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Definition:Pointwise Supremum From ProofWiki 2 Also known as 3 Also defined as Let $S$ be a set, and let $\left({T, \preceq}\right)$ be an ordered set. Let $\left({f_i}\right)_{i \in I}, f_i: S \to T$ be an $I$-indexed collection of mappings. Suppose that for all $s \in S$, it holds that: $\displaystyle \sup_{i \mathop \in I} f_i \left({s}\right) \in T$ where the supremum is taken in $T$. Then the pointwise supremum of $\left({f_i}\right)_{i \in I}$, denoted $\displaystyle \sup_{i \mathop \in I} f_i: S \to T$, is defined by: $\displaystyle \left({\sup_{i \mathop \in I} f_i}\right) \left({s}\right) := \sup_{i \mathop \in I} f_i \left({s}\right)$ where the latter supremum is again taken in $T$. By assumption, this supremum is guaranteed to exist. Thence it can be seen that pointwise supremum is an instance of a pointwise operation. Because of the way $\displaystyle \sup_{i \mathop \in I} f_i$ is defined, there is usually no need to distinguish between the left- and right-hand side of the definition. Thus $\displaystyle \sup_{i \mathop \in I} f_i \left({s}\right)$ is commonly used instead of $\displaystyle \left({\sup_{i \mathop \in I} f_i}\right) \left({s}\right)$. Also defined as Sometimes the imposition that all suprema exist in $T$ is considered too strong. In these cases, some suitable extension of $\preceq$ to a suitable ordered set $\overline T$ may be created, in which the suprema do exist. This $\overline T$ is then set to be the codomain of the pointwise supremum. For example, this is done on Pointwise Supremum of Real-Valued Functions. Pointwise Supremum of Real-Valued Functions, taking $T$ to be $\R$ Pointwise Supremum of Extended Real-Valued Functions, taking $T$ to be the extended real numbers $\overline{\R}$ Retrieved from "https://proofwiki.org/w/index.php?title=Definition:Pointwise_Supremum&oldid=178646" Definitions/Order Theory Definitions/Mapping Theory Random proof ProofWiki.org Proof Index Definition Index Symbol Index Axiom Index Proofread Articles Wanted Proofs Tidy Articles Improvements Invited Proposed Mergers Maintenance Needed This page was last modified on 5 March 2014, at 16:32 and is 0 bytes About ProofWiki	CommonCrawl
\begin{document} \pagestyle{myheadings} \markboth{P. -V. Koseleff, D. Pecker}{{\em Chebyshev diagrams for rational knots}} \title{Chebyshev diagrams for rational knots} \author{P. -V. Koseleff, D. Pecker} \maketitle \begin{abstract} We show that every rational knot $K$ of crossing number $N$ admits a polynomial parametrization $x=T_a(t), \, y = T_b(t), z = C(t)$ where $T_k(t)$ are the Chebyshev polynomials, $a=3$ and $b+ \deg C = 3N.$ We show that every rational knot also admits a polynomial parametrization with $a=4$. If $C (t)= T_c(t)$ is a Chebyshev polynomial, we call such a knot a harmonic knot. We give the classification of harmonic knots for $a \le 4.$ \par\noindent {\bf keywords:}{ Polynomial curves, Chebyshev curves, rational knots, continued fractions}\\ {\bf Mathematics Subject Classification 2000:} 14H50, 57M25, 11A55, 14P99 \end{abstract} \begin{center} \parbox{12cm}{\small \tableofcontents } \end{center} \section{Introduction} We study the polynomial parametrization of knots, viewed as non singular space curves. Vassiliev proved that any knot can be represented by a polynomial embedding ${\bf R} \to {\bf R} ^3 \subset {\bf S}_3 $ (\cite{Va}). Shastri (\cite{Sh}) gave another proof of this theorem, he also found explicit parametrizations of the trefoil and of the figure-eight knot (see also \cite{Mi}). \par\noindent We shall study polynomial embeddings of the form $x=T_a(t), \, y= T_b(t), \, z=C(t)$ where $a$ and $b$ are coprime integers and $T_n(t)$ are the classical Chebyshev polynomials defined by $T_n(\cos t) = \cos nt$. The projection of such a curve on the $xy$-plane is the Chebyshev curve ${\cal C}(a,b): T_b(x)=T_a(y)$ which has exactly $\frac 12 (a-1)(b-1)$ crossing points (\cite{Fi,P1,P2}). We will say that such a knot has the Chebyshev diagram ${\cal C}(a,b)$. \par\noindent We observed in \cite{KP1} that the trefoil can be parametrized by Chebyshev polynomials: $x=T_3(t);\, y=T_4(t);\,z= T_5(t)$. This led us to study Chebyshev knots in \cite{KP3}. \begin{definition} A knot in ${\bf R} ^3 \subset {{\bf S}}^3$ is the Chebyshev knot ${\cal C}(a,b,c,\varphi)$ if it admits the one-to-one parametrization $$ x=T_a(t); \ y=T_b(t) ; \ z=T_c(t + \varphi) $$ where $t \in {\bf R}$, $a$ and $b$ are coprime integers, $c$ is an integer and $\varphi$ is a real constant. \par\noindent When $\varphi=0$ and $a,b,c$ are coprime, it is denoted by ${\rm H}(a,b,c)$ and is called a harmonic knot. \end{definition} We proved that any knot is a Chebyshev knot. Our proof uses theorems on braids by Hoste, Zirbel and Lamm (\cite{HZ}), and a density argument. In a joint work with F. Rouillier (\cite{KPR}), we developed an effective method to enumerate all the knots ${\cal C}(a,b,c,\varphi), \varphi \in {\bf R}$ where $a=3$ or $a=4$, $a$ and $b$ coprime. \par\noindent Chebyshev knots are polynomial analogues of Lissajous knots that admit a parametrization of the form $$ x=\cos (at ); \ y=\cos (bt + \varphi) ; \ z=\cos (ct + \psi) $$ where $ 0 \le t \le 2 \pi $ and where $ a, b, c$ are pairwise coprime integers. These knots, introduced in \cite{BHJS}, have been studied by V. F. R. Jones, J. Przytycki, C. Lamm, J. Hoste and L. Zirbel. Most known properties of Lissajous knots are deduced from their symmetries (see \cite{BDHZ,Cr,HZ,JP,La1}). \par\noindent The symmetries of harmonic knots, obvious from the parity of Chebyshev polynomials, are different from those of Lissajous. For example, the figure-eight knot which is amphicheiral but not a Lissajous knot, is the harmonic knot $ {\rm H}(3,5,7).$ \par\noindent We proved that the harmonic knot ${\rm H}(a,b, ab-a-b)$ is alternate, and deduced that there are infinitely many amphicheiral harmonic knots and infinitely many strongly invertible harmonic knots. We also proved (\cite{KP3}) that the torus knot $T(2, 2n+1)$ is the harmonic knot $ {\rm H}(3,3n+2,3n+1)$. \par\noindent In this article, we give the classification of the harmonic knots ${\rm H}(a,b,c)$ for $a \le 4.$ We also give explicit polynomial parametrizations of all rational knots. The diagrams of our knots are Chebyshev curves of minimal degrees with a small number of crossing points. The degrees of the height polynomials are small. \par\noindent In section {\bf \ref{cf}.} we recall the Conway notation for rational knots, and the computation of their Schubert fractions with continued fractions. We observe that Chebyshev diagrams correspond to continued fractions of the form $[\pm 1, \ldots, \pm 1]$ when $a=3$ and of the form $[\pm 1,\pm 2, \ldots, \pm 1, \pm 2]$ when $a=4$. We show results on our continued fraction expansion: \par\noindent {\bf Theorem \ref{th1}.}\\ {\em Every rational number $r$ has a unique continued fraction expansion $r = [e_1, e_2, \ldots, e_n ]$, $e_i= \pm 1$, where there are no two consecutive sign changes in the sequence $(e_1,\ldots, e_n ).$ } \par\noindent We have a similar theorem for continued fractions of the form $r = [\pm 1, \pm 2, \ldots, \pm 1, \pm 2]$. We provide a formula for the crossing number of the corresponding knots. Then we study the matrix interpretation of these continued fraction expansions. As an application, we give optimal Chebyshev diagrams for the torus knots $T(2,N)$, the twist knots ${\cal T}_n$, the generalized stevedore knots and some others. \par\noindent In section {\bf \ref{harmonic}.} we describe the harmonic knots ${\rm H}(a,b,c)$ where $a \le 4.$ We begin with a careful analysis of the nature of the crossing points, giving the Schubert fractions of ${\rm H} ( 3,b,c) $ and ${\rm H}(4,b,c).$ Being rather long, the proofs of these results will be given in the last paragraph. We deduce the following algorithmic classification theorems. \par\noindent {\bf Theorem \ref{h3bc}.}\\ {\em Let $K= {\rm H}(3,b,c)$. There exists a unique pair $(b',c')$ such that (up to mirror symmetry) $$ K = {\rm H} (3,b',c'), \ b'<c'< 2b', \ b' \not\equiv c' \Mod 3.$$ The crossing number of $K$ is $ \Frac 13 ( b'+c')$.\\ The Schubert fractions $ \Frac \alpha \beta $ of $K$ are such that $\beta ^2 \equiv \pm 1 \Mod \alpha.$ } \par\noindent {\bf Theorem \ref{h4bc}.}\\ {\em Let $K= {\rm H}(4,b,c).$ There exists a unique pair $(b',c') $ such that (up to mirror symmetry) $$ K= {\rm H}(4,b',c'), \ b' < c' < 3b', \ b' \not\equiv c' \Mod 4.$$ The crossing number of $K$ is $ \Frac 14 ( 3b'+c'-2)$.\\ $K$ has a Schubert fraction $ \Frac \alpha \beta $ such that $ \beta ^2 \equiv \pm 2 \Mod \alpha.$ } \par\noindent We notice that the trefoil is the only knot which is both of form ${\rm H}(3,b,c)$ and ${\rm H}(4,b,c).$ We remark that the $6_1$ knot (the stevedore knot) is not a harmonic knot ${\rm H}( a,b,c), \ a \le 4.$ \par\noindent In section {\bf \ref{diagrams}.} we find explicit polynomial parametrizations of all rational knots. We first compute the optimal Chebyshev diagrams for $a=3$ and $a=4$. Then we define a height polynomial of small degree. More precisely: \par\noindent {\bf Theorem \ref{gauss3}.}\\ {\em Every rational knot of crossing number $N$ can be parametrized by $x=T_3(t), y= T_b(t), z= C(t)$ where $b + \deg C = 3N$. Furthermore, when the knot is amphicheiral, $b$ is odd and we can choose $C$ to be an odd polynomial.} \par\noindent In the same way we show: {\em Every rational knot of crossing number $N$ can be parametrized by $x=T_4(t), y= T_b(t), z= C(t)$} where $b$ is odd and $C$ is an odd polynomial. \par\noindent As a consequence, we see that any rational knot has a representation $K \subset {\bf R}^3$ such that $K$ is symmetrical about the $y$-axis (with reversed orientation). It clearly implies the classical result: {\em every rational knot is strongly invertible}. \par\noindent We give polynomial parametrizations of the torus knots $T(2,2n+1).$ We also give the first polynomial parametrizations of the twist knots and the generalized stevedore knots. We conjecture that the lexicographic degrees of our polynomials are minimal (among odd or even polynomials). \section{Continued fractions and rational Chebyshev knots}\label{cf} A two-bridge knot (or link) admits a diagram in Conway's normal form. This form, denoted by $C(a_1, a_2, \ldots, a_n)$ where $a_i$ are integers, is explained by the following picture (see \cite{Con}, \cite{Mu} p. 187). \psfrag{a}{\small $a_1$}\psfrag{b}{\small $a_2$} \psfrag{c}{\small $a_{n-1}$}\psfrag{d}{\small $a_{n}$} \begin{figure} \caption{Conway's normal forms} \label{conways3} \end{figure} The number of twists is denoted by the integer $\abs{a_i}$, and the sign of $a_i$ is defined as follows: if $i$ is odd, then the right twist is positive, if $i$ is even, then the right twist is negative. On Fig. \ref{conways3} the $a_i$ are positive (the $a_1$ first twists are right twists). \par\noindent \begin{examples} The trefoil has the following Conway's normal forms $C(3)$, $C(-1,-1,-1)$, $C(4, -1)$ and $C(1,1,-1,-1 ).$ The diagrams in Figure \ref{trefoils} clearly represent the same trefoil. \end{examples} \def{$\scriptstyle{+}$}{{\small $+$}} \def{\small $-$}{{\small $-$}} \begin{figure} \caption{Diagrams of the standard trefoil} \label{trefoils} \end{figure} \par\noindent The two-bridge links are classified by their Schubert fractions $$ \Frac {\alpha}{\beta} = a_1 + \Frac{1} {a_2 + \Frac {1} {a_3 + \Frac{1} {\cdots +\Frac 1{a_n}}}}= [ a_1, \ldots, a_n], \quad \alpha >0. $$ We shall denote $S \bigl( \Frac {\alpha}{\beta} \bigr)$ a two-bridge link with Schubert fraction $ \Frac {\alpha}{\beta}.$ The two-bridge links $ S (\Frac {\alpha} {\beta} )$ and $ S( \Frac {\alpha ' }{\beta '} )$ are equivalent if and only if $ \alpha = \alpha' $ and $ \beta' \equiv \beta ^{\pm 1} ( {\rm mod} \ \alpha).$ The integer $ \alpha$ is odd for a knot, and even for a two-component link. If $K= S (\Frac {\alpha}{\beta} ),$ its mirror image is $ \overline{K}= S ( \Frac {\alpha}{- \beta} ).$ We shall study knots with a Chebyshev diagram ${\cal C} (3,b) : \ x= T_3(t), y= T_b(t).$ It is remarkable that such a diagram is already in Conway normal form (see Figure \ref {conways3}). Consequently, the Schubert fraction of such a knot is given by a continued fraction of the form $ [ \pm 1, \pm 1, \ldots,\pm 1 ].$ For example the only diagrams of Figure \ref{trefoils} which may be Chebyshev are the second and the last (in fact they are Chebyshev). \par\noindent Figure \ref{t7} shows a typical example of a knot with a Chebyshev diagram. \def{$\scriptstyle{+}$}{{$\scriptstyle{+}$}} \def{\small $-$}{{\small $-$}} \begin{figure} \caption{A Chebyshev diagram of the torus knot $T(2,7)$} \label{t7} \end{figure} \par\noindent This knot is defined by $ x= T_3(t), \, y= T_{10}(t), \, z = -T_{11}(t).$ Its $xy$-projection is in the Conway normal form $ C(-1,-1,-1, 1,1,1, -1,-1,-1).$ Its Schubert fraction is then $\Frac{7}{-6}$ and this knot is the torus knot $T(2, 7)=S(\Frac{7}{-6}) = S(7)$. \par\noindent We shall also need to study knots with a diagram illustrated by the following picture. \psfrag{a}{\small $b_1$}\psfrag{b}{\small $a_1$}\psfrag{c}{\small $c_1$} \psfrag{d}{\small $b_n$}\psfrag{e}{\small $a_n$}\psfrag{f}{\small $c_n$} \begin{figure} \caption{A knot isotopic to $C(b_1,a_1+c_1,b_2,a_2+c_2,\ldots, b_n,a_n+c_n)$} \label{conway4} \end{figure} In this case, the $a_i$ and the $c_i$ are positive if they are left twists, the $b_i$ are positive if they are right twists (on our figure $a_i, b_i, c_i $ are positive). Such a knot is equivalent to a knot with Conway's normal form $C(b_1, a_1+c_1, b_2, a_2 +c_2, \ldots, b_n, a_n+c_n )$ (see \cite {Mu} p. 183-184). \par\noindent We shall study the knots with a Chebyshev diagram ${\cal C}(4,k): x= T_4(t), \, y= T_ k(t)$. In this case we get diagrams of the form illustrated by Figure \ref{conway4}. Consequently, such a knot has a Schubert fraction of the form $[ b_1, d_1, b_2, d_2, \ldots, b_n, d_n ]$ with $b_i=\pm 1$, $d_i = \pm 2$ or $d_i=0.$ Once again, the situation is best explained by typical examples. Figure \ref{diag4} represents two knots with the same Chebyshev diagram ${\cal C}(4,5): x= T_4(t), \ y= T_5(t)$. A Schubert fraction of the first knot is $ \Frac 52 = [1,0,1,2]$, it is the figure-eight knot. A Schubert fraction of the second knot is $ \Frac 7 {-4} = [-1,-2,1,2]$, it is the twist knot $5_2$. \begin{figure} \caption{Knots with the Chebyshev diagram ${\cal C}(4,5)$} \label{diag4} \end{figure} \subsection{Continued fractions} Let $\alpha,\beta$ be relatively prime integers. Then $ \Frac {\alpha }{\beta}$ admits the continued fraction expansion $\Frac{\alpha}{\beta} = [q_1, q_2, \ldots, q_n ] $ if and only if there exist integers $r_i$ such that $$ \left \{ \begin{array}{rcl} \alpha &=& q_1 \beta +r_2,\\ \beta &=& q_2 r_2 + r_3,\\ &\vdots&\\ r_{n-2} &=& q_{n-1} r_{n-1} + r_n,\\ r_{n-1} &=& q_n r_n. \end{array} \right . $$ The integers $q_i$ are called the quotients of the continued fraction. Euclidean algorithms provide various continued fraction expansions which are useful to the study of two-bridge knots (see \cite{BZ,St}). \begin{definition} Let $r >0$ be a rational number, and $r=[q_1, \ldots, q_n ] $ be a continued fraction with $q_i >0.$ The {\em crossing number } of $r$ is defined by $\hbox{\rm cn}\,(r)= q_1 + \cdots + q_n. $ \end{definition} \begin{remark} When $q_i$ are positive integers, the continued fraction expansion $[q_1, q_2, \ldots, q_n ] $ is unique up to $[q_n]=[q_{n}-1,1]$. $\hbox{\rm cn}\,(\Frac\alpha\beta)$ is the crossing number of the knot $K= S\bigl ( \Frac\alpha\beta \bigr) $. It means that it is the minimum number of crossing points for all diagrams of $K$ (\cite{Mu}). \end{remark} \par\noindent We shall be interested by algorithms where the sequence of remainders is not necessarily decreasing anymore. In this case, if $\Frac\alpha\beta = [a_1, \ldots, a_n]$, we have $\hbox{\rm cn}\,(\Frac\alpha\beta) \leq \sum_{k=1}^n \abs{a_i}$. \begin{definition} A continued fraction $ [ a_1,a_2, \ldots, a_n]$ is regular if it has the following\\properties: $$ a_i \neq 0, \, a_{n-1} a_n >0, \hbox{ and } a_i a_{i+1} <0 \Rightarrow a_{i+1} a_{i+2} >0, \ i = 1, \ldots, n-2.$$ If $a_1a_2>0$ she shall say that the continued fraction is biregular. \end{definition} \begin{proposition}\label{bireg} Let $\Frac \alpha\beta = [a_1, \ldots, a_n]$ be a biregular continued fraction. Its crossing number is \[ \hbox{\rm cn}\,(\Frac \alpha\beta) = \sum_{k=1}^n \abs{a_i} - \sharp \{i, a_ia_{i+1}<0\}. \label{biregf} \] \end{proposition} {\em Proof}. We prove this result by induction on the number of sign changes $k = \sharp \{i, a_ia_{i+1}<0\}$. If $k$ is 0, then $K$ is alternate and the result is true. If $k>0$ let us consider the first change of sign. The Conway normal form of $K$ is $[x,a,b,-c,-d,-y]$ where $a,b,c,d$ are positive integers and $x$ is a sequence (possibly empty) of positive integers and $y$ is a sequence of integers. We have $ [x,a,b,-c,-d,-y] = [x,a,b-1,1,c-1,d,y]$. \begin{itemize}\itemsep -2pt plus 1pt minus 1pt \item Suppose $(b-1)(c-1)>0$, then the sum of absolute values has decreased by 1 and the number of changes of sign has also decreased by 1. \item Suppose $b=1, c \not = 1$ (resp. $c=1, b\not = 1$). Then $[x,a,b,-c,-d,-y] = [x,a,0,1,c-1,d,y] =[x,a+1,c-1,d,y]$. (resp. $[x,a-1,c+1,d,y]$). The sum of absolute values has decreased by 1 and the number of changes of sign has also decreased by 1. \item Suppose $b=c=1$. Then $[x,a,b,-c,-d,-y] = [x,a,0,1,0,d,y] =[x,a+d+1,y]$. The sum of absolute values has decreased by 1 and the number of changes of sign has also decreased by 1. \hbox{} $\Box$ \end{itemize} Note that Formula (\ref{biregf}) is also true in other cases. For instance, Formula (\ref{biregf}) still holds when $a_1, \ldots,a_n$ are non zero even integers (see \cite{St}). \par\noindent We shall now use the basic (subtractive) Euclidean algorithm to get continued fractions of the form $ [ \pm 1, \pm 1, \ldots, \pm 1 ]$. \subsection{Continued fractions ${\mathbf{[\pm 1,\pm 1,\ldots,\pm 1]}}$} We will consider the following homographies: \[ P : x \mapsto [1,x] = 1+\Frac 1x, \, M : x \mapsto [1,-1,-x] = \Frac{1}{1+x}. \label{pm} \] Let $E$ be the set of positive real numbers. We have $P(E) = ]1, \infty [$ and $M(E)= ]0, 1 [.$ $P(E)$ and $M(E)$ are disjoint subsets of $E.$ \begin{theorem}\label{th1} Let $ \Frac \alpha\beta >0 $ be a rational number. There is a unique regular continued fraction such that $$\Frac \alpha\beta = [1, e_2, \ldots, e_n], \, e_i= \pm 1.$$ Furthermore, $\alpha>\beta$ if and only if $[e_1, e_2, \ldots, e_n]$ is biregular. \end{theorem} {\em Proof}. Let us prove the existence by induction on the height $h(\Frac\alpha\beta)=\alpha+\beta$. \begin{itemize}\itemsep -2pt plus 1pt minus 1pt \item If $h=2$ then $\Frac \alpha \beta = 1 = [1]$ and the result is true. \item If $ \alpha > \beta,$ we have $\Frac \alpha \beta = P(\Frac \beta {\alpha - \beta}) = [1,\Frac \beta {\alpha - \beta }]. $ Since $h( \Frac \beta {\alpha - \beta } ) < h ( \Frac \alpha \beta )$, we get our regular continued fraction for $r$ by induction. \item If $ \beta > \alpha$ we have $ \Frac \alpha \beta = M( \Frac {\beta - \alpha } \alpha ) = [1, -1, -\Frac {\beta - \alpha} \alpha ]. $ And we also get a regular continued fraction for $r.$ \end{itemize} Conversely, let $r$ be defined by the regular continued fraction $ r = [ 1, r_2, \ldots, r_n]$, $r_i = \pm 1$, $n \ge 2 $. Let us prove, by induction on the length $n$ of the continued fraction, that $r>0$ and that $r >1$ if and only if $r_2 = 1$. \begin{itemize}\itemsep -2pt plus 1pt minus 1pt \item If $r_2=1$ we have $r=P([1,r_3,\ldots,r_n] ),$ and by induction $r\in P(E)$ and then $r>1$. \item If $r_2=-1,$ we have $r_3=-1$ and $r= M([1, -r_4,\ldots, -r_n])$. By induction, $r\in M(E) = ]0,1[$. \end{itemize} The uniqueness is now easy to prove. Let $ r = [1, r_2, \ldots, r_n] = [1, r'_2, \ldots, r'_{n'} ].$ \begin{itemize}\itemsep -2pt plus 1pt minus 1pt \item If $r>1$ then $r_2=r'_2=1$ and $[1,1, r_3, \ldots, r_n ] =[1, 1, r'_3, \ldots r'_{n'} ]$. Consequently, $[1, r_3, \ldots, r_n] = [1, r'_3, \ldots, r'_{n'} ],$ and by induction $r_i=r'_i$ for all $i.$ \item If $r < 1,$ then $r_2=r_3=r'_2=r'_3=-1$ and $[1,-1,-1, r_4, \ldots, r_n] =[1,-1,-1, r'_4,\ldots, r'_{n'} ].$ Then, $ [ 1, -r_4, \ldots, -r_n] = [ 1, -r'_4, \ldots, -r'_{n'} ]$ and by induction $ r_i=r'_i$ for all $i.$ \hbox{} $\Box$ \end{itemize} \begin{definition} Let $\Frac\alpha\beta>0$ be the regular continued fraction $[e_1, \ldots, e_n]$, $e_i=\pm 1$. We will denote its length $n$ by $\ell(\Frac\alpha\beta)$. Note that $\ell(-\Frac\alpha\beta) = \ell(\Frac\alpha\beta)$. \end{definition} \begin{examples}\label{9297} Using our algorithm we obtain $$ \begin{array}{rcl} \Frac 97&=& [1,\Frac 72] = [1,1,\Frac 25] = [1,1,1,-1, -\Frac 32] = [1,1,1,-1, -1, -\Frac 21] = [1,1,1,-1, -1, -1, -1],\\[10pt] \Frac 92&=& [1,\Frac 27] = [1,1,-1,-\Frac 52] = [1,1,-1,-1,-\Frac 23] = [1,1,-1,-1,-1,1,\Frac 12] \\ &=& [1,1,-1,-1,-1,1,1,-1,-1] = [4,2]. \end{array} $$ \begin{figure} \caption{Diagrams of the knot $6_1 = S(\Frac 97)$ and its mirror image $S(\Frac 92)$ } \label{k61} \end{figure} \end{examples} We will rather use the notation $$ \Frac 97 = P^2 M P^3 (\infty), \, \Frac 92 = P MP M^2 P (\infty). $$ We get $\ell (\Frac 97) = 7$, $\ell (\Frac 92) = 9$. The crossing numbers of these fractions are $\hbox{\rm cn}\,(\Frac 97) = \hbox{\rm cn}\,([1,3,2]) = 6 = 7-1$ and $\hbox{\rm cn}\,(\Frac 92) = \hbox{\rm cn}\,([4,2]) = 6 = 9-3$. If the fractions $\Frac 97$ and $\Frac 92$ have the same crossing number, it is because the knot $S(\Frac 97)$ is the mirror image of $S(\Frac 92)$. In order to get a full description of two-bridge knots we shall need a more detailed study of the homographies $P$ and $M$. \par\noindent \begin{proposition} The multiplicative monoid ${\mathbf{G}}=\langle P,M \rangle$ is free. The mapping $g: G \mapsto G(\infty)$ is a bijection from ${\mathbf{G}} \cdot P$ to ${\bf Q}_{>0}$ and $g(P\cdot {\mathbf{G}} \cdot P) = {\bf Q}_{>1}$, the set of rational numbers greater than $1.$ \end{proposition} {\em Proof}. Suppose that $PX=MX'$ for some $X,X'$ in ${\mathbf{G}}$. Then we would have $PX(1)=MX'(1) \in P(E)\bigcap M(E) = \emptyset$. Clearly, this means that ${\mathbf{G}}$ is free. Similarly, from $P(\infty)=1$, we deduce that the mapping $G \mapsto G \cdot P (\infty)$ is injective. From Theorem \ref{th1} and $P(\infty)=1$, we deduce that $g$ is surjective. \hbox{} $\Box$ \begin{remark}\label{rl} Let $r=G(\infty)=[e_1, \ldots, e_n], \ e_i=\pm1, $ be a regular continued fraction. It is easy to find the unique homography $G \in {\mathbf{G}}\cdot P$ such that $r=G(\infty)$. Consider the sequence $(e_1, \ldots, e_n)$. For any $i$ such that $e_ie_{i+1}<0$, replace the couple $(e_i,e_{i+1})$ by $M,$ and then replace each remaining $e_i$ by $P$. \par\noindent Let $G= P^{p_1} M^{m_1} \cdots M^{m_k} P^{p_{k+1}}$. Let $p= p_1 + \cdots + p_{k+1}$ be the degree of $G$ in $P$ and $m = m_1 + \cdots + m_k$ its degree in $M$. Then we have $n = \ell(r) = p + 2m$ and $\hbox{\rm cn}\,(r) = p+m$. \end{remark} We shall consider matrix notations for many proofs. We will consider $$\cmatrix{\alpha\cr\beta} = P^{p_1} M^{m_1} \cdots M^{m_k} P^{p_{k+1}} \cmatrix{1\cr 0}, \quad P = \cmatrix{1&1\cr1&0},\quad M =\cmatrix{0&1\cr1&1}. $$ \begin{lemma}\label{length} Let $\Frac{\alpha}{\beta} = [e_1, \ldots, e_n]$ be a regular continued fraction ($e_i = \pm 1$). We have \begin{itemize}\itemsep -2pt plus 1pt minus 1pt \item $n\equiv 2 \Mod 3$ if and only if $\alpha$ is even and $\beta$ is odd. \item $n\equiv 0 \Mod 3$ if and only if $\alpha$ is odd and $\beta$ is even. \item $n\equiv 1 \Mod 3$ if and only if $\alpha$ and $\beta$ are odd. \end{itemize} \end{lemma} {\em Proof}. Let us write $\cmatrix{\alpha\cr\beta} = P^{p_1} M^{m_1} \cdots M^{m_k} P^{p_{k+1}} \cmatrix{1\cr 0}$. We have (from remark \ref{rl}) $n= p+2m.$ Since $P^3 \equiv {\mathbf{Id}} $ and $M \equiv P^2 \Mod 2$, we get $P^{p_1} M^{m_1} \cdots M^{m_k} P^{p_{k+1}} \equiv P^n \Mod 2$ \begin{itemize}\itemsep -2pt plus 1pt minus 1pt \item[] If $n\equiv 2 \Mod 3$, then $\cmatrix{\alpha\cr \beta} \equiv M \cmatrix{1\cr 0} \equiv \cmatrix{0\cr 1} \Mod 2$. \item[] If $n\equiv 1 \Mod 3$, then $\cmatrix{\alpha\cr \beta} \equiv P \cmatrix{1\cr 0} \equiv \cmatrix{1\cr 1} \Mod 2$. \item[] If $n\equiv 0 \Mod 3$, then $\cmatrix{\alpha\cr \beta} \equiv \cmatrix{1\cr 0} \Mod 2$. \hbox{} $\Box$ \end{itemize} \begin{definition}\\ We define on ${\mathbf{G}}$ the anti-homomorphism $G\mapsto \overline G$ with $\overline M=M$, $\overline P=P$. \\ We define on ${\mathbf{G}}$ the homomorphism $G \mapsto \hat G$ with $\hat M = P, \, \hat P = M$. \end{definition} \begin{proposition}\label{gg} Let $\alpha>\beta$ and consider $\Frac\alpha\beta=PGP(\infty)$ and $N=\hbox{\rm cn}\,(\Frac\alpha\beta)$. Let $\beta'$ such that $0<\beta'<\alpha$ and $\beta\beta'\equiv (-1)^{N-1}\Mod\alpha$. Then we have $$ \Frac\beta\alpha = M\hat G P(\infty),\quad \Frac\alpha{\alpha-\beta} = P \hat G P(\infty), \quad \Frac\alpha{\beta'} = P \overline G P(\infty).$$ We also have $$ \ell (\Frac {\beta}{\alpha}) + \ell( \Frac {\alpha}{\beta }) = 3 N -1, \quad \ell ( \Frac {\alpha}{\alpha - \beta } ) + \ell ( \Frac {\alpha}{\beta}) = 3 N -2, \quad \ell ( \Frac {\alpha}{\beta'} ) = \ell( \Frac {\alpha}{\beta }). $$ \end{proposition} {\em Proof}. We use matrix notations for this proof. Let us consider $$PGP=\cmatrix{\alpha&\beta'\cr\beta&\alpha'}= P^{p_1} M^{m_1} \cdots M^{m_k} P^{p_{k+1}}.$$ From $\det P = \det M = -1$, we obtain $\alpha\alpha' - \beta\beta' = (-1)^N$. Let $A =\cmatrix{a&c\cr b&d}$ be a matrix such that $0\leq c \leq a$ and $0\leq d \leq b$. From $PA = \cmatrix{a+c&b+d\cr a&c}$ and $MA = \cmatrix{b&d\cr a+b&c+d}$ we deduce that $PGP$ satisfies $0<\alpha'<\beta$ and $0<\beta'<\alpha$. We therefore conclude that, $\beta'$ is the integer defined by $0<\beta'<\alpha$, $\beta\beta'\equiv (-1)^{N-1} \Mod\alpha$. By transposition we deduce that $$\cmatrix{\alpha&\beta\cr\beta'&\alpha'} = P^{p_{k+1}} M^{m_{k}} \cdots M^{m_1} P^{p_1} = P \overline G P,$$ which implies $\Frac{\alpha}{\beta'} = P \overline G P(\infty)$. \par\noindent Let us introduce $J = \cmatrix{0&1\cr1&0}$. We have $J^2 = {\mathbf{Id}}, M=JPJ$ and $P=JMJ.$ Therefore $$ \cmatrix{\beta\cr\alpha} = J \cmatrix{\alpha\cr\beta}= M^{p_1} P^{m_1} \cdots P^{m_k} M^{p_{k+1}-1} J P \cmatrix{1 \cr 0}= M^{p_1} P^{m_1} \cdots P^{m_k} M^{p_{k+1}-1} P \cmatrix{1 \cr 0}, $$ that is $\Frac\beta\alpha = M\hat G P (\infty) $. \par\noindent Then, $\cmatrix{\alpha \cr \alpha - \beta } = PM^{-1} \cmatrix{ \beta \cr \alpha } = P \hat G P \cmatrix{1 \cr 0}.$ That is $ \Frac {\alpha}{ \alpha - \beta} = P \hat G P ( \infty ).$ \par\noindent Relations on lengths are derived from the previous relations and remark \ref{rl}. \hbox{} $\Box$ \par\noindent We deduce \begin{proposition}\label{palin} Let $G \in {\mathbf{G}} $ and $\Frac\alpha\beta = [e_1, \ldots, e_n] = PGP(\infty).$ Let $K = S( \Frac \alpha\beta )$ and $N= \hbox{\rm cn}\, (K)$. The following properties are equivalent: \begin{enumerate}\itemsep -2pt plus 1pt minus 1pt \item $G$ is palindromic (i.e. $\overline G = G$). \item the sequence of sign changes in $[e_1, \ldots, e_n]$ is palindromic (i.e. $e_ie_{i+1} = e_{n-i}e_{n-i+1}$). \item $\beta^2 \equiv (-1)^{N-1}\Mod \alpha$. \end{enumerate} Moreover we have \begin{itemize}\itemsep -2pt plus 1pt minus 1pt \item $\beta^2 \equiv -1 \Mod \alpha$ (i.e. $K= \overline{K}$ is amphicheiral) if and only if $N$ is even and $G = \overline G$. Furthermore, the length $n=\ell( \frac \alpha \beta) $ is even and the sequence $[e_1, \ldots, e_n]$ is palindromic (i.e. $e_i = e_{n-i+1}$). \item $\beta^2\equiv 1 \Mod\alpha$ if and only if $N$ is odd and $G = \overline G$ or $N$ is even and $\hat G = {\overline G}$ (in this case $K$ is a two-component link). \end{itemize} \end{proposition} {\em Proof}. From Remark \ref{rl}, it is straightforward that if $[\varepsilon_1, \ldots, \varepsilon_n] = PGP(\infty)$ then $P\overline G P(\infty) = \varepsilon_n [\varepsilon_n, \ldots, \varepsilon_1]$. We deduce that $G=\overline G$ is palindromic if and and only if the sequence of sign changes in $[e_1, \ldots, e_n]$ is palindromic. Let $0<\beta'<\alpha$ such that $\beta'\beta \equiv (-1)^{N-1}\Mod\alpha$. We have from the previous proposition: $\Frac\alpha{\beta'} = P \overline{G}P ( \infty )$ We thus deduce that $G=\overline G$ is equivalent to $\beta = \beta'$, that is $\beta^2\equiv(-1)^{N-1}\Mod\alpha$. Suppose now that $\beta^2\equiv 1\Mod\alpha$. If $N$ is even then $\beta'=\alpha-\beta$, that is $P\overline GP = P\hat G P$ and $\overline G=\hat G$. We have $p+2m = m + 2 p-2$ and then $2n =2(p+2m) = 3N-2$. This implies $n\equiv 2\Mod 3$. By Lemma \ref{length}, $\alpha$ is even and $K$ is a two-component link. If $N$ is odd then $\beta'=\beta$ and $G=\overline G$ by the first part of our proof. Suppose now that $\beta^2\equiv -1 \Mod\alpha$. If $N$ is odd then $\beta'=\alpha-\beta$ and by the same argument we should have $n=3N-n-2,$ which would imply that $N$ is even. We deduce that amphicheiral rational links have even crossing numbers and from $\beta'=\beta$ we get $G=\overline G$. The crossing number $N=m+p$ is even and $G$ is palindromic so $m$ and $p$ are both even. Consequently $n= p+2m$ is even and the number of sign changes is even. We thus have $e_n=1$ and $(e_n, \ldots, e_1)=(e_1, \ldots, e_n)$. \hbox{} $\Box$ \par\noindent We deduce also a method to find a minimal Chebyshev diagram for any rational knot. \begin{proposition}\label{ell} Let $K$ be a two-bridge knot with crossing number $N$. \begin{enumerate}\itemsep -2pt plus 1pt minus 1pt \item There exists $\Frac{\alpha}{\beta}>1$ such that $K=S(\pm \Frac\alpha\beta)$ and $n=\ell(\Frac{\alpha}{\beta}) < \frac 32 N-1$. \item There exists a biregular sequence $(e_1, \ldots, e_n)$, $e_i=\pm 1$, such that $K=C(e_1,\ldots,e_n)$. \item If $K=C(\varepsilon_1,\ldots,\varepsilon_m)$, $\varepsilon_i=\pm1$, then $m\geq n\geq N$. \end{enumerate} \end{proposition} Let $K$ be a two-bridge knot. Let $r=\Frac\alpha\beta>1$ such that $K=S(r)$. We have $\overline K = S(r')$ where $r'=\Frac{\alpha}{\alpha-\beta}$. From Proposition \ref{gg} and Proposition \ref{bireg}, we have $\ell(r)+\ell(r')=3N-2$ and therefore $N\leq \min(\ell(r),\ell(r')) < \frac 32 N$. From Lemma \ref{length}, we have $\ell(r)\not \equiv 2 \Mod 3$ so $\ell(r)\not=\ell(r')$ and $\min(\ell(r),\ell(r'))<\frac 32 N -1$. We may suppose that $n=\ell(r)$. Let $r= \pm [e_1, \ldots, e_n]$, then $C(e_1,\ldots,e_n)$ is a Conway normal form for $K$ with $n<\frac 32 N -1$. This Conway normal form is a Chebyshev diagram ${\cal C}(3,n+1): x= T_3(t),\, y=T_{n+1}(t)$. \par\noindent Let us consider $\gamma$ such that $\beta\gamma \equiv 1 \Mod\alpha$ and $0<\gamma<\alpha$. Let $\rho = \Frac{\alpha}{\gamma}$ and $\rho' = \Frac{\alpha}{\alpha-\gamma}$. We have $K = S(\rho)$ and $\overline K = S(\rho')$ and from Proposition \ref{gg}: $\min(\ell(\rho),\ell(\rho'))=\min(\ell(r),\ell(r'))$. Suppose that $K=C(\varepsilon_1, \ldots, \varepsilon_m)$ then $K = S(x)$ where $x=[\varepsilon_1, \ldots, \varepsilon_m]$. We thus deduce that $x=\Frac{\alpha}{\beta+k\alpha}$ or $x' = \Frac{\alpha}{\gamma+k\alpha}$ where $k \in {\bf Z}$. \begin{enumerate}\itemsep -2pt plus 1pt minus 1pt \item[--] If $k=2p>0$ then we have $x=(MP)^p r$ so $m=\ell(x)=\ell(r)+3p>\ell(r)$. \item[--] If $k=2p+1>0$ then $x=(MP)^p M (\Frac 1r )$ so $\ell(x)=\ell(1/r)+3p+2=\ell(r')+3p+3>\ell(r')$. \item[--] If $k=-(2p+1)<0$ then $-x=(MP)^p (r')$ so $\ell(x)=\ell(-x)=\ell(r')+3p>\ell(r')$. \item[--] If $k=-2p>0$ then $-x=(MP)^{p-1} M (\Frac 1{r'})$ so $\ell(x)=\ell(1/r')+3p-1=\ell(r)+3p>\ell(r)$. \end{enumerate} If $x'=\Frac{\alpha}{\gamma+k\alpha}$, we obtain the same relations. We deduce that $m\geq\min(\ell(r),\ell(r'))$. \hbox{} $\Box$ \begin{remark}[Computing the minimal Chebyshev diagram $\mathbf{{\cal C}(3,b)}$]\label{minb}\\ Let $K=S(\Frac\alpha\beta)$ with $\Frac\alpha\beta = PGP(\infty)$. The condition $\ell(\Frac\alpha\beta)$ is minimal, that is $\ell(\Frac\alpha\beta) < \frac 32 N -1$, is equivalent to $p\geq m+3$ where $p=\deg_P(PGP)$ and $m=\deg_M(PGP)$. In this case $b=\ell(\Frac\alpha\beta)+1$ is the smallest integer such that $K=S(\Frac\alpha\beta)$ has a Chebyshev diagram $x=T_3(t),\, y=T_b(t)$. If $p<m+3$, using Proposition \ref{gg}, $K$ has a Chebyshev diagram ${\cal C}(3,b')$ with $b' = 3N - b < \frac 32 N$. This last diagram is minimal. \end{remark} \begin{example}[Torus knots]\label{tok} The Schubert fraction of the torus knot $T(2,2k+1)$ is $2k+1$. We have $ PM(x) = x+2,$ and then $ (PM)^k (x) = x+2k, (PM)^kP (x)= 2k+1+ \Frac 1x.$ So we get the continued fraction of length $3k+1$: $ 2k+1 = (PM)^kP(\infty)$. This shows that the torus knot $T(2, 2k+1)$ has a Chebyshev diagram $ {\cal C}( 3, 3k+2).$ This is not a minimal diagram. \par\noindent On the other hand, we get $(PM)^{k-1}P^2(\infty) = 2k$ so $\Frac{2k+1}{2k} = P(PM)^{k-1}P^2(\infty) >1$. This shows that the torus knot $T(2, 2k+1)$ has a Chebyshev diagram $ {\cal C}( 3,3k+1).$ This diagram is minimal by Remark \ref{minb}. We will see that $T(2, 2k+1)$ is in fact a harmonic knot. \end{example} \begin{example}[Twist knots]\label{twk} The twist knot ${\cal T}_{n}$ is defined by ${\cal T}_n = S(n+\Frac{1}{2})$. \par\noindent From $P^3( x) = \Frac{3x+2}{2x+1}$, we get the continued fraction of length $3k+3$: $\Frac {4k+3}2= (PM)^kP^3 (\infty).$ This shows that the twist knot ${\cal T}_{2k+1}$ has a Chebyshev diagram ${\cal C} ( 3, 3k+4),$ which is minimal by Remark \ref{minb}. \par\noindent We also deduce that $P(PM)^{k-1}P^3 (\infty) = P \left(\frac 12 (4k-1)\right ) = \Frac{4k+1}{4k-1}$. This shows that the twist knot ${\cal T}_{2k}$ has a minimal Chebyshev diagram ${\cal C} (3,{3k+2}).$ \par\noindent We shall see that these knots are not harmonic knots for $a=3$ and we will give explicit bounds for their polynomial parametrizations. \end{example} \begin{example}[Generalized stevedore knots]\label{sk3} The generalized stevedore knot ${\cal S}_k$ is defined by ${\cal S}_k = S(2k+2+\Frac{1}{2k})$. We have $$ (MP)^k = \cmatrix{1&0\cr 2k&1}, \, (PM)^k =\cmatrix{1&2k\cr 0&1} $$ so $2k+2+\Frac{1}{2k} = (PM)^{k+1}(MP)^k (\infty)$. This shows that the stevedore knots have a Chebyshev diagram ${\cal C}(3,6k+4)$. It is not minimal and we see, using Remark \ref{minb}, that the stevedore knot ${\cal S}_k$ also has a minimal Chebyshev diagram ${\cal C}(3,6k+2)$. Moreover, using Proposition \ref{gg}, we get $$ \Frac{(k+1)^2}{(k+1)^2-2k} = P^2 (MP)^k (PM)^{k-1} P^2 (\infty). $$ \end{example} \subsection{Continued fractions ${\mathbf{[\pm 1,\pm 2,\ldots,\pm 1,\pm 2]}}$} Let us consider the homographies $A(x)=[1,2,x] =\Frac{3x+1}{2x+1}$, $B(x)=[1,-2,-x]=\Frac{x+1}{2x+1}$, $S(x)=-x$. We shall also use the classical matrix notation for these homographies $$ A = \cmatrix{3&1\cr2&1}, \, B = \cmatrix{1&1\cr2&1}, \, S = \cmatrix{1&0\cr0&-1}. $$ \begin{lemma}\label{nn} The monoid ${\mathbf{\Gamma}} = \langle (AS)^k A, (AS)^{k+1}B, B(SA)^k, \ B(SA)^{k+1}SB, \ k\geq 0\rangle$ is free. Let $ {\mathbf{\Gamma}} ^* $ be the subset of elements of $ {\mathbf{\Gamma}}$ that are not of the form $ M \cdot (AS)^{k+1} B, $ or $ M\cdot B(SA)^{k+1}SB$. There is an injection $h : {\mathbf{\Gamma}} ^* \to {\bf Q}_{>0}$ such that $h(G) = G(\infty)$. \end{lemma} {\em Proof}. Let us denote $E= {\bf R} ^* _+ \bigcup \{ \infty \}= ]0, \infty].$ We will describe $G(E)$ for any generator $G$ of $ {\mathbf{\Gamma}}.$ Let $C=AS =\cmatrix{3&-1\cr2&-1}$. We get $C^2 = 2C+{\mathbf{Id}},$ and then $C^{k+2}A = 2C^{k+1}A+C^kA$. From $CA = ASA =\cmatrix{7&2\cr4&1}$ we deduce by induction that $ (AS)^k A (E) \subset ]1, \infty [ .$ Similarly, we get $ (AS)^{k+1} B(E) \subset ]1, \infty], \ B(SA)^k(E)\subset [ 0, 1[, $ and $ B( SA) ^{k+1}SB (E) \subset ]0, 1[ .$ \par\noindent Now, we shall prove that if $G$ and $G'$ are distinct generators of $ {\mathbf{\Gamma}},$ the relation $ G \cdot M = G' \cdot M' $ is impossible. This is clear in cases where $G(E)$ and $G'(E)$ are disjoint. Suppose that $ ( AS)^k AM= (AS)^{k'} BM' .$ If $k<k',$ we would get $ M= S \Bigl( ( AS) ^{k''} BM' \Bigr) = S M'' ,$ which is impossible because $ S M''$ has some negative coefficients. If $ k \ge k'$, we get $ (AS)^{k''} AM= BM'$, which is also impossible because $ B(E)$ and $ ( AS)^{k''} A (E) $ are disjoint. The proof of the impossibility of $ B (SA)^k M= B(SA)^{k'+1} SB M' $ is analogous to the preceding one. This completes the proof that $ {\mathbf{\Gamma}}$ is free. \par\noindent Now, let us prove that $ M \neq M', \ M, M' \in {\mathbf{\Gamma}} ^$ implies $ M(\infty) \neq M'(\infty).$ There are two cases that are not obvious. If $ (AS)^kA M(\infty)= (AS)^{k'} BM' ( \infty) , \ M, M' \in {\mathbf{\Gamma}} ^ .$ If $ k< k'$, we get $ M(\infty) = S ( M'' ( \infty)) ,$ which is impossible since $ M(\infty) \in ]0, \infty[, $ and $ SM'' (\infty) \in ] -\infty , 0[ .$\\ If $ k \ge k',$ we get $ BM'(\infty) = ( AS)^{k''} A M (\infty) ,$ which is also impossible because $BM'(\infty) <1$ and $ (AS)^{k''} AM( \infty) >1.$ The proof of the impossibility of $B (SA)^k M (\infty) = B(SA)^{k'+1} SB M' (\infty) $ is analogous to the preceding one. This completes the proof of the injectivity of $ h.$ \hbox{} $\Box$ \begin{theorem} \label{cf1212} Let $r=\Frac\alpha\beta>0$ be a rational number. Then $r$ has a continued fraction expansion $\Frac\alpha\beta = [1,2 e_2,\ldots,e_{2n-1},2 e_{2n}]$, $e_i = \pm 1$, if and only if $\beta$ is even. Furthermore, we can suppose that there are no three consecutive sign changes. In this case the continued fraction is unique, and $\alpha > \beta$ if and only if $e_1= e_2= 1$. \end{theorem} {\em Proof}. Let us suppose that $\Frac\alpha\beta =[1,\pm2,\ldots,\pm1,\pm2]$. It means that $\cmatrix{\alpha\cr\beta} = G \cmatrix{1\cr0}$ where $G \in \langle A,B,S\rangle$. But we have $A\equiv B \equiv \cmatrix{1&1\cr0&1} \Mod 2$, $S \equiv {\mathbf{Id}} \Mod 2$. Consequently, $\cmatrix{\alpha\cr\beta} \equiv \cmatrix{1&1\cr0&1}^k \cmatrix{1\cr0} \equiv \cmatrix{1\cr0}\Mod 2$ and $\beta$ is even. \par\noindent Let us suppose that $\beta$ is even. We shall use induction on the height $h(\Frac\alpha\beta) = \max(\abs\alpha,\abs\beta)$. \begin{itemize}\itemsep -2pt plus 1pt minus 1pt \item If $h(\Frac\alpha\beta)=2,$ then $\alpha=1$ and $\beta=2$ and we have $r=[1,-2]=B(\infty)$. \item We have two cases to consider \begin{enumerate}\itemsep -2pt plus 1pt minus 1pt \item[--] If $\alpha>\beta>0$ then we write $ \Frac\alpha\beta = [1,2,-1,2,\Frac{\alpha-2\beta}{\beta}]= ASB \left (\Frac{\alpha-2\beta}{\beta}\right ). $ We have $h(\Frac{\alpha-2\beta}{\beta}) < h(\Frac\alpha\beta)$ and we conclude by induction. \item[--] If $\beta > \alpha >0$ we write $ \Frac\alpha\beta = [1,-2,\Frac{\alpha-\beta}{2\alpha-\beta}]= B \left (\Frac{\beta - \alpha}{2\alpha-\beta}\right ). $ From $\abs{2\beta-\alpha}\leq \beta$ we have $h(\Frac{\alpha-\beta}{2\alpha-\beta})<h(\Frac\alpha\beta)$ and we conclude by induction. \end{enumerate} \end{itemize} The existence of a continued fraction $[ 1, \pm 2, \ldots, \pm 1, \pm2]$ is proved. \par\noindent Using the identities $BSBS(x)=[1,-2,1,-2,x]=x$ and $[2,-1,2,-1,x]=x,$ we can delete all subsequences with three consecutive sign changes in our sequence. We deduce also that $\cmatrix{\alpha\cr\beta} = G \cmatrix{1\cr 0}$ where $G \in \langle A,S,B\rangle$. Furthermore, from $BSB(x) = [1,-2,1,-2,-x]$ we see that $G$ contains no $BSB$. We also see from $ASBSA(x) = [1,2,-1,2,-1,-2,-x]$ that $G$ contains no $SBS$. Consequently $G $ is an element of $ {\mathbf{\Gamma}} ^* $ and, by Lemma \ref{nn}, the continued fraction $\Frac\alpha\beta = [1,\pm2,\ldots,\pm1,\pm2]$ is unique. \hbox{} $\Box$ \begin{example}[Torus knots]\label{tnh4}\\ Since $BSA = \cmatrix{1&0\cr4&1}, \ $ we get by induction $$ A (BSA)^k = \cmatrix{4k+3&1\cr 4k+2&1}, \, A (BSA)^k B = \cmatrix{4k+5&4k+4\cr 4k+4&4k+3}. $$ We deduce the following continued fractions $$ \Frac {4k+3}{4k+2}= [ 1,2, \underline{1,-2,1,2}, \ldots, \underline{1, -2,1,2} ], \ \quad \Frac {4k+5}{4k+4}= [1,2, \underline{1, -2, 1,2}, \ldots, \underline{1,-2,1,2}, 1, -2 ] $$ of length $4k+2$ and $4k+4.$ Since the knot with Schubert fraction $ \Frac {N}{N-1}$ is the torus knot $\overline{T}(2,N),$ we see that this knot admits a Chebyshev diagram $ x= T_4(t), y= T_N (t).$ \end{example} \begin{example}[Twist knots]\label{twk4}\\ The twist knot is the knot ${\cal T}_m= S( \Frac {2m+1} 2 ) = S ( \Frac {2m+1}{m+1} )= \overline{S}(\Frac{2m+1}m).$ We have the continued fractions $$ \Frac{8k+1}{4k} = (ASB) (BSA)^k ( \infty ), \ \quad \Frac{8k+5}{4k+2} = (ASB)( BSA)^k B( \infty ). $$ We deduce that ${\cal T}_{2n}$ has a Chebyshev diagram $ x= T_4(t), \, y= T_ {2n+5}(t).$ We get similarly $$ \Frac {8k+7}{4k+4} = ASA (BSA)^k ( \infty ), \ \Frac {8k+3}{4k+2} = ASA (BSA)^{k-1} B( \infty ), \ $$ and we deduce that ${\cal T}_{2n+1}$ has a Chebyshev diagram $x= T_4(t), \, y= T_{2n+3} (t).$ \end{example} \begin{example}[Generalized stevedore knots]\label{sk4}\\ The generalized stevedore knot ${\cal S}_m$ is defined by ${\cal S}_m= S(2m+2 + \Frac 1{2m}), \ m \ge 1.$ The stevedore knot is ${\cal S}_1= \overline{6}_1$. We have $ \overline{{\cal S}}_m = S \Bigl(\Frac{(2m+1)^2}{2m+2}\Bigr) $ and the continued fractions $$ \Frac {(4k+1)^2}{4k+2} = (ASB)^{2k} (BSA)^k B (\infty), \ \Frac {(4k-1)^2}{4k} = (ASB)^{2k-1}(BSA)^k (\infty ). $$ Consequently, the stevedore knot ${\cal S}_m$ has a Chebyshev diagram ${\cal C}(4,6m+3)$. \end{example} These examples show that our continued fractions are not necessarily regular. In fact, the subsequences $ \pm (2,-1,2) $ of the continued fraction correspond bijectively to factors $(ASB).$ \par\noindent There is a formula to compute the crossing number of such knots. \begin{proposition}\label{ilot} Let $\Frac{\alpha}{\beta} = [e_1,2e_2,\ldots,e_{2n-1},2e_{2n}]$, $e_i=\pm 1$, where there are no three consecutive sign changes and $e_1=e_2$. We say that $i$ is an islet in $[a_1,a_2,\ldots,a_{n}]$ when $$\abs{a_i}=1, \, a_{i-1}=a_{i+1}=-2a_i.$$ We denote by $\sigma$ the number of islets. We have \[ \hbox{\rm cn}\,(\Frac\alpha\beta) = \sum_{k=1}^n \abs{a_i} - \sharp \{i, a_ia_{i+1}<0\} - 2 \sigma. \label{ilotf} \] \end{proposition} {\em Proof}. By induction on the number of double sign changes $k=\sharp\{i, e_{i-1}e_{i}<0, \, e_{i}e_{i+1}<0\}$. If $k=0$, the sequence is biregular and Formula (\ref{ilotf}) is true by Proposition \ref{bireg}. Suppose $k\geq 1$. First, we have $a_1 a_2 >0$. Then $\Frac\alpha\beta =\pm [x,a,b,-c,d,e,y]$ where $[x,a,b]$ is a biregular sequence and $a,b,c,d,e>0$. We have $[x,a,b,-c,d,e,y] = [x,a,b-1,1,c-1,-d,-e,-y] $. \begin{itemize}\itemsep -2pt plus 1pt minus 1pt \item If $c = 2$ (there is no islet at $c$), then we have $b=d=1$ and $[x,a,b,-c,d,e,y] = [x,a+1,c-1,-d,-e,-y]$. The sum of the absolute values has decreased by 1, as has the number of sign changes, $\sigma$ is unchanged. \item If $c=1$ (there is an islet at $c$) then $[x,a,b,-c,d,e,y] = [x,a,b-1,-(d-1),-e,-y]$. The sum of absolute values has decreased by 3, the number of sign changes by 1, $\sigma$ by 1. \end{itemize} In both cases, $\sum_{k=1}^n \abs{a_i} - \sharp \{i,a_ia_{i+1}<0\} - 2\sharp\{i, e_{2i}e_{2i+1}<0, \, e_{2i+1}e_{2i+2}<0\}$ remains unchanged while $k$ has decreased by 1. \hbox{} $\Box$ \par\noindent We shall need a specific result for biregular fractions $[\pm 1,\pm 2,\ldots, \pm 1,\pm 2]$. \begin{proposition} \label{palin4} Let $r= \Frac \alpha \beta$ be a rational number given by a biregular continued fraction of the form $ r= [e_1, 2e_2, e_3, 2e_4, \ldots e_{2m-1}, 2 e_{2m} ], \, e_1=1, \, e_i= \pm 1.$ If the sequence of sign changes is palindromic, i.e. if $e_k e_{k+1} = e_{2m-k} e_{2m-k+1},$ we have $\beta^2 \equiv \pm 2 \Mod \alpha.$ \end{proposition} {\em Proof}. From Theorem \ref{cf1212}, and because $\Frac\alpha\beta = [ 1, 2e_2, e_3, 2e_4, \ldots e_{2m-1}, 2 e_{2m} ]$ is regular, we have $\Frac\alpha\beta = G (\infty)$ where $G \in \langle B, (AS)^k A, \, k \geq 0 \rangle \subset {\mathbf{\Gamma}}$. \par\noindent We shall consider the mapping (analogous to matrix transposition) $$\varphi: \cmatrix {a & b \cr c & d }\mapsto \cmatrix { a & \Frac c2 \cr 2b & d }.$$ We have $ \varphi(XY)= \varphi (Y) \varphi (X), \ \varphi(A)=A, \ \varphi(B)= B$ and $\varphi((AS)^k A) = (AS)^k A$. \par\noindent Let us show that $G$ is a palindromic product of terms $A_k= (AS)^k A$ and $B$. By induction on $s = \sharp\{i, e_{i}e_{i+1}<0\}.$ If $s=0$ then $G = A^m$. Let $k = \min \{i, e_{i}=-e_{i+1}\}$. If $k=2p$ then $G = A^p G'$ and $G'\in S \cdot {\mathbf{\Gamma}}$. We have $e_1 = \ldots = e_{2p}$ and $e_{2(m-p)+1} = \ldots = e_{2m}$ that is $G = A^p S G' S A^p$. The subsequence $(-e_{2p+1}, \ldots, -e_{2m-2p})$ is still palindromic and corresponds to $G'(\infty)$. By regularity $e_{2p+1}= e_{2p+2},$ and we conclude by induction. If $k=2p+1$ we have $G=A^p B G' B A^p$ and we conclude also by induction. \par\noindent Hence $ \varphi (G)= G,$ and since $G\cmatrix {1 \cr 0 } = \cmatrix {\alpha \cr \beta },$ we see that $G$ has the form $G= \cmatrix{ \alpha & \gamma \cr \beta & \lambda },$ with $ \beta = 2 \gamma.$ Using the fact that $ \det (G)= \pm 1,$ we get $ \beta ^2 \equiv \pm 2 \Mod\alpha.$ \hbox{} $\Box$ \section{The harmonic knots $\mathbf{{\rm H}(a,b,c)}$}\label{harmonic} In this paragraph we shall study Chebyshev knots with $ \varphi= 0.$ Comstock (1897) found the number of crossing points of the harmonic curve parametrized by $x=T_a(t), y=T_b(t), z=T_c(t).$ In particular, he proved that this curve is non-singular if and only if $ a,b,c$ are pairwise coprime integers (\cite{Com}). Such curves will be named harmonic knots ${\rm H}(a, b,c)$. \par\noindent We shall need the following result proved in \cite{JP,KP3} \begin{proposition} Let $a $ and $b$ be coprime integers. The $\frac 12 (a-1)(b-1)$ double points of the Chebyshev curve $x= T_a(t), y= T_b(t)$ are obtained for the parameter pairs $$ t= \cos \bigl( \Frac ka + \Frac hb \bigr) \pi, \ s = \cos \bigl( \Frac ka - \Frac hb \bigr) \pi, $$ where $h,k$ are positive integers such that $ \Frac ka + \Frac hb < 1.$ \end{proposition} Using the symmetries of Chebyshev polynomials, we see that this set of parameters is symmetrical about the origin. We will write $ x \sim y$ when $\sign x = \sign y.$ We shall need the following result proved in \cite{KP3}. \begin{lemma}\label{sign} Let ${\rm H}(a,b,c)$ be the harmonic knot: $x=T_a(t),\, y=T_b(t),\, z=T_c(t)$. A crossing point of parameter $ t = \cos \left (\Frac ka + \Frac hb\right ) \pi, \ $ is a right twist if and only if $$D = \Bigl( z(t)-z(s) \Bigr) x'(t) y'(t) >0$$ where $$ z(t)-z(s) = T_c(t)-T_c(s) = -2 \sin \Bigl( \Frac {ch}{b} \pi \Bigr) \sin \Bigl( \Frac {ck}a \pi \Bigr). $$ and $$ x'(t) y'(t) \sim (-1)^{h+k} \sin \Bigl( \Frac {ah}b \pi \Bigr) \sin \Bigl( \Frac {bk}a \pi \Bigr). $$ \end{lemma} From this lemma we immediately deduce \begin{corollary}\label{cprime} Let $a,b,c$ be coprime integers. Suppose that the integer $c'$ verifies $ c' \equiv c \Mod{2a} $ and $ c' \equiv -c \Mod{2b}.$ Then the knot ${\rm H}(a,b,c')$ is the mirror image of ${\rm H}( a,b,c).$ \end{corollary} {\em Proof}. At each crossing point we have $ T_{c'}(t) - T_{c'}(s) = - \Bigl( T_c(t) - T_c (s) \Bigr). $ \hbox{} $\Box$ \begin{corollary} Let $a,b,c$ be coprime integers. Suppose that the integer $c$ is of the form $c= \lambda a + \mu b$ with $\lambda, \mu >0$. Then there exists $c'< c $ such that ${\rm H}( a,b,c)= \overline{{\rm H}} (a,b,c') $ \end{corollary} {\em Proof}. Let $c'=\abs{\lambda a - \mu b}.$ The result follows immediately from corollary \ref{cprime} \hbox{} $\Box$ \par\noindent In a recent paper, G. and J. Freudenburg have proved the following stronger result. {\em There is a polynomial automorphism $ \Phi$ of $ {\bf R}^3$ such that $ \Phi ({\rm H}(a,b,c)) = {\rm H}(a,b,c').$} They also conjectured that the knots ${\rm H}(a,b,c), \ a<b<c, \ c \neq \lambda a + \mu b, \ \lambda, \mu >0 $ are different knots (\cite{FF}, Conjecture 6.2). \subsection{The harmonic knots ${\mathbf{{\rm H}(3,b,c)}}$ } The following result is the main step in the classification of the harmonic knots ${\rm H}( 3,b,c)$. \begin{theorem}\label{h3} Let $b=3n+1, \ c= 2b-3 \lambda, \, (\lambda, b)=1.$ The Schubert fraction of the knot ${\rm H} (3, b,c )$ is $$ \Frac {\alpha}{\beta} = [e_{1}, e_{2}, \ldots, e_{3n}], \hbox{ where } e_k = \sign {\sin k \theta} \hbox{ and } \theta = \Frac {\lambda}b \pi. $$ If $0< \lambda < \Frac {b}{2}$, its crossing number is $N= b - \lambda = \Frac {b+c}3,$ and we have $\beta^2 \equiv \pm 1 \Mod \alpha.$ \end{theorem} {\em Proof}. Will be given in section \ref{proofs}, p. \pageref{h3proof}. \begin{corollary}\label{hh3} The knots ${\rm H}(3,b,c)$ where $\Frac c2 <b<2c, \ b \equiv 1 \Mod 3, \ c \equiv 2 \Mod 3$ are different knots (even up to mirroring). Their crossing number is given by $b+c=3N.$ \end{corollary} {\em Proof}. Let $K={\rm H}(3,b,c)$ and $\Frac \alpha \beta>1$ be its biregular Schubert fraction given by Theorem \ref{h3}. From Prop \ref{ell}, $\min(b,c)$ is the minimum length of any Chebyshev diagram of $K$ and $\max(b,c)=3N-\min(b,c)$. The pair $(b,c)$ is uniquely determined. \hbox{} $\Box$ The following result gives the classification of harmonic knots ${\rm H}(3,b,c)$. \begin{theorem}\label{h3bc} Let $K= {\rm H}(3,b,c).$ There exists a unique pair $(b',c')$ such that (up to mirror symmetry) $$ K = {\rm H}(3,b',c'), \ b'<c'<2b', \ b'+c' \equiv 0 \Mod 3. $$ The crossing number of $K$ is $\frac 13 (b'+c'),$ its fractions $ \Frac \alpha \beta $ are such that $ \beta^2 \equiv \pm 1 \Mod \alpha.$ Furthermore, there is an algorithm to find the pair $(b',c').$ \end{theorem} {\em Proof}. Let $K= {\rm H}(a,b,c)$ We will show that if the pair $(b,c)$ does not satisfy the condition of the theorem, then it is possible to reduce it. If $c<b$ we consider ${\rm H}(3,c,b)=\overline {\rm H}(3,b,c)$. If $b \equiv c \Mod 3,$ we have $ c= b + 3 \mu, \ \mu>0.$ Let $c'= \abs{b-3\mu}$. We have $c'\equiv \pm c \Mod {2b}$ and $c' \equiv\mp c \Mod 6.$ By Lemma \ref{cprime}, we see that $ K= \overline {\rm H}(3,b,c')$ and we get a smaller pair. If $b \not\equiv c \Mod 3$ and $c>2b,$ we have $c=2b + 3\mu, \ \mu>0.$ Let $c'= \abs{2b-3 \mu}$. Similarly, we get $ K= \overline{\rm H}(3,b,c')$. This completes the proof of existence. This uniqueness is a direct consequence of Corollary \ref{hh3}. \hbox{} $\Box$ \begin{remark} Our theorem gives a positive answer to the Freudenburg conjecture for $a=3.$ \end{remark} \subsection{Examples} As applications of Proposition \ref{bireg}, let us deduce the following results (already in \cite{KP3}) \begin{corollary} The harmonic knot $ {\rm H} (3, 3n+2, 3n+1) $ is the torus knot $ T(2, 2n+1).$ \end{corollary} {\em Proof}. The harmonic knot $ K= {\rm H} (3, 3n+1, 3n+2 )$ is obtained for $ b= 3n+1,$ $c= 2b -3\lambda, \ \lambda= n,$ $ \theta = \Frac n {3n+1} \pi.$ If $j=1,2,$ or $ 3,$ and $ k=0,\ldots, n-1$ we have $(3k+j) \theta = k \pi + \Frac {jk-n}{3n+1},$ hence $ \sign{ \sin (3k+j) \theta } = (-1)^k $, so that the Schubert fraction of $K$ is $$ [1,1,1, -1,-1,-1, \ldots, (-1)^{n+1}, (-1)^{n+1}, (-1)^{n+1} ] = \Frac{2n+1} {2n} \thickapprox -(2n+1). $$ We see that $K$ is the mirror image of $ T(2,2n+1),$ which completes the proof. \hbox{} $\Box$ \par\noindent It is possible to parameterize the knot $T(2, 2n+1)$ by polynomials of the same degrees and a diagram with only $2n+1$ crossings (\cite{KP2}). However, our Chebyshev parametrizations are easier to visualize. We conjecture that these degrees are minimal (see also \cite{RS,FF,KP1}). \begin{corollary} The harmonic knot ${\rm H}(3,b,2b-3)$ ($b\not \equiv 0 \Mod 3$) is alternate and has crossing number $b-1$. \end{corollary} {\em Proof}. For this knot we have $ \lambda=1, \ \theta = \Frac \pi b.$ The Schubert fraction is given by the continued fraction of length $b-1$: [1,1, \ldots,1] = $\Frac{F_{b}}{F_{b-1}}$ where $F_n$ are the Fibonacci numbers ($F_0=0, F_1=1, \ldots$). J. C. Turner named these knots Fibonacci knots (\cite{Tu}). \hbox{} $\Box$ \par\noindent The two previous examples describe infinite families of harmonic knots. They have a Schubert fraction $\Frac\alpha\beta$ with $\beta^2 \equiv 1 \Mod\alpha$ (torus knots) or with $\beta^2 \equiv - 1\Mod\alpha$ (Fibonacci knots with odd $b$). There is also an infinite number of two-bridge knots with $\beta^2= \pm 1 \Mod\alpha$ that are not harmonic. \begin{proposition} The knots (or links) $K_n = S(\Frac{5F_{n+1}}{F_{n+1}+F_{n-1}}),\, n>1$ are not harmonic knots ${\rm H}(3,b,c)$. Their crossing number is $n+4$ and we have $(F_{n+1}+F_{n-1})^2 \equiv (-1)^{n+1} \Mod{5F_{n+1}} $. \end{proposition} {\em Proof}. Using the fact that $P^n = \cmatrix{F_{n+1}&F_n\cr F_n&F_{n-1}}$ we deduce that $$ P M P^n M P = \cmatrix{5F_{n+1}&F_{n+1}+F_{n-1}\cr F_{n+1}+F_{n-1}&F_{n-1}}. $$ Taking determinants, we get $(F_{n+1}+F_{n-1})^2 \equiv (-1)^{n+1} \Mod{5F_{n+1}} $. We also have $$ \Frac{5F_{n+1}}{F_{n+1}+F_{n-1}} = P M P^n M P(\infty)= [1,1,\underbrace{-1,\ldots,-1}_{n+2},1,1]. $$ Since $ n+2 \ge 4,$ it cannot be of the form $[\sign{\sin \theta},\sign{\sin 2\theta}, \ldots, \sign{\sin k\theta}]$. If $n\equiv 2\Mod 3$, $K_n$ is a two-component link. If $n\equiv 1\Mod 6$ or $n\equiv 3\Mod 6$, the Schubert fraction $\Frac\alpha\beta$ satisfies $\beta^2\equiv 1\Mod\alpha$. If $n\equiv 0\Mod 6$ or $n\equiv 4\Mod 6$, $K_n$ is amphicheiral. \hbox{} $\Box$ \subsection{The harmonic knots ${\mathbf{{\rm H}(4,b,c)}}.$} The following result will allow us to classify the harmonic knots of the form ${\rm H}(4,b,c).$ \begin{theorem}\label{h4} Let $b, c$ be odd integers such that $ b \not \equiv c \Mod 4.$ The Schubert fraction of the knot $K= {\rm H}(4,b,c)$ is given by the continued fraction $$ \Frac \alpha\beta = [ e_1, 2e_2, e_3, 2e_4, \ldots, e_{b-2}, 2e_{b-1} ], \, e_j= -\sign{\sin (b-j) \theta}, \, \theta = \Frac {3b-c} {4b} \pi.$$ If $b< c < 3b,$ this fraction is biregular, the crossing number of $K$ is $ N= \Frac { 3b+c-2}4,$ and $\beta ^2 \equiv \pm 2 \Mod 3.$ \end{theorem} {\em Proof}. Will be given in section \ref{proofs}, p. \pageref{h4proof}.\hbox{} $\Box$ \par\noindent We are now able to classify the harmonic knots of the form ${\rm H}(4,b,c)$. \begin{theorem}\label{h4bc}\\ Let $K= {\rm H}(4,b,c).$ There is a unique pair $(b',c')$ such that (up to mirroring) $$K= {\rm H}(4,b',c'), \ b'<c'<3b', \ b' \not\equiv c' \Mod 4.$$ The crossing number of $K$ is $\frac 14 (3b'+c'-2). $ $K$ has a Schubert fraction $\Frac \alpha \beta $ such that\\ $ \beta^2 \equiv \pm 2 \Mod \alpha.$ Furthermore, there is an algorithm to find $(b', c').$ \end{theorem} {\em Proof}. Let us first prove the uniqueness of this pair. Let $K= {\rm H}(4,b,c)$ with $ b<c<3b, \ c \not\equiv b \Mod 4.$ By theorem \ref{h4}, $K$ has a Schubert fraction $ \Frac \alpha \beta = \pm [ 1, \pm2, \ldots, \pm 1, \pm 2] $ of length $b-1$ with $\beta$ even, $ -\alpha < \beta < \alpha,$ and $ \beta ^2 \equiv \pm 2 \Mod \alpha.$ The other fraction of $K$ is $\Frac \alpha{\beta'}$, where $\beta'$ is even and $ \abs{\beta'}<\alpha $ and $ \beta \beta ' \equiv 1 \Mod \alpha.$ If $ \alpha > 3,$ we cannot have $ {\beta'}^2 \equiv \pm 2 \Mod \alpha.$ We deduce that $b$ is uniquely determined by $K$: $b=\ell ( \Frac \alpha \beta ) +1 $ where $\Frac \alpha \beta$ is a Schubert fraction of $K$ such that $\beta^2 \equiv \pm 2 \Mod \alpha.$ Since we also have $3 b+c -2= 4 \hbox{\rm cn}\,(K),$ we conclude that $(b,c)$ is uniquely determined by $K.$ \par\noindent Now, let us prove the existence of the pair $(b',c').$ Let $K= {\rm H}(4,b,c), \ b<c$. We have only to show that if the pair $(b,c)$ does not satisfy the condition of the theorem, then it is possible to reduce it. \par\noindent If $ c \equiv b \Mod 4, $ then $c= b + 4 \mu, \ \mu >0.$ Let $c'= \abs{b - 4 \mu}$. Then $K= \overline {\rm H}( 4, b, c' )$, and the pair $(b,c')$ is smaller than $(b,c).$ \par\noindent If $ c \not\equiv b \Mod 4 $ and $c>3b,$ we have $c= 3b + 4 \mu, \ \mu >0.$ Let $c' = \abs{3b-4 \mu}$. We have, $K= \overline{\rm H}(4,b,c')$ with $(b,c')$ smaller than $(b,c).$ This completes the proof. \hbox{} $\Box$ \begin{remark} Our theorem gives a positive answer to the Freudenburg conjecture for $a=4.$ \end{remark} We also see that the only knot belonging to the two families of knots $ {\rm H}(3,b,c)$ and ${\rm H}(4,b,c)$ is the trefoil $ {\rm H}(3, 4,5 )= \overline {\rm H}(4,3,5).$ \begin{example}[$\mathbf{{\rm H}(4,2k-1,2k+1)}$]\\ From Theorem \ref{h4}, we know that ${\rm H}(4,2k-1,2k+1)$ has crossing number $2k-1$. Using this theorem, the Conway sequence of this knot is $ [e_1,2e_2, \ldots, e_{2k-3},2e_{2k-2}]$, where $$e_j = -\sign{\sin((2k-1-j)\Frac{(k-1)\pi}{2k-1})} = (-1)^{k+1} \sign{\sin(\Frac{j(k-1)\pi}{2k-1})} = (-1)^{k+\pent{j+1}2}. $$ We deduce that the Schubert fraction of ${\rm H}(4,2k-1,2k+1)$ is $$ \Frac{\alpha_k}{\beta_k} = (-1)^{k+1} [1,2,-1,-2,1,2,\ldots, (-1)^{k},2(-1)^{k}] = (-1)^{k+1} (AS)^{k-2}A (\infty). $$ Using recurrence formula in Lemma \ref{nn}, we deduce that $$ \begin{array}{rcl} \alpha_2 = 3, & \alpha_3=7, & \alpha_{k+2} = 2 \alpha_{k+1}+\alpha_k\\ \beta_2 = -2, & \beta_3=4, & \abs {\beta_{k+2}} = 2 \abs {\beta_{k+1}}+\abs {\beta_k}. \end{array} $$ Let us consider the homography $ G(x)= [2,x],$ and its matrix $ G = \cmatrix { 2 & 1 \cr 1 & 0 }$. Let the sequence $u_k$ be defined by $$ G^k = \cmatrix { u_{k+1} & u_k \cr u_k & u_{k-1}}.$$ The sequence $u_k$ verifies the same recurrence formula $ u_{k+2}= 2 u_{k+1} + u_k. $ We deduce $\alpha _k = u_k + u_{k-1}, \abs { \beta _k } = 2 u_{k-1}.$ We also have $$ P^2 G^{k-2} P ( \infty ) = \cmatrix { 2 & 1 \cr 1 & 1 } \, \cmatrix { u_{k-1} & u _ {k-2} \cr u_{k-2} & u _{k-3} } \, \cmatrix {1 \cr 1 } = \cmatrix { u_k + u_{k-1} \cr 2 u_k } = \Frac {\alpha _k }{ \abs { \beta _k } }. $$ Finally, we get $r_k = (-1)^{k+1} [1,1,\underbrace{2, \ldots 2}_{k-2},1]$. \end{example} \begin{example}[The twist knots]\label{twh4} The knots ${\cal T}_n$ are not harmonic knots ${\rm H}(4,b,c) $ for $n>3.$ \end{example} {\em Proof}. The Schubert fractions of ${\cal T}_n= S( n+ \Frac 12 ) $ with an even denominator are $ \Frac {2n+1}2,$ and $ \Frac{2n+1}{-n} $ or $ \Frac {2n+1} {n+1} $ according to the parity of $n.$ The only such fractions verifying $ \beta ^2 \equiv \pm 2 \Mod \alpha $ are $ \Frac 32, \, \Frac 74, \, \Frac 94.$ The first two are the Schubert fractions of the trefoil and the $\overline{5}_2$ knot, which are harmonic for $a=4.$ We have only to study the case of $ 6_1 = S ( \Frac 9{4} ).$ We have $ \Frac 9{4} = [ 1,2, -1,2, 1,-2, 1,2 ].$ Since this fraction is not biregular, we see that $6_1$ is not of the form ${\rm H}(4,b,c).$ \hbox{} $\Box$ \par\noindent But there also exist infinitely many rational knots whose Schubert fractions $\Frac\alpha\beta$ satisfy $\beta^2\equiv -2 \Mod\alpha$ that are not harmonic for $a=4$. \begin{proposition} The knots $S(n+\Frac{1}{2n})$ are not harmonic knots ${\rm H}(4,b,c)$ for $n>1$. Their crossing number is $3n$ and their Schubert fractions $\Frac\alpha\beta = \Frac{2n^2+1}{2n}$ satisfy $\beta^2\equiv -2 \Mod \alpha$. \end{proposition} {\em Proof}. If $n=2k$, we deduce from $(ASB)^k(x) = 2k+x$ and $(BSA)^k (\infty)= \Frac{1}{4k}$, that $$n+\Frac{1}{2n} = (ASB)^k (BSA)^k(\infty).$$ If $n=2k+1$, we use (see the torus knots, example \ref{tnh4}) $\Frac{2n+1}{2n} = A(BSA)^k(\infty)$, so $$n+\Frac{1}{2n} = n-1 + \Frac{2n+1}{2n} = (ASB)^k A(BSA)^k(\infty).$$ For $n>1$ these continued fractions are not biregular, and since $ \beta ^2 \equiv -2 \Mod \alpha$, they do not correspond to harmonic knots ${\rm H}(4,b,c)$. \hbox{} $\Box$ \section{Chebyshev diagrams of rational knots}\label{diagrams} \begin{definition} We say that a knot in ${\bf R} ^3 \subset {{\bf S}}^3$ has a Chebyshev diagram ${\cal C}(a,b)$, if $a$ and $b$ are coprime and the Chebyshev curve $${\cal C}(a,b): x=T_a(t); \ y=T_b(t)$$ is the projection of some knot which is isotopic to $K$. \end{definition} \subsection{Chebyshev diagrams with $a=3$} Using the previous results of our paper (Proposition \ref{ell}) we have \begin{theorem} Let $K$ be a two-bridge knot with crossing number $N$. There is an algorithm to determine the smallest $b$ such that $K$ has a Chebyshev diagram ${\cal C}(3,b)$ with $N < b < \Frac 32 N$. \end{theorem} {\em Proof}. Let $\Frac \alpha \beta > 1$ and $ \Frac \alpha{\alpha - \beta}>1 $ be Schubert fractions of $K$ and $\overline{K}.$ By Proposition \ref{ell}, $b=\min\Bigl( \ell (\Frac \alpha \beta),\ell (\Frac \alpha {\alpha - \beta})\Bigr)+1$ has the required property. \hbox{} $\Box$ \begin{definition} Let ${\cal D}(K)$ be a diagram of a knot having crossing points corresponding to the parameters $ t_1, \ldots, t_{2m} $. The Gauss sequence of ${\cal D}(K)$ is defined by $ g_k = 1 $ if $t_k$ corresponds to an overpass, and $ g_k = -1$ if $t_k$ corresponds to an underpass. \end{definition} \begin{theorem}\label{gauss3} Let $ K$ be a two-bridge knot of crossing number $N.$ Let $x= T_3(t),\ y =T_b(t)$ be the minimal Chebyshev diagram of $K$. Let $c$ denote the number of sign changes in the corresponding Gauss sequence. Then we have $$ b+c=3N.$$ \end{theorem} {\em Proof}. Let $s$ be the number of sign changes in the Conway normal form of $K.$ By Proposition \ref{bireg} we have $ N= b-1-s.$ From this we deduce that our condition is equivalent to $3s+c = 2b-3.$ Let us prove this assertion by induction on $s.$ If $s=0$ then the diagram of $K$ is alternate, and we deduce $c= 2(b-1) -1=2b-3.$ Let $C(e_1,e_2, \ldots, e_{b-1}) $ be the Conway normal form of $K.$ We may suppose $e_1=1$. We shall denote by $M_1, \ldots, M_{b-1}$ the crossing points of the diagram, and by $ x_1< x_2 < \cdots < x_{b-1} $ their abscissae. Let $e_k$ be the first negative coefficient in this form. By the regularity of the sequence we get $e_{k+1}<0,$ and $ 3 \le k \le b-1.$ Let us consider the knot $K'$ defined by its Conway normal form $$ K' = C(e_1,e_2, \ldots, e_{k-1}, -e_k, -e_{k+1}, \ldots, - e_{b-1}).$$ We see that the number of sign changes in the Conway sequence of $K'$ is $s'= s-1.$ By induction, we get for the knot $K'$: $3s'+c'= 2b-3.$ The plane curve $x=T_3(t), \ y= T_b(t) $ is the union of three arcs where $x(t)$ is monotonic. Let $ \Gamma$ be one of these arcs. $\Gamma$ contains (at least) one point $M_k$ or $M_{k+1}.$ \psfrag{g1}{\large$\mathbf{\Gamma_1}$} \psfrag{g2}{\large$\mathbf{\Gamma_2}$} \psfrag{g3}{\large$\mathbf{\Gamma_3}$} \psfrag{k1}{$M_{k-1}$} \psfrag{k2}{$M_k$} \psfrag{k3}{$M_{k+1}$} \begin{figure} \caption{The modification of Gauss sequences} \label{gaussf} \end{figure} Let $j$ be the first integer in $\{ k, k+1 \}$ such that $M_j$ is on $\Gamma$, and let $j_- < j$ be the greatest integer such that $ M_{j_-} \in \Gamma.$ In figure \ref{gaussf}, we have for $\Gamma_1$: $j=k, j^- = k-1$, for $\Gamma_2$: $j=k, j^- = k-2$, for $\Gamma_3$: $j=k+1, j^- = k-1.$ On each arc $\Gamma$, there is a sign change in the Gauss sequence iff the corresponding Conway signs are equal. Then, since the Conway signs $ s(M_{j_-} )$ and $s(M_j)$ are different, we see that the corresponding Gauss signs are equal. Now, consider the modifications in the Gauss sequences when we transform $K$ into $K'$. Since the the Conway signs $s( M_h), \, h\ge k $ are changed, we see that we get one more sign change on every arc $\Gamma$. Thus the number of sign changes in the Gauss sequence of $K'$ is $c' = c+3.$ We get $3s+c= 3(s'+1) + c'-3= 3s'+c' = 2b-3,$ which completes our induction proof. \hbox{} $\Box$ \begin{corollary}\label{degc3} Let $K$ be a two-bridge knot with crossing number $N$. Then there exist $b,c$, $b+c=3N$, and an polynomial $C$ of degree $c$ such that the knot $x=T_3(t), \, y=T_b(t), \, z=C(t)$ is isotopic to $K$. If $K$ is amphicheiral, then $b$ is odd, and the polynomial $C(t)$ can be chosen odd. \end{corollary} {\em Proof}. Let $b=n+1$ be the smallest integer such that $K$ has a Chebyshev diagram $x=T_3(t), \, y= T_b(t)$. By our theorem \ref{gauss3}, the Gauss sequence $(g(t_1), \ldots, g(t_{2n}))$ of this diagram has $c=3N-b$ sign changes. We choose $C$ such that $C(t_i)g(t_i)>0$ and we can realize it by choosing the roots of $C$ to be $\frac 12(t_{i}+t_{i+1})$ where $g(t_i)g(t_{i+1})<0$. If $K$ is amphicheiral, then $b$ is odd and the Conway form is palindromic by Proposition \ref{palin}. Then our Chebyshev diagram is symmetrical about the origin. We see that the Gauss sequence is odd: $g(t_h) = -g(-t_h)$. This implies that the polynomial $C(t)$ is odd. \hbox{} $\Box$ \begin{remark} This corollary gives a simple proof of a famous theorem of Hartley and Kawauchi: {\em every amphicheiral rational knot is strongly negative amphicheiral} (\cite{HK,Kaw}). \end{remark} \begin{example}[The knot $6_1$]\label{6_1-3}\\ The knot $\overline{6}_1= S(\Frac 92)$ is not harmonic with $a=4.$ It is not even harmonic with $a=3$ because $2^2 \not \equiv \pm 1 \Mod 9$. Its crossing number is $6$. In the example \ref{9297}, we get $\ell(\Frac 92)=9,\, \ell(\Frac 97)=7$. $b=8$ is the minimal value for which $x=T_3(t), \, y=T_8(t)$ is a Chebyshev diagram for $\overline{6}_1$. The Gauss sequence associated to the Conway form $\overline{6}_1 = C(-1,-1,-1,1,1,1,1)$ has exactly 10 sign changes. It is precisely $$[1, -1, -1, 1, -1, 1, -1, -1, 1, -1, 1, 1, -1, 1].$$ We can check that $$ x=T_3(t),\, y=T_8(t),\, z= \left( 8\,t+7 \right) \left( 5\,t-4 \right) \left( 15\,{t}^{2}-14 \right) \left( 2\,{t}^{2}-1 \right) \left( 3\,{t}^{2}-1 \right) \left( 15\,{t}^{2}-1 \right) $$ is a parametrization of $\overline{6}_1$ of degree $(3,8,10)$. In \cite{KP3} we gave the Chebyshev parametrization $6_1= {\cal C}(3,8,10,\frac{1}{100})$. We will give another parametrization in example \ref{6_1-4}. \begin{figure} \caption{The knot $6_1$} \label{61_3} \end{figure} \end{example} \subsection{Chebyshev diagrams with $a=4$} It is also possible to get Chebyshev diagrams of the form ${\cal C}(4,b)$. The following result is analogous to the Theorem \ref{gauss3}. \begin{theorem}\label{gauss4} Let $K$ be a two-bridge knot of crossing number $N$ and Schubert fraction $\Frac \alpha \beta$, $\beta$ even. Let $\Frac \alpha \beta = \pm [1,\pm2, \ldots,\pm1, \pm 2 ]$ be a continued fraction expansion of minimal length $b-1,$ and $\sigma$ be the number of islets (subsequences of the form $ \pm ( 2, -1, 2 )$) in this expansion. Let $x= T_4(t),\, y = T_b(t) $ be the corresponding Chebyshev diagram of $K$ and $c$ be the number of sign changes in the corresponding Gauss sequence of $K$. Then we have $$ 3b+c-2 = 4N + 12 \sigma. $$ \end{theorem} {\em Proof}. Let $s$ be the number of sign changes in the given continued fraction. Since $N= \frac 32 (b-1) -s -2 \sigma $ the formula is equivalent to $3b+c-2 = 6( b-1) -4s -8 \sigma + 12 \sigma, $ that is $$ 4 s +c - 4 \sigma = 3b -4. $$ We shall prove this formula by induction on $s.$ If $s=0,$ the knot is alternate. We have $ c = 3b-4, \ \sigma =0,$ and the formula is true. We shall need precise notations. Let $ M_1, M_2,N_2, M_3, M_4, N_4,\ldots, M_{b-1}, N_{b-1} $ be the crossing points where $ x(M_k)=x_k, \ x(N_{2k})= x_{2k}, $ and $ x_1< x_2< \cdots < x_{b-1}.$ The plane curve $x= T_4(t), \ y= T_b(t) $ is the union of four arcs where $x(t)$ is monotonic (see the following figures). On each arc there is a sign change in the Gauss sequences iff the corresponding Conway signs are equal. Let $ C( e_1,2e_2, \ldots, e_{b-2}, 2 e_{b-1} ), \ e_i= \pm1$ be the Conway form of $K.$ Let $k$ be the first integer such that $ e_{k-1} e_k <0.$ We have three cases to consider. \par\noindent {\bf $\mathbf{k}$ is odd, and $\mathbf{e_k e_{k+1} <0}$.}\\ In this case, $ ( 2e_{k-1}, e_k, 2e_{k+1})= \pm (2, -1, 2) $ is an islet. Let us consider the knot $K'$ obtained by changing only the sign of $e_k.$ The number of sign changes in the Conway sequence of $K'$ is $s'=s-2.$ By induction we get for $K':$ $4s' +c' -4 \sigma' = 3b-4.$ The number of islets of $K'$ is $ \sigma ' = \sigma -1.$ Let us look the modification of Gauss sequences. There are only two arcs containing $ M_k.$ On each of these arcs there is no sign change in the Gauss sequence of $K,$ and then there are two sign changes in the Gauss sequence of $K'.$ Consequently, $c'= c+4.$ \psfrag{g1}{\large$\mathbf{\Gamma_1}$} \psfrag{g2}{\large$\mathbf{\Gamma_2}$} \psfrag{g3}{\large$\mathbf{\Gamma_3}$} \psfrag{g4}{\large$\mathbf{\Gamma_4}$} \psfrag{a1}{$M_{k-1}$} \psfrag{c1}{$N_{k-1}$} \psfrag{b2}{$M_{k}$} \psfrag{a2}{$M_{k+1}$} \psfrag{c2}{$N_{k+1}$} \begin{figure}\label{dgauss41} \end{figure} Finally, we get $4s +c -4 \sigma = 4 ( s'+2) + (c'-4) -4( \sigma ' +1) = 4s' +c' -4 \sigma ' = 3b-4,$ which completes the proof in this case. \par\noindent {\bf $\mathbf{k}$ is even, and $\mathbf{e_k e_{k+1} <0}$.}\\ In this case there are two crossing points $ M_k$ and $N_k$ with $ x(M_k)=x(N_k)= x_k.$ Each arc contains one of these points. Let us consider the knot $K'$ obtained by changing only the sign of $e_k.$ The number of sign changes in the Conway sequence of $K'$ is $s' = s-2.$ By induction the formula is true for $K'.$ \psfrag{b0}{$M_{k-1}$} \psfrag{a1}{$M_{k}$} \psfrag{c1}{$N_{k}$} \psfrag{b2}{$M_{k+1}$} \psfrag{a2}{} \psfrag{c2}{} \begin{figure}\label{dgauss42} \end{figure} On each arc the Gauss sequence gains two sign changes, so that $c'=c+8.$ Since we have $\sigma ' = \sigma,$ we get $ 4 s + c - 4 \sigma = 4 ( s'+2) + (c'-8) -4 \sigma ' = 3b-4.$ \par\noindent {\bf The case $\mathbf{e_k e_{k+1} >0}$.}\\ \begin{figure}\label{dgauss43} \end{figure} In this case we consider the knot $K'$ obtained by changing the signs of $e_j, \ j \ge k.$ For $K'$ we have $ s' = s-1,$ and by induction $ 4 s' + c' - 4 \sigma' = 3b-4.$\\ On each of the four arcs the Gauss sequence gains one sign change, and then $c'=c+4.$ Since $ \sigma ' = \sigma,$ we conclude $$4 s+c- 4 \sigma= 4 (s'+1) + (c'-4) - 4 \sigma ' = 4 s' + c' - 4 \sigma ' = 3b-4.$$ This completes the proof of the last case. \hbox{} $\Box$ \begin{corollary}\label{degc4} Let $K$ be a two-bridge knot of crossing number $N$ and Schubert fraction $\Frac \alpha \beta$, $\beta$ even. Let $\Frac \alpha \beta = \pm [1,\pm2, \ldots,\pm1, \pm 2 ]$ be a continued fraction expansion of minimal length $b-1$ and $\sigma$ be the number of islets (subsequences of the form $ \pm ( 2, -1, 2 )$) in this expansion. There exists an odd polynomial $C(t)$ of degree $c$ such that $3b+c-2 = 4N + 12 \sigma$ and such that the knot defined by $x=T_4(t),\, y=T_b(t), \, z=C(t)$ is isotopic to $K$. \end{corollary} {\em Proof}. This proof is similar to the proof of Corollary \ref{degc3}. We must bear in mind that in this case, the Gauss sequence is odd: $g(t_i) = -g(-t_i)$ and $t_{3(b-1)+1-i} = -t_i$. \hbox{} $\Box$ \par\noindent Corollary \ref{degc4} gives an algorithm to represent any rational knot as a polynomial knot which is rotationally symmetric around the $y$-axis. This gives a strong evidence for the following classical result. \begin{corollary} Every rational knot is strongly invertible. \end{corollary} \begin{example}[The stevedore knot $\overline{6}_1$]\label{6_1-4} The stevedore knot $6_1$ is $S(\Frac 92)=S(-\Frac 94)$. We get $\Frac 94 = (ASB)(BSA)(\infty) = [1,2,-1,2,1,-2,1,2]$. We deduce that it can be parametrized by $x=T_4(t),\ y= T_9(t), \ z=C(t)$ where $\deg C = 11$. We find $$ C(t) = t \left( 34\,{t}^{2}-33 \right) \left( 2\,{t}^{2}-1 \right) \left( 3 \,{t}^{2}-1 \right) \left( 4\,{t}^{2}-1 \right) \left( 6\,{t}^{2}-1 \right). $$ On the other hand, we find $S(\Frac 9{14}) = [1,-2,-1,-2,-1,2] = BASB(\infty)$. We find that $6_1$ can also be represented by polynomials of degrees $(4,7,9)$: $$ x=T_4(t),\ y= T_7(t), \ z=t \left( 10\,{t}^{2}-9 \right) \left( 4\,{t}^{2}-3 \right) \left( 4 \,{t}^{2}-1 \right) \left( 6\,{t}^{2}-1 \right). $$ \begin{figure}\label{61_4} \end{figure} \end{example} \subsection{Examples} In this section, we give several examples of polynomial parametrizations of rational knots with Chebyshev diagrams ${\cal C}(3,b)$ and ${\cal C}(4,b').$ \subsubsection{Parametrizations of the torus knots} The torus knot $T(2,N), \ N=2n+1$ is the harmonic knot $\overline{\rm H}(3,3n+1,3n+2)$. The torus knot $T(2, N)$ can be parametrized by $x=T_4(t), \, y= T_N (t), \, z= C(t),$ where $C(t)$ is an odd polynomial of degree $\deg (C) = N+2=2n+3.$ \par\noindent {\em Proof}. We have already seen (example \ref{tnh4}) that $T(2,N)$ has a Chebyshev diagram $ x= T_4(t), y= T_N(t)$. Since there is no islet in the corresponding continued fractions, we see that the number of sign changes in the Gauss sequence is $c= N+2.$ By symmetry of the diagram, we can find an odd polynomial of degree $c$ giving this diagram. \hbox{} $\Box$ \par\noindent In both cases, the diagrams have the same number of crossing points : $\frac{3}2{(N-1)}=3n$. \par\noindent As an example, we obtain for $\overline{T}(2,5)$: $$ x = T_4(t),\, y=T_5(t), \, z = t \left( 2\,{t}^{2}-1 \right) \left( 3\, {t}^{2}-1 \right) \left( 5\,{t}^{2}-4 \right). $$ In this case the Chebyshev diagram has exactly 6 crossing points as it is for ${\rm H}(3,7,8)$. Note that we also obtain $ T(2,5)=S(\Frac 56)$: $$ x = T_4(t),\, y=T_7(t), \, z =t \left( 21\,{t}^{2}-20 \right) \left( 4\,{t}^{2}-1 \right). $$ We therefore obtain parametrizations of degrees $(4,5,7)$ or $(4,7,5)$. \begin{figure} \caption{Diagrams of the torus knot $\overline{T}(2,5)$ and its mirror image} \label{K51} \end{figure} \subsubsection{Parametrizations of the twist knots} The twist knot ${\cal T}_m= S(m + \Frac 12 ) $ has crossing number $m+2$. We have seen (example \ref{twh4}) that the only twist knots that are harmonic for $a=4$ are the trefoil and the $\overline{5}_2$ knot. The knot ${\cal T} _m$ is not harmonic for $a=3$ because $2^2 \not \equiv \pm 1 \Mod{2m+1}$ except when $m=2$ (the figure-eight knot) or $m=1$ (trefoil). From example \ref{twk}, we know that: \begin{itemize}\itemsep -2pt plus 1pt minus 1pt \item ${\cal T}_{2k+1}$ can be parametrized by $x=T_3(t), \, y=T_{3k+4}, \, z=C(t)$ where $\deg(C)=3k+5$. \item ${\cal T}_{2k}$ can be parametrized by $x=T_3(t), \, y=T_{3k+2}, \, z=C(t)$ where $\deg(C)=3k+4.$ \end{itemize} Using results of \ref{twk4}, we deduce other parametrizations \begin{itemize}\itemsep -2pt plus 1pt minus 1pt \item ${\cal T}_{2k+1}$ can be represented by $x=T_4(t), \ y= T_{2k+3} (t),\ z= C(t)$ where $C(t)$ is an odd polynomial of degree $2k+5.$ \item ${\cal T}_{2k}$ can be represented by $ x= T_4(t), y= T_{2k+5} (t), z= C(t)$ where $C(t)$ is an odd polynomial of degree $2k+7$. \end{itemize} {\em Proof}. The proof is very similar to the preceding one, except that there is an islet in the continued fractions for $2k$ even. \hbox{} $\Box$ \par\noindent Note that Chebyshev diagrams we obtain ($a=3$ or $a=4$) for ${\cal T}_{2k+1}$ have the same number of crossing points: $3k+3$. \begin{example}[The figure-eight knot] ${\cal T}_2$ is the figure-eight knot. \begin{figure} \caption{The figure-eight knot} \label{K41} \end{figure} Note that we obtain the figure-eight knot as the harmonic knot ${\rm H}(3,5,7)$ or as a Chebyshev knot $$x=T_4(t), \, y=T_7(t), z= t \left( 10\,{t}^{2}-9 \right) \left( 4\,{t}^{2}-3 \right) \left( 3 \,{t}^{2}-2 \right) \left( 2\,{t}^{2}-1 \right). $$ But, considering $S(-5/8)$, we obtain a better parametrization $$x=T_4(t), \, y = T_5(t), \, z= t \left( 11\,{t}^{2}-10 \right) \left( 5\,{t}^{2}-4 \right) \left( 5 \,{t}^{2}-1 \right).$$ \end{example} \begin{example}[The 3-twist knot] ${\cal T}_3$ is the 3-twist knot $\overline{5}_2$. It is the harmonic knot ${\rm H}(4,5,7).$ It can also be parametrized by $$x=T_3(t), \ y=T_7(t),\ z= t \left( 4\,t+3 \right) \left( 3\,t+1 \right) \left( 6\,t-5 \right) \left( 12\,{t}^{2}-11 \right) \left( 2\,{t}^{2}-1 \right)$$ \begin{figure} \caption{Diagrams of the 3-twist knot $\overline{5}_2$} \label{K52} \end{figure} \end{example} \subsubsection{Parametrizations of the generalized stevedore knots} The stevedore knot ${\cal S}_m= S (2m+2+ \Frac 1{2m} )$ can be represented by $ x=T_3(t), \ y= T_{6m+2}(t), \ z = C(t) $ where $C(t)$ is a polynomial of degree $6m+4$. The stevedore knot ${\cal S}_m= S (2m+2+ \Frac 1{2m} )$ can be represented by $ x=T_4(t), \ y= T_{6m+3}(t), \ z = C(t) $ where $C(t)$ is an odd polynomial of degree $ c= 10m +1.$ \par\noindent {\em Proof}. The case $a=3$ is deduced from \ref{sk3} and Corollary \ref{degc3}. The case $a=4$ is a consequence of Theorem \ref{gauss4}. In this case $b=6m+3$, and the crossing number is $N= 4m+2.$ For $m=2k-1$ the number of islets in $ \Frac { (4k-1)^2 }{4k} = (ASB)^{2k-1}(BSA)^k (\infty) $ is $\sigma = 2k-1=m.$ For $m= 2k,$ we also find $\sigma= m.$ Consequently we get $ 3( 6m+3)+c -2 = 4 (4m+2)+ 12 m,$ that is, $c= 10 m +1.$ The rest of the proof is analogous to the preceding ones. \hbox{} $\Box$ \begin{remark} There is an algorithm to determine minimal Chebyshev diagrams for $a=3$ (Remark \ref{minb} and Prop. \ref{ell}). When $a=4$, we can determine Chebyshev diagrams using Theorem \ref{cf1212} but we do not know yet if they are minimal (consider for example $S(\Frac 9{14})$ and $S(-\Frac 58)$). \end{remark} \section{Proofs of theorems \ref{h3} and \ref{h4}}\label{proofs} \subsection{Proof of Theorem \ref{h3}} \label{h3proof} We study here the diagram of ${\rm H}(3,b,c)$ where $b=3n+1$ and $c=2b-3\lambda$. The crossing points of the plane projection of ${\rm H}(3,b,c)$ are obtained for pairs of values $(t,s)$ where $t= \cos \bigl( \Frac m{3b} \pi \bigr), \ s = \cos \bigl( \Frac {m'}{3b} \pi \bigr).$ For $k = 0, \ldots, n-1$, let us consider \begin{itemize}\itemsep -2pt plus 1pt minus 1pt \item $A_{k}$ obtained for $m=3 k +1, \ m'= 2b-m.$ \item $B_{k}$ obtained for $ m = 3 k +2, \ m'= 2b + m$. \item $C_{k}$ obtained for $ m = 2b - 3k - 3, \ m'= 4b-m$. \end{itemize} Then we have \begin{itemize}\itemsep -2pt plus 1pt minus 1pt \item $x(A_{k}) = \cos \bigl( \Frac {3 k +1}b \pi \bigr)$, \ $y(A_k) = \frac 12 (-1)^k$. \item $x(B_{k} )= \cos \bigl( \Frac {3 k +2}b \pi \bigr)$, \ $y(B_k) = \frac 12 (-1)^{k+1}$. \item $x(C_{k}) = \cos \bigl( \Frac {3 k +3}b \pi \bigr)$, \ $y(C_k) = \frac 12 (-1)^{k}$. \end{itemize} \psfrag{a0}{\small $A_0$}\psfrag{b0}{\small $B_0$}\psfrag{c0}{\small $C_0$} \psfrag{a1}{\small $A_1$} \psfrag{an}{\small $A_{n-1}$}\psfrag{bn}{\small $B_{n-1}$}\psfrag{cn}{\small $C_{n-1}$} \begin{figure} \caption{${\rm H}(3,3n+1,c)$, $n$ even} \label{dh3} \end{figure} Hence our $3n$ points satisfy $$x(A_{k-1}) > x(B_{k-1}) >x(C_{k-1}) > x( A_{k}) > x(B_{k}) > x ( C_{k} ),\ k=1, \ldots, n-1.$$ Using the identity $T'_a( \cos \tau ) = a \Frac{ \sin a \tau}{\sin \tau },$ we get $ x'(t)y'(t) \sim \sin \bigl( \Frac mb \pi \bigr) \sin \bigl( \Frac m3 \pi \bigr). $ We obtain \begin{itemize}\itemsep -2pt plus 1pt minus 1pt \item[] for $A_k$: $ \begin{array}[t]{rcl} {x'(t)y'(t)}&\sim& {\sin ( \Frac {3 k +1}b \pi ) \sin ( \Frac {3 k +1}3 \pi )} \sim (-1)^{k}. \end{array}$ \item[] for $B_k$: $ \begin{array}[t]{rcl} x'(t)y'(t)&\sim& {\sin \bigl( \Frac {3 k +2}b \pi \bigr) \sin \bigl( \Frac{3k+2}3 \pi \bigr)} \sim (-1)^{k}. \end{array}$ \item[] for $C_k$: $ \begin{array}[t]{rcl} {x'(t)y'(t)}&\sim& {\sin ( \Frac {2b -3 k -3}b \pi ) \sin (\Frac {2b-3 k -3}3 \pi )}\\ &\sim& - {\sin (\Frac{3k+3}b \pi ) \sin ( -\Frac {3k+1}3 \pi ) } \sim (-1)^{k}. \end{array}$ \end{itemize} The following identity will be useful in computing the sign of $z(t)-z(s).$ $$ T_c(t) - T_c(s) = 2 \sin \Bigl( \Frac{c}{6b}(m'-m) \pi \Bigr) \sin \Bigl( \Frac{c}{6b}(m+m') \pi \Bigr). $$ We have, with $c=2b -3\lambda$, $\theta= \Frac{\lambda}b \pi,$ (and $b=3n+1$ ), \begin{itemize}\itemsep -2pt plus 1pt minus 1pt \item[] for $A_k$: $ z(t)-z(s) = -2 \sin c \Frac{\pi}3 \sin \Bigl( c \Frac{m-b}{3b} \pi \Bigr).$ But \[ \sin c \Frac {\pi}3 = \sin \Bigl( \Frac {6n+2-3 \lambda}3 \pi \Bigr) = (-1)^{\lambda} \sin \Frac {2 \pi }3,\label{c} \] and $$ \sin \Bigl( c \Frac{b-m}{3b} \pi \Bigr)= \sin \Bigl( (2- \Frac {3 \lambda}b ) \, \Frac {b-m}3 \pi \Bigr) = \sin \bigl( \Frac {\lambda}b (m-b) \pi \bigr) = (-1)^{\lambda} \sin ( 3k+1) \theta.$$ We deduce that $z(t) - z(s) \sim \sin (3k+1) \theta$. Finally, we obtain $$ \sign{D(A_k)}= (-1)^k\sign{\sin (3k+1) \theta}.$$ \item for $B_k$: $ z(t)-z(s) = 2 \sin c \Frac{\pi}{3} \sin \bigl( \Frac cb \,. \, \Frac{b+m}{3} \pi\bigr). $ We have \[ \sin \bigl( \Frac cb \cdot \Frac {b+m}3 \pi \bigr) &=& \sin \bigl( ( 2-\Frac {3\lambda}b ) \Frac {b+m}3 \pi \bigr) \nonumber \\ &=& -\sin \bigl( \Frac {\lambda}b (b+m) \pi \bigr) = (-1)^{\lambda+1} \sin (3k+2) \theta. \nonumber \] Then, using Equation \ref{c}, we get $z(t) - z(s) \sim - \sin( 3k+2) \theta$, and finally $$ \sign{D(B_k)} = (-1)^{k+1} \sign{\sin(3k+2) \theta}. $$ \item for $C_k$: $ \begin{array}[t]{rcl} z(t)-z(s) &\sim& \sin \Frac {2 c }3 \pi \sin \bigl( \Frac cb ( k+1) \pi \bigr)\\ &\sim& \sin \Frac {4 \pi } 3 \sin \bigl( (2- \Frac{3\lambda}b )( k+1) \pi \bigr) \sim \sin (3k+3) \theta. \end{array}$\\ We obtain $$ \sign{ D(C_k)} = (-1)^k \sign{ \sin (3k+3) \theta }. $$ \end{itemize} These results give the Conway normal form. If $n$ is odd, the Conway's signs of our points are $$ \begin{array}{rcccl} s(A_k) &\sim& (-1)^k D( A_k ) &\sim& { \sin (3k+1) \theta },\\ s(B_k)&\sim& (-1)^{k+1} D(B_k) &\sim& { \sin (3k+2) \theta }, \\ s(C_k) &\sim& (-1)^k D(C_k) &\sim& { \sin (3k+3) \theta }. \end{array} $$ In this case our result follows, since the fractions $ [e_1, e_2, \ldots, a_{3n} ] $ and $(-1)^{3n+1} [a_{3n}, \ldots, a_1] $ define the same knot. If $n$ is even, the Conway's signs are the opposite signs, and we also get the Schubert fraction of our knot. \par\noindent Since $ 0<\theta < \Frac {\pi}2 $, we see that there are not two consecutive sign changes in our sequence. We also see that the first two terms are of the same sign, and so are the last two terms. The Conway normal form is biregular and the total number of sign changes in this sequence is $\lambda -1 $: the crossing number of our knot is then $b- \lambda.$ Finally, we get $ \beta ^2 \equiv \pm 1 $ by Proposition \ref{palin}. \hbox{} $\Box$ \subsection{Proof of Theorem \ref{h4}} \label{h4proof} The crossing points of the plane projection of ${\rm H}= {\rm H} (4,b,c)$ are obtained for parameter pairs $ ( t, s) $ where $ t=\cos \bigl( \Frac m{4b} \pi \bigr), \ s= \cos \bigl( \Frac {m'}{4b} \pi \bigr).$ We shall denote $ \lambda = \Frac {3b-c}4,$ ( or $ c= 3b - 4 \lambda$) and $ \theta = \Frac {\lambda}b \pi.$ We will consider the two following cases. \par\noindent {\bf The case $\mathbf{b = 4n+1}$.} \par\noindent For $k=0, \ldots, n-1$, let us consider the following crossing points \begin{itemize}\itemsep -2pt plus 1pt minus 1pt \item $A_k$ corresponding to $ m=4k+1, \ m'= 2b-m,$ \item $B_k$ corresponding to $ m= 4k+2, \ m'= 4b-m,$ \item $C_k$ corresponding to $ m= 4k+3, \ m' = 2b+m,$ \item $ D_k$ corresponding to $ m = 2b - 4(k+1), \ m' = 4b-m.$ \end{itemize} Then we have \begin{itemize}\itemsep -2pt plus 1pt minus 1pt \item $ x( A_k)= \cos \bigl( \Frac {4k+1}b \pi \bigr), \ \ y(A_k) = (-1)^k \cos \Frac \pi 4 \neq 0, $ \item $ x(B_k) = \cos \bigl( \Frac {4k+2}b \pi \bigr), \ \ y(B_k)=0,$ \item $ x( C_k) = \cos \bigl( \Frac {4k+3}b \pi \bigr), \ \ y(C_k) = (-1)^k \cos \Frac {3 \pi }4 \neq 0, $ \item $x(D_k) = \cos \bigl( \Frac {4k+4}b \pi \bigr), \ \ y(D_k)=0.$ \end{itemize} \psfrag{a0}{\small $A'_0$}\psfrag{b0}{\small $B_0$}\psfrag{c0}{\small $C_0$}\psfrag{d0}{\small $D_0$} \psfrag{an}{\small $A_{n-1}$}\psfrag{bn}{\small $B_{n-1}$}\psfrag{cn}{\small $C'_{n-1}$}\psfrag{dn}{\small $D_{n-1}$} \psfrag{aa0}{\small $A_0$}\psfrag{cc0}{\small $C'_0$} \psfrag{aan}{\small $A'_{n-1}$}\psfrag{ccn}{\small $C_{n-1}$} \begin{figure} \caption{${\rm H}(4,4n+1,c)$, $n$ even} \label{dh4} \end{figure} Hence our $4n$ points satisfy \$x(A_{k-1}) > x(B_{k-1}) > x( C_{k-1}) > x( D_{k-1})>&&\\ \quad\quad x(A_k) > x( B_k) > x(C_k ) > x( D_k),&& k=1, \ldots, n-1. \$ We remark that these points together with the symmetric points $A'_k$ (resp. $C'_k$) of $A_k$ (resp. $C_k$) with respect to the $y-$axis form the totality of the crossing points. The Conway sign of a crossing point $M$ is $s(M)= \sign{D(M)}$ if $y(M)=0,$ and $s(M)=- \sign{D(M)}$ if $y(M) \neq 0.$ By symmetry, we have $s(A'_k) = s(A_k)$ and $s(C'_k) = s(C_k)$ because symmetric points correspond to opposite parameters. The Conway form of ${\rm H}$ is then (see paragraph \ref{cf}) : $$ C\Bigl( s (D_{n-1}), 2s(C_{n-1}), s(B_{n-1}),2s( A_{n-1}), \ldots, s (B_0), 2s (A_0) \Bigr). $$ Using the identity $ T' _a ( \cos \tau ) = a \Frac { \sin a \tau }{ \sin \tau},$ we get $ x'(t) y'(t)\sim \sin \bigl( \Frac mb \pi \bigr) \sin \bigl( \Frac {m}4 \pi \bigr).$ Consequently, \begin{itemize}\itemsep -2pt plus 1pt minus 1pt \item For $A_k$ we have $ x'(t) y'(t) \sim \sin \bigl( \Frac {4k+1}b \pi \bigr) \sin \bigl( \Frac {4k+1}4 \pi \bigl) \sim(-1)^k.$ \item Similarly, for $B_k$ and $C_k$ we get $ x'(t) y'(t) \sim (-1)^k. $ \item For $D_k$ we get $ x'(t) y'(t)\sim \sin \bigl( \Frac {2b-4k-4}b \pi \bigr) \sin \bigl( \Frac {2b-4k-4}4 \pi \bigr)\sim (-1)^{k+1}.$ \end{itemize} On the other hand, at the crossing points we have $$ z(t) - z(s) = 2 \sin \Bigl( \Frac c{8b} (m'-m) \pi \Bigr) \, \sin \Bigl( \Frac c{8b} (m+m') \pi \Bigr). $$ We obtain the signs of our crossing points, with $c = 3b-4 \lambda, \ \theta= \Frac {\lambda}b,t.$ \begin{itemize}\itemsep -2pt plus 1pt minus 1pt \item For $A_k$ we get: $ z(t )-z(s)= 2 \sin \Frac cb (n-k) \pi\, \sin c \Frac {\pi}4.$\\ We have $ \sin c \Frac {\pi}4 = \sin \Frac { 12 n +3 - 4 \lambda}4 \pi = (-1)^{n+ \lambda} \sin \Frac {3 \pi }4 \sim (-1)^{n+ \lambda}$\\ and also $ \begin{array}[t]{rcl} \sin \bigl( \Frac cb (n-k) \pi \bigr) &=& \sin \Bigl( \bigl( 3 - \Frac {4 \lambda}b \bigr) \bigl( n-k) \pi \Bigr)\\ &=& (-1)^{n+k} \sin \Bigl( \Frac {4k-4n}b \lambda \pi \Bigr) = (-1) ^{ n+k+ \lambda} \sin ( 4k+1) \theta \end{array}$.\\ Consequently, the sign of $A_k$ is $$ s(A_k) = - \sign{\sin ( 4 k+1) \theta }.$$ \item For $B_k$, we have: $z(t) - z(s) = 2 \sin \bigl( \Frac cb ( 2n-k) \pi \bigr) \sin c \Frac {\pi}2 = - 2 \sin \bigl( \Frac cb ( 2n-k) \pi \bigr).$\\ But $ \begin{array}[t]{rcl} \sin \bigl( \Frac cb ( 2n-k) \pi \bigr) &=& \sin \Bigl( \bigl( 3 - \Frac{4 \lambda}b \bigr) \bigl( 2n-k \bigr) \pi \Bigr) \\ &=& (-1)^{k} \sin \Bigl( \Frac {\lambda}b ( 4k-8n) \pi \Bigr) = (-1)^k \sin (4k+2) \theta. \end{array}$.\\ Therefore the sign of $B_k$ is $$ s (B_k) = - \sign{\sin (4k+2) \theta}.$$ \item For $C_k$: $ z(t) - z(s) = 2 \sin \bigl( \Frac c4 \pi \bigr) \sin \bigl( \Frac cb (n+k+1) \pi \bigr). $\\ We know that $ \sin \Frac {c \pi} 4 \sim (-1)^{n+ \lambda}$. Let us compute the second factor $$ \begin{array}{rcl} \sin \Bigl( \bigl( 3 - \Frac {4 \lambda}b \bigr) \bigl( n+k+1 \bigr) \pi\Bigr) &=& (-1)^{n+k} \sin \Bigl( \Frac {\lambda}b \bigl( 4n+4k+4 \bigr) \pi \Bigr) \\ &=&(-1)^{n+k} \sin \Bigl( \Frac {\lambda}b ( b+4k+3) \pi \Bigr) \\ &=& ( -1)^{n+k+ \lambda} \sin (4k+3) \theta. \end{array}$$ Hence $$ s( C_k) = - \sign{\sin( 4k+3) \theta}.$$ \item For $D_k$: $ \begin{array}[t]{rcl} z(t) -z(s)&=& 2 \sin \bigl( \Frac cb (k+1) \pi \bigr) \sin \bigl( c \Frac {\pi}2 \bigr)\\ &=& 2 \sin \Bigl( \bigl( 3- \Frac {4 \lambda }b \bigr) \bigl( k+1 \bigr) \pi \Bigr) = (-1)^{k} \sin ( 4k+4) \theta . \end{array}$.\\ We conclude $$s (D_k)= - \sign{\sin( 4k+1) \theta}.$$ \end{itemize} This completes the computation of our Conway normal form of $H$ in this first case. \par\noindent {\bf The case $\mathbf{b = 4n+3}$.}\\ Here, the diagram is different. Let us consider the following $4n+2$ crossing points. For $k= 0, \ldots, n$ \begin{itemize}\itemsep -2pt plus 1pt minus 1pt \item $ A_k$ corresponding to $ m=4k+1, \ m' = 2b+m,$ \item $B_k$ corresponding to $ m=4k+2, \ m' = 4b-m.$ \end{itemize} For $k=0, \ldots, n-1$ \begin{itemize}\itemsep -2pt plus 1pt minus 1pt \item $ C_k$ corresponding to $ m=4k+3, \ m' =2b-m,$ \item $ D_k$ corresponding to $ m = 2b + 4 (k+1), \ m'=4b-m.$ \end{itemize} These points are chosen so that $$ x(A_0) > x(B_0)> x(C_0) > x(D_0) > \cdots > x(D_{n-1}) > x( A_n) > x(B_n), $$ and we have $\sign{x'(t) y'(t)}= (-1)^k.$ \begin{itemize}\itemsep -2pt plus 1pt minus 1pt \item For $A_k$ we get $$ z(t)-z(s)= 2 \sin \bigl( c \Frac {\pi}4 \bigr) \sin \bigl( \Frac cb ( n+k+1) \pi \bigr). $$ We easily get $ \sign{\sin c \Frac {\pi}4 } = (-1)^{n+ \lambda}.$ We also get $$ \begin{array}{rcl} \sin \bigl( \Frac cb ( n+k+1) \pi \bigr)&=& \sin \Bigl( \bigl( 3 - \Frac {4 \lambda}b \bigr) \bigl( n+k+1 \bigr) \pi \Bigr)\\ &=& (-1)^{n+k} \sin \bigl( \Frac {\lambda}b ( b+4k+1) \pi \bigr) = (-1)^{n+k+\lambda} \sin ( 4k+1) \theta. \end{array} $$ Hence the sign of $A_k$ is $$ s(A_k)= - \sign{\sin (4k+1) \theta}. $$ \item For $B_k$ we get $$ z(t)-z(s)= 2 \sin \bigl( \Frac cb ( 2n+1-k) \pi \bigr) \sin c \Frac {\pi}2. $$ We have $ \sin \bigl( c \Frac {\pi}2 \bigr) =1 >0, $ and $$ \begin{array}{rcl} \sin \bigl( \Frac cb (2n+1-k) \pi \bigr)&=& \sin \Bigl( \bigl( 3- \Frac { 4 \lambda}b \bigr) \bigl( 2n+1 -k \bigr) \pi \Bigr)\\ &=&(-1)^{k+1} \sin \bigl( \Frac {\lambda}b ( 4k-8n-4) \pi \bigr) = (-1)^{k+1} \sin (4k+2) \theta. \end{array} $$ Then, the sign of $B_k$ is $$ s(B_k) = - \sign{\sin(4k+2) \theta}. $$ \item For $ C_k$ we have $$ z(t) - z(s) = 2 \sin \bigl( \Frac cb (n-k) \pi \bigr) \sin c \Frac {\pi}4. $$ We get $$ \begin{array}{rcl} \sin \bigl( \Frac cb (n-k) \pi \bigr) &=& \sin \Bigl( \bigl( 3- \Frac {4 \lambda}b \bigr) \bigl( n-k \bigr) \pi \Bigr) \\ &=& (-1)^{n+k} \sin \bigl( \Frac {4k-4n}b \lambda \pi \bigr)= (-1)^{n+k+ \lambda} \sin (4k+3) \theta. \end{array} $$ The sign of $C_k$ is then $$ s( C_k) = - \sign{\sin ( 4k+3) \theta}. $$ \item For $D_k$ we get $$ z(t)-z(s)= 2 \sin \bigl( -\Frac{c}b (k+1) \pi \bigr) \sin c \Frac {\pi}2. $$ We have $ \sin c \Frac {\pi}2 >0.$ We also have $$ \begin{array}{rcl} \sin \bigl( -\Frac cb (k+1) \pi ) &=& \sin \Bigl( \bigl( \Frac {4 \lambda}b -3 \bigr) \bigl( k+1 \bigr) \pi \Bigr) (-1)^{k+1} \sin(4k+4) \theta. \end{array} $$ Consequently, the sign of $D_k$ is $$ s(D_k)= - \sign{\sin(4k+4) \theta }. $$ \end{itemize} This concludes the computation of the Conway normal form of ${\rm H}(4,b,c).$ \par\noindent If $ b<c< 3b,$ we get $ \lambda < \Frac b2,$ and then $ \theta < \Frac {\pi} 2.$ Consequently, our sequence is biregular. Furthermore, the total number of sign changes is $ \lambda-1.$ We conclude that the crossing number is $ N= \Frac { 3 (b-1) }2 - (\lambda -1)= \Frac {3b+c-2}4.$ The fact that $ \beta ^2 \equiv \pm2 \Mod \alpha $ is a consequence of Proposition \ref{palin4}. \hbox{} $\Box$ \par\noindent \hrule width 5cm height 2pt \par\noindent Pierre-Vincent Koseleff, \\ \'Equipe-project INRIA Salsa \& Universit{\'e} Pierre et Marie Curie (UPMC-Paris 6)\\ e-mail: {\tt [email protected]} \par\noindent Daniel Pecker, \\ Universit{\'e} Pierre et Marie Curie (UPMC-Paris 6)\\ e-mail: {\tt [email protected]} \end{document}	arXiv
\begin{definition}[Definition:Algebraic Element of Field Extension/Definition 2] Let $E / F$ be a field extension. Let $\alpha \in E$. $\alpha$ is '''algebraic over $F$''' {{iff}} the evaluation homomorphism $F \sqbrk X \to K$ at $\alpha$ is not injective. \end{definition}	ProofWiki
Making Mathematics Count Making Mathematics Count is the title of a report on mathematics education in the United Kingdom (U.K.). The report was written by Adrian Smith as leader of an "Inquiry into Post–14 Mathematics Education", which was commissioned by the UK Government in 2002. The report recommended an increase in mathematics schooling; the report recommended that statistics be taught as part of the natural sciences rather than as part of the mathematics curriculum. Inquiry and report Making Mathematics Count is the title of a report on mathematics education in the United Kingdom (U.K.). The report[1] was written by Adrian Smith as leader of an "Inquiry into Post–14 Mathematics Education", which was commissioned by the UK Government in 2002. The purpose of the Inquiry was: "To make recommendations on changes to the curriculum, qualifications and pedagogy for those aged 14 and over in schools, colleges and higher education institutions to enable those students to acquire the mathematical knowledge and skills necessary to meet the requirements of employers and of further and higher education."[2] Publication of the report was followed two years later by a conference of 241 delegates, who included mathematics teachers, college lecturers, as well as university mathematicians, head teachers, local authority consultants and advisers, and other mathematics professionals. There is a report of the conclusions of this conference,[3] which was intended to bring together policymakers and practitioners to share information and discuss ways in which changes in mathematics education could be implemented to benefit schools, teachers and students. Influence The Smith report has influenced debate on U.K. educational policy.[4] A particular concern of the report was where and how statistics should be taught: the report recommended that statistics should be embedded in application subjects and taught by teachers of those subjects where it is applied. The government decision was that statistics teaching should remain within the mathematics curriculum. A more recent report for the Royal Statistical Society, The Future of Statistics in our Schools and Colleges retains this view.[5] Predecessor reports The report's title recalls the Cockcroft report Mathematics Counts which addressed some of the same issues but was compiled 2 decades earlier, instigated by Callaghan and submitted under the Thatcher government.[6] Notes 1. Adrian Smith (2004) 2. Adrian Smith (2004, p. 2) 3. Royal Society (2006) Making Mathematics Count - Two Years On, Advisory Committee on Mathematics Education, ISBN 0-85403-627-X 4. T. M. F. Smith & Staetsky (2007, p. 622): Smith, T. M. F.; Staetsky, L. (2007). "The teaching of statistics in UK universities". Journal of the Royal Statistical Society, Series A. 170 (3): 581–622. doi:10.1111/j.1467-985X.2007.00482.x. 5. Royal Statistical Society Future of Statistics in our Schools and Colleges Archived 2013-10-08 at the Wayback Machine 6. Cockroft report, Mathematics Counts References • Smith, Adrian (2004). Making mathematics count: The report of Professor Adrian Smith's inquiry into post-14 mathematics education. London, England: The Stationery Office. Mathematics in the United Kingdom Organizations and Projects • International Centre for Mathematical Sciences • Advisory Committee on Mathematics Education • Association of Teachers of Mathematics • British Society for Research into Learning Mathematics • Council for the Mathematical Sciences • Count On • Edinburgh Mathematical Society • HoDoMS • Institute of Mathematics and its Applications • Isaac Newton Institute • United Kingdom Mathematics Trust • Joint Mathematical Council • Kent Mathematics Project • London Mathematical Society • Making Mathematics Count • Mathematical Association • Mathematics and Computing College • Mathematics in Education and Industry • Megamaths • Millennium Mathematics Project • More Maths Grads • National Centre for Excellence in the Teaching of Mathematics • National Numeracy • National Numeracy Strategy • El Nombre • Numbertime • Oxford University Invariant Society • School Mathematics Project • Science, Technology, Engineering and Mathematics Network • Sentinus Maths schools • Exeter Mathematics School • King's College London Mathematics School • Lancaster University School of Mathematics • University of Liverpool Mathematics School Journals • Compositio Mathematica • Eureka • Forum of Mathematics • Glasgow Mathematical Journal • The Mathematical Gazette • Philosophy of Mathematics Education Journal • Plus Magazine Competitions • British Mathematical Olympiad • British Mathematical Olympiad Subtrust • National Cipher Challenge Awards • Chartered Mathematician • Smith's Prize • Adams Prize • Thomas Bond Sprague Prize • Rollo Davidson Prize	Wikipedia
Results for 'Abhishek Majhi' Abhishek Majhi A Logico-Linguistic Inquiry Into the Foundations of Physics: Part 1.Abhishek Majhi - forthcoming - Axiomathes (NA):1-46.details Physical dimensions like "mass", "length", "charge", represented by the symbols [M], [L], [Q], are not numbers, but used as numbers to perform dimensional analysis in particular, and to write the equations of physics in general, by the physicist. The law of excluded middle falls short of explaining the contradictory meanings of the same symbols. The statements like "m tends to 0", "r tends to 0", "q tends to 0", used by the physicist, are inconsistent on dimensional grounds because "m", "r", (...) "q" represent quantities with physical dimensions of [M], [L], [Q] respectively and "0" represents just a number—devoid of physical dimension. Consequently, due to the involvement of the statement "q tends to 0'', where q is the test charge" in the definition of electric field leads to either circular reasoning or a contradiction regarding the experimental verification of the smallest charge in the Millikan–Fletcher oil drop experiment. Considering such issues as problematic, by choice, I make an inquiry regarding the basic language in terms of which physics is written, with an aim of exploring how truthfully the verbal statements can be converted to the corresponding physico-mathematical expressions, where "physico-mathematical" signifies the involvement of physical dimensions. Such investigation necessitates an explanation by demonstration of "self inquiry", "middle way", "dependent origination", "emptiness/relational existence", which are certain terms that signify the basic tenets of Buddhism. In light of such demonstration I explain my view of "definition"; the relations among quantity, physical dimension and number; meaninglessness of "zero quantity" and the associated logico-linguistic fallacy; difference between unit and unity. Considering the importance of the notion of electric field in physics, I present a critical analysis of the definitions of electric field due to Maxwell and Jackson, along with the physico-mathematical conversions of the verbal statements. The analysis of Jackson's definition points towards an expression of the electric field as an infinite series due to the associated "limiting process" of the test charge. However, it brings out the necessity of a postulate regarding the existence of charges, which nevertheless follows from the definition of quantity. Consequently, I explain the notion of undecidable charges that act as the middle way to resolve the contradiction regarding the Millikan–Fletcher oil drop experiment. In passing, I provide a logico-linguistic analysis, in physico-mathematical terms, of two verbal statements of Maxwell in relation to his definition of electric field, which suggests Maxwell's conception of dependent origination of distance and charge ) and that of emptiness in the context of relative vacuum. This work is an appeal for the dissociation of the categorical disciplines of logic and physics and on the large, a fruitful merger of Eastern philosophy and Western science. Nevertheless, it remains open to how the reader relates to this work, which is the essence of emptiness. (shrink) Physics in Natural Sciences Logic, Philosophy and Physics: A Critical Commentary on the Dilemma of Categories.Abhishek Majhi - forthcoming - Axiomathes:1-17.details I provide a critical commentary regarding the attitude of the logician and the philosopher towards the physicist and physics. The commentary is intended to showcase how a general change in attitude towards making scientific inquiries can be beneficial for science as a whole. However, such a change can come at the cost of looking beyond the categories of the disciplines of logic, philosophy and physics. It is through self-inquiry that such a change is possible, along with the realization of the (...) essence of the middle that is otherwise excluded by choice. The logician, who generally holds a reverential attitude towards the physicist, can then actively contribute to the betterment of physics by improving the language through which the physicist expresses his experience. The philosopher, who otherwise chooses to follow the advancement of physics and gets stuck in the trap of sophistication of language, can then be of guidance to the physicist on intellectual grounds by having the physicist's experience himself. In course of this commentary, I provide a glimpse of how a truthful conversion of verbal statements to physico-mathematical expressions unravels the hitherto unrealized connection between Heisenberg uncertainty relation and Cauchy's definition of derivative that is used in physics. The commentary can be an essential reading if the reader is willing to look beyond the categories of logic, philosophy and physics by being 'nobody'. (shrink) Natural Sciences, Misc in Natural Sciences Complexity of a Problem of Energy Efficient Real-Time Task Scheduling on a Multicore Processor.Abhishek Mishra & Anil Kumar Tripathi - 2016 - Complexity 21 (1):259-267.details Influence of Family Structure on Child Health: Evidence From India.Abhishek Kumar & Faujdar Ram - 2013 - Journal of Biosocial Science 45 (5):577-599.details SummaryThis paper examines the association between family structure and child health in India using the third round of the National Family Health Survey, conducted during 2005–06. Two important child health indicators – underweight and full immunization – are used as dependent variables. Descriptive and multivariate statistics are deployed to establish the relationship between family structure and child health. The results of the descriptive statistics show that children who belong to a non-nuclear family have better nutritional status and higher immunization coverage (...) than those in nuclear families. Children living with siblings have worse health status than those living without siblings for both the outcomes. Multivariate analysis shows that family structure has a small effect on the two child health outcomes, which is no longer significant after adjusting for socioeconomic measures and region. However, number of siblings is significantly and negatively associated with the nutritional status of children and full immunization coverage, even after other socio-demographic and geographic factors are controlled for. Along with family structure, parent's educational attainment, age of the mother and household economic status are significant determinants of underweight and full immunization. (shrink) Influence of Ethical Ideology on Job Stress.Abhishek Shukla & Rajeev Srivastava - 2017 - Asian Journal of Business Ethics 6 (2):233-254.details The relationship between ethical ideology and job stress appears to be complex. This study is based on a model presented by Forsyth, showing two dimensions that play an important role in ethical evaluation and behavior. Based on a survey of 561 employees of hotel industry in India, ethical ideologies were found to be negatively associated with job stress. The data were analyzed using Pearson correlations and multiple regressions. The result showed that relativism is negatively correlated with job stress. Further, it (...) has been established that idealism and relativism interacted in such a way that there is a negative relationship between idealism and job stress when relativism is low and positive relationship when relativism is high. The findings imply that ethical ideology adversely influences the job stress in the organization. (shrink) Is Economic Inequality in Family Planning in India Associated with the Private Sector?Abhishek Kumar, Anrudh K. Jain, Kumudha Aruldas, Arupendra Mozumdar, Ankita Shukla, Rajib Acharya, Faujdar Ram & Niranjan Saggurti - forthcoming - Journal of Biosocial Science:1-12.details This study examined the pattern of economic disparity in the modern contraceptive prevalence rate among women receiving contraceptives from the public and private health sectors in India, using data from all four rounds of the National Family Health Survey conducted between 1992–93 and 2015–16. The mCPR was measured for currently married women aged 15–49 years. A concentration index was calculated and a pooled binary logistic regression analysis conducted to assess economic disparity in modern contraceptive use between the public and private (...) health sectors. The analyses were stratified by rural–urban place of residence. The results indicated that mCPR had increased in India over time. However, in 2015–16 only half of women – 48% – were using any modern contraceptive in India. Over time, the economic disparity in modern contraceptive use reduced across both public and private health sectors. However, the extent of the disparity was greater when women obtained the services from the private sector: the value of the concentration index for mCPR was 0.429 when obtained from the private sector and 0.133 when from the public sector in 2015–16. Multivariate analysis confirmed a similar pattern of the economic disparity across public and private sectors. Economic disparity in the mCPR has reduced considerably in India. While the economic disparity in 2015–16 was minimal among those accessing contraceptives from the public sector, it continued to exist among those receiving services from the private sector. While taking appropriate steps to plan and monitor private sector services for family planning, continued and increased engagement of public providers in the family planning programme in India is required to further reduce the economic disparity among those accessing contraceptive services from the private sector. (shrink) Transparent AI: Reliabilist and Proud.Abhishek Mishra - 2021 - Journal of Medical Ethics 47 (5):341-342.details Durán et al argue in 'Who is afraid of black box algorithms? On the epistemological and ethical basis of trust in medical AI'1 that traditionally proposed solutions to make black box machine learning models in medicine less opaque and more transparent are, though necessary, ultimately not sufficient to establish their overall trustworthiness. This is because transparency procedures currently employed, such as the use of an interpretable predictor,2 cannot fully overcome the opacity of such models. Computational reliabilism, an alternate approach to (...) adjudicating trustworthiness that goes beyond transparency solutions, is argued to be a more promising approach. CR can bring the benefits of traditional process reliabilism in epistemology to bear on this problem of model trustworthiness. Durán et al 's explicitly reliabilist epistemology to assess the trustworthiness of black box models is a timely addition to current transparency-focused approaches in the literature. Their delineation of the epistemic from the ethical also serves the debate by clarifying the nature of the different problems. However, their overall account underestimates the epistemic value of certain transparency-enabling approaches by conflating different types of opacity and also oversimplifies transparency-advocating arguments in the literature. First, it is unclear why Durán et al consider transparency approaches as insufficient to overcome epistemic opacity, if heiraccount of opacity is the traditional one from the machine learning literature: opacity stemming from the mismatch between mathematical optimisation in high dimensionality that is characteristic of machine learning and the demands of human-scale reasoning and styles of semantic interpretation.3 …. (shrink) Biomedical Ethics in Applied Ethics Ontology, Epistemology, and Multimethod Research in Political Science.Abhishek Chatterjee - 2013 - Philosophy of the Social Sciences 43 (1):73-99.details Epistemologies and research methods are not free of metaphysics. This is to say that they are both, supported by (or presumed by), and support (or presume) fundamental ontologies. A discussion of the epistemological foundations of "multimethod" research in the social sciences—in as much as such research claims to unearth "causal" relations—therefore cannot avoid the ontological presuppositions or implications of such a discussion. But though there isn't necessarily a perfect correspondence between ontology, epistemology, and methodology, they do constrain each other. As (...) such it is possible to make methodological choices that are at odds with one's (implicit) ontology or argue from an ontology that is inconsistent one's choice of methods.Yet lack of recognition of this fact has hampered methodological discussions in political science, especially with respect to the discussion on the merits of multimethod research. The ontology implicitly accepted in such discussions is "reductionist" and "regularist," that is, one that respectively defines causes in terms of noncausal relations and states of affair and affirms that such noncausal relations are regularities in nature. This article will argue that any attempt to fit "multimethod" research (where "multimethod" signifies some combination of inferential statistics and case studies) within this narrow ontology is destined to fail since such a metaphysics logically cannot accord case studies a necessary or sufficient role in the in the establishment of causal relations. However, there are metaphysical positions within the ambit of an empiricist philosophy of science that can accommodate multiple methods without contradiction. The article discusses two such ontologies and suggests ways in which they might allow the establishment of a coherent epistemological foundation for multimethod research, however, within a decidedly empiricist philosophy of science. (shrink) Social Epistemology, Misc in Epistemology Social Ontology, Misc in Social and Political Philosophy Does Type of Household Affect Maternal Health? Evidence From India.Nandita Saikia & Abhishek Singh - 2009 - Journal of Biosocial Science 41 (3):329-353.details The Duhem-Quine Problem for Equiprobable Conjuncts.Abhishek Kashyap & Vikram S. Sirola - 2019 - Studies in History and Philosophy of Science Part A 75:43-50.details Debating Humanitarian Intervention: Should We Try to Save Strangers? By Fernando R. Tesón and Bas van der Vossen: Oxford and New York: Oxford University Press, 2017.Abhishek Choudhary - 2019 - Human Rights Review 20 (3):395-396.details Duhem-Quine Problem for Equiprobable Conjuncts.Abhishek Kashyap & Vikram S. Sirola - forthcoming - Studies in History and Philosophy of Science Part A.details Sibling- and Family-Level Clustering of Underweight Children in Northern India.Abhishek Singh, P. Arokiasamy, Jalandhar Pradhan, Kshipra Jain & Sangram Kishor Patel - 2017 - Journal of Biosocial Science 49 (3):348-363.details Peace Agreements by Nina Caspersen: Cambridge and Malden, MA: Polity, 2017.Abhishek Choudhary - 2018 - Human Rights Review 19 (3):411-412.details John Perry on Cognitive Significance.R. C. Majhi - 1997 - Indian Philosophical Quarterly 24 (2):225-236.details Consciousness and Materialism in Philosophy of Mind The Wnt Transcriptional Switch: TLE Removal or Inactivation?Aravinda-Bharathi Ramakrishnan, Abhishek Sinha, Vinson B. Fan & Ken M. Cadigan - 2018 - Bioessays 40 (2):1700162.details Many targets of the Wnt/β-catenin signaling pathway are regulated by TCF transcription factors, which play important roles in animal development, stem cell biology, and oncogenesis. TCFs can regulate Wnt targets through a "transcriptional switch," repressing gene expression in unstimulated cells and promoting transcription upon Wnt signaling. However, it is not clear whether this switch mechanism is a general feature of Wnt gene regulation or limited to a subset of Wnt targets. Co-repressors of the TLE family are known to contribute to (...) the repression of Wnt targets in the absence of signaling, but how they are inactivated or displaced by Wnt signaling is poorly understood. In this mini-review, we discuss several recent reports that address the prevalence and molecular mechanisms of the Wnt transcription switch, including the finding of Wnt-dependent ubiquitination/inactivation of TLEs. Together, these findings highlight the growing complexity of the regulation of gene expression by the Wnt pathway. Wnt targets are regulated by a "switch" where TLE co-repressors are displaced from TCFs by β-catenin, leading to activation of transcription. Recent reports challenge this view and suggest that additional processes, including exchange of specific TCF proteins and Wnt-dependent ubiquitination of TLEs, play important roles in the Wnt transcriptional switch. (shrink) Genetics and Molecular Biology in Philosophy of Biology Emotional Speech Processing: Disentangling the Effects of Prosody and Semantic Cues.Marc D. Pell, Abhishek Jaywant, Laura Monetta & Sonja A. Kotz - 2011 - Cognition and Emotion 25 (5):834-853.details G.P. Rao, Humanising Management: Transformation Through Human Values. New Delhi: Ane Books, 2010, P. 219, Rs 595.00.Abhishek Goel - 2010 - Journal of Human Values 16 (2):196-200.details Neonatal Mortality in the Empowered Action Group States of India: Trends and Determinants.Perianayagam Arokiasamy & Abhishek Gautam - 2008 - Journal of Biosocial Science 40 (2):183-201.details Freedom and Liberty in Social and Political Philosophy A Cross Sectional Study of the Patient′s Awareness and Understanding Toward Legal Nature of Informed Consent in a Dental Hospital in Rural Haryana.Abhishek Singh, Anu Bhardwaj, Rajnish Jindal, Prassana Mithra, D. R. Rajesh & Adiba Siddique - 2012 - Journal of Education and Ethics in Dentistry 2 (1):25.details Autonomy in Applied Ethics in Applied Ethics Concordance Between Partners in Desired Waiting Time to Birth for Newlyweds in India.Abhishek Singh & Stan Becker - 2012 - Journal of Biosocial Science 44 (1):57-71.details SummaryExamining waiting time to birth among newlywed couples is likely to provide insights into the desire for spacing births among newlywed husbands and wives. Data from the Indian National Family Health Survey of 2005–06 are used to examine the desired waiting time to birth among newlywed couples. The dependent variable is spousal concordance on desired waiting times. Overall 65% of couples have concordant desired waiting times. Among discordant couples, wives were more likely to want to wait longer than their husbands. (...) Couples from richer wealth quintiles were more likely than couples from the poorest quintile to have concordant desired waiting times. Muslims were less likely than Hindus to have concordant desires. There is a need for spacing contraceptive methods among newlyweds in India. This may have implications for the Indian Family Planning Programme, which to date has largely focused on sterilization. Programmes need to include newlywed husbands to promote use of spacing methods. (shrink) Is Antenatal Care Effective in Improving Maternal Health in Rural Uttar Pradesh? Evidence From a District Level Household Survey.Faujdar Ram & Abhishek Singh - 2006 - Journal of Biosocial Science 38 (4):433-448.details Feminism: Mothering in Philosophy of Gender, Race, and Sexuality Urban Poverty and Utilization of Maternal and Child Health Care Services in India.Ravi Prakash & Abhishek Kumar - 2013 - Journal of Biosocial Science 45 (4):433-449.details New Evidence on the Impact of the Quality of Prenatal Care on Neonatal and Infant Mortality in India.Ashish Kumar Upadhyay, Abhishek Singh & Swati Srivastava - 2020 - Journal of Biosocial Science 52 (3):439-451.details Evidence on the impact of the quality of prenatal care on childhood mortality is limited in developing countries, including India. Therefore, using nationally representative data from the latest round of the National Family Health Survey, this study examined the impact of the quality of prenatal care on neonatal and infant mortality in India using a multivariable binary logistic regression model. The effect of the essential components of prenatal care services on neonatal and infant mortality were also investigated. The results indicate (...) that improvement in the quality of prenatal care is associated with a decrease in neonatal and infant mortality in India. Tetanus toxoid vaccination, consumption of iron–folic acid tablets during pregnancy and having been weighed during pregnancy were statistically associated with a lower risk of neonatal and infant mortality. Educating women on pregnancy complications was also associated with a lower risk of neonatal mortality. No effect of blood pressure examination, blood test and examination of the abdomen during pregnancy were found on either of the two indicators of childhood mortality. Although the coverage of prenatal care has increased dramatically in India, the quality of prenatal care is still an area of concern. There is therefore a need to ensure high-quality prenatal care in India. (shrink) Predictors of the Diets Consumed by Adolescent Girls, Pregnant Women and Mothers with Children Under Age Two Years in Rural Eastern India.Sayeed Unisa, Abhishek Saraswat, Arti Bhanot, Abdul Jaleel, Rabi N. Parhi, Sourav Bhattacharjee, Apollo Purty, Sudhira Rath, Babita Mohapatra, Avinash Lumba, Sonali Sinha, Nita Kejrewal, Neeraj Agrawal, Vikas Bhatia, Manisha Ruikar & Vani Sethi - forthcoming - Journal of Biosocial Science:1-20.details Adolescents, pregnant women and mothers of children under 2 years of age are in stages of life characterized by higher nutritional demands. The study measured the dietary diversity of 17,680 adolescent girls, pregnant women and mothers of children under age 2 years in the eastern Indian states of Bihar, Chhattisgarh and Odisha using data from the Swabhimaan baseline survey conducted in 2016. The association of women's mean Dietary Diversity Scores with socioeconomic, health and nutrition service indicators was assessed. The sampled (...) population was socioeconomically more vulnerable than the average Indian population. There was not much variation in the types of foods consumed daily across target groups, with diet being predominantly cereal and vegetable based. Nearly 30% of the mothers had low Dietary Diversity Scores, compared with 25% of pregnant women and 24% of adolescent girls. In each target group, more than half of the respondents were unable to meet the Minimum Dietary Diversity score of at least five of ten food groups consumed daily. Irrespective of their background characteristics, mean Dietary Diversity Scores were significantly lower in Bihar than in Chhattisgarh and Odisha for all target groups. Having at least 6 years of education, belonging to a relatively rich household and possessing a ration card predicted mean dietary diversity. Project interventions of participatory women's group meetings improved mean Dietary Diversity Scores for mothers and adolescent girls. Considering the association between poverty and dietary diversity, the linkage between girls and women and nutrition-focused livelihoods and supplementary nutrition programmes needs to be tested. (shrink) A Note on Identification in Discrete Choice Models with Partial Observability.Mogens Fosgerau & Abhishek Ranjan - 2017 - Theory and Decision 83 (2):283-292.details This note establishes a new identification result for additive random utility discrete choice models. A decision-maker associates a random utility Uj+mj\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$U_{j}+m_{j}$$\end{document} to each alternative in a finite set j∈1,…,J\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$j\in \left\{ 1,\ldots,J\right\} $$\end{document}, where U=U1,…,UJ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf {U}=\left\{ U_{1},\ldots,U_{J}\right\} $$\end{document} is unobserved by the researcher and random with an unknown joint distribution, while the perturbation m=m1,…,mJ\documentclass[12pt]{minimal} \usepackage{amsmath} (...) \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf {m}=\left $$\end{document} is observed. The decision-maker chooses the alternative that yields the maximum random utility, which leads to a choice probability system m→Pr1\|m,…,PrJ\|m\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf { m\rightarrow }\left,\ldots,\Pr \left \right) $$\end{document}. Previous research has shown that the choice probability system is identified from the observation of the relationship m→Pr1\|m\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathbf {m}\rightarrow \Pr \left $$\end{document}. We show that the complete choice probability system is identified from observation of a relationship m→∑j=1sPrj\|m\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf {m}\rightarrow \sum _{j=1}^{s}\Pr \left $$\end{document}, for any sshrink) Honor Killing: Where Pride Defeats Reason.Tanuj Kanchan, Abhishek Tandon & Kewal Krishan - 2016 - Science and Engineering Ethics 22 (6):1861-1862.details Honor killings are graceless and ferocious murders by chauvinists with an antediluvian mind. These are categorized separately because these killings are committed for the prime reason of satisfying the ego of the people whom the victim trusts and always looks up to for support and protection. It is for this sole reason that honor killings demand strict and stern punishment, not only for the person who committed the murder but also for any person who contributed or was party to the (...) act. A positive change can occur with stricter legislation and changes in the ethos of the society we live in today. (shrink) Technology Ethics in Applied Ethics Is Unintended Birth Associated with Physical Intimate Partner Violence? Evidence From India.Srinivas Goli, Abhishek Gautam, Md Juel Rana, Harchand Ram, Dibyasree Ganguly, Tamal Reja, Priya Nanda, Nitin Datta & Ravi Verma - 2020 - Journal of Biosocial Science 52 (6):907-922.details A growing number of studies have tested the association between intimate partner violence and the unintendedness of pregnancy or birth, and most have suggested that unintendedness of pregnancy is a cause of IPV. However, about nine in every ten women face violence after delivering their first baby. This study examined the effects of the intendedness of births on physical IPV using data from the National Family Health Survey. The multivariate logistic regression model analysis found that, compared with women with no (...) unwanted births, physical IPV was higher among those women who had unwanted births, followed by those who had mistimed births, even after adjusting for several women's individual and socioeconomic characteristics. Thus, the reduction of women with mistimed and unwanted births could reduce physical IPV in India. The study highlights the unfinished agenda of family planning in the country and argues for the need to integrate family planning and Reproductive, Maternal and Child Health Care services to yield multi-sectoral outcomes, including the elimination of IPV. (shrink) The Plausibility and Significance of Underdetermination Arguments.Vikram S. Sirola & Abhishek Kashyap - 2019 - Journal of the Indian Council of Philosophical Research 36 (2):339-356.details Underdetermination of theory choice claims that empirical evidence fails to provide sufficient grounds for choosing a theory over its rivals. We explore the epistemological and methodological significance of this thesis by utilising a classificatory scheme to situate three arguments that purport to establish its plausibility. Proponents of these three arguments, W.V.O Quine, John Earman, and Kyle Stanford, use different premises to arrive at the conclusion that theory choice is empirically underdetermined and their classification along the proposed schema brings out the (...) variety in underdetermination arguments and the historical trajectory of the thesis. Although the epistemological significance of underdetermination—it is seen as undermining the doctrine of scientific realism—is widely discussed in the literature, Quine understood the acceptance of the thesis as interrogating the attitude one is justified in adopting towards rival theories. We argue that sticking with one's own theory in the face of underdetermination leads to a distinct type of disagreement, which we present as the methodological significance of underdetermination. The examination of the methodological significance of underdetermination allows us to propose weak underdetermination as a philosophically interesting variant of underdetermination. (shrink) Underdetermination of Theory by Data, Misc in General Philosophy of Science Rights Vis-À-Vis Duties and Contemporary Human Rights Debate.Sudhir Singh & Abhishek Kumar - 2021 - Journal of the Indian Council of Philosophical Research 38 (3):389-396.details Most of the theories of rights propounded by philosophers, right from the beginning till the twentieth century, conceive rights either as a claim against the state or an obligation upon the state. Certainly such a conception has had something to do with the prevailing social, political and economic systems of the time concerned. Social, political and economic systems also had a particular relationship amongst them. Change in individual and social perspectives, values, priorities and beliefs has affected the philosophy of right. (...) From the ages of Locke and Hobbes when natural right was taken in obvious terms to the times of communitarians like Michael Sandel and thinkers like Ronald Dworkin the term "Right" has earned many dimensions. Progress and changing arrangements of systems complicates the status of a philosophical theory of rights propounded at a particular time, in the sense that its losses its teeth in any new found milieu. This paper evaluates the notion of abstract universalism of individual rights in the light of Gandhian notion of duty. (shrink) Governing AI-Driven Health Research: Are IRBs Up to the Task?Phoebe Friesen, Rachel Douglas-Jones, Mason Marks, Robin Pierce, Katherine Fletcher, Abhishek Mishra, Jessica Lorimer, Carissa Véliz, Nina Hallowell, Mackenzie Graham, Mei Sum Chan, Huw Davies & Taj Sallamuddin - 2021 - Ethics and Human Research 2 (43):35-42.details Many are calling for concrete mechanisms of oversight for health research involving artificial intelligence (AI). In response, institutional review boards (IRBs) are being turned to as a familiar model of governance. Here, we examine the IRB model as a form of ethics oversight for health research that uses AI. We consider the model's origins, analyze the challenges IRBs are facing in the contexts of both industry and academia, and offer concrete recommendations for how these committees might be adapted in order (...) to provide an effective mechanism of oversight for health‐related AI research. (shrink) Ethics of Artificial Intelligence, Misc in Philosophy of Cognitive Science Medical Ethics, Misc in Applied Ethics Medical Research Ethics in Applied Ethics Scientific Research Ethics in Applied Ethics Hand Gesture Recognition and Classification by Discriminant and Principal Component Analysis Using Machine Learning Techniques.Sauvik Das Gupta, Souvik Kundu, Rick Pandey, Rahul Ghosh, Rajesh Bag & Abhishek Mallik - 2012 - In Zdravko Radman (ed.), The Hand. MIT Press.details Philosophy of Psychology in Philosophy of Cognitive Science Post-Sterilization Autonomy Among Young Mothers in South India.Saseendran Pallikadavath, Irudaya Rajan, Abhishek Singh, Reuben Ogollah & Samantha Page - 2015 - Journal of Biosocial Science 47 (1):75-89.details SummaryThis study examined the post-sterilization autonomy of women in south India in the context of early sterilization and low fertility. Quantitative data were taken from the third round of the National Family Health Survey carried out in 2005–06, and qualitative data from one village each in Kerala and Tamil Nadu during 2010–11. The incident rate ratios and thematic analysis showed that among currently married women under the age of 30 years, those who had been sterilized had significantly higher autonomy in (...) household decision-making and freedom of mobility compared with women who had never used any modern family planning method. Early age at sterilization and low fertility enables women to achieve the social status that is generally attained at later stages in the life-cycle. Policies to capitalize on women's autonomy and free time resulting from early sterilization and low fertility should be adopted in south India. (shrink) Descriptive Modeling of the Dynamical Systems and Determination of Feedback Homeostasis at Different Levels of Life Organization.G. N. Zholtkevych, K. V. Nosov, Yu G. Bespalov, L. I. Rak, M. Abhishek & E. V. Vysotskaya - 2018 - Acta Biotheoretica 66 (3):177-199.details The state-of-art research in the field of life's organization confronts the need to investigate a number of interacting components, their properties and conditions of sustainable behaviour within a natural system. In biology, ecology and life sciences, the performance of such stable system is usually related to homeostasis, a property of the system to actively regulate its state within a certain allowable limits. In our previous work, we proposed a deterministic model for systems' homeostasis. The model was based on dynamical system's (...) theory and pairwise relationships of competition, amensalism and antagonism taken from theoretical biology and ecology. However, the present paper proposes a different dimension to our previous results based on the same model. In this paper, we introduce the influence of inter-component relationships in a system, wherein the impact is characterized by direction (neutral, positive, or negative) as well as its (absolute) value, or strength. This makes the model stochastic which, in our opinion, is more consistent with real-world elements affected by various random factors. The case study includes two examples from areas of hydrobiology and medicine. The models acquired for these cases enabled us to propose a convincing explanation for corresponding phenomena identified by different types of natural systems. (shrink) Biological Modeling in Philosophy of Biology Surface Anchoring Effect on Guest–Host Ferroelectric Liquid Crystal Response Time – an Electro-Optical Investigation.R. Manohar, Kamal Kumar Pandey, Satya Prakash Yadav, Abhishek Kumar Srivastava & Abhishek Kumar Misra - 2010 - Philosophical Magazine 90 (34):4529-4539.details Birth Order, Stage of Infancy and Infant Mortality in India.S. K. Mishra, Bali Ram, Abhishek Singh & Awdhesh Yadav - 2018 - Journal of Biosocial Science 50 (5):604-625.details Share of Current Unmet Need for Modern Contraceptive Methods Attributed to Past Users of These Methods in India.Ankita Shukla, Anrudh K. Jain, Rajib Acharya, F. Ram, Arupendra Mozumdar, Abhishek Kumar, Subrato Mondal & Niranjan Saggurti - 2021 - Journal of Biosocial Science 53 (3):407-418.details Despite persistent efforts, unmet need for contraceptives in India has declined only slightly from 14% to 13% between 2005–06 and 2015–16. Many women using a family planning method discontinue it without switching to another method and continue to have unmet need. This study quantified the share of current unmet need for modern contraceptive methods attributed to past users of these methods in India. Data were drawn from two rounds of the National Family Health Survey conducted in 2005–06 and 2015–16. Using (...) information on women with current unmet need, and whether they used any modern method in the past, the share of past users with current unmet need for modern methods was calculated. Bivariate and multivariate analyses were performed. Among 46 million women with unmet need, 11 million were past users of modern methods in 2015–16. The share of current unmet need attributed to past users of modern contraceptive methods declined from 27% in 2005–06 to 24% in 2015–16. Share of current unmet need attributed to past users was associated with reversible method use. This share rose with increased use of modern reversible methods. With the Indian family planning programme's focus on increasing modern reversible method use, the share of unmet need attributed to past users of modern methods is likely to increase in the future. The programme's emphasis on continuation of contraceptive use, along with bringing in new users, could be one of the key strategies for India to achieve the FP2020 goals. (shrink) Database Creation and Dialect-Wise Comparative Analysis of Prosodic Features for Punjabi Language.Shipra J. Arora & Rishipal Singh - 2019 - Journal of Intelligent Systems 29 (1):1275-1282.details The paper represents a Punjabi corpus in the agriculture domain. There are various dialects in the Punjabi language and the main concentration is on major dialects, i.e. Majhi, Malwai and Doabi for the present study. A speech corpus of 125 isolated words is taken into consideration. These words are uttered by 100 speakers, i.e. 60 Malwi dialect speakers, 20 Majhi dialect speakers and 20 Doabi dialect speakers. Tonemes, adhak and nasal words are selected from the corpus. Recordings have (...) been processed through two mediums. The paper also elaborates some distinctive features of the corpus. This corpus is of quite significance for the speech recognition system. Prosodic characteristics such as intonation, rhythm and stress create a crucial impact on the speech recognition system. These characteristics vary from language to language as well as various dialects of a language. This paper portrays a comparative analysis of isolated words prosodic features of Malwi, Majhi and Doabi dialects of Punjabi language. Analysis is done using the PRAAT tool. Pitch, intensity, formant I and formant II values are extracted for toneme, adhak, nasal and nasal words. For all kinds of words, there is a significant variation in pitch, intensity, formant I and formant II values of male and female speakers of Malwi, Majhi and Doabi dialects. A detailed analysis has been discussed throughout this paper. (shrink)	CommonCrawl
Sally Elizabeth Carlson Sally Elizabeth Carlson (October 2, 1896 – November 1, 2000) was an American mathematician,[1] the first woman and one of the first two people to obtain a doctorate in mathematics from the University of Minnesota.[1][2] Sally Elizabeth Carlson Born(1896-10-02)October 2, 1896 Minneapolis, Minnesota DiedNovember 1, 2000(2000-11-01) (aged 104) NationalityAmerican Academic background Alma materUniversity of Minnesota ThesisThe Convergence of Certain Methods of Closest Approximation (1924) Doctoral advisorDunham Jackson Academic work DisciplineMathematics Sub-disciplineFunctional analysis InstitutionsUniversity of Minnesota Notable studentsMargaret P. Martin Early life and education Carlson was born in Minneapolis to a large working-class family of Swedish immigrants. She became her high school valedictorian in 1913, graduated from the University of Minnesota in 1917, and earned a master's degree there in 1918. After teaching mathematics for two years, she returned to graduate study in 1920, and completed her Ph.D. at Minnesota in 1924. Both students were supervised by Dunham Jackson;[1] Carlson's dissertation, in functional analysis, was On The Convergence of Certain Methods of Closest Approximation.[3] Career and contributions She joined the Minnesota faculty, and remained there until her retirement in 1965 as a full professor.[1] She has no record of supervising doctoral dissertations,[3] and published little research after the work of her own dissertation. However, she supervised several master's students, and was described as a mentor by Margaret P. Martin, who completed her Ph.D. at Minnesota in 1944.[4] Recognition Carlson won a Distinguished Teacher Award at Minnesota.[1] After her 2000 death, the library of the University of Minnesota memorialized her in an exhibit, titled "Elizabeth Carlson, notable alumna".[1] References 1. Green, Judy; LaDuke, Jeanne (2008), "Carlson, Elizabeth", Pioneering Women in American Mathematics: The Pre-1940 PhD's, History of Mathematics, vol. 34, American Mathematical Society, The London Mathematical Society, pp. 153–154, ISBN 978-0-8218-4376-5 2. Riddle, Larry (June 2, 2016), "The First Ph.D.'s", Biographies of Women Mathematicians, Agnes Scott College, retrieved 2017-11-18 3. Sally Elizabeth Carlson at the Mathematics Genealogy Project 4. Murray, Margaret A. M. (2001), Women Becoming Mathematicians: Creating a Professional Identity in Post-World War II America, MIT Press, p. 100, ISBN 9780262632461 Authority control International • VIAF National • Germany Academics • MathSciNet • Mathematics Genealogy Project • zbMATH	Wikipedia
\begin{definition}[Definition:Field Adjoined Element] {{delete\|think it fits more naturally here}} Let $E/F$ be a field extension, $\alpha \in E$. Then: :$F[\alpha] $ denotes the smallest subring of $E$ containing $F \cup \alpha$. :$F(\alpha) $ denotes the smallest subfield of $E$ containing $F \cup \alpha$. We say this as '''$F$ adjoined with $\alpha$'''. Category:Definitions/Field Theory \end{definition}	ProofWiki
A Novel Method for Fast and Efficient Measurement of Diffusion Tensor Size and Shape Distributions Grant Yang1,2 and Jennifer McNab2 1Electrical Engineering, Stanford University, Stanford, CA, United States, 2Radiology, Stanford University, Stanford, CA, United States We demonstrate through simulations and empirical data that it is possible to simultaneously estimate the variance of the voxel-wise diffusion tensor shape and size distributions using efficient isotropic and linear diffusion encodings on a whole-body clinical MRI scanner with whole-brain coverage at 3mm isotropic resolution in under 2 minutes. Characterizing the complex microenvironments in the brain is a primary challenge of diffusion MRI. Recent advances in gradient waveform design and modeling have enabled increased specificity to changes in compartment shape and size through advanced diffusion waveforms1-3(Fig. 1a-c) and modeling4-8. The diffusion tensor distribution (DTD) model6 extends the diffusion tensor (DT) model9 by adding a fourth-order tensor($$$\mathbb{C}$$$), describing the covariance of the distribution of DTs. Rotationally invariant measures of size variation (CMD) and microscopic diffusion anisotropy (μFA)(Fig. 1d) are derived from the 21 independent elements of $$$\mathbb{C}$$$. We demonstrate that CMD and μFA can be derived from the orientation averaged $$$\mathbb{C}_{iso}$$$ with fewer measurements than required to fit the full DTD model. The DTD model approximates the diffusion signal using a fourth-order diffusion covariance tensor: $$S(\mathbf{B}){\approx}S_0{\exp}(-{\langle}\mathbf{B},\langle{\mathbf{D}\rangle\rangle+\frac{1}{2}\langle}\mathbb{B},\mathbb{C}\rangle)\hspace{3cm}(1)$$ where $$$\mathbf{B}=\int_0^\tau\mathbf{q}(t)\mathbf{q}^\mathrm{T}\,d\tau$$$ is the b-tensor, $$${\langle}\mathbf{D}{\rangle}$$$ is the average DT, $$$\mathbb{B}=\mathbf{B}^{\otimes2}$$$, and $$$\mathbb{C}\in\mathbb{R}^{3\times3\times3\times3}$$$ is the diffusion covariance tensor. Equation 1 can be written in matrix vector form: $$\left(\begin{array}{c}{\log}S_1\\\vdots \\{\log}S_m\end{array}\right)=\left( \begin{array}{ccc}1&-\mathbf{b}_1^\mathrm{T}&\frac{1}{2}\mathbb{b}_1^\mathrm{T} \\\vdots&\vdots&\vdots\\1& -\mathbf{b}_m^\mathrm{T}&\frac{1}{2}\mathbb{b}_m^\mathrm{T} \end{array}\right)\left(\begin{array}{ccc}\log S_0&\langle\mathbf{d}\rangle&\mathbb{c}\end{array}\right)^\mathrm{T}\hspace{3cm}(2)$$ where $$$\mathbf{b}=\left(\begin{array}{cccccc}b_{xx}&b_{yy} & b_{zz}&\sqrt{2}b_{yz}&\sqrt{2}b_{xz} &\sqrt{2} b_{xy}\end{array}\right)^\mathrm{T}$$$, $$$\mathbb{b}\in\mathbb{R}^{21\times1}$$$ contains the independent elements of $$$\mathbf{b}\mathbf{b}^{\mathrm{T}}$$$, $$$ \mathbf{d}=\left(\begin{array}{cccccc}d_{xx}&d_{yy}&d_{zz}&\sqrt{2}d_{yz}& \sqrt{2}d_{xz}&\sqrt{2}d_{xy}\end{array}\right)^\mathrm{T}$$$ are independent elements of $$$\langle \mathbf{D}\rangle$$$, and $$$\mathbb{c}\in\mathbb{R}^{21\times1}$$$ contains independent elements of $$$\mathbb{C}$$$. However, since CMD and μFA are rotationally invariant, they can be estimated from $$$\mathbb{C}$$$ averaged over all orientations ( $$$\mathbb{C}_{iso}$$$). Since $$$\mathbb{C}_{iso}$$$ describes an isotropic medium, the same signal ($$$S_{iso}$$$) is expected from all diffusion encoding orientations. Therefore $$$\mathbb{b}_{linear}^\mathrm{T}\mathbb{c}_{iso}=\alpha $$$ and $$$\mathbb{b}_{planar}^\mathrm{T}\mathbb{c}_{iso}=\alpha$$$. This can be rewritten as the homogeneous equation: $$$\mathbf{A}\left(\begin{array}{ccc}\mathbb{c}_{iso}&\alpha&\beta\end{array}\right)=0$$$. The null space of $$$\mathbf{A}$$$ can be spanned by two vectors and fully parameterizes $$$\mathbb{c}_{iso}$$$ such that $$$\mathbb{c}_{iso}=\mathbf{V}\mathbf{v}$$$, where $$$\mathbf{V}\in\mathbb{R}^{21\times2}$$$, $$$ \mathbf{v}\in\mathbb{R}^{2\times1}$$$. Therefore, Equation 2 simplifies to: $$\left(\begin{array}{c}{\log}S_{iso,1}\\\vdots\\{\log}S_{iso,m}\end{array}\right)=\left( \begin{array}{ccc}1&-b_1&\frac{1}{2}b_1^2\mathbf{r}_1\\\vdots&\vdots&\vdots\\1&-b_m&\frac{1}{2}b_m^2\mathbf{r}_m\end{array}\right)\left(\begin{array}{ccc}\log S_0 &\mathrm{MD}&\mathbf{v}\end{array}\right)^\mathrm{T} \hspace{3cm}(3)$$ , where $$$\mathbf{r}_n = \mathbb{b}_n^\mathrm{T}\mathbf{V}/\mathrm{b}_n^2$$$ and $$$S_{iso, n}$$$ can be computed for anisotropic mediums by averaging over multiple orientations. The advantages of Equation 3 over Equation 2 include: 1) it is computationally and numerically advantageous to fit 4 variables compared to 28, 2) estimating $$$S_{iso}$$$ requires fewer encoding directions compared to estimating $$$\mathbb{C}$$$ and 3) both CMD and μFA can be estimated using only linear and isotropic b-tensors, thereby avoiding ellipsoidal or planar b-tensors, which require longer TEs. The diffusion environments shown in Figure 3a-c were simulated with the diffusion encoding scheme described in Table 1. Rician noise was introduced to the signal to produce 10,000 noisy measurements with SNRs of 15, 50, and 100 for the non-diffusion weighted signal. Equation 3 was fitted to the noisy signal and the mean and standard deviation of the CMD and μFA were computed6. Orientation averaging to approximate $$$S_{iso}$$$ was tested by simulating the worst-case scenario (Daxial=3μm2/ms, Dradial=0μm2/ms). The coefficient of variation(COV) of the diffusion signal across 150 tensor orientations equally spaced on a sphere was computed for diffusion schemes containing 6 to 60 diffusion directions. Two healthy subjects were scanned with IRB approval using a 3T whole-body MR system(Premier, GE Healthcare) equipped with a 32-channel head coil(Nova Medical). Each subject was examined with linear and isotropic diffusion encoding sequences(Fig. 1) using diffusion acquisition schemes shown in Table 1. The eddy current and motion corrected10,11 data was fit to Equation 3 to compute CMD and μFA6. Figure 2 quantifies the rotational invariance achievable using a given number of linear diffusion encodings. Figure 3d-i display μFA and CMD estimates from simulated diffusion signals from linear and isotropic diffusion encodings. Figure 4 shows full brain estimates of μFA and CMD at $$$3\times3\times3$$$mm3 resolution with acquisition times under 2min. Even in the worst-case scenario, a COV under 4% can be achieved with only 12 linear diffusion encodings. This allows estimation of μFA and CMD with 15 (12 linear+B0+isotropic with 2 b-values) measurements compared to the 28 measurements needed to estimate the full DTD. Note that the matrix from Equation 2 is not full rank with any number of linear and isotropic diffusion encodings, and therefore fitting the full DTD is intractable. Both the simulated and in vivo results demonstrate robust estimation of μFA and CMD. The μFA and CMD contrast in the brain is consistent with prior work6, but furthers these efforts by dramatically reducing the scan time and improving brain coverage. Our in vivo results indicate that robust estimates can be achieved with less than 12 linear encodings, and therefore it is advantageous to acquire additional averages of isotropic diffusion encodings to equalize the SNR across linear and isotropic encodings. We demonstrate the simultaneous estimation of compartment shape and size dispersion using efficient isotropic and linear diffusion encodings. Funding provided by an NSF-GRFP, NIH: R01-NS095985, S10-RR026351, P41-EB015891, GE Healthcare and the Dana Foundation. 1. Sjolund J, Szczepankiewicz F, Nilsson M, Topgaard D, Westin CF, Knutsson H. Constrained optimization of gradient waveforms for generalized diffusion encoding. J Magn Reson 2015;261:157-68. 2. Mitra PP. Multiple wave-vector extensions of the NMR pulsed-field-gradient spin-echo diffusion measurement. Phys Rev B Condens Matter 1995;51(21):15074-15078. 3. Eriksson S, Lasic S, Topgaard D. Isotropic diffusion weighting in PGSE NMR by magic-angle spinning of the q-vector. J Magn Reson 2013;226:13-8. 4. Jespersen SN, Lundell H, Sonderby CK, Dyrby TB. Orientationally invariant metrics of apparent compartment eccentricity from double pulsed field gradient diffusion experiments. NMR Biomed 2013;26(12):1647-62. 5. Lawrenz M, Koch MA, Finsterbusch J. A tensor model and measures of microscopic anisotropy for double-wave-vector diffusion-weighting experiments with long mixing times. J Magn Reson 2010;202(1):43-56. 6. Westin CF, Knutsson H, Pasternak O, Szczepankiewicz F, Ozarslan E, van Westen D, Mattisson C, Bogren M, O'Donnell LJ, Kubicki M and others. Q-space trajectory imaging for multidimensional diffusion MRI of the human brain. Neuroimage 2016;135:345-62. 7. Szczepankiewicz F, Lasic S, van Westen D, Sundgren PC, Englund E, Westin CF, Stahlberg F, Latt J, Topgaard D, Nilsson M. Quantification of microscopic diffusion anisotropy disentangles effects of orientation dispersion from microstructure: applications in healthy volunteers and in brain tumors. Neuroimage 2015;104:241-52. 8. Lasič S, Szczepankiewicz F, Eriksson S, Nilsson M, Topgaard D. Microanisotropy imaging: quantification of microscopic diffusion anisotropy and orientational order parameter by diffusion MRI with magic-angle spinning of the q-vector. Frontiers in Physics 2014;2:11. 9. Pierpaoli C, Jezzard P, Basser PJ, Barnett A, Di Chiro G. Diffusion tensor MR imaging of the human brain. Radiology 1996;201(3):637-48. 10. Andersson JL, Sotiropoulos SN. An integrated approach to correction for off-resonance effects and subject movement in diffusion MR imaging. Neuroimage 2016;125:1063-78. 11. Andersson JL, Skare S, Ashburner J. How to correct susceptibility distortions in spin-echo echo-planar images: application to diffusion tensor imaging. Neuroimage 2003;20(2):870-88. 12. Yang G, Tian Q, Leuze C, Wintermark M, McNab J. Visualizing Axonal Damage in Multiple Sclerosis Using Double Diffusion Encoding MRI in a Clinical Setting. Proc. Intl. Soc. Mag. Reson. Med. 25 (2017) 2017. Figure 1: Q-space trajectory imaging employs time varying diffusion waveforms to produce isotropic (a) planar (b) and linear (c) diffusion encoding as described by the appearance of the resulting b-tensor. Red, green, and blue gradient waveforms represent orthogonal gradient axes. By varying the b-tensor shape, information can be obtained about the diffusion tensor distribution (d), which is inaccessible using only conventional linear diffusion encoding. The diffusion tensor distribution can be used to disentangle microstructural shape and size distributions (d). Pulse sequence parameters for the linear and isotropic diffusion encoding pulse sequences used in this study (e). Table 1: Diffusion acquisition schemes for simulated and acquired data. The table specifies the number of measurements for each b-tensor shape and magnitude. Measurements for linear diffusion encodings are uniformly spaced over a sphere. Figure 2: The coefficient of variation over 150 orientations of a stick diffusion tensor with axial diffusivity 3μm2/ms after orientation averaging of a linear diffusion encoding scheme consisting of 6-60 orientations equally spaced over a sphere with b=2ms/μm2. Figure 3: Simulation results for voxels with net isotropic diffusion and mean diffusivity of 1 μm2/ms. Estimated μFA (d,e,f) and CMD (g,h,i) for (a) a change in tensor shape with no size variation, (b) a change in size variation with isotropic tensors, and (c) a change in size variation for elongated tensors. Figure 4: Maps of microscopic fractional anisotropy (μFA) and the normalized size variance (CMD) measured using linear and isotropic diffusion encoding pulse sequences. The reduced sampling requirements of modeling the orientation averaged diffusion signal enables whole-brain mapping in 1 minute 26 seconds.	CommonCrawl
