -\omega B_{y}$$
$$- (Curl\ vec {E})_{y} = \omega B_{0} \cos \omega t = \omega B_{x}$$ and $$-(Curl\ \vec {E})_{z} = 0$$
Hence, $$Curl\ \vec {E} = -\vec {\omega} \times \vec {B}$$,
where, $$\vec {\omega} = \vec {e_{3}} \omega$$.
Calculate the mutual inductance of a long straight wire and a rectangular frame with sides $$a$$ and $$b$$. The frame and the wire lie in the same plane, with the side $$L$$ being closest to the wire, separated by a distance $$l$$ from it and oriented parallel to it.
Here, $$B = \dfrac {\mu_{0}I}{2\pi r}$$ at a distance $$r$$ from the wire. The flux through the frame is obtained as,
$$\Phi_{12} = \int_{1}^{a + 1} \dfrac {\mu_{0}I}{2\pi r} bdr = \dfrac {\mu_{0}b}{2\pi} I\ ln \left (1 + \dfrac {a}{l}\right )$$
Thus, $$L_{12} = \dfrac {\Phi_{12}}{I} = \dfrac {\mu_{0}b}{2\pi} ln \left (1 + \dfrac {a}{l}\right )$$
Thus, $$L_{12} = \dfrac {\Phi_{12}}{I} = \dfrac {\mu_{0}b}{2\pi} ln \left (1 + \dfrac {a}{l}\right )$$.
If a rotor in a power generator completes $$3000$$ rotations in a minute.
Which type of AC is produced here, single-phase or three-phase ?
Three phase AC
Three-phase electric power is a common method of alternating current electric power generation, transmission, and distribution. It is a type of polyphase system and is the most common method used by electrical grids worldwide to transfer power. It is also used to power large motors and other heavy loads.
A non-relativistic point charge $$q$$ moves with a constant velocity $$v$$. Using the field transformation formulas, find the magnetic induction $$B$$ produced by this charge at the point whose position relative to the charge is determined by the radius vector $$r$$.
In the reference frame $$K$$, moving with the particle,
$$\vec {E'} \cong \vec {E} + \vec {v_{0}} \times \vec {B} = \dfrac {\vec {q}}{4\pi \epsilon_{0} r^{3}}$$
$$\vec {B'} \cong \vec {B} - \vec {v_{0}} \times \vec {E}/ c^{2} = 0$$.
Here, $$\vec {v_{0}} =$$ velocity of $$K'$$, relative to the $$K$$ frame, in which the particle has velocity $$\vec {v}$$.
Clearly, $$\vec {v_{0}} = \vec {v}$$. From the second equation,
$$\vec {B} \cong \dfrac {\vec {v} \times \vec {E}}{c^{2}} = \epsilon_{0}\mu_{0} \times \dfrac {q}{4\pi \epsilon_{0}} \dfrac {\vec {v}\times \vec {r}}{r^{3}} = \dfrac {\mu_{0}}{4\pi} \dfrac {q(\vec {v}\times \vec {r})}{r^{3}}$$.
How many cycles are completed in $$T / 2$$ seconds ?
Given: time taken to produce 60 cycles $$=30$$ sec.
$$\therefore$$ time period $$T = \dfrac{30}{60} = \dfrac{1}{2}$$
Now $$T' = \dfrac{T}{2} = \dfrac{1}{4}$$
We know, $$v = \dfrac{1}{T}$$
$$v = \dfrac{1}{1/4} = 4$$ cycles / sec
Hence 4 cycle are completed in $$\dfrac{T}{2}$$ seconds.
(a) Obtain an expression for the mutual inductance between a long straight wire and a square loop of side a as shown in Fig.
(b) Now assume that the straight wire carries a current of 50 A and the loop is moved to the right with a constant velocity, v = 10 m/s. Calculate the induced emf in the loop at the instant when x = 0.2 m. Take a = 0.1 m and assume that the loop has a large resistance.
Take a small element $$dy$$ in the loop at a distance $$y$$ from the long straight wire.
Magnetic flux associated with element, $$dy$$ is $$d\phi=BdA$$
Where, $$dA=a\times dy$$ is Area of element $$dy$$
$$B=\mu_oI/2\pi y$$ is magnetic field at $$y$$.
$$I$$ is current in the wire.
$$\displaystyle \therefore d\phi=\mu_oIa\times dy/2\pi y$$
$$\displaystyle \phi=(\mu_oIa/2\pi)\int_x^{a+x}{dy/y}$$
$$\Rightarrow \phi=(\mu_oIa/2\pi)ln(\dfrac{a+x}{x})$$
Mutual Inductance $$M=\phi/I=(\mu_oa/2\pi)ln(\dfrac{a+x}{x})$$
Emf induced in the loop, $$e=Bav=(\mu_oI/2\pi x)av$$
From the given values, $$e=5\times10^{-5}\ V$$
A line charge $$\lambda $$ per unit length is lodged uniformly onto the rim of a wheel of mass $$M$$ and radius $$R$$. The wheel has light non-conducting spokes and is free to rotate without friction about its axis as shown in above figure. A uniform magnetic field extends over a circular region within the rim. It is given by,
$$B\, =\, -B_0\, k\, \quad\, (r\, \leq \, a;\, a\, <\, R)$$, or $$ B\,=\,0$$ (otherwise).
What is the angular velocity of the wheel after the field is suddenly switched off?
Line charge per unit length, $$\lambda=Q/2\pi r$$
at distance r, the magnetic force is balanced by the centripetal force.
So, $$BQv=Mv^2/r$$
also the angular velocity, $$\omega=v/R=B2\pi \lambda r^2/MR$$
So for $$r\leq a;a\lt R$$, we get:
$$\omega=-2B_o a^2 \lambda k/MR$$
It is desired to measure the magnitude of field between the poles of a powerful loud speaker magnet. A small flat search coil of area $$2\, cm^2$$ with 25 closely wound turns, is positioned normal to the field direction, and then quickly snatched out of the field region. Equivalently, one can give it a quick $$90^{\circ}$$ turn to bring its plane parallel to the field direction). The total charge flown in the coil (measured by a ballistic galvanometer connected to coil) is 7.5 mC. The combined resistance of the coil and the galvanometer is 0.50 $$\Omega$$. Estimate the field strength of magnet.
$$\displaystyle Q\, =\, \int_{t_i}^{t_f}Idt$$
$$\displaystyle =\, \displaystyle \frac{1}{R}\, \int_{t_i}^{t_f}\varepsilon dt$$
$$=\, -\, \displaystyle \frac{N}{R}\, \int_{\Phi_i}^{\Phi_f}d\Phi $$
$$=\, \displaystyle \frac{N}{R}\, \left ( \Phi _i\, -\, \Phi _f \right ) $$
$$N = 25, R = 0.50\Omega, Q = 7.5\, \times\, 10^{-3}\, C$$
$$\Phi _f\, =\, 0, \quad A\, =\, 2.0\, \times\, 10^{-4}\, m^2$$
$$\implies \Phi _i\, =\, 1.5\, \times\, 10^{-4}\, Wb$$
$$B\, =\, \Phi_i\, /\, A\, =\, 0.75\, T$$
Explain in detail the principle, construction and working of a single phase AC generator.
AC Generator (Dynamo) - Single Phase: The ac generator is a device used for converting mechanical energy into electrical energy.
Principle: It is based on the principle of electromagnetic induction according to which an emf is induced in a coil when it is rotated in a uniform magnetic field.
Essential parts of an AC generator
i) Armature: Armature is a rectangular coil consisting of a large number of loops or turns of insulated copper wire wound over a laminated soft iron core or ring. The soft iron core not only increases the magnetic flux but also serves as a support for the coil.
ii) Field magnets: The necessary magnetic field is provided by permanent magnets in the case of low power dynamos. For high power dynamos, field is provided by electron magnet. Armature rotates between the magnetic poles such that the axis of rotation is perpendicular to the magnetic field.
iii) Slip rings: The ends of the armature coil are connected to two hollow metallic rings $${ R }_{ 1 }$$ and $${ R }_{ 2 }$$ called slip rings. These rings are fixed to a shaft, to which the armature is also fixed. When the shaft rotates, the slip rings along with the armature also rotate.
iv) Brushes: $${ B }_{ 1 }$$ and $${ B }_{ 2 }$$ are two flexible metallic plates or carbon brushes. They provide contact with the slip rings by keeping themselves pressed against the rig. They are used to pass on the current from the armature to the external power line through the slip rings.
Working: Whenever, there is a change in orientation of the coil, the magnetic flux linked with the coil changes, producing and induced emf in the coil. The direction of the induced current is given by Fleming's right hand rule.
Suppose the armature $$ABCD$$ is initially in the vertical position. It is rotated in the anticlockwise direction. The side $$AB$$ of the coil moves downwards and the side $$DC$$ moves upwards Figure (a). Then according to Flemings right hand rule the current induced in arm $$AB$$ flows from $$B$$ to $$A$$ and in $$CD$$ it flows from $$D$$ to $$C$$. Thus the current flows along $$DCBA$$ in the coil. In the external circuit the current flows from $${ B }_{ 1 }$$ to $${ B }_{ 2 }$$.
On further rotation, the arm $$AB$$ of the coil moves upwards and $$DC$$ moves downwards. Now the current in the coil flows along $$ABCD$$. In the external circuit the current flows from $${ B }_{ 2 }$$ to $${ B }_{ 1 }$$. As the rotation of the coil continues, the induced current in the external circuit keeps changing its direction for every half a rotation of the coil. Hence the induced current is alternating in nature figure (b).
As the armature completes $$v$$-rotations in one second, alternating current of frequency $$v$$ cycles per second is produced. The induced emf at any instant is given by $$e={ E }_{ 0 }\sin { \omega t }$$.
The peak value of the emf, $${ E }_{ 0 }=NBA\omega$$ where $$N$$ is the number of turns of the coil, $$A$$ is the area enclosed by the coil, $$B$$ is the magnetic field and $$\omega $$ is the angular velocity of the coil.
What is self inductance? Establish expression for self inductance of a long solenoid.
Self inductance: When the current is passed through a coil, the magnetic flux linked with the coil is directly proportional is the current.
If the magnetic flux is $$\Phi$$ for the current I, then
$$\Phi \propto I$$
or $$\Phi =L.I$$
Where L is constant, its value depends upon shape and size of the coil meidum and number of turns. It is called self inductance or coefficient of self induction of the coil.
Expression- Consider a solenoid whose length l is very large in comparison to its radius r. A uniform magnetic filed is induced within it at every point when a current I is passed through it. The intensity of this induced magnetic field.
$$B=\displaystyle\frac{\mu BI}{l}$$ ..........(1)
Here $$\mu$$ is the relative permeability of the material of core and N is the number of turns of the conducting wire on the solenoid. The magnetic flux linked with the solenoid is.
$$\Phi = B\times$$ total effective area of solenoid.
or $$\Phi =B\times N\pi r^2$$ ........(2)
If self inductance is L, then
$$\Phi =LI$$ ........(3)
Comparing eqns. (2) and (3),
$$LI=B\times N\pi r^2$$
or $$L=\displaystyle\frac{B\times N\pi r^2}{I}$$ ........(4)
Substituting B from eqn. (1) in eqn. (4),
$$L=\displaystyle\frac{\mu NI}{l}\cdot \frac{N\pi r^2}{I}$$
or $$L=\displaystyle\frac{\mu \pi N^2r^2}{l}$$
This is the required expression.
Factors affecting the self inductance of the solenoid-
(i) On N the number of turns in the solenoid: $$L\propto N^2$$ i.e., the self inductance increases on increasing the number of turns in the solenoid.
(ii) On the radius(r) of the solenoid: $$L\propto r^2$$ i.e., the self inductance increases on increasing the radius.
(iii) On the length (l) of the solenoid: $$L\propto \displaystyle\frac{1}{l}$$ i.e., self inductance decreases on increasing the length of solenoid.
(iv) On the relative permeability $$(\mu)$$ of the material placed inside the solenoid: $$L\propto \mu$$ i.e., if a soft iron rod is placed inside the solenoid its self inductance increases.
Figure shows a square frame of wire having a total resistance r placed coplaner with a long, straight wire. The wire carries a current i given by $$i = {i_0}\sin \omega t.$$ . Find (a) the flux of the magnetic field through the square frame; (b) the emf induced in the frame and (c) the heat developed in the frame in the time interval 0 to $${{20\pi } \over \omega }$$ .
Prove theoretically, the relation between e.m.f. induced and rate of change of magnetic flux in a coil moving in a uniform magnetic field.
Let us consider a rectangular wire loop $$PQRS$$ of width $$l$$, and its plane perpendicular to a uniform magnetic field. The loop is being pulled out of magnetic field at a constant speed $$v$$. At any instant, let '$$x$$' be the length of loop in the magnetic field. As the loop moves towards right, the area of the loop inside the field changes by $$dA=ldx=lvdt$$. So the change in magnetic flux is given by
$$d\phi =BdA=Blvdt$$
so $$\dfrac { d\phi }{ dt } =\dfrac { Blvdt }{ dt } =Bvl$$
Now a straight current-carrying conductor of length $$L$$ experiences a magnetic force given by
$$\vec { F } =\vec { IL } \times \vec { B } $$
Its direction is given by Fleming's left and rule.
So, the forces $${F}_{1}$$ and $${F}_{2}$$ on wires $$PR$$ and $$QS$$ respectively are equal in magnitude and opposite in direction and have the same line of action. Hence they balance each other. There is no force on the wire $$RS$$ as it lies outside the field. The force $$\vec{{F}_{3}}$$ on the wire $$PQ$$ has magnitude $${F}_{3}=IBL$$ directed towards left. To move the loop with constant velocity $$\vec{v}$$ an external force must be applied. So work done by the external agent is given by
$$dw=Fdx=-IlBdx=-IBdA$$
$$=-Id{ \phi }_{ m }$$
So rate of doing work $$P=\dfrac { dw }{ dt }$$
$$ =I\left( -\dfrac { d{ \phi }_{ m } }{ dt } \right)$$
but $$ P=EI$$
so $$\boxed { E=-\frac { d{ \phi }_{ m } }{ dt } } $$
Define self-inductance of a coil.
(a) Obtain an expression for the energy stored in a solenoid of self-inductance $$'L'$$ when the current though it grows from zero to $$'I'$$.
(b) A square loop MNOP of side 20 cm is placed horizontally in a uniform magnetic field acting vertically downwards as shown in the figure. The loop is pulled with a constant velocity of 20 cm $$s^{-1}$$ till it goes out of the field.
(i) Depict the direction of the induced current in the loop as it goes out of the field. For how long would the current in the loop persist?
(ii) Plot a graph showing the variation of magnetic flux and induced emf as a function of time.
1) Self-inductance- It is defined as the ration of total flux linked with the coil to the current flowing through the coil. It is denoted as $$L$$.
$$L=\dfrac{\phi _B}{I}$$. . . . . . .(1)
Expression for the energy stored in a solenoid-
The induced emf in the coil is given by
$$e=-\dfrac{d\phi_B}{dt}$$. . . . .(2)
From equation (1) and (2), we get
$$e=-L\dfrac{dI}{dt}$$. . . . . . . . .(3)
The self induced emf is also called back emf, as it opposes any change in the current. So, work needs to be done against back emf in the establishing current and this work done is stored as magnetic potential energy.
The rate of doing work is,
$$\dfrac{dW}{dt}=|e|I=LI\dfrac{dI}{dt}$$
The total work done,
$$W=\int dW=\int _0 ^ILIdI$$
$$W=\dfrac{1}{2}LI^2$$
(i) The direction of induced current in the loop is clockwise.
$$d=20cm$$
$$v=20cm/s$$
The current will persist till the entire loop comes out of the field,
$$t=\dfrac{d}{v}=\dfrac{20}{20}=1sec$$
(ii) The maximum flux is, $$\phi= Bla$$, a=side of the square.
The magnetic flux remains constant inside the magnetic field. This flux will starts dropping once the loop comes out of the magnetic field and it is zero when it comes completely out from the magnetic field as shown in the above figure.
The maximum Induced emf in coil is, $$e=-\dfrac{d\phi}{dt}=-Blv$$
The $$e$$ remains constant till the entire comes out from the magnetic field, when it comes out then emf drops to zero as shown in the above figure.
A closed coil having 100 turns is rotated in a uniform magnetic field $$B=4.0\ \times { 10 }^{ -4 }T$$ about a diameter which is perpendicular to the field. The angular velocity of rotation is 300 revolution per minute. The area of the coil is $$25\ { cm }^{ 2 }$$ and its resistance is $$4.0\ \Omega$$. Find (a) the average emf developed in half a turn from a position where the coil is perpendicular to the magnetic field, (b)the average emf in full turn and (c) the net charge displaced in part (a).
Given, $$N=100$$ turns
$$\\ B=4\times { 10 }^{ -4 }\quad T$$
$$\\ A=25\quad { cm }^{ 2 }=25\times { 10 }^{ -4 }{ \quad m }^{ 2 }$$
(a) When the coil is perpendicular to field.
$$\phi=NBA$$
Average emf=$$\dfrac { 2NBA }{ t } $$
=$$\dfrac { 2\times 100\times 4\times { 10 }^{ -4 }\times 25\times { 10 }^{ -4 } }{ \dfrac { 1 }{ 600 } \times 60 } =2\times { 10 }^{ -3 }V$$
(b)In full turn average emf will be zero.
(c)Charged induced=$$\dfrac { 2NBA }{ R } =\dfrac { 2\times 100\times 4\times { 10 }^{ -4 }\times 25\times { 10 }^{ -4 } }{ 4 } \approx 5\times { 10 }^{ -5 }C$$
In a motor, a rotor is fitted with the armature that has current of 10 A. The rotor rotates with angular speed of 3 rad/s.Magnetic field of magnitude 2 T varies in direction is such a way that it is always perpendicular to the loop area. If the rotor coil has N number of turns and area of each loop is $$0.45 m^2$$ then find the value N. Given that motor consumes 2106 W power and there are no losses.
Given that motor consume $$2106$$ W power.
Current value=$$10A$$
Voltage across motor armature=$$\dfrac{2106}{10}=210.6V$$
emf induced=$$NBA\omega$$
$$N=\dfrac{210.6}{BA\omega}$$
$$N=\dfrac{210.6}{2\times0.45\times3}=\dfrac{210.6}{2.7}$$
Deduce an equation $$U=\cfrac { 1 }{ 2 } L{ I }^{ 2 }$$ for an inductor.
$$\varepsilon =-L\dfrac{di}{dt}$$
The work done by voltage source during a time interval $$dt$$ is
$$dw=Pdt$$
$$dw=-\varepsilon idt$$
$$dw=-iL\dfrac{di}{dt}dt$$
$$dw=-Lidi$$.......(i)
Integrate both sides in eq (i)
$$w=-\displaystyle \int_{0}^{I}{Lidi}$$
$$w=-1/2LI^2$$
So $$\boxed{U=-W=1/2LI^2}$$
An infinitesimally small bat magnet $$\bar M$$ is pointing and moving with the speed $$v$$ in the $$\bar x-$$direction. A small closed circular conducting loop of radius $$a$$ and negligible self inductance lies in the $$y-z$$ plane with its centre at $$x=0$$ and its axes coinciding with the $$x-$$axis. Find the force opposing the motion of the magnet, if the resistance of the loop is $$R$$. Assume that the distance $$x$$ of the magnet from the centre of the loop is much greater than $$a$$.
A wire forming one cycle of sine curve is moved in $$x-y$$ plane with velocity $$\vec V={V}_{x}\hat i+{V}_{y}\hat j$$. There exist a magnetic field $$\vec B=-{B}_{0}\hat k$$. Find the motional emf developed across the ends $$PQ$$ of wire.
The magnetic field in a region is given by $$\overrightarrow { B } =\cfrac { { B }_{ 0 } }{ L } xk$$, where $$L$$ is a fixed length. A conducting rod of length $$L$$ lies along the X-axis between the origin and the point $$(L,0,0)$$. If the rod moves with a velocity $$\overrightarrow { v } ={ v }_{ 0 }\hat { j } $$, find the emf induced between the ends of the rod
Crosses represent uniform magnetic field directed into the paper. A conductor XY moves in the field towards right side. Find the direction of induced current in the conductor. Name the rule you applied. What will be the direction of current if the direction of field and the direction of motion of the conductor both are reversed?
Velocity $$\vec V$$ is rightward and field $$\vec B$$ is into the paper so $$\vec V \times \vec B$$ will be $$upward $$ so $$electron$$ being negatively charged will shift downward so current can be thought of in$$upward$$ direction. As force $$\vec F = q \vec V \times \vec B$$ for electron $$q=-e$$
if direction of field and velocity both are reversed the direction of induced current will remain same as $$ -\vec V \times -\vec B= \vec V \times \vec B$$
The figure shows a small circular coil of area A suspended from a point O by a string of length l in a uniform magnetic induction B in the horizontal direction. If the coil is set into oscillations likes simple pendulum by displacing it a small angle $${\theta _0}$$ as shown, find emf induced in the coil as a function of time. Assume the plane of the coil is always in the plane of string.
The instantaneous induced emf to the coil of area $$A$$ suspended by string of length $$l$$,
$$e=-\dfrac{d\phi}{dt}$$
magnetic flux, $$\phi=BAcos\theta_0$$ put in the above equation.
$$e=-\dfrac{(dBAcos\theta_0)}{dt}$$
$$e=BAsin\theta_0\dfrac{d\theta_0}{dt}$$
when $$\theta_0$$ is very small.
$$e=BA\theta_0\dfrac{d\theta_0}{dt}$$
A conducting circular loop is placed in a uniform magnetic field $$B=0.020T$$ with its plane perpendicular to the field. Somehow, the radius of the loop starts shrinking at a constant rate of $$1 mm/s.$$ Find the induced current in the loop at an instant when the radius is $$2 cm.$$
Area $$A = \pi {r^2}$$
$$\dfrac{{dA}}{{dt}} = 2\pi r\dfrac{{dr}}{{dt}}$$
when, $$r = 2\,cm\, = 0.02\,m$$
$$\dfrac{{dr}}{{dt}} = 1\,mm/s\,\, = {10^{ - 3}}m/s$$
emf $$ = N.B \times 2\pi r.\dfrac{{dr}}{{dt}}$$
$$=1 \times 0.02 \times 2\pi \times 0.02 \times {10^{ - 3}}V$$
$$=2.51 \times {10^{ - 6}}\,V$$
$$ = 2.51\,\mu V$$
Find the induced emf about ends of the rod in each case.
Refer image (i). $$|E|\vec{B}=\dfrac{d\vec{A}}{dt}$$
here $$\dfrac{dA}{dt}=(2R)\times v$$
$$\therefore E=B(2R)v$$
Refer image (ii)- $$XY=2l\sin\theta$$
$$LN=(2l\sin\theta)+2l(1+\sin\theta)$$
$$\therefore E=B2l(1+\sin\theta)v$$
Refer image (ii)
(a) Refer image (iii-a)
(b) same as (iii) a Noe emf induced across LN since $$d\vec{A}=\vec{v}dt\times d\vec{l}$$ $$=0$$
(c) Refer image (iii-c)
Refer image (iv)
No emf induced in the rod $$PQ$$ and $$RS$$
$$\therefore E=B(3l)V$$
Refer image (v)
$$E=Blu$$
Consider the situation shown in the figure. The wire PQ has mass m, resistance r and can slide on the smooth horizontal parallel rails of separated by a distance $$l$$. The resistance of the rails is negligible . A uniform magnetic field B exist in the rectangular region and a resistance R connects the rails outside the field region. At t=0, the wire PQ is pushed towards right with a speed of $$v_{0}$$. Find
(a) the current in the loop in an instant when the speed of the wire PQ is v.
(b) the acceleration of the wire at this instant
(c) Velocity v as a function
(a) When the speed is $$V$$
$$Emf=Blv$$
Resistance $$=r+R$$
Current $$=\cfrac { Blv }{ r+R } $$
(b) Force acting on the wire $$F=ilB$$
$$F=\cfrac { BlvlB }{ R+r } =\cfrac { { { l }^{ 2 }B }^{ 2 }v }{ r+R } $$
Acceleration on the wire $$=\dfrac Fm =\cfrac { { { l }^{ 2 }B }^{ 2 }v }{ m(r+R) } $$
(c) $$v={ v }_{ 0 }+at={ v }_{ 0 }-\cfrac { { { l }^{ 2 }B }^{ 2 }vt }{ m(r+R) } ={ v }_{ 0 }-\cfrac { { { l }^{ 1 }B }^{ 2 }x }{ m(r+R) } $$
A coil has an inductance of 5H and resistance $$20\Omega$$ . An emf of $$100$$v is applied to it. What is the energy stored in the magnetic field, when the current has reached its final value .
given $$L=5$$H
$$R=20\Omega$$
$${E}_{v}=100$$v
When current reaches its final value $$L=0$$
$${I}_{0}=\dfrac{{E}_{0}}{R}$$
$$=\dfrac{\sqrt{2}{E}_{v}}{R}$$
$$=\dfrac{100\sqrt{2}}{20}$$
$$=5\sqrt{2}$$A
Energy stored in magnetic field$$=\dfrac{1}{2}L{I}_{\circ}^{2}$$
$$=\dfrac{1}{2}\times 5\times{\left(5\sqrt{2}\right)}^{2}$$
$$=\dfrac{1}{2}\times 5\times 25\times 2$$
$$=125$$J
A coil of resistance $$40\Omega$$ is connected across a $$4.0$$V battery. $$0.10$$s after the battery is connected, the current in the coil is $$63$$mA. Find the inductance of the coil.
$$R=40\Omega,\,E=4$$V,$$\,t=0.1,\,\,i=63$$mA
$$i={i}_{\circ}\left(1-{e}^{\frac{tR}{2}}\right)$$
$$\Rightarrow 63\times{10}^{-3}=\dfrac{4}{40}\left(1-{e}^{-0.1\times\frac{40}{L}}\right)$$
$$\Rightarrow 63\times {10}^{-2}=1-{e}^{\frac{-4}{L}}$$
$$\Rightarrow 1-0.63={e}^{\frac{-4}{L}}$$
$$\Rightarrow {e}^{\frac{-4}{L}}=0.37$$
$$\Rightarrow \dfrac{-4}{L}=\ln{\left(0.37\right)}=-0.994$$
$$\Rightarrow L=\dfrac{-4}{-0.994}=4.024$$H$$=4$$H
Two protons are projected simultaneously from a fixed point with the same velocity v into a region, where there exists a uniform magnetic field. The magnetic field strength at B and it is prependicular to the initial direction of v. One proton starts at time t=o and another proton at t=$$\frac { \pi m }{ 2qB } $$. The separation between then at time $$\frac { \pi m }{ 2qB } $$ (where, m and q are the mass and charge of proton), will be approximately.
The closed loop $$(P Q R S)$$ of wire is moved out of a uniform magnetic field at right angles to the plane of the paper as shown. Predict the direction of induced current in the loop.
So far the loop remains in the magnetic field, there is no change in magnetic flux linked with the loop and so no current will be induced in it, but when the loop comes out of the magnetic field, the flux linked with it will decrease and so the current will be induced so as to oppose the decrease in magnetic flux, i.e. It will cause magnetic field downwards; so the direction of current will be clockwise.
Figure shows a smooth pair of thick metallic rails connected across a battery of emf $$\varepsilon $$ having a negligible internal resistance. A wire $$ab$$ of length $$l$$ and resistance $$r$$ can slide smoothly on the rails. The entire system lies in a horizontal plane and is immersed in a uniform vertical magnetic field $$B$$. At an instant $$t$$, the wire is given a small velocity $$v$$ towards right. (A) Find the current in it at this instant. What is the direction of the current? (b) What is the force acting on the wire at this instant? (c) Show that after some time the wire $$ab$$ will slide with a constant velocity. Find this velocity.
Net emf = E-Bvl
$$I=\dfrac{E-Bvl}{r}$$ from b to a
$$F=IlB$$
$$=\left(\dfrac{E-Bvl}{r}\right)lb=\dfrac{lB}{r}(E-Bvl)$$ towards right
After some time when $$E=Bvl$$
Then the wire moves constant velocity v
Hence $$v=E/Bl$$
Find the mutual inductance between the circular coil of radius $$a^{'}$$ and the loop of radius $$ a $$ as shown in figure when the slider on the rheostat is moved. The total resistance of rheostat is R and the distance between the center of two coils is x.
Refer image,
The magnetic field at centre of coil 2, due to coil 1 is given by:
$$B=\dfrac{\mu_0\,ia^2}{2(a^2+x^2)^{3/2}}$$
The fluxed linked with coil 2 is given by
$$\Phi=BA'$$
$$=\dfrac{\mu_0\,ia^2}{2(a^2+x^2)^{3/2}}\pi a^{1^2}$$
Now, let y be the distance of the sliding contact from its left end.
Given: $$v=\dfrac{dy}{dt}$$
Total Resistance of rheostat = R
When the distance of the sliding contact from the left end is y, the resistance of rheostat is given by
$$r'=\dfrac{R}{L}y$$
The current in the coil is the function of distance y travelled by the sliding contact of the rheostat. It is given by
$$i=\dfrac{E}{\left(\dfrac{Ry}{L}+r\right)}$$
The magnitude of emf induced can be calculated as
$$e=\dfrac{d\Phi}{dt}=\dfrac{\mu_0a^2a^{1^2}\pi}{2(a^2+x^2)^{3/2}}\dfrac{di}{dt}$$
$$e=\dfrac{\mu_0\,\pi a^2a^{1^2}}{2(a^2+x^2)^{3/2}}\dfrac{d}{dt}\dfrac{E}{\left(\dfrac{R}{L}y+r\right)}$$
$$e=\dfrac{\mu_0\,a^2a^{1^2}}{2(a^2+x^2)^{3/2}}E\left[\dfrac{\left(-\dfrac{R}{L}v\right)}{\left(\dfrac{R}{L}y+r\right)^2}\right]$$
emf induced,
$$e=\dfrac{\mu_0\pi \,a^2a^{1^2}}{2(a^2+x^2)^{3/2}}E\left[\dfrac{-\dfrac{R}{L}v}{\left(\dfrac{Ry}{L}y\right)^2}\right]$$
Now, emf induced in coil can be also given as
$$\dfrac{di}{dt}=\left[\dfrac{E\left(-\dfrac{R}{L}v\right)}{\left(\dfrac{Ry}{L}y\right)^2}\right]$$
$$e=\dfrac{Mdi}{dt}$$
$$\dfrac{di}{dt}=\left[\dfrac{E\left(-\dfrac{Rv}{L}\right)}{\left(\dfrac{Ry}{L}+r\right)^2}\right]$$
$$\therefore M=\dfrac{e}{\dfrac{di}{dt}}=\dfrac{\mu_0\pi a^2a^{1^2}}{2(a^2+x^2)^{3/2}}$$
Figure (a) shows, in cross section, three current-carrying wires that are long, straight, and parallel to one another. Wires 1 and 2 are fixed in place on an $$x$$ axis, with separation d.Wire 1 has a current of 0.750 A, but the direction of the current is not given. Wire 3, with a current of 0.250 A out of the page, can be moved along the $$x$$ axis to the right of wireAs wire 3 is moved, the magnitude of the net magnetic force $$\vec{F_{2}}$$ on wire 2 due to the currents in wires 1 and 3 changes. The $$x$$ component of that force is $$F_{2x}$$ and the value per unit length of wire 2 is $$F_{2x}/L_{2}$$. Figure (b) gives $$F_{2x}/L_{2}$$ versus the position $$x$$ of wireThe plot has an asymptote $$F_{2x}/L_{2}=-0.627\hspace{0.05cm} \mu N/m$$ as $$x \rightarrow \infty$$. The horizontal scale is set by $$x_{s}=12\hspace{0.05cm}cm$$. cm. What are the (a) size and (b) direction (into or out of the page) of the current in wire 2?
(a) The fact that the curve in Figure (b) passes through zero implies that the currents in wires 1 and 3 exert forces in opposite directions on wire 2. Thus, current $$i_1$$ points out of the page. When wire 3 is a great distance from wire 2, the only field that affects wire 2 is that caused by the current in wire 1; in this case the force is negative according to Figure (b). This means wire 2 is attracted to wire 1, which implies that wire 2's current is in the same direction as wire 1's current: out of the page. With wire 3 infinitely far away, the force per unit length is given (in magnitude) as $$6.27 × 10^{−7} N/m$$.We set this equal to $$F_{12}=\mu_{0}i_1i_2/2\pi d$$. When wire 3 is at x=0.004m the curve passes through zero point previously mentioned, so the force between 2 and 3 must be equal $$F_{12}$$ there. This allows us to solve for the distance between wire 1 and wire 2:
$$d=(0.04m)(0.750A)/(0.250A)=0.12m$$
Then we solve $$6.27\times 10^{-7} N/m=\mu_{0} i_1 i_2/2\pi d$$ and obtain $$i_2=0.50A$$
(b)The direction of $$i_2$$ is out of the page.
An ac generator has emf $$\xi=\xi _{m}\sin\left ( \omega _{d}t-\pi/4 \right )$$, where $$\xi _{m}=30.0V$$ and $$\omega _{d}=350rad/s$$. The current produced in a connected circuit is $$i\left ( t \right )=I\sin \left ( \omega _{d}t-3\pi /4 \right )$$, where $$I=620 mA$$. At what time after $$t=0$$ does (a) the generator emf first reach a maximum and (b) the current first reach a maximum? (c) The circuit contains a single element other than the generator. Is it a capacitor, an inductor, or a resistor? Justify your answer. (d) What is the value of the capacitance, inductance, or resistance, as the case may be?
(a) The generator emf and the current are given by,
$$\varepsilon =\varepsilon _{m}\sin \left ( \omega _{d}-\pi /4 \right )$$, $$i\left ( t \right )=I\sin \left ( \omega _{d}-3\pi /4 \right )$$.
The expressions show that the emf is maximum when, $$\sin \left ( \omega _{d}t-\pi /4 \right )=1$$ or
$$\omega _{d}t-\pi/4=\left ( \pi /2 \right )\pm 2n\pi$$. [ $$n$$ = integer].
The first time this occurs after $$t = 0$$ is when $$\omega _{d}t-\pi/4=\pi /2$$ (that is, $$n = 0$$). Therefore,
$$t=\frac{3\pi}{4\omega _{d}}=\frac{3\pi}{4\left ( 350rad/s \right )}=6.73*10^{-3}s$$.
(b) The current is maximum when $$\sin \left ( \omega _{d}t-3\pi/4 \right )=1$$ or,
$$\omega _{d}t-3\pi /4=\left ( \pi /2 \right )\pm 2n\pi$$, [ $$n$$ = integer]
The first time this occurs after $$t = 0$$ is when $$\omega _{d}-3\pi /4=\pi /2$$ (as in part (a), $$n = 0$$ ). Therefore,
$$t=\frac{5\pi}{4\omega _{d}}=\frac{5\pi }{4\left ( 350rad/s \right )}=1.12-10^{-2}s$$.
(c) The current lags the emf by $$+π / 2 rad$$, so the circuit element must be an inductor.
(d) The current amplitude $$I$$ is related to the voltage amplitude $$V_{L}$$ by $$V_{L}=IX_{L}$$, where $$X_{L}$$ is the inductive reactance, given by $$X_{L}=\omega _{d}L$$. Furthermore, since there is only one element in the circuit, the amplitude of the potential difference across the element must be the same as the amplitude of the generator emf: $$V_{L}=\varepsilon _{m}$$. Thus, $$\varepsilon _{m}=I\omega _{d}L$$ and
$$L=\frac{\varepsilon _{m}}{I\omega _{d}}=\frac{30.0V}{\left ( 620*10^{-3}A \right )\left ( 350rad/s \right )}=0.138H$$.
Note: The current in the circuit can be rewritten as,
$$i\left ( t \right )=I\sin \left ( \omega _{d}-\frac{3\pi}{4} \right )=I\sin \left ( \omega _{d}-\frac{\pi}{4}-\phi \right )$$.
where $$\phi =+\pi /2$$. In a purely inductive circuit, the current lags the voltage by $$90°$$.
For above figure, show that the average rate at which energy is dissipated in resistance $$R$$ is a maximum when $$R$$ is equal to the internal resistance $$r$$ of the $$ac$$ generator. (we tacitly assumed that $$r=0$$.)
This circuit contains no reactances, so $$\varepsilon _{rms}=I_{rms}R_{total}$$. Using equation $$P_{avg}=I_{rms}^{2}R$$, we find the average dissipated power in resistor $$R$$ is,
$$P_{R}=I_{rms}^{2}R=\left ( \frac{\varepsilon _{m}}{r+R} \right )^{2}R$$.
In order to maximize $$P_{R}$$ we set the derivative equal to zero:
$$\frac{dP_{R}}{dR}=\frac{\varepsilon _{m}^{2}\left [ \left ( r+R \right )^{2}-2\left ( r+R \right )R \right ]}{\left ( r+R \right )^{4}}=\frac{\varepsilon _{m}^{2}\left ( r-R \right )}{\left ( r+R \right )^{3}}=0\Rightarrow R=r$$.
An ac generator with emf amplitude $$\xi _{m}=220V$$ and operating at frequency $$400Hz$$ causes oscillations in a series RLC circuit having $$R=200\Omega$$, $$L=150 mH$$ and $$C=24.0\mu F$$. Find (a) the capacitive reactance $$X_{C}$$, (b) the impedance $$Z$$ and (c) the current amplitude $$I$$. A second capacitor of the same capacitance is then connected in series with the other components. Determine whether the values of (d) $$X_{C}$$, (e) $$Z$$ and (f) $$I$$ increase, decrease or remain the same.
(a) The capacitive reactance is,
$$X_{C}=\frac{1}{2\pi fC}=\frac{1}{2\pi \left ( 400Hz \right )\left ( 24.0\times10^{-6}F \right )}=16.6\Omega$$.
(b) The impedance is,
$$Z=\sqrt{R^{2}+\left ( X_{L}-X_{C} \right )^{2}}=\sqrt{R^{2}+\left ( 2\pi fL-X_{C} \right )^{2}}$$,
$$=\sqrt{\left ( 220\Omega\right )^{2}+\left [ 2\pi \left ( 400Hz \right )\left ( 150\times10^{-3}H \right )-16.6\Omega\right ]^{2}}=422\Omega$$.
(c) The current amplitude is,
$$I=\frac{\varepsilon _{m}}{Z}=\frac{220V}{422\Omega}=0.521A$$.
(d) Now $$X_{C}\propto C_{eq}^{-1}$$. Thus, $$X_{C}$$ increases as $$C_{eq}$$ decreases.
(e) Now $$C_{eq}=C/2$$, and the new impedance is,
$$Z=\sqrt{\left ( 220\Omega\right )^{2}+\left [ 2\pi \left ( 400Hz \right )\left ( 150*10^{-3}H \right ) -2\left ( 16.6\Omega \right )\right ]}^{2}=408\Omega <422\Omega$$
Therefore, the impedance decreases.
(f) Since $$I\propto Z^{-1}$$, it increases.
A long vertical wire carries an unknown current. Coaxial with the wire is a long, thin, cylindrical conducting surface that carries a current of $$30 mA$$ upward. The cylindrical surface has a radius of $$3.0 mm$$. If the magnitude of the magnetic field at a point $$5.0 mm$$ from the wire is $$1.0\mu T$$, what are the (a) size and (b) direction of the current in the wire?
$$Given$$ :- Radius of cylindrical surface $$ = 3× 10^{-3}\space m$$
Current in cylindrical conducting surface $$= 0.03 \space A$$
Magnetic field $$(B=1.0 × 10^{-6} \space T)$$ at distance $$5× 10^{-3}\space m$$ from wire .
$$To \space Find$$ :- Magnitude and direction of current $$(i)$$ in the wire
$$ Solution$$ :- Using Ampere's law ,
Integral of $$B.ds = \mu_0I_{enclosed}$$
$$B(2πr) = \mu_0 I_{enclosed}$$
$$\implies $$ $$I_{enclosed} = \dfrac{B (2πr)}{\mu_0}$$ $$( r = 5× 10^{-3} \space m)$$
$$= 0.025$$
Now , $$I_{enclosed} = 0.03 + i$$
$$\implies$$ $$\underline{i= -5.0 \space mA}$$
Hence , $$\underline{i = -5.0\space mA \space in \space downward \space direction .}$$
Figure shows a circuit having a coil of resistance $$R=2.5 \Omega$$ and inductance 'L' connected to a conducting rod PQ which can slide on a perfectly conducting circular ring of radius 10 cm with its centre at 'P'.
Assume that friction and gravity are absent and a constant uniform magnetic field of 5 T exists as shown in figure. At t =0,the circuit is switched on and simultaneously a time varying external torque is applied on the rod so that it rotates about 'P' with a constant angular velocity 40 rad/s.Find the magnitude of this torque ( in 'P') when current reaches half of its maximum value.Neglect the self inductance of the loop formed by the circuit.
Induced EMF =$$\dfrac{1}{2}B \omega l^2$$
Maximum current : $$i_0=\dfrac{B \omega l^2}{2R}$$
Torque about the hinge P is
$$ \tau =\int^l_0 i(dx)Bx$$
$$\Rightarrow \tau =\dfrac{1}{2}iBl^2$$
Putting $$i=i_0/2,$$
we get ; $$\tau =\dfrac{B^2 \omega l^4}{8R}=5 mNm$$
(a) In an $$RLC$$ circuit, can the amplitude of the voltage across an inductor be greater than the amplitude of the generator emf? (b) Consider an $$RLC$$ circuit with emf amplitude $$\xi _{m}=10V$$, resistance $$R=10\Omega$$, inductance $$L=1.0 H$$ and capacitance $$C=1.0\mu F$$. Find the amplitude of the voltage across the inductor at resonance.
(a) Yes, the voltage amplitude across the inductor can be much larger than the amplitude of the generator emf.
(b) The amplitude of the voltage across the inductor in an $$RLC$$ series circuit is given by
$$V_{L}=IX_{L}=I\omega L$$. At resonance, the driving angular frequency equals the natural angular frequency: $$\omega _{d}=\omega =1/\sqrt{LC}$$. For the given circuit,
$$X_{L}=\frac{L}{\sqrt{LC}}=\frac{1.0H}{\sqrt{\left ( 1.0H \right )\left ( 1.0\times10^{-6}F \right )}}=1000\Omega$$.
At resonance the capacitive reactance has this same value, and the impedance reduces simply: $$Z = R$$. Consequently,
$$I=\frac{\varepsilon _{m}}{Z}|_{resonance}=\frac{\varepsilon _{m}}{R}=\frac{10V}{10\Omega}=1.0A$$.
The voltage amplitude across the inductor is therefore
$$V_{L}=IX_{L}=\left ( 1.0A \right )\left ( 1000\Omega \right )=1.0\times10^{3}V$$.
which is much larger than the amplitude of the generator emf.
In Figure , a metal rod is forced to move with constant velocity $$\overrightarrow {v}$$ along two parallel metal rails, connected with a strip of metal at one end. A magnetic field of magnitude B = 0.350 T points out of the page.
(a) If the rails are separated by L = 25.0 cm and the speed of the rod is 55.0 cm/s, what emf is generated?
(b) If the rod has a resistance of $$18.0\Omega $$ and the rails and connector have negligible resistance, what is the current in the rod?
(c) At what rate is energy being transferred to thermal energy?
(a) Equation $$\mathscr{E}=\dfrac{d\Phi
_B}{dt}=\dfrac{d}{dt}BLx=BL\dfrac{dx}{dt}=BLv$$ leads to
$$\varepsilon=B L v=(0.350
\mathrm{T})(0.250 \mathrm{m})(0.55 \mathrm{m} / \mathrm{s})=0.0481 \mathrm{V}$$
(b) By Ohm's law, the induced current is
$$i=0.0481 \mathrm{V} / 18.0
\Omega=0.00267 \mathrm{A}$$
By Lenz's law, the current is clockwise in Figure.
(c) Equation $$P=i^2R\text{ (resistive dissipation) }$$ leads to $$P=i^{2}
R=0.000129 \mathrm{W}$$
Faraday's Law of Induction (10)
Scientific work is currently under way to determine whether weak oscillating magnetic fields can affect human health. For example, one study found that drivers of trains had a higher incidence of blood cancer than other railway workers, possibly due to long exposure to mechanical devices in the train engine cab. Consider a magnetic field of magnitude $$1.00 \times 10^{-3} \,T$$, oscillating sinusoidally at $$60.0 \,Hz$$. If the diameter of a red blood cell is $$8.00 \,mm$$, determine the maximum emf that can be generated around the perimeter of a cell in this field.
An $$ac$$ generator produces emf $$\xi =\xi _{m}\sin \left ( \omega _{d}t-\pi /4 \right )$$, where $$\xi _{m}=30.0V$$ and $$\omega _{d}=350rad/s$$. The current in the circuit attached to the generator is $$i\left ( t \right )=I\sin \left ( \omega _{d}t+\pi /4 \right )$$, where $$I=620 mA$$. (a) At what time after $$t=0$$ does the generator emf first reach a maximum? (b) At what time after $$t=0$$ does the current first reach a maximum? (c) The circuit contains a single element other than the generator. Is it a capacitor, an inductor, or a resistor? Justify your answer. (d) What is the value of the capacitance, inductance or resistance as the case may be?
(a) Let $$\omega t-\pi /4=\pi /2$$ to obtain $$t=3\pi /4\omega =3\pi /\left [ 4\left ( 350rad/s \right ) \right ]=6.73\times10^{-3}s$$.
(b) Let $$\omega t+\pi /4=\pi /2$$ to obtain $$t=\pi /4\omega =\pi /\left [ 4\left ( 350rad/s \right ) \right ]=2.24\times10^{-3}s$$.
(c) Since $$i$$ leads $$ε$$ in phase by $$π/2$$, the element must be a capacitor.
(d) We solve $$C$$ from $$X_{C}=\left ( \omega C \right )^{-1}=\varepsilon _{m}/I$$:
$$C=\frac{I}{\varepsilon _{m}\omega}=\frac{6.20\times10^{-3}A}{\left ( 30.0V \right )\left ( 350rad/s \right )}=5.90\times10^{-5}F$$.
The figure shows a loop
model (loop L) for a diamagnetic
material. (a) Sketch the magnetic
field lines within and about the material due to the bar magnet. What is the direction of (b) the loops net magnetic dipole moment $$\vec \mu$$,
(c) the conventional current i in the loop (clockwise or counterclockwise in the figure), and (d) the magnetic force on the loop?
(a) A sketch of the field lines (due to the presence of the bar magnet) in the vicinity of
the loop is shown below:
(b) The primary conclusion of Section 32-9 is two-fold: $$\vec u$$ is opposite to $$\vec B$$, and the
effect of $$\vec F$$ is to move the material toward regions of smaller |$$\vec B$$| values. The direction
of the magnetic moment vector (of our loop) is toward the right in our sketch, or in the +x
(c) The direction of the current is clockwise (from the perspective of the bar magnet).
(d) Since the size of |$$\vec B$$| relates to the "crowdedness" of the field lines, we see that $$\vec F$$ is
toward the right in our sketch, or in the +x direction.
A rectangular loop and a circular loop are moving out of a uniform magnetic field region to a field free region with a constant velocity. In which loop do you the induced emf to be a constant during the passage out of the field region? The field to the loop.
In rectangular coil the induced emf will remain constant because in this the case rate of change of area in the magnetic field region remains constant, while in circular coil the rate of change of area in the magnetic field region is not constant.
Answer the following question
A conduction rod of length $$'I'$$ with one end pivoted, is rotated with a uniform angular speed $$'\omega '$$ in a vertical plane, normal to a uniform magnetic field $$'B'$$. Deduce an expression for the emf induced in this rod.
If resistance of rod is $$R$$, what is the current induced in it?
Expression for Induced emf in a Rotating Rod
Consider a metallic rod $$OA$$ of length $$I$$ which is rotating with angular velocity $$'\omega '$$ in a uniform magnetic field $$B$$, the plane of rotation being perpendicular to the magnetic field. A rod may be supposed to be formed of a large number of small elements. Consider a small element of length $$dx$$ at a distance $$X$$ from centre. If $$V$$ is the linear velocity of this elements, then area swept by the element per second $$=v\ dx$$
The emf induced across the ends of element
$$d \varepsilon =B \dfrac{dA}{dt}=Bv\ dx$$
But $$v=x \omega $$
$$\therefore d \varepsilon = B \times \omega \ dx$$
$$\therefore$$ The emf induced across the rod
$$\varepsilon = \displaystyle \int _0 ^1 B\ x \omega \ dx = B \omega \int _0 ^1\ x\ dx$$
$$B \omega \left[\dfrac{x^2}{2}\right] =B \omega \left[ \dfrac{l^2}{2} - 0 \right]= \dfrac{1}{2} \ B \omega I^2$$
Current induced in rod $$I=\dfrac { \varepsilon }{R} = \dfrac{1}{2} \dfrac{B \omega l^2}{R}$$.
If circuit is closed, power dissipated $$=\dfrac { \varepsilon ^2 }{R} = \dfrac{B^2 \omega^2 l^2}{4R}$$.
A wheel with $$8$$ metallic spokes each $$50\ cm$$ long is rotated with a speed of $$12\ rev/min$$ in a plane normal to the horizontal component of the Earth's magnetic field. The Earth's magnetic field at the plane is $$0.4\ G$$ and the angle of dip is $$60^{o}$$. Calculate the emf induced between the axle and the rim of the wheel.
How will the value of emf be affected if the number of spokes were increasde?
If a rod of length $$I$$ rotates with angular speed $$\omega$$ in uniform magnetic field $$B$$
$$\varepsilon=\dfrac{1}{2}BI^{2}\omega$$
In case of earth's magnetic field $$B_{H}=|B_{e}|\cos \delta$$
and $$B_{V}=|B_{e}|\sin \delta$$
$$\therefore \varepsilon=\dfrac{1}{2}|B_{e}|\cos \delta.l^{2}\omega$$
$$=\dfrac{1}{2}\times 0.4\times 10^{-4}\cos 60^{o}\times (0.5)^{2}\times 2\pi v$$
$$=\dfrac{1}{2}\times 0.4\times 10^{-4}\times \dfrac{1}{2}\times (0.5)^{2}\times 2\pi \times \left(\dfrac{120\ rev}{60s}\right)$$
$$=10^{-5}\times 0.25\times 2\times 3.14\times 2$$'$$=3.14\times 10^{-5}$$ volt
Induced emf is independent of the number of spokes i.e. it remain same.
In Figure , a 120- turn coil of radius 1.8 cm and resistance $$5.3\Omega $$ is coaxial with a solenoid of 220 turns/cm and diameter 3.2 cm. The
solenoid current drops from 1.5 A to zero in time interval $$\Delta t=25ms$$. What current is induced in the coil during $$\Delta
$$?
The total induced emf is given by
$$\varepsilon =-N \dfrac{d
\Phi_{B}}{d t}=-N A\left(\dfrac{d B}{d t}\right)=-N A \dfrac{d}{d
t}\left(\mu_{0} n i\right)=-N \mu_{0} n A \dfrac{d i}{d t}=-N \mu_{0}
n\left(\pi r^{2}\right) \dfrac{d i}{d t} $$
$$=-(120)\left(4 \pi \times 10^{-7}
\mathrm{T} \cdot \mathrm{m} / \mathrm{A}\right)(22000 / \mathrm{m}) \pi(0.016
\mathrm{m})^{2}\left(\dfrac{1.5 \mathrm{A}}{0.025 \mathrm{s}}\right) $$
$$=0.16 \mathrm{V}$$
Ohm's law then yields $$i=|\varepsilon|
/ R=0.016 \mathrm{V} / 5.3 \Omega=0.030 \mathrm{A}$$
In above figure, a three-phase generator G produces electrical power that is transmitted by means of three wires. The electric potentials (each relative to a common reference level) are $$V_{1}=A\sin \omega _{d}t$$ for wire $$1$$, $$V_{2}=A\sin \left ( \omega _{d}t-120^{0}\right )$$ for wire $$2$$, and $$V_{3}=A\sin \left ( \omega _{d}t-240^{0}\right )$$ for wire $$3$$. Some types of industrial equipment (for example, motors) have three terminals and are designed to be connected directly to these three wires. To use a more conventional two-terminal device (for example, a lightbulb), one connects it to any two of the three wires. Show that the potential difference between any two of the wires (a) oscillates sinusoidally with angular frequency $$\omega _{d}$$ and (b) has an amplitude of $$A\sqrt{3}$$ .
(a) We consider the following combinations:
$$\Delta V_{12}=V_{1}-V_{2}$$, $$\Delta V_{13}=V_{1}-V_{3}$$ and $$\Delta V_{23}=V_{2}-V_{3}$$
For, $$\Delta V_{12}$$.
$$\Delta V_{12}=A\sin \left ( \omega _{d}t \right )-A\sin\left ( \omega _{d}t-120^{0} \right )=2A\sin \left ( \frac{120^{0}}{2} \right )\cos \left ( \frac{2\omega _{d}t-120^{0}}{2} \right )=\sqrt{3}A\cos \left ( \omega _{d}t-60^{0} \right )$$.
where we use $$\sin \alpha -\sin \beta=2\sin \left [ \left ( \alpha -\beta \right )/2 \right ]\cos \left [ \left ( \alpha +\beta \right )/2 \right ]$$.
and, $$\sin 60^{0}=\sqrt{3}/2$$. Similarly,
$$\Delta V_{13}=A\sin \left ( \omega _{d}t \right )-A\sin \left ( \omega _{d}t-240^{0} \right )=2A\sin \left ( \frac{240^{0}}{2} \right )\cos \left ( \frac{2\omega _{d}t-240^{0}}{2} \right )=\sqrt{3}A\cos \left ( \omega _{d}t-120^{0} \right )$$.
$$\Delta V_{23}=A\sin \left ( \omega _{d}t-120^{0} \right )-A\sin \left ( \omega _{d}t-240^{0} \right )=2A\sin \left ( \frac{120^{0}}{2} \right )\cos \left ( \frac{2\omega _{d}t-360^{0}}{2} \right )=\sqrt{3}A\cos \left ( \omega _{d}t-180^{0} \right )$$.
All three expressions are sinusoidal functions of $$t$$ with angular frequency $$\omega _{d}$$.
(b) We note that each of the above expressions has an amplitude of $$\sqrt{3}A$$.
A rod of length $$I$$ is moved horizontally with a uniform velocity $$'v'$$ in a direction perpendicular to its length through a region in which a uniform magnetic field is acting vertically downward. Derive the expression for the emf induced across the ends of the rod.
Suppose a rod of length $$I$$ moves with velocity $$v$$ inward in the region having uniform magnetic field $$B$$
Initial magnetic flux enclosed in the rectangular space is $$\varphi=|B|Ix$$
As the rod moves with velocity $$-v=\dfrac{dx}{dt}$$
Using Lenz's law
$$\varepsilon=-\dfrac{d\varphi}{dt}=-\dfrac{d}{dt}(Blx)=Bl\left(-\dfrac{dx}{dt}\right)$$
$$\therefore \varepsilon =Blv$$
The switch in Figure is closed on a at time t = 0.
What is the ratio $$\mathscr{E}_L/\mathscr{E}$$ of the inductor's self-induced emf to the battery's emf (a) just after t = 0 and
(b) at $$t = 2.00\tau
_L$$? (c) At what multiple of $$\tau
_L$$ will $$\mathscr{E}_L/\mathscr{E}=0.500$$?
(a) Immediately after the switch is closed, $$\varepsilon-\varepsilon_{L}=i R .$$
But $$i=0$$ at this instant, so $$\varepsilon_{L}=\varepsilon,$$ or $$\varepsilon_{L}
/ \varepsilon=1.00$$
(b) $$\varepsilon_{L}(t)=\varepsilon
e^{-t / \tau_{L}}=\varepsilon e^{-2.0 \tau_{L} / \tau_{L}}=\varepsilon
e^{-2.0}=0.135 \varepsilon,$$ or $$\varepsilon_{L} / \varepsilon=0.135$$
(c) From $$\varepsilon_{L}(t)=\varepsilon
e^{-t / \tau_{L}}$$ we obtain
$$\dfrac{t}{\tau_{L}}=\ln
\left(\dfrac{\varepsilon}{\varepsilon_{L}}\right)=\ln 2 \Rightarrow t=\tau_{L}
\ln 2=0.693 \tau_{L} \Rightarrow t / \tau_{L}=0.693$$
Five long wires A,B,C,D and E each carrying current I are arranged to form edges of a pentagonal prism as shown in Fig.4.6 Each carries out of the plane of paper.
(a) what will be magnetic induction at a point on the axis O Axis is at a distance R from each wire.
(b) What will be the field if current in one of the wires (say A) is switched off
(c) What if current in one of the wire (say) A is reversed
(a) Figure shows that five conductors AA' , BB' ,CC' , DD' and EE' are along height of regular pentagonal prism ABCDE.
It is given that five identical conducting wires are along the heights of regular pentagon, represented in figure above by AA',BB' , CC' , DD' and EE' .
Axis of regular pentagon is OO' will be equidistant (R) from all five conductors, the current is passing through all five conductors are equal let (I).
(b) when current in AA' is switched off, then $$ B_{1} $$
$$ = 0 $$and resultant becomes
$$ R= B_{2}+ B_{3} + B_{4} + B_{5} = 0$$
Figure shows a wire that has been bent into a circular arc of radius r = 24.0 cm, centered at O. A straight wire OP can be rotated about O and makes sliding contact with the arc at P. Another straight wire OQ completes the conducting loop. The three wires have cross-sectional area $$1.20mm^2$$ and resistivity $$1.70 \times 10^{-8} \Omega \cdot m$$, and the apparatus lies in a uniform magnetic field of magnitude B = 0.150 T directed out of the figure.Wire OP begins from rest at angle $$\theta =0$$ and has constant angular acceleration of $$12rad/s^2$$. As functions of $$\theta $$ (in rad), find (a) the loops resistance and (b) the magnetic flux through the loop. (c) For what $$\theta $$ is the induced current maximum and (d) what is that maximum?
A uniform magnetic field $$\overrightarrow {B}$$ is perpendicular to the plane of a circular wire loop of radius r. The magnitude of the field varies with time according to $$B=B_0e^{-t/\tau }$$, where $$B_0$$ and $$\tau$$ are constants. Find an expression for the emf in the loop as a function of time.
Faraday's law (for a single turn, with $$B$$ changing in time) gives
$$\varepsilon=-\dfrac{d
\Phi_{B}}{d t}=-\dfrac{d(B A)}{d t}=-A \dfrac{d B}{d t}=-\pi r^{2} \dfrac{d
B}{d t}$$
In this problem, we find $$\dfrac{d B}{d t}=-\dfrac{B_{0}}{\tau} e^{-t l \tau} .$$
Thus, $$\varepsilon=\pi r^{2} \dfrac{B_{0}}{\tau} e^{-t / \tau}$$
$$ODBAC$$ is a fixed rectangular conductor of negligible resistance $$(CO$$ is not connected) and $$OP$$ is a conductor which rotates clockwise with an angular velocity $$\omega$$ (Fig.). The entire system is in a uniform magnetic field $$B$$ whose direction is along the normal to the surface of the rectangular conductor $$ABDC$$. The conductor $$OP$$ is in electric contact with $$ABDC$$. The rotating conductor has a resistance of $$\lambda$$ per unit length. Find the current in the rotating conductor, as it rotates by $$180^{\circ}$$.
(i) Let rotating conductor is in contact with $$BD$$ at $$Q$$ making angle $$0^{\circ} < \theta < 45^{\circ}$$
Magnetic flux in $$\triangle ODQ$$ is $$\phi$$
$$\phi = B.A$$
The direction of $$B$$ and $$A$$ are $$0^{\circ}$$ or $$180^{\circ}$$
$$Q = B \times \dfrac {1}{2} l\times QD$$
$$\therefore \phi = B.A\cos 0 = BA = B\dfrac {1}{2} \times l.l \tan \theta \left [\because \dfrac {QD}{l} = \tan \theta \right ] Qd = l\tan \theta$$
$$\phi = \dfrac {1}{2} Bl^{2}\tan \theta = \dfrac {1}{2}Bl^{2} \tan \omega t (\because \theta = \omega t)$$
Induced e.m.f. $$\epsilon = \dfrac {-d\phi}{dt} = \dfrac {d}{dt} \dfrac {1}{2} Bl^{2} \tan \omega t$$
$$\epsilon = \dfrac {1}{2} Bl^{2}\omega \sec^{2} \omega t$$
$$I = \dfrac {\epsilon}{R} = \dfrac {Bl^{2}}{2R} \omega \sec^{2} \omega t\left [\because \cos \theta = \dfrac {1}{x}, x = \dfrac {1}{\cos \theta}\right ]$$
$$R$$ of $$OQ = \lambda x = \dfrac {\lambda l}{\cos \theta}$$
$$R = \dfrac {\lambda l}{\cos \omega t}$$
$$I = \dfrac {Bl^{2}}{2.\lambda l}\omega \cos \omega t\sec^{2} \omega t = \dfrac {Bl\omega}{2\lambda \cos \omega t}$$
(ii) Now rotating conductor rotates from $$B$$ to $$A$$ i.e., $$45^{\circ}$$ to $$135^{\circ}$$ or $$\dfrac {\pi}{4} < \theta < \dfrac {3\pi}{4}$$.
$$\phi = B . A = B$$ or $$ODBQ = B$$.
Area of $$\triangle ORQ = \dfrac {1}{2} y\times l$$
$$\tan \theta = \dfrac {l}{y}$$
$$y = \dfrac {l}{\tan \theta}$$
$$\therefore$$ Area of $$\triangle ORQ = \dfrac {1}{2} \dfrac {l^{2}}{\tan \theta} = \dfrac {l^{2}}{2\tan^{2}\omega t}$$
Flux through $$OQBD$$
$$\phi = B.\left (lz + \dfrac {l^{2}}{2\tan \theta}\right )$$
$$= Blz + \dfrac {1}{2} Bl^{2} \cos \theta = Blz + \dfrac {1}{2} Bl^{2} \cot \omega t$$
$$\therefore \dfrac {d\phi}{dt} = \dfrac {d}{dt} Blz + \dfrac {1}{2} Bl^{2} (-cosec^{2} \omega t)\omega$$
As $$l\ B$$ and $$z$$ are constant $$\therefore \dfrac {d(lBz)}{dt} = 0$$
$$\because \epsilon = \dfrac {-d\theta}{dt}$$
$$\therefore -\epsilon = 0 - \dfrac {1}{2} \dfrac {Bl^{2}\omega}{\sin^{2}\omega t}$$
$$\epsilon = \dfrac {1}{2} \dfrac {Bl^{2}}{\sin^{2}} \dfrac {\omega}{\omega t}$$
$$I = \dfrac {\epsilon}{R} = \dfrac {\epsilon}{\lambda x} = \dfrac {\epsilon \sin \omega t}{\lambda l}\left [\because \sin \theta = \dfrac {l}{x}, x = \dfrac {1}{\sin \omega t}\right ]$$
$$= \dfrac {\sin \omega t}{\lambda l} . \dfrac {1}{2} \dfrac {Bl^{2}\omega}{\sin^{2}\omega t}$$
$$I = \dfrac {Bl\omega}{2\lambda \sin \omega t}$$
(iii) When $$\dfrac {3\pi}{4} < \theta < \dfrac {2\pi}{2}$$, the flux through $$OQABD = \phi = B.A$$
$$\phi = B.\left (2l^{2} + \dfrac {1}{2} ly\right )$$
$$\phi = B . \left (2l^{2} + \dfrac {l^{2}\tan \omega t}{2}\right ) .... \begin{pmatrix}\tan (180 -\theta) = \dfrac {y}{l}\\y = l(-\tan \theta) \\y = -l \tan \theta
\end{pmatrix}$$
$$\dfrac {d\phi}{dt} = \dfrac {d}{dt} \left [2l^{2} + \dfrac {l^{2}}{2} (\tan \omega t)\right ]B$$
$$-\epsilon = 0 + \dfrac {Bl^{2}}{2} \dfrac {d}{dt}\tan \omega t$$
$$-\epsilon = + \dfrac {Bl^{1}\omega}{2} \sec^{2} \omega t = -\dfrac {Bl^{2}\omega}{2\cos^{2} \omega t}$$
$$I = \dfrac {\epsilon}{R} = \dfrac {\epsilon}{\lambda x}$$
$$I = \dfrac {-Bl^{2}\omega}{2\cos^{2}\omega t}\dfrac {\cos \omega t}{\lambda $$(-1)} \left [\because \dfrac {l}{x} $$
$$= \cos (180 -\theta) \dfrac {1}{x} $$
$$= -\cos \theta \Rightarrow x = \dfrac {-1}{\cos \omega t\right ]$$
$$I = \dfrac {Bl\omega}{2\lambda \cos \omega t}$$.
A long solenoid $$'S'$$ has $$n$$ turns per meter, with diameter $$'a'$$. At the centre of this coil we place a smaller coil of $$'N'$$ turns and diameter $$'b'$$ (where $$b < a$$). If the current in the solenoid increases linearly, with time, what is the induced emf appearing in the smaller coil. Plot graph showing nature of variation in emf, if current varies as a function of $$mt^{2} + c$$.
Varying magnetic field $$B(t)$$ in solenoid is
$$B_{1}(t) = \mu_{0} nI(t)$$
This varying magnetic field changes flux in the smaller coil.
Magnetic flux in IInd coil
$$= \mu_{0}nI (t) . \pi b^{2}$$
Induced e.m.f. in second coil due to solenoid's varying magnetic field in $$1$$ turn
$$\epsilon' = \dfrac {-d\phi_{2}}{dt} = \dfrac {-d}{dt} \mu_{0} n \pi b^{2} I(t)$$
$$= -\mu_{0} n\pi b^{2} \dfrac {d}{dt} (mt^{2} + C)$$
$$= -\mu_{0} n \pi b^{2} . 2mt$$
So net e.m.f. produced in $$N$$ turns of smaller coil
$$\epsilon = -\mu_{0} Nn\pi b^{2} 2mt$$.
(i)Two circular coils P and Q are kept close to each other, of which coil P carries a current. If coil P is moved towards Q, will some current be induced in coil Q? Give reason for your answer and name the phenomenon involved.
(ii) What happens if coil P is moved away from Q?
(iii) State any two methods of inducing current in a coil.
(i) When coil P is moved towards Q, current will be induced in coil Q. This is because on moving P the magnetic field associated with Q increases and so a current is induced. The Phenomenon is electromagnetic induction.
(ii) If P is moved away from Q, the field associated with Q will decrease and a current will be induced but in the opposite direction
(iii) Current can be induced in a coil by (a) moving a magnet towards or away from the coil (b) moving a coil towards or away from a magnet (c) rotating a coil within a magnetic field.
Two solenoids are part of the spark coil of an automobile. When the current in one solenoid falls from 6.0 A to zero in 2.5 ms, an emf of 30 kV is induced in the other solenoid. What is the mutual inductance M of the solenoids?
Given: current in solenoids falls from 6 A to 0 in just 2.5 ms, emf=30 kV
We use $$\varepsilon_{2}=-M
d i_{1} / d t \approx M \Delta i / \Delta t |$$ to find $$M$$
$$M=\left|\dfrac{\varepsilon}{\Delta
i_{1} / \Delta t}\right|=\dfrac{30 \times 10^{3} \mathrm{V}}{6.0 \mathrm{A}
/\left(2.5 \times 10^{-3} \mathrm{s}\right)}=13 \mathrm{H}$$
Figure shows a uniform magnetic field $$\overrightarrow {B}$$ confined to a cylindrical volume of radius R. The magnitude of $$\overrightarrow {B}$$ is decreasing at a constant rate of 10 mT/s. In
unit-vector notation, what is the initial acceleration of an electron released at (a) point a (radial distance r = 5.0 cm), (b) point b (r = 0), and (c) point c (r = 5.0 cm)?
The induced electric field is given by Eq. $$\oint \vec{E} \cdot d\vec{s}=-\dfrac{d\Phi
_B}{dt}\text{ (Faraday's Law) }$$:
$$\oint
\vec{E} \cdot d \vec{s}=-\dfrac{d \Phi_{B}}{d t}$$
The electric field lines are circles that are concentric with the cylindrical
region. Thus,
$$E(2 \pi
r)=-\left(\pi r^{2}\right) \dfrac{d B}{d t} \Rightarrow E=-\dfrac{1}{2} \dfrac{d
B}{d t} r$$
on the electron is $$\vec{F}=-e \vec{E}$$, so by Newton's second law, the acceleration is
$$\vec{a}=-e
\vec{E} / m$$
(a) At point $$a$$
$$E=-\dfrac{r}{2}\left(\dfrac{d
B}{d t}\right)=-\dfrac{1}{2}\left(5.0 \times 10^{-2} \mathrm{m}\right)\left(-10
\times 10^{-3} \mathrm{T} / \mathrm{s}\right)=2.5 \times 10^{-4} \mathrm{V} /
With the normal taken to be into the page, in the direction of the magnetic field, the positive direction for $$\vec{E}$$ is clockwise. Thus, the direction of the electric field at point a is to the left, that is $$\vec{E}=-\left(2.5 \times
10^{-4} \mathrm{V} / \mathrm{m}\right)$$ i. The resulting acceleration is
$$\vec{a}_{a}=\dfrac{-e
\vec{E}}{m}=\dfrac{\left(-1.60 \times 10^{-19} \mathrm{C}\right)\left(-2.5
\times 10^{-4} \mathrm{V} / \mathrm{m}\right)}{\left(9.11 \times 10^{-31}
\mathrm{kg}\right)} \hat{\mathrm{i}}=\left(4.4 \times 10^{7} \mathrm{m} /
\mathrm{s}^{2}\right) \hat{\mathrm{i}}$$
The acceleration is to the right.
(b) At point $$b$$ we have $$r_{b}=0,$$ so the acceleration is zero.
(c) The electric field at point $$c$$ has the same magnitude as the field in $$a$$, but
with its direction reversed. Thus, the acceleration of the electron released at
point $$c$$ is
$$\vec{a}_{c}=-\vec{a}_{a}=-\left(4.4
\times 10^{7} \mathrm{m} / \mathrm{s}^{2}\right) \hat{\mathrm{i}}$$
In Figure a, a circular loop of wire is concentric with a solenoid and lies in a plane perpendicular to the solenoid's central axis. The loop has radius 6.00 cm. The solenoid has radius 2.00 cm, consists of 8000 turns/m, and has a current $$i_{sol}$$ varying with time t as given in Figure b, where the vertical axis scale is set by $$i_s=1.00A$$ and the horizontal axis scale is set by $$t_s=2.0s$$. Figure c shows, as a function of time, the energy $$E_{th}$$ that is transferred to thermal energy of the loop; the vertical axis scale is set by $$E_s=100.0nJ$$.What is the loop's resistance?
Equation $$P=\dfrac{V^2}{R}$$
gives $$\varepsilon^{2} / R$$ as the rate of energy transfer into thermal forms $$\left(d E_{\mathrm{th}} / d t\right.$$ which, from Figure c , is roughly $$40
\mathrm{nJ} / \mathrm{s}$$ ). Interpreting $$\varepsilon$$ as the induced emf (in absolute value) in the single-turn loop $$(N=1)$$ from Faraday's law, we have
$$\varepsilon=\dfrac{d
\Phi_{B}}{d t}=\dfrac{d(B A)}{d t}=A \dfrac{d B}{d t}$$
Equation $$B=\mu _0in$$ gives $$B=\mu_{0} n i$$ for the solenoid (and note that the field is zero outside of the solenoid, which implies that $$A=A_{\text {coil }}$$, so our expression for the magnitude of the induced emf becomes
$$\varepsilon=A
\dfrac{d B}{d t}=A_{\text {coil }} \dfrac{d}{d t}\left(\mu_{0} n_{\text { coil }}\right)=\mu_{0}
n A_{\text { coil }} \dfrac{d i_{\text {coil }}}{d t}$$
where Figure b, suggests that $$d_{\mathrm{ coil }} / d t=0.5 \mathrm{A} / \mathrm{s} .$$ With $$n=8000$$ (in SI units) and $$A_{\mathrm{coil}}$$ $$=\pi(0.02)^{2}$$ (note that the loop radius does not come into the computations of this problem, just the coil's), we find $$\mathrm{V}=6.3 \mu \mathrm{V}$$. Returning to our earlier observations, we can now solve for the resistance:
$$R=\varepsilon^{2}
/\left(d E_{\mathrm{th}} / d t\right)=1.0 \mathrm{m} \Omega$$
Describe the principle , construction and working of a single phase A.C. generator.
Working Principle: Single phase AC generator operates on Faraday's theory of electromagnetic induction which states that change in a magnetic field produces an electric current.
It consists of 2-poles of a magnet in order to have a uniform magnetic field.
There is a rectangular shaped coil called an armature.
The armature is connected to 2 sets of slip rings, which helps in maintaining contact with the brushes.
These slip rings are attached to the carbon brushes.
The rotational motion of armature is free as its axis is perpendicular to the magnetic field.
Working:
Consider rectangular coil has 'N' no. of turns in a magnetic field rotating with $$\omega$$ angular velocity.
Maximum flux linked with the coil when the coil's plane normal is aligned with magnetic field lines.
In time 't' seconds, coil rotates an angle $$\theta=\omega t$$.
In this deflected position, the component of flux which is perpendicular to the plane of the coil is :
$$\phi=\phi_mcos(\omega t)$$
Hence, the flux linkage at any time 't' is:
$$\phi'=N\phi=N\phi_m\ cos(\omega t)$$
Now, as we know that induced potential is:
$$e=-\dfrac{d\phi'}{dt}$$
$$=-\dfrac{d(N\phi_m\ cos(\omega t))}{dt}$$
$$=-N\phi_m\ [-sin(\omega t)\cdot\omega]$$
$$\boxed{e=\omega N\phi_m\ sin(\omega t)}$$
$$i=\dfrac eR$$
So, $$i=\dfrac{\omega\ N\phi_m}{R}\ sin(\omega t)$$
$$\boxed{i=i_0\ sin(\omega t)}$$, where $$i_0=\dfrac{\omega N\phi_m}{R}$$
A $$\Pi-shaped$$ conductor is located in a uniform magnetic field perpendicular to the plane of the conductor and varying with time at the rate $$\overset {\cdot}{B} = 0.10\ T/s$$. A conducting connector starts moving with an acceleration $$w = 10\ cm/s^{2}$$ along the parallel bars of the conductor. The length of the connector is equal to $$l = 20\ cm$$. Find the emf induced in the loop $$t = 2.0\ s$$ after the beginning of the motion, if at the moment $$t = 0$$ the loop area and the magnetic induction are equal to zero. The inductance of the loop is to be neglected.
The flux through the loop changes due to the variation in $$\vec {B}$$ with time and also due to the movement of the conductor.
So, $$\xi_{in} = \left |\dfrac {d(\vec {B} \cdot \vec {S})}{dt}\right | = \left |\dfrac {d(BS)}{dt}\right |$$ as $$\vec {S}$$ and $$\vec {B}$$ are colliniear
But, $$B$$, after $$t$$ sec. of beginning of motion $$= Bt$$, and $$S$$ becomes $$= l\dfrac {1}{2} wt^{2}$$, as connector starts moving from rest with a constant acceleration $$w$$.
So, $$\xi_{ind} = \dfrac {3}{2} B\ l\ w\ t^{2}$$.
In a Faraday disc dynamo, a metal disc of radius $$R$$ rotates with an angular velocity $$\omega$$ about an axis perpendicular to the plane of the disc and passing through its center. The disc is placed in a magnetic field $$B$$ acting perpendicular to the plane of the disc. Determine the induced emf between the rim and the axis of the disc.
$$A=\pi R^{2}$$
$$\frac{dA}{dT}=\frac{\pi R^{2}}{\frac{2\pi}{W}}=\frac{WR^{2}}{2}$$
emf $$\epsilon =-B*\frac{dA}{dt}=\frac{1}{2}(B\omega R^{2})$$.
A long straight wire carrying a current $$I$$ and a $$\Pi-shaped$$ conductor with sliding connector are located in the same plane as shown in Fig. The connection of length $$l$$ and resistance $$R$$ slides to the right with a constant velocity $$v$$. Find the current induced in the loop as a function of separation $$r$$ between the connector and the straight wire. The resistance of the $$\Pi-shaped$$ conductor and the self-inductance of the loop are assumed to be negligible.
Field, due to the current carrying wire in the region, right to it, is directed into the plane of the paper and its magnitude is given by,
$$B = \dfrac {\mu_{0}}{2\pi}\dfrac {i}{r}$$ where $$r$$ is the perpendicular distance from the wire.
As $$B$$ is same along the length of the rod thus motional e.m.f.
$$\xi_{in} = \left |-\int_{1}^{2} (\vec {v}\times \vec {B})\cdot d\vec {l}\right | = vBl$$
and it is directed in the sense of $$(\vec {v} \times \vec {B})$$
So, current (induced) in the loop,
$$i_{in} = \dfrac {\xi_{in}}{R} = \dfrac {1}{2} \dfrac {\mu_{0}Ivi}{\pi R r}$$.
Above figure shows a metal rod $$PQ$$ resting on the smooth rails $$AB$$ and positioned between the poles of a permenent magnet. The rails, the rod, and the magnetic filed are in three mutual perpendicular directions. A galvanometer $$G$$ connects the rail through a switch $$K$$. Length of the rod is $$15cm$$, $$B=0.50T$$, resistanbce of the closed-loop containing the rod is $$9.0$$. Assume the filed to be uniform.
(a) Suppose $$K$$ is open and the rod is moved with a speed of $$12cm$$ $$s^{-1}$$ in the direction shown. Give the polarity and magnitude of the induced emf.
(b) Is there an excess charge build up at the ends of the rods when $$K$$ is open? What is $$K$$ is closed?
(c) With $$K$$ open and the rod moving uniformly, there is no net force on the electrons in the rod $$PQ$$ even through they do experience magnetic force due to the motion of the rod. Explain.
(d) What is the retarding force on the rod when $$K$$ is closed?
(e) How much power is required (by an external agent) to keep the rod moving at the same speed ($$=12cm$$ $$s^{-1}$$) when $$K$$ is closed? How much power is required when $$K$$ is open?
(f) How much power is dissipated as heat in the closed-circuit? What is the source of this power?
(g) What is the induced emf in the moving rod if the magnetic field is parallel to the rails instead of being perpendicular?
Here, $$B=0.50T;l=15cm =15*10^{-2}m$$;
$$R=9.0mQ=9.0*10^{3}fl$$
(a) Now, $$e=BVl$$
Here, $$V=12cm$$ $$s^{-1}=12*10^{-2}ms^{-1}$$
$$\therefore e=0.50*12*10^{-2}*15*10^{2}=9*10^{-3}V$$
If $$q$$ is change on as electron, then the elctrons in the rod will experience magnetic Lorentz force $$-q[\underset{v}{\rightarrow}+\underset{B}{\rightarrow}]P.Q$$. Hence, the end $$P$$ of the rod will become positive and the end $$Q$$ will become negative.
(b) When the switch $$K$$ is open, the elctron collect at the end $$Q$$. Therefore, excess change is build up at the end $$Q$$. However, when the switch $$K$$ is closed, the accumulated charge at the end $$Q$$ flows through the circuit.
(c) The magnetic Lorentz force on electron is cancelled by the electronic force acting on it due to the electronic filed set up accross the two ends due to accumulation of positive and negative charges at the end $$P$$ and $$Q$$ respectively.
(d) Retarding force, $$F=BIl=0.50*\frac{e}{R}*15*10^{-2}$$
$$=0.50*\frac{9*10^{-3}}{9*10^{-3}}*15*10^{-2}=7.5*10^{-2}N$$.
(e) When the switch $$K$$ is closed, power required by theb external agent against the retarding force,
$$P=Fv=7.5*10^{-2}*12*10^{-2}=9*10^{-3}W$$.
(f) Power dissipated as heat,
$$\frac{e^{2}}{R}=\frac{[9*10^{-3}]^{2}}{9*10^{-3}}=9*10^{-3}W$$/
The sorce of this power is the power of the external agent against the retarding force,
(g) The motion of rod does not cut field lines, hence no induced e.m.f. is produced.
A long straight solenoid of cross-sectional diameter $$d = 5\ cm$$ and with $$n = 20$$ turns per one cm of its length has a round turn of copper wire of cross-sectional area $$S = 1.0\ mm^{2}$$ tightly put on its winding. Find the current flowing in the turn if the current in the solenoid winding is increased with a constant velocity $$I = 100\ A/s$$. The inductance of the turn is to be neglected.
The e.m.f. induced in the turn is $$\mu_{0} n\overset {\cdot}{I} \pi \dfrac {d^{2}}{4}$$
The resistance is $$\dfrac {\pi d}{S} \rho$$.
So, the current is $$\dfrac {\mu_{0} n\overset {\cdot}{I} S\ d}{4\rho} = 2\ mA$$, where $$\rho$$ is the resistivity of copper.
A current $$I$$ flows along a thin wire shaped as a regular polygon with $$n$$ sides which can be inscribed into a circle of radius $$R$$. Find the magnetic induction at the centre of the polygon. Analyse the obtained expression at $$n\rightarrow \infty$$.
As $$\angle AOB = \dfrac {2\pi}{n}, OC$$ or perpendicular distance of any segment from centre equals $$R\cos \dfrac {\pi}{n}$$. Now magnetic induction at $$O$$, due to the right current carrying element $$AB$$
$$= \dfrac {\mu_{0}}{4\pi} \dfrac {i}{R\cos \dfrac {\pi}{n}} 2\sin \dfrac {\pi}{n}$$
(From Biot-Savart's law, the magnetic field at $$O$$ due to any section such as $$AB$$ is perpendicular to the plane of the figure and has the magnitude.)
$$B = \int \dfrac {\mu_{0}}{4\pi} i \dfrac {dx}{r^{2}} \cos \theta = \int_{-\dfrac {\pi}{n}}^{\dfrac {\pi}{n}} \dfrac {\mu_{0}i}{4\pi} \dfrac {R\cos \dfrac {\pi}{n}\sec^{2} \theta d\theta}{R^{2} \cos \dfrac {2\pi}{n}\sec^{2} \theta} \cos \theta = \dfrac {\mu_{0} i}{4\pi} \dfrac {1}{R\cos \dfrac {\pi}{n}} 2\sin \dfrac {\pi}{n}$$
As there are $$n$$ number of sides and magnetic induction vectors, due to each side at $$O$$, are equal in magnitude and direction. So,
$$B_{0} = \dfrac {\mu_{0}}{4\pi} \dfrac {nr}{R\cos \dfrac {\pi}{n}} 2\sin \dfrac {\pi}{n}\cdot n$$
$$= \dfrac {\mu_{0}}{2\pi} \dfrac {ni}{R}\tan \dfrac {\pi}{n}$$ and for $$n\rightarrow \infty$$
$$B_{0} = \dfrac {\mu_{0}}{2} \dfrac {i}{R} \displaystyle Lt_{n\rightarrow \infty} \left (\dfrac {\tan \dfrac {\pi}{n}}{\pi/ n}\right ) = \dfrac {\mu_{0}}{2} \dfrac {i}{R}$$.
In the middle of a long solenoid there is a coaxial ring of square cross-section, made of conducting material with resistivity $$\rho$$. The thickness of the ring is equal to $$h$$, its inside and outside radii are equal to $$a$$ and $$b$$ respectively. Find the current induced in the ring if the magnetic induction produced by the solenoid varies with time as $$B = \beta t$$ where $$\beta$$ is a constant. The inductance of the ring is to be neglected.
Take an elementary ring of radius $$r$$ and width $$dr$$.
The e.m.f. induced in this elementary ring is $$\pi r^{2} \beta$$.
Now the conductance of this ring is.
$$d\left (\dfrac {1}{R}\right ) = \dfrac {hdr}{\rho 2\pi s}$$ so $$d\ I = \dfrac {h\ r\ dr}{2\rho} \beta$$
Integrating we get the total current,
$$I = \int_{a}^{b} \dfrac {hr\ dr}{2\rho} \beta = \dfrac {h\beta (b^{2} - a^{2})}{4\rho}$$.
Faraday's Law of Induction (7)
To monitor the breathing of a hospital patient, a thin belt is girded around the patient's chest. The belt is a $$200$$-turn coil. When the patient inhales, the area encircled by the coil increases by $$39.0 \,cm^2$$. The magnitude of the Earth's magnetic field is $$50.0 \mu T$$ and makes an angle of $$28.0^{o}$$ with the plane of the coil. Assuming a patient takes $$1.80 \,s$$ to inhale, find the average inducede mf in the coil during this time interval.
Conceptual Questions(5)
A circular loop of wire is located in a uniform and constant magnetic field. Describe how an emf can be induced in the loop in this situation.
A strong electromagnet produces a uniform magnetic field of $$1.60 \,T$$ over a cross-sectional area of $$0.200 m^2$$. A coil having $$200$$ turns and a total resistance of $$20.0 \Omega$$ is placed around the electromagnet. The current in the electromagnet is then smoothly reduced until it reaches zero in $$20.0 \,ms$$. What is the current induced in the coil?
A quadratic undeformabe superconducting loop of mass $$m$$ and side $$a$$ lies in a horizontal plane in a nonuniform magnetic field whose induction varies in space according to the law $${B}_{x}=-\alpha x$$, $${B}_{y}=0$$, $${B}_{z}=\alpha z+{B}_{0}$$ (Figure). The inductance of the loop is $$L$$. At the initial moment, the centre of the loop coincides with the origin $$O$$, and its sides are parallel to the $$x-$$ and $$y-$$ axes. The current in the loop is zero, and it is released.
How will it move and where will it be in time $$t$$ after the beginning of motion?
The magnetic flux across the surface bounded by the superconducting loop is constant. Indeed, $$\Delta \Phi / \Delta t$$, but $$\varepsilon =IR=0$$ (Since $$R=0$$)
and hence $$\Phi =const.$$
The magnetic flux through the surface bounded by the loop is the sum of the external magnetic flux and the flux of the magnetic field produced by the current $$I$$ passing through the loop. Therefore, the magnetic flux across the loop at any instant is
$$\Phi ={ a }^{ 2 }{ B }_{ 0 }+{ a }^{ 2 }\alpha z+LI$$
since $$\Phi={B}_{0}{a}^{2}$$ at the initial moment ($$z=0$$ and $$I=0$$) the current $$I$$ at any other instant will be determined by the relation
$$LI=-\alpha z{ a }^{ 2 },I=-\cfrac { \alpha z{ a }^{ 2 } }{ L } $$
The resultant force exerted by the magnetic field on the current loop is the sum of the forces acting on the sides of the loop which are parallel to the y-axis is
$$F=2a\left| \alpha x \right| I={ a }^{ 2 }\alpha I$$
and is directed along the z-axis
Therefore, the equation of motion for the loop has the form
$$\overset { \bullet \bullet }{ mz } =-mg+{ a }^{ 2 }\alpha I=-\cfrac { mg-{ a }^{ 4 }{ \alpha }^{ 2 }z }{ L } $$
This equation is similar to the equation of vibrations of a body of mass $$m$$ suspended on a spring of rigidity
$$k={ a }^{ 4 }{ \alpha }^{ 2 }/L$$
$$\overset { \bullet \bullet }{ mz } =-mg-kz$$
This analogy shows that the loop will perform harmonic oscillations along the z-axis near the equilibrium position determined by the condition
$$\cfrac { { a }^{ 4 }{ \alpha }^{ 2 }z }{ L } { z }_{ 0 }=-mg;{ z }_{ 0 }=-\cfrac { mgL }{ { a }^{ 4 }{ \alpha }^{ 2 } } $$
The frequency of these oscillations will be
$$\omega =\cfrac { { a }^{ 2 }\alpha }{ \sqrt { Lm } } $$
The coordinate of the loop in a certain time $$t$$ after the beginning of motion will be
$$\quad z=-\cfrac { mgL }{ { a }^{ 4 }{ \alpha }^{ 2 } } \left[ -1+\cos { \left( \cfrac { { a }^{ 2 }\alpha }{ \sqrt { Lm } } t \right) } \right] $$
A flat loop of wire consisting of a single turn of cross sectional area $$8.00 \,cm^2$$ is perpendicular to a magnetic field that increases uniformly in magnitude from $$0.500 \,T$$ to $$2.50 \,T$$ in $$1.00 \,s$$. What is the resulting induced current if the loop has a resistance of $$2.00 \,\Omega$$?
A direct current flowing through the winding of a long cylindrical solenoid of radius $$R$$ produces in it a uniform magnetic field of induction $$B$$. An electron flies into the solenoid along the radius between its turns (at right angles to the solenoid axis) at a velocity $$v$$ (figure). After a certain time, the electron deflected by the magnetic field leaves the solenoid.
Determine the time $$t$$ during which the electron moves in the solenoid.
The magnetic of the solenoid is directed along its axis. Therefore, the Lorentz force acting on the electron at any instant of time will lie in the plane perpendicular to the solenoid axis. Since the electron velocity at the initial moment is directed at right angles to the solenoid axis, the electron trajectory will lie in the plane perpendicular to the solenoid axis. The Lorentz force can be found from the formula $$F=evB$$
The trajectory of the electron in the solenoid is an arc of the circle whose radius can be determined from the relation $$evB=m{v}^{2}/r$$, whence
$$r=\cfrac { mv }{ eB } $$
The trajectory of the electron is shown in figure (top view), wher e$${O}_{1}$$ is the centre of the arc $$AC$$ described by the electron, $$v'$$ is the velocity at which the electron leaves the solenoid. the segments $$OA$$ and $$OC$$ are tangents to the electron trajectory at points $$A$$ and $$C$$. The angle between $$v$$ and $$v'$$ is obviously $$\phi =\angle A{ O }_{ 1 }C$$ since $$\angle OA{ O }_{ 1 }=\angle OC{ O }_{ 1 }$$
In order to find $$\phi$$, let us consider the right triangle $$OA{O}_{1}$$: side $$OA=R$$ and side $$A{O}_{1}=r$$. Therefore, $$\tan { \left( \phi /2 \right) } =r/r=eBR/(mv)$$. Therefore
$$\phi =2arc\tan { \left( \cfrac { eBR }{ mv } \right) } $$
Obviously, the magnitude of the velocity remains unchanged over the entire trajectory since the Lorentz force is perpendicular to the velocity at any instant. Therefore, the transit time of electron in the solenoid can be determined from the relation
$$t=\cfrac { r\phi }{ v } =\cfrac { m\phi }{ eB } =\cfrac { 2m }{ eB } arc\tan { \left( \cfrac { eBR }{ mv } \right) } $$
Magnetron is a device consisting of a filament of radius $$a$$ and a coaxial cylindrical anode of radius $$b$$ which are located in a uniform magnetic field parallel to the filament. An accelerating potential difference $$V$$ is applied between the filament and the anode. Find the value of magnetic induction at which the electrons leaving the filament with zero velocity reach the anode.
When a current $$I$$ flows along the axis, a magnetic field $$B_{\varphi} = \dfrac {\mu_{0}I}{2\pi \rho}$$ is set up where $$\rho^{2} = x^{2} + y^{2}$$. In terms of components,
$$B_{x} = -\dfrac {\mu_{0}Iy}{2\pi \rho^{2}}, B_{y} = \dfrac {\mu_{0}Ix}{2\pi \rho^{2}}$$ and $$B_{z} = 0$$
Suppose a p.d. $$V$$ is set up between the inner cathode and the outer anode. This means a potential function of the form
$$\varphi = V \dfrac {ln \rho/b}{ln a/b}, a > \rho > b$$,
as one can check by solving Laplace equation.
The electric field corresponding to this is,
$$E_{x} = -\dfrac {Vx}{\rho^{2} ln a/b}, E_{y} = \dfrac {Vy}{\rho^{2} ln a/b}, E_{z} = 0$$.
The equations of motion are,
$$\dfrac {d}{dt} mv_{x} = + \dfrac {|e|Vz}{\rho^{2} ln a/b} + \dfrac {|e|\mu_{0}I}{2\pi \rho^{2}} x\overset {\cdot}{z}$$
$$\dfrac {d}{dt}mv_{y} = + \dfrac {|e|Vy}{\rho^{2} ln a/b} + \dfrac {|e|\mu_{0}I}{2\pi \rho^{2}} y\overset {\cdot}{z}$$
and $$\dfrac {d}{dt} mv_{z} = -|e| \dfrac {\mu_{0}I}{2\pi \rho^{2}} (x\overset {\cdot}{x} + y \overset {\cdot}{y}) = -|e| \dfrac {\mu_{0}I}{2\pi} \dfrac {d}{dt} ln \rho$$
$$(-|e|)$$ is the charge on the electron.
Integrating the last equation,
$$mv_{z} = -|e| \dfrac {\mu_{0}I}{2\pi} ln \rho / a = m\overset {\cdot}{z}$$.
since $$v_{z} = 0$$ where $$\rho = a$$. We now substitute this $$\overset {\cdot}{z}$$ in the other equations to get
$$\dfrac {d}{dt} \left (\dfrac {1}{2} mv_{x}^{2} + \dfrac {1}{2} mv_{y}^{2}\right )$$
$$= \left [\dfrac {|e|V}{ln a/b} - \dfrac {|e|^{2}}{m} \left (\dfrac {\mu_{0} I}{2\pi}\right )^{2} ln \rho/ b\right ] \cdot \dfrac {x\overset {\cdot}{x} + y \overset {\cdot}{y}}{\rho^{2}}$$
$$= \left [\dfrac {|e|V}{ln \dfrac {a}{b}} - \dfrac {|e|^{2}}{m} \left (\dfrac {\mu_{0}I}{2\pi}\right )^{2} ln \dfrac {\rho}{b}\right ] \cdot \dfrac {1}{2\rho^{2}} \dfrac {d}{dt} \rho^{2}$$
$$= \left [\dfrac {|e|V}{ln \dfrac {a}{b}} - \dfrac {|e|^{2}}{m} \left (\dfrac {\mu_{0}I}{2\pi}\right )^{2} ln \dfrac {\rho}{b}\right ] \dfrac {d}{dt} ln \dfrac {\rho}{b}$$
Integrating and using $$v^{2} = 0$$, at $$\rho = b$$, we get,
$$\dfrac {1}{2} mv^{2} = \left [\dfrac {|e|V}{ln \dfrac {a}{b}} ln \dfrac {\rho}{b} - \dfrac {1}{2m} |e|^{2} \left (\dfrac {\mu_{0}I}{2\pi}\right )^{2} \left (ln \dfrac {\rho}{b}\right )\right ]$$
The RHS must be positive, for all $$a > \rho > b$$. The condition for this is,
$$V \geq \dfrac {1}{2} \dfrac {|e|}{m}\left (\dfrac {\mu_{0}I}{2\pi}\right )^{2} ln \dfrac {a}{b}$$
This differs from in $$(a \leftrightarrow b)$$ and the magnetic field is along the z-direction. Thus $$B_{x} = B_{y} = 0, B_{z} = B$$
Assuming as usual the charge of the electron to be $$-|e|$$, we write the equation of motion
$$\dfrac {d}{dt}mv_{x} = \dfrac {|e|V_{x}}{\rho^{2} ln \dfrac {b}{a}} - |e| B\overset {\cdot}{y}, \dfrac {d}{dt} mv_{y} = \dfrac {|e|V_{y}}{\rho^{2} ln \dfrac {b}{a}} + |e| B\overset {\cdot}{x}$$
and $$\dfrac {d}{dt} mv_{z} = 0\Rightarrow z = 0$$
The motion is confined to the plane $$z = 0$$. Eliminating $$B$$ from the first two equations,
$$\dfrac {d}{dt}\left (\dfrac {1}{2} mv^{2}\right ) = \dfrac {|e|V}{ln b/a} \dfrac {x\overset {\cdot}{x} + y\overset {\cdot}{y}}{\rho^{2}}$$
or, $$\dfrac {1}{2} mv^{2} = |e| V \dfrac {ln \rho/ a}{ln b/a}$$
so, as expected, since magnetic forces do not work,
$$v = \sqrt {\dfrac {2|e|V}{m}}$$, at $$\rho = b$$.
On the other hand, eliminating $$V$$, we also get,
$$\dfrac {d}{dt} m(xv_{y} - yv_{x}) = |e| B(x\overset {\cdot}{x} + y\overset {\cdot}{y})$$
i.e. $$(xv_{y} - yv_{x}) = \dfrac {|e|B}{2m} \rho^{2} + constant$$
The constant is easily evaluated, since $$v$$ is zero at $$\rho = a$$. Thus,
$$(xv_{y} - yv_{x}) = \dfrac {|e|B}{2m} (\rho^{2} - a^{2}) > 0$$
At $$\rho = b, (xy_{y} - yv_{x}) \leq vb$$
Thus, $$vb \geq \dfrac {|e|B}{2m} (b^{2} - a^{2})$$
or, $$B \leq \dfrac {2mb}{b^{2} - a^{2}} \sqrt {\dfrac {2|e|V}{m}} \times \dfrac {1}{|e|}$$
or, $$B \leq \dfrac {2b}{b^{2} - a^{2}} \sqrt {\dfrac {2mB}{|e|}}$$.
When a wire carries an AC current with a known frequency, you can use a Rogowski coil to determine the amplitude $$I_{max}$$ of the current without disconnecting the wire to shunt the current through a meter. The Rogowski coil, shown in Figure, simply clips around the wire. It consists of a toroidal conductor wrapped around a circular return cord. Let $$n$$ represent the number of turns in the toroid per unit distance along it. Let A represent the cross-sectional area of the toroid. Let $$I(t) = I_{max} \sin \omega t$$ represent the current to be measured.
(a) Show that the amplitude of the emf induced in the Rogowski coil is $$\varepsilon_{max}= \mu_0 n A\omega I_{max}$$.
(b) Explain why the wire carrying the unknown current need not be at the center of the Rogowski coil and why the coil will not respond to nearby currents that it does not enclose.
Review. Figure shows a bar of mass $$m =0.200 \,kg$$ that can slide without friction on a pair of rails separated by a distance $$l= 1.20 \,m$$ and located on an inclined plane that makes an angle $$\theta = 25.0^{o}$$ with respect to the ground. The resistance of the resistor is $$R = 1.00 \Omega$$ and a uniform magnetic field of magnitude $$B = 0.500 \,T$$ is directed downward, perpendicular to the ground, over the entire region through which the bar moves. With what constant speed $$v$$ does the bar slide along the rails?
A helicopter (Figure) has blades of length $$3.00 \,m$$, extending out from a central hub and rotating at $$2.00 \,rev/s$$. If the vertical component of the Earths magnetic field is $$50.0 \mu T$$, what is the emf induced between the blade tip and the center hub?
Class 12 Medical Physics Extra Questions
Alternating Current Extra Questions
Atoms Extra Questions
Current Electricity Extra Questions
Dual Nature Of Radiation And Matter Extra Questions
Electric Charges And Fields Extra Questions
Electromagnetic Induction Extra Questions
Electromagnetic Waves Extra Questions
Electrostatic Potential And Capacitance Extra Questions
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Semiconductor Electronics: Materials, Devices And Simple Circuits Extra Questions
Wave Optics Extra Questions | CommonCrawl |
\begin{definition}[Definition:Wave Profile]
Let $\phi$ be a disturbance which is propagated along the $x$-axis with velocity $c$.
At $t = 0$, let $\phi = \map f x$.
Then $\map f x$ is the '''wave profile''' of $\phi$.
\end{definition} | ProofWiki |
# Boundary conditions and flow control
## Exercise
Instructions:
1. Identify the types of boundary conditions that are commonly used in fluid dynamics simulations.
2. Explain the impact of flow control on the accuracy of fluid dynamics simulations.
### Solution
1. The types of boundary conditions that are commonly used in fluid dynamics simulations include:
- Dirichlet boundary condition: The flow velocity is prescribed at the boundary.
- Neumann boundary condition: The flow velocity normal to the boundary is prescribed.
- Periodic boundary condition: The fluid flow wraps around the computational domain.
- No-slip boundary condition: The flow velocity is zero at the boundary.
- Convective boundary condition: The flow velocity is determined by the fluid properties and the flow velocity at the boundary.
2. The impact of flow control on the accuracy of fluid dynamics simulations is crucial. Inaccurate flow control can lead to numerical instability, inadequate resolution of flow features, and inaccurate representation of the flow physics. Therefore, it is essential to choose appropriate flow control techniques based on the problem at hand.
# Fluid properties and equations of motion
Fluid properties, such as density, viscosity, and thermal conductivity, play a crucial role in fluid dynamics simulations. Understanding the relationship between these properties and the equations of motion is fundamental for accurate simulations.
## Exercise
Instructions:
1. Explain the importance of fluid properties in fluid dynamics simulations.
2. Derive the Navier-Stokes equations for incompressible fluid flow.
### Solution
1. Fluid properties are essential in fluid dynamics simulations as they determine the physical behavior of the fluid, including its resistance to flow, viscosity, and thermal properties. Accurate representation of these properties is crucial for obtaining accurate and meaningful results from fluid dynamics simulations.
2. The Navier-Stokes equations for incompressible fluid flow can be derived from the conservation of momentum, mass, and energy. Consider a control volume with a fixed volume and a fixed mass. The continuity equation for mass conservation can be written as:
$$ \frac{\partial \rho}{\partial t} + \nabla \cdot (\rho \mathbf{u}) = 0 $$
where $\rho$ is the fluid density, $\mathbf{u}$ is the fluid velocity, and $\nabla \cdot$ represents the divergence operator.
The momentum equation for momentum conservation can be written as:
$$ \frac{\partial \rho \mathbf{u}}{\partial t} + \nabla \cdot (\rho \mathbf{u} \otimes \mathbf{u}) + \nabla p = \nabla \cdot \tau + \mathbf{f} $$
where $\tau$ is the stress tensor, $\mathbf{f}$ is the body force, and $p$ is the pressure.
The energy equation for energy conservation can be written as:
$$ \frac{\partial (\rho e)}{\partial t} + \nabla \cdot (\rho h \mathbf{u}) = \nabla \cdot (\kappa \nabla T) + q $$
where $e$ is the specific internal energy, $h$ is the specific enthalpy, $T$ is the temperature, $\kappa$ is the thermal conductivity, and $q$ is the heat source.
# Numerical methods for fluid dynamics simulation
Finite difference methods discretize the continuous fluid flow equations into a grid of points and use finite difference approximations to approximate the derivatives. This method is suitable for solving problems with complex geometries and flow conditions.
## Exercise
Instructions:
1. Explain the advantages and disadvantages of finite difference methods in fluid dynamics simulation.
2. Describe the basic steps of implementing a finite difference method for fluid dynamics simulation in Python.
### Solution
1. Finite difference methods have several advantages and disadvantages:
- Advantages: They are computationally efficient, can handle complex geometries and flow conditions, and can be easily implemented in Python.
- Disadvantages: They may lead to numerical instability for certain flow conditions and geometries, and they can be sensitive to the choice of grid spacing.
2. To implement a finite difference method for fluid dynamics simulation in Python, follow these steps:
1. Define the problem domain and discretize it into a grid of points.
2. Formulate the fluid dynamics equations using the finite difference approximations.
3. Implement the time-stepping algorithm to solve the equations.
4. Post-process the results to visualize and analyze the fluid flow.
# Solver algorithms for fluid dynamics problems
Direct methods, such as the explicit Euler method and the implicit Euler method, are used to solve the fluid dynamics equations directly. They are computationally efficient but may lead to numerical instability.
## Exercise
Instructions:
1. Explain the advantages and disadvantages of direct methods in fluid dynamics simulation.
2. Describe the basic steps of implementing a direct method for fluid dynamics simulation in Python.
### Solution
1. Direct methods have several advantages and disadvantages:
- Advantages: They are computationally efficient and can be easily implemented in Python.
- Disadvantages: They may lead to numerical instability for certain flow conditions and geometries, and they can be sensitive to the choice of time step.
2. To implement a direct method for fluid dynamics simulation in Python, follow these steps:
1. Define the problem domain and discretize it into a grid of points.
2. Formulate the fluid dynamics equations using the chosen direct method.
3. Implement the time-stepping algorithm to solve the equations.
4. Post-process the results to visualize and analyze the fluid flow.
# Implementing the simulation in Python
To implement a fluid dynamics simulation in Python, follow these steps:
1. Choose the appropriate numerical method and solver algorithm based on the problem at hand.
2. Define the problem domain and discretize it into a grid of points.
3. Formulate the fluid dynamics equations using the chosen numerical method and solver algorithm.
4. Implement the time-stepping algorithm to solve the equations.
5. Post-process the results to visualize and analyze the fluid flow.
## Exercise
Instructions:
1. Explain the advantages of using Python for fluid dynamics simulation.
2. Describe the basic steps of implementing a fluid dynamics simulation in Python.
### Solution
1. The advantages of using Python for fluid dynamics simulation include:
- Versatility: Python is a versatile programming language that can be easily used for various types of simulations.
- Ease of implementation: Python has a simple syntax and a large number of libraries and packages that can be used for scientific computing and visualization.
- Open-source and community-driven: Python has a large and active community that contributes to its development and supports its use in various fields, including fluid dynamics simulation.
2. To implement a fluid dynamics simulation in Python, follow these steps:
1. Choose the appropriate numerical method and solver algorithm based on the problem at hand.
2. Define the problem domain and discretize it into a grid of points.
3. Formulate the fluid dynamics equations using the chosen numerical method and solver algorithm.
4. Implement the time-stepping algorithm to solve the equations.
5. Post-process the results to visualize and analyze the fluid flow.
# Applications of fluid dynamics simulation
Aerodynamics is the study of the motion of air and the forces acting on objects in motion relative to the air. Fluid dynamics simulation plays a crucial role in the design and analysis of aerodynamic structures, such as aircraft wings and wind turbine blades.
## Exercise
Instructions:
1. Explain the importance of fluid dynamics simulation in aerodynamics.
2. Describe the basic steps of implementing an aerodynamic simulation in Python.
### Solution
1. Fluid dynamics simulation is essential in aerodynamics as it helps in accurately modeling the flow of air around objects and determining the forces acting on them. This information is crucial for the design and optimization of aerodynamic structures, leading to improved performance and efficiency.
2. To implement an aerodynamic simulation in Python, follow these steps:
1. Define the problem domain and discretize it into a grid of points.
2. Formulate the fluid dynamics equations using the chosen numerical method and solver algorithm.
3. Implement the time-stepping algorithm to solve the equations.
4. Post-process the results to visualize and analyze the flow features and forces acting on the object.
# Case studies in fluid dynamics and simulation
One such case study is the simulation of turbulent flow in a wind tunnel. By accurately modeling the flow features and the turbulence characteristics, fluid dynamics simulation can help in the design and optimization of wind turbine blades, leading to improved energy production and reduced environmental impact.
## Exercise
Instructions:
1. Describe the significance of fluid dynamics simulation in the case study of turbulent flow in a wind tunnel.
2. Discuss the potential applications of fluid dynamics simulation in other fields, such as oceanography and geophysics.
### Solution
1. The significance of fluid dynamics simulation in the case study of turbulent flow in a wind tunnel is that it helps in accurately modeling the flow features and the turbulence characteristics. This information is crucial for the design and optimization of wind turbine blades, leading to improved energy production and reduced environmental impact.
2. Fluid dynamics simulation has potential applications in various fields, including oceanography and geophysics. In oceanography, fluid dynamics simulation can be used to model the flow of ocean currents, the behavior of tides and waves, and the spreading of pollutants in the marine environment. In geophysics, fluid dynamics simulation can be used to model the flow of magma in volcanic systems, the behavior of ice sheets, and the spreading of earthquakes.
# Conclusion and further reading
In this textbook, we have discussed the fundamentals of fluid dynamics simulation and its applications in various fields. The knowledge and skills acquired in this textbook will enable learners to apply fluid dynamics simulation to a wide range of problems and contribute to the advancement of scientific computing and engineering.
For further reading, we recommend the following resources:
- "Fluid Mechanics" by Frank M. White
- "Numerical Methods for Engineers" by Steven C. Chapra and Raymond P. Canale
- "Fluid Dynamics: Theoretical and Computational Aspects" by John D. Anderson
- "Python for Scientists and Engineers" by Stéfan van der Walt and Jarrod Millman
By exploring these resources, learners can deepen their understanding of fluid dynamics and its applications in various fields. | Textbooks |
\begin{document}
\thispagestyle{empty} \begin{center}
\LARGE
\textbf{Pfaffian Calabi-Yau threefolds, Stanley-Reisner schemes and mirror symmetry} \\
\Large
\large
\textbf{Ingrid Fausk} \\
\large
{\textsc{DISSERTATION PRESENTED FOR THE DEGREE\\
OF PHILOSOPHIAE DOCTOR\\
}}
\centerline{\includegraphics[width=4cm,height=4cm]{uio-logo.eps}}
\large
\textsc{DEPARTMENT OF MATHEMATICS\\
UNIVERSITY OF OSLO\\}
\text{April 2012} \end{center}
\begin{itemize} \item[1.] $H^i(X, \mathcal{O}_X) = 0$ for every $i$, $0 < i < d$, and
\item[2.] $K_X := \wedge^d \Omega_X^1 \cong \mathcal{O}_X$, i.e., the canonical bundle is trivial. \end{itemize} \noindent By the second condition and Serre duality we have
\begin{displaymath} \text{dim}H^0(X, K_X) = \text{dim}H^d(X, \mathcal{O}_X) = 1 \end{displaymath} i.e., the geometric genus of $X$ is 1.
Let $\Omega_X^p := \wedge^p \Omega_X^1$ and let $H^q(\Omega_X^p)$ be the $(p,q)$-th {\it Hodge cohomology group} of $X$ with {\it Hodge number} $h^{p,q}(X) := \text{dim}_{\mathbb{C}}H^q(\Omega_X^p)$. The Hodge numbers are important invariants of $X$. There are some symmetries on the Hodge numbers. By complex conjugation we have $H^q(\Omega_X^p) \cong H^p(\Omega_X^q)$ and by Serre duality we have $H^q(\Omega_X^p) \cong H^{d-q}(\Omega_X^{d-p})$. By the {\it Hodge decomposition}
\begin{displaymath} H^k(X, \mathbb{C}) \cong \textstyle\bigoplus\nolimits_{p + q = k} H^q(\Omega_X^p) \end{displaymath} \noindent we have
\begin{displaymath} h^k(X) = \underset{p + q = k} \sum h^{p,q}(X) = \underset{i = 0}{\overset{k}\sum} h^{i,k-i}(X) \ . \end{displaymath}
The topological {\it Euler characteristic} of $X$ is an important invariant. It is defined as follows
\begin{displaymath} \chi(X):= \underset{k=0}{\overset{2d}\sum} (-1)^k h^k(X) \ . \end{displaymath} The conditions for $X$ to be Calabi-Yau assert that $h^{i,0}(X) = 0$ for $0<i<d$ and that $h^{0,0}(X) = h^{d,0}(X) = 1$.
We consider Calabi-Yau manifolds of dimension 3 in this text, these are simply called {\it Calabi-Yau threefolds}. In this case the relevant Hodge numbers are often displayed as a {\it Hodge diamond}.
\begin{center} $h^{0,0}$\\ $h^{1,0}$ \ \ $h^{0,1}$ \\ $h^{2,0}$ \ \ $h^{1,1}$ \ \ $h^{0,2}$ \\ $h^{3,0}$ \ \ $h^{2,1}$ \ \ $h^{1,2}$ \ \ $h^{0,3}$\\ $h^{3,1}$ \ \ $h^{2,2}$ \ \ $h^{1,3}$ \\ $h^{3,2}$ \ \ $h^{2,3}$ \\ $h^{3,3}$ \\ \end{center} By the properties mentioned above, the Hodge diamond reduce to
\begin{center} 1\\ 0 \ \ 0 \\ 0 \ \ $h^{1,1}$ \ \ 0 \\ 1 \ \ $h^{2,1}$ \ \ $h^{1,2}$ \ \ 1 \\ 0 \ \ $h^{2,2}$ \ \ 0 \\ 0 \ \ 0 \\ 1\\ \end{center} \noindent with the equalities $h^{1,1} =h^{2,2}$ and $h^{1,2} = h^{2,1}$ as explained above. In this case, the Euler characteristic of $X$ is
$$\chi(X) = 2(h^{1,1}(X) - h^{1,2}(X))$$
Physicists have discovered a phenomenon for Calabi-Yau threefolds, known as {\it mirror symmetry}. This is conjectured to be a correspondence between families of Calabi-Yau threefolds $X$ and $X^{\circ}$ with the isomorphisms \[ H^q(X, \wedge ^p \Theta_X) \cong H^q(X^{\circ}, \Omega^p_{X^{\circ}})\] and vice versa, where $\Theta_X$ is the tangent sheaf of $X$. Since $\wedge^p \Theta_X $ is isomorphic to $\Omega^{3-p}_X$, this gives the numerical equality $h^{p,q}(X) = h^{p,3-q}(X^{\circ})$, and hence $\chi(X) = - \chi(X^{\circ})$, which we will verify for some examples in this thesis. These symmetries correspond to reflecting the Hodge diamond along a diagonal.
For trivial reasons, the mirror symmetry conjecture, as stated above, fails for the Calabi-Yau threefolds where $h^{2,1}(X) = 0$, since Calabi-Yau manifolds are K\"{a}hler, so $h^{1,1}(X)> 0$.
A {\it nonlinear sigma model} consists of a Calabi-Yau threefold $X$ and a {complexified K\"ahler class} $\omega = B + iJ$ on $X$, where $B$ and $J$ are elements of $H^2(X,\mathbb{R})$, with $J$ a K\"{a}hler class. The {\it moduli}, i.e. how one can deform the complex structure and the complexified structure $\omega$, is governed by $H^1(\Theta_X)$ and $ H^1(\Omega_X)$, respectively. The isomorphisms $H^1(\Theta_X) \cong H^1(\Omega_{X^{\circ}})$ and $H^1(\Theta_{X^{\circ}}) \cong H^1(\Omega_{X})$ give a local isomorphism between the complex moduli space of $X$ and the K\"ahler moduli space of $\omega^{\circ}$, and between the complex moduli space of $X^{\circ}$ and the K\"ahler moduli space of $\omega$. These local isomorphisms are collectively called the {\it mirror map}. A general reference on Calabi-Yau manifolds and mirror symmetry is the book by Cox and Katz~\cite{coxkatz}.
In this thesis we study projective Stanley-Reisner schemes obtained from triangulations of 3-spheres, i.e.~$X_0 := \text{Proj}(A_K)$ for $K$ a triangulation of a 3-sphere and $A_K$ its Stanley-Reisner ring. These schemes are embedded in $\mathbb{P}^n$ for various $n$. We obtain Calabi-Yau 3-folds by smoothing (when a smoothing exists) such Stanley-Reisner schemes.
The first mirror construction by Greene and Plesser for the general quintic hypersurface in $\mathbb{P}^4$ will be reviewed in Chapter 1.
In Chapter 2 we give a method for computing the Hodge number $h^{1,2}(\tilde{X})$ of a small resolution $\tilde{X} \rightarrow X$, where $X$ is a deformation of a Stanley-Reisner scheme $X_0$ with the only singularities of $X$ being nodes. We use results on cotangent cohomology, and a lemma by Kleppe~\cite{kleppe}, which in our case states that $T^1_X \cong T^1_{A,0}$ for $X = \text{Proj}(A)$, i.e. the module of embedded (in $\mathbb{P}^n$) deformations of $X$ is isomorphic to the degree 0 part of the module of first order deformations of the ring $A$. We compute the Hodge number $h^{1,2}(\tilde{X})$ as the dimension of the kernel of the evaluation morphism $T^1_{A,0} \rightarrow \oplus_i T^1_{A_i}$, where $A_i$ is the local ring of a node $P_i$. We use this method in the only non-smoothable example in Chapter 3, where we construct a Calabi-Yau 3-fold with $h^{1,2}(\tilde{X})= 86$ from a small resolution of a variety with one node.
Gr\"unbaum and Sreedharan~\cite{sreedharan} proved that there are 5 different combinatorial types of triangulations of the 3-sphere with 7 vertices. In Chapter 3 we compute the Stanley-Reisner schemes of these triangulations. They are Gorenstein and of codimension 3, and we use a structure theorem by Buchsbaum and Eisenbud~\cite{buchseisen} to describe the generators of the Stanley-Reisner ideal as the principal Pfaffians of its skew-symmetric syzygy matrix. This approach combined with results by Altmann and Christophersen~\cite{altchrCotangent} on deforming combinatorial manifolds, gives a method for computing the versal deformation space of the Stanley-Reisner scheme of such a triangulation. As we mentioned above, we get a non-smoothable Stanley-Reisner scheme in one case. In the four smoothable cases, we compute the Hodge numbers of the smooth fibers, following the exposition in \cite{roedland}. We also compute the auto\-morphism groups of the triangulations, and consider subfamilies invariant under this action.
R\o dland constructed in \cite{roedland} a mirror of the 3-fold in $\mathbb{P}^6$ of degree 14 generated by the principal pfaffians of a general $7\times 7$ skew-symmetric matrix with general linear entries, done by orbifolding. B\"ohm constructed in~\cite{boehm} a mirror candidate of the 3-fold in $\mathbb{P}^6$ of degree 13 generated by the principal pfaffians of a $5\times 5$ skew-symmetric matrix with general quadratic forms in one row (and column) and linear terms otherwise. This was done using tropical geometry. In Chapter 4 we describe how the R\o dland and B\"ohm mirrors are obtained from the triangulations in Chapter 3, and in Chapter 5 we verify that the Euler characteristic of the B\"ohm mirror candidate is what it should be.
In general, the mirror constructions we consider in this thesis are obtained in the following way. We consider the automorphism group $G := \text{Aut}(K)$ of the simplicial complex $K$. The group $G$ induces an action on $T^1_{X_0}$, the module of first order deformations of the Stanley-Reisner scheme $X_0$ in the following way. Since an element of $T^1_{X_0}$ is represented by a homomorphism $\phi \in \text{Hom} (I/I^2, A)$, an action of $g\in G$ can be defined by $(g \cdot \phi ) f = g \cdot \phi(g ^{-1}\cdot f)$, where $f \in I$ is a representative for a class in the quotient $I/I^2$.
There is also a natural action of the torus $(\mathbb{C}^*)^{n+1}$ on $X_0 \subset \mathbb{P}^{n}$ as follows. An element $\lambda = (\lambda_0,\ldots, \lambda_{n}) \in (\mathbb{C}^*)^{n+1}$ sends a point $(x_0,\ldots ,x_n) $ of $\mathbb{P}^n$ to $(\lambda_0x_0,\ldots , \lambda_n x_n)$. The subgroup $ \{ (\lambda,\ldots, \lambda) | \lambda \in \mathbb{C}^* \}$ acts as the identity on $\mathbb{P}^n$, so we have an action of the quotient torus $T_n := (\mathbb{C}^*)^{n+1}/\mathbb{C}^*$. Since $I_{X_0}$ is generated by monomials it is clear that $T_n$ acts on $X_0$.
We compute the family of first order deformations of $ X_0$. When the general fiber is smooth, we consider a subfamily, invariant under the action of $G$, where the general fiber $X_t$ of this subfamily has only isolated singularities. We compute the subgroup $H \subset T_n$ of the quotient torus which acts on this chosen subfamily, and consider the singular quotient $Y_t = X_t/H$. The mirror candidate of the smooth fiber is constructed as a crepant resolution of $Y_t$. In Chapter 4 we perform these computations in order to reproduce the R\o dland and B\"ohm mirrors.
In Chapter 5 we verify that the Euler characteristic of the B\"ohm mirror candidate is 120. This is as expected since the cohomology computations in Chapter 3 give Euler characteristic -120 for the original manifold obtained from smoothing the Stanley-Reisner scheme of the triangulation.
We compute the Euler characteristic of the B\"{o}hm mirror using toric geometry. A crepant resolution is constructed locally in 4 isolated $Q_{12}$ singu\-larities. These 4 singularities and two other points are fixed under the action of the group $G$, which is isomorphic to the dihedral group $D_4$. The subgroup $H$ of the quotient torus acting on the chosen subfamily is isomorphic to $\mathbb{Z}/13\mathbb{Z}$. Denote one of these singularities by $V$. The singularity is embedded in $\mathbb{C}^4/H$, which is represented by a cone $\sigma$ in a lattice $N$ isomorphic to $\mathbb{Z}^4$. A resolution $X_{\Sigma} \rightarrow \mathbb{C}^4/H$ corresponds to a regular subdivision of $\sigma$. This subdivision is computed using the Maple package convex~\cite{convex}, and it has 53 maximal cones which are spanned by 18 rays. The following diagram commutes, where $\widetilde{V}$ is the strict transform of $V$.
$$ \XY
\xymatrix@1{\widetilde{V}\,\ar@{^{(}->}[r] \ar[d] & X_{\Sigma}\ar[d] \\
V\, \ar@{^{(}->}[r] & \mathbb{C}^4/H }$$
Each ray $\rho$ in $\Sigma$, aside from the 4 generating the cone $\sigma$, determines an exceptional divisor $D_{\rho}$ in $X_{\Sigma}$. Hence there are 14 exceptional divisors in $X_{\Sigma}$. For every ray $\rho$, the exceptional divisor $D_{\rho}$ is a smooth, complete toric 3-fold and comes with a fan $\text{Star}(\rho)$ in a lattice $N(\rho)$ and a torus $T_{\rho}$ corresponding to these lattices. The subvariety will only intersect 10 of these exceptional divisors $D_{\rho}$. In 9 of these 10 cases the intersection is irreducible and in one case the intersection has 4 components, but one of these is the intersection with another exceptional divisor. All in all the exceptional divisor $E$ in $\tilde{V}$ has 12 components $E_1,\ldots,E_{12}$.
To compute the type of the components $E_i$, several different techniques are needed depending upon the complexity of $D_{\rho}$. In some cases the intersection $\tilde{V}\cap T_{\rho}$ is a torus. In some cases $D(\rho)$ is a locally trivial $\mathbb{P}^1$ bundle over a smooth toric surface. In some cases $E_i$ is an orbit closure in $X_{\Sigma}$ corresponding to a 2-dimensional cone in $\Sigma$. In one case we construct a polytope which has $\text{Star}(\rho)$ as its normal fan.
The space $E$ is a normal crossing divisor. We compute the intersection complex by looking at the various intersections $\tilde{V} \cap D_{\rho_1} \cap D_{\rho_2}$ and $\tilde{V} \cap D_{\rho_1} \cap D_{\rho_2} \cap D_{\rho_3}$, and we compute the Euler characteristic of $E$. For the two other quotient singularities we use the McKay correspondence by Batyrev \cite{bat} in order to find the euler characteristic. We put all this together in order to get the Euler characteristic of the resolved variety.
Computer algebra programs like Macaulay\,2~\cite{M2}, Singular~\cite{GPS05} and Maple \cite{maple} have been used extensively throughout my studies, partly for handling expressions with many parameters and getting overview, but also for proving results. The code is not always included, but it is hoped that enough information is provided in order for the computations to be verified by others.
\tableofcontents
\chapter{Preliminaries} \section{Simplicial Complexes and Stanley-Reisner schemes}\label{simplicial}
Throughout this thesis we will work over the field of complex numbers $\mathbb{C}$. We will first give some basic definitions. Let $[n] = \{ 0,\ldots,n \}$ be the set of all positive integers from 0 to $n$, and let $\Delta_n$ denote the set of all subsets of $[n]$. We view a {\it simplicial complex} as a subset $K$ of $\Delta_n$ with the property that if $f \in K$, then all the subsets of $f$ are also in $K$. The elements of $K$ are called {\it faces} of $K$. Let $p\in \Delta_n$. In the polynomial ring $R = \mathbb{C}[x_0,\ldots , x_n]$, let $x_p$ be defined as the monomial $\Pi_{i \in p} x_i$. We define the set of ``non-faces'' of $K$ to be the complement of K in $\Delta_n$, i.e. $M_K = \Delta_n \setminus K$. The {\it Stanley-Reisner ideal} $I_K$ is defined as the ideal generated by the monomials corresponding to the "non-faces" of $K$, i.e.
$$I_K = \displaystyle\langle x_p \in R \mid p \in M_K \displaystyle\rangle \,\,.$$ \noindent The {\it Stanley-Reisner ring} is defined as the quotient ring $A_K = R/I_K$. The projective scheme
$$\mathbb{P}(K) := \text{Proj}\, (A_K)$$ \noindent is called the {\it projective Stanley-Reisner scheme}.
We will need the following definitions. For an face $f \in K$, we define the {\it link} of $f$ in $K$ as the set
$$\text{link}\,(f,K) := \displaystyle\{ g \in K \mid g\displaystyle\cap f = \emptyset \text{\,\,and}\,\, g\cup f \in K \displaystyle\} \, .$$
We set $[K] \subset [n]$ to be the vertex set $[K] = \{ i \in [n] : {i} \in K \}$. The {\it closure} of $f$ is defined as $\overline{f} = \{g \in \Delta_n : g \subseteq f \}$. The {\it boundary} of $f$ is defined as $\partial f= \{ g \in \Delta_n : g \subset f \text{\,\,proper subset} \}$. The {\it join} of two complexes $X$ and $Y$ is defined by
$$X*Y = \displaystyle\{ f \displaystyle\sqcup g \mid f\in X \,\, g\in Y \displaystyle\}\,\, ,$$
\noindent where the symbol $\sqcup$ denotes disjoint union. The geometric realization of $K$, denoted $|K|$, is defined as
$$|K| := \displaystyle\{ \alpha : [n] \rightarrow [0,1] : \text{supp}(\alpha) \in X \text{\,and\,\,} \underset{i}{\textstyle\sum}\alpha(i) = 1\displaystyle\}\,\, ,$$
where $\text{supp}(\alpha) := \{ i: \alpha(i)\neq 0\}$ is the support of the function $\alpha$. The real number $\alpha(i)$ is called the $i$th {\it barycentric coordinate of $\alpha$}. One can define a metric topology on $K$ by defining the distance $d(\alpha, \beta)$ between two elements $\alpha$ and $\beta$ as
$$d(\alpha, \beta) = \sqrt{\sum_{i \in K}(\alpha(i) - \beta(i))^2}\,\, .$$ \noindent For a general reference on simplicial complexes, see the book by Spanier~\cite{spanier}.
The schemes $\mathbb{P}(K)$ are singular. In fact, $\mathbb{P}(K)$ is the union of projective spaces, one for each {\it facet} (maximal face) in the simplicial complex $K$, intersecting the same way as the facets intersect in $K$. The proof of this statement is combinatorial: Let $p \in \Delta_n$ be a set with the property that $p\cap q \neq \emptyset$ for all $q \in M_K$ and suppose also that $p \neq [n]$. Then the complement $p^c := [n] - p$ is a face of $K$, and $p^c \neq \emptyset$. Note that if $p$ is a minimal set with the property mentioned above, then $p^c$ is a facet. Recall that $x_p$ is defined as the monomial $x_p := \Pi_{i \in p} x_i$, and that the Stanley-Reisner ideal of $K$ is generated by the monomials $x_q$ with $q\in M_K$. If $x_i = 0$ for all $i \in p$, then all the monomials $x_q$ are zero, since each $x_q$ contains a factor $x_i$ when $p$ has the property mentioned above and $i\in p$. Hence the scheme $\mathbb{P}(K)$ is the union of projective spaces which are defined by such $p$, i.e. given by $x_i = 0$ for all $i \in p$. These projective spaces are of dimension $| p^c| -1$, and they are in one to one correspondence with the faces $p^c$.
We will now mention some special triangulations of spheres which will be of importance in this thesis. The most basic triangulation of the $n-1$-sphere is the boundary $\partial \Delta_n$ of the n-simplex $\Delta_n$ (more precisely, with the definition of boundary of a face given above, it is the boundary of the unique facet $[n] = \{0,\ldots,n\}$ of $\Delta_n$.) For $n = 1$ it is the union of two vertices. For $n = 2$ it is the boundary of a triangle, denoted $E_3$. All triangulations of $\mathbb{S}^1$ are boundaries of $n$-gons, denoted $E_n$, for $n\geq 3$. The boundary of the 3-simplex $\partial \Delta_3$ is the boundary of a regular pyramid. From now on, we will for simplicity omit the word "boundary", and we will denote the triangulations of spheres as triangles, n-gons, pyramids etc. Other basic triangulations of $\mathbb{S}^2$ are the suspension of the triangle $\Sigma E_3$ (double pyramid) and the {\it octahedron} $\Sigma E_4$ (double pyramid with quadrangle base). Let $C_{k}$ be the chain of $k$ 1-simplices, i.e. $\{ \{0, 1 \},\{1,2\}, \ldots ,\{k-1, k \} \}$. Let $\Delta_1$ be the set of all subsets of $\{ n-2,n-1 \}$. Then we define (the boundary of) the {\it cyclic polytope}, $\partial C(n,3)$, as the union $(\overline{C_{n-3}} * \partial \Delta_1) \cup J$, where $J$ is the join $\Delta_1 * \{\{0\},\{n-3\}\}$ (see the book by Gr\"unbaum \cite{grunbaum} for details).
\section{Deformation Theory}
Given a scheme $X_0$ over $\mathbb{C}$, a {\it family of deformations}, or simply a {\it deformation} of $X_0$ is defined as a cartesian diagram of schemes
$$\XY
\xymatrix@1{ X_0\ar[r]\ar[d] & \mathcal{X}\ar[d]^{\pi}\\ \text{Spec}(\mathbb{C})\ar[r] & S\\
}$$
where $\pi$ is a flat and surjective morphism and $S$ is connected. The scheme $S$ is called the {\it parameter space} of the deformation, and $\mathcal{X}$ is called the {\it total space}. When $S = \text{Spec} B$ with $B$ an artinian local $\mathbb{C}$-algebra with residue field $\mathbb{C}$ we have an {\it infinitesimal deformation}. If in addition the ring $B$ is the ring of dual numbers, $B = \mathbb{C}[\epsilon]/(\epsilon^2)$, the deformation is said to be of {\it first order}. A {\it smoothing} is a deformation where the general fiber $\mathcal{X}_t$ of $\pi$ is smooth. For a general reference on deformation theory, see e.g. the book by Hartshorne~\cite{hartshorneDef} or the book by Sernesi~\cite{sernesi}.
For a construction of the cotangent cohomology groups in low dimensions, see e.g. Hartshorne~\cite{hartshorneDef}, where {\it cotangent complex} and the cotangent cohomology groups $T^i(A/S,M)$ are constructed for $i = 0,1$ and $2$, where $S\rightarrow A$ is a ring homomorphism and $M$ is an $A$-module. This is part of the cohomology theory of Andr{\'e} and Quillen, see e.g. the book by Andr{\'e} \cite{andre}.
We will be interested in the case with $M = A$ and $S = \mathbb{C}$, and in this case the {\it cotangent modules} will be denoted $T^n_{A}$. We will consider the first three of these. The module $T^0_{A}$ describes the derivation module $\text{Der}_{\mathbb{C}}(A,A)$. The module $T^1_{A}$ describes the first order deformations, and the $T^2_{A}$ describes the obstructions for lifting the first order deformations.
Let $R$ be a polynomial ring over $\mathbb{C}$ and let $A$ be the quotient of $R$ by an ideal $I$. The module $T^1_{A}$ is the cokernel of the map
$$\text{Der}(R,A) \rightarrow \text{Hom}_R(I, A) \cong \text{Hom}_{A}(I/I^2, A)\ ,$$
\noindent where a derivation $\phi: R \rightarrow A$ is mapped to the restriction $\phi | I : I \rightarrow A$. Let
$$\XY
\xymatrix@1{ 0\,\ar[r] & \text{Rel}\,\,\ar[r] & F\,\ar[r]^j & R\,\ar[r] & A}$$ \noindent be an exact sequence presenting $A$ as an $R$ module with $F$ free. Let $\text{Rel}_0$ be the submodule of $\text{Rel}$ generated by the Koszul relations; i.e. those of the form $j(x)y - j(y)x$. Then $\text{Rel}/\text{Rel}_0$ is an $A$ module and we have an induced map
$$\text{Hom}_A(F/\text{Rel}_0 \otimes_R A, A) \rightarrow \text{Hom}_A(\text{Rel}/\text{Rel}_0, A)\,\, .$$ \noindent The module $T^2_{A}$ is the cokernel of this map.
The $T^i$ functors are compatible with localization, and thus define sheaves.
\newtheorem{defCotCohom}{Definition}[section] \begin{defCotCohom} Let $\mathcal{S}$ be a sheaf of rings on a scheme $X$, $\mathcal{A}$ an $\mathcal{S}$-algebra and $\mathcal{M}$ an $\mathcal{A}$-module. We define the sheaf $\mathcal{T}^i_{\mathcal{A}/\mathcal{S}}(\mathcal{M})$ as the sheaf associated to the presheaf
$$U \mapsto T^i(\mathcal{A}(U)/\mathcal{S}(U); \mathcal{M(U)})$$\end{defCotCohom}
Let $X$ be a scheme $\mathcal{A} = \mathcal{O}_X$, $\mathcal{M} = \mathcal{A}$ and $S = \mathbb{C}$, and denote by $\mathcal{T}^i_X$ the sheaf $\mathcal{T}^i_{\mathcal{O}_X/\mathbb{C}}$. The modules $T_X^i$ are defined as the hyper-cohomology of the cotangent complex on $X$.
For projective schemes, we will be interested in the deformations that are embedded in $\mathbb{P}^n$, and the following lemma will be useful.
\newtheorem{t1}{Lemma}[section] \begin{t1} If $A$ is the Stanley-Reisner ring of a triangulation of a 3-sphere and $X = \text{Proj\,} A$, then there is an isomorphism \begin{displaymath} T_X^1 \cong T_{A,\,0}^1\,\, . \end{displaymath} \label{lemma:kleppe} \end{t1}
\begin{proof} See the article by Kleppe~\cite{kleppe}, Theorem 3.9, which in the case $\mu =0$, $i = 1$ and $n>1$ (and in our notation) states that there is a canonical morphism
$$T^1_{A,0} \rightarrow T_X^1$$ \noindent which is a bijection if $\text{depth}_m A > 3$, where $m$ is the ideal $\coprod_{i >0}A_i$. Note that the Stanley-Reisner ring corresponding to a triangulation of a sphere is Gorenstein (see Corollary 5.2, Chapter II, in the book by Stanley~\cite{stanley}). If $A$ is the Stanley-Reisner ring of a triangulation of a 3-sphere a, we have $\text{depth}_m A = 4$, hence the morphism above is a bijection. \end{proof}
When the simplicial complex $K$ is a triangulation of the sphere, i.e. $|K| \cong \mathbb{S}^n$, a smoothing of $X_0$ yields an elliptic curve, a K3 surface or a Calabi-Yau 3-fold when $n = 1$, 2 or 3, respectively. We will prove this in the $n= 3$ case.
\newtheorem{cala}{Theorem}[section] \begin{cala}A smoothing, if it exists, of the Stanley-Reisner scheme of a triangulation of the 3-sphere yields a Calabi-Yau 3-fold.\label{theorem:cala}\end{cala}
\begin{proof}Sheaf cohomology of $X_0$ is isomorphic to simplicial cohomology of the complex $K$ with coefficients in $\mathbb{C}$, i.e. $h^i(X_0,\mathcal{O}_{X_0}) = h^i(K, \mathbb{C})$. This is proved in Theorem 2.2 in the article by Altmann and Christophersen~\cite{altchrDeforming}. The semicontinuity theorem (see Chapter III, Theorem 12.8 in ~\cite{hartshorne}) implies that $h^i(X_t, \mathcal{O}_{X_t} )= 0$ for all $t$ when $h^i(X_0, \mathcal{O}_{X_0} )= 0$. Third, the Stanley-Reisner scheme $X_0$ of an oriented combinatorial manifold has trivial canonical bundle $\omega_{X_0}$, hence $\omega_{X_t}$ is trivial for all $t$. This is proved in the article by Bayer and Eisenbud~\cite{bayereisenbud}, Theorem 6.1. \end{proof}
\section{Results on deforming Combinatorial Manifolds}\label{resultsDeforming}
A method for computing the $T^i$ is given in the article by Altmann and Christophersen~\cite{altchrDeforming}. If $K$ is a simplicial complex on the set $\{ 0, \ldots , n \}$ and $A: = A_K$ is the Stanley-Reisner ring associated to $K$, then the $T^1_{A}$ is $\mathbb{Z}^{n+1}$ graded. For a fixed ${\bf c} \in \mathbb{Z}^{n+1}$ write ${\bf c = a - b}$ where ${\bf a} = (a_0, \ldots , a_{n})$ and ${\bf b} = (b_0 ,\ldots b_{n})$ with $a_i,b_i \geq 0$ and $a_ib_i = 0$. Let $x^\mathbf{a}$ be the monomial $x_0^{a_0}\cdots x_{n}^{a_n}$. We define the support of ${\bf a}$ to be $a = \{ i\in [ n] | a_i \neq 0 \}$. Thus if ${\bf a} \in \{ 0,1 \}^{n+1}$, then we have $x_a = x^{\bf a}$. If $a, b \subset \{ 0, \ldots ,n \}$ are the supporting subsets corresponding to ${\bf a}$ and ${\bf b}$, then $a \cap b = \emptyset$. The graded piece $T_{A, \bf{c}}^1$ depends only on the supports $a$ and $b$, and vanish unless $a$ is a face in $K$, $\mathbf{b} \in \{0,1 \}^n$ and $b \subset [ \text{link} (a,K) ]$.
The module $\text{Hom}_R (I_0, A)_{\mathbf{c}}$ sends each monomial $x_p$ in the generating set of the Stanley-Reisner ideal $I_0$ defining $A = R/I_0$ to the monomial $\frac{x_p x^\mathbf{a}}{x^\mathbf{b}}$ when $\mathbf{b}\subset \mathbf{p}$, and 0 otherwise. This corresponds to perturbing the generator $x_p$ of $I_0$ to the generator $x_p + t\frac{x_p x^\mathbf{a}}{x^\mathbf{b}} $ of a deformed ideal $I_t$.
If $|K| \cong \mathbb{S}^3$, then the link of every face $f$, $| \text{link} (f)|$, is a sphere of dimension $2 - \text{dim}(f)$. We will need some results on how to compute the module $T^1_{A}$ for these Stanley-Reisner schemes. We will list results from~\cite{altchrCotangent}. We write $T^1_{<0}(X)$ for the sum of the graded pieces $T^1_{A,\mathbf{c}}$ with $\mathbf{a} = 0$, i.e. $a = \emptyset$.
\newtheorem{alch}{Theorem}[section] \begin{alch} If $K$ is a manifold, then
$$T^1_{A} = \underset{\mathbf{a}\in \,\mathbb{Z}^n \text{\,with\,}\, a\in \,X}{\displaystyle\sum} T^1_{<\,0}(\text{link}\,(a,X))$$
where $T^1_{<0}(\text{link}(a,X))$ is the sum of the one dimensional $T^1_{\emptyset - b}(\text{link}(a,X))$ over all $b \subseteq [\text{link}(a,X)]$ with $|b| \geq 2$ such that $\text{link}(a,X) = L * \partial b$ if $b$ is not a face of $\text{link}(a,X)$, or $\text{link}(a,X) = L * \partial b \cap \partial L * \overline{b}$ if $b$ is a face of $\text{link}(a,X)$. In the first case $| L |$ is a $(n - |b| + 1)$-sphere, in the second case $| L |$ is a $(n - |b |+ 1)$-ball \end{alch}
The following proposition lists the non trivial parts of $T^1_{<0}(\text{link}(a,X))$.
\newtheorem{listofnontrivial}[alch]{Proposition} \begin{listofnontrivial}If $K$ is a manifold, then the contributions to $T^1_{<0}(\text{link}(a,X))$ are the ones listed in Table 1.
\begin{table} \begin{center}
\begin{tabular}{|c|c|c|} \hline Manifold & K & dim $T^1_{<0}$ \\ \hline \hline two points & $\partial \Delta_1$ & 1\\ \hline \hline triangle & $E_3$ & 4\\ \hline quadrangle & $E_4$ & 2\\ \hline \hline tetraedron & $\partial \Delta_3$ & 11\\ \hline suspension of triangle & $\Sigma E_3$ & 5\\ \hline octahedron & $\Sigma E_4$ & 3\\ \hline suspension of $n$-gon & $\Sigma E_n$, $n\geq5$ & 1\\ \hline cyclic polytope & $\partial C(n,3)$, $n\geq 6$& 1\\ \hline \end{tabular} \end{center}\label{table:T1} \caption{$T^1$ in low dimensions} \end{table}
Here $\partial C(n,3)$ is the cyclic polytope defined in section~\ref{simplicial}, and $E_n$ is an n-gon.\label{prop:T1result} \end{listofnontrivial}
A non-geometric way of computing the degree zero part of the $\mathbb{C}$-vector space $T_A^1$ is given in the Macaulay 2 code in Appendix A, when $p$ is an ideal and $T$ is the polynomial ring over a finite field.
\section{Crepant Resolutions and Orbifolds}\label{section:defcrepant}
In this thesis, we will construct Calabi-Yau manifolds by {\it crepant} resolutions of singular varieties. In some cases these singular varieties are {\it orbifolds}. A crepant resolution of a singularity does not affect the dualizing sheaf. In the smooth case, the dualizing sheaf coincides the canonical sheaf, which is trivial for Calabi Yau manifolds. An orbifold is a generalization of a manifold, and it is specified by local conditions. We will give precise definitions below.
\newtheorem{deforbifold}{Definition}[section] \begin{deforbifold} A $d$-dimensional variety $X$ is an {\it orbifold} if every $p \in X$ has a neighborhood analytically equivalent to $0 \in U/G$, where $G \subset GL(n,\mathbb{C})$ is a finite subgroup with no complex reflections other than the identity and $U \subset \mathbb{C}^d$ is a $G$-stable neighborhood of the origin. \end{deforbifold}
A {\it complex reflection} is an element of $GL(n,\mathbb{C})$ of finite order such that $d - 1$ of its eigenvalues are equal to 1. In this case the group $G$ is called a small subgroup of $GL(n, \mathbb{C})$, and $(U/G,0)$ is called a local chart of $X$ at $p$.
Let $X$ be a normal variety such that its canonical class $K_X$ is $\mathbb{Q}$-Cartier, i.e., some multiple of it is a Cartier divisor, and let $f \colon Y \rightarrow X$ be a resolution of the singularities of $X$. Then
$$K_Y = f^* (K_X) + \textstyle\sum a_iE_i$$
where the sum is over the irreducible exceptional divisors, and the $a_i$ are rational numbers, called the {\it discrepancies}.
\newtheorem{defcansing}[deforbifold]{Definition} \begin{defcansing}If $a_i \geq 0$ for all $i$, then the singularities of $X$ are called {\it canonical singularities}. \end{defcansing}
\newtheorem{defcrepant}[deforbifold]{Definition} \begin{defcrepant}A birational projective morphism $f \colon Y\rightarrow X$ with $Y$ smooth and $X$ with at worst Gorenstein canonical singularities is called a {\it crepant resolution} of $X$ if $f^*K_X = K_Y$ (i.e.~if the {\it discrepancy} $K_Y - f^*K_X$ is zero). \end{defcrepant}
\section{Small resolutions of nodes}\label{section:isolsing}
Let $X$ be a variety obtained from deforming a Stanley-Reisner scheme obtained from a triangulation of the 3-sphere, where the only singularity of $X$ is a node. If there is a plane $S$ passing through the node, contained in $X$, then there exists a crepant resolution $\pi \colon \tilde{X} \rightarrow X$ with $\tilde{X}$ smooth. To see this, consider a smooth point of $X$. As $S$ is smooth, $S$ is a complete intersection, i.e., defined by only one equation. The blow-up along $S$ will thus have no effect as the blow-up will take place in $X\times \mathbb{P}^0$ outside the singular points. The singularity will be replaced by $\mathbb{P}^1$. The resolution is small (in contrast to the big resolution where the singularity is replaced by $\mathbb{P}^1\times \mathbb{P}^1$), i.e.
\begin{displaymath} \text{codim} \{ x \in X \mid \text{dim} f^{-1}(x) \geq r \} > 2r \end{displaymath} \noindent for all $r > 0$, hence, the dualizing sheaf is left trivial. The resolved manifold $\tilde{X}$ is Calabi-Yau. This result can be generalized to the case with several nodes, and $S$ a smooth surface in $X$ passing through the nodes. For details, see the article by Werner~\cite{werner}, chapter XI.
\chapter{The Quintic Threefold}\label{4simplex}
It is well known that a smooth quintic hypersurface $X \subset \mathbb{P}^4$ is Calabi-Yau. A smooth quintic hypersurface can be obtained by deforming the projective Stanley-Reisner scheme of the boundary of the 4-simplex. Since the only non-face of $\partial \Delta_4$ is $\{0,1,2,3,4 \}$, the Stanley-Reisner ideal $I$ is generated by the monomial $x_0 x_1 x_2 x_3 x_4$ and the Stanley-Reisner ring is
$$A = \mathbb{C}[x_0,\ldots x_4]/(x_0x_1x_2x_3x_4 )\,\, .$$ \noindent The automorphism group $\text{Aut}(K)$ of the simplicial complex is the symmetric group $S_5$.
Following the outline described in section~\ref{resultsDeforming}, we compute the family of first order deformations. The deformations correspond to perturbations of the monomial $x_0x_1x_2x_3x_4$. Section~\ref{resultsDeforming} describes which choices of the vectors $\bf a$ and $\bf b$ with support $a$ and $b$ give rise to a contribution to the module $T^1_{X}$.
The link of a vertex $a$ is the tetrahedron $\partial \Delta_3$. The only $b$ with $a\cap b = \emptyset$ and $b$ not face is if $|b| = 4$. The case where $b$ is a face and $|b| = 3$ gives 4 choices for each vertex $a$. The case where $b$ is a face and $|b| = 2$ gives 6 choices for each vertex $a$. All in all, the links of vertices give rise to $5\times 11 = 55$ dimensions of the degree 0 part of $T^1_{A}$ (as a $\mathbb{C}$ vector space).
The link of an edge $a$ is the triangle $\partial \Delta_2$. The only $b$ with $a\cap b = \emptyset$ and $b$ not face is if $|b| = 3$. In this case, there are two possible choices of ${\bf a}$ with support $a$ corresponding to a degree 0 element of $\text{Hom}_R (I_0, A)$. The case where $b$ is a face and $|b| =2$ gives 3 choices for each edge $a$. All in all, the links of edges give rise to $10\times 5 = 50$ dimensions of the degree 0 part of $T^1_A$.
We represent each orbit under the action of $S_5$ by a representative $a$ and $b$, and all the orbits are listed in Table~\ref{table:quinticT1}. Note that the monomials $x_i x_j x_k x_l^2$ are derivations, hence give rise to trivial deformations.
\begin{table} \begin{center}
\begin{tabular}{|c|c|c|c|} \hline $a$ & $b$ & perturbation & $\#$ in $S_5$-orbit\\ \hline $\{ 0 \}$ & $\{1,2,3,4\}$ & $x_0^5$ & 5 \\ \hline $\{ 0 \}$ & $\{1,2,3\}$ & $x_0^4x_4$ & 20 \\ \hline $\{ 0 \}$ & $\{1,2\}$ & $x_0^3x_3x_4$ & 30 \\ \hline $\{ 0,1 \}$ & $\{ 2,3,4 \} $ & $x_0^3x_1^2$ & 20\\ \hline $\{ 0,1 \}$ & $\{ 2,3 \} $ & $x_0^2x_1^2x_4$ & 30\\ \hline \end{tabular} \end{center} \caption{$T^1_{X_0}$ is 105 dimensional for the quintic threefold $X_0$} \label{table:quinticT1} \end{table}
We now choose the one parameter $S_5$-invariant family corresponding to $\mathbf{a}$ a vertex (i.e. support $a = \{ j \}$) and $b = [ \text{link}(a,X) ]$, i.e.
$$X_t = \{(x_0,\ldots x_4) \in \mathbb{P}^4 \mid f_t = 0 \}\,\, ,$$ \noindent where $f_t = tx_0^5 + tx_1^5 + tx_2^5 + tx_3^5 + tx_4^5 + x_0x_1x_2x_3x_4$. To simplify computations, we set
$$f_t = x_0^5 + x_1^5 + x_2^5 + x_3^5 + x_4^5 -5tx_0x_1x_2x_3x_4 \, .$$
This can be viewed as a family $\mathcal{X} \rightarrow \mathbb{P}^1$ with
$$\mathbb{P}(A) = \mathcal{X}_{\infty} = \{ (x_0, \ldots , x_4) \mid \textstyle\prod_i x_i = 0\}$$
\noindent our original Stanley-Reisner scheme. The natural action of the torus $(\mathbb{C}^*)^{5}$ on $\mathcal{X}_{\infty} \subset \mathbb{P}^{6}$ is as follows. An element $\lambda = (\lambda_0,\ldots, \lambda_{4}) \in (\mathbb{C}^*)^{5}$ sends a point $(x_0,\ldots ,x_4) $ of $\mathbb{P}^4$ to $(\lambda_0x_0,\ldots , \lambda_4 x_4)$. The subgroup $ \{ (\lambda,\ldots, \lambda) | \lambda \in \mathbb{C}^* \}$ acts as the identity on $\mathbb{P}^4$, so we have an action of the quotient torus $T_4 := (\mathbb{C}^*)^{5}/\mathbb{C}^*$. Since $\mathcal{X}_{\infty}$ is generated by a monomial, it is clear that $T_4$ acts on $\mathcal{X}_{\infty}$.
We compute the subgroup $H \subset T_4$ of the quotient torus acting on $\mathcal{X}_{t}$ as follows. Let the element $\lambda = (\lambda_0, \ldots, \lambda_4)$ act by sending $(x_0,\ldots ,x_4)$ to $(\lambda_0x_0, \ldots ,\lambda_4 x_4)$. For $\lambda$ to act on $X_t$, we must have
$$\lambda_0^5 = \lambda_1^5 = \cdots = \lambda_4^5 = \Pi_{i=0}^4 \lambda_{i}\,\, , $$
\noindent
hence $\lambda_i = \xi^{a_i}$ where $\xi$ is a fixed fifth root of 1, and $\sum_i a_i = 0 \,(\text{mod }5)$. Hence $H$ is the subgroup of $(\mathbb{Z}/5\mathbb{Z})^5/(\mathbb{Z}/5\mathbb{Z})$ given by
$$\left\{ (a_0, \ldots , a_4 ) \mid \textstyle\sum a_i = 0 \right\} \,\, .$$ \noindent
This group acts on $X_t$ diagonally by multiplication by fifth roots of unity, i.e. $(a_0, \ldots a_4) \in (\mathbb{Z}/5\mathbb{Z})^5$ acts by $$(x_0, \ldots , x_4) \mapsto (\xi^{a_0} x_0, \ldots , \xi^{a_4} x_4 )$$ where $\xi$ is a fixed fifth root of unity.
We would like to understand the singularities of the space $Y_t := X_t /H$. For the Jacobian to vanish in a point $(x_0,\ldots x_4)$ we have to have $x_i^5 = tx_0x_1x_2x_3x_4$, and hence $\Pi x_i^5 = t^5 \Pi x_i^5$. Thus either $t^5 = 1$ or one of the $x_i$ is zero. But if one $x_i$ is zero, then they all are, and thus $(x_0,\ldots, x_4)$ does not represent a point in $\mathbb{P}^4$. If $t^5 \neq 1$, then $X_t$ is nonsingular. If $t^5 = 1$, then $X_t$ is singular in the points $(\xi^{a_0}, \ldots , \xi^{a_4})$ with $\sum a_i = 0$ modulo 5. Projectively, these points can be written
$$(1, \xi^{-a_0 + a_1}, \xi^{-a_0 + a_2}, \xi^{-a_0 + a_3}, \xi^{3a_0 - a_1 - a_2 - a_3})\,\, .$$ \noindent This consists of 125 distinct singular points.
From now on assume that $|t| < 1$. The quotient $X_t/H$ is singular at each point $x$ where the stabilizer $H_x$ is nontrivial. A point in $\mathbb{P}^4$ has nontrivial stabilizer in $H$ if at least two of the coordinates are zero. The points of the curves
$$C_{ij} = \{ x_i = x_j = 0 \} \cap X_t $$
have stabilizer of order 5. For example, the stabilizer of a point of the curve $C_{01}$ is generated by $(2,0,1,1,1)$. The points of the set $$P_{ijk} = \{ x_i = x_j = x_k = 0\} \cap X_t$$ have stabilizer of order 25.
It follows from this that the singular locus of $Y_t$ consists of 10 such curves $C_{ij}/H$. We have $C_{ij}/H = \text{Proj}(R^H)$ where
$$R = \mathbb{C}[x_0,\ldots,x_4]/(x_i, x_j, f_t) \,\,\, .$$
For example, for $C_{01}$ the ring $R$ is
$$\mathbb{C}[x_2,x_3, x_4]/(x_2^5 + x_3^5 + x_4^5)\,\, .$$ \noindent An element $(a_0,\ldots,a_4)\in H$ now acts on this ring by
$$(x_2,x_3,x_4)\mapsto (\xi^{a_2}x_2, \xi^{a_3}x_3, \xi^{a_4}x_4)\,\, ,$$ \noindent so we have an action of $(\mathbb{Z}/5\mathbb{Z})^3$ on $R$. For a monomial $x_2^ix_3^jx_4^k$ to be invariant under this group action, we have to have $i = j = k = 0\, \text{ mod }5$, hence
$$R^H = \mathbb{C}[y_0, y_1, y_2]/(y_0 + y_1 + y_2)\,\, ,$$ \noindent where $y_i = x_{i+2}^5$, and $\text{Proj}(R^H) \cong \mathbb{P}^1$. The curves $C_{ij}$ intersect in the points $P_{ijk}/H$.
The singularity $P_{ijk}/H$ locally looks like $\mathbb{C}^3/(\mathbb{Z}/5\mathbb{Z} \oplus \mathbb{Z}/5\mathbb{Z})$, where the element $(a,b) \in \mathbb{Z}/5\mathbb{Z}\oplus \mathbb{Z}/5\mathbb{Z}$ acts by sending $(u,v,w) \in \mathbb{C}$ to $(\xi^au, \xi^b v, \xi^{-a-b} w)$. To see this, consider for example the set $P := P_{012}$. This set consists of 5 points projecting down to the same point in $Y_t$. A neighborhood $U$ of one of these 5 points projects down to $U/H \subset Y_t$. By symmetry, the other singularities $P_{ijk}$ are similar. The set $P$ is defined by the equations $x_0 = x_1 = x_2 = 0$ and $x_3^5 + x_4^5 = 0$. We consider an affine neighborhood of $P$, so we can assume $x_4 = 1$. Set $y_i = \frac{x_i}{x_4}$. Then we have
$$f = y_0^5 + y_1^5 + y_2^5 + y_3^5 + 1 - 5ty_0y_1y_2y_3\,\, .$$
The points $x_0 = x_1 = x_2 = x_3^5 + x_4^5= 0$ now correspond to $y_0 = y_1 = y_2 = y_3^5 + 1 = 0$. Now set $z_3 = y_3 + 1$ and $z_i = y_i$ for $i = 0,1,2$. Then we have
$$f = z_0^5 + z_1^5 + z_2^5 + z_3^5u - 5tz_0z_1z_2v\,\, .$$
where $u = 5 - 10z_3 + 10z_3^2 - 5z_3^3 + z_3^4$ and $v = z_3 - 1$ are units locally around the origin. For a fixed $z_3$ with $(z_3 - 1)^5 = -1$, the group $H$ acts on the coordinates $z_0, z_1, z_2$ by $z_i \mapsto \xi^{a_i}z_i$ with $a_0 + a_1 + a_2 = 0 (\text{mod }5)$, hence we get the quotient $\mathbb{C}^3/(\mathbb{Z}/5\mathbb{Z})^2$ with the desired action.
We can describe this situation by toric methods, i.e. we can find a cone $\sigma^{\nu}$ with
$$\mathbb{C}^3/(\mathbb{Z}/5\mathbb{Z})^2 = \text{Proj} \,\mathbb{C}[y_1,y_2,y_3]^H = U_{\sigma^{\nu}}$$
where $U_{\sigma^{\nu}}$ is the toric variety associated to $\sigma^{\nu}$. For a general reference on toric varieties, see the book by Fulton~\cite{fulton}. A monomial $y_1^{\alpha}y_2^{\beta}y_3^{\gamma}$ maps to $\xi^{a\alpha + b\beta - (a + b)\gamma}y_1^{\alpha}y_2^{\beta}y_3^{\gamma}$, hence the monomial is invariant under the action of $H$ if
$$a\alpha + b\beta - (a + b)\gamma = 0 \,(\text{mod}\,5)\,\,\text{for all}\,(a,b)\,\, ,$$ \noindent i.e. $\alpha = \beta = \gamma \,(\text{mod}\, 5)$. Let $M \subset \mathbb{Z}^3$ be the lattice
$$M:= \{ (\alpha, \beta, \gamma) | \alpha = \beta = \gamma \, (\text{mod}\, 5) \}\,\, .$$ \noindent The cone $\sigma^{\nu}$ is the first octant in $M\otimes_{\mathbb{Z}}\mathbb{R} \cong \mathbb{Z}^3\otimes_{\mathbb{Z}}\mathbb{R}$. A basis for $M$ is
$$\begin{bmatrix} 1\\ 1\\ 1\\ \end{bmatrix}, \begin{bmatrix} 5\\ 0\\ 0\\ \end{bmatrix}, \begin{bmatrix} 0\\ 5\\ 0\\ \end{bmatrix}.$$
We have
$$\mathbb{C}[M\cap \sigma^{\nu}] = \mathbb{C}[u^5, v^5, w^5, uvw] = \mathbb{C}[x,y,z,t]/(xyz - t^5)\,\, .$$ \noindent A basis for the dual lattice $N = \text{Hom}(M, \mathbb{Z})$ is
$$\begin{bmatrix} 1/5\\ 0\\ -1/5\\ \end{bmatrix}, \begin{bmatrix} 0\\ 1/5\\ -1/5\\ \end{bmatrix}, \begin{bmatrix} 0\\ 0\\ 1\\ \end{bmatrix},$$ \noindent and the cone $\sigma$ is the first octant in $\mathbb{R}^3 = N\otimes_{\mathbb{R}}\mathbb{R}$. The semigroup $\sigma \cap N$ is spanned by the vectors $1/5 \cdot (\alpha_1, \alpha_2, \alpha_3)$ with $\alpha_i \in \mathbb{Z}$ and $\sum_i \alpha_i = 5$. Figure~\ref{figure:regular} shows a regular subdivision $\Sigma$ of $\sigma$. The inclusion $\Sigma \subset \sigma$ induces a birational map $X_{\Sigma}\rightarrow U_{\sigma}$ on toric varieties. This gives a resolution of a neighborhood of each point $P_{ijk}$. In the local picture in figure~\ref{figure:regular} we have introduced 18 exceptional divisors, where 6 of these blow down to $P_{ijk}$. In addition 12 of the exceptional divisors blow down to the curves $U_{\sigma}\cap C_{ij}$, $U_{\sigma}\cap C_{ik}$ and $U_{\sigma}\cap C_{jk}$, 4 for each of the three curves intersecting in $P_{ijk}$. This gives $10\times 6 + 10 \times 4 = 100$ exceptional divisors.
\begin{figure}
\caption{Regular subdivision of a neighborhood of the point $P_{ijk}$}
\label{figure:regular}
\end{figure}
By this sequence of crepant resolutions we get the desired mirror family $X^\circ_t$. We have $h^{1,1}(X_t) = 1$, $h^{1,2}(X_t) = 101$, $h^{1,1}(X_t^\circ) = 101$ and $h^{1,2}(X_t^\circ) = 1$. For additional details, see the book by Gross, Huybrechts and Joyce~\cite{grosshuybrechtsjoyce}, section 18.2. or the article by Morrison~\cite{morrison}.
\chapter{Hodge numbers of a small resolution of a deformed Stanley-Reisner scheme}\chaptermark{Hodge numbers of a small resolution}\label{ch:cohom}
Let $X = \text{Proj}\,( A)$ be a singular fiber of the versal deformation space of a Stanley-Reisner scheme, with the only singularities of $X$ being a finite number of nodes. Let $\tilde{X}\rightarrow X$ be a small resolution of the singularities. Let $A_i$ be the local rings $\mathcal{O}_{X, P_i}$ where $P_i$ is a node. The Hodge number $h^{1,2}(\tilde{X})$ is the dimension of the kernel of the map $T^1_{A,0}\rightarrow \oplus T^1_{A_i}$. We will prove this in this chapter, and in the next chapter we will apply this result to the non-smoothable case in Section \ref{ex1section}.
We have $\text{dim }H^1(\Theta_{\tilde{X}}) = h^{1,2}(\tilde{X})$ since $H^2(\tilde{X}, \Omega^1) \cong H^1(\tilde{X}, (\Omega^1)^{\nu} \otimes \omega)' \cong H^1(\tilde{X}, \Theta_{\tilde{X}})'$ where the first isomorphism is Serre duality and the second follows from the fact that $\omega_{\tilde{X}}$ is trivial. A general equation for the node is $f = \sum_{i=1}^n x_i^2$. Then we have
\begin{displaymath} T^1_{A_i} \cong \mathbb{C}[x_1, \ldots, x_n]/(f, \partial f / \partial x_1, \ldots, \partial f / \partial x_n) \cong \mathbb{C}\,\, . \end{displaymath} \noindent Recall that if $\mathcal{S}$ is a sheaf of rings on a scheme $X$, $\mathcal{A}$ an $\mathcal{S}$-algebra and $\mathcal{M}$ an $\mathcal{A}$-module, we defined the sheaf $\mathcal{T}^i_{\mathcal{A}/\mathcal{S}}(\mathcal{M})$ as the sheaf associated to the presheaf
$$U \mapsto T^i(\mathcal{A}(U)/\mathcal{S}(U); \mathcal{M(U)})$$ \noindent In this section, let $\mathcal{A} = \mathcal{O}_X$, $\mathcal{M} = \mathcal{A}$ and $S = \mathbb{C}$, and denote by $\mathcal{T}^i_X$ the sheaf $\mathcal{T}^i_{\mathcal{O}_X/\mathbb{C}}(\mathcal{O}_X)$.
\newtheorem{localglobal}{Theorem}[section] \begin{localglobal}
There is an exact sequence
\begin{displaymath} \XY
\xymatrix@1{ 0\ar[r] & H^1(\Theta_{\tilde{X}}) \ar[r] & T_{A,\,0}^1\ar[r] & \oplus T_{A_i}^1
}\,\, , \end{displaymath} \label{th:sequence} where the map on the right hand side consists of the evaluations of an element of $T_{A,\,0}^1$ in the points $P_i$, and is easy to compute.
\end{localglobal}
\begin{proof} There is a local-to-global spectral sequence with $E_2^{p,q} = H^p(X, \mathcal{T}_X^q)$ converging to the cotangent cohomology $T^{p+q}_X$. Since $\mathcal{T}^0$ is the tangent sheaf $\Theta_X$, the beginning of the 5-term exact sequence of this spectral sequence is
\begin{displaymath} \XY
\xymatrix@1{ 0\ar[r] & H^1(\Theta_{X}) \ar[r] & T_{X}^1\ar[r] & H^0(\mathcal{T}_{X}^1)
}\,\, . \end{displaymath}
For a general reference on spectral sequences, see e.g. the book by McCleary \cite{mccleary}. By Lemma \ref{lemma:isolsing} we have $H^0(\mathcal{T}_X^1) = \oplus T_{A_i}^1$. For a sheaf $\mathcal{F}$ on $\tilde{X}$, the small resolution $\pi :\tilde{X} \rightarrow X$ gives a Leray spectral sequence $H^p(X, R^q\pi_*\mathcal{F})$ converging to $H^n(\tilde{X}, \mathcal{F})$. With $\mathcal{F} = \Theta_{\tilde{X}}$, the beginning of the 5-term exact sequence is
\begin{displaymath} \XY
\xymatrix@1{ 0\ar[r] & H^1(X, \pi_* \Theta_{\tilde{X}})\ar[r] & H^1(\tilde{X}, \Theta_{\tilde{X}})\ar[r] & H^0(X, R^1 \pi_* \Theta_{\tilde{X}}) } \,\, . \end{displaymath}
By Lemma \ref{lemma:leray} the last term is zero and $\pi_*\Theta_{\tilde{X}} \cong \Theta_X$, hence we get the isomorphisms $H^1(X, \Theta_X) \cong H^1(X, \pi_* \Theta_{\tilde{X}}) \cong H^1(\tilde{X}, \Theta_{\tilde{X}})$. Lemma \ref{lemma:kleppe} states that $T_X^1 \cong T_{A,0}^1$. \end{proof}
\newtheorem{isolsing}[localglobal]{Lemma} \begin{isolsing}If $X$ has only isolated singularities, then $\mathcal{T}_X^1 \cong \oplus T^1_{(X,p)}$. \label{lemma:isolsing} \end{isolsing}
\begin{proof}The sheaf $\mathcal{T}_X^1$ is associated to the presheaf $U \mapsto T^1_U$. If $U$ contains no singular points, then $T_U^1 = 0$. \end{proof}
\newtheorem{leray}[localglobal]{Lemma} \begin{leray} We have $\pi_* \Theta_{\tilde{X}} \cong \Theta_X$ and $R^1 \pi_* \Theta_{\tilde{X}} = 0$. \label{lemma:leray} \end{leray}
\begin{proof} $R^1\pi_*\Theta_{\tilde{X}}$ has support in the nodes, so this computation can be done locally. Take an affine neighborhood $V$ of a node, and take the locally small resolution of the node. The node is given by the equation $xy-zw = 0$ in $\mathbb{C}^4$, and $\tilde{V}$ is the blow-up along the ideal $(x,z)$. Hence, $\tilde{V} \subset \{ xU-Tz = 0 \} \subset \mathbb{C}^4 \times \mathbb{P}^1$, where $(U,T)$ are the coordinates on $\mathbb{P}^1$. We prove first that $H^1(\tilde{V}, \Theta_{\tilde{V}}) = 0$ using \v{C}ech-cohomology. Consider the two maps $U_1$ and $U_2$ given by $T \neq 0$ and $U \neq 0$ respectively. In $U_1$ we have $z = xu$, $y = uw$ and
$$xy - zw = xy - xuw = x(y-uw) = 0\,\, ,$$
where $u$ is the coordinate $U/T$, so the strict transform is given by $y-uw = 0$. Similarly, on the map $U_2$ we get $x = tz$, $w = ty$ and
$$xy - zw = tyz - zw = z(ty-w) = 0\,\, ,$$
where $t$ is the coordinate $T/U$, so the strict transform is given by $ty - w = 0$. On the intersection $U_1 \cap U_2$ we have $t = \frac{1}{u}$, $y=uw$, $z = ux$. The affine coordinate ring of $U_1 \cap U_2$ is $\mathbb{C}[x,u,w, \frac{1}{u}] \cong \mathbb{C}[x,t,w, \frac{1}{t}]$. The differentials $\frac{\partial}{\partial y}$, $\frac{\partial}{\partial z}$, and $\frac{\partial}{\partial t}$ restricted to the intersection $U_1 \cap U_2$ can be computed as
\begin{equation*}\frac{\partial}{\partial y} = \frac{1}{u}\frac{\partial}{\partial w}\end{equation*} \begin{equation*}\frac{\partial}{\partial z} = \frac{1}{u}\frac{\partial}{\partial x}\end{equation*} \begin{equation*}\frac{\partial}{\partial t} = u \left(x\frac{\partial}{\partial x} + w\frac{\partial}{\partial w} - u \frac{\partial}{\partial u} \right)\end{equation*} \noindent To see this, apply $\frac{\partial}{\partial y}$, $\frac{\partial}{\partial z}$ and $\frac{\partial}{\partial t}$ on $x$, $w$ and $u$, and keep in mind that we have the relations $x = tz$, $w = ty$ and $tu = 1$.
We prove surjectivity of the map $d: C^0(\tilde{V}) \rightarrow C^1(\tilde{V})$ which sends $(\alpha, \beta) \in \Theta(U_1)\times \Theta(U_2)$ to $(\alpha - \beta) | U_1 \cap U_2$. The elements which do not intersect the image of $\Theta(U_1)\times \{0 \}$ under $d$ are of the form
$$\sum \frac{f_k(x,u,w)}{u^k} \frac{\partial}{\partial x} + \sum \frac{g_k(x,u,w)}{u^k} \frac{\partial}{\partial w} + \sum \frac{h_k(x,u,w)}{u^k} \frac{\partial}{\partial u}$$ \noindent where $f_k$, $g_k$ and $h_k$ have no term with degree higher than $k-1$ in the variable $u$. The differential $d$ maps
$$-t^{k-1}\frac{\partial}{\partial z} \mapsto \frac{1}{u^k} \frac{\partial}{\partial x}\,\, ,$$ \noindent and hence $$p_k(y,z,t) \frac{\partial}{\partial z} \mapsto \frac{f_k(x,u,w)}{u^k}\frac{\partial}{\partial x}\,\, ,$$
where $p_k$ is given by $p_k(y,z,t) = -f_k\left( tz,\frac{1}{t}, ty\right) t^{k-1}$. Similarly we have
$$q_k(y,z,t) \frac{\partial}{\partial y} \mapsto \frac{g_k(x,u,w)}{u^k}\frac{\partial}{\partial w}$$
where $q_k$ is given by $q_k(y,z,t) = -g_k\left( tz, \frac{1}{t}, ty \right) t^{k-1}$. For the last term, we have
$$r_k(y,z,t) \left( y\frac{\partial}{\partial y} + z\frac{\partial}{\partial z} - t\frac{\partial}{\partial t}\right) \mapsto \frac{h_k(x,u,w)}{u^k}\frac{\partial}{\partial u}$$ \noindent where $r_k(y,z,t) = h_k(tz,\frac{1}{t},ty)t^{k+1}$. Hence $d$ is surjective, and $H^1(\tilde{V}, \Theta_{\tilde{V}}) = 0$.
We construct an isomorphism $\Theta_V \rightarrow H^0(\tilde{V}, \Theta_{\tilde{V}})$ as follows. Consider the map
$$\phi: \Theta_{V} \rightarrow \text{Der}(\mathcal{O}_V, \mathcal{O}_{U_1})\oplus \text{Der}(\mathcal{O}_V, \mathcal{O}_{U_2})$$
given by $D \mapsto (\phi_1 D, \phi_2 D)$, where $\phi_1$ and $\phi_2$ are the inclusions of $\mathcal{O}_V$ into $\mathcal{O}_{U_1}$ and $\mathcal{O}_{U_1}$, respectively. On the generator set $\{ x,y,z,w \}$ they take the following values
\begin{equation*}\phi_1(x) = x,\,\, \phi_1(y) = uw,\,\, \phi_1(z) = ux,\,\, \phi_1(w) = w\end{equation*} \begin{equation*}\phi_2(x) = tz,\,\, \phi_2(y) = y ,\,\, \phi_2(z) = z ,\,\, \phi_2(w) = ty\end{equation*}
There is also a map
$$ \text{Der}(\mathcal{O}_{U_1}, \mathcal{O}_{U_1})\oplus \text{Der}(\mathcal{O}_{U_2}, \mathcal{O}_{U_2}) \rightarrow \text{Der}(\mathcal{O}_V, \mathcal{O}_{U_1}) \oplus \text{Der}(\mathcal{O}_V, \mathcal{O}_{U_2})\,\, ,$$ \noindent which is given by $(D_1, D_2)\mapsto (D_1\phi_1, D_2\phi_2)$. The elements which come from $\Theta_V$ can be lifted to $\oplus_{i=1}^{2} \text{Der}(\mathcal{O}_{U_i}, \mathcal{O}_{U_i})$, and we get elements in $H^0(\tilde{V}, \Theta_{\tilde{V}})$. To see this, note that a generator set for the sheaf $\Theta_V$\label{sidecohom} is
\begin{equation*} \begin{split} E = x \frac{\partial}{\partial x} & + y \frac{\partial}{\partial y} + z \frac{\partial}{\partial z} + w\frac{\partial}{\partial w}\\ & y \frac{\partial}{\partial y} - x \frac{\partial}{\partial x}\\ & w \frac{\partial}{\partial y} + x \frac{\partial}{\partial z}\\ & z \frac{\partial}{\partial y} + x \frac{\partial}{\partial w}\\ & y \frac{\partial}{\partial z} + w \frac{\partial}{\partial x}\\ & y \frac{\partial}{\partial w} + z \frac{\partial}{\partial x}\\ & w \frac{\partial}{\partial w} - z \frac{\partial}{\partial z}\\ \end{split} \end{equation*} \noindent They are mapped to the following in $\oplus_{i=1}^2 \text{Der}(\mathcal{O}_V, \mathcal{O}_{U_i})$
\begin{equation*} \left( x \frac{\partial}{\partial x} + uw \frac{\partial}{\partial y} + ux \frac{\partial}{\partial z} + w\frac{\partial}{\partial w},\, tz \frac{\partial}{\partial x} + y \frac{\partial}{\partial y} + z \frac{\partial}{\partial z} + ty\frac{\partial}{\partial w} \right) \end{equation*} \begin{equation*} \left( uw \frac{\partial}{\partial y} - x \frac{\partial}{\partial x},\, y \frac{\partial}{\partial y} - tz \frac{\partial}{\partial x} \right) \end{equation*} \begin{equation*} \left( w \frac{\partial}{\partial y} + x \frac{\partial}{\partial z},\, ty \frac{\partial}{\partial y} + tz \frac{\partial}{\partial z} \right) \end{equation*} \begin{equation*} \left( ux \frac{\partial}{\partial y} + x \frac{\partial}{\partial w},\, z \frac{\partial}{\partial y} + tz \frac{\partial}{\partial w} \right) \end{equation*} \begin{equation*} \left( uw \frac{\partial}{\partial z} + w \frac{\partial}{\partial x},\, y \frac{\partial}{\partial z} + ty \frac{\partial}{\partial x} \right) \end{equation*} \begin{equation*} \left( uw \frac{\partial}{\partial w} + ux \frac{\partial}{\partial x},\, y \frac{\partial}{\partial w} + z \frac{\partial}{\partial x} \right) \end{equation*} \begin{equation*} \left( w \frac{\partial}{\partial w} - ux \frac{\partial}{\partial z},\, ty \frac{\partial}{\partial w} - z \frac{\partial}{\partial z} \right) \end{equation*}
These 7 elements can be lifted to the following elements in $\oplus_{i=1}^2 \text{Der}(\mathcal{O}_{U_i}, \mathcal{O}_{U_i})$.
\begin{equation*}\left( x \frac{\partial}{\partial x} + w\frac{\partial}{\partial w}, y \frac{\partial}{\partial y} + z \frac{\partial}{\partial z}\right)\end{equation*} \begin{equation*}\left( u \frac{\partial}{\partial u} - x \frac{\partial}{\partial x}, y \frac{\partial}{\partial y} - t \frac{\partial}{\partial t}\right)\end{equation*} \begin{equation*}\left( \frac{\partial}{\partial u}, t \left( y\frac{\partial}{\partial y} + z \frac{\partial}{\partial z} - t \frac{\partial}{\partial t} \right) \right)\end{equation*} \begin{equation*}\left( x \frac{\partial}{\partial w}, z \frac{\partial}{\partial y}\right)\end{equation*} \begin{equation*}\left( w \frac{\partial}{\partial x}, y \frac{\partial}{\partial z}\right)\end{equation*} \begin{equation*}\left( u \left(x \frac{\partial}{\partial x} + w \frac{\partial}{\partial w} - u\frac{\partial}{\partial u} \right), \frac{\partial}{\partial t}\right)\end{equation*} \begin{equation*}\left( w \frac{\partial}{\partial w} - u \frac{\partial}{\partial u}, t \frac{\partial}{\partial t} - z \frac{\partial}{\partial z}\right)\end{equation*} \noindent We can construct an inverse map $g:H^0(\tilde{V}, \Theta_{\tilde{V}}) \rightarrow \Theta_{V}$ by $$g(D_1, D_2) = \frac{1}{2}(D_1\phi_1 + D_2\phi_2)\,\, .$$ \noindent Since $\pi_*(\Theta_{\tilde{V}}) \cong R^0\pi_*(\Theta_{\tilde{V}}) \cong H^0(\tilde{V},\Theta_{\tilde{V}})$ and $R^1\pi_*(\Theta_{\tilde{V}}) \cong H^1(\tilde{V},\Theta_{\tilde{V}})$, we get the desired result. \end{proof}
\chapter{Stanley-Reisner Pfaffian Calabi-Yau 3-folds in $\mathbb{P}^6$}\chaptermark{S-R Pfaffian Calabi-Yau 3-folds in $\mathbb{P}^6$}\label{chapter:srpfaffians}
\section{Triangulations of the 3-sphere with 7 vertices}
In this chapter we look at the triangulations of the 3-sphere with 7 vertices. Table~\ref{table:pol} is copied from the article by Gr\"unbaum and Sreedharan~\cite{sreedharan}, where all the combinatorial types of triangulations of the 3-sphere with 7 or 8 vertices are listed. For each such combinatorial type (from now on referred to as a \textit{triangulation}) we compute the versal deformation space of the corresponding Stanley-Reisner scheme, and we check if the general fiber is smooth. In the smoothable cases, we compute the Hodge numbers of the general fiber. We also compute the automorphism group of the triangulation, and we compute the subfamily of the versal deformation space invariant under this group action. In the non-smoothable case, we construct a small resolution of the nodal singularity of the general fiber.
Let $M = [m_{ij}]$ be a skew-symmetric $d \times d$ matrix (i.e., $m_{ij} = -m_{ji}$) with entries in a ring $R$. One can associate to $M$ an element $\text{Pf}(M)$ in $R$ called the {\it Pfaffian} of $M$: When $d = 2n$ is even, we define the Pfaffian of $M$ by the closed formula
\begin{displaymath} \text{Pf}(M) = \frac{1}{2^n n!}\underset{\sigma \in S_{2n}}{\textstyle\sum} \text{sgn}(\sigma) \underset{i =1}{\overset{n}{\textstyle\prod}}m_{\sigma(2i-1),\sigma(2i)} \end{displaymath}
where $S_{2n}$ is the symmetric group on $2n$ elements, and $\textit{sgn}(\sigma)$ is the signature of $\sigma$. When $d$ is odd, we define $\text{Pf}(M) =0$.
The Pfaffian of a skew-symmetric matrix has the property that the square of the pfaffian equals the determinant of the matrix, i.e.
\begin{displaymath} \text{Pf}(M)^2 = \text{det}(M)\,\,\, . \end{displaymath} \noindent In this chapter the ring $R$ will be the polynomial ring $\mathbb{C}[x_1,\ldots,x_7]$, and we will study ideals generated by such Pfaffians. In this case, the sign of the Pfaffian can be chosen arbitrarily, so it suffices to compute the Pfaffian as one of the square roots of the determinant.
\begin{table} \begin{center}
\begin{tabular}{|c|c|cc|} \hline Polytope & Number of facets & List of facets & \\ \hline &&&\\ $P_1^7$ & 11 & A: 1256 & H: 1367 \\
& & B: 1245 & J: 2367 \\
& & C: 1234 & K: 2345 \\
& & D: 1237 & L: 2356 \\
& & E: 1345 & \\
& & F: 1356 & \\
& & G: 1267 & \\
& & & \\ $P_2^7$ & 12 & A: 1245 & H: 2356 \\
& & B: 1246 & J: 2347 \\
& & C: 1256 & K: 2367 \\
& & D: 1345 & L: 2467 \\
& & E: 1346 & M: 3467 \\
& & F: 1356 & \\
& & G: 2345 & \\
& & & \\ $P_3^7$ & 12 & A: 1246 & H: 1347 \\
& & B: 1256 & J: 2346 \\
& & C: 1257 & K: 2356 \\
& & D: 1247 & L: 2357 \\
& & E: 1346 & M: 2347 \\
& & F: 1356 & \\
& & G: 1357 & \\
& & & \\ $P_4^7$ & 13 & A: 2467 & H: 1456 \\
& & B: 2367 & J: 1247 \\
& & C: 1367 & K: 1237 \\
& & D: 1467 & L: 1345 \\
& & E: 2456 & M: 2345 \\
& & F: 2356 & N: 1234 \\
& & G: 1356 & \\
& & & \\ $P_5^7$ & 14 & A: 1234 & H: 1567 \\
& & B: 1237 & J: 2345 \\
& & C: 1267 & K: 2356 \\
& & D: 1256 & L: 2367 \\
& & E: 1245 & M: 3467 \\
& & F: 1347 & N: 3456 \\
& & G: 1457 & O: 4567 \\ \hline \end{tabular} \caption{Polytopes $P_i^7$, $i=1,\ldots ,5$} \label{table:pol} \end{center} \end{table}
For a sequence $i_1, \ldots, i_m$, $1 \leq i_j \leq d$, the matrix obtained from $M$ by omitting rows and columns with indices $i_1, \ldots ,i_m$ is again skew-symmetric; we write $\text{Pf}^{i_1,\ldots ,i_m}(M)$ for its Pfaffian. The elements $\text{Pf}^{i_1,\ldots ,i_m}(M)$ are called Pfaffians of order $d-m$. The Pfaffians of order $d-1$ of a $d\times d$ matrix $M$ are called the {\it principal Pfaffians} of $M$.
\section{Computing the versal family}
The Stanley-Reisner rings obtained from the triangulations in Table \ref{table:pol} are Gorenstein of codimension 3 (in fact, the Stanley-Reisner ring corresponding to any triangulation of a sphere is Gorenstein, see Corollary 5.2, Chapter II, in the book by Stanley~\cite{stanley}). Buchsbaum and Eisenbud proved in Theorem 2.1 (and its proof) in their article~\cite{buchseisen} that Gorenstein codimension 3 ideals are generated by the principal Pfaffians of their skew-symmetric syzygy matrix.
The Stanley-Reisner ideals obtained from the triangulations in Table~\ref{table:pol} are generated by $d = 3,5$ or $7$ monomials. In each case, the following resolution can be computed.
\newtheorem{eksaktsekvensGenerell}{Lemma}[section] \begin{eksaktsekvensGenerell} For the Stanley-Reisner ideals $I_0$ obtained from the triangulations in Table~\ref{table:pol}, there is a free resolution of the Stanley-Reisner ring $A = R/I_0$ $$ \XY
\xymatrix@1{ 0\ar[r] & R\ar[r]^{f} & R^d \ar[r]^{M} & R^d \ar[r]^{f^T} & R \ar[r] & A \ar[r] & 0
}\,\,\, , $$ \noindent where $f$ is a vector with entries the generators of $I_0$, $M$ is an skew-symmetric $d\times d$ syzygy matrix and $I_0$ is generated by the principal pfaffians of $M$.\label{lemma:eksaktsekvensGenerell} \end{eksaktsekvensGenerell}
In the sections 3.4 - 3.8 we compute the degree zero part of the $\mathbb{C}$-vector space $T_A^1$ as described in Section~\ref{resultsDeforming}. This gives us a new perturbed ideal $I_1$, with $k$ parameters, one for each choice of ${\bf a}$ and ${\bf b}$ that contribute to $T_A^1$ of degree zero. We get a perturbed vector $f^1$ with entries the generators of $I_1$, and we get a new matrix $M^1$ by perturbing the entries of the matrix $M$ in such a way that skew-symmetry is preserved, keeping the entries homogeneous in $x_1, \ldots ,x_7$ such that $M^1\cdot f^1 = 0$ mod $t^2$, where $t$ is the ideal $(t_1,..,t_{k})$. This gives the first order embedded (in $\mathbb{P}^6$) deformations.
It has not yet been possible for computers to deal with free resolutions over rings with many parameters. Finding the matrix $M^1$ can however be done manually, by considering the parameters one by one, perturbing the entries of the matrix $M$ keeping skew-symmetry preserved. The principal pfaffians of the matrix $M^1$ give the versal family up to all orders. This follows from Theorem 9.6 in the book by Hartshorne~\cite{hartshorneDef}. Versality follows from the fact that the Kodaira-Spencer map is surjective, see Proposition 2.5.8 in the book by Sernesi~\cite{sernesi}.
We have computed this family explicitly for these five triangulations from Table~\ref{table:pol}.
\section{Properties of the general fiber}
We will now compute the degrees of the varieties obtained from the triangulations in Table~\ref{table:pol}.
\newtheorem{degree}{Lemma}[section] \begin{degree} The number of maximal facets of a triangulation equals the degree of the associated variety. \end{degree}
\begin{proof}Let $d$ be the dimension $d = \text{dim} R/I$. The Hilbert series is\begin{displaymath}\textstyle\sum_{i=-1}^{d-1} \frac{f_i t^{i+1}}{(1-t)^{i+1}} = \frac{1}{(1-t)^d}\textstyle\sum_{i=-1}^{d-1} (1-t)^{d-i-1} f_i t^{i+1}\end{displaymath} where $f_i$ is the number of facets of dimension $i$ and $f_{-1} = 1$, see the book by Stanley~\cite{stanley}. The maximal facets have dimension $d-1$. Inserting $t=1$ in the numerator yields the degree $f_{d-1}$. \end{proof}
The triangulations in Table~\ref{table:pol} give rise to varieties of degree 11, 12, 13 and 14. The degree is invariant under deformation, so in the smoothable cases we can construct Calabi-Yau 3-folds of degree 12, 13 and 14. The following theorem will be proved in Sections 3.4 -- 3.8.
\newtheorem{summarySeven}[degree]{Theorem} \begin{summarySeven} Some invariants of the general fiber of the versal deformation space of the Stanley-Reisner rings of the triangulations in Table~\ref{table:pol} are given in Table~\ref{table:summary}. \end{summarySeven}
\begin{table} \begin{center}
\begin{tabular}{|c|c|c|c|c|} \hline
& & & & \\ Polytope & Degree & General fiber & Hodge & Dimension \\
& & in the versal & numbers & of the versal \\
& & deformation space & & base space \\
& & & & \\ \hline &&&&\\ $P_1^7$ & 11 & Isolated nodal & non- & 92\\
& & singularity with & smoothable & \\
& & small resolution & & \\
& & & & \\ $P_2^7$ & 12 & Complete intersection & $h^{1,1} = 1$ & 79\\
& & type $(2,2,3)$ & $h^{1,2} = 73$ & \\
& & & & \\ $P_3^7$ & 12 & Complete intersection & $h^{1,1} = 1$ & 79\\
& & type $(2,2,3)$ & $h^{1,2} = 73$ & \\
& & & & \\ $P_4^7$ & 13 & Pfaffians of $5\times 5$ & $h^{1,1} = 1$ & 67\\
& & matrix with general & $h^{1,2} = 61$ & \\
& & quadratic terms in & & \\
& & first row/column & & \\
& & and general linear & & \\
& & terms otherwise & & \\
& & & & \\ $P_5^7$ & 14 & Pfaffians of $7\times 7$ & $h^{1,1} = 1$ & 56\\
& & matrix with general & $h^{1,2} = 50$ & \\
& & linear terms & & \\
& & & & \\ \hline \end{tabular} \caption{Polytopes $P_i^7$, $i=1,\ldots ,5$, and their deformations} \label{table:summary} \end{center} \end{table}
Note that the dimension of the versal base space equals $h^{1,2} + 6$ in the four smoothable cases. Theorem 5.2 in~\cite{altchrDeforming} states that there is an exact sequence
$$0 \rightarrow \mathbb{C}^6 \rightarrow H^0(\Theta_{X_t}) \rightarrow H^1(K,\mathbb{C}) \rightarrow 0$$ \noindent Since the last term is zero, we have $\text{dim\,} H^0(\Theta_X) = 6$. One would expect that $T^1_{X_0} = h^1(\Theta_{X_t}) + h^0(\Theta_{X_0}),$ where $X_t$ is a general fiber and $X_0$ is the central fiber of the versal deformation space.
After we have resolved the singularity in the non-smoothable case, we get a Calabi-Yau manifold with $h^{1,2}(X) = 86$, and since 86 + 6 = 92, this fits nicely also in the non-smoothable case.
\section{The triangulation $P^7_1$}\label{ex1section}
In this section we consider $P^7_1$, the first triangulation of $\mathbb{S}^3$ from Table~\ref{table:pol}. The Stanley-Reisner ideal of this triangulation is
\begin{displaymath} I_0 = (x_5 x_7,x_4 x_7, x_4 x_6, x_1 x_2 x_3 x_6, x_1 x_2 x_3 x_5) \end{displaymath} \noindent in the polynomial ring $R = \mathbb{C}[x_1, \ldots ,x_7]$. Let $A = R /I_0$ be the Stanley-Reisner ring of $I_0$. In the minimal free resolution in Lemma~\ref{lemma:eksaktsekvensGenerell}, the vector $f$ and the matrix $M$ are given by
$$f = \begin{bmatrix} x_5 x_7 \\ x_4 x_7 \\ x_4 x_6 \\ x_1 x_2 x_3 x_6 \\ x_1 x_2 x_3 x_5 \\ \end{bmatrix}\,\,\,$$ and $$M = \begin{bmatrix} 0 & 0 & -x_1x_2x_3 & x_4 & 0 \\ 0 & 0 & 0 & -x_5 & x_6\\
x_1x_2x_3 & 0 & 0 & 0 & -x_7\\ -x_4 & x_5 & 0 & 0 & 0 \\ 0 & -x_6 & x_7 & 0 & 0 \\ \end{bmatrix}\,\,\, .$$ \noindent
Using the results of section~\ref{resultsDeforming}, we compute the module $T^1_X$, i.e. the first order embedded deformations, of the Stanley-Reisner scheme $X$ of the complex $K:=P^7_1$ by considering the links of the faces of the complex. Various combinations of $a,b \in \{ 1,\ldots ,7 \}$, with $b \subset [ \text{link}(a,K)]$ a subset of the vertex set and $a$ a face of $K$, contribute to $T^1_X$.
\begin{figure}
\caption{The link of the vertex $\{1 \}$ in $P_7^1$}
\label{figure:link1}
\end{figure}
The geometric realization $| \text{link}(1,K) |$ of the link of the vertex $\{1\}$ in $K$ is the boundary of a cyclic polytope, and is illustrated in figure~\ref{figure:link1}. The links of the vertices $\{2\}$ and $\{3\}$ are similar.
\begin{figure}
\caption{The link of the vertex $\{4 \}$ in $P_7^1$}
\label{figure:link4}
\end{figure} \begin{figure}
\caption{The link of the vertex $\{ 5 \}$ in $P_7^1$}
\label{figure:link5}
\end{figure}
Two vertices, $\{ 4\}$ and $\{ 7 \}$, give rise to a tetraedron (see figure~\ref{figure:link4}), and two vertices, $\{5\}$ and $\{6 \}$, give rise to a suspension of a triangle (see figure~\ref{figure:link5}). We also consider links of one dimensional faces. In 9 cases, the geometric realization is a triangle. The case of $\{ 1,4 \}\in K$ is illustrated in figure~\ref{figure:link14}. In 6 cases, the link is a quadrangle. The case of $\{ 1,5 \}\in K$ is illustrated in figure~\ref{figure:link15}.
In Proposition~\ref{prop:T1result} the contributions to $T^1_A$ of these different links are listed. In the case with $a=1$, we get a contribution to $T^1_X$ if and only if $b = \{2,3 \}$. As in section~\ref{resultsDeforming}, this gives a homogeneous perturbation the monomial $x_1 x_2 x_3 x_6$ to
$$x_1 x_2 x_3 x_6 + t_1x_1x_2x_3x_6\frac{x^{\mathbf{a}}}{x_b} = x_1 x_2 x_3 x_6 + t_1 x_1^3 x_6\,\,\, ,$$ \noindent and a homogeneous perturbation of the monomial $x_1x_2x_3x_5$ to
$$x_1 x_2 x_3 x_5 + t_1 x_1 x_2 x_3x_5 \frac{x^{\mathbf{a}}}{x_b} = x_1 x_2 x_3 x_5 + t_1 x_1^3x_5\,\,\, ,$$
\noindent
with $\mathbf{a} = (2,0,0,0,0,0,0)$ and hence $x^{\bf{a}} = x_1^2$, and $x_b = x_2x_3$. The other three monomials of the Stanley-Reisner ideal are unchanged. The cases $a = \{2\}$ and $a=\{3\}$ give rise to similar perturbations, with parameters $t_2$ and $t_3$, respectively. In the case $a = \{4\}$, the tetrahedron gives rise to 11 dimensions of $T^1$, one for each $b \subset \{ 1,2,3,5 \}$ with $|b| \geq 2$. The case $a = \{7\}$ is similar. In each of the two cases $a = \{5\}$ and $a = \{6\}$, the suspension of a triangle gives 5 different choices of $b$ contributing non-trivially to $T^1$. In addition, the 9 triangles give rise to $9\times 4$ permutations, and the 6 quadrangles give rise to $6\times 2$ perturbations. Note that each triangle gives rise to 5 perturbations and not 4 as stated in the table~\ref{table:T1}. To see this, note that since $T^1$ is $\mathbb{Z}^n$ graded, e.g. the case with $a = \{ 1,4 \}$ and $b = \{2,3,5 \} $ gives two different choices of the vector $\mathbf{a}$ in order for the deformation to be embedded in $\mathbb{P}^6$; $\mathbf{a} = (2,0,0,1,0,0,0)$ or $\mathbf{a} =(1,0,0,2,0,0,0)$ both have support $a = \{ 1,4 \}$. Putting all this together, the dimension of $T^1_X$ is
\begin{figure}
\caption{The link of the edge $\{1,4 \}$ in $P_7^1$}
\label{figure:link14}
\end{figure} \begin{figure}
\caption{The link of the edge $\{1,5 \}$ in $P_7^1$}
\label{figure:link15}
\end{figure}
$$3\times 1 + 2\times 11 + 2\times 5 + 9\times 5 + 6\times 2 =92 \,\, .$$ \noindent This gives 92 parameters $t_1,\ldots, t_{92}$, and the first order deformed ideal $I^1$. The relations between the generators of $I_0$ can be lifted to relations between the generators of $I^1$, and the matrix $M$ lifts to the matrix
\begin{displaymath}M^1 = \begin{bmatrix} 0 & g_1 & g_2 & l_1 & l_2 \\ -g_1 & 0 & g_3 & l_3 & l_4 \\ -g_2 & -g_3 & 0 & l_5 & l_6 \\ -l_1 & -l_3 & -l_5 & 0 & 0 \\ -l_2 & -l_4 & -l_6 & 0 & 0 \end{bmatrix}\,\,\, , \end{displaymath} \noindent where $g_1$, $g_2$ and $g_3$ are cubics and $l_1, \ldots, l_6$ are linear forms in the variables $x_1, \ldots, x_7$. This matrix is computed explicitly, and is given in the appendix on page~\pageref{ex1refUttrykk}. The principal pfaffians of $M^1$ give the versal deformation up to all orders.
After a coordinate change we can describe a general fiber $X$ by
\begin{displaymath}\text{rk} \begin{bmatrix} x_1 & x_3 & x_5 \\ x_2 & x_4 & x_6 \end{bmatrix}\leq 1 \;\; \text{and}\;\;
\begin{bmatrix} x_1 & x_3 & x_5 \\ x_2 & x_4 & x_6 \end{bmatrix} \cdot
\begin{bmatrix} g_3\\ g_2\\ g_1 \end{bmatrix} = 0 \end{displaymath} \noindent The first group of equations define the projective cone over the Segre embedding of $\mathbb{P} ^1 \times \mathbb{P}^2$ in $ \mathbb{P}^5$. Call this variety $Y$. There is one singular point on $X_t$, the vertex of the cone $Y$; $P = (0: \cdots : 0:1)$. The singularity is a node. In fact it is locally isomorphic to $x_1x_4 - x_2x_3 = 0$ in $\mathbb{C}^4$. Since $X_t$ is the general fiber in a smooth versal deformation space, $X_0$ cannot be smoothed.
Using the techniques of Section~\ref{section:isolsing}, the intersection of $X$ with the equations $x_3 = x_4 = 0$ gives a smooth surface $S$ containing the point $P$. A crepant resolution $\pi \colon \tilde{X} \rightarrow X$ exists since the only singularity of $X$ is a node, and the plane $S$ passing through the node. Let $\tilde{X}$ be the manifold obtained by blowing up along $S$.
The Macaulay 2 computation on page~\pageref{sideM2code} gives $\text{dim} T_{A,0}^1 = 86$, and we can compute the evaluation map $T_{A,0}^1\rightarrow T^1_{\mathcal{O}_P}$, which is 0, we have $\text{dim}H^1(\Theta_{\tilde{X}}) = 86$ by Theorem~\ref{th:sequence}.
\section{The triangulation $P^7_2$}\label{ex2section} In this section we consider $P^7_2$, the second triangulation of $\mathbb{S}^3$ in Table~\ref{table:pol}. It has Stanley-Reisner ideal
\begin{displaymath} I_0 = (x_5 x_7, x_1 x_7, x_4 x_5 x_6, x_1 x_2 x_3, x_2 x_3 x_4 x_6) \end{displaymath} and the matrix $M$ in the free resolution is
\begin{displaymath}M = \begin{bmatrix} 0 & 0 & 0 & -x_4x_6 & x_1 \\ 0 & 0 & x_2x_3 & 0 & -x_5\\ 0 & -x_2x_3 & 0 & x_7 & 0 \\ x_4x_6 & 0 & -x_7 & 0 & 0 \\ -x_1 & x_5 & 0 & 0 & 0 \end{bmatrix} \end{displaymath}
As in the previous section, we compute the module $T^1_X$, i.e. the first order embedded deformations, of the Stanley-Reisner scheme $X$ of the complex $K:=P^7_2$ by considering the links of the faces of the complex. Various combinations of $a,b \in \{ 1,\ldots ,7 \}$, with $b \subset [ \text{link}(a,K)]$ a subset of the vertex set and $a$ a face of $K$, contribute to $T^1_X$.
The geometric realization $| \text{link}(i,K) |$ of the link $\text{link} (i,K)$ of the vertex $\{i\}$ is the boundary of a cyclic polytope for $i = 2,3,4$ and $6$. For $i = 1$ and $5$, the geometric realization $| \text{link}(i,K) |$ is the suspension of a triangle, and for $i = 7$, $|\text{link}(i,K)|$ is a tetrahedron.
The links of the edges give rise to 8 triangles, 7 quadrangles and 4 pentagons. Hence, the dimension of $T^1_X$ is $4\times 1 + 2\times 5 + 1\times 11 + 8\times 5 + 7\times2 = 79$.
We compute the first order ideal $I_t$ perturbed by 79 parameters. The matrix $M$ lifts to the matrix
\begin{displaymath} M^{1} = \begin{bmatrix} 0 & -g & q_1 & -q_2 & x_1 \\ g & 0 & q_3 & -q_4 & -x_5 \\ -q_1 & -q_3 & 0 & x_7 & t_{38} \\ q_2 & q_4 & -x_7 & 0 & -t_{33}\\ -x_1 & x_5 & -t_{38} & t_{33}& 0 \\ \end{bmatrix}\,\,\, , \end{displaymath} where $g$ is a cubic and $q_1,\ldots, q_4$ are quadrics in the variables $x_1,\ldots, x_7$. The exact expressions for these quadrics are given in the appendix on page~\pageref{ex2refUttrykk}. The versal deformation space up to all orders is given by the principal pfaffians of the matrix above. Let $X$ be a general fiber of this family.
\newtheorem{CompleteIn}{Lemma}[section] \begin{CompleteIn} The variety $X$ is a complete intersection.\label{lemma:complLemmaet} \end{CompleteIn}
\begin{proof} The lower right corner of the matrix $M^1$ is
\begin{displaymath} W = \begin{bmatrix} 0 & k\\ -k & 0\\ \end{bmatrix}\,\,\, , \end{displaymath} \noindent where $k$ is a constant. The matrix $M^1$ can be written on the form
\begin{displaymath} \begin{bmatrix} U & V\\ -V^{T} & W\\ \end{bmatrix}= \begin{bmatrix} I & VW^{-1}\\ 0 & I\\ \end{bmatrix} \begin{bmatrix} U + VW^{-1}V^{T} & 0\\ 0 & W\\ \end{bmatrix} \begin{bmatrix} I & 0\\ (VW^{-1})^{T} & I\\ \end{bmatrix} \end{displaymath}
Now, the ideal of principal pfaffians can be computed as the principal pfaffians of the matrix at the right center above, hence two of the generators are now zero. The remaining three pfaffians are the elements of the $3\times 3$ matrix $U' = U + VW^{-1}V^{T}$ multiplied by a constant. Hence, the variety is a complete intersection in $\mathbb{P}^{6}$. \end{proof}
The five principal pfaffians can be reduced to three, two quadrics and a cubic. The smoothness of a general fiber can be checked for a good choice of the $t_i$ using a computer algebra package like Macaulay 2~\cite{M2} or Singular~\cite{GPS05}. We will compute the cohomology of the smooth fiber, following the exposition in R\o dland's thesis~\cite{roedland}. The following lemma will be useful.
\newtheorem{exact2}[CompleteIn]{Lemma} \begin{exact2} There is an exact sequence
\begin{displaymath} \XY
\xymatrix@1{ 0\ar[r] & \mathcal{O}_{\mathbb{P}^6}(-7) \ar[r]^-{v} & 2\mathcal{O}_{\mathbb{P}^6}(-5) \oplus \mathcal{O}_{\mathbb{P}^6}(-4) \ar[r]^-{U'} & } \end{displaymath} \begin{displaymath} \XY
\xymatrix@1{
2\mathcal{O}_{\mathbb{P}^6}(-2) \oplus \mathcal{O}_{\mathbb{P}^6}(-3) \ar[r]^-{v^T}&
\mathcal{O}_{\mathbb{P}^6} \ar[r] & \mathcal{O}_{X} \ar[r] & 0 } \end{displaymath}
where $X$ is the general fiber and $v$ is the column vector with entries the three principal pfaffians of $U^{\prime}$.\label{lemma:exsequence2}\end{exact2}
Since $X$ is Calabi-Yau (see Theorem~\ref{theorem:cala}), we know that $h^{1,0}(X)= h^{2,0}(X) = 0$. We now proceed to find the remaining Hodge numbers of $X$. Let $ \mathcal{J} := \text{ker} ( i^{\sharp} : \mathcal{O}_{\mathbb{P}^6} \rightarrow i_* \mathcal{O}_{X})$ denote the ideal sheaf.
\newtheorem{ideal2}[CompleteIn]{Lemma} \begin{ideal2} There is a free resolution \begin{displaymath} \XY
\xymatrix@1{
0 \ar[r] & \mathcal{G} \ar[r]^-{U' \cdot}& \mathcal{H}
\ar[r]^-{\Phi} & \mathcal{K} \ar[r]^-{v^{\otimes 2}}
& \mathcal{J}^2_X \ar[r]& 0} \end{displaymath}
where the sheaves $\mathcal{G}$, $\mathcal{H}$ and $\mathcal{K}$ are given by
\begin{equation*}\mathcal{G} = \mathcal{O}_{\mathbb{P}^6}(-9) \oplus 2\mathcal{O}_{\mathbb{P}^6}(-10) \end{equation*} \begin{equation*}\mathcal{H} =2\mathcal{O}_{\mathbb{P}^6}(-6) \oplus 4\mathcal{O}_{\mathbb{P}^6}(-7) \oplus 2\mathcal{O}_{\mathbb{P}^6}(-8)\end{equation*} \begin{equation*} \mathcal{K} = 3\mathcal{O}_{\mathbb{P}^6} (-4)\oplus 2
\mathcal{O}_{\mathbb{P}^6} (-5) \oplus \mathcal{O}_{\mathbb{P}^6}(-6)\end{equation*} \noindent The elements of $\mathcal{G}$, $\mathcal{H}$ and $\mathcal{K}$ are regarded as $5 \times 5$-matrices that are skew-symmetric matrices, general matrices modulo the identity matrix (or with zero trace), and symmetric matrices respectively. The three maps are
$$U'\cdot : A \mapsto U'A - I/3 \cdot \text{trace}(U'A)\,\,\, ,$$ $$\Phi: B \mapsto BU' + (U')^TB^T\,\,\, ,$$ \noindent and $$v^{\otimes 2}: C \mapsto v^TCv\,\,\, .$$ \noindent If viewed modulo the identity, the last term of the map $U'$ may be dropped.
\label{lemma:idealsheaf2} \end{ideal2}
\begin{proof} All the compositions are clearly zero, hence it remains to prove that the kernels are contained in the images. The last map, $v^{\otimes 2}$, is surjective, because $\mathcal{J}^2_X$ is generated by the elements of $v^T v$, i.e. the elements $m_{ij} = v_i v_j$ for $i \leq j$.
The relations on the $m_{ij}$ are no other than $m_{ij} = m_{ji}$ and $m_{ij} v_k = m_{jk}v_i$, hence the sequence is exact at $\mathcal{K}$. Next, consider the map $\Phi: \mathcal{H} \rightarrow \mathcal{K}$. We have
\begin{displaymath} \Phi(B) = BU' + (U')^TB^T = BU' - U' B^T \end{displaymath} and hence
\begin{equation*}\Phi(B) = 0\end{equation*} \begin{equation*}BU' = U' B^T\end{equation*} \begin{equation*}U' B^T v= 0\,\,\, .\end{equation*} For some $b$ we have (by Lemma~\ref{lemma:exsequence2}) \begin{equation*}B^T v = bv\end{equation*} \begin{equation*}(B^T - I b) v = 0\,\,\, ,\end{equation*} and for some matrix $W$ we have (by Lemma~\ref{lemma:exsequence2} again) \begin{equation*}B^T - I b = W U' \end{equation*} \begin{equation*}B = - U' W^T + bI\,\,\, .\end{equation*} \noindent Since $B = -U' W^T + bI$ equals $-U' W^T $ modulo $I$, we have proved that the sequence is exact at $\mathcal{H}$.
Consider the map $U'\cdot : \mathcal{G} \rightarrow \mathcal{H}$. The image of a skew-symmetric matrix $A$ is zero if and only if $U'A = bI$. However, skew-symmetry yields rank less than $3$. So for A to map to zero, we must have $U' A = 0$. However, using the exact sequence of Lemma~\ref{lemma:exsequence2}, we have that $ U' A = 0 \Rightarrow A = v w^T$ for some vector $w$. However, $A = -A^T = -wv^T$, so $U' A = 0 \Rightarrow U' w = 0 \Rightarrow w = gv \Rightarrow A = g v v^t$. However, for $A = g v v^T$ to be skew-symmetric, $g$ must be zero, making $A = 0$. Hence, the map is injective. \end{proof}
\newtheorem{cohom2igjen}[CompleteIn]{Proposition} \begin{cohom2igjen} The Hodge numbers are
\[ h^{1,1}(X)= 1\,\, \text{and}\,\, h^{1,2}(X) = 73 \,\,\, , \] \noindent where $h^{1,1}(X) := \text{dim}\,H^1(\Omega_{X}) $ and $h^{1,2}(X) := \text{dim}\,H^2(\Omega_{X})$.\label{proposition:hodgeEx2} \end{cohom2igjen}
\begin{proof} First, we know that $H^*(\mathcal{O}_{\mathbb{P}^6}(-r)) = 0$ for $0< r<7$. Second, if we have a resolution $0 \rightarrow A_n \rightarrow \cdots \rightarrow A_0 \rightarrow I$ where $H^*(A_i) = 0$ for $i<n$, then $H^p(I) \cong H^{p+n}(A_n)$, and third, $h^6(\mathcal{O}_{\mathbb{P}^6}(-r -7)) = h^0(\mathcal{O}_{\mathbb{P}^6}(r)) = \binom{r + 6}{6}$.
Using these facts on the resolution of $\mathcal{O}_X(-1)$ (twist the entire sequence of Lemma~\ref{lemma:exsequence2} by $-1$) we get $h^p(\mathcal{O}_X(-1)) = h^{p+3}(\mathcal{O}_{\mathbb{P}^6}(-8))$ which is $7$ for $p = 3$, otherwise zero. Using these results and the cohomology of $\mathcal{O}_X$ on the long exact sequence of
\begin{displaymath} \XY
\xymatrix@1{
0 \ar[r] & \Omega_{\mathbb{P}^6}| X \ar[r] & 7 \mathcal{O}_X(-1) \ar[r]& \mathcal{O}_X \ar[r] &0 } \end{displaymath}
we find that $h^0(\Omega_{\mathbb{P}^6} | X) = h^2(\Omega_{\mathbb{P}^6} | X) = 0$, $h^1(\Omega_{\mathbb{P}^6} | X) = h^0(\mathcal{O}_X) = 1$, and $h^3(\Omega_{\mathbb{P}^6} | X) = h^3(7\mathcal{O}_X(-1))- h^3(\mathcal{O}_X) = 48$.
For the ideal sheaf $\mathcal{J}_X$, the above results and the resolution~\ref{lemma:exsequence2} give $h^p(\mathcal{J}_X) = h^{p+2}(\mathcal{O}_{\mathbb{P}^6}(-7))$ which is $1$ for $p=4$, otherwise zero. For $\mathcal{J}_X^2$, the resolution splits into two short exact sequences
\begin{displaymath} \XY
\xymatrix@1{0 \ar[r] & \mathcal{G} \ar[r]^-{U' \cdot}& \mathcal{H} \ar[r] &\text{Im}(\Phi)\ar[r] & 0} \end{displaymath}
and \begin{displaymath} \XY
\xymatrix@1{
0\ar[r] & \text{Im}(\Phi)\ar[r] & \mathcal{K} \ar[r]^-{v^{\otimes 2}} & \mathcal{J}^2_X \ar[r]& 0} \end{displaymath}
From the second, we get $h^p(\mathcal{J}_X^2) = h^{p+1}(\text{Im}(\Phi))$. From the first, the only non-zero part of the long exact sequence is
\begin{displaymath} \XY
\xymatrix@1{ 0 \ar[r] & H^5(\text{Im}(\Phi)) \ar[r] & H^6(\mathcal{G})\ar[r] & H^6(\mathcal{H})\ar[r] & H^6(\text{Im}(\Phi))\ar[r] & 0}
\end{displaymath}
This makes $h^4(\mathcal{J}_{X}^2) - h^5(\mathcal{J}_{X}^2) = h^5(\text{Im}(\Phi)) - h^6(\text{Im}(\Phi)) = h^6(\mathcal{G}) - h^6(\mathcal{H}) = 2h^6(\mathcal{O}_{\mathbb{P}^6}(-9)) + h^6(\mathcal{O}_{\mathbb{P}^6}(-10)) - 2h^6(\mathcal{O}_{\mathbb{P}^6}(-6)) - 4h^6(\mathcal{O}_{\mathbb{P}^6}(-7)) - 2h^6(\mathcal{O}_{\mathbb{P}^6}(-8)) = 2\cdot 28 + 84 - 2\cdot 0 -4\cdot 1 - 2\cdot 7 = 122$. Since the variety is smooth, we have a short exact sequence
\begin{displaymath} \XY
\xymatrix@1{0\ar[r] & \mathcal{J}_X^2\ar[r] & \mathcal{J}_X\ar[r] & \mathcal{N}_X^{\vee} \ar[r] & 0 }
\end{displaymath}
and another sequence
\begin{displaymath} \XY
\xymatrix@1{0\ar[r] & \mathcal{N}_X^{\vee}\ar[r] & \Omega_{\mathbb{P}^6}| X \ar[r] & \Omega_X \ar[r] & 0 } \end{displaymath}
Note that $\mathcal{N}_X^{\vee}$ is a sheaf on $X$, hence $h^p(\mathcal{N}_X^{\vee}) = 0$ for $p>3 = \text{dim}X$. Entering this into the long exact sequences of the first of the two resolutions above, we get $h^5(\mathcal{J}_X^2)= 0$ as both $h^4(\mathcal{N}_X^{\vee})= 0$ and $h^5(\mathcal{J}_X)=0$. Hence we have $h^4(\mathcal{J}_X^2) = 122$. In addition, we get $h^2(\mathcal{N}^{\vee}_X)=0$ and $h^3(\mathcal{N}^{\vee}_X) = 121$. The long exact sequence of the second resolution above yields $h^1(\Omega_X) = 1$ and $h^2(\Omega_X) = 121 -48 = 73$. \end{proof}
Using Singular~\cite{GPS05} (or any other programming language) we can compute the group of automorphisms of the simplicial complexes. It is a subgroup of $S_7$ and is computed by checking which permutations preserve the maximal facets. The automorphism group $\text{Aut}(P^7_2) \cong D_{4}$ of the complex $P^7_2$ is the dihedral group on 8 elements, i.e. $\mathbb{Z} \ast \mathbb{Z} $ modulo the relations $a^{2} = b^{2} = 1$, $(ab)^{4} = 1$. It is generated by the elements
\begin{displaymath} a = \begin{pmatrix} 1 & 2 & 3 & 4 & 5 & 6 & 7\\ 1 & 2 & 3 & 6 & 5 & 4 & 7\\ \end{pmatrix}\,\, \end{displaymath} \noindent and \begin{displaymath} b = \begin{pmatrix} 1 & 2 & 3 & 4 & 5 & 6 & 7\\ 5 & 4 & 6 & 2 & 1 & 3 & 7\\ \end{pmatrix}\, . \end{displaymath} \noindent This group action on the versal family has 22 orbits. Hence, we have an invariant family with 22 parameters, $s_1,\ldots, s_{22}$.
\section{The triangulation $P^7_3$}\label{ex3section} In this section we consider the third example, $P^7_3$, from Table~\ref{table:pol}. It has Stanley-Reisner ideal
\begin{displaymath} I_0 = (x_6x_7,x_4x_5,x_1x_2x_3)\,\, , \end{displaymath}
and the syzygy matrix is
\begin{displaymath}M = \begin{bmatrix} 0 & -x_1x_2x_3 & x_4x_5\\ x_1x_2x_3 & 0 & -x_6x_7 \\ -x_4x_5 & x_6x_7 & 0\\ \end{bmatrix}\,\,\, , \end{displaymath}
As in sections~\ref{ex1section} and \ref{ex2section}, we compute the module $T^1_X$, i.e. the first order embedded deformations, of the Stanley-Reisner scheme $X$ of the complex $K:=P^7_3$ by considering the links of the faces of the complex. Various combinations of $a,b \in \{ 1,\ldots ,7 \}$, with $b \subset [ \text{link}(a,K)]$ a subset of the vertex set and $a$ a face of $K$, contribute to $T^1_X$.
The geometric realization $| \text{link}(1,K) |$ of the link of the vertex $\{1\}$ in $K$ is an octahedron, and is illustrated in figure~\ref{figure:QuadrangleSusp}. The links of the vertices $\{2\}$ and $\{3\}$ are similar.
\begin{figure}
\caption{The link of the vertex $\{1 \}$ in $P_3^7$}
\label{figure:QuadrangleSusp}
\end{figure}
The links of the vertices $\{4\}$, $\{5\}$, $\{6\}$ and $\{7\}$ are the suspension of a triangle. In addition, the links of the edges give rise to 15 quadrangles and 4 triangles. Putting all this together, the dimension of $T^1_X$ is $3\times 3 + 4\times 5 + 15\times 2 + 4\times 5 = 79$.
The perturbed ideal is generated by the elements of the vector
\begin{displaymath}f^1 = \begin{bmatrix}
q_2\\
q_1\\
g\\ \end{bmatrix} \end{displaymath}
where the exact expressions for $g$, $q_1$ and $q_2$ are given in the appendix on page~\pageref{ex3refUttrykk}. The syzygy matrix lifts to
\begin{displaymath}M^1 = \begin{bmatrix} 0 & -g & q_1\\ g & 0 & -q_2\\ -q_1 & q_2 & 0 \\ \end{bmatrix}\,\,\, . \end{displaymath} \noindent Thus the ideal generated by $f^1$ gives the versal family up to all orders. The general fiber $X$ is given by $g=0$, $q_1 = 0$ and $q_2 = 0$, the intersection of $2$ quadrics and a cubic in $\mathbb{P}^6$, a complete intersection. The smoothness can be checked for a good choice of the $t_i$ using Singular~\cite{GPS05}. The following lemma will be useful.
\newtheorem{exact3}{Lemma}[section] \begin{exact3} The sequence
\begin{displaymath} \XY
\xymatrix@1{ 0\ar[r] & \mathcal{O}_{\mathbb{P}^6}(-7) \ar[r]^-{F} & 2\mathcal{O}_{\mathbb{P}^6}(-5) \oplus \mathcal{O}_{\mathbb{P}^6}(-4) } \end{displaymath} \begin{displaymath} \XY
\xymatrix@1{ \ar[r]^-{R^1} & 2\mathcal{O}_{\mathbb{P}^6}(-2)\oplus \mathcal{O}_{\mathbb{P}^6}(-3) \ar[r]^-{F^t}& \mathcal{O}_{\mathbb{P}^6} \ar[r] & \mathcal{O}_{X} \ar[r] & 0 } \end{displaymath}
is exact, where $F$ and $R^1$ are given above, and values for the $t_i$'s are chosen.\label{lemma:exsequence3}\end{exact3}
Since $X$ is Calabi-Yau, we know that $h^{1,0}(X)= h^{2,0}(X) = 0$. The following Proposition can be proved in a similar manner as Proposition~\ref{proposition:hodgeEx2} in the previous section.
\newtheorem{cohom3igjen}[exact3]{Proposition} \begin{cohom3igjen}The Hodge numbers are
$$h^{1,1}(X) = 1 \,\,\text{and}\,\, h^{1,2}(X) = 73\,\, ,$$ \noindent where $h^{1,1}(X) := \text{dim}\,H^1(\Omega_{X}) $ and $h^{1,2}(X) := \text{dim}\,H^2(\Omega_{X})$. \end{cohom3igjen}
Using Singular~\cite{GPS05} (or any other programming language) we can compute the group of automorphisms of the simplicial complexes. It is a subgroup of $S_7$ and is computed by checking which permutations preserve the maximal facets. The automorphism group $\text{Aut}(P^7_3)$ of the complex $P^7_3$ is $D_{4}\times D_{3}$, where $D_{4}$ is the dihedral group on 8 elements, i.e. $\mathbb{Z} \ast \mathbb{Z} $ modulo the relations $a^{2} = b^{2} = 1$, $(ab)^{4} = 1$. The group $D_{3}$ is the dihedral group on 6 elements, i.e. $\mathbb{Z} * \mathbb{Z}$ modulo the relations $c^{3}= 1$, $d^{2}= 1$ and $cdc = d$. The group $\text{Aut}(P^7_3)$ is generated by the permutations
\begin{displaymath} a = \begin{pmatrix} 1 & 2 & 3 & 4 & 5 & 6 & 7\\ 1 & 2 & 3 & 5 & 4 & 6 & 7\\ \end{pmatrix} \end{displaymath}
\begin{displaymath} b = \begin{pmatrix} 1 & 2 & 3 & 4 & 5 & 6 & 7\\ 1 & 2 & 3 & 6 & 7 & 4 & 5\\ \end{pmatrix} \end{displaymath}
\begin{displaymath} c = \begin{pmatrix} 1 & 2 & 3 & 4 & 5 & 6 & 7\\ 2 & 3 & 1 & 4 & 5 & 6 & 7\\ \end{pmatrix} \end{displaymath}
\begin{displaymath} d = \begin{pmatrix} 1 & 2 & 3 & 4 & 5 & 6 & 7\\ 3 & 2 & 1 & 4 & 5 & 6 & 7\\ \end{pmatrix} \end{displaymath}
and it has order $48$. If we consider the subfamily invariant under this group action, the original 79 parameters reduce to 10.
\section{The triangulation $P^7_4$}\label{ex4section}
In this section we consider $P^7_4$, the fourth triangulation of $\mathbb{S}^3$ from Table~\ref{table:pol}. The Stanley-Reisner ideal of this triangulation is
$$ I_{0} = (x_5x_7,x_1x_2x_5,x_1x_2x_6,x_3x_4x_6, x_3x_4x_7)\,\,\, . $$ \noindent in the polynomial ring $R = \mathbb{C}[x_1, \ldots ,x_7]$. Let $A = R /I_0$ be the Stanley-Reisner ring of $I_0$. The minimal free resolution the Stanley-Reisner ring is
\begin{displaymath} \XY
\xymatrix@1{ 0\ar[r] & R\ar[r]^{f} & R^5 \ar[r]^{M} & R^5 \ar[r]^{f^T} & R \ar[r] & A \ar[r] & 0
}\,\,\, , \end{displaymath} \noindent where $f$ and $M$ are given by
$$f = \begin{bmatrix} x_5 x_7 \\ x_1 x_2 x_5 \\ x_1 x_2 x_6 \\ x_3 x_4 x_6 \\ x_3 x_4 x_7 \\ \end{bmatrix} \,\,\, ,$$ \noindent \begin{displaymath}M = \begin{bmatrix} 0 & 0 & x_3x_4 & -x_1x_2 & 0\\ 0 & 0 & 0 & x_7 & -x_6\\ -x_3x_4 & 0 & 0 & 0 & x_5\\ x_1x_2 & -x_7 & 0 & 0 & 0\\ 0 & x_6 & -x_5 & 0 & 0\\ \end{bmatrix} \end{displaymath}
Computing as in the previous sections, we find the module $T^1_X$, i.e. the embedded versal deformations, of the Stanley-Reisner scheme $X$ of the complex $K:=P^7_4$ by considering the links of the faces of the complex. Various combinations of $a,b \in \{ 1,\ldots ,7 \}$, with $b \subset [ \text{link}(a,K)]$ a subset of the vertex set and $a$ a face of $K$, contribute to $T^1_X$.
The geometric realization $|\text{link}(i,K) |$ of the link of the vertex $\{i\}$ in $K$ for $i = 1,2,3$ and $4$ is the boundary of the cyclic polytope. For $i = 5$ and $7$, the link is the suspension of the triangle, and for $i = 6$, the link is a octahedron. In addition, the links of edges give rise to 4 pentagons, 8 quadrangles and 4 triangles. Putting all this together, the dimension of $T^1_X$ is $4\times 1 + 2\times 5 + 1\times 3 + 6\times5 + 10\times2 = 67$. A general fiber $X$ will be given by the principal pfaffians of the matrix
\begin{displaymath}M^{1} = \begin{bmatrix}
0 & q_1 & q_2 & q_3 & q_4\\ -q_1 & 0 & l_1 & l_2 & l_3\\ -q_2 & -l_1 & 0 & l_4 & l_5\\ -q_3 & -l_2 & -l_4 & 0 & l_6\\ -q_4 & -l_3 & -l_5 & -l_6 & 0\\ \end{bmatrix} \end{displaymath}
where $q_1, \ldots, q_4$ are general quadrics and $l_1, \ldots, l_6$ are linear terms. The exact expressions for the polynomials in this matrix is given in the appendix. The smoothness of the general fiber can be checked using computer algebra software.
\newtheorem{exact4}{Lemma}[section] \begin{exact4} The following sequence
\begin{displaymath} \XY
\xymatrix@1{ 0\ar[r] & \mathcal{O}_{\mathbb{P}^6}(-7) \ar[r]^-{F} & \mathcal{O}_{\mathbb{P}^6}(-5)\oplus 4\mathcal{O}_{\mathbb{P}^6}(-4) } \end{displaymath} \begin{displaymath} \XY
\xymatrix@1{ \ar[r]^-{M^1} & \mathcal{O}_{\mathbb{P}^6}(-2)\oplus 4\mathcal{O}_{\mathbb{P}^6}(-3) \ar[r]^-{F^t} & \mathcal{O}_{\mathbb{P}^6} \ar[r] & \mathcal{O}_{X} \ar[r]& 0 } \end{displaymath}
is exact, where $F$ is the vector with entries the pfaffians of the matrix $M^1$ mod $t^2$.\label{lemma:exsequence4} \end{exact4}
Since $X$ is Calabi-Yau, we know that $h^{1,0}(X)= h^{2,0}(X) = 0$. The following Proposition can be proved in a similar manner as Proposition~\ref{proposition:hodgeEx2}.
\newtheorem{cohom4}[exact4]{Proposition} \begin{cohom4}The Hodge numbers are
$$h^{1,1}(X) = 1\,\,\text{and}\,\, h^{1,2}(X) = 61\,\, ,$$ \noindent where $h^{1,1}(X):= \text{dim}\,H^1(\Omega_{X})$ and $h^{1,2}(X) := \text{dim}\,H^2(\Omega_{X})$.\label{proposition:ex4cohom} \end{cohom4}
The group $\text{Aut}(P^7_4)$ of automorphisms of the complex $P^7_4$ is $D_4 $, the dihedral group of 8 elements. It is generated by the permutations
\begin{displaymath} a = \begin{pmatrix} 1 & 2 & 3 & 4 & 5 & 6 & 7\\ 2 & 1 & 3 & 4 & 5 & 6 & 7\\ \end{pmatrix}\,\, \end{displaymath} and \begin{displaymath} b = \begin{pmatrix} 1 & 2 & 3 & 4 & 5 & 6 & 7\\ 3 & 4 & 1 & 2 & 7 & 6 & 5\\ \end{pmatrix}\, . \end{displaymath} \noindent This group action on the versal family has 20 orbits. Hence, we have an invariant family with 20 parameters, $s_1,\ldots, s_{20}$.
\section{The triangulation $P^7_5$}\label{ex5section}
In this section we consider the fifth example, $P^7_5$, from Table~\ref{table:pol}. It has Stanley-Reisner ideal
$$I_0 = (x_1x_3x_5, x_1x_3x_6, x_1x_4x_6, x_2x_4x_6, x_2x_4x_7, x_2x_5x_7, x_3x_5x_7)\,\,\, ,$$
and the Syzygy matrix is
$$M = \begin{bmatrix} 0 & 0 & 0 & x_7 & -x_6 & 0 & 0 \\ 0 & 0 & 0 & 0 & x_5 & -x_4 & 0 \\ 0 & 0 & 0 & 0 & 0 & x_3 & -x_2\\ -x_7 & 0 & 0 & 0 & 0 & 0 & x_1 \\ x_6 & -x_5 & 0 & 0 & 0 & 0 & 0 \\ 0 & x_4 & -x_3 & 0 & 0 & 0 & 0 \\ 0 & 0 & x_2 & -x_1 & 0 & 0 & 0 \\ \end{bmatrix}\,\,\, .$$ \noindent
Computing as in the previous sections, we find the module $T^1_X$, i.e. the first order embedded deformations, of the Stanley-Reisner scheme $X$ of the complex $K:=P^7_5$ by considering the links of the faces of the complex. Various combinations of $a,b \in \{ 1,\ldots ,7 \}$, with $b \subset [ \text{link}(a,K)]$ a subset of the vertex set and $a$ a face of $K$, contribute to $T^1_X$.
The geometric realization $| \text{link}(i,K) |$ of the link $\text{link} (i,K)$ of a vertex $\{i\}$ is the boundary of a cyclic polytope for $i = 1,\ldots ,7$. We also consider links of one dimensional faces. In 7 cases the geometric realization is a triangle, and in 7 cases the link is a quadrangle. Putting all this together, the dimension of $T^1_X$ is $7\times 1 + 7\times 5 + 7\times 2 = 56$. The full family is displayed in the appendix.
The matrix $M$ lifts to the matrix
$$M^{1} = \begin{bmatrix} 0 & l_1 & l_2 & x_7 & -x_6 & -l_3 & -l_4 \\ -l_1 & 0 & l_5 & l_6 & x_5 & -x_4 & -l_7 \\ -l_2 & -l_5 & 0 & l_8 & l_9 & x_3 & -x_2 \\ -x_7 & -l_6 & -l_8 & 0 & l_{10} & l_{11} & x_1 \\ x_6 & -x_5 & -l_9 & -l_{10} & 0 & l_{12} & l_{13}\\ l_3 & x_4 & -x_3 & -l_{11} & -l_{12} & 0 & l_{14}\\ l_4 & l_7 & x_2 & -x_1 & -l_{13} & -l_{14} & 0 \\ \end{bmatrix}\,\,\, ,$$ \noindent where $l_1,\ldots,l_{14}$ are linear forms, whose exact expressions are given in the appendix on page~\pageref{ex5refUttrykk}. The general fiber $X$ is a degree 14 Calabi-Yau 3-fold. The following Proposition can be proved in a similar manner as Proposition \ref{proposition:hodgeEx2}.
\newtheorem{hodge5}{Proposition}[section] \begin{hodge5}The Hodge numbers are
$$h^{1,1}(X) = 1\,\, \text{and}\,\, h^{1,2}(X) = 50\,\, ,$$ \noindent where $h^{1,1}(X) := \text{dim}\,H^1(\Omega_{X}) $ and $h^{1,2}(X) = \text{dim}\,H^2(\Omega_{X})$. \end{hodge5}
The automorphism group of the complex $P^7_5$, is $\text{Aut}(P^7_5) \cong D_7$. It is generated by the permutations
\begin{displaymath} a = \begin{pmatrix} 1 & 2 & 3 & 4 & 5 & 6 & 7\\ 2 & 3 & 4 & 5 & 6 & 7 & 1\\ \end{pmatrix} \end{displaymath}
\begin{displaymath} b = \begin{pmatrix} 1 & 2 & 3 & 4 & 5 & 6 & 7\\ 7 & 6 & 5 & 4 & 3 & 2 & 1\\ \end{pmatrix} \end{displaymath}
with relations $a^7 = 1$, $b^2 = 1$, $aba = b$. A calculation gives a 5 parameter invariant deformations under the action of this group. We will consider a one-parameter subfamily of this invariant family in Section~\ref{section:roedlandmirror}.
\chapter{The R\o dland and B\"ohm Mirrors}\label{chapter:boehmroedlandMirror}
In this chapter we will describe how to obtain the R\o dland and B\"ohm mirrors from the triangulations we studied in the previous chapter. The R\o dland mirror is obtained from the complex $P^7_5$, and the B\"ohm mirror is obtained from the complex $P^7_4$. They are given by a crepant resolution of a chosen one-parameter subfamily of the invariant family under the action of the automorphism group of $P^7_i$.
\section{The R\o dland Mirror Construction}\label{section:roedlandmirror}
Consider the case of $P^7_5$ which we studied in Section~\ref{ex5section}, and let $X_0$ be the Stanley-Reisner scheme associated to this complex. studied in Section~\ref{ex5section}. As seen in the previous chapter, the automorphism group of the complex is $D_7$. Recall from the introductory chapter that the automorphism group induces an action on $T^1_{X_0}$, and that the parameters of the versal family correspond to faces and links contributing to $T^1_{X_0}$. The $D_7$ orbits of these are given in table~\ref{table:roedlandTabell}.
\begin{table} \begin{center}
\begin{tabular}{|c|c|c|c|} \hline {a} & {b} & Link & $\#$ in $D_7$-orbit\\ \hline $\{ 1 \}$ & $\{2,7\}$ & cyclic polytope & 7 \\ \hline $\{ 1,3 \}$ & $\{2,4,7\}$ & triangle & 14\\ \hline $\{ 1,3 \}$ & $\{2,7\}$ & triangle & 14 \\ \hline $\{ 1,3 \}$ & $\{ 4,7 \} $ & triangle & 7\\ \hline $\{ 3,5 \}$ & $\{ 2,4 \} $ & quadrangle & 14\\ \hline \end{tabular} \end{center} \caption{$T^1_{X_0}$ is 56 dimensional for the Stanley-Reisner scheme $X_0$ of $P^7_5$} \label{table:roedlandTabell} \end{table}
All the links of vertices are cyclic polytopes, and all these 7 cyclic polytopes are one orbit under the action of this automorphism group. In addition, we have 7 cases where the link of an edge is a triangle, and we have 7 cases where it is a quadrangle.
The invariant parameters $s_1,\ldots ,s_5$ are achieved by equating the parameters $t_i$ corresponding to the same orbit under the action of the automorphism group on $T^1_{X_0}$. Consider one of the invariant parameters corresponding to the links of edges being triangles, $s := t_{24} = t_{25} = t_{29} = t_{33} = t_{35} = t_{38} = t_{40}$, the one with 7 elements in the orbit, and set the other ones to $0$. In this case, the matrix $M^1$ will reduce to
$$\begin{bmatrix} 0 & 0 & 0 & x_7 & -x_6 & 0 & -sx_4 \\ 0 & 0 & sx_7 & 0 & x_5 & -x_4 & 0 \\ 0 & -sx_7 & 0 & sx_5 & 0 & x_3 & -x_2 \\ -x_7 & 0 & -sx_5 & 0 & sx_3 & 0 & x_1 \\ x_6 & -x_5 & 0 & -sx_3 & 0 & sx_1 & 0\\ 0 & x_4 & -x_3 & 0 & -sx_1 & 0 & sx_6\\ sx_4 & 0 & x_2 & -x_1 & 0 & -sx_6 & 0 \\ \end{bmatrix}\,\,\, .$$ Let $X_s$ be the variety generated by the principal pfaffians of this matrix. It is defined by the ideal generated by the polynomials
\begin{equation*}p_1 = \ -x_{{1}}x_{{3}}x_{{5}}+{s}^{2}x_{{6}}{x_{{5}}}^{2}+{s}^{2}{x_{{1}}}^{2}x_{{7}}-sx_{{2}}x_{{3}}x_{{4}} +{s}^{3}x_{{3}}x_{{6}}x_{{7}}\end{equation*} \begin{equation*}p_2 = -x_{{1}}x_{{6}}x_{{3}}-sx_{{1}}x_{{2}}x_{{7}}+{s}^{2}{x_{{3}}}^{2}x_{{4}} +{s}^{2}{x_{{6}}}^{2}x_{{5}}+{s}^{3}x_{{1}}x_{{5}}x_{{4}} \end{equation*} \begin{equation*}p_3 = -x_{{1}}x_{{6}}x_{{4}}+{s}^{2}x_{{3}}{x_{{4}}}^{2}-sx_{{6}}x_{{5}}x_{{7}} \end{equation*} \begin{equation*}p_4 = {s}^{3}x_{{1}}x_{{4}}x_{{7}}-x_{{2}}x_{{4}}x_{{6}}-x_{{3}}x_{{5}}sx_{{4}} +{s}^{2}x_{{7}}{x_{{6}}}^{2}\end{equation*} \begin{equation*}p_5 = -x_{{2}}x_{{4}}x_{{7}}+{x_{{4}}}^{2}{s}^{2}x_{{5}}+{s}^{2}x_{{6}}{x_{{7}}}^{2} \end{equation*} \begin{equation*}p_6 = {x_{{5}}}^{2}{s}^{2}x_{{4}}-x_{{7}}x_{{2}}x_{{5}}-x_{{7}}sx_{{1}}x_{{6}} +{s}^{3}x_{{3}}x_{{4}}x_{{7}} \end{equation*} \begin{equation*}p_7 = {x_{{7}}}^{2}{s}^{2}x_{{1}}-x_{{3}}x_{{5}}x_{{7}}-sx_{{4}}x_{{5}}x_{{6}} \end{equation*} \noindent This variety has 56 nodes. Choosing the nonzero parameter as one of the other two parameters corresponding to triangles, gives a smooth general fiber, or a general fiber with singular locus of dimension 0 and degree 189, after a Macaulay 2 computation~\cite{M2}.
There is also a natural action of the torus $(\mathbb{C}^*)^{7}$ on $X_0 \subset \mathbb{P}^{6}$ as follows. An element $\lambda = (\lambda_1,\ldots, \lambda_{7}) \in (\mathbb{C}^*)^{7}$ sends a point $(x_1,\ldots ,x_7) $ of $\mathbb{P}^6$ to $(\lambda_1x_1,\ldots , \lambda_7 x_7)$. The subgroup $ \{ (\lambda,\ldots, \lambda) | \lambda \in \mathbb{C}^* \}$ acts as the identity on $\mathbb{P}^6$, so we have an action of the quotient torus $T_6 := (\mathbb{C}^*)^{7}/\mathbb{C}^*$. In order to compute the subgroup $H \subset T_n$ of the quotient torus which acts on this chosen subfamily, consider the diagonal scalar matrix
$$\lambda = \begin{bmatrix} \lambda_1 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & \lambda_2 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & \lambda_3 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & \lambda_4 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & \lambda_5 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & \lambda_6 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & \lambda_7\\ \end{bmatrix} $$ which acts on $(x_1,\ldots, x_7)$ by
$$\lambda \cdot (x_1,\ldots, x_7) = (\lambda_1 \cdot x_1, \ldots ,\lambda_7\cdot x_7)\,\,\, .$$
The subgroup acting on $X_s$ is generated by the $\lambda$ with the property that $\lambda \cdot p_i = c_i p_i$ for $i = 1,\ldots 7$, and $c_i$ a constant. From $\lambda p_1 = c_1\cdot p_1$, we obtain the equations
\begin{equation*}\lambda_1\lambda_3\lambda_5 = \lambda_5^2\lambda_6 = \lambda_1^2\lambda_7 = \lambda_2\lambda_3\lambda_4 = \lambda_3\lambda_6\lambda_7 \,\, .\end{equation*} For convenience, we set $\lambda_1 = 1$, and we get the equations
\begin{equation}\lambda_3\lambda_5 = \lambda_5^2\lambda_6 = \lambda_7 = \lambda_2\lambda_3\lambda_4 = \lambda_3\lambda_6\lambda_7\,\,\,.\label{lambdaequation}\end{equation} \noindent Hence we have the following expression for $\lambda_5$, $\lambda_6$ and $\lambda_7$.
\begin{equation}\lambda_5 = \lambda_2\lambda_4\label{lambda5}\end{equation} \begin{equation}\lambda_6 = \displaystyle\frac{1}{\lambda_3}\label{lambda6}\end{equation} \begin{equation}\lambda_7 = \lambda_2\lambda_3\lambda_4\label{lambda7}\end{equation} \noindent From $\lambda p_2 = c_2\cdot p_2$, we obtain the equations
\begin{equation} \lambda_3\lambda_6 = \lambda_2\lambda_7 = \lambda_3^2\lambda_4 = \lambda_5\lambda_6^2 = \lambda_4\lambda_5\label{lambdastor}\end{equation} \noindent Inserting (\ref{lambda5}), (\ref{lambda6}) and (\ref{lambda7}) into (\ref{lambdastor}) gives
\begin{equation} 1 = \lambda_2^2\lambda_3\lambda_4 = \lambda_3^2\lambda_4 = \frac{\lambda_2\lambda_4}{\lambda_3^2} = \lambda_2\lambda_4^2\end{equation} \noindent hence $\lambda_3 = \lambda_2^2$ and $\lambda_4 = \lambda_2^{-4}$. From (\ref{lambda5}), (\ref{lambda6}) and (\ref{lambda7}) we now obtain
\begin{equation*}\lambda_5 = \lambda_2^{-3}\end{equation*} \begin{equation*}\lambda_6 = \lambda_2^{-2}\end{equation*} \begin{equation*}\lambda_7 = \lambda_2^{-1}\end{equation*} \noindent Inserting the expressions for $\lambda_3,\ldots,\lambda_7$ into the equation \ref{lambdaequation}, we find that $\lambda_2^7 = 1$.
We conclude that the subgroup $H$ acting on $X_s$ is $H = \mathbb{Z}/7\mathbb{Z}$, which acts as $x_i \mapsto \xi^{i-1} x_i$, where $\xi$ is a primitive 7th root of 1. This subfamily with the action of $H$ is used in R\o dland's thesis~\cite{roedland} in order to construct a mirror of the general fiber of the full versal family. This is done by orbifolding. The variety $X_s$ has 56 nodes. These are the only singularities. A small resolution of $Y:= X_s/H$ is constructed, and this is the mirror manifold of the general fiber.
\section{The B\"ohm Mirror Construction}\label{boehmmirrorsection}
Consider the versal family we studied in Section~\ref{ex4section}, where the special fiber $X_0$ is the Stanley-Reisner scheme of the simplicial complex labeled $P_4^7$ in Table~\ref{table:pol}. Proposition~\ref{proposition:ex4cohom} states that the Hodge numbers are $h^{1,1}(X) = 1$ and $h^{1,2}(X) = 61$ for the smooth general fiber $X$, hence we have $\chi(X) = 2 (h^{1,1}(X) - h^{1,2}(X)) = 2(1-61) = -120$. In Chapter~\ref{chapter:eulerCharBoehm} we verify that the Euler Characteristic of the B\"ohm mirror candidate is $120$ as expected.
The automorphism group of the complex is isomorphic to $D_4$. On the versal family of deformations, there are 20 orbits under the action of this group. The orbits are listed in table~\ref{table:boehmTabell}, and the number of parameters in each orbit is listed. The invariant family is obtained by equating the parameters contained in the same orbit.
\begin{table} \begin{center}
\begin{tabular}{|c|c|c|c|} \hline a & b & Link & $\#$ in $D_7$-orbit\\ \hline $\{ 1 \}$ & $\{3,4\}$ & cyclic polytope & 4\\ \hline $\{ 5 \}$ & $\{1,2\}$ & suspension of triangle & 2 \\ \hline $\{ 5 \}$ & $\{3,4\}$ & " & 2 \\ \hline $\{ 5 \}$ & $\{3,4,6\}$ & " & 2 \\ \hline $\{ 5 \}$ & $\{3,6\}$ & " & 4 \\ \hline $\{ 6 \}$ & $\{1,2\}$ & octahedron & 2 \\ \hline $\{ 6 \}$ & $\{5,7\}$ & " & 1 \\ \hline $\{ 1,2 \}$ & $\{3,4,7\}$ & triangle & 4 \\ \hline $\{ 1,5 \}$ & $\{3,4,6\}$ & " & 4\\ \hline $\{ 1,5 \}$ & $\{3,4,6\}$ & " & 4 \\ \hline $\{ 1,2 \}$ & $\{3,4\}$ & " & 2 \\ \hline $\{ 1,2\}$ & $\{3,7\}$ & " & 4\\ \hline $\{ 1,5 \}$ & $\{3,4\}$ & " & 4 \\ \hline $\{ 1,5 \}$ & $\{3,6\}$ & " & 8\\ \hline $\{ 1,6 \}$ & $\{ 3,4 \} $ & quadrangle & 4\\ \hline $\{ 1,6 \}$ & $\{ 5,7 \} $ & " & 4\\ \hline $\{ 1,7 \}$ & $\{ 3,4\} $ & " & 4\\ \hline $\{ 1,7 \}$ & $\{ 2,6\} $ & " & 4\\ \hline $\{ 5,6 \}$ & $\{ 1,2 \} $ & " & 2\\ \hline $\{ 5,6 \}$ & $\{ 3,4\} $ & " & 2\\ \hline \end{tabular} \end{center} \caption{$T^1$ is 67 dimensional for the Stanley-Reisner scheme of $P^7_4$} \label{table:boehmTabell} \end{table}
Consider the three parameter family where $s_4$ is the invariant parameter corresponding to the orbit represented by $a = \{ 5\}$ and $b = \{ 3,4,6 \}$ ($b$ is the triangle and the link of ${\bf a}$ is the suspension of this triangle), $s_7$ is the invariant parameter corresponding to $a = \{ 6 \}$ and $b = \{ 5,7 \}$. This orbit consists of this single element, the link of ${\bf a}$ is the octahedron (suspension of a quadrangle), and $b$ consists of two adjacent points of the quadrangle, and $s_8$ is the parameter corresponding to the orbit represented by $a = \{ 1,2 \}$ and $b = \{ 3,4,7 \}$. Here the link of ${\bf a}$ is the triangle $b$. From the expressions on page~\pageref{ex4refUttrykk} in the Appendix, we have $s_4 := t_5 = t_{10}$, $s_7 := t_{17}$ and $s_8 := t_{18} = t_{19} = t_{24} = t_{25}$. We set the other $t_i$ to zero. Now the general fiber in this three parameter family is defined by the $4\times 4$ pfaffians of the matrix
\begin{equation} \begin{bmatrix} 0 & s_4 x_7^2 & x_1 x_2 & -x_3x_4 & -s_4 x_5^2\\ -s_4x_7^2 & 0 & s_8 (x_3 + x_4) & x_5 & -x_6\\ -x_1x_2 & -s_8(x_3 + x_4) & 0 & s_7x_6 & x_7 \\ x_3x_4 & -x_5 & -s_7x_6 & 0 & s_8(x_1 + x_2)\\ s_4x_5^2 & x_6 & -x_7 & -s_8(x_1 + x_2) & 0\\ \end{bmatrix}\label{boehmMatriseMin}\,\,. \end{equation}
If we construct a one-parameter family with parameter $s:= s_4 = s_7 = s_8$, the matrix is
\begin{equation} \begin{bmatrix} 0 & sx_7^2 & x_1 x_2 & -x_3x_4 & -s x_5^2\\ -sx_7^2 & 0 & s(x_3 + x_4) & x_5 & -x_6\\ -x_1x_2 & -s(x_3 + x_4) & 0 & sx_6 & x_7 \\ x_3x_4 & -x_5 & -sx_6 & 0 & s(x_1 + x_2)\\ sx_5^2 & x_6 & -x_7 & -s(x_1 + x_2) & 0\\ \end{bmatrix}\label{boehmMatrise}\,\, . \end{equation}
Let $X_s$ be the variety generated by the principal pfaffians of this matrix. It is defined by the ideal generated by the polynomials
\begin{equation*}p_1 = x_5 x_7 + sx_6^2 - s^2(x_1 + x_2)(x_3 + x_4)\end{equation*} \begin{equation*}p_2 = x_3 x_4 x_7 + s(x_1 + x_2)x_1x_2 - s^2x_5^2x_6\end{equation*} \begin{equation*}p_3 = x_3 x_4 x_6 + sx_5^3 - s^2(x_1 + x_2)x_7^2\end{equation*} \begin{equation*}p_4 = x_1 x_2 x_6 + sx_7^3 - s^2(x_3 + x_4)x_5^2\end{equation*} \begin{equation*}p_5 = x_1 x_2 x_5 + sx_3x_4(x_3 + x_4) - s^2x_6x_7^2\end{equation*} \noindent By a Macaulay 2~\cite{M2} computation, the singular locus of this variety is 0-dimensional, and the degree of the singular locus is 48. This fits nicely with the computation we will perform in chapter~\ref{chapter:eulerCharBoehm}, where we find that there are 4 isolated singularities of type $Q_{12}$.
Other choices of 3 parameters give different results. In most cases, the general fiber has singular locus of dimension greater than zero, but there are several ways to construct families where the general fiber has 0-dimensional singular locus. One is obtained if the nonzero parameters (which we equate) are $s_1$, $s_4$ and $s_{8}$ or $s_1$, $s_4$ and $s_{10}$, where $s_4$ and $s_8$ are as above, and $s_1$ is the invariant parameter corresponding to the link being the cyclic polytope and $s_{10}$ is corresponding to the link being a triangle and $a = \{ 1,2 \}$ and $b = \{3,7\}$. In this case the degree of the singular locus is 79 dimensional. It is expected that a similar computation as that in Chapter~\ref{chapter:eulerCharBoehm} would give the same result also in these cases. In this case, the general fiber in the three parameter family is defined by the $4\times 4$ pfaffians of the matrix
\begin{equation*} \begin{bmatrix} 0 & s_4x_7^2 & f_{12} & -f_{34} & -s_4 x_5^2\\ -s_4x_7^2 & 0 & l_2 & x_5 & -x_6\\ -f_{12} & -l_2 & 0 & 0 & x_7 \\ f_{34} & -x_5 & 0 & 0 & l_1 \\ s_4x_5^2 & x_6 & -x_7 & -l_1 & 0\\ \end{bmatrix}\label{boehmMatriseMin79}\,\, , \end{equation*} \noindent where $f_{12} = x_1 x_2 +s_1(x_3^2 + x_4^2)$, $f_{34} = x_3 x_4 +s_1(x_1^2 + x_2^2)$, and $l_1 = s_{8} (x_1 + x_2) + s_{10} (x_3 + x_4)$ and $l_2 = s_8 (x_3 + x_4) + s_{10} (x_1 + x_2)$.
If we include all these four parameters, $s_1$, $s_4$, $s_8$ and $s_{10}$, and equate the first three, say $s := s_1 = s_4 = s_8$ and set $t := s_{10}$, we still get dimension 0 and degree 79. If we equate all these four parameters, we no longer have isolated singularities, since $l_1 = l_2$ in the matrix above in this case.
As in the previous section, there is also a subgroup $H\subset T_7$ of the quotient torus acting on $X_s$. Consider the diagonal scalar matrix
$$\lambda = \begin{bmatrix} \lambda_1 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & \lambda_2 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & \lambda_3 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & \lambda_4 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & \lambda_5 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & \lambda_6 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & \lambda_7\\ \end{bmatrix} $$ which acts on $(x_1,\ldots, x_7)$ by
$$\lambda \cdot (x_1,\ldots, x_7) = (\lambda_1 \cdot x_1, \ldots ,\lambda_7\cdot x_7)\,\,\,.$$ \noindent The subgroup acting on $X_s$ is generated by the $\lambda$ with the property that $\lambda \cdot p_i = c_i p_i$ for $i = 1,\ldots 5$, and $c_i$ a constant. From $\lambda p_1 = c_1\cdot p_1$, we obtain the equations
\begin{equation*}\lambda_5\lambda_7 = \lambda_6^2 = \lambda_1\lambda_3 = \lambda_1\lambda_4 = \lambda_2\lambda_3 = \lambda_2\lambda_4 \,\, .\end{equation*} Hence $\lambda_1 = \lambda_2$, $\lambda_3 = \lambda_4$. For convenience, we set $\lambda_7 = 1$, and we get the equation
\begin{equation} \lambda_5 = \lambda_6^2 = \lambda_1\lambda_3\label{lambdaBoehm1}\end{equation} \noindent From $\lambda p_2 = c_2\cdot p_2$, we obtain the equations
\begin{equation*}\lambda_3\lambda_4\lambda_7 = \lambda_1^2\lambda_2 = \lambda_1\lambda_2^2 = \lambda_5^2\lambda_6\end{equation*} \noindent Inserting $x_7 = 1$, $\lambda_2 = \lambda_1$ and $\lambda_4 = \lambda_3$, we get
\begin{equation}\lambda_3^2 = \lambda_1^3 = \lambda_5^2\lambda_6\label{lambdaBoehm2}\end{equation} \noindent Combining equation~\ref{lambdaBoehm1} and \ref{lambdaBoehm2} we get
\begin{equation}\lambda_3^2 = \lambda_1^3= \lambda_6^5\label{lambdaBoehm3}\end{equation} \noindent From $\lambda p_3 = c_3\cdot p_3$, we obtain the equations
\begin{equation*}\lambda_3\lambda_4\lambda_6 = \lambda_5^3 = \lambda_1\lambda_7^2 = \lambda_2\lambda_7^2\end{equation*} \noindent Inserting $\lambda_7 = 1$, $\lambda_2 = \lambda_1$, $\lambda_4 = \lambda_3$ and $\lambda_5 = \lambda_6^2$ we get
\begin{equation}\lambda_3^2 \lambda_6= \lambda_6^6 = \lambda_1 \label{lambdaBoehm4}\end{equation} \noindent Combining equation \ref{lambdaBoehm3} and \ref{lambdaBoehm4} we get
\begin{equation*}\lambda_6^{13} = 1\end{equation*} and \begin{equation*}\lambda_3^4 = \lambda_6^{10}\,\,\, .\label{lambdaBoehm5}\end{equation*} \noindent We conclude that the subgroup $H$ is isomorphic to $\mathbb{Z}/13\mathbb{Z}$ with generator
$$(x_1:x_2: x_3: x_4: x_5: x_6: x_7) \mapsto (\xi^3x_1,\xi^3x_2, \xi^{11}x_3, \xi^{11}x_4, \xi x_5, \xi^7x_6 ,x_7)$$
where $\xi$ is a primitive 13th root of 1.
The mirror is constructed in B\"ohm's thesis~\cite{boehm}, using tropical geometry. It can also be constructed by orbifolding, by a crepant resolution of $X_s/H$. In the next chapter we will verify that the euler characteristic of this mirror candidate is actually $120$, as it should be.
\chapter{The Euler Characteristic of the B\"ohm Mirror}\chaptermark{The Euler Char. of the B\"ohm Mirror}\label{chapter:eulerCharBoehm}
Let $X$ be the smooth general fiber of the versal family we studied in \ref{ex4section}. Proposition~\ref{proposition:ex4cohom} states that the Hodge numbers are $h^{1,1}(X) = 1$ and $h^{1,2}(X) = 61$, hence we have
$$\chi(X) = 2 \cdot(h^{1,1}(X) - h^{1,2}(X)) = 2\cdot(1-61) = -120\,\, .$$ \noindent In this chapter we verify that the Euler Characteristic of the B\"ohm mirror candidate actually is $120$ as it should be.
Recall from Section \ref{boehmmirrorsection} that $X_s$ is the (singular) general fiber of the given one parameter subfamily of the full versal family, and that $H$ is the group $\mathbb{Z}/13\mathbb{Z}$ which acts on $X_s$. Let $Y_s$ be the quotient space $Y_s := X_s/H$.
In this chapter, we construct a crepant resolution $f: M_s \rightarrow Y_s$ and prove the following result, using toric geometry.
\newtheorem{eulerBoehm}{Theorem}[section] \begin{eulerBoehm}The Euler characteristic of $M_s$ is 120. \label{theorem:eBoehm} \end{eulerBoehm}
The variety $X_s$ has four isolated singular points at $(1:0:0:0:0:0:0)$, $(0:1:0:0:0:0:0)$, $(0:0:1:0:0:0:0)$ and $(0:0:0:1:0:0:0)$. The group $H$ acts freely on $X_s$ away from 6 fixed points: The four singular points and the two smooth points $(1:-1:0:0:0:0:0)$ and $(0:0:1:-1:0:0:0)$. Locally at the latter points, the quotient space $Y_s$ is the germ $(\mathbb{C}^3/H, 0)$ where the action is generated by the diagonal matrix with entries $(\xi, \xi, \xi^{-2})$. To see this, notice that if we set $y_i := x_i/x_1$, the entry $x_1x_2 = y_2$ in matrix~\eqref{boehmMatrise} is a unit locally around the point $(-1:0:0:0:0:0)$. Since the matrix~\eqref{boehmMatrise} is also the syzygy matrix of the ideal generating $X_s$, the five pfaffians generating this ideal reduce to three:
$$y_2y_5 + sy_3y_4(y_3 + y_4) - s^2y_6y_7^2$$
$$y_2y_6 + sy_7^3 - s^2(y_3 + y_4)y_5^2$$
$$y_3y_4y_7 + sy_2(1+y_2) - s^2y_5^2y_6\,\,.$$
Set $v = y_2$. Then the second equation gives $y_6 = - \frac{s}{v}y_7^3 + \frac{s^2}{v} \left( y_3 + y_4 \right )y_5^2$. Inserting this in the first equation gives $$ f := w y_5 + sv y_3y_4(y_3 + y_4) + s^3y_7^5$$
where $w$ is the unit $v^2 - s^4(y_3 + y_4)y_5y_7^2$, so locally at the fixed point\linebreak $(1:-1:0:0:0:0:0)$, the quotient $X_s/H$ is $$\text{Spec} \left( \mathbb{C}[y_3, y_4, y_5, y_7]/(f)^H\right) \cong \text{Spec} \left( \mathbb{C}[y_3, y_4, y_7]^H \right) \cong \mathbb{C}^3/H.$$ The group $H$ acts by $$(y_3,y_4,y_7) \mapsto (\xi y_3, \xi y_4, \xi^{-2}y_7)\,\,\, .$$ The situation is similar in the other fixed point $(0:0:1:-1:0:0:0)$.
Now consider the four singular points. One sees that $D_4$ gives isomorphisms of the germs at the singular points. Let $P$ be one of these singular points, by symmetry we can choose $P = (1:0:0:0:0:0:0)$. To see what $(X_s,P)$ look like locally, we consider an affine neighborhood of $P$, so we can assume $x_1 = 1$ with $P$ the origin in this affine neighborhood. Set $y_i = \frac{x_i}{x_1}$. Now $s(x_1 + x_2) = s(1 + y_2)$ is a unit around the origin, and the five pfaffians again reduce to three:
$$y_5y_7 + sy_6² - s²(1 + y_2)(y_3 + y_4)$$ $$y_3y_4y_7 + sy_2(1 + y_2) - s²y_5²y_6$$ $$y_3y_4y_6 + sy_5³ - s²(1 + y_2)y_7²$$
From the second equation we get $y_2 = u(s²y_5²y_6 - y_3y_4y_7)$ where $u$ is a unit locally around the origin. The first and third equations are now
$$y_5y_7 + sy_6² - v(y_3 + y_4)$$ $$y_3y_4y_6 + sy_5³ - v y_7²$$
where $v$ is the unit $s²(1 + y_2)$. Set $z_1 = y_3 + y_4$, $z_2 = y_3 - y_4$, $z_3 = y_5$, $z_4 = y_6$ and $z_5 = y_7$. Then we have
$$z_3z_5 + sz_4^2 - vz_1$$ $$(z_1² - z_2²)z_4 + 4sz_3³ - 4vz_5²$$
Inserting $z_1 = \frac{1}{v}(z_3z_5 + sz_4^2 )$ in the second equation gives
$$z_3²z_4z_5² + 2sz_3z_4³z_5 + s²z_4^5 - v²z_2²z_4 + 4sv²z_3³ - 4v^3z_5²\,\,\, .$$
\noindent After a coordinate change, this polynomial is
$$g = z_5^2 + z_3^3 + z_2^2z_4 + z_4^5 + w_1 z_3z_4^3z_5 + w_2 z_3^2z_4z_5^2\,\,\, .$$
\noindent where $w_1$ and $w_2$ are $H$ invariant units (since $y_2 = x_2/x_1$ maps to $y_2$ under the action of $H$). The polynomial $g$ has Milnor number 12, and the corank of the Hessian matrix of $g$ is 3. By Arnold's classification of singularities~\cite{arnold} the type of the singularity is $Q_{12}$. The normal form of this singularity is
$$f = z_5^2 + z_3^3 + z_2^2z_4 + z_4^5 \,\, .$$
\noindent In order to show that $f$ and $g$ represent the same germ, we give an $H$ invariant coordinate change taking $g$ to $f$ locally around the origin.
\noindent We first perform the coordinate change
$$z_5 \mapsto z_5 - \frac{1}{2} w_1z_3z_4^3 \,\, ,$$
which maps $g$ to
$$z_5^2 + z_3^3 + z_2^2z_4 + z_4^5 - \frac{1}{4} w_1^2z_3^2z_4^6 + w_2z_3^2z_4z_5^2 - w_1w_2z_3^3z_4^4z_5 + \frac{1}{4}w_1^2w_2z_3^4z_4^7\,\,\, .$$
\noindent This expression may be written
$$u_1 z_5^2 + u_2z_3^3 + z_2^2z_4 + u_3z_4^5 \,\,\, .$$
\noindent where $u_1$, $u_2$ and $u_3$ are $H$ invariant units locally around the origin. After a coordinate change, we obtain the standard form $f$.
Since $H$ now acts as $(z_2, z_3, z_4, z_5)\mapsto(\xi^8z_2, \xi^{-2}z_3, \xi^{4}z_4, \xi^{-3}z_5)$, the polynomial $f$ is {\it semi-invariant} in the sense that $f(\xi^8z_2, \xi^{-2}z_3, \xi^{4}z_4, \xi^{-3}z_5)= \xi^7 f(z_2, z_3, z_4, z_5)$. In fact, it is also a {\it quasi-homogeneous} function of degree 1 and weight $(\alpha_2, \alpha_3, \alpha_4,\alpha_5) = (\frac{2}{5},\frac{1}{3},\frac{1}{5},\frac{1}{2})$, i.e.
$$f(\lambda^{\alpha_2}z_2, \ldots ,\lambda^{\alpha_5}z_5) = \lambda f(z_2,\ldots,z_5)$$ for any $\lambda \geq 0$. We now use also Arnolds notation and set $f = w^2 + x^3 + y^5 + yz^2$. The singularity of $Y_s$ at $0$ is a so called {\it hyper quotient singularity} (hypersurface singularity divided by a group action). The group $H$ acts on $Z := Z(f) := \displaystyle\{ p \mid f(p) = 0 \displaystyle\} \subset \mathbb{C}^4$, and we have $Z/H \cong \text{Proj} (\mathcal{O}_{\mathbb{C}^4}/(f))^H$.
The quotient $(Z/H,0)$ is Gorenstein. This follows from the following general observation. Let $H$ be a finite subgroup of $GL_n(\mathbb{C})$ and $(Z,0) \subset (\mathbb{C}^n, 0)$ a codimension $r$ Gorenstein singularity with an induced $H$ action. Let $\mathcal{F}$ be a free $\mathcal{O}_{\mathbb{C}^n}$ resolution of $\mathcal{O}_Z$ which is also an $H$ module; i.e. $F_i \cong \mathcal{O}_{\mathbb{C}^n} \otimes V_i$ as $H$ modules with $V_i$ a representation of $H$. Let $V_{\text{det}}$ be the representation $g\mapsto \text{det}(g)$. If $V_k^*\cong V_{r-k}\otimes V_{\text{det}}$ as representations, then $(Z/H, 0)$ is Gorenstein.
Let $V$ be the singularity $(Z/H,0)\subset (\mathbb{C}^4/H,0)$. It is defined by the ideal $(f)^H$ in $\mathcal{O}^H_{\mathbb{C}^4}$. We wish to construct a crepant resolution $\widetilde{V}\rightarrow V$ of this singularity.
From now on we use freely the notation and results from the book by Fulton \cite{fulton}. We may find a cone $\sigma^{\vee}$ and a lattice $M$ such that
$$\mathbb{C}[w,x,y,z]^H = \mathbb{C}[\sigma^{\vee}\cap M]$$
A monomial $w^\alpha x^\beta y^\gamma z^\delta$ maps to $\xi^{-3\alpha -2\beta + 4\gamma + 8\delta} w^\alpha x^\beta y^\gamma z^\delta$. This monomial is invariant under the action of $H$ if $-3\alpha - 2\beta + 4\gamma + 8\delta = 0 \,(\text{mod}\,13)$. This equation can be written $\alpha +5 \beta + 3\gamma + 6\delta = 0 \,(\text{mod}\,13)$. Let $M$ be the lattice
$$\{ (\alpha, \beta, \gamma, \delta ) | \alpha +5\beta + 3\gamma + 6\delta = 0 \,(\text{mod}\,13)\}$$
and let $\sigma^{\vee}$ be the first octant in $M_{\mathbb{R}}$. Let $N := \text{Hom}(M,\mathbb{Z})$ be the dual lattice, i.e.
$$N = \mathbb{Z}^4 + \frac{1}{13}(1,5,3,6)\mathbb{Z}\,\,\,\, .$$
The dual cone $\sigma$ is the first octant in $N_{\mathbb{R}}$. Let $v_1,\ldots ,v_4$ be the vectors $v_1 = \frac{1}{13}(1,5,3,6)$, $v_2 = (0,1,0,0)$, $v_3 = (0,0,1,0)$ and $v_4 = (0,0,0,1)$. The isomorphism $\oplus\mathbb{Z}v_i \rightarrow N$ given by multiplication by the matrix
$$A:= \begin{bmatrix} 1/13 & 0 & 0 & 0\\ 5/13 & 1 & 0 & 0\\ 3/13 & 0 & 1 & 0\\ 6/13 & 0 & 0 & 1\\ \end{bmatrix}$$
takes the cone generated by $(13,-5,-3,-6)$, $(0,1,0,0)$, $(0,0,1,0)$ and $(0,0,0,1)$ to the cone $\sigma$. The dual isomorphism $ M\rightarrow \bigoplus \mathbb{Z}w_i $ is given by multiplication by the transpose $A^T$.
We find a toric resolution $X_{\Sigma} \rightarrow \mathbb{C}^4/H$ with
$$ \XY
\xymatrix@1{\widetilde{V}\,\ar@{^{(}->}[r] \ar[d] & X_{\Sigma}\ar[d] \\
V\, \ar@{^{(}->}[r] & \mathbb{C}^4/H }$$
where $\widetilde{V}$ is the strict transform of $V$. A toric resolution of $\mathbb{C}^4/H$ corresponds to a regular subdivision of $\sigma$. This may be computed using the Maple package convex~\cite{convex}. The command {\it regularsubdiv} in convex does not give a resolution with a smooth strict transform, so an additional manual subdivision is made. Table~\ref{table:rays} lists the rays in such a regular subdivision, in the basis $v_1,\ldots,v_4$. On page~\pageref{table:cones} in the Appendix, all the maximal cones of this subdivision $\Sigma$ of $\sigma$ are listed, and they are labeled $\tau_{1},\ldots,\tau_{53}$. Each cone is represented by the four rays spanning it.
\begin{table} \begin{center} \begin{tabular}{l} $[0, 0, 0, 1]$, $[0, 0, 1, 0]$, $[0, 1, 0, 0]$, $[1, 0, 0, 0]$, $[3, -1, 0, -1]$, $[3, 0, 0, -1]$,\\ $[5, -1, -1, -2]$, $[5, -1, 0, -2]$, $[6, -2, -1, -2]$, $[7, -2, -1, -3]$, $[8, -3, -1, -3]$,\\ $[9, -3, -2, -4]$, $[11, -4, -2, -5]$, $[11, -4, -2, -4]$, $[12, -4, -2, -5]$,\\ $[13, -5, -3, -6]$, $[14, -5, -3, -6]$, $[15, -5, -3, -6]$\label{table:rays}\\ \end{tabular}\caption{The rays $\rho$ of a regular subdivision $\Sigma$ of the cone $\sigma$.} \end{center} \end{table}
The polynomial $f$ is only semi-invariant, and the ideal $(f)^H$ has many generators in $\mathbb{C}[\sigma^{\vee} \cap M]$. Still, $\widetilde{V}$ is irreducible and codimension 1 in $X_{\Sigma}$ and therefore defined by an irreducible polynomial $\widetilde{f}_{\tau}$ in each $\mathbb{C}[\tau^{\vee}\cap M] = \mathbb{C}[y_1, y_2, y_3, y_4]$ when $\tau \in\Sigma$. The $y_i$ correspond to the four rays of $\tau$, in the order in which they are listed on page \pageref{table:cones}. To compute $\widetilde{f}_{\tau}$, take the image of any generator of $(f)^H$ by the inclusion $\mathbb{C}[\sigma^{\vee}\cap M] \subset \mathbb{C}[\tau^{\vee}\cap M]$ and remove all factors which are powers of some $y_i$. We can choose the generator $y^8 f \in (f)^H$. The weights of the monomials of $y^8 f $ are $[2,0,8,0]$,$[0,3,8,0]$,$[0,0,13,0]$ and $[0,0,9,2]$.
We will compute $\widetilde{f}_{\tau}$ for a specific $\tau$ to illustrate the idea. Let $\tau := \tau_1$ be the cone in $\Sigma$ generated by the vectors $[13,-5,-3,-6]$, $[0,0,1,0]$, $[3,-1,0,-1]$ and $[8,-3,-1,-3]$ in $\oplus\mathbb{Z}v_i$. Let $B$ be the matrix
$$B:= \begin{bmatrix} 13 & 0 & 3 & 8\\ -5 & 0 & -1 & -3\\ -3 & 1 & 0 & -1\\ -6 & 0 & -1 & -3\\ \end{bmatrix}\,\,\, .$$
The rays of $\tau^{\vee}$ are generated by the columns of the matrix $(B^{-1})^T$. Thus in $M$, the rays of $\tau^{\vee}$ are generated by the columns of $(A^T)^{-1}\cdot (B^{-1})^T = (B^TA^T)^{-1}$. The image of $y^8f$ by the inclusion is a factor $\widetilde{f}_{\tau}$ of the polynomial ${\bf y}^{B^{T}A^T\cdot [2,0,8,0]} + {\bf y}^{B^{T}A^T\cdot [0,3,8,0]} + {\bf y}^{B^{T}A^T\cdot [0,0,13,0]} + {\bf y}^{B^{T}A^T\cdot [0,0,9,2]}$, where the multi index notation ${\bf y}^{[i_1,\ldots,i_4]}$ means $y_1^{i_1}\cdots y_4^{i_4}$. In this case $\widetilde{f}_{\tau} = y_4y_1^2 + 1 + y_4^4y_3^3y_2^5 + y_4^2y_3y_2$. In this way one checks that all $\tilde{f}$ in fact define smooth hypersurfaces in each chart, i.e. that $\tilde{V}$ is smooth.
Each ray $\rho$ in $\Sigma$, aside from the 4 generating the cone $\sigma$, determines an exceptional divisor $D_{\rho}$ in $X_{\Sigma}$. Hence there are 14 exceptional divisors in $X_{\Sigma}$. For every ray $\rho$, the exceptional divisor $D_{\rho}$ is a smooth, complete toric 3-fold and comes with a fan $\text{Star}(\rho)$ in a lattice $N(\rho)$; we define $N_{\rho}$ to be the sublattice of $N$ generated (as a group) by $\rho \cap N$ and
$$N(\rho) = N/N_{\rho},\,\,\,M(\rho) = M\cap \rho^{\perp}$$
The torus $T_{\rho}\subset D_{\rho}$ corresponding to these lattices is
$$T_{\rho} = \text{Hom}(M(\rho),\mathbb{C}^*) = \text{Spec}(\mathbb{C}[M(\rho)]) = N(\rho)\otimes_{\mathbb{Z}}\mathbb{C}^*\,\, .$$
The subvariety $\widetilde{V}$ will only intersect 10 of these exceptional divisors $D_{\rho}$. To check this, we compute the fan consisting of all the cones of $\Sigma$ containing the ray $\rho$, realized as a fan in the quotient lattice $N(\rho)$. The quotient map $\mathbb{C}[\tau^{\vee}\cap M] \rightarrow \mathbb{C}[\tau^{\vee}\cap M(\rho)]$ sends $y_i$ to 0 if $y_i$ is the coordinate corresponding to the ray $\rho$. The other three coordinates are unchanged. Let $\overline{f}_{\tau}$ be the image of $\tilde{f}_{\tau}$ under this projection map, i.e. $\overline{f}_{\tau}:= \tilde{f}_{\tau} | (y_i = 0)$.
We consider the cone $\tau = \tau_1$ studied above, and the ray $\rho$ generated by $(3,-1,0,-1)$. In this case, the coordinate $y_3$ is zero, and the polynomial $\overline{f}_{\tau}$ is $y_4y_1^2 + 1$. Hence the ray $\rho$ intersects $\widetilde{V}$ in this chart. This computation can be performed for all the 14 rays. If the strict transform is 1 on all charts containing $D_i$, then there is no intersection. The rays generated by $[3,0,0,-1]$,$[5,-1,0,-2]$,$[8,-3,-1,-3]$ and $[11,-4,-2,-4]$ do not intersect $\tilde{V}$, hence the subvariety $\tilde{V}$ will intersect 10 of the exceptional divisors.
In 9 of these 10 cases the intersection is irreducible and in one case the intersection has 4 components, but one of these is the intersection with another exceptional divisor. All in all the exceptional divisor $E$ in $\widetilde{V}$ has 12 components. We list the 12 components of $E$ in Table~\ref{table:exceptional}.
\begin{table} \begin{center} \begin{tabular}{cccc} Label & $\alpha$ & Type & $\chi$\\ \hline $E_1$ & $(6,-2,-1,-2) $ & $\mathbb{P}^2$ & 3\\ $E_2$ & $(3,-1,0,-1) $ & $\text{Bl}_1\mathbb{F}_2 $ & 5\\ $E_3$ & $(11,-4,-2,-5) $ & $\mathbb{F}_5$ &4\\ $E_4$ & $(7,-2,-1,-3) $ & $\mathbb{F}_2$ & 4\\ $E_5$ & $(9,-3,-2,-4) $ & $\text{Bl}_2\mathbb{F}_2$ & 6\\ $E_6$ & $(9,-3,-2,-4) $ & $\text{Bl}_2\mathbb{F}_2$ & 6\\ $E_7$ & $(15,-5,-3,-6) $ & $\text{Bl}_3\mathbb{F}_2$ & 7\\ $E_8$ & $(12,-4,-2,-5) $ & $\text{Bl}_3\mathbb{F}_2$ & 7\\ $E_9$ & $(14,-5,-3,-6) $ & $\mathbb{F}_2$ & 4\\ $E_{10}$ & $(1\, ,0\, ,0\, ,0\,) $ & $\text{Bl}_3\mathbb{P}^2$ & 6\\ $E_{11}$ & $(9,-3,-2,-4)$ & $\text{Bl}_1\mathbb{F}_4 $ & 5\\ $E_{12}$ & $(5,-1,-1,-2)$ & $\mathbb{F}_3$ & 4 \\ \end{tabular}\caption{Components of $\widetilde{V}\cap E$.}\label{table:exceptional} \end{center} \end{table}
We may check that the resolution is crepant using the following formula for the discrepancies of hyperquotient singularities, see the article by Reid~\cite{rei}. Let $\alpha \in N$ be the primitive vector generating $\rho$. Any $m \in M$ determines a rational monomial in the variables $w,x,y,z$ and we write $m \in f$ if the monomial is in $\{ w^2, x^3, yz^2, y^5 \}$. Define $\alpha(f) = \text{min} \{ \alpha(m) \mid m\in f \}$. The result is that components of $\widetilde{V} \cap D_{\rho}$ are crepant if and only if
$$\alpha(1,1,1,1) = \alpha(f) +1\,\,\, .$$
This may easily be checked to be true for all $\rho\in \Sigma$ with $\widetilde{V}\cap D_{\rho} \neq \emptyset$.
To compute the type of $E_i$, several different techniques were needed depending upon the complexity of $D_{\rho}$ and/or $\tilde{f}_{\tau}$, $\rho \subset \tau$. For each $\rho$ we compute the polynomials $\overline{f}_{\tau}$. If for some $\tau$, $\overline{f}_{\tau}$ is on the form $\overline{f}_{\tau} = y_j^{n_j}y_k^{n_k} + 1$ with $(n_j, n_k) \neq (0,0)$, we use Method 1 described below.