Input to the European Particle Physics Strategy Update 2018-2020 1 November 2018 to 19 December 2018 Submit Input Submitted Input Open Symposium 5. A European Data Science Institute for Fundamental Physics Instrumentation and computing In order to facilitate the deployment of modern data science technologies (e.g., Deep Learning) into theoretical and experimental research in high energy physics, we suggest that the creation of a European Data Science Institute for Fundamental Physics is included among the recommendations of the European Strategy group. Such an institute would facilitate the development of... 112. A European Strategy Towards Finding Axions and Other WISPs Dark matter and dark sector (accelerator and non-accelerator dark matter, dark photons, hidden sector, axions) Since the last update of the European strategy on particle physics (ESPP) the interest in hypothetical very weakly interacting slim particles, dubbed WISPs, has gained significant momentum. Searches for WISPs with masses below about 1 eV require new approaches beyond accelerator experiments. This document summarizes the physics case, the experimental status and its prospects 57. A high precision neutrino beam for a new generation of short baseline experiments Neutrino physics (accelerator and non-accelerator) The current generation of short baseline neutrino experiments is approaching intrinsic source limitations in the knowledge of flux, initial neutrino energy and flavor. A dedicated facility based on conventional accelerator techniques and existing infrastructures designed to overcome these impediments would have a remarkable impact on the entire field of neutrino oscillation physics. It would... 70. A memorandum by the Global Neutrino Network as input to the update of the European Strategy for Particle Physics The Global Neutrino Network is an association of the neutrino telescope projects targeting the investigation of cosmic and atmospheric neutrinos in the energy range from GeV to beyond PeV. The main scientific objectives are the exploration of high-energy cosmic neutrinos from non-thermal astrophysical sources (neutrino astronomy), the investigation of fundamental questions of particle physics... 143. A New QCD Facility at the M2 beam line of the CERN SPS Strong interactions (perturbative and non-perturbative QCD, DIS, heavy ions) This document summarises the physics interest, sensitivity reach and competitiveness of a future general-purpose fixed-target facility for Particle Physics research. Based upon the versatile M2 beam line of the CERN SPS, a great variety of measurements is proposed to address fundamental issues of Quantum Chromodynamics. 110. A next-generation LHC heavy-ion experiment The present document discusses plans for a compact, next-generation multi-purpose detector at the LHC as a follow-up to the present ALICE experiment. The aim is to build a nearly massless barrel detector consisting of truly cylindrical layers based on curved wafer-scale ultra-thin silicon sensors with MAPS technology, featuring an unprecedented low material budget of 0.05$\%$ X$_0$ per layer,... 81. A View on the European Strategy for Particle Physics Other (communication, outreach, strategy process, technology transfer, individual contributions,…) Worldwide, the particle physics and accelerator community is very actively working towards the next major facility. We emphasize the need to evaluate and compare progress on linear and circular e+e- collider designs in the 100 to 360 GeV centre-of-mass energy and their respective performance potential for making the optimal choice. 7. Advanced LinEar collider study GROup (ALEGRO) Input Accelerator Science and Technology Advanced and Novel Accelerators (ANAs) can provide acceleration gradients orders of magnitude greater than conventional accelerator technologies, and hence they have the potential to provide a new generation of more compact, high-energy machines. Four technologies are of particular interest, all of which rely on the generation of a wakefield which contains intense electric fields suitable for... 105. An Open Lab for the Development of Technical Superconductors Superconductivity is a core technology that has fueled the progress in high-energy physics (HEP) accelerators, from the Tevatron of the early 1980's to the Large Hadron Collider of the late 2010's [1,2,3,4]. The engineering knowledge of superconducting materials in the form of composite wires, tapes, or thin layers finds application in high-field accelerator magnets [5], very large... 84. APPEC Contribution to the European Particle Physics Strategy APPEC strongly supports and encourages the prospects of an even stronger synergy between Astroparticle Physics and Cosmology with Particle Physics. Areas of synergy between the above domains are identified: dark matter and dark energy; multi-messenger astrophysics, with the recently achieved discoveries of gravitational waves and the extraordinary opportunities also offered by gamma-ray and... 149. APS Division of Particles and Fields Response to European Strategy Group Call for White Papers: Community Planning and Science Drivers National road maps This white paper describes the community strategic planning process organized by APS DPF, and summarizes U.S. particle physics community input on activities and aspirations. This is the first of two documents, covering the five P5 Science Drivers. 150. APS Division of Particles and Fields Response to European Strategy Group Call for White Papers: Tools for Particle Physics The U.S. particle physics strategy process is summarized in a companion white paper that also describes U.S. activities related to the five P5 science drivers. Additional activities within the U.S. particle physics program that are critical to progress in our field are described here. 58. AWAKE++: the AWAKE Acceleration Scheme for New Particle Physics Experiments at CERN The AWAKE experiment reached all planned milestones during Run 1 (2016-18), notably the demonstration of strong plasma wakes generated by proton beams and the acceleration of externally injected electrons to multi-GeV energy levels in the proton driven plasma wakefields. During Run 2 (2021 - 2024) AWAKE aims to demonstrate the scalability and the acceleration of electrons to high energies... 35. AWAKE: On the path to Particle Physics Applications Proton-driven plasma wakefield acceleration allows the transfer of energy from a proton bunch to a trailing bunch of particles, the 'witness' particles, via plasma electrons. The AWAKE experiment at CERN is pursuing a demonstration of this scheme using bunches of protons from the CERN SPS. Assuming continued success of the AWAKE program, high energy electron or muon beams will become... 122. Belgian national input to the EPP Strategy update This document summarizes the input of the Belgian scientists to the European Strategy Group (ESG) in the context of the upcoming European strategy update for particle physics research. The Belgian HEP community (permanent researchers from particle and astroparticle physics) met in Brussels on September 12th, 2018. During this meeting we took stock of our current activities, evaluated our... 139. Birmingham Particle Physics Group Submission This document is sent on behalf of the academic faculty members of the Birmingham Particle Physics Group. We begin by summarising our view of the most significant physics questions driving our field and go on to argue how we think they can best be addressed through new facilities, before ending with a few comments on maintaining our community and its wider impact. 157. Canadian Submission to the European Particle Physics Strategy Update This document provides input from the Canadian subatomic physics community to the European Particle Physics Strategy 2020-2025 and is based on the 2017-21 Canadian Subatomic Physics Long Range Plan (www.subatomicphysics.ca). This submission provides an overview of the major projects in which the Canadian community is participating, and discusses connections to the European Particle Physics... 155. CEA-Irfu contribution to the 2020 update of the European Strategy for Particle Physics This document summarizes the scientific and technological position of CEA-Irfu teams on the main projects for particle physics, including neutrino physics and QGP physics. The achievement of the HL-LHC and its scientific exploitation is the immediate priority. The crucial question is therefore the definition of the next generation of collider able to pursue an ambitious search for new physics. 29. CEPC Input to the ESPP 2018 - Physics and Detector Large experiments and projects The Higgs boson, discovered in 2012 by the ATLAS and CMS Collaborations at the Large Hadron Collider (LHC), plays a central role in the Standard Model. Measuring its properties precisely will advance our understandings of some of the most important questions in particle physics, such as the naturalness of the electroweak scale and the nature of the electroweak phase transition. The Higgs boson... 51. CEPC Input to the ESPP 2018 -Accelerator The discovery of the Higgs boson at CERN's Large Hadron Collider (LHC) in July 2012 raised new opportunities for a large-scale accelerator. Due to the low mass of the Higgs, it is possible to produce it in the relatively clean environment of a circular electron–positron collider with reasonable luminosity, technology, cost and power consumption. The Higgs boson is a... 54. CERN's view on Knowledge Transfer as input for the European Strategy on Particle Physics update Please refer to the document attached. 25. Charged Lepton Flavour Violation using Intense Muon Beams at Future Facilities Flavour Physics and CP violation (quarks, charged leptons and rare processes) Charged-lepton flavour-violating (cLFV) processes offer deep probes for new physics with discovery sensitivity to a broad array of new physics models — SUSY, Higgs Doublets, Extra Dimensions, and, particularly, models explaining the neutrino mass hierarchy and the matter-antimatter asymmetry of the universe via leptogenesis. The most sensitive probes of cLFV utilize high-intensity muon beams... 38. COMET The search for charged lepton flavour violation (CLFV) has an enormous discovery potential in probing new physics Beyond the Standard Model (BSM). The observation of a CLFV transition would be an undeniable sign of the presence of BSM physics which goes beyond non-zero masses for neutrinos. Furthermore, CLFV measurements can provide a way to distinguish between different BSM models, which may... 22. Communicating particle physics matters Public and political support for particle physics is essential for sustaining the long-term future of the field - whether this is for attracting young people into STEM careers, gaining support from local communities for building new experiments, or for securing government funding for new and existing experiments. The importance of communicating particle physics has long been recognised by the... 67. Community Support for A Fixed-Target Programme for the LHC This contribution aims at promoting the ground-breaking physics programme accessible with the multi-TeV LHC proton and ion beams used in the fixed-target mode. It can be realised in a parasitic mode for the LHC complex using existing detectors like those of the LHCb and ALICE collaborations or new dedicated systems during the LHC lifetime. It contains a brief description of the different... 14. Complex NEVOD for multi-component investigations of cosmic rays in the record-wide energy range 10^10 – 10^19 eV The wide energy range covered by the complex NEVOD is determined by a unique combination of detectors and installations that have no analogues in the world. The energy region from few GeV to ~ 100 GeV is covered by muon hodoscope URAGAN with an area of 46 m^2. The region from several tens of GeV to several tens of TeV is covered by the Cherenkov water calorimeter with a volume of 2000 m^3 with... 48. Conclusions of the Town Meeting: Relativistic Heavy Ion Collisions This text summaries the consensus view of the scientific community on priorities in the field of relativistic heavy ion collisions, as expressed by the 421 registered participants of the Town Meeting held on 24 October 2018 at CERN. 15. Contribution of the french physics society The French physics society and its division Champs et Particules, after a large consultation of its members, has reached a consensus, and proposes a list of hierarchized items for the European strategy for particle physics. It is motivated by the search of new physics and our belief that Europe must continue playing the leading role in this domain. 36. Dark Sector Physics with a Primary Electron Beam Facility at CERN This input is a summary of an Expression of Interest (SPSC-EOI-018) submitted to the CERN Scientific Committee for the SPS accelerator, SPSC. A primary electron beam facility is proposed with the main motivations being (i) dark sector experiments, and (ii) to enable a suite of development projects in acceleration technology. The facility would deliver a beam to a Light Dark Matter eXperiment,... 126. Deep Underground Neutrino Experiment (DUNE) The 2013 European Strategy for Particle Physics (ESPP) identified the long-baseline neutrino programme as one of the four scientific objectives that require international collaboration. This strong recommendation led to the formation of DUNE as an international collaboration in 2015. The DUNE Collaboration now includes CERN and 14 of its member states, with approximately 400 European... 87. Development of the Micro-Pattern Gaseous Detector Technologies: an overview of the CERN-RD51 Collaboration RD51 is a well-established collaboration with the aim to develop Micro-Pattern Gaseous Detector (MPGD) technologies, to support experiments using this technology, and to disseminate the technology within particle physics and in other fields. Originally created for a five-year term in 2008, RD51 was extended for a third five years term beyond 2018. The rich portfolio of MPGD projects, under... 8. DM follies and the 'Krisis' of particle physics This document has been prepared as an input for the discussions on an European Strategy for Particles Physics. 68. ECFA Detector Panel Report The ECFA Detector Panel is a European Committee composed of detector physicists from a variety of experimental communities in particle and astro-particle physics. Its primary role is to review early stage detector R&D for programmes in these areas that are not yet linked to a host or leading laboratory with its own established review mechanisms. The role and composition of the panel make it... 123. Electric dipole moment community input to the update of the ESPP We describe the status and prospect of the European community of electric dipole moment searches for the input to the update of the European strategy for particle physics. 74. Electron Ion Collider Accelerator Science and Technology - Designs, R&D and Synergies with European research in Accelerators A U.S.-based Electron-Ion Collider (EIC) has recently been endorsed by the U.S. National Academies of Sciences, Engineering, and Medicine (NAS). This brings the realization of such a collider another step closer, after its earlier recommendation in the 2015 Long-Range Plan for U.S. nuclear science of the Nuclear Science Advisory Committee "as the highest priority for new facility construction... 140. Energy frontier lepton-hadron colliders, vector-like quarks/leptons, preons and so on Beyond the Standard Model at colliders (present and future) First of all, an importance of the LHC/FCC based energy frontier lepton-hadron and photon-hadron colliders is emphasised. Then arguments favoring existence of new heavy isosinglet down-type quarks and vector-like isosinglet or isodoublet leptons are presented, following by historical arguments favoring new (preonic) level of matter. The importance of Super-Charm factory and GeV energy proton... 131. Enhancing the LBNF/DUNE Physics Program The Long-Baseline Neutrino Facility (LBNF) offers a unique opportunity for neutrino physics due to the high intensity (anti)neutrino beam with a broad energy spectrum. The possibility to collect unprecedented exposures alleviates one of the primary limitations of past neutrino experiments. An experimental technique has been recently proposed to achieve a control of the configuration, chemical... 83. Ensuring the future of particle physics in a more sustainable world As the European particle physics community engages in an updated strategy for the coming years, we present three recommendations that will ensure a more sustainable future for the field of particle physics in view of climate change. The text and recommendations are signed by 314 particle physicists, representing a diverse set of the community. 39. EPIC: Exploiting the Potential of ISOLDE at CERN The user's community of ISOLDE, CERN's radioactive ion beam (RIB) facility, has been steadily growing in the last 10-15 years, thanks to the increasing range of research fields that opened up when post-accelerated radioactive beams and more isotopes became available. The demand for beam time therefore outnumbers the current production capabilities. The EPIC project takes full advantage of the... 144. European Particle Physics Strategy: Input from UK National Laboratories This document supplements the main scientific input from the UK particle physics community, and sets out the position of the UK national laboratories on points of European collaboration and organisation. 159. Exploring the Energy Frontier with Deep Inelastic Scattering at the LHC The Large Hadron Collider determines the energy frontier of experimental collider physics for the next two decades. Following the current luminosity upgrade, the LHC can be further upgraded with a high energy, intense electron beam such that it becomes a twin-collider facility, in which ep operates concurrently with pp. A joint ECFA, CERN and NuPECC initiative led to a detailed conceptual... 94. FASER: ForwArd Search ExpeRiment at the LHC FASER, the ForwArd Search ExpeRiment, is a proposed experiment dedicated to searching for light, extremely weakly-interacting particles at the LHC. Such particles may be produced in the LHC's high-energy collisions in large numbers in the far-forward region and then travel long distances through concrete and rock without interacting. They may then decay to visible particles in FASER, which... 18. Feasibility Study for an EDM Storage Ring This project exploits charged particles confined as a storage ring beam (proton, deuteron, possibly $^3$He) to search for an intrinsic electric dipole moment (EDM, $\vec d$) aligned along the particle spin axis. Statistical sensitivities can approach $10^{-29}$e$\cdot$cm. The challenge will be to reduce systematic errors to similar levels. The ring will be adjusted to preserve the spin... 41. Further searches of the Higgs scalar sector Electroweak physics (physics of the W, Z, H bosons, of the top quark, and QED) Recent decades have witnessed remarkable confirmations of the Standard Model (SM) describing the Electro-Weak and Strong Interactions. The experimental discovery of the Higgs boson Ho at the CERN/LHC has crowned a success of the SM and calls for further studies on this newly observed sector. New and more precise activities are needed in order to extend more precisely this new discovery... 104. Future Challenges in Particle Physics Education and Outreach This document is meant to serve as input from the IPPOG Collaboration for the open call for the European Particle Physics Strategy Update 2020. It emphasises the strategic relevance of concerted, global outreach activities in particle physics today and beyond when envisaging new large-scale projects. 133. Future Circular Collider - The Hadron Collider (FCC-hh) This report describes a novel research infrastructure, based on a hadron collider with centre-of-mass collision energy of 100 TeV, collecting an integrated luminosity a factor of 5 or more larger than the HL-LHC. It will extend the current energy frontier by almost an order of magnitude. The mass reach for direct discovery will reach several tens of TeV, and allow, for example, to produce new... 136. Future Circular Collider - The High-Energy LHC (HE-LHC) This report contains the description of a novel research infrastructure based on a high-energy hadron collider, which extends the current energy frontier by almost a factor 2 (27 TeV collision energy) and an integrated luminosity of at least a factor of 3 larger than the HL-LHC. In connection with four experimental detectors, this infrastructure will deepen our understanding of the origin of... 135. Future Circular Collider - The Integrated Programme (FCC-int) The most effective and comprehensive approach to thoroughly explore the open questions in modern particle physics is a staged research programme, integrating in sequence lepton (FCC-ee) and hadron (FCC-hh) collision programmes, to achieve an exhaustive understanding of the Standard Model and of electroweak symmetry breaking, and to maximize the potential for the discovery of phenomena beyond... 132. Future Circular Collider - The Lepton Collider (FCC-ee) This report contains the description of a novel research infrastructure based on a highest-luminosity energy frontier electron-positron collider (FCC-ee) to address the open questions of modern physics. It will be a general precision instrument for the continued in-depth exploration of nature at the smallest scales, optimised to study with high precision the Z, W, Higgs and top particles, with... 52. Future colliders - Linear and circular The completion of the Standard Model of particle physics by the discovery of a light Higgs boson at the LHC in 2012 triggered the debate about the best way forward to discover physics beyond the Standard Model. At the eve of the update of the European Strategy of particle physics, this article summarises the motivation and status of the different collider projects in particle physics at the... 62. Future Dark Matter Searches with Low-Radioactivity Argon We present the case for the DarkSide-Argo program for direct dark matter searches with low-radioactivity argon. The immediate objective is the DarkSide-20k two-phase liquid argon detector, currently under construction at the Gran Sasso laboratory (LNGS). DarkSide-20k will have ultra-low backgrounds, with the ability to measure its backgrounds in situ, and sensitivity to WIMP-nucleon cross... 37. Future of Heavy Ion Physics at Colliders the Germany ALICE community points out a future direction with a new nearly massless detector 45. Future Opportunities in Accelerator-based Neutrino Physics This document summarizes the conclusions of the Neutrino Town Meeting held at CERN in October 2018 to review the neutrino field at large with the aim of defining a strategy for accelerator-based neutrino physics in Europe. The importance of the field across its many complementary components is stressed. Recommendations are presented regarding the accelerator based neutrino physics, pertinent... 89. Future strategies for the discovery and the precise measurement of the Higgs self coupling The European Strategy for Particle Physics (ESSP) submitted in 2013 a deliberation document to the CERN council explaining that a lepton collider with "energies of 500 GeV or higher could explore the Higgs properties further, for example the [Yukawa] coupling to the top quark, the [trilinear] self-coupling and the total width.". In view of the forthcoming ESPP update in 2020,... 6. Gamma Factory for CERN This contribution discusses the possibility of creating novel research tools at CERN by producing and storing highly relativistic atomic beams in its high-energy storage rings, and by exciting their atomic degrees of freedom by lasers to produce high-energy photon beams. Their intensity would be, by several orders of magnitude, higher than those of the presently operating light sources, in the... 119. GRAND: Giant Radio Array for Neutrino Detection input for the European Particle Physics Strategy Update 2020 The Giant Radio Array for Neutrino Detection (GRAND) is a planned large-scale observatory of ultra-high-energy (UHE) cosmic particles — cosmic rays, gamma rays, and neutrinos with energies exceeding $10^8$ GeV. Its ultimate goal is to solve the long-standing mystery of the origin of UHE cosmic rays. It will do so by detecting an unprecedented number of UHECRs and by looking with unmatched... 64. GRAVITATIONAL WAVES IN THE EUROPEAN STRATEGY FOR PARTICLE PHYSICS This document briefly describes some of the scientific and technological synergies that are possible between the nascent field of Gravitational Waves (GWs) and High Energy Particle Physics (HEPP). It is submitted by the ET steering committee under the supervision of GWIC-3G (a team of the Gravitational Wave International Committee (GWIC)) as contribution to the European Strategy for Particle... 115. Hadron Physics Opportunities in Europe On November 20 and November 21 this working group has met in order to discuss and summarize future Hadron Physics Opportunities in Europe. The work of this working group is based on the Long Range Plan of the Nuclear Physics European Collaboration Committee from November 27, 2017 (NuPECC is an Expert Committee of the European Science Foundation). Furthermore it is based on a series of four... 46. Heavy-flavour production in relativistic heavy-ion collisions and development of novel generation of extra-low-material-budget Vertex Detectors for future experiments at CERN and JINR One of the key requirements to be met by the future experimental installations like ALICE is to increase the accuracy of secondary vertices reconstruction in order to meet the challenging task of high precision studies in relativistic heavy-ion collisions of such rare processes like heavy-flavour production. This task requires the further reduction of the existing values of material budget... 53. HEP Computing Evolution High Energy Physics (HEP) has demonstrated a unique capability with the global computing infrastructure for LHC, achieving the management of data at the many-hundred-Petabyte scale, and providing access to the entire community in a manner that is largely transparent to the end user. Other HEP experiments have expressed a desire to benefit from this infrastructure and organization, and recent... 44. HFLAV input to the update of the European Strategy for Particle Physics The Heavy Flavor Averaging Group provides with this document input to the European Strategy for Particle Physics. Research in heavy-flavor physics is an essential component of the particle-physics program, both within and beyond the Standard Model. To fully realize the potential of the field, we believe the strategy should include strong support for the ongoing experimental and theoretical... 116. IN2P3 contribution for the update of European Strategy for Particle Physics The "Institut de Physique Nucléaire et de Physique des Particules" (IN2P3) comprises 25 laboratories located in major universities. IN2P3 is in charge of coordinating nuclear, particle and astro-particles physics in France. The number of people working at IN2P3 is 3200, about half of them PhDs. This represents 600 CNRS physicists, 400 professors from universities and 1500 engineers,... 138. INFN National Scientific Committee for High Energy Particle Physics with Accelerators High energy colliders offer the opportunity to widen our investigation of sub-nuclear phenomena to the highest unexplored energy regimes and shortest interaction scales. They consequently have the strongest potential for discovering new heavy particles and forces, allowing more accurate precision tests of Standard Model objects, and detecting new elusive signatures like the ones associated... 21. Initial contribution of the INFN Hadron Physics Community INFN has a strong tradition in high-energy hadron physics, both in the heavy-ion sector and in the deep-inelastic-scattering sector, with important participation in international programmes, that also include relevant and specific contributions in terms of dedicated detectors. In this context, it is recognized that high centre-of-mass energy is a fundamental handle for the future... 59. Initial INFN input on the update of the European Strategy for Particle Physics: software and computing The INFN sees two major areas of application for high energy physics computing in the time scale relevant for the Strategy Update: the exploitation of high performance computing (HPC) and the use of Quantum Computing. The first one can be realized within few years, with high energy physics experiments becoming major users of HPC. Quantum computing is still at the level of research and... 26. Initial INFN input on the update of the European Strategy for Particle Physics This document contains some initial input from INFN to the update of the European Strategy for Particle Physics. It does not aim at providing a comprehensive overview of the INFN position on all the relevant aspects of the Strategy: this will be defined with the other actors all along the update process, from the Open Symposium of May 2019 to the Strategy Drafting Session of January 2020. It... 165. Initial INFN input on the update of the European Strategy for Particle Physics - The Astroparticle Physics Commission 2 This document contains some initial input from the INFN Astroparticle Physics Commission 2 (CSN2) to the update of the European Strategy for Particle Physics. It does not aim at providing a comprehensive overview of the CSN2 activities nor to cover all possible relevant aspects of the Strategy: this will be defined with the other actors all along the update process, from the Open Symposium... 76. Input from J-PARC to the update of the European Strategy for Particle Physics Current research activities of Japan Proton Accelerator Research Complex (J-PARC) and the future prospects are summarized with emphasis on the particle physics experiments. 97. Input from the DARWIN collaboration to the European Strategy for Particle Physics The DARWIN collaboration (www.darwin-observatory.org) is planning to build the "ultimate" underground-based direct detection dark matter detector, with a dark matter sensitivity limited only by irreducible neutrino backgrounds. The core of the detector will have a 40 ton liquid xenon target instrumented as a dual-phase time projection chamber. The large xenon target, the exquisitely low... 166. Input from the Netherlands for the European Strategy for Particle Physics – Update 2020 Input from the Netherlands for the European Strategy for Particle Physics – Update 2020 31. Input from the Spanish Particle Physics Community Input to the update of the European Strategy for Particle Physics provided by the Spanish Scientific Particle Physics community. It contains two files, the main document plus an addendum 80. Input of Joint Institute for Nuclear Research This document summarizes the discussions of representatives of laboratories of the Joint Institute for Nuclear Research (intergovernmental, international organization in Dubna) concerning the European Strategy for Particle Physics Update. The document reflects the forward view of JINR scientists to the development of PP in Europe based on long and successful history of CERN - JINR... 40. Input of Nuclear Physics Section, Division of Physical Sciences of the Russian Academy of Sciences to European Strategy for Particle Physics Update See attached file 61. Input to the ESPP process from the community of Danish high-energy and nuclear experimentalists and theorists. This documents presents the interests and priorities of the Danish high energy and nuclear physics community based on the town meeting held by the National Center for CERN research (NICE) in June 2018 in Middelfart, DK. 130. Input to the Strategy Process from LNF-INFN Input to the Strategy Process from LNF-INFN 88. Inputs to European Strategy Update 2018-2020 by the Czech particle physics community Although the Standard Model has been very successful in predicting and interpreting current measurements in particle physics, it has become clear that it cannot answer all the outstanding questions. To resolve the remaining issues new theories have been developed and further measurements are needed. The experience shows that diverse and complementary scientific program is the right approach to... 34. Israeli Input to the European Strategy for Particle Physics European Strategy for Particle Physics, Israeli Input Z. Citron, Y. Kats (BGU), E. Kuflik (HU), L. Barak, T. Volansky (TAU), E. Kajomovitz, Y. Shadmi (Technion), S. Bressler, G. Perez (WIS, coordinator) The enclosed document is based on a compilation of theoretical concepts and experimental methods solicited from the Israeli community by a steering committee with... 63. Japan's Updated Strategy for Future Projects in Elementary Particle Physics In September 2017, the Japanese high energy physics community, JAHEP (Japan Association of High Energy Physicists), updated the strategy for future particle physics by endorsing the Final Report of the Committee on Future Projects in High Energy Physics. The Final Report summarizes Japan's future strategy as follows: In 2012, not only was a Higgs boson with a mass of 125 GeV discovered... 153. KLEVER: An experiment to measure BR(KL -> pi0 nu anti-nu) at the CERN SPS Precise measurements of the branching ratios for the flavor-changing neutral current decays $K\to\pi\nu\bar{\nu}$ can provide unique constraints on CKM unitarity and, potentially, evidence for new physics. It is important to measure both decay modes, $K^+\to\pi^+\nu\bar{\nu}$ and $K_L\to\pi^0\nu\bar{\nu}$, since different new physics models affect the rates for each channel differently. The... 30. Large-scale neutrino detectors: input for the 2020 update of the European Strategy for Particle Physics from the Institute for Nuclear Research of the Russian Academy of Sciences We propose a multi-purpose neutrino observatory comprising two very large detectors solving different problems at the intersection of particle physics, astrophysics and Earth science. Baikal-GVD will work jointly with KM3NET and IceCube in the Global Neutrino Network, aiming at the detection and study of high-energy astrophysical neutrinos. The new Baksan neutrino telescope (NBNT) will inherit... 85. LPNHE scientific perspectives for the European Strategy for Particle Physics This note summarizes the activities and the scientific and technical perspectives of the Laboratoire de Physique Nucleaire et de Hautes Energies (LPNHE) at Sorbonne University, Paris. Although the ESPP is specifically aimed at particle physics, we discuss in this note in parallel the three scientific lines developed at LPNHE (Particle Physics, Astroparticles, Cosmology), first with the current... 161. MAGIS-1K: A 1000 m Atom Interferometer Device for Searches in Dark Matter and Gravity Waves MAGIS-1K: A 1000 m Atom Interferometer Device for Searches in Dark Matter and Gravity Waves Jonathon Coleman, University of Liverpool, Merseyside, L69 7ZE, UK. On behalf of the MAGIS-100 collaboration This document is responding to the call for input collaboration to the update of the European Strategy for Particle Physics from the MAGIS collaboration. Atom interferometry can be... 75. MATHUSLA The observation of long-lived particles at the LHC would reveal physics beyond the Standard Model could account for the many open issues in our understanding of our universe and conceivably point to a more complete theory of the fundamental interactions. Such long-lived particle signatures are fundamentally motivated and can appear in virtually every theoretical construct that addresses the... 114. Monte Carlo event generators for high energy particle physics event simulation Monte Carlo event generators (MCEGs) are the indispensable workhorses of particle physics, bridging the gap between theoretical ideas and first-principles calculations on the one hand, and the complex detector signatures and data of the experimental community on the other hand. All collider physics experiments are dependent on simulated events by MCEG codes such as Herwig, Pythia, Sherpa,... 120. Muon Colliders Muon colliders have a great potential for high-energy physics. They can offer collisions of point-like par- ticles at very high energies, since muons can be accelerated in a ring without limitation from synchrotron radiation. However, the need for high luminosity faces technical challenges which arise from the short muon lifetime at rest and the difficulty of producing large numbers of muons... 124. Neutrino Beam from Protvino to KM3NeT/ORCA The Protvino accelerator facility located in the Moscow region, Russia, is in a good position to enable a rich experimental research program in the field of neutrino physics. Of particular interest is the possibility to direct a neutrino beam from Protvino towards the KM3NeT/ORCA detector which is currently under construction in the Mediterranean sea 40 km offshore Toulon, France. Such an... 151. New physics searches with heavy-ion collisions at the LHC This document summarizes proposed searches for new physics accessible in the heavy-ion mode at the LHC, both through hadronic and ultraperipheral $\gamma\gamma$ interactions, and that have a competitive or, even, unique discovery potential compared to standard proton-proton collision studies. Illustrative examples include searches of new particles --- such as axion-like pseudoscalars, radions,... 2. Newtonian Test of the Standard Model Isaac Newton's book 'Opticks' from the 18th century includes a number of hypotheses on the structure of matter. Most of the hypotheses were confirmed during the 19th and 20th centuries at the scales of nucleons, nuclei, atoms, molecules and macromolecules. Conflicts appear however at the scale of quarks and gluons according to the Standard Model of particle physics. The confirmations at the... 148. Nuclear physics and the European Particle Physics Strategy Update 2018 This document provides input to the update of the European Strategy for Particle Physics in fields that are related to Nuclear Physics as described in the NuPECC Long Range Plan 2017. 93. Nuclotron-based Ion Collider Facility at JINR (NICA Complex) NICA Application; NICA Addendum 154. nuSTORM at CERN: Executive Summary The Neutrinos from Stored Muons, nuSTORM, facility has been designed to deliver a definitive neutrino-nucleus scattering programme using beams of (anti-)electron and (anti-)muon neutinos from the decay of muons confined within a storage ring. The facility is unique, it will be capable of storing muon beams with a central momentum of between 1 GeV/c and 6 GeV/c and a momentum spread of 16\%.... 95. On the Prospect and Vision of Ultra-High Gradient Plasma Accelerators for High Energy Physics Plasma accelerators generate accelerating fields that are up to 1,000 times higher than fundamentally possible in RF accelerators. They therefore offer a promising alternative path to the high-energy frontier. In 2012 the European Strategy Preparatory Group received for the first time detailed input about the prospects and promise of plasma accelerators, a 15 page report provided by the... 158. Opportunities in Accelerator-based Neutrino Physics in Japan This document provides the inputs to the ongoing European Particle Physics Strategy based on the opportunities offered by the accelerator-based neutrino programme in Japan to which the 2013 European Strategy recommended due to the "Rapid progress in neutrino oscillation physics, with significant European involvement", "CERN should develop a neutrino programme to pave the way for a substantial... 121. Particle Physics and Related Topics at the Paul Scherrer Institute We highlight the specific and unique opportunities that PSI offers for particle physics. The input outlines mid- and longterm projects and asks for support and cooperation of the international community. It exemplifies the benefits of the particle physics activities for other fields and also the return from developments elsewhere into the particle physics program. Sharing of facilities, e.g.,... 50. Particle physics applications of the AWAKE acceleration scheme The AWAKE experiment had a very successful Run 1 (2016-8), demonstrating proton-driven plasma wakefield acceleration for the first time, through the observation of the modulation of a long proton bunch into micro-bunches and the acceleration of electrons up to 2 GeV in 10 m of plasma. The aims of AWAKE Run 2 (2021-4) are to have high-charge bunches of electrons accelerated to high energy,... 86. PARTICLE PHYSICS AT PIK REACTOR COMPLEX The Standard Model provides estimations on neutron EDM (nEDM) value on the level inaccessible for the modern experiment: $10^{–30} – 10^{–33}$ е$\cdot$сm. CP-violation (and nEDM) arises only in the second order of smallness on the weak interaction constant. SМ fails to account for baryon asymmetry of Universe. Search for nEDM is expected to be search for some phenomena beyond the... 55. Particle Physics in Finland Abstract: In the report current plans and some ideas for future are reported for particle physics and related fields in Finland. 20. PBC Conventional Beams Executive Summary This document summarises the main conclusions of the Conventional Beams Working group, which has analysed the beam related and technical requirements and requests in the proposals to the Physics Beyond Colliders study for the North Area at the CERN SPS. We present results from studies on feasibility, requirements, compatibility between proposals and, where possible, the order of magnitude of... 60. PBC technonlogy subgroup report This document contains results of the work of the PBC technology subgroup set up by the Physics Beyond Collider Working Group. Goal of the technology subgroup set by PBC: Exploration and evaluation of possible technological contributions of CERN to non-accelerator projects possibly hosted elsewhere: survey of suitable experimental initiatives and their connection to and potential... 147. PERLE : A High Power Energy Recovery Facility for Europe The efficient recovery of power, to re-excite cavities from the used beam, was proposed in 1965. Major advances in superconducting RF technology, as quantified by cavity quality factors Q0 in excess of 1010, and the consideration of multi-turn recirculator passages, have opened the door to the green generation of high energy, high brightness, high current electron beams. The facility PERLE,... 47. Physics opportunities for a fixed-target programme in the ALICE experiment A fixed-target programme in the ALICE experiment using the LHC proton and lead beams offers many physics opportunities related to the parton content of the nucleon and nucleus at high-x, the nucleon spin and the Quark-Gluon Plasma. We investigate two solutions that would allow ALICE to run in a fixed-target mode: the internal solid target coupled to a bent crystal and the internal gas target.... 125. Polish input to the Update of European Strategy for Particle Physics 2020-2025 This is a position paper which aims at describing the current status and plans of Polish PP research community for the coming Update of ESPP 2020. The document does not discuss some subject closely related to PP in Poland like the Polish research in Astroparticle Physic and Nuclear Physics, and education, outreach and technology transfer issues. The worldwide activities of PP in the... 100. Precision calculations for high-energy collider processes For the field of particle physics to benefit from the immense body of data from present and possible future colliders, it is essential to be able to accurately relate the experimental observations to the underlying Lagrangian. This is the case whether searching for physics beyond the Standard Model, exploring the many facets of the Higgs sector, or pinning down fundamental parameters such as... 49. Precision experiments at electron-positron collider Super Charm-Tau Factory This document describes research program of Budker INP (Novosibirsk) on high energy physics for the next two decades based on the flagship project of the electron-positron collider Super Charm-Tau (SCT) factory. The SCT factory is designed to operate in the center-of-mass energy range from 2 to 6 GeV with peak luminosity of 10**35 cm-2s−1 above 4 GeV. Longitudinal polarization of the electron... 13. Proposal from the NA61/SHINE Collaboration for the update of the European Strategy for Particle Physics Based on the success of the currently running program and motivated by new physics needs the NA61/SHINE Collaboration proposes to continue the measurements of hadron and nuclear fragment production properties in reactions induced by hadron and ion beams after the CERN Long Shutdown 2 (LS2). These measurements are requested by heavy ion, cosmic ray and neutrino communities and they include:... 92. PROSPECT OF THE IN2P3 COMMUNITY INVOLVED IN THE ILC PROJECT A large community of the French national funding agency IN2P3/CNRS has been in- volved since two decades in the developments addressing a linear electron-positron collider at the energy frontier, within an international collaboration involving nearly 100 laboratories around the world. This long term involvement has progressively generated a concrete project named International Linear Collider... 9. Prospects for exploring the Dark Sector physics and rare processes with NA64 at the CERN SPS The CERN SPS offers a unique opportunity for exploring new physics due to the availability of high-quality and high-intensity secondary beams. In the 2016-18 runs, the NA64 experiment has successfully performed sensitive searches for Dark Sector and other rare processes in missing energy events using high energy electron interactions in an active dump. The NA64 Collaboration plans to... 163. Quantum Chromodynamics: Theory - Input for the European Particle Physics Strategy Update This contribution highlights the role of QCD theory in studying the physics of the Standard Model and beyond, as well as opportunities and challenges, as an input to the preparation of the European Particle Physics Strategy Update (EPPSU). 128. Quantum Computing for High Energy Physics High Energy Physics (HEP) is once again facing a "requirement wall" in all dimensions of Information and Communication Technology (ICT). Considering the current estimations of computing needs, the constrained budget that will be devoted to HEP computing, and the projected evolution of the actual capacity delivered by technology to HEP computing, a shortage of an order of magnitude in the next... 28. REDTOP: Rare Eta Decays with a TPC for Optical Photons The REDTOP experiment is primarily intended to look for new violations of the basic symmetries. It aims to improve the sensitivity level of key physics conservation laws by several orders of magnitude beyond those of previous experiments. In doing so, it will open doorways for possible Physics Beyond the Standard Model including dark matter and energy, and/or new forces. The REDTOP... 106. Research and Development for Near Detector Systems Towards Long Term Evolution of Ultra-precise Long-baseline Neutrino Experiments. With the discovery of non-zero value of $\theta_{13}$ mixing angle, the next generation of long-baseline neutrino (LBN) experiments offers the possibility of obtaining statistically significant samples of muon and electron neutrinos and anti-neutrinos with large oscillation effects. In this document we intend to highlight the importance of Near Detector facilities in LBN experiments to both... 43. Research Plans of the Norwegian Particle, Astroparticle and Nuclear Physics Communities till 2025 Norwegian particle physics, heavy-ion physics, nuclear physics, astroparticle physics, particle, astroparticle and cosmology theory present their research program for the next period of the European Strategy Update. 108. Research Strategy of the Austrian Particle Physics Community In the following, Austria's particle physicists outline their current research program on particle physics and define their visions and plans. The overarching goal is to contribute to the understanding of known particle physics as well as the discovery of physics beyond the standard model and its theoretical understanding. This is achieved through Austria's involvement in major international... 73. Romanian input to the European Particle Physics Strategy Update 2018-2020 The document represents a synthesis of the input provided by research groups from Romanian National Institutes and Universities which are currently participating in the CERN Scientific Programme. The Romanian groups are focused on key areas of particle physics with large discovery potential, precision measurements and searches for new physics. Development of new detectors or performance... 1. Searches for Sterile Neutrinos at CERN: Contribution to European Strategy for Particle Physics These are some suggestions concerning ongoing and future tests that can be performed at CERN to search for possible sterile neutrinos and set constraints on their masses and mixings. In addition to European participation in neutrino oscillation experiments, in searches for neutrinoless double beta decay, and in KATRIN, it is emphasized here that sensitive searches for sterile... 78. Slovenian input to the European Strategy for Particle Physics Update 2018–2020 This document reflects the view of the Slovenian HEP community on its engagement in current and future particle physics projects. It represents the input of a relatively small particle physics community to the European Strategy for Particle Physics. 129. SPS Beam Dump Facility The proposed Beam Dump Facility (BDF) is foreseen to be located at the North Area of the SPS. It is designed to be able to serve both beam dump like and fixed target experiments. The SPS and the new facility would offer unique possibilities to enter a new era of exploration at the intensity frontier. Possible options include searches for very weakly interacting particles predicted by Hidden... 69. Statement by the German Astroparticle Physics Community as input to the European Strategy for Particle Physics The German Committee for Astroparticle Physics, KAT, is the elected representation of all astroparticle physicists working at German research institutions of the Helmholtz Association, Max Planck Society and at German universities. This paper comprises the statements of the German astroparticle physics community to the 2020 update of the European Strategy for Particle Physics. The three... 33. Statement by the German Particle Physics Community as Input to the Update of the European Strategy for Particle Physics The German Committee for Particle Physics (KET), arranged, jointly with the German Committees for astroparticle physics (KAT) and for hadronic and nuclear physics (KHuK), a series of workshops to discuss status and future plans of particle physics and neighbouring fields. KET has extracted central statements and strategic proposals from the joint declaration and hereby submits them to the 2020... 117. Statement of the Pierre Auger Collaboration as input for the European Particle Physics Strategy Update 2018 - 2020 There is large overlap in the research interests of the European particle physics community and the Pierre Auger Collaboration and there is a remarkable complementarity of the measurement principles and information accessible at accelerators and the Auger Observatory. With this statement, we provide input to the European Particle Physics Strategy Update 2018 - 2020. We strongly encourage very... 17. Status and perspectives of the neutron time-of-flight facility n_TOF at CERN Since the start of its operation in 2001, based on an idea of Prof. Carlo Rubbia[1], the neutron time-of-flight facility of CERN, n_TOF, has become one of the most forefront neutron facilities in the world for wide-energy spectrum neutron cross section measurements. Thanks to the combination of excellent neutron energy resolution and high instantaneous neutron flux available in the two... 167. Status of Fermilab's Neutrino Facilities Fermilab is the only laboratory in the world that operates two accelerator-based neutrino beams simultaneously. These intense neutrino sources enable an important collection of experi- ments that are studying neutrinos over both short and long distances, allowing the experiments to address questions such as the neutrino mass ordering, whether additional (sterile) neutrinos exist, and whether... 96. STFC input to the "European Strategy for Particle Physics" related to Innovation/Technology Transfer, Knowledge Exchange The infrastructure and research related to Particle Physics plays an important role in creating a more sustainable society by actively pushing the frontiers of human understanding. Whilst positively contributing to the EU's social and economic growth through innovation and Knowledge Transfer. In this short paper, STFC outlines its innovation strategy, in the context of Particle Physics, and... 16. Strategic R&D Programme on Technologies for Future Experiments Instrumentation is a key ingredient for progress in experimental high energy physics. The Experimental Physics Department of CERN has defined a strategic R&D (Research and Development) programme on technologies for future experiments. Provided the required resources can be made available, it will start in 2020 and initially extend over five years. The selection of topics and the established... 90. Study of hard and electromagnetic processes at CERN-SPS energies: an investigation of the high-$\mu_{\mathbf{B}}$ region of the QCD phase diagram The exploration of the phase diagram of Quantum ChromoDynamics (QCD) is carried out by studying ultrarelativistic heavy-ion collisions. The energy range covered by the CERN SPS ($\sqrt{s_{\rm \scriptscriptstyle{NN}}} \sim 6\text{--}17$~GeV) is ideal for the investigation of the region of the phase diagram corresponding to finite baryochemical potential ($\mu_{\rm B}$), and has been little... 127. Submission from the Swedish Particle Physics Community to the European Strategy for Particle Physics Update 2018-2020 A national submission from the particle physics community (experiment and theory) in Sweden is given. The status and plans of the community are described and are used to inform a set of recommendations to the Strategy Update. In addition to recommendations of direct relevance to ongoing and planned Swedish research, the community's views on proposed major facilities and the general direction... 82. Submission to the European Strategy from University of Liverpool Experimental Particle Physics Group The Particle Physics Group at the University of Liverpool has prepared this submission to the European Strategy Update. This was performed as a consultation process across the whole group including academics, research scientists, engineers, and research associates. 142. Swiss input for the discussion on the European Strategy for Particle Physics The position of the Swiss particle physics community towards the update for the European Strategy for Particle Physics compiled by the Swiss Institute for Particle Physics (CHIPP). 99. Synergies between a U.S.-based Electron-Ion Collider and the European research in Particle Physics This document is submitted as input to the European Strategy for Particle Physics Update (ESPPU). The U.S.-based Electron-Ion Collider (EIC) project recently received strong endorsement by the U.S. National Academies of Sciences, Engineering, and Medicine, bringing its realization another step closer. A large group of European scientists is already involved in the EIC project. Currently, more... 91. Synthesis on the ELN Project by A.Zichichi 102. TauFV: a fixed-target experiment to search for flavour violation in tau decays TauFV is a fixed-target experiment designed to search for Lepton Flavour Violation in tau decays, utilising a high energy, high intensity proton beam impinging on a distributed system of targets. In the baseline proposal, TauFV will be situated on the high-intensity beam line the Beam-Dump Facility of the SPS, upstream of the SHiP experiment. Its performance will rely on advanced detector... 103. The "DIS and Related Subjects" Strategy Document: Fundamental Science from Lepton-Hadron Scattering The diverse community of scientists involved in Deep Inelastic Scattering includes about 2000 experi- mental and theoretical physicists worldwide and envisages projects such as the EIC, LHeC, FCC-eh and VHEeP as future lepton-hadron scattering facilities. The proposed facilities will address fundamental ques- tions in strong interaction / QCD physics, including a first-ever tomographic mapping... 11. The Belle II experiment at SuperKEKB: input to the European Particle Physics Strategy (update 2018-2020) In the present document we outline some reflections related to the next mid-term particle physics strategy: the need for complementary approach among various frontiers in searches and interpretations of NP; an experimental initiative in the aforementioned efforts, especially after the Higgs boson discovery at the LHC; the specific advantages of $e^+e^-$ over hadron colliders; and the... 24. THE BIENNIAL AFRICAN SCHOOL ON FUNDAMENTAL PHYSICS AND APPLICATIONS We have established a biennial school in Africa, on fundamental physics and its applications (ASP). We find that fundamental physics provides excellent motivation for students of science. The aim of the school is to build capacity to harvest, interpret, and exploit the results of current and future physics experiments and to increase proficiency in related applications. The school is based on... 146. 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\begin{document} \title[Perturbation of Lyapunov-Subcenter-Manifolds]{Global Persistence of Lyapunov-Subcenter-Manifolds as Spectral Submanifolds under Dissipative Perturbations} \begin{abstract} For a nondegenerate analytic system with a conserved quantity, a classic result by Lyapunov guarantees the existence of an analytic manifold of periodic orbits tangent to any two-dimensional, elliptic eigenspace of a fixed point satisfying nonresonance conditions. These two dimensional manifolds are referred to as Lyapunov Subcenter Manifolds (LSM).\\ Numerical and experimental observations in the nonlinear vibrations literature suggest that LSM's often persist under autonomous, dissipative perturbations. These perturbed manifolds are useful since they provide information of the asymptotics of the convergence to equilibrium. \\ In this paper, we formulate and prove precise mathematical results on the persistence of LSMs under dissipation. We show that, for Hamiltonian systems under mild non-degeneracy conditions on the perturbation, for small enough dissipation, there are analytic invariant manifolds of the perturbed system that approximate (in the analytic sense) the LSM in a fixed neighborhood. We provide examples that show that some non-degeneracy conditions on the perturbations are needed for the results to hold true. \\ We also study the dependence of the manifolds on the dissipation parameter. If $\varepsilon$ is the dissipation parameter, we show that the manifolds are real analytic in $(-\varepsilon_0, \varepsilon_0) \setminus \{0\} $ and $C^\infty $ in $(-\varepsilon_0, \varepsilon_0)$. We construct explicit asymptotic expansions in powers of $\varepsilon$ (which presumably do not converge).\\ Finally, we present applications of our results to a mechanical systems. \end{abstract} \author[R.de la Llave]{Rafael de la Llave} \thanks{R.L. Supported in part by NSF grant DMS-1800241} \address{School of Mathematics, Georgia Institute of Technology, Atlanta, Georgia 30332-0160} \email{[email protected]} \author[F. Kogelbauer]{Florian Kogelbauer} \address{Institute for Mechanical Systems, ETH Z\"urich, Leonhardstrasse 21, 8092 Z\"{u}rich, Switzerland} \email{[email protected]} \keywords{Lyapunov-Subcenter-Manifold, Spectral Submanifold, Perturbative Analysis, Singular pertubations, Slow manifolds} \maketitle \today \section{Introduction} As shown first by Lyapunov \cite[\S 42 p. 376]{liapounoff1907probleme}, given an analytic ODE with a non-degenerate conserved quantity (for example, a Hamiltonian system), near a fixed point whose linearization contains a pair of complex conjugate imaginary eigenvalues which do not resonate with the other eigenvalues (see precise definition later), we can find a one dimensional family of periodic orbits (hence a two dimensional invariant manifold) tangent at the origin to the eigenspace corresponding to the the pair of imaginary eigenvalues. \\ One can think of these families of periodic orbits as a nonlinear analogue of the periodic orbits predicted by the linearization, i.e., the harmonic oscillator. These two-dimensional manifolds are often referred to as\textit{ Lyapunov subcenter manifolds} (LSMs) because they are submanifolds of the full center manifold of the fixed point.\\ The proof of \cite{liapounoff1907probleme}, is based on constructing perturbative series starting at the origin and then showing that they converge. The notation of \cite{liapounoff1907probleme} may require some effort for modern readers and the formal statement is only for Hamiltonian systems. See \cite{kelley1967liapounov, MoserZ, MeyerH} or later in this paper for modern proofs based on the elementary implicit function theorem.\\ More modern proofs for Hamiltonian systems based on estimating series can be found in \cite{Moulton20}, \cite[II \S16 p. 104]{SiegelM} or on normal form theory \cite{kelley1969analytic}. The paper \cite{moser1958generalization} contains a generalization of the convergence of the normal forms in the saddle-center case. Results (based on variational methods) on the existence of periodic orbits that do not require non-resonance assumptions but assume positive definiteness on the conserved quantity appear in \cite{weinstein73, moser1976periodic}. The papers \cite{Duistermaat1972, SchmidtS73} use averaging methods to prove generalizations of Lyapunov theorems for resonant systems. A useful review of these results, along with further material, is in \cite{sijbrand1985properties}. Most of the results above on families of quasi-periodic solutions work for systems with finite differentiability.\\ None of the above results, however, apply under the addition of the slightestdissipative perturbation. Such small dissipative perturbations are present in applications to mechanical vibrations. Perturbation arguments based on normal hyperbolicity for the continuation of LSMs as invariant sets are also inapplicable, given that LSMs are not normally hyperbolic. The addition of a small dissipation to a system is a very singular perturbation since even the smallest perturbation destroys all periodic orbits completely.\\ There is, however, a strong indication that remnants of LSMs continue to organize the dynamics of mechanical systems under the addition of small damping and forcing to their conservative limits in a domain whose size is independent of the perturbation. Specifically, dissipative backbone curves (observed periodic response amplitudes near resonances plotted as a function of an external forcing frequency) of such systems are virtually indistinguishable from their conservative backbone curves (amplitudes of periodic orbits in LSMs plotted as a function of their frequency) for small enough damping. The paper \cite{TOUZE2006958} illustrate this relationship numerically, but also shows that conservative and dissipative backbone curves start deviating noticeably from each other for larger damping values.\\ An experimental technique, the \emph{force appropriation method} \cite{PEETERS2011486, PEETERS20111227} is directly based on the observation that a periodic orbit on an LSM survives unchanged when an external forcing is selected to cancel out damping exactly along the orbit. The paper \cite{PEETERS2011486, PEETERS20111227} demonstrate that close enough to the fixed point, such a forcing can be approximately constructed for small enough linear damping. This provides an intuitive explanation for the observed closeness of conservative and dissipative backbone curves under small linear damping and harmonic forcing, at least near the unforced equilibrium. \\ Another experimental technique, the \emph{resonance decay method}, cf. \cite{KERSCHEN2006505} and \cite{PEETERS2011486}, also applies periodic external forcing to the lightly damped conservative system, then tunes the forcing frequency to reach a locally maximal amplitude for the system. At this frequency, the forcing is subsequently turned off, and the instantaneous amplitude-frequency diagram of the resulting decaying oscillations is taken to be as an approximation of the backbone curve of the conservative system. Again, this approach implicitly assumes that after the forcing is turned off, solutions evolve close to an LSM of the conservative system. \\ Independent of these developments, the recent theory of \emph{spectral submanifolds} (SSMs) offers an extension of the LSM concept to dissipative systems, cf. \cite{Haller2016, Kogelbauer2018, Szalai20160759}. Inspired by the nonlinear normal mode concept of \cite{SHAW1994319} and based on the abstract invariant manifold results of \cite{de1997invariant, Cab2003pam,Cab2003,CABRE2005444}, the theory of SSMs guarantees the existence of a unique analytic, two-dimensional invariant manifold tangent to any two-dimensional, nonresonant eigenspace of a linearly asymptotically stable fixed point. Such eigenspaces arise from two-dimensional center subspaces of conservative systems under small dissipative perturbations. The question we address in this paper is whether indeed LSM continue into SSMs.\\ A mathematically trivial (and not very interesting physically) theory of persistence of LSMs is to observe that, if the eigenvalue pair corresponding to the LSM, becomes dissipative, it has to remain non-resonant. Then, we can obtain a SMM because of the theory of \cite{de1997invariant, Cab2003, CABRE2005444}. Since the coefficients of the expansion are obtained recursively by algebraic operations, we obtain easily that the Taylor coefficients of the SSM converge to that of the LSM as the dissipation goes to zero.\\ The physical shortcomings of the simple theory explained above come from the observation that, since the contraction along the manifold is becoming weaker as the dissipation becomes weaker, the theory of \cite{de1997invariant, Cab2003,CABRE2005444} only guarantees the existence of a manifold whose size goes to zero as the dissipation vanishes. The physical usefulness of manifolds whose size goes to zero with the dissipation is rather tenuous. See also Remark~\ref{rem:easy}.\\ Therefore, the question we address in this paper is the existence of SMMs whose size is independent of the value of the dissipation parameter and which converges to the LSM in a fixed domain in the sense of convergence of analytic manifolds. We present two results. Assuming that the system is Hamiltonian, the first results shows in great generality that there are asymptotic expansions in powers of the dissipation of invariant manifolds. In the second result,also for Hamiltonian systems and under some further assumptions in the first order perturbation, we show that there are indeed invariant manifolds of the system which, when the dissipation goes to zero, approximate (in the sense of real analytic manifolds) the LSM in a neighborhood of fixed size independent of the dissipation.\\ We will also present examples that show that some version of the non-degeneracy assumptions we make are necessary. This seems to be in accordance to the physical experiments.\\ in the proof, we will see that -- as customary in singular perturbation theory -- one needs to make assumptions on the leading effects of the perturbation. In our case, the assumptions are rather mild, but we show examples that show that without any assumptions, we could have that the domains of the analytic SMM decreases to zero as the dissipation vanishes (similar results happen for manifolds of finite order regularity). \\ In this paper we will concentrate specially in situations where the unperturbed manifold is not hyperbolic. The perturbed manifolds that arise in our construction will be slow manifolds that violate the standard rate conditions of Normally Hyperbolic manifolds (NHIM) in \cite{Fenichel74, Mane78} and therefore, their persistence properties are not based on the theory of Normally Hyperbolic Invariant Manifolds. They use essentially the fact that the manifolds are attached to a fixed point. The mathematical theory of slow manifolds and, a fortiori, the theory of SSMs, is very subtle. Before the mathematical theory was settled, several of the complications and puzzlements \footnote{Many of the puzzles on the theory of the slow manifolds arise from the fact that some uniqueness results are based on long term behavior and others are based on regularity near the origin. The two conditions used to produce uniqueness lead to different manifolds. In particular, the slow manifolds of \cite{Fraser98,MaasP92} are, generically different since the former is based in smooth expansions at the origin and the later in expansions on asymptotic behavior at infinity. Both of them are unique under the appropriate conditions, which are, however incompatible in many systems. See the discussion of this point in \cite{de1997invariant}.} the difficulties of the theory of slow manifolds were mapped out lucidly in \cite{lorenz1986existence,lorenz1987nonexistence,lorenz1992slow}.\\ Since we will have to rely on the theory of subcenter manifolds, we also refer to the pioneering papers on slow manifolds \cite{de1995irwin, irwin1980new} and \cite{poschel1986invariant} for invariant submanifolds of center manifold under nonresonance conditions (which are not satisfied by Hamiltonian systems). We refer to \cite{CABRE2005444} for a more extensive review of the literature on slow invariant manifolds up to 15 years ago.\\ Other similar singular perturbation theories have been considered in the literature. Indeed, we will use a method similar to those used in \cite{calleja2017domains}, for small dissipations proportional to velocity and \cite{calleja2012construction, calleja2017response}, for strong dissipation and forcing. The techniques used in small dissipation problems are closely related to the techniques used for parabolic manifolds, which can be considered heuristically as systems with an infinitesimal dissipation. See, for example, \cite{baldoma2004exponentially} or \cite{baldoma2007parameterization} which, as this paper, is based on the parametrization method. In those papers (as in the present one) the technique is to first develop a formal perturbation theory that produces an approximate solution, then develop an a-posteriori theory that produces a true solution. In our case, this technique is crucial to obtain results in neighborhoods of uniform size. In the theoretical results, the approximate solutions in the a-posteriori theorem are those produced by the asymptotic expansions, but one could take as well the solutions produced by a numerical method. \subsection{Organization of the paper} \label{sec:organization} n section 2, we recall the classical results on the LSM, including their proves. We also present a special coordinate system for Hamiltonian systems, that will prove useful in the later calculations.\\ Sections 3 and 4 contain the new results of the paper. An informal statement of our main result, Theorem \ref{mainThm}, is included in Section~\ref{sec:statement}. The formal result is included as Theorem~\ref{mainprop}. The proof of Theorem~\ref{mainprop} is completed in two steps in Section~\ref{sec:proof}. \\ In a first step, taken up in Section~\ref{sec:approximate}, we obtain formal asymptotic expansions for the SSM. These expansions may be of practical interest since they give approximations of the manifolds and of the dynamics up to order $\|\varepsilon\|^{N+1}$ ($\varepsilon$ being a parameter that measures the strength of the dissipation) domains of size ${\mathcal O}(1)$. These expansions give very detailed information on the convergence to equilibrium under weak dissipation.\\ The second step of the proof of Theorem~\ref{mainprop}, taken up Section~\ref{sec:fixedpoint}, shows that near prediction of the formal expansions there is a true SSM. This is obtained by reformulating the problem of invariance as a fixed-point problem for a functional acting an appropriate function space. Even if the contraction is weak, we can obtain a fixed point starting the iterative step from the approximate solution obtained in the first stage.\\ In Section~\ref{sec:examples}, we provide some examples that show that some of our assumptions on the survival of LSMs as SSMs cannot be completely omitted. We present examples where there is no analytic (or even differentiable of high order) convergence in domains of size ${\mathcal O}(1)$.\\ Finally, in Section~\ref{sec:applications}, we illustrate our results on a concrete mechanical example. \subsection{Notation} \def{\tilde B}^{n}_\delta{{\tilde B}^{n}_\delta} Let \begin{equation} B_{\delta}^{n}=\{x\in{\mathbb R}^{n}:\,\|x\|<\delta\}, \end{equation} denote the $n$-dimensional ball of radius $\delta$ around the origin of ${\mathbb R}^n$. For a sufficiently small $\tau > 0$, we will denote \begin{equation}\label{deftau} {\tilde B}^{n}_\delta = \{x \in {\mathbb C}^{n}\| d(x, B_\delta^{n}) < \tau\}, \end{equation} where, here, $d$ is the standard Euclidean distance. In the following, we will not write explicitly the parameter $\tau$ to avoid cluttering the notation.\\ Since the proofs we present will be based on soft methods (the implicit function theorem and contraction mappings in function spaces), the arguments for real values carry over to complex values as well. Of course, we need to take care of making sure that the domains match.\footnote{Even if the complex values of variables of parameters may not have a direct physical interpretation, they are indispensable to discuss analyticity properties. Of course, the physically relevant real values are particular cases of the complex ones and the results stated for complex sets apply to the real sets of physical interest. We will assume that the functions, even though they are defined for complex values, give real results for real arguments.}\\ For any Lipschitz continuous function $f:U\to{\mathbb R}^n$, defined on some open subset $U\subseteq{\mathbb R}^m$, we will denote its Lipschitz constant on $U$ as $\Lip(f)$.\\ For a matrix $A$, denote the full spectrum of $A$ by $\sigma(A)$. For a collection of eigenvalues $\lambda_{1},...,\lambda_{n}$ of an $n\times n$ matrix $A$, we denote the (generalized) eigenspace associated with $\lambda_{1},..,\lambda_{k}$ as \begin{equation} \mathrm{eig}(\lambda_{1},...,\lambda_{j}):= {\rm Span} \{v\in{\mathbb C}^{n}:(A-\lambda_kI)^lv=0\text{ for }1\leq k\leq n\text{ and some } l\geq 1\},\label{specss} \end{equation} which we will call a \textit{spectral subspace}. We write $\\|A\\|$ for the operator norm of $A$.\\ As it is well known, when $A$ is a real matrix and the sets of eigenvalues contains the complex conjugates of all the its members, i.e. $\lambda_j^* \in \{ \lambda_m\}_{m = 1}^n$, then $\mathrm{eig}(\lambda_1, \cdots, \lambda_k)$ restricts to a real subspace of ${\mathbb R}^n$.\\ For two functions $f,g:{\mathbb R}^{n}\to{\mathbb R}$, we write $f(x)=\mathcal{O}(g(x))$ if there exists a constant $C>0$ such that $\|f(x)\|\leq C\|g(x)\|$ in a neighborhood of $x = 0$.\\ Let \begin{equation}\label{Cdomain} {\mathcal C}_{\theta}=\{\varepsilon\in{\mathbb C} \|\, \Re(\varepsilon) > 0\|, \, \|\Im(\varepsilon)\|<\theta \|\Re(\varepsilon)\|\}, \end{equation} for some $\theta>0$, be a cone of complex numbers with width $\theta$\footnote{The domain ${\mathcal C}_{\theta}$ will play an important role in some of our results. The formal expansions in the dissipation parameter will be valid in domains of the form ${\mathcal C}_{\theta}$. Note that the domains ${\mathcal C}_{\theta}$ do not contain any ball centered at the origin, so that we do not show that the expansions converge.}. \subsection{Spaces of functions} Our upcoming contraction mapping arguments will require a careful definition of function spaces and norms. Specifically, we define for analytic functions $K:{\tilde B}^{n}_\delta\to{\mathbb C}^n$ with $D^jK(0)=0, \text{ for }j=0,1,...,d-1$, \begin{equation}\label{analyticnorm} \\| K \\|_{{\mathcal A}_{\delta,d} } = \sup_{z \in {\tilde B}^{n}_\delta \setminus\{0\} } \|z\|^{-d } \| K(z)\|, \end{equation} Let \begin{equation}\label{defA} \mathcal{A}_{\delta,d} := \{K:{\tilde B}^{n}_\delta\to{\mathbb C}^n: K\text{ is analytic }, D^jK(0)=0, \text{ for } j=0,1,...,d-1, \\| K\\|_{\mathcal{A}_{\delta, d}} <\infty \}, \end{equation} a space of bounded analytic functions defined in ${\tilde B}^{n}_\delta$ with vanishing derivatives at the origin up to order $d$. We endow these spaces with the norm \eqref{analyticnorm}, which turns $\mathcal{A}_{\delta,d}$ into a complex Banach space. Equivalently, the norm $\\| K\\|_{{\mathcal A}_{\delta,d}}$ can be defined as the smallest constant $C\geq0$ for which $\|K(z)\| \leq C \|z\|^{d}$.\\ \begin{remark} The proof that the space $\mathcal{A}_{\delta,d}$ is complete under \eqref{analyticnorm} is included for completeness. We argue that, given a Cauchy sequence $\{K_n\}_{n\in{\mathbb N}}$ in $\mathcal{A}_{\delta,d}$, it is also a Cauchy sequence in $C^0$ and, by the completeness in of $C^0$ it has a $C^0$-limit, which we denote as $K$. This limit will be analytic because the uniform limit of analytic functions is analytic. Moreover, since $\|K_n(x) \| \le C \|x\|^{d}$, we conclude that $K$ is $\mathcal{A}_{\delta, d}$. Because $K_n$ is Cauchy, we know that given $\varepsilon > 0 $ we can find $N_0(\varepsilon)\in\mathbb{N}$ so that if $n, m > N_0$, then $\| K_n(x) - K_m(x) \le \varepsilon \|x\|^d$. Taking limits in $m$ to $\infty$, we conclude that for $n > N_0$, we have $\| K_n(x) - K(x) \| \le \varepsilon \|x\|^d$. \end{remark} Since we are interested in real-valued parametrizations of the invariant manifolds, we define \begin{equation}\label{defAreal} \mathcal{A}_{\delta,d}^{real}=\{K\in\mathcal{A}_{d,\delta}: K \text{ takes real values for real arguments} \}, \end{equation} which defines a linear subspace of the space $\mathcal{A}_{\delta,d}$. Since $\mathcal{A}_{\delta,d}^{real}$ is a closed linear subspace of $\mathcal{A}_{\delta,d}$, it is a Banach space as well. \begin{remark}[Contraction properties of composition] Weighted norms such as \eqref{analyticnorm} have been found useful in proofs dealing with weak contractions, cf. \cite{CABRE2005444,Cab2003,de1997invariant}. The relevant property of these weighted norms is that if $s: {\tilde B}^{n}_\delta\rightarrow {\tilde B}^{n}_\delta$, $s(0) =0$ is a contraction, i.e., $\Lip(s) < 1$, the operator $K\rightarrow K\circ s$ is an even stronger contraction in such norms. Indeed, \begin{equation} \|K\circ s (z) \| \le \|s(z)\|^{d} \\| K\\|_{{\mathcal A}_{\delta,d}} \leq \Lip(s)^{d}\|z\|^{d} \\| K\\|_{{\mathcal A}_{\delta,d}}. \end{equation} Hence, if $s$ is a contraction fixing the origin, we obtain \begin{equation} \label{goodcontraction} \\| K\circ s \\|_{{\mathcal A}_{\delta,d}} \leq \Lip(s)^{d} \\| K \\|_{{\mathcal A}_{\delta,d}}. \end{equation} If $\{K^n_\varepsilon(z)\}_{n\in{\mathbb N}}$ is a sequence of functions analytic jointly in $(z,\varepsilon)$ and in $z$ for fixed $\varepsilon$, and the sequence $\{K^n_\varepsilon(z)\}_{n\in{\mathbb N}}$ converges in ${\mathcal A}_{\delta,d}$ to $K_\varepsilon$, then $K_\varepsilon$ is also jointly analytic in $(z,\varepsilon)$, cf. \cite[Chapter III]{hille1996functional}. \end{remark} \section{Lyapunov Subcenter Manifolds} In this section, we review some classical results on the theory of Lyapunov subcenter manifolds. For more details and other variants of the results, we refer the reader to \cite{liapounoff1907probleme, MoserZ, MeyerH, Duistermaat1972,kelley1967liapounov,kelley1969analytic,moser1976periodic}.\\ Specifically, we consider a differential equations of the form \begin{equation} \dot{X}=LX+N(X)+ \varepsilon CX + \varepsilon G_\varepsilon(X) \equiv F_\varepsilon(X) , \label{main} \end{equation} for an unknown $X:(0,\infty)\to{\mathbb R}^n, t\mapsto X(t)$ and $\varepsilon\geq 0$.\\ The $n\times n$-matrix $L$ and the analytic nonlinearity $N(X)= O(\|X\|^2)$ constitute the unperturbed system, while the $n\times n$ matrix $C$ and the analytic nonlinearity $G_\varepsilon(X)=\mathcal{O}(\|X\|^2)$ constitute the perturbation, i.e., we regard equation \eqref{main} as a perturbation of the system \begin{equation} \dot{X}=LX+N(X),\label{unpert} \end{equation} on which we make the following assumptions. \begin{assumption}\label{AssLSC} \noindent \begin{enumerate} \item The matrix $L$ is semi-simple (i.e. diagonalizable) \item The matrix $L$ has a pair of complex conjugate eigenvalues with zero real-part, i.e., \begin{equation} \{\pm\mathrm{i}\omega_0\}\subset\sigma(L), \end{equation} for some $\omega_0>0$. \item The remaining $n-2$ eigenvalues of $L$, which we denote by $\{\mu_{k}\}_{1\leq k\leq n-2}$, are \textit{non-resonant} with the eigenvalues $\pm\mathrm{i}\omega_0$, i.e., \begin{equation}\label{nonresunpert} \frac{\mu_{k}}{\mathrm{i}\omega_0}\notin{\mathbb Z}, \end{equation} for all $1\leq k\leq n-2$. In particular, $0\notin\mathrm{spec}(L)$. \item There exists an analytic first integral to equation \eqref{unpert}, i.e., there exists an analytic $I:{\mathbb R}^n\to{\mathbb R}$ such that for any solution $t\mapsto X(t)$, \begin{equation} \frac{d}{dt}I(X(t))=0.\label{FirstInt} \end{equation} We will assume that the function $I$ is normalized to $I(0)=0$ (without loss of generality), satisfies $\nabla I(0)=0$ and its second derivative at the origin is non-degenerate (without loss of generality, positive definite) on the eigenspace associated with $\pm\mathrm{i}\omega_0$, i.e., \begin{equation} D^{2}I(0)(Y,Y)>0, \label{posdef} \end{equation} for all $Y\in\mathbb{R}^n, Y \in \mathrm{eig}(\pm \mathrm{i} \omega_0)$. \end{enumerate} \end{assumption} \begin{remark} Because of assumption \eqref{posdef}, the energy $I$ is equivalent to $\|x\|^2$ near the origin in the LSM. Also, the orbits of the unperturbed system \eqref{unpert} stay on level set of $I$ by assumption \eqref{FirstInt}. Therefore, inside the LSM, we can define \textit{action-angle coordinates}, which are geometrically equivalent to $\Big(\|x\|^2, \text{Arg}(x)\Big)$, cf. \cite{arnol2013mathematical}. In these coordinates, the orbits of the unperturbed system are just circles. \end{remark} \begin{remark} The assumption that $L$ is semi-simple will not play an important role. It is not used in the Lyapunov subcenter theorem nor on the existence of asymptotic expansions. For the proof of existence of spectral submanifolds, only two consequences are used, namely, the analytic dependence of the dissipation parameter $\varepsilon$ and the persistence of certain non-resonance conditions. If they can be verified by other means, we do not need semisimplicity of $L$. \end{remark} \begin{remark} For the Lyapunov subcenter theorem, we do not need to impose any restriction on the eigenvalues $\mu_k$ except the nonresonance. They could be imaginary or have non-zero real part. \footnote{Note, however, that the preservation of $I$ imposes some restrictions. If some eigenvalues have positive or negative real parts, the conserved quantity has to be degenerate along the eigenspaces related to these eigenvalues.} As for our results, the (formal) expansions will not require any restrictions on $\mu_k$ beyond the non-resonance with $i\omega_0$. The results on convergence, i.e., on the existence of a true solution to the invariance equation, presented in this paper will require that the $\mu_k$ are imaginary as well as a further non-resonance condition and the assumption that the unperturbed system is Hamiltonian.\\ Note that, for a general system, we can always reduce to the case of imaginary eigenvalues by taking a restriction to the center manifold. Using the center manifold reduction, however, requires dealing with the problem of non-uniqueness of the center manifold and that it is only finitely differentiable. These problems will require different techniques. We hope to come back to them in a subsequent paper. \end{remark} \begin{remark} In our convergence results we will also assume that the system is Hamiltonian. This is a natural assumption for the applications to mechanical systems. From the mathematical point of view, this leads to some uniform estimates. See Lemma~\ref{uniform}. In Example~\ref{non-uniform}, we show that in the case that the system is energy preserving but not Hamiltonian, the uniform estimates in Lemma~\ref{uniform} may be false. It seems that, in this case, the results of convergence may be false and that there are new phenomena that may appear. Again, dealing with these new phenomena will require new techniques. \end{remark} We let \begin{equation} \begin{split} & X_{1}:=\mathrm{eig}(\pm\omega_0\mathrm{i}),\\ & X_{2}:=\mathrm{eig}\Big(\{\mu_{k}\}_{1\leq k\leq n-2}\Big). \end{split} \end{equation} Using the spectral projections, we can decompose the phase space as $X=X_{1}\oplus X_{2}$. The linear spaces $X_{1}$ and $X_{2}$ are invariant under $L$, i.e., $L(X_{1})\subseteq X_{1}$ and $L(X_{2})\subseteq X_{2}$. We let $\pi_{X_{1}}:X\to X_{1}$ and $\pi_{X_{2}}:X\to X_{2}$ be the corresponding projections (cf. Figure \ref{ImgLSC}) and define \begin{equation} L_{1}:=\pi_{X_{1}}L,\quad L_{2}:=\pi_{X_{2}}L. \end{equation} For later computations, we will denote the variables in the spaces $X_{1}$ and $X_{2}$ as \begin{equation} (x,y)\in X_{1}\oplus X_{2}={\mathbb R}^n, \end{equation} with $\dim(x)=2$ and $\dim(y)=n-2$.\\ The following theorem summarizes the classical existence results on a one-parameter family of periodic solutions close to the trivial solution $X=0$ of equation \eqref{unpert}. \begin{theorem}\label{exLSC} Let system \eqref{unpert} satisfy parts (2), (3), (4) of Assumption \ref{AssLSC}. Then, there exists a two-dimensional, invariant manifold $M_{0}$ tangent to the spectral subspace $X_{1}$. The manifold $M_{0}$ is analytic and filled with a one-parameter family of periodic orbits. \end{theorem} We refer to the invariant manifold $M_{0}$ as a \textit{Lyapunov subcenter manifold} (LSM). There exists a constant $\delta>0$, such that, locally around the origin, we can describe this LSM as the graph of an analytic function $w_{0}:B_{\delta}^{2}\to{\mathbb R}^{n-2}$, as illustrated in Figure \ref{ImgLSC}.\\ Note that the fact that an orbit is periodic is a topological property, so that the set is unique under topological properties. \begin{figure} \caption{LSM tangent to the two-dimensional spectral subspace $X_{1}$, represented as the graph of an analytic function $x\protect\mapsto w_{0}(x)$ with radius of convergence $\delta$. The manifold is unique and filled with periodic orbits.} \label{ImgLSC} \end{figure} \begin{proof} This result was first proved in \cite[p. 352]{liapounoff1907probleme}. See also \cite{kelley1967liapounov,MoserZ, MeyerH} for similar arguments as the argument presented here. This proof does not assume that the system is Hamiltonian (only that it has a conserved quantity) and it allows that the system has repeated eigenvalues or Jordan blocks, some of which could be stable/unstable.\\ We introduce a small parameter $\nu$ and scale the variables in \eqref{unpert}. Writing $x = \nu u$ we, see that \eqref{unpert} is equivalent to \begin{equation}\label{scaled} u' = L u + \nu^{-1} N(\nu u). \end{equation} Because $N$ vanishes up to second order, we have $\nu^{-1}N(\nu u) = \nu \tilde N_\nu(u) $, so that \eqref{scaled} has a well defined limit as $\nu$ tends to $0$. We also observe that if \eqref{unpert} preserves $I$, then \eqref{scaled} preserves $I_\nu(u) = \nu^{-2} I (\nu u)$. Note that $I_\nu$ has a well defined limit $I_0(u) = D^2I(0)(u,u)$.\\ We start by studying the limit $\nu = 0 $ of \eqref{unpert}. In the two dimensional space $X^1$, the flow is just a rigid rotation with period $T \equiv 2 \pi/\omega_0$. We note that the spectrum of $\exp( TL)$ is the exponential of the spectrum of $T L$, that is \begin{equation} \label{eigenvalues} \sigma( \exp{TL}) = \left\{ \{\exp( 2 \pi \mathrm{i} \mu_k/\omega_0)\}_{k= 1}^{n-2}, 1, 1\right\}. \end{equation} We write the two dimensional real plane $X_1$ corresponding to the eigenvalues $\pm \mathrm{i} \omega_0$ as the set of points $(x_1, x_2)$ and we will denote the points in the complementary spectral space as $y$.\\ If we consider the return map $R_{0,E}$ of to the co-dimension one plane $x_1 = 0$ restricted to a level surface of the conserved quantity\footnote{ We will refer to this conserved quantity as the energy since this is what happens in many problems. It also allows to use names such as ``energy surface'' for the level sets etc.} we see that the spectrum of the return map restricted to an energy surface, cf. \eqref{eigenvalues}, is just $\{\exp( 2 \pi \mathrm{i} \mu_k/\omega_0)\}_{k= 1}^{n-2}$ since the two eigenvalues $1$ of $\exp{LT}$ correspond respectively to the translation along the flow (eliminated by the return map) and the translation along the energy (eliminated by taking the energy surface). The non-resonance assumption of the theorem tells that $\mu_k/\omega_0$ is not an integer, hence, the eigenvalues of the return map restricted to the energy surface are not $1$. Now we observe that $R_{\nu,E}$ depends analytically on $\nu$ for $\nu$ small.\\ The previous observations amount to the fact that writing points in the axis $x_1 = 0$ say that $R_{0, E}(0, x_2(E), 0) = (0, x_2(E), 0)$. Furthermore, $\partial_y R_{0, E}(0,x_2(E), y) \|_{y = 0} - \Id$ is an invertible matrix. Hence, applying the finite dimensional, implicit function theorem \cite{Lang1999, Dieudonne,Meyerimplicit} we obtain that, for small $\nu$, we can find analytic families of fixed-points of the return map $R_{\nu, E}$ indexed by $\nu$ and $E$. This argument also shows that the manifold is analytic everywhere, including at zero. Indeed, it is an interesting exercise to compute the coefficients of the expansion at zero of the manifold.\\ Using the scaling, it is not difficult to show that the family is tangent at zero to the eigenspace. One can also observe that the implicit function theorem allows to compute the derivatives of the manifold at the origin. We leave the details to the reader, see also \cite{Moulton20}. Also the local uniqueness statements are those of the standard implicit function theorem. The periodic orbits are locally unique in the energy surface. \end{proof} \begin{remark} The method of proof of Lyapunov center theorem presented here has been generalized to infinite dimensional systems \cite{Bambusi00} or systems with symmetry in \cite{BuonoLM05,CallejaDG18}. It would be interesting to study the effects of adding dissipation to these models. \end{remark} \begin{remark} Since the proof above is based only on the implicit function theorem, it applies also to finitely differentiable systems. If the vector field is $C^\ell$, $\ell \ge 1$, we get a $C^\ell$ manifold, see \cite{kelley1967liapounov}. \end{remark} \begin{remark} To set up the analyticity results, it is convenient to examine what happens for complex values of the variables and the parameters.\\ We observe that if we consider now $x \in {\mathbb C}^2$, the scaling arguments still work and the flow is transversal to $x_1 = 0$ in the complex sense, entailing the return time to be a complex variable. The periodic orbit will consist of the orbit for complex times in a neighborhood of the path joining $0$ to the complex period. The union of all these periodic orbits covers ${\tilde B}^{n}_\delta$.\\ The singular nature of the dissipative perturbations, is also apparent in the complex interpretation. Once we add a dissipation, using Sternberg theorem \cite{sternberg1957local}, we know that the exponentially contracting orbit conjugate to an exponential, hence a complex periodic orbit. So, the family of periodic orbits in the conservative system bifurcate into a single complex periodic orbit. This clearly indicates the singular nature of the problem. \end{remark} For later calculations, we introduce a normalization of the vector field $F_0(X)$ such that the dynamics on the LSM are just given by constant-phase rotations. This will simplify subsequent arguments.\\ Let $T=T(I)$, only depending upon the energy, be the first return time of the periodic orbit with energy $I$ to a line of section and define $\Omega(I)=\frac{2\pi}{T(I)}$. Then, the dynamics on the LSM associated to the system \begin{equation}\label{suspension} \dot{X}=\frac{\omega_0}{\Omega(I)}F_0(X), \end{equation} is just given by rigid rotations with frequency $\omega_0$.\\ Note that, multiplying a vector field by an scalar, does not change the invariant manifolds, the periodic orbits or the conserved quantity. In the Hamiltonian case, if we multiply the vector field by a function of the Hamiltonian, we obtain a Hamiltonian vector field.\footnote{If $X = J \nabla H$, then, for any function $\alpha: {\mathbb R}\rightarrow {\mathbb R}$, we have $\alpha(H) X = J \alpha(H)\nabla H = J \nabla \beta(H)$ where $\beta' = \alpha$. Hence, the time-scaled vector field is also Hamiltonian} \begin{remark} The advantage of multiplying the vector field is that it is obvious that the derivative of the flow restricted to the Lyapunov manifold is just a rotation (hence modulus $1$). This, of course, could be obtained also by defining a new system of coordinates. \\ The normalization \eqref{suspension} is an explicit application of the standard suspension construction explained e.g. in \cite{katokh}, showing that, for a compact manifold, the special flow with respect to the first return time is equivalent to the suspension flow. The construction presented makes the constructions more explicit. \end{remark} We will need the following lemma in later calculations when perturbing from the LSM. \begin{lemma}\label{uniform} Assume that system \eqref{unpert}, normalized according to \eqref{suspension}, satisfies Assumptions \eqref{AssLSC}. Choose coordinates $(x,y)$, $x\in\mathbb{R}^2$ and $y\in\mathbb{R}^{n-2}$, such that the LSM corresponds to the plane $\{y=0\}$ and let $\phi=\phi^{T_0}$ be the time-$T_0$ map of system \eqref{suspension}, for $T_0=\frac{2\pi}{\omega_0}$. Then we have \begin{equation}\label{time1y} \phi(x,y)=(x+B(x)y,A(x)y)+\mathcal{O}(\|y\|^2), \end{equation} for some $n-2\times n-2$-dimensional matrix function $A$ and some $2\times n-2$-dimensional matrix function $B$ depending only on the energy.\\ In case that the flow is Hamiltonian and that $A(0)$ has only simple eigenvalues with modulus one, we have that the eigenvalues of the matrix $A(x)$ are of modulus one for all $\|x\| \in \mathbb{R}^2$ with $\|x\| \le \delta$, for some $\delta>0$\footnote{This $\delta$, for which the coordinates \eqref{time1y} with all eigenvalues of $A$ having modulus one, will be the fundamental domain of existence for the later perturbation argument.}. \end{lemma} \begin{proof} We can take coordinates in which the LSM corresponds to $\{ y= 0\}$. After the normalization \eqref{suspension}, all the periodic orbits have period $T_0$. By assumption, the Jacobian $D\phi(x,0)$ is a symplectic matrix. It is well known that for symplectic matrices, the inverse of the eigenvalues are also eigenvalues, since, if a symplectic matrix has simple eigenvalues in the unit circle, all the nearby matrices have an eigenvalue in the unit circle too.\\ Our assumption that $A(0)$ has simple eigenvalues implies that $D\phi(0,0)$ has a double eigenvalue $1$ and all the other eigenvalues are simple in the unit circle. Hence, for small enough $x$, we see that $D\phi(x,0)$ has to have $n-2$ eigenvalues on the unit circle. The double eigenvalue $1$ could, in principle bifurcate, but it does not because of the invariance of the LSM and the conservation of energy inside of the LSM. \end{proof} \section{The main theorem} \label{sec:statement} \subsection{An analytical formulation of the problem} In this section, we translate the geometric problem of invariant manifolds into a functional analysis problem by following the idea of the parameterization method \cite{Cab2003,CABRE2005444,haro2016parameterization}. Given a vector field $F_\varepsilon$ (satisfying the hypothesis of the subsequent theorem) we will seek an embedding $K_\varepsilon: B_\delta^2 \rightarrow {\mathbb R}^n$ (which extends to an embedding defined on $\tilde B_\delta^2$) and another vector field $R_\varepsilon:B_\delta^2 \rightarrow {\mathbb R}^2$ (which also extends to $\tilde B^2_\delta$) in such a way that \begin{equation}\label{CH} F_\varepsilon(K_\varepsilon(x)) = DK_\varepsilon(x)R_\varepsilon(x), \end{equation} with $K_\varepsilon(0) = 0$ and $R_\varepsilon(0) = 0$.\\ The equation \eqref{CH} will be the centerpiece of our analysis. Note that the geometric meaning is that the range of $K_\varepsilon$ is invariant under the flow of $X_\varepsilon$, i.e., the vector field $F_\varepsilon$ at one point in the range is tangent to the range. The vector field $R_\varepsilon$ is then a representation of the dynamics on the manifold. \begin{remark} Since $K_\varepsilon$ is an embedding, it can be used to follow several turns of the manifold which are very different from being a graph. There are numerical examples \cite{haro2016parameterization,kalies2018analytic,van2016parameterization} in which the same parameterization can be used to follow a large area containing turns and folds of the manifold in some model examples such as the Lorenz equations. The fact that the proofs are based on a contraction mapping argument allows to justify rigorously any method that produces approximate solutions, e.g., numerical computations. Using the contraction mapping theorem, we can show that if some function moves a very small amount by the application of the operator, then there is a fixed point at a distance from the approximate solution comparable to the distance of the approximate solution to its iterate. This allows to validate numerical calculations rigorously. \end{remark} \begin{remark} \label{underdetermined} Equations \eqref{CH} are highly under-determined. We have already remarked in \eqref{suspension} that we can change the time multiplying the vector field by a scalar function without affecting the invariant manifolds or the conserved quantities, but there are other sources of undeterminacy as well. In fact, any change of variables in the reference disk leads to a parameterization of the same manifold. Indeed, if $K_\varepsilon, R_\varepsilon$ are a solution of \eqref{CH} and $h_\varepsilon$ is a local diffeomorphism $h_\varepsilon(0) = 0$, we have \begin{equation} \begin{split} F_\varepsilon \circ K_\varepsilon \circ h_\varepsilon &= (DK_\varepsilon R_\varepsilon) \circ h_\varepsilon = D(K_\varepsilon) \circ h_\varepsilon \, R_\varepsilon \circ h_\varepsilon \\ &= D( K_\varepsilon \circ h_\varepsilon) (Dh_\varepsilon)^{-1} R_\varepsilon \circ h_\varepsilon \end{split} \end{equation} In other words, $\tilde K_\varepsilon =K_\varepsilon \circ h_\varepsilon$, $\tilde R_\varepsilon = (Dh_\varepsilon)^{-1} R_\varepsilon \circ h_\varepsilon$ is also a solution of \eqref{CH}. One can show, however, that, up to this family of transformations, the manifold is unique among the $d$-times differentiable ones, where $d$ is a number that depends on the spectral properties of $DF_\varepsilon(0)$, cf. \cite{de1997invariant,Cab2003,CABRE2005444}.\\ We will take advantage of this underdeterminacy to impose some normalization conditions on the parametrization $K_\varepsilon$ and the vector field on the manifold $R_\varepsilon$. From the computational point of view, the underdeterminacy of equation \eqref{CH} can be used to construct more efficient algorithms for the computation of invariant manifolds as well, cf. \cite{haro2016parameterization}. Furthermore, \cite{sternberg1957local} shows that, in the absence of resonances, there is a system of coordinates in which $R_\varepsilon$ is linear. In our situation, however, this cannot be achieved due to the singular nature of the problem. A linear vector field on the invariant manifold, as described in \cite{sternberg1957local}, would lead to singularities in the expansions as $\varepsilon\to 0$. By allowing higher order $z$-terms in $R_\varepsilon$ we can avoid the singularities of the change of variables leading to the linearization. \end{remark} We will make the following assumption on the linear part of the perturbed system: \begin{assumption}\label{Asspert} The matrix $L+\varepsilon C$ has a pair of complex conjugate eigenvalues $\lambda^{\pm}_\varepsilon$, perturbing from the nonresonant eigenvalues in Assumption \ref{AssLSC}, such that \begin{equation} \lambda^\pm_\varepsilon = \pm\mathrm{i}\omega_0 + (- \alpha \pm \mathrm{i} \alpha_I) \varepsilon + O(\varepsilon^2), \end{equation} for some $\alpha>0$ and $\alpha_I\in\mathbb{R}$. \end{assumption} \begin{remark} We note that the eigenvalues depend differentiably on $\varepsilon$ at $\varepsilon = 0$ as a consequence of the nonresonance condition. \\ If $\alpha < 0$, we obtain similar results by switching the direction of time. The content of Assumption~\ref{Asspert} is that $\alpha \ne 0$. We will see that if $\alpha = 0$, the conclusions of the main theorem may be false and there may fail to be an SSM of size one in an neighborhood, see Example~\ref{firstexample}. On the other hand, the quantity $\alpha_I$, i.e., the change of the (pseudo)-frequency induced by the dissipation, does not play any role in our analysis.\\ We also note that Assumption~\ref{Asspert} is a condition on the perturbation, not on the unperturbed problem which is very typical for singular perturbation problems.\\ In practical problems, verifying Assumption~\ref{Asspert} is an easy task, since it only involves checking the first order perturbation theory for eigenvalues of a finite dimensional matrix. \end{remark} \subsubsection{Contraction properties of the perturbed linear part} \label{pertlin} Let $X_{1}^{\varepsilon}$ be the eigenspace of $L+\varepsilon C$ that perturbs from $X_{1}$, i.e., $X_{1}^{\varepsilon}\to X_{1}$ as $\varepsilon\to 0$, and let $X_{2}^{\varepsilon}$ be its spectral complement. Since $L+\varepsilon C$ is semi-simple by assumption for $\varepsilon$ small enough, we again have that $X=X_{1}^{\varepsilon}\oplus X_{2}^{\varepsilon}$. Since $L$ does not have repeated eigenvalues, the remaining eigenvalues $\{\mu_k\}_{1\leq k\leq n-2}$ continue to a family of eigenvalues $\{\mu^\varepsilon_k\}_{1\leq k\leq n-2}$, which is differentiable at $\varepsilon = 0$ and the corresponding spectral subspace and the spectral projection are differentiable in $\varepsilon$, satisfying \begin{equation}\label{Oepsspace} \begin{split} &X_1^\varepsilon=X_1+\mathcal{O}(\varepsilon),\qquad X_2^\varepsilon=X_2+\mathcal{O}(\varepsilon),\\ & \pi_{X_1^\varepsilon}=\pi_{X_1}+\mathcal{O}(\varepsilon),\qquad \pi_{X_2^\varepsilon}=\pi_{X_2}+\mathcal{O}(\varepsilon), \end{split} \end{equation} where $X_1^\varepsilon=\mathrm{eig}(\lambda^{\pm}_\varepsilon)$, for $\varepsilon$ small enough. All this follows from spectral perturbation theory for matrices, cf. \cite{kato1995perturbation}, \cite[p. 396 Theorem 1]{lancaster1985theory}. Indeed, due to the nonresonance condition \eqref{nonres}, the eigenvalues $\lambda_\varepsilon^{\pm}$ necessarily have algebraic multiplicity one, which then implies the $\mathcal{O}(\varepsilon)$-dependence in $X_1^\varepsilon$. \begin{remark} Generally, if a matrix $L$ has a repeated eigenvalue $\lambda$, only the weaker relation \begin{equation} \mathrm{eig}(\lambda_\varepsilon)=\mathrm{eig}(\lambda)+\mathcal{O}(\varepsilon^{1/p}), \end{equation} for some $p>0$, holds, cf. \cite{lancaster1985theory}, p.402, Theorem 1. Here, $p$ is the length of the Jordan block associated with $\lambda$. Indeed, e.g., the matrix \begin{equation} \left(\begin{matrix} 1 & 1\\ \varepsilon & 1 \end{matrix}\right), \end{equation} has spectrum $\{1\pm\sqrt{\varepsilon}\}$ and the corresponding eigenspaces are spanned by the vectors $(1, \pm \sqrt{\varepsilon})$. Note that the eigenvalues move $\sqrt \varepsilon$ in this case, i.e., much faster than $\varepsilon$ and that the angles between the eigenspaces are also small. The later may cause problems if we need to consider the projections over these spaces, since they will have a norm that grows as $\varepsilon$ goes to zero. \end{remark} Let $\beta$ be defined as in Lemma \ref{Dyeps} and let $\alpha$ be the linear contraction rate as defined in Assumption \ref{Asspert}. Then, there exists a number $d$ such that \begin{equation}\label{defL} \beta<d\alpha. \end{equation} Note that if the condition \eqref{defL} is satisfied for some $d$, it is satisfied for all larger ones. Any of those will work for our subsequent considerations. We will choose the parameter $d$ in the definition of the space $\mathcal{A}_{\delta,d}$ such that \eqref{defL} is satisfied, i.e., depending upon the ratio between the linear contraction rate on the parametrization space of the LSM and a parameter depending on the second derivative of the flow map.\\ \subsection{Formulation of the main theorem} In this section, we introduce some fundamental parameters that and formulate our main theorem. For the formulation and the proof, we need the following lemma on the time-$T_0$ map for $y$-values of order $\varepsilon$. \begin{lemma}\label{Dyeps} Assume that system \eqref{unpert}, normalized according to \eqref{suspension}, satisfies Assumptions \eqref{AssLSC} and assume that the flow is Hamiltonian, such that the normal form \eqref{time1y} holds for all $\|x\|<\delta$. Let $\phi_\varepsilon^t$ be the flow map of the perturbed system. Then, the Jacobian of the inverse of the time-$T_0$ map of the perturbed system \eqref{main}, which we denote as $\phi_\varepsilon^{-1}$, satisfies \begin{equation}\label{estDphi-1} \\|D\phi^{-1}_\varepsilon(x,\varepsilon y)\\|=1+\beta\varepsilon+\mathcal{O}(\varepsilon^2), \end{equation} for $\varepsilon>0$, for all $\|x\|<\delta$, $\|y\|<\eta$, where $\eta>0$ independent on $\varepsilon$. Here, $\beta$ is such that \begin{equation} \\|D_yD\phi_0(x,0)\\|\leq \beta, \end{equation} for all $\|x\|<\delta$. \end{lemma} \begin{proof} The proof is an immediate consequence of Lemma \ref{uniform}. Indeed, thanks to the form of $\phi$ in \eqref{time1y} and the simplicity of the eigenvalues of $A$ for small enough $x$ (non of which is equal to one), we can find matrices $Q$, depending on $x$, such that $D\phi$ can be block-diagonalized as \begin{equation} D\phi(x,y)=\left(\begin{matrix} 1 & Q(x)\\ 0 & 1 \end{matrix}\right)\left(\begin{matrix} 1 & 0\\ 0 & A(x) \end{matrix}\right)\left(\begin{matrix} 1 & Q(x)\\ 0 & 1 \end{matrix}\right)^{-1}+\mathcal{O}(\|y\|), \end{equation} just by choosing $Q(x):=(A-1)^{-1}B$. Therefore, since all eigenvalues of $A$ are simple for $\|x\|<\delta$, it follows from Taylor-expanding $D\phi_\varepsilon$ in $y$ that \begin{equation} \\|D\phi_\varepsilon(x,\varepsilon y)\\|=1+\beta\varepsilon+\mathcal{O}(\varepsilon^2), \end{equation} for all $\|x\|<\delta$ and $\|y\|<\eta$, for some $\eta$ independent on $\varepsilon$. By the standard implicit function theorem, the claim follows. \end{proof} The set up for our main theorem involves four small parameters: \begin{itemize} \item The parameter $\delta$, which controls the domain of definition of the LSM (and also the domain of definition of the SSM in the perturbation). This parameter will be independent on the dissipation $\varepsilon$. \item The parameter $\theta$, which controls the aperture of the cone of complex values for $\varepsilon$, domain considered. \item The parameter $\tau$ which controls the size of the complex extension. \item The parameter $\varepsilon_0$, which is the maximum value of $\|\varepsilon\|$ (as a complex number) for which the results are valid. \end{itemize} Along the proof, we will specify some smallness conditions on these quantities. It will be important that all these parameters (in particular $\delta$) can be chosen uniformly in the value of the dissipation, $\varepsilon$, so that we obtain results for $\delta$, $\theta$ and $\tau$, which are uniform in $\varepsilon$.\\ We are now ready to formulate informally our main result. \begin{theorem}\label{mainThm} Consider the system \begin{equation}\label{maininthm} \dot{X}=LX+N(X)+ \varepsilon CX + \varepsilon G_\varepsilon(X) \equiv F_\varepsilon(X), \end{equation} under Assumptions \ref{AssLSC} and Assumption \ref{Asspert}. Assume further that the unperturbed system is Hamiltonian, that the matrix $L$ does not have repeated eigenvalues and that all the eigenvalues are imaginary.\\ Then, for $\varepsilon$ sufficiently small, there exists an invariant, analytic, two-dimensional manifold $M_{\varepsilon}$ around the origin for the perturbed system \eqref{maininthm}. As $\varepsilon$ converges to zero, the manifold $M_\varepsilon$ converges -- in the sense of analytic manifolds -- to the LSM (of the unperturbed system) in a domain which is independent of $\varepsilon$. Moreover, we have explicit asymptotic expansions to any order in $\varepsilon$, uniformly valid in an $\varepsilon$-independent domain. The manifold is unique among the invariant manifolds that are sufficiently differentiable at the origin. \end{theorem} The precise formulation of the main result is the following Theorem~\ref{mainprop}. \begin{theorem}\label{mainprop} Consider the system \begin{equation}\label{maininthm2} \dot{X}=LX+N(X)+ \varepsilon CX + \varepsilon G_\varepsilon(X) \equiv F_\varepsilon(X) , \end{equation} under Assumptions \ref{AssLSC} and Assumption \ref{Asspert}. Assume further that the unperturbed system is Hamiltonian, that the matrix $L$ does not have repeated eigenvalues and that all the eigenvalues are imaginary satisfying \begin{equation}\label{no-resonances} \mu_k - \mu_l \ne m \, 2 \pi \mathrm{i} \omega_0 \quad \forall k,l = 1,\ldots n-2, k \ne l \forall m \in \mathbb{Z}. \end{equation} Let $\delta$ be the maximal radius for which the flow map of the unperturbed part of system \eqref{maininthm} can be written according to Lemma \ref{uniform} and let $\beta$ be the minimal upper bound of the inequality \begin{equation} \\|D_yD\phi_0(x,0)\\|\leq \beta, \end{equation} for all $\|x\|<\delta$, as in Lemma \ref{Dyeps}. Choose any $d$ that satisfies \begin{equation} \beta<d\alpha, \end{equation} where $\alpha$ is as in Assumption \ref{Asspert}.\\ Then, there is a sequence of functions $K_j: B_\delta \rightarrow {\mathbb C}^n$ (extending to functions from ${\tilde B}^{n}_\delta$) and $R_j: B_\delta \rightarrow B_\delta$ (extending also to functions from ${\tilde B}^{n}_\delta$), such that $K_j(0)=0$, $R_j(0)=0$, $DK_0(0)$ being the embedding from ${\mathbb R}^2$ to the real part $X^1_\varepsilon$ and such that \begin{equation} DR_0(0) = \begin{pmatrix} 0 & \omega_0 \\ -\omega_0 & 0\end{pmatrix}, \end{equation} with the following properties: \begin{enumerate} \item The sequences $K_j$ and $R_j$ solve the equation \eqref{CH} in the sense of formal power series, i.e., for any $N \in {\mathbb N}$, setting \begin{equation} \begin{split} K^{\le N}_\varepsilon(z)=\sum_{j=0}^N K_j(z) \varepsilon^j,\quad R^{\le N}_\varepsilon(z) = \sum_{j=0}^N R_j(z) \varepsilon^j, \end{split} \end{equation} we have that \begin{equation}\label{firstconclusion} \\| F_\varepsilon(K_\varepsilon^{\le N}(\cdot )) - DK_\varepsilon^{\le N} (\cdot)R_\varepsilon^{\le N}(\cdot) \\|_{{\mathcal A}_{\delta,d}} \le C_N \|\varepsilon\|^{N+1}, \end{equation} for all $\varepsilon$ small enough. \item For $N \ge d$, where again $d$ is as in \eqref{defL}, there exists a unique $K_\varepsilon \in \mathcal{A}^{real}_{\delta, d}$ such that \begin{equation} F_\varepsilon(K_\varepsilon(z)) = DK_\varepsilon(z)R^{\le N}_\varepsilon(z) \end{equation} for all $\varepsilon \in {\mathcal C}_\theta$ with $\varepsilon$ and $\theta$ small enough. \item Furthermore, for $\varepsilon \in \mathcal{C}_\theta$ (in particularly for all $\varepsilon >0$) small enough, we have \begin{equation}\label{secondconclusion} \\| K^{\le N}_\varepsilon - K_\varepsilon \\|_{{\mathcal A}_{\delta,d}} \le C \|\varepsilon\|^{N}, \end{equation} for some constant $C>0$. \end{enumerate} \end{theorem} \begin{remark} The polynomials $K^{\le N}_\varepsilon$ and $R^{\le N}_\varepsilon$, satisfying the invariance equation up to order $N$ in $\varepsilon$, i.e., with a small error, will be obtained through formal calculations. The approximate solutions will satisfy the invariance equations for all $\varepsilon$, small enough, in a complex ball. In fact, the Assumption that the system is Hamiltonian is not needed for the formal calculations. If we want, however, to have formal solutions that also satisfy the invariance equation to a sufficiently high order in $x$, we have to assume a normal form of the kind \eqref{normalform}, i.e., a suitable non-resonance condition.\\ The smallness condition in $\|\varepsilon\|$ needed for the second conclusion in Theorem \eqref{mainprop}, however, may depend upon $N$. This is a reasonable limitation since we do not expect that the formal sums $\hat{K}_\varepsilon = \sum_{j=0}^\infty K_\varepsilon^j$ and $\hat{R}_\varepsilon =\sum_{j=0}^\infty R_\varepsilon^j$ to converge in the sense of series of analytic functions. In practice, one can choose an $N$ which is optimal for the goals at hand. \end{remark} \begin{remark} \label{rem:negativeeps} The approximate solutions $K^{\le N}_\varepsilon$ and $R^{\le N}_\varepsilon$ solve the invariance equation up to a very small error for all complex $\varepsilon$ small, in particular, also for $\varepsilon< 0$ and small. Then, these give approximately unstable manifolds. Whether these ghost manifolds correspond to actual invariant manifolds, is not obvious. In the case that the $\mu_k$ are all imaginary, by reversing the direction of time and changing the sign of $\varepsilon$ we can can apply Theorem~\ref{mainprop} to obtain slow unstable manifolds for $\varepsilon \in -\mathcal{C}_\theta$.\\ These unstable SSMs for $\varepsilon< 0$ are smooth continuations of the stable SSMs for $\varepsilon>0$. They are of physical interest, for example, in systems with active media or in periodic perturbations. \end{remark} For subsequent arguments, it will be important that the approximate solutions obtained in the first conclusion of Theorem \eqref{mainprop} solve equation \eqref{CH} approximately in a neighborhood of the origin of size $\delta$, which is independent of $\varepsilon$. The procedure to find the solutions will be a global perturbation argument that depends on the known solutions of \eqref{CH} for $\varepsilon = 0$ given by the LSM.\\ In a second step, we will show that these approximate solutions can be corrected to true solutions. We take $R^{\le N}_\varepsilon$ as the solution, but we need to correct $K^{\le N}_\varepsilon$. Hence, in the second step, the only unknown is $K^\varepsilon$. Again, we note that these functions have to be defined in domains whose size is uniform as the dissipation goes to zero.\\ These two steps are achieved by different methods. The calculation of the approximate solutions in the first step is done using a formal expansion based in a global averaging method. They provide approximations on a fixed neighborhood of $x$. The correction of the approximate solution into a true solution is based on transforming equation \eqref{CH} into a fixed-point problem in a small ball in an appropriately chosen function space, centered at the approximate solution.\\ The fact that we have to divide the proof into two different stages is very typical of singular perturbation theories. In the first stage, we get some perturbation that gets us a flimsy foothold in a neighborhood of the problem and then we switch to a more effective method. The second stage includes hypothesis on what is the outcome of the first stage. \begin{remark} Note that in the second step of the argument, we need some more assumptions. Notably, we use the condition \eqref{no-resonances} and the fact that the unperturbed system is Hamiltonian. This enters because in the contraction argument, we use Lemma~\ref{uniform}. \end{remark} \begin{remark} The condition \eqref{no-resonances} is equivalent to assuming that $\exp( T_0 \mu_k)$ are all different complex numbers, which is one of the conditions of Lemma~\ref{uniform}. Note that the condition \eqref{no-resonances} can be verified considering only a finite number of $m$. It suffices to verify that there are no repetitions, $\|m\| \le \frac{1}{\omega_0}\max_{k \ne l} \| \mu_k - \mu_l\|$. \end{remark} \begin{remark}\label{goodanalysis} Both $K_\varepsilon, R_\varepsilon$ are unknowns of the full problem. In the first stage, we deal with both unkowns to find approximate solutions. In the second step, we take $R_\varepsilon = R^{\le N}_\varepsilon$, so that the only unknown in the second state is the $K_\varepsilon$. This is possible because we take advantage of the underdeterminacy of the equation. See Remark~\ref{underdetermined}. The fact that the second stage -- mathematically the most delicate -- has only one unknown, is an important advantage. \end{remark} \begin{remark}We note that, by the non-resonance conditions, the normal forms up to order $d$ for any solution are determined. By the linearization theorem in \cite{sternberg1957local}, any possible $R_\varepsilon$ can be written as $R_\varepsilon=(Dh_\varepsilon)^{-1}R_\varepsilon^{\leq N}\circ h_\varepsilon$ in a neighborhood. Hence, taking advantage of \eqref{underdetermined}, it would suffice to take $R_\varepsilon^{\leq N}$ in a neighborhood. Of course, being conjugate in a neighborhood is not enough for our purposes, since we want that the flow is defined in a neighborhood uniform in $\varepsilon$ and $h_\varepsilon$ may fail to do so. The $R_\varepsilon^{\leq N}$ is a good candidate to use since it is defined in a uniform neighborhood. In summary: There is not going to be any advantage to find $R_\varepsilon$ in small scale, but there is a global advantage to keep $R_\varepsilon^{\leq N}$ . \end{remark} \begin{figure} \caption{The perturbation of the LSM (in green) tangent to the two-dimensional perturbed subspace $X_{1}^{\varepsilon}$, represented as the graph of $w_{0}$ plus a small perturbation $x\protect\mapsto v(x,\varepsilon)$.} \label{ImgPert} \end{figure} \section{Proof of the Main Theorem} \label{sec:proof} \subsection{Outline of the Proof} \label{sec:outlineproof} In Section~\ref{sec:elementary}, we will derive some immediate consequences of the Assumption ~\ref{Asspert}. After that, in Section~\ref{sec:elementary}, we perform some preliminary transformations, such as a partial normal form, which will simplify the calculations. In Section~\ref{sec:approximate} we will construct the approximate solutions claimed in part 1 of Theorem~\ref{mainThm}. Again, we emphasize that the main difficulty is that we need to get solutions in a domain of size $\delta> 0$ which is independent of the dissipation parameter $\varepsilon$, so, it has to be a globally defined perturbation expansion.\\ Finally in Section~\ref{sec:fixedpoint} we will reformulate the problem of existence of solutions of \eqref{CH} as a fixed-point problem for an operator defined on the ${\mathcal A}_{\delta,d}^{real}$ spaces introduced before and show that it is a contraction in a small neighborhood of the approximate solution.\\ One subtle point of the contraction argument is that the contraction will be rather weak. Indeed, the contraction rate will be $ l = 1 - C \|\varepsilon\| + {\mathcal O}(\|\varepsilon\|^2)$, for some $C>0$.\\ This weak contraction nevertheless suffices because the approximate solution provided in the first conclusion of Theorem \eqref{mainprop} solves the equation with very high accuracy $O( \|\varepsilon\|^{N+1})$ (in a domain independent of $\varepsilon$). Then, applying the contraction mapping theorem, we get that the difference between the approximation and the solution is $\mathcal{O}( \frac{1}{1 -l} \|\varepsilon\|^{N+1}) = \|\varepsilon\|^N)$ measured in an appropriate norm for globally defined functions. \subsection{Domain of attraction, preliminary changes of variables and normalizations} \label{sec:elementary} In this section, we collect some rather elementary results that follow from Assumption~\ref{Asspert}. It seems that these conclusions are the only uses of Assumption~\ref{Asspert} in the proof. Hence, any assumption that leads to them can be used. \subsubsection{Global domain of attraction} \label{sec:global} The following lemma will be a consequence of Assumption~\ref{Asspert}. It will be important for the transformation of the invariance equation \eqref{CH} into a fixed point problem. \begin{lemma} \label{prop:domain} Fix any vector field $R_\varepsilon$ that perturbs from a vector field $R_0$ as in the Lyapunov Subcenter Manifold Theorem \ref{exLSC}, satisfying Assumption \ref{Asspert}.\\ Then, for all $\varepsilon \in {\mathcal C}_\theta$ (in particular for all $\varepsilon>0$), we can find a complex domain ${\tilde B}^{n}_\delta$, which is mapped into itself by the forward flow of $R_\varepsilon$. \end{lemma} \begin{proof} Choose again coordinates $(x,y)$ such that the LSM corresponds to the plane $\{y=0\}$. Using the normalization \eqref{suspension} on the LSM, we can assume without loss of generality, that the flow on the LSM is given by a rigid rotation with period $T_0$ and let $r_0:=r_0^{T_0}$ be the corresponding time-$T_0$-map, i.e., $r_0(x)=x$. In particular, we have that $D^2r_0(x)=0$, for all $\|x\|< \delta$. Denoting the time-$T_0$ map of the perturbed vector field $R_\varepsilon$ as $r_\varepsilon$, it immediately follows that \begin{equation}\label{D2r} D^2r_\varepsilon(x)=\mathcal{O}(\varepsilon), \end{equation} for all $\|x\|<\delta$.\\ Let $\Lambda_\varepsilon$ be the linear part the time-$T_0$ map of the flow generated by $R_\varepsilon$, i.e., \begin{equation} \Lambda_\varepsilon=Dr_\varepsilon(0). \end{equation} By Assumption \ref{Asspert}, we obtain that \begin{equation} \sigma( \Lambda_\varepsilon)=e^{\pm\mathrm{i}\omega_0-\alpha\varepsilon \pm \mathrm{i} \alpha_I \varepsilon+ {\mathcal O}(\varepsilon^2)}. \end{equation} If $\varepsilon \in {\mathcal C}_\theta$, $\varepsilon$ is a perturbation of $\|\varepsilon\|$ in the sense that $\varepsilon - \|\varepsilon\| = O(\theta)\|\varepsilon\|$ and we can change $\varepsilon$ into $\|\varepsilon\|$ up to an error which is controlled by $\theta$. It follows that \begin{equation} \label{estLambda} \begin{split} & \| \Lambda_\varepsilon\| = 1 - (\alpha + {\mathcal O}(\theta)) \|\varepsilon\| +{\mathcal O}(\|\varepsilon\|^2), \\ & \| \Lambda_\varepsilon^{-1}\| = 1+ (\alpha + {\mathcal O}(\theta)) \|\varepsilon\| +{\mathcal O}(\|\varepsilon\|^2), \\ \end{split} \end{equation} also because of Assumption~\ref{Asspert}.\\ We can therefore estimate \begin{equation} \begin{split} \|Dr_\varepsilon(x)\|&\leq\|Dr_\varepsilon(0)\|+\|Dr_\varepsilon(x)-Dr_\varepsilon(0)\|\\ &\leq 1 - (\alpha + {\mathcal O}(\theta)) \|\varepsilon\| +{\mathcal O}(\|\varepsilon\|^2)+\|x\|\mathcal{O}(\varepsilon)\\ &\leq 1- (\alpha + {\mathcal O}(\theta) +\mathcal{O}(\delta))\|\varepsilon\|+{\mathcal O}(\|\varepsilon\|^2), \end{split} \end{equation} for $\|x\|<\delta$, where we have used the (complex) mean-value theorem, as well as the estimates \eqref{estLambda} and \eqref{D2r}. In particular, if $\delta$ and $\theta$ are small enough, the time-$T_0$ map $r_\varepsilon$ is a contraction with contraction factor \begin{equation}\label{defgamma} \gamma:=1- (\alpha - {\mathcal O}(\theta) -\mathcal{O}(\delta))\|\varepsilon\|+{\mathcal O}(\|\varepsilon\|^2)<1. \end{equation} Therefore, the time -$T_0$ map of the flow in a ball of radius $\delta$ has a derivative which agrees with the above up to $ {\mathcal O}(\delta)$ in $B^2_\delta$ and up to ${\mathcal O}(\delta) + {\mathcal O}(\tau)$ in ${\tilde B}^{n}_\delta$, cf. (\ref{deftau}). Hence, we can get that the spectrum is bounded away from $1$ for all $\delta$ sufficiently small and the conditions of smallness for $\delta$ can be taken independently of $\varepsilon$. \end{proof} \begin{remark} Notice that the above argument essentially uses that the real part of the contraction factor changes with a leading order comparable with $\|\varepsilon\|$. We anticipate (see Section~\ref{sec:examples}) that, if the contraction was moving more slowly and the nonlinear terms were moving still with $\|\varepsilon\|$, it would be possible to find periodic orbits in arbitrary small neighborhoods, for small $\|\varepsilon\|$. In \cite{de1997invariant}, it was observed that these periodic orbits provide an obstruction to the existence of manifolds with sufficiently high differentiability, in particular, analytic manifolds.\\ Of course, even if the real part of the contraction changed more slowly that $\|\varepsilon\|$, say ${\mathcal O}(\|\varepsilon\|^2)$, we could recover the result of the global domain of attraction by assuming properties of the non-linear terms and the rest of the proof could go through.\\ In many practical problems, the dissipation is a global phenomenon that does not get stopped by the non-linear terms, hence in many practical systems, the conclusions of Proposition~\ref{prop:domain} hold even if Assumption~\ref{Asspert} does not hold (but other global assumptions do). It would be interesting to formulate other general physically meaningful assumptions that account for these phenomena. \end{remark} \subsubsection{Preliminary changes of variables} \label{changes} In this section, we perform some analytic changes of variables that simplify our formulas. Under Assumption \ref{no-resonances}, we have that \begin{equation} \label{nonres} \| k \lambda_\varepsilon - \mu^\varepsilon_j \| \ge \kappa > 0, \quad \forall k \in {\mathbb Z}, \quad \|\varepsilon \| \le \varepsilon_0, \end{equation} for some $\varepsilon_0,\kappa>0$ and $1\leq j\leq n-2$.\\ This follows because the imaginary parts for $ k \lambda_\varepsilon - \mu^\varepsilon_j $ differ by a constant for $\|k\| > d$. We can check that for $k$ small, we cannot generate any new resonances. For large $k$ they cannot be generated either, since then the imaginary parts of $k \lambda_\varepsilon$ and $\mu^\varepsilon_j$ are very different. First, we can adjust our coordinates $(x,y)\in X_1\oplus X_2$ such that the LSM invariant for $F_0$ corresponds to one of the coordinates. That is to say, in this coordinate system, we have that the embedding is just $K_0(x) = (x, 0) $. The plane $\{y=0\}$ then defines an invariant manifold for the vector field \begin{equation} F_0(x, y) = ( R_0(x), \tilde{A}(x,y)), \end{equation} where $(x,y)\mapsto \tilde{A}(x,y)$, the direction transversal to the $R_0$ field that satisfies $\tilde{A}(x,0) = 0$.\\ We can therefore arrange that \begin{equation}\label{normalization} DK_\varepsilon(0) = \Pi_{X_1}, \end{equation} is a fixed isometric embedding from ${\mathbb R}^2$ (or ${\mathbb C}^2$) into the invariant space (which we arrange to be the first components of the space). The normalization \eqref{normalization} indicates that the embedding $K_\varepsilon$ will be in the affine space $\Pi_{X_1} + \mathcal{A}_{\delta,d}^{real}$.\\ Note that, in particular, in these coordinates, $DK_\varepsilon(0)$ will be independent of $\varepsilon$. In the same vein, we can arrange that $D R_\varepsilon(0)$ is the constant map corresponding to the eigenvalues $\lambda^{\pm}_\varepsilon$. In contrast with $DK_\varepsilon(0)$, $DR_\varepsilon(0)$ does depend on $\varepsilon$. We can also make sure that the conserved quantity on the LSM is just given by $\|x\|^2$. From the results in \cite{kelley1969analytic,SiegelM}, we know that there exists a system of coordinates, such that $R_0$ takes the form \begin{equation}\label{R0} R_0(x)=\left(\begin{matrix} 0 & \Omega(\|x\|^2)\\ -\Omega(\|x\|^2) & 0 \end{matrix}\right)x, \end{equation} with $\Omega(0)=\omega_0$, for $\omega_0\in{\mathbb R}$, being the linear frequency of the Lyapunov mode.\\ Proceeding as in the theory of normal forms \cite{murdock2003normal,takens1971partially}, for any $N\in{\mathbb N}$, we can take advantage of the assumed absence of resonances, cf. \eqref{nonres}, and change variables polynomially in $x$ and analytically in $\varepsilon$, such that, separating the variables into $x$, the tangent to the LSM and $y$, the tangent to the complementary space, we have \begin{equation} \label{normalform} F_\varepsilon(x,y) = (L + \varepsilon C)(x,y) + A_\varepsilon(x)y + {\mathcal O}(\|y\|^2\|x\|^{N+1}), \end{equation} for some matrix $A_\varepsilon$. Note that the above normal form can be done uniformly for all $\varepsilon$ sufficiently small. \subsection{Approximate solutions to the invariance equation \eqref{CH}} \label{sec:approximate} \subsubsection{Construction of approximate solutions} In this section, we construct the sequences $K_j, R_j$ introduced in Theorem~\ref{mainprop}, based on perturbation theory (generalized averaging theory). We again recall that the important feature is that the size of the domain where the approximation is obtained is independent of the size of the dissipative parameter.\\ The perturbation theory we use is valid with uniform bounds in a neighborhood in the $x$ variables which is independent of $\varepsilon$. The perturbation theory uses essentially the assumption that there is a LSM consisting of periodic orbits and is basically a mildly sophisticated version of the averaging method on periodic orbits. Note that this global perturbation theory is very different from the perturbation theory based on normal forms which just matches expansions near the origin of coordinates; these local expansions are very good near the origin, but the region they describe depends on $\varepsilon$. For our purposes, it is crucial that we can obtain estimates in a region of $x$ which is independent of $\varepsilon$. \\ We remark that the methods used in this section are rather elementary extensions of averaging theory and that they work just as well for finitely differentiable vector fields. By assumption, equation \eqref{CH} is satisfied for $\varepsilon=0$ by the Lyapunov Subcenter Theorem, i.e., \begin{equation} F_0(K_0(x))=DK_0(x)R_0(x). \end{equation} Formally expanding $F_\varepsilon$, $K_\varepsilon$ and $R_\varepsilon$ in $\varepsilon$, \begin{equation}\label{approx} \begin{split} &F_\varepsilon(x)=\sum_{j=0}^NF_j(x)\varepsilon^j+\mathcal{O}(\varepsilon^{N+1}),\\ &K_\varepsilon(x)=\sum_{j=0}^NK_j(x)\varepsilon^j+\mathcal{O}(\varepsilon^{N+1}),\\ &R_\varepsilon(x)=\sum_{j=0}^NR_j(x)\varepsilon^j+\mathcal{O}(\varepsilon^{N+1}),\\ &F_0(0)=0,\quad K_0(0)=0,\quad R_0(0)=0, \end{split} \end{equation} we see that equation \eqref{CH} at order $\varepsilon$ becomes \begin{equation}\label{Oeps} DF_0(K_0(x))K_1(x)+F_1(K_0(x))=DK_0(x)R_1(x)+DK_1(x)R_0(x). \end{equation} The terms $F_0$ and $F_1$ are given by the right-hand side of the differential equation, while $K_0$ and $R_0$ are given by the shape and the dynamics of the LSC, respectively. The unknowns of \eqref{Oeps} are $K_1$, $R_1$ which are, respectively, the first order corrections to the shape of the invariant manifold and to the dynamics on it.\\ More generally, we claim (see the justification below) that expanding to higher order in $\varepsilon$ and matching terms of order $\varepsilon$, we are led to \begin{equation}\label{Oepsn} (DF_0)(K_0(x)) K_n(x)-(DK_n)(x)\, R_0(x)=F_n(K_0(x))+DK_0(x)R_n(x)+S_n(x), \end{equation} where $S_n(x)$ is a polynomial expression involving $K_1,...,K_{n-1}$ and $R_1,...,R_{n-1}$, as well as their derivatives.\\ We will study the equations \eqref{Oepsn} recursively. We will show that if $K_1,...,K_{n-1}$ and $R_1,...,R_{n-1}$ -- and hence $S_n$ -- are known, we can find $(K_n, R_n)$ solving \eqref{Oepsn}.\\ Hence, we will consider \eqref{Oepsn} as an equation for $K_n$, $R_n$ when all the other quantities are known. We anticipate that the solutions $K_n, R_n$ will not be unique, which is consistent with the fact that the equations \eqref{CH} are underdetermined, cf. Remark~\ref{underdetermined}. To prove \eqref{Oepsn}, we first note that \begin{equation} \begin{split} \left.\frac{\partial^n}{\partial\varepsilon^n}\Big(DK_\varepsilon(x)R_\varepsilon(x)\Big)\right\|_{\varepsilon=0} &=\left.\sum_{j=0}^n\frac{\partial^j DK_\varepsilon}{\partial\varepsilon^j}(x) \frac{\partial^{n-j}R_\varepsilon}{\partial\varepsilon^{n-j}}(x)\right\|_{\varepsilon=0}\\ &=DK_n(x)R_0(x)+DK_0(x)R_n(x)+\sum_{j=1}^{n-1} DK_j(x)R_{n-j}(x). \end{split} \end{equation} Next, we prove inductively that \begin{equation} \frac{\partial^n}{\partial\varepsilon^n}F_\varepsilon(K_\varepsilon)=\frac{\partial^n F_\varepsilon}{\partial\varepsilon^n}(K_\varepsilon)+\frac{\partial F_\varepsilon}{\partial x}\frac{\partial^nK_\varepsilon}{\partial\varepsilon^n}+P_n(K_\varepsilon,F_\varepsilon,...), \end{equation} where $P_n$ is a polynomial in $F_\varepsilon$, $K_\varepsilon$ and their derivatives up to order $n-1$ in $\varepsilon$. For $n=1$, we obtain \eqref{Oeps}. Assuming the formula for $n$, we obtain \begin{equation} \begin{split} \frac{\partial^{n+1}}{\partial\varepsilon^{n+1}}F_\varepsilon(K_\varepsilon)&=\frac{\partial}{\partial\varepsilon}\left(\frac{\partial^n F_\varepsilon}{\partial\varepsilon^n}(K_\varepsilon)+\frac{\partial F_\varepsilon}{\partial x}\frac{\partial^nK_\varepsilon}{\partial\varepsilon^n}+P_n(K_\varepsilon,F_\varepsilon,...)\right)\\ &=\frac{\partial^{n+1} F_\varepsilon}{\partial\varepsilon^{n+1}}(K_\varepsilon)+\frac{\partial^{n+1} F_\varepsilon}{\partial\varepsilon^n\partial x}(K_\varepsilon)\frac{\partial K_\varepsilon}{\partial\varepsilon}+\frac{\partial F_\varepsilon}{\partial x}\frac{\partial^{n+1}K_\varepsilon}{\partial\varepsilon^{n+1}}\\ &+\frac{\partial^2 F_\varepsilon}{\partial x\partial\varepsilon}\frac{\partial^nK_\varepsilon}{\partial\varepsilon^n}+\frac{\partial^2 F_\varepsilon}{\partial x^2}\frac{\partial^nK_\varepsilon}{\partial\varepsilon^n}\frac{\partial K_\varepsilon}{\partial\varepsilon}+DP_n(K_\varepsilon,F_\varepsilon,...)\cdot\left(\frac{\partial K_\varepsilon}{\partial\varepsilon},\frac{\partial F_\varepsilon}{\partial\varepsilon},...\right), \end{split} \end{equation} where the last bracket contains derivatives in $\varepsilon$ up to order $n$. This proves \eqref{Oepsn}. Now we turn to analyzing \eqref{Oepsn}. Equation \eqref{Oepsn} defines a linear, first-order system of PDEs of the form \begin{equation}\label{cohomology} M(x)K_n(x)-DK_n(x)R_0(x) + \eta(x) = 0, \end{equation} where the unknowns are $K_n$ and all the other elements, i.e., $M$, $R_0$ and $\eta$, are known with $K_n:{\mathbb R}^2\to{\mathbb R}^N$, $R_0:{\mathbb R}^2\to{\mathbb R}^2$ and $\eta:{\mathbb R}^2\to{\mathbb R}^N$ (again, all of them extend to a complex domain). In our case, we have that \begin{equation}\label{etadefined} \begin{split} &\eta(x)= - DK_0(x)R_n(x)-F_n(K_0(x))-S_n(x),\quad M(x)=DF_0(K_0(x)). \end{split} \end{equation} The coefficients in the left-hand side of equation \eqref{cohomology} are the same for all $n$, i.e., $M(x)$ and $R_0(x)$ do not depend upon $n$.\\ We will develop a theory for general $\eta$ and discover that, to have a solution, $\eta$ has to satisfy some constraints. For our problem, $\eta$ contains the unknown $R_n$. Hence, we will determine $R_n$ so that $\eta$ satisfies the compatibility conditions for the existence of $K_n$ and hence, we can determine $K_n$. Similar procedures (one of the unknowns is determined so that compatibility conditions are met) happen in many perturbative theories in mechanics, cf. \cite{BogoliubovM, Giacaglia}. They seem to have originated in the perturbative expansions in Celestial Mechanics. We will apply this procedure only a finite number of times (bigger than $d$ in \eqref{defL}). Our only goal is to produce an approximate solution and we do not need (indeed we do not expect) that the series converges. \subsection{Study of the cohomology equation \eqref{cohomology}} \label{sec:cohomology} In this section, we analyze the cohomology equation \eqref{cohomology}. The main result is to identify the obstructions in $\eta$ for the existence of solutions $K_n$. Later, we will study how to apply this obstructions to find $R_n$ solving \eqref{Oepsn}.\\ Notice that \eqref{cohomology} simply says that if $x(t)$ is a solution of $\dot x = R_0(x)$ then \begin{equation}\label{periodic} D K_n(x(s)) R_0(x(s)) = \frac{d}{ds} K_n(x(s)) = M(x(s)) K_n(x(s)) + \eta(x(s)) \end{equation} The solutions of $\dot x = R_0(x)$ are precisely the periodic Lyapunov orbits. Hence, \eqref{periodic} is a linear equation with periodic coefficients and periodic forcing. If $K_n$ has to be a function of the point $x(s)$ it has to be a periodic function of time of the same period as the orbit $x(s)$.\\ Hence, we study the system: \begin{equation}\label{char} \begin{split} &\frac{dx}{ds}(s)=R_0(x(s)),\\ &\frac{du}{ds}(s)= \eta(x(s))+ M(x(s))u(s). \end{split} \end{equation} The first equation in \eqref{char} can be written in polar coordinates as \begin{equation}\label{polar} \begin{split} &\frac{d\rho}{ds}(s)=0,\\ &\frac{d\theta}{ds}=-\Omega(\rho(s)^2), \end{split} \end{equation} for $x(s)= \Big(\rho(s)\cos(\theta(s),\rho(s)\sin{\theta(s)}\Big)$ and its solution is given by \begin{equation} x(s)=\rho_0\Big(\cos(\theta_0-\Omega(\rho_0^2)s),\sin(\theta_0-\Omega(\rho_0^2)s)\Big). \end{equation} Note that the equations \eqref{char} are just the equations of variation around Lyapunov orbits subject to some forcing. See \cite{JorbaV98,Masdemont05,Capinski12} for numerical treatments of \eqref{char} in celestial mechanics. Note also that equations at all orders are equations of the same form.\\ Since the trajectory of $s\mapsto x(s)$ is periodic of period $T(\rho_0)=\frac{2\pi}{\Omega(\rho_0^2)}$ in order for $s\mapsto u(s)$ to define a function of $x(s)$, the solution to the second equation in \eqref{char} must be periodic.\\ More precisely, for a given $\rho_0$, we have to find a condition on the function $s\mapsto\eta(x(s;\rho_0))$, such that the equation \begin{equation}\label{charu} \begin{split} &\frac{du}{ds}(s;\rho_0)= \eta(x(s;\rho_0))+M(x(s;\rho_0))u(s),\\ &u(0;\rho_0)=u_0(\rho_0), \end{split} \end{equation} has a $T$-periodic solution. Since the equation is a non-homogeneous linear equation, the standard variation of parameters formula gives \begin{equation}\label{perturbation} u(t)=\Phi(t;t_0)\left[u_0 + \int_{t_0}^t \Phi(s;t_0)^{-1}\eta(s)\, ds\right], \end{equation} where $\Phi(t;t_0)$ is the fundamental solution of the non-autonomous homogeneous problem \begin{equation}\label{homogeneous} \frac{d}{dt} \Phi(t; t_0) = M(x(t)) \Phi(t,t_0); \quad \Phi(t_0; t_0) = { \rm Id}\end{equation} For typographical reasons, we omit the dependence on $\rho_0$. \\ To have a periodic solution, i.e., $u(t+T)=u(t)$, it suffices to show that $u(t_0)=u(t_0+T)$, where $T$ is the period of the Lyapunov orbit or, explicitly, \begin{equation}\label{eqy0} u_0 =\Phi(t_0+T;t_0)\left[u_0 + \int_{t_0}^{t_0+ T} \Phi(s;t_0)^{-1}\eta(s)\, ds\right]. \end{equation} Rearranging \eqref{eqy0} gives \begin{equation}\label{eqy0rearranged} \begin{split} \left[ \Phi(t_0+T;t_0) - { \rm Id}\right] u_0 &= \Phi(t_0+T;t_0)\int_{t_0}^{t_0+ T} \Phi(s;t_0)^{-1}\eta(s)\, ds \\ & = \int_{t_0}^{t_0 + T} \Phi(t_0 +T, s) \eta(s) \, ds. \end{split} \end{equation} Since $\Phi(t_0 + T, t_0)$ is the linearization of the unperturbed flow under the unperturbed flow, we see that the spectrum of $\Phi(t_0 + T, t_0)$ will contain two eigenvalues $1$ (one of them corresponding to the direction of the flow and another one corresponding to the conservation of the energy).\\ To analyze the $n-2$ remaining Lyapunov exponents, we observe that, if we fix a sufficiently small neighborhood in the Lyapunov manifold, the matrix $M(x(s;\rho_0))$ will be a small perturbation of the constant matrix $L$ and that the period $T$ is close to $ 2 \pi/\omega_0$. Hence the spectrum of $\Phi(t_0 + T; t_0)$ will be close to the spectrum of $\exp( \frac{2 \pi}{\omega_0} L)$.\\ Putting the two remarks together, we conclude that in a neighborhood of the origin, the spectrum of $\Phi(t_0 + T; t_0)$ contains two eigenvalues which are exactly $1$ and the remaining $n -2$ are close to $\exp( 2 \pi \mathrm{i} \frac{\mu_k}{\omega_0})$, which is bounded away from one because of the non-resonace assumptions for Lyapunov orbits.\\ Equation \eqref{eqy0rearranged}, therefore, can be solved if and only if, the right-hand side has no components over the eigenspaces corresponding to the eigenvalues $1$ (identified before as the direction of the flow and the gradient of the energy). This is the obstruction in the solution of $K_n$. In the following, we ill show that we can choose the $R_n$'s so that this is soluble.\\ We proceed as in the proof of the Lyapunov theorem and take a surface of section in a coordinate axis in the $X_1$ space. This eliminates the eigenvalue $1$ corresponding to the flow. Another way to interpret this is to observe that all the points in the periodic orbit are solutions, so that we always get a one-dimensional family of solutions.\\ On the other hand, for the existence of a solution of \eqref{eqy0rearranged}, there is a true obstruction for $\eta$. We need that the projection to the right-hand side along the direction where the energy vanishes. Note that this is a linear function in $\eta$. We will deal with this obstruction in the next paragraph. \subsection{Algorithm for the iterative step of perturbative expansions} \label{sec:iterativestep} Using the theory of the cohomology equation as derived in the previous section, we can device an algorithm to solve to solve recursively the equations for $K_n, R_n$. To do so, we first determine $R_n$ so that $\eta$ in the right-hand side of \eqref{etadefined} satisfies the constraints needed for the existence of $K_n$. Then, we determine $K_n$ using the formulas \eqref{cohomology}. Note that in this selection, the dependence on the periodic orbit becomes very important.\\ By choosing the energy $I$ as one coordinate, one coefficient of the matrix $M$ is identically zero, due to energy conservation in the unperturbed system. Also, in our system of coordinates, the matrix $DK_0$ is the identity, so that, using \eqref{etadefined}, the condition for the existence of a periodic orbit becomes \begin{equation}\label{obstructionsolved} \int_0^{\frac{2 \pi}{\Omega(\rho_0)}} \Pi_E R_n ( x(s, \rho_0) )\, ds = -\int_0^{\frac{2 \pi}{\Omega(\rho_0)}}\Pi_E (F_n( x(s, \rho_0) + S_n(x(s; \rho_0) )\, ds, \end{equation} where $\Pi_E$ denotes the projection in the direction of the energy with respect to the eigenvalues.\\ Due to the underdetermined nature of the the invariance equation \eqref{CH}, we may choose several functions $R_n$. For the sake of simplicity, we choose it to be constant and obtain \begin{equation} \label{average} R_n (\rho_0) = -\frac{1}{T(\rho_0)}\int_0^{T(\rho_0)}\Pi_E (F_n( x(s, \rho_0) + S_n(x(s; \rho_0) )\, ds. \end{equation} \begin{remark} For $n=1$, \eqref{average} recovers the results of the well known averaging method or the Melnikov theory. We can, therefore, interpret $R_n$ for $n>1$ as higher order extensions of Melnikov's method.\\ \end{remark} \begin{remark} The case of $n = 1$ in the above derivation can also be obtained by more familiar averaging arguments or fast/slow variables. Since these methods are more familiar in the mechanical systems community, we outline them here.\\ We observe that the conserved quantity of the unperturbed system is an slow variable for the perturbed system, as it evolves with a speed $O(\varepsilon)$. Denoting by $x^\varepsilon(t)$ the orbits of the perturbed system, we see that \[ \begin{split} \frac{d}{dt} I( x^\varepsilon(t) ) &= (\nabla I )(x^\varepsilon(t)) \cdot F_\varepsilon( x^\varepsilon(t))\\ &= (\nabla I )(x^\varepsilon(t)) \cdot ( L x^\varepsilon(t) + N(x^\varepsilon(t) ) + \varepsilon ( C x^\varepsilon (t) + G(x^\varepsilon(t)) ) \\ &= \varepsilon (\nabla I )(x^\varepsilon(t)) ( C x^\varepsilon (t) + G(x^\varepsilon(t)) ). \end{split} \] The change of energy on a cycle is then given by \[ \begin{split} \int_0^T \frac{d}{dt} I( x^\varepsilon(t) ) &= \varepsilon \int_{0}^T(\nabla I )(x^\varepsilon(t)) ( C x^\varepsilon (t) + G(x^\varepsilon(t)) ) \\ &= \varepsilon \int_{0}^T(\nabla I )(x^0(t)) ( C x^0 (t) + G(x^0(t)) ) + O(\varepsilon^2) . \end{split} \] where we used that, by the smooth dependence on parameters, during the finite interval $[0,T]$, we have $\|x^\varepsilon(t) - x^0(t)\| = O(\varepsilon)$.\\ Hence, we approximate, at first order, the evolution of of the energy over a cycle by the average. \\ If we seek for the invariant manifold to be given by selecting the normal variables as a function of the energy, we see that since this function will be of order $\varepsilon$, the invariant manifold will be obtained by selecting the normal variables to be periodic. \end{remark} \subsubsection{Some analytic considerations} Now we finish the proof of the first conclusion in Theorem~\ref{mainprop}. We examine carefully the formal solutions obtained in the previous section and obtain the desired estimates in the appropriate function space. Since the procedure is going to be applied a finite number of times, we will not need very detailed estimates.\\ We proceed by induction, assuming that the $K_1,\ldots, K_{n-1}$ and $R_1,\ldots, R_{n-1}$ are in the appropriate spaces and we want to conclude the same for the $K_n$ and $R_n$. \\ First of all, we argue that $K_n$ and $R_n$ are (complex) differentiable away from the origin. The differentiability of the average is clear. The differentiability of $y_0$ -- the initial condition in the transversal section -- with respect to $\rho$ follows from the fact that it is a solution of the implicit equation \eqref{eqy0rearranged} whose coefficients depend differentiability on the radius $\rho_0$. The differentiability with respect to the angle is clear for $R_n$ and for $K_n$ it follows because it solves a differential equation.\\ This shows that the function $R_n$ is also differentiable at $\rho_0 =0$. For the function $K_n$, we argue similarly. We note that the choice of $y_0$ is also differentiable as a function of $\rho_0$ (we are inverting a matrix which is clearly differentiable). Then, the propagation \eqref{cohomology} is also differentiable along the angle. Again, we use the assumption that the forcing terms vanish to ${\mathcal O}(\rho^{d }) $. This can, indeed, be achieved thanks to the normal form \eqref{normalform} and by noting that, since $F_n$ only has terms of order $d$ and higher, also the composition of $F_n$ with a function hat does not have any constant terms is of order $d$ or higher. By the same token, any algebraic function that involves terms of this form has the desired property, implying that $S_n$ vanishes up to order ${\mathcal O}(\rho^{d }) $.\\ Finally, to obtain the estimates claimed in Theorem \eqref{mainprop}, we find that the recursive solution procedure gave functions $K_1, \ldots, K_N$ as well as functions $R_1,\ldots R_N$, which are uniformly differentiable. Also, the function \begin{equation} \label{residual} F_\varepsilon \circ K^{\le N} - DK^{\le N} R^{\le N}, \end{equation} is differentiable in $\varepsilon$ for fixed $x \in B_\delta^2$.\\ Since the functions $K^{\le N}$ and $R^{\le N}$ have been chosen to match the derivatives with respect to $\varepsilon$ of the invariance equation up to order $N$, the first conclusion of Theorem~\ref{mainprop} follows. \begin{remark} \label{rem:finiteregularityexpansion} We note that the results obtained here apply also to the case that $F_\varepsilon$ is only finitely differentiable, jointly in $\varepsilon$ and $x$. If the function is $C^\ell$ jointly in $\varepsilon$ and $x$ we see that the equations at order $j$ involve $C^{\ell-j}$ functions and the algebraic functions of the previously computed solutions. The solutions are obtained using only soft arguments such as implicit function theorems and hence, the solutions are as smooth as the right hand side. Therefore, by induction, we obtain that the $K_j$'s and $R_j$'s are $C^{k -j}$ and that the of the expansion up to order $N$ is $O(\|\varepsilon\|^{N+1})$ small in the sense of $C^{\ell -N -2}$. \end{remark} \subsection{An alternative approach to the theory of the cohomology equation and its analytic estimates using Fourier series} \label{sec:Fourier} In several applications, it is convenient to develop an approach for \eqref{cohomology} based on Fourier series \cite{JorbaV98,Masdemont05}. We recall that a function $\phi(x)$ is analytic in a neighborhood of the origin if and only if it admits an expansion \begin{equation} \phi(x) = \sum_{n_1, n_2 \in {\mathbb N}} \phi_{n_1,n_2} x_1^{n_1} x_2^{n_2}. \end{equation} Using polar coordinates $x_1 = \rho \cos(\theta), x_2 = \rho \sin(\theta)$ we see that \begin{equation} \phi(x) = \sum_{n_1, n_2 \in {\mathbb N}} \phi_{n_1,n_2} \rho^{n_1+ n_2} \cos(\theta)^{n_1} \sin(\theta)^{n_2} = :\sum_{n \in {\mathbb N}} \rho_n \sum_{k \in {\mathbb Z}, \|k\| \le n} e^{\mathrm{i} k \theta} = : \sum_{k \in {\mathbb Z}} \phi_k(\rho) e^{\mathrm{i} k \theta}, \end{equation} where $\phi_k(\rho)$, $\rho_n$ and $\phi_{n_1,n_2}$ are related through the binomial theorem and finite summation.\\ As it is well known, functions are analytic in a non-trivial domain if and only if the coefficients decrease exponentially. A certain exponential rate in the decrease of the coefficients implies analyticity in a domain and analyticity in a domain implies an exponential rate of decrease of the coefficients. The conditions are not exactly symmetric, but this does not matter for us, since we will only use the procedure a finite number of times.\\ To study \eqref{cohomology}, we can observe that, for each fixed value of $\rho$, the equation \eqref{charu} is a linear periodic equation in the time variable $s$. For sufficiently small $\rho$, the matrix $M$ is a perturbation of a constant coefficient equation. It follows from Floquet theory \cite{Chicone06} that for a fixed $\rho$, we can perform a linear, $T$-periodic change of variables in such a way that the matrix becomes independent of $s$. \footnote{Of course, in the general Floquet theory, we may need to make a $2T$-periodic change of variables, but in our case, for small $\rho$ the differential equation is a perturbation of the constant equation with the matrix $L$ as a right-hand side, so that the reducibility matrix is periodic and depends analytically on the parameter $\rho$.} Furthermore, the change of variables can be chosen in a way which depends analytically on $\rho$.\\ Hence, the equation \eqref{cohomology} is equivalent to \begin{equation} 2 \pi \mathrm{i} \Omega(\rho) k \phi_k(\rho) - A(\rho) \phi_k(\rho) = \eta_k(\rho). \end{equation} For $k \ne 0$, the above equation can be solved because $2\pi \mathrm{i} \Omega(\rho) k \notin\sigma(A(\rho))$ for all $\rho$ in an small neighborhood. Hence, we just set \begin{equation} \phi_k(\rho) = (2 \pi \mathrm{i} \Omega(\rho) k \phi_k(\rho) - A(\rho) \phi_k(\rho))^{-1} \eta_k(\rho). \end{equation} For $k=0$, the equation amounts to \begin{equation} A(\rho) \phi_0(\rho) = \eta_0(\rho). \end{equation} As indicated, in Section~\ref{changes} we have that there is an eigenvalue zero of $A(\rho)$ corresponding to the change of energy and, by the non-resonance assumption and the perturbation arguments, this is the only zero eigenvalue. Hence we obtain, again, that the obstruction is just that the average of the change of energy of $\eta$ vanishes.\\ There is a constant $C>0$ such that \begin{equation} \\|2 \pi \mathrm{i} \Omega(\rho) k \phi_k(\rho) - A(\rho) \phi_k(\rho))^{-1}\\| \le C. \end{equation} Therefore, if $\eta$ satisfies the obstruction and is an analytic function a domain, then the solution of \eqref{cohomology} is analytic in a slightly smaller domain. \begin{remark} The above Fourier analysis procedure also works for finitely differentiable functions, but the results are weaker than those obtained by the method of integral equations. We know that if $\phi$ is $C^\ell$, then the Fourier coefficients satisfy $\|\phi_n\| \le C n^{-\ell}$ for some constant $C>0$. The approximate converse is that if $\|\phi_n\| \le C n^{-\ell- \tau}$ for some $\tau > 1$ then $\phi \in C^\ell$.\\ Hence, by working with Fourier coefficients to analyze \eqref{cohomology}, we obtain estimates with $ 1+ \tau$ derivatives less. For the purposes of this section, this is not a fatal loss since we only need to apply it a finite number of times. On the other hand, it would be a very useless estimate for a fixed point argument. One can, however, avoid this shortcoming by working in Sobolev spaces. A comparison between Fourier methods and integral formulas for closely related problems appears in \cite{HuguetL13}. \end{remark} \subsection{The Fixed-Point Argument} \label{sec:fixedpoint} Throughout this section, we will assume that the unperturbed vector field has been normalized according to \eqref{normalization} and that we have chosen coordinates $(x,y)\in\mathbb{R}^2\times\mathbb{R}^{n-2}$, such that the LSM corresponds to the invariant plane $\{y=0\}$.\\ We start by transforming equation \eqref{CH} into an equivalent form, suitable for a fixed point argument. Let $\phi^t_\varepsilon$, either defined as a function $\phi_\varepsilon^t:{\mathbb R}^n\to{\mathbb R}^n$ or $\phi_\varepsilon^t:{\mathbb C}^n\to{\mathbb C}^n$, be the flow map associated to the vector field $F_\varepsilon$, i.e., \begin{equation} \frac{d}{dt} \phi^t_\varepsilon = F_\varepsilon \circ \phi^t_\varepsilon, \qquad \phi^0_\varepsilon = {\rm Id}. \end{equation} Analogously, let $r^t_\varepsilon$, either defined as a function $r_{\varepsilon}^t:{\mathbb R}^n\to{\mathbb R}^n$ or $r_{\varepsilon}^t:{\mathbb C}^n\to{\mathbb C}^n$, be the flow associated to the vector field $R_\varepsilon$, i.e., \begin{equation} \frac{d}{dt} r^t_{\varepsilon} = R_\varepsilon \circ r^t_\varepsilon, \qquad r^0_\varepsilon = \text{Id}. \end{equation} To simplify notation, we denote the corresponding time-$T_0$ maps, cf. \eqref{normalization}, as $\phi_\varepsilon = \phi^{T_0}_\varepsilon$ and $r_\varepsilon = r^{T_0}_\varepsilon$. \\ The invariance equation \eqref{CH} is then equivalent to \begin{equation}\label{invflow} \phi^t_\varepsilon(K_\varepsilon(x))=K_\varepsilon(r^t_\varepsilon(x)), \end{equation} for all $t\geq 0$.\\ Consider equation \eqref{invflow} only for $t = T_0$ and rewrite it as \begin{equation} \label{CH2} K_\varepsilon(x)=\phi_\varepsilon^{-1} \circ K_\varepsilon(r_\varepsilon(x)). \end{equation} Later, we will show that the solution of \eqref{CH2} also solves \eqref{invflow} and, hence \eqref{CH}. To establish existence of solution of \eqref{CH} we will show that the operator defined by the right-hand side of \eqref{CH2} is a contraction in a ball around the approximate solution produced in Part 1) of Theorem~\ref{mainThm}.\\ As we already have found an approximate solution $K_\varepsilon^{\leq N}(x)$ in both $x$ and $\varepsilon$ \eqref{approx}, we can reformulate \eqref{invflow} as \begin{equation} K_\varepsilon^{\leq N}(x)+K^{>N}_\varepsilon(x)=\phi_\varepsilon^{-1}\Big(K_\varepsilon^{\leq N}(r_\varepsilon(x))+K_\varepsilon^{>N}(r_\varepsilon(x))\Big). \end{equation} That is to say, $K^{>N}_\varepsilon$ should be a fixed point of the $\varepsilon$-dependent functional \begin{equation}\label{Taudefined} \mathcal{T}_{\varepsilon}({\hat K})(x)= \phi_\varepsilon^{-1}\circ\Big(K_\varepsilon^{\leq N}(r_\varepsilon(x))+ {\hat K}(r_\varepsilon(x))\Big) - K_\varepsilon^{\leq N}(x). \end{equation} We emphasize that the approximate solution $K_\varepsilon^{\leq N}$ has been obtained on all $x$ in a neighborhood which is independent of $\varepsilon$ since we have just integrate along periodic orbits.\\ We will first show that for all $\varepsilon \in {\mathcal C}_\theta$, $\mathcal{T}_\varepsilon$ defined in \eqref{Taudefined}, maps a ball in ${\mathcal A}^{real}_{\delta, d}$ to itself and is a contraction. After that, we will study the dependence of the fixed point on $\varepsilon$ and show that the fixed point is analytic in $\varepsilon$ for $\varepsilon \in {\mathcal C}_\theta$ and that, near zero, it has an asymptotic expansion. We also recall, see Remark~\ref{rem:negativeeps}, that, by reversing the time, we can also study the case $\varepsilon \in -{\mathcal C}_\theta$. \\ In particular, we will obtain that, if we consider $\varepsilon \in {\mathbb R}$ -- the physically more interesting case -- we have that the fixed point as a function of $\varepsilon$ is real analytic for $\varepsilon\in {\mathbb R} \setminus \{0\} $ and $C^\infty$ at $\varepsilon = 0$. First, we claim that $\mathcal{T}_\varepsilon:B_{\mathcal{A}_{\delta,d}^{real}}^\sigma\subseteq\mathcal{A}_{\delta,d}^{real}\to\mathcal{A}_{\delta,d}^{real}$, for \begin{equation} B_{\mathcal{A}_{\delta,d}^{real}}^\sigma=\{K\in\mathcal{A}_{\delta,d}^{real}:\\|K\\|_{\mathcal{A}_{\delta,d}}<\sigma\}, \end{equation} is well-defined.\\ Indeed, by the second statement in Assumption \ref{Asspert}, ${\tilde B}^{n}_\delta$ is mapped into itself by $r_\varepsilon$ for all $\varepsilon \in {\mathcal C}_\theta$. Therefore $K(r_\varepsilon(x))$ is well-defined for $\|x\|<\delta$. We also note that $D( \phi^{-t}_\varepsilon \circ K^{<N} \circ r_\varepsilon^t)(0) = \Id$ and that if $K$ vanishes to high order, we get that $ \mathcal{T}_\varepsilon(K)$ also satisfies the normalization of the derivatives \eqref{normalization}.\\ To see that $\mathcal{T}_\varepsilon(K(x))={\mathcal O}(\|x\|^{d})$, it suffices to employ the coordinate system presented in Section \ref{changes} and the fact that $K_\varepsilon^{\leq N}$ solves the invariance equation up to order $d$ in $x$. Clearly, $\mathcal{T}_\varepsilon(K)\|_{{\mathbb R}^2}\subseteq{\mathbb R}^N$.\\ Since $K_\varepsilon^{\leq N}$ is an approximate solution to \eqref{invflow} up to oder $\|\varepsilon\|^N$, it follow that \begin{equation} \\|\mathcal{T}_\varepsilon(0)\\|_{\mathcal{A}_{\delta,d}}={\mathcal O}(\|\varepsilon\|^N). \end{equation} We will assume that the size of the ball $B_{\mathcal{A}_{\delta,d}^{real}}^\sigma$ in function space is $\sigma=\varepsilon^M$, i.e., \begin{equation} \hat{K}(x)=\mathcal{O}(\varepsilon^M), \end{equation} for all $\|x\|<\delta$ and for some $M>1$, indicating that the correction to the formal expansions will be small. The exact value of $M$ will be determined in the course of the proof.\\ To estimate the contraction rate of the functional $\mathcal{T}_\varepsilon$, we calculate \begin{equation} \begin{split} \\|\mathcal{T}_\varepsilon({\hat K}_1)&-\mathcal{T}_\varepsilon({\hat K}_2)\\|_{\mathcal{A}_{\delta,d}} =\sup_{z\in {\tilde B}^{n}_\delta}\|z\|^{-d}\left\|\phi_\varepsilon^{-1}\Big(K_\varepsilon^{\leq N}(r_\varepsilon(z))+{\hat K}_1(r_\varepsilon(z))\Big)-\phi_\varepsilon^{-1}\Big(K_\varepsilon^{\leq N}(r_\varepsilon(z))+{\hat K}_2(r_\varepsilon(z))\Big)\right\|\\ &\leq \Lip(\phi_\varepsilon^{-1})\sup_{z\in {\tilde B}^{n}_\delta}\|z\|^{-d}\|{\hat K}_1(r_\varepsilon(z))-{\hat K}_2(r_\varepsilon(z))\|\\ &\leq \Big[1+(\beta+\mathcal{O}(\theta))\|\varepsilon\|+\mathcal{O}(\|\varepsilon\|^2)\Big]\sup_{z\in {\tilde B}^{n}_\delta}\|z\|^{-d}\|{\hat K}_1(r_\varepsilon(z))-{\hat K}_2(r_\varepsilon(z))\|. \end{split} \end{equation} Here, we have used Lemma \ref{uniform} together with \eqref{estDphi-1} and the fact that, in our coordinate system, \begin{equation} K^{\leq N}_\varepsilon(z)+\hat{K}(z)=K_0(z)+\mathcal{O}(\varepsilon)=(z,0)+\mathcal{O}(\varepsilon). \end{equation} Also, we have used again that $\varepsilon=(1+\mathcal({\theta}))\|\varepsilon\|$. To proceed, we estimate the contraction rate of the perturbed reduced dynamics as \begin{equation} \begin{split} \\|\mathcal{T}_\varepsilon({\hat K}_1)&-\mathcal{T}_\varepsilon({\hat K}_2)\\|_{\mathcal{A}_{\delta,d}}\\ &\leq\Big[1+(\beta+\mathcal{O}(\theta))\|\varepsilon\|+\mathcal{O}(\|\varepsilon\|^2)\Big]\sup_{z\in {\tilde B}^{n}_\delta}\|z\|^{-d}\|r_\varepsilon(z)\|^{d}\|r_\varepsilon(z)\|^{-d}\|{\hat K}_1(r_\varepsilon(z))-{\hat K}_2(r_\varepsilon(z))\|\\ &\leq \Big[1+(\beta+\mathcal{O}(\theta))\|\varepsilon\|+\mathcal{O}(\|\varepsilon\|^2)\Big]\gamma^d\sup_{z \in {\tilde B}^{n}_\delta}\|z\|^{-d}\|z\|^{d}\sup_{z \in {\tilde B}^{n}_\delta}\|r_\varepsilon(z)\|^{-d}\|{\hat K}_1(r_\varepsilon(z))-{\hat K}_2(r_\varepsilon(z))\|\\ &\leq \Big[1+(\beta+\mathcal{O}(\theta))\|\varepsilon\|+\mathcal{O}(\|\varepsilon\|^2)\Big]\gamma^{d}\\|{\hat K}_1-{\hat K}_2\\|_{\mathcal{A}_{\delta,d}}, \end{split} \end{equation} where $\gamma$ is the contraction factor introduced in \eqref{defgamma}.\\ Expanding the contraction factor to leading orders in $\|\varepsilon\|$, we obtain \begin{equation} \\|\mathcal{T}_\varepsilon({\hat K}_1)-\mathcal{T}_\varepsilon({\hat K}_2)\\|_{\mathcal{A}_{\delta,d}} \leq \Big( 1 +(\beta+\mathcal{O}(\theta)-d[\alpha - {\mathcal O}(\theta) -\mathcal{O}(\delta)])\|\varepsilon\|+\mathcal{O}(\|\varepsilon\|^2 )\Big) \\|{\hat K}_1-{\hat K}_2\\|_{\mathcal{A}_{\delta,d}}, \end{equation} which, by \eqref{defL}, implies that $\mathcal{T}_\varepsilon$ is a contraction for $\varepsilon$, $\theta$ and $\delta$ small enough and $d$ big enough such that \begin{equation} \beta+\mathcal{O}(\theta)-d[\alpha - {\mathcal O}(\theta) -\mathcal{O}(\delta)]<0. \end{equation} \begin{remark}\label{rem:uniformcontraction} It will be important for future applications to observe that, after we fix the $\theta$, $\delta$ and $\beta$ the contraction is uniform for all values of $\varepsilon \in {\mathcal C}_\theta$ that satisfy $0 < a_- < \| \varepsilon\| < a_+$, for some constants $a_-$ and $a_+$. Of course, these uniform rate of contraction becomes close to $1$ as $a_-$ converges to zero and we cannot obtain a uniform contraction for all $\varepsilon \in {\mathcal C}_\theta$. \end{remark} To see that $\mathcal{T}_\varepsilon\Big(B_{\mathcal{A}_{\delta,d}^{real}}^\sigma\Big)\subseteq B_{\mathcal{A}_{\delta,d}^{real}}^\sigma$, for some $\sigma=\sigma(\varepsilon)>0$, we observe that, for $\varepsilon$ sufficiently small, \begin{equation}\label{sigmaest} \begin{split} \\|\mathcal{T}_\varepsilon({\hat K})\\|_{\mathcal{A}_{\delta,d}}&=\\|\mathcal{T}_\varepsilon({\hat K})-\mathcal{T}_\varepsilon(0)+\mathcal{T}_\varepsilon(0)\\|_{\mathcal{A}_{\delta,d}}\\ &\leq \\|\mathcal{T}_\varepsilon({\hat K})-\mathcal{T}_\varepsilon(0)\\|_{\mathcal{A}_{\delta,d}}+\\|\mathcal{T}_\varepsilon(0)\\|_{\mathcal{A}_{\delta,d}}\\ &\leq \Big( 1 +\Big[\beta+\mathcal{O}(\theta)-d[\alpha - {\mathcal O}(\theta) -\mathcal{O}(\delta)]\Big]\|\varepsilon\|+\mathcal{O}(\|\varepsilon\|^2 )\Big)\\|{\hat K}\\|_{\mathcal{A}_{\delta,d}}+\mathcal{O}(\varepsilon^N)\\ &\leq \sigma\Big( 1 +\Big[\beta+\mathcal{O}(\theta)-d[\alpha - {\mathcal O}(\theta) -\mathcal{O}(\delta)]\Big]\|\varepsilon\|+\mathcal{O}(\|\varepsilon\|^2 )\Big)+C_2\|\varepsilon\|^N\\ &\leq \sigma, \end{split} \end{equation} for some $C_2>0$. To ensure the last inequality in \eqref{sigmaest}, it suffices to take $\varepsilon$ sufficiently small and \begin{equation} \sigma\geq\frac{C_2 \|\varepsilon\|^{N-1}}{d[\alpha - {\mathcal O}(\theta) -\mathcal{O}(\delta)]-\beta-\mathcal{O}(\theta)-\mathcal{O}(\|\varepsilon\|^2)}>0. \end{equation} To finish the proof of Theorem~\ref{mainThm}, we have to show that the solutions of \eqref{CH2} also solve \eqref{invflow}. Note that the main difference between \eqref{CH2} and \eqref{invflow} is that \eqref{invflow} is only the evolution of $r_\varepsilon$ for one time, while \eqref{CH2} involves the evolution for all times. To achieve this, we use an argument coming from \cite{Cab2003}. If $K_\varepsilon$ is a solution of \eqref{CH2}, for any $s \in{\mathbb R}$ sufficiently small, we have that \begin{equation} \begin{split} \phi^s_\varepsilon \circ K_\varepsilon \circ r^{-s}_\varepsilon & = \phi^s_\varepsilon \circ \phi_\varepsilon^{-1} \circ K_\varepsilon \circ r_\varepsilon \circ r^{-s}_\varepsilon\\ & = \phi_\varepsilon^{-1} \circ \phi^s_\varepsilon \circ K_\varepsilon \circ r^{-s}_\varepsilon \circ r_\varepsilon \\ & = \mathcal{T}_\varepsilon(\phi^s \circ K_\varepsilon \circ r^{-s}_\varepsilon), \end{split} \end{equation} by the flow property of $\phi_\varepsilon^s$. Hence, we obtain that $\phi^s_\varepsilon \circ K_\varepsilon \circ r^s_\varepsilon$ is also a fixed point of $\mathcal{T}_\varepsilon$ and it also satisfies the normalization conditions specified in our main theorem. For $s$ small enough, $\phi^s_\varepsilon \circ K_\varepsilon \circ r^s_\varepsilon$, will be in the domain of uniqueness of the fixed point theorem. Therefore, we obtain that there exists an interval of $s$ such that \begin{equation} \phi^s_\varepsilon \circ K_\varepsilon \circ r^{-s}_\varepsilon = K_\varepsilon, \end{equation} which implies \eqref{invflow}.\\ To prove the analyticity in $\varepsilon$ for $\varepsilon \in {\mathcal C}_\theta$, we recall that the contraction properties are the same for all the $\varepsilon \in {\mathcal C}_\theta$ such that $0 < a_- < \|\varepsilon\| < a_+$, for some $a_-,a_+>0$, and that around any point, we can find a set ${\mathcal U}$ -- without of loss of generality inside the previous one -- so that for all values in this set, there is a ball of radius $\sigma > 0$ in ${\mathcal A}^{real}_{d,\delta}$ that gets mapped into itself, see Remark~\ref{rem:uniformcontraction}.\\ We also observe that if $\tilde K_\varepsilon$ is analytic in $\varepsilon$ for $\varepsilon$ in such a domain, we also obtain that $\mathcal{T}_\varepsilon( \tilde K_\varepsilon)$ is also analytic in $\varepsilon$ -- it suffices to apply the chain rule for derivatives. Putting these two remarks together, we obtain that $\mathcal{T}^n_\varepsilon(0)$ is a sequence of uniformly converging analytic functions and hence their limit is analytic in the domain. Therefore, the fixed-point depends analytically on $\varepsilon$. \begin{remark} The contraction argument also works for the case that the $F_\varepsilon$ is $C^\ell$ if $\ell > d$. The Lipschitz constants of the operators $\mathcal{T}_\varepsilon$ acting on $C^r$-spaces vanishing to order $d$ are estimated in \cite[Proposition 3.2]{de1997invariant} or \cite{Cab2003}. Hence we obtain that the invariant manifolds produced in Theorem~\ref{mainprop} are locally unique under the condition that the manifolds are invariant and $C^r$ for $r>d$. Note also that, remembering Remark~\ref{rem:finiteregularityexpansion}, we obtain, rather straightforwardly, an analogue for finite regularity of the existence results claimed in Theorem~\ref{mainprop}.\\ The results of Theorem~\ref{mainprop} on regularity with respect to $\varepsilon$ or with respect to parameters seem to be also true, but they seem to require substantial work (which we will not undertake here). \end{remark} \begin{remark} As a note for experts (which most readers may want to postpone) we note that the operator $\mathcal{T}_\varepsilon$ is differentiable in $\hat K$ in $C^\ell$ spaces (note that this is not the case with the operators used in the graph transform approach to NHIHM \cite{Fenichel73}). To obtain differentiability with respect to parameters, the main problem is that, in spaces of finite differentiability, the operator $\mathcal{T}_\varepsilon$ is not differentiable with respect to $\varepsilon$., since the formal derivative with respect to $\varepsilon$ would include a term $D \phi^{-1}_\varepsilon\circ ( K_\varepsilon^{\leq N} + {\hat K}) \circ r^1_\varepsilon D ( K_\varepsilon^{\leq N} + {\hat K})\circ r^1_\varepsilon D_\varepsilon r^1_\varepsilon $. Indeed, this derivative involves $D{\hat K}$, so that the derivative in a certain $C^\ell$ space would involve a term that can only be controlled in $C^{\ell +1}$. This is a well known problem for operators involving composition in the left \cite{LlaveO99}. As a consequence, the standard implicit function theorem does not apply and one needs to develop more sophisticated methods.\\ For the problem of non-resonant manifolds, a very detailed study of the differentiability with respect to parameters, developing specialized implicit function theorems, appears in \cite{Cab2003pam}. The results of \cite{Cab2003pam} show that the differentiable manifold will be continuously differentiable -- in a rather subtle sense -- for $\varepsilon \in (0, \varepsilon_0)$. Since it satisfies the asymptotic expansions, Theorem~\ref{mainprop} shows also it is differentiable at $\varepsilon = 0$. The question of continuity of the derivative with respect to $\varepsilon$ at $\varepsilon = 0$ is significantly more subtle and, as far as we know, does not follow from the literature above. \end{remark} \section{Some mathematical examples} \label{sec:examples} In this section, we present some examples that show that some assumptions of Theorem \eqref{mainThm} are necessary to guarantee the existence of an invariant manifold with the specified properties and having a size independent of the dissipation parameter. \\ In particular, we show that if the eigenvalues of the linear part depend upon $\varepsilon^p$, for some $p>1$, existence of a sufficiently differentiable invariant manifold is no longer guaranteed. Indeed, the $\varepsilon$-dependence of the eigenvalue reflects the singular nature of the problem. Notice that these examples show that \eqref{defL} is not enough to guarantee the existence of an SMM of size independent of the dissipation. One also seems to need some global information on the shape of the dissipation.\\ We will first describe in Example~\ref{firstexample} a very special system. After we understand its basic properties, we will show how to modify the system slightly so that there are obstructions for the existence of SMM of size $1$. \begin{example}\label{firstexample} Consider the following system in polar coordinates $\rho,\theta$ and $u, \psi$. \begin{equation}\label{example1} \begin{split} &\rho' = -\varepsilon^2\rho + \varepsilon \rho^2,\\ &\theta' = \omega,\\ &u' = a \varepsilon^2 u, \\ &\psi' = \gamma, \end{split} \end{equation} for $(\rho,\theta,u, \psi)\in{\mathbb R}^+\times {\mathbb S}^1\times{\mathbb R}^+\times {\mathbb S}^1$ and $a \in {\mathbb R}_+$ a number which we will adjust to get obstructions to the regularity.\\ Equation \eqref{example1} corresponds to a polynomial vector field in Cartesian coordinates, being a mere rotation for $\varepsilon=0$. The linear part of the system \eqref{example1} changes only by $\varepsilon^2$, thus violating assumption \ref{Asspert}. If $\gamma$ is not an integer multiple of $\omega$, we can verify the hypothesis of LSM. Indeed, in this very simple example, the LSM is just the plain described by $\rho$ and $\theta$. For $\varepsilon=0$, the energy is $\rho^2+u^2$, which is non-degenerate. The equation for $\psi$ plays no role and we can avoid it in much of the analysis. For simplicity, we have chosen in the first of \eqref{example1} only a perturbation which is quadratic in $\rho$, which leads to some coincidences which are not really relevant for the analysis and could be removed by adding more complicated nonlinearities. The key point is that the nonlinear term is affected by $\varepsilon$ and takes the trajectories out of the origin while the linear term is affected by $\varepsilon^2$ and takes the trajectories in, even if very slowly. For $\varepsilon> 0$, the origin is a stable, hyperbolic fixed-point. The local stable manifold of the hyperbolic fixed-point is precisely the plane $u,\theta$. Moreover, there exists an unstable limit cycle at $\rho=\varepsilon$, $u = 0$, which we denote as $P$. The Lyapunov exponent of $P$ in the $\rho$-direction is given by $\varepsilon^2$, while in the $u$-direction, it is given by $a\varepsilon^2$ (the variable $\psi$ makes that the eigenvalues of the return map in the periodic orbit are not just the Lyapunov exponents but that they acquire a phase.)\\ Now that we have understood the geometry of \eqref{example1}, we can construct perturbations that do not have any analytic SMMs near the LSM. Note that the perturbations we construct now are perturbations of the function $F_\varepsilon(x)$. That is, we will change the function of two variables $\varepsilon$ and $x$, keeping, of course, the normalizations. As a concrete way to understand the perturbations of the family we can imagine adding an extra parameter $\nu$ and that we add terms containing $\nu\varepsilon$ to the family. \begin{remark} Note that the analysis so far already shows that the proofs of existence of SMM based on the graph transform will have a problem even to get started, cf. Example 5. By the unperturbed center flow, there is no domain of size $1$ in $u$ that gets mapped into itself. In analytic regularity, we cannot cut off and extend, so that the graph transform proofs for analytic manifolds have problems even being formulated. Even for $C^2$-regularity, we cannot find domains that are mapped to themselves.\\ Of course, the failure of a method of proof does not automatically imply the failure of the conclusions (e.g. the folding of the manifold gives problems to graph methods but poses no problem to parameterization methods), but it certainly gives a hint of the problems. Later, we will see that one can exclude the existence of SMMs of high enough regularity. \end{remark} We start by observing that for any perturbation, the origin will be hyperbolic and that the only invariant manifold near the origin close to the stable manifold of the unperturbed system is the unstable manifold. Hence, any SMM has to coincide with the stable manifold near the origin. We also note that any possible SSM of the perturbation has to go through the periodic orbit $P$. This is because the periodic orbit $P$ has a basin of repulsion of size $\varepsilon$. Any invariant object that does not include $P$ has to be out of the basin of repulsion.\\ The key observation is that, since $P$ has eigenvalues that do not resonate with those of the complement, there is a non-resonant invariant manifold near $P$. This manifold is unique under the assumption that it is sufficiently differentiable, cf. \cite{de1997invariant, Cab2003,CABRE2005444}. We note that the space spanned by the non-resonant eigenvectors is close to the tangent of the LSM. This non-resonant manifold is persistent under small perturbations. We observe that the non-resonant manifold near $P$ is the only candidate for an invariant manifold close to the LSM for $\varepsilon = 0$. Hence, it is the only candidate for being the SMM of size $1$ near the LSM which is sufficiently differentiable. Therefore, we have two conditions that the SSM or size of order $1$ has to satisfy: It has to agree with the stable manifold near the origin and it has to be the non-resonant manifold near $P$.\\ The coincidence of these two manifolds is an infinite codimension phenomenon in the space of maps. If they indeed coincided, a generic small perturbation will break this coincidence, as the perturbation theory for these two manifolds lead to very different Melnikov functions, cf. \cite{de1997invariant} for more details. Some similar examples for parabolic manifolds with numerical computations appear in \cite{baldoma2007parameterization}. Notice also that the above examples show that finite differentiable manifolds of size of order one cannot exist for general perturbations that do not satisfy our assumptions. Note also that the obstructions cannot be easily seen by studying just the jet of the manifold at the origin. One can also consider slightly more complicated perturbations for the $u$-equation in \eqref{example1} so that one can get several attracting/hyperbolic periodic orbits appearing with the perturbation. \end{example} \begin{remark} The key to make the example \eqref{example1} work is that the non-linear terms push out of the origin with a coefficient $\varepsilon$ while the linear terms push in with a coefficient $\varepsilon^2$. The possibility of the example disappears if one adds the extra assumption that all the terms push in. It would be interesting to investigate if indeed under some more strict global assumption one can recover the result of existence of SSMs. This result would be physically interesting since many models of physical interest seem to satisfy the extra assumption. \end{remark} \begin{example}\label{non-uniform} Consider a system of the following form, \begin{equation}\label{hyper} \begin{split} &\dot{x}=Jx-2\|y\|^2\text{tr}\Gamma(\|x\|^2)\frac{x}{\|x\|^2},\\ &\dot{y}=\Gamma(\|x\|^2)y, \end{split} \end{equation} where $x\in\mathbb{R}^2$, $y\in\mathbb{R}^{2k}, k>1$, $J$ being the standard symplectic matrix and \begin{equation} \Gamma(\|x\|^2)=\mathrm{diag}(\Gamma_1(\|x\|^2),...,\Gamma_k(\|x\|^2)), \end{equation} for $2\times 2$ matrices $\Gamma_1,...,\Gamma_k$ such that $\text{tr}(\Gamma(0))=0$ as well as $D_x\text{tr}(\Gamma(0))=0$, implying that $\frac{x\text{tr}\Gamma(\|x\|^2)}{\|x\|^2}$ is differentiable at $x=0$. By construction, the scalar function $I(x,y)=\|x\|^2+\|y\|^2$ is a conserved quantity for system \eqref{hyper}. Clearly, at the same time, system \eqref{hyper} need not be Hamiltonian. If the matrices $\Gamma_1,...,\Gamma_k$ are now chosen in such a way that $\text{tr}(\Gamma(x))\neq 0$ for $x\neq 0$, then system \eqref{hyper} can admit hyperbolic directions, showing that the assumption a symplectic map in Lemma \ref{uniform} cannot be omitted. The LSM is thus, in this example, a normally hyperbolic invariant manifold with weakening hyerbolicity as $x\to 0$. Note also that the hyperbolicity properties can change in $\|x\|\approx\varepsilon$ in this model. \end{example} \begin{remark} The examples above exclude not only the existence of analytic manifolds of size of order one but they also exclude the existence of $C^\ell$ manifolds of size of order $1$ for large enough $\ell$. One question that deserves more exploration is whether one can get invariant manifolds with very low regularity even in the cases where we cannot get very differentiable manifolds. \end{remark} \begin{remark} \label{rem:easy} The examples above allow us to compare the results obtained in this paper with other possible alternatives and will serve the experts to understand some of our choices.\ To study general perturbations of LSMs (even those not satisfying Assumption~\ref{Asspert}), we could apply the theory \cite{de1997invariant, Cab2003,CABRE2005444} and, for every value of $\varepsilon \in {\mathbb C}$ leading to $\Re(\lambda_\varepsilon) < 0$, produce an analytic invariant manifold. Since the coefficients can be computed recursively, it is easy to see that these manifolds converge to the LSM in the sense that each of the derivatives at zero converge. It seems that one could obtain similar results using \cite{Moulton20}. Going through the proofs in \cite{de1997invariant, Cab2003,CABRE2005444}, one can see that the fact that the contraction at the origin is weak for small $\varepsilon$ results in the straightforward reading of the results applying only to functions defined in smaller domains as $\varepsilon$ goes to zero.\\ The examples above show that this decrease of the domain is not an artifact of the proof. Indeed, to obtain manifolds defined in a domain of size independent of $\varepsilon$ we need not only to take advantage of the conserve quantity for the unperturbed case, but also of some sort of global information on the perturbations such as Lemma~\ref{prop:domain}. This seems to require that the Floquet multipliers remain elliptic in a neighborhood.\\ Proofs based only on local information of power series near the origin will only produce results in small domains unless one take advantage of subtle cancellations that will depend on other hypothesis. The physical meaning of the manifolds produced in this paper is clear since they serve as long paths that guide the systems to equilibria \cite{GorbanK05, Smooke91}. We do not know what could be the physical meaning of the manifolds obtained by power matching (in the cases when they do not coincide with those in the theorem here). \end{remark} \section{Applications and examples} \label{sec:applications} Next, we compare Theorem \ref{mainThm} to the following theorem on the existence of two-dimensional spectral submanifolds for autonomous dynamical systems in \cite{Haller2016}:\\ \begin{theorem}\label{SSM} Consider a two-dimensional spectral subspace $X_{1}$ of the linearization $L+\varepsilon C$ of equation (\ref{main}) associated with the eigenvalues $\lambda_{\varepsilon},\overline{\lambda}_{\varepsilon}$. Assume that $\text{Re}(\sigma(L+\varepsilon C))<0$ and assume that the non-resonance condition \begin{equation} k\lambda_{\varepsilon}+l\overline{\lambda}_{\varepsilon}\neq\mu,\label{nonresSSM} \end{equation} for all $k,l\in{\mathbb Z}$ and all eigenvalues $\mu\in\sigma(L+\varepsilon\tilde{L})\setminus\{\lambda_{\varepsilon},\overline{\lambda}_{\varepsilon}\}$, holds. Then, there exists a two-dimensional, analytic, invariant manifold $W_{\varepsilon}$, tangent to the spectral subspace $X_{1}$ around the trivial solution $X=0$. \end{theorem} The proof of the above theorem is based on a theory of invariant spectral manifolds in Banach spaces derived in \cite{Cab2003pam}, \cite{Cab2003} and \cite{CABRE2005444}. Since the non-resonance condition in eq. \eqref{nonresSSM} is weaker than the non-resonance condition \eqref{nonres}, a spectral submanifold guaranteed by Theorem \ref{SSM} may not converge to the LSM of Theorem \ref{exLSC} for $\varepsilon\to 0$, as the following example shows. \begin{example}(Not all SSMs are perturbed LSMs.) Consider the dynamical system \begin{equation} \begin{split} & \dot{\xi}=\left(\begin{matrix}-\varepsilon\lambda & 1\\ -1 & -\varepsilon\lambda \end{matrix}\right)\xi,\\ & \dot{\eta}=\left(\begin{matrix}-\varepsilon\mu & \alpha\\ -\alpha & -\varepsilon\mu \end{matrix}\right)\eta+F(\xi), \end{split} \label{ex1} \end{equation} for the functions $t\mapsto\xi(t),\eta(t)\in{\mathbb R}^{2}$, some nonlinearity $F(\xi)=\mathcal{O}(\|\xi\|^{2})$ and the parameters $\lambda,\mu\in{\mathbb R}^{+},\varepsilon>0,$ as well as $\alpha\in{\mathbb Z}$. The spectrum of the linearization is given by $\{-\varepsilon\lambda\pm\mathrm{i}\alpha,-\varepsilon\mu\pm\mathrm{i}\alpha\}$. We assume that $\lambda$ and $\mu$ are independent over the integers, i.e., that there is no $n\in{\mathbb Z}$ such that $\lambda=n\mu$. We conclude from Theorem \ref{SSM} the existence of an analytic, two-dimensional, invariant manifold tangent to the $\xi$-plane for any $\varepsilon>0$. For small enough $\xi$, we can write $\eta$ as a function of $\xi$, i.e., $\eta(t)=H(\xi(t))$, for $H(\xi)=\sum_{\|n\|=2}^{\infty}H_{n}\xi^{n}$, $\xi=(\xi_{1},\xi_{2}),n=(n_{1},n_{2})$. Substituting the expression for $\eta$ as a function of $\xi$ into equation (\ref{ex1}) and expanding $F(\xi)=\sum_{\|n\|=2}^{\infty}F_{n}\xi^{n}$, we obtain at order $\|\xi\|^{2}$: \begin{equation} \left(\begin{matrix}B_{\varepsilon} & 0 & I_{2}\\ 0 & B_{\varepsilon} & -I_{2}\\ -2I_{2} & 2I_{2} & B_{\varepsilon} \end{matrix}\right)\left(\begin{array}{c} H_{20}\\ H_{02}\\ H_{11} \end{array}\right)=\left(\begin{array}{c} F_{20}\\ F_{02}\\ F_{11} \end{array}\right),\label{inverse} \end{equation} where we $B_{\varepsilon}=\left(\begin{matrix}\varepsilon(2\lambda-\mu) & \alpha\\ -\alpha & \varepsilon(2\lambda-\mu) \end{matrix}\right)$ and $I_{2}$ denotes the $(2\times2)$-identity matrix. For the choice $\alpha=2$, the inverse of the matrix in the right-hand side of (\ref{inverse}) will contain terms of order $\varepsilon^{-1}$. Therefore, assuming that $(F_{20},F_{02},F_{11})\neq(0,0,0)$, we find that the $\mathcal{O}(\|\xi\|^{2})$-terms in the expansion of $H$ will blow up as $\varepsilon\to0$. Indeed, we use the inversion formula for block-matrices, \begin{equation} \left(\begin{matrix}A & B\\ C & D \end{matrix}\right)^{-1}=\left(\begin{matrix}A^{-1}+A^{-1}B\Delta^{-1}CA^{-1} & -A^{-1}B\Delta^{-1}\\ -\Delta^{-1}CA^{-1} & \Delta^{-1} \end{matrix}\right),\label{blockinv} \end{equation} with matrices $A,B,C,D$ of arbitrary dimensions, $A$ invertible and $\Delta=D-CA^{-1}B$, to show that the inverse of (\ref{inverse}) contains an entry of order $\varepsilon^{-1}$. Focusing on the lower-right corner in (\ref{blockinv}), we obtain \begin{small} \begin{equation} \begin{split}\Delta & =\left(B_{\varepsilon}-(-2I_{2},2I_{2})\left(\begin{matrix}B_{\varepsilon}^{-1} & 0\\ 0 & B_{\varepsilon}^{-1} \end{matrix}\right)\left(\begin{array}{c} I_{2}\\ -I_{2} \end{array}\right)\right)\\ & =(B_{\varepsilon}+4B_{\varepsilon}^{-1})\\ & =\left(\begin{matrix}\varepsilon(2\lambda-\mu)+4\frac{\varepsilon(2\lambda-\mu)}{\varepsilon^{2}(2\lambda-\mu)^{2}+\alpha^{2}} & \alpha-4\frac{\alpha}{\varepsilon^{2}(2\lambda-\mu)^{2}+\alpha^{2}}\\[0.2cm] -\alpha+4\frac{\alpha}{\varepsilon^{2}(2\lambda-\mu)^{2}+\alpha^{2}} & \varepsilon(2\lambda-\mu)+4\frac{\varepsilon(2\lambda-\mu)}{\varepsilon^{2}(2\lambda-\mu)^{2}+\alpha^{2}} \end{matrix}\right)\\ & =(\varepsilon^{2}(2\lambda-\mu)^{2}+\alpha^{2})^{2}\left(\begin{matrix}\varepsilon(2\lambda-\mu)\Big(\varepsilon^{2}(2\lambda-\mu)^{2}+\alpha^{2}+4\Big) & \varepsilon^{2}(2\lambda-\mu)^{2}\alpha+(\alpha^{3}-4\alpha)\\ -\varepsilon^{2}(2\lambda-\mu)^{2}\alpha+(4\alpha-\alpha^{3}) & \varepsilon(2\lambda-\mu)\Big(\varepsilon^{2}(2\lambda-\mu)^{2}+\alpha^{2}+4\Big) \end{matrix}\right). \end{split} \label{Deltainv} \end{equation} \end{small} For $\alpha=2$, the $(\alpha^{3}-4\alpha)$-contribution vanishes and $\Delta$ scales like $\varepsilon$ in its entries, which implies that $\Delta^{-1}$ scales like $\varepsilon^{-1}$ in its entries. This shows that the SSMs does not converge to an LSM in this example (even in the sense of convergence of Taylor coefficients). \end{example} \begin{remark} Note that in this example, the $\varepsilon = 0$ case is resonant and does not verify the hypothesis of the Lyapunov center theorem. Indeed, it is easy to show matching powers of a possible expansion that, indeed, we obtain some equations that have no solution and, hence, that in this case there is no LSM.\\ Therefore the SSM for small dissipation is not generated by the LSM, and the SMM are created by the dissipation and have no chance of surviving in the limit of zero dissipation. As we mentioned in the introduction, as soon as the eigenvalues move into non-resonance, the theory of \cite{Cab2003} guarantees the existence of SSMs whose Taylor coefficients converge to those of the LSM. Hence, the situation exemplified in the example -- no LSM -- is the only one when one could get failure of convergence of the Taylor coefficients.\\ One question that would be interesting to study is what is the domain of convergence of the SSM is in this example. Since the Taylor coefficients blow up, Cauchy estimates indicate that the (complex) domains when the parameterization has size $1$ have to go to zero but we do not know the rate and we do not know what is the nature of the singularities.\\ Extensions of Lyapunov theorem to resonant eigenvalues have been worked out in \cite{weinstein73, moser1976periodic,Duistermaat1972, SchmidtS73} using variational methods and averaging methods. It would also be interesting to apply the results in this case. \end{remark} \subsection{Dissipative perturbations of Hamiltonian systems} In the following, we discuss the existence of an SSM, smoothly perturbed from an LSM, for nearly conservative systems relevant in application to mechanics.\\ Consider the dynamical system \begin{equation} M\ddot{q}+\varepsilon C\dot{q}+Kq+\nabla V(q)=0,\label{mech1} \end{equation} for $t\mapsto q(t)\in{\mathbb R}^{n}$, $n\in{\mathbb N}$, where $M$ is a positive definite mass matrix, $K$ is a positive definite stiffness matrix, $C$ is a positive definite damping matrix and $V:{\mathbb R}^{n}\to{\mathbb R}$ a twice continuously-differentiable function such that $V(0)=0$ and $\nabla V(0)=0$. Introducing the generalized momentum $p:=M\dot{q}$, we can rewrite system \eqref{mech1} as a first-order, dissipatively perturbed Hamiltonian system of the form \begin{equation} \left(\begin{array}{c} \dot{q}\\ \dot{p} \end{array}\right)=\left(\begin{matrix}0 & M^{-1}\\ -K & -\varepsilon CM^{-1} \end{matrix}\right)\left(\begin{array}{c} q\\ p \end{array}\right)-\left(\begin{array}{c} 0\\ \nabla V(q) \end{array}\right).\label{mech2} \end{equation} For $\varepsilon=0$, equation \eqref{mech1} admits the Hamiltonian \begin{equation} H(q,p)=\frac{1}{2}p\cdot M^{-1}p+\frac{1}{2}q\cdot Kq+V(q),\label{eq:hamiltonian} \end{equation} such that $\frac{d}{dt}H((q(t),p(t)))=0$ for any solution $t\mapsto(q(t),p(t))$ to (\ref{mech2}) with $\varepsilon=0$. The spectrum of the linearization of (\ref{mech2}) for $\varepsilon=0$ is given by $\pm\mathrm{i}\sqrt{\sigma(M^{-1}K)}=:\{\pm\mathrm{i}\lambda_{k}\}_{1\leq k\leq n}$, which is purely imaginary thanks to the positive-definiteness of $M$ and $K$. To apply Theorem \ref{exLSC}, we assume that the first eigenvalue of $M^{-1}K$ is non-resonant with the other eigenvalues, i.e., \begin{equation} \frac{\lambda_{k}}{\lambda_{1}}\notin{\mathbb Z}, \end{equation} for $2\leq k\leq n$. We also have to assume that $\nabla^{2}V(Y,Y)>0$ for all $Y\in\mathrm{eig}(\pm\mathrm{i}\lambda_{1})$. With these conditions, Assumption \ref{AssLSC} is satisfied.\\ Assumption \ref{Asspert} is satisfied for any positive definite damping matrix $C$ for which the linear part in (\ref{mech2}) has no repeated eigenvalues. Indeed, the Hamiltonian acts as a Lyapunov function for system \eqref{mech1}, as $H(0,0)=0$ and \begin{equation} \dot{H}(q,p)=\nabla H\cdot(\dot{q},\dot{p})^{T}=-\varepsilon M^{-1}p\cdot CM^{-1}p<0, \end{equation} by the positive-definiteness of $C$. Therefore, the origin is asymptotically stable and, for $\delta'$ and $\varepsilon$ small enough. \begin{example}(Nonlinear Elastic Pendulum with Air Drag) Based on Duistermaat \cite{Duistermaat1972}, consider a two-dimensional elastic pendulum with mass $m>0$ under the influence of gravity $g>0$. At the initial position, the mass is assumed to be located at the origin $(0,0)$ and the length of the spring is given by $l_{0}$. We assume a linear spring constant $k>0$ and a cubic spring constant $K>0$, so that the potential energy at the position $(q_{1},q_{2})$ is given by \begin{equation} U(q)=U(q_{1},q_{2})=mgq_{2}+k[q_{1}^{2}+(l_{0}-q_{2})^{2}]+K[q_{1}^{2}+(l_{0}-q_{2})^{2}]^{2}+U_{0},\label{pot} \end{equation} for $U_{0}=-(kl_{0}^{2}+Kl_{0}^{4})$ (cf. Fig. \ref{ImgElastic}). \begin{figure} \caption{Elastic pendulum under the influence of gravity. The initial position is assumed to be at $(0,0)$, while the initial length is assumed to be $l_{0}$.} \label{ImgElastic} \end{figure} The Hamiltonian \eqref{eq:hamiltonian} takes the specific form \begin{equation} H(q,p)=\frac{1}{2m}\|p\|^{2}+U(q),\label{Ham} \end{equation} and $H(0,0)=0$ and $\nabla H(0,0)=0$ hold when we choose our parameters such that $2kl_{0}+4Kl_{0}^{3}=mg$. Assuming a constant, small air drag $\varepsilon$ (which we consider as the linearization of the general, quadratic air resistance), the equations of motions take the form \begin{equation} \begin{split}\left(\begin{array}{c} \dot{q}_{1}\\ \dot{q}_{2}\\ \dot{p}_{1}\\ \dot{p}_{2} \end{array}\right)= & \left(\begin{matrix}0 & 0 & \frac{1}{m} & 0\\ 0 & 0 & 0 & \frac{1}{m}\\ -(2k+4Kl_{0}^{2}) & 0 & -\frac{\varepsilon}{m} & 0\\ 0 & -(2k+12Kl_{0}^{2}) & 0 & -\frac{\varepsilon}{m} \end{matrix}\right)\left(\begin{array}{c} q_{1}\\ q_{2}\\ p_{1}\\ p_{2} \end{array}\right)\\ & \quad-\left(\begin{array}{c} 0\\ 0\\ 4Kq_{1}(q_{1}^{2}-2l_{0}q_{2}+q_{2}^{2})\\ 4Kq_{2}(q_{2}^{2}-3l_{0}q_{2}+q_{1}^{2})-4Kl_{0}q_{1}^{2} \end{array}\right). \end{split} \label{pend} \end{equation} For $\varepsilon$ sufficiently small, the eigenvalues of the linearization of (\ref{pend}) are given by \begin{equation} \left\{ \frac{-\varepsilon\pm\mathrm{i}\sqrt{2k+4Kl_{0}^{2}-\varepsilon^{2}}}{2m},\frac{-\varepsilon\pm\mathrm{i}\sqrt{2k+12Kl_{0}^{2}-\varepsilon^{2}}}{2m}\right\} , \end{equation} with the $(q_{1},p_{1})$-plane and the $(q_{2},p_{2})$-plane as corresponding eigenspaces for any $\varepsilon\geq0$.\\ Assumption \ref{AssLSC}/(3) is therefore satisfied for both pairs of complex conjugate eigenvalues and their corresponding eigenspaces for $\varepsilon=0$. To ensure the existence of LSMs for these eigenspaces, we also require that \begin{equation} \frac{k+2l_{0}^{2}K}{k+6l_{0}^{2}K}\notin{\mathbb Z}, \end{equation} which holds for a generic choice of $k$ and $K$, assuming that $l_{0}$ is chosen according to the normalization of the Hamiltonian. We will now construct a second-order polynomial approximation to the invariant manifold tangent to the $(q_{1},p_{1})$-plane and its corresponding polynomial dynamics, which are guaranteed to exists by Theorem \ref{mainThm}.\\ To this end, we expand $w:U\subset{\mathbb R}^{2}\times[0,\varepsilon_{0}]\to{\mathbb R}^{2}$, for some $\varepsilon_{0}>0$, up to order two \begin{equation} w(\varepsilon,q_{1},p_{1})=w_{20}(\varepsilon)q_{1}^{2}+w_{11}(\varepsilon)q_{1}p_{1}+w_{02}(\varepsilon)p_{1}^{2}+\mathcal{O}((\|q\|+\|p\|)^{3}),\label{ansatz1} \end{equation} for $\varepsilon\mapsto w_{20}(\varepsilon),w_{11}(\varepsilon),w_{02}(\varepsilon)\in{\mathbb R}^{2}$, and assume that $(q_{2}(t),p_{2}(t))=w(\varepsilon,q_{1}(t),q_{2}(t))$. Substituting the ansatz (\ref{ansatz1}) into equation \eqref{pend} and solving for powers in $q_{1}$ and $p_{1}$, we obtain \begin{equation} \begin{split} & w_{20}(\varepsilon)=\kappa_{\varepsilon}\left(\begin{array}{c} -2Kl_{0}(-\varepsilon^{2}(k+6Kl_{0}^{2})-6k^{2}m+8kKl_{0}^{2}m+8K^{2}l_{0}^{4}m)\\ -16\varepsilon Kl_{0}(k+2Kl_{0}^{2})^{2}m \end{array}\right),\\[0.3cm] & w_{11}(\varepsilon)=\kappa_{\varepsilon}\left(\begin{array}{c} 16\varepsilon Kl_{0}(k+2Kl_{0}^{2})\\ -4Kl_{0}(\varepsilon^{2}(k-2Kl_{0}^{2})+2(3k^{2}+20kKl_{0}^{2}+12K^{2}l_{0}^{4})m) \end{array}\right),\\[0.3cm] & w_{02}(\varepsilon)=\kappa_{\varepsilon}\left(\begin{array}{c} 8Kl_{0}(3k+2Kl_{0}^{2})\\ 8\varepsilon Kl_{0}(-k+2Kl_{0}^{2}) \end{array}\right), \end{split} \label{wij} \end{equation} where \begin{equation} \kappa_{\varepsilon}=\frac{1}{(k+6Kl_{0}^{2})(-\varepsilon^{2}(k-2Kl_{0}^{2})+2(3k+2Kl_{0}^{2})^{2}m)}. \end{equation} The approximate dynamics on the SSM (i.e., perturbed LSM) are given by \begin{equation} \left(\begin{array}{c} \dot{x}\\ \dot{y} \end{array}\right)=\left(\begin{array}{c} \frac{2}{m}y\\ -2(k+2Kl_{0}^{2})x-\frac{\varepsilon y}{m}+4K\kappa_{\varepsilon}^{2}x\left(x^{2}+l_{0}^{2}(\alpha_{\varepsilon}+\beta_{\varepsilon}xy+\gamma_{\varepsilon}x^{2}+\delta_{\varepsilon}y^{2})^{2}\right) \end{array}\right),\label{red1} \end{equation} where \begin{equation} \begin{split} & \alpha_{\varepsilon}:=-(k+6Kl_{0}^{2})(-\varepsilon^{2}(k-2Kl_{0}^{2})+2(3k+2Kl_{0}^{2})^{2}m),\\ & \beta_{\varepsilon}:=16\varepsilon K(k+2Kl_{0}^{2}),\\ & \gamma_{\varepsilon}:=2K(\varepsilon^{2}(k+6Kl_{0}^{2})+2(3k^{2}-4kKl_{0}^{2}-4K^{2}l_{0}^{4})m),\\ & \delta_{\varepsilon}:=8K(3k+2Kl_{0}^{2}). \end{split} \end{equation} Typical phase portraits for the unperturbed and for the perturbed system (\ref{red1}) are depicted in Figures \ref{Red1Fig} and \ref{Red1epsFig}. \begin{figure} \caption{The unperturbed dynamics on the LSM for\\ $l_0=1$, $m=0.1$, $k=0.1414$ and $K=0.4$. } \label{Red1Fig} \caption{The perturbed dynamics on the SSM for $l_0=1$,\\ $m=0.1$, $k=0.1414$, $K=0.4$ and $\varepsilon=0.1$.} \label{Red1epsFig} \end{figure} \end{example} \subsection{Conclusion and Further Perspectives} We have proved that, adding dissipation (satisfying some mild non-degeneracy conditions) to a Hamiltonian system, an analytic, two-dimensional Lyapunov Subcenter Manifold (LSM) perturbs to an analytic spectral submanifold (SSM) in a neighborhood of size or order one.\\ As a consequence, the corresponding reduced dynamics on the SSM are $\varepsilon$-close to the dynamics of the LSM in a neighborhood of size one. We have also illustrated our results on several examples.\\ It would be useful to extend the present results to time-periodic or quasi-periodic perturbations and thereby relate experimentally observed backbone curves under sinusoidal excitation to the backbone curve of the conservative, unforced limit of the system. We refer to \cite{breunung2018explicit} for theoretical and numerical discussion of backbone curve calculations based on the SSM technique for damped systems. Of course, periodic perturbations lead to new phenomena such as resonances that lead to different behaviors and will require new formulations of results.\\ In view of Example~\ref{firstexample} it may also be interesting to study the possibility of existence of $C^\ell$ manifolds of size ${\mathcal O}(1)$ with $\ell < d$. Indeed, in many cases, it has been observed that the manifolds that guide the convergence to equilibrium are of low regularity \cite{MaasP92, Smooke91, WarnatzMD96, GorbanK05}.\\ We also think it would be interesting to consider results in PDE adding dissipation to the results in \cite{Bambusi00} or in systems with symmetry \cite{BuonoLM05}, keeping in mind that the dissipation may also break some symmetries.\\ It seems that the analyticity domains in the dissipation established here can be improved to parabolic domains $\{\varepsilon \in {\mathbb C} \, \\| \|\Im(\varepsilon)\| \le B \Re(\varepsilon)^2\}$. It would be interesting to characterize the optimal analyticity domains and in particular, show that they do not contain a circle centered at the origin.\\ We also refer to upcoming results by Szalai \cite{Szalai2018conservative}, in which a perturbation theory based on different techniques has been announced. \section*{Acknowledgments} We thank George Haller for formulating the problem and suggesting it to us as well as for many conversations and continued interest and encouragement. We also thank Robert Szalai for conversations and for pointing out several mistakes in a previous version.\\ We would also like to thank the anonymous reviewers for several useful comments and suggestions. \end{document}	arXiv