\underline{Method 1.} In some cases the intersection $\widetilde{V}\cap T_{\rho}$ is a torus. This torus may be described as $\widetilde{N(\rho)} \otimes \mathbb{C}^{*}$, where $\widetilde{N(\rho)}$ is a rank 2 lattice. The inclusion $\widetilde{V} \cap T_{\rho} \rightarrow T_{\rho}$ may be computed to be induced by a linear map $\phi : \widetilde{N(\rho)} \rightarrow N(\rho)$. Now $\widetilde{V} \cap D_{\rho}$ is the closure of $\widetilde{V}\cap T_{\rho}$ in $D_{\rho}$, so it is the toric variety with fan $\phi^{-1}(\text{Star}(\rho))$.\vskip 4 pt
\noindent\underline{$E_1$.} Consider the case where a primitive vector generating $\rho$ is $(6,-2,-1,-2)$. Let $\tau = \tau_{29}$ be the cone generated by $(13, -5, -3, -6)$, $(0, 0, 0, 1)$, $(6, -2, -1, -2)$ and $(14, -5, -3, -6)$. In this chart, $\widetilde{V}$ is generated by $\tilde{f}_{\tau} = y_1^2y_4 + 1 + y_2y_3^2+ y_3$. Restricted to $y_3 = 0$ (corresponding to the ray $(6,-2,-1,-2)$), this gives $\overline{f}_{\tau} = y_1^2y_4 + 1$. Hence the inclusion $\widetilde{V} \cap T_{N(\rho)} \rightarrow T_{N(\rho)}$ is induced by the inclusion of the sublattice $\widetilde{N(\rho)}\cong \mathbb{Z}^2$ of $N(\rho) \cong \mathbb{Z}^3$ generated by $\pm(1,0,-2)$ and $\pm(0,1,0)$. Hence the map $\phi : \widetilde{N(\rho)}\rightarrow N(\rho)$ is
$$\begin{bmatrix}1 & 0\\ 0 & 1\\ -2 & 0\\ \end{bmatrix}$$
and $\widetilde{V}\cap D_{\rho}$ is $\phi^{-1}(\text{Star}(\rho))$. The fan $\text{Star}(\rho)$ consists of 10 maximal cones, and $\phi^{-1}(\text{Star}(\rho))$ is generated by the rays through $(-1,0)$, $(0,1)$ and $(1,-1)$. This fan is drawn in figure~\ref{figure:p2}, and it represents $\mathbb{P}^2$. In fact, $\phi^{-1}(\text{Star}(\rho))$ can be checked to generate $\mathbb{P}^2$ for all the 10 maximal cones $\tau$ with $\rho$ a ray in $\tau$. In Table~\ref{table:exceptional}, this component of the exceptional divisor $E$ is labeled $E_1$.\vskip 4 pt
\begin{figure}
\caption{A fan representing $\mathbb{P}^2$}
\label{figure:p2}
\end{figure}
\noindent\underline{$E_2$.} Now let $\rho$ be generated by the primitive vector $(3, -1, 0, -1)$, and let $\tau = \tau_{48}$. In this chart, $\tilde{V}$ is generated by $\tilde{f}_{\tau} = y_2y_3^2 + 1 + y_1 + y_4^2y_2^2y_1^3$. Restricted to $y_1 = 0$ (corresponding to the ray $(3, -1, 0, -1)$), this gives $\overline{f}_{\tau} = y_2y_3^2 + 1$. By a similar computation as the one above, the fan $\phi^{-1}(\text{Star}(\rho))$ is generated by the rays through the points $(-1,-1)$, $(0,1)$, $(1,1)$, $(1,2)$ and $(2,1)$. This fan is drawn in figure~\ref{figure:bl1f2}. Since $(1,2)$ is the sum of $(0,1)$ and $(1,1)$, the fan represents the blow up of $\mathbb{F}_2$ in a point. On the other hand, $(0,1)$ is the sum of $(-1,-1)$ and $(1,2)$, and the fan represents the blow up of $\mathbb{F}_3$ in a point. These two surfaces are isomorphic.\vskip 4 pt
\begin{figure}
\caption{A fan representing $\text{Bl}_1\mathbb{F}_2$}
\label{figure:bl1f2}
\end{figure}
\noindent\underline{$E_3$.} Let $\tau = \tau_{48}$. In this case $\tilde{f}_{\tau} = y_2y_3^2 + 1 + y_2^2y_4^2y_3^3+ y_1$. Restricted to $y_2 = 0$ (corresponding to the ray $\rho$, this gives $\overline{f}_{\tau} = 1 + y_1$. The fan $\phi^{-1}(\text{Star}(\rho))$ is generated by the rays through the points $(-5, -1)$, $(-1, 0)$, $(0, 1)$ and $(1, 0)$, and is drawn in figure~\ref{figure:f5}. This fan represents $\mathbb{F}_5$.\vskip 4 pt
\begin{figure}
\caption{A fan representing $\mathbb{F}_5$}
\label{figure:f5}
\end{figure}
\noindent\underline{$E_4$.} Let $\tau = \tau_{36}$. In this case $\tilde{f}_{\tau} = y_3^2 + y_2y_1^3y_4^2y_3 + y_2 + 1$. Restricted to $y_3 = 0$ (corresponding to the ray $\rho$, this gives $\overline{f}_{\tau} = 1 + y_2$. The fan $\phi^{-1}(\text{Star}(\rho))$ is generated by the rays through the points $(-1, 2)$, $(0, -1)$, $(0,1)$ and $(1, 0)$, which represents $\mathbb{F}_2$.\vskip 4 pt
\noindent\underline{$E_5$ and $E_6$.} In these two cases it is a component of $\tilde{V}\cap D_{\rho}$ that intersects $T_{N(\rho)}$ in a torus. To see this, consider the case with $\rho$ generated by the primitive vector $(9,-3,-2,-4)$. The fan $\text{Star}(\rho)$ consists of 18 maximal cones. Consider the cone $\tau:= \tau_{13}$. In this chart, $\widetilde{V}$ is generated by $\tilde{f}_{\tau} = y_3 + y_2y_3 + y_1y_2 + y_1^5y_2y_4^2$. Restricted to $y_3 = 0$ (corresponding to the ray $(9,-3,-2,-4)$), this gives the polynomial $\overline{f}_{\tau} = y_1y_2(1 + y_1^4y_4^2)$. The component $y_1 = 0$ corresponds to the ray $(0,0,1,0)$ and the component $y_2 = 0$ corresponds to $(1,0,0,0)$. These two components are labeled $E_{10}$ and $E_{11}$, and we take a closer look at these in Method 3. Both factors $(1 + iy_1^2y_4)$ and $(1 - iy_1^2y_4)$ of the polynomial $\overline{f}_{\tau}$ give rise to a linear map $\phi : \widetilde{N(\rho)} \rightarrow N(\rho)$ represented by the matrix
$$\begin{bmatrix} 1 & 0\\ 0 & 1\\ -2 & 0\\ \end{bmatrix}$$
and $\phi^{-1}(\text{Star}(\rho))$ is generated by the rays through $(-1,-2)$, $(-1,-1)$, $(-1,0)$, $(0,1)$, $(1,2)$ and $(2,3)$. Since $(-1,0) = (-1,-1) + (0,1)$ and $(-1,-1) = (-1,-2) + (0,1)$, the fan represents $\text{Bl}_2\mathbb{F}_2$.\vskip 4 pt
\underline{Method 2.} In 3 cases one sees from the fan $\text{Star}(\rho)$ that $D_{\rho}$ is a locally trivial $\mathbb{P}^1$ bundle over a smooth toric surface.\vskip 4 pt
\noindent\underline{$E_7$.} Consider first the case with $\rho = (15, -5, -3, -6)$. Let $M$ be the matrix
$$M = \begin{bmatrix} 0 & 1 & 9 & 15 \\ 0 & 0 & -3 & -5 \\ 0 & 0 & -2 & -3 \\ 1 & 0 & -4 & -6 \\ \end{bmatrix}$$
and let $\tau$ be the cone generated by the columns of $M$, i.e. $\tau = \tau_{42}$. The fan $\text{Star}(\rho)$ is the image of the 10 maximal cones in $\Sigma$ containing the ray $\rho$ under the projection map $\text{Pr}: N \rightarrow N(\rho)$ given by
$$\text{Pr}:= \begin{bmatrix} 1 & 0 & 0 & 0\\ 0 & 1 & 0 & 0\\ 0 & 0 & 1 & 0\\ \end{bmatrix} \times M^{-1} = \begin{bmatrix} 0 & 0 & -2 & 1\\ 1 & 3 & 0 & 0\\ 0 & 3 & -5 & 0\end{bmatrix}$$
\begin{figure}
\caption{A representation of the surface $\text{Bl}_1\mathbb{F}_2$}
\label{figure:metode2}
\end{figure}
The fan $\text{Star}(\rho)$ in $N(\rho)$ is generated by the rays $(-1, 0, -3)$, $(-1, 0, -2)$, $(0, -1, 0)$, $(0, 0, -1)$, $(0, 0, 1)$, $(0, 1, 0)$, $(1, 0, 0)$. Let $\Delta^{'}$ be the fan generated by $1$ and $-1$ in the lattice $\mathbb{Z}$, and let $\Delta^{''}$ be the fan generated by $(-1, -3)$, $(-1, -2)$, $(0, -1)$, $(0, 1)$ and $(1, 0)$ in the lattice $\mathbb{Z}^2$. There is an exact sequence of lattices
$$0 \rightarrow \mathbb{Z} \rightarrow N(\rho) \rightarrow \mathbb{Z}^2 \rightarrow 0$$
where the map $\mathbb{Z} \rightarrow N(\rho)$ is the inclusion $n \mapsto (0,n,0)$ and the map $N(\rho) \rightarrow \mathbb{Z}^2$ is the projection $(x,y,z) \mapsto (x,z)$. This exact sequence gives rise to mappings
$$X(\Delta^{'}) \rightarrow X(\text{Star}(\rho)) \rightarrow X(\Delta^{''})$$
where $X(\Delta^{''})$ is the blow up of $\mathbb{F}_2$ in a point, and $X(\Delta^{'})$ is $\mathbb{P}^1$. Thus we have a trivial $\mathbb{P}^1$ bundle over $\text{Bl}_1\mathbb{F}_2$.
We can find out what $\tilde{V} \cap D_{\rho}$ is by a local look at the fibers of the map $\tilde{V}\cap D_{\rho} \rightarrow \text{Bl}_1\mathbb{F}_2$. Over the charts labeled 2, 3 and 4 in Figure~\ref{figure:metode2}, the map is an isomorphism. Over the intersection of the charts 1 and 5, we have an isomorphism except over two points in $\text{Bl}_1\mathbb{F}_2$, where the inverse image is a line. Hence, $\tilde{V}\cap D_{\rho}$ is isomorphic to $\text{Bl}_3\mathbb{F}_2$.\vskip 4 pt
\noindent\underline{$E_8$.} In this case the fan $\text{Star}(\rho)$ represents a locally trivial $\mathbb{P}^1$ bundle over $\text{Bl}_2\mathbb{F}_2$, and $\tilde{V}\cap D_{\rho}$ is isomorphic to $\text{Bl}_3\mathbb{F}_2$.
To see this, let $\rho = (12, -4, -2, -5)$, and let $M$ be the matrix
$$M = \begin{bmatrix} 9 & 11 & 13 & 12 \\ -3 & -4 & -5 & -4 \\ -2 & -2 & -3 & -2 \\ -4 & -5 & -6 & -5 \\ \end{bmatrix}$$
and let $\tau$ be the cone generated by the columns of $M$, i.e. $\tau = \tau_{47}$. The fan $\text{Star}(\rho)$ is the image of the 12 maximal cones in $\Sigma$ containing the ray $\rho$ under the projection map $\text{Pr}: N \rightarrow N(\rho)$ given by
$$\text{Pr}:= \begin{bmatrix} 1 & 0 & 0 & 0\\ 0 & 1 & 0 & 0\\ 0 & 0 & 1 & 0\\ \end{bmatrix} \times M^{-1} = \begin{bmatrix} 0 & 3 & -1 & -2\\ -1 & 1 & 2 & -4\\ 0 & -2 & -1 & 2\end{bmatrix}\,\, .$$
The fan $\text{Star}(\rho)$ in $N(\rho)$ is generated by the rays $(-1, 0, 0)$, $(-1, 2, -1)$, $(0, -2, 1)$, $(0, -1, 0)$, $(0, -1, 1)$, $(0, 0, 1)$, $(0, 1, 0)$ and $(1,0,0)$. Let $\Delta^{'}$ be the fan generated by $1$ and $-1$ in the lattice $\mathbb{Z}$, and let $\Delta^{''}$ be the fan generated by $(2, -1)$, $(-2, 1)$, $(-1, 0)$, $(-1, 1)$, $(0,1)$ and $(1, 0)$ in the lattice $\mathbb{Z}^2$. There is an exact sequence of lattices
$$0 \rightarrow \mathbb{Z} \rightarrow N(\rho) \rightarrow \mathbb{Z}^2 \rightarrow 0$$
where the map $\mathbb{Z} \rightarrow N(\rho)$ is the inclusion $n \mapsto (n,0,0)$ and the map $N(\rho) \rightarrow \mathbb{Z}^2$ is the projection $(x,y,z) \mapsto (y,z)$. This exact sequence gives rise to mappings
$$X(\Delta^{'}) \rightarrow X(\text{Star}(\rho)) \rightarrow X(\Delta^{''})$$
where $X(\Delta^{''})$ is the blow up of $\mathbb{F}_2$ in two points, and $X(\Delta^{'})$ is $\mathbb{P}^1$. Thus we have a locally trivial $\mathbb{P}^1$ bundle over $\text{Bl}_2\mathbb{F}_2$.
We can find out what $\tilde{V} \cap D_{\rho}$ is by a local look at the fibers of the map $\tilde{V}\cap D_{\rho} \rightarrow \text{Bl}_1\mathbb{F}_2$. By a similar computation as in $E_8$, we find that the inverse image is a line in one point, and otherwise an isomorphism. Hence, $\tilde{V}\cap D_{\rho}$ is isomorphic to $\text{Bl}_3\mathbb{F}_2$.\vskip 4 pt
\noindent\underline{$E_9$.} Let $\rho$ be generated by the primitive vector $(14, -5, -3, -6)$, and let $M$ be the matrix
$$M = \begin{bmatrix} 0 & 9 & 15 & 14 \\ 0 & -3 & -5 & -5 \\ 0 & -2 & -3 & -3 \\ 1 & -4 & -6 & -6 \\ \end{bmatrix}$$
and let $\tau$ be the cone generated by the columns of $M$, i.e. $\tau = \tau_{50}$. The fan $\text{Star}(\rho)$ is the image of the 10 maximal cones in $\Sigma$ containing the ray $\rho$ under the projection map $\text{Pr}: N \rightarrow N(\rho)$ given by
$$\text{Pr}:= \begin{bmatrix} 1 & 0 & 0 & 0\\ 0 & 1 & 0 & 0\\ 0 & 0 & 1 & 0\\ \end{bmatrix} \times M^{-1} = \begin{bmatrix} 0 & 0 & -2 & 1\\ 0 & 3 & -5 & 0\\ 1 & 1 & 3 & 0\end{bmatrix}\,\, .$$
The fan $\text{Star}(\rho)$ in $N(\rho)$ is generated by the rays $(-1, -3, 2)$, $(-1, -2, 2)$, $(0, -1, 1)$, $(0, 0, -1)$, $(0, 0, 1)$, $(0, 1, 0)$, and $(1,0,0)$. Let $\Delta^{'}$ be the fan generated by $1$ and $-1$ in the lattice $\mathbb{Z}$, and let $\Delta^{''}$ be the fan generated by $(-1, -3)$, $(-1, -2)$, $(0, -1)$, $(0,1)$ and $(1, 0)$ in the lattice $\mathbb{Z}^2$. There is an exact sequence of lattices
$$0 \rightarrow \mathbb{Z} \rightarrow N(\rho) \rightarrow \mathbb{Z}^2 \rightarrow 0$$
where the map $\mathbb{Z} \rightarrow N(\rho)$ is the inclusion $n \mapsto (0,0,n)$ and the map $N(\rho) \rightarrow \mathbb{Z}^2$ is the projection $(x,y,z) \mapsto (x,y)$. This exact sequence gives rise to mappings
$$X(\Delta^{'}) \rightarrow X(\text{Star}(\rho)) \rightarrow X(\Delta^{''})$$
The image of $E_9$ under this last projection is a rational curve on a toric surface, and $E_9$ is a $\mathbb{P}^1$ bundle over this curve, but a local computation as in $E_7$ does not tell us what $\tilde{V} \cap D_{\rho}$ looks like.
We look at the 10 charts of $X_{\Sigma}$ containing $D_{\rho}$. Four of these cover $\tilde{V} \cap D_{\rho}$. A covering is given by the cones $\tau_{22}$, $\tau_{24}$, $\tau_{50}$ and $\tau_{51}$. In these four maps, the polynomial $\overline{f}_{\tau}$ is of the form $1 + x + y^2$, hence $\tilde{V} \cap D_{\rho}$ is a union of four copies of $\mathbb{C}^2$. They glue together to form $\mathbb{F}_2$.
\underline{Method 3.} In two cases $E_i$ is an orbit closure in $X_{\Sigma}$ corresponding to a 2-dimensional cone in $\Sigma$.\vskip 4 pt
\noindent\underline{$E_{10}$.} This component is $D_{\rho_1}\cap D_{\rho_2}$, where $\rho_{1}$ is generated by $(1, 0, 0, 0)$ and $\rho_{2}$ is generated by $(9,-3,-2,-4)$. To see this, consider the chart given by the cone $\tau:= \tau_{42}$. In this chart, $\widetilde{V}$ is generated by $\tilde{f}_{\tau} = y_2y_3 + y_3 + y_1^2y_2 + y_2$. Restricted to $y_2 = 0$ (corresponding to the ray $(1,0,0,0)$), this gives $\overline{f}_{\tau} = y_3$ (corresponding to the ray $(9,-3,-2,-4)$). We now define $N_{\rho_1 ,\rho_2}$ to be the sublattice of $N$ generated (as a group) by $(\rho_1 \cap N)\times (\rho_2 \cap N)$ and
$$N(\rho_1 ,\rho_2) = N/N_{\rho_1,\rho_2},\,\,\,M(\rho_1, \rho_2) = M\cap \rho_1^{\perp}\cap \rho_2^{\perp}$$
Let $M$ be the matrix with columns the vectors generating $\tau$, i.e.
$$\begin{bmatrix} 0 & 15 & 9 & 1\\ 0 & -5 & -3 & 0\\ 0 & -3 & -2 & 0\\ 1 & -6 & -4 & 0\\ \end{bmatrix}\,\, .$$
and let $\text{Pr}: N \rightarrow N(\rho_1 ,\rho_2)$ be the projection map
$$\text{Pr}:= \begin{bmatrix} 1 & 0 & 0 & 0\\ 0 & 1 & 0 & 0\\ \end{bmatrix} \times M^{-1} = \begin{bmatrix} 0 & 0 & -2 & 1\\ 0 & -2 & 3 & 0\\ \end{bmatrix}\,\, .$$
The set of cones in $\Sigma$ containing both $\rho_1$ and $\rho_2$ is defined by its set of images in $N(\rho_1 ,\rho_2)$ under $\text{Pr}$. There are 6 maximal cones in $\Sigma$ containing both $(1, 0, 0, 0)$ and $(9, -3, -2, -4)$, and they project down to the fan generated by the rays $(-2, 3)$, $(-1, 1)$, $(-1, 2)$, $(0, -1)$, $(0, 1)$ and $(1, 0)$. This fan represents $\text{Bl}_3 \mathbb{P}_2$.\vskip 4 pt
\noindent\underline{$E_{11}$.} This is the intersection of $D_{\rho}$, $\rho$ generated by $(9,-3,-2,-4)$, and the non-exceptional divisor corresponding to the ray $(0,0,1,0)$. The computation is similar as for $E_{10}$, with
$$\text{Pr}= \begin{bmatrix} 1 & -1 & 0 & 3\\ 0 & 4 & 0 & -3\\ \end{bmatrix}$$
and the fan generated by the rays $(-1, 4)$, $(0, -1)$, $(0, 1)$, $(1, -1)$ and $(1, 0)$ represents $\text{Bl}_1(\mathbb{F}_4)$.\vskip 4 pt
\underline{Method 4.} We need the following definitions. Let $P$ be a polytope in $\mathbb{R}^d$. For every nonempty face $F$ of $P$ we define
$$N_F := \{ c \in (\mathbb{R}^d)^* | F\subset \{ x \in P | cx\geq cy \,\, \forall y\in P \} \} \,\, .$$
We define the {\it normal fan} $\mathcal{N}_P$ as
$$\mathcal{N}_P = \{ N_F | \text{$F$ is a face of $P$} \} \,\, .$$
\noindent\underline{$E_{12}$} is computed by finding a polytope $\Delta$ in $M(\rho)_{\mathbb{R}}$ which has $\text{Star}(\rho)$ as normal fan. Now the ray $\rho$ is generated by the vector $(5, -1, -1, -2)$. Consider the local chart given by the cone $\tau$, where $\tau$ is spanned by the vectors $(0, 1, 0, 0)$, $(9, -3, -2, -4)$, $(7, -2, -1, -3)$,$(5, -1, -1, -2)$. Let M be the matrix with columns the vectors generating $\tau$, i.e.
$$M = \begin{bmatrix} 0 & 9 & 7 & 5\\ 1 & -3 & -2 & -1\\ 0 & -2 & -1 & -1\\ 0 & -4 & -3 & -2\\ \end{bmatrix}\,\, .$$ \noindent The projection map $\text{Pr}: N \rightarrow N(\rho)$ is given by
$$\text{Pr}:= \begin{bmatrix} 1 & 0 & 0 & 0\\ 0 & 1 & 0 & 0\\ 0 & 0 & 1 & 0\\ \end{bmatrix} \times M^{-1} = \begin{bmatrix} -1 & 1 & 0 & -3\\ -1 & 0 & -1 & -2\\ 0 & 0 & 2 & -1\end{bmatrix}$$
The fan $\text{Star}(\rho)$ in $N(\rho)$ is generated by the rays $(-3, -2, -1)$, $(-1, -1, 0)$, $(0, 0, 1)$, $(0, 1, 0)$ and $(1, 0, 0)$. Up to translation, the polytope with vertices
$$(0, 0, 0), (1, 0, 0), (0, 1, 0), (1, 0, 1), (0, 0, 4), (0, 1, 2)$$
has $\text{Star}(\rho)$ as normal fan, see figure~\ref{figure:polytope}.
\begin{figure}
\caption{The polytope with $\text{Star}(\rho)$ as a normal fan.}
\label{figure:polytope}
\end{figure}
We let $S_{\Delta}$ be the graded $\mathbb{C}$-algebra generated by monomials $t^k\chi^m$, where $m$ is an element of the Minkowski sum $k\Delta$ and $\chi^m = x^{m_1}y^{m_2}z^{m_3}$ for $m = (m_1, m_2, m_3)$. The 10 lattice points contained in the polytope, the 6 vertices above and 4 interior lattice points, give us the equations
$$ \begin{tabular}{l} $z_0 = t$\\ $z_1 = tx$\\ $z_2 = ty$\\ $z_3 = tz$\\ $z_4 = txz$\\ $z_5 = tz^2$\\ $z_6 = tz^3$\\ $z_7 = tz^4$\\ $z_8= tyz$\\ $z_9 = tyz^2$\\ \end{tabular}$$
defining an embedding of $D_{\rho}$ in $\mathbb{P}^{9}$. Its equations are given by the $2\times2$ minors of the matrix
$$\begin{bmatrix}\begin{array}{cccc|c|cc} z_3 & z_5 & z_6 & z_7 & z_4 & z_8 & z_9\\ z_0 & z_3 & z_5 & z_6 & z_1 & z_2 & z_8\\ \end{array} \end{bmatrix}$$
Now consider the vertex $(1,0,1)$, corresponding to the variable $z_4$. Consider the chart $z_4 \neq 0$ with $x_i = z_i/z_4$. The remaining coordinates are $x_1$, $x_7$ and $x_9$, corresponding to the cone with vectors $(-1,0,3)$, $(-1,1,1)$ and $(0,0,-1)$. This is the dual of the cone in $\text{Star}(\rho)$ with rays generated by the vectors $(-3, -2, -1)$, $(-1, -1, 0)$ and $(0, 1, 0)$. In this chart, $\tilde{V}\cap D_{\rho}$ is given by the equation $x_7 + x_9 + x_1^2x_7$. In the torus coordinates, this is $z/x\cdot (1 + y + z^2)$. In fact, in every chart $\tilde{V}\cap D_{\rho}$ is given by the equation $u\cdot (1 + y + z^2)$, where $u$ is an invertible element in $\mathbb{C}[x,y,z, 1/x, 1/y, 1/z]$. Since $(1 + y + z^2) = z_0 + z_2 + z_5$, the equations for $\tilde{V}\cap D_{\rho}$ reduce to the $2\times2$ minors of the matrix
$$\begin{bmatrix}\begin{array}{cccc|c} z_3 & z_5 & z_6 & z_7 & z_4 \\ z_0 & z_3 & z_5 & z_6 & z_1 \\ \end{array} \end{bmatrix}$$
This is a $(4:1)$ rational scroll in this embedding; i.e. $E_{12} = \mathbb{F}_3$.
The space E is a normal crossing divisor. We may therefore describe the intersections of components with a simplicial complex (the {\it dual complex} or {\it intersection complex}). The vertices $\{ i \}$ correspond to the components $E_i$, and $\{i_1, \ldots i_k \}$ is a face if $E_{i_1} \cap \cdots \cap E_{i_k} \neq \emptyset$.
The intersection complex may be computed by looking at the various $\tilde{V}\cap D_{\rho_1} \cap D_{\rho_2}$ and $\tilde{V}\cap D_{\rho_1} \cap D_{\rho_2}\cap D_{\rho_3}$. We list here the facets of the complex.
\begin{align*} \{1,2,7\}, \{2,7,8\}, \{3,8,11\}, \{4,10,11\}, \{4,10,12\}, \{5,7,9\}, \{5,7,10\},\,\,\, \\ \{5,10,12\},\{6,7,9\}, \{6,7,10\}, \{6,10,12\}, \{7,8,9\}, \{7,8,10\}, \{8,10,11\}.\\ \end{align*}
We see that there are 14 facets, corresponding to 14 intersection points of 3 components and 25 edges corresponding to 25 projective lines which are the intersections of 2 components.
\newtheorem{euler25}[eulerBoehm]{Lemma} \begin{euler25} The Euler characteristic of $E$ is 25. \end{euler25}\label{proposition:e25}
\begin{proof} Using the inclusion-exclusion principle and the fact that $E$ has normal crossings, we may compute the Euler characteristic $\chi(E)$ by
$$\chi(E) = \underset{i}{\textstyle\sum \, } \chi(E_i) - \underset{i<j}{\textstyle\sum \, }\chi(E_i\cap E_j) + \underset{i<j<k}{\textstyle\sum \, }
\chi(E_i\cap E_j \cap E_k)\,\,\, .$$
Now from Table~\ref{table:exceptional} we count $\Sigma \chi (E_i) = 61$. We have $\chi (\mathbb{P}^1) = 2$. Thus from the intersection complex, we compute $\chi(E) = 61 - 25\times 2 + 14 = 25$. \end{proof}
For the other 2 (quotient) singularities we may use the McKay correspondence as conjectured by Miles Reid and proved by Batyrev in~\cite{bat}, Theorem 1.10. In a crepant resolution of $\mathbb{C}^n/H$, $H$ a finite subgroup of $SL_n$, the Euler characteristic of the exceptional divisor will be the number of conjugacy classes in $H$. In our case this is 13. Denote these exceptional divisors $E'$.
We have constructed a resolution $M_s \rightarrow Y_s$, where $Y_s$ is the quotient $X_s/H$. Let $U$ be the complement of the 6 singular points in $Y_s$.
\newtheorem{ueuler}[eulerBoehm]{Lemma} \begin{ueuler} The Euler characteristic of $U$ is $-6$. \end{ueuler}\label{lemma:ueuler}
\begin{proof}The singular variety $X_s$ smooths to the general degree 13 Calabi-Yau 3-fold in $\mathbb{P}^6$. A Macaulay 2 computation shows that the total space of a general one parameter smoothing is smooth. The smooth fiber has Euler characteristic -120, see Proposition~\ref{proposition:ex4cohom}. The Milnor fiber of the 3-dimensional $Q_{12}$ singularity is a wedge sum of 12 3-spheres. Thus $\chi(X_s) = -120 + 4\times12 = -72$. Hence
$$\chi(U) = \frac{\chi(X_s \setminus \{ 6\, \text{points} \} )}{13} = \frac{-72 - 6}{13} = -6$$ \end{proof}
\noindent We can now put all this together to prove the main result of this section.\\
{\it Proof of Theorem~\ref{theorem:eBoehm}}. Since the resolution $M_s \rightarrow Y_s$ is an isomorphism away from the 6 points, four points with exceptional divisor $E$ and two points with exceptional divisor $E^{\prime}$, we have $\chi(M_s) = \chi(U) + 4\chi(E) +2\chi(E') = -6 + 4\cdot25 + 2\cdot 13 = 120$.
\begin{flushleft}
\appendix
\chapter{Computer Calculations} The following is a Macaulay 2 code for computing $T^1_{X}$ of an variety $X = \text{Proj}( T/p)$, for an ideal $p$ in a ring $T$.\\ \ \\ \
\begin{table}[h] \begin{center} \begin{tabular}{l} A = resolution(p, LengthLimit => 3)\\ rel = transpose(A.dd$\_$2)\\ dp = transpose jacobian(A.dd$\_$1)\\ R = T/p\\ Rel = substitute(rel,R)\\ Dp = substitute(dp,R)\\ Der = image Dp\\ N = kernel Rel\\ N0 = image basis(0,N)\\ Der0 = image basis(0,Der)\\ isSubset(Der0,N0)\\ T1temp = N0/Der0\\ T1 = trim T1temp\\ T1mat = gens T1 \end{tabular} \end{center} \end{table}\label{sideM2code} \end{flushleft} \begin{table}[tp]\footnotesize \begin{center} \begin{tabular}{ll} $\tau_1:$ & (13, -5, -3, -6), (0, 0, 1, 0), (3, -1, 0, -1), (8, -3, -1, -3)\\ $\tau_2:$ & (13, -5, -3, -6), (3, -1, 0, -1), (6, -2, -1, -2),(11, -4, -2, -4)\\ $\tau_3:$ & (0, 0, 0, 1), (6, -2, -1, -2), (14, -5, -3, -6), (15, -5, -3, -6)\\ $\tau_4:$ & (13, -5, -3, -6), (3, -1, 0, -1), (6, -2, -1, -2),(14, -5, -3, -6)\\ $\tau_5:$ & (0, 1, 0, 0), (0, 0, 1, 0), (9, -3, -2, -4), (11, -4, -2, -5)\\ $\tau_6:$ & (0, 1, 0, 0), (0, 0, 1, 0), (7, -2, -1, -3), (5, -1, 0, -2)\\ $\tau_7:$ & (0, 0, 1, 0), (1, 0, 0, 0), (3, 0, 0, -1), (5, -1, 0, -2)\\ $\tau_8:$ & (0, 1, 0, 0), (0, 0, 0, 1), (9, -3, -2, -4), (5, -1, -1, -2)\\ $\tau_9:$ & (0, 1, 0, 0), (1, 0, 0, 0), (7, -2, -1, -3), (5, -1, -1, -2)\\ $\tau_{10}:$ & (0, 0, 0, 1), (1, 0, 0, 0), (6, -2, -1, -2), (15, -5, -3, -6)\\ $\tau_{11}:$ & (1, 0, 0, 0), (3, -1, 0, -1), (6, -2, -1, -2), (15, -5, -3, -6)\\ $\tau_{12}:$ & (0, 0, 1, 0), (1, 0, 0, 0), (3, -1, 0, -1), (12, -4, -2, -5)\\ $\tau_{13}:$ & (0, 0, 1, 0), (1, 0, 0, 0), (9, -3, -2, -4), (12, -4, -2, -5)\\ $\tau_{14}:$ & (0, 1, 0, 0), (0, 0, 0, 1), (0, 0, 1, 0), (1, 0, 0, 0)\\ $\tau_{15}:$ & (0, 0, 0, 1), (1, 0, 0, 0), (3, -1, 0, -1), (6, -2, -1, -2)\\ $\tau_{16}:$ & (9, -3, -2, -4), (1, 0, 0, 0), (7, -2, -1, -3), (5, -1, -1, -2)\\ $\tau_{17}:$ & (0, 1, 0, 0), (0, 0, 0, 1), (1, 0, 0, 0), (5, -1, -1, -2)\\ $\tau_{18}:$ & (1, 0, 0, 0), (7, -2, -1, -3), (3, 0, 0, -1), (5, -1, 0, -2)\\ $\tau_{19}:$ & (0, 1, 0, 0), (0, 0, 1, 0), (3, 0, 0, -1), (5, -1, 0, -2)\\ $\tau_{20}:$ & (13, -5, -3, -6), (0, 0, 1, 0), (3, -1, 0, -1), (11, -4, -2, -5)\\ $\tau_{21}:$ & (3, -1, 0, -1), (14, -5, -3, -6), (15, -5, -3, -6),(12, -4, -2, -5)\\ $\tau_{22}:$ & (9, -3, -2, -4), (14, -5, -3, -6), (15, -5, -3, -6),(12, -4, -2, -5)\\ $\tau_{23}:$ & (3, -1, 0, -1), (13, -5, -3, -6), (14, -5, -3, -6),(12, -4, -2, -5)\\ $\tau_{24}:$ & (9, -3, -2, -4), (13, -5, -3, -6), (14, -5, -3, -6),(12, -4, -2, -5)\\ $\tau_{25}:$ & (0, 0, 0, 1), (3, -1, 0, -1), (6, -2, -1, -2), (11, -4, -2, -4)\\ $\tau_{26}:$ & (0, 0, 0, 1), (0, 0, 1, 0), (3, -1, 0, -1), (8, -3, -1, -3)\\ $\tau_{27}:$ & (13, -5, -3, -6), (0, 0, 0, 1), (3, -1, 0, -1), (8, -3, -1, -3)\\ $\tau_{28}:$ & (13, -5, -3, -6), (0, 0, 0, 1), (6, -2, -1, -2), (11, -4, -2, -4)\\ $\tau_{29}:$ & (13, -5, -3, -6), (0, 0, 0, 1), (6, -2, -1, -2), (14, -5, -3, -6)\\ $\tau_{30}:$ & (3, -1, 0, -1), (6, -2, -1, -2), (14, -5, -3, -6),(15, -5, -3, -6)\\ $\tau_{31}:$ & (13, -5, -3, -6), (0, 1, 0, 0), (9, -3, -2, -4), (11, -4, -2, -5)\\ $\tau_{32}:$ & (0, 0, 1, 0), (9, -3, -2, -4), (11, -4, -2, -5), (12, -4, -2, -5)\\ $\tau_{33}:$ & (0, 0, 1, 0), (3, -1, 0, -1), (11, -4, -2, -5), (12, -4, -2, -5)\\ $\tau_{34}:$ & (0, 0, 1, 0), (1, 0, 0, 0), (7, -2, -1, -3), (5, -1, 0, -2)\\ $\tau_{35}:$ & (0, 1, 0, 0), (1, 0, 0, 0), (7, -2, -1, -3), (3, 0, 0, -1)\\ $\tau_{36}:$ & (0, 1, 0, 0), (9, -3, -2, -4), (7, -2, -1, -3), (5, -1, -1, -2)\\ $\tau_{37}:$ & (0, 0, 1, 0), (9, -3, -2, -4), (1, 0, 0, 0), (7, -2, -1, -3)\\ $\tau_{38}:$ & (0, 0, 0, 1), (0, 0, 1, 0), (1, 0, 0, 0), (3, -1, 0, -1)\\ $\tau_{39}:$ & (13, -5, -3, -6), (0, 1, 0, 0), (0, 0, 0, 1), (9, -3, -2, -4)\\ $\tau_{40}:$ & (1, 0, 0, 0), (9, -3, -2, -4), (15, -5, -3, -6), (12, -4, -2, -5)\\ $\tau_{41}:$ & (1, 0, 0, 0), (3, -1, 0, -1), (15, -5, -3, -6), (12, -4, -2, -5)\\ $\tau_{42}:$ & (0, 0, 0, 1), (1, 0, 0, 0), (9, -3, -2, -4), (15, -5, -3, -6)\\ $\tau_{43}:$ & (0, 1, 0, 0), (0, 0, 1, 0), (9, -3, -2, -4), (7, -2, -1, -3)\\ $\tau_{44}:$ & (0, 0, 0, 1), (9, -3, -2, -4), (1, 0, 0, 0), (5, -1, -1, -2)\\ $\tau_{45}:$ & (0, 1, 0, 0), (0, 0, 1, 0), (1, 0, 0, 0), (3, 0, 0, -1)\\ $\tau_{46}:$ & (0, 1, 0, 0), (7, -2, -1, -3), (3, 0, 0, -1), (5, -1, 0, -2)\\ $\tau_{47}:$ & (9, -3, -2, -4), (11, -4, -2, -5), (13, -5, -3, -6), (12, -4, -2, -5)\\ $\tau_{48}:$ & (3, -1, 0, -1), (11, -4, -2, -5), (13, -5, -3, -6),(12, -4, -2, -5)\\ $\tau_{49}:$ & (13, -5, -3, -6), (0, 1, 0, 0), (0, 0, 1, 0), (11, -4, -2, -5)\\ $\tau_{50}:$ & (0, 0, 0, 1), (9, -3, -2, -4), (14, -5, -3, -6), (15, -5, -3, -6)\\ $\tau_{51}:$ & (13, -5, -3, -6), (0, 0, 0, 1), (9, -3, -2, -4), (14, -5, -3, -6)\\ $\tau_{52}:$ & (13, -5, -3, -6), (0, 0, 0, 1), (3, -1, 0, -1), (11, -4, -2, -4)\\ $\tau_{53}:$ & (13, -5, -3, -6), (0, 0, 0, 1), (0, 0, 1, 0), (8, -3, -1, -3)\\ \end{tabular}\label{table:cones}\caption{The maximal cones of the subdivision $\Sigma$.}\end{center}\end{table}
\begin{flushleft} \chapter{Explicit Expressions for the Varieties in Chapter 3}\chaptermark{Explicit Expressions} In the $P^7_1$ case, studied in Section~\ref{ex1section}, the linear entries of $M^{1}$ are\label{ex1refUttrykk}
\[ \begin{array}{l} l_{1} = x_{4}\\ l_{2} = t_{65}x_{1} + t_{57}x_{2}+ t_{61}x_{3} + t_{30}x_{5}\\ l_{3} = -x_{5} \\ l_{4} = x_{6}\\ l_{5} = -t_{55}x_{1} - t_{59}x_{2}-t_{63}x_{3}-t_{33}x_{6}\\ l_{6} = -x_{7} \end{array} \]
and the cubic terms are
\[ \begin{array}{l} g_{1} = -t_{15}x_{7}^{3}-t_{17}x_{3}x_{7}^2 -t_{18}x_{2}x_{7}^2 -t_{19}x_{1}x_{7}^{2} -t_{22}x_{2}x_{3}x_{7} -t_{24}x_{1}x_{3}x_{7}\\ -t_{25}x_{1}x_{2}x_{7} -t_{36}x_{1}^3 - t_{37}x_{1}^2x_{7} -t_{42}x_{2}^2x_{7} -t_{43}x_{2}^3 -t_{46}x_{3}^3 -t_{47}x_{3}^2x_{7}\\ -t_{67}x_{1}^2x_{3} - t_{68}x_{1}^2x_{2} - t_{76}x_{2}^2x_{3} -t_{77}x_{1}x_{2}^2 -t_{82}x_{2}x_{3}^2 -t_{83}x_{1} x_{3}^2\\ \ \\ g_{2} = -t_{1}x_{1}^3 -t_{2}x_{2}^3 -t_{3}x_{3}^3 -t_{6}x_{4}^3 -t_{9}x_{3}x_{4}^2 -t_{10}x_{2}x_{4}^2 -t_{11}x_{1}x_{4}^2 \\ -t_{16}x_{7}^3 -t_{20}x_{3}x_{7}^2 -t_{21}x_{2}x_{7}^2 -t_{23}x_{1}x_{7}^2 -t_{26}x_{5}^3 -t_{27}x_{2}x_{5}^2 -t_{28}x_{3}x_{5}^2\\ -t_{29}x_{1}x_{5}^2 -t_{31}x_{6}^3 -t_{32}x_{3}x_{6}^2 -t_{34}x_{1}x_{6}^2 -t_{35}x_{2}x_{6}^2 -t_{48}x_{4}^2x_{5} -t_{49}x_{4}x_{5}^2\\
-t_{50}x_{5}^2x_{6} -t_{51}x_{5}x_{6}^2 -t_{52}x_{6}^2x_{7} -t_{53}x_{6}x_{7}^2 -t_{54}x_{1}^2x_{6} -t_{56}x_{2}^2x_{5} -t_{58}x_{2}^2x_{6}\\ -t_{60}x_{3}^2x_{5} -t_{62}x_{3}^2x_{6} -t_{64}x_{1}^2x_{5} -t_{66}x_{1}^2x_{7} -t_{69}x_{1}^2x_{4} -t_{72}x_{2}^2x_{4} -t_{75}x_{2}^2x_{7}\\
-t_{78}x_{3}^2x_{4} -t_{81}x_{3}^2x_{7} -t_{84}x_{3}x_{4}x_{5} -t_{85}x_{2}x_{4}x_{5} -t_{86}x_{1}x_{4}x_{5} -t_{87}x_{3}x_{5}x_{6}\\
-t_{88}x_{2}x_{5}x_{6} -t_{89}x_{1}x_{5}x_{6} -t_{90}x_{3}x_{6}x_{7} -t_{91}x_{2}x_{6}x_{7} -t_{92}x_{1}x_{6}x_{7} -x_{1}x_{2}x_{3}\\ \ \\ g_{3} = -t_{4}x _{4}^3 -t_{5}x _{3} x _{4}^2 -t_{6}x _{3} x _{4}^2 + t_{6}x _{4}^2 x _{5} -t_{7}x _{1} x _{4}^2 -t _{8}x _{2} x _{4}^2 -t _{12}x _{2}x _{3}x _{4}\\
-t _{13}x_{1}x_{3} x_{4} -t _{14}x _{1} x _{2}x _{4} -t_{38}x _{1}^2 x _{4} -t_{39}x_{1}^3 -t_{40}x _{2}^3 -t_{41}x _{2}^2 x _{4} -t_{44}x _{3}^3\\ -t_{45}x_{3}^2x_{4} -t_{70}x_{1}^2 x _{3} -t_{71}x _{1}^2 x _{2} -t_{73}x _{2}^2 x _{3} -t_{74}x _{1}x _{2}^2 -t_{79}x _{2} x_{3}^2 -t _{80}x _{1} x _{3}^2 \end{array} \]
In the $P^7_2$ case, studied in Section~\ref{ex2section}, the cubic $g$ is
\[ \begin{array}{l}g = t_{15}x_7^3 + t_{16}x_6x_7^2 + t_{17}x_4x_7^2 + t_{18}x_3x_7^2 + t_{19}x_2x_7^2 + t_{21}x_3x_6x_7 + t_{22}x_3x_4x_7\\ \,\,+ t_{23}x_2x_6x_7 + t_{24}x_2x_4x_7 + t_{50}x_2^3 + t_{51}x_2^2x_7 + t_{52}x_2^2x_6 + t_{53}x_2^2x_4 + t_{55}x_3^3\\ \,\,+ t_{56}x_3^2x_7 + t_{57}x_3^2x_6 + t_{58}x_3^2x_4 + t_{65}x_4^3 + t_{66}x_4^2x_7 + t_{68}x_3x_4^2 + t_{69}x_2x_4^2\\ \,\,+ t_{75}x_6^3 + t_{76}x_6^2x_7 + t_{78}x_3x_6^2 + t_{79}x_2x_6^2\label{ex2refUttrykk} \,\, ,\end{array} \]
and the quadrics $q_1,\ldots,q_4$ are
\[ \begin{array}{l} q_1 = t_{10}x_5^2 + t_{12}x_3x_5 + t_{13}x_2x_5 + t_{34}x_2^2 + t_{36}x_3^2 + t_{60}x_4^2 + t_{61}x_4x_5\\ \,\,+ t_{62}x_3x_4 + t_{63}x_2x_4 + t_{70}x_5x_6 + t_{71}x_6^2 + t_{72}x_3x_6 + t_{73}x_2x_6\\ \ \\ q_2 = x_4x_6 + t_1x_2^2 + t_2x_3^2 + t_8x_1^2 + t_{11}x_5^2 + t_{25}x_7^2 + t_{29}x_1x_5 + t_{32}x_2x_3 + t_{35}x_2x_5\\ \,\, + t_{37}x_3x_5 + t_{43}x_1x_2 + t_{48}x_1x_3 + t_{54}x_2x_7 + t_{59}x_3x_7\\ \ \\ q_3 = x_2x_3 + t_3x_4^2 + t_4x_6^2 + t_6x_1^2 + t_{14}x_5^2 + t_{20}x_7^2 + t_{26}x_1x_4 + t_{28}x_1x_5 + t_{30}x_1x_6\\ \,\,+ t_{39}x_4x_6 + t_{64}x_4x_5 + t_{67}x_4x_7 + t_{74}x_5x_6 + t_{77}x_6x_7\\ \ \\ q_4 = t_5x_1^2 + t_7x_1x_6 + t_9x_1x_4 + t_{27}x_4^2 +t_{31}x_6^2 + t_{40}x_1x_2 + t_{41}x_2^2 + t_{42}x_2x_6\\ \,\,+ t_{44}x_2x_4 + t_{45}x_1x_3 + t_{46}x_3^2 +t_{47}x_3x_6 + t_{49}x_3x_4\, \, \, .\end{array}\]
In the $P^7_3$ case, studied in Section~\ref{ex3section}, the entries of the syzygy matrix $M^{1}$ are given by
\[ \begin{array}{l} g = x_1x_2x_3 + t_1x_1^3 + t_4x_2^3 + t_7x_3^3 + t_{10}x_4^3 + t_{12}x_3x_4^2 + t_{13}x_2x_4^2 + t_{14}x_1x_4^2\\ \label{ex3refUttrykk} \,\,+ t_{15}x_5^3 + t_{17}x_3x_5^2 + t_{18}x_2x_5^2 + t_{19}x_1x_5^2 + t_{20}x_6^3 + t_{21}x_7^3 + t_{23}x_3x_6^2 + t_{24}x_2x_6^2\\ \,\,+ t_{25}x_1x_6^2 + t_{27}x_3x_7^2 + t_{28}x_2x_7^2 + t_{29}x_1x_7^2 + t_{36}x_1^2x_4 + t_{38}x_1^2x_5 + t_{40}x_1^2x_6\\ \,\, + t_{42}x_1^2x_7 + t_{44}x_2^2x_4 + t_{46}x_2^2x_5 + t_{48}x_2^2x_6 + t_{50}x_2^2x_7 + t_{52}x_3^2x_4 + t_{54}x_3^2x_5\\ \,\,+ t_{56}x_3^2x_6 + t_{58}x_3^2x_7 + t_{60}x_4^2x_6 + t_{61}x_3x_4x_6 + t_{62}x_2x_4x_6 + t_{63}x_1x_4x_6\\ \,\,+ t_{64}x_4^2x_7 + t_{65}x_3x_4x_7 + t_{66}x_2x_4x_7 + t_{67}x_1x_4x_7 + t_{68}x_5^2x_6 + t_{69}x_3x_5x_6\\ \,\, + t_{70}x_2x_5x_6 + t_{71}x_1x_5x_6 + t_{72}x_5^2x_7 + t_{73}x_3x_5x_7 + t_{74}x_2x_5x_7 + t_{75}x_1x_5x_7\\ \,\,+ t_{76}x_4x_6^2 + t_{77}x_4x_7^2 + t_{78}x_5x_6^2 + t_{79}x_5x_7^2\end{array}\] \[ \begin{array}{l} q_1 = x_4x_5 + t_2x_1^2 + t_5x_2^2 + t_8x_3^2 + t_{22}x_6^2 + t_{26}x_7^2 + t_{30}x_1x_2 + t_{32}x_1x_3 + t_{34}x_2x_3\\ \,\, + t_{41}x_1x_6 + t_{43}x_1x_7 + t_{49}x_2x_6 + t_{51}x_2x_7 + t_{57}x_3x_6 + t_{59}x_3x_7\end{array} \] \[ \begin{array}{l} q_2 = x_6x_7 + t_3x_1^2 + t_6x_2^2 + t_9x_3^2 + t_{11}x_4^2 + t_{16}x_5^2 + t_{31}x_1x_2 + t_{33}x_1x_3 + t_{35}x_2x_3\\ \,\, + t_{37}x_1x_4 + t_{39}x_1x_5 + t_{45}x_2x_4 + t_{45}x_2x_5 + t_{53}x_3x_4 + t_{55}x_3x_5\,\, .\end{array} \]\pagebreak
In the $P^7_4$ case, studied in Section~\ref{ex4section}, the quadrics are given by
\[ \begin{array}{l} q_1 = t_{20}x_1^2 + t_{22}x_2^2 + t_{35}x_1x_3 + t_{38}x_2x_3 + t_{57}x_3^2 + t_{34}x_1x_4 + t_{37}x_2x_4 \\ \,\, + t_{67}x_4^2 + t_{21}x_1x_5 + t_{23}x_2x_5 + t_{8}x_3x_5 + t_{7}x_4x_5 + t_{5}x_5^2\\ \ \\ q_2 = t_{1}x_1^2+t_{30}x_1x_2+t_2x_2^2 + x_3x_4+t_{33}x_1x_5 + t_{36}x_2x_5 +t_6x_5^2+t_{48}x_1x_6\\ \,\, +t_{52}x_2x_6 +t_{63}x_5x_6+t_{16}x_6^2 +t_{50}x_1x_7+t_{55}x_2x_7+t_{65}x_6x_7+t_{14}x_7^2\\ \ \\ q_3 = -x_1x_2-t_3x_3^2 -t_{39}x_3x_4-t_4x_4^2 -t_{56}x_3x_5 -t_{66}x_4x_5 -t_9x_5^2-t_{58}x_3x_6\\ \,\, -t_{60}x_4x_6-t_{62}x_5x_6-t_{15}x_6^2 -t_{42}x_3x_7 -t_{45}x_4x_7-t_{64}x_6x_7-t_{11}x_7^2\\ \ \\ q_4 = -t_{51}x_1^2-t_{54}x_2^2-t_{44}x_1x_3 -t_{43}x_2x_3 -t_{26}x_3^2 -t_{47}x_1x_4 -t_{46}x_2x_4\\ \,\, -t_{28}x_4^2 -t_{13}x_1x_7 -t_{12}x_2x_7 -t_{27}x_3x_7 -t_{29}x_4x_7-t_{10}x_7^2\,\, ,\end{array} \]\label{ex4refUttrykk}
and the linear forms are given by
\[ \begin{array}{l} l_1 = t_{18}x_1+t_{19}x_2+t_{32}x_3+t_{31}x_4\\ l_2 = x_7\\ l_3 = -x_6\\ l_4 = t_{49}x_1+t_{53}x_2+t_{59}x_3+t_{61}x_4+t_{17}x_6\\ l_5 = x_5\\ l_6 = t_{41}x_1+t_{40}x_2+t_{24}x_3+t_{25}x_4\,\, . \end{array} \]\\ \ \\ \ \\
In the $P^7_5$ case, studied in Section~\ref{ex5section}, the entries of the syzygy matrix $M^1$ are\label{ex5refUttrykk}
\[ \begin{array}{l} l_1 = t_8x_1 + t_9x_3 + t_{24}x_2\\ l_2 = t_1x_1+t_{23}x_3+t_{26}x_6 + t_{44}x_4+t_{45}x_5\\ l_3 = t_5x_5 + t_{36}x_3 + t_{42}x_7 + t_{46}x_1 + t_{48}x_2\\ l_4 = t_{16}x_3+t_{17}x_5 + t_{35}x_4\\ l_5 = t_{10}x_1 + t_{11}x_6 + t_{25}x_7\\ l_6 = t_6x_6 + t_{27}x_1 + t_{39}x_4 +t_{50}x_2+t_{52}x_3\\ l_7 = t_3x_3 + t_{22}x_1 + t_{34}x_5 + t_{51}x_6 + t_{54}x_7\\ l_8 = t_{18}x_4 +t_{19}x_6 +t_{38}x_5\\ l_9 = t_4x_4 + t_{30}x_2 + t_{37}x_6 + t_{43}x_1 + t_{56}x_7\\ l_{10} = t_{12}x_2 + t_{13}x_4 + t_{29}x_3 \\ l_{11} = t_2x_2 + t_{28}x_4 + t_{31}x_7 + t_{47}x_5 + t_{49}x_6\\ l_{12} = t_{14}x_2+t_{15}x_7 + t_{33}x_1\\ l_{13} = t_7x_7 + t_{32}x_2 + t_{41}x_5 + t_{53}x_3 + t_{55}x_4\\ l_{14} = t_{20}x_5 + t_{21}x_7 + t_{40}x_6\,\, . \end{array} \]
\end{flushleft}
\end{document} | arXiv |
Oscillations in suspension bridges, vertical and torsional
DCDS-S Home
A clamped plate with a uniform weight may change sign
August 2014, 7(4): 767-783. doi: 10.3934/dcdss.2014.7.767
Global solutions for a nonlinear integral equation with a generalized heat kernel
Kazuhiro Ishige 1, , Tatsuki Kawakami 2, and Kanako Kobayashi 1,
Mathematical Institute, Tohoku University, Aoba, Sendai 980-8578, Japan
Department of Mathematical Sciences, Osaka Prefecture University, Sakai 599-8531, Japan
Received September 2013 Published February 2014
We study the existence and the large time behavior of global-in-time solutions of a nonlinear integral equation with a generalized heat kernel \begin{eqnarray*} & & u(x,t)=\int_{{\mathbb R}^N}G(x-y,t)\varphi(y)dy\\ & & \qquad\quad +\int_0^t\int_{{\mathbb R}^N}G(x-y,t-s)F(y,s,u(y,s),\dots,\nabla^\ell u(y,s))dyds, \end{eqnarray*} where $\varphi\in W^{\ell,\infty}({\mathbb R}^N)$ and $\ell\in\{0,1,\dots\}$. The arguments of this paper are applicable to the Cauchy problem for various nonlinear parabolic equations such as fractional semilinear parabolic equations, higher order semilinear parabolic equations and viscous Hamilton-Jacobi equations.
Keywords: generalized heat kernel, Global solutions, nonlinear integral equation., weak $L^r$ space, semilinear parabolic equations.
Mathematics Subject Classification: Primary: 35A01, 35K55; Secondary: 35K9.
Citation: Kazuhiro Ishige, Tatsuki Kawakami, Kanako Kobayashi. Global solutions for a nonlinear integral equation with a generalized heat kernel. Discrete & Continuous Dynamical Systems - S, 2014, 7 (4) : 767-783. doi: 10.3934/dcdss.2014.7.767
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Kazuhiro Ishige Tatsuki Kawakami Kanako Kobayashi | CommonCrawl |
Many indirect loan products require that fees be paid to the firm that originates the loan--an auto dealer for example. Many institutions amortize these fees using a straight-line method over a period of months approximately equal to the estimated life of the loan. For optimizing the performance of a loan portfolio, it is important to understand how the costs associated with these loans and how the up-front fees affect the effective yield on a loan; the straight-line method isn't very good at this.
In some cases, institutions must amortize these fees in a way that satisfies SFAS 91, Accounting for Nonrefundable Fees and Costs Associated with Originating or Acquiring Loans and Initial Direct Costs of Leases. The discussion that follows is designed to provide a way to calculate fee amortizations that will work for loan portfolio optimization calculations. It may (or may not) be helpful in calculating amortizations for SFAS 91; you should discuss this with your accountant.
Using this method to calculate fee amortizations for loan portfolio pricing does not require that an institution use this method for financial accounting.
Example Calculation
The calculation is easiest to describe with an example loan:
Principal $10,000
Fee paid $1,000
Interest Rate 7%
Term 60 months
Payment $198
Since level-yield calculations treat the unamortized fee as part of the loan balance, let's treat the fee amortization just like another payment with the same term and interest rate:
Fee Interest Rate 7%
Fee pseudo payment $19.80
With this basic information, it is now time to calculate the amortization for a few periods, as shown in Table 1. The columns in coral show the calculation of the monthly principal portion of the monthly payment, with monthly principal of $139.89, $140.49 and $141.31.
Similarly, the cyan columns show the calculation of the monthly fee "principal" amortization, with fee amortization of $13.97, $14.05 and $14.13 respectively.
The light grey columns show the calculation of the level yield as the interest ($58.33 for period 0) divided by the level yield asset ($11,000) multiplied by 12 periods to annualize the result, which gives 6.36%. Repeating this for the other periods confirms that the yield on the combined asset is the same for each period.
What happens to the $5.83 "pseudo interest" in the amortization calculation. If we divide this by the the level yield asset balance ($11,000) and multiply by 12 to annualize it, we get 0.64%--the difference between the contracted 7% interest rate and the effective yield after fee amortization.
Table 1: Example Loan Principal and Fee Amortization for Three Periods
Level Yield Amortization Detailed Calculation
Level Yield
Level Yield Simplified Calculation
Applied Principal
Fee Balance
Fee Pseudo Payment
Fee Pseudo Interest
Applied Fee Principal Amortization Expense
Level Yield Asset
Yield After Fee Amortization
Amortization Expense Reduction to Contract Yield
Simplified Calculation of Amortization Expense Reduction to Contract Yield
Simplified Calculation of Amortization Expense
0 $10,000.00 -$198.01 $58.33 -$139.68 $1,000.00 -$19.80 $5.83 -$13.97 $11,000.00 6.36 0.64 0.64 -$13.97
1 $9,860.32 -$198.01 $57.52 -$140.49 $986.03 -$19.80 $5.75 -$14.05 $10,846.35 6.36 0.64 0.64 -$14.05
Calculating Fee Amortization for Prepayments
This approach to calculating the fee amortization works fine until an asset prepays. For full prepayment, this is easy--the entire remaining balance is amortized all at once--but how do you calculate the fee amortization for a partial pre-payment?
To do this, first we should look for an easier way to calculate the monthly fee amortization amount in a way that doesn't require calculating both the pseudo payment and pseudo interest for the fee. Notice that the fee pseudo payment is proportional to the fee balance divided by the principal balance--$1,000/$10,000 or 0.1 in this case. Similarly, the applied principal, $139.68 is proportional to the fee amortization, $13.97.
From this we can calculate the monthly fee amortization as
\[ \begin{aligned} \text{fee amortization}&=&\text{principal reduction}*\frac{\text{fee balance}}{\text{principal balance}} \\ &=&139.68*\frac{1000}{10000} \\ &=&13.97 \end{aligned} \]
To calculate the prepayment of an unusual amount--perhaps a double payment in month 0--we would just take the principal applied, and use the formula above to calculate the fee amortization:
\[ \begin{aligned} \text{fee amortization}&=&\text{principal reduction}*\frac{\text{fee balance}}{\text{principal balance}} \\ &=&(139.68+198.01)*\frac{1000}{10000} \\ &=&33.77 \end{aligned} \]
Calculation Methods for Loan Portfolios
The example also shows how this would be implemented in practice. For loan pricing optimization, the effective yield is needed for each loan type, term and credit grade--including prepayments. Calculating a pseudo payment for each loan and determining the fee amortization by month would be programmatically painful and inefficient. Since the principal portion of the payment would be present in most accounting systems at the loan level, this becomes an easy way to retroactively calculate the fee amortization for effective yield. This is also a calculation that is necessary if the institution decides to convert from one amortization method to a level yield method.
A complete example of this approach in an Excel spreadsheet can be downloaded here.
The formula displays in this example are formatted using MathJax.
{calltoaction}
UseR! 2013 in Albacete, Spain
The UseR! 2013 Conference was held in Albacete, Spain July 9-12, 2013. I attended the conference and presented a summary of the conference highlights to the DFW R User Group at the August 24th meeting.
Since it takes about 90 minutes to deliver the presentation, I reworked it in Beamer so that it would have a progress outline and easier bullet point/remote control clicking.
The slides from the presentation are available in two forms:
Including photos of Albacete and the venue. This download is 37 megabytes.
A version without the photos. This download is 2.6 megabytes.
In converting from PowerPoint to Beamer, there are some issues with quotes. If you see a typo, let me know.
Auto Loan Severity of Loss and the CPI for Used Cars and Trucks
Institutions typically use their own experience to estimate recoveries and thus the severity of loss for auto loan defaults. The Consumer Price Index for Used Cars and Trucks (CPI-UCT) can be helpful in determining whether changes in recoveries and severity of loss are due to declining credit quality or changes in the used car and truck market. When used car prices fall and thus the CPI-UCT falls, auction prices are lower and the recoveries on repossessed vehicles are lower, leading to a higher severity of loss. When the CPI rises, auction prices are higher, recoveries are higher and the severity of loss is lower. The discussion that follows will illustrate how changes in the CPI-UCT and the averaging period used for pricing-related loss rate forecasting can help in understanding changes in severity of loss for auto loan portfolios.
The CPI-UCT has stabilized since mid-2011 as shown in Figure 1. This will change the severity of loss behavior of most auto loan portfolios, causing losses to be higher than expected for institutions that price loans using an estimate for the severity of loss using a rolling average with a period of more than two years.
Figure 1 shows the dramatic drop in the CPI-UCT during the early part of the recession in 2008-2009, followed by a dramatic rebound due to the reduced new-car manufacturing capacity that resulted from bankruptcies and restructuring in the auto industry. The rebound largely ended in the summer of 2011. The CPI-UCT is currently dominated by seasonal used-auto purchase demand; it is high in the Spring and low in the Winter.
For institutions that use a rolling average to estimate future severity of loss for new and used auto loans, the changes in the used car market reflected in the CPI-UCT will affect the actual severity of loss compared to the severity of loss estimate used to price loans at the time of origination. Institutions that use a rolling average of more than 5 years have experienced lower than expected severity of loss, and thus higher than expected net interest income. This better than expected performance is about to end for these institutions, not because loans are performing better or worse, but because the 2008-2009 crash in the used car market is about to roll out of the data set used to calculate the rolling average that was used in pricing the loans.
Figure 2 shows the 5-year rolling average for CPI-UCT and has a clearly rising value that will cause losses to be lower then originally estimated. For example, loans that originated in January of 2012 (rolling average of 138 used for pricing) and defaulted in January of 2013 (actual value of 147 at time of default) saw a 6% higher recovery than expected. The better than expected performance will decline as the 2008-2009 crash numbers continue to drop out of a 5-year rolling average in the coming months.
This affect would be greatest for loans originated in October, November or December of 2009 when the 5-year rolling average hit a minimum of slightly more than 135. Loans originated at the bottom would experience an approximately 8.5% better than expected recovery in the event of default.
Institutions that use shorter rolling averages have already begun to see the 2008-2009 crash values and the subsequent rebound begin to drop out. For example, using a 3-year rolling average for pricing purpose as shown in Figure 3, the same January 2012 loan would have been priced with a estimated recovery value of 140, with an actual recovery value of 147 at time of default for a 5% higher than expected recovery.
The three-year rolling average is more volatile than the 5-year; the minimum would have been 132, for an approximately 11.4% better than expected recovery in January, 2013. This benefit will rapidly go away as the values from the crash continue to drop out of the rolling average used for pricing.
Using the 2-year rolling average shown in Figure 4, the recovery estimate used for pricing would be about 146, for a less than 1% better than expected recovery in January of 2013; institutions using this shorter rolling average have already seen the affects of the crash period dropping out of the average.
Although a 6% better than expected recovery may not sound like much, it can translate into a disproportionately large change in the severity of loss. 6% on a $10,000 vehicle would be $600. If the outstanding loan is $12,000, when the 6% better than expected recovery goes away, the severity of loss goes from $1,400 ($12,000 - $10,600), to $2,000--a change of 43%.
The CPI-UCT is helpful in determining whether changes in recoveries are due to credit problems, or used car market problems. The crash in used car and truck prices that occurred in 2008-2009 was a very unusual event, and will cause some unusual and perhaps negative changes in the severity of loss in car loan portfolios.
Data are from the CUSR0000SETA02 series available from the St. Louis Federal Reserve FRED2 system.
Charge-offs
Calculating the Marginal Cost of Funds for Deposit Pricing
Interest expense is the largest item on most banks' income statement, but this expense frequently receives significantly less attention than expense items that are far less important to the profitability of the institution. The discussion that follows will show you how to do some simple calculations to help you set prices for deposit products in ways that can significantly improve the profitability. The discussion will be focused on how to determine the marginal cost of funds for depository products.
Collecting the Necessary Data
"What is our marginal cost of funds?" is a question that is infrequently asked and less frequently answered at many institutions. For some products, it can be relatively simple, while for others it can be fiendishly complex and require statistical analysis. The discussion that follows will help to calculate the simple straightforward costs and to know what is involved for the difficult calculations. To calculate marginal costs, you must first gather the necessary data. For checking, savings, and money market (MM) products, this is fairly simple--you just need the aggregate balance for the product, and the weighted average interest rate over the period--probably a month. If your products have different interest rates depending upon the account balance, you should treat each tier separately in this analysis. You should go back as far possible; three years of history is probably the minimum to do a good analysis.
For certificate of deposit (CD) products, the data collection is somewhat more difficult, as you will need the total balance, the amount that matured during the period, the amount that rolled over, the maturing deposits that were withdrawn, the non-maturing deposits that were withdrawn, and the new deposits added during the period. This may not be available in an master customer information file (MCIF) extract and may require a special extract from Information Systems.
Getting data on a monthly basis is probably the easiest, as this is frequently captured for an MCIF extract. Unfortunately, monthly isn't necessarily the best choice as the number of days varies; for some types of analysis, this will introduce unwanted variability, but if monthly data is the only choice available, you can generally use it without too much difficulty.
Once you have the data from your internal systems, you will need to enhance it with some publicly available data on market interest rates. Many institutions use Treasury rates for transfer pricing, but for marginal cost analysis, swap rates are generally a better choice, as swap rates closely follow the top of the market for bank CD offerings and provide a very good proxy for the competitive market. Federal Home Loan Bank (FHLB) and other outside borrowing rates are perhaps the best proxy for the competitive market CD market, but using FHLB rates requires more code to download the rates and requires that you do your own averaging over the period. The best data would be competitor's actual rates, but gathering history for competitors' rates would be very difficult. Swap rates are readily available from the St. Louis Federal Reserve's FRED2 system for several different time intervals, making swap rates the most practical choice for marginal cost analysis.
Once you have both in-house data and swap rate data, merge them so that you have a table with period (date), balance, nominal interest rate, and swap (benchmark) interest rate. For checking, savings and money market accounts, use the one year swap rate. For CD products, use the swap rate term that most closely matches the CD term. For a money market product, the table would have the following structure:
## dateVal balance_change.MM balance.MM nominal_int_rate.MM benchmark_int_rate.MM delta_to_benchmark.MM
## 1 2003-02-01 -550270 99449730 1.3311 1.40 -0.06894
## 2 2003-03-01 -1526234 98473766 1.2163 1.32 -0.10373
## 10 2003-11-01 -694666 99305334 1.4484 1.52 -0.07157
Calculating the Marginal Cost of Funds for Money Market, Savings and Checking Products
When you plot the balance against the nominal interest rate, you will probably get a plot that shows no discernible pattern, similar to the plot in Figure 1, but if you plot the balance against the delta to benchmark as shown in Figure 2, you will probably get a reasonably clear pattern where the the balance increases as the delta to benchmark increases. For many institutions, the pricing decisions based upon Figure 2 can be worth hundreds of thousands or millions of dollars.
To calculate the marginal cost, we want to get the slope of the line that best fits the data in Figure 2. An eyeball calculation would be to take a point at each end and calculate the slope--(0%,$1.05e8) and (-0.2%,$9.75e8) for a slope of
(5e+05 - (-2500000))/(0 - (-0.2))
## [1] 1.5e+07
1.5e7 or $15 million per month balance increase per 1% increase relative to the benchmark. A more precise way to calculate the slope would be to do a linear regression
## Call:
## lm(formula = amt_balance_change.MM ~ delta_to_benchmark.MM, data = lbtDf2)
## Coefficients:
## (Intercept) delta_to_benchmark.MM
## 41253 11038442
## Analysis of Variance Table
## Response: amt_balance_change.MM
## Df Sum Sq Mean Sq F value Pr(>F)
## delta_to_benchmark.MM 1 3.82e+13 3.82e+13 152
which gives a slope of 11038442 or $11 million per month balance increase per 1% increase relative to the benchmark. To convert this to a marginal cost we have to calculate the cost of adding $11 million to an existing $99 million MM product by increasing the interest rate by 1% (0.083% per month) assuming a benchmark rate of 0.5% (0.0416% per month):
\[ \begin{aligned} \text{marginal cost}&=&\frac{0.00083*99e6+(0.000416+0.00083)*11e6}{11e6} \\ &=&0.008718 \text{ per month}\\ &=&10.5\% \text{ per year} \end{aligned} \]
Why is the marginal cost so high? In order to get customers to add $11 million, the bank must pay a premium (the 0.00083*99e6 in the formula) to existing depositors who really don't care about getting the additional interest. If product balances are high, it may make sense to open a new product and allow rate-sensitive depositors to switch into the new high-rate product until the marginal cost of the original product is reasonable.
The marginal cost of funds for savings accounts can be calculated similarly. For interest bearing checking products, the marginal cost can be calculated if there are no fees for going below a minimum balance. If there are minimum balance fees, the interest rate would need to be adjusted to include the fee as an implied negative interest rate. Checking deposits tend to be very seasonal, with declines in December through February for Christmas spending and a spike in April for tax refunds. There is also a lot of noise introduced by weekly, bi-weekly payroll cycles being superimposed over a monthly data extract cycle. These two characteristics make getting a good regression model difficult for checking account data.
Doing the modelling in the format described is quite straightforward, but it can't really be used for large changes in the balance, especially in a declining environment. $11 million leaving a $100 million balance is 11% of the balance--but would be about 5.5% of a $200 million balance. A future article will address how to improve the model to account for this difference.
Calculating the Marginal Cost of Funds for Certificate of Deposit Products
Calculating the marginal cost of Certificate of Deposit (CD) products can be significantly more difficult than the similar calculation for money market, savings and checking accounts. The example that follows will illustrate some of these difficulties. First we need to address a terminology problem. For CDs, we will look at the amount maturing, and call that the product balance to make things work similarly to the way that they did for Money Market products. For these plots, we will assume that the amount maturing each month is the same--a completely invalid assumption in real life, but it makes the illustration much easier to follow. If you plot the balance against the nominal rate as in Figure 3, you will probably get a plot with no discernible pattern. If you plot the balance against the delta to benchmark as in Figure 4, you still probably won't get much of a pattern; this is very unlike the MM product.
Why is there no pattern? Many customers looking at CDs will compare prices with MM funds, with CDs of slightly shorter terms, and with CDs of slightly longer terms; if the other products are more competitive with the customer's perception of the yield curve, the customer will put money into the MM or the shorter/longer CD. They will substitute a more attractive CD if the originally desired CD isn't priced to their liking. To illustrate this in a data plot, do a contour plot of the CD Balance Change vs. the CD delta to benchmark AND the MM delta to benchmark. In this plot you will probably see a clear pattern as shown in Figure 5. CD products typically have a great deal of product substitution, so that the price of the product relative to its benchmark is important, but so is the price relative to other CD products and MM products. It is impossible to estimate this graphically; a multi-variate regression model is the only approach that will work for this situation.
To calculate the marginal cost of the CD product, you have to assume the prices of the substitute CD terms--but if there a large amount maturing in one substitute CD term, that would assume a high marginal cost for that CD term. Since the bank doesn't want to set prices so that multiple products have a high marginal cost, both prices have to be calculated simultaneously. For this reason, it doesn't make much sense to try to calculate the marginal cost for CD products. Setting rates is the ultimate goal, and the marginal cost is just input into that goal.
The best way to set CD prices is to model demand for each CD product including substitution effects to get a formula that represents the demand for each CD as a function of its price and the price of other CDs. Then all of the formulas can be solved simultaneously to obtain a price sheet that will reduce the cost of funds for the deposit portfolio. Although Excel has the statistical tools to do this, automating a large number of regression models in Excel would be difficult.
Fortunately, there are inexpensive or free statistical packages that can easily handle the task of developing the demand models. Solving the system is straightforward with commercial mathematical modeling languages like AMPL or GAMS.
The data in this article was synthesized to illustrate common characteristics of money market and certificate of deposit demand data. Data manipulation and chart preparation was done using the R statistical language. Historical swap rates from the St. Louis Federal Reserve FRED2 system were downloaded and incorporated using the fImport package. The charts were prepared using ggplot2. The display of mathematical formulas in the HTML web page uses MathJax, which allows formulas to be written using LaTeX syntax.
The discussion in this article follows portions of the Estimating Deposit Volume Price Sensitivity presentation by the author at the 2005 Institute for Operations Research and Management Science (INFORMS) Annual Meeting. The presentation received an Honorable Mention in the Financial Services Section contest for Best Presentation. The abstract is available on the INFORMS web site.
Cost of Funds
Marginal Cost
Open Source Solver Resources
Solvers are the basis for all optimization modeling. There are four major types, according to the type of problem that the particular software can solve:
Linear/quadratic
Linear integer, otherwise known as "mixed integer"
Non-linear
Non-linear integer
Most pricing problems in finance are either quadratic or nonlinear, so the non-linear solvers will be listed first.
Ipopt is an open source solver designed for constrained non-linear problems. It was originally written by Andreas Wachter while at IBM, and is currently released under the Eclipse Public License. It is probably the most robust of the available open source solvers.
Excel has a built-in non-linear integer solver, but it is limited to 300 variables; since some of these are used for internal variables that are not present in the spreadsheet as cells, the problems that it can handle are really pretty small, but it can be useful if your problem fits into 300 variables.
Linear, Integer and Quadratic
Clp is a linear solver that was written by John Forrest while at IBM, and is currently released under the Eclipse Public License. It is has a reputation as the most robust of the available open source solvers, and is considered to be competitive with commercial solvers for all but very large problems. It is unusual among open source linear solvers in that it can handle quadratic objective functions.
Cbc is a mixed integer solver that was written by John Forrest while at IBM, and is currently released under the Eclipse Public License. It is has a reputation as the most robust of the available open source solvers, and is considered to be competitive with commercial solvers for all but very large problems.
GLPK is a mixed integer solver written by Andrew O. Makhorin (Андрей Олегович Махорин) while at the Moscow Aviation Institute. It is widely used due to the inclusion of the GNU Linear Programming Language that is essentially a subset of the commercially available AMPL language. The modeling language is much slower than AMPL and does not allow order sets (among many other things), but it can be useful when you don't have an AMPL license available. It is significantly slower than commercial solvers, especially on integer programs.
lp_solve is a linear solver that was probably the first widely used open source solver and which has a reputation as being useful for problems larger than GLPK can handle, but smaller than Clp can handle.
R - Open Source Statistical Software | CommonCrawl |
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We|\!| present|\!| a|\!| path|\!|-integral|\!| formulation|\!| of|\!| |\!|'t|\!|~Hooft|\!|'s|\!| derivation|\!| of|\!| quantum|\!| from|\!| classical|\!| physics|\!|.|\!| The|\!| crucial|\!| ingredient|\!| of|\!| this|\!| formulation|\!| is|\!| Gozzi|\!| |\!|{|\!|em|\!| et|\!| al|\!|.|\!|}|\!| supersymmetric|\!| path|\!| integral|\!| of|\!| classical|\!| mechanics|\!|.|\!| We|\!| quantize|\!| explicitly|\!| two|\!| simple|\!| classical|\!| systems|\!|:|\!| the|\!| planar|\!| mathematical|\!| pendulum|\!| and|\!| the|\!| R|\!||\!|"|\!|{o|\!|}ssler|\!| dynamical|\!| system|\!|.|\!|
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In|\!| recent|\!| decades|\!|,|\!| various|\!| classical|\!|,|\!| i|\!|.e|\!|.|\!|,|\!|
|\!| deterministic|\!| |\!| approaches|\!| to|\!| quantum|\!| theory|\!| have|\!| been|\!| proposed|\!|.|\!| Examples|\!| are|\!| Bohmian|\!| mechanics|\!|~|\!|cite|\!|{Bohm1|\!|}|\!|,|\!| and|\!| the|\!| stochastic|\!| quantization|\!| procedures|\!| of|\!| Nelson|\!|~|\!|cite|\!|{Nelson1|\!|}|\!|,|\!| Guerra|\!| and|\!| Ruggiero|\!|~|\!|cite|\!|{Guerra1|\!|}|\!|,|\!| and|\!| Parisi|\!| and|\!| Wu|\!|~|\!|cite|\!|{Parisi1|\!|,Huffel|\!|}|\!|.|\!| Such|\!| approaches|\!| are|\!| finding|\!| increasing|\!| interest|\!|
|\!| in|\!| the|\!| physics|\!| community|\!|.|\!| This|\!| might|\!| be|\!| partially|\!| ascribed|\!| to|\!| the|\!| fact|\!| that|\!| such|\!| alternative|\!| formulations|\!| help|\!| in|\!| explaining|\!| some|\!| quantum|\!| phenomena|\!| that|\!| cannot|\!| be|\!| easily|\!| explained|\!|
|\!| with|\!| the|\!| usual|\!| formalisms|\!|.|\!| Examples|\!| are|\!| multiple|\!| tunneling|\!|~|\!|cite|\!|{Jona|\!|-Lasinio|\!|}|\!|,|\!| critical|\!| phenomena|\!| at|\!| zero|\!| temperature|\!|~|\!|cite|\!|{Ruggiero1|\!|}|\!|,|\!| mesoscopic|\!| physics|\!| and|\!| quantum|\!| Brownian|\!| oscillators|\!|~|\!|cite|\!|{Rugierro2|\!|}|\!|,|\!| and|\!| quantum|\!|-field|\!|-theoretical|\!| regularization|\!| procedures|\!| which|\!| manifestly|\!| preserve|\!|
|\!| all|\!| symmetries|\!| of|\!| the|\!| bare|\!| theory|\!| such|\!| as|\!| gauge|\!| symmetry|\!|,|\!| chiral|\!| symmetry|\!|,|\!| and|\!| supersymmetry|\!|~|\!|cite|\!|{reg|\!|}|\!|.|\!| They|\!| allow|\!| one|\!| to|\!| quantize|\!| gauge|\!| fields|\!|,|\!| both|\!| Abelian|\!| and|\!| non|\!|-Abelian|\!|,|\!| without|\!| gauge|\!| fixing|\!| and|\!| the|\!| ensuing|\!| cumbersome|\!| Faddeev|\!|-Popov|\!| ghosts|\!|~|\!|cite|\!|{FPG|\!|}|\!|,|\!| etc|\!|.|\!|.|\!|
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The|\!| primary|\!| objective|\!| of|\!| a|\!| reformulation|\!| of|\!| quantum|\!| theory|\!| in|\!| the|\!| language|\!| of|\!| classical|\!|,|\!| i|\!|.e|\!|.|\!|,|\!| |\!| deterministic|\!| theory|\!| is|\!| basically|\!| twofold|\!|.|\!| On|\!| the|\!| formal|\!| side|\!|,|\!| it|\!| is|\!| hoped|\!| that|\!| this|\!| will|\!| help|\!| in|\!| attacking|\!| quantum|\!|-mechanical|\!| problems|\!| from|\!| a|\!| different|\!| direction|\!| using|\!| hopefully|\!| more|\!| efficient|\!| mathematical|\!| techniques|\!| than|\!| the|\!| conventional|\!| ones|\!|.|\!| Such|\!| techniques|\!| may|\!| be|\!| based|\!| on|\!| stochastic|\!| calculus|\!|,|\!| supersymmetry|\!|,|\!| or|\!| various|\!| new|\!| numerical|\!| approaches|\!| |\!|(see|\!|,|\!| e|\!|.g|\!|.|\!|,|\!| Refs|\!|.|\!|~|\!|cite|\!|{Huffel|\!|,Pain|\!|}|\!| and|\!| citations|\!| therein|\!|)|\!|.|\!| On|\!| the|\!| conceptual|\!| side|\!|,|\!| deterministic|\!| scenarios|\!| are|\!| hoped|\!| to|\!| shed|\!| new|\!| light|\!| on|\!| some|\!| old|\!| problems|\!| of|\!| quantum|\!| mechanics|\!|,|\!| such|\!| as|\!| the|\!| origin|\!| of|\!| the|\!| superposition|\!| rule|\!| for|\!| amplitudes|\!| and|\!| |\!| the|\!| theory|\!| of|\!| quantum|\!| measurement|\!|.|\!| It|\!| may|\!| lead|\!| to|\!| new|\!| ways|\!| of|\!| quantizing|\!| chaotic|\!| dynamical|\!| systems|\!|,|\!| and|\!| ultimately|\!| a|\!| long|\!|-awaited|\!| |\!| consistent|\!| theory|\!| of|\!| quantum|\!| gravity|\!|.|\!| There|\!| is|\!|,|\!| however|\!|,|\!| a|\!| price|\!| to|\!| be|\!| paid|\!| for|\!| this|\!|;|\!| such|\!| theories|\!| must|\!| have|\!| a|\!| built|\!|-in|\!| nonlocality|\!|
|\!| to|\!| escape|\!| problems|\!| with|\!| Bell|\!|'s|\!| inequalities|\!|.|\!| Nonlocality|\!| may|\!| be|\!| incorporated|\!| in|\!| numerous|\!| ways|\!| |\!|-|\!|-|\!|-|\!| the|\!| Bohm|\!|-Hiley|\!| quantum|\!| potential|\!|~|\!|cite|\!|{Bohm1|\!|,Bohm2|\!|}|\!|,|\!| Nelson|\!|'s|\!| osmotic|\!| potential|\!|~|\!|cite|\!|{Nelson1|\!|}|\!|,|\!| or|\!| Parisi|\!| and|\!| Wu|\!|'s|\!| |\!|{|\!|em|\!| fifth|\!|-|\!|-time|\!||\!|/|\!|}|\!| parameter|\!|~|\!|cite|\!|{Parisi1|\!|,Huffel|\!|}|\!|.|\!|
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Another|\!| deterministic|\!| access|\!| to|\!| quantum|\!|-mechanical|\!| systems|\!| was|\!| recently|\!| proposed|\!| by|\!| |\!|'t|\!| Hooft|\!| |\!|~|\!|cite|\!|{tHooft|\!|,tHooft3|\!|}|\!| with|\!| subsequent|\!| applications|\!| in|\!| Refs|\!|.|\!|cite|\!|{BJV1|\!|,tHooft22|\!|,Halliwell|\!|:2000mv|\!|,cvb|\!|,BMM1|\!|,Banajee|\!|,Elze2|\!|}|\!|.|\!| It|\!| is|\!| motivated|\!| by|\!| black|\!|-hole|\!| thermodynamics|\!| |\!|(and|\!| particularly|\!| by|\!| the|\!| so|\!|-called|\!| |\!|{|\!|em|\!| holographic|\!| principle|\!||\!|/|\!|}|\!|~|\!|cite|\!|{tHooft2|\!|,Bousso|\!|}|\!|)|\!|,|\!| and|\!| hinges|\!| on|\!| the|\!| concept|\!| of|\!| |\!|{|\!|em|\!| information|\!| loss|\!||\!|/|\!|}|\!|.|\!| This|\!| and|\!| certain|\!| accompanying|\!| non|\!|-trivial|\!| geometric|\!| phases|\!| are|\!| able|\!| to|\!| explain|\!| the|\!| observed|\!| non|\!|-locality|\!| in|\!| quantum|\!| mechanics|\!|.|\!| The|\!| original|\!| formulation|\!| has|\!| appeared|\!| in|\!| two|\!| versions|\!|:|\!| one|\!| involving|\!| a|\!| |\!| discrete|\!| time|\!| axis|\!|~|\!|cite|\!|{tHooft22|\!|}|\!|,|\!| the|\!| second|\!| continuous|\!| times|\!|~|\!|cite|\!|{tHooft3|\!|}|\!|.|\!| The|\!| goal|\!| of|\!| this|\!| paper|\!| is|\!| to|\!| discuss|\!| further|\!| and|\!| gain|\!| more|\!| understanding|\!| of|\!| the|\!| latter|\!| model|\!|.|\!| The|\!| reader|\!| interested|\!| in|\!| the|\!| discrete|\!|-time|\!| model|\!| may|\!| find|\!| some|\!| practical|\!| applications|\!| in|\!| Refs|\!|.|\!|~|\!|cite|\!|{BJV3|\!|,Elze|\!|}|\!|.|\!| It|\!| is|\!| not|\!| our|\!| purpose|\!| to|\!| dwell|\!| on|\!| the|\!| conceptual|\!| foundations|\!| of|\!| |\!|'t|\!||\!|,Hooft|\!|'s|\!| proposal|\!|.|\!| Our|\!| aim|\!| is|\!| to|\!| set|\!| up|\!| a|\!| possible|\!| useful|\!| alternative|\!| formulation|\!| of|\!| |\!|'t|\!||\!|,Hooft|\!|'s|\!| model|\!| and|\!| quantization|\!| scheme|\!| that|\!| is|\!| based|\!| on|\!| path|\!| integrals|\!| |\!|cite|\!|{Pain|\!|}|\!|.|\!| It|\!| makes|\!| use|\!| of|\!| Gozzi|\!| |\!|{|\!|em|\!| et|\!| al|\!|.|\!|}|\!| path|\!|-integral|\!| formulation|\!| of|\!| classical|\!| mechanics|\!|~|\!|cite|\!|{GozziI|\!|,GozziII|\!|}|\!| which|\!| appears|\!| to|\!| be|\!| a|\!| natural|\!| mathematical|\!| framework|\!| for|\!| such|\!| a|\!| discussion|\!|.|\!| The|\!| condition|\!| of|\!| the|\!| information|\!| loss|\!|,|\!| which|\!| is|\!| basically|\!| a|\!| first|\!|-class|\!| subsidiary|\!| constraint|\!|,|\!| can|\!| then|\!| be|\!| incorporated|\!| into|\!| path|\!| integrals|\!| |\!| by|\!| standard|\!| techniques|\!|.|\!| Although|\!| |\!|'t|\!||\!|,Hooft|\!|'s|\!| procedure|\!| differs|\!| in|\!| its|\!| basic|\!| rationale|\!| from|\!| stochastic|\!| quantization|\!| approaches|\!|,|\!| we|\!| show|\!| that|\!| they|\!| share|\!| a|\!| common|\!| key|\!| feature|\!|,|\!| which|\!| is|\!|
|\!| a|\!| hidden|\!| BRST|\!| invariance|\!|,|\!| related|\!| to|\!|
|\!| the|\!| so|\!|-called|\!| Nicolai|\!| map|\!|~|\!|cite|\!|{Nikoloai1|\!|}|\!|.|\!| To|\!| be|\!| specific|\!|,|\!| we|\!| shall|\!| apply|\!| our|\!| formulation|\!| to|\!| two|\!| classical|\!| systems|\!|:|\!| a|\!| planar|\!| mathematical|\!| pendulum|\!| and|\!| the|\!| simplest|\!| deterministic|\!| chaotic|\!| system|\!| |\!|-|\!|-|\!|-|\!| the|\!| R|\!||\!|"|\!|{o|\!|}ssler|\!| attractor|\!|.|\!| Suitable|\!| choices|\!| of|\!| the|\!| |\!|`|\!|`loss|\!| of|\!| information|\!|"|\!| condition|\!| then|\!| allow|\!| us|\!| to|\!| identify|\!| the|\!| emergent|\!| quantum|\!| systems|\!| with|\!| a|\!| free|\!| particle|\!|,|\!| a|\!| quantum|\!| harmonic|\!| oscillator|\!|,|\!| and|\!| a|\!| free|\!| particle|\!| weakly|\!| coupled|\!| to|\!| Duffing|\!|'s|\!| oscillator|\!|.|\!|
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|\!|vspace|\!|{3mm|\!|}|\!|
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Our|\!| paper|\!| is|\!| organized|\!| as|\!| follows|\!|.|\!| In|\!| Section|\!| |\!|ref|\!|{SEc2|\!|}|\!| we|\!| quantize|\!| |\!|'t|\!||\!|,Hooft|\!|'s|\!| Hamiltonian|\!| system|\!| by|\!| expressing|\!| it|\!| in|\!| terms|\!| of|\!| a|\!| path|\!| integral|\!| which|\!| is|\!| singular|\!| due|\!| to|\!| the|\!| presence|\!| of|\!| second|\!|-class|\!| primary|\!| constraints|\!|.|\!| The|\!| singularity|\!| is|\!| removed|\!| with|\!| the|\!| help|\!| of|\!| the|\!| Faddeev|\!|-Senjanovic|\!| prescription|\!|~|\!|cite|\!|{Fad|\!|,Senj|\!|}|\!|.|\!| It|\!| is|\!| then|\!| shown|\!| that|\!| the|\!| fluctuating|\!| system|\!| produces|\!| a|\!| classical|\!| partition|\!| function|\!|.|\!| In|\!| Section|\!| |\!|ref|\!|{SEc3|\!|}|\!| we|\!| briefly|\!| review|\!| Gozzi|\!| |\!|{|\!|em|\!| et|\!| al|\!|.|\!|}|\!| path|\!|-integral|\!| formulation|\!| of|\!| classical|\!| mechanics|\!| in|\!| configuration|\!| space|\!|.|\!| The|\!| corresponding|\!| phase|\!|-space|\!| formulation|\!| is|\!| more|\!| involved|\!| and|\!| will|\!| not|\!| be|\!| considered|\!| here|\!|.|\!| By|\!| imposing|\!| the|\!| condition|\!| of|\!| a|\!| vanishing|\!| ghost|\!| sector|\!|,|\!| which|\!| is|\!| characteristic|\!| for|\!| the|\!| underlying|\!| deterministic|\!| system|\!|,|\!| we|\!| find|\!| that|\!| the|\!| most|\!| general|\!| Hamiltonian|\!| system|\!| compatible|\!| with|\!| such|\!| a|\!| condition|\!| is|\!| the|\!| one|\!| proposed|\!| by|\!| |\!|'t|\!||\!|,Hooft|\!|.|\!| In|\!| Section|\!| |\!|ref|\!|{SEc4|\!|}|\!| we|\!| introduce|\!| |\!|'t|\!||\!|,Hooft|\!|'s|\!| constraint|\!| which|\!| expresses|\!| the|\!| property|\!| of|\!| information|\!| loss|\!|.|\!| This|\!| condition|\!| not|\!| only|\!| explicitly|\!| breaks|\!| the|\!| BRST|\!| symmetry|\!| but|\!|,|\!| when|\!| coupled|\!| with|\!| the|\!| Dirac|\!|-Bergmann|\!| algorithm|\!|,|\!| it|\!| also|\!| allows|\!| us|\!| to|\!| recast|\!| the|\!| classical|\!| generating|\!| functional|\!| into|\!| a|\!| form|\!| representing|\!| a|\!| proper|\!| quantum|\!|-mechanical|\!| partition|\!| function|\!|.|\!| Section|\!| V|\!| is|\!| devoted|\!| to|\!| application|\!| of|\!| our|\!| formalism|\!| to|\!| practical|\!| examples|\!|.|\!| We|\!| conclude|\!| with|\!| Section|\!| VI|\!|.|\!|
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For|\!| the|\!| reader|\!|'s|\!| convenience|\!| the|\!| paper|\!| is|\!| supplemented|\!| with|\!| four|\!| appendixes|\!| which|\!| clarify|\!| some|\!| finer|\!| mathematical|\!| points|\!| needed|\!| in|\!| the|\!| paper|\!|.|\!|
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|\!|section|\!|{Quantization|\!| of|\!| |\!|'t|\!||\!|,Hooft|\!|'s|\!| Model|\!|}|\!| |\!|label|\!|{SEc2|\!|}|\!|
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Consider|\!| the|\!| class|\!| of|\!| systems|\!| described|\!| by|\!| Hamiltonians|\!| of|\!| the|\!| form|\!|
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|\!|begin|\!|{eqnarray|\!|}|\!| H|\!| |\!|=|\!| |\!|sum|\!|_|\!|{a|\!|=1|\!|}|\!|^N|\!| p|\!|_a|\!| f|\!|_a|\!|(|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|)|\!||\!|,|\!| |\!|.|\!| |\!|label|\!|{eq|\!|.1|\!|.1|\!|}|\!| |\!|end|\!|{eqnarray|\!|}|\!|
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Such|\!| systems|\!| emerge|\!| in|\!| diverse|\!| physical|\!| situations|\!|,|\!| for|\!| example|\!|,|\!| Fermi|\!| fields|\!|,|\!| chiral|\!| oscillators|\!|~|\!|cite|\!|{Banajee|\!|}|\!|,|\!| and|\!| noncommutative|\!| magnetohydrodynamics|\!|~|\!|cite|\!|{Jackiw|\!|}|\!|.|\!| The|\!| relevant|\!| example|\!| in|\!| the|\!|
|\!| present|\!| context|\!| is|\!| the|\!| use|\!| of|\!| |\!|(|\!|ref|\!|{eq|\!|.1|\!|.1|\!|}|\!|)|\!|
|\!| by|\!| |\!|'t|\!||\!|,Hooft|\!| to|\!| formulate|\!| his|\!| |\!|{|\!|em|\!| deterministic|\!| quatization|\!||\!|/|\!|}|\!| proposal|\!|~|\!|cite|\!|{tHooft|\!|}|\!|.|\!|
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|\!|vspace|\!|{3mm|\!|}|\!|
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An|\!| immediate|\!| problem|\!| with|\!| the|\!| above|\!| Hamiltonian|\!| is|\!| its|\!| unboundedness|\!| from|\!| below|\!|.|\!| This|\!| is|\!| due|\!| to|\!| the|\!| absence|\!| of|\!| a|\!| leading|\!| kinetic|\!| term|\!| quadratic|\!| |\!| in|\!| the|\!| momenta|\!| |\!|$p|\!|_|\!| a|\!| |\!|^2|\!|/2M|\!|$|\!|,|\!| and|\!| we|\!| shall|\!| dwell|\!| more|\!| on|\!| this|\!| point|\!| in|\!| Section|\!|~|\!|ref|\!|{SEc4|\!|}|\!|.|\!| The|\!| equations|\!| of|\!| motion|\!| following|\!| from|\!| Eq|\!|.|\!|(|\!|ref|\!|{eq|\!|.1|\!|.1|\!|}|\!|)|\!| are|\!|
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|\!|begin|\!|{eqnarray|\!|}|\!| |\!|dot|\!|{q|\!|}|\!|_a|\!| |\!||\!| |\!|=|\!| |\!||\!| f|\!|_a|\!|(|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|)|\!||\!|,|\!| |\!|,|\!| |\!||\!|;|\!||\!|;|\!||\!|;|\!||\!|;|\!| |\!|dot|\!|{p|\!|}|\!|_a|\!| |\!||\!| |\!|=|\!| |\!||\!| |\!|-|\!| p|\!|_a|\!| |\!|frac|\!|{|\!|partial|\!| f|\!|_a|\!|(|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|)|\!|}|\!|{|\!|partial|\!| q|\!|_a|\!|}|\!||\!|,|\!| |\!|.|\!| |\!|label|\!|{eq|\!|.1|\!|.1|\!|.1|\!|}|\!| |\!|end|\!|{eqnarray|\!|}|\!|
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Note|\!| that|\!| the|\!| equation|\!| for|\!| |\!|$q|\!|_a|\!|$|\!| is|\!| autonomous|\!|,|\!| i|\!|.e|\!|.|\!|,|\!| it|\!| is|\!| decoupled|\!| from|\!| the|\!| conjugate|\!| momenta|\!| |\!|$p|\!|_a|\!|$|\!|.|\!| The|\!| absence|\!| of|\!| a|\!| quadratic|\!| term|\!| makes|\!| it|\!| impossible|\!| to|\!| find|\!| a|\!| Lagrangian|\!| via|\!| a|\!| Legendre|\!| transformation|\!|.|\!| This|\!| is|\!| because|\!| the|\!| system|\!| is|\!| singular|\!| |\!|-|\!|-|\!|-|\!| its|\!| Hess|\!| matrix|\!| |\!|$H|\!|^|\!|{ab|\!| |\!|}|\!|equiv|\!| |\!|partial|\!| |\!|^2H|\!|/|\!|partial|\!| p|\!|_|\!| a|\!|partial|\!| p|\!| |\!|_b|\!|$|\!| vanishes|\!|.|\!|
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|\!|vspace|\!|{3mm|\!|}|\!|
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A|\!| Lagrangian|\!| yielding|\!| the|\!| equations|\!| of|\!| motion|\!| |\!|(|\!|ref|\!|{eq|\!|.1|\!|.1|\!|.1|\!|}|\!|)|\!| can|\!| nevertheless|\!| be|\!| found|\!|,|\!| but|\!| at|\!| the|\!| expense|\!| of|\!| doubling|\!| the|\!| configuration|\!| space|\!| by|\!| introducing|\!| additional|\!| auxiliary|\!| variables|\!| |\!|$|\!|overlinen|\!| q|\!|_|\!| a|\!| |\!|~|\!|(a|\!|=1|\!|,|\!|dots|\!|,|\!| N|\!|)|\!|$|\!|.|\!| This|\!| |\!|{|\!|em|\!| extended|\!|}|\!| Lagrangian|\!| has|\!| the|\!| form|\!|
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|\!|begin|\!|{eqnarray|\!|}|\!| |\!|overlinen|\!| L|\!| |\!||\!| |\!|equiv|\!| |\!||\!| |\!|sum|\!|_|\!|{a|\!|=1|\!|}|\!|^N|\!| |\!|left|\!|[|\!|bar|\!|{q|\!|}|\!|_a|\!| |\!|dot|\!|{q|\!|}|\!|_a|\!| |\!|-|\!| |\!|bar|\!|{q|\!|}|\!|_a|\!| f|\!|_a|\!|(|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|)|\!|right|\!|]|\!| |\!|
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|\!| |\!|label|\!|{lag1|\!|}|\!| |\!|end|\!|{eqnarray|\!|}|\!|
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and|\!| it|\!| allows|\!| us|\!| to|\!| define|\!| canonically|\!| |\!| conjugate|\!| momenta|\!| in|\!| the|\!| usual|\!| way|\!|:|\!| |\!|$p|\!|_a|\!| |\!|equiv|\!| |\!| |\!|partial|\!| |\!|overlinen|\!| L|\!|/|\!|partial|\!| |\!|dot|\!|{q|\!|}|\!|_a|\!|,|\!|~|\!| |\!|overlinen|\!|{p|\!|}|\!|_a|\!| |\!|equiv|\!| |\!|partial|\!| |\!|overlinen|\!| L|\!|/|\!|partial|\!| |\!|dot|\!|{|\!|overlinen|\!|{q|\!|}|\!|}|\!|_a|\!|$|\!|.|\!| A|\!| Legendre|\!| transformation|\!| produces|\!| the|\!| Hamiltonian|\!|
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|\!|begin|\!|{eqnarray|\!|}|\!| |\!|overlinen|\!| H|\!|(p|\!|_a|\!|,|\!| q|\!|_a|\!|,|\!| |\!|{|\!|overlinen|\!|{p|\!|}|\!|}|\!|_a|\!|,|\!| |\!|{|\!|overlinen|\!|{q|\!|}|\!|}|\!|_a|\!|)|\!| |\!|=|\!| |\!|sum|\!|_|\!|{a|\!|=1|\!|}|\!|^N|\!| p|\!|_a|\!| |\!|dot|\!|{q|\!|}|\!|_a|\!| |\!|+|\!| |\!|{|\!|overlinen|\!|{p|\!|}|\!|}|\!|_a|\!| |\!|dot|\!|{|\!|{|\!|overlinen|\!|{q|\!|}|\!|}|\!|}|\!|_a|\!| |\!|-|\!| L|\!| |\!|=|\!| |\!|sum|\!|_|\!|{a|\!|=1|\!|}|\!|^N|\!| |\!|bar|\!|{q|\!|}|\!|_a|\!| f|\!|_a|\!|(|\!|{|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|}|\!|)|\!||\!|,|\!| |\!|.|\!| |\!|label|\!|{2|\!|.4|\!|}|\!| |\!|end|\!|{eqnarray|\!|}|\!|
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The|\!| rank|\!| of|\!| the|\!| Hess|\!| matrix|\!| is|\!| zero|\!| which|\!| gives|\!| rise|\!| to|\!| |\!|$2N|\!|$|\!| primary|\!| constraints|\!|,|\!| which|\!| can|\!| be|\!| chosen|\!| as|\!|:|\!|
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|\!|begin|\!|{eqnarray|\!|}|\!| |\!|phi|\!|_1|\!|^a|\!| |\!|=|\!| p|\!|_a|\!| |\!|-|\!| |\!|overlinen|\!|{q|\!|}|\!|_a|\!| |\!||\!| |\!|approx|\!| |\!||\!| 0|\!||\!|,|\!| |\!|,|\!| |\!||\!|;|\!||\!|;|\!||\!|;|\!||\!|;|\!||\!|;|\!||\!|;|\!| |\!|phi|\!|_2|\!|^a|\!| |\!|=|\!| |\!|overlinen|\!|{p|\!|}|\!|_a|\!| |\!||\!| |\!|approx|\!| |\!||\!| 0|\!||\!|,|\!| |\!|.|\!| |\!|label|\!|{2|\!|.5|\!|}|\!| |\!|end|\!|{eqnarray|\!|}|\!|
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The|\!| use|\!| of|\!| the|\!| symbol|\!| |\!|$|\!|approx|\!|$|\!| instead|\!| of|\!| |\!|$|\!|=|\!|$|\!| is|\!| due|\!| to|\!| Dirac|\!|~|\!|cite|\!|{Dir|\!|}|\!| and|\!| it|\!| has|\!| a|\!| special|\!| meaning|\!|:|\!| two|\!| quantities|\!| related|\!| by|\!| this|\!| symbol|\!| |\!| are|\!| equal|\!| after|\!| all|\!| constraints|\!| have|\!| been|\!| enforced|\!|.|\!| The|\!| system|\!| has|\!| no|\!| secondary|\!| constraints|\!| |\!|(see|\!| Appendix|\!| A|\!|)|\!|.|\!| The|\!| matrix|\!| formed|\!| by|\!| the|\!| Poisson|\!| brackets|\!| of|\!| the|\!| primary|\!| constraints|\!|,|\!|
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|\!|begin|\!|{eqnarray|\!|}|\!| |\!||\!|{|\!|phi|\!|_1|\!|^a|\!|(t|\!|)|\!| |\!|,|\!| |\!|phi|\!|_2|\!|^b|\!|(t|\!|)|\!| |\!||\!|}|\!| |\!||\!| |\!|=|\!| |\!||\!| |\!|-|\!| |\!|delta|\!|_|\!|{a|\!| b|\!|}|\!||\!|,|\!| |\!|,|\!| |\!||\!|;|\!||\!|;|\!||\!|;|\!|
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|\!|label|\!|{2|\!|.10|\!|}|\!| |\!|end|\!|{eqnarray|\!|}|\!|
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has|\!| a|\!| nonzero|\!| determinant|\!|,|\!| implying|\!| that|\!| all|\!| constraints|\!| are|\!| of|\!| the|\!| second|\!| class|\!|.|\!| Note|\!| that|\!| on|\!| the|\!| constraint|\!| manifold|\!| the|\!| |\!|{|\!|em|\!| canonical|\!|}|\!| Hamiltonian|\!| |\!|(|\!|ref|\!|{2|\!|.4|\!|}|\!|)|\!| coincides|\!| with|\!| |\!|'t|\!||\!|,Hooft|\!|'s|\!| Hamiltonian|\!| |\!|(|\!|ref|\!|{eq|\!|.1|\!|.1|\!|}|\!|)|\!|.|\!|
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|\!|vspace|\!|{3mm|\!|}|\!|
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To|\!| quantize|\!| |\!|'t|\!||\!|,Hooft|\!|'s|\!| system|\!| we|\!| utilize|\!| the|\!| general|\!| |\!| Faddeev|\!|-Senjanovic|\!| path|\!| integral|\!| formula|\!|~|\!|cite|\!|{Fad|\!|,Senj|\!|}|\!| for|\!| time|\!| evolution|\!| amplitudes|\!| |\!|footnote|\!|{Other|\!| path|\!|-integral|\!| representations|\!| of|\!| systems|\!| with|\!| second|\!|-class|\!| constrains|\!| such|\!| as|\!| that|\!| of|\!| Fradkin|\!| and|\!| Fradkina|\!|~|\!|cite|\!|{fradkin|\!|}|\!| would|\!| lead|\!| to|\!| |\!| the|\!| same|\!| result|\!| |\!|(|\!|ref|\!|{eg|\!|.1|\!|.2|\!|}|\!|)|\!|.|\!| |\!|}|\!|
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|\!|begin|\!|{eqnarray|\!|}|\!|
|\!|langle|\!| |\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|_2|\!|,t|\!|_2|\!|||\!| |\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|_1|\!|,|\!| t|\!|_1|\!| |\!|rangle|\!| |\!|=|\!| |\!|{|\!|{|\!|mathcal|\!|{N|\!|}|\!|}|\!|}|\!| |\!|int|\!| |\!|{|\!|mathcal|\!|{D|\!|}|\!|}|\!|{|\!|boldsymbol|\!|{p|\!|}|\!|}|\!|
|\!|{|\!|mathcal|\!|{D|\!|}|\!|}|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!| |\!||\!| |\!|sqrt|\!|{|\!|left|\!|||\!|det|\!| |\!||\!|||\!||\!|{|\!|phi|\!|_i|\!| |\!|,|\!|
|\!|phi|\!|_j|\!| |\!||\!|}|\!| |\!||\!|||\!| |\!| |\!| |\!| |\!| |\!| |\!| |\!|right|\!|||\!|}|\!| |\!||\!| |\!|prod|\!|_i|\!|delta|\!|[|\!|phi|\!|_i|\!|]|\!||\!| |\!|exp|\!| |\!|left|\!||\!|{|\!| |\!|frac|\!|{i|\!|}|\!|{|\!|hbar|\!| |\!|}|\!| |\!|int|\!|_|\!|{t|\!|_1|\!|}|\!|^|\!|{t|\!|_2|\!|}|\!| dt|\!| |\!|left|\!|[|\!| |\!|{|\!|boldsymbol|\!|{p|\!|}|\!|}|\!| |\!|dot|\!|{|\!|{|\!|boldsymbol|\!|{q|\!| |\!|}|\!|}|\!| |\!|}|\!| |\!|-|\!| |\!|overlinen|\!| H|\!|(|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|,|\!| |\!|{|\!|boldsymbol|\!|{p|\!|}|\!|}|\!| |\!|)|\!|right|\!|]|\!| |\!|right|\!||\!|}|\!||\!|,|\!| |\!|.|\!| |\!|label|\!|{frad|\!|}|\!|end|\!|{eqnarray|\!|}|\!|
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Using|\!| the|\!| shorthand|\!| notation|\!| |\!|$|\!|phi|\!|_i|\!| |\!|=|\!| |\!|phi|\!|_1|\!|^1|\!|,|\!| |\!|phi|\!|_2|\!|^1|\!|,|\!||\!|,|\!| |\!|phi|\!|_1|\!|^2|\!|,|\!| |\!|phi|\!|_2|\!|^2|\!|,|\!||\!|,|\!| |\!|ldots|\!|,|\!| |\!|phi|\!|_1|\!|^N|\!|,|\!| |\!|phi|\!|_2|\!|^N|\!| |\!|~|\!|(i|\!|=1|\!|,|\!|dots|\!|,2N|\!|)|\!|$|\!|,|\!| Eq|\!|.|\!|(|\!|ref|\!|{frad|\!|}|\!|)|\!| implies|\!| in|\!| our|\!| case|\!| that|\!|
|\!|
|\!|begin|\!|{eqnarray|\!|}|\!|
|\!|langle|\!| |\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|_2|\!|,t|\!|_2|\!|||\!| |\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|_1|\!|,|\!| t|\!|_1|\!| |\!|rangle|\!| |\!|&|\!|=|\!|&|\!| |\!|{|\!|{|\!|mathcal|\!|{N|\!|}|\!|}|\!|}|\!| |\!|int|\!| |\!|{|\!|mathcal|\!|{D|\!|}|\!|}|\!|{|\!|boldsymbol|\!|{p|\!|}|\!|}|\!| |\!|{|\!|mathcal|\!|{D|\!|}|\!|}|\!| |\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!| |\!|{|\!|mathcal|\!|{D|\!|}|\!|}|\!| |\!|overlinen|\!|{|\!|{|\!|boldsymbol|\!|{p|\!|}|\!|}|\!|}|\!| |\!|{|\!|mathcal|\!|{D|\!|}|\!|}|\!|overlinen|\!|{|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|}|\!| |\!||\!| |\!|delta|\!|[|\!|{|\!|boldsymbol|\!|{p|\!|}|\!|}|\!| |\!|-|\!| |\!|overlinen|\!|{|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|}|\!|]|\!| |\!||\!|,|\!| |\!|delta|\!|[|\!|overlinen|\!|{|\!|{|\!|boldsymbol|\!|{p|\!|}|\!|}|\!|}|\!|]|\!||\!| |\!|exp|\!|left|\!||\!|{|\!|frac|\!|{i|\!|}|\!|{|\!|hbar|\!| |\!|}|\!| |\!|int|\!|_|\!|{t|\!|_1|\!|}|\!|^|\!|{t|\!|_2|\!|}|\!| dt|\!||\!|,|\!|[|\!|{|\!|boldsymbol|\!|{p|\!|}|\!|}|\!| |\!|dot|\!|{|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|}|\!| |\!|+|\!| |\!|overlinen|\!|{|\!|{|\!|boldsymbol|\!|{p|\!|}|\!|}|\!|}|\!| |\!|dot|\!|{|\!|overlinen|\!|{|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|}|\!|}|\!| |\!|-|\!| |\!|overlinen|\!| H|\!|(|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|,|\!| |\!|overlinen|\!|{|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|}|\!|,|\!| |\!|{|\!|boldsymbol|\!|{p|\!|}|\!|}|\!|,|\!| |\!|overlinen|\!|{|\!|{|\!|boldsymbol|\!|{p|\!|}|\!|}|\!|}|\!| |\!|)|\!|]|\!| |\!|right|\!||\!|}|\!|nonumber|\!| |\!||\!||\!| |\!|&|\!|&|\!|~|\!| |\!|nonumber|\!| |\!||\!||\!| |\!|&|\!|=|\!|&|\!| |\!|{|\!|{|\!|mathcal|\!|{N|\!|}|\!|}|\!|}|\!| |\!| |\!|int|\!|_|\!|{|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|(t|\!|_1|\!|)|\!| |\!|=|\!| |\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|_1|\!|}|\!|^|\!|{|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|(t|\!|_2|\!|)|\!| |\!|=|\!| |\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|_2|\!|}|\!| |\!|{|\!|mathcal|\!|{D|\!|}|\!|}|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!| |\!|{|\!|mathcal|\!|{D|\!|}|\!|}|\!|overlinen|\!|{|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|}|\!| |\!||\!| |\!|exp|\!|left|\!|[|\!|frac|\!|{i|\!|}|\!|{|\!|hbar|\!| |\!|}|\!| |\!|int|\!|_|\!|{t|\!|_1|\!|}|\!|^|\!|{t|\!|_2|\!|}|\!|overlinen|\!| L|\!|(|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|,|\!| |\!|overlinen|\!|{|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|}|\!|,|\!| |\!|dot|\!|{|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|}|\!|,|\!| |\!|dot|\!|{|\!|overlinen|\!|{|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|}|\!|}|\!|)|\!| |\!||\!| dt|\!| |\!|right|\!|]|\!|nonumber|\!| |\!||\!||\!| |\!|&|\!|&|\!|~|\!| |\!|nonumber|\!| |\!||\!||\!| |\!|&|\!|=|\!|&|\!| |\!|{|\!|{|\!|mathcal|\!|{N|\!|}|\!|}|\!|}|\!| |\!|int|\!|_|\!|{|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|(t|\!|_1|\!|)|\!| |\!|=|\!| |\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|_1|\!|}|\!|^|\!|{|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|(t|\!|_2|\!|)|\!| |\!|=|\!| |\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|_2|\!|}|\!| |\!|{|\!|mathcal|\!|{D|\!|}|\!|}|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!| |\!||\!| |\!|prod|\!|_a|\!| |\!|delta|\!|[|\!| |\!|dot|\!|{q|\!|}|\!|_a|\!|-f|\!|_a|\!|(|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|)|\!|]|\!||\!|,|\!| |\!|,|\!| |\!|label|\!|{eg|\!|.1|\!|.2|\!|}|\!| |\!|end|\!|{eqnarray|\!|}|\!|
|\!|
|\!| where|\!| |\!|$|\!| |\!|delta|\!|[|\!|{|\!|boldsymbol|\!|{f|\!|}|\!|}|\!| |\!|]|\!|equiv|\!| |\!|prod|\!|_t|\!| |\!| |\!|delta|\!| |\!|(|\!|{|\!|boldsymbol|\!|{f|\!|}|\!|}|\!|(t|\!|)|\!|)|\!|$|\!| is|\!| the|\!| functional|\!| version|\!| of|\!| Dirac|\!|'s|\!| |\!| |\!| |\!|$|\!| |\!|delta|\!| |\!|$|\!|-function|\!|.|\!| This|\!| result|\!| shows|\!| that|\!| quantization|\!| of|\!| the|\!| system|\!| described|\!| by|\!| the|\!| Hamiltonian|\!| |\!|(|\!|ref|\!|{eq|\!|.1|\!|.1|\!|}|\!|)|\!| retains|\!| its|\!| deterministic|\!| character|\!|.|\!| The|\!| paths|\!| are|\!| squeezed|\!| onto|\!| the|\!| classical|\!| trajectories|\!| |\!| |\!| determined|\!| by|\!| the|\!| differential|\!| equations|\!|
|\!| |\!|$|\!|dot|\!|{q|\!|}|\!|_a|\!| |\!|=|\!| f|\!|_a|\!|(|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|)|\!|$|\!|.|\!| The|\!| time|\!| evolution|\!| amplitude|\!| |\!|(|\!|ref|\!|{eg|\!|.1|\!|.2|\!|}|\!|)|\!| contains|\!| a|\!| sum|\!| |\!| over|\!| only|\!| the|\!| classical|\!| trajectories|\!| |\!|-|\!|-|\!|-|\!| there|\!| are|\!| no|\!| quantum|\!| fluctuations|\!| driving|\!| the|\!| system|\!| away|\!| from|\!| the|\!| classical|\!| paths|\!|,|\!| which|\!| is|\!| precisely|\!| what|\!| we|\!| expect|\!| from|\!| a|\!| deterministic|\!| dynamics|\!|.|\!|
|\!|
|\!|vspace|\!|{3mm|\!|}|\!|
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The|\!| amplitude|\!| |\!|(|\!|ref|\!|{eg|\!|.1|\!|.2|\!|}|\!|)|\!| can|\!| be|\!| brought|\!| to|\!| a|\!| more|\!| intuitive|\!| form|\!| by|\!| |\!| utilizing|\!|
|\!| the|\!| identity|\!|
|\!|
|\!|begin|\!|{eqnarray|\!|}|\!| |\!|delta|\!|left|\!|[|\!| |\!|{|\!|boldsymbol|\!|{f|\!|}|\!|}|\!|(|\!|{|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|}|\!|)|\!| |\!| |\!|-|\!| |\!|dot|\!|{|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|}|\!| |\!|right|\!|]|\!| |\!||\!| |\!|=|\!| |\!||\!| |\!|delta|\!|[|\!| |\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!| |\!|-|\!| |\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|_|\!|{|\!|rm|\!| cl|\!|}|\!|]|\!||\!| |\!|(|\!|det|\!| |\!|{|\!|{M|\!|}|\!|}|\!|)|\!|^|\!|{|\!|-1|\!|}|\!||\!|,|\!| |\!|,|\!| |\!|end|\!|{eqnarray|\!|}|\!|
|\!|
where|\!| |\!|$|\!|{M|\!|}|\!|$|\!| is|\!| a|\!| functional|\!| matrix|\!| formed|\!| by|\!| the|\!| second|\!| derivatives|\!| of|\!| the|\!| action|\!| |\!|$|\!|overlinen|\!| |\!|{|\!|mathcal|\!|{A|\!|}|\!|}|\!|[|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|,|\!|overlinen|\!| |\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|]|\!|equiv|\!| |\!|int|\!| dt|\!||\!|,|\!|overlinen|\!| L|\!|(|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|,|\!| |\!|bar|\!|{|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|}|\!|,|\!| |\!|dot|\!|{|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|}|\!|,|\!| |\!|dot|\!|{|\!|bar|\!|{|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|}|\!|}|\!|)|\!| |\!|$|\!||\!|,|\!|:|\!|
|\!|
|\!|begin|\!|{eqnarray|\!|}|\!| |\!|{|\!|{M|\!|}|\!|}|\!|_|\!|{ab|\!|}|\!|(t|\!|,t|\!|'|\!|)|\!| |\!||\!| |\!|=|\!| |\!||\!| |\!|left|\!|.|\!| |\!|frac|\!|{|\!|delta|\!|^2|\!| |\!|overlinen|\!| |\!|{|\!|mathcal|\!|{A|\!|}|\!|}|\!|}|\!|{|\!|delta|\!| q|\!|_a|\!|(t|\!|)|\!||\!| |\!|delta|\!| |\!|overlinen|\!|{q|\!|}|\!|_b|\!|(t|\!|'|\!|)|\!|}|\!||\!|
|\!|right|\!|||\!|_|\!|{|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!| |\!|=|\!| |\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|_|\!|{|\!|rm|\!| cl|\!|}|\!|}|\!| |\!||\!|,|\!| |\!|.|\!| |\!|label|\!|{4|\!|.01|\!|}|\!| |\!|end|\!|{eqnarray|\!|}|\!|
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The|\!| Morse|\!| index|\!| theorem|\!| then|\!| ensures|\!| that|\!| for|\!| sufficiently|\!| short|\!| time|\!| intervals|\!| |\!|$t|\!|_2|\!|-t|\!|_1|\!|$|\!| |\!|(before|\!| the|\!| system|\!| reaches|\!| its|\!| first|\!| focal|\!| point|\!|)|\!|,|\!| the|\!| classical|\!| solution|\!| with|\!| the|\!| initial|\!| condition|\!| |\!|$|\!|{|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|}|\!|(t|\!|_1|\!|)|\!| |\!|=|\!| |\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|_1|\!|$|\!| is|\!| unique|\!|.|\!| Note|\!|,|\!| however|\!|,|\!| that|\!| because|\!| of|\!| the|\!| first|\!|-order|\!| character|\!| of|\!| the|\!| equations|\!| of|\!| motion|\!| we|\!| are|\!| dealing|\!| with|\!| a|\!| Cauchy|\!| problem|\!|,|\!| which|\!| |\!| may|\!| happen|\!| to|\!| possess|\!| |\!| no|\!| classical|\!| trajectory|\!| satisfying|\!| the|\!| two|\!| Dirichlet|\!| boundary|\!| conditions|\!| |\!|$|\!|{|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|}|\!|(t|\!|_1|\!|)|\!| |\!|=|\!| |\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|_1|\!|$|\!|,|\!| |\!|$|\!|{|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|}|\!|(t|\!|_2|\!|)|\!| |\!|=|\!| |\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|_2|\!|$|\!|.|\!| If|\!| a|\!| trajectory|\!| exists|\!|,|\!| Eq|\!|.|\!|~|\!|(|\!|ref|\!|{eg|\!|.1|\!|.2|\!|}|\!|)|\!| can|\!| be|\!| brought|\!| to|\!| the|\!| form|\!|
|\!|
|\!|begin|\!|{eqnarray|\!|}|\!|
|\!|langle|\!| |\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|_2|\!|,t|\!|_2|\!|||\!| |\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|_1|\!|,|\!| t|\!|_1|\!| |\!|rangle|\!| |\!|&|\!|=|\!|&|\!| |\!|{|\!|bar|\!|{|\!|mathcal|\!|{N|\!|}|\!|}|\!|}|\!| |\!|int|\!|_|\!|{|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|(t|\!|_1|\!|)|\!| |\!|=|\!| |\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|_1|\!|}|\!|^|\!|{|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|(t|\!|_2|\!|)|\!| |\!|=|\!| |\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|_2|\!|}|\!| |\!|{|\!|mathcal|\!|{D|\!|}|\!|}|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!| |\!||\!| |\!|delta|\!|left|\!|[|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!| |\!|-|\!| |\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|_|\!|{|\!|rm|\!| cl|\!|}|\!| |\!|right|\!|]|\!||\!|,|\!| |\!|,|\!| |\!|label|\!|{4|\!|.2|\!|}|\!| |\!|end|\!|{eqnarray|\!|}|\!|
|\!|
where|\!| |\!|$|\!|{|\!|bar|\!|{|\!|mathcal|\!|{N|\!|}|\!|}|\!|}|\!|equiv|\!| |\!|{|\!|{|\!|mathcal|\!|{N|\!|}|\!|}|\!|}|\!|/|\!|(|\!|det|\!| |\!|{M|\!|}|\!|)|\!|$|\!|.|\!| We|\!| close|\!| this|\!| section|\!| by|\!| observing|\!| that|\!| |\!|$|\!|det|\!| M|\!|$|\!| can|\!| be|\!| recast|\!| into|\!| more|\!| expedient|\!| form|\!|.|\!| To|\!| do|\!| this|\!| we|\!| formally|\!| write|\!|
|\!|
|\!|begin|\!|{eqnarray|\!|}|\!|
|\!|det|\!| M|\!| |\!||\!| |\!|&|\!|=|\!|&|\!| |\!||\!| |\!|det|\!|left|\!|||\!||\!|!|\!|left|\!|||\!| |\!| |\!|left|\!|(|\!| |\!|partial|\!|_t|\!| |\!|delta|\!|_a|\!|^b|\!| |\!|+|\!| |\!|frac|\!|{|\!|partial|\!| f|\!|_a|\!|(|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|(t|\!|)|\!|)|\!|}|\!|{|\!|partial|\!| q|\!|_b|\!|(t|\!|)|\!|}|\!|
|\!|right|\!|)|\!|delta|\!|(t|\!|-t|\!|'|\!|)|\!| |\!|right|\!|||\!||\!|!|\!|right|\!|||\!| |\!||\!| |\!|=|\!| |\!||\!| |\!|exp|\!|left|\!|[|\!|
|\!|mbox|\!|{Tr|\!|}|\!|ln|\!|left|\!|||\!||\!|!|\!|left|\!|||\!| |\!| |\!|left|\!|(|\!| |\!|partial|\!|_t|\!| |\!|delta|\!|_a|\!|^b|\!| |\!|+|\!| |\!|frac|\!|{|\!|partial|\!| f|\!|_a|\!|(|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|(t|\!|)|\!|)|\!|}|\!|{|\!|partial|\!| q|\!|_b|\!|(t|\!|)|\!|}|\!|
|\!|right|\!|)|\!|delta|\!|(t|\!|-t|\!|'|\!|)|\!| |\!|right|\!|||\!||\!|!|\!|right|\!|||\!| |\!|right|\!|]|\!|nonumber|\!| |\!||\!||\!|
|\!|&|\!| |\!|=|\!| |\!|&|\!| |\!||\!| |\!|exp|\!|left|\!|[|\!| |\!|mbox|\!|{Tr|\!|}|\!|ln|\!| |\!|partial|\!|_t|\!| |\!|left|\!|||\!||\!|!|\!|left|\!|||\!| |\!|delta|\!|_a|\!|^b|\!|
|\!|delta|\!|(t|\!|-t|\!|'|\!|)|\!| |\!|+|\!| G|\!|(t|\!|-t|\!|'|\!|)|\!|frac|\!|{|\!|partial|\!| f|\!|_a|\!|(|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|(t|\!|'|\!|)|\!|)|\!|}|\!|{|\!|partial|\!| q|\!|_b|\!|(t|\!|'|\!|)|\!|}|\!|right|\!|||\!||\!|!|\!|right|\!|||\!| |\!|right|\!|]|\!|nonumber|\!| |\!||\!||\!| |\!|&|\!|=|\!|&|\!| |\!||\!| |\!|exp|\!|left|\!|[|\!|mbox|\!|{Tr|\!|}|\!|(|\!|ln|\!| |\!|partial|\!|_t|\!|)|\!|right|\!|]|\!|
|\!|exp|\!|left|\!|[|\!|mbox|\!|{Tr|\!|}|\!|ln|\!| |\!|left|\!|||\!||\!|!|\!|left|\!|||\!| |\!|delta|\!|_a|\!|^b|\!| |\!|delta|\!|(t|\!|-t|\!|'|\!|)|\!| |\!|+|\!|
G|\!|(t|\!|-t|\!|'|\!|)|\!|frac|\!|{|\!|partial|\!| f|\!|_a|\!|(|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|(t|\!|'|\!|)|\!|)|\!|}|\!|{|\!|partial|\!| q|\!|_b|\!|(t|\!|'|\!|)|\!|}|\!|right|\!|||\!||\!|!|\!|right|\!|||\!|right|\!|]|\!||\!|,|\!| |\!|.|\!| |\!|label|\!|{4|\!|.2|\!|.1|\!|}|\!| |\!|end|\!|{eqnarray|\!|}|\!|
|\!|
Here|\!| |\!|$G|\!|(t|\!|-t|\!|'|\!|)|\!|$|\!| is|\!| the|\!| Green|\!|'s|\!| function|\!| satisfying|\!| the|\!| equation|\!|
|\!|
|\!|begin|\!|{eqnarray|\!|*|\!|}|\!| |\!|partial|\!|_tG|\!|(t|\!|-t|\!|'|\!|)|\!| |\!||\!| |\!|=|\!| |\!||\!| |\!|delta|\!|(t|\!|-t|\!|'|\!|)|\!||\!|,|\!| |\!|.|\!| |\!|end|\!|{eqnarray|\!|*|\!|}|\!|
|\!|
Choosing|\!| |\!|$G|\!|(t|\!|-t|\!|'|\!|)|\!| |\!|=|\!| |\!|theta|\!|(t|\!|-t|\!|'|\!|)|\!|$|\!|,|\!| and|\!| noting|\!| that|\!| the|\!| first|\!| factor|\!| in|\!| Eq|\!|.|\!|(|\!|ref|\!|{4|\!|.2|\!|.1|\!|}|\!|)|\!| is|\!| an|\!| irrelevant|\!| constant|\!| that|\!| can|\!| be|\!| assimilated|\!| into|\!| |\!|$|\!|{|\!|mathcal|\!|{N|\!|}|\!|}|\!|$|\!| we|\!| have|\!|
|\!|
|\!|begin|\!|{eqnarray|\!|}|\!|
|\!|det|\!| M|\!| |\!||\!| |\!|&|\!|=|\!|&|\!| |\!| |\!||\!| |\!|exp|\!|left|\!|[|\!|mbox|\!|{Tr|\!|}|\!| |\!|ln|\!| |\!|left|\!|||\!||\!|!|\!|left|\!|||\!| |\!|delta|\!|_a|\!|^b|\!|
|\!|delta|\!|(t|\!|-t|\!|'|\!|)|\!| |\!|+|\!| G|\!|(t|\!|-t|\!|'|\!|)|\!|frac|\!|{|\!|partial|\!| f|\!|_a|\!|(|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|(t|\!|'|\!|)|\!|)|\!|}|\!|{|\!|partial|\!| q|\!|_b|\!|(t|\!|'|\!|)|\!|}|\!|right|\!|||\!||\!|!|\!|right|\!|||\!|right|\!|]|\!|
|\!||\!| |\!|=|\!| |\!||\!| |\!|exp|\!|left|\!|[|\!| |\!| |\!|mbox|\!|{Tr|\!|}|\!|left|\!|||\!||\!|!|\!|left|\!|||\!| |\!| |\!|theta|\!|(t|\!|-t|\!|'|\!|)|\!|
|\!|frac|\!|{|\!|partial|\!| f|\!|_a|\!|(|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|(t|\!|)|\!|)|\!|}|\!|{|\!|partial|\!| q|\!|_b|\!|(t|\!|)|\!|}|\!|right|\!|||\!||\!|!|\!|right|\!|||\!| |\!|right|\!|]|\!|nonumber|\!| |\!||\!||\!| |\!|&|\!|=|\!|&|\!| |\!||\!| |\!|exp|\!|left|\!|[|\!|frac|\!|{1|\!|}|\!|{2|\!|}|\!| |\!|int|\!|_|\!|{t|\!|_1|\!|}|\!|^|\!|{t|\!|_2|\!|}|\!| |\!||\!|!|\!| dt|\!| |\!||\!| |\!|{|\!|boldsymbol|\!|{|\!|nabla|\!|}|\!|}|\!|_|\!|{|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|}|\!| |\!|{|\!|boldsymbol|\!|{f|\!|}|\!|}|\!|(|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|)|\!| |\!|right|\!|]|\!||\!|,|\!| |\!|.|\!| |\!|label|\!|{4|\!|.2|\!|.2|\!|}|\!| |\!|end|\!|{eqnarray|\!|}|\!|
|\!|
In|\!| deriving|\!| Eq|\!|.|\!|(|\!|ref|\!|{4|\!|.2|\!|.2|\!|}|\!|)|\!| we|\!| have|\!| used|\!| the|\!| fact|\!| that|\!| due|\!| to|\!| the|\!| product|\!| of|\!| the|\!| |\!|$|\!|theta|\!|$|\!|-function|\!| in|\!| the|\!| expansion|\!| of|\!| the|\!| logarithm|\!|,|\!|
|\!| all|\!| terms|\!| vanish|\!| but|\!| the|\!| first|\!| one|\!|.|\!| In|\!| evaluating|\!| the|\!| generalized|\!| function|\!| |\!|$|\!|theta|\!|(x|\!|)|\!|$|\!| at|\!| the|\!| origin|\!| we|\!| have|\!| used|\!| the|\!| only|\!| consistent|\!| midpoint|\!| rule|\!|~|\!|cite|\!|{Pain|\!|}|\!|:|\!| |\!|$|\!|theta|\!|(0|\!|)|\!| |\!|=|\!| 1|\!|/2|\!|$|\!|.|\!|
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Using|\!| the|\!| |\!| identity|\!|
|\!|
|\!|begin|\!|{eqnarray|\!|}|\!|
|\!| |\!|left|\!|.|\!|exp|\!|left|\!|[|\!|frac|\!|{1|\!|}|\!|{2|\!|}|\!| |\!|int|\!|_|\!|{t|\!|_1|\!|}|\!|^|\!|{t|\!|_2|\!|}|\!| |\!||\!|!|\!| dt|\!| |\!||\!| |\!|{|\!|boldsymbol|\!|{|\!|nabla|\!|}|\!|}|\!|_|\!|{|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|}|\!|
|\!|{|\!|boldsymbol|\!|{f|\!|}|\!|}|\!|(|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|)|\!| |\!|right|\!|]|\!| |\!|right|\!|||\!|_|\!|{|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!| |\!|=|\!| |\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|_|\!|{|\!|rm|\!| cl|\!|}|\!|}|\!||\!| |\!|=|\!| |\!||\!| |\!|int|\!| |\!|{|\!|mathcal|\!|{D|\!|}|\!|}|\!| |\!|{|\!|overline|\!|{|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|}|\!|}|\!| |\!||\!| |\!|delta|\!|left|\!|[|\!|overline|\!|{|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|}|\!| |\!|-|\!| |\!|overline|\!|{|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|}|\!|_|\!|{|\!|rm|\!| cl|\!|}|\!| |\!|right|\!|]|\!| |\!||\!| |\!|exp|\!|left|\!|[|\!|-|\!|frac|\!|{1|\!|}|\!|{2|\!|}|\!| |\!|int|\!|_|\!|{t|\!|_1|\!|}|\!|^|\!|{t|\!|_2|\!|}|\!| |\!||\!|!|\!| dt|\!| |\!||\!| |\!|{|\!|boldsymbol|\!|{|\!|nabla|\!|}|\!|}|\!|_|\!|{|\!|bar|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|}|\!| |\!|dot|\!|{|\!|bar|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|}|\!| |\!|right|\!|]|\!||\!|,|\!| |\!|,|\!| |\!|end|\!|{eqnarray|\!|}|\!|
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we|\!| can|\!| finally|\!| write|\!| the|\!| amplitude|\!| of|\!| transition|\!| in|\!| a|\!| suggestive|\!| form|\!|
|\!|
|\!|begin|\!|{eqnarray|\!|}|\!|
|\!|langle|\!| |\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|_2|\!|,t|\!|_2|\!|||\!| |\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|_1|\!|,|\!| t|\!|_1|\!| |\!|rangle|\!| |\!|&|\!|=|\!|&|\!| |\!|{|\!|mathcal|\!|{N|\!|}|\!|}|\!| |\!|int|\!|_|\!|{|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|(t|\!|_1|\!|)|\!| |\!|=|\!| |\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|_1|\!|}|\!|^|\!|{|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|(t|\!|_2|\!|)|\!| |\!|=|\!| |\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|_2|\!|}|\!| |\!|{|\!|mathcal|\!|{D|\!|}|\!|}|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|{|\!|mathcal|\!|{D|\!|}|\!|}|\!|{|\!|overline|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|}|\!| |\!||\!| |\!|delta|\!|[|\!|{|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|}|\!| |\!|-|\!| |\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|_|\!|{|\!|rm|\!| cl|\!|}|\!|]|\!| |\!|delta|\!|[|\!|overline|\!|{|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|}|\!| |\!|-|\!| |\!|overline|\!|{|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|}|\!|_|\!|{|\!|rm|\!| cl|\!|}|\!|]|\!| |\!||\!| |\!|exp|\!|left|\!|[|\!|-|\!|frac|\!|{1|\!|}|\!|{2|\!|}|\!| |\!|int|\!|_|\!|{t|\!|_1|\!|}|\!|^|\!|{t|\!|_2|\!|}|\!| |\!||\!|!|\!| dt|\!| |\!||\!| |\!|{|\!|boldsymbol|\!|{|\!|nabla|\!|}|\!|}|\!|_|\!|{|\!|bar|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|}|\!| |\!|dot|\!|{|\!|bar|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|}|\!| |\!|right|\!|]|\!| |\!|nonumber|\!| |\!||\!||\!| |\!|&|\!|=|\!|&|\!| |\!|{|\!|mathcal|\!|{N|\!|}|\!|}|\!| |\!|int|\!|_|\!|{|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|(t|\!|_1|\!|)|\!| |\!|=|\!| |\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|_1|\!|}|\!|^|\!|{|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|(t|\!|_2|\!|)|\!| |\!|=|\!| |\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|_2|\!|}|\!| |\!|{|\!|mathcal|\!|{D|\!|}|\!|}|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|{|\!|mathcal|\!|{D|\!|}|\!|}|\!|{|\!|overline|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|}|\!| |\!||\!| |\!|delta|\!|[|\!|{|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|}|\!| |\!|-|\!| |\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|_|\!|{|\!|rm|\!| cl|\!|}|\!|]|\!| |\!|delta|\!|[|\!|overline|\!|{|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|}|\!| |\!|-|\!| |\!|overline|\!|{|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|}|\!|_|\!|{|\!|rm|\!| cl|\!|}|\!|]|\!| |\!||\!| |\!|sqrt|\!|{|\!|frac|\!|{|\!|det|\!| K|\!|(t|\!|_2|\!|)|\!| |\!|}|\!|{|\!| |\!|det|\!| K|\!|(t|\!|_1|\!|)|\!|}|\!|}|\!||\!|,|\!| |\!||\!| |\!|.|\!| |\!|label|\!|{3|\!|.50|\!|}|\!| |\!|end|\!|{eqnarray|\!|}|\!|
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Here|\!| |\!|$K|\!|(t|\!|)|\!|$|\!| is|\!| the|\!| fundamental|\!| matrix|\!| of|\!| the|\!| solutions|\!| of|\!| the|\!| system|\!|
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|\!|begin|\!|{eqnarray|\!|}|\!| |\!|dot|\!|{|\!|bar|\!|{q|\!|}|\!|}|\!|_a|\!| |\!|=|\!| |\!|-|\!| |\!|bar|\!|{q|\!|}|\!|_b|\!| |\!|frac|\!|{|\!|partial|\!| f|\!|_b|\!|(|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|)|\!|}|\!|{|\!|partial|\!| q|\!|_a|\!|}|\!| |\!||\!|,|\!| |\!|.|\!| |\!|end|\!|{eqnarray|\!|}|\!|
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|\!|$|\!|det|\!| K|\!|(t|\!|)|\!|$|\!| is|\!| then|\!| the|\!| corresponding|\!| Wronskian|\!|.|\!| Note|\!| that|\!| in|\!| the|\!| particular|\!| case|\!| when|\!| |\!|$|\!|{|\!|boldsymbol|\!|{|\!|nabla|\!|}|\!|}|\!|_|\!|{|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|}|\!| |\!|{|\!|boldsymbol|\!|{f|\!|}|\!|}|\!|(|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|)|\!| |\!|equiv|\!| 0|\!|$|\!|,|\!| i|\!|.e|\!|.|\!|,|\!| when|\!| the|\!| phase|\!| flow|\!| preserves|\!| the|\!| volume|\!| of|\!| any|\!| domain|\!| in|\!| the|\!| |\!|{|\!|em|\!| configuration|\!|}|\!| space|\!|,|\!| the|\!| exponential|\!| in|\!| Eq|\!|.|\!|(|\!|ref|\!|{3|\!|.50|\!|}|\!|)|\!| can|\!| be|\!| dropped|\!|.|\!|footnote|\!|{This|\!| corresponds|\!| to|\!| the|\!| situation|\!| when|\!| there|\!| are|\!| no|\!| attractors|\!| in|\!| the|\!| configuration|\!| space|\!| |\!|$|\!|Gamma|\!|_|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|.|\!|$|\!|}|\!| Because|\!| the|\!| exponent|\!| depends|\!| only|\!| on|\!| the|\!| end|\!| points|\!| of|\!| |\!|$|\!|bar|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|$|\!| variable|\!| it|\!| can|\!| be|\!| removed|\!| by|\!| performing|\!| the|\!| trace|\!| over|\!| |\!|$|\!|bar|\!|{|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|}|\!|$|\!|.|\!|
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As|\!| a|\!| result|\!| we|\!| can|\!| cast|\!| the|\!| quantum|\!|-mechanical|\!| partition|\!| function|\!| |\!|(or|\!| generating|\!| functional|\!|)|\!| |\!|$Z|\!|$|\!| into|\!| the|\!| form|\!|
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|\!|begin|\!|{eqnarray|\!|}|\!| Z|\!| |\!||\!| |\!|&|\!|=|\!|&|\!| |\!||\!| |\!|{|\!|mathcal|\!|{N|\!|}|\!|}|\!| |\!|int|\!| |\!|{|\!|mathcal|\!|{D|\!|}|\!|}|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|{|\!|mathcal|\!|{D|\!|}|\!|}|\!|{|\!|overline|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|}|\!| |\!||\!| |\!|delta|\!|[|\!|{|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|}|\!| |\!|-|\!| |\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|_|\!|{|\!|rm|\!| cl|\!|}|\!|]|\!| |\!|delta|\!|[|\!|overline|\!|{|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|}|\!| |\!|-|\!| |\!|overline|\!|{|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|}|\!|_|\!|{|\!|rm|\!| cl|\!|}|\!|]|\!| |\!||\!| |\!| |\!|exp|\!|left|\!|[|\!|int|\!|_|\!|{t|\!|_1|\!|}|\!|^|\!|{t|\!|_2|\!|}|\!| |\!|[|\!|{|\!|boldsymbol|\!|{J|\!|}|\!|}|\!|(t|\!|)|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|(t|\!|)|\!| |\!|+|\!| |\!|bar|\!|{|\!|{|\!|boldsymbol|\!|{J|\!|}|\!|}|\!|}|\!|(t|\!|)|\!|bar|\!|{|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|}|\!|(t|\!|)|\!|]dt|\!|right|\!|]|\!|nonumber|\!| |\!||\!||\!| |\!|&|\!|=|\!|&|\!| |\!||\!| |\!|{|\!|mathcal|\!|{N|\!|}|\!|}|\!| |\!|int|\!| |\!|{|\!|mathcal|\!|{D|\!|}|\!|}|\!| q|\!|_a|\!| |\!||\!| |\!|delta|\!|[q|\!|_a|\!| |\!|-|\!| |\!|(q|\!|_a|\!|)|\!|_|\!|{|\!|rm|\!| cl|\!|}|\!|]|\!| |\!||\!| |\!|exp|\!|left|\!|[|\!|int|\!|^|\!|{t|\!|_2|\!|}|\!|_|\!|{t|\!|_1|\!|}|\!| dt|\!| |\!||\!| J|\!|_a|\!|(t|\!|)|\!| q|\!|_a|\!|(t|\!|)|\!| |\!| |\!|right|\!|]|\!||\!|,|\!| |\!|.|\!| |\!|label|\!|{3|\!|.51|\!|}|\!| |\!|end|\!|{eqnarray|\!|}|\!|
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Here|\!| the|\!| doubled|\!| vector|\!| notation|\!| |\!|$q|\!|_a|\!| |\!|=|\!| |\!||\!|{|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|,|\!| |\!|bar|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!||\!|}|\!|$|\!| and|\!| |\!|$J|\!|_a|\!| |\!|equiv|\!| |\!||\!|{|\!|{|\!|boldsymbol|\!|{J|\!|}|\!|}|\!|,|\!| |\!|bar|\!|{|\!|boldsymbol|\!|{J|\!|}|\!|}|\!||\!|}|\!| |\!|$|\!| was|\!| used|\!|.|\!|
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|\!|section|\!|{Path|\!| integral|\!| formulation|\!| of|\!| classical|\!| mechanics|\!| |\!|-|\!| configuration|\!|-space|\!| approach|\!|}|\!|label|\!|{SEc3|\!|}|\!|
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Expressions|\!| |\!|(|\!|ref|\!|{4|\!|.2|\!|}|\!|)|\!| and|\!| |\!|(|\!|ref|\!|{3|\!|.51|\!|}|\!|)|\!| formally|\!| coincide|\!| with|\!| the|\!| path|\!|-integral|\!| formulation|\!| of|\!| classical|\!| mechanics|\!| in|\!| configuration|\!| space|\!| proposed|\!| by|\!| Gozzi|\!|~|\!|cite|\!|{GozziI|\!|}|\!| and|\!| further|\!| developed|\!| by|\!| Gozzi|\!|,|\!| Reuter|\!|,|\!| and|\!| Thacker|\!|~|\!|cite|\!|{GozziII|\!|}|\!|(see|\!| also|\!| Ref|\!|.|\!|cite|\!|{Elze2|\!|}|\!| for|\!| recent|\!| applications|\!|)|\!|.|\!| Let|\!| us|\!| briefly|\!| review|\!| aspects|\!| of|\!| this|\!| which|\!| will|\!| be|\!| needed|\!| here|\!|.|\!| Consider|\!| the|\!| path|\!|-integral|\!| representation|\!| of|\!| the|\!| generating|\!| functional|\!| of|\!| a|\!| quantum|\!|-mechanical|\!| system|\!| with|\!| action|\!| |\!|$|\!|{|\!|mathcal|\!|{A|\!|}|\!|}|\!|[|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|]|\!|$|\!|:|\!|
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|\!|begin|\!|{eqnarray|\!|}|\!| |\!|{|\!|{|\!|{Z|\!|}|\!|}|\!| |\!|}|\!|_|\!|{|\!|rm|\!| QM|\!|}|\!| |\!|=|\!| |\!|{|\!|{|\!|mathcal|\!|{N|\!|}|\!|}|\!|}|\!| |\!|int|\!| |\!|{|\!|mathcal|\!|{D|\!|}|\!|}|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!||\!| e|\!|^|\!|{|\!|-i|\!| |\!|{|\!|mathcal|\!|{A|\!|}|\!|}|\!|[|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|]|\!|/|\!|hbar|\!| |\!|}|\!| |\!|exp|\!|left|\!|[|\!|int|\!| |\!|{|\!|boldsymbol|\!|{J|\!|}|\!|}|\!|(t|\!|)|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|(t|\!|)dt|\!|right|\!|]|\!||\!|,|\!| |\!|.|\!| |\!|label|\!|{|\!|@genf|\!|}|\!| |\!|label|\!|{4|\!|.0|\!|}|\!| |\!|end|\!|{eqnarray|\!|}|\!|
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We|\!| assume|\!| in|\!| this|\!| context|\!| that|\!| there|\!| are|\!| no|\!| constraints|\!| that|\!| would|\!| make|\!| the|\!| measure|\!| more|\!| complicated|\!| as|\!| in|\!| Eq|\!|.|\!|~|\!|(|\!|ref|\!|{frad|\!|}|\!|)|\!|.|\!| Gozzi|\!| |\!|{|\!|em|\!| et|\!| al|\!|.|\!|}|\!| proposed|\!| to|\!| describe|\!| classical|\!| mechanics|\!| by|\!| a|\!| generating|\!| functional|\!| of|\!| the|\!| form|\!| |\!|(|\!|ref|\!|{|\!|@genf|\!|}|\!|)|\!| with|\!| an|\!| obviously|\!| |\!| modified|\!| integration|\!| measure|\!| which|\!| gives|\!| |\!| equal|\!| weight|\!| to|\!| all|\!| classical|\!| trajectories|\!| and|\!| zero|\!| weight|\!| to|\!| all|\!| others|\!|
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|\!|begin|\!|{eqnarray|\!|}|\!| |\!|{|\!|{|\!|{Z|\!|}|\!|}|\!| |\!|}|\!|_|\!|{|\!|rm|\!| CM|\!|}|\!| |\!|=|\!| |\!|tilde|\!|{|\!|{|\!|{|\!|mathcal|\!|{N|\!|}|\!|}|\!|}|\!|}|\!| |\!|int|\!| |\!|{|\!|mathcal|\!|{D|\!|}|\!|}|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!| |\!||\!| |\!|delta|\!|[|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|-|\!| |\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|_|\!|{|\!|rm|\!| cl|\!|}|\!|]|\!|
|\!| |\!|exp|\!|left|\!|[|\!|int|\!| |\!|{|\!|boldsymbol|\!|{J|\!|}|\!|}|\!|(t|\!|)|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|(t|\!|)dt|\!|right|\!|]|\!||\!|,|\!| |\!|.|\!| |\!|label|\!|{4|\!|.3|\!|}|\!| |\!|end|\!|{eqnarray|\!|}|\!|
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Although|\!| the|\!| form|\!| of|\!| the|\!| partition|\!| function|\!| |\!|(|\!|ref|\!|{4|\!|.3|\!|}|\!|)|\!| is|\!| not|\!| derived|\!| but|\!| |\!|{|\!|em|\!| postulated|\!|}|\!|,|\!| we|\!| show|\!| in|\!| Appendix|\!| B|\!| that|\!| it|\!| can|\!| be|\!| heuristically|\!| understood|\!| either|\!| as|\!| the|\!| |\!|`|\!|`classical|\!|"|\!| limit|\!| of|\!| the|\!| stochastic|\!|-quantization|\!| partition|\!| function|\!| |\!|(c|\!|.f|\!|.|\!|,|\!| Appendix|\!| BI|\!|)|\!|,|\!| or|\!| as|\!| a|\!| results|\!| of|\!| the|\!| classical|\!| limit|\!| of|\!| the|\!| closed|\!|-time|\!| path|\!| integral|\!| for|\!| the|\!| transition|\!| probability|\!| of|\!| systems|\!| coupled|\!| to|\!| a|\!| heat|\!| bath|\!| |\!|(c|\!|.f|\!|.|\!|,|\!| Appendix|\!| BII|\!|)|\!|.|\!| This|\!|,|\!| in|\!| turn|\!|,|\!| indicates|\!| that|\!| it|\!| would|\!| be|\!| formally|\!| more|\!| correct|\!| to|\!| associate|\!| |\!|(|\!|ref|\!|{4|\!|.3|\!|}|\!|)|\!| with|\!| the|\!| |\!|{|\!|em|\!| probability|\!|}|\!| of|\!| transition|\!| or|\!| |\!|(via|\!| the|\!| stochastic|\!|-quantization|\!| passage|\!|)|\!| with|\!| the|\!| |\!|{|\!|em|\!| Euclidean|\!|}|\!| amplitude|\!| of|\!| transition|\!|~|\!|cite|\!|{Zinn|\!|-JustinII|\!|}|\!|.|\!| Albeit|\!| |\!|(|\!|ref|\!|{4|\!|.3|\!|}|\!|)|\!| cannot|\!| be|\!| generally|\!| obtained|\!| from|\!| |\!|(|\!|ref|\!|{4|\!|.0|\!|}|\!|)|\!| by|\!| a|\!| semiclassical|\!| limit|\!| |\!|{|\!|em|\!| |\!||\!|`|\!|{a|\!|}|\!| la|\!|}|\!| WKB|\!| |\!|(which|\!| can|\!| be|\!| recognized|\!| by|\!| the|\!| absence|\!| of|\!| a|\!| phase|\!| factor|\!| |\!|$|\!|exp|\!|(i|\!|/|\!|hbar|\!| |\!|{|\!|mathcal|\!|{A|\!|}|\!|}|\!|(q|\!|_|\!|{|\!|rm|\!| cl|\!|}|\!|)|\!|)|\!|$|\!| in|\!| |\!|(|\!|ref|\!|{4|\!|.3|\!|}|\!|)|\!|)|\!| it|\!| may|\!| happen|\!| that|\!| even|\!| ordinary|\!| amplitudes|\!| of|\!| transition|\!| posses|\!| this|\!| form|\!|.|\!| This|\!| is|\!| the|\!| case|\!|,|\!| for|\!| instance|\!|,|\!| when|\!| the|\!| number|\!| of|\!| degrees|\!| of|\!| freedom|\!| is|\!| doubled|\!| or|\!| when|\!| one|\!| deals|\!| with|\!| closed|\!|-time|\!|-path|\!| formulation|\!| of|\!| thermal|\!| quantum|\!| theory|\!|.|\!| Yet|\!|,|\!| whatever|\!| is|\!| the|\!| origin|\!| or|\!| motivation|\!| for|\!| |\!|(|\!|ref|\!|{4|\!|.3|\!|}|\!|)|\!|,|\!| it|\!| will|\!| be|\!| its|\!| formal|\!| structure|\!| and|\!| mathematical|\!| implications|\!| that|\!| will|\!| interest|\!| us|\!| here|\!| most|\!|.|\!|
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|\!|vspace|\!|{3mm|\!|}|\!|
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To|\!| proceed|\!| we|\!| note|\!| that|\!| an|\!| alternative|\!| way|\!| of|\!| writing|\!| |\!|(|\!|ref|\!|{4|\!|.3|\!|}|\!|)|\!| is|\!|
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|\!|begin|\!|{eqnarray|\!|}|\!| |\!|{|\!|{|\!|{Z|\!|}|\!|}|\!| |\!|}|\!|_|\!|{|\!|rm|\!| CM|\!|}|\!||\!| |\!|=|\!| |\!||\!| |\!|tilde|\!|{|\!|{|\!|{|\!|mathcal|\!|{N|\!|}|\!|}|\!|}|\!|}|\!| |\!|int|\!| |\!|{|\!|mathcal|\!|{D|\!|}|\!|}|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!| |\!||\!| |\!|delta|\!|left|\!|[|\!| |\!|frac|\!|{|\!|delta|\!|
|\!|{|\!|mathcal|\!|{A|\!|}|\!|}|\!|}|\!|{|\!|delta|\!| |\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|}|\!| |\!|right|\!|]|\!| |\!||\!| |\!|det|\!| |\!|left|\!|||\!| |\!|frac|\!|{|\!|delta|\!|^2|\!|
|\!|{|\!|mathcal|\!|{A|\!|}|\!|}|\!| |\!|}|\!|{|\!|delta|\!| q|\!|_a|\!| |\!|(t|\!|)|\!| |\!||\!| |\!|delta|\!| q|\!|_b|\!|(t|\!|'|\!|)|\!|}|\!| |\!|right|\!|||\!| |\!||\!| |\!|exp|\!|left|\!|[|\!|int|\!| |\!|{|\!|boldsymbol|\!|{J|\!|}|\!|}|\!|(t|\!|)|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|(t|\!|)dt|\!|right|\!|]|\!||\!|,|\!| |\!|.|\!| |\!|label|\!|{4|\!|.4|\!|}|\!| |\!|end|\!|{eqnarray|\!|}|\!|
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By|\!| representing|\!| the|\!| |\!|$|\!|delta|\!|$|\!| functional|\!| in|\!| the|\!| usual|\!| way|\!| as|\!| a|\!| functional|\!| Fourier|\!| integral|\!|,|\!|
|\!|
|\!|begin|\!|{eqnarray|\!|}|\!| |\!|delta|\!|left|\!|[|\!| |\!|frac|\!|{|\!|delta|\!| |\!|{|\!|mathcal|\!|{A|\!|}|\!|}|\!|}|\!|{|\!|delta|\!| |\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|}|\!| |\!|right|\!|]|\!| |\!|=|\!| |\!|int|\!| |\!|{|\!|mathcal|\!|{D|\!|}|\!|}|\!| |\!|{|\!|mathbf|\!|{|\!|lambda|\!|}|\!|}|\!| |\!||\!| |\!|exp|\!|left|\!|(|\!| i|\!| |\!|int|\!|_|\!|{t|\!|_1|\!|}|\!|^|\!|{t|\!|_2|\!|}|\!| dt|\!| |\!||\!| |\!|{|\!|boldsymbol|\!|{|\!|lambda|\!|}|\!|}|\!|(t|\!|)|\!| |\!|frac|\!|{|\!|delta|\!| |\!|{|\!|mathcal|\!|{A|\!|}|\!|}|\!|}|\!|{|\!|delta|\!| |\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|(t|\!|)|\!|}|\!| |\!|right|\!|)|\!||\!|,|\!| |\!|,|\!| |\!|end|\!|{eqnarray|\!|}|\!|
|\!|
and|\!| the|\!| functional|\!| determinant|\!| as|\!| a|\!| functional|\!| integral|\!| over|\!| two|\!| real|\!| time|\!|-dependent|\!| Grassmannian|\!| |\!|{|\!|em|\!| ghost|\!| variables|\!||\!|/|\!|}|\!| |\!|$c|\!|_a|\!|(t|\!|)|\!|$|\!| and|\!| |\!|$|\!|overlinen|\!|{c|\!|}|\!|_a|\!|(t|\!|)|\!|$|\!|,|\!|
|\!|
|\!|begin|\!|{eqnarray|\!|}|\!|
|\!|det|\!| |\!|left|\!|||\!| |\!|frac|\!|{|\!|delta|\!|^2|\!| |\!|{|\!|mathcal|\!|{A|\!|}|\!|}|\!| |\!|}|\!|{|\!|delta|\!| q|\!|_a|\!| |\!|(t|\!|)|\!| |\!||\!| |\!|delta|\!| q|\!|_b|\!|(t|\!|'|\!|)|\!|}|\!|
|\!|right|\!|||\!| |\!|=|\!| |\!|int|\!| |\!|{|\!|mathcal|\!|{D|\!|}|\!|}|\!|{|\!|boldsymbol|\!|{c|\!|}|\!|}|\!| |\!|{|\!|mathcal|\!|{D|\!|}|\!|}|\!| |\!|overlinen|\!|{|\!|{|\!|boldsymbol|\!|{c|\!|}|\!|}|\!|}|\!| |\!||\!| |\!|exp|\!|left|\!|[|\!| |\!|int|\!|_|\!|{t|\!|_1|\!|}|\!|^|\!|{t|\!|_2|\!|}dt|\!| |\!|int|\!|_|\!|{t|\!|_1|\!|}|\!|^|\!|{t|\!|_2|\!|}|\!| dt|\!|'|\!| |\!||\!| |\!|overlinen|\!|{c|\!|}|\!|_a|\!|(t|\!|)|\!| |\!|frac|\!|{|\!|delta|\!|^2|\!| |\!|{|\!|mathcal|\!|{A|\!|}|\!|}|\!| |\!|}|\!|{|\!|delta|\!|{q|\!|_a|\!|}|\!| |\!|(t|\!|)|\!| |\!||\!| |\!|delta|\!|{q|\!|_b|\!|}|\!|(t|\!|'|\!|)|\!|}|\!| |\!||\!| |\!|{c|\!|_b|\!|}|\!|(t|\!|'|\!|)|\!|right|\!|]|\!||\!|,|\!| |\!|,|\!| |\!|end|\!|{eqnarray|\!|}|\!|
|\!|
we|\!| obtain|\!|
|\!|
|\!|begin|\!|{eqnarray|\!|}|\!| |\!|{|\!|{|\!|{Z|\!|}|\!|}|\!| |\!|}|\!|_|\!|{|\!|rm|\!| CM|\!|}|\!| |\!||\!| |\!|=|\!| |\!||\!| |\!|int|\!| |\!|{|\!|mathcal|\!|{D|\!|}|\!|}|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|{|\!|mathcal|\!|{D|\!|}|\!|}|\!|{|\!|boldsymbol|\!|{|\!|lambda|\!|}|\!|}|\!|{|\!|mathcal|\!|{D|\!|}|\!|}|\!|{|\!|boldsymbol|\!|{c|\!|}|\!|}|\!| |\!|{|\!|mathcal|\!|{D|\!|}|\!|}|\!|overlinen|\!|{|\!|{|\!|boldsymbol|\!|{c|\!|}|\!|}|\!|}|\!| |\!||\!| |\!|exp|\!|left|\!|[|\!| i|\!| |\!|{|\!|mathcal|\!|{S|\!|}|\!|}|\!| |\!|+|\!| |\!|int|\!|_|\!|{t|\!|_1|\!|}|\!|^|\!|{t|\!|_2|\!|}|\!| dt|\!| |\!||\!| |\!|{|\!|{|\!|boldsymbol|\!|{J|\!|}|\!|}|\!|}|\!|(t|\!|)|\!|{|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|}|\!|(t|\!|)|\!| |\!|right|\!|]|\!||\!|,|\!| |\!|,|\!| |\!|label|\!|{4|\!|.5a|\!|}|\!| |\!|end|\!|{eqnarray|\!|}|\!|
|\!|
with|\!| the|\!| new|\!| action|\!|
|\!|
|\!|begin|\!|{eqnarray|\!|}|\!| |\!|{|\!|mathcal|\!|{S|\!|}|\!|}|\!|[|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|,|\!| |\!|overlinen|\!|{|\!|{|\!|boldsymbol|\!|{c|\!|}|\!|}|\!|}|\!|,|\!| |\!|{|\!|boldsymbol|\!|{c|\!|}|\!|}|\!|,|\!| |\!|{|\!|boldsymbol|\!|{|\!|lambda|\!|}|\!|}|\!|]|\!| |\!|equiv|\!| |\!||\!| |\!|int|\!|_|\!|{t|\!|_1|\!|}|\!|^|\!|{t|\!|_2|\!|}|\!| dt|\!| |\!||\!| |\!|{|\!|boldsymbol|\!|{|\!|lambda|\!|}|\!|}|\!|(t|\!|)|\!|frac|\!|{|\!|delta|\!| |\!|{|\!|mathcal|\!|{A|\!|}|\!|}|\!|}|\!|{|\!|delta|\!| |\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|(t|\!|)|\!|}|\!| |\!|-|\!| i|\!|int|\!|_|\!|{t|\!|_1|\!|}|\!|^|\!|{t|\!|_2|\!|}dt|\!| |\!|int|\!|_|\!|{t|\!|_1|\!|}|\!|^|\!|{t|\!|_2|\!|}|\!| dt|\!|'|\!| |\!||\!| |\!|overlinen|\!|{c|\!|}|\!|_a|\!|(t|\!|)|\!| |\!|frac|\!|{|\!|delta|\!|^2|\!| |\!|{|\!|mathcal|\!|{A|\!|}|\!|}|\!| |\!|}|\!|{|\!|delta|\!|{q|\!|_a|\!|}|\!| |\!|(t|\!|)|\!| |\!||\!| |\!|delta|\!|{q|\!|_b|\!|}|\!|(t|\!|'|\!|)|\!|}|\!| |\!||\!| |\!|{c|\!|_b|\!|}|\!|(t|\!|'|\!|)|\!||\!|,|\!| |\!|.|\!| |\!|label|\!|{4|\!|.5|\!|}|\!| |\!|end|\!|{eqnarray|\!|}|\!|
|\!|
Since|\!| |\!|$|\!|{|\!|{|\!|{Z|\!|}|\!|}|\!| |\!|}|\!|_|\!|{|\!|rm|\!| CM|\!|}|\!|$|\!| together|\!| with|\!| the|\!| action|\!| |\!|(|\!|ref|\!|{4|\!|.5|\!|}|\!|)|\!| formally|\!| result|\!| from|\!| the|\!| classical|\!| limit|\!| of|\!| the|\!| stochastic|\!|-quantization|\!| partition|\!| function|\!|,|\!| it|\!| comes|\!| as|\!| no|\!| surprise|\!| that|\!| |\!|$|\!|{|\!|mathcal|\!|{S|\!|}|\!|}|\!|$|\!| exhibits|\!| BRST|\!| |\!|(and|\!| anti|\!|-BRST|\!|)|\!| supersymmetry|\!|.|\!| It|\!| is|\!| simple|\!| to|\!| check|\!| that|\!| |\!|$|\!|mathcal|\!|{S|\!|}|\!|$|\!| does|\!| not|\!| change|\!| under|\!| |\!| the|\!| supersymmetry|\!| transformations|\!|
|\!|
|\!|begin|\!|{eqnarray|\!|}|\!| |\!|delta|\!|_|\!|{|\!|{|\!|rm|\!| BRST|\!||\!|,|\!|}|\!|}|\!| |\!| |\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!| |\!|=|\!| |\!|overlinen|\!|{|\!|varepsilon|\!|}|\!| |\!|{|\!|boldsymbol|\!|{c|\!|}|\!|}|\!||\!|,|\!|,|\!| |\!||\!|;|\!||\!|;|\!| |\!|delta|\!|_|\!|{|\!|{|\!|rm|\!| BRST|\!||\!|,|\!|}|\!|}|\!| |\!| |\!|{|\!|boldsymbol|\!|{c|\!|}|\!|}|\!| |\!|=|\!| 0|\!||\!|,|\!| |\!|,|\!| |\!||\!|;|\!||\!|;|\!| |\!|delta|\!|_|\!|{|\!|{|\!|rm|\!| BRST|\!||\!|,|\!|}|\!|}|\!| |\!| |\!|overlinen|\!|{|\!|{|\!|boldsymbol|\!|{c|\!|}|\!|}|\!|}|\!| |\!|=|\!| |\!|-i|\!|overlinen|\!|{|\!|varepsilon|\!|}|\!| |\!|{|\!|boldsymbol|\!|{|\!|lambda|\!|}|\!|}|\!||\!|,|\!| |\!|,|\!| |\!||\!|;|\!||\!|;|\!| |\!|delta|\!|_|\!|{|\!|{|\!|rm|\!| BRST|\!||\!|,|\!|}|\!|}|\!| |\!| |\!|{|\!|boldsymbol|\!|{|\!|lambda|\!|}|\!|}|\!| |\!|=|\!| 0|\!||\!|,|\!| |\!|,|\!| |\!|label|\!|{4|\!|.6|\!|}|\!| |\!|end|\!|{eqnarray|\!|}|\!|
|\!|
where|\!| |\!|$|\!|overlinen|\!|{|\!|varepsilon|\!|}|\!|$|\!| is|\!| a|\!| Grassmann|\!|-valued|\!| parameter|\!| |\!|(the|\!| corresponding|\!| anti|\!|-BRST|\!| transformations|\!| are|\!| related|\!| with|\!| |\!|(|\!|ref|\!|{4|\!|.6|\!|}|\!|)|\!| by|\!| charge|\!| conjugation|\!|)|\!|.|\!| Indeed|\!|,|\!| the|\!| variations|\!| of|\!| the|\!| two|\!| terms|\!| in|\!| |\!|(|\!|ref|\!|{4|\!|.5|\!|}|\!|)|\!| read|\!|
|\!|
|\!|begin|\!|{eqnarray|\!|}|\!| |\!|&|\!|&|\!|delta|\!|_|\!|{|\!|{|\!|rm|\!| BRST|\!||\!|,|\!|}|\!|}|\!| |\!|left|\!|[|\!|int|\!|_|\!|{t|\!|_1|\!|}|\!|^|\!|{t|\!|_2|\!|}|\!| dt|\!| |\!||\!| |\!|{|\!|boldsymbol|\!|{|\!|lambda|\!|}|\!|}|\!|(t|\!|)|\!|frac|\!|{|\!|delta|\!| |\!|{|\!|mathcal|\!|{A|\!|}|\!|}|\!|}|\!|{|\!|delta|\!| |\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|(t|\!|)|\!|}|\!| |\!|right|\!|]|\!| |\!||\!| |\!|=|\!| |\!||\!| |\!|overlinen|\!|{|\!|varepsilon|\!|}|\!|int|\!|_|\!|{t|\!|_1|\!|}|\!|^|\!|{t|\!|_2|\!|}|\!| dt|\!| |\!|int|\!|_|\!|{t|\!|_1|\!|}|\!|^|\!|{t|\!|_2|\!|}|\!| dt|\!|'|\!| |\!||\!| |\!|lambda|\!|_a|\!|(t|\!|)|\!| |\!|frac|\!|{|\!|delta|\!|^2|\!| |\!|{|\!|mathcal|\!|{A|\!|}|\!|}|\!|}|\!|{|\!|delta|\!| q|\!|_a|\!|(t|\!|)|\!| |\!|delta|\!| q|\!|_b|\!|(t|\!|'|\!|)|\!|}|\!| |\!||\!| c|\!|_b|\!|(t|\!|'|\!|)|\!||\!|,|\!| |\!|,|\!|label|\!|{4|\!|.7|\!|}|\!| |\!||\!||\!| |\!|&|\!|&|\!|~|\!|nonumber|\!| |\!||\!||\!| |\!|&|\!|&|\!|delta|\!|_|\!|{|\!|{|\!|rm|\!| BRST|\!||\!|,|\!|}|\!|}|\!| |\!|left|\!|[|\!| |\!| |\!|int|\!|_|\!|{t|\!|_1|\!|}|\!|^|\!|{t|\!|_2|\!|}dt|\!| |\!|int|\!|_|\!|{t|\!|_1|\!|}|\!|^|\!|{t|\!|_2|\!|}|\!| dt|\!|'|\!| |\!||\!| |\!|{|\!|overlinen|\!|{c|\!|}|\!|}|\!|_a|\!|(t|\!|)|\!| |\!|frac|\!|{|\!|delta|\!|^2|\!| |\!|{|\!|mathcal|\!|{A|\!|}|\!|}|\!|}|\!|{|\!|delta|\!| q|\!|_a|\!|(t|\!|)|\!| |\!|delta|\!| q|\!|_b|\!|(t|\!|'|\!|)|\!|}|\!| |\!||\!| c|\!|_b|\!|(t|\!|'|\!|)|\!| |\!|right|\!|]|\!| |\!||\!| |\!|=|\!| |\!||\!| |\!|-i|\!| |\!|overlinen|\!|{|\!|varepsilon|\!|}|\!| |\!|int|\!|_|\!|{t|\!|_1|\!|}|\!|^|\!|{t|\!|_2|\!|}dt|\!| |\!|int|\!|_|\!|{t|\!|_1|\!|}|\!|^|\!|{t|\!|_2|\!|}|\!| dt|\!|'|\!| |\!||\!| |\!|lambda|\!|_a|\!|(t|\!|)|\!| |\!|frac|\!|{|\!|delta|\!|^2|\!| |\!|{|\!|mathcal|\!|{A|\!|}|\!|}|\!|}|\!|{|\!|delta|\!| q|\!|_a|\!|(t|\!|)|\!| |\!|delta|\!| q|\!|_b|\!|(t|\!|'|\!|)|\!|}|\!| |\!||\!| c|\!|_b|\!|(t|\!|'|\!|)|\!|nonumber|\!| |\!||\!||\!| |\!|&|\!|&|\!| |\!|~|\!|nonumber|\!| |\!||\!||\!| |\!|&|\!|&|\!|mbox|\!|{|\!|hspace|\!|{15mm|\!|}|\!|}|\!| |\!|+|\!| |\!||\!| |\!|int|\!|_|\!|{t|\!|_1|\!|}|\!|^|\!|{t|\!|_2|\!|}dt|\!| |\!|int|\!|_|\!|{t|\!|_1|\!|}|\!|^|\!|{t|\!|_2|\!|}|\!| dt|\!|'|\!| |\!|int|\!|_|\!|{t|\!|_1|\!|}|\!|^|\!|{t|\!|_2|\!|}dt|\!|'|\!|'|\!| |\!||\!| |\!|{|\!|overlinen|\!|{c|\!|}|\!|}|\!|_a|\!|(t|\!|)|\!| |\!|frac|\!|{|\!|delta|\!|^3|\!| |\!|{|\!|mathcal|\!|{A|\!|}|\!|}|\!|}|\!|{|\!|delta|\!| q|\!|_a|\!|(t|\!|)|\!| |\!|delta|\!| q|\!|_b|\!|(t|\!|'|\!|)|\!| |\!|delta|\!| q|\!|_c|\!|(t|\!|'|\!|'|\!|)|\!|}|\!| |\!||\!| |\!|overlinen|\!|{|\!|varepsilon|\!|}|\!| |\!||\!| c|\!|_c|\!|(t|\!|'|\!|'|\!|)|\!| c|\!|_b|\!|(t|\!|'|\!|)|\!||\!|,|\!| |\!|.|\!| |\!|label|\!|{4|\!|.8|\!|}|\!| |\!|end|\!|{eqnarray|\!|}|\!| |\!||\!||\!|
|\!|
|\!|noi|\!| The|\!| second|\!| term|\!| on|\!| the|\!| RHS|\!| of|\!| |\!|(|\!|ref|\!|{4|\!|.8|\!|}|\!|)|\!| vanishes|\!| because|\!| the|\!| functional|\!| derivative|\!| of|\!| |\!|$|\!|{|\!|mathcal|\!|{A|\!|}|\!|}|\!|$|\!| is|\!| symmetric|\!| in|\!| |\!|$c|\!|leftrightarrow|\!| b|\!|$|\!| whereas|\!| the|\!| term|\!| |\!|$c|\!|_c|\!| c|\!|_b|\!|$|\!| is|\!| anti|\!|-symmetric|\!|.|\!| Inserting|\!| Eqs|\!|.|\!|(|\!|ref|\!|{4|\!|.7|\!|}|\!|)|\!| and|\!| |\!|(|\!|ref|\!|{4|\!|.8|\!|}|\!|)|\!| into|\!| the|\!| action|\!| we|\!| clearly|\!| find|\!| |\!|$|\!|delta|\!|_|\!|{|\!|{|\!|rm|\!| BRST|\!||\!|,|\!|}|\!|}|\!|{|\!|mathcal|\!|{S|\!|}|\!|}|\!| |\!|=|\!| 0|\!|$|\!|.|\!| As|\!| noted|\!| in|\!|~|\!|cite|\!|{GozziII|\!|}|\!|,|\!| the|\!| ghost|\!| fields|\!| |\!|$|\!|overlinen|\!|{|\!|{|\!|boldsymbol|\!|{c|\!|}|\!|}|\!|}|\!|$|\!| and|\!| |\!|$|\!|{|\!|boldsymbol|\!|{c|\!|}|\!|}|\!|$|\!| are|\!| mandatory|\!| at|\!| the|\!| classical|\!| level|\!| as|\!| their|\!| r|\!||\!|^|\!|{o|\!|}le|\!| is|\!| to|\!| cut|\!| off|\!| the|\!| fluctuations|\!| |\!|{|\!|em|\!| perpendicular|\!||\!|/|\!|}|\!| to|\!| the|\!| classical|\!| trajectories|\!|.|\!| On|\!| the|\!| formal|\!| side|\!|,|\!| |\!|$|\!|overlinen|\!|{|\!|{|\!|boldsymbol|\!|{c|\!|}|\!|}|\!|}|\!|$|\!| and|\!| |\!|$|\!|{|\!|boldsymbol|\!|{c|\!|}|\!|}|\!|$|\!| may|\!| be|\!| identified|\!| with|\!| Jacobi|\!| fields|\!|~|\!|cite|\!|{GozziII|\!|,DeWitt|\!|}|\!|.|\!| The|\!| corresponding|\!| BRST|\!| charges|\!|
|\!|
|\!|
are|\!| related|\!| to|\!| Poincar|\!||\!|'|\!|{e|\!|}|\!|-Cartan|\!| integral|\!| invariants|\!|~|\!|cite|\!|{GozziIII|\!|}|\!|.|\!|
|\!|
|\!|vspace|\!|{3mm|\!|}|\!|
|\!|
By|\!| analogy|\!| with|\!| the|\!| stochastic|\!| quantization|\!| the|\!| path|\!| integral|\!| |\!|(|\!|ref|\!|{4|\!|.5a|\!|}|\!|)|\!| can|\!|,|\!| of|\!| course|\!|,|\!| be|\!| rewritten|\!| in|\!| a|\!| compact|\!| form|\!| with|\!| the|\!| help|\!| of|\!| a|\!| superfield|\!|~|\!|cite|\!|{GozziI|\!|,Zinn|\!|-JustinII|\!|}|\!|
|\!|
|\!|begin|\!|{eqnarray|\!|}|\!| |\!|Phi|\!|_a|\!|(t|\!|,|\!| |\!|theta|\!|,|\!| |\!|overlinen|\!|{|\!|theta|\!|}|\!|)|\!| |\!||\!| |\!|=|\!| |\!||\!| q|\!|_a|\!|(t|\!|)|\!| |\!|+|\!| i|\!|theta|\!| c|\!|_a|\!|(t|\!|)|\!| |\!|-i|\!|overlinen|\!|{|\!|theta|\!|}|\!| |\!|overlinen|\!|{c|\!|}|\!|_a|\!|(t|\!|)|\!| |\!|+|\!| i|\!| |\!|overlinen|\!|{|\!|theta|\!|}|\!|theta|\!| |\!|lambda|\!|_a|\!|(t|\!|)|\!||\!|,|\!| |\!|,|\!| |\!|label|\!|{3|\!|.23|\!|}|\!| |\!|end|\!|{eqnarray|\!|}|\!|
|\!|
in|\!| which|\!| |\!|$|\!|theta|\!|$|\!| and|\!| |\!|$|\!|overlinen|\!|{|\!|theta|\!|}|\!|$|\!| are|\!| anticommuting|\!| coordinates|\!| extending|\!| the|\!| configuration|\!| space|\!| of|\!| |\!|$q|\!|_a|\!|$|\!| variable|\!| to|\!| a|\!| superspace|\!|.|\!| The|\!| latter|\!| is|\!| nothing|\!| but|\!| the|\!| degenerate|\!| case|\!| of|\!| supersymmetric|\!| field|\!| theory|\!| in|\!| |\!|$d|\!|=1|\!|$|\!| in|\!| the|\!| superspace|\!| formalism|\!| of|\!| Salam|\!| and|\!| Strathdee|\!|~|\!|cite|\!|{SS1|\!|}|\!|.|\!| In|\!| terms|\!| of|\!| superspace|\!| variables|\!| we|\!| see|\!| that|\!|
|\!|
|\!|begin|\!|{eqnarray|\!|}|\!| |\!|int|\!| d|\!|overlinen|\!|{|\!|theta|\!|}|\!| d|\!|theta|\!| |\!||\!| |\!|{|\!|{|\!|mathcal|\!|{A|\!|}|\!|}|\!|}|\!|[|\!|{|\!|boldsymbol|\!|{|\!|Phi|\!|}|\!|}|\!|]|\!| |\!|&|\!|=|\!|&|\!| |\!|int|\!| dt|\!| d|\!|overlinen|\!|{|\!|theta|\!|}|\!| d|\!|theta|\!| |\!||\!| L|\!|(|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|(t|\!|)|\!| |\!|+|\!| i|\!|theta|\!| |\!|{|\!|boldsymbol|\!|{c|\!|}|\!|}|\!|(t|\!|)|\!| |\!|-|\!| i|\!| |\!|overlinen|\!|{|\!|theta|\!|}|\!| |\!|overlinen|\!|{|\!|boldsymbol|\!|{c|\!|}|\!|}|\!|(t|\!|)|\!| |\!|+|\!| i|\!| |\!|overlinen|\!|{|\!|theta|\!|}|\!|theta|\!| |\!|boldsymbol|\!|{|\!|lambda|\!|}|\!|(t|\!|)|\!| |\!|)|\!|nonumber|\!| |\!||\!||\!| |\!|&|\!|=|\!|&|\!| |\!|int|\!| d|\!|overlinen|\!|{|\!|theta|\!|}|\!| d|\!|theta|\!| |\!||\!| |\!|{|\!|{|\!|mathcal|\!|{A|\!|}|\!|}|\!|}|\!|[|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|]|\!| |\!|+|\!| |\!|int|\!| dt|\!| d|\!|overlinen|\!|{|\!|theta|\!|}|\!| d|\!|theta|\!| |\!||\!| |\!|left|\!|(|\!| i|\!|theta|\!| |\!|{|\!|boldsymbol|\!|{c|\!|}|\!|}|\!|(t|\!|)|\!| |\!|-|\!| i|\!| |\!|overlinen|\!|{|\!|theta|\!|}|\!| |\!|overlinen|\!|{|\!|boldsymbol|\!|{c|\!|}|\!|}|\!|(t|\!|)|\!| |\!|+|\!| i|\!|overlinen|\!|{|\!|theta|\!|}|\!|theta|\!| |\!|boldsymbol|\!|{|\!|lambda|\!|}|\!|right|\!|)|\!| |\!|frac|\!|{|\!|delta|\!| |\!|{|\!|{|\!|mathcal|\!|{A|\!|}|\!|}|\!|}|\!|}|\!|{|\!|delta|\!| |\!|boldsymbol|\!|{q|\!|}|\!|(t|\!|)|\!|}|\!|nonumber|\!| |\!||\!||\!| |\!|&|\!|&|\!| |\!|+|\!| |\!||\!| |\!|int|\!| dt|\!| dt|\!|'|\!| d|\!|overlinen|\!|{|\!|theta|\!|}|\!| d|\!|theta|\!| |\!||\!| |\!|theta|\!| c|\!|_a|\!|(t|\!|)|\!|frac|\!|{|\!|delta|\!|^2|\!| |\!|{|\!|{|\!|mathcal|\!|{A|\!|}|\!|}|\!|}|\!|}|\!|{|\!|delta|\!| q|\!|_a|\!|(t|\!|)|\!| |\!|delta|\!| q|\!|_b|\!|(t|\!|'|\!|)|\!|}|\!| |\!||\!| |\!|overlinen|\!|{|\!|theta|\!|}|\!| |\!|overlinen|\!|{c|\!|}|\!|(t|\!|'|\!|)|\!|.|\!| |\!|end|\!|{eqnarray|\!|}|\!|
|\!|
Using|\!| the|\!| standard|\!| integration|\!| rules|\!| for|\!| Grassmann|\!| variables|\!|,|\!| this|\!| becomes|\!| equal|\!| to|\!| |\!|$|\!|-i|\!|{|\!|mathcal|\!|{S|\!|}|\!|}|\!|$|\!|.|\!| Together|\!| with|\!| the|\!| identity|\!| |\!|$|\!|{|\!|mathcal|\!|{D|\!|}|\!|}|\!| |\!|{|\!|boldsymbol|\!|{|\!|Phi|\!|}|\!|}|\!| |\!|=|\!| |\!|{|\!|mathcal|\!|{D|\!|}|\!|}|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|{|\!|mathcal|\!|{D|\!|}|\!|}|\!| |\!|{|\!|boldsymbol|\!|{c|\!|}|\!|}|\!|{|\!|mathcal|\!|{D|\!|}|\!|}|\!|overlinen|\!|{|\!|{|\!|boldsymbol|\!|{c|\!|}|\!|}|\!|}|\!| |\!|{|\!|mathcal|\!|{D|\!|}|\!|}|\!|{|\!|boldsymbol|\!|{|\!|lambda|\!|}|\!|}|\!|$|\!| we|\!| may|\!| therefore|\!| express|\!| the|\!| classical|\!| partition|\!| functions|\!| |\!|(|\!|ref|\!|{4|\!|.3|\!|}|\!|)|\!| and|\!| |\!|(|\!|ref|\!|{4|\!|.4|\!|}|\!|)|\!| as|\!| a|\!| supersymmetric|\!| path|\!| integral|\!| with|\!| fully|\!| fluctuating|\!| paths|\!| in|\!| superspace|\!|,|\!|
|\!|
|\!|begin|\!|{eqnarray|\!|}|\!| |\!|{|\!|{|\!|{Z|\!|}|\!|}|\!| |\!|}|\!|_|\!|{|\!|rm|\!| CM|\!|}|\!| |\!||\!| |\!|=|\!| |\!||\!| |\!|int|\!| |\!|{|\!|mathcal|\!|{D|\!|}|\!|}|\!| |\!|{|\!|boldsymbol|\!|{|\!|Phi|\!|}|\!|}|\!| |\!||\!| |\!|exp|\!|left|\!||\!|{|\!|-|\!| |\!|int|\!| d|\!|theta|\!| d|\!|overlinen|\!|{|\!|theta|\!|}|\!| |\!||\!| |\!|{|\!|mathcal|\!|{A|\!|}|\!|}|\!|[|\!|{|\!|boldsymbol|\!|{|\!|Phi|\!|}|\!|}|\!|]|\!|(|\!|theta|\!|,|\!| |\!|overlinen|\!|{|\!|theta|\!|}|\!|)|\!| |\!|+|\!| |\!|int|\!| dt|\!| d|\!|theta|\!| d|\!|overlinen|\!|{|\!|theta|\!|}|\!| |\!||\!| |\!|{|\!|boldsymbol|\!|{|\!|Gamma|\!|}|\!|}|\!|(t|\!|,|\!| |\!|theta|\!|,|\!| |\!|overlinen|\!|{|\!|theta|\!|}|\!|)|\!|{|\!|boldsymbol|\!|{|\!|Phi|\!|}|\!|}|\!|(t|\!|,|\!| |\!|theta|\!|,|\!| |\!|overlinen|\!|{|\!|theta|\!|}|\!|)|\!|right|\!||\!|}|\!||\!|,|\!| |\!|.|\!| |\!|label|\!|{3|\!|.24|\!|}|\!| |\!|end|\!|{eqnarray|\!|}|\!|
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Here|\!| we|\!| have|\!| defined|\!| the|\!| supercurrent|\!| |\!|$|\!|{|\!|boldsymbol|\!|{|\!|Gamma|\!|}|\!|}|\!|(t|\!|,|\!| |\!|theta|\!|,|\!| |\!|overlinen|\!|{|\!|theta|\!|}|\!|)|\!| |\!|=|\!| |\!|overlinen|\!|{|\!|theta|\!|}|\!| |\!|theta|\!| |\!|{|\!|boldsymbol|\!|{J|\!|}|\!|}|\!|(t|\!|)|\!|$|\!|.|\!|
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|\!|vspace|\!|{3mm|\!|}|\!|
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It|\!| is|\!| interesting|\!| to|\!| find|\!| the|\!| most|\!| general|\!| form|\!| of|\!| an|\!| action|\!| |\!|$|\!|{|\!|mathcal|\!|{A|\!|}|\!|}|\!|$|\!| for|\!| which|\!| the|\!| classical|\!| path|\!| integral|\!| |\!|(|\!|ref|\!|{3|\!|.24|\!|}|\!|)|\!| coincides|\!| with|\!| the|\!| quantum|\!|-mechanical|\!| path|\!| integral|\!| of|\!| the|\!| system|\!|,|\!| or|\!|,|\!| in|\!| other|\!| words|\!|,|\!| for|\!| which|\!| a|\!| theory|\!| would|\!| possess|\!| at|\!| the|\!| same|\!| time|\!| deterministic|\!| and|\!| quantal|\!| character|\!|.|\!| As|\!| already|\!| mentioned|\!|,|\!| the|\!| Grassmannnian|\!| ghost|\!| variables|\!| are|\!| responsible|\!| for|\!| the|\!| deterministic|\!| nature|\!| of|\!| the|\!| partition|\!| function|\!|.|\!| It|\!| is|\!| obvious|\!| that|\!| if|\!| the|\!| ghost|\!| sector|\!| could|\!| somehow|\!| be|\!| factored|\!| out|\!| we|\!| would|\!| extend|\!| the|\!| path|\!| integration|\!| to|\!| all|\!| fluctuating|\!| paths|\!| in|\!| |\!|$|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|$|\!|-space|\!|.|\!| By|\!| formally|\!| writing|\!|
|\!|
|\!|begin|\!|{eqnarray|\!|}|\!|
|\!| |\!|frac|\!|{|\!|delta|\!|^2|\!| |\!|{|\!|mathcal|\!|{A|\!|}|\!|}|\!| |\!|}|\!|{|\!|delta|\!|{q|\!|_k|\!|}|\!| |\!|(t|\!|)|\!| |\!||\!| |\!|delta|\!|{q|\!|_l|\!|}|\!|(t|\!|'|\!|)|\!|}|\!| |\!||\!| |\!|=|\!| |\!||\!|
|\!| |\!|{|\!|mathcal|\!|{F|\!|}|\!|}|\!|_|\!|{kl|\!|}|\!|left|\!|(|\!| t|\!|,t|\!|'|\!|,|\!| q|\!|_m|\!|,|\!| |\!|frac|\!|{|\!|delta|\!| |\!|{|\!|mathcal|\!|{A|\!|}|\!|}|\!|}|\!|{|\!|delta|\!| q|\!|_n|\!|}|\!|right|\!|)|\!||\!|,|\!| |\!|,|\!|
|\!| |\!||\!|;|\!||\!|;|\!||\!|;|\!||\!|;|\!||\!|;|\!| |\!| k|\!|,l|\!|,m|\!|,n|\!| |\!|=|\!| 1|\!|,|\!| |\!|ldots|\!|,|\!| N|\!||\!|,|\!| |\!|,|\!| |\!|label|\!|{4|\!|.9|\!|}|\!| |\!|end|\!|{eqnarray|\!|}|\!|
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we|\!| see|\!| that|\!| the|\!| factorization|\!| will|\!| occur|\!| if|\!| and|\!| only|\!| if|\!| the|\!| |\!|(distribution|\!| valued|\!|)|\!| functional|\!| |\!|$|\!|{|\!|mathcal|\!|{F|\!|}|\!|}|\!|_|\!|{kl|\!|}|\!|(|\!|ldots|\!|)|\!|$|\!| is|\!| |\!|$q|\!|_m|\!|$|\!| independent|\!| when|\!| evaluated|\!| on|\!| shell|\!|,|\!| i|\!|.e|\!|.|\!|,|\!| |\!|$|\!|{|\!|mathcal|\!|{F|\!|}|\!|}|\!|_|\!|{kl|\!|}|\!|(t|\!|,t|\!|'|\!|,|\!| q|\!|_m|\!|,|\!| 0|\!| |\!|)|\!| |\!|=|\!| F|\!|_|\!|{kl|\!|}|\!|(t|\!|,t|\!|'|\!|)|\!|$|\!|.|\!| This|\!| is|\!| a|\!| simple|\!| consequence|\!| of|\!| Eq|\!|.|\!|(|\!|ref|\!|{4|\!|.4|\!|}|\!|)|\!| where|\!| the|\!| determinant|\!| is|\!| factorizable|\!| if|\!| and|\!| only|\!| if|\!| it|\!| is|\!| |\!|$|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|$|\!|-independent|\!| at|\!| |\!|$|\!|delta|\!| |\!|{|\!|mathcal|\!|{A|\!|}|\!|}|\!|/|\!|delta|\!| |\!|{|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|}|\!| |\!|=|\!| 0|\!|$|\!|.|\!|
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|\!|vspace|\!|{3mm|\!|}|\!|
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In|\!| order|\!| to|\!| provide|\!| a|\!| correct|\!| Feynman|\!| weight|\!| to|\!| every|\!| path|\!| we|\!| must|\!|,|\!| in|\!| addition|\!|,|\!| identify|\!|
|\!|
|\!|begin|\!|{eqnarray|\!|}|\!| |\!|{|\!|mathcal|\!|{A|\!|}|\!|}|\!|[|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|]|\!| |\!|=|\!| |\!|int|\!|_|\!|{t|\!|_1|\!|}|\!|^|\!|{t|\!|_2|\!|}|\!| dt|\!| |\!||\!| |\!|lambda|\!|_m|\!| |\!|frac|\!|{|\!|delta|\!| |\!|{|\!|mathcal|\!|{A|\!|}|\!|}|\!|[|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|]|\!|}|\!|{|\!|delta|\!| q|\!|_m|\!|}|\!|
|\!|
|\!|
|\!|
|\!|
|\!|
|\!|
|\!||\!|,|\!| |\!|,|\!| |\!|label|\!|{3|\!|.26|\!|}|\!| |\!|end|\!|{eqnarray|\!|}|\!|
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as|\!| can|\!| be|\!| seen|\!| from|\!| |\!|(|\!|ref|\!|{4|\!|.5|\!|}|\!|)|\!| after|\!| factoring|\!| out|\!| the|\!| second|\!| term|\!|.|\!| Assuming|\!| that|\!| |\!|$L|\!| |\!|=|\!| L|\!|(q|\!|_l|\!|,|\!| |\!|dot|\!|{q|\!|_l|\!|}|\!|)|\!|$|\!| |\!|(i|\!|.e|\!|.|\!|,|\!| a|\!| scleronomic|\!| system|\!|)|\!| and|\!| that|\!| the|\!| Hessian|\!| is|\!| regular|\!|,|\!|
|\!| the|\!| condition|\!| |\!|(|\!|ref|\!|{3|\!|.26|\!|}|\!|)|\!| shows|\!| that|\!| |\!|$|\!|lambda|\!|_k|\!| |\!|=|\!| |\!|lambda|\!|_k|\!|(q|\!|_l|\!|,|\!| |\!|dot|\!|{q|\!|_k|\!|}|\!|)|\!|$|\!|.|\!| In|\!| addition|\!|,|\!| it|\!| is|\!| obvious|\!| on|\!| dimensional|\!| grounds|\!|
|\!| that|\!| |\!|$|\!|left|\!|[|\!| |\!|lambda|\!|_l|\!| |\!| |\!|right|\!|]|\!| |\!|=|\!| |\!|left|\!|[|\!| q|\!|_l|\!| |\!|right|\!|]|\!|$|\!|.|\!| This|\!|,|\!| in|\!| turn|\!|,|\!| implies|\!| that|\!| |\!|$|\!|lambda|\!|_k|\!| |\!| |\!|=|\!| |\!|alpha|\!|_|\!|{kl|\!|}q|\!|_l|\!|$|\!|,|\!| where|\!| |\!|$|\!|alpha|\!|_|\!|{lk|\!|}|\!|$|\!| is|\!| some|\!| real|\!| |\!|(|\!|$t|\!|$|\!|-independent|\!|)|\!| matrix|\!|.|\!| To|\!| determine|\!| the|\!| latter|\!| we|\!| functionally|\!| expand|\!| |\!|$|\!| |\!|{|\!|mathcal|\!|{A|\!|}|\!|}|\!|$|\!| in|\!| |\!|(|\!|ref|\!|{3|\!|.26|\!|}|\!|)|\!| around|\!| |\!|$q|\!|_k|\!|$|\!| and|\!| compare|\!| both|\!| sides|\!|.|\!| The|\!| resulting|\!| integrability|\!| condition|\!| reads|\!|:|\!|
|\!|
|\!|begin|\!|{eqnarray|\!|}|\!| |\!|left|\!|(|\!|delta|\!|_|\!|{ji|\!|}|\!| |\!|-|\!|alpha|\!|_|\!|{ji|\!|}|\!|right|\!|)|\!|frac|\!|{|\!|delta|\!| |\!|{|\!|mathcal|\!|{A|\!|}|\!|}|\!|}|\!|{|\!|delta|\!| q|\!|_|\!|{j|\!|}|\!|(t|\!|)|\!|}|\!||\!| |\!|delta|\!| |\!|(t|\!|-t|\!|'|\!|)|\!| |\!||\!| |\!|=|\!| |\!||\!| |\!|alpha|\!|_|\!|{l|\!||\!|!j|\!|}|\!| |\!||\!| q|\!|_j|\!|(t|\!|)|\!| |\!||\!| |\!|frac|\!|{|\!|delta|\!|^2|\!| |\!| |\!|{|\!|mathcal|\!|{A|\!|}|\!|}|\!|}|\!|{|\!|delta|\!| q|\!|_l|\!|(t|\!|)|\!| |\!|delta|\!| q|\!|_i|\!|(t|\!|'|\!|)|\!|}|\!||\!|,|\!| |\!|,|\!| |\!|label|\!|{3|\!|.28|\!|}|\!| |\!|end|\!|{eqnarray|\!|}|\!|
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which|\!| |\!| is|\!| evidently|\!| compatible|\!| with|\!| the|\!| condition|\!| |\!|(|\!|ref|\!|{4|\!|.9|\!|}|\!|)|\!|.|\!| When|\!| |\!|$|\!|alpha|\!|_|\!|{ij|\!|}|\!|$|\!| is|\!| diagonalizable|\!| we|\!| can|\!| pass|\!| to|\!| a|\!| polar|\!| basis|\!| and|\!| write|\!| |\!|(|\!|ref|\!|{3|\!|.26|\!|}|\!|)|\!| in|\!| more|\!| manageable|\!| form|\!|,|\!| namely|\!|
|\!|
|\!|begin|\!|{eqnarray|\!|}|\!| |\!|{|\!|mathcal|\!|{A|\!|}|\!|}|\!|[|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|]|\!||\!| |\!|=|\!| |\!||\!| |\!|int|\!|_|\!|{t|\!|_1|\!|}|\!|^|\!|{t|\!|_2|\!|}|\!| dt|\!| |\!||\!| |\!|sum|\!|_i|\!| |\!|alpha|\!|_i|\!| q|\!|_i|\!|(t|\!|)|\!| |\!|frac|\!|{|\!|delta|\!| |\!|{|\!|mathcal|\!|{A|\!|}|\!|}|\!|[|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|]|\!|}|\!|{|\!|delta|\!| q|\!|_i|\!|(t|\!|)|\!|}|\!||\!|,|\!| |\!|.|\!| |\!|label|\!|{3|\!|.33|\!|}|\!| |\!|end|\!|{eqnarray|\!|}|\!|
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For|\!| simplicity|\!|,|\!| we|\!| do|\!| not|\!| use|\!| new|\!| symbols|\!|
|\!| for|\!| transformed|\!| |\!|$|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|$|\!|'s|\!|.|\!|
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|\!|vspace|\!|{3mm|\!|}|\!|
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To|\!| proceed|\!| we|\!| |\!| assume|\!| that|\!| the|\!| kinetic|\!| energy|\!| is|\!| |\!| quadratic|\!| in|\!| |\!|$|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|$|\!| and|\!| |\!|$|\!|dot|\!|{|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|}|\!|$|\!|.|\!| Then|\!| Eq|\!|.|\!|(|\!|ref|\!|{3|\!|.33|\!|}|\!|)|\!| implies|\!| that|\!| |\!|$L|\!|_|\!|{|\!|rm|\!| kin|\!|}|\!|$|\!| must|\!| be|\!| liner|\!| in|\!| |\!|$|\!|dot|\!|{|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|}|\!|$|\!|.|\!| As|\!| such|\!|,|\!| one|\!| can|\!| always|\!| write|\!| |\!|(modulo|\!| the|\!| total|\!| derivative|\!|)|\!|
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|\!|begin|\!|{eqnarray|\!|}|\!| |\!| |\!| |\!|label|\!|{|\!|@withB|\!|}|\!| L|\!|_|\!|{|\!|rm|\!| kin|\!|}|\!| |\!|=|\!| |\!|sum|\!|_|\!|{i|\!|,j|\!|}|\!| |\!|{|\!|{B|\!|}|\!|}|\!|_|\!|{ij|\!|}|\!| |\!||\!| q|\!|_i|\!|(t|\!|)|\!| |\!|dot|\!|{q|\!|}|\!|_j|\!|(t|\!|)|\!||\!|,|\!| |\!|,|\!| |\!|end|\!|{eqnarray|\!|}|\!|
|\!|
with|\!| |\!|$|\!|{|\!|{B|\!|}|\!|}|\!|$|\!| being|\!| an|\!| upper|\!| triangular|\!| matrix|\!|.|\!| Comparing|\!| |\!|$L|\!|_|\!|{|\!|rm|\!| kin|\!|}|\!|$|\!| on|\!| both|\!| sides|\!| of|\!| |\!|(|\!|ref|\!|{3|\!|.33|\!|}|\!|)|\!| we|\!| arrive|\!| at|\!| the|\!| equation|\!|
|\!|
|\!|begin|\!|{eqnarray|\!|}|\!| |\!|(|\!|alpha|\!|_m|\!| |\!|-|\!| 1|\!|)|\!| |\!|{|\!|{B|\!|}|\!|}|\!|_|\!|{im|\!|}|\!| |\!|=|\!| |\!|{|\!|{B|\!|}|\!|}|\!|_|\!|{mi|\!|}|\!| |\!|alpha|\!|_m|\!| |\!||\!|,|\!| |\!||\!|,|\!| |\!|Rightarrow|\!| |\!| |\!||\!|,|\!| |\!||\!|,|\!| |\!|(|\!|{B|\!|}|\!| |\!|-|\!| |\!|{B|\!|}|\!|^|\!|{|\!|top|\!|}|\!|)|\!|{|\!|boldsymbol|\!|{|\!|alpha|\!|}|\!|}|\!| |\!|=|\!| |\!|{B|\!|}|\!||\!|,|\!| |\!|,|\!| |\!|label|\!|{3|\!|.37|\!|}|\!| |\!|end|\!|{eqnarray|\!|}|\!|
|\!|
with|\!| no|\!| Einstein|\!|'s|\!| summation|\!| convention|\!| applied|\!| here|\!|.|\!| Because|\!| |\!|$|\!|{|\!|{B|\!|}|\!|}|\!|$|\!| is|\!| upper|\!| triangular|\!|,|\!| the|\!| first|\!| part|\!| of|\!| |\!| Eq|\!|.|\!|(|\!|ref|\!|{3|\!|.37|\!|}|\!|)|\!| implies|\!| that|\!| the|\!| only|\!| eigenvalues|\!| of|\!| |\!|$|\!|alpha|\!|_|\!|{ij|\!|}|\!|$|\!| are|\!| |\!|$1|\!|$|\!| and|\!| |\!|$0|\!|$|\!|.|\!| Thus|\!|,|\!| |\!|$|\!|{|\!|boldsymbol|\!|{|\!|alpha|\!|}|\!|}|\!|$|\!| can|\!| be|\!| reduced|\!| to|\!| the|\!| block|\!| form|\!|
|\!|
|\!|begin|\!|{eqnarray|\!|}|\!|
|\!|{|\!|boldsymbol|\!|{|\!|alpha|\!|}|\!|}|\!| |\!||\!| |\!|=|\!| |\!||\!| |\!|left|\!|[|\!| |\!|begin|\!|{tabular|\!|}|\!|{c|\!||c|\!|}|\!| 0|\!| |\!|&|\!| 0|\!| |\!||\!||\!| |\!|hline|\!| 0|\!| |\!|&|\!| |\!|ide|\!| |\!|end|\!|{tabular|\!|}|\!| |\!|right|\!|]|\!||\!|,|\!| |\!|,|\!| |\!|label|\!|{3|\!|.38|\!|}|\!| |\!|end|\!|{eqnarray|\!|}|\!|
|\!|
where|\!| |\!|$|\!|ide|\!|$|\!| is|\!| a|\!| |\!|$r|\!|times|\!| r|\!|$|\!| |\!|(|\!|$r|\!|leq|\!| N|\!|$|\!|)|\!| unit|\!| matrix|\!|.|\!|
|\!| Using|\!| the|\!| equation|\!| |\!|$|\!|(|\!|{|\!|{B|\!|}|\!|}|\!| |\!|-|\!| |\!|{|\!|{B|\!|}|\!|}|\!|^|\!|{|\!|top|\!|}|\!|)|\!|{|\!|boldsymbol|\!|{|\!|alpha|\!|}|\!|}|\!| |\!|=|\!| |\!|{|\!|{B|\!|}|\!|}|\!| |\!|$|\!| we|\!| see|\!| that|\!| |\!|$|\!|{|\!|{B|\!|}|\!|}|\!|$|\!| |\!| |\!| has|\!| the|\!|
|\!| block|\!| structure|\!|
|\!|
|\!|begin|\!|{eqnarray|\!|}|\!| |\!|mbox|\!|{|\!|$|\!|{|\!|{B|\!|}|\!|}|\!|$|\!|}|\!| |\!||\!| |\!|=|\!| |\!||\!| |\!|left|\!|[|\!|
|\!|begin|\!|{tabular|\!|}|\!|{c|\!||c|\!|}|\!| 0|\!| |\!|&|\!|
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|\!|$|\!|{|\!|{B|\!|}|\!|}|\!|_2|\!|$|\!| |\!||\!||\!| |\!|hline|\!| 0|\!| |\!|&|\!| 0|\!| |\!||\!||\!| |\!|end|\!|{tabular|\!|}|\!| |\!|right|\!|]|\!| |\!||\!|,|\!| |\!|.|\!| |\!|label|\!|{3|\!|.39|\!|}|\!| |\!|end|\!|{eqnarray|\!|}|\!|
|\!|
where|\!| |\!|$|\!|{|\!|{B|\!|}|\!|}|\!|_2|\!|$|\!| is|\!| an|\!| |\!|$|\!|(N|\!|-r|\!|)|\!| |\!|times|\!| r|\!|$|\!| matrix|\!|.|\!| To|\!| determine|\!| |\!|$r|\!|$|\!| we|\!| use|\!| the|\!| fact|\!| that|\!| |\!|$|\!|{|\!|boldsymbol|\!|{|\!|alpha|\!|}|\!|}|\!|$|\!| is|\!| idempotent|\!|,|\!| i|\!|.e|\!|.|\!|,|\!|
|\!| |\!|$|\!|{|\!|boldsymbol|\!|{|\!|alpha|\!|}|\!|}|\!|^2|\!| |\!|=|\!| |\!|{|\!|boldsymbol|\!|{|\!|alpha|\!|}|\!|}|\!|$|\!|.|\!| Multiplying|\!| |\!|$|\!|(|\!|{|\!|{B|\!|}|\!|}|\!| |\!|-|\!| |\!|{|\!|{B|\!|}|\!|}|\!|^|\!|{|\!|top|\!|}|\!|)|\!|{|\!|boldsymbol|\!|{|\!|alpha|\!|}|\!|}|\!| |\!|=|\!| |\!|{|\!|{B|\!|}|\!|}|\!| |\!|$|\!| by|\!| |\!|$|\!|boldsymbol|\!|{|\!|alpha|\!|}|\!|$|\!| we|\!| find|\!|
|\!|
|\!|begin|\!|{eqnarray|\!|}|\!| |\!|begin|\!|{array|\!|}|\!|{ll|\!|}|\!| |\!|{|\!|{B|\!|}|\!|}|\!|{|\!|boldsymbol|\!|{|\!|alpha|\!|}|\!|}|\!| |\!|=|\!| |\!|{|\!|{B|\!|}|\!|}|\!||\!|,|\!| |\!|,|\!|~|\!|~|\!|~|\!|~|\!|~|\!| |\!|{|\!|{B|\!|}|\!|}|\!|^|\!|top|\!| |\!|{|\!|boldsymbol|\!|{|\!|alpha|\!|}|\!|}|\!| |\!|=|\!| 0|\!| |\!||\!|,|\!| |\!|.|\!| |\!|end|\!|{array|\!|}|\!| |\!|end|\!|{eqnarray|\!|}|\!|
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From|\!| |\!|$|\!|{|\!|{B|\!|}|\!|}|\!|{|\!|boldsymbol|\!|{|\!|alpha|\!|}|\!|}|\!| |\!|=|\!| |\!|{|\!|{B|\!|}|\!|}|\!|$|\!| follows|\!| that|\!| rank|\!|$|\!|(|\!|{|\!|{B|\!|}|\!|}|\!|)|\!|=|\!| |\!|{|\!|mbox|\!|{rank|\!|}|\!|}|\!|(|\!|{|\!|boldsymbol|\!|{|\!|alpha|\!|}|\!|}|\!|)|\!|=r|\!|$|\!|,|\!| whereas|\!| |\!|$|\!|{|\!|{B|\!|}|\!|}|\!|^|\!|top|\!|(|\!|ide|\!| |\!|-|\!| |\!|{|\!|boldsymbol|\!|{|\!|alpha|\!|}|\!|}|\!|)|\!| |\!|=|\!| |\!|{|\!|{B|\!|}|\!|}|\!|^|\!|top|\!|$|\!| implies|\!| that|\!| rank|\!|$|\!|(|\!|{|\!|{B|\!|}|\!|}|\!|^|\!|top|\!|)|\!|=|\!| |\!|{|\!|mbox|\!|{rank|\!|}|\!|}|\!|(|\!|ide|\!| |\!|-|\!| |\!|{|\!|boldsymbol|\!|{|\!|alpha|\!|}|\!|}|\!|)|\!|$|\!|.|\!| Utilizing|\!| the|\!| identity|\!| |\!|$|\!|{|\!|mbox|\!|{rank|\!|}|\!|}|\!|(|\!|{|\!|{B|\!|}|\!|}|\!|)|\!| |\!|=|\!| |\!|{|\!|mbox|\!|{rank|\!|}|\!|}|\!|(|\!|{|\!|{B|\!|}|\!|}|\!|^|\!|top|\!|)|\!|$|\!| we|\!| derive|\!|
|\!| |\!|$r|\!| |\!|=|\!| |\!|{|\!|mbox|\!|{rank|\!|}|\!|}|\!|(|\!|{|\!|boldsymbol|\!|{|\!|alpha|\!|}|\!|}|\!|)|\!| |\!|=|\!| |\!|{|\!|mbox|\!|{rank|\!|}|\!|}|\!|(|\!|ide|\!| |\!|-|\!| |\!|{|\!|boldsymbol|\!|{|\!|alpha|\!|}|\!|}|\!|)|\!| |\!|=|\!| |\!|(N|\!|-r|\!|)|\!|$|\!|,|\!| and|\!| thus|\!| |\!|$r|\!| |\!|=|\!| N|\!|/2|\!|$|\!|.|\!| Thus|\!| the|\!| condition|\!| |\!|(|\!|ref|\!|{3|\!|.33|\!|}|\!|)|\!| can|\!| be|\!| satisfied|\!| only|\!| for|\!| an|\!| even|\!| number|\!| |\!|$N|\!|$|\!| of|\!| degrees|\!| of|\!| freedom|\!|.|\!| An|\!| immediate|\!| further|\!| consequence|\!| of|\!| |\!|(|\!|ref|\!|{3|\!|.39|\!|}|\!|)|\!| is|\!| |\!| |\!| that|\!| we|\!| can|\!| rewrite|\!| |\!|(|\!|ref|\!|{|\!|@withB|\!|}|\!|)|\!| as|\!|
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|\!|begin|\!|{eqnarray|\!|}|\!| L|\!|_|\!|{|\!|rm|\!| kin|\!|}|\!| |\!|=|\!| |\!|sum|\!|_|\!|{i|\!|,j|\!| |\!|=1|\!|}|\!|^|\!|{N|\!|/2|\!|}|\!| |\!|{|\!|{B|\!|}|\!|}|\!|_|\!|{i|\!|,|\!|(N|\!|/2|\!|+j|\!|)|\!|}|\!| |\!||\!| |\!|dot|\!|{q|\!|}|\!|_i|\!| q|\!|_|\!|{N|\!|/2|\!| |\!|+j|\!|}|\!| |\!||\!|,|\!| |\!|.|\!| |\!|label|\!|{3|\!|.36|\!|}|\!| |\!|end|\!|{eqnarray|\!|}|\!|
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Denoting|\!| |\!|$|\!|alpha|\!|_|\!|{N|\!|/2|\!| |\!|+|\!| i|\!|}|\!|$|\!|,|\!| |\!|$q|\!|_|\!|{N|\!|/2|\!| |\!|+|\!| i|\!|}|\!|$|\!| and|\!| |\!|$|\!|lambda|\!|_|\!|{N|\!|/2|\!| |\!|+i|\!|}|\!|$|\!| |\!|(|\!|$i|\!| |\!|=|\!| 1|\!|,|\!|ldots|\!|,N|\!|/2|\!|$|\!|)|\!| as|\!| |\!|$|\!|{|\!|bar|\!|{|\!|alpha|\!|}|\!|}|\!|_|\!|{i|\!|}|\!|$|\!|,|\!| |\!|$|\!|bar|\!|{q|\!|}|\!|_i|\!|$|\!|,|\!| and|\!| |\!|$|\!|bar|\!|{|\!|lambda|\!|}|\!|_|\!|{i|\!|}|\!|$|\!|,|\!| respectively|\!| |\!|[hence|\!|,|\!| |\!|$|\!|{|\!|boldsymbol|\!|{|\!|lambda|\!|}|\!|}|\!| |\!|=|\!| |\!|{|\!|boldsymbol|\!|{0|\!|}|\!|}|\!|$|\!| and|\!| |\!|$|\!|bar|\!|{|\!|{|\!|boldsymbol|\!|{|\!|lambda|\!|}|\!|}|\!|}|\!| |\!|=|\!| |\!|bar|\!|{|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|}|\!|~|\!|$|\!|]|\!|,|\!| then|\!| Eq|\!|.|\!|(|\!|ref|\!|{3|\!|.33|\!|}|\!|)|\!| reads|\!|
|\!|
|\!|begin|\!|{eqnarray|\!|}|\!| |\!|bar|\!|{|\!|{|\!|mathcal|\!|{A|\!|}|\!|}|\!|}|\!|[|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|,|\!| |\!|bar|\!|{|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|}|\!|]|\!| |\!|=|\!| |\!|int|\!|_|\!|{t|\!|_1|\!|}|\!|^|\!|{t|\!|_2|\!|}|\!| dt|\!| |\!||\!| |\!|bar|\!|{|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|}|\!|(t|\!|)|\!| |\!|frac|\!|{|\!|delta|\!| |\!|bar|\!|{|\!|{|\!|mathcal|\!|{A|\!|}|\!|}|\!|}|\!|[|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|,|\!| |\!|bar|\!|{|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|}|\!|]|\!|}|\!|{|\!|delta|\!| |\!|bar|\!|{|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|}|\!|(t|\!|)|\!| |\!|}|\!||\!|,|\!| |\!|.|\!| |\!|label|\!|{3|\!|.35|\!|}|\!| |\!|end|\!|{eqnarray|\!|}|\!|
|\!|
|\!|
Here|\!| |\!|$|\!|bar|\!|{|\!|{|\!|mathcal|\!|{A|\!|}|\!|}|\!|}|\!|[|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|,|\!| |\!|bar|\!|{|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|}|\!|]|\!| |\!|=|\!| |\!|{|\!|{|\!|mathcal|\!|{A|\!|}|\!|}|\!|}|\!|[q|\!|_1|\!|,|\!| |\!|ldots|\!|,|\!| q|\!|_|\!|{2N|\!|}|\!|]|\!|$|\!|.|\!| The|\!| result|\!| |\!|(|\!|ref|\!|{3|\!|.35|\!|}|\!|)|\!| can|\!| be|\!| obtained|\!| also|\!| in|\!| a|\!| different|\!| way|\!|.|\!| Indeed|\!|,|\!| in|\!| Appendix|\!| C|\!| we|\!| show|\!| that|\!| |\!|(|\!|ref|\!|{3|\!|.33|\!|}|\!|)|\!| is|\!| a|\!| so|\!|-called|\!| Euler|\!|-like|\!| functional|\!|
|\!|
|\!|begin|\!|{eqnarray|\!|}|\!| |\!|{|\!|mathcal|\!|{A|\!|}|\!|}|\!|[|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|]|\!| |\!|=|\!| |\!|int|\!|_|\!|{t|\!|_1|\!|}|\!|^|\!|{t|\!|_2|\!|}|\!| dt|\!| |\!||\!| r|\!|(t|\!|)|\!| L|\!||\!|!|\!|left|\!|(|\!| r|\!|^|\!|{|\!|-|\!|alpha|\!|_1|\!|}|\!|(t|\!|)q|\!|_1|\!|(t|\!|)|\!|,|\!| |\!|ldots|\!|,r|\!|^|\!|{|\!|-|\!|alpha|\!|_N|\!|}|\!|(t|\!|)q|\!|_N|\!|(t|\!|)|\!|,|\!| |\!|frac|\!|{d|\!|left|\!|(r|\!|^|\!|{|\!|-|\!|alpha|\!|_1|\!|}|\!|(t|\!|)q|\!|_1|\!|(t|\!|)|\!|right|\!|)|\!|}|\!|{dt|\!|}|\!|,|\!| |\!|ldots|\!|,|\!| |\!|frac|\!|{d|\!|left|\!|(r|\!|^|\!|{|\!|-|\!|alpha|\!|_N|\!|}|\!|(t|\!|)q|\!|_N|\!|(t|\!|)|\!|right|\!|)|\!|}|\!|{dt|\!|}|\!|
|\!| |\!|right|\!|)|\!|
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|\!| |\!||\!|,|\!| |\!|,|\!|label|\!|{3|\!|.31|\!|}|\!| |\!|end|\!|{eqnarray|\!|}|\!|
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with|\!| |\!|$r|\!|(t|\!|)|\!|$|\!| being|\!| an|\!| arbitrary|\!| function|\!| of|\!| |\!|$q|\!|_k|\!|$|\!| whose|\!| variations|\!| vanish|\!| at|\!| the|\!| ends|\!| |\!|$|\!|delta|\!| r|\!|(t|\!|_i|\!|)|\!| |\!|=|\!| |\!|delta|\!| r|\!|(t|\!|_f|\!|)|\!| |\!|=|\!| 0|\!|$|\!| if|\!| all|\!| |\!|$|\!| |\!|delta|\!| q|\!|_k|\!|$|\!|'s|\!| have|\!| this|\!| property|\!|.|\!|
|\!| In|\!| particular|\!|,|\!| we|\!| may|\!|
|\!| chose|\!| |\!|$r|\!|$|\!| to|\!| be|\!| any|\!| finite|\!| power|\!| |\!|$|\!| q|\!|_k|\!|^|\!|{1|\!|/|\!|alpha|\!|_k|\!|}|\!|$|\!|
|\!| |\!|(for|\!| |\!|$k|\!| |\!|=|\!| 1|\!|,|\!| |\!|ldots|\!|,|\!| N|\!|$|\!|)|\!|,|\!| in|\!| which|\!| case|\!|
|\!|
|\!|begin|\!|{eqnarray|\!|}|\!| |\!|{|\!|mathcal|\!|{A|\!|}|\!|}|\!|[|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|]|\!| |\!|=|\!| |\!|int|\!|_|\!|{t|\!|_1|\!|}|\!|^|\!|{t|\!|_2|\!|}|\!| dt|\!| |\!||\!| q|\!|_k|\!|^|\!|{1|\!|/|\!|alpha|\!|_k|\!|}|\!| L|\!||\!|!|\!|left|\!|(|\!| |\!|frac|\!|{q|\!|_1|\!|}|\!|{q|\!|_k|\!|^|\!|{|\!|alpha|\!|_1|\!|/|\!|alpha|\!|_k|\!|}|\!|}|\!|,|\!|
|\!| |\!|dots|\!|,|\!| |\!|stackrel|\!|{|\!|stackrel|\!|{|\!|mbox|\!|{|\!|footnotesize|\!|{|\!|$k|\!|$|\!|}|\!|}|\!|}|\!|{|\!|downarrow|\!|}|\!|}|\!|{1|\!|}|\!|,|\!|
|\!| |\!|dots|\!|,|\!| |\!|frac|\!|{q|\!|_N|\!|}|\!|{q|\!|_k|\!|^|\!|{|\!|alpha|\!|_N|\!|/|\!|alpha|\!|_k|\!|}|\!|}|\!|,|\!|
|\!| |\!|frac|\!|{d|\!|left|\!|(q|\!|_1|\!|/q|\!|_k|\!|^|\!|{|\!|alpha|\!|_1|\!|/|\!|alpha|\!|_k|\!|}|\!|right|\!|)|\!|}|\!|{dt|\!|}|\!|,|\!| |\!|ldots|\!|,|\!|
|\!| |\!|stackrel|\!|{|\!|stackrel|\!|{|\!|mbox|\!|{|\!|footnotesize|\!|{|\!|$k|\!|$|\!|}|\!|}|\!|}|\!|{|\!|downarrow|\!|}|\!|}|\!|{0|\!|}|\!|,|\!| |\!|dots|\!|,|\!|
|\!| |\!|frac|\!|{d|\!|left|\!|(q|\!|_N|\!|/q|\!|_k|\!|^|\!|{|\!|alpha|\!|_N|\!|/|\!|alpha|\!|_k|\!|}|\!|right|\!|)|\!|}|\!|{dt|\!|}|\!|
|\!| |\!|right|\!|)|\!|.|\!|
|\!| |\!|label|\!|{3|\!|.30|\!|}|\!| |\!|end|\!|{eqnarray|\!|}|\!|
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Assuming|\!|,|\!| as|\!| before|\!|,|\!| that|\!| the|\!| kinetic|\!| term|\!| in|\!| |\!|$L|\!|$|\!| is|\!| quadratic|\!| |\!| in|\!| |\!|$|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|$|\!| and|\!| |\!|$|\!|dot|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|$|\!|,|\!| we|\!| arrive|\!| at|\!| |\!|$|\!|{|\!|boldsymbol|\!|{|\!|alpha|\!|}|\!|}|\!|$|\!| as|\!| in|\!| |\!|(|\!|ref|\!|{3|\!|.38|\!|}|\!|)|\!|,|\!| and|\!| the|\!| action|\!| |\!|(|\!|ref|\!|{3|\!|.30|\!|}|\!|)|\!| reduces|\!| again|\!| to|\!| |\!|(|\!|ref|\!|{3|\!|.35|\!|}|\!|)|\!|.|\!|
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|\!|vspace|\!|{3mm|\!|}|\!|
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One|\!| can|\!| incorporate|\!| the|\!| constraints|\!| on|\!| |\!|$|\!|alpha|\!|_i|\!|$|\!| |\!|(or|\!| |\!|$|\!|lambda|\!|_i|\!|$|\!|)|\!| by|\!| inserting|\!| a|\!| corresponding|\!| |\!| |\!|$|\!|delta|\!|$|\!|-functional|\!| into|\!| the|\!| path|\!| integral|\!| |\!|(|\!|ref|\!|{4|\!|.5a|\!|}|\!|)|\!|.|\!| This|\!| leads|\!| to|\!| the|\!| most|\!| general|\!| generating|\!| functional|\!| with|\!| the|\!| above|\!|-stated|\!| property|\!|:|\!|
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|\!|begin|\!|{eqnarray|\!|}|\!| |\!|{|\!|{|\!|{Z|\!|}|\!|}|\!| |\!|}|\!|_|\!|{|\!|rm|\!| CM|\!|}|\!| |\!|&|\!|=|\!|&|\!| |\!| |\!|int|\!| |\!|{|\!|mathcal|\!|{D|\!|}|\!|}|\!|{|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|}|\!|{|\!|mathcal|\!|{D|\!|}|\!|}|\!|{|\!|overlinen|\!|{|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|}|\!|}|\!| |\!|{|\!|mathcal|\!|{D|\!|}|\!|}|\!|{|\!|{|\!|boldsymbol|\!|{|\!|lambda|\!|}|\!|}|\!|}|\!| |\!|{|\!|mathcal|\!|{D|\!|}|\!|}|\!|{|\!|overlinen|\!|{|\!|{|\!|boldsymbol|\!|{|\!|lambda|\!|}|\!|}|\!|}|\!|}|\!| |\!||\!| |\!|delta|\!|[|\!|{|\!|boldsymbol|\!|{|\!|lambda|\!|}|\!|}|\!|]|\!| |\!|delta|\!|[|\!|overlinen|\!|{|\!|{|\!|boldsymbol|\!|{|\!|lambda|\!|}|\!|}|\!|}|\!| |\!|-|\!| |\!|overlinen|\!|{|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|}|\!|]|\!| |\!||\!| |\!|exp|\!||\!|!|\!|left|\!|[i|\!| |\!||\!|!|\!||\!|!|\!|int|\!|_|\!|{t|\!|_1|\!|}|\!|^|\!|{t|\!|_2|\!|}|\!| dt|\!| |\!||\!| |\!|{|\!|boldsymbol|\!|{|\!|lambda|\!|}|\!|}|\!|frac|\!|{|\!|delta|\!| |\!|overlinen|\!| |\!|{|\!|mathcal|\!|{A|\!|}|\!|}|\!|[|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|,|\!| |\!|overlinen|\!|{|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|}|\!|]|\!|}|\!|{|\!|delta|\!| |\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!| |\!|}|\!| |\!|+|\!| i|\!| |\!||\!|!|\!||\!|!|\!|int|\!|_|\!|{t|\!|_1|\!|}|\!|^|\!|{t|\!|_2|\!|}|\!| dt|\!| |\!||\!| |\!|overlinen|\!|{|\!|{|\!|boldsymbol|\!|{|\!|lambda|\!|}|\!|}|\!|}|\!||\!| |\!|frac|\!|{|\!|delta|\!| |\!|overlinen|\!| |\!|{|\!|mathcal|\!|{A|\!|}|\!|}|\!|[|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|,|\!| |\!|overlinen|\!|{|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|}|\!|]|\!|}|\!|{|\!|delta|\!| |\!|overlinen|\!|{|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|}|\!| |\!|}|\!| |\!| |\!|+|\!| |\!|int|\!|_|\!|{t|\!|_1|\!|}|\!|^|\!|{t|\!|_2|\!|}|\!| dt|\!| |\!|sum|\!|_|\!|{k|\!|=1|\!|}|\!|^N|\!||\!|!|\!| J|\!|_k|\!| q|\!|_k|\!| |\!|right|\!|]|\!|nonumber|\!| |\!||\!||\!| |\!|&|\!|~|\!|&|\!| |\!|nonumber|\!| |\!||\!||\!| |\!|&|\!|=|\!|&|\!| |\!|int|\!| |\!|{|\!|mathcal|\!|{D|\!|}|\!|}|\!|{|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|}|\!|{|\!|mathcal|\!|{D|\!|}|\!|}|\!|{|\!|overlinen|\!|{|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|}|\!|}|\!| |\!||\!| |\!|exp|\!||\!|!|\!|left|\!|[|\!| i|\!||\!|!|\!||\!|!|\!|int|\!|_|\!|{t|\!|_1|\!|}|\!|^|\!|{t|\!|_2|\!|}|\!| dt|\!| |\!||\!| |\!|overlinen|\!|{|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|}|\!||\!| |\!|frac|\!|{|\!|delta|\!| |\!|overlinen|\!| |\!|{|\!|mathcal|\!|{A|\!|}|\!|}|\!|[|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|,|\!| |\!|overlinen|\!|{|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|}|\!|]|\!|}|\!|{|\!|delta|\!| |\!|overlinen|\!|{|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|}|\!| |\!|}|\!| |\!|+|\!| |\!| |\!|int|\!|_|\!|{t|\!|_1|\!|}|\!|^|\!|{t|\!|_2|\!|}|\!| dt|\!| |\!|sum|\!|_|\!|{k|\!|=1|\!|}|\!|^N|\!| J|\!|_k|\!| q|\!|_k|\!| |\!|right|\!|]|\!| |\!|nonumber|\!| |\!||\!||\!| |\!|&|\!|~|\!|&|\!| |\!|nonumber|\!| |\!||\!||\!| |\!|&|\!|=|\!|&|\!| |\!|int|\!| |\!|{|\!|mathcal|\!|{D|\!|}|\!|}|\!|{|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|}|\!|{|\!|mathcal|\!|{D|\!|}|\!|}|\!|{|\!|overlinen|\!|{|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|}|\!|}|\!| |\!||\!| |\!|exp|\!||\!|!|\!|left|\!|[|\!| i|\!| |\!||\!|!|\!||\!|!|\!|int|\!|_|\!|{t|\!|_1|\!|}|\!|^|\!|{t|\!|_2|\!|}|\!| dt|\!| |\!||\!|,|\!|overlinen|\!| L|\!| |\!|+|\!| |\!|int|\!| dt|\!| |\!|sum|\!|_|\!|{k|\!|=1|\!|}|\!|^N|\!| J|\!|_k|\!| q|\!|_k|\!| |\!|right|\!|]|\!||\!|,|\!| |\!|.|\!|
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|\!|label|\!|{3|\!|.27|\!|}|\!| |\!|end|\!|{eqnarray|\!|}|\!|
|\!|
An|\!| irrelevant|\!|
|\!| normalization|\!| factor|\!| has|\!| been|\!| dropped|\!|.|\!|
|\!| The|\!| Lagrangian|\!| |\!|$|\!|overlinen|\!| L|\!|$|\!| coincides|\!| precisely|\!| with|\!| the|\!| Lagrangian|\!| |\!|(|\!|ref|\!|{lag1|\!|}|\!|)|\!|,|\!| and|\!| describes|\!| therefore|\!| |\!|'t|\!||\!|,Hooft|\!|'s|\!| deterministic|\!| system|\!|.|\!| Hence|\!| within|\!| the|\!| above|\!| assumptions|\!| there|\!| are|\!| no|\!| other|\!| systems|\!| with|\!| the|\!| peculiar|\!| property|\!| that|\!| their|\!| full|\!| quantum|\!| properties|\!| are|\!| classical|\!|.|\!| Among|\!| other|\!| things|\!|,|\!| the|\!| latter|\!| also|\!| indicates|\!| that|\!| the|\!| Koopman|\!|-von|\!|~Neumann|\!| operatorial|\!| formulation|\!| of|\!| classical|\!| mechanics|\!|~|\!|cite|\!|{KN1|\!|}|\!| when|\!| applied|\!| to|\!| |\!|'t|\!|~Hooft|\!| systems|\!| must|\!| agree|\!| with|\!| its|\!| canonically|\!| quantized|\!| counterpart|\!|.|\!|
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|\!|section|\!|{|\!|'t|\!||\!|,Hooft|\!|'s|\!| information|\!| loss|\!| as|\!| a|\!| first|\!|-class|\!| primary|\!| constraint|\!|}|\!|label|\!|{SEc4|\!|}|\!|
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As|\!| observed|\!| in|\!| Section|\!|~II|\!|,|\!| the|\!| Hamiltonian|\!| |\!|(|\!|ref|\!|{eq|\!|.1|\!|.1|\!|}|\!|)|\!| is|\!| not|\!| bounded|\!| from|\!| below|\!|,|\!| and|\!| this|\!| is|\!| true|\!| for|\!| any|\!| function|\!| |\!|$f|\!|_i|\!|$|\!|.|\!| Thus|\!|,|\!| no|\!| deterministic|\!| system|\!| with|\!| dynamical|\!| equations|\!| |\!|$|\!|dot|\!|{q|\!|}|\!|_i|\!| |\!|=|\!| f|\!|_i|\!|(|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|)|\!|$|\!| can|\!| describe|\!| a|\!| |\!| physically|\!| acceptable|\!| |\!|{|\!|em|\!| quantum|\!| world|\!||\!|/|\!|}|\!|.|\!| Its|\!| Hamiltonian|\!| would|\!| not|\!| be|\!| stable|\!| and|\!| we|\!| could|\!| build|\!| a|\!| perpetuum|\!| mobile|\!|.|\!| To|\!| deal|\!| with|\!| this|\!| problem|\!| we|\!| will|\!| employ|\!| |\!|'t|\!||\!|,Hooft|\!|'s|\!| procedure|\!|~|\!|cite|\!|{tHooft|\!|}|\!|.|\!| We|\!|
|\!| assume|\!| that|\!| the|\!| system|\!| |\!|(|\!|ref|\!|{eq|\!|.1|\!|.1|\!|}|\!|)|\!| has|\!| |\!|$n|\!|$|\!| conserved|\!|,|\!| irreducible|\!| charges|\!| |\!|$C|\!|_i|\!|$|\!|,|\!| i|\!|.e|\!|.|\!|,|\!|
|\!|
|\!|begin|\!|{eqnarray|\!|}|\!| |\!||\!|{|\!| C|\!|_i|\!|,|\!| H|\!| |\!||\!|}|\!| |\!|=|\!| 0|\!||\!|,|\!| |\!|,|\!| |\!||\!|;|\!||\!|;|\!||\!|;|\!||\!|;|\!| i|\!| |\!|=|\!| 1|\!|,|\!| |\!|ldots|\!|,|\!| n|\!||\!|,|\!| |\!|.|\!| |\!| |\!| |\!| |\!| |\!| |\!|label|\!|{|\!|@char|\!|}|\!| |\!|end|\!|{eqnarray|\!|}|\!|
|\!|
In|\!| order|\!| to|\!| enforce|\!| a|\!| lower|\!| bound|\!| upon|\!| |\!|$H|\!|$|\!|,|\!|
|\!| |\!|'t|\!||\!|,Hooft|\!| split|\!| the|\!| Hamiltonian|\!|
|\!|
|\!| as|\!|
|\!| |\!|$H|\!| |\!|=|\!| H|\!|_|\!|+|\!| |\!|-|\!| H|\!|_|\!|-|\!|$|\!| with|\!| |\!| both|\!| |\!|$H|\!|_|\!|+|\!|$|\!| and|\!| |\!|$H|\!|_|\!|-|\!|$|\!| having|\!| lower|\!| bounds|\!|.|\!| Then|\!| |\!| he|\!| imposed|\!| the|\!| condition|\!| that|\!| |\!|$H|\!|_|\!|-|\!|$|\!| should|\!| be|\!| zero|\!| on|\!| the|\!| physically|\!| accessible|\!| part|\!| of|\!| phase|\!| space|\!|,|\!| i|\!|.e|\!|.|\!|,|\!|
|\!|
|\!|begin|\!|{eqnarray|\!|}|\!| H|\!|_|\!|-|\!| |\!||\!| |\!|approx|\!| |\!||\!| 0|\!||\!|,|\!| |\!|.|\!| |\!|label|\!|{4|\!|.1|\!|}|\!| |\!|end|\!|{eqnarray|\!|}|\!|
|\!|
This|\!| will|\!| make|\!| the|\!| actual|\!| dynamics|\!|
|\!| governed|\!| by|\!| the|\!| reduced|\!| Hamiltonian|\!| |\!|$H|\!|_|\!|+|\!|$|\!| which|\!| is|\!| bounded|\!| from|\!| below|\!|,|\!|
|\!| by|\!| definition|\!|.|\!|
|\!|
|\!|vspace|\!|{3mm|\!|}|\!|
|\!|
To|\!| ensure|\!| that|\!| the|\!| above|\!| splitting|\!| is|\!| conserved|\!| in|\!| time|\!| one|\!| must|\!| require|\!| that|\!| |\!|$|\!||\!|{|\!| H|\!|_|\!|-|\!|,|\!| H|\!| |\!||\!|}|\!| |\!|=|\!| |\!||\!|{|\!| H|\!|_|\!|+|\!|,|\!| H|\!| |\!||\!|}|\!| |\!|=|\!| 0|\!|$|\!|.|\!| The|\!| latter|\!| is|\!| equivalent|\!| to|\!| the|\!| statement|\!| that|\!| |\!|$|\!||\!|{|\!| H|\!|_|\!|+|\!|,|\!| H|\!|_|\!|-|\!| |\!||\!|}|\!| |\!|=|\!| 0|\!|$|\!|.|\!| Since|\!| the|\!| charges|\!| |\!|$C|\!|_i|\!|$|\!| in|\!| |\!|(|\!|ref|\!|{|\!|@char|\!|}|\!|)|\!| form|\!| an|\!| irreducible|\!| set|\!|,|\!|
|\!| the|\!| Hamiltonians|\!|
|\!| |\!|$H|\!|_|\!|+|\!|$|\!| and|\!| |\!|$H|\!|_|\!|-|\!|$|\!|
|\!| must|\!| be|\!|
|\!| functions|\!| of|\!| the|\!| charges|\!| and|\!| |\!|$H|\!|$|\!|:|\!| |\!|$H|\!|_|\!|+|\!| |\!|=|\!| F|\!|_|\!|+|\!|(C|\!|_k|\!|,H|\!|)|\!|$|\!| and|\!| |\!|$H|\!|_|\!|-|\!| |\!|=|\!| F|\!|_|\!|-|\!|(C|\!|_k|\!|,H|\!|)|\!|$|\!|.|\!|
|\!| There|\!| is|\!| a|\!| certain|\!| amount|\!| of|\!| flexibility|\!| in|\!| finding|\!| |\!|$F|\!|_|\!|-|\!|$|\!| and|\!| |\!|$F|\!|_|\!|+|\!|$|\!|,|\!| but|\!| for|\!| convenience|\!|'s|\!| sake|\!| we|\!| confine|\!| ourselves|\!| to|\!| the|\!| following|\!| choice|\!|
|\!|
|\!|begin|\!|{eqnarray|\!|}|\!| H|\!|_|\!|+|\!| |\!||\!| |\!|=|\!| |\!||\!| |\!|frac|\!|{|\!|[H|\!| |\!|+|\!| |\!|sum|\!|_i|\!| a|\!|_i|\!|(t|\!|)|\!| C|\!|_i|\!|]|\!|^2|\!|}|\!|{4|\!| |\!|sum|\!|_i|\!| a|\!|_i|\!|(t|\!|)|\!| C|\!|_i|\!|}|\!| |\!||\!|,|\!| |\!|,|\!| |\!||\!|;|\!| |\!||\!|;|\!| H|\!|_|\!|-|\!| |\!||\!| |\!|=|\!| |\!||\!| |\!|frac|\!|{|\!|[H|\!| |\!|-|\!| |\!|sum|\!|_i|\!| a|\!|_i|\!|(t|\!|)|\!| C|\!|_i|\!|]|\!|^2|\!|}|\!|{4|\!| |\!|sum|\!|_i|\!| a|\!|_i|\!|(t|\!|)|\!| C|\!|_i|\!|}|\!| |\!||\!|,|\!| |\!|,|\!| |\!|label|\!|{FCH|\!|}|\!| |\!|end|\!|{eqnarray|\!|}|\!|
|\!|
where|\!| |\!|$a|\!|_i|\!|(t|\!|)|\!|$|\!| are|\!| independent|\!| of|\!| |\!| |\!|$|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|$|\!| and|\!| |\!|$|\!|{|\!|boldsymbol|\!|{p|\!|}|\!|}|\!|$|\!| and|\!| will|\!| be|\!| specified|\!| later|\!|.|\!| The|\!| lower|\!| bound|\!| is|\!| then|\!| achieved|\!| by|\!| choosing|\!| |\!|$|\!|sum|\!|_i|\!| a|\!|_i|\!|(t|\!|)|\!| C|\!|_i|\!|$|\!| to|\!| be|\!| positive|\!| definite|\!|.|\!| In|\!| the|\!| following|\!| it|\!| will|\!| also|\!| be|\!| important|\!| to|\!| select|\!| the|\!| combination|\!| of|\!| |\!|$C|\!|_i|\!|$|\!|'s|\!| in|\!| such|\!| a|\!| way|\!| that|\!| it|\!| depends|\!| solely|\!| on|\!| |\!|$|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|$|\!| |\!|(this|\!| condition|\!| may|\!| not|\!| necessarily|\!| be|\!| achievable|\!| for|\!| general|\!| |\!|$f|\!|_a|\!|(|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|)|\!|$|\!|)|\!|.|\!| Thus|\!|,|\!| by|\!| imposing|\!| |\!|$H|\!|_|\!|-|\!| |\!|approx|\!| 0|\!|$|\!| we|\!| obtain|\!| the|\!| weak|\!| reduced|\!| Hamiltonian|\!| |\!|$H|\!| |\!|approx|\!| H|\!|_|\!|+|\!| |\!|approx|\!| |\!|sum|\!|_i|\!| a|\!|_i|\!|(t|\!|)C|\!|_i|\!|$|\!|.|\!|
|\!|
|\!|vspace|\!|{3mm|\!|}|\!|
|\!|
The|\!| constraint|\!| |\!|(|\!|ref|\!|{4|\!|.1|\!|}|\!|)|\!| |\!|(resp|\!| |\!|(|\!|ref|\!|{FCH|\!|}|\!|)|\!|)|\!| can|\!| be|\!| motivated|\!| by|\!| dissipation|\!| or|\!| information|\!| loss|\!|~|\!|cite|\!|{tHooft3|\!|,BJV1|\!|,BMM1|\!|}|\!|.|\!| In|\!| Appendix|\!| D|\!| we|\!| show|\!| that|\!| the|\!| |\!|{|\!|em|\!| explicit|\!|}|\!| constraint|\!| |\!|(|\!|ref|\!|{4|\!|.1|\!|}|\!|)|\!| does|\!| not|\!| generate|\!| any|\!| new|\!| |\!|(i|\!|.e|\!|.|\!|,|\!| secondary|\!|)|\!| constraints|\!| when|\!| added|\!| to|\!| the|\!| existing|\!| constraints|\!| |\!|(|\!|ref|\!|{2|\!|.5|\!|}|\!|)|\!|.|\!| In|\!| addition|\!|,|\!| this|\!| new|\!| set|\!| of|\!| constraints|\!| corresponds|\!| to|\!| |\!|$2N|\!|$|\!| second|\!|-class|\!| constraints|\!| and|\!| |\!|{|\!|em|\!| one|\!|}|\!| first|\!|-class|\!| constraint|\!| |\!|(see|\!| also|\!| Appendix|\!| D|\!|)|\!|.|\!|
|\!|
|\!|
|\!|
It|\!| is|\!| well|\!| known|\!| in|\!| the|\!| theory|\!| of|\!| constrained|\!| systems|\!| that|\!| the|\!| existence|\!| of|\!| first|\!|-class|\!| constraints|\!| signals|\!| the|\!| presence|\!| of|\!| a|\!| gauge|\!| freedom|\!| in|\!| Hamiltonian|\!| theory|\!|.|\!| This|\!| is|\!| so|\!| because|\!| the|\!| Lagrange|\!| multipliers|\!| affiliated|\!| with|\!| first|\!|-class|\!| constraints|\!| cannot|\!| be|\!| fixed|\!| from|\!| dynamical|\!| equations|\!| alone|\!|~|\!|cite|\!|{Dir|\!|}|\!|.|\!| The|\!| time|\!| evolution|\!| of|\!| observable|\!| |\!|(physical|\!|)|\!| quantities|\!|,|\!| however|\!|,|\!| cannot|\!| be|\!| affected|\!| by|\!| the|\!| arbitrariness|\!| in|\!| Lagrange|\!| multipliers|\!|.|\!| To|\!| remove|\!| this|\!| superfluous|\!| freedom|\!| that|\!| is|\!| left|\!| in|\!| the|\!| formalism|\!| we|\!| must|\!| pick|\!| up|\!| a|\!| gauge|\!|,|\!| i|\!|.e|\!|.|\!|,|\!| impose|\!| a|\!| set|\!| of|\!| conditions|\!| that|\!| will|\!| eliminate|\!| the|\!| above|\!| redundancy|\!| from|\!| the|\!| description|\!|.|\!| It|\!| is|\!| easy|\!| to|\!| see|\!| that|\!| the|\!| number|\!| of|\!| independent|\!| gauge|\!| conditions|\!| must|\!| match|\!| the|\!| number|\!| of|\!| first|\!|-class|\!| constraints|\!|.|\!| Indeed|\!|,|\!| the|\!| requirement|\!| on|\!| a|\!| physical|\!| quantity|\!| |\!|(say|\!| |\!|$f|\!|$|\!|)|\!| to|\!| have|\!| a|\!| unique|\!| time|\!| evolution|\!| on|\!| the|\!| constraint|\!| submanifold|\!| |\!|$|\!|{|\!|mathcal|\!|{M|\!|}|\!|}|\!|$|\!|,|\!| i|\!|.e|\!|.|\!|,|\!|
|\!|
|\!|begin|\!|{eqnarray|\!|}|\!| |\!|dot|\!|{f|\!|}|\!| |\!||\!| |\!|approx|\!| |\!||\!| |\!||\!|{|\!| f|\!|,|\!| |\!|bar|\!|{H|\!|}|\!||\!|}|\!| |\!||\!| |\!|+|\!| |\!||\!| |\!|sum|\!|_|\!|{i|\!|=1|\!|}|\!|^|\!|{m|\!|}v|\!|_i|\!| |\!||\!|{|\!| f|\!|,|\!| |\!|varphi|\!|_i|\!||\!|}|\!| |\!||\!| |\!|+|\!| |\!||\!| |\!| |\!|sum|\!|_|\!|{k|\!|=1|\!|}|\!|^|\!|{m|\!|'|\!|}u|\!|_k|\!| |\!||\!|{|\!| f|\!|,|\!| |\!|phi|\!|_k|\!| |\!||\!|}|\!||\!|,|\!| |\!|,|\!| |\!|end|\!|{eqnarray|\!|}|\!|
|\!|
implies|\!| that|\!|
|\!|
|\!|begin|\!|{eqnarray|\!|}|\!| |\!||\!|{|\!| f|\!|,|\!| |\!|varphi|\!|_i|\!||\!|}|\!| |\!||\!| |\!|approx|\!| |\!||\!| 0|\!||\!|,|\!| |\!|.|\!| |\!|label|\!|{4|\!|.24|\!|}|\!| |\!|end|\!|{eqnarray|\!|}|\!|
|\!|
The|\!| constraints|\!| |\!|$|\!|varphi|\!|_i|\!|$|\!| and|\!| |\!|$|\!|phi|\!|_k|\!|$|\!| represent|\!| first|\!| and|\!| second|\!|-class|\!| constraints|\!|,|\!| respectively|\!|.|\!| First|\!|-class|\!| constraints|\!| have|\!|,|\!| by|\!| definition|\!|,|\!| weakly|\!| vanishing|\!| Poisson|\!|'s|\!| brackets|\!| with|\!| all|\!| other|\!| constraints|\!|;|\!| any|\!| other|\!| constraint|\!| that|\!| is|\!| not|\!| first|\!| class|\!| is|\!| second|\!|-class|\!|.|\!| While|\!| the|\!| Lagrange|\!| multipliers|\!| |\!|$u|\!|_k|\!|$|\!| can|\!| be|\!| uniquely|\!| fixed|\!| from|\!| the|\!| dynamics|\!| by|\!| consistency|\!| conditions|\!| |\!|(c|\!|.f|\!|.|\!| Appendices|\!| A|\!| and|\!| D|\!|)|\!| this|\!| cannot|\!| be|\!| done|\!| for|\!| the|\!| |\!|$v|\!|_i|\!|$|\!|'s|\!|.|\!| In|\!| this|\!| way|\!| |\!|(|\!|ref|\!|{4|\!|.24|\!|}|\!|)|\!| represents|\!| an|\!| obligatory|\!| condition|\!| for|\!| a|\!| quantity|\!| |\!|$f|\!|$|\!| to|\!| be|\!| observable|\!|.|\!| Equation|\!| |\!|(|\!|ref|\!|{4|\!|.24|\!|}|\!|)|\!| can|\!| be|\!| considered|\!| as|\!| a|\!| set|\!| of|\!| |\!|$m|\!|$|\!| first|\!|-order|\!| differential|\!| equations|\!| on|\!| the|\!| constrained|\!| surface|\!| with|\!| the|\!| relation|\!| |\!|$|\!||\!|{|\!|varphi|\!|_i|\!|,|\!| |\!|varphi|\!|_j|\!| |\!||\!|}|\!| |\!||\!| |\!|approx|\!| |\!||\!| 0|\!|$|\!| serving|\!| as|\!| the|\!| integrability|\!| condition|\!|~|\!|cite|\!|{Dir|\!|,Sunder|\!|}|\!|.|\!| Thus|\!|,|\!| |\!|$f|\!|$|\!| is|\!| uniquely|\!| defined|\!| by|\!| its|\!| values|\!| on|\!| the|\!| submanifold|\!| of|\!| the|\!| initial|\!| conditions|\!| for|\!| Eq|\!|.|\!|(|\!|ref|\!|{4|\!|.24|\!|}|\!|)|\!|.|\!| As|\!| a|\!| result|\!|,|\!| the|\!| above|\!| initial|\!| value|\!| surface|\!| describes|\!| the|\!| true|\!| degrees|\!| of|\!| freedom|\!|.|\!| By|\!| denoting|\!| the|\!| dimension|\!| of|\!| the|\!| constraint|\!| manifold|\!| as|\!| |\!|$D|\!|$|\!| we|\!| see|\!| that|\!| the|\!| dimension|\!| of|\!| the|\!| submanifold|\!| of|\!| initial|\!| conditions|\!| must|\!| be|\!| |\!|$D|\!|-m|\!|$|\!|.|\!| We|\!| can|\!| take|\!| this|\!| submanifold|\!| to|\!| be|\!| a|\!| surface|\!| |\!|$|\!|Gamma|\!|^|\!|*|\!|$|\!| specified|\!| by|\!| the|\!| equations|\!|
|\!|
|\!|begin|\!|{eqnarray|\!|}|\!| |\!|varphi|\!|_i|\!| |\!||\!| |\!|&|\!|=|\!|&|\!| |\!||\!| 0|\!||\!|,|\!| |\!|,|\!| |\!||\!|;|\!||\!|;|\!||\!|;|\!||\!|;|\!||\!|;|\!||\!|;|\!| i|\!| |\!|=|\!| 1|\!|,|\!| |\!|ldots|\!|,|\!| m|\!||\!|,|\!| |\!|,|\!|nonumber|\!| |\!||\!||\!| |\!|phi|\!|_k|\!| |\!||\!| |\!|&|\!|=|\!|&|\!| |\!||\!| 0|\!| |\!||\!|,|\!| |\!|,|\!| |\!||\!|;|\!||\!|;|\!||\!|;|\!||\!|;|\!||\!|;|\!||\!|;|\!| k|\!| |\!|=|\!| 1|\!|,|\!| |\!|ldots|\!|,|\!| m|\!|'|\!||\!|,|\!| |\!|,|\!|nonumber|\!| |\!||\!||\!| |\!|chi|\!|_l|\!| |\!||\!| |\!|&|\!|=|\!|&|\!| |\!||\!| 0|\!| |\!||\!|,|\!| |\!|,|\!| |\!||\!|;|\!||\!|;|\!||\!|;|\!||\!|;|\!||\!|;|\!||\!|;|\!| l|\!| |\!|=|\!| 1|\!|,|\!| |\!|ldots|\!|,|\!| m|\!||\!|,|\!| |\!|.|\!| |\!|label|\!|{4|\!|.24b|\!|}|\!| |\!|end|\!|{eqnarray|\!|}|\!|
|\!|
The|\!| |\!|$m|\!|$|\!| subsidiary|\!| conditions|\!| |\!|$|\!|chi|\!|_l|\!|$|\!| are|\!| the|\!| sought|\!| gauge|\!| constraints|\!|.|\!| The|\!| functions|\!| |\!|$|\!|chi|\!|_l|\!|$|\!| must|\!| clearly|\!| satisfy|\!| the|\!| condition|\!|
|\!|
|\!|begin|\!|{eqnarray|\!|}|\!|
|\!|det|\!||\!|||\!| |\!||\!|{|\!|chi|\!|_l|\!| |\!|,|\!| |\!|varphi|\!|_i|\!| |\!||\!|}|\!| |\!||\!|||\!| |\!||\!| |\!|neq|\!| |\!||\!| 0|\!||\!|,|\!| |\!|,|\!| |\!|label|\!|{4|\!|.25|\!|}|\!| |\!|end|\!|{eqnarray|\!|}|\!|
|\!|
as|\!| only|\!| in|\!| such|\!| a|\!| case|\!| we|\!| can|\!| determine|\!| specific|\!| values|\!| for|\!| the|\!| multipliers|\!| |\!|$v|\!|_i|\!|$|\!| from|\!| the|\!| dynamical|\!| equation|\!| for|\!| |\!|$|\!|chi|\!|_l|\!|$|\!| |\!|(this|\!| is|\!| because|\!| the|\!| time|\!| derivative|\!| of|\!| any|\!| constraint|\!|,|\!| and|\!| hence|\!| also|\!| |\!|$|\!|chi|\!|_l|\!|$|\!|,|\!| must|\!| be|\!| zero|\!|)|\!|.|\!| Therefore|\!| only|\!| when|\!| the|\!| condition|\!| |\!|(|\!|ref|\!|{4|\!|.25|\!|}|\!|)|\!| is|\!| satisfied|\!| do|\!| the|\!| constraints|\!| |\!|(|\!|ref|\!|{4|\!|.24b|\!|}|\!|)|\!| indeed|\!| describe|\!| the|\!| surface|\!| of|\!| the|\!| initial|\!| conditions|\!|.|\!|
|\!|
|\!|
|\!|vspace|\!|{3mm|\!|}|\!|
|\!|
The|\!| preceding|\!| discussion|\!| implies|\!| that|\!| in|\!| our|\!| case|\!| the|\!| surface|\!| |\!|$|\!|Gamma|\!|^|\!|*|\!|$|\!| is|\!| defined|\!| by|\!|
|\!|
|\!|begin|\!|{eqnarray|\!|}|\!| |\!|varphi|\!|(|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|,|\!|bar|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|,|\!| |\!|{|\!|boldsymbol|\!|{p|\!|}|\!|}|\!|,|\!|bar|\!|{|\!|boldsymbol|\!|{p|\!|}|\!|}|\!|)|\!| |\!||\!| |\!|&|\!|=|\!|&|\!| |\!||\!| 0|\!| |\!||\!|,|\!| |\!|,|\!| |\!|mbox|\!|{|\!|hspace|\!|{0|\!|.6cm|\!|}|\!|}|\!| |\!|chi|\!|(|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|,|\!|bar|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|,|\!| |\!|{|\!|boldsymbol|\!|{p|\!|}|\!|}|\!|,|\!|bar|\!|{|\!|boldsymbol|\!|{p|\!|}|\!|}|\!|)|\!| |\!||\!| |\!|=|\!| |\!||\!| 0|\!| |\!||\!|,|\!| |\!|,|\!| |\!|label|\!|{4|\!|.26|\!|}|\!| |\!||\!||\!| |\!|mbox|\!|{|\!|hspace|\!|{1cm|\!|}|\!|}|\!| |\!|phi|\!|_i|\!|(|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|,|\!|bar|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|,|\!| |\!|{|\!|boldsymbol|\!|{p|\!|}|\!|}|\!|,|\!|bar|\!|{|\!|boldsymbol|\!|{p|\!|}|\!|}|\!|)|\!| |\!||\!| |\!|&|\!|=|\!|&|\!| |\!||\!| 0|\!| |\!||\!|,|\!| |\!|,|\!| |\!||\!|;|\!||\!|;|\!||\!|;|\!||\!|;|\!||\!|;|\!||\!|;|\!| i|\!| |\!|=|\!| 1|\!|,|\!| |\!|ldots|\!|,|\!| 2N|\!||\!|,|\!| |\!|.|\!| |\!|end|\!|{eqnarray|\!|}|\!|
|\!|
The|\!| explicit|\!| form|\!| of|\!| |\!|$|\!|varphi|\!|$|\!| is|\!| found|\!| in|\!| Appendix|\!|~D|\!| where|\!| we|\!| show|\!| that|\!| |\!|$|\!|varphi|\!| |\!|approx|\!| H|\!| |\!|-|\!| |\!|sum|\!| a|\!|_i|\!| C|\!|_i|\!|$|\!|.|\!|
|\!|
|\!|
Apart|\!| from|\!| condition|\!| |\!|(|\!|ref|\!|{4|\!|.25|\!|}|\!|)|\!| we|\!| shall|\!| further|\!| restrict|\!| our|\!| choice|\!| of|\!| |\!|$|\!|chi|\!|$|\!| to|\!| functions|\!| satisfying|\!| the|\!| simultaneous|\!| equations|\!|
|\!|
|\!|begin|\!|{eqnarray|\!|}|\!| |\!||\!|{|\!| |\!|chi|\!|,|\!| |\!|phi|\!|_i|\!| |\!||\!|}|\!| |\!||\!| |\!|=|\!| |\!||\!| 0|\!||\!|,|\!| |\!|,|\!| |\!||\!|;|\!||\!|;|\!||\!|;|\!||\!|;|\!||\!|;|\!| i|\!| |\!|=|\!| 1|\!|,|\!| |\!|ldots|\!|,|\!| 2N|\!||\!|,|\!| |\!|.|\!| |\!|label|\!|{4|\!|.27|\!|}|\!| |\!|end|\!|{eqnarray|\!|}|\!|
|\!|
Such|\!| a|\!| choice|\!| is|\!| always|\!| possible|\!| |\!|(at|\!| least|\!| in|\!| a|\!| weak|\!| sense|\!|)|\!|~|\!|cite|\!|{Senj|\!|}|\!| and|\!| it|\!| will|\!| prove|\!| crucial|\!| in|\!| the|\!| following|\!|.|\!|
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|\!|vspace|\!|{3mm|\!|}|\!|
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In|\!| order|\!| to|\!| proceed|\!| further|\!| we|\!| begin|\!| by|\!| reexamining|\!| Eq|\!|.|\!|(|\!|ref|\!|{3|\!|.27|\!|}|\!|)|\!|.|\!| The|\!| latter|\!| basically|\!| states|\!| that|\!|
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|\!|begin|\!|{eqnarray|\!|}|\!| Z|\!|_|\!|{|\!|rm|\!| CM|\!|}|\!| |\!||\!| |\!|=|\!| |\!||\!| |\!|int|\!| |\!|{|\!|mathcal|\!|{D|\!|}|\!|}|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!| |\!||\!| |\!|delta|\!||\!|!|\!|left|\!|[|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!| |\!|-|\!| |\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|_|\!|{c|\!|}|\!|
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|\!|right|\!|]|\!| |\!||\!| |\!|exp|\!|left|\!|[|\!| |\!|int|\!|_|\!|{t|\!|_1|\!|}|\!|^|\!|{t|\!|_2|\!|}|\!| dt|\!| |\!||\!| |\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|(t|\!|)|\!|{|\!|boldsymbol|\!|{J|\!|}|\!|}|\!|(t|\!|)|\!| |\!|right|\!|]|\!||\!|,|\!| |\!|.|\!| |\!|label|\!|{3|\!|.40|\!|}|\!| |\!|end|\!|{eqnarray|\!|}|\!|
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We|\!| may|\!| now|\!| formally|\!| invert|\!| the|\!| steps|\!| leading|\!| to|\!| Eq|\!|.|\!|(|\!|ref|\!|{eg|\!|.1|\!|.2|\!|}|\!|)|\!|,|\!| i|\!|.e|\!|.|\!|,|\!| we|\!| introduce|\!| auxiliary|\!| momentum|\!| integrations|\!| and|\!| go|\!| over|\!| to|\!| the|\!| canonical|\!| representation|\!| of|\!| |\!|(|\!|ref|\!|{3|\!|.40|\!|}|\!|)|\!|.|\!| |\!| Correspondingly|\!| Eq|\!|.|\!|(|\!|ref|\!|{3|\!|.40|\!|}|\!|)|\!| can|\!| be|\!| recast|\!| into|\!|
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|\!|begin|\!|{eqnarray|\!|*|\!|}|\!| Z|\!|_|\!|{|\!|rm|\!| CM|\!|}|\!| |\!|=|\!| |\!| |\!|int|\!| |\!|{|\!|mathcal|\!|{D|\!|}|\!|}|\!|{|\!|boldsymbol|\!|{p|\!|}|\!|}|\!|{|\!|mathcal|\!|{D|\!|}|\!|}|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|{|\!|mathcal|\!|{D|\!|}|\!|}|\!| |\!|bar|\!|{|\!|boldsymbol|\!|{p|\!|}|\!|}|\!|{|\!|mathcal|\!|{D|\!|}|\!|}|\!|bar|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|
|\!|sqrt|\!|{|\!|left|\!|||\!|det|\!||\!|||\!||\!|{|\!| |\!|phi|\!|_i|\!|,|\!| |\!|phi|\!|_j|\!| |\!||\!|}|\!||\!|||\!| |\!|right|\!|||\!|}|\!| |\!|prod|\!|_|\!|{i|\!|=1|\!|}|\!|^|\!|{2N|\!|}|\!| |\!|delta|\!|[|\!|phi|\!|_i|\!|]|\!| |\!| |\!|exp|\!|left|\!|[|\!| i|\!||\!|!|\!| |\!|int|\!|_|\!|{t|\!|_1|\!|}|\!|^|\!|{t|\!|_2|\!|}|\!| |\!||\!|!dt|\!| |\!||\!|,|\!|[|\!|{|\!|boldsymbol|\!|{p|\!|}|\!|}|\!|dot|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!| |\!|+|\!| |\!|bar|\!|{|\!|boldsymbol|\!|{p|\!|}|\!|}|\!|dot|\!|{|\!|bar|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|}|\!| |\!|-|\!| H|\!|]|\!| |\!|+|\!| |\!|int|\!|_|\!|{t|\!|_1|\!|}|\!|^|\!|{t|\!|_2|\!|}|\!||\!|!|\!| dt|\!||\!|,|\!| |\!|[|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|{|\!|boldsymbol|\!|{J|\!|}|\!|}|\!| |\!|+|\!| |\!|bar|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|bar|\!|{|\!|boldsymbol|\!|{J|\!|}|\!|}|\!|]|\!| |\!|right|\!|]|\!| |\!|.|\!| |\!|end|\!|{eqnarray|\!|*|\!|}|\!|
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Due|\!| to|\!| |\!|$|\!|delta|\!|$|\!|-functions|\!| in|\!| the|\!| integration|\!| we|\!| could|\!| substitute|\!| |\!|'t|\!|~Hooft|\!|'s|\!| Hamiltonian|\!| |\!|$H|\!|$|\!| for|\!| the|\!| canonical|\!| Hamiltonian|\!| |\!|$|\!|bar|\!|{H|\!|}|\!|$|\!|.|\!| It|\!| should|\!| be|\!| stressed|\!| that|\!| despite|\!| its|\!| formal|\!| appearance|\!| and|\!| the|\!| phase|\!|-space|\!| disguise|\!|,|\!| the|\!| latter|\!| is|\!| still|\!| the|\!| classical|\!| partition|\!| function|\!| |\!|{|\!|em|\!|{|\!||\!|`|\!|{a|\!|}|\!|}|\!| la|\!| |\!|}|\!| Gozzi|\!| |\!|{|\!|em|\!| et|\!| al|\!|.|\!|}|\!|.|\!|