There are $n$ heaps of sticks and two players who move alternately. On each move, a player chooses a non-empty heap and removes $1$, $2$, or $3$ sticks. The player who removes the last stick wins the game. Your task is to find out who wins if both players play optimally. The first line contains an integer $n$: the number of heaps. The next line has $n$ integers $x_1,x_2,\ldots,x_n$: the number of sticks in each heap. For each test case, print "first" if the first player wins the game and "second" if the second player wins the game.	CommonCrawl
Papers from TCC 2018 Search Problems: A Cryptographic Perspective Moni Naor Encrypted Computation Two-Round MPC: Information-Theoretic and Black-Box Abstract Sanjam Garg Yuval Ishai Akshayaram Srinivasan We continue the study of protocols for secure multiparty computation (MPC) that require only two rounds of interaction. The recent works of Garg and Srinivasan (Eurocrypt 2018) and Benhamouda and Lin (Eurocrypt 2018) essentially settle the question by showing that such protocols are implied by the minimal assumption that a two-round oblivious transfer (OT) protocol exists. However, these protocols inherently make a non-black-box use of the underlying OT protocol, which results in poor concrete efficiency. Moreover, no analogous result was known in the information-theoretic setting, or alternatively based on one-way functions, given an OT correlations setup or an honest majority.Motivated by these limitations, we study the possibility of obtaining information-theoretic and "black-box" implementations of two-round MPC protocols. We obtain the following results:Two-round MPC from OT correlations. Given an OT correlations setup, we get protocols that make a black-box use of a pseudorandom generator (PRG) and are secure against a malicious adversary corrupting an arbitrary number of parties. For a semi-honest adversary, we get similar information-theoretic protocols for branching programs.New NIOT constructions. Towards realizing OT correlations, we extend the DDH-based non-interactive OT (NIOT) protocol of Bellare and Micali (Crypto'89) to the malicious security model, and present new NIOT constructions from the Quadratic Residuosity Assumption (QRA) and the Learning With Errors (LWE) assumption.Two-round black-box MPC with strong PKI setup. Combining the two previous results, we get two-round MPC protocols that make a black-box use of any DDH-hard or QRA-hard group. The protocols can offer security against a malicious adversary, and require a PKI setup that depends on the number of parties and the size of computation, but not on the inputs or the identities of the participating parties.Two-round honest-majority MPC from secure channels. Given secure point-to-point channels, we get protocols that make a black-box use of a pseudorandom generator (PRG), as well as information-theoretic protocols for branching programs. These protocols can tolerate a semi-honest adversary corrupting a strict minority of the parties, where in the information-theoretic case the complexity is exponential in the number of parties. Impossibility of Order-Revealing Encryption in Idealized Models Abstract Mark Zhandry Cong Zhang An Order-Revealing Encryption (ORE) scheme gives a public procedure by which two ciphertexts can be compared to reveal the order of their underlying plaintexts. The ideal security notion for ORE is that only the order is revealed—anything else, such as the distance between plaintexts, is hidden. The only known constructions of ORE achieving such ideal security are based on cryptographic multilinear maps and are currently too impractical for real-world applications.In this work, we give evidence that building ORE from weaker tools may be hard. Indeed, we show black-box separations between ORE and most symmetric-key primitives, as well as public key encryption and anything else implied by generic groups in a black-box way. Thus, any construction of ORE must either (1) achieve weaker notions of security, (2) be based on more complicated cryptographic tools, or (3) require non-black-box techniques. This suggests that any ORE achieving ideal security will likely be somewhat inefficient.Central to our proof is a proof of impossibility for something we call information theoretic ORE, which has connections to tournament graphs and a theorem by Erdös. This impossibility proof will be useful for proving other black box separations for ORE. Perfect Secure Computation in Two Rounds Abstract Benny Applebaum Zvika Brakerski Rotem Tsabary We show that any multi-party functionality can be evaluated using a two-round protocol with perfect correctness and perfect semi-honest security, provided that the majority of parties are honest. This settles the round complexity of information-theoretic semi-honest MPC, resolving a longstanding open question (cf. Ishai and Kushilevitz, FOCS 2000). The protocol is efficient for $${\mathrm {NC}}^1$$NC1 functionalities. Furthermore, given black-box access to a one-way function, the protocol can be made efficient for any polynomial functionality, at the cost of only guaranteeing computational security.Technically, we extend and relax the notion of randomized encoding to specifically address multi-party functionalities. The property of a multi-party randomized encoding (MPRE) is that if the functionality g is an encoding of the functionality f, then for any (permitted) coalition of players, their respective outputs and inputs in g allow them to simulate their respective inputs and outputs in f, without learning anything else, including the other outputs of f. A Ciphertext-Size Lower Bound for Order-Preserving Encryption with Limited Leakage Abstract David Cash Cong Zhang We consider a security definition of Chenette, Lewi, Weis, and Wu for order-revealing encryption (ORE) and order-preserving encryption (OPE) (FSE 2016). Their definition says that the comparison of two ciphertexts should only leak the index of the most significant bit on which they differ. While their work could achieve order-revealing encryption with short ciphertexts that expand the plaintext by a factor $$\approx 1.58$$, it could only find order-preserving encryption with longer ciphertexts that expanded the plaintext by a security-parameter factor. We give evidence that this gap between ORE and OPE is inherent, by proving that any OPE meeting the information-theoretic version of their security definition (for instance, in the random oracle model) must have ciphertext length close to that of their constructions. We extend our result to identify an abstract security property of any OPE that will result in the same lower bound. Two-Round Adaptively Secure Multiparty Computation from Standard Assumptions Abstract Fabrice Benhamouda Huijia Lin Antigoni Polychroniadou Muthuramakrishnan Venkitasubramaniam We present the first two-round multiparty computation (MPC) protocols secure against malicious adaptive corruption in the common reference string (CRS) model, based on DDH, LWE, or QR. Prior two-round adaptively secure protocols were known only in the two-party setting against semi-honest adversaries, or in the general multiparty setting assuming the existence of indistinguishability obfuscation (iO).Our protocols are constructed in two steps. First, we construct two-round oblivious transfer (OT) protocols secure against malicious adaptive corruption in the CRS model based on DDH, LWE, or QR. We achieve this by generically transforming any two-round OT that is only secure against static corruption but has certain oblivious sampleability properties, into a two-round adaptively secure OT. Prior constructions were only secure against semi-honest adversaries or based on iO.Second, building upon recent constructions of two-round MPC from two-round OT in the weaker static corruption setting [Garg and Srinivasan, Benhamouda and Lin, Eurocrypt'18] and using equivocal garbled circuits from [Canetti, Poburinnaya and Venkitasubramaniam, STOC'17], we show how to construct two-round adaptively secure MPC from two-round adaptively secure OT and constant-round adaptively secure MPC, with respect to both malicious and semi-honest adversaries. As a corollary, we also obtain the first 2-round MPC secure against semi-honest adaptive corruption in the plain model based on augmented non-committing encryption (NCE), which can be based on a variety of assumptions, CDH, RSA, DDH, LWE, or factoring Blum integers. Finally, we mention that our OT and MPC protocols in the CRS model are, in fact, adaptively secure in the Universal Composability framework. Ciphertext Expansion in Limited-Leakage Order-Preserving Encryption: A Tight Computational Lower Bound Abstract Gil Segev Ido Shahaf Order-preserving encryption emerged as a key ingredient underlying the security of practical database management systems. Boldyreva et al. (EUROCRYPT '09) initiated the study of its security by introducing two natural notions of security. They proved that their first notion, a "best-possible" relaxation of semantic security allowing ciphertexts to reveal the ordering of their corresponding plaintexts, is not realizable. Later on Boldyreva et al. (CRYPTO '11) proved that any scheme satisfying their second notion, indistinguishability from a random order-preserving function, leaks about half of the bits of a random plaintext.This unsettling state of affairs was recently changed by Chenette et al. (FSE '16), who relaxed the above "best-possible" notion and constructed a scheme satisfying it based on any pseudorandom function. In addition to revealing the ordering of any two encrypted plaintexts, ciphertexts in their scheme reveal only the position of the most significant bit on which the plaintexts differ. A significant drawback of their scheme, however, is its substantial ciphertext expansion: Encrypting plaintexts of length m bits results in ciphertexts of length $$m \cdot \ell $$ bits, where $$\ell $$ determines the level of security (e.g., $$\ell = 80$$ in practice).In this work we prove a lower bound on the ciphertext expansion of any order-preserving encryption scheme satisfying the "limited-leakage" notion of Chenette et al. with respect to non-uniform polynomial-time adversaries, matching the ciphertext expansion of their scheme up to lower-order terms. This improves a recent result of Cash and Zhang (TCC '18), who proved such a lower bound for schemes satisfying this notion with respect to computationally-unbounded adversaries (capturing, for example, schemes whose security can be proved in the random-oracle model without relying on cryptographic assumptions). Our lower bound applies, in particular, to schemes whose security is proved in the standard model. Towards Tight Security of Cascaded LRW2 Abstract Bart Mennink The Cascaded LRW2 tweakable block cipher was introduced by Landecker et al. at CRYPTO 2012, and proven secure up to $$2^{2n/3}$$ queries. There has not been any attack on the construction faster than the generic attack in $$2^n$$ queries. In this work we initiate the quest towards a tight bound. We first present a distinguishing attack in $$2n^{1/2}2^{3n/4}$$ queries against a generalized version of the scheme. The attack is supported with an experimental verification and a formal success probability analysis. We subsequently discuss non-trivial bottlenecks in proving tight security, most importantly the distinguisher's freedom in choosing the tweak values. Finally, we prove that if every tweak value occurs at most $$2^{n/4}$$ times, Cascaded LRW2 is secure up to $$2^{3n/4}$$ queries. One-Message Zero Knowledge and Non-malleable Commitments Abstract Nir Bitansky Huijia Lin We introduce a new notion of one-message zero-knowledge (1ZK) arguments that satisfy a weak soundness guarantee—the number of false statements that a polynomial-time non-uniform adversary can convince the verifier to accept is not much larger than the size of its non-uniform advice. The zero-knowledge guarantee is given by a simulator that runs in (mildly) super-polynomial time. We construct such 1ZK arguments based on the notion of multi-collision-resistant keyless hash functions, recently introduced by Bitansky, Kalai, and Paneth (STOC 2018). Relying on the constructed 1ZK arguments, subexponentially-secure time-lock puzzles, and other standard assumptions, we construct one-message fully-concurrent non-malleable commitments. This is the first construction that is based on assumptions that do not already incorporate non-malleability, as well as the first based on (subexponentially) falsifiable assumptions. Continuous NMC Secure Against Permutations and Overwrites, with Applications to CCA Secure Commitments Abstract Ivan Damgård Tomasz Kazana Maciej Obremski Varun Raj Luisa Siniscalchi Non-Malleable Codes (NMC) were introduced by Dziembowski, Pietrzak and Wichs in ICS 2010 as a relaxation of error correcting codes and error detecting codes. Faust, Mukherjee, Nielsen, and Venturi in TCC 2014 introduced an even stronger notion of non-malleable codes called continuous non-malleable codes where security is achieved against continuous tampering of a single codeword without re-encoding.We construct information theoretically secure CNMC resilient to bit permutations and overwrites, this is the first Continuous NMC constructed outside of the split-state model.In this work we also study relations between the CNMC and parallel CCA commitments. We show that the CNMC can be used to bootstrap a Self-destruct parallel CCA bit commitment to a Self-destruct parallel CCA string commitment, where Self-destruct parallel CCA is a weak form of parallel CCA security. Then we can get rid of the Self-destruct limitation obtaining a parallel CCA commitment, requiring only one-way functions. Smooth NIZK Arguments Abstract Charanjit S. Jutla Arnab Roy We introduce a novel notion of smooth (-verifier) non- interactive zero-knowledge proofs (NIZK) which parallels the familiar notion of smooth projective hash functions (SPHF). We also show that the single group element quasi-adaptive NIZK (QA-NIZK) of Jutla and Roy (CRYPTO 2014) and Kiltz and Wee (EuroCrypt 2015) for linear subspaces can be easily extended to be computationally smooth. One important distinction of the new notion from SPHFs is that in a smooth NIZK the public evaluation of the hash on a language member using the projection key does not require the witness of the language member, but instead just requires its NIZK proof.This has the remarkable consequence that if one replaces the traditionally employed SPHFs with the novel smooth QA-NIZK in the Gennaro-Lindell paradigm of designing universally-composable password- authenticated key-exchange (UC-PAKE) protocols, one gets highly efficient UC-PAKE protocols that are secure even under adaptive corruption. This simpler and modular design methodology allows us to give the first single-round asymmetric UC-PAKE protocol, which is also secure under adaptive corruption in the erasure model. Previously, all asymmetric UC-PAKE protocols required at least two rounds. In fact, our protocol just requires each party to send a single message asynchronously. In addition, the protocol has short messages, with each party sending only four group elements. Moreover, the server password file needs to store only one group element per client. The protocol employs asymmetric bilinear pairing groups and is proven secure in the (limited programmability) random oracle model and under the standard bilinear pairing assumption SXDH. Best Possible Information-Theoretic MPC Abstract Shai Halevi Yuval Ishai Eyal Kushilevitz Tal Rabin We reconsider the security guarantee that can be achieved by general protocols for secure multiparty computation in the most basic of settings: information-theoretic security against a semi-honest adversary. Since the 1980s, we have elegant solutions to this problem that offer full security, as long as the adversary controls a minority of the parties, but fail completely when that threshold is crossed. In this work, we revisit this problem, questioning the optimality of the standard notion of security. We put forward a new notion of information-theoretic security which is strictly stronger than the standard one, and which we argue to be "best possible." This notion still requires full security against dishonest minority in the usual sense, and adds a meaningful notion of information-theoretic security even against dishonest majority.We present protocols for useful classes of functions that satisfy this new notion of security. Our protocols have the unique feature of combining the efficiency benefits of protocols for an honest majority and (most of) the security benefits of protocols for dishonest majority. We further extend some of the solutions to the malicious setting. Round-Optimal Fully Black-Box Zero-Knowledge Arguments from One-Way Permutations Abstract Carmit Hazay Muthuramakrishnan Venkitasubramaniam In this paper, we revisit the round complexity of designing zero-knowledge (ZK) arguments via a black-box construction from minimal assumptions. Our main result implements a 4-round ZK argument for any language in $$\textsf {NP}$$ NP, based on injective one-way functions, that makes black-box use of the underlying function. As a corollary, we also obtain the first 4-round perfect zero-knowledge argument for $$\textsf {NP}$$ NP based on claw-free permutations via a black-box construction and 4-round input-delayed commit-and-prove zero-knowledge argument based on injective one-way functions. Secure Certification of Mixed Quantum States with Application to Two-Party Randomness Generation Abstract Frédéric Dupuis Serge Fehr Philippe Lamontagne Louis Salvail We investigate sampling procedures that certify that an arbitrary quantum state on n subsystems is close to an ideal mixed state $$\varphi ^{\otimes n}$$ for a given reference state $$\varphi $$, up to errors on a few positions. This task makes no sense classically: it would correspond to certifying that a given bitstring was generated according to some desired probability distribution. However, in the quantum case, this is possible if one has access to a prover who can supply a purification of the mixed state.In this work, we introduce the concept of mixed-state certification, and we show that a natural sampling protocol offers secure certification in the presence of a possibly dishonest prover: if the verifier accepts then he can be almost certain that the state in question has been correctly prepared, up to a small number of errors.We then apply this result to two-party quantum coin-tossing. Given that strong coin tossing is impossible, it is natural to ask "how close can we get". This question has been well studied and is nowadays well understood from the perspective of the bias of individual coin tosses. We approach and answer this question from a different—and somewhat orthogonal—perspective, where we do not look at individual coin tosses but at the global entropy instead. We show how two distrusting parties can produce a common high-entropy source, where the entropy is an arbitrarily small fraction below the maximum. Round Optimal Black-Box "Commit-and-Prove" Abstract Dakshita Khurana Rafail Ostrovsky Akshayaram Srinivasan Motivated by theoretical and practical considerations, an important line of research is to design secure computation protocols that only make black-box use of cryptography. An important component in nearly all the black-box secure computation constructions is a black-box commit-and-prove protocol. A commit-and-prove protocol allows a prover to commit to a value and prove a statement about this value while guaranteeing that the committed value remains hidden. A black-box commit-and-prove protocol implements this functionality while only making black-box use of cryptography.In this paper, we build several tools that enable constructions of round-optimal, black-box commit and prove protocols. In particular, assuming injective one-way functions, we design the first round-optimal, black-box commit-and-prove arguments of knowledge satisfying strong privacy against malicious verifiers, namely:Zero-knowledge in four rounds and,Witness indistinguishability in three rounds. Prior to our work, the best known black-box protocols achieving commit-and-prove required more rounds.We additionally ensure that our protocols can be used, if needed, in the delayed-input setting, where the statement to be proven is decided only towards the end of the interaction. We also observe simple applications of our protocols towards achieving black-box four-round constructions of extractable and equivocal commitments.We believe that our protocols will provide a useful tool enabling several new constructions and easy round-efficient conversions from non-black-box to black-box protocols in the future. Provable Time-Memory Trade-Offs: Symmetric Cryptography Against Memory-Bounded Adversaries Abstract Stefano Tessaro Aishwarya Thiruvengadam We initiate the study of symmetric encryption in a regime where the memory of the adversary is bounded. For a block cipher with n-bit blocks, we present modes of operation for encryption and authentication that guarantee security beyond$$2^n$$ encrypted/authenticated messages, as long as (1) the adversary's memory is restricted to be less than $$2^n$$ bits, and (2) the key of the block cipher is long enough to mitigate memory-less key-search attacks. This is the first proposal of a setting which allows to bypass the $$2^n$$ barrier under a reasonable assumption on the adversarial resources.Motivated by the above, we also discuss the problem of stretching the key of a block cipher in the setting where the memory of the adversary is bounded. We show a tight equivalence between the security of double encryption in the ideal-cipher model and the hardness of a special case of the element distinctness problem, which we call the list-disjointness problem. Our result in particular implies a conditional lower bound on time-memory trade-offs to break PRP security of double encryption, assuming optimality of the worst-case complexity of existing algorithms for list disjointness. Topology-Hiding Computation Beyond Semi-Honest Adversaries Abstract Rio LaVigne Chen-Da Liu-Zhang Ueli Maurer Tal Moran Marta Mularczyk Daniel Tschudi Topology-hiding communication protocols allow a set of parties, connected by an incomplete network with unknown communication graph, where each party only knows its neighbors, to construct a complete communication network such that the network topology remains hidden even from a powerful adversary who can corrupt parties. This communication network can then be used to perform arbitrary tasks, for example secure multi-party computation, in a topology-hiding manner. Previously proposed protocols could only tolerate passive corruption. This paper proposes protocols that can also tolerate fail-corruption (i.e., the adversary can crash any party at any point in time) and so-called semi-malicious corruption (i.e., the adversary can control a corrupted party's randomness), without leaking more than an arbitrarily small fraction of a bit of information about the topology. A small-leakage protocol was recently proposed by Ball et al. [Eurocrypt'18], but only under the unrealistic set-up assumption that each party has a trusted hardware module containing secret correlated pre-set keys, and with the further two restrictions that only passively corrupted parties can be crashed by the adversary, and semi-malicious corruption is not tolerated. Since leaking a small amount of information is unavoidable, as is the need to abort the protocol in case of failures, our protocols seem to achieve the best possible goal in a model with fail-corruption.Further contributions of the paper are applications of the protocol to obtain secure MPC protocols, which requires a way to bound the aggregated leakage when multiple small-leakage protocols are executed in parallel or sequentially. Moreover, while previous protocols are based on the DDH assumption, a new so-called PKCR public-key encryption scheme based on the LWE assumption is proposed, allowing to base topology-hiding computation on LWE. Furthermore, a protocol using fully-homomorphic encryption achieving very low round complexity is proposed. Classical Proofs for the Quantum Collapsing Property of Classical Hash Functions Abstract Serge Fehr Hash functions are of fundamental importance in theoretical and in practical cryptography, and with the threat of quantum computers possibly emerging in the future, it is an urgent objective to understand the security of hash functions in the light of potential future quantum attacks. To this end, we reconsider the collapsing property of hash functions, as introduced by Unruh, which replaces the notion of collision resistance when considering quantum attacks. Our contribution is a formalism and a framework that offers significantly simpler proofs for the collapsing property of hash functions. With our framework, we can prove the collapsing property for hash domain extension constructions entirely by means of decomposing the iteration function into suitable elementary composition operations. In particular, given our framework, one can argue purely classically about the quantum-security of hash functions; this is in contrast to previous proofs which are in terms of sophisticated quantum-information-theoretic and quantum-algorithmic reasoning. On the Power of Amortization in Secret Sharing: d-Uniform Secret Sharing and CDS with Constant Information Rate Abstract Benny Applebaum Barak Arkis Consider the following secret-sharing problem. Your goal is to distribute a long file s between n servers such that $$(d-1)$$ (d-1)-subsets cannot recover the file, $$(d+1)$$ (d+1)-subsets can recover the file, and d-subsets should be able to recover s if and only if they appear in some predefined list L. How small can the information ratio (i.e., the number of bits stored on a server per each bit of the secret) be?We advocate the study of such d-uniform access structures as a useful scaled-down version of general access structures. Our main result shows that, for constant d, any d-uniform access structure admits a secret sharing scheme with a constant asymptotic information ratio of $$c_d$$ cd that does not grow with the number of servers n. This result is based on a new construction of d-party Conditional Disclosure of Secrets (CDS) for arbitrary predicates over n-size domain in which each party communicates at most four bits per secret bit.In both settings, previous results achieved a non-constant information ratio that grows asymptotically with n, even for the simpler (and widely studied) special case of $$d=2$$ d=2. Moreover, our multiparty CDS construction yields the first example of an access structure whose amortized information ratio is constant, whereas its best-known non-amortized information ratio is sub-exponential, thus providing a unique evidence for the potential power of amortization in the context of secret sharing.Our main result applies to exponentially long secrets, and so it should be mainly viewed as a barrier against amortizable lower-bound techniques. We also show that in some natural simple cases (e.g., low-degree predicates), amortization kicks in even for quasi-polynomially long secrets. Finally, we prove some limited lower-bounds, point out some limitations of existing lower-bound techniques, and describe some applications to the setting of private simultaneous messages. Static-Memory-Hard Functions, and Modeling the Cost of Space vs. Time Abstract Thaddeus Dryja Quanquan C. Liu Sunoo Park A series of recent research starting with (Alwen and Serbinenko, STOC 2015) has deepened our understanding of the notion of memory-hardness in cryptography—a useful property of hash functions for deterring large-scale password-cracking attacks—and has shown memory-hardness to have intricate connections with the theory of graph pebbling. Definitions of memory-hardness are not yet unified in the somewhat nascent field of memory-hardness, however, and the guarantees proven to date are with respect to a range of proposed definitions. In this paper, we observe two significant and practical considerations that are not analyzed by existing models of memory-hardness, and propose new models to capture them, accompanied by constructions based on new hard-to-pebble graphs. Our contribution is two-fold, as follows. First, existing measures of memory-hardness only account for dynamic memory usage (i.e., memory read/written at runtime), and do not consider static memory usage (e.g., memory on disk). Among other things, this means that memory requirements considered by prior models are inherently upper-bounded by a hash function's runtime; in contrast, counting static memory would potentially allow quantification of much larger memory requirements, decoupled from runtime. We propose a new definition of static-memory-hard function (SHF) which takes static memory into account: we model static memory usage by oracle access to a large preprocessed string, which may be considered part of the hash function description. Static memory requirements are complementary to dynamic memory requirements: neither can replace the other, and to deter large-scale password-cracking attacks, a hash function will benefit from being both dynamic-memory-hard and static-memory-hard. We give two SHF constructions based on pebbling. To prove static-memory-hardness, we define a new pebble game ("black-magic pebble game"), and new graph constructions with optimal complexity under our proposed measure. Moreover, we provide a prototype implementation of our first SHF construction (which is based on pebbling of a simple "cylinder" graph), providing an initial demonstration of practical feasibility for a limited range of parameter settings. Secondly, existing memory-hardness models implicitly assume that the cost of space and time are more or less on par: they consider only linear ratios between the costs of time and space. We propose a new model to capture nonlinear time-space trade-offs: e.g., how is the adversary impacted when space is quadratically more expensive than time? We prove that nonlinear tradeoffs can in fact cause adversaries to employ different strategies from linear tradeoffs.Please refer to the full version of our paper for all results, proofs, appendices, and implementation details [DLP18]. Traitor-Tracing from LWE Made Simple and Attribute-Based Abstract Yilei Chen Vinod Vaikuntanathan Brent Waters Hoeteck Wee Daniel Wichs A traitor tracing scheme is a public key encryption scheme for which there are many secret decryption keys. Any of these keys can decrypt a ciphertext; moreover, even if a coalition of users collude, put together their decryption keys and attempt to create a new decryption key, there is an efficient algorithm to trace the new key to at least one the colluders.Recently, Goyal, Koppula and Waters (GKW, STOC 18) provided the first traitor tracing scheme from LWE with ciphertext and secret key sizes that grow polynomially in $$\log n$$, where n is the number of users. The main technical building block in their construction is a strengthening of (bounded collusion secure) secret-key functional encryption which they refer to as mixed functional encryption (FE).In this work, we improve upon and extend the GKW traitor tracing scheme:We provide simpler constructions of mixed FE schemes based on the LWE assumption. Our constructions improve upon the GKW construction in terms of expressiveness, modularity, and security.We provide a construction of attribute-based traitor tracing for all circuits based on the LWE assumption. Information-Theoretic Secret-Key Agreement: The Asymptotically Tight Relation Between the Secret-Key Rate and the Channel Quality Ratio Abstract Daniel Jost Ueli Maurer João Ribeiro Information-theoretic secret-key agreement between two parties Alice and Bob is a well-studied problem that is provably impossible in a plain model with public (authenticated) communication, but is known to be possible in a model where the parties also have access to some correlated randomness. One particular type of such correlated randomness is the so-called satellite setting, where uniform random bits (e.g., sent by a satellite) are received by the parties and the adversary Eve over inherently noisy channels. The antenna size determines the error probability, and the antenna is the adversary's limiting resource much as computing power is the limiting resource in traditional complexity-based security. The natural assumption about the adversary is that her antenna is at most Q times larger than both Alice's and Bob's antenna, where, to be realistic, Q can be very large.The goal of this paper is to characterize the secret-key rate per transmitted bit in terms of Q. Traditional results in this so-called satellite setting are phrased in terms of the error probabilities $$\epsilon _A$$ϵA, $$\epsilon _B$$ϵB, and $$\epsilon _E$$ϵE, of the binary symmetric channels through which the parties receive the bits and, quite surprisingly, the secret-key rate has been shown to be strictly positive unless Eve's channel is perfect ($$\epsilon _E=0$$ϵE=0) or either Alice's or Bob's channel output is independent of the transmitted bit (i.e., $$\epsilon _A=0.5$$ϵA=0.5 or $$\epsilon _B=0.5$$ϵB=0.5). However, the best proven lower bound, if interpreted in terms of the channel quality ratio Q, is only exponentially small in Q. The main result of this paper is that the secret-key rate decreases asymptotically only like $$1/Q^2$$1/Q2 if the per-bit signal energy, affecting the quality of all channels, is treated as a system parameter that can be optimized. Moreover, this bound is tight if Alice and Bob have the same antenna sizes.Motivated by considering a fixed sending signal power, in which case the per-bit energy is inversely proportional to the bit-rate, we also propose a definition of the secret-key rate per second (rather than per transmitted bit) and prove that it decreases asymptotically only like 1/Q. Secure Computation Using Leaky Correlations (Asymptotically Optimal Constructions) Abstract Alexander R. Block Divya Gupta Hemanta K. Maji Hai H. Nguyen Most secure computation protocols can be effortlessly adapted to offload a significant fraction of their computationally and cryptographically expensive components to an offline phase so that the parties can run a fast online phase and perform their intended computation securely. During this offline phase, parties generate private shares of a sample generated from a particular joint distribution, referred to as the correlation. These shares, however, are susceptible to leakage attacks by adversarial parties, which can compromise the security of the secure computation protocol. The objective, therefore, is to preserve the security of the honest party despite the leakage performed by the adversary on her share.Prior solutions, starting with n-bit leaky shares, either used 4 messages or enabled the secure computation of only sub-linear size circuits. Our work presents the first 2-message secure computation protocol for 2-party functionalities that have $$\varTheta (n)$$ circuit-size despite $$\varTheta (n)$$-bits of leakage, a qualitatively optimal result. We compose a suitable 2-message secure computation protocol in parallel with our new 2-message correlation extractor. Correlation extractors, introduced by Ishai, Kushilevitz, Ostrovsky, and Sahai (FOCS–2009) as a natural generalization of privacy amplification and randomness extraction, recover "fresh" correlations from the leaky ones, which are subsequently used by other cryptographic protocols. We construct the first 2-message correlation extractor that produces $$\varTheta (n)$$-bit fresh correlations even after $$\varTheta (n)$$-bit leakage.Our principal technical contribution, which is of potential independent interest, is the construction of a family of multiplication-friendly linear secret sharing schemes that is simultaneously a family of small-bias distributions. We construct this family by randomly "twisting then permuting" appropriate Algebraic Geometry codes over constant-size fields. Information-Theoretic Broadcast with Dishonest Majority for Long Messages Abstract Wutichai Chongchitmate Rafail Ostrovsky Byzantine broadcast is a fundamental primitive for secure computation. In a setting with n parties in the presence of an adversary controlling at most t parties, while a lot of progress in optimizing communication complexity has been made for $$t < n/2$$t<n/2, little progress has been made for the general case $$t<n$$t<n, especially for information-theoretic security. In particular, all information-theoretic secure broadcast protocols for $$\ell $$ℓ-bit messages and $$t<n$$t<n and optimal round complexity $${\mathcal {O}}(n)$$O(n) have, so far, required a communication complexity of $${\mathcal {O}}(\ell n^2)$$O(ℓn2). A broadcast extension protocol allows a long message to be broadcast more efficiently using a small number of single-bit broadcasts. Through broadcast extension, so far, the best achievable round complexity for $$t<n$$t<n setting with the optimal communication complexity of $${\mathcal {O}}(\ell n)$$O(ℓn) is $${\mathcal {O}}(n^4)$$O(n4) rounds.In this work, we construct a new broadcast extension protocol for $$t<n$$t<n with information-theoretic security. Our protocol improves the round complexity to $${\mathcal {O}}(n^3)$$O(n3) while maintaining the optimal communication complexity for long messages. Our result shortens the gap between the information-theoretic setting and the computational setting, and between the optimal communication protocol and the optimal round protocol in the information-theoretic setting for $$t<n$$t<n. Two-Message Statistically Sender-Private OT from LWE Abstract Zvika Brakerski Nico Döttling We construct a two-message oblivious transfer (OT) protocol without setup that guarantees statistical privacy for the sender even against malicious receivers. Receiver privacy is game based and relies on the hardness of learning with errors (LWE). This flavor of OT has been a central building block for minimizing the round complexity of witness indistinguishable and zero knowledge proof systems, non-malleable commitment schemes and multi-party computation protocols, as well as for achieving circuit privacy for homomorphic encryption in the malicious setting. Prior to this work, all candidates in the literature from standard assumptions relied on number theoretic assumptions and were thus insecure in the post-quantum setting. This work provides the first (presumed) post-quantum secure candidate and thus allows to instantiate the aforementioned applications in a post-quantum secure manner.Technically, we rely on the transference principle: Either a lattice or its dual must have short vectors. Short vectors, in turn, can be translated to information loss in encryption. Thus encrypting one message with respect to the lattice and one with respect to its dual guarantees that at least one of them will be statistically hidden. Oblivious Transfer in Incomplete Networks Abstract Varun Narayanan Vinod M. Prabahakaran Secure message transmission and Byzantine agreement have been studied extensively in incomplete networks. However, information theoretically secure multiparty computation (MPC) in incomplete networks is less well understood. In this paper, we characterize the conditions under which a pair of parties can compute oblivious transfer (OT) information theoretically securely against a general adversary structure in an incomplete network of reliable, private channels. We provide characterizations for both semi-honest and malicious models. A consequence of our results is a complete characterization of networks in which a given subset of parties can compute any functionality securely with respect to an adversary structure in the semi-honest case and a partial characterization in the malicious case. Adaptively Secure Distributed PRFs from $\mathsf {LWE}$ Abstract Benoît Libert Damien Stehlé Radu Titiu In distributed pseudorandom functions (DPRFs), a PRF secret key SK is secret shared among N servers so that each server can locally compute a partial evaluation of the PRF on some input X. A combiner that collects t partial evaluations can then reconstruct the evaluation F(SK, X) of the PRF under the initial secret key. So far, all non-interactive constructions in the standard model are based on lattice assumptions. One caveat is that they are only known to be secure in the static corruption setting, where the adversary chooses the servers to corrupt at the very beginning of the game, before any evaluation query. In this work, we construct the first fully non-interactive adaptively secure DPRF in the standard model. Our construction is proved secure under the $$\mathsf {LWE}$$ assumption against adversaries that may adaptively decide which servers they want to corrupt. We also extend our construction in order to achieve robustness against malicious adversaries. Injective Trapdoor Functions via Derandomization: How Strong is Rudich's Black-Box Barrier? Abstract Lior Rotem Gil Segev We present a cryptographic primitive $$\mathcal {P}$$ P satisfying the following properties:Rudich's seminal impossibility result (PhD thesis '88) shows that $$\mathcal {P}$$ P cannot be used in a black-box manner to construct an injective one-way function. $$\mathcal {P}$$ P can be used in a non-black-box manner to construct an injective one-way function assuming the existence of a hitting-set generator that fools deterministic circuits (such a generator is known to exist based on the worst-case assumption that $$\text{ E } = \text{ DTIME }(2^{O(n)})$$ E=DTIME(2O(n)) has a function of deterministic circuit complexity $$2^{\Omega (n)}$$ 2Ω(n)).Augmenting $$\mathcal {P}$$ P with a trapdoor algorithm enables a non-black-box construction of an injective trapdoor function (once again, assuming the existence of a hitting-set generator that fools deterministic circuits), while Rudich's impossibility result still holds. The primitive $$\mathcal {P}$$ P and its augmented variant can be constructed based on any injective one-way function and on any injective trapdoor function, respectively, and they are thus unconditionally essential for the existence of such functions. Moreover, $$\mathcal {P}$$ P can also be constructed based on various known primitives that are secure against related-key attacks, thus enabling to base the strong structural guarantees of injective one-way functions on the strong security guarantees of such primitives.Our application of derandomization techniques is inspired mainly by the work of Barak, Ong and Vadhan (CRYPTO '03), which on one hand relies on any one-way function, but on the other hand only results in a non-interactive perfectly-binding commitment scheme (offering significantly weaker structural guarantees compared to injective one-way functions), and does not seem to enable an extension to public-key primitives.The key observation underlying our approach is that Rudich's impossibility result applies not only to one-way functions as the underlying primitive, but in fact to a variety of "unstructured" primitives. We put forward a condition for identifying such primitives, and then subtly tailor the properties of our primitives such that they are both sufficiently unstructured in order to satisfy this condition, and sufficiently structured in order to yield injective one-way and trapdoor functions. This circumvents the basic approach underlying Rudich's long-standing evidence for the difficulty of constructing injective one-way functions (and, in particular, injective trapdoor functions) based on seemingly weaker or unstructured assumptions. A Simple Construction of iO for Turing Machines Abstract Sanjam Garg Akshayaram Srinivasan We give a simple construction of indistinguishability obfuscation for Turing machines where the time to obfuscate grows only with the description size of the machine and otherwise, independent of the running time and the space used. While this result is already known [Koppula, Lewko, and Waters, STOC 2015] from $$i\mathcal {O}$$ for circuits and injective pseudorandom generators, our construction and its analysis are conceptually much simpler. In particular, the main technical component in the proof of our construction is a simple combinatorial pebbling argument [Garg and Srinivasan, EUROCRYPT 2018]. Our construction makes use of indistinguishability obfuscation for circuits and $$\mathrm {somewhere\, statistically\, binding\, hash\, functions}$$ . Enhancements are Blackbox Non-trivial: Impossibility of Enhanced Trapdoor Permutations from Standard Trapdoor Permutations Abstract Mohammad Hajiabadi Trapdoor permutations (TDP) are a fundamental primitive in cryptography. Several variants of this notion have emerged as a result of different applications. However, it is not clear whether these variants can be based on the standard notion of TDPs.We study the question of whether enhanced trapdoor permutations can be based on classical trapdoor permutations. The main motivation of our work is in the context of existing TDP-based constructions of oblivious transfer and non-interactive zero knowledge protocols, which require enhancements to the classical TDP notion. We prove that these enhancements are non-trivial, in the sense that there does not exist fully blackbox constructions of enhanced TDPs from classical TDPs.On the technical side, we show that the enhanced TDP security of any construction in the random TDP oracle world can be broken via a polynomial number of queries to the TDP oracle as well as a weakening oracle, which provides inversion with respect to randomness. We also show that the standard one-wayness of the random TDP oracle stays intact in the presence of this weakening oracle. Succinct Garbling Schemes from Functional Encryption Through a Local Simulation Paradigm Abstract Prabhanjan Ananth Alex Lombardi We study a simulation paradigm, referred to as local simulation, in garbling schemes. This paradigm captures simulation proof strategies in which the simulator consists of many local simulators that generate different blocks of the garbled circuit. A useful property of such a simulation strategy is that only a few of these local simulators depend on the input, whereas the rest of the local simulators only depend on the circuit.We formalize this notion by defining locally simulatable garbling schemes. By suitably realizing this notion, we give a new construction of succinct garbling schemes for Turing machines assuming the polynomial hardness of compact functional encryption and standard assumptions (such as either CDH or LWE). Prior constructions of succinct garbling schemes either assumed sub-exponential hardness of compact functional encryption or were designed only for small-space Turing machines.We also show that a variant of locally simulatable garbling schemes can be used to generically obtain adaptively secure garbling schemes for circuits. All prior constructions of adaptively secure garbling that use somewhere equivocal encryption can be seen as instantiations of our construction. FE and iO for Turing Machines from Minimal Assumptions Abstract Shweta Agrawal Monosij Maitra We construct Indistinguishability Obfuscation ($$\mathsf {iO}$$) and Functional Encryption ($$\mathsf {FE}$$) schemes in the Turing machine model from the minimal assumption of compact $$\mathsf {FE}$$ for circuits ($$\mathsf {CktFE}$$). Our constructions overcome the barrier of sub-exponential loss incurred by all prior work. Our contributions are:1.We construct $$\mathsf {iO}$$ in the Turing machine model from the same assumptions as required in the circuit model, namely, sub-exponentially secure $$\mathsf {FE}$$ for circuits. The previous best constructions [6, 41] require sub-exponentially secure $$\mathsf {iO}$$ for circuits, which in turn requires sub-exponentially secure $$\mathsf {FE}$$ for circuits [5, 15].2.We provide a new construction of single input $$\mathsf {FE}$$ for Turing machines with unbounded length inputs and optimal parameters from polynomially secure, compact $$\mathsf {FE}$$ for circuits. The previously best known construction by Ananth and Sahai [7] relies on $$\mathsf {iO}$$ for circuits, or equivalently, sub-exponentially secure $$\mathsf {FE}$$ for circuits.3.We provide a new construction of multi-input $$\mathsf {FE}$$ for Turing machines. Our construction supports a fixed number of encryptors (say k), who may each encrypt a string $$\mathbf {x}_i$$ of unbounded length. We rely on sub-exponentially secure $$\mathsf {FE}$$ for circuits, while the only previous construction [10] relies on a strong knowledge type assumption, namely, public coin differing inputs obfuscation. Our techniques are new and from first principles, and avoid usage of sophisticated $$\mathsf {iO}$$ specific machinery such as positional accumulators and splittable signatures that were used by all relevant prior work [6, 7, 41]. Certifying Trapdoor Permutations, Revisited Abstract Ran Canetti Amit Lichtenberg The modeling of trapdoor permutations has evolved over the years. Indeed, finding an appropriate abstraction that bridges between the existing candidate constructions and the needs of applications has proved to be challenging. In particular, the notions of certifying permutations (Bellare and Yung, 96), enhanced and doubly enhanced trapdoor permutations (Goldreich, 04, 08, 11, Goldreich and Rothblum, 13) were added to bridge the gap between the modeling of trapdoor permutations and needs of applications. We identify an additional gap in the current abstraction of trapdoor permutations: Previous works implicitly assumed that it is easy to recognize elements in the domain, as well as uniformly sample from it, even for illegitimate function indices. We demonstrate this gap by using the (Bitansky-Paneth-Wichs, 16) doubly-enhanced trapdoor permutation family to instantiate the Feige-Lapidot-Shamir (FLS) paradigm for constructing non-interactive zero-knowledge (NIZK) protocols, and show that the resulting proof system is unsound. To close the gap, we propose a general notion of certifiably injective doubly enhanced trapdoor functions (DECITDFs), which provides a way of certifying that a given key defines an injective function over the domain defined by it, even when that domain is not efficiently recognizable and sampleable. We show that DECITDFs suffice for instantiating the FLS paradigm; more generally, we argue that certifiable injectivity is needed whenever the generation process of the function is not trusted. We then show two very different ways to construct DECITDFs: One is via the traditional method of RSA/Rabin with the Bellare-Yung certification mechanism, and the other using indistinguishability obfuscation and injective pseudorandom generators. In particular the latter is the first candidate injective trapdoor function, from assumptions other than factoring, that suffices for the FLS paradigm. Finally we observe that a similar gap appears also in other paths proposed in the literature for instantiating the FLS paradigm, specifically via verifiable pseudorandom generators and verifiable pseudorandom functions. Closing the gap there can be done in similar ways to the ones proposed here. On the Security Loss of Unique Signatures Abstract Andrew Morgan Rafael Pass We consider the question of whether the security of unique digital signature schemes can be based on game-based cryptographic assumptions using linear-preserving black-box security reductions—that is, black-box reductions for which the security loss (i.e., the ratio between "work" of the adversary and the "work" of the reduction) is some a priori bounded polynomial. A seminal result by Coron (Eurocrypt'02) shows limitations of such reductions; however, his impossibility result and its subsequent extensions all suffer from two notable restrictions: (1) they only rule out so-called "simple" reductions, where the reduction is restricted to only sequentially invoke "straight-line" instances of the adversary; and (2) they only rule out reductions to non-interactive (two-round) assumptions. In this work, we present the first full impossibility result: our main result shows that the existence of any linear-preserving black-box reduction for basing the security of unique signatures on some bounded-round assumption implies that the assumption can be broken in polynomial time. The MMap Strikes Back: Obfuscation and New Multilinear Maps Immune to CLT13 Zeroizing Attacks Abstract Fermi Ma Mark Zhandry All known multilinear map candidates have suffered from a class of attacks known as "zeroizing" attacks, which render them unusable for many applications. We provide a new construction of polynomial-degree multilinear maps and show that our scheme is provably immune to zeroizing attacks under a strengthening of the Branching Program Un-Annihilatability Assumption (Garg et al., TCC 2016-B).Concretely, we build our scheme on top of the CLT13 multilinear maps (Coron et al., CRYPTO 2013). In order to justify the security of our new scheme, we devise a weak multilinear map model for CLT13 that captures zeroizing attacks and generalizations, reflecting all known classical polynomial-time attacks on CLT13. In our model, we show that our new multilinear map scheme achieves ideal security, meaning no known attacks apply to our scheme. Using our scheme, we give a new multiparty key agreement protocol that is several orders of magnitude more efficient that what was previously possible.We also demonstrate the general applicability of our model by showing that several existing obfuscation and order-revealing encryption schemes, when instantiated with CLT13 maps, are secure against known attacks. These are schemes that are actually being implemented for experimentation, but until our work had no rigorous justification for security. On the Complexity of Fair Coin Flipping Abstract Iftach Haitner Nikolaos Makriyannis Eran Omri A two-party coin-flipping protocol is $$\varepsilon $$ε-fair if no efficient adversary can bias the output of the honest party (who always outputs a bit, even if the other party aborts) by more than $$\varepsilon $$ε. Cleve [STOC '86] showed that r-round o(1 / r)-fair coin-flipping protocols do not exist. Awerbuch et al. [Manuscript '85] constructed a $$\varTheta (1/\sqrt{r})$$Θ(1/r)-fair coin-flipping protocol, assuming the existence of one-way functions. Moran et al. [Journal of Cryptology '16] constructed an r-round coin-flipping protocol that is $$\varTheta (1/r)$$Θ(1/r)-fair (thus matching the aforementioned lower bound of Cleve [STOC '86]), assuming the existence of oblivious transfer.The above gives rise to the intriguing question of whether oblivious transfer, or more generally "public-key primitives", is required for an $$o(1/\sqrt{r})$$o(1/r)-fair coin flipping. This question was partially answered by Dachman-Soled et al. [TCC '11] and Dachman-Soled et al. [TCC '14], who showed that restricted types of fully black-box reductions cannot establish $$o(1/\sqrt{r})$$o(1/r)-fair coin-flipping protocols from one-way functions. In particular, for constant-round coin-flipping protocols, [10] yields that black-box techniques from one-way functions can only guarantee fairness of order $$1/\sqrt{r}$$1/r.We make progress towards answering the above question by showing that, for any constant , the existence of an $$1/(c\cdot \sqrt{r})$$1/(c·r)-fair, r-round coin-flipping protocol implies the existence of an infinitely-often key-agreement protocol, where c denotes some universal constant (independent of r). Our reduction is non black-box and makes a novel use of the recent dichotomy for two-party protocols of Haitner et al. [FOCS '18] to facilitate a two-party variant of the attack of Beimel et al. [FOCS '18] on multi-party coin-flipping protocols. Return of GGH15: Provable Security Against Zeroizing Attacks Abstract James Bartusek Jiaxin Guan Fermi Ma Mark Zhandry The GGH15 multilinear maps have served as the foundation for a number of cutting-edge cryptographic proposals. Unfortunately, many schemes built on GGH15 have been explicitly broken by so-called "zeroizing attacks," which exploit leakage from honest zero-test queries. The precise settings in which zeroizing attacks are possible have remained unclear. Most notably, none of the current indistinguishability obfuscation (iO) candidates from GGH15 have any formal security guarantees against zeroizing attacks.In this work, we demonstrate that all known zeroizing attacks on GGH15 implicitly construct algebraic relations between the results of zero-testing and the encoded plaintext elements. We then propose a "GGH15 zeroizing model" as a new general framework which greatly generalizes known attacks.Our second contribution is to describe a new GGH15 variant, which we formally analyze in our GGH15 zeroizing model. We then construct a new iO candidate using our multilinear map, which we prove secure in the GGH15 zeroizing model. This implies resistance to all known zeroizing strategies. The proof relies on the Branching Program Un-Annihilatability (BPUA) Assumption of Garg et al. [TCC 16-B] (which is implied by PRFs in $$\mathsf {NC}^1$$ secure against $$\mathsf {P}/\mathsf {poly}$$) and the complexity-theoretic p-Bounded Speedup Hypothesis of Miles et al. [ePrint 14] (a strengthening of the Exponential Time Hypothesis). Game Theoretic Notions of Fairness in Multi-party Coin Toss Abstract Kai-Min Chung Yue Guo Wei-Kai Lin Rafael Pass Elaine Shi Coin toss has been extensively studied in the cryptography literature, and the well-accepted notion of fairness (henceforth called strong fairness) requires that a corrupt coalition cannot cause non-negligible bias. It is well-understood that two-party coin toss is impossible if one of the parties can prematurely abort; further, this impossibility generalizes to multiple parties with a corrupt majority (even if the adversary is computationally bounded and fail-stop only).Interestingly, the original proposal of (two-party) coin toss protocols by Blum in fact considered a weaker notion of fairness: imagine that the (randomized) transcript of the coin toss protocol defines a winner among the two parties. Now Blum's notion requires that a corrupt party cannot bias the outcome in its favor (but self-sacrificing bias is allowed). Blum showed that this weak notion is indeed attainable for two parties assuming the existence of one-way functions.In this paper, we ask a very natural question which, surprisingly, has been overlooked by the cryptography literature: can we achieve Blum's weak fairness notion in multi-party coin toss? What is particularly interesting is whether this relaxation allows us to circumvent the corrupt majority impossibility that pertains to strong fairness. Even more surprisingly, in answering this question, we realize that it is not even understood how to define weak fairness for multi-party coin toss. We propose several natural notions drawing inspirations from game theory, all of which equate to Blum's notion for the special case of two parties. We show, however, that for multiple parties, these notions vary in strength and lead to different feasibility and infeasibility results. The Security of Lazy Users in Out-of-Band Authentication Abstract Moni Naor Lior Rotem Gil Segev Faced with the threats posed by man-in-the-middle attacks, messaging platforms rely on "out-of-band" authentication, assuming that users have access to an external channel for authenticating one short value. For example, assuming that users recognizing each other's voice can authenticate a short value, Telegram and WhatApp ask their users to compare 288-bit and 200-bit values, respectively. The existing protocols, however, do not take into account the plausible behavior of users who may be "lazy" and only compare parts of these values (rather than their entirety).Motivated by such a security-critical user behavior, we study the security of lazy users in out-of-band authentication. We start by showing that both the protocol implemented by WhatsApp and the statistically-optimal protocol of Naor, Segev and Smith (CRYPTO '06) are completely vulnerable to man-in-the-middle attacks when the users consider only a half of the out-of-band authenticated value. In this light, we put forward a framework that captures the behavior and security of lazy users. Our notions of security consider both statistical security and computational security, and for each flavor we derive a lower bound on the tradeoff between the number of positions that are considered by the lazy users and the adversary's forgery probability.Within our framework we then provide two authentication protocols. First, in the statistical setting, we present a transformation that converts any out-of-band authentication protocol into one that is secure even when executed by lazy users. Instantiating our transformation with a new refinement of the protocol of Naor et al. results in a protocol whose tradeoff essentially matches our lower bound in the statistical setting. Then, in the computational setting, we show that the computationally-optimal protocol of Vaudenay (CRYPTO '05) is secure even when executed by lazy users – and its tradeoff matches our lower bound in the computational setting. Achieving Fair Treatment in Algorithmic Classification Abstract Fairness in classification has become an increasingly relevant and controversial issue as computers replace humans in many of today's classification tasks. In particular, a subject of much recent debate is that of finding, and subsequently achieving, suitable definitions of fairness in an algorithmic context. In this work, following the work of Hardt et al. (NIPS'16), we consider and formalize the task of sanitizing an unfair classifier $$\mathcal {C}$$C into a classifier $$\mathcal {C}'$$C′ satisfying an approximate notion of "equalized odds" or fair treatment. Our main result shows how to take any (possibly unfair) classifier $$\mathcal {C}$$C over a finite outcome space, and transform it—by just perturbing the output of $$\mathcal {C}$$C—according to some distribution learned by just having black-box access to samples of labeled, and previously classified, data, to produce a classifier $$\mathcal {C}'$$C′ that satisfies fair treatment; we additionally show that our derived classifier is near-optimal in terms of accuracy. We also experimentally evaluate the performance of our method. Is There an Oblivious RAM Lower Bound for Online Reads? Abstract Mor Weiss Daniel Wichs Oblivious RAM (ORAM), introduced by Goldreich and Ostrovsky (JACM 1996), can be used to read and write to memory in a way that hides which locations are being accessed. The best known ORAM schemes have an $$O(\log n)$$ overhead per access, where $$n$$ is the data size. The work of Goldreich and Ostrovsky gave a lower bound showing that this is optimal for ORAM schemes that operate in a "balls and bins" model, where memory blocks can only be shuffled between different locations but not manipulated otherwise. The lower bound even extends to weaker settings such as offline ORAM, where all of the accesses to be performed need to be specified ahead of time, and read-only ORAM, which only allows reads but not writes. But can we get lower bounds for general ORAM, beyond "balls and bins"?The work of Boyle and Naor (ITCS '16) shows that this is unlikely in the offline setting. In particular, they construct an offline ORAM with $$o(\log n)$$ overhead assuming the existence of small sorting circuits. Although we do not have instantiations of the latter, ruling them out would require proving new circuit lower bounds. On the other hand, the recent work of Larsen and Nielsen (CRYPTO '18) shows that there indeed is an $$\varOmega (\log n)$$ lower bound for general online ORAM.This still leaves the question open for online read-only ORAM or for read/write ORAM where we want very small overhead for the read operations. In this work, we show that a lower bound in these settings is also unlikely. In particular, our main result is a construction of online ORAM where reads (but not writes) have an $$o(\log n)$$ overhead, assuming the existence of small sorting circuits as well as very good locally decodable codes (LDCs). Although we do not have instantiations of either of these with the required parameters, ruling them out is beyond current lower bounds. Upgrading to Functional Encryption Abstract Saikrishna Badrinarayanan Dakshita Khurana Amit Sahai Brent Waters The notion of Functional Encryption (FE) has recently emerged as a strong primitive with several exciting applications. In this work, we initiate the study of the following question: Can existing public key encryption schemes be "upgraded" to Functional Encryption schemes without changing their public keys or the encryption algorithm? We call a public-key encryption scheme with this property to be FE-compatible. Indeed, assuming ideal obfuscation, it is easy to see that every CCA-secure public-key encryption scheme is FE-compatible. Despite the recent success in using indistinguishability obfuscation to replace ideal obfuscation for many applications, we show that this phenomenon most likely will not apply here. We show that assuming fully homomorphic encryption and the learning with errors (LWE) assumption, there exists a CCA-secure encryption scheme that is provably not FE-compatible. We also show that a large class of natural CCA-secure encryption schemes proven secure in the random oracle model are not FE-compatible in the random oracle model.Nevertheless, we identify a key structure that, if present, is sufficient to provide FE-compatibility. Specifically, we show that assuming sub-exponentially secure iO and sub-exponentially secure one way functions, there exists a class of public key encryption schemes which we call Special-CCA secure encryption schemes that are in fact, FE-compatible. In particular, each of the following popular CCA secure encryption schemes (some of which existed even before the notion of FE was introduced) fall into the class of Special-CCA secure encryption schemes and are thus FE-compatible:1.[CHK04] when instantiated with the IBE scheme of [BB04].2.[CHK04] when instantiated with any Hierarchical IBE scheme.3.[PW08] when instantiated with any Lossy Trapdoor Function. Perfectly Secure Oblivious Parallel RAM Abstract T.-H. Hubert Chan Kartik Nayak Elaine Shi We show that PRAMs can be obliviously simulated with perfect security, incurring only $$O(\log N \log \log N)$$ blowup in parallel runtime, $$O(\log ^3 N)$$ blowup in total work, and O(1) blowup in space relative to the original PRAM. Our results advance the theoretical understanding of Oblivious (Parallel) RAM in several respects. First, prior to our work, no perfectly secure Oblivious Parallel RAM (OPRAM) construction was known; and we are the first in this respect. Second, even for the sequential special case of our algorithm (i.e., perfectly secure ORAM), we not only achieve logarithmic improvement in terms of space consumption relative to the state-of-the-art, but also significantly simplify perfectly secure ORAM constructions. Third, our perfectly secure OPRAM scheme matches the parallel runtime of earlier statistically secure schemes with negligible failure probability. Since we remove the dependence (in performance) on the security parameter, our perfectly secure OPRAM scheme in fact asymptotically outperforms known statistically secure ones if (sub-)exponentially small failure probability is desired. Our techniques for achieving small parallel runtime are novel and we employ special expander graphs to derandomize earlier statistically secure OPRAM techniques—this is the first time such techniques are used in the constructions of ORAMs/OPRAMs. Impossibility of Simulation Secure Functional Encryption Even with Random Oracles Abstract Shashank Agrawal Venkata Koppula Brent Waters In this work we study the feasibility of achieving simulation security in functional encryption (FE) in the random oracle model. Our main result is negative in that we give a functionality for which it is impossible to achieve simulation security even with the aid of random oracles.We begin by giving a formal definition of simulation security that explicitly incorporates the random oracles. Next, we show a particular functionality for which it is impossible to achieve simulation security. Here messages are interpreted as seeds to a (weak) pseudorandom function family F and private keys are ascribed to points in the domain of the function. On a message s and private key x one can learn F(s, x). We show that there exists an attacker that makes a polynomial number of private key queries followed by a single ciphertext query for which there exists no simulator.Our functionality and attacker access pattern closely matches the standard model impossibility result of Agrawal, Gorbunov, Vaikuntanathan and Wee (CRYPTO 2013). The crux of their argument is that no simulator can succinctly program in the outputs of an unbounded number of evaluations of a pseudorandom function family into a fixed size ciphertext. However, their argument does not apply in the random oracle setting since the oracle acts as an additional conduit of information which the simulator can program. We overcome this barrier by proposing an attacker who decrypts the challenge ciphertext with the secret keys issued earlier without using the random oracle, even though the decryption algorithm may require it. This involves collecting most of the useful random oracle queries in advance, without giving the simulator too many opportunities to program.On the flip side, we demonstrate the utility of the random oracle in simulation security. Given only public key encryption and low-depth PRGs we show how to build an FE system that is simulation secure for any poly-time attacker that makes an unbounded number of message queries, but an a-priori bounded number of key queries. This bests what is possible in the standard model where it is only feasible to achieve security for an attacker that is bounded both in the number of key and message queries it makes. We achieve this by creating a system that leverages the random oracle to get one-key security and then adapt previously known techniques to boost the system to resist up to q queries.Finally, we ask whether it is possible to achieve simulation security for an unbounded number of messages and keys, but where all key queries are made after the message queries. We show this too is impossible to achieve using a different twist on our first impossibility result. Fine-Grained Secure Computation Abstract Matteo Campanelli Rosario Gennaro This paper initiates a study of Fine Grained Secure Computation: i.e. the construction of secure computation primitives against "moderately complex" adversaries. We present definitions and constructions for compact Fully Homomorphic Encryption and Verifiable Computation secure against (non-uniform) $$\mathsf {NC}^1$$ adversaries. Our results do not require the existence of one-way functions and hold under a widely believed separation assumption, namely $$\mathsf {NC}^{1}\subsetneq \oplus \mathsf {L}/ {\mathsf {poly}}$$ . We also present two application scenarios for our model: (i) hardware chips that prove their own correctness, and (ii) protocols against rational adversaries potentially relevant to the Verifier's Dilemma in smart-contracts transactions such as Ethereum. Watermarking PRFs Under Standard Assumptions: Public Marking and Security with Extraction Queries Abstract Willy Quach Daniel Wichs Giorgos Zirdelis A software watermarking scheme can embed some information called a mark into a program while preserving its functionality. No adversary can remove the mark without damaging the functionality of the program. Cohen et al. (STOC '16) gave the first positive results for watermarking, showing how to watermark certain pseudorandom function (PRF) families using indistinguishability obfuscation (iO). Their scheme has a secret marking procedure to embed marks in programs and a public extraction procedure to extract the marks from programs; security holds even against an attacker that has access to a marking oracle. Kim and Wu (CRYPTO '17) later constructed a PRF watermarking scheme under only the LWE assumption. In their scheme, both the marking and extraction procedures are secret, but security only holds against an attacker with access to a marking oracle but not an extraction oracle. In fact, it is possible to completely break the security of the latter scheme using extraction queries, which is a significant limitation in any foreseeable application.In this work, we construct a new PRF watermarking scheme with the following properties. The marking procedure is public and therefore anyone can embed marks in PRFs from the family. Previously we had no such construction even using obfuscation.The extraction key is secret, but marks remain unremovable even if the attacker has access to an extraction oracle. Previously we had no such construction under standard assumptions.Our scheme is simple, uses generic components and can be instantiated under many different assumptions such as DDH, Factoring or LWE. The above benefits come with one caveat compared to prior work: the PRF family that we can watermark depends on the public parameters of the watermarking scheme and the watermarking authority has a secret key which can break the security of all of the PRFs in the family. Since the watermarking authority is usually assumed to be trusted, this caveat appears to be acceptable. No-signaling Linear PCPs Abstract Susumu Kiyoshima In this paper, we give a no-signaling linear probabilistically checkable proof (PCP) system for polynomial-time deterministic computation, i.e., a PCP system for $${\mathcal {P}}$$P such that (1) the PCP oracle is a linear function and (2) the soundness holds against any (computational) no-signaling cheating prover, who is allowed to answer each query according to a distribution that depends on the entire query set in a certain way. To the best of our knowledge, our construction is the first PCP system that satisfies these two properties simultaneously.As an application of our PCP system, we obtain a 2-message scheme for delegating computation by using a known transformation. Compared with existing 2-message delegation schemes based on standard cryptographic assumptions, our scheme requires preprocessing but has a simpler structure and makes use of different (possibly cheaper) standard cryptographic primitives, namely additive/multiplicative homomorphic encryption schemes. Registration-Based Encryption: Removing Private-Key Generator from IBE Abstract Sanjam Garg Mohammad Hajiabadi Mohammad Mahmoody Ahmadreza Rahimi In this work, we introduce the notion of registration-based encryption (RBE for short) with the goal of removing the trust parties need to place in the private-key generator in an IBE scheme. In an RBE scheme, users sample their own public and secret keys. There will also be a "key curator" whose job is only to aggregate the public keys of all the registered users and update the "short" public parameter whenever a new user joins the system. Encryption can still be performed to a particular recipient using the recipient's identity and any public parameters released subsequent to the recipient's registration. Decryption requires some auxiliary information connecting users' public (and secret) keys to the public parameters. Because of this, as the public parameters get updated, a decryptor may need to obtain "a few" additional auxiliary information for decryption. More formally, if n is the total number of identities and $$\mathrm {\kappa }$$κ is the security parameter, we require the following.Efficiency requirements: (1) A decryptor only needs to obtain updated auxiliary information for decryption at most $$O(\log n)$$O(logn) times in its lifetime, (2) each of these updates are computed by the key curator in time $${\text {poly}}(\mathrm {\kappa },\log n)$$poly(κ,logn), and (3) the key curator updates the public parameter upon the registration of a new party in time $${\text {poly}}(\mathrm {\kappa },\log n)$$poly(κ,logn). Properties (2) and (3) require the key curator to have random access to its data.Compactness requirements: (1) Public parameters are always at most $${\text {poly}}(\mathrm {\kappa },\log n)$$poly(κ,logn) bit, and (2) the total size of updates a user ever needs for decryption is also at most $${\text {poly}}(\mathrm {\kappa },\log n)$$poly(κ,logn) bits.We present feasibility results for constructions of RBE based on indistinguishably obfuscation. We further provide constructions of weakly efficient RBE, in which the registration step is done in $${\text {poly}}(\mathrm {\kappa },n)$$poly(κ,n), based on CDH, Factoring or LWE assumptions. Note that registration is done only once per identity, and the more frequent operation of generating updates for a user, which can happen more times, still runs in time $${\text {poly}}(\mathrm {\kappa },\log n)$$poly(κ,logn). We leave open the problem of obtaining standard RBE (with $${\text {poly}}(\mathrm {\kappa },\log n)$$poly(κ,logn) registration time) from standard assumptions. Exploring Crypto Dark Matter: Abstract Dan Boneh Yuval Ishai Alain Passelègue Amit Sahai David J. Wu Pseudorandom functions (PRFs) are one of the fundamental building blocks in cryptography. Traditionally, there have been two main approaches for PRF design: the "practitioner's approach" of building concretely-efficient constructions based on known heuristics and prior experience, and the "theoretician's approach" of proposing constructions and reducing their security to a previously-studied hardness assumption. While both approaches have their merits, the resulting PRF candidates vary greatly in terms of concrete efficiency and design complexity.In this work, we depart from these traditional approaches by exploring a new space of plausible PRF candidates. Our guiding principle is to maximize simplicity while optimizing complexity measures that are relevant to cryptographic applications. Our primary focus is on weak PRFs computable by very simple circuits—specifically, depth-2$$\mathsf {ACC}^0$$ circuits. Concretely, our main weak PRF candidate is a "piecewise-linear" function that first applies a secret mod-2 linear mapping to the input, and then a public mod-3 linear mapping to the result. We also put forward a similar depth-3 strong PRF candidate.The advantage of our approach is twofold. On the theoretical side, the simplicity of our candidates enables us to draw many natural connections between their hardness and questions in complexity theory or learning theory (e.g., learnability of $$\mathsf {ACC}^0$$ and width-3 branching programs, interpolation and property testing for sparse polynomials, and new natural proof barriers for showing super-linear circuit lower bounds). On the applied side, the piecewise-linear structure of our candidates lends itself nicely to applications in secure multiparty computation (MPC). Using our PRF candidates, we construct protocols for distributed PRF evaluation that achieve better round complexity and/or communication complexity (often both) compared to protocols obtained by combining standard MPC protocols with PRFs like AES, LowMC, or Rasta (the latter two are specialized MPC-friendly PRFs).Finally, we introduce a new primitive we call an encoded-input PRF, which can be viewed as an interpolation between weak PRFs and standard (strong) PRFs. As we demonstrate, an encoded-input PRF can often be used as a drop-in replacement for a strong PRF, combining the efficiency benefits of weak PRFs and the security benefits of strong PRFs. We conclude by showing that our main weak PRF candidate can plausibly be boosted to an encoded-input PRF by leveraging standard error-correcting codes. On Basing Search SIVP on NP-Hardness Abstract ★ Best Student Paper Tianren Liu The possibility of basing cryptography on the minimal assumption $$\mathbf{NP }\nsubseteq \mathbf{BPP }$$ NP⊈BPP is at the very heart of complexity-theoretic cryptography. The closest we have gotten so far is lattice-based cryptography whose average-case security is based on the worst-case hardness of approximate shortest vector problems on integer lattices. The state-of-the-art is the construction of a one-way function (and collision-resistant hash function) based on the hardness of the $$\tilde{O}(n)$$ O~(n)-approximate shortest independent vector problem $${\textsf {SIVP}}_{\tilde{O}(n)}$$ SIVPO~(n).Although $${\textsf {SIVP}}$$ SIVP is NP-hard in its exact version, Guruswami et al. (CCC 2004) showed that $${\textsf {gapSIVP}}_{\sqrt{n/\log n}}$$ gapSIVPn/logn is in $$\mathbf{NP } \cap \mathbf{coAM }$$ NP∩coAM and thus unlikely to be $$\mathbf{NP }$$ NP-hard. Indeed, any language that can be reduced to $${\textsf {gapSIVP}}_{\tilde{O}(\sqrt{n})}$$ gapSIVPO~(n) (under general probabilistic polynomial-time adaptive reductions) is in $$\mathbf{AM } \cap \mathbf{coAM }$$ AM∩coAM by the results of Peikert and Vaikuntanathan (CRYPTO 2008) and Mahmoody and Xiao (CCC 2010). However, none of these results apply to reductions to search problems, still leaving open a ray of hope: can $$\mathbf{NP }$$ NPbe reduced to solving search SIVP with approximation factor $$\tilde{O}(n)$$ O~(n)?We eliminate such possibility, by showing that any language that can be reduced to solving search $${\textsf {SIVP}}$$ SIVP with any approximation factor $$\lambda (n) = \omega (n\log n)$$ λ(n)=ω(nlogn) lies in AM intersect coAM. On the Structure of Unconditional UC Hybrid Protocols Abstract Mike Rosulek Morgan Shirley We study the problem of secure two-party computation in the presence of a trusted setup. If there is an unconditionally UC-secure protocol for f that makes use of calls to an ideal g, then we say that freduces tog (and write $$f \sqsubseteq g$$). Some g are complete in the sense that all functions reduce to g. However, almost nothing is known about the power of an incomplete g in this setting. We shed light on this gap by showing a characterization of $$f \sqsubseteq g$$ for incomplete g.Very roughly speaking, we show that f reduces to g if and only if it does so by the simplest possible protocol: one that makes a single call to ideal g and uses no further communication. Furthermore, such simple protocols can be characterized by a natural combinatorial condition on f and g.Looking more closely, our characterization applies only to a very wide class of f, and only for protocols that are deterministic or logarithmic-round. However, we give concrete examples showing that both of these limitations are inherent to the characterization itself. Functions not covered by our characterization exhibit qualitatively different properties. Likewise, randomized, superlogarithmic-round protocols are qualitatively more powerful than deterministic or logarithmic-round ones.	CommonCrawl
\begin{document} \title{A particle micro-macro decomposition based numerical scheme for collisional kinetic equations in the diffusion scaling.} \footnotetext[1] {INRIA Rennes - Bretagne Atlantique, IPSO team \& Laboratoire de Math\'ematiques Jean Leray, CNRS UMR 6629, Universit\'e de Nantes, France. E-mail: \url{[email protected]}} \footnotetext[2] {INRIA Rennes - Bretagne Atlantique, IPSO team \& Institut de Recherche Math\'ematiques de Rennes, CNRS UMR 6625, Universit\'e de Rennes 1, France \& ENS Rennes. E-mail: \url{[email protected]}} \footnotetext[3] {Institut de Recherche Math\'ematiques de Rennes, CNRS UMR 6625, Universit\'e de Rennes 1, France \& ENS Rennes \& INRIA Rennes - Bretagne Atlantique, IPSO team. E-mail: \url{[email protected]}} \begin{abstract} In this work, we derive particle schemes, based on micro-macro decomposition, for linear kinetic equations in the diffusion limit. Due to the particle approximation of the micro part, a splitting between the transport and the collision part has to be performed, and the stiffness of both these two parts prevent from uniform stability. To overcome this difficulty, the micro-macro system is reformulated into a continuous PDE whose coefficients are no longer stiff, and depend on the time step $\Delta t$ in a consistent way. This non-stiff reformulation of the micro-macro system allows the use of standard particle approximations for the transport part, and extends the work in \cite{ccl} where a particle approximation has been applied using a micro-macro decomposition on kinetic equations in the fluid scaling. Beyond the so-called asymptotic-preserving property which is satisfied by our schemes, they significantly reduce the inherent noise of traditional particle methods, and they have a computational cost which decreases as the system approaches the diffusion limit. \end{abstract} \section{Introduction} Particle systems appearing in plasma physics or radiative transfer can be described at different scales. When the system is far from its thermodynamical equilibrium, a kinetic description is necessary. Particles are then represented by a distribution function $f$ which depends on time $t\geq 0$, position $x\in\mathbb{R}^d$ and velocity $v\in V\subset\mathbb{R}^d$, $d\geq 1$. The distribution $f\left(t,x,v\right)$ satisfies a collisional kinetic equation. Particle methods are often used for simulating kinetic problems, especially in realistic $3$-dimensional situations, $d=3$. However, they are affected by numerical noise due to their probabilistic character. A simple way to reduce this noise is to increase the number of particles, but then the numerical cost increases as well. Other standard kinetic descriptions, as phase space grid methods, may require too much memory in the two or three dimensional framework. Otherwise, macroscopic descriptions depending only on $t$ and $x$ can be sufficient if the system stays near its thermodynamical equilibrium, and are less expensive since their unknown does not depend on the velocity variable anymore. Beside the noisy character of standard particle methods, there is an additional difficulty in kinetic descriptions which is linked to the presence of various scales in the system. Multi-scale phenomena may indeed appear in plasma devices or radiative transfer applications, depending on some physical parameters as for example the mean free path of particles or the Knudsen number denoted here by $\varepsilon$. This multi-scale character is often represented by stiff terms in the kinetic equation, and the general challenge is to construct efficient numerical methods for these multiscale kinetic equations: this means that, without numerically resolving the stiffness, the numerical method must solve accurately the kinetic regime, must have the right asymptotics in the high-stiffness limit (the so-called asymptotic preserving property) and its computational cost should decrease as the system approaches the equilibrium (a time diminishing property). Note that direct numerical methods whose parameters resolve the smallest scale of size $\varepsilon$ are impossible to use, since they automatically involve an extremely high computational cost. Several strategies have been proposed to overcome this strong constraint. Domain decomposition methods can be applied when we have different regions with different values of the scaling parameter, see \cite{dd,gjl, tallec, tiwari}. When the different scales are less clearly delimited, we have to develop kinetic schemes that naturally reduce to good approximations of the macroscopic problem when the system goes near its equilibrium, and overcome the stiffness. Such schemes are often called Asymptotic-Preserving (AP), see \cite{jin,bt,klar,dp,cl,larsen,lemou-note,mlb,lm,jpt,np,buet}. Mainly, the numerical cost remains comparable to the one of the non-stiff kinetic problem, even when $\varepsilon\ll 1$. Our goal is to design an efficient AP scheme, {\em using particles}, for the following kinetic radiative transport equation (RTE) in the diffusion scaling \begin{equation} \partial_t f +\frac{1}{\varepsilon} v\partial_x f = \frac{1}{\varepsilon^2}(\rho M - f), \;\; f(t=0, x, v)=f_0(x, v), \label{eq:etrbgk} \end{equation} where $x\in \Omega\subset\mathbb{R}$, $\rho(t, x)=\frac{1}{2}\int_V f(t, x, v) \textnormal{d} v$, $V=[-1, 1]$, $M=1$ and $f_0(x, v)$ is a given initial condition. Periodic boundary conditions are considered. It is well-known (see \cite{larsen, dgp}) that when $\varepsilon$ goes to zero, the distribution function $f(t, x, v)$ converges towards $\bar{\rho}(t, x) M(v)$, where $\bar{\rho}$ satisfies the following diffusion equation \begin{equation} \partial_t \bar{\rho} -\frac{1}{3} \partial_{xx} \bar{\rho} = 0, \;\; \bar{\rho}(t=0, x) = \frac{1}{2}\int_V f_0(x, v) \textnormal{d} v. \label{eq:diff} \end{equation} An extension to the Vlasov-Poisson-BGK case is presented in Section \ref{ssec:e}. The kinetic equation is coupled to a Poisson equation for the electric field denoted by $E(t, x)$. More precisely, we consider \begin{eqnarray} \partial_t f +\frac{1}{\varepsilon} v\partial_x f + \frac{1}{\varepsilon}E\partial_v f= \frac{1}{\varepsilon^2}(\rho M - f), \label{eq:vlasovbgk} \\ \partial_x E=\rho-1\label{eq:vlasovbgkpoisson}, \end{eqnarray} where $x\in \Omega\subset\mathbb{R}$, $\rho(t, x)=\int_V f(t, x, v) \textnormal{d} v$, $V=\mathbb{R}$, $M\left(v\right)=\frac{1}{\sqrt{2\pi}}\exp\left(-\frac{v^2}{2}\right)$ is the absolute Maxwellian and we consider periodic boundary conditions. Note that an additional condition $\int_\Omega E \textnormal{d} x=0$ is imposed to obtain a well-posed problem. When $\varepsilon$ goes to zero, the asymptotic model is a drift-diffusion equation satisfied by $\bar{\rho}(t, x)$ (see \cite{bat}) \begin{equation} \partial_t \bar{\rho} - \partial_x(\partial_x \bar{\rho}-\bar{E}\bar{\rho}) = 0, \;\; \bar{\rho}(t=0, x) =\int_\mathbb{R} f_0(x, v) \textnormal{d} v, \label{eq:drift-diff} \end{equation} where $\bar{E}$ is linked to $\bar{\rho}$ by the Poisson equation $\partial_x\bar{E}=\bar{\rho}-1$. The strategy will be the use of the micro-macro decomposition (see \cite{liu, lm,bennoune,cl}). It consists in writing the distribution function as the sum of the equilibrium part and a rest. One can then derive a system of two equations: a kinetic one for the rest $g(t, x, v)$ and a macroscopic one for the equilibrium $\rho(t, x) M(v)$. AP micro-macro schemes for (\ref{eq:etrbgk}) have been proposed in \cite{lm,bennoune,cl}. These schemes consist in a semi-implicit phase space grid method for the kinetic part, coupled to a classical spatial grid method for the macro part. Our strategy in this work follows the strategy of \cite{ccl} in the case of a fluid scaling: we use particles to sample the kinetic part whereas an Eulerian solver is used to discretize the macro unknown. The main motivation of this strategy lies in the fact that the micro part $g$ converges to zero when $\varepsilon$ goes to zero, so that a very few number of particles can sample it. As a consequence, in this regime, the cost of the global micro-macro solver is almost the same as the cost of an asymptotic solver for \eqref{eq:diff}. In this work, we focus on a diffusion type scaling (as in \eqref{eq:etrbgk} or \eqref{eq:vlasovbgk}) so that an additional scale is involved compared to the fluid scaling considered in \cite{ccl}. In \cite{lm, cl}, a diffusion scaling was studied, but using a fully grid based solver. Hence, the stiffest term (of order $1/\varepsilon^2$) is considered implicit in time in the micro equation, which enables to stabilize the transport term (of order $1/\varepsilon$) and then to derive an AP scheme for \eqref{eq:etrbgk} and \eqref{eq:vlasovbgk}. The use of particles for the micro part prevents from a similar strategy since a splitting between the transport term (of order $1/\varepsilon$) and the source term needs to be done. Then, a uniform stable scheme is hard to obtain in this context. To overcome this difficulty, a suitable formulation of the original model \eqref{eq:etrbgk} is performed so that the stiff transport term $(1/\varepsilon) \, v\partial_x g$ becomes $\varepsilon/\Delta t (1-e^{-\Delta t/\varepsilon^2}) v\partial_x g$. This reformulation is correct up to $\Delta t^2$ (for fixed $\varepsilon>0$) and has the good behavior when $\varepsilon$ goes to zero (for fixed $\Delta t>0$). This formulation is the starting point of the design of micro-macro-particle based numerical schemes which enjoy the AP property and for which the numerical cost diminishes as $\varepsilon$ goes to zero. This approach is extended to the second-order (in time) and to the Vlasov-Poisson-BGK case \eqref{eq:vlasovbgk}-\eqref{eq:vlasovbgkpoisson}. The sequel of the paper is organized as follows. In Section \ref{sec:1storder_00}, we recall the formal derivation of the asymptotic model of \eqref{eq:vlasovbgk} and \eqref{eq:etrbgk}. The first-order (in time) reformulation of \eqref{eq:etrbgk} is presented in Subsection \ref{sec:1storder} and its Lagrangian discretization in Subsection \ref{ssec:pic}. Its extension to a second-order in time model is detailed in Section \ref{sec:2ndorder_0}: the continuous model is presented in Subsection \ref{sec:2ndorder} and its discretization is developed in Subsection \ref{subsec:ordre2_discr}. Section \ref{ssec:e} proposes an extension of our strategy to the Vlasov-Poisson-BGK system. Finally, Section \ref{sec:numres} is devoted to numerical simulations. \section{Diffusion asymptotics} \setcounter{equation}{0} \label{sec:1storder_00} In this section, we recall the main steps of the derivation of the model obtained from \eqref{eq:vlasovbgk} when $\varepsilon$ goes to zero. To do so, we consider the micro-macro decomposition (see \cite{lm,bennoune,liu}) of $f$: $f(t, x, v)=\rho(t, x) M(v) + g(t, x, v)$, with $\rho(t, x)=\langle f\rangle$, $M(v)$ is the Maxwellian equilibrium and the rest $g$ satisfies $\langle g\rangle=0$. Here $\langle f \rangle =\int_V f(v)\textnormal{d} v$, with $V=\mathbb{R}$ and we also use the notation $\Pi f = \langle f\rangle M$. The following micro-macro model is equivalent to the original model \eqref{eq:vlasovbgk} \begin{equation} \label{eq:micromacro_initial_E} \left\{ \begin{aligned} \partial_t \rho &+ \frac{1}{\varepsilon}\partial_x \langle v g \rangle = 0,\\ \partial_t g &+ \frac{1}{\varepsilon}(I-\Pi) \left[v \partial_x (\rho M+g) + E\partial_v (\rho M+g) \right]= -\frac{1}{\varepsilon^2}g. \end{aligned} \right. \end{equation} Since $\langle vM\rangle=0$ and $\partial_v M=-vM$, the micro equation can be rewritten as \begin{equation} \label{micro} \partial_t g + \frac{1}{\varepsilon}\left[ vM\partial_x \rho - vME\rho + (I-\Pi)(v\partial_x g +E\partial_v g) \right]= -\frac{1}{\varepsilon^2}g. \end{equation} When $\varepsilon$ goes to zero, one gets from \eqref{micro}: $g=-\varepsilon (vM\partial_x \rho - vME\rho) + {\cal O}(\varepsilon^2)$. Then the macro equation becomes $$ \partial_t \rho - \partial_x \left[ \langle v^2 M\rangle \partial_x \rho -\langle v^2 M\rangle E\rho\right] = {\cal O}(\varepsilon), $$ which gives, using $\langle v^2 M\rangle=1$ the following drift-diffusion equation satisfied by the limit $\bar{\rho}$ \begin{equation} \partial_t \bar{\rho} - \partial_{xx} \bar{\rho} + \partial_x (\bar{E}\bar{\rho})=0, \;\; \partial_x \bar{E} = \bar{\rho}-1. \label{eq:ddlimit} \end{equation} The same calculations enables to derive the micro-macro model equivalent to \eqref{eq:etrbgk} \begin{equation} \label{eq:micromacro_initial} \left\{ \begin{aligned} \partial_t \rho &+ \frac{1}{\varepsilon}\partial_x \langle v g \rangle = 0,\\ \partial_t g &+ \frac{1}{\varepsilon}(I-\Pi) \left[v \partial_x (\rho M+g)\right]= -\frac{1}{\varepsilon^2}g, \end{aligned} \right. \end{equation} from which we derive the corresponding asymptotic model $$ \partial_t \bar{\rho} - \frac{1}{3}\partial_{xx} \bar{\rho}=0, $$ with $M=1$, $V=\left[-1,1\right]$ and $\langle f\rangle =\frac{1}{2}\int_V f \textnormal{d} v$. \section{First-order in time reformulation and its discretization} \setcounter{equation}{0} \label{sec:1storder_0} In this part, a first-order reformulation of the micro part is proposed, which enables to avoid the stiff transport term in space. The strategy is presented in the case of the equation \eqref{eq:etrbgk} and its corresponding micro-macro model \eqref{eq:micromacro_initial}. \subsection{First-order in time reformulation} \label{sec:1storder} We start with (\ref{eq:etrbgk}) (with periodic boundary condition in space) and consider the micro-macro decomposition of $f=\rho + g$ (here $M(v)=1$ for all $v\in [-1, 1]$) and the micro-macro model \eqref{eq:micromacro_initial}. First, we rewrite the micro part of \eqref{eq:micromacro_initial} as \begin{equation} \label{eq:expg} \partial_t (e^{t/\varepsilon^2} g) = - \frac{e^{t/\varepsilon^2} }{\varepsilon}\mathcal{F}\left(\rho,g\right), \end{equation} where $\mathcal{F}\left(\rho,g\right)$ is given by \begin{equation} \label{def:F} \mathcal{F}\left(\rho,g\right) = v\partial_x \rho + v\partial_x g - \partial_x \langle vg\rangle. \end{equation} We denote $\Delta t>0$ the time step, $t^n=n\Delta t$ with $n\in\mathbb{N}$. Then, a second step consists in integrating \eqref{eq:expg} on $[t^n, t^{n+1}]$ to get $$ g(t^{n+1}) = e^{-\Delta t/\varepsilon^2} g(t^n) - \varepsilon(1-e^{-\Delta t/\varepsilon^2}) \mathcal{F}\left(\rho(t^n),g(t^n)\right) + {\cal O}(\Delta t^2). $$ To derive a continuous (in time) equation, we make appear a discrete time derivative on the left-hand side \begin{equation} \label{micro2} \frac{g(t^{n+1}) -g(t^n)}{\Delta t}= \frac{e^{-\Delta t/\varepsilon^2} -1}{\Delta t}g(t^n) - \varepsilon \frac{1-e^{-\Delta t/\varepsilon^2}}{\Delta t} \mathcal{F}\left(\rho(t^n),g(t^n)\right) + {\cal O}(\Delta t^2), \end{equation} which can be rewritten, up to terms of order ${\cal O}(\Delta t^2)$, as \begin{equation} \partial_t g(t^n)= \frac{e^{-\Delta t/\varepsilon^2} -1}{\Delta t}g(t^n) - \varepsilon \frac{1-e^{-\Delta t/\varepsilon^2}}{\Delta t}\mathcal{F}\left(\rho(t^n),g(t^n)\right),~\forall n. \end{equation} We finally obtain the first-order reformulation of \eqref{eq:etrbgk} \begin{align} \partial_t \rho &+ \frac{1}{\varepsilon}\partial_x \langle v g \rangle = 0,\label{eq:mm_macro}\\ \partial_t g&= \frac{e^{-\Delta t/\varepsilon^2} -1}{\Delta t}g - \varepsilon \frac{1-e^{-\Delta t/\varepsilon^2}}{\Delta t} \mathcal{F}\left(\rho,g\right), \label{eq:mm_micro} \end{align} with $\mathcal{F}\left(\rho,g\right)$ given by \eqref{def:F}. We remark that the micro equation does not contain any stiff term and then has a suitable form for a numerical discretization using particles. Moreover, this first-order reformulation satisfies the following properties. \begin{itemize} \item Consistency: For all fixed $\varepsilon >0$, equation (\ref{eq:mm_micro}) is consistent with the initial micro equation (\ref{micro}) as $\Delta t$ goes to zero. \item Asymptotic behaviour: For all fixed $\Delta t >0$, as $\varepsilon$ goes to zero, we get from (\ref{eq:mm_micro}) $g=-\varepsilon v \partial_x \rho+ O(\varepsilon^2)$, which injected in the macro equation (\ref{eq:mm_macro}) provides the limit model \eqref{eq:diff}. \end{itemize} \subsection{Lagrangian discretization}\label{ssec:pic} This subsection is devoted to the derivation of an AP-particle based numerical scheme for (\ref{eq:mm_macro})-(\ref{eq:mm_micro}). \paragraph{Explicit in time discretization\\} We propose now a Lagrangian discretization of (\ref{eq:mm_micro}). More precisely, we adopt a particle method, see \cite{birdsall}, and consider a set of $N_p\in\mathbb{N}$ macro particles. The position of particle $k$, $1\leq k\leq N_p$, is denoted by $x_k(t)\in\Omega= \left[0,L_x\right]$, with $L_x>0$, its velocity by $v_k(t)\in V=\left[-1,1\right]$ and its weight by $\omega_k(t)\in\mathbb{R}$. Let $L_v=\|V\|=2$. The function $g$ is then assumed to be of the form \begin{equation} g\left(t,x,v\right)=\sum_{k=1}^{N_p}\omega_k(t)\delta(x-x_k(t))\delta(v-v_k(t)), \label{eq:diracmasses} \end{equation} where $\delta$ denotes the Dirac mass function. Weights $\omega_k(t)$ are related to the distribution function $g$ through \begin{equation}\label{eq:poids} \omega_k(t)=g(t,x_k(t),v_k(t))\frac{L_xL_v}{N_p}. \end{equation} Initially, particles are randomly distributed in the phase-space domain $\left[0,L_x\right]\times V$ and their weights are computed following (\ref{eq:poids}). The density $\rho$ is computed on a uniform spatial grid defined by $\mbox{x}_i=i\Delta x$, $i=0,\dots,N_x$, $N_x\in\mathbb{N}^\star$ and $\Delta x=L_x/N_x$. We denote by $\rho_i^n$ the approximation at time $t^n=n\Delta t$ and position $\mbox{x}_i$ of $\rho(t^n,\mbox{x}_i)$, with $\Delta t>0$ the time step. Moreover, $g^n(x,v)\approx g(t^n, x, v)$, $x_k^n\approx x_k(t^n)$, $v_k^n\approx v_k(t^n)$ and $w_k^n\approx w_k(t^n)$. Let us remark that $\dot{v}_k=0$, so that the velocities $v_k(t)$ are constant in time and we will note $v_k^n=v_k^0=:v_k$ for all $n$. Our goal is then to extend the particle discretization of \cite{ccl} to diffusion scaling. To that purpose, we exploit the reformulation (\ref{eq:mm_micro}). As already said in \cite{ccl}, we have to use a splitting procedure between the transport part and the source part. Then, the (first-order) splitting writes \begin{itemize} \item start with an initial repartition of the $N_p$ particles $(x_k^0, v_k^0)$, with $\omega_k^0=g(t=0, x_k^0, v_k^0)L_x L_v/N_p$, \item solve the transport part \begin{equation} \partial_t g = \varepsilon \frac{1-e^{-\Delta t/\varepsilon^2}}{\Delta t} v\partial_x g, \end{equation} with the (non stiff) characteristics \begin{equation} \label{carx} \dot{x}_k = \varepsilon \frac{1-e^{-\Delta t/\varepsilon^2}}{\Delta t} v_k, \end{equation} \item solve the source part \begin{equation} \partial_t g = \frac{e^{-\Delta t/\varepsilon^2} -1}{\Delta t}g- \varepsilon\frac{1-e^{-\Delta t/\varepsilon^2}}{\Delta t} \left[ v\partial_x \rho - \partial_x \langle vg\rangle \right], \end{equation} using the equation satisfied by the weights \begin{equation} \label{weight} \dot{\omega}_k = \frac{e^{-\Delta t/\varepsilon^2} -1}{\Delta t}\omega_k - \varepsilon \frac{1-e^{-\Delta t/\varepsilon^2}}{\Delta t} \left[ v_k (\partial_x \rho(x_k) - \partial_x \langle vg \rangle (x_k) \right]\frac{L_x L_v}{N_p}. \end{equation} \end{itemize} Now, we detail the time discretization of the two steps. First, (\ref{carx}) is approximated by a simple forward Euler scheme \begin{equation} \label{carxd} x_k^{n+1} = x_k^n + \varepsilon (1-e^{-\Delta t/\varepsilon^2})v_k. \end{equation} Second, we compute the last term in \eqref{weight}. The term $\langle v g \rangle$ is approximated on the spatial grid $\mbox{x}_i$ using \begin{equation} \label{momg} \langle v g \rangle(\mbox{x}_i) \approx \sum_{k=1}^{N_{p}} \omega^n_k B_\ell(\mbox{x}_i-x_k^{n+1}) v_k, \end{equation} where $B_\ell \geq 0$ is a B-spline function of order $\ell$: \begin{equation} \label{bspline} B_\ell(x)=(B_0 * B_{\ell -1})(x), \;\; \mbox{ with } \;\; B_0(x)= \left\{ \begin{array}{llcc} \frac{1}{\Delta x} & \mbox{ if } \|x\|<\Delta x/2, \\ 0 & \mbox{ else}. \end{array} \right. \end{equation} We then approximate the equation on the weights (\ref{weight}) using a first-order explicit integrator \begin{equation} \label{weightd} \omega^{n+1}_k = e^{-\Delta t/\varepsilon^2} \omega_k^n - \varepsilon (1-e^{-\Delta t/\varepsilon^2}) \left[ \alpha_k^n + \beta_k^n\right], \end{equation} with \begin{equation} \label{poids_rhs} \alpha_k^n= v_k \partial_x \rho^n(x_k^{n+1}) \frac{L_x L_v}{N_p}~~~\textrm{and}~~~\beta_k^n=- \partial_x \langle vg \rangle (x_k^{n+1}) \frac{L_x L_v}{N_p}. \end{equation} To compute $\alpha_k^n$ (resp. $\beta_k^n$), since $\rho^n$ (resp. $\langle v g \rangle$) is known on the spatial grid, we approximate $\partial_x \rho^n$ (resp. $\partial_x\langle v g \rangle$) by centered finite differences and evaluate at $x_k^{n+1}$ using an interpolation with B-spline functions, for example $$ \partial_x \rho^n(x_k^{n+1})\approx\sum_{i=1}^{N_x}\frac{\rho_{i+1}^n-\rho_{i-1}^n}{2\Delta x}B_\ell(\mbox{x}_i-x_k^{n+1}). $$ Finally, the macro equation (\ref{eq:mm_macro}) is advanced through \begin{equation} \rho_i^{n+1} = \rho_i^n -\frac{\Delta t}{\varepsilon} \frac{\langle vg^{n+1}\rangle_{i+1}-\langle vg^{n+1}\rangle_{i-1}}{2\Delta x}, \label{macrod0} \end{equation} where $\langle v g^{n+1}\rangle_i$ is computed using \eqref{momg}. Then, we have the following proposition. \begin{prop} The scheme given by (\ref{carxd})-(\ref{weightd})-(\ref{macrod0}) enjoys the AP property, \textit{i.e.} it satisfies the following properties \begin{itemize} \item for fixed $\varepsilon>0$, the scheme is a first-order (in time) approximation of the original model (\ref{eq:etrbgk}), \item for fixed $\Delta t>0$, the scheme degenerates into an explicit first-order (in time) scheme of \eqref{eq:diff}. \end{itemize} \label{prop:lag_1st_exp} \end{prop} \begin{proof} The consistency follows directly from standard approximation. For the asymptotic behavior, when $\varepsilon$ goes to zero, we get $\omega_k^{n+1} = -\varepsilon \alpha_k^n+{\cal O}(\varepsilon^2)$ (since $\omega_k^n={\cal O}(\varepsilon)$ $\forall n\geq 1$). Computing the momentum of $g^{n+1}$ means that we use (\ref{momg}) with $g^{n+1}$, or in the limit regime \begin{eqnarray} \langle v g^{n+1} \rangle_i &\approx & -\varepsilon \sum_{k=1}^{N_{p}} \alpha^n_k B_\ell(\mbox{x}_i-x_k^{n+1}) v_k^{n+1} + {\cal O}(\varepsilon^2), \nonumber\\ &\approx & -\varepsilon\left[ \langle v^2 \rangle \partial_x \rho^n \right]\|_{x=\mbox{x}_i} + {\cal O}(\varepsilon^2) \nonumber\\ &\approx &-\varepsilon \frac{1}{3} \partial_x \rho_i^n + {\cal O}(\varepsilon^2). \end{eqnarray} Injecting in the macro equation (\ref{eq:mm_macro}) then leads to a consistent discretization of \eqref{eq:diff}. \end{proof} \begin{remark}[Preservation of the micro-macro structure.] As detailed in \cite{ccl}, we have to correct the particle' weights in order to preserve at the numerical level the micro-macro structure. Indeed, the micro-macro decomposition technique uses the zero-mean property $\langle g\rangle=0$. But nothing guarantees that this property is satisfied at the discrete level (on the weights $\omega_k$). That is why we have to correct the weights, by applying a discrete approximation of the operator $\left(I-\Pi\right)$ to each weight $\omega_k$, which is consistent with the continuous model. We do not give the details of this correction (called \textit{projection step} in following algorithms) and refer the reader to \cite{ccl}. \end{remark} \paragraph{Implicit time discretization\\} In this part, we want to make the previous scheme (\ref{carxd})-(\ref{weightd})-(\ref{macrod0}) degenerate into an implicit time discretization of \eqref{eq:diff} (as done in \cite{lemou-note, cl}). To do so, we decompose (\ref{weightd}) into two parts so that the macro flux becomes \begin{eqnarray} \langle v g^{n+1} \rangle_i &= & - \varepsilon (1-e^{-\Delta t/\varepsilon^2}) \sum_{k=1}^{N_{p}} \alpha^n_k B_\ell(\mbox{x}_i-x_k^{n+1}) v_k +h_i^n,\nonumber \end{eqnarray} with $B_\ell$ given by \eqref{bspline} and $$ h_i^n= e^{-\Delta t/\varepsilon^2}\sum_{k=1}^{N_{p}} \omega^n_k B_\ell(\mbox{x}_i-x_k^{n+1}) v_k -\varepsilon (1-e^{-\Delta t/\varepsilon^2}) \sum_{k=1}^{N_{p}} \beta^n_k B_\ell(\mbox{x}_i-x_k^{n+1}) v_k. $$ Since $\alpha_k^n$ is the weight of $v\partial_x \rho^n$ (see \eqref{poids_rhs}) and $\langle v^2\rangle=1/3$, we can write \begin{eqnarray} \langle v g^{n+1} \rangle_i &\approx & - \varepsilon (1-e^{-\Delta t/\varepsilon^2}) \frac{1}{3}\partial_x \rho^n_i +h_i^n, \end{eqnarray} so that the macro scheme can be \begin{eqnarray} \rho_i^{n+1} &=& \rho_i^n + \Delta t (1-e^{-\Delta t/\varepsilon^2}) \frac{1}{3}\frac{ \rho^n_{i+1}-2\rho^n_i+\rho^n_{i-1} }{\Delta x^2}-\frac{\Delta t}{\varepsilon}\frac{h_{i+1}^n-h_{i-1}^n}{2\Delta x}. \label{eq:lag_manoeuvre} \end{eqnarray} We can go further by considering now the diffusion term implicit in time to get \begin{eqnarray} \rho_i^{n+1} &=& \rho_i^n + \Delta t (1-e^{-\Delta t/\varepsilon^2}) \frac{1}{3}\frac{ \rho^{n+1}_{i+1}-2\rho^{n+1}_i+\rho^{n+1}_{i-1} }{\Delta x^2}-\frac{\Delta t}{\varepsilon}\frac{h_{i+1}^n-h_{i-1}^n}{2\Delta x}. \label{macrod0_cn} \end{eqnarray} We can write the following proposition. \begin{prop} The scheme given by (\ref{carxd})-(\ref{weightd})-(\ref{macrod0_cn}) enjoys the AP property, \textit{i.e.} it satisfies the following properties \begin{itemize} \item for fixed $\varepsilon>0$, the scheme is a first-order (in time) approximation of the original model (\ref{eq:etrbgk}), \item for fixed $\Delta t>0$, the scheme degenerates into an implicit first-order (in time) scheme of \eqref{eq:diff}. \end{itemize} \label{prop:lag_1st_imp} \end{prop} \begin{remark} Instead of an implicit scheme, it is possible to use a Crank-Nicolson method. \end{remark} The scheme is finally summarized in the following algorithm. \begin{algo}~~ \begin{itemize} \item Initialize $(x_k^0, v_k^0)$, $\omega_k^0$ and $\rho_i^0$. At each time step: \item 1) Advance micro part: \begin{itemize} \item advance the characteristics with (\ref{carxd}), \item compute $\langle v g\rangle$ with (\ref{momg}), \item advance the equation on the weights with (\ref{weightd}). \end{itemize} \item 2) Projection step: compute $(I-\Pi)g^{n+1}$ using \cite{ccl}. \item 3) Advance macro part: \begin{itemize} \item compute $\langle v g^{n+1}\rangle$ with (\ref{momg}), \item compute $\rho^{n+1}$ with (\ref{macrod0_cn}). \end{itemize} \end{itemize} \label{algo:lag_1st} \end{algo} \section{Second-order in time reformulation and its discretization}\label{sec:2ndorder_0} \setcounter{equation}{0} This section is devoted to the derivation of a second-order scheme for the micro-macro system (\ref{eq:micromacro_initial}). As for the first-order scheme, we will first reformulate the microscopic equation (\ref{micro}) in order to suppress stiff terms (see Subsection \ref{sec:1storder} for the first-order case), and then discretize the obtained micro-macro model to get an AP efficient numerical scheme (see Subsection \ref{ssec:pic} for the first-order case). \subsection{Second-order reformulation}\label{sec:2ndorder} Let start from the following (equivalent) reformulation of the micro part of \eqref{eq:micromacro_initial} $$ \partial_t\left(e^{t/\varepsilon^2}g\right)=-\frac{e^{t/\varepsilon^2}}{\varepsilon}\mathcal{F}\left(\rho(t),g(t)\right), $$ where $\mathcal{F}\left(\rho,g\right)$ is defined by \eqref{def:F}. We now integrate with respect to $t\in [t^n, t^{n+1}]$ and use a second-order mid-point quadrature $$ g(t^{n+1})=e^{-\Delta t/\varepsilon^2}g(t^{n})-\frac{\Delta te^{-\Delta t/2\varepsilon^2}}{\varepsilon}\mathcal{F}\left(\rho(t^{n+1/2}),g(t^{n+1/2})\right)+\mathcal{O}\left(\Delta t^3\right). $$ To derive a continuous (in time) equation, we make appear a discrete time derivative on the left-hand side $$ \frac{g(t^{n+1})-g(t^{n})}{\Delta t} =\frac{e^{-\Delta t/\varepsilon^2}-1}{\Delta t}g(t^{n})-\frac{e^{-\Delta t/2\varepsilon^2}}{\varepsilon}\mathcal{F}\left(\rho(t^{n+1/2}),g(t^{n+1/2})\right)+\mathcal{O}\left(\Delta t^2\right). $$ We now look for a continuous (in time) equation for which the previous relation is a second numerical scheme. To do so, we perform Taylor expansions of the different terms at $t^{n+1/2}$ \begin{equation}\label{eq:approx_o2_1} \partial_t g(t^{n+1/2}) \!=\!\frac{e^{-\Delta t/\varepsilon^2}\!\!-\!1}{\Delta t} \!\! \left(g(t^{n+1/2})-\frac{\Delta t}{2}\partial_tg(t^{n+1/2})\right) -\frac{e^{-\Delta t/2\varepsilon^2}}{\varepsilon}\mathcal{F}\!\left(\rho(t^{n+1/2}),g(t^{n+1/2})\right) +\mathcal{O}\left(\Delta t^2\right)\!. \end{equation} Finally, the microscopic equation of \eqref{eq:micromacro_initial} is reformulated up to the second-order by $$ \partial_t g=\frac{2}{\Delta t}\frac{e^{-\Delta t/\varepsilon^2}-1}{e^{-\Delta t/\varepsilon^2}+1}g-\frac{2}{\varepsilon}\frac{e^{-\Delta t/2\varepsilon^2}}{e^{-\Delta t/\varepsilon^2}+1}\mathcal{F}\left(\rho,g\right), $$ and we can now consider the second-order reformulated micro-macro system \begin{align} \partial_t \rho &+ \frac{1}{\varepsilon}\partial_x \langle v g \rangle = 0,\label{eq:mm_macro_2nd}\\ \partial_tg&=\frac{2}{\Delta t}\frac{e^{-\Delta t/\varepsilon^2}-1}{e^{-\Delta t/\varepsilon^2}+1}g-\frac{2}{\varepsilon}\frac{e^{-\Delta t/2\varepsilon^2}}{e^{-\Delta t/\varepsilon^2}+1} \left[ v\partial_x \rho + v\partial_x g - \partial_x \langle vg\rangle \right]. \label{eq:mm_micro_2nd} \end{align} \subsection{Time discretization}\label{subsec:ordre2_discr} We are now interested in the construction of an AP scheme for system (\ref{eq:mm_macro_2nd})-(\ref{eq:mm_micro_2nd}), based on a second-order Runge-Kutta (RK2) method for the time discretization and a Lagrangian method for the phase space discretization of the micro part. The RK2 is based on a prediction step on $\Delta t/2$ (first-order) and a correction step on $\Delta t$. Then, a second-order (in time) scheme for (\ref{eq:mm_macro_2nd})-(\ref{eq:mm_micro_2nd}) would read \begin{align} \mbox{Prediction step on $\Delta t/2$}&\nonumber\\ \begin{split} g^{n+1/2}=&g^n+\frac{e^{-\Delta t/\varepsilon^2}-1}{e^{-\Delta t/\varepsilon^2}+1}\; g^n-\frac{\Delta t}{\varepsilon}\frac{e^{-\Delta t/2\varepsilon^2}}{e^{-\Delta t/\varepsilon^2}+1} \mathcal{F}\left(\rho^n,g^n\right), \end{split}\label{gnpdemi_0}\\ \rho^{n+1/2} =&\rho^n- \frac{\Delta t}{2\varepsilon}\partial_x \langle v g^{n+1/2} \rangle,\label{rhonpdemi_0}\\ \mbox{Correction step on $\Delta t$}\hphantom{/2}&\nonumber\\ \begin{split} g^{n+1}=&g^n+2\frac{e^{-\Delta t/\varepsilon^2}-1}{e^{-\Delta t/\varepsilon^2}+1}\; \widetilde{g}-\frac{2\Delta t}{\varepsilon}\frac{e^{-\Delta t/2\varepsilon^2}}{e^{-\Delta t/\varepsilon^2}+1} \mathcal{F}\left(\rho^{n+1/2},g^{n+1/2}\right), \end{split}\label{gnp1_0}\\ \rho^{n+1} =&\rho^n- \frac{\Delta t}{\varepsilon}\partial_x \langle v g^{n+1/2} \rangle.\label{rhonp1_0} \end{align} We still have to fix $\widetilde{g}$ in \eqref{gnpdemi_0} in order to get a second-order scheme and to ensure the convergence of $g^{n+1}$ to zero as $\varepsilon$ goes to zero. It turns out the choice $\widetilde{g}=\frac{g^n+g^{n+1}}{2}$ ensures the two conditions. Indeed, we get for the correction step of the micro part $$ g^{n+1}=e^{-\Delta t/\varepsilon^2}g^n-\frac{\Delta t}{\varepsilon}e^{-\Delta t/2\varepsilon^2} \mathcal{F}\left(\rho^{n+1/2},g^{n+1/2}\right). $$ Up to now, the micro part converges exponentially fast to zero (when $\varepsilon$ goes to zero), so that the asymptotic behavior of the scheme is $\rho^{n+1} = \rho^n$. Hence, the last but not the least step consists in modifying the macro part to capture the correct asymptotic limit. As done in Section \ref{sec:1storder_0}, we will modify the discretization in order to make appear the diffusion term directly in the macro part \eqref{rhonp1_0}. However, the modification has to be of order $\Delta t^3$ for a fixed $\varepsilon>0$ to not spoil the second-order accuracy of the scheme. The correction we propose consists in computing $\rho_i^{n+1}$ using a Crank-Nicolson scheme for the diffusion weighted which is of order ${\cal O}(\Delta t^3)$ for fixed $\varepsilon>0$ and which degenerates to $\Delta t$ when $\varepsilon$ goes to zero \begin{equation} \rho_i^{n+1}=\rho_i^n- \frac{\Delta t}{\varepsilon}\partial_x \langle v g^{n+1/2} \rangle_i+\Delta t(1-e^{-\Delta t/\varepsilon^2})^2\frac{1}{3}\partial_{xx}\left(\frac{\rho_i^{n+1}+\rho_i^n}{2}\right). \label{eq:rho_ordre2_man} \end{equation} \subsection{Lagrangian discretization}\label{ssec:pic_o2} We consider the same notations as in Subsection \ref{ssec:pic} and detail here the Lagrangian discretization of the micro-macro system (\ref{eq:mm_macro_2nd})-(\ref{eq:mm_micro_2nd}). In the prediction step \eqref{gnpdemi_0}, we compute $x_k^{n+1/2}$ with a forward Euler integrator \begin{equation} x_k^{n+1/2}=x_k^n+\frac{\Delta t}{\varepsilon}\frac{e^{-\Delta t/2\varepsilon^2}}{e^{-\Delta t/\varepsilon^2}+1}v_k, \label{eq:cara_pred} \end{equation} and advance the weights with \begin{equation} \begin{split} w_k^{n+1/2}=\frac{2e^{-\Delta t/\varepsilon^2}}{e^{-\Delta t/\varepsilon^2}+1}w_k^n-\frac{\Delta t}{\varepsilon}\frac{e^{-\Delta t/2\varepsilon^2}}{e^{-\Delta t/\varepsilon^2}+1}\left[ v_k \partial_x \rho^n(x_k^n)- \partial_x \langle v_k g^n(x_k^n)\rangle \right]\frac{L_xL_v}{N_p}. \end{split}\label{eq:poids_pred} \end{equation} We end this prediction step by computing the flux $\langle v g^{n+1/2}\rangle$ with \eqref{momg} to get the density \begin{equation} \rho_i^{n+1/2}=\rho_i^n- \frac{\Delta t}{2\varepsilon}\partial_x \langle v g^{n+1/2} \rangle_i. \label{eq:rho_pred} \end{equation} Now in the correction step, we compute the position at $t^{n+1}$ with \begin{equation} x_k^{n+1}=x_k^n+\frac{\Delta t}{\varepsilon}e^{-\Delta t/2\varepsilon^2}v_k, \label{eq:cara_corr} \end{equation} then the weights are given by \begin{equation} w_k^{n+1}=e^{-\Delta t/\varepsilon^2}w_k^n-\frac{\Delta t}{\varepsilon}e^{-\Delta t/2\varepsilon^2}\left[ v_k \partial_x \rho^{n+1/2}(x_k^{n+1/2}) - \partial_x \langle v_kg^{n+1/2}(x_k^{n+1/2})\rangle \right]\frac{L_xL_v}{N_p}. \label{eq:poids_corr} \end{equation} Now, using \eqref{eq:rho_ordre2_man} in the last step, we have \begin{equation} \rho_i^{n+1}=\rho_i^n- \frac{\Delta t}{\varepsilon}\partial_x \langle v g^{n+1/2} \rangle_i+\Delta t(1-e^{-\Delta t/\varepsilon^2})^2\frac{1}{3}\partial_{xx}\left(\frac{\rho_i^{n+1}+\rho_i^n}{2}\right), \label{eq:rho_corr} \end{equation} where $\langle v g^{n+1/2} \rangle_i$ is computed using \eqref{momg}. We finally have the following result. \begin{prop} The scheme given by \eqref{eq:cara_pred}-\eqref{eq:poids_pred}-\eqref{eq:rho_pred}-\eqref{eq:cara_corr}-\eqref{eq:poids_corr}-\eqref{eq:rho_corr} enjoys the AP property, \textit{i.e.} it satisfies the following properties \begin{itemize} \item for fixed $\varepsilon>0$, the scheme is a second-order (in time) approximation of the original model (\ref{eq:etrbgk}), \item for fixed $\Delta t>0$, the scheme degenerates into an implicit second-order (in time) scheme of \eqref{eq:diff}. \end{itemize} \label{prop:lag_2st} \end{prop} \begin{proof} When $\varepsilon\to 0$, we get from (\ref{eq:poids_pred}) $w_k^{n+1/2}\to 0$ exponentially fast and then $\langle v g^{n+1/2} \rangle_i\to 0$. By injecting it in the macro equation (\ref{eq:rho_corr}), we have at the limit $\rho_i^{n+1}=\rho_i^n+\frac{\Delta t}{3}\partial_{xx}\left(\frac{\rho_i^{n+1}+\rho_i^n}{2}\right)$, which is a Crank-Nicolson discretization of the diffusion equation \eqref{eq:diff}. \end{proof} \begin{remark} Let us emphasize that the moments $\langle \cdot \rangle$ have to be computed with B-spline functions of order $\ell\geq 1$ in order to obtain a second-order in time scheme. Taking $\ell=0$ would lead to space discontinuities preventing the time scheme to be of second-order. \end{remark} \begin{remark} The projection step being by construction a first-order step, we do it in the prediction step. \end{remark} The scheme is finally summarized in the following algorithm. \begin{algo}~~ \begin{itemize} \item Initialization of $(x_k^0, v_k^0)$, $\omega_k^0$ and $\rho_i^0$. At each time step: \textbf{Prediction step: from $t^n$ to $t^{n+1/2}$.} \item 1) Advance micro part: \begin{itemize} \item advance the characteristics with (\ref{eq:cara_pred}), \item compute $\langle v g\rangle$ with (\ref{momg}) and B-spline functions of order $\ell\geq 1$, \item advance the equation on the weights with (\ref{eq:poids_pred}). \end{itemize} \item 2) Projection step: compute $(I-\Pi)g^{n+1/2}$ using \cite{ccl}. \item 3) Advance macro part: \begin{itemize} \item compute $\langle v g^{n+1/2}\rangle$ with (\ref{momg}) and B-spline functions of order $\ell\geq 1$, \item compute the density with (\ref{eq:rho_pred}). \end{itemize} \textbf{Correction step: from $t^n$ to $t^{n+1}$.} \item 4) Advance micro part: \begin{itemize} \item advance the characteristics with (\ref{eq:cara_corr}), \item compute $\langle v g\rangle$ with (\ref{momg}) and B-spline functions of order $\ell\geq 1$, \item advance the equation on the weights with (\ref{eq:poids_corr}). \end{itemize} \item 5) Advance macro part with (\ref{eq:rho_corr}). \end{itemize} \label{algo:2ndorder} \end{algo} \section{Extension to the Vlasov-Poisson-BGK case}\label{ssec:e} \setcounter{equation}{0} This section is devoted to the extension of our method to kinetic equation making appear an electric field in the velocity direction. We consider the Vlasov-Poisson-BGK system in the diffusion scaling \begin{eqnarray} \partial_t f +\frac{1}{\varepsilon} v\partial_x f + \frac{1}{\varepsilon}E\partial_v f= \frac{1}{\varepsilon^2}(\rho M - f), \label{eq:vlasovbgk_2} \\ \partial_xE=\rho-1,\label{eq:poisson}\\ \int_\Omega E\textnormal{d} x=0,~\forall t\geq 0,\label{eq:zeromean} \end{eqnarray} where $x\in \Omega=\left[0,L_x\right]\subset\mathbb{R}$, $\rho(t,x)=\int_\mathbb{R} f(t,x,v) \textnormal{d} v$ and $M\left(v\right)=\frac{1}{\sqrt{2\pi}}\exp\left(-\frac{v^2}{2}\right)$ is the absolute Maxwellian. Let $f_0\left(x,v\right)=f\left(t=0,x,v\right)$ the initial distribution function and let consider periodic boundary conditions in $x$: $f\left(t,0,v\right)=f\left(t,L_x,v\right)$, $\forall~v\in V$, $E\left(t,0\right)=E\left(t,L_x\right)$ $\forall t\geq 0$. We can extend our schemes to this problem by adapting the computations of Subsections \ref{sec:2ndorder} and \ref{subsec:ordre2_discr}. We do not give all the details of the computations but insist on difficulties coming from the electric field term and write the resulting schemes. The second-order reformulated micro-macro system corresponding to (\ref{eq:vlasovbgk}) is \begin{align} \partial_t \rho &+ \frac{1}{\varepsilon}\partial_x \langle v g \rangle = 0,\label{eq:mm_macro_2nd_bis}\\ \partial_tg&=\frac{2}{\Delta t}\frac{e^{-\Delta t/\varepsilon^2}-1}{e^{-\Delta t/\varepsilon^2}+1}g-\frac{2}{\varepsilon}\frac{e^{-\Delta t/2\varepsilon^2}}{e^{-\Delta t/\varepsilon^2}+1} \left[ vM\partial_x \rho + v\partial_x g - \partial_x \langle vg\rangle M -vME\rho +E\partial_v g \right]. \label{eq:mm_micro_2nd_bis} \end{align} The limit model is here the drift-diffusion equation coupled to Poisson equation (\ref{eq:ddlimit}). Equipped with this system (\ref{eq:mm_macro_2nd_bis})-(\ref{eq:mm_micro_2nd_bis}), we construct the following Lagrangian, second-order in time scheme for (\ref{eq:vlasovbgk_2})-(\ref{eq:poisson})-(\ref{eq:zeromean}). In the prediction step, characteristics are solved through \begin{equation} x_k^{n+1/2}=x_k^n+\frac{\Delta t}{\varepsilon}\frac{e^{-\Delta t/2\varepsilon^2}}{e^{-\Delta t/\varepsilon^2}+1}v_k^n,~~~v_k^{n+1/2}=v_k^n+\frac{\Delta t}{\varepsilon}\frac{e^{-\Delta t/2\varepsilon^2}}{e^{-\Delta t/\varepsilon^2}+1}E^n(x_k^n), \label{eq:characwithE} \end{equation} the weights evolve with \begin{equation} \begin{split} w_k^{n+1/2}=\frac{2e^{-\Delta t/\varepsilon^2}}{e^{-\Delta t/\varepsilon^2}+1}w_k^n-\frac{\Delta t}{\varepsilon}\frac{e^{-\Delta t/2\varepsilon^2}}{e^{-\Delta t/\varepsilon^2}+1}\left[ v_k^nM(v_k^n)\partial_x \rho^n(x_k^n)- \partial_x \langle v_k^ng^n(x_k^n)\rangle M(v_k^n)\right. \\ \left.-v_k^nM(v_k^n)E^n(x_k^n)\rho^n(x_k^n) \right]\frac{L_xL_v}{N_p} \end{split} \label{eq:poidswithE} \end{equation} and the macro equation is advanced with \begin{equation} \rho_i^{n+1/2}=\rho_i^n- \frac{\Delta t}{2\varepsilon}\partial_x \langle v g^{n+1/2} \rangle_i+\frac{\Delta t}{2}(1-e^{-\Delta t/\varepsilon^2})\partial_{x}\left(\partial_x\left(\frac{\rho_i^{n+1/2}+\rho_i^n}{2}\right)-E_i^{n}\rho_i^{n}\right). \label{eq:rhowithE} \end{equation} In the correction step, characteristics are solved through \begin{equation} x_k^{n+1}=x_k^n+\frac{2\Delta t}{\varepsilon}\frac{e^{-\Delta t/2\varepsilon^2}}{e^{-\Delta t/\varepsilon^2}+1}v_k^{n+1/2},~~~v_k^{n+1}=v_k^n+\frac{2\Delta t}{\varepsilon}\frac{e^{-\Delta t/2\varepsilon^2}}{e^{-\Delta t/\varepsilon^2}+1}E^{n+1/2}(x_k^{n+1/2}), \label{eq:characwithE2} \end{equation} the weights evolve with \begin{equation} \begin{split} w_k^{n+1}=e^{-\Delta t/\varepsilon^2}w_k^n-\frac{\Delta t}{\varepsilon}e^{-\Delta t/2\varepsilon^2}\left[ v_k^{n+1/2}M(v_k^{n+1/2})\partial_x \rho^{n+1/2}(x_k^{n+1/2})~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\right. \\ \left.- \partial_x \langle v_k^{n+1/2}g^{n+1/2}(x_k^{n+1/2})\rangle M(v_k^{n+1/2})~~~~~~~~~~~~~~~~~~~~~~~~~\right. \\ \left.-v_k^{n+1/2}M(v_k^{n+1/2})E^{n+1/2}(x_k^{n+1/2})\rho^{n+1/2}(x_k^{n+1/2}) \right]\frac{L_xL_v}{N_p}. \end{split} \label{eq:poidswithE2} \end{equation} and the macro equation is advanced with \begin{equation} \rho_i^{n+1}=\rho_i^n- \frac{\Delta t}{\varepsilon}\partial_x \langle v g^{n+1/2} \rangle_i+\Delta t(1-e^{-\Delta t/\varepsilon^2})^2\partial_{x}\left(\partial_x\left(\frac{\rho_i^{n+1}+\rho_i^n}{2}\right)-E_i^{n+1/2}\rho_i^{n+1/2}\right). \label{eq:vp_lag_2nd_man} \end{equation} The limit has been directly written in the macroscopic equation and the diffusion term is managed by a Crank-Nicolson method, in the prediction as well as in the correction step. We have the following proposition. \begin{prop} The scheme given by \eqref{eq:characwithE}-\eqref{eq:poidswithE}-\eqref{eq:rhowithE}-\eqref{eq:characwithE2}-\eqref{eq:poidswithE2}-\eqref{eq:vp_lag_2nd_man} enjoys the AP property, \textit{i.e.} it satisfies the following properties \begin{itemize} \item for fixed $\varepsilon>0$, the scheme is a second-order (in time) approximation of the original model (\ref{eq:vlasovbgk}), \item for fixed $\Delta t>0$, the scheme degenerates into an implicit second-order (in time) scheme of \eqref{eq:drift-diff}. \end{itemize} \end{prop} The scheme is finally summarized in the following algorithm. \begin{algo}~~ \begin{itemize} \item Initialize $(x_k^0, v_k^0)$, $\omega_k^0$, and $\rho_i^{0}$. \item Compute $E_i^{0}$ thanks to FFT or finite differences. At each time step: \textbf{Prediction step: from $t^n$ to $t^{n+1/2}$.} \item 1) Advance micro part: \begin{itemize} \item advance the characteristics with (\ref{eq:characwithE}), \item compute $\langle v g\rangle$ with (\ref{momg}) and B-spline functions of order $\ell\geq 1$, \item advance the equation on the weights with (\ref{eq:poidswithE}). \end{itemize} \item 2) Projection step: compute $(I-\Pi)g^{n+1/2}$ using \cite{ccl}. \item 3) Advance macro part: \begin{itemize} \item compute $\langle v g^{n+1/2}\rangle$ with (\ref{momg}) and B-spline functions of order $\ell\geq 1$, \item compute $\rho_i^{n+1/2}$ with (\ref{eq:rhowithE}), \item compute $E_i^{n+1/2}$ thanks to FFT or finite differences. \end{itemize} \textbf{Correction step: from $t^n$ to $t^{n+1}$.} \item 4) Advance micro part: \begin{itemize} \item advance the characteristics with (\ref{eq:characwithE2}), \item compute $\langle v g\rangle$ with (\ref{momg}) and B-spline functions of order $\ell\geq 1$, \item advance the equation on the weights with (\ref{eq:poidswithE2}). \end{itemize} \item 5) Advance macro part: \begin{itemize} \item compute $\rho_i^{n+1}$ with (\ref{eq:vp_lag_2nd_man}), \item compute $E_i^{n+1}$ thanks to FFT or finite differences. \end{itemize} \end{itemize} \label{algo:vp_lag_2nd} \end{algo} \begin{remark} We propose to use an upwind discretization of the derivative $\partial_x\left(E_i^{n+1/2}\rho_i^{n+1/2}\right)$: $$ \partial_x\left(E_i^{n+1/2}\rho_i^{n+1/2}\right)\approx\frac{E_i^{n+1/2,+}\rho_i^{n+1/2}+E_i^{n+1/2,-}\rho_{i+1}^{n+1/2}-E_{i-1}^{n+1/2,+}\rho_{i-1}^{n+1/2}-E_{i-1}^{n+1/2,-}\rho_i^{n+1/2}}{\Delta x}, $$ where the standard notations $u^+=\mathrm{max}(u,0)$ and $u^-=\mathrm{min}(u,0)$ are used. The same discretization is done for $\partial_x\left(E_i^{n}\rho_i^{n}\right)$ in the prediction step. \end{remark} \section{Numerical results}\label{sec:numres} \setcounter{equation}{0} This section is devoted to some numerical experiments comparing the here designed micro-macro model with particles (denoted by MiMa-Part-1 for the first-order scheme and by MiMa-Part-2 for the second-order scheme) to the micro-macro Eulerian model (denoted by MiMa-Grid), the moment guided method (denoted by Moment G.) and (i) the Full PIC method in kinetic regimes ($\varepsilon$ of order 1) or (ii) the limit scheme in diffusion regime ($\varepsilon\ll 1$). MiMa-Part-1 corresponds to Proposition \ref{prop:lag_1st_imp} and MiMa-Part-2 corresponds to Proposition \ref{prop:lag_2st}. The micro-macro Eulerian scheme is presented in Appendix \ref{subsec:eulerian}. The moment guided method was first presented in \cite{ddp} and is adapted to our context in Appendix \ref{app:mgm}. The Full PIC method \cite{birdsall} consists in applying the particle representation (\ref{eq:diracmasses}) to the whole function $f$ (and not only to the perturbation $g$) and to solve the characteristics and the equations on weights coming from equation \eqref{eq:etrbgk} or \eqref{eq:vlasovbgk}. In the sequel, we consider three families of test cases: radiative transport equation (RTE) test cases with periodic boundary conditions (see equation (\ref{eq:etrbgk})) in Subsection \ref{ssec:RTEperio}, Vlasov-Poisson-BGK test cases (see equations (\ref{eq:vlasovbgk})-(\ref{eq:vlasovbgkpoisson})) of Landau damping type in Subsection \ref{ssec:landau} and two-stream instability (TSI) test cases in Subsection \ref{ssec:tsi}. \subsection{RTE with periodic boundary conditions}\label{ssec:RTEperio} We consider the RTE test case given by the initial condition \begin{equation} f\left(t=0,x,v\right)=1+\cos\left(2\pi\left(x+\frac{1}{2}\right)\right),~~~x\in\left[0,1\right], v\in\left[-1,1\right], \end{equation} with $M\left(v\right)=1,~\forall v\in\left[-1,1\right]$, and periodic boundary conditions in $x$. We propose here to verify numerically the convergence of MiMa-Part-2 (presented in Subsection \ref{ssec:pic_o2}), Lagrangian in space, of second-order in time and with an implicit treatment of the diffusion term (see Algorithm \ref{algo:2ndorder}). In Figure \ref{fig:etr_ordre2_1}, we plot the error in $L^\infty$ norm of the density $\rho$ at time $T=0.1$ as a function of $\Delta t$ (from $10^{-4}$ to $0.1$) for the following parameters: $N_x=16$, $N_p=100$. For $\varepsilon=1$, $0.5$ and $0.1$, the reference solution is computed with MiMa-Part-2 using the same parameters but with $\Delta t=10^{-7}$. Whereas for $\varepsilon=10^{-6}$, the reference is a numerical solution of the diffusion equation (computed on a space grid, with $N_x=16$ and $\Delta t=10^{-7}$). In Figure \ref{fig:etr_ordre2_cloche}, the error in $L^\infty$ norm is now represented as a function of $\varepsilon$ for different values of $\Delta t$: $10^{-1}$, $10^{-2}$, $10^{-3}$ and $10^{-4}$. \begin{minipage}[t]{0.48\textwidth} \begin{figure} \caption{Error in $L^\infty$ norm of $\rho$ at time $T=0.1$ as a function of $\Delta t$ for $N_x=16$, $N_p=100$, $\varepsilon=1$, $0.5$, $0.1$ and $10^{-6}$.} \label{fig:etr_ordre2_1} \end{figure} \end{minipage} ~~ \begin{minipage}[t]{0.48\textwidth} \begin{figure} \caption{Error in $L^\infty$ norm of $\rho$ at time $T=0.1$ as a function of $\varepsilon$ for $N_x=16$, $N_p=100$, $\Delta t=10^{-1}$, $10^{-2}$, $10^{-3}$ and $10^{-4}$.} \label{fig:etr_ordre2_cloche} \end{figure} \end{minipage} ~~ These plots confirm the fact that MiMa-Part-2 is second-order accurate in time for any fixed $\varepsilon>0$, and also when $\varepsilon\to 0$. However, for intermediate regimes (for instance $\varepsilon=0.1$), order reduction is observed. This is a classical observation for AP schemes. Note that similar behaviour is obtained with $L^2$ norm. In Figure \ref{fig:etr_ordre2_ap}, we verify the AP property of the MiMa-Part-2 scheme and plot the density $\rho(T=0.1,x)$ as a function of $x$ for different values of $\varepsilon$: $1$, $0.25$, $10^{-2}$ and $10^{-6}$. We take fixed parameters: $N_x=64$, $\Delta t=10^{-3}$ and $N_p=10^4$. We compare the solutions obtained by MiMa-Part-2 to a numerical solution of the diffusion equation (computed on a space grid, with $N_x=512$ and $\Delta t=\Delta x^2$) and see that the $\varepsilon$-dependent solutions come closer to the diffusion one when $\varepsilon$ decreases. Moreover, we illustrate in Figure \ref{fig:etr_ordre2_npart} the fact that the cost of our method is very small at the limit. For that, we plot the density $\rho(T=0.1,x)$ as a function of $x$ for $\varepsilon=10^{-6}$ ($N_x=64$ and $\Delta t=10^{-2}$) and see that $N_p=100$ is sufficient to represent in a good way (without noise) the density. The numerical cost is then very close to the one of the asymptotic model. \begin{minipage}[t]{0.48\textwidth} \begin{figure} \caption{AP property. Density $\rho(T=0.1,x)$ for $\varepsilon=1$, $0.25$, $10^{-2}$ and $10^{-6}$. $N_x=64$, $\Delta t=10^{-3}$ and $N_p=10^4$. Comparison with the diffusion solution.} \label{fig:etr_ordre2_ap} \end{figure} \end{minipage} ~~ \begin{minipage}[t]{0.48\textwidth} \begin{figure} \caption{Cost at the limit. Density $\rho(T=0.1,x)$ for $\varepsilon=10^{-6}$, $N_x=64$, $\Delta t=10^{-2}$ and $N_p=10^4$ or $N_p=100$. Comparison with the diffusion solution.} \label{fig:etr_ordre2_npart} \end{figure} \end{minipage} ~~ \subsection{Landau damping}\label{ssec:landau} In this subsection, we present Landau damping test cases in both regimes (kinetic when $\varepsilon=\mathcal{O}(1)$ and diffusive when $\varepsilon\to 0$). The initial distribution function is given by \begin{equation} f\left(t=0,x,v\right)=\frac{1}{\sqrt{2\pi}}\textrm{exp}\left(-\frac{v^2}{2}\right)\left(1+\alpha\cos\left(kx\right)\right),~~~x\in\left[0,\frac{2\pi}{k}\right],~v\in \mathbb{R}, \end{equation} with the wave number $k=0.5$ and $\alpha=0.05$. For the micro-macro model (\ref{eq:vlasovbgk}), the initial condition is $\rho\left(t=0,x\right)=1+\alpha\cos\left(kx\right)$ and $g\left(t=0,x,v\right)=0$. For the limit drift-diffusion equation (\ref{eq:drift-diff}), we have $\rho\left(t=0,x\right)=1+\alpha\cos\left(kx\right)$. We first verify the order in time of the MiMa-Part-2 scheme detailed in Section \ref{ssec:e} and plot in Figure \ref{fig:vp_ordre2_1} the error in $L^\infty$ norm of the density $\rho$ at time $T=0.1$ as a function of $\Delta t$ (from $10^{-4}$ to $0.1$) for the following parameters: $N_x=16$, $N_p=100$. For $\varepsilon=1$, $0.5$ and $0.1$, the reference solution is computed with MiMa-Part-2 using the same parameters but with $\Delta t=10^{-7}$. Whereas for $\varepsilon=10^{-6}$ the reference is a numerical solution of the drift-diffusion equation (computed on a space grid, with $N_x=16$ and $\Delta t=10^{-7}$). Results are similar to the RTE case: the second-order in time is preserved for big and small values of $\varepsilon$ but not for intermediate regimes. We are now interested in more qualitative tests by considering the time history of the electric energy $\mathcal{E}\left(t\right)=\sqrt{\int_0^{L_x} E\left(t,x\right)^2\textnormal{d} x}$ in semi-logarithmic scale for different values of $\varepsilon$. We compare the results obtained by MiMa-Part-2 (detailed in Algorithm \ref{algo:vp_lag_2nd}) to other schemes: MiMa-part-1, Moment G. and Full PIC (for $\varepsilon$ of order 1) or the scheme for the drift-diffusion model (for small values of $\varepsilon$). We expect that the number of particles that is necessary to represent in a good way the perturbation $g$ in MiMa-Part methods decreases when $\varepsilon$ diminishes and consider $N_p=10^{5}$ if $\varepsilon\geq 0.5$, $N_p=10^{4}$ if $\varepsilon=0.1$ and $N_p=10^{2}$ if $\varepsilon=10^{-4}$. For comparison, we take the same $N_p$ for moment guided and Full PIC methods. Results for $\varepsilon=10$ are given in Figure \ref{figLandaukin10}. For the four particle methods, we take $\Delta t=0.1$, and $N_x=128$. With the same parameters, results for $\varepsilon=1$ are presented in Figure \ref{figLandaukin}. For $\varepsilon=0.5$, we consider $\Delta t=0.01$ and $N_x=256$ for the four particle methods. Results are given in Figure \ref{figLandauinter}. For these three values of $\varepsilon$, the reference is given by MiMa-Grid with $N_x=N_v=512$ and $\Delta t=\Delta x^2\approx 6\times 10^{-4}$. First, we note that the behaviour of the electric energy is well described during time by micro-macro schemes (MiMa-Part and MiMa-Grid). As observed in \cite{ccl}, the Full PIC method suffers from numerical noise. This is due to the probabilistic character of particle methods (for instance the random initialization of particles). To reduce this noise, we should consider more particles, which would increase the numerical cost. As expected, the moment guided method gives better results than the Full PIC one, but suffers however also from this noise. In MiMa-Part schemes, only the perturbation $g$ is represented by particles (not the whole distribution function $f$), that is why for the same $N_p$, the noise is lower, which enables to capture the reference solution for large times. In addition, MiMa-Part-2 is closer to the reference MiMa-Grid than the first-order version MiMa-Part-1. \begin{minipage}[t]{0.48\textwidth} \begin{figure} \caption{Error in $L^\infty$ norm of $\rho$ at time $T=0.1$ as a function of $\Delta t$ for $N_x=16$, $N_p=100$, $\varepsilon=1$, $0.5$, $0.1$ and $10^{-6}$.} \label{fig:vp_ordre2_1} \end{figure} \end{minipage} ~~ \begin{minipage}[t]{0.48\textwidth} \begin{figure} \caption{Time history of the electric energy, $\varepsilon=10$. $\Delta t=0.1$, $N_x=128$ and $N_p=10^5$ for the four particle methods.} \label{figLandaukin10} \end{figure} \end{minipage} \begin{minipage}[t]{0.48\textwidth} \begin{figure} \caption{Time history of the electric energy, $\varepsilon=1$. $\Delta t=0.1$, $N_x=128$ and $N_p=10^5$ for the four particle methods.} \label{figLandaukin} \end{figure} \end{minipage} ~~ \begin{minipage}[t]{0.48\textwidth} \begin{figure} \caption{Time history of the electric energy, $\varepsilon=0.5$. $\Delta t=0.01$, $N_x=256$ and $N_p=10^5$ for the four particle methods.} \label{figLandauinter} \end{figure} \end{minipage} ~~ For smaller values of $\varepsilon$, we compare the four AP schemes (MiMa-Part-1, MiMa-Part-2, MiMa-Grid and Moment G.) to the limit scheme. Results for $\varepsilon=0.1$ are given in Figure \ref{figLandaudif}. Parameters are the following: $\Delta t=10^{-3}$ and $N_x=128$ for particle methods, $\Delta t=0.1\Delta x^2\approx 3.5\times 10^{-3}$ and $N_x=N_v=64$ for MiMa-Grid. We observe that MiMa-Part-2 is the best method since it almost coincides with the reference MiMa-Grid method. Finally, results for $\varepsilon=10^{-4}$ are given in Figure \ref{figLandaudiflim}, where we have $\Delta t=10^{-2}$ and $N_x=128$ for particle methods, $\Delta t=0.1\Delta x^2\approx 3.5\times 10^{-3}$ and $N_x=N_v=64$ for MiMa-Grid. The asymptotic regime is well recovered by all these AP methods. \begin{minipage}[t]{0.48\textwidth} \begin{figure} \caption{Time history of the electric energy, $\varepsilon=0.1$. $\Delta t=10^{-3}$, $N_x=128$ and $N_p=10^4$ for the four particle methods.} \label{figLandaudif} \end{figure} \end{minipage} ~~ \begin{minipage}[t]{0.48\textwidth} \begin{figure} \caption{Time history of the electric energy, $\varepsilon=10^{-4}$. $\Delta t=0.01$, $N_x=128$ and $N_p=100$ for the four particle methods.} \label{figLandaudiflim} \end{figure} \end{minipage} ~~ As in \cite{ccl}, we remark that few particles are sufficient in the particle-micro-macro schemes to describe in a good way the solution when $\varepsilon$ is small. The cost is then reduced at the limit. \subsection{Two stream instability}\label{ssec:tsi} We propose now a study in which the perturbation $g$ is not zero initially and consider the Two-Stream Instability (TSI) test case in both regimes (kinetic and diffusive). The initial distribution function is given by \begin{equation} f\left(t=0,x,v\right)=\frac{1}{\sqrt{2\pi}}v^2\textrm{exp}\left(-\frac{v^2}{2}\right)\left(1+\alpha\cos\left(kx\right)\right),~~~x\in\left[0,\frac{2\pi}{k}\right],~v\in\mathbb{R}, \end{equation} with the wave number $k=0.5$ and $\alpha=0.05$. The initial condition for the micro-macro model (\ref{eq:vlasovbgk}) is $\rho\left(t=0,x\right)=1+\alpha\cos\left(kx\right)$ and $g\left(t=0,x,v\right)=\frac{1}{\sqrt{2\pi}}\left(v^2-1\right)\textrm{exp}\left(-\frac{v^2}{2}\right)\left(1+\alpha\cos\left(kx\right)\right)$. For the limit drift-diffusion equation (\ref{eq:drift-diff}), we have as in the Landau damping case $\rho\left(t=0,x\right)=1+\alpha\cos\left(kx\right)$. We first verify the order in time of the MiMa-Part-2 scheme detailed in Section \ref{ssec:e} and plot in Figure \ref{fig:tsi_ordre2_1} the error in $L^\infty$ norm of the density $\rho$ at time $T=0.1$ as a function of $\Delta t$ (from $10^{-4}$ to $0.1$) for the following parameters: $N_x=16$, $N_p=100$. For $\varepsilon=1$, $0.5$ and $0.1$, the reference solution is computed with MiMa-Part-2 using the same parameters but with $\Delta t=10^{-7}$. Whereas for $\varepsilon=10^{-6}$, the reference is a numerical solution of the drift-diffusion equation (computed on a space grid, with $N_x=16$ and $\Delta t=10^{-7}$). As for the RTE and the Landau damping cases, the second-order in time is preserved for big and small values of $\varepsilon$ but not for intermediate regimes. We are now interested in the time evolution of the electric energy $\mathcal{E}\left(t\right)=\sqrt{\int_0^{L_x} E\left(t,x\right)^2\textnormal{d} x}$ in all regimes. Results for $\varepsilon=10$ are given in Figure \ref{figTSIkin10}. For the four particle methods, we take $N_p=10^{6}$, $\Delta t=0.1$, and $N_x=128$. By taking $N_p=10^{5}$, $\Delta t=0.1$, and $N_x=128$ for particle methods, we obtain results for $\varepsilon=1$ presented in Figure \ref{figTSIkin}. For $\varepsilon=0.5$, we consider $N_p=10^{5}$, $\Delta t=0.01$ and $N_x=256$ for the four particle methods. Results are given in Figure \ref{figTSIinter}. For these three values of $\varepsilon$, the reference is given by MiMa-Grid with $N_x=N_v=512$ and $\Delta t=\Delta x^2\approx 6\times 10^{-4}$. The behaviour of the electric energy is well described during time by micro-macro schemes (MiMa-Part and MiMa-Grid). As previously, the Full PIC method, as well as moment guided method suffer from numerical noise. \begin{minipage}[t]{0.48\textwidth} \begin{figure} \caption{Error in $L^\infty$ norm of $\rho$ at time $T=0.1$ as a function of $\Delta t$ for $N_x=16$, $N_p=100$, $\varepsilon=1$, $0.5$, $0.1$ and $10^{-6}$.} \label{fig:tsi_ordre2_1} \end{figure} \end{minipage} ~~ \begin{minipage}[t]{0.48\textwidth} \begin{figure} \caption{Time history of the electric energy, $\varepsilon=10$. $\Delta t=0.1$, $N_x=128$ and $N_p=10^6$ for the four particle methods.} \label{figTSIkin10} \end{figure} \end{minipage} \begin{minipage}[t]{0.48\textwidth} \begin{figure} \caption{Time history of the electric energy, $\varepsilon=1$. $\Delta t=0.1$, $N_x=128$ and $N_p=10^5$ for the four particle methods.} \label{figTSIkin} \end{figure} \end{minipage} ~~ \begin{minipage}[t]{0.48\textwidth} \begin{figure} \caption{Time history of the electric energy, $\varepsilon=0.5$. $\Delta t=0.01$, $N_x=256$ and $N_p=10^5$ for the four particle methods.} \label{figTSIinter} \end{figure} \end{minipage} ~~ To illustrate the efficiency of the method, we plot $f(T=5,x,v)$ obtained by the reference MiMa-Grid, by MiMa-Part-2 and by Full PIC for $\varepsilon=10$ on Figure \ref{fig3Deps10} and for $\varepsilon=0.5$ on Figure \ref{fig3Deps05}. For MiMa-Grid and MiMa-Part-2, $f$ is reconstruted from $g$, $\rho$ and $M$, whereas the approximation of $f$ is directly given by the Full PIC scheme. The numerical parameters are the same as previously (see comments on Figures \ref{figTSIkin10} and \ref{figTSIinter}). On Figure \ref{fig3Deps10}, we observe that the result obtained by MiMa-Part-2 and Full PIC are in good agreement with MiMa-Grid; however, some numerical noise can be distinguished on $f$ obtained by the Full PIC method. On Figure \ref{fig3Deps05} ($\varepsilon=0.5$), we can see clearly that the level of the noise is higher for Full PIC, which prevents it from giving good results. On the contrary, MiMa-Part-2 produces good results compared to MiMa-Grid, since the noise only affects the micro part $g$, which is small in this regime. \begin{figure} \caption{Representation of $f(T=5,x,v)$, $\varepsilon=10$. Distribution function $f$ reconstructed from MiMa-Grid on the left, from MiMa-Part-2 on the middle and obtained by Full PIC on the right. $\Delta t=0.1$, $N_x=128$ and $N_p=10^6$ for both particle methods. $N_x=N_v=512$ and $\Delta t\approx 6\times 10^{-4}$ for MiMa-Grid.} \label{fig3Deps10} \end{figure} \begin{figure} \caption{Representation of $f(T=5,x,v)$, $\varepsilon=0.5$. Distribution function $f$ reconstructed from MiMa-Grid on the left, from MiMa-Part-2 on the middle and obtained by Full PIC on the right. $\Delta t=0.01$, $N_x=256$ and $N_p=10^5$ for both particle methods. $N_x=N_v=512$ and $\Delta t\approx 6\times 10^{-4}$ for MiMa-Grid.} \label{fig3Deps05} \end{figure} For smaller values of $\varepsilon$, we compare the four AP schemes (MiMa-Part-1, MiMa-Part-2, MiMa-Grid and Moment G.) to the limit scheme. Results for $\varepsilon=10^{-1}$ are given in Figure \ref{figTSIdif}. Parameters are the following: $\Delta t=10^{-3}$ , $N_x=128$ and $N_p=10^4$ for particle methods, $\Delta t=0.1\Delta x^2\approx 3.5\times 10^{-3}$ and $N_x=N_v=64$ for MiMa-Grid. As in the Landau damping case, MiMa-Part-2 gives the best result comparing to the reference MiMa-Grid. Finally, results for $\varepsilon=10^{-4}$ are given in Figure \ref{figTSIdiflim}, where we have $\Delta t=10^{-2}$, $N_x=128$ and $N_p=100$ for particle methods, $\Delta t=0.1\Delta x^2\approx 3.5\times 10^{-3}$ and $N_x=N_v=64$ for MiMa-Grid. The asymptotic regime is well recovered by all these AP methods. \begin{minipage}[t]{0.48\textwidth} \begin{figure} \caption{Time history of the electric energy, $\varepsilon=0.1$. $\Delta t=10^{-3}$, $N_x=128$ and $N_p=10^4$ for the four particle methods.} \label{figTSIdif} \end{figure} \end{minipage} ~~ \begin{minipage}[t]{0.48\textwidth} \begin{figure} \caption{Time history of the electric energy, $\varepsilon=10^{-4}$. $\Delta t=0.01$, $N_x=128$ and $N_p=100$ for the four particle methods.} \label{figTSIdiflim} \end{figure} \end{minipage} \section{Conclusion}\label{sec:ccl} \setcounter{equation}{0} In this paper, we have presented new micro-macro models for the kinetic radiative transport equation (RTE), as well as for the Vlasov-Poisson-BGK system, in the diffusion scaling with periodic boundary conditions. First-order in time and second-order in time models are derived, and their Lagrangian discretizations are detailed. The obtained schemes are proved to degenerate into implicit discretizations of the limit model (the diffusion equation in the RTE case and the drift-diffusion equation in the Vlasov-Poisson-BGK case) when $\varepsilon\to 0$. This asymptotic property is shown in the numerical results too. Moreover, thanks to the use of particle methods for the microscopic equation, the numerical cost is reduced when $\varepsilon$ diminishes. Finally, compared to a standard PIC method (where $f$ is represented by particles, and not $g$), the numerical noise is reduced. In future works, we would like to extend the Monte-Carlo approach proposed for the hydrodynamic limit of Vlasov-BGK in \cite{cdl} to the diffusion and the drift-diffusion limits. \appendix \section{Time discretization for Eulerian schemes }\label{subsec:eulerian} \setcounter{equation}{0} We present the time discretization of (\ref{eq:micromacro_initial}) having in spirit a Eulerian discretization of the phase space. Obviously, the numerical scheme proposed in \cite{bennoune, cl, lm} works well. Now, (\ref{eq:mm_micro}) also provides a numerical scheme that we will exploit in this appendix. Let us consider staggered grids in the phase-space domain and adopt the following notations: $\mbox{x}_i=i\Delta x$ and $\mbox{x}_{i+1/2}=i\Delta x+\Delta x/2$, $i\in\mathbb{N}$, define two grids in space and $\mbox{v}_j=j\Delta v$, $j\in\mathbb{N}$, defines a grid in velocity, where $\Delta x$ (resp. $\Delta v$) is the step in space (resp. in velocity). Time is also discretized with a time step $\Delta t$ and we note $t^n=n\Delta t$, $n\in\mathbb{N}$. The density $\rho$ is discretized on the first space grid: $\rho_i^n$ approximates $\rho(t^n,\mbox{x}_i)$, whereas the perturbation $g$ is discretized on the second one: $g_{i+1/2,j}^n$ approximates $g(t^n,\mbox{x}_{i+1/2},\mbox{v}_j)$. Let an approximation $D$ of the spatial derivative, the numerical scheme we propose consists in computing $g_{i+1/2,j}^{n+1}$ with \begin{eqnarray} g^{n+1}_{i+1/2, j} &=& e^{-\Delta t/\varepsilon^2} g^{n+1}_{i+1/2, j} - \varepsilon(1-e^{-\Delta t/\varepsilon^2}) \left[ v_j \frac{\rho^n_{i+1}-\rho_i^n}{\Delta x} \right. \nonumber\\ \label{s2} && \left. + (I-\Pi)\left(v^+_j (D^-_x g^n)_{i+1/2, j}+v^-_j (D^+_x g^n)_{i+1/2, j} \right) \right], \nonumber\\ \end{eqnarray} where $(\Pi h)_{i+1/2, j} = (\sum_j h_{i+1/2, j}\Delta v)$, and then to compute $\rho_i^{n+1}$ with \begin{equation} \label{macrod} \rho_i^{n+1} = \rho_i^n -\frac{\Delta t}{\varepsilon} \sum_j \left(v_j \frac{g^{n+1}_{i+1/2, j}-g^{n+1}_{i-1/2, j}}{\Delta x} \right)\Delta v. \end{equation} \begin{prop} The scheme given by (\ref{s2})-(\ref{macrod}) enjoys the AP property, \textit{i.e.} it satisfies the following properties \begin{itemize} \item for fixed $\varepsilon>0$, the scheme is a first-order (in time) approximation of the original model (\ref{eq:etrbgk}), \item for fixed $\Delta t>0$, the scheme degenerates into an explicit first-order (in time) scheme of \eqref{eq:diff}. \end{itemize} \label{prop:euler_1st_exp} \end{prop} \begin{proof} We observe easily that when $\varepsilon$ goes to zero, (\ref{s2}) gives $$ g^{n+1}_{i+1/2, j} = -\varepsilon v_j \frac{\rho^n_{i+1}-\rho_i^n}{\Delta x} +{\cal O}(\varepsilon^2), $$ which, injected in the time discretization (\ref{macrod}) for $\rho$, gives up to terms of order ${\cal O}(\varepsilon^2)$ $$ \rho_i^{n+1} = \rho_i^n + \Delta t \left(\sum_j v^2_j \Delta v\right) \frac{\rho^n_{i+1}-2\rho_i^n+\rho^n_{i-1}}{\Delta x^2} . $$ Since $\sum_j v^2_j \Delta v$ is an approximation of $\int_{-1}^1 v^2 \textnormal{d} v=1/3$, we obtain a consistent discretization of the diffusion equation. \end{proof} Proposition \ref{prop:euler_1st_exp} is of big interest for impliciting the diffusion term $\partial_{xx}\rho$. Indeed, let us rewrite \eqref{s2} as follows $$ g_{i+1/2,j}^{n+1} = - \varepsilon(1-e^{-\Delta t/\varepsilon^2}) v_j \frac{\rho^n_{i+1}-\rho_i^n}{\Delta x} + h_{i+1/2,j}, $$ with $h_{i+1/2,j}=e^{-\Delta t/\varepsilon^2} g^{n+1}_{i+1/2, j} - \varepsilon(1-e^{-\Delta t/\varepsilon^2}) \left[ (I-\Pi)\left(v^+_j (D^-_x g^n)_{i+1/2, j}+v^-_j (D^+_x g^n)_{i+1/2, j} \right) \right].$ Injecting this relation into the macro part, we get \begin{eqnarray} \rho^{n+1}_i &=& \rho_i^n + \Delta t (1-e^{-\Delta t/\varepsilon^2})\sum_j ( v_j^2)\Delta v \frac{\rho^n_{i+1}-2\rho_i^n + \rho^n_{i-1}}{\Delta x^2} - \frac{\Delta t}{\varepsilon} \sum_j \left( v_j \frac{h_{i+1/2, j}-h_{i-1/2, j}}{\Delta x}\right) \Delta v. ~~\nonumber\\ \label{eq:euler_1st_manoeuvre} \end{eqnarray} Since $h_{i+1/2, j}= {\cal O}(\varepsilon^2)$ as $\varepsilon$ goes to zero after two iterations, the asymptotic preserving property is ensured. Moreover, the diffusion term can now be chosen as implicit, so that the macro equation becomes \begin{eqnarray} \rho^{n+1}_i &=& \rho_i^n + \Delta t (1-e^{-\Delta t/\varepsilon^2}) \sum_j ( v_j^2)\Delta v \frac{\rho^{n+1}_{i+1}-2\rho_i^{n+1} + \rho^{n+1}_{i-1}}{\Delta x^2}- \frac{\Delta t}{\varepsilon} \sum_j \left( v_j \frac{h_{i+1/2, j}-h_{i-1/2, j}}{\Delta x}\right) \Delta v,~~\nonumber\\ \label{macrod_cn} \end{eqnarray} and the scheme is now free from the usual diffusion condition on the time step. The algorithm finally writes \begin{algo}~~ \begin{itemize} \item Initialize $g_{i+1/2,j}^0$ and $\rho_i^0$. At each time step: \item Advance micro part with (\ref{s2}). \item Advance macro part with (\ref{macrod_cn}). \end{itemize} \label{algo:euler_1st} \end{algo} And we have the following result. \begin{prop} The scheme given by (\ref{s2})-(\ref{macrod_cn}) enjoys the AP property, \textit{i.e.} it satisfies the following properties \begin{itemize} \item for fixed $\varepsilon>0$, the scheme is a first-order (in time) approximation of the original model (\ref{eq:etrbgk}), \item for fixed $\Delta t>0$, the scheme degenerates into an implicit first-order (in time) scheme of \eqref{eq:diff}. \end{itemize} \label{prop:euler_1st_imp} \end{prop} We do not present here the extension to the Vlasov-Poisson-BGK case, but it is straightforward. \section{Moment guided}\label{app:mgm} \setcounter{equation}{0} In this section, we present the adaptation of the moment guided particle method proposed in \cite{ddp} to our context. For the sake of simplicity, we present it in the RTE case but note that these computations can also be extended to the Vlasov-Poisson-BGK case, without difficulty. The kinetic equation on $f$ has to be reformulated to avoid the singularity linked to the transport term. To do that, we proceed as previously, but from (\ref{eq:etrbgk}). Indeed, we rewrite equation (\ref{eq:etrbgk}) as $$ \partial_t (e^{t/\varepsilon^2}f) = \frac{e^{t/\varepsilon^2}}{\varepsilon}\left[-v\partial_x f + \frac{1}{\varepsilon}\rho\right], $$ and we integrate between $t^n$ and $t^{n+1}$ to get $$ f(t^{n+1}) = e^{-\Delta t/\varepsilon^2}f(t^n) - \frac{e^{-t^{n+1}/\varepsilon^2}}{\varepsilon} \int_{t^n}^{t^{n+1}} e^{t/\varepsilon^2} \left[v\partial_x f - \frac{1}{\varepsilon}\rho\right] \textnormal{d} t. $$ We make the following approximation $$ f^{n+1} = e^{-\Delta t/\varepsilon^2}f^n - \varepsilon (1-e^{-\Delta t/\varepsilon^2}) \left[v\partial_x f^n - \frac{1}{\varepsilon}\rho^{n}\right], $$ where $f^n\approx f(t^n)$ and $\rho^{n}\approx \rho(t^{n})$, $\forall n$. Making appear the discrete time derivative enables to write $$ \frac{f^{n+1} -f^n}{\Delta t}= \frac{e^{-\Delta t/\varepsilon^2}-1}{\Delta t}f^n - \varepsilon \frac{1-e^{-\Delta t/\varepsilon^2}}{\Delta t} \left[v\partial_x f^n - \frac{1}{\varepsilon}\rho^{n}\right], $$ which we approximate by \begin{equation} \partial_t f= \frac{e^{-\Delta t/\varepsilon^2}-1}{\Delta t}f - \varepsilon \frac{1-e^{-\Delta t/\varepsilon^2}}{\Delta t} \left[v\partial_x f - \frac{1}{\varepsilon}\rho\right]. \label{eq:appB} \end{equation} Following the spirit of the moment guided method (see \cite{ddp}), this equation is coupled with the macro one, that is \begin{eqnarray} \partial_t \rho+\frac{1}{\varepsilon}\partial_x\langle vf\rangle&=&0,\\ \partial_t f+\varepsilon \frac{1-e^{-\Delta t/\varepsilon^2}}{\Delta t}v\partial_x f&=& \frac{e^{-\Delta t/\varepsilon^2}-1}{\Delta t}f + \frac{1-e^{-\Delta t/\varepsilon^2}}{\Delta t} \rho. \end{eqnarray} To derive an AP scheme for this latter system satisfied by $(\rho,f)$, we adapt the strategy presented in \cite{ddp} to our diffusion framework. To do so, we first remark that $\langle vf\rangle=\langle vg\rangle$ and using the expression of $g$ obtained by (\ref{eq:mm_micro}), we get the following approximation for the macro flux (considered implicit in time) $$ \frac{1}{\varepsilon}\partial_x\langle vg^{n+1}\rangle = - (1-e^{-\Delta t/\varepsilon^2})\partial_{xx}\rho^n+\frac{1}{\varepsilon}e^{-\Delta t/\varepsilon^2}\partial_x\langle vg^n\rangle. $$ Then, we get the following scheme for $\rho$ \begin{equation} \rho^{n+1}=\rho^n+\Delta t(1-e^{-\Delta t/\varepsilon^2})\partial_{xx}\rho^n-\frac{\Delta t}{\varepsilon}e^{-\Delta t/\varepsilon^2}\partial_x\langle vf^n\rangle. \label{eq:rhomg} \end{equation} A Lagrangian method can be used to approximate the equation on $f$. As for the micro-macro scheme, we use a splitting procedure \begin{itemize} \item solve $\partial_t f + \varepsilon \frac{1-e^{-\Delta t/\varepsilon^2}}{\Delta t} v\partial_x f = 0$ \item solve $\partial_t f = \frac{e^{-\Delta t/\varepsilon^2} -1}{\Delta t}f+ \frac{1-e^{-\Delta t/\varepsilon^2}}{\Delta t} \rho$. \end{itemize} To do that, the transport part is solved with the (non stiff) characteristics \begin{equation} \label{carxf} \dot{x}_k(t) = \varepsilon \frac{1-e^{-\Delta t/\varepsilon^2}}{\Delta t} v_k(t). \end{equation} The source part is solved using the equation satisfied by the weights \begin{equation} \label{weightf} \dot{\omega}_k(t) = \frac{e^{-\Delta t/\varepsilon^2} -1}{\Delta t}\omega_k(t) + \frac{1-e^{-\Delta t/\varepsilon^2}}{\Delta t} \rho(t, x_k(t)). \end{equation} The last step consists in matching the moment of $f^{n+1}$ obtained by the particle method with $\rho^{n+1}$ obtained with (\ref{eq:rhomg}). This can be done using the techniques proposed in \cite{ccl}. Indeed, considering the function $g=f-\rho$, its weight can be written as $$ \gamma_k = \omega_k - \beta_k, \;\; \mbox{ with } \;\; \beta_k=\rho(x_k)\frac{L_xL_v}{N_{p}}. $$ Then, we apply the discrete version of $(I-\Pi)$ to the weights $\gamma_k$ as in \cite{ccl} $$ \omega_k^{new} = \beta_k +(I-\Pi)( \omega_k - \beta_k). $$ \paragraph{Acknowledgements}~~\\ N. Crouseilles and M. Lemou are supported by the French ANR project MOONRISE ANR-14-CE23-0007-01 and by the Enabling Research EUROFusion project CfP-WP14-ER-01/IPP-03. A. Crestetto is supported by the French ANR project ACHYLLES ANR-14-CE25-0001. \end{document}	arXiv