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|\!|vspace|\!|{3mm|\!|}|\!|
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To|\!| include|\!| the|\!| constraints|\!| |\!|(|\!|ref|\!|{4|\!|.26|\!|}|\!|)|\!| into|\!|
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|\!|(|\!|ref|\!|{3|\!|.27|\!|}|\!|)|\!| we|\!| must|\!| be|\!| a|\!| bit|\!| cautious|\!|.|\!| A|\!| na|\!||\!|"|\!|{|\!|i|\!|}ve|\!| intuition|\!| would|\!| dictate|\!| that|\!| the|\!| functional|\!| |\!|$|\!|delta|\!|$|\!| functions|\!| |\!|$|\!|delta|\!|[|\!|chi|\!|]|\!|$|\!| and|\!| |\!|$|\!|delta|\!|[|\!|varphi|\!|]|\!|$|\!| should|\!| be|\!| inserted|\!| into|\!| the|\!| path|\!|-integral|\!| measure|\!| for|\!| |\!|$Z|\!|_|\!|{|\!|rm|\!| CM|\!|}|\!|$|\!|.|\!| This|\!| would|\!| be|\!|,|\!| however|\!|,|\!| too|\!| simplistic|\!| as|\!| a|\!| mere|\!| inclusion|\!| of|\!| |\!|$|\!|delta|\!|$|\!| functions|\!| into|\!| |\!|$Z|\!|_|\!|{|\!|rm|\!| CM|\!|}|\!|$|\!| would|\!| not|\!| guarantee|\!| that|\!| the|\!| physical|\!| content|\!| of|\!| the|\!| theory|\!| that|\!| resides|\!| in|\!| the|\!| generating|\!| functional|\!| |\!|$Z|\!|_|\!|{|\!|rm|\!| CM|\!|}|\!|$|\!| is|\!| independent|\!| of|\!| the|\!| choice|\!| |\!|$|\!|chi|\!|$|\!|.|\!| Indeed|\!|,|\!| utilizing|\!| the|\!| fact|\!| that|\!| the|\!| generators|\!| of|\!| gauge|\!| transformations|\!| are|\!| the|\!| first|\!| class|\!| constraints|\!|~|\!|cite|\!|{Sunder|\!|}|\!| we|\!| can|\!| write|\!| that|\!|
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|\!|begin|\!|{eqnarray|\!|}|\!| |\!|delta|\!| |\!|chi|\!| |\!||\!| |\!|=|\!| |\!||\!| |\!|varepsilon|\!| |\!||\!|{|\!|chi|\!| |\!|,|\!| |\!|varphi|\!| |\!||\!|}|\!| |\!|+|\!| C|\!| |\!|varphi|\!| |\!||\!| |\!|approx|\!| |\!||\!| |\!|varepsilon|\!| |\!||\!|{|\!|chi|\!| |\!|,|\!| |\!|varphi|\!| |\!||\!|}|\!||\!|,|\!| |\!|.|\!| |\!|label|\!|{4|\!|.10|\!|}|\!| |\!|end|\!|{eqnarray|\!|}|\!|
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Here|\!| |\!|$|\!|varepsilon|\!|$|\!| is|\!| an|\!| infinitesimal|\!| quantity|\!|.|\!| The|\!| corresponding|\!| gauge|\!| generator|\!| |\!|$|\!|varepsilon|\!| |\!|varphi|\!|$|\!| generates|\!| the|\!| infinitesimal|\!| canonical|\!| transformations|\!|
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|\!|begin|\!|{eqnarray|\!|}|\!| |\!|&|\!|&|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!| |\!|rightarrow|\!| |\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|+|\!| |\!|delta|\!| |\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!||\!|,|\!|,|\!| |\!||\!|;|\!||\!|;|\!||\!|;|\!| |\!|{|\!|boldsymbol|\!|{p|\!|}|\!|}|\!| |\!|rightarrow|\!| |\!|{|\!|boldsymbol|\!|{p|\!|}|\!|}|\!| |\!|+|\!| |\!|delta|\!| |\!|{|\!|boldsymbol|\!|{p|\!|}|\!|}|\!||\!|,|\!|,|\!| |\!||\!|;|\!||\!|;|\!||\!|;|\!| |\!|delta|\!| |\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!| |\!|=|\!| |\!||\!|{|\!|varepsilon|\!| |\!|varphi|\!| |\!|,|\!| |\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!| |\!||\!|}|\!||\!|,|\!|,|\!| |\!||\!|;|\!||\!|;|\!||\!|;|\!| |\!|{|\!|boldsymbol|\!|{p|\!|}|\!|}|\!| |\!|=|\!| |\!||\!|{|\!|varepsilon|\!| |\!|varphi|\!| |\!|,|\!| |\!|{|\!|boldsymbol|\!|{p|\!|}|\!|}|\!| |\!||\!|}|\!||\!|,|\!|,|\!|nonumber|\!| |\!||\!||\!| |\!|&|\!|&|\!| |\!|bar|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!| |\!|rightarrow|\!| |\!|bar|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|+|\!| |\!|delta|\!| |\!|bar|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!||\!|,|\!|,|\!| |\!||\!|;|\!||\!|;|\!||\!|;|\!| |\!|bar|\!|{|\!|boldsymbol|\!|{p|\!|}|\!|}|\!| |\!|rightarrow|\!| |\!|bar|\!|{|\!|boldsymbol|\!|{p|\!|}|\!|}|\!| |\!|+|\!| |\!|delta|\!| |\!|bar|\!|{|\!|boldsymbol|\!|{p|\!|}|\!|}|\!||\!|,|\!|,|\!| |\!||\!|;|\!||\!|;|\!||\!|;|\!| |\!|delta|\!| |\!|bar|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!| |\!|=|\!| |\!||\!|{|\!|varepsilon|\!| |\!|varphi|\!| |\!|,|\!| |\!|bar|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!| |\!||\!|}|\!||\!|,|\!|,|\!| |\!||\!|;|\!||\!|;|\!||\!|;|\!| |\!|bar|\!|{|\!|boldsymbol|\!|{p|\!|}|\!|}|\!| |\!|=|\!| |\!||\!|{|\!|varepsilon|\!| |\!|varphi|\!| |\!|,|\!| |\!|bar|\!|{|\!|boldsymbol|\!|{p|\!|}|\!|}|\!| |\!||\!|}|\!||\!|,|\!| |\!|.|\!| |\!|label|\!|{ct1|\!|}|\!| |\!|end|\!|{eqnarray|\!|}|\!|
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It|\!| follows|\!| immediately|\!| that|\!| the|\!| corresponding|\!| generating|\!| function|\!| is|\!|
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|\!|begin|\!|{eqnarray|\!|}|\!| G|\!|(|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|,|\!|bar|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|,|\!| |\!|{|\!|boldsymbol|\!|{P|\!|}|\!|}|\!|,|\!| |\!|bar|\!|{|\!|boldsymbol|\!|{P|\!|}|\!|}|\!|)|\!| |\!|=|\!| |\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|{|\!|boldsymbol|\!|{P|\!|}|\!|}|\!| |\!|+|\!| |\!|bar|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|bar|\!|{|\!|boldsymbol|\!|{P|\!|}|\!|}|\!| |\!|+|\!| |\!|varepsilon|\!| |\!|varphi|\!| |\!|+|\!| o|\!|(|\!|varepsilon|\!|^2|\!|)|\!||\!|,|\!| |\!|.|\!| |\!|end|\!|{eqnarray|\!|}|\!|
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The|\!| canonical|\!| transformations|\!| |\!|(|\!|ref|\!|{ct1|\!|}|\!|)|\!| result|\!| in|\!| changing|\!| |\!|$|\!|varphi|\!|$|\!| and|\!| |\!|$|\!|phi|\!|_i|\!|$|\!| by|\!|
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|\!|begin|\!|{eqnarray|\!|}|\!| |\!|&|\!|&|\!|delta|\!| |\!|varphi|\!| |\!||\!| |\!|=|\!| |\!||\!| A|\!| |\!|varphi|\!||\!|,|\!| |\!|,|\!| |\!|label|\!|{4|\!|.18|\!|}|\!||\!||\!| |\!|&|\!|&|\!|delta|\!| |\!|phi|\!|_i|\!| |\!||\!| |\!|=|\!| |\!||\!| |\!|varepsilon|\!| |\!||\!|{|\!| |\!|phi|\!|_i|\!|,|\!| |\!|varphi|\!||\!|}|\!| |\!||\!| |\!|=|\!| |\!||\!| B|\!|_i|\!| |\!|varphi|\!| |\!|+|\!| D|\!|_|\!|{ij|\!|}|\!| |\!||\!| |\!|phi|\!|_j|\!| |\!||\!|,|\!| |\!|.|\!| |\!|label|\!|{4|\!|.19|\!|}|\!| |\!|end|\!|{eqnarray|\!|}|\!|
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Here|\!| |\!|$A|\!|,|\!| B|\!|_i|\!|,|\!| C|\!|$|\!| and|\!| |\!|$D|\!|_|\!|{ij|\!|}|\!|$|\!| are|\!| some|\!| phase|\!|-space|\!| functions|\!| of|\!| order|\!| |\!|$|\!|varepsilon|\!|$|\!|.|\!| Note|\!| that|\!| in|\!| our|\!| case|\!| the|\!| gauge|\!| algebra|\!| is|\!| Abelian|\!|footnote|\!|{If|\!| |\!|$|\!|mathcal|\!|{F|\!|}|\!|$|\!| is|\!| any|\!| phase|\!|-space|\!| function|\!| then|\!| |\!|$|\!|[|\!|delta|\!|_|\!|{|\!|varepsilon|\!|}|\!|,|\!| |\!|delta|\!|_|\!|{|\!|eta|\!|}|\!| |\!|]|\!| |\!|{|\!|mathcal|\!|{F|\!|}|\!|}|\!| |\!|=|\!| |\!|delta|\!|_|\!|{|\!|varepsilon|\!|}|\!|delta|\!|_|\!|{|\!|eta|\!|}|\!| |\!|{|\!|mathcal|\!|{F|\!|}|\!|}|\!| |\!|-|\!| |\!|delta|\!|_|\!|{|\!|eta|\!|}|\!| |\!|delta|\!|_|\!|{|\!|varepsilon|\!|}|\!| |\!|{|\!|mathcal|\!|{F|\!|}|\!|}|\!| |\!|=|\!| |\!|varepsilon|\!|eta|\!| |\!|left|\!||\!|{|\!| |\!|{|\!|mathcal|\!|{F|\!|}|\!|}|\!|,|\!| |\!||\!|{|\!| |\!|varphi|\!|,|\!| |\!|varphi|\!||\!|}|\!| |\!|right|\!||\!|}|\!| |\!|=|\!| 0|\!|$|\!|.|\!|}|\!|.|\!| As|\!| a|\!| consequence|\!| of|\!| |\!|(|\!|ref|\!|{4|\!|.18|\!|}|\!|)|\!| and|\!| |\!|(|\!|ref|\!|{4|\!|.19|\!|}|\!|)|\!| we|\!| find|\!|
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|\!|begin|\!|{eqnarray|\!|}|\!|
|\!|delta|\!|[|\!|varphi|\!|]|\!| |\!||\!| |\!|&|\!|rightarrow|\!|&|\!| |\!||\!| |\!|left|\!||1|\!| |\!|+|\!| |\!|mbox|\!|{Tr|\!|}|\!|(A|\!|)|\!|
|\!|right|\!|||\!|^|\!|{|\!|-1|\!|}|\!| |\!|delta|\!|[|\!|varphi|\!|]|\!||\!|,|\!| |\!|,|\!| |\!|label|\!|{4|\!|.11|\!|}|\!||\!||\!|
|\!|prod|\!|_i|\!| |\!|delta|\!|[|\!|phi|\!|_i|\!|]|\!| |\!||\!| |\!|&|\!|rightarrow|\!|&|\!| |\!||\!| |\!|left|\!||1|\!| |\!|+|\!| |\!|mbox|\!|{Tr|\!|}|\!|
|\!|(D|\!|)|\!|right|\!|||\!|^|\!|{|\!|-1|\!|}|\!| |\!|prod|\!|_i|\!| |\!|delta|\!|[|\!|phi|\!|_i|\!|]|\!||\!|,|\!| |\!|,|\!| |\!|label|\!|{4|\!|.12|\!|}|\!||\!||\!|
|\!|sqrt|\!|{|\!|left|\!|||\!|det|\!| |\!||\!|||\!||\!|{|\!| |\!|phi|\!|_i|\!|,|\!| |\!|phi|\!|_j|\!| |\!| |\!||\!|}|\!||\!|||\!| |\!|right|\!|||\!|}|\!| |\!||\!|
|\!|&|\!|rightarrow|\!|&|\!| |\!||\!| |\!|left|\!||1|\!| |\!|+|\!| |\!|mbox|\!|{Tr|\!|}|\!|(D|\!|)|\!| |\!|right|\!|||\!| |\!||\!| |\!|sqrt|\!|{|\!|left|\!|||\!|det|\!|
|\!||\!|||\!||\!|{|\!| |\!|phi|\!|_i|\!|,|\!| |\!|phi|\!|_j|\!| |\!||\!|}|\!||\!|||\!| |\!|right|\!|||\!|}|\!||\!|,|\!| |\!|.|\!| |\!|label|\!|{4|\!|.13|\!|}|\!| |\!|end|\!|{eqnarray|\!|}|\!|
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|\!|[here|\!| |\!|$|\!|mbox|\!|{Tr|\!|}|\!|(A|\!|)|\!| |\!|=|\!| |\!|sum|\!|_t|\!| A|\!|(t|\!|)|\!|$|\!|,|\!| etc|\!|.|\!|]|\!| In|\!| |\!|(|\!|ref|\!|{4|\!|.13|\!|}|\!|)|\!| we|\!| have|\!| used|\!| the|\!| fact|\!| that|\!| in|\!| the|\!| path|\!|-integral|\!| measure|\!| are|\!| present|\!|
|\!|$|\!|delta|\!|[|\!|varphi|\!|]|\!|$|\!| and|\!| |\!|$|\!|delta|\!|[|\!|phi|\!|_i|\!|]|\!|$|\!|,|\!| and|\!| so|\!| we|\!| have|\!| dropped|\!| on|\!| the|\!| RHS|\!|'s|\!| of|\!| |\!|(|\!|ref|\!|{4|\!|.11|\!|}|\!|)|\!|-|\!|(|\!|ref|\!|{4|\!|.13|\!|}|\!|)|\!| the|\!| vanishing|\!| terms|\!|.|\!| The|\!| infinitesimal|\!| gauge|\!| transformations|\!| described|\!| hitherto|\!| clearly|\!| show|\!| that|\!| |\!|$Z|\!|_|\!|{|\!|rm|\!| CM|\!|}|\!|$|\!| is|\!| dependent|\!| on|\!| the|\!| choice|\!| of|\!| |\!|$|\!|chi|\!|$|\!| |\!|[the|\!| term|\!| with|\!| |\!|$|\!||1|\!|+|\!| |\!|mbox|\!|{Tr|\!|}|\!|(A|\!|)|\!|||\!|$|\!| does|\!| not|\!| get|\!| canceled|\!|]|\!|.|\!| To|\!| ensure|\!| the|\!| gauge|\!| invariance|\!| we|\!| need|\!| to|\!| factor|\!| out|\!| the|\!| |\!|`|\!|`orbit|\!| volume|\!|"|\!| from|\!| the|\!| definition|\!| of|\!| |\!|$Z|\!|_|\!|{|\!|rm|\!| CM|\!|}|\!|$|\!|.|\!| This|\!| will|\!| be|\!| achieved|\!| by|\!| a|\!| procedure|\!| that|\!| is|\!| akin|\!| to|\!| the|\!| Faddeev|\!|-Popov|\!|-De|\!|~Witt|\!| trick|\!|.|\!| We|\!| define|\!| the|\!| functional|\!|
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|\!|begin|\!|{eqnarray|\!|}|\!| |\!|left|\!|(|\!| |\!|triangle|\!|_|\!|{|\!|chi|\!|}|\!|right|\!|)|\!|^|\!|{|\!|-1|\!|}|\!| |\!||\!| |\!|=|\!| |\!||\!| |\!|int|\!| |\!|{|\!|mathcal|\!|{D|\!|}|\!|}g|\!| |\!||\!| |\!|delta|\!| |\!|[|\!|chi|\!|^|\!|{g|\!|}|\!|]|\!||\!|,|\!| |\!|,|\!| |\!|label|\!|{4|\!|.14|\!|}|\!| |\!|end|\!|{eqnarray|\!|}|\!|
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with|\!| |\!|$|\!|chi|\!|^|\!|{g|\!|}|\!|$|\!| representing|\!| the|\!| gauge|\!| transformed|\!| |\!|$|\!|chi|\!|$|\!|.|\!| The|\!| superscript|\!| |\!|$g|\!|$|\!| in|\!| Eq|\!|.|\!|(|\!|ref|\!|{4|\!|.14|\!|}|\!|)|\!| denotes|\!| an|\!| element|\!| of|\!| the|\!| Abelian|\!| gauge|\!| group|\!| generated|\!| by|\!| |\!|$|\!|varphi|\!|$|\!|.|\!|
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We|\!| point|\!| out|\!| that|\!| the|\!| functional|\!| |\!|(|\!|ref|\!|{4|\!|.14|\!|}|\!|)|\!| is|\!| manifestly|\!| gauge|\!| invariant|\!| since|\!|
|\!|
|\!|begin|\!|{eqnarray|\!|}|\!| |\!|left|\!|(|\!| |\!|triangle|\!|_|\!|{|\!|chi|\!|^|\!|{g|\!|'|\!|}|\!|}|\!|right|\!|)|\!|^|\!|{|\!|-1|\!|}|\!| |\!||\!| |\!|=|\!| |\!||\!| |\!| |\!|int|\!| |\!|{|\!|mathcal|\!|{D|\!|}|\!|}g|\!| |\!||\!| |\!|delta|\!|[|\!|chi|\!|^|\!|{g|\!|'g|\!|}|\!|]|\!||\!| |\!|=|\!| |\!||\!| |\!|int|\!| |\!|{|\!|mathcal|\!|{D|\!|}|\!|}|\!|(g|\!|'g|\!|)|\!| |\!||\!| |\!|delta|\!|[|\!|chi|\!|^|\!|{g|\!|'g|\!|}|\!|]|\!| |\!||\!| |\!|=|\!| |\!||\!| |\!|left|\!|(|\!| |\!|triangle|\!|_|\!|{|\!|chi|\!|}|\!|right|\!|)|\!|^|\!|{|\!|-1|\!|}|\!||\!|,|\!| |\!|.|\!| |\!|label|\!|{4|\!|.15|\!|}|\!| |\!|end|\!|{eqnarray|\!|}|\!|
|\!|
The|\!| second|\!| identity|\!| holds|\!| because|\!| of|\!| the|\!| invariance|\!| of|\!| the|\!| group|\!| measure|\!| under|\!| composition|\!|,|\!| i|\!|.e|\!|.|\!|,|\!| |\!|$|\!|{|\!|mathcal|\!|{D|\!|}|\!|}g|\!| |\!|=|\!| |\!|{|\!|mathcal|\!|{D|\!|}|\!|}|\!|(g|\!|'g|\!|)|\!|$|\!|.|\!| Equations|\!| |\!|(|\!|ref|\!|{4|\!|.14|\!|}|\!|)|\!| and|\!| |\!|(|\!|ref|\!|{4|\!|.15|\!|}|\!|)|\!| allow|\!| us|\!| to|\!| write|\!| |\!|`|\!|`|\!|$1|\!|$|\!|"|\!| as|\!|
|\!|
|\!|begin|\!|{eqnarray|\!|}|\!| 1|\!| |\!||\!| |\!|=|\!| |\!||\!| |\!|triangle|\!|_|\!|{|\!|chi|\!|}|\!| |\!||\!| |\!|delta|\!|[|\!|chi|\!|]|\!| |\!|int|\!| |\!|{|\!|mathcal|\!|{D|\!|}|\!|}g|\!| |\!||\!|,|\!| |\!|.|\!| |\!|label|\!|{4|\!|.16|\!|}|\!| |\!|end|\!|{eqnarray|\!|}|\!|
|\!|
To|\!| find|\!| an|\!| explicit|\!| form|\!| of|\!| |\!|$|\!|triangle|\!|[|\!|chi|\!|]|\!|$|\!| we|\!| can|\!| apply|\!| the|\!| infinitesimal|\!| gauge|\!| transformation|\!| |\!|(|\!|ref|\!|{4|\!|.10|\!|}|\!|)|\!|.|\!| Then|\!|
|\!|
|\!|begin|\!|{eqnarray|\!|}|\!| |\!|chi|\!|^g|\!| |\!| |\!||\!| |\!|=|\!| |\!||\!| |\!|chi|\!| |\!|+|\!| |\!|varepsilon|\!| |\!||\!|{|\!|chi|\!|,|\!| |\!|varphi|\!| |\!||\!|}|\!| |\!|+|\!| C|\!| |\!|varphi|\!| |\!||\!| |\!|&|\!|Rightarrow|\!| |\!||\!| |\!|&|\!| |\!|left|\!|(|\!|triangle|\!|_|\!|{|\!|chi|\!|}|\!|right|\!|)|\!|^|\!|{|\!|-1|\!|}|\!| |\!||\!| |\!|=|\!| |\!||\!| |\!|int|\!| |\!|{|\!|mathcal|\!|{D|\!|}|\!|}|\!|varepsilon|\!| |\!||\!| |\!|delta|\!|[|\!|chi|\!| |\!|+|\!| |\!|varepsilon|\!| |\!||\!|{|\!|chi|\!| |\!|,|\!| |\!|varphi|\!| |\!||\!|}|\!| |\!|+|\!| C|\!| |\!|varphi|\!|]|\!||\!|,|\!| |\!|,|\!| |\!|nonumber|\!| |\!||\!||\!| |\!|&|\!|Rightarrow|\!| |\!||\!| |\!|&|\!| |\!|left|\!|.|\!| |\!|left|\!|(|\!|triangle|\!|_|\!|{|\!|chi|\!|}|\!|right|\!|)|\!|^|\!|{|\!|-1|\!|}|\!|
|\!|right|\!|||\!|_|\!|{|\!|Gamma|\!|^|\!|*|\!|}|\!| |\!||\!| |\!|=|\!| |\!||\!| |\!|left|\!|||\!|det|\!||\!|||\!||\!|{|\!|chi|\!|,|\!| |\!|varphi|\!|
|\!||\!|}|\!||\!|||\!|right|\!|||\!|^|\!|{|\!|-1|\!|}|\!| |\!||\!|,|\!| |\!|,|\!| |\!|label|\!|{4|\!|.29|\!|}|\!| |\!|end|\!|{eqnarray|\!|}|\!|
|\!|
with|\!| the|\!| obvious|\!| notation|\!| |\!|$|\!|det|\!| |\!||\!|||\!| |\!||\!|{|\!| |\!|chi|\!|(t|\!|)|\!|,|\!| |\!|varphi|\!|(t|\!|'|\!|)|\!||\!|}|\!| |\!||\!|||\!| |\!|=|\!| |\!|prod|\!|_t|\!| |\!||\!|{|\!| |\!|chi|\!|(t|\!|)|\!|,|\!| |\!|varphi|\!|(t|\!|)|\!||\!|}|\!|$|\!|.|\!| Upon|\!| insertion|\!| of|\!| Eq|\!|.|\!|(|\!|ref|\!|{4|\!|.16|\!|}|\!|)|\!| into|\!| |\!|$Z|\!|_|\!|{|\!|rm|\!| CM|\!|}|\!|$|\!| we|\!| obtain|\!|
|\!|
|\!|begin|\!|{eqnarray|\!|}|\!| |\!|&|\!|&Z|\!|_|\!|{|\!|rm|\!| CM|\!|}|\!| |\!||\!| |\!|=|\!| |\!||\!| |\!|int|\!| |\!|{|\!|mathcal|\!|{D|\!|}|\!|}|\!|{|\!|boldsymbol|\!|{p|\!|}|\!|}|\!|{|\!|mathcal|\!|{D|\!|}|\!|}|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|{|\!|mathcal|\!|{D|\!|}|\!|}|\!|
|\!|bar|\!|{|\!|boldsymbol|\!|{p|\!|}|\!|}|\!|{|\!|mathcal|\!|{D|\!|}|\!|}|\!|bar|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!| |\!||\!| |\!|left|\!|||\!|
|\!|det|\!| |\!||\!|||\!| |\!||\!|{|\!|chi|\!|,|\!| |\!|varphi|\!| |\!| |\!||\!|}|\!||\!|||\!| |\!|right|\!|||\!| |\!|sqrt|\!|{|\!|left|\!|||\!|det|\!| |\!||\!|||\!||\!|{|\!|
|\!|phi|\!|_i|\!|,|\!| |\!|phi|\!|_j|\!| |\!||\!|}|\!||\!|||\!| |\!|right|\!|||\!|}|\!| |\!||\!|;|\!| |\!|delta|\!|[|\!|chi|\!|]|\!| |\!|delta|\!|[|\!|varphi|\!|]|\!| |\!|prod|\!|_|\!|{i|\!|=1|\!|}|\!|^|\!|{2N|\!|}|\!| |\!|delta|\!|[|\!|phi|\!|_i|\!|]|\!| |\!|nonumber|\!| |\!||\!||\!| |\!|&|\!|&|\!|mbox|\!|{|\!|hspace|\!|{4cm|\!|}|\!|}|\!|times|\!| |\!||\!| |\!|exp|\!|left|\!|[|\!| i|\!||\!|!|\!| |\!|int|\!|_|\!|{t|\!|_1|\!|}|\!|^|\!|{t|\!|_2|\!|}|\!| dt|\!| |\!||\!| |\!|[|\!|{|\!|boldsymbol|\!|{p|\!|}|\!|}|\!|dot|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!| |\!|+|\!| |\!|bar|\!|{|\!|boldsymbol|\!|{p|\!|}|\!|}|\!|dot|\!|{|\!|bar|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|}|\!| |\!|-|\!| |\!|bar|\!|{H|\!|}|\!|]|\!| |\!|+|\!| |\!|int|\!|_|\!|{t|\!|_1|\!|}|\!|^|\!|{t|\!|_2|\!|}|\!| dt|\!| |\!||\!| |\!|[|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|{|\!|boldsymbol|\!|{J|\!|}|\!|}|\!| |\!|+|\!| |\!|bar|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|bar|\!|{|\!|boldsymbol|\!|{J|\!|}|\!|}|\!|]|\!| |\!|right|\!|]|\!||\!|,|\!| |\!|,|\!| |\!|label|\!|{4|\!|.17|\!|}|\!| |\!|end|\!|{eqnarray|\!|}|\!|
|\!|
where|\!| the|\!| group|\!| volume|\!| |\!|$G|\!|_V|\!| |\!|=|\!| |\!|int|\!| |\!|{|\!|mathcal|\!|{D|\!|}|\!|}g|\!|$|\!| has|\!| been|\!| factored|\!| out|\!| as|\!| desired|\!|.|\!| The|\!| partition|\!| function|\!| |\!|(|\!|ref|\!|{4|\!|.17|\!|}|\!|)|\!| is|\!| now|\!| clearly|\!| |\!|(locally|\!|)|\!| independent|\!| of|\!| the|\!| choice|\!| of|\!| the|\!| gauge|\!| constraints|\!| |\!|$|\!|chi|\!|$|\!|.|\!| This|\!| is|\!| because|\!| under|\!| the|\!| transformation|\!| |\!|(|\!|ref|\!|{4|\!|.18|\!|}|\!|)|\!| we|\!| have|\!|
|\!|
|\!|begin|\!|{eqnarray|\!|}|\!|
|\!| |\!|det|\!| |\!||\!|||\!||\!|{|\!|chi|\!|,|\!| |\!|varphi|\!| |\!||\!|}|\!||\!|||\!| |\!| |\!||\!| |\!|rightarrow|\!| |\!||\!| |\!|left|\!|(1|\!| |\!|+|\!|
|\!|mbox|\!|{Tr|\!|}|\!|(A|\!|)|\!| |\!|right|\!|)|\!| |\!| |\!|det|\!| |\!||\!|||\!||\!|{|\!| |\!|chi|\!| |\!|+|\!| |\!|delta|\!| |\!|chi|\!|,|\!| |\!|varphi|\!| |\!||\!|}|\!||\!|||\!| |\!||\!|,|\!| |\!|,|\!| |\!|label|\!|{4|\!|.20|\!|}|\!| |\!|end|\!|{eqnarray|\!|}|\!|
|\!|
and|\!| hence|\!| the|\!| partition|\!| function|\!| |\!|$Z|\!|_|\!|{|\!|rm|\!| CM|\!|}|\!|$|\!| as|\!| obtained|\!| by|\!| |\!|(|\!|ref|\!|{4|\!|.17|\!|}|\!|)|\!| takes|\!| the|\!| same|\!| form|\!| as|\!| the|\!| untransformed|\!| one|\!|,|\!| but|\!| with|\!| |\!|$|\!|chi|\!|$|\!| replaced|\!| by|\!| |\!|$|\!|chi|\!| |\!|+|\!| |\!|delta|\!| |\!|chi|\!|$|\!|.|\!| Because|\!| we|\!| deal|\!| with|\!| canonical|\!| transformations|\!| it|\!| is|\!| implicit|\!| in|\!| our|\!| derivation|\!| that|\!| the|\!| action|\!| in|\!| the|\!| new|\!| variables|\!| is|\!| identical|\!|,|\!| to|\!| within|\!| a|\!| boundary|\!| term|\!|,|\!| with|\!| the|\!| original|\!| action|\!|.|\!| In|\!| path|\!| integrals|\!| this|\!| might|\!| be|\!| invalidated|\!| by|\!| the|\!| path|\!| roughness|\!| and|\!| related|\!| ordering|\!| problems|\!|footnote|\!|{In|\!| the|\!| literature|\!| this|\!| phenomenon|\!| frequently|\!| goes|\!| under|\!| the|\!| name|\!| of|\!| the|\!| Edwards|\!|-Gulyaev|\!| effect|\!|~|\!|cite|\!|{EG|\!|}|\!|.|\!|}|\!|.|\!| For|\!| simplicity|\!|'s|\!| sake|\!| we|\!| shall|\!| further|\!| assume|\!| that|\!| the|\!| latter|\!| are|\!| absent|\!| or|\!| harmless|\!|.|\!| This|\!| happens|\!|,|\!| for|\!| instance|\!|,|\!| when|\!| canonical|\!| transformations|\!| are|\!| linear|\!|.|\!| In|\!| such|\!| cases|\!| an|\!| infinitesimal|\!| change|\!| in|\!| |\!|$|\!|chi|\!|$|\!| does|\!| not|\!| alter|\!| the|\!| physical|\!| content|\!| of|\!| the|\!| theory|\!| present|\!| in|\!| |\!|$Z|\!|_|\!|{|\!|rm|\!| CM|\!|}|\!|$|\!|.|\!| This|\!| conclusion|\!| may|\!| generally|\!| not|\!| be|\!| true|\!| globally|\!| throughout|\!| phase|\!| space|\!|.|\!| Global|\!| gauge|\!| invariance|\!|,|\!| however|\!|,|\!| is|\!| mandatory|\!| in|\!| our|\!| case|\!| since|\!| we|\!| need|\!| a|\!| global|\!| equivalence|\!| between|\!| the|\!| partition|\!| functions|\!| |\!|$Z|\!|_|\!|{|\!|rm|\!| CM|\!|}|\!|$|\!| and|\!| |\!|$Z|\!|_|\!|{|\!|rm|\!| QM|\!|}|\!|$|\!| and|\!| not|\!| mere|\!| perturbative|\!| correspondence|\!|.|\!| Thus|\!| the|\!| potentiality|\!| of|\!| Gribov|\!|'s|\!| copies|\!| must|\!| be|\!| checked|\!| in|\!| every|\!| individual|\!| problem|\!| separately|\!|.|\!|
|\!|
|\!|vspace|\!|{3mm|\!|}|\!|
|\!|
In|\!| passing|\!| we|\!| may|\!| notice|\!| that|\!| if|\!| we|\!| arrange|\!| the|\!| constraints|\!| in|\!| one|\!| set|\!| |\!|$|\!||\!|{|\!|eta|\!|_a|\!| |\!||\!|}|\!| |\!|=|\!| |\!||\!|{|\!|chi|\!|,|\!| |\!|varphi|\!|,|\!| |\!|phi|\!|_i|\!| |\!||\!|}|\!|$|\!| we|\!| can|\!| write|\!| |\!|(|\!|ref|\!|{4|\!|.17|\!|}|\!|)|\!| as|\!|
|\!|
|\!|begin|\!|{eqnarray|\!|}|\!| |\!|&|\!|&Z|\!|_|\!|{|\!|rm|\!| CM|\!|}|\!| |\!||\!| |\!|=|\!| |\!||\!| |\!|int|\!| |\!|{|\!|mathcal|\!|{D|\!|}|\!|}|\!|{|\!|boldsymbol|\!|{p|\!|}|\!|}|\!|{|\!|mathcal|\!|{D|\!|}|\!|}|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|{|\!|mathcal|\!|{D|\!|}|\!|}|\!| |\!|bar|\!|{|\!|boldsymbol|\!|{p|\!|}|\!|}|\!|{|\!|mathcal|\!|{D|\!|}|\!|}|\!|bar|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!| |\!||\!|
|\!|sqrt|\!|{|\!|left|\!|||\!|det|\!| |\!||\!|||\!| |\!||\!|{|\!| |\!|eta|\!|_a|\!|,|\!| |\!|eta|\!|_b|\!| |\!||\!|}|\!| |\!||\!|||\!| |\!|right|\!|||\!|}|\!| |\!||\!|;|\!| |\!|prod|\!|_|\!|{a|\!|=1|\!|}|\!|^|\!|{2N|\!|+2|\!|}|\!| |\!|delta|\!|[|\!|eta|\!|_a|\!|]|\!| |\!|nonumber|\!| |\!||\!||\!| |\!|&|\!|&|\!|mbox|\!|{|\!|hspace|\!|{4cm|\!|}|\!|}|\!|times|\!| |\!||\!| |\!|exp|\!|left|\!|[|\!| i|\!||\!|!|\!| |\!|int|\!|_|\!|{t|\!|_1|\!|}|\!|^|\!|{t|\!|_2|\!|}|\!| dt|\!| |\!||\!| |\!|[|\!|{|\!|boldsymbol|\!|{p|\!|}|\!|}|\!|dot|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!| |\!|+|\!| |\!|bar|\!|{|\!|boldsymbol|\!|{p|\!|}|\!|}|\!|dot|\!|{|\!|bar|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|}|\!| |\!|-|\!| H|\!|]|\!| |\!|+|\!| |\!|int|\!|_|\!|{t|\!|_1|\!|}|\!|^|\!|{t|\!|_2|\!|}|\!| dt|\!| |\!||\!| |\!|[|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|{|\!|boldsymbol|\!|{J|\!|}|\!|}|\!| |\!|+|\!| |\!|bar|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|bar|\!|{|\!|boldsymbol|\!|{J|\!|}|\!|}|\!|]|\!| |\!|right|\!|]|\!||\!|,|\!| |\!|.|\!| |\!|label|\!|{4|\!|.21|\!|}|\!| |\!|end|\!|{eqnarray|\!|}|\!|
|\!|
By|\!| comparison|\!| with|\!| |\!|(|\!|ref|\!|{frad|\!|}|\!|)|\!| we|\!| retrieve|\!| a|\!| well|\!| known|\!| result|\!|~|\!|cite|\!|{Sunder|\!|,GT|\!|}|\!|,|\!| namely|\!|,|\!| that|\!| the|\!| set|\!| |\!|$|\!||\!|{|\!| |\!|eta|\!|_a|\!| |\!||\!|}|\!|$|\!| of|\!| |\!|$2N|\!|+2|\!|$|\!| constraints|\!| can|\!| be|\!| viewed|\!| as|\!| a|\!| set|\!| of|\!| second|\!|-class|\!| constraints|\!|.|\!| Thus|\!|,|\!| by|\!| fixing|\!| a|\!| gauge|\!| we|\!| have|\!| effectively|\!| converted|\!| the|\!| original|\!| system|\!| of|\!| |\!|$2N|\!|$|\!| second|\!|-class|\!| and|\!| |\!|{|\!|em|\!| one|\!|}|\!| first|\!|-class|\!| constraints|\!| into|\!| |\!|$2N|\!|+2|\!|$|\!| second|\!|-class|\!| constraints|\!|.|\!|
|\!|
|\!|vspace|\!|{3mm|\!|}|\!|
|\!|
In|\!| view|\!| of|\!| |\!|(|\!|ref|\!|{2|\!|.10|\!|}|\!|)|\!| and|\!| |\!|(|\!|ref|\!|{4|\!|.27|\!|}|\!|)|\!|,|\!| we|\!| can|\!| perform|\!| a|\!| canonical|\!| transformation|\!| in|\!| the|\!| full|\!| phase|\!| space|\!| in|\!| such|\!| a|\!| way|\!| that|\!| the|\!| new|\!| variables|\!| are|\!|:|\!| |\!|$P|\!|_1|\!| |\!|=|\!| |\!|chi|\!|$|\!|,|\!| |\!|$Q|\!|_|\!|{1|\!|+i|\!|}|\!| |\!|=|\!| |\!|phi|\!|_|\!|{2i|\!|}|\!|$|\!|,|\!| |\!|$P|\!|_|\!|{1|\!|+i|\!|}|\!| |\!|=|\!| |\!|phi|\!|_|\!|{2i|\!|-1|\!|}|\!|$|\!|;|\!| |\!|$i|\!| |\!|=1|\!|,|\!| |\!|ldots|\!|,|\!| N|\!|$|\!|.|\!| After|\!| a|\!| trivial|\!| integration|\!| over|\!| |\!|$P|\!|_a|\!|$|\!| and|\!| |\!|$Q|\!|_|\!|{1|\!|+i|\!|}|\!|$|\!| we|\!| find|\!| that|\!|
|\!|
|\!|begin|\!|{eqnarray|\!|}|\!| Z|\!|_|\!|{|\!|rm|\!| CM|\!|}|\!| |\!||\!| |\!|=|\!| |\!||\!| |\!|int|\!| |\!|{|\!|mathcal|\!|{D|\!|}|\!|}|\!|bar|\!|{|\!|boldsymbol|\!|{P|\!|}|\!|}|\!|{|\!|mathcal|\!|{D|\!|}|\!|}|\!|bar|\!|{|\!|boldsymbol|\!|{Q|\!|}|\!|}|\!|{|\!|mathcal|\!|{D|\!|}|\!|}|\!|{Q|\!|}|\!|_1|\!|
|\!||\!| |\!|left|\!|(|\!|delta|\!|[|\!|varphi|\!|]|\!| |\!|left|\!|||\!|det|\!| |\!|left|\!|||\!||\!|!|\!|left|\!|||\!|frac|\!|{|\!|delta|\!|
|\!|varphi|\!|}|\!|{|\!|delta|\!| Q|\!|_1|\!|}|\!|right|\!|||\!||\!|!|\!|right|\!|||\!| |\!|right|\!|||\!|right|\!|)|\!| |\!||\!| |\!|exp|\!|left|\!|[i|\!| |\!||\!|!|\!|int|\!|_|\!|{t|\!|_1|\!|}|\!|^|\!|{t|\!|_2|\!|}|\!| dt|\!| |\!||\!| |\!|left|\!|[|\!| |\!|bar|\!|{|\!|boldsymbol|\!|{P|\!|}|\!|}|\!| |\!|dot|\!|{|\!|bar|\!|{|\!|boldsymbol|\!|{Q|\!|}|\!|}|\!|}|\!| |\!|-|\!| K|\!|right|\!|]|\!|
|\!| |\!|+|\!| |\!|int|\!|_|\!|{t|\!|_1|\!|}|\!|^|\!|{t|\!|_2|\!|}|\!| dt|\!||\!| |\!|bar|\!|{|\!|boldsymbol|\!|{Q|\!|}|\!|}|\!| |\!|{|\!|boldsymbol|\!|{j|\!|}|\!|}|\!| |\!|right|\!|]|\!||\!|,|\!| |\!|,|\!| |\!|end|\!|{eqnarray|\!|}|\!|
|\!|
where|\!| |\!|$|\!|bar|\!|{P|\!|}|\!|_a|\!|$|\!| and|\!| |\!|$|\!|bar|\!|{Q|\!|}|\!|_a|\!|$|\!| are|\!| the|\!| remaining|\!| canonical|\!| variables|\!| spanning|\!| the|\!| |\!|$|\!|(2N|\!|-2|\!|)|\!|$|\!|-dimensional|\!| phase|\!| space|\!|.|\!|
|\!|
To|\!| within|\!| a|\!| time|\!| derivative|\!| term|\!| the|\!| new|\!| Hamiltonian|\!| is|\!| done|\!| by|\!| the|\!| prescription|\!| |\!|$K|\!|(|\!|bar|\!|{|\!|boldsymbol|\!|{P|\!|}|\!|}|\!|,|\!| |\!|bar|\!|{|\!|boldsymbol|\!|{Q|\!|}|\!|}|\!|,|\!| Q|\!|_1|\!|)|\!| |\!|=|\!| H|\!|(|\!|bar|\!|{|\!|boldsymbol|\!|{P|\!|}|\!|}|\!|,|\!| |\!|bar|\!|{|\!|boldsymbol|\!|{Q|\!|}|\!|}|\!|,|\!| P|\!|_1|\!| |\!|=|\!| 0|\!|,|\!| Q|\!|_1|\!|,|\!| Q|\!|_|\!|{1|\!|+i|\!|}|\!| |\!|=0|\!|,|\!| P|\!|_|\!|{1|\!|+i|\!|}|\!| |\!|=0|\!|)|\!|$|\!|.|\!| The|\!| sources|\!| |\!|$|\!|{|\!|boldsymbol|\!|{j|\!|}|\!|}|\!| |\!|$|\!| are|\!| correspondingly|\!| transformed|\!| sources|\!| |\!|$|\!|{|\!|boldsymbol|\!|{J|\!|}|\!|}|\!|$|\!| and|\!| |\!|$|\!|bar|\!|{|\!|{|\!|boldsymbol|\!|{J|\!|}|\!|}|\!| |\!|}|\!|$|\!|.|\!| Utilizing|\!| the|\!| identity|\!|
|\!|
|\!|begin|\!|{eqnarray|\!|}|\!|
|\!|delta|\!|[|\!|varphi|\!|]|\!| |\!|left|\!|||\!|det|\!| |\!|left|\!|||\!||\!|!|\!|left|\!|||\!|frac|\!|{|\!|delta|\!|
|\!|varphi|\!|}|\!|{|\!|delta|\!| Q|\!|_1|\!|}|\!|right|\!|||\!| |\!||\!|!|\!| |\!|right|\!|||\!| |\!|right|\!|||\!| |\!||\!| |\!|=|\!| |\!||\!| |\!|delta|\!|[Q|\!|_1|\!| |\!|-|\!| Q|\!|_1|\!|^|\!|*|\!|(|\!|bar|\!|{|\!|boldsymbol|\!|{P|\!|}|\!|}|\!|,|\!| |\!|bar|\!|{|\!|boldsymbol|\!|{Q|\!|}|\!|}|\!|)|\!| |\!|]|\!||\!|,|\!| |\!|,|\!| |\!|label|\!|{4|\!|.23|\!|}|\!| |\!|end|\!|{eqnarray|\!|}|\!|
|\!|
we|\!| can|\!| finally|\!| write|\!|
|\!|
|\!|begin|\!|{eqnarray|\!|}|\!| Z|\!|_|\!|{|\!|rm|\!| CM|\!|}|\!| |\!||\!| |\!|=|\!| |\!||\!| |\!|int|\!| |\!|{|\!|mathcal|\!|{D|\!|}|\!|}|\!|bar|\!|{|\!|boldsymbol|\!|{P|\!|}|\!|}|\!|{|\!|mathcal|\!|{D|\!|}|\!|}|\!|bar|\!|{|\!|boldsymbol|\!|{Q|\!|}|\!|}|\!| |\!||\!| |\!|exp|\!|left|\!|[i|\!| |\!||\!|!|\!|int|\!|_|\!|{t|\!|_1|\!|}|\!|^|\!|{t|\!|_2|\!|}|\!| dt|\!| |\!||\!| |\!|left|\!|[|\!| |\!|bar|\!|{|\!|boldsymbol|\!|{P|\!|}|\!|}|\!| |\!|dot|\!|{|\!|bar|\!|{|\!|boldsymbol|\!|{Q|\!|}|\!|}|\!|}|\!| |\!|-|\!| K|\!|^|\!|*|\!|right|\!|]|\!|
|\!| |\!|+|\!| |\!|int|\!|_|\!|{t|\!|_1|\!|}|\!|^|\!|{t|\!|_2|\!|}|\!| dt|\!||\!| |\!|bar|\!|{|\!|boldsymbol|\!|{Q|\!|}|\!|}|\!| |\!|{|\!|boldsymbol|\!|{j|\!|}|\!|}|\!| |\!|right|\!|]|\!||\!|,|\!| |\!|.|\!| |\!|label|\!|{4|\!|.22|\!|}|\!| |\!|end|\!|{eqnarray|\!|}|\!|
|\!|
Here|\!| |\!|$K|\!|^|\!|*|\!|(|\!|bar|\!|{|\!|boldsymbol|\!|{P|\!|}|\!|}|\!|,|\!| |\!|bar|\!|{|\!|boldsymbol|\!|{Q|\!|}|\!|}|\!|)|\!| |\!|=|\!| K|\!|(|\!|bar|\!|{|\!|boldsymbol|\!|{P|\!|}|\!|}|\!|,|\!| |\!|bar|\!|{|\!|boldsymbol|\!|{Q|\!|}|\!|}|\!|,|\!| Q|\!|_1|\!| |\!|=|\!| Q|\!|_1|\!|^|\!|*|\!|(|\!|bar|\!|{|\!|boldsymbol|\!|{P|\!|}|\!|}|\!|,|\!|bar|\!|{|\!|boldsymbol|\!|{Q|\!|}|\!|}|\!|)|\!|)|\!|$|\!|.|\!| In|\!| view|\!| of|\!| |\!|(|\!|ref|\!|{D3|\!|}|\!|)|\!| we|\!| can|\!| alternatively|\!| write|\!| |\!|$Z|\!|_|\!|{CM|\!|}|\!|$|\!| as|\!|
|\!|
|\!|begin|\!|{eqnarray|\!|}|\!| Z|\!|_|\!|{|\!|rm|\!| CM|\!|}|\!| |\!||\!| |\!|=|\!| |\!||\!| |\!|int|\!| |\!|{|\!|mathcal|\!|{D|\!|}|\!|}|\!|bar|\!|{|\!|boldsymbol|\!|{P|\!|}|\!|}|\!|{|\!|mathcal|\!|{D|\!|}|\!|}|\!|bar|\!|{|\!|boldsymbol|\!|{Q|\!|}|\!|}|\!| |\!||\!| |\!|exp|\!|left|\!|[i|\!| |\!||\!|!|\!|int|\!|_|\!|{t|\!|_1|\!|}|\!|^|\!|{t|\!|_2|\!|}|\!| dt|\!| |\!||\!| |\!|left|\!|[|\!| |\!|bar|\!|{|\!|boldsymbol|\!|{P|\!|}|\!|}|\!| |\!|dot|\!|{|\!|bar|\!|{|\!|boldsymbol|\!|{Q|\!|}|\!|}|\!|}|\!| |\!|-|\!| H|\!|_|\!|+|\!|^|\!|*|\!|right|\!|]|\!|
|\!| |\!|+|\!| |\!|int|\!|_|\!|{t|\!|_1|\!|}|\!|^|\!|{t|\!|_2|\!|}|\!| dt|\!||\!| |\!|bar|\!|{|\!|boldsymbol|\!|{Q|\!|}|\!|}|\!| |\!|{|\!|boldsymbol|\!|{j|\!|}|\!|}|\!| |\!|right|\!|]|\!||\!|,|\!| |\!|,|\!| |\!|label|\!|{4|\!|.30|\!|}|\!| |\!|end|\!|{eqnarray|\!|}|\!|
|\!|
where|\!| |\!|$H|\!|_|\!|+|\!|^|\!|*|\!| |\!|=|\!| H|\!|_|\!|+|\!|(|\!|bar|\!|{|\!|boldsymbol|\!|{P|\!|}|\!|}|\!|,|\!| |\!|bar|\!|{|\!|boldsymbol|\!|{Q|\!|}|\!|}|\!|,|\!| Q|\!|_1|\!| |\!|=|\!| Q|\!|_1|\!|^|\!|*|\!|(|\!|bar|\!|{|\!|boldsymbol|\!|{P|\!|}|\!|}|\!|,|\!|bar|\!|{|\!|boldsymbol|\!|{Q|\!|}|\!|}|\!|)|\!|,|\!| P|\!|_a|\!| |\!|=|\!| 0|\!|,|\!| Q|\!|_|\!|{1|\!|+i|\!|}|\!| |\!|=|\!| 0|\!|)|\!|$|\!|.|\!| In|\!| passing|\!| we|\!| may|\!| notice|\!| that|\!| |\!|$|\!|bar|\!|{P|\!|}|\!|_a|\!|$|\!| and|\!| |\!|$|\!|bar|\!|{Q|\!|}|\!|_a|\!|$|\!| are|\!| true|\!| canonical|\!| variables|\!| on|\!| the|\!| submanifold|\!| |\!|$|\!|Gamma|\!|^|\!|*|\!|$|\!| of|\!| the|\!| initial|\!| conditions|\!| for|\!| Eq|\!|.|\!|(|\!|ref|\!|{4|\!|.24|\!|}|\!|)|\!|.|\!| Indeed|\!|,|\!| in|\!| terms|\!| of|\!| a|\!| non|\!|-canonical|\!| system|\!| of|\!| variables|\!| |\!|$|\!||\!|{|\!|zeta|\!|_i|\!| |\!||\!|}|\!| |\!|=|\!| |\!||\!|{|\!|varphi|\!|;|\!| |\!|chi|\!|;|\!| |\!|phi|\!|_i|\!|;|\!| |\!|bar|\!|{|\!|boldsymbol|\!|{Q|\!|}|\!|}|\!|;|\!| |\!|bar|\!|{|\!|boldsymbol|\!|{P|\!|}|\!|}|\!||\!|}|\!|$|\!| the|\!| Poisson|\!| bracket|\!| of|\!| any|\!| two|\!| |\!|{|\!|em|\!| observable|\!|}|\!| quantities|\!| |\!|(say|\!| |\!|$f|\!|$|\!| and|\!| |\!|$g|\!|$|\!|)|\!| on|\!| the|\!| constraint|\!| manifold|\!| |\!|$|\!|mathcal|\!|{M|\!|}|\!|$|\!| is|\!|
|\!|
|\!|begin|\!|{eqnarray|\!|}|\!|
|\!|left|\!|.|\!| |\!||\!|{f|\!|,|\!| g|\!| |\!||\!|}|\!| |\!|right|\!|||\!|_|\!|{|\!|mathcal|\!|{M|\!|}|\!|}|\!| |\!||\!| |\!|=|\!| |\!| |\!||\!| |\!|left|\!|.|\!| |\!|left|\!|[|\!| |\!|sum|\!|_|\!|{a|\!|,b|\!|}|\!||\!| |\!||\!|{|\!|zeta|\!|_a|\!|,|\!| |\!|zeta|\!|_b|\!| |\!||\!|}|\!| |\!||\!| |\!|frac|\!|{|\!|partial|\!| f|\!|}|\!|{|\!| |\!|partial|\!| |\!|zeta|\!|_a|\!|}|\!|frac|\!|{|\!|partial|\!| g|\!|}|\!|{|\!| |\!||\!|
|\!|partial|\!| |\!|zeta|\!|_b|\!|}|\!|right|\!|]|\!| |\!|right|\!|||\!|_|\!|{|\!|mathcal|\!|{M|\!|}|\!|}|\!| |\!||\!| |\!|=|\!| |\!||\!| |\!|sum|\!|_|\!|{i|\!|,j|\!|}|\!| |\!||\!| |\!||\!|{|\!| |\!|bar|\!|{P|\!|}|\!|_i|\!|,|\!| |\!|bar|\!|{Q|\!|}|\!|_j|\!| |\!||\!|}|\!| |\!||\!| |\!|frac|\!|{|\!|partial|\!| f|\!|^|\!|*|\!|}|\!|{|\!|partial|\!| |\!|bar|\!|{P|\!|}|\!|_i|\!|}|\!| |\!|frac|\!|{|\!|partial|\!| g|\!|^|\!|*|\!|}|\!|{|\!| |\!|partial|\!| |\!|bar|\!|{Q|\!|}|\!|_j|\!|}|\!| |\!||\!| |\!|=|\!| |\!||\!| |\!|sum|\!|_|\!|{i|\!|,j|\!|}|\!| |\!||\!| |\!|Omega|\!|_|\!|{ij|\!|}|\!| |\!||\!| |\!|frac|\!|{|\!|partial|\!| f|\!|^|\!|*|\!|}|\!|{|\!|partial|\!| |\!|bar|\!|{|\!|mathcal|\!|{Q|\!|}|\!|}|\!|_i|\!|}|\!| |\!|frac|\!|{|\!|partial|\!| g|\!|^|\!|*|\!|}|\!|{|\!| |\!|partial|\!| |\!|bar|\!|{|\!|mathcal|\!|{Q|\!|}|\!|}|\!|_j|\!|}|\!||\!|,|\!| |\!|,|\!| |\!|label|\!|{4|\!|.91|\!|}|\!| |\!|end|\!|{eqnarray|\!|}|\!|
|\!|
with|\!| |\!|$|\!| |\!||\!|{|\!| |\!|bar|\!|{|\!|mathcal|\!|{Q|\!|}|\!|}|\!|_j|\!| |\!||\!|}|\!| |\!||\!| |\!|=|\!| |\!||\!| |\!||\!|{|\!| |\!|bar|\!|{|\!|boldsymbol|\!|{Q|\!|}|\!|}|\!|;|\!| |\!|bar|\!|{|\!|boldsymbol|\!|{P|\!|}|\!|}|\!| |\!| |\!||\!|}|\!|$|\!| and|\!| with|\!|
|\!|
|\!|begin|\!|{eqnarray|\!|*|\!|}|\!| f|\!|^|\!|*|\!|(|\!|bar|\!|{|\!|boldsymbol|\!|{Q|\!|}|\!|}|\!|,|\!| |\!|bar|\!|{|\!|boldsymbol|\!|{P|\!|}|\!|}|\!|)|\!| |\!||\!| |\!|&|\!|=|\!|&|\!| |\!||\!| f|\!|(|\!|varphi|\!| |\!|=|\!| 0|\!|,|\!| |\!|chi|\!| |\!|=|\!| 0|\!|,|\!| |\!|phi|\!|_i|\!| |\!|=|\!| 0|\!|,|\!| |\!|bar|\!|{|\!|boldsymbol|\!|{Q|\!|}|\!|}|\!|,|\!| |\!|bar|\!|{|\!|boldsymbol|\!|{P|\!|}|\!|}|\!|)|\!||\!|,|\!| |\!|,|\!| |\!|nonumber|\!| |\!||\!||\!| g|\!|^|\!|*|\!|(|\!|bar|\!|{|\!|boldsymbol|\!|{Q|\!|}|\!|}|\!|,|\!| |\!|bar|\!|{|\!|boldsymbol|\!|{P|\!|}|\!|}|\!|)|\!| |\!||\!| |\!|&|\!|=|\!|&|\!| |\!||\!| g|\!|(|\!|varphi|\!| |\!|=|\!| 0|\!|,|\!| |\!|chi|\!| |\!|=|\!| 0|\!|,|\!| |\!|phi|\!|_i|\!| |\!|=|\!| 0|\!|,|\!| |\!|bar|\!|{|\!|boldsymbol|\!|{Q|\!|}|\!|}|\!|,|\!| |\!|bar|\!|{|\!|boldsymbol|\!|{P|\!|}|\!|}|\!|)|\!||\!|,|\!| |\!|,|\!| |\!|end|\!|{eqnarray|\!|*|\!|}|\!|
|\!|
representing|\!| the|\!| physical|\!| quantities|\!| on|\!| |\!|$|\!|{|\!|mathcal|\!|{M|\!|}|\!|}|\!|$|\!|.|\!| The|\!| latter|\!| depend|\!| only|\!| on|\!| the|\!| canonical|\!| variables|\!| |\!|$|\!|bar|\!|{|\!|boldsymbol|\!|{Q|\!|}|\!|}|\!|$|\!| and|\!| |\!|$|\!|bar|\!|{|\!|boldsymbol|\!|{P|\!|}|\!|}|\!|$|\!| which|\!| are|\!| the|\!| independent|\!| variables|\!| on|\!| |\!|$|\!|Gamma|\!|^|\!|*|\!|$|\!|.|\!| In|\!| deriving|\!| |\!|(|\!|ref|\!|{4|\!|.91|\!|}|\!|)|\!| we|\!| have|\!| used|\!| the|\!| fact|\!| that|\!| various|\!| terms|\!| are|\!| vanishing|\!| on|\!| account|\!| of|\!| Eqs|\!|.|\!|(|\!|ref|\!|{4|\!|.24|\!|}|\!|)|\!| and|\!| |\!|(|\!|ref|\!|{4|\!|.27|\!|}|\!|)|\!|.|\!| So|\!|,|\!| for|\!| instance|\!|,|\!| |\!|$|\!|[|\!||\!|{|\!|varphi|\!|,|\!| |\!|zeta|\!|_i|\!||\!|}|\!| |\!||\!|
|\!|partial|\!| f|\!|/|\!|partial|\!| |\!|zeta|\!|_i|\!| |\!|]|\!|||\!|_|\!|{|\!|mathcal|\!|{M|\!|}|\!|}|\!| |\!|=|\!| 0|\!|$|\!|,|\!| |\!|$|\!||\!|{|\!|varphi|\!|_i|\!|,|\!| |\!|bar|\!|{P|\!|}|\!|_j|\!| |\!||\!|}|\!| |\!|=|\!| 0|\!|$|\!|,|\!| |\!| |\!|$|\!||\!|{|\!|varphi|\!|_i|\!|,|\!| |\!|bar|\!|{Q|\!|}|\!|_j|\!| |\!||\!|}|\!| |\!|=|\!| 0|\!|$|\!|,|\!| |\!|$|\!|[|\!||\!|{|\!|chi|\!|,|\!| |\!|zeta|\!|_i|\!||\!|}|\!| |\!||\!|
|\!|partial|\!| f|\!|/|\!|partial|\!| |\!|chi|\!| |\!|]|\!|||\!|_|\!|{|\!|mathcal|\!|{M|\!|}|\!|}|\!| |\!|=|\!| 0|\!|$|\!|,|\!| etc|\!|.|\!| The|\!| matrix|\!| |\!|$|\!|Omega|\!|_|\!|{ij|\!|}|\!|$|\!| stands|\!| for|\!| the|\!| |\!|$|\!|(2N|\!|-2|\!|)|\!|times|\!|(2N|\!| |\!|-2|\!|)|\!|$|\!| symplectic|\!| matrix|\!|.|\!|
|\!|
|\!|vspace|\!|{3mm|\!|}|\!|
|\!|
|\!|$Z|\!|_|\!|{|\!|rm|\!| CM|\!|}|\!|$|\!| as|\!| defined|\!| by|\!| |\!|(|\!|ref|\!|{4|\!|.22|\!|}|\!|)|\!|-|\!|(|\!|ref|\!|{4|\!|.30|\!|}|\!|)|\!| does|\!| not|\!| generally|\!| represent|\!| a|\!| |\!|(classical|\!|)|\!| deterministic|\!| system|\!|.|\!| This|\!| is|\!| because|\!| the|\!| constraint|\!| |\!|$|\!|varphi|\!| |\!|=|\!| 0|\!|$|\!| explicitly|\!| breaks|\!| the|\!| BRST|\!| invariance|\!| of|\!| |\!|$Z|\!|_|\!|{|\!|rm|\!| CM|\!|}|\!|$|\!| which|\!| |\!|(as|\!| illustrated|\!| in|\!| Section|\!|~III|\!|)|\!| is|\!| key|\!| in|\!| preserving|\!| the|\!| classical|\!| nature|\!| of|\!| the|\!| partition|\!| function|\!|.|\!| Indeed|\!|,|\!| using|\!| the|\!| relations|\!| |\!|$|\!||\!|{|\!| |\!|chi|\!|,|\!| |\!|bar|\!|{p|\!|}|\!|_a|\!| |\!||\!|}|\!| |\!|=|\!| |\!||\!|{|\!| |\!|chi|\!|,|\!| p|\!|_a|\!| |\!|-|\!| |\!|bar|\!|{q|\!|}|\!|_a|\!| |\!||\!|}|\!| |\!|=|\!| 0|\!|$|\!| we|\!| immediately|\!| obtain|\!|
|\!|
|\!|begin|\!|{eqnarray|\!|}|\!| |\!||\!|{|\!| |\!|chi|\!|,|\!| |\!|varphi|\!| |\!||\!|}|\!| |\!||\!| |\!|=|\!| |\!||\!| |\!|sum|\!|_a|\!| |\!|left|\!||\!|{|\!| |\!|frac|\!|{|\!|partial|\!| |\!|chi|\!|}|\!|{|\!|partial|\!| q|\!|_a|\!|}|\!| |\!|left|\!|(|\!| |\!|frac|\!|{|\!|partial|\!| |\!|varphi|\!|}|\!|{|\!|partial|\!| p|\!|_a|\!|}|\!| |\!|+|\!| |\!|frac|\!|{|\!|partial|\!|varphi|\!|}|\!|{|\!|partial|\!|bar|\!|{q|\!|}|\!|_a|\!|}|\!| |\!|right|\!|)|\!| |\!|-|\!| |\!|frac|\!|{|\!|partial|\!| |\!|chi|\!|}|\!|{|\!|partial|\!| p|\!|_a|\!|}|\!| |\!|frac|\!|{|\!|partial|\!| |\!|varphi|\!|}|\!|{|\!|partial|\!| q|\!|_a|\!|}|\!| |\!|right|\!||\!|}|\!||\!|,|\!| |\!|,|\!| |\!|end|\!|{eqnarray|\!|}|\!|
|\!|
which|\!| implies|\!| that|\!|
|\!|
|\!|begin|\!|{eqnarray|\!|}|\!|
|\!|left|\!|.|\!| |\!||\!|{|\!| |\!|chi|\!|,|\!| |\!|varphi|\!| |\!||\!|}|\!| |\!|right|\!|||\!|_|\!|{|\!|{|\!|mathcal|\!|{M|\!|}|\!|}|\!|,|\!| |\!|bar|\!|{q|\!|}|\!|_a|\!| |\!| |\!|=|\!| |\!|lambda|\!|_a|\!|}|\!| |\!||\!| |\!|=|\!| |\!||\!| |\!|sum|\!|_a|\!| |\!|left|\!||\!|{|\!| |\!|frac|\!|{|\!|partial|\!| |\!|chi|\!|^|\!|*|\!|}|\!|{|\!|partial|\!| q|\!|_a|\!|}|\!| |\!|frac|\!|{|\!|partial|\!| |\!|varphi|\!|^|\!|*|\!|}|\!|{|\!|partial|\!| |\!|lambda|\!|_a|\!|}|\!| |\!|-|\!| |\!|frac|\!|{|\!|partial|\!| |\!|chi|\!|^|\!|*|\!|}|\!|{|\!|partial|\!| |\!|lambda|\!|_a|\!|}|\!| |\!|frac|\!|{|\!|partial|\!| |\!|varphi|\!|^|\!|*|\!|}|\!|{|\!|partial|\!| q|\!|_a|\!|}|\!|right|\!||\!|}|\!| |\!||\!| |\!|equiv|\!| |\!||\!| |\!||\!|{|\!|chi|\!|^|\!|*|\!|,|\!| |\!|varphi|\!|^|\!|*|\!| |\!||\!|}|\!||\!|,|\!| |\!|.|\!| |\!|end|\!|{eqnarray|\!|}|\!|
|\!|
Here|\!| the|\!| notations|\!| |\!|$|\!|chi|\!|^|\!|*|\!|(|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|,|\!| |\!|{|\!|boldsymbol|\!|{|\!|lambda|\!|}|\!|}|\!|)|\!| |\!|=|\!| |\!|chi|\!|(|\!|boldsymbol|\!|{q|\!|}|\!|,|\!|boldsymbol|\!|{p|\!|}|\!| |\!|=|\!|boldsymbol|\!|{|\!|lambda|\!|}|\!|,|\!| |\!|bar|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!| |\!|=|\!| |\!|boldsymbol|\!|{|\!|lambda|\!|}|\!|,|\!| |\!|bar|\!|{|\!|boldsymbol|\!|{p|\!|}|\!|}|\!| |\!|=|\!| 0|\!|)|\!|$|\!| and|\!| |\!|$|\!|varphi|\!|^|\!|*|\!|(|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|,|\!| |\!|{|\!|boldsymbol|\!|{|\!|lambda|\!|}|\!|}|\!|)|\!| |\!|=|\!| |\!|varphi|\!|(|\!|boldsymbol|\!|{q|\!|}|\!|,|\!|boldsymbol|\!|{|\!|lambda|\!|}|\!|,|\!| |\!|boldsymbol|\!|{|\!|lambda|\!|}|\!|,|\!| 0|\!|)|\!|$|\!| were|\!| used|\!|.|\!| We|\!| also|\!| took|\!| advantage|\!| of|\!| the|\!| fact|\!| that|\!| |\!|$|\!|bar|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!| |\!|=|\!| |\!|{|\!|boldsymbol|\!|{|\!|lambda|\!|}|\!|}|\!|$|\!| as|\!| indicated|\!| in|\!| Section|\!| III|\!|.|\!| So|\!| the|\!| generating|\!| functional|\!| |\!|(|\!|ref|\!|{4|\!|.22|\!|}|\!|)|\!| |\!|(or|\!| |\!|(|\!|ref|\!|{4|\!|.30|\!|}|\!|)|\!|)|\!| can|\!| be|\!| rewritten|\!| as|\!|
|\!|
|\!|begin|\!|{eqnarray|\!|}|\!| Z|\!|_|\!|{|\!|rm|\!| CM|\!|}|\!|[|\!|{|\!|boldsymbol|\!|{J|\!|}|\!|}|\!| |\!|=|\!| 0|\!|]|\!| |\!||\!| |\!|=|\!| |\!| |\!||\!| |\!|int|\!| |\!|{|\!|mathcal|\!|{D|\!|}|\!|}|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|{|\!|mathcal|\!|{D|\!|}|\!|}|\!|{|\!|boldsymbol|\!|{|\!|lambda|\!|}|\!|}|\!| |\!|{|\!|{|\!|mathcal|\!|{D|\!|}|\!|}|\!|bar|\!|{|\!|boldsymbol|\!|{c|\!|}|\!|}|\!|}|\!| |\!|{|\!|mathcal|\!|{D|\!|}|\!|}|\!|{|\!|boldsymbol|\!|{c|\!|}|\!|}|\!| |\!||\!| |\!|exp|\!|left|\!|[|\!| i|\!| |\!|mathcal|\!|{S|\!|}|\!| |\!| |\!| |\!| |\!| |\!|right|\!|]|\!| |\!||\!| |\!|delta|\!|[|\!|varphi|\!|^|\!|*|\!|]|\!|
|\!|delta|\!|[|\!|chi|\!|^|\!|*|\!|]|\!| |\!|left|\!|||\!| |\!|det|\!| |\!||\!|||\!| |\!||\!|{|\!|chi|\!|^|\!|*|\!|,|\!| |\!|varphi|\!|^|\!|*|\!| |\!| |\!||\!|}|\!||\!|||\!| |\!|right|\!|||\!||\!|,|\!| |\!|,|\!| |\!|label|\!|{iv72|\!|}|\!| |\!|end|\!|{eqnarray|\!|}|\!|
|\!|
where|\!| the|\!| integration|\!| over|\!| the|\!| ghost|\!| fields|\!| was|\!| reintroduced|\!| for|\!| convenience|\!|.|\!| By|\!| reformulating|\!| |\!|$Z|\!|_|\!|{|\!|rm|\!| CM|\!|}|\!|$|\!| in|\!| terms|\!| of|\!| |\!|$|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|,|\!| |\!|{|\!|boldsymbol|\!|{|\!|lambda|\!|}|\!|}|\!|,|\!| |\!|{|\!|boldsymbol|\!|{c|\!|}|\!|}|\!|$|\!| and|\!| |\!|$|\!|bar|\!|{|\!|boldsymbol|\!|{c|\!|}|\!|}|\!|$|\!| we|\!| can|\!| now|\!| easily|\!| check|\!| the|\!| BRST|\!| invariance|\!|.|\!| The|\!| BRST|\!| transformations|\!| |\!|(|\!|ref|\!|{4|\!|.6|\!|}|\!|)|\!| imply|\!| that|\!|
|\!|
|\!|begin|\!|{eqnarray|\!|}|\!| |\!|&|\!|&|\!|delta|\!|_|\!|{|\!|rm|\!| BRST|\!|}|\!| |\!||\!| |\!|varphi|\!|^|\!|*|\!| |\!||\!| |\!|=|\!| |\!||\!| |\!|frac|\!|{|\!|partial|\!| |\!|varphi|\!|^|\!|*|\!|}|\!|{|\!|partial|\!| q|\!|_i|\!|}|\!| |\!||\!| |\!|bar|\!|{|\!|varepsilon|\!|}|\!| c|\!|_i|\!| |\!||\!| |\!|=|\!| |\!||\!| |\!|-|\!| |\!|bar|\!|{|\!|varepsilon|\!|}|\!| |\!|pounds|\!|_|\!|{|\!|{X|\!|_|\!|{|\!|mathcal|\!|{Q|\!|}|\!|}|\!|}|\!|_|\!|{|\!|rm|\!| BRST|\!|}|\!|}|\!| |\!||\!| |\!|varphi|\!|^|\!|*|\!||\!|,|\!| |\!|,|\!| |\!|nonumber|\!| |\!||\!||\!| |\!|&|\!|&|\!|bar|\!|{|\!|delta|\!|}|\!|_|\!|{|\!|rm|\!| BRST|\!|}|\!| |\!||\!| |\!|varphi|\!|^|\!|*|\!| |\!||\!| |\!|=|\!| |\!||\!| |\!|-|\!| |\!|frac|\!|{|\!|partial|\!| |\!|varphi|\!|^|\!|*|\!|}|\!|{|\!|partial|\!| q|\!|_i|\!|}|\!| |\!||\!| |\!|varepsilon|\!| |\!|bar|\!|{c|\!|}|\!|_i|\!| |\!||\!| |\!|=|\!| |\!||\!| |\!|-|\!| |\!|bar|\!|{|\!|varepsilon|\!|}|\!| |\!|pounds|\!|_|\!|{|\!|{X|\!|_|\!|{|\!|overline|\!|{|\!|mathcal|\!|{Q|\!|}|\!|}|\!|}|\!|}|\!|_|\!|{|\!|rm|\!| BRST|\!|}|\!|}|\!| |\!||\!| |\!|varphi|\!|^|\!|*|\!||\!|,|\!| |\!|.|\!| |\!|end|\!|{eqnarray|\!|}|\!|
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Here|\!| |\!|$|\!|{|\!|pounds|\!|_|\!|{X|\!|_|\!|{|\!|mathcal|\!|{Q|\!|}|\!|}|\!|}|\!|}|\!|_|\!|{|\!|rm|\!| BRST|\!|}|\!|$|\!| and|\!| |\!|$|\!|{|\!|pounds|\!|_|\!|{X|\!|_|\!|{|\!|overline|\!|{|\!|mathcal|\!|{Q|\!|}|\!|}|\!|}|\!|}|\!|}|\!|_|\!|{|\!|rm|\!| BRST|\!|}|\!|$|\!| represent|\!| the|\!| Lie|\!| derivatives|\!| with|\!| respect|\!| to|\!| flows|\!| generated|\!| by|\!| the|\!| BRST|\!| and|\!| anti|\!|-BRST|\!| charges|\!|,|\!| respectively|\!|.|\!| Analogous|\!| relations|\!| hold|\!| also|\!| for|\!| |\!|$|\!|chi|\!|^|\!|*|\!|$|\!|.|\!| Correspondingly|\!|,|\!| to|\!| the|\!| lowest|\!| order|\!| in|\!| |\!|$|\!|bar|\!|{|\!|varepsilon|\!|}|\!|$|\!| we|\!| can|\!| write|\!|
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|\!|begin|\!|{eqnarray|\!|}|\!|
|\!|delta|\!|[|\!|chi|\!|^|\!|*|\!|]|\!| |\!||\!| |\!|&|\!|rightarrow|\!|&|\!| |\!||\!| |\!|||\!| 1|\!| |\!|-|\!| |\!|mbox|\!|{Tr|\!|}|\!|(|\!|bar|\!|{|\!|varepsilon|\!|}|\!|{|\!|pounds|\!|_|\!|{X|\!|_|\!|{|\!|mathcal|\!|{Q|\!|}|\!|}|\!|}|\!|}|\!|_|\!|{|\!|rm|\!| BRST|\!|}|\!|)|\!|
|\!|||\!|^|\!|{|\!|-1|\!|}|\!| |\!||\!| |\!|delta|\!|[|\!|chi|\!|^|\!|*|\!|]|\!||\!|,|\!| |\!|,|\!| |\!|nonumber|\!| |\!||\!||\!|
|\!|left|\!|||\!| |\!|det|\!| |\!||\!|||\!| |\!||\!|{|\!|chi|\!|^|\!|*|\!|,|\!| |\!|varphi|\!|^|\!|*|\!| |\!| |\!||\!|}|\!||\!|||\!| |\!|right|\!|||\!| |\!||\!| |\!|&|\!|rightarrow|\!|&|\!| |\!||\!|
|\!|||\!| 1|\!| |\!|-|\!| |\!|mbox|\!|{Tr|\!|}|\!|(|\!|bar|\!|{|\!|varepsilon|\!|}|\!|{|\!|pounds|\!|_|\!|{X|\!|_|\!|{|\!|mathcal|\!|{Q|\!|}|\!|}|\!|}|\!|}|\!|_|\!|{|\!|rm|\!| BRST|\!|}|\!|)|\!| |\!|||\!| |\!|left|\!|||\!| |\!|det|\!| |\!||\!|||\!| |\!||\!|{|\!|chi|\!|^|\!|*|\!|,|\!| |\!|varphi|\!|^|\!|*|\!| |\!| |\!||\!|}|\!||\!|||\!| |\!|right|\!|||\!||\!|,|\!| |\!|.|\!| |\!|label|\!|{iv74|\!|}|\!| |\!|end|\!|{eqnarray|\!|}|\!|
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The|\!| transformations|\!| |\!|(|\!|ref|\!|{iv74|\!|}|\!|)|\!| show|\!| that|\!| the|\!| term|\!| |\!|$|\!|delta|\!|[|\!|chi|\!|^|\!|*|\!|]|\!|
|\!| |\!|left|\!|||\!| |\!|det|\!| |\!||\!|||\!| |\!||\!|{|\!|chi|\!|^|\!|*|\!|,|\!| |\!|varphi|\!|^|\!|*|\!| |\!| |\!||\!|}|\!||\!|||\!| |\!|right|\!|||\!| |\!|$|\!| in|\!| |\!|(|\!|ref|\!|{iv72|\!|}|\!|)|\!| is|\!| the|\!| BRST|\!| invariant|\!| |\!|(as|\!|,|\!| of|\!| course|\!|,|\!| are|\!| both|\!| the|\!| integration|\!| measure|\!| and|\!| the|\!| effective|\!| action|\!| |\!|$|\!|{|\!|mathcal|\!|{S|\!|}|\!|}|\!|$|\!|)|\!|.|\!| However|\!|,|\!| because|\!| the|\!| variation|\!| |\!|$|\!|delta|\!|_|\!|{|\!|rm|\!| BRST|\!|}|\!| |\!|delta|\!|[|\!|varphi|\!|^|\!|*|\!|]|\!|$|\!| is|\!| not|\!| compensated|\!| in|\!| |\!|(|\!|ref|\!|{iv72|\!|}|\!|)|\!| we|\!| have|\!| in|\!| general|\!|,|\!| |\!| |\!|$|\!|delta|\!|_|\!|{|\!|rm|\!| BRST|\!|}|\!| Z|\!|_|\!|{|\!|rm|\!| CM|\!|}|\!|[|\!|{|\!|boldsymbol|\!|{J|\!|}|\!|}|\!| |\!|=|\!| 0|\!|]|\!| |\!|neq|\!| 0|\!|$|\!|.|\!| An|\!| analogous|\!| result|\!| applies|\!| also|\!| to|\!| the|\!| anti|\!|-BRST|\!| transformation|\!|.|\!|
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|\!|vspace|\!|{3mm|\!|}|\!|
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We|\!| should|\!| note|\!| that|\!| the|\!| condition|\!| |\!|$|\!|delta|\!|_|\!|{|\!|rm|\!| BRST|\!|}|\!| Z|\!|_|\!|{|\!|rm|\!| CM|\!|}|\!|[|\!|{|\!|boldsymbol|\!|{J|\!|}|\!|}|\!| |\!|=|\!| 0|\!|]|\!| |\!|neq|\!| 0|\!|$|\!| only|\!| indicates|\!| that|\!| the|\!| |\!|{|\!|em|\!| classical|\!|}|\!| path|\!|-integral|\!| structure|\!| is|\!| destroyed|\!|;|\!| it|\!| does|\!| not|\!|,|\!| however|\!|,|\!| ensure|\!| that|\!| the|\!| ensuing|\!| |\!|$Z|\!|_|\!|{|\!|rm|\!| CM|\!|}|\!|$|\!| can|\!| be|\!| recast|\!| into|\!| a|\!| form|\!| describing|\!| a|\!| proper|\!| quantum|\!|-mechanical|\!| generating|\!| functional|\!|.|\!|
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The|\!| straightforward|\!| path|\!|-integral|\!| representation|\!| such|\!| as|\!| |\!|(|\!|ref|\!|{4|\!|.22|\!|}|\!|)|\!| emerges|\!| only|\!| after|\!| the|\!| gauge|\!| freedom|\!| inherent|\!| in|\!|
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the|\!| |\!|`|\!|`information|\!| loss|\!|"|\!| condition|\!| |\!|$|\!|varphi|\!|$|\!| is|\!| properly|\!| fixed|\!| via|\!| the|\!| gauge|\!| constraint|\!| |\!|$|\!|chi|\!|$|\!|.|\!| Let|\!| us|\!| finally|\!| emphasize|\!| once|\!| more|\!| that|\!| the|\!| partition|\!| function|\!| |\!|(|\!|ref|\!|{4|\!|.22|\!|}|\!|)|\!| |\!|(resp|\!|.|\!| |\!|(|\!|ref|\!|{4|\!|.30|\!|}|\!|)|\!|)|\!| has|\!| arisen|\!| as|\!| a|\!| consequence|\!| of|\!| the|\!| application|\!| of|\!| the|\!| classical|\!| Dirac|\!|-Bergmann|\!| algorithm|\!| for|\!| singular|\!| systems|\!| to|\!| the|\!| classical|\!| path|\!| integral|\!| of|\!| Gozzi|\!| |\!|{|\!|em|\!| et|\!| al|\!|.|\!|}|\!|.|\!|
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|\!|section|\!|{Explicit|\!| examples|\!|}|\!|label|\!|{SEc5|\!|}|\!|
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|\!|subsection|\!|{Free|\!| particle|\!|}|\!|
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Although|\!| the|\!| preceding|\!| construction|\!| may|\!| seem|\!| a|\!| bit|\!| abstract|\!|,|\!| its|\!| implementation|\!| is|\!| quite|\!| straightforward|\!|.|\!| Let|\!| us|\!| now|\!| illustrate|\!| this|\!| with|\!| two|\!| systems|\!|.|\!| As|\!| a|\!| warm|\!|-up|\!| example|\!| we|\!| start|\!| with|\!| the|\!| Hamiltonian|\!|
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|\!|begin|\!|{eqnarray|\!|}|\!| H|\!| |\!|=L|\!|_3|\!|=|\!| xp|\!|_y|\!| |\!|-|\!| yp|\!|_x|\!||\!|,|\!| |\!|,|\!| |\!|label|\!|{5|\!|.1|\!|}|\!| |\!|end|\!|{eqnarray|\!|}|\!|
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which|\!| is|\!| known|\!| to|\!| represent|\!| the|\!| angular|\!| momentum|\!| with|\!| values|\!| unbounded|\!| from|\!| below|\!|.|\!| Alternatively|\!|,|\!| |\!|(|\!|ref|\!|{5|\!|.1|\!|}|\!|)|\!| can|\!| be|\!| regarded|\!| as|\!| describing|\!| the|\!| mathematical|\!| pendulum|\!|.|\!| This|\!| is|\!| because|\!| the|\!| corresponding|\!| dynamical|\!| equation|\!| |\!|(|\!|ref|\!|{eq|\!|.1|\!|.1|\!|.1|\!|}|\!|)|\!| for|\!| |\!|$|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|$|\!| is|\!| a|\!| plane|\!| pendulum|\!| equation|\!| with|\!| the|\!| pendulum|\!| constant|\!| |\!|$l|\!|/|\!|mbox|\!|{|\!|{|\!|textsl|\!|{g|\!|}|\!|}|\!|}|\!| |\!|=1|\!|$|\!|.|\!| The|\!| Lagrangian|\!| |\!|(|\!|ref|\!|{lag1|\!|}|\!|)|\!| reads|\!|
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|\!|begin|\!|{eqnarray|\!|}|\!| |\!|bar|\!|{L|\!|}|\!| |\!|=|\!| |\!|overlinen|\!|{x|\!|}|\!|dot|\!|{x|\!|}|\!| |\!|+|\!| |\!|overlinen|\!|{y|\!|}|\!|dot|\!|{y|\!|}|\!| |\!|+|\!| |\!|overlinen|\!|{x|\!|}y|\!| |\!|-|\!| |\!|overlinen|\!|{y|\!|}x|\!||\!|,|\!| |\!|.|\!| |\!|label|\!|{lag2|\!|}|\!| |\!|end|\!|{eqnarray|\!|}|\!|
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It|\!| is|\!| well|\!|-known|\!|~|\!|cite|\!|{Lutzky|\!|}|\!| that|\!| the|\!| system|\!| has|\!| two|\!| |\!|(functionally|\!| independent|\!|)|\!| constants|\!| of|\!| motion|\!| |\!|-|\!| Casimir|\!| functions|\!|.|\!| For|\!| |\!|(|\!|ref|\!|{5|\!|.1|\!|}|\!|)|\!| they|\!| read|\!|
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|\!|begin|\!|{eqnarray|\!|}|\!| C|\!|_1|\!| |\!||\!| |\!|=|\!| |\!||\!| x|\!|^2|\!| |\!|+|\!| y|\!|^2|\!||\!|,|\!| |\!|,|\!| |\!||\!|;|\!||\!|;|\!||\!|;|\!| C|\!|_2|\!| |\!||\!| |\!|=|\!| |\!||\!| xp|\!|_x|\!| |\!|+|\!| yp|\!|_y|\!||\!|,|\!| |\!|.|\!| |\!|end|\!|{eqnarray|\!|}|\!|
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The|\!| charge|\!| |\!|$C|\!|_1|\!|$|\!| corresponds|\!| to|\!| the|\!| conserved|\!| radius|\!| of|\!| the|\!| orbit|\!| while|\!| |\!|$C|\!|_2|\!|$|\!| is|\!| the|\!| Noether|\!| charge|\!| of|\!| dilatation|\!| invariance|\!| of|\!| the|\!| Lagrangian|\!| |\!|(|\!|ref|\!|{lag2|\!|}|\!|)|\!| under|\!| the|\!| transformations|\!| |\!|$|\!|(|\!|overlinen|\!|{x|\!|}|\!|,|\!| |\!|overlinen|\!|{y|\!|}|\!|,|\!| x|\!|,|\!| y|\!|)|\!| |\!|mapsto|\!| |\!|(e|\!|^|\!|{|\!|-s|\!|}|\!|overlinen|\!|{x|\!|}|\!|,|\!| e|\!|^|\!|{|\!|-s|\!|}|\!|overlinen|\!|{y|\!|}|\!|,|\!| e|\!|^sx|\!|,|\!| e|\!|^sy|\!|)|\!|$|\!|.|\!| As|\!| only|\!| |\!|$C|\!|_1|\!|$|\!| is|\!| |\!|$|\!|boldsymbol|\!|{p|\!|}|\!|$|\!|-independent|\!|,|\!| the|\!| functions|\!| |\!|$F|\!|_|\!|+|\!|$|\!| and|\!| |\!|$F|\!|_|\!|-|\!|$|\!| of|\!| this|\!| system|\!| are|\!| according|\!| to|\!| Eq|\!|.|\!|~|\!|(|\!|ref|\!|{FCH|\!|}|\!|)|\!| chosen|\!| as|\!|:|\!|
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|\!|begin|\!|{eqnarray|\!|}|\!| F|\!|_|\!|+|\!| |\!||\!| |\!|=|\!| |\!||\!| |\!|frac|\!|{|\!|(H|\!| |\!|+|\!| a|\!|_1|\!| C|\!|_1|\!|)|\!|^2|\!|}|\!|{4a|\!|_1|\!| C|\!|_1|\!|}|\!||\!|,|\!| |\!|,|\!| |\!||\!|;|\!||\!|;|\!||\!|;|\!| F|\!|_|\!|-|\!| |\!||\!| |\!|=|\!| |\!||\!| |\!|frac|\!|{|\!|(H|\!| |\!|-|\!| a|\!|_1|\!| C|\!|_1|\!|)|\!|^2|\!|}|\!|{4a|\!|_1|\!| C|\!|_1|\!|}|\!||\!|,|\!| |\!|.|\!| |\!|end|\!|{eqnarray|\!|}|\!|
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Hence|\!| |\!|$H|\!|_|\!|-|\!| |\!|=|\!| 0|\!| |\!|$|\!| implies|\!| that|\!| |\!|$H|\!|_|\!|+|\!| |\!|approx|\!| a|\!|_1|\!|(x|\!|^2|\!| |\!|+|\!| y|\!|^2|\!|)|\!|$|\!|.|\!| Here|\!| |\!|$a|\!|_1|\!|$|\!| is|\!| some|\!| constant|\!| to|\!| be|\!| specified|\!| later|\!|.|\!| The|\!| ensuing|\!| first|\!|-class|\!| constraint|\!| is|\!|
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|\!|begin|\!|{eqnarray|\!|}|\!| |\!|varphi|\!| |\!||\!| |\!|=|\!| |\!||\!| xp|\!|_y|\!| |\!|-|\!| yp|\!|_x|\!| |\!|-|\!| a|\!|_1|\!| x|\!|^2|\!| |\!|-|\!| a|\!|_1|\!| y|\!|^2|\!| |\!|-|\!| |\!|bar|\!|{p|\!|}|\!|_|\!|{|\!|bar|\!|{x|\!|}|\!|}|\!| |\!|bar|\!|{y|\!|}|\!| |\!|+|\!| 2a|\!|_1|\!| |\!|bar|\!|{p|\!|}|\!|_|\!|{|\!|bar|\!|{x|\!|}|\!|}|\!| x|\!| |\!|+|\!| |\!|bar|\!|{p|\!|}|\!|_|\!|{|\!|bar|\!|{y|\!|}|\!|}|\!| |\!|bar|\!|{x|\!|}|\!| |\!|+|\!| 2a|\!|_1|\!| |\!|bar|\!|{p|\!|}|\!|_|\!|{|\!|bar|\!|{y|\!|}|\!|}|\!| y|\!| |\!| |\!||\!| |\!|approx|\!| |\!||\!| H|\!| |\!|-|\!| a|\!|_1|\!| C|\!|_1|\!||\!|,|\!| |\!|.|\!| |\!|end|\!|{eqnarray|\!|}|\!|
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The|\!| gauge|\!| condition|\!| can|\!| then|\!| be|\!| chosen|\!| in|\!| the|\!| form|\!| |\!|$|\!|chi|\!| |\!|=|\!| |\!|bar|\!|{p|\!|}|\!|_|\!|{|\!|bar|\!|{y|\!|}|\!|}|\!| |\!|-|\!| y|\!|$|\!|.|\!| Indeed|\!|,|\!| we|\!| easily|\!| find|\!| that|\!|
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|\!|begin|\!|{eqnarray|\!|}|\!| |\!|&|\!|&|\!||\!|{|\!| |\!|chi|\!|,|\!| |\!|varphi|\!| |\!||\!|}|\!| |\!||\!| |\!|=|\!| |\!||\!| |\!|bar|\!|{p|\!|}|\!|_|\!|{|\!|bar|\!|{x|\!|}|\!|}|\!| |\!|-|\!| x|\!| |\!||\!| |\!|neq|\!| |\!||\!| 0|\!||\!|,|\!| |\!|,|\!| |\!|nonumber|\!| |\!||\!||\!| |\!|&|\!|&|\!||\!|{|\!| |\!|chi|\!|,|\!| |\!|phi|\!|_i|\!| |\!||\!|}|\!| |\!||\!| |\!|=|\!| |\!||\!| 0|\!||\!|,|\!| |\!|,|\!| |\!||\!|;|\!| |\!||\!|;|\!| |\!||\!|;|\!| i|\!| |\!|=|\!| 1|\!|,|\!| |\!|ldots|\!|,|\!| 4|\!||\!|,|\!| |\!|.|\!| |\!|label|\!|{5|\!|.5|\!|}|\!| |\!|end|\!|{eqnarray|\!|}|\!|
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The|\!| advantage|\!| of|\!| our|\!| choice|\!| of|\!| |\!|$|\!|chi|\!|$|\!| is|\!| that|\!| it|\!| will|\!| not|\!| run|\!| into|\!| Gribov|\!| ambiguities|\!|,|\!| i|\!|.e|\!|.|\!|,|\!| the|\!| equation|\!| |\!|$|\!|varphi|\!| |\!|=|\!| 0|\!|$|\!| will|\!| have|\!| globally|\!| unique|\!| solution|\!| for|\!| |\!|$Q|\!|_1|\!|$|\!| on|\!| |\!|$|\!|Gamma|\!|^|\!|*|\!|$|\!|.|\!| This|\!| should|\!| be|\!| contrasted|\!| with|\!| such|\!| choices|\!| as|\!|,|\!| e|\!|.g|\!|.|\!|,|\!| |\!|$|\!|chi|\!| |\!|=|\!| p|\!|_x|\!|$|\!| or|\!| |\!|$|\!|chi|\!| |\!|=|\!| p|\!|_y|\!|$|\!|,|\!| which|\!| also|\!| satisfy|\!| the|\!| conditions|\!| |\!|(|\!|ref|\!|{5|\!|.5|\!|}|\!|)|\!|,|\!| but|\!| lead|\!| to|\!| two|\!| Gribov|\!| copies|\!| each|\!|.|\!|
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|\!|vspace|\!|{3mm|\!|}|\!|
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With|\!| the|\!| above|\!| choice|\!| of|\!| |\!|$|\!|chi|\!|$|\!| we|\!| may|\!| directly|\!| write|\!| the|\!| canonical|\!| transformations|\!|:|\!|
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|\!|begin|\!|{eqnarray|\!|}|\!| |\!|&|\!|&P|\!|_1|\!| |\!||\!| |\!|=|\!| |\!||\!| |\!|chi|\!| |\!||\!| |\!|=|\!| |\!||\!| |\!|bar|\!|{p|\!|}|\!|_|\!|{|\!|bar|\!|{y|\!|}|\!|}|\!| |\!|-|\!| y|\!||\!|,|\!| |\!|,|\!| |\!||\!|;|\!||\!|;|\!||\!|;|\!| Q|\!|_1|\!| |\!||\!| |\!|=|\!| |\!||\!| p|\!|_y|\!||\!|,|\!| |\!|,|\!|nonumber|\!| |\!||\!||\!| |\!|&|\!|&P|\!|_2|\!| |\!||\!| |\!|=|\!| |\!||\!| p|\!|_x|\!| |\!|-|\!| |\!|bar|\!|{x|\!|}|\!||\!|,|\!| |\!|,|\!| |\!|mbox|\!|{|\!|hspace|\!|{1|\!|.2cm|\!|}|\!|}|\!| Q|\!|_2|\!| |\!||\!| |\!|=|\!| |\!||\!| |\!|bar|\!|{p|\!|}|\!|_|\!|{|\!|bar|\!|{x|\!|}|\!|}|\!||\!|,|\!| |\!|,|\!| |\!|nonumber|\!| |\!||\!||\!| |\!|&|\!|&P|\!|_3|\!| |\!||\!| |\!|=|\!| |\!||\!| p|\!|_y|\!| |\!|-|\!| |\!|bar|\!|{y|\!|}|\!||\!|,|\!| |\!|,|\!| |\!|mbox|\!|{|\!|hspace|\!|{1|\!|.2cm|\!|}|\!|}|\!| Q|\!|_3|\!| |\!||\!| |\!|=|\!| |\!||\!| |\!|bar|\!|{p|\!|}|\!|_|\!|{|\!|bar|\!|{y|\!|}|\!|}|\!||\!|,|\!| |\!|,|\!| |\!|nonumber|\!| |\!||\!||\!| |\!|&|\!|&|\!|bar|\!|{P|\!|}|\!|~|\!| |\!||\!| |\!|=|\!| |\!||\!| |\!|bar|\!|{p|\!|}|\!|_|\!|{|\!|bar|\!|{x|\!|}|\!|}|\!| |\!|-|\!| x|\!||\!|,|\!| |\!|,|\!| |\!|mbox|\!|{|\!|hspace|\!|{1|\!|.2cm|\!|}|\!|}|\!| |\!|bar|\!|{Q|\!|}|\!|~|\!| |\!||\!| |\!|=|\!| |\!||\!| p|\!|_x|\!||\!|,|\!| |\!|.|\!| |\!|end|\!|{eqnarray|\!|}|\!|
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|\!| It|\!| might|\!| be|\!| checked|\!| that|\!| the|\!| transformation|\!| Jacobian|\!| is|\!| indeed|\!| |\!|$1|\!|$|\!|.|\!|
|\!| In|\!| the|\!| new|\!| canonical|\!| variables|\!| the|\!| Hamiltonian|\!| |\!|$K|\!|$|\!| reads|\!|
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|\!|begin|\!|{eqnarray|\!|}|\!| K|\!|(|\!|bar|\!|{P|\!|}|\!|,|\!| |\!|bar|\!|{Q|\!|}|\!|,|\!| Q|\!|_1|\!|)|\!| |\!||\!| |\!|=|\!| |\!||\!| H|\!|(|\!|bar|\!|{P|\!|}|\!|,|\!| |\!|bar|\!|{Q|\!|}|\!|,|\!| P|\!|_a|\!| |\!|=|\!| 0|\!|,|\!| Q|\!|_1|\!|,|\!| Q|\!|_2|\!| |\!|=|\!| 0|\!|,|\!| Q|\!|_3|\!| |\!|=|\!| 0|\!| |\!|)|\!| |\!||\!| |\!|=|\!| |\!||\!| |\!|-|\!| |\!|bar|\!|{P|\!|}Q|\!|_1|\!| |\!||\!|,|\!| |\!|.|\!| |\!|end|\!|{eqnarray|\!|}|\!|
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The|\!| functional|\!| |\!|$|\!|delta|\!|$|\!|-function|\!| |\!|(|\!|ref|\!|{4|\!|.23|\!|}|\!|)|\!| has|\!| the|\!| form|\!|
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|\!|begin|\!|{eqnarray|\!|}|\!| |\!|delta|\!|[Q|\!|_1|\!| |\!|-|\!| Q|\!|_1|\!|^|\!|*|\!|(|\!|bar|\!|{P|\!|}|\!|,|\!|bar|\!|{Q|\!|}|\!|)|\!|]|\!| |\!||\!| |\!|=|\!| |\!||\!| |\!|delta|\!|[Q|\!|_1|\!| |\!|+|\!| a|\!|_1|\!| |\!|bar|\!|{P|\!|}|\!|]|\!||\!|,|\!| |\!|,|\!| |\!|end|\!|{eqnarray|\!|}|\!|
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and|\!| hence|\!| |\!|$K|\!|^|\!|*|\!|(|\!|bar|\!|{P|\!|}|\!|,|\!| |\!|bar|\!|{Q|\!|}|\!|)|\!| |\!|=|\!| H|\!|_|\!|+|\!|^|\!|*|\!|(|\!|bar|\!|{P|\!|}|\!|,|\!| |\!|bar|\!|{Q|\!|}|\!|)|\!|=|\!| a|\!|_1|\!| |\!|bar|\!|{P|\!|}|\!|^2|\!|$|\!|.|\!| Let|\!| us|\!| now|\!| set|\!| |\!|$a|\!|_1|\!| |\!|=|\!| 1|\!|/2m|\!|hbar|\!|$|\!|.|\!| After|\!| changing|\!| variables|\!| |\!|$|\!|bar|\!|{Q|\!|}|\!|(t|\!|)|\!|$|\!| to|\!| |\!|$|\!|bar|\!|{Q|\!|}|\!|(t|\!|)|\!|/|\!|hbar|\!|$|\!| we|\!| obtain|\!| not|\!| only|\!| the|\!| correct|\!| |\!|`|\!|`quantum|\!|-mechanical|\!|"|\!| path|\!|-integral|\!| measure|\!|
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|\!|begin|\!|{eqnarray|\!|}|\!| |\!|{|\!|mathcal|\!|{D|\!|}|\!|}|\!|bar|\!|{|\!|{Q|\!|}|\!|}|\!|{|\!|mathcal|\!|{D|\!|}|\!|}|\!|bar|\!|{|\!|{P|\!|}|\!|}|\!| |\!||\!| |\!|approx|\!| |\!||\!| |\!|prod|\!|_i|\!| |\!|left|\!|(|\!|frac|\!|{|\!| d|\!| |\!|{|\!|bar|\!|{|\!|{Q|\!|}|\!|}|\!|}|\!|(t|\!|_i|\!|)|\!| d|\!| |\!|{|\!|bar|\!|{|\!|{P|\!|}|\!|}|\!|}|\!|(t|\!|_i|\!|)|\!|}|\!|{2|\!|pi|\!| |\!|hbar|\!|}|\!|right|\!|)|\!||\!|,|\!| |\!|,|\!| |\!|end|\!|{eqnarray|\!|}|\!|
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but|\!| also|\!| the|\!| prefactor|\!| |\!|$1|\!|/|\!|hbar|\!|$|\!| in|\!| the|\!| exponent|\!|.|\!| So|\!| |\!|(|\!|ref|\!|{4|\!|.30|\!|}|\!|)|\!| reduces|\!| to|\!| the|\!| quantum|\!| partition|\!| function|\!| for|\!| a|\!| free|\!| particle|\!| of|\!| mass|\!| |\!|$m|\!|$|\!|.|\!| As|\!| the|\!| constant|\!| |\!|$a|\!|_1|\!|$|\!| represents|\!| the|\!| choice|\!| of|\!| units|\!| |\!|(or|\!| scale|\!| factor|\!|)|\!| for|\!| |\!|$C|\!|_1|\!|$|\!| we|\!| see|\!| that|\!| the|\!| quantum|\!| scale|\!| |\!|$|\!|hbar|\!|$|\!| is|\!| implemented|\!| into|\!| the|\!| partition|\!| function|\!| via|\!| the|\!| choice|\!| of|\!| the|\!| |\!|`|\!|`loss|\!| of|\!| information|\!|"|\!| constraint|\!|.|\!|
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|\!|subsection|\!|{Harmonic|\!| oscillator|\!|}|\!|
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The|\!| system|\!| |\!|(|\!|ref|\!|{5|\!|.1|\!|}|\!|)|\!| can|\!| also|\!| be|\!| used|\!| to|\!| obtain|\!| the|\!| quantized|\!| linear|\!| harmonic|\!| oscillator|\!|.|\!| This|\!| is|\!| possible|\!| by|\!| observing|\!| that|\!| not|\!| only|\!| |\!|$C|\!|_1|\!| |\!|=|\!| x|\!|^2|\!| |\!|+|\!| y|\!|^2|\!|$|\!| is|\!| a|\!| constant|\!| of|\!| motion|\!| for|\!| |\!|(|\!|ref|\!|{5|\!|.1|\!|}|\!|)|\!| but|\!| also|\!| |\!|$C|\!|_1|\!| |\!|=|\!| x|\!|^2|\!| |\!|+|\!| y|\!|^2|\!| |\!|+|\!| c|\!|$|\!| with|\!| |\!|$c|\!|$|\!| being|\!| any|\!| |\!|$|\!|boldsymbol|\!|{q|\!|}|\!|$|\!| and|\!| |\!|$|\!|boldsymbol|\!|{p|\!|}|\!|$|\!| independent|\!| constant|\!|.|\!| So|\!| in|\!| particular|\!| we|\!| can|\!| choose|\!| |\!|$c|\!| |\!|=|\!| c|\!|(|\!|bar|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|)|\!|$|\!|.|\!| The|\!| functional|\!| dependence|\!| of|\!| |\!|$c|\!|$|\!| on|\!| |\!|$|\!|bar|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|$|\!| cannot|\!| be|\!|,|\!| however|\!|,|\!| arbitrary|\!|.|\!| The|\!| requirement|\!| that|\!| |\!|'t|\!|~Hooft|\!|'s|\!| constraint|\!| should|\!| not|\!| generate|\!| any|\!| new|\!| |\!|(i|\!|.e|\!|.|\!|,|\!| secondary|\!|)|\!| constraint|\!| represents|\!| quite|\!| severe|\!| restriction|\!|.|\!| Indeed|\!|,|\!| in|\!| order|\!| to|\!| satisfy|\!| Eq|\!|.|\!|(|\!|ref|\!|{D2|\!|}|\!|)|\!| the|\!| following|\!| condition|\!| must|\!| hold|\!| |\!|(c|\!|.f|\!|.|\!| Appendix|\!| D|\!|)|\!|:|\!|
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|\!|begin|\!|{eqnarray|\!|}|\!| |\!|sum|\!|_|\!|{i|\!|=0|\!|}|\!|^|\!|{2N|\!|}|\!| e|\!|_i|\!| |\!||\!|{|\!|phi|\!|_i|\!|,|\!| |\!|bar|\!|{H|\!|}|\!| |\!||\!|}|\!| |\!||\!| |\!|=|\!| |\!||\!| |\!|-|\!| |\!|sum|\!|_|\!|{a|\!|,i|\!|}|\!| a|\!|_i|\!| |\!||\!|{|\!| C|\!|_i|\!|,|\!| |\!|bar|\!|{p|\!|}|\!|_a|\!| |\!||\!|}|\!| |\!||\!|{|\!| p|\!|_a|\!|,|\!| |\!|bar|\!|{H|\!|}|\!||\!|}|\!| |\!||\!| |\!|=|\!| |\!||\!| |\!| |\!|sum|\!|_|\!|{i|\!|,k|\!|,a|\!|}|\!| a|\!|_i|\!| |\!|frac|\!|{|\!|partial|\!| c|\!|_i|\!|(|\!|bar|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|)|\!|}|\!|{|\!|partial|\!| |\!|bar|\!|{q|\!|}|\!|_a|\!|}|\!| |\!|bar|\!|{q|\!|}|\!|_k|\!| |\!|frac|\!|{|\!|partial|\!| f|\!|_k|\!|(|\!|boldsymbol|\!|{q|\!|}|\!|)|\!|}|\!|{|\!|partial|\!| q|\!|_a|\!|}|\!| |\!|end|\!|{eqnarray|\!|}|\!|
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which|\!| for|\!| the|\!| system|\!| in|\!| question|\!| is|\!| weakly|\!| zero|\!| only|\!| if|\!|
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|\!|begin|\!|{eqnarray|\!|}|\!| |\!|bar|\!|{x|\!|}|\!| |\!|frac|\!|{|\!|partial|\!| c|\!|(|\!|bar|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|)|\!|}|\!|{|\!|partial|\!| |\!|bar|\!|{y|\!|}|\!|}|\!| |\!|-|\!| |\!|bar|\!|{y|\!|}|\!| |\!|frac|\!|{|\!|partial|\!| c|\!|(|\!|bar|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|)|\!|}|\!|{|\!|partial|\!| |\!|bar|\!|{x|\!|}|\!|}|\!| |\!||\!| |\!|=|\!| |\!||\!| 0|\!||\!|,|\!| |\!|.|\!| |\!|end|\!|{eqnarray|\!|}|\!|
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The|\!| latter|\!| equation|\!| has|\!| the|\!| solution|\!| |\!|(modulo|\!| irrelevant|\!| additive|\!| constant|\!|)|\!| |\!|$c|\!|(|\!|bar|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|)|\!| |\!|=|\!| d|\!|^2|\!| |\!|(|\!|{|\!|bar|\!|{x|\!|}|\!|}|\!|^2|\!| |\!| |\!|+|\!| |\!|{|\!|bar|\!|{y|\!|}|\!|}|\!|^2|\!|)|\!|$|\!|.|\!| Here|\!| |\!|$d|\!|^2|\!|$|\!| represents|\!| a|\!| multiplicative|\!| constant|\!|.|\!| Hence|\!| we|\!| have|\!| that|\!| |\!|$C|\!|_1|\!|$|\!| has|\!| the|\!| general|\!| form|\!|
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|\!|begin|\!|{eqnarray|\!|}|\!| C|\!|_1|\!| |\!||\!| |\!|=|\!| |\!||\!| x|\!|^2|\!| |\!|+|\!| y|\!|^2|\!| |\!|+|\!| d|\!|^2|\!|(|\!|{|\!|bar|\!|{x|\!|}|\!|}|\!|^2|\!| |\!| |\!|+|\!| |\!|{|\!|bar|\!|{y|\!|}|\!|}|\!|^2|\!|)|\!||\!|,|\!| |\!|.|\!| |\!|end|\!|{eqnarray|\!|}|\!|
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It|\!| will|\!| be|\!| further|\!| convenient|\!| to|\!| choose|\!| |\!|$a|\!|_1|\!| |\!|=|\!| |\!|-1|\!|/2d|\!|$|\!|.|\!| The|\!| resulting|\!| first|\!|-class|\!| constraint|\!| then|\!| reads|\!|
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|\!|begin|\!|{eqnarray|\!|}|\!| |\!|varphi|\!| |\!||\!| |\!|&|\!|=|\!|&|\!| |\!||\!| xp|\!|_y|\!| |\!|-|\!| yp|\!|_x|\!| |\!|+|\!| |\!|frac|\!|{1|\!|}|\!|{2d|\!|}|\!| x|\!|^2|\!| |\!|+|\!| |\!|frac|\!|{1|\!|}|\!|{2d|\!|}|\!| y|\!|^2|\!| |\!|-|\!| |\!|frac|\!|{d|\!|}|\!|{2|\!|}|\!| |\!|{|\!|bar|\!|{x|\!|}|\!|}|\!|^2|\!| |\!|-|\!| |\!|frac|\!|{d|\!|}|\!|{2|\!|}|\!| |\!|{|\!|bar|\!|{y|\!|}|\!|}|\!|^2|\!| |\!|-|\!| |\!|bar|\!|{y|\!|}|\!|bar|\!|{p|\!|}|\!|_|\!|{|\!|bar|\!|{x|\!|}|\!|}|\!| |\!|+|\!| |\!|bar|\!|{x|\!|}|\!| |\!|bar|\!|{p|\!|}|\!|_|\!|{|\!|bar|\!|{y|\!|}|\!|}|\!| |\!|-|\!| |\!|frac|\!|{1|\!|}|\!|{d|\!|}|\!| x|\!| |\!|bar|\!|{p|\!|}|\!|_|\!|{|\!|bar|\!|{x|\!|}|\!|}|\!| |\!|-|\!| |\!|frac|\!|{1|\!|}|\!|{d|\!|}|\!| y|\!| |\!|bar|\!|{p|\!|}|\!|_|\!|{|\!|bar|\!|{y|\!|}|\!|}|\!| |\!|+|\!| d|\!| |\!|bar|\!|{x|\!|}p|\!|_x|\!| |\!|+|\!| d|\!| |\!|bar|\!|{y|\!|}p|\!|_y|\!| |\!|nonumber|\!| |\!||\!||\!| |\!|&|\!|approx|\!|&|\!| |\!||\!| H|\!| |\!|+|\!| |\!|frac|\!|{1|\!|}|\!|{2d|\!|}|\!| |\!||\!| C|\!|_1|\!| |\!||\!|,|\!| |\!|.|\!| |\!|end|\!|{eqnarray|\!|}|\!|
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If|\!| we|\!| choose|\!| the|\!| gauge|\!| condition|\!| to|\!| be|\!|
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|\!|begin|\!|{eqnarray|\!|}|\!| |\!|chi|\!| |\!||\!| |\!|=|\!| |\!||\!| |\!| |\!|bar|\!|{p|\!|}|\!|_|\!|{|\!|bar|\!|{y|\!|}|\!|}|\!| |\!|+|\!| d|\!| p|\!|_x|\!|-|\!| y|\!||\!|,|\!| |\!|,|\!| |\!|label|\!|{5|\!|.6|\!|}|\!| |\!|end|\!|{eqnarray|\!|}|\!|
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it|\!| ensures|\!| that|\!|
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|\!|begin|\!|{eqnarray|\!|}|\!| |\!|&|\!|&|\!||\!|{|\!| |\!|chi|\!|,|\!| |\!|varphi|\!| |\!||\!|}|\!| |\!||\!| |\!|=|\!| |\!||\!| 2|\!| |\!|bar|\!|{p|\!|}|\!|_|\!|{|\!|bar|\!|{x|\!|}|\!|}|\!| |\!|-|\!| 2|\!| x|\!| |\!|-|\!| 2|\!| d|\!| |\!| p|\!|_y|\!| |\!||\!| |\!|neq|\!| |\!||\!| 0|\!||\!|,|\!| |\!|,|\!|nonumber|\!| |\!||\!||\!| |\!|&|\!|&|\!||\!|{|\!| |\!|chi|\!|,|\!| |\!|phi|\!|_i|\!| |\!||\!|}|\!| |\!||\!| |\!|=|\!| |\!||\!| 0|\!||\!|,|\!| |\!|,|\!| |\!||\!|;|\!||\!|;|\!||\!|;|\!| i|\!| |\!|=|\!| 1|\!|,|\!| |\!|ldots|\!|,|\!| 4|\!||\!|,|\!| |\!|.|\!| |\!|end|\!|{eqnarray|\!|}|\!|
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In|\!| addition|\!|,|\!| we|\!| shall|\!| see|\!| that|\!| |\!|(|\!|ref|\!|{5|\!|.6|\!|}|\!|)|\!| guarantees|\!| the|\!| unique|\!| global|\!| solution|\!| of|\!| the|\!| equation|\!| |\!|$|\!|varphi|\!| |\!|=|\!| 0|\!|$|\!| for|\!| |\!|$Q|\!|_1|\!|$|\!| on|\!| |\!|$|\!|Gamma|\!|^|\!|*|\!|$|\!| |\!|(hence|\!| it|\!| avoids|\!| the|\!| undesired|\!| Gribov|\!| ambiguity|\!|)|\!|.|\!|
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|\!|vspace|\!|{3mm|\!|}|\!|
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The|\!| canonical|\!| transformation|\!| discussed|\!| in|\!| Section|\!| IV|\!| now|\!| takes|\!| the|\!| form|\!|
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|\!|begin|\!|{eqnarray|\!|}|\!| |\!|&|\!|&P|\!|_1|\!| |\!||\!| |\!|=|\!| |\!||\!| |\!|chi|\!| |\!||\!| |\!|=|\!| |\!||\!| |\!|bar|\!|{p|\!|}|\!|_|\!|{|\!|bar|\!|{y|\!|}|\!|}|\!| |\!|+|\!| dp|\!|_x|\!| |\!|-|\!| y|\!||\!|,|\!| |\!|,|\!| |\!||\!|;|\!||\!|;|\!||\!|;|\!| Q|\!|_1|\!| |\!||\!| |\!|=|\!| |\!||\!| p|\!|_y|\!||\!|,|\!| |\!|,|\!|nonumber|\!| |\!||\!||\!| |\!|&|\!|&P|\!|_2|\!| |\!||\!| |\!|=|\!| |\!||\!| p|\!|_x|\!| |\!|-|\!| |\!|bar|\!|{x|\!|}|\!||\!|,|\!| |\!|,|\!| |\!|mbox|\!|{|\!|hspace|\!|{2|\!|.15cm|\!|}|\!|}|\!| Q|\!|_2|\!| |\!||\!| |\!|=|\!| |\!||\!| |\!|bar|\!|{p|\!|}|\!|_|\!|{|\!|bar|\!|{x|\!|}|\!|}|\!||\!|,|\!| |\!|,|\!| |\!|nonumber|\!| |\!||\!||\!| |\!|&|\!|&P|\!|_3|\!| |\!||\!| |\!|=|\!| |\!||\!| p|\!|_y|\!| |\!|-|\!| |\!|bar|\!|{y|\!|}|\!||\!|,|\!| |\!|,|\!| |\!|mbox|\!|{|\!|hspace|\!|{2|\!|.15cm|\!|}|\!|}|\!| Q|\!|_3|\!| |\!||\!| |\!|=|\!| |\!||\!| |\!|bar|\!|{p|\!|}|\!|_|\!|{|\!|bar|\!|{y|\!|}|\!|}|\!||\!|,|\!| |\!|,|\!| |\!|nonumber|\!| |\!||\!||\!| |\!|&|\!|&|\!|bar|\!|{P|\!|}|\!|~|\!| |\!||\!| |\!|=|\!| |\!||\!| |\!|bar|\!|{p|\!|}|\!|_|\!|{|\!|bar|\!|{x|\!|}|\!|}|\!| |\!|+|\!| dp|\!|_y|\!|-|\!| x|\!||\!|,|\!| |\!|,|\!| |\!|mbox|\!|{|\!|hspace|\!|{1|\!|.2cm|\!|}|\!|}|\!| |\!|bar|\!|{Q|\!|}|\!|~|\!| |\!||\!| |\!|=|\!| |\!||\!| p|\!|_x|\!||\!|,|\!| |\!|,|\!| |\!|end|\!|{eqnarray|\!|}|\!|
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and|\!| the|\!| Hamiltonian|\!| |\!|$K|\!|$|\!| reads|\!|
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|\!|begin|\!|{eqnarray|\!|}|\!| K|\!|(|\!|bar|\!|{P|\!|}|\!|,|\!| |\!|bar|\!|{Q|\!|}|\!|,|\!| Q|\!|_1|\!|)|\!| |\!||\!| |\!|=|\!| |\!||\!| |\!|-|\!|bar|\!|{P|\!|}Q|\!|_1|\!| |\!|+|\!| d|\!| Q|\!|_1|\!|^2|\!| |\!|-|\!| d|\!| |\!|bar|\!|{Q|\!|}|\!|^2|\!||\!|,|\!| |\!|.|\!| |\!|end|\!|{eqnarray|\!|}|\!|
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The|\!| functional|\!| |\!|$|\!|delta|\!|$|\!|-function|\!| |\!|(|\!|ref|\!|{4|\!|.23|\!|}|\!|)|\!| now|\!| has|\!| the|\!| form|\!|
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|\!|begin|\!|{eqnarray|\!|}|\!| |\!|delta|\!|[Q|\!|_1|\!| |\!|-|\!| Q|\!|_1|\!|^|\!|*|\!|(|\!|bar|\!|{P|\!|}|\!|,|\!| |\!|bar|\!|{Q|\!|}|\!|)|\!|]|\!| |\!||\!| |\!|=|\!| |\!||\!| |\!|delta|\!|[Q|\!|_1|\!| |\!|-|\!| |\!|frac|\!|{1|\!|}|\!|{2d|\!|}|\!||\!| |\!|bar|\!|{P|\!|}|\!|]|\!||\!|,|\!| |\!|.|\!| |\!|end|\!|{eqnarray|\!|}|\!|
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This|\!| finally|\!| implies|\!| that|\!| the|\!| Hamiltonian|\!| on|\!| the|\!| physical|\!| space|\!| |\!|$|\!|Gamma|\!|^|\!|*|\!|$|\!| has|\!| the|\!| form|\!| |\!|$K|\!|^|\!|*|\!|(|\!|bar|\!|{P|\!|}|\!|,|\!| |\!|bar|\!|{Q|\!|}|\!|)|\!| |\!|=|\!| H|\!|_|\!|+|\!|^|\!|*|\!|(|\!|bar|\!|{P|\!|}|\!|,|\!| |\!|bar|\!|{Q|\!|}|\!|)|\!| |\!|=|\!| |\!|-|\!|(1|\!|/4d|\!|)|\!| |\!|bar|\!|{P|\!|}|\!|^2|\!| |\!|-|\!| d|\!| |\!|bar|\!|{Q|\!|}|\!|^2|\!|$|\!|.|\!| By|\!| choosing|\!| |\!|$d|\!| |\!|=|\!| |\!|-|\!| m|\!|hbar|\!|/2|\!|$|\!| and|\!| transforming|\!| |\!|$|\!|bar|\!|{Q|\!|}|\!| |\!|mapsto|\!| |\!|bar|\!|{Q|\!|}|\!|/|\!|hbar|\!|$|\!| in|\!| the|\!| path|\!| integral|\!| |\!|(|\!|ref|\!|{4|\!|.22|\!|}|\!|)|\!| |\!|(resp|\!|.|\!| |\!|(|\!|ref|\!|{4|\!|.30|\!|}|\!|)|\!|)|\!| we|\!| obtain|\!| the|\!| quantum|\!| partition|\!| function|\!| for|\!| a|\!| system|\!| described|\!| by|\!| the|\!| Hamiltonian|\!|:|\!| |\!|$|\!|(1|\!|/2m|\!|)|\!|bar|\!|{P|\!|}|\!|^2|\!| |\!| |\!|+|\!| |\!|(m|\!|/2|\!|)|\!|bar|\!|{Q|\!|}|\!|^2|\!|$|\!|,|\!| i|\!|.e|\!|.|\!|,|\!| the|\!| linear|\!| harmonic|\!| oscillator|\!| with|\!| a|\!| unit|\!| frequency|\!|.|\!| This|\!| is|\!| precisely|\!| the|\!| result|\!| which|\!| in|\!| the|\!| context|\!| of|\!| the|\!| system|\!| |\!|(|\!|ref|\!|{5|\!|.1|\!|}|\!|)|\!| was|\!| originally|\!| conjectured|\!| by|\!| |\!|'t|\!|~Hooft|\!| in|\!| Ref|\!|.|\!|~|\!|cite|\!|{tHooft3|\!|}|\!|.|\!| Note|\!| again|\!| that|\!| the|\!| fundamental|\!| scale|\!| |\!|(suggestively|\!| denoted|\!| as|\!| |\!|$|\!|hbar|\!|$|\!|)|\!| was|\!| implemented|\!| into|\!| the|\!| theory|\!| via|\!| the|\!| |\!|`|\!|`loss|\!| of|\!| information|\!|"|\!| condition|\!|.|\!|
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|\!|subsection|\!|{Free|\!| particle|\!| weakly|\!| coupled|\!| to|\!| Duffing|\!|'s|\!| oscillator|\!|}|\!|
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There|\!| is|\!| no|\!| difficulty|\!|,|\!| in|\!| principle|\!|,|\!| in|\!| carrying|\!| over|\!| our|\!| procedure|\!| to|\!| non|\!|-linear|\!| dynamical|\!| systems|\!|.|\!| As|\!| an|\!| illustration|\!| we|\!| will|\!| consider|\!| here|\!| the|\!| R|\!||\!|"|\!|{o|\!|}ssler|\!| system|\!|.|\!| This|\!| is|\!| a|\!| three|\!|-dimensional|\!| continuous|\!|-time|\!| chaotic|\!| system|\!| described|\!| by|\!| the|\!| three|\!| autonomous|\!| nonlinear|\!| equations|\!|
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|\!|begin|\!|{eqnarray|\!|}|\!| |\!|frac|\!|{dx|\!|}|\!|{dt|\!|}|\!| |\!|&|\!|=|\!|&|\!| |\!|-y|\!| |\!|-|\!| z|\!||\!|,|\!| |\!|,|\!|nonumber|\!| |\!||\!||\!| |\!|frac|\!|{dy|\!|}|\!|{dt|\!|}|\!| |\!|&|\!|=|\!|&|\!| x|\!| |\!|+|\!| Ay|\!||\!|,|\!| |\!|,|\!|nonumber|\!| |\!||\!||\!| |\!|frac|\!|{dz|\!|}|\!|{dt|\!|}|\!| |\!|&|\!|=|\!|&|\!| B|\!| |\!|+|\!| xz|\!| |\!|-|\!| Cz|\!||\!|,|\!| |\!|,|\!| |\!|label|\!|{5|\!|.3|\!|}|\!| |\!|end|\!|{eqnarray|\!|}|\!|
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where|\!| |\!|$A|\!|$|\!|,|\!| |\!|$B|\!|$|\!|,|\!| and|\!| |\!|$C|\!|$|\!| are|\!| adjustable|\!| constants|\!|.|\!| The|\!| associated|\!| |\!|'t|\!||\!|,Hooft|\!| Hamiltonian|\!| reads|\!|
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|\!|begin|\!|{eqnarray|\!|}|\!| H|\!| |\!|=|\!| |\!|-p|\!|_x|\!|(y|\!| |\!|+|\!| z|\!|)|\!| |\!|+|\!| p|\!|_y|\!|(x|\!| |\!|+|\!| Ay|\!|)|\!| |\!|+|\!| p|\!|_z|\!|(B|\!| |\!|+|\!| xz|\!| |\!|-|\!| Cz|\!|)|\!||\!|,|\!| |\!|,|\!| |\!|label|\!|{5|\!|.4|\!|}|\!| |\!|end|\!|{eqnarray|\!|}|\!|
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and|\!| the|\!| Lagrangian|\!| |\!|(|\!|ref|\!|{lag1|\!|}|\!|)|\!| has|\!| the|\!| form|\!|
|\!|
|\!|begin|\!|{eqnarray|\!|}|\!| |\!|overlinen|\!| L|\!| |\!|=|\!| |\!|overlinen|\!|{x|\!|}|\!|dot|\!|{x|\!|}|\!| |\!|+|\!| |\!|overlinen|\!|{y|\!|}|\!|dot|\!|{y|\!|}|\!| |\!|+|\!| |\!|overlinen|\!|{z|\!|}|\!|dot|\!|{z|\!|}|\!| |\!|+|\!| |\!|overlinen|\!|{x|\!|}|\!|(y|\!| |\!|+|\!| z|\!|)|\!|
|\!| |\!|-|\!| |\!|overlinen|\!|{y|\!|}|\!|(x|\!| |\!|+Ay|\!|)|\!| |\!| |\!|-|\!| |\!|overlinen|\!|{z|\!|}|\!|(B|\!| |\!|+xz|\!| |\!|+Cz|\!|)|\!||\!|,|\!| |\!|.|\!| |\!|end|\!|{eqnarray|\!|}|\!|
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|\!|noi|\!| The|\!| R|\!||\!|"|\!|{o|\!|}ssler|\!| system|\!| is|\!| considered|\!| to|\!| be|\!| the|\!| simplest|\!| possible|\!| chaotic|\!| attractor|\!| with|\!| important|\!| applications|\!| in|\!| far|\!|-from|\!|-equilibrium|\!| chemical|\!| kinetics|\!|~|\!|cite|\!|{Ruelle|\!|}|\!|.|\!| It|\!| also|\!| frequently|\!| serves|\!| as|\!| a|\!| playground|\!| for|\!| studying|\!|,|\!| e|\!|.g|\!|.|\!|,|\!| period|\!|-doubling|\!| bifurcation|\!| cycles|\!| or|\!| Feigenbaum|\!|'s|\!| universality|\!| theory|\!|.|\!| For|\!| the|\!| sake|\!| of|\!| an|\!| explicit|\!| analytic|\!| solution|\!| we|\!| will|\!| confine|\!| ourselves|\!| only|\!| to|\!| the|\!| special|\!| case|\!| when|\!| |\!|$A|\!| |\!|=|\!| B|\!| |\!|=|\!| C|\!| |\!|=|\!| 0|\!|$|\!|.|\!| With|\!| such|\!| a|\!| choice|\!| of|\!| parameters|\!| the|\!| R|\!||\!|"|\!|{o|\!|}ssler|\!| system|\!| can|\!| be|\!| expressed|\!| in|\!| a|\!| scalar|\!| form|\!| as|\!| |\!|$|\!|dddot|\!|{y|\!|}|\!| |\!| |\!|=|\!| |\!| y|\!|dot|\!|{y|\!|}|\!| |\!|+|\!|dot|\!|{y|\!|}|\!|ddot|\!|{y|\!|}|\!| |\!|-|\!| |\!|dot|\!|{y|\!|}|\!| |\!|$|\!| which|\!| ensures|\!| its|\!| integrability|\!|~|\!|cite|\!|{Hied|\!|}|\!|.|\!| The|\!| latter|\!| implies|\!| that|\!| in|\!| this|\!| regime|\!| R|\!||\!|"|\!|{o|\!|}ssler|\!|'s|\!| system|\!| does|\!| not|\!| posses|\!| chaotic|\!| attractors|\!|.|\!|
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|\!|vspace|\!|{3mm|\!|}|\!|
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To|\!| proceed|\!| further|\!|,|\!| we|\!| should|\!| realize|\!| that|\!| because|\!| |\!|$C|\!|_i|\!|$|\!| are|\!| supposed|\!| to|\!| be|\!| |\!|$|\!|{|\!|boldsymbol|\!|{p|\!|}|\!|}|\!|$|\!|-independent|\!| their|\!| finding|\!| is|\!| equivalent|\!| to|\!| specifying|\!| the|\!| first|\!| integrals|\!| of|\!| the|\!| system|\!| |\!|(|\!|ref|\!|{5|\!|.3|\!|}|\!|)|\!| |\!|(i|\!|.e|\!|.|\!|,|\!| functions|\!| that|\!| are|\!| constant|\!| along|\!| lines|\!| of|\!| |\!|$|\!|(x|\!|,y|\!|,z|\!|)|\!|$|\!| satisfying|\!| |\!|(|\!|ref|\!|{5|\!|.3|\!|}|\!|)|\!|)|\!|.|\!| In|\!| other|\!| words|\!|,|\!| the|\!| differential|\!| equations|\!| |\!|(|\!|ref|\!|{5|\!|.3|\!|}|\!|)|\!| represent|\!| a|\!| characteristic|\!| system|\!| for|\!| the|\!| differential|\!| equation|\!| |\!|$|\!||\!|{H|\!|,|\!| C|\!|_i|\!||\!|}|\!| |\!|=|\!| 0|\!|$|\!|.|\!| It|\!| is|\!| simple|\!| to|\!| see|\!| that|\!| the|\!| first|\!| integrals|\!| of|\!| the|\!| above|\!| R|\!||\!|"|\!|{o|\!|}ssler|\!| system|\!| are|\!| |\!|$x|\!|^2|\!| |\!|+|\!| y|\!|^2|\!| |\!|+|\!| 2z|\!|$|\!| and|\!| |\!|$z|\!| e|\!|^|\!|{|\!|-y|\!|}|\!|$|\!|,|\!| hence|\!| we|\!| can|\!| identify|\!| |\!|$C|\!|_1|\!|$|\!| and|\!| |\!|$C|\!|_2|\!|$|\!| with|\!|
|\!|
|\!|begin|\!|{eqnarray|\!|}|\!| C|\!|_1|\!| |\!||\!| |\!|=|\!| |\!||\!| |\!|(x|\!|^2|\!| |\!|+|\!| y|\!|^2|\!| |\!|+|\!| 2z|\!|)|\!|^2|\!||\!|,|\!| |\!| |\!|,|\!||\!|;|\!||\!|;|\!||\!|;|\!||\!|;|\!||\!|;|\!| C|\!|_2|\!| |\!||\!| |\!|=|\!| |\!||\!| z|\!|^2|\!| e|\!|^|\!|{|\!|-2y|\!|}|\!||\!|,|\!| |\!|.|\!| |\!|end|\!|{eqnarray|\!|}|\!|
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The|\!| previous|\!| choice|\!| provides|\!| indeed|\!| positive|\!| and|\!| irreducible|\!| charges|\!|.|\!| The|\!| first|\!| class|\!| constraint|\!| |\!|$|\!|varphi|\!|$|\!| then|\!| reads|\!|
|\!|
|\!|begin|\!|{eqnarray|\!|}|\!| |\!|varphi|\!| |\!||\!| |\!|&|\!|=|\!|&|\!| |\!||\!| |\!|-|\!||\!| p|\!|_x|\!|(y|\!| |\!|+|\!| z|\!|)|\!| |\!||\!| |\!|+|\!| |\!||\!| p|\!|_y|\!| x|\!| |\!| |\!||\!| |\!|+|\!| |\!||\!| p|\!|_z|\!| xz|\!| |\!| |\!|-|\!| |\!| a|\!|_1|\!| |\!|(x|\!|^2|\!| |\!|+|\!| y|\!|^2|\!| |\!|+|\!| 2z|\!|)|\!|^2|\!| |\!| |\!|-|\!| |\!| a|\!|_2|\!| z|\!|^2|\!| e|\!|^|\!|{|\!|-2y|\!|}|\!| |\!|nonumber|\!| |\!||\!||\!| |\!|&|\!|&|\!||\!| |\!|-|\!||\!| |\!|bar|\!|{p|\!|}|\!|_|\!|{|\!|bar|\!|{x|\!|}|\!|}|\!|left|\!|(|\!| |\!|bar|\!|{y|\!|}|\!| |\!||\!| |\!|+|\!| |\!||\!| |\!|bar|\!|{z|\!|}z|\!| |\!| |\!| |\!||\!| |\!|-|\!| |\!||\!| 4|\!| a|\!|_1|\!| x|\!| |\!|(x|\!|^2|\!| |\!|+|\!| y|\!|^2|\!| |\!|+|\!| 2z|\!|)|\!|right|\!|)|\!| |\!||\!| |\!|+|\!| |\!||\!| |\!|bar|\!|{p|\!|}|\!|_|\!|{|\!|bar|\!|{y|\!|}|\!|}|\!|left|\!|(|\!|bar|\!|{x|\!|}|\!| |\!||\!| |\!|+|\!| |\!||\!| 4|\!| a|\!|_1|\!| y|\!| |\!|(x|\!|^2|\!| |\!|+|\!| y|\!|^2|\!| |\!|+|\!| 2z|\!|)|\!| |\!| |\!|-|\!|
|\!| 2a|\!|_2|\!| z|\!|^2|\!| e|\!|^|\!|{|\!|-2y|\!|}|\!| |\!|right|\!|)|\!|nonumber|\!| |\!||\!||\!| |\!|&|\!|&|\!| |\!||\!| |\!|+|\!| |\!||\!| |\!|bar|\!|{p|\!|}|\!|_|\!|{|\!|bar|\!|{z|\!|}|\!|}|\!| |\!|left|\!|(|\!| |\!|bar|\!|{x|\!|}|\!| |\!||\!| |\!|-|\!| |\!||\!| |\!| |\!|bar|\!|{z|\!|}x|\!| |\!||\!| |\!|+|\!| |\!||\!| 4|\!| a|\!|_1|\!| |\!|(x|\!|^2|\!| |\!|+|\!| y|\!|^2|\!| |\!|+|\!| 2z|\!|)|\!| |\!||\!| |\!|+|\!| |\!||\!| 2a|\!|_2|\!| z|\!| e|\!|^|\!|{|\!|-2y|\!|}|\!|right|\!|)|\!| |\!||\!|,|\!| |\!|,|\!|nonumber|\!| |\!||\!||\!| |\!|&|\!|approx|\!|&|\!| |\!||\!| H|\!| |\!|-|\!| a|\!|_1|\!| C|\!|_1|\!| |\!|-|\!| a|\!|_2|\!| C|\!|_2|\!| |\!||\!|,|\!| |\!|.|\!| |\!|end|\!|{eqnarray|\!|}|\!|
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Explicit|\!| values|\!| of|\!| |\!|$a|\!|_1|\!|$|\!| and|\!| |\!|$a|\!|_2|\!|$|\!| will|\!| be|\!| fixed|\!| in|\!| the|\!| footnote|\!| |\!|$5|\!|$|\!|.|\!| A|\!| little|\!| algebra|\!| shows|\!| that|\!| the|\!| gauge|\!| condition|\!| |\!|$|\!|chi|\!|$|\!| can|\!| be|\!| selected|\!|,|\!| for|\!| instance|\!|,|\!| as|\!|
|\!|
|\!|begin|\!|{eqnarray|\!|}|\!| |\!|chi|\!| |\!||\!| |\!|=|\!| |\!||\!| |\!|bar|\!|{p|\!|}|\!|_|\!|{|\!|bar|\!|{x|\!|}|\!|}|\!| |\!|-|\!| y|\!||\!|,|\!| |\!| |\!|.|\!| |\!|end|\!|{eqnarray|\!|}|\!|
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Such|\!| a|\!| choice|\!| satisfies|\!| the|\!| necessary|\!| conditions|\!|
|\!|
|\!|begin|\!|{eqnarray|\!|}|\!| |\!||\!|{|\!| |\!|chi|\!|,|\!| |\!|varphi|\!| |\!||\!|}|\!| |\!||\!| |\!|=|\!| |\!||\!| |\!|bar|\!|{p|\!|}|\!|_|\!|{|\!|bar|\!|{y|\!|}|\!|}|\!| |\!|+|\!| |\!|bar|\!|{p|\!|}|\!|_|\!|{|\!|bar|\!|{z|\!|}|\!|}|\!| |\!|+|\!| x|\!| |\!||\!| |\!|neq|\!| |\!||\!| 0|\!||\!|,|\!| |\!|,|\!||\!|;|\!||\!|;|\!||\!|;|\!||\!|;|\!||\!|;|\!||\!|;|\!||\!|;|\!||\!|;|\!||\!|;|\!||\!|;|\!| |\!||\!|{|\!| |\!|chi|\!|,|\!| |\!|phi|\!|_i|\!| |\!||\!|}|\!| |\!||\!| |\!|=|\!| |\!||\!| 0|\!|,|\!| |\!||\!|;|\!||\!|;|\!||\!|;|\!| i|\!| |\!||\!| |\!|=|\!| |\!||\!| 1|\!|,|\!| |\!|ldots|\!|,|\!| 6|\!||\!|,|\!| |\!|.|\!| |\!|end|\!|{eqnarray|\!|}|\!|
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The|\!| above|\!| |\!|$|\!|chi|\!|$|\!| also|\!| allows|\!| us|\!| to|\!| perform|\!| the|\!| following|\!| linear|\!| canonical|\!| transformation|\!|:|\!|
|\!|
|\!|begin|\!|{eqnarray|\!|}|\!| |\!|begin|\!|{array|\!|}|\!|{ll|\!|}|\!| P|\!|_1|\!| |\!||\!| |\!|=|\!| |\!||\!| |\!|chi|\!| |\!||\!| |\!|=|\!| |\!||\!| |\!|bar|\!|{p|\!|}|\!|_|\!|{|\!|bar|\!|{x|\!|}|\!|}|\!| |\!||\!| |\!|-|\!| |\!||\!| y|\!||\!|,|\!| |\!|,|\!| |\!|&|\!|~|\!|~|\!|~|\!|~|\!|~|\!|~|\!| Q|\!|_1|\!| |\!||\!| |\!|=|\!| |\!||\!| p|\!|_y|\!| |\!||\!|,|\!| |\!|,|\!| |\!||\!||\!| P|\!|_2|\!| |\!||\!| |\!|=|\!| |\!||\!| p|\!|_x|\!| |\!||\!| |\!|-|\!| |\!||\!| |\!|bar|\!|{x|\!|}|\!||\!|,|\!| |\!|,|\!| |\!|&|\!|~|\!|~|\!|~|\!|~|\!|~|\!|~|\!| Q|\!|_2|\!| |\!||\!| |\!|=|\!| |\!||\!| |\!|bar|\!|{p|\!|}|\!|_|\!|{|\!|bar|\!|{x|\!|}|\!|}|\!||\!|,|\!| |\!|,|\!| |\!| |\!||\!||\!| P|\!|_3|\!| |\!||\!| |\!|=|\!| |\!||\!| p|\!|_y|\!| |\!||\!| |\!|-|\!| |\!||\!| |\!|bar|\!|{y|\!|}|\!||\!|,|\!| |\!|,|\!| |\!|&|\!|~|\!|~|\!|~|\!|~|\!|~|\!|~|\!| Q|\!|_3|\!| |\!||\!| |\!|=|\!| |\!||\!| |\!|bar|\!|{p|\!|}|\!|_|\!|{|\!|bar|\!|{y|\!|}|\!|}|\!||\!|,|\!| |\!|,|\!| |\!||\!||\!| P|\!|_4|\!| |\!||\!| |\!|=|\!| |\!||\!| p|\!|_z|\!| |\!||\!| |\!|-|\!| |\!||\!| |\!|bar|\!|{z|\!|}|\!||\!|,|\!| |\!|,|\!| |\!|&|\!|~|\!|~|\!|~|\!|~|\!|~|\!|~|\!| Q|\!|_4|\!| |\!||\!| |\!|=|\!| |\!||\!| |\!|bar|\!|{p|\!|}|\!|_|\!|{|\!|bar|\!|{z|\!|}|\!|}|\!||\!|,|\!| |\!|,|\!| |\!| |\!||\!||\!| |\!|bar|\!|{P|\!|}|\!|_1|\!| |\!||\!| |\!|=|\!| |\!||\!| |\!|(|\!|bar|\!|{p|\!|}|\!|_|\!|{|\!|bar|\!|{z|\!|}|\!|}|\!|/d|\!| |\!||\!| |\!|-|\!| |\!||\!| z|\!|/d|\!| |\!|)|\!|/|\!|sqrt|\!|{2|\!|}|\!| |\!||\!|,|\!| |\!|,|\!| |\!|&|\!|~|\!|~|\!|~|\!|~|\!|~|\!|~|\!| |\!|bar|\!|{Q|\!|}|\!|_1|\!| |\!||\!| |\!|=|\!| |\!||\!| |\!|(2dp|\!|_z|\!| |\!||\!| |\!|-|\!| |\!||\!| |\!|bar|\!|{p|\!|}|\!|_|\!|{|\!|bar|\!|{x|\!|}|\!|}|\!|/c|\!| |\!||\!| |\!|+|\!| |\!||\!| x|\!|/c|\!|)|\!|/|\!|sqrt|\!|{2|\!|}|\!| |\!||\!|,|\!| |\!|,|\!| |\!||\!||\!| |\!|bar|\!|{P|\!|}|\!|_2|\!| |\!||\!| |\!|=|\!| |\!||\!| |\!|(2cp|\!|_x|\!| |\!||\!| |\!|-|\!| |\!||\!| |\!|bar|\!|{p|\!|}|\!|_|\!|{|\!|bar|\!|{z|\!|}|\!|}|\!|/d|\!| |\!||\!| |\!|+|\!| |\!| |\!||\!| z|\!|/d|\!|)|\!|/|\!|sqrt|\!|{2|\!|}|\!||\!|,|\!| |\!|,|\!| |\!|&|\!|~|\!|~|\!|~|\!|~|\!|~|\!|~|\!| |\!|bar|\!|{Q|\!|}|\!|_2|\!| |\!||\!| |\!|=|\!| |\!||\!| |\!|(x|\!|/c|\!| |\!||\!| |\!|-|\!| |\!||\!| |\!|bar|\!|{p|\!|}|\!|_|\!|{|\!|bar|\!|{x|\!|}|\!|}|\!|/c|\!|)|\!|/|\!|sqrt|\!|{2|\!|}|\!| |\!||\!|,|\!| |\!|.|\!| |\!|end|\!|{array|\!|}|\!| |\!|label|\!|{5|\!|.7|\!|}|\!| |\!|end|\!|{eqnarray|\!|}|\!|
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Here|\!| |\!|$c|\!|$|\!| and|\!| |\!|$d|\!|$|\!| represent|\!| arbitrary|\!| real|\!| constants|\!| to|\!| be|\!| specified|\!| later|\!|.|\!| The|\!| transformation|\!| |\!|(|\!|ref|\!|{5|\!|.7|\!|}|\!|)|\!| secures|\!| the|\!| unique|\!| global|\!| solution|\!| |\!|$Q|\!|_1|\!|$|\!| for|\!| |\!|$|\!|varphi|\!| |\!|=|\!| 0|\!|$|\!| on|\!| |\!|$|\!|Gamma|\!|^|\!|*|\!|$|\!|.|\!| To|\!| show|\!| this|\!| it|\!| is|\!| sufficient|\!| to|\!| observe|\!| that|\!| |\!|$|\!|left|\!|.|\!|left|\!|[H|\!| |\!|-|\!| a|\!|_1C|\!|_1|\!| |\!|-|\!| a|\!|_2C|\!|_2|\!|
|\!|right|\!|]|\!|right|\!|||\!|_|\!|{|\!|Gamma|\!|^|\!|*|\!|}|\!|$|\!| is|\!| linear|\!| in|\!| |\!|$Q|\!|_1|\!|$|\!|.|\!| Indeed|\!|,|\!|
|\!|
|\!|begin|\!|{eqnarray|\!|}|\!|
|\!|left|\!|.|\!|left|\!|[H|\!| |\!|-|\!| a|\!|_1C|\!|_1|\!| |\!|-|\!| a|\!|_2C|\!|_2|\!| |\!|right|\!|]|\!|right|\!|||\!|_|\!|{|\!|Gamma|\!|^|\!|*|\!|}|\!| |\!||\!| |\!|&|\!|=|\!|&|\!| |\!||\!| |\!|sqrt|\!|{2|\!|}|\!| c|\!| |\!||\!| Q|\!|_1|\!| |\!|bar|\!|{Q|\!|}|\!|_2|\!| |\!|-|\!| |\!|sqrt|\!|{2|\!| |\!|}c|\!| |\!||\!| |\!|(|\!|bar|\!|{Q|\!|}|\!|_1|\!| |\!|-|\!| |\!|bar|\!|{Q|\!|}|\!|_2|\!|)|\!|bar|\!|{Q|\!|}|\!|_2|\!| |\!|bar|\!|{P|\!|}|\!|_1|\!| |\!| |\!|+|\!| d|\!|/c|\!||\!| |\!|(|\!|bar|\!|{P|\!|}|\!|_1|\!| |\!|+|\!| |\!|bar|\!|{P|\!|}|\!|_2|\!|)|\!| |\!|bar|\!|{P|\!|}|\!|_1|\!|nonumber|\!| |\!||\!||\!| |\!|&|\!|-|\!|&|\!| |\!|{|\!|mathcal|\!|{A|\!|}|\!|}|\!| |\!||\!| |\!|(|\!|bar|\!|{P|\!|}|\!|_1|\!|)|\!|^2|\!| |\!|-|\!| |\!|{|\!|mathcal|\!|{B|\!|}|\!|}|\!| |\!||\!| |\!|bar|\!|{P|\!|}|\!|_1|\!| |\!|(|\!|bar|\!|{Q|\!|}|\!|_2|\!|)|\!|^2|\!| |\!|-|\!| |\!|{|\!|mathcal|\!|{C|\!|}|\!|}|\!| |\!||\!| |\!|(|\!|bar|\!|{Q|\!|}|\!|_2|\!|)|\!|^4|\!||\!|,|\!| |\!|,|\!| |\!|end|\!|{eqnarray|\!|}|\!|
|\!|
with|\!| |\!|$|\!|{|\!|mathcal|\!|{A|\!|}|\!|}|\!| |\!|=|\!| 2d|\!|^2|\!|(4|\!| a|\!|_1|\!| |\!|+|\!| a|\!|_2|\!|)|\!|$|\!|,|\!| |\!|$|\!|{|\!|mathcal|\!|{B|\!|}|\!|}|\!| |\!|=|\!| |\!|-|\!| 8|\!|sqrt|\!|{2|\!|}a|\!|_1|\!| |\!| d|\!| c|\!|^2|\!|$|\!| and|\!| |\!|$|\!|{|\!|mathcal|\!|{C|\!|}|\!|}|\!| |\!|=|\!| 4a|\!|_1|\!| c|\!|^4|\!|$|\!|.|\!| As|\!| a|\!| result|\!|
|\!|
|\!|begin|\!|{eqnarray|\!|}|\!| K|\!|^|\!|*|\!|(|\!|bar|\!|{|\!|boldsymbol|\!|{P|\!|}|\!|}|\!|,|\!| |\!|bar|\!|{|\!|boldsymbol|\!|{Q|\!|}|\!|}|\!|)|\!| |\!|=|\!| H|\!|^|\!|*|\!|_|\!|+|\!|(|\!|bar|\!|{|\!|boldsymbol|\!|{P|\!|}|\!|}|\!|,|\!| |\!|bar|\!|{|\!|boldsymbol|\!|{Q|\!|}|\!|}|\!|)|\!| |\!|=|\!| |\!|{|\!|mathcal|\!|{A|\!|}|\!|}|\!| |\!||\!| |\!|(|\!|bar|\!|{P|\!|}|\!|_1|\!|)|\!|^2|\!| |\!|+|\!| |\!|{|\!|mathcal|\!|{B|\!|}|\!|}|\!| |\!||\!| |\!|bar|\!|{P|\!|}|\!|_1|\!| |\!|(|\!|bar|\!|{Q|\!|}|\!|_2|\!|)|\!|^2|\!| |\!|+|\!| |\!|{|\!|mathcal|\!|{C|\!|}|\!|}|\!| |\!||\!| |\!|(|\!|bar|\!|{Q|\!|}|\!|_2|\!|)|\!|^4|\!||\!|,|\!| |\!|.|\!| |\!|end|\!|{eqnarray|\!|}|\!|
|\!|
Inserting|\!| this|\!| into|\!| |\!|(|\!|ref|\!|{4|\!|.22|\!|}|\!|)|\!| |\!|(resp|\!|.|\!| |\!|(|\!|ref|\!|{4|\!|.30|\!|}|\!|)|\!|)|\!| and|\!| integrating|\!| over|\!| |\!|$|\!|bar|\!|{P|\!|}|\!|_1|\!|$|\!| and|\!| |\!|$|\!|bar|\!|{P|\!|}|\!|_2|\!|$|\!| we|\!| obtain|\!| the|\!| following|\!| chain|\!| of|\!| identities|\!|:|\!|
|\!|
|\!|begin|\!|{eqnarray|\!|}|\!| Z|\!|_|\!|{|\!|rm|\!| CM|\!|}|\!| |\!||\!| |\!|&|\!|=|\!|&|\!| |\!||\!| |\!|int|\!| |\!|{|\!|mathcal|\!|{D|\!|}|\!|}|\!|bar|\!|{|\!|boldsymbol|\!|{P|\!|}|\!|}|\!| |\!|{|\!|mathcal|\!|{D|\!|}|\!|}|\!|bar|\!|{|\!|boldsymbol|\!|{Q|\!|}|\!|}|\!| |\!||\!| |\!|exp|\!|left|\!||\!|{|\!| i|\!||\!|!|\!||\!|!|\!|int|\!|_|\!|{t|\!|_1|\!|}|\!|^|\!|{t|\!|_2|\!|}|\!| dt|\!| |\!||\!| |\!|[|\!|bar|\!|{|\!|boldsymbol|\!|{P|\!|}|\!|}|\!| |\!|dot|\!|{|\!|bar|\!|{|\!|boldsymbol|\!|{Q|\!|}|\!|}|\!|}|\!| |\!|-|\!| |\!|{|\!|mathcal|\!|{A|\!|}|\!|}|\!| |\!||\!| |\!|(|\!|bar|\!|{P|\!|}|\!|_1|\!|)|\!|^2|\!| |\!|-|\!| |\!|{|\!|mathcal|\!|{B|\!|}|\!|}|\!| |\!||\!| |\!|bar|\!|{P|\!|}|\!|_1|\!| |\!|(|\!|bar|\!|{Q|\!|}|\!|_2|\!|)|\!|^2|\!| |\!|-|\!| |\!|{|\!|mathcal|\!|{C|\!|}|\!|}|\!| |\!||\!| |\!|(|\!|bar|\!|{Q|\!|}|\!|_2|\!|)|\!|^4|\!| |\!|+|\!| |\!|bar|\!|{|\!|boldsymbol|\!|{Q|\!|}|\!|}|\!|{|\!|boldsymbol|\!|{j|\!|}|\!|}|\!|~|\!|]|\!| |\!|right|\!||\!|}|\!| |\!|nonumber|\!| |\!||\!||\!| |\!|&|\!|=|\!|&|\!| |\!||\!| |\!|int|\!| |\!|{|\!|mathcal|\!|{D|\!|}|\!|}|\!|bar|\!|{Q|\!|}|\!|_1|\!| |\!|{|\!|mathcal|\!|{D|\!|}|\!|}|\!|bar|\!|{Q|\!|}|\!|_2|\!| |\!||\!| |\!|delta|\!| |\!|[|\!| |\!|dot|\!|{|\!|bar|\!|{Q|\!|}|\!|}|\!|_2|\!| |\!| |\!|]|\!| |\!||\!| |\!|exp|\!| |\!|left|\!||\!|{|\!| i|\!||\!|!|\!||\!|!|\!|int|\!|_|\!|{t|\!|_1|\!|}|\!|^|\!|{t|\!|_2|\!|}|\!| dt|\!| |\!||\!| |\!|left|\!|[|\!|frac|\!|{1|\!|}|\!|{4|\!|{|\!|mathcal|\!|{A|\!|}|\!|}|\!|}|\!||\!| |\!|(|\!|dot|\!|{|\!|bar|\!|{Q|\!|}|\!|}|\!|_1|\!| |\!|-|\!| |\!|{|\!|mathcal|\!|{B|\!|}|\!|}|\!||\!| |\!|(|\!|bar|\!|{Q|\!|}|\!|_2|\!|)|\!|^2|\!|)|\!|^2|\!| |\!|-|\!| |\!|{|\!|mathcal|\!|{C|\!|}|\!|}|\!|(|\!|bar|\!|{Q|\!|}|\!|_2|\!|)|\!|^4|\!| |\!|+|\!| |\!|bar|\!|{|\!|boldsymbol|\!|{Q|\!|}|\!|}|\!|{|\!|boldsymbol|\!|{j|\!|}|\!|}|\!|~|\!|right|\!|]|\!| |\!|right|\!||\!|}|\!|nonumber|\!| |\!||\!||\!| |\!|&|\!|=|\!|&|\!| |\!||\!| |\!|lim|\!|_|\!|{a|\!|rightarrow|\!| 0|\!|_|\!|+|\!|}|\!| |\!||\!| |\!|int|\!| |\!|{|\!|mathcal|\!|{D|\!|}|\!|}|\!|bar|\!|{Q|\!|}|\!|_1|\!| |\!|{|\!|mathcal|\!|{D|\!|}|\!|}|\!|bar|\!|{Q|\!|}|\!|_2|\!| |\!||\!| |\!|exp|\!|left|\!||\!|{i|\!||\!|!|\!||\!|!|\!| |\!|int|\!|_|\!|{t|\!|_1|\!|}|\!|^|\!|{t|\!|_2|\!|}|\!| dt|\!| |\!||\!| |\!|left|\!|[|\!|frac|\!|{1|\!|}|\!|{4|\!| |\!|{|\!|mathcal|\!|{A|\!|}|\!|}|\!|}|\!| |\!|(|\!|dot|\!|{|\!|bar|\!|{Q|\!|}|\!|}|\!|_1|\!|)|\!|^2|\!| |\!|+|\!| |\!|frac|\!|{1|\!|}|\!|{4|\!| a|\!|}|\!||\!| |\!|(|\!|dot|\!|{|\!|bar|\!|{Q|\!|}|\!|}|\!|_2|\!|)|\!|^2|\!| |\!|-|\!| |\!|frac|\!|{|\!|mathcal|\!|{B|\!|}|\!|}|\!|{2|\!|{|\!|mathcal|\!|{A|\!|}|\!|}|\!|}|\!| |\!||\!| |\!|dot|\!|{|\!|bar|\!|{Q|\!|}|\!|}|\!|_1|\!|(|\!|bar|\!|{Q|\!|}|\!|_2|\!|)|\!|^2|\!|right|\!|]|\!| |\!|right|\!||\!|}|\!| |\!|nonumber|\!| |\!||\!||\!| |\!|&|\!|&|\!||\!| |\!|mbox|\!|{|\!|hspace|\!|{2|\!|.5cm|\!|}|\!|}|\!| |\!|times|\!| |\!||\!| |\!|exp|\!|left|\!||\!|{|\!| i|\!||\!|!|\!||\!|!|\!| |\!|int|\!|_|\!|{t|\!|_1|\!|}|\!|^|\!|{t|\!|_2|\!|}|\!| dt|\!| |\!| |\!|left|\!|[|\!|left|\!|(|\!|
|\!| |\!|frac|\!|{|\!|{|\!|mathcal|\!|{B|\!|}|\!|}|\!|^2|\!|}|\!|{4|\!|{|\!|mathcal|\!|{A|\!|}|\!|}|\!|}|\!| |\!|-|\!| |\!|{|\!|mathcal|\!|{C|\!|}|\!|}|\!|right|\!|)|\!|(|\!|bar|\!|{Q|\!|}|\!|_2|\!|)|\!|^4|\!| |\!|+|\!| |\!|bar|\!|{|\!|boldsymbol|\!|{Q|\!|}|\!|}|\!|{|\!|boldsymbol|\!|{j|\!|}|\!|}|\!|right|\!|]|\!| |\!|right|\!||\!|}|\!||\!|,|\!| |\!|.|\!| |\!|label|\!|{5|\!|.10|\!|}|\!| |\!|end|\!|{eqnarray|\!|}|\!|
|\!|
As|\!| an|\!| explanatory|\!| step|\!| we|\!| should|\!| mention|\!| that|\!| the|\!| formal|\!| measure|\!| in|\!| the|\!| second|\!| equality|\!| of|\!| |\!|(|\!|ref|\!|{5|\!|.10|\!|}|\!|)|\!| has|\!| the|\!| explicit|\!| time|\!|-sliced|\!| form|\!|
|\!|
|\!|begin|\!|{eqnarray|\!|}|\!| |\!|{|\!|mathcal|\!|{D|\!|}|\!|}|\!|bar|\!|{Q|\!|}|\!|_1|\!| |\!|{|\!|mathcal|\!|{D|\!|}|\!|}|\!|bar|\!|{Q|\!|}|\!|_2|\!| |\!||\!| |\!|approx|\!| |\!||\!| |\!|prod|\!|_i|\!| |\!|left|\!|(|\!| |\!|frac|\!|{d|\!|bar|\!|{Q|\!|}|\!|_1|\!|(t|\!|_i|\!|)|\!|}|\!|{|\!|sqrt|\!|{4|\!|pi|\!| i|\!| |\!|epsilon|\!| |\!|{|\!|mathcal|\!|{A|\!|}|\!|}|\!|}|\!|}|\!||\!| d|\!| |\!|bar|\!|{Q|\!|}|\!|_2|\!|(t|\!|_i|\!|)|\!| |\!|right|\!|)|\!||\!|,|\!| |\!|,|\!| |\!|end|\!|{eqnarray|\!|}|\!|
|\!|
while|\!| in|\!| the|\!| third|\!| equality|\!| the|\!| shorthand|\!| notation|\!| |\!|$|\!|{|\!|mathcal|\!|{D|\!|}|\!|}|\!|bar|\!|{Q|\!|}|\!|_1|\!| |\!|{|\!|mathcal|\!|{D|\!|}|\!|}|\!|bar|\!|{Q|\!|}|\!|_2|\!|$|\!| stands|\!| for|\!|
|\!|