Wu–Yang dictionary In topology and high energy physics, the Wu–Yang dictionary refers to the mathematical identification that allows to translate back and forth between the concepts of gauge theory and those of differential geometry. It was devised by Tai Tsun Wu and C. N. Yang in 1975 when studying the relation between electromagnetism and fiber bundle theory.[1] This dictionary has been credited as bringing mathematics and theoretical physics closer together.[2] A crucial example of the success of the dictionary is that it allowed to understand Paul Dirac's monopole quantization in terms of Hopf fibrations.[3] History In 1975, theoretical physicists Tsun Wu and C. N. Yang working in Stony Brook University, published a paper on the mathematical framework of electromagnetism and the Aharonov–Bohm effect in terms of fiber bundles. A year later, mathematician Isadore Singer came to visit and brought a copy back to the University of Oxford.[2][4][5] Singer showed the paper to Michael Atiyah and other mathematicians, sparking a close collaboration between physicists and mathematicians.[2] Yang also recounts a conversation that he had with one of the mathematicians that founded fiber bundle theory, Shiing-Shen Chern:[2] In 1975, impressed with the fact that gauge fields are connections on fiber bundles, I drove to the house of Shiing-Shen Chern in El Cerrito, near Berkeley. (I had taken courses with him in the early 1940s when he was a young professor and I an undergraduate student at the National Southwest Associated University in Kunming, China. That was before fiber bundles had become important in differential geometry and before Chern had made history with his contributions to the generalized Gauss–Bonnet theorem and the Chern classes.) We had much to talk about: friends, relatives, China. When our conversation turned to fiber bundles, I told him that I had finally learned from Jim Simons the beauty of fiber-bundle theory and the profound Chern-Weil theorem. I said I found it amazing that gauge fields are exactly connections on fiber bundles, which the mathematicians developed without reference to the physical world. I added ‘this is both thrilling and puzzling, since you mathematicians dreamed up these concepts out of nowhere.’ He immediately protested, ‘No, no. These concepts were not dreamed up. They were natural and real.' Description Summarized version The Wu-Yang dictionary relates terms in particle physics with terms in mathematics, specifically fiber bundle theory. Many versions and generalization of the dictionary exist. Here is an example of a dictionary, which puts each physics term next to its mathematical analogue:[6] Physics Mathematics Potential Connection Field tensor (interaction) Curvature Field tensor-potential relation Structural equation Gauge transformation Change of bundle coordinates Gauge group Structure group Original version for electromagnetism Wu and Yang considered the description of an electron traveling around a cylinder in the presence of a magnetic field inside the cylinder (outside the cylinder the field vanishes i.e. $f_{\mu \nu }=0$). According to the Aharonov–Bohm effect, the interference patterns shift by a factor $\exp(-i\Omega /\Omega _{0})$, where $\Omega $ is the magnetic flux and $\Omega _{0}$ is the magnetic flux quantum. For two different fluxes a and b, the results are identical if $\Omega _{a}-\Omega _{b}=N\Omega _{0}$, where $N$ is an integer. We define the operator $S_{ab}$ as the operator that brings the electron wave function from one configuration to the other $\psi _{b}=S_{ba}\psi _{a}$. For an electron that takes a path from point P to point Q, we define the phase factor as $\Phi _{PQ}=\exp \left(-{\frac {i}{\Omega _{0}}}\int _{P}^{Q}A_{\mu }\mathrm {d} x^{\mu }\right)$, where $A_{\mu }$ is the electromagnetic four-potential. For the case of a SU2 gauge field, we can make the substitution $A_{\mu }=ib_{\mu }^{k}X_{k}$, where $X_{k}=-i\sigma _{k}/2$ are the generators of SU2, $\sigma _{k}$ are the Pauli matrices. Under these concepts, Wu and Yang showed the relation between the language of gauge theory and fiber bundles, was codified in following dictionary:[2][7][8] Wu–Yang dictionary (1975) Gauge field terminology Bundle terminology Gauge (or global gauge) Principal coordinate fiber bundle Gauge type Principal fiber bundle Gauge potential $b_{\mu }^{k}$ Connection on principal fiber bundle $S_{ba}$ Transition function Phase factor $\Phi _{QP}$ Parallel displacement Field strength $f_{\mu \nu }^{k}$ Curvature Source $J_{\mu }^{k}$ ? Electromagnetism Connection in a U1(1) bundle Isotopic spin gauge field Connection in a SU2 bundle Dirac's monopole quantization Classification in a U1(1) bundle according to first Chern class Electromagnetism without monopole Connection on a trivial a U1(1) bundle Electromagnetism with monopole Connection on a nontrivial a U1(1) bundle See also • 't Hooft–Polyakov monopole • Wu–Yang monopole References 1. Wu, Tai Tsun; Yang, Chen Ning (1975-12-15). "Concept of nonintegrable phase factors and global formulation of gauge fields". Physical Review D. 12 (12): 3845–3857. doi:10.1103/PhysRevD.12.3845. ISSN 0556-2821. 2. Poo, Mu-ming; Chao, Alexander Wu (2020-01-01). "Conversation with Chen-Ning Yang: reminiscence and reflection". National Science Review. 7 (1): 233–236. doi:10.1093/nsr/nwz113. ISSN 2095-5138. PMC 8288855. PMID 34692035. 3. Woit, Peter (5 April 2008). "Stony Brook Dialogues in Mathematics and Physics". Not even wrong blog. Retrieved 2023-03-14. 4. Wells, Raymond O'Neil; Weyl, Hermann (1988). The Mathematical Heritage of Hermann Weyl. American Mathematical Soc. ISBN 978-0-8218-1482-6. 5. Freed, Daniel S. (2021). "Isadore Singer Transcended Mathematical Boundaries". Quanta Magazine. 6. Zeidler, Eberhard (2008-09-03). Quantum Field Theory II: Quantum Electrodynamics: A Bridge between Mathematicians and Physicists. Springer Science & Business Media. ISBN 978-3-540-85377-0. 7. Boi, Luciano (2004). "Geometrical and topological foundations of theoretical physics: from gauge theories to string program". International Journal of Mathematics and Mathematical Sciences. 2004 (34): 1777–1836. doi:10.1155/S0161171204304400. ISSN 0161-1712. 8. Wells, Raymond O'Neil; Weyl, Hermann (1988). The Mathematical Heritage of Hermann Weyl. American Mathematical Soc. ISBN 978-0-8218-1482-6.	Wikipedia
Convolutional Neural Network Based Multi-feature Fusion for Non-rigid 3D Model Retrieval Hui Zeng* , Yanrong Liu* , Siqi Li* , JianYong Che and Xiuqing Wang* Corresponding Author: Hui Zeng* ([email protected]) Hui Zeng, Beijing Engineering Research Center of Industrial Spectrum Imaging, School of Automation and Electrical Engineering, University ofScience and Technology Beijing, Beijing, China, [email protected] Yanrong Liu, Beijing Engineering Research Center of Industrial Spectrum Imaging, School of Automation and Electrical Engineering, University ofScience and Technology Beijing, Beijing, China, [email protected] Siqi Li, Beijing Engineering Research Center of Industrial Spectrum Imaging, School of Automation and Electrical Engineering, University ofScience and Technology Beijing, Beijing, China, [email protected] JianYong Che, The Tiantan Park Management Office, Beijing, China, [email protected] Xiuqing Wang**, Vocational Technical Institute, Hebei Normal University, Shijiazhuang, China, [email protected] Received: May 3 2017 Revision received: June 22 2017 Abstract: This paper presents a novel convolutional neural network based multi-feature fusion learning method for non-rigid 3D model retrieval, which can investigate the useful discriminative information of the heat kernel signature (HKS) descriptor and the wave kernel signature (WKS) descriptor. At first, we compute the 2D shape distributions of the two kinds of descriptors to represent the 3D model and use them as the input to the networks. Then we construct two convolutional neural networks for the HKS distribution and the WKS distribution separately, and use the multi-feature fusion layer to connect them. The fusion layer not only can exploit more discriminative characteristics of the two descriptors, but also can complement the correlated information between the two kinds of descriptors. Furthermore, to further improve the performance of the description ability, the cross-connected layer is built to combine the low-level features with high-level features. Extensive experiments have validated the effectiveness of the designed multi-feature fusion learning method. Keywords: Convolutional Neural Network , HKS , Multi-Feature Fusion , Non-rigid 3D Model , WKS In recent years, with the rapid development of the computer technology and the multimedia technology, more and more 3D models have been used in many research fields such as face recognition, object recognition, self-driving car, virtual reality, biology and 3D game. Efficient 3D model retrieval has becomes a research hot spot in the field of computer vision. Generally, 3D model retrieval includes the following three steps: model preprocessing, feature extraction, and similarity matching [1]. Among these steps, the 3D feature extraction is the most key step and it plays a decisive role in retrieval results. So it has attracted more and more researchers' attentions. According to the type of the 3D models, the existing 3D feature extraction methods can be divided into two categories: rigid 3D model based method and non-rigid 3D model based method. Most of the existing methods are designed for the rigid 3D model, such as 3D shape contexts descriptor [2], local surface patch (LSP) descriptor [3], THRIFT descriptor [4], spin image (SI) descriptor [5,6], normal histogram (NormHist) [7], rotational projection statistics (RoPS) descriptor [8], fast point feature histogram (FPFH) [9], signature histograms of orientations (SHOT) [10,11], and so on. Although the above methods have achieved good retrieval results, they are designed for rigid 3D models and not suitable for non-rigid 3D models. All the above features are not invariant to the non-rigid deformations of the 3D models. To solve this problem, the researchers began to study the 3D feature extraction method for 3D non-rigid model. Sun et al. [12] proposed the heat kernel signature (HKS) descriptor to describe the local characteristics of the non-rigid 3D models. It is based on diffusion scale-space analysis and characterized by the heat transfer process of the 3D surface. The HKS descriptor is invariant under isometric deformations and stable under perturbations of the model. It has achieved good performance in the application of non-rigid 3D model retrieval [13-15]. However, it is sensitive to the scale changes of the 3D model. Aubry et al. [16] proposed the wave kernel signature (WKS) descriptor to describe the non-rigid 3D model, which describes the average probability of quantum mechanics at a position on a non-rigid 3D model surface. The WKS descriptor explains the relationship between the points on the different spatial scales and the rest of the model surface, and it has a better discriminative ability than the HKS descriptor. Up to now, the HKS descriptor and the WKS descriptor have been used in non-rigid 3D model retrieval separately, and the relationships between them have not been fully investigated. To further improve the retrieval performance, it is necessary to study the effective method to fusion the two kinds of descriptors. Recently, a large number of deep learning methods have been proposed for feature extraction in the field of computer vision, such as auto-encoder, Convolutional Neural Network (CNN), restricted Boltzmann machine (RBM), Deep Belief Networks (DBN), etc. Many experiments have proved that the deep learning based feature extraction methods are more effective. The most widely used deep learning methods in image analysis field is the CNN, which was proposed in 1998 by LeCun et al. [17] to identify handwritten numbers. In 2012, the CNN's accuracy beyond the second nearly 10% in the competition ImageNet. An important feature of the CNN is that it is very similar to our visual system, and the feature is extracted hierarchically. The high-level feature is a combination of low-level features. From low to high, the features are more and more abstract, and this is more conducive to the representation of semantics. Compared with the previous manual designed features, CNN can be automatically extracted more appropriate features, which can greatly improve the recognition performance. So the CNN has been used in more and more areas. For example, the deep learning methods have been used for 3D shape retrieval. Xie et al. [18] proposed a deep shape descriptor for 3D shape retrieval. Firstly, the multiscale shape distribution features are computed. Then a set of discriminative auto-encoders are trained to extract high-level shape features at different scales. Finally, the outputs from hidden layers of the auto-encoders are concatenated to for the shape descriptor. According to our knowledge, the CNNbased 3D retrieval methods have not been fully investigated. In this paper, we proposed a CNN based multi-feature fusion learning method for non-rigid 3D model retrieval. It can combine the effective discriminative information of the HKS descriptor and the WKS descriptor, which not only includes the time domain information of the HKS descriptor but also makes full use of the frequency domain information of the WKS descriptor. Our experimental results have testified that our proposed method can improve the retrieval performance than the single descriptor based method and other state-of-theart methods. The rest of the paper is organized as follows. In Section 2, the related works including the HKS descriptor and the WKS descriptor are reviewed. Section 3 gives our proposed CNN-based multi- feature fusion learning method. The non-rigid 3D model retrieval experimental results are presented in Section 4, and some concluding remarks are listed in Section 5. 2. Related Works For the non-rigid 3D shape retrieval, it is extremely important to select the appropriate feature descriptor. Due to the wide existence of non-rigid deformation, the extracted features must have strong robustness. In this paper, we select the HKS descriptor and the WKS descriptor to describe the 3D local structures, which can describe the 3D local patch from different views and complement with each other. 2.1 HKS Descriptor Sun et al. [12] proposed the HKS descriptor to describe the local structure of the 3D model. The HKS descriptor is constructed based on the heat diffusion process of the surface of the model, which abandoned the heat kernel's spatial information and only left the time domain information. Considering a 3D model M as a Riemannian manifold, there is the following heat diffusion equation: [TeX:] $$\left( \Delta _ { M } + \frac { \partial } { \partial t } \right) u ( x , t ) = 0$$ where, ∆M is the positive semi-definite Laplace-Beltrami operator of M and t is time parameter. The solution u(x,t) of Eq. (1) is the amount of heat on the surface at point x in time t. When the [TeX:] $$u ( x , 0 ) = \delta ( x - y )$$ is defined as the initial conditions, the solution set of the heat diffusion equation is called heat kernel kt(x,y), which can be written as follows: [TeX:] $$k _ { t } ( x , y ) = \sum _ { k = 1 } ^ { \infty } e ^ { - \lambda , t } \phi _ { i } ( x ) \phi _ { i } ( y )$$ where, x and y are the points of the 3D model, [TeX:] $$\lambda _ { i } \geq 0 \text { is the } i ^ { \text { th } }$$ eigenvalue and [TeX:] $$\phi _ { i } ( x ) \text { is the } i ^ { \text { th } }$$ eigenfunction of the Laplace-Beltrami operator [TeX:] $$\Delta _ { M } , \text { satisfying } \Delta _ { M } \phi _ { i } = \lambda _ { i } \phi _ { i }$$.The heat kernel signature of the point x of the 3D model at time t can be expressed as [12]: [TeX:] $$h ( x , t ) = k _ { t } ( x , x ) = \sum _ { k = 1 } ^ { \infty } e ^ { - \lambda , t } \phi _ { i } ^ { 2 } ( x )$$ Then the HKS descriptor of the point x can be obtained by computing its corresponding heat kernel signatures at time sequence. The HKS feature has many excellent characteristics: isometric invariance, multi-scale, and time parameters can be used for representing the small distortion on the model. The heat kernel can be regarded as the transfer density function of the Brownian motion on the fluid, so the local distortion of the model surface will not cause much influence on the heat kernel. In addition, HKS features also have some shortcomings. For example, the HKS descriptor is simplified by the heat kernel. Although the search efficiency is improved, but the experimental results show that the heat kernel can be affected by the spatial domain. So discarding the spatial domain information is a limitation of HKS characteristics. Secondly, the HKS descriptor is not invariant to the scale of the model. The HKS descriptor mainly contains the low frequency information of the 3D model, ignoring the high frequency information, so it is not suitable for high precision matching. Furthermore, the time parameter t of the HKS descriptor is not directly related to the intrinsic attributes of the 3D model itself. Therefore, for the HKS descriptor, there are some limitations for describing the 3D models. 2.2 WKS Descriptor Aubry et al. [16] proposed the WKS descriptor for characterizing pints on non-rigid 3D shapes. Given a model M, the WKS descriptor is represented by measuring the average probability of the particles to be measured at each vertex at an energy level. The energy of the particles is related to the frequency, so the effects of different frequencies can be clearly distinguished. The difference between the WKS descriptor and the HKS descriptor is that the WKS descriptor uses the following Schrodinger equation instead of the heat diffusion equation: [TeX:] $$\frac { \partial \Psi } { \partial t } ( x , t ) = i \Delta \Psi ( x , t )$$ where Δ is the Laplace–Beltrami operator of the 3D model, (x,t) is the wave function. Then the WKS descriptor can be defined by the following formula: [TeX:] $$W K S ( x , E ) = \sum _ { k = 0 } ^ { \infty } \phi _ { k } ^ { 2 } ( x ) f _ { E } ^ { 2 } \left( E _ { k } \right)$$ where [TeX:] $$f _ { E } ^ { 2 } \left( E _ { k } \right)$$ is the energy probability distribution. In order to select the appropriate energy distribution, let [TeX:] $$f _ { E } ^ { 2 }$$ be the Gaussian distribution, the energy scale [TeX:] $$e = \log \left( E _ { k } \right)$$, there are following formula: [TeX:] $$\left\{ \begin{array} { l } { W K S ( x , \cdot ) : \mathrm { R } \rightarrow R }, \\ { W K S ( x , e ) = C _ { e } \sum _ { k = 0 } ^ { \infty } \phi _ { k } ^ { 2 } ( x ) \exp \left( \frac { - \left( e - \log E _ { k } \right) ^ { 2 } } { 2 \sigma ^ { 2 } } \right) } \end{array} \right.$$ where [TeX:] $$C _ { e } = \left( \sum _ { k = 0 } ^ { \infty } \exp \left( \frac { - \left( e - \log E _ { k } \right) ^ { 2 } } { 2 \sigma ^ { 2 } } \right) \right) ^ { - 1 }$$ Like the HKS descriptor, the WKS descriptor also can be looked as an application of a set of filters with the frequency responses [TeX:] $$f _ { E } ^ { 2 } \left( E _ { k } \right)$$. But the HKS descriptor only uses low-pass filters, and the WKS descriptor uses different frequencies to separate different scales. Compared with the HKS descriptor, the WKS descriptor describes the 3D local patch from a different view. The WKS descriptor not only contains low-frequency information, but also contains high-frequency information. The WKS descriptor is invariant to the non-rigid transformations, and it is stable under perturbations of the 3D shape. So the WKS descriptor is suitable for analyzing 3D shapes undergoing non-rigid deformations. 3. Convolutional Neural Network-Based Multi-feature Fusion In this section, we firstly introduce the extraction method of the HKS distribution and the WKS distribution. Then the architecture of the multi-feature fusion learning networks and the network optimization are described. 3.1 The HKS Distribution and the WKS Distribution For each vertex of the 3D model, we can calculate its corresponding HKS descriptor and corresponding WKS descriptor separately. Because the numbers of the vertices of different 3D models are different, the HKS descriptors and the WKS descriptors can't be used as the input of the CNN directly. In this paper, we use the shape distribution to describe the 3D model [19], which refers to a probability distribution sampled from a shape function and can be used as the input of the CNN. For the HKS descriptor, we use the multi-scale shape distribution proposed by Xie et al. [18] to obtain the input of the network. It is a statistics probability distribution of the HKS descriptor of the 3D model. For each scale, we compute the histogram of the HKS descriptor to form the shape distribution. The detailed construction method can be found in [18]. For the WKS descriptor, we can construct the multi-scale shape distribution at each energy using similar method. Here, different scale denotes different energy. In this paper, The HKS multi-scale shape distribution matrix is 128×96, and the number of the diffusion times is 96 and the number of the discrete HKS descriptor values is 128. The WKS multi-scale shape distribution matrix is also 128×96, and the number of the energy values is 96 and the number of the discrete WKS descriptor values is 128. All the above parameters are determined by experiments. Fig. 1 shows the HKS multi-scale shape distributions of the human models and the ant models with different poses. From Fig. 1 we can see that the multi-scale shape distributions are different for different classes, and the 3D models of the same class have similar multi-scale shape distributions. Although these 3D models of the same class have different postures and the details of the shape distributions are different, their main features have been captured by the multi-scale shape distributions. Fig. 2 shows the WKS multi-scale shape distributions of the human models and the ant models with different poses. From Fig. 2 we can see that the distributions have not clear differences for different models. This is because that the WKS descriptor is represented by the average probabilities of quantum particles of different energy levels. The scale of the WKS multi-scale shape distribution denotes the energy, and the 3D models with different postures of the same class usually correspond different energies. So it is necessary to extract more discriminative features from the HKS multi-scale shape distributions and the WKS multi-scale shape distributions, and study the effective fusion method to further improve the retrieval performance. The HKS multi-scale shape distributions of human models (a) and ant models (b). The WKS multi-scale shape distributions of human models (a) and ant models (b). 3.2 The Architecture of the Multi-feature Fusion Learning Inspired by the multi-modal deep learning method for RGB-D object recognition [20], we proposed a conventional neural network based multi-feature fusion learning architecture for non-rigid 3D model retrieval. Our proposed multi-feature fusion learning architecture includes two CNNs and a multifeature fusion layer. At first, two CNNs are built to learn the HKS and WKS features, and the inputs of them are the HKS and WKS distributions. As shown in Fig. 3, each CNN consists of three convolutional layers (C1, C2, C3), three pooling layers (S1, S2, S3), one cross-connected layer and two fully-connected layers. Then the last pool layer of each CNNs is combined with the penultimate pool layer into a crossconnected layer, which can fully utilize the characteristics of hidden layer to further improve the retrieval performance. The cross-connected layer is composed of S2 layer and S3 layer. Finally, the multi-feature fusion layer is used to fuse the two descriptor of the fully-connected layers. It combines the two kinds of features with two feature transformation matrix Q1 and Q2. Our multi-feature fusion network, where "conv", "pool", "cc", "fc1" and "fc2" represent "convolutional", "mean-pooling", "cross-connected layers", "first fully-connected layer of the HKS/WKS feature" and "second fully-connected layer of the HKS/WKS feature" respectively. Let A be the output of the second fully-connected layer of the HKS feature, where [TeX:] $$A = \left[ a _ { 1 } , a _ { 2 } , \cdots , a _ { N } \right] \in R ^ { M \times N }$$, where M is the dimension of the activations and N is the number of the training samples. Similarly, B is the output of the second fully-connected layer of the WKS feature, and [TeX:] $$B = \left[ b _ { 1 } , b _ { 2 } , \cdots , b _ { N } \right] \in R ^ { M \times N }$$. The transformation matrixes of the two kinds of features are Q1 and Q2, where [TeX:] $$a _ { i } ^ { \prime } = Q _ { 1 } a _ { i } , b _ { i } ^ { \prime } = Q _ { 2 } b _ { i } , \text { and } a _ { i } ^ { \prime } \text { and } b _ { i } ^ { \prime }$$ are weighted by k1 and k2 to form the final learned features. In the process of learning, we simultaneously optimize the matrixes 1Q , 2 Q and the weights k1 and k2. 3.3 Network Optimization In order to learn Q1 and Q2 for the two kinds of features to obtain better representations, we consider the discriminative information and related information of the two kinds of features. Our objective is to learn the fusion features to minimize the distances between same-class samples and maximize the distances between different-class samples. So the objective function is defined as follows: [TeX:] $$\begin{array} { l } { \min _ { \left\{ Q _ { 1 } , Q _ { 2 } , k _ { 1 } , k _ { 2 } \right\} } F = k _ { 1 } D _ { 1 } \left( Q _ { 1 } \right) + k _ { 2 } D _ { 2 } \left( Q _ { 2 } \right) + \lambda C \left( Q _ { 1 } , Q _ { 2 } \right) } \\ { \text { subject to } k _ { 1 } + k _ { 2 } = 1 , k _ { 1 } \geq 0 , k _ { 2 } \geq 0 , \lambda > 0 } \end{array}$$ where D1 and D2 represent the discriminative terms, C denotes the correlated term, and λ is the weight between the discriminative terms and the related term. Here the discriminative term D1 is defined as: [TeX:] $$D _ { 1 } \left( Q _ { 1 } \right) = \sum _ { i j } h \left( t _ { 1 } - y _ { i j } \left( u _ { 1 } - d _ { Q _ { 1 } } \left( a _ { i } , a _ { j } \right) \right) \right)$$ where [TeX:] $$h ( x ) = \max ( 0 , x ) , \text { and } d _ { Q _ { 1 } }$$ represents the distance of transformed feature [TeX:] $$Q _ { 1 } a _ { i } \text { and } Q _ { 1 } a _ { j }$$. The distance [TeX:] $$d _ { Q _ { 1 } }$$ can be computed as: [TeX:] $$d _ { Q _ { 1 } } \left( a _ { i } , a _ { j } \right) = \left( Q _ { 1 } a _ { i } - Q _ { 1 } a _ { j } \right) ^ { T } \cdot \left( Q _ { 1 } a _ { i } - Q _ { 1 } a _ { j } \right)$$ If ai and aj are the features from the same class, the distance [TeX:] $$d _ { Q _ { 1 } }$$ should be smaller than a given threshold [TeX:] $$u _ { 1 } - t _ { 1 } \left( u _ { 1 } > t _ { 1 } > 0 \right) . \text { If } a _ { i } \text { and } a _ { j }$$ are the features from different classes, the distance [TeX:] $$d _ { Q _ { 1 } }$$ should be larger than a given threshold [TeX:] $$u _ { 1 } + t _ { 1 }$$. So we can conclude that the distances of the same and different class have the following constraint: [TeX:] $$y _ { i j } \left( u _ { 1 } - d _ { Q _ { t } } \left( a _ { i } , a _ { j } \right) > t _ { 1 }\right.$$ For the label yij, we define it as follows: when ai and aj are from the same class, yij = 1. when ai and aj are from different classes, yij = -1. For the WKS feature bi, we have similar definitions and constraints. The correlated term is used to exploit the complementary information of the two kinds of features, and it is defined by the difference of pairwise distances between the transformed HKS features and the transformed WKS features. The correlated term can be described as: [TeX:] $$C \left( Q _ { 1 } , Q _ { 2 } \right) = \sum _ { i j } \left( \sqrt { d _ { Q _ { 1 } } \left( a _ { i } , a _ { j } \right) } - \sqrt { d _ { Q _ { 2 } } \left( b _ { i } , b _ { j } \right) } \right) ^ { 2 }$$ For the two 3D models from the same class, their corresponding distance [TeX:] $$d _ { Q _ { 1 } }$$ is small and their corresponding distance [TeX:] $$d _ { Q _ { 2 } }$$ is also small. For the two 3D models from different classes, their corresponding distance [TeX:] $$d _ { Q _ { 1 } }$$ is large and their corresponding distance [TeX:] $$d _ { Q _ { 2 } }$$ is also large. So we can minimize the correlated term [TeX:] $$C \left( Q _ { 1 } , Q _ { 2 } \right)$$ to optimize the transform matrixes [TeX:] Q _ { 1 } \text { and } Q _ { 2 }. In this paper, we use an alternating optimization approach to obtain the optimal solution for Equation (7). At first, Q1 and Q2 are fixed while k1 and k2 are optimized. Secondly, k1 and k2 are fixed while Q1 and Q2 are optimized. Furthermore, in order to avoid suboptimal results and increase the non-linearity, we adopt the strategy proposed in [20] and modify the objective function to: [TeX:] $$\begin{array} { l } { \min _ { \left\{ Q _ { 1 } , Q _ { 2 } , k _ { 1 } , k \right\} } F = k _ { 1 } ^ { p } D _ { 1 } \left( Q _ { 1 } \right) + k _ { 2 } ^ { p } D _ { 2 } \left( Q _ { 2 } \right) + \lambda C \left( Q _ { 1 } , Q _ { 2 } \right) } \\ { \text { subject to } k _ { 1 } + k _ { 2 } = 1 , k _ { 1 } \geq 0 , k _ { 2 } \geq 0 , \lambda > 0 } \end{array}$$ where p > 1 . In Eq. (12), we use [TeX:] $$k _ { 1 } ^ { p } \text { and } k _ { 2 } ^ { p }$$ instead of k1 and k2, this will balance the weights of the two kinds of features. Then we can construct the following Lagrangian function: [TeX:] $$L ( k , \eta ) = k _ { 1 } ^ { p } D _ { 1 } + k _ { 2 } ^ { p } D _ { 2 } + \lambda C - \eta \left( k _ { 1 } + k _ { 2 } - 1 \right)$$ By setting [TeX:] $$\frac { \partial L ( k , \eta ) } { \partial k } \text { and } \frac { \partial L ( k , \eta ) } { \partial \eta }$$ to 0, the weight [TeX:] $$k _ { m } ( m = 1,2 )$$ can be updated as: [TeX:] $$k _ { m } = \frac { \left( \frac { 1 } { D _ { m } } \right) ^ { \frac { 1 } { p - 1 } } } { \sum _ { m = 1 } ^ { 2 } \left( \frac { 1 } { D _ { m } } \right) ^ { \frac { 1 } { p - 1 } } }$$ And then use the back propagation algorithm to update the transform matrix, Q1 and Q2. The derivative of Q1 can be calculated as follows: [TeX:] $$\frac { \partial F } { \partial Q _ { 1 } } = 2 Q _ { 1 } \left[ k _ { 1 } ^ { p } \sum _ { i j } y _ { i j } h ^ { \prime } \left( t _ { 1 } - y _ { i j } \left( u _ { 1 } - d _ { Q _ { 1 } } \left( a _ { i } , a _ { j } \right) \right) \right) A _ { i j } ^ { 1 } + \lambda \sum _ { i j } \left( 1 - \sqrt { \frac { d _ { Q _ { 2 } } \left( b _ { i } , b _ { j } \right) } { d _ { Q _ { 1 } } \left( a _ { i } , a _ { j } \right) } } \right) A _ { i j } ^ { 1 } \right]$$ where [TeX:] $$A _ { i j } ^ { 1 } = \left( a _ { i } - a _ { j } \right) \left( a _ { i } - a _ { j } \right) ^ { T }$$. So the transform matrix Q1 can be updated using the following equation: [TeX:] $$Q _ { 1 } = Q _ { 1 } - \beta \frac { \partial F } { \partial Q _ { 1 } }$$ The updating method of the transform matrix Q2 is similar to Q1. In the process of the back-propagation, the derivatives of D and C for ai and bi is used. For ai, they can be described as follows: [TeX:] $$\frac { \partial D _ { 1 } } { \partial a _ { j } } = \sum _ { j } y _ { i j } \left( Q _ { 1 } ^ { T } Q _ { 1 } + Q _ { 1 } Q _ { 1 } ^ { T } \right) \left( a _ { i } - a _ { j } \right) h ^ { \prime } \left( t _ { 1 } - y _ { i j } \left( u _ { 1 } - d _ { Q _ { 1 } } \left( a _ { i } , a _ { j } \right) \right) \right)$$ [TeX:] $$\frac { \partial C } { \partial a _ { j } } = \sum _ { j } \left( Q _ { 1 } ^ { T } Q _ { 1 } + Q _ { 1 } Q _ { 1 } ^ { T } \right) \left( a _ { i } - a _ { j } \right) \sqrt { \frac { d Q _ { 1 } \left( a _ { i } , a _ { j } \right) - d Q _ { 2 } \left( b _ { i } , b _ { j } \right) } { d Q _ { 1 } \left( a _ { i } , a _ { j } \right) } }$$ Likewise for bi to compute the derivatives of D and C. The proposed multi-feature fusion learning method can be listed as follows: (i) For each training sample, compute the HKS descriptor and the WKS descriptor of each vertex; (ii) Compute the HKS multi-scale shape distribution and the WKS multi-scale shape distribution of each 3D model; (iii) Randomly initialize each conventional neural network, and then pre-training to get pretrained ai and bi.respectively. iv) Perform an alternating method to optimize Qi, Q2, ki and k2 : Fix Q1 and Q2, firstly, update k1 and k2, and then fix k1 and k2, update Q1 and Q2. (v) Perform back-propagation: fix Q1, Q2, k1 and k2, update the parameters of the conventional neural networks with gradient descent method. (vi) Repeat (iv)–(v) until convergence or reach the maximum number of iterations. In this paper, the McGill 3D shape benchmark is used for non-rigid 3D model retrieval experiments to evaluate the performance of our proposed method [20]. This benchmark contains 255 non-rigid 3D models from 10 different categories, including: ant, crab, spectacle, hand, human, octopus, plier, snake, spider and teddy-bear. Each category has 20–30 3D models. Among these models, there are rotational transformation, scale transformation and non-rigid deformation. In our non-rigid 3D model retrieval experiments, 15 3D models per class are randomly selected for training and the others for querying. Some example 3D models are shown in Fig. 4. The experimental environment was i7-6700 CPU 3.40 GHz 12.0G memory Lenovo computer with MATLAB R2014a. To evaluate the retrieval performance, we use the following measures: nearest neighbor (NN), the first tier (FT), the second tier (ST) and the discounted cumulative gain (DCG). In this experiments, the HKS descriptor and the WKS descriptor of each vertex are firstly computed. Secondly, the HKS multi-scale shape distribution and the WKS multi-scale shape distribution are calculated, which can be used as the inputs of the two neural networks. Then the two conventional neural networks and the fusion layer are built. Table 1 gives a detailed description of each conventional neural network, including the type of each layer, the size of the convolutional kernel, the stride and the output size of each layer. The activation function of all convolutional layers and fully-connected layers are ReLU. One of the advantages of the ReLU function is that it can learn the features faster. The other advantage is the biology of rationality, which is unilateral. Compared with sigmoid function and tanh function, the ReLU function conforms to the characteristics of biological neurons. In this paper, the pre-training methods are used to initialize the networks, which can make the network converge faster. The two conventional neural networks are trained independently with the HKS distribution and the WKS distribution. The learning rate is set to 0.05, Qm is initialized as an identity of 1000×1000, and k is initialized as [0.5, 0.5]. The [TeX:] $$u _ { 1 } , u _ { 2 } , t _ { 1 } , t _ { 2 } , p , \beta , \lambda$$ were set as 700, 500, 70, 50, 2, 0.0003, 0.009, respectively. In the process of learning, we use the stochastic gradient descent to update the parameters. Example 3D models of the McGill 3D shape benchmark. Description of each conventional neural network To validate the effectiveness of the multi-feature fusion, we compare the retrieval results of our proposed method with single feature based method. For the HKS descriptor, we firstly compute the HKS multi-scale shape distribution. Then the conventional neural network is built and its structure is the same as the single conventional neural network used in the multi-feature fusion learning method. Likewise for the WKS descriptor. Table 2 gives the retrieval results of the single-feature based methods and the multi-feature fusion method. From Table 2 we can see that compared with the single-feature based methods, our proposed multi-feature fusion learning method has better performance with the NN, FT, ST, and DCG measures. It shows that the fusion layer can learn more useful information. So compared with single-feature based method, using the HKS descriptor and the WKS descriptor simultaneously and our proposed conventional neural network based fusion method can improve the retrieval results effectively. Then we compare our proposed method to the covariance descriptor based method [21], the graphbased method [22], the PCA-based VLAT method [23], the hybrid BOW method [24], hybrid 2D/3D method [25], the CBoFHKS method [26] and the discriminative auto-encoder based shape descriptor (DASD) method [18]. Table 3 gives the retrieval results of our proposed multi-feature fusion learning method and other methods. From Table 3 we can see that our proposed method has the best performance with the FT, ST, and DCG measures and have comparable performance with NN measures. So our proposed method is more robust to non-rigid deformations and it has achieved very competitive results compared with other methods. Retrieval results compared with single-feature based methods Retrieval results compared with other methods In this paper, we have proposed a novel multi-feature fusion learning method for non-rigid 3D model retrieval. Firstly, the HKS descriptor and the WKS descriptor are computed. Then the corresponding HKS multi-scale shape distribution and the WKS multi-scale shape distribution are constructed to be used as the inputs of the conventional neural networks. Finally the conventional neural networks based multi-feature fusion learning framework is built to obtain the fusion feature. The contribution of this paper is that we use the cross-connected layer to combine the low-level features with high-level features, and the fusion layer can learn not only the discriminative characteristics of the two kinds of descriptors but also the correlated information between them. So our proposed fusion feature can make full use of the effective information of the HKS descriptor and the WKS descriptor, and it has achieved a better retrieval performance. This paper is supported by the National Natural Science Foundation of China (No. 61375010 and 61503224), the Fundamental Research Funds for the China Central Universities of USTB (No. FRF-BD- 16-005A), Beijing Key Discipline Development Program (No. XK100080537). She received B.S. and M.S. degrees from Shandong University in 2001 and 2004, respectively, and received the Ph.D. degree from National Laboratory of Pattern Recognition, Institute of Automation, Chinese Academy of Sciences in 2007. She is currently an associate professor at School of Automation and Electrical Engineering, University of Science and Technology Beijing, China. Her main research interests include computer vision, pattern recognition and machine learning. 1 Y. Matsuda, N. Miura, A. Nagasaka, H. Kiyomizu, T. Miyatake, "Finger-vein authentication based on deformation-tolerant feature-point matching," Machine Vision and Applications, 2016, vol. 27, no. 2, pp. 237-250. doi:[[[10.1007/s00138-015-0745-3]]] 2 A. Frome, D. Huber, R. Kolluri, T. Bulow, J. Malik, "Recognizing objects in range data using regional point descriptors," in Proceedings of the European Conference on Computer Vision, Prague, Czech Republic, 2004;pp. 224-237. doi:[[[10.1007/978-3-540-24672-5_18]]] 3 H. Chen, B. Bhanu, "3D free-form object recognition in range images using local surface patches," Pattern Recognition Letters, 2007, vol. 28, no. 10, pp. 1252-1262. doi:[[[10.1016/j.patrec.2007.02.009]]] 4 A. Flint, A. Dick, A. 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Ohbuchi, et al., "SHREC'10 Track: non-rigid 3D shape retrieval," in Proceedings of the Eurographics Workshop on 3D Object Retrieval, Zurich, Switzerland, 2015;pp. 107-120. doi:[[[10.2312/3DOR/3DOR10/101-108]]] Layer Type Patch size Stride feature maps Output size x Input 128×96 C1 Convolution 5×5 1 6 124×92 S1 Mean pooling 2×2 2 6 62×46 C2 Convolution 5×5 1 12 58×42 S2 Mean pooling 2×2 2 12 29×21 cc Cross-connected 1 9548 fc1 Fully-connected 1 3000 ff Feature fusion layer 1 2000 NN FT ST DCG HKS 0.819 0.622 0.744 0.827 WKS 0.914 0.775 0.866 0.914 Multi-feature fusion learning method 0.971 0.905 0.981 0.963 Methods NN FT ST DCG Covariance method [21] 0.977 0.732 0.818 0.937 Graph-based method [22] 0.976 0.741 0.911 0.933 PCA-based VLAT [23] 0.969 0.658 0.781 0.894 Hybrid BOW [24] 0.957 0.635 0.790 0.886 Hybrid 2D/3D [25] 0.925 0.557 0.698 0.850 CBoFHKS [26] 0.901 0.778 0.876 0.891 DASD [18] 0.988 0.782 0.834 0.955	CommonCrawl
\begin{document} \title{A Unified Contraction Analysis of a Class of Distributed Algorithms for Composite Optimization } \author{Jinming Xu, Ying Sun, Ye Tian, and Gesualdo Scutari\thanks{School of Industrial Engineering, Purdue University, West-Lafayette, IN, USA. Emails: \texttt{<xu1269, sun578, tian110, gscutari>} \texttt{@purdue.edu.} This work has been supported by the USA NSF Grants CIF 1632599 and CIF 1564044; and the ARO Grant W911NF1810238.}} \maketitle \begin{abstract} We study distributed composite optimization over networks: agents minimize the sum of a smooth (strongly) convex function--the agents' sum-utility--plus a nonsmooth (extended-valued) convex one. We propose a general algorithmic framework for such a class of problems and provide a unified convergence analysis leveraging the theory of operator splitting. Our results unify several approaches proposed in the literature of distributed optimization for special instances of our formulation. Distinguishing features of our scheme are: (i) when the agents' functions are strongly convex, the algorithm converges at a {\it linear} rate, whose dependencies on the agents' functions and the network topology are {\it decoupled}, matching the typical rates of centralized optimization; (ii) the step-size does not depend on the network parameters but only on the optimization ones; and (iii) the algorithm can adjust the ratio between the number of communications and computations to achieve the {\it same} rate of the centralized proximal gradient scheme (in terms of computations). This is the first time that a distributed algorithm applicable to {\it composite} optimization enjoys such properties. \end{abstract} \IEEEpeerreviewmaketitle \section{Introduction} We study distributed multi-agent optimization over networks, modeled as undirected static graphs. Agents aim at solving \begin{equation}\label{prob:dop_nonsmooth_same} \min_{x\in\mathbb{R}^{d}} F(x)+G(x),\quad F(x)\triangleq {\frac{1}{m}}\sum_{i=1}^m f_i(x),\tag{P} \end{equation} where $f_i:\mathbb{R}^d\to \mathbb{R}$ is the cost-function of agent $i$, assumed to be smooth, (strongly) convex and known only to the agent; and $G:\mathbb{R}^d\to \mathbb{R}\cup\{-\infty, \infty\}$ is a nonsmooth, convex (extended-value) function, which can be used to enforce shared constraints or some specific structure on the solution, such as sparsity.\\ \indent Our focus is on the design of distributed algorithms for Problem \eqref{prob:dop_nonsmooth_same} that provably converge at a {\it linear} rate. When $G=0$, several distributed schemes have been proposed in the literature enjoying such a property; examples include EXTRA~\cite{shi2015extra}, AugDGM~\cite{xu2015augmented}, NEXT~\cite{di2016next}, SONATA \cite{YingMAPR,sun2019convergence}, DIGing~\cite{nedich2016achieving}, NIDS~\cite{li2017decentralized}, Exact Diffusion~\cite{yuan2018exact_p1}, MSDA~\cite{scaman17optimal}, and the distributed algorithms in \cite{qu2017harnessing},\cite{jakovetic2018unification}, and \cite{mansoori2019general}. When $G\neq 0$ results are scarce; to our knowledge, the only two schemes available in the literature achieving linear rate for \eqref{prob:dop_nonsmooth_same} are SONATA \cite{sun2019convergence} and the distributed proximal gradient algorithm \cite{alghunaim2019linearly}. The aforementioned algorithms apparently look different; no unified convergence analysis can be inferred; and, in most of the cases, step-size bounds and convergence rate seem quite conservative. This naturally suggests the following two questions: \begin{itemize} \item[{\bf (Q1)}] Can one unify the design and analysis of distributed algorithms in the setting \eqref{prob:dop_nonsmooth_same}? \item[{\bf (Q2)}] Can one match the linear convergence rate of the centralized proximal-gradient algorithm applied to \eqref{prob:dop_nonsmooth_same}? \end{itemize} Recent efforts toward a better understanding of the taxonomy of distributed algorithms (question Q1) are the following: \cite{jakovetic2018unification} provides a connection between EXTRA and DIGing; \cite{Scoy-canonical18} provides a canonical representation of some of the distributed algorithms above--NIDS and Exact-Diffusion are proved to be equivalent; and \cite{Sundararajan17} provide an automatic (numerical) procedure to prove linear rate of some classes of distributed algorithms. These efforts model only first order algorithms applicable to Problem \eqref{prob:dop_nonsmooth_same} {\it with $G=0$} and employing a {\it single} round of communication and gradient computation. Because of that, in general, they cannot achieve the rate of the centralized gradient algorithm (addressing thus Q2). Works partially addressing Q2 are the following: MSDA~\cite{scaman17optimal} uses multiple communication steps to achieve the lower complexity bound of \eqref{prob:dop_nonsmooth_same} when $G=0$; and the algorithms in \cite{van2019distributed} and \cite{li2017decentralized} achieve linear rate and can adjust the number of communications performed at each iteration to match the rate of the centralized gradient descent. However it is not clear how to extend (if possible) these methods and their convergence analysis to the more general composite (i.e., $G\neq 0$) setting \eqref{prob:dop_nonsmooth_same}. \\\indent This paper aims at addressing Q1 and Q2 in the general setting \eqref{prob:dop_nonsmooth_same}. Our major contributions are the following: 1) We propose a general primal-dual distributed algorithmic framework that subsumes several existing ATC- and CTA-based distributed algorithms; 2) A sharp linear convergence rate is proved (when $G\neq 0$) developing an operator contraction-based analysis. By product, our convergence results apply also to the algorithms in \cite{shi2015extra,li2017decentralized,yuan2018exact_p1,qu2017harnessing,di2016next,xu2015augmented}, which so far have been studied in isolation; 3) For ATC forms of our schemes, the dependencies of the linear rate on the agents' functions and the network topology are {\it decoupled}, matching the typical rates for the centralized optimization and the consensus averaging. This is a major departure from existing analyses, which do not show such a clear separation, and complements the results in \cite{li2017decentralized} applicable only to smooth instances of \eqref{prob:dop_nonsmooth_same}. Furthermore, convergence is established under a proper choice of the step-size, whose upper bound does {\emph not} depend on the network parameters but only on the optimization ones (Lipschitz constants of the gradients and strongly convexity constants); and 4) The proposed scheme can naturally adjusts the ratio between the number of communications and computations to achieve the {\it same} rate of the centralized proximal gradient scheme (in terms of computations). Chebyshev acceleration can also be employed to significantly reduce the number of communication steps per computation. Because of space limitation, all the proofs are available as supporting material in the technical report \cite{XuSunScutariJ}.\\ \begin{comment} \begin{itemize} \item [1)] \alert{A general algorithmic framework proposed based on operator splitting methods, which subsumes many existing algorithms in the literature; the proposed general scheme naturally allows for multiple communication steps at each iteration, facilitating acceleration mechanism;} \item [2)] \alert{Chebyshev acceleration is empoyed to significantly reduce the number of communication steps needed to balance computation-communication so as to achieve the same rate of centralized proximal-gradient method in terms of worst case (proximal-)gradient evaluations; this represents the first result for Problem \eqref{prob:dop_nonsmooth_same}, which complements~\cite{li2017decentralized,van2019distributed};} \item [3)] \alert{A sharp linear convergence result is obtained based on a unified operator contraction analysis; it provides a unified rate for \cite{shi2015extra,li2017decentralized,yuan2018exact_p1,qu2017harnessing,di2016next,xu2015augmented}, which so far have been studied separately; the analysis is different in its nature from most existing primal-dual and small gain methods which usually do not provide tight bounds, especially for Problem~\eqref{prob:dop_nonsmooth_same}.} \end{itemize} \end{comment} \noindent {\bf Notations}: $\mathbb{N}_+$ is the set of positive integer numbers; $\mathbb{S}^m$ is the set of $\mathbb{R}^{m\times m}$ symmetric matrices while $\mathbb{S}_+^m$ (resp. $\mathbb{S}_{++}^m$) is the set of positive semidefinite (resp. definite) matrices in $\mathbb{S}^m$. $\mathbb{P}_K$ denotes the set of (real) monic polynomials of order $K$. {Unless otherwise indicated, column vectors are denoted by lower-case letters while upper-case letters are used for matrices (with the exception of $L$ in Assumption 1 to conform with conventional notation). The symbols $1_m$ and $0_m$ denote the $m$-length column vectors of all ones and all zeros, respectively. The $\mathbf{0}_m$ denotes the $m\times m$ zero matrix; $I_m$ denotes the identity matrix in $\mathbb{R}^{m\times m}$; $\J\triangleq 1_m\,1_m^\top/m$ is the projection matrix onto $1_m$.} With a slight abuse of notation, $I$ will denote either the identity matrix or the identity operator on the space under consideration. We use $\Null{\cdot}$ [resp. $\Span{\cdot}$] to denote the null space (resp. range space) of the matrix argument. For any $X,Y\in\mathbb{R}^{m\times d}$, let $\innprod{X}{Y}\triangleq\text{trace}(X^\top Y)$ while we write $\norm{X}$ for $\norm{X}_F$; the same notation is used for vectors, treated as special cases. Given ${G}\in \mathbb{S}_+^n$, $\innprod{X}{Y}_\G\triangleq\innprod{\G\x}{\y}$ {and} $\norm{\x}_\G\triangleq\sqrt{\innprod{\x}{\x}_{\G}}.$ The eigenvalues of a symmetric matrix ${A}\in \mathbb{R}^{m\times m}$ are denoted by $\lambda_i(\A)$, $i=1,\ldots, m$, and arranged in increasing order. For $x\in \mathbb{R},$ we denote $x_+ = \max(x,0).$ \section{Problem Statement} We study Problem~\eqref{prob:dop_nonsmooth_same} under the following assumption. \begin{assum}\label{assum:Lipschiz_gradient} Each local cost function $f_i:\mathbb{R}^d\rightarrow\mathbb{R}$ is $\mu$-strongly convex and $L$-smooth; and $G:\mathbb{R}^d\rightarrow\mathbb{R}\cup\{\pm\infty\}$ is proper, closed and convex. Define $\kappa\triangleq L/\mu$. \end{assum} Note that Assumption \ref{assum:Lipschiz_gradient} also accounts for the case where $f_i$ is convex and $G$ is $\mu$-strongly convex.\\ \noindent {\bf Network model:} Agents are embedded in a network, modeled as an undirected, static graph $\Gh=(\Vx,\Eg)$, where $\Vx$ is the set of nodes (agents) and $\{i,j\}\in\Eg$ if there is an edge (communication link) between node $i$ and $j$. We make the blanket assumption that $\Gh$ is connected. We introduce the following matrices associated with $\mathcal{G}$, which will be used to build the proposed distributed algorithms. \begin{dfn}[Gossip matrix] \label{dfn:weight_matrix} A matrix $\W\triangleq [W_{ij}]\in \mathbb{R}^{m\times m}$ is said to be compliant to the graph $\Gh=(\Vx,\Eg)$ if $W_{ij} \neq 0$ for $\{i,j\}\in\Eg$, and $W_{ij}=0$ otherwise. The set of such matrices is denoted by $\mathcal{W}_\mathcal{G}$. \end{dfn} \begin{dfn}[$K$-hop gossip matrix] \label{dfn:k-hop_weight_matrix} Given $K\in\mathbb{N}_+,$ a matrix $ {W}'\in\mathbb{R}^{m\times m}$ is said to be a $K$-hop gossip matrix associated to $\Gh=(\Vx,\Eg)$ if $ {W}'=P_K(\W)$, for some $\W\in\mathcal{W}_\Gh$, where $P_K(\cdot)\in\mathbb{P}_K$. \end{dfn} Note that, if $\W\in \mathcal{W}_\Gh$, using $W_{ij}$ to linearly combine information between agent $i$ and $j$ corresponds to performing a single communication between the two agents ($i$ and $j$ are immediate neighbors). Using a $K$-hop matrix ${W}'=P_K(\W)$ requires instead $K$ consecutive rounds of communications among immediate neighbors for the aforementioned weighting process to be implemented in a distributed way (note that the zero-pattern of $ {W}'$ is in general not compliant with $\mathcal{G}$). $K$-hop weight matrices are crucial to employ acceleration of the communication step, which will be a key ingredient to exploit the tradeoff between communications and computations (cf.~Sec.~\ref{sec:tradeoff}).\\ \noindent{\textbf{A saddle-point reformulation:}} Our path to design distributed solution methods for \eqref{prob:dop_nonsmooth_same} is to solve a saddle-point reformulation of \eqref{prob:dop_nonsmooth_same} via general proximal splitting algorithms that are implementable over $\mathcal{G}$. Following a standard path in the literature, we introduce local copies $x_i\in \mathbb{R}^d$ (the $i$-th one is owned by agent $i$) of $x$ and functions \begin{equation} f(\x)\triangleq\sum_{i=1}^mf_i(x_i)\quad \text{and}\quad g(\x)\triangleq\sum_{i=1}^m G(x_i), \end{equation} with $\x\triangleq [x_1,\ldots, x_m]^\top\in\mathbb{R}^{m\times d}$; \eqref{prob:dop_nonsmooth_same} can be then rewritten as \begin{equation}\label{prob:dop_nonsmooth_same_augmented} \min_{\x\in\mathbb{R}^{m\times d}} f(\x)+g(\x),~{\text{s.t.}}~ \sqrt{{C}}\x=\zeros, \end{equation} where $C$ satisfies the following assumption: \begin{assum}\label{assum:cond_C} $C \in \mathbb{S}^m_+$ and $\Null{{C}}=\Span{1}$. \end{assum} Under this condition, the constraint $\sqrt{ {C}}\x=\zeros$ enforces a consensus among $x_i$'s and thus \eqref{prob:dop_nonsmooth_same_augmented} is equivalent to \eqref{prob:dop_nonsmooth_same}. \\ \indent In the setting above, \eqref{prob:dop_nonsmooth_same_augmented} is equivalent to its KKT conditions: there exists $\x^\star \in \mathcal{S}_{\text{KKT}}$, where $\mathcal{S}_{\text{KKT}}$ is defined as \begin{align}\label{eq:kkt_conditions} \mathcal{S}_{\texttt{KKT}} \triangleq & \left\{ \x \in \mathbb{R}^{m\times d} \, \big\vert\, \exists\, \y \in \mathbb{R}^{m\times d} \text{ such that} \right.\nonumber\\ & \left. \sqrt{{C}}\x=\zeros, \quad \nabla f(X)+ \sqrt{{C}}\y \in-\partial g(\x)\right\}, \end{align} where $\nabla f(\x)\triangleq [\nabla f_1(x_1),\nabla f_2(x_2),...,\nabla f_m(x_m)]^\top$ and $\partial g(\x)$ denotes the subdifferential of $g$ at $\x$. We have the following. \begin{comment} \begin{lem}\label{lemma_eq_KKT} Consider Problems \eqref{prob:dop_nonsmooth_same} and \eqref{prob:dop_nonsmooth_same_augmented} [along with its KKT conditions (\ref{eq:kkt_conditions})] under Assumptions 1 and 2; $x^\star\in \mathbb{R}^d$ is an optimal solution of \eqref{prob:dop_nonsmooth_same} if and only if there exists $\y'^\star\in \mathbb{R}^{m\times d}$ such that $(x^\star \,\mathbf{1},\y'^\star)$ satisfies \eqref{eq:kkt_conditions}. \end{lem} \end{comment} \begin{lem}\label{lemma_eq_KKT} Consider Problem \eqref{prob:dop_nonsmooth_same} under Assumptions 1 and 2; $x^\star\in \mathbb{R}^d$ is an optimal solution of \eqref{prob:dop_nonsmooth_same} if and only if ${1}_m x^{\star\top} \in \mathcal{S}_{\texttt{KKT}}$. \end{lem} Building on Lemma \ref{lemma_eq_KKT}, in the next section, we propose a general distributed algorithm for \eqref{prob:dop_nonsmooth_same} based on a suitably defined operator splitting solving the KKT system \eqref{eq:kkt_conditions}. \begin{comment} To solve Problem~\eqref{prob:dop_nonsmooth_same}, we consider the following saddle-point formulation: \begin{equation}\label{prob:dop_nonsmooth_same_saddle_point} \min_{\x\in\mathbb{R}^{m\times d}} \max_{\y\in\mathbb{R}^{m\times d}}f(\x)+g(\x)+\innprod{\x}{\y}-\iota_{\mathcal{C}^\perp}(\y), \end{equation} where $\y\in \mathbb{R}^{m\times d}$ is the vector of dual variable. \alert{Note that, differently from more classical saddle-point reformulations of \eqref{prob:dop_nonsmooth_same} used in the literature of distributed optimization \textcolor{red}{[ref]}, the proposed one is ``graph-independent''--\eqref{prob:dop_nonsmooth_same_saddle_point} does not contain any matrix compatible with $\mathcal{G}$ that enforces consensus on the local copies $x_i$'s. This will lead to more general distributed algorithms.} \\\indent By Assumption~\ref{assum:solution_set}, strong duality holds and \eqref{prob:dop_nonsmooth_same_saddle_point} is equivalent to its KKT conditions: there exists $(\x^\star,\y^\star)\in \mathbb{R}^{m\times d}\times \mathbb{R}^{m\times d}$ satisfying \begin{equation}\label{eq:kkt_conditions} \x^\star\in\Span{\ones},~\ones^\top\y^\star=0,~\nabla f(\x^\star)+\y^\star\in-\partial g(\x^\star), \end{equation} where $\nabla f(\x)\triangleq [\nabla f_1(x_1)^\top,\nabla f_2(x_2)^\top,...,\nabla f_m(x_m)^\top]^\top$; $\mathcal{C}^\perp$ is the orthogonal complement to $\mathcal{C}$; and $\partial g(\mathbf{x})$ denotes the subdifferential of $g$ at $\x$. A direct inspection of the KKT conditions of \eqref{prob:dop_nonsmooth_same} and \eqref{eq:kkt_conditions} shows that \eqref{prob:dop_nonsmooth_same} and \eqref{prob:dop_nonsmooth_same_saddle_point} are equivalent in the following sense. \begin{lem}\label{lemma_eq_KKT} Consider Problems \eqref{prob:dop_nonsmooth_same} and \eqref{prob:dop_nonsmooth_same_saddle_point} under Assumptions 1 and 2; $x^\star\in \mathbb{R}^d$ is an optimal solution of \eqref{prob:dop_nonsmooth_same} if and only if there exists $\mathbf{y}^\star\in \mathbb{R}^{m\times d}$ such that $(x^\star \,\mathbf{1},\mathbf{y}^\star)$ satisfies \eqref{eq:kkt_conditions}. \end{lem} We can then focus on Problem \eqref{prob:dop_nonsmooth_same_saddle_point} without loss of generality. In the next section we propose a general primal-dual proximal gradient algorithm solving \eqref{prob:dop_nonsmooth_same_saddle_point}, which is implementable on the network $\mathcal{G}$. \end{comment} \begin{comment} ...... It is not difficult to see that, by introducing local copies, the above problem \eqref{prob:dop_nonsmooth_same} is equivalent to \begin{equation}\label{prob:dop_nonsmooth_same_augmented} \min_{\x\in\mathbb{R}^{m\times d}} f(\x)+g(\x)+\iota_\mathcal{C}(\x), \end{equation} where $f(\x)\triangleq\frac{1}{m}\sum_{i=1}^mf_i(x_i)$, $g(\x)\triangleq\frac{1}{m}\sum_{i=1}^mG(x_i)$ and $\iota_\mathcal{C}$ is the indicator function defined as \[ \iota_\mathcal{C}(\x)\triangleq \begin{cases} 0,~\text{if}~\x \in\mathcal{C}\\ \infty,~\text{otherwise}. \end{cases} \] with $\mathcal{C}\triangleq\Span{\ones}$ representing the consensus space. From the optimality condition, we can easily obtain the optimal solution set of the problem~\eqref{prob:dop_nonsmooth_same_augmented} which can be defined as follows \[\mathcal{S}\triangleq\{\x\in\mathbb{R}^{m\times d}\|\x\in\Span{\ones},~\ones^\top(\nabla f(\x)+\q)=\zeros,~\q\in\partial g(\x)\} \] where $\nabla f(\x)=[\nabla f_1(x_1)\top,\nabla f_2(x_2)\top,...,\nabla f_m(x_m)\top]^\top$ is the collective gradient vector; $\q$ is the subgradient vector of $g$ evaluated at $\x$ and $\partial g$ is the subdifferential at $\x$. \end{comment} \section{A General Primal-Dual Proximal Algorithm}\label{sec:alg-des} The proposed general primal-dual proximal algorithm reads \begin{subequations}\label{alg:g-pd-ATC} \begin{align} \x^k&=\prox{\gamma g}{\z^k}, \label{alg:g-pd-ATC_z}\\ \z^{k+1}&=\A\x^k-\gamma\B\nabla f(\x^k)-\y^k, \label{alg:g-pd-ATC_x}\\ \y^{k+1}&=\y^k+{\C}\z^{k+1}, \label{alg:g-pd-ATC_y} \end{align} \end{subequations} with $\z^0\in \mathbb{R}^{m\times d}$ and $\y^0\in {\texttt{span}}({C})$. In \eqref{alg:g-pd-ATC_z}, $\prox{\gamma g}{\x}\triangleq\text{arg}\min_{\y}g(\y)+\frac{1}{2\gamma}\norm{\x-\y}^2$ is the standard proximal operator, which accounts for the nonsmooth term. Eq. \eqref{alg:g-pd-ATC_z} represents the update of the primal variables, where $\A,\B\in \mathbb{R}^{m\times m}$ are suitably chosen weight matrices, and $\gamma>0$ is the step-size. Finally, \eqref{alg:g-pd-ATC_y} represents the update of the dual variables. Note that there is no loss of generality in initializing $\y^0\in {\texttt{span}}({C})$, as any $\y$ in (\ref{eq:kkt_conditions}) is so (unless all the $f_i$ share a common minimizer). \\\indent Define the set $\mathcal{S}_{\texttt{Fix}} \triangleq \big\{ \x \in \mathbb{R}^{m\times d} \, \big\vert\, \C\x=0 \text{ and } 1^\top(I-\A)\x+\gamma \, 1^\top\B\nabla f(\x)\in - \gamma\, 1^\top\partial g(\x) \big\}$. It is not difficult to check that any fixed point $(X^\star, Z^\star, Y^\star)$ of Algorithm \eqref{alg:g-pd-ATC} satisfies $X^\star \in \mathcal{S}_{\texttt{Fix}}$. The following are {\it necessary} and sufficient conditions on $\A$ and $\B$ for $X^\star \in \mathcal{S}_{\texttt{Fix}}$ to be the solution of \eqref{prob:dop_nonsmooth_same_augmented}. \begin{assum}\label{assum:cond_A_B} The weight matrices $\A,\B \in \mathbb{R}^{m\times m}$ satisfy: $1^\top\A \,1=m$, and $1^\top\B = 1^\top$. \end{assum} \begin{lem}[\!\cite{XuSunScutariJ}] Under Assumption~\ref{assum:cond_C}, $\mathcal{S}_\texttt{KKT}=\mathcal{S}_\texttt{Fix}$ if and only if $\A,\B$ satisfy Assumption~\ref{assum:cond_A_B}. \end{lem} \subsection{Connections with existing distributed algorithms} \label{Sec:special-cases} Algorithm~\eqref{alg:g-pd-ATC} contains a gamut of distributed (and centralized) schemes, corresponding to different choices of the weight matrices $\A,\B,$ and $\C$; any $A,B,C\in \mathcal{W}_{\mathcal{G}}$ leads to distributed implementations. The use of general matrices $\A$ and $\B$ (rather the more classical choices $\A=\B$ or $\B=I$) permits to model for the first time in a unified algorithmic framework both ATC- and CTA-based updates; this includes several existing distributed algorithms proposed for special cases of (P), as briefly discussed next; see \cite{XuSunScutariJ} for more examples. Rewrite Algorithm~\eqref{alg:g-pd-ATC} in the following equivalent form: \begin{equation}\label{eq:g-pd-ATC_eliminate_y} \z^{k+2}=(\I-\C)\z^{k+1}+\A(\x^{k+1}-\x^k)-\gamma\B(\nabla f(\x^{k+1})-\nabla f(\x^k)). \end{equation} When $G=0$, the above update reduces to \begin{equation}\label{eq:g-pd-ATC_eliminate_y_G=0} \x^{k+2}=(\I-\C+\A)\x^{k+1}-\A\x^k-\gamma\B(\nabla f(\x^{k+1})-\nabla f(\x^k)). \end{equation} It is not difficult to check that the schemes in \cite{shi2015extra,li2017decentralized,yuan2018exact_p1,di2016next,xu2015augmented,nedich2016achieving,qu2017harnessing,jakovetic2018unification,mansoori2019general,alghunaim2019linearly} are all special cases of \eqref{eq:g-pd-ATC_eliminate_y} or \eqref{eq:g-pd-ATC_eliminate_y_G=0} and thus of Algorithm \eqref{alg:g-pd-ATC}--Table \ref{tab:connections_to_existing_algs} shows the proper parameter setting to establish the equivalence, where $\W\in \mathcal{W}_\mathcal{G}$ is the weight matrix used in the target distributed algorithms, see \cite{XuSunScutariJ} for more details. \begin{table} \renewcommand{1.2}{1.2} \scriptsize\centering \setlength\tabcolsep{2pt} \begin{tabu}{lc} {\small \bf Algorithm} &$\A~~\|~~\B~~\|~~\C$\\ \tabucline[1.2pt]\\ EXTRA~\cite{shi2015extra} & $\frac{1}{2}(\I+\W)~~\|~~\I~~\|~~\frac{1}{2}(\I-\W)$\\ NIDS~\cite{li2017decentralized}/Exact Diffusion~\cite{yuan2018exact_p1} & $\frac{1}{2}(\I+\W)~~\|~~\frac{1}{2}(\I+\W)~~\|~~\frac{1}{2}(\I-\W)$ \\ NEXT~\cite{di2016next}/AugDGM~\cite{xu2015augmented} & $\W^2~~\|~~\W^2~~\|~~(\I-\W)^2$ {} \\ DIGing~\cite{nedich2016achieving}/ \cite{qu2017harnessing} & $\W^2~~\|~~\I~~\|~~(\I-\W)^2$ {} \\ \cite{jakovetic2018unification} & $b\W^2+(1-b)\W~~\|~~\I~~\|~~b\W^2-(1+b)\W+\I$ {}\\ \cite{mansoori2019general} &$\W^K~~\|~~\sum_{i=1}^{K-1}{\W^i}~~\|~~\W-\W^K$ {}\\ \hline \cite{alghunaim2019linearly} ($G\neq 0$) &$\W~~\|~~\I~~\|~~\alpha(\I-\W)$ \\ \end{tabu}\caption{Connections with existing distributed algorithms. All the schemes but ours and \cite{alghunaim2019linearly} apply only to \eqref{prob:dop_nonsmooth_same} with $G=0$.} \label{tab:connections_to_existing_algs} \end{table} \section{Convergence Analysis} We establish linear rate of Algorithm \eqref{alg:g-pd-ATC} under the following assumption (along with Assumption \ref{assum:cond_A_B}).\begin{comment} \begin{assum}\label{assum:conditions_convergence} Let $\mathbf{E}\triangleq\I-\C-\A$. Define $p(\A,\C,\sigma)\triangleq 1/\lambda_{\min}(\I+\sigma\sqrt{\C}\A^{-1}\sqrt{\C}-\J)$. We assume \begin{equation}\label{eq:conditions_rate} \left \{\begin{aligned} &\C\succeq\frac{\sigma}{1-\sigma}\mathbf{E}\A^{-1}\mathbf{E},~p(\A,\C,\sigma)<1,\,\,\text{for}~\sigma\in[0,1) & \text{if}\quad\mathbf{E}\neq\zeros;\\ &p(\A,\C,1)<1,\,\, &\text{if}\quad\mathbf{E}=\zeros. \end{aligned}\right.\end{equation} \end{assum}\end{comment} \begin{assum}\label{assum:conditions_convergence} The weight matrices $A\in \mathbb{R}^{m\times m},\, {B\in \mathbb{S}^{m}}$ and $C\in \mathbb{S}_+^{m} $ satisfy: i) $A = BD$ for some $-I \prec D \preceq I$; ii) $0 \prec I-C$; iii) $B$ and $C$ commute; and iv) $B^2 \prec \frac{(L+\mu)^2}{\left( L\lambda_{\text{max}}(D)- \mu\lambda_{\text{min}}(D) \right)^2} (I-C) $. \end{assum} Assumption~\ref{assum:conditions_convergence} together with Assumption 3 are quite mild and satisfied by a variety of algorithms; for instance, this is the case for all the schemes in Table I (see \cite{XuSunScutariJ} for more details). In particular, the commuting property is trivially satisfied when $B, C\in P_K(W)$, for some given $W\in \mathcal{W}_{\mathcal G}$ (as in Table I). Also, one can show that condition iv) is {\it necessary} to achieve linear rate. \begin{thm} \label{thm:contraction_T_c_T_f} Consider Problem~\eqref{prob:dop_nonsmooth_same} under Assumption~\ref{assum:Lipschiz_gradient}, whose optimal solution is $x^\star$. Let $\{(\x^k,\z^k,\y^k)\}_{k\geq 0}$ be the sequence generated by Algorithm \eqref{alg:g-pd-ATC} under Assumptions \ref{assum:cond_C} and \ref{assum:cond_A_B} and step-size \begin{align} & \frac{1}{\mu} \left( \lambda_{\max}(D) - \lambda_{\max}\left( B^2(I-C)^{-1}\right)^{- {1}/{2}}\right)_+ <\gamma \\ & \quad < \frac{1}{L} \left( \lambda_{\min}(D) + \lambda_{\max}\left( B^2(I-C)^{-1}\right)^{- {1}/{2}}\right). \end{align} Then $\norm{\x^k-1x^{\star\top}}^2=\mathcal{O}(\lambda^k)$, with \begin{equation} \label{def_lambda} \lambda \triangleq \max \left(q^2 \lambda_{\max}(B^2 (I-C)^{-1}), ~1-\lambda_2(C) \right)<1, \end{equation} and \begin{align}\label{eq:def_q} q \triangleq & \max\left(\abs{\lambda_{\min}(D)-\gamma L}, ~ \abs{\lambda_{\max}(D)-\gamma \mu} \right) . \end{align} The optimal step-size is $\gamma^\star \triangleq \frac{\lambda_{\max}(D)+\lambda_{\min}(D)}{L+\mu}$ leading to the smallest $q^\star \triangleq \frac{L\lambda_{\max}(D)- \mu\lambda_{\min}(D)}{L+\mu}$, and thus the optimal rate. \end{thm} \iffalse \textcolor{red}{\[ V^{k+1}\leq\lambda V^k, \quad with \quad \lambda\triangleq\max\left\{p(\A,\C,\sigma),\frac{q(\A,\B)^2}{\eta(\B^{-1}\A)}\right\} \] and \begin{equation}\label{eq:contraction-factors} \hspace{-0.3cm}\begin{cases} \begin{aligned} &p(\A,\C,\sigma)\triangleq\frac{1}{\lambda_2(\I+\sigma\sqrt{\C}\A^{-1}\sqrt{\C})}& \\ &q(\A,\B)\triangleq\frac{\kappa-\eta(\B^{-1}\A)}{\kappa+\eta(\B^{-1}\A)},\,\,\,\eta(\B^{-1}\A)\triangleq\frac{\lambda_{\min}(\B^{-1}\A)}{\lambda_{\max}(\B^{-1}\A)}. \end{aligned} \end{cases} \end{equation}} \fi \begin{col} \label{col:contraction_overall_optimal_choice} Under the same setting as Theorem~\ref{thm:contraction_T_c_T_f}, let $B^2\preceq I-C$ and $\A=\B$, so that $D=I,~\gamma^\star =\frac{2}{L+\mu}$. Then, the rate reduces to \begin{equation}\label{rate-separation} \lambda=\max\left\{ \bracket{\frac{\kappa-1}{\kappa+1}}^2, ~ 1-\lambda_2(\C) \right\}. \end{equation} \end{col} Note that the lower bound condition on the step-size in Theorem \ref{thm:contraction_T_c_T_f} nulls when $B^2(I-C)^{-1}\preceq I$ (since $\lambda_{\max}(D)=1$). Theorem \ref{thm:contraction_T_c_T_f} and Corollary \ref{col:contraction_overall_optimal_choice} provide a unified set of convergence conditions for CTA- and ATC-based distributed algorithms. We refer to \cite{XuSunScutariJ} for a detailed discussion of several special instances. Here, we mainly comment Algorithm \eqref{alg:g-pd-ATC} in the setting of Corollary \ref{col:contraction_overall_optimal_choice}. This special instance enjoys two desirable properties, namely: \textbf{(i) rate-separation:} The rate (\ref{rate-separation}) is determined by the worst rate between the one due to the communication $[1-\lambda_2(\C)]$ and that of the optimization $[((\kappa-1)/(\kappa+1))^2]$. This separable structure is the key enabler for our distributed scheme to achieve the convergence rate of the centralized proximal gradient algorithm applied to Problem~\eqref{prob:dop_nonsmooth_same}--see Sec.~\ref{sec:tradeoff}; and \textbf{(ii) network-independent step-size:} The step-size in Corollary \ref{col:contraction_overall_optimal_choice} does not depend on the network parameters but only on the optimization and its value coincides with the optimal step-size of the centralized proximal-gradient algorithm. This is a major advantage over current distributed schemes applicable to \eqref{prob:dop_nonsmooth_same} (with $G\neq 0$) and complements the results in \cite{li2017decentralized}, whose algorithm however cannot deal with the non-smooth term $G$ and use a non-optimal step-size. \begin{comment} \textcolor{blue}{........... REWRITE MAYBE MOVE AFTER THEOREM\\ Several choices are possible for such matrices. For instance, one can set $\A=\B=\W$ and $\C=\I-\W$, with $\W\in\mathcal{W}_\mathcal{G}$, which corresponds to one round of communication per iteration among neighboring nodes. When the graph is not well connected and communication cost is relative low, multiple communication steps might be preferred. This can be accommodated, e.g., choosing $\A=\B=\W^K$ and $\C=\I-\W^K$, where $K$ is the number of communication rounds carried out at each iteration. To further accelerate information mixing one can also use polynomial filters to build the weight matrix; a widely used class is the Chebyshev polynomials~\cite{auzinger2011iterative}, corresponding to $\A=\B=P_K(\W)$ and $\C=\I-P_K(\W)$, with $P_K(1)=1$ to ensure the doubly stochasticity of $\A$ and $\B$, and $P_K\in\mathbb{P}_K$ is certain $K$-order polynomial. Note that $P_K$ is chosen so that the second largest eigenvalue $\lambda_{m-1}(P_K(\W))$ is minimized, attaining the fastest linear mixing over $\mathcal{G}$ (cf. Sec.~\ref{sec:tradeoff} and \cite[Corollary 6.3]{auzinger2011iterative}). } \end{comment} \section{ Communication and computation trade-off} \label{sec:tradeoff} In this section we build on the rate separation property in Corollary 4 to show how to choose the matrices $A,\,B$ and $C$ to achieve the same rate of the centralized proximal gradient algorithm, possibly using multiple (finite) rounds of communications. \\\indent Note that $\rho_{\texttt{opt}}\triangleq (\kappa-1)/(\kappa+1)$ is the rate of the centralized proximal-gradient algorithm applied to Problem~\eqref{prob:dop_nonsmooth_same}, under Assumption 1. This means that if the network is ``well connected'', specifically $1-\lambda_2(\C)\leq \rho_{\texttt{opt}}^2$, the proposed algorithm with the choice of $A,\,B$ and $C$ under consideration already converges at the \emph{desired} linear rate $\rho_{\texttt{opt}}$. On the other hand, when $1-\lambda_2(\C)>\rho_{\texttt{opt}}^2$, one can still achieve the centralized rate $\rho_{\texttt{opt}}$ by enabling multiple (finite) rounds of communications per proximal gradient evaluations. We discuss next two strategies to reach this goal, namely: 1) performing multiple rounds of plain consensus using each time the same weight matrix; and 2) employing acceleration via Chebyshev polynomials. \\\noindent\textbf{1) Multiple rounds of consensus:} Given a weight matrix $W\in \mathcal{W}_{\mathcal{G}}$ (i.e., compatible with $\mathcal{G}$), we consider two possible choices of $\A,\B,\C$ satisfying Corollary \ref{col:contraction_overall_optimal_choice} and leading to distributed algorithms. \textbf{Case 1:} Suppose $W\in\mathbb{S}^m_{++}$. We set $\A=\B=\I-\C=W$, which implies $\B^2\preceq \I-\C$ (cf. Corollary~\ref{col:contraction_overall_optimal_choice}). The resulting algorithm implemented using (\ref{eq:g-pd-ATC_eliminate_y}) or (\ref{eq:g-pd-ATC_eliminate_y_G=0}) will require one communication exchange per gradient evaluation. Note that this setting subsumes most existing primal-dual methods such as NIDS~\cite{li2017decentralized}/Exact Diffusion~\cite{yuan2018exact_p1}. If $W$ in the setting above is replaced by $\W^K$, with $K>1$, this corresponds to run $K$ rounds of consensus per computation, each round using $\W$. Denote $\rho_{\texttt{com}}\triangleq\lambda_{\max}(\W-\J)$; we have $1-\lambda_2(\C)=\lambda_{\max}(W^K-\J)=\rho^K_{\texttt{com}}$. The value of $K$ is chosen to minimize the resulting rate $\lambda$ [cf. (\ref{rate-separation})], i.e., such that $\rho_{\texttt{com}}^K\leq\rho_{\texttt{opt}}^2$, which leads to $K=\lceil\log_{\rho_{\texttt{com}}}({\rho^2_{\texttt{opt}}})\rceil$. \textbf{Case 2:} Consider now the case $\W\in\mathbb{S}^m$ and $\text{det}(W)\neq 0$. We can set $\A^2=\B^2=\I-\C=W^2$, so that Corollary~\ref{col:contraction_overall_optimal_choice} still applies. With this choice, every update in (\ref{eq:g-pd-ATC_eliminate_y}) or (\ref{eq:g-pd-ATC_eliminate_y_G=0}) will call for two communication exchanges per gradient evaluation. To reach the centralized rate $\rho_{\texttt{opt}}^2$, the optimal $K$ can be still found as $1-\lambda_2(\C)=(\lambda_{\max}(\A^{2K}-\J))=(\lambda_{\max}(\A-\J))^{{2K}}\leq \rho_{\texttt{opt}}^2$.\\ \noindent \textbf{2) Chebyshev acceleration:} To further reduce the number of communication steps, we can leverage Chebyshev acceleration~\cite{auzinger2011iterative}. Specifically, in the setting of Case 2 above, we set $\A=\P_K(\W)$ and $P_K(1)=1$ (the latter is to ensure the double stochasticity of $\A$), with $P_K\in\mathbb{P}_K$. This leads to $1-\lambda_2(C)=\lambda_{\max}(\A^2-\J)$. {The idea of Chebyshev acceleration is to find the ``optimal'' polynomial $P_K$ such that $\lambda_{\max}(\A^2-\J)$ is minimized, i.e., $\rho_C\triangleq\min_{P_K\in \mathbb{P}_K,P_K(1)=1}\max_{t\in[-\rho_{\texttt{com}},\rho_{\texttt{com}}]} \abs{P_K(t)}$. The optimal solution of this problem is $P_K(x)={T_K(\frac{x}{\rho_{\texttt{com}}})}/{T_K(\frac{1}{\rho_{\texttt{com}}})}$ \cite[Theorem 6.2]{auzinger2011iterative}, with $\alpha'=-\rho_{\texttt{com}}$, $\beta'=\rho_{\texttt{com}},\gamma'=1$ (which are certain parameters therein), where $T_K$ is the $K$-order Chebyshev polynomials that can be computed in a distributed manner via the following iterates \cite{auzinger2011iterative,scaman17optimal}: $T_{k+1}=2\xi T_k(\xi)-T_{k-1}(\xi),$ $k\geq 1$, with $T_0(\xi)=1$, $T_1(\xi)=\xi$. Also, invoking \cite[Corollary 6.3]{auzinger2011iterative}, we have $\rho_C=\frac{2c^K}{1+c^{2K}}$ where $c=\frac{\sqrt{\vartheta}-1}{\sqrt{\vartheta}+1},\vartheta=\frac{1+\rho_{\texttt{com}}}{1-\rho_{\texttt{com}}}$. Thus, the minimum value of $K$ that leads to $\rho_C\leq\rho^2_{\texttt{opt}}$ can be obtained as $K=\lceil\log_c^{ 1/\rho^2_{\texttt{opt}}+\sqrt{1/\rho^4_{\texttt{opt}}-1} }\rceil$. Note that to be used in the setting above, $A$ must be returned as nonsingular. \begin{figure}\label{fig_first_case} \label{fig_second_case} \label{fig_second_case} \label{fig_tradeoff} \end{figure} \\\indent In Fig.~\ref{fig_tradeoff} we plot the minimum number $K$ of communication steps needed to achieve the rate of the centralized gradient as a function of $\rho_{\texttt{com}}$ and $\rho^2_{\texttt{opt}}$. Since only one computation is performed per iteration, this adjusts the ratio between the number of communications and computations. We compare our algorithm in the setting of Case 2 above, using $\A=\W^K$ or Chebyshev acceleration $\A=P_K(\W)$, with the distributed scheme in \cite{van2019distributed}. The figure shows that (i) Chebyshev acceleration helps to reduce the number of communications to sustain a given rate; and (ii) when $\rho_{\texttt{opt}}$ is close to $1$ ($\kappa$ is ``large''), {both instances of the proposed scheme need much less communication steps to attain the centralized rate than that in \cite{van2019distributed}. More specifically, to match the rate $\rho_{opt}$, one needs to run at least $K$ number of communications such that: \[ \begin{aligned} \rho_{com}^K= \begin{cases} \rho_{opt}^2,&\text{[this work]};\\ \frac{\sqrt{1+\rho_{opt}}-\sqrt{1-\rho_{opt}}}{2}, &\text{\cite{van2019distributed}}. \end{cases} \end{aligned} \] When $\rho_{opt}\rightarrow 0$, we have $({\sqrt{1+\rho_{opt}}-\sqrt{1-\rho_{opt}}})/{2}\approx {\rho_{opt}}/{2}$. Thus, $\rho_{opt}^2\leq {\rho_{opt}}/{2}$, since $\rho_{opt}\ll{1}/{2}$; hence, the scheme in \cite{van2019distributed} needs less number of communications than the proposed algorithm in the aforementioned setting. On the other hand, when $\rho_{opt}\rightarrow 1$, we have $({\sqrt{1+\rho_{opt}}-\sqrt{1-\rho_{opt}}})/{2}\approx {\rho_{opt}}/{\sqrt{2}}$. In this case, $ {\rho_{opt}}/{\sqrt{2}}\leq {1}/{\sqrt{2}}\leq \rho_{opt}^2$; hence, our scheme require less communications than that in \cite{van2019distributed}. Moreover, since $({\sqrt{1+\rho_{opt}}-\sqrt{1-\rho_{opt}}})/{2}\leq {1}/{\sqrt{2}}<1$, when $\rho_{com}\rightarrow 1$, the scheme in \cite{van2019distributed} will need significantly more communication to match the centralized optimal rate. \section{Conclusion} We proposed a unified distributed algorithmic framework for composite optimization problems over networks; the algorithm includes many existing schemes as special cases. Linear rate was proved, leveraging a contraction operator-based anaysis. Under a proper choice of the design parameters, the rate dependency on the network and cost functions can be decoupled, which allowed us to determine the minimum number of communication steps needed to match the rate of centralized (proximal)-gradient methods. \appendix \section{Convergence Analysis} We provide here a sketch of the proof of Theorem \ref{thm:contraction_T_c_T_f}; see \cite{XuSunScutariJ} for more details. Assumptions 2 and 3 are tacitly assumed hereafter. \noindent {\bf Step 1: Auxiliary sequence and operator splitting:} Lemma \ref{lem-tranformation} below interpretes \eqref{alg:g-pd-ATC} as the fixed-point iterate of a suitably defined composition of contractive and nonexpansive operators. \begin{lem}[\!\cite{XuSunScutariJ}]\label{lem-tranformation} Given the sequence $\{(\z^k, \x^k, \y^k)\}_k$ generated by Algorithm \eqref{alg:g-pd-ATC}, define $U^k \triangleq [({\z}^k)^\top, ({\y}^k)^\top]^\top$. There holds \[ U^{k}= \begin{bmatrix}B & 0\\ 0 & B\sqrt{C} \end{bmatrix} \underset{\widetilde{U}^k}{\underbrace{\begin{bmatrix}\widetilde{\z}^{k}\\ \sqrt{C}\widetilde{\y}^{k} \end{bmatrix}}}, \] with $\{\widetilde{U}^k\}_k$ defined by the following dynamics \begin{align} \widetilde{U}^{k+1} = \underbrace{ \begin{bmatrix} (D- \gamma \nabla f)\circ \text{prox}_{\gamma g} \circ B & -\sqrt{C}\\ \sqrt{C}(D- \gamma \nabla f)\circ \text{prox}_{\gamma g} \circ B & I-C \end{bmatrix}}_{T} \widetilde{U}^k,\quad k\geq 1, \end{align} and the initialization $\widetilde{\z}^{1} = \widetilde{\y}^{1} = (D - \gamma \nabla f) (\x^{0})$. Furthermore, the operator $T$ can be decomposed as \begin{align} T= \underbrace{ \begin{bmatrix} I & -\sqrt{C}\\ \sqrt{C} & I-C \end{bmatrix}}_{\triangleq T_C} \underbrace{ \begin{bmatrix} D- \gamma \nabla f & 0\\ 0 & I \end{bmatrix}}_{\triangleq T_f} \underbrace{ \begin{bmatrix} \text{prox}_{\gamma g} & 0 \\ 0 & I \end{bmatrix}}_{\triangleq T_g} \underbrace{ \begin{bmatrix} B & 0 \\ 0 & I \end{bmatrix}}_{\triangleq T_B}, \end{align} where $T_C$ and $T_B$ are the operators associated with communications while $T_f$ and $T_g$ are the gradient and proximal operators, respectively. Finally, every fixed point $\widetilde{U}^\star\triangleq [\widetilde{Z}^\star,\sqrt{C}\widetilde{Y}^\star]$ of $T$ is such that $B \widetilde{Z}^\star= 1 x^{\star \top}\in \mathcal{S}_{\texttt{Fix}}$. \end{lem} Building on Lemma \ref{lem-tranformation}, the proof of Theorem \ref{thm:contraction_T_c_T_f} reduces to showing $\\| \widetilde{\z}^k- \widetilde{\z}^\star\\|=\mathcal{O}(\lambda^k)$. To do so, Step 2 below studies the contraction (nonexpansive) properties of single operators composing $T$ while Step 3 chains these properties showing that $T$ is {$\lambda$-contractive with respect to a suitable norm}. \noindent \textbf{Step 2: On the properties of $T_C$, $T_f$, $T_g$ and $T_B$.} We summarize next the main properties of the aforementioned operators; proofs of the results below can be found in \cite{XuSunScutariJ}. We will use the following notation: given $X \in \mathbb{R}^{2m\times d}$, we denote by $(X)_u$ and $(X)_\ell$ its upper and lower $m\times d$ matrix-block. \begin{lem} The operator $\T_c$ satisfies \label{lem:contraction_T_c} \[ \norm{\T_C\, X-\T_C \,Y }^2_{\Lambda_C} = \norm{X-Y }^2_{V_C},\quad \forall X,Y\in\mathbb{R}^{2m\times d}, \] where $\Lambda_C \triangleq \diag(\I-C, \I)$ and $V_C\triangleq\diag(\I,\I-C)$. \end{lem} \begin{comment} \begin{proof} \begin{align} \begin{bmatrix} I & \sqrt{C}\\ -\sqrt{C} & I-C \end{bmatrix} \begin{bmatrix} I-C & 0\\ 0 & I \end{bmatrix} \begin{bmatrix} I & -\sqrt{C}\\ \sqrt{C} & I-C \end{bmatrix} =\begin{bmatrix} I & 0\\ 0 & I-C \end{bmatrix}. \end{align} \end{proof}\end{comment} \begin{lem}\label{lem:contraction_T_f} With $q$ defined in in Th. \ref{thm:contraction_T_c_T_f}, $\T_f$ satisfies: $\forall X, Y \in\mathbb{R}^{2m\times d}$, \begin{equation} \norm{(\T_f \,X)_u - (\T_f\,Y)_u}^2\leq q^2 \norm{(X)_u- (Y)_u}^2\,\,\text{and}\,\, (\T_f\,X)_\ell = (X)_\ell. \end{equation} \end{lem} \begin{comment} \begin{proof}For $X \in \mathbb{R}^{m\times d}$, we have \begin{align} & DX-\gamma \nabla f(X) = X - \gamma \left( \nabla f(X) + \frac{1}{\gamma} (I-D)X \right) \\ & = X - \gamma \, \nabla \left( f(X) + \frac{1}{2\gamma} \norm{X}^2_{I-D} \right). \end{align} With the definition $f_D\triangleq f(X) + \frac{1}{2\gamma} \norm{X}^2_{I-D},$ $f_D$ is $L' \triangleq L+\frac{1}{\gamma} \lambda_{\text{max}}(I-D)$-smooth and $\mu' \triangleq \mu+\frac{1}{\gamma} \lambda_{\text{min}}(I-D)$-strongly convex. Then following similar proof of \cite[Lemma~10]{qu2017harnessing}, we have for $\gamma < \frac{2}{L'}$, i.e., $\gamma < \frac{1+\lambda_{\text{min}}(D)}{L}$, \begin{align} & \norm{(D-\gamma \nabla f)(X) - (D-\gamma \nabla f)(Y)} \\ \leq & \max\left((1-\gamma L'), ~ (1-\gamma \mu') \right) \norm{X-Y} \\ = & \max\left(\abs{\lambda_{\text{min}}(D)-\gamma L}, ~ \abs{\lambda_{\text{max}}(D)-\gamma \mu} \right) \norm{X-Y} . \end{align} Thus choosing $\gamma = \frac{\lambda_{\text{max}}(D)+\lambda_{\text{min}}(D)}{L+\mu}$, we get the the optimal rate as $q = \frac{L\lambda_{\text{max}}(D)- \mu\lambda_{\text{min}}(D)}{L+\mu}.$ Note that $q=\frac{L-\mu}{L+\mu}$ when $D=I$. \iffalse \begin{align} & \innprod{X-Y}{\nabla f_D(X) - \nabla f_D(Y)} \\ & \geq \frac{L' \mu'}{L'+ \mu'} \norm{X-Y}^2 + \frac{1}{L'+ \mu'} \norm{\nabla f_D(X) - \nabla f_D(Y)}^2. \end{align} Therefore \begin{align} & \norm{(D-\gamma \nabla f)(X) - (D-\gamma \nabla f)(Y)}^2 \\ = & \norm{X-Y}^2 - 2\gamma \innprod{X-Y}{\nabla f_D(X) - \nabla f_D(Y)}\\ & +\gamma^2 \norm{\nabla f_D(X) - \nabla f_D(Y)}^2 \\ \leq & \left( 1- \frac{2\gamma\,L' \mu'}{L'+ \mu'} \right) \norm{X-Y}^2 \\ & + \left( \gamma^2 - \frac{2\gamma}{L'+ \mu'}\right) \norm{\nabla f_D(X) - \nabla f_D(Y)}^2. \end{align} Thus when $\gamma = \frac{2}{L' + \mu'},$ i.e., $\gamma = \frac{\lambda_{\text{max}}(D)+\lambda_{\text{min}}(D)}{L+\mu}$, we get the rate \begin{align} q =& \frac{L'-\mu'}{L'+\mu'} = \left(L-\mu + (L+\mu) \frac{\lambda_{\text{max}}(D)-\lambda_{\text{min}}(D)}{\lambda_{\text{max}}(D)+\lambda_{\text{min}}(D)} \right) \bigg/\\ & \left( L+ \mu + (L+\mu) \frac{2-\lambda_{\text{max}}(D)-\lambda_{\text{min}}(D)}{\lambda_{\text{max}}(D)+\lambda_{\text{min}}(D)}\right). \end{align} (\alert{This result is exactly the same as that of the note using the operator analysis}) \fi \end{proof}\end{comment} \begin{lem} $\T_g$ satisfies: $\forall X, Y\in\mathbb{R}^{2m\times d}$, \label{lem:contraction_T_g} \begin{equation} \norm{(\T_g\, X)_u-(\T_g\, Y)_u}^2\leq \norm{(X)_u-(Y)_u}^2 \,\,\text{and}\,\, (\T_g\,X)_\ell = (X)_\ell. \end{equation} \end{lem} \begin{lem} The operator $\T_B$ satisfies: \label{lem:contraction_T_B} \begin{equation} \norm{(\T_B\, X)_u}^2 = \norm{(X)_u}_{B^2}^2,\quad (\T_g\,X)_\ell = (X)_\ell,\quad \forall X \in\mathbb{R}^{2m\times d}. \end{equation} \end{lem} \noindent \textbf{Step 3: Chaining Lemmata 6-9.} Define the matrices $Q_f \triangleq \diag(q^2 \I,\,\I)$ and $\Lambda_B=\diag(B^2, \I);$ the contraction property of $T$ are implied by the following chain: $\forall \, X, Y\in\mathbb{R}^{2m\times d}$ with $X_\ell,Y_\ell \in \Span{\sqrt{C}}$, \begin{align} & \norm{T\,X- T\,Y}_{\Lambda_C}^2 \stackrel{Lm.~\ref{lem:contraction_T_c}}{=}\norm{T_f \circ T_g \circ T_B\,(X- Y)}_{V_C}^2 \\ & \stackrel{Lm.~\ref{lem:contraction_T_f}}{\leq} \norm{ T_g \circ T_B\,(X- Y)}_{V_C Q_f}^2 \stackrel{Lm.~\ref{lem:contraction_T_g}}{\leq} \norm{ T_B\,(X- Y)}_{V_C Q_f}^2 \\ & \stackrel{Lm.~\ref{lem:contraction_T_B}}{=} \norm{X- Y}_{V_C Q_f \Lambda_B}^2 \stackrel{()}{\leq} \lambda \, \norm{X- Y}_{\Lambda_C}^2, \end{align} where $V_C Q_f \Lambda_B = \diag(q^2 B^2, I-C)$, $\lambda$ is defined in (\ref{def_lambda}); and () is due to the following two facts: i) $ q^2 \\|(Z)_u\\|^2_{B^2}=q^2 \\|(I-C)^{\frac{1}{2}}(Z)_u\\|^2_{B^2(I-C)^{-1}} \leq q^2 \lambda_{\text{max}} (B^2(I-C)^{-1})\\|(I-C)^{\frac{1}{2}}(Z)_u\\|^2 = q^2 \lambda_{\text{max}} (B^2(I-C)^{-1})\norm{(Z)_u}^2_{I-C},$ for all $\,(Z)_u \in \mathbb{R}^{m\times d}$; and ii) $X_\ell,Y_\ell \in \Span{\sqrt{C}}$. \end{document} \subsection{Temporary convergence analysis} From \eqref{alg:g-pd-ATC}, we have \[ \z^{k+1}=(\I-\C)\z^{k}+\A(\x^{k}-\x^{k-1})-\gamma\B(\nabla f(\x^{k})-\nabla f(\x^{k-1})). \] Applying the above recursively leads to \begin{align} & \z^{k+1} \\ = & \sum_{t=1}^k (\I-\C)^{k-t} \left( \A(\x^{t}-\x^{t-1})-\gamma\B(\nabla f(\x^{t})-\nabla f(\x^{t-1}) \right) \\ & + (\I-\C)^k \left( \A\x^{0} - \gamma\B \nabla f(\x^0) \right) \\ \stackrel{()}{=} & B \bigg(\sum_{t=1}^k (\I-\C)^{k-t} \left( D(\x^{t}-\x^{t-1})-\gamma(\nabla f(\x^{t})-\nabla f(\x^{t-1}) \right) \\ & + (\I-\C)^k \left( D\x^{0} - \gamma \nabla f(\x^0) \right) \bigg), \end{align} where $()$ is due to that $A = BD$ and $B$ commutes with $C.$ We define the following variable \begin{align} & \widetilde{\z}^{k+1} \\ \triangleq & \sum_{t=1}^k (\I-\C)^{k-t} \left( D (\x^{t}-\x^{t-1})-\gamma (\nabla f(\x^{t})-\nabla f(\x^{t-1}) \right) \\ & + (\I-\C)^k \left( D\x^{0} - \gamma \nabla f(\x^0) \right) \\ = & \sum_{t=1}^k (\I-\C)^{k-t} \left( (D - \gamma \nabla f) (\x^{t})- (D - \gamma \nabla f)(\x^{t-1}) \right) \\ & + (\I-\C)^k (D - \gamma \nabla f) (\x^{0}) \\ = & \sum_{t=0}^k (\I-\C)^{k-t} (D - \gamma \nabla f) (\x^{t}) \\ & - \sum_{t=0}^{k-1} (\I-\C)^{k-1-t} (D - \gamma \nabla f) (\x^{t}), \quad \forall k\geq 1;\\ & \widetilde{\z}^{1} = (D - \gamma \nabla f) (\x^{0}). \end{align} Then we have $\z^k = B \widetilde{\z}^k,\,\forall k \geq 1.$ We further define for $k\geq 0$ \begin{align} \widetilde{\y}^{k+1} \triangleq \sum_{t=1}^{k+1} \widetilde{\z}^t = \sum_{t=0}^k (\I-\C)^{k-t} (D - \gamma \nabla f) (\x^{t}). \end{align} It is clear from the definition of $\widetilde{\z}$ and $\widetilde{\y}$ that \begin{align} \begin{bmatrix} \widetilde{\z}^{k+1}\\ \widetilde{\y}^{k+1} \end{bmatrix} = \begin{bmatrix} (D- \gamma \nabla f)\circ \text{prox}_{\gamma g} \circ B & -C\\ (D- \gamma \nabla f)\circ \text{prox}_{\gamma g} \circ B & I-C \end{bmatrix} \begin{bmatrix} \widetilde{\z}^{k}\\ \widetilde{\y}^{k} \end{bmatrix} \end{align} Then for the following variable \[ U^k = \begin{bmatrix} \widetilde{\z}^k \\ \sqrt{C}\widetilde{\y}^k \end{bmatrix}, \] we have the dynamics \begin{align} U^{k+1} = \underbrace{ \begin{bmatrix} (D- \gamma \nabla f)\circ \text{prox}_{\gamma g} \circ B & -\sqrt{C}\\ \sqrt{C}(D- \gamma \nabla f)\circ \text{prox}_{\gamma g} \circ B & I-C \end{bmatrix}}_{T} U^k. \end{align} Then the transition matrix can be decomposed as four operators \begin{align} T= \underbrace{ \begin{bmatrix} I & -\sqrt{C}\\ \sqrt{C} & I-C \end{bmatrix}}_{\triangleq T_C} \underbrace{ \begin{bmatrix} D- \gamma \nabla f & 0\\ 0 & I \end{bmatrix}}_{\triangleq T_f} \underbrace{ \begin{bmatrix} \text{prox}_{\gamma g} & 0 \\ 0 & I \end{bmatrix}}_{\triangleq T_g} \underbrace{ \begin{bmatrix} B & 0 \\ 0 & I \end{bmatrix}}_{\triangleq T_B}. \end{align} \end{document} \section{Supporting Material} In this appendix, we collect the proofs of all the main results presented in the paper. \subsection{Supporting lemmas} \begin{lem}[Contraction of $\mathbf{T}_c$] \label{lem:contraction_T_c} Consider the consensus operator $\mathbf{T}_c$ as defined in~\eqref{alg:multi-cons-one-grad_operator}. Suppose Assumptions~\ref{assum:cond_A_B} and \ref{assum:conditions_convergence} hold. Then, \[ \norm{\mathbf{T}_c\mathbf{a}-\mathbf{T}_c\mathbf{b}}^2_{\boldsymbol{\Lambda_c}\mathbf{H}}\leq \norm{\mathbf{a}-\mathbf{b}}^2_\mathbf{H},\quad \forall \mathbf{a},\mathbf{b}\in\mathbb{R}^{m\times d}\times \mathcal{C}^\perp, \] where $\boldsymbol{\Lambda}_c=\diag(\I,\frac{1}{p(\A,\C,\sigma)}\I)$ and $\mathcal{C}=\Span{\ones}$. \end{lem} \begin{lem}[Contraction of $\mathbf{T}_f$] \label{lem:contraction_T_f} Consider an operator $\mathbf{T}'_f$ with $\mathbf{T}'_f\triangleq\I-\gamma\B^{-1}\A\nabla f$. Let $f$ be $L$-smooth and $\mu$-strongly convex. Suppose that $\gamma=\frac{2}{L'+\mu'}$, where $L'=\lambda_{\max}(\B^{-1}\A)L$ and $\mu'=\lambda_{\min}(\B^{-1}\A)\mu$. Then, we have \[ \norm{\mathbf{T}'_f\x-\mathbf{T}'_f\y}^2_{\B^{-1}\A}\leq q^2\norm{\x-\y}^2_{\B^{-1}\A},\quad \forall \x,\y\in\mathbb{R}^{m\times d} \] where $q=\frac{\kappa-\eta}{\kappa+\eta}$ with $\kappa=\frac{L}{\mu}$ being the condition number of $f$ and $\eta\triangleq\frac{\lambda_{\min}(\B^{-1}\A)}{\lambda_{\max}(\B^{-1}\A)}$ being the eigengap of $\B^{-1}\A$. \end{lem} \begin{lem}[Contraction of $\mathbf{T}_g$] \label{lem:contraction_T_g} Consider an operator $\mathbf{T}'_g$ with $\mathbf{T}'_g\triangleq(\I+\gamma\partial g)^{-1}$. Let the function $g$ be proper, closed, and convex. Then, we have \[ \norm{\mathbf{T}'_g\x-\mathbf{T}'_g\y}^2_{\B^{-1}\A}\leq \frac{1}{\eta(\B^{-1}\A)}\norm{\x-\y}^2_{\B^{-1}\A},\quad \forall \x,\y\in\mathbb{R}^{m\times d}. \] where $\eta\triangleq\frac{\lambda_{\min}(\B^{-1}\A)}{\lambda_{\max}(\B^{-1}\A)}$ is the eigengap of $\B^{-1}\A$ as defined above. \end{lem} \subsection{Proofs of the main results} \begin{proof}[Proof of Lemma \ref{lem:inclusion_equivalence}] It is not difficult to see that \eqref{eq:inclusion_II} can be rewritten explicitely as \begin{subequations}\label{eq:inclusion_II_expanded} \begin{align} \sqrt{\C}\y'+\nabla f(\x)&\in-\B^{-1}\A\partial g(\x), \label{eq:inclusion_II_expanded_x}\\ \zeros\in-\sqrt{\C}\z,~\z&\in(\I+\gamma\partial g)(\x). \label{eq:inclusion_II_expanded_y} \end{align} \end{subequations} Multiplying both sides of \eqref{eq:inclusion_II_expanded_x} by $\ones$, we have \begin{equation}\label{eq:inclusion_II_kkt_x} \ones^\top\nabla f(\x)\in-\ones^\top\partial g(\x) \end{equation} where we have used the fact that $\ones^\top\A=\ones^\top\B=\ones^\top$ by Assumption~\ref{assum:cond_A_B}. In addition, since $\Null{\C}=\Span{\ones}$ by Assumption~\ref{assum:cond_A_B}, we have $\z\in\Span{\ones}$ which implies that $\x\in\Span{\ones}$ due to the uniqueness of proximal mapping of $(\I+\gamma\partial g)^{-1}$. This, together with \eqref{eq:inclusion_II_kkt_x}, lead to the same solution set of the KKT conditions~\eqref{eq:kkt_conditions} and thus also that of the problem~\eqref{prob:inclusion_g}, which completes the proof. \end{proof} \begin{proof}[Proof of Lemma~\ref{lem:contraction_T_c}] Let $\mathbf{u}=\mathbf{T}_c\mathbf{a}$ and $\mathbf{v}=\mathbf{T}_c\mathbf{b}$. Then, by the definition of $\mathbf{T}_c$ we have \begin{equation}\label{eq:expanded_T_c} \mathbf{H}(\mathbf{a}-\mathbf{u})=\mathbf{M}\mathbf{u},~\mathbf{H}(\mathbf{b}-\mathbf{v})=\mathbf{M}\mathbf{v}. \end{equation} Noticing that $\mathbf{M}=\mathbf{M}_1+\mathbf{M}_2$, with (recalling that $\mathbf{D}=\B^{-1}\mathbf{E}$) \[ \mathbf{M}_1= \begin{bmatrix} \B^{-1}\mathbf{E} &\zeros\\ \zeros &\zeros \end{bmatrix} , \mathbf{M}_2 =\begin{bmatrix} \zeros &\sqrt{\C}\\ -\sqrt{\C} &\zeros \end{bmatrix}, \] where $\mathbf{M}_1\succeq \zeros$ and $\mathbf{M}_2$ is skew-symmetric by Assumption~\ref{assum:cond_A_B}, is therefore monotone, we thus have $\innprod{\mathbf{a}-\mathbf{b}}{\mathbf{u}-\mathbf{v}}_{\mathbf{H}}\geq\norm{\mathbf{u}-\mathbf{v}}^2_{\mathbf{H}}$ by \eqref{eq:expanded_T_c}, which further implies that \begin{equation}\label{eq:average_operator_consensus} \norm{\mathbf{u}-\mathbf{v}}^2_\mathbf{H}\leq\norm{\mathbf{a}-\mathbf{b}}^2_\mathbf{H}-\norm{(\mathbf{a}-\mathbf{b})-(\u-\v)}^2_\mathbf{H}. \end{equation} Since $\mathbf{H}$ is invertible, from \eqref{eq:expanded_T_c} we also have \[ \mathbf{H}^\frac{1}{2}((\mathbf{a}-\mathbf{b})-(\mathbf{u}-\mathbf{v}))=\mathbf{H}^{-\frac{1}{2}}\mathbf{M}(\u-\v) \] Taking the square norm of both sides leads to \[ \norm{(\mathbf{a}-\mathbf{b})-(\mathbf{u}-\mathbf{v})}_\mathbf{H}^2=\norm{\u-\v}_{\mathbf{M}^\top\mathbf{H}^{-1}\mathbf{M}}^2 \] Thus, using \eqref{eq:average_operator_consensus} we further obtain \begin{equation}\label{eq:contraction_proximalpoint} \norm{\u-\v}^2_\mathbf{H+\mathbf{M}^\top\mathbf{H}^{-1}\mathbf{M}}\leq\norm{\mathbf{a}-\mathbf{b}}^2_\mathbf{H}. \end{equation} Also, noticing that \[ \begin{aligned} &\mathbf{H+\mathbf{M}^\top\mathbf{H}^{-1}\mathbf{M}}\\ &= \begin{bmatrix} \frac{1}{\gamma}(\B^{-1}\A+\B^{-1}\mathbf{E}\A^{-1}\mathbf{E} +\B^{-1}\C) &\mathbf{E}\A^{-1}\sqrt{\C}\\ \sqrt{\C}\A^{-1}\mathbf{E} &\gamma(\B+\sqrt{\C}\A^{-1}\B\sqrt{\C}) \end{bmatrix} \\ &= \begin{bmatrix} \frac{1}{\gamma}\B^{-1}\A &\zeros\\ \zeros &\gamma(\B+\sigma\sqrt{\C}\A^{-1}\B\sqrt{\C}) \end{bmatrix} \\ &~~~+ \begin{bmatrix} \frac{1}{\gamma}(\B^{-1}\mathbf{E}\A^{-1}\mathbf{E}+\B^{-1}\C) &\mathbf{E}\A^{-1}\sqrt{\C}\\ \sqrt{\C}\A^{-1}\mathbf{E} &\gamma(1-\sigma)\sqrt{\C}\A^{-1}\B\sqrt{\C} \end{bmatrix} \end{aligned} \] where, for the last term, since $\C\succeq\frac{\sigma}{1-\sigma}\mathbf{E}\A^{-1}\mathbf{E}$ with $\sigma\in(0,1)$ when $\mathbf{E}\neq\zeros$ by Assumption~\ref{assum:conditions_convergence} and thus \[ \begin{aligned} &\frac{1}{\gamma}(\B^{-1}\mathbf{E}\A^{-1}\mathbf{E}+\B^{-1}\C)\\ &~~~-\frac{1}{\gamma(1-\sigma)}\mathbf{E}\A^{-1}\sqrt{\C}\sqrt{\C^\dagger}\A\B^{-1}\sqrt{\C^\dagger}\sqrt{\C}\A^{-1}\mathbf{E}\succeq\zeros \end{aligned} \] and, knowing that $\gamma(1-\sigma)\sqrt{\C}\A^{-1}\B\sqrt{\C}\succeq \zeros$ and $\Span{\sqrt{\C}\A^{-1}\mathbf{E}}\subset\Span{\sqrt{\C}\A^{-1}\B\sqrt{\C}}$ by Assumption~\ref{assum:cond_A_B}, using the generalized Schur complement~\cite[Th.~1.20]{zhang2006schur} we have \[ \begin{bmatrix} \frac{1}{\gamma}(\B^{-1}\mathbf{E}\A^{-1}\mathbf{E}+\B^{-1}\C) &\mathbf{E}\A^{-1}\sqrt{\C}\\ \sqrt{\C}\A^{-1}\mathbf{E} &\gamma(1-\sigma)\sqrt{\C}\A^{-1}\B\sqrt{\C} \end{bmatrix} \succeq 0 \] which, clearly, also holds for the case of $\mathbf{E}=0$ with $\sigma=1$. Now, partitioning $\mathbf{a}=[\mathbf{a}_x^\top,\mathbf{a}_y^\top]^\top,\mathbf{b}=[\mathbf{b}_x^\top,\mathbf{b}_y^\top]^\top,\mathbf{u}=[\mathbf{u}_x^\top,\mathbf{u}_y^\top]^\top,\mathbf{v}=[\mathbf{v}_x^\top,\mathbf{v}_y^\top]^\top$, using~\eqref{eq:contraction_proximalpoint} leads to \[ \begin{aligned} &\frac{1}{\gamma}\norm{\mathbf{a}_x-\mathbf{b}_x}^2_{\B^{-1}\A}+\gamma\norm{\mathbf{a}_y-\mathbf{b}_y}^2_\B\\ &=\frac{1}{\gamma}\norm{\u_x-\v_x}^2_{\B^{-1}\A}+\gamma\norm{\u_y-\v_y}^2_{\B+\sigma\sqrt{\C}\A^{-1}\B\sqrt{\C}}\\ &\overset{(a)}{\geq} \frac{1}{\gamma}\norm{\u_x-\v_x}^2_{\B^{-1}\A}+\frac{1}{p(\A,\C,\sigma)}\gamma\norm{\u_y-\v_y}^2_\B \end{aligned} \] where (a) is due to the fact that $\v_y,\u_y\in\Span{\C}$. Note also that $p(\A,\C,\sigma)=\lambda_2(\I+\sigma\sqrt{\C}\A^{-1}\sqrt{\C})>1$ since $\Null{\C}=\Span{\ones}$ such that $\lambda_2(\C)>0$. We thus complete the proof. \end{proof} \begin{proof}[Proof of Lemma~\ref{lem:contraction_T_f}] Let $\mathbf{Q}=\B^{-1}\A,~\x=\sqrt{\mathbf{Q}}\x',~f'(\x)=f(\sqrt{\mathbf{Q}}\x)$. Given two points $\x_1,\x_2\in\mathbb{R}^{m\times d}$, we have \[ \begin{aligned} &\innprod{\x_1-\x_2}{\nabla f(\x_1)-\nabla f(\x_2)}\\ &=\innprod{\sqrt{\mathbf{Q}}\x'_1-\sqrt{\mathbf{Q}}\x'_2}{\nabla f(\sqrt{\mathbf{Q}}\x'_1)-\nabla f(\sqrt{\mathbf{Q}}\x'_2)}\\ &=\innprod{\x'_1-\x'_2}{\nabla f'(\x'_1)-\nabla f'(\x'_2)}\\ &\overset{(a)}{\geq}\frac{L'\mu'}{L'+\mu'}\norm{\x'_1-\x'_2}^2+\frac{1}{L'+\mu'}\norm{\nabla f'(\x'_1)-\nabla f'(\x'_2)}^2\\ &=\frac{L'\mu'}{L'+\mu'}\norm{\x_1-\x_2}^2_{\mathbf{Q}^{-1}}+\frac{1}{L'+\mu'}\norm{\nabla f(\x_1)-\nabla f(\x_2)}^2_{\mathbf{Q}} \end{aligned} \] where $(a)$ is due to \cite[Lem.~3.11]{bubeck2015theory}. Thus, knowing that $\gamma=\frac{2}{L'+\mu'}$, for the gradient operator $\mathbf{T}'_f$ we further have \[ \begin{aligned} &\norm{\x_1-\gamma\mathbf{Q}\nabla f(\x_1)-\x_2+\gamma\mathbf{Q}\nabla f(\x_2)}^2_{\mathbf{Q}^{-1}}\\ &=\norm{\x_1-\x_2}^2_{\mathbf{Q}^{-1}}-2\gamma\innprod{\x_1-\x_2}{\nabla f(\x_1)-\nabla f(\x_2)}\\ &~~~+\gamma^2\norm{\nabla f(\x_1)-\nabla f(\x_2)}^2_{\mathbf{Q}}\\ &\leq q^2\norm{\x_1-\x_2}^2_{\mathbf{Q}^{-1}}\\ &~~~-(\frac{2\gamma}{L'+\mu'}-\gamma^2)\norm{\nabla f(\x_1)-\nabla f(\x_2)}^2_{\mathbf{Q}}\\ &\leq q^2\norm{\x_1-\x_2}^2_{\mathbf{Q}^{-1}} \end{aligned} \] which completes the proof \end{proof} \begin{proof}[Proof of Lemma~\ref{lem:contraction_T_g}] Let $\mathbf{Q}=\B^{-1}\A$. Then, we have \[ \begin{aligned} \norm{\mathbf{T}'_g\x_1-\mathbf{T}'_g\x_2}^2_{\mathbf{Q}^{-1}}&\leq \lambda_{\max}({\mathbf{Q}^{-1}})\norm{\mathbf{T}'_g\x_1-\mathbf{T}'_g\x_2}^2\\ &\leq\lambda_{\max}({\mathbf{Q}^{-1}})\norm{\x_1-\x_2}^2\\ &\leq\frac{1}{\eta(\mathbf{Q}^{-1})}\norm{\x_1-\x_2}^2_{\mathbf{Q}^{-1}} \end{aligned} \] where we have used the nonexpansive property of $\mathbf{T}'_g$ to obtain the second inequality. We thus complete the proof. \end{proof} \begin{proof}[Proof of Theorem~\ref{thm:contraction_T_c_T_f}] Invoking Lemmas~\ref{lem:contraction_T_c}, \ref{lem:contraction_T_f} and~\ref{lem:contraction_T_g}, we have \[ \begin{aligned} &\norm{\mathbf{T}_c\mathbf{T}_f\mathbf{T}_g\p'^k-\mathbf{T}_c\mathbf{T}_f\mathbf{T}_g\p'^\star}^2_{\boldsymbol{\Lambda}_c\mathbf{H}}\\ &\leq\norm{\mathbf{T}_f\mathbf{T}_g\p'^k-\mathbf{T}_f\mathbf{T}_g\p'^\star}^2_\mathbf{H}\\ &\leq \frac{1}{\gamma}\norm{\mathbf{T}_f'\mathbf{T}_g'\z^k-\mathbf{T}_f'\mathbf{T}_g'\z^\star}^2_{\B^{-1}\A}+\gamma\norm{\y'^k-\y'^\star}^2_\B\\ &\leq \frac{q^2}{\eta}\frac{1}{\gamma}\norm{\z^k-\z^\star}^2_{\B^{-1}\A}+\gamma\norm{\y'^k-\y'^\star}^2_\B \end{aligned} \] The above relation also leads to \[ \begin{aligned} &\frac{1}{\gamma}\norm{\z^{k+1}-\z^\star}^2_{\B^{-1}\A}+\frac{1}{p(\A,\C,\sigma)}\gamma\norm{\y'^{k+1}-\y'^\star}^2_{\B}\\ &~~~~~~\leq \frac{q^2}{\eta}\frac{1}{\gamma}\norm{\z^k-\z^\star}_{\B^{-1}\A}^2+\gamma\norm{\y'^k-\y'^\star}^2_{\B}, \end{aligned} \] which, together with the definition of $V_k$, setting $\lambda$ as the maximum of $p(\A,\C,\sigma)$ and $q^2/\eta$ completes the proof. \end{proof} \subsection{Connections to existing algorithms} \noindent {\bf 1) EXTRA \cite{shi2015extra}:} EXTRA solves \eqref{prob:dop_nonsmooth_same} with $G=0$, and reads \begin{equation}\label{eq:EXTRA} \x^{k+2}=(\I+\W)\x^{k+1}-\tilde{\W}\x^k-\gamma(\nabla f(\x^{k+1})-\nabla f(\x^k)), \end{equation} where $\W,\tilde{\W}$ are two design weight matrices satisfying $({\I+\W})/2\succeq\tilde{\W}\succeq\W$ and $\tilde{\W}\succ 0$. In practice, for better performance, one chooses $\tilde{\W}=\frac{1}{2}(\I+\W)$. Clearly, \eqref{eq:EXTRA} is an instance of \eqref{eq:g-pd-ATC_eliminate_y_G=0} [and thus \eqref{alg:g-pd-ATC}], with $\A=\tilde{\W}$, $\B=\I,$ and $\C=\tilde{\W}-\W$. Notice that such $\A,\B,\C$ satisfy Assumption~\ref{assum:cond_A_B}. \noindent {\bf 2) NIDS~\cite{li2017decentralized}~/~Exact diffusion~\cite{yuan2018exact_p1,yuan2018exact_p2}:} The NIDS (Exact Diffusion) algorithm applies to \eqref{prob:dop_nonsmooth_same} with $G=0$, and reads \begin{equation} \x^{k+2}=\frac{\I+\W}{2}(2\x^{k+1}-\x^k-\gamma(\nabla f(\x^{k+1})-\nabla f(\x^k))), \end{equation} which is an instance of our general scheme, with $\A=\B=({\I+\W})/{2}$ and $\C=\frac{\I-\W}{2}$. Note that Assumption~\ref{assum:cond_A_B} holds. \noindent {\bf 3) NEXT~\cite{di2016next} \& AugDGM~\cite{xu2015augmented}:} The gradient tracking-based algorithms NEXT and AugDGM applied to \eqref{prob:dop_nonsmooth_same} with $G=0$, are: \begin{subequations}\label{alg:NEXT/AugDGM} \begin{align} \x^{k+1}&=\mathbf{W}(\x^k-\gamma\y^k), \label{alg:g-gradtrack-ATC_x}\\ \y^{k+1}&=\mathbf{W}(\y^k+\nabla f(\x^{k+1})-\nabla f(\x^k)).\label{alg:g-gradtrack-ATC_y} \end{align} \end{subequations} Solving for the y-variable, \eqref{alg:NEXT/AugDGM} can be rewritten as: \begin{equation}\label{eq:g-gradtrack-ATC_eliminate_y} \x^{k+2}=2\mathbf{W}\x^{k+1}-\mathbf{W}^2\x^k-\gamma\mathbf{W}^2(\nabla f(\x^{k+1})-\nabla f(\x^k)). \end{equation} Clearly (\ref{eq:g-gradtrack-ATC_eliminate_y}) is an instance of our scheme \eqref{alg:g-pd-ATC}, with $\A=\B=\W^2,\C=(\I-\W)^2$. Notice that distributed gradient tracking schemes in the so-called CTA form are also special cases of Algorithm \eqref{alg:g-pd-ATC}. For instance, one can show that the DIGing algorithm~\cite{nedich2016achieving} corresponds to the setting $\A=\W^2,\B=\I,$ and $\C=(\I-\W)^2$. \noindent {\bf 4) General primal-dual scheme~\cite{jakovetic2018unification,mansoori2019general}}: A general distributed primal-dual algorithm was proposed in~\cite{jakovetic2018unification} for \eqref{prob:dop_nonsmooth_same} with $G=0$: \begin{subequations}\label{alg:Jakovetic} \begin{align} \x^{k+1}&=\mathbf{W}\x^k-\gamma(\nabla f(\x^k)+\y^k),\label{alg:Jakovetic_x}\\ \y^{k+1}&=\y^k-(\I-\W)(\nabla f(\x^k)+\y^k-\B'\x^k),\label{alg:Jakovetic_y} \end{align} \end{subequations} with $\B'=b\I$ or $\B'=~b\W$ for some positive constant $b>0$. Eliminating the $y$-variables, \eqref{alg:Jakovetic} reduces to \[ \x^{k+2}=2\W\x^{k+1}-(\W^2-\gamma(\I-\W)\B')\x^k-\gamma(\nabla f(\x^{k+1})-\nabla f(\x^k)), \] which corresponds to the proposed algorithm, with $\A=\W^2+\gamma(\I-\W)\B',\B=\I,\C=(\I-\W)^2+\gamma(\I-\W)\B'$. Note that Assumption~\ref{assum:cond_A_B} holds for the above choice of $\B'$.\\\indent Similarly, building on a general augmented Lagrangian, another general primal-dual algorithm was proposed in~\cite{mansoori2019general} for \eqref{prob:dop_nonsmooth_same} with $G=0$, which reads \begin{subequations}\label{alg:ermin} \begin{align} \x^{k+1}&=(\I-\alpha\B')^K\x^k-\alpha\C'(\nabla f(\x^k)+\A'^\top\y^k), \label{alg:ermin_x}\\ \y^{k+1}&=\y^k+\beta\A'\x^{k+1}, \label{alg:ermin_y} \end{align} \end{subequations} where $\C'=\sum_{i=1}^{K-1}(\I-\alpha\B')^i$, with $K$ being the number of communication step performed at each iteration. Eliminating $\y$ yields \[ \begin{aligned} \x^{k+2}&=(\I+(\I-\alpha\B')^K-\alpha\beta\C'\A'^\top\A')\x^{k+1}\\ &~~~-(\I-\alpha\B')^K\x^k-\alpha\C'(\nabla f(\x^{k+1})-\nabla f(\x^k)) \end{aligned} \] which corresponds to Algorithm \eqref{alg:g-pd-ATC} with $\A=(\I-\alpha\B')^K,\B=\C',\C=\alpha\beta\C'\A'^\top\A'$. Notice that, letting $\W=\I-\alpha\B'$ and $\B'=\beta\A'^\top\A'$, we have $\A=\W^K,\B=\sum_{i=1}^{K-1}\W^i$ and $\C=(\I-\W)\sum_{i=1}^{K-1}\W^i=\W-\W^K$, which satisfy Assumption~\ref{assum:cond_A_B}.\\ \noindent {\bf 6) Decentralized proximal algorithm~\cite{alghunaim2019linearly} }: A proximal algorithm is proposed to solve~\eqref{prob:dop_nonsmooth_same}, which reads \[ \z^{k+2}=(\I-\alpha\B')\z^{k+1}+(\I-\B')(\x^{k+1}-\x^k)-\gamma(\nabla f(\x^{k+1})-\nabla f(\x^k)). \] which corresponds to Algorithm \eqref{alg:g-pd-ATC} with $\A=\I-\B',\B=\I,\C=\alpha\B'$. Choosing $\W=\I-\B'$ and thus $\A=\W,\B=\I$ and $\C=\alpha(\I-\W)$, which clearly satisfy Assumption~\ref{assum:cond_A_B}. \begin{comment} \subsection{Communication and computation trade-off} ....... Suppose that we have a normalized budget $C_cd_c+C_gd_g=1$, where $C_c$ denotes the cost for carrying out a consensus step and $C_g$ the cost for a gradient step. Our objective is, given certain amount of budget, to properly allocate $d_c$ and $d_g$ such that the distance of the final solution to the optimum (or the rate in each interval) is minimized, i.e., \[ \min_{d_c,d_g} \boldsymbol{\lambda}=\max\{q^{2d_g},\eta(\A,\B)\},~{\bf s.t.}~C_cd_c+C_gd_g=1 \] The minmum of the above problem is attained only when \[ d_g\log q^{2}=\log\eta(\A,\B) \] which implies that the optimal ratio has nothing to do with the cost of consensus steps and gradient steps and thus can be determined quite easily. \subsubsection{The case when $\A=\W^{d_c},\B=\I-\W^{d_c}$} Consider a special case of $\A=\W^{d_c},\B=\I-\W^{d_c}$. Then, we have $\eta(\A,\B)=\rho^{d_c}$ with $\rho\triangleq\boldsymbol{\lambda}_{\max}(\W-\J)$. As a result, we have \[ \frac{d_g}{d_c}=\frac{\log \rho}{\log q^{2}} \] ...... \[ \min_{} comm. cost~{\bf s.t.}~1/\lambda_{\min}(\I+\sigma\sqrt{\C}\A^{-1}\sqrt{\C}-\J)\geq q^2 \] \section{Tightness of contraction and optimal paramter selection} There are several parameters to be determined in the existing algorithms. In this section, we will discuss how to make the optimal parameter section such that the contraction is maximized and can not be improved without further properties on the cost functions and the graph. \[ \text{arg} \max_{\boldsymbol{\Lambda},\boldsymbol{\Lambda}_\tau} (1-\delta(\T))~{\bf s.t.}~\nabla f\in \mathcal{F}^L_\mu, \partial G\in \mathcal{F}_\mu,\B^T\Lambda_\tau\B\in\mathcal{G}_\eta \] \subsection{Transferring to operator splitting form} We now attempt to transform the proposed general scheme~\eqref{alg:g-pd-ATC} into an operator factorization form that are composed by the aforementioned averaged operators. \noindent To see this, adding \eqref{alg:g-pd-ATC_y} with \eqref{alg:g-pd-ATC_x}, we have \begin{equation}\label{alg:g-pd-ATC_added_x} \begin{aligned} (\I-\C)\z^{k+1}=\A\x^k-\gamma\B\nabla f(\x^k)-\y_{k+1} \end{aligned} \end{equation} Now, multiplying $\B^{-1}$ from both sides of \eqref{alg:g-pd-ATC_added_x} and introducing a new variable $\y_k$ such that $\y_k=\B\sqrt{\C}\y'_k$, then \eqref{alg:g-pd-ATC_y} becomes \[\B\sqrt{\C}\y'^{k+1}=\B\sqrt{\C}\y'^k+\C\z^{k+1} \] Now, assuming that $\y'^0\in{\bf range}(\C)$ such that $\y'^k\in{\bf range}(\C),\forall k\geq 0$ and knowing from Assumption~\ref{assum:cond_A_B} that $\B$ and $\C$ commutes, we further have \[\B\y'^{k+1}=\B\y'^k+\sqrt{\C}\z^{k+1} \] which, together with \eqref{alg:g-pd-ATC_added_x}, we can rewrite \eqref{alg:g-pd-ATC_x} and \eqref{alg:g-pd-ATC_y} in a matrix form as follows \begin{equation} \begin{aligned} \bracket{ \begin{bmatrix} \B^{-1}\A &\zeros\\ \zeros &\B \end{bmatrix} + \begin{bmatrix} \B^{-1}\mathbf{E} &\sqrt{\C}\\ -\sqrt{\C} &\zeros \end{bmatrix} } \begin{bmatrix} \z^{k+1}\\ \y'_{k+1} \end{bmatrix} =\\ \begin{bmatrix} \B^{-1}\A-\gamma\nabla f &\zeros\\ \zeros &\B \end{bmatrix} \begin{bmatrix} \x^k\\ \y'_k \end{bmatrix} \end{aligned} \end{equation} where $\mathbf{E}\triangleq\I-\C-\A$. Knowing the fact that $\x^k=(\I+\gamma\partial g)^{-1}(\z^k)$ from \eqref{alg:g-pd-ATC_z} and $\mathbf{H}$ is invertible, the above relation can be further rewritten in a compact form as follows, \begin{equation}\label{alg:multi-cons-one-grad_operator} \mathbf{p}^{k+1}=(\I+\H^{-1}\mathbf{M})^{-1}(\I-\H^{-1}\mathbf{F})(\I+\G)^{-1}\mathbf{p}^k \end{equation} where $\mathbf{p}^k=[{\z^k}^\top,{\y'^k}^\top]^\top$ and \[ \begin{array}{ll} \mathbf{H}\triangleq\begin{bmatrix} \B^{-1}\A &\zeros\\ \zeros &\B \end{bmatrix} , &\mathbf{M}\triangleq \begin{bmatrix} \B^{-1}\mathbf{E} &\sqrt{\C}\\ -\sqrt{\C} &\zeros \end{bmatrix} ,\\ \mathbf{F}\triangleq\begin{bmatrix} \gamma\nabla f &\zeros\\ \zeros &\zeros \end{bmatrix} , &\mathbf{G}\triangleq\begin{bmatrix} \gamma\partial g &\zeros\\ \zeros &\zeros \end{bmatrix}. \end{array} \] Let $\mathbf{T}_c=(\I+\H^{-1}\mathbf{M})^{-1},\mathbf{T}_f=\I-\H^{-1}\mathbf{F},\mathbf{T}_g=(\I+\mathbf{G})^{-1}$ be the three averaged operators corresponding to consensus, gradient descent and regualization, respectively. The above iterates \eqref{alg:multi-cons-one-grad_operator} can be rewritten in a decoupled form as follows \[ \begin{cases} \begin{aligned} \mathbf{p}^{k+\frac{1}{2}}&=\mathbf{T}_f\mathbf{T}_g\mathbf{p}^k,~~~~\text{(Computation)}\\ \mathbf{p}^{k+1}&=\mathbf{T}_c\mathbf{p}^{k+\frac{1}{2}},~~~~\text{(Communication)} \end{aligned} \end{cases} \] where $\mathbf{p}^{k+\frac{1}{2}}$ is the introduced intermediate variable. \begin{rem} ......The above transformation to the form of three averaged operators is unique. \end{rem} \subsection{Proof of Lemma \ref{lemma_KKT_limit_point}} TBD: From \eqref{alg:g-pd-ATC}, we have $\C\z^\star=\zeros,~\z^\star=\A\x^\star-\gamma\B\nabla f(\x^\star)-\y^\star,~\text{and}~\z^\star-\x^\star\in\gamma\partial g(\x^\star)$. Thus, since $\Null{\C}=\Span{\ones}$ by Assumption~\ref{assum:cond_A_B}, we have $\z^\star\in\Span{\ones}$. Also, since $\ones^\top\A=\ones^\top,\ones^\top\B=\ones^\top$ again by Assumption~\ref{assum:cond_A_B}, together with the fact that $\ones^\top\y^\star=0$ we have $\ones^\top(\z^\star-\x^\star)=-\gamma\ones^\top\nabla f(\x^\star)$. Then, letting $\gamma\q^\star=\z^\star-\x^\star$, we obtain \[ \begin{aligned} \ones^\top(\nabla f(\x^\star)+\q^\star)=\zeros,~\q^\star\in\partial g(\x^\star) \end{aligned} \] and also $\x^\star\in\Span{\ones}$ due to the uniqueness of proximal mapping of $(\I+\gamma\partial g)^{-1}$ given $\z^\star\in\Span{\ones}$, which further implies the KKT conditions and in turn $\x^\star\in\mathcal{S}$. \alert{The other direction to be added!!!} $\square$ \subsection{Preliminaries} \begin{dfn}[Monotone operator~\cite{bauschke2011convex}] An operator $\mathbf{T}:\mathcal{H}\rightarrow2^\mathcal{H}$ is said to be $\alpha$-monotone if for $\forall \mathbf{a},\mathbf{b}\in\mathcal{H}$ we have $\innprod{\mathbf{T}\mathbf{a}-\mathbf{T}\mathbf{b}}{\mathbf{a}-\mathbf{b}}_{\mathbf{H} }\geq \alpha\norm{\mathbf{a}-\mathbf{b}}^2_\mathbf{H}$. \end{dfn} \begin{dfn}[Cocoercive operator~\cite{bauschke2011convex}] An operator $\mathbf{T}:\mathcal{H}\rightarrow2^\mathcal{H}$ is said to be $\beta$-cocoercive if for $\forall \mathbf{a},\mathbf{b}\in\mathcal{H}$ we have $\innprod{\mathbf{T}\mathbf{a}-\mathbf{T}\mathbf{b}}{\mathbf{a}-\mathbf{b}}_\mathbf{H}\geq \beta\norm{\mathbf{T}\mathbf{a}-\mathbf{T}\mathbf{b}}^2_\mathbf{H}$. \end{dfn} \begin{dfn}[$\alpha$-averaged operator~\cite{bauschke2011convex}] An operator $\mathbf{T}:\mathcal{H}\rightarrow2^\mathcal{H}$ is said to be $\alpha$-averaged in $\mathbf{H}$-space if for $\forall \mathbf{a},\mathbf{b}\in\mathcal{H}$ we have \[ \norm{\mathbf{T}\mathbf{a}-\mathbf{T}\mathbf{b}}^2_\mathbf{H}\leq\norm{\mathbf{a}-\mathbf{b}}^2_\mathbf{H}-\frac{\alpha}{1-\alpha}\norm{(\I-\mathbf{T})\mathbf{a}-(\I-\mathbf{T})\mathbf{b}}^2_\mathbf{H}. \] \end{dfn} It is well known that averaged operators are one of the basic operators that enjoys nice firmly nonexpansive property. The following two lemmas show that the well known ``backward'' operator building on monotone operator and the ``forward ''operator building on cocoercive operator, respectively, is, indeed, $\frac{1}{2}$averaged, which are the backbone to our subsequent contraction analysis. \begin{lem}[Backward Operator] Let $\mathbf{M}$ be $\alpha$-monotone in $\H$-space. Then, we have $(\I+\mathbf{M})^{-1}$ is $\frac{1}{2}$-averaged in $\H$-space. \end{lem} \begin{lem}[Forward Operator] Let $\mathbf{F}$ be $\beta$-cocoercive in $\mathbf{H}$-space. Then, we have $\I-\frac{1}{\beta}\mathbf{F}$ is $\frac{1}{2}$-averaged in $\H$-space. \end{lem} \noindent{\bf Forward-Backward Splitting}: Consider an inclusion problem \[ \zeros\in(\mathbf{M}+\mathbf{F}) \] where $\mathbf{M}$ is a monotone operator while $\mathbf{F}$ is a cocoercive operator. The above problem can be solved by the forward-backward splitting technique via the following operator \[ \mathbf{T}=(\H+\mathbf{M})^{-1}(\H-\mathbf{F}) \] where $\mathbf{T}=\mathbf{T}_1\mathbf{T}_2$ with $\mathbf{T}_1=(\H+\mathbf{M})^{-1}\H$ and $\mathbf{T}_2=(\I-\mathbf{H}^{-1}\mathbf{F})$. The operator $\mathbf{T}_1$ is regarded as a backward step that performs an proximal operation (also known as proximal point method) towards the operator $\mathbf{M} $ in~$\mathbf{H}$-space while $\mathbf{T}_2$ is referred to as the forward step that performs a search (e.g., gradient search for convex optimization) towards $\mathbf{F}$. When $\mathbf{M}$ is (maximally) monotone and $\mathbf{F}$ is cocoercive, it can be shown that the fixed point of the operator $\fix(\mathbf{T})$ will converge to ${\bf Zer}(\mathbf{M}+\mathbf{F})$, solving the above inclusion. \textcolor{red}{...Jimmy, if you start from the inclusion as you did above, this section looks disconnected to the next one and does not help to understand how from (4) you go to (14) and (15). The steps in the next section still looks mysterious as in the next section, to obtain (14), you start from the algorithm (4) and not the KKT condition} \alert{Prof Aldo, the logical flow here is to leverage operator factorization to transfer the algorithm into a form composed by several averaged operators which are the very basic operator that enjoys very nice properties, such as firmly nonexpansiveness.} \begin{rem} It is easy to see that the composition of averaged operators is also $\frac{1}{2}$-averaged. We show next that the general scheme, indeed, can be represented as a composition of several such averaged operators. \end{rem} \end{comment} \end{document}	arXiv
\begin{document} \begin{frontmatter} \title{On the moments and the interface of the symbiotic branching model} \runtitle{On the symbiotic branching model} \begin{aug} \author[A]{\fnms{Jochen} \snm{Blath}\ead[label=e1]{[email protected]}}, \author[A]{\fnms{Leif} \snm{D\"oring}\corref{}\thanksref{t2}\ead[label=e2]{[email protected]}} and \author[B]{\fnms{Alison} \snm{Etheridge}\ead[label=e3]{[email protected]}} \runauthor{J. Blath, L. D\"oring and A. Etheridge} \affiliation{Technische Universit\"at Berlin, Technische Universit\"at Berlin\break and University of Oxford} \address[A]{J. Blath\\ L. D\"oring\\ Institut f\"ur Mathematik\\ Technische Universit\"at Berlin\\ Stra\ss e des 17 Juni 136\\ 10623 Berlin\\ Germany\\ \printead{e1}\\ \phantom{E-mail: }\printead{e2}} \address[B]{A. Etheridge\\ Department of Statistics\\ University of Oxford\\ 1, South Parks Road\\ Oxford OX1 3TG\\ United Kingdom\\ \printead{e3}} \end{aug} \thankstext{t2}{Supported in part by the DFG International Research Training Group ``Stochastic Models of Complex Processes'' and the Berlin Mathematical School.} \received{\smonth{7} \syear{2009}} \revised{\smonth{2} \syear{2010}} \begin{abstract} In this paper we introduce a critical curve separating the asymptotic behavior of the moments of the symbiotic branching model, introduced by Etheridge and Fleischmann [\textit{Stochastic Process. Appl.} \textbf{114} (2004) 127--160] into two regimes. Using arguments based on two different dualities and a classical result of Spitzer [\textit{Trans. Amer. Math. Soc.} \textbf{87} (1958) 187--197] on the exit-time of a planar Brownian motion from a wedge, we prove that the parameter governing the model provides regimes of bounded and exponentially growing moments separated by subexponential growth. The moments turn out to be closely linked to the limiting distribution as time tends to infinity. The limiting distribution can be derived by a self-duality argument extending a result of Dawson and Perkins [\textit{Ann. Probab.} \textbf{26} (1998) 1088--1138] for the mutually catalytic branching model. As an application, we show how a bound on the $35$th moment improves the result of Etheridge and Fleischmann [\textit{Stochastic Process. Appl.} \textbf{114} (2004) 127--160] on the speed of the propagation of the interface of the symbiotic branching model. \end{abstract} \begin{keyword}[class=AMS] \kwd[Primary ]{60K35} \kwd[; secondary ]{60J80}. \end{keyword} \begin{keyword} \kwd{Symbiotic branching model} \kwd{mutually catalytic branching} \kwd{stepping stone model} \kwd{parabolic Anderson model} \kwd{moment duality} \kwd{self-duality} \kwd{propagation of interface} \kwd{exit distribution}. \end{keyword} \end{frontmatter} \section{Introduction}\label{sec:intro} In 2004, Etheridge and Fleischmann \cite{EF04} introduced a stochastic spatial model of two interacting populations known as the symbiotic branching model, parametrized by a parameter $\varrho\in[-1,1]$ governing the correlation between the two driving noises. The model can be considered in three different spatial setups which we now explain. First, the continuous-space symbiotic branching model is given by the system of stochastic partial differential equations \begin{equation}\label{eqn:spde}\quad {\mathrm{cSBM}(\varrho,\kappa)}_{u_0, v_0}\dvtx \cases{ \dfrac{\partial}{\partial t}u_t(x) = \dfrac{1}{2}\Delta u_t(x) + \sqrt{ \kappa u_t(x) v_t(x)} \,dW^1_t(x),\vspace{2pt}\cr \dfrac{\partial}{\partial t}v_t(x) = \dfrac{1}{2}\Delta v_t(x) + \sqrt{ \kappa u_t(x) v_t(x)} \,dW^2_t(x),\vspace{2pt}\cr u_0(x) \ge0, \qquad x \in\mathbb{R},\cr v_0(x) \ge0, \qquad\hspace{1.1pt} x \in\mathbb{R},} \end{equation} where $\Delta$ denotes the Laplace operator and $\kappa> 0$ is a fixed constant known as the branching rate. $\mathbf{{W}} = ({W}^1, {W}^2)$ is a pair of correlated standard Gaussian white noises on $\mathbb{R}_+ \times\mathbb{R}$ with correlation $\varrho \in[-1,1]$, that is, the unique Gaussian process with covariance structure \begin{eqnarray}\label{correlation} \mathbb{E} [ W^1_{t_1}(A_1)W^1_{t_2}(A_2) ] &=& (t_1\wedge t_2) \ell(A_1\cap A_2),\\ \mathbb{E} [ W^2_{t_1}(A_1)W^2_{t_2}(A_2) ] &=& (t_1\wedge t_2) \ell(A_1\cap A_2),\\ \mathbb{E} [ W^1_{t_1}(A_1)W^2_{t_2}(A_2) ] &=& \varrho (t_1\wedge t_2) \ell(A_1\cap A_2), \end{eqnarray} where $\ell$ denotes Lebesgue measure, $A_1,A_2\in\mathcal{B}(\mathbb{R})$ and $t_1,t_2 \geq0$. Note that we work with a white noise $\mathbf{{W}}$ in the sense of Walsh \cite{W86}. Solutions of this model have been considered rigorously in the framework of the corresponding martingale problem in Theorem 4 of \cite{EF04}, which states that, under suitable conditions on the initial conditions $u_0(\cdot), v_0(\cdot)$, a solution exists for all $\varrho\in[-1, 1]$. The martingale problem is well posed for all $\varrho\in[-1,1)$, which implies the strong Markov property except in the boundary case $\varrho=1$. For a discrete spatial version we consider the system of interacting diffusions on $\mathbb{Z}^d$, with values in $\mathbb{R}_{\geq0}$, defined by the coupled stochastic differential equations \begin{equation}\label{eq:dsbm} \mathrm{dSBM}(\varrho,\kappa)_{u_0, v_0} \dvtx \cases{ d u_t(i)=\Delta u_t(i) \,dt + \sqrt{\kappa u_t(i)v_t(i)} \,dB^1_t(i),\vspace{2pt}\cr d v_t(i)=\Delta v_t(i) \,dt + \sqrt{\kappa u_t(i)v_t(i)} \,dB^2_t(i),\vspace{2pt}\cr u_0(i) \ge0, \qquad i \in\mathbb{Z}^d,\cr v_0(i) \ge0, \qquad\hspace{1.1pt} i \in\mathbb{Z}^d,} \end{equation} where now $ \{B^1(i),B^2(i) \}_{i \in\mathbb{Z}^d}$ is a family of standard Brownian motions with covariances given by \begin{equation}\label{eq:dcv} [ B_{\cdot}^n(i),B_{\cdot}^m(j)]_t = \cases{ \varrho t, &\quad $i=j$ and $n \neq m$,\cr t, &\quad $i=j$ and $n=m$, \cr 0, &\quad otherwise.} \end{equation} In the discrete case, $\Delta$ denotes the discrete Laplacian \[ \Delta u_t(i)=\sum_{\|k-i\|=1}\frac{1}{2d}\bigl(u_t(k)-u_t(i)\bigr). \] Note that in this paper we denote by $[N_{\cdot},M_{\cdot}]_t$ the cross-variation of two martingales $N,M$. This is to avoid confusion with $\langle f,g\rangle$ which will be defined to be the sum (resp., integral) of the product of $f$ and $g$. Finally, the nonspatial symbiotic branching model is defined by the stochastic differential equations \[ \mathrm{SBM}(\varrho,\kappa)_{u_0, v_0} \dvtx \cases{ du_t=\sqrt{\kappa u_tv_t} \,dB^1_t,\cr dv_t=\sqrt{\kappa u_tv_t} \,dB^2_t,\cr u_0\geq0,\cr v_0\geq0.} \] Again, the noises are correlated with $[B^1_{\cdot},B^2_{\cdot }]_t=\varrho t$. This simple toy-model (see also \cite{R95} and \cite {DFX05}) can be analyzed quite simply and will be used to prove properties of the spatial models. \begin{convention} From time to time we skip the dependence on $\varrho, \kappa, u_0$ and $v_0$ if there is no ambiguity. Solutions of $\mathrm{cSBM}, \mathrm {SBM}$ and $\mathrm{dSBM}$ for $d\leq2$ are called symbiotic branching processes in the recurrent case whereas solutions of $\mathrm{dSBM}$ for $d\geq3$ are called symbiotic branching processes in the transient case. \end{convention} Interestingly, symbiotic branching models include well-known spatial models from different branches of probability theory. In the discrete spatial case (and analogously in continuous-space) interacting diffusions of the type \begin{equation} \label{int} d w_t(i)=\Delta w_t(i) \,dt + \sqrt{\kappa f(w_t(i))} \,dB_t(i) \end{equation} have been studied extensively in the literature. Some important examples are the following: \begin{example}\label{ex1} The stepping stone model from mathematical genetics: $f(x)= x(1-x)$. \end{example} \begin{example}\label{ex2} The parabolic Anderson model (with Brownian potential) from mathematical physics: $f(x)= x^2$. \end{example} \begin{example} \label{ex3} The super random walk from probability theory: $f(x)= x$. \end{example} For the super random walk, $\kappa$ is the branching rate which in this case is time--space independent. In \cite{DP98}, a two-type model based on two super random walks with time--space dependent branching was introduced. The branching rate for one species is proportional to the value of the other species. More precisely, the authors considered \begin{eqnarray} d u_t(i)&=&\Delta u_t(i) \,dt + \sqrt{\kappa u_t(i)v_t(i)}\, dB^1_t(i),\\ d v_t(i)&=&\Delta v_t(i) \,dt + \sqrt{\kappa u_t(i)v_t(i)}\, dB^2_t(i), \end{eqnarray} where now $ \{B^1(i),B^2(i) \}_{i \in\mathbb{Z}^d}$ is a family of independent standard Brownian motions. Solutions are called mutually catalytic branching processes. In the following years, properties of this model were well studied (see, e.g., \cite{CK00} and \cite {CDG04}). The corresponding continuous-space version was also treated in \cite{DP98}. For correlation $\varrho=0$, solutions of the symbiotic branching model are obviously solutions of the mutually catalytic branching model. The case $\varrho=-1$ with the additional assumption $u_0+v_0\equiv1$ corresponds to the stepping stone model. To see this, observe that in the perfectly negatively correlated case $B^1(i)=-B^2(i)$ which implies that the sum $u+v$ solves a discrete heat equation and with the further assumption $u_0+v_0\equiv1$ stays constant for all time. Hence, for all $t\geq0$, $u(t,\cdot)\equiv1-v(t,\cdot)$, which shows that $u$ is a solution of the stepping stone model with initial condition $u_0$ and $v$ is a solution with initial condition $v_0$. Finally, suppose $w$ is a solution of the parabolic Anderson model, then, for $\varrho=1$, the pair $(u,v):=(w,w)$ is a solution of the symbiotic branching model with initial conditions \mbox{$u_0=v_0=w_0$}. The purpose of this and the accompanying paper \cite{AD09} is to understand the nature of the symbiotic branching model better. How does the model depend on the correlation $\varrho$? Are properties of the extremal cases $\varrho\in\{-1,0,1\}$ inherited by some parts of the parameter space? Since the longtime behavior of the super random walk, stepping stone model, mutually catalytic branching model and parabolic Anderson model is very different, one might guess that the parameter space $[-1,1]$ can be divided into disjoint subsets corresponding to different regimes. The focus of \cite{AD09} is second moment properties. In the discrete setting, but with a more general setup, growth of second moments is analyzed in detail. A moment duality is used to reduce the problem to moment generating functions and Laplace transforms of local times of discrete-space Markov processes. A precise analysis of those is used to derive intermittency and aging results which show that different regimes occur for $\varrho<0$, $\varrho=0$ and $\varrho>0$. In contrast to \cite{AD09}, the present paper is not restricted to second moment properties. The aim is to understand the pathwise behavior of symbiotic branching processes better. \begin{remark}\label{rem:migration} In this paper, we restrict ourselves to the simplest setups which already provide the full variety of results. For the discrete spatial model we thus restrict ourselves to the discrete Laplacian instead of allowing more general transitions. This is not necessary; see \cite {DP98} or \cite{CDG04} for a construction of solutions and main properties for more general underlying migration mechanisms in the case $\varrho=0$. Furthermore, we mainly restrict ourselves to homogeneous initial conditions and remark where results hold more generally. Here, for nonnegative real numbers we denote by $\mathbf{u}$ the constant functions $u(\cdot)\equiv u$. \end{remark} The paper is organized as follows: our main results are presented in Section \ref{subsec:smr}. Before proving the results, we collect basic properties of the symbiotic branching models and discuss the dualities that we need. This is carried out in Section \ref{sec:bpd}. The final sections are devoted to the proofs. In Section \ref{sec:comnvlaw}, proofs of the longtime convergence in law are given, and in Section \ref{sec:moments} we discuss the longtime behavior of moments. Finally, in Section \ref{sec:wavespeed} we show how to use the results of Section \ref{sec:moments} to strengthen the main result of \cite{EF04}. \section{Results}\label{subsec:smr} Before stating the main results, we briefly recall from \cite{EF04} that the state space of $\mathrm{cSBM}$ is given by pairs of tempered functions, that is, pairs of functions contained in \[ M_{\mathrm{tem}}= \Bigl\{u \| u\dvtx\mathbb{R}\to\mathbb{R}_{\geq0}, \lim_{\|x\|\rightarrow\infty }u(x)\phi_{\lambda}(x)\mbox{ exists and }\Vert u\phi_{\lambda }\Vert _{\infty }<\infty\ \forall\lambda<0 \Bigr\}, \] where $\phi_{\lambda}(x)=e^{\lambda\|x\|}$, and we think of $M_{\mathrm{tem}}$ as being topologized by the metric given in \cite{EF04}, equation (13), yielding a Polish space. The state space for $\mathrm{dSBM}$ is similar. It was not discussed in \cite{EF04} and so we present details in Section \ref{sec:bpd}. \subsection{Convergence in law} We begin with a result, generalizing Theorem 1.5 of~\cite{DP98}, on the longtime behavior of the laws of symbiotic branching processes in the recurrent case. \begin{proposition}\label{prop:convlaw} Suppose $(u_t,v_t)$ is a spatial symbiotic branching process in the recurrent case with $\varrho\in(-1,1)$, $\kappa>0$ and initial conditions $u_0=\mathbf{u}, v_0=\mathbf{v}$. Let $B^1$ and $B^2$ be two Brownian motions with covariance \[ [B^1_{\cdot},B^2_{\cdot}]_t=\varrho t,\qquad t\ge0, \] and initial conditions $B_0^1=u, B_0^2=v$. Further, let \[ \tau=\inf\{t\geq0\dvtx B^1_t B^2_t=0 \} \] be the first exit time of the correlated Brownian motions $B^1, B^2$ from the upper right quadrant. Then, weakly in $M_{\mathrm{tem}}^2$, \[ \mathbb{P}^{\mathbf{u},\mathbf{v}}[(u_t,v_t)\in\cdot]\Rightarrow P^{u,v}[(\bar B^1_{\tau },\bar B^2_{\tau})\in\cdot] \] as $t\to\infty$. Here, $(\bar B^1_\tau, \bar B^2_\tau)$ denotes the pair of constant functions on $\mathbb{R}$, respectively, $\mathbb{Z}^d$ ($d=1,2$) taking the values of the stopped Brownian motions $(B^1_\tau, B^2_\tau)$. \end{proposition} In particular, the proposition shows ultimate extinction of one species in law. \begin{remark}\label{all} For simplicity, Proposition \ref{prop:convlaw} is formulated for constant initial conditions even though the result holds more generally. Theorem 1.5 of \cite{DP98} (the case $\varrho=0$) was extended in \cite{CKP00} to nondeterministic initial conditions: for fixed $u,v\geq0$ let ${\mathcal M}_{u,v}$ be the set of probability measures $\nu $ on $M_{\mathrm{tem}}^2$ such that \begin{equation} \sup_{x \in\mathbb{R}} \int\bigl(a^2(x) + b^2(x)\bigr) \,d\nu(a,b)<\infty \end{equation} and \begin{equation}\qquad \lim_{t \to\infty} \int\bigl[ \bigl(P_ta(x)- u\bigr)^2 + \bigl(P_tb(x)-v\bigr)^2\bigr]\, d\nu (a,b) =0 \qquad\mbox{for all } x \in\mathbb{R}. \end{equation} Here, $(P_t)$ denotes the transition semigroup of Brownian motion (the definition for the discrete case is similar). The proof of \cite{CKP00} can also be applied to $\varrho\neq0$ and, thus, Proposition \ref {prop:convlaw} holds in the same way for initial distributions $\nu\in \mathcal{M}_{u,v}$. \end{remark} The restriction to $\varrho\in(-1,1)$ arises from our method of proof which exploits a self-duality of the process which gives no information for $\varrho\in\{-1,1\}$. Let us briefly discuss the behavior of the limiting distributions in the boundary cases $\varrho\in\{-1, 1\}$ which are well known in the literature and fit neatly into our result. First, suppose $(w_t)$ is a solution of the stepping stone model (see Example \ref{ex1}) and $w_0\equiv w \in[0,1]$. It was proved in \cite{S80} that \begin{equation} \label{eq:shigalimit} \mathcal{L}^{\mathbf{w}}(w_t)\stackrel{t\rightarrow\infty }{\Rightarrow} w \delta_{\mathbf{1}} +(1-w) \delta_{\mathbf{0}}, \end{equation} where $\delta_{\mathbf{1}}$ (resp., $\delta_{\mathbf{0}}$) denotes the Dirac distribution concentrated on the constant function $\mathbf{1}$ (resp., $\mathbf0$). This can be reformulated in terms of perfectly anti-correlated Brownian motions $(B^1, B^2)$ as before: for $\varrho=-1$, the pair $(B^1, B^2)$ takes values only on the straight line connecting $(0,1)$ and $(1,0)$, and stops at the boundaries. Hence, the law of $(B^1_{\tau }, B^2_{\tau})$ is a mixture of $\delta_{(0,1)}$ and $\delta_{(1,0)}$ and the probability of hitting $(1,0)$ is equal to the probability of a one-dimensional Brownian motion started in $w \in[0,1]$ hitting $1$ before $0$, which is $w$, and hence matches (\ref{eq:shigalimit}). Second, let $(w_t)$ be a solution of the parabolic Anderson model with Brownian potential (see Example \ref{ex2}) and constant initial condition $w_0 \equiv w \geq0$. In \cite{r7} it was shown that \begin{equation} \mathcal{L}^{\mathbf{w}}(w_t)\stackrel{t\rightarrow\infty }{\Rightarrow} \delta_{\mathbf{0}}. \end{equation} As discussed above, when viewed as a symbiotic branching process with $\varrho=1$, this implies \begin{equation} \mathcal{L}^{\mathbf w, {\mathbf w}}(u_t,v_t)\stackrel{t\rightarrow \infty }{\Rightarrow} \delta_{\mathbf{0}, \mathbf{0}}. \end{equation} From the viewpoint of two perfectly positive-correlated Brownian motions, we obtain the same result since they simple move on the diagonal dissecting the upper right quadrant until they eventually get absorbed in the origin, that is, $(B^1_\tau, B^2_\tau)=(0,0)$ almost surely. To summarize, we have seen that the weak longtime behavior (in the recurrent case) of the classical models connected to symbiotic branching is appropriately described by correlated Brownian motions hitting the boundary of the upper right quadrant. \subsection{Nonalmost-sure behavior} In contrast to extinction in law, the almost-sure behavior is very different. In the recurrent case for the mutually catalytic branching model, Cox and Klenke \cite{CK00} showed that, almost surely, there is no longtime local extinction of any type, but in fact the locally predominant type changes infinitely often. It is not hard to see that the same is true for symbiotic branching with $\varrho\in(-1,1)$. We do not give a proof since it follows from Proposition \ref{prop:convlaw} along the same lines as in \cite{CK00}. \begin{proposition}\label{prop:pb} Let $\varrho\in(-1,1)$, $\kappa>0$ and suppose $(u_t,v_t)$ is a spatial symbiotic branching process in the recurrent case with initial distribution $u_0=\mathbf{u}, v_0=\mathbf{v}$. Then, for all $(u', v') \in\{(x, 0) \dvtx x\in\mathbb{R}_{\geq0}\} \cup\{ (0,y) \dvtx y\in\mathbb{R}_{\geq0}\}$ and $K \subset\mathbb{R}$ bounded, \[ \mathbb{P}^{\mathbf{u},\mathbf{v}} \Bigl[{\liminf_{t\rightarrow\infty} \sup_{x\in K}} \Vert(u_t(x), v_t(x))-(u', v')\Vert=0 \Bigr] =1, \] respectively, for $K\subset\mathbb{Z}^d$ bounded, \[ \mathbb{P}^{\mathbf{u},\mathbf{v}} \Bigl[ {\liminf_{t\rightarrow\infty} \sup_{k\in K}} \Vert(u_t(k), v_t(k))-(u', v')\Vert=0 \Bigr] =1. \] \end{proposition} Again, as in Remark \ref{all}, the result holds for random initial conditions of the class $\mathcal{M}_{u,v}$. Note that Proposition \ref {prop:pb} depends strongly on the spatial structure since in the nonspatial model almost sure convergence holds (see Proposition \ref {prop:sconv}). \subsection{Longtime behavior of moments} In \cite{AD09} the second moments of symbiotic branching processes are analyzed. This particular case admits a detailed study since a moment duality (see Lemma \ref{la:mdual}) has a particularly simple structure which allows one to reduce the study of the moments to that of moment generating functions and Laplace transforms of local times. Here we are interested in the behavior of moments as $t$ tends to infinity. The two available dualities (self-duality and moment duality) are combined in two steps. First, a self-duality argument combined with an equivalence between bounded moments of the exit time distribution and of the exit point distribution for correlated Brownian motions stopped on exiting the first quadrant is used to understand the effect of $\varrho$. It turns out that for any $p>1$ there are critical values, independent of $\kappa$, dividing regimes in which the moments $\mathbb{E}^{1,1}[u_t^p]$, $\mathbb{E} ^{\mathbf{1},\mathbf{1}}[u_t(k)^p]$ and $\mathbb{E}^{\mathbf{1},\mathbf{1}}[u_t(x)^p]$ are bounded in $t$ or grow to infinity. Second, for $p\in\mathbb{N}$, a perturbation argument combined with the first step and a moment duality is used to analyze the growth to infinity in more detail. The following critical curve captures the effect of $\varrho$. Note that the definition is independent of $\kappa$ which will become important in the second step. \begin{definition} \label{def:cc} We define the critical curve of symbiotic branching models to be the real-valued function $p\dvtx(-1,1) \to\mathbb{R}^+$, given by \begin{equation}\label{criticalcurve} p(\varrho)=\frac{\pi}{{\pi}/{2}+\arctan({\varrho }/({\sqrt {1-\varrho^2}}))}. \end{equation} Its inverse will be denoted by $\varrho(p)$ for $p>1$. \end{definition} The critical curve is plotted in Figure \ref{fig:cl}. Here, $\varrho (35)$ and $\varrho(2)$ are marked. Thirty-fifth moments are the key for \begin{figure} \caption{The critical curve $p(\varrho), \varrho\in(-1,1)$.} \label{fig:cl} \end{figure} the improved wavespeed result below and the special case $\varrho(2)=0$ is discussed in \cite{AD09}. We will see in Section \ref{sec:moments} that this curve is closely connected with the exit distribution of $(B^1_\tau, B^2_\tau)$ from the upper right quadrant which appeared in Proposition \ref{prop:convlaw} above. The first main theorem states that the critical curve separates two regimes (independently of $\kappa $): that of bounded moments and that of unbounded moments. \begin{theorem}\label{thm:mc} Suppose $(u_t,v_t)$ is a symbiotic branching process $($either nonspatial, continuous space or discrete space in arbitrary dimension$)$ with initial conditions $u_0=v_0=\mathbf{1}$. If $\varrho\in (-1,1)$, then, for any $\kappa>0$, the following hold for $p>1$: \begin{longlist} \item In the recurrent case, \[ \quad\varrho<\varrho(p)\quad\Leftrightarrow\quad\mathbb{E} ^{1,1}[u_t^p], \mathbb{E}^{\mathbf{1},\mathbf{1}}[u_t(k)^p] \mbox{ and } \mathbb{E}^{\mathbf{1} ,\mathbf{1}} [u_t(x)^p ] \mbox{ are bounded in }t. \] \item In the transient case, \[ \varrho<\varrho(p) \quad\Rightarrow\quad\mathbb{E}^{\mathbf{1},\mathbf{1}} [u_t(k)^p ] \mbox{ is bounded in }t. \] \end{longlist} Due to symmetry the same holds for $\mathbb{E}^{1,1}[v_t^p]$, $\mathbb{E}^{\mathbf{1},\mathbf{1} }[v_t(k)^p]$ and $\mathbb{E}^{\mathbf{1},\mathbf{1}}[v_t(x)^p]$. \end{theorem} Note that the theorem provides information about all positive real moments, not just integer moments. In the area below the critical curve in Figure \ref{fig:cl}, the moments remain bounded. On and above the critical curve, in the recurrent case, the moments grow to infinity. \begin{remark}\label{shift} For $\varrho=-1$ the curve could be extended with $p(-1)=\infty$. In terms of the previous theorem this makes sense since for $\varrho=-1$, symbiotic branching processes with initial conditions $u_0=v_0=\mathbf{1}$ are bounded by $2$. This is justified by a simple observation: for initial conditions $u_0=v_0\equiv1/2$ symbiotic branching processes with $\varrho=-1$ are solutions of the stepping stone model and, hence, bounded by $1$. Uniqueness in law of solutions implies that solutions $(u_t,v_t)$ with initial conditions $(cu_0,cv_0)$ are equal in law to solutions $c$ times solutions with initial conditions $(u_0,v_0)$. \end{remark} With this first understanding of the effect of $\varrho$ on moments, we may discuss integer moments for the discrete-space model in more detail. Let us first recall some known results for solutions $(w_t)$ of the parabolic Anderson model (see Example \ref{ex2}) where only the parameter $\kappa$ appears. Using It\^o's lemma, one sees that $m(t,k_1,\ldots,k_n):=\mathbb{E}^{\mathbf{1}}[w_t(k_1)\cdots w_t(k_n)]$ solves the (discrete-space) partial differential equation \[ \frac{\partial}{\partial t}m(t,k_1,\ldots,k_n)=\Delta m(t,k_1,\ldots ,k_n)+V(k_1,\ldots,k_n)m(t,k_1,\ldots,k_n) \] with homogeneous initial conditions. Here, the potential $V$ is given by \[ V(k_1,\ldots,k_n)=\kappa\sum_{1\leq i<j\leq n}\delta_0(k_i-k_j). \] Since $H=-\Delta-V$ is an $n$-particle Schr\"odinger operator, many properties are known from the physics literature. In particular, it is well known that in the recurrent case (the potential is nonnegative) exponential growth of solutions holds for any $\kappa>0$. By contrast, in the transient case the discrete Laplacian requires a stronger perturbation before we see exponential growth. Intuitively from the particle picture this should be true since the potential $V$ only increases solutions if particles meet, which occurs less frequently in the transient case. For the transient case (see, e.g., \cite{CM94} or \cite{GdH07} for more precise results), there is a decreasing sequence $\kappa(n)$ such that \[ \mathbb{E}^{\mathbf{1}}[w_t(k)^n]\mbox{ is bounded in }t \quad\Leftrightarrow\quad\kappa<\kappa(n) \] and for the Lyapunov exponents \[ \gamma_n(\kappa):=\lim_{t\rightarrow\infty}\frac{1}{t}\log\mathbb{E} ^{\mathbf{1} }[w_t(k)^n]>0 \quad\Leftrightarrow\quad\kappa>\kappa(n). \] These results can be proved with the $n$-particle path-integral representation in which solutions are expressed as \[ m(t,k_1,\ldots,k_n)=\mathbb{E}\bigl[e^{\kappa\int_0^t V(X_s^1,\ldots,X_s^n) \,ds} \bigr], \] where $(X^1_t),\ldots,(X^n_t)$ are independent simple random walks started in\break $k_1,\ldots,k_n$. Coming back to the symbiotic branching model, we ask whether or not the $n$th Lyapunov exponents \[ \gamma_{n}(\varrho,\kappa):=\lim_{t\rightarrow\infty}\frac {1}{t}\log\mathbb{E} ^{\mathbf{1},\mathbf{1}}[u_t(k)^{n}] \] exist and in which cases $\gamma_n(\varrho,\kappa)$ is strictly positive. As for the parabolic Anderson model, there is a system of partial differential equations describing the moments (see Proposition 16 of \cite{EF04} for the continuous-space model) and an $n$-particle path-integral representation of the moments. In addition to the independent motion, the particles now carry a color which randomly changes if particles of the same color stay at the same site (see Lemma \ref{la:mdual}). With $L_t^=$ denoting collision times of particles of same color and $L_t^{\neq}$ denoting collision times of particles of different colors, the path-integral representation of moments reads \[ \mathbb{E}^{\mathbf{1},\mathbf{1}}[u_t(k)^n]=\mathbb{E}\bigl[e^{\kappa(L_t^=+\varrho L_t^{\neq})} \bigr]. \] This representation is more involved than the path-integral representation for the parabolic Anderson model since, in addition to the motion of particles, a second stochastic mechanism is included. Nonetheless, we use it to prove the following theorem which reveals that even in the recurrent case a nontrivial transition occurs. \begin{theorem}\label{thm:im} For solutions of $\mathrm{dSBM}(\varrho,\kappa)_{\mathbf{1},\mathbf{1}}$, in any dimension, the following hold for $n\in\mathbb{N}, n>1$: \begin{longlist} \item$\gamma_{n}(\varrho,\kappa)$ exists for any $\varrho \in[-1,1]$, $\kappa>0$, \item$\gamma_{n}(\varrho(n),\kappa)=0$ for any $\kappa>0$, \item for any $\varrho>\varrho(n)$ there is a critical $\kappa(n)$ such that $\gamma_{n}(\varrho,\kappa)>0$ if $\kappa >\kappa(n)$. \end{longlist} \end{theorem} Combined with Theorem \ref{thm:mc}, parts (ii) and (iii) emphasize the ``criticality'' of the critical curve: for $\varrho<\varrho(n)$, moments stay bounded, for $\varrho=\varrho(n)$ moments grow subexponentially fast to infinity, and for $\varrho>\varrho(n)$ moments grow exponentially fast if $\kappa$ is large enough. \begin{remark} As discussed above, for the parabolic Anderson model it is natural that in the transient case perturbing the critical case does not immediately yield exponential growth, whereas perturbing the recurrent case does immediately lead to exponential growth. It is clear that in the transient case the gap in (iii) of Theorem \ref{thm:im} is really necessary: for small $\kappa$ moments of the parabolic Anderson model are bounded. Since moments of symbiotic branching are dominated by moments of the parabolic Anderson model (see Lemma \ref{la:mdual}), for small $\kappa$ moments are bounded for all $\varrho$. \end{remark} In the case $p\notin\mathbb{N}$ there seems to be no reason why exponential growth should fail. Unfortunately, in this case there is no moment duality and hence the most useful tool to analyze exponential growth is not available. \begin{conj} \label{conj:1} In the recurrent case the moment diagram for symbiotic branching (Figure \ref{fig:cl}) describes the moments as follows: pairs $(\varrho ,p)$ below the critical curve correspond precisely to bounded moments, pairs at the critical curve correspond to moments which grow subexponentially fast to infinity and pairs above to the critical curve correspond to exponentially growing moments. \end{conj} A deeper understanding of the Lyapunov exponents as functions of $\varrho,\kappa$ remains mainly open (for an upper bound see Proposition \ref{up}). For second moments [$\varrho(2)=0$] this is carried out in \cite{AD09}. It is shown that exponential growth holds for $\varrho>0$ and arbitrary $\kappa>0$ in the recurrent case, whereas only for $\kappa>2/(\varrho G_{\infty}(0,0))$ in the transient case. Here $G_{\infty}$ denotes the Green function of the simple random walk. The exponential (and subexponential) growth rates were analyzed in detail by Tauberian theorems. A direct application of Theorem \ref{thm:im} is so-called intermittency of solutions. One says a spatial system with Lyapunov exponents $\gamma _p$ is $p$-intermittent if \[ \frac{\gamma_p}{p}<\frac{\gamma_{p+1}}{p+1}. \] Intermittent systems concentrate on few peaks with extremely high intensity (see~\cite{GM90}). The results above show that as $\varrho$ tends to $-1$, solutions (at least for large $\kappa$) are $p$-intermittent for $p$ tending to infinity. This holds since for fixed $\varrho$, the $p$th moments are bounded if $(\varrho,p)$ lies below the critical curve. Increasing $p$ (and $\kappa$ if necessary) there is a first $p$ such that the $p$th Lyapunov exponent is positive. Intermittency for higher exponents suggests that the effect gets weaker. This is to be expected since for $\varrho=-1$ solutions with homogeneous initial conditions are bounded and, hence, solutions do not produce high peaks at all. Making this effect more precise, in particular combined with the effect of Proposition \ref{prop:convlaw}, is an interesting task for the future. \subsection{Speed of propagation of the interface} Let us conclude with a direct application of the moment bounds. Here, we will be concerned with an improved upper bound on the speed of the propagation of the interface of continuous-space symbiotic branching processes which served to some extent as the motivation for this work. To explain this, we need to introduce the notion of the interface of continuous-space symbiotic branching processes introduced in \cite{EF04}. \begin{definition}\label{def:ifc} The interface at time $t$ of a solution $(u_t,v_t)$ of the symbiotic branching model $\mathrm{cSBM}(\varrho,\kappa)_{u_0, v_0}$ with $\varrho\in[-1,1]$ is defined as \[ \mathrm{Ifc}_t = \operatorname{cl} \{x\dvtx u_t(x) v_t(x) > 0 \}, \] where $\operatorname{cl}(A)$ denotes the closure of the set $A$ in $\mathbb{R}$. \end{definition} In particular, we will be interested in complementary Heaviside initial conditions \[ u_0(x) = \mathbf{1}_{\mathbb{R}^-}(x) \quad\mbox{and}\quad v_0(x) = \mathbf{1}_{\mathbb{R} ^+}(x),\qquad x \in\mathbb{R}. \] The main question addressed in \cite{EF04} is whether for the above initial conditions the so-called compact interface property holds, that is, whether the interface is compact at each time almost surely. This is answered affirmatively in Theorem 6 in \cite{EF04}, together with the assertion that the interface propagates with at most linear speed, that is, for each $\varrho\in[-1,1]$ there exists a constant $c>0$ and a finite random-time $T_0$ so that almost surely for all $T \ge T_0$ \[ \bigcup_{t \le T} \operatorname{Ifc}_t \subseteq[-cT, cT ]. \] Heuristically, due to the scaling property of the symbiotic branching model (Lemma 8 of \cite{EF04}) one expects that the interface should move with a square-root speed. Indeed, with the help of Theorem \ref {thm:mc} one can strengthen their result, at least for sufficiently small $\varrho$, to obtain almost square-root speed. \begin{theorem}\label{cor:wavespeed} Suppose $(u_t,v_t)$ is a solution of $\mathrm{cSBM}(\varrho,\kappa )_{1_{\mathbb{R}^-}, 1_{\mathbb{R}^+}}$ with $\varrho< \varrho(35)$ and $\kappa>0$. Then there is a constant $C>0$ and a finite random-time $T_0$ such that almost surely \[ \bigcup_{t\leq T} \operatorname{Ifc}_t \subseteq\bigl[-C\sqrt{T\log (T)},C\sqrt{T\log(T)} \bigr] \] for all $T>T_0$. \end{theorem} The restriction to $\varrho<\varrho(35)$ is probably not necessary and only caused by the technique of the proof. Though $\varrho(35)\approx -0.9958$ is rather close to $-1$, the result is interesting. It shows that sub-linear speed of propagation is not restricted to situations in which solutions are uniformly bounded as they are for $\varrho=-1$. The proof is based on the proof of \cite{EF04} for linear speed which carries over the proof of \cite{T95} for the stepping stone model to nonbounded processes. We are able to strengthen the result by using a better moment bound which is needed to circumvent the lack of uniform boundedness. \begin{remark}\label{r} We believe that, at least for $\varrho\leq0$, the speed of propagation should be at most $C'\sqrt t$, for some suitable constant $C'$, that is, for all $T$ greater than some $T'>0$, \[ \bigcup_{t\leq T} \operatorname{Ifc}_t \subseteq\bigl[-C'\sqrt {T},C'\sqrt {T} \bigr]. \] However, it seems unclear how to obtain such a refinement of Theroem \ref{cor:wavespeed} based on our moment results and the method of \cite{T95} (resp., \cite{EF04}). As subexponential bounds of higher moments cannot be avoided (see the proof of the fluctuation term estimate Lemma \ref{la:ma}), our results on the behavior of higher moments show that at present, in light of Conjecture \ref{conj:1}, one can only hope for stronger results for very small $\varrho$. To overcome this limitation, new methods need to be employed. The authors think that a possible approach could be based on the scaling property (Lemma 8 of \cite{EF04}) and recent results by Klenke and Oeler \cite{KO09}. Recall that the scaling property states that if $(u_t, v_t)$ is a solution to $\operatorname{cSBM}(\varrho, \kappa)_{u_0, v_0}$, then \[ (u_t(x)^K, v_t(x)^K ) := \bigl(u_{Kt}\bigl(\sqrt K x\bigr), v_{Kt}\bigl(\sqrt K x\bigr) \bigr),\qquad x \in\mathbb{R}, K >0, \] is a solution to $\operatorname{cSBM}(\varrho, K \cdot\kappa )_{u^K_0, v^K_0}$ (with suitably transformed initial states $u^K_0, v^K_0$). In other words, a diffusive time--space rescaling leads to the original model with a suitably increased branching rate $\kappa$. Klenke and Oeler \cite{KO09} show that, at least for the mutually catalytic model in discrete space, a nontrivial limiting process as $\kappa\to\infty $ exists. This limit is called ``infinite rate mutually catalytic branching process'' (see also \cite{KM09a,KM09b} for a further discusion). In particular, in Corollary 1.2 of \cite{KO09} they claim that, under suitable assumptions, a nontrivial interface for the limiting process exists, which would in turn predict a square-root speed of propagation in our case. However, to make this approach rigorous is beyond the scope of the present paper. \end{remark} \begin{remark}[(Shape of the interface)] Note that our results give only limited information about the shape of the interface. For the case $\varrho=-1$, that is, with locally constant total population size, it is shown in \cite{MT97} that there exists a unique stationary interface law, which may therefore be interpreted as a ``stationary wave'' whose position fluctuates at the boundaries, according to \cite{T95}, like a Brownian motion, hence explaining the square-root speed (note that for both results, suitable bounds on fourth mixed moments are required). However, for $\varrho> -1$, the population sizes of the interface are expected to fluctuate significantly and it seems unclear how this affects the shape and speed of the interface, in particular the formation of a ``stationary wave.'' The significance of fourth mixed moments might even lead to a phase-transition in $\varrho$. This gives rise to many interesting open questions. \end{remark} \section{Basic properties and duality}\label{sec:bpd} In this section we review the setting and properties of the discrete-space model, whereas for continuous-space we refer to~\cite{EF04}. Note that instead of using the state space of tempered functions alternatively we may use a suitable Liggett--Spitzer space. As the results are only presented for the discrete Laplacian this does not play a crucial role. For a discussion of the mutually catalytic branching model in the Liggett--Spitzer space see \cite{CDG04}. \subsection{Basic properties} \label{subsec:bp} For functions $f,g\dvtx\mathbb{Z}^d \to\mathbb{R}$ we abbreviate $\langle f,g\rangle =\sum _kf(k)g(k)$. With $\phi_{\lambda}(k)=e^{\lambda\|k\|}$ the space of pairs of tempered sequences is defined by \[ M^2_{\mathrm{tem}}= \{(u,v) \| u,v\dvtx\mathbb{Z}^d\rightarrow\mathbb{R}_{\geq0} , \langle u,\phi_{\lambda}\rangle,\langle v,\phi_{\lambda}\rangle <\infty\ \forall\lambda<0 \}. \] The space of continuous paths is denoted by \[ \Omega_{\mathrm{tem}}=C(\mathbb{R}_{\geq0},M_{\mathrm{tem}}^2). \] Similarly, the space of pairs of rapidly decreasing sequences is defined by \[ M^2_{\mathrm{rap}}= \{(u,v) \| u,v\dvtx\mathbb{Z}^d\rightarrow\mathbb{R}_{\geq0} , \langle u,\phi_{\lambda}\rangle,\langle v,\phi_{\lambda}\rangle <\infty\ \forall\lambda>0 \} \] and the corresponding path space by \[ \Omega_{\mathrm{rap}}=C(\mathbb{R}_{\geq0},M_{\mathrm{rap}}^2). \] Weak solutions are defined as in \cite{DP98} for $\varrho=0$. In much the same way as for Theorems 1.1 and 2.2 of \cite{DP98}, we obtain existence and the Green-function representation. \begin{proposition}\label{prop:bp1} Suppose $(u_0,v_0)\in M_{\mathrm{tem}}^2$ (resp., $M^2_{\mathrm{rap}}$), $\varrho\in [-1,1]$ and $\kappa>0$. Then there is a weak solution of $\mathrm {dSBM}(\varrho,\kappa)_{u_0,v_0}$ such that $(u_t,v_t)\in\Omega_{\mathrm{tem}}$ (resp., $\Omega_{\mathrm{rap}}$) and for all $(\phi,\psi)\in M^2_{\mathrm{rap}}$ (resp., $M^2_{\mathrm{tem}}$) \begin{eqnarray}\label{12} \langle u_t,\phi\rangle&=&\langle u_0, P_t\phi\rangle+\sum_{j\in\mathbb{Z}^d}\int_0^t P_{t-s}\phi(j)\sqrt {\kappa u_s(j)v_s(j)} \,dB^1_s(j),\\ \label{13} \langle v_t,\psi\rangle&=&\langle v_0, P_t\psi\rangle+\sum_{j\in\mathbb{Z}^d}\int_0^t P_{t-s}\psi(j)\sqrt {\kappa u_s(j)v_s(j)} \,dB^2_s(j), \end{eqnarray} where $P_tf(k)=\sum_{j\in\mathbb{Z}^d}p_t(j,k)f(j)$ is the semigroup associated to the simple random walk. In particular, we have \begin{eqnarray} \label{14} u_t(k)&=&P_tu_0(k)+\sum_{j\in\mathbb{Z}^d}\int_0^tp_{t-s}(j,k)\sqrt{\kappa u_s(j)v_s(j)} \,dB^1_s(j),\\ \label{15} v_t(k)&=&P_tv_0(k)+\sum_{j\in\mathbb{Z}^d}\int_0^tp_{t-s}(j,k)\sqrt{\kappa u_s(j)v_s(j)} \,dB^2_s(j). \end{eqnarray} The covariation structure of the Brownian motions is given by (\ref{eq:dcv}). \end{proposition} In fact, (\ref{12}), (\ref{13}) can be seen as the discrete-space versions of the martingale problem of Definition 3 in \cite{EF04}. Further, (\ref{14}), (\ref{15}) are the discrete-space versions of the convolution form given in Corollary 20 of \cite{EF04}. For the proofs of the longtime behavior of laws and moments, the key step is to transfer to the total mass processes $\langle u_t,\mathbf{1} \rangle, \langle v_t,\mathbf{1}\rangle$. To this end, in a similar way to Proposition \ref{prop:bp1}, we define \[ M_{F}^2= \{(u,v) \| u,v\dvtx\mathbb{Z}^d\rightarrow\mathbb{R}_{\geq0}, \langle u,1\rangle,\langle v,1\rangle<\infty\} \] and \[ \Omega_{F}=C(\mathbb{R}_{\geq0},M_{F}^2). \] For summable initial conditions we obtain the following crucial martingale characterization. \begin{proposition}\label{prop:tmmart} If $(u_0,v_0)\in M^2_F$, then each solution of $\mathrm{dSBM}(\varrho ,\kappa)_{u_0,v_0}$ has the following properties: $(u_t,v_t)\in\Omega_F$ and $\langle u_t,\mathbf{1}\rangle, \langle v_t,\mathbf{1} \rangle$ are nonnegative, continuous, square-integrable martingales with square-functions \[ [\langle u_{\cdot},\mathbf{1}\rangle]_t= [\langle v_{\cdot},\mathbf{1} \rangle]_t=\kappa\int_0^t\langle u_s,v_s\rangle \,ds \] and \[ [\langle u_{\cdot},\mathbf{1}\rangle,\langle v_{\cdot},\mathbf{1}\rangle ]_t=\varrho\kappa\int_0^t\langle u_s,v_s\rangle \,ds. \] \end{proposition} We omit the proofs since they are basically standard. The only step where one needs to be careful is the existence proof. As usual for such models one first restricts the space to bounded subsets (boxes) of $\mathbb{Z} ^d$, where standard Markov process theory applies. Enlarging the boxes one obtains a sequence of processes which are shown to converge to a limiting process solving $\mathrm{dSBM}(\varrho,\kappa)_{u_0,v_0}$. To prove tightness of the approximating sequence, the moments need to be bounded uniformly in the size of the boxes. Here, more care than for $\varrho=0$ in \cite{DP98} is needed. The uniform moment bound can, for instance, be achieved using a colored particle moment duality for each box similar to the one of Lemma \ref{la:mdual}. \subsection{Dualities}\label{subsec:dualities} The symbiotic branching model exhibits an exceptionally rich duality structure, providing powerful tools for the analysis of the longtime properties. \subsubsection{Colored particle moment dual}\label{subsec:cpd} We now recall the two-colors particle moment-duality introduced in Section 3.1 of \cite{EF04}. Since the dual Markov process is presented rigorously in \cite{EF04} we only sketch the pathwise behavior. To find a suitable description of the mixed moment \[ \mathbb{E}^{u_0,v_0}[u_t(k_1)\cdots u_t(k_{n})v_t(k_{n+1})\cdots v_t(k_{n+m})], \] $n+m$ particles are located in $\mathbb{Z}^d$. Each particle moves as a continuous-time simple random walk independent of all other particles. At time $0$, $n$ particles of color $1$ are located at positions $k_1,\ldots,k_n$ and $m$ particles of color $2$ are located at positions $k_{n+1},\ldots,k_{n+m}$. For each pair of particles, one of the pair changes color when the time the two particles have spent in the same site, while both have same color, first exceeds an (independent) exponential time with parameter $\kappa$. Let \begin{eqnarray} L_t^=&=&\mbox{total collision time of all pairs of same colors up to time }t,\\ L_t^{\neq}&=&\mbox{total collision time of all pairs of different colors up to time }t,\\ l^1_t(a)&=&\mbox{number of particles of color }1\mbox{ at site }a\mbox{ at time $t$},\\ l^2_t(a)&=&\mbox{number of particles of color }2\mbox{ at site }a\mbox{ at time $t$},\\ (u_0,v_0)^{l_t}&=&\prod_{a\in\mathbb{Z}^d}u_0(a)^{l_t^1(a)}v_0(a)^{l_t^2(a)}. \end{eqnarray} Note that since there are only $n+m$ particles, the infinite product is actually a finite product and hence well defined. The following lemma is taken from Section~3 of~\cite{EF04}. \begin{lemma}\label{la:mdual} Let $(u_t,v_t)$ be a solution of $\mathrm{dSBM}(\varrho,\kappa )_{u_0,v_0}$, $\kappa>0$ and $\varrho\in[-1,1]$. Then, for any $k_i\in\mathbb{Z} ^d$, $t\geq0$, \[ \mathbb{E}^{u_0,v_0}[u_t(k_1)\cdots u_t(k_{n})v_t(k_{n+1})\cdots v_t(k_{n+m})]=\mathbb{E}\bigl[(u_0,v_0)^{l_t}e^{\kappa(L_t^=+\varrho L_t^{\neq })} \bigr], \] where the dual process behaves as explained above. \end{lemma} Note that for homogeneous initial conditions $u_0=v_0=\mathbf{1}$, the first factor in the expectation of the right-hand side equals $1$. In the special case $\varrho=1$, \mbox{$u_0=v_0=\mathbf{1}$} Lemma \ref{la:mdual} was already stated in \cite{CM94}, reproved in \cite{GdH07} and used to analyze the Lyapunov exponents of the parabolic Anderson model. For $\varrho\neq1$, the difficulty of the dual process is based on the two stochastic effects: on the one hand, one has to deal with collision times of random walks which were analyzed in \cite{GdH07}; additionally, particles have colors either $1$ or $2$ which change dynamically. \begin{remark}\label{rdual} Similar dualities hold for $\mathrm{cSBM}$ and $\mathrm{SBM}$. For continuous-space, the random walks are replaced by Brownian motions and the collision times of the random walks by collision local times of the Brownian motions (see Section~4.1 in \cite{EF04}). The simplest case is the nonspatial symbiotic branching model where the particles stay at the same site and local times are replaced by real times (see Theorem 3.2 of \cite{R95} or Proposition A5 of \cite{DFX05}). \end{remark} \subsubsection{Self-duality}\label{subsec:sduality} Mytnik \cite{M99} introduced a self-duality for the continuous-space mutually catalytic branching model to obtain uniqueness of solutions of the corresponding martingale problem. This can be extended to symbiotic branching models for $\varrho\in(-1,1)$ as shown in Proposition 5 of \cite{EF04}. The discrete-space self-duality for $\varrho=0$ was proved in Theorem 2.4 of \cite{DP98}. We first need more spaces of sequences: \[ E= \{ (x,y)\dvtx(x,\|y\|)\in M^2_{\mathrm{tem}}, \|y(k)\|\leq x(k)\ \forall k\in\mathbb{Z} ^d \} \] and \[ \tilde E= \{ (x,y)\in E\dvtx x\in M_{\mathrm{rap}} \}\supset\{ (x,y)\in E\dvtx x\mbox{ has bounded support} \}=\tilde{E}_f. \] In the sequel, the space $E$ and its subspaces will be used for $(x,y)=(u_t+v_t,u_t-v_t)$. The duality function for $\varrho\in(-1,1)$ maps $ E \times\tilde E$ to $\mathbb{C}$ via \begin{equation}\label{eq:selfdf2} H (u,v,\tilde{u}, \tilde{v} )=\exp\bigl(-\sqrt{1-\varrho}\langle u,\tilde{u} \rangle+i\sqrt{1+\varrho}\langle v,\tilde{v} \rangle\bigr). \end{equation} With this definition the generalized Mytnik duality states: \begin{lemma}\label{la:sduality} For $\varrho\in(-1,1)$, $\kappa>0$, $(u_0,v_0)\in M^2_{\mathrm{tem}}$ and $(\tilde u_0, \tilde v_0)\in M^2_{\mathrm{rap}}$ let $(u_t,v_t)$ be a solution of $\mathrm{dSBM}(\varrho,\kappa)_{u_0, v_0}$ and $(\tilde {u}_t,\tilde {v}_t)$ be a solution of $\mathrm{dSBM}(\varrho,\kappa)_{\tilde u_0, \tilde v_0}$. Then the following holds: \begin{eqnarray} && \mathbb{E}^{u_0,v_0} [H(u_t+v_t,u_t-v_t,\tilde u_0+\tilde v_0, \tilde u_0-\tilde v_0) ]\\ &&\qquad=\mathbb{E}^{\tilde{u}_0,\tilde{v}_0} [H(u_0+v_0,u_0-v_0,\tilde{u}_t+\tilde v_t,\tilde u_t-\tilde{v}_t) ]. \end{eqnarray} \end{lemma} Analogously, the self-duality relation holds for the nonspatial model with duality function \[ H^0(u,v,\tilde u,\tilde v)=\exp\bigl(-\sqrt{1-\varrho}u\tilde{u}+i\sqrt {1+\varrho}v\tilde{v} \bigr), \] mapping $(\mathbb{R}_{\geq0}\times\mathbb{R}_{\geq0})^2$ to $\mathbb{C}$. \section{Weak longtime convergence}\label{sec:comnvlaw} In this section we discuss weak longtime convergence of symbiotic branching models and prove Proposition \ref{prop:convlaw}. We proceed in two steps: first, we prove convergence in law to some limit law following the proof of \cite{DP98} for $\varrho=0$. Second, to characterize the limit law for the spatial models, we reduce the problem to the nonspatial model. \begin{proposition}\label{prop:wconv} Let $\varrho\in(-1,1), \kappa>0$ and $(u_t,v_t)$ a solution of either $\mathrm{cSBM}(\varrho,\kappa)_{u_0, v_0}$ or $\mathrm {dSBM}(\varrho,\kappa)_{u_0, v_0}$ with initial conditions $u_0=\mathbf{u}, v_0=\mathbf{v}$. Then, as $t \to\infty$, the law of $(u_t,v_t)$ converges weakly on $M_{\mathrm{tem}}^2$ to some limit $(u_{\infty},v_{\infty})$. \end{proposition} \begin{pf} The proof is only given for the discrete spatial case and the continuous case is completely analogous. Let us first recall the strategy of \cite{DP98} for $\varrho=0$ which can also be applied with the generalized self-duality required here. Convergence of $(u_t,v_t)$ in $M_{\mathrm{tem}}^2$ follows from convergence of $(u_t+v_t,u_t-v_t)$ in $E$. Using Lemma 2.3(c) of \cite{DP98}, it suffices to show convergence of $\mathbb{E}^{\mathbf{u},\mathbf{v}}[H(u_t+v_t,u_t-v_t,\phi, \psi)]$ for all $(\phi,\psi )\in \tilde E_f$. Furthermore, the limit $(u_{\infty},v_{\infty})$ is uniquely determined by $\mathbb{E}^{\mathbf{u},\mathbf{v}}[H(u_{\infty}+v_{\infty },u_{\infty }-v_{\infty},\phi,\psi)]$ (see Lemma 2.3(b) of \cite{DP98}). Hence, it suffices to show convergence of \begin{equation}\label{456} \mathbb{E}^{\mathbf{u},\mathbf{v}}[H(u_t+v_t,u_t-v_t,\phi,\psi)]=\mathbb{E}^{\mathbf{u},\mathbf{v}} \bigl[e^{-\sqrt {1-\varrho}\langle u_t+v_t,\phi\rangle+i\sqrt{1+\varrho}\langle u_t-v_t,\psi\rangle} \bigr],\hspace{-32pt} \end{equation} for all $(\phi,\psi)\in\tilde{E}_f$. Note that the technical condition of Lemma 2.3(c) of \cite{DP98} is fullfilled since due to Proposition \ref{prop:bp1} \[ \mathbb{E}^{\mathbf{u},\mathbf{v}}[\langle u_t+v_t,\phi_{-\lambda}\rangle]=(u+v)\langle \mathbf{1} ,P_t \phi_{-\lambda}\rangle<C<\infty. \] To ensure convergence of (\ref{456}) we employ the generalized Mytnik self-duality of Lemma \ref{la:sduality} with $\tilde{u}_0:=\frac {\phi +\psi}{2},\tilde{v}_0:=\frac{\phi-\psi}{2}$: \begin{eqnarray}\label{eq:dualconv} &&\mathbb{E}^{\mathbf{u},\mathbf{v}} \bigl[e^{-\sqrt{1-\varrho} \langle u_t+v_t,\phi\rangle +i\sqrt{1+\varrho} \langle u_t-v_t,\psi\rangle} \bigr] \nonumber\\ &&\qquad=\mathbb{E}^{u_0,v_0} \bigl[e^{-\sqrt{1-\varrho}\langle u_t+v_t,\tilde {u}_0+\tilde{v}_0\rangle +i\sqrt{1+\varrho}\langle u_t-v_t,\tilde{u}_0-\tilde{v}_0\rangle} \bigr] \nonumber\\[-8pt]\\[-8pt] &&\qquad=\mathbb{E}^{\tilde{u}_0,\tilde{v}_0} \bigl[e^{-\sqrt{1-\varrho}\langle u_0+v_0, \tilde{u}_t+\tilde{v}_t\rangle+i\sqrt{1+\varrho}\langle u_0-v_0, \tilde{u}_t-\tilde{v}_t\rangle} \bigr]\nonumber\\ &&\qquad=\mathbb{E}^{\tilde{u}_0,\tilde{v}_0} \bigl[e^{-\sqrt{1-\varrho}(u+v)\langle \mathbf{1}, \tilde{u}_t+\tilde{v}_t\rangle+i\sqrt{1+\varrho}(u-v)\langle \mathbf{1}, \tilde{u}_t-\tilde{v}_t\rangle} \bigr].\nonumber \end{eqnarray} By assumption, $\tilde u_0, \tilde v_0$ have compact support and hence by Proposition \ref{prop:tmmart} the total-mass processes $\langle\mathbf{1}, \tilde{u}_t \rangle$ and $\langle\mathbf{1}, \tilde {v}_t \rangle$ are nonnegative martingales. By the martingale convergence theorem $\langle\mathbf{1}, \tilde{u}_t \rangle$ and $\langle\mathbf{1}, \tilde {v}_t \rangle$ converge almost surely to finite limits denoted by $\langle\mathbf{1}, \tilde{u}_{\infty} \rangle$, $\langle\mathbf {1},\tilde {v}_{\infty} \rangle$. Finally, the dominated convergence theorem implies convergence of the right-hand side of (\ref{eq:dualconv}) to \begin{equation}\label{eq:elimit} \mathbb{E}^{\tilde u_0,\tilde v_0} \bigl[e^{-\sqrt{1-\varrho}(u+v)\langle \mathbf{1},\tilde{u}_{\infty}+\tilde{v}_{\infty}\rangle+i\sqrt {1+\varrho}(u-v) \langle\mathbf{1},\tilde{u}_{\infty}-\tilde{v}_{\infty}\rangle} \bigr]. \end{equation} Combining the above, we have proved convergence of \[ \mathbb{E}^{\mathbf{u},\mathbf{v}} \bigl[e^{-\sqrt{1-\varrho} \langle u_t+v_t,\phi\rangle +i\sqrt{1+\varrho}\langle u_t-v_t,\psi\rangle} \bigr], \] which ensures weak convergence of $(u_t,v_t)$ in $M_{\mathrm{tem}}^2$ to some limit which is uniquely determined by (\ref{eq:elimit}). \end{pf} Again, as in Remark \ref{all}, the previous proposition can be proved for nondeterministic initial conditions as in \cite{CKP00}. The rest of this section is devoted to identifying the limit $(u_{\infty },v_{\infty})$ in the recurrent case. Before completing the proof of Theorem \ref{prop:convlaw} we discuss a version of Knight's extension of the Dubins--Schwarz theorem (see \cite{KS98}, 3.4.16) for nonorthogonal continuous local martingales. \begin{lemma}\label{la:eds} Let $(N_t)$ and $(M_t)$ be continuous local martingales with $N_0=M_0=0$ almost surely. Assume further that, for $t \ge0$, \[ [ M_{\cdot},M_{\cdot}]_t=[ N_{\cdot},N_{\cdot}]_t \quad\mbox{and}\quad [ M_{\cdot},N_{\cdot}]_t=\varrho[ M_{\cdot},M_{\cdot}]_t\qquad \mbox{a.s.}, \] where $\varrho\in[-1,1]$. If $[ M_{\cdot},M_{\cdot}]_{\infty }=\infty$ a.s., then \[ (B^1_t,B^2_t):=\bigl(M_{T(t)},N_{T(t)}\bigr) \] is a pair of Brownian motions with covariances $[ B^1_{\cdot },B^2_{\cdot }]_t=\varrho t$, where \begin{equation}\label{eq:timeshift} T(t)=\inf\{s\dvtx[ M_{\cdot},M_{\cdot}]_s>t \}. \end{equation} \end{lemma} \begin{pf} It follows from the Dubins--Schwarz theorem that $B^1,B^2$ are each Brownian motions. Further, by the definition of $T(t)$ we obtain the claim \[ [ B^1_{\cdot}, B^2_{\cdot}]_t=[ M_{\cdot}, N_{\cdot}]_{T(t)}=\varrho[ M_{\cdot},M_{\cdot}]_{T(t)}=\varrho t. \] \upqed\end{pf} \begin{remark} If $T^:=[ M_{\cdot},M_{\cdot}]_{\infty}<\infty$ the situation becomes slightly more delicate but one can use a local version of Lemma \ref {la:eds}. Indeed, define, for $t \ge0$, \begin{equation}\label{eq:delicate} B^1_t:= \cases{ M_{T(t)}, &\quad for $t < T^$,\cr M_{T^}, &\quad for $t \geq T^$,} \end{equation} where the time-change $T$ is given in (\ref{eq:timeshift}) and define $B^2$ analogously for $N$ (recall that $[M_{\cdot},M_{\cdot}]_t=[ N_{\cdot},N_{\cdot}]_t$). Then the processes $B^1, B^2$ are Brownian motions stopped at time $T^$. The covariance is again given by \[ [ B^1_{\cdot}, B^2_{\cdot}]_{t\wedge T^} = \varrho(t\wedge T^),\qquad t \ge0. \] \end{remark} For the rest of this section let $B^1,B^2$ be standard Brownian motions with covariance \begin{equation} [ B^1_{\cdot},B^2_{\cdot}]_t=\varrho t \end{equation} started in $u,v$, denote their expectations by $E^{u,v}$, and let \[ \tau=\inf\{t\dvtx B^1_tB^2_t=0 \}. \] The above discussion can now be used to understand the longtime behavior of symbiotic branching processes. We start by giving a proof for the nonspatial symbiotic branching model and then modify the proof to capture the corresponding result for the spatial models. \begin{proposition}\label{prop:sconv} Let $(u_t,v_t)$ be a solution of $\mathrm{SBM}(\varrho,\kappa)_{u,v}$. Then, as $t\to\infty$, $(u_t,v_t)$ converges almost surely to some $(u_{\infty},v_{\infty})$. Furthermore, $\mathcal{L}^{u,v}(u_{\infty },v_{\infty})=\mathcal{L}^{u,v}(B^1_{\tau},B^2_{\tau})$ with $B^1_{\tau },B^2_{\tau}$ from Proposition \ref{prop:convlaw}. \end{proposition} \begin{pf} Solutions of the nonspatial symbiotic branching model are nonnegative martingales and hence converge almost surely. This implies the first part of the claim and it only remains to characterize the limit. Obviously, the $L^2$-martingales $(u_t),(v_t)$ satisfy the cross-variation structure assumptions of Lemma~\ref{la:eds} and, thus, $(u_{t},v_{t})= (B^1_{T^{-1}(t)}, B^2_{T^{-1}(t)} )$. To obtain the result, we need to check that $T^{-1}(\infty)= \tau$. By definition of $\mathrm{SBM}$, the time-change is given by \begin{equation}\label{eq:invtime} T^{-1}(t) = [ u_\cdot, u_\cdot]_t = \biggl[ \int_0^\cdot\sqrt{\kappa u_s u_s} \,dB^1_s, \int_0^\cdot\sqrt{\kappa u_s v_s} \,dB^1_s \biggr]_t = \kappa\int_0^t u_s v_s \,ds.\hspace{-32pt} \end{equation} To see that $T^{-1}(\infty)=\tau<\infty$, first note that $T^{-1}(t) \le\tau$ for all $t\geq0$. This is true since $u_t=B^1_{T^{-1}(t)}, v_t=B^2_{T^{-1}(t)}$ and solutions of $\mathrm{SBM}$ are nonnegative. To argue that $T^{-1}(t)$ increases to $\tau$, more care is needed. Since the martingales converge almost surely, $T^{-1}(t)$ converges to some value $a\leq\tau$. Suppose $a<\tau$, then $(u_t,v_t)$ converges to some $(x,y)$ with $x,y>0$. This yields a contradiction since $T^{-1}(t)=\kappa\int_0^tu_sv_s \,ds$ would increase to infinity. Hence, almost surely, \[ (u_t,v_t)= \bigl(B^1_{T^{-1}(t)},B^2_{T^{-1}(t)} \bigr) \stackrel {t\rightarrow \infty}{\rightarrow}\bigl(B^1_{T^{-1}(\infty)},B^2_{T^{-1}(\infty )}\bigr)=(B^1_{\tau},B^2_{\tau}). \] \upqed\end{pf} In particular, the proof of Proposition \ref{prop:sconv} provides an important relation for $(B^1_{\tau},B^2_{\tau})$. As remarked below Lemma \ref{la:sduality}, the self-duality also works in the nonspatial model: \begin{eqnarray} &&\mathbb{E}^{u_0,v_0}\bigl[e^{-\sqrt{1-\varrho}(u_t+v_t)(\tilde{u}_0+\tilde{v}_0 )+i\sqrt{1+\varrho}(u_t-v_t)(\tilde{u}_0-\tilde{v}_0 )}\bigr]\\ &&\qquad=\mathbb{E}^{\tilde{u}_0,\tilde{v}_0}\bigl[e^{-\sqrt{1-\varrho }(u_0+v_0)(\tilde {u}_t+\tilde{v}_t)+i\sqrt{1+\varrho}(u_0-v_0)(\tilde{u}_t-\tilde {v}_t )}\bigr], \end{eqnarray} where both $(u_t,v_t)$ and $(\tilde{u}_t,\tilde{v}_t)$ are solutions of $\mathrm{SBM}(\varrho,\kappa)$ with different initial conditions. As shown in the proof of Proposition \ref{prop:sconv}, $(u_t,v_t)$ [resp., $(\tilde{u}_t,\tilde{v}_t)$] converges almost surely to $(B^1_{\tau },B^2_{\tau})$ with initial condition $(u_0,v_0)$ [resp., $(\tilde u_0,\tilde v_0$)]. Using dominated convergence, this shows the following duality relation for $(B^1_{\tau},B^2_{\tau})$ when started in initial conditions $(u,v)$, $(\tilde{u},\tilde{v})$: \begin{eqnarray}\label{dual0} &&E^{u,v} [H^0(B^1_{\tau}+B^2_{\tau},B^1_{\tau}-B^2_{\tau},\tilde u+\tilde v,\tilde u-\tilde v) ]\nonumber\\[-8pt]\\[-8pt] &&\qquad=E^{\tilde{u},\tilde{v}} [H^0(B^1_{\tau}+B^2_{\tau},B^1_{\tau }-B^2_{\tau},u+v,u-v) ].\nonumber \end{eqnarray} \begin{pf}{Proof of Proposition \ref{prop:convlaw}} Again, the proof is only presented in the discrete spatial setting since the continuous case is analogous. We retain the notation of the proof of Proposition \ref{prop:wconv} where we showed that, as $t$ tends to infinity, \begin{eqnarray} &&\mathbb{E}^{\mathbf{u},\mathbf{v}} \bigl[e^{-\sqrt{1-\varrho} \langle u_t+v_t,\phi\rangle +i\sqrt{1+\varrho} \langle u_t-v_t,\psi\rangle} \bigr]\\ &&\qquad\rightarrow\mathbb{E}^{({\phi+\psi})/{2},({\phi-\psi})/{2}} \bigl[e^{-\sqrt {1-\varrho}(u+v)\langle\mathbf{1},\tilde{u}_{\infty}+\tilde {v}_{\infty }\rangle+i\sqrt{1+\varrho}(u-v) \langle\mathbf{1}, \tilde{u}_{\infty}-\tilde{v}_{\infty}\rangle} \bigr]. \end{eqnarray} Let us specify the limit law as for the nonspatial symbiotic branching process. As seen in Proposition \ref{prop:tmmart} the total-mass processes $\bar u_t :=\langle\tilde{u}_t,\mathbf{1}\rangle$ and $\bar v_t:=\langle\tilde{v}_t,\mathbf{1}\rangle$ are nonnegative continuous $L^2$-martingales with cross-variations $[\bar u_{\cdot}, \bar v_{\cdot } ]_t=\varrho[\bar u_{\cdot}, \bar u_{\cdot}]_t=\varrho[\bar v_{\cdot }, \bar v_{\cdot}]_t, t\ge0$. Thus, by Lemma \ref{la:eds}, reasoning as in (\ref{eq:invtime}), $(\bar u_t,\bar v_t)= (B^1_{T^{-1}(t)},B^2_{T^{-1}(t)} )$, where $B^1,B^2$ are Brownian motions started in $\bar u_0= \langle\frac{\phi+\psi}{2},\mathbf{1} \rangle$, $\bar v_0= \langle\frac{\phi-\psi}{2},\mathbf{1}\rangle$ with covariance $[B^1_{\cdot}, B^2_{\cdot}]_t=\varrho t$ and $T^{-1}(t) =\kappa\int_0^t \langle u_s,v_s\rangle \,ds$. Again, we need to show that $T^{-1}(\infty) = \tau$. This is much more subtle than in the nonspatial case since the quadratic variation might level off even if both total-mass processes $\bar u_t$, $\bar v_t$ are strictly positive. In \cite{DP98} it was shown that for $\varrho=0$, almost surely, this does not happen in the recurrent case [cf. the proof of their Theorem~1.2(b)]. Their proof can be used directly for $\varrho\in(-1,1)$. Hence, almost surely, \begin{equation}\label{eq:barconv} (\langle\tilde{u}_t,\mathbf{1}\rangle,\langle\tilde{v}_t,\mathbf{1}\rangle) \stackrel {t\rightarrow\infty}{\rightarrow} (B^1_{\tau}, B^2_{\tau}). \end{equation} Combining the above discussion with (\ref{eq:elimit}), we are able to determine the limit. First, we derived \begin{eqnarray} &&\mathbb{E}^{\mathbf{u},\mathbf{v}} \bigl[e^{-\sqrt{1-\varrho} \langle u_t+v_t,\phi\rangle +i\sqrt{1+\varrho}\langle u_t-v_t,\psi\rangle} \bigr]\\ &&\qquad\stackrel{t\rightarrow\infty}{\rightarrow}E^{\langle({\phi +\psi })/{2},\mathbf{1}\rangle,\langle({\phi-\psi})/{2},\mathbf{1} \rangle} \bigl[e^{-\sqrt{1-\varrho}(u+v)( B^1_{\tau}+B^2_{\tau}) +i\sqrt {1+\varrho}(u-v) ( B^1_{\tau}-B^2_{\tau})} \bigr]. \end{eqnarray} To use Lemma 2.3(c) of \cite{DP98} we manipulate the right-hand side using (\ref{dual0}): \begin{eqnarray} &&E^{\langle({\phi+\psi})/{2},\mathbf{1}\rangle,\langle({\phi -\psi })/{2},\mathbf{1} \rangle} \bigl[e^{-\sqrt{1-\varrho}(u+v)( B^1_{\tau}+B^2_{\tau}) +i\sqrt {1+\varrho}(u-v) ( B^1_{\tau}-B^2_{\tau})} \bigr]\\ &&\qquad=E^{\langle({\phi+\psi})/{2},\mathbf{1}\rangle,\langle({\phi -\psi })/{2},\mathbf{1} \rangle} [H^0 ({B^1_{\tau}}+{B^2_{\tau}},{B^1_{\tau}}-{B^2_{\tau }},u+v,u-v ) ]\\ &&\qquad=E^{u,v} [H^0 ({B^1_{\tau}}+{B^2_{\tau}},{B^1_{\tau}}-{B^2_{\tau }},\langle\phi,\mathbf{1}\rangle,\langle\psi,\mathbf{1}\rangle) ]\\ &&\qquad=E^{u,v} [H (\bar{B}^1_{\tau}+\bar{B}^2_{\tau},\bar{B}^1_{\tau }-\bar{B}^2_{\tau},\phi,\psi) ], \end{eqnarray} where, as in Proposition \ref{prop:convlaw}, $\bar{B}_{\tau}^1$ (resp., $\bar{B}_{\tau}^2$) denotes the constant function taking only the (random) value $B_{\tau}^1$ (resp., ${B}_{\tau}^2$). In total we have \[ \mathbb{E}^{\mathbf{u},\mathbf{v}} [H(u_t+v_t,u_t-v_t,\phi,\psi) ] \stackrel{t\rightarrow\infty}{\rightarrow}E^{u,v} [H(\bar {B}^1_{\tau }+\bar{B}^2_{\tau},\bar{B}^1_{\tau}-\bar{B}^2_{\tau},\phi,\psi) ], \] which implies weak convergence in $M_{\mathrm{tem}}^2$ of $(u_t,v_t)$ to $(\bar {B}^1_{\tau},\bar{B}^2_{\tau})$ by Lemma 2.3(c) of \cite{DP98}. \end{pf} \section{Moments}\label{sec:moments} In this section we prove Theorems \ref{thm:mc} and \ref{thm:im}. Before giving the proofs we prove an equivalence for moments of correlated Brownian motions. \subsection{Moments of the exit-point and exit-time distribution of correlated Brownian motions in a quadrant}\label{subsec:exitmoments} Let $\varrho\in(-1,1)$, $u,v>0$ and $B^1, B^2$ be Brownian motions started in $u,v$ with \begin{equation}\label{eq:bmcor} \langle B^1_{\cdot}, B^2_{\cdot}\rangle_t=\varrho t. \end{equation} The starting points of Brownian motions will be indicated by superscripts in probabilities and expectations. Further, let \begin{equation}\label{eq:tauagain} \tau^B=\inf\{t\geq0\dvtx B^1_{t} B^2_{t}=0 \}. \end{equation} \begin{theorem}\label{thm:theo2} Let $p > 0$ and $u, v > 0$. Under the above assumptions, the following conditions are equivalent: \begin{longlist} \item \[ p<\frac{\pi}{{\pi}/{2}+\arctan({\varrho}/({\sqrt {1-\varrho ^2}}) )}, \] \item \[ E^{u,v} [(\tau^B)^{{p}/{2}} ]<\infty, \] \item \[ E^{u,v} [ \|(B^1_{\tau^B},B^2_{\tau^B}) \|^p ]<\infty. \] \end{longlist} \end{theorem} \begin{pf} We start with the proof of the equivalence of (i) and (ii). Define a cone in the plane with angle $\theta\in(0, 2 \pi)$ by \[ C(\varphi)= \{re^{i\phi}\dvtx r\geq0, 0\leq\phi\leq\varphi\} \] and denote its boundary by $\partial C(\varphi)$. Note that with this definition, the positive real line is always contained in $C(\varphi)$. Further, we define, for $\varrho\in(-1,1)$, a sector in $\mathbb{R}^2$ by \[ S(\varrho) = \biggl\{(x,y)\in\mathbb{R}^2 \dvtx x\geq0, y\geq-\frac{\varrho }{\sqrt{1-\varrho^2}}x \biggr\} \] and denote by $\partial S(\varrho)$ its boundary. Note that this time, the positive imaginary axis is always in $S(\varrho)$ and that the angle of the sector at the origin is given by \[ \theta:=\frac{\pi}{2}+\arctan\biggl(\frac{\varrho}{\sqrt{1-\varrho ^2}} \biggr). \] To transform the correlated Brownian motions $B^1,B^2$ to planar Brownian motion we use the simple fact that $W^1:=B^1,W^2:= (\frac {B^2-\varrho B^1}{\sqrt{1-\varrho^2}} )$ defines a pair of independent Brownian motions started in $u, (\frac{v-\varrho u}{\sqrt{1-\varrho^2}} )$ satisfying $(B^1,B^2)=(W^1,\varrho W^1+\sqrt{1-\varrho^2}W^2)$. By the definition of $S(\varrho)$, the planar Brownian motion $(W^1,W^2)$ started in $ (u, (\frac {v-\varrho u}{\sqrt{1-\varrho^2}} ) )$ hits $\partial S(\varrho)$ if and only if the correlated Brownian motions $B^1,B^2$ started in $u,v$ hit $\partial C (\frac{\pi}{2} )$. Hence, for $\tau^B$ as in (\ref{eq:tauagain}), we have \begin{equation}\label{tt} \tau^B=\tau^W:=\inf\{t\geq0 \dvtx(W^1_t,W^2_t) \in\partial S(\varrho ) \}. \end{equation} Since planar Brownian motion is rotation invariant, $S(\varrho)$ may be rotated to agree with the cone $C(\theta)$, without changing the exit time. Obviously, with the corresponding rotated initial conditions, the law of the first exit time $\tau _{C(\theta)}$ from the cone $C(\theta)$ agrees with the law of $\tau ^W$. For planar Brownian motion in a cone $C(\theta)$ it is well known (see \cite{S58}, Theorem 2) that \begin{equation} \label{124} E^{x,y} \bigl[ \bigl(\tau_{C(\theta)} \bigr)^{p/2} \bigr] < \infty \quad\Leftrightarrow\quad p<\frac{\pi}{\theta}, \end{equation} independently of $x,y$. (\ref{tt}) and (\ref{124}) now imply the equivalence of (i) and (ii) and independence of $u,v$. The proof of the equivalence of (i) and (iii) is via conformal transformation of the cone $C(\theta)$ to the upper half-plane. Indeed, we are going to calculate the densities of the exit-point distributions \begin{equation}\label{h} P^{u,v} (B^1_{\tau^B}=0,B^2_{\tau^B}\geq y ),\qquad P^{u,v} ( B^1_{\tau^B}\leq x,B^2_{\tau^B}=0 ). \end{equation} We proceed in three steps: after reducing to independent Brownian motions in $S(\varrho)$ as for the exit time, we rotate $S(\varrho)$ to $C(\theta)$ and, finally, stretch the cone to end up with the upper half-plane. Recall that the first exit of $(B^1,B^2)$ happens at position $(0,y)\in\partial C(\frac{\pi}{2})$ if and only if the first exit of $(W^1,W^2)$ takes place at $ (0,\frac{y}{\sqrt{1-\varrho^2}} )\in \partial S(\varrho)$. Hence, (\ref{h}) transforms to \begin{eqnarray}\label{e1} &&P^{u,v} (B^1_{\tau^B}=0,B^2_{\tau^B}\geq y ) \nonumber\\[-8pt]\\[-8pt] &&\qquad= P^{u,({v-\varrho u})/{\sqrt{1-\varrho^2}}} \biggl(W^1_{\tau ^W}=0,W^2_{\tau^W}\geq\frac{y}{\sqrt{1-\varrho^2}} \biggr).\nonumber \end{eqnarray} In a similar fashion one obtains \begin{eqnarray}\label{e2} &&P^{u,v} ( B^1_{\tau^B}\leq x,B^2_{\tau^B}=0 )\nonumber\\[-8pt]\\[-8pt] &&\qquad= P^{u, ({v-\varrho u})/{\sqrt{1-\varrho^2}}} \biggl(W^1_{\tau^W}\leq x,W^2_{\tau^W}=-\frac{\varrho}{\sqrt{1-\varrho ^2}}W^1_{\tau^W} \biggr).\nonumber \end{eqnarray} We represent the transformed initial conditions $(z_1,z_2)= (u,\frac {v-\varrho u}{\sqrt{1-\varrho^2}} )\in S(\varrho)$ in polar coordinates, that is, \begin{eqnarray} z_1&=&\sqrt{u^2+\frac{(v-\varrho u)^2}{1-\varrho^2}} \cos\biggl(\arctan\biggl(\frac{v-\varrho u}{u\sqrt{1-\varrho^2}} \biggr) \biggr),\\ z_2&=&\sqrt{u^2+\frac{(v-\varrho u)^2}{1-\varrho^2}} \sin\biggl(\arctan\biggl(\frac{v-\varrho u}{u\sqrt{1-\varrho^2}} \biggr) \biggr). \end{eqnarray} For the rotation we add the angle $\arctan(\frac{\varrho}{\sqrt {1-\varrho^2}} )$ to get the new initial condition. Finally, to map\vspace{2pt} the cone $C(\theta)$ conformally to the upper half-plane $\mathbb{H}$, we apply the map $ z\mapsto z^{\pi/\theta}$ which maps $C(\theta)$ onto $\mathbb{H}$. Using conformal invariance of Brownian motion (e.g., Lemma 7.19 of \cite{MP09}), the problem is reduced to the computation of the exit distribution of planar (time-changed) Brownian motion from the upper half-plane. Indeed, due to the random time change the (almost surely finite) exit time changes but not the distribution of the exit points, which is Cauchy (see Theorem 2.37 of \cite{MP09}). Thus, to obtain the distribution of the exit points explicitly it, only remains to specify the transformed initial condition $\tilde{z}_1,\tilde{z}_2$, which is given by \begin{eqnarray} \tilde{z}_1 &=& \biggl( u^2 +\frac{(v-\varrho u)^2}{1-\varrho^2} \biggr)^{{\pi}/({2\theta})}\\ &&{}\times\cos\biggl( \frac{\pi}{\theta} \biggl( \arctan \biggl({\frac{v-\varrho u}{\sqrt{1-\varrho^2}u}} \biggr) + \arctan\biggl( \frac{\varrho}{\sqrt{1-\varrho^2}} \biggr) \biggr)\biggr),\\% \tilde{z}_2 &=& \biggl( u^2 +\frac{(v-\varrho u)^2}{1-\varrho^2} \biggr)^{{\pi}/({2\theta})}\\ &&{}\times\sin\biggl( \frac{\pi}{\theta} \biggl( \arctan \biggl( {\frac{v-\varrho u}{\sqrt{1-\varrho^2}u}} \biggr) + \arctan\biggl( \frac{\varrho}{\sqrt{1-\varrho^2}} \biggr) \biggr)\biggr). \end{eqnarray} Now, let $\tilde{W}^1,\tilde{W}^2$ be two independent Brownian motions with $\tilde W^1_0=\tilde z_1, \tilde W^2_0=\tilde z_2$ and \[ \tau^{\tilde W}:= \inf\{ t> 0\dvtx\tilde{W}^2_t=0 \}. \] Then, by (\ref{e1}), (\ref{e2}), \begin{eqnarray} P^{u,v} ( B^1_{\tau^B}=0,B^2_{\tau^B}\geq y )&=& P^{u, ({v-\varrho u})/{\sqrt{1-\varrho^2}}} \biggl(W^1_{\tau^W}=0,W^2_{\tau^W}\geq \frac{y}{\sqrt{1-\varrho^2}} \biggr)\\ &=& P^{\tilde{z}_1,\tilde{z}_2} \biggl(\tilde{W}^1_{\tau^{\tilde W}}\leq - \biggl(\frac{y}{\sqrt{1-\varrho^2}} \biggr)^{{\pi/\theta}} \biggr),\\ P^{u,v} ( B^1_{\tau^B}\leq x,B^2_{\tau^B}=0 )&=&P^{u,( {v-\varrho u})/{\sqrt{1-\varrho^2}}} \biggl(W^1_{\tau^W}\leq x,W^2_{\tau ^W}=-\frac{\varrho}{\sqrt{1-\varrho^2}}W^1_{\tau^W} \biggr)\\ &=&P^{\tilde{z}_1,\tilde{z}_2} \biggl( 0\leq\tilde{W}^1_{\tau^{\tilde W}}\leq\biggl(x \biggl(1+\frac{\varrho^2}{1-\varrho^2} \biggr)^{1/2} \biggr)^{{\pi/\theta}} \biggr)\\ &=&P^{\tilde{z}_1,\tilde{z}_2} \biggl( 0\leq\tilde{W}^1_{\tau^{\tilde W}}\leq\biggl(\frac{x}{\sqrt{1-\varrho^2}} \biggr)^{{\pi}/{\theta}} \biggr). \end{eqnarray} Explicit manipulations of the Cauchy distribution yield \begin{eqnarray} \label{h1} &&P^{u,v} ( B^1_{\tau^B}=0,B^2_{\tau^B}\geq y )\nonumber\\[-8pt]\\[-8pt] &&\qquad=\int_{y}^{\infty} \frac{1}{\pi\tilde{z}_2\sqrt{1-\varrho ^2}^{\pi /\theta}}\frac{{\pi}/{\theta}r^{{\pi}/{\theta}-1} }{1+ (({ ({r}/{\sqrt{1-\varrho^2}} )^{{\pi }/{\theta}} + \tilde{z}_1})/{\tilde{z}_2} )^2 } \,dr,\nonumber\\ \label{h2} &&P^{u,v} ( B^1_{\tau^B}\leq x ,B^2_{\tau^B}=0 )\nonumber\\[-8pt]\\[-8pt] &&\qquad=\int_0^x\frac{1}{\pi\tilde{z}_2\sqrt{1-\varrho^2}^{\pi/\theta}} \frac{ {\pi/\theta} r^{{\pi/\theta}-1} }{1+ ( ({ ({r}/{\sqrt{1-\varrho^2}} )^{{\pi /\theta}}- \tilde{z}_1})/{\tilde{z}_2} )^2} \,dr.\nonumber \end{eqnarray} Finally, noting that $\int_0^{\infty}\frac{x^{p+\alpha -1}}{1+x^{2\alpha }} \,dx<\infty$ if and only if $p<\alpha$, we deduce from (\ref{h1}) and (\ref{h2}) that \[ E^{u,v} [\|(B^1_{\tau^B},B^2_{\tau^B})\|^p ]<\infty\quad\mbox{if and only if} \quad p<\frac{\pi}{\theta}. \] \upqed\end{pf} \subsection[Proof of Theorem 2.5]{Proof of Theorem \protect\ref{thm:mc}}\label{subsec:proofthmmc} The proof relies on a combination of the self-duality based technique of the proof of Proposition \ref{prop:pb} and the close relation between the moments of the exit-time and exit-point distribution of correlated Brownian motions obtained in Theorem \ref{thm:theo2}. \begin{pf}{Proof of Theorem \ref{thm:mc}} We proceed in several steps. First, the result for the nonspatial model is proved and thereafter the results for the discrete-space and the continuous-space models. Finally, we present the argument in the transient case. In the following we use the definition of $B^1,B^2$ and $\tau$ from Proposition \ref{prop:convlaw}. \textit{Step} 1. Suppose $(u_t,v_t)$ is a solution of $\mathrm {SBM}(\varrho,\kappa)_{1,1}$. ``$\Rightarrow$'': We first assume $\varrho<\varrho(p)$, in which case Theorem \ref{thm:theo2} implies\break $E^{1,1}[\tau^{p/2}]<\infty$. As argued in the proof of Proposition \ref{prop:sconv}, $u_t$ is a nonnegative martingale and due to the same arguments satisfies $\mathbb{E}^{1,1} [[u_{\cdot}]_{t}^{p/2} ]\leq E^{1,1}[\tau^{p/2}]<\infty$ for all $t\geq0$ and $\kappa>0$. Considering $\bar u_t=u_t-u_0=u_t-1$, we apply the Burkholder--Davis--Gundy inequality to get \begin{eqnarray} \mathbb{E}^{1,1}[u_t^p]&=&\mathbb{E}^{1,1}[(\bar u_t+1)^p]\\ &=&\mathbb{E}^{1,1}\bigl[\mathbf{1}_{\{\bar u_t\leq1\}}(\bar u_t+1)^p\bigr]+\mathbb{E}^{1,1}\bigl[\mathbf{1} _{\{ \bar u_t>1\}}(\bar u_t+1)^p\bigr]\\ &\leq& C_p+C_p\mathbb{E}^{1,1}[{\bar u_t}^p]\\ &\leq& C_p+C_p\mathbb{E}^{1,1}\Bigl[\sup_{0\leq s\leq t} \bar u_s^p\Bigr]\\ &\leq& C_p+C'_p\mathbb{E}^{1,1}[[\bar u_{\cdot}]_t^{p/2}]<\infty \end{eqnarray} independently of $t$ and $\kappa$. ``$\Leftarrow$.'' Conversely, for $\varrho\geq\varrho(p)$, Theorem \ref{thm:theo2} implies that $E^{1,1}[(B^1_{\tau})^p]=\infty$. Using Fatou's lemma and almost sure convergence of $u_t$ to $B^1_{\tau}$, the proof for the nonspatial case is finished with \[ \liminf_{t\rightarrow\infty}\mathbb{E}^{1,1}[u_t^p]\geq\mathbb{E}^{1,1}[u_{\infty }^p]=E^{1,1}[(B_{\tau})^p]=\infty. \] Again, this lower bound is independent of $\kappa$. \textit{Step} 2. The proof for $\mathrm{dSBM}({\varrho,\kappa })_{\mathbf{1} ,\mathbf{1} }$ is started by reducing the moments for homogeneous initial conditions to finite initial conditions. Indeed, employing Lem\-ma~\ref {la:sduality} with $\phi=\psi=\frac{\theta}{2}\mathbf{1}_{k}$, where $\mathbf{1}_{k}$ denotes the indicator function of site $k \in\mathbb{Z}^d$, gives \begin{eqnarray} \mathbb{E}^{\mathbf{1},\mathbf{1}} \bigl[ e^{-\sqrt{1-\varrho} \theta(u_t(k)+v_t(k))} \bigr] &=&\mathbb{E}^{\mathbf{1},\mathbf{1}} \bigl[e^{-\sqrt{1-\varrho}\langle u_{t}+v_{t},\phi +\psi \rangle} \bigr]\\ &=&\mathbb{E}^{\phi,\psi} \bigl[e^{-\sqrt{1-\varrho}\langle\mathbf{1}+\mathbf{1},\tilde {u}_{t}+\tilde{v}_{t}\rangle} \bigr]\\ &=&\mathbb{E}^{\mathbf{1}_{k},\mathbf{1}_{k}} \bigl[e^{-\sqrt{1-\varrho}\theta\langle\mathbf{1}, \tilde {u}_{t}+\tilde{v}_{t}\rangle} \bigr], \end{eqnarray} where we used the argument of Remark \ref{shift}. Note that, due to our choice of initial conditions, the complex part of the self-duality vanishes. Since the above is a Laplace transform identity, we have \[ \mathcal{L}^{\mathbf{1},\mathbf{1}} \bigl(u_t(k)+v_t(k) \bigr)=\mathcal{L}^{\mathbf{1}_k, \mathbf{1}_k} (\langle\mathbf{1}, \tilde{u}_t \rangle+\langle\mathbf{1}, \tilde{v}_t \rangle) \] and hence \begin{equation}\label{23} \mathbb{E}^{\mathbf{1},\mathbf{1}} \bigl[ \bigl(u_t(k)+v_t(k)\bigr)^p \bigr] = \mathbb{E}^{\mathbf{1}_k,\mathbf{1}_k} [ (\langle\mathbf{1}, \tilde{u}_t \rangle+ \langle\mathbf{1}, \tilde{v}_t \rangle )^p ]. \end{equation} We are now prepared to finish the proof of the theorem for the discrete case. ``$\Rightarrow$.'' Suppose $\varrho<\varrho(p)$. Let $M_t=\langle \mathbf{1}, \tilde{u}_t \rangle+\langle\mathbf{1}, \tilde{v}_t \rangle$, which due to Lemma \ref{prop:tmmart} is a square-integrable martingale with quadratic variation \[ [M_\cdot]_t = [\langle\mathbf{1}, \tilde{u}_{\cdot}\rangle]_t+ [\langle\mathbf{1}, \tilde{v}_{\cdot}\rangle]_t+2 [\langle\mathbf{1}, \tilde {u}_{\cdot} \rangle,\langle\mathbf{1}, \tilde{v}_{\cdot} \rangle]_t=(2+2\varrho) [\langle\mathbf{1}, \tilde{u}_{\cdot} \rangle]_t. \] To apply the Burkholder--Davis--Gundy inequality, we switch again from $M$ to $\bar M_t=M_t-M_0$, which is a martingale null at zero. Hence, \[ \mathbb{E}^{\mathbf{1}_k, \mathbf{1}_k} [M_t^p ]=\mathbb{E}^{\mathbf{1}_k, \mathbf{1}_k} [(\bar{M}_t+M_0)^p ] \leq C_p+C_p\mathbb{E}^{\mathbf{1}_k, \mathbf{1}_k} [\bar{M}_t^p ]. \] Then we get from (\ref{23}) and the Burkholder--Davis--Gundy inequality \begin{eqnarray} \mathbb{E}^{\mathbf{1},\mathbf{1}} \bigl[\bigl(u_t(k)+v_t(k)\bigr)^p \bigr] &\leq& C_p+C_p\mathbb{E}^{\mathbf{1}_k, \mathbf{1}_k} [\bar{M}_t^p ]\\ &\leq& C_p+C_p\mathbb{E}^{\mathbf{1}_k,\mathbf{1}_k} \Bigl[ \sup_{0\leq s\leq t} \bar M_s^p \Bigr]\\ &\leq& C_p+C'_p \mathbb{E}^{\mathbf{1}_k, \mathbf{1}_k} [[\bar M_{\cdot }]_t^{p/2} ] \\ &=&C_p+ C'_p(2+2\varrho)^{p/2} \mathbb{E}^{\mathbf{1}_k,\mathbf{1}_k} [[\langle \mathbf{1} , \tilde{u}_{\cdot} \rangle]_t^{p/2} ] \end{eqnarray} for some constants $C_p,C'_p$ independent of $t$ and $\kappa$. As in the proof of Theorem \ref{prop:convlaw}, the random time change which makes the pair of total masses a pair of correlated Brownian motions is bounded by $\tau$, that is, $ [\langle\mathbf{1}, \tilde{u}_{\cdot} \rangle ]_t\leq\tau$ for all $t\geq0$. This yields by Theorem \ref{thm:theo2} \[ \mathbb{E}^{\mathbf{1},\mathbf{1}} [u_t(k)^p ]\leq\mathbb{E}^{\mathbf{1},\mathbf{1}} \bigl[\bigl(u_t(k)+v_t(k)\bigr)^p \bigr]\leq C_p+C'_p(2+2\varrho)^{p/2} E^{1, 1} [\tau^{p/2} ]<\infty. \] ``$\Leftarrow$.'' Suppose $\varrho\geq\varrho(p)$. As in the proof of Theorem \ref{prop:convlaw} we use the almost sure convergence of $(\langle\mathbf{1},\tilde{u}_t \rangle,\langle\mathbf{1}, \tilde{v}_t \rangle)$ to $(B^1_{\tau}, B^2_{\tau})$. Combining this with Fatou's lemma gives \begin{eqnarray} \liminf_{t\rightarrow\infty} \mathbb{E}^{\mathbf{1}_k,\mathbf{1}_k} [(\langle \mathbf{1}, \tilde{u}_t \rangle+ \langle\mathbf{1}, \tilde{v}_t \rangle)^p ] &\geq&\liminf_{t\rightarrow\infty}\mathbb{E}^{\mathbf{1}_k,\mathbf{1}_k} [\langle\mathbf{1}, \tilde{u}_t \rangle^p ]\\ &\geq&\mathbb{E}^{\mathbf{1}_k,\mathbf{1}_k} \Bigl[\liminf_{t\rightarrow\infty}\langle \mathbf{1}, \tilde{u}_t \rangle^p \Bigr] \\ &=&E^{1,1} [(B^1_{\tau})^p ]. \end{eqnarray} The right-hand side is infinite due to Theorem \ref{thm:theo2} and hence $ \mathbb{E}^{\mathbf{1}_k,\mathbf{1}_k} [(\langle\mathbf{1}, \tilde{u}_t \rangle+ \langle\mathbf{1}, \tilde{v}_t \rangle)^p ]$ diverges. Equation (\ref{23}) now shows that $\mathbb{E}^{\mathbf{1},\mathbf{1}}[(u_t(k)+v_t(k))^p]$ also grows without bound. Since symbiotic branching processes are nonnegative, this is also true for $\mathbb{E}^{\mathbf{1},\mathbf{1}}[u_t(k)^p]$ as can be seen as follows: \begin{eqnarray} \mathbb{E}^{\mathbf{1},\mathbf{1}}\bigl[\bigl(u_t(k)+v_t(k)\bigr)^p\bigr] &\leq&\mathbb{E}^{\mathbf{1},\mathbf{1}}\bigl[(2u_t(k))^p \mathbf{1}_{\{u_t(k)\geq v_t(k)\}}\bigr]\\ &&{} + \mathbb{E} ^{\mathbf{1},\mathbf{1} }\bigl[(2v_t(k))^p \mathbf{1}_{\{u_t(k)<v_t(k)\}}\bigr]\\ &\leq&2^p\mathbb{E}^{\mathbf{1},\mathbf{1}}[u_t(k)^p]+ 2^p\mathbb{E}^{\mathbf{1},\mathbf{1}}[v_t(k)^p]\\ &=& 2^{p+1}\mathbb{E}^{\mathbf{1},\mathbf{1}}[u_t(k)^p], \end{eqnarray} where we used Lemma \ref{la:mdual} to see that $\mathbb{E}^{\mathbf{1},\mathbf{1} }[u_t(k)^p]=\mathbb{E} ^{\mathbf{1},\mathbf{1}}[v_t(k)^p]$. \textit{Step} 3. The proof for $\mathrm{cSBM}(\varrho,\kappa)_{\mathbf{1} ,\mathbf{1}}$ is slightly more involved since we cannot use the indicator $\mathbf{1}_{x}$ to get $u_t(x)=\langle u_t,\mathbf{1}_{x}\rangle$, where now $\langle f,g\rangle =\int_{\mathbb{R}}f(x)g(x) \,dx$. Instead we use a standard smoothing procedure. For fixed $x\in\mathbb{R}$ let \[ p_{\varepsilon}(y)=\frac{1}{\sqrt{2\pi\varepsilon}}e^{- {(x-y)^2}/({2\varepsilon})}, \] where we skip the dependence on $x$. The main part is to show that \begin{eqnarray}\label{z} &&\bigl\Vert\bigl(u_t(x)+v_t(x)\bigr)-(\langle u_t,p_{\varepsilon}\rangle+\langle v_t,p_{\varepsilon}\rangle)\bigr\Vert_{L^p} \nonumber\\[-8pt]\\[-8pt] &&\qquad\leq\Vert u_t(x)-\langle u_t,p_{\varepsilon}\rangle \Vert_{L^p}+\Vert v_t(x)-\langle v_t,p_{\varepsilon}\rangle\Vert_{L^p} \stackrel {\varepsilon\rightarrow0}{\rightarrow} 0,\nonumber \end{eqnarray} which implies \begin{equation}\label{z2} \lim_{\varepsilon\rightarrow0}\Vert\langle u_t,p_{\varepsilon }\rangle +\langle v_t,p_{\varepsilon}\rangle\Vert_{L^p}=\Vert u_t(x)+v_t(x)\Vert_{L^p}. \end{equation} Due to symmetry we only consider $\Vert u_t(x)-\langle u_t,p_{\varepsilon }\rangle\Vert_{L^p}$. To prove (\ref{z}) we first observe that, due to the Green function representation provided in Corollary 19 of~\cite{EF04}, \begin{eqnarray} &&\Vert u_t(x)-\langle u_t,p_{\varepsilon}\rangle\Vert_{L^p}\\ &&\qquad= \biggl\Vert P_tu_0(x)-\langle P_{t+\varepsilon},u_0\rangle+\int_0^t\int_{\mathbb{R} }p_{t-s}(x-b)M(ds,db) \\ &&\qquad\quad\hspace{85.8pt}{} - \int_0^t\int_{\mathbb{R}}P_{t-s}p_{\varepsilon}(x-b) M(ds,db) \biggr\Vert_{L^p}, \end{eqnarray} where $M(ds,db)$ is a zero-mean martingale measure with quadratic variation \[ \biggl[\int_0^{\cdot}\int_{\mathbb{R}}f(s,b)M(ds,db) \biggr]_t=\kappa\int_0^t\int _{\mathbb{R}}f^2(s,b)u_s(b)v_s(b) \,ds\,db \] for test functions $f$ such that the integral is well defined (see Lemma 18 of \cite{EF04} for details). For homogeneous initial conditions, the first difference vanishes and it suffices to concentrate on the difference of the stochastic integrals. By the Burkholder--Davis--Gundy inequality the difference of the integrals can be estimated as \begin{eqnarray} &&\mathbb{E}\biggl[ \biggl( \int_0^t\int_{\mathbb{R}}p_{t-s}(x-b)M(ds,db)-\int_0^t\int_{\mathbb{R} }P_{t-s}p_{\varepsilon}(x-b)M(ds,db) \biggr)^{p} \biggr]\\ &&\qquad\leq C\kappa^{p/2}\mathbb{E}\biggl[ \biggl( \int_0^t\int_{\mathbb{R} }\bigl(p_{t-s}(x-b)-p_{\varepsilon+t-s}(x-b)\bigr)^2u_s(b)v_s(b) \,ds\,db \biggr)^{p/2} \biggr]. \end{eqnarray} Now expanding $(p_{t-s}(x-b)-p_{\varepsilon+t-s}(x-b))^2 u_s(b)v_s(b)$ as \begin{eqnarray} &&\bigl(p_{t-s}(x-b)-p_{\varepsilon+t-s}(x-b) \bigr)^{2(p-1)/p}\\ &&\qquad{}\times\bigl(p_{t-s}(x-b)-p_{\varepsilon+t-s}(x-b) \bigr)^{2/p} u_s(b)v_s(b), \end{eqnarray} we get the upper bound (taking the expectation under the integral is valid since the integrands are nonnegative) \begin{eqnarray} &&C\kappa^{p/2} \biggl[ \biggl(\int_0^t\int_{\mathbb{R}} \bigl(p_{t-s}(x-b)-p_{\varepsilon+t-s}(x-b) \bigr)^2 \,ds \,db \biggr)^{p-1}\\ &&\qquad\hspace{10.2pt}{}\times\int_0^t\int_{\mathbb{R}} \bigl(p_{t-s}(x-b)-p_{\varepsilon+t-s}(x-b) \bigr)^2\mathbb{E}[(u_s(b)v_s(b))^{p} ] \,ds\,db \biggr], \end{eqnarray} where we have used that, for $f, g \in L^p$, \[ \biggl(\int\bigl(f^{2(p-1)/p}\bigr)(f^{2/p}g) \,dx \biggr)^p\leq\biggl(\int f^2 \,dx \biggr)^{p-1}\int f^2g^p \,dx \] by H\"older's inequality. As in \cite{EF04}, page 153, the second term can now be bounded from above by a constant depending only on $p$ and $t$. The first factor can be estimated by $\varepsilon^{(p-1)/2}$ due to \cite{r9}, Lemma 6.2. Hence, for fixed $p>1, x\in\mathbb{R}$ and $t\geq0$, (\ref{z}) holds and thus we obtain (\ref{z2}). The rest of the proof is similar to the discrete case but slightly more technical. Since $p_{\varepsilon}(x-\cdot)$ is rapidly decreasing, we have \begin{eqnarray} \mathbb{E}^{\mathbf{1},\mathbf{1}} \bigl[ e^{-2\theta\sqrt{1-\varrho}\langle u_t+v_t, p_{\varepsilon}\rangle} \bigr] &=&\mathbb{E}^{\theta p_{\varepsilon},\theta p_{\varepsilon}} \bigl[ e^{-2\sqrt {1-\varrho}\langle\mathbf{1}, \tilde{u}_t+\tilde{v}_t \rangle} \bigr]\\ &=&\mathbb{E}^{ p_{\varepsilon}, p_{\varepsilon}} \bigl[e^{-\sqrt{1-\varrho }2\theta \langle\mathbf{1}, \tilde{u}_t+\tilde{v}_t \rangle} \bigr]. \end{eqnarray} Thus, we get \[ \mathcal{L}^{\mathbf{1},\mathbf{1}} (\langle u_t+v_t,p_{\varepsilon}\rangle )=\mathcal{L}^{p_{\varepsilon},p_{\varepsilon}} (\langle\mathbf{1}, \tilde {u}_t+\tilde{v}_t \rangle) \] and in particular \[ \mathbb{E}^{\mathbf{1},\mathbf{1}} [(\langle u_t+v_t,p_{\varepsilon}\rangle)^p ] =\mathbb{E}^{p_{\varepsilon},p_{\varepsilon}} [(\langle\mathbf{1}, \tilde{u}_t \rangle +\langle\mathbf{1}, \tilde{v}_t \rangle)^p ]. \] We may now finish the proof in a similar way to the discrete case. ``$\Rightarrow$.'' Due to (\ref{z2}) we are done if we can bound $\mathbb{E} ^{\mathbf{1},\mathbf{1}}[\langle u_t+v_t,p_{\varepsilon}\rangle^p]$ independently of $\varepsilon>0$ and $t\geq0$. This can be done as before: $\langle \mathbf{1}, \tilde{u}_t \rangle$ and $\langle\mathbf{1}, \tilde{v}_t \rangle$ are random time-changed correlated Brownian motions with initial conditions $\langle\mathbf{1}, p_{\varepsilon} \rangle=1$ for all $\varepsilon>0$. Using, as before, the auxiliary martingale \[ \bar M_t=\langle\mathbf{1}, \tilde{u}_t \rangle+\langle\mathbf {1},\tilde {v}_t\rangle-\langle\mathbf{1},\tilde{u}_0 \rangle- \langle \mathbf{1}, \tilde{v}_0\rangle, \] we obtain (as in the discrete case) with the help of the Burkholder--Davis--Gundy inequality \begin{eqnarray} \mathbb{E}^{\mathbf{1},\mathbf{1}} \bigl[\bigl(u_t(x)+v_t(x)\bigr)^p \bigr] &=&\lim_{\varepsilon\rightarrow0}\mathbb{E}^{\mathbf{1},\mathbf{1}} [\langle u_t+v_t,p_{\varepsilon}\rangle^{p} ]\\ &=&\lim_{\varepsilon\rightarrow0} \mathbb{E}^{p_{\varepsilon},p_{\varepsilon}} [\langle\mathbf{1},\tilde {u}_t+\tilde{v}_t \rangle^p ]\\ &\leq& C_p+C_p\lim_{\varepsilon\rightarrow0} \mathbb{E}^{ p_{\varepsilon},p_{\varepsilon}} [\bar{M}_t^p ]\\ &\leq& C_p+C'_p\lim_{\varepsilon\rightarrow0} \mathbb{E}^{p_{\varepsilon}, p_{\varepsilon}} [[ \bar{M}_{\cdot }]^{p/2}_{t} ]\\ &\leq& C_p+ C'_p(2+2\varrho)^{p/2}E^{1,1}[\tau^{p/2}]. \end{eqnarray} The positive constants $C_p, C'_p$ are independent of $\varepsilon$ and $t$, whereas $\bar M$ and the random time change $[\bar M_{\cdot}]_t$ do depend on $\varepsilon$. However, the bound $[\bar{M}_{\cdot }]_{t}\leq \tau$ holds for all $\varepsilon>0$ and $t\geq0$ since $B^1_0=B^2_0=\langle\mathbf{1},p_{\varepsilon}\rangle=1$. For $\varrho <\varrho (p)$ the right-hand side is finite by Theorem \ref{thm:theo2} and independent of $t\geq0$. Since $\mathbb{E}^{\mathbf{1},\mathbf{1}} [u_t(x)^p ]\leq\mathbb{E}^{\mathbf{1},\mathbf{1}} [(u_t(x)+v_t(x))^p ] $, the first direction is shown. ``$\Leftarrow$.'' First note that by translation invariance of initial condition, spatial motion and white noise \[ \mathbb{E}^{\mathbf{1},\mathbf{1}}\bigl[\bigl(u_t(x)+v_t(x)\bigr)^p\bigr]=\mathbb{E}^{\mathbf{1},\mathbf{1}}\bigl[\bigl(u_t(y)+v_t(y)\bigr)^p\bigr] \] for fixed time $t\geq0$ and arbitrary spatial positions $x,y\in\mathbb{R}$ implying that \[ \mathbb{E}^{\mathbf{1},\mathbf{1}}\bigl[\bigl(u_t(x)+v_t(x)\bigr)^p\bigr]=\int_{x-1/2}^{x+1/2}\mathbb{E}^{\mathbf{1} ,\mathbf{1} }\bigl[\bigl(u_t(y)+v_t(y)\bigr)^p\bigr]\, dy. \] Using Fubini's theorem and Jensen's inequality we obtain for $p>1$ the lower bound \begin{eqnarray} \mathbb{E}^{\mathbf{1},\mathbf{1}}\bigl[\bigl(u_t(x)+v_t(x)\bigr)^p\bigr]&\geq&\mathbb{E}^{\mathbf{1},\mathbf{1}} \biggl[ \biggl(\int _{x-1/2}^{x+1/2}\bigl(u_t(y)+v_t(y)\bigr) \,dy \biggr)^p \biggr]\\ &=&\mathbb{E}^{\mathbf{1},\mathbf{1}}\bigl[\bigl\langle u_t+v_t,1_{(x-1/2,x+1/2)}\bigr\rangle^p\bigr]. \end{eqnarray} We now choose an arbitrary nonnegative (nontrivial) smooth function $f$ with support contained in $(x-1/2,x+1/2)$ that is bounded by $1$ and integrates to some $c \in(0,1)$, say. A lower bound is now given by \begin{eqnarray} \mathbb{E}^{\mathbf{1},\mathbf{1}}\bigl[\bigl(u_t(x)+v_t(x)\bigr)^p\bigr]&\geq&\mathbb{E}^{\mathbf{1},\mathbf{1}}[\langle u_t+v_t,f\rangle ^p]\\ &=&\mathbb{E}^{f,f}[\langle\tilde u_t+\tilde v_t,\mathbf{1}\rangle^p]\\ &\geq&\mathbb{E}^{f,f}[\langle\tilde u_t,\mathbf{1}\rangle^p], \end{eqnarray} where we utilized for the equality the self-duality of Proposition 5 of \cite{EF04}. Finally, as in the discrete case, Fatou's lemma and the martingale convergence theorem imply \[ \liminf_{t\to\infty}\mathbb{E}^{\mathbf{1},\mathbf{1}}\bigl[\bigl(u_t(x)+v_t(x)\bigr)^p\bigr]\geq E^{c,c}[(B^1_{\tau})^p]=\infty \] by Theorem \ref{thm:theo2} and due to nonnegativity of solutions as well \[ \liminf_{t\rightarrow\infty}\mathbb{E}^{\mathbf{1},\mathbf{1}} [u_t(x)^p]=\infty \] proving the claim. \textit{Step} 4. The first direction of the above proof for $\mathrm {dSBM}(\varrho,\kappa)_{\mathbf{1},\mathbf{1}}$ also works for the transient case since $\mathbb{E}^{\mathbf{1}_k,\mathbf{1}_k} [[\bar{M}_{\cdot}]^{p/2}_{\infty} ]\leq E^{1,1}[\tau^{p/2}]$ is independent of recurrence/transience. \end{pf} \subsection[Proof of Theorem 2.7]{Proof of Theorem \protect\ref{thm:im}} We now study the ``criticality'' of the critical curve in more detail. As a preliminary result (mixed) moments of the nonspatial model are analyzed. The idea is to combine three different techniques: the martingale argument which led to Theorem \ref{thm:mc} for $\mathbb{E} ^{1,1}[u_t^n]$, a perturbation argument based on the moment duality which allows us to deduce exponential increase/decrease of $\mathbb{E} ^{1,1}[u_t^{n-1}v_t]$, and finally moment equations which yield exponential increase/decrease for all mixed moments $\mathbb{E}^{1,1}[u_t^{n-m}v_t^m]$. \begin{proposition}\label{0} The following hold for nonspatial symbiotic branching processes: \begin{itemize}[(1)] \item[(1)] For all $\kappa>0$ and $n\in\mathbb{N}$: \begin{itemize} \item[$\bullet$] $\mathbb{E}^{1,1}[u_t^n]$ grows to a finite constant if $\varrho <\varrho(n)$, \item[$\bullet$] $\mathbb{E}^{1,1}[u_t^n]$ grows subexponentially fast to infinity if $\varrho=\varrho(n)$, \item[$\bullet$] $\mathbb{E}^{1,1}[u_t^n]$ grows exponentially fast if $\varrho>\varrho(n)$. \end{itemize} \item[(2)] For all $\kappa>0$, $n\in\mathbb{N}$ and $m=1,\ldots,n-1$: \begin{itemize} \item[$\bullet$] $\mathbb{E}^{1,1}[u_t^{n-m}v_t^m]$ decreases exponentially fast if $\varrho<\varrho(n)$, \item[$\bullet$] $\mathbb{E}^{1,1}[u_t^{n-m}v_t^m]$ neither grows exponentially fast nor decreases exponentially fast if $\varrho=\varrho(n)$, \item[$\bullet$] $\mathbb{E}^{1,1}[u_t^{n-m}v_t^m]$ grows exponentially fast if $\varrho >\varrho(n)$. \end{itemize} \end{itemize} \end{proposition} \begin{pf} \textit{Step} 1. Martingale arguments based on the connection of moments of exit times and exit points of correlated Brownian motions were carried out in the proof of Theorem \ref{thm:mc}. This led to the first part of (1). Applying H\"older's inequality with $p=\frac {n}{n-m}$, $q=\frac{n}{m}$, we get the bound \begin{equation}\label{df} \mathbb{E}^{1,1}[u_t^{n-m}v_t^m]\leq\mathbb{E}^{1,1}[u_t^n]^{(n-m)/n}\mathbb{E} ^{1,1}[v_t^n]^{m/n}=\mathbb{E}^{1,1}[u_t^n] \end{equation} by symmetry. This implies that for $\varrho<\varrho(n)$ all mixed moments stay bounded as well. \textit{Step} 2. We apply the moment duality for the nonspatial model as explained in Remark \ref{rdual}. Combining the duality with the martingale argument of the first step we can understand the case $\varrho<\varrho(n)$ for mixed moments in a simple way. Note that for mixed moments the dual process starts with $n-m$ particles of one color and $m$ particles of the other color at time $0$. Note that for mixed moments $L_t^{\neq}\geq t$, since there is always at least one pair of different color. Now suppose $\varrho<\varrho(n)$, then for $0<\varepsilon <\varrho(n)-\varrho$ we get \begin{eqnarray} \mathbb{E}^{1,1}[u_t^{n-m}v_t^m]&=&\mathbb{E}\bigl[e^{\kappa(L_t^=+\varrho L_t^{\neq })}\bigr]=\mathbb{E} \bigl[e^{\kappa(L_t^=+(\varrho+\varepsilon) L_t^{\neq})}e^{-\kappa \varepsilon L_t^{\neq}}\bigr]\\ &\leq&\mathbb{E}\bigl[e^{\kappa(L_t^=+(\varrho+\varepsilon) L_t^{\neq })}\bigr]e^{-\kappa\varepsilon t}. \end{eqnarray} Since the first factor of the right-hand side is just the moment $\mathbb{E} ^{1,1}[u_t^{n-m}v_t^m]$ for $\varrho+\varepsilon$ strictly smaller than $\varrho(n)$, this is bounded for all $t$ and $\kappa$. Hence, for $\varrho<\varrho(n)$ all mixed moments decrease exponentially fast proving the first part of (2). Note that since $u_t^n$ is a submartingale, the moment $\mathbb{E}^{1,1}[u_t^n]$ is nondecreasing. For $\varrho=\varrho(n)$ we first consider the pure moments. Again, for the critical case, Theorem \ref{thm:mc} implies \[ \mathbb{E}\bigl[e^{\kappa(L_t^=+(\varrho(n)-\varepsilon)L_t^{\neq })}\bigr]<C(\varepsilon )<\infty \] for all $\varepsilon>0$ and $t\geq0$. With the crude estimate $L_t^{\neq }\leq{n\choose2}t$ we get \[ C(\varepsilon)>\mathbb{E}\bigl[e^{\kappa(L_t^=+\varrho(n) L_t^{\neq })}e^{-\kappa \varepsilon L_t^{\neq}}\bigr]\geq\mathbb{E}\bigl[e^{\kappa(L_t^=+\varrho(n)L_t^{\neq })} \bigr]e^{-\kappa\varepsilon{n\choose2}t}. \] Since $\varepsilon$ is arbitrary this implies subexponential growth to infinity of $\mathbb{E}^{1,1} [u_t(k)^n ]$ at the critical point. Hence, the second part of (1) is proven and combined with (\ref{df}) so is the upper bound of the second part of (2). \textit{Step} 3. A direct application of It\^o's lemma and Fubini's theorem yields \[ \mathbb{E}^{1,1}[u_t^n]=1+\kappa\pmatrix{n\cr2}\int_0^t\mathbb{E} ^{1,1}[u_s^{n-1}v_s] \,ds. \] Since we already know from the martingale arguments that $\mathbb{E} ^{1,1}[u_t^n]$ increases to infinity in the critical case, the mixed moment $\mathbb{E}[u_t^{n-1}v_t]$ cannot decrease exponentially fast proving the lower bound of part two of (2). Furthermore, with the same arguments as above, for $\varrho>\varrho(n)$, this leads to \[ \mathbb{E}^{1,1}[u_t^{n-1}v_t]=\mathbb{E}\bigl[e^{\kappa(L_t^=+\varrho(n) L_t^{\neq })}e^{\kappa(\varrho-\varrho(n))L_t^{\neq}} \bigr]\geq\mathbb{E}\bigl[e^{\kappa (L_t^=+\varrho(n) L_t^{\neq})} \bigr]e^{\kappa(\varrho-\varrho(n))t}. \] Since the first factor of the right-hand side equals $\mathbb{E}[u_t^{n-1}v_t]$ at the critical point, it does not decrease exponentially fast. Hence, the product increases exponentially fast. In particular, due to (\ref {df}), this also implies the third part of (1). Now it only remains to prove exponential increase for the other mixed moments. Again, using It\^o's lemma and Fubini's theorem yields the following moment equations for the mixed moments: \begin{eqnarray} \mathbb{E}^{1,1}[u_t^{n-2}v_t^2] &=&1+\kappa\int_0^t\mathbb{E}^{1,1}[u_s^{n-1}v_s] \,ds+\varrho(n-2)\kappa\int _0^t\mathbb{E}^{1,1}[u_s^{n-2}v_s^2] \,ds\\ &&{} + \pmatrix{{n-2}\cr2 }\kappa\int_0^t\mathbb{E}^{1,1}[u_s^{n-3}v_s^3] \,ds \end{eqnarray} and similarly for all other mixed moments. Since we already know that\break $\mathbb{E}^{1,1}[u_t^{n-1}v_t]$ grows exponentially fast in $t$, this implies exponential growth of $\mathbb{E}^{1,1}[u_t^{n-2}v_t^2]$. Iterating this argument gives exponential growth of all mixed moments for $\varrho >\varrho(n)$. This shows the third part of (2) and the proof is finished. \end{pf} Now it only remains to prove Theorem \ref{thm:im}, where some ideas for the nonspatial case are recycled. \begin{pf}{Proof of Theorem \ref{thm:im}} First, due to Lemma \ref{la:mdual}, for homogeneous initial conditions, the moments of $u_t(k)$ and $v_t(k)$ are equal for all $t\geq0$. For the existence of the Lyapunov exponents we use a standard subadditivity argument. Hence, it suffices to show \[ \mathbb{E}^{\mathbf{1},\mathbf{1}}[u_{t+s}(k)^n] \leq\mathbb{E}^{\mathbf{1},\mathbf{1}} [u_{t}(k)^n] \mathbb{E} ^{\mathbf{1},\mathbf{1}} [u_{s}(k)^n]. \] Using Lemma \ref{la:mdual}, we reduce the problem to $\mathbb{E}[e^{\kappa (L_{t}^=+\varrho L_{t}^{\neq})} ]$, where the dual process $(n_t)$ starts with $n$ particles of the same color all placed at site $k$. By the tower property and the strong Markov property, we obtain \[ \mathbb{E}^{n_0} \bigl[e^{\kappa(L_{t+s}^=+\varrho L_{t+s}^{\neq})} \bigr]=\mathbb{E} ^{n_0} \bigl[e^{\kappa(L_{t}^=+\varrho L_{t}^{\neq})}\mathbb{E}^{n_t} \bigl[e^{\kappa(L_{s}^=+\varrho L_{s}^{\neq})} \bigr] \bigr]. \] We are done if we can show that \begin{equation} \mathbb{E}^{n'} \bigl[e^{\kappa(L_{s}^=+\varrho L_{s}^{\neq})} \bigr]\leq\mathbb{E} ^{n_0} \bigl[e^{\kappa(L_{s}^=+\varrho L_{s}^{\neq})} \bigr] \end{equation} for any given initial configuration $n'$ of the dual process consisting of $n$ particles. The general initial conditions of the dual process consist of $n^1$ particles of one color and $n^2$ particles of the other color ($n^1+n^2=n$) distributed arbitrarily in space at positions $k_1,\ldots,k_{n}$. Using the duality relation of Lemma \ref{la:mdual}, we obtain \begin{eqnarray} \mathbb{E}^{n'} \bigl[e^{\kappa(L_{s}^=+\varrho L_{s}^{\neq})} \bigr]&=&\mathbb{E}^{\mathbf{1},\mathbf{1} }[u_s(k_1) \cdot\cdot\cdot u_s(k_{n^1})v_s(k_{n^1+1})\cdot\cdot\cdot v_s(k_{n^1+n^2})]\\ &\leq&\mathbb{E}^{\mathbf{1},\mathbf{1}} [u_s(k)^{n} ]=\mathbb{E}^{n_0} \bigl[e^{\kappa (L_{s}^=+\varrho L_{s}^{\neq})} \bigr], \end{eqnarray} where, in the penultimate step, we have used the generalized H\"older inequality. Having established existence of the Lyapunov exponents, we now turn to the more interesting question of positivity. The boundedness for $\varrho<\varrho(n)$ in Theorem~\ref{thm:mc} immediately implies that in this case $\gamma(\varrho,\kappa)=0$. Now suppose $\varrho =\varrho (n)$, that is, $(\varrho,n)$ lies on critical curve. We use the perturbation argument which we already used for the nonspatial case combined with Lemma \ref{la:mdual} and Theorem~\ref {thm:mc} to prove that in this case moments only grow subexponentially fast. This implies that the Lyapunov exponents are zero. Again we switch from $\mathbb{E}^{\mathbf{1},\mathbf{1}}[u_t(k)^n]$ to $\mathbb{E}[e^{\kappa (L_t^=+\varrho L_t^{\neq})} ]$, where the dual process is started with all particles at the same site and the same color. Since moments below the critical curve are bounded, we can proceed as for the nonspatial model. For any $\varepsilon>0$, we get \begin{eqnarray} \infty&>&C(\varepsilon)>\mathbb{E}\bigl[e^{\kappa(L_t^=+\varrho L_t^{\neq })}e^{-\kappa\varepsilon L_t^{\neq}} \bigr] \geq\mathbb{E}\bigl[e^{\kappa(L_t^=+\varrho L_t^{\neq})} \bigr]e^{-\kappa \varepsilon{n\choose2}t} \\ &\geq&\mathbb{E}^{\mathbf{1},\mathbf{1}} [u_t(k)^n ]e^{-\kappa\varepsilon{n\choose 2}t}, \end{eqnarray} where we estimated the collision time of particles of different colors by the collision time of all particles which is bounded from above by ${n\choose2}t$. Since $\varepsilon$ on the right-hand side is arbitrary, $\gamma(\varrho,\kappa)$ cannot be positive. Finally, we assume $\varrho>\varrho(n)$. The idea is to reduce the problem to the nonspatial case which we already discussed in Proposition \ref{0}. Actually, we prove more than stated in the theorem since we also show that mixed moments $\mathbb{E}^{\mathbf{1},\mathbf{1} }[u_t(k)^{n-m}v_t(k)^m]$ grow exponentially fast. For $m=1,\ldots,n-1$ the perturbation argument leads to \[ \mathbb{E}^{\mathbf{1},\mathbf{1}}[u_t(k)^{n-m}v_t(k)^m]=\mathbb{E}\bigl[e^{\kappa(L_t^=+\varrho L_t^{\neq})} \bigr]=\mathbb{E}\bigl[e^{\kappa(L_t^=+\varrho(n) L_t^{\neq })}e^{\kappa(\varrho-\varrho(n))L_t^{\neq}} \bigr]. \] The idea is to obtain a lower bound by conditioning on the event that all particles have not changed their spatial positions before time $t$ (but, of course, have changed their colors). Under this condition the particle dual is precisely the particle dual of the nonspatial model. More precisely, we get the lower bound \begin{eqnarray} &&\mathbb{E} \bigl[e^{\kappa(L_t^=+\varrho(n) L_t^{\neq})}e^{\kappa(\varrho -\varrho(n))L_t^{\neq}};\mbox{no spatial change of particles before time } t \bigr]\\ &&\qquad=\mathbb{E}\bigl[e^{\kappa(L_t^=+\varrho(n) L_t^{\neq})}e^{\kappa(\varrho -\varrho(n))L_t^{\neq}} \|\mbox{no spatial change of particles before time } t \bigr]\\ &&\qquad\quad{} \times\mathbb{P}[\mbox{no spatial change of particles before time }t ]\\ &&\qquad=\mathbb{E}\bigl[e^{\kappa(L_t^=+\varrho(n) L_t^{\neq})}e^{\kappa(\varrho -\varrho(n))L_t^{\neq}} \|\mbox{no spatial change of particles} \bigr]e^{-nt}, \end{eqnarray} where the final equality is valid since the event $\{$no spatial change of particles before time $t\}$ has probability $e^{-nt}$. This is true since the event is precisely the event that $n$ independent exponential clocks with parameter $1$ did not ring before time $t$. For $1\leq m\leq n-1$ there is always at least one pair of particles of different colors and, hence, we get the lower bound \[ \mathbb{E}\bigl[e^{\kappa(L_t^=+\varrho(n) L_t^{\neq})}\|\mbox{no spatial change of particles until time } t \bigr]e^{\kappa(\varrho-\varrho(n))t}e^{-nt}, \] which equals \[ \mathbb{E}^{1,1}[u_t^{n-m}v_t^m]e^{\kappa(\varrho-\varrho(n))t}e^{-nt} \] for a nonspatial symbiotic branching process with critical correlation $\varrho=\varrho(n)$. Choosing $\kappa$ such that $\kappa(\varrho -\varrho(n))>n$ the result now follows from Proposition~\ref{0}. \end{pf} As mentioned in the course of the proof, we actually proved that for $\varrho>\varrho(n)$ and $m=0,\ldots,n$ \[ \mathbb{E}^{\mathbf{1},\mathbf{1}}[u_t(k)^{n-m}(k)v_t^m(k)] \] grows exponentially in $t$. As for the nonspatial model one could ask whether, and if so how fast, mixed moments decrease for $\varrho <\varrho (n)$. For the second moments it was shown in \cite{AD09} that for $\varrho<\varrho(2)=0$ \[ \mathbb{E}^{\mathbf{1},\mathbf{1}}[u_t(k)v_t(k)]\approx \cases{ \dfrac{1}{\sqrt{t}}, &\quad $d=1$,\cr \dfrac{1}{\log(t)}, &\quad $d=2$,\vspace{2pt}\cr 1, &\quad $d\geq3$,} \] where $\approx$ denotes weak asymptotic equivalence as $t\to\infty$. It would be interesting to see whether or not different rates of decrease appear for moments. A detailed quantitative study of the Lyapunov exponents as functions of $\varrho$ and $\kappa$ has so far only been carried out for second moments (see \cite{AD09}). In contrast to the parabolic Anderson model, where higher Lyapunov exponents are well studied (see \cite{GdH07}), we do not have much insight. Only a first upper bound for the Lyapunov exponents in $\kappa$ and the distance to the critical curve can be obtained from the perturbation argument of the previous proof. \begin{proposition}\label{up} If $\varrho>\varrho(n)$, then $\gamma_n(\varrho,\kappa)\leq\kappa {n\choose2}(\varrho-\varrho(n))$. \end{proposition} \begin{pf} By Lemma \ref{la:mdual} and Theorem \ref{thm:mc} for $\varrho >\varrho (n)$, there are constants $C(\varepsilon)$ such that \begin{eqnarray} C(\varepsilon)&>&\mathbb{E}\bigl[{e^{\kappa(L_t^=+(\varrho-(\varrho-\varrho (n))-\varepsilon)L_t^{\neq})}} \bigr]\\ &=&\mathbb{E}\bigl[e^{\kappa(L_t^=+\varrho L_t^{\neq})}e^{-\kappa (\varrho-\varrho(n)+\varepsilon)L_t^{\neq}} \bigr]\\ &\geq&\mathbb{E}\bigl[e^{\kappa(L_t^=+\varrho L_t^{\neq})} \bigr]e^{-\kappa (\varrho-\varrho(n)+\varepsilon) {n\choose2}t}. \end{eqnarray} Hence, for all $\varepsilon>0$ \[ \mathbb{E}^{\mathbf{1},\mathbf{1}}[u_t(k)^n]\leq C(\varepsilon)e^{\kappa(\varrho -\varrho (n)+\varepsilon) {n\choose2}t}, \] yielding the result. \end{pf} \section{Speed of propagation of the interface} \label{sec:wavespeed} In this section we show how to use the moment bounds of Theorem \ref {thm:mc} to obtain an improved upper bound on the speed of propagation of the interface as defined in Definition \ref{def:ifc}. We will only sketch the crucial parts in the proof of Theorem 6 of \cite{EF04} that need modification. Note that the method used here is based on Mueller's ``dyadic grid technique'' introduced in \cite{M91}. \begin{pf}{Proof of Theorem \ref{cor:wavespeed}} To prove that the interface will eventually be contained in \[ \bigl[-C\sqrt{T\log(T)},C\sqrt{T\log(T)} \bigr] \] (for suitable $C>0$), by symmetry, it suffices to show that the right endpoint of the interface \[ R(u_t):=\sup\{x\in\mathbb{R}\| u_t(x)>0 \} \] up to time $T$ can eventually be bounded by $C\sqrt{T\log(T)}$. To this end we define \[ A_n:= \Bigl\{\sup_{t\leq n}R(u_t)>C\sqrt{n\log(n)} \Bigr\} \] and show that, for suitably chosen $C$, $\mathbb{P}^{1_{\mathbb{R}^-},1_{\mathbb{R}^+}} (\limsup_{n\in\mathbb{N}} A_n )=0$. By the Borel--Cantelli lemma, this follows from \begin{equation}\label{abc} \sum_{n=0}^{\infty}\mathbb{P}^{1_{\mathbb{R}^-},1_{\mathbb{R}^+}}(A_n)<\infty. \end{equation} In the following we modify the arguments of \cite{EF04} to obtain an upper bound for $\mathbb{P}^{1_{\mathbb{R}^-},1_{\mathbb{R}^+}}(A_n)$ which is sumamble over $n$. \begin{lemma}\label{la:m} For any integer $n$ there is a finite constant $c_n$ such that for $\varrho<\varrho(4n-1)$ \[ \mathbb{E}^{1_{\mathbb{R}^-},1_{\mathbb{R}^+}}[(u_t(x)v_t(x))^n]\leq c_n\sqrt{P_t1_{\mathbb{R} ^-}(x)},\qquad x\in\mathbb{R}, t\geq0. \] \end{lemma} \begin{pf} First recall from (87) of \cite{EF04} that $\mathbb{E}^{1_{\mathbb{R}^-},1_{\mathbb{R} ^+}}[u_t(x)]=P_t1_{\mathbb{R}^-}(x)$. We now use H\"older's inequality and Theorem \ref{thm:mc} to reduce the mixed moment to the first moment: \begin{eqnarray} && \mathbb{E}^{1_{\mathbb{R}^-},1_{\mathbb{R}^+}} [(u_t(x)v_t(x))^n ]\\ &&\qquad=\mathbb{E}^{1_{\mathbb{R}^-},1_{\mathbb{R}^+}} [u_t(x)^{1/2}u_t(x)^{n-1/2}v_t(x)^n ]\\ &&\qquad\leq(\mathbb{E}^{1_{\mathbb{R}^-},1_{\mathbb{R}^+}} [u_t(x) ] )^{1/2} (\mathbb{E}^{\mathbf{1} ,\mathbf{1}} [u_t(x)^{4n-1} ] )^{({2n-1})/({8n-2})}\\ &&\qquad\quad{}\times (\mathbb{E}^{\mathbf{1},\mathbf{1}} [v_t(x)^{4n-1} ] )^{{n}/({4n-1})}. \end{eqnarray} This follows from the generalized H\"older inequality with exponents $2,(8n-2)/(2n-1)$ and $(4n-1)/n$. The first factor yields the heat flow and Theorem~\ref{thm:mc} shows that the latter two factors are bounded by constants for $\varrho<\varrho(4n-1)$. \end{pf} We now strengthen the estimate of Lemma 23 of \cite{EF04} of the stochastic part \[ N_t(b)=\int_0^t\int_{\mathbb{R}}p_{t-s}(b-a)M(ds,da) \] of the convolution representation of solutions of Corollary 20 of \cite{EF04}. \begin{lemma}\label{la:ma} For $\varrho<\varrho(35)$ there is a constant $C_3$ such that for $\varepsilon\in(0,1)$, $A,T\geq1$, the following estimate holds: \[ \mathbb{P}^{1_{\mathbb{R}^-},1_{\mathbb{R}^+}} \bigl(\|N_t(b)\|\geq\varepsilon\mbox{ for some } t\leq T\mbox{ and }b\geq A \bigr)\leq C_3 \varepsilon^{-18}\frac {T^{22}}{\sqrt{A}}p_{2T}(A). \] \end{lemma} \begin{pf} The proof is along the same lines of \cite{EF04} replacing only in (116) the weaker (exponentially growing) moment bound of \cite{EF04} by our stronger (bounded) moment bound. In the following we sketch the arguments to show where the moments appear. Before performing the ``dyadic grid technique,'' increments of $N_t$ need to be estimated. First, by definition \begin{eqnarray} &&\mathbb{E}^{1_{\mathbb{R}^-},1_{\mathbb{R}^+}}[\|N_t(a)-N_{t'}(a')\|^{2q}]\\ &&\qquad=\mathbb{E}^{1_{\mathbb{R} ^-},1_{\mathbb{R} ^+}} \biggl[ \biggl\|\int_0^t\int_{\mathbb{R}}\bigl(p_{t-s}(b-a)-p_{t'-s}(b-a')\bigr)M(ds,db) \biggr\|^{2q} \biggr], \end{eqnarray} which by Burkholder--Davis--Gundy and H\"older's inequality gives the upper bound \begin{eqnarray} &&C_1 \biggl\|\int_0^t\int_{\mathbb{R}}[p_{t-s}(b-a)-p_{t'-s}(b-a')]^2 \,db \,ds \biggr\|^{q-1}\\ &&\qquad{}\times\int_0^t\int_{\mathbb{R}}[p_{t-s}(b-a)-p_{t'-s}(b-a')]^2 \mathbb{E}^{1_{\mathbb{R} ^-},1_{\mathbb{R}^+}} [(u_s(b)v_s(b))^q ] \,db \,ds. \end{eqnarray} Using Lemma \ref{la:m} and classical heat kernel estimates we can derive (see the calculation on pages 153, 154 of \cite{EF04}) the upper bound \begin{eqnarray} && \mathbb{E}^{1_{\mathbb{R}^-},1_{\mathbb{R}^+}}[\|N_t(a)-N_{t'}(a')\|^{2q}]\\ &&\qquad\leq C_2\bigl((\|t'-t\|^{1/2}+\|a'-a\|)\wedge t^{1/2}\bigr)^{q-1} \bigl(\sqrt {tP_t1_{\mathbb{R}^-}(a)}+\sqrt{t'P_{t'}1_{\mathbb{R}^-}(a')} \bigr). \end{eqnarray} This upper bound corresponds to (119) of \cite{EF04} where they have an additional exponentially growing factor coming from their moment bound. The dyadic grid technique can now be carried out as in \cite{EF04}, choosing $q=9$, without carrying along their exponential factor. Hence, we may delete the exponential term from their final estimate (110). Note that the necessity of $\varrho<\varrho(35)$ comes from our choice $q=9$ and Lemma \ref{la:m}. \end{pf} The following lemma corresponds to Proposition 24 of \cite{EF04}. \begin{lemma} If $\varrho<\varrho(35)$ then, for some constants $C_4, C_5$, the following estimate holds for $T\geq1$ and $r\geq C_4 \sqrt{T}$: \[ \mathbb{P}^{1_{\mathbb{R}^-},1_{\mathbb{R}^+}} \Bigl(\sup_{t\leq T}R(u_t)>r \Bigr)\leq C_5 T^{22} p_{16T}(r). \] \end{lemma} \begin{pf} All we need to do is to argue that Proposition 24 of \cite{EF04} is valid for $r\geq C_4 \sqrt{T}$ instead of $r\geq9^4(1\vee\kappa) T$. We perform the same decomposition and note that the estimates of Step 2 of \cite{EF04} are already given for $r\geq C_4 \sqrt{T}$ if $C_4$ is large enough. The only trouble occurs in their Step 3. Up to the estimate (154), this step works for $r\geq C_4 \sqrt{T}$ but here their (weaker) Lemma 23 produces an exponential in $T$. More precisely, they need to justify \[ e^{9^5\kappa^2 T/c}\frac{T^{22}}{\sqrt{r}}p_{8T}(r)\leq T^{22}p_{16T}(r), \] which is only valid for $r\geq9^4 (1\vee\kappa)T$. As our Lemma \ref {la:ma} avoids the exponential on the left-hand side the estimate holds for $r\geq C_4\sqrt{T}$ with suitably chosen $C_4$ and $C_5$. \end{pf} The significant distinction of the previous lemma to the result of \cite{EF04} is that the inequality is not only valid for $r\geq 9^4(1\vee\kappa)T$ but for $r\geq C_4\sqrt{T}$. At this point one might hope to obtain a square-root upper bound for the growth of the interface but this fails in the final step in which we validate (\ref{abc}): \begin{eqnarray} \sum_{n=0}^{\infty}\mathbb{P}^{1_{\mathbb{R}^-},1_{\mathbb{R}^+}}(A_n)&\leq&\sum _{n=0}^{\infty }C_5 n^{22} p_{16n} \bigl(C\sqrt{n\log(n)} \bigr)\\ &=&\sum_{n=0}^{\infty}C_5 n^{22} \frac{1}{\sqrt{\pi32n}}e^{- {C^2n\log(n)}/({32n})}\\ &=&\frac{C_5}{\sqrt{32\pi}}\sum_{n=0}^{\infty}n^{22-C^2/32-1/2}, \end{eqnarray} which is finite for $C$ large enough. \end{pf} \section*{Acknowledgments} This work is part of the Ph.D. thesis of the second author who would like to thank the students and faculty from TU Berlin for many discussions. The authors would like to express their gratitude to an anonymous referee for a very careful reading of the manuscript and for pointing out and correcting an error in an earlier version of Theorem \ref{cor:wavespeed}. \printaddresses \end{document}	arXiv
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Schwefel ORCID: orcid.org/0000-0002-4304-64691,3, Stéphane Coen ORCID: orcid.org/0000-0001-5605-59061,2, Miro Erkintalo ORCID: orcid.org/0000-0001-7753-70071,2 & Stuart G. Murdoch ORCID: orcid.org/0000-0002-9169-94721,2 Communications Physics volume 5, Article number: 123 (2022) Cite this article Frequency combs Solitons The nonlinear scattering of a linear optical wave from a conservative soliton has been widely studied in optical fibers as a mechanism for nonlinear frequency conversion. Here we extend this analysis to consider the scattering of an externally injected probe wave from a dissipative cavity soliton circulating in a Kerr microresonator. We demonstrate, both theoretically and experimentally, that this nonlinear interaction can be harnessed for useful expansion of the soliton frequency comb via the formation of a secondary idler comb. We explore the physics of the process, showing that the phase detuning of the injected probe from a cavity resonance plays a key role in setting the central frequency of the idler comb, thus providing a convenient parameter through which to control the spectral envelope of that comb. Our results elucidate the dynamics that govern the interactions between dissipative Kerr cavity solitons and externally injected probe waves, and could prove useful in the design of future Kerr frequency comb systems by enabling the possibility to provide high-power comb lines in a specified spectral region simply through the injection of a suitably chosen probe. Microresonator frequency combs (microcombs) offer an attractive pathway towards the realization of miniaturized, low-power coherent optical frequency combs1,2,3. Applications of these chip-scale devices have already been demonstrated across fields ranging from telecommunications4 and spectroscopy5 to remote sensing6,7 and medical diagnostics8. In the vast majority of systems, the resonators are driven with a monochromatic laser, and frequency combs form through the third-order optical Kerr nonlinearity1. In such systems, coherent comb states are underpinned by the excitation of ultra-short localized structures known as temporal cavity solitons9 – also referred to as dissipative Kerr solitons2,10. The spectral properties of Kerr microcombs are dominantly set by the material and geometric parameters of the microresonator. A key comb property is its spectral bandwidth, which (to leading-order) scales as11 $${{\Delta }}f\propto \sqrt{\frac{\gamma {P}_{{{{{{{{\rm{in}}}}}}}}}{{{{{{{\mathcal{F}}}}}}}}}{\| {\beta }_{2}\| }},$$ where γ, ${{{{{{{\mathcal{F}}}}}}}}$, and β2 are the resonator's Kerr nonlinearity coefficient, finesse, and group-velocity dispersion (GVD) at the driving wavelength, and Pin is the driving power, respectively. The large ${{{{{{{\mathcal{F}}}}}}}}$ and γ of microresonators are thus highly conducive to the realization of combs with large spectral bandwidth. In addition, the dispersion of microresonators can be engineered to provide further increase in the attainable comb bandwidth. This typically involves both lowering the resonator's GVD coefficient ∣β2∣, and optimizing its third-order dispersion so as to allow additional spectral extension through the formation of dispersive waves that are phase-matched to the pump12,13,14. Very recently, a new approach has been put forward that allows for the spectral characteristics and extent of microcombs to be controlled even after the fabrication of the resonator (whereupon its material and geometric properties are set). Specifically, the injection of a frequency-shifted probe field alongside the main comb generating pump has been shown to permit controllable expansion of the comb spectrum via two distinct mechanisms15,16,17. In the first mechanism15,16, a cavity soliton excited by the pump field imparts a nonlinear phase shift on the intracavity field of the frequency-shifted probe. This induces a frequency comb structure around the probe frequency via cross-phase modulation (XPM). In the second mechanism, frequency components of the soliton comb are spectrally translated through the four-wave-mixing process of nonlinear Bragg scattering17, again giving rise to a comb structure around the frequency-shifted probe field. Both mechanisms result in the generation of a secondary comb around the probe frequency, with line spacing equal to the spacing of the original soliton comb, yet they rely on fundamentally different physical phenomena: incoherent XPM15,16 and coherent Bragg scattering17. Recent experiments have unequivocally demonstrated the application potential of Bragg-scattering spectral expansion: a coherent Kerr microcomb with an expanded bandwidth of 1.6 octaves was reported in17. However, a number of questions remain open with regards to the physics that underpin the phenomenon. For instance, earlier studies18,19 have shown that nonlinear Bragg scattering of a frequency comb can be understood as the frequency-domain description of a particular soliton-linear wave interaction20,21,22,23 that has been extensively studied in the context of conservative (single-pass) nonlinear fibre optics24,25,26,27. This raises the question as to whether the simple phase-matching conditions known to govern the process in the single-pass case hold predictive power in the resonator context. In addition, nonlinear Bragg scattering is intrinsically a coherent FWM (four-wave mixing) process, and could therefore be envisaged to depend upon the linear detuning of the probe wave from a cavity resonance; however, no discussion of the role of detuning has hitherto been presented. Here we theoretically and experimentally study the spectral extension of soliton microcombs via coherent FWM Bragg-scattering. We show that the simple phase-matching condition that underpins soliton-linear wave interactions in single-pass systems19 remains valid in the resonator context, provided however that the phase detuning of the probe wave is appropriately accounted for. Indeed, we find that this detuning plays a key role in the process, providing a convenient parameter through which to control the spectral characteristics of the comb extension. We perform experiments in a magnesium-fluoride (MgF2) micro-disk resonator, and demonstrate the generation of frequency tunable, low-noise idler combs that possess identical line spacing, and the same low-noise characteristics, as the driving cavity soliton comb. We further highlight the flexibility of this comb expansion technique by swapping the spectral locations of the cavity soliton pump and the probe wave whilst still maintaining the required phasematching – and hence comb extension. Our results provide significant insights into the spectral extension of soliton microcombs, and highlight the intimate linkage between Bragg scattering spectral extension and soliton-linear wave interactions that have been studied widely in the context of nonlinear fiber optics and supercontinuum generation18,19,20,21,22,23,24,25,26,27. Whilst our experimental results are obtained in a monolithic crystalline microresonator we expect these results to find useful application across all microresonator platforms. Soliton linear-wave scattering We begin by briefly recounting how interactions between solitons and weak linear waves can enact resonant energy transfer to new frequencies in Kerr media20,21,22,23,28,29,30,31. Considering a superposition field E = S + p, where S and p represent the soliton and the probe, respectively, the Kerr nonlinearity ∣E∣2E will after linearization with respect to p yield three terms: ${\left\|S\right\|}^{2}S$, $2{\left\|S\right\|}^{2}p$, and S2p*. Each of these three terms can drive resonant energy transfer to new waves30, provided that those waves are phase-matched to one of the driving terms. In fact, the first term is responsible for the generation of the standard dispersive waves that are directly phasematched with the soliton28,29. On the other hand, the second and third terms represent nonlinear mixing between the soliton and the weak probe wave, and can drive additional phasematched radiation processes20,21,22,23,31. In particular, the $2{\left\|S\right\|}^{2}p$ interaction has been shown to drive both the incoherent XPM and coherent FWM Bragg scattering interactions22. Earlier studies have shown that the soliton-linear wave interaction can be understood in the frequency domain as a cascade of individual FWM Bragg scattering processes18,19. Specifically, pairs of discrete frequency components of a periodic train of solitons can act as pumps that drive a Bragg scattering cascade that translates the incident linear wave to a new idler frequency [see Fig. 1(a), (b)]. Remarkably, this cascade can be phase-matched even though none of the elementary FWM processes are phase-matched18,19,32; moreover, the phase-matching condition of the entire cascade is (approximately) the same as that of the soliton-linear wave interaction driven by the nonlinear polarization term $2{\left\|S\right\|}^{2}p$. In what follows, we show that this result extends to resonator configurations by demonstrating that the spectral extension of soliton microcombs via FWM Bragg scattering obeys the phasematching condition of the pertinent time-domain soliton-linear wave interaction. Fig. 1: Soliton-linear wave scattering. a Schematic illustration of the interaction between a soliton at ωs and a weak linear wave (LW) at ωp. The interaction enables the flow of energy to a phase-matched idler wave at ωi. b Frequency domain description of the soliton-linear wave interaction in (a), showing how the energy flow can be described as a cascade of nonlinear Bragg-scattering events. Components of the newly-generated idler comb are all shown in purple for clarity. The idler comb possesses the same free spectral range = D1/(2π) but is offset from the soliton comb (by dω). c, d visualize the phasematching of the process in single-pass and resonator configurations, respectively. c In single-pass configurations, higher-order dispersion is required for phasematching. d The extra degree of freedom provided by the probe detuning δp relaxes the phasematching condition in resonators. Soliton linear-wave phasematching To derive the soliton-linear wave phasematching condition in a resonator configuration, we note that the phase accumulated by the linear idler wave at frequency ωi over one round trip is: ϕi = β(ωi)L − ωitR, where β(ω), L, and tR are respectively the propagation constant, length, and round trip time of the resonator. Phase-matching is achieved when ϕi = ϕp + 2πm, where ϕp = − ωptR is the phase accumulated by the externally-injected probe wave (at the input to the resonator) and m is an integer33. Expanding the propagation constant β(ω) as a Taylor-series around the soliton pump, ωs, we obtain $$\hat{D}({\omega }_{{{{{{{{\rm{i}}}}}}}}}-{\omega }_{{{{{{{{\rm{s}}}}}}}}})L={\delta }_{{{{{{{{\rm{p}}}}}}}}}+\hat{D}({\omega }_{{{{{{{{\rm{p}}}}}}}}}-{\omega }_{{{{{{{{\rm{s}}}}}}}}})L+2\pi q,$$ where δp is the phase detuning of the injected probe from the cavity resonance closest to it, and the reduced dispersion $$\hat{D}(\omega -{\omega }_{{{{{{{{\rm{s}}}}}}}}})=\beta (\omega )-\beta ({\omega }_{{{{{{{{\rm{s}}}}}}}}})-\frac{{t}_{{{{{{{{\rm{R}}}}}}}}}}{L}(\omega -{\omega }_{{{{{{{{\rm{s}}}}}}}}})$$ $$=\mathop{\sum}\limits_{k\ge 2}\frac{{\beta }_{k}}{k!}{\left(\omega -{\omega }_{{{{{{{{\rm{s}}}}}}}}}\right)}^{k}.$$ The coefficient q = m − m0 with m0 the mode index of the resonance closest to ωp describes phase-matching to higher-order resonant sidebands that will not be considered in this work; in what follows, we set q = 0. We note that the analysis presented above considers only linear contributions to the idler's phasematched frequency. Cross phase modulation between the idler wave and the other intracavity fields present will contribute an additional nonlinear contribution to this shift14,34. The magnitude of this nonlinear shift can be estimated by comparing the difference between the idler frequency shift obtained from a generalized Lugiato-Lefever equation (LLE) simulation that includes all nonlinear contributions (see Methods) and that predicted by Eq. (2). For the experimental parameters used in this work, we find the difference between these two values to be only of the order of a few percent. For this reason we omit these terms from our phasematching analysis. Equation (2) is akin to the phasematching condition for Bragg-type soliton-linear wave interactions in single-pass fibre configurations19, but with the additional probe detuning δp accounting for the resonant nature of the system. Importantly, the presence of δp significantly relaxes the phasematching requirements: whilst higher-order dispersion is required in single-pass configurations, in resonators phasematching can be achieved even with a purely quadratic dispersion profile [see Fig. 1(c), (d)]. Explicitly, for a resonator with second-order dispersion only, the (angular) frequency detuning Ωi = ωi − ωs of the newly generated idler field from the soliton pump satisfies: $${{{\Omega }}}_{{{{{{{{\rm{i}}}}}}}}}^{2}={{{\Omega }}}_{{{{{{{{\rm{p}}}}}}}}}^{2}+\frac{2{\delta }_{{{{{{{{\rm{p}}}}}}}}}}{{\beta }_{2}L},$$ where Ωp = ωp − ωs. It is worth noting that Eq. (5) predicts two phasematched idler frequencies symmetrically located either side of the cavity soliton pump. However, the FWM Bragg cascade to the idler frequency located on the opposite side of the cavity soliton to the probe will require a cascade with substantially more steps and hence be less efficient. Indeed, for the parameters used in this paper, only the idler frequency closest to the probe frequency is observed in simulations and experiments. For different resonator parameters, however, it may be possible to efficiently drive both cascades resulting in new spectral components generated on both sides of the cavity soliton17. Illustrative simulations To confirm the analysis above, we performed numerical simulations using the generalized LLE (see Methods), and compared the simulations to predictions based on Eq. (5). Figure 2 shows results from simulations that use parameters similar to the experiments that will follow (see figure caption). In particular, the false colour plot in Fig. 2(a) shows the spectral envelope of the intracavity field as a function of the probe phase detuning δp normalized to the resonance half-width (Δp = δp/α), where α corresponds to half the intracavity power lost per roundtrip. To obtain this plot, at each probe detuning a cavity soliton is first excited (at Ω = ω − ωs = 0) and allowed to stabilize before the probe field is turned on. Also, in Fig. 2(c)–(f) we show lineplots of selected spectra in more detail, whilst Fig. 2(g)–(j) show the corresponding temporal intensity profile of the intracavity field over a single roundtrip. Fig. 2: Numerical simulation of soliton-linear wave interactions in a Kerr microresonator. a Pseudo-color plot shows the intracavity spectrum as a function of the probe detuning normalized to half the cavity linewidth, Δp = δp/α. The x-axis denotes the frequency offset from the soliton-generating pump, Ω = ω − ωs. Ωs, Ωp, and Ωi respectively denote angular frequency detunings of the soliton (Ωs = 0), the probe, and the idler. b Circles show the peak frequency detuning of the idler comb as a function of the linear probe detuning Δp as extracted from the simulation data in (a). Solid red curve in (b) shows the phasematched frequency predicted by Eq. (5). Dashed horizontal lines in (a), (b) delineate the region within which the probe undergoes modulational instability and prohibits the soliton-linear wave interaction. c–f show individual spectra from (a) at selected probe detunings as indicated, with dashed vertical lines indicating the phasematched idler frequencies predicted by Eq. (5). g–j show the temporal intensity of the intracavity fields corresponding to the spectra shown in (c–f). The simulation results were obtained using the Lugiato-Lefever equation as described in Methods, with parameters similar to the experiments that will follow: ${{{{{{{\mathcal{F}}}}}}}}=3.1\times 1{0}^{4}$, γ = 1.4 × 10−3W−1m−1, β2 = − 3.2 × 10−27s2m−1, Pin,i = ∣Ein,i∣2 = 80 mW, with i = (s, p), and θ = 1 × 10−4. In addition to the broadband soliton field around Ω = 0 and the injected probe field at Ωp, a third wave whose position changes continuously with the probe detuning is clearly visible. This corresponds to the idler comb that is generated via the soliton-linear wave interaction described above. Indeed, in Fig. 2(b) we plot the peak frequency shift of the idler comb as a function of the probe detuning, superimposed with the theoretical prediction of the phasematched frequency given by Eq. (5). The agreement between the simulated idler frequency and the theoretical prediction is very good, validating the use of the linearised phasematching expression of Eq. (5) for our experimental parameters. When the probe detuning is sufficiently close to its own resonance [with Δp ∈ ( − 3, 11)], our simulations predict that the soliton-linear wave interaction is interrupted. This occurs because the probe itself undergoes modulation instability within this range of detunings, as clearly evidenced by the characteristic modulation instability spectral features observed in Fig. 2(a). For most detunings within this range, the large intensity fluctuations of modulation instability prevent the persistence of the cavity soliton altogether39,40 and accordingly prohibit any soliton-linear wave interaction. However, for a narrow sliver of detunings [with Δp ∈ ( − 3, 1)], our simulations predict that the cavity soliton can coexist with the modulation instability state associated with the probe, and in this case a linear wave interaction can take place, but with the noise from the probe's modulation instability transferred to both the soliton and the idler combs, resulting in low-coherent states. This noise-like modulation instability structure can be clearly seen in the spectral and temporal traces obtained at Δp = − 1 plotted in Fig. 2(d), (h). The incoherence arising from modulation instability is in stark contrast with the results seen at the other detunings shown in Fig. 2(c, g), (e, i) and (f, j). Here, simulations confirm that both the soliton and idler combs are coherent as required for useful spectral extension, and stable temporal structures are observed with the newly generated idler wave manifesting itself in the time domain as an oscillatory tail on the soliton's leading edge. At this point, we note that the simulations presented here consider equal driving powers for the soliton and external probe fields. Simulations and theoretical considerations show that the efficiency of the soliton-linear wave cascade is set by the soliton pump power only18. Varying the external probe power will thus result in the power in the idler comb simply scaling proportionally. The exact value of probe power used does, however, play an important role in the range of probe detunings Δp over which modulation instability is observed in the probe field, with larger probe powers resulting in wider regions of modulation instability. For experimental demonstration, we use a setup that is built around an MgF2 micro-disk with a free spectral range (FSR) of 58.4 GHz (see Methods) 41,42,43. The micro-disk is driven by three continuous-wave lasers: A C-band pump laser at 1550 nm, an L-band pump laser 1582 nm, and a third laser at 1534 nm configured to act as an auxiliary pump that provides thermal compensation during the cavity soliton excitation44,45. The two pump lasers are coupled to the same mode family, and they can interchangeably act as the soliton-generating pump and the probe field. The auxillary laser (wavelength fixed at 1534 nm) is coupled to a different mode-family of the resonator to ensure it does not participate in the soliton-linear wave dynamics. We first set the wavelength of the L-band pump to 1582 nm and excite a cavity soliton at that wavelength. The remaining C-band laser acts the probe, and is tuned into resonance close to 1550 nm from the blue-detuned (negative Δp) side. Figure 3(a)–(c) show the comb spectra measured at the output of the resonator as the probe detuning is scanned into resonance. We can clearly observe an idler peak that moves towards the probe as the detuning Δp is increased. The idler peak tunes continuously until we reach the modulation instability region of the probe field, at which point the cavity soliton ceases to exist and the soliton-linear wave interaction is halted – in accordance with our simulations [see Fig. 2(a), (b)]. We then reset the experiment such that a new cavity soliton is excited at 1582 nm, and tune the probe field into resonance from the red-detuned (positive Δp) side. Figure 3(d)–(f) show the resulting spectra, and we again observe the appearance of an idler peak that continuously tunes towards the probe, now as Δp is decreased. These results are in qualitative agreement with the phasematching prediction of Eq. (5), with positive (and negative) probe detunings observed to generate idler peaks at frequencies that are below (and above) the original probe frequency. Fig. 3: Experimental spectra with the soliton-generating pump at 1582nm and the probe at 1550nm. a–c Blue curves show microcomb spectra measured at the resonator output for different negative values of the probe detuning, such that the idler is generated at wavelengths shorter than the probe. Orange curves show corresponding results from simulations and the dashed vertical lines show the phasematched wavelength predicted by Eq. (5). d–f show spectra as in (a–c) but for different positive values of the probe detuning, such that the idler is generated at wavelengths longer than the probe. g–l Radio-frequency (RF) spectra corresponding to (a–f). The beat frequencies in the RF spectra correspond to the offset between the soliton and the idler combs, dω/(2π). Labels A, I, P, and S on the top respectively indicate the auxiliary laser used for thermal compensation, the idler, the probe, and the soliton. Figure 3 (g)–(l) show the low-frequency RF (radio frequency) spectra of each microcomb state shown in Fig. 3(a)–(f). All of the spectra are clean with no excess noise, indicating that we are indeed observing low-noise, stable frequency combs. Yet, each RF spectrum does exhibit a single sharp RF tone. This is because, whilst the idler comb inherits the line spacing of the soliton comb, the two are offset from one another [see Fig. 1(b)], resulting in beating between adjacent lines in the region where the combs overlap. The beat frequency dν corresponds to the separation between the probe frequency and the soliton comb line closest to it. This can be written in terms of the soliton and probe detunings and the dispersive shift of the probe resonance as: $$\frac{d\omega }{2\pi }=d\nu \approx \left[{\delta }_{{{{{{{{\rm{p}}}}}}}}}-{\delta }_{{{{{{{{\rm{s}}}}}}}}}+\frac{{\beta }_{2}L{{{\Omega }}}_{{{{{{{{\rm{p}}}}}}}}}^{2}}{2}\right]\frac{{{{{{{{\rm{FSR}}}}}}}}}{2\pi }.$$ In our experiments, the RF beat tones range from 30 to 140 MHz, and are hence far too small to be resolved optically. Indeed, the optical comb spectra shown in Fig. 3 were recorded at an optical spectrum analyzer (OSA) resolution of 5 GHz, and a careful examination of these traces reveals only a single set of equispaced comb lines. Likewise, further RF intensity-noise measurements made using an electronic spectrum analyzer reveal only a low-noise background out to a frequency of 10 GHz (in addition to the low-frequency RF beat-notes already observed in Fig. 3). Combined, these measurements confirm that the output spectrum is indeed composed of two individual low-noise combs: the first, the cavity soliton comb, and the second, the idler comb extending from the probe frequency to the phasematched idler peak, and spectrally offset from the cavity soliton comb. The ability to measure the RF beat frequency between the soliton and the idler comb allows the two combs to be "stitched" together, i.e., the absolute frequencies of the idler comb lines to be directly related to the frequencies of the cavity soliton comb. In addition, when combined with measurements of the spectral position of the idler wave, the beat frequencies allow us to accurately estimate the parameters of the entire experiment (see Methods). Using parameters obtained in this manner (see caption of Fig. 2) in LLE simulations, we find excellent agreement with our experiments, as shown by the orange solid curves in Fig. 3(a)–(f). We observe particularly how the position and magnitude of the newly generated idler combs are very well reproduced by the simulations, and we also note that the idler positions are well predicted by the phasematched frequency given by Eq. (5). The soliton and probe fields are interchangeable: we can exchange their roles by selecting appropriate detunings and still generate a phase-matched idler comb. To show this, we set the detuning of the 1550 nm laser such that it generates a cavity soliton, and use the 1582 nm laser as the external probe. Figure 4(a)–(c) show the output comb spectra as the probe field is continuously tuned into resonance from the blue-detuned (i.e., negative Δp) side, whilst Fig. 4(d)–(f) show the corresponding RF spectra. The observed spectra are qualitatively identical to those obtained when the 1582 nm pump generates the cavity soliton. Moreover, we again find excellent agreement between the experimentally recorded spectra and the numerical results obtained from the LLE (red traces). The only adjustment to the simulation parameters used in Figs. 2 and 3 was a change to the value of the group-velocity dispersion coefficient β2 = − 2.9 × 10−27s2m−1 due to the change in pump wavelength and residual third-order dispersion. The small spectral features visible in the experimentally measured spectra at ~1505 and 1515 nm are a result of mode crossings at these wavelengths46. They play no significant role in the soliton-linear wave interaction and are not seen in our LLE simulations that considers only a single resonator mode family. Finally, we note in closing this section that for all the results presented here, the idler comb remains spectrally offset from the soliton comb (δν ≠ 0). It is possible to envisage different experimental conditions where δν could be set to zero. In this case, one would observe an interference between the contributions of the soliton and idler waves at each individual comb line. As the two lasers used to generate the soliton and probe driving fields are free-running independent lasers, this interference would be expected to vary on the timescale set by the two lasers' coherence times. Fig. 4: Experimental spectra with positions of the soliton and probe waves interchanged. a–c The soliton is here generated at 1550 nm whilst the probe sits at 1582 nm. d–f Radio-frequency (RF) spectra corresponding to (a–c). The results shown are for different negative values of the probe detuning. Orange curves show results from numerical simulations. Labels A, I, P, and S on the top respectively indicate the auxiliary laser used for thermal compensation, the idler, the probe, and the soliton. The results presented in Figs. 3 and 4 offer strong experimental support for the soliton-linear wave phasematching theory presented in the previous section. In addition to dynamics involving a single cavity soliton and the probe wave, our experiments show that more complex interactions can also take place. First, the soliton-linear wave interaction can occur when the resonator hosts several solitons. Indeed, in Fig. 5(a) we show comb spectra recorded at the resonator output with two temporally separated cavity solitons centred around a 1550 nm pump interacting with a probe field at 1582 nm; Fig. 5(b) shows the corresponding spectrum with the roles of the soliton-generating pump and the probe reversed. In both cases, a strong spectral modulation with a period of 4 nm can be observed, indicating the presence of two cavity solitons in the cavity separated by 2 ps. This spectral modulation manifests itself across both the cavity soliton and the idler combs, thus providing further evidence that the two combs are temporally locked to each other. Also shown in Fig. 5(a), (b) are theoretical ${{{{{\rm{sech}}}}}}^{2}$ profiles of the cavity soliton spectral envelope in the absence of the probe wave. This shows how the soliton-linear wave interaction leads to a substantial increase in comb intensity around the probe. Finally, in accordance with the simulations shown in Fig. 2, our experiments show that a narrow region of probe detunings exist where the probe can undergo modulational instability whilst still allowing the cavity soliton and the idler comb to persist. Figure 5(c) shows the measured optical spectrum recorded in this regime, superimposed with corresponding simulations, and we indeed observe characteristic (i) modulation instability sidebands around the 1550 nm probe wavelength and (ii) soliton ${{{{{\rm{sech}}}}}}^{2}$ profile around the 1582 nm cavity soliton pump. Moreover, the RF spectrum of this state [see Fig. 5(d)] shows clearly elevated level of intensity noise, as expected due to the chaotic nature of the modulation instability state. Fig. 5: Additional soliton-linear wave dynamics. Blue curves in (a), (b) show experimentally measured spectra at the resonator output when a two-cavity soliton state interacts with a probe. In (a) the solitons are at 1550 nm and the probe at 1583 nm, whilst in (b) the roles are swapped. The orange curves in (a), (b) show theoretically predicted cavity soliton envelopes in the absence of the probe wave, highlighting how the interaction leads to substantial spectral extension. c Blue and orange curves respectively show experimental and simulated optical spectra in the narrow parameter regime where the probe at 1550 nm undergoes modulation instability yet permits the cavity soliton at 1582 nm to persist. d shows the radio-frequency spectrum corresponding to (c), highlighting the low-coherence of the state. Labels A, I, P, and S respectively indicate the auxiliary laser used for thermal compensation, the idler, the probe, and the soliton. To summarize, we have shown that the nonlinear interaction between a cavity soliton and an externally injected probe can engender spectral extension of a Kerr microcomb via the formation of a secondary idler comb. We have shown that the process is underpinned by a simple phasematching condition derived from the time-domain soliton-linear wave interaction picture, revealing that the probe field's linear detuning plays a key role in controlling the phasematched frequency – and hence the spectral characteristics of the idler comb. We obtain a simple linear phasematching expression for the frequency shift of the phasematched idler wave and find that, for our parameters, it agrees well with the numerical shifts predicted by a full LLE simulation. The idler and its generating soliton comb have the same line spacing and low-noise characteristics, but are spectrally offset from one another; measurement of the RF beat frequency between the two combs allows the frequencies of the idler comb lines to be directly related to the cavity soliton comb. Combined, these properties make soliton-linear wave scattering an attractive candidate for the spectral expansion of Kerr combs, enabling comb power to be enhanced at desired spectral locations simply through the injection of an appropriate external probe field. We anticipate this ability could find useful application in many areas of microcomb research, including spectroscopy, frequency metrology and optical frequency synthesis. Finally, we close by emphasizing that the soliton-linear wave interaction described in our work represents the time-domain description of cascaded FWM Bragg scattering18,19, which has been linked to microcomb spectral extension in recent works completed in parallel with our study17. Experimental setup Our setup is built around an MgF2 micro-disk shaped via diamond point turning and then hand-polished to achieve a measured finesse of ${{{{{{{\mathcal{F}}}}}}}}\approx 3.1\times 1{0}^{4}$ at 1550 nm. The disk has a minor radius of 30 μm, and a major radius of 600 μm, yielding an free-spectral range of 58.4 GHz. A micron diameter fiber taper is used to couple the driving fields to and from the resonator. The microdisk is driven by three optical fields derived from two C-band external-cavity-lasers (ECL), and one L-band ECL. Each laser is amplified by an erbium-doped-fiber amplifier, then filtered and coupled to the optical taper using fiber wavelength-division-multiplexers. One of the C-band lasers and the L-band laser are coupled to the same mode family, and they can interchangeably act as the soliton-generating pump and the probe field. The remaining C-band laser (wavelength fixed at 1534 nm) is coupled to a different mode-family of the resonator and configured to act as an auxiliary pump that provides thermal compensation during the cavity soliton excitation. Throughout our measurements, the powers of the soliton and probe lasers were set to 80 mW whilst that of the auxiliary pump was set to 150 mW. Bichromatically-pumped LLE Simulations Our experimental observations are modelled by the Lugiato-Lefever equation (LLE)15,35,36,37,38: $${t}_{R}\frac{\partial E(t,\tau )}{\partial t}=\left[-\alpha -i{\delta }_{{{{{{{{\rm{s}}}}}}}}}-i\frac{{\beta }_{2}L}{2}\frac{{\partial }^{2}}{\partial {\tau }^{2}}+i\gamma L\| E{\| }^{2}\right]E+\sqrt{\theta }{E}_{{{{{{{{\rm{in}}}}}}}}}(t,\tau ).$$ Here E(t, τ) describes the slowly-varying intracavity electric field envelope, t is a "slow" time that describes the evolution of E(t, τ) at the scale of the photon lifetime, τ is a corresponding "fast" time that describes the envelope's profile over a single roundtrip, $\alpha =\pi /{{{{{{{\mathcal{F}}}}}}}}$ is half of the power lost per round trip (and equal to the resonance half-width), δs is the phase detuning of the soliton-generating pump from the cavity resonance closest to it, and θ is the input power coupling coefficient. The driving field Ein(t, τ) accounts for the bi-chromatic pumping and is defined as $${E}_{{{{{{{{\rm{in}}}}}}}}}(t,\tau )={E}_{{{{{{{{\rm{in}}}}}}}},{{{{{{{\rm{s}}}}}}}}}+{E}_{{{{{{{{\rm{in,p}}}}}}}}}{e}^{-i{{{\Omega }}}_{{{{{{{{\rm{p}}}}}}}}}\tau +i\left({\delta }_{{{{{{{{\rm{p}}}}}}}}}-{\delta }_{{{{{{{{\rm{s}}}}}}}}}+\frac{{\beta }_{2}L}{2}{{{\Omega }}}_{{{{{{{{\rm{p}}}}}}}}}^{2}\right)\frac{t}{{t}_{R}}},$$ where Ein,s and Ein,p are the pump amplitudes with units of W1/2 at the soliton and probe frequencies ωs and ωp, respectively. All simulation results presented in the paper were obtained by numerically integrating Eq. (7) using the split-step Fourier method with parameters quoted in the caption of Fig. 2 (or elsewhere in the main text). Fitting procedure The following describes the procedure used to fit our simulations to our experimental data, allowing direct comparisons to be drawn. First, the change in probe detuning between each spectrum (shown in Fig. 2) can be calculated from the change in the measured RF beat notes, Δdν, as Δδp = 2π ⋅ (Δdν/FSR); second, the GVD coefficient β2 can be extracted by fitting the change in idler position with the change in probe detuning: ${{\Delta }}{\Omega }_{{{{{{{{\rm{i}}}}}}}}}={({\beta }_{2}L{{{\Omega }}}_{{{{{{{{\rm{i}}}}}}}}})}^{-1}{{\Delta }}{\delta }_{{{{{{{{\rm{p}}}}}}}}}$; third, γ and δs can be estimated by fitting numerical simulations to the cavity soliton spectrum, and to the measured value of δp at which the probe undergoes modulation instability. These procedures yield the parameters quoted in the caption of Fig. 2. The data that support the findings of this study are available from the corresponding author upon reasonable request. Code availability The code that supports the findings of this study are available from the corresponding author upon reasonable request. Pasquazi, A. et al. 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Nonlinear and quantum optics with whispering gallery resonators. J. Opt. 18, 123002 (2016). Sayson, N. L. B. et al. Octave-spanning tunable parametric oscillation in crystalline Kerr microresonators. Nat. Photonics 13, 701–706 (2019). Zhang, S. et al. Sub-milliwatt-level microresonator solitons with extended access range using an auxiliary laser. Optica 6, 206–212 (2019). Lu, Z. et al. Deterministic generation and switching of dissipative Kerr soliton in a thermally controlled micro-resonator. AIP Advances 9, 025314 (2019). Herr, T. et al. Mode Spectrum and Temporal Soliton Formation in Optical Microresonators. Phys. Rev. Lett. 113, 123901 (2014). This work was supported by the Marsden Fund and the Rutherford Discovery Fellowships of the Royal Society of New Zealand. Department of Physics, University of Auckland, Auckland, 1010, New Zealand Pierce C. Qureshi, Vincent Ng, Farhan Azeem, Luke S. Trainor, Harald G. L. Schwefel, Stéphane Coen, Miro Erkintalo & Stuart G. Murdoch The Dodd-Walls Centre for Photonic and Quantum Technologies, Dunedin, New Zealand Pierce C. Qureshi, Vincent Ng, Stéphane Coen, Miro Erkintalo & Stuart G. Murdoch Department of Physics, University of Otago, Dunedin, 3016, New Zealand Farhan Azeem, Luke S. Trainor & Harald G. L. Schwefel Pierce C. Qureshi Vincent Ng Farhan Azeem Luke S. Trainor Harald G. L. Schwefel Stéphane Coen Miro Erkintalo Stuart G. Murdoch P.C.Q. performed all the experiments. V.N. performed numerical modelling of resonators. F.A., L.S.T., and H.G.L.S. fabricated the resonators. M.E. and S.C. contributed to theoretical interpretation of the results. M.E. and S.G.M. wrote the manuscript. All authors contributed to discussing and interpreting the results. Correspondence to Stuart G. Murdoch. Communications Physics thanks Dmitry Skryabin, Naoya Kuse and Xingyuan (Mike) Xu for their contribution to the peer review of this work. Peer reviewer reports are available. Peer Review File Qureshi, P.C., Ng, V., Azeem, F. et al. Soliton linear-wave scattering in a Kerr microresonator. Commun Phys 5, 123 (2022). https://doi.org/10.1038/s42005-022-00903-5 Microresonator Frequency Combs: New Horizons Communications Physics (Commun Phys) ISSN 2399-3650 (online)	CommonCrawl
\begin{document} \begin{frontmatter} \title{The compressed word problem in relatively hyperbolic groups} \author{Derek Holt and Sarah Rees\\ \textit{Dedicated to Patrick Dehornoy, our collaborator and friend, from whom we learned a lot}} \date{13th July 2021, Warwick} \begin{abstract} We prove that the compressed word problem in a group that is hyperbolic relative to a collection of free abelian subgroups is solvable in polynomial time. \end{abstract} \end{frontmatter} \section{Introduction} The main result of \cite{HLS} is that the compressed word problem in a hyperbolic group is solvable in polynomial time. Here we generalise this result to a group hyperbolic relative to a set of free abelian subgroups. \begin{theoremA}\label{thm:main} The compressed word problem for a group that is hyperbolic relative to a collection of free abelian subgroups is solvable in polynomial time. \end{theoremA} We prove the theorem by extending the arguments of \cite{HLS} from hyperbolic to relatively hyperbolic groups. Our principal source for results about this class of groups is \cite{AC}. In particular, we use the automaticity of groups of that type, proved in \cite[Theorem 7.7]{AC}; however we need to construct a new asynchronously automatic structure, with particular properties that we need. We believe that our result remains true if we substitute arbitrary (finitely generated) abelian groups for free abelian groups in the theorem statement but (as we shall explain after the definition of the \emph{components} of a word in Section~\ref{sec:relhyp}) extending the proof in that way would result in further technical difficulties in a proof that is already highly technical, so we decided not to attempt it here. It might be more interesting to try to extend the result to groups hyperbolic with respect to a collection of virtually abelian subgroups, but we are not currently able to extend our methods to cover that case in general. We introduce basic concepts and notation, including the definitions of straight line programs ({\sc slp}\xspace) and the compressed word problem, in Section~\ref{sec:defs_notation}, which follows this introduction. Section~\ref{sec:relhyp} contains the definition of relatively hyperbolic groups (following Osin \cite{Osin}) and properties of those that we shall need in the article. Section~\ref{sec:slp_background} provides basic background material on {\sc slp}s\xspace, Section~\ref{sec:slps_ab} gives some results for {\sc slp}s\xspace associated with finitely generated abelian groups, Section~\ref{sec:gammahat_geom} examines the geometry of the compressed Cayley graph $\widehat{\Gamma}$ of a relatively hyperbolic group $G$, and Section~\ref{sec:slps_relhyp_props} establishes some results for {\sc slp}s\xspace associated with such a group. The main result of the article, Theorem~\ref{thm:main}, is proved across the final two sections, Sections~\ref{sec:convert} and \ref{sec:slextcslp}, as the concatenations of two theorems, Theorem~\ref{thm:slextcslp} and Theorem~\ref{thm:convert}. The authors would like to thank Saul Schleimer, for introducing us to the study of the compressed word problem, especially for hyperbolic groups \cite{HLS}, and Yago Antolin, for some very helpful discussions on the properties of relatively hyperbolic groups and their automatic structures. We are also grateful to an anonymous referee for a careful reading of the paper and several helpful comments and suggestions. \section{Definitions and notation} \label{sec:defs_notation} To a large extent (but not entirely) our notation and definitions follow \cite{HLS}. \subsection{Words} For a finite set $\Sigma$ (which we call an {\em alphabet}), we define a {\em word} $w$ over $\Sigma$ to be a string $x_0\cdots x_{n-1}$, where each $x_i$ is in $\Sigma$. We denote by $\|w\|$ the length $n$ of $w$, by $\epsilon$ the empty word (of length 0), and for $0 \leq i < j \leq n$ we denote by $w[i:j)$ the substring $x_i\cdots x_{j-1}$, which we call a {\em subword} of $w$ (in \cite{HLS}, and elsewhere, such a substring is called a {\em factor}). We abbreviate the prefix $w[0:j)$ of $w$ as $w[:j)$, its suffix $w[i:n)$ as $w[i:)$, and $w[i:i+1)=x_i$ as $w[i]$, and also consider $\epsilon$ to be a subword of $w$. For words $v,w \in \Sigma^$, we write $v=w$ if $v$ and $w$ are equal as strings and, when $\Sigma$ is a generating set of a group $G$, we write $v=_G w$ if $v$ and $w$ represent the same element of $G$. All group generating sets $\Sigma$ in this article will be assumed to be inverse closed (that is, $x \in \Sigma \Rightarrow x^{-1} \in \Sigma$). Suppose that $\Sigma$ is an ordered finite generating set for a group $G$ and that $w \in \Sigma^$ represents an element $g \in G$. Then we define $\mathsf{slex}(w)$ (or $\mathsf{slex}_\Sigma(w))$ to be the shortlex minimal word representing $g$; that is, $\mathsf{slex}(w)$ is the lexicographically least among the shortest words that represent the same group element as $w$. \subsection{Straight-line programs} Let $\Sigma$ be a finite alphabet and $V$ a finite set with $V \cap \Sigma = \emptyset$. Let $\rho: V \to (V \cup \Sigma)^$ be a map and extend the definition of $\rho$ to $(V \cup \Sigma)^$ by defining $\rho(a)=a$ for all $a \in \Sigma \cup \{\epsilon\}$ and $\rho(uv) = \rho(u)\rho(v)$ for all $u,v \in (V \cup \Sigma)^$. We define the associated binary relation $\succeq$ on $V$ by $A \succeq B$ whenever the symbol $B$ occurs within the string $\rho^k(A)$, for some $k \geq 0$. We define a {\em straight-line program} ({\sc slp}\xspace for short) over an alphabet $\Sigma$ to be a triple $\mathcal{G} = (V,S,\rho)$, with $S \in V$ and $\rho: V \to (V \cup \Sigma)^$ a map such that the associated binary relation $\succeq$ on $V$ is acyclic, that is the corresponding directed graph contains no directed cycles. The set $V$ is called the set of {\em variables} of $\mathcal{G}$, and $S$ is called the {\em start variable}. Where necessary, we write $V_\mathcal{G}$, $S_\mathcal{G}$, $\rho_\mathcal{G}$, rather than simply $V,S,\rho$. An {\sc slp}\xspace $\mathcal{G}$ is naturally associated with a context-free grammar $(V,\Sigma, S, P)$, where $P$ is the set of all productions $A \to \rho(A)$ with $A \in V$, and we will often use the name $\mathcal{G}$ also for this grammar. It follows from the definition of an {\sc slp}\xspace that this associated grammar derives exactly one terminal word, which we call the {\em value} of $\mathcal{G}$ and denote by $\mathsf{val}(\mathcal{G})$. {\sc slp}s\xspace are used to provide succinct representations of words that contain many repeated substrings. For instance, the word $(ab)^{2^n}$ is the value of the {\sc slp}\xspace $\mathcal{G} = (\{A_0, \ldots,A_n\},\rho,A_0)$ with $\rho(A_n) = ab$ and $\rho(A_{i-1}) = A_i A_i$ for $0 < i \leq n$. We provide more background on {\sc slp}s\xspace in Section~\ref{sec:slp_background}. \subsection{The compressed word problem} The {\em compressed word problem} for a finitely generated group $G$ with the finite symmetric generating set $\Sigma$ is the following decision problem: \begin{description} \item[Input:] an {\sc slp}\xspace $\mathcal{G}$ over the alphabet $\Sigma$. \item [Question:] does $\mathsf{val}(\mathcal{G})$ represent the group identity of $G$? \end{description} It is an easy observation that the computational complexity of the compressed word problem for $G$ is independent of the choice of generating set $\Sigma$; more precisely, if $\Sigma'$ is another finite symmetric generating set for $G$, then the compressed word problem for $G$ with respect to $\Sigma$ is log-space reducible to the compressed word problem for $G$ with respect to $\Sigma'$ \cite[Lemma~4.2]{Loh14}. So, when proving that the compressed word problem for $G$ is solvable in polynomial time, we are free to choose whichever finite symmetric generating set of $G$ is most convenient for the purpose. \subsection{Fellow travelling and automatic groups} Suppose that $\Gamma=\Gamma(G,\Sigma)$ is the Cayley graph for a group $G$ with finite symmetric generating set $\Sigma$. Let $v,w$ be words over $\Sigma$, and let $\gamma_v,\gamma_w$ be the paths traced out in $\Gamma$ by $v,w$ from the identity vertex $1$ of $\Gamma$. For $0 \le i < \|v\|$, we denote the vertex of $\Gamma$ labelled $v[i]$ by $\gamma_v[i]$, and similarly for $w[i]$ and $\gamma_w[i]$. (So, in particular, $\gamma_v[0]=\gamma_w[0]$ is the identity vertex.) For the following definition, we define $\gamma_v[i] := \gamma_v[\|v\|-1]]$ for integers $i \ge \|v\|$. We say that the words $v$ and $w$ {\em fellow travel} at distance $k$ (or, more briefly, {\em $k$-fellow travel}) if $d_\Gamma(\gamma_v[i],\gamma_w[i]) \le k$ for all $i \ge 0$. In other words, the distance in $\Gamma$ between the vertices on $\gamma_v$ and $\gamma_w$ at the ends of subpaths traced out by the subwords $v[:i)$ and $w[:i)$ of $v$ and $w$ is at most $k$. In this situation we also say that the paths $\gamma_v$ and $\gamma_w$ $k$-fellow travel. We can extend this terminology to paths $\gamma_v,\gamma_w$ that do not start at the same vertex in $\Gamma$. In particular, if two such paths $k$-fellow travel, then their start points and also their end points are at distance at most $k$ from each other. We say that the words $v$ and $w$ {\em asynchronously fellow travel} at distance $k$ if we can choose sequences $(i_0=0,i_1,\ldots,i_n=\|v\|-1)$ and $(j_0=0,j_1,\ldots,j_n=\|w\|-1)$, with $i_{t+1}\in \{i_t,i_t+1\}$, $j_{t+1} \in \{j_t,j_t+1\}$ for each $t$, such that $d_\Gamma(\gamma_v[i_t],\gamma_w[j_t])\leq k$ for each $t$ with $0 \le t \le n$, and again we may also apply this terminology to the paths that they trace out in $\Gamma$. We call the pairs of vertices $\gamma_v[i_t],\gamma_w[j_t]$ \emph{corresponding vertices} in the fellow travelling. Note that a vertex on one of the paths can have more than one corresponding vertex on the other path, but in that case the corresponding vertices are the vertices lying on a contiguous subpath of the other path. We say that $G$ is {\em automatic} if there exists a finite state automaton $A$ over $\Sigma$ for which every element of $G$ has at least one representative word in $L(A)$, as well as an integer $k$ such that, whenever $v,w \in L(A)$ and either $v=_G w$ or $v=_G wx$ with $x \in \Sigma$, then $v,w$ fellow travel at distance $k$. We say that $G$ is {\em asynchronously automatic} if the same is true but with an asynchronous fellow traveller property, and {\em (asynchronously) biautomatic} if, in addition, for words $u,w$ that satisfy $xu =_G w$ with $x \in \Sigma$, that paths traced out by $u,w$ from the vertices $x$ and $1$, respectively, (asynchronously) fellow travel at distance $k$. We call the pair $(A,k)$ an {\em automatic structure} (or {\em asynchronously automatic structure}) for $G$. An automatic structure $(A,k)$ is called {\em geodesic} or {\em shortlex} if each word in $L(A)$ is a representative of minimal length, or minimal within the shortlex word ordering, of the element that it represents, respectively. We say that $(A,k)$ is a {\em structure with uniqueness} if it contains a unique representative of each element of $G$. We refer to \cite{ECHLPT} for basic properties of automatic structures and automatic groups. \section{Relatively hyperbolic groups}\label{sec:relhyp} The purpose of this section is to give the definition of a relatively hyperbolic group and to list the properties that we shall need in this article. The properties we need are proved in the article \cite{AC}, and build on results of \cite{Osin}. We have used Osin's definition of relatively hyperbolicity; it is proved in \cite[Theorem 1.5]{Osin} that (for finitely generated groups, as in our case) this is equivalent to the definition of \cite{Bowditch}, also to the definition of \cite{Farb} combined with the Coset Penetration Property (see below), called strong relative hyperbolicity in \cite{Farb}. Below we have (essentially) used notation and statements from \cite{AC}. We suppose that $\Sigma$ is a finite generating set for a group $G$, and that $\{H_i : i \in \Omega\}$ is a finite collection of subgroups of a $G$, which we call the collection of \emph{parabolic subgroups} of $G$. Define $\mathcal{H} := \bigcup_{i \in \Omega} (H_i \setminus \{1\})$, and $\widehat{\Sigma} := \Sigma \cup \mathcal{H}$. We let $\Gamma:=\Gamma(G,\Sigma)$ and $\widehat{\Gamma}=\widehat{\Gamma}(G,\widehat{\Sigma})$ be the Cayley graphs for $G$ over $\Sigma$ and $\widehat{\Sigma}$, respectively. (So $\widehat{\Gamma}$ has the same vertices as $\Gamma$ but more edges than $\Gamma$.) We call a word over $\Sigma$ (or $\widehat{\Sigma}$) \emph{geodesic} if it labels a geodesic path in $\Gamma$ (or $\widehat{\Gamma}$). Following \cite[Definition 2.5]{AC} and \cite[Section 1.2]{Osin}, we define $F$ to be the free product of groups \[ F:= (\ast_{ i \in \Omega} H_i) \ast F(\Sigma) \] and suppose that a finite subset $R$ of $F$ exists whose normal closure in $F$ is the kernel of the natural map from $F$ to $G$; in that case we say that $G$ has the {\em finite presentation} \[ \left\langle X \cup \left. \bigcup_{i \in \Omega} H_i\, \right\vert R \right\rangle \] {\em relative to} the collection of subgroups $\{ H_i : i \in \Omega \}$. Now if $u$ is a word over $\widehat{\Sigma}$ that represents the identity in $G$, then $u$ is equal within $F$ to a product of the form \[ \prod_{j=1}^n f_j r_j^{\eta_j} f_j^{-1}, \] with $r_j \in R, f_j \in F$ and $\eta_j = \pm 1 $ for each $j$. The smallest possible value of $n$ in any such expression of this type for $u$ is called the {\em relative area} of $u$, denoted by $\mathsf{Area}_\mathsf{rel}(u) $. We say that $G$ is \emph{hyperbolic relative to} the collection of subgroups $\{H_i\}$ if it has a finite relative presentation as above and a constant $C \geq 0$ such that \[ \mathsf{Area}_\mathsf{rel}(u) \leq C \|u\| \] for all words $u$ over $\widehat{\Sigma}$ that represent the identity in $G$. We note that if $G$ is relatively hyperbolic then the graph $\widehat{\Gamma}$ is Gromov-hyperbolic. Note also that, by \cite[Proposition 2.36]{Osin}, the intersection $H_i \cap H_j$ for $i \ne j$ is finite. The notation and results that follow are all taken from \cite{AC}. Given a path $p$ in $\widehat{\Gamma}$, we say that the path $p$ \emph{penetrates} the left coset $gH_i$ if $p$ contains an edge labelled by an element of $H_i$ that connects two vertices of $gH_i$. An $H_i$--\emph{component} of such a path is defined to be a non-empty maximal subpath of $p$ that is labelled by a word in $H_i^$. Two components $s$ and $r$ (not necessarily of the same path) are {\em connected} if both are $H_i$-components for some $H_i$, and if the start points of both paths lie in the same left coset $gH_i$ of $H_i$. A path $p$ is said to \emph{backtrack} if $p=p'srs'p''$ where $s,s'$ are $H_i$--components, and the word labelling $r$ represents an element of $H_i$; if no such decomposition of $p$ exists, then $p$ is \emph{without backtracking}. A path $p$ is said to \emph{vertex backtrack} if it contains a subpath of length greater than 1 labelled by a word that represents an element of some $H_i$; otherwise $p$ is said to be \emph{without vertex backtracking}. We note that if a path does not vertex backtrack then it does not backtrack and all of its components have length 1. We denote the start and end points of a path $p$ in $\widehat{\Gamma}$ by $p_-$ and $p_+$, respectively, and say that paths $p$, $q$ in $\widehat{\Gamma}$ are \emph{$k$-similar} if $\max\{d_\Gamma(p_-,q_-),d_\Gamma(p_+,q_+)\} \le k$. The following fundamental result about $k$-similar paths in $\widehat{\Gamma}$, proved as \cite[Theorem 3.23]{Osin}, is also stated as \cite[Theorem 2.8]{AC}. \begin{proposition}\label{prop:bcpp} \cite[Theorem 3.23]{Osin}. (Bounded Coset Penetration Property) Let $G$ be relatively hyperbolic, as above. For any $\lambda \ge 1$, $c \ge 0$, $k \ge 0$, there exists a constant $\mathsf e = \mathsf e(\lambda,c,k)$ such that, for any two $k$-similar paths $p$ and $q$ in $\widehat{\Gamma}$ that are $(\lambda,c)$--quasigeodesics and do not backtrack, the following conditions hold. \begin{mylist} \item[(1)] The sets of vertices of $p$ and $q$ are contained in the closed $\mathsf e$-neighbourhoods of each other in $\Gamma$. \item[(2)] Suppose that, for some $i$, $s$ is an $H_i$-component of $p$ with $d_\Gamma(s_-,s_+) > \mathsf e$; then there exists an $H_i$-component of $q$ that is connected to $s$. \item[(3)] Suppose that $s$ and $t$ are connected $H_i$-components of $p$ and $q$, respectively. Then $s$ and $t$ are $\mathsf e$-similar. \end{mylist} \end{proposition} The next three results are derived in \cite{AC} from the Bounded Coset Penetration Property. We define the \emph{components} of a word $w \in \Sigma^$ to be the nonempty subwords of $w$ of maximal length that lie in $(\Sigma \cap H_i)^$ for some parabolic subgroup $H_i$. In general, since $H_i \cap H_j$ is finite for $i \ne j$, it is possible for the end of one component to overlap the beginning of the next, where the overlapping generators lie in a finite intersection. In this paper, we shall generally be assuming that $H_i \cap H_j$ is trivial for $i \ne j$, in which case the components are necessarily disjoint. This holds in particular when the $H_i$ are free abelian groups, and it is the main reason why we have not attempted to generalise our main theorem to groups that are hyperbolic relative to arbitrary finitely generated abelian groups. We strongly believe that such a generalisation would be possible, but it might involve significant additional technicalities, of a similar nature to those involved in \cite{AC}. Let $w := \alpha_0u_1\alpha_1u_2 \cdots u_n \alpha_n$, where the $u_j$ are its components. Then, following \cite[Construction 4.1]{AC}, we define the {\em derived word} $\hat{w} := \alpha_0h_1\alpha_1h_2 \cdots h_n \alpha_n \in \widehat{\Sigma}^$, where each $h_j$ is the element of a parabolic subgroup represented by $u_j$. So the components of paths in $\Gamma$ and $\widehat{\Gamma}$ labelled by $w$ and $\hat{w}$ are labelled by the subwords $u_i$ and $h_i$ of $w$ and $\hat{w}$, respectively. A word over $\Sigma$ is said to have a \emph{parabolic shortening} if, for some $i$, it has a component over $\Sigma \cap H_i$ that is non-geodesic; otherwise it has {\em no parabolic shortenings}. \begin{proposition} \label{prop:genset} \cite[Lemma 5.3, Theorems 7.6, 7.7]{AC} Let $G$ be a finitely generated group, hyperbolic with respect to a family of subgroups $\{H_i\}_{i \in \Omega}$, and let $\Sigma'$ be a finite generating set of $G$. Then there exist $\lambda \ge 1$, $c \ge 0$ and a finite subset $\mathcal{H}'$ of $\mathcal{H}$ such that, for every finite, ordered generating set $\Sigma$ of $G$ with $\Sigma' \cup \mathcal{H}' \subseteq \Sigma \subseteq \Sigma' \cup \mathcal{H}$ for which each $H_i$ has a geodesic biautomatic structure over $\Sigma \cap H_i$, we have: \begin{mylist} \item[(i)] $G$ has a geodesic biautomatic structure over $\Sigma$ which is a shortlex structure if the structures on $H_i$ are shortlex; \item[(ii)] there exists a finite set $\Phi$ of non-geodesic words over $\Sigma$ such that, for each word $w \in \Sigma^$ with no parabolic shortenings and no subwords in $\Phi$, the word $\hat{w} \in (\Sigma \cup \mathcal{H})^$ is a $(\lambda,c)$--quasigeodesic without vertex backtracking. \end{mylist} \end{proposition} For the remainder of this paper, given a relatively hyperbolic group $G$ and its finite generating set $\Sigma$, we shall say that $(G,\Sigma)$ is {\em suitable for parabolic geodesic biautomaticity} if \begin{mylist} \item[(i)] $H_i \cap H_j = \{1\}$ for all $i \ne j$; \item[(ii)] each parabolic subgroup $H_i$ has a geodesic biautomatic structure over $\Sigma \cap H_i$, and there exist $\lambda$ and $c$ such that the conclusions (i) and (ii) of Proposition \ref{prop:genset} hold for $G$ and $\Sigma$. \end{mylist} So this applies in particular when the parabolic subgroups $H_i$ are free abelian, as in the hypothesis of our main result. We are now in a position to state and prove a corollary to the above proposition. \begin{corollary}\label{cor:genset} Let $(G,\Sigma)$ be relatively hyperbolic and suitable for parabolic geodesic biautomaticity. If $w \in \Sigma^$ is a geodesic word that represents an element of $H_i$ for some $i$, then $w \in (\Sigma \cap H_i)^$. In particular, we have $H_i = \langle \Sigma \cap H_i \rangle$ for each $i$. \end{corollary} \begin{proof} Since $w$ is geodesic, it cannot contain subwords in $\Phi$ or have parabolic shortenings. The fact that $\hat{w}$ is without vertex backtracking implies that $\|\hat{w}\|=1$ and hence that $w$ is a word over $\Sigma \cap H_i$. Since every element of $H_i$ can be represented by some geodesic word $w$, $H_i$ is generated by $\Sigma \cap H_i$. \end{proof} \begin{lemma} \label{lem:qgeo} Let $(G,\Sigma)$ be relatively hyperbolic and suitable for parabolic geodesic biautomaticity, and let $w \in \Sigma^$ be a geodesic word. Then there exists a constant $\lambda \ge 1$ such that $\hat{w}$ labels a $(\lambda,0)$--quasigeodesic path in $\widehat{\Gamma}$ that does not vertex backtrack. \end{lemma} \begin{proof} It follows from Proposition~\ref{prop:genset} that $\hat{w}$ labels a $(\lambda,c)$--quasigeodesic path for some $\lambda \ge 1$ and $c \ge 0$ and, since $w$ is geodesic, it cannot represent $1_G$ unless $w =\epsilon$, and so by increasing $\lambda$ if necessary we may assume that $c=0$. \end{proof} Unfortunately, the biautomatic structure for $G$ that is given by Proposition \ref{prop:genset} does not appear to have all of the properties that we need in the proofs of our main results. For that we need a structure in which $\hat{w}$ is a geodesic for words $w$ in the language. Since no such structure appears in the literature, we need to establish its existence here. It turns out that this structure could be asynchronous, but that will be adequate for our purposes. \begin{proposition}\label{prop:ftqd} Let $(G,\Sigma)$ be relatively hyperbolic and suitable for parabolic geodesic biautomaticity, and let $\Gamma:=\Gamma(G,\Sigma)$, $\widehat{\Gamma}:=\widehat{\Gamma}(G,\widehat{\Sigma})$. Let $u,v, w_1,w_2$ be words over $\Sigma$ satisfying $w_1u = _G vw_2$, with $\|w_1\|, \|w_2\| \le k$ for some $k\geq 0$ and, in quadrilaterals in $\Gamma$ and $\widehat{\Gamma}$ whose sides are labelled by the words in that equation, let $p$ and $\hat{p}$ be paths (in $\Gamma$ and $\widehat{\Gamma}$) labelled by $u$ and $\hat{u}$, and let $q$ and $\hat{q}$ be the paths labelled by $v$ and $\hat{v}$. Suppose that \begin{mylist} \item[(a)] for each parabolic subgroup $H_i$, all $H_i$-components of both $u$ and $v$ lie in the specified geodesic biautomatic structure; \item[(b)] for some $\lambda \ge 1$ and $c \ge 0$, the paths $\hat{p}$ and $\hat{q}$ are $(\lambda,c)$-quasigeodesics that do not backtrack. \end{mylist} Then \begin{mylist} \item[(i)] there is a constant $\mathsf e' = \mathsf e'(\lambda,c,k)$ such that the paths $p$ and $q$ $\mathsf e'$-fellow travel in $\Gamma$, in such a way that those vertices of $p$ that are also vertices of $\hat{p}$ have at least one corresponding vertex on $q$ that is also a vertex of $\hat{q}$, and vice versa; \item[(ii)] there is a constant $k' = k'(\lambda,c,k)$ such that, whenever two vertices $b_1$ and $b_2$ on $q$ (or $p$) are both at $\Gamma$-distance at most $\mathsf e'$ from the same vertex on $p$ (or $q$), then the distance in the path $q$ (or $p$) between $b_1$ and $b_2$ is at most $k'$; \item[(iii)] we have $\|u\| \le (k'+1)\|v\|$ and $\|v\| \le (k'+1)\|u\|$. \end{mylist} \end{proposition} \begin{proof} Applying Proposition~\ref{prop:bcpp} to the paths $\hat{p}$ and $\hat{q}$, we choose $\mathsf e_1 \geq \mathsf e(\lambda,c,k)$ such that also $\mathsf e_1 \ge k$, and then choose $\mathsf e_2 \geq \mathsf e(\lambda,c,\mathsf e_1)$ such that $\mathsf e_2 \ge \mathsf e_1$. For a subword $u[i:j)$ of $u$, we denote by $p[i:j)$ the subpath of $p$ labelled by $u[i:j)$. So, if $u[i:j)$ is a component of $u$, then $p[i:j)$ is a component of $p$. Let $u[i_1:j_1)$,\,$u[i_2:j_2),$\,$\ldots,$\,$ u[i_t:j_t)$ with $i_1 < i_2 < \cdots < i_t$ be the components of $u$ of length greater that $\mathsf e_2$, and let the corresponding $\mathsf e_1$-similar components of $v$ to which they are connected be $v[k_1:l_1)$,\,$v[k_2:l_2)$,\,$\ldots$,\,$v[k_t:l_t)$. (Since $\hat{v}$ does not backtrack, these components are uniquely defined.) \begin{figure} \caption{Fellow travelling quasigeodesics} \label{fig:redshort} \end{figure} We claim that $k_1 < k_2 < \cdots < k_t$; that is, the connected components on $v$ occur in the same order as their corresponding components on $u$. We shall prove by induction on $r$ that $k_1 < k_2 <\cdots < k_r$ for all $1 \le r \le t$. It is vacuously true for $i=1$, so suppose that it is true for some value of $r$. Since $\mathsf e_1 \ge k$, the subpaths $\widehat{p[j_r:)}$ and $\widehat{q[l_r:)}$ of $\hat{p}$ and $\hat{q}$ are $\mathsf e_1$-similar $(\lambda,c)$-quasigeodesics that do not backtrack and, by choice of $\mathsf e_2$, the component $u[i_{r+1}:j_{r+1})$ of $u$ is connected to a component of $v[l_r:)$. But since $\hat{v}$ does not backtrack, this component must be $v[k_{r+1}:l_{r+1})$, so $k_r < l_r \le k_{r+1}$, which completes the induction and establishes the claim. See Figure \ref{fig:redshort}. Since the component subwords of $u$ and $v$ lie in a geodesic biautomatic structure, the paths $p[i_r:j_r)$ and $q[i_r:j_r)$ must $\mathsf e_1L_1$-fellow travel for $1 \le r \le t$, where $L_1$ is the maximum of the fellow travelling constants for the biautomatic structures of the parabolic subgroups $H_i$. On the other hand, the subpaths $\widehat{p'}$ and $\widehat{q'}$ of $\hat{p}$ and $\hat{q}$, where $p':=p[j_{r-1}:i_r)$ and $q':=q[l_{r-1}:k_r)$, are $\mathsf e_1$-similar for $1 \le r \le t+1$ (where for convenience we define $j_0=k_0=0$, $i_{t+1} = \|u\|$ and $k_{t+1} = \|v\|$), so they are contained in closed $\mathsf e_2$-neighborhoods of each other. Now the components of $u[j_{r-1}:i_r)$ have length at most $\mathsf e_2$, and those of $v[l_{r-1}:k_r)$ have length at most $\mathsf e_2' :=\mathsf e_2+ 2\mathsf e_1$ (since otherwise they would be connected to a component of $u[j_{r-1}:i_r)$, which would have length greater than $\mathsf e_2$). So the paths $p[j_{r-1}:i_r)$ and $q[l_{r-1}:k_r)$ are $(\lambda\mathsf e_2',c\mathsf e_2')$-quasigeodesics that lie within $(\mathsf e_2+\mathsf e_2')$-neighborhoods of each other. By the argument used in the proof of \cite[Proposition 3.1]{HR}, it follows that these paths $L_2$-fellow travel for some constant $L_2$ that depends only on $\lambda$, $k$ and $c$. Furthermore, by increasing $L_2$ by at most $\mathsf e_2$ we can ensure that all vertices on either of these paths that are also vertices of $\hat{p}$ or $\hat{q}$ have at least one corresponding vertex with the same property. So we have proved (i) with $\mathsf e' = \max(\mathsf e_1 L_1,L_2)$. Now let $b_1$ and $b_2$ be vertices on $q$ as in (ii), and assume that $b_2$ is further along $q$ than $b_1$. Then $d_\Gamma(b_1,b_2) \le 2 \mathsf e'$. Let $v' \in \Sigma^$ be a geodesic word with $v' =_G v$, and let $q'$ and $\widehat{q'}$ be the paths in $\Gamma$ and $\widehat{\Gamma}$ labelled by $v'$ and $\widehat{v'}$, respectively. Then $\widehat{q'}$ is quasigeodesic by Lemma~\ref{lem:qgeo}, and by applying (i) to $v$ and $v'$ with suitable constants, we find that $v$ and $v'$ $L_3$-fellow travel for some constant $L_3$. Let $b_1'$ and $b_2'$ be vertices of $q$ that correspond to $b_1$ and $b_2$ in the fellow-travelling. Then $d_\Gamma(b_1',b_2') = d_{q'}(b_1',b_2') \le 2(\mathsf e' + L_3)$. Now any two vertices $c_1$ and $c_2$ of $q$ that lie between $b_1$ and $b_2$ are each at distance at most $L_3$ from some vertex of $q'$ that lies between $b_1'$ and $b_2'$, so we have $d_\Gamma(c_1,c_2) \le 2\mathsf e' + 4 L_3$. So, in particular, since the components of $q$ are geodesics in $\Gamma$, the subpath of $q$ between $b_1$ and $b_2$ cannot have any components of length greater than $2\mathsf e' + 4 L_3$. So this subpath of $q$ is a quasigeodesic for suitable constants, and (ii) now follows. We get a corresponding result with $p$ and $q$ interchanged, and if this results in a larger constant $k'$, then we replace $k'$ by this larger value. Now (iii) follows directly from (ii). \end{proof} \begin{corollary}\label{cor:wdfsa} Let $(G,\Sigma)$ be relatively hyperbolic and suitable for parabolic geodesic biautomaticity. Let $g_1,g_2 \in G$ and let $\lambda \ge 1$ and $c \ge 0$. Then there is a deterministic 2-tape finite state automaton ${\mathcal{A}} = {\mathcal{A}}(\lambda,c,g_1,g_2)$ such that: \begin{mylist} \item[(i)] for all $u,v$ with $(u,v) \in L({\mathcal{A}})$, we have $g_1u =_G vg_2$; \item[(ii)] for all $u, v \in \Sigma^$ with $g_1u =_G vg_2$ that satisfy hypotheses (a) and (b) in the statement of Proposition~\ref{prop:ftqd}, we have $(u,v) \in L({\mathcal{A}})$. \end{mylist} \end{corollary} \begin{proof} Let $w_1,w_2 \in \Sigma^$ be geodesic words defining $g_1$ and $g_2$, and let $k = \max(\|w_1\|,\|w_2\|)$, and let $\mathsf e' = \mathsf e'(\lambda,c,k)$ and $k' = k'(\lambda,c,k)$ be the constants defined in Proposition~\ref{prop:ftqd}\,(i) and (ii). We define a 2-tape asynchronous {\em word-difference} automaton ${\mathcal{A}}'$, following the recipe of \cite[Definition 2.3.3]{ECHLPT}. The states ${\mathcal{A}}'$ are elements of $G$ of $\Sigma$-length at most $\mathsf e'$, with start state $g_1$ and single accepting state $g_2$, and transitions $g^{(x,y)} = x^{-1}gy$ for all $x,y \in \Sigma \cup \{\epsilon\}$ such that $g$ and $x^{-1}gy$ are both states of ${\mathcal{A}}'$. Then ${\mathcal{A}}'$ clearly satisfies (i) and it satisfies (ii) by Proposition~\ref{prop:ftqd}\,(i). But since ${\mathcal{A}}'$ contains $\epsilon$-transitions, it is non-deterministic. Proposition~\ref{prop:ftqd}\,(ii) enables us to define a deterministic 2-tape automaton ${\mathcal{A}}$ with $L({\mathcal{A}}) = L({\mathcal{A}}')$. After reading a prefix $u_1$ of $u$, we know that the lengths of the prefixes $v_1$ of $v$ for which $v_1^{-1}g_1u_1$ has $\Sigma$-length at most $\mathsf e'$ can differ by at most $k'$. So ${\mathcal{A}}$ can read the longest such prefix $v_1$ and remember all shorter such prefixes. After reading each letter of $u_1$, ${\mathcal{A}}$ need never read forward more than $k'$ letters of $v_2$ to maintain this information. If at any time there are no eligible prefixes $v_1$ of $v$ then ${\mathcal{A}}$ stops and rejects $(u,v)$. \end{proof} \begin{proposition}\label{prop:automatic} Let $(G,\Sigma)$ be relatively hyperbolic, and suitable for parabolic geodesic biautomaticity. Define $\mathcal{L}$ to be the set of all words $u \in \Sigma^$ such that the derived word $\hat{u} \in \widehat{\Sigma}^$ is geodesic and such that, for each parabolic subgroup $H_i$, all $H_i$-components of $u$ lie in the specified geodesic biautomatic structure. Then $\mathcal{L}$ is the language of an asynchronous biautomatic structure for $G$. Furthermore, for any ordering of $\Sigma$ with associated lexicographical ordering $\le_\mathsf{lex}$ of $\Sigma^$, the language $\mathcal{L}_0 = \{ u \in \mathcal{L} : u =_G v, v \in \mathcal{L} \Rightarrow u \le_\mathsf{lex} v \}$ is an asynchronous biautomatic structure for $G$ with uniqueness. \end{proposition} \begin{proof} If we can prove that $\mathcal{L}$ is a regular language, then the first claim will follow immediately from Corollary~\ref{cor:wdfsa}. (The hypothesis that $\hat{u}$ is geodesic for $u \in \mathcal{L}$ implies that $\hat{u}$ does not backtrack.) The required property that all $H_i$-components of words in $\mathcal{L}$ lie in the specified geodesic biautomatic structure is certainly testable by a finite state automaton, so we may restrict our attention to words that satisfy it. For all $u \in \mathcal{L}$, we have $u_1 \in \mathcal{L}$ for all prefixes $\widehat{u_1}$ of $\hat{u}$. So if $u \in \Sigma^$ with $u \not\in \mathcal{L}$, then $\hat{u}$ has a shortest prefix $u_1$ such that $\widehat{u_1}$ is a prefix of $\hat{u}$ (i.e. such that $u_1$ is a union of complete components of $u$ and subwords in $(\Sigma \setminus \mathcal{H})^$) and $u_1 \not\in \mathcal{L}$. Since the maximal proper prefix of $\widehat{u_1}$ is geodesic, $\widehat{u_1}$ labels a $(1,2)$-quasigeodesic in $\widehat{\Gamma}$, and it is not hard to see that $\widehat{u_1}$ cannot backtrack. Let $v_1 \in \mathcal{L}$ with $v_1 =_G u_1$. Then $(u_1,v_1) \in L({\mathcal{A}})$, where ${\mathcal{A}} := {\mathcal{A}}(1,2,1_G,1_G)$ as defined in Corollary~\ref{cor:wdfsa}. Furthermore, if $(u_2,v_2)$ is a prefix of $(u_1,v_1)$ in an accepting path of $(u_1,v_1)$ through ${\mathcal{A}}$, then the $\Sigma$-length and hence also the $\widehat{\Sigma}$-length of the element of $G$ defined by $u_2^{-1}v_2$ is bounded. So ${\mathcal{A}}$ can be modified to keep track of the difference between the $\widehat{\Sigma}$-lengths of these pairs of prefixes $(u_2,v_2)$, and hence it can detect that $\|\widehat{u_1}\| > \|\widehat{v_1}\|$. So a word $u \in \Sigma^$ lies in $\mathcal{L}$ if and only if its components satisfy the required condition, and if it has no prefix $u_1$ consisting of a union of complete components of $u$ and subwords in $(\Sigma \setminus \mathcal{H})^$ for which there exists $v_1$ with $(u_1,v_1) \in L({\mathcal{A}})$ and $\|\widehat{u_1}\| > \|\widehat{v_1}\|$. So $\mathcal{L}$ is regular by \cite[Lemma 7.1.5]{ECHLPT}. The final statement follows immediately from \cite[Theorem 7.3.2]{ECHLPT} (together with its proof, in which $\mathcal{L}_0$ is defined as we have done so here). \end{proof} \begin{proposition}\label{prop:nfwdlenbd} There is a constant $\mathsf{D}$ (depending on $G$ and $\Sigma$) such that $\|\mathsf{nf}(w)\| \le \mathsf{D} \|w\|$ for any $w \in \Sigma^$. \end{proposition} \begin{proof} It is sufficient to prove this when $w$ is a geodesic word. Then by Lemma~\ref{lem:qgeo} $\hat{w}$ is a $(\lambda,0)$-quasigeodesic for some $\lambda \ge 1$, and the result follows by applying Proposition~\ref{prop:ftqd}\,(iii) with $u=w$ and $v=\mathsf{nf}(w)$ and $w_1=w_2 = \epsilon$. \end{proof} \begin{remark}\label{rem:subclosure} The languages $\mathcal{L}$ and $\mathcal{L}_0$ are not necessarily closed under subwords, but if $u \in \mathcal{L}$ or $u \in \mathcal{L}_0$, and $u_1$ is a subword that contains only complete components of $u$ together with subwords in $\Sigma \setminus \mathcal{H})^$, then $u_1 \in \mathcal{L}$ or $\mathcal{L}_0$. \end{remark} \begin{remark} It is not difficult to show that the paths in $\Gamma$ labelled by words in $\mathcal{L}$ and $\mathcal{L}_0$ are quasigeodesics, but we shall not need that property. \end{remark} \section{Background on straight line programs} \label{sec:slp_background} Let $\mathcal{G}=(V,S,\rho)$ be a straight line program over an alphabet $\Sigma$. For a variable $A \in V$, the word $\rho(A)$ is called the {\em right-hand side} of $A$. We define the {\em size} of $\mathcal{G}$ to be the total length of all right-hand sides: $\|\mathcal{G}\| := \sum_{A \in V} \|\rho(A)\|$. It follows from the acyclicity condition that each variable $A$ in $V$ has a well defined {\em height}, $\mathsf{h}(A)$, namely the smallest positive integer $r$ for which $\rho^r(A) \in \Sigma^$. By removing from $\mathcal{G}$ all variables that do not occur in any image $\rho^k(S)$ for $k \in \mathbb{N}$, we obtain (in time that is a polynomial function of $\|\mathcal{G}\|$) an {\sc slp}\xspace in which $S$ is the only variable of maximal height. We call such an {\sc slp}\xspace\ {\em trimmed}; we will often want to assume this property for an {\sc slp}\xspace. Suppose that $A$ is a variable of height $r$ in $\mathcal{G}$. We define $\mathsf{val}_{\mathcal{G}}(A):=\rho^r(A) \in \Sigma^$, and observe that $\mathsf{val}_{\mathcal{G}}(S)= \mathsf{val}(\mathcal{G})$. We also define a (trimmed) {\sc slp}\xspace $\mathcal{G}_A := (V_A,A,\rho_A)$ over $\Sigma$, the {\em restriction} of $\mathcal{G}$ to $A$, with start variable $A$, set of variables $V_A$ consisting of all $B \in V$ that appear within $\rho^k(A)$ for some $k \geq 0$, and map $\rho_A$ defined to be the restriction of $\rho$ to $V_A$. We note that for any $B \in V_A$, $\mathsf{val}_\mathcal{G}(B)=\mathsf{val}_{\mathcal{G}_A}(B)$, and in particular $\mathsf{val}(\mathcal{G}_A) = \mathsf{val}_{\mathcal{G}}(A)$. If every right-hand side (except that of $\rho(S)=\epsilon$, if it occurs) has the form $a \in \Sigma$ or $BC$ with $B,C \in V$ (that is, the grammar associated with $\mathcal{G}$ is in Chomsky normal form), then we shall say that $\mathcal{G}$ itself is in {\em Chomsky normal form}. We shall often want to assume this property. Given {\sc slp}s\xspace $\mathcal{G}_1$ and $\mathcal{G}_2$ over a single alphabet $\Sigma$, we denote by $\mathcal{G}_1\mathcal{G}_2$ the {\sc slp}\xspace with $\mathsf{val}(\mathcal{G}_1\mathcal{G}_2)$ equal to the concatenation $\mathsf{val}(\mathcal{G}_1)\mathsf{val}(\mathcal{G}_2)$, which we derive from $\mathcal{G}_1$ and $\mathcal{G}_2$ by adding to their disjoint union a single variable $S_{\mathcal{G}_1\mathcal{G}_2}$ and the single production $S_{\mathcal{G}_1\mathcal{G}_2}\rightarrow S_{\mathcal{G}_1}S_{\mathcal{G}_2}$. We extend this definition to a concatenation of any number of {\sc slp}s\xspace, and also to a concatenation of {\sc slp}s\xspace and words over $\Sigma$, so that a concatenation of the form $\mathcal{G} = u_0\mathcal{G}_1u_1\cdots \mathcal{G}_ku_k$, where each $\mathcal{G}_i$ is a {\sc slp}\xspace and each $u_i$ a possibly empty word over $\Sigma$ is formed by the addition of a single production \[ S_\mathcal{G} \rightarrow u_0S_{\mathcal{G}_1}u_1S_{\mathcal{G}_2}u_2\cdots S_{\mathcal{G}_k}u_k \] to the disjoint union of the productions of the {\sc cslp}s\xspace $\mathcal{G}_i$. We will make use of a number of results from the literature, which we collect together here as a single proposition: \begin{proposition} \label{prop:slp_results} Let $\mathcal{G}=(V,S,\rho)$ be a straight line program over a finite alphabet $\Sigma$. \begin{mylist} \item[(i)] An algorithm exists that transforms $\mathcal{G}$ into an {\sc slp}\xspace $\mathcal{G}'$ in Chomsky normal form with $\|\mathcal{G}'\| \le 2\|\mathcal{G}\|$ and $\mathsf{val}(\mathcal{G}) = \mathsf{val}(\mathcal{G}')$, in time that is a linear function of $\|\mathcal{G}\|$; see for example \cite[Proposition 3.8]{Loh14}. \item[(ii)] We have $\|\mathsf{val}(\mathcal{G})\| \leq 3^{\|\mathcal{G}\|/3}$ \cite[proof of Lemma 1]{CLLLPPSS05}. \item[(iii)] The length $\|\mathsf{val}(\mathcal{G})\|$ can be computed in time that is a polynomial function of $\|\mathcal{G}\|$ \cite[Proposition 3.9]{Loh14}. \item[(iv)] For $0 \leq i \leq j \leq \|\mathsf{val}(\mathcal{G})\|$, an {\sc slp}\xspace with value $\mathsf{val}(\mathcal{G})[i:j)$ can be computed in time that is a polynomial function of $\|\mathcal{G}\|$ \cite[Proposition 3.9]{Loh14}. \item[(v)] Given a deterministic finite state automaton $M$ over the alphabet $\Sigma$ and an {\sc slp}\xspace $\mathcal{G}$ over the alphabet $\Sigma$, it can be determined in time that is a polynomial function of $\|\mathcal{G}\|$ whether $\mathsf{val}(\mathcal{G}) \in L(M)$ \cite[Theorem 3.11]{Loh14}. \item[(vi)] Given two {\sc slp}s\xspace $\mathcal{G}$ and $\mathcal{H}$, it can be checked in time that is a polynomial function of $\|\mathcal{G}\|+\|\mathcal{H}\|$ whether $\mathsf{val}(\mathcal{G}) = \mathsf{val}(\mathcal{H})$ \cite{Pla94}. \end{mylist} \end{proposition} \begin{definition}\label{def:root} Let $\mathcal{G}=(V,S,\rho)$ be an {\sc slp}\xspace over an alphabet $\Sigma$, with value $w_1uw_2$. We say that $u$ has a {\em root}, $A$ in $V$, if for some $k$ and $\ell=\mathsf{h}(S)-k$, we have $\rho^k(S)=\alpha A \beta$, where $\alpha,\beta \in (V \cup \Sigma)^$, $\rho^\ell(\alpha)=w_1$, $\rho^\ell(\beta)=w_2$, and $\rho^\ell(A)=u$. \end{definition} \subsection{ Extensions of {\sc slp}s\xspace}\label{subsec:extend_slps} It will sometimes be convenient in our proofs to make use of particular extensions of {\sc slp}s\xspace, namely cut-{\sc slp}s\xspace and tethered-{\sc slp}s\xspace. Cut-{\sc slp}s\xspace (which we shall abbreviate as {\sc cslp}s\xspace) are defined analogously to {\sc slp}s\xspace, but in addition a cut-{\sc slp}\xspace may contain right-hand sides that are written in the form $B[:i)$ or $B[i:)$ for a variable $B$ and an integer $i \geq 0$ \cite{Loh14}; {\sc cslp}s\xspace are used in the construction of a polynomial time algorithm for the compressed word problem of a free group in \cite{Loh06siam}, where they are called {\em composition systems}. We call $[:i)$ and $[i:)$ cut operators. It is convenient to allow cut operators of the form $B[i:j)$ for $0 \le i < j$, which we can achieve as the combination of two cut operators $B[:j)[i:)$. If $\rho(A) = B[i:j)$ in a {\sc cslp}\xspace $\mathcal{G}$, then we define $\mathsf{val}_{\mathcal{G}}(A)$ to be the string $\mathsf{val}_{\mathcal{G}}(B)[i:j)$. Given a {\sc cslp}\xspace $\mathcal{G}$, we denote by $\mathcal{G} [i:j)$ the {\sc cslp}\xspace with value $\mathsf{val}(\mathcal{G})[i:j)$ that we derive from $\mathcal{G}$ by adding to $\mathcal{G}$ a single variable $S_{\mathcal{G}[i:j)}$ (the start variable of $\mathcal{G} [i:j)$) and the single production $S_{\mathcal{G}[i:j)}\rightarrow S_{\mathcal{G}}$ to the {\sc cslp}\xspace $\mathcal{G}$. In a {\sc cslp}\xspace, as in a {\sc slp}\xspace, we require that the associated binary relation is acyclic. We define concatenations of {\sc cslp}s\xspace analogously to concatenations of {\sc slp}s\xspace. The following results can be found in the literature; the second follows from the first together with Proposition~\ref{prop:slp_results}\,(vi).\\ \begin{proposition}\label{prop:cslps} \begin{mylist} \item[(i)] From a given {\sc cslp}\xspace $\mathcal{G}$ one can compute, in time that is a polynomial function of $\|\mathcal{G}\|$, an {\sc slp}\xspace $\mathcal{G}'$ such that $\mathsf{val}(\mathcal{G}) = \mathsf{val}(\mathcal{G}')$ \cite{Hag00}; see also \cite[Theorem 3.14]{Loh14}. \item[(ii)] Given {{\sc cslp}\xspace}s $\mathcal{G}_1$ and $\mathcal{G}_2$, it can be checked in polynomial time whether $\mathsf{val}(\mathcal{G}_1) = \mathsf{val}(\mathcal{G}_2)$ \end{mylist} \end{proposition} As in \cite{HLS}, the proof of the main theorem also involves extensions to {\sc slp}s\xspace and {\sc cslp}s\xspace that we call {\sc tslp}s\xspace and {\sc tcslp}s\xspace, respectively (T stands for ``tethered''). These extensions only make sense when the alphabet $\Sigma$ is the (inverse closed) generating set of a group $G$, and when $G$ is equipped with a normal form $\mathsf{nf}(w)$ for words $w \in \Sigma^$; that is, $\mathsf{nf}(w) \in \Sigma^$, $\mathsf{nf}(w) =_G w$, and, for $v,w \in \Sigma^$, $v=_Gw \Rightarrow \mathsf{nf}(v) = \mathsf{nf}(w)$. We extend the definition of a {\sc slp}\xspace or {\sc cslp}\xspace to that of a {\sc tslp}\xspace or {\sc tcslp}\xspace over such an alphabet $\Sigma$ by allowing additional right-hand sides that are expressions of the form $B\langle \alpha,\beta \rangle$ with a variable $B$, and strings $\alpha,\beta$ over $\Sigma$, each of length at most $J$, where $J = J_\mathcal{T} \in \mathbb{N}$ is a parameter of the {\sc tslp}\xspace or {\sc tcslp}\xspace $\mathcal{T}$. Given a right-hand side $\rho(A)=B\langle \alpha,\beta \rangle$ in a {\sc tslp}\xspace or {\sc tcslp}\xspace $\mathcal{T}$, we define \[\mathsf{val}_{\mathcal{T}}(A) := \mathsf{nf}(\alpha \, \mathsf{val}_{\mathcal{T}}(B) \, \beta^{-1}).\] In {\sc tslp}s\xspace and {\sc tcslp}s\xspace, as with {\sc slp}s\xspace and {\sc cslp}s\xspace, we require that the associated binary relations are acyclic, and we define concatenations of {\sc tslp}s\xspace and {\sc tcslp}s\xspace analogously to concatenations of {\sc slp}s\xspace. We define the {\em size} of a variable $A$ in a {\sc tcslp}\xspace $\mathcal{T}$ to be the total number of occurrences of symbols from $\Sigma \cup V$ in $\rho(A)$, and the size of $\mathcal{T}$ is obtained by taking the sum over all variables. As we did for {\sc slp}s\xspace, for a {\sc tcslp}\xspace $\mathcal{U}$ over $\Sigma$, we define the height, $\mathsf{h}(A)$, of a variable $A$ of $\mathcal{U}$ to be the smallest integer $k$ such that $\rho_{\mathcal{U}}^k(A) \in \Sigma^$. Proposition~\ref{prop:slp_results}\,(i) extends to {\sc tcslp}s\xspace: for a given {\sc tcslp}\xspace $\mathcal{U}$ over $\Sigma$ with $\mathsf{val}(\mathcal{U}) \ne \epsilon$, we can in linear time construct a {\sc tcslp}\xspace $\mathcal{U}'$ with the same value in which $\|\mathcal{U}'\| \le 2 \|\mathcal{U}\|$ and all right hand sides $\rho_{\mathcal{U}'}(A)$ that lie in $(V_{\mathcal{U}'} \cup \Sigma)^$ have the form $a \in \Sigma$ or $BC$ with $B,C \in V_{\mathcal{U}'}$. We say that the {\sc tcslp}\xspace (or {\sc tslp}\xspace) $\mathcal{T}$ is {\em $\mathsf{nf}$-reduced} if for every variable $A$ the word $\mathsf{val}_{\mathcal{T}}(A)$ is $\mathsf{nf}$-reduced; that is $\mathsf{nf}(\mathsf{val}_{\mathcal{T}}(A)) = \mathsf{val}_{\mathcal{T}}(A)$. In Section~\ref{sec:convert} we shall prove Theorem~\ref{thm:convert} that, given a {\sc tcslp}\xspace $\mathcal{T}$ over a suitably chosen generating set $\Sigma$ of a group that is hyperbolic relative to a collection of free abelian subgroups (and which satisfies suitable conditions relating to the `splitting of components', explained in Definition~\ref{def:split}), we can, in time that is a polynomial function of $\|\mathcal{T}\|$ (depending on $J_\mathcal{T}$), compute an {\sc slp}\xspace $\mathcal{G}$ with $\mathsf{val}(\mathcal{T}) = \mathsf{val}(\mathcal{G})$. This result will be a vital component of our main result. Its proof will require the following generalisation of Proposition~\ref{prop:slp_results}\,(ii) to $\mathsf{nf}$-reduced {\sc tcslp}\xspace\,s. \begin{proposition}\label{prop:tcslplenbd} Let $\mathcal{T}=(V,S,\rho)$ be an $\mathsf{nf}$-reduced {\sc tcslp}\xspace. Then there is a constant $J'$ (depending on $\Sigma$ and $J_\mathcal{T}$) such that $\|\mathsf{val}(\mathcal{T})\| \leq (J')^{\|\mathcal{T}\|}$ \end{proposition} \begin{proof} As we saw above, there is a {\sc tcslp}\xspace $\mathcal{T}' = (V',S,\rho')$ with $\mathsf{val}(\mathcal{T}') = \mathsf{val}(\mathcal{T})$, $\|\mathcal{T}'\| \le 2\|\mathcal{T}\|$ in which all right hand sides $\rho'(A)$ that lie in $(V' \cup \Sigma)^$ have the form $a \in \Sigma$ or $BC$ with $B,C \in V'$. If $\rho(A) = B\langle \alpha,\beta \rangle$ with $\|\alpha\|,\|\beta\| \le J_\mathcal{T}$ then $\rho'(A) = \rho(A)$ and, by \ref{prop:ftqd}\,(iii), we have $\|\mathsf{val}(A)\| \le (k'+1)\mathsf{val}(B)\|$ with $k' = k'(1,0,J_\mathcal{T})$. We claim that $\mathsf{val}_{\mathcal{T}'}(A) \le \max(2,k'+1)^{\mathsf{h}(A)}$ for all $A \in V'$. Since $\mathsf{h}(S) \le \|\mathcal{T}'\| \le 2\|\mathcal{T}\|$ this will prove the proposition with $J' = \max(2,k'+1)^2$. The proof of the claim is by induction on $\mathsf{h}(A)$, and the base case $\mathsf{h}(A)=0$ is clear. Otherwise, if $\rho'(A) = BC$ then $\|\mathsf{val}_{\mathcal{T}'}(A)\| = \|\mathsf{val}_{\mathcal{T}'}(B)\|+\|\mathsf{val}_{\mathcal{T}'}(C)\|$ with $\mathsf{h}(B),\mathsf{h}(C) \le \mathsf{h}(A)$; if $\rho'(A) = B[i:)$ or $B[:j)$, then $\|\mathsf{val}_{\mathcal{T}'}(A)\| \le \|\mathsf{val}_{\mathcal{T}'}(B)\|$; and if $\rho'(A) = B\langle \alpha,\beta \rangle$ with $\|\alpha\|,\|\beta\| \le J_\mathcal{T}$ then, as saw above, $\|\mathsf{val}_{\mathcal{T}'}(A)\| \le (k'+1)\|\mathsf{val}_{\mathcal{T}'}(B)\|$. In all three cases the claim follows immediately from the inductive hypothesis. \end{proof} \section{Some constructions of {\sc slp}s\xspace for abelian groups} \label{sec:slps_ab} We need a few results about {\sc slp}s\xspace for finitely generated abelian groups, which will be the parabolic subgroups of our relatively hyperbolic groups. \begin{lemma} \label{lem:ab_polytime} Let $G$ be a finitely generated abelian group with generating set $X =\{x_1^{\pm 1},\ldots,x_k^{\pm 1}\}$, and let $\mathcal{G}$ be an {\sc slp}\xspace over $X$. Then, in time that is polynomial in $\|\mathcal{G}\|$, we can \begin{mylist} \item[(i)] compute the vector $(n_1,n_2,\cdots,n_k)$, defined by $\mathsf{val}(\mathcal{G}) =_G \prod_{i=1}^t x_i^{n_i}$ (where the integers $n_i$ are output as binary numbers), \item[(ii)] construct an {\sc slp}\xspace $\mathcal{G}'$ with $\mathsf{val}(\mathcal{G}') = \prod_{i=1}^t x_i^{n_i}$ and $\|\mathcal{G}'\| \le \max(4k\log_2(\|\mathsf{val}(\mathcal{G}')\|),1)$. \end{mylist} \end{lemma} \begin{proof} The fact that we can compute the integers $n_i$ in polynomial time follows from Proposition \ref{prop:slp_results}\,(ii) together with the fact that we can perform addition and subtraction on integers of absolute value at most $N$ in time $O(\log N)$. For part (ii), note first that for any $x \in X$ and integer $n>0$, we can define an {\sc slp}\xspace with value $x^n$ and size at most $\max(3 \log_2(n),1)$ by expressing $n$ in binary and introducing a variable $A_i$ with size 2 and value $x^{2^i}$ for each $i$ with $2^i< n$. For example, with $n=14$, we write $14 = 2^1 + 2^2 + 2^3$, and define the {\sc slp}\xspace $(\{S,A_1,A_2,A_3\},S,\rho)$ with $\rho(S)=A_1A_2A_3$, $\rho(A_1)=xx$, $\rho(A_2)=A_1A_1$, $\rho(A_3)=A_2A_2$, which has value $x^{14}$ and size 9. So we can construct an {\sc slp}\xspace $\mathcal{G}'$ with value $\mathsf{val}(\mathcal{G})$ and size at most $(3\log_2(\|\mathsf{val}(\mathcal{G}')\|)+1)k \le \max(4k\log_2(\|\mathsf{val}(\mathcal{G}')\|),1)$. \end{proof} \begin{proposition}\label{prop:abslp} Let $G = \langle X \rangle$ be a finitely generated abelian group. Then $X$ is contained in a finite subset $Y$ of $G$ for which, given an {\sc slp}\xspace $\mathcal{G}$ over $Y$, we can construct an {\sc slp}\xspace $\mathcal{G}'$ over $Y$ with $\mathsf{val}(\mathcal{G}') = \mathsf{slex}_Y(\mathsf{val}(\mathcal{G}))$ and $\|\mathcal{G}'\| \le \max(4\|Y\|\log_2(\|\mathsf{val}(\mathcal{G}')\|),1)$ in time that is a polynomial function of $\|\mathcal{G}\|$. \end{proposition} \begin{proof} Choose generators $z_1,\ldots,z_r,z_{r+1},\ldots,z_s$ of the cyclic direct factors of $G$, where $z_i$ has infinite order if and only if $i \le r$. We can write each element $x$ of $X$ as $\prod_{i=1}^s z_i^{{n_i(x)}}$ with ${n_i(x)} \in \mathbb{Z}$; note that the set of all ${n_i(x)}$ is finite, and so its elements can be regarded as constants. Let $e$ be the exponent of the torsion subgroup of $G$, and let $M := e(\max_{x \in X} \sum_{i=1}^r \|{n_i(x)}\|+1)$. Now we define $Y$ to be the union of the three sets $\{ z_i^{\pm M}: 1 \le i \le r\}$, $\{ z_i^{\pm 1}: 1 \le i \le s\}$, and $X$; we order $Y$ so that \begin{mylist} \item[$\bullet$] pairs of mutually inverse generators are adjacent in the ordering; and \item[$\bullet$] the generators $z_i^{\pm M}$ come first in the ordering, with $z_1^M< z_1^{-M} < \cdots < z_r^M < z_r^{-M}$. \end{mylist} We denote by $y_i$ and $y_i^{-1}$ the elements in positions $2i-1$ and $2i$ of this ordered set of generators. Suppose that there are $2t$ such generators in total. We claim that, in any geodesic word $\prod_{i=1}^t y_i^{n_i}$ with $n_i \in \mathbb{Z}$, we have $\|n_k\| < M$ for all $k>r$. To see this, given such a geodesic word, suppose that $r < k \le t$. If $y_k= z_j$ for some $j$, then we could replace $y_k^M$ by the shorter word $y_j$ if $j \le r$ and by the empty word if $j>r$; so we must have $\|n_k\| < M$. Otherwise, we have $y_k \in X$, and then $x := y_k =_G \prod_{i=1}^s z_i^{n_i(x)}$ and (because $e\|M$), $x^M =_G \prod_{i=1}^r z_i^{Mn_i(x)}=_G \prod_{i=1}^r y_i^{n_i(x)}$. This last word is shorter than $x^M$, since $M > \sum_{i=1}^r \|n_i(x)\|$ So replacing $x^M$ by $\prod_{i=1}^r y_i^{n_i(x)}$ would be a reduction in length, and hence again we must have $\|n_k\| < M$. Now it follows from Lemma~\ref{lem:ab_polytime} that, in time polynomial in $\|\mathcal{G}\|$, we can write $g := \mathsf{val}(\mathcal{G})$ as an integer vector over the generators in $Y$, and hence as a product $\prod_{i=1}^t y_i^{n_i} \in G$. Now we want to compute $\mathsf{slex}(g)$. If any $\|n_k\| \ge M$ for $k > r$, then we write $n_k = qM + n_k'$ with $\|n_k'\| < M$ and $\mathrm{sgn}(n_k) = \mathrm{sgn}(n_k')$ and replace the expression for $g$ by an equivalent expression in which $y_k^{qM}$ is replaced by a shorter word, as described in the preceding paragraph. This involves integer arithmetic and can be done in time polynomial in the sizes of the integers. So we may assume that $g = \prod_{i=1}^t y_i^{n_i} \in G$ with $\|n_k\| < M$ for all $k>r$. Now suppose that $\mathsf{slex}(g) = \prod_{i=1}^t y_i^{n_i'}$. Then $\prod_{i=1}^t y_i^{(n_i-n_i')} =1$. and since $\|n_k'\|,\|n_k\| < M$, we have $\|n_k-n_k'\| < 2M$ for all $k>r$. Since $y_1,\ldots,y_r$ are free generators, there are no nontrivial relations in $G$ that involve only $y_1,\ldots,y_r$, and so the number of lists of integers $m_1,\ldots,m_t$ for which $\prod_{i=1}^t y_i^{m_i} =1$ and $\|m_i\| < M$ is at most $(2M)^{t-r}$. So we can consider each of these equations in turn, and thereby find all possible values of $n_i'$. From these, we select the shortlex least representative of $g$. We can now use Proposition \ref{lem:ab_polytime} to construct the required {\sc slp}\xspace $\mathcal{G}'$. \end{proof} \section{Examining the geometry of $\widehat{\Gamma}$} \label{sec:gammahat_geom} For the remainder of this article, $G$ will be a group that is hyperbolic relative to a collection of free abelian subgroups $H_i$, and $\mathcal{H}$ the set of non-identity elements of those subgroups, as in Section~\ref{sec:relhyp}. By \cite[Theorem 4.3.1]{ECHLPT}, abelian groups are shortlex automatic with respect to any finite ordered generating set, and a little thought shows that any automatic structure for an abelian group must be biautomatic. So the hypotheses of shortlex biautomaticity of the subgroups $H_i$ in Proposition~\ref{prop:genset} are satisfied for all choices of finite, ordered generating sets of the $H_i$. \subsection{Fixing the generating set $\Sigma$ and constant $\delta$} \label{subsec:relhyp_param1} Given a generating set $\Sigma'$ for $G$, we define, for each $i$, $X_i := (\Sigma' \cup \mathcal{H}') \cap H_i$, where $\mathcal{H}'$ is the finite subset of $\mathcal{H}$ whose existence is guaranteed by Proposition~\ref{prop:genset}. It follows from Corollary~\ref{cor:genset} that, for each $i$, $X_i$ generates $H_i$. Now we select finite subsets $Y_i \supset X_i$ as in Proposition~\ref{prop:abslp}, and define $\Sigma := \Sigma' \cup \mathcal{H}' \cup \bigcup_{i\in \Omega} Y_i.