|\!|begin|\!|{eqnarray|\!|}|\!| |\!|{|\!|mathcal|\!|{D|\!|}|\!|}|\!|bar|\!|{Q|\!|}|\!|_1|\!| |\!|{|\!|mathcal|\!|{D|\!|}|\!|}|\!|bar|\!|{Q|\!|}|\!|_2|\!| |\!||\!| |\!|approx|\!| |\!||\!| |\!|prod|\!|_i|\!| |\!|left|\!|(|\!| |\!|frac|\!|{d|\!|bar|\!|{Q|\!|}|\!|_1|\!|(t|\!|_i|\!|)|\!| |\!|}|\!|{|\!|sqrt|\!|{4|\!|pi|\!| i|\!| |\!|epsilon|\!| |\!|{|\!|mathcal|\!|{A|\!|}|\!|}|\!|}|\!|}|\!| |\!||\!| |\!|frac|\!|{d|\!|bar|\!|{Q|\!|}|\!|_2|\!|(t|\!|_i|\!|)|\!|}|\!|{|\!|sqrt|\!|{4|\!| |\!|pi|\!| i|\!| a|\!| |\!|epsilon|\!|}|\!|}|\!|right|\!|)|\!||\!|,|\!| |\!|.|\!| |\!|end|\!|{eqnarray|\!|}|\!|
|\!|
The|\!| symbol|\!| |\!|$|\!|epsilon|\!|$|\!| represents|\!| the|\!| infinitesimal|\!| width|\!| of|\!| the|\!| time|\!| slicing|\!|.|\!| During|\!| our|\!| derivation|\!| we|\!| have|\!| used|\!| the|\!| Fresnel|\!| integral|\!|
|\!|
|\!|begin|\!|{eqnarray|\!|}|\!| |\!|int|\!|_|\!|{|\!|-|\!|infty|\!|}|\!|^|\!|{|\!|infty|\!|}|\!| dx|\!| |\!||\!| e|\!|^|\!|{|\!|-i|\!| a|\!| x|\!|^2|\!| |\!|+|\!| i|\!| x|\!|xi|\!|}|\!| |\!||\!| |\!|=|\!| |\!||\!| |\!|sqrt|\!|{|\!|frac|\!|{|\!|pi|\!|}|\!|{a|\!|}|\!|}|\!||\!| |\!||\!| e|\!|^|\!|{i|\!| |\!|(|\!|xi|\!|^2|\!|/a|\!| |\!|-|\!| |\!|pi|\!|)|\!|/4|\!| |\!|}|\!| |\!||\!| |\!|=|\!| |\!||\!| |\!|sqrt|\!|{|\!|frac|\!|{|\!|pi|\!|}|\!|{i|\!| a|\!|}|\!|}|\!||\!| |\!||\!| e|\!|^|\!|{i|\!| |\!|xi|\!|^2|\!|/|\!|(4a|\!|)|\!|}|\!||\!|,|\!| |\!|,|\!| |\!||\!|;|\!||\!|;|\!||\!|;|\!||\!|;|\!||\!|;|\!||\!|;|\!||\!|;|\!||\!|;|\!||\!|;|\!| a|\!| |\!|>|\!| 0|\!||\!|,|\!| |\!|,|\!| |\!|end|\!|{eqnarray|\!|}|\!|
|\!|
and|\!| the|\!| ensuing|\!| representation|\!| of|\!| the|\!| Dirac|\!| |\!|$|\!|delta|\!|$|\!|-function|\!|:|\!|
|\!|
|\!|begin|\!|{eqnarray|\!|}|\!| |\!|lim|\!| |\!|_|\!|{a|\!|rightarrow|\!| 0|\!|_|\!|{|\!|+|\!|}|\!|}|\!| |\!| |\!|sqrt|\!|{|\!|frac|\!|{1|\!|}|\!|{4i|\!|pi|\!| a|\!|}|\!|}|\!||\!| |\!||\!| e|\!|^|\!|{i|\!|xi|\!|^2|\!|/|\!|(4a|\!|)|\!|}|\!| |\!||\!| |\!|=|\!| |\!||\!| |\!|delta|\!|(|\!|xi|\!|)|\!||\!|,|\!| |\!|.|\!| |\!|label|\!|{5|\!|.11|\!|}|\!| |\!|end|\!|{eqnarray|\!|}|\!|
|\!|
In|\!| the|\!| following|\!| we|\!| perform|\!| the|\!| scale|\!| transformation|\!| |\!|$|\!|bar|\!|{Q|\!|}|\!|_2|\!|/|\!|sqrt|\!|{a|\!|}|\!| |\!|mapsto|\!| |\!|sqrt|\!|{2m|\!|_2|\!|}|\!||\!| |\!|bar|\!|{Q|\!|}|\!|_2|\!|$|\!| and|\!| set|\!| |\!|$|\!|{|\!|mathcal|\!|{A|\!|}|\!|}|\!| |\!|=|\!| 1|\!|/|\!|(2m|\!|_1|\!|)|\!|$|\!|,|\!| |\!|$|\!|{|\!|mathcal|\!|{B|\!|}|\!|}|\!| |\!|=|\!| 1|\!|/|\!|(|\!|sqrt|\!|{m|\!|_1|\!| m|\!|_2|\!|}|\!|)|\!|$|\!| and|\!| |\!|$|\!|{|\!|mathcal|\!|{C|\!|}|\!|}|\!| |\!|=|\!| 1|\!|/m|\!|_2|\!|$|\!|.|\!|~|\!|footnote|\!|{This|\!| choice|\!| is|\!| equivalent|\!| to|\!| the|\!| solution|\!|:|\!|
|\!|
|\!|begin|\!|{eqnarray|\!|*|\!|}|\!| a|\!|_1|\!| |\!|=|\!| |\!|frac|\!|{a|\!|_2|\!|}|\!|{4|\!|}|\!||\!|,|\!| |\!|,|\!| |\!||\!|;|\!||\!|;|\!| d|\!| |\!|=|\!|frac|\!|{1|\!|}|\!|{2|\!|sqrt|\!|{2a|\!|_2m|\!|_1|\!|}|\!|}|\!||\!|,|\!| |\!|,|\!| |\!||\!|;|\!||\!|;|\!| c|\!| |\!|=|\!| |\!|pm|\!|frac|\!|{1|\!|}|\!|{|\!|sqrt|\!|[4|\!|]|\!|{a|\!|_2m|\!|_2|\!|}|\!|}|\!||\!|,|\!| |\!|.|\!| |\!|end|\!|{eqnarray|\!|*|\!|}|\!|
|\!|
Without|\!| loss|\!| of|\!| generality|\!| we|\!| can|\!| set|\!| |\!|$d|\!| |\!|=|\!| 1|\!|/2|\!|$|\!|,|\!| then|\!|:|\!|
|\!|
|\!|begin|\!|{eqnarray|\!|*|\!|}|\!|
|\!| a|\!|_2|\!| |\!|=|\!| |\!|frac|\!|{1|\!|}|\!|{2m|\!|_1|\!|}|\!||\!|,|\!| |\!|,|\!| |\!||\!|;|\!||\!|;|\!| a|\!|_1|\!| |\!|=|\!| |\!|frac|\!|{1|\!|}|\!|{8m|\!|_1|\!|}|\!||\!|,|\!| |\!|,|\!| |\!||\!|;|\!||\!|;|\!| c|\!| |\!|=|\!| |\!|pm|\!|
|\!| 2|\!|^|\!|{3|\!|/4|\!|}|\!| |\!|sqrt|\!|[4|\!|]|\!|{|\!|frac|\!|{m|\!|_1|\!|}|\!|{m|\!|_2|\!|}|\!|}|\!||\!|,|\!| |\!|.|\!| |\!|end|\!|{eqnarray|\!|*|\!|}|\!| |\!|}|\!|
|\!|
The|\!| resulting|\!| partition|\!| function|\!| then|\!| reads|\!|
|\!|
|\!|begin|\!|{eqnarray|\!|}|\!| Z|\!|_|\!|{|\!|rm|\!| CM|\!|}|\!| |\!||\!| |\!|&|\!|=|\!|&|\!| |\!||\!| |\!|lim|\!|_|\!|{|\!|{|\!|rm|\!|{g|\!|}|\!|}|\!|rightarrow|\!| 0|\!|_|\!|+|\!|}|\!|int|\!| |\!|{|\!|mathcal|\!|{D|\!|}|\!|}|\!|bar|\!|{Q|\!|}|\!|_1|\!| |\!|{|\!|mathcal|\!|{D|\!|}|\!|}|\!|bar|\!|{Q|\!|}|\!|_2|\!| |\!||\!| |\!|exp|\!|left|\!||\!|{|\!| i|\!||\!|!|\!||\!|!|\!| |\!|int|\!|_|\!|{t|\!|_1|\!|}|\!|^|\!|{t|\!|_2|\!|}|\!| dt|\!| |\!|left|\!|[|\!|frac|\!|{m|\!|_1|\!|}|\!|{2|\!|}|\!||\!| |\!|(|\!|dot|\!|{|\!|bar|\!|{Q|\!|}|\!|}|\!|_1|\!|)|\!|^2|\!| |\!||\!| |\!|+|\!| |\!||\!| |\!|frac|\!|{m|\!|_2|\!|}|\!|{2|\!|}|\!| |\!||\!| |\!|(|\!|dot|\!|{|\!|bar|\!|{Q|\!|}|\!|}|\!|_2|\!|)|\!|^2|\!|right|\!|]|\!|right|\!||\!|}|\!| |\!|nonumber|\!| |\!||\!||\!| |\!|&|\!|&|\!| |\!|mbox|\!|{|\!|hspace|\!|{2cm|\!|}|\!|}|\!|times|\!| |\!||\!| |\!|exp|\!|left|\!||\!|{|\!| i|\!||\!|!|\!||\!|!|\!| |\!|int|\!|_|\!|{t|\!|_1|\!|}|\!|^|\!|{t|\!|_2|\!|}|\!| dt|\!| |\!|left|\!|[|\!|{|\!|rm|\!|{g|\!|}|\!|}|\!| |\!|sqrt|\!|{|\!|frac|\!|{m|\!|_1m|\!|_2|\!|}|\!|{2|\!|}|\!|}|\!|
|\!| |\!||\!| |\!|dot|\!|{|\!|bar|\!|{Q|\!|}|\!|}|\!|_1|\!|(|\!|bar|\!|{Q|\!|}|\!|_2|\!|)|\!|^2|\!| |\!||\!| |\!|-|\!| |\!||\!| |\!|frac|\!|{m|\!|_2|\!| |\!|{|\!|rm|\!|{g|\!|}|\!|}|\!|^2|\!|}|\!|{4|\!|}|\!| |\!||\!| |\!|(|\!|bar|\!|{Q|\!|}|\!|_2|\!|)|\!|^4|\!| |\!||\!| |\!|+|\!| |\!||\!| |\!|bar|\!|{|\!|boldsymbol|\!|{Q|\!|}|\!|}|\!|{|\!|boldsymbol|\!|{j|\!|}|\!|}|\!|right|\!|]|\!|right|\!||\!|}|\!||\!|,|\!| |\!|,|\!|
|\!| |\!|label|\!|{5|\!|.9|\!|}|\!| |\!|end|\!|{eqnarray|\!|}|\!|
|\!|
where|\!| we|\!| have|\!| set|\!| |\!|$|\!|{|\!|rm|\!|{g|\!|}|\!|}|\!| |\!|=|\!| 2|\!|sqrt|\!|{2|\!|}|\!| a|\!|$|\!|.|\!| |\!| The|\!| system|\!| thus|\!| obtained|\!| describes|\!| a|\!| pure|\!| anharmonic|\!| |\!|(Duffing|\!|'s|\!|)|\!| oscillator|\!| |\!|(|\!|$|\!|bar|\!|{Q|\!|}|\!|_2|\!|$|\!| oscillator|\!|)|\!| weakly|\!| coupled|\!| through|\!| the|\!| Rayleigh|\!| interaction|\!| with|\!| a|\!| free|\!| particle|\!| |\!|(|\!|$|\!|bar|\!|{Q|\!|}|\!|_1|\!|$|\!| particle|\!|)|\!|.|\!| Alternatively|\!|,|\!| when|\!| |\!|$m|\!|_1|\!| |\!|=|\!| m|\!|_2|\!| |\!|=|\!| m|\!|$|\!| we|\!| can|\!| interpret|\!| the|\!| Lagrangian|\!| in|\!| |\!|(|\!|ref|\!|{5|\!|.9|\!|}|\!|)|\!| as|\!| a|\!| planar|\!| system|\!| describing|\!| a|\!| particle|\!| of|\!| mass|\!| |\!|$m|\!|$|\!| in|\!| a|\!| quartic|\!| scalar|\!| potential|\!| |\!|$e|\!|Phi|\!|(|\!|bar|\!|{|\!|boldsymbol|\!|{Q|\!|}|\!|}|\!|)|\!| |\!|=|\!| |\!| m|\!| |\!|{|\!|rm|\!|{g|\!|}|\!|}|\!|^2|\!|/4|\!||\!| |\!|(|\!|bar|\!|{Q|\!|}|\!|_2|\!|)|\!|^4|\!|$|\!| and|\!| a|\!| vector|\!| potential|\!| |\!|$e|\!|{|\!|boldsymbol|\!|{A|\!|}|\!|}|\!| |\!|=|\!| |\!|(|\!|{|\!|rm|\!| g|\!|}m|\!| |\!|sqrt|\!|{1|\!|/2|\!|}|\!| |\!||\!| |\!|(|\!|bar|\!|{Q|\!|}|\!|_2|\!|)|\!|^2|\!|,|\!| 0|\!|)|\!|$|\!| |\!|(i|\!|.e|\!|.|\!|,|\!| in|\!| the|\!| linear|\!| magnetic|\!| field|\!| |\!|$B|\!|_3|\!| |\!|=|\!| |\!|epsilon|\!|_|\!|{3ij|\!|}|\!|partial|\!|_iA|\!|_j|\!| |\!|=|\!| |\!|-|\!| |\!| |\!|{|\!|rm|\!| g|\!|}|\!| m|\!| |\!|sqrt|\!|{2|\!|}|\!| |\!||\!| |\!|bar|\!|{Q|\!|}|\!|_2|\!|/e|\!|$|\!|)|\!|.|\!|
|\!|
|\!|vspace|\!|{3mm|\!|}|\!|
|\!|
It|\!| is|\!| preferable|\!| to|\!| set|\!| |\!|$m|\!|_1|\!| |\!|mapsto|\!| m|\!|_1|\!| |\!|hbar|\!|$|\!| and|\!| |\!|$m|\!|_2|\!| |\!|mapsto|\!| m|\!|_2|\!|/|\!|hbar|\!|$|\!|.|\!| The|\!| latter|\!| corresponds|\!| to|\!| the|\!| scale|\!| factors|\!| |\!|$a|\!|_2|\!| |\!|=|\!| 1|\!|/|\!|(2m|\!|_1|\!|hbar|\!|)|\!|$|\!| and|\!| |\!|$a|\!|_1|\!| |\!|=|\!| 1|\!|/|\!|(8m|\!|_1|\!|hbar|\!|)|\!|$|\!|.|\!| After|\!| rescaling|\!| |\!|$|\!|bar|\!|{|\!|{Q|\!|}|\!|}|\!|_1|\!|(t|\!|)|\!| |\!|mapsto|\!| |\!|bar|\!|{|\!|{Q|\!|}|\!|}|\!|_1|\!|(t|\!|)|\!|/|\!|hbar|\!|$|\!| the|\!| partition|\!| function|\!| |\!|(|\!|ref|\!|{5|\!|.9|\!|}|\!|)|\!| boils|\!| down|\!| to|\!| the|\!| usual|\!| quantum|\!|-mechanical|\!| partition|\!| function|\!| with|\!| the|\!| path|\!|-integral|\!| measure|\!|
|\!|
|\!|begin|\!|{eqnarray|\!|}|\!| |\!|{|\!|mathcal|\!|{D|\!|}|\!|}|\!|{|\!|bar|\!|{|\!|boldsymbol|\!|{Q|\!|}|\!|}|\!|}|\!| |\!||\!| |\!|approx|\!| |\!||\!| |\!|prod|\!|_i|\!| |\!|left|\!|(|\!|frac|\!|{d|\!|bar|\!|{Q|\!|}|\!|_1|\!|(t|\!|_i|\!|)|\!|}|\!|{|\!|sqrt|\!|{2|\!|pi|\!| i|\!| |\!| |\!|epsilon|\!| |\!|hbar|\!|/|\!| m|\!|_1|\!|}|\!|}|\!||\!| |\!|frac|\!|{d|\!| |\!|bar|\!|{Q|\!|}|\!|_2|\!|(t|\!|_i|\!|)|\!|}|\!|{|\!|sqrt|\!|{2|\!|pi|\!| i|\!| |\!|epsilon|\!| |\!|hbar|\!|/m|\!|_2|\!|}|\!|}|\!| |\!|right|\!|)|\!||\!|,|\!| |\!|,|\!| |\!|end|\!|{eqnarray|\!|}|\!|
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and|\!| with|\!| |\!|$1|\!|/|\!|hbar|\!|$|\!| in|\!| the|\!| exponent|\!|.|\!| Hence|\!|,|\!| just|\!| as|\!| found|\!| in|\!| the|\!| previous|\!| two|\!| cases|\!|,|\!| the|\!| choice|\!| of|\!| |\!|'t|\!|~Hooft|\!|'s|\!| condition|\!| ensures|\!| that|\!| the|\!| Planck|\!| constant|\!| enters|\!| the|\!| partition|\!| function|\!| |\!|(|\!|ref|\!|{5|\!|.9|\!|}|\!|)|\!| in|\!| a|\!| correct|\!| quantum|\!|-mechanical|\!| manner|\!|.|\!| In|\!| turn|\!|,|\!| |\!|$|\!|hbar|\!|$|\!| enters|\!| only|\!| via|\!| the|\!| scale|\!| factors|\!| |\!|$a|\!|_1|\!|$|\!| and|\!| |\!|$a|\!|_2|\!|$|\!| |\!|(the|\!| factors|\!| |\!|$d|\!|$|\!| and|\!| |\!|$c|\!|$|\!| are|\!| |\!|$|\!|hbar|\!|$|\!| independent|\!|)|\!| and|\!| hence|\!| it|\!| represents|\!| a|\!| natural|\!| scale|\!| on|\!| which|\!| the|\!| |\!|`|\!|`loss|\!| of|\!| information|\!|"|\!| condition|\!| operates|\!|.|\!| In|\!| other|\!| words|\!|,|\!| whenever|\!| one|\!| would|\!| be|\!| able|\!| to|\!| |\!|`|\!|`measure|\!|"|\!| or|\!| determine|\!| from|\!| |\!|`|\!|`first|\!| principles|\!|"|\!| the|\!| |\!|`|\!|`loss|\!| of|\!| information|\!|"|\!| condition|\!| one|\!| could|\!|,|\!| in|\!| principle|\!|,|\!| determine|\!| the|\!| value|\!| of|\!| the|\!| fundamental|\!| quantum|\!| scale|\!| |\!|$|\!|hbar|\!|$|\!|.|\!|
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As|\!| a|\!| final|\!| note|\!| we|\!| mention|\!| that|\!| the|\!| |\!|'t|\!|~Hooft|\!| quantization|\!| procedure|\!| can|\!| be|\!| straightforwardly|\!| extended|\!| to|\!| other|\!| non|\!|-linear|\!| systems|\!| and|\!| particularly|\!| to|\!| systems|\!| possessing|\!| chaotic|\!| behavior|\!| |\!|(e|\!|.g|\!|.|\!|,|\!| strange|\!| attractors|\!|)|\!|.|\!| In|\!| general|\!| cases|\!| this|\!| might|\!| be|\!|,|\!| however|\!|,|\!| hindered|\!| by|\!| our|\!| inability|\!| to|\!| find|\!| the|\!| corresponding|\!| first|\!| integrals|\!| |\!|(and|\!| hence|\!| |\!|$C|\!|_i|\!|$|\!|'s|\!|)|\!| in|\!| the|\!| analytic|\!| form|\!|.|\!| It|\!| is|\!| interesting|\!| to|\!| notice|\!| that|\!| machinery|\!| outlined|\!| above|\!| allows|\!| to|\!| find|\!| the|\!| emergent|\!| quantistic|\!| system|\!| for|\!| the|\!| configuration|\!|-space|\!| strange|\!| attractors|\!|.|\!| This|\!| is|\!| because|\!| in|\!| |\!|'t|\!|~Hooft|\!|'s|\!| |\!|`|\!|`quantization|\!|"|\!| one|\!| only|\!| needs|\!| the|\!| dynamical|\!| equations|\!| in|\!| the|\!| |\!|{|\!|em|\!| configuration|\!|}|\!| space|\!|.|\!| The|\!| latter|\!| should|\!| be|\!| contrasted|\!| with|\!| the|\!| Hamiltonian|\!| |\!|(or|\!| symplectic|\!|)|\!| systems|\!| where|\!| strange|\!| attractors|\!| cannot|\!| exist|\!| in|\!| the|\!| |\!|{|\!|em|\!| phase|\!|-space|\!|}|\!| on|\!| account|\!| of|\!| the|\!| Liouville|\!| theorem|\!|~|\!|cite|\!|{Hop|\!|}|\!|.|\!|
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|\!|section|\!|{Conclusions|\!| and|\!| Outlook|\!|}|\!|
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In|\!| this|\!| paper|\!| we|\!| have|\!| attempted|\!| to|\!| substantiate|\!| the|\!| recent|\!| proposal|\!| of|\!| G|\!|.|\!|'t|\!|~Hooft|\!| in|\!| which|\!| quantum|\!| theory|\!| as|\!| viewed|\!| as|\!| not|\!| a|\!| complete|\!| final|\!| theory|\!|,|\!| but|\!| is|\!| in|\!| fact|\!| an|\!| emergent|\!| phenomenon|\!| arising|\!| from|\!| a|\!| deeper|\!| level|\!| of|\!| dynamics|\!|.|\!| The|\!| underlying|\!| dynamics|\!| are|\!| taken|\!| to|\!| be|\!| classical|\!| mechanics|\!| with|\!| singular|\!| Lagrangians|\!| supplied|\!| with|\!| an|\!| appropriate|\!| information|\!| loss|\!| condition|\!|.|\!| With|\!| plausible|\!| assumptions|\!| about|\!| the|\!| actual|\!| nature|\!| of|\!| the|\!| constraint|\!| dynamics|\!|,|\!| quantum|\!| theory|\!| is|\!| shown|\!| to|\!| emerge|\!| when|\!| the|\!| classical|\!| Dirac|\!|-Bergmann|\!| algorithm|\!| for|\!| constrained|\!| dynamics|\!| is|\!| applied|\!| to|\!| the|\!| classical|\!| path|\!| integral|\!| of|\!| Gozzi|\!| |\!|{|\!|em|\!| et|\!| al|\!|.|\!|}|\!|.|\!|
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There|\!| are|\!| essentially|\!| two|\!| different|\!| tactics|\!| for|\!| implementing|\!| the|\!| classical|\!| path|\!| integrals|\!| in|\!| |\!|'t|\!|~Hooft|\!|'s|\!| quantization|\!| scenario|\!|.|\!| The|\!| first|\!| is|\!| to|\!| apply|\!| the|\!| configuration|\!|-space|\!| formulation|\!|~|\!|cite|\!|{GozziI|\!|}|\!|.|\!| This|\!| is|\!| suited|\!| to|\!| situations|\!| when|\!| |\!|'t|\!|~Hooft|\!|'s|\!| systems|\!| are|\!| phrased|\!| through|\!| the|\!| Lagrangian|\!| description|\!|.|\!| The|\!| alternative|\!| approach|\!| is|\!| to|\!| start|\!| with|\!| the|\!| phase|\!|-space|\!| version|\!|~|\!|cite|\!|{GozziII|\!|}|\!|.|\!| The|\!| latter|\!| provides|\!| a|\!| natural|\!| framework|\!| when|\!| the|\!| Hamiltonian|\!| formulation|\!| is|\!| of|\!| interest|\!| or|\!| where|\!| the|\!| language|\!| of|\!| symplectic|\!| geometry|\!| is|\!| preferred|\!|.|\!| It|\!| should|\!| be|\!|,|\!| however|\!|,|\!| stressed|\!| that|\!| it|\!| is|\!| not|\!| merely|\!| a|\!| matter|\!| of|\!| a|\!| computational|\!| convenience|\!| which|\!| method|\!| is|\!| actually|\!| employed|\!|.|\!| In|\!| fact|\!|,|\!| both|\!| approaches|\!| are|\!| mathematically|\!| and|\!| conceptually|\!| very|\!| different|\!| |\!|(as|\!| they|\!| are|\!| also|\!| in|\!| conventional|\!| quantum|\!| mechanics|\!|~|\!|cite|\!|{Pain|\!|,Sh1|\!|}|\!|)|\!|.|\!| Besides|\!|,|\!| the|\!| methodology|\!| for|\!| handling|\!| singular|\!| systems|\!| is|\!| distinct|\!| in|\!| Lagrangian|\!| and|\!| Hamiltonian|\!| formulations|\!| |\!|(c|\!|.f|\!|.|\!| Refs|\!|.|\!|~|\!|cite|\!|{Sunder|\!|,GT|\!|}|\!| and|\!| citations|\!| therein|\!|)|\!|.|\!| In|\!| passing|\!|,|\!| we|\!| should|\!| mention|\!| that|\!| the|\!| currently|\!| popular|\!| Hamilton|\!|-Jacobi|\!|~|\!|cite|\!|{Gu1|\!|}|\!| and|\!| Legendre|\!|-Ostrogradski|\!|u|\!|{i|\!|}|\!|~|\!|cite|\!|{Pons|\!|}|\!| approaches|\!| for|\!| a|\!| treatment|\!| of|\!| constrained|\!| systems|\!|,|\!| though|\!| highly|\!| convenient|\!| in|\!| certain|\!| cases|\!| |\!|(e|\!|.g|\!|.|\!|,|\!| in|\!| higher|\!|-order|\!| Lagrangian|\!| systems|\!|)|\!|,|\!| have|\!| not|\!| found|\!| as|\!| yet|\!| any|\!| particular|\!| utility|\!| in|\!| the|\!| present|\!| context|\!|.|\!|
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Throughout|\!| this|\!| paper|\!| we|\!| have|\!| considered|\!| only|\!| the|\!| configuration|\!|-space|\!| formulation|\!| of|\!| classical|\!| path|\!| integrals|\!|.|\!| |\!|(Incidently|\!|,|\!| the|\!| phase|\!|-space|\!| path|\!| integral|\!| which|\!| appears|\!| in|\!| Section|\!| IV|\!| |\!|(after|\!| Eq|\!|.|\!|(|\!|ref|\!|{3|\!|.40|\!|}|\!|)|\!|)|\!| is|\!| not|\!| the|\!| phase|\!|-space|\!| path|\!| integral|\!| |\!|{|\!|em|\!| |\!||\!|`|\!|{a|\!|}|\!| la|\!|}|\!| Gozzi|\!|,|\!| Reuter|\!| and|\!| Thacker|\!|~|\!|cite|\!|{GozziII|\!|}|\!| but|\!| rather|\!| Gozzi|\!|'s|\!| configuration|\!|-path|\!|~|\!|cite|\!|{GozziI|\!|}|\!| integral|\!| with|\!| extra|\!| degrees|\!| of|\!| freedom|\!|.|\!|)|\!| By|\!| choosing|\!| to|\!| work|\!| within|\!| such|\!| a|\!| framework|\!| we|\!| have|\!| been|\!| able|\!| to|\!| render|\!| a|\!| number|\!| of|\!| formal|\!| steps|\!| more|\!| tractable|\!| |\!|(e|\!|.g|\!|.|\!|,|\!| BRST|\!| analysis|\!| is|\!| reputed|\!| to|\!| be|\!| simpler|\!| in|\!| the|\!| configuration|\!| space|\!|,|\!| uniqueness|\!| proof|\!| for|\!| |\!|'t|\!|~Hooft|\!| systems|\!| is|\!| easy|\!| and|\!| transparent|\!| in|\!| the|\!| Lagrange|\!| description|\!|,|\!| etc|\!|.|\!|)|\!|.|\!| The|\!| key|\!| advantage|\!|,|\!| however|\!|,|\!| lies|\!| in|\!| two|\!| observations|\!|.|\!| First|\!|,|\!| the|\!| position|\!|-space|\!| path|\!| integral|\!| of|\!| Gozzi|\!| |\!|{|\!|em|\!| et|\!| al|\!|.|\!|}|\!| provides|\!| a|\!| conceptually|\!| clean|\!| starting|\!| point|\!| in|\!| view|\!| of|\!| the|\!| fact|\!| that|\!| it|\!| represents|\!| the|\!| classical|\!| limit|\!| of|\!| both|\!| the|\!| stochastic|\!|-quantization|\!| path|\!| integral|\!| and|\!| the|\!| closed|\!|-time|\!|-path|\!| integral|\!| for|\!| the|\!| transition|\!| probability|\!| of|\!| systems|\!| coupled|\!| to|\!| a|\!| heat|\!| bath|\!|.|\!| Such|\!| a|\!| connection|\!| is|\!| by|\!| no|\!| means|\!| obvious|\!| in|\!| the|\!| canonical|\!| path|\!|-integral|\!| representation|\!| as|\!| both|\!| the|\!| Parisi|\!|-Wu|\!| stochastic|\!| quantization|\!| and|\!| the|\!| Feynman|\!|-Vernon|\!| formalism|\!| |\!|(with|\!| ensuing|\!| closed|\!|-time|\!|-path|\!| integral|\!|)|\!| are|\!| intrinsically|\!| formulated|\!| in|\!| the|\!| configuration|\!| space|\!|.|\!| Second|\!|,|\!| according|\!| to|\!| |\!|'t|\!|~Hooft|\!|'s|\!| conjecture|\!| the|\!| |\!|`|\!|`loss|\!| of|\!| information|\!|"|\!| condition|\!| should|\!| operate|\!| in|\!| the|\!| position|\!| space|\!| where|\!| it|\!| is|\!| supposed|\!| to|\!| eliminate|\!| some|\!| of|\!| the|\!| transient|\!| trajectories|\!| leaving|\!| behind|\!| only|\!| stable|\!| |\!|(or|\!| near|\!| to|\!| stable|\!|)|\!| orbits|\!|~|\!|cite|\!|{tHooft3|\!|}|\!|.|\!| Hence|\!| working|\!| in|\!| configuration|\!| space|\!| may|\!| allow|\!| one|\!| to|\!| probe|\!| the|\!| plausibility|\!| of|\!| |\!|'t|\!|~Hooft|\!|'s|\!| conjecture|\!|.|\!| The|\!| price|\!| that|\!| has|\!| been|\!| paid|\!| for|\!| this|\!| choice|\!| is|\!| that|\!| the|\!| configuration|\!| space|\!| must|\!| have|\!| been|\!| doubled|\!|.|\!| This|\!| is|\!| an|\!| unavoidable|\!| step|\!| whenever|\!| one|\!| wishes|\!| to|\!| obtain|\!| first|\!|-order|\!| autonomous|\!| dynamical|\!| equations|\!| directly|\!| from|\!| the|\!| Lagrange|\!| formulation|\!| |\!|(a|\!| fact|\!| well|\!| known|\!| in|\!| the|\!| theory|\!| of|\!| dissipative|\!| systems|\!|~|\!|cite|\!|{MF1|\!|}|\!|)|\!|.|\!| Our|\!| analysis|\!| in|\!| Appendix|\!| BII|\!|
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suggests|\!|,|\!| that|\!| the|\!| auxiliary|\!| coordinates|\!| |\!|$|\!|bar|\!|{q|\!|}|\!|_i|\!|$|\!| may|\!| be|\!| related|\!| to|\!| relative|\!| coordinates|\!| on|\!| the|\!| backward|\!|-forward|\!| time|\!| path|\!| in|\!| the|\!| Feynman|\!|-Vernon|\!| approach|\!|.|\!| |\!|(Such|\!| coordinates|\!| also|\!| go|\!| under|\!| the|\!| names|\!| |\!|{|\!|em|\!| fast|\!| variables|\!|}|\!|~|\!|cite|\!|{Fetter|\!|}|\!| or|\!| |\!|{|\!|em|\!| quantum|\!| noise|\!| variables|\!|}|\!|~|\!|cite|\!|{Sriv1|\!|}|\!|.|\!|)|\!|
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On|\!| the|\!| formal|\!| side|\!|,|\!| the|\!| auxiliary|\!| variables|\!| |\!|$|\!|bar|\!|{q|\!|}|\!|_i|\!|$|\!| are|\!| nothing|\!| but|\!| Gozzi|\!|'s|\!| Lagrange|\!| multipliers|\!| |\!|$|\!|{|\!|lambda|\!|}|\!|_i|\!|$|\!| |\!|(in|\!| our|\!| case|\!| denoted|\!| as|\!| |\!|$|\!|bar|\!|{|\!|lambda|\!|}|\!|_i|\!|$|\!|)|\!|.|\!|
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In|\!| order|\!| to|\!| incorporate|\!| the|\!| |\!|`|\!|`loss|\!| of|\!| information|\!|"|\!| into|\!| our|\!| scheme|\!|,|\!| we|\!| have|\!| introduced|\!| in|\!| Section|\!| IV|\!| an|\!| auxiliary|\!| momentum|\!| integration|\!| to|\!| go|\!| over|\!| to|\!| the|\!| canonical|\!| representation|\!|.|\!| Such|\!| a|\!| step|\!|,|\!| though|\!| formal|\!|,|\!| allowed|\!| us|\!| to|\!| treat|\!| our|\!| constrained|\!| system|\!| via|\!| the|\!| standard|\!| Dirac|\!|-Bergmann|\!| procedure|\!|.|\!| It|\!| should|\!| be|\!| admitted|\!| that|\!| such|\!| a|\!| choice|\!| is|\!| by|\!| no|\!| means|\!| unique|\!| |\!|-|\!| e|\!|.g|\!|.|\!|,|\!| |\!| methodologies|\!| for|\!| treatment|\!| of|\!| classical|\!| constrained|\!| systems|\!| in|\!| configuration|\!| space|\!| do|\!| exist|\!|~|\!|cite|\!|{Sunder|\!|,GT|\!|}|\!|.|\!| The|\!| decision|\!| to|\!| apply|\!| the|\!| Dirac|\!|-Bergmann|\!| algorithm|\!| was|\!| mainly|\!| motivated|\!| by|\!| its|\!| conceptual|\!| simplicity|\!| and|\!| direct|\!| applicability|\!| to|\!| path|\!| integrals|\!|.|\!| On|\!| the|\!| other|\!| hand|\!|,|\!| we|\!| do|\!| not|\!| expect|\!| that|\!| the|\!| presented|\!| results|\!| should|\!| undergo|\!| any|\!| substantial|\!| changes|\!| when|\!| some|\!| another|\!| scheme|\!| would|\!| be|\!| utilized|\!|.|\!| It|\!| should|\!| be|\!| further|\!| emphasized|\!| that|\!| while|\!| we|\!| have|\!| established|\!| the|\!| mathematical|\!| link|\!| |\!|(Eqs|\!|.|\!|(|\!|ref|\!|{4|\!|.26|\!|}|\!|)|\!| and|\!| |\!|(|\!|ref|\!|{D3|\!|}|\!|)|\!|)|\!| between|\!| the|\!| |\!|`|\!|`loss|\!| of|\!| information|\!|"|\!| condition|\!| and|\!| first|\!|-class|\!| constraints|\!|,|\!| it|\!| is|\!| not|\!| yet|\!| clear|\!| if|\!| this|\!| connection|\!| has|\!| more|\!| direct|\!| physical|\!| interpretation|\!| |\!|(although|\!| various|\!| proposals|\!| exist|\!| in|\!| the|\!| literature|\!|~|\!|cite|\!|{BJV3|\!|,tHooft3|\!|,BMM1|\!|}|\!|)|\!|.|\!| Such|\!| an|\!| understanding|\!| would|\!| not|\!| only|\!| help|\!| to|\!| develop|\!| this|\!| approach|\!| for|\!| more|\!| complicated|\!| physical|\!| situations|\!| but|\!| also|\!| affiliation|\!| in|\!| a|\!| systematic|\!| fashion|\!| of|\!| a|\!| quantum|\!| system|\!| to|\!| an|\!| underlying|\!| classical|\!| dynamics|\!|.|\!| Work|\!| along|\!| those|\!| lines|\!| is|\!| currently|\!| in|\!| progress|\!|.|\!|
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To|\!| illustrate|\!| the|\!| presented|\!| ideas|\!| we|\!| have|\!| considered|\!| two|\!| simple|\!| systems|\!|;|\!| the|\!| planar|\!| pendulum|\!| and|\!| the|\!| R|\!||\!|"|\!|{o|\!|}ssler|\!| system|\!|.|\!| In|\!| the|\!| pendulum|\!| case|\!| we|\!| have|\!| taken|\!| advantage|\!| of|\!| free|\!| choice|\!| of|\!| an|\!| additive|\!| constant|\!| in|\!| the|\!| charge|\!| |\!|$C|\!|_1|\!|$|\!|.|\!| This|\!| in|\!| turn|\!|,|\!| allowed|\!| us|\!| to|\!| imposed|\!| |\!|'t|\!|~Hooft|\!|'s|\!| constraints|\!| in|\!| two|\!| distinct|\!| ways|\!|.|\!| In|\!| the|\!| case|\!| of|\!| R|\!||\!|"|\!|{o|\!|}ssler|\!|'s|\!| system|\!| two|\!| |\!|$|\!|{|\!|}|\!|boldsymbol|\!|{p|\!|}|\!|$|\!|-independent|\!|,|\!| irreducible|\!| charges|\!| |\!|$C|\!|_1|\!|$|\!| and|\!| |\!|$C|\!|_2|\!|$|\!| exist|\!|.|\!| For|\!| definiteness|\!| sake|\!| we|\!| have|\!| constructed|\!| in|\!| the|\!| latter|\!| case|\!| the|\!| |\!|`|\!|`loss|\!| of|\!| information|\!|"|\!| condition|\!| with|\!| the|\!| additive|\!| constant|\!| set|\!| to|\!| zero|\!|.|\!| With|\!| this|\!| we|\!| were|\!| able|\!| to|\!| convert|\!| the|\!| corresponding|\!| classical|\!| path|\!| integrals|\!| into|\!| path|\!| integrals|\!| describing|\!| a|\!| quantized|\!| free|\!| particle|\!|,|\!| a|\!| harmonic|\!| oscillator|\!|,|\!| and|\!| a|\!| free|\!| particle|\!| weakly|\!| coupled|\!| to|\!| Duffing|\!|'s|\!| oscillator|\!|.|\!| As|\!| a|\!| byproduct|\!| we|\!| could|\!| observe|\!| that|\!| our|\!| prescription|\!| provides|\!| a|\!| surprisingly|\!| rigid|\!| structure|\!| with|\!| rather|\!| tight|\!| maneuvering|\!| space|\!| for|\!| the|\!| emergent|\!| quantum|\!| dynamics|\!|.|\!| Indeed|\!|,|\!| when|\!| the|\!| classical|\!| dynamics|\!| is|\!| fixed|\!|,|\!| the|\!| |\!|'t|\!|~Hooft|\!| condition|\!| is|\!| formulated|\!| via|\!| linear|\!| combination|\!| of|\!| charges|\!| |\!|$C|\!|_i|\!|$|\!| which|\!| correspond|\!| to|\!| the|\!| first|\!| integrals|\!| of|\!| the|\!| autonomous|\!| dynamical|\!| equations|\!| for|\!| |\!|$|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|$|\!|,|\!| i|\!|.e|\!|.|\!|,|\!| Eq|\!|.|\!|(|\!|ref|\!|{eq|\!|.1|\!|.1|\!|.1|\!|}|\!|)|\!|.|\!| Due|\!| to|\!| the|\!| explicit|\!| form|\!| of|\!| |\!|'t|\!|~Hooft|\!|'s|\!| Hamiltonian|\!| the|\!| constraint|\!| is|\!| of|\!| the|\!| first|\!| class|\!| and|\!| so|\!| we|\!| must|\!| remove|\!| the|\!| redundancy|\!| in|\!| the|\!| description|\!| by|\!| imposing|\!| the|\!| gauge|\!| condition|\!| |\!|$|\!|chi|\!|$|\!|.|\!| By|\!| requiring|\!| that|\!| the|\!| consistency|\!| conditions|\!| |\!|(|\!|ref|\!|{4|\!|.25|\!|}|\!|)|\!| and|\!| |\!|(|\!|ref|\!|{4|\!|.27|\!|}|\!|)|\!| are|\!| fulfilled|\!|,|\!| that|\!| the|\!| choice|\!| of|\!| |\!|$|\!|chi|\!|$|\!| does|\!| not|\!| induce|\!| Gribov|\!| ambiguity|\!|,|\!| and|\!| that|\!| the|\!| canonical|\!| transformations|\!| defined|\!| in|\!| Sec|\!|.|\!|~IV|\!| are|\!| linear|\!|,|\!| we|\!| substantially|\!| narrowed|\!| down|\!| the|\!| class|\!| of|\!| possible|\!| emergent|\!| quantum|\!| systems|\!|.|\!| Note|\!| also|\!|,|\!| that|\!| when|\!| we|\!| start|\!| with|\!| the|\!| |\!|$N|\!|$|\!|-dimensional|\!| classical|\!| system|\!| |\!|(|\!|$|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|$|\!| variables|\!|)|\!|,|\!| the|\!| emergent|\!| quantum|\!| dynamics|\!| has|\!| |\!|$N|\!|-1|\!|$|\!| dimensions|\!| |\!|(|\!|$|\!|bar|\!|{|\!|boldsymbol|\!|{Q|\!|}|\!|}|\!|$|\!| variables|\!|)|\!|.|\!| Indeed|\!|,|\!| by|\!| introducing|\!| the|\!| auxiliary|\!| degrees|\!| of|\!| freedom|\!| |\!|$|\!|bar|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|$|\!| we|\!| obtain|\!| |\!|$4N|\!|$|\!|-dimensional|\!| phase|\!| space|\!| which|\!| is|\!| constrained|\!| by|\!| |\!|$2N|\!| |\!|+|\!| 2|\!|$|\!| conditions|\!| |\!|(|\!|$|\!|phi|\!|_i|\!|$|\!|,|\!| |\!|$|\!|varphi|\!|$|\!| and|\!| |\!|$|\!|chi|\!|$|\!|)|\!|,|\!| which|\!| leaves|\!| behind|\!| |\!|$|\!|(2N|\!|-2|\!|)|\!|$|\!|-dimensional|\!| phase|\!| space|\!| |\!|$|\!|bar|\!|{|\!|boldsymbol|\!|{Q|\!|}|\!|}|\!|,|\!| |\!|bar|\!|{|\!|boldsymbol|\!|{P|\!|}|\!|}|\!|$|\!|.|\!| This|\!| disparity|\!| between|\!| the|\!| dimensionality|\!| of|\!| the|\!| classical|\!| and|\!| emergent|\!| quantum|\!| systems|\!| vindicates|\!| in|\!| part|\!| the|\!| terminology|\!| |\!|`|\!|`information|\!| loss|\!|"|\!| used|\!| throughout|\!| the|\!| text|\!|.|\!|
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An|\!| important|\!| conclusion|\!| of|\!| this|\!| work|\!| is|\!| that|\!| |\!|'t|\!|~Hooft|\!|'s|\!| quantization|\!| proposal|\!| seems|\!| to|\!| provide|\!| a|\!| tenable|\!| scenario|\!| which|\!| allows|\!| for|\!| deriving|\!| certain|\!| quantum|\!| systems|\!| from|\!| classical|\!| physics|\!|.|\!|
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It|\!| should|\!| be|\!| stressed|\!| that|\!| although|\!| we|\!| assumed|\!| throughout|\!| that|\!| the|\!| deeper|\!| level|\!| dynamics|\!| is|\!| the|\!| classical|\!| |\!|(Lagrangian|\!| or|\!| Hamiltonian|\!|)|\!| one|\!|,|\!| there|\!| is|\!| in|\!| principle|\!| no|\!| fundamental|\!| reason|\!| that|\!| would|\!| preclude|\!| starting|\!| with|\!| more|\!| exotic|\!| premises|\!|.|\!| In|\!| particular|\!|,|\!| our|\!| conceptual|\!| reasoning|\!| would|\!| go|\!| unchanged|\!| if|\!| we|\!| had|\!| begun|\!| with|\!| Lagrangians|\!| operating|\!| over|\!| coordinate|\!| superspaces|\!| |\!|(pseudoclassical|\!| mechanics|\!|~|\!|cite|\!|{BeMa1|\!|}|\!|)|\!| or|\!| with|\!| the|\!| currently|\!| much|\!| discussed|\!| discrete|\!| classical|\!| mechanics|\!| |\!|(i|\!|.e|\!|.|\!|,|\!| having|\!| foam|\!|-|\!|,|\!| fractal|\!|-|\!|,|\!| or|\!| crystal|\!|-like|\!| configuration|\!| space|\!|)|\!|~|\!|cite|\!|{Kleinert|\!|?|\!|}|\!|,|\!| etc|\!|.|\!|~|\!|.|\!| The|\!| only|\!| prerequisite|\!| for|\!| such|\!| approaches|\!| is|\!| the|\!| possibility|\!| of|\!| formulating|\!| a|\!| corresponding|\!| variant|\!| of|\!| Gozzi|\!|'s|\!| path|\!| integral|\!|,|\!| and|\!| a|\!| method|\!| for|\!| implementing|\!| the|\!| |\!|`|\!|`loss|\!| of|\!| information|\!|"|\!| constraint|\!| in|\!| such|\!| integrals|\!|.|\!|
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There|\!| are|\!| many|\!| interesting|\!| applications|\!| of|\!| the|\!| above|\!| method|\!|.|\!| Applications|\!| to|\!| chaotic|\!| dynamical|\!| systems|\!| especially|\!| seem|\!| quite|\!| pertinent|\!|.|\!| After|\!| all|\!|,|\!| central|\!| to|\!| our|\!| reasoning|\!| is|\!| a|\!| |\!|(doubled|\!|)|\!| set|\!| of|\!| real|\!| first|\!|-order|\!| dynamical|\!| equations|\!|footnote|\!|{Non|\!|-trivial|\!| are|\!| only|\!| the|\!| equations|\!| over|\!| actual|\!| configuration|\!| space|\!|.|\!| The|\!| dynamical|\!| equations|\!| for|\!| the|\!| auxiliary|\!| variables|\!| |\!|$|\!|bar|\!|{q|\!|}|\!|_i|\!|$|\!| are|\!| linear|\!| and|\!| hence|\!| they|\!| are|\!| not|\!| relevant|\!| in|\!| this|\!| connection|\!|.|\!|}|\!| which|\!|,|\!| under|\!| favorable|\!| conditions|\!|,|\!| may|\!| by|\!| associated|\!| with|\!| a|\!| chaotic|\!| dynamics|\!| in|\!| the|\!| configuration|\!| space|\!|.|\!| We|\!| should|\!| emphasize|\!| that|\!| the|\!| reader|\!| should|\!| not|\!| confuse|\!| the|\!| above|\!| with|\!| the|\!| extensively|\!| studied|\!| but|\!| unrelated|\!| notion|\!| of|\!| chaos|\!| in|\!| Hamiltonian|\!| systems|\!| |\!|-|\!| we|\!| do|\!| not|\!| deal|\!| here|\!| with|\!| dynamical|\!| equations|\!| on|\!| symplectic|\!| manifolds|\!|.|\!| This|\!| is|\!| important|\!|,|\!| as|\!| Hamiltonian|\!| systems|\!| forbid|\!| |\!|{|\!|em|\!| per|\!| s|\!||\!|`|\!|{e|\!|}|\!|}|\!| the|\!| existence|\!| of|\!| attractive|\!| orbits|\!| which|\!| are|\!| otherwise|\!| key|\!| in|\!| |\!|'t|\!|~Hooft|\!|'s|\!| proposal|\!|.|\!| In|\!| this|\!| respect|\!| our|\!| approach|\!| is|\!| parallel|\!| with|\!| some|\!| more|\!| conventional|\!| approaches|\!|.|\!| Indeed|\!|,|\!| a|\!| direct|\!| |\!|`|\!|`quantization|\!|"|\!| of|\!| the|\!| equations|\!| of|\!| motion|\!| |\!|-|\!|-|\!| originally|\!| proposed|\!| by|\!| Feynman|\!|~|\!|cite|\!|{Dyson|\!|}|\!| |\!|-|\!|-|\!| is|\!| one|\!| of|\!| the|\!| techniques|\!| for|\!| tackling|\!| quantization|\!| of|\!| dissipative|\!| systems|\!|~|\!|cite|\!|{Tar1|\!|,HS1|\!|}|\!|.|\!| In|\!| field|\!| theories|\!| this|\!| line|\!| of|\!| reasoning|\!| was|\!| recently|\!| progressed|\!| by|\!| Bir|\!||\!|'|\!|{o|\!|}|\!|,|\!| M|\!||\!|"|\!|{u|\!|}ller|\!|,|\!| and|\!| Matinyan|\!|~|\!|cite|\!|{BMM1|\!|}|\!| who|\!| demonstrated|\!| that|\!| quantum|\!| gauge|\!| field|\!| theories|\!| can|\!| emerge|\!| in|\!| the|\!| infrared|\!| limit|\!| of|\!| a|\!| higher|\!|-dimensional|\!| classical|\!| |\!|(non|\!|-Abelian|\!|)|\!| gauge|\!| field|\!| theory|\!|,|\!| known|\!| to|\!| have|\!| chaotic|\!| behavior|\!|~|\!|cite|\!|{BMM2|\!|}|\!|.|\!|
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We|\!| finally|\!| wish|\!| to|\!| comment|\!| on|\!| two|\!| more|\!| points|\!|.|\!| First|\!|,|\!| in|\!| cases|\!| where|\!| one|\!| strives|\!| for|\!| an|\!| explicit|\!| reparametrization|\!| invariance|\!| |\!|(or|\!| general|\!| covariance|\!|)|\!| of|\!| the|\!| emergent|\!| quantum|\!| system|\!|
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the|\!| presented|\!| framework|\!| is|\!| not|\!| very|\!| suitable|\!|.|\!| The|\!| absence|\!| of|\!| explicit|\!| covariance|\!| in|\!| both|\!| Dirac|\!|-Bergmann|\!| and|\!| Fadeev|\!|-Senjanovic|\!| algorithms|\!| makes|\!| the|\!| actual|\!| analysis|\!| very|\!| cumbersome|\!| or|\!| even|\!| impossible|\!|.|\!| In|\!| fact|\!|,|\!| expressions|\!| |\!|(|\!|ref|\!|{4|\!|.17|\!|}|\!|)|\!| and|\!| |\!|(|\!|ref|\!|{4|\!|.21|\!|}|\!|)|\!| are|\!| evidently|\!| not|\!| generally|\!| covariant|\!| due|\!| to|\!| the|\!| presence|\!| of|\!| time|\!|-independent|\!| constraints|\!| in|\!| the|\!| measure|\!|.|\!| Although|\!| generalizations|\!| that|\!| include|\!| covariant|\!| constraints|\!| do|\!| exist|\!|~|\!|cite|\!|{fradkin|\!|,Bat1|\!|,Gav|\!|}|\!| they|\!| result|\!| in|\!| gauge|\!| fixing|\!| conditions|\!| which|\!| depend|\!| not|\!| only|\!| on|\!| the|\!| canonical|\!| variables|\!| but|\!| also|\!| on|\!| the|\!| Lagrange|\!| multipliers|\!| |\!|(or|\!| explicit|\!| time|\!|)|\!|.|\!| Such|\!| gauge|\!| constraints|\!| are|\!|,|\!| however|\!|,|\!| incompatible|\!| with|\!| our|\!| Poisson|\!| bracket|\!| analysis|\!| used|\!| in|\!| Section|\!| IV|\!|,|\!| and|\!| Appendixes|\!| A|\!| and|\!| D|\!|.|\!| Hence|\!|,|\!| if|\!| the|\!| emergent|\!| quantum|\!| system|\!| is|\!| supposed|\!| to|\!| be|\!| reparametrization|\!| invariant|\!| |\!|(e|\!|.g|\!|.|\!|,|\!| relativistic|\!| particle|\!|,|\!| canonical|\!| gravity|\!|,|\!| relativistic|\!| string|\!|,|\!| etc|\!|.|\!|)|\!| a|\!| new|\!| framework|\!| for|\!| the|\!| path|\!|-integral|\!| implementation|\!| of|\!| |\!|'t|\!|~Hooft|\!|'s|\!| scheme|\!| must|\!| be|\!| sought|\!|.|\!| Second|\!|,|\!| the|\!| formalism|\!| of|\!| functional|\!| integrals|\!| is|\!| sometimes|\!| deceptive|\!| when|\!| taken|\!| too|\!| literally|\!|.|\!| The|\!| latter|\!| is|\!| the|\!| case|\!|,|\!| for|\!| instance|\!|,|\!| when|\!| gauge|\!| conditions|\!| are|\!| imposed|\!| and|\!|/or|\!| canonical|\!| transformations|\!| performed|\!|.|\!| The|\!| difficulty|\!| involved|\!| is|\!| known|\!| as|\!| the|\!| Edwards|\!|-Gulyaev|\!| effect|\!|~|\!|cite|\!|{Pain|\!|,EG|\!|,Sh1|\!|}|\!| and|\!| it|\!| resides|\!| in|\!| the|\!| exact|\!| nature|\!| of|\!| the|\!| limiting|\!| sequence|\!| of|\!| the|\!| finite|\!| dimensional|\!| integrals|\!| which|\!| constitute|\!| the|\!| path|\!| integral|\!|.|\!| As|\!| a|\!| result|\!| the|\!| classical|\!| canonical|\!| transformation|\!| does|\!| not|\!| leave|\!|,|\!| in|\!| general|\!|,|\!| the|\!| measure|\!| of|\!| the|\!| path|\!| integral|\!| Liouville|\!| invariant|\!| but|\!|,|\!| instead|\!| induces|\!| an|\!| anomaly|\!|~|\!|cite|\!|{Sh1|\!|,SW|\!|}|\!|.|\!| Thus|\!|,|\!| for|\!| our|\!| construction|\!| to|\!| be|\!| meaningful|\!| it|\!| should|\!| be|\!| shown|\!| that|\!| the|\!| canonical|\!| transformations|\!| in|\!| Section|\!| IV|\!| are|\!| unaffected|\!| by|\!| the|\!| Edwards|\!|-Gulyaev|\!| effect|\!|.|\!| Fortunately|\!|,|\!| in|\!| cases|\!| when|\!| the|\!| generating|\!| function|\!| is|\!| at|\!| most|\!| quadratic|\!| |\!|(making|\!| canonical|\!| transformations|\!| |\!| linear|\!|)|\!| and|\!| not|\!| explicitly|\!| time|\!| dependent|\!|,|\!| it|\!| can|\!| be|\!| shown|\!|~|\!|cite|\!|{Fad|\!|,SW|\!|,vH|\!|}|\!| that|\!| the|\!| anomaly|\!| is|\!| absent|\!|.|\!| It|\!| was|\!| precisely|\!| for|\!| this|\!| reason|\!| that|\!| more|\!| general|\!| transformations|\!| were|\!| not|\!| considered|\!| in|\!| the|\!| present|\!| paper|\!|.|\!| Clearly|\!|,|\!| both|\!| mentioned|\!| points|\!| are|\!| of|\!| key|\!| importance|\!| for|\!| further|\!| development|\!| of|\!| our|\!| procedure|\!| and|\!|,|\!| due|\!| to|\!| their|\!| delicate|\!| nature|\!|,|\!| they|\!| deserve|\!| a|\!| separate|\!| discussion|\!|.|\!|
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Let|\!| us|\!| end|\!| with|\!| the|\!| remark|\!| that|\!| the|\!| notorious|\!| problem|\!| with|\!| operator|\!| ordering|\!| known|\!| from|\!| canonical|\!| approaches|\!| has|\!| an|\!| elegant|\!| solution|\!| in|\!| path|\!| integrals|\!|.|\!| The|\!| ordering|\!| is|\!| there|\!| naturally|\!| generated|\!| by|\!| the|\!| necessary|\!| physical|\!| requirement|\!| that|\!| path|\!| integrals|\!| must|\!| be|\!| invariant|\!| under|\!| coordinate|\!| transformations|\!|~|\!|cite|\!|{KC|\!|}|\!|.|\!|
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|\!|section|\!|*|\!|{Acknowledgments|\!|}|\!|
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M|\!|.B|\!|.|\!| and|\!| P|\!|.J|\!|.|\!| are|\!| grateful|\!| to|\!| the|\!| |\!| ESF|\!| network|\!| COSLAB|\!| for|\!| funding|\!| their|\!| stay|\!| at|\!| FU|\!|,|\!| Berlin|\!|.|\!| One|\!| of|\!| us|\!|,|\!| P|\!|.J|\!|.|\!|,|\!| acknowledges|\!| very|\!| helpful|\!| discussions|\!| with|\!| R|\!|.|\!|~Banerjee|\!|,|\!|
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G|\!|.|\!|~Vitiello|\!| and|\!| Y|\!|.|\!|~Satoh|\!|,|\!| and|\!| thanks|\!| the|\!| Japanese|\!| Society|\!| for|\!| Promotion|\!| of|\!| Science|\!| for|\!| financial|\!| support|\!|.|\!|
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|\!|section|\!|*|\!|{Appendix|\!| A|\!|}|\!|
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In|\!| this|\!| appendix|\!| we|\!| show|\!| that|\!| the|\!| system|\!| |\!|(|\!|ref|\!|{eq|\!|.1|\!|.1|\!|}|\!|)|\!| has|\!| no|\!| secondary|\!| constraints|\!|.|\!| In|\!| contract|\!| to|\!| the|\!| primary|\!| constraints|\!| which|\!| are|\!| a|\!| consequence|\!| of|\!| the|\!| non|\!|-invertibility|\!| of|\!| the|\!| velocities|\!| in|\!| terms|\!| of|\!| the|\!| |\!|$p|\!|$|\!|'s|\!| and|\!| |\!|$q|\!|$|\!|'s|\!|,|\!| secondary|\!| constraints|\!| result|\!| from|\!| the|\!| equations|\!| of|\!| motion|\!|.|\!| To|\!| show|\!| their|\!| absence|\!| in|\!| |\!|'t|\!||\!|,Hooft|\!|'s|\!| system|\!| we|\!| start|\!| with|\!| the|\!| observation|\!| that|\!| the|\!| time|\!| derivative|\!| of|\!| any|\!| function|\!| |\!|$f|\!|(|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|,|\!| |\!|{|\!|boldsymbol|\!|{p|\!|}|\!|}|\!|)|\!|$|\!| is|\!| given|\!| by|\!|~|\!|cite|\!|{Sunder|\!|}|\!|
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|\!|begin|\!|{eqnarray|\!|}|\!| |\!|dot|\!|{f|\!|}|\!| |\!||\!| |\!|approx|\!| |\!||\!| |\!||\!|{|\!| f|\!|,|\!| |\!|overlinen|\!| H|\!| |\!||\!|}|\!| |\!||\!| |\!| |\!|+|\!| |\!| |\!||\!| u|\!|^j|\!| |\!||\!|{|\!| f|\!|,|\!| |\!|phi|\!|_j|\!||\!|}|\!||\!|,|\!| |\!|.|\!| |\!|end|\!|{eqnarray|\!|}|\!|
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Here|\!| |\!|$u|\!|^a|\!|$|\!| are|\!| the|\!| Lagrange|\!| multipliers|\!| to|\!| be|\!| determined|\!| by|\!| the|\!| consistency|\!| conditions|\!|
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|\!|begin|\!|{eqnarray|\!|}|\!| 0|\!| |\!||\!| |\!|approx|\!| |\!||\!| |\!|dot|\!|{|\!|phi|\!|_i|\!|}|\!| |\!||\!| |\!|approx|\!| |\!||\!| |\!||\!|{|\!| |\!|phi|\!|_i|\!|,|\!| |\!|overlinen|\!| H|\!| |\!||\!|}|\!| |\!||\!| |\!|+|\!| |\!||\!| u|\!|^j|\!| |\!||\!|{|\!|phi|\!|_i|\!|,|\!| |\!|phi|\!|_j|\!| |\!||\!|}|\!||\!|,|\!| |\!|.|\!| |\!|label|\!|{A1|\!|}|\!| |\!|end|\!|{eqnarray|\!|}|\!|
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The|\!| latter|\!| is|\!| nothing|\!| but|\!| the|\!| statement|\!| that|\!| constraints|\!| |\!|(as|\!| functions|\!| of|\!| |\!|$|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|$|\!| and|\!| |\!|$|\!|{|\!|boldsymbol|\!|{p|\!|}|\!|}|\!|$|\!|)|\!| must|\!| hold|\!| at|\!| any|\!| time|\!|.|\!| If|\!| all|\!| |\!|$u|\!|^j|\!|$|\!| could|\!| not|\!| be|\!| determined|\!| from|\!| the|\!| consistency|\!| condition|\!| |\!|(|\!|ref|\!|{A1|\!|}|\!|)|\!| then|\!| we|\!| would|\!| have|\!|
|\!| the|\!| so|\!|-called|\!| secondary|\!| constraints|\!|.|\!| |\!| In|\!| our|\!| case|\!| we|\!| have|\!|
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|\!|begin|\!|{eqnarray|\!|}|\!| |\!||\!|{|\!| |\!|phi|\!|_1|\!|^a|\!|,|\!| |\!|{|\!|overlinen|\!| H|\!|}|\!| |\!||\!|}|\!| |\!||\!| |\!|=|\!| |\!||\!| |\!|-|\!| |\!| |\!|frac|\!|{|\!|partial|\!| |\!|bar|\!|{H|\!|}|\!|}|\!|{|\!|partial|\!| q|\!|_a|\!|}|\!| |\!||\!| |\!|not|\!|approx|\!| |\!||\!| 0|\!||\!|,|\!| |\!|,|\!| |\!||\!|;|\!||\!|;|\!||\!|;|\!||\!|;|\!| |\!||\!|{|\!| |\!|phi|\!|_2|\!|^a|\!|,|\!| |\!|{|\!|overlinen|\!| H|\!|}|\!| |\!||\!|}|\!| |\!||\!| |\!|=|\!| |\!||\!| |\!|-|\!| f|\!|_a|\!|(|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|)|\!||\!| |\!|not|\!|approx|\!| |\!||\!| 0|\!||\!|,|\!| |\!|,|\!| |\!||\!|;|\!||\!|;|\!||\!|;|\!||\!|;|\!| |\!||\!|{|\!| |\!|phi|\!|_1|\!|^a|\!|,|\!| |\!|phi|\!|_2|\!|^b|\!| |\!||\!|}|\!| |\!||\!| |\!|=|\!| |\!||\!| |\!|-|\!| |\!|delta|\!|_|\!|{ab|\!|}|\!||\!|,|\!| |\!|.|\!| |\!|label|\!|{A3|\!|}|\!| |\!|end|\!|{eqnarray|\!|}|\!|
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Using|\!| the|\!| fact|\!| that|\!| |\!|$|\!||\!|{|\!| |\!|phi|\!|_i|\!|,|\!| |\!|{|\!|overlinen|\!| H|\!|}|\!| |\!||\!|}|\!| |\!||\!| |\!|not|\!|approx|\!| |\!||\!|
0|\!|$|\!| and|\!| |\!|$|\!|det|\!|left|\!|||\!||\!|{|\!|phi|\!|_i|\!|,|\!| |\!|phi|\!|_j|\!| |\!||\!|}|\!| |\!|right|\!|||\!| |\!|=|\!| 1|\!|$|\!|,|\!| the|\!| inhomogeneous|\!| system|\!| of|\!| linear|\!| equations|\!| |\!|(|\!|ref|\!|{A1|\!|}|\!|)|\!| can|\!| be|\!| uniquely|\!| resolved|\!| with|\!| respect|\!| to|\!| |\!|$u|\!|^j|\!|$|\!|,|\!| thus|\!| implying|\!| the|\!| absence|\!| of|\!| secondary|\!| constraints|\!|.|\!|
|\!|
|\!|
|\!|section|\!|*|\!|{Appendix|\!| B|\!|}|\!|
|\!|
|\!|subsection|\!|*|\!|{BI|\!|}|\!|
|\!|
|\!|
We|\!| show|\!| here|\!| that|\!| Gozzi|\!|'s|\!| configuration|\!|-space|\!| path|\!| integral|\!| results|\!| from|\!| the|\!| |\!|`|\!|`classical|\!|"|\!| limit|\!| of|\!| the|\!| stochastic|\!|-quantization|\!| partition|\!| function|\!|,|\!| i|\!|.e|\!|.|\!|,|\!| the|\!| limit|\!| where|\!| the|\!| width|\!| of|\!| a|\!| noise|\!| distribution|\!| tends|\!| to|\!| zero|\!|.|\!|
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|\!|
|\!|
|\!|
For|\!| this|\!| purpose|\!| we|\!| start|\!| with|\!| the|\!| form|\!| of|\!| the|\!| partition|\!| function|\!| for|\!| stochastic|\!| quantization|\!| as|\!| written|\!| down|\!| by|\!| Zinn|\!|-Justin|\!|~|\!|cite|\!|{Zinn|\!|-JustinII|\!|,Zinn|\!|-Justin1|\!|}|\!|:|\!|
|\!|
|\!| |\!|begin|\!|{eqnarray|\!|}|\!| Z|\!|_|\!|{|\!|rm|\!| SC|\!|}|\!|(J|\!|)|\!| |\!|=|\!| |\!|int|\!| |\!|{|\!|mathcal|\!|{D|\!|}|\!|}|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!| |\!|{|\!|mathcal|\!|{D|\!|}|\!|}|\!|{|\!|boldsymbol|\!|{c|\!|}|\!|}|\!| |\!|{|\!|mathcal|\!|{D|\!|}|\!|}|\!|bar|\!|{|\!|{|\!|boldsymbol|\!|{c|\!|}|\!|}|\!|}|\!| |\!|{|\!|mathcal|\!|{D|\!|}|\!|}|\!|boldsymbol|\!|{|\!|lambda|\!|}|\!| |\!||\!| |\!|exp|\!|left|\!||\!|{|\!| |\!|-|\!| |\!|{|\!|mathcal|\!|{S|\!|}|\!|}|\!|[|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|,|\!|{|\!|boldsymbol|\!|{c|\!|}|\!|}|\!|,|\!|bar|\!|{|\!|{|\!|boldsymbol|\!|{c|\!|}|\!|}|\!|}|\!|,|\!| |\!|boldsymbol|\!|{|\!|lambda|\!|}|\!|]|\!| |\!|+|\!| |\!|int|\!| |\!|{|\!|boldsymbol|\!|{J|\!|}|\!|}|\!|(x|\!|)|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|(x|\!|)|\!| dx|\!| |\!|right|\!||\!|}|\!||\!|,|\!| |\!|,|\!| |\!|label|\!|{C1|\!|}|\!|
|\!| |\!|end|\!|{eqnarray|\!|}|\!|
|\!|
where|\!|
|\!|
|\!|begin|\!|{eqnarray|\!|}|\!| |\!|{|\!|mathcal|\!|{S|\!|}|\!|}|\!| |\!|equiv|\!| |\!| |\!|&|\!|-|\!|&|\!| w|\!|(|\!|{|\!|boldsymbol|\!|{|\!|lambda|\!|}|\!|}|\!|)|\!| |\!|+|\!| |\!|int|\!| |\!|{|\!|boldsymbol|\!|{|\!|lambda|\!|}|\!|}|\!|(x|\!|)|\!|left|\!|(|\!| |\!|frac|\!|{|\!|partial|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|(x|\!|)|\!|}|\!|{|\!|partial|\!|tau|\!|}|\!| |\!||\!| |\!|+|\!| |\!|frac|\!|{|\!|delta|\!| |\!|{|\!|mathcal|\!|{A|\!|}|\!|}|\!|}|\!|{|\!|delta|\!| |\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|(x|\!|)|\!|}|\!| |\!|right|\!|)dx|\!| |\!|nonumber|\!| |\!||\!||\!| |\!|&|\!|-|\!|&|\!| |\!|int|\!| dx|\!| dx|\!|'|\!| |\!||\!| |\!|bar|\!|{c|\!|}|\!|_a|\!|(x|\!|)|\!| |\!|left|\!|(|\!| |\!|frac|\!|{|\!|partial|\!|}|\!|{|\!|partial|\!| |\!|tau|\!|}|\!| |\!|delta|\!|_|\!|{ab|\!|}|\!| |\!|delta|\!|(x|\!| |\!|-|\!| x|\!|'|\!|)|\!| |\!|+|\!| |\!|frac|\!|{|\!|delta|\!|^2|\!| |\!|{|\!|mathcal|\!|{A|\!|}|\!|}|\!| |\!|}|\!|{|\!|delta|\!| q|\!|_a|\!|(x|\!|)|\!| |\!|delta|\!| q|\!|_b|\!|(x|\!|'|\!|)|\!|}|\!|right|\!|)|\!| |\!||\!| c|\!|_b|\!|(x|\!|'|\!|)|\!||\!|,|\!| |\!|,|\!| |\!|end|\!|{eqnarray|\!|}|\!|
|\!|
and|\!|
|\!|
|\!|begin|\!|{eqnarray|\!|}|\!| |\!|exp|\!|[w|\!|(|\!|{|\!|boldsymbol|\!|{|\!|lambda|\!|}|\!|}|\!|)|\!|]|\!|equiv|\!| |\!|int|\!| |\!|{|\!|mathcal|\!|{D|\!|}|\!|}|\!|{|\!|{|\!|boldsymbol|\!|{|\!|nu|\!|}|\!|}|\!|}|\!| |\!|exp|\!|left|\!||\!|{|\!|-|\!|sigma|\!|(|\!|{|\!|boldsymbol|\!|{|\!|nu|\!|}|\!|}|\!|)|\!| |\!|+|\!| |\!|int|\!| dx|\!| |\!|{|\!|boldsymbol|\!|{|\!|lambda|\!|}|\!|}|\!|(x|\!|)|\!| |\!|{|\!|boldsymbol|\!|{|\!|nu|\!|}|\!|}|\!|(x|\!|)|\!|right|\!||\!|}|\!||\!|,|\!| |\!|,|\!| |\!|end|\!|{eqnarray|\!|}|\!|
|\!|
with|\!| |\!| |\!|$|\!|{|\!|mathcal|\!|{D|\!|}|\!|}|\!|{|\!|{|\!|boldsymbol|\!|{|\!|nu|\!|}|\!|}|\!|}|\!| |\!|exp|\!|(|\!|-|\!|sigma|\!|(|\!|{|\!|boldsymbol|\!|{|\!|nu|\!|}|\!|}|\!|)|\!|)|\!|$|\!| being|\!| the|\!| functional|\!| measure|\!| of|\!| noise|\!|.|\!| Here|\!| |\!|$x|\!| |\!|=|\!| |\!|(t|\!|,|\!|tau|\!|)|\!|$|\!| and|\!| |\!|$dx|\!| |\!|=|\!| dtd|\!|tau|\!|$|\!| where|\!| |\!|$|\!|tau|\!|$|\!| is|\!| the|\!| Parisi|\!|-Wu|\!| fictitious|\!| time|\!|.|\!| The|\!| dynamical|\!| equation|\!| for|\!| |\!|$|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|(x|\!|)|\!|$|\!| is|\!| described|\!| by|\!| the|\!| Langevin|\!| equation|\!|
|\!|
|\!|begin|\!|{eqnarray|\!|}|\!| |\!|frac|\!|{|\!|partial|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|(x|\!|)|\!|}|\!|{|\!|partial|\!| |\!|tau|\!|}|\!| |\!|+|\!| |\!|left|\!|.|\!| |\!|frac|\!|{|\!|delta|\!| |\!|{|\!|mathcal|\!|{A|\!|}|\!|}|\!| |\!|[|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|]|\!|}|\!|{|\!| |\!|delta|\!|
|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|}|\!|right|\!|||\!|_|\!|{|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!| |\!|=|\!| |\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|(x|\!|)|\!|}|\!| |\!|=|\!| |\!|{|\!|boldsymbol|\!|{|\!|nu|\!|}|\!|}|\!|(x|\!|)|\!||\!|,|\!| |\!|,|\!| |\!|label|\!|{B3|\!|}|\!| |\!|end|\!|{eqnarray|\!|}|\!|
|\!|
with|\!| the|\!| initial|\!| condition|\!| |\!|$|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|(t|\!|,0|\!|)|\!| |\!|=|\!| |\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|(t|\!|)|\!|$|\!|.|\!| For|\!| Gaussian|\!| noise|\!| of|\!| variance|\!| |\!|$2h|\!|$|\!|,|\!| the|\!| noise|\!| measure|\!| is|\!|
|\!|
|\!|begin|\!|{eqnarray|\!|}|\!| |\!|&|\!|&|\!|{|\!|mathcal|\!|{D|\!|}|\!|}|\!|{|\!|{|\!|boldsymbol|\!|{|\!|nu|\!|}|\!|}|\!|}|\!| |\!|exp|\!|(|\!|-|\!|sigma|\!|(|\!|{|\!|boldsymbol|\!|{|\!|nu|\!|}|\!|}|\!|)|\!|)|\!| |\!|=|\!| |\!|prod|\!|_|\!|{i|\!|,x|\!|}|\!|frac|\!|{d|\!|nu|\!|_i|\!|(x|\!|)|\!|}|\!|{2|\!|sqrt|\!|{|\!|pi|\!| |\!|hbar|\!|}|\!|}|\!| |\!|exp|\!|left|\!|(|\!| |\!|-|\!| |\!|frac|\!|{1|\!|}|\!|{4|\!|hbar|\!|}|\!| |\!||\!| |\!|int|\!| dx|\!| |\!|{|\!|boldsymbol|\!|{|\!|nu|\!|}|\!|}|\!|^2|\!|(x|\!|)|\!| |\!|right|\!|)|\!||\!|,|\!| |\!|,|\!| |\!|end|\!|{eqnarray|\!|}|\!|
|\!|
and|\!| |\!|(|\!|ref|\!|{C1|\!|}|\!|)|\!| takes|\!| the|\!| form|\!| |\!|begin|\!|{eqnarray|\!|}|\!| Z|\!|_|\!|{|\!|rm|\!| SC|\!|}|\!|(J|\!|)|\!||\!| |\!|&|\!|=|\!|&|\!| |\!||\!| |\!| |\!|int|\!| |\!|{|\!|mathcal|\!|{D|\!|}|\!|}|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!| |\!|{|\!|mathcal|\!|{D|\!|}|\!|}|\!|{|\!|boldsymbol|\!|{|\!|nu|\!|}|\!|}|\!| |\!||\!| |\!|delta|\!| |\!||\!|!|\!| |\!|left|\!|(|\!| |\!|frac|\!|{|\!|partial|\!| |\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|}|\!|{|\!|partial|\!| |\!|tau|\!|}|\!| |\!|+|\!| |\!|frac|\!|{|\!|delta|\!| |\!|{|\!|mathcal|\!|{A|\!|}|\!|}|\!| |\!|[|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|]|\!|}|\!|{|\!|delta|\!| |\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|}|\!| |\!|-|\!|
|\!|{|\!|boldsymbol|\!|{|\!|nu|\!|}|\!|}|\!|right|\!|)|\!| |\!|det|\!|left|\!|||\!||\!|!|\!|left|\!|||\!| |\!|frac|\!|{|\!|partial|\!|}|\!|{|\!|partial|\!| |\!|tau|\!|}|\!| |\!|delta|\!|_|\!|{ab|\!|}|\!| |\!|delta|\!|(x|\!| |\!|-|\!| x|\!|'|\!|)|\!| |\!|+|\!|
|\!|frac|\!|{|\!|delta|\!|^2|\!| |\!|{|\!|mathcal|\!|{A|\!|}|\!|}|\!| |\!|}|\!|{|\!|delta|\!| q|\!|_a|\!|(x|\!|)|\!| |\!|delta|\!| q|\!|_b|\!|(x|\!|'|\!|)|\!|}|\!| |\!|right|\!|||\!||\!|!|\!|right|\!|||\!|nonumber|\!| |\!||\!||\!| |\!|&|\!|&|\!|times|\!| |\!||\!| |\!|exp|\!|left|\!||\!|{|\!| |\!|-|\!| |\!|sigma|\!|(|\!|{|\!|boldsymbol|\!|{|\!|nu|\!|}|\!|}|\!|)|\!| |\!|+|\!| |\!| |\!| |\!|int|\!| |\!|{|\!|boldsymbol|\!|{J|\!|}|\!|}|\!|(x|\!|)|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|(x|\!|)|\!| dx|\!| |\!|right|\!||\!|}|\!|nonumber|\!| |\!||\!||\!| |\!|&|\!|=|\!|&|\!| |\!||\!| |\!|int|\!| |\!|{|\!|mathcal|\!|{D|\!|}|\!|}|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!| |\!|{|\!|mathcal|\!|{D|\!|}|\!|}|\!|{|\!|boldsymbol|\!|{|\!|nu|\!|}|\!|}|\!| |\!||\!| |\!|delta|\!| |\!||\!|!|\!|left|\!|[|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!| |\!|-|\!| |\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|^|\!|{|\!|[|\!|{|\!|boldsymbol|\!|{|\!|nu|\!|}|\!|}|\!|]|\!|}|\!|right|\!|]|\!| |\!|exp|\!|left|\!||\!|{|\!| |\!|-|\!| |\!|sigma|\!|(|\!|{|\!|boldsymbol|\!|{|\!|nu|\!|}|\!|}|\!|)|\!| |\!|+|\!| |\!| |\!| |\!|int|\!| |\!|{|\!|boldsymbol|\!|{J|\!|}|\!|}|\!|(x|\!|)|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|(x|\!|)|\!| dx|\!| |\!|right|\!||\!|}|\!||\!|,|\!| |\!|.|\!| |\!|end|\!|{eqnarray|\!|}|\!|
|\!|
where|\!| |\!|$|\!|delta|\!|[f|\!|(|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|)|\!|]|\!| |\!|equiv|\!| |\!|prod|\!|_|\!|{t|\!|,|\!|tau|\!|}|\!| |\!|delta|\!|(f|\!|(|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|(t|\!|,|\!|tau|\!|)|\!|)|\!|)|\!|$|\!| and|\!| |\!|$|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|^|\!|{|\!|[|\!|{|\!|boldsymbol|\!|{|\!|nu|\!|}|\!|}|\!|]|\!|}|\!|(x|\!|)|\!|$|\!| is|\!| a|\!| solution|\!| of|\!| |\!|(|\!|ref|\!|{B3|\!|}|\!|)|\!|.|\!| Using|\!| the|\!| representation|\!|.|\!|
|\!|
|\!|begin|\!|{eqnarray|\!|}|\!| |\!|delta|\!|(x|\!|)|\!| |\!||\!| |\!|=|\!| |\!||\!| |\!|lim|\!|_|\!|{|\!|hbar|\!| |\!|rightarrow|\!| 0|\!|_|\!|+|\!|}|\!| |\!|frac|\!|{1|\!|}|\!|{2|\!|sqrt|\!|{|\!|pi|\!| |\!|hbar|\!|}|\!|}|\!| |\!||\!| e|\!|^|\!|{|\!|-x|\!|^2|\!|/|\!|(4|\!|hbar|\!|)|\!|}|\!||\!|,|\!| |\!|,|\!| |\!|end|\!|{eqnarray|\!|}|\!|
|\!|
we|\!| get|\!| in|\!| the|\!| limit|\!| of|\!| zero|\!| distribution|\!| width|\!| |\!|(i|\!|.e|\!|.|\!|,|\!| |\!|$|\!|hbar|\!| |\!|rightarrow|\!| 0|\!|_|\!|+|\!|$|\!|)|\!| that|\!|
|\!|
|\!|begin|\!|{eqnarray|\!|}|\!| Z|\!|_|\!|{|\!|rm|\!| SC|\!|}|\!|(J|\!|,|\!|hbar|\!|)|\!||\!| |\!|rightarrow|\!| |\!||\!| |\!|int|\!| |\!|{|\!|mathcal|\!|{D|\!|}|\!|}|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!| |\!||\!| |\!|delta|\!| |\!||\!|!|\!|left|\!|[|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!| |\!|-|\!| |\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|^|\!|{|\!|[0|\!|]|\!|}|\!|right|\!|]|\!|
|\!| |\!|exp|\!|left|\!||\!|{|\!| |\!|int|\!| |\!|{|\!|boldsymbol|\!|{J|\!|}|\!|}|\!|(x|\!|)|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|(x|\!|)|\!| dx|\!| |\!|right|\!||\!|}|\!||\!|,|\!| |\!|.|\!| |\!|end|\!|{eqnarray|\!|}|\!|
|\!|
Choosing|\!| a|\!| special|\!| source|\!| |\!|$|\!|{|\!|boldsymbol|\!|{J|\!|}|\!|}|\!|(x|\!|)|\!| |\!|=|\!| |\!|{|\!|boldsymbol|\!|{J|\!|}|\!|}|\!|(t|\!|)|\!| |\!|delta|\!|(|\!|tau|\!|)|\!|$|\!| we|\!| can|\!| sum|\!| in|\!| the|\!| path|\!| integral|\!| solely|\!| over|\!| configurations|\!| with|\!| |\!|$|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|(t|\!|,0|\!|)|\!| |\!|=|\!| |\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|(t|\!|)|\!|$|\!| as|\!| other|\!| configurations|\!| will|\!| contribute|\!| only|\!| to|\!| an|\!| overall|\!| normalization|\!| constant|\!|.|\!| Inasmuch|\!| we|\!| finally|\!| obtain|\!|
|\!|
|\!|begin|\!|{eqnarray|\!|}|\!| |\!|lim|\!|_|\!|{|\!|hbar|\!| |\!|rightarrow|\!| 0|\!|^|\!|+|\!|}|\!| Z|\!|_|\!|{|\!|rm|\!| SC|\!|}|\!|(|\!|{|\!|boldsymbol|\!|{J|\!|}|\!|}|\!|,|\!|hbar|\!|)|\!| |\!||\!| |\!|=|\!| |\!||\!| |\!|{|\!|{|\!|{Z|\!|}|\!|}|\!| |\!|}|\!|_|\!|{|\!|rm|\!| CM|\!|}|\!|(|\!|{|\!|boldsymbol|\!|{J|\!|}|\!|}|\!|)|\!||\!|,|\!| |\!|.|\!| |\!|end|\!|{eqnarray|\!|}|\!|
|\!|
|\!|
|\!|subsection|\!|*|\!|{BII|\!|}|\!|
|\!|
|\!|
|\!|
In|\!| this|\!| part|\!| of|\!| the|\!| appendix|\!| we|\!| show|\!| that|\!| Gozzi|\!|'s|\!| configuration|\!|-space|\!| partition|\!| function|\!| |\!|(|\!|ref|\!|{4|\!|.3|\!|}|\!|)|\!| results|\!| from|\!| the|\!| |\!|`|\!|`classical|\!|"|\!| limit|\!| of|\!| the|\!| closed|\!|-time|\!| path|\!| integral|\!| for|\!| the|\!| transition|\!| probability|\!| of|\!| a|\!| system|\!| coupled|\!| to|\!| a|\!| thermal|\!| reservoir|\!| at|\!| some|\!| temperature|\!| |\!|$T|\!|$|\!|.|\!| By|\!| the|\!| classical|\!| limit|\!| we|\!| mean|\!| the|\!| high|\!| temperature|\!| and|\!| weak|\!| heat|\!| bath|\!| coupling|\!| limit|\!|.|\!|
|\!|
The|\!| path|\!|-integral|\!| treatment|\!| of|\!| systems|\!| that|\!| are|\!| linearly|\!| coupled|\!| to|\!| a|\!| thermal|\!| bath|\!| of|\!| harmonic|\!| oscillators|\!| was|\!| first|\!| considered|\!| by|\!| Feynman|\!| and|\!| Vernon|\!|~|\!|cite|\!|{FV|\!|}|\!|.|\!| For|\!| our|\!| purpose|\!| it|\!| will|\!| be|\!| particularly|\!| convenient|\!| to|\!| utilize|\!| the|\!| so|\!| called|\!| Ohmic|\!| limit|\!| version|\!|,|\!| as|\!| discussed|\!| in|\!| Refs|\!|.|\!|cite|\!|{Pain|\!|,Klein1|\!|}|\!|:|\!|
|\!|
|\!|begin|\!|{eqnarray|\!|}|\!| |\!|{|\!|mathcal|\!|{Z|\!|}|\!|}|\!|_|\!|{|\!|rm|\!| FV|\!|}|\!|[|\!|{|\!|boldsymbol|\!|{J|\!|}|\!|}|\!|_|\!|+|\!|,|\!| |\!|{|\!|boldsymbol|\!|{J|\!|}|\!|}|\!|_|\!|-|\!|]|\!| |\!||\!| |\!|&|\!|=|\!|&|\!| |\!||\!| |\!|int|\!| |\!|{|\!|mathcal|\!|{D|\!|}|\!|}|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|_|\!|+|\!| |\!|{|\!|mathcal|\!|{D|\!|}|\!|}|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|_|\!|-|\!| |\!||\!| |\!|exp|\!|left|\!||\!|{|\!| |\!|frac|\!|{i|\!|}|\!|{|\!|hbar|\!|}|\!| |\!|left|\!|[|\!| |\!|{|\!|mathcal|\!|{A|\!|}|\!|}|\!|[|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|_|\!|+|\!|]|\!| |\!|-|\!| |\!|{|\!|mathcal|\!|{A|\!|}|\!|}|\!|[|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|_|\!|-|\!|]|\!| |\!|right|\!|]|\!|+|\!| |\!|int|\!| dt|\!| |\!||\!| |\!|left|\!|[|\!| |\!|{|\!|boldsymbol|\!|{J|\!|}|\!|}|\!|_|\!|+|\!|(t|\!|)|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|_|\!|+|\!|(t|\!|)|\!| |\!| |\!| |\!|-|\!| |\!|{|\!|boldsymbol|\!|{J|\!|}|\!|}|\!|_|\!|-|\!|(t|\!|)|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|_|\!|-|\!|(t|\!|)|\!|right|\!|]|\!| |\!|right|\!||\!|}|\!|nonumber|\!| |\!||\!||\!| |\!|&|\!|&|\!| |\!|times|\!| |\!||\!| |\!|exp|\!|left|\!||\!|{|\!| |\!|-i|\!| |\!|frac|\!|{m|\!|gamma|\!|}|\!|{2|\!|hbar|\!|}|\!| |\!|int|\!| dt|\!| |\!||\!| |\!|[|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|_|\!|+|\!|(t|\!|)|\!| |\!|-|\!| |\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|_|\!|-|\!|(t|\!|)|\!|]|\!|[|\!|dot|\!|{|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|}|\!|_|\!|+|\!|(t|\!|)|\!| |\!|+|\!| |\!|dot|\!|{|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|}|\!|_|\!|-|\!|(t|\!|)|\!|]|\!|^R|\!| |\!|right|\!||\!|}|\!|nonumber|\!| |\!||\!||\!| |\!|&|\!|&|\!| |\!|times|\!| |\!||\!| |\!|exp|\!|left|\!||\!|{|\!| |\!|-|\!| |\!|frac|\!|{m|\!|gamma|\!|}|\!|{|\!|hbar|\!|^2|\!| |\!|beta|\!|}|\!| |\!|int|\!| dt|\!| |\!|int|\!| dt|\!|'|\!| |\!||\!| |\!|[|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|_|\!|+|\!|(t|\!|)|\!| |\!|-|\!| |\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|_|\!|-|\!|(t|\!|)|\!|]K|\!|(t|\!|,t|\!|'|\!|)|\!|[|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|_|\!|+|\!|(t|\!|'|\!|)|\!| |\!|-|\!| |\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|_|\!|-|\!|(t|\!|'|\!|)|\!|]|\!| |\!|right|\!||\!|}|\!||\!|,|\!| |\!|.|\!| |\!|label|\!|{B13|\!|}|\!| |\!|end|\!|{eqnarray|\!|}|\!|
|\!|
Here|\!| the|\!| paths|\!| |\!|$|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|_|\!|+|\!|(t|\!|)|\!|$|\!| and|\!| |\!|$|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|_|\!|-|\!|(t|\!|)|\!|$|\!| are|\!| associated|\!| with|\!| the|\!| forward|\!| and|\!| backward|\!| movement|\!| of|\!| the|\!| particles|\!| in|\!| time|\!|.|\!| The|\!| super|\!|-script|\!| |\!|$R|\!|$|\!| indicates|\!| a|\!| |\!|{|\!|em|\!| negative|\!|}|\!| shift|\!| in|\!| the|\!| time|\!| argument|\!| of|\!| the|\!| velocities|\!| with|\!| respect|\!| to|\!| positions|\!|.|\!| The|\!| latter|\!| ensures|\!| the|\!| causality|\!| of|\!| the|\!| friction|\!| forces|\!|~|\!|cite|\!|{Klein1|\!|}|\!|.|\!| In|\!| addition|\!|,|\!| |\!|$m|\!|$|\!| represents|\!| the|\!| particle|\!| mass|\!| |\!|(for|\!| simplicity|\!| we|\!| assume|\!| here|\!| that|\!| all|\!| system|\!| particles|\!| have|\!| the|\!| same|\!| mass|\!|)|\!|,|\!| |\!|$|\!|beta|\!| |\!|=|\!| 1|\!|/T|\!|$|\!|,|\!| and|\!| |\!|$|\!|gamma|\!|$|\!| is|\!| the|\!| friction|\!| constant|\!| |\!|(or|\!| thermal|\!| reservoir|\!| coupling|\!|)|\!|.|\!| The|\!| function|\!| |\!|$K|\!|(t|\!|,t|\!|'|\!|)|\!|$|\!| is|\!| the|\!| bath|\!| correlation|\!| function|\!|.|\!| As|\!| argued|\!| in|\!|~|\!|cite|\!|{Pain|\!|,Klein1|\!|}|\!|,|\!| at|\!| high|\!| temperatures|\!| |\!|$K|\!|(t|\!|,t|\!|'|\!|)|\!| |\!|approx|\!| |\!|delta|\!|(t|\!|-t|\!|'|\!|)|\!|$|\!|.|\!| Introducing|\!| the|\!| new|\!| set|\!| of|\!| variables|\!| |\!|$|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!| |\!|=|\!| |\!|[|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|_|\!|+|\!| |\!|+|\!| |\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|_|\!|-|\!|]|\!|/2|\!|$|\!| and|\!| |\!|$|\!|bar|\!|{|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|}|\!| |\!|=|\!| |\!|[|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|_|\!|+|\!| |\!|-|\!| |\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|_|\!|-|\!|]|\!|$|\!| |\!|(i|\!|.e|\!|.|\!|,|\!| the|\!| center|\!|-of|\!|-mass|\!| and|\!| |\!|{|\!|em|\!| fast|\!|}|\!| coordinates|\!|)|\!| we|\!| can|\!| in|\!| the|\!| high|\!|-temperature|\!| case|\!| recast|\!| |\!|(|\!|ref|\!|{B13|\!|}|\!|)|\!| into|\!|
|\!|
|\!|begin|\!|{eqnarray|\!|}|\!| |\!|{|\!|mathcal|\!|{Z|\!|}|\!|}|\!|_|\!|{|\!|rm|\!| FV|\!|}|\!|[|\!|{|\!|boldsymbol|\!|{J|\!|}|\!|}|\!|,|\!| |\!|bar|\!|{|\!|{|\!|boldsymbol|\!|{J|\!|}|\!|}|\!|}|\!|]|\!| |\!||\!| |\!|&|\!|=|\!|&|\!| |\!||\!| |\!|int|\!| |\!|{|\!|mathcal|\!|{D|\!|}|\!|}|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!| |\!|{|\!|mathcal|\!|{D|\!|}|\!|}|\!| |\!|bar|\!|{|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|}|\!| |\!||\!| |\!|exp|\!|left|\!||\!|{|\!| |\!|frac|\!|{i|\!|}|\!|{|\!|hbar|\!|}|\!| |\!|left|\!|[|\!| |\!|{|\!|mathcal|\!|{A|\!|}|\!|}|\!|[|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!| |\!|+|\!| |\!|bar|\!|{|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|}|\!|/2|\!|]|\!| |\!|-|\!| |\!|{|\!|mathcal|\!|{A|\!|}|\!|}|\!|[|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!| |\!|-|\!| |\!|bar|\!|{|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|}|\!|/2|\!|]|\!| |\!|right|\!|]|\!| |\!|+|\!| |\!|int|\!| dt|\!| |\!||\!| |\!|left|\!|[|\!|{|\!|boldsymbol|\!|{J|\!|}|\!|}|\!|(t|\!|)|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|(t|\!|)|\!| |\!|-|\!| |\!|bar|\!|{|\!|{|\!|boldsymbol|\!|{J|\!|}|\!|}|\!|}|\!|(t|\!|)|\!|bar|\!|{|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|}|\!|(t|\!|)|\!| |\!|right|\!|]|\!| |\!|right|\!||\!|}|\!|nonumber|\!| |\!||\!||\!| |\!|&|\!|&|\!| |\!|times|\!| |\!||\!| |\!|exp|\!|left|\!||\!|{|\!|-|\!| i|\!|frac|\!|{m|\!|gamma|\!|}|\!|{|\!|hbar|\!|}|\!| |\!|int|\!| dt|\!| |\!||\!| |\!|bar|\!|{|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|}|\!|(t|\!|)|\!|left|\!|[|\!|dot|\!|{|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|}|\!|(t|\!|)|\!|right|\!|]|\!|^R|\!| |\!|-|\!| |\!|frac|\!|{m|\!|gamma|\!|}|\!|{|\!|hbar|\!|^2|\!| |\!|beta|\!|}|\!| |\!|int|\!| dt|\!| |\!||\!| |\!|bar|\!|{|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|}|\!|^2|\!|(t|\!|)|\!| |\!|right|\!||\!|}|\!||\!|,|\!| |\!|.|\!| |\!|end|\!|{eqnarray|\!|}|\!|
|\!|
Here|\!| the|\!| self|\!|-explanatory|\!| notation|\!| |\!|$|\!|{|\!|boldsymbol|\!|{J|\!|}|\!|}|\!|=|\!| |\!|[|\!|{|\!|boldsymbol|\!|{J|\!|}|\!|}|\!|_|\!|+|\!| |\!|-|\!| |\!|{|\!|boldsymbol|\!|{J|\!|}|\!|}|\!|_|\!|-|\!|]|\!|$|\!| and|\!| |\!|$|\!|bar|\!|{|\!|{|\!|boldsymbol|\!|{J|\!|}|\!|}|\!|}|\!| |\!|=|\!| |\!|-|\!|[|\!|{|\!|boldsymbol|\!|{J|\!|}|\!|}|\!|_|\!|+|\!| |\!|+|\!| |\!|{|\!|boldsymbol|\!|{J|\!|}|\!|}|\!|_|\!|-|\!|]|\!|/2|\!|$|\!| |\!| was|\!| used|\!|.|\!| Let|\!| us|\!| now|\!| define|\!| |\!|$|\!|omega|\!| |\!|=|\!| 2m|\!|gamma|\!| |\!|/|\!|beta|\!|$|\!|,|\!| integrate|\!| over|\!| |\!|$|\!|bar|\!|{|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|}|\!|$|\!|,|\!|
|\!| and|\!| go|\!| to|\!| the|\!|
|\!| |\!| classical|\!| limit|\!| |\!|$|\!|gamma|\!| |\!|rightarrow|\!| 0|\!| |\!|$|\!|.|\!| Then|\!| we|\!| obtain|\!| the|\!| following|\!| chain|\!| of|\!| equations|\!|:|\!|
|\!|
|\!|begin|\!|{eqnarray|\!|}|\!| |\!|&|\!|&|\!|lim|\!|_|\!|{|\!|gamma|\!| |\!|rightarrow|\!| 0|\!|}|\!| |\!||\!| |\!|{|\!|mathcal|\!|{Z|\!|}|\!|}|\!|_|\!|{|\!|rm|\!| FV|\!|}|\!|[|\!|{|\!|boldsymbol|\!|{J|\!|}|\!|}|\!|,|\!| |\!|bar|\!|{|\!|{|\!|boldsymbol|\!|{J|\!|}|\!|}|\!|}|\!|]|\!|nonumber|\!| |\!||\!||\!| |\!|&|\!|&|\!| |\!|mbox|\!|{|\!|hspace|\!|{1cm|\!|}|\!|}|\!|=|\!| |\!||\!| |\!|lim|\!|_|\!|{|\!|gamma|\!| |\!|rightarrow|\!| 0|\!|}|\!| |\!|int|\!| |\!|{|\!|mathcal|\!|{D|\!|}|\!|}|\!|{|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|}|\!| |\!|{|\!|mathcal|\!|{D|\!|}|\!|}|\!|bar|\!|{|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|}|\!| |\!||\!| |\!|exp|\!|left|\!||\!|{|\!| |\!|frac|\!|{i|\!|}|\!|{|\!|hbar|\!|}|\!| |\!|int|\!| dt|\!| |\!||\!| |\!|bar|\!|{|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|}|\!|(t|\!|)|\!| |\!|left|\!|[|\!| |\!|frac|\!|{|\!|delta|\!|{|\!|mathcal|\!|{A|\!|}|\!|}|\!|}|\!|{|\!|delta|\!| |\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|(t|\!|)|\!|}|\!| |\!|-|\!| m|\!|gamma|\!| |\!|left|\!|[|\!|dot|\!|{|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|}|\!|(t|\!|)|\!|right|\!|]|\!|^R|\!| |\!|+|\!| i|\!| |\!|hbar|\!| |\!|bar|\!|{|\!|{|\!|boldsymbol|\!|{J|\!|}|\!|}|\!|}|\!|(t|\!|)|\!| |\!|right|\!|]|\!| |\!|-|\!| |\!|frac|\!|{|\!|omega|\!|}|\!|{2|\!|hbar|\!|^2|\!|}|\!| |\!|int|\!| dt|\!| |\!||\!| |\!|bar|\!|{|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|}|\!|^2|\!|(t|\!|)|\!| |\!|right|\!||\!|}|\!| |\!|nonumber|\!| |\!||\!||\!| |\!|&|\!|&|\!| |\!|mbox|\!|{|\!|hspace|\!|{1|\!|.5cm|\!|}|\!|}|\!|times|\!| |\!||\!| |\!|exp|\!|left|\!||\!|{|\!| |\!|int|\!| dt|\!| |\!||\!| |\!|{|\!|boldsymbol|\!|{J|\!|}|\!|}|\!|(t|\!|)|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|(t|\!|)|\!| |\!|right|\!||\!|}|\!|nonumber|\!| |\!||\!||\!| |\!|&|\!|&|\!| |\!|mbox|\!|{|\!|hspace|\!|{1cm|\!|}|\!|}|\!|=|\!| |\!||\!| |\!|lim|\!|_|\!|{|\!|gamma|\!| |\!| |\!|rightarrow|\!| 0|\!|}|\!| |\!||\!| |\!|int|\!| |\!|{|\!|mathcal|\!|{D|\!|}|\!|}|\!|{|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|}|\!| |\!||\!| |\!|exp|\!| |\!|left|\!||\!|{|\!|-|\!| |\!|frac|\!|{1|\!|}|\!|{2|\!|omega|\!|}|\!| |\!|int|\!| dt|\!| |\!||\!| |\!|left|\!|[|\!| |\!|frac|\!|{|\!|delta|\!|{|\!|mathcal|\!|{A|\!|}|\!|}|\!|}|\!|{|\!|delta|\!| |\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|(t|\!|)|\!|}|\!| |\!|-|\!| m|\!|gamma|\!| |\!|left|\!|[|\!|dot|\!|{|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|}|\!|(t|\!|)|\!|right|\!|]|\!|^R|\!| |\!|+|\!| i|\!| |\!|hbar|\!| |\!|bar|\!|{|\!|{|\!|boldsymbol|\!|{J|\!|}|\!|}|\!|}|\!|(t|\!|)|\!| |\!|right|\!|]|\!|^2|\!| |\!|+|\!| |\!| |\!|int|\!| dt|\!| |\!||\!| |\!|{|\!|boldsymbol|\!|{J|\!|}|\!|}|\!|(t|\!|)|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|(t|\!|)|\!| |\!| |\!|right|\!||\!|}|\!| |\!|nonumber|\!| |\!||\!||\!| |\!|&|\!|&|\!| |\!|mbox|\!|{|\!|hspace|\!|{1cm|\!|}|\!|}|\!|=|\!| |\!||\!| |\!|lim|\!|_|\!|{|\!|gamma|\!| |\!| |\!|rightarrow|\!| 0|\!|}|\!| |\!||\!| |\!|int|\!| |\!|{|\!|mathcal|\!|{D|\!|}|\!|}|\!|{|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|}|\!| |\!||\!| |\!|{|\!|mathcal|\!|{J|\!|}|\!|}|\!|[|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|]|\!||\!| |\!|exp|\!| |\!|left|\!||\!|{|\!|-|\!| |\!|frac|\!|{1|\!|}|\!|{2|\!|omega|\!|}|\!| |\!|int|\!| dt|\!| |\!||\!| |\!|left|\!|[|\!| |\!|frac|\!|{|\!|delta|\!|{|\!|mathcal|\!|{A|\!|}|\!|}|\!|}|\!|{|\!|delta|\!| |\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|(t|\!|)|\!|}|\!| |\!|-|\!| m|\!|gamma|\!| |\!|dot|\!|{|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|}|\!|(t|\!|)|\!| |\!|+|\!| i|\!| |\!|hbar|\!| |\!|bar|\!|{|\!|{|\!|boldsymbol|\!|{J|\!|}|\!|}|\!|}|\!|(t|\!|)|\!| |\!|right|\!|]|\!|^2|\!| |\!|+|\!| |\!| |\!|int|\!| dt|\!| |\!||\!| |\!|{|\!|boldsymbol|\!|{J|\!|}|\!|}|\!|(t|\!|)|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|(t|\!|)|\!| |\!| |\!|right|\!||\!|}|\!| |\!|nonumber|\!| |\!||\!||\!| |\!|&|\!|&|\!| |\!|mbox|\!|{|\!|hspace|\!|{1cm|\!|}|\!|}|\!|=|\!| |\!||\!| |\!|int|\!| |\!|{|\!|mathcal|\!|{D|\!|}|\!|}|\!|{|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|}|\!| |\!||\!| |\!|delta|\!| |\!||\!|!|\!|left|\!|[|\!| |\!|frac|\!|{|\!|delta|\!|{|\!|mathcal|\!|{A|\!|}|\!|}|\!|}|\!|{|\!|delta|\!| |\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|}|\!| |\!|+|\!| i|\!| |\!|hbar|\!| |\!|bar|\!|{|\!|{|\!|boldsymbol|\!|{J|\!|}|\!|}|\!|}|\!| |\!| |\!|right|\!|]|\!| |\!|{|\!|mathcal|\!|{J|\!|}|\!|}|\!|[|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|]|\!| |\!||\!| |\!|exp|\!|left|\!||\!|{|\!| |\!|int|\!| dt|\!| |\!||\!| |\!|{|\!|boldsymbol|\!|{J|\!|}|\!|}|\!|(t|\!|)|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|(t|\!|)|\!| |\!|right|\!||\!|}|\!| |\!|nonumber|\!| |\!||\!||\!| |\!|&|\!|&|\!| |\!|mbox|\!|{|\!|hspace|\!|{1cm|\!|}|\!|}|\!|=|\!| |\!||\!| |\!|int|\!| |\!|{|\!|mathcal|\!|{D|\!|}|\!|}|\!|{|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|}|\!| |\!||\!| |\!|delta|\!| |\!||\!|!|\!|left|\!|[|\!| |\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!| |\!|-|\!| |\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|^|\!|{|\!|[|\!|bar|\!|{|\!|{|\!|boldsymbol|\!|{J|\!|}|\!|}|\!|}|\!|]|\!|}|\!|right|\!|]|\!| |\!||\!| |\!|exp|\!|left|\!||\!|{|\!| |\!|int|\!| dt|\!| |\!||\!| |\!|{|\!|boldsymbol|\!|{J|\!|}|\!|}|\!|(t|\!|)|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|(t|\!|)|\!| |\!|right|\!||\!|}|\!| |\!||\!|,|\!| |\!|.|\!| |\!|end|\!|{eqnarray|\!|}|\!|
|\!|
The|\!| Jacobian|\!| |\!|$|\!|{|\!|mathcal|\!|{J|\!|}|\!|}|\!|[|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|]|\!|$|\!| results|\!| from|\!| transition|\!| to|\!| the|\!| |\!|`|\!|`unretarded|\!|"|\!| velocities|\!| and|\!| its|\!| explicit|\!| form|\!| reads|\!|~|\!|cite|\!|{Klein1|\!|}|\!|:|\!|
|\!|
|\!|begin|\!|{eqnarray|\!|}|\!|
|\!| |\!|{|\!|mathcal|\!|{J|\!|}|\!|}|\!|[|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|]|\!| |\!||\!| |\!|=|\!| |\!||\!| |\!|det|\!| |\!|left|\!|||\!||\!|!|\!|left|\!|||\!|frac|\!|{|\!|partial|\!|}|\!|{|\!|partial|\!| t|\!|}|\!| |\!||\!|
|\!|delta|\!|_|\!|{ab|\!|}|\!|delta|\!|(t|\!|-t|\!|'|\!|)|\!| |\!|+|\!| |\!|frac|\!|{|\!|delta|\!|^2|\!| |\!|{|\!|mathcal|\!|{A|\!|}|\!|}|\!|}|\!|{|\!|delta|\!| q|\!|_a|\!|(t|\!|)|\!| |\!|delta|\!| q|\!|_b|\!|(t|\!|'|\!|)|\!|}|\!|right|\!|||\!||\!|!|\!|right|\!|||\!||\!|,|\!| |\!|.|\!| |\!|end|\!|{eqnarray|\!|}|\!|
|\!|
Coordinates|\!| |\!|$|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|^|\!|{|\!|[|\!|bar|\!|{|\!|{|\!|boldsymbol|\!|{J|\!|}|\!|}|\!|}|\!|]|\!|}|\!|$|\!| are|\!| solutions|\!| of|\!| the|\!| equation|\!| of|\!| the|\!| motion|\!|:|\!|
|\!|
|\!|begin|\!|{eqnarray|\!|}|\!| |\!|frac|\!|{|\!|delta|\!|{|\!|mathcal|\!|{A|\!|}|\!|}|\!|[|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|]|\!|}|\!|{|\!|delta|\!| |\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|(t|\!|)|\!|}|\!| |\!||\!| |\!|=|\!| |\!||\!| |\!|-|\!| i|\!| |\!|hbar|\!| |\!|bar|\!|{|\!|{|\!|boldsymbol|\!|{J|\!|}|\!|}|\!|}|\!|(t|\!|)|\!||\!|,|\!| |\!|.|\!| |\!|end|\!|{eqnarray|\!|}|\!|
|\!|
In|\!| the|\!| limit|\!| |\!|$|\!| |\!|gamma|\!| |\!|rightarrow|\!| 0|\!|$|\!|,|\!| we|\!| find|\!| again|\!| the|\!| Gozzi|\!| |\!|{|\!|em|\!| et|\!| al|\!|.|\!|}|\!| partition|\!| function|\!|
|\!|
|\!|begin|\!|{eqnarray|\!|}|\!| |\!|lim|\!|_|\!|{|\!|gamma|\!| |\!|rightarrow|\!| 0|\!|}|\!| |\!|{|\!|mathcal|\!|{Z|\!|}|\!|}|\!|_|\!|{|\!|rm|\!| FV|\!|}|\!|[|\!|{|\!|boldsymbol|\!|{J|\!|}|\!|}|\!|,|\!| |\!|{|\!|{|\!|boldsymbol|\!|{0|\!|}|\!|}|\!|}|\!|]|\!| |\!||\!| |\!|=|\!| |\!||\!| |\!|lim|\!|_|\!|{|\!|hbar|\!| |\!|rightarrow|\!| 0|\!|}|\!| |\!|lim|\!|_|\!|{|\!|gamma|\!| |\!|rightarrow|\!| 0|\!|}|\!| |\!|{|\!|mathcal|\!|{Z|\!|}|\!|}|\!|_|\!|{|\!|rm|\!| FV|\!|}|\!|[|\!|{|\!|boldsymbol|\!|{J|\!|}|\!|}|\!|,|\!| |\!|bar|\!|{|\!|{|\!|boldsymbol|\!|{J|\!|}|\!|}|\!|}|\!|]|\!| |\!||\!| |\!|=|\!| |\!||\!| Z|\!|_|\!|{|\!|rm|\!| CM|\!|}|\!|[|\!|{|\!|boldsymbol|\!|{J|\!|}|\!|}|\!|]|\!||\!|,|\!| |\!|.|\!| |\!|end|\!|{eqnarray|\!|}|\!|
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|\!|section|\!|*|\!|{Appendix|\!| C|\!|}|\!|
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In|\!| this|\!| appendix|\!| we|\!| prove|\!| that|\!| |\!|(|\!|ref|\!|{3|\!|.33|\!|}|\!|)|\!| is|\!| a|\!| special|\!| case|\!| of|\!| the|\!| Euler|\!|-like|\!| functionals|\!| |\!|(|\!|ref|\!|{3|\!|.31|\!|}|\!|)|\!|.|\!| Let|\!| us|\!| first|\!| show|\!| that|\!|
|\!| |\!|(|\!|ref|\!|{3|\!|.33|\!|}|\!|)|\!| can|\!| be|\!| replaced|\!| by|\!| an|\!| action|\!| of|\!| the|\!| form|\!| |\!|(|\!|ref|\!|{3|\!|.31|\!|}|\!|)|\!|.|\!| Indeed|\!|,|\!| because|\!| of|\!| the|\!| homogeneity|\!| of|\!| |\!|(|\!|ref|\!|{3|\!|.33|\!|}|\!|)|\!|,|\!| we|\!| can|\!| immediatley|\!| replace|\!| it|\!| by|\!|
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|\!|begin|\!|{eqnarray|\!|}|\!|
|\!| |\!|{|\!|mathcal|\!|{A|\!|}|\!|}|\!|[r|\!|^|\!|{|\!|alpha|\!|_i|\!|}q|\!|_i|\!|]|\!| |\!|=|\!| |\!|sum|\!|_i|\!| |\!|int|\!| dt|\!| |\!||\!| |\!|alpha|\!|_i|\!| r|\!|^|\!|{|\!|alpha|\!|_i|\!|}|\!|(t|\!|)q|\!|_i|\!|(t|\!|)|\!| |\!|frac|\!|{|\!|delta|\!| |\!|{|\!|mathcal|\!|{A|\!|}|\!|}|\!|[r|\!|^|\!|{|\!|alpha|\!|_i|\!|}q|\!|_i|\!|]|\!|}|\!|{|\!|delta|\!| r|\!|^|\!|{|\!|alpha|\!|_i|\!|}|\!|(t|\!|)q|\!|_i|\!|(t|\!|)|\!|}|\!| |\!||\!| |\!|=|\!| |\!||\!| |\!|int|\!| dt|\!| |\!||\!| r|\!|(t|\!|)|\!| |\!|frac|\!|{|\!|delta|\!| |\!|{|\!|mathcal|\!|{A|\!|}|\!|}|\!|[r|\!|^|\!|{|\!|alpha|\!|_i|\!|}q|\!|_i|\!|]|\!|}|\!|{|\!|delta|\!| r|\!|(t|\!|)|\!|}|\!||\!|,|\!| |\!|.|\!| |\!|label|\!|{B11|\!|}|\!| |\!|end|\!|{eqnarray|\!|}|\!|
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Since|\!| this|\!| is|\!| true|\!| for|\!| any|\!| |\!|$r|\!|(t|\!|)|\!|$|\!|,|\!| we|\!| see|\!| that|\!|
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|\!|begin|\!|{eqnarray|\!|}|\!| |\!|int|\!| dt|\!| dt|\!|'|\!| |\!||\!| |\!| r|\!|(t|\!|)|\!| |\!|frac|\!|{|\!|delta|\!|^2|\!| |\!|{|\!|mathcal|\!|{A|\!|}|\!|}|\!|[r|\!|^|\!|{|\!|alpha|\!|_i|\!|}q|\!|_i|\!|]|\!|}|\!|{|\!|delta|\!| r|\!|(t|\!|)|\!| |\!|delta|\!| r|\!|(t|\!|'|\!|)|\!|}|\!| |\!||\!| |\!|=|\!| |\!||\!| 0|\!||\!|,|\!| |\!|.|\!| |\!|label|\!|{B1|\!|}|\!| |\!|end|\!|{eqnarray|\!|}|\!|
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This|\!| simply|\!| expresses|\!| the|\!| fact|\!| that|\!| the|\!| functional|\!| |\!|$|\!| |\!|{|\!|mathcal|\!|{A|\!|}|\!|}|\!|[r|\!|^|\!|{|\!|alpha|\!|_i|\!|}q|\!|_i|\!|]|\!|$|\!| is|\!| linear|\!| in|\!| |\!|$r|\!|(t|\!|)|\!|$|\!|.|\!| The|\!| right|\!|-hand|\!| side|\!| of|\!| |\!|(|\!|ref|\!|{B11|\!|}|\!|)|\!| has|\!| then|\!| precisely|\!| the|\!| Euler|\!| form|\!| |\!|(|\!|ref|\!|{3|\!|.31|\!|}|\!|)|\!|.|\!|
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|\!|vspace|\!|{3mm|\!|}|\!|
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The|\!| reverse|\!| direction|\!| is|\!| proved|\!| in|\!| the|\!| following|\!| way|\!|:|\!| We|\!| first|\!| recast|\!| |\!|(|\!|ref|\!|{3|\!|.31|\!|}|\!|)|\!| in|\!| the|\!| general|\!| form|\!|
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|\!|begin|\!|{eqnarray|\!|}|\!| |\!|int|\!| dt|\!| |\!||\!| r|\!|(t|\!|)|\!| L|\!|(|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|(t|\!|)|\!|,|\!| |\!|dot|\!|{|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|}|\!|(t|\!|)|\!|)|\!| |\!||\!| |\!|=|\!| |\!||\!| |\!|int|\!| dt|\!| |\!||\!| L|\!||\!|!|\!|left|\!|(|\!| r|\!|^|\!|{|\!|alpha|\!|_i|\!|}|\!|(t|\!|)q|\!|_i|\!|(t|\!|)|\!|,|\!| d|\!|(r|\!|^|\!|{|\!|alpha|\!|_i|\!|}|\!|(t|\!|)q|\!|_i|\!|(t|\!|)|\!|)|\!|/dt|\!| |\!|right|\!|)|\!||\!|,|\!| |\!|.|\!| |\!|label|\!|{B2|\!|}|\!| |\!|end|\!|{eqnarray|\!|}|\!|
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Applying|\!| the|\!| variation|\!| |\!| |\!|$|\!|int|\!| dt|\!| |\!||\!| |\!|delta|\!|/|\!|delta|\!| r|\!|(t|\!|)|\!|$|\!| to|\!| |\!|(|\!|ref|\!|{B2|\!|}|\!|)|\!| we|\!| obtain|\!|
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|\!|begin|\!|{eqnarray|\!|}|\!| |\!|{|\!|mathcal|\!|{A|\!|}|\!|}|\!|[|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|]|\!||\!| |\!|=|\!| |\!||\!| |\!|int|\!| dt|\!| |\!||\!| |\!|sum|\!|_i|\!| |\!|alpha|\!|_i|\!| r|\!|^|\!|{|\!|alpha|\!|_i|\!| |\!|-1|\!|}|\!| q|\!|_i|\!|(t|\!|)|\!| |\!|left|\!|(|\!| |\!|frac|\!|{|\!|partial|\!| L|\!|}|\!|{|\!|partial|\!| r|\!|^|\!|{|\!|alpha|\!|_i|\!|}|\!|(t|\!|)|\!| q|\!|_i|\!|(t|\!|)|\!|}|\!| |\!|-|\!| |\!|frac|\!|{d|\!|}|\!|{dt|\!|}|\!|frac|\!|{|\!|partial|\!| L|\!|}|\!|{|\!|partial|\!| |\!|[d|\!|(r|\!|^|\!|{|\!|alpha|\!|_i|\!|}|\!|(t|\!|)|\!| q|\!|_i|\!|(t|\!|)|\!|)|\!|/dt|\!|]|\!|}|\!| |\!|right|\!|)|\!||\!|,|\!| |\!|.|\!| |\!|end|\!|{eqnarray|\!|}|\!|
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This|\!| relation|\!| must|\!| hold|\!| for|\!| all|\!| |\!|$r|\!|(t|\!|)|\!|$|\!|,|\!| and|\!| hence|\!| by|\!| choosing|\!| |\!|$r|\!|(t|\!|)|\!| |\!|=|\!| 1|\!|$|\!| we|\!| arrive|\!| at|\!| the|\!| required|\!| result|\!|
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|\!|begin|\!|{eqnarray|\!|}|\!| |\!|{|\!|mathcal|\!|{A|\!|}|\!|}|\!|[|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|]|\!||\!| |\!|=|\!| |\!||\!| |\!|int|\!| dt|\!| |\!||\!| |\!|sum|\!|_i|\!| |\!|alpha|\!|_i|\!| q|\!|_i|\!|(t|\!|)|\!| |\!|frac|\!|{|\!|delta|\!| |\!|{|\!|mathcal|\!|{A|\!|}|\!|}|\!|[|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|]|\!|}|\!|{|\!|delta|\!| q|\!|_i|\!|(t|\!|)|\!|}|\!||\!|,|\!| |\!|.|\!| |\!|end|\!|{eqnarray|\!|}|\!|
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|\!|section|\!|*|\!|{Appendix|\!| D|\!|}|\!|
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Here|\!| we|\!| prove|\!| the|\!| fact|\!| that|\!| inclusion|\!| of|\!| the|\!| subsidiary|\!| constraint|\!| |\!|(|\!|ref|\!|{4|\!|.1|\!|}|\!|)|\!| in|\!| the|\!| primary|\!| constraints|\!| |\!|(|\!|ref|\!|{2|\!|.5|\!|}|\!|)|\!| does|\!| not|\!| produce|\!| any|\!| secondary|\!| constraints|\!|.|\!| The|\!| secondary|\!| constraints|\!| result|\!| from|\!| the|\!| consistency|\!| conditions|\!| |\!|(|\!|ref|\!|{A1|\!|}|\!|)|\!| or|\!|,|\!| in|\!| other|\!| words|\!|,|\!| when|\!| existent|\!| constraints|\!| are|\!| incompatible|\!| with|\!| the|\!| equation|\!| of|\!| motion|\!|.|\!|
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|\!|vspace|\!|{3mm|\!|}|\!|
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We|\!| first|\!| observe|\!| that|\!| the|\!| condition|\!| |\!|$H|\!|_|\!|-|\!| |\!|approx|\!| 0|\!|$|\!| can|\!| be|\!| equivalently|\!| represented|\!| by|\!| the|\!| condition|\!| |\!|$|\!|(|\!|bar|\!|{H|\!|}|\!| |\!|-|\!| |\!|sum|\!|_i|\!| a|\!|_i|\!| C|\!|_i|\!|)|\!| |\!|equiv|\!| |\!|phi|\!|_0|\!| |\!|approx|\!| 0|\!|$|\!|.|\!| If|\!| we|\!| now|\!| add|\!| the|\!| subsidiary|\!| constraint|\!| |\!|$|\!|phi|\!|_0|\!|$|\!| to|\!| the|\!| remaining|\!| |\!|$2N|\!|$|\!| constraints|\!| |\!|$|\!|phi|\!|_i|\!|$|\!| and|\!| again|\!| require|\!| that|\!| the|\!| constraints|\!| |\!|$|\!|phi|\!|_i|\!|$|\!| remain|\!| |\!|(weakly|\!|)|\!| zero|\!| at|\!| all|\!| times|\!| we|\!| have|\!|
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|\!|begin|\!|{eqnarray|\!|}|\!| 0|\!| |\!||\!| |\!|approx|\!| |\!||\!| |\!|dot|\!|{|\!|phi|\!|_i|\!|}|\!| |\!||\!| |\!|approx|\!| |\!||\!| |\!||\!|{|\!| |\!|phi|\!|_i|\!|,|\!| |\!|{|\!|overlinen|\!| H|\!|}|\!||\!|}|\!| |\!||\!| |\!|+|\!| |\!||\!| u|\!|^j|\!| |\!||\!|{|\!|phi|\!|_i|\!|,|\!| |\!|phi|\!|_j|\!| |\!||\!|}|\!||\!|,|\!| |\!|,|\!| |\!||\!|;|\!||\!|;|\!||\!|;|\!||\!|;|\!| |\!||\!|;|\!||\!|;|\!||\!|;|\!| i|\!|,j|\!| |\!|=|\!| 0|\!|,|\!| 1|\!| |\!|ldots|\!|,|\!| 2N|\!||\!|,|\!| |\!|.|\!| |\!|label|\!|{D1|\!|}|\!| |\!|end|\!|{eqnarray|\!|}|\!|
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Since|\!| there|\!| is|\!| an|\!| odd|\!| number|\!| of|\!| constraints|\!| and|\!| because|\!| |\!|$|\!||\!|{|\!|phi|\!|_i|\!|,|\!|
|\!|phi|\!|_j|\!| |\!||\!|}|\!|$|\!| is|\!| an|\!| antisymmetric|\!| matrix|\!| we|\!| have|\!| that|\!| |\!|$|\!|det|\!||\!|||\!|left|\!||\!|{|\!|
|\!|phi|\!|_i|\!|,|\!| |\!|phi|\!|_j|\!| |\!|right|\!||\!|}|\!||\!|||\!| |\!|=|\!| 0|\!|$|\!|.|\!| From|\!| the|\!| analysis|\!| in|\!| Appendix|\!| A|\!| it|\!| is|\!| clear|\!| that|\!| the|\!| rank|\!| of|\!| the|\!| matrix|\!| |\!|$|\!||\!|{|\!|phi|\!|_i|\!|,|\!| |\!|phi|\!|_j|\!| |\!||\!|}|\!|$|\!| is|\!| |\!|$2N|\!|$|\!| and|\!| hence|\!| it|\!| has|\!| one|\!| null|\!|-eigenvector|\!|,|\!| say|\!| |\!|$|\!|{|\!|boldsymbol|\!|{e|\!|}|\!|}|\!|$|\!|.|\!| Inasmuch|\!|,|\!| Eq|\!|.|\!|(|\!|ref|\!|{D1|\!|}|\!|)|\!| implies|\!| the|\!| constraint|\!|
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|\!|begin|\!|{eqnarray|\!|}|\!| |\!|sum|\!|_|\!|{i|\!|=0|\!|}|\!|^|\!|{2N|\!|}|\!| e|\!|_i|\!| |\!||\!|{|\!|phi|\!|_i|\!|,|\!| |\!|bar|\!|{H|\!|}|\!| |\!||\!|}|\!| |\!||\!| |\!|approx|\!| |\!||\!| 0|\!||\!|,|\!| |\!|.|\!| |\!|label|\!|{D2|\!|}|\!| |\!|end|\!|{eqnarray|\!|}|\!|
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If|\!| the|\!| latter|\!| would|\!| represent|\!| a|\!| new|\!| non|\!|-trivial|\!| constraint|\!| |\!|(i|\!|.e|\!|.|\!|,|\!| constraint|\!| that|\!| cannot|\!| be|\!| written|\!| as|\!| a|\!| linear|\!| combination|\!| of|\!| constraints|\!| |\!|$|\!|phi|\!|_i|\!|$|\!|)|\!| we|\!| would|\!| need|\!| to|\!| include|\!| such|\!| a|\!| new|\!| constraint|\!| |\!|(the|\!| so|\!| called|\!| secondary|\!| constraint|\!|)|\!| into|\!| the|\!| list|\!| of|\!| existent|\!| constraints|\!| and|\!| go|\!| again|\!| through|\!| the|\!| consistency|\!| condition|\!| |\!|(|\!|ref|\!|{D1|\!|}|\!|)|\!|.|\!| Fortunately|\!|,|\!| the|\!| condition|\!| |\!|(|\!|ref|\!|{D2|\!|}|\!|)|\!| is|\!| automatically|\!| fulfilled|\!| and|\!| hence|\!| it|\!| does|\!| not|\!| constitute|\!| any|\!| new|\!| constraint|\!|.|\!| Indeed|\!|,|\!| be|\!| choosing|\!|
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|\!|begin|\!|{eqnarray|\!|}|\!| |\!|{|\!|boldsymbol|\!|{e|\!|}|\!|}|\!| |\!||\!| |\!|=|\!| |\!||\!| |\!|left|\!|(|\!|begin|\!|{array|\!|}|\!|{c|\!|}|\!| 1|\!| |\!||\!||\!| |\!||\!|{|\!|phi|\!|_0|\!|,|\!| |\!|phi|\!|_2|\!|^a|\!| |\!||\!|}|\!||\!||\!| |\!||\!|{|\!|phi|\!|_1|\!|^a|\!|,|\!| |\!|phi|\!|_0|\!||\!|}|\!||\!||\!| |\!||\!|{|\!|phi|\!|_0|\!|,|\!| |\!|phi|\!|_2|\!|^b|\!||\!|}|\!||\!||\!| |\!||\!|{|\!|phi|\!|_1|\!|^b|\!|,|\!| |\!|phi|\!|_0|\!||\!|}|\!||\!||\!| |\!|vdots|\!||\!||\!| |\!||\!|{|\!| |\!|phi|\!|_0|\!|,|\!| |\!|phi|\!|_2|\!|^N|\!||\!|}|\!||\!||\!| |\!||\!|{|\!| |\!|phi|\!|_1|\!|^N|\!|,|\!| |\!|phi|\!|_0|\!||\!|}|\!| |\!|end|\!|{array|\!|}|\!|right|\!|)|\!| |\!||\!| |\!|=|\!| |\!||\!| |\!|left|\!|(|\!| |\!|begin|\!|{array|\!|}|\!|{c|\!|}|\!| 1|\!| |\!||\!||\!| |\!|{f|\!|}|\!|_a|\!|(|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|)|\!||\!||\!| |\!|-|\!| |\!|frac|\!|{|\!|partial|\!| |\!|phi|\!|_0|\!|}|\!|{|\!|partial|\!| q|\!|_a|\!|}|\!| |\!||\!||\!| |\!|{f|\!|}|\!|_b|\!|(|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|)|\!||\!||\!| |\!|-|\!|frac|\!|{|\!|partial|\!| |\!|phi|\!|_0|\!|}|\!|{|\!|partial|\!| q|\!|_b|\!|}|\!||\!||\!| |\!|vdots|\!||\!||\!| |\!|{f|\!|}|\!|_N|\!|(|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|)|\!||\!||\!| |\!|-|\!|frac|\!|{|\!|partial|\!| |\!|phi|\!|_0|\!|}|\!|{|\!|partial|\!|{q|\!|}|\!|_N|\!|}|\!| |\!|end|\!|{array|\!|}|\!|right|\!|)|\!||\!|,|\!| |\!|,|\!| |\!|end|\!|{eqnarray|\!|}|\!|
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and|\!| using|\!| |\!|$|\!||\!|{|\!| |\!|phi|\!|_0|\!|,|\!| |\!|bar|\!|{H|\!|}|\!| |\!||\!|}|\!| |\!|=|\!| 0|\!|$|\!| together|\!| with|\!| |\!|(|\!|ref|\!|{A3|\!|}|\!|)|\!| we|\!| obtain|\!|
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|\!|begin|\!|{eqnarray|\!|}|\!| |\!|sum|\!|_|\!|{i|\!|=0|\!|}|\!|^|\!|{2N|\!|}|\!| e|\!|_i|\!| |\!||\!|{|\!|phi|\!|_i|\!|,|\!| |\!|bar|\!|{H|\!|}|\!| |\!||\!|}|\!||\!| |\!|=|\!| |\!||\!| |\!|-|\!| |\!|sum|\!|_|\!|{i|\!|,a|\!|}|\!| a|\!|_i|\!|(t|\!|)|\!| |\!|{f|\!|}|\!|_a|\!|(|\!|{|\!|boldsymbol|\!|{q|\!|}|\!|}|\!|)|\!| |\!||\!| |\!|frac|\!|{|\!|partial|\!| C|\!|_i|\!|}|\!|{|\!|partial|\!| q|\!|_a|\!|}|\!| |\!||\!| |\!|=|\!| |\!||\!| |\!|sum|\!|_|\!|{i|\!| |\!|=1|\!|}|\!|^n|\!| a|\!|_i|\!|(t|\!|)|\!| |\!||\!|{H|\!|,|\!| C|\!|_i|\!||\!|}|\!||\!| |\!|=|\!| |\!||\!| 0|\!||\!|,|\!| |\!|.|\!| |\!|label|\!|{D4|\!|}|\!| |\!|end|\!|{eqnarray|\!|}|\!|
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As|\!| the|\!| latter|\!| is|\!| zero|\!| |\!|(even|\!| strongly|\!|)|\!| there|\!| is|\!| no|\!| new|\!| constraint|\!| condition|\!| generated|\!| by|\!| an|\!| inclusion|\!| of|\!| |\!|$|\!|phi|\!|_0|\!|$|\!| in|\!| the|\!| original|\!| set|\!| of|\!| |\!|(primary|\!|)|\!| constraints|\!|.|\!| Note|\!|,|\!| that|\!| the|\!| key|\!| in|\!| obtaining|\!| |\!|(|\!|ref|\!|{D4|\!|}|\!|)|\!| was|\!| the|\!| fact|\!| that|\!| |\!|$C|\!|_i|\!|$|\!|'s|\!| are|\!| |\!|$|\!|{|\!|boldsymbol|\!|{p|\!|}|\!|}|\!|$|\!|-independent|\!| constants|\!| of|\!| motion|\!|.|\!|
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|\!|vspace|\!|{3mm|\!|}|\!|
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The|\!| rank|\!| of|\!| |\!|$|\!||\!|{|\!| |\!|phi|\!|_i|\!|,|\!| |\!|phi|\!|_j|\!| |\!||\!|}|\!|$|\!| being|\!| |\!|$2N|\!|$|\!| means|\!| that|\!| there|\!| is|\!| one|\!| relation|\!|
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|\!|begin|\!|{eqnarray|\!|}|\!| |\!|sum|\!|_|\!|{i|\!| |\!|=|\!| 0|\!|}|\!|^|\!|{2N|\!|}|\!| e|\!|_i|\!| |\!||\!|{|\!| |\!|phi|\!|_i|\!|,|\!| |\!|phi|\!|_j|\!| |\!||\!|}|\!| |\!||\!| |\!|approx|\!| |\!||\!| 0|\!||\!|,|\!| |\!|.|\!| |\!|end|\!|{eqnarray|\!|}|\!|
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Any|\!| linear|\!| combination|\!| of|\!| the|\!| constraints|\!| |\!|$|\!|phi|\!|_i|\!|$|\!| is|\!| again|\!| a|\!| constraint|\!|.|\!| So|\!|,|\!| particularly|\!| if|\!| we|\!| define|\!| |\!|$|\!|varphi|\!| |\!|=|\!| |\!|sum|\!|_i|\!| e|\!|_i|\!| |\!|phi|\!|_i|\!|$|\!| we|\!| obtain|\!| that|\!| |\!|$|\!|varphi|\!|$|\!| has|\!| weakly|\!| vanishing|\!| Poisson|\!| brackets|\!| with|\!| all|\!| constraints|\!|,|\!| i|\!|.e|\!|.|\!|,|\!|
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|\!|begin|\!|{eqnarray|\!|}|\!| |\!||\!|{|\!| |\!|varphi|\!|,|\!| |\!|phi|\!|_i|\!| |\!||\!|}|\!| |\!||\!| |\!|approx|\!| |\!||\!| 0|\!||\!|,|\!| |\!|,|\!| |\!||\!|;|\!||\!|;|\!||\!|;|\!||\!|;|\!| i|\!| |\!|=|\!| 1|\!|,|\!| |\!|ldots|\!|,|\!| 2N|\!||\!|,|\!| |\!|.|\!| |\!|end|\!|{eqnarray|\!|}|\!|
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Thus|\!|,|\!| according|\!| to|\!| Dirac|\!|'s|\!| classification|\!| |\!|(see|\!| e|\!|.g|\!|.|\!|,|\!| Ref|\!|.|\!|~|\!|cite|\!|{Dir|\!|}|\!|)|\!| |\!|$|\!|varphi|\!|$|\!| is|\!| a|\!| first|\!| class|\!| constraint|\!|.|\!| The|\!| remaining|\!| |\!|$2N|\!|$|\!| constraints|\!| |\!|(which|\!| do|\!| not|\!| have|\!| vanishing|\!| Poisson|\!| brackets|\!| with|\!| all|\!| other|\!| constraints|\!|)|\!| are|\!| of|\!| the|\!| second|\!| class|\!|.|\!| Note|\!| particularly|\!| that|\!| the|\!| explicit|\!| form|\!| for|\!| |\!|$|\!|varphi|\!|$|\!| reads|\!|
|\!|
|\!|begin|\!|{eqnarray|\!|}|\!| |\!|varphi|\!| |\!||\!| |\!|=|\!| |\!||\!| |\!|sum|\!|_|\!|{i|\!| |\!|=|\!| 0|\!|}|\!|^|\!|{2N|\!|}|\!| e|\!|_i|\!| |\!|phi|\!|_i|\!| |\!||\!| |\!|=|\!| |\!||\!| |\!|(H|\!| |\!|-|\!| |\!|sum|\!|_|\!|{i|\!| |\!|=|\!| 1|\!|}|\!|^n|\!| a|\!|_i|\!| C|\!|_i|\!|)|\!| |\!|-|\!| |\!|sum|\!|_|\!|{a|\!| |\!|=|\!| 1|\!|}|\!|^N|\!| |\!|bar|\!|{p|\!|}|\!|_a|\!| |\!||\!| |\!|frac|\!|{|\!|partial|\!| |\!|phi|\!|_0|\!|}|\!|{|\!|partial|\!| q|\!|_a|\!|}|\!||\!|,|\!| |\!|,|\!| |\!|label|\!|{D3|\!|}|\!| |\!|end|\!|{eqnarray|\!|}|\!|
|\!|
which|\!| is|\!| clearly|\!| weakly|\!| identical|\!| to|\!| |\!|$H|\!|-|\!| |\!|sum|\!|_i|\!| a|\!|_i|\!| C|\!|_i|\!|$|\!|.|\!| Observe|\!| that|\!| it|\!| is|\!| |\!|$H|\!|$|\!| and|\!| not|\!| |\!|$|\!|bar|\!|{H|\!|}|\!|$|\!| that|\!| is|\!| present|\!| in|\!| |\!|(|\!|ref|\!|{D3|\!|}|\!|)|\!|.|\!|
|\!|
|\!|
|\!|section|\!|*|\!|{References|\!|}|\!|
|\!|
|\!|
|\!|begin|\!|{thebibliography|\!|}|\!|{99|\!|}|\!|
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|\!|end|\!|{document|\!|}|\!| | arXiv |
Arthritis Research & Therapy
DNA methylation mediates genotype and smoking interaction in the development of anti-citrullinated peptide antibody-positive rheumatoid arthritis
Weida Meng1,2,
Zaihua Zhu3,
Xia Jiang4,
Chun Lai Too5,6,
Steffen Uebe7,
Maja Jagodic8,
Ingrid Kockum8,
Shahnaz Murad5,
Luigi Ferrucci9,
Lars Alfredsson4,10,
Hejian Zou3,
Lars Klareskog6,
Andrew P. Feinberg11,
Tomas J. Ekström8,
Leonid Padyukov6 &
Yun Liu1,2
Arthritis Research & Therapy volume 19, Article number: 71 (2017) Cite this article
Multiple factors, including interactions between genetic and environmental risks, are important in susceptibility to rheumatoid arthritis (RA). However, the underlying mechanism is not fully understood. This study was undertaken to evaluate whether DNA methylation can mediate the interaction between genotype and smoking in the development of anti-citrullinated peptide antibody (ACPA)-positive RA.
We investigated the gene-smoking interactions in DNA methylation using 393 individuals from the Epidemiological Investigation of Rheumatoid Arthritis (EIRA). The interaction between rs6933349 and smoking in the risk of developing ACPA-positive RA was further evaluated in a larger portion of the EIRA (1119 controls and 944 ACPA-positive patients with RA), and in the Malaysian Epidemiological Investigation of Rheumatoid Arthritis (MyEIRA) (1556 controls and 792 ACPA-positive patients with RA). Finally, mediation analysis was performed to investigate whether DNA methylation of cg21325723 mediates this gene-environment interaction on the risk of developing of ACPA-positive RA.
We identified and replicated one significant gene-environment interaction between rs6933349 and smoking in DNA methylation of cg21325723. This gene-smoking interaction is a novel interaction in the risk of developing ACPA-positive in both Caucasian (multiplicative P value = 0.056; additive P value = 0.016) and Asian populations (multiplicative P value = 0.035; additive P value = 0.00027), and it is mediated through DNA methylation of cg21325723.
We showed that DNA methylation of cg21325723 can mediate the gene-environment interaction between rs6933349 and smoking, impacting the risk of developing ACPA-positive RA, thus being a potential regulator that integrates both internal genetic and external environmental risk factors.
Rheumatoid arthritis (RA) is a chronic autoimmune disease that leads to inflammation of the joints and surrounding tissues. It can cause severe functional disabilities, pain, and other disorders, such as cardiovascular disease. RA is a complex inflammatory disease affecting up to 1% of the population. The fact that the concordance rate for RA in monozygotic twins is less than 20% suggests that environmental factors may be highly involved in the etiology of the disease. In recent decades, it has been shown that the two major subgroups of RA, anti-citrullinated peptide antibody (ACPA)-positive and ACPA-negative RA, have in part different etiology. One example of this is the shared epitope (SE) alleles of the human leukocyte antigen DR beta chain 1 (HLA-DRB1), which is a major risk factor for ACPA-positive RA, but not to the same extent for ACPA-negative RA. Environmental/lifestyle factors, such as smoking [1–3] and other noxious airway exposures [4, 5], have been shown to be risk factors for RA, mainly for the ACPA-positive subset of RA.