$ Our construction ensures that $\Sigma \cap H_i = Y_i$. For the remainder of this article, we use this generating set $\Sigma$ for $G$, and denote $\Sigma \cap H_i$ by $\Sigma_i$; we note that $(G,\Sigma)$ is suitable for parabolic geodesic biautomaticity. For convenience we assume that the elements of $\Sigma$ represent distinct elements of $G$ and that no element of $\Sigma$ represents the identity element. Furthermore, we define $\mathcal{L}_0$ to be the asynchronous automatic structure of $G$ with uniqueness that is defined in Proposition \ref{prop:automatic}, where we use the shortlex biautomatic structure on $H_i$ over $\Sigma_i$. We shall call words in $\mathcal{L}_0$ {\it $\mathcal{L}_0$-reduced} and, for $v \in \Sigma^$, we denote the unique element $u \in \Sigma^$ with $u =_G v$ by $\mathsf{nf}(v)$. We shall use this normal form in all of our {\sc tcslp}s\xspace over $\Sigma$. Our assumptions on $\Sigma$ ensure that $\mathsf{nf}(a) = a$ for all $a \in \Sigma$. Now that we have fixed $\Sigma$, we can also fix the associated Cayley graphs $\Gamma$ and $\widehat{\Gamma}$. We know from the definition of relative hyperbolicity that $\widehat{\Gamma}$ is Gromov hyperbolic, and we fix the constant $\delta>0$ such that $\widehat{\Gamma}$ is a $\delta$-hyperbolic space; that is, all geodesic triangles in $\widehat{\Gamma}$ are $\delta$-thin (and hence also $\delta$-slim) as defined in \cite[Chapter 1]{AL}. We assume these choices for $\Sigma$, $\Gamma$, $\widehat{\Gamma}$ and $\delta$ for the remainder of this article. \subsection{Properties of some geodesic triangles and quadrilaterals in $\widehat{\Gamma}$} \label{subsec:Gammahat_triangles} \begin{proposition}\label{prop:backup} Let $u,v,w \in \Sigma^$ be words over our selected generating set for $G$ with $v =_G uw$, where $\hat{u}$ and $\hat{v}$ are geodesic words over $\widehat{\Sigma}$, and $\|w\| \le \kappa$ for some $\kappa$. Then constants $K_1(\kappa)$ and $L_1(\kappa)$ exist such that, for any vertex $d$ on the path in $\widehat{\Gamma}$ labelled by $\hat{u}$ and at distance at least $K_1(\kappa)$ from the end of that path, there is a vertex $e$ of $\widehat{\Gamma}$ on the path labelled by $\hat{v}$, with $d_\Gamma(d,e)\leq L_1(\kappa)$. \end{proposition} \begin{proof} Since $\hat{u}$ labels a geodesic path in $\widehat{\Gamma}$ and $\|\hat{w}\| \le \|w\| \le \kappa$, we know that $\hat{u}\hat{w}$ labels a $(1,2\kappa)$--quasigeodesic path in $\widehat{\Gamma}$. Our aim is to replace $\hat{u}\hat{w}$ by a $G$-equivalent word $t \in \widehat{\Sigma}^$ that also labels a $(1,2\kappa)$--quasigeodesic path, and does not backtrack, and whose prefix of length $\|\hat{u}\|-K_1(\kappa)$ matches the corresponding length prefix of $\hat{u}$ for some constant $K_1(\kappa)$. The result will then follow directly from Theorem~\ref{prop:bcpp}\,(1), applied to $t$ and $\hat{v}$. If $\hat{u}\hat{w}$ backtracks, then it contains a subword of length greater than $1$ that represents an element of $H_i$ for some $i \in \Omega$. Since such elements have length at most $1$ over $\widehat{\Sigma}^$, and $\hat{u}\hat{w}$ labels a $(1,2\kappa)$--quasigeodesic path, any such subword has length at most $K_1(\kappa):= 1+2\kappa$. Furthermore, since $\hat{u}$ does not vertex backtrack, such a subword must intersect the suffix $\hat{w}$ nontrivially. So it does not intersect the prefix $\hat{u}(:\|\hat{u}\|-K_1(\kappa)]$ of $u$. So, after replacing any such subwords by $G$-equivalent words of length $1$ over $\Gamma^$, the resulting word $t$ does not backtrack, and has the desired property. \end{proof} Our next result can be proved by two applications of Proposition~\ref{prop:backup}, and we omit the details. \begin{lemma}\label{lem:backup2} Let $u,v,w_1,w_2 \in \Sigma^$ be words over $\Sigma$ with $w_1u =_G vw_2$, and suppose that $\hat{u}$ and $\hat{v}$ are geodesic words, and that $\|w_1\|, \|w_2\| \le \kappa$ for some $\kappa$. Then constants $K_2(\kappa)$ and $L_2(\kappa)$ exist, such that, for any vertex $d$ on the path in $\widehat{\Gamma}$ labelled by $\hat{u}$ and at distance at least $K_2(\kappa)$ from the beginning and the end of that path, there is a vertex $e$ of $\widehat{\Gamma}$ on the path labelled by $\hat{v}$, with $d_\Gamma(d,e)\leq L_2(\kappa)$. \end{lemma} \begin{proposition}\label{prop:backup2} There is a linear function $\mathsf{f}:\mathbb{N} \to \mathbb{N}$ and a constant $L'$, with the following property. Suppose that $u,v,w_1,w_2$ are words over our selected generating set $\Sigma$ for $G$, with $w_1u =_G vw_2$, and suppose that $\hat{u}$ and $\hat{v}$ are geodesic words. Consider a quadrilateral in $\widehat{\Gamma}$ with sides labelled $\widehat{w_1},\hat{u},\widehat{w_2},\hat{v}$. Then, for any vertex $d$ on the path labelled by $\hat{u}$ in that quadrilateral, and at distance at least $\mathsf{f}(\max\{\|w_1\|,\|w_2\|\})$ from the beginning and the end points of that path, there is a vertex $e$ of the path labelled by $\hat{v}$ in the quadrilateral, with $d_\Gamma(d,e)\leq L'$. \end{proposition} \begin{proof} We recall the constant $\delta$ defined in Section~\ref{subsec:relhyp_param1}. In the quadrilateral defined in the statement, replace the sides labelled by $\widehat{w_1}$ and $\widehat{w_2}$ by geodesic paths $p_1$ and $p_2$ between their endpoints to give a geodesic quadrilateral in $\widehat{\Gamma}$. So $\|p_i\| \le \|\widehat{w_i}\| \le \|w_i\|$ for $i=1,2$. Since $\widehat{\Gamma}$ is $\delta$-hyperbolic, any vertex on any side of this quadrilateral is at distance at most $2\delta$ in $\widehat{\Gamma}$ from a vertex on one of the other three sides. Now for a vertex $d$ on the path labelled $\hat{u}$ that is at distance $\ell$ along $\hat{u}$ from the start point, all vertices on the path $p_1$ are at distance at least $\ell- \|p_1\| \ge \ell - \|w_1\|$ from $d$ in $\widehat{\Gamma}$. So, if $\ell \ge \|w_1\| + 2\delta+1$, then $d$ cannot be at distance at most $2\delta$ from a vertex on $p_1$. Similarly, if the distance of $d$ on $\hat{u}$ from the end point of $\hat{u}$ is greater than $\|w_2\| + 2\delta+1$, then it cannot be at distance at most $2\delta$ from a vertex on $p_2$. So if both of those conditions on $d$ hold, then $d$ must be at distance at most $2\delta$ in $\widehat{\Gamma}$ from a vertex $e$ on the side of the quadrilateral labelled $\hat{v}$. Now, by Lemma~\ref{lem:backup2}, there are constants $K_2:=K_2(2\delta)$ and $L_2:=L_2(2\delta)$ depending only on $G$ and $\Sigma$, such that any vertex $d$ on the path labelled $\hat{u}$ that is at distance at least $\|w_1\| + 2\delta+1+K_2$ and $\|w_2\| + 2\delta+1+K_2$ from the start and end points of that path, respectively, is at distance at most $L_2$ in $\Gamma$ from a vertex on the side of the quadrilateral labelled $\hat{v}$. This proves the proposition with $L'=L_2$. \end{proof} \subsection{Fixing the constant $L$} \label{subsec:relhyp_param2} We define $L$ to be the larger of the constants $L_1(\delta)$ and $L'$ that were defined in Propositions \ref{prop:backup} and~\ref{prop:backup2}, respectively. We will refer to $L$ repeatedly in the final two sections of the paper. \section{Some constructions of {\sc slp}s\xspace for relatively hyperbolic groups} \label{sec:slps_relhyp_props} The proof of our main result (which is split across Theorems ~\ref{thm:slextcslp} and~\ref{thm:convert}) will need some technical results relating to {\sc slp}s\xspace for our relatively hyperbolic group $G$ over our selected generating set $\Sigma$. In general these results will be applied to sub-{\sc slp}s\xspace of the {\sc slp}\xspace that is input and the {\sc slp}s\xspace that are derived from it within the above two theorems. \begin{proposition}\label{prop:comproot} Let $\mathcal{G}$ be an {\sc slp}\xspace over our selected generating set $\Sigma$ for $G$, and let $w:=\mathsf{val}(\mathcal{G})$. Then, in time polynomial in $\|\mathcal{G}\|$, we can construct an {\sc slp}\xspace $\mathcal{G}'$ in Chomsky normal form with value $w$ such that, for all variables $A$ of $\mathcal{G}'$, all components of $\mathsf{val}(A)$ have roots in $\mathcal{G}'_A$. Furthermore, for each parabolic subgroup $H_i$, we can compute a list of those variables $A$ of $\mathcal{G}'$ for which $\mathsf{val}_{\mathcal{G}'}(A) \in \Sigma_i^$. \end{proposition} \begin{proof} We may assume that $\mathcal{G}$ is trimmed and in Chomsky normal form. We construct $\mathcal{G}'=(V',S,\rho')$ from $\mathcal{G}=(V,S,\rho)$ by modifying the map $\rho$ on some of the variables in $V$, while also introducing some new variables that are needed within those new images for $\rho$. For each $A \in V$, we will have $\mathsf{val}_{\mathcal{G}'}(A)=\mathsf{val}_{\mathcal{G}}(A)$. In addition, for each $A \in V$, every component of $\mathsf{val}_{\mathcal{G}'}(A)$ will have a root in $\mathcal{G}'_A$. We put the variables of $V$ into increasing order of height, and consider them in that order. The {\sc slp}\xspace $\mathcal{G}'$ will be the last of a sequence of {\sc slp}s\xspace $\mathcal{G}_0=\mathcal{G}, \mathcal{G}_1,\ldots,\mathcal{G}_n$, where $n=\|V\|$, and $\mathcal{G}_i=(V_i,S,\rho_i)$ will be made from $\mathcal{G}_{i-1}$ by considering the $i$-th variable, $A_i$, in the list. We have $V_i \supset V_{i-1}$ and form $\rho_i$ as a modification of $\rho_{i-1}$. The {\sc slp}\xspace $\mathcal{G}_i$ might not be in Chomsky normal form, since, for some variables $A'$, $\rho_i(A')$ could be a string of any number of variables $\geq 0$, possibly the empty string, But it will be clear from our construction that this is the only obstruction to $\mathcal{G}_i$ being in Chomsky normal form. We describe the construction of the sequence $\mathcal{G}_1,\ldots,\mathcal{G}_n$, and prove by induction on $i$ that, for every variable $A'$ of $\mathcal{G}_{i}$ not in the sequence $A_{i+1},\ldots,A_n$, every component of $\mathsf{val}_{\mathcal{G}_i}(A')$ has a root in $(\mathcal{G}_i)_{A'}$. To prove the $i$-th inductive step we need to verify the existence of appropriate roots both for the variable $A_i$ and for any new variables that are defined in the construction of $\mathcal{G}_i$ from $\mathcal{G}_{i-1}$. For each such $A_i$ and all new variables, we can record whether their values lies in $\Sigma_i^$ for some $i$. If $A_i$ has height one, then $\rho(A_i)$ has length at most 1, with $A_i$ as its root. So no modification is necessary, and $\mathcal{G}_i=\mathcal{G}_{i-1}$. So now suppose that $A_i$ has height greater than one. For notational convenience we define $A:= A_i$. The construction of $\mathcal{G}_i$ from $\mathcal{G}_{i-1}$ will involve the definition of some new variables, and of the images of these and of $A$ under $\rho_i$; the images under $\rho_i$ and $\rho_{i-1}$ of all other variables will match. Since $\mathcal{G}$ is in Chomsky normal form, $\rho_{i-1}(A)=\rho(A)$, and $\mathsf{h}(A)>1$, we have variables $B,C \in V$ with $\rho_{i-1}(A)=BC$. By the inductive hypothesis, every component of $\mathsf{val}_{\mathcal{G}}(A)$ that lies entirely within $\mathsf{val}_{\mathcal{G}}(B)$ or $\mathsf{val}_{\mathcal{G}}(C)$ has a root in $(\mathcal{G}_{i-1})_B$ or $(\mathcal{G}_{i-1})_C$ (respectively), and hence in $\mathcal{G}_{i-1}$. So no modification is necessary unless $\mathsf{val}_{\mathcal{G}}(A)$ contains a component $u=u_1u_2$ for which $\mathsf{val}_{\mathcal{G}}(B)=v_1u_1$ and $\mathsf{val}_{\mathcal{G}}(C)=u_2v_2$ with $u_1,u_2 \neq \epsilon$. We suppose that $u$ is such a component. We note that (since, as discussed earlier when we defined components of words, any two components of $\mathsf{val}_{\mathcal{G}}(A)$ are disjoint) any other component of $\mathsf{val}_{\mathcal{G}}(A)$ is a subword of either $v_1$ or $v_2$, and hence of $\mathsf{val}_{\mathcal{G}}(B)$ or $\mathsf{val}_{\mathcal{G}}(C)$. By the induction hypothesis, $u_1,u_2$ have roots $D_1,D_2$ in $(\mathcal{G}_{i-1})_B$ and $(\mathcal{G}_{i-1})_C$ respectively. By finding the rightmost variable in $\rho^k(B)$ for $k=1,2,\ldots$ and using our records of which variables have values in $\Sigma_i^$ for some $i$, we can locate the variable $D_1$ in polynomial time, and similarly $D_2$. (In fact we do that in any case to establish whether we are in the situation where modification is required.) \begin{figure} \caption{Adjusting the {\sc slp}\xspace below $A$} \label{fig:adjustment} \end{figure} Now we introduce a new variable $D$, and define $\rho_i(D):=D_1D_2$. We also introduce new variables $B',C'$ and set $\rho_i(A)=B'DC'$. We want to define $\rho_i(B')$ and $\rho_i(C')$ to ensure that $\mathsf{val}_{\mathcal{G}_i}(B')=v_1$, and $\mathsf{val}_{\mathcal{G}_i}(C')=v_2$, and this motivates the definitions that now follow. First we note the existence of a sequence of variables $B_0=B,B_1,\ldots, B_r=D_1$ in $V_{i-1}$, for which each $B_{j+1}$ is the last letter of $\rho_{i-1}(B_j)$; that is $\rho_{i-1}(B_j)=\beta_j B_{j+1}$, for some $\beta_j \in V_{i-1}^$. Similarly, we find a sequence of variables $C_0=C,C_1,\ldots, C_s=D_2$ in $V_{i-1}$, such that, for each $j$, $\rho_{i-1}(C_j)=C_{j+1}\gamma_j$, for some $\gamma_j \in V_{i-1}^$. We define $B'_0:=B'$, $C'_0:=C'$ and introduce new variables $B'_j$ for each $j=1,\ldots,r-1$, and $C_j'$ for each $j=1,\ldots,s-1$. We define \[ V_i := V_{i-1} \cup \{B'_0,B'_1,\ldots B'_{r-1},C_0',C'_1,\ldots C'_{s-1},D\}.\] Then we define \[ \rho_i(B'_0):=\beta_0B'_1,\quad \rho_i(B'_j):=\beta_jB'_{j+1}\ \hbox{\rm for}\,j=1,\ldots,r-2,\,\quad \rho_i(B'_{r-1}):=\beta_{r-1},\] and similarly \[ \rho_i(C'_0):=\gamma_0C'_1,\quad \rho_i(C'_j):=\gamma_jC'_{j+1}\ \hbox{\rm for}\,j=1,\ldots,s-2,\,\quad \rho_i(C'_{s-1}):=\gamma_{s-1}.\] and $\rho_i(E):= \rho_{i-1}(E)$ for all variables $E \in V_{i-1}\setminus \{ A \}$. Figure~\ref{fig:adjustment} illustrates this adjustment of $\mathcal{G}_{i-1}$ to $\mathcal{G}_i$. This completes our construction of $\mathcal{G}_i$, which is certainly acyclic. Our definition of new variables and of their images under $\rho_i$ was designed to ensure that $\mathsf{val}_{\mathcal{G}_i}(A_i)=\mathsf{val}_{\mathcal{G}}(A_i)$. We note that $\mathsf{h}_{\mathcal{G}_i}(B') = \mathsf{h}_{\mathcal{G}'_{i-1}}(B)$, $\mathsf{h}_{\mathcal{G}_i}(C') = \mathsf{h}_{\mathcal{G}_{i-1}}(C)$, and $\mathsf{h}_{\mathcal{G}_i}(D) \le \mathsf{h}_{\mathcal{G}_{i-1}}(A_i)$ (equality can occur here if $D_1=B$ or $D_2=C$). So $\mathsf{h}_{\mathcal{G}_i}(A_i) \le \mathsf{h}_{\mathcal{G}_{i-1}}(A_i)+1$, for each $i$, and hence $\mathsf{h}_{\mathcal{G}_i}(S) \le \mathsf{h}_{\mathcal{G}_{i-1}}(S)+1$. We need to verify our claim that, for all variables $A' \in \mathcal{G}_i$ that are not in the sequence $A_{i+1},\ldots,A_n$, every component of $\mathsf{val}_{\mathcal{G}_i}(A')$ has a root in $(\mathcal{G}_i)_{A'}$. This is true for all $A' \in V_{i-1} \setminus \{A_i,A_{i+1},\ldots,A_n\}$, because $\rho_i(A')=\rho_{i-1}(A')$ and hence $\mathsf{val}{_{\mathcal{G}_i}}(A')=\mathsf{val}_{\mathcal{G}}(A')$ for all such $A'$. Our construction of $\mathcal{G}_i$ was designed to ensure that our claim is true for $A'=A_i$, and it is clearly true for $A'=D,D_1$ and $D_2$. Finally, for the new variables $B_j'$ and $C_j'$, the components of $\mathsf{val}_{\mathcal{G}_i}(B_j')$ and $\mathsf{val}_{\mathcal{G}_i}(C_j')$ are components of $\mathsf{val}_{\mathcal{G}_{i-1}}(B_j)$ and $\mathsf{val}_{\mathcal{G}_{i-1}}(C_j)$ respectively and, by the inductive hypothesis, they have roots in $(\mathcal{G}_{i-1})_{B_j}$ and $(\mathcal{G}_{i-1})_{C_j}$. It follows immediately from the definitions of $\rho_i(B_j')$ and $\rho_i(C_j')$ that these components have roots in $(\mathcal{G}_{i})_{B_j'}$ and $(\mathcal{G}_{i})_{C_j'}$. After the process is complete, we have $\mathsf{h}_{\mathcal{G}'}(S) \le n+\mathsf{h}_{\mathcal{G}}(S) \le 2n$. So, during each of the $n$ steps of this process the number of variables we have added has been at most than $4n$, and hence the process is bounded by a polynomial in $\|V\|$, so certainly by a polynomial in $\|\mathcal{G}\|$. We complete the proof by putting the final {\sc slp}\xspace into Chomsky normal form. Since this involves only the addition of new variables, it does not affect the property that all components of $w$ have roots. \end{proof} \begin{definition}\label{def:split} Let $w$ be a word over $\Sigma$ and suppose that $u:=w[i:j)$ is a subword of $w$. We say that $u$ {\em splits a component} of $w$ if $u$ starts or ends part way through a component of $w$, but is not a subword of a single component; otherwise we say that $u$ {\em splits no components} of $w$. \end{definition} So if $u$ splits no component of $w$, then either $u$ is a proper subword of a component of $w$, or $u$ is a concatenation of components of $w$ and of subwords in $(\Sigma \setminus \mathcal{H})^$. In that second case, there exist integers $k,l$ with $k<l$ such that $\hat{u} = \hat{w}[k:l)$; we shall sometimes choose to write $w[[k:l))$ rather than $w[i:j)$ as a notation for this subword $u$ of $w$. Now let $\mathcal{G}=(V,S,\rho)$ be an {\sc slp}\xspace, {\sc cslp}\xspace, {\sc tslp}\xspace or {\sc tcslp}\xspace for the group $G$, and let $w:= \mathsf{val}(\mathcal{G})$. We say that $\mathcal{G}$ {\em splits no components} (or is {\em non-splitting}) if for any variable $A$ of $\mathcal{G}$, whenever $\mathsf{val}(A)$ occurs as a subword of $w$ with root $A$, then that that subword splits no components of $w$. Now suppose that $A,B$ are variables in a {\sc cslp}\xspace $\mathcal{G}$ and that $\rho(A)=B[i:j)$ (so that $\mathsf{val}_\mathcal{G}(A)=\mathsf{val}_\mathcal{G}(B)[i:j)$). We say that the cut operator $B[i:j)$ {\em splits a component} if the subword $w_B[i:j) = w_A$ splits a component of $w$, or if it is a proper subword of a component of $w$; otherwise $B[i:j)$ is called {\em non-splitting}. Notice that for cut operators to be non-splitting, we are also excluding the possibility of $w_A$ being a proper subword of a component of $w_B$. If $B[i:j)$ is a non-splitting cut-operator and $k,l$ are the integers defined by $\mathsf{val}(B)[i:j) = \mathsf{val}(B)[[k:l))$, it is often convenient to specify the cut operator in terms of $k$ and $l$ rather than $i$ and $j$, that is, as $B[[k:l))$. In this case, we say that the cut operator is {\em specified relative to compression}. In general the {\sc slp}s\xspace (and {\sc cslp}s\xspace, {\sc tslp}s\xspace and {\sc tcslp}s\xspace) that we shall construct in this article, as well as cut operators within the {\sc tcslp}s\xspace, will not split components, and the cut operators will be specified relative to compression. \begin{remark}\label{rem:subword} Suppose that $\mathcal{G}$ is an {\sc slp}\xspace as in the conclusion of Proposition~\ref{prop:comproot} (where it is called $\mathcal{G}'$); that is, for each variable $A$ of $\mathcal{G}$, every component of $\mathsf{val}(A)$ has a root in $\mathcal{G}_A$. Then $\mathcal{G}$ splits no components. Moreover, if $\mathcal{G}$ is trimmed and $\mathsf{val}(\mathcal{G})$ is $\mathsf{nf}$-reduced, then so is $\mathsf{val}(A)$, for every variable $A$ of $\mathcal{G}$. \end{remark} \begin{proof} Suppose that the subword $u = \mathsf{val}(A)$ of $w := \mathsf{val}(\mathcal{G})$ splits a component of $w$; so we have $u = u_1u_2$ with $u_1$ and $u_2$ nonempty, where either $u_1$ is a proper suffix or $u_2$ is a proper prefix of a component $v$ of $w$. But, by assumption, $v$ has a root $B$ in $\mathcal{G}$, and these occurrences of $\mathsf{val}(A)$ and $\mathsf{val}(B)$ in $w$ overlap without one being a subword of the other, which is not possible. The final assertion now follows from Remark~\ref{rem:subclosure}. \end{proof} In order to build {\sc cslp}s\xspace that do not split components, we shall need the following result. \begin{corollary} \label{cor:gamlen} Let $\mathcal{G}$ be an {\sc slp}\xspace over our selected generating set $\Sigma$ for $G$, let $w := \mathsf{val}(\mathcal{G})$, and suppose that every component of $w$ has a root in $\mathcal{G}$ (and hence $\mathcal{G}$ splits no components). Then given $k,l$ with $0\leq k<l\leq \|\hat{w}\|$, in time that is polynomial in $\|\mathcal{G}\|$, we can compute the integers $i,j$ with $w[i:j) = w[[k:l))$. Conversely, given $i$ and $j$ such that the subword $w[i:j)$ of $w$ is a union of complete components of $w$ and subwords in $(\Sigma \setminus \mathcal{H})^$, we can compute $k$ and $l$ with $w[[k:l))=w[i:j)$, in polynomial time. \end{corollary} \begin{proof} By regarding the roots of components of $w$ as new terminals, we can regard $\mathcal{G}$ as an {\sc slp}\xspace $\widehat{\mathcal{G}}$ over some finite subset of the infinite alphabet $\widehat{\Sigma}$ with $\mathsf{val}({\widehat{\mathcal{G}}})=\hat{w}$. Our lists of variables $A$ of $\mathcal{G}$ for which $\mathsf{val}(A) \in \Sigma_i^$ for some $i$ enable us to identify such variables. We can then use Proposition \ref{prop:slp_results}\,(iv) to compute {\sc slp}s\xspace over $\widehat{\Sigma}$ with values $\hat{w}[:l)$ and $\hat{w}[k:l)$. Then by reinterpreting them as {\sc slp}s\xspace over $\Sigma$, we can (by Proposition \ref{prop:slp_results}\,(iii)) compute their lengths and thereby compute $i$ and $j$. For the converse, we compute {\sc slp}s\xspace for $w[:j)$ and $w[i:j)$, regard then as {\sc slp}s\xspace with values $\hat{w}[:l)$ and $\hat{w}[k:l)$ over a finite subset of $\widehat{\Sigma}$, and then compute their lengths. \end{proof} \begin{lemma} \label{lem:ftwd} Let $\mathcal{G}$ be an {\sc slp}\xspace over our selected generating set $\Sigma$ for $G$, and let $w := \mathsf{val}(\mathcal{G})$. Suppose that the components of $w$ are all shortlex reduced and that $\hat{w}$ is a $(\lambda,c)$-quasigeodesic that does not backtrack for some constants $\lambda \ge 1$ and $c \ge 0$. Then, in time polynomial in $\|\mathcal{G}\|$, we can construct an {\sc slp}\xspace with value $\mathsf{nf}(w)$. \end{lemma} \begin{proof} We assume that $\mathcal{G}$ is trimmed and in Chomsky normal form and that all of its components have roots. So, in particular, for all variables $A$ of $\mathcal{G}$, $\mathsf{val}_\mathcal{G}(A)$ arises (possibly multiple times) as a subword of $w$. The hypotheses allow us to apply Proposition \ref{prop:ftqd} with $v = w$, $u = \mathsf{nf}(w)$, and $w_1=w_2=1$. Let $p$ and $q$ be the paths in $\Gamma$ that are labelled by $\mathsf{nf}(w)$ and $w$, and $\hat{p}$ and $\hat{q}$ the corresponding paths in $\widehat{\Gamma}$. Then $p$ and $q$ $\mathsf e'$-fellow travel, where $\mathsf e'$ is a constant that depends only on $G$, $\Sigma$, $\lambda$ and $c$, and all vertices on $\hat{q}$ have at least one corresponding vertex on $\hat{p}$. As in the proof of Proposition \ref{prop:ftqd}, we define $\mathsf e_1 := \mathsf e(\lambda,c,0)$ and $\mathsf e_2 := \mathsf e(\lambda,c,\mathsf e_1)$, where $\mathsf e()$ is as defined in Proposition~\ref{prop:bcpp} Now consider some instance of $\mathsf{val}(A)$ as a subword $w_1$ of $w$ that is derived from $A$ and labels a subpath $q_1$ of $q$. As we observed in Remark~\ref{rem:subword}, either (this instance of) $w_1$ is a concatenation of complete components of $w$ and subwords in $(\Sigma \setminus \mathcal{H})^$, or else $w_1 \in \Sigma_i^$ for some $i$ and $w_1$ is a proper subword of a component of $w$. In the first of these cases (Case 1), the start and end vertices of subpath $q_1$ of $q$ are also vertices of $\hat{q}$, and the fact that vertices on $\hat{q}$ have corresponding vertices on $\hat{p}$ implies that there exist $\alpha,\beta \in \Sigma^$ with $\|\alpha\|,\|\beta\| \le \mathsf e'$ such that $\mathsf{nf}(\alpha\mathsf{val}_\mathcal{G}(A)\beta^{-1})$ is a subword of $\mathsf{nf}(w)$ that labels a subpath $p_1$ of $p$ whose start and end vertices are also vertices of $\hat{p}$. (Note that we are using Remark~\ref{rem:subclosure} here, which ensures that the subword of $\mathsf{nf}(w)$ labelled by this subpath of $p$ is $\mathsf{nf}$-reduced.) In the second case we see from the proof of Proposition \ref{prop:ftqd} that, if (Case 2a) the component of $w$ of which $w_1$ is a proper subword has length greater than $\mathsf e_2$, then there exist $\alpha,\beta \in \Sigma_i^$ with $\|\alpha\|,\|\beta\| \le \mathsf e'$ such that $\mathsf{nf}(\alpha\mathsf{val}_\mathcal{G}(A)\beta^{-1})$ labels a subword of a connected component of $\mathsf{nf}(w)$. (Here we using the fact that the connected components are shortlex reduced words and so their subwords are also shortlex reduced and hence $\mathsf{nf}$-reduced.) In Cases 1 and 2a, we say that this instance of the subword $w_1=\mathsf{val}_\mathcal{G}(A)$ of $w$ {\em corresponds to} the subpath of $p$ that is labelled by $\mathsf{nf}(\alpha\mathsf{val}_\mathcal{G}(A)\beta^{-1})$. If $w_1$ is a proper subword of a component of $w$ of length at most $\mathsf e_2$ (Case 2b), then we do not attempt to define a corresponding subpath of $p$. We shall construct an {\sc slp}\xspace $\mathcal{S}$ with value $\mathsf{nf}(w)$. As usual, we do this by processing the variables of $\mathcal{G}$ in order of increasing height. For each variable $A$ of $\mathcal{G}$ and each pair of words $\alpha,\beta \in \Sigma^$ with $\|\alpha\|,\|\beta\| \le \mathsf e'$, we attempt to define a variable $A_{\alpha,\beta}$ of $\mathcal{S}$ with $\mathsf{val}_{\mathcal{S}}(A_{\alpha,\beta}) = \mathsf{nf}(\alpha\mathsf{val}_\mathcal{G}(A)\beta^{-1})$. We shall not necessarily succeed in doing this for every such $\alpha,\beta$, but we shall do so at least for each $\alpha,\beta$ for which $\mathsf{val}(A)$ corresponds to $\mathsf{nf}(\alpha\mathsf{val}_\mathcal{G}(A)\beta^{-1})$ as described above. After carrying out this process for all variables $A$ of $\mathcal{G}$, we complete the proof by letting the start variable of $\mathcal{S}$ be $S_{\epsilon,\epsilon}$ for $S = S_\mathcal{G}$. Suppose first that either $A$ has height $0$, or that $\mathsf{val}(A) \in \Sigma_i^$ for some $i$ and $\|\mathsf{val}(A)\| \le \mathsf e_2$. Then we compute $\mathsf{nf}(\alpha \mathsf{val}(A) \beta^{-1})$ for all $\alpha,\beta \in \Sigma^$ with $\|\alpha\| ,\|\beta\|\leq \mathsf e'$. Since $\|\alpha \mathsf{val}(A) \beta\|$ is bounded above by the constant $\mathsf e_2+2\mathsf e'$, we can do this in time bounded by a constant, by using the word acceptor in the asynchronous automatic structure of $G$ to enumerate the words in the accepted language in order of increasing length until one is found that is equal in $G$ to $\alpha \mathsf{val}(A) \beta^{-1}$, and then define $A_{\alpha,\beta}$ with $\rho_{\mathcal{S}}(A_{\alpha,\beta}) = \mathsf{nf}(\alpha \mathsf{val}(A) \beta^{-1})$. Otherwise, we have $\rho_\mathcal{G}(A) = BC$ for variables $B$ and $C$ of $\mathcal{G}$ that we have processed already. We proceed as follows for each pair $\alpha,\beta \in \Sigma^$ with $\|\alpha\|,\|\beta\| \le \mathsf e'$. For all words $\gamma \in \Sigma^$ with $\|\gamma\| \le \mathsf e'$ for which we have defined variables $B_{\alpha,\gamma}$ and $C_{\gamma,\beta}$ for $\mathcal{S}$, we check whether the word $\mathsf{val}_{\mathcal{S}}(B_{\alpha,\gamma})\mathsf{val}_{\mathcal{S}}(C_{\gamma,\beta})$ lies in the language of the word acceptor, which we can do in polynomial time by Proposition~\ref{prop:slp_results}\,(v). If so, then we introduce the new variable $A_{\alpha,\beta}$ for $\mathcal{S}$, define $\rho_{\mathcal{S}}(A_{\alpha,\beta}) = B_{\alpha,\gamma} C_{\gamma,\beta}$, and move on to the next pair of words $\alpha,\beta$. \begin{figure} \caption{Subwords of fellow travelling words $w$ and $\mathsf{nf}(w)$} \label{fig:ftsubwd} \end{figure} We need to show that we succeed in defining $A_{\alpha,\beta}$ whenever $A$ corresponds to $\mathsf{nf}(\alpha\mathsf{val}_\mathcal{G}(A)\beta^{-1})$ as described above. Let $w_1 := \mathsf{val}_\mathcal{G}(A)$, $w_2 := \mathsf{val}_\mathcal{G}(B)$ and $w_3 := \mathsf{val}_\mathcal{G}(C)$, so $w_1=w_2w_3$. Since we are assuming that $\|\mathsf{val}_\mathcal{G}(A)\| > \mathsf e_2$, all instances of $w_1$ as subwords of $w$ must lie in Case 1 or Case 2a, and (since all components of $\mathcal{G}$ have roots) we see then that the same applies to the instances of $w_2$ and $w_3$ that arise from the derivation $\rho(A) = BC$. So the vertex of $\Gamma$ corresponding to the end of the subword $\mathsf{val}_\mathcal{G}(B)$ is at distance at most $\mathsf e'$ from a vertex in the corresponding subword $u_1 := \mathsf{nf}(\alpha\mathsf{val}_\mathcal{G}(A)\beta^{-1})$ of $\mathsf{nf}(w)$ such that the associated subwords $u_2$ and $u_3$ of $u_1$ correspond to $w_2$ and $w_3$, respectively. See Figure \ref{fig:ftsubwd}. In other words, there exists $\gamma \in \Sigma^$ with $\|\gamma\| \le \mathsf e'$ such that $\mathsf{val}_\mathcal{G}(B)$ corresponds to $\mathsf{nf}(\alpha \mathsf{val}_\mathcal{G}(B) \gamma^{-1})$ and $\mathsf{val}_\mathcal{G}(C)$ corresponds to $\mathsf{nf}(\gamma \mathsf{val}_\mathcal{G}(C) \beta^{-1})$, in which case the variables $B_{\alpha,\gamma}$ and $C_{\gamma,\beta}$ have been defined, and so we successfully define $A_{\alpha,\beta}$ with $\rho_{\mathcal{S}}(A_{\alpha,\beta}) = B_{\alpha,\gamma} C_{\gamma,\beta}$. \end{proof} Note that the {\sc slp}\xspace $\mathcal{S}$ that is constructed in the above lemma can have size up to $(\mathsf e')^2$ times the size of $\mathcal{G}$. The following result, guarantees a size that is bounded as a function of the length of the derived word $\hat{w}$. \begin{proposition}\label{prop:slpshort} Let $\mathcal{G}$ be an {\sc slp}\xspace over our selected generating set $\Sigma$ for $G$, and let $w:=\mathsf{val}(\mathcal{G})$. Then we can construct an {\sc slp}\xspace $\mathcal{S}$ with value equal to $\mathsf{nf}(w)$ and with size at most $\max(\mathsf{C}\|\hat{w}\|\log(\|w\|),1)$ in time that is bounded by $\mathsf{j}(\|\hat{w}\|,\|\mathcal{G}\|)$, for some constant $\mathsf{C}$ and some (increasing) polynomial function $\mathsf{j}()$. \end{proposition} \begin{proof} The proof is by induction on $\|\hat{w}\|$. We shall not attempt to specify the constant $\mathsf{C}$ in the statement precisely, but it would be possible to do so by following the calculation at the end of the proof. This constant will depend on $\|\Sigma\|$ and on the constant $\mathsf{D}$ from Proposition~\ref{prop:nfwdlenbd}. Similarly we shall not attempt to specify the polynomial $\mathsf{j}()$ in the statement, but merely observe that all steps in the proof can be done in time polynomial in $\|\hat{w}\|$ and $\log(\|w\|)$, and that there are $\|\hat{w}\|$ steps in the inductive proof. By Proposition \ref{prop:comproot} we may assume that all components of $\mathcal{G}$ have roots. There is nothing to prove if $\|\hat{w}\|=0$. Otherwise, we can write $\hat{w} = \hat{v}\hat{z}$ with $\|\hat{z}\|=1$, where also $w = vz$, and so $w=_G vz$, and we can assume that we have found an {\sc slp}\xspace $\mathcal{S}_v$ with $\mathsf{val}(\mathcal{S}_v)=\mathsf{nf}(v)$, where the size of $\mathcal{S}_v$ is bounded as required. We denote by $v_{\mathsf{nf}}$ the word $\mathsf{nf}(v) = \mathsf{val}(\mathcal{S}_v)$ and by $w_{\mathsf{nf}}$ the word $\mathsf{nf}(w)$. Then $w_{\mathsf{nf}} =_G v_{\mathsf{nf}} z$. Suppose first that $z$ represents an element of $H_i$ for some component subgroup $H_i$ (so $z \in \Sigma_i^$) and that $v_{\mathsf{nf}}$ ends in a letter from $\Sigma_i$. Then let $y$ be the $H_i$-component of $v_{\mathsf{nf}}$ that contains this letter; so we have $v_{\mathsf{nf}}=uy$ and $\widehat{v_{\mathsf{nf}}} = \hat{u}\hat{y}$, and it follows from the description of $\mathcal{L}$ in Proposition \ref{prop:automatic} that $\mathsf{nf}(u) = u$. The word $u$ might not be a prefix of $w$, but $w=_G u(yz)$, and $\hat{u}$ does not end in a letter from $\Sigma_i$. In this situation, we replace $v_{\mathsf{nf}}$ by $u$, derive $\mathcal{S}_u$ with value $u=\mathsf{nf}(u)$ from $\mathcal{S}_v$, and replace $z$ by $yz$. So in any case, we may now assume that, if $z \in\Sigma_i^$ for some $i$, then $v_{\mathsf{nf}}$ does not end in a letter from $\Sigma_i$. Let $z_{\mathsf{nf}} := \mathsf{nf}(z)$. So $z_{\mathsf{nf}} = \mathsf{slex}(z)$ when $z \in\Sigma_i^$; otherwise $z$ consists of a generator that does not lie in any parabolic subgroup and, since we are assuming that the elements of $\Sigma$ represent distinct elements of $G$, we have $\|z\|=1$ and $z_{\mathsf{nf}} = z$. In the former case we can compute an {\sc slp}\xspace $\mathcal{S}_z$ with $\mathsf{val}(\mathcal{S}_z) = z_{\mathsf{nf}}$ and $\|\mathcal{S}_z\| \le \max(4\|\Sigma_i\|\log_2(z_{\mathsf{nf}})),1)$ by Proposition \ref{prop:abslp}. So we can in any case compute an {\sc slp}\xspace $\mathcal{G}$ with value $v_{\mathsf{nf}} z_{\mathsf{nf}}$. Now the path $\hat{p}$ in $\widehat{\Gamma}$ labelled by $\widehat{v_{\mathsf{nf}} z_{\mathsf{nf}}} = \widehat{v_{\mathsf{nf}}}\widehat{z_{\mathsf{nf}}}$, as an extension of a geodesic by an element of $\widehat{\Sigma}$, is a $(1,2)$--quasigeodesic. We claim that $\hat{p}$ does not backtrack. For suppose that it does. Then the final edge of $\hat{p}$ must penetrate the same $H_{i}$-coset as one of the other edges of $\hat{p}$, for some index $i$, and then we must have $z \in H_i$. So some suffix of $\widehat{v_{\mathsf{nf}}}$ represents an element of $H_i$ and, since the word $\widehat{v_{\mathsf{nf}}}$ is geodesic, this suffix must have length $1$. But then $\widehat{v_{\mathsf{nf}}}$ ends in a letter in $\Sigma_i$, contrary to assumption. We can now apply Lemma \ref{lem:ftwd} to the {\sc slp}\xspace $\mathcal{G}$ with value $v_{\mathsf{nf}} z_{\mathsf{nf}}$ and compute an {\sc slp}\xspace $\mathcal{S}_0$ with value $w_{\mathsf{nf}}:=\mathsf{nf}(v_{\mathsf{nf}} z)$. By Proposition~\ref{prop:comproot}, we can modify $\mathcal{S}_0$ as necessary (in polynomial time), to ensure that all of its components have roots. We now explain how to modify $\mathcal{S}_0$ such that its size is bounded by $\mathsf{C}\|\hat{w}\|\log(\|w\|)$ for some constant $\mathsf{C}$. Let $w_{\mathsf{nf}}[i_r:j_r)$ for $1 \le r \le t$ be the components of $w_{\mathsf{nf}}$, and let $A_r$ be the root of $w_{\mathsf{nf}}[i_r:j_r)$ in $\mathcal{S}_0$. Then by Lemma~\ref{lem:ab_polytime} we can construct an {\sc slp}\xspace $\mathcal{S}_r$ with $\mathsf{val}(\mathcal{S}_r) = \mathsf{val}(A_r)$ and $\|\mathcal{S}_r\| \le \max(4\|\Sigma\|\log_2(\|\mathsf{val}(\mathcal{S}_r)\|)\|,1)$, and define $B_r := S_{\mathcal{S}_r}$. Note that $\|\mathcal{S}_r\|$ is bounded by a multiple of $\log(\|w_{\mathsf{nf}}\|)$, which is itself bounded by a multiple of $\log(\|w\|)$ by Proposition~\ref{prop:nfwdlenbd}. We now define $\mathcal{S}$, whose variables consist of $S_\mathcal{S}$ together with all of the variables of each $\mathcal{S}_r$ with $r \geq 1$. For each $A \in \mathcal{S}_r$, we define $\rho_\mathcal{S}(A) := \rho_{\mathcal{S}_r}(A)$, and further \[ \rho(S_{\mathcal{S}}) = w_{\mathsf{nf}}[j_0:i_1)B_1w_{\mathsf{nf}}[j_1:i_2)B_2 \cdots w_{\mathsf{nf}}[j_{t-1}:i_t) B_tw_{\mathsf{nf}}[j_t,i_{t+1}), \] where $j_0:=0$ and $i_{t+1} := \|w_{\mathsf{nf}}\|$. Then the {\sc slp}\xspace $\mathcal{S}$ satisfies the required bound on its size. \end{proof} \section{Converting {\sc tcslp}s\xspace and {\sc tslp}s\xspace to {\sc slp}s\xspace} \label{sec:convert} The proof of our main result falls into two parts, the first part constructing, from the input {\sc slp}\xspace $\mathcal{G}$, a {\sc tcslp}\xspace that defines the $\mathsf{nf}$-reduced representative $\mathsf{nf}(\mathsf{val}(\mathcal{G}))$ of its value, and then the second part converting that {\sc tcslp}\xspace into a {\sc slp}\xspace with the same value; this is the same strategy as was applied to prove the corresponding result \cite[Theorem 6.7]{HLS} for hyperbolic groups, and our proofs of the component results are based on the proofs in \cite{HLS}. The construction of a {\sc tcslp}\xspace accepting $\mathsf{nf}(\mathsf{val}(\mathcal{G}))$ is described in the final section of the paper, Section~\ref{sec:slextcslp}. This section is devoted to the proof of the following conversion theorem: \begin{theorem} \label{thm:convert} Let $G$ be a group hyperbolic relative to a collection of free abelian subgroups, and suppose that a generating set $\Sigma$ for $G$, and integer $L$ are selected as in Sections ~\ref{subsec:relhyp_param1}, \ref{subsec:relhyp_param2}. Let $\mathcal{T}$ be an $\mathsf{nf}$-reduced non-splitting {\sc tcslp}\xspace for $G$ over $\Sigma$, with $J_\mathcal{T} \le L$. Suppose further that each cut operator of $\mathcal{T}$ is non-splitting and specified relative to compression. Then we can construct, in time polynomial in $\|\mathcal{T}\|$, an $\mathsf{nf}$-reduced {\sc slp}\xspace $\mathcal{S}$ over $\Sigma$ with $\mathsf{val}(\mathcal{S}) = \mathsf{val}(\mathcal{T})$, whose size is bounded by a polynomial in $\|\mathcal{T}\|$. \end{theorem} We choose to stress the polynomial bound on the size of $\mathcal{S}$, although it follows from the polynomial bound on time. The proof of Theorem~\ref{thm:convert} is split into the two results that follow, Propositions~\ref{prop:tslp-slp} and~\ref{prop:tcslp-tslp}. The first of these computes from a given $\mathsf{nf}$-reduced {\sc tslp}\xspace $\mathcal{U}$ an {\sc slp}\xspace $\mathcal{S}$ with the same value, and the second computes an $\mathsf{nf}$-reduced {\sc tslp}\xspace $\mathcal{U}$ from an $\mathsf{nf}$-reduced {\sc tcslp}\xspace $\mathcal{T}$. This follows the strategy of the proof of the corresponding result in \cite{HLS}, and our proofs adapt those of the components of that proof. There are complications in our proofs that do not arise in \cite{HLS}, resulting partly from the fact that we are using Proposition \ref{prop:backup2} rather than \cite[Lemma 4.4]{HLS}, and partly because many of the upper bounds on lengths of words are bounds on their lengths as words over $\widehat{\Sigma}$ rather than over $\Sigma$. \begin{proposition}\label{prop:tslp-slp} Let $G,\Sigma$ and $L$ be as in Theorem~\ref{thm:convert}. Then, given an $\mathsf{nf}$-reduced non-splitting {\sc tslp}\xspace $\mathcal{U}$ for $G$ over $\Sigma$ with $J_\mathcal{U} \le L$, we can construct, in time polynomial in $\|\mathcal{U}\|$, an {\sc slp}\xspace $\mathcal{S}$ over $\Sigma$ with $\mathsf{val}(\mathcal{S}) = \mathsf{val}(\mathcal{U})$, whose size is bounded by a polynomial function of $\|\mathcal{U}\|$. \end{proposition} \begin{proof} Let $\mathsf{f}$ be the linear function in the conclusion of Proposition \ref{prop:backup2}. Modifying $\mathsf{f}$ as necessary, we can assume that $\mathsf{f}$ is an increasing function with $\mathsf{f}(n) \ge n$ for all $n$. The result is trivial if $\mathsf{val}(\mathcal{U}) = \epsilon$, so suppose not. We defined the height of a variable in $\mathcal{U}$ in Section~\ref{subsec:extend_slps}. We also pointed out that the right hand sides $\rho(A)$ that lie in $(V_{\mathcal{U}} \cup \Sigma)^$ may be assumed to have the form $a \in \Sigma$ or $BC$ with $B,C \in V_{\mathcal{U}}$, and we shall assume that here. We define the {\em tether-height} $\mathsf{t}(A)$ and {\em tether-depth} $\mathsf{d}(A)$ of a variable $A$, via \begin{eqnarray} \mathsf{t}(A)&:=&\left \{ \begin{array}{l} 0 \\ \max\{\mathsf{t}(B),\mathsf{t}(C)\}\\ \mathsf{t}(B)+1 \end{array} \right . \, \hbox{\rm if} \, \begin{array}{l} \rho_\mathcal{U}(A)\in \Sigma^\\ \rho_\mathcal{U}(A)=BC\\ \rho_\mathcal{U}(A) = B \langle \alpha,\beta \rangle, \end{array} \\ \mathsf{d}(A) &:=& \mathsf{t}(S) - \mathsf{t}(A) +1,\end{eqnarray} where $S=S_\mathcal{U}$. By removing unused variables, we may assume that $S$ has maximal height and maximal tether-height, so that every variable has positive tether-depth. In the proof, it is convenient to assume that $A$, $B$ and $C$ have the same tether-depths in all productions of type $A \to BC$. To achieve this, we can increase the tether-depth of any variable $A$, if necessary, by introducing a new redundant variable $X$ together with a production $X \to A\langle \epsilon,\epsilon \rangle$. This will not affect the maximality of the height and tether-height of $S$. We process the variables of $\mathcal{U}$ in order of increasing height. Since the number of variables of $\mathcal{U}$ is certainly bounded by $\|\mathcal{U}\|$, in order to get the bounds we need on time and space, it will be sufficient to bound the time spent processing each variable $A$, together with the length of right-hand sides added during each such step. As we process each variable $A$ of $\mathcal{U}$ we shall either define a copy of $A$ within $\mathcal{S}$ or a set of bounded size of new variables for $\mathcal{S}$ that are associated with $A$, and for each new variable of $\mathcal{S}$ that we introduce we shall define its right-hand side in $\mathcal{S}$. Our construction will ensure the following, for any variable $A$ of $\mathcal{U}$, where $w:= \mathsf{val}_\mathcal{U}(A)$. (Recall that, since $\mathcal{U}$ is $\mathsf{nf}$-reduced, $w=\mathsf{nf}(w)$.) \begin{mylist} \item[(i)] If $\|\hat{w}\| \le (8\,\mathsf{d}(A) + 1)\mathsf{f}(2L)$, then $\mathcal{S}$ contains a copy of $A$, and the {\sc slp}\xspace $\mathcal{S}_A$ is computed from $\mathcal{U}_A$ in time bounded by $\mathsf{j}(\|\hat{w}\|,\|\mathcal{U}\|)$, and has size at most $\mathsf{C}\|\hat{w}\|\log(\|w\|)$, where $\mathsf{C}$ and $\mathsf{j}$ are the constant and polynomial of Proposition ~\ref{prop:slpshort}. Note that Proposition \ref{prop:tcslplenbd} implies that the size of $\mathcal{S}_A$ is bounded by a polynomial function of the input size $\|\mathcal{U}\|$ in this case. \item[(ii)] If $\|\hat{w}\| > (8\,\mathsf{d}(A) + 1)\mathsf{f}(2L)$, then $w$ decomposes as a concatenation $\ell_Aw'r_A$ with $\|\widehat{w'}\| \ge \mathsf{f}(2L)$, and \[4\mathsf{f}(2L)\mathsf{d}(A) \le \|\widehat{\ell_A}\|,\|\widehat{r_A}\| \le (4\mathsf{d}(A) + \mathsf{h}(A))\mathsf{f}(2L).\] As $A$ is processed, new variables $A_\ell$ and $A_r$ are adjoined to $\mathcal{S}$ as the roots of {\sc slp}s\xspace with values $\ell_A,r_A$ (with size at most $\mathsf{C}\|\widehat{\ell_A}\|\log(\|\ell_A\|)$ and $\mathsf{C}\|\widehat{r_A}\|\log(\|r_A\|)$), as well as variables $A_{\alpha,\beta}$ as the roots of {\sc slp}s\xspace with values $\mathsf{nf}(\alpha w' \beta^{-1})$, for each $\alpha,\beta \in \Sigma^$ with $\|\alpha\|,\|\beta\| \le L$. Each of the subwords $\ell_A$, $w'$ and $r_A$ of $w$ is a union of complete components of $w$ and subwords in $(\Sigma \setminus \mathcal{H})^$, and is in normal form. \end{mylist} Suppose that, while processing the variables of $\mathcal{U}$ in increasing order of height, we have reached the variable $A$ of $\mathcal{U}$. Let $w:= \mathsf{val}_\mathcal{U}(A)$. We consider three different possible cases. {\bf Case 1.} $\rho_{\mathcal{U}}(A) \in \Sigma$. We define a variable $A$ within $\mathcal{S}$, and define $\rho_\mathcal{S}(A) := a$. {\bf Case 2.} $\rho_\mathcal{U}(A)=BC$ for variables $B,C$ of $\mathcal{U}$. Recall that we are assuming that $\mathsf{d}(A),\mathsf{d}(B),\mathsf{d}(C)$ are all equal; let $\mathsf{d}$ be their common value. Let $u := \mathsf{val}_\mathcal{U}(B)$, $v:=\mathsf{val}_\mathcal{U}(C)$, so that $w=uv$. Since we are assuming that $\mathcal{U}$ is non-splitting, either $uv \in \Sigma_i^$ for some $i$, in which case $\|\hat{u}\|,\|\hat{v}\|,\|\hat{w}\|\le 1$, or $\hat{w} = \hat{u}\hat{v}$. Suppose first (Case 2.1) that $\|\hat{u}\|,\|\hat{v}\| > (8\mathsf{d} + 1) \mathsf{f}(2L)$ (so we do not have $uv \in \Sigma_i^$). Then, when we processed the variables $B,C$, we computed {\sc slp}s\xspace with values $\ell_B, r_B, \ell_C, r_C$ such that: \begin{mylist} \item[(i)] $4\mathsf{d} \mathsf{f}(2L) \le \|\widehat{\ell_B}\|$, $\|\widehat{r_B}\| \le (4\mathsf{d} + \mathsf{h}(B))\mathsf{f}(2L)$, \item[(ii)] $4\mathsf{d} \mathsf{f}(2L) \le \|\widehat{\ell_C}\|$, $\|\widehat{r_C}\| \le (4\mathsf{d} + \mathsf{h}(C))\mathsf{f}(2L)$, \item[(iii)] $u = \ell_B u' r_B$ and $v = \ell_C v' r_C$ where $\|\widehat{u'}\|, \|\widehat{v'}\| \ge \mathsf{f}(2L)$. \end{mylist} Moreover, for all words $\eta,\theta \in \Sigma^$ with $\|\eta\|,\|\theta\| \le L$, we defined variables $B_{\eta,\theta}$ and $C_{\eta,\theta}$ in $\mathcal{S}$ whose values in $\mathcal{S}$ are $\mathsf{nf}(\eta u' \theta^{-1})$ and $\mathsf{nf}(\eta v' \theta^{-1})$, respectively. We now define $\ell_A := \ell_B$, $r_A := r_C$, and $w' := u' r_B \ell_C v'$; so that $w = \ell_A w' r_A$. Note that, since $\mathsf{d}(A) = \mathsf{d}(B) = \mathsf{d}(C) = \mathsf{d}$ and $\mathsf{h}(B),\mathsf{h}(C) \le \mathsf{h}(A)$, we have (from (i) and (ii)) the length constraints $4\mathsf{f}(2L) \mathsf{d}(A) \le \|\widehat{\ell_A}\|$, $\|\widehat{r_A}\| \le (4\,\mathsf{d}(A) + \mathsf{h}(A))\mathsf{f}(2L)$. And from the lower bounds on $\|\widehat{u'}\|, \|\widehat{v'}\|$ in (iii) we can deduce the required bound $\|\widehat{w'}\| \geq \mathsf{f}(2L)$. \begin{figure} \caption{Case 2.1} \label{fig:case2.1} \end{figure} We already have {\sc slp}s\xspace with values $\ell_A$ and $r_A$. It remains to define the right-hand sides for the variables $A_{\alpha,\beta}$ for all $\alpha,\beta \in \Sigma^$ with $\|\alpha\|,\|\beta\| \le L$. For each such $\alpha,\beta$, and for all $\eta,\theta \in \Sigma^$ with $\|\eta\|, \|\theta\| \le L$, we compute (using Proposition \ref{prop:slpshort}) an {\sc slp}\xspace $\mathcal{Z}$ with value $z := \mathsf{nf}( \eta r_B \ell_C \theta^{-1} )$; since $\hat{z}$ has length bounded by a constant multiple of $\|V_\mathcal{U}\|$, this computation takes time bounded by a polynomial in $\|\mathcal{U}\|$, and $\|\mathcal{Z}\|$ is similarly bounded. Then we check (in polynomial time) using Proposition~\ref{prop:slp_results}(v) whether the word \begin{eqnarray} \mathsf{val}_{\mathcal{S}}(B_{\alpha,\eta}) \, z \, \mathsf{val}_{\mathcal{S}}(C_{\theta,\beta}) &=& \mathsf{nf}(\alpha u' \eta^{-1}) \, \mathsf{nf}(\eta r_B \ell_C \theta^{-1}) \, \mathsf{nf}(\theta v' \beta^{-1}) \end{eqnarray} is $\mathsf{nf}$-reduced, in which case it is the word $\mathsf{nf}(\alpha u' r_B \ell_C v' \beta^{-1})= \mathsf{nf}(\alpha w'\beta^{-1})$; see Figure \ref{fig:case2.1}. We claim that there must be at least one pair $\eta,\theta$ for which this holds. To see this, we apply Proposition \ref{prop:backup2} twice. First we apply it to the quadrilateral with sides labelled $\alpha$, $w'$, $\beta$, $\mathsf{nf}(\alpha w' \beta^{-1})$, using $\|\widehat{u'}\|, \|\widehat{v'}\| \ge \mathsf{f}(2L) \ge \mathsf{f}(L)$, to define $\eta$, and then to the quadrilateral with sides labelled $\eta$, $r_B\ell_Cv'$, $\beta$, $\mathsf{nf} (\eta r_B\ell_Cv'\beta^{-1})$ using $\|\widehat{r_B\ell_C}\|, \|\widehat{v'}\| \ge \mathsf{f}(L)$ to define $\theta$. Proposition~\ref{prop:backup2} ensures that $\|\theta\|, \|\eta\|\leq L'$, and since in Section~\ref{subsec:relhyp_param2} we chose $L\geq L'$, we certainly have $\|\theta\|,\|\eta\| \leq L$. We then include the variables of the {\sc slp}\xspace $\mathcal{Z}$ within $\mathcal{S}$ and define $\rho_\mathcal{S}(A_{\alpha,\beta}) := B_{\alpha,\eta} \, S_\mathcal{Z} \, C_{\theta,\beta}$. Suppose next (Case 2.2) that $\|\hat{u}\| > (8\mathsf{d} + 1)\mathsf{f}(2L)$ and $\|\hat{v}\| \le (8\mathsf{d} + 1)\mathsf{f}(2L)$ (so again we do not have $uv \in \Sigma_i^$). (Case 2.3 where $\|\hat{u}\| \le (8\mathsf{d} + 1)\mathsf{f}(2L)$ and $\|\hat{v}\| > (8\mathsf{d} + 1)\mathsf{f}(2L)$ is similar, and we shall omit the details.) Then we have already computed an {\sc slp}\xspace with value $v$, and {\sc slp}s\xspace for words $\ell_B$ and $r_B$ where: \begin{mylist} \item[(i)] $4\mathsf{d} \mathsf{f}(2L) \le \|\widehat{\ell_B}\|, \|\widehat{r_B}\| \le (4\mathsf{d} + \mathsf{h}(B))\mathsf{f}(2L)$, \item[(ii)] $u = \ell_B u' r_B$ for a word $u'$ with $\|\widehat{u'}\| \ge \mathsf{f}(2L)$. \end{mylist} Moreover, for all words $\eta,\theta \in \Sigma^$ with $\|\eta\|,\|\theta\| \le L$, we have defined variables $B_{\eta,\theta}$ with value $\mathsf{nf}(\eta u' \theta^{-1})$. If $\|\hat{v}\| \leq \mathsf{f}(2L)$, then we set $\ell_A := \ell_B$, $w':=u'$, and $r_A := r_B v$, so that $A_\ell=B_\ell$ and $A_r$ is the root of an {\sc slp}\xspace with value $r_Bv$, and we define $\rho_\mathcal{S}(A_{\alpha,\beta}) := B_{\alpha,\beta}$ for all $\alpha,\beta \in \Sigma^$ with $\|\alpha\|,\|\beta\| \le L$; so the step takes polynomial time and adds right-hand sides of bounded total length. In this case, we have $4\mathsf{d} \mathsf{f}(2L) \le \|\widehat{\ell_A}\| \le (4\mathsf{d} + \mathsf{h}(B))\mathsf{f}(2L) \le (4\mathsf{d} + \mathsf{h}(A))\mathsf{f}(2L)$ and $4\mathsf{d} \mathsf{f}(2L) \le \|\widehat{r_A}\| \le (4\mathsf{d} + \mathsf{h}(B) + 1)\mathsf{f}(2L) \le (4\mathsf{d} + \mathsf{h}(A))\mathsf{f}(2L)$, as required. So now assume that $\|\hat{v}\| > \mathsf{f}(2L)$. Again, we set $\ell_A := \ell_B$. Since we are assuming that $\mathcal{U}$ is non-splitting and $\rho(A) = BC$, an occurrence of $\mathsf{val}_\mathcal{U}(B)$ as a prefix of an occurrence of $\mathsf{val}_\mathcal{U}(A)$ cannot split a component, and so we have $\widehat{r_Bv}= \widehat{r_B}\hat{v}$. Since $\|\widehat{r_B v}\| \ge (4\mathsf{d} +1)\mathsf{f}(2L)$ we can define $r_A$ as the suffix of $r_B v$ with $\|\widehat{r_A}\| = 4\mathsf{d} \mathsf{f}(2L)$; that is, $r_B v = y r_A$ for some word $y$. Since $\widehat{r_Bv}=\widehat{r_B}\hat{v}$ and $\|\widehat{r_B}\| \ge 4\mathsf{d} \mathsf{f}(2L) = \|\widehat{r_A}\|$, we have $\|\hat{y}\| = \|\widehat{r_B}\|+\|\hat{v}\|-\|\widehat{r_A}\| \ge \|\hat{v}\| > \mathsf{f}(2L)$. (Recall that $r_B$ and $v$ are in normal form, and hence $\widehat{r_B}$ and $\hat{v}$ are geodesic.) Then $w = \ell_Aw'r_A$ with $w' = u'y$. This satisfies the required bounds on the lengths of $\widehat{\ell_A}$, $\widehat{r_A}$ and $\widehat{w'}$. We can use Proposition \ref{prop:comproot} and Corollary \ref{cor:gamlen} to define {\sc slp}s\xspace with values $y$ and $r_A$, in time polynomial in $\|\mathcal{U}\|$; we set $A_r$ to be the root of the second of these. \begin{figure} \caption{Case 2.2} \label{fig:case2.2} \end{figure} It remains to define the right-hand sides for the variables $A_{\alpha,\beta}$ for all words $\alpha,\beta \in \Sigma^$ with $\|\alpha\|, \|\beta\| \le L$. Now, for all $\eta \in \Sigma^$ with $\|\eta\| \le L$, we compute (using Proposition \ref{prop:slpshort}) an {\sc slp}\xspace $\mathcal{Z}$ with value $z := \mathsf{nf}( \eta y \beta^{-1})$; this is done in polynomial time and $\mathcal{Z}$ has polynomially bounded size. Then we check whether the word \[ \mathsf{val}_{\mathcal{S}}(B_{\alpha,\eta}) \, z = \mathsf{nf}(\alpha u' \eta^{-1}) \, \mathsf{nf}(\eta y \beta^{-1}) \] is $\mathsf{nf}$-reduced, in which case it is $\mathsf{nf}(\alpha u' y \beta^{-1}) = \mathsf{nf}(\alpha w'\beta^{-1})$; see Figure \ref{fig:case2.2}. Since $\|\hat{u}\|>(8\mathsf{d}+1)\mathsf{f}(2L)>\mathsf{f}(L)$ and $\|\hat{y}\| > \mathsf{f}(2L) \ge \mathsf{f}(L)$, Proposition~\ref{prop:backup2} implies that there must be at least one such $\eta$ with $\|\eta\| \leq L'\leq L$. We include the variables of the {\sc slp}\xspace $\mathcal{Z}$ within $\mathcal{S}$ and define $\rho_\mathcal{S}(A_{\alpha,\beta}) := B_{\alpha,\eta} \, S_\mathcal{Z}$. Finally (Case 2.4), suppose that $\|\hat{u}\|, \|\hat{v}\| \le (8\mathsf{d} + 1)\mathsf{f}(2L)$ (and hence $\|\hat{w}\| \le 2(8\mathsf{d}+1)\mathsf{f}(2L)$). Note that we could now have $w \in \Sigma_i^$ for some $i$. In this case, we have already computed {\sc slp}s\xspace with values $u$ and $v$ when we processed $B,C$. If $\|\hat{w}\| \le (8\mathsf{d}+ 1)\mathsf{f}(2L)$ (which is certainly true when $w \in \Sigma_i^$) then we can define an {\sc slp}\xspace for $w$ as the concatenation of those for $u$ and $v$ and attach it to $\mathcal{S}$ with $A$ as its root. We can then use Proposition \ref{prop:slpshort} to ensure that this {\sc slp}\xspace satisfies the required bound on its size. Otherwise we factorise $w$ as $w = \ell_A w' r_A$ with $\|\widehat{\ell_A}\| = \|\widehat{r_A}\| = 4\mathsf{d} \mathsf{f}(2L) $, and thus $(8\mathsf{d}+ 2)\mathsf{f}(2L) \ge \|\widehat{w'}\| \ge \mathsf{f}(2L)$. We use Proposition \ref{prop:comproot} and Corollary \ref{cor:gamlen} to define {\sc slp}s\xspace with values $\ell_A$ and $r_A$, and attach those to $\mathcal{S}$. For $\alpha,\beta \in \Sigma^$ with $\|\alpha\|,\|\beta\| \le L$ we compute (using Proposition \ref{prop:slpshort}), an {\sc slp}\xspace $\mathcal{Z}$ with value $\mathsf{nf}(\alpha w' \beta^{-1})$; Then we attach $\mathcal{Z}$ within $\mathcal{S}$ with $A_{\alpha,\beta}$ as its root, so that $\mathsf{val}_\mathcal{S}(A_{\alpha,\beta}) = \mathsf{nf}(\alpha w' \beta^{-1})$. This computation takes time bounded by a polynomial function of its input size which, as a result of our size restriction on the {\sc slp}s\xspace $\mathcal{S}_B$ and $\mathcal{S}_C$ is in turn bounded by a polynomial function of $\|\mathcal{U}\|$; so the same applies to the size of $\mathcal{Z}$. {\bf Case 3.} $\rho_\mathcal{U}(A) = B\langle \sigma,\tau \rangle$ for a variable $B$ of $\mathcal{U}$ and words $\sigma,\tau \in \Sigma^$ with $\|\sigma\|,\|\tau\| \le L$. Let $u := \mathsf{val}_{\mathcal{U}}(B)$ and $w := \mathsf{val}_{\mathcal{U}}(A) = \mathsf{nf}(\sigma u \tau^{-1})$. Let $\mathsf{d} := \mathsf{d}(B)$. Then $\mathsf{d}(A) = \mathsf{d}-1 \ge 1$, and so $\mathsf{d} \geq 2$. If $\|\hat{u}\| \le (8\mathsf{d} +1) \mathsf{f}(2L)$ (Case 3.1) , then we have already computed an {\sc slp}\xspace with value $u$. In that case, we proceed much as we did in Case~2.4, as follows. Using Proposition \ref{prop:slpshort}, we construct an {\sc slp}\xspace for $\sigma u \tau^{-1}$ as the concatenation of three {\sc slp}s\xspace (the word $\sigma$, our {\sc slp}\xspace for $u$, and the word $\tau$), and then construct from this an {\sc slp}\xspace $\mathcal{Z}$ for $w$. This computation takes time bounded by $\mathsf{j}((8\mathsf{d}+1)\mathsf{f}(2L)+2L)$ and $\mathcal{Z}$ has size bounded by $\mathsf{C}\|\hat{w}\|\log(\|w\|)$. If $\|\hat{w}\| \le (8 \mathsf{d} + 1)\mathsf{f}(2L)$, then we just attach $\mathcal{Z}$ to $\mathcal{S}$ with $A$ as its root. Otherwise, we factorise $w$ as $w=\ell_Aw'r_A$ and, just as in the second part of Case~2.4, we define {\sc slp}s\xspace with values $\ell_A$ and $r_A$, and attach those to $\mathcal{S}$ and, for each $\alpha,\beta \in \Sigma^$ with $\|\alpha\|,\|\beta\| \leq L$, we compute an {\sc slp}\xspace with value $\mathsf{nf}(\alpha w'\beta^{-1})$, and attach it to $\mathcal{S}$ with the new variable $A_{\alpha,\beta}$ as its root, so that $\mathsf{val}_{\mathcal{S}}(A_{\alpha,\beta}) = \mathsf{nf}(\alpha w' \beta^{-1})$. Otherwise (Case 3.2), we have $\|\hat{u}\| > (8\mathsf{d} + 1) \mathsf{f}(2L)$. We have computed {\sc slp}s\xspace for words $\ell_B, r_B$ such that $4\mathsf{d} \mathsf{f}(2L) \le \|\widehat{\ell_B}\|, \|\widehat{r_B}\| \le (4\mathsf{d} + \mathsf{h}(B))\mathsf{f}(2L)$ and $u = \ell_B u' r_B$ for a word $u'$ with $\|\widehat{u'}\| \ge \mathsf{f}(2L)$. Moreover, for all words $\eta,\theta \in \Sigma^$ with $\|\eta\|,\|\theta\| \le L$, we have defined variables $B_{\eta,\theta}$ with value $\mathsf{nf}(\eta u' \theta^{-1})$. We check for all $\eta,\theta \in \Sigma^$ with $\|\eta\|,\|\theta\| \le L$ whether \[ \mathsf{nf}(\sigma \ell_B \eta^{-1}) \mathsf{val}_{\mathcal{S}}(B_{\eta,\theta}) \mathsf{nf}(\theta r_B \tau^{-1}) = \mathsf{nf}(\sigma \ell_B \eta^{-1}) \mathsf{nf}(\eta u' \theta^{-1}) \mathsf{nf}(\theta r_B \tau^{-1}) \] is $\mathsf{nf}$-reduced, in which case it is $\mathsf{nf}(\sigma \ell_B u' r_B \tau^{-1}) = \mathsf{nf}(\sigma u \tau^{-1}) =w$. Since $\|\widehat{\ell_B}\|,\|\widehat{r_B}\|,\|\widehat{u'}\| \ge \mathsf{f}(L)$, we can show that such words $\eta,\theta$ exist by two applications of Proposition~\ref{prop:backup2} as we did in Case 2.1. Let $s := \mathsf{nf}(\sigma \ell_B \eta^{-1})$ and $t := \mathsf{nf}(\theta r_B \tau^{-1})$, so that $w=s\,\mathsf{nf}(\eta u'\theta^{-1}) t$. We deduce from the triangle inequality that $\|\widehat{\ell_B}\| \leq \|\widehat{\sigma}\|+\|\hat{s}\|+\|\hat{\eta}\| \leq \|\hat{s}\|+2L$ and similarly that $\|\widehat{r_B}\| \leq \|\hat{t}\|+2L$, and hence we have $\|\hat{s}\|,\, \|\hat{t}\| \ge 4\mathsf{d} \mathsf{f}(2L) -2L \ge (4 \mathsf{d}-1) \mathsf{f}(2L)\geq 7\mathsf{f}(2L)$. (Recall that $\mathsf{d}\geq 2$.) Hence we can factorise these words as $s = vx$ and $t = yz$ with $\|\hat{v}\| = \|\hat{z}\| = 4(\mathsf{d}-1) \mathsf{f}(2L) = 4\mathsf{d}(A) \mathsf{f}(2L) \ge 4\mathsf{f}(2L)$, and $\|\hat{x}\|, \,\|\hat{y}\| \ge 3\mathsf{f}(2L)$. We define $\ell_A := v$ and $r_A := z$; these words satisfy the required bounds on their lengths. Note that $\mathsf{val}_{\mathcal{U}}(A) = \mathsf{nf}(\sigma u \tau^{-1}) = \ell_A w' r_A$ with $w' := x \, \mathsf{nf}(\eta u' \theta^{-1}) y$, and $\|\widehat{w'}\| \ge 6\mathsf{f}(2L) \ge \mathsf{f}(2L)$. As in earlier cases we can apply Proposition \ref{prop:comproot} and Corollary \ref{cor:gamlen} to compute {\sc slp}s\xspace with values $v=\ell_A$, $x$, $y$, $z=r_A$, and $w'$. \begin{figure} \caption{Case 3} \label{fig:case3} \end{figure} It remains to define the right-hand sides of the variables $A_{\alpha,\beta}$ with values $\mathsf{nf}(\alpha w' \beta^{-1})$ for all words $\alpha,\beta \in \Sigma^$ with $\|\alpha\|,\|\beta\| \le L$. The lower bounds on the lengths of $\hat{v},\hat{x},\hat{y},\hat{z}$ allow us to apply Proposition \ref{prop:backup2} to the quadrilaterals with sides labelled $\sigma, \ell_B, \eta, vx$ and $\theta, r_B, \tau, yz$, respectively. We can compute in polynomial time words $\mu,\nu \in \Sigma^$ with $\|\mu\|,\|\nu\| \le L$ and factorisations $\ell_B = v'x'$, $r_B = y'z'$ such that $\sigma v' =_G v \mu$, $x \eta =_G \mu x'$, $\theta y' =_G y\nu$, and $\nu z' =_G z \tau$. By the triangle inequality, the words $x'$ and $y'$ satisfy $\|\widehat{x'}\|,\,\|\widehat{y'}\| \ge \mathsf{f}(2L)$. Now consider the quadrilateral with sides labelled $x'u'y'$, $\mathsf{nf}(\alpha\mu)$, $\mathsf{nf}(\beta\nu)$, and $\mathsf{nf}(\alpha\mu x'u'y'\nu^{-1}\beta^{-1})$. Since $\|\widehat{x'}\|,\,\|\widehat{y'}\|, \|\widehat{u'}\| \ge \mathsf{f}(2L)$ and $\|\mathsf{nf}(\alpha\mu)\|, \|\mathsf{nf}(\nu\beta)\| \leq 2L$, we can again make two applications of Proposition \ref{prop:backup2} to show that there exist words $\chi,\psi \in \Sigma^$ with $\|\chi\|,\,\|\psi\| \le L' \le L$ such that the word \begin{eqnarray} \lefteqn{\mathsf{nf}(\alpha\mu x' \chi^{-1})\, \mathsf{val}_{\mathcal{S}}(B_{\chi,\psi}) \, \mathsf{nf}(\psi y' \nu^{-1}\beta^{-1}) =}\\ && \mathsf{nf}(\alpha\mu x' \chi^{-1})\, \mathsf{nf}(\chi u' \psi^{-1}) \,\mathsf{nf}(\psi y' \nu^{-1}\beta^{-1}) \end{eqnarray} is $\mathsf{nf}$-reduced, in which case the above word is $\mathsf{nf}(\alpha\mu x' u' y' \nu^{-1}\beta^{-1}) = \mathsf{nf}(\alpha w' \beta^{-1})$; see Figure \ref{fig:case3}. As before, we can find these words $\chi,\psi$ in polynomial time. Finally (using Proposition \ref{prop:slpshort}), we define {\sc slp}s\xspace $\mathcal{Y}$ and $\mathcal{Z}$ with values $\mathsf{nf}(\alpha\mu x' \chi^{-1})$ and $\mathsf{nf}(\psi y' \nu^{-1}\beta^{-1})$, include their variables within $\mathcal{U}$, and then define $\rho_\mathcal{S}(A_{\alpha,\beta}) := S_\mathcal{Y}\, B_{\chi,\psi} \, S_\mathcal{Z}.$ This concludes the definition of the right-hand sides for the variables $A_{\alpha,\beta}$. We complete the definition of the {\sc slp}\xspace $\mathcal{S}$ by adding a start variable $S_\mathcal{S}$ to $\mathcal{S}$ and setting $\rho_\mathcal{S}(S_\mathcal{S}) := S_\ell S_{\epsilon,\epsilon} S_r$, where $S := S_\mathcal{U}$ is the start variable of $\mathcal{U}$ and $S_\ell$ and $S_r$ are the variables with values $\ell_S$ and $r_S$. This ensures $\mathsf{val}(\mathcal{S}) = \ell_S \, \mathsf{nf}(s')\, r_S$, where $s'$ is such that $\ell_S s' r_S = \mathsf{val}(\mathcal{U})$. But we are assuming that $\mathcal{U}$ is $\mathsf{nf}$-reduced, so $s'$ is also $\mathsf{nf}$-reduced and we get $\mathsf{val}(\mathcal{S}) = \ell_S \, \mathsf{nf}(s') \, r_S = \ell_S s' r_S = \mathsf{val}(\mathcal{U})$. \end{proof} \begin{proposition}\label{prop:tcslp-tslp} Let $G,\Sigma$ and $L$ be as in Theorem~\ref{thm:convert}. Then, given an $\mathsf{nf}$-reduced non-splitting {\sc tcslp}\xspace $\mathcal{T}$ for $G$ over $\Sigma$ with $J_\mathcal{T} \le L$, such that each of its cut operators is non-splitting and specified relative to compression, we can compute in polynomial time an $\mathsf{nf}$-reduced non-splitting {\sc tslp}\xspace $\mathcal{U}$ with $J_{\mathcal{U}} \le L$ and $\mathsf{val}(\mathcal{U}) = \mathsf{val}(\mathcal{T})$, whose size is bounded by a polynomial function of $\|\mathcal{T}\|$. \end{proposition} \begin{proof} We follow the proof of \cite[Lemma 6.5]{HLS}. The idea of the proof is taken from \cite{Hag00}, where it is shown that a {\sc cslp}\xspace can be transformed in polynomial time into an {\sc slp}\xspace with the same value. Let $\mathcal{T} = (V_\mathcal{T},S_\mathcal{T},\rho_\mathcal{T})$ be the input TCSLP. As discussed in Section~\ref{subsec:extend_slps}, we can assume that all right-hand sides from $(V_\mathcal{T} \cup \Sigma)^$ are of the form $a \in \Sigma$ or $BC$ with $B,C \in V_\mathcal{T}$. Consider a variable $A$ such that $\rho_\mathcal{T}(A) = B[[:i))$; the case that $\rho_\mathcal{T}(A) = B[[i:))$ can be dealt with analogously. By considering the variables in order of increasing height, we can assume that no cut operator occurs in the right-hand side of any variable $A'$ with $\mathsf{h}(A') < \mathsf{h}(A)$. Using Proposition \ref{prop:tslp-slp} we can compute an {\sc slp}\xspace with value $\mathsf{val}_{\mathcal{T}}(B)$ and then use Corollary~\ref{cor:gamlen} to compute $n_B:= \|\widehat{\mathsf{val}_{\mathcal{T}}(B)}\|$ in polynomial time. Now we show how to eliminate the cut operator in $\rho_\mathcal{T}(A)$. This involves adding at most $\mathsf{h}(\mathcal{T})$ new variables to the TCSLP. Moreover the height of the {\sc tcslp}\xspace after the cut elimination will still be bounded by $\mathsf{h}(\mathcal{T})$. Hence, the final {\sc tslp}\xspace has at most $\mathsf{h}(\mathcal{T}) \cdot \|V\|$ variables, and its size is polynomially bounded. The idea of the cut elimination is to push the cut operator towards variables of lesser height. For this, we need to consider the various possibilities for the right-hand side of $B$. {\bf Case 1.} $\rho_\mathcal{T}(B) = a \in \Sigma$. If $i = 1$ we define $\rho_\mathcal{U}(A) := a$, and if $i = 0$ we define $\rho_\mathcal{U}(A) := \epsilon$. {\bf Case 2.} $\rho_\mathcal{T}(B) = CD$ with $C,D \in V$. Since we are assuming that the cut operator in $\rho_\mathcal{T}(A) = B[[i:))$ is non-splitting, $\mathsf{val}_\mathcal{U}(B)$ cannot consist of a single component, and so $\mathsf{val}_\mathcal{U}(C)$ and $\mathsf{val}_\mathcal{U}(D)$ must consist of complete components of $\mathsf{val}_\mathcal{U}(B)$ together with subwords in $(\Sigma \setminus \mathcal{H})^$. Define $n_C := \|\widehat{\mathsf{val}_\mathcal{T}(C)}\|$. If $i \leq n_C$ then we define $\rho_\mathcal{U}(A) := C[[:i))$. If $i > n_C$ then we define $\rho_\mathcal{U}(A) := CX$ for a new variable $X$ and set $\rho_\mathcal{U}(X) := D[[:i-n_C))$. We then continue with the elimination of the remaining cut operator in $C[[:i))$ or in $D[[:i-n_C))$. {\bf Case 3.} $\rho_\mathcal{T}(B) = C\langle \alpha,\beta \rangle$ with $C \in V$ and $\alpha,\beta \in \Sigma^$ with $\|\alpha\|,\|\beta\| \le J_{\mathcal{T}} \le L$. Let $u := \mathsf{val}_{\mathcal{T}}(C)$, $v := \mathsf{val}_{\mathcal{T}}(B)=_G \alpha u\beta^{-1}$, and $v_1=\mathsf{val}_{\mathcal{T}}(A)$. So $\|\widehat{v_1}\| = i$ and, where we write $v=v_1v_2$, our condition on the non-splitting of cut operators ensures that not only $v_1$ but also $v_2$ must consist of complete components of $v$ together with subwords in $(\Sigma \setminus \mathcal{H})^$; so $\|\widehat{v_2}\|=n_B-i$. Thus, we have $\mathsf{val}_{\mathcal{T}}(A) = v_1$ and $v = \mathsf{nf}(\alpha u \beta^{-1})$. By Proposition~\ref{prop:tslp-slp} we can assume that we have computed {\sc slp}s\xspace with values $u$ and $v$ and we may assume by Proposition \ref{prop:comproot} that all components of $u$ and $v$ have roots in these {\sc slp}s\xspace. Suppose first that $\|\widehat{v_1}\|,\|\widehat{v_2}\| \ge \mathsf{f}(L)$. (This corresponds to Case 3.3 of the proof of \cite[Lemma 6.5]{HLS}.) Then by Proposition \ref{prop:backup2} there exists a factorisation $u = u_1 u_2$ and $\eta \in \Sigma^$ with $\|\eta\| \le L$ such that $v_1 =_G \alpha u_1 \eta^{-1}$ and $v_2 =_G \eta u_2 \beta^{-1}$. We note that $\|\hat{\alpha}\|$, $\|\hat{\beta}\|$, $\|\hat{\eta}\|\le L$, and applying the triangle inequality in a quadrilateral within $\widehat{\Gamma}$ with geodesic sides labelled by $\hat{\alpha}$, $\widehat{v_1}$, $\hat{\eta}$, $\widehat{u_1}$, we see that $i-2L \leq \|\widehat{u_1}\| \leq i + 2L$, and so we can find such a factorisation of $u$ in polynomial time by computing {\sc slp}s\xspace for the words $u[[:j))$ and $u[[j:))$ for every integer $j$ with $\|i-j\|\leq 2L$. Then we apply Proposition~\ref{prop:tslp-slp} and compute {\sc slp}s\xspace for the words $w_1 := \mathsf{nf}(\alpha u[[:j)) \eta^{-1})$ and $w_2 := \mathsf{nf}(\eta u[[j:)) \beta^{-1})$ for every $\eta \in \Sigma^$ with $\|\eta\| \le L$. Finally we can, by using Proposition \ref{prop:slp_results}\,(vi), check whether $v_1 = w_1$ and $v_2 = w_2$ (we are guaranteed to find at least one such $j$ and $\eta$), and we define $\rho_\mathcal{U}(A) := X \langle \alpha,\eta\rangle$ for a new variable $X$ and set $\rho_\mathcal{U}(X) := C[[:j))$. We then continue with the elimination of the cut operator in $C[[:j))$. The other two cases are a little more complicated than in \cite{HLS}. Suppose first that $\|\widehat{v_1}\| < \mathsf{f}(L)$. Then by Proposition \ref{prop:slpshort}, in polynomial time we can compute an {\sc slp}\xspace $\mathcal{T}_A$ with $\mathsf{val}(\mathcal{T}_A) = \mathsf{val}(A) = v_1$. We include the variables and productions of $\mathcal{T}_A$ as part of the {\sc tslp}\xspace $\mathcal{U}$ that we are constructing, and define $\rho_\mathcal{U}(A) := S_{\mathcal{T}_A}$. Otherwise we have $\|\widehat{v_1}\| \ge \mathsf{f}(L)$ and $\|\widehat{v_2}\| < \mathsf{f}(L)$. Let $k:= \|\hat{v}\| - \mathsf{f}(L)$. Then, as above, we can use Proposition \ref{prop:tcslp-tslp} together with Proposition \ref{prop:backup2} to find $l$ and a word $\eta \in \Sigma^$ with $\|\eta\| \le L$ such that $v[[:k)) =_G \alpha u[[:l)) \eta^{-1}$. Now we introduce a new variable $X$ with $\rho_\mathcal{U}(X) = C[[:l))$. But in addition, using Proposition \ref{prop:slpshort} we compute, in polynomial time, an {\sc slp}\xspace $\mathcal{T}_A$ with $\mathsf{val}(\mathcal{T}_A) = v[[k:i))$, include the variables and productions of $\mathcal{T}_A$ in $\mathcal{U}$, and and define $\rho_\mathcal{U}(A) := X \langle \alpha,\eta \rangle S_{\mathcal{T}_A}$. Then $\mathsf{val}(A) = v[[:k))v[[k:i)) = v[[:i)) = v_1$ as required. As before, we continue with the elimination of the cut operators below $C[[:l))$. Since each elimination of a cut operator in $\rho(A)$ can lead to further such eliminations in the variables below $A$, the total number of such eliminations in the processing of $\mathcal{T}$ is bounded by $\mathsf{h}(\mathcal{T})^2$. \end{proof} With the completion of the proofs of Propositions~\ref{prop:tslp-slp} and ~\ref{prop:tcslp-tslp} we have now completed the proof of Theorem~\ref{thm:convert}. The following corollary of that theorem will be used in the final section. \begin{corollary}\label{cor:addtail} Let $\mathcal{G}$ be an {\sc slp}\xspace over $\Sigma$ for which $w := \mathsf{val}(\mathcal{G})$ is $\mathsf{nf}$-reduced, and let $v \in \Sigma^$ have length at most $L$. Then we can, in polynomial time, compute an {\sc slp}\xspace $\mathcal{S}$ over $\Sigma$ with $\mathsf{val}(\mathcal{S}) = \mathsf{nf}(wv)$. \end{corollary} \begin{proof} From $\mathcal{G}$, we can immediately define a {\sc tslp}\xspace $\mathcal{T}$ with $\mathsf{val}(\mathcal{T}) = \mathsf{nf}(wv)$ and $J_\mathcal{T} = L$ by adjoining a new start variable $S_\mathcal{T}$ together with the production $\rho(S_\mathcal{T}) = S_\mathcal{G} \langle \epsilon,v^{-1} \rangle$. We can then construct $\mathcal{S}$ with $\mathsf{val}(\mathcal{S}) = \mathsf{nf}(wv)$ in polynomial time, by Theorem~\ref{thm:convert}. \end{proof} \section{The final step.} \label{sec:slextcslp} Since a word represents the identity element if and only if its $\mathsf{nf}$-reduction is the empty word, the main theorem, Theorem~\ref{thm:main}, follows immediately from the combination of the following Theorem~\ref{thm:slextcslp} with Theorem~\ref{thm:convert}. Hence the proof of Theorem~\ref{thm:slextcslp} is our final step. \begin{theorem}\label{thm:slextcslp} Let $G$ be a group hyperbolic relative to a collection of free abelian subgroups, and suppose that a generating set $\Sigma$ for $G$, and integer $L$ are selected as in Sections ~\ref{subsec:relhyp_param1}, \ref{subsec:relhyp_param2}. Let $\mathcal{G}$ be an {\sc slp}\xspace for $G$ over $\Sigma$. Then we can construct, in polynomial time, a non-splitting $\mathsf{nf}$-reduced {\sc tcslp}\xspace $\mathcal{T}$ with $\mathsf{val}(\mathcal{T}) = \mathsf{nf}(\mathsf{val}(\mathcal{G}))$ and $J_\mathcal{T} \le L$, where each cut operator of $\mathcal{T}$ is non-splitting and specified relative to compression. \end{theorem} The following lemma, which is a special case of Theorem ~\ref{thm:slextcslp}, will be used within its proof, and applied to sub-{\sc slp}s\xspace of the {\sc slp}\xspace $\mathcal{G}$ within the statement of the theorem. Since the proof of the lemma involves similar but simpler arguments to that of the theorem, it is convenient to defer its proof. \begin{lemma}\label{lem:geo2slextcslp} Let $G$, $\Sigma$ and $\mathcal{G}$ be as in Theorem ~\ref{thm:slextcslp}, and assume also that $\widehat{\mathsf{val}(\mathcal{G})}$ is a geodesic word. Then in polynomial time we can construct a non-splitting $\mathsf{nf}$-reduced {\sc tcslp}\xspace $\mathcal{T}$ with $\mathsf{val}(\mathcal{T}) = \mathsf{nf}(\mathsf{val}(\mathcal{T}))$ and $J_\mathcal{T} \le L$, where each cut operator of $\mathcal{T}$ is non-splitting and specified relative to compression. \end{lemma} \begin{proofof}{Theorem~\ref{thm:slextcslp}} We know from Proposition \ref{prop:genset} that $G$ is asynchronously automatic over $\Sigma$ so, by Proposition~\ref{prop:slp_results}\,(v), we can test in polynomial time whether the words defined by {\sc slp}s\xspace over $\Sigma$ are $\mathsf{nf}$-reduced. We may assume by Proposition~\ref{prop:slp_results} that the given {\sc slp}\xspace $\mathcal{G}= (V_\mathcal{G},S_\mathcal{G},\rho_\mathcal{G})$ is trimmed and in Chomsky normal form, and by Proposition \ref{prop:comproot} that all components of $\mathsf{val}(\mathcal{G})$ have roots. The proof follows the same strategy as that of \cite[Theorem 6.7]{HLS}, but the presence of the parabolic subgroups gives rise to complications. Our aim is to construct a {\sc tcslp}\xspace $\mathcal{T}=(V_\mathcal{T},S_\mathcal{T},\rho_\mathcal{T})$ with value $\mathsf{nf}(\mathsf{val}(\mathcal{G}))$ that satisfies $J_\mathcal{T} \le L$, where $L$ is the constant defined in Section \ref{subsec:relhyp_param2}. We consider the variables of $\mathcal{G}$ in order of increasing height; $V_\mathcal{T}$ will contain a copy $A$ of each variable $A$ of $\mathcal{G}$, together with some auxiliary variables. We build $\mathcal{T}$ piece by piece, starting with $V_\mathcal{T}$ empty. At each stage, as we consider the variable $A$, we add a copy of $A$ and possibly some other new variables to $V_\mathcal{T}$, and define the image of $\rho_\mathcal{T}$ on each of those so as to make $\mathsf{val}_{\mathcal{T}}(A) = \mathsf{nf}(\mathsf{val}_\mathcal{G}(A))$. Our construction of {\sc tcslp}s\xspace rather than {\sc cslp}s\xspace will ensure a polynomial bound on the number of new variables we add at each stage and on the size of $\mathcal{T}$. That the condition on cut-operators holds will be clear from the construction. If the variable $A$ under consideration has height one, then $\rho_\mathcal{G}(A)=a$, for some $a \in \Sigma$; in that case, we simply define $\rho_\mathcal{T}(A)= \mathsf{nf}(a) = a$. So from now on we suppose that $\mathsf{h}(A)>1$, in which case $\rho_\mathcal{G}(A)=BC$, for variables $B$, $C$ of height less than $\mathsf{h}(A)$. Since we have already processed the variables $B$ and $C$, we know that $\mathcal{T}$ already contains sub-{\sc tcslp}s\xspace $\mathcal{T}_B$ and $\mathcal{T}_C$, with start variables $B_\mathcal{T}$ and $C_\mathcal{T}$, and with $\mathsf{val}(\mathcal{T}_B) = v_1 := \mathsf{nf}(\mathsf{val}_\mathcal{G}(B))$, $\mathsf{val}(\mathcal{T}_C) = v_2 := \mathsf{nf}(\mathsf{val}_\mathcal{G}(C))$, and $J_{\mathcal{T}_B},J_{\mathcal{T}_C} \le L$. By Theorem~\ref{thm:convert}, we can construct, in polynomial time, {\sc slp}s\xspace $\mathcal{S}_B$ and $\mathcal{S}_C$, with the same values as $\mathcal{T}_B$ and $\mathcal{T}_C$. As was the case for $\mathcal{G}$, we can ensure that $\mathcal{S}_B$ and $\mathcal{S}_C$ are in Chomsky normal form, and contain roots for all components of $v_1$ and $v_2$, respectively. We consider a triangle in the Cayley graph $\Gamma = \Gamma(G,\Sigma)$ whose sides are paths labelled by $v_1$, $v_2$ and the $\mathsf{nf}$-reduced representative $v_3$ of the element $v_1v_2$. Our aim is to construct $\mathcal{T}_A$ as a {\sc tcslp}\xspace with value $v_3$. Let $a,b$ and $c$ be the vertices of this triangle with sides from $a$ to $b$, $b$ to $c$ and $a$ to $c$ labelled by $v_1,v_2,v_3$. So the sides labelled $\widehat{v_1},\widehat{v_2},\widehat{v_3}$ in the corresponding triangle in $\widehat{\Gamma}$ are geodesic. So, since $\widehat{\Gamma}$ is a $\delta$-hyperbolic space, there are meeting vertices $d_1,d_2,d_3$, with $d_i$ on $\gamma_{\widehat{v_i}}$ ($i=1,2,3$) and $d_{\widehat{\Gamma}}(d_i,d_j) \leq \delta$ for $i \neq j$; see Figure~\ref{fig:hyptri}. Now let $K := K_1(\delta)$ as defined in Proposition~\ref{prop:backup}, and recall that the constant $L$ defined in Section \ref{subsec:relhyp_param2} satisfies $L \ge L_1(\delta)$. Now we apply Proposition \ref{prop:backup} to the sections of $\gamma_{\widehat{v_1}}$ and $\gamma_{\widehat{v_2}}$ that join $b$ to $d_1$ and $d_2$, and so deduce that, for any vertex $b_1$ of $\widehat{\Gamma}$ on $\gamma_{\widehat{v_1}}$ between $d_1$ and $b$ and distance on $\gamma_{\widehat{v_1}}$ at least $K$ from $d_1$, there exists a vertex $b_2$, on $\gamma_{\widehat{v_2}}$, with $d_{\Gamma}(b_1,b_2) \leq L$. We shall call vertices $b_1,b_2$ of $\widehat{\Gamma}$ with $d_{\Gamma}(b_1,b_2) \leq L$ that lie on two different sides of the triangle \emph{corresponding vertices}. Note that although the particular corresponding vertices $b_1,b_2$ whose existence we have just shown are found within the sections of $\gamma_{\widehat{v_1}}$ and $\gamma_{\widehat{v_2}}$ that join $b$ to $d_1$ and $d_2$, in general, corresponding vertices might be found past either or both of $d_1$ and $d_2$ on those paths. We claim that it is possible to decide in polynomial time whether a vertex $b_1$ on $\gamma_{\widehat{v_1}}$ has a corresponding vertex $b_2$ on $\gamma_{\widehat{v}_2}$ and, if so, find $b_2$ together with the label in $\Sigma^$ of a path of length at most $L$ in $\Gamma$ joining $b_1$ to $b_2$. The justification for this claim is as follows. Let $l$ be the distance in $\widehat{\Gamma}$ from $b$ to $b_1$. It is straightforward to define an {\sc slp}\xspace $\overline{S_B}$ with value $v_1^{-1}=\mathsf{val}(\mathcal{S}_B)^{-1}$. The word $v_1^{-1}$ might not be $\mathsf{nf}$-reduced but its derived word $\widehat{v_1}^{-1}$ is geodesic, so we can use Lemma \ref{lem:geo2slextcslp} followed by Theorem~\ref{thm:convert} to construct an {\sc slp}\xspace with value equal to $(\mathsf{nf}(v_1^{-1}))[[:l))$. Since the number of words over $\Sigma$ of length at most $L$ is bounded above by the constant $\|\Sigma\|^{L+1}$, we can in polynomial time, by Corollary \ref{cor:addtail}, compute {\sc slp}s\xspace with values $\mathsf{nf}(\overline{\mathcal{S}}_B[[:l)) \eta)$ for all words $\eta \in \Sigma^$ of length at most $L$. For each of these, we compute the length $l'$ of its derived word, and then check whether its value is equal to $\mathsf{val}(\mathcal{S}_C[[:l')))$. If so, then the vertex $b_2$ at distance $l'$ from $b$ in $\Gamma$ corresponds to $b_1$, and $\eta$ is the required path label from $b_1$ to $b_2$. Furthermore all such corresponding vertices $b_2$ are found by this procedure. We now consider two possible situations, as follows. In Case 1, there is either a vertex $a'$ on $\gamma_{\widehat{v_2}}$ that corresponds to the vertex $a$ of $\gamma_{\widehat{v_1}}$, or there is a vertex $c'$ on $\gamma_{\widehat{v_1}}$ that corresponds to the vertex $c$ of $\gamma_{\widehat{v_2}}$. In Case 2, no such vertices $a'$ or $c'$ exist. By the claim above, we can check which case we are in. Case 2 is more difficult, so we shall deal with that first and provide a brief description of the argument for Case 1 at the end. So suppose that we are in Case 2. Now the vertex $b$ of $\gamma_{\widehat{v_1}}$ has a corresponding vertex ($b$ itself) on $\gamma_{\widehat{v_2}}$, but the vertex $a$ has no corresponding vertex on $\gamma_{\widehat{v_2}}$. We need to find corresponding vertices $b_1$ and $b_2$ on $\gamma_{\widehat{v_1}}$ and $\gamma_{\widehat{v_2}}$ with the additional property that the vertex $b'_1$ that is at distance 1 from $b_1$ in $\widehat{\Gamma}$, on $\gamma_{\widehat{v_1}}$, between $b_1$ and $a$, has no corresponding vertex on $\gamma_{\widehat{v_2}}$. We do this, as in the proof of \cite[Theorem 6.7]{HLS}, using the technique of {\em binary search} to find $b_1$ by testing whether various vertices that we call $b_t$ have corresponding vertices; that is, where $l_0$ is the length of $\gamma_{\widehat{v_1}}$, we first test if the vertex $b_t$ at distance $l_0/2$ from $b$ along $\gamma_{\widehat{v_1}}$ has a corresponding vertex on $\gamma_{\widehat{v_2}}$. If $b_t$ has a corresponding vertex, and the next vertex $b_t'$ on $\gamma_{\widehat{v_1}}$ has no corresponding vertex on $\gamma_{\widehat{v_2}}$, then we set $b_1 := b_t$ and set $b_1' := b_t'$, Otherwise, our next choice for $b_t$ is either the vertex at distance $l_0/4$ from $b$ along $\gamma_{\widehat{v_1}}$ (when our first choice for $b_t$ has no corresponding vertex) or at distance $3l_0/4$ from $b$ (when our first choices for both $b_t$ and $b_t'$ have corresponding vertices). We continue searching in this way, each time in one half of the previous interval of $\gamma_{\widehat{v_1}}$, until we find the vertices $b_1,b_2,b_1'$ that satisfy the required conditions. Note that the time taken for this search is logarithmic in the length of $\widehat{v_1}$ and so polynomial in the size of its defining {\sc slp}\xspace $\mathcal{S}_B$. If $b_1'$ were distance greater than $K$ from $d_1$ along $\gamma_{\widehat{v_1}}$ within the section joining $d_1$ to $b$, then Proposition~\ref{prop:backup} would imply the existence of a corresponding vertex for $b_1'$ on $\gamma_{\widehat{v_2}}$; but we know there is no such vertex. It follows that \[d_{\gamma_{\widehat{v_1}}}(a,b_1') \leq d_{\gamma_{\widehat{v_1}}}(a,d_1) + K-1, \ \hbox{\rm and}\ d_{\gamma_{\widehat{v_1}}}(a,b_1) \leq d_{\gamma_{\widehat{v_1}}}(a,d_1) + K.\] We claim that $$d_{\gamma_{\widehat{v_2}}}(c,b_2) \le d_{\gamma_{\widehat{v_2}}}(c,d_2) + 3L+\delta.$$ Since $d_{\widehat{\Gamma}}(d_1,d_2) \le \delta$ and $d_{\widehat{\Gamma}}(b_1,b_2) \le L$, this follows from the triangle inequality if $d_{\widehat{\Gamma}}(d_1,b_1) \le 2L$. So suppose not. Then, since $L\ge K$, $b_1$ must be between $a$ and $d_1$ on $\widehat{v_1}$; note that this position for $b_1$ is {\em not} as suggested in Figure~\ref{fig:hyptri}. But, by Proposition~\ref{prop:backup}, if $d_{\gamma_{\widehat{v_2}}}(c,b_2) > d_{\gamma_{\widehat{v_2}}}(c,d_2) + K$, then there is a point $b_1''$ on $\widehat{v_1}$ between $b$ and $d_1$ with $d_{\Gamma}(b_2,b_1'') \le L$, and hence also $d_{\widehat{\Gamma}}(b_2,b_1'') \le L$. But then we have $2L \ge d_{\widehat{v_1}}(b_1,b_1'') \ge d_{\widehat{v_1}}(b_1,d_1) > 2L$, a contradiction. So $d_{\gamma_{\widehat{v_2}}}(c,b_2) \le d_{\gamma_{\widehat{v_2}}}(c,d_2) + K$ and, since $K < 3L+\delta$, the claim also holds in this case. So we have \begin{eqnarray} d_{\Gamma}(b_1,b_2) \leq L,\quad d_{\gamma_{\widehat{v_1}}}(a,b_1) &\leq& d_{\gamma_{\widehat{v_1}}}(a,d_1) + K, \\ d_{\gamma_{\widehat{v_2}}}(c,b_2) &\leq& d_{\gamma_{\widehat{v_2}}}(c,d_2) + 3L + \delta. \end{eqnarray} Now it follows from Proposition~\ref{prop:backup} that any vertex on $\gamma_{\widehat{v_1}}$ between $a$ and $d_1$ that is at distance at least $K$ along the curve from $d_1$, must have a corresponding vertex on $\gamma_{\widehat{v_3}}$ that lies between $a$ and $d_3$. So now define $a_1$ to be the vertex on $\gamma_{\widehat{v_1}}$ between $a$ and $d_1$ that is distance $2K$ along the curve from $b_1$ (if there is no such vertex, then define $a_1$ to be $a$). Then there is a vertex $a_3$ on $\gamma_{\widehat{v_3}}$ between $a$ and $d_3$ that corresponds to $a_1$. \begin{figure} \caption{The hyperbolic triangle} \label{fig:hyptri} \end{figure} Similarly let $c_2$ be the vertex on $\gamma_{\widehat{v_2}}$ between $c$ and $d_2$ that is distance $K+3L+\delta$ along the curve from $b_2$ (if there is no such vertex, then define $c_2$ to be $c$). Then there is a vertex $c_3$ on $\gamma_{\widehat{v_3}}$ that corresponds to $c_2$. There exist words $\zeta$, $\eta$ and $\theta$ over $\Sigma$ each of length at most $L$, that label paths in $\Gamma$ from $a_3$ to $a_1$, $b_1$ to $b_2$, and $c_3$ to $c_2$. We know $\eta$ already, because (by the claim above) we found it when we defined $b_1$ and $b_2$; to progress further, we need to find $\zeta$ and $\theta$. We find these through an exhaustive search process, as we shall now describe. We generate all possible word pairs $(\zeta,\theta)$ (that is, word pairs of lengths at most $L$), and check their validity, until we find a solution. So suppose that $(\zeta,\theta)$ is a candidate pair. Define $k_1,l_1,k_2,l_2$ to be the integers for which $v_1[[k_1:l_1))$ labels the path in $\Gamma$ from $a_1$ to $b_1$ and $v_2[[k_2:l_2))$ the path from $b_2$ to $c_2$; then $v_1[[k_1:l_1)) = \mathsf{val}(\mathcal{S}_B[[k_1:l_1)))$ and $v_2[[k_2:l_2)) = \mathsf{val}(\mathcal{S}_C[[k_2:l_2)))$. The $\mathsf{nf}$-reduced representative of $v_1[[:k_1))\zeta^{-1}$ is the value of the {\sc tcslp}\xspace $\mathcal{S}_B[[:k_1)) \langle \epsilon,\zeta \rangle$, and we can find an {\sc slp}\xspace $\mathcal{S}_1$ with the same value in polynomial time, by Theorem~\ref{thm:convert}. The word $\zeta v_1[[k_1:l_1))\eta v_2[[k_2:l_2))\theta^{-1}$ has length at most $3K+6L+\delta$ over $\widehat{\Sigma}$, and is the value of the {\sc cslp}\xspace $\zeta\mathcal{S}_B[[k_1:l_1))\eta\mathcal{S}_C[[k_2:l_2))\theta^{-1}$. By Proposition \ref{prop:slpshort} applied with $\kappa = 3K+6L+\delta$, we can (in polynomial time) find an {\sc slp}\xspace $\mathcal{S}_2$ with value $\mathsf{nf}(\zeta v_1[[k_1:l_1))\eta v_2[[k_2:l_2))\theta^{-1}).$ The $\mathsf{nf}$-reduced representative of $\theta v_2[[l_2:))$ is the value of the {\sc tcslp}\xspace $\mathcal{S}_C[[l_2:)) \langle \theta,\epsilon \rangle$, and, again, we can find an {\sc slp}\xspace $\mathcal{S}_3$ with the same value in polynomial time. Now the word $\mathsf{val}(\mathcal{S}_1\mathcal{S}_2\mathcal{S}_3)$ represents the same element of $G$ as each of the words $v_1v_2$ and $v_3$, and will be equal as a word to $v_3$ if and only if it is $\mathsf{nf}$-reduced. By Proposition~\ref{prop:slp_results}\,(v), we can test in polynomial time whether $\mathsf{val}(\mathcal{S}_1\mathcal{S}_2\mathcal{S}_3)$ is $\mathsf{nf}$-reduced; if it is, then $\mathcal{S}_1\mathcal{S}_2\mathcal{S}_3$ is an {\sc slp}\xspace for $v_3$. We define a {\sc tcslp}\xspace $\mathcal{T}_1:= \mathcal{T}_B[[:k_1)) \langle \epsilon,\zeta \rangle$ as an extension of $\mathcal{T}_B$ by a single variable $S_{\mathcal{T}_1}$, and similarly $\mathcal{T}_3 := \mathcal{T}_C[[l_2:)) \langle \theta,\epsilon \rangle$ as an extension of $\mathcal{T}_C$. The {\sc tcslp}s\xspace $\mathcal{T}_1$ and $\mathcal{T}_3$ have the same values as $\mathcal{S}_1,\mathcal{S}_3$, respectively, and the concatenation $\mathcal{T}_1\mathcal{S}_2\mathcal{T}_3$ is a {\sc tcslp}\xspace with value $v_3$. We set our copy of $A$ within $\mathcal{T}$ to be the start variable of that {\sc tcslp}\xspace, and define $\rho_\mathcal{T}(A)$ accordingly. (The reason that we do not simply adjoin the {\sc slp}s\xspace $\mathcal{S}_1$ and $\mathcal{S}_3$ to $\mathcal{T}$ is that, if we did that, then we would be unable to prove that the $\|\mathcal{T}\|$ remains bounded throughout the complete process by a polynomial function of $\|\mathcal{G}\|$. This explains why we needed to introduce the concepts of {\sc tslp}\xspace and {\sc tcslp}\xspace.) We shall now briefly describe the corresponding argument in Case 1, which is similar to that for Case 2, but simpler. Suppose that there is a vertex $c'$ on $\gamma_{\widehat{v_1}}$ that corresponds to the vertex $c$ of $\gamma_{\widehat{v_2}}$ - the other case is similar. Then, as explained earlier, we can locate $c'$, we can find $\eta \in \Sigma^$ that labels a path of length at most $L$ in $\Gamma$ from $c$ to $c'$, and we can calculate the integer $k_1$ such that $v_1[[:k_1))$ is the prefix of $v_1$ from $a$ to $c'$. Let $a_1$ be the vertex on $\gamma_{\widehat{v_1}}$ that is between $a$ and $c'$ and is at distance $K$ in $\gamma_{\widehat{v_1}}$ from $c'$ (or $a_1=a$ if there is no such vertex), and let $k_2$ be its distance from $a$ along $v_1$. Then by Proposition~\ref{prop:backup}, $a_1$ has a corresponding vertex $a_3$ on $\gamma_{\widehat{v_3}}$. Let $\zeta \in \Sigma^$ be the label of a path of length at most $L$ in $\Gamma$ from $a_3$ to $a_1$. Then $$v_3 = \mathsf{nf}(v_1[[:k_2))\zeta^{-1})\mathsf{nf}(\zeta v_1[[k_2:k_1))\eta^{-1})$$ and, much as in Case 2, we can find $\zeta$ by exhaustive search, then define the {\sc tcslp}\xspace $\mathcal{T}$ as the concatenation $\mathcal{T}_1\mathcal{S}_2$, where $\mathsf{val}(\mathcal{T}_1) = \mathsf{nf}(v_1[[:k_2)\zeta^{-1}))$, and $\mathsf{val}(\mathcal{S}_2) = \mathsf{nf}(\zeta v_1[[k_2:k_1))\eta^{-1})$. In both Cases 1 and 2, we observe that all of the variables of $\mathcal{T}_B$ and $\mathcal{T}_C$ were defined during the processing of other variables, but a copy of $A$, the variables $S_{\mathcal{T}_1}$ and $S_{\mathcal{T}_2}$ and also the variables of $\mathcal{S}_2$ are added to $V_\mathcal{T}$ during the processing of the variable $A$ of $\mathcal{G}$, and their images under $\rho_T$ are correspondingly defined. Since the length of $\mathcal{S}_2$ as a word over $\widehat{\Sigma}$ is bounded by the constant $3K+6L+\delta$ we know from Proposition~\ref{prop:slpshort} that its size is bounded by a constant multiple of $\log(\|\mathsf{val}(\mathcal{S}_2)\|)$ and hence, by Proposition~\ref{prop:slp_results}\,(ii), of $\|\mathcal{G}\|$. So after all variables of $\mathcal{G}$ have been processed, the size of the final {\sc tcslp}\xspace $\mathcal{T}$ is bounded by a quadratic function of $\mathcal{G}$. This completes the proof. \end{proofof} \begin{proofof}{Lemma \ref{lem:geo2slextcslp}} As we observed in Remark~\ref{rem:subword}, since all components of $w = \mathsf{val}(\mathcal{G})$ already have roots, it follows from Proposition~\ref{prop:comproot} that for each variable $A$ of $\mathcal{G}$, all occurrences of $\mathsf{val}(A)$ as subwords of $w$ that are derived from $A$ consist either of complete components of $w$ together with subwords in $(\Sigma \setminus \mathcal{H})^*$, or of subwords of a component. So the assumption that $\hat{w}$ is a geodesic word implies that $\widehat{\mathsf{val}_\mathcal{G}(A)}$ is also geodesic for all variables $A$ of $\mathcal{G}$. We follow the proof of Theorem \ref{thm:slextcslp}. When we consider the variable $A$ with $\rho(A) = BC$, we know that $\widehat{\mathsf{val}_\mathcal{G}(A)} = \widehat{\mathsf{val}_\mathcal{G}(B)}\widehat{\mathsf{val}_\mathcal{G}(C)}$ is geodesic, and so the concatenation of the consecutive sides $ab$ and $bc$ of the associated hyperbolic triangle in $\widehat{\Gamma}$, which is labelled by the word $\widehat{uv}=\hat{u}\hat{v}$, is a geodesic path in $\Gamma$. Proposition \ref{prop:backup} now implies that the vertices $a_1$ and $c_2$ on the two paths at distance $K$ from $b$ in $\widehat{\Gamma}$ have corresponding vertex on $\gamma_{\widehat{v_3}}$. After defining $a_1$ and $c_2$ in this way, the rest of the proof is the same as the proof of Theorem \ref{thm:slextcslp}, with $\eta = \epsilon$. But it is of course important to point out that we have not used the conclusion of the lemma in its proof! \end{proofof} \end{document}	arXiv
A matrix whose all elements are arranged in a row is called a Row matrix. Row matrix is one type of matrix and it is also called as a row vector. In this type of matrix, all elements are arranged in only one row but in different columns. $M$ is a matrix in general form and it is a row matrix of order $1 \times n$. It can also be expressed in simple form. In this case of a Row vector , each element is displayed in one row. So, the row $i = 1$. Similarly, the number of rows $m = 1$. Therefore, a row matrix can be displayed in simple but in general form as follows. The following matrices are best examples for a row matrix. $A$ is a row matrix of order $1 \times 1$. Only one element is arranged in one row and one column in this matrix. $B$ is a row matrix of order $1 \times 2$. In this matrix, two elements are arranged in one row but in two columns. $C$ is a row matrix of order $1 \times 3$. Three elements are arranged in one row but in three columns in this matrix. $D$ is a row matrix of order $1 \times 4$. In this matrix, four elements are arranged in one row but in four columns. All the row matrices are single row matrices but they have common shape and it is a rectangle. Hence, the row matrices are known as rectangular matrices.	CommonCrawl
Quantization commutes with reduction In mathematics, more specifically in the context of geometric quantization, quantization commutes with reduction states that the space of global sections of a line bundle L satisfying the quantization condition[1] on the symplectic quotient of a compact symplectic manifold is the space of invariant sections of L. This was conjectured in 1980s by Guillemin and Sternberg and was proven in 1990s by Meinrenken[2][3] (the second paper used symplectic cut) as well as Tian and Zhang.[4] For the formulation due to Teleman, see C. Woodward's notes. See also • Geometric invariant theory Notes 1. This means that the curvature of the connection on the line bundle is the symplectic form. 2. Meinrenken 1996 3. Meinrenken 1998 4. Tian & Zhang 1998 References • Guillemin, V.; Sternberg, S. (1982), "Geometric quantization and multiplicities of group representations", Inventiones Mathematicae, 67 (3): 515–538, Bibcode:1982InMat..67..515G, doi:10.1007/BF01398934, MR 0664118, S2CID 121632102 • Meinrenken, Eckhard (1996), "On Riemann-Roch formulas for multiplicities", Journal of the American Mathematical Society, 9 (2): 373–389, doi:10.1090/S0894-0347-96-00197-X, MR 1325798. • Meinrenken, Eckhard (1998), "Symplectic surgery and the Spinc-Dirac operator", Advances in Mathematics, 134 (2): 240–277, arXiv:dg-ga/9504002, doi:10.1006/aima.1997.1701, MR 1617809. • Tian, Youliang; Zhang, Weiping (1998), "An analytic proof of the geometric quantization conjecture of Guillemin–Sternberg", Inventiones Mathematicae, 132 (2): 229–259, Bibcode:1998InMat.132..229T, doi:10.1007/s002220050223, MR 1621428, S2CID 119943992. • Woodward, Christopher T. (2010), "Moment maps and geometric invariant theory", Les cours du CIRM, 1 (1): 55–98, arXiv:0912.1132, Bibcode:2009arXiv0912.1132W, doi:10.5802/ccirm.4	Wikipedia
\begin{document} \title{The delta-nabla calculus of variations\thanks{Accepted for publication (02/December/2009) in \emph{Fasciculi Mathematici}.}} \author{Agnieszka B. Malinowska$^{1, 2}$\\ \texttt{[email protected]} \and Delfim F. M. Torres$^{1}$\\ \texttt{[email protected]}} \date{$^1$Department of Mathematics\\ University of Aveiro\\ 3810-193 Aveiro, Portugal\\[0.3cm] $^2$Faculty of Computer Science\\ Bia{\l}ystok University of Technology\\ 15-351 Bia\l ystok, Poland} \maketitle \begin{abstract} The discrete-time, the quantum, and the continuous calculus of variations have been recently unified and extended. Two approaches are followed in the literature: one dealing with minimization of delta integrals; the other dealing with minimization of nabla integrals. Here we propose a more general approach to the calculus of variations on time scales that allows to obtain both delta and nabla results as particular cases. \noindent \textbf{Keywords}: calculus of variations; Euler-Lagrange equations; time scales. \noindent \textbf{2010 Mathematics Subject Classification:} 49K05, 39A12, 34N05. \end{abstract} \section{Introduction} The calculus of variations on time scales was introduced by M.~Bohner using the delta derivative and integral \cite{B:04}: to extremize a functional of the form \begin{equation} \label{eq:Pd} \mathcal{J}_\Delta(y) = \int_a^b L\left(t,y^\sigma(t),y^\Delta(t)\right) \Delta t \, . \end{equation} Motivated by applications in economics \cite{A:B:L:06,A:U:08}, a different formulation for the problems of the calculus of variations on time scales has been considered, which involve a functional with a nabla derivative and a nabla integral \cite{A:T,Atici:comparison,NM:T}: \begin{equation} \label{eq:Pn} \mathcal{J}_\nabla(y) = \int_a^b L\left(t,y^\rho(t),y^\nabla(t)\right) \nabla t \, . \end{equation} Formulations \eqref{eq:Pd} and \eqref{eq:Pn} are consistent in the sense that results obtained \emph{via} delta and nabla approaches are similar among them and similar to the classical results of the calculus of variations. An example of this is given by the time scale versions of the Euler-Lagrange equations: if $y \in C_{\textrm{rd}}^2$ is an extremizer of \eqref{eq:Pd}, then $y$ satisfies the delta-differential equation \begin{equation} \label{eq:el:d} \frac{\Delta}{\Delta t} \partial_{3}L\left(t,y^\sigma(t),{y}^\Delta(t)\right) = \partial_{2}L\left(t,y^\sigma(t),{y}^\Delta(t)\right) \end{equation} for all $t \in [a,b]^{\kappa^2}$ \cite{B:04}; if $y \in C_{\textrm{ld}}^2$ is an extremizer of \eqref{eq:Pn}, then $y$ satisfies the nabla-differential equation \begin{equation} \label{eq:el:n} \frac{\nabla}{\nabla t} \partial_{3}L\left(t,y^\rho(t),{y}^\nabla(t)\right) = \partial_{2}L\left(t,y^\rho(t),{y}^\nabla(t)\right) \end{equation} for all $t \in [a,b]_{\kappa^2}$ \cite{NM:T}, where we use $\partial_{i}L$ to denote the standard partial derivative of $L(\cdot,\cdot,\cdot)$ with respect to its $i$th variable, $i = 1,2,3$. In the classical context $\mathbb{T} = \mathbb{R}$ one has \begin{equation} \label{eq:Pc} \mathcal{J}_\Delta(y) = \mathcal{J}_\nabla(y) = \int_a^b L\left(t,y(t),y'(t)\right) dt \end{equation} and both \eqref{eq:el:d} and \eqref{eq:el:n} coincide with the standard Euler-Lagrange equation: if $y \in C^2$ is an extremizer of the integral functional \eqref{eq:Pc}, then \begin{equation} \frac{d}{d t} \partial_{3}L\left(t,y(t),y'(t)\right) =\partial_{2}L\left(t,y(t),y'(t)\right) \end{equation} for all $t \in [a,b]$. However, the problems of extremizing \eqref{eq:Pd} and \eqref{eq:Pn} are intrinsically different, in the sense that is not possible to obtain the nabla results as corollaries of the delta ones and \emph{vice versa}. Indeed, if admissible functions $y$ are of class $C^2$ then (\textrm{cf.} \cite{G:G:S:05}) \begin{equation} \mathcal{J}_\Delta(y) = \int_a^b L\left(t,y^\sigma(t),y^\Delta(t)\right) \Delta t = \int_a^b L\left(\rho(t),(y^\sigma)^\rho(t),y^\nabla(t)\right) \nabla t \end{equation} while \begin{equation} \mathcal{J}_\nabla(y) = \int_a^b L\left(t,y^\rho(t),y^\nabla(t)\right) \nabla t = \int_a^b L\left(\sigma(t),(y^\rho)^\sigma(t),y^\Delta(t)\right) \Delta t \end{equation} and one easily see that functionals \eqref{eq:Pd} and \eqref{eq:Pn} have a different nature and are not compatible with each other. In this paper we introduce a more general formulation of the calculus of variations that includes, as trivial examples, the problems with functionals $\mathcal{J}_\Delta(y)$ and $\mathcal{J}_\nabla(y)$ that have been previously studied in the literature. Our main result provides an Euler-Lagrange necessary optimality type condition (\textrm{cf.} Theorem~\ref{thm:mr}). \section{Our goal} Let $\mathbb{T}$ be a given time scale with $a, b \in \mathbb{T}$, $a < b$, and $\left(\mathbb{T} \setminus \{a,b\}\right)\cap [a,b] \ne \emptyset$; $L_{\Delta}(\cdot,\cdot,\cdot)$ and $L_{\nabla}(\cdot,\cdot,\cdot)$ be two given smooth functions from $\mathbb{T} \times \mathbb{R}^2$ to $\mathbb{R}$. The results here discussed are trivially generalized for admissible functions $y : \mathbb{T}\rightarrow\mathbb{R}^n$ but for simplicity of presentation we restrict ourselves to the scalar case $n=1$. We consider the delta-nabla integral functional \begin{equation} \label{eq:P} \begin{split} \mathcal{J}(y) &= \left(\int_a^b L_{\Delta}\left(t,y^\sigma(t),y^\Delta(t)\right) \Delta t\right) \cdot \left(\int_a^b L_{\nabla}\left(t,y^\rho(t),y^\nabla(t)\right) \nabla t\right)\\ &= \int_a^b \int_a^b \left[ L_{\Delta}\left(t,y^\sigma(t),y^\Delta(t)\right) \cdot L_{\nabla}\left(\tau,y^\rho(\tau),y^\nabla(\tau)\right) \right] \Delta t \nabla \tau \, . \end{split} \end{equation} \begin{remark} \label{obs} In the particular case $L_\nabla \equiv \frac{1}{b-a}$ functional \eqref{eq:P} reduces to \eqref{eq:Pd} (\textrm{i.e.}, $\mathcal{J}(y) = \mathcal{J}_\Delta(y)$); in the particular case $L_\Delta \equiv \frac{1}{b-a}$ functional \eqref{eq:P} reduces to \eqref{eq:Pn} (\textrm{i.e.}, $\mathcal{J}(y) = \mathcal{J}_\nabla(y)$). \end{remark} Our main goal is to answer the following question: \emph{What is the Euler-Lagrange equation for $\mathcal{J}(y)$ defined by \eqref{eq:P}?} For simplicity of notation we introduce the operators $[y]$ and $\{y\}$ defined by $[y](t) = \left(t,y^\sigma(t),y^\Delta(t)\right)$ and $\{y\}(t) = \left(t,y^\rho(t),y^\nabla(t)\right)$. Then, \begin{gather} \mathcal{J}_\Delta(y) = \int_a^b L_\Delta[y](t) \Delta t \, , \quad \mathcal{J}_\nabla(y) = \int_a^b L_\nabla\{y\}(t) \nabla t \, , \\ \mathcal{J}(y) = \mathcal{J}_\Delta(y) \mathcal{J}_\nabla(y) = \int_a^b \int_a^b L_{\Delta}[y](t) L_{\nabla}\{y\}(\tau) \Delta t \nabla \tau \, . \end{gather} \section{Preliminaries to the calculus of variations} Similar to the classical calculus of variations, integration by parts will play an important role in our delta-nabla calculus of variations. If functions $f,g : \mathbb{T}\rightarrow\mathbb{R}$ are delta and nabla differentiable with continuous derivatives, then the following formulas of integration by parts hold \cite{B:P:01}: \begin{equation} \label{intBP} \begin{split} \int_{a}^{b}f^\sigma(t) g^{\Delta}(t)\Delta t &=\left.(fg)(t)\right\|_{t=a}^{t=b} -\int_{a}^{b}f^{\Delta}(t)g(t)\Delta t \, , \\ \int_{a}^{b}f(t)g^{\Delta}(t)\Delta t &=\left.(fg)(t)\right\|_{t=a}^{t=b} -\int_{a}^{b}f^{\Delta}(t)g^\sigma(t)\Delta t \, , \\ \int_{a}^{b}f^\rho(t)g^{\nabla}(t)\nabla t &=\left.(fg)(t)\right\|_{t=a}^{t=b} -\int_{a}^{b}f^{\nabla}(t)g(t)\nabla t \, ,\\ \int_{a}^{b}f(t)g^{\nabla}(t)\nabla t &=\left.(fg)(t)\right\|_{t=a}^{t=b} -\int_{a}^{b}f^{\nabla}(t)g^\rho(t)\nabla t \, . \end{split} \end{equation} The following fundamental lemma of the calculus of variations on time scales involving a nabla derivative and a nabla integral has been proved in \cite{NM:T}. \begin{lemma}{\rm (The nabla Dubois-Reymond lemma \cite[Lemma~14]{NM:T}).} \label{DBRL:n} Let $f \in C_{\textrm{ld}}([a,b], \mathbb{R})$. If $$ \int_{a}^{b} f(t)\eta^{\nabla}(t)\nabla t=0 \quad \mbox{for all $\eta \in C_{\textrm{ld}}^1([a,b], \mathbb{R})$ with $\eta(a)=\eta(b)=0$} \, , $$ then $f(t) = c$ on $t\in [a,b]_\kappa$ for some constant $c$. \end{lemma} Lemma~\ref{DBRL:d} is the analogous delta version of Lemma~\ref{DBRL:n}: \begin{lemma}{\rm (The delta Dubois-Reymond lemma \cite{B:04}).} \label{DBRL:d} Let $g\in C_{\textrm{rd}}([a,b], \mathbb{R})$. If $$\int_{a}^{b} g(t) \eta^\Delta(t)\Delta t=0 \quad \mbox{for all $\eta \in C_{\textrm{rd}}^1$ with $\eta(a)=\eta(b)=0$} \, ,$$ then $g(t)=c$ on $[a,b]^\kappa$ for some $c\in\mathbb{R}$. \end{lemma} Proposition~\ref{prop:rel:der} gives a relationship between delta and nabla derivatives. \begin{proposition}{\rm (Theorems~2.5 and 2.6 of \cite{A:G:02}).} \label{prop:rel:der} (i) If $f : \mathbb{T} \rightarrow \mathbb{R}$ is delta differentiable on $\mathbb{T}^\kappa$ and $f^\Delta$ is continuous on $\mathbb{T}^\kappa$, then $f$ is nabla differentiable on $\mathbb{T}_\kappa$ and \begin{equation} \label{eq:chgN_to_D} f^\nabla(t)=\left(f^\Delta\right)^\rho(t) \quad \text{for all } t \in \mathbb{T}_\kappa \, . \end{equation} (ii) If $f : \mathbb{T} \rightarrow \mathbb{R}$ is nabla differentiable on $\mathbb{T}_\kappa$ and $f^\nabla$ is continuous on $\mathbb{T}_\kappa$, then $f$ is delta differentiable on $\mathbb{T}^\kappa$ and \begin{equation} \label{eq:chgD_to_N} f^\Delta(t)=\left(f^\nabla\right)^\sigma(t) \quad \text{for all } t \in \mathbb{T}^\kappa \, . \end{equation} \end{proposition} \begin{remark} Note that, in general, $f^\nabla(t) \ne f^\Delta\left(\rho(t)\right)$ and $f^\Delta(t) \ne f^\nabla\left(\sigma(t)\right)$. In Proposition~\ref{prop:rel:der} the assumptions on the continuity of $f^\Delta$ and $f^\nabla$ are crucial. \end{remark} \begin{proposition}{\rm (\cite[Theorem~2.8]{A:G:02}).} \label{eq:prop} Let $a, b \in\mathbb{T}$ with $a \le b$ and let $f$ be a continuous function on $[a, b]$. Then, \begin{equation} \begin{split} \int_a^b f(t)\Delta t &= \int_a^{\rho(b)} f(t)\Delta t + (b - \rho(b))f^\rho(b) \, , \\ \int_a^b f(t)\Delta t &= (\sigma(a) - a) f(a) + \int_{\sigma(a)}^b f(t)\Delta t \, , \\ \int_a^b f(t)\nabla t &= \int_a^{\rho(b)} f(t)\nabla t + (b - \rho(b)) f(b) \, , \\ \int_a^b f(t)\nabla t &= (\sigma(a) - a) f^\sigma(a) + \int_{\sigma(a)}^b f(t)\nabla t \, . \end{split} \end{equation} \end{proposition} We end our brief review of the calculus on time scales with a relationship between the delta and nabla integrals. \begin{proposition}{\rm (\cite[Proposition~7]{G:G:S:05}).} If function $f : \mathbb{T} \rightarrow \mathbb{R}$ is continuous, then for all $a, b \in \mathbb{T}$ with $a < b$ we have \begin{gather} \int_a^b f(t) \Delta t = \int_a^b f^\rho(t) \nabla t \, , \label{eq:DtoN}\\ \int_a^b f(t) \nabla t = \int_a^b f^\sigma(t) \Delta t \, . \label{eq:NtoD} \end{gather} \end{proposition} \section{Main Result} We consider the problem of extremizing the variational functional \eqref{eq:P} subject to given boundary conditions $y(a) = \alpha$ and $y(b) = \beta$: \begin{equation} \label{problem:P} \begin{gathered} \mathcal{J}(y) = \left(\int_a^b L_{\Delta}[y](t) \Delta t\right) \left(\int_a^b L_{\nabla}\{y\}(t) \nabla t\right) \longrightarrow \textrm{extr} \\ y(\cdot) \in C_{\diamond}^1 \\ y(a) = \alpha \, , \quad y(b) = \beta \, , \end{gathered} \end{equation} where $C_{\diamond}^1$ denote the class of functions $y : [a,b]\rightarrow\mathbb{R}$ with $y^\Delta$ continuous on $[a,b]^\kappa$ and $y^\nabla$ continuous on $[a,b]_\kappa$. Before presenting the Euler-Lagrange equations for problem \eqref{problem:P} we introduce the definition of weak local extremum. \begin{definition} We say that $\hat{y}\in C_{\diamond}^{1}([a,b], \mathbb{R})$ is a weak local minimizer (respectively weak local maximizer) for problem \eqref{problem:P} if there exists $\delta >0$ such that $\mathcal{J}(\hat{y})\leq \mathcal{J}(y)$ (respectively $\mathcal{J}(\hat{y}) \geq \mathcal{J}(y)$) for all $y \in C_{\diamond}^{1}([a,b], \mathbb{R})$ satisfying the boundary conditions $y(a) = \alpha$, $y(b) = \beta$, and $\parallel y - \hat{y}\parallel_{1,\infty} < \delta$, where $$\parallel y\parallel_{1,\infty}:= \parallel y^{\sigma}\parallel_{\infty} + \parallel y^{\rho}\parallel_{\infty} + \parallel y^{\Delta}\parallel_{\infty} + \parallel y^{\nabla}\parallel_{\infty}$$ and $\parallel y\parallel_{\infty} :=\sup_{t \in [a,b]_{\kappa}^{\kappa}}\mid y(t) \mid$. \end{definition} Theorem~\ref{thm:mr} gives two different forms for the Euler-Lagrange equation on time scales associated with the variational problem \eqref{problem:P}. \begin{theorem}{\rm (The general Euler-Lagrange equations on time scales).} \label{thm:mr} If $\hat{y} \in C_{\diamond}^1$ is a weak local extremizer of problem \eqref{problem:P}, then $\hat{y}$ satisfies the following delta-nabla integral equations: \begin{multline} \label{eq:EL1} \mathcal{J}_\nabla(\hat{y}) \left(\partial_3 L_\Delta[\hat{y}](\rho(t)) -\int_{a}^{\rho(t)} \partial_2 L_\Delta[\hat{y}](\tau) \Delta\tau\right)\\ + \mathcal{J}_\Delta(\hat{y}) \left(\partial_3 L_\nabla\{\hat{y}\}(t) -\int_{a}^{t} \partial_2 L_\nabla\{\hat{y}\}(\tau) \nabla\tau\right) = \text{const} \quad \forall t \in [a,b]_\kappa \, ; \end{multline} \begin{multline} \label{eq:EL2} \mathcal{J}_\nabla(\hat{y}) \left(\partial_3 L_\Delta[\hat{y}](t) -\int_{a}^{t} \partial_2 L_\Delta[\hat{y}](\tau) \Delta\tau\right)\\ + \mathcal{J}_\Delta(\hat{y}) \left(\partial_3 L_\nabla\{\hat{y}\}(\sigma(t)) -\int_{a}^{\sigma(t)} \partial_2 L_\nabla\{\hat{y}\}(\tau) \nabla\tau\right) = \text{const} \quad \forall t \in [a,b]^\kappa \, . \end{multline} \end{theorem} \begin{remark} In the classical context (\textrm{i.e.}, when $\mathbb{T} = \mathbb{R}$) the necessary conditions \eqref{eq:EL1} and \eqref{eq:EL2} coincide with the Euler-Lagrange equations recently given in \cite{Pedregal}. \end{remark} \begin{proof} Suppose that $\mathcal{J}$ has a weak local extremum at $\hat{y}$. We consider the value of $\mathcal{J}$ at nearby functions $\hat{y} + \varepsilon \eta$, where $\varepsilon\in \mathbb{R}$ is a small parameter, $\eta \in C_{\diamond}^{1}([a,b],\mathbb{R})$ with $\eta(a)=\eta(b)=0$. Thus, function $\phi(\varepsilon) = \mathcal{J}(\hat{y} + \varepsilon \eta)$ has an extremum at $\varepsilon = 0$. Using the first-order necessary optimality condition $\left.\phi'(\varepsilon)\right\|_{\varepsilon = 0} = 0$, \begin{multline} \label{eq:prf:+} \mathcal{J}_\nabla(\hat{y}) \int_a^b \left(\partial_2 L_\Delta[\hat{y}](t) \eta^\sigma(t) + \partial_3 L_\Delta[\hat{y}](t) \eta^\Delta(t)\right) \Delta t \\ + \mathcal{J}_\Delta(\hat{y}) \int_a^b \left(\partial_2 L_\nabla\{\hat{y}\}(t) \eta^\rho(t) + \partial_3 L_\nabla\{\hat{y}\}(t) \eta^\nabla(t)\right) \nabla t = 0 \, . \end{multline} Let $A(t) = \int_a^t \partial_2 L_\Delta[\hat{y}](\tau) \Delta\tau$ and $B(t) = \int_a^t \partial_2 L_\nabla\{\hat{y}\}(\tau) \nabla\tau$. Then, $A^\Delta(t) = \partial_2 L_\Delta[\hat{y}](t)$, $B^\nabla(t) = \partial_2 L_\nabla\{\hat{y}\}(t)$, and the first and third integration by parts formula in \eqref{intBP} tell us, respectively, that \begin{equation} \begin{split} \int_a^b \partial_2 L_\Delta[\hat{y}](t) \eta^\sigma(t) \Delta t &= \int_a^b A^\Delta(t) \eta^\sigma(t) \Delta t = \left. A(t) \eta(t)\right\|_{t=a}^{t=b} - \int_a^b A(t) \eta^\Delta(t) \Delta t\\ &= - \int_a^b A(t) \eta^\Delta(t) \Delta t \end{split} \end{equation} and \begin{equation} \begin{split} \int_a^b \partial_2 L_\nabla\{\hat{y}\}(t) \eta^\rho(t) \nabla t &= \int_a^b B^\nabla(t) \eta^\rho(t) \nabla t = \left. B(t) \eta(t)\right\|_{t=a}^{t=b} - \int_a^b B(t) \eta^\nabla(t) \nabla t\\ &= - \int_a^b B(t) \eta^\nabla(t) \nabla t \, . \end{split} \end{equation} If we denote $f(t) = \partial_3 L_\Delta[\hat{y}](t) - A(t)$ and $g(t) = \partial_3 L_\nabla\{\hat{y}\}(t) - B(t)$, then we can write the necessary optimality condition \eqref{eq:prf:+} in the form \begin{equation} \label{eq:prf:+:aftIP} \mathcal{J}_\nabla(\hat{y}) \int_a^b f(t) \eta^\Delta(t) \Delta t + \mathcal{J}_\Delta(\hat{y}) \int_a^b g(t) \eta^\nabla(t) \nabla t = 0 \, . \end{equation} We now split the proof in two parts: we prove \eqref{eq:EL1} transforming the delta integral in \eqref{eq:prf:+:aftIP} to a nabla integral by means of \eqref{eq:DtoN}; we prove \eqref{eq:EL2} transforming the nabla integral in \eqref{eq:prf:+:aftIP} to a delta integral by means of \eqref{eq:NtoD}. By \eqref{eq:DtoN} the necessary optimality condition \eqref{eq:prf:+:aftIP} is equivalent to \begin{equation} \int_a^b \left(\mathcal{J}_\nabla(\hat{y}) f^\rho(t) (\eta^\Delta)^\rho(t) + \mathcal{J}_\Delta(\hat{y}) g(t) \eta^\nabla(t)\right) \nabla t = 0 \end{equation} and by \eqref{eq:chgN_to_D} to \begin{equation} \label{eq:bef:FL1} \int_a^b \left(\mathcal{J}_\nabla(\hat{y}) f^\rho(t) + \mathcal{J}_\Delta(\hat{y}) g(t)\right) \eta^\nabla(t) \nabla t = 0 \, . \end{equation} Applying Lemma~\ref{DBRL:n} to \eqref{eq:bef:FL1} we prove \eqref{eq:EL1}: \begin{equation} \mathcal{J}_\nabla(\hat{y}) f^\rho(t) + \mathcal{J}_\Delta(\hat{y}) g(t) = c \quad \forall t \in [a,b]_\kappa \, , \end{equation} where $c$ is a constant. By \eqref{eq:NtoD} the necessary optimality condition \eqref{eq:prf:+:aftIP} is equivalent to $\int_a^b \left(\mathcal{J}_\nabla(\hat{y}) f(t) \eta^\Delta(t) + \mathcal{J}_\Delta(\hat{y}) g^\sigma(t) \left(\eta^\nabla\right)^\sigma(t)\right) \Delta t = 0$ and by \eqref{eq:chgD_to_N} to \begin{equation} \label{eq:bef:FL2} \int_a^b \left(\mathcal{J}_\nabla(\hat{y}) f(t) + \mathcal{J}_\Delta(\hat{y}) g^\sigma(t)\right) \eta^\Delta(t) \Delta t = 0 \, . \end{equation} Applying Lemma~\ref{DBRL:d} to \eqref{eq:bef:FL2} we prove \eqref{eq:EL2}: \begin{equation} \mathcal{J}_\nabla(\hat{y}) f(t) + \mathcal{J}_\Delta(\hat{y}) g^\sigma(t) = c \quad \forall t \in [a,b]^\kappa \, , \end{equation} where $c$ is a constant. \end{proof} \begin{corollary} Let $L_\Delta\left(t,y^\sigma,y^\Delta\right) = L_\Delta(t)$ and $\mathcal{J}_\Delta(\hat{y}) \ne 0$ (this is true, \textrm{e.g.}, for $L_\Delta \equiv \frac{1}{b-a}$ for which $\mathcal{J}_\Delta = 1$; \textrm{cf.} Remark~\ref{obs}). Then, $\partial_2 L_\Delta = \partial_3 L_\Delta = 0$ and the Euler-Lagrange equation \eqref{eq:EL1} takes the form \begin{equation} \label{cor:EL1} \partial_3 L_\nabla\{\hat{y}\}(t) -\int_{a}^{t} \partial_2 L_\nabla\{\hat{y}\}(\tau) \nabla\tau = \text{const} \quad \forall t \in [a,b]_\kappa \, . \end{equation} \end{corollary} \begin{remark} If $\hat{y} \in C_{\textrm{ld}}^2$, then nabla-differentiating \eqref{cor:EL1} we obtain the Euler-Lagrange differential equation \eqref{eq:el:n} as proved in \cite{NM:T}: \begin{equation} \frac{\nabla}{\nabla t} \partial_3 L_\nabla\{\hat{y}\}(t) - \partial_2 L_\nabla\{\hat{y}\}(t) = 0 \quad \forall t \in [a,b]_{\kappa^2} \, . \end{equation} \end{remark} \begin{corollary} Let $L_\nabla\left(t,y^\rho,y^\nabla\right) = L_\nabla(t)$ and $\mathcal{J}_\nabla(\hat{y}) \ne 0$ (this is true, \textrm{e.g.}, for $L_\nabla \equiv \frac{1}{b-a}$ for which $\mathcal{J}_\nabla = 1$; \textrm{cf.} Remark~\ref{obs}). Then, $\partial_2 L_\nabla = \partial_3 L_\nabla = 0$ and the Euler-Lagrange equation \eqref{eq:EL2} takes the form \begin{equation} \label{cor:EL2} \partial_3 L_\Delta[\hat{y}](t) -\int_{a}^{t} \partial_2 L_\Delta[\hat{y}](\tau) \Delta\tau = \text{const} \quad \forall t \in [a,b]^\kappa \, . \end{equation} \end{corollary} \begin{remark} If $\hat{y} \in C_{\textrm{rd}}^2$, then delta-differentiating \eqref{cor:EL2} we obtain the Euler-Lagrange differential equation \eqref{eq:el:d} as proved in \cite{B:04}: \begin{equation} \frac{\Delta}{\Delta t} \partial_3 L_\Delta[\hat{y}](t) - \partial_2 L_\Delta[\hat{y}](t) = 0 \quad \forall t \in [a,b]^{\kappa^2} \, . \end{equation} \end{remark} \begin{example} \label{ex:first:simp:ex} Let $\mathbb{T}$ be a time scale with $0$, $\xi \in \mathbb{T}$, $0 < \xi$, and $\left(\mathbb{T}\setminus \{0,\xi\}\right)\cap [0,\xi] \ne \emptyset$. Consider the problem \begin{equation} \label{ex:1} \begin{gathered} \textrm{minimize} \quad \mathcal{J}(y)=\left(\int_{0}^{\xi}(y^\Delta(t))^2\Delta t\right) \left(\int_{0}^{\xi}\left(y^\nabla(t))^2\right)\nabla t\right) \, ,\\ y(0)=0, \quad y(\xi)=\xi \, . \end{gathered} \end{equation} Since $L_{\Delta}=(y^\Delta)^2$ and $L_{\nabla}=(y^\nabla)^2$, we have $\partial_2L_{\Delta}=0$, $\partial_3L_{\Delta}=2y^\Delta$, $\partial_2L_{\nabla}=0$, and $\partial_3L_{\nabla}=2y^\nabla$. Using equation \eqref{eq:EL2} of Theorem~\ref{thm:mr} we get the following delta-nabla differential equation: \begin{equation}\label{ex:2} 2Ay^{\Delta}(t)+2By^{\nabla}(\sigma(t))=C, \end{equation} where $C\in\mathbb{R}$ and $A$, $B$ are the values of functionals $\mathcal{J}_{\nabla}$ and $\mathcal{J}_{\Delta}$ in a solution of problem \eqref{ex:1}, respectively. From \eqref{eq:chgD_to_N} we can write equation \eqref{ex:2} in the form \begin{equation} \label{ex:3} 2Ay^{\Delta}(t)+2By^\Delta=C. \end{equation} Observe that $A+B$ cannot be equal to $0$. Thus, solving equation \eqref{ex:3} subject to the boundary conditions $y(0)=0$ and $y(\xi)=\xi$ we get $y(t)=t$ as a candidate local minimizer for the problem \eqref{ex:1}. \end{example} \section{Conclusion} A general necessary optimality condition for problems of the calculus of variations on time scales has been given. The proposed calculus of variations extends the problems with delta derivatives considered in \cite{B:T:08,B:04} and analogous nabla problems \cite{A:T,NM:T} to more general cases described by the product of a delta and a nabla integral. Minimization of functionals given by the product of two integrals were considered by Euler himself, and are now receiving an increasing interest because of their nonlocal properties and their applications in economics \cite{Pedregal}. \section*{Acknowledgments} Work supported by the {\it Centre for Research on Optimization and Control} (CEOC) from the ``Funda\c{c}\~{a}o para a Ci\^{e}ncia e a Tecnologia'' (FCT), cofinanced by the European Community Fund FEDER/POCI 2010. Agnieszka Malinowska is also supported by Bia{\l}ystok University of Technology, via a project of the Polish Ministry of Science and Higher Education ``Wsparcie miedzynarodowej mobilnosci naukowcow''. \end{document}	arXiv
What is the marginal posterior distribution? Asked 1 month ago Based on this question: How to build a Bayesian regression model of a response that is a Gaussian mixture Consider the mixture of normal, $$y_j\sim (N(0,\sigma_1))^{\pi}(N(0, \sigma_2))^{1-\pi}, j=1,2,3,4.$$ My question is what is the conditional distribution of $y_j\|\sigma_1^2, \sigma_2^2, z$. r bayesian mixed-model gaussian-mixture $\begingroup$ If I understand you correctly, then there is something wrong if H is four dimensional while Y is 100 dimensional. $\endgroup$ – Benjamin Christoffersen Nov 30 '20 at 8:52 $\begingroup$ @BenjaminChristoffersen Sorry, that is 100. $\endgroup$ – user261225 Nov 30 '20 at 8:57 Note: In this simplified linear model, the OLS estimator $\hat\beta(y)$ is a sufficient statistic, meaning that the posterior on the parameters is the same given $y$ and given $\hat\beta(y)$. Left graph is a (directed acyclic) graph representing the dependence structure in the model. Right graph is the so-called moral graph associated with it (where parents are linked). It is most useful to find conditional dependencies for building a Gibbs sampler, as a node is independent of everything else given its neighbours, i.e. parents and children. For instance, $\beta$ only depends on $y$, $z$, $X$, and $\sigma=(\sigma_1,\sigma_2)$, but not on $\pi$. $$ \beta\| z, \sigma_1, \sigma_2, y\sim f(\beta\| z, \sigma_1, \sigma_2,y)\propto f(\beta\| z, \sigma_1, \sigma_2)\times f(y\|,\beta,X) $$ Similarly, $z$ only depends on $\pi$, $\sigma$, and $\beta$, and not on $y$. And at last $\pi$ solely depends on $z$,$$f(\pi\|z,\ldots,y)=f(\pi\|z)$$ When considering the full conditional of one component of $\beta$, like $\beta_1$, the density satisfies $$f(\beta_1\|\beta_{-1},z, \sigma_1, \sigma_2, y)\sim f(\beta_1\| z, \sigma_1, \sigma_2,y)\propto f(\beta\| z, \sigma_1, \sigma_2,y)$$ which only depends on $z_1$ (and not $z_2,z_3,z_4$): $$f(\beta_1\|\beta_{-1},z, \sigma_1, \sigma_2, y)\sim f(\beta_1\| z, \sigma_1, \sigma_2,y)\propto f(\beta_1\| z_1, \sigma_1, \sigma_2)\times f(y\|X,\beta)$$ Although this should be considered as a separate question, here are the details when running a full conditional Gibbs sampler on $\beta$: At step 0, start with an arbitrary vector $\beta^{(0)}$ (for instance, the OLS $\hat\beta(y)$, and $\pi^{(0)}$, and generate $z^{(0)}$ from its full conditional distribution. At step t, given the current state $\beta^{(t)},\sigma^{(t)},z^{(t)},\pi^{(t)}$ of the parameter, do update $\beta_1^{(t)}$ into $\beta_1^{(t+1)}$ by simulating from $$f(\beta_1\|\beta_2^{(t)},\beta_3^{(t)},\beta_4^{(t)}, z^{(t)}, \sigma^{(t)},y)\propto f(\beta_1\|z_1^{(t)}, \sigma^{(t)})\times f(y\|\beta_1,\beta_2^{(t)},\beta_3^{(t)},\beta_4^{(t)})$$ update $\beta_2^{(t)}$ into $\beta_2^{(t+1)}$ by simulating from $$f(\beta_2\|\beta_1^{(t+1)},\beta_3^{(t)},\beta_4^{(t)}, z^{(t)}, \sigma^{(t)},y)\propto f(\beta_2\|z_2^{(t)}, \sigma^{(t)})\times f(y\|\beta_1^{(t+1)},\beta_2,\beta_3^{(t)},\beta_4^{(t)})$$ update $\beta_3^{(t)}$ into $\beta_3^{(t+1)}$ by simulating from $$f(\beta_3\|\beta_1^{(t+1)},\beta_2^{(t+1)},\beta_4^{(t)}, z^{(t)}, \sigma^{(t)},y)\propto f(\beta_3\|z_3^{(t)}, \sigma^{(t)})\times f(y\|\beta_1^{(t+1)},\beta_2^{(t+1)},\beta_3,\beta_4^{(t)})$$ update $\beta_4^{(t)}$ into $\beta_4^{(t+1)}$ by simulating from $$f(\beta_4\|\beta_1^{(t+1)},\beta_2^{(t+1)},\beta_3^{(t+1)}, z^{(t)}, \sigma^{(t)},y)\propto f(\beta_4\|z_4^{(t)}, \sigma^{(t)})\times f(y\|\beta_1^{(t+1)},\beta_2^{(t+1)},\beta_3^{(t+1)},\beta_4)$$ Xi'anXi'an $\begingroup$ Thank you very much! $\endgroup$ – user261225 Nov 30 '20 at 19:09 How to build a Bayesian regression model of a response that is a Gaussian mixture Gaussian Mixture: is this plot right? Gibbs sampling for Multivariate: how to update? Gaussian Mixture Model - marginal likelihood Gaussian mixture as a prior of gaussian Ponderate two gaussian mixture Skewness of fitted mixture not correct? MLE of the mixture parameter in mixing two normal densities Simulate from a truncated mixture normal distribution Is the posterior distribution on means in a Bayesian Gaussian mixture model with symmetric priors Gaussian? Generate Posterior predictive distribution at every step in the MCMC chain for a hierarchical regression model Interpretation of posterior distribution for Gelman's Rat Example What is the interpretation of the weights in the GMM? Bayesian inference with false models: to what does it converge?	CommonCrawl
\begin{definition}[Definition:Irrational Number] An '''irrational number''' is a real number which is not rational. That is, an '''irrational number''' is one that can not be expressed in the form $\dfrac p q$ such that $p$ and $q$ are both integers. The set of '''irrational numbers''' can therefore be expressed as $\R \setminus \Q$, where: :$\R$ is the set of real numbers :$\Q$ is the set of rational numbers :$\setminus$ denotes set difference. \end{definition}	ProofWiki
\begin{definition}[Definition:Strictly Midpoint-Convex] Let $f$ be a real function defined on a real interval $I$. $f$ is '''strictly midpoint-convex''' {{iff}}: :$\forall x, y \in I : f \left({\dfrac {x + y} 2}\right) < \dfrac {f \left({x}\right) + f \left({y}\right)} 2$ \end{definition}	ProofWiki
The magnetic ray transform on Anosov surfaces May 2015, 35(5): 1767-1800. doi: 10.3934/dcds.2015.35.1767 On the integrability of polynomial vector fields in the plane by means of Picard-Vessiot theory Primitivo B. Acosta-Humánez 1, , J. Tomás Lázaro 2, , Juan J. Morales-Ruiz 3, and Chara Pantazi 2, Department of Mathematics, Universidad del Atlántico and Intelectual.Co, Barranquilla, Colombia Departament de Matemàtica Aplicada I, Universitat Politècnica de Catalunya, Barcelona, Spain, Spain Department of Applied Mathematics, Technical University of Madrid, Madrid, Spain Received September 2013 Revised September 2014 Published December 2014 We study the integrability of polynomial vector fields using Galois theory of linear differential equations when the associated foliations is reduced to a Riccati type foliation. In particular we obtain integrability results for some families of quadratic vector fields, Liénard equations and equations related with special functions such as Hypergeometric and Heun ones. The Poincaré problem for some families is also approached. Keywords: Poincaré problem, rational first integral, Riccati equation, Differential Galois theory, integrating factor, Liouvillian solution., Liénard equation, Darboux theory of integrability. Mathematics Subject Classification: Primary: 12H05; Secondary: 32S6. Citation: Primitivo B. Acosta-Humánez, J. Tomás Lázaro, Juan J. Morales-Ruiz, Chara Pantazi. On the integrability of polynomial vector fields in the plane by means of Picard-Vessiot theory. Discrete & Continuous Dynamical Systems - A, 2015, 35 (5) : 1767-1800. doi: 10.3934/dcds.2015.35.1767 P. B. Acosta-Humánez, Galoisian Approach to Supersymmetric Quantum Mechanics,, PhD. Thesis, (2009). Google Scholar P. B. Acosta-Humánez, Galoisian Approach to Supersymmetric Quantum Mechanics. 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Intra-cluster and inter-period correlation coefficients for cross-sectional cluster randomised controlled trials for type-2 diabetes in UK primary care James Martin1, Alan Girling1, Krishnarajah Nirantharakumar1, Ronan Ryan1, Tom Marshall1 & Karla Hemming1 Trials volume 17, Article number: 402 (2016) Cite this article Clustered randomised controlled trials (CRCTs) are increasingly common in primary care. Outcomes within the same cluster tend to be correlated with one another. In sample size calculations, estimates of the intra-cluster correlation coefficient (ICC) are needed to allow for this nonindependence. In studies with observations over more than one time period, estimates of the inter-period correlation (IPC) and the within-period correlation (WPC) are also needed. This is a retrospective cross-sectional study of all patients aged 18 or over with a diagnosis of type-2 diabetes, from The Health Improvement Network (THIN) database, between 1 October 2007 and 31 March 2010. We report estimates of the ICC, IPC, and WPC for typical outcomes using unadjusted and adjusted generalised linear mixed models with cluster and cluster by period random effects. For binary outcomes we report on the proportions scale, which is the appropriate scale for trial design. Estimated ICCs were compared to those reported from a systematic search of CRCTs undertaken in primary care in the UK in type-2 diabetes. Data from 430 general practices, with a median [IQR] number of diabetics per practice of 241 [150–351], were analysed. The ICC for HbA1c was 0.032 (95 % CI 0.026–0.038). For a two-period (each of 12 months) design, the WPC for HbA1c was 0.035 (95 % CI 0.030–0.040) and the IPC was 0.019 (95 % CI 0.014–0.026). The difference between the WPC and the IPC indicates a decay of correlation over time. Following dichotomisation at 7.5 %, the ICC for HbA1c was 0.026 (95 % CI 0.022–0.030). ICCs for other clinical measurements and clinical outcomes are presented. A systematic search of ICCs used in the design of CRCTs involving type-2 diabetes with HbA1c (undichotomised) as the outcome found that published trials tended to use more conservative ICC values (median 0.047, IQR 0.047–0.050) than those reported here. These estimates of ICCs, IPCs, and WPCs for a variety of outcomes commonly used in diabetes trials can be useful for the design of CRCTs. In studies with observations taken at different time-points, the correlation of observations may decay over time, as reflected in lower values for the IPC than for the ICC. The IPC and WPC estimates are the first reported for UK primary care data. Diabetes is an important public health issue [1] and an increasing number of clinical trials are being conducted to improve care for patients with diabetes. Increasingly, interventions aimed at improving the quality of care are evaluated using cluster randomised controlled trials (CRCTs) [2–5]. Whilst observations used in the evaluation may still be made at the individual level, randomisation at the cluster level (such as GP surgery) will often be necessary [5–7] and is increasingly being used [8]. In CRCTs patients within the same cluster tend to more similar than patients from differing clusters [7, 9]. Thus, the observations within a cluster may not be independent, and the design and analysis of CRCTs should acknowledge this [5, 10–13]. Important outcomes in trials of diabetes include clinical measurements, such as glycosylated haemoglobin (HbA1c) (both as a continuous and dichotomised outcome) [14], body mass index (BMI) [15], cholesterol [16], blood pressure [17], or the incidence of macrovascular and microvascular outcomes [18, 19]. Sample size calculations for an individually randomised controlled trial (RCT) are relatively straightforward, but for a CRCT it is necessary to account for the nonindependence [10–12]. A design effect can be used to inflate the sample size of an RCT to that required in a CRCT [9, 20]. For a trial with equal cluster sizes, the design effect is calculated as: $$ 1+\left(m-1\right)\rho . $$ Here m is the cluster size and ρ is the correlation between patients within a cluster [21]. This correlation has important implications for the sample size required [22, 23]. The majority of CRCTs have a parallel design. That is to say, clusters are allocated to either intervention or control. However, increasingly, the value of alternative cluster designs is being appreciated. Some alternative designs include the cluster cross-over [24], the stepped wedge [25, 26], and the dog-leg [27, 28]. In these alternative designs repeated cross-sectional samples are taken from each cluster over multiple time periods. It is becoming increasingly recognised that observations from the same cluster and same period are likely to be more highly correlated than observations in the same cluster but at different periods [29–32]. This leads to the notion of a within-period cluster correlation (WPC) and an inter-period cluster correlation (IPC). Unfortunately, there is little or no empirical literature to inform likely values for these parameters at the design stage [28, 29]. For a trial to be powered correctly, an accurate estimate of the correlation of observations within a cluster is required. In the past, many type-2 diabetes trials in primary care have failed to report this correlation, forcing many planned trials to use ad hoc values at the design stage [33]. This leads to inaccurate sample size estimates and (sometimes) to underpowered trials. Typically, this correlation is assumed to be time independent – and a single intra-cluster correlation coefficient (ICC) is used in the sample size calculation. This assumption may not always be valid. For designs with observations taken over multiple time periods, estimates of the WPC and IPC are vital in the sample size calculation [28, 29]. These can be obtained from routinely collected data, in a similar way to ordinary ICCs [34, 35]. Our objective here is to estimate ICCs for typical trial outcomes related to type-2 diabetes using anonymised patient data from The Health Improvement Network database [36]. We additionally report estimates of the WPC and the IPC for a subset of continuous outcomes. Finally, we review previous CRCTs in type-2 diabetes to compare the ICCs estimated in this paper to those previously used. Correlation of observations in a cluster trial The quantity ρ in Eq. 1 is defined as the correlation between two randomly selected observations within the same cluster. Typically, an assumption is made that this correlation is independent of the timing of the observations. This property is consistent with a decomposition of the total variance into two independent components representing variation between clusters and between subjects (within clusters). In view of this, the ICC can be defined as the proportion of the variance that is attributable to the between-cluster variance, given as: $$ \frac{{\sigma_b}^2}{{\sigma_b}^2+{\sigma_w}^2}, $$ where σ b 2 and σ w 2 represent the between- and within-cluster variance components. Cluster trials are typically analysed using a multilevel linear model. If the correlation between observations in a cluster is independent of when they are taken, an approach using the ratio of variances is a simple method to estimate the ICC. This approach is taken throughout the paper whenever an estimated ICC is reported. Time-dependent correlation In some contexts, a model based on the assumption of time-independent correlations is flawed. An alternative model can be fitted to the data by splitting time into a number of (equal) periods. In this formulation, constant correlations are assumed: (1) for any two observations in the cluster from the same time period (WPC); and (2) for any two observations from the same cluster in different time periods (IPC). These assumptions are consistent with a variance-decomposition into three independent components: between clusters (σ e 2); between time periods (within clusters) (σ c 2); and between subjects (within time period and cluster) (σ t 2). Now, the WPC is the correlation of observations between two patients in the same cluster from the same time period. This can be calculated as: $$ \frac{{\sigma_c}^2+{\sigma_t}^2}{{\sigma_c}^2+{\sigma_e}^2+{\sigma_t}^2}. $$ The IPC is the correlation of observations between two patients in the cluster from different time periods, and is calculated as: $$ \frac{{\sigma_c}^2}{{\sigma_c}^2+{\sigma_e}^2+{\sigma_t}^2}. $$ In this framework, the correlation, ρ, between two randomly selected observations within the same cluster is given by a within-cluster correlation (WCC) defined by: $$ WCC=IPC+\frac{1}{n_{tp}}\left(WPC-IPC\right). $$ Here n tp is the number of time periods in the study. It is assumed that each time period contains an equal number of observations. The ratio of the IPC to the WPC is known as the cluster autocorrelation (CA), which is the correlation between the cluster level mean outcome over time [28]. The cluster autocorrelation has been established as key to sample size formula for studies with a repeated cross-sectional design [37]. We present estimates of the CA alongside the IPC and WPC. In the absence of period effects, the CA = 1, indicating that the time-dependent model is unnecessary. In this setting, WCC = WPC = IPC. Otherwise it follows from the definitions that WPC > WCC > IPC. Correlation of binary outcomes In the context of a clinical trial, data are often dichotomous – recording the presence or absence of a particular clinical outcome. The ICC that appears in the design effect is then defined as the correlation between two binary outcomes from two patients in the same cluster. In such cases, sample size calculations will typically entail a normal approximation to the binomial distribution which describes the number of positive outcomes in a sample of fixed size. Nevertheless the analysis of dichotomous outcomes in cluster trials is often conducted via a multilevel logistic model. In such models the observed binary outcome may be conceptualised as having arisen by dichotomising a continuous latent scale. When these models are fitted in some analysis packages (e.g. Stata) a type of ICC is presented which relates not to the observed binary outcomes but to this unobservable latent scale. It takes the form: $$ \frac{{\sigma_b}^2}{{\sigma_b}^2+{\pi}^2/3}, $$ where σ b 2 is the between-cluster component of variance on the latent scale and the term π 2/3 is associated with the logistic distribution used to generate the binary model. Since this version of the ICC refers to the unobservable latent scale, rather than the correlation between the binary outcomes of two patients from within the same cluster, this ICC should not be used directly to compute design effects for sample size calculations. In principle, a latent ICC from a logistic regression model can be converted to a natural ICC on the proportion scale for the raw binary data, taking account of the prevalence of the outcome – see, for example, the table presented by Eldridge et al. [21]. Throughout this paper we maintain the distinction between a natural ICC on the proportion scale and a latent ICC for binary data. It is the natural ICC on the proportion scale that contributes to the calculation of design effects. Outcome variables The aim was to investigate the correlation of all routinely recorded variables that might be clinically relevant to a trial undertaken in type-2 diabetes. The outcome variables were divided into three categories: clinical measures, medication, and clinical outcomes. Clinical measures included HbA1c, systolic blood pressure, diastolic blood pressure, BMI, total cholesterol level, and high-density lipoprotein (HDL) cholesterol level. Medication measurements involved insulin and other hypoglycaemic medications. The clinical outcomes were a first diagnosis of: atrial fibrillation, chronic kidney disease, chronic obstructive pulmonary disease (COPD), ischaemic heart disease (IHD), peripheral vascular disease, and stroke. Patients who had suffered an event prior to the study were excluded from the analysis for that outcome. Dichotomisation of continuous outcomes In practice, many trials use dichotomised values of continuous outcome measures [38, 39], and so we generated dichotomised values for each continuous outcome. A threshold value of 7.5 % was chosen for HbA1c as NICE guidelines state that 7.5 % indicates inadequate control [40], in addition to being used in previous studies [41]. Multiple recommendations have been made that total cholesterol levels should be below 4.0 mmol/L and HDL cholesterol levels be above 1.2 mmol/L [42, 43]. Two relevant cut-points were used for both systolic blood pressure and BMI. For systolic blood pressure, a value of 140 mmHg is the upper limit recommended for patients with type-2 diabetes [40]. A lower value of 130 mmHg is the target that health care professionals aim to reduce systolic blood pressure to in patients who suffer from kidney and eye problems, or those who have suffered a stroke [40]. Two cut-points were chosen for BMI to correspond to the categories of overweight (25 kg/m2) and moderately obese (30 kg/m2). Measurement periods A cross-sectional sample of measurements taken over a 15-month period was used (1 January 2009 to 31 March 2010), to reflect the NICE quality and outcomes framework (QOF) [44], which monitors measurements taken for patients over a 15-month period. To estimate the IPC and WPC an additional 15 months (1 October 2007 to 31 December 2008) of data is used to estimate the time-dependent correlation, creating two 15-month time periods. Since the measuring unit of HbA1c changed in 2009 from % to mmol/mol, the consistency in reporting is likely to be poor around this time. In view of this, we consider a slight variation, and a cross-sectional sample of measurements taken over a 12-month period was used (1 January 2008 to 31 December 2008). An additional 12 months (1 January 2007 to 31 December 2007) of data contributes towards the estimation of the IPC and WPC. The Health Improvement Network The retrospective cross-section of patients with type-2 diabetes was formed using data from The Health Improvement Network (THIN) database [36]. Participating general practices contributed anonymised demographics, prescribing information, and clinical data for more than 3.7 million patients throughout the UK. All practices used the Vision computer system. All patients over 18 years of age were included if a diagnosis of type-2 diabetes, indicated by the appropriate 'Read codes', was made before the study index date. Read codes are a coded thesaurus of clinical terms that are used in the recording of patient data in primary care electronic medical records in the UK. The general practices were required to have been using the Vision computer system for a minimum of a 1 year period prior to the study index date, and to have an acceptable mortality reporting (AMR) date (an indicator of practice quality) [45]. The included population was summarised by describing both patient and practice characteristics using appropriate summary statistics. General practice characteristics include the total number of practices, location (country) of the practice, and practice inclusion size (the number of patients from each practice satisfying the entry criteria). Patient characteristics (of the included population) were age (years), gender, location (country of residence), and deprivation quintiles. We also summarised potential trial outcomes using suitable summary statistics. Outcomes included clinical measures, onset of clinical outcomes, and the prescription of medication. Although the HbA1c variable exhibits skewness, both mean and median values were given as it is assumed to be normally distributed in many trials. Variation across practices in mean (or median) clinical measures, clinical outcomes, and the prescription of medication, was summarised by reporting the interquartile range (IQR) of the practice mean (or median) values. Statistical models Generalised linear mixed models were used to estimate the ICCs with cluster (general practice) modelled as the random effect. Both adjusted and unadjusted ICCs were estimated, with adjustments made for age, sex, location, and deprivation quintiles. All clinical measures were presented in both continuous and dichotomised form. For continuous outcomes, a mixed-effects linear model was fitted and the ICC was estimated as the ratio of the between-cluster variance (of the outcome) to the total variance of the outcome. For binary outcomes, a mixed-effects linear model was fitted to estimate the natural ICC on the proportion scale, whilst a mixed-effects logistic regression was fitted to estimate the latent ICC. To estimate the WPC, IPC, and CA, a generalised linear mixed model was used, with two random effects – one for cluster (general practice) and one for a cluster by period interaction. All analysis was performed using Stata 13 (StataCorp, College Station, TX, USA). Linear models were fitted using the mixed command, and logistic models fitted using the melogit command. Estimates of the ICC, WPC, and IPC were produced using the estat function. Search of previous CRCTs A systematic search of previous CRCTs investigating diabetes in primary care in the UK was carried out in order to compare the results from this analysis to values used in previous CRCTs. The following sources were used: Medline (1950 to week 2 of May 2013), Medline InProcess (May 2013), and Google Scholar (May 2013). The searches were conducted in May 2013. The following phrases were used: type-II diabetes, type-2 diabetes, diabetes mellitus, diabetes mellitus non-insulin-dependent, adult-onset diabetes mellitus, cluster trial, clustered trial, cluster analysis, cluster analyses, clustering, disease clustering, cluster RCT, and cluster randomised (randomized) controlled trial. The search was limited to the English language. Studies from all fields of research were included if they described a CRCT that had taken place, or was planned to take place, that used UK general practices as the unit of randomisation. Studies were included if at least one of the trial outcomes were: HbA1c levels, systolic blood pressure, diastolic blood pressure, BMI, total cholesterol, HDL cholesterol, the prescription of insulin, or the onset of microvascular and macrovascular outcomes. Since the focus is on the ICCs used in the design of a CRCT, all trials in which individuals were the unit of randomisation were excluded from the study. All trials that did not take place in the UK were also excluded since ICC estimates may be affected by the country in which the trial is taking place. All trials with unspecified outcomes were excluded. Trials that aim to prevent the onset of diabetes were also excluded. Any duplicate or follow-on publications from the same trial were included as a single study. Titles and abstracts retrieved from the search process were screened to obtain relevant trials. Full articles were then read and classified as either included or excluded. All included articles were then used for data extraction. The extracted information consisted of: study authors, outcome used, value of ICC used in the sample size calculation, standard deviation used in the sample size calculation (where appropriate), and the ICC estimated from the trial data (if reported). Analysis of THIN data A summary of patient and practice characteristics is given in Table 1. A total of 112,633 patients from 430 practices covering all areas of the UK, were included in the study. The socioeconomic status was fairly balanced across the categories. The median value of HbA1c (%) (7.05) was lower than the mean value (7.35), highlighting the positive skewness that is exhibited by the variable. Atrial fibrillation was the most common clinical outcome (1.06 %), whilst chronic kidney disease was the least common (0.35 %). Table 1 Summary of study population (THIN database) by practice and patient-level characteristics Table 2 summarises the proportion of patients whose clinical measures exceed the dichotomised value of the outcomes. Of the participants with a recording for HbA1c, over one third (34.2 %) had an HbA1c % exceeding 7.5 %. It was also found that over one half (57.2 %) exceeded the target systolic blood pressure of 130 mmHg whilst approximately one quarter (25.2 %) exceeded 140 mmHg. A large proportion (83.1 %) of the population were categorised as being overweight (>25 kg/m2) (34.8 %), obese (>30 kg/m2) (27.3 %), or morbidly obese (>35 kg/m2) (21.0 %). Table 2 Summary statistics for clinical measures of included patients from THIN database in binary form The variation of both the clinical outcomes and clinical measures across practices is given in Table 3. The interquartile range represents the practice mean outcome for the central 50 % of practices. ICC estimates and corresponding standard errors (SE) for clinical measures of continuous nature are given in Table 4 and compared further in Fig. 1. For clinical measurements, in continuous form, the ICCs had a median of 0.026 [IQR 0.020–0.032] and were similar when adjusting for confounding factors (median 0.025, IQR 0.020–0.029). The ICC for HbA1c was estimated to be 0.032 (SE 0.003) when using an unadjusted model and 0.032 (SE 0.003) after adjustment for patient-level factors. Table 3 Summary of the variation of practice average values from included patients from THIN database Table 4 Intra-cluster correlation coefficients (ICCs) for continuous outcomes for included patients from THIN database Box plot highlighting the median, interquartile range, and range of the intra-cluster correlation coefficients (ICCs) that were estimated for continuous and binary clinical outcomes from both linear and logistic models (n = number of outcomes that had estimate of the ICC) After dichotomising, the ICCs of clinical measures had a median latent ICC of 0.037 [IQR 0.023–0.055] and a median natural ICC on the proportion scale of 0.028 [IQR 0.018–0.039]. Clinical outcomes had a median latent ICC of 0.094 [IQR 0.027–0.136] and a median natural ICC on the proportion scale of 0.003 [IQR 0.001–0.005]. When comparing two clinical outcomes with similar prevalence, it is expected that the outcome with a larger IQR of the practice average would have a larger ICC. This is consistent with the larger natural and latent ICCs (Table 5) that are associated with COPD compared to IHD, both of which have a prevalence of around 1 % (Table 1). Figure 1 further highlights that latent ICCs were larger than natural ICCs on the proportion scale for binary outcomes, but also that the range of latent ICCs is higher than natural ICCs. Table 5 Intra-cluster correlation coefficients (ICCs) for binary outcomes for included patients from THIN database Estimates of the WPC, IPC, and CA for the two-period study design are given in Table 6. For HbA1c, the correlation between two patients during the same (12-month) time period (WPC) was estimated at 0.035 (SE 0.003). The correlation between two patients at different (12-month) time periods (IPC) is 0.019 (SE 0.003). There is evidence to suggest that the variance component related to time period is non-zero, and so the correlation of observations seems to decay over time. Excluding HbA1c, in the two-period (each of 15 months) design, the decay of correlation is further highlighted by the median WPC (0.021, IQR 0.021–0.032) and median IPC (0.018, IQR 0.013–0.021). Table 6 Estimates of the within-period and inter-period correlation for included patients from THIN database from two consecutive periods The median cluster autocorrelation (excluding HbA1c) is 0.649 [IQR 0.612–0.692], with total cholesterol having the smallest value – indicating that correlation of total cholesterol observations for patients in different time periods is much smaller than the correlation of observations in the same time period. Adjusting for covariates had some impact on correlation estimates. For total cholesterol, the CA in the adjusted model (0.281) was much lower than the unadjusted model (0.486). Conversely, HbA1c had much higher CA in the adjusted model (0.747) than in the unadjusted model (0.612). Systematic search Our search strategy found 133 relevant articles. From this, 70 articles were of irrelevant outcome or trial type (individually randomised design, genetics of diabetes, cross-sectional studies, etc.), 36 were excluded due to the population of the trials (not of UK origin), 7 articles were screening programmes, 6 aimed to prevent diabetes, and 2 articles were excluded as they measure prevalence of diabetes. Of the 12 trials remaining, 3 duplicates were removed, leaving 9 articles that met the inclusion criteria (see Additional file 1). One CRCT used the cluster as unit of randomisation but did not use an ICC when calculating sample size [46]. Of the remaining eight CRCTs, two CRCTs [39, 47] used multiple outcomes and calculated sample sizes for each outcome of relevance. Seven CRCTs [14, 39, 47–51] used HbA1c as an outcome measure, three [38, 39, 47] used systolic blood pressure, and two [39, 47] used cholesterol. However, cholesterol was not used as a sole outcome measure, only as secondary measure alongside both HbA1c and blood pressure. Of these eight CRCTs, two [38, 39] used a binary outcome, and seven [14, 39, 47–51] used a continuous outcome (one used both a binary and continuous outcome [39]). The median [IQR] ICC used to power the study for trials in which HbA1c % was the primary outcome was 0.047 [0.047–0.05] (Table 7). The two CRCTs [39, 47] in which total cholesterol (mmol/L) was the main outcome used 0.047 and 0.06 (binary outcome) as the ICC whilst the three CRCTs using blood pressure (mmHg) as the main outcome [38, 39, 47] used ICCs of 0.001 (binary outcome), 0.02 (binary outcome), and 0.035. The standard deviation of HbA1c % used was reported in six trials [14, 39, 47, 49–51], of which the mean value was 1.7. The results of this paper found a similar standard deviation of 1.4 for HbA1c %, whereas the ICC found by this paper was lower (0.032 versus 0.047). Table 7 Summary of systematic search of intra-cluster correlation coefficients (ICCs) used in previous trials Only three trials reported ICCs from their analysis [14, 38, 48]. Two trials reported ICCs for HbA1c % [14, 48], with ICCs of 0.0253 and 0.02 (95 % CI 0.00–0.08), and one trial [38] reported an ICC for blood pressure of 0.035. For the two trials that reported the ICC, the reported value was lower than the value used in the initial sample size calculation, whilst for blood pressure the reported value was notably higher. However, for the trial that estimated an ICC for blood pressure [38], it was not clear what method was used to estimate this value. Using THIN database, we have estimated ICCs for a variety of outcomes associated with type-2 diabetes. We are the first to report time-dependent correlations, the IPC and WPC, which can be used in the design of cluster cross-over and stepped wedge CRCTs. For binary outcomes, we reported both the latent ICC (an ICC from a logistic model) and the natural ICC on the proportion scale (an ICC from a linear model). These results are primarily applicable for planned CRCTs aimed at the general practice level in the UK, but in the absence of other estimates, may be useful more widely. We found that the ICC for HbA1c used in the design of trials tended to be larger than that estimated here. Intra-cluster correlation coefficients ICCs were calculated for continuous and dichotomous clinical measurements and outcomes, using both adjusted and unadjusted models. This includes ICCs for continuous outcomes and ICCs for binary outcomes. Upon adjusting for age, sex, location, and deprivation quintiles, the ICCs were generally similar to the ICCs estimated from the unadjusted models (HbA1c 0.032 versus 0.032). Adjusting for confounding factors also had minimal impact on the standard error of the ICCs (HbA1c 0.003 versus 0.003). There was a noticeable difference between natural ICCs and latent ICCs for binary outcomes. Latent ICCs estimated for clinical events were much larger than their corresponding natural ICC. Similar results were found by Wu et al. [52], who found that ICCs were smaller when modelled using linear regression than logistic regression. For binary outcomes it is important to note that natural ICCs (an ICC from a linear model) are smaller for cases in which the prevalence's are low [35, 53]. Here all clinical outcomes chosen were rare events and consequently had small prevalence's. Since the dichotomised values were chosen to reflect typical values in relation to type-2 diabetes, the prevalence's of these were naturally larger – resulting in a larger ICC. Due to the importance of the prevalence on the natural ICC, care should be taken to ensure that an appropriate ICC is used. If the prevalence in a planned trial differs greatly from the prevalence used here, sample size calculations using the natural ICC from these results may be inaccurate Since latent ICCs for dichotomous outcomes, are estimated using logistic regression, they are on a log-odds scale and so are defined on a different scale to a natural ICC [35, 52]. A latent ICC estimated in this manner will refer to an unobservable latent scale, rather than the correlation of observations within a cluster, and so would not be a relevant ICC for use in the design stage of a trial. Eldridge et al. [21] provide a table that allows some ICCs on this logistic scale to be converted into a natural ICC for a selection of prevalence's. Previous trials Many authors discuss the most appropriate methods and models that should be used to model ICCs in situations in which the outcome is binary [35, 52, 54], and there are numerous cases in which previous authors have correctly estimated ICCs for binary outcomes using linear models for future trialists to use [34, 55–57]. However, there are still some situations where a logistic model is used [58–60]. The differences between the natural ICC and the latent ICC are also considered by Merlo et al. [61] who note that since the natural ICC depends on the prevalence of the outcome; any comparisons made regarding the magnitude of clustering should be made using the latent ICC. We agree that that care should be taken when using the natural ICC to describe the extent of clustering in a trial with binary outcomes; however, we cannot recommend that the latent ICC is used directly in the design of future trials. The number of previous cluster trials involving type-2 diabetes that have reported ICCs from their results is rather small, which will leave future trialists using ad hoc values or conservative values. The ICCs found in this paper were smaller than that often used in trials, but more consistent with the ICCs that were reported from the results of previous trials. The ICC for HbA1c %, the most common outcome in a trial involving type-2 diabetes, was found to be 0.032 (SD 0.003). Trials in which the primary outcome is binary should use an ICC from a linear model when estimating a required sample size, and not one obtained from a logistic model, even if the data will be analysed using a logistic model. Inter-period correlation coefficients It is emerging that cluster designs require not only estimates of within-cluster correlation measures, but some value of how this correlation decays over time [29, 62]. We have attempted in part to address this issue and are the first to provide estimates of the inter-period correlation and the within-period correlation alongside ICCs. However, we have only provided these estimates for continuous outcomes and we have only provided estimates assuming a cross-sectional study design. Clearly, many studies use a cohort design and many studies contain a primary outcome that is dichotomous in nature. However, estimation of correlation coefficients for binary outcomes are more complex due to the change of scale; and adding a cohort structure would increase complexity, as it would also be necessary to allow for within-person correlation. The IPC and WPC may also be reported as the CA. It has been established that the sample size is directly impacted by the CA [37]. No guidelines exist for reasonable values of the CA, but values of 0.8 and 1.0 have previously been used [28, 63]. Here we have shown that for our study design, the CA may be smaller than these estimates. Ignoring the IPC and CA in sample size calculations may lead to incorrect estimates of the required number of clusters in a CRT [29] or to underpowered studies [28]. Studies in which the IPC differs to the WPC should ensure that the estimates of ρ for use in Eq. 1 stem from the WCC estimated via Eq. 3, and not from an ICC estimated by Eq. 2. It has been established that the ICC, IPC, and CA are necessary for sample size calculations for CRCTs. However, there is opportunity for future research into the IPC and the impact of time between observations in the model for CRCTs. It is perhaps naïve to assume a fixed correlation between observations in a cluster trial regardless of the time between these. Instead, this correlation should depend on time, and this length of time may be important. It is not known what impact changing the length of time period or the length of the study period would have on the IPC. Additionally, the IPC used to direct a sample size calculation should be calculated from a dataset using a similar time period and study length. The motivating idea behind additional correlation types is repeated cross-sectional designs such as the cluster cross-over design and the stepped wedge design. However, these results may indicate that sample size in parallel CRCTs should also acknowledge that correlation may be time-dependent. Future research is likely to show that recognising the decay in correlation over time in the model would increase power in parallel designs. There are limitations that may arise from using routine data from general practices. It is not always possible to distinguish between follow-up care for a first clinical event (e.g. myocardial infarction) from a second event as they may have been coded in an identical manner. This means that patients who had suffered an event prior to the study inclusion period would have to be excluded from the analysis. There is also the possibility of misclassification as type-2 diabetes rather than type-1 diabetes due to coding errors, which could lead to younger patients being included in the study unintentionally. Since THIN dataset consists of data from general practices only, the results can only be adjusted for variables that are recorded by the practice. The quality of service may vary between practices and so there may be situations in which clinical measures are monitored in different intervals which, along with quality of reporting and recording of measurements, could lead to an inconsistency. Although the reporting of clinical measures during the 15-month cross-section that was chosen as the inclusion period was high, the length of the cross-section may not accurately represent the length of trials in practice. An estimate of the ICC is vital when calculating the sample size requirement in a pretrial calculation [21]. We estimated ICCs for a range of clinical outcomes related to type-2 diabetes that would be useful for planning a trial in UK primary care. The primary outcome used in type-2 diabetes trials is often HbA1c, for which we estimated an ICC of 0.032. We have also illustrated how the methodology described here could be extended for other outcomes or disease settings. For binary outcomes, the results show careful consideration is needed when estimating the ICC. This is because, in a trial with a dichotomous outcome, the ICC used at the design stage should refer to the variation in the observed data rather than the underlying logistic scale. Despite the analysis of binary outcomes being usually conducted via a logistic regression model, the latent ICC obtained from such model should not be used for sample size calculations. Rather, the ICC used in the design stage of a trial should be estimated from a linear mixed model on the natural scale. In cluster trials with repeated cross-sections, observations are taken over multiple time periods. It is likely that observations within a cluster within the same time period are more highly correlated than observations from different time periods. The inter-period correlation and within-period correlation provides an estimate of how this correlation deteriorates over time. We are the first to report estimates of the IPC and WPC and we have illustrated how these differ from the ICC. It may be important to acknowledge the degeneration of correlation over time in repeated cross-sectional studies. 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J Clin Epidemiol. 2004;57(8):785–94. doi:10.1016/j.jclinepi.2003.12.013. Roudsari B, Fowler R, Nathens A. Intracluster correlation coefficient in multicenter childhood trauma studies. Inj Prev. 2007;13(5):344–7. doi:10.1136/ip.2007.015313. Thompson DM, Fernald DH, Mold JW. Intraclass correlation coefficients typical of cluster-randomized studies: estimates from the Robert Wood Johnson Prescription for Health projects. Ann Fam Med. 2012;10(3):235–40. doi:10.1370/afm.1347. Kul S, Vanhaecht K, Panella M. Intraclass correlation coefficients for cluster randomized trials in care pathways and usual care: hospital treatment for heart failure. BMC Health Serv Res. 2014;14:84. doi:10.1186/1472-6963-14-84. Moineddin R, Matheson FI, Glazier RH. A simulation study of sample size for multilevel logistic regression models. BMC Med Res Methodol. 2007;7:34. doi:10.1186/1471-2288-7-34. Turner RM, Omar RZ, Thompson SG. Bayesian methods of analysis for cluster randomized trials with binary outcome data. Stat Med. 2001;20(3):453–72. Merlo J, Chaix B, Ohlsson H, Beckman A, Johnell K, Hjerpe P, et al. A brief conceptual tutorial of multilevel analysis in social epidemiology: using measures of clustering in multilevel logistic regression to investigate contextual phenomena. J Epidemiol Community Health. 2006;60(4):290–7. doi:10.1136/jech.2004.029454. Giraudeau B, Ravaud P, Donner A. Sample size calculation for cluster randomized cross-over trials. Stat Med. 2008;27(27):5578–85. doi:10.1002/sim.3383. Hussey MA, Hughes JP. Design and analysis of stepped wedge cluster randomized trials. Contemp Clin Trials. 2007;28(2):182–91. doi:10.1016/j.cct.2006.05.007. JM is supported by a University of Birmingham-funded PhD. KH and AG acknowledge financial support for the submitted work from the National Institute for Health Research (NIHR) Collaborations for Leadership in Applied Health Research and Care (CLAHRC) for West Midlands. KH and AG also acknowledge financial support from the Medical Research Council (MRC) Midland Hub for Trials Methodology Research (grant number G0800808). KH, AG, and JM conceived of the study. JM carried out the data analysis of THIN data, conducted the systematic search, extracted data and performed the data analysis. JM wrote the first draft of the manuscript. KN, RR, TM, AG, and KH commented on drafts of the manuscript and provided an interpretation of the results. All authors read and approved the final manuscript. Institute of Applied Health Research, University of Birmingham, Birmingham, B15 2TT, UK , Alan Girling , Krishnarajah Nirantharakumar , Ronan Ryan , Tom Marshall & Karla Hemming Search for James Martin in: Search for Alan Girling in: Search for Krishnarajah Nirantharakumar in: Search for Ronan Ryan in: Search for Tom Marshall in: Search for Karla Hemming in: Correspondence to James Martin. Additional file Flow diagram of included trials for systematic search of trials undertaken in primary care in type-2 diabetes. (PNG 17 kb) Martin, J., Girling, A., Nirantharakumar, K. et al. Intra-cluster and inter-period correlation coefficients for cross-sectional cluster randomised controlled trials for type-2 diabetes in UK primary care. Trials 17, 402 (2016) doi:10.1186/s13063-016-1532-9 DOI: https://0-doi-org.brum.beds.ac.uk/10.1186/s13063-016-1532-9 Inter-period correlation coefficient Cluster randomised trial By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate. Please note that comments may be removed without notice if they are flagged by another user or do not comply with our community guidelines. General enquiries: [email protected]	CommonCrawl
rational function identity I just had to make use of an elementary rational function identity (below). The proof is a straightforward exercise, but that isn't the point. First, "my" identity is almost surely not original, but I don't have a reference for it. Perhaps someone knows it (like a lost cat without a collar) or, more likely, could spot this as a special case of a more general identity. Second, the obvious proof is not much of an explanation: a combinatorial identity often arises for a conceptual reason, and I'd be happy to hear if anyone sees mathematics behind this one. Let $f(x_1,\ldots,x_n)=\prod_{p=1}^n\big(\sum_{i=p}^n x_i\big)^{-1}$. Then $$ f(x_1,\ldots,x_n)+f(x_2,x_1,x_3,\ldots,x_n)+\cdots+f(x_2,\ldots,x_n,x_1)=\big(\sum_{i=1}^n x_i\big)/x_1\cdot f(x_1,\ldots,x_n), $$ where $x_1$ appears as the $i$th argument to $f$ in the $i$th summand on the left side, for $1\leq i\leq n$. But why? co.combinatorics reference-request Graham DenhamGraham Denham $\begingroup$ +1 for the lost cat. See, mathematics is not only about finding black cats in dark rooms - sometimes it is about finding the owner. $\endgroup$ – darij grinberg Aug 31 '11 at 13:55 $\begingroup$ Also on math.se: math.stackexchange.com/questions/4220/…. $\endgroup$ – Yuval Filmus Nov 15 '15 at 22:53 I have seen a cat of a similar breed in the representation theory of symmetric groups. Out of habit, let me quote a lemma attributed to Littlewood in Donald Knutson, $\lambda$-rings and the Representation Theory of the Symmetric Group, Springer 1973 (LNM #308), Chapter III, section 2, p. 149: $\sum\limits_{\sigma\in S_n} f\left(x_{\sigma\left(1\right)},x_{\sigma\left(2\right)},...,x_{\sigma\left(n\right)}\right) = \frac{1}{x_1x_2...x_n}$. At the moment, neither does this cat imply yours, nor the other way round. But can we cross them? Let me try. The left paw side of your cat is $\sum\limits_{\sigma\in \mathrm{Sh}\left(1,n-1\right)} f\left(x_{\sigma^{-1}\left(1\right)},x_{\sigma^{-1}\left(2\right)},...,x_{\sigma^{-1}\left(n\right)}\right)$, where $\mathrm{Sh}\left(a,b\right)$ is defined as the subgroup $\left\lbrace \sigma \in S_{a+b} \mid \sigma\left(1\right) < \sigma\left(2\right) < ... < \sigma\left(a\right) \text{ and } \sigma\left(a+1\right) < \sigma\left(a+2\right) < ... < \sigma\left(a+b\right) \right\rbrace$ of the symmetric group $S_{a+b}$. (The elements of this subgroup $\mathrm{Sh}\left(a,b\right)$ are known as $\left(a,b\right)$-shuffles.) Now I suspect tat $\sum\limits_{\sigma\in \mathrm{Sh}\left(a,b\right)} f\left(x_{\sigma^{-1}\left(1\right)},x_{\sigma^{-1}\left(2\right)},...,x_{\sigma^{-1}\left(a+b\right)}\right) = f\left(x_1,x_2,...,x_a\right) f\left(x_{a+1},x_{a+2},...,x_{a+b}\right)$ for any $a$ and $b$ and any $x_i$. This generalizes your cat. Does it generalize Littlewood's? Yes, at least if we generalize it even further, to the so-called $\left(a_1,a_2,...,a_k\right)$-multishuffles (which are permutations $\sigma\in S_{a_1+a_2+...+a_k}$ increasing on each of the intervals $\left[a_i+1,a_{i+1}\right]$, where $a_0=0$ and $a_{k+1}=n$). This is not much of a generalization, since it follows from the $\left(a,b\right)$-shuffle version by induction over $k$, but applying it to $\left(1,1,...,1\right)$-multishuffles (which are simply all the elements of $S_n$) yields Littlewood's cat. Now I see that Littlewood's cat even follows from yours, if we notice that every permutation $\sigma\in S_n$ can be written uniquely as a product $t_1t_2...t_{n-1}$, where each of the $t_k$ moves the $k$ some places to the right. (This is one of the stupid sorting algorithms.) Oh, and I don't have a proof of my cat, but it can catch mice, so it's a good cat, isn't it? darij grinbergdarij grinberg $\begingroup$ I enjoy the cat comparisons. They will get a fifth vote from my voting account. (If the moderators let me, I would add 6 more to it.) Gerhard "Yes, I'm A Cat Person" Paseman, 2011.08.31 $\endgroup$ – Gerhard Paseman Aug 31 '11 at 16:20 $\begingroup$ Yes, that's an excellent cat!! So Darij conjectures that, more generally, $f$ satisfies this "shuffle-coproduct" identity for any $(a,b)$. (Which, Frédéric points out, makes $f$ into a "symmetral mould", if I understand correctly, but I'm kind of fuzzy about operads.) Is it perhaps possible to prove this using, say, some clever trick like Tom proposed for the $(1,n-1)$ case? $\endgroup$ – Graham Denham Aug 31 '11 at 19:17 $\begingroup$ Actually this cat is no longer a conjecture, because the proof is completely straightforward: Every $\sigma \in \mathrm{Sh}\left(a,b\right)$ satisfies either $\sigma^{-1}\left(1\right)=1$ or $\sigma^{-1}\left(1\right)=a+1$. Thus, the sum splits into two parts, each of which can be handled by induction (once for $\left(a-1,b\right)$ instead of $\left(a,b\right)$, and once for $\left(a,b-1\right)$ instead of $\left(a,b\right)$). $\endgroup$ – darij grinberg Aug 31 '11 at 22:32 $\begingroup$ Like every proof related to shuffles, this proof is very simple but nigh impossible to formalize. We really need a reasonable shuffle calculus. Maybe dendriform dialgebras can be of use here. $\endgroup$ – darij grinberg Aug 31 '11 at 22:34 $\begingroup$ For me, they come from the shuffle Hopf algebra. ;) $\endgroup$ – darij grinberg Sep 1 '11 at 7:32 This property ( or rather the generalized version by Darij using (a,b)-shuffles ) means that f is what is called a "symmetral mould" in the context of Ecalle's theory of moulds. There is a related notion of "alternal mould" where the right hand side is 0 rather than a product of two f. Here is just one reference among many : page 591 of Jean Ecalle; Bruno Vallet The arborification-coarborification transform: analytic, combinatorial, and algebraic aspects This may not be transparent when looking at this article. Maybe page 2 of my article The anticyclic operad of moulds would be more clear, but it only defines "alternal moulds". The symmetral property is really a property of sequence of functions $f_n$, with $f_n$ a function of $n$ variables $x_1,\dots,x_n$. The notions of alternal and symmetral moulds, when considered under some specific point of view, turn into the notion of primitive and group-like element in a Hopf algebra. F. C.F. C. I'm not sure whether my answer is conceptual in your sense, but here is a relatively short proof. First of all, your definition of $f$ suggests the notation $$s_p := \sum_{i=p}^n x_i.$$ Now consider the following telescopic sum: \begin{equation}\label{eq} (1 - z_2) + z_2(1 - z_3) + z_2 z_3 (1 - z_4) + \dotsm + z_2 \dotsm z_{n-1} (1 - z_n) + z_2 \dotsm z_n = 1. \quad () \end{equation} For each $i \in \{2,\dots,n\}$, take $$z_i = \frac{s_i}{x_1 + s_i},$$ hence $$1 - z_i = \frac{x_1}{x_1 + s_i},$$ and plug this into the telescopic sum $()$. Divide both sides of the equation by $x_1 \cdot s_2 s_3 \dotsm s_n$ to get the desired expression. Tom De MedtsTom De Medts $\begingroup$ That's nicer than my argument (namely: just rewrite in terms of $s_i$'s, clear denominators, and collapse). It's possible, I guess, that my identity can only be seen as an obscured form of (*), in which case I should not expect too much from it. But I'll remain optimistic for a while. $\endgroup$ – Graham Denham Aug 31 '11 at 14:26 $\begingroup$ You say the definition of f suggests the notation s_i = ..., but s_i makes no sense: the i on the right side is the index of summation. You meant s_p, not s_i. $\endgroup$ – KConrad Sep 2 '11 at 12:50 $\begingroup$ @KConrad: Sorry for the obvious typo; I will correct it. Thanks for noticing :) $\endgroup$ – Tom De Medts Sep 3 '11 at 17:31 A simple proof of the Sh(a,b) cat, using iterated integrals, is as follows. Note that $$ f(x_1,\ldots,x_n)=\int_{1>t_1>\cdots>t_n>0} dt_1\cdots dt_n \ t_1^{x_1-1}\cdots t_n^{x_n-1}\ . $$ Littlewood's identity follows from changing variables using the permutation so as to keep the integrand fixed. Then one has a sum of simplices (corresponding to all possible relative orderings of the variables) which recombines into a cube of integration $[0,1]^n$. The proof of the Sh(a,b) identity follows the same idea. Here the total volume of integration is a product of simplices which is broken into a union of simplices. This is probably well known to people working with moulds, operads, etc. An additional remark: Littlewood's identity follows from Lemma II.2 in my article "Trees forests and jungles: a botanical garden for cluster expansions" with V. Rivasseau. To see this, extract the coefficient of the highest degree monomial in the v variables (notations of that article), then specialize the u variables to the case where $u_{i, i+1}=x_i$ and all other pair variables are zero (killing all edges of the complete graph which are not in a `spanning chain'). The Lemma in our article is related to many other topics in mathematical physics such as the Wilson-Polchinski renormalization group equation, see e.g. these slides. Abdelmalek AbdesselamAbdelmalek Abdesselam $\begingroup$ So Graham was right, it was a stray cat from the land of topology. (Incidentally, Damien Calaque answered another question of mine about shuffles using your integral: mathoverflow.net/questions/63923/… .) $\endgroup$ – darij grinberg Sep 1 '11 at 19:35 $\begingroup$ I think the cat is too feral to belong to one particular land... $\endgroup$ – Abdelmalek Abdesselam Sep 1 '11 at 19:40 $\begingroup$ A geometric argument! That's very nice too. $\endgroup$ – Graham Denham Sep 3 '11 at 14:03 Not the answer you're looking for? Browse other questions tagged co.combinatorics reference-request or ask your own question. Can the stupid salesman do better? How many expressions can be formed with $k$ commutative and associative functions? Lagrange interpolation vs homogeneous symmetric polynomials? How to recover $k$ lost items in binary data $x_1,x_2,x_3 \dots,x_n$ via only XOR operator?	CommonCrawl
\begin{document} \newcommand{\emph{i.e.}\@\xspace}{\emph{i.e.}\@\xspace} \newcommand{\emph{e.g.}\@\xspace}{\emph{e.g.}\@\xspace} \newcommand{\mathbb{E}}{\mathbb{E}} \newcommand{\mathbb{V}}{\mathbb{V}} \newcommand{\mathbbm{1}}{\mathbbm{1}} \newcommand{\bold{G}}{\bold{G}} \newcommand{\bold{Y}}{\bold{Y}} \newcommand{\bm{y}}{\bm{y}} \newcommand{\bm{x}}{\bm{x}} \newcommand{\bm{w}}{\bm{w}} \newcommand{\bold{I}}{\bold{I}} \newcommand{\textsf{R}\xspace}{\textsf{R}\xspace} \newcommand{\bm{\beta}}{\bm{\beta}} \newcommand{\bm{\epsilon}}{\bm{\epsilon}} \title{Inference for feature selection using the Lasso with high-dimensional data} \author[1]{Kasper Brink-Jensen} \author[2]{Claus Thorn Ekstr\o m} \affil[1]{Department of Mathematical Sciences, University of Copenhagen} \affil[2]{Department of Biostatistics, University of Copenhagen} \maketitle \begin{abstract} \textbf{Motivation:} Penalized regression models such as the Lasso have proved useful for variable selection in many fields --- especially for situations with high-dimensional data where the numbers of predictors far exceeds the number of observations. These methods identify and rank variables of importance but do not generally provide any inference of the selected variables. Thus, the variables selected might be the ``most important'' but need not be significant. We propose a significance test for the selection found by the Lasso. \textbf{Results:} We introduce a procedure that computes inference and $p$-values for features chosen by the Lasso. This method rephrazes the null hypothesis and uses a randomization approach which ensures that the error rate is controlled even for small samples. We demonstrate the ability of the algorithm to compute $p$-values of the expected magnitude with simulated data using a multitude of scenarios that involve various effects strengths and correlation between predictors. The algorithm is also applied to a prostate cancer dataset that has been analyzed in recent papers on the subject. The proposed method is found to provide a powerful way to make inference for feature selection even for small samples and when the number of predictors are several orders of magnitude larger than the number of observations. \textbf{Availability:} The algorithm is implemented in the MESS package in \textsf{R}\xspace and is freely available. \textbf{Contact:} \href{[email protected]}{[email protected]} \end{abstract} \section{Introduction} Molecular technologies have reached a state where it is possible to quickly and cheaply measure the status of millions of nucleotides, mRNAs, genes, proteins, or metabolites simultaneously and repeatedly over time. The availability of these massive data ideally allows for complex and detailed modeling of the underlying biological system but they also pose a serious potential multiple testing problem because the number of covariates are typically orders of magnitude larger than the number of observations. Recently, investigators are starting to combine several of these high-dimensional dataset which has increased the demand for analysis methods that accommodates these vast data \citep{nie2006integrated,plosone2013,su:etal:2011,kamb:etal:2011}. As data sets increase in size, methods that promote sparse results have become more important and popular. \citet{hastie2005elements} and \citet{hesterberg2008least} provide an overview of the development of these methods of which the Lasso (least absolute shrinkage and selection operator) is the most widely used \citep{tibshirani1996regression}. While all these estimators enforce sparsity (since only a fraction of the variables are believed to be influential) and perform variable selection, they lack the inference from traditional statistics, such as $p$-values and confidence intervals. As a consequence, these models will always identify a set of the most important predictors even if none or only some of them are significant. To address this issue several recent papers propose techniques that help asses the importance of the selected predictors. \citet{Lockhart:2013fk} developed a method that use a covariance test statistic approach to compute $p$-values for parameters obtained from a Lasso regression. With this method $p$-values can be calculated for all predictors in a data set if there are more samples than predictors, $p<n$. In the case of $p>n$ it is not possible to use the covariance test without specifying an estimate of $\sigma^2$, the error standard deviation. \citet{meinshausen2009p} presents another approach where the data is split into two groups. The Lasso is applied to one group, after which the variables selected by the Lasso are used as predictors to obtain $p$-values from an ordinary least squares (OLS) regression on the other group. \citet{Wu2009} suggest to first use the Lasso to select a set of relevant variables which are then fitted to a non-penalized model where from which the $p$-values can be calculated. In applied sciences we find often that variable selection is used for screening to determine which predictors such as genes to focus on, in further (possibly costly) analyses. A useful tool in this setting would be a model that allowed us to perform variable selection and assign a $p$-value to evaluate the evidence of the most important predictors found from the variable selection procedure. We wish to propose an approach for variable selection and inference for the Lasso that works directly when $p>n$ rather than choosing variables with Lasso and transferring these to an OLS domain for inference. Our approach uses randomization/resampling to infer the significance of the $k$th chosen predictors from a Lasso model. Thus, our focus is on using the Lasso to not only generate new (biological) hypotheses but also to evaluate the evidence for those hypotheses. The paper is structured as follows: in the next section we present a method to both identify features and for computing $p$-values of those features obtained from a Lasso regression model. Potential problems and possible extensions are also discussed. The simulation section presents results from a series of simulation studies that show the power of the proposed method under various conditions and for situations both with and without correlated predictors. In addition we show how well the method identifies the most important of the predictors and apply the proposed method to a dataset on prostate cancer. Finally, the proposed method is discussed. Note that while we use the Lasso in the following the proposed procedure can essentially be applied in combination with any sparse feature selection approach (e.g., elastic net, ridge regression). \section{Methods} We consider the situation where we have a quantitative response vector, $\bm{y} \in \mathbb{R}^{n}$ and a set of $p$ quantitative predictors $x_{i1}, \ldots, x_{ip}$ for each observation $i \in \{1, \ldots, n \}$. The number of predictors can be much larger than the number of observations, $p\gg n$ and we are interesting in identifying predictors that are associated with the response vector $\bm{y}$. Suppose that $\bm{y} = X\bm{\beta} + \bm{\epsilon}$ where the noise is assumed to be Gaussian, $\bm{\epsilon}\sim N(0, \sigma^2 \bold{I})$. The Lasso is a penalized version of the multiple regression model where sparsity of the parameter vector $\bm{\beta}\in \mathbb{R}^{p}$ is achieved by adding a penalty term to solve \begin{equation} \hat{\bm{\beta}}=\underset{\bm{\beta} \in \mathbb{R}^p}{\operatorname{arg\,min}} \left\{ \frac{1}{2}\\|\bm{y}-X\bm{\beta}\\|^{2}_{2}+\lambda\\|\bm{\beta}\\|_{1} \right\}, \label{eq:lasso} \end{equation} where $\lambda\ge0$ is the regularization parameter \citep{tibshirani1996regression}. $\lambda$ controls the amount of shrinkage of $\hat{\beta}$ such that $\lambda =\infty$ corresponds to setting all parameters to 0, while $\lambda=0$ corresponds to an ordinary least squares multiple regression model with no restrictions on the parameter vector. The choice of $\lambda$ will be discussed below. The Lasso can be used for variable selection for high-dimensional data and produces a list of selected non-zero predictor variables. \citet{Meinshausen2009} show that while the Lasso may not recover the full sparsity pattern when $p\gg n$ and when the irrepresentable condition is not fulfilled (e.g., if highly correlated predictor variables are present) it still distinguishes between important predictors (i.e., those with sufficiently large coefficients) and those which are zero with high probability under relaxed conditions. In many high-dimensional molecular genetic datasets it is reasonable to assume that there is just a small number of genes that influence the outcome while the vast majority of genes are irrelevant for the outcome. This common setup is also relevant in other situations and it forms the basis of the following. When the relative number of truly non-zero predictors is small then the results from \citet{Meinshausen2009} also show that the Lasso selects the non-zero parameters as well as ``not too many additional zero entries of $\bm{\beta}$''. Here, we seek to identify which of the selected predictors that we believe to be non-zero. The predictors found by Lasso are generally ranked by their importance, so that a high value for $\lambda$ will include the most important predictors, decreasing $\lambda$ will include other less important predictors \citep{efron2004least, hastie2005elements, hesterberg2008least}. We wish to determine which of the features identified by the Lasso that are indeed true positives. Let $\beta_{(1)}, \ldots, \beta_{(k)}$ be the ordered absolute coefficients of the $k$ non-zero predictors that are selected by the Lasso. To ensure that the effects of the predictors are comparable across variables we assume that each predictor has been standardized by dividing by their standard deviation. This is not a restriction when determining the importance of selected variables as the internal structure of the predictors (and their pairwise correlations) is preserved \citep{efron2004least}. Essentially, this approach mimics the adaptive Lasso of \citet{Zou2006} and ensures that the coefficients are equally penalized by the $\ell_1$ penalty in \eqref{eq:lasso}. The variables can then be ordered according to their absolute size such that $\|\beta_{(1)}\| \geq \|\beta_{(2)}\| \geq \cdots \geq \|\beta_{(p)}\|$. We are interested in testing hypotheses of the form \begin{equation} H_0 : \beta_{(k)} = 0, \label{eq:null} \end{equation} where $\beta_{(k)}$ is the $k$th most important feature. Note that contrary to classical hypothesis testing we are not testing hypotheses about \emph{specific} predictors but we are testing whether the $k$th identified parameter is equal to zero. As a test statistic for the $k$th selected parameter we use the corresponding coefficient obtained from the Lasso, $\|\beta_{(k)}\|$. In order to derive the distribution of the $k$th identified predictor we permute response vector $\bm{y}$ to remove any associations to the set of predictors while retaining the individual structure and correlations among the set of predictors \citep{Manly2006}. This randomization is undertaken a large number of times and for each randomization we run the Lasso on the permuted data to obtain a distribution of coefficients for the $k$th identified predictor. Note that this approach might result in different variables being selected for each permutation and this matches the null hypothesis \eqref{eq:null} where the focus is on finding out whether the $k$th identified predictor is significant. The algorithm for testing the null hypothesis \eqref{eq:null} using a randomization approach can be summarized as follows: \begin{enumerate} \item Start by scaling the predictors so their effects are comparable, i.e., $$X_j = X_j / \sqrt{\mathbb{V} (X_j)},$$ where $X_j$ is the $j$th column of the design matrix $X$. \item Fit a Lasso regression model to the original data and extract the coefficients for the selected predictors, $\hat{\beta}_{(1)}, \hat{\beta}_{(2)}, \ldots$. \item Let $B$ be the (large) number of randomizations to perform to determine the distribution of the coefficients under the null hypothesis. For each $b\in B$ do the following: \begin{enumerate} \item Permute the response vector, $\bm{y}^b$, and fit a new Lasso model using $\bm{y}^b$ as the response. \item Extract the coefficients for the predictors for this model, $\hat{\beta}_{(1)}^b, \hat{\beta}_{(2)}^b, \ldots$. \end{enumerate} \item For each feature, $k$, we can compare the size of the coefficient identified in the original dataset, $\|\hat{\beta}_{(k)}\|$ to the distribution of the coefficients found for the $k$th feature for the permuted data, $\|\hat{\beta}_{(k)}^1\|, \ldots, \|\hat{\beta}_{(k)}^B\|$. The $p$-value for the $k$th feature is then the fraction of coeffients under the null that are larger than or equal to $\|\hat{\beta}_{(k)}\|$: \begin{equation} p\mbox{-value} = \frac{1 + \sum_{b=1}^B \mathbbm{1}_{ \|\hat{\beta}_{(k)}^b\| \geq \|\hat{\beta}_{(k)}\| }}{B+1} \label{formula:pval} \end{equation} \end{enumerate} Note that by construction the $p$-value defined by \eqref{formula:pval} controls the error rate since we are simulating the distribution on the null hypothesis. \subsection{Feature selection and collinearity} In applications such as genetics, the number of predictors, $p$, is often several orders of magnitude larger than $n$ and we are typically more concerned with identifying \emph{some} associations rather than finding \emph{all} associations since any relationships found will often be the focus of subsequent specific experiments to verify the genetic function. Thus, our interest is on hypothesis generation rather than testing a specific hypothesis: is it possible to identity one or a few potentially interesting or relevant predictors, in a scenario with little or no knowledge of the individual predictors. It is well-known that the Lasso has a weakness when dealing with correlated predictors: it will only select one of a group of collinear predictors to represent their effect on the response \citep{zou2005regularization} and the coefficient for the selected variable will have high variance \citep{meinshausen2008hierarchical}. Collinearity poses a problem if we are intersted in making inferences about specific predictors because the corresponding standard error becomes unstable. However, in the present setup collinearity among predictors means that while we may not be able to identify all predictors that are associated with the outcome, the size of the coefficients for the variable that are selected essentially represent the maximum effect size of the cluster of correlated predictors. In particular, if $p\gg n$ then we may be interested in just finding \emph{one} association between a predictor and the response in which case our primary focus is on the null hypothesis \begin{equation} H_0 : \beta_{(1)} = 0. \label{eq:null1} \end{equation} \subsection{Choice of penalty parameter $\lambda$} The Lasso involves a penalty parameter, $\lambda$, that determines the amount of shrinkage that is applied to the parameters. In our approach, we recommend using the same procedure for estimating $\lambda$, that the reseacher would typically use together with a Lasso regularization model. Typically, $\lambda$ is determined through some form of $K$-fold cross-validation where $\lambda$ is chosen such that it minimizes the cross-validation error. \begin{equation} CV(\lambda) = \frac1K \sum_{i=1}^N \|y_i - \hat{f}^{-\kappa(i)}(\bm{x}_i,\lambda)\|, \end{equation} where $\bm{x}_i$ is the set of covariates for individual $i$, $\hat{f}^{-\kappa(i)}$ is the expected value from the Lasso regression based on the folds/parts of the data that does not contain observation $i$. There are two considerations to consider here: First, we know that we are in a situation where the majority of the variables are unrelated to the outcome. Thus we want to ensure, that the majority (or possibly all) of the variables are shrunk to zero. This is particularly true under the null hypothesis where there is no association between the variables and the outcome due to randomization. Secondly, we want to ensure that when there is indeed an association between some of the variables and the outcome then we want $\lambda$ to be small enough that the effect of the associated variables are not shrunk too much. In the following we have worked with a 10-fold cross-validation using the mean absolute error as loss function since our analyses have shown that this gives good stable results. Using the mean absolute error instead of the traditional mean squared error makes the cross-validations less sensitive to spurious extreme fits which is often the case under the null, where none of the variables are associated with the outcome. However, other approaches would be directly applicable as well at the researchers discretion. One possible approach here, that would substantially reduce the computation time would be to use the same $\lambda$ for the randomization samples as was used in the original dataset. That ensures that we only have to do cross-validation once (for the original data) and not for each randomization. Also, the estimate of $\lambda$ under the null hypohtesis (where the randomization approach assumes that \emph{none} of the predictors are associated with the outcome) can be rather unstable since there really is no obvious minimum for the cross validation procedure to reach. \subsection{Including external predictors} In some situations it is of interest to use the Lasso to select predictors from among a subset of the predictors. For example, some predictors might be related to the design of the experiment (and we wish to force them to be part of the modeling) while the remaining predictors contains the variables we wish to identify/select from because we have no prior knowledge about them. Thus, we consider two sets of predictors, $\tilde{X}$ and $X'$ such that $X = [\tilde{X}\|X']$ where $\tilde{X}$ are the predictors that we wish to force into the model while $X'$ are the predictors from which we wish to make a selection. If we apply the Lasso to $X$ we will select variables from the full set of predictors which means that some of the potential confounders that we wish to include are disregarded in the modeling. There are two obvious alternatives to just using the classical Lasso in this situation. One is to make a two-step procedure where we first fit a model using only the predictors in $\tilde{X}$, i.e., \begin{equation} \bm{y} = \tilde{X}\tilde{\bm{\beta}} + \bm{\epsilon}. \label{eq:stepprocedure} \end{equation} From \eqref{eq:stepprocedure} we extract the residuals, $\bm{r} = \bm{y} - \tilde{X}\hat{\tilde{\bm{\beta}}}$, and use the residuals as outcomes in combination with the procedure described above. This approach ensures that the (marginal) effects of the predictors in $\tilde{X}$ are removed and and features that are subsequently found are conditional on these marginal effects. The disadvantage is that the effects of the predictors in $\tilde{X}$ and $X'$ are assumed to be working independently of each other, which may not be realistic. The adaptive (or weighted) Lasso of \citet{Zou2006} uses a weighted penalty resembling \eqref{eq:lasso} \begin{equation} \hat{\bm{\beta}}=\underset{\bm{\beta} \in \mathbb{R}^p}{\operatorname{arg\,min}} \left\{ \frac{1}{2}\\|\bm{y}-X\bm{\beta}\\|^{2}_{2}+\lambda\\|\bm{w}\bm{\beta}\\|_{1} \right\}, \label{eq:adaptivelasso} \end{equation} where $\bm{w}$ is a vector of non-negative weights for each predictor. Typically, the individual weights are set to $w_j = 1/\|\hat{\beta_j'}\|^\nu$, where $\hat{\beta_j'}$ is the univariate regression coefficient estimate of parameter $j$ and $\nu> 0$. If a weight is set to zero then the corresponding predictor will not be penalized. Thus if the weights of the parameters in $\tilde{X}$ are set to zero then the corresponding variables enter the model without being shrunk towards zero. \subsection{Family-wise error rate for multiple inference results} The $p$-value obtained from \eqref{formula:pval} controls the error rate by construction as mentioned above. However, the error rate control is on a per-hypothesis basis so if we are only aiming on making inference for the ``best'' selected predictor --- corresponding to the hypothesis \eqref{eq:null1} --- then the proposed procedure yield the correct error level. However, in some situations it is of interest to extract inference on as many predictors as possible and simultaneously control the family-wise error rate. We can still use \eqref{formula:pval} to obtain individual $p$-values for the first feature, the second feature, etc., and Holm's step-down procedure can control the family-wise error rate in those situations \citep{Holm1979}. In its original form, Holm's procedure orders the $p$-values and makes sequential comparisons from the smallest $p$-value to the largest until the first occurrence of a null hypothesis that fails to be rejected. All other subsequent hypotheses are also not rejected. Here we suggest a slightly different version of Holm's procedure. The $p$-value of the $k$'th selected feature is compared against $$\frac{\alpha}{m+1-k},$$ where $\alpha$ is the overall significance level desired and $m$ is the number of hypotheses tested. The significance level for each feature is identical to the level used by Holm's procedure but unlike that we do not order our $p$-values according to size but test them in the order that they are selected by our procedure. This ensures that we still control the family-wise error rate (by the exact same arguments as used by \citet{Holm1979}) since it becomes more difficult to reject hypotheses by \emph{not} ordering them according to the size of the $p$-value. Also, since we keep the order of the features it prevents us from ending with a result where, say, the first and third selected features are significant but the second selected feature is not. \section{Results} \subsection{Simulation studies} \label{sec:results} To examine the validity of our procedure, we performed simulation studies to assess the performance of the sensitivity and specificity. For each setting, we simulated 100 data sets with only $50$ observations generated under the linear model, $$ \bm{y} = X\bm{\beta} + \bm{\epsilon}, $$ where $\bm{\epsilon} \sim N(0, \bold{I})$ and where $X_{ij} \sim N(0,1), \; i=1,\ldots, 50, \; j=1, \ldots, p$. The unit variance is not really a restriction in this case as we start the analysis algorithm by standardizing each predictor (as outlined in the methods section above). The number of predictors were varied from 1000 to 250000 to represent various datasets. Initially we assume that there is just a single predictor that is associated with the response and we vary the corresponding regression coefficient, $\beta$, from 0 to 1.5. This corresponds to a partial correlation coefficient between the predictor and the outcome ranging from 0 to 0.83. With these data we follow the procedure described above where each combination of predictors, $p$, and associated coefficient, $\beta$, is run 100 times and for each combination we use 100 permutations to determine the power of the feature identified as the most important corresponding to the hypothesis $\beta_{(1)} = 0$. In all computations a significance level of $\alpha=0.05$ was used. In order to see how collinearity influences the power we run simulations with the same association between a single predictor and the outcome but we let that predictor be part of a cluster of ten correlated predictors such that $\mbox{cor}(X_i, X_{i'}) = \rho$. Here, $\rho=0.5$ or $\rho=0.95$ to represent moderately and highly correlated predictors. We first focus on two primary abilities of the model: the power to identify the first selected predictor and the impact of correlated predictors in the model. The simulation results are shown in Figure \ref{fig:pvals}. The simulations show three clear results: 1) for moderate to high effect size (i.e., when $\beta$ is 1--1.5) we obtain a high power even when the number of predictors is large, 2) when the causal gene is part of a correlated cluster of predictors then the power is higher than if the gene is independent, and 3) generally there is a fall in power as the number of predictors increase although this drop appears to stabilize and level off as the set of predictors increases. The power is 100\% for an effect size of 1 or 1.5 (the lines overlay each other in Figure~\ref{fig:pvals}) except for the independent predictors. That means that even when there is a ratio of predictors to observations, $p/n$, of 5000 then we are still able to identify a signal in the data even with just 50 observations. An effect size of 1 corresponds to an effect of 1 standard deviation so we are even able to identify the presence of a gene with realistic biogical effects. For correlated data we see that the power is \emph{larger} than the power observed from independent predictors. This is caused by the fact that the causal predictor is part of a cluster; with correlated predictors, then each of the predictors in the cluster has a probability of showing an improved association to the outcome just by chance. Thus, we essentially improve our power to detect \emph{one} of the predictors in the cluster at the cost of perhaps not detecting the true one but at least one that is highly correlated to it. If the causal predictor was a singleton and not part of a cluster of correlated predictors then the power results resemble the results from the uncorrelated data (results not shown). Initially the power drops for all situations as the number of predictors increase (making it harder to identify that there is in fact a true association among one of the predictors and the outcome) but the overall power seems fairly constant once the number of predictors reach around 50000 (of which just one is associated with the outcome). Not surprisingly, this drop is more noticeable when the true association has a moderate effect than when the effect is smaller (where the power is consistently low) or larger (where the power is consistently high). Figure \ref{fig:pvals} also shows that we are generally able to control the family-wise error rate at the desired significance level (0.05). Perhaps most surprisingly is the consistent high power for moderate to high effect sizes ($\beta=1$ or $\beta=1.5$) considering this is based on just 50 observations. \begin{figure} \caption{Power to detect the first identified predictor selected by the Lasso for varying number of predictors and varying effect size of the predictor for a dataset containing 50 observations. Each point is the average from 100 simulations, each using 100 randomizatons to compute the $p$-value. The black lines correspond to completely independent predictors, the red lines are when there is some collinearity between a group of predictors while the blue lines represent the situation with a group of highly correlated predictors.} \label{fig:pvals} \end{figure} Figure~\ref{fig:pvals} focused on the evidence of the first/most important predictor. Once that has been identified we would like to determine the power to detect the second most important predictor and its power. In our simulations we assumed that there was always one causal predictor with a corresponding parameter of $\beta_1 = 1.5$ and we varied the effect of another predictor, $\beta_2$, with values ranging from 0 to 1.5 as previously. Following the setup above we also allow for correlations and assume that the two predictors are actually part of the same cluster. \begin{figure} \caption{Power to detect the \emph{second} identified predictor selected by the Lasso for varying number of predictors and varying effect size of the predictor for a dataset containing 50 observations. The dataset contains two predictors from the same group/cluster that are associated with the response, one of which is fixed with an effect size of 1.5. Each point is the average from 100 simulations, each using 100 randomizatons to compute the $p$-value. The black lines correspond to completely independent predictors, the red lines are when there is some collinearity between a group of predictors while the blue lines represent the situation with a group of highly correlated predictors.} \label{fig:2ndpvals} \end{figure} Figure~\ref{fig:2ndpvals} shows the power to detect the \emph{second most important} variable when there are two potential causal variables from the same cluster. Overall, we see the same trends as we saw for the most important variable in Figure~\ref{fig:pvals} except that the power is generally lower. Looking at the solid line corresponding to the largest effect of the second most important variable we see a marked decline in power as the number of predictors increase if the predictors are all independent. The decline is not very surprising as both of the two causal predictors have the same effect size (both $1.5$) so the second most important variable will by design be the smallest of those two (after sampling error has been added). Thus, due to random variation, the estimated effect of the second is likely to be slightly smaller than the true effect (ie., biased downward) and contrary the effect of the most important variable will be slightly biased upward. However, the drop in power for the second most important variable is still substantial even for high effect sizes except when the predictors are correlated. Figure~\ref{fig:2ndpvals} also shows that the error rate is controlled around the 5\% level for independent predictors when there is only one predictor that is truly associated to the response (black dotted line). As for the results from the first identified predictor, Figure~\ref{fig:pvals}, we get that correlated predictors increase the power to identify subsequent predictors. As highlighted in the methods section, the proposed procedure computes the evidence for the identified features but does not directly provide significance inference for a particular predictor. In practice, however, it is also of interest how well the Lasso approach identifies the predictors, that are truly associated with the response. While this has been investigated in several papers \citep{efron2004least,hastie2005elements, hesterberg2008least,meinshausen2009p}, we include results using the same simulation setup as described above for direct comparison with the simulation results shown in Figure \ref{fig:pvals}. The results are shown in Figure \ref{fig:detecttrue} which clearly shows that the precision increases with increasing signal strength and decreases when the true predictor is part of a cluster of predictors. In the latter case, it is generally one of the the predictors from the correlated cluster that is identified, so while we may not identify the specific variable we do identify the cluster (results not shown). \begin{figure} \caption{Feature identification precision for the Lasso to identify the single true underlying (most significant) predictor for varying number of predictors and effect sizes. Each point is the average from 100 simulations. The black lines correspond to completely independent predictors, the red lines are when there is some collinearity between a group of predictors while the blue lines represent the situation with a group of highly correlated predictors.} \label{fig:detecttrue} \end{figure} \subsection{Application to prostate cancer data} We apply our proposed method to the prostate cancer example from \citet{Lockhart:2013fk}, a subset of 67 samples and 8 predictors from a larger dataset. The outcome is the logarithm of a prostate-specific antigen and we want to determine the association between the outcome and any predictors. Data can be found at \verb+www-stat.stanford.edu/~tibs/ElemStatLearn/+. The setup for this dataset is simpler than the situation mentioned in methods section since $p<n$ so it is possible to analyze all predictors simultaneously by using, say, a classical multiple regression model. We compare the results from our proposed method with three other analyses approaches: simple multiple regression, using the Lasso in combination with the covariance test statistic of \citet{Lockhart:2013fk} and the multi-split method of \citet{meinshausen2009p}. The multi-split method essentially consists of variant of 2-fold cross-validation: First, Lasso regression is applied to a part of the dataset to select a list of predictors, and secondly an ordinary multiple regression (using the selected predictors) is applied to the remaining data to obtain $p$-values for the selected predictors. Note that unlike our proposed approach, both the multiple regression, the Lasso-covariance test, and the multi-split method are all designed to test hypotheses about each specific predictor. However, the multi-split method is vulnerable to the selection obtained from the first split which gives rise to rather substantial variance of the $p$-values in this dataset. Splitting the sample repeatedly and getting a set of $p$-values has also been suggested by the authors, but ``it is not obvious, though, how to combine and aggregate the results.'' \citep{meinshausen2009p}. Here, we have just averaged the $p$-values obtained for each of the sets (with non-selected predictors getting a $p$-value of 1). \begin{table}[t] \centering \caption{Table of $p$-values for each predictor obtained from simple linear regression (LR), multiple regression (OLS), the multi-split method (SPLIT), the covariance test method (COV) and the proposed method (RAND). Last column shows the marginal Pearson correlation between each predictor and the outcome. Predictors are listed in order they are selected by the randomization procedure.} \label{tab:pval} \begin{tabular}{rrrrrrc} \hline Predictor &LR $p$ & OLS $p^\ast$ & COV $p$ & SPLIT $p$ & RAND $p$ & Correlation with outcome \\ \hline lcavol &0.00 & 0.00 & 0.00 & 0.00 & 0.00& 0.73 \\ svi & 0.00 & 0.05& 0.17 & 0.29 & 0.34 & 0.56 \\ lweight & 0.00 & 0.00 &0.05 & 0.06 & 0.14& 0.49 \\ lcp & 0.00 & 0.09&0.05 & 0.89 & 0.07 & 0.49\\ pgg45 & 0.00 &0.24& 0.35 & 0.86 & 0.01& 0.45 \\ lbph & 0.03 & 0.05&0.92 & 0.51 & 0.01& 0.26\\ age & 0.06 & 0.14& 0.65 & 0.92 & 0.04 & 0.23 \\ gleason & 0.00 & 0.88& 0.97 & 0.97 & 0.58& 0.34 \\ \hline \multicolumn{6}{l}{$^\ast$ $p$-values for OLS are shown after backward elimination.} \end{tabular} \end{table} Table~\ref{tab:pval} shows that the three methods that accommodate multiple predictors and test hypotheses about each specific predictor (OLS, multi-split and the covariance test) generally give the same results and all identify the same predictor, \emph{lcavol}, as being the most important. The same result is obtained with our proposed method, where the first selected variable is highly significant (and turns out to be \emph{lcavol}). The simple marginal regression analyses show that virtually all predictors are associated with the response, but the other approaches reduce the number of significant variables partly due to correlation among the predictors. Our proposed randomization test identifies slightly more variables than the covariance test (and a few more than the multi-split procedure) and seems to lie somewhere between the ordinary multiple regression model and the covariance test. It is only the covariance test that places any real emphasis on the \emph{lcp} predictor in part because of a high degree of collinearity to \emph{lcavol} (the correlation coefficient between \emph{lcavol} and \emph{lcp} is $0.692$, which is the second largest among the predictors). The largest pairwise correlation is between \emph{pgg45} and \emph{gleason} and is 0.757 but the association to the outcome is less for those two variables so the impact of their collinearity on the $p$-values is less noticeable. \section{Discussion} We have proposed an algorithm to aid in making inference and subsequently relevant hypotheses concerning data with (many) more predictors than samples. It greatly extends the results from shrinkage regression methods to be able to assign a $p$-value to findings from regularized regression methods that combine variable selection and estimation. When the number of predictors $p$ is larger than then sample size $n$, regularized regression methods can be used to identify a sparse model and to provide stable parameter estimates. The standard Lasso suffers from several problems in this regard, in particular that it is not consistent in cariable selection and that the limiting distribution of the estimates are non-standard and cannot be directly derived. To overcome these shortcomings \citet{fan:li:2001} and \citet{Zou2006} have presented versions of regularized regression that not only have asymptotic oracle properties but also have consistency in variable selection and asymptotic normality of the estimators. However, inference based on these improved regularization methods for \emph{finite samples} generally perform poorly even when the effects are large \citep{minn:etal:2011}. An obvious difference between the $p$-values obtained from ordinary least squares or from other regularized regression inference approaches is that we pursue investigating a null hypothesis that is not concerned with a specific (set of) predictor(s) but is concerned solely with evaluating the evidence that the results obtained from regularized regression are stronger than what would be expected if none of the predictors were associated with the response. Classical OLS-style $p$-values are extremely useful but it can be argued that for the majority of problems in genomics --- where there is often little or no prior information about \emph{any} of the possible predictors --- the focus is on hypothesis generation and variable selection and not on testing specific hypotheses. Hence our focus (and the proposed procedure) is on hypothesis generation and not on inferences about specific variables. Our simulations show that --- even with just 50 observations --- the proposed procedure has substantial power to identify at least one associated predictor among a set of 50--250 thousand predictors without serious performance decline (Figure \ref{fig:pvals}). This suggests that not only can we can identify relevant predictors within a large set of irrelevant predictors --- we can also attach a level of significance so we can evaluate whether our findings are indeed likely to be true (and relevant) predictors, that are associated to the outcome. Our results extend to the situation where there are multiple causal variables (Figure~\ref{fig:2ndpvals}) although the power declines slightly more rapidly with the number of predictors when testing the importance of subsequent features. The results regarding the second most important variable shown in Figure~\ref{fig:2ndpvals} was based on two causal variables in the same cluster. If we run the same analyzes where the two causal variables are in two different clusters we get essentially the same results as shown for the uncorrelated data in Figure~\ref{fig:2ndpvals} (results not shown). We have run the same simulations with a reduced dataset of 30 observations instead of 50 observations and while the power obviously is lower we see the same overall trends as shown above. Thus, even with fairly small datasets we have decent power to identify a variable that truly is associated to the outcome. It can be argued that for genetic data where many genes may follow a similar pattern of expression and thus be correlated, Lasso is not an obvious choice for variable selection. If the purpose of the selection is indeed to identify a whole cluster of co-expressed genes, other shrinkage algorithms could be better suited. However if the genes selected by the Lasso are seen as the most influential representative of a cluster there are numerous ways to identify others in the same cluster. Since the whole nature of our approach is not to identify specific variables this is not a problem \emph{per se}. In fact, Figures~\ref{fig:pvals} and \ref{fig:2ndpvals} show that the power to detect \emph{something} from a clustered set of predictors rise dramatically so clustered predictors will just increase our power to identify at least one of the predictors from the cluster. When comparing methods applied to the prostate dataset, it was clear that all four methods (the proposed approach, ordinary least squares, the covariance test statistic, and the split method) are able to identify the variable lcavol as among the most important and a small $p$-value is assigned to this predictor. Note that while the four methods differ somewhat in their results they also test different versions of the null hypothesis. The split method consistenly assigns a small $p$-value to the lcavol predictor but the remaining seven predictors obtain $p$-values estimated with very large uncertainties. The randomization approach generally yields lowest $p$-values which is partly due to the fact that the variables are indeed mostly correlated with the outcome, and partly due to that the null hypothesis is different (in particular it assumes that \emph{none} of the predictors are associated with the outcome). We believe the proposed method could also be applied in for instance Genome-Wide Association Studies (GWAS), where Lasso is already widely used, but significance testing typically done outside the Lasso domain, as done by \cite{Wu2009}. \section{Conclusion} We have presented a method that can be used to make inference about variable selection results from regularized regression models such as the Lasso. We show that it has high power to infer evidence that the selected features are not chance findings --- even when the number of predictors is several orders of magnitude larger than the number of observations, i.e., when $p>n$. The method controls the family-wise error rate and can essentially be used with any regularization method. The proposed method is relevant for situations where regularized regression methods is used as part of the statistical modeling to identify features for subsequent analyses. \input{arxiv.bbl} \end{document}	arXiv
\begin{document} \title{Decoherence~of~Quantum~Fields: Pointer~States~and~Predictability} \preprint{LA--UR 95--3364} \author{J.R.~Anglin and W.H.~Zurek} \address{Theoretical~Astrophysics, T-6, Mail~Stop~B288, Los~Alamos~National~Laboratory, Los~Alamos, NM~87545} \date{\today} \maketitle \abstract{We study environmentally induced decoherence of an electromagnetic field in a homogeneous, linear, dielectric medium. We derive an independent oscillator model for such an environment, which is sufficiently realistic to encompass essentially all of linear physical optics. Applying the ``predictability sieve'' to the quantum field, and introducing the concept of a ``quantum halo'', we recover the familiar dichotomy between background field configurations and photon excitations around them. We are then able to explain why a typical linear environment for the electromagnetic field will effectively render the former classically distinct, but leave the latter fully quantum mechanical. Finally, we suggest how and why quantum matter fields should suffer a very different form of decoherence.} \pacs{} \draft \section{Introduction} Decoherence and environmentally induced superselection have been studied extensively in the system composed of a single harmonic oscillator linearly coupled to a bath of independent oscillators\cite{GSI,CaldeiraLeggett,HPZ}. This system has generally been presented as a conveniently solvable model of value in investigating fundamental problems of principle, such as the issues of dissipation in quantum mechanics\cite{Schwinger,Ullersma}, or of emergence of classical behaviour in open systems\cite{UnruhZurek,ZHP}. In this paper we point out that this simple system actually constitutes a realistic description of a quantum electromagnetic field propagating in a linear dielectric medium. The mechanisms of decoherence identified in single oscillator models can therefore be applied straightforwardly to electrodynamics. The particular aspect of decoherence that we consider is the selection, by the environment and its coupling to the system, of a preferred basis of {\it pointer states}\cite{Zurek1}. We find that the linear interaction of the electromagnetic field with the environment implies that the pointer states of the quantum field are coherent states. While single oscillator models often tend to suggest the interpretation of coherent states as localized particles, in the case of the field they are not localized photon packets at all: the pointer states of the quantum electromagnetic field are in fact background field configurations. There are also, however, many experiments which reveal the existence of photons; and so we examine decoherence in our model more carefully, to determine how it is that photons can be robust despite propagating through an environment. We are led to associate with every pointer state a {\it quantum halo} of states that are not effectively distinguished from it by the environment, and we show that excitations of a few photons above a background field are typically states within such a quantum halo. The paper is organized as follows. The following section presents our model system, and derives a description of a typical dielectric medium as a bath of independent oscillators, from the assumption that such a medium will contain a large number of molecules within a volume on the scale of the smallest electromagnetic wavelength under study. We then specialize considerably to the case of ultraweak Ohmic dissipation at ultrahigh temperatures. In Section II, we take advantage of this simplification to derive several exact results concerning the pointer states of our system. Our third section then discusses quantum halos. Section IV then summarizes our results, and briefly suggests why decoherence may be expected to affect matter fields much differently from linearly coupled systems such as the electromagnetic field. \section{The Model} The system we will study will be an electromagnetic field in 3+1 dimensions. We quantize the field in Coulomb gauge, in a box of linear dimension $2L$, and couple it to molecules composing a dielectric medium inside the box: \begin{equation}\label{Ls} {\cal L}_{\cal S} = {1\over2}\sum_{s=1}^2\sum_{\vec k=-\vec K}^{\vec K} \Bigl[\dot{A}_{\vec{k},s}^2 - \omega(k^2) A_{\vec{k},s}^2 + g\sum_n \dot{A}_{\vec{k},s} e^{-i{\pi\over L}\vec{k}\cdot\vec{x}_n} j_{n,s}\Bigr] \;. \end{equation} We let $\vec{k}$ label the Fourier modes and $s$ the polarization states; $\vec{K} \equiv (L\Gamma,L\Gamma,L\Gamma)$, where $\Gamma$ is an ultraviolet cut-off wave number, above which we consider the gauge field to decouple from the medium (or at least to interact with it in such a way that there will be negligible effect on the field modes below the cut-off). The quantities $j_{n,s}$ represent the electric dipole interaction with molecules located at positions $\vec{x}_n$; $g$ is the coupling strength of this interaction, which is assumed to be the same for all $n$ and to be small. Turning now to the environment, we will initially assume merely that it consists of a large number of molecules, which interact with each other only via the gauge field coupling presented above, and which are located at the points $\vec{x}_n$. We will neglect the motions of the molecules (with consequences that may be easily remedied, as discussed below), and consider only their internal energies: \begin{equation}\label{hame} \hat{H}_{\cal E} = \sum_{n} \hat{H}_n\;, \end{equation} where $\hat{H}_n$ have some arbitrary discrete spectra. We will {\it not} assume that the environment actually consists of independent harmonic oscillators, but instead we will derive the fact that a generic environment may be treated as such, in the limit of large $N$\cite{FeynmanVernon}. The $N$ that must be large is the number of molecules within a volume on the cut-off scale; we will therefore require that the number density of the medium satisfy $d>>\Gamma^3$. For an ideal gas at room temperature and atmospheric pressure, $d\simeq 10^3\Gamma^3$ corresponds to a cut-off of electromagnetic modes in the high ultraviolet range ($\lambda \sim 10$ nm). In solids or liquids we might perhaps handle somewhat shorter wavelengths, but our derivation of the independent oscillator model as a large $N$ approximation to a general non-conducting environment must be expected to break down in the X-ray band ($\lambda \sim 1$ nm). We will treat the medium as an unobserved environment, and describe only the state of the electromagnetic field, using the reduced density matrix formed by tracing over the states of the environmental molecules. If we assume that the initial state is a direct product of field and medium states, then we can obtain the evolution of the reduced density matrix from the path integral propagator \begin{equation}\label{rdmprop} \rho[A,A';t] = \int\!{\cal D}A{\cal D}A'\, \rho[A,A';0]\,e^{{i\over\hbar}\Bigl(S[A]-S[A']\Bigr)}\,F[A,A';t]\;, \end{equation} where $F[A,A']$ is the influence functional\cite{FeynmanVernon} describing the effect of environmental molecules on the electromagnetic field. With the Hamiltonian (\ref{hame}), and a thermal initial state for all the molecules, the influence functional is given by \begin{eqnarray}\label{IFdef} F[A,A';t] &=& {\rm Tr}_{\cal E}\Bigl(T \exp\bigl[-{i\over\hbar}\sum_{n,\vec{k},s}\int_0^t\!dt'\, \dot{A}_{\vec{k},s}(t') e^{-i{\pi\over L}\vec{k}\cdot\vec{x}_n}\hat{\jmath}_{n,s}(t')\bigr] \nonumber\\ & &\qquad\times\; \exp\bigl[-\sum_{n}\beta_n\hat H_n}\; \bar Te^{{i\over\hbar}\sum_{n,\vec{k},s}\int_0^t\!dt'\, \dot{A}'_{\vec{k},s}(t') e^{-i{\pi\over L}\vec{k}\cdot\vec{x}_n}\hat{\jmath}_{n,s}(t')\bigr] \Bigr)\;, \end{eqnarray} where $A_{\vec{k},s}(t)$ and $A'_{\vec{k},s}(t)$ are c-numbers in the path integral for the field, but $\hat{\jmath}_{n,s}(t)$ is the dipole moment operator of the $n$th molecule, in the interaction picture. $T$ and $\bar T$ denote time-ordering and anti-time-ordering, respectively, while $\beta_n$ is the usual inverse temperature, which we allow to vary from place to place in the environment. The trace is to be taken over the states of the environment only. We can now reduce this very general influence functional to the special form of an independent oscillator model, by implementing our large $N$ approximation. We divide the box of volume $8L^3$ into cells of volume $\alpha^3\Gamma^{-3}$, where $\alpha$ is a number much smaller than one. Within the cell $C$ centred at the point $\vec{x}_c$ there will be a large number $N(\vec{x}_c) = d(\vec{x}_c)\alpha^3\Gamma^{-3}$ of molecules.\footnote{ The appearance of $\alpha$ here would seem to lower, perhaps by an order of magnitude, the maximum frequencies up to which our analysis will be accurate. As we will discuss below, however, it is easy to dispense with $\alpha$, which is only present to ensure that $e^{i\vec{k}\cdot\vec{x}}$ varies negligibly within a cell.} By using time-dependent perturbation theory in the interaction picture, keeping explicitly only terms up to second order in $g$, and zeroth order in $\alpha$, we can obtain a simple form for the influence functional for a single cell of dielectric medium: \begin{eqnarray}\label{approx} F[A,A';t] &=& \prod_{\vec{x}_{n}\in C}\Bigl( 1 - {g^2\over\hbar^2}\bigl(\sum_j e^{-\beta E_j}\bigr)^{-1} \sum_{l,l'}\vert J_{ll'}\vert^2\nonumber\\ &&\qquad\times\sum_{\vec{k},s} \int_0^t\!dt'\int_0^2\!dt''\,e^{-\beta_c E_l} [\dot{A}_{\vec{k},s}(t') - \dot{A}_{\vec{k},s}(t')]\nonumber\\ & &\qquad\qquad\times [\dot{A}_{\vec{k},s}^(t'') e^{-i\omega_{lm}(t'-t'')} - \dot{A}_{\vec{k},s}'^(t'') e^{i\omega_{lm}(t'-t'')}]\Bigr)\;. \end{eqnarray} Here $J_{ll'}$ are unpolarized matrix elements of the dipole moment operator, {\it i.e.}, we assume unpolarized scattering from individual molecules, so that at the initial time $t=0$ \begin{equation} \langle E_l\vert\hat{\jmath}_{n,s}\hat{\jmath}_{n,s'}\vert E_{m}\rangle = \delta_{s,s'} \sum_{m} J_{lm}J_{ml'} \;. \end{equation} There are no terms linear in $g$, because we take our molecules to have no preferred orientation of their dipole moments: $\langle\hat{\jmath}_{n,s}\rangle = 0$. And we assume that the initial state of the environment is a direct product of single-molecule thermal states, with every molecule in a cell having the same initial temperature $(k_B\beta_c)^{-1}$. {}From the last line of (\ref{approx}) we discard all but the leading terms in $N(\vec{x}_c)$, then put all the cells together and smooth out the cell structure by defining interpolated density and inverse temperature fields $d(\vec{x}), \beta(\vec{x})$. We can even allow the molecular composition of the environment to vary from cell to cell, so that the entire form of $I_{eff}$ is spatially dependent as well. We find that the influence functional for the dielectric medium is that of a set of independent harmonic oscillators at every point in the box, (which we can now allow to become infinite): \begin{eqnarray}\label{optics} F[A,A';t] &=& \exp\Bigl( -{g^2\over2\hbar}\sum_s \int\!d^3x\,d(x) \int_0^\infty\!{d\omega\over\omega}\,I\bigl(\beta(x),\omega;x\bigr) \nonumber\\ &&\qquad\times\int_0^t\!dt'\int_0^{t'}\!dt''\, [\dot{A} - \dot{A}]_{t'}\bigl([\dot{A} - \dot{A}]_{t''} \coth{\hbar\beta(x)\omega\over2}\cos\omega (t'-t'')\nonumber\\ &&\qquad\qquad\qquad\qquad\qquad -i[\dot{A} + \dot{A}]_{t''}\sin\omega (t'-t'')\bigr)\Bigr)\;, \end{eqnarray} where the spectral density of the effective bath of independent oscillators is the (generally) temperature dependent quantity \begin{equation}\label{Ieff} I\bigl(\beta(x),\omega;x\bigr) = {4\omega\sinh{\hbar\beta(x)\omega\over2} \sum_{l,m}\vert J_{lm}(x)\vert^2\; \delta\bigl(\omega- {E_l(x)-E_m(x)\over\hbar}\bigr) \over\hbar\sum_{l}e^{-\beta(x) E_l(x)}}\;. \end{equation} This effective environmental model describes physical optics in linear dielectric media, at all frequencies below the cut-off, and for all field strengths below thresholds for current generation. The failure of our model to describe conductors and non-linear media is clearly due to our neglect of charge motion and higher-order terms in ${1\over N}$, and so our recovery of linear optics is not simply a co-incidence. In the important and prevalent cases where free charges and non-linear effects are negligible, our result is indeed physically sound, even though our derivation may have appeared somewhat naive. In particular, our assignment of fixed positions to the molecules is certainly a very crude treatment, especially for gases; but our results can be checked by comparison with a more sophisticated analysis, in which the molecules are treated as an ideal gas whose initial state is described by a grand canonical ensemble. The only additional effects one finds are thermal broadening of the molecular spectra, and a Gaussian cut-off on the effective coupling of field modes with energies on the scale of the temperature (reflecting the smaller number of molecules possessing kinetic energies in this range). This more sophisticated analysis must assume that the gas is dilute, so that quantum statistics are not significant, as well as that $d\Gamma^{-3} >> 1$. It is worth noting that the fuller analysis does not require $d\alpha^3\Gamma^{-3} >>1$ for some small $\alpha$: the delocalization of the molecules will itself smear out the phases $e^{i\vec{k}\cdot\vec{x}_n}$, so that the smallest volume containing very many molecules need only be on the cut-off scale, and not so much smaller still that $e^{i\vec{k}\cdot\vec{x}_n}$ varies negligibly across it. This effect of the fuller treatment can be incorporated in an approach like ours above, by making the $x_n$ into stochastic variables, which fluctuate over distances on the order of $\Gamma^{-1}$. In the influence functional, we can then take the ensemble averages of all the locations, and obtain Eqn. (\ref{optics}) even when $\alpha\to 1$. It is thus evident that the delocalization of molecules that obviates $\alpha$ need not be coherent. For solids and liquids delocalization is not so obviously sufficient to eliminate $\alpha$, but since they are denser, we can retain $\alpha$ and still achieve a cut-off in the high UV range. Thermal broadening and cut-offs can also be incorporated phenomenologically, and so we have presented the cruder analysis with fixed molecular positions, in order to more clearly make the physical point that large numbers of molecules within a cut-off volume leads to effectively linear behaviour of an environment. (It is also in aid of this demonstration that we have been careful to employ the infra-red regulator $L$, for if we had assumed from the start a countable number of molecules and a continuum of field modes below any UV cut-off, we could never have achieved the correct high ratio of molecules to modes. In this instance, the IR regulator is not just mathematical pedantry, but is actually necessary to express some important physics.) Since thermal motion and various sources of dissipation on the molecular excitations will broaden the spectral lines, we will assume that $I(\omega,\beta;x)$ is a continuous function of $\omega$ --- though it may have sharp peaks around strong absorption lines. This will have the unphysical effect of giving the environment an infinite specific heat capacity, so that radiative heating and cooling will be neglected; but for most optical phenomena, and for the subjects discussed in this paper, this will not be important. The model we have arrived at encompasses all the physics of reflection and refraction, and absorption. It provides \begin{equation}\label{ImK} {\rm Im}K(\omega,\beta;x) = {\pi g^2\over2\Omega} I(\omega,\beta;x)\;, \end{equation} where $n(\omega,\beta;x)\equiv\sqrt{K(\omega,\beta;x)}$ is the complex index of refraction. The real part of $K$ is given, as it should be for a linear medium, by the Kramers-Kronig relation \begin{equation}\label{K-K} {\rm}K(\omega,\beta;x) = 1 + {2\over\pi}\int_0^\infty\!d\omega'\, {\omega'\over \omega'^2-\omega^2}\,{\rm Im}K(\omega',\beta;x)\;, \end{equation} taking the Cauchy principal part of the integral. (The formal derivation of these results is straightforward; the relation between the quantum theory and classical optics will be clarified in the remainder of this paper.) Our model also describes thermal radiation, albeit without heating or cooling of sources and sinks. Nevertheless, for simplicity in the remainder of this paper we will assume perfect spatial homogeneity. In this limit, the Fourier modes of the field decouple, even though they interact with the environment. Each field mode thus constitutes a harmonic oscillator linearly coupled to its own private bath of independent oscillators, with a continuous spectral density. And so we obtain a conclusion which will allow us to apply the results of many apparently idealized studies of decoherence to a real and important physical phenomenon: electrodynamics in a homogeneous linear dielectric medium is, within the physically tenable assumptions and approximations we have made, a realization of harmonic quantum Brownian motion in the independent oscillator model. \section{Pointer states} Having mapped our field theoretic problem onto the problem of harmonic Brownian motion in an independent oscillator environment, we are now able to determine the pointer states of the field, in a straightforward way. We first review a clear-cut procedure for identifying pointer states: the {\it predictability sieve}\cite{Zurek2,ZHP}. We extend slightly the argument of Ref. \cite{ZHP}, in which certain squeezed states are shown to minimize linear entropy, and also to yield the smallest von Neumann entropy generation among all Gaussian initial states. Here we show that these same states actually minimize von Neumann entropy against unrestricted variations of the initial states. Pointer states are those states which are preferred by decoherence, in a process that may be termed ``environmentally induced superselection''. A generic quantum state will tend to evolve into a probabilistic mixture of pointer states. The suppression of quantum interference between these states makes the parameter space of pointer states the natural phase space of the classical limit of the quantum system in question. The predictability sieve identifies the pointer states by demanding that the environmentally induced splitting of a quantum state into non-interfering branches be stable: the branches must not rapidly branch in their turn. A pointer state must remain as pure as possible despite environmental decoherence. A concrete expression of this requirement is that pointer states minimize the growth of the entropy. We therefore wish to use our propagator (\ref{rdmprop}) to compute the reduced density operator $\hat{\rho}(t)$ that evolves from some pure initial state with wave function $\psi_i$. From this we will obtain the von Neumann entropy $S(t) = - \rm{Tr} \hat{\rho}(t)\ln\hat{\rho}(t)$ of this density operator, as a functional of $\psi_i$. Extremizing $S(t)$ with respect to variations of $\psi_i$ then identifies those initial states that acquire the least entropy by time $t$. Since we must ensure that our variations maintain the normalization of the initial state, we must solve the constrained variational problem \begin{equation}\label{Smin} \rm{Tr}\Bigl[ (\ln\hat{\rho} + 1){\delta\hat{\rho}\over\delta\psi_i} = \lambda \psi_i^\Bigr]\;, \end{equation} for some Lagrange multiplier $\lambda$. In general, entropy evolves in a complicated way during Brownian motion, and this procedure becomes too difficult; but since we are concerned here with decoherence, and not with such other effects as dissipation and thermalization, we select the simple model which has Ohmic spectral density and in which the dissipation rate $\gamma\to 0$. We let the temperature become infinite, such that $\gamma T$ remains finite, and the environmental noise becomes white. In this limit, decoherence for a single oscillator is characterized by the dimensionless quantity \begin{equation} D\equiv 8{\gamma k_B T\over \hbar\Omega^2}\;, \end{equation} where $\Omega$ is the frequency of the Brownian oscillator --- which in our case is a Fourier mode of the quantum field, so that $\Omega = ck$. Since all our field modes decouple, we will first focus on a single mode, and write $A$ and $A'$ without subscripts to refer to its amplitude. (To avoid complex numbers, we will assume that we are discussing Fourier sine and cosine modes, and rectangular polarizations.) The single-mode part of the density matrix propagator, in this weak coupling, high temperature limit, is \begin{eqnarray}\label{1prop} \rho(A,A';t)&=&{\Omega\over2\pi\hbar\sin\Omega t} \int\!dA_idA'_i\,\Bigl(\rho(A_i,A'_i;0)\nonumber\\ & &\qquad\times\exp\Bigl[{i\Omega\over2\hbar\sin\Omega t} [(A^2-A'^2 + A_i^2 - A_i'^2)\cos\Omega t - 2(AA_i - A'A_i')]\Bigr]\nonumber\\ & &\qquad\times\;\exp-{\Omega D\over4\hbar\sin^2\Omega t} \Bigl[\bigl((A-A')^2+(A_i-A_i')^2\bigr) (\Omega t -\sin\Omega t\cos\Omega t)\nonumber\\ & &\qquad\qquad\qquad - 2(A-A')(A_i-A_i') (\Omega t\cos\Omega t - \sin\Omega t)\Bigr]\;. \end{eqnarray} The mixed state density matrix that evolves from any initial squeezed state, according to (\ref{1prop}), can be diagonalized explicitly. We present the results for an arbitrary squeezed state in the Appendix; here we quote only a particularly relevant special case, namely the one-parameter family of initial states with $\rho(A,A';0) = \psi(A,\tau)\psi^(A',\tau)$ for \begin{equation}\label{psiAt} \psi(A,\tau) = Z e^{-{\Omega\over2\hbar}\sigma(\tau)A^2}\;. \end{equation} Here $Z$ is a normalization constant, and \begin{eqnarray}\label{sigma} \|\sigma(\tau)\|^2 &=& {2\Omega\tau + \sin2\Omega\tau\over 2\Omega\tau - \sin2\Omega\tau}\nonumber\\ {\rm Im}\bigl(\sigma(\tau)\bigr) &=& {2\sin^2\Omega\tau\over 2\Omega\tau - \sin2\Omega\tau}\;. \end{eqnarray} The quantity $\|\sigma(\tau)\|^{-1}$ is the ``squeezing factor'' for these states. For a given final time $t$, we will consider the initial state $\psi(A,\tau)\vert_{\tau=t}$. By the final time, this state will have evolved into a state with the density matrix \begin{eqnarray}\label{rhot} \rho(A,A';t)&=& \sqrt{\Omega{\rm Re}(\sigma)\over\pi\hbar\Lambda} \exp-\Bigl({\Omega{\rm Re}(\sigma)\over4\hbar\Lambda}\Bigl[(A+A')^2 + \Lambda^2 (A-A')^2\nonumber\\ & &\qquad\qquad\qquad\qquad -2i\bigl(D\sin^2\Omega t + {\rm Im}(\sigma)\bigr)(A^2-A'^2)\Bigr]\Bigr)\nonumber\\ &=& {2\over\Lambda +1} e^{i{\Omega{\rm Re}(\sigma)\over2\hbar\Lambda} \bigl(D\sin^2\Omega t + {\rm Im}(\sigma)\bigr)(A^2-A'^2)} \sum_{n=0}^\infty \Bigl({\Lambda-1\over\Lambda+1}\Bigr)^n\phi_n(A)\phi^* n(A') \;, \end{eqnarray} where $\Lambda \equiv 1 + D\sqrt{(\Omega t)^2 - \sin^2\Omega t}$. The $\phi_n$ happen to be the energy eigenfunctions of a harmonic oscillator with natural frequency $\omega = \Omega {\rm Re}(\sigma)$: \begin{equation}\label{eigen} -\hbar^2{d^2\ \over dA^2}\phi_n(A) + [\Omega{\rm Re}(\sigma)]^2 A^2\phi_n(A) = (2n+1)\hbar\Omega{\rm Re}(\sigma)\phi_n(A)\;. \end{equation} This precise form of $\rho(A,A';t)$ has the convenient property that \begin{eqnarray}\label{larry} \langle A\|\ln\hat{\rho}(t)\|A'\rangle &=& \exp\Bigl[i{\Omega{\rm Re}(\sigma)\over2\hbar\Lambda}\bigl(D\sin^2\Omega t + {\rm Im}(\sigma)\bigr)(A^2-A'^2)\Bigr]\nonumber\\ & &\qquad\qquad\qquad\times \Bigl(C_1 - C_2\Bigl[\hbar^2{d^2\ \over dA'^2} - \bigl(\Omega{\rm Re}(\sigma)\bigr)^2 A'^2\Bigr]\Bigr) \,\delta(A-A')\;, \end{eqnarray} where $C_1$ and $C_2$ are constants that may readily be computed from Eqn. (\ref{rhot}). We can also determine from Eqn. (\ref{1prop}) the operator valued functional ${\delta\hat{\rho}(t)\over\delta\psi_i(A_i)}$, for any $\psi_i$. Even where the initial state is our special squeezed state $\psi(A_i,\tau)\vert_{\tau=t}$, carefully chosen with regard to the final time $t$, this operator variation is somewhat complicated. Its diagonal matrix elements, though, are quite simple: \begin{equation}\label{curly} \langle A\|{\delta\hat{\rho}(t)\over\delta\psi_i(A_i)}\|A\rangle = \sqrt{\Omega\over2\pi\hbar\eta(t)} \psi^(A_i,t)\,e^{-{\Omega\over2\hbar\eta(t)} [A-(\cos\Omega t - i\sigma^(t)\sin\Omega t)A_i]^2}\;, \end{equation} where $\eta(t)\equiv \sigma^\sin^2\Omega t +{D\over2}(\Omega t - \sin\Omega t\cos\Omega t) - i\sin\Omega t\cos\Omega t$. It is easy to see that the property (\ref{curly}) reflects conservation of ${\rm Tr}\hat{\rho}$. The variation also has another property, much less trivial (and with a much more tedious derivation): \begin{eqnarray}\label{moe} \lefteqn{\Bigl[\bigl(\hbar^2{d^2\ \over dA^2} - [\Omega{\rm Re}(\sigma)]^2 A^2\bigr) e^{i{\Omega{\rm Re}(\sigma)\over2\hbar\Lambda} \bigl(D\sin^2\Omega t + {\rm Im}(\sigma)\bigr)(A^2-A'^2)}\, \langle A\| {\delta\hat{\rho}(t)\over\delta\psi_i(A_i)} \|A'\rangle \Bigr]_{A=A'}}\nonumber\\ &=& \Bigl(C_3(t)\Bigl[A -A_i(\cos\Omega t - i\sigma^\sin\Omega t)\Bigr]-C_4(t) A\Bigr)\nonumber\\ & &\qquad\qquad\times \Bigl[A -A_i(\cos\Omega t - i\sigma^\sin\Omega t)\Bigr] \nonumber\\ &&\qquad\qquad\qquad\qquad\times\, \exp-\Bigl[{\Omega\over2\hbar\eta(t)} [A-(\cos\Omega t - i\sigma^(t)\sin\Omega t)A_i]^2\Bigr] \;, \end{eqnarray} where $C_3(t)$ and $C_4(t)$ are functions whose exact form will be irrelevant to our discussion. Combining Eqns. (\ref{larry}), (\ref{curly}), and (\ref{moe}), we find that \begin{eqnarray} \lefteqn{\int\!dAdA'\,\langle A'\|[1+\ln\hat{\rho}(t)]\|A\rangle \langle A\|{\delta\hat{\rho}(t)\over\delta\psi_i(A_i)}\|A'\rangle} \nonumber\\ &=& [1+C_1(t) -{\hbar\eta(t)\over\Omega}C_2(t)C_3(t)]\psi^*(A_i,t)\;. \end{eqnarray} This is the constrained Euler-Lagrange equation (\ref{Smin}); the initial state $\psi(A,t)$ of Eqn. (\ref{psiAt}) therefore minimizes the entropy of the reduced density matrix at time $t$. This is identical to the result obtained in Ref. \cite{ZHP} for the initial state which minimizes linear entropy at time $t$. {}From Eqn. (\ref{1prop}), it is apparent that the phase space translation \begin{equation} \|\psi_i\rangle \to e^{{i\over\hbar}a\hat{p}_A} e^{-{i\over\hbar}b\hat{A}} \|\psi_i\rangle\;, \end{equation} where $\hat{p}_A$ is the canonical momentum operator conjugate to $\hat{A}$, leads to a unitary transformation of the density operator at time $t$: \begin{eqnarray}\label{V} \hat{\rho}(t) &\to& \hat{V}\hat{\rho}(t)\hat{V}^\dagger\nonumber\\ \hat{V} &=& e^{{i\over\hbar}({p\over\Omega}\sin\Omega t + x\cos\Omega t)\hat{p}_A} e^{-{i\over\hbar}(x\cos\Omega t - p\Omega\sin\Omega t)\hat{A}} \;. \end{eqnarray} The entropy of the state at time $t$ is thus invariant under such phase space translations of the initial state. Therefore, the two-parameter set of initial wave functions $e^{-{i\over\hbar}bA}\psi(A-a,t)$ also minimize $S(t)$. We conjecture that these are the only such minimizing states. There is thus no initial pure state which will have minimum entropy at all times. However, after a few dynamical times, the squeezing and the imaginary part of $\sigma(t)$ become steadily less significant. Also, the state which instantaneously minimizes $S(t)$ oscillates back and forth, over time, around unsqueezed coherent states. It is therefore clear that the least mixing initial states, on average over a few dynamical times, are the coherent states. While it is only our special limit $\gamma\to 0$, $T\to\infty$ that has allowed us to implement the predictability sieve analytically, calculations in other models\cite{Gallis}, as well as general arguments\cite{ZHP}, support the conclusion that coherent states can be considered the natural pointer states for harmonic oscillators coupled linearly to an environment. From the more general analysis of our first section, it then follows that they are the natural pointer states of an electromagnetic field mode in a linear medium. Since the field modes are decoupled, an initial direct product state of all modes will evolve into a final direct product of mixed states, for which the total entropy will be the sum of the individual entropies. It is therefore clear that the generalization of Equation (\ref{Smin}) to all $8(K+1)^3$ decoupled field modes is solved by a direct product of such squeezed states, and that coherent states of all $8(K+1)^3$ oscillators are the optimum pointer states for the field in a homogeneous medium. Furthermore, it follows trivially from Equation (\ref{V}) that the c-number parameters $x(t),p(t)$ labelling the pointer states obey the classical equations of motion. As long as environmental noise is not so strong that the Gaussian peak in (\ref{rhot}) becomes too broad too fast, it is clear that classical mechanics provides a good effective description of the evolution of the pointer states. (Of course, the existence of classical histories follows so trivially from our instantaneous definition of pointer states only because the dynamics of our model is linear.) While coherent states of single oscillators are typically interpreted as localized particles, a coherent state of a quantum field is a vacuum state displaced by an `external' or `background' field configuration. The localization associated with decoherence occurs not in the positions of particles, but in the amplitudes of field modes. In this way one can understand that the classical physics which emerges from quantum electrodynamics, in the presence of a linear environment environment, will naturally be a field theory and not a many-body particle theory. We emphasize that this result is a significant addition to the observation that one can obtain field equations as classical limits of quantum dynamics. After all, the equation of motion for a quantum harmonic oscillator is exactly the same as it is in the classical case, but this does nothing towards providing a classical interpetation for a ``Schr\"odinger's Cat'' state. One must consider decoherence in order to establish the crucial additional point that the pointer states of the quantum system, in the presence of an environment, behave in a sufficiently classical manner. In the present problem, we have done this, and observed that the emergent pointer states are classical field configurations --- a fact which is empirically familiar, but does not follow at all from the free quantum field theory. We have therefore made contact, via the predictability sieve, between decoherence in quantum Brownian motion and the standard field theoretic notion of a classical background field. We have confirmed that such background fields really do behave classically, in that quantum interference between distinct background field configurations is rapidly eliminated by a dissipative medium, and that the coherent quantum states labelled by these background fields are themselves the states least affected by decoherence. We now turn to the other side of the field theoretic coin, and consider how photons excited above a background field may be affected by the environment. \section{Quantum halos} The first point to be made is that our model for the environment is not intended to describe a sensitive detector. It is a very poor model for an ultra-high-gain amplifier, such as is required to detect single quanta. So while our discussion concerns the emergence of classical electrodynamics, we do not really address quantum measurement itself. Leaving aside the issue of actually detecting photons, however, we still have a point to address. Before a photon reaches such a special environment as a film plate, we know from several classic experiments that it maintains quantum coherence, despite propagating through air or other media that are described by our model. A naive application of one-particle results to the case of a photon might make this seem problematic, but in fact the explanation is very simple. We have found that coherent states are decohered least, on average over several dynamical times, of all initial pure states. For a single harmonic oscillator evolving under (\ref{1prop}), it is well known that a ``Schr\"odinger's Cat state'' \begin{equation} \|\psi\rangle = c_1 e^{-{i\over\hbar}p_1\hat{A}} e^{{i\over\hbar}x_1\hat{p}_A}\|0\rangle + c_2 e^{-{i\over\hbar}p_2\hat{A}} e^{{i\over\hbar}x_2\hat{p}_A}\|0\rangle \end{equation} formed by superposing two coherent states, {\it well separated in $(a,b)$-space}, will decohere thoroughly and rapidly. At a time $t={2n\pi\over\Omega}$, the reduced density matrix that has evolved from this initial state will be \begin{eqnarray}\label{rho2npi} \rho(Q,Q';{2n\pi\over\Omega}) &=& \Bigl[{M\Omega\over\hbar\pi (1+2n\pi D)}\Bigr] \sum_{i,j=1}^2 \exp \Bigl(-{M\Omega\over4\hbar (1+2n\pi D)} {\cal R}_{ij}\Bigr) \nonumber\\ {\cal R}_{ij} &=& \Bigl[Q+Q' - x_i - x_j\Bigr]^2 + (1+2n\pi D)^2\Bigl[Q-Q' - {x_i-x_j\over1+2n\pi D}\Bigr]^2 \nonumber\\ & &\qquad -{4i\over M\Omega}\Bigl[Qp_i - Q'p_j + n\pi D (Q-Q')(p_i+p_j) - {x_ip_i - x_jp_j\over2}\Bigr]\nonumber\\ & &\qquad + 2n\pi D\Bigl[(x_i-x_j)^2 + \bigl({1\over M\Omega}\bigr)^2(p_i-p_j)^2\Bigr]\;. \end{eqnarray} {}From the last line of (\ref{rho2npi}), we can infer that the timescale for decoherence of the two pointer states states is \begin{equation}\label{tD} t_D = \Bigl[\Omega D (\Delta^2 -1)\Bigr]^{-1}\;, \end{equation} where \begin{equation}\label{Delta} \Delta^2 \equiv {M\Omega\over2\hbar}\Bigl[\bigl(x_1-x_2\bigl)^2 + \bigl({p_1-p_2\over M\Omega}\bigr)^2\Bigr]\;. \end{equation} It is obvious that equation (\ref{tD}) makes sense only when $\Delta^2 >1$. (We will consider below what happens to a superposition of two orthogonal states whose wave functions are concentrated within a phase-space distance of order $\Delta^2 =1$.) It is also obvious that during processes that occur over timescales shorter than some $t_{max}$, quantum coherence between two superposed states will {\it not} decay significantly, if the Wigner functions for the two states are concentrated within a phase space disc of radius $\sim 2\sqrt{t_{max}/\Omega D}$. This rather elementary fact is of considerable conceptual importance, as it clearly exhibits the limitations of environmental decoherence. From it, we can deduce a succinct refinement of our formulation of environmental-induced superselection, introducing a new term that complements the notion of a `pointer state': {\it Every pointer state is surrounded, in Hilbert space, by a `quantum halo' of states which are not sharply distinguished from it by the environment.} The size of the quantum halo of a pointer state is in general a function both of the strength of environmental noise, and of the maximum timescale over which it is allowed to act. However, there is an upper bound to this timescale, past which the whole notion of environmentally-induced superselection breaks down anyway, and neither pointer states nor quantum halos are particularly meaningful. We can deduce from Eqn. (\ref{rhot}) that the entropy for an initially coherent state after $n$ periods of motion is \begin{equation}\label{S0} S({n\pi\over\Omega}) = \bigl(1+{n\pi D\over2}\bigr) \ln\bigl(1+{n\pi D\over2}\bigr) - {n\pi D\over2}\ln\bigl({n\pi D\over2}\bigr)\;. \end{equation} When $nD={2\over\pi}$, the entropy even for a pointer state is equal to that of a statistical mixture of four equally probable pure states. It is clear that decoherence this severe does not produce superselection, but merely swamps the system with environmental noise. For our pointer states to be meaningful, therefore, we must have $D\Omega t<<1$. This means that, as long as decoherence is mild enough to be achieving superselection instead of mere randomization, the quantum halo surrounding a coherent state is bound to extend to at least a radius $\Delta \sim 1$. Even this minimal halo supports a two dimensional subspace of states: the first excited energy eigenstate of the oscillator resides within the quantum halo of the ground state (see Figure 1), and a similar halo state may easily be found for any coherent state. Generalizing straightforwardly from the single oscillator to the electromagnetic field in a homogeneous medium, we can conclude that every background field configuration is surrounded by a quantum halo of photons. This explains why a dielectric medium does little to eliminate quantum interference in a double slit experiment, and why propagation through an environment will not necessarily destroy the long-range entanglements of an EPR pair. \section{Conclusion} The pointer states of the quantum electromagnetic field, propagating in a homogeneous linear dielectric medium, are coherent states. When decoherence is not so strong that it merely swamps the field with noise, coherent states evolve almost freely. The pointer states therefore behave as classical field configurations, evolving under the classical equations of motion. We have therefore provided an insight into the emergence of classical electromagnetism from quantum electrodynamics. Each pointer state of a quantum field is surrounded, in Hilbert space, by a quantum halo --- a set of states which are negligibly decohered from the pointer state over whatever time period is of interest. When the environmental noise is weak enough that it does not significantly degrade the pointer states themselves, this quantum halo is large enough to contain at least a few particles, excited above the background classical field configuration represented by the pointer state. We have thus recovered the familiar field-theoretic dichotomy between background classical fields and $N$-particle excitations. The relative immunity of the particle excitations to decoherence, in comparison with the strong decoherence of superpositions of distinct pointer states, explains the co-existence of effective classical electrodynamics and coherent propagation of photons. The $n$-particle excitations are not localized by our homogeneous environment. All localization occurs in the space of Fourier mode amplitudes, and not in position space. This result is consistent with the ``indications'' arrived at in the studies by K\"ubler and Zeh\cite{KublerZeh}, and by Kiefer\cite{Kiefer}; but it is in strong contrast with what one might expect decoherence to do, based on a naive translation of the particles studied in quantum Brownian motion into field quanta. Although a linear dielectric medium does not have the avalanche instability of a cloud chamber, one would still look to the concept of decoherence for a general explanation of why electrons, for example, should behave in the classical limit as localized particles. Indeed, we do expect that this is the case: the linearly coupled field we have analyzed in this paper differs in an essential way from the environmental coupling of a typical matter field. For matter fields, the interaction Hamiltonian with an environment tends to be bilinear, rather than linear, in creation and annihilation operators. The crude rule of thumb, that pointer states should be eigenstates of operators that approximately commute with the interaction Hamiltonian, suggests then that the pointer states for matter fields should be $n$-particle states rather than coherent states. And while a photon can only impart information to a localized environmental degree of freedom by being absorbed by it, material particles can scatter, surviving the information transfer without having to rely on a rare recurrence event to re-emit them. This observation supplements the usual reference to the statistics of fermions and bosons, since even a charged scalar field would be expected to have particle, rather than field, pointer states. Finally, we point out that the field-like nature of a Bose condensate of atoms must be examined with proper consideration for the dynamical origin of the chemical potential, which can be considered to mimic a linear interaction (capable of creating or annihilating particles) with an unobserved environment. \section{Appendix} The reduced density matrix $\rho(Q,Q';t)$ which evolves under the propagator (\ref{1prop}), from an initial squeezed state \begin{equation} \langle Q\|\psi_I\rangle = \Bigl({M\Omega{\rm Re}(C)\over\pi\hbar}\Bigr)^{1\over4} e^{-{M\Omega\over2\hbar}C Q^2} \end{equation} with complex $C$, is given by \begin{equation} \rho(Q,Q';t) = \sqrt{{M\Omega{\rm Re}(C)\over\pi\hbar}} \sqrt{\alpha(t)} e^{-{M\Omega\over4\hbar}\alpha(t)[(Q+Q')^2 + \beta(t)(Q-Q')^2 -2i\lambda(t)(Q^2-Q'^2)]}\;. \end{equation} The dimensionless functions $\alpha$, $\beta$, and $\lambda$ are defined as \begin{eqnarray} [\alpha(t)]^{-1} &\equiv& [{\rm Re}(C)]^2\sin^2\Omega t + D{\rm Re}(C)(\Omega t - \sin\Omega t\cos\Omega t) + [{\rm Im}(C)\sin\Omega t - \cos\Omega t]^2\;;\nonumber\\ \beta(t) &\equiv& {\rm Re}(C)[1 + D^2(\Omega^2t^2 - \sin^2\Omega t)] + D \|C\|^2 (\Omega t - \sin\Omega t\cos\Omega t)\nonumber\\ &&\qquad\qquad\qquad + D(\Omega t +\sin\Omega t\cos\Omega t) - 2D{\rm Im}(C)\sin^2\Omega t\;;\nonumber\\ \lambda(t) &\equiv& [\|C\|^2 - 1]\sin\Omega t \cos\Omega t - {\rm Im}(C)\cos2\Omega t + D{\rm Re}(C)\sin^2\Omega t\;. \end{eqnarray} Final states evolving from other initial squeezed states may be obtained trivially from the result we exhibit, by applying translation operators as discussed in Section III. \begin{references} \bibitem{GSI} H. Grabert, P. Schramm, and G.-L. Ingold, {\em Phys. Rep.} {\bf 168}, 115 (1988), and references therein. \bibitem{CaldeiraLeggett} A.O. Caldeira and A.J. Leggett, {\em Physica} {\bf A 121}, 587 (1983), and references therein. \bibitem{HPZ} B.L. Hu, J.P. Paz, and Y. Zhang, {\em Phys. Rev.} {\bf D 45}, 2843 (1992), and references therein. \bibitem{Schwinger} J. Schwinger, {\em J. Math. Phys} {\bf 2}, 407 (1961). \bibitem{Ullersma} P. Ullersma, {\em Physica} {\bf 32}, 27 (1966). \bibitem{UnruhZurek} W.G. Unruh and W.H. Zurek, {\em Phys. Rev.} {\bf D 40}, 1071 (1989). \bibitem{ZHP} W.H. Zurek, S. Habib, and J.P. Paz, {\em Phys. Rev. Lett.} {\bf 70}, 1187 (1993). \bibitem{Zurek1} W.H. Zurek, {\em Phys. Rev} {\bf D 24}, 1516 (1981); {\em Phys. Rev.} {\bf D 26}, 1862 (1982). \bibitem{KublerZeh} O. K\"ubler and H.D. Zeh, {\em Ann. Phys.} {\bf 76}, 405 (1973). \bibitem{Kiefer} Claus Kiefer, {\em Phys. Rev.} {\bf D46}, 1658 (1992). \bibitem{FeynmanVernon} R.P. Feynman and F.L. Vernon, jr., {\em Ann. Phys. (N.Y.)} {\bf 24}, 118 (1963). \bibitem{Zurek2} W.H. Zurek, in {\em The Physics of Time Asymmetry}, ed. J.J. Halliwell, J. Perez-Mercader, and W.H. Zurek (Cambridge, 1992); an earlier version may be found in {\em Prog. Theor. Phys.} {\bf 89}, 281-312 (1992). \bibitem{Gallis} M.R. Gallis, in Proceedings of the 4th Drexel Symposium on Quantum Nonintegrability. \end{references} \eject \proclaim Figure Captions. Figure 1 Plots of the reduced density matrix $\rho(Q,Q';t)$ at $t=0$ (figs. 1a and 1a') and at $t = 2n\pi\Omega^{-1}$, with $D$ chosen so that $2n\pi D = 0.05$ (figs. 1b and 1b'). The horizontal axes measure $Q$ and $Q'$, in units of $\hbar\over M\Omega$. Figures 1a and 1b show the evolution of a superposition of two coherent states $\sqrt{2}\cos(\hat{p}/\sqrt{\hbar M\Omega})\|0\rangle$. It is clear that decoherence is much advanced in Fig. 1b, but that the two diagonal peaks are essentially intact, as they represent pointer states. In contrast, Figures 1a' and 1b' show the evolution of an initial superposition of energy eignenstates, ${1\over\sqrt2}(\|0\rangle + \|1\rangle)$. The first excited state lies within the quantum halo of the ground state, and the superposition has not suffered any discernible loss of coherence. \end{document}	arXiv
Issues ▾ Issues For authors ▾ For authors Statistics ▾ Statistics About JASSS Contact JASSS Home > 23 (4), 1 Model Exploration of an Information-Based Healthcare Intervention Using Parallelization and Active Learning Chaitanya Kaligotlaa, Jonathan Ozika, Nicholson Colliera, Charles M. Macala, Kelly Boydb, Jennifer Makelarskib, Elbert S. Huangb and Stacy T. Lindaub aArgonne National Laboratory and the Consortium for Advanced Science and Engineering at the University of Chicago, United States; bThe University of Chicago, United States Other articles by these authors In JASSS From Google Journal of Artificial Societies and Social Simulation 23 (4) 1 <https://www.jasss.org/23/4/1.html> DOI: 10.18564/jasss.4379 Save citation... Text HTML BibTex Ref. Manager EndNote ProCite Received: 05-Mar-2019 Accepted: 15-Jul-2020 Published: 31-Oct-2020 This paper describes the application of a large-scale active learning method to characterize the parameter space of a computational agent-based model developed to investigate the impact of CommunityRx, a clinical information-based health intervention that provides patients with personalized information about local community resources to meet basic and self-care needs. The diffusion of information about community resources and their use is modeled via networked interactions and their subsequent effect on agents' use of community resources across an urban population. A random forest model is iteratively fitted to model evaluations to characterize the model parameter space with respect to observed empirical data. We demonstrate the feasibility of using high-performance computing and active learning model exploration techniques to characterize large parameter spaces; by partitioning the parameter space into potentially viable and non-viable regions, we rule out regions of space where simulation output is implausible to observed empirical data. We argue that such methods are necessary to enable model exploration in complex computational models that incorporate increasingly available micro-level behavior data. We provide public access to the model and high-performance computing experimentation code. Keywords: Agent-Based Modeling, Model Exploration, High-Performance Computing, Active Learning Other articles with these keywords In JASSS From Google CommunityRx (CRx), developed with the support of a Health Care Innovation Award from the U.S. Center for Medicare and Medicaid Innovation (CMMI), is an information-based health intervention designed to improve population health by systematically connecting people to a broad range of community-based resources for health-maintenance, wellness, disease management and care-giving (Lindau et al. 2016, 2019). The CRx program was initially implemented on Chicago's South Side and covered 16 ZIP codes, encompassing an area of 106 square miles with a population of 1.08M people. Two studies were conducted between 2012-2016, a large observational study and a pragmatic clinical trial, to evaluate the impact of the CRx intervention on health, healthcare utilization, and self-efficacy. The CRx intervention is centered around the "HealtheRx" (or HRx), a 3-page printed list of community resources personalized to a patient's characteristics, location, and diagnoses (Lindau et al. 2016). Community resources prescribed in the HRx include resources to address basic needs (e.g., food and housing), physical and mental wellness (e.g., fitness, counseling), disease management (e.g., smoking cessation, weight loss), and care-giving (e.g., respite care for a person with dementia.) The goal of the information intervention via the HRx is to effect an increase in utilization in community resources and, ultimately, health. The HRx represented the primary vector of information diffusion regarding community resources, while the diffusion of this information into the patient's social and interaction networks represented the secondary diffusion vectors. An information-based intervention like CRx spreads non-linearly across the population via the individuals' networks, to other people who did not initially receive the intervention and, in turn, can impact their behaviors by affecting their decision-making related to utilizing community resources. Accurate assessment of such multi-level interventions necessitates moving beyond traditional methods like prospective trials (Kaligotla et al. 2018; Lindau et al. 2016). A computational agent-based model (ABM) was thus developed to investigate the impact of the CRx intervention. The development of the CRx ABM was informed by empirical data from the CRx studies and other observational, experimental, and expert informant sources (Kaligotla et al. 2018). This paper describes the characterization of parameter space of the CRx ABM through an implementation of an Active Learning (AL) (Settles 2012) model exploration (ME) algorithm at high-performance computing (HPC) scales using the EMEWS framework (Ozik et al. 2016), as a step towards model validation. Building valid computational models at scale Our broad research goal is to integrate clinical trial methodology with agent-based modeling, to enable in silico experimentation at scale, to inform clinical trial design and evaluation, and to amplify thereby the impact of individual-level clinical trials of information-based population health interventions. A challenge in using ABMs to analyze interventions or run computational experiments is in the validation that these computational models adequately represent the dynamics of interest. Building validated models requires a robust characterization of model parameter spaces to facilitate the calibration of model outputs against empirical observations. The characterization of parameter spaces, however, is often difficult to achieve in practice for complex, expensive-to-run models with large parameter spaces. Sufficient characterization of high-dimensional parameter spaces is usually infeasible using brute force techniques that attempt to span the space with an a priori defined set of points. Instead, adaptive, or sequential, heuristic techniques can be used: techniques that strategically sample larger parameter spaces by selectively applying computational budgets to more important regions, e.g., History Matching (Holden et al. 2018; Williamson et al. 2013), Gaussian process surrogate models (Baker et al. 2020; Kennedy & O'Hagan 2001), and stochastic equilibrium models (Flötteröd et al. 2011). These sequential model exploration techniques, however, are difficult to implement in a generalizable way and at scale. The computational complexity involved in building validated models (in terms of effort and computational power) impedes the broader applicability of ABMs and simulation models in general. In this paper, we describe our progress towards building a validated CRx ABM through large-scale model exploration techniques. We describe a random forest meta-model to iteratively classify and characterize the input parameter space into potentially viable and non-viable regions with respect to observed empirical data, as the initial step in validating our model. This stepwise tightening approach is similar to history matching (Holden et al. 2018) and is driven by the goal of reducing computational costs associated with the analysis of complex models. We also describe the parallelization techniques and frameworks that enable our model exploration approach on HPC resources. The rest of the paper is organized as follows: Section 2 describes the development and implementation of the CRx ABM. Section 3 introduces the AL workflow we implement to calibrate the CRx ABM against empirical observations. Section 4 describes parallelization techniques that enable model performance gains needed for efficient model exploration, followed by a description of the HPC AL workflow implemented with the EMEWS framework. Section 5 demonstrates how the AL algorithm characterizes the large CRx ABM parameter space. We conclude and highlight future research in Section 6. CRx ABM and Implementation This section provides an overview of the CRx ABM and its implementation. We created a synthetic environment with statistical equivalence to the geographical region that corresponds to the CRx study area, on which we model the diffusion of information and its effect on agent decision behavior. A detailed description of the model is available in Kaligotla et al. (2018). Synthetic population The synthetic environment in the CRx model consists of 3 entities - a population of agents ($\mathbf{P}$), resources ($\mathbf{R}$), and clinics ($\mathbf{C}$). Using SPEW (Synthetic Population and Ecosystems of the World) data from (Gallagher et al. 2018), we create a population of 802,191 agents with static sociodemographic characteristics matching the population of the South and West sides of Chicago (ages 16 and older since the HRx information sheet for patients younger than 16 was typically given to an accompanying adult), each of whom are assigned to a household, and work or school location. Additionally, using CRx and MAPSCorps data from www.mapscorps.org (Makelarski et al. 2013), we build a synthetic physical environment in which agents move, consisting of 4,903 unique places that provide services, including health-related services. The set of places also includes health care clinics where an agent could receive a HRx and exchange information with other agents about community resources. Agents perform an activity during each 1-hour time-step of a 24-hour activity schedule. Agents are randomly assigned an activity schedule each simulated day based on their sociodemographic characteristics. The activity schedules of the agents were developed using the American Time Use Survey (ATUS) dataset (https://www.bls.gov/tus/data.htm). Activities are mapped onto relevant services that are associated with geo-located resources in ($\mathbf{R}$). This approach allows us to connect two separate data sets, SPEW and ATUS, and maintain a stochastic variability of the population activities and use of resources based on matching demographic characteristics. Agent decision behavior We model an agent's health maintenance behavior through an agent-based decision model. Agents can encounter two general types of activities in their activity schedules. The first type of activities are related to health maintenance behaviors, which correspond to types of services that are frequently referred to by HRxs. These activities provide a choice in behavior - either to decide in favor of engaging in a health maintenance related activity (use of wellness or health promoting community resource) in question, denoted by Decision A, or not, denoted by Decision B. The second type of agent activities does not contain any decision-making component. The agent will simply do the activity in question at the specified time if a relevant service type is known to the agent. We model the use of a resource $j \in \mathbf{R}$, by agent $i \in \mathbf{P}$, at time $t\in {0,1,2,\dots,T}$ as a decision model whose functional form is given by: $$\begin{cases} \textit{Decision A} & \text{If}\ \frac{\beta_{i,j}^t}{(\gamma_j \times \delta^t_{i,j})} > \alpha_i \\ \textit{Decision B} & \text{otherwise} \end{cases}: \ \alpha, \beta, \gamma , \delta \in (0,1)$$ (1) Here $\alpha_i$, the agent activation score, denotes an individual agent's intrinsic threshold for health maintenance activities, resource score $\beta_{i,j}^t$ denotes an agent's dynamic level of knowledge of the particular resource and its associated benefits, $\gamma_j$ is a measure of resource inertia, the inherent difficulty associated with performing an activity at a resource, and $\delta_{i,j}^t$ represents the distance threshold between an individual agent and the location for a resource/activity. An agent thus chooses to perform a health related activity, Decision A, only when their activation threshold $\alpha_i$ is exceeded by a function of their perceived characteristics of resource $j$ at time $t$. Agent activation score values ($\alpha_i$ ) are derived from Table 1 in (Skolasky et al. 2011), stochastically matched to individual agents via demographic characteristics. An agent with a high activation score implies a higher level of difficulty in performing tasks, and ceteris paribus, is therefore less likely to perform health related activities. The resource score ( $\beta_{i,j}^t$) is dynamic (explained in the following subsection) and evolves over time based on "dosing," a term we use in this paper to refer to an agent's exposure to information about a particular resource at a specific time. The higher the resource score, the higher the likelihood of an agent performing an activity. Delta ($\delta_{i,j}^t$) accounts for a "location effect" with respect to an agent's decision – the higher the distance to an activity, the lower the likelihood of an agent performing an activity. We derive resource-specific thresholds for what we classify as low-medium-high resource distances using survey data from Garibay et al.(2014). Information diffusion We model the dynamic diffusion of information as a function of agent interactions, the source of information dosing, and the evolution of an agent's knowledge with respect to the information. Agents following their activity schedules will find themselves geographically co-located with other agents. All co-located agents interact and exchange information about resources with some probability. We model this probabilistic information exchange as dependent on an agent's propensity to share information, and the amenability of the activity itself to information sharing, e.g., the propensity to share information while an agent's activity is "socializing and communicating with others" will be higher than an activity like"doing aerobics." We associate a scaled propensity score for different types of activities representing the different likelihoods of an agent sharing information with other agents who are co-located at that time – none (e.g., sleeping), low (e.g., doing aerobics), medium (e.g., grocery shopping), and high (e.g., socializing and communicating with others). The propensity scores and their mapping to the activity types were obtained from expert surveys. Different sources for information dosing are considered in our model (to account for the relative trustworthiness of the source of information) and are denoted by a set $X\in \{\text{Doctor, Nurse, PSR, Use, Peer}\}$. $\epsilon_{x} \in (0,1), x\in X$ is a dosing parameter representing the effect of each dosing source's relative trustworthiness on an agent's $\beta$. $n_{ij}^{t}=1$ denotes the instance of information dosing for agent $i$ about resource $j$, at time $t$ by some source $x\in X$. An agent's evolving level of knowledge for a particular resource and its associated benefits is described through the time evolution of $\beta_{i,j}^t$, given by Equation 2, where $\lambda$ is a decay parameter over time, representing the effect of a resource receding from an agent's attention, possibly replaced with knowledge about other resources, and $\epsilon_{x} \in (0,1)$, the dosing parameter for the dosing source's effect on recall, and $n_{ij}^{t}=1$ denotes the instance of information dosing for agent $i$ about resource $j$, at time $t$ by a dosing source $X$. $$\forall i, \forall j, \ \beta_{i,j}^{0} =f_{\beta}(I), \ \beta_{i,j}^{t} = \begin{cases} \lambda\times\left(\beta_{i,j}^{t-1}\right)^{\epsilon_{x} (1-\beta_{i,j}^{t-1}) + \beta_{i,j}^{t-1}} & \text{If} \ n_{ij}^{t}=1 \\ \lambda\times\left(\beta_{i,j}^{t-1}\right) & \text{otherwise} \end{cases} : \lambda \in (0,1) , x\in \{1,2,\dots,5\} $$ (2) Initialization of $\beta$ follows from a function $f_{\beta}(I)$, which considers the implicit assumption that an agent (at time 0) knows between 10 and 100 resources in a near (1-mile radius) distance around their home location, and 1 to 5 resources each in the medium (less than 3 miles) and far (more than 3 miles) distances. Each of these initial known resources has a $\beta$ score equal to $\kappa$, while all other resources are unknown to the agent, hence having an effective $\beta$ score of 0. The CRx intervention was modeled by replicating the original algorithms that generated an HRx based on patient demographics, home address, health and social conditions, and preferred language (Kaligotla et al. 2018). CRx model implementation The CRx model is implemented in C++ using the Repast for High Performance Computing (Repast HPC) (Collier & North 2013) and the Chicago Social Interaction Model (chiSIM) (Macal et al. 2018) toolkits. Repast HPC is a C++-based agent-based model framework for implementing distributed ABMs using MPI (Message Passing Interface). chiSIM, built on Repast HPC, is a framework for implementing models that simulate the mixing of a synthetic population. chiSIM itself is a generalization of a model of community associated methicillin-resistant Staphylococcus aureus (CA-MRSA) (Macal et al. 2014). In the chiSIM-based CRx model, each agent in the simulated population resides in a place (a household). Places are created on a process and remain there. Agents move among the processes according to their activity profiles. When an agent selects a next place to move to, the agent may stay on its current process, or it may have to move to another process if its next place is not on the agent's current process. A load-balancing algorithm is applied to the synthetic population to create an efficient distribution of agents and places, minimizing this computationally expensive cross-process movement of agents and balancing the number of agents on each process (Collier et al. 2015). The CRx model and the workflow code used to implement the parameter space characterization experiments are publicly available (at the following URL: https://github.com/jozik/community-rx). Model Exploration using Active Learning Model calibration refers to the process of fitting simulation model output to observed empirical data by identifying values for the set of calibration parameters (Kennedy & O'Hagan 2001). We note that we use the term model exploration to refer to the family of approaches used for characterizing model parameter spaces, including model calibration. Model calibration techniques described in the extant literature generally fall into two approaches: direct calibration methods and model-based methods (Xu 2017). Direct calibration methods generally utilize direct search methods, e.g., stochastic approximation (Yuan et al. 2012) or discrete optimization (Xu et al. 2014), to explore the parameter space and identify calibration values that minimize differences between model outputs and empirical observations. Model-based methods, including Bayesian calibration methods, make use of surrogate models, e.g., Gaussian process (Kennedy & O'Hagan 2001) or stochastic equilibrium models (Flötteröd et al. 2011), to combine empirical observations with prior knowledge, and to obtain a posterior distribution on a model's calibration parameters. Xu (2017) provides an overview of the merits and limitations of each of these approaches. Baker et al. ( 2020) provides a comprehensive review of Gaussian process surrogate models used for calibrating stochastic simulators. The method described in this paper falls into sequential model-based approaches. The increasing availability of computational resources has emboldened advances in the exploration and calibration of simulation models using new approaches. Han et al. (2009), for instance, introduce a statistical methodology for simultaneously determining empirically observable and unobservable parameters in settings where data are available. Reuillon et al.( 2015) also describes an automated calibration process, computing the effect of each calibration parameter on overall agent-based model behavior, independently from others. Particularly relevant to large scale parameter space exploration, Lamperti et al. (2018) describes a combination of machine learning and adaptive sampling to build a surrogate meta-model that combines model simulation with output analysis to calibrate their ABM. Another multi-method example of large-scale parameter space exploration is the use of dimension reduction techniques, common in climate science. (Higdon et al. 2008), for instance, describe the combination of dimension reduction techniques and regression models to enable computation at scale. Also common in this class of model calibration is the use of history matching as an alternative to probabilistic matching (Williamson et al. 2013). Like traditional calibration, history matching identifies regions of parameter space that result in acceptable matches between simulation output and empirically observed data. This method has been used to calibrate complex simulators across domains - oil reservoir modeling (Craig et al. 1997), epidemiology (Andrianakis et al. 2015), and climate modeling (Edwards et al. 2011; Holden et al. 2018) Our primary goal in this paper is to characterize the parameter space of the CRx ABM, i.e., identify parameter value combinations where model output is potentially compatible with empirical data. The model exploration techniques described in this section partition the parameter space into potentially viable and non-viable regions. This approach has the benefit of allowing subsequent analyses to focus only on a subset of the parameter space that is likely to yield empirically representative behaviors. The stepwise tightening of viability constraints is not unique to our work but is a common ingredient in sequential parameter sampling algorithms such as sequential Approximate Bayesian Computing (ABC) approaches (Hartig et al. 2011; Holden et al. 2018; Rutter et al. 2019). Holden et al. ( 2018), for instance, describes an approach where history matching is initially performed to rule out regions of space that are implausible, as sequential Approximate Bayesian Computing (ABC) approaches without the use of history matching are computationally costly. The approach we describe in this Section is similarly the first step in validating our model while keeping within finite computational budgets. In the following paragraphs, we describe the empirical targets for the output of the CRx ABM, define the parameter space of the model, and then introduce an AL workflow to characterize the model's parameter space. Empirical targets for model output To identify viable outputs from the CRx model, we use empirical data for total weekly visit volumes to 10 select clinics over three years for patients age 16 years and older, obtained from Lindau et al. ( 2016). To control for the natural variation in the observed weekly clinic visit volumes, we manually select stable time regions (periods without visible ramp up or ramp down variations in weekly visit volumes) for each of the 10 clinics, from which we measure the visit statistics (mean and standard deviation) to serve as the target outputs for the same clinics in the CRx ABM. Figure 1 depicts the observed empirical visit volumes for one exemplar clinic, showing the selected stable region. Manual selection of stable regions allows us to avoid tail events, which are outside the scope of our model, as are the seasonality of clinic visits . Table 1 details the mean and standard deviation for stable weekly visit volumes for the 10 clinics of interest, and represent the targeted outputs for the CRx ABM. The 10 clinics were selected from the set of CRx model clinics $\mathbf{C}$, based on the availability of empirical data from Lindau et al. (2016), after ignoring clinics attached to schools (which have a significant population under 16 and thus out of our model scope), clinics with too few weekly visits to obtain robust statistics ($< 20$), and emergency rooms (which included patient visits for reasons other than health maintenance, and thus were out of scope). Figure 1. Weekly visits volumes for clinic 1273871. X axis represents week number since reference date (4 Mar 2013) of first observation. Stable period considered for parameter space characterization indicated in by orange bracket. Table 1: Targets for Model Output: Empirical Data of Weekly Clinic Visit Volumes for 10 Clinics by people living in the 16 ZIP codes of the CRx study region. Clinic Code Mean of Weekly Visits Standard Deviation of Weekly Visits 1273871 267.14 74.75 1274377 28.60 11.14 The objective function used to characterize the model output calculates the $z$ score ($z=\frac{x-\mu}{\sigma}$) from the model-generated mean clinic visits for each of the 10 clinics during the third week of the model run, against the empirically observed weekly mean and standard deviation ($\mu,\sigma$) of visits for these clinics (as specified in Table 1). The model exhibited stable behavior by week 3 (steady-state output seen in the number of weekly visits to the 10 selected clinics,) and thus, we calculate the $z$ score using data from that period. 1 Considering week 3 instead of a later week, also allowed us to fit in more model evaluations within the same computational budget. A threshold condition was defined using the z score within which the model outputs are deemed to adequately resemble empirically observed weekly clinic volumes. The threshold condition used was a $z$ score for each of the 10 clinics in the range of +4 and -4 for the third week of the model run, i.e., $-4 \geq \frac{x_i^t-\mu_i}{\sigma_i} \geq +4$ where $x$ is the total number of agents who visited clinic $i$ (Clinic_Code in Table 1) in week $t=3$, and ($\mu_i,\sigma_i$) are the empirically observed mean and standard deviation for clinic $i$, as detailed in Table 1. Note that while empirical observations in Table 1 imply that 0 visits to some of these clinics would individually result in a $z$ score within $($ -4,+4 $)$ , it, however, does not satisfy our threshold condition. The implemented threshold condition necessitates that all 10 clinics individually satisfy the $z$ score threshold condition. Figure 16 in Appendix D shows the weekly visit numbers across all clinics across all 173 potentially viable parameter combinations; it is observed that while a low number of visits (<10) account for approximately 20% of observations, there are no zero visits during the week for any of the 10 clinics despite the use of the z-score threshold of +/-4. Figure 2 shows the marginal distributions for the z-scores for each of the 10 clinics across all 173 potentially viable points (refer Figure 17 in Appendix D for marginal distributions of actual visit numbers recorded by the simulation for each of the 10 clinics across all potentially viable parameterizations.) Given our stated goal of characterizing the parameter space into potentially viable and non-viable regions as the initial part of a stepwise tightening of viability constraints, this threshold condition results in a restricted parameter space for subsequent, more constrained model calibration. 2 Appendix D includes additional discussions on the sensitivity of the $z$ score threshold on potentially viable parameterizations, as well as the stepwise tightening of viability constraints. Figure 2. Histograms of Z-score distribution for each of the 10 clinics across all 173 viable points. CRx ABM parameter space and discretization Clinic visits by agents within the CRx ABM occur through the decision calculus described in Section 2.4. The implementation of the CRx ABM has several parameters that have the potential to influence agents' clinic visits (refer to Table 5 in the appendix for a complete list of parameters used in the CRx ABM). For this study, we sub-selected the five most potentially influential parameters in Table 2, based on their potential to directly affect an agent's decision calculus in Equations 1 and 2, and for which we could not directly infer values from the extant literature. Given the targets for model output, potentially viable regions within the parameter space of the CRx model are regions representing the valid combinations of input parameter values that satisfy the threshold condition defined above. Given our inability to infer parameter values from empirical data, we utilize an AL model exploration method described in the following sections to identify potentially viable parameter combinations. The AL algorithm that we implemented utilizes a discretization of the parameter space. The algorithm selects from unique parameter points (combinations of parameter values across all dimensions) for evaluation and prediction. We selected 15 discrete points along each parameter dimension to provide a sufficient level of granularity with respect to the model output targets and to be able to show uncertainty gradients between the non-viable and potentially viable regions while keeping within an easily manageable memory footprint for the AL algorithm. 3 For each combination of the five parameters in Table 2 (which represent a unique point in parameter space), the CRx model produces weekly clinic visits for each of the 3 simulated weeks. Table 2: Varied CRx Parameters dosing.decay rate of knowledge attrition $0.9910$ $0.9994$ $0.0006$ dosing.peer dosing from network peer $0.8000$ $0.9500$ $0.1000$ gamma.med resource inertia of performing an moderate level of activity $1.0000$ $3.0000$ $0.1430$ propensity.multiplier multiplier applied to propensity of information sharing $0.5000$ $1.5000$ $0.0714$ delta.multiplier multiplier applied to distance threshold $\delta_l$, $\delta_m$ , $\delta_h$ $0.5000$ $1.5000$ $0.0714$ Adaptive sampling methods for ME Given the targets for model output and threshold condition for determining potentially viable regions for model exploration, the challenge becomes one of computational feasibility. As described earlier, the CRx ABM parameter space consists of $15^5$ points (each point corresponding to a unique combination of parameter values) across 5 dimensions. Characterizing this parameter space into potentially viable and non-viable regions thus presents a computational challenge – evaluating 759,375 points by brute force methods or producing a sufficiently dense covering of the space is computationally prohibitive. Instead, we strategically sample the parameter space by selectively applying computational budgets to more important regions. There do exist multiple sampling methods to deal with large parameter spaces, each, however, present their own problems. Grid search methods (dividing the parameter space into equivalent grids), while naturally parallel, suffer from evaluations being made independent of each other, which in high-dimensional parameter spaces can result in high computational costs without corresponding information value. Bergstra & Bengio (2012), for instance, provide empirical and theoretical proof that even random selection is superior to grid selection in high-parameter optimization. Random search, however, is generally not as good as the sequential combination of manual and grid search (Larochelle et al. 2007) or other adaptive methods (Bergstra & Bengio 2012). Other commonly used a priori sampling methods include orthogonal sampling (Garcia 2000) and Latin Hypercube Sampling (LHS) (Tang 1993). While both methods ensure that sampled points (for evaluation) are representative of overall variability, orthogonal sampling ensures that each subspace is evenly sampled. However, these methods also suffer in high-dimensional spaces, where checking parameter combinations and interactions across all dimensions are computationally costly. The challenge of computational feasibility in large parameter spaces often results in computational compromises, like restricting the fidelity, size or scale of the model, or limiting the types of computational experiments that are undertaken. An adaptive approach to ME is thus necessary for models with large multi-dimensional parameter spaces, like the CRx ABM. While a number of different adaptive methods, including evolutionary algorithms and ABC could potentially be used (Ozik et al. 2018), an AL approach, described in the next section, maps naturally to the problem. This approach is generalizable and, as we demonstrate in this paper, computationally feasible with new approaches to HPC-scale techniques. Active learning for ME We use an AL (Settles 2012) approach to characterize the large parameter space of the CRx ABM, using meta-models (Cevik et al. 2016). The AL method combines the adaptive design of experiments (Jin et al. 2002) and machine learning, to iteratively sample and characterize the parameter space of the model (Xu et al. 2007). In this work, we use the EMEWS framework (described later in Section 4.1), to integrate an R-based AL algorithm, enabling its application at HPC scales, as reported in (Wozniak et al. 2018). We use an uncertainty sampling strategy in our AL approach. We employ a random forest classifier on already evaluated points and choose subsequent samples close to the classification boundary, i.e., where the uncertainty between classes is maximal, to exploit the information of the classifier. With the availability of concurrency on HPC systems, samples at each iteration of the AL procedure are batch collected (and evaluated) in parallel. Candidate points are first clustered, and an individual point is chosen from this cluster. This approach decreases the overlap in reducing classification uncertainty and ensures a level of diversity in the sampled points (Xu et al. 2007). Each sampling iteration also includes randomly sampled points, striking a balance between the exploitation of information that the classifier provides with an exploration of the parameter space to prevent an incomplete meta-model due to premature convergence. The pseudo-code for our AL algorithm is shown in Figure 3. [AL] Parallel evaluations of the objective function F(), the CRx model simulation, are performed in lines 11 and 19 over a sampling of parameter space. At each iteration, the sampled results are fed into the random forest classifier R (lines 13 and 21). At the end of the workflow, the final meta-model predictions are generated for the unevaluated parts of the parameter space. Figure 3. Pseudo-code of implemented AL algorithm Enabling Model Exploration using EMEWS and Parallelization EMEWS (Extreme-scale Model Exploration with Swift) As ABMs improve their ability to model complex processes and interventions, while also incorporating increasingly disparate data sources, there is a concurrent need to develop methods for robustly characterizing model behaviors. The EMEWS framework (Ozik et al. 2016) was a response to the ubiquitous need for better approaches to large-scale model exploration on HPC resources. EMEWS, built on top of the Swift/T parallel scripting language (Wozniak et al. 2013) enables the user to directly plugin: 1) native-coded models (i.e., without the need to port or re-code) and 2) existing ME algorithms implementing potentially complex iterating logic (e.g., Active Learning Settles 2012). The models can be implemented as external applications accessed directly by Swift/T (for fast invocation), which is the method that was used in this work. However, applications can also be invoked via command-line executables, or Python, R, Julia, JVM, and other language applications via Swift/T multi-language scripting capabilities. ME algorithms can be expressed in the popular data analytics languages Python and R. We implemented an R-based Active Learning ME algorithm for this work. Despite its flexibility, an EMEWS workflow is highly performant and scalable to the largest cutting-edge HPC resources. This large-scale computation capability provided us the ability to efficiently characterize the parameter space of the CRx ABM and discover parameter space regions compatible with empirical data. Parallelization to enable model exploration The CRx model is a computationally expensive application, primarily due to the interactions between the large number of agents and places. At each time step, every agent in each place can potentially share information with every other person in that place. In the absence of any parallelization, this entails iterating over each place in the model in order to iterate over all the agents in that place so that each person can share information with every other person in that place. In order to mitigate this expense and reduce the run time such that adaptive model exploration was feasible, we parallelized the model both across and within compute cores. Inter-process parallelization The model was parallelized across compute processes using the chiSIM framework (Macal et al. 2018). chiSIM-based models such as CRx are MPI applications in which the agent population and the agents' potential locations are distributed across compute processes, that is, across MPI ranks. Each process corresponds to an MPI rank and is identified with a rank id. Agents move across ranks while places remain fixed to a particular rank. A chiSIM model's time step consists, at the very least, of each agent determining where (i.e., what place) it moves to next and then moving to that place, possibly moving between process ranks in doing so. Once in that place, some model specific co-location dependent behavior occurs, for example, disease infection dynamics, or in the case of CRx, information sharing. The model code runs in parallel across processes and thus the iteration over each of the 482,945 places in the model described above becomes n number of parallel iterations over p number of places, where n is the number of process ranks, and p is the number of places on that rank. The success of this cross-process distribution is dependent on how well the process loads are balanced. Too many places or agents on a process results in a longer per time step run time for that process and other processes sitting idle while that slower process finishes. In addition, we also want to minimize the potentially expensive cross-process movement that occurs when agents move between places on different process ranks. Our load balancing strategy described in detail in Collier et al. (2015), uses the Metis (Karypis & Kumar 1999) graph partitioning software to allocate ranks to processes, balancing the number of agents per rank while minimizing cross-process movement. The movement of agents between places can be conceptualized as a network where each place is vertex in the network and where an edge between two places is created by agent movement between those two vertices. It is this network that Metis partitions. In previous work, (Macal et al. 2014) and (Ozik et al. 2018), this movement was dictated by relatively static agent schedules and the network was created directly from those schedules without having to run the model itself. In the CRx case, agents' movement schedules are selected at random every simulated day from a demographically appropriate set of schedules. In the absence of a static schedule , we ran the model for 14 simulated days in order to capture the stochastic variation in agent movement, logging all individual agent movements between places. These data were then used to create the weighted place-to-place network for Metis to partition using the strategy outlined in Collier et al. (2015). We experimented with distributing the model over different numbers of ranks - 1, 8, 16, 32, 64 and 128 - running the model for a simulated week and recording the run time to execute each simulated day. These experiments were performed on Bebop, an HPC cluster managed by the Laboratory Computing Resource Center at Argonne National Laboratory (further details on the Bebop cluster are given in Section 4.4). The results can be seen in Figure 4 . Performance does not scale linearly, which is to be expected given the overhead of moving agents between process ranks. The intention here is to achieve good enough performance such that the model exploration can be run within the maximum job run time, memory, and node count constraints of our target HPC machines. In that respect, the 64 or 128 rank configurations are sufficient. We utilize the 128 rank configuration for the AL workflow described below. Figure 4. Distributed Model Run Times for a Simulated Week We were also interested in how the model distribution related to the geographic distribution of places across process ranks. This was prompted by previous attempts in the literature to distribute ABMs geographically (Lettieri et al. 2015), where it was determined that geographically based model distribution presented difficulties in achieving performance gains based on the uneven density of agent populations and, as a result, agent activities. In Figure 5 we show all places across Chicago, including households, schools, workplaces, and services, assigned to each process rank in the 128 rank configuration (while we restricted the synthetic population to 16 zip codes, we included places across all Chicago locations). While there are differences in the patterns observed across ranks, one cannot readily detect any process-specific geographic clustering. This observation strengthens the argument against regarding geographic extents as natural partitions for load balancing when complex travel patterns across geographic regions are involved. However, we do know that the information on the HRxs distributed to our agents is based on geographic proximity to the agent household location. Hence, we could expect that, through geographically local information exchanges between agents engaging in geographically local activities at service providers, the information about service providers could retain some locality. In fact, we do see a correspondence between the geographic location of a service and its process rank in Figure 6, using the 8 rank configuration for clarity. This suggests that the location of services and the information flow about them produce correlations in agent movement between them that, in turn, create useful partitions for model load balancing. Figure 5. Locations of households, schools, workplaces, and service providers across all Chicago locations, assigned to each process rank in the 128 rank load balancing configuration. Figure 6. Geographic locations of service providers colored by their assigned process rank in the 8 rank load balancing configuration. Intra-process parallelization In addition to parallelizing the model by distributing it across processes, we also experimented with intra-process parallelization by multi-threading the model using the OpenMP (Dagum & Menon 1998) application programming interface. In distributing the model across processes, we effectively reduced the number of agents and places on each process. And thus, the loop iteration through all the places in the model that occurs every time step was that much smaller (i.e., faster) and occurred in parallel. Intra-process parallelization with threads used the openMP pragma directive #pragma omp parallel for to iterate through this loop in parallel. An omp parallel for essentially divides the range spanned by the loop into some number of sub-ranges, each sub-range then executes the code within the loop on a separate parallel thread. However, this code executed in the loop iteration discussed above contains many random draws from a shared random stream. In a multi-threaded context, this could, and in fact most certainly would, lead to race conditions, stream corruption and application crashes, for example, when one thread is drawing from the stream while another is simultaneously updating the stream after its random draw. There are sophisticated strategies for implementing parallel random streams in simulations (see for example Freeth et al. 2012), but we implemented a simpler solution. OpenMP provides a function call to retrieve a running thread's unique numeric id. On model initialization we created a number of random streams equal to the number of threads and associated each with a thread's unique id, allowing each thread to retrieve its own random number stream by thread id. We ran the multi-threaded version of the model, again distributed over different numbers of ranks, setting the number of threads to 6, and using two different threading schedule directives: static (the default) and dynamic . With the static directive each thread is assigned a chunk of the total number of iterations in a fixed fashion and iterations are divided equally among threads. With the dynamic directive, each thread is assigned an iteration when that thread becomes available for work. Six threads were chosen as the best tradeoff between how many cores to allocate to a process for individual threads and how many processes to pack into a HPC node (a collection of cores). The results of these runs can be seen in Figure 7. In all of the cases, we can see that the dynamic directive runs are the fastest for the same number of ranks and that the difference between static and dynamic run times is especially prominent in those runs that were distributed over smaller number of ranks. This difference is likely due to the non-uniform distribution of agents among places. For example, a typical household may have 4 or so agents in that household, while a larger workplace or clinic may have much more. Given that the length of time it takes to iterate through all the places assigned to a particular thread is dependent on the number of agents in those places, a thread that contains places with larger numbers of agents takes longer to complete when compared to a thread with places with fewer numbers of people. The dynamic directive naturally load balances the place iteration by assigning a place to a thread when a thread is available for work. Consequently, while places that require a longer runtime will occupy a thread for a while, other threads are still capable of accepting new places, resulting in a more even spread of work across all the threads. The drawback of the dynamic directive is that the order of random draws is now no longer predictable but rather dependent on run time and the threading library, and as a result, we cannot be sure that runs using the same random seed will necessarily produce the same results. We hope to examine this issue of run reproducibility under different threading regimes (static, dynamic, and others) in future work. Figure 7. Threaded Model Run Times for a Simulated Week AL Results In this section, we first evaluate the relative importance of 5 model parameters, describe the evolution of the AL algorithm across iterations, present the parameter space characterization results of the CRx ABM and report on the meta-model performance. The AL runs were performed on the Cray XE6 Beagle at the University of Chicago, hosted at Argonne National Laboratory, and on the Bebop cluster, managed by the Laboratory Computing Resource Center at Argonne National Laboratory. Beagle has 728 nodes, each with two AMD Operton 6300 processors, each having 16 cores, for a total of 32 cores per node. Each node has 64 GB of RAM. Bebop has 1024 nodes comprised of 672 Intel Broadwell processors with 36 cores per node and 128 GB of RAM and 372 Intel Knights Landing processors with 64 cores per node and 96 GB of RAM. Additional development was done on the Midway2 cluster managed by the Research Computing Center at the University of Chicago. Midway2 has 400 nodes, each with 28 cores and 64 GB of memory. The AL parameter space characterization runs were executed in 4 rounds. The first round consisted of an LHS sweep used to seed the first AL round (see Section 5.4), followed by 3 AL rounds. The second and third AL rounds were restarted from the serialized final state of the AL algorithm from the previous round. Each of the 4 rounds was run on 215 nodes with 6 computational processes per node. The LHS sweep ran for 85 hours. AL Rounds 1 and 2 ran for 48 hours, and the 3rd AL round for 47 hours for a total of 228 hours (1,764,720 total core hours) for all 4 rounds. Each model run was distributed over 128 processes and was allocated 4 threads per process using the static threading directive. The combination of inter- and intra-node parallelism provides an estimated 18-fold runtime speedup from the base undistributed and unthreaded model. Parameter importance The random forest classifier is an ensemble of decision trees, where each tree is trained on a subset of the data and votes on the classification of each observation, variable importance can be calculated from the characteristics of decision trees. Two commonly used measures for importance are: the mean accuracy decrease, which is a measure of mean classification error, and the mean decrease in Gini, which is a measure of the weighted average of a variable's total decrease in node impurity (which translates into a particular predictor variable's role in partitioning the data into the defined classes). A higher Gini decrease value indicates higher variable importance and vice versa. Table 3 shows an analysis based on these two metrics for our results. Table 3: Random forest evaluation of parameter importance. assigned dimension id accuracydecrease Ginidecrease gamma.med d1 0.306 244.660 delta.multiplier d2 0.309 210.093 propensity.multiplier d3 0.184 115.937 dosing.decay d4 0.230 112.647 dosing.peer d5 0.174 117.695 Based on these results we assign the following designations in order of relative parameter importance: $d1 := gamma.med$, $d2 := delta.multiplier$, $d3 := propensity.multiplier$ , $d4 := dosing.decay$ and $d5 := dosing.peer$. We use this relative ordering of parametric importance to organize the visualizations of our results in the following sections. Characterization of CRx ABM parameter space with active learning The AL algorithm was seeded with the results of 240 evaluations where the sampled points (to evaluate) were selected via a Latin Hypercube Sampling. At each iteration step, 20 new points are evaluated, 10 points close to the classification boundary (exploitation) and 10 randomly sampled points (exploration); the class (i.e., potential viability or non-viability) of each parameter point is determined using its z-score as previously described (to statistically match the empirically observed clinic visits). Following the evaluations, the random forest model is updated with this new class information and generates predictions for out-of-sample (non-evaluated) points across the remaining parameter space. Following 58 iterations, the AL algorithm evaluated a total of 1400 unique points. The characterization (evaluated points and predictions for out-of-sample points) of this multi-dimensional parameter space following the AL workflow after 58 iterations is shown in Figure 8. Figure 8 provides a view into the 5-dimensional parameter space through a grid of 2D plots. Each cell in the grid plots the two most relevant dimensions, gamma.med (d1, x-axis) and delta.multiplier (d2, y-axis), against each other. The propensity.multiplier (d3) is varied across the overall rows and dosing.decay (d4) across the overall columns. The final dimension, dosing.peer (d5), is kept constant. Figure 8 thus represents a slice of 5-dimensional parameter space with 225 snapshots of d1 $\times$ d2 across 15 unique values each for d3 and d4, for a fixed unique value of d5 (0.907). Evaluated points are shown in red/green dots indicating non-viable / potentially viable points. Out of sample predictions are shaded as blue (non-viable), orange (potentially viable), and black (equiprobable), where the color gradient between blue to black to orange denotes the varying probability of classification. Figure 8. Characterization of the 5-dimensional parameter space at a fixed dosing.peer value. Each individual panel (square) depicts a 2-dimensional space of dimensions d1 (bottom x-axis, in black) $\times$ d2 (left y-axis in black), across dimension d3 (right y-axis in red) and d4 (upper x-axis in blue), for a unique value of d5 (dosing.peer = 0.907). Red/green dots indicate evaluated (non-viable/potentially viable) points in parameter space, and blue/orange regions correspond to out-of-sample predictions for non-viable/potentially viable regions while black represents equiprobable prediction. Evolution of parameter space characterization across AL iterations Figure 9 shows the progression of the AL algorithm for the subset of panels shown in Figure 8 within the dashed bounding box, demonstrating the evolution of evaluated points and the corresponding random forest model predictions for out-of-sample points. We see in Figure 9 that as the AL progresses across the four sets of panels (representing almost-equally spaced iterations 0, 19, 38 and 58), the initial prediction boundary (black regions) is gradually refined as additional points are evaluated across the iterations, while the meta-model prediction for the rest of the parameter space (blue/orange regions depicting the non-viable / potentially viable predictions) is clarified with less uncertainty in the model prediction. Figure 9. Progression of the AL workflow across a 5x5 slice of the parameter space described in Figure 8 for iterations 0, 19, 38, and 58. Each individual panel (square) depicts a 2-dimensional space of dimensions d1 (bottom x-axis, in black) $\times$ d2 (left y-axis in black), across dimension d3 (right y-axis in red) and d4 (upper x-axis in blue), for a unique value of d5 (dosing.peer = 0.907). Red/green dots indicate points in the parameter space that were evaluated as non-viable/potentially viable points. Yellow halos indicate newly selected points for evaluation since the previous panel iteration. Blue/orange regions correspond to out-of-sample meta-model predictions for evaluating the rest of the parameter space into non-viable/ potentially viable regions. Black regions indicate uncertainty between class predictions. Next we consider the panel labeled A in the set of panels corresponding to iteration 0 in Figure 9 – shown in Figure 10 , across iterations 0, 19, 38 and 58. At iteration 0 and 19 in Figure 10, while no point was yet positively evaluated, we still see an evolution of the meta-model predictions for out-of-sample regions. There is a positive evaluation by iteration 38 (green point with yellow halo), and we observe that as a result of the positive evaluation, the meta-model prediction for the potentially viable regions (orange) in the parameter space around this point has been refined with increased certainty (increased color intensity). Figure 11 features the individual panel labeled B in Figure 9, also across iterations 0, 19, 38 and 58. We see new points in iteration 0 and iteration 58 evaluated as non-viable (red dots with yellow halo). It is also observed that regions of parameter space with high uncertainty (black regions) decrease in width and area, sharpening the distinction between non-viable and potentially viable (blue/orange) regions. Figure 11 thus shows reduced uncertainty about the non-viable regions following a negative evaluation. Figure 10. Selective panel from Figure 9 showing progression of meta-model predictions across iteration 0, 19, 38 and 58, following a positive evaluation. Potentially viable regions are predicted with more certainty as the shaded regions representing uncertainty in meta-model predictions are reduced across iterations. The same color scheme as Figure 9 is used. Figure 11. Selective panel from Figure 9 showing progression of meta-model predictions across iteration 0, 19, 38 and 58, following a negative evaluation. Non-viable regions are predicted with reduced uncertainty following a negative evaluation. The same color scheme as Figure 9 is used. Parameter space evaluation and meta-model performance The AL algorithm evaluated a total of 1400 unique points in the discretized parameter space (out of a total of 759,375 possible points) after 58 iterations. Of these,173 points were classified as potentially viable, and are presented in Figure 12, with point sizes indicating the number of unique points at each collapsed 2-dimensional space point. One aspect to note is the strong correlation exhibited between parameters d1 and d2 and the sharp boundary within the parameter space that this produces. This observation shows an inverse relationship between the parameterization of resource inertia for moderate activities (d1) and the multiplier applied to the distance threshold (d2). This observed relationship suggests that this interplay between activity difficulty and distance to a resource is the primary driver within the set of five selected model parameters in determining resource use at the population level. Table 6 lists all 173 points. Figure 12. Points represent parameter combinations of 173 potentially viable points from 1400 evaluations after 58 iterations in 2 dimensions per panel. The size of each point in each panel corresponds to the frequency (N) of points across the remaining 3 dimensions. All points listed in Table 6. To analyze the performance of the random forest meta-model in classifying out-of-sample (non-evaluated) parameter space, we train the random forest algorithm on the evaluated points. Using a 3-fold cross-validation training method, we determine the Positive Predictive Value (PPV) measure of our meta-model prediction (Altman & Bland 1994a, 1994b). PPV is the proportion of true positives to the total positive classifications, i.e., $PPV=\frac{\text{True Positive}}{\text{True Positive} + \text{False Positive}}$. As PPV increases, we increase our certainty that a point classified as $X1$ is a True Positive. The classification of a point in the parameter space is a mapping function $f:s\to\{X0,X1\}$, of a probability score $s$ assigned by the trained meta-model, to a classification class ($X0$ or $X1$), based on a threshold measure (Lipton et al. 2014) $p_t$ , such that $f = X1 \ \text{for} \ s\geq p_t, X0 \ \text{otherwise}$. Table 4 shows the PPV against different thresholds as well as the expected number of $X1$ classified points in the non-evaluated parameter space, for each threshold. We observe that higher thresholds result in higher PPV, but with smaller total numbers of points classified as $X1$. As the threshold increases, the meta-model generates more False Negatives in exchange for a higher rate of True Positives. We chose a threshold of 0.9 for the meta-model. Table 4: Threshold: Positive Predictive Value (PPV) and corresponding expected number of $X1$ classified points (potentially viable) in out-of-sample (non-evaluated) parameter space. Positive Predictive Value Number of $X1$ Points 0.50 0.480 30056 0.80 0.740 5392 As mentioned earlier, the random forest model predicted potentially viable points from the out-of-sample region (the remaining 757,975 non-evaluated points) with an associated probability, ranging from 0.0 (non-viable) to 0.5 (equiprobability of classification) to 1.0 (potentially viable). 2212 potentially viable points are predicted from the out-of-sample region when considering a 0.9 threshold probability; these are shown in Figure 13. Figure 13. Points represent parameter combinations of 2212 potentially viable points inferred by the meta-model with probability $\geq 0.90$ from out-of-sample region after 58 iterations in 2 dimensions per panel. The size of each point in each panel corresponds to the frequency (N) of points across the remaining 3 dimensions. The set of 173 potentially viable points ( Figure 12) and 2,212 predicted potentially viable points (Figure 13) identifies the region in the parameter space of the CRx model where further investigations are warranted. That is, the main contribution of these points is in characterizing the rest of the parameter space as non-viable, thereby avoiding the expenditure of unnecessary computing resources on other regions, i.e., those not likely to yield empirically corresponding model outputs. These subsequent analyses can use, e.g., stepwise tightening of viability thresholds or surrogate model-based optimization approaches to yield model outputs that are better aligned with empirical observations. As the model calibration is improved, the CRx ABM can increasingly be applied to CRx intervention scenarios as a computational laboratory to conduct in silico experimentation and drive implementation policy (for instance, identifying the most effective delivery mode, or ideal number of clinics to effect an increase in resource recall). As techniques for building ABMs become more sophisticated and the ABMs become more complex, it has become increasingly difficult to produce trusted tools that can be used to aid in decision and policy making. This is partly due to the computational expense of running large (e.g., urban-scale) models and partly due to the size of the model parameter spaces that need to be explored to obtain a robust understanding of the possible model behaviors. These two issues exacerbate each other when iterative parameter space sampling techniques are utilized to strategically sample the parameter spaces since iterative algorithms require at least some form of sequential evaluation of potentially long-running simulations. In this work, we presented an urban-scale distributed ABM of information diffusion built using the Repast HPC and chiSIM frameworks. Describing the application of a large-scale AL method, we characterize the parameter space of the CRx ABM to identify a sampling of potentially viable points, while characterization of the rest of the parameter space as non-viable. In the process, we developed intra-node and inter-node parallelization and load balancing schemes to enable an estimated 18-fold runtime speedup. The load balancing was done using the place-to-place agent-movement network, which produced geographic clustering of service providers, understood to be a demonstration of the local nature of resource information sharing between the agents. With the increased efficiency achieved through parallelization in running individual models, we were able to consider an iterative algorithm for characterizing the model parameter space. We exploited the polyglot and pluggable architecture of EMEWS framework and implemented an R-based random forest Active Learning algorithm, and ran it on 215 nodes of a HPC cluster to calibrate the CRx ABM. The run produced 173 potentially viable parameter combinations through direct evaluation and an additional 2212 parameter combinations inferred to be potentially viable, with an estimated 86% PPV, by a random forest model trained on the evaluated points. The parameter space characterization that was enabled by our approach represents one component of a comprehensive and iterative model development and validation cycle. Relative to the extant literature on model calibration (Section 3.1) and sequential sampling (Section 3.11), this paper presents the use of Active Learning as part of sequential model-based calibration methods, as seen in (Edwards et al. 2011; Holden et al. 2018), for large scale parameter space exploration. The combination of AL and parallelization techniques (described in Section 4.2) enables model exploration in simulators with large parameter spaces and helps mitigate (Bellman 1961)'s curse of dimensionality. Another important aspect for model validation is the establishing of plausibility for the mechanisms (Edmonds et al. 2019) at multiple model scales, including the parameterization of agent attributes, decision making, and behaviors. The conception of the CRx ABM originated after a discovery by experts in medicine about the spread of information, that was not adequately captured, assessed, or analyzed using individual-level models. The demonstrable and empirical need for an ABM ensured a close collaboration. In jointly developing the CRx ABM, we have repeatedly incorporated assessments of domain experts on model design, mechanisms, and results, and, perhaps most importantly, these inputs have been sought and incorporated from the early model design stage. The CRx ABM is an additional analytic tool for public health officials who are adopting new technologies such as geospatial statistics and predictive modeling. Potential users of these models include other academics, local and state public health departments, federal agencies such as the Centers for Disease Control and Prevention, health care systems, and health care payers. In the highly decentralized public health system of the U.S., open-source analytic tools that can capture and evaluate the interaction between service entities and community members are needed more than ever. One limitation of the present work is the potential bias resulting from excluding seasonality in agent behaviors to account for fluctuations in empirical clinic visit data. Furthermore, in order to simplify the scope of the parameter space characterization and produce meaningful results with the computational allocations we had access to, the five model parameters in Table 2 were selected based on expert opinions on model parameters deemed to be most relevant to agent clinic visits. However, this process may have omitted additional parameters with significant effects. For future work, the CRx ABM (with the potentially viable parameter points) can be used as a starting point on which to focus computing resources to tighten the viability thresholds and identify better-calibrated points, allowing us to use the CRx ABM to expand our investigations into the effects of information spread on population health outcomes. These analyses may also find that structural modifications are needed to align the model with empirical data more closely. Additional model exploration algorithms can also be examined, particularly those that have the ability to incorporate stochastic noise from model replicates, e.g., sequential design with Gaussian Processes (Binois et al. 2018), for producing robust uncertainty estimates of model outputs. Additionally, in order to be able to include larger parameter spaces for our computational experiments, the application of dimensional reduction techniques can be investigated, e.g., active subspaces (Constantine 2015). Finally, we aim to continue expanding our approaches to improved model efficiency and look at reproducibility under different intra-node threading regimes. The methods we describe in this paper enable computation at HPC scale. A direct potential application of such methods is particularly relevant to public health emergencies like the COVID-19 pandemic, where empirically-informed large scale models can help aid and inform public health authorities in decision-making. Our current research efforts extending some of the methods described in this paper and in (Ozik et al. 2018) involve modeling the spread of COVID-19 in Chicago. A subset of the authors (Macal, Ozik, Collier, Kaligotla) built the CityCOVID ABM, also based on the ChiSIM framework (Macal et al. 2018), which incorporates COVID-19 epidemiology and spread among the 2.7 Million people in the City of Chicago and across 1.2 Million geographical locations. We calibrate our model with daily reported hospitalizations and deaths. Model results are being provided to the City of Chicago and Cook County Public Health Departments as well as the Illinois Governor's COVID-19 Modeling Task Force. Our work contributes to the growing need for empirically-informed models using granular, local, and time-sensitive data to support decision-making during and in advance of public health and other crises. Figures 14 and 15 in Appendix D depict model output. Figure 14 shows the z scores from uncalibrated simulated weekly visits for each of the 10 clinics over 8 weeks, while Figure 15 compares the distribution of the total number of weekly visits in Week 3 and Week 4 of the calibrated simulation output. See, e.g., Rutter et al. (2019) for a description of the benefits of employing a stepwise narrowing of thresholds approach instead of narrower initial thresholds. While this discretization was chosen for the current work, it is possible that either a less or a more granular representation could provide useful insights as well. Investigations of optimal discretization for the CRx model is a possible future area of work. Model Documentation A comprehensive model description of the CRx ABM is described in (Kaligotla et al. 2018). The CRx ABM model is implemented using the Repast for High Performance Computing (Repast HPC) (Collier & North 2013) and the Chicago Social Interaction Model (chiSIM) (Macal et al. 2018) toolkits. The CRx ABM model code and the workflow code used to implement the parameter space characterization experiments are publicly available (at the following URL: https://github.com/jozik/community-rx). Research reported in this publication was supported by the National Institute on Aging of the National Institutes of Health R01AG047869 (ST Lindau., PI). This content is solely the responsibility of the authors and does not necessarily represent the official views of the National Institutes of Health's National Institute on Aging. This work is also supported by the U.S. Department of Energy under contract number DE-AC02-06CH11357. This work was completed in part with resources provided by the Research Computing Center at the University of Chicago (the Midway2 cluster), the Laboratory Computing Resource Center at Argonne National Laboratory (the Bebop cluster), and the University of Chicago (the Beagle supercomputer). Conflicts of Interest: ST Lindau directed a Center for Medicare and Medicaid Innovation Health Care Innovation Award (1C1CMS330997-03) called CommunityRx. This award required the development of a sustainable business model to support the model test after award funding ended. To this end, S. T. Lindau is the founder and co-owner of NowPow LLC and President of MAPSCorps, 501c3. Neither The University of Chicago nor The University of Chicago Medicine is endorsing or promoting any NowPow or MAPSCorps entity or its business, products, or services. A: Model parameters Table 5: List of CRx model parameters with values and ranges. Values / Range alpha $(\alpha)$ Activation level $\in(0,1)$ Derived from (Skolasky et al. 2011) delta $(\delta_{l},\delta_{m},\delta_{h})$ Distance threshold (low,med,high) $\delta_{l}=1;\ \ \delta_{m}=2;\ \ \delta_{h}=3$ $delta.multiplier$ multiplier applied to distance threshold $\in(0.5,\dots,1.5)$ in increments of $(0.0714)$ gamma.low $\left(\gamma_{low}\right)$ resource inertia of performing an easy activity $=1$ gamma.med $\left(\gamma_{med}\right)$ resource inertia of performing a moderate activity $\in(1,\dots,3)$ in increments of $(0.143)$ gamma.high $\left(\gamma_{high}\right)$ resource inertia of performing a difficult activity $=3$ propensity $(pr_{none},pr_{l},pr_{m},pr_{h})$ propensity of information sharing (none,low,medium,high) $pr_{none}=0.0001, pr_{l}=0.005,\ \ pr_{m}=0.025,\ \ pr_{h}=0.01$ $propensity.multiplier$ multiplier applied to $p.score$ $\in(0.5,\dots,1.5)$ in increments of $(0.0714)$ $dosing.decay$ $(\lambda)$ Rate of knowledge attrition $\in(0.9910,\dots,0.9994)$ in increments of $(0.0006)$ $dosing.doctor$ $\left(\epsilon_{doctor}\right)$ dosing from doctor $0.05$ $dosing.nurse$ $\left(\epsilon_{nurse}\right)$ dosing from nurse $0.15$ $dosing.psr$ $\left(\epsilon_{psr}\right)$ dosing from patient service representative $0.25$ $dosing.use$ $\left(\epsilon_{use}\right)$ dosing from use of resource $0.2$ $dosing.peer$ $\left(\epsilon_{peer}\right)$ dosing from network peer $\in(0.8,\dots,0.95)$ in increments of $(0.1)$ $dosing.kappa$ $\left(\kappa\right)$ dosing value for initialization $0.02$ B: Set of potentially viable points Table 6: Unique parameter combinations (numbered 1 - 173) that correspond to empirical observations via direct evaluation of 1400 points after 58 iterations. Unique Parameter Combination ID 2 1.43 1.07 0.93 0.99 10 1.29 1.36 1.29 1.00 100 1.86 0.93 0.71 1.00 C: Animation The following animation illustrates the progression of the AL workflow across the 5x5 slice of the parameter space described in Figure 9 across iteration runs 0 through 58. Each individual panel (square) depicts a 2-dimensional space of dimensions d1 (bottom x-axis, in black) $\times$ d2 (left y-axis in black), across dimension d3 (right y-axis in red) and d4 (upper x-axis in blue), for a unique value of d5 (dosing.peer = 0.907).Yellow halos indicate newly selected points for evaluation since the previous iteration. Red/green dots indicate points in the parameter space which were evaluated as non-viable/potentially viable points. Blue/orange regions correspond to out-of-sample meta-model predictions for evaluating the rest of the parameter space into non-viable/potentially viable regions. Black regions indicate uncertainty between class predictions. Figure 14. Progression of the AL workflow across the 5x5 slice of the parameter space described in Figure 9 across iteration runs 0 through 58. Each individual panel (square) depicts a 2-dimensional space of dimensions d1 (bottom x-axis, in black) x d2 (left5 (dosing.peer = 0.907). The same color scheme as Figure 9 is used. D: Analysis of simulation output In this section, we provide figures of various simulation outputs and corresponding analyses. We also include additional discussions on the sensitivity of the z score threshold on potentially viable parameterizations, as well as the stepwise tightening of viability constraints. Figure 15. Sample uncalibrated output from simulated weekly visits for each of the 10 clinics over 8 weeks, showing associated z-score for each clinic on x-axis and time in weeks on y-axis. Figure 15. Histogram for the total number of weekly visits in Week 3 and Week 4 across 173 potentially viable points for all 10 clinics. Weekly visits shown on x-axis and frequency shown on y-axis. Figure 17. Histogram of clinic visits across all clinics for all 173 potentially viable points. Figure 18. Histograms of weekly visits to each of the 10 clinics across all 173 viable points. A note on Z-score sensitivity and potentially viable parameterizations: We analyze the sensitivity of potentially viable points to different z-score thresholds, under current settings. Figure 19 shows the number of viable parameter points as the z-score threshold is reduced. Note, however, that this set is a result of the +/-4 threshold setting for the Active Learning algorithm from the first run of the simulation. If the threshold condition were initially tighter (say +/-3 or +/-2,) we expect the resulting set of potentially viable points to be different. Figure 19 below represents a set of points from which fall within z=+/4 and z=+/-2 range. Figure 19. Sensitivity of the total number of viable parameterizations to z-score threshold. A note on sensitivity of z-score thresholds: Another approach to a tighter fit (reduced z-score thresholds) between simulation output and empirical data in our simulation could be to consider a more selective subset of clinics used for history matching. E.g., dropping clinic 1273871 from consideration (where 0 visits result in a z-score between +/-4,) results in 82 points at z +/- 3 and points at z +/- 2. The sensitivity of the number of viable points to different z-score criteria under this exemplar restricted scenario is shown below in Figure 20. Figure 20. Number of viable points across Z-score criteria when clinic 1273871 is omitted from the objective function. A note on the stepwise tightening of viability constraints: The aim of our paper is a methodological case study in model exploration techniques, and our approach represents the first step in a stepwise tightening towards identifying parameter value combinations where model output is potentially compatible with empirical data. The model exploration techniques we use to partition the parameter space into potentially viable and non-viable regions is methodologically similar to history matching as described in Holden et al. (2018), where non-viable regions are selectively removed in expensive simulators. Our stepwise tightening approach is driven by the goal of reducing computational costs compared to using standalone approaches like ABC. ALTMAN, D. G., & Bland, J. M. (1994a). Diagnostic tests. 1: Sensitivity and specificity. British Medical Journal, 308(6943), 1552. [doi:10.1136/bmj.308.6943.1552] ALTMAN, D. G., & Bland, J. M. (1994b). 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Op-Amp-Applications A circuit is said to be linear, if there exists a linear relationship between its input and the output. Similarly, a circuit is said to be non-linear, if there exists a non-linear relationship between its input and output. Op-amps can be used in both linear and non-linear applications. The following are the basic applications of op-amp − Inverting Amplifier Non-inverting Amplifier This chapter discusses these basic applications in detail. An inverting amplifier takes the input through its inverting terminal through a resistor $R_{1}$, and produces its amplified version as the output. This amplifier not only amplifies the input but also inverts it (changes its sign). The circuit diagram of an inverting amplifier is shown in the following figure − Note that for an op-amp, the voltage at the inverting input terminal is equal to the voltage at its non-inverting input terminal. Physically, there is no short between those two terminals but virtually, they are in short with each other. In the circuit shown above, the non-inverting input terminal is connected to ground. That means zero volts is applied at the non-inverting input terminal of the op-amp. According to the virtual short concept, the voltage at the inverting input terminal of an op-amp will be zero volts. The nodal equation at this terminal's node is as shown below − $$\frac{0-V_i}{R_1}+ \frac{0-V_0}{R_f}=0$$ $$=>\frac{-V_i}{R_1}= \frac{V_0}{R_f}$$ $$=>V_{0}=\left(\frac{-R_f}{R_1}\right)V_{t}$$ $$=>\frac{V_0}{V_i}= \frac{-R_f}{R_1}$$ The ratio of the output voltage $V_{0}$ and the input voltage $V_{i}$ is the voltage-gain or gain of the amplifier. Therefore, the gain of inverting amplifier is equal to $-\frac{R_f}{R_1}$. Note that the gain of the inverting amplifier is having a negative sign. It indicates that there exists a 1800 phase difference between the input and the output. A non-inverting amplifier takes the input through its non-inverting terminal, and produces its amplified version as the output. As the name suggests, this amplifier just amplifies the input, without inverting or changing the sign of the output. The circuit diagram of a non-inverting amplifier is shown in the following figure − In the above circuit, the input voltage $V_{i}$ is directly applied to the non-inverting input terminal of op-amp. So, the voltage at the non-inverting input terminal of the op-amp will be $V_{i}$. By using voltage division principle, we can calculate the voltage at the inverting input terminal of the op-amp as shown below − $$=>V_{1} = V_{0}\left(\frac{R_1}{R_1+R_f}\right)$$ According to the virtual short concept, the voltage at the inverting input terminal of an op-amp is same as that of the voltage at its non-inverting input terminal. $$=>V_{1} = V_{i}$$ $$=>V_{0}\left(\frac{R_1}{R_1+R_f}\right)=V_{i}$$ $$=>\frac{V_0}{V_i}=\frac{R_1+R_f}{R_1}$$ $$=>\frac{V_0}{V_i}=1+\frac{R_f}{R_1}$$ Now, the ratio of output voltage $V_{0}$ and input voltage $V_{i}$ or the voltage-gain or gain of the non-inverting amplifier is equal to $1+\frac{R_f}{R_1}$. Note that the gain of the non-inverting amplifier is having a positive sign. It indicates that there is no phase difference between the input and the output. A voltage follower is an electronic circuit, which produces an output that follows the input voltage. It is a special case of non-inverting amplifier. If we consider the value of feedback resistor, $R_{f}$ as zero ohms and (or) the value of resistor, 1 as infinity ohms, then a non-inverting amplifier becomes a voltage follower. The circuit diagram of a voltage follower is shown in the following figure − In the above circuit, the input voltage $V_{i}$ is directly applied to the non-inverting input terminal of the op-amp. So, the voltage at the non-inverting input terminal of op-amp is equal to $V_{i}$. Here, the output is directly connected to the inverting input terminal of opamp. Hence, the voltage at the inverting input terminal of op-amp is equal to $V_{0}$. According to the virtual short concept, the voltage at the inverting input terminal of the op-amp is same as that of the voltage at its non-inverting input terminal. So, the output voltage $V_{0}$ of a voltage follower is equal to its input voltage $V_{i}$. Thus, the gain of a voltage follower is equal to one since, both output voltage $V_{0}$ and input voltage $V_{i}$ of voltage follower are same.	CommonCrawl