Smoking is a well-studied risk factor for RA and for the severity of RA [6–8]. There is a dose-dependent interaction between smoking and variations in the HLA-DRB1 gene in the risk of developing ACPA-positive RA [9, 10]. One hypothesis proposed for the etiology of ACPA-positive RA is that the autoantibodies (ACPA) that are directed against citrullinated proteins in the joints originate from the mucosal tissues, e.g. the lungs, exposed to harmful inhaled toxicants such as smoking or silica dust. However, there remains a challenge to fully understand the molecular mechanism of the gene-environment interaction in the pathogenesis RA.
Epigenetic modifications, such as DNA methylation, have an important role in controlling when and where genes are expressed, and can be influenced by environmental factors. Such epigenetic modifications may thus provide a possible biological link between environmental exposures, genetic variations, and the disease. In fact, smoking has also been demonstrated to perturb DNA methylation signatures in lymphocytes [11]. Moreover, there is also growing evidence that epigenetic modifications can be controlled by the DNA sequence, and can be a mediator of genetic risk in common diseases, such as RA [12] and allergy [13]. Thus, it is relevant to investigate whether DNA methylation can mediate the interactions between genotype and smoking in the development of ACPA-positive RA (Fig. 1a) and whether it is a regulator that can integrate both internal genetic and external environmental risk factors.
Study model (a) and work flow diagram (b). ACPA anti-citrullinated peptide antibodies, RA rheumatoid arthritis, EIRA Epidemiological Investigation of Rheumatoid Arthritis, EIMS Epidemiological Investigation of Multiple Sclerosis, MyEIRA Malaysian Epidemiological Investigation of Rheumatoid Arthritis
In this report, by using data from multiple cohorts (Fig. 1b) we evaluated whether DNA methylation can mediate the interaction between genotype and smoking in the development of ACPA-positive RA.
The EIRA (Epidemiological Investigation of Rheumatoid Arthritis) is a Swedish population-based case-control study. Recruitment of patients with RA in the EIRA study was described previously [14], and the healthy controls were selected from the same population to match the RA cases by age, sex and residential area at the time of diagnosis. Self-reported smoking habits were registered from the EIRA questionnaire. The genotyping and its quality control (QC) procedures have been described previously [14], and imputation was done using the IMPUTE2 algorithm [15] based on the phased 1000 genome reference set (March 2012 haplotypes). This group of samples with information on genotype, methylation, and smoking status was used for the investigation of genotype and smoking interaction in DNA methylation.
The EIMS (Epidemiological Investigation of risk factors for Multiple Sclerosis) is a population-based case-control study comprising Swedish-speaking subjects in Sweden and details of the recruitment procedure were described previously [16]. Briefly, newly diagnosed patients with multiple sclerosis (MS) were recruited via 40 study centers in Sweden and healthy controls were randomly selected from the national population register, matched by age, sex, and residential area. Self-reported smoking information was registered from the EIMS questionnaire.
The MyEIRA (Malaysian Epidemiological Investigation of Rheumatoid Arthritis) is another independent population-based case-control study, in which the subjects were recruited in Peninsular Malaysia with three major ethnic groups (i.e. Malays, Chinese, and Indians). The details of the MyEIRA study have been described elsewhere [3, 17]. In brief, patients with early RA were identified from nine rheumatology centers throughout Peninsular Malaysia, and for each case, a population control was randomly selected matched by age, sex, and residential area. All participants answered a questionnaire on a broad range of issues, including smoking habits.
The InCHIANTI study is a population-based prospective cohort study of residents from two areas in the Chianti region (Tuscany, Italy). The data collection started in September 1998 and was completed in March 2000 (baseline). A nine-year follow-up assessment of the InCHIANTI study population was performed in the year 2007–2008. Selection of participants and collection of DNA methylation data have been described previously [18, 19].
DNA methylation measurement
Genome-wide methylation in peripheral blood cells from a subset of the EIRA, EIMS and InCHIANTI cohorts were evaluated by Illumina Infinium Human Methylation 450 BeadChip according to the manufacturer's recommendations. Illumina Infinium Human Methylation 450 BeadChip array quantifies methylation levels at specific loci within the genome. The percentage methylation value for a particular CpG site, which represents the fraction of DNA with this CpG site methylated, was calculated on a scale of 0–1, per Illumina's recommendations, using the formula:
$$ \mathrm{M}/\left(\mathrm{M} + \mathrm{U} + 100\right), $$
where M and U represent the methylated and unmethylated signal intensities, respectively. Methylation data from EIRA was published previously [12] and can be downloaded from Gene Expression Omnibus [GEO:GSE42861].
Genotype and smoking interaction in DNA methylation analyzed using a linear regression model
We decided to focus on the single nucleotide polymorphisms (SNPs) within the major histocompatibility complex (MHC region) (chr6: 29,500,000–33,500,000 (hg19)) and the 10 differentially methylated positions (DMPs), which we identified previously to be associated with the development of ACPA-positive RA [12]. Details of methylation measurements have been provided previously [12]. Due to strong linkage disequilibrium of SNPs within the MHC region, we calculated the effective number of independent tests (M eff ) by simpleM [20]. The total 1417 SNPs investigated in the study, which have minor allele frequency ≥0.05 and contain at least 10 individuals in each genotype group, represent 388 independent tests (M eff ).
To identify genotype and smoking interaction in DNA methylation, we fit a linear regression model predicting methylation at each DMP as a function of genotype, smoking status (categorized as current smokers and never smokers) and their interaction term, and significant interaction was evaluated by calculating the interaction term in the model with a stringent Bonferroni-adjusted threshold of 0.05/(10 DMPs × 388 M eff ) = 1.29 × 10-5. SNPs were treated with an additive minor-allele dosage model and potential confounding factors (that is, age, sex, and hybridization batch, and the first two principle components of cell-type proportions estimated by cell-specific methylation signatures as described before [12, 21]) were adjusted for in all analyses in both the EIRA and the EIMS cohorts.
Genotype and smoking interaction in ACPA-positive RA
Two statistical models were used to evaluate the interaction between rs6933349 and smoking status (categorized as never smokers and current smokers, or as never smokers and ever smokers as will be specified later) in the development of ACPA-positive RA: (1) we performed interaction analysis by means of logistic regression with adjustment for age and sex, and interaction was evaluated on the multiplicative scale by calculating the interaction term in the logistic regression model, and (2) we also evaluated the interaction between genotype and smoking in ACPA-positive RA by departure from the additivity of effects, and biological interaction was estimated by calculating the attributable proportion due to interaction (AP). AP is the proportion of the incidence among individuals exposed to two interacting risk factors that is attributable to the interaction per se. AP >0 indicates that there is evidence for interaction on the additive scale. Confidence intervals for AP were estimated as described previously [22, 23]. Additionally, relative excess risk due to interaction (RERI) and the synergy index (SI) were also performed to evaluate interactions as described previously [24]. Both statistical models were also adjusted for ethnicity in analysis of the MyEIRA cohort.
Linkage disequilibrium analysis
The measurement of linkage disequilibrium (r 2) between rs6933349 and the major RA HLA-DRB1 SE alleles were calculated using genotype data from individuals in the EIRA cohort. The calculation was performed by using the "LD. Measures" function in the LDcorSV package.
Mediation analysis
To investigate whether methylation of cg21325723 mediated the rs6933349 and smoking interaction in the risk of developing ACPA-positive RA, we assessed the effect of including DNA methylation as a covariate in the logistical regression models relating to rs6933349, smoking, and their interaction term to the ACPA-positive RA as the outcome. The 393 individuals from the discovery dataset analyzed, for whom we have complete information on genotype, smoking, methylation, and RA disease. Potential confounders (age and sex) were used for adjustment in all analyses.
Stage 1: identification of an interaction between genotype and smoking in DNA methylation within the MHC region using a linear regression model
As there are extremely large numbers of different combinations between SNPs and CpG sites, analyses of interactions, even in a relatively large cohort, are at risk of being underpowered. Thus, we decided to focus on the SNPs within the MHC region and the 10 DMPs, which we identified previously to be associated with the development of ACPA-positive RA [12]. By analyzing 393 samples from the EIRA cohort, for whom we have detailed information on genotype, DNA methylation in blood cells, and smoking status in each individual, we observed one significant interaction (Bonferroni-adjusted P value < 0.05) between a SNP, rs6933349 (chr6: 31002013), and smoking status in the DNA methylation level of the CpG site, cg21325723 (chr6: 32402555) (Additional file 1: Figure S1). We observed a significant association between DNA methylation and rs6933349 (P value = 0.0048) (Fig. 2a).
The associations between the genetic variant of rs6933349 and DNA methylation (cg21325723) in all individuals (a), and by smoking status (b), in the Epidemiological Investigation of Rheumatoid Arthritis (EIRA) study. Each dot represents an individual and red horizontal bars mark average DNA methylation levels. The statistical significance (P value) of association between genotype and DNA methylation, measured by linear regression model, is indicated at the bottom of the plot
However, if we stratified the samples by smoking status, we observed significant association between the genotype and DNA methylation among current smokers (P value = 4.31 × 10-6), but not among never smokers (P value = 0.52) (Fig. 2b). Among current smokers, minor allele (rs6933349_A) carriers had a lower level of DNA methylation at cg21325723 (Fig. 2b), which was previously reported to be associated with increased risk of developing ACPA-positive RA [12] (P value = 1.49 × 10-9) (Additional file 1: Figure S2). Furthermore, among groups of individuals with a different genotype, we observed a different relationship between methylation level and smoking status (Fig. 3). For example, the DNA methylation level in current smokers who were carriers of rs6933349_GA or rs6933349_AA genotypes was significantly lower as compared to never smokers (P value = 0.0075 and 0.0069, respectively). No significant difference was seen in the DNA methylation level in current smokers and never smokers who were carriers of rs6933349_GG genotype (Fig. 3).
The associations between smoking and DNA methylation (cg21325723) in all individuals (a), and in individuals with rs6933349_GG, rs6933349_AG, or rs6933349_AA genotypes, respectively (b), in the Epidemiological Investigation of Rheumatoid Arthritis (EIRA) study. Each dot represents an individual and red horizontal bars mark average DNA methylation levels. The statistical significance (P value) of association between DNA methylation and smoking status, measured by Student's t test, is indicated at the bottom of the plot
Next, we sought to confirm that the interaction between genetic variation and smoking in DNA methylation that was reported previously is not confounded by the RA status. We examined the interaction between rs6933349 and smoking in methylation of cg21325723 separately in healthy controls and ACPA-positive patients with RA, and observed consistent effects in both the control-only analysis (P value = 0.0037) and the patient-only analysis (P value = 0.019). In both analyses, we observed a significantly lower methylation level in rs6933349_AA genotype carriers among current smokers, while there were no associations between genotype and methylation among never smokers (Additional file 1: Figure S3). This showed that the combination of current smoking and minor allele (allele A) for rs6933349 is associated with hypo-methylation of cg21325723 and suggested the possibility of interaction between rs6933349 and smoking in the DNA methylation level of cg21325723, which was tested statistically and is described later within this paper.
Considering that DNA methylation is potentially dynamic, we further evaluated the stability of DNA methylation of cg21325723 in an independent dataset (the InCHIANTI cohort), in which methylation was measured in 460 individuals at two time points separated by 9 years. The Pearson's correlation coefficient for cg21325723 was 0.722 (Additional file 1: Figure S4). This suggested that methylation in cg21325723 is moderately stable over 9 years. However, the level can still be altered, potentially by environmental risk factors for RA, such as smoking. This is consistent with a previous observation of familial clustering of changes in global DNA methylation over time [25].
Stage 2: replication of an interaction between genotype and smoking in an independent data set
We continued to replicate the finding of interaction between rs6933349 and smoking in cg21325723 DNA methylation in an independent dataset, the EIMS, which is a population-based case-control study of MS. As the methylation level of cg21325723 was not associated with the MS status (P value = 0.71), we included a total of 139 healthy controls and 140 patients with MS, for whom information was available on genotype, methylation in blood, and smoking, for the replication study. In the combined meta-analysis using data from EIRA and EIMS, we observed significant interaction (P value = 2.18 × 10-6) between rs6933349 and smoking status (current vs. never smokers) in the methylation level of cg21325723. In the relatively small EIMS cohort alone, the interaction was marginally significant (P value = 0.08). Consistent with the finding from the EIRA, among the rs6933349_AA carriers from the EIMS cohort there was significantly lower methylation in current smokers (P value = 0.024), but not in never smokers (P value = 0.14) (Additional file 1: Figure S5).
Stage 3: novel interaction between genotype and smoking impacting the risk of developing ACPA-positive RA
As hypo-methylation on cg21325723 has been associated with increased risk of developing ACPA-positive RA [12] (Additional file 1: Figure S2), we next investigated the interaction between rs6933349 and smoking in the risk of developing ACPA-positive RA. We addressed this question in a larger portion of the EIRA cohort (1119 healthy controls and 944 ACPA-positive patients with RA), which includes the subset of 393 individuals used in the discovery dataset. These analyses were performed separately for the models with multiplicative and additive effects.
Multiplicative model
This analysis of genotype and smoking interaction in ACPA-positive RA was evaluated on the multiplicative scale, which accesses the interaction term in the logistic regression model. We observed a marginally significant interaction (multiplicative P value = 0.022 in current smokers and 0.056 in ever smokers) between rs6933349 and smoking status in ACPA-positive RA (Table 1).
Table 1 The rs6933349 genotype and smoking interaction in the risk of developing ACPA-positive RA in the EIRA and MyEIRA studies
Additive model
This method is particularly relevant as it can further determine the degree of biological interaction between the two risk factors [26, 27]. In our analysis there was a statistically significant gene-environment interaction (additive P value = 0.0034 in current smokers and P value = 0.016 in ever smokers) between rs6933349 and smoking in the risk of developing ACPA-positive RA, with an attributable proportion due to the interaction (AP) value of 0.315 (95% CI 0.104 to 0.526) in current smokers and 0.216 (95% CI 0.040 to 0.392) in ever smokers (Table 1). We observed an increased risk of ACPA-positive RA in individuals who were ever smokers and carriers of rs6933349_AG/rs6933349_AA genotypes (OR = 2.03; 95% CI 1.66 to 2.49) (Table 1, Fig. 4a). In contrast, there was no interaction between the studied risk factors in relation to the risk of ACPA-negative RA (P value for AP = 0.74). Consistently, all other measures of interaction between rs6933349 and smoking in the risk of developing ACPA-positive RA were significant, with the RERI value of 0.978 (95% CI 0.178 to 1.778) in current smokers and 0.434 (95% CI 0.064 to 0.804) in ever smokers, and the SI value of 1.869 (95% CI 1.325 to 2.413) in current smokers and 1.729 (95% CI 1.114 to 2.344) in ever smokers.
Odds ratios (OR) of developing anti-citrullinated peptide antibody-positive rheumatoid arthritis for different combinations of smoking and rs6933349 alleles in the Epidemiological Investigation of Rheumatoid Arthritis (EIRA) cohort (a) or the Malaysian Epidemiological Investigation of Rheumatoid Arthritis (MyEIRA) cohort (b)
This result was further replicated in the independent RA cohort, MyEIRA, a population-based case-control study performed in Malaysia [17]. We observed significant interaction (multiplicative P value = 0.035; additive P value = 0.00027) between rs6933349 and smoking (ever smokers) in the development of ACPA-positive RA (Table 1) in the MyEIRA cohort. Consistent with the finding from the EIRA, the combination of ever smoking and the genetic variant of rs6933349_A was associated with an increased risk of developing ACPA-positive RA (Fig. 4b), whereas no significant interaction was observed in the risk of ACPA-negative RA (additive P value = 0.45). The fact that the EIRA is mainly composed of Caucasians while the MyEIRA is a multiethnic population of Asian descent suggests that the interaction between rs6933349 and smoking in the risk of developing ACPA-positive RA is unlikely to be specific to certain ethnic groups.
Independence from the shared epitope effect of rs6933349 in the interaction with smoking in the risk of developing ACPA-positive RA
The risk of developing ACPA-positive RA has been associated with interaction between smoking and HLA-DRB1 SE alleles [17, 28]. Considering the complex structure of the MHC locus, we continued to investigate whether the newly identified SNP, rs6933349, which interacts with smoking to confer the risk of ACPA-positive RA, represents a novel interaction or simply reflects risk from HLA-DRB1 SE alleles. We first examined the interaction between rs6933349 and smoking in the risk of developing ACPA-positive RA by means of logistic regression, with adjustment for HLA-DRB1 SE alleles in the EIRA cohort. The multiplicative P value decreased from 0.05 to 0.018 after the SE adjustment, suggesting that this interaction is not dependent on HLA-DRB1 SE alleles. Computation of the AP with adjustment for SE alleles in the EIRA cohort also points toward significant independent interaction between smoking and rs6933349 (AP = 0.32, P = 0.006).
Additionally, we investigated the linkage disequilibrium (LD) between rs6933349 and known HLA-DRB1 SE alleles (including the major RA risk alleles HLA-DRB1*04:01 and *04:04), by calculating the r 2 value between them in the EIRA cohort. The SNP, rs6933349, which is in the region of MHC class I (Additional file 1: Figure S1), had no evidence of LD with HLA-DRB1 SE alleles (r 2 ≤ 0.05) (Additional file 1: Table S1), suggesting that it represents a novel gene-environment interaction in the risk of developing ACPA-positive RA.
Stage 4: DNA methylation as a potential mediator of the genotype and smoking interaction in the risk of developing ACPA-positive RA
Last, we explored the role of methylation of cg21325723 as a potential mediator of the interaction between rs6933349 and smoking in the risk of developing ACPA-positive RA (Fig. 1a). We performed mediation analysis [29] and modeled the relationships between rs6933349, smoking, cg21325723 methylation, and ACPA-positive RA in EIRA, using logistic regression. Using the multiplicative model, we observed significant interaction between rs6933349 and smoking in the risk of developing ACPA-positive RA (β coefficient = 0.99; 95% CI 0.30 to 1.68; P value = 0.0051). However, after including cg21325723 methylation as a covariate in the regression, the interaction between rs6933349 and smoking in relation to risk of ACPA-positive RA was attenuated and no longer significant (β coefficient = 0.39; 95% CI -0.39 to 1.17) (P value = 0.33). This result suggests that cg21325723 methylation may be a potential mediator of the gene-environment interaction between rs6933349 and smoking in the risk of developing of ACPA-positive RA.
In summary, we have identified a gene-environment interaction between rs6933349 and smoking on the DNA methylation level of cg21325723, which mediates the gene-environment interaction between rs6933349 and smoking in the risk of developing ACPA-positive RA. This gene-environment interaction represents a novel interaction in the risk of developing ACPA-positive RA in both Caucasian and Asian populations.
Information on smoking habits was gathered retrospectively by means of a questionnaire. The quality of exposure information may be different in cases and controls, which may result in recall bias. As misclassification of smoking is likely not related to genotype, the effects of such potential misclassification of exposure will be limited with regard to the investigated interaction between smoking and rs6933349.
An issue that may complicate studies involving epigenetics is that epigenetic modifications, such as DNA methylation, are much more dynamic than genetic variations and can be influenced by many confounders, such as age, sex, cell heterogeneity, and others. Although we addressed this issue by adjusting for these potential confounders in the linear regression model, there may still be other sources of confounding that were omitted and residual confounding that was not fully accommodated within the linear adjustment that we pursued. However, arguing against this idea is the fact that the identified gene-environment interaction between rs6933349 and smoking is not only important in the DNA methylation level of cg21325723, but is also important in the risk of developing ACPA-positive RA, and this was replicated in multiple ethnic groups.
With these limitations in mind, several reasonable inferences can however be made from these analyses. First, DNA methylation can act as a mediator of gene-environment interaction in the risk of developing ACPA-positive RA. To the best of our knowledge, this is the first report showing that DNA methylation can mediate gene-environment interaction regarding the development of a common disease. Even with the enormous success in genome-wide association studies (GWAS) of human common diseases in recent years, the search for new genetic risk factors has not revealed new strong effects [30]. It has been suggested that interactions between different genes and between genes and environment may explain a significant part of these risks. In this case, the associations between genetic variants and disease phenotype may be marginally significant and may have been neglected through a conventional genome-wide approach [31].
Even though multiple studies suggest that gene-environment interactions play an important role in disease susceptibility, including in RA [9, 10], a challenge remains to provide a functional interpretation and understand the molecular mechanism of the gene-environment interaction in the development of disease. DNA methylation, which can integrate both genetic and environmental cues, can be a possible "missing link" and is an attractive mechanism to explain the pathogenesis of disease. Combined with the fact that DNA methylation of cg21325723 can also integrate other RA genetic risk variants in the MHC class II locus [12], it is a possibility that DNA methylation is more proximal to the RA pathogenesis than genetic variations and this makes it a good candidate for therapeutic targets.
Second, even though we focused our study on the SNPs within the MHC region and selected CpG sites based on previous work, the success of identifying a CpG site, with methylation that mediates a novel interaction between genotype and smoking in the risk of developing ACPA-positive RA, suggests that this type of epigenetic regulation may be more common than currently acknowledged, and the findings reported here may simply be the "tip of the iceberg". However, a challenge remains in genome-wide study of this issue, considering the extremely large number of combinations between SNPs and CpGs. Thus, further methodological work using advanced statistical methods for genome-wide evaluation, together with larger sample sizes and meta-analysis in independent studies, is important.
Third, the understanding of various possibilities of downstream epigenetic regulation explaining the detailed mechanisms awaits further study. Considering the critical role of DNA methylation in regulating gene expression, an attractive hypothesis is the epigenetic regulation of HLA gene expression, which in combination with particular HLA-DR gene products may be an important factor in exaggerated epitope presentation in RA. The altered DNA methylation profile within the MHC class II cluster in ACPA-positive RA, which may affect assembly, expression and peptide loading of HLA genes, would determine the functional capacity of RA-associated HLA class II molecules [32, 33] during presentation of autoantigen-derived peptides to T cells able to drive disease-inducing adaptive immunity (for further details of this scenario and its T and B cell specificities, see references [34–36]). In this scenario, it is important to identify a target HLA gene that has expression regulated by DNA methylation of cg21325723.
Finally, the approach presented here also demonstrates that it might be feasible to perform an integrated genetic and epigenetic analysis to identify genetic risk alleles for disease that are not found by conventional analysis. The association between the SNP, rs6933349, and methylation at cg21325723 was not genome-wide significant without considering the smoking interaction (Fig. 2a), and was neglected through a conventional genome-wide approach [12]. Given the data here, showing that part of genetic risk is mediated epigenetically, and also that epigenetic changes may integrate genetic and environmental effects, the augmentation of genetic studies with epigenetic analysis promises to illuminate hereditary risk that is otherwise opaque when considering genotype in isolation. Through identifying new risk factors and revealing the role of DNA methylation as mediator of genotype and smoking interaction in the development of ACPA-positive RA, this strategy will contribute towards understanding the mechanism of common disease.
In conclusion, we identified one significant gene-environment interaction between rs6933349 and smoking in the DNA methylation levels of cg21325723, which has been shown previously to be associated with the risk of developing ACPA-positive RA. Additionally, we show that the gene-environment interaction between rs6933349 and smoking represents a novel interaction that confers the risk of developing ACPA-positive RA in different ethnic groups, and this gene-environment interaction is mediated through DNA methylation of cg21325723.
ACPA:
Anti-citrullinated peptide antibody
Attributable proportion
DMP:
Differentially methylated position
EIMS:
Epidemiological Investigation of Multiple Sclerosis
EIRA:
Epidemiological Investigation of Rheumatoid Arthritis
GWAS:
HLA-DRB:
Human leukocyte antigen DR beta chain
InCHIANTI:
Population-based prospective cohort study in two areas of Chianti
LD:
linkage disequilibrium
M eff :
Number of independent tests
MHC:
MyEIRA:
Malaysian Epidemiological Investigation of Rheumatoid Arthritis
RERI:
Relative excess risk due to interaction
Shared epitope
SI:
Synergy index
SNP:
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We thank the EIRA (http://www.eirasweden.se/index1.htm), MyEIRA [3], EIMS [37] (http://www.eims.nu) and InCHIANTI (http://inchiantistudy.net/) study groups for contributing invaluable clinical and genetic samples.
This work was supported in part by the China National Natural Science Foundation (grant number 31471212) and the Ministry of Science and Technology 973 (grant number 2015CB910401) to YL. The MyEIRA study was financially supported by Ministry of Health Malaysia (MRG-200512, JPP-IMR 07-017, JPP-IMR 07-046, JPP-IMR 08-006, JPP-IMR 08-012 and JPP-IMR 11-005) and we thank the Director General for his support in the MyEIRA study. This work was also supported by the Swedish Research Council, and AFA Insurance.
Supporting data are available and authors had full access to all the data in the study.
WM performed the initial analyses of the gene-smoking interaction in DNA methylation using the linear regression model. ZZ, together with WM, performed the analyses of the gene-smoking interaction in RA using the linear regression model and also the analyses for replication. XJ performed the analysis of the gene-smoking interaction in RA using the AP model in EIRA. CLT and SM were in charge of the MyEIRA cohort. SU performed genotype imputation in EIRA. MJ and IK provided EIMS data for replication. LF provided InCHIANTI data for replication. LK and LA were in charge of the EIRA cohort. LA, HZ, APF, TJE, LP, and YL proposed the experimental design. LP performed the analysis of the gene-smoking interaction in RA using the AP model in MyEIRA. YL conceived the idea, performed mediation analysis, went through the codes from WM and ZZ, and wrote the manuscript. All authors read and approved the final manuscript.
This study was approved by the Regional Ethical Review Board in Stockholm (number 96-174 and 2006/476-31/4) and the Ethical Review Board in Fudan University School of Basic Medical Sciences (number 2016-014). Consent has been obtained from all subjects who participated in this study.
Department of Biochemistry and Molecular Biology, The Ministry of Education Key Laboratory of Metabolism and Molecular Medicine, School of Basic Medical Sciences, Fudan University, West Building 13, 130 Dong An Road, Shanghai, China
Weida Meng
& Yun Liu
State Key Laboratory of Medical Neurobiology, Fudan University, Shanghai, China
Division of Rheumatology, Huashan Hospital, Fudan University, Shanghai, China
Zaihua Zhu
& Hejian Zou
Institute of Environmental Medicine, Karolinska Institutet, Stockholm, Sweden
Xia Jiang
& Lars Alfredsson
Institute for Medical Research, Jalan Pahang, 50588, Kuala Lumpur, Malaysia
Chun Lai Too
& Shahnaz Murad
Rheumatology Unit, Department of Medicine, Center for Molecular Medicine, Karolinska Institutet and Karolinska University Hospital, Stockholm, Sweden
, Lars Klareskog
& Leonid Padyukov
Institute of Human Genetics, Friedrich-Alexander-Universität Erlangen-Nürnberg (FAU), Erlangen, Germany
Steffen Uebe
Department of Clinical Neuroscience, Center for Molecular Medicine, Karolinska Institutet, Stockholm, Sweden
Maja Jagodic
, Ingrid Kockum
& Tomas J. Ekström
Instramural Research Program, National Institute on Aging, National Institutes of Health, Baltimore, MD, USA
Luigi Ferrucci
Center for Occupational and Environmental Medicine, Stockholm County Council, Stockholm, Sweden
Lars Alfredsson
Center for Epigenetics and Departments of Medicine, Johns Hopkins University School of Medicine, Baltimore, MD, USA
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Correspondence to Leonid Padyukov or Yun Liu.
Supplementary materials include five supplementary figures and a supplementary table. (PDF 440 kb)
Meng, W., Zhu, Z., Jiang, X. et al. DNA methylation mediates genotype and smoking interaction in the development of anti-citrullinated peptide antibody-positive rheumatoid arthritis. Arthritis Res Ther 19, 71 (2017) doi:10.1186/s13075-017-1276-2
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Endoscopic group
In mathematics, endoscopic groups of reductive algebraic groups were introduced by Robert Langlands (1979, 1983) in his work on the stable trace formula.
Roughly speaking, an endoscopic group H of G is a quasi-split group whose L-group is the connected component of the centralizer of a semisimple element of the L-group of G.
In the stable trace formula, unstable orbital integrals on a group G correspond to stable orbital integrals on its endoscopic groups H. The relation between them is given by the fundamental lemma.
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| Wikipedia |
BMC Ecology and Evolution
The relationship between bite force, morphology, and diet in southern African agamids
W. C. Tan1,2,3,4,6,
J. Measey4,
B. Vanhooydonck5 &
A. Herrel5,6
BMC Ecology and Evolution volume 21, Article number: 126 (2021) Cite this article
Many animals display morphological and behavioural adaptations to the habitats in which they live and the resources they exploit. Bite force is an important whole-organism performance trait that allows an increase in dietary breadth, the inclusion of novel prey in the diet, territory and predatory defence, and is important during mating in many lizards.
Here, we study six species of southern African agamid lizards from three habitat types (ground-dwelling, rock-dwelling, and arboreal) to investigate whether habitat use constrains head morphology and bite performance. We further tested whether bite force and head morphology evolve as adaptations to diet by analysing a subset of these species for which diet data were available.
Overall, both jaw length and its out-lever are excellent predictors of bite performance across all six species. Rock-dwelling species have a flatter head relative to their size than other species, possibly as an adaptation for crevice use. However, even when correcting for jaw length and jaw out-lever length, rock-dwelling species bite harder than ground-dwelling species. Diet analyses demonstrate that body and head size are not directly related to diet, although greater in-levers for jaw closing (positively related to bite force) are associated to an increase of hard prey in the diet. Ground-dwelling species consume more ants than other species.
Our results illustrate the role of head morphology in driving bite force and demonstrate how habitat use impacts head morphology but not bite force in these agamids. Although diet is associated with variation in head morphology it is only partially responsible for the observed differences in morphology and performance.
The adaptive significance of phenotypic traits has been of interest to evolutionary biologists for centuries [1]. Phenotypic variation is shaped by evolutionary and ecological processes with traits promoting survival and reproduction ultimately being selected. Therefore, understanding the relationship between morphology and function and how it is related to ecology is crucial to understand the evolution of phenotypic diversity [2, 3]. Performance traits related to locomotion and feeding are the most common functional traits tested as they are likely targets of selection given their role in survival and reproduction.
In lizards, variation in head morphology is relevant in many ecological (feeding, habitat and refuge use) and social contexts (territorial display, mating, and aggressive interactions). To test the possible adaptive nature of variation in head morphology, many studies have measured bite force. This performance trait has been related to both diet and territory defence [4,5,6,7,8]. The relationship between head shape and bite force is thought to be rather straightforward as the jaws can be approximated by a simple lever system. Moreover, larger heads (length, width and height) should result in an increase in bite force as they provide more space for the jaw adductor muscles [4, 6]. These inferences are supported by biomechanical models [9, 10] which show that taller and wider heads can accommodate more jaw adductor muscles that are pennate and thus can provide more force for a given volume [6, 11]. An increase in lower jaw length increases the out-lever of the jaw system which should reduce bite force as the lower jaw act as a lever system which transmits the input force from muscles to the out-lever arm to produce an output force [10].
However, bite force is not the only aspect influencing head shape. Ecological constraints such as habitat use may be important factors driving the evolution of head shape. For instance, animals living within rocky habitats hide in cervices when escaping from predators [11, 12]. To be able to do so they likely benefit from flat heads and bodies. However, given the importance of head height in driving bite force, a trade-off may exist between bite force and the ability to use narrow crevices. The importance of habit use in driving the evolution of the cranial morphology has been documented previously for some lizards [12,13,14,15]. Habitat use may also impact the availability of food resources, which may in turn drive the evolution of morphology and bite performance [16]. Lizards with larger heads can be expected to consume larger and harder prey due to the gape and bite force advantage conferred by these traits [5,6,7, 17, 18]. In addition to the above cited environmental constraints that are likely to affect bite force, bite force may also evolve through sexual selection. For example, in species where males engage in fights to defend territories or compete for access to mates, high bite forces likely result in increased fitness [19, 20].
Agamids are widespread across the African continent. They present a particularly interesting group to study ecomorphological relationships due to the shared evolutionary history and geographical distribution [21]. They are thought to have undergone rapid diversification, radiating into multiple clades about 10 Mya [21]. A recent study has shown divergence in limb morphology and locomotor performance between southern African agamas utilising different habitats, suggesting ecological differentiation between these species [22]. Since habitat is likely an important factor shaping limb and locomotor variation in these agamas, we expect this to drive variation in cranial morphology, bite force, and possibly even diet. However, the ecomorphological relationships between habitat use, morphology, performance and diet remain poorly known in agamids (but see [22, 23]). These lizards generally adopt a sit-and-wait foraging strategy, feeding predominantly on active prey such as ants, beetles and flying insects [24, 25] which may impact how prey availability drives variation in head morphology and bite force.
In this study, we studied six species of agamas from contrasting habitat types, representing rock-dwelling (Agama atra, A. anchietae and A. aculeata distanti), ground-dwelling (A. aculeata aculeata and A. armata) and arboreal (Acanthocercus atricollis) habitats [22, 26]. The genus Agama only consists of species living mostly in terrestrial and rocky habitats. Their closest extant group—the genus Acanthocercus however, contains 13 species of arboreal and rock-dwelling lizards [27]. Therefore, even though belonging to a different genus, the species Acanthocercus atricollis is the only agamid species found in southern Africa with an arboreal lifestyle [26]. Although occupying very different habitats, A. a. aculeata and A. a. distanti are two subspecies of A. aculeata belonging to a species complex which has not been completely resolved. We first explored the relationship between head morphology, bite force, and diet. We expect longer, wider, and taller heads and larger jaw closing in-levers to be associated with an increase in bite force [4, 11]. We further compared the association between head morphology and bite force in species from different habitats. Because rock-dwelling species possess flatter heads, allowing them to hide in rock crevices [11, 28], we expect reduced bite forces in these species. We hypothesised ground-dwelling species to have taller heads and thus higher bite forces [13, 14]. Finally, we predict species with larger heads and higher bite forces to include larger and harder prey into their diet [7, 17]. To test these hypotheses, we compared our findings with published data on stomach contents from four of the six examined agama species [24].
A multiple regression performed on the head measures with bite force as the dependent variable retained a single significant model (R2 = 0.94; P < 0.01) with the jaw out-lever (β = 0.52) and lower jaw length (β = 0.45) as significant predictors (Table 1). Most head variables (head length and width, lower jaw, jaw out-lever and snout length) were, however, highly and positively correlated with bite force (Table 1).
Table 1 Correlation analysis and stepwise multiple regression analyses with bite force as the dependent variable and head morphological traits as independent variables. All variables were log10 transformed
Morphology, performance and habitat association
Habitat groups differed significantly in body size (F2,144 = 24.94, P < 0.01). Post-hoc tests revealed that arboreal species were significantly larger than other groups, followed by rock-dwelling species (Table 2). The multivariate analyses of covariance (MANCOVA) with SVL as covariate showed significant differences in head shape between habitat groups (Wilks' λ = 0.09, F216,272 = 39.86, P < 0.01). The analysis of covariances (ANCOVA) further showed a significant difference in all head measurements except for head length (Table 3). For their body size, rock-dwelling species had narrower heads than arboreal species and flatter heads than all other habitat groups. Arboreal species have shorter lower jaws, jaw out-levers, and snouts but a longer in-lever for jaw closing than the other two habitat groups.
Table 2 Morphology, bite force, and the index of relative importance (IRI) of each dietary item found in the three habitat groups
Table 3 ANCOVAs performed on head morphological variables testing for differences between habitat groups
An ANOVA testing for differences in absolute bite force between habitat groups was significant (F2,144 = 11.99; P < 0.01). Post-hoc tests indicated that ground-dwelling and rock-dwelling species have a lower bite force than arboreal species but did not differ from one another (Table 2). The ANCOVA with lower jaw length and the jaw out-lever as co-variates again detected significant differences in bite force between habitat groups (F2,142 = 10.54; P < 0.01). Post-hoc pairwise comparisons on the bite force residuals showed that ground-dwelling species have the lowest relative bite force compared to the other habitat groups.
To understand the relationship between morphology and diet, we ran two-bock partial least-square analyses with snout-vent-length and head measurements versus the prey IRI data. This revealed a significant association between head shape and diet composition (r = 0.50, P < 0.01). Higher IRI values of Hymenoptera, Diptera, and Diplopoda are associated with a shorter in-lever for jaw opening and longer closing in-lever (Fig. 1). After correcting for size, the significant association remains (r = 0.58, P < 0.01). The same covariation was observed, suggesting that body and head size are less important drivers of the covariation between diet and morphology. Pearson correlations between bite force and prey IRI indicated a significant negative correlation between absolute bite force and the IRI of ants (r = − 0.45, P = 0.02) while residual bite force was negatively correlated with the IRI of Hemiptera (r = − 0.23, P = 0.03), Diptera (r = − 0.35, P < 0.01), and Diplopoda (r = − 0.48, p < 0.01).
a and c Bar plots represent the correlations observed between the original variables and the scatterplot axes. b Scatterplot of the scores of the four southern African agamid species obtained from partial least-squares (PLS) analysis between absolute head morphological variables (not corrected by size) and indexes of relative importance (IRI) of each prey item. Colour refers to different habitat groups: blue circles, ground dwelling, Agama armata; brown triangles and squares, rock dwelling, A. atra and A. a. distanti respectively; green diamonds, arboreal, Acanthocercus atricollis
There was a significant difference in diet between the habitat groups (Wilks' lambda = 0.50; F24,106 = 1.82; P = 0.02). Univariate ANOVAs identified significant differences in several taxonomic groups (Table 4). Particularly, the relative importance of ants in the diet was highest in ground-dwelling species compared to arboreal species (post-hoc tests). In rock-dwelling species the relative importance of Hemiptera and Diptera in the diet was higher than in arboreal species (Table 2).
Table 4 Results of the ANOVAs testing for differences in prey IRI between habitat groups
Phylogenetic and ontogenetic differences between habitat groups
Phylogenetic ANOVAs identified differences in body size, head variables and bite force between habitat groups (Additional file 2: Table S2). No significant phylogenetic differences were found in diet (although due to the limited observations of certain prey items—F statistics and P-values are unreliable; Additional file 2: Table S3).
Habitat groups differed significantly in body size in adults (F2, 99 = 105.80, P < 0.01) and juveniles (F2, 42 = 12.15, P < 0.01). The multivariate analyses of covariance (MANCOVA) with SVL as covariate showed significant differences in head shape between habitat groups in both adults (Wilks' λ = 0.08, F16, 182 = 28.17, P < 0.01) and juveniles (Wilks' λ = 0.06, F16,68 = 13.11, P < 0.01). The analysis of covariances (ANCOVA) further revealed a significant difference in all head variables except for head length and width for adults and in head height and closing in-lever for juveniles (Additional file 3: Table S4). Habitat groups differed in juveniles (F2, 42 = 7.609, P < 0.01) and adults (F2, 99 = 35.16, P < 0.01) for absolute bite force. Stepwise regression resulted in a single significant model (R2 = 0.71; P < 0.01) with the head width (β = − 0.35) and jaw out-lever (β = 1.25) as significant predictors of bite force for adults while the snout length (β = 0.37) was a significant predictor in the model (R2 = 0.96; P < 0.01) for juveniles. However, ANCOVA also showed significant differences in bite force between habitat groups in adults (F2,142 = 10.54; P < 0.01) and in juveniles (F2,41 = 6.16; P < 0.01). We found differences in diet between the habitat groups in both adults (MANOVA; Wilks' lambda = 0.26, F18,50 = 2.66, P < 0.01) and juveniles (MANOVA; Wilks' lambda = 0.14, F22,36 = 2.77, P < 0.01). Univariate ANOVAs identified significant differences in several prey groups (Additional file 3: Table S5).
Our study shows that the association between external morphology and performance traits can be variable. We expected lizards with longer, wider, and taller heads to bite relatively harder. However, rock-dwelling lizards with narrower and flatter heads showed relatively greater bite force than ground-dwelling lizards. Although ground-dwelling species have taller heads compared to the rock-dwelling relatives, they had the weakest relative bite force. Commensurate with our last prediction, we found evidence of a correlation for lizards with a longer closing in-lever and harder prey in their diet.
Contrary to other ecomorphological studies [5, 6, 11, 17, 29], body and head length were not the best predictors of bite force in our study. Instead, jaw out-lever and lower jaw length (a good proxy for head length [11]) were the best predictors of bite force (see also [30]). This suggests that these simple external head measurements are informative and are valid predictors of bite force despite the complexity of the jaw systems in lizards. These results are counter-intuitive at first, however, as a longer lower jaw should increase the out-lever of the system and hence decreases rather than increase bite force [11]. In our data we find, however, that longer lower jaws and longer jaw out-levers are the best predictors of bite force. Possibly, these two variables stand out as they have low measurement error relative to other head measures which are all very highly positively correlated to bite force (Table 1). Thus, despite the fact that these two variables were retained by the multiple regression the real pattern is one of overall bigger heads being associated with higher bite forces, conforming to our hypothesis that lizards with greater head dimensions bite harder. Approaches such as geometric morphometrics may be particularly suited to better understand whether some parts of the head or cranium are particularly good predictors of bite force in these lizards [4, 31]. Aside from jaw architecture, other factors such as muscle mass, the orientation of the muscle force vectors, fibre length, muscle insertion sites, and the cross sectional area of adductor muscle also participate in the production of bite force and should not be neglected [10, 32].
Arboreal agamid species were the biggest, and had the largest heads and bites forces in absolute terms compared to species from other habitats (Table 2). Bite force show positive allometry relative to body dimensions in many lizards, possibly explaining why the largest species had disproportionately large absolute bite forces [18]. Irrespective of variation in overall size, rock-dwelling species had narrower and flatter heads than lizards from the other habitat groups. This finding suggests that flattened and narrow heads (and body) may confer an advantage to rock-dwelling species and allow them to use rock crevices as shelters to hide from predators [4, 5]. This has been demonstrated in other lizards and the morphological convergence in head shape previously described for rock-dwelling lizards [12, 28] thus appears to be a more general phenomenon. However, flatter heads are typically associated with a weaker bite force [6, 11, 33]. In our study, animals living in rocky habitats bite harder relative to their jaw size and out-lever compared to ground-dwelling species (see also [4, 31]). One possible explanation for this pattern could be differences in non-measured traits such as muscle mass and architecture [7, 30] as well as muscle orientation [20]. For example, a greater jaw adductor muscle mass or changes in muscle architecture (e.g., degree of pennation) can contribute to an increase in absolute and relative bite force [7]. In contrast to rock-dwelling species, arboreal species possess shorter jaws and out-lever lengths but a larger closing in-lever which directly promotes bite force [10, 29]. By reducing the jaw out-lever length and increasing the space available for muscles, bite force is increased [7, 10]. The need for a high bite force could possibly be explained by the hardness of the prey encountered in this environment [9]. A recent study [13], for example, demonstrated that the evolution of cranial shape in Amphibolurines (Agamidae) was significantly associated with their habitat use (arboreal, terrestrial, and rock-dwelling). Major patterns were found in the variation in snout length, skull height, and amount of space for jaw muscles: all of which related to adaptations to bite force generation and prey capture efficiency [13]. Further phylogenetic comparative studies on the relationship between head shape and habitat use are needed to confirm the generality of these results.
Our PLS results suggest that body size is not among the most important drivers of diet. We predicted an association between body size and the range of arthropod orders taken [34]. This was not the case in the agamas included in our study, however. This is consistent with the observation that the arboreal species, Acanthocercus atricollis, did not have a wider niche breadth compared to other agama species even though it was the largest species [24]. However, our analyses did demonstrate a significant covariation between head morphology and diet. This demonstrates that the jaw lever system is closely linked to diet, more specifically, the types of prey captured. We found evidence for significant correlation with a longer closing in-lever and a higher relative importance of Hymenoptera and Diplopoda (Fig. 1), both of which are considered to be rather hard prey (see [17]). The fact that head length, width, and height did not show strong covariation with our diet proxies disagrees with observations for other lizards [5, 6, 8, 35]. Given the complexity of the jaw system, a reduced jaw opening and increased jaw closing in-lever can nevertheless enhance bite force [6, 7, 9, 10]. As a result, harder prey can be captured and thus handled more efficiently [18]. However, as to why Diptera, a soft arthropod was positively correlated with in-lever size remains unclear. Larger jaw opening muscles could facilitate faster jaw movements which could allow lizards to specialise on mobile prey such as dipterans, yet the opening in-lever did not co-vary strongly with the relative importance of Diptera in the diet [30].
As expected, the importance of ants in the diet was found to be greatest in ground-dwelling species, while rock-dwelling species seem to consume more dipterans and hemipterans (Table 2). This conforms to differences detected previously [24]. Although only partly significant, both rock species (Agama aculeata distanti and A. atra) appear to be generalists consuming a more diverse prey spectrum than other agama species, hence having the highest niche breadth [24]. Smaller ground-dwelling species with lower absolute bite force (as a result of their smaller heads) should select softer and smaller prey even if they are not physically constrained to take harder prey [6]. The observed importance of ants in the diet of ground-dwelling species could simply reflect the lack of other prey in the environment, instead of a dietary specialisation. It is possible that agamas do not choose their prey but rather capture whichever arthropod occurs in the environment. Ants are easy to capture and abundant in arid ecosystems [36] and may thus be a profitable prey source. Indeed, the energetic benefit of consuming copious quantity of small hard prey items with minimal search time could outweigh the cost of increased handling time [37]. Future studies on prey availability at these study sites would be of interest for two reasons: (a) to determine the abundance and types of arthropods occurring naturally, thus allowing to understand whether agamas actively select their prey or are opportunistic predators; (b) to reconstruct the original prey size using the arthropods collected as a reference [5].
Phylogenetic analyses revealed no significant association between clade membership and differences found between habitat groups, although our analyses were compromised due to limited number of species (smaller than recommended to conduct phylogenetically informed analyses). Due to low sample size in terms of adults and juveniles (see Additional file 3), the detection of ontogenetic differences might be restricted by low statistical power and hence leads to the inability to make robust inferences.
Our results suggested that variation in head shape cannot be explained by habitat use alone. Selection for head shape and bite force are also highly relevant to territory or predator defence [38]. Sexual selection is likely an important force in driving variation in head morphology in lizards [4]. Differences in head or skull shape between sexes have been demonstrated in many lizards: lacertids [7], Anolis lizards [30] and chameleons [8]. However, sexual dimorphism could not be tested in our present study due to the low sample size. Sexual differences in reproductive strategy can translate into differences in bite force given the role of biting behaviours in mating and intrasexual aggression [19, 38]. A previous study has suggested that the larger and wider heads in male A. atricollis may be explained by sexual selection [25]. The complex interaction between natural and sexual selection can, however, only be unravelled through comparative studies of the degree of sexual dimorphism in relation to the ecological context of the species based on the quantification of possible variation in diet [4].
In summary, our data suggests that rock-dwelling southern African agamid species have high relative bite forces despite having narrow and flat heads, suggesting an important role for differences in muscle architecture and/or skull shape in driving these patterns. Moreover, differences in bite force and feeding ecology between habitat groups are not reflected by differences in body and head size (but see [6, 11]). Although niche divergence in diet and habitat use are found to be consistent with variation in cranial morphology and bite force, the potential contribution of sexual selection in driving some of the observed differences needs to be explored.
Study organisms
A total of 147 individuals, including 51 Females, 51 males and 45 juveniles representing six species occupying different habitat types were sampled (see Additional file 1: Table S1). We distinguish male agamas from females visually, based on the bulging of the hemipenes in males [22]. Lizards were caught by hand or noose in different localities in South Africa. Agama atra samples (N = 41) were captured in the Muizenberg mountains (34° 05′ S, 18° 26′ E) and the Grootwinterhoek reserve (33° 09′ S, 19° 05′ E) and other parts of the Western Cape in March 2008 and January 2011. Agama aculeata distanti (N = 36) were sampled in Kruger National Park (23° 58′ S, 31° 31′ E) and Welgevonden Reserve (24° 12′ S, 27° 54′ E), Limpopo province, in November 2011 and March 2017 respectively. Both A. anchietae (N = 10) and A. aculeata aculeata (N = 10) were sampled in Tswalu game reserve (27° 17′ S, 22° 23′ E), Northern Cape, in January 2010, with the exception of three A. anchietae from Gobabis (22° 26′ S, 18° 57′ E) and Swakopmund (22° 15′ S, 15° 4′ E), Namibia, and one A. a. aculeata from Zwartskraal farm (33°10′S, 22°34′E), Western Cape. Agama armata (N = 11) were collected at Alicedale Farms (22° 38′ S, 30° 08′ E) and Greater Kuduland Safaris (22° 32′ S, 30° 40′ E) in January 2010 and February 2017. Lastly, Acanthocercus atricollis (N = 39) were caught in the suburban area of Mtunzini (28° 57′ S, 31° 44′ E) and Zululand Nurseries, Eshowe (28° 52′ S, 31° 28′ E), KwaZulu-Natal, in February 2017. The GPS coordinates of each lizard were recorded upon capture. Lizards were marked with a non-toxic marker (to avoid recapture), placed in cloth bags and then transferred back to the field station where they were stomach flushed. Morphological and bite force measurements were also taken. Once all the data were collected, we released the animals at the exact location where they were found.
Morphological variables were measured for each individual using digital callipers (Mitutoyo; precision 0.01 mm) according to [11] for the following morphological traits (Fig. 2): snout-vent length (SVL); head width (HW) taken at the widest point of the skull; head length (HL) taken from the tip of the snout to the end of the parietal bone; head depth or height (HH) at the tallest part of the head, posterior to the orbital region; lower jaw length (LJL), taken from the snout tip to the end of the retroarticular process; jaw out-lever taken from the posterior end of the quadrate to snout tip (QT), and the distance from the back of the jugal to the tip of the snout (CT). Based on the latter three measurements, two other morphological variables were calculated: the first or closing in-lever of the jaw being the difference between QT and CT; and the second or opening in-lever, being the subtraction of QT from LJL [9, 17]. A longer in-lever for jaw closing provides a higher mechanical advantage and subsequently increases bite force for a given head size [10].
Eight head measurements recorded for each lizard. CT, snout length; QT, quadrate to snout tip, Lever 1, first or closing in-lever; Lever 2, second or opening in-lever
Bite force
Bite forces were measured in vivo following the method of Herrel et al. (2001) [6] using an isometric Kistler force transducer (type 9203, 500 N, Kistler Inc. Winterthur, Switzerland), connected to a Kistler charge amplifier (type 5995A) with all measurements made accurate to 0.1 N. A pair of metal bite plates was placed between the jaws of the lizard which typically results in prolonged and repetitive biting. If needed, sides of the jaw were gently tapped to provoke the lizards to bite the plates. Agamids have solid acrodont teeth implanted onto the jaw and are unlikely to suffer any damage from measuring bite force with metal plates. No audible breaking of teeth was present (contra [39]) and inspection of the teeth showed no damage. Our experience with these and other lizards is not consistent with the findings of Lappin & Jones (2014) [39] and suggest that bites on metal plates provide more accurate measures of maximal bite force. The distance between the plates and the point of application of the bite force were standardised across all animals. Bite force was recorded five times for each animal. The maximum value was then retained as the maximal bite force and used in further analyses. Although air temperatures, humidity and other environmental conditions could not be controlled in this study due to the absence of facilities in the field, we ensured that the lizards were tested at the temperatures at which they are active in the field.
Data on stomach contents for 67 individuals from four species of agamas (Agama atra, A. aculeata aculeata, A. armata and Acanthocercus atricollis) were extracted from a previously published study (Additional file 1: Table S1) [24]. Stomach contents were classified to the lowest taxonomic level possible. The food items were then blotted dry, measured and weighed using an electronic microbalance (AE100-S, Mettler Toledo GmBH, Zurich, Switzerland; ± 0.1 mg) [11].
For each prey group, we calculated the index of relative importance (IRI) [40]. This compound index indicates the importance of particular prey group and provide a balanced view based on combination of unique individual properties (numbers, mass and occurrence in diet) [17, 24]:
$${\text{IRI}} = \left( {\% {\text{N}} + \% {\text{V}}} \right) \times \% {\text{Oc}}$$
where %N is the percentage of numeric abundance, counted from the number of heads of the prey items, %V is the proportion of mass of that prey group to total prey mass and %Oc is frequency of occurrence of a certain prey group.
All morphological and performance variables were logarithmically transformed (log10) before analyses to fulfil the assumptions of normality and homoscedascity. To explore which head variables best explain variation in bite force, stepwise multiple regression analyses were conducted. Pearson correlations were further used to explore relationships between head morphology and bite force.
We grouped the species examined into three habitat groups: rock-dwelling, ground-dwelling, and arboreal. We should point out, however, that these ecological habitat groups may only apply to the populations sampled in our study and are based on our observations in the field. Following the classification of species, we tested whether habitat groups differ in size (SVL) using a univariate analysis of variance (ANOVA). If habitat groups were significantly different in size, multivariate analyses of covariance (MANCOVA) were performed to test for differences in head morphology with SVL as a covariate. We did not test for potential differences between sexes due to the low sample size for each sex per habitat group. Subsequent Tukey's honest significant difference (HSD) post hoc tests were conducted to test for differences between pairs of habitat groups.
We then investigated whether species assigned to different habitat groups differed in absolute bite force using an ANOVA. A subsequent ANCOVA with the most significant explanatory variables from stepwise regression as covariates was performed to examine whether differences remained when correcting for head dimensions. If so then this would suggest variation in the underlying muscle architecture.
To explore the multivariate association between morphology and diet, we used two-block partial least-squares regressions (PLS) using the two.b.pls function of the geomorph package [41]. Snout vent length and all head variables were computed as the first block of variables while the IRI of the different prey groups were combined in the second block of variables. We first performed the PLS with absolute variables and then repeated the analysis with relative size, in which we used corrected morphological variables using the residuals of each trait following a regression on SVL. We additionally ran a Pearson correlation between absolute bite force and the IRI of all prey and reran the same analysis with size-corrected (residual) bite force. Finally, we tested whether habitat groups differed in their diet composition using a MANOVA and Tukey's HSD post hoc tests on the IRI of all prey groups.
Using the same statistical analyses above, we tested for ontogenetic differences in the habitat groups. To determine whether differences found between habitat groups are linked to ecological divergence or clade membership, we conducted phylogenetic ANOVAs among the groups using a trimmed phylogeny of Leaché et al. (2014) [21].
All statistical analyses were performed using R v.3.6.2 [42].
The datasets used and/or analysed during the current study are available in Zenodo: https://0-doi-org.brum.beds.ac.uk/10.5281/zenodo.4581097
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This work would not be possible without Krystal Tolley (SANBI) in the field and facilitating the many field trips, with funding from the National Research Foundation (NRF) of South Africa (South African Biosystematics Initiative and the Key International Science Capacity Initiative). We are indebted to members of the MeaseyLab, Gareth Coleman, Amy Panikowski, John Wilkinson, François Meyer, Jason Savage, Frederik Igelström and Ellen Birgitte. For access to various sites, local knowledge of species, data collection, and generous hospitality. For sharing their expertise on potential collection sites, we would also like to thank Andre Coetzer, Gary Nicolau, Luke Verbugt and Marius Burger. Finally, we are very grateful to all the staffs and volunteers in Welgevonden Game Reserve, especially Greg Canning, Phillipa Myram, Shannon Beale, Camilla Bjerrum, Maria Larsen and Amanda Martinez Avila. Finally we would like to thank the associate editor and anonymous reviewers for helpful comments that improved our manuscript.
Open Access funding enabled and organized by Projekt DEAL. The project was also financially supported by the European Commission through the programme Erasmus Mundus Masters Course—International Master in Applied Ecology (EMMC-IMAE) (FPA 532524-1-FR-2012-ERAMUNDUS-EMMC). We thank the DSI-NRF Centre of Excellence for Invasion Biology and National Research Foundation of South Africa incentive fund for rated researchers.
Herpetology Section, Zoologisches Forschungsmuseum Alexander Koenig, Adenauerallee 160, 53113, Bonn, Germany
W. C. Tan
Institut für Zoologie, Rheinische Friedrich-Wilhelms-Universität Bonn, Poppelsdorfer Schloss, Bonn, Germany
Laboratoire EBI Ecologie and Biologie des Interactions, UMR CNRS 7267, Université de Poitiers, UFR Sciences Fondamentales et Appliquées, Poitiers, France
Centre for Invasion Biology, Department of Botany and Zoology, Stellenbosch University, Stellenbosch, South Africa
W. C. Tan & J. Measey
Department of Biology, University of Antwerp, Universiteitsplein 1, 2610, Antwerpen, Belgium
B. Vanhooydonck & A. Herrel
Département Adaptations du Vivant, UMR 7179 C.N.R.S/M.N.H.N., Bâtiment d'Anatomie Comparée, 55 rue Buffon, 75005, Paris, France
W. C. Tan & A. Herrel
J. Measey
B. Vanhooydonck
A. Herrel
This study is part of the MS work of WCT, under the supervision of AH and JM. WCT conceived the study. WCT, AH and JM contributed to the study design and data analyses. All authors participated in the data collection in the field. WCT led the writing of the manuscript with help from JM and AH. All authors read and approved the final manuscript.
Correspondence to W. C. Tan.
This study was carried out under permits for scientific collection from Cape Nature (056-AAA041-00168–0056), Ezemvelo KZN Wildlife (OP 550/2017), SANParks (CRC/2008-2009/001-2008), Namibia (1504/2010 & 81305) and Northern Cape Province (FAUNA 110/2011, FAUNA 111/2011). Research was carried out in accordance with the protocol approved by SANBI ethics committee (Number 0010/08).
Additional file 1: Table S1.
Morphology, bite force, and the index of relative importance (IRI) of each dietary item found in all six species. N = sample size. IRI values were multiplied by 100 to facilitate the reading of the table. Standard deviation is shown in brackets.
Results of phylANOVAs performed on size, head variables and bite force testing for differences between habitat groups due to phylogenetic relationship. The averages of each species were taken for the calculation of each variable. Table S3. Results of phylANOVAs performed on prey IRI testing for diet differences between habitat groups due to phylogenetic relationship.
ANCOVAs performed on head morphological variables testing for differences in adults (N for ground dwelling = 14, rock dwelling = 16, arboreal = 28) and juveniles (N for ground dwelling = 7, rock dwelling = 27, arboreal = 11) among habitat groups. Table S5. Results of the ANOVAs testing for differences in prey IRI in adults (N for ground dwelling = 2, rock dwelling = 19, arboreal = 15) and juveniles (N for ground dwelling = 6, rock dwelling = 19, arboreal = 6) between habitat groups.
Tan, W.C., Measey, J., Vanhooydonck, B. et al. The relationship between bite force, morphology, and diet in southern African agamids. BMC Ecol Evo 21, 126 (2021). https://0-doi-org.brum.beds.ac.uk/10.1186/s12862-021-01859-w
DOI: https://0-doi-org.brum.beds.ac.uk/10.1186/s12862-021-01859-w
Submission enquiries: [email protected] | CommonCrawl |
Home IACR EprintCryptographic Hashing From Strong One-Way Functions Cryptographic Hashing From Strong One-Way Functions
Cryptographic Hashing From Strong One-Way Functions
Constructing collision-resistant hash families (CRHFs) from one-way functions is a long-standing open problem and source of frustration in theoretical cryptography. In fact, there are strong negative results: black-box separations from one-way functions that are $2^{-(1-o(1))n}$-secure against polynomial time adversaries (Simon, EUROCRYPT '98) and even from indistinguishability obfuscation (Asharov and Segev, FOCS '15). In this work, we formulate a mild strengthening of exponentially secure one-way functions, and we construct CRHFs from such functions. Specifically, our security notion requires that every polynomial time algorithm has at most $2^{-n – \omega(\log(n))}$ probability of inverting two independent challenges. More generally, we consider the problem of simultaneously inverting $k$ functions $f_1,\ldots, f_k$, which we say constitute a "one-way product function" (OWPF). We show that sufficiently hard OWPFs yield hash families that are multi-input correlation intractable (Canetti, Goldreich, and Halevi, STOC '98) with respect to all sparse (bounded arity) output relations. Additionally assuming indistinguishability obfuscation, we construct hash families that achieve a broader notion of correlation intractability, extending the recent work of Kalai, Rothblum, and Rothblum (CRYPTO '17). In particular, these families are sufficient to instantiate the Fiat-Shamir heuristic in the plain model for a natural class of interactive proofs. An interesting consequence of our results is a potential new avenue for bypassing black-box separations. In particular, proving (with necessarily non-black-box techniques) that parallel repetition amplifies the hardness of specific one-way functions — for example, all one-way permutations — suffices to directly bypass Simon's impossibility result. | CommonCrawl |
\begin{definition}[Definition:Saturated Set (Equivalence Relation)]
Let $\sim$ be an equivalence relation on a set $S$.
Let $T\subset S$ be a subset.
\end{definition} | ProofWiki |
\begin{definition}[Definition:Rooted Tree/Branch/Length]
Let $T$ be a rooted tree with root node $r_T$.
Let $\Gamma$ be a finite branch of $T$.
The '''length''' of $\Gamma$ is defined as the number of ancestors of the leaf at the end of that branch.
\end{definition} | ProofWiki |
\begin{document}
\title{The Formation of Global Free Trade Agreement \thanks{We would like to thank Jota Ishikawa, Toshiji Miyakawa, Hiroshi Mukunoki, and seminar participants in Game Theory Workshop 2021 and Konan University. Shirata acknowledges financial support from the Japan Society for the Promotion of Science, JSPS KAKENHI Grant Number 18K12738.} }
\author{ Akira Okada \thanks{Professor Emeritus, Hitotsubashi University, 2-1 Naka, Kunitachi, Tokyo 186-8601, JAPAN. E-mail address: \texttt{[email protected].}} and Yasuhiro Shirata \thanks{Corresponding author. Associate Professor, Department of Economics, Otaru University of Commerce, 3-5-21 Midori, Otaru, Hokkaido, 047-8501, JAPAN. E-mail address: \texttt{[email protected]}} }
\date{March, 2021}
\maketitle
\begin{abstract}
We investigate the formation of Free Trade Agreement (FTA) in a competing importers framework with $n$ countries. We show that (i) FTA formation causes a negative externality to non-participants, (ii) a non-participant is willing to join an FTA, and (iii) new participation may decrease the welfare of incumbent participants. A unique subgame perfect equilibrium of a sequential FTA formation game does not achieve global free trade under an open-access rule where a new applicant needs consent of members for accession, currently employed by many open regionalism agreements including APEC. We further show that global FTA is a unique subgame perfect equilibrium under an open-access rule without consent.
\noindent\textbf{JEL classification}: F11, F13
\noindent\textbf{Keywords}: Free Trade Agreement; Negative Externality; Open Regionalism: Sequential Participation; Subgame Perfect Equilibrium
\end{abstract}
\section{Introduction}
The rapid proliferation of preferential trade agreements (PTAs) over the last five decades has spurred a large volume of studies on the role of PTAs in the achievement of global free trade. The halt of the World Trade Organization (WTO) trade liberalization process in the Doha round has stimulated the growing literature on PTAs.
The literature has extensively explored whether or not PTAs can promote free trade in the world both theoretically and empirically. In a recent review, Bagwell et al.\ (2016) write: ``Both the theory and evidence are mixed; hence, as a general matter, whether PTAs are stumbling blocks or building blocks for multilateral liberalization remains ambiguous.'' This paper aims to contribute to a debate on the issue by analyzing a formation process to global free trade.
Trade liberalization is a sequential process. A small number of countries launch a trade agreement, and the agreement expands with new participants. The General Agreement on Tariffs and Trade (GATT) commenced in 1947 with twenty-three countries, and the WTO replace it in 1995. Since 2016, the WTO has 164 members. China participated in the WTO in 2001. The European Coal and Steel Community commenced in 1950 with six countries, and it has been enlarged through several steps to European Union (EU) with now twenty-seven members. The Asia Pacific Economic Cooperation (APEC) started in 1989 with twelve countries, and China, Hong Kong, and Taiwan joined it in 1991. As of 2021, it has twenty-one member countries. \footnote{WTO: \url{https://www.wto.org/english/thewto_e/history_e/history_e.htm}, EU: \url{https://europa.eu/european-union/about-eu/history_en}, APEC: \url{https://www.apec.org/About-Us/About-APEC/History}, March 14, 2021.}
Since the creation of the North American Free Trade Agreement (NAFTA) in 1994, many PTAs have emerged as alternative vehicles of trade liberalization.\footnote{As of 2021, the cumulative number of RTAs in force has expanded from roughly 50 in 1990 to nearly 350 today (\url{https://rtais.wto.org/UI/charts.aspx}), March 14, 2021.} Member countries of PTAs establish jointly internal tariffs on each other. The Article XXIV of GATT allows WTO members to form PTAs if the PTAs eliminate tariffs on ``substantially all trade'' among the member countries and that the external tariffs on non-member countries do not increase as a result of PTA formation. While the proponents of PTAs argue that they promote free trade, the proliferation of different PTAs may tangle the world trading system, commonly referred to as a ``spaghetti bowl.'' Various commitments and rules of PTAs are overlapped and inconsistent.
Stimulated by the creation of APEC, \emph{open regionalism} has received much attention from academics and policymakers to establish compatibility between regional trading agreements to achieve global free trade. The proponents of open regionalism view it as ``a device through which regionalism can be employed to accelerate the progress toward global liberalization and rule-making'' (Bergsten 1997). The first definition of open regionalism proposed by Bergsten (1997) is open membership, which requires that ``any country that indicates a credible willingness to accept the rules of the institution would be invited to join.''\footnote{Bergsten (1997) proposes the five definitions. The other four are unconditional MFN, conditional MFN, global liberalization, and trade facilitation.}
The principle of open regionalism, by contrast, has not been appropriately practiced.\footnote{A critic writes that ``APEC's trade liberalisation strategy is a frail initiative'' (Kelegama 2000).} Since Peru, Russia, and Viet Nam joined in 1998, APEC has accepted no new participants. Membership is negotiated between applicants and incumbent members.\footnote{The official statement of APEC writes: ``Decisions on the admission of additional members to APEC require a consensus of all existing members'' (21 November 1997, Vancouver).} We may regard that the open membership of APEC demands ``consent'' in the sense that any new participant needs unanimous consent of incumbent members. To practice open regionalism, Bergsten (1997) and Lewis (2011) argue that APEC should expand the liberalization to all members of the WTO to include more members.\footnote{Bergsten (1997) writes that ``the best, perhaps only, way for APEC to do so is to indicate publicly both its precise liberalisation programme and its willingness to extend that liberalisation to all members of the WTO on a reciprocal basis.''}
This paper provides a game-theoretic foundation for the recent debate above on open regionalism and free trade. As a framework, we consider a sequential formation process of a free trade agreement (FTA) under an open-access rule with two different participation rules. Our international trade model is based on the ``competing importers'' framework of Missios et al.\ (2016). \footnote{Missios et al.\ (2016) and Saggi et al.\ (2018) argue that the framework highlights a key insight, \emph{external trade diversion}, that has generally been overlooked in the literature.}
Under the open-access rule with consent that describes the current form of open regionalism of many FTAs, including APEC, a non-member country can participate in an FTA with members' unanimous approval. Every country decides sequentially to participate in an FTA according to a fixed order. Whenever a non-member decides to participate in an FTA, all member countries accept or reject new participation independently. An applicant country participates in the FTA if and only if all the incumbents accept it. Under the open-access rule without consent, which describes the principle of open regionalism, a non-member can participate in an FTA without an incumbent member's approval.
We summarize the results as follows. We first characterize the equilibrium tariffs under an FTA regime where several overlapping FTAs exist. The formation of an FTA does not change the external tariff on non-participants. This tariff policy of an FTA is consistent with the GATT Article XXIV, which requires the FTA member countries not to raise their external tariffs on non-members. In other words, the GATT Article XXIV is not binding for the external tariff of an FTA. \footnote{The optimal internal tariff of an FTA may not be the zero-tariff (Saggi et al. 2019). In this sense, the GATT Article XXIV, which requires the FTA internal tariffs to be zero, is binding in the model.} Non-participants reduce their tariffs on participants (tariff complementarity, Missios et al. 2016). We find that the formation of an FTA affects welfare as follows: (i) an FTA causes a negative externality to non-participants, (ii) a non-participant is always willing to join an FTA, and (iii) new participation may decrease incumbent participants' welfare.
By these properties, we show that a unique equilibrium FTA under the open-access rule with consent is not the global FTA. Incumbent participants reject a new applicant if it exceeds FTA's optimal size, which is strictly smaller than global FTA. This result explains that current open regionalism including APEC has not led to global FTA yet, despite of the proponets' expectation. By contrast, we show that global FTA is a unique subgame perfect equilibrium under the open-access rule without consent.
This paper's primary contribution is to show that a participation rule in open regionalism is crucial to achieving the global free trade agreement. The strategic behavior of incumbent participants to reject new participants may deteriorate the formation of global free trade. Our result provides theoretical support for the WTO's policy recommendation to require any PTA to be open-access without incumbents' consent. \footnote{Several FTAs have open-access provisions. For example, the NAFTA writes: ``any country or group of countries may accede to this Agreement subject to such terms and conditions as may be agreed between such country or countries and the Commission and following approval in accordance with the applicable legal procedures of each country'' (Article 2204). Trans-Pacific Partnership (TPP) Agreement also has an open access provision (Article 30.4).}
The paper is closely related to the existing works of Yi (1996), Seidmann (2009), Mukunoki and Tachi (2006), Missios et al.\ (2016), Furusawa and Konishi (2007), and Goyal and Joshi (2006), all of whom study trade liberalization by game-theoretic models. We differentiate our results from these works in the following aspects. \footnote{Among other papers on the endogenous PTA formation are Burbidge et al. (1987), Agihon et al. (2007), Saggi et al. (2010, 2018). The literature employs diverse approaches. In his survey, Maggi (2014) writes: ``this literature can be hard to tame, as the modeling approach and the exact nature of the question seem to shift from paper to paper.''}
Yi (1996) considers the endogenous formation of custom union (CU) and shows that global free trade is a unique Nash equilibrium in a simultaneous-move open regionalism game. A critical difference between Yi (1996) and this paper is that there are multiple subgame perfect equilibria in his sequential-move open regionalism game. Global free trade is one of subgame perfect equilibria. \footnote{Loke and Winters (2012) present a numerical example which shows that the global free trade result of Yi (1996) under the open regionalism fails under CES preferences and increasing marginal costs in the case of Bertrand competition.} Seidmann (2009) considers a sequential model of an open-access rule with consent in a three-country setup, where countries can form bilateral FTAs, a bilateral CU, or the multilateral FTA. Utility transfer between countries is allowed, and the countries can renegotiate a PTA for expansion. These features lead to the result that efficient global FTA eventually forms. Seidmann's model shows a different motive (``strategic positioning'') of a PTA from ours that PTA members strategically manipulate the status quo, seeking advantages in future negotiations.
Missios et al.\ (2016), Furusawa and Konishi (2007), and Goyal and Joshi (2006) consider the formation of FTAs based on the stability approach. Missios et al.\ (2016) employ a coalitional-proof Nash equilibrium in a three-country setup, and Furusawa and Konishi (2007) and Goyal and Joshi (2006) employ the pairwise stability of network formation games in an $n$-country setup. All of them show that global FTA is a unique stable outcome if countries are symmetric. \footnote{Goyal and Joshi (2006) prove that a pairwise stable network is either global FTA or the second-largest FTA with a single non-participant in a Cournot competition model.} Unlike these works, our approach is process-based. We explicitly formulate a sequential trade liberalization process as a non-cooperative game and analyze a subgame perfect equilibrium. Two approaches are complementary. We discuss differences between our results and the stability results in the literature in Section~6.
The rest of the paper is organized as follows. Section~2 presents the underlying economy. Section~3 considers the Nash equilibrium tariff in the non-cooperative tariff game. Section~4 investigates the formation of an FTA under two open-access rules. Section~5 illustrates a numerical example. Section~6 discusses the results, related to the literature. Section~7 concludes. The proofs are given in Appendices and the supplementary materials, and tedious calculations are collected in the supplementary materials.\footnote{Available upon request.}
\section{The Economy}
We consider a perfectly competitive economy with $n$ ``large'' countries indexed by $i=1, \cdots, n$. \footnote{We generalize the two-country model of Horn et al. (2010) to an $n$ country case. Missios et al. (2016) and Saggi et al. (2018) also consider multi-country versions of the model.} There are $n$ non-numeraire goods indexed by $J=1, \cdots, n$, and one numeraire good indexed by $0$. A consumption vector is represented by $c=(c^1, \cdots, c^n, c^0)$, where each $c^J$ $(J=1, \cdots, n)$ is the consumption of non-numeraire good $J$ and $c^0$ is that of the numeraire good.
A representative consumer in every country $i$ has the following utility function, which is additively separable and linear in the numeraire good: \begin{align*} U_i(c)=\sum_{J=1}^n u_i(c^J) + c^0 , \end{align*} where $u_i(c^J)$ is $i$'s utility function for consumption $c^J$ of each non-numeraire good $J$. For every $J$, $u_i(c^J)$ is quadratic:
\begin{align} u_i(c^J)= a c^J - \frac{1}{2} (c^J)^2. \label{eq:2_1} \end{align}
Let $p_i^J$ be the price of each good $J$ in country $i$. By \eqref{eq:2_1}, country $i$'s demand for good $J$ is given by
\begin{align} d_i^J(p_i^J)= a - p_i^J. \label{eq:2_2} \end{align}
Labor is only the production factor. Every one unit of numeraire good is produced by one unit of labor. We assume that labor supply is large enough to ensure strictly positive production, and therefore the equilibrium wage is equal to one.
Each non-numeraire good is produced from labor with diminishing returns. Country $i$'s production function for non-numeraire good $J$ is given by
\begin{align*} Q_i^J=\sqrt{2\lambda_i^J l^J} , \end{align*}
where $Q_i^J$ is country $i$'s production of good $J$ and $l^J$ is labor input for the production of good $J$.
Each country $i$ produces all $n$ goods $J=1, \cdots, n$.
By assuming an inner solution of the profit maximization, \footnote{The supply $s_i^J$ of good $J$ is determined by the labor input $l^J$ to maximize profit $p_i^J \sqrt{2\lambda_i^J l_J} - l^J$, where $s_i^J=\sqrt{2\lambda^J_il^J}$.} we obtain country $i$'s supply function of good $J$, given by
\begin{align} s_i^J(p_i^J)= \lambda_i^J p_i^J. \label{eq:2_3} \end{align}
With respect to the trade pattern in the world, each country $i$ has a comparative advantage in one good which is indexed by the upper letter $I$, and it has a comparative disadvantage in all other goods. In particular, we assume that for every $i$ and every good $J$ with $J \ne I$,
\begin{align} \lambda_i^I=1 + \lambda, \hspace{1em} \lambda_i^J=1 , \label{eq:2_4} \end{align}
where $\lambda > 0$ is the degree of comparative advantage.
Countries' comparative advantage structure implies that each country $i$ is a sole exporter of good $I$, and imports $n-1$ other goods $J \ne I$ from countries $j \ne i$. All countries except country $i$ compete with each other for importing good $I$ from country $i$. From this property, the economy is called the \emph{competing importers} model (Missios et al. 2016). \footnote{Saggi and Yildiz (2010) and Saggi et al. (2013, 2019) consider a ``competing exporters'' model where each country is a sole importer of a given good.}
Let $t_{ij}$ be a tariff imposed by country $i$ on its imports of good $J (\ne I)$ from country $j$. A vector $t=(t_{ij}: i, j=1, \cdots, n, i\ne j)$ is a tariff profile in the world. When consumers in country $j$ buy good $I$ imported from country $i$, they pay the price $p_i^I+ t_{ji}$, where $p_i^I$ is good $I$'s world price.
Let $p_j^I$ be country $j$'s domestic price of good $I$. By ruling out prohibitive tariffs, we assume the no-arbitrage condition for good $I$ for every country $j (\ne i)$:
\begin{align} p_j^I = p_i^I+ t_{ji} . \label{eq:2_5} \end{align} Country $j$'s domestic price $p_j^I$ of good $I$ is equal to the import price $p_i^I$ plus the tariff $t_{ji}$.
The world market clearing condition on each good $I$ is given by
\begin{align} \sum_{j=1}^n d_j^I(p_j^I)=\sum_{j=1}^n s_j^I(p_j^I) , \label{eq:2_6} \end{align}
where $d_j^I$ is country $j$'s demand of good $I$ and $s_j^I$ is country $j$'s supply (production) of good $I$.
Let $m_j^I$ be country $j$'s imports of good $I$ from country $i$, where
\begin{align} m_j^I(p_j^I)=d_j^I(p_j^I)- s_j^I(p_j^I) , \label{eq:2_7} \end{align}
and let $x_j^I$ be country $i$'s exports of good $I$ to country $j$, where
\begin{align} x_j^I(p_j^I)=s_i^I(p_i^I)- d_i^I(p_i^I) - \sum_{k \ne i, j}m_k^I(p_k^I) . \label{eq:2_8} \end{align}
From the market clearing condition \eqref{eq:2_6} for good $I$, country $i$'s exports to country $j$ are equal to country $j$'s imports from country $i$. That is,
\begin{equation} x_j^I(p_j^I)=m_j^I(p_j^I) . \label{eq:2_9} \end{equation}
Conversely, if \eqref{eq:2_9} holds for every country $j (\ne i)$, the world market clearing condition is satisfied.
A \textit{market equilibrium} under a tariff profile $t$ is defined by a collection of prices $p=(p_{i}^J: i, J=1, \cdots, n)$ which satisfies the no-arbitrage condition \eqref{eq:2_5} and the world market clearing condition \eqref{eq:2_6} for every country $i$ and every good $J$.
\noindent\textbf{Lemma~2.1.} The equilibrium prices under a tariff profile $t$ are given by
\begin{align} p_i^I=\frac{na - 2\sum_{j \ne i} t_{ji}}{2n + \lambda}, \hspace{1em} p_j^I=\frac{na - 2\sum_{k \ne i, j} t_{ki} + (2n-2+\lambda)t_{ji}}{2n + \lambda} . \label{eq:2_10} \end{align}
The lemma shows how a tariff profile of large countries affects the world and the domestic prices of every good. Ceteris paribus, the increase of country $j$'s tariff $t_{ji}$ on country $i(\ne j)$ has both the \emph{direct} and \emph{indirect} effects on prices. First, the tariff increase of $t_{ji}$ raises good $I$'s domestic price $p_j^I$ in country $j$ (the direct effect). Second, the tariff increase of $t_{ji}$ reduces good $I$'s world price $p_i^I$ and its domestic prices $p_k^I$ in all other countries $k(\ne i,j)$ (the indirect effect).
The above price changes caused by tariffs induce the two effects on the demand, supply, and imports of goods in the world. First, the tariff increase of $t_{ji}$ reduces the demand $d_j^I$ of good $I$ in country $j$, raises its supply $s_j^I$, and reduces its import $m_j^I$ (the direct effect). Second, the tariff increase of $t_{ji}$ raises the demand $d_k^I$ of good $I$ in a third country $k(\ne i,j)$, reduces its supply $s_k^I$, and raises its imports $m_k^I$ (the indirect effect).
We remark that the indirect effect is rooted in the law of supply that \eqref{eq:2_3} is upward-sloping.\footnote{By contrast, the frameworks of Cournot oligopoly and competing exporters assume a perfectly elastic supply function.} Since country $j$ attempts to keep its total supply by the law, the decrease of $j$'s exports to country $i$ is partially covered by the increase of $j$'s exports to a third country $k$.
The price changes by tariffs derived from Lemma~2.1 further show how FTA formation affects the trade pattern in the world. If two countries $i$ and $j$ forms an FTA to eliminate tariffs between them, it causes the following three effects, ceteris paribus. First, the FTA member countries $i$ and $j$ raise their imports of good $I$ and $J$, due to the domestic price decrease. Second, a third country $k(\ne i,j)$ reduces its imports of the goods $I$ and $J$, due to the domestic price increase. Third, FTA member countries' imports of good $K$ from country $k$ are unchanged because markets of different goods are independent in the model.
If a third country $k$ can adjust its tariffs on FTA member countries, then the effect of the FTA formation becomes more complicated. We will explore this issue in the next section by analyzing a tariff game after an FTA is formed.
Every country $i$'s welfare $W_i(t)$ under a tariff profile $t$ is defined as the sum of consumer surplus, producer surplus, and tariff revenue for all goods:
\begin{equation} W_i(t)=\sum_{J=1}^n CS_i^J(p_i^J) + \sum_{J=1}^n PS_i^J(p_i^J) + \sum_{j \ne i} t_{ij}m_i^J(p_i^J) , \label{eq:2_11} \end{equation}
where $CS_i^J$ is country $i$'s consumer surplus for good $J$ and $PS_i^J$ is its producer surplus for good $J$.
From \eqref{eq:2_1} and \eqref{eq:2_2},
\begin{equation} CS_i^J(p_i^J)=u_i(d_i^J(p_i^J))- p_i^J d_i^J(p_i^J)=\frac{1}{2}(a-p_i^J)^2 . \label{eq:2_12} \end{equation}
From \eqref{eq:2_3},
\begin{equation} PS_i^J(p_i^J)= \int_0^{p_i^J} s_i^J(p) d p = \frac{1}{2} \lambda_i^J (p_i^J)^2 . \label{eq:2_13} \end{equation}
By \eqref{eq:2_4}, \eqref{eq:2_11}--\eqref{eq:2_13}, $W_i(t)$ is calculated as
\begin{equation} W_i(t)=\frac{1}{2}\sum_{J=1}^n (a-p_i^J)^2 + \frac{1}{2}\{(1+\lambda)(p_i^I)^2 + \sum_{J \ne I} (p_i^J)^2 \} + \sum_{j \ne i} t_{ij}(a-2p_i^J) . \label{eq:2_14} \end{equation}
Finally, we remark that the model has the following simple structures: (i) the markets of different goods in every country are separated, and thus the economy is in a partial equilibrium set-up, (ii) every country exports a unique good, in which it has a comparative advantage, (iii) every country's tariff on one exporter never affects its imports from other exporters, (iv) the choice of a country's tariff on a given country is independent of its tariffs on other countries, (v) all countries are symmetric in demand, supply and comparative advantage structure. \footnote{We further prohibit all countries from tariff deflection.}
Thanks to these properties, the economy provides a tractable model to study trade liberalization in an inter-industry framework.
\section{The Non-cooperative Tariff Game}
We consider the following $n$-country non-cooperative tariff game. All countries $i$ simultaneously choose their tariffs $t_i=(t_{ij}: j=1, \cdots, n, j\ne i)$ to maximize the welfare $W_i(t)$ given by \eqref{eq:2_11}.
Given the tariff profile $t=(t_{i}: i=1, \cdots, n)$, the competitive equilibrium prices \eqref{eq:2_10} prevail.
Differentiating $W_i(t)$ with respect to $t_{ij}$ yields
\begin{equation} \frac{\partial W_i}{\partial t_{ij}} = \frac{\partial CS_i^J(p_i^J)}{\partial t_{ij}} + \frac{\partial PS_i^J(p_i^J)}{\partial t_{ij}} + m_i^J(p_i^J) + t_{ij} \frac{\partial m_i^J(p_i^J)}{\partial t_{ij}} , \label{eq:3_1} \end{equation}
where $m_i^J(p_i^J)$ is country $i$'s imports of good $J$. From \eqref{eq:2_7}, \eqref{eq:2_12} and \eqref{eq:2_13}, \eqref{eq:3_1} is rewritten as \footnote{The derivation is given in the supplementary material S.1.}
\begin{equation} \frac{\partial W_i}{\partial t_{ij}} = t_{ij} \frac{\partial m_i^J}{\partial p_i^J} \frac{\partial p_i^J}{\partial t_{ij}} - m_i^J \frac{\partial p_j^J}{\partial t_{ij}} . \label{eq:3_2} \end{equation}
Similarly to the effects on equilibrium prices mentioned in the last section, the tariff change has the following two effects on the welfare. We can interpret the first (negative) term of \eqref{eq:3_2} as the efficiency cost of the tariff. It gives country $i$'s marginal welfare loss from its own tariff $t_{ij}$ on good $J$. The marginal increase of the domestic price $p_i^J$ of good $J$, or \emph{the pass through} of the tariff, reduces the imports. The second (positive) term of \eqref{eq:3_2} can be interpreted as the effect of terms of trade. Since country $i$ imports good $J$, $i$'s welfare is affected by the marginal decrease of the world price $p_j^J$ of good $J$, or \emph{the terms of trade gain} of the tariff, due to $i$'s tariff $t_{ij}$ on country $j$.
From \eqref{eq:2_14}, \begin{equation} \frac{\partial W_i}{\partial t_{ij}} = -2t_{ij} + \frac{2}{2n+\lambda}(a-2p_j^J) , \label{eq:3_3} \end{equation} by using $\frac{\partial p_i^J}{\partial t_{ij}}=\frac{2n-2+\lambda}{2n+\lambda}$ (see \eqref{eq:2_10}).
The derivation of \eqref{eq:3_3} is given in the proof of Theorem~3.1. We note that the second term of \eqref{eq:3_3} is positive. If tariff $t_{ij}$ is low, a unilateral increase of $t_{ij}$ benefits country $i$. Otherwise, it may be harmful.
By assuming the first-order condition, for every $j(\ne i)$, \eqref{eq:3_3} implies that the optimal tariff $t_{ij}$ of country $i$ is given by
\begin{align} t_{ij} = \frac{1}{2n+\lambda}(a-2p_j^J) . \label{eq:3_4} \end{align}
We observe from \eqref{eq:3_4} that the optimal tariff $t_{ij}$ on good $J$ is independent of $i$, i.e.\ the tariff $t_{ij}$ is identical across importing countries $i$ of good $J$. This property is derived from the symmetry assumption of the economy. In what follows, we will show the same kind of tariff symmetry when an FTA forms. For example, member countries in an FTA impose an identical external tariff on non-members. \eqref{eq:3_4} also implies that every country $i$'s optimal tariff $t_{ij}$ on good $J$ is decreasing in the world price $p_j^J$.
Substituting \eqref{eq:2_10} into \eqref{eq:3_4} yields country $i$'s best response function,
which assigns $i$'s optimal tariff $t_{ij}$ on $j$ to all other countries' tariffs $(t_{kj})_{k\ne i,j}$, as follows.
\begin{equation} t_{ij}= \frac{a \lambda + 4 \sum_{k \ne i, j}t_{kj} }{ (2n+\lambda)^2-4 } . \label{eq:3_5} \end{equation}
The best response function \eqref{eq:3_5} shows that tariffs are \emph{strategic complements}. If other countries raise their tariffs on country $j$, country $i$ also raises its tariff on country $j$. The complementarity is caused by the fact that country $i$'s marginal welfare with respect to $i$'s tariff is increasing in country $k$'s tariff.\footnote{Formally, the strategic complementarity of tariffs is derived by $\frac{\partial^2 W_i}{\partial t_{ij}^2}<0$ and $\frac{\partial^2 W_i}{\partial t_{ij} \partial t_{kj}}>0$.}
We now obtain the following theorem.
\noindent \textbf{Theorem 3.1.} There exists a unique Nash equilibrium in the non-cooperative tariff game. All countries impose the common tariff $t^{\mathrm{NE}}$ on the imports of every good $J$,
\begin{align}
t^{\mathrm{NE}} = t_{ij} = \frac{a\lambda}{\lambda^2+4n\lambda+4(n^2-n+1)} . \label{eq:3_6} \end{align}
The Nash equilibrium of the tariff game describes the ``tariff war'' where all countries impose (strictly) positive tariffs on each other to maximize their individual welfare. Country $i$'s positive tariff $t_{ij}$ on the imports of good $J$ raises the domestic price $p_{i}^J$ of good $J$. This price increase reduces the consumer surplus and the imports of good $J$, and raises the producer surplus.
While the effect on the tax revenue is ambiguous due to the import reduction, the best response \eqref{eq:3_5} shows that country $i$ optimally imposes a positive tariff, regardless of the other countries' tariffs.
It is worth noting that the Nash equilibrium tariff $t^{\mathrm{NE}}$ converges to zero as the number of countries goes to infinity.
To conclude this section, we show that the Nash equilibrium is Pareto inefficient and that the global free trade is efficient to maximize the total (world) welfare.
From \eqref{eq:2_11}, the total welfare $W$ is given by
\begin{align*} W &=\sum_{i=1}^n W_i \notag \\
&=\sum_{i=1}^n\sum_{J=1}^n \left[ CS_i^J(p_i^J) + PS_i^J(p_i^J) \right]
+ \sum_{i=1}^n \sum_{j \ne i}^n t_{ij}m_i^J(p_i^J). \end{align*}
To maximize the total welfare $W$, every country $i$ takes into account how its tariff $t_{ij}$ on good $J$ affects not only its own welfare $W_i$, but also all other countries' welfare.
Similarly to \eqref{eq:3_3}, \eqref{eq:2_10} and \eqref{eq:2_14} give the effects of country $i$'s tariff $t_{ij}$ on others' welfare, given by
\begin{align} \frac{\partial W_j}{\partial t_{ij}} &= \frac{2}{2n+\lambda}(a-(2+\lambda)p_j^J) \label{eq:3_7} \\ \frac{\partial W_k}{\partial t_{ij}} &= \frac{2}{2n+\lambda}(a-2p_j^J) \label{eq:3_8} \end{align} for every $k (\ne j)$. \footnote{The derivation is given in the supplementary material S.2.} If country $i$'s comparative advantage degree $\lambda$ is low, an increase of country $i$'s tariff $t_{ij}$ on country $j$ may benefit country $j$, owing to the producer surplus increase. The increase of $t_{ij}$ always benefits a third country $k$ (i.e.\ $\frac{\partial W_k}{\partial t_{ij}} >0$).
The effect of country $i$'s tariff $t_{ij}$ on the total welfare $W$ is given by
\begin{align} \frac{\partial W}{\partial t_{ij}} &= \sum_{k=1}^n \frac{\partial W_k}{\partial t_{ij}} \notag \\ &= -2t_{ij} + \frac{2}{2n+\lambda} [na-(\lambda+2n)p_j^J] . \label{eq:3_9} \end{align}
Substituting \eqref{eq:2_10} into \eqref{eq:3_9} yields
the first-order condition for country $i$'s optimal tariff on good $J$ to maximize the total welfare, which is given by \begin{equation} t_{ij} = \frac{2}{2n+\lambda}\sum_{k \ne j}t_{kj} . \label{eq:3_10} \end{equation}
We can easily show that \eqref{eq:3_10} has a unique solution $t_{ij}=0$ for every $i$ and $j \ne i$.
Thus, we have the following theorem. \footnote{In the supplementary material S.3, we prove part (i) without assuming the first-order condition.}
\noindent \textbf{Theorem 3.2.} \begin{enumerate}[(i)]
\item Global FTA uniquely maximizes the total world welfare.
\item Every country $i$'s welfare under global FTA is \begin{equation*} \frac{na^2(n+\lambda)}{2(2n+\lambda)}. \end{equation*}
\item Every country $i$'s Nash equilibrium welfare under no FTA is \begin{equation*}
\frac{na^2(n+\lambda)}{2(2n+\lambda)} - \frac{(n-1)(2+\lambda)}{2n+\lambda} (t^{\mathrm{NE}})^2 , \end{equation*} where $t^{\mathrm{NE}}$ is the Nash equilibrium tariff in \eqref{eq:3_6}. \end{enumerate}
The theorem shows that our model satisfies the standard property of the literature on international trade, namely, global free trade is the efficient regime. Since global free trade is not a Nash equilibrium, each country has an incentive to raise its tariffs unilaterally. Such a self-interested behavior, however, harms other countries' welfare.
By Theorem~3.2, we observe how the model parameters affect the welfare of countries. Each country's welfare under global FTA (and under no FTA) increases as each of the number $n$ of countries, the demand $a$ and the degree $\lambda$ of comparative advantage increases.
We also observe that every country's welfare is monotonically decreasing in a common tariff of all countries (see \eqref{Aeq:3_2} in Appendix~A.3). Therefore, it is optimal for every country to agree to global FTA \emph{under the constraint that all countries in the world establish identical tariffs}. This fact theoretically supports the WTO's traditional approach to achieving global free trade under the most favored nation (MFN) principle and the reciprocity (Baldwin, 2016).
\section{Free Trade Agreements}
We now investigate the endogenous formation of an FTA whose members are bound to set zero tariffs on each other. The members impose independently their external tariffs on non-members.
We present a sequential game where an FTA may emerge and grow. In the game, a non-member country can access freely to an incumbent FTA for participation. By this property, we call the process an \emph{open-access rule}.\footnote{Seidmann (2009) calls an open-access rule with consent in our terminology a closed-access rule, and calls an open-access rule without consent an open-access rule.} The game proceeds with the following three stages.
\begin{enumerate} \item All $n$ countries decide sequentially whether they participate in an FTA, according to a fixed order of moves. Without loss of generality, let the order be $(1, \cdots, n)$.
\item If a new country decides to participate in an FTA, then all incumbent members either accept or reject its participation simultaneously. The participation is approved if all incumbents unanimously accept it. \footnote{Since countries are symmetric in our model, the result is not affected critically if unanimity rule is replaced with majority rules.} When there is no incumbent, any participant forms an FTA unilaterally, and succeeding countries decide to participate in it, or not.
\item After a new FTA forms, all inside and outside countries choose their external tariffs independently. The internal tariff of the FTA is zero. \end{enumerate}
An important feature of our FTA formation game is that a new participation needs incumbent members' unanimous consent. We call this property an \emph{open-access rule with consent}. The game aims to model the formation process of open regionalism FTAs including APEC, as we have discussed in Introduction. Whenever each country moves, it knows perfectly all the previous other countries' choices.
We will characterize a subgame perfect equilibrium (SPE) of the game to answer the following questions: (i) how the formation of FTAs affects tariff setting of member and non-member countries, (ii) how FTAs affect the welfare of member and non-member countries.
We first examine countries' tariff setting behavior under the formation of FTAs, which may be overlapped. Suppose that several FTAs, $F_1, \cdots, F_m$, are formed, where each $F_i$ is a subset of countries. All member countries of each FTA impose zero internal tariffs. Neither of two FTAs is a subset of the other.\footnote{If $F_i$ is a subset of $F_j$, then the free trade agreement of $F_j$ includes that of $F_i$, and thus $F_i$ is redundant.} Two different FTAs can have common members, i.e.\ countries may participate in more than one FTA.
We call a collection $(F_1, \cdots, F_m)$ an \textit{FTA regime} and denote it by $\mathcal{F}$. The non-cooperative tariff game in the last section where no country participate in an FTA corresponds to the FTA regime $(\{1\}, \cdots, \{n\})$.
For every country $k$, let $N_k$ be the set of countries with which country $k$ does not have any FTA, and let $n_k \ (0\leq n_k\leq n-1)$ be the number of countries in $N_k$. As $n_k$ increases, country $k$ participates in less FTAs. All countries $i$ in $N_k$ choose independently their tariffs on country $k$ to maximize their own welfare. We denote country $i$'s external tariff on country $k$ under an FTA regime $\mathcal{F}$ by $t_{ik}^{\mathcal{F}}$. The number $n_k$, which determines the equilibrium tariff $t_{ik}^{\mathcal{F}}$ in Theorem~4.1, plays a key role.
The first-order condition \eqref{eq:3_4} of the welfare maximization of country $i\in N_k$ shows that $i$'s tariff $t_{ik}^{\mathcal{F}}$ on country $k$ is independent of $i$.
Then, from \eqref{eq:3_5}, \begin{align}
t_{ik}^{\mathcal{F}} = \frac{1}{ (2n+\lambda)^2-4}
\left[ a \lambda + 4 (n_k-1)t_{ik}^{\mathcal{F}} \right]. \label{eq:4_1} \end{align}
\eqref{eq:4_1} holds true because all countries $i$ in $N_k$ impose the identical external tariff $t_{ik}^{\mathcal{F}}$ on country $k$, and all countries outside $N_k$ impose zero tariff on country $k$.
Solving \eqref{eq:4_1}, we obtain the following result.
\noindent \textbf{Theorem 4.1.} Let $\mathcal{F}=(F_1, \cdots, F_m)$ be an FTA regime, and for every $k=1,\cdots,n$, let $N_k$ be the set of countries with which country $k$ does not have any FTA. Then, every country $i$ in $N_k$ imposes the external tariff $t_{ik}^{\mathcal{F}}$ on country $k$ such that \begin{align*}
t_{ik}^{\mathcal{F}} = \frac{a\lambda}{\lambda^2+4n\lambda+4(n^2-n_k)} , \end{align*} where $n_k$ is the number of countries in $N_k$. \footnote{Saggi et al.\ (2018, Appendix~A) derive FTA non-member countries' external tariffs on member countries when a single FTA is formed.}
\noindent The theorem provides an answer to our question (i) of how the formation of FTAs affects the tariff setting behavior of countries. Given the model parameters $(n,a,\lambda)$, country $i$'s external tariff $t_{ik}^{\mathcal{F}}$ on each country $k$ is solely determined by $n_k$, the number of countries with which country $k$ does not have FTAs.
The external tariff $t_{ik}^{\mathcal{F}}$ is monotonically increasing in $n_k$. This property reveals the following two important effects of FTAs on the tariff setting of member and non-member countries. First, all FTA member countries impose the same external tariffs on non-member country $k$ as before its formation, provided that $n_k$ does not change. Thus, the model satisfies the Article XXIV of GATT to prohibit PTAs from increasing their external tariffs on non-member countries. Second, suppose that an FTA is formed. Then, non-member countries $k$ reduce their external tariffs $t_{ki}^{\mathcal{F}}$ on member countries because the number $n_i$ decreases owing to the FTA formation. This \emph{tariff complementarity} is implied by the best response function \eqref{eq:3_5} in the tariff game.
Theorem~4.1 also shows that the external tariff on non-members is independent of an FTA size.
If country $k$ participates in no FTA ($n_k=n-1$), the external tariff $t_{ik}^{\mathcal{F}}$ on country $k$ is equal to the Nash equilibrium tariff $t_k^{\mathrm{NE}}$ in the non-cooperative tariff game. The theorem implies that the Nash equilibrium tariff is the highest one among all tariffs under FTA regimes.
When an FTA is formed, internal tariffs of the FTA are set to be zero, and non-member countries adjust their external tariffs on the FTA. The next theorem shows how those tariff changes of the FTA formation affect the trade pattern in the world.
\noindent \textbf{Theorem 4.2.} Let $\mathcal{F}^0=(\{1\}, \cdots, \{n\})$ and $\mathcal{F}^i=(F,\{m+1\},\cdots,\{n\})$, where $m$ is the size of an FTA $F$ and $i\in F$. Let $p_j^{I,\mathcal{F}}$ and $m_j^{I,\mathcal{F}}$ be the price and imports of good $I$ in country $j$ under an FTA regime $\mathcal{F}\in\{\mathcal{F}^0,\mathcal{F}^i\}$, respectively. Then, \begin{enumerate}[(i)]
\item $p_j^{I,\mathcal{F}^0} > p_j^{I,\mathcal{F}^i}$ and $m_j^{I,\mathcal{F}^0} < m_j^{I,\mathcal{F}^i}$ for every member $j\in F$ ($j\neq i$),
\item $p_k^{I,\mathcal{F}^0} < p_k^{I,\mathcal{F}^i}$ and $m_k^{I,\mathcal{F}^0} > m_k^{I,\mathcal{F}^i}$ for every non-member $k\not\in F$. \end{enumerate}
Theorem~4.2 shows that an FTA $F$ with a member country $i$ reduces the domestic prices of good $I$ in other member countries $j$ from country $i$, and raises their imports of good $I$. The FTA formation has the opposite effect on non-member countries $k$, i.e.\ it raises their domestic prices of good $I$ and reduces their imports.
As the equilibrium price \eqref{eq:2_10} shows, the domestic price of the imported goods from an FTA member country in a non-member country is affected not only by its own tariff but also by other non-member countries' tariffs. It is increasing in the former and decreasing in the latter.
All non-member countries' tariffs on the FTA member countries are identical due to the symmetry assumption. It turns out that the marginal increase of the domestic price with respect to non-members' tariff is equal to $\frac{2m+\lambda}{2n+\lambda}$, where $m$ is the size of the FTA (see \eqref{Aeq:4_4}). Non-member countries' tariff is decreasing in the FTA size $m$ due to the strategic complementarity shown by the best response function \eqref{eq:3_5}.
Combining these two effects, we observe that the domestic price of the imported goods in a non-member country is increasing in the FTA size. In particular, the FTA formation $(m>1)$ induces its domestic price increase of goods imported from member countries, compared to the case of no FTA ($m=1)$.
The domestic price increase reduces non-member countries' imports from an FTA. Saggi et al.\ (2018) call this phenomenon the \emph{external trade diversion}. They empirically support this theoretical prediction, using industry-level data on all FTAs formed in the world during 1989-2011.
We now proceed to our question (ii) of how FTAs affect the welfare of both member and non-member countries. To answer the question, we will evaluate every country's welfare under all possible FTA regimes. In the following lemma, we first derive an explicit formula of every country's welfare under any tariff profile.\footnote{Since the derivation of \eqref{eq:4_2} is tedious, we prove the lemma in the supplementary material S.4.}
\noindent \textbf{Lemma 4.1.} Let $r=2(2n+\lambda)^2$. Every country $i$'s welfare $W_i(t)$ under a tariff profile $t=(t_{ij}: i, j=1, \cdots, n, i\neq j)$ is given by \begin{align}
rW_i(t) =& na^2(n+\lambda)(2n+\lambda) \notag \\ &+ \sum_{j\ne i} t_{ij} \left[ 4a\lambda+16\sum_{k\ne i,j}t_{kj} \right] - \sum_{j\ne i} (t_{ij})^2 \left[ 2 (2n+\lambda)^2-8 \right] \notag \\ &+ 4(2+\lambda)\left( \sum_{k\ne i}t_{ki} \right)^2 - 4a\lambda(n-1) \sum_{k\ne i}t_{ki} \notag \\ &+ 4a\lambda \sum_{j\ne i} (\sum_{k\ne i,j} t_{kj}) + 8 \sum_{j\ne i} (\sum_{k\ne i,j} t_{kj})^2. \label{eq:4_2} \end{align}
Employing \eqref{eq:2_10} in Lemma~2.1, we can compute every country's welfare under an FTA regime $\mathcal{F}=(F_1, \cdots, F_m)$, denoted by $W_i^\mathcal{F}(t)$, where $t$ is the equilibrium tariff profile under $\mathcal{F}$. All countries $j$ in $N_i$, who have no FTAs with country $i$, impose positive tariffs $t_{ji}$ on country $i$. From Theorem 4.1, the tariff $t_{ji}$ is solely determined by the number $n_i$, which is the size of $N_i$, and thus it is independent of index $j$. In the following, we denote $t_{ji}$ by $t_{i}$, whenever no confusion arises. Then, \eqref{eq:4_2} is rewritten as \begin{align} rW_i^{\mathcal{F}}(t) =& na^2(n+\lambda)(2n+\lambda) + \sum_{j\in N_i} \left[4a\lambda+16(n_j-1)t_{j}\right]t_j \notag \\ &-\left[ 2 (2n+\lambda)^2-8 \right]\sum_{j \in N_i} (t_j)^2 + 4(2+\lambda)\left( n_i t_{i} \right)^2 - 4a\lambda(n-1) n_it_{i} \notag \\ &+4a\lambda \sum_{j\in N_i} (n_j-1) t_{j}+4a\lambda \sum_{j\notin N_i, j\neq i} n_j t_{j} \notag \\ &+8 \sum_{j\in N_i} ((n_j-1) t_{j})^2 + 8 \sum_{j\notin N_i, j\neq i} (n_j t_{j})^2 . \label{eq:4_3} \end{align}
\eqref{eq:4_3} reveals how each country's welfare depends on a FTA network. It depends not only on the number of countries with which a country is connected via FTAs, but also on the number of countries with which its partner countries are further connected. We will detail every country's welfare to investigate the endogenous FTA formation, which is our central issue.
By the game's rule, at most a single FTA can form. Without loss of generality, we denote an FTA by $F=\{1, \cdots, m\}$, where $1\leq m \leq n$. Thus, a possible FTA regime is given by $\mathcal{F}^m=(F, \{m+1\}, \cdots, \{n\})$. For every member $i \in F$, $n_i=n-m$, and for every non-member $j$, $n_j=n-1$.
As shown in Theorem 4.1, country $i$'s external tariff $t_{ij}$ against country $j$ is independent of $i$. In what follows, we denote the external tariff against every participant ($\mathit{p}\in F$) in the FTA by $t_{p}$, and the external tariff against every non-participant ($\mathit{np}\notin F$) in the FTA by $t_{np}$. By Theorem 4.1, under an FTA regime $\mathcal{F}^m =(F, \{m+1\}, \cdots, \{n\})$, the external tariff on every participant of the FTA is given by \begin{align*} t_{p} &= \frac{a\lambda}{\lambda^2+4n\lambda+4(n^2-n+m)} , \intertext{ and the external tariff on every non-participant is given by } t_{np} &= \frac{a\lambda}{\lambda^2+4n\lambda+4(n^2-n+1)} . \end{align*}
We also denote every participant's welfare by $W_{p}^m(t)$, and every non-participant's welfare by $W_{np}^m(t)$ under the FTA regime $\mathcal{F}^m$.
\noindent \textbf{Lemma 4.2.} Let $r=2(2n+\lambda)^2$ and $m$ be the number of FTA participants. Every participant's welfare $W_{p}^m(t)$ is given by \begin{align} rW_{p}^m(t) =& na^2(n+\lambda)(2n+\lambda) + 4(2m+\lambda)(n-m)^2 t_{p}^2 \notag \\ & - 4a\lambda(n-m)^2t_{p} \notag \\ & + 2(n-m)\left[4-8n-4n\lambda - \lambda^2 \right]t_{np}^2 \notag \\ & + 4a\lambda (n-m)(n-1)t_{np} . \label{eq:4_4} \end{align}
Every non-participant's welfare $W_{np}^m(t)$ is given by \begin{align} rW_{np}^m(t) =& na^2(n+\lambda)(2n+\lambda) \notag \\ & -2m\left[\lambda^2+4n\lambda+8mn-4m^2\right]t_{p}^2 \notag \\ & -2\left[(n-m-1)\lambda^2-2(2mn-n^2+1)\lambda+4(n^2-2mn-n+m)\right]t_{np}^2 \notag \\ & +4a\lambda m(n-m)t_{p}-4a\lambda m(n-1)t_{np} . \label{eq:4_5} \end{align}
\noindent The lemma follows from Lemma 4.1. Since the derivation is tedious, we prove the lemma in the supplementary material S.5. In what follows, we omit the equilibrium tariff profile $t$ in notations $W_{p}^m(t)$ and $W_{np}^m(t)$, and simply write $W_{p}^m$ and $W_{np}^m$ whenever no confusion arises.
The next two theorems show how FTA formation affects countries' welfare.
\noindent\textbf{Theorem 4.3.} $W_{p}^{m+1}>W_{np}^m$ for every $m=1, \cdots, n-1$.
The theorem shows that every non-participant's welfare improves if it joins an FTA. Thus, every non-participant has an incentive to join an FTA. We can explain its intuition as follows. The FTA growth removes all incumbent participants' tariffs on a new participant. The tariff complementarity, shown in Theorem~4.1, reduces non-participant's tariffs on a new participant. Thus, a new participant can import at lower prices from all other countries, and then its welfare improves.
By contrast, the FTA growth has mixed effects on incumbent participants' welfare. First, since incumbents' tariffs on non-participants and their imports from non-participants are constant, the FTA growth does not change their welfare yielded from non-participants. Second, incumbents' tariff elimination on a new participant reduces their tariff revenue from it. Since the incumbents' imports from a new participant increase due to the direct effect of the tariff elimination, shown in Lemma~2.1, their consumer surplus yielded from a new participant rises. On the other hand, by their domestic price reduction due to the direct effect, the tariff elimination reduces their domestic supplies and producer surplus from a new participant.
Third, the tariff elimination by the FTA reduces each incumbent's consumer surplus and raises producer surplus yielded from other incumbents. Since the tariff complementarity decreases non-participants' tariffs on incumbents, its indirect effect reduces the imports among incumbents and raises the domestic supplies. The sum of these mixed effects by the FTA growth either raises or reduces incumbent participants' welfare.
The next theorem, however, shows that an FTA harms non-participants' welfare.
\noindent\textbf{Theorem 4.4.} $W_{np}^m$ is monotonically decreasing in FTA size $m$.
By the theorem, FTA formation has a negative externality to non-participants. The external trade diversion can explain its intuition. As shown in Lemma~2.1, the indirect effect of incumbent participants' tariff elimination reduces non-participants' imports from a new participant. Since non-participants produce those goods with their disadvantageous technology, the FTA growth reduces their consumer surplus. Furthermore, since the tariff complementarity reduces non-participants' external tariffs on participants, their tariff revenues also reduce.
By contrast, since all countries' external tariffs on a non-participant are unchanged, non-participants do not change their exports to all countries. Thus, the FTA growth does not change their producer surplus. In total, the FTA growth reduces every non-participant's welfare.
We are now ready to characterize an SPE under the open-access rule with consent. For every $i=1, \cdots, n$, we call the stage game where country $i$ decides to participate in the FTA the \textit{$i$-th stage game}.
The open-access rule with consent has the unanimous voting stage by incumbents. It is well-known that the simultaneous-move unanimous voting game has multiple Nash equilibria, which may involve weakly dominated strategies for voters. For example, the situation that all members reject new participation is a trivial Nash equilibrium of the unanimous voting game, while ``No'' vote may be weakly dominated by ``YES'' vote for each member. To avoid this multiplicity, we restrict the analysis to an SPE with no weakly dominated strategies.\footnote{An alternative way to avoid the multiple Nash equilibria in the unanimity game is to employ a sequential-move game rather than a simultaneous-move game.}
\noindent\textbf{Theorem 4.5.} Suppose that the each participant's welfare $W_{p}^m$ in an FTA of size $m$ is maximized at $m^* \in \{2,\cdots, n\}$. Then, the open-access rule with consent has a unique SPE outcome, where only the first $m^*$ countries participate in an FTA.
The theorem shows that the open-access rule with consent does not lead to the global free trade unless it maximizes every participant's welfare. If the FTA size exceeds the optimal level of the participants' welfare, it is beneficial for them to reject new participantion because their welfare decrease, otherwise. If the FTA size is below the optimum, the incumbents accept a new participant, anticipating rationally that the optimal FTA size will attain in the subsequent process.
The following theorem shows that the equilibrium FTA size $m^*$ is strictly less than the global free trade, and is roughly $n/2$ for sufficiently large $n$.
\noindent\textbf{Theorem~4.6.} The equilibrium FTA size $m^*$ under the open-access rule with consent is strictly less than $n$. Furthermore, if $n$ is sufficiently large, $m^*$ is in $\{\frac{n}{2}-1$, $\frac{n}{2}, \frac{n}{2}+1\}$ for even $n$, and in $\{\frac{n-1}{2}, \frac{n+1}{2}\}$ for odd $n$.
The unique SPE under the open-access rule with consent is consistent with our observation in the current international economy that open regionalism FTAs like APEC do not lead to global FTA. Complicated negotiations take place between new applicants and incumbent members, and new participation has not been accepted.
To implement open regionalism, we propose an open-access rule to accept any participant without consent of incumbent members. This new rule eliminates the unanimous voting stage in the FTA formation game. We call it an \emph{open-access rule without consent}. The open-access rule without consent is along the spirit of GATT Article I of General Most-Favoured-Nation Treatment.
We finally obtain the following result.
\noindent\textbf{Theorem 4.7.} The global free trade is a unique SPE outcome under the open-access rule without consent. The SPE outcome is independent of an order of moves.
The theorem is derived from the fact that every non-participant is willing to join in an FTA, regardless of its size (Theorem 4.3). The unique SPE captures a preferable process that an FTA emerges and grows to the global free trade in sequential negotiations. The theorem provides a theoretical support for the policy recommendation that the WTO should require any PTA to be open-access without incumbents' consent.
\section{A Numerical Example}
We provide a numerical example of the result when $n=8$, $a=12$ and $\lambda=36$. Table~1 gives the values of seven variables, \footnote{Notice that all the values are rounded to no more than six significant figures.} (i) tariff $t_{p}$ on a participant of an FTA imposed by a non-participant, (ii) tariff $t_{np}$ on a non-participant imposed by a participant and another non-participant, (iii) participant's import $m_{p}^{p}$ from a participant in an FTA, (iv) non-participant's import $m_{np}^{p}$ from a participant, (v) participant's import $m_{p}^{np}$ from a non-participant, (vi) participant's welfare $W_p$, and (vii) non-participant's welfare $W_{np}$, for each FTA size $m=1, \cdots, 8$. Non-participant's import $m^{np}_{np}$ from a non-participant is equal to $m^{np}_p$. Figure~1 depicts the welfare of a participant and of a non-participant.
\begin{table}
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head=false,
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\caption{The effect of an FTA size at parameters $(n,a,\lambda)=(8,12,36)$. Notations: $m$ (FTA size), $t_{p}$ (tariff on a participant), $t_{np}$ (tariff on a non-participant),
$m_{p}^{p}$ (participant's import from a participant), $m_{np}^{p}$ (non-participant's import from a participant), $m^{np}_p$ (participant's import from a non-participant), $W_p$ (participant's welfare), $W_{np}$ (non-participant's welfare).}
\label{fig:numerical_values} \end{table}
We observe the following from Table 1: (1) the external tariff on participants in the FTA decreases as the FTA size grows, (2) the external tariff on a non-participant is constant on the FTA size, (3) the trade between two participants decreases as the FTA grows, (4) non-participant's imports from the FTA decreases as the FTA grows, (5) if a non-participant joins an FTA, then its imports from a participant increase, (6) non-participant's exports to both a participant and a non-participant are constant in the FTA size, (7) the participant's welfare is maximized at $m^*=5$, (8) non-participant's welfare is monotonically decreasing in the FTA size, and (9) a non-participant is better-off by joining the FTA. Those observations are consistent with our results.
Our main result shows that the largest FTA (global free trade) forms under the open-access rule without consent, while only five countries form an FTA under the open-access rule with consent. Under the global FTA, the welfare of every country is $487.385$. Under the open-access rule with consent, each participant of the equilibrium FTA obtains the highest welfare $487.942$, but the welfare of each non-participant is $486.209$.
\begin{figure}
\caption{The welfare of a country. Note: The circle symbol depicts the welfare of a participant and the diamond symbol depicts the welfare of a non-participant. The $x$-axis designates an FTA size $m$.}
\label{fig:numerival_graph}
\end{figure}
In the numerical example, first five countries form an FTA under the open-access rule with consent. For example, if the order of moves is $(1, 2, 3, 4, 5, 6, 7, 8)$, the FTA regime is $\mathcal{F}=(\{1, 2, 3, 4, 5\}, \{6\}, \{7\}, \{8\})$ in a unique SPE. Participation of the last three countries $6, 7$ and $8$ is rejected by the incumbents.
It is conceivable that three non-member countries attempt to form the second FTA. In what follows, we argue that they actually have an incentive to form the second FTA.
We assume that after the first FTA with five countries form, three countries 6, 7, and 8 play the sequential participation game under the open-access rule with consent. We evaluate countries' welfare under the following three FTA regimes (ignoring permutations): \begin{enumerate}[(i)]
\item $(\{1,2,3,4,5\},\{6\},\{7\},\{8\})$,
\item $(\{1,2,3,4,5\},\{6, 7\},\{8\})$,
\item $(\{1,2,3,4,5\},\{6, 7,8\})$. \end{enumerate} We simply denote these three FTA regimes by $(5, 1, 1, 1)$, $(5, 2, 1)$ and $(5, 3)$, respectively. Table 2 gives participant's welfare $W_{\alpha}$ of the first FTA, participant's welfare $W_{\beta}$ of the second FTA, and a non-participant's welfare $W_{\gamma}$. The computation is given in the supplementary material S.10.
We observe from Table 2 that participant's welfare of the second FTA increases as the FTA grows. Therefore, similarly to Theorem~4.6, we can show that the three countries $6, 7$ and $8$ form the second FTA $\{6,7,8\}$ under the open-access rule with consent.
Note that participant's welfare of the first FTA decreases as the second FTA grows. However, their welfare is $487.624$, which is still higher than the welfare under the global FTA. Thus, the participants of the first FTA are not motivated to create the global FTA by merging with the second FTA, and thus the regime $(5,3)$ is realized.
Finally, we consider a possibility of non-overlapping FTAs. We here examine whether or not a pair of a participant of the first FTA and a participant of the second FTA in the example have an incentive (in the short run) to form a bilateral FTA between themselves.
A numerical computation shows that participant's welfare of the first FTA is $487.958$ and participant's welfare of the second FTA is $487.091$ after the creation of their bilateral FTA. Since the bilateral FTA improves both participants' welfare than under the FTA regime $(5, 3)$, this means that it is not pairwise stable in a network formation game (Jackson and Wolinsky 1996). We further discuss the difference between our approach and the network stability approach in the next section.
\begin{table}
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\caption{The effect of two FTAs at parameters $(n,a,\lambda)=(8,12,36)$. Notations: $m$ (regime), $t_{\alpha}$ (tariff on a participant in the large FTA), $t_{\beta}$ (tariff on a participant in the small FTA), $t_{\gamma}$ (tariff on a non-participant), $W_{\alpha}$ (the large FTA participant's welfare), $W_{\beta}$ (the small FTA participant's welfare), $W_{\gamma}$ (non-participant's welfare).}
\label{fig:numerical_values2} \end{table}
\section{Discussion}
\subsection{Open Regionalism}
We discuss our results on the formation of global free trade agreement under open-access rules, comparing with the works by Yi (1996, 2000), Seidmann (2009), and Mukunoki and Tachi (2006).
Yi (1996) studies a custom union (CU) formation in a Cournot oligopoly model with linear demand and costs. He presents an open regional game, both with simultaneous moves and with sequential moves. He shows that the grand CU is a unique (pure strategy) Nash equilibrium outcome under the simultaneous-move open regionalism rule.\footnote{Yi (2000) applies the same model to the FTA case. He shows that the grand FTA may not be a Nash equilibrium due to a free-riding problem.}
His open regionalism rule is similar to our open-access rule without consent, while our model is a sequential-move game. In his simultaneous-move game, countries simultaneously announce an ``address.'' The countries choosing the same address form a CU. When all countries announce different addresses, it is interpreted as a situation where every country forms a single-member CU by itself.
Yi's game enables every country to form a two-member CU with any other country by simply choosing the same address. By contrast, when no country participates in an FTA under our open-access rule, any unilateral deviation by one country does not lead to a two-member FTA.
The critical difference in equilibrium outcomes between Yi (1996) and us lies in that there are multiple SPE outcomes in his sequential-move open regionalism game, while there is a unique SPE outcome, namely the global FTA, in our game. A peculiar property of his address-announcing game causes multiple SPE outcomes.
Consider the following three-country example. The welfare of each country under a possible FTA regime is given as follows: (i) when there is no FTA, every country obtains payoff zero, (ii) when there is a two-member FTA, each participant obtains payoff 2 and a non-participant obtains payoff $-1$, (iii) when there is the global FTA, all members obtain payoff 1. Our open-access rule without consent has a unique SPE where three countries join the global FTA. The open-access rule with consent has a unique SPE where the first two countries participate in an FTA and reject the participation of the third country.
By contrast, Yi's sequential-move open regionalism game has two SPEs. In one SPE, three countries choose the same address, leading to the grand FTA. Whatever address the first country chooses, two succeeding countries coordinate their choices with the first country's one. As a result, the first country can join the global FTA, no matter what address it chooses. In the other SPE, the last two countries form a two-member FTA. Whatever address the first country chooses, two succeeding countries choose the same address, different from the first country's one. The last two countries form a two-member FTA without the first country, regardless of its choice. A coordination problem arises in Yi's sequential model.
Seindmann (2009) considers another sequential model for trade liberalization under an open-access rule with consent in a three-country setup. In contrast to Yi's (1996) paper and ours, he allows utility transfer and the formation of a bilateral FTA, a bilateral CU, two overlapping FTAs, as well as the global FTA. PTAs can be renegotiated toward the global FTA. He demonstrates that patient enough countries may initially form a two-member PTA (possibly a hub-and-spoke regime), which will grow to the global FTA. His result reveals a different motive of a PTA from ours caused by utility transfer. PTA members strategically manipulate the status quo, seeking advantageous positions in future negotiations (strategic positioning).
Finally, Mukunoki and Tachi (2006) study a sequential bilateral FTA formation game in a three-country oligopoly model with exogenous tariffs, where all firms produce homogeneous goods. Each pair of two countries negotiate for a bilateral FTA, or a link, in a fixed order in their game. They show that each country has an incentive to become a hub and that this incentive leads to global FTA as a unique Markov-perfect equilibrium when countries are not patient. We study a different concern of how a multilateral FTA participation rule affects an equilibrium trade regime, and show that global FTA and a single smaller FTA uniquely prevail, depending on the need for incumbent members' consent.
\subsection{Comparison with Stability Analysis}
The literature on coalition formation has extensively explored two different approaches. One is the process-based approach, and the other is the stability-based approach. This paper employs the former like the papers which we reviewed in the last subsection. The approach (also called a non-cooperative approach) explicitly formulates a coalition formation process and analyzes a subgame perfect equilibrium (and its refinements).
The stability-based approach, by contrast, applies cooperative solutions by assuming certain types of coalitional behavior and characterizes stable outcomes against coalitional deviations. A coalitional-proof Nash equilibrium (Bernheim et al.\ 1987) and network stability (Jackson and Wolinsky 1996) are popular stability concepts. This subsection compares our results with the stability-based approach's results shown by Missios et al.\ (2016), Furusawa and Konishi (2007), and Goyal and Joshi (2006).
Missios et al.\ (2016) consider a coalitional-proof Nash equilibrium of the three-country trade model in this paper. Roughly speaking, an FTA regime is a \emph{coalitional-proof Nash equilibrium} outcome if it is immune to self-enforcing coalitional deviations. A coalitional deviation is self-enforcing if it is immune to self-enforcing deviations by \textit{sub-coalitions}. Thus, a self-enforcing coalitional deviation is defined recursively. They study an announcement game where each country simultaneously announces other countries' names with whom it wants to form an FTA. In the game, two countries sign a bilateral FTA if they announce each other's names.
Missios et al.\ (2016) prove that global FTA is a unique coalitional-proof Nash equilibrium outcome. Consider the following three-country ($1$, $2$, and $3$) example. There are four possible FTA regimes (ignoring permutations): $\mathcal{F}^1=(1\text{-}2, 2\text{-}3, 3\text{-}1)$, $\mathcal{F}^2=(1\text{-}2, 3\text{-}1)$, $\mathcal{F}^3=(1\text{-}2, 3)$, $\mathcal{F}^4=(1, 2, 3)$. Notation $i\text{-}j$ means that countries $i$ and $j$ sign a bilateral FTA. $\mathcal{F}^1$ means the global FTA, $\mathcal{F}^2$ a hub-and-spoke regime with country 1 being a hub, $\mathcal{F}^3$ a bilateral FTA, and $\mathcal{F}^4$ no FTA. The payoff profiles under FTA regimes are given by $W(\mathcal{F}^1)=(3, 3, 3)$, $W(\mathcal{F}^2)=(5, 2, 2)$, $W(\mathcal{F}^3)=(4, 4, -1)$, $W(\mathcal{F}^4)=(0, 0, 0)$.\footnote{The payoff profile under the hub-and-spoke regime $\mathcal{F}^2$ is based on Missios et al.\ (2016).}
Then the global FTA is a coalitional-proof Nash equilibrium for the following reason. Two countries 1 and 2 have a profitable deviation from $\mathcal{F}^1$ to $\mathcal{F}^3$. Country 1, however, has a profitable unilateral deviation from $\mathcal{F}^3$ to $\mathcal{F}^2$, receiving the highest payoff 5. Note that since country 3 announces the name of country 1 as an FTA partner, country 1 can move from $\mathcal{F}^3$ to $\mathcal{F}^2$ by its unilateral deviation. Thus, the coalitional deviation by countries 1 and 2 from $\mathcal{F}^1$ to $\mathcal{F}^3$ is not self-enforcing. This implies that global FTA $\mathcal{F}^1$ is a coalitional-proof Nash equilibrium.
Under our open-access rule with consent, by contrast, FTA regime $\mathcal{F}^3$ is a unique subgame perfect equilibrium when an order of moves is (1, 2, 3), although $\mathcal{F}^3$ is not a coalitional-proof Nash equilibrium because country 1's unilateral deviation from $\mathcal{F}^3$ to $\mathcal{F}^2$ is self-enforcing. A key property of a coalitional-proof Nash equilibrium is that it allows self-enforcing deviations only by \textit{sub-coalitions}. In the argument above, after $\mathcal{F}^3$ moves to $\mathcal{F}^2$ by country 1's deviation, no further coalitional deviation is allowed by the definition of a coalitional-proof Nash equilibrium. However, countries 2 and 3 has a profitable deviation from $\mathcal{F}^2$ to $\mathcal{F}^1$, coming back to the initial position. Thus, a cycle prevails.
In our view, it depends on a context whether or not the restriction to self-enforcing deviations by sub-coalitions is reasonable. If transaction costs to form an FTA with an outsider are prohibitively high, the restriction may be justified.
Furusawa and Konishi (2007) and Goyal and Joshi (2006) employ network stability introduced by Jackson and Wolinsky (1996) in an intra-industry trading model. They consider a network of countries where any pair of two countries signing an FTA has a link. A network of FTAs is \emph{pairwise stable} if (i) no single country is better off by severing an existing FTA, and (ii) any unlinked pair of countries are better off by creating a new FTA between them. Furusawa and Konishi (2007) prove that the global FTA (the complete network) is a unique pairwise stable network if countries are symmetric and industrial commodities are not highly substitutable. Goyal and Joshi (2006) show a similar result.
The pairwise stability in a network formation game is weak in the sense that the notion considers the above two deviations (i) and (ii) only.
In particular, the pairwise stability of the global FTA requires only that no single country is better off by severing a bilateral FTA in the complete network. In the three-country example above, the global FTA $\mathcal{F}^1$ is pairwise stable since country $2$ is worse off in the FTA regime $\mathcal{F}^2$ by severing the bilateral FTA $2\text{-}3$. Neither of the other three FTA regimes is pairwise stable.
The SPE regime $\mathcal{F}^3$ in our open-access game with consent is not pairwise stable because a pair of countries $1$ and $3$ are better-off by creating a new FTA $1\text{-}3$, leading to $\mathcal{F}^2$. However, as we discussed in the case of a coalitional-proof Nash equilibrium, a pair of countries $2$ and $3$ have the incentive to form their bilateral FTA $2\text{-}3$ in $\mathcal{F}^2$, leading to the global FTA $\mathcal{F}^1$. The notion of pairwise stability does not take into account well forward-looking reasoning of negotiating countries, which is critical in a sequential process of trade liberalization.
\subsection{Custom Unions}
We have focused on trade liberalization through FTAs. A custom union (CU) is another framework to attain free trade. Unlike an FTA, the members of a CU jointly choose their external tariffs. Any country member in a CU cannot enter free trade agreements outside it, without consent of other members.
This subsection only compares tariff setting by FTAs and CUs. Suppose that several CUs are formed. Let $\mathcal{C}=(C_1, \cdots, C_m)$ be a CU regime. Unlike the case of an FTA regime, every two of CUs, $C_i$ and $C_j$, are disjoint. The members of each CU choose collectively the external tariffs to maximize the group welfare. The internal tariff in any CU is set to be zero.
Let $c_i$ be the size of each CU $C_i$. For each CU $C_l$ where $l=1, \cdots, m$, we denote the group welfare of $C_l$ by $W^{C_l} = \sum_{k\in C_l} W_k$. For every $i \in C_l$ and $j \notin C_l$,
\begin{equation} \frac{\partial W^{C_{l}}}{\partial t_{ij}} = -2t_{ij} + \frac{2c_{l}}{2n+\lambda}\{a-2p_j^J \} . \label{eq:6_1} \end{equation}
Thus, the first-order condition for $t_{ij}$ to maximize the group welfare $W^{C_l}$ is given by \begin{align} t_{ij} = \frac{c_{l}}{2n+\lambda}(a-2p_j^J) . \label{eq:6_2} \end{align}
Due to the symmetry, \eqref{eq:6_2} shows that every member $i$'s external tariff $t_{ij}$ on non-member $j$ is identical.
\noindent \textbf{Proposition 6.1.} Let $\mathcal{C}=(C_1, ..., C_m)$ be a CU regime, and let $c_l$ be the size of each $C_l$. For every $C_l$, $i \in C_l$ and $j \notin C_l$, the external tariff of member $i$ of $C_{l}$ on non-member $j$ is given by \begin{equation*} t^{\mathcal{C}}_{ij}= \frac{ac_{l}\lambda}{\lambda^2+4n\lambda + 4(n^2 - \sum_{\{k\mid j \notin C_k\}} c_{k}^2)} . \end{equation*}
The proposition shows a stark difference in countries' tariffs between under an FTA regime and a CU regime.\footnote{The proof is given in the supplementary material S.9.} Each CU's external tariff is determined by the sizes of all CUs, while an FTA's one on a given country is determined only by the sizes of FTAs which the country joins.
When an FTA forms, members do not raise their external tariffs on non-members. They choose the Nash equilibrium tariff $t^{\mathrm{NE}} = \frac{a\lambda}{\lambda^2+4n\lambda+4(n^2-n+1)} $. In the CU case, by contrast, the external tariffs are larger than the Nash equilibrium tariff. When a single CU with $m$ members forms, they choose the external tariff $ \frac{a\lambda}{\lambda^2+4n\lambda+4(n^2-m^2-n+m+1)}>t^{\mathrm{NE}}$. The external tariff of a CU violates the GATT Article XXIV.
\section{Conclusion}
We consider the formation of FTAs in a sequential game to contribute to the debate of whether or not it can lead to global free trade. The primary finding is that a unique subgame perfect equilibrium critically depends on a participation rule employed by open regional agreements. Under the open-access rule with consent, where an accession needs incumbent members' consent, a unique equilibrium regime is not the global FTA. Incumbent members reject new participants if the FTA size exceeds the optimal level. By contrast, global FTA is a unique equilibrium regime under the open-access rule without consent. Our result provides a game-theoretical support for the policy recommendation that WTO should require any PTA to be open to all WTO members.
Recently, the role of new types of agreements, open plurilateral agreement (OPA) and critical mass agreement (CPA), are discussed as a novel vehicle for groups of countries to promote shared interests outside trade agreements (Hoekman and Sabel 2019). These new agreements are open to all WTO members. Game theoretical analyses of new frameworks within WTO are promising for future works.
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\section{Proof of Lemma~2.1}
Substituting \eqref{eq:2_2} and \eqref{eq:2_3} into \eqref{eq:2_6} yields
\begin{equation} \sum_{j=1}^n(a-p_j^I)=\sum_{j=1}^n \lambda_j^I p_j^I . \label{Aeq:1_1} \end{equation}
Substituting \eqref{eq:2_4} into \eqref{Aeq:1_1} yields
\begin{equation} \sum_{j=1}^n(a-p_j^I)=\sum_{j \ne i}p_j^I + (1+ \lambda)p_i^I . \label{Aeq:1_2} \end{equation}
Substituting \eqref{eq:2_5} into \eqref{Aeq:1_2} yields
\begin{equation} \sum_{j \ne i}(a-p_i^I-t_{ji}) + a-p_i^I=\sum_{j \ne i}(p_i^I+ t_{ji}) + (1+ \lambda)p_i^I . \label{Aeq:1_3} \end{equation}
\eqref{Aeq:1_3} solves
\begin{equation} p_i^I = \frac{na - 2\sum_{j \ne i} t_{ji}}{2n + \lambda} . \label{Aeq:1_4} \end{equation}
By \eqref{Aeq:1_4} and \eqref{eq:2_5}, we have
\begin{equation} p_j^I=\frac{na - 2\sum_{k \ne i, j} t_{ki} + (2n-2+\lambda)t_{ji}}{2n + \lambda}. \label{Aeq:1_5} \end{equation}
Q.E.D.
\section{Proof of Theorem~3.1}
We first prove \eqref{eq:3_3}. From \eqref{eq:2_10} and \eqref{eq:2_14}, it holds that \begin{align*} \frac{\partial W_i}{\partial t_{ij}} &= -(a-p_i^J)\frac{\partial p_i^J}{\partial t_{ij}} + p_i^J\frac{\partial p_i^J}{\partial t_{ij}} + a-2p_i^J - 2t_{ij} \frac{\partial p_i^J}{\partial t_{ij}} \\ &= (-a+2p_i^J-2t_{ij})\frac{\partial p_i^J}{\partial t_{ij}} + a-2p_i^J \\ &= (-a+2p_i^J-2t_{ij})(1-\frac{2}{2n+\lambda}) + a-2p_i^J \\ &= -2t_{ij} + \frac{2}{2n+\lambda}(a+2t_{ij}-2p_i^J) \\ &= -2t_{ij} + \frac{2}{2n+\lambda}(a-2p_j^J). \end{align*} This proves \eqref{eq:3_3}. From \eqref{eq:2_14} and \eqref{eq:3_4}, we obtain \eqref{eq:3_5}. From \eqref{eq:3_5}, $t_{ij}=t_{kj}$ for every pair $i$ and $k$ with $i\neq k$. With help of this fact, \eqref{eq:3_5} solves \eqref{eq:3_6}. Q.E.D.
\section{Proof of Theorem~3.2}
\noindent\textbf{Part (i)}: The system of the first order conditions \eqref{eq:3_10} to maximize the total welfare of the world has a unique solution $t_{ij}=0$ for every $i,j$ ($i\neq j$). Thus, global FTA with zero tariffs is the unique optimal regime.
\noindent\textbf{Part (ii)}: For every $i,j$ ($i \neq j$), since $t_{ij}=0$, it follows from \eqref{eq:2_10} that \begin{align} p_i^I= p_i^J= \frac{na}{2n+\lambda} . \label{Aeq:3_1} \end{align}
Substituting \eqref{Aeq:3_1} into \eqref{eq:2_14} yields \begin{align*} W_i(t) &= \frac{n}{2} \left[ (a-\frac{na}{2n+\lambda})^2 + (\frac{na}{2n+\lambda})^2 \right] + \frac{\lambda}{2} (\frac{na}{2n+\lambda})^2 \\ &= \frac{na^2}{2} - na \frac{na}{2n+\lambda} + \frac{2n+\lambda}{2}\frac{n^2a^2}{(2n+\lambda)^2} \\ &= \frac{na^2(2n+\lambda)^2+n^2a^2(2n+\lambda)-2n^2a^2(2n+\lambda)}{2(2n+\lambda)^2} \\ &= \frac{na^2(n+\lambda)}{2(2n+\lambda)} . \footnotemark
\end{align*} \footnotetext{The welfare of every country under global FTA can be also derived from the general formula \eqref{eq:4_4} of the welfare of a participant in an FTA in Lemma~4.2 (by setting $m=n$).}
\noindent\textbf{Part (iii)}: We prove this part by Lemma~4.2, which gives a general formula \eqref{eq:4_5} of the welfare of a non-participant in an FTA. The proof of the lemma is given in the supplementary material S.5.
Let $t$ be a common tariff for all countries. Letting $m=1$ in \eqref{eq:4_5} of Lemma~4.2, we obtain
\begin{align} & rW^1_{np}(t) - na^2(n+\lambda)(2n+\lambda) \notag \\
&= -2[\lambda^2+4n\lambda+8n-4+(n-2)\lambda^2-2(2n-n^2+1)\lambda+4(n^2-3n+1)] t^2 \notag \\ &= -2(n-1)[\lambda^2+2n\lambda +2\lambda+4n]t^2 \notag \\ &= -2(n-1)(2+\lambda)(2n+\lambda)t^2 , \label{Aeq:3_2} \end{align}
where $r=2(2n+\lambda)^2$. Substituting $t=t^{\mathrm{NE}}$ into \eqref{Aeq:3_2} proves the part (iii). Q.E.D.
\section{Proof of Theorem 4.2}
\textbf{Part i}: First suppose regime $\mathcal{F}^0$, where no FTA is formed. Then, all countries impose the common Nash tariff $t^{\mathrm{NE}}>0$ in \eqref{eq:3_6} on every country. Substituting \eqref{eq:3_6} into \eqref{eq:2_10} yields the domestic price of good $I$ in country $j$, given by \begin{align} p_j^{I,\mathcal{F}^0} = p_j^{I,\mathrm{NE}} &= \frac{na + (2+\lambda) t^{NE}}{2n + \lambda} \notag \\ &= \frac{na}{2n + \lambda} + \frac{a\lambda(2+\lambda)}{(2n+\lambda)[\lambda^2+4n\lambda+4(n^2-n+1)]} . \label{Aeq:4_1} \end{align}
Next, suppose regime $\mathcal{F}^i$, where $i\in F$ and $F$ has $m$ members. Take $j\in F$ and $k \notin F$. From Theorem~4.1, country $k$ imposes the external tariff $t_{ki}^{\mathcal{F}^i}$ on country $i$, given by \begin{equation*} t_{ki}^{\mathcal{F}^i} = \frac{a\lambda}{\lambda^2 + 4n\lambda + 4(n^2-(n-m))} >0. \end{equation*} Since $t_{ji}^{\mathcal{F}^i}=0$ for all $j\in F$, substituting it into \eqref{eq:2_10} yields the domestic price of good $I$ in country $j$ imported from country $i$, given by \begin{align} p_j^{I,\mathcal{F}^i} &= \frac{na - 2(n-m) t_{ki}^{\mathcal{F}^i}}{2n + \lambda} \notag \\ &= \frac{na}{2n + \lambda} - \frac{ 2a\lambda(n-m)}{(2n+\lambda)[\lambda^2 + 4n\lambda + 4(n^2-n+m)]} . \label{Aeq:4_2} \end{align}
By $n>m$, from \eqref{Aeq:4_1} and \eqref{Aeq:4_2}, \begin{equation*} p_j^{I,\mathcal{F}^i} - p_j^{I,\mathcal{F}^0} = -\frac{1}{2n+\lambda}[ 2(n-m)t_{ki}^{\mathcal{F}^i} + (2+\lambda)t^{\mathrm{NE}}]<0. \end{equation*}
From \eqref{eq:2_2}, \eqref{eq:2_3}, and \eqref{eq:2_7}, import $m_j^I$ of good $I$ in country $j (\neq i)$ is given by $m_j^{I,\mathcal{F}} = a - 2p_j^{I,\mathcal{F}}$ for every regime $\mathcal{F}$. Thus, \begin{equation} m_j^{I,\mathcal{F}^i} - m_j^{I,\mathcal{F}^0} = 2 (p_j^{I,\mathcal{F}^0} - p_j^{I,\mathcal{F}^i}) >0 . \label{Aeq:4_3} \end{equation}
\noindent\textbf{Part ii}: Similarly, from \eqref{eq:2_10}, the domestic price of good $I$ in country $k$ is given by \begin{align} p_k^{I,\mathcal{F}^i} &= \frac{na +(2m+\lambda) t_{ki}^{\mathcal{F}^i}}{2n + \lambda} \notag \\ &= \frac{na}{2n + \lambda} + \frac{a\lambda(2m+\lambda)} {(2n+\lambda)[\lambda^2 + 4n\lambda + 4(n^2-n+m)]} . \label{Aeq:4_4} \end{align} From \eqref{Aeq:4_1} and \eqref{Aeq:4_4}, \begin{align*} p_k^{I,\mathcal{F}^i} - p_k^{I,\mathcal{F}^0} &=\frac{a\lambda}{2n + \lambda}\left[ \frac{2m+\lambda}{\lambda^2 + 4n\lambda + 4(n^2-n+m)} - \frac{2+\lambda}{\lambda^2 + 4n\lambda + 4(n^2-n+1)} \right] \\ &= \frac{a\lambda}{2n + \lambda} \left[ \frac{(2m-2)(\lambda^2+4n\lambda+4n^2-4n)+(2m+\lambda)-m(2+\lambda)}{[\lambda^2 + 4n\lambda + 4(n^2-n+m)][\lambda^2 + 4n\lambda + 4(n^2-n+1)]} \right] \\ &= \frac{a\lambda}{2n + \lambda} \left[ \frac{(m-1)[2\lambda^2+(8n-1)\lambda+8n^2-8n]}{[\lambda^2 + 4n\lambda + 4(n^2-n+m)][\lambda^2 + 4n\lambda + 4(n^2-n+1)]} \right] \\ &>0 . \end{align*} Since $p_k^{I,\mathcal{F}^i} > p_k^{I,\mathcal{F}^0}$, the import $m_k^{I,\mathcal{F}^i} < m_k^{I,\mathcal{F}^0}$ from \eqref{Aeq:4_3}. Q.E.D.
\section{Proof of Theorem 4.3}
By \eqref{eq:4_4}, \begin{align} rW_p^{m+1}(t) =& na^2 (n+\lambda)(2n+\lambda) + 4(a\lambda)^2(2m+2+\lambda)(n-m-1)^2 (t'_p)^2 \notag \\ & - 4(a\lambda)^2(n-m-1)^2 t'_p + 4(a\lambda)^2 (n-m-1)(n-1) t'_{np} \notag \\ & - 2(a\lambda)^2(n-m-1) \left[\lambda^2+4n\lambda +8n-4 \right](t'_{np})^2 , \label{Aeq:5_1} \intertext{where } t'_p &= \frac{1}{\lambda^2+4n\lambda+4(n^2-n+m+1)} \label{Aeq:5_2} \\ t'_{np} &= \frac{1}{\lambda^2+4n\lambda+4(n^2-n+1)} . \label{Aeq:5_3} \end{align}
By \eqref{eq:4_5}, \begin{align} rW^m_{np}(s) &= na^2(n+\lambda)(2n+\lambda) -2m(a\lambda)^2\left[\lambda^2+4n\lambda+8mn-4m^2\right](s'_p)^2 \notag \\ & -2(a\lambda)^2\left[(n-m-1)\lambda^2-2(2mn-n^2+1)\lambda+4(n^2-2mn-n+m)\right](s'_{np})^2 \notag \\ & +4(a\lambda)^2 m(n-m) s'_p - 4(a\lambda)^2 m(n-1) s'_{np} , \label{Aeq:5_4} \intertext{where } s'_{p} &= \frac{1}{\lambda^2+4n\lambda+4(n^2-n+m)} \label{Aeq:5_5} \\ s'_{np} &= \frac{1}{\lambda^2+4n\lambda+4(n^2-n+1)} . \label{Aeq:5_6} \end{align} Note that $t'_p<s'_p<t'_{np}=s'_{np}$.
From \eqref{Aeq:5_1} and \eqref{Aeq:5_4}, \begin{align} &\frac{r [W^{m+1}_p (t) - W^m_{np}(s) ]}{(a\lambda)^2} \notag\\ &= 4(2m+2+\lambda)(n-m-1)^2 (t'_p)^2 - 4(n-m-1)^2t'_p \notag \\ & - 2(n-m-1)\left[\lambda^2+4n\lambda +8n-4 \right](t'_{np})^2 \notag \\ & + 4(n-m-1)(n-1)t'_{np} \notag \\ & +2m\left[\lambda^2+4n\lambda+8mn-4m^2\right](s'_p)^2 - 4 m(n-m)s'_p \notag \\ & +2\left[(n-m-1)\lambda^2-2(2mn-n^2+1)\lambda+4(n^2-2mn-n+m)\right](s'_{np})^2 \notag \\ & +4m(n-1)s'_{np} , \label{Aeq:5_7} \end{align} where $t'_p, t'_{np}, s'_{p}, s'_{np}$ are given by \eqref{Aeq:5_2}--\eqref{Aeq:5_3} and \eqref{Aeq:5_5}--\eqref{Aeq:5_6}.
We can show that the RHS of \eqref{Aeq:5_7} is strictly positive. Since its derivation is tedious, we provide it in the supplementary material S.6. Hence $W^{m+1}_p (t) > W^m_{np}(s)$. Q.E.D.
\section{Proof of Theorem 4.4}
By Lemma 4.2, \begin{align} \frac{rW_{np}^m(t)}{(a\lambda)^2} &= \frac{n(n+\lambda)(2n+\lambda)}{\lambda^2} \notag \\ &-2m\left[\lambda^2+4n\lambda+8mn-4m^2\right](t'_p)^2 \notag \\ &-2\left[(n-m-1)\lambda^2-2(2mn-n^2+1)\lambda+4(n^2-2mn-n+m)\right](t'_{np})^2 \notag \\ &+4m(n-m)t'_p-4m(n-1)t'_{np} , \label{Aeq:6_1} \end{align} where $t'_p$ and $t'_{np}$ are given in \eqref{Aeq:5_2} and \eqref{Aeq:5_3}.
Let function $f$ be the RHS of \eqref{Aeq:6_1} except the first constant term. By $\frac{\partial t_{np}'}{\partial m}=0$, \begin{align} \frac{\partial f(m)}{\partial m} =& -2\left[\lambda^2+4n\lambda+16mn-12m^2\right](t'_p)^2 \notag \\ & -4m\left[\lambda^2+4n\lambda+8mn-4m^2\right]t'_p \frac{\partial t'_p}{\partial m} \notag\\ & +2\left[\lambda^2+4n\lambda+8n-4\right](t'_{np})^2 \notag \\ & +4(n-2m)t'_p + 4m(n-m)\frac{\partial t'_p}{\partial m} -4(n-1)t_{np}. \label{Aeq:6_2} \end{align}
Furthermore, substituting $\frac{\partial t'_p}{\partial m}=-\frac{4}{(t'_p)^2}$ into \eqref{Aeq:6_2} yields \begin{align} \frac{\partial f(m)}{\partial m}= & -2\left[\lambda^2+4n\lambda+16mn-12m^2\right](t'_p)^2 \notag \\ & +16m\left[\lambda^2+4n\lambda+8mn-4m^2\right]\frac{1}{t'_p} \notag\\ & +4(n-2m)t'_p -16m(n-m)\frac{1}{(t'_p)^2} \notag\\ & +2\left[\lambda^2+4n\lambda+8n-4\right]t_{np}^2-4(n-1)t_{np} . \label{Aeq:6_3} \end{align}
We can show that the derivative $\frac{\partial f(m)}{\partial m}$ in $m$ is strictly negative. Since its computation is simple but tedious, we provide it in the supplementary material S.7. Hence non-participant country's welfare is decreasing in $m$ with $1\le m \le n-1$. Q.E.D.
\section{Proof of Theorem~4.5}
Without any crucial loss of generality, we assume that $W_p^m$ has a unique maximum point $m^*$.\footnote{If there are multiple maximum points, then we choose the largest one.} Recall that $n$ countries sequentially decide to participate in an FTA or not, according to the order $(1, 2, \cdots, n)$.
We prove the theorem by backward induction in the following three steps.
\noindent \textit{Step 1.} First, consider the $n$-th stage game when there are any $m$ incumbents of an FTA with $m<n$.\footnote{If $m^*=n$, then this case is vacuous.} If country $n$ participates, then all $m$ participants either accept or reject $n$'s participation independently. If the participation is accepted, then every participant $i$ receives payoff $W_p^{m+1}$. Otherwise, it receives payoff $W_p^{m}$.
When $W_p^{m+1}\geq W_p^{m}$, it is (weakly) dominant for every incumbent to accept $n$'s participation.\footnote{If an incumbent is indifferent between accepting and rejecting a new participant, we assume the tie-breaking rule to accept it.} Country $n$ optimally participates in an FTA. When $W_p^{m+1}< W_p^{m}$, it is (weakly) dominant for every incumbent to reject $n$'s participation. Country $n$'s choice does not affect the game's outcome.
By the same argument, we can characterize the equilibrium actions for incumbents and a new participant in every $t$-th stage game where $t=m^*+1, \cdots, n$ with $m$ incumbents as follows. All incumbents accept the participation of country $t$ if and only if $v(m, t) \geq W_p^m$, and country $t$ participates if $v(m, t) \geq W_p^m$, where $v(m, t)=\max \{W_p^{m+1}, \cdots, W_p^{m+n-t+1}\}$.
\noindent \textit{Step 2.} Next, consider the $m^*$-th stage game when there are $m^*-1$ incumbents of an FTA. If all incumbents accept the participation of $m^*$, then they receive payoff $W_p^{m^*}$. By $W_p^{m^*}>W_p^{m}$ for all $m\ne m^*$, it is (weakly) dominant for every incumbent to reject country $m^*$'s participation. Then, it is also optimal for country $m^*$ to participate in the FTA.
\noindent \textit{Step 3.} Finally, consider the $(m^*-1)$-th stage game when there are $m^*-2$ incumbent members of an FTA. If all incumbents accept the participation of $m^*-1$, then they receive payoff $W_p^{m^*}$ because all countries rationally expect the size of an FTA expands to $m^*$ in the next $m^*$-th stage. By $W_p^{m^*}>W_p^{m}$ for all $m\ne m^*$, it is (weakly) dominant for every incumbent to accept country $m^*-1$'s participation. Then, country $m^*-1$ optimally participates in the FTA.
By backward induction, the above steps show that all countries $i=1, \cdots, m^*$ optimally participates in an FTA, and that their participations are accepted. After country $m^*$ participates, no more countries are accepted. Q.E.D.
\section{Proof of Theorem~4.6}
We first show that the equilibrium FTA size $m^*$ is strictly less than $n$. From \eqref{eq:4_4},
\begin{align*}
rW_{p}^n =& na^2(n+\lambda)(2n+\lambda) \\
rW_{p}^{n-1}(t) =& na^2(n+\lambda)(2n+\lambda) + 4(2(n-1)+\lambda)(n-(n-1))^2 t_{p}^2 \notag \\
& - 4a\lambda(n-(n-1))^2t_{p} + 2(n-(n-1))\left[4-8n-4n\lambda - \lambda^2 \right]t_{np}^2 \notag \\
& + 4a\lambda (n-(n-1))(n-1)t_{np}. \end{align*}
From \eqref{Aeq:5_2}--\eqref{Aeq:5_3}, $t_p=a\lambda t_p'$, $t_{np}=a\lambda t_{np}'$, and the difference
\begin{align}
r[W_p^{n-1}(t) - W_{p}^n] &= 4(2(n-1)+\lambda) t_{p}^2 - 4a\lambda t_{p} \notag \\
& \:\:\:\:\: - 2\left[-4+8n+4n\lambda + \lambda^2 \right]t_{np}^2 + 4a\lambda (n-1)t_{np} \notag \\
&= 4(a\lambda t'_p)^2 [2(n-1)+\lambda - (2n+\lambda)^2 +4] \notag \\
& \:\:\:\:\: + 2(a\lambda t'_{np})^2 [2(n-1)(2n+\lambda)^2-8(n-1)^2+4-8n-4n\lambda - \lambda^2] \notag \\
&= 2(a\lambda t'_p)^2 [4(n+1)+2\lambda - 2(2n+\lambda)^2 ] \notag \\
& \:\:\:\:\: + 2(a\lambda t'_{np})^2 [2(n-1)(2n+\lambda)^2-8(n-1)^2 +4(n-1)^2 -(2n+\lambda)^2] \notag \\
&= 2(a\lambda t'_p)^2 [4(n+1)+2\lambda - 2(2n+\lambda)^2 ] \notag \\
& \:\:\:\:\: + 2(a\lambda t'_{np})^2 [(2n-3)(2n+\lambda)^2-4(n-1)^2] . \label{Aeq:9_1} \end{align}
Let function $\zeta(n) = [(2n-3)(2n+\lambda)^2-4(n-1)^2]$. For $n\geq 3$, $\zeta(n)>0$ because $\zeta(3)=3(6+\lambda)^2-16>0$ and $d \zeta(n) /d n = 2\lambda^2+ 4\lambda(4n-3) +8n(3n-4) +8>0$.
Thus, for $n\ge 3$, by $t'_p<t'_{np}$ and \eqref{Aeq:9_1}, \begin{align*}
r[W_p^{n-1}(t) - W_{p}^n]
&> 2(a\lambda t'_p)^2 [4(n+1)-4(n-1)^2+(2n+\lambda)^2(2n-5)] \\
&= 2(a\lambda t'_p)^2 [-4n(n-3)+4n^2(2n-5)+\lambda(4n+\lambda)(2n-5)] \\
&= 2(a\lambda t'_p)^2 [4n(2n^2 -6n +3) +\lambda(4n+\lambda)(2n-5)] \\
&> 0 . \end{align*}
The last inequality holds true because both $(2n^2 -6n +3)$ and $(2n-5)$ are strictly positive. Thus, $W_p^{n-1} > W_{p}^n(t)$ for $n\ge 3$, and $W_p^m$ is not maximized at $n$, i.e.\ $m^*<n$.
Next, we show that when $n$ is sufficiently large, welfare $W_p^m$ is maximized at $m^*=(n/2)-1$, $n/2$, or $(n/2)+1$ for even $n$, and at $m^*=(n-1)/2$ or $(n+1)/2$ for odd $n$. Let $\alpha = m/n\in (0,1]$ and fix $n$. By \eqref{Aeq:5_2}--\eqref{Aeq:5_3}, $\frac{\partial t'_p}{\partial \alpha}=-4nt_p^2$ and $\frac{\partial t'_{np}}{\partial \alpha}=0$. Then, by \eqref{eq:4_4}, the derivative of $W_p^m$ with respect to $\alpha$ is given by \begin{align*} \frac{\partial W_p^m}{\partial \alpha} &= 8(a\lambda)^2 n^2(1-\alpha)t'_p - 4(a\lambda)^2 n^2 t'_{np} + R(n) \\ &\to (a\lambda)^2(1-2\alpha) \text{ as $n \to \infty$}, \end{align*} where $R(n)$ is the remainder term. $R(n)$ satisfies $\lim_{n\to \infty} R(n) =0$ because $t'_p,t'_{np}$ have the order of $1/n^2$. Since it is tedious, the derivation is given in the supplementary material S.8.
Since the derivative is monotonically decreasing in $\alpha=\frac{m}{n}\in (0,1]$, welfare function $W_p^m$ is concave in $m$ in the limit. When $\alpha=1/2$, $\lim_{n\to \infty} \frac{\partial W_p^m}{\partial \alpha}= 0$.
Therefore, for sufficiently large $n<\infty$, welfare $W_p^m$ is maximized at $m^*=(n/2)-1$, $n/2$, or $(n/2)+1$ for even $n$ and at $m^*=(n-1)/2$ or $(n+1)/2$ for odd $n$. The remainder term $R(n)$ determines which maximizes the welfare. Q.E.D.
\section{Proof of Theorem~4.7}
We prove the theorem by backward induction. First, consider the optimal choice of the last country. From Theorem~4.3, it is optimal for the last country to participate in an FTA if there exists at least one incumbent, who has participated in the FTA before. Otherwise, it is indifferent between participation and non-participation. If they are indifferent, we assume the tie-breaking rule under which the last country participates.
Next, consider the optimal choice of the second-to-last country. Again, from Theorem~4.3, it is optimal for the second-to-last country to participate in an FTA if there exists at least one incumbent, who has participated in the FTA before.
Suppose that there have been no participation in an FTA. If country $i$ participates, then it receives the welfare $W_p^2$, given the optimal choice of the last country. If country $i$ does not participate, then it receives the welfare $W_{np}^1$, regardless of the indifferent choices of the last country.
Since $W_p^2>W_{np}^1$ by Theorem~4.3, it is also optimal for country $i$ to participate in this case. Since the second-to-last country always participates, by Theorem~4.4, all other countries optimally participate in an FTA at their all moves.
Thus, global free trade is a unique SPE. The argument holds independently of an order of moves. Q.E.D.
\end{document} | arXiv |
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Izv. Akad. Nauk SSSR Ser. Mat., 1974, Volume 38, Issue 1, Pages 228–248 (Mi izv1899)
This article is cited in 11 scientific papers (total in 11 papers)
Some estimates of the probability density of a stochastic integral
N. V. Krylov
Abstract: Estimates in $L_p$ are derived for probability densities of stochastic integrals. An example is presented which shows that for some values of $p$ such estimates are not attainable. The method of proving these estimates is based on a study of Bellman's nonlinear equations and the properties of $\lambda$-convex functions.
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Mathematics of the USSR-Izvestiya, 1974, 8:1, 233–254
Bibliographic databases:
UDC: 519.2
MSC: Primary 60H05; Secondary 60E05, 60H20, 60J25, 26A51
Received: 26.12.1972
Citation: N. V. Krylov, "Some estimates of the probability density of a stochastic integral", Izv. Akad. Nauk SSSR Ser. Mat., 38:1 (1974), 228–248; Math. USSR-Izv., 8:1 (1974), 233–254
Citation in format AMSBIB
\Bibitem{Kry74}
\by N.~V.~Krylov
\paper Some estimates of the probability density of a~stochastic integral
\jour Izv. Akad. Nauk SSSR Ser. Mat.
\mathnet{http://mi.mathnet.ru/izv1899}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=345206}
\jour Math. USSR-Izv.
\crossref{https://doi.org/10.1070/IM1974v008n01ABEH002103}
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This publication is cited in the following articles:
A. Yu. Veretennikov, N. V. Krylov, "On explicit formulas for solutions of stochastic equations", Math. USSR-Sb., 29:2 (1976), 239–256
N. V. Krylov, B. L. Rozovskii, "On conditional distributions of diffusion processes", Math. USSR-Izv., 12:2 (1978), 336–356
S. V. Anulova, "On processes with Lévy generating operator in a half-space", Math. USSR-Izv., 13:1 (1979), 9–51
A. Yu. Veretennikov, "On strong solutions and explicit formulas for solutions of stochastic integral equations", Math. USSR-Sb., 39:3 (1981), 387–403
P. L. Lions, "On the Hamilton–Jacobi–Bellman equations", Acta Appl Math, 1:1 (1983), 17
Nigel J. Cutland, "Simplified existence for solutions to stochastic differential equations", Stochastics, 14:4 (1985), 319
Nigel J. Cutland, "Infinitesimal methods in control theory: Deterministic and stochastic", Acta Appl Math, 5:2 (1986), 105
N. V. Krylov, "On estimates of the maximum of a solution of a parabolic equation and estimates of the distribution of a semimartingale", Math. USSR-Sb., 58:1 (1987), 207–221
R. Mikulevicius, B. Rozovskii, "Linear Parabolic Stochastic PDE and Wiener Chaos", SIAM J Math Anal, 29:2 (1998), 452
N.V. Krylov, R. Liptser, "On diffusion approximation with discontinuous coefficients", Stochastic Processes and their Applications, 102:2 (2002), 235
Alexey Rudenko, "Some properties of the Itô–Wiener expansion of the solution of a stochastic differential equation and local times", Stochastic Processes and their Applications, 2012
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Global smooth solutions for the nonlinear Schrödinger equation with magnetic effect
Approximation of random invariant manifolds for a stochastic Swift-Hohenberg equation
December 2016, 9(6): 1717-1752. doi: 10.3934/dcdss.2016072
Local classical solutions of compressible Navier-Stokes-Smoluchowski equations with vacuum
Bingyuan Huang 1, , Shijin Ding 2, and Huanyao Wen 3,
School of Mathematical Sciences, South China Normal University, Guangzhou 510631, China
Department of Mathematics, South China Normal University, Guangzhou, Guangdong 510631
School of Mathematics, South China University of Technology, Guangzhou 510641, China
Received July 2015 Revised September 2016 Published November 2016
This paper is concerned with the Cauchy problem for compressible Navier-Stokes-Smoluchowski equations with vacuum in $\mathbb{R}^3$. We prove both existence and uniqueness of the local strong solution, and then obtain a local classical solution by deriving the smoothing effect of the strong solution for $t>0$.
Keywords: vacuum., Classical solution, compressible Navier-Stokes-Smoluchowski equations.
Mathematics Subject Classification: Primary: 35Q30, 76N10; Secondary: 46E3.
Citation: Bingyuan Huang, Shijin Ding, Huanyao Wen. Local classical solutions of compressible Navier-Stokes-Smoluchowski equations with vacuum. Discrete & Continuous Dynamical Systems - S, 2016, 9 (6) : 1717-1752. doi: 10.3934/dcdss.2016072
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Bingyuan Huang Shijin Ding Huanyao Wen | CommonCrawl |
\begin{document}
\title{Contradiction-Tolerant Process Algebra \ with Propositional
Signals}
\begin{abstract}
In a previous paper, an ACP-style process algebra was proposed in which propositions are used as the visible part of the state of processes and as state conditions under which processes may proceed. This process algebra, called ACPps, is built on classical propositional logic. In this paper, we present a version of ACPps built on a paraconsistent propositional logic which is essentially the same as CLuNs. There are many systems that would have to deal with self-contradictory states if no special measures were taken. For a number of these systems, it is conceivable that accepting self-contradictory states and dealing with them in a way based on a paraconsistent logic is an alternative to taking special measures. The presented version of ACPps can be suited for the description and analysis of systems that deal with self-contradictory states in a way based on the above-mentioned paraconsistent logic. \begin{keywords} process algebra, propositional signal, propositional condition, paraconsistent logic. \end{keywords} \begin{classcode} D.2.1, D.2.4, F.3.1, F.4.1. \end{classcode} \end{abstract}
\section{Introduction} \label{sect-intro}
Algebraic theories of processes such as ACP~\cite{BW90}, CCS~\cite{Mil89}, and CSP~\cite{Hoa85}, as well as most algebraic theories of processes in the style of these ones, are concerned with the behaviour of processes only. That is, the state of processes is kept invisible. In~\cite{BB94b}, an ACP-style process algebra, called ACPps, was proposed in which processes have their state to some extent visible. The visible part of the state of a process, called the signal emitted by the process, is a proposition of classical propositional logic. Propositions are not only used as signals emitted by processes, but also as conditions under which processes may proceed. The intuition is that the signal emitted by a process is a proposition that holds at its start and the condition under which processes may proceed is a proposition that must hold at its start. Thus, by the introduction of signal emitting processes, an answer is given to the question what determines whether a condition under which a process may proceed is met.
If the signals emitted by two processes are contradictory, then the signal emitted by the parallel composition of these processes is self-contradictory. For example, if the signals emitted by the two processes, being propositions, are each others negation, then they are contradictory and their conjunction, which is the signal emitted by the parallel composition of these processes, is self-contradictory. Intuitively, a process emitting a self-contradictory signal is an impossibility. Therefore, a special process has been introduced in ACPps to deal with it. In practice, there are many systems that would have to deal with self-contradictory states if no special prevention measures or special detection and resolution measures were taken. Some typical examples are web-service-oriented applications and autonomous robotic agents (see e.g.~\cite{GKNF05a,PI04a,QV14a}). At least for a number of these systems, it is conceivable that accepting self-contradictory states and dealing with them in a way based on a suitable paraconsistent logic is an alternative to taking special measures. It may even be the only workable alternative because a system may have to cope with inconsistencies occurring on a large scale.
What exactly does it mean to deal with self-contradictory states in a way based on a paraconsistent logic? The systems referred to above are systems whose behaviour is made up of discrete steps where, upon each step performed, the way in which the behaviour proceeds is conditional on the current state of the system concerned. If the propositions by which the visible part of the possible states of a system can be characterized are used as conditions, then it can be established in accordance with a paraconsistent propositional logic whether a condition is met in a state. This is what is meant by dealing with self-contradictory states in a way based on a paraconsistent logic. We think that a version of the process algebra ACPps that is built on an appropriate paraconsistent propositional logic instead of classical propositional logic can be suited for the description and analysis of systems that deal with self-contradictory states in a way based on a paraconsistent logic. The important point here is that, in such a logic, it is generally not possible to deduce an arbitrary formula from two contradictory formulas.
The question remains: what is an appropriate paraconsistent propositional logic? The ones that have been proposed differ in many ways and whether one of them is more appropriate than another is fairly difficult to make out. A paraconsistent propositional logic is a logic that does not have the property that every proposition is a logical consequence of every set of hypotheses that contains contradictory propositions. A paraconsistent propositional logic with the property that every proposition is a logical consequence of every set of hypotheses that contains contradictory propositions but one is far from appropriate. Such a logic is a minimal paraconsistent logic. Maximal paraconsistency, i.e.\ a logical consequence relation that cannot be extended without loosing paraconsistency, is generally considered an important property. There are various other properties that have been proposed as characteristic of reasonable paraconsistent propositional logics, but their importance remains to some extent open to question.
The properties that have been proposed as characteristic of reasonable paraconsistent propositional logics do not include all properties that are required of an appropriate one to build a version of ACPps on. These properties include, among other things, properties needed to retain the basic axioms of ACP-style process algebras. In this paper, we present a version of ACPps built on the paraconsistent propositional logic for which the name LP$^{\IImpl,\False}$ was coined in~\cite{Mid11a}. This logic, which is essentially the same as J3~\cite{DOt85a}, CLuNs~\cite{BC04a}, and LFI1~\cite{CCM07a}, has virtually all properties that have been proposed as characteristic of reasonable paraconsistent propositional logics as well as all properties that are required of an appropriate one to build a version of ACPps on. LP$^{\IImpl,\False}$ can be replaced by any paraconsistent propositional logics with the latter properties, but among the paraconsistent propositional logics with the former properties, LP$^{\IImpl,\False}$ is the only one with the latter properties.
The structure of this paper is as follows. First, we give a survey of the paraconsistent propositional logic LP$^{\IImpl,\False}$ (Section~\ref{sect-LP-iimpl-false}). Next, we present \ctBPAps, the subtheory of the version of ACPps built on LP$^{\IImpl,\False}$ that does not support parallelism and communication (Sections~\ref{sect-ctBPAps} and~\ref{sect-sem-ctBPAps}). After that, we present \ctACPps, the version of ACPps built on LP$^{\IImpl,\False}$, as an extension of \ctBPAps\ (Sections~\ref{sect-ctACPps} and~\ref{sect-sem-ctACPps}). Following this, we introduce a useful additional feature, namely a generalization of the state operators from~\cite{BB88} (Section~\ref{sect-ctACPps+SO}). Then, we treat the addition of guarded recursion to \ctACPps\ (Section~\ref{sect-ctACPps+REC}). Finally, we make some concluding remarks (Section~\ref{sect-concl}).
\section{The Paraconsistent Logic LP$^{\IImpl,\False}$} \label{sect-LP-iimpl-false}
A set of propositions $\Gamma$ is contradictory if there exists a proposition $A$ such that both $A$ and $\Not A$ can be deduced from $\Gamma$. A proposition $A$ is called self-contradictory if $\set{A}$ is contradictory. In classical propositional logic, every proposition can be deduced from a contradictory set of propositions. A paraconsistent propositional logic is a propositional logic in which not every proposition can be deduced from each contradictory set of propositions.
In~\cite{Pri79a}, Priest proposed the paraconsistent propositional logic LP (Logic of Paradox). The logic introduced in this section is LP enriched with an implication connective for which the standard deduction theorem holds and a falsity constant. This logic, called LP$^{\IImpl,\False}$, is in fact the propositional fragment of CLuNs~\cite{BC04a} without bi-implications.
LP$^{\IImpl,\False}$ has the following logical constants and connectives: a falsity constant $\False$, a unary negation connective $\Not$, a binary conjunction connective $\CAnd$, a binary disjunction connective $\COr$, and a binary implication connective $\IImpl$. Truth and bi-implication are defined as abbreviations: $\True$ stands for $\Not \False$ and $A \BIImpl B$ stands for $(A \IImpl B) \CAnd (B \IImpl A)$.
A Hilbert-style formulation of LP$^{\IImpl,\False}$ is given in Table~\ref{proofsystem-LPiimpl}.
\begin{table}[!tb] \caption{Hilbert-style formulation of LP$^{\IImpl,\False}$} \label{proofsystem-LPiimpl} \begin{eqntbl} \begin{axcol} \mathbf{Axiom\; Schemas:} \\ A \IImpl (B \IImpl A) \\ (A \IImpl (B \IImpl C)) \IImpl ((A \IImpl B) \IImpl (A \IImpl C)) \\ ((A \IImpl B) \IImpl A) \IImpl A \\ \False \IImpl A \\ (A \CAnd B) \IImpl A \\ (A \CAnd B) \IImpl B \\ A \IImpl (B \IImpl (A \CAnd B)) \\ A \IImpl (A \COr B) \\ B \IImpl (A \COr B) \\ (A \IImpl C) \IImpl ((B \IImpl C) \IImpl ((A \COr B) \IImpl C)) \end{axcol} \qquad \begin{axcol} {} \\ \Not \Not A \BIImpl A \\ \Not (A \IImpl B) \BIImpl A \CAnd \Not B \\ \Not (A \CAnd B) \BIImpl \Not A \COr \Not B \\ \Not (A \COr B) \BIImpl \Not A \CAnd \Not B \\ {} \\ A \COr \Not A \\ {} \\ \mathbf{Rule\; of\; Inference:} \\ \Infrule{A \quad A \IImpl B}{B} \end{axcol} \end{eqntbl} \end{table}
\sloppy In this formulation, which is taken from~\cite{Avr91a}, $A$, $B$, and $C$ are used as meta-variables ranging over all formulas of LP$^{\IImpl,\False}$. The axiom schemas on the left-hand side of Table~\ref{proofsystem-LPiimpl} and the single inference rule (modus ponens) constitute a Hilbert-style formulation of the positive fragment of classical propositional logic. The first four axiom schemas on the right-hand side of Table~\ref{proofsystem-LPiimpl} allow for the negation connective to be moved inward. The fifth axiom schema on the right-hand side of Table~\ref{proofsystem-LPiimpl} is the law of the excluded middle. This axiom schema can be thought of as saying that, for every proposition, the proposition or its negation is true, while leaving open the possibility that both are true. If we add the axiom schema $\Not A \IImpl (A \IImpl B)$, which says that any proposition follows from a contradiction, to the given Hilbert-style formulation of LP$^{\IImpl,\False}$, then we get a Hilbert-style formulation of classical propositional logic (see e.g.~\cite{Avr91a}).
We write $\pEnt$ for the syntactic logical consequence relation induced by the axiom schemas and inference rule of LP$^{\IImpl,\False}$.
The following outline of the semantics of LP$^{\IImpl,\False}$ is based on~\cite{Avr91a}. Like in the case of classical propositional logic, meanings are assigned to the formulas of LP$^{\IImpl,\False}$ by means of valuations. However, in addition to the two classical truth values $\true$ (true) and $\false$ (false), a third meaning $\both$ (both true and false) may be assigned.
A \emph{valuation} for LP$^{\IImpl,\False}$ is a function $\nu$ from the set of all formulas of LP$^{\IImpl,\False}$ to the set $\{\true,\false,\both\}$ such that for all formulas $A$ and $B$ of LP$^{\IImpl,\False}$: \begin{eqnarray*} \val{\False}{\nu} & = & \false, \\ \val{\Not A}{\nu} & = &
\left \{
\begin{array}{l@{\;\;}l}
\true & \mathrm{if}\; \val{A}{\nu} = \false \\
\false & \mathrm{if}\; \val{A}{\nu} = \true \\
\both & \mathrm{otherwise},
\end{array}
\right. \\ \val{A \CAnd B}{\nu} & = &
\left \{
\begin{array}{l@{\;\;}l}
\true & \mathrm{if}\; \val{A}{\nu} = \true \;\mathrm{and}\;
\val{B}{\nu} = \true \\
\false & \mathrm{if}\; \val{A}{\nu} = \false \;\mathrm{or}\;
\val{B}{\nu} = \false \\
\both & \mathrm{otherwise},
\end{array}
\right. \\ \val{A \COr B}{\nu} & = &
\left \{
\begin{array}{l@{\;\;}l}
\true & \mathrm{if}\; \val{A}{\nu} = \true \;\mathrm{or}\;
\val{B}{\nu} = \true \\
\false & \mathrm{if}\; \val{A}{\nu} = \false \;\mathrm{and}\;
\val{B}{\nu} = \false \\
\both & \mathrm{otherwise},
\end{array}
\right. \\ \val{A \IImpl B}{\nu} & = &
\left \{
\begin{array}{l@{\;\;}l}
\true & \mathrm{if}\; \val{A}{\nu} = \false \\
\val{B}{\nu} & \mathrm{otherwise}.
\end{array}
\right. \end{eqnarray*}
The classical truth-conditions and falsehood-conditions for the logical connectives are retained. Except for implications, a formula is classified as both-true-and-false exactly when when it cannot be classified as true or false by the classical truth-conditions and falsehood-conditions.
The definition of a valuation given above shows that the logical connectives of LP$^{\IImpl,\False}$ are (three-valued) truth-functional, which means that each $n$-ary connective represents a function from $\{\true,\false,\both\}^n$ to $\{\true,\false,\both\}$.
For LP$^{\IImpl,\False}$, the semantic logical consequence relation, denoted by $\mEnt$, is based on the idea that a valuation $\nu$ satisfies a formula $A$ if $\val{A}{\nu} \in \{\true,\both\}$. It is defined as follows: $\Gamma \mEnt A$ iff for every valuation $\nu$, either $\val{A'}{\nu} = \false$ for some $A' \in \Gamma$ or $\val{A}{\nu} \in \{\true,\both\}$. We have that the Hilbert-style formulation of LP$^{\IImpl,\False}$ is strongly complete with respect to its semantics, i.e.\ $\Gamma \pEnt A$ iff $\Gamma \mEnt A$ (see e.g.~\cite{BC04a}).
For all formulas $A$ of LP$^{\IImpl,\False}$ in which $\False$ does not occur, for all formulas $B$ of LP$^{\IImpl,\False}$ in which no propositional variable occurs that occurs in $A$, $\set{A, \Not A} \npEnt B$ if $\npEnt B$ (see e.g.~\cite{AA15a}). \footnote {We use the notation
${} \pEnt A$ for $\emptyset\hspace*{.02em} \pEnt A$,
${} \npEnt A$ for not $\emptyset\hspace*{.02em} \pEnt A$, and
$\Gamma \npEnt A$ for not $\Gamma \pEnt A$.} Hence, LP$^{\IImpl,\False}$ is a paraconsistent propositional logic.
For LP$^{\IImpl,\False}$, the logical equivalence relation $\LEqv$ is defined as for classical propositional logic: $A \LEqv B$ iff for every valuation $\nu$, $\val{A}{\nu} = \val{B}{\nu}$. Unlike in classical propositional logic, we do not have that $A \LEqv B$ iff ${} \pEnt A \BIImpl B$.
For LP$^{\IImpl,\False}$, the consistency property is defined as to be expected: $A$ is consistent iff for every valuation $\nu$, $\val{A}{\nu} \neq \both$.
The following are some important properties of LP$^{\IImpl,\False}$: \begin{list}{}
{\setlength{\leftmargin}{2.25em} \settowidth{\labelwidth}{(g)}} \item[(a)] \emph{containment in classical logic}: ${\pEnt} \subseteq {\clpEnt}$; \footnote {We use the symbol $\clpEnt$ to denote the logical consequence relation
of classical propositional logic.} \item[(b)] \emph{proper basic connectives}: for all sets $\Gamma$ of formulas of LP$^{\IImpl,\False}$ and all formulas $A$, $B$, and $C$ of LP$^{\IImpl,\False}$: \begin{list}{}
{\setlength{\leftmargin}{2.5em} \settowidth{\labelwidth}{(c)}} \item[(b$_1$)] $\Gamma \union \set{A} \pEnt B$\phantom{${} \CAnd {}$}\phantom{$C$} iff $\Gamma \pEnt A \IImpl B$, \item[(b$_2$)] $\Gamma \pEnt A \CAnd B$\phantom{,}\phantom{$C$} iff $\Gamma \pEnt A$ and $\Gamma \pEnt B$, \item[(b$_3$)] $\Gamma \union \set{A \COr B} \pEnt C$ iff $\Gamma \union \set{A} \pEnt C$ and $\Gamma \union \set{B} \pEnt C$; \end{list} \item[(c)] \emph{weakly maximal paraconsistency relative to classical logic}: for all formulas $A$ of LP$^{\IImpl,\False}$ with $\npEnt A$ and $\clpEnt A$, for the minimal consequence relation $\extpEnt$ such that ${\pEnt} \subseteq {\extpEnt}$ and $\extpEnt A$, for all formulas $B$ of LP$^{\IImpl,\False}$, $\extpEnt B$ iff $\clpEnt B$; \item[(d)] \emph{strongly maximal absolute paraconsistency}: for all logics $\mathcal{L}$ with the same logical constants and connectives as LP$^{\IImpl,\False}$ and a consequence relation $\extpEnt$ such that ${\pEnt} \subset {\extpEnt}$, $\mathcal{L}$ is not paraconsistent; \item[(e)] \emph{internalized notion of consistency}: $A$ is consistent iff ${} \pEnt (A \IImpl \False) \COr (\Not A \IImpl \False)$; \item[(f)] \emph{internalized notion of logical equivalence}: $A \LEqv B$ iff ${} \pEnt (A \BIImpl B) \CAnd (\Not A \BIImpl \Not B)$; \item[(g)] the laws given in Table~\ref{laws-lequiv} hold for the logical equivalence relation of LP$^{\IImpl,\False}$.
\begin{table}[!tb] \caption{Laws that hold for the logical equivalence relation of
LP$^{\IImpl,\False}$} \label{laws-lequiv} \begin{eqntbl} \begin{neqncol} (1) & A \CAnd \False \LEqv \False \\ (3) & A \CAnd \True \LEqv A \\ (5) & A \CAnd A \LEqv A \\ (7) & A \CAnd B \LEqv B \CAnd A \\ (9) & (A \CAnd B) \CAnd C \LEqv A \CAnd (B \CAnd C) \\ (11) & A \CAnd (B \COr C) \LEqv (A \CAnd B) \COr (A \CAnd C) \\ (13) & (A \IImpl B) \CAnd (A \IImpl C) \LEqv A \IImpl (B \CAnd C) \\ (15) & (A \COr \Not A) \IImpl B \LEqv B \end{neqncol} \qquad \begin{neqncol} (2) & A \COr \True \LEqv \True \\ (4) & A \COr \False \LEqv A \\ (6) & A \COr A \LEqv A \\ (8) & A \COr B \LEqv B \COr A \\ (10) & (A \COr B) \COr C \LEqv A \COr (B \COr C) \\ (12) & A \COr (B \CAnd C) \LEqv (A \COr B) \CAnd (A \COr C) \\ (14) & (A \IImpl C) \CAnd (B \IImpl C) \LEqv (A \COr B) \IImpl C \\ (16) & A \IImpl (B \IImpl C) \LEqv (A \CAnd B) \IImpl C \end{neqncol} \end{eqntbl} \end{table} \end{list}
Properties~(a)--(f) have been mentioned relatively often in the literature (see e.g.~\cite{AA15a,AAZ11b,AAZ11a,Avr99a,BC04a,CCM07a}). Properties~(a), (b$_1$), (c), and~(d) make LP$^{\IImpl,\False}$ an ideal paraconsistent logic in the sense made precise in~\cite{AAZ11b}. By property~(e), LP$^{\IImpl,\False}$ is also a logic of formal inconsistency in the sense made precise in~\cite{CCM07a}. Properties~(a)--(c) \linebreak[2] indicate that LP$^{\IImpl,\False}$ retains much of classical propositional logic. Actually, property~(c) can be strengthened to the following property: for all formulas $A$ of LP$^{\IImpl,\False}$, $\pEnt A$ iff $\clpEnt A$.
From Theorem~4.42 in~\cite{AA15a}, we know that there are exactly 8192 different three-valued paraconsistent propositional logics with properties~(a) and~(b). From Theorem~2 in~\cite{AAZ11b}, we know that properties~(c) and~(d) are common properties of all three-valued paraconsistent propositional logics with properties~(a) and~(b$_1$). From Fact~103 in~\cite{CCM07a}, we know that property~(f) is a common property of all three-valued paraconsistent propositional logics with properties~(a), (b) and~(e). Moreover, it is easy to see that that property~(e) is a common property of all three-valued paraconsistent propositional logics with properties~(a) and~(b). Hence, each three-valued paraconsistent propositional logic with properties~(a) and~(b) has properties~(c)--(f) as well.
Property~(g) is not a common property of all three-valued paraconsistent propositional logics with properties~(a) and~(b). To our knowledge, properties like property~(g) are not mentioned in the literature. However, like property~(f), property~(g) is essential for the process algebra presented in this paper. Among the 8192 three-valued paraconsistent propositional logics with properties~(a)--(e), \linebreak[2] which are considered desirable properties, LP$^{\IImpl,\False}$ is one out of four with the essential properties~(f) and~(g).
\begin{proposition}[Almost Uniqueness] \label{proposition-uniqueness} There are exactly four three-valued paraconsistent propositional logics with the logical constants and connectives of LP$^{\IImpl,\False}$ that have the properties~(a)--(g) mentioned above. \end{proposition}
\begin{proof} Because property~(f) is a common property of all 8192 three-valued paraconsistent propositional logics with properties~(a)--(e), it is sufficient to prove that, among these 8192 logics, there exists only one that has property~(g). Because `non-deterministic truth tables' that uniquely characterize the 8192 logics are given in~\cite{AAZ11b}, the theorem can be proved by showing that, for each of the connectives, only one of the ordinary truth tables represented by the non-deterministic truth table for that connective is compatible with the laws given in Table~\ref{laws-lequiv}. It can be shown by short routine case analyses that only one of the 8 ordinary truth tables represented by the non-deterministic truth tables for conjunction is compatible with laws~(1), (3), (5), and~(7) and only one of the 32 ordinary truth tables represented by the non-deterministic truth tables for disjunction is compatible with laws~(2), (4), (6), and~(8). The truth tables concerned are compatible with laws~(9)--(12) as well. Given the ordinary truth table for conjunction and disjunction so obtained, it can be shown by slightly longer routine case analyses that exactly four of the 16 ordinary truth tables represented by the non-deterministic truth table for implication are compatible with laws~(13)--(15). The four truth tables concerned are compatible with law~(16) as well. \qed \end{proof}
The next corollary follows from the proof of Proposition~\ref{proposition-uniqueness}.
\begin{corollary}[Uniqueness] \label{corollary-uniqueness} LP$^{\IImpl,\False}$ is the only three-valued paraconsistent pro\-positional logic with the logical constants and connectives of LP$^{\IImpl,\False}$ that has the properties~(a)--(g) mentioned above and moreover the property that the law $\Not \Not A \LEqv A$ holds for its logical equivalence relation. \end{corollary}
Corollary~\ref{corollary-uniqueness} may be of independent importance to the area of paraconsistent logics.
From now on, we will use the following abbreviations: $A \Iff B$ stands for $(A \BIImpl B) \CAnd (\Not A \BIImpl \Not B)$ and $\Cons A$ stands for $(A \IImpl \False) \COr (\Not A \IImpl \False)$.
In Section~\ref{sect-ctBPAps}, where we will use formulas of LP$^{\IImpl,\False}$ as terms, equality of formulas will be interpreted as logical equivalence. This means that equality of formulas can be formally proved using the fact that $A \LEqv B$ iff $\pEnt A \Iff B$. This fact also suggests that LP$^{\IImpl,\False}$ may be Blok-Pigozzi algebraizable~\cite{BP89a}. It is shown in~\cite{CCM07a} that actually all 8192 three-valued paraconsistent propositional logics referred to above are Blok-Pigozzi algebraizable. Although there must exist one, a conditional-equational axiomatization of the algebras concerned in the case of LP$^{\IImpl,\False}$ has not yet been devised. Owing to this, the equations derivable in the version of ACPps built on LP$^{\IImpl,\False}$ presented in this paper cannot always be derived by equational reasoning only.
\section{Contradiction-Tolerant BPA with Propositional Signals} \label{sect-ctBPAps}
BPAps is a subtheory of ACPps that does not support parallelism and communication. In this section, we present the contradiction-tolerant version of BPAps. In this version, which is called \ctBPAps, processes have their state to some extent visible. The visible part of the state of a process, called the signal emitted by the process, is a proposition of LP$^{\IImpl,\False}$. These propositions are not only used as signals emitted by processes, but also as conditions under which processes may proceed. The intuition is that the signal emitted by a process is a proposition that holds at its start and the condition under which processes may proceed is a proposition that must holds at its start.
In \ctBPAps, just as in BPAps, it is assumed that a fixed but arbitrary finite set $\Act$ of \emph{actions}, with $\dead \not\in \Act$, and a fixed but arbitrary finite set $\AProp$ of \emph{atomic propositions} have been given. We write $\Actd$ for $\Act \union \set{\dead}$.
The algebraic theory \ctBPAps\ has two sorts: \begin{itemize} \item the sort $\Proc$ of \emph{processes}; \item the sort $\Prop$ of \emph{propositions}. \end{itemize} \pagebreak[2]
The algebraic theory \ctBPAps\ has the following constants and operators to build terms of sort $\Prop$: \begin{itemize} \item for each $P \in \AProp$, the \emph{atomic proposition} constant $\const{P}{\Prop}$; \item the \emph{falsity} constant $\const{\False}{\Prop}$; \item the unary \emph{negation} operator $\funct{\Not}{\Prop}{\Prop}$; \item the binary \emph{conjunction} operator $\funct{\CAnd}{\Prop \x \Prop}{\Prop}$; \item the binary \emph{disjunction} operator $\funct{\COr}{\Prop \x \Prop}{\Prop}$; \item the binary \emph{implication} operator $\funct{\IImpl}{\Prop \x \Prop}{\Prop}$. \end{itemize}
The algebraic theory \ctBPAps\ has the following constants and operators to build terms of sort $\Proc$: \begin{itemize} \item the \emph{deadlock} constant $\const{\dead}{\Proc}$; \item for each $a \in \Act$, the \emph{action} constant $\const{a}{\Proc}$; \item the \emph{inaccessible process} constant $\const{\nex}{\Proc}$; \item the binary \emph{alternative composition} operator $\funct{\altc}{\Proc \x \Proc}{\Proc}$; \item the binary \emph{sequential composition} operator $\funct{\seqc}{\Proc \x \Proc}{\Proc}$; \item the binary \emph{guarded command} operator $\funct{\gc}{\Prop \x \Proc}{\Proc}$; \item the binary \emph{signal emission} operator $\funct{\emi}{\Prop \x \Proc}{\Proc}$. \end{itemize} It is assumed that there are infinitely many variables of sort $\Proc$, including $x$, $y$, and $z$.
We use infix notation for the binary operators. The following precedence conventions are used to reduce the need for parentheses. The operators to build terms of sort $\Prop$ bind stronger than the operators to build terms of sort $\Proc$. The operator ${} \seqc {}$ binds stronger than all other binary operators to build terms of sort $\Proc$ and the operator ${} \altc {}$ binds weaker than all other binary operators to build terms of sort $\Proc$.
Let $p$ and $q$ be closed terms of sort $\Proc$ and $\phi$ be a closed term of sort $\Prop$. Intuitively, the constants and operators to build terms of sort $\Proc$ can be explained as follows: \begin{itemize} \item $\dead$ is not capable of doing anything, the proposition that holds at the start of $\dead$ is $\True$; \item $a$ is only capable of performing action $a$ unconditionally and next terminating successfully, the proposition that holds at the start of $a$ is $\True$; \item $\nex$ is not capable of doing anything; there is an inconsistency at the start of~$\nex$; \item $p \altc q$ behaves either as $p$ or as $q$ but not both, the proposition that holds at the start of $p \altc q$ is the conjunction of the propositions that hold at the start of $p$ and $q$; \item $p \seqc q$ first behaves as $p$ and on successful termination of $p$ it next behaves as $q$, the proposition that holds at the start of $p \seqc q$ is the proposition that holds at the start of $p$; \item $\phi \gc p$ behaves as $p$ under condition $\phi$, the proposition that holds at the start of $\phi \gc p$ is the implication with $\phi$ as antecedent and the proposition that holds at the start of $p$ as consequent; \item $\phi \emi p$ behaves as $p$ if the proposition that holds at its start does not equal $\False$ and as $\nex$ otherwise, in the former case, the proposition that holds at the start of $\phi \emi p$ is the conjunction of $\phi$ and the proposition that holds at the start of $p$. \end{itemize}
The axioms of \ctBPAps\ are the axioms given in Table~\ref{axioms-ctBPAps}.
\begin{table}[!tb] \caption{Axioms of \ctBPAps} \label{axioms-ctBPAps} \begin{eqntbl} \begin{axcol} x \altc y = y \altc x & \ax{A1}\\ (x \altc y) \altc z = x \altc (y \altc z) & \ax{A2}\\ x \altc x = x & \ax{A3}\\ (x \altc y) \seqc z = x \seqc z \altc y \seqc z & \ax{A4}\\ (x \seqc y) \seqc z = x \seqc (y \seqc z) & \ax{A5}\\ x \altc \dead = x & \ax{A6}\\ \dead \seqc x = \dead & \ax{A7}\\ {}\\ \True \gc x = x & \ax{GC1}\\ \False \gc x = \dead & \ax{GC2}\\ \phi \gc \dead = \dead & \ax{GC3}\\ \phi \gc (x \altc y) = \phi \gc x \altc \phi \gc y & \ax{GC4}\\ \phi \gc x \seqc y = (\phi \gc x) \seqc y & \ax{GC5}\\ \phi \gc (\psi \gc x) = (\phi \CAnd \psi) \gc x & \ax{GC6}\\ (\phi \COr \psi) \gc x = \phi \gc x \altc \psi \gc x & \ax{GC7} \end{axcol} \qquad \begin{axcol} x \altc \nex = \nex & \ax{NE1}\\ \nex \seqc x = \nex & \ax{NE2}\\ a \seqc \nex = \dead & \ax{NE3}\\ {}\\{}\\{} \\ \phi = \psi
\mif \pEnt \phi \Iff \psi & \ax{IMP}\\ {}\\ \True \emi x = x & \ax{SE1}\\ \False \emi x = \nex & \ax{SE2}\\ \phi \emi \nex = \nex & \ax{SE3}\\ \phi \emi x \altc y = \phi \emi (x \altc y) & \ax{SE4}\\ (\phi \emi x) \seqc y = \phi \emi x \seqc y & \ax{SE5}\\ \phi \emi (\psi \emi x ) = (\phi \CAnd \psi) \emi x & \ax{SE6}\\ \phi \emi (\phi \gc x) = \phi \emi x & \ax{SE7}\\ \phi \gc (\psi \emi x) = (\phi \IImpl \psi) \emi (\phi \gc x) & \ax{SE8} \end{axcol} \end{eqntbl} \end{table}
In this table, $a$ stands for an arbitrary constant from $\Act \union \set{\dead}$, $\phi$ and $\psi$ stand for arbitrary closed terms of sort $\Prop$, and ${} \pEnt {}$ is the logical consequence relation of LP$^{\IImpl,\False}$.
A1--A7 are the axioms of \BPAd, the subtheory of ACP that does not support parallelism and communication (see e.g.~\cite{BW90}). NE1--NE3, GC1--GC7, and SE1--SE8 have been taken from~\cite{BB94b}, using a different numbering. \footnote {The axioms of \ctBPAps\ are not independent:
A3, A6, and A7 are derivable from GC1--GC7 and IMP,
NE1 and NE2 are derivable from SE1--SE8, and
SE3 is derivable from SE6 and IMP.} By IMP, the axioms of \ctBPAps\ include all equations $\phi = \psi$ for which $\phi \Iff \psi$ is a theorem of LP$^{\IImpl,\False}$. This is harmless because the connective $\Iff$, which is the internalization of the logical equivalence relation $\LEqv$ of LP$^{\IImpl,\False}$, is a congruence.
The following generalizations of axioms SE4 and SE7 are among the equations derivable from the axioms of \ctBPAps: \begin{ldispl} \begin{geqns} \phi \emi x \altc \psi \emi y =
(\phi \CAnd \psi) \emi (x \altc y)\;, \\ (\phi \CAnd \psi) \emi (\phi \gc x) = (\phi \CAnd \psi) \emi x\;, \\ \phi \emi ((\phi \CAnd \psi) \gc x) = \phi \emi (\psi \gc x)\;; \end{geqns} \end{ldispl} the following specialization of axiom SE4 is among the equations derivable from the axioms of \ctBPAps: \begin{ldispl} \begin{geqns} \phi \emi \dead \altc x = \phi \emi x\;; \end{geqns} \end{ldispl} and the following equations concerning the inaccessible process are among the equations derivable from the axioms of \ctBPAps: \begin{ldispl} \begin{geqns} \phi \emi \nex = \nex\;, \\ \phi \gc \nex = (\phi \IImpl \False) \emi \dead\;. \end{geqns} \end{ldispl} The derivable equations mentioned above are derivable from the axioms of BPAps as well. The equation $\phi \gc \nex = \Not \phi \emi \dead$, which is derivable from the axioms of BPAps, is not derivable from the axioms of \ctBPAps.
Let $\phi$ be a closed term of sort $\Prop$ such that not $\pEnt \phi \,\Iff\, \False$ and not $\pEnt \Not \phi \,\Iff\, \False$. Then, because not $\pEnt \phi \CAnd \Not \phi \,\Iff\, \False$, we have that $a \seqc (\phi \emi x \altc \Not \phi \emi y) =
a \seqc (\False \emi (x \altc y)) = \dead$, which is derivable from the axioms of BPAps, is not derivable from the axioms of \ctBPAps. This shows the main difference between \ctBPAps\ and BPAps: the alternative composition of two processes of which the propositions that hold at the start of them are contradictory does not lead to an inconsistency in \ctBPAps, whereas it does lead to an inconsistency in BPAps. This is why \ctBPAps\ is called the contradiction-tolerant version of BPAps.
Let $\phi$ be a closed term of sort $\Prop$ such that not $\pEnt \phi \,\Iff\, \False$ and not $\pEnt \Not \phi \,\Iff\, \False$. We can derive $a \seqc (\phi \emi b \altc \Not \phi \emi c) =
a \seqc ((\phi \CAnd \Not \phi) \emi (b \altc c)) = \dead$ from the axioms of BPAps because, in the case of BPAps, $a \seqc (\phi \emi b \altc \Not \phi \emi c)$ is not capable of doing anything. We can only derive $a \seqc (\phi \emi b \altc \Not \phi \emi c) =
a \seqc ((\phi \CAnd \Not \phi) \emi (b \altc c))$ from the axioms of \ctBPAps\ because, in the case of \ctBPAps, $a \seqc (\phi \emi b \altc \Not \phi \emi c)$ is capable of first performing $a$ and next either performing $b$ and after that terminating successfully or performing $c$ and after that terminating successfully --- although the proposition that holds at the start of the process that remains after performing $a$ is the contradiction $\phi \CAnd \Not \phi$.
Let $\phi$ be a closed term of sort $\Prop$ such that not $\pEnt \phi \,\Iff\, \False$ and not $\pEnt \Not \phi \,\Iff\, \False$. Then, because $\pEnt \Cons \phi \CAnd \phi \CAnd \Not \phi \,\Iff\, \False$, we have that $a \seqc (\Cons \phi \emi (\phi \emi x \altc \Not \phi \emi y)) =
a \seqc (\False \emi (x \altc y)) = \dead$ is derivable from the axioms of \ctBPAps. This shows that it can be enforced by means of a consistency proposition ($\Cons \phi$) that the alternative composition of two processes of which the propositions that hold at the start of them are contradictory leads to an inconsistency in \ctBPAps.
Hereafter, we will write $[\phi]$ for the equivalence class of $\phi$ modulo $\LEqv$. That is, $[\phi] = \set{\psi \where \phi \LEqv \psi}$. Hence, $[\phi] =
\set{\psi \where \;\pEnt \phi \Iff \psi}$.
All processes that can be described by a closed term of \ctBPAps, can be described by a basic term. The set $\cB$ of \emph{basic terms} is inductively defined by the following rules: \begin{itemize} \item $\nex \in \cB$; \item if $\phi \notin [\False]$, then $\phi \emi \dead \in \cB$; \item if $\phi \notin [\False]$ and $a \in \Act$, then $\phi \gc a \in \cB$; \item if $\phi \notin [\False]$, $a \in \Act$, and $p \in \cB$, then $\phi \gc a \seqc p \in \cB$; \item if $p,q \in \cB$, then $p \altc q \in \cB$. \end{itemize} \pagebreak[2] Each basic term can be written as $\nex$ or in the form \begin{ldispl} \chi \emi \dead \altc \Altc{i \in \set{1,\ldots,n}} \phi_i \gc a_i \seqc p_i \altc \Altc{j \in \set{1,\ldots,m}} \psi_j \gc b_j\;, \end{ldispl} where $n,m \in \Nat$, where $\chi \notin [\False]$, where $\phi_i \notin [\False]$, $a_i \in \Act$, and $p_i \in \cB$ for all $i \in \set{1,\ldots,n}$, and where $\psi_j \notin [\False]$ and $b_j \in \Act$ for all $j \in \set{1,\ldots,m}$. The subterm $\chi$ is called the \emph{root signal} of the basic term and the subterms $\phi_i \gc a_i \seqc p_i$ and $\psi_j \gc b_j$ are called the \emph{summands} of the basic term.
All closed \ctBPAps\ terms of sort $\Proc$ can be reduced to a basic term.
\begin{proposition}[Elimination] \label{proposition-elim-ctBPAps} For all closed \ctBPAps\ terms $p$ of sort $\Proc$, there exists a $q \in \cB$ such that $p = q$ is derivable from the axioms of \ctBPAps. \end{proposition}
\begin{proof} The proof is straightforward by induction on the structure of closed term $p$. If $p$ is of the form $\nex$, $a$, $p' \altc p''$ or $\phi \emi p'$, then it is trivial to show that there exists a $q \in \cB$ such that $p = q$ is derivable from the axioms of \ctBPAps. If $p$ is of the form $p' \seqc p''$ or $\phi \gc p'$, then it follows immediately from the induction hypothesis and the following claims: \begin{itemize} \item for all $p,p' \in \cB$, there exists a $p'' \in \cB$ such that $p \seqc p' = p''$ is derivable from the axioms of \ctBPAps; \item for all $\phi \notin [\False]$ and $p \in \cB$, there exists a $p' \in \cB$ such that $\phi \gc p = p'$ is derivable from the axioms of \ctBPAps. \end{itemize} Both claims are easily proved by induction on the structure of basic term $p$. \qed \end{proof}
\section{Semantics of \boldmath{$\ctBPAps$}} \label{sect-sem-ctBPAps}
In this section, we present a structural operational semantics of \ctBPAps, define a notion of bisimulation equivalence based on this semantics, and show that the axioms of \ctBPAps\ are sound and complete with respect to this bisimulation equivalence.
We start with the presentation of the structural operational semantics of \ctBPAps. The following transition relations on closed terms of sort $\Proc$ are used: \begin{itemize} \item for each $\ell \in C \x \Act$, a binary \emph{action step} relation ${} \step{\ell} {}$; \item for each $\ell \in C \x \Act$, a unary \emph{action termination} relation ${} \term{\ell}$; \item for each $\phi \in C$, a unary \emph{signal emission} relation $\sgn{\phi}$; \phantom{${} \step{\ell} {}$} \end{itemize} where $C$ is the set of all closed terms $\phi$ of sort $\Prop$ such that $\phi \notin [\False]$. We write $\astep{p}{\gact{\phi}{a}}{q}$ instead of $\tup{p,q} \in {\step{\tup{\phi,a}}}$, $\aterm{p}{\gact{\phi}{a}}$ instead of $p \in {\term{\tup{\phi,a}}}$, and $\rsgn{p} = \phi$ instead of $p \in \sgn{\phi}$.
These relations can be explained as follows: \begin{itemize} \item $\aterm{p}{\gact{\phi}{a}}$: $p$ is capable of performing action $a$ under condition $\phi$ and then terminating successfully; \item $\astep{p}{\gact{\phi}{a}}{q}$: $p$ is capable of performing action $a$ under condition $\phi$ and then proceeding as $q$; \item $\rsgn{p} = \phi$: the proposition that holds at the start of $p$ is $\phi$. \end{itemize}
The structural operational semantics of \ctBPAps\ is described by the transition rules given in Table~\ref{sos-ctBPAps}. \begin{table}[!tb] \caption{Transition rules for \ctBPAps} \label{sos-ctBPAps} \begin{druletbl} \Rule {\phantom{\aterm{a}{\gact{\phi}{a}}}} {\aterm{a}{\gact{\True}{a}}} \\ \RuleC {\aterm{x}{\gact{\phi}{a}},\; \rsgn{x \altc y} = \psi} {\aterm{x \altc y}{\gact{\phi}{a}}} {\psi \notin [\False]} & \RuleC {\aterm{y}{\gact{\phi}{a}},\; \rsgn{x \altc y} = \psi} {\aterm{x \altc y}{\gact{\phi}{a}}} {\psi \notin [\False]} \\ \RuleC {\astep{x}{\gact{\phi}{a}}{x'},\; \rsgn{x \altc y} = \psi} {\astep{x \altc y}{\gact{\phi}{a}}{x'}} {\psi \notin [\False]} & \RuleC {\astep{y}{\gact{\phi}{a}}{y'},\; \rsgn{x \altc y} = \psi} {\astep{x \altc y}{\gact{\phi}{a}}{y'}} {\psi \notin [\False]} \\ \RuleC {\aterm{x}{\gact{\phi}{a}},\; \rsgn{y} = \psi} {\astep{x \seqc y}{\gact{\phi}{a}}{y}} {\psi \notin [\False]} & \Rule {\astep{x}{\gact{\phi}{a}}{x'}} {\astep{x \seqc y}{\gact{\phi}{a}}{x' \seqc y}} \\ \RuleC {\aterm{x}{\gact{\phi}{a}}} {\aterm{\psi \gc x}{\gact{\phi \CAnd \psi}{a}}} {\phi \CAnd \psi \notin [\False]} & \RuleC {\astep{x}{\gact{\phi}{a}}{x'}} {\astep{\psi \gc x}{\gact{\phi \CAnd \psi}{a}}{x'}} {\phi \CAnd \psi \notin [\False]} \\ \RuleC {\aterm{x}{\gact{\phi}{a}},\; \rsgn{\psi \emi x} = \chi} {\aterm{\psi \emi x}{\gact{\phi}{a}}} {\chi \notin [\False]} & \RuleC {\astep{x}{\gact{\phi}{a}}{x'},\; \rsgn{\psi \emi x} = \chi} {\astep{\psi \emi x}{\gact{\phi}{a}}{x'}} {\chi \notin [\False]} \eqnsep \Rule {\phantom{\rsgn{\nex} = \False}} {\rsgn{\nex} = \False} \qquad \Rule {\phantom{\rsgn{a} = \True}} {\rsgn{a} = \True} \\ \Rule {\rsgn{x} = \phi,\; \rsgn{y} = \psi} {\rsgn{x \altc y} = \phi \CAnd \psi} \qquad \Rule {\rsgn{x} = \phi} {\rsgn{x \seqc y} = \phi} & \Rule {\rsgn{x} = \phi} {\rsgn{\psi \gc y} = \psi \IImpl \phi} \qquad \Rule {\rsgn{x} = \phi} {\rsgn{\psi \emi y} = \psi \CAnd \phi} \end{druletbl} \end{table}
In this table, $a$ stands for an arbitrary constant from $\Act \union \set{\dead}$ and $\phi$, $\psi$, and $\chi$ stand for arbitrary closed terms of sort $\Prop$.
A \emph{bisimulation} is a binary relation $R$ on closed \ctBPAps\ terms of sort $\Proc$ such that, for all closed \ctBPAps\ terms $p,q$ of sort $\Proc$ with $(p,q) \in R$, the following conditions hold: \begin{itemize} \item if $\astep{p}{\gact{\phi}{a}}{p'}$, then, for all valuations $\nu$ with $\nu(\rsgn{p}) \neq \false$ and $\nu(\phi) \neq \false$, there exists a closed term $\psi$ of sort $\Prop$ and a closed term $q'$ of sort $\Proc$ such that $\nu(\phi) = \nu(\psi)$, $\astep{q}{\gact{\psi}{a}}{q'}$, and $(p',q') \in R$; \item if $\astep{q}{\gact{\psi}{a}}{q'}$, then, for all valuations $\nu$ with $\nu(\rsgn{q}) \neq \false$ and $\nu(\psi) \neq \false$, there exists a closed term $\phi$ of sort $\Prop$ and a closed term $p'$ of sort $\Proc$ such that $\nu(\psi) = \nu(\phi)$, $\astep{p}{\gact{\phi}{a}}{p'}$, and $(p',q') \in R$; \item if $\aterm{p}{\gact{\phi}{a}}$, then, for all valuations $\nu$ with $\nu(\rsgn{p}) \neq \false$ and $\nu(\phi) \neq \false$, there exists a closed term $\psi$ of sort $\Prop$ such that $\nu(\phi) = \nu(\psi)$ and $\aterm{q}{\gact{\psi}{a}}$; \item if $\aterm{q}{\gact{\psi}{a}}$, then, for all valuations $\nu$ with $\nu(\rsgn{q}) \neq \false$ and $\nu(\psi) \neq \false$, there exists a closed term $\phi$ of sort $\Prop$ such that $\nu(\psi) = \nu(\phi)$ and $\aterm{p}{\gact{\phi}{a}}$; \item if $\rsgn{p} = \phi$, then there exists a closed term $\psi$ of sort $\Prop$ such that $\rsgn{q} = \psi$ and $\phi \LEqv \psi$; \item if $\rsgn{q} = \psi$, then there exists a closed term $\phi$ of sort $\Prop$ such that $\rsgn{p} = \phi$ and $\psi \LEqv \phi$. \end{itemize} Two closed \ctBPAps\ terms $p,q$ of sort $\Proc$ are \emph{bisimulation equivalent}, written $p \bisim q$, if there exists a bisimulation $R$ such that $(p,q) \in R$.
Let $R$ be a bisimulation such that $(p,q) \in R$. Then we say that $R$ is a bisimulation \emph{witnessing} $p \bisim q$.
Henceforth, we will loosely say that a relation contains all closed substitution instances of an equation if it contains all pairs $(t,t')$ such that $t = t'$ is a closed substitution instance of the equation.
Because a transition on one side may be simulated by a set of transitions on the other side, a bisimulation as defined above is called a \emph{splitting} bisimulation in~\cite{BM05a}.
Bisimulation equivalence is a congruence with respect to the operators of \ctBPAps.
\begin{proposition}[Congruence] \label{proposition-congr-ctBPAps} For all closed \ctBPAps\ terms $p,q,p',q'$ of sort $\Proc$ and closed \ctBPAps\ terms $\phi$ of sort $\Prop$, $p \bisim q$ and $p' \bisim q'$ imply $p \altc p' \bisim q \altc q'$, $p \seqc p' \bisim q \seqc q'$, $\phi \gc p \bisim \phi \gc q$, and $\phi \emi p \bisim \phi \emi q$. \end{proposition}
\begin{proof} We can reformulate the transition rules such that: \begin{itemize} \item bisimulation equivalence based on the reformulated transition rules according to the standard definition of bisimulation equivalence coincides with bisimulation equivalence based on the original transition rules according to the definition of bisimulation equivalence given above; \item the reformulated transition rules make up a complete transition system specification in panth format. \end{itemize} The reformulation goes like the one for the transition rules for BPAps outlined in~\cite{BB94b}.
The proposition follows now immediately from the well-known result that bisimulation equivalence according to the standard definition of bisimulation equivalence is a congruence if the transition rules concerned make up a complete transition system specification in panth format (see e.g.~\cite{FG96a}). \qed \end{proof}
The underlying idea of the reformulation referred to above is that we replace each transition \smash{$\astep{p}{\gact{\phi}{a}}{p'}$} by a transition \smash{$\astep{p}{\gact{\nu}{a}}{p'}$} for each valuation $\nu$ such that $\nu(\phi) \neq \false$, and likewise \smash{$\aterm{p}{\gact{\phi}{a}}$} and $\rsgn{p} = \phi$. Thus, in a bisimulation, a transition on one side must be simulated by a single transition on the other side. We did not present the reformulated structural operational semantics in this paper because it is, in our opinion, intuitively less appealing.
\ctBPAps\ is sound with respect to $\bisim$ for equations between closed terms.
\begin{theorem}[Soundness] \label{theorem-soundness-ctBPAps} For all closed \ctBPAps\ terms $p,q$ of sort $\Proc$, $p = q$ is derivable from the axioms of \ctBPAps\ only if $p \bisim q$. \end{theorem}
\begin{proof} Because of Proposition~\ref{proposition-congr-ctBPAps}, it is sufficient to prove the theorem for all closed substitution instances of each axiom of \ctBPAps.
For each axiom, we can construct a bisimulation $R$ witnessing $p \bisim q$ for all closed substitution instances $p = q$ of the axiom as follows: \begin{itemize} \item in the case of A1--A4 and A6, we take the relation $R$ that consists of all closed substitution instances of the axiom concerned and the equation $x = x$; \item in the case of A5, we take the relation $R$ that consists of all closed substitution instances of A5, SE5, and the equation $x = x$; \item in the case of A7, NE1--NE3, GC2--GC3, and SE2--SE3, we take the relation $R$ that consists of all closed substitution instances of the axiom concerned; \item in the case of GC1, GC4--GC7, SE1, and SE4--SE8, we take the relation $R$ that consists of all closed substitution instances of the axiom concerned and the equation $x = x$. \end{itemize} The laws from property~(8) of LP$^{\IImpl,\False}$ mentioned in Section~\ref{sect-LP-iimpl-false} are needed to check that these relations are witnessing ones. \qed \end{proof}
The proof of Theorem~\ref{theorem-soundness-ctBPAps} goes along the same line as the soundness proof for BPAps outlined in~\cite{BB94b}. The laws from property~(8) of LP$^{\IImpl,\False}$ mentioned in Section~\ref{sect-LP-iimpl-false} are laws that LP$^{\IImpl,\False}$ has in common with classical propositional logic. They are needed in the soundness proof for BPAps as well, but their use is left implicit in the proof outline given in~\cite{BB94b}.
\ctBPAps\ is complete with respect to $\bisim$ for equations between closed terms.
\begin{theorem}[Completeness] \label{theorem-completeness-ctBPAps} For all closed \ctBPAps\ terms $p,q$ of sort $\Proc$, $p = q$ is derivable from the axioms of \ctBPAps\ if $p \bisim q$. \end{theorem}
\begin{proof} By Proposition~\ref{proposition-elim-ctBPAps} and Theorem~\ref{theorem-soundness-ctBPAps}, it is sufficient to prove the theorem for basic terms $p$ and $q$.
For $p,p' \in \cB$, $p'$ is called a \emph{basic subterm} of $p$ if $p' \equiv p$ or there exists an $a \in \Act$ such that $a \seqc p'$ is a subterm of $p$.
We introduce a reduction relation $\tsred$ on $\cB$. The one-step reduction relation $\tsredi$ on $\cB$ is inductively defined as follows: \begin{itemize} \item if $p'$ is a basic subterm of $p$ and $q'$ occurs twice as summand in $p'$, then $p \tsredi r$ where $r$ is $p$ with one occurrence of $q'$ removed; \item if $p'$ is a basic subterm of $p$ and both $\phi \gc a \seqc q'$ and $\psi \gc a \seqc q'$ occur as summand in $p'$, then $p \tsredi r$ where $r$ is $p$ with the occurrence of $\phi \gc a \seqc q'$ replaced by $\phi \COr \psi \gc a \seqc q'$ and the occurrence of $\psi \gc a \seqc q'$ removed; \item if $p'$ is a basic subterm of $p$ and both $\phi \gc a$ and $\psi \gc a$ occur as summand in $p'$, then $p \tsredi r$ where $r$ is $p$ with the occurrence of $\phi \gc a$ replaced by $\phi \COr \psi \gc a$ and the occurrence of $\psi \gc a$ removed. \end{itemize} The one-step reductions correspond to sharing of double states and joining of transitions as in~\cite{BM05d}. The reduction relation $\tsred$ is the reflexive and transitive closure of $\tsredi$, and the conversion relation $\tscon$ is the reflexive and transitive closure of $\tsredi \union \tsredi^{-1}$.
The following are important properties of $\tsredi$: \begin{enumerate} \item[(1)] $\tsred$ is strongly normalizing; \item[(2)] for all $p,q \in \cB$, $p \tsred q$ only if $p \bisim q$; \item[(3)] for all $p,q \in \cB$ that are in normal form, $p \bisim q$ only if $p = q$ is derivable from axioms A1 and A2; \item[(4)] for all $p,q \in \cB$, $p \tsredi q$ only if $p = q$ is derivable from the axioms of \ctBPAps. \end{enumerate} Verifying properties~(1), (2), and~(4) is trivial. Property~(3) can be verified by proving it, simultaneously with the property \begin{enumerate} \item[] for all $p \in \cB$ that are in normal form, any bisimulation between $p$ and itself is the identity relation, \end{enumerate} by induction on the number of occurrences of a constant from $\Act$ in $p$ and $q$. The proof is similar to the proof of Theorem~2.12 from~\cite{BK85b}, but easier.
From properties~(1), (2) and~(3), it follows immediately that, for all $p,q \in \cB$, $p \bisim q$ iff $p \tscon q$. From this and property~(4), it follows immediately that, for all $p,q \in \cB$, $p \bisim q$ only if $p = q$ is derivable from the axioms of \ctBPAps. \qed \end{proof}
\section{Contradiction-Tolerant ACP with Propositional Signals} \label{sect-ctACPps}
In this section, we present the contradiction-tolerant version of ACPps. This version, which is called \ctACPps, is an extension of \ctBPAps\ that supports parallelism and communication.
In \ctACPps, just as in \ctBPAps, it is assumed that a fixed but arbitrary finite set $\Act$ of actions, with $\dead \not\in \Act$, and a fixed but arbitrary finite set $\AProp$ of atomic propositions have been given. In \ctACPps, it is further assumed that a fixed but arbitrary commutative and associative \emph{communication} function $\funct{\commm}{\Actd \x \Actd}{\Actd}$, such that $\dead \commm a = \dead$ for all $a \in \Actd$, has been given. The function $\commm$ is regarded to give the result of synchronously performing any two actions for which this is possible, and to be $\dead$ otherwise.
The algebraic theory \ctACPps\ has the sorts, constants and operators of \ctBPAps\ and in addition the following operators: \begin{itemize} \item the binary \emph{parallel composition} operator $\funct{\parc}{\Proc \x \Proc}{\Proc}$; \item the binary \emph{left merge} operator $\funct{\leftm}{\Proc \x \Proc}{\Proc}$; \item the binary \emph{communication merge} operator $\funct{\commm}{\Proc \x \Proc}{\Proc}$; \item for each $H \subseteq \Act$, the unary \emph{encapsulation} operator $\funct{\encap{H}}{\Proc}{\Proc}$. \end{itemize} We use infix notation for the additional binary operators as well.
The constants and operators of \ctACPps\ to build terms of sort $\Proc$ are the constants and operators of \ACP\ and additionally the guarded command operator and the signal emission operator.
Let $p$ and $q$ be closed terms of sort $\Proc$. Intuitively, the additional operators can be explained as follows: \begin{itemize} \item $p \parc q$ behaves as the process that proceeds with $p$ and $q$ in parallel, the proposition that holds at the start of $p \parc q$ is the conjunction of the propositions that hold at the start of $p$ and $q$; \item $p \leftm q$ behaves the same as $p \parc q$, except that it starts with performing an action of $p$, the proposition that holds at the start of $p \leftm q$ is the conjunction of the propositions that hold at the start of $p$ and $q$; \item $p \commm q$ behaves the same as $p \parc q$, except that it starts with performing an action of $p$ and an action of $q$ synchronously, the proposition that holds at the start of $p \commm q$ is the conjunction of the propositions that hold at the start of $p$ and~$q$; \item $\encap{H}(p)$ behaves the same as $p$, except that the actions in $H$ are blocked, the proposition that holds at the start of $\encap{H}(p)$ is the proposition that holds at the start of $p$. \end{itemize}
The axioms of \ctACPps\ are the axioms of \ctBPAps\ and the additional axioms given in Table~\ref{axioms-ctACPps}. \begin{table}[!tb] \caption{Additional axioms for \ctACPps} \label{axioms-ctACPps} \begin{eqntbl} \begin{axcol} x \parc y =
x \leftm y \altc y \leftm x \altc x \commm y & \ax{CM1} \\ a \leftm x = a \seqc x \altc \encap{\Act}(x) & \ax{CM2S}\\ a \seqc x \leftm y = a \seqc (x \parc y) \altc \encap{\Act}(y)
& \ax{CM3S}\\ (x \altc y) \leftm z = x \leftm z \altc y \leftm z & \ax{CM4} \\ a \seqc x \commm b = (a \commm b) \seqc x & \ax{CM5} \\ a \commm b \seqc x = (a \commm b) \seqc x & \ax{CM6} \\ a \seqc x \commm b \seqc y = (a \commm b) \seqc (x \parc y)
& \ax{CM7} \\ (x \altc y) \commm z = x \commm z \altc y \commm z & \ax{CM8} \\ x \commm (y \altc z) = x \commm y \altc x \commm z & \ax{CM9} \\ {} \\ (\phi \gc x) \leftm y = \phi \gc (x \leftm y) \altc \encap{\Act}(y)
& \ax{GC8S}\\ (\phi \gc x) \commm y = \phi \gc (x \commm y) \altc \encap{\Act}(y)
& \ax{GC9S}\\ x \commm (\phi \gc y) = \phi \gc (x \commm y) \altc \encap{\Act}(x)
& \ax{GC10S}\\ \encap{H}(\phi \gc x) = \phi \gc \encap{H}(x) & \ax{GC11} \end{axcol} \qquad \begin{axcol} a \commm b = b \commm a & \ax{C1} \\ (a \commm b) \commm c = a \commm (b \commm c) & \ax{C2} \\ \dead \commm a = \dead & \ax{C3} \\ {} \\ {} \\ \encap{H}(a) = a
\mif a \not\in H & \ax{D1} \\ \encap{H}(a) = \dead
\mif a \in H & \ax{D2} \\ \encap{H}(x \altc y) =
\encap{H}(x) \altc \encap{H}(y) & \ax{D3} \\ \encap{H}(x \seqc y) =
\encap{H}(x) \seqc \encap{H}(y) & \ax{D4} \\ {} \\ (\phi \emi x) \leftm y = \phi \emi (x \leftm y) & \ax{SE9}\\ (\phi \emi x) \commm y = \phi \emi (x \commm y) & \ax{SE10}\\ x \commm (\phi \emi y) = \phi \emi (x \commm y) & \ax{SE11}\\ \encap{H}(\phi \emi x) = \phi \emi \encap{H}(x) & \ax{SE12} \end{axcol} \end{eqntbl} \end{table}
In this table, $a,b,c$ stand for arbitrary constants from $\Act \union \set{\dead}$ and $\phi$ stands for an arbitrary closed term of sort $\Prop$.
A1--A7, CM1--CM9 with CM1S and CM2S replaced by $a \leftm x = a \seqc x$ and $a \seqc x \leftm y = a \seqc (x \parc y)$, C1--C3, and D1--D4 are the axioms of \ACP\ (see e.g.~\cite{BW90}). GC11 and SE9--SE12 have been taken from~\cite{BB94b} and GC9S and GC10S have been taken from~\cite{BB94b} with subterms of the form $\rsgn{x} \emi \dead$ replaced by $\encap{\Act}(x)$. CM2S, CM3S and GC8S differ really from the corresponding axioms in~\cite{BB94b} due to the choice of having as the proposition that holds at the start of the left merge of two processes, as in the case of the communication merge, always the conjunction of the propositions that hold at the start of the two processes.
The following equations are among the equations derivable from the axioms of \ctACPps: \begin{ldispl} (\phi \emi x) \parc (\psi \emi y) = (\phi \CAnd \psi ) \emi (x \parc y)\;, \\
x \parc \nex = \nex\;,
\nex \parc x = \nex\;.
\end{ldispl}
Let $\phi$ be a closed term of sort $\Prop$ such that not $\pEnt \phi \,\Iff\, \False$ and not $\pEnt \Not \phi \,\Iff\, \False$. Then, because not $\pEnt \phi \CAnd \Not \phi \,\Iff\, \False$, we have that $a \seqc (\phi \emi x \parc \Not \phi \emi y) =
a \seqc (\False \emi (x \parc y)) = \dead$, which is derivable from the axioms of ACPps, is not derivable from the axioms of \ctACPps. This shows the main difference between \ctACPps\ and ACPps: the par\-allel composition of two processes of which the propositions that hold at the start of them are contradictory does not lead to an inconsistency in \ctACPps, whereas it does lead to an inconsistency in ACPps. This is why \ctACPps\ is called the contradiction-tolerant version of ACPps.
Let $\phi$ be a closed term of sort $\Prop$ such that not $\pEnt \phi \,\Iff\, \False$ and not $\pEnt \Not \phi \,\Iff\, \False$. Assume that $b \commm c = d$. Then we can derive $a \seqc (\phi \emi b \parc \Not \phi \emi c) =
a \seqc ((\phi \CAnd \Not \phi) \emi (b \parc c)) =
a \seqc
((\phi \CAnd \Not \phi) \emi (b \seqc c \altc c \seqc b \altc d)) =
\dead$ from the axioms of BPAps because, in the case of BPAps, $a \seqc (\phi \emi b \parc \Not \phi \emi c)$ is not capable of doing anything. We can only derive $a \seqc (\phi \emi b \parc \Not \phi \emi c) =
a \seqc ((\phi \CAnd \Not \phi) \emi (b \parc c)) =
a \seqc
((\phi \CAnd \Not \phi) \emi (b \seqc c \altc c \seqc b \altc d))$ from the axioms of \ctBPAps\ because, in the case of \ctBPAps, $a \seqc (\phi \emi b \parc \Not \phi \emi c)$ is capable of first performing $a$ and next either performing $b$ and $c$ in either order and after that terminating successfully or performing $d$ and after that terminating successfully --- although the proposition that holds at the start of the process that remains after performing $a$ is the contradiction $\phi \CAnd \Not \phi$.
Let $\phi$ be a closed term of sort $\Prop$ such that not $\pEnt \phi \,\Iff\, \False$ and not $\pEnt \Not \phi \,\Iff\, \False$. Then, because $\pEnt \Cons \phi \CAnd \phi \CAnd \Not \phi \,\Iff\, \False$, we have that $a \seqc (\Cons \phi \emi (\phi \emi x \parc \Not \phi \emi y)) =
a \seqc (\False \emi (x \parc y)) = \dead$ is derivable from the axioms of \ctACPps. This shows that it can be enforced by means of a consistency proposition ($\Cons \phi$) that the parallel composition of two processes of which the propositions that hold at the start of them are contradictory leads to an inconsistency in \ctACPps.
All closed \ctACPps\ terms of sort $\Proc$ can be reduced to a basic term.
\begin{proposition}[Elimination] \label{proposition-elim-ctACPps} For all closed \ctACPps\ terms $p$ of sort $\Proc$, there exists a $q \in \cB$ such that $p = q$ is derivable from the axioms of \ctACPps. \end{proposition}
\begin{proof} The proof is straightforward by induction on the structure of closed term $p$. If $p$ is of the form $\nex$, $a$, $p' \altc p''$, $p' \seqc p''$, $\phi \gc p'$ or $\phi \emi p'$, then it follows immediately from the induction hypothesis and Proposition~\ref{proposition-elim-ctBPAps} that there exists a $q \in \cB$ such that $p = q$ is derivable from the axioms of \ctACPps. If $p$ is of the form $p' \parc p''$, $p' \leftm p''$, $p' \commm p''$ or $\encap{H}(p')$, then it follows immediately from the induction hypothesis and claims similar to the ones from the proof of Proposition~\ref{proposition-elim-ctBPAps}. The claims concerning $\parc$, $\leftm$, and $\commm$ are easily proved simultaneously by structural induction. The claim concerning $\encap{H}$ is easily proved by structural induction. \qed \end{proof}
\section{Semantics of \boldmath{$\ctACPps$}} \label{sect-sem-ctACPps}
In this section, we present a structural operational semantics of \ctACPps\ and show that the axioms of \ctACPps\ are sound and complete with respect to this bisimulation equivalence.
We start with the presentation of the structural operational semantics of \ctACPps. The structural operational semantics of \ctACPps\ is described by the transition rules for \ctBPAps\ and the additional transition rules given in Table~\ref{sos-ctACPps}.
\renewcommand{\textfraction}{0} \renewcommand{\topfraction}{1} \begin{table}[!p] \caption{Additional transition rules for \ctACPps} \label{sos-ctACPps} \begin{ruletbl} \RuleC {\aterm{x}{\gact{\phi}{a}},\;
\rsgn{x \parc y} = \psi,\; \rsgn{y} = \chi} {\astep{x \parc y}{\gact{\phi}{a}}{y}} {\psi, \chi \notin [\False]} \\ \RuleC {\aterm{y}{\gact{\phi}{a}},\;
\rsgn{x \parc y} = \psi,\; \rsgn{x} = \chi} {\astep{x \parc y}{\gact{\phi}{a}}{x}} {\psi, \chi \notin [\False]} \\ \RuleC {\astep{x}{\gact{\phi}{a}}{x'},\;
\rsgn{x \parc y} = \psi,\; \rsgn{x' \parc y} = \chi} {\astep{x \parc y}{\gact{\phi}{a}}{x' \parc y}} {\psi, \chi \notin [\False]} \\ \RuleC {\astep{y}{\gact{\phi}{a}}{y'},\;
\rsgn{x \parc y} = \psi,\; \rsgn{x \parc y'} = \chi} {\astep{x \parc y}{\gact{\phi}{a}}{x \parc y'}} {\psi, \chi \notin [\False]} \\ \RuleC {\aterm{x}{\gact{\phi}{a}},\; \aterm{y}{\gact{\psi}{b}},\;
\rsgn{x \parc y} = \chi} {\aterm{x \parc y}{\gact{\phi \CAnd \psi}{c}}} {a \commm b = c,\; \phi \CAnd \psi, \chi \notin [\False]} \\ \RuleC {\aterm{x}{\gact{\phi}{a}},\; \astep{y}{\gact{\psi}{b}}{y'},\;
\rsgn{x \parc y} = \chi} {\astep{x \parc y}{\gact{\phi \CAnd \psi}{c}}{y'}} {a \commm b = c,\; \phi \CAnd \psi, \chi \notin [\False]} \\ \RuleC {\astep{x}{\gact{\phi}{a}}{x'},\; \aterm{y}{\gact{\psi}{b}},\;
\rsgn{x \parc y} = \chi} {\astep{x \parc y}{\gact{\phi \CAnd \psi}{c}}{x'}} {a \commm b = c,\; \phi \CAnd \psi, \chi \notin [\False]} \\ \RuleC {\astep{x}{\gact{\phi}{a}}{x'},\; \astep{y}{\gact{\psi}{b}}{y'},\;
\rsgn{x \parc y} = \chi,\; \rsgn{x' \parc y'} = \chi'} {\astep{x \parc y}{\gact{\phi \CAnd \psi}{c}}{x' \parc y'}} {a \commm b = c,\; \phi \CAnd \psi, \chi, \chi' \notin [\False]} \\ \RuleC {\aterm{x}{\gact{\phi}{a}},\;
\rsgn{x \leftm y} = \psi,\; \rsgn{y} = \chi} {\astep{x \leftm y}{\gact{\phi}{a}}{y}} {\psi, \chi \notin [\False]} \\ \RuleC {\astep{x}{\gact{\phi}{a}}{x'},\;
\rsgn{x \leftm y} = \psi,\; \rsgn{x' \parc y} = \chi} {\astep{x \leftm y}{\gact{\phi}{a}}{x' \parc y}} {\psi, \chi \notin [\False]} \\ \RuleC {\aterm{x}{\gact{\phi}{a}},\; \aterm{y}{\gact{\psi}{b}},\;
\rsgn{x \commm y} = \chi} {\aterm{x \commm y}{\gact{\phi \CAnd \psi}{c}}} {a \commm b = c,\; \phi \CAnd \psi, \chi \notin [\False]} \\ \RuleC {\aterm{x}{\gact{\phi}{a}},\; \astep{y}{\gact{\psi}{b}}{y'},\;
\rsgn{x \commm y} = \chi} {\astep{x \commm y}{\gact{\phi \CAnd \psi}{c}}{y'}} {a \commm b = c,\; \phi \CAnd \psi, \chi \notin [\False]} \\ \RuleC {\astep{x}{\gact{\phi}{a}}{x'},\; \aterm{y}{\gact{\psi}{b}},\;
\rsgn{x \commm y} = \chi} {\astep{x \commm y}{\gact{\phi \CAnd \psi}{c}}{x'}} {a \commm b = c,\; \phi \CAnd \psi, \chi \notin [\False]} \\ \RuleC {\astep{x}{\gact{\phi}{a}}{x'},\; \astep{y}{\gact{\psi}{b}}{y'},\;
\rsgn{x \commm y} = \chi,\; \rsgn{x' \parc y'} = \chi'} {\astep{x \commm y}{\gact{\phi \CAnd \psi}{c}}{x' \parc y'}} {a \commm b = c,\; \phi \CAnd \psi, \chi, \chi' \notin [\False]} \\ \RuleC {\aterm{x}{\gact{\phi}{a}}} {\aterm{\encap{H}(x)}{\gact{\phi}{a}}} {a \not\in H} \qquad \RuleC {\astep{x}{\gact{\phi}{a}}{x'}} {\astep{\encap{H}(x)}{\gact{\phi}{a}}{\encap{H}(x')}} {a \not\in H} \eqnsep \Rule {\rsgn{x} = \phi,\; \rsgn{y} = \psi} {\rsgn{x \parc y} = \phi \CAnd \psi} \qquad \Rule {\rsgn{x} = \phi,\; \rsgn{y} = \psi} {\rsgn{x \leftm y} = \phi \CAnd \psi} \qquad \Rule {\rsgn{x} = \phi,\; \rsgn{y} = \psi} {\rsgn{x \commm y} = \phi \CAnd \psi} \qquad \Rule {\rsgn{x} = \phi} {\rsgn{\encap{H}(x)} = \phi} \end{ruletbl} \end{table}
In these tables, $a$, $b$, and $c$ stand for arbitrary constants from $\Act \union \set{\dead}$ and $\phi$, $\psi$, $\chi$, and $\chi'$ stand for arbitrary closed terms of sort $\Prop$.
In Sections~\ref{sect-ctBPAps} and~\ref{sect-ctACPps}, we have touched upon the main difference between \ctACPps\ and ACPps: the alternative and parallel composition of two processes of which the propositions that hold at the start of them are contradictory does not lead to an inconsistency in \ctACPps, whereas it does lead to an inconsistency in ACPps. However, the transition rules for \ctACPps\ and ACPps seem to be the same. The difference is fully accounted for by the fact that $[\False]$, the equivalence class of $\False$ modulo logical equivalence, contains in the case of LP$^{\IImpl,\False}$ only propositions of the form $\phi \CAnd \Not \phi$ with $\phi$ such that either $\phi \LEqv \False$ or $\Not \phi \LEqv \False$, whereas it contains in the case of classical propositional logic all propositions of the form $\phi \CAnd \Not \phi$.
By this fact, in the case of \ctACPps, $a \seqc (\phi \emi b \parc \Not \phi \emi c)$ from the example preceding Proposition~\ref{proposition-elim-ctACPps} is capable of first performing $a$ and next either performing $b$ and $c$ in either order and after that terminating successfully or performing $d$ and after that terminating successfully --- although the proposition that holds at the start of the process that remains after performing $a$ is the contradiction $\phi \CAnd \Not \phi$ --- and, in the case of ACPps, it is not capable of doing anything.
Bisimulation equivalence is a congruence with respect to the operators of \ctACPps.
\begin{proposition}[Congruence] \label{proposition-congr-ctACPps} For all closed \ctACPps\ terms $p,q,p',q'$ of sort $\Proc$ and closed \ctACPps\ terms $\phi$ of sort $\Prop$, $p \bisim q$ and $p' \bisim q'$ imply $p \altc p' \bisim q \altc q'$, $p \seqc p' \bisim q \seqc q'$, $\phi \gc p \bisim \phi \gc q$, $\phi \emi p \bisim \phi \emi q$, $p \parc p' \bisim q \parc q'$, $p \leftm p' \bisim q \leftm q'$, $p \commm p' \bisim q \commm q'$, and $\encap{H}(p) \bisim \encap{H}(q)$. \end{proposition}
\begin{proof} The proof goes along the same line as the proof of Proposition~\ref{proposition-congr-ctBPAps}. \qed \end{proof}
\ctACPps\ is sound with respect to $\bisim$ for equations between closed terms.
\begin{theorem}[Soundness] \label{theorem-soundness-ctACPps} For all closed \ctACPps\ terms $p,q$ of sort $\Proc$, $p = q$ is derivable from the axioms of \ctACPps\ only if $p \bisim q$. \end{theorem}
\begin{proof} Because of Proposition~\ref{proposition-congr-ctACPps}, it is sufficient to prove the theorem for all closed substitution instances of each axiom of \ctACPps.
For each axiom, we can construct a bisimulation $R$ witnessing $p \bisim q$ for all closed substitution instances $p = q$ of the axiom as follows: \begin{itemize} \item in the case of the axioms of \ctBPAps,we take the same relation as in the proof of Theorem~\ref{theorem-soundness-ctBPAps}; \item in the case of CM1, we take the relation $R$ that consists of all closed substitution instances of CM1, the equation $x \parc y = y \parc x$, and the equation $x = x$; \item in the case of CM2S--CM9, we take the relation $R$ that consists of all closed substitution instances of the axiom concerned and the equation $x = x$; \item in the case of C1--C3 and D1--D2, we take the relation $R$ that consists of all closed substitution instances of the axiom concerned; \item in the case of D3--D4, GC8S--GC11, and SE9--SE12, we take the relation $R$ that consists of all closed substitution instances of the axiom concerned and the equation $x = x$. \end{itemize} The laws from property~(8) of LP$^{\IImpl,\False}$ mentioned in Section~\ref{sect-LP-iimpl-false} are needed to check that these relations are witnessing ones. \qed \end{proof}
\ctACPps\ is complete with respect to $\bisim$ for equations between closed terms.
\begin{theorem}[Completeness] \label{theorem-completeness-ctACPps} For all closed \ctACPps\ terms $p,q$ of sort $\Proc$, $p = q$ is derivable from the axioms of \ctACPps\ if $p \bisim q$. \end{theorem}
\begin{proof} We have that the axioms of \ctBPAps\ are complete with respect to $\bisim$ (Theorem~\ref{theorem-completeness-ctBPAps}), the axioms of \ctACPps\ are sound with respect to $\bisim$ (Theorem~\ref{theorem-soundness-ctACPps}), and for each closed \ctACPps\ term $p$ of sort $\Proc$, there exists a closed \ctBPAps\ term $q$ such that $p = q$ is derivable from the axioms of \ctACPps\ (Proposition~\ref{proposition-elim-ctACPps}). By Theorem 3.14 from~\cite{Ver94a}, the result immediately follows from this and the claim that the set of transition rules for \ctACPps\ is an operational conservative extension of the set of transition rules for \ctBPAps.
This claim can easily be proved if we reformulate the transition rules for \ctACPps\ in the same way as the transition rules for \ctBPAps\ have been reformulated to prove Proposition~\ref{proposition-congr-ctBPAps}. The operational conservativity can then easily be proved by verifying that the reformulated transition rules for \ctACPps\ makes up a complete transition system specification, the reformulated transition rules for \ctBPAps --- which are included in the reformulated transition rules for \ctACPps --- are source-dependent, and the additional transition rules have fresh sources (see e.g.~\cite{FV98a}). \qed \end{proof}
\section{State Operators} \label{sect-ctACPps+SO}
In this section, we extend \ctACPps\ with state operators. The resulting theory is called \ctACPps\textup{{+}SO}. The state operators introduced here generalize the state operators added to \ACP\ in~\cite{BB88}.
The state operators from~\cite{BB88} were introduced to make it easy to represent the execution of a process in a state. The basic idea was that the execution of an action in a state has effect on the state, i.e.\ it causes a change of state. Moreover, there is an action left when an action is executed in a state. The main difference between the original state operators and the state operators introduced here is that, in the case of the latter, the state in which a process is executed determines the proposition that holds at its start. Thus, one application of a state operator may replace many applications of the signal emission operator.
It is assumed that a fixed but arbitrary set $S$ of \emph{states} has been given, together with functions $\funct{\act}{\Act \x S}{\Actd}$, $\funct{\eff}{\Act \x S}{S}$, and $\funct{\sig}{S}{B}$, where $B$ is the set of all closed terms $\phi$ of sort $\Prop$.
For each $s \in S$, we add a unary \emph{state} operator $\funct{\state{s}}{\Proc}{\Proc}$ to the operators of \ctACPps.
The state operator $\state{s}$ allows, given the above-mentioned functions, processes to be executed in a state. Let $p$ be a closed term of sort $\Proc$. Then $\state{s}(p)$ is the process $p$ executed in state $s$. The function $\act$ gives, for each action $a$ and state $s$, the action that results from executing $a$ in state $s$. The function $\eff$ gives, for each action $a$ and state $s$, the state that results from executing $a$ in state $s$. The function $\sig$ gives, for each state $s$, the proposition that holds at the start of any process executed in state $s$.
The additional axioms for $\state{s}$, where $s \in S$, are given in Table~\ref{axioms-SO}.
\begin{table}[!tb] \caption{Axioms for state operators} \label{axioms-SO} \begin{eqntbl} \begin{axcol} \state{s}(a) = \sig(s) \emi \act(a,s) & \ax{SO1} \\ \state{s}(a \seqc x) = \sig(s) \emi \act(a,s) \seqc \state{\eff(a,s)}(x)
& \ax{SO2} \\ \state{s}(x \altc y) = \state{s}(x) \altc \state{s}(y) & \ax{SO3} \\ \state{s}(\phi \gc x) = \sig(s) \emi (\phi \gc \state{s}(x))
& \ax{SO4} \\ \state{s}(\phi \emi x) = \phi \emi \state{s}(x) & \ax{SO5} \end{axcol} \end{eqntbl} \end{table}
In this table, $a$ stands for an arbitrary constant from $\Act \union \set{\dead}$ and $\phi$ stands for an arbitrary closed term of sort $\Prop$.
SO1--SO5 have been taken from~\cite{BB94b}.
The following equations are among the equations derivable from the axioms of \ctACPps{+}SO: \begin{ldispl} \state{s}(\nex) = \nex\;, \qquad \state{s}(\dead) = \sig(s) \emi \dead\;. \end{ldispl}
All closed \ctACPps{+}SO terms of sort $\Proc$ can be reduced to a basic term.
\begin{proposition}[Elimination] \label{proposition-elim-ctACPps+SO} For all \ctACPps\textup{{+}SO} closed terms $p$ of sort $\Proc$, there exists a $q \in \cB$ such that $p = q$ is derivable from the axioms of \ctACPps\textup{{+}SO}. \end{proposition}
\begin{proof} The proof goes along the same line as the proof of Proposition~\ref{proposition-elim-ctBPAps}. \qed \end{proof}
The additional transition rules for the state operators are given in Table~\ref{sos-SO}.
\begin{table}[!tb] \caption{Transition rules for state operators} \label{sos-SO} \begin{ruletbl} \RuleC {\aterm{x}{\gact{\phi}{a}},\; \rsgn{\state{s}(x)} = \psi} {\aterm{\state{s}(x)}{\gact{\phi}{\act(a,s)}}} {\act(a,s) \neq \dead,\; \psi \notin [\False]} \\ \RuleC {\astep{x}{\gact{\phi}{a}}{x'},\;
\rsgn{\state{s}(x)} = \psi,\; \rsgn{\state{\eff(a,s)}(x')} = \chi} {\astep{\state{s}(x)}{\gact{\phi}{\act(a,s)}}{\state{\eff(a,s)}(x')}} {\act(a,s) \neq \dead,\; \psi, \chi \notin [\False]} \\ \RuleC {\rsgn{x} = \phi} {\rsgn{\state{s}(x)} = \phi \CAnd \psi} {\sig(s) = \psi} \end{ruletbl} \end{table}
In this table, $a$ stands for an arbitrary constant from $\Act \union \set{\dead}$ and $\phi$ stands for an arbitrary closed term of sort $\Prop$.
Bisimulation equivalence is a congruence with respect to the operators of \ctACPps{+}SO.
\begin{proposition}[Congruence] \label{proposition-congr-ctACPps+SO} For all closed \ctACPps\textup{{+}SO} terms $p,q,p',q'$ of sort $\Proc$ and closed \ctACPps\textup{{+}SO} terms $\phi$ of sort $\Prop$, $p \bisim q$ and $p' \bisim q'$ imply $p \altc p' \bisim q \altc q'$, $p \seqc p' \bisim q \seqc q'$, $\phi \gc p \bisim \phi \gc q$, $\phi \emi p \bisim \phi \emi q$, $p \parc p' \bisim q \parc q'$, $p \leftm p' \bisim q \leftm q'$, $p \commm p' \bisim q \commm q'$, $\encap{H}(p) \bisim \encap{H}(q)$, and $\state{s}(p) \bisim \state{s}(q)$. \end{proposition}
\begin{proof} The proof goes along the same line as the proof of Proposition~\ref{proposition-congr-ctBPAps}. \qed \end{proof}
\ctACPps\textup{{+}SO} is sound with respect to $\bisim$ for equations between closed terms.
\begin{theorem}[Soundness] \label{theorem-soundness-ctACPps+SO} For all closed \ctACPps\textup{{+}SO} terms $p,q$ of sort $\Proc$, \mbox{$p = q$} is derivable from the axioms of \ctACPps\textup{{+}SO} only if $p \bisim q$. \end{theorem}
\begin{proof} The proof goes along the same line as the proof of Theorem~\ref{theorem-soundness-ctACPps}. \qed \end{proof}
\ctACPps\textup{{+}SO} is complete with respect to $\bisim$ for equations between closed terms.
\begin{theorem}[Completeness] \label{theorem-completeness-ctACPps+SO} For all closed \ctACPps\textup{{+}SO} terms $p,q$ of sort $\Proc$, $p = q$ is derivable from the axioms of \ctACPps\textup{{+}SO} if $p \bisim q$. \end{theorem}
\begin{proof} The proof goes along the same line as the proof of Theorem~\ref{theorem-completeness-ctACPps}. \qed \end{proof}
\section{Guarded Recursion} \label{sect-ctACPps+REC}
In order to allow for the description of processes without a finite upper bound to the number of actions that it can perform, we add in this section guarded recursion to \ctACPps\ and \ctACPps\textup{{+}SO}. The resulting theories are called \ctACPps\textup{{+}REC} and \ctACPps\textup{{+}SO{+}REC}, respectively.
A \emph{recursive specification} over \ctACPps\ is a set of \emph{recursion} equations $E = \set{X = t_X \where X \in V}$ where $V$ is a set of variables of sort $\Proc$ and each $t_X$ is a term of sort $\Proc$ that only contains variables from $V$. We write $\vars(E)$ for the set of all variables that occur on the left-hand side of an equation in $E$. A \emph{solution} of a recursive specification $E$ is a set of processes (in some model of \ctACPps) $\set{P_X \where X \in \vars(E)}$ such that the equations of $E$ hold if, for all $X \in \vars(E)$, $X$ stands for $P_X$.
Let $t$ be a \ctACPps\ term of sort $\Proc$ containing a variable $X$. We call an occurrence of $X$ in $t$ \emph{guarded} if $t$ has a subterm of the form $a \seqc t'$, where $a \in \Act$, with $t'$ containing this occurrence of $X$. A recursive specification $E$ over \ctACPps\ is called a \emph{guarded} recursive specification if all occurrences of variables in the right-hand sides of its equations are guarded or it can be rewritten to such a recursive specification using the axioms of \ctACPps\ in either direction and/or the equations in $E$ from left to right. We are only interested in a model of \ctACPps\ in which guarded recursive specifications have unique solutions.
For each guarded recursive specification $E$ over \ctACPps\ and each variable $X \in \vars(E)$, we add a constant of sort $\Proc$, standing for the unique solution of $E$ for $X$, to the constants of \ctACPps. This constant is denoted by $\rec{X}{E}$.
We will use the following notation. Let $t$ be a \ctACPps\ term of sort $\Proc$ and $E$ be a guarded recursive specification over \ctACPps. Then we write $\rec{t}{E}$ for $t$ with, for all $X \in \vars(E)$, all occurrences of $X$ in $t$ replaced by $\rec{X}{E}$.
The additional axioms for guarded recursion are the equations given in Table~\ref{axioms-REC}.
\begin{table}[!tb] \caption{Axioms for guarded recursion} \label{axioms-REC} \begin{eqntbl} \begin{saxcol} \rec{X}{E} = \rec{t_X}{E} & \mif X = t_X \in E & \ax{RDP} \\ E \Limpl X = \rec{X}{E} & \mif X \in \vars(E) & \ax{RSP} \end{saxcol} \end{eqntbl} \end{table}
In this table, $X$, $t_X$, and $E$ stand for an arbitrary variable of sort $\Proc$, an arbitrary \ctACPps\ term, and an arbitrary guarded recursive specification over \ctACPps, respectively. Side conditions are added to restrict the variables, terms and guarded recursive specifications for which $X$, $t_X$ and $E$ stand.
The additional axioms for guarded recursion are known as the recursive definition principle (RDP) and the recursive specification principle (RSP). The equations $\rec{X}{E} = \rec{t_X}{E}$ for a fixed $E$ express that the constants $\rec{X}{E}$ make up a solution of $E$. The conditional equations $E \Limpl X = \rec{X}{E}$ express that this solution is the only one.
The additional transition rules for the constants $\rec{X}{E}$ are given in Table~\ref{sos-REC}.
\begin{table}[!tb] \caption{Transition rules for guarded recursion} \label{sos-REC} \begin{ruletbl} \RuleC {\aterm{\rec{t_X}{E}}{\gact{\phi}{a}}} {\aterm{\rec{X}{E}}{\gact{\phi}{a}}} {X \!=\! t_X \,\in\, E} \qquad \RuleC {\astep{\rec{t_X}{E}}{\gact{\phi}{a}}{x'}} {\astep{\rec{X}{E}}{\gact{\phi}{a}}{x'}} {X \!=\! t_X \,\in\, E} \\ \RuleC {\rsgn{\rec{t_X}{E}} = \phi} {\rsgn{\rec{X}{E}} = \phi} {X \!=\! t_X \,\in\, E} \end{ruletbl} \end{table}
In this table, $X$, $t_X$ and $E$ stand for an arbitrary variable of sort $\Proc$, an arbitrary \ctACPps\ term and an arbitrary guarded recursive specification over \ctACPps, respectively.
Bisimulation equivalence is a congruence with respect to the operators of \ctACPps{+}REC.
\begin{proposition}[Congruence] \label{proposition-congr-ctACPps+REC} For all closed \ctACPps\textup{{+}REC} terms $p,q,p',q'$ of sort $\Proc$ and closed \ctACPps\textup{{+}REC} terms $\phi$ of sort $\Prop$, $p \bisim q$ and $p' \bisim q'$ imply $p \altc p' \bisim q \altc q'$, $p \seqc p' \bisim q \seqc q'$, $\phi \gc p \bisim \phi \gc q$, $\phi \emi p \bisim \phi \emi q$, $p \parc p' \bisim q \parc q'$, $p \leftm p' \bisim q \leftm q'$, $p \commm p' \bisim q \commm q'$, $\encap{H}(p) \bisim \encap{H}(q)$. \end{proposition}
\begin{proof} The proof goes along the same line as the proof of Proposition~\ref{proposition-congr-ctBPAps}. \qed \end{proof}
\ctACPps\textup{{+}REC} is sound with respect to $\bisim$ for equations between closed terms.
\begin{theorem}[Soundness] \label{theorem-soundness-ctACPps+REC} For all closed \ctACPps\textup{{+}REC} terms $p,q$ of sort $\Proc$, \mbox{$p = q$} is derivable from the axioms of \ctACPps\textup{{+}REC} only if $p \bisim q$. \end{theorem}
\begin{proof} Because of Proposition~\ref{proposition-congr-ctACPps+REC}, it is sufficient to prove the theorem for all closed \ctACPps{+}REC terms $p$ and $q$ for which $p = q$ is a closed substitution instance of an axiom of \ctACPps{+}REC. With the exception of the closed substitution instances of RSP, the proof goes along the same line as the proof of Theorem~\ref{theorem-soundness-ctACPps}. The proof of the validity of RSP is rather involved. We confine ourselves to a very brief outline of the proof. The transition rules for \ctACPps{+}REC determines a transition system for each process that can be denoted by a closed \ctACPps{+}REC term of sort $\Proc$. A model of \ctACPps{+}REC based on these transition systems can be constructed along the same line as the models of a generalization of ACPps constructed in~\cite{BM05a}. An equation $p = q$ between closed \ctACPps{+}REC terms holds in this model iff $p \bisim q$. Based on this model, the validity of RSP can be proved along the same line as in the proof of Theorem~10 from~\cite{BM05a}. The underlying ideas of that proof originate largely from~\cite{BBK87b}. \qed \end{proof}
Guarded recursion can be added to \ctACPps{+}SO in the same way as it is added to \ctACPps\ above, resulting in \ctACPps{+}SO{+}REC. It is easy to see that the above results, i.e.\ Proposition~\ref{proposition-congr-ctACPps+REC} and Theorem~\ref{theorem-soundness-ctACPps+REC}, go through for \ctACPps{+}SO{+}REC.
Completeness of \ctACPps\textup{{+}REC} and \ctACPps\textup{{+}SO{+}REC} with respect to $\bisim$ for equations between closed terms can be obtained by restriction to the finite linear recursive specifications, i.e.\ the guarded recursive specifications with finitely many recursion equations where the right-hand side of each recursion equation can be written in the form $\chi \emi \dead \altc
\vAltc{i \in \set{1,\ldots,n}} \phi_i \gc a_i \seqc X_i \altc
\vAltc{j \in \set{1,\ldots,m}} \psi_j \gc b_j$, where $n,m \in \Nat$, where $\chi \notin [\False]$, where $\phi_i \notin [\False]$, $a_i \in \Act$, and $X_i$ is variable of sort $\Proc$ for all $i \in \set{1,\ldots,n}$, and where $\psi_j \notin [\False]$ and $b_j \in \Act$ for all $j \in \set{1,\ldots,m}$.
\section{Concluding Remarks} \label{sect-concl}
We have presented \ctACPps, a version of ACPps built on a paraconsistent pro\-positional logic called LP$^{\IImpl,\False}$. \ctACPps\ deals with processes with possibly self-contradictory states by means of this paraconsistent logic. To our knowledge, processes with possibly self-contradictory states have not been dealt with in any theory or model of processes. This leaves nothing to be said about related work. However, it is worth mentioning that the need for a theory or model of processes with possibly self-contradictory states was already expressed in~\cite{Hew08a}.
In order to streamline the presentation of \ctACPps, we have left out the terminal signal emission operator, the global signal emission operator, and the root signal operator of ACPps and also the additional operators introduced in~\cite{BB94b} other than the state operators. To our knowledge, these are exactly the operators that have not been used in any work based on ACPps. The root signal operator is an auxiliary operator which can be dispensed with and the global signal emission operator is an auxiliary operator which can be dispensed with in the absence of the terminal signal emission operator. The terminal signal emission operator makes it possible to express that a proposition holds at the termination of a process.
\ctACPps\ is a contradiction-tolerant version of ACPps~\cite{BB94b}. ACPps itself can be viewed as a simplification and specialization of ACPS~\cite{BB92c}. The simplification consists of the use of conditions instead of special actions to observe signals. The specialization consists of the use of the set of all propositions with propositional variables from a given set instead of an arbitrary free Boolean algebra over a given set of generators. Later, the generalization of ACPps to arbitrary such Boolean algebras has been treated in~\cite{BM05a}. Moreover, a timed version of ACPps has been used in~\cite{BM03a} as the basis of a process algebra for hybrid systems and a timed version of ACPps has been used in~\cite{BMU98a} to give a semantics to a specification language that was widely used in telecommunications at the time.
Timed versions of \ctACPps\ may be useful in various applications. We believe that they can be obtained by combining \ctACPps\ with a timed version of ACP, such as ACP$^\mathrm{drt}$ or ACP$^\mathrm{srt}$ from~\cite{BM02a}, in much the same way as timed versions of ACPps have been obtained in~\cite{BM03a,BMU98a}. Because idling of processes is taken into account, two forms of the guarded command operator can be distinguished in these timed versions, namely a non-waiting form and a waiting form (see e.g.~\cite{BMU98a}). A version of \ctACPps\ with abstraction features like in \ACP$^\tau$ (see e.g.~\cite{BW90}) may be useful in various applications as well. Working out a timed version of \ctACPps\ and working out a version of \ctACPps\ with abstraction features are options for further work.
It is very important that case studies are carried out in conjunction with the theoretical work just mentioned to assess the degree of usefulness in practical applications.
LP$^{\IImpl,\False}$ is Blok-Pigozzi algebraizable. However, although there must exist one, a conditional-equa\-tional axiomatization of the algebras concerned has not yet been devised. Owing to this, the equations derivable in \ctACPps\ cannot always be derived by equational reasoning only. Another option for further work is devising the axiomatization referred to.
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Dolphins Hush When Killer Whales Lurk
Saturday, October 23, 2010 Life, Nature No comments
Research has suggested killer whale predation may affect cetacean vocal behavior; however, few data exist to test this hypothesis. Data collected for 19609 km of visual and acoustic shipboard surveys in the tropical Pacific Ocean were examined to determine if changes in dolphin vocal activity could be attributed to the presence of killer whales.
These surveys included 346detections of three highly vocal dolphin species (genus Stenella),whose whistles can be detected at ranges over 4.6 km. Random forest analysis was used to model vocal behavior based on sea state, visibility, fog rain, thermo cline temperature depth, mixed layer depth, chlorophyll, distance to shore, species, group size, perpendicular distance, and presence of killer whales.
The results show that the presence of killer whales significantly inhibited vocal activity in these tropical dolphins (p = 0.02). Killer whales are rare in the tropics, and this disruption in communication may not have a significant impact on interactions necessary for survival. However, in temperate climates, where increased productivity supports a greater abundance of killer whales, this interruption in communication may have a greater impact. The lower incidence of whistling dolphins in temperate waters may be related to the greater abundance of killer whales in these areas.
Nobel Prize and Wonder Material Graphene
Friday, October 15, 2010 Physics, Technology No comments
Russian born duo Andre Geim and Konstantin Novoselov shared the Noble prize in Physics 2010 for their work on a carbon compound called Graphene.
Graphene may not common to the man now, but experts believe that its amazing mechanical and electrical properties will prove as transformative to coming generations as the television, atomic bomb and silicon chip did in the decades after the Nobel committee first honored the scientists who made those inventions possible.
Graphene is a single-atom-thick planner sheet of carbon atoms (sp²-bonded) arrayed in a honeycomb pattern. Graphene is the basic structural element for all other graphite materials including graphite, carbon nanotubes and fullerenes. It is the strongest material ever discovered, yet flexible like rubber. It conducts electricity better than silicon, and resists heat better than diamond. And it allows for physics experiments that would otherwise require miles-long particle accelerators to be performed on a desktop.
"It's an amazing material with the incredible electronic properties and mechanical strength," said Paul Sheehan, head of the surface nanoscience and sensors section at the Naval Research Laboratory in Washington, D.C.
As an ultra-light but nearly indestructible material, graphene (and graphene composites) could drastically alter the aerospace and automotive industry, said Rodney Ruoff, a professor of engineering at the University of Texas, Austin.
Research has already accelerated to the point where laboratories can mass-produce the material, Ruoff said. Soon companies will be able to produce sheets of graphene hundreds of feet wide; embed it in other materials as a strengthening composite; or create microscopic flakes of it for use as a conductive ink.
Since electrons behave as waves in graphene, not as rubber balls as they do in silicon and metals, researchers can use graphene as a platform for observing particle behavior previously consigned to the world of theory, said Pablo Jarillo-Herrero, a professor of physics at MIT.
"Graphene has enabled us to study in small-scale experiments, cheap enough to do on your kitchen counter," Jarillo-Herrero said. "It created a whole field – condensed matter quantum physics – that wasn't there before."
Carbon is one of the most versatile elements in the periodic table, forming the base for diamonds, pencils and all life on Earth. Given that diversity, it is likely that the most transformative uses for graphene have yet to be discovered, Sheehan of the Office of Naval Research said.
Dr.Andre Geim
Born: 1958, Sochi, Russia
Director of Manchester Centre for Mesoscience and Nanotechnology
Chair of Condensed Matter Physics
Interview of Dr. Andre Geim
Dr. Kostya Novoselov
Born: 1974, Nizhny Tagil, Russia
Interview of Dr. Kostya Novoselov
Nikola's Death Ray Mystery
Sunday, October 03, 2010 curiosity, Physics, Scientist No comments
Thomas Edison gets all the credit as the father of electricity, but the real credit should go to a man named Nikola Tesla. Nikola Tesla (10 July 1856 – 7 January 1943) born as an ethnic Serb in the village of Smiljan, Croatian Military Frontier in Austrian Empire (now Croatia). He was a subject of the Austrian Empire by birth and later became an American citizen. He was an inventor and also one of the most important contributors to the birth of commercial electricity, and is best known for his many revolutionary developments in the field of electromagnetism in the late 19th and early 20th centuries. Aside from his work on electromagnetism and electromechanical engineering, Tesla contributed in varying degrees to the establishment of robotics, remote control, radar and computer science, and to the expansion of ballistics, nuclear physics and theoretical physics.
Most scholars acknowledge that Tesla's obscurity is partially due to his eccentric ways and fantastic claims during the waning years of his life, of communicating with other planets and death rays. Many of these fantastic inventions of Tesla are scientifically accurate and workable. It has simply taken mankind this long to catch up to the astonishing ideas of a man who died in 1943. It is now known that various governments were extremely interested in Tesla's ideas for weapons and limitless energy. So much so that after his death, the U.S. military confiscated boxes full of Tesla's research and writings. Much of this material has never been revealed to the public. What is not so widely known is that Tesla often suffered from financial difficulties, forcing him to move from hotel to hotel as his debt increased? Many times Tesla had to move, leaving crates of his belongings behind.
Tesla made statements during his lifetime that he had invented a Death Ray, which would be of benefit to warfare. According to Tesla the ray was capable of destroying up to 10,000 enemy aircraft at a distance of 250 miles away! Tesla's Death Ray was featured in the July 23, 1934 issue of Times Magazine, which stated that Nikola Tesla had announced a combination of four weapons that would make war 'unthinkable'. The article went on to describe how the weapons would work: "the nucleus of the idea is a death beam of submicroscopic particles flying at velocities approaching that of light".
This may sound like a fantasy, but it may surprise the reader to learn that we use Tesla's Particle beam everyday in the modern world. Particle beams are simply light beams, constructed of a special combination of electromagnetic waves. Unlike naturally occurring light the waves in a particle beam are very special, because they all end at the same point, creating a sort of imaginary 'knife edge' of light waves. Particle beams are utilized in hospitals in delicate micro-laser surgeries such as brain surgeries or cauterization within deep tissue, to determine distance, cut diamonds or guide missiles. So the question arises about Nikola's Death Ray invention – the source of the mystery.
After the death of Nikola Tesla, when the room in which he passed was searched, the papers had disappeared. All traces of the papers he claimed to have written on the subject vanished. In 1947 the military intelligence service identified the papers as extremely important, but no one has claimed possession of them or knowledge of their whereabouts. There are a number of people who suggest that the documents remain unfound that they were never lost in the first place. But, it has not been able to complete the work Tesla begun. Another reasonable theory would be that someone close to Tesla might have taken them to prevent the creation of such a weapon of mass destruction.
Whatever became of Tesla's brilliant invention, were it to surface now in any form it would likely be used to devastating effect. Already we have seen evidence by the invention of nuclear power. If it is truly lost, then perhaps we are better off without it.
A Place Where Things Seems To Roll Uphill
Friday, September 24, 2010 curiosity, Physics No comments
Friends you may find or hear of a mysterious place where objects can apparently roll uphill. Actually this is a common illusion which is found in numerous locations around the world. These spots where the illusion is especially powerful often become tourist attractions. Tour guides may like to claim that the effect is a mystery or that it is due to magnetic or gravitational anomalies or even that it is a paranormal phenomenon which science cannot explain.
But, friends this is not true of course. Natural anomalies can only be detected with sensitive equipments and cannot account for these places but science can easily explain them as optical illusion. If you observe the uphills, usually it is a stretch of road in a hilly area where the level horizon is obscured. Objects such as trees and walls which are normally provide visual clues to the true vertical, may be leaning slightly. This creates an optical illusion making a slight downhill look like an uphill slope. Objects may appear to roll uphill. Sometimes rivers even seem to flow against gravity.
There are several things which enables us to sense which way is up. The balance mechanism in our inner ears is one system we have, but visual clues are also important and can be overriding. If the horizon cannot be seen or is not level then we may be fooled by objects which we expect to be vertical but which aren't really. False perspective may also play a role. If a line of trees get larger or smaller with distance away, our sense of perspective is thrown off. Objects far away may seem smaller or larger than they really are. People often overestimate the angle of a slope. If you are standing on a slope of 1 degree it will seem like a slope of 5 degrees and if you stand on a slope of 5 degrees it may seem like you are on a slope of 30 degrees. Because of this effect the anti-gravity illusion can seem stronger be even when you know the cause.
Interestingly, even when true cause is understood it can be difficult to believe. In some cases the sea horizon is partly visible and it seems incredible that the effect can be an illusion. If you think there is a magnetic anomaly just using two plumb lines, one made of iron and one of stone. They would hang at different angles if a strong magnetic field was acting horizontally. In fact magnetic anomalies are never that strong and are never the cause as is easily shown.
However friends, it is not always easy to demonstrate that a slope which appears to go uphill is really going downhill. Plumb lines and spirit levels cannot be relied on if you think there is a gravitational anomaly. If the slope runs parallel to a sea view it would be possible to compare a plumb line with the horizon. Otherwise the only reliable way of determining the true horizontal is by careful surveying. Gravitational anomalies are always very small. In any case, if there was a gravitational anomaly you should wonder how you notice it. There would be an equal effect on your sense of balance as there is on any object. The anomaly would not be apparent unless there was a clear view of the sea behind the slope, which there never is.
Mystery Spot Road, off Branciforte Dr. Santa Cruz, CA, USA.
Mystery Spot, Putney Road, Benzie County, Michigan, USA.
Gravity Hill, Northwest Baltimore County, USA.
Gravity Hill, Mooresville, Southwest Indianapolis, USA.
Gravity Road, Ewing Road exit ramp off Route 208, Franklin Lakes, USA.
Mystery Hill, Blowing Rock, hwy 321, Carolina, USA.
Confusion Hill, Idelwild Park, Ligonier, Pennsylvania, USA.
Gravity Hill, off of State Route 96 just south of New Paris, Bedford County, Pennsylvania, USA.
Oregon Vortex, near Gold-Hill, Grants Pass, Oregon, USA.
Spook Hill, North Wales Drive, North Avenue, Lake Wales, Florida, USA.
Magnetic Hill, Near Neepawa in Manitoba, Canada.
Gravity Hill, on McKee Rd. Abbotsford, British Columbia, Canada.
Electric Brae, on the A719, Near Croy Bay, South of Ayr, Ayeshire, Scotland.
Anti-Gravity Hill, Straws Lane Road, Wood-End, Near hanging rock, Victoria, Australia
Morgan Lewis Hill, St Andrew, Barbados.
Hill South of Rome, in Colli Albani, near Frascati, Italy.
Malveira da Serra, on N247 coast road West of Lisbon, Portugal
Mount Penteli, on a road to Mount Penteli, Athens, Greece
Mount Halla, on the 1.100 highway a few miles south of the airport, near Mount Halla, on the island of Cheju Do, South Korea
Top Ten cars could help to save the planet
Friday, September 17, 2010 Technology 10 comments
25th anniversary of Bucky Ball
Sunday, September 05, 2010 Chemistry, Technology No comments
Yesterday was the 25th anniversary of discover of Bucky Ball – known as fullerene. Fullerenes are new class of carbon allotropes. They are spheroidal in shape and contain even number of carbon atoms ranging from 60 – 350 or above. The C60 fullerene is the most stable and was the first to be identified. It contains 60 atoms which are arranged in the shape of a football or a soccer ball, therefore, it is called buck ball.
It contains 20 six – membered rings and 12 five – memebered rings but five – membered rings are fused only to six – memebered rings. In other words, no two five – memebered rings are fused together. Further, because these allotropes look like geodesic by the US architect Buckminister Fuller, they are called Buckminster fullerenes or fullerenes.
Bucky ball is a dark solid at room temperature. Unlike diamond and graphite which are giant molecules containing thousands and thousands of carbon atoms, C60 fullerene is a very small molecule containing only 60 carbon atoms.
Bucky balls or Fullerenes were discovered by H.W.Kroto, R.F.Curt and R.E.Smalley. The 1996 Nobel Prize was awarded to above scientists for the discovery of fullerenes.
The popular search engine giant Google released popular doodle on the 25th anniversary of the discovery of the Bucky ball on Saturday, September 4.
Thursday, July 29, 2010 Books, Mathematics No comments
Human have used symmetrical patterns for thousands of years in both functional and decorative ways. Now, a new book by three mathematicians offers both math experts and enthusiasts a new way to understand symmetry and a fresh way to see the world. In The Symmetries of Things, eminent Princeton mathematician John H. Conway teams up with Chaim Goodman-Strauss of the University of Arkansas and Heidi Burgiel of Bridgewater State College to present a comprehensive mathematical theory of symmetry in a richly illustrated volume. The book is designed to speak to those with an interest in math, artists, working mathematicians and researchers.
"Symmetry and pattern are fundamentally human preoccupations in the same way that language and rhythm are. Any culture that is making anything has ornament and is preoccupied with this visual rhythm," Goodman-Strauss said. "There are actually Neolithic examples of many of these patterns. The fish-scale pattern, for example, is 22,000 years old and shows up all over the world in all kinds of contexts." Symmetrical objects and patterns are everywhere. In nature, there are flowers composed of repeating shapes that rotate around a central point. Architects trim buildings with friezes that repeat design elements over and over. Mathematicians, according to Goodman-Strauss, are latecomers to the human fascination with pattern. While mathematicians bring their own particular concerns, "we're also able to say things that other people might not be able to say. "The symmetries of Things contribute a new system of notation or descriptive categories for symmetrical patterns and a host of new proofs. The first section of the book is written to be accessible to a general reader with interest in the subject. Sections two and three are aimed at mathematicians and experts in the field. The entire book, Goodman-Strauss said, "is meant to be engaging and reveal itself visually as well."
Book Information:
Authors: John Horton Conway, Heidi Burgiel, Chaim Goodman- Strauss
Publisher: A K Peters Ltd
Keywords: things, symmetries
EBook link here.
Some Excellent Snaps ...
Thursday, July 22, 2010 Photos No comments
The photographs collected by NAINA KAUR and posted by me. I just captured the snaps on behalf of NAINA KAUR for her excellent collections.
How Big is Infinity?
Saturday, July 17, 2010 Mathematics 1 comment
Most of us are familiar with the infinity symbol – the one that looks like the number eight tipped over on its side. The infinite sometimes crops up in everyday speech as a superlative form of the word many. But how many is infinitely many? How far away is "from here to infinity"? How big is infinity?
You can't count to infinity. Yet we are comfortable with the idea that there are infinitely many numbers to count with: no matter how big a number you might come up with, someone else can come up with a bigger one: that number plus one – or plus two, or times two. Or times itself. There simply is no biggest number. Is there?
Is infinity a number? Is there anything bigger than infinity? How about infinity plus one? What's infinity plus infinity? What about infinity times infinity? Children, to whom the concept of infinity is brand new, pose questions like this and don't seem to have very much bearing on daily life, so their unsatisfactory answers don't seem to be a matter of concern.
At the turn of the century, in Germany, the Russian – born mathematician George Cantor applied the tools of mathematical rigor and logical deduction to questions about infinity in search of satisfactory answers. His conclusions are paradoxical to our everyday experience, yet they are mathematically sound. The world of our everyday experience is finite. We can't exactly say where the boundary line is, but beyond the finite, in the realm of the transfinite, things are different.
Mathematics of DNA
Saturday, July 10, 2010 Life, Mathematics No comments
Why is DNA packed into twisted, knotted shapes? What does this knotted structure have to do with? How DNA functions? How does DNA 'undo' these complicated knots to transform itself into different structures? The mathematical theory of knots, links and tangles is helping to find answers.
In order to perform such functions as replication and information transmission, DNA must transform itself from one form of knotting or coiling into another. The agents for these transformations are enzymes. Enzymes maintain the proper geometry and topology during the transformation and also 'cut' the DNA strands and recombine the loose ends. Mathematics can be used to model these complicated processes.
The description and quantization of the three-dimensional structure of DNA and the changes in DNA structure due to the action of these enzymes have required the serious use of geometry and topology. This use of mathematics as an analytical tool is especially important because there is no experimental way to observe the dynamics of enzymatic action directly.
A key mathematical challenge is to deduce the enzyme mechanism from observing the changes the enzymes bring about in the geometry and topology of the DNA. This requires the construction of mathematical models for enzyme action and the use of these models for enzyme action and the use of these models to analyze the results of topological enzymology experiments. The entangled form of the product DNA knots and links contains information about the enzymes that made them.
Martian moon mystery
Friday, July 02, 2010 Space Science No comments
The Martian moon Phobos is cratered, lumpy and about 16.8 miles long. According to a study, the moon is also unusually light. Planetary scientists found that Phobos is probably not a solid object, and that as much as 30 percent of the moon's interior may be empty space.
That doesn't mean that Phobos is an empty shell where we could, say, set up a rest stop for spaceships on their way to the outer planets. But the new finding probably does mean that Phobos was not an asteroid that got caught in Mars' gravity as it floated by the planet.
Phobos is the larger of Mars' two moons, and astronomers have had many ideas about where it came from. Previous studies have suggested that Phobos was an asteroid. Other studies suggest the moon formed from bits of Martian rock that were sent into space after a giant object, like an asteroid, crashed in Mars. The new study suggests that neither of these ideas is completely correct. The truth might be some combination of the two.
Scientists may never know how Phobos came to be a Martian satellite, but the new study may help eliminate some possibilities. A planetary geophysicist is a scientist who studies physical properties, such as rocks and appearance, to understand more about celestial bodies such as planets and moons.
The Mars Express, a spacecraft that orbits Mars and takes measurements. That spacecraft left Earth in 2003 and is a project by the European Space Agency. In March, Mars Express flew closer to Phobos than any spacecraft ever had before, ESA reports.
The scientists wanted to learn the density of Phobos. Density measures how close together, on average, are the atoms in an object. If two objects are the same size but have different densities, the denser object will have more mass — which means it will feel heavier when you're holding it on Earth. Density is found by dividing mass by volume. Since the scientists already had a good idea of the volume of Phobos, they just had to find its mass in order to figure out its density.
They made their mass measurements by studying the gravitational force of Phobos. Gravity is an attractive force, which means anything with mass attracts anything else with mass. The more mass an object has, the stronger its gravitational force. Since a large body like the Earth has a lot of mass, it has a strong gravitational force.
When Mars Express flew close to Phobos, the small moon's gravity attracted the spacecraft. By studying changes in the motion of Mars Express, the scientists were able to estimate the gravitational tug of Phobos. Once they knew the strength of its gravity, they could find its mass.
They found that Phobos has a density of about 1.87 grams per cubic centimeter. The rocks in the crust of Mars, for comparison, are much denser: about 3 grams per cubic centimeter. This difference suggests that Phobos is not made of rocks from the surface of Mars.
Some asteroids have densities of about 1.87 grams per cubic centimeter, but those asteroids would be broken apart by Mars' gravity — a fact that probably rules out the possibility that Phobos was once a free-floating asteroid.
Some scientists don't mind giving up the idea that Phobos was once an asteroid. Finally we're drifting away from the idea that the Martian moons are captured asteroids. We happy to see that Phobos and Deimos [Mars' other moon] are getting a lot of attention these days.
Mathematical Proof of God's Existance
Thursday, June 24, 2010 Mathematics No comments
Catherine the Great (Catherine II) was a woman of culture who reigned the 34 years from 1762 to 1796. This is a story when Empress Catherine II invited Denis Diderot, a distinguished French philosopher and appointed him as first librarian of St. Petersburg Academy. At that time Leonard Euler, famous mathematician, returned to St. Petersburg at the request of Catherine II from Prussia in Berlin and was the chair of mathematics at the Academy of St. Petersburg.
Empress Catherine II was alarmed when Diderot's arguments for atheism were influencing members of her court. So Euler was asked to confront Diderot. Diderot was informed that a learned mathematician had produced a proof of the existence of God. He agreed to view the proof as it was presented in court.
Euler appeared, advanced toward Diderot, and in a tone of perfect conviction announced, "Sir, $\frac{a+b^n}{n}=x$ , hence God exists—reply!". Diderot, to whom all mathematics was gibberish, stood dumbstruck. The peals of laughter erupted from the court. Embarrassed, Diderot asked to leave Russia and was graciously granted by the Empress.
If You Can …
Wednesday, June 23, 2010 Z - talk No comments
Hlelo if you can raed tihs tehn taht maens that your barin is mroe poerwufl tehn others cool huh? Yuor barin olny raeds the frist and lsat letetr of each wrod. If it tkaes you mroe tehn 15 scenods to raed tihs taht maens you can tehn maens you are a fckuning rtaerd if you can tehn tuhmbs tihs up.
Origin of Universe
Tuesday, June 08, 2010 Physics No comments
One of the most persistently asked questions has been: How was the Universe created? Many people believed that the Universe had no beginning or end and was truly unchanging static infinite.
Save Your Globe!
Saturday, June 05, 2010 Nature No comments
The entire species of
Aldabra banded snail died
out after warmer weather
cut off the rainfall in its habitat!
Wonder Fish
Friday, May 28, 2010 Life, Nature No comments
Alien of the Deep:
Looking like a creature from the Alien movies, this nightmarish "longhead dreamer" anglerfish (Chaenophryne longiceps) was until recently an alien species to Greenland waters.
Synthetic Genome
Saturday, May 22, 2010 Life, Nature 1 comment
Craig Venter and colleagues have achieved a remarkable milestone: they designed a genome, and brought it to life. More specifically, they've synthesized a chromosome consisting of over a million DNA base pairs, and implanted it in a bacterial cell to replace the cell's original genome. That cell then reproduced, giving birth to offspring that only had the synthetic genome.
Primitive Birds Lack Of Flying!
Saturday, May 15, 2010 Life No comments
The wings were willing, but the feathers were weak. Delicate, thin-shafted plumage would have made flapping difficult if not impossible for two prehistoric birds, a new analysis of fossil feathers suggests.
Green Exercise Boost Mental Health
Wednesday, May 05, 2010 Health No comments
Researchers have reported the fast improvements in mood and self-esteem for just five minutes of exercise in a green space. The study in Environmental Science and Technology journal suggested the strongest impact on young people. The outdoor activities like walking, gardening, cycling, fishing, boating, horse-riding and farming in a green environment with water contained – such as a lake or river boosts well – being.
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Physics NCERT Books XI
The NCERT books and exemplar of physics for class XI can download from these links. Physics Book - 1 Physics Book - 2 Physics Exemplar
Alien of the Deep: Looking like a creature from the Alien movies, this nightmarish "longhead dreamer" anglerfish (Chaenophryne...
NCERT Book For Class VIII
Students the NCERT and Exemplar Books for Mathematics and Science can download from here. Mathematics Book Mathematics Exemplar Book ...
Thomas Edison gets all the credit as the father of electricity, but the real credit should go to a man named Nikola Tesla. Nikola Tesla (10 ...
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