Finite difference A finite difference is a mathematical expression of the form f (x + b) − f (x + a). If a finite difference is divided by b − a, one gets a difference quotient. The approximation of derivatives by finite differences plays a central role in finite difference methods for the numerical solution of differential equations, especially boundary value problems. The difference operator, commonly denoted $\Delta $ is the operator that maps a function f to the function $\Delta [f]$ defined by $\Delta [f](x)=f(x+1)-f(x).$ A difference equation is a functional equation that involves the finite difference operator in the same way as a differential equation involves derivatives. There are many similarities between difference equations and differential equations, specially in the solving methods. Certain recurrence relations can be written as difference equations by replacing iteration notation with finite differences. In numerical analysis, finite differences are widely used for approximating derivatives, and the term "finite difference" is often used as an abbreviation of "finite difference approximation of derivatives".[1][2][3] Finite difference approximations are finite difference quotients in the terminology employed above. Finite differences were introduced by Brook Taylor in 1715 and have also been studied as abstract self-standing mathematical objects in works by George Boole (1860), L. M. Milne-Thomson (1933), and Károly Jordan (1939). Finite differences trace their origins back to one of Jost Bürgi's algorithms (c. 1592) and work by others including Isaac Newton. The formal calculus of finite differences can be viewed as an alternative to the calculus of infinitesimals.[4] Basic types Three basic types are commonly considered: forward, backward, and central finite differences.[1][2][3] A forward difference, denoted $\Delta _{h}[f],$ of a function f is a function defined as $\Delta _{h}[f](x)=f(x+h)-f(x).$ Depending on the application, the spacing h may be variable or constant. When omitted, h is taken to be 1; that is, $\Delta [f](x)=\Delta _{1}[f](x)=f(x+1)-f(x).$ A backward difference uses the function values at x and x − h, instead of the values at x + h and x: $\nabla _{h}[f](x)=f(x)-f(x-h)=\Delta _{h}[f](x-h).$ Finally, the central difference is given by $\delta _{h}[f](x)=f(x+{\tfrac {h}{2}})-f(x-{\tfrac {h}{2}})=\Delta _{h/2}[f](x)+\nabla _{h/2}[f](x).$ Relation with derivatives Finite difference is often used as an approximation of the derivative, typically in numerical differentiation. The derivative of a function f at a point x is defined by the limit. $f'(x)=\lim _{h\to 0}{\frac {f(x+h)-f(x)}{h}}.$ If h has a fixed (non-zero) value instead of approaching zero, then the right-hand side of the above equation would be written ${\frac {f(x+h)-f(x)}{h}}={\frac {\Delta _{h}[f](x)}{h}}.$ Hence, the forward difference divided by h approximates the derivative when h is small. The error in this approximation can be derived from Taylor's theorem. Assuming that f is twice differentiable, we have ${\frac {\Delta _{h}[f](x)}{h}}-f'(x)=O(h)\to 0\quad {\text{as }}h\to 0.$ The same formula holds for the backward difference: ${\frac {\nabla _{h}[f](x)}{h}}-f'(x)=O(h)\to 0\quad {\text{as }}h\to 0.$ However, the central (also called centered) difference yields a more accurate approximation. If f is three times differentiable, ${\frac {\delta _{h}[f](x)}{h}}-f'(x)=O\left(h^{2}\right).$ The main problem with the central difference method, however, is that oscillating functions can yield zero derivative. If f (nh) = 1 for n odd, and f (nh) = 2 for n even, then f ′(nh) = 0 if it is calculated with the central difference scheme. This is particularly troublesome if the domain of f is discrete. See also Symmetric derivative. Authors for whom finite differences mean finite difference approximations define the forward/backward/central differences as the quotients given in this section (instead of employing the definitions given in the previous section).[1][2][3] Higher-order differences In an analogous way, one can obtain finite difference approximations to higher order derivatives and differential operators. For example, by using the above central difference formula for f ′(x + h/2) and f ′(x − h/2) and applying a central difference formula for the derivative of f ′ at x, we obtain the central difference approximation of the second derivative of f: Second-order central $f''(x)\approx {\frac {\delta _{h}^{2}[f](x)}{h^{2}}}={\frac {{\frac {f(x+h)-f(x)}{h}}-{\frac {f(x)-f(x-h)}{h}}}{h}}={\frac {f(x+h)-2f(x)+f(x-h)}{h^{2}}}.$ Similarly we can apply other differencing formulas in a recursive manner. Second order forward $f''(x)\approx {\frac {\Delta _{h}^{2}[f](x)}{h^{2}}}={\frac {{\frac {f(x+2h)-f(x+h)}{h}}-{\frac {f(x+h)-f(x)}{h}}}{h}}={\frac {f(x+2h)-2f(x+h)+f(x)}{h^{2}}}.$ Second order backward $f''(x)\approx {\frac {\nabla _{h}^{2}[f](x)}{h^{2}}}={\frac {{\frac {f(x)-f(x-h)}{h}}-{\frac {f(x-h)-f(x-2h)}{h}}}{h}}={\frac {f(x)-2f(x-h)+f(x-2h)}{h^{2}}}.$ More generally, the nth order forward, backward, and central differences are given by, respectively, Forward $\Delta _{h}^{n}[f](x)=\sum _{i=0}^{n}(-1)^{n-i}{\binom {n}{i}}f{\bigl (}x+ih{\bigr )},$ or for h = 1, $\Delta ^{n}[f](x)=\sum _{i=0}^{n}{\binom {n}{i}}(-1)^{n-i}f(x+i)$ Backward $\nabla _{h}^{n}[f](x)=\sum _{i=0}^{n}(-1)^{i}{\binom {n}{i}}f(x-ih),$ Central $\delta _{h}^{n}[f](x)=\sum _{i=0}^{n}(-1)^{i}{\binom {n}{i}}f\left(x+\left({\frac {n}{2}}-i\right)h\right).$ These equations use binomial coefficients after the summation sign shown as (n i ) . Each row of Pascal's triangle provides the coefficient for each value of i. Note that the central difference will, for odd n, have h multiplied by non-integers. This is often a problem because it amounts to changing the interval of discretization. The problem may be remedied taking the average of δn[ f ](x − h/2) and δn[ f ](x + h/2). Forward differences applied to a sequence are sometimes called the binomial transform of the sequence, and have a number of interesting combinatorial properties. Forward differences may be evaluated using the Nörlund–Rice integral. The integral representation for these types of series is interesting, because the integral can often be evaluated using asymptotic expansion or saddle-point techniques; by contrast, the forward difference series can be extremely hard to evaluate numerically, because the binomial coefficients grow rapidly for large n. The relationship of these higher-order differences with the respective derivatives is straightforward, ${\frac {d^{n}f}{dx^{n}}}(x)={\frac {\Delta _{h}^{n}[f](x)}{h^{n}}}+O(h)={\frac {\nabla _{h}^{n}[f](x)}{h^{n}}}+O(h)={\frac {\delta _{h}^{n}[f](x)}{h^{n}}}+O\left(h^{2}\right).$ Higher-order differences can also be used to construct better approximations. As mentioned above, the first-order difference approximates the first-order derivative up to a term of order h. However, the combination ${\frac {\Delta _{h}[f](x)-{\frac {1}{2}}\Delta _{h}^{2}[f](x)}{h}}=-{\frac {f(x+2h)-4f(x+h)+3f(x)}{2h}}$ approximates f ′(x) up to a term of order h2. This can be proven by expanding the above expression in Taylor series, or by using the calculus of finite differences, explained below. If necessary, the finite difference can be centered about any point by mixing forward, backward, and central differences. Polynomials For a given polynomial of degree n ≥ 1, expressed in the function P(x), with real numbers a ≠ 0 and b and lower order terms (if any) marked as l.o.t.: $P(x)=ax^{n}+bx^{n-1}+l.o.t.$ After n pairwise differences, the following result can be achieved, where h ≠ 0 is a real number marking the arithmetic difference:[5] $\Delta _{h}^{n}[P](x)=ah^{n}n!$ Only the coefficient of the highest-order term remains. As this result is constant with respect to x, any further pairwise differences will have the value 0. Base case Let Q(x) be a polynomial of degree 1: $\Delta _{h}[Q](x)=Q(x+h)-Q(x)=[a(x+h)+b]-[ax+b]=ah=ah^{1}1!$ This proves it for the base case. Inductive step Let R(x) be a polynomial of degree m − 1 where m ≥ 2 and the coefficient of the highest-order term be a ≠ 0. Assuming the following holds true for all polynomials of degree m − 1: $\Delta _{h}^{m-1}[R](x)=ah^{m-1}(m-1)!$ Let S(x) be a polynomial of degree m. With one pairwise difference: $\Delta _{h}[S](x)=[a(x+h)^{m}+b(x+h)^{m-1}+{\text{l.o.t.}}]-[ax^{m}+bx^{m-1}+{\text{l.o.t.}}]=ahmx^{m-1}+{\text{l.o.t.}}=T(x)$ As ahm ≠ 0, this results in a polynomial T(x) of degree m − 1, with ahm as the coefficient of the highest-order term. Given the assumption above and m − 1 pairwise differences (resulting in a total of m pairwise differences for S(x)), it can be found that: $\Delta _{h}^{m-1}[T](x)=ahm\cdot h^{m-1}(m-1)!=ah^{m}m!$ This completes the proof. Application This identity can be used to find the lowest-degree polynomial that intercepts a number of points (x, y) where the difference on the x-axis from one point to the next is a constant h ≠ 0. For example, given the following points: xy 14 4109 7772 102641 136364 We can use a differences table, where for all cells to the right of the first y, the following relation to the cells in the column immediately to the left exists for a cell (a + 1, b + 1), with the top-leftmost cell being at coordinate (0, 0): $(a+1,b+1)=(a,b+1)-(a,b)$ To find the first term, the following table can be used: xyΔyΔ2yΔ3y 1 4 4 109105 7 772663558 10 264118691206648 13 636437231854648 This arrives at a constant 648. The arithmetic difference is h=3, as established above. Given the number of pairwise differences needed to reach the constant, it can be surmised this is a polynomial of degree 3. Thus, using the identity above: $648=a\cdot 3^{3}\cdot 3!=a\cdot 27\cdot 6=a\cdot 162$ Solving for a, it can be found to have the value 4. Thus, the first term of the polynomial is 4x3. Then, subtracting out the first term, which lowers the polynomial's degree, and finding the finite difference again: xyΔyΔ2y 1 4 − 4(1)3 = 4 − 4 = 0 4 109 − 4(4)3 = 109 − 256 = −147−147 7 772 − 4(7)3 = 772 − 1372 = −600−453−306 10 2641 − 4(10)3 = 2641 − 4000 = −1359−759−306 13 6364 − 4(13)3 = 6364 − 8788 = −2424−1065−306 Here, the constant is achieved after only two pairwise differences, thus the following result: $-306=a\cdot 3^{2}\cdot 2!=a\cdot 18$ Solving for a, which is −17, the polynomial's second term is −17x2. Moving on to the next term, by subtracting out the second term: xyΔy 1 0 − (−17(1)2) = 0 + 17 = 17 4 −147 − (−17(4)2) = −147 + 272 = 125108 7 −600 − (−17(7)2) = −600 + 833 = 233 108 10 −1359 − (−17(10)2) = −1359 + 1700 = 341 108 13 −2424 − (−17(13)2) = −2424 + 2873 = 449 108 Thus the constant is achieved after only one pairwise difference: $108=a\cdot 3^{1}\cdot 1!=a\cdot 3$ It can be found that a = 36 and thus the third term of the polynomial is 36x. Subtracting out the third term: xy 1 17 − 36(1) = 17 − 36 = −19 4 125 − 36(4) = 125 − 144 = −19 7 233 − 36(7) = 233 − 252 = −19 10 341 − 36(10) = 341 − 360 = −19 13 449 − 36(13) = 449 − 468 = −19 Without any pairwise differences, it is found that the 4th and final term of the polynomial is the constant -19. Thus, the lowest-degree polynomial intercepting all the points in the first table is found: $4x^{3}-17x^{2}+36x-19$ Arbitrarily sized kernels Main article: Finite difference coefficient Further information: Five-point stencil Using linear algebra one can construct finite difference approximations which utilize an arbitrary number of points to the left and a (possibly different) number of points to the right of the evaluation point, for any order derivative. This involves solving a linear system such that the Taylor expansion of the sum of those points around the evaluation point best approximates the Taylor expansion of the desired derivative. Such formulas can be represented graphically on a hexagonal or diamond-shaped grid.[6] This is useful for differentiating a function on a grid, where, as one approaches the edge of the grid, one must sample fewer and fewer points on one side. The details are outlined in these notes. The Finite Difference Coefficients Calculator constructs finite difference approximations for non-standard (and even non-integer) stencils given an arbitrary stencil and a desired derivative order. Properties • For all positive k and n $\Delta _{kh}^{n}(f,x)=\sum \limits _{i_{1}=0}^{k-1}\sum \limits _{i_{2}=0}^{k-1}\cdots \sum \limits _{i_{n}=0}^{k-1}\Delta _{h}^{n}\left(f,x+i_{1}h+i_{2}h+\cdots +i_{n}h\right).$ • Leibniz rule: $\Delta _{h}^{n}(fg,x)=\sum \limits _{k=0}^{n}{\binom {n}{k}}\Delta _{h}^{k}(f,x)\Delta _{h}^{n-k}(g,x+kh).$ In differential equations Main article: Finite difference method An important application of finite differences is in numerical analysis, especially in numerical differential equations, which aim at the numerical solution of ordinary and partial differential equations. The idea is to replace the derivatives appearing in the differential equation by finite differences that approximate them. The resulting methods are called finite difference methods. Common applications of the finite difference method are in computational science and engineering disciplines, such as thermal engineering, fluid mechanics, etc. Newton's series The Newton series consists of the terms of the Newton forward difference equation, named after Isaac Newton; in essence, it is the Gregory–Newton interpolation formula[7] (named after Isaac Newton and James Gregory), first published in his Principia Mathematica in 1687,[8] [9] namely the discrete analog of the continuous Taylor expansion, $f(x)=\sum _{k=0}^{\infty }{\frac {\Delta ^{k}[f](a)}{k!}}\,(x-a)_{k}=\sum _{k=0}^{\infty }{\binom {x-a}{k}}\,\Delta ^{k}[f](a),$ which holds for any polynomial function f and for many (but not all) analytic functions. (It does not hold when f is exponential type $\pi $. This is easily seen, as the sine function vanishes at integer multiples of $\pi $; the corresponding Newton series is identically zero, as all finite differences are zero in this case. Yet clearly, the sine function is not zero.) Here, the expression ${\binom {x}{k}}={\frac {(x)_{k}}{k!}}$ is the binomial coefficient, and $(x)_{k}=x(x-1)(x-2)\cdots (x-k+1)$ is the "falling factorial" or "lower factorial", while the empty product (x)0 is defined to be 1. In this particular case, there is an assumption of unit steps for the changes in the values of x, h = 1 of the generalization below. Note the formal correspondence of this result to Taylor's theorem. Historically, this, as well as the Chu–Vandermonde identity, $(x+y)_{n}=\sum _{k=0}^{n}{\binom {n}{k}}(x)_{n-k}\,(y)_{k},$ (following from it, and corresponding to the binomial theorem), are included in the observations that matured to the system of umbral calculus. Newton series expansions can be superior to Taylor series expansions when applied to discrete quantities like quantum spins (see Holstein–Primakoff transformation), bosonic operator functions or discrete counting statistics.[10] To illustrate how one may use Newton's formula in actual practice, consider the first few terms of doubling the Fibonacci sequence f = 2, 2, 4, ... One can find a polynomial that reproduces these values, by first computing a difference table, and then substituting the differences that correspond to x0 (underlined) into the formula as follows, ${\begin{matrix}{\begin{array}{\|c\|\|c\|c\|c\|}\hline x&f=\Delta ^{0}&\Delta ^{1}&\Delta ^{2}\\\hline 1&{\underline {2}}&&\\&&{\underline {0}}&\\2&2&&{\underline {2}}\\&&2&\\3&4&&\\\hline \end{array}}&\quad {\begin{aligned}f(x)&=\Delta ^{0}\cdot 1+\Delta ^{1}\cdot {\dfrac {(x-x_{0})_{1}}{1!}}+\Delta ^{2}\cdot {\dfrac {(x-x_{0})_{2}}{2!}}\quad (x_{0}=1)\\\\&=2\cdot 1+0\cdot {\dfrac {x-1}{1}}+2\cdot {\dfrac {(x-1)(x-2)}{2}}\\\\&=2+(x-1)(x-2)\\\end{aligned}}\end{matrix}}$ For the case of nonuniform steps in the values of x, Newton computes the divided differences, $\Delta _{j,0}=y_{j},\qquad \Delta _{j,k}={\frac {\Delta _{j+1,k-1}-\Delta _{j,k-1}}{x_{j+k}-x_{j}}}\quad \ni \quad \left\{k>0,\;j\leq \max \left(j\right)-k\right\},\qquad \Delta 0_{k}=\Delta _{0,k}$ the series of products, ${P_{0}}=1,\quad \quad P_{k+1}=P_{k}\cdot \left(\xi -x_{k}\right),$ and the resulting polynomial is the scalar product,[11] $f(\xi )=\Delta 0\cdot P\left(\xi \right)$ . In analysis with p-adic numbers, Mahler's theorem states that the assumption that f is a polynomial function can be weakened all the way to the assumption that f is merely continuous. Carlson's theorem provides necessary and sufficient conditions for a Newton series to be unique, if it exists. However, a Newton series does not, in general, exist. The Newton series, together with the Stirling series and the Selberg series, is a special case of the general difference series, all of which are defined in terms of suitably scaled forward differences. In a compressed and slightly more general form and equidistant nodes the formula reads $f(x)=\sum _{k=0}{\binom {\frac {x-a}{h}}{k}}\sum _{j=0}^{k}(-1)^{k-j}{\binom {k}{j}}f(a+jh).$ Calculus of finite differences The forward difference can be considered as an operator, called the difference operator, which maps the function f to Δh[ f ].[12][13] This operator amounts to $\Delta _{h}=T_{h}-I,$ where Th is the shift operator with step h, defined by Th[ f ](x) = f (x + h), and I is the identity operator. The finite difference of higher orders can be defined in recursive manner as Δn h ≡ Δh(Δn − 1 h ) . Another equivalent definition is Δn h = [Th − I]n . The difference operator Δh is a linear operator, as such it satisfies Δh[αf + βg](x) = α Δh[ f ](x) + β Δh[g](x). It also satisfies a special Leibniz rule indicated above, Δh(f (x)g(x)) = (Δhf (x)) g(x+h) + f (x) (Δhg(x)). Similar statements hold for the backward and central differences. Formally applying the Taylor series with respect to h, yields the formula $\Delta _{h}=hD+{\frac {1}{2!}}h^{2}D^{2}+{\frac {1}{3!}}h^{3}D^{3}+\cdots =\mathrm {e} ^{hD}-I,$ where D denotes the continuum derivative operator, mapping f to its derivative f ′. The expansion is valid when both sides act on analytic functions, for sufficiently small h. Thus, Th = ehD, and formally inverting the exponential yields $hD=\ln(1+\Delta _{h})=\Delta _{h}-{\tfrac {1}{2}}\,\Delta _{h}^{2}+{\tfrac {1}{3}}\,\Delta _{h}^{3}-\cdots .$ This formula holds in the sense that both operators give the same result when applied to a polynomial. Even for analytic functions, the series on the right is not guaranteed to converge; it may be an asymptotic series. However, it can be used to obtain more accurate approximations for the derivative. For instance, retaining the first two terms of the series yields the second-order approximation to f ′(x) mentioned at the end of the section Higher-order differences. The analogous formulas for the backward and central difference operators are $hD=-\ln(1-\nabla _{h})\quad {\text{and}}\quad hD=2\operatorname {arsinh} \left({\tfrac {1}{2}}\,\delta _{h}\right).$ The calculus of finite differences is related to the umbral calculus of combinatorics. This remarkably systematic correspondence is due to the identity of the commutators of the umbral quantities to their continuum analogs (h → 0 limits), $\left[{\frac {\Delta _{h}}{h}},x\,T_{h}^{-1}\right]=[D,x]=I.$ A large number of formal differential relations of standard calculus involving functions f (x) thus map systematically to umbral finite-difference analogs involving f (xT−1 h ) . For instance, the umbral analog of a monomial xn is a generalization of the above falling factorial (Pochhammer k-symbol), $~(x)_{n}\equiv \left(xT_{h}^{-1}\right)^{n}=x(x-h)(x-2h)\cdots {\bigl (}x-(n-1)h{\bigr )},$ so that ${\frac {\Delta _{h}}{h}}(x)_{n}=n(x)_{n-1},$ hence the above Newton interpolation formula (by matching coefficients in the expansion of an arbitrary function f (x) in such symbols), and so on. For example, the umbral sine is $\sin \left(x\,T_{h}^{-1}\right)=x-{\frac {(x)_{3}}{3!}}+{\frac {(x)_{5}}{5!}}-{\frac {(x)_{7}}{7!}}+\cdots $ As in the continuum limit, the eigenfunction of Δh/h also happens to be an exponential, ${\frac {\Delta _{h}}{h}}(1+\lambda h)^{\frac {x}{h}}={\frac {\Delta _{h}}{h}}e^{\ln(1+\lambda h){\frac {x}{h}}}=\lambda e^{\ln(1+\lambda h){\frac {x}{h}}},$ and hence Fourier sums of continuum functions are readily mapped to umbral Fourier sums faithfully, i.e., involving the same Fourier coefficients multiplying these umbral basis exponentials.[14] This umbral exponential thus amounts to the exponential generating function of the Pochhammer symbols. Thus, for instance, the Dirac delta function maps to its umbral correspondent, the cardinal sine function, $\delta (x)\mapsto {\frac {\sin \left[{\frac {\pi }{2}}\left(1+{\frac {x}{h}}\right)\right]}{\pi (x+h)}},$ and so forth.[15] Difference equations can often be solved with techniques very similar to those for solving differential equations. The inverse operator of the forward difference operator, so then the umbral integral, is the indefinite sum or antidifference operator. Rules for calculus of finite difference operators Analogous to rules for finding the derivative, we have: • Constant rule: If c is a constant, then $\Delta c=0$ • Linearity: if a and b are constants, $\Delta (a\ f+b\ g)=a\ \Delta f+b\ \Delta g$ All of the above rules apply equally well to any difference operator as to Δ, including δ and ∇. • Product rule: ${\begin{aligned}\Delta (fg)&=f\,\Delta g+g\,\Delta f+\Delta f\,\Delta g\\[4pt]\nabla (fg)&=f\,\nabla g+g\,\nabla f-\nabla f\,\nabla g\end{aligned}}$ • Quotient rule: $\nabla \left({\frac {f}{g}}\right)=\left.\left(\det {\begin{bmatrix}\nabla f&\nabla g\\f&g\end{bmatrix}}\right)\right/\left(g\cdot \det {\begin{bmatrix}g&\nabla g\\1&1\end{bmatrix}}\right)$ or $\nabla \left({\frac {f}{g}}\right)={\frac {g\,\nabla f-f\,\nabla g}{g\cdot (g-\nabla g)}}$ • Summation rules: ${\begin{aligned}\sum _{n=a}^{b}\Delta f(n)&=f(b+1)-f(a)\\\sum _{n=a}^{b}\nabla f(n)&=f(b)-f(a-1)\end{aligned}}$ See references.[16][17][18][19] Generalizations • A generalized finite difference is usually defined as $\Delta _{h}^{\mu }[f](x)=\sum _{k=0}^{N}\mu _{k}f(x+kh),$ where μ = (μ0, …, μN) is its coefficient vector. An infinite difference is a further generalization, where the finite sum above is replaced by an infinite series. Another way of generalization is making coefficients μk depend on point x: μk = μk(x), thus considering weighted finite difference. Also one may make the step h depend on point x: h = h(x). Such generalizations are useful for constructing different modulus of continuity. • The generalized difference can be seen as the polynomial rings R[Th]. It leads to difference algebras. • Difference operator generalizes to Möbius inversion over a partially ordered set. • As a convolution operator: Via the formalism of incidence algebras, difference operators and other Möbius inversion can be represented by convolution with a function on the poset, called the Möbius function μ; for the difference operator, μ is the sequence (1, −1, 0, 0, 0, …). Multivariate finite differences Finite differences can be considered in more than one variable. They are analogous to partial derivatives in several variables. Some partial derivative approximations are: ${\begin{aligned}f_{x}(x,y)&\approx {\frac {f(x+h,y)-f(x-h,y)}{2h}}\\f_{y}(x,y)&\approx {\frac {f(x,y+k)-f(x,y-k)}{2k}}\\f_{xx}(x,y)&\approx {\frac {f(x+h,y)-2f(x,y)+f(x-h,y)}{h^{2}}}\\f_{yy}(x,y)&\approx {\frac {f(x,y+k)-2f(x,y)+f(x,y-k)}{k^{2}}}\\f_{xy}(x,y)&\approx {\frac {f(x+h,y+k)-f(x+h,y-k)-f(x-h,y+k)+f(x-h,y-k)}{4hk}}.\end{aligned}}$ Alternatively, for applications in which the computation of f is the most costly step, and both first and second derivatives must be computed, a more efficient formula for the last case is $f_{xy}(x,y)\approx {\frac {f(x+h,y+k)-f(x+h,y)-f(x,y+k)+2f(x,y)-f(x-h,y)-f(x,y-k)+f(x-h,y-k)}{2hk}},$ since the only values to compute that are not already needed for the previous four equations are f (x + h, y + k) and f (x − h, y − k). See also • Discrete calculus • Divided differences • Finite-difference time-domain method (FDTD) • Finite volume method • FTCS scheme • Gilbreath's conjecture • Sheffer sequence • Summation by parts • Time scale calculus • Upwind differencing scheme for convection References 1. Paul Wilmott; Sam Howison; Jeff Dewynne (1995). The Mathematics of Financial Derivatives: A Student Introduction. Cambridge University Press. p. 137. ISBN 978-0-521-49789-3. 2. Peter Olver (2013). Introduction to Partial Differential Equations. Springer Science & Business Media. p. 182. ISBN 978-3-319-02099-0. 3. M Hanif Chaudhry (2007). Open-Channel Flow. Springer. p. 369. ISBN 978-0-387-68648-6. 4. Jordán, op. cit., p. 1 and Milne-Thomson, p. xxi. Milne-Thomson, Louis Melville (2000): The Calculus of Finite Differences (Chelsea Pub Co, 2000) ISBN 978-0821821077 5. "Finite differences of polynomials". February 13, 2018. 6. Fraser, Duncan C. (January 1, 1909). "On the Graphic Delineation of Interpolation Formulæ". Journal of the Institute of Actuaries. 43 (2): 235–241. doi:10.1017/S002026810002494X. Retrieved April 17, 2017. 7. Burkard Polster/Mathologer (2021). " Why don't they teach Newton's calculus of 'What comes next?' " on YouTube 8. Newton, Isaac, (1687). Principia, Book III, Lemma V, Case 1 9. Iaroslav V. Blagouchine (2018). "Three notes on Ser's and Hasse's representations for the zeta-functions" (PDF). Integers (Electronic Journal of Combinatorial Number Theory). 18A: 1–45. arXiv:1606.02044. 10. König, Jürgen; Hucht, Fred (2021). "Newton series expansion of bosonic operator functions". SciPost Physics. 10 (1): 007. arXiv:2008.11139. Bibcode:2021ScPP...10....7K. doi:10.21468/SciPostPhys.10.1.007. S2CID 221293056. 11. Richtmeyer, D. and Morton, K.W., (1967). Difference Methods for Initial Value Problems, 2nd ed., Wiley, New York. 12. Boole, George, (1872). A Treatise On The Calculus of Finite Differences, 2nd ed., Macmillan and Company. On line. Also, [Dover edition 1960] 13. Jordan, Charles, (1939/1965). "Calculus of Finite Differences", Chelsea Publishing. On-line: 14. Zachos, C. (2008). "Umbral Deformations on Discrete Space-Time". International Journal of Modern Physics A. 23 (13): 2005–2014. arXiv:0710.2306. Bibcode:2008IJMPA..23.2005Z. doi:10.1142/S0217751X08040548. S2CID 16797959. 15. Curtright, T. L.; Zachos, C. K. (2013). "Umbral Vade Mecum". Frontiers in Physics. 1: 15. arXiv:1304.0429. Bibcode:2013FrP.....1...15C. doi:10.3389/fphy.2013.00015. S2CID 14106142. 16. Levy, H.; Lessman, F. (1992). Finite Difference Equations. Dover. ISBN 0-486-67260-3. 17. Ames, W.F. (1977). Numerical Methods for Partial Differential Equations. New York, NY: Academic Press. Section 1.6. ISBN 0-12-056760-1. 18. Hildebrand, F.B. (1968). Finite-Difference Equations and Simulations. Englewood Cliffs, NJ: Prentice-Hall. Section 2.2. 19. Flajolet, Philippe; Sedgewick, Robert (1995). "Mellin transforms and asymptotics: Finite differences and Rice's integrals" (PDF). Theoretical Computer Science. 144 (1–2): 101–124. doi:10.1016/0304-3975(94)00281-M. • Richardson, C. H. (1954): An Introduction to the Calculus of Finite Differences (Van Nostrand (1954) online copy • Mickens, R. E. (1991): Difference Equations: Theory and Applications (Chapman and Hall/CRC) ISBN 978-0442001360 External links • "Finite-difference calculus", Encyclopedia of Mathematics, EMS Press, 2001 [1994] • Table of useful finite difference formula generated using Mathematica • D. 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Finite element approximation of nonlocal dynamic fracture models Ground state homoclinic orbits for a class of asymptotically periodic second-order Hamiltonian systems March 2021, 26(3): 1653-1673. doi: 10.3934/dcdsb.2020177 Dynamic aspects of Sprott BC chaotic system Marcos C. Mota and Regilene D. S. Oliveira , Departamento de Matemática, Instituto de Ciências Matemáticas e de Computação, Universidade de São Paulo, Avenida Trabalhador São–carlense, 400, Centro, 13.566-590, São Carlos, SP, Brazil * Corresponding author: [email protected] Communicated by Dongmei Xiao Received April 2019 Published June 2020 Fund Project: This study was financed in part by the Coordenação de Aperfeiçoamento de Pessoal de Níıvel Superior - Brasil (CAPES) - Finance Code 001 and FAPESP grant number 2017/20854-5 In this paper we study global dynamic aspects of the quadratic system $ \dot x = yz,\quad \dot y = x-y,\quad \dot z = 1-x(\alpha y+\beta x), $ $ (x,y,z) \in \mathbb R^3 $ $ \alpha, \beta \in[0,1] $ are two parameters. It contains the Sprott B and the Sprott C systems at the two extremes of its parameter spectrum and we call it Sprott BC system. Here we present the complete description of its singularities and we show that this system passes through a Hopf bifurcation at $ \alpha = 0 $ . Using the Poincaré compactification of a polynomial vector field in $ \mathbb R^3 $ we give a complete description of its dynamic on the Poincaré sphere at infinity. We also show that such a system does not admit a polynomial first integral, nor algebraic invariant surfaces, neither Darboux first integral. Keywords: Sprott BC chaotic system, Hopf bifurcation, Poincaré compactification, invariant algebraic surfaces, Darboux integrability. Mathematics Subject Classification: Primary: 34C05, 34C45; Secondary: 37D10, 37D45. Citation: Marcos C. Mota, Regilene D. S. Oliveira. Dynamic aspects of Sprott BC chaotic system. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1653-1673. doi: 10.3934/dcdsb.2020177 [1] D. Bleecker and G. Csordas, Basic Partial Differential Equations, International Press, Cambridge, MA, 1996. doi: 10.1201/9781351070089. Google Scholar C. J. Christopher, Invariant algebraic curves and conditions for a centre, Proc. Roy. Soc. Edinburgh Sect. 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Local behavior of orbits around the finite singularities of Sprott B (in 1(A)) and Sprott C (in 1(B)) systems Figure Options Download full-size image Download as PowerPoint slide Figure 2. Phase portrait of system (3) on the Poincaré sphere. In Figure 2(A) there exist two closed curves filled up with singularities and one pair of distinguished singularities. These distinguished singularities possess two parabolic attractor sectors and two parabolic repelling sectors. In Figure 2(B) there exist one closed curve filled up with singularities and one pair of center type singularities Figure 3. Phase portrait of system (3) on the Poincaré sphere. In Figure 3(A) there exist a pair of cusp type singularities and a pair of node type singularities (being one attractor and other repelling). In Figure 3(B) there exist a pair of saddles, a pair of centers and a pair of nodes (being one attractor and other repelling) Kerioui Nadjah, Abdelouahab Mohammed Salah. 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\begin{document} \selectlanguage{english} \title{On a Newton filtration for functions on a curve singularity} \begin{abstract} In a previous paper, there was defined a multi-index filtration on the ring of functions on a hypersurface singularity corresponding to its Newton diagram generalizing (for a curve singularity) the divisorial one. Its Poincar\'e series was computed for plane curve singularities non-degenerate with respect to their Newton diagrams. Here we use another technique to compute the Poincar\'e series for plane curve singularities without the assumption that they are non-degenerate with respect to their Newton diagrams. We show that the Poincar\'e series only depends on the Newton diagram and not on the defining equation. \end{abstract} \section{Introduction} In \cite{EG1, EG2}, there were defined two multi-index filtrations on the ring ${\cal O}_{{\mathbb C}^n,0}$ of germs of holomorphic functions in $n$ variables associated to a Newton diagram $\Gamma$ in ${\mathbb R}^n$ and to a germ of an analytic function $f : ({\mathbb C}^n,0) \to ({\mathbb C},0)$ with this Newton diagram. We assumed that the function $f$ was non-degenerate with respect to its Newton diagram $\Gamma$. These filtrations are essentially filtrations on the ring ${\cal O}_{V,0}={\cal O}_{{\mathbb C}^n,0}/(f)$ of germs of functions on the hypersurface singularity $V=\{f=0\}$. They correspond to the quasihomogeneous valuations on the ring ${\cal O}_{{\mathbb C}^n,0}$ defined by the facets of the diagram $\Gamma$. These facets correspond to some components of the exceptional divisor of a toric resolution of the germ $f$ constructed from the diagram $\Gamma$. Such a component defines the corresponding divisorial valuation on the ring ${\cal O}_{{\mathbb C}^n,0}$. For $n\ge 3$ (and for a $\Gamma$-non-degenerate $f$) these valuations induce divisorial valuations on the ring ${\cal O}_{V,0}$ and define the corresponding multi-index filtration on it. The filtration defined in \cite{EG1} was regarded as a certain ``simplification'' of the divisorial one. This appeared not to be the case. For example, a general formula for the Poincar\'e series of this filtration is not known even for the number of variables $n=2$. For Newton diagrams of special type, A.~Lemahieu identified this filtration with a so called embedded filtration on ${\cal O}_{V,0}$ \cite{L}. In \cite{L}, a formula for the Poincar\'{e} series of the embedded filtration for a hypersurface singularity was given. H.~Hamm studied the embedded filtration and the corresponding Poincar\'{e} series for complete intersection singularities \cite{H}. In \cite{EG2}, there was given an ``algebraic'' definition of the divisorial valuation corresponding to a Newton diagram (for $n\ge 3$) somewhat similar to the definition in \cite{EG1}. Roughly speaking, the difference consists in using the ring ${\cal O}_{{\mathbb C}^n,0}[x_1^{-1}, \ldots, x_n^{-1}]$ instead of ${\cal O}_{{\mathbb C}^n,0}$. For $n=2$, this definition does not give, in general, a valuation, but an order function (see the definition below). For a $\Gamma$-non-degenerate $f\in {\cal O}_{{\mathbb C}^2,0}$, this order function was described as a ``generalized divisorial valuation'' defined by the divisorial valuations corresponding to all the points of intersection of the resolution (normalization) $\widetilde{V}$ of the curve $V$ with the corresponding component of the exceptional divisor. This permitted to apply the technique elaborated in \cite{IJM} and to compute the corresponding Poincar\'e series. (This technique has no analogue which could be applied to degenerate $f$, or to the case $n>2$, or to the filtration defined in \cite{EG1}.) In this case the Poincar\'e series depends only on the Newton diagram $\Gamma$ and does not depend on the function $f$ with $\Gamma_f=\Gamma$. The definitions in \cite{EG1} and \cite{EG2} make also sense for functions $f$ degenerate with respect to their Newton diagrams. Here we compute the Poincar\'e series of the filtration introduced in \cite{EG2} for $n=2$ directly from the definition without the assumption that $f$ is non-degenerate with respect to the Newton diagram. We show that the answer is the same as in \cite[Corollary 1]{EG2} for non-degenerate $f$. Thus, for $n=2$, the Poincar\'e series of this filtration depends only on the Newton diagram $\Gamma$. One can speculate that the same holds for $n\ge 3$ and for the Poincar\'e series of the filtration defined in \cite{EG1}. We hope that some elements of the technique used here can be applied to the case $n\ge 3$ and/or to the filtration defined in \cite{EG1} as well. One motivation to study (multi-variable) Poincar\'e series of filtrations comes from the fact that they are sometimes related or even coincide with appropriate monodromy zeta functions or with Alexander polynomials (see e.g.\ \cite{IJM}). We show that the obtained formula for the Poincar\'e series has a relation to the (multi-variable) Alexander polynomial of a collection of functions. \section{Filtrations associated to Newton diagrams} Let $(V,0)$ be a germ of a complex analytic variety and let ${\cal O}_{V,0}$ be the ring of germs of holomorphic functions on $(V,0)$. A map $v: {\cal O}_{V,0}\to {\mathbb Z}_{\ge 0}\cup \{+\infty\}$ is an {\em order function} on ${\cal O}_{V,0}$ if $v(\lambda g)=v(g)$ for a non-zero $\lambda \in {\mathbb C}$ and $v(g_1+g_2)\ge \min\{v(g_1), v(g_2)\}$. (If, moreover, $v(g_1g_2)=v(g_1)+v(g_2)$, the map $v$ is a {\em valuation} on ${\cal O}_{V,0}$.) A collection $\{v_1, v_2, \ldots, v_r\}$ of order functions on ${\cal O}_{V,0}$ defines a multi-index filtration on ${\cal O}_{V,0}$: \begin{equation}\label{filtration} J({\underline{\upsilon}}):=\{g\in {\cal O}_{V,0}: \underline{v}(g)\ge{\underline{\upsilon}}\} \end{equation} for ${\underline{\upsilon}}=(\upsilon_1, \ldots, \upsilon_r)\in{\mathbb Z}_{\ge 0}^r$, $\underline{v}(g)=(v_1(g), \ldots, v_r(g))$, ${\underline{\upsilon}}'=(\upsilon'_1, \ldots, \upsilon'_r)\ge {\underline{\upsilon}}''=(\upsilon''_1, \ldots, \upsilon''_r)$ iff $\upsilon'_i\ge \upsilon''_i$ for all $i=1, \ldots, r$. (It is convenient to assume that the equation (\ref{filtration}) defines the subspaces $J({\underline{\upsilon}})\subset {\cal O}_{V,0}$ for all ${\underline{\upsilon}}\in{\mathbb Z}^r$.) The {\em Poincar\'e series} $P_{\{v_i\}}(\underline{t})$ ($\underline{t}=(t_1, \ldots, t_r)$) of the filtration (\ref{filtration}) can be defined as \begin{equation}\label{Poincare} P_{\{v_i\}}(\underline{t}):= \frac { \left(\sum_{{\underline{\upsilon}}\in{\mathbb Z}^r}\dim(J({\underline{\upsilon}})/J({\underline{\upsilon}}+\underline{1}))\underline{t}^{\underline{\upsilon}}\right)\prod_{i=1}^r(t_i-1) } {(t_1t_2\cdots t_r-1)}\,, \end{equation} where $\underline{1}=(1,\ldots, 1)\in{\mathbb Z}^r$, $\underline{t}^{\underline{\upsilon}}=t_1^{\upsilon_1}\cdots t_r^{\upsilon_r}$ (see e.g. \cite{IJM}; it is defined when the dimensions of all the factor spaces $J({\underline{\upsilon}})/J({\underline{\upsilon}}+\underline{1})$ are finite). In \cite{IJM} it was explained that the Poincar\'e series (\ref{Poincare}) is equal to the integral with respect to the Euler characteristic \begin{equation}\label{integral} P_{\{v_i\}}(\underline{t})=\int_{{\mathbb P}{\cal O}_{V,0}}\underline{t}^{\underline{v}(g)}d\chi \end{equation} over the projectivization ${\mathbb P}{\cal O}_{V,0}$ of the space ${\cal O}_{V,0}$. (In the integral $t_i^{+\infty}$ has to be assumed to be equal to zero.) Let $f\in{\cal O}_{{\mathbb C}^n,0}$ be a function germ with the Newton diagram $\Gamma=\Gamma_f\subset{\mathbb R}^n$, $V:=\{f=0\}$. Let $\gamma_i$, $i=1, \ldots, r$, be (all) the facets of the diagram $\Gamma$ and let $\ell_i(\bar{k})=c_i$ be the reduced equation of the hyperplane containing the facet $\gamma_i$. One has $\ell_i(\bar{k})=\sum_{j=1}^n \ell_{ij}k_j$ ($\bar{k}=(k_1, \ldots, k_n)$), where $\ell_{ij}$ are positive integers, $\gcd{(\ell_{i1}, \ldots, \ell_{in})=1}$. For $g\in {{\cal O}}_{{\mathbb C}^n,0}[x_1^{-1},\ldots, x_n^{-1}]$, $g=\sum\limits_{\bar{k}\in{\mathbb Z}^n}a_{\bar{k}}{\bar{x}}^{\bar{k}}$ ($\bar{x}=(x_1, \ldots, x_n)$), let $$ u_i(g):=\min_{{\bar{k}}:a_{\bar{k}}\ne0}\ell_i({\bar{k}})\,. $$ One can see that $u_i$ is a valuation on ${{\cal O}}_{{\mathbb C}^n,0}\subset {{\cal O}}_{{\mathbb C}^n,0}[x_1^{-1},\ldots, x_n^{-1}]$. For a Newton diagram $\Lambda$ in ${\mathbb R}^n$, let $$ u_i(\Lambda):=\min_{\bar{k} \in \Lambda} \ell_i(\bar{k}). $$ (It is also equal to $u_i(g)$ for any germ $g$ with the Newton diagram $\Lambda$.) Let $g_{\gamma_i}(\bar{x}):=\sum\limits_{\bar{k}:\ell_i(\bar{k})=u_i(g)}a_{\bar{k}}{\bar{x}}^{\bar{k}}$. The following two collections of order functions on ${{\cal O}}_{{\mathbb C}^n,0}$ corresponding to the pair $(\Gamma,f)$ were defined in \cite{EG1} and \cite{EG2} respectively: \begin{eqnarray} v'_i(g) & := & \sup_{h\in {\cal O}_{{\mathbb C}^n,0}} u_i(g+hf)\,, \\ v''_i(g) & := & \sup_{h\in {\cal O}_{{\mathbb C}^n,0}[x_1^{-1}, \ldots, x_n^{-1}]} u_i(g+hf)\,. \end{eqnarray} ($v'_i$ and $v''_i$ are, in general, not valuations, at least when $n=2$ or when $f$ is degenerate with respect to its Newton diagram $\Gamma$.) They can be considered as order functions on the ring ${{\cal O}}_{V,0}={{\cal O}}_{{\mathbb C}^n,0}/(f)$ as well. (These order functions and moreover the corresponding Poincar\'e series are, in general, different.) Assume that the function $f$ is non-degenerate with respect to its Newton diagram $\Gamma$ and let $p:(X,D)\to({\mathbb C}^n,0)$ be a toric resolution of $f$ corresponding to the Newton diagram $\Gamma$. The facets $\gamma_1$, \dots, $\gamma_r$ of $\Gamma$ correspond to some components (say, $E_1$, \dots, $E_r$) of the exceptional divisor $D$. Let $\widetilde{V}$ be the strict transform of the hypersurface singularity $V$ (it is a smooth complex manifold) and let ${\cal E}_i:=\widetilde{V}\cap E_i$, $i=1, \ldots, r$. For $n\geq 3$, the set ${\cal E}_i$ is an irreducible component of the exceptional divisor ${\cal D}=D\cap\widetilde{V}$ of the resolution $p_{\vert\widetilde{V}}:(\widetilde{V}, {\cal D})\to (V,0)$. The divisorial valuation $v_{{\cal E}_i}$ on ${{\cal O}}_{V,0}$ defined by this component coincides with $v''_i$: see \cite{EG2}. For $n=2$, the set ${\cal E}_i$ is, in general, reducible (if the integer length $s_i$ of the facet (edge) $\gamma_i$ is greater than 1). Let ${\cal E}_i=\bigcup\limits_{j=1}^{s_i}{\cal E}_i^{(j)}$ be the decomposition into the irreducible components (${\cal E}_i^{(j)}$ are points on the curve $\widetilde V$). One can show that in this case $v''_i(g)=\min\limits_j v_{{\cal E}_i^{(j)}}(g)$, where $v_{{\cal E}_i^{(j)}}$ are the corresponding divisorial valuations on ${{\cal O}}_{V,0}$. This order function $v_i''$ can be regarded as a generalized divisorial valuation. \section{The Poincar\'e series} Let $\Gamma$ be a Newton diagram in ${\mathbb R}^2$ with the facets (edges) $\gamma_1$, \dots, $\gamma_r$ and let $f$ be a function germ $({\mathbb C}^2,0)\to ({\mathbb C}, 0)$ with the Newton diagram $\Gamma$. One can see that $f=x^ay^b\prod\limits_{i=1}^r f_i$, where $f_i$ is such that $f_{\gamma_i}=\lambda_i{\bar{x}}^{{\bar{k}}_i}(f_i)_{\gamma_i}$ for certain $\lambda_i \in {\mathbb C}^$ and $\bar{k}_i \in {\mathbb Z}_{\ge 0}^2$. The Newton diagram $\Gamma_i$ of the germ $f_i$ consists of one segment congruent (by a shift; in particular, parallel) to the facet $\gamma_i$ with the vertices on the coordinate lines in ${\mathbb R}^2$. Let ${\underline M}_i=\underline{u}(\Gamma_i)$, i.e. ${\underline M}_i=(M_{i1}, \ldots, M_{ir})$, where $M_{ij}=u_j(\Gamma_i)$. (One can see that ${\underline M}_i=s_i\underline{m}_i$ in the notations of \cite{EG2}.) \begin{theorem} \label{theo1} One has \begin{equation} \label{main} P_{\{v''_i\}}(\underline{t})=\frac{\prod\limits_{i=1}^r(1-\underline{t}^{{\underline M}_i})}{(1-\underline{t}^{\underline{u}(x)})(1-\underline{t}^{\underline{u}(y)})}\,. \end{equation} \end{theorem} \begin{corollary} For the number of variables $n=2$ the Poincar\'e series $P_{\{v''_i\}}(\underline{t})$ depends only on the Newton diagram $\Gamma$ and does not depend on $f$ with $\Gamma_f=\Gamma$. \end{corollary} For the proof of Theorem~\ref{theo1} we need some auxiliary statements. We first introduce some notation. For a Newton diagram $\Lambda$ in ${\mathbb R}^2$, let $\Sigma_{\Lambda}$ be the corresponding Newton polygon: $\Sigma_{\Lambda}= \bigcup\limits_{\bar{q} \in\Lambda} \left( \bar{q} +{\mathbb R}_{\ge0}^2 \right)$. Let ${\cal O}^{\Lambda}$ be the set of functions $g\in{\cal O}_{{\mathbb C}^2,0}$ with the Newton diagram $\Gamma_g=\Lambda$. For ${\underline{\upsilon}}\in{\mathbb Z}_{\ge0}^r$, let ${\cal O}^{\Lambda}_{{\underline{\upsilon}}}= \{g\in {\cal O}^{\Lambda}: {\underline{v}}''(g)={\underline{\upsilon}}\}$. The set ${\cal O}_{{\mathbb C}^2,0}\setminus\{0\}$ is the disjoint union of the sets ${\cal O}^{\Lambda}$ over all diagrams $\Lambda$. According to (\ref{integral}) one has \begin{equation} \label{partition} P_{\{v''_i\}} (\underline{t}) = \sum_{\Lambda} \int_{{\mathbb P}{\cal O}^{\Lambda}}\underline{t}^{{\underline{v}}''(g)}d\chi\,. \end{equation} For $g\in{\cal O}^\Lambda$, one has $v''_i(g)\ge u_i(\Lambda)$. We shall first show that the integrals in (\ref{partition}) can be restricted only to functions $g \in {\mathbb P}{\cal O}^{\Lambda}$ with $\underline{v}''(g) = \underline{u}(\Lambda)$. \begin{proposition}\label{bad_functions} For a Newton diagram $\Lambda$ in ${\mathbb R}^2$, let ${\underline{\upsilon}}\in{\mathbb Z}_{\ge0}^r$ be such that ${\underline{\upsilon}}> \underline{u}(\Lambda)$, i.e. $\upsilon_i\ge u_i(\Lambda)$ for all $i=1, \ldots, r$ and $\upsilon_i > u_i(\Lambda)$ for some $i$. Then the set ${\mathbb P}{\cal O}^\Lambda_{{\underline{\upsilon}}}$ has Euler characteristic equal to zero. \end{proposition} \begin{remark} The direct analogue of this proposition does not hold for the filtration defined by the order functions $\{ v_i' \}$. As an example one can take $f(x,y)=y^5+xy^2+x^2y+x^5$ with the Newton diagram $\Gamma$ with the set of vertices $\{ (0,5), (1,2), (2,1), (5,0) \}$. One has $\ell_1(\bar{k})=3k_x+k_y$, $\ell_2(\bar{k}) = k_x+k_y$, $\ell_3(\bar{k})=k_x+3k_y$. Let $\Lambda$ be the Newton diagram with the set of vertices $\{ (0,5), (1,2)\}$. One has $\underline{u}(\Lambda)=(5,3,7)$. Let us take ${\underline{\upsilon}} = (7,3,7)$. One can see that for the order functions $\{ v_i' \}$ the set ${\cal O}^\Lambda_{{\underline{\upsilon}}}$ consists of the germs $g(x,y)=\sum a_{ij}x^iy^j$ from ${\cal O}^\Lambda$ with $a_{05}=a_{12} \neq 0$ and $a_{06}=a_{13}=0$. This gives $\chi({\mathbb P}{\cal O}^\Lambda_{{\underline{\upsilon}}})=1$. For the order functions $\{ v_i'' \}$ the set ${\cal O}^\Lambda_{{\underline{\upsilon}}}$ consists of the germs with $a_{05}=a_{12} \neq 0$, $a_{06}=a_{13}=0$ and $a_{07} - a_{14}+a_{21} \neq 0$. This gives $\chi({\mathbb P}{\cal O}^\Lambda_{{\underline{\upsilon}}})=0$ in accordance with Proposition~\ref{bad_functions}. \end{remark} For the proof of Proposition~\ref{bad_functions} we need two lemmas. Let ${\cal O}^\Lambda$ be non-empty. For $g \in {\cal O}^\Lambda$ with $\underline{v}''(g)={\underline{\upsilon}}$ and for $i=1, \ldots, r$, one can find $h(=h_i) \in {\cal O}_{{\mathbb C}^2,0}[x^{-1}, y^{-1}]$ such that the Newton diagram of $g+hf$ lies in the (closed) half-plane $H_i=\{\bar{k}:\ell_i(\bar{k})\ge \upsilon_i\}$, but there are no $h$ for which the Newton diagram of $g+hf$ lies in the open half-plane $\{\bar{k}:\ell_i(\bar{k})> \upsilon_i\}$. Let $\Lambda^$ be the union of the compact edges of the (infinite) polygon $\Sigma_\Lambda^= \bigcap\limits_{i=1}^r H_i \cap \Sigma_\Lambda$, where $\Sigma_\Lambda$ is the Newton polygon corresponding to $\Lambda$. ($\Lambda^$ is not, in general, a Newton diagram since it may have non-integral vertices. Nevertheless we shall use the name ``diagram" for it.) \begin{lemma} \label{lemma1} In the situation described above, there exists an index $i$ $(1\le i\le r)$ such that $\Lambda^$ has an edge $\delta_i$ parallel to $\gamma_i$ and (strictly) longer than $\gamma_i$. \end{lemma} \begin{proof} We shall prove that there exists an edge of the diagram $\Lambda^$ which is (strictly) longer than the edge of $\Lambda$ parallel to it. This implies that this edge is parallel to a certain edge $\gamma_i$ of the diagram $\Gamma$ and is longer than it. Since we assumed ${\cal O}^{\Lambda}$ being non-empty, all edges of $\Lambda^$ are parallel to edges of $\Lambda$. Let $a_0 < a_1 < \ldots < a_\sigma$ be the $k_x$-coordinates of all the vertices of $\Lambda$. Let $b_0 \leq b_1 \leq \ldots \leq b_\sigma$ be the $k_x$-coordinates of the corresponding vertices of $\Lambda^$, i.e.\ $[b_{i-1}, b_i]$ is the projection of the segment of $\Lambda^$ parallel to the segment of $\Lambda$ projected to $[a_{i-1},a_i]$ ($b_{i-1}$ and $b_i$ may coincide). One can see that $a_0=b_0$ and $a_\sigma \leq b_\sigma$. Then either $b_i=a_i$ for all $i=0,1, \ldots , \sigma$ or $[a_{i-1},a_i] \subsetneq [b_{i-1}, b_i]$ for some $i \in \{ 0,1, \ldots, \sigma \}$. But the first case cannot happen since $\Lambda^\neq \Lambda$. \end{proof} We shall also use the following generalized version of the division with remainder for Laurent polynomials. \begin{lemma}\label{division} Let $p(z)$ and $q(z)$ be Laurent polynomials in $z$. Assume that ${\rm supp}\, q$ has length $s$, i.e. $q(z)=\sum\limits_{i=0}^s b_iz^{d+i}$ with $b_0\ne 0$, $b_s\ne 0$, and let $d'$ be an integer. Then the polynomial $p(z)$ has a unique representation of the form $p(z)=q(z)a(z) + r(z)$ with $r(z)=\sum\limits_{i=0}^{s-1} c_iz^{d'+i}$ . \end{lemma} \begin{proof}{Proof of Proposition~\ref{bad_functions}} Let $i$ be as in Lemma~\ref{lemma1}. Let the integer length of $\gamma_i$ be equal to $s_i$. Then the segment $\delta_i$ contains at least $s_i$ integer points. Let $Q_1, \ldots, Q_{s_i}$ be $s_i$ consecutive integer points on the segment $\delta_i$. Let $g\in{\cal O}^\Lambda$ be such that $\underline{v}''(g)={\underline{\upsilon}}$ ($>\underline{u}(\Lambda)$) and let $\widetilde{g}=g+hf$ be a Laurent polynomial such that ${\rm supp}\,{\widetilde{g}}\subset H_i=\{\ell_i(\bar{k})\ge \upsilon_i\}$. Lemma~\ref{division} implies that $\widetilde{g}_{\gamma_i}(x,y)=f_{\gamma_i}(x,y)p_i(x,y) + r_i(x,y)$ where ${\rm supp}\,{r}\subset \{Q_1, \ldots, Q_{s_i}\}$ and $r_i\ne 0$ (otherwise $v_i(g)>\upsilon_i=u_i(\Lambda^)$). (Let us recall that ${\rm supp}\, f_{\gamma_i}$ consists of $s_i+1$ consecutive points on the line containing $\gamma_i$.) Moreover the polynomial $r_i$ depends only on $g$ and does not depend on the choice of $\widetilde{g}$ (i.e.\ on the choice of $h$). One can see that all functions $g'$ of the form $g+(\lambda -1)r_i$ with $\lambda \ne 0$ lie in ${\cal O}^{\Lambda}$ and satisfy the condition $v''_i(g')=v''_i(g)$. Thus the set ${\mathbb P}{\cal O}^{\Lambda}_{{\underline{\upsilon}}}$ is fibred by ${\mathbb C}^$-families and therefore its Euler characteristic is equal to zero. \end{proof} Proposition~\ref{bad_functions} implies that \begin{equation}\label{int_good} P_{\{ v''_i\}} (\underline{t}) = \sum_{\Lambda} \int_{{\mathbb P}{\cal O}^{\Lambda}_{\underline{u}(\Lambda)}}\underline{t}^{{\underline{v}}''(g)}d\chi\,. \end{equation} \begin{proposition}\label{bad_diagrams} Suppose that a Newton diagram $\Lambda$ contains an edge $\delta$ not congruent to any edge of $\Gamma$, i.e. either not parallel to all the edges $\gamma_i$, or parallel to one of them, but of another length. Then $\chi({\mathbb P}{\cal O}^{\Lambda}_{\underline{u}(\Lambda)})=0$. \end{proposition} \begin{proof} Assume first that the edge $\delta$ is either not parallel to all the edges $\gamma_i$, or it is parallel to $\gamma_i$, but is shorter than $\gamma_i$. Let $\bar{q}=(q_x, q_y)$ and $\bar{q}'=(q'_x, q'_y)$, $q_x>q'_x$, be the vertices of the edge $\delta$ and let $\Lambda'$ be the set of points $\bar{k}$ in $\Lambda$ with $k_x\ge q_x$. A function germ $g\in {\cal O}^{\Lambda}_{\underline{u}(\Lambda)}$ can be represented as $g_1+g_2$, where ${\rm supp}\,g_1\subset \Lambda'$, ${\rm supp}\,g_2\subset \Sigma_\Lambda\setminus\Lambda'$. (Note that $g_1\ne 0$ and $g_2\ne 0$.) One can see that all the functions of the form $g_1+\lambda g_2$ with $\lambda\ne 0$ lie in ${\cal O}^{\Lambda}_{\underline{u}(\Lambda)}$. Thus the set ${\mathbb P}{\cal O}^{\Lambda}_{\underline{u}(\Lambda)}$ is fibred by ${\mathbb C}^$-families and therefore its Euler characteristic is equal to zero. Now assume that the edge $\delta$ of the diagram $\Lambda$ is parallel to $\gamma_i$ and is longer than it. Let $\bar{q}=(q_x, q_y)$ and $\bar{q}'=(q'_x, q'_y)$, $q_x>q'_x$, (respectively $\bar{q}_0=(q_{0x}, q_{0y})$ and $\bar{q}'_0=(q'_{0x}, q'_{0y})$, $q_{0x}>q'_{0x}$) be the vertices of the edge $\delta$ (respectively of the edge $\gamma_i$) and let $\Lambda'$ be defined as above: $\Lambda'=\{\bar{k}\in \Lambda: k_x\ge q_x\}$. Let $g(\bar{x})=\sum a_{\bar{k}}{\bar{x}}^{\bar{k}} \in {\cal O}^{\Lambda}_{\underline{u}(\Lambda)}$, $f(\bar{x})=\sum c_{\bar{k}}{\bar{x}}^{\bar{k}}$ ($\bar{x}=(x,y)$) and let $$ g_1(\bar{x})=g_{\Lambda'}(\bar{x})-a_{\bar{q}}{\bar{x}}^{\bar{q}}+(a_{\bar{q}}/c_{\bar{q}_0}) f_{\gamma_i}(\bar{x})\cdot {\bar{x}}^{\bar{q}-\bar{q}_0}, $$ where $g_{\Lambda'}(\bar{x})=\sum\limits_{\bar{k}\in\Lambda'} a_{\bar{k}}{\bar{x}}^{\bar{k}}$, $g_2=g-g_1$. One has ${\rm supp}\,g_1\subset \Lambda'\cup (\bar{q},\bar{q}')$, ${\rm supp}\,g_2\subset \Sigma_\Lambda\setminus\Lambda'$, $f_{\gamma_i} {\not\|} \, (g_2)_{\gamma_i}$ (in ${\cal O}_{{\mathbb C}^2,0}[x^{-1}, x^{-1}]$), where $(\bar{q},\bar{q}')$ denotes the open line segment connecting the two points. All the functions of the form $g_1+\lambda g_2$ with $\lambda\ne 0$ lie in ${\cal O}^{\Lambda}_{\underline{u}(\Lambda)}$. Thus the set ${\mathbb P}{\cal O}^{\Lambda}_{\underline{u}(\Lambda)}$ is again fibred by ${\mathbb C}^$-families and therefore its Euler characteristic is equal to zero. \end{proof} \begin{proposition}\label{value} Let the Newton diagram $\Lambda$ consist (only) of segments congruent to $\gamma_i$ for $i\in I\subset \{1, \ldots, r\}$. Then $\chi({\mathbb P}{\cal O}^{\Lambda}_{\underline{u}(\Lambda)})=(-1)^{\# I}$. \end{proposition} \begin{proof} For $I=\emptyset$, the statement is obvious. Let $I\ne\emptyset$. Let ${\mathbb P}{\cal O}^{\Lambda}_{i}$ be the set of functions $g\in{\mathbb P}{\cal O}^{\Lambda}_{\underline{u}(\Lambda)}$ with $f_{\gamma_i} \| g_{\gamma_i}$ (in ${\cal O}_{{\mathbb C}^2,0}[x^{-1}, x^{-1}]$). One has $$ {\mathbb P}{\cal O}^{\Lambda}_{\underline{u}(\Lambda)}= {\mathbb P}{\cal O}^{\Lambda}\setminus \bigcup_{i\in I}{\mathbb P}{\cal O}^{\Lambda}_{i}\,. $$ Therefore \begin{equation} \label{Euler} \chi({\mathbb P}{\cal O}^{\Lambda}_{\underline{u}(\Lambda)})= \chi({\mathbb P}{\cal O}^{\Lambda})+ \sum_{I'\subset I, I'\ne\emptyset}(-1)^{\# I'}\chi \left(\bigcap_{i\in I'}{\mathbb P}{\cal O}^{\Lambda}_{i} \right)\,. \end{equation} Let $\bar{q}_i$, $i=0,1, \ldots , \# I$, be the vertices of the diagram $\Lambda$. The set ${\mathbb P}{\cal O}^{\Lambda}$ consists of functions $g(\bar{x})=\sum a_{\bar{k}} \bar{x}^{\bar{k}}$ with $a_{\bar{q}_i} \neq 0$ for $i=0,1, \ldots, \# I$ and $a_{\bar{k}} = 0$ for $\bar{k} \not\in \Sigma^{\Lambda}$. Its Euler characteristic is equal to zero. Assume that $I' \subsetneq I$, $I' \neq \emptyset$. Let $[\bar{q}, \bar{q}']$ be an edge of $\Lambda$ congruent to $\gamma_i$, $i \in I \setminus I'$ ($\bar{q} = (q_x,q_y)$, $\bar{q}'=(q_x',q_y')$, $q_x>q_y'$). Let $\Lambda'= \{ \bar{k} \in \Lambda \, : \, k_x \geq q_x \}$. For $g \in \bigcap_{i \in I'} {\mathbb P}{\cal O}_i^{\Lambda}$, let $g_1(\bar{x})=g_{\Lambda'}(\bar{x})$, $g_2=g-g_1$. All the functions of the form $g_1+ \lambda g_2$ with $\lambda \neq 0$ belong to $\bigcap_{i \in I'} {\mathbb P}{\cal O}_i^{\Lambda}$. Thus $\bigcap_{i \in I'} {\mathbb P}{\cal O}_i^{\Lambda}$ is fibred by ${\mathbb C}^\ast$-families and therefore $\chi(\bigcap_{i \in I'} {\mathbb P}{\cal O}_i^{\Lambda})=0$. Let $f_I(\bar{x}):= \prod_{i \in I} f_i(\bar{x})$. The intersection $\bigcap_{i \in I} {\mathbb P}{\cal O}_i^{\Lambda}$ (the Euler characteristic of which corresponds to $I'=I$ in (\ref{Euler})) consists of the functions $g \in {\mathbb P}{\cal O}_{{\mathbb C}^2,0}$ such that $g_\Lambda(\bar{x}) = \lambda \bar{x}^{\bar{a}} (f_I)_{\Gamma_{f_I}}$ where $\Gamma_{f_I}$ is the Newton diagram of $f_I$, $\lambda \neq 0$ and $\bar{x}^{\bar{a}}$ is a certain monomial. Therefore \[ \chi \left(\bigcap_{i \in I} {\mathbb P}{\cal O}_i^{\Lambda} \right)=1. \] \end{proof} \begin{proof}{Proof of Theorem~\ref{theo1}} Propositions~\ref{bad_functions} and \ref{bad_diagrams} imply that \begin{equation} \label{finalPS} P_{\{v_i'' \}}(\underline{t})= \sum_\Lambda \int_{{\mathbb P}{\cal O}^{\Lambda}_{\underline{u}(\Lambda)}}\underline{t}^{{\underline{v}}''(g)}d\chi \end{equation} where the sum runs over all diagrams $\Lambda$ consisting only of edges congruent to some of the edges $\gamma_i$ of the diagram $\Lambda$. Let the edges of $\Lambda$ be congruent to the edges $\gamma_i$ with $i \in I=I(\Lambda)$. Proposition~\ref{value} implies that the summand in (\ref{finalPS}) corresponding to such a diagram $\Lambda$ is equal to $(-1)^{\# I} \underline{t}^{\underline{u}(\Lambda)}$. All the diagrams of this sort are obtained from the diagrams $\Gamma_I=\Gamma_{f_I}$ by shifts by non-negative integral vectors $\bar{k}$, i.e.\ $\Lambda= \bar{k} + \Gamma_I$. One has $\underline{u}(\Lambda)=\underline{\ell}(\bar{k}) + \sum_{i \in I} {\underline M}_i$. Therefore \begin{eqnarray} P_{\{v_i'' \}}(\underline{t}) & = & \sum_{\bar{k} \in {\mathbb Z}^2_{\geq 0}} \sum_{I \subset \{ 1, \ldots , r \}} (-1)^{\# I} \underline{t}^{\underline{\ell}(\bar{k}) + \sum_{i \in I} {\underline M}_i} \\ & = & \left( \sum_{\bar{k} \in {\mathbb Z}^2_{\geq 0}} \underline{t}^{\underline{\ell}(\bar{k})} \right) \cdot \left( \sum_{I \subset \{ 1, \ldots , r \}} (-1)^{\# I} \underline{t}^{\sum_{i \in I} {\underline M}_i} \right) \\ & = & \frac{\prod\limits_{i=1}^r(1-\underline{t}^{{\underline M}_i})}{(1-\underline{t}^{\underline{u}(x)})(1-\underline{t}^{\underline{u}(y)})} . \end{eqnarray} \end{proof} \section{Relation with an Alexander polynomial} One can see that the equation (\ref{main}) gives the Poincar\'e series $P_{\{v''_i\}}(\underline{t})$ as a finite product/ratio of ``cyclotomic'' binomials of the form $(1-\underline{t}^{\underline M})$ with ${\underline M}\in{\mathbb Z}_{> 0}^r$. This looks similar to the usual A'Campo type expressions for monodromy zeta functions or for Alexander polynomials of algebraic links \cite{EN}. Here we shall describe a relation between the Poincar\'e series (\ref{main}) and a certain Alexander polynomial. A notion of the multi-variable Alexander polynomial for a finite collection of germs of functions on $({\mathbb C}^n,0)$ was defined in \cite{S}: see Proposition 2.6.2 therein. (In \cite{S} it is called the (multi-variable) zeta function.) In a somewhat more precise form this definition can be found in \cite{GDC}. (The definition in \cite{GDC} gives the one for a collection of functions if one considers the corresponding principal ideals.) As above, let $\Gamma$ be a Newton diagram in ${\mathbb R}^2$ with the edges $\gamma_1$, \dots, $\gamma_r$ of integer lengths $s_1, \ldots, s_r$ and let $p:(X,D)\to({\mathbb C}^2, 0)$ be a toric modification of $({\mathbb C}^2, 0)$ corresponding to the diagram $\Gamma$. For $i=1, \ldots, r$, let $\widetilde{C}_i$ be a germ of a smooth curve on $X$ transversal to the component $E_i$ of the exceptional divisor $D$. Let $C_i=p(\widetilde{C}_i)$ be the image of $\widetilde{C}_i$ in $({\mathbb C}^2, 0)$ and let $L_i=C_i\cap S^3_\varepsilon$ be the corresponding knot in the 3 sphere $S^3_\varepsilon=S^3_\varepsilon(0)$ for $\varepsilon > 0$ small enough. The curve $C_i$ can be defined by an equation $g_i=0$ where $g_i$ is a function germ $({\mathbb C}^2, 0)\to({\mathbb C}, 0)$ with the Newton diagram consisting of one segment parallel to $\gamma_i$, with the (integer) length 1 and with the vertices on the coordinate lines. Let $\Delta_{\underline{g}}(\underline{t})$ and $\Delta_{\underline{g}^{\underline{s}}}(\underline{t})$ be the Alexander polynomials of the collections of functions $\underline{g}=(g_1, \ldots , g_r)$ and $\underline{g}^{\underline{s}}=(g_1^{s_1}, \ldots , g_r^{s_r})$ respectively. The polynomial $\Delta_{\underline{g}}(\underline{t})$ is the classical Alexander polynomial $\Delta^L(\underline{t})$ of the link $L = \bigcup L_i$ (see e.g.\ \cite{EN}). A one-variable analogue of $\Delta_{\underline{g}^{\underline{s}}}(\underline{t})$ is considered in \cite[I.5]{EN} as the Alexander polynomial of the multilink $L(\underline{s})=\bigcup s_iL_i$. One has $$ \Delta_{\underline{g}}(\underline{t})=\frac{\prod_{i=1}^r (1-\underline{t}^{\underline{m}_i})}{(1-\underline{t}^{\underline{u}(x)})(1-\underline{t}^{\underline{u}(y)})}\,, $$ $$ \Delta_{\underline{g}^{\underline{s}}}(\underline{t})= \frac{\prod_{i=1}^r (1-\underline{t}^{{\underline{s}} \, \underline{m}_i})}{(1-\underline{t}^{\underline{u}(x)})(1-\underline{t}^{\underline{u}(y)})}\,, $$ where $\underline{s}\, \underline{m}_i=(s_1m_{1i}, s_2m_{2i}, \ldots, s_rm_{ri})$. The main result of \cite{IJM} says that $\Delta_{\underline{g}}(\underline{t})=\Delta^L(\underline{t})$ coincides with the Poincar\'e series of the filtration corresponding to the Newton diagram of the function $\prod_{i=1}^r g_i$. Let the {\em reduced Poincar\'e series} of the filtration defined by $\{ v_i'' \}$ be $$ \widetilde{P}_{\{ v_i''\}} ( \underline{t}) := P_{\{ v_i''\}} ( \underline{t}) / P_{\{ u_i\}}(\underline{t}), \quad P_{\{ u_i\}}(\underline{t})= \frac{1}{(1-\underline{t}^{\underline{u}(x)})(1-\underline{t}^{\underline{u}(y)})} $$ being the Poincar\'e series of the filtration defined by the quasihomogeneous valuations $\{ u_i \}$ on ${\cal O}_{{\mathbb C}^2,0}$. One has \begin{equation} \label{redPS} \widetilde{P}_{\{ v_i''\}} ( \underline{t}) = \prod\limits_{i=1}^r (1- \underline{t}^{s_i \underline{m}_i}). \end{equation} Let $$ \widetilde{\Delta}_{\underline{g}^{\underline{s}}} ( \underline{t}) := \Delta_{\underline{g}^{\underline{s}}} ( \underline{t}) / \Delta^x_{\underline{g}^{\underline{s}}} ( \underline{t}) \cdot \Delta^y_{\underline{g}^{\underline{s}}} ( \underline{t}) $$ where $\Delta^x_{\underline{g}^{\underline{s}}} ( \underline{t})$ and $\Delta^y_{\underline{g}^{\underline{s}}} ( \underline{t})$ are the Alexander polynomials of the sets of functions $\underline{g}^{\underline{s}}= \{ g_1^{s_1}, \ldots , g_r^{s_r} \}$ restricted to the coordinate axes ${\mathbb C}_x$ and ${\mathbb C}_y$ respectively. One can regard $\widetilde{\Delta}_{\underline{g}^{\underline{s}}} ( \underline{t})$ as the Alexander polynomial of the set of functions $\underline{g}^{\underline{s}}$ restricted to the complex torus $({\mathbb C}^)^2 \subset {\mathbb C}^2$. One has \begin{equation} \label{Alexander} \widetilde{\Delta}_{\underline{g}^{\underline{s}}} ( \underline{t}) = \prod\limits_{i=1}^r (1- \underline{t}^{\underline{s}\, \underline{m}_i}). \end{equation} One can see that a relation between (\ref{redPS}) and (\ref{Alexander}) can be described in the following way. Consider products of $r$ ordered cyclotomic binomials in $r$ variables. Such a product $$ \prod\limits_{i=1}^r (1-\underline{t}^{{\underline N}_i}), \quad {\underline N}_i=(N_{i1}, \ldots , N_{ir}), $$ can be described by the corresponding $r \times r$-matrix $N:=(N_{ij})$. The transposition of the matrix induces an involution on the set of products of this sort. One can see that this involution maps the product (\ref{redPS}) for the Poincar\'e series to the product (\ref{Alexander}) for the Alexander polynomial. \noindent Leibniz Universit\"{a}t Hannover, Institut f\"{u}r Algebraische Geometrie,\\ Postfach 6009, D-30060 Hannover, Germany \\ E-mail: [email protected]\\ \noindent Moscow State University, Faculty of Mechanics and Mathematics,\\ Moscow, GSP-1, 119991, Russia\\ E-mail: [email protected] \end{document}	arXiv
\begin{document} \title{Microlocal sheaves and quiver varieties} \begin{dedication} \`A Vadim Schechtman pour son 60-\`eme anniversaire \end{dedication} \tableofcontents \addtocounter{section}{-1} \section{Introduction.} The goal of this paper is to relate two classes of symplectic manifolds of great importance in Representation Theory and to put them into a common framework. \vskip .2cm \Par{ Moduli of local systems on Riemann surfaces.} First, let $X$ be a compact oriented $C^\infty$ surface and $G$ be a reductive algeraic group. The moduli space $\operatorname{LS}_G(X)$ of $G$-local systems on $X$ is naturally a symplectic manifold \cite{goldman}, with the symplectic structure given by the cohomological pairing. As shown by Atiyah-Bott, $\operatorname{LS}_G(X)$ can be obtained as the Hamiltonian reduction of an infinite-dimensional flat symplectic space formed by all $G$-connections, with the Lie algebra-valued moment map given by the curvature. Alternatively, $\operatorname{LS}_G(X)$ can be obtained as a Hamiltonian reduction of a finite-dimensional symplectic space but at the price of passing to the { multiplicative theory}: replacing the Lie algebra-valued moment map by a group-valued one \cite{AMM}. The variety $\operatorname{LS}_G(X)$ and its versions associated to surfaces with punctures, marked points etc. form fundamental examples of cluster varieties \cite{FG}, and their quantization is interesting from many points of view. We will be particularly interested in the case $G=GL_n$, in which case local systems form an abelian category. \vskip .2cm \Par{ Quiver varieties.} The second class is formed by the Nakajima quiver varieties \cite{N}. Given a finite oriented graph $Q$, the corresponding quiver varieties can be seen as symplectic reductions of the cotangent bundles to the moduli spaces of representations of $Q$ with various dimension vectors. Passing to the cotangent bundle has the effect of ``doubling the quiver": introducing, for each arrow $\xymatrix{ i \ar[r] &j}$ of $Q$, a new arrow $\xymatrix {i & \ar[l] j} $ in the opposite direction. Interestingly, one also has the ``multiplicative" versions of quiver varieties defined by Crawley-Boevey and Shaw \cite{CBS} and Yamakawa \cite{yamakawa}. They can be constructed by performing the Hamiltonian reduction but using the group-valued moment map. It is these multiplicative versions that we will consider in this paper. \vskip .2cm \Par { Relation to perverse sheaves.} It turns out that both these classes can be put under the same umbrella of {\em varieties arising from classification of perverse sheaves}. From the early days of the theory \cite{BBD}, a lot of effort has been spent on finding descriptions of various categories of perverse sheaves as representation categories of some explicit quivers with relations. In all of these cases, the quivers have the following remarkable property: {\em their arrows come in pairs of opposites} $ \xymatrix{ i \ar@<.4ex>[r]& \ar@<.4ex>[l] j } $. This reflects the fact that any category of perverse sheaves has a perfect duality (Verdier duality). The diagram (representation of the quiver) corresponding to the dual perverse sheaf $\mathcal{F}^\bigstar$ is obtained from the diagram corresponding to $\mathcal{F}$ by dualizing both the spaces and (up to a minor twist, cf. \cite[(II.3.4)]{Ma}) the arrows, thus interchanging the elements of each pair of opposites. We see therefore a conceptual reason for a possible relationship between perverse sheaves and quiver varieties. The relation between perverse sheaves and $\operatorname{LS}_{GL_n}(X)$ is even more immediate: local systems are nothing but perverse sheaves without singularities, so ``moduli spaces of perverse sheaves" are natural objects to look at. \vskip .2cm \Par { Microlocal sheaves.} However, to make the above relations precise, we need to use a generalization of perverse sheaves: {\em microlocal sheaves}. These objects can be thought as modules over a (deformation) quantization of a symplectic manifold $S$ supported in a given Lagrangian subvariety $X$, see \cite{KS-DQ}. The case $S=T^M$ being the cotangent bundle to a manifold $M$ and $X$ being conic, corresponds to the usual theory of holonomic $\mathcal{D}$-modules and perverse sheaves. However, for our applications it is important to consider the case when $X$ is compact. In this paper we need only the simplest case when $X$ is an algebraic curve over $\mathbb{C}$ which is allowed to have nodal singularities. In this case microlocal sheaves can be defined in a very elementary way as perverse sheaves on the normalization satisfying a Fourier transform condition near each self-intersection point. The relation with quiver varieties appears when we take $X$ to be a union of projective lines whose intersection graph is our ``quiver" $Q$ (with orientation ignored). If we consider only ``smooth" microlocal sheaves (no singularities other than the nodes), we get a natural analog of the concept of a local system for nodal curves. In particular, for a compact $X$ we consider such microlocal sheaves as objects of a triangulated category $D\mathcal{M}(X,\emptyset)$ of {\em microlocal complexes}, and we show in Thm. \ref{thm:CY} that it has the {\em 2-Calabi-Yau property}, extending the Poincar\'e duality for local systems: \[ R\operatorname{Hom} (\mathcal{F}, \mathcal{G})^ \,\,\simeq \,\, R\operatorname{Hom}(\mathcal{G}, \mathcal{F})[2]. \] This gives an intrinsic reason to expect that the ``moduli spaces" parametrizing microlocal sheaves or complexes, are symplectic, in complete analogy with Goldman's picture \cite{goldman} for local systems. We discuss the related issues in \S \ref{sec:preprogen}D and give a more direct construction of such spaces in \S \ref{sec:framed} by using quasi-Hamiltonian reduction. \vskip .2cm \Par{Relation to earlier work.} An earlier attempt to relate (multiplicative) quiver varieties and D-module type objects (i.e., to invoke the Riemann-Hilbert correspondence) was made by D. Yamakawa \cite{yamakawa}. Although his construction is quite different from ours and is only applicable to quivers of a particular shape, it was one of the starting points of our inverstigation. \vskip .2cm More recently, a Riemann-Hilbert type interpretation of multiplicative preprojective algebras was given by W.~Crawley-Boevey \cite{CB}. His setup is in fact quite close to ours (although we learned of his paper only after most of our constructions have been formulated). In particular, the datum of a ``Riemann surface quiver with non-interfering arrows", a central concept of \cite{CB}, is equivalent to the datum of a nodal curve $X$: the normalization $\widetilde X$ is then the corresponding Riemann surface, and the pairwise identifications of the points of $\widetilde X$ needed to get $X$, form a Riemann surface quiver. From our point of view, the construction of \cite{CB} can be seen as leading to an explicit description, in terms of D-module type data, of ``smooth" microlocal sheaves on a nodal curve, see Theorem \ref{thm:micro-dr}. \vskip .2cm Considering a nodal curve $X$ as the basic object, has the advantage of putting the situation, at least heuristically, into the general framework of deformation quantization (DQ-)modules. In particular, one can consider for $X$ a projective curve with more complicated singularities, realized as a (necessarily Lagrangian) subvariety in a holomorphic symplectic surface. The general theory of \cite{KS-DQ} suggests that moduli spaces of ``smooth" microlocal sheaves in this situation will produce interesting symplectic varieties. Further, passing to higher-dimensional projective singular Lagrangian varieties $X$, one expects to get shifted symplectic varieties, as suggested by the Calabi-Yau property of DQ-modules \cite[Cor. 6.2.5]{KS-DQ} and the general theory of \cite{kontsevich-soibelman} and \cite{PTVV}. \vskip .2cm \Par { Acknowledgements.} We are grateful to A. Alekseev, Y. Brunebarbe, T. Dyckerhoff, V. Ginzburg, P. Schapira, Y. Soibelman and G. Williamson for useful discussions and correspondence. The work of M.K. was supported by the World Premier International Research Center Initiative (WPI), MEXT, Japan and by the Max-Planck Institute f\"ur Mathematik, Bonn, Germany. \vskip .3cm \eject \eject \section{Microlocal sheaves on nodal curves}\label{sec:microsheaves} \sparagraph{Topological definitions.} Let $X$ be a nodal curve over $\mathbb{C}$, i.e., an algebraic, quasi-projective curve whose only singularities are transversal self-intersection points (also known as {\em nodes}, or ordinary double points). For a node $x\in X$ we denote two ``branches" of $X$ near $x$ (defined up to permutation) by $B'$ and $B''$. More precisely, we think of $B'$ and $B''$ as small disks meeting at $x$. Alternatively, let $\varpi: \widetilde X\to X$ be the normalization of $X$. Then $\varpi^{-1}(x)=\{x', x''\}$ consists of two points, and we define $\widetilde {B}', \widetilde {B}''$ as the neighborhoods of $x'$ and $x''$ in $\widetilde X$. We can then identify canonically $B' = \widetilde{B}'$, $B''=\widetilde{B}''$. We note that the Zariski tangent space to $X$ at a node $x$ is 2-dimensional: \[ T _xX \,\,=\,\, T_x B' \oplus T_x B''. \] \begin{Defi} A {\em duality structure} on $X$ is a datum, for each node $x$, of a symplectic structure $\omega_x$ on the 2-dimensional vector space $T_xX$. \end{Defi} Alternatively, a duality structure at a node $x$ can be considered as a datum of isomorphisms \[ {\varepsilon}'_x: T_x B' \to T^_x B'', \quad {\varepsilon}_x'': T_x B'' \to T^_x B' \] such that $({\varepsilon}''_x)^* = - {\varepsilon}'_x$. \begin{ex}\label{ex:SS} (a) Suppose $X$ embedded into a holomorphic symplectic surface $(S,\omega)$. Then the restrictions of $\omega$ to all the nodes of $X$ give a duality structure on $X$. Note that any duality structure on $X$ can be obtained in this way. Indeed, we first consider a neighborhood $\widetilde S$ of the zero section in the cotangent bundle $T^\widetilde X$. Then for any node $x\in X$ with $\varpi^{-1}(x)=\{x', x''\}$, we identify the neighborhoods $U'$ of $x'$ and $U''$ of $x''$ in $\widetilde S$ by an appropriate symplectomorphism so that the intersection of $U'$ with the zero section of $T^\widetilde S$ becomes identified with the intersection of $U''$ with the fiber of $T^\widetilde S$ over $x''$ and vice versa. \vskip .1cm (b) Situations when $X$ is naturally embedded into an {\em algebraic} symplectic surface $S$, provide a richer structure. The best known examples are provided by $S$ being the minimal resolution of a Kleinian singularity $\mathbb{C}^2/G$, where $G$ is a finite subgroup in $SL_2(\mathbb{C})$. In this case $X$ is a union of projective lines, with the intersection graph being a Dynkin diagram of type ADE. \end{ex} Let $X$ be a nodal curve with a duality structure. For each node $x\in X$ we can identify $B'$ and $B''$ with open disks in $T_xB'$ and $T_x B''$ or, equivalently, in $T_x\widetilde{B}'$ and $T_x {\widetilde B}''$ respectively. Such identifications are unique up to contractible spaces of choices. Let $D^b(\widetilde {B}', x')$ be the full subcategory in $D^b_{\operatorname{constr}}(\widetilde {B}')$ formed by complexes whose cohomology sheaves are locally constant outside $x'$, and similarly for $D^b(\widetilde {B}'', x'')$. Let $\on{Perv}(\widetilde{B}', x')\subset D^b(\widetilde {B}', x')$ and $\on{Perv}(\widetilde{B}'', x'')\subset D^b(\widetilde {B}'', x'')$ be the full (abelian) subcategories formed by perverse sheaves. The above identifications with the disks in the tangent spaces together with the isomorphisms ${\varepsilon}', {\varepsilon}''$ give rise to geometric Fourier(-Sato) transforms which are equivalences of pre-triangulated categories \begin{equation}\label{eq:FT-1} \xymatrix{ D^b(\widetilde{B}', x') \ar@<.7ex>[rr]^{\operatorname{FT}'}&& \ar@<.7ex>[ll]^{\operatorname{FT}''} D^b(\widetilde{B}'', x''), } \end{equation} which are canonically inverse to each other and restrict to equivalence of abelian categories \begin{equation}\label{eq:FT-2} \xymatrix{ \on{Perv}(\widetilde{B}', x') \ar@<.7ex>[rr]^{\operatorname{FT}'}&& \ar@<.7ex>[ll]^{\operatorname{FT}''} \on{Perv}(\widetilde{B}'', x''). } \end{equation} \begin{rem} The fact that $\operatorname{FT}'$ and $\operatorname{FT}''$ are precisely inverse to each other, comes from the requirement that ${\varepsilon}'_x$ and ${\varepsilon}''_x$ are the negatives of the transposes of each other, rather than exact transposes. We recall that the ``standard" Fourier-Sato transform for a $\mathbb{C}$-vector space $E$ is an equivalence (\cite{KS}, Ch. 3) \[ \operatorname{FT}_E: D^b_{\operatorname{mon}}(E) \to D^b_{\operatorname{mon}}(E^) \] ($D^b_{\operatorname{mon}}$ means the derived category of $\mathbb{C}$-monodromic constructible complexes). In this setting $\operatorname{FT}_{E^}$ is not canonically inverse to $\operatorname{FT}_E$: the composition $\operatorname{FT}_{E^}\circ \operatorname{FT}_E$ is canonically identified with $(-1)^$, the pullback with respect to the antipodal transformation $(-1): E\to E$. \end{rem} \begin{Defi}\label{def:micro-complex} A {\em microlocal complex} $\mathcal{F}$ on $X$ is a datum consisting of: \begin{enumerate} \item [(1)] A $\mathbb{C}$-constructible complex $\widetilde\mathcal{F}$ on $\widetilde X$. \item[(2)] For each node $x\in X$, quasi-isomorphisms of constructible complexes \[ \alpha': \widetilde\mathcal{F}\|_{\widetilde{B}'} \longrightarrow \operatorname{FT}''\left( \widetilde\mathcal{F}\|_{\widetilde{B}''}\right),\quad \alpha'': \widetilde\mathcal{F}\|_{\widetilde{B}''} \longrightarrow \operatorname{FT}'\left( \widetilde\mathcal{F}\|_{\widetilde{B}'}\right), \] inverse to each other. \end{enumerate} A {\em microlocal sheaf} on $X$ is a microlocal complex $\mathcal{F}$ such that $\widetilde\mathcal{F}$ is a perverse sheaf on $\widetilde X$. \end{Defi} A {\em morphism} of microlocal complexes (resp. microlocal sheaves) $\mathcal{F}\to\mathcal{G}$ is a morphism of constructible complexes (resp. perverse sheaves) $\widetilde\mathcal{F}\to\widetilde\mathcal{G}$ on $\widetilde X$ compatible with the identifications $\alpha', \alpha''$. In this way we obtain a pre-triangulated category $D\mathcal{M}(X)$ formed by microlocal complexes on $X$ and an abelian subcategory $\mathcal{M}(X)$ formed by microlocal sheaves. For a finite subset of smooth points $A\subset X_{\operatorname{sm}}$ we denote by $D\mathcal{M}(X, A)\subset D\mathcal{M}(X)$ the full subcategory formed by microlocal complexes $\mathcal{F}$ such that $\widetilde\mathcal{F}$ is smooth (i.e., each cohomology sheaf of it is a local system) outside of $\varpi^{-1}(A)$. Let $\mathcal{M}(X,A)$ be the intersection of $\mathcal{M}(X)$ with $D\mathcal{M}(X,A)$. \begin{rems}\label{rems:DQ} (a) Suppose $\k=\mathbb{C} (\hskip -.07cm ( h)\hskip -.07cm )$ is the field of Laurent series in one variable $h$ with complex coefficients. Assume that $X$ is embedded into a symplectic surface $(S,\omega)$, as in Example \ref{ex:SS}. As shown in \cite{KS-DQ}, $S$ admits a {\em deformation quantization algebroid} $\mathcal{A}_S$, which locally can be viewed as a sheaf of $\mathbb{C} [\hskip -.04cm [ h]\hskip -.04cm ]$-algebras whose reduction modulo $h$ is identified with $\mathcal{O}_S$ and whose first order commutators are given by the Poisson bracket of $\omega$. One also has the $h$-localized algebroid $\mathcal{A}_S^{\operatorname{loc}}=\mathcal{A}_S \otimes_{\mathbb{C} [\hskip -.04cm [ h]\hskip -.04cm ]} \mathbb{C}(\hskip -.07cm ( h )\hskip -.07cm )$. \vskip .2cm The category $D\mathcal{M}(X, \emptyset)$ can be compared with the category $D^b_{\operatorname{gd}, X}(\mathcal{A}_S^{\operatorname {loc}})$ of complexes of $\mathcal{A}_X^{\operatorname{loc}}$-modules whose cohomology modules are coherent, algebraically good \cite[2.7.2]{KS-DQ} modules supported on $X$. More precisely, each smooth (not necessarily closed) Lagrangian $\mathbb{C}$-submanifold (i.e., a smooth complex curve) $\Lambda\subset S$, gives a simple holonomic $\mathcal{A}_X^{\operatorname{loc}}$-module $\mathcal{O}_\Lambda$, and we have the ``$\Lambda$-Riemann-Hilbert functor" \[ R\underline\Hom_{\mathcal{A}_S^{\operatorname {loc}}}(-, \mathcal{O}_\Lambda): D^b_{\operatorname{gd}, X}(\mathcal{A}_S^{\operatorname {loc}})\longrightarrow D^b_{\operatorname{constr}}(\Lambda). \] Taking for $\Lambda$ various smooth branches of $X$, we associate to an object $\mathcal{N}$ of $D^b_{\operatorname{gd}, X}(\mathcal{A}_S^{\operatorname {loc}})$ a constructible complex $\widetilde\mathcal{F}$ on $\widetilde X$. If $\mathcal{N}$ is a single module in degree 0, then $\widetilde\mathcal{F}$ is a perverse sheaf. When two branches meet at a point (node $x$ of $X$), the corresponding Riemann-Hilbert functors are, near $x$, related to each other by the Fourier transform, thus leading to Definition \ref {def:micro-complex}. \vskip .2cm (b) A particularly interesting algebraic case is provided by $S$ being the minimal resolution of a Kleinian singularity, see Example \ref{ex:SS}(b). In this case quantizations of $S$ exist algebraically in finite form (not just over power series in $h$), see \cite{Bo}. It is therefore interesting to compare their modules with microlocal sheaves on Dynkin chains of $\mathbb{P}^1$'s. \end{rems} Let $X$ be a nodal curve with duality structure and $A\subset X_{\operatorname{sm}}$ a finite subset of smooth points. Let us form a new, noncompact nodal curve \[ X_A \,\,=\,\, X \,\cup \, \bigcup_{a\in A} T^_aX \] by attaching each cotangent line $T^_aX$ to $X$ at the point $a$ which becomes a new node. The symplectic structure on $T^ X_{\operatorname{sm}}$ gives a duality structure at each new node. \begin{prop}\label{prop:A>empty} We have canonical equivalences\[ D\mathcal{M}(X,A) \,\,\simeq \,\, D\mathcal{M}(X_A,\emptyset), \quad \mathcal{M}(X,A) \,\,\simeq \,\, \mathcal{M}(X_A,\emptyset). \] \end{prop} \noindent{\sl Proof:} We identify the normalization of $X_A$ as \[ \widetilde X_A\,\,=\,\, \widetilde X \, \sqcup \,\bigsqcup_{a\in A} T^_aX. \] To each microlocal complex $\mathcal{F}$ on $X$ we associate a microlocal complex $\mathcal{F}_A$ on $X_A$ given by \[ \widetilde\mathcal{F}_A\|_{\widetilde X}=\widetilde\mathcal{F},\quad \widetilde\mathcal{F}_A\|_{T^_aX} = \mu_a(\mathcal{F}), \] where $\mu_a(\mathcal{F})$ is the microlocalization of $\mathcal{F}$ at $a$, i.e., the Fourier transform of the specialization of $\mathcal{F}$ at $a$ \cite{KS}. The definition gives the Fourier transform identifications for $\widetilde\mathcal{F}_A$. This defines the desired equivalence. \qed \sparagraph{The Calabi-Yau property.} Important for us will be the following. \begin{thm}\label{thm:CY} Let $X$ be a compact nodal curve over $\mathbb{C}$ equipped with a duality structure. Then $D\mathcal{M}(X,\emptyset)$ is a Calabi-Yau dg-category of dimension 2. In other words, for any $\mathcal{F}, \mathcal{G}\in D\mathcal{M}(X,\emptyset)$ we have a canonical quasi-isomorphism of complexes of $\k$-vector spaces \[ R\operatorname{Hom}(\mathcal{F}, \mathcal{G})^* \simeq R\operatorname{Hom}(\mathcal{G}, \mathcal{F})[2]. \] \end{thm} \begin{ex} For $X$ smooth, the category $\mathcal{M}(X,\emptyset)$ consists of local systems on $X$, and $D\mathcal{M}(X)$ consists of complexes with locally constant cohomology. Theorem \ref{thm:CY} in this case reduces to the Poincar\'e duality for local systems on a compact oriented topological surface. \end{ex} \begin{rem} Consider the situation of Remark \ref{rems:DQ}(a). For a compact symplectic manifold $S$ of any dimension $d$, Corollary 6.2.5 of \cite{KS-DQ} gives that $D^b_{\operatorname{gd}}(\mathcal{A}_S^{\operatorname {loc}})$, the category of all complexes of $\mathcal{A}_S^{\operatorname {loc}}$-modules with coherent and algebraically good cohomology, is a Calabi-Yau category over $\mathbb{C}(\hskip -.07cm ( h)\hskip -.07cm )$ of dimension $d$. This result can be seen as a noncommutative lifting of the classical Serre duality for coherent $\mathcal{O}_S$-modules. \vskip .2cm If $S$ is non-compact, then restricting the support to a given compact subvariety $X$ allows one to preserve the duality, cf. \cite[Cor. 3.3.4]{KS-DQ}. In particular, when $S$ is a symplectic surface, and $X\subset S$ is a compact nodal curve, $D^b_{\operatorname{gd}, X}(\mathcal{A}_S^{\operatorname {loc}})$ is a Calabi-Yau category over $\mathbb{C}(\hskip -.07cm ( h)\hskip -.07cm )$ of dimension 2. Our Theorem \ref{thm:CY} can be seen as a topological analog of this fact. \end{rem} \noindent {\bf Proof of Theorem \ref{thm:CY}.} Let $\mathcal{F}, \mathcal{G}\in D\mathcal{M}(X,\emptyset)$. For any open set $U\subset X$ (in the classical topology) we have the complex of vector spaces \[ R\operatorname{Hom}_{D\mathcal{M} (U, \emptyset)}(\mathcal{F}\|_U, \mathcal{G}\|_U) \,\,\,\in \,\,\, D^b\on{Vect}_\k. \] Taken for all $U$, these complexes can be thought as forming a complex of sheaves which we denote \[ \Mc\underline\Hom (\mathcal{F}, \mathcal{G}) \,\,\,\in \,\,\, D^b_{\operatorname{constr}} (X), \] so that, in a standard way, we have \[ R\operatorname{Hom}_{D\mathcal{M}(X,\emptyset)}(\mathcal{F}, \mathcal{G}) \,\,=\,\, R\Gamma(X, \Mc\underline\Hom(\mathcal{F}, \mathcal{G})). \] Our statement will follow from the Poincar\'e-Verdier duality on the compact space $X$, if we establish the following. \begin{prop}\label{eq:Mhom} For any nodal curve $X$ (compact or not) with duality structure and any microlocal complexes $\mathcal{F},\mathcal{G} \in D\mathcal{M}(X,\emptyset)$ we have a canonical identification \[ \mathbb{D}_X \Mc\underline\Hom(\mathcal{F}, \mathcal{G}) \,\,\simeq \,\, \Mc\underline\Hom(\mathcal{G}, \mathcal{F})[2]. \] \end{prop} To prove the proposition, we compare the bifunctor $\Mc\underline\Hom$ with the microlocal Hom bifunctor of \cite{KS} which we recall. Let $M$ be a smooth manifold and $\pi: T^M\to M$ be its cotangent bundle. For any two complexes of sheaves $F, G$ on $M$. Kashiwara and Schapira \cite{KS} defined a complex of sheaves \[ \mu\underline{\operatorname{Hom}}(F, G) \,\,\in \,\, D^b\on{Sh}_{T^M} \] so that \[ \begin{gathered} R\underline\Hom(F, G) \,\,=\,\, R\pi_\bigl( \mu\underline{\operatorname{Hom}}(F, G)\bigr), \\ R\operatorname{Hom}_{D^b\on{Sh}_M}(F, G) \,\,=\,\, R\Gamma\bigl(T^M, \mu\underline{\operatorname{Hom}}(F, G)\bigr). \end{gathered} \] \begin{lem}\label{lem:mhom} Assume that $M$ is a complex manifold and $F,G\inD^b_{\operatorname{constr}}(M)$. Then we have a canonical identification \[ \mathbb{D}_{T^M} \bigl( \mu\underline{\operatorname{Hom}}(F,G)\bigr) \,\,\simeq \,\, \mu\underline{\operatorname{Hom}}(G,F) . \] \end{lem} \noindent {\sl Proof:} This is a particular case of Proposition 8.4.14(ii) of \cite{KS}. \qed \vskip .2cm We now deduce Proposition \ref{eq:Mhom} from Lemma \ref{lem:mhom}. \begin{Defi} Call a subset $Z\subset X$ {\em unibranched}, if $Z$ is the image, under the normalization map $\varpi: \widetilde X\to X$, of an open (in the classical topology) subset $\widetilde Z$ such that the restriction $\varpi\|_{\widetilde Z}: \widetilde Z\to Z$ is a bijection. \end{Defi} Note that a unibranched subset $Z$ is a complex analytic curve which may not be open in $X$, if it passes through some nodes (in which case it contains only one branch near each node it passes through). For a microlocal complex $\mathcal{F}$ on $X$ and a unibranched $Z\subset X$ we have a well-defined constructible complex \[ \mathcal{F}\|\|_Z\ \,\, := \,\, (\varpi\|_{\widetilde Z})_ \widetilde \mathcal{F} \,\, \in\,\, D^b_{\operatorname{constr}}(Z). \] Assume that $X$ is embedded into a symplectic surface $S$ and let $U$ be a neighborhood of $Z$ in $S$. Then we can make the following identifications: \begin{itemize} \item[(1)] $U$ can be identified with a neighborhood of $Z$ in $T^Z$ so that $Z$ becomes identified with the zero section $T^_ZZ$. \item[(2)] If we denote the nodes of $X$ contained in $Z$, by $x_i, i\in I$, then $U\cap Z$ can be identified with the union of $T^_Z Z$ and of some neighborhoods of 0 in the fibers $T^_{x_i} Z$. \item[(3)] Let $\mathcal{F}, \mathcal{G}$ be two microlocal complexes on $X$. Then, under the above identifications, we have an isomorphism \[ \Mc\underline\Hom(\mathcal{F}, \mathcal{G})\|_{U\cap Z}\,\,\simeq \,\, \mu\underline{\operatorname{Hom}}(\mathcal{F}\|\|_Z, \mathcal{G}\|\|_Z)\|_{U\cap Z}. \] \end{itemize} Further, because of the Fourier transform identifications in the definition of a microlocal complex, the identifications in (3) are compatible for different unibranched sets passing through a given node. Therefore the identifications (3) allow us to glue the identifications of Lemma \ref{lem:mhom} to a canonical identification as in Proposition \ref{eq:Mhom}. This proposition and Theorem \ref{thm:CY} are now proved. \eject \section {Microlocal sheaves: de Rham description.}\label{sec:dr} We now give a $\mathcal{D}$-module type description of microlocal sheaves, relating our approach with that of \cite{CB}. \sparagraph{Formulations.} Let $X$ be a nodal curve with the set of nodes $D$ and its preimage $\widetilde D=\varpi^{-1}(D)\subset \widetilde X$. By an {\em orientation} of $X$ we mean a choice, for each node $x$, of the order $(x'<x'')$ on the two element set of preimages $\varpi^{-1}(x) = \{x', x''\}$. We denote by \[ \Re^{-1}[0,1) \,\,=\,\, [0,1) + i\mathbb{R} \,\,\subset \,\, \mathbb{C} \] the standard fundamental domain for $\mathbb{C}/\mathbb{Z}$. Let $Y$ be a smooth algebraic curve over $\mathbb{C}$ (not necessarily compact) and $Z\subset Y$ a finite subset. We recall, see, e.g. \cite{Ma}, the concept of a {\em logarithmic connection} (along $Z$) on an algebraic vector bundle $\mathcal{E}$ on $Y$. Such a connection $\nabla$ can be viewed as an algebraic differential operator $\nabla: \mathcal{E}\to \mathcal{E}\otimes\Omega^1_Y(\log Z)$. It has a well-defined {\em residue} $\operatorname{Res}_z(\nabla)\in \operatorname{End}(\mathcal{E}_z)$ at each $z\in Z$. For a noncompact $Y$ there is a concept of a {\em regular} logarithmic connection (having regular singularities at the infinity of $Y$). \begin{Defi} Let $X$ be a nodal curve over $\mathbb{C}$, not necessarily compact, with orientation. A {\em de Rham microlocal sheaf} (without singularities) on $X$ is a datum of: \begin{enumerate} \item[(1)] A vector bundle $\mathcal{E}$ on $\widetilde X$, together with a regular logarithmic connection $\nabla$ along $\widetilde D$. \item[(2)] For each node $x\in D$ with preimages $x', x''\in\widetilde D$ (order given by the orientation), two linear operators \[ \xymatrix{ \mathcal{E}_{x'} \ar@<.7ex>[r]^{u_x}& \ar@<.7ex>[l]^{v_x} \mathcal{E}_{x''} } \] such that: \item[(3)] $\operatorname{Res}_{x'}(\nabla) = v_x u_x, \quad \operatorname{Res}_{x''}(\nabla) = -u_x v_x$; \item[(4)] All eigenvalues of $v_xu_x$ and $-u_x v_x$ lie in $\Re^{-1}[0,1)$. \end{enumerate} The category of de Rham microlocal sheaves on $X$ without singularities will be denoted by $\mathcal{M}_{\operatorname{dR}}(X,\emptyset)$. \end{Defi} \begin{rems} A de Rham microlocal sheaf is a particular case $(\lambda=0)$ of a $\lambda$-connection of \cite{CB}, but with additional restriction (4). \end{rems} \begin{thm}\label{thm:micro-dr} Take the base field $\k=\mathbb{C}$. Assume that $X$ is equipped with both an orientation and a duality structure. Then $\mathcal{M}_{\operatorname{dR}}(X,\emptyset)$ is equivalent to $\mathcal{M}(X,\emptyset)$. \end{thm} \sparagraph{ Riemann-Hilbert correspondence.} In order to prove Theorem \ref{thm:micro-dr}, we recall two classical results about the Riemann-Hilbert correspondence. First, let $Y$ be a smooth curve over $\mathbb{C}$ and $Z\subset Y$ a finite subset. A regular logarithmic connection $\nabla: \mathcal{E}\to \mathcal{E}\otimes\Omega^1_Y(\log Z)$ will be called {\em canonical}, all eigenvalues of all $\operatorname{Res}_z(\nabla)$, $z\in Z$, lie in $\Re^{-1}[0,1)$. In this case $(E,\nabla)$ is obtained by the {\em Deligne canonical extension} from its restriction to $Y-Z$, see \cite{Ma}. We denote by $\operatorname{Conn}_{\operatorname{can}}^{\operatorname{reg}}(Y,Z)$ the category of vector bundles with regular canonical connections. \begin{prop}\label{prop:can} The category $\operatorname{Conn}_{\operatorname{can}}^{\operatorname{reg}}(Y,Z)$ is equivalence to $\operatorname{LS}(Y-Z)$, the category of local systems on $Y-Z$. The equivalence is obtained by restricting $(\mathcal{E},\nabla)$ to $Y-Z$ and taking the sheaf of covariantly constant sections. \qed \end{prop} \begin{prop}\label{prop:I} \cite{Ka}\cite[(II.2.1)]{Ma} Let $\mathfrak{I}$ be the category of diagrams of finite-dimensional $\mathbb{C}$-vector spaces \[ H= \bigl\{ \xymatrix{ E \ar@<.7ex>[r]^{u}& \ar@<.7ex>[l]^{v} F }\bigr\} \] s.t. all eigenvalues of $uv$ and $vu$ lie in $\Re^{-1}[0,1)$. Then $\mathfrak{I}$ is equivalent to $\on{Perv}(\mathbb{C},0)$. The equivalence takes an object $ H\in\mathfrak{I}$ to the $\mathcal{D}_\mathbb{C}$-module $M_H$ with the space of generators $E\oplus F$ and relations \[ \begin{gathered} x\cdot f = v(f), \,\,\, f\in F, \\ {d\over dx} \cdot e = u(e), \,\, e\in E, \end{gathered} \] and then to the de Rham complex of $M_H$. \qed \end{prop} \sparagraph{Fourier transform and RH.} Recall \cite{Ma} that the Fourier-Sato transform on $\on{Perv}(\mathbb{C},0)$ corresponds, at the $\mathcal{D}$-module level, to passing from passing from the generators $x, {d\over dx}$ of the Weyl algebra of differential operators to new generators \[ p = -{d\over dx}, \,\,\, {d\over dp} = x, \] so that \[ \left[ {d\over dp}, \, p\right] \,\,=\,\, \left[ {d\over dx}, \, x\right] \,\,=\,\, 1. \] This implies: \begin{cor}\label{cor:FT-I} The effect of the Fourier-Sato transform on $\mathfrak{I}$ is the functor \[ \operatorname{FT}_\mathfrak{I}: \,\,\, H = \bigl\{ \xymatrix{ E \ar@<.7ex>[r]^{u}& \ar@<.7ex>[l]^{v} F } \bigr\} \quad \buildrel{}\over\longmapsto \quad \hat H = \bigl\{\xymatrix{ F \ar@<.7ex>[r]^{v}& \ar@<.7ex>[l]^{-u} E }\bigr\}. \quad\quad\quad \qed \] \end{cor} Therefore we can reformulate Proposition \ref{prop:I} in a more ``microlocal" form \begin{prop} Let $C=\{ x p=0\} \subset\mathbb{C}^2$ be the coordinate cross with the orientation defined by putting the $x$-branch before the $p$-branch. Then $\mathcal{M}_{\operatorname{dR}}(C,\emptyset)$ is equivalent to $\on{Perv}(\mathbb{C},0) \simeq \mathcal{M}(C,\emptyset)$. \end{prop} \noindent {\sl Proof:} For a diagram $H\in\mathfrak{I}$, the $\mathcal{D}_\mathbb{C}$-module $M_H$ becomes $\mathcal{O}$-coherent on $\mathbb{C}-\{0\}$, and is identified with the following bundle with connection: \[ \mathcal{E}_{H}^0 \,\,=\,\,\biggl( E\otimes\mathcal{O}_{\mathbb{C}-\{0\}}, \,\nabla = d- (vu) {dx\over x}\biggr). \] Therefore the Deligne canonical extension of $\mathcal{E}^0_H$ to $\mathbb{C}$ is the logarithmic connection \[ \mathcal{E}_{H} \,\,=\,\,\biggl( E\otimes\mathcal{O}_{\mathbb{C}}, \,\nabla = d- (vu) {dx\over x}\biggr). \] Similarly for the Fourier transformed diagram $\hat H$ which gives a bundle with logarithmic connection on $\mathbb{C}$ which we view as the other branch of $C$ with coordinate $p$: \[ \mathcal{E}_{\hat H} \,\,=\,\,\biggl( E\otimes\mathcal{O}_{\mathbb{C}}, \,\nabla = d+ (uv)) {d p\over p}\biggr). \] This means that the data $(\mathcal{E}_H, \mathcal{E}_{\hat H}, u,v)$ form an object of $\mathcal{M}_{\operatorname{dR}}(C,\emptyset)$. So we get a functor $\mathfrak{I}\to \mathcal{M}_{\operatorname{dR}}(C,\emptyset)$. The fact that it is an equivalence, is verified in a standard way. \qed \vskip .2cm Theorem \ref {thm:micro-dr} is now obtained by gluing together the descriptions given by Proposition \ref{prop:can} over $X_{\operatorname{sm}}$ and by Proposition \ref{prop:I} near the nodes of $X$. \qed \eject \section{Twisted microlocal sheaves}\label{sec:twisted-micro} \sparagraph{\bf Motivation: twisted $\mathcal{D}$-modules and sheaves.} Let $X$ be a smooth algebraic variety over $\mathbb{C}$. We recall \cite{BB} that to each class $t\in H^1_{\operatorname{Zar}} \bigl(X, \bigl\{\Omega^1_X \buildrel d\over \to \Omega^{2, \operatorname{cl}}_X\bigr\}\bigr)$ there corresponds a sheaf of rings of {\em twisted differential operators} on $X$ which we denote $\mathcal{D}_X^t$. Recall further that the first Chern class can be understood as a homomorphism \[ c_1: \operatorname{Pic}(X) \longrightarrow H^1_{\operatorname{Zar}} \bigl(X, \bigl\{\Omega^1_X \buildrel d\over \to \Omega^{2, \operatorname{cl}}_X\bigr\}\bigr). \] If $\mathcal{L}$ is a line bundle on $X$, then we have an explicit model: \[ \mathcal{D}_x^{c_1(\mathcal{L})} \,\,=\,\,\operatorname{Diff}(\mathcal{L}, \mathcal{L}) \] is the sheaf formed by differential operators from sections of $\mathcal{L}$ to sections of $\mathcal{L}$. For a compact $X$, the image of $c_1$ is typically an integer lattice in a complex vector space and the sheaves $\mathcal{D}_X^t$ can be seen as interpolating between the $\operatorname{Diff}(\mathcal{L}, \mathcal{L})$ for different $\mathcal{L}$. We recall a particular explicit instance of this interpolation. Given a line bundle $\mathcal{L}$ on $X$, we denote by $\mathcal{L}^\circ$ the total space of $\mathcal{L}$ minus the zero section, so $p:\mathcal{L}^\circ\to X$ is a $\mathbb{C}^$-torsor over $X$. We denote by $\theta$ the Euler vector field ``$x \partial/\partial x$" on $\mathcal{L}^\circ$, i.e., the infinitesimal generator of the $\mathbb{C}^$-action. Thus $\theta$ is a global section of $\mathcal{D}_{\mathcal{L}^\circ}$. \begin{prop}\label{prop:lambda} Let $\lambda\in\mathbb{C}$. Then \[ \mathcal{D}_X^{\lambda c_1(\mathcal{L})} \,\,\simeq \,\, p_\biggl( \mathcal{D}_{\mathcal{L}^\circ} \biggl/ \mathcal{D}_{\mathcal{L}^\circ}(\theta-\lambda) \mathcal{D}_{\mathcal{L}^\circ}\biggr). \qed \] \end{prop} We now discuss the consequences of Proposition \ref{prop:lambda} for the Riemann-Hilbert correspondence for twisted $\mathcal{D}$-modules. \vskip .2cm On the $\mathcal{D}$-module side, the concepts of holonomic and regular $\mathcal{D}_X^t$-modules are defined in the same way as in the untwisted case. We denote by $\mathcal{D}_X^{t}-\operatorname{Mod}_{\operatorname{h.r.}}$ the category of holonomic regular $\mathcal{D}_X^t$-modules, and by $D^b_{\operatorname{h.r.}}(\mathcal{D}_X^{t}-\operatorname{Mod})$ the derived category formed by complexes with holonomic regular cohomology modules. \vskip .2cm On the sheaf side, choose $q\in\k^$. Let $\mathcal{L}$ be a line bundle on $X$. We denote by $\on{Sh}^{\mathcal{L}, q}(X)$ the category of sheaves on $\mathcal{L}^\circ$ whose restriction on each fiber of $p$ is a local system with scalar monodromy $q\cdot\operatorname{Id}$. Let $D^b(X)^{\mathcal{L},q}$ be the bounded derived category of $\on{Sh}^{\mathcal{L}, q}(X)$. We denote by $D^b_{\operatorname{constr}}(X)^{\mathcal{L}, q} \subset D^b(X)^{\mathcal{L}, q}$ the full subcategory formed by complexes with $\mathbb{C}$-constructible cohomology sheaves, and $\on{Perv}^{\mathcal{L}, q}(X)\subset D^b_{\operatorname{constr}}(X)^{\mathcal{L}, q}$ the full subcategory of perverse sheaves. Proposition \ref{prop:lambda} implies the following. \begin{cor} Take the base field $\k=\mathbb{C}$. Let $\mathcal{L}$ be a line bundle on $X$ and $\lambda\in\mathbb{C}$. We have an anti-equivalence of (pre-)triangulated categories and a compatible anti-equivalence of abelian categories \[ D^b_{\operatorname{h.r.}}(\mathcal{D}_X^{\lambda c_1(\mathcal{L})}-\operatorname{Mod}) \to D^b_{\operatorname{constr}}(X)^{\mathcal{L}, e^{2\pi i \lambda}}, \quad \mathcal{D}_X^{\lambda c_1(\mathcal{L})}-\operatorname{Mod}_{\operatorname{h.r.}}\to \on{Perv}^{\mathcal{L}, e^{2\pi i\lambda}}(X). \quad \] \end{cor} \begin{rem} For example, if $\lambda=n$ is an integer, then the monodromy comes out to be trivial, and we get that $\mathcal{D}_X^{\lambda c_1(\mathcal{L})}-\operatorname{Mod}_{\operatorname{h.r.}}$ is anti-equivalent to $\on{Perv}_X$. This can also be seen directly, as $D_X^{n c_1(\mathcal{L})}= \operatorname{Diff}(\mathcal{L}^{\otimes n}, \mathcal{L}^{\otimes n})$ and so we have the ``solution functor" associating to any module $\mathcal{M}$ over this sheaf of rings the complex \[ \operatorname{Sol}(\mathcal{M}) \,\,=\,\, R\underline\Hom_{\operatorname{Diff}(\mathcal{L}^{\otimes n}, \mathcal{L}^{\otimes n})} (\mathcal{M}, \mathcal{L}^{\otimes n}). \] This complex is perverse, and the functor $\operatorname{Sol}$ establishes the desired anti-equivalence. \end{rem} We will also consider the ``universal twist" situation by not requiring the monodromy to be a fixed scalar multiple of 1 and working instead with monodromic sheaves and complexes on $\mathcal{L}^\circ$. That is, we consider the derived category $D^b_{\operatorname{mon}}(\mathcal{L}^\circ)$ defined as the full subcategory in $D^b\on{Sh}(\mathcal{L}^\circ)$ formed by $\mathbb{C}$-monodromic complexes. Inside it, let $D^b_{\operatorname{constr}}(X)^\mathcal{L}$ be the full triangulated subcategory of $\mathbb{C}$-constructible $\mathbb{C}$-monodromic complexes and $\on{Perv}(X)^\mathcal{L}$ the abelian subcategory of perverse sheaves on $\mathcal{L}^\circ$ which are $\mathbb{C}$-monodromic. Note that the natural functor $D^b (X)^{\mathcal{L}, q}\to D^b_{\operatorname{constr}}(X)^\mathcal{L}$ is not fully faithful. In the $\mathcal{D}$-module picture this correponds to the fact that the derived pullback functor on modules corresponding to the projection of sheaves of rings $\mathcal{D}_{\mathcal{L}^\circ}\to \mathcal{D}_{\mathcal{L}^\circ}/(\theta-\lambda)$ is not fully faithful. \vskip .2cm \sparagraph {\bf Twisted microlocal sheaves.} We now modify the above and apply it to the case when $X$ is a nodal curve. So let $X$ be a nodal curve over $\mathbb{C}$ with the normalization map $\varpi: \widetilde X\to X$, as in \S \ref{sec:microsheaves}. We denote by $D\subset X$ the set of nodes, and by $\widetilde D\subset X$ its preimage under $\varpi$. For any node $x$ we choose a small analytic neighborhood $U=U_x = B'\cup_x B''$ of $x$. Here $B', B''$ are two branches of $X$ near $x$ which we identify with their preimages $\widetilde B', \widetilde B''\subset\widetilde X$. Let $\mathcal{L}$ be a line bundle on $X$. We denote by $\widetilde \mathcal{L} = \varpi^{}(\mathcal{L})$ its pullback to $\widetilde X$ and by $\widetilde p: \widetilde\mathcal{L}^\circ\to\widetilde X$ the projection. For each node $x$ we choose an {\em almost-trivialization} of $\mathcal{L}$ over $U_x$, by which we mean an identification of $\mathcal{L}\|_{U_x}$ with the trivial line bundle with fiber $\mathcal{L}_x$ or, equivalently, an identification of $\mathbb{G}_m$-torsors \begin{equation}\label{eq:almost} \mathcal{L}^\circ\|_{U_x} \longrightarrow U_x\times\mathcal{L}_x^\circ \end{equation} (Note that the space of almost-trivializations is contractible.) The isomorphism \eqref{eq:almost} gives rise to the {\em relative}, or (fiberwise with respect to the projection to $\mathcal{L}^\circ_x$) Fourier transforms which are quasi-inverse equivalences of triangulated categories \[ \xymatrix{ D^b(\widetilde{B}', x')^{\widetilde\mathcal{L}} \ar@<.7ex>[r]^{\operatorname{FT}'}& \ar@<.7ex>[l]^{\operatorname{FT}''} D^b(\widetilde{B}'', x'')^{\widetilde\mathcal{L}}, } \quad\quad \xymatrix{ D^b(\widetilde{B}', x')^{\widetilde \mathcal{L}, q} \ar@<.7ex>[r]^{\hskip -.7cm \operatorname{FT}'}& \ar@<.7ex>[l]^{\hskip -.7cm \operatorname{FT}''} D^b(\widetilde{B}'', x'')^{\widetilde \mathcal{L}, q}, \,\, q\in\k^. } \] They induce similar equivalences of abelian categories of twisted perverse sheaves. \begin{Defi}\label{def:tw-micro} Let $q\in\k^$. (a) An $\mathcal{L}$-twisted, resp. $(\mathcal{L}, q)$-twisted microlocal complex on $X$ is a datum $\mathcal{F}$ consisting of: \begin{enumerate} \item[(1)] An object $\widetilde\mathcal{F}^\circ$ of $D^b_{\operatorname{constr}}(\widetilde X)^{\widetilde\mathcal{L}}$, resp. of $D^b_{\operatorname{constr}}(\widetilde X)^{\widetilde\mathcal{L}, q}$ \item[(2)] For each node $x\in D$ with the two branches $B', B''$ as above, isomorphisms \[ \operatorname{FT}'(\widetilde\mathcal{F}^\circ \|_{\widetilde p^{-1}(\widetilde B')}) \longrightarrow \widetilde\mathcal{F}^\circ \|_{\widetilde p^{-1}(\widetilde B'')}, \quad \operatorname{FT}''(\widetilde\mathcal{F}^\circ \|_{\widetilde p^{-1}(\widetilde B'')}) \longrightarrow \widetilde\mathcal{F}^\circ \|_{\widetilde p^{-1}(\widetilde B')}, \] inverse to each other. \end{enumerate} \vskip .2cm \noindent (b) An $\mathcal{L}$-twisted, resp. $(\mathcal{L}, q)$-twisted microlocal sheaf is an $\mathcal{L}$-twisted, resp. $(\mathcal{L}, q)$-twisted microlocal complex such that $\widetilde \mathcal{F}^\circ$ is a perverse sheaf on $\widetilde\mathcal{L}^\circ$. \end{Defi} As before, for any finite subset $A\subset X$ of smooth points we denote by $D\mathcal{M}^{\mathcal{L}}(X,A)$, resp. $D\mathcal{M}^{\mathcal{L}, q}(X,A)$ the pre-triangulated dg-category formed by $\mathcal{L}$-twisted, resp. $(\mathcal{L}, q)$-twisted microlocal complexes $\mathcal{F}$ on $X$ such that $\widetilde\mathcal{F}^\circ$ has locally constant cohomology outside of the preimage of $A$ in $\widetilde\mathcal{L}^\circ$. By $\mathcal{M}^\mathcal{L}(X,A)$, resp. $\mathcal{M}^{\mathcal{L}, q}(X,A)$ we denote the full (abelian) subcategory in $D\mathcal{M}^{\mathcal{L}}(X,A)$, resp. $D\mathcal{M}^{\mathcal{L}, q}(X,A)$ formed by $q$-twisted microlocal sheaves. \sparagraph{Calabi-Yau properties.} Theorem \ref{thm:CY} generalizes to the twisted case as follows. \begin{thm}\label{thm:CY-twist} Assume $X$ is a compact nodal curve with a duality structure, and $(X_i)_{i\in I}$ be its irreducible components. Let $\mathcal{L}$ be a line bundle on $X$ with an almost-trivialization on a neighborhood of each node. Then: \begin{enumerate} \item[(a)] $D\mathcal{M}^\mathcal{L}(X,\emptyset)$ is a Calabi-Yau category of dimension 3. \item[(b)] For any $q\in\k^$ we have that $D\mathcal{M}^{\mathcal{L}, q}(X,\emptyset)$ is a Calabi-Yau category of dimension 2. \end{enumerate} \end{thm} \begin{ex} If $X$ is a smooth projective curve of genus $g$, then part (a) corresponds to the Poincar\'e duality on the compact 3-manifold $\mathcal{L}^\circ/\mathbb{R}^_+$, the circle bundle on $X$ associated to $\mathcal{L}$. \end{ex} \noindent{\sl Sketch of proof of Theorem \ref {thm:CY-twist}:} It is obtained by arguments similar to those for Theorem \ref{thm:CY}. That is, for any two objects $\mathcal{F}, \mathcal{G}$ of the category $D\mathcal{M}^\mathcal{L}(X,\emptyset)$ resp. $D\mathcal{M}^{\mathcal{L},q}(X,\emptyset)$ we introduce a constructible complex $\mathcal{M}\underline\Hom^\mathcal{L}(\mathcal{F},\mathcal{G})$ resp. $\mathcal{M}\underline\Hom^{\mathcal{L},q}(\mathcal{F},\mathcal{G})$ whose complex of global sections over $X$ is identified with $R\operatorname{Hom}(\mathcal{F},\mathcal{G})$ in the corresponding category. The statement then follows from canonical identifications \[ \begin{gathered} \mathbb{D} \mathcal{M}\underline\Hom^\mathcal{L}(\mathcal{F},\mathcal{G}) \,\,\simeq \,\,\mathcal{M}\underline\Hom^\mathcal{L}(\mathcal{G},\mathcal{F})[3], \\ \mathbb{D} \mathcal{M}\underline\Hom^{\mathcal{L},q}(\mathcal{F},\mathcal{G}) \,\,\simeq \,\, \mathcal{M}\underline\Hom^{\mathcal{L}, q}(\mathcal{G},\mathcal{F})[2]. \end{gathered} \] These identifications are obtained by comparing the bifunctor $\mathcal{M}\underline\Hom^\mathcal{L}$ with the bifuctor $\mu\underline\Hom$ of \cite{KS} applied to constructible complexes on manifolds of the form $\mathcal{L}^\circ\|_Z$, where $Z$ is a unibranched subset of $X$. \qed \eject \eject \section{ Multiplicative preprojective algebras }\label{sec:prepro} \sparagraph {\bf The definitions.} We recall the definition of multiplicative preprojective algebras, following \cite{CBS} \cite{yamakawa}. \begin{conv}\label{conv:alg-cat} There is a very close correspondence between: \begin{itemize} \item[(1)] $\k$-linear categories $\mathcal{C}$ with finitely many objects. \item[(2)] Their {\em total algebras} \[ \Lambda_\mathcal{C} \,\,=\,\,\bigoplus_{x,y\in\operatorname{Ob}(\mathcal{C})}\operatorname{Hom}_\mathcal{C}(x,y). \] \end{itemize} For instance, each object $x\in\mathcal{C}$ gives an idempotent ${\bf 1}_x\in\Lambda_\mathcal{C}$, left $\Lambda_\mathcal{C}$-modules are the same as covariant functors $\mathcal{C}\to\operatorname\on{Vect}_\k$, and so on. For this reason we will not make a notational distinction between objects of type (1) and (2), thus, for example, speaking about objects of an algebra $\Lambda$ and morphisms between them (meaning objects and morphisms of a category $\mathcal{C}$ such that $\Lambda=\Lambda_\mathcal{C}$). \end{conv} Let $\Gamma$ be a {\em quiver}, i.e., finite oriented graph, with the set of vertices $I$ and the set of arrows $E$, so we have the source and target maps $s,t: E\to I$. We fix a total ordering $<$ on $E$. \begin{Defi} Let $\underline q=(q_i)_{i\in I}\in(\mathbb{C}^)^I$. The {\em multiplicative preprojective algebra} $\Lambda^{\underline q}(\Gamma)$ is defined by generators and relations as follows: \begin{enumerate} \item[(0)] $\operatorname{Ob}(\Lambda^{\underline q}(\Gamma)) = I$. In particular, for each $i\in I$ we have the identity morphism ${\bf 1}_i: i\to i$. \item[(1)] For each arrow $h\in E$ there are two generating morphisms $a_h: s(h)\to t(h)$ and $b_h: t(h)\to s(h)$. We impose the condition that \[ {\bf 1}_{t(h)} + a_h b_h: t(h)\to t(h), \quad {\bf 1}_{s(h)} + b_h a_h: s(h)\to s(h) \] are invertible, i.e., introduce their formal inverses. \item[(2)] We further impose the following relations: for each $i\in I$, \[ \prod_{h\in E: t(h)=i} ({\bf 1}_i + a_h b_h) \prod_{h\in E, s(h)=i} ({\bf 1}_i + b_h a_h)^{-1} \,\,=\,\, q_i {\bf 1}_i, \] where the factors in each product are ordered using the chosen total order $<$ on $E$. \end{enumerate} \end{Defi} It was proven in \cite[Th. 1.4]{CBS} that up to an isomorphism, $\Lambda^{\underline q}(\Gamma)$ is independent on the choice of the order $<$, as well as on the choice of orientation of edges of $\Gamma$. \vskip .2cm \sparagraph {\bf Microlocal sheaves on rational curves.} Let now $X$ be a compact nodal curve over $\mathbb{C}$ with the set of components $X_i, i\in I$. We then have the {\em intersection graph} $\Gamma_X$ of $X$. By definition, this is an un-oriented graph with the set of vertices $I$ and as many edges from $i$ to $j$ as there are intersection points of $X_i$ and $X_j$. In particular, for $i=j$ we put as many loops as there are self-intersection points of $X_i$. We now choose an orientation of $\Gamma_X$ and an ordering of the arrows in an arbitrary way, thus making it into a quiver, so that the above constructions apply to $\Gamma_X$. Note that an orientation of $\Gamma_X$ is the same as an orientation of $X$ in the sense of \S \ref {sec:dr}A. Let $\mathcal{L}$ be a line bundle on $X$. We keep the notation of \S \ref{sec:twisted-micro}. Let $d_i = \deg(\varpi^{}_i\mathcal{L})\in\mathbb{Z}$. For $q\in\k^$ we denote $q^{\deg(\mathcal{L})}= (q^{d_i})_{i\in I}$. \begin{thm}\label{thm:prepro} Assume that all the components $X_i$ are rational, i.e., the normalizations $\widetilde X_i$ are isomorphic to $\mathbb{P}^1$. Then the category $\mathcal{M}^{\mathcal{L}, q}(X,\emptyset)$ is equivalent to the category of finite-dimensional modules over $\Lambda^{q^{\deg(\mathcal{L})}} (\Gamma_X)$. \end{thm} \sparagraph {\bf Perverse sheaves on a disk: the $(\Phi, \Psi)$-description.} The proof of Theorem \ref{thm:prepro} is based on a conceptual interpretation of the factors ${\bf 1}_i + a_h b_h$ and $({\bf 1}_i+ b_h a_h)^{-1}$ entering the defining relations of $\Lambda^{\underline q}(\Gamma)$. We observe that such expressions describe the {\em monodromies of perverse sheaves on a disk}. More precisely, let $B$ be an open disk in the complex plane containing a point $y$. Let $\overline B$ be an ``abstract" closed disk containing $B$ as its interior. We denote $\on{Perv}(B,y)$ the category of perverse sheaves on $B$ smooth everywhere except possibly $y$. Note that for any $\mathcal{F}\in\on{Perv}(B,y)$, the restriction of $\mathcal{F}$ to $B-\{y\}$ is a local system in degree 0 and so extends, by direct image, to a local system on $\overline B-\{y\}$. So we can think of $\mathcal{F}$ as a complex of sheaves on $\overline B$, whose restriction to $\overline B-\{y\}$ is quasi-isomorphic to a local system in degree 0. In particular, for each $z\in \overline{B}-\{y\}$ we have a single vector space $\mathcal{F}_z$, the stalk of $\mathcal{F}$ at $z$. We have the following classical result \cite{beil-gluing} \cite{galligo-GM}. \begin{prop}\label{prop:GGM} (a) Let $\mathfrak{J}$ be the category of diagrams of finite-dimensional $\k$-vector spaces \[ \xymatrix{ \Phi \ar@<.5ex>[r]^a& \ar@<.5ex>[l]^b \Psi } \] such that the operator $T_\Psi={\bf 1}_\Psi+ ab$ is invertible. For such a diagram the operator $T_\Phi = {\bf 1}_\Phi+ba$ is invertible as well. The category $\on{Perv}(B,y)$ is equivalent to $\mathfrak{J}$. \vskip .2cm (b) Explicitly, an equivalence in (a) is obtained by choosing a boundary point $z\in\partial B$ and joining it with a simple arc $K$ with $y$. After such choices the vector spaces corresponding to $\mathcal{F}\in\on{Perv}(B,y)$ are found as \[ \Psi=\Psi(\mathcal{F}) = \mathcal{F}_z = \mathbb{H}^0(K-\{y\}, \mathcal{F}), \quad \Phi = \Phi(\mathcal{F}) = \mathbb{H}^1_K(B,\mathcal{F}). \] The operator $T_\Psi$ is the anti-clockwise monodromy of the local system $\mathcal{F}\|_{B-\{y\}}$ around $y$. \qed \end{prop} The space $\Psi(\mathcal{F})$ and $\Phi(\mathcal{F})$ are referred to as the spaces of {\em nearby} and {\em vanishing cycles} of $\mathcal{F}$ at $y$ (with respect to the choice of an arc $K$). \vskip .2cm \sparagraph {\bf Fourier transform in the $(\Phi, \Psi)$-description.} Let $L$ be a 1-dimensional $\mathbb{C}$-vector space, $L^=\operatorname{Hom}_\mathbb{C}(L,\mathbb{C})$ be the dual space, with the canonical pairing \[ (z,w)\mapsto \langle z,w\rangle: L\times L^\longrightarrow\mathbb{C}. \] Let $K$ be a half-line in $L$ originating at $0$, and \[ K^* \,\,=\,\,\bigl\{ w\in L^: \,\, \langle z,w\rangle \in\mathbb{R}_{\geq 0}, \,\, \forall z\in K\bigl\} \] be the dual half-line in $L^$. We can consider $K$ as a simple arc in $L$ joining $0$ with the infinity of $L$, and similarly with $K^$. Therefore the choices of $K$ and $K^$ give identifications of the categories $\on{Perv}(L,0)$ and $\on{Perv}(L^, 0)$ with the categories of diagrams as in Proposition \ref{prop:GGM}. \begin{prop}\label{prop:cluster} Under the identifications of Proposition \ref{prop:GGM}, the Fourier-Sato transform \[ \operatorname{FT}: \on{Perv}(L,0) \longrightarrow \on{Perv}(L^, 0) \] corresponds to the functor $\operatorname{FT}_\mathfrak{J}$ which takes \[ \bigl\{ \xymatrix{ \Phi \ar@<.5ex>[r]^a& \ar@<.5ex>[l]^b \Psi } \bigr\} \,\, \buildrel \operatorname{FT}_\mathfrak{J}\over\longmapsto \,\, \bigl\{ \xymatrix{ \Psi \ar@<.5ex>[r]^{a'}& \ar@<.5ex>[l]^{b'} \Phi } \bigr\}, \] where $(a',b')$ are related to $(a,b)$ by the ``cluster transformation" \[ \begin{cases} a'=-b, \\ b' = a({\bf 1}_\Phi + ba)^{-1}. \end{cases} \qed \] \end{prop} \begin{cor}\label{cor:inverse} In the situation of Proposition \ref{prop:cluster} we have \[ {\bf 1}+a'b'\,\, =\,\, ({\bf 1}+ba)^{- 1}. \] \end{cor} Note that this corollary prevents us from having a naive statement of the kind ``Fourier transform interchanges $\Phi$ with $\Psi$ and $a$ with $b$". \vskip .2cm \noindent {\sl Proof of Proposition \ref{prop:cluster}:} We first establish the identifications \begin{equation}\label{eq:phi-ft-psi} \Psi(\operatorname{FT}(\mathcal{F})) \,\,\simeq \,\,\Phi(\mathcal{F}). \end{equation} Let $K^\dagger\subset L$ be the half-plane formed by $z$ such that $\Re \langle z,w\rangle \geq 0$ for each $w\in K^$. From the definition of $\operatorname{FT}$, see \cite{KS}, \S 3.7 and the fact that $\mathcal{F}$ is $\mathbb{C}^$-monodromic, we see that $\Psi(\operatorname{FT}(\mathcal{F}))$, i.e., the stalk of $\operatorname{FT}(\mathcal{F})$ at a generic point of the ray $K^$, is equal to the vector space $\mathbb{H}^1_{K^\dagger}(L,\mathcal{F})$. But $K^\dagger$ contains $K$ and can be contracted to it without changing the cohomology with support for any $\mathcal{F}\in\on{Perv}(L,0)$. This means that $\Psi(\operatorname{FT}(\mathcal{F})) \simeq \mathbb{H}^1_{K}(L,\mathcal{F}) = \Psi(\mathcal{F})$. \vskip .2cm Next, we prove the Corollary \ref{cor:inverse}. Note that rotating $K$ in $L$ anti-clockwise results in rotating $K^$ in $L^$ clockwise. So the monodromy on $\Phi(\mathcal{F})$ obtained by rotating $K$ in the canonical way given by the complex structure (i.e., anti-clockwise), is the inverse of the monodromy on $\Psi(\operatorname{FT}(\mathcal{F}))= \Phi(\mathcal{F})$ obtained by rotating $K^$ in the same canonical way (i.e., also anti-clockwise). This establishes the corollary. \vskip .2cm We now prove Proposition \ref{prop:cluster} in full generality by using the approach of \cite{beil-gluing}. We identify $\on{Perv}(L,0)$ with $\mathfrak{J}$ throughout. Note that $m=(T_\Phi, T_\Psi)$ defines an automorphism of the identity functor of $\mathfrak{J}=\on{Perv}(L,0)$ called the {\em monodromy operator}. Further, $\on{Perv}(L,0)$ splits into a direct sum of abelian categories \[ \on{Perv}(L,0) \,\,=\,\, \on{Perv}(L,0)_u\oplus \on{Perv}(L,0)_{nu}. \] Here $m$ acts unipotently on every object ${\mathcal F}\in \on{Perv}(L,0)_u$ (equivalently, on $\Phi({\mathcal F})$, $\Psi({\mathcal F})$ for ${\mathcal F}\in \on{Perv}(L,0)_u$), while ${\bf 1}-m$ is invertible on every object ${\mathcal F}\in \on{Perv}(L,0)_{nu}$ . We construct the isomorphism claimed in Proposition \ref{prop:cluster} separately for ${\mathcal F}\in \on{Perv}(L,0)_u$ and ${\mathcal F}\in \on{Perv}(L,0)_{nu}$. \vskip .2cm Assume first that ${\mathcal F}\in \on{Perv}(L,0)_{nu}$. Notice that for ${\mathcal F}\in \on{Perv}(L,0)_{nu}$ the maps $a:\Phi({\mathcal F})\to \Psi({\mathcal F})$ and $b:\Psi({\mathcal F})\to \Phi({\mathcal F})$ are invertible. This means that either of the two functors ${\mathcal F}\mapsto (\Psi({\mathcal F}), T_\Psi)$, ${\mathcal F}\mapsto (\Phi({\mathcal F}), T_\Phi)$ is an equivalence between $\on{Perv}(L,0)_{nu}$ and the category of vector spaces with an automorphism which does not have eigenvalue one. Thus in this case it suffices to construct a functorial isomorphism $\Phi({\mathcal F})\cong \Psi(\operatorname{FT} ({\mathcal F}))$ sending the automorphism $T_\Phi$ to $T_\Psi^{-1}$. This reduces to Corollary \ref{cor:inverse}. \vskip .2cm We now consider ${\mathcal F}\in \on{Perv}(L,0)_u$. Notice that the category $\on{Perv}(L,0)_u$ has, up to isomorphism, two irreducible objects, $\mathbb{L}_0 = \underline\k_0[-1]$ and $\mathbb{L}_1=\underline\k_L$ (the sky-scraper at zero and the constant sheaf). Let $\Pi_0$, $\Pi_1$ be projective covers of $\mathbb{L}_0, \mathbb{L}_1$, which are projective objects in the category of pro-objects \[ \operatorname{Pro} \bigl( \on{Perv}(L,0)_u\bigr) \,\,\subset \,\, \operatorname{Fun}\bigl( \on{Perv}(L,0)_u, \on{Vect}_\k\bigr)^{\operatorname{op}}. \] They are defined uniquely up to an isomorphism. Moreover, any exact functor from $ \on{Perv}(L,0)_u$ to vector spaces sending $\mathbb{L}_1$ (resp. $\mathbb{L}_0$) to zero and $\mathbb{L}_0$ (resp. $\mathbb{L}_1$) to a one dimensional space is isomorphic, in the sense of viewing pro-objects as functors above, to $\Pi_0$ (resp. $\Pi_1$). This means that there exist isomorphisms of functors $\on{Perv}(L,0)\to\on{Vect}_\k$ \[ \operatorname{Hom}(\Pi_0, - ) \,\,\cong \Phi,\quad \operatorname{Hom}(\Pi_0, - )\,\,\cong \,\, \Psi. \] We fix such isomorphisms. Proposition \ref{prop:GGM} implies that $\operatorname{End}(\Pi_0)\simeq \mathbf{k} [[(m-1)]]\simeq \operatorname{End}(\Pi_1)$ while each of the spaces $\operatorname{Hom}(\Pi_0, \Pi_1)$, $\operatorname{Hom}(\Pi_1,\Pi_0)$ is a free rank one module over $\mathbf{k} [[(m-1)]]$ generated respectively by elements $a$, $b$. Since $\operatorname{FT}$ interchanges $\mathbb{L}_0$ and $\mathbb{L}_1$, we have \begin{equation}\label{eq:phi-psi-pi} \operatorname{FT}(\Pi_0)\,\,\simeq \,\, \Pi_1, \quad \operatorname{FT}(\Pi_1)\,\,\simeq \,\, \Pi_0. \end{equation} Furthermore, the isomorphism \eqref{eq:phi-ft-psi} sending $m_{\mathcal F}$ to $m_{\operatorname{FT}({\mathcal F})}^{-1}$ shows that for some (hence for any) choice of the isomorphisms $\operatorname{FT}(\Pi_0)\cong \Pi_1$ the automorphim $\operatorname{FT}(m)$ of the left hand side corresponds to the automorphism $m^{-1}$ of the right hand side. It follows that an isomorphism $\operatorname{FT}(\Pi_1)\cong \Pi_0$ also sends $\operatorname{FT}(m)$ to $m^{-1}$. We can choose the isomorphisms \eqref{eq:phi-psi-pi} in such a way that the map $\operatorname{FT}(a)$ becomes compatible with $-b$. This is clear since both elements generate the corresponding free rank one modules over $\mathbf{k} [[(m-1)]]$. Then we see that $\operatorname{FT}(b)$ corresponds to $a({\bf 1}_{\Pi_0}+ba)^{-1}$, this implies the statement. \qed \begin{rem} In the last paragraph of the proof we made a choice of isomorphisms \eqref{eq:phi-psi-pi} satisfying certain requirements. We have earlier constructed an isomorpism of functors \eqref{eq:phi-ft-psi}. Combining it with the canonical isomorphism $\operatorname{FT}^2({\mathcal F})=(-1)^({\mathcal F})$ we can (upon making a binary choice of a homotopy class of a path connecting the ray $K$ to the ray $-K$) produce a canonical isomorphism $\Psi({\mathcal F})\simeq \Phi(\operatorname{FT}({\mathcal F}))$. These two isomorphism of functors yield isomorphisms of representing objects. We do not claim however that these isomorphisms satisfy our requirements. They provide another (isomorphic but different) functor on the category of linear algebra data of Proposition \ref{prop:GGM}; it may be interesting to work it out explicitly. \end{rem} \begin{rem} In the case $\k=\mathbb{C}$ one can deduce the proposition from the infinitesimal description $\on{Perv}(\mathbb{C},0)\simeq \mathfrak{I}$ (Proposition \ref{prop:I}), where the Fourier transform functor $\operatorname{FT}_\mathfrak{I}: \mathfrak{I}\to \mathfrak{I}$ is given by Corollary \ref{cor:FT-I}: \begin{equation}\label{eq:FT-I} \bigl\{ \xymatrix{ E \ar@<.5ex>[r]^{u}& \ar@<.5ex>[l]^{v} F}\bigr\} \,\,\longmapsto \,\, \bigl\{ \xymatrix{ E'=F \ar@<.5ex>[r]^{u'}& \ar@<.5ex>[l]^{v'} F'=E}\bigr\} , \quad u'=v, v'=-u. \end{equation} Since both $\mathfrak{I}$ and $\mathfrak{J}$ describe $\on{Perv}(\mathbb{C},0)$, we get an identification $\mathfrak{I}\to \mathfrak{J}$ which was given explicitly by Malgrange \cite[(II.3.2)]{Ma} as follows: \begin{equation}\label{eq:Mal-corr} \begin{gathered} \bigl\{ \xymatrix{ E \ar@<.5ex>[r]^{u}& \ar@<.5ex>[l]^{v} F}\bigr\} \,\,\longmapsto \,\, \bigl\{ \xymatrix{ \Phi= \ar@<.5ex>[r]^{a}& \ar@<.5ex>[l]^{b} \Psi=F}\bigr\} \\ \begin {cases} a=u,\\ b= \varphi(vu)\cdot v, \quad \varphi(z) = (e^{2\pi i z} -1)/z. \end{cases} \end{gathered} \end{equation} By inverting \eqref{eq:Mal-corr} (i.e., finding $u$ and $v$ through $a$ and $b$), and then applying \eqref{eq:Mal-corr} to $u', v'$ given by \eqref{eq:FT-I}, we get an object $\bigl\{ \xymatrix{ \Psi \ar@<.5ex>[r]^{a^\dagger}& \ar@<.5ex>[l]^{b^\dagger} \Phi}\bigr\} $ which turns out to be isomorphic to $\bigl\{ \xymatrix{ \Psi \ar@<.5ex>[r]^{a'}& \ar@<.5ex>[l]^{b'} \Phi } \bigr\}$ by conjugation with an explicit invertible function of ${\bf 1}_\Phi+ba)$. \end{rem} \vskip .2cm \sparagraph {\bf Proof of Theorem \ref{thm:prepro}}. We start with an almost obvious model case of one projective line $Y\simeq \mathbb{P}^1$. Suppose we are given a point $z\in Y$ which will serve as an ``origin" and a further set of $N$ points $A=\{y_1, \cdots, y_N\}$ which we position on the boundary of a closed disk $B$ containing $z$, in the clockwise order. Choose a system of simple arcs $K_\nu$ joining $z$ with $y_\nu$ and not intersecting outside of $z$. Let $\mathcal{L} $ be a line bundle of degree $d$ on $\mathbb{P}^1$ and let $q\in\k^$. \begin{lem}\label{lem:twisted} The category $\on{Perv}^{(\mathcal{L}, q)}(Y,A)$ is equivalent to the category of diagrams consisting of vector spaces $\Psi, \Phi_1, \cdots, \Phi_N$ and linear maps \[ \bigl\{ \xymatrix{ \Phi_\nu \ar@<.5ex>[r]^{a_\nu}& \ar@<.5ex>[l]^{b_\nu} \Psi } \bigr\}, \quad \nu=1,\cdots, N, \] such that each ${\bf 1}_\Psi + a_\nu b_\nu$ is invertible and \[ \prod_{\nu=1}^N ({\bf 1}_\Psi+a_\nu b_\nu) \,\,=\,\, q^d {\bf 1}_\Psi. \] \end{lem} \noindent {\sl Proof:} We first consider the untwisted case: $q=1$ or, equivalently, no $\mathcal{L}$. In this case the statement follows at once from Proposition \ref{prop:GGM}. Indeed, choose thin neighborhoods $U_\nu$ of $K_\nu$ (thus containing $z$ and $y_\nu$ which are topologically disks and let $U=\bigcup U_\nu$. We can assume that $Y$ is, topologically, a disk as well. An object $\mathcal{F}\in\on{Perv}(Y,A)$ can be seen as consisting of perverse sheaves $\mathcal{F}_\nu$ on $U_\nu$ which are glued together into a global perverse sheaf on $Y$. Each $\mathcal{F}_\nu$ is described by a diagram $ \bigl\{ \xymatrix{ \Phi_\nu \ar@<.5ex>[r]^{a_\nu}& \ar@<.5ex>[l]^{b_\nu} \Psi_\nu } \bigr\},$ To glue the $\mathcal{F}_\nu$ together, we need, first, to identify all the $\Psi_\nu$ with each other, i.e., with a single vector space $\Psi$. This will give a perverse sheaf $\mathcal{F}_U$ on $U$. In order for $\mathcal{F}_U$ to extend to a perverse sheaf on $Y=\mathbb{C}\mathbb{P}^1$, it is necessary and sufficient that the monodromy of $\mathcal{F}_U$ along the boundary $\partial U$ of $U$ be trivial, in which case the extension is unique up to a unique isomorphism. To identify this condition explicitly, let $\gamma_\nu$ be a loop in $Y$ beginning at $z$, going towards $y_\nu$ along $K_\nu$, then circling around $y_\nu$ anti-clockwise and returning back to $z$ along the same path. Then $\partial U$ can be represented, up to homotopy, by the composite loop $\gamma = \gamma_1 \gamma_2\cdots\gamma_N$ and the monodromy of $\mathcal{F}_\nu$ around $\gamma_\nu$ is $1+a_\nu b_\nu$. In the twisted case, choose a trivialization of $\mathcal{L}$ over $U$, so that we have the projections \[ U\buildrel \alpha\over\longleftarrow \mathcal{L}^\circ\|_U \buildrel\beta\over\longrightarrow \mathbb{C}^* \] Let $\widetilde z$ be the vector in the fiber of $\mathcal{L}$ over $z$ such that $\beta(\widetilde z)=1$. Let \[ \widetilde\gamma = \gamma\times\{ 1\} \,\,\subset U\times \mathbb{C}^* \simeq \mathcal{L}^\circ\|_U \] be the lift of $\gamma$ with respect to the trivialization. Since $\gamma$ does not meet $A$, we can regard $\widetilde\gamma$ as a loop in $\mathcal{L}^\circ\|_{Y-A}$, beginning and ending at $\widetilde z$. Note that the line bundle $\mathcal{L}$ is trivial over $Y-A$ as well, and so \[ \pi_1\bigl( \mathcal{L}^\circ\|_{Y-A}, \widetilde z\bigr) \,\,=\,\, \mathbb{Z}\cdot \zeta, \] where $\zeta$ is the counterclockwise loop in the fiber $\mathcal{L}^\circ\|_z$. Under this identification, the element represented by $\widetilde\gamma$, is equal to $d\cdot\zeta$. Now, using our trivialization, we have an equivalence \[ M: \on{Perv}(U,A) \longrightarrow \on{Perv}^{(\mathcal{L}, q)}(U,A), \quad \mathcal{F} \mapsto \alpha^\mathcal{F} \otimes_\mathbf k\beta^\mathcal{E}_q, \] where $\mathcal{E}_q$ is the 1-dimensional local system on $\mathbb{C}^$ with monodromy $q$. An object $\mathcal{F}$ of $\on{Perv}(U,A)$ is described by a diagram of \[ \bigl\{ \xymatrix{ \Phi_\nu \ar@<.5ex>[r]^{a_\nu}& \ar@<.5ex>[l]^{b_\nu} \Psi } \bigr\}, \quad \nu=1, \cdots, N \] as before. The possibility of extending $M(\mathcal{F})$ from $\mathcal{L}^\circ\|_U$ to the whole of $\mathcal{L}^\circ$ is equivalent to the monodromy around $\zeta \in \pi_1(\mathcal{L}\|_{Y-A},\widetilde z)$ being equal to $q\cdot {\bf 1}$. In view of the equality $\widetilde\gamma = d\cdot\zeta$, this gives precisely the condition of the lemma. \qed \vskip .3cm The proof of Theorem \ref{thm:prepro} is now obtained by gluing together the descriptions of Lemma \ref{lem:twisted}, using Proposition \ref{prop:cluster} and Corollary \ref{cor:inverse}. More precisely, we apply the lemma to each $(Y_i, A_i)$, $i\in I$, where the $Y_i=\widetilde X_i\buildrel\varpi_i\over\to X$, $i\in I$ are the components of the normalization $\widetilde X$ of $X$, and $A_i = \widetilde D\cap \widetilde X_i$. We recall that $\widetilde D\subset \widetilde X$ is the preimage of the set of nodes $D\subset X$. We put $\mathcal{L}_i = \varpi_i^\mathcal{L}$ so that $d_i=\deg(\mathcal{L}_i)$. Choose an orientation of the intersection graph $\Gamma = \Gamma_X$, or, equivalently, an ordering $(x', x'')$ of the pair of preimages of each node $x\in D$. We will label these preimages by the arrows $h$ of $\Gamma$, i.e., denote them by \[ x'_h \in A_{s(h)} \subset Y_{s(h)}, \quad x''_h \in A_{t(h)} \subset Y_{t(h)}, \quad h\in E. \] Thus $A_i$ consists of \[ x'_h, \,\, s(h) = i \text{ and } \,\, x''_{h}, \,\, t(h)=i. \] We choose a base point $z_i$ in each $Y_i$ and position the elements of $A_i$ on the boundary of a disk around $z_i$, so that, in the clockwise order, we have first the $x''_h, t(h)=i$ (according to the order $<$ on $E$) and then the $x'_h, s(h)=i$ (again according to $<$). We join $z_i$ with the elements of $A_i$ simple arcs meeting only at $z_i$. An object $\mathcal{F}_i\in\on{Perv}^{(\mathcal{L}_i, q_i)}(Y_i, A_i)$ is then described by a diagram consisting of one space $\Psi_i$ together with spaces $\Phi_{x'_h}$, $s(h)=i$ and $\Phi_{x''_h}$, $t(h)=i$ together with the maps \[ \bigl\{ \xymatrix{ \Phi_{x'_h} \ar@<.5ex>[r]^{a'_{h}}& \ar@<.5ex>[l]^{b'_{h}} \Psi_i } \bigr\}, \,\, s(h)=i, \quad \bigl\{ \xymatrix{ \Phi_{x''_h} \ar@<.5ex>[r]^{a''_h}& \ar@<.5ex>[l]^{b''_h} \Psi_i } \bigr\},\,\,\, t(h)=i \] so that the condition of the lemma reads: \begin{equation}\label{eq:perv-product} \prod_{t(h)=i} ({\bf 1}+ a''_h b''_h) \prod_{s(h)=i} ({\bf 1}+ a'_h b'_h) \,\,\,=\,\,\, q^{d_i}\cdot {\bf 1}. \end{equation} In order to glue the $\mathcal{F}_i$ into one twisted microlocal sheaf on $X$, we need to specify an identification of Fourier transforms at each node $x$. This means that $\Psi_i$ (which is identified with the space of nearby cycles of $\mathcal{F}_i$ at each $x'_h$, $s(h)=i$ and each $x''_h, t(h)=i$) becomes identified with the space of vanishing cycles of $\mathcal{F}_{t(h)}$ at $x''_h$ for $s(h)=i$ and of $\mathcal{F}_{s(h)}$ at $x'_h$ for $t(h)=i$. Therefore all the linear algebra data reduce to the vector spaces $V_i=\Psi_i$ and linear operators \[ \begin{gathered} a_h: V_{s(h)} = \Psi_{s(h)} \simeq \Phi_{x''_h}(\mathcal{F}_{t(h)}) \buildrel a''_h\over\longrightarrow \Psi_{t(h)} = V_{t(h)}, \\ b_h: V_{t(h)} = \Psi_{t(h)} \buildrel b''_h \over\longrightarrow \Phi_{x''_h}(\mathcal{F}_{t(h)} \simeq \Psi_{s(h)} = V_{s(h)}, \end{gathered} \] where $\simeq$ stands for the identifications given by the Fourier transform. This means that we do not use the simply primed $a'_h, b'_h$, expressing them through $a''_h, b''_h$ by Proposition \ref{prop:cluster}. After this reduction, the conditions \eqref{eq:perv-product} coincide, in view of Corollary \ref{cor:inverse}, with the defining relations of the multiplicative preprojective algebra. \qed \eject \section{Preprojective algebras for general nodal curves}\label{sec:preprogen} Theorem \ref{thm:prepro} can be extended to the case of arbitrary compact nodal curves by introducing an appropriate analog of preprojective algebras (PPA). In this section we present this analog and discuss possible further generalizations to differential graded (dg-) case and their consequences for the symplectic structure of moduli spaces. Throughout the paper we use the notation \[ [\alpha, \beta] \,\,=\,\, \alpha\beta \alpha^{-1}\beta^{-1} \] to denote the group commutator. \sparagraph{Higher genus PPA.} Let $X$ be a compact nodal curve over $\mathbb{C}$. As before we denote by $D$ the set of nodes of $X$, by $X_i, i\in I$ the irreducible components of $X$ and by $\widetilde X_i\subset \widetilde X \buildrel\varpi\over\to X$ the normalizations of $X_i$ and $X$. Let $\mathcal{L}$ be a line bundle on $X$ and $\widetilde\mathcal{L} = \varpi^\mathcal{L}$. We denote by: \[ \begin{gathered} g_i = \text{ the genus of } \widetilde X_i, \quad d_i = \deg(\widetilde \mathcal{L}\|_{\widetilde X_i}), \quad \widetilde D_i = \varpi^{-1}(D)\cap \widetilde X_i. \end{gathered} \] We choose an orientation of $X$, i.e., a total order $x' < x''$ on each 2-element set $\varpi^{-1}(x), x\in D$, see \S \ref {sec:dr}A. For each node $x\in D$ we denote by $s(x)\in I$ the label of the irreducible component containing $x'$, and by $t(x)$ the label of the component containing $x''$. We also choose a total order on the set $D$. \begin{Defi}\label{def:hg-ppa} Let $X, \mathcal{L}$ as above be given and $q\in\k^$. The {\em preprojective $(X,\mathcal{L})$-algebra} $\Lambda^{\mathcal{L}, q} (X)$ is defined by generators and relations as follows: \begin{enumerate} \item[(0)] Objects $i\in I$. \item[(1)] For each node $x\in D$, two generating morphisms $a_x: s(x)\to t(x)$ and $b_x: t(x)\to s(x)$. We impose the condition that \[ {\bf 1}_{t(h)} + a_h b_h: t(h)\to t(h), \quad {\bf 1}_{s(h)} + b_h a_h: s(h)\to s(h) \] are invertible, i.e., introduce their formal inverses. \item[(1')] For each $i\in I$ there are generating morphisms \[ \alpha^i_\nu, \beta^i_\nu, \,\,\, i=1, \cdots, g_i, \] which are required to be invertible. \item[(2)] For each $i\in I$ we impose a relation \[ \begin{gathered} \prod_{x\in D: t(h)=i} ({\bf 1}_i + a_x b_x) \prod_{x\in D, s(x)=i} ({\bf 1}_i + b_x a_x)^{-1} \prod_{\nu=1}^{g_i} [\alpha^i_\nu, \beta^i_\nu] \,\,= \,\, q^{d_i} {\bf 1}_i. \end{gathered} \] Here the factors in the first two products are ordered using the chosen total order $<$ on $D$. \end{enumerate} \end{Defi} \begin{exas} (a) If all $X_i$ are rational, then $\Lambda^{\mathcal{L}, q}(C)$ reduces to the multiplicative preprojective algebra associated to the quiver $\Gamma_X$, and parameters $q^{d_i}$, see \S \ref{sec:prepro}. (b) If $X$ is smooth irreducible of genus $g>0$, then the fundamental group $\pi_1(X)$ has a universal central extension $\widetilde\pi_1(X)$ given by generators and relations as follows \[ \widetilde\pi_1(X) \,\, = \,\, \biggl\langle \alpha_1, \cdots, \alpha_g, \beta_1, \cdots, \beta_g, {\mathbf q}\, \biggl\| \,\,\, \prod_{\nu=1}^g [\alpha_\nu \beta_\nu] = {\mathbf q}, \,\, [\alpha_i,{\mathbf q}] = [\beta_i, {\mathbf q}]=1 \biggr\rangle. \] In this case $\Lambda^{\mathcal{L}, q}(X)$ is a quotient of the group algebra of $\widetilde\pi_1(X)$ by the relation ${\mathbf q}=q^d$. \end{exas} \begin{thm}\label{thm:higher-pre} The abelian category $\mathcal{M}^{\mathcal{L},q}(X,\emptyset)$ is equivalent to the category of finite-dimensional modules over $\Lambda^{\mathcal{L},q}(X)$. \end{thm} \sparagraph{Proof of Theorem \ref{thm:higher-pre}.} It is similar to that of Theorem \ref{thm:prepro}. We first consider the following model case. Let $Y$ be a smooth, compact, irreducible curve of genus $g$ together with finite subset $A=\{y_1, \cdots, y_N\} \subset Y$. Let $\mathcal{L}$ be a line bundle over $Y$ of degree $d$. Define a $\k$-algebra $\Lambda^{\mathcal{L},q}(Y,A)$ by generators and relations as follows; \begin{enumerate} \item[(0)] Objects $\psi$, $\phi_1, \cdots, \phi_N$. \item[(1)] Generating morphisms \[ \begin{gathered} a_\lambda: \phi_\lambda\to \psi, \quad b_\lambda: \psi\to\phi_\lambda,\quad \lambda = 1,\cdots, N; \\ \alpha_\nu, \beta_\nu: \psi\to\psi, \,\,\, \nu=1, \cdots, g. \end{gathered} \] We require that \[ {\bf 1}_\psi + a_\lambda b_\lambda, \,\,\, {\bf 1}_{\phi_\lambda}+b_\lambda a_\lambda, \,\,\, a_\nu, b_\nu, h_\mu \] be invertible, i.e., introduce their formal inverses. \item[(2)] One relation \[ \prod_{\lambda=1}^N ({\bf 1}_\psi + a_\lambda b_\lambda) \prod_{\nu=1}^g [\alpha_\nu, \beta_\nu] \,\,=\,\, q^d {\bf 1}_\psi. \] \end{enumerate} \begin{lem}\label{lem:higher-perv} The abelian category $\on{Perv}^{\mathcal{L}}(Y, A)$ is equivalent to the category of finite-dimensional $\Lambda^\mathcal{L}(Y,A)$-modules. \end{lem} \noindent {\sl Proof:} Completely analogous to that of Lemma \ref{lem:twisted}. We choose a base point $p\in Y-A$, realize $\alpha_i$ and $\beta_i$ as the standard A- and B-loops based at $p$ and choose simple arcs $K_\lambda$ jointing $p$ with $y_\lambda$ so that they do not intersect except at $p$ and follow each other in the clockwise order. Conjugating with $K_\lambda$ a small loop around $y_\lambda$, we get a loop $h_\lambda$ based at $p$, and we can choose the $K_\lambda$ to follow the system of $\alpha_i, \beta_i$ in the clockwise order so that in $\pi_1(Y-A, p)$ we have the relation \[ \prod_{\lambda=1}^N h_\lambda \prod_{\nu=1}^g [\alpha_\nu, \beta_\nu] \,\,=\,\, 1. \] Let $D$ be a disk containing all the paths $K_\lambda$, so $\mathcal{L}$ is trivial over $D$. The lemma is obtained by gluing the category of perverse sheaves on $D$ and that of (twisted) local systems on $X-D$. \qed Theorem \ref{thm:higher-pre} is now obtained by gluing the descriptions of Lemma \ref{lem:higher-perv} for $(Y,A)=(\widetilde X_i, \widetilde D_i)$ for various $i$. \qed \sparagraph{Remarks on derived PPA.} The algebra $\Lambda^{\mathcal{L},q}(X)$ has a derived analog. This is a dg-algebra $L\Lambda^{\mathcal{L},q}(X)$ with the same generators $a_x, b_x, \alpha^i_\nu, \beta^i_\nu$ as $\Lambda^{\mathcal{L},q}(X)$ (considered in degree $0$) with the same conditions of invertibility but instead of imposing relations in Definition \ref {def:hg-ppa}, we introduce new free generators of degree $-1$ whose differentials are put to be the differences between the LHS and RHS of the relations. The symbol $L$ is used to signify the left derived functor. Thus $\Lambda^{\mathcal{L},q}(X)$ is the $0$th cohomology algebra of $L\Lambda^{\mathcal{L},q}(X)$. \vskip .2cm It seems very likely that the triangulated category $D\mathcal{M}^{\mathcal{L},q}(X)$ can be identified with the derived category formed by finite-dimensional dg-modules over $L\Lambda^{\mathcal{L},q}(X)$ (with quasi-isomorphisms inverted). In view of Theorem \ref{thm:CY} we can then expect that $D\mathcal{M}^{\mathcal{L},q}(X)$ is a Calabi-Yau dg-algebra of dimension 2. In other word, we expect that, denoting $L = L\Lambda^{\mathcal{L},q}(X)$, there is a quasi-isomorphism of $L$-bimodules \begin{equation}\label{eq:hoch} \gamma: L\to L^! := R\operatorname{Hom}_{L\otimes L^{\operatorname{op}}}(L, L\otimes L^{\operatorname{op}})[2], \quad \text{such that} \quad \gamma = \gamma^![2], \end{equation} see \cite{Gi}, Def. 3.2.3. In general, $L\Lambda^{\mathcal{L},q}(X)$ is not quasi-isomorphic to $\Lambda^{\mathcal{L},q}(X)$, which explains the following example. \begin{ex} Let $X$ be the union of two projective lines meeting transversely, let $\mathcal{L}$ be trivial and $q=1$. Then $D\mathcal{M}(X,\emptyset)$ is a Calabi-Yau category of dimension 2, while $\mathcal{M}(X,\emptyset)$ has infinite cohomological dimension. Indeed, $\mathcal{M}(X,\emptyset)$ is identified with the category of modules over the multiplicative preprojective algebra corresponding to the quiver $A_2$; this algebra has two objects $1,2$ generating morphisms $a:1\to 2$ and $b:2\to 1$ subject to the relations $ab=ba=0$. \end{ex} We can also define the {\em universal higher genus PPA} (derived as well as non-derived) by replacing $q\in\k^$ in the above by an indeterminate ${\mathbf q}$ and working over the Laurent polynomial ring $\k[{\q^{\pm 1}}]$. We denote the corresponding (dg-) algebras by $L\Lambda^\mathcal{L} (X)$ and $\Lambda^\mathcal{L}(X)$. Because of the 1-dimensionality of $\k[{\q^{\pm 1}}]$, we expect that $L\Lambda^\mathcal{L} (X)$, considered as a dg-algebra over $\k$, is 3-Calabi-Yau, rather than 2-Calabi-Yau. \begin{ex} If $X$ is a smooth projective curve of genus $g>0$, then $\Lambda^\mathcal{L}(X)$ is the group algebra of the fundamental group of $\mathcal{L}^\circ$. Now, $\mathcal{L}^\circ$ is homotopy equivalent to a circle bundle over $X$, which is a compact, apsherical, oriented 3-manifold. By \cite{Gi}, Cor. 6.1.4 this implies that $\Lambda^\mathcal{L}(X)$ is a (non-dg) 3-Calabi-Yau algebra. Further, in this case $L\Lambda^\mathcal{L} (X)$ is quasi-isomorphic to $\Lambda^\mathcal{L}(X)$ by \cite{Gi}, Thm. 5.3.1. \end{ex} \sparagraph{Remarks on moduli spaces.} Assume $\operatorname{char}(\k)=0$. We would like to view the symplectic nature of (multiplicative) quiver varieties as yet another manifestation of the following general principle, which also encompasses the approaches of \cite{goldman} and \cite{Mu} to local systems (resp. coherent sheaves) on topological (resp. K3 or abelian) surfaces. \begin{CYP}\label{CYP} If $\mathcal{C}$ is a Calabi-Yau category of dimension 2, then $\mathfrak{M}$, the ``moduli space" of objects in $\mathcal{C}$, has a canonical symplectic structure. After identifying the ``tangent space" to $\mathfrak{M}$ at the point corresponding to object $E$, with $\operatorname{Ext}^1_\mathcal{C}(E,E)$, the symplectic form is given by the {\em cohomological pairing} \[ \operatorname{Ext}_\mathcal{C}^1(E,E) \otimes \operatorname{Ext}_\mathcal{C}^1(E,E) \buildrel\cup\over\longrightarrow \operatorname{Ext}^2_\mathcal{C}(E,E) \buildrel \operatorname{tr}\over\longrightarrow \k, \] where $\operatorname{tr}$ corresponds, via the Calabi-Yau isomorphism, to the embedding $\k\to\operatorname{Hom}_\mathcal{C}(E,E)$. \end{CYP} This principle, along with a generalization to $N$-Calabi-Yau categories for any $N$, was formulated in \cite{kontsevich-soibelman} \S 10 and made precise in a formal neighborhood context. A wider, more global, interpretation would be as follows. \vskip .2cm \noindent {\bf ``Space":} understood in the sense of derived algebraic geometry \cite{lurie-DGA} \cite{TVe}, as a {\em derived stack}. Informally, a derived stack $\mathfrak{Y}$ can be seen as a nonlinear (curved) analog of a cochain complex of $\k$-vector spaces, in the same sense in which a manifold can be seen as a curved analog of a single vector space. In particular, for a $\k$-point $y\in\mathfrak{Y}$ we have the {\em tangent dg-space} $T^\bullet_y\mathfrak{Y}$, which is a cochain complex. The {\em amplitude} of $\mathfrak{Y}$ is an integer interval $[a,b]$ such that $H^i T^\bullet_y\mathfrak{Y}=0$ for $i\notin [a,b]$ and all $y$. Given a morphism $f: Y\to Z$ of smooth affine algebraic varieties over $\k$ and a $\k$-point $z\in Z$, we have the {\em derived preimage} $Rf^{-1}(z)$, which is a derived stack (scheme) of amplitude $[0,1]$, see \cite{CFK} for elementary treatment. \vskip .2cm \noindent {\bf ``Moduli":} understood as the derived stack $\mathfrak{M}_\mathcal{C}$ of moduli of objects in a dg-category $\mathcal{C}$ defined in \cite{TV}. Under good conditions on $\mathcal{C}$, each object $E$ gives a $\k$-point $[E]\in\mathfrak{M}_\mathcal{C}$ and we have the Kodaira-Spencer quasi-isomorphism \[ T^\bullet_{[E]} \mathfrak{M}_\mathcal{C} \,\,\simeq \,\, R\operatorname{Hom}_\mathcal{C}(E,E)[1]. \] \noindent {\bf ``Symplectic":} understood in the sense of \cite{PTVV}. That is, the datum of a symplectic form on a derived stack $\mathfrak{Y}$ includes not only pairings on the tangent dg-spaces $T^\bullet_y \mathfrak{Y}$ but also higher homotopies for the de Rham differentials of such pairings. \vskip .2cm \noindent {\bf ``2-Calabi-Yau":} In order for the approach of \cite{kontsevich-soibelman} to be applicable, even at the formal level, we need not only canonical identifications $R\operatorname{Hom}(E,F)^ \simeq R\operatorname{Hom}(F,E)[2]$ but a finer structure: a class in the Hochschild cohomology of $\mathcal{C}$ inducing these identifications. For instance, if $\mathcal{C}$ is the derived category of dg-modules over a dg-algebra $L$, we need an isomorphism $\gamma$ as in \eqref{eq:hoch}, i.e., $L$ should have a structure of a Calabi-Yau dg-algebra in the sense of \cite{Gi}. For the categories of deformation quantization modules, Hochschild cohomology classes of this nature were constructed in \cite{KS-DQ} Thm. 6.3.1. \vskip .2cm While there is every reason to expect the validity of Principle \ref{CYP} in this setting, this has not yet been established. The case of $\mathcal{C} = D\mathcal{M}(X,\emptyset) = D^b_{\text {loc. const}}(X)$ for a smooth compact $X$ follows from the results of \cite{PTVV}, as in this case $\mathfrak{M}_\mathcal{C}$ is interpreted in terms of mapping stacks to the $(-2)$-shifted symplectic stacks $BGL_N$. This interpretation does not apply to $D\mathcal{M}(X,\emptyset)$ for a general compact nodal curve $X$. So we cannot use Principle \ref{CYP} to construct ``symplectic moduli spaces of microlocal sheaves". In the next section we present an alternative, more direct approach via quasi-Hamiltonian reduction. \eject \section{Framed microlocal sheaves and multiplicative quiver varieties}\label{sec:framed} \sparagraph {\bf Motivation.} Recall \cite{N} that the original setting of Nakajima Quiver Varieties $M_\Gamma(V,W)$ involves two types of vector spaces associated to vertices $i$ of quiver $\Gamma$: \begin{enumerate} \item[(1)] The ``color" spaces $V_i$ which are {\em gauged}, i.e., we perform the Hamiltonian reduction by the group $GL(V)=\prod GL(V_i)$ in order to arrive at $M_\Gamma(V,W)$. \item[(2)] The ``flavor" spaces $W_i$ which are {\em fixed}, in the sense that $M_\Gamma(V,W)$ depends on $W$ functorially. In particular, it has a Hamiltonian action of the group $GL(W)=\prod GL(W_i)$. The setting of preprojective algebras (whose multiplicative version was reviewed in \S \ref{sec:prepro}), corresponds to the case when $W_i=0$. \end{enumerate} \noindent In this section we explain a geometric framework allowing us to introduce such flavor spaces in a more general context of microlocal sheaves. For simplicity we restrict the discussion to untwisted microlocal sheaves. \vskip .2cm \sparagraph {\bf Microlocal sheaves framed at $\infty$.} Let $Y$ be a quasi-projective nodal curve over $\mathbb{C}$ with a duality structure. We assume that $Y=\overline {Y} -\infty$, where $\overline Y$ is a compact nodal curve and $\infty = \{\infty_j\}_{j\in J}$ is a finite set of smooth points. Let \[ Y^\partial \,\,=\,\,\operatorname{Bl}_{\infty}(\overline Y) \,\,=\,\, Y \sqcup C, \quad C \,\,=\,\, \bigsqcup_{j\in J} C_j \] be the real blowup of $\overline Y$ at $\infty$. Thus $Y^\partial$ is a compact topological space obtained by adding to $Y$ the circles $C_j$, so that each $C_j=S^1_{\infty_j}\overline Y$ is the circle of real directions of $\overline Y$ at $\infty_j$. Note that in a neighborhood of $C$, the space $Y^\partial$ is naturally a 2-dimensional oriented $C^\infty$-manifold with boundary $C$. We choose a base point $p_j$ in each $C_j$. Any microlocal sheaf $\mathcal{F}$ on $Y$ is a local system in degree 0 near $\infty$. Thus it extends canonically (by direct image) to a complex of sheaves $\mathcal{F}^\partial$ on $Y^\partial$ which is a local system in degree 0 near $C$. In particular, it gives rise to finite-dimensional $\k$-vector spaces $\mathcal{F}_{p_j}$, defined as the stalks of $\mathcal{F}^\partial$ at $p_j$. We denote by \[ \mathfrak{m}_j(\mathcal{F}): \mathcal{F}_{p_j}\longrightarrow\mathcal{F}_{p_j} \] the anti-clockwise monodromy of $\mathcal{F}^\partial$ around $C_j$ \begin{Defi}\label{def:framed} Let $W=(W_j)_{j\in J}$ be a family of finite-dimensional $\k$-vector spaces. By a $W$-{\em framed microlocal sheaf} on $Y$ we mean a datum consisting of a microlocal sheaf $\mathcal{F}\in\mathcal{M}(Y,\emptyset)$ together with isomorphisms $\phi_j: \mathcal{F}_{p_j} \to W_j$, $j\in J$. We denote by $\mathcal{M}(Y)_W$ the category (groupoid) formed by $W$-framed microlocal sheaves on $Y$ and their isomorphisms (identical on $W$). \end{Defi} \begin{prop}\label{prop:affine-mod} Assume that $Y$ is an affine nodal curve with a duality structure, i.e., there is at least one puncture on each irreducible component. Then: \begin{itemize} \item[(a)] There exists a smooth affine algebraic $\k$-variety $\mathfrak{M}(Y)_W$ (the {\em moduli space of $W$-framed microlocal sheaves}) such that isomorphism classes of objects of $\mathcal{M}(Y)_W$ are in bijection with $\k$-points of $\mathfrak{M}(Y)_W$. \item[(b)] The group $GL(W)=\prod GL(W_j)$ acts on $\mathfrak{M}(Y)_W$ by change of the framing. Taking the monodromies around the $C_j$ gives an equivariant morphism (which we call the {\em moment map}) \[ \mathfrak{m} = (\mathfrak{m}_j)_{j\in J}: \mathfrak{M}(Y)_W \longrightarrow GL(W). \] \end {itemize} \end{prop} \noindent {\sl Proof:} (a) We analyze the data of a $W$-framed microlocal sheaf directly on $\widetilde X$, as in the previous section. These data reduce to a collection of linear operators between the $W_j$ such that certain expressions formed out of them are invertible but, since each $\widetilde X_i$ is affine, subject to no other relations. This means that $\mathfrak{M}(Y)_W$ is realized as an open subset in the product of sufficiently many copies of affine spaces $\operatorname{Hom}(W_j, W_{j'})$. (b) Obvious. \qed \begin{ex}[(Smooth Riemann surface)]\label{ex:RS} (a) Let $\overline Y$ be a smooth projective curve of genus $g$. Choose one point $\infty\in\overline Y$ and put $Y=\overline Y-\{\infty\}$, so that $\|J\|=1$. Accordingly, we choose one base point $p\in Y$ near $\infty$ in the sense explained above. A microlocal sheaf $\mathcal{F}\in\mathcal{M}(Y,\emptyset)$ is just a local system on $Y$. So we fix one vector space $W$ and denote $G=GL(W)$. A $W$-framed microlocal sheaf is just a homomorphism $\pi_1(Y, p)\to G$. As well known, $\pi_1(Y,p)$ is a free group on $2g$ generators $\alpha_1, \cdots, \alpha_g, \beta_1, \cdots, \beta_g$ which correspond to the $a$- and $b$-cycles on the compact curve $\overline Y$. Therefore $\mathfrak{M}(Y)_W = G^{2g}$ is the product of $2g$ copies of $g$. The $G$-action on $\mathfrak{M}(Y)_W$ is by simultaneous conjugation. The moment map has the form \[ \mathfrak{m}: G^{2g} \longrightarrow G, \quad (A_1, \cdots, A_g, B_1, \cdots, B_g) \,\mapsto \, \prod_{\nu=1}^g [A_\nu, B_\nu], \] so $\mathfrak{m}^{-1}(e) = \operatorname{Hom}(\pi_1(\overline Y, \infty), G)$ is the set of local systems on the compactified curve, trivialized at $\infty$. \vskip .2cm (b) More generally, let $Y$ be an arbitrary smooth curve, compactified to $\overline Y$ by a finite set of punctures $\infty_j, j\in J$. Then $\mathfrak{M}(Y)_W$ is the space of representations of $\pi_1(Y, \{\infty_j\}_{j\in J})$, the fundamental groupoid of $Y$ with respect to the set of base points $\infty_j$. This is the setting of \cite {AMM}, \S 9.2, see also \cite{Boa}, Thm. 2.5. \end{ex} \begin{ex}[(Coordinate cross)]\label{ex:cross} Let $Y=\{(x_1,x_2)\in\mathbb{A}^2 \| \,\, x_1 x_2=0\}$ be the union of two affine lines meeting transversely. Then $\overline Y$ is the union of two projective lines meeting transversely and $\infty$ consists of two punctures. Accordingly, we have two marked points on $Y^\partial$, denote them $p_1$ and $p_2$. Given a family of two vector spaces $W=(W_1, W_2)$, the stack $\mathfrak{M}(Y)_W$ is the affine algebraic variety known as the {\em van den Bergh's quasi-Hamiltonian space}, see \cite{vdB} and \cite[\S 2.4]{Boa}: \[ \mathfrak{M}(Y)_W \,\,=\,\, \mathcal{B}(W_1, W_2) \,\, := \,\, \bigl\{ \xymatrix{ W_1 \ar@<.5ex>[r]^{a}& \ar@<.5ex>[l]^{b} W_2 } \bigl\| 1+ab \text{ is invertible}\bigr\}. \] Note that $1+ba$ is also invertible on $\mathcal{B}(W_1, W_2)$. \end{ex} \begin{ex}[(Microlocal sheaves with framed $\Phi$)]\label{ex:fr-phi} Let $X$ be a compact nodal curve with a duality structure, and $A\subset X$ be a finite subset of smooth points. Form a new curve $Y=X_A$, as in Proposition \ref{prop:A>empty}. Then $\mathcal{M}(Y)_W$ can be seen as the category parametrizing microlocal sheaves on $X$ which are allowed singularities at $A$, but are equipped with a $W$-framing of their vanishing cycles at each such singular point. To emphasize it, we denote this category by $\mathcal{M}(X,A)_W$. \end{ex} \begin{ex}[(Multiplicative quiver varieties)] We now specialize the above example further. Let $X$ be a compact nodal curve with irreducible components $X_i, i\in I$. Assume, as in \S \ref{sec:prepro}, that each $X_i$ is a rational curve, i.e., that the normalization $\widetilde X_i$ is isomorphic to $\mathbb{P}^1$. Choose the set $A$ consisting of precisely one smooth point $a_i$ on each $X_i$. Let $W=(W_i)_{i\in I}$ be a family of $\k$-vector spaces. Thus the topological structure of $(X,A)$ is determined by the graph $\Gamma$ of intersections of irreducible components of $X$, in particular, $I$ is the set of vertices of $\Gamma$. We will write $X=X_\Gamma$ to indicate this dependence. \end{ex} \begin{prop} In the situation just described, $\mathcal{M}(X,A)_W$ is equivalent to the category which parametrizes linear algebra data consisting of: \begin{enumerate} \item[(1)] Collections of vector spaces $V=(V_i)_{i\in I}$; \item[(2)] Linear maps \[ \begin{gathered} a_h: V_{s(h)}\to V_{t(h)}, \quad b_h: V_{t(h)}\to V_{s(h)}, \quad h\in E, \\ u_i: V_i\to W_i, \quad v_i: W_i\to V_i, \quad i\in I, \end{gathered} \] such that all the maps \[ ({\bf 1}+a_h b_h),\,\, ({\bf 1}+b_h a_h), \,\, ({\bf 1}+u_iv_i), \,\, ({\bf 1}+v_iu_i) \] are invertible, and \item[(3)] For each $i\in I$ we have the identity \[ ({\bf 1}_{V_i} + v_i u_i) \prod_{h\in E, \, t(h)=i} ({\bf 1}_{V_i} + a_h b_h) \prod_{h\in E, \, s(h)=i} ({\bf 1}_{V_i} + b_h a_h)^{-1} \,\,=\,\, {\bf 1}_{V_i}. \] \end{enumerate} These data are considered modulo isomoprhisms of the $V_i$. \end{prop} \noindent{\sl Proof:} Completely analogous to that of Theorem \ref{thm:prepro} and we leave it to the reader. \qed \vskip .2cm The moduli spaces of semistable objects of $\mathcal{M}(X,A)_W$ (\ defined as GIT quotients) as well as their analogs for twisted sheaves are the {\em multiplicative quiver varieties} (MQV) as defined in \cite{yamakawa}. \begin{ex}[(Higher genus MQV)]\label{ex:hi-MQV} In the interpretation of the previous example, we associated to a graph $\Gamma$ a nodal curve $X_\Gamma$ with all components rational. In particular, the number $g_i$ of loops at a vertex $i\in\Gamma$ was interpreted as the number of self-intersection points of the corresponding rational curve $X_i$. We can also associate to $\Gamma$ a nodal curve $X'_\Gamma$ in a different way, by taking the component $X'_i$ associated to $i$ to be of genus $g_i$ (and interpreting other edges of $\Gamma$ as intersection points of the $X'_i$). Choose the set $A$ to consist of one point on each irreducible component of $X'_\Gamma$. This defines a datum $(X'_\Gamma, A)$ uniquely up to a diffeomorphism. We will refer to the moduli spaces of objects of $\mathcal{M}(X'_\Gamma, A)_W$ (defined as GIT quotients) as {\em higher genus multiplicative quiver varieties} associated to $\Gamma$. Note that one can also consider their twisted versions, involving twisted microlocal sheaves. \end{ex} \sparagraph {\bf Quasi-Hamiltonian $G$-spaces.} Here we review the main points of the theory of group valued moment maps \cite{AMM}. For simplicity we work in the complex algebraic situation, not that of compact Lie groups. Let $G$ be a reductive algebraic group over $\mathbb{C}$, with Lie algebra $\mathfrak{g}$. We denote by \[ \theta^L = g^{-1}dg, \,\, \theta^R = (dg)g^{-1} \,\,\in \,\,\Omega^1(G, \mathfrak{g}) \] the standard left and right invariant $\mathfrak{g}$-valued 1-forms on $G$. We fix an invariant symmetric bilinear form $(-,-)$ on $\mathfrak{g}$. It gives rise to the bi-invariant scalar 3-form (the {\em Cartan form}) \[ \eta = {1\over 12} (\theta^L, [\theta^L, \theta^L]) \,\,=\,\, {1\over 12} (\theta^R, [\theta^R, \theta^R])\,\,\in \,\,\Omega^3(G). \] For a $G$-manifold $M$ and $\xi\in\mathfrak{g}$ we denote by $\partial_\xi$ the vector field on $M$ corresponding to $\xi$ by the infinitesimal action. \begin{Defi} \label{def:AMM} \cite{AMM} A quasi-Hamiltonian $G$-space is a smooth algebraic variety $M$ with $G$-action, together with a $G$-invariant 2-form $\omega\in\Omega^2(M)^G$ and a $G$-equivariant map $\mathfrak{m}: M\to G$ (the {\em group valued moment map}) such that: \begin{enumerate} \item[(QH1)] The differential of $\omega$ satisfies $d\omega = -\mathfrak{m}^\chi$. \item[(QH2)] The map $\mathfrak{m}$ satisfies, for each $\xi\in\mathfrak{g}$, the condition \[ i_{\partial_\xi} \omega \,\,= {1\over 2} \mathfrak{m}^(\theta^L+ \theta^R, \xi). \] Here $(\theta^L+ \theta^R, \xi)$ is the scalar 1-form on $G$ obtained by pairing the $\mathfrak{g}$-valued form $\theta^L+ \theta^R$ and the element $\xi\in\mathfrak{g}$ via the scalar product $(-,-)$. \item[(QH3)] For each $x\in M$, the kernel of the 2-form $\omega_x$ on $T_xM$ is given by \[ \operatorname{Ker}(\omega_x) \,\,=\,\, \bigl\{ \partial_\xi (x), \,\,\xi\in \operatorname{Ker}(\operatorname{Ad}_{\mathfrak{m}(x)} +{\bf 1}\bigr\}. \] \end{enumerate} \end{Defi} Given a quasi-Hamiltonian $G$-space $(M,\omega, \mathfrak{m})$, the {\em quasi-Hamiltonian reduction} of $M$ by $G$ is, classically \cite{AMM}, the quotient \[ M\slash \hskip -.12cm \slash \hskip -.12cm \slash G \,\,=\,\, \mathfrak{m}^{-1}(e)^{\operatorname{sm}} / G, \] where $\mathfrak{m}^{-1}(e)^{\operatorname{sm}}$ is the smooth locus of the scheme-theoretic preimage $\mathfrak{m}^{-1}(e)$ or, more precisely, the open part formed by those points $m$, for which $d_m\mathfrak{m}$ is surjective. \begin{thm}\label{thm:qred} \cite{AMM} For any quasi-Hamiltonian $G$-space $M$ the quotient $M\slash \hskip -.12cm \slash \hskip -.12cm \slash G$ is a smooth orbifold (i.e., Deligne-Mumford stack) with a canonical symplectic structure. \qed \end{thm} \begin{rem} Using the language of derived stacks allows one to formulate Theorem \ref{thm:qred} in a more flexible way, without restricting to the locus of smooth points. More precisely, we can form the smooth derived stack of amplitude $[-1,1]$ \[ [M\slash \hskip -.12cm \slash \hskip -.12cm \slash G]^{\operatorname{der}} \,\,=\,\, R\mathfrak{m}^{-1}(e) \slash \hskip -.12cm \slash G. \] Here $R\mathfrak{m}^{-1}(e)$ is the derived preimage of $e$, a smooth derived scheme of amplitude $[0,1]$. Further, the symbol $- \slash \hskip -.12cm \slash G$ means stacky quotient by $G$. The analog of Theorem \ref{thm:qred} is then that $[M\slash \hskip -.12cm \slash \hskip -.12cm \slash G]^{\operatorname{der}}$ is a symplectic derived stack which contains $M\slash \hskip -.12cm \slash \hskip -.12cm \slash G$ as an open part. \end{rem} The following is the main result of this section. \begin{thm}\label{thm:MW-quasi} Let $Y$ be an affine nodal curve, and $W=(W_j)$ as before. The smooth algebraic variety $\mathfrak{M}(Y)_W$ has a natural structure of a quasi-Hamiltonian $GL(W)$-space with the moment map $\mathfrak{m}=\mathfrak{m}_W$ given by the monodromies (Proposition \ref{prop:affine-mod}(b)). \end{thm} \begin{rem} This result provides a more direct approach to the ``moduli space" of microlocal sheaves on a compact nodal curve, in particular, to the symplectic structure on this space. Indeed, the set-theoretic quotient $\mathfrak{m}_W^{-1}(e)/GL(W)$ parametrizes microlocal sheaves $\mathcal{F}$ on the compact curve $\overline Y$ such that the dimensions of the stalk of $\mathcal{F}$ at $\infty_j$ is equal to ${\rm{dim}} W_j$. Thus we can {\em define} the derived stack \[ \mathfrak{M}(\overline Y, \emptyset) \,\,=\,\, \bigsqcup_{W}\,\, [\mathfrak{M}(Y)_W\slash \hskip -.12cm \slash \hskip -.12cm \slash GL(W)]^{\operatorname{der}}, \] the disjoint union over all possibe choices of $({\rm{dim}} W_j)_{j\in J}$. Alternatively, one can consider the Poisson variety obtained as the spectrum of the algebra of $GL(W)$-invariant functions on $\mathfrak{M}(Y)_W$, cf. \cite{Boa}, Prop. 2.8. \end{rem} \vskip .2cm In the case of a smooth curve $Y$, see Example \ref{ex:RS}(b), a proof of Theorem \ref{thm:MW-quasi} was given in \cite[\S 9.3] {AMM} using a procedure called {\em fusion} which allows one to construct complicated quasi-Hamiltonian spaces from simpler ones. We use the same strategy but allow one more type of ``building block" in the fusion construction. \vskip .2cm \sparagraph {\bf Fusion of quasi-Hamiltonian spaces.} We now briefly review the necessary concepts. \begin{thm}[\cite{AMM}]\label{thm:fusion} Let $M$ be a quasi-Hamiltonian $G\times G\times H$-space, with moment map $\mathfrak{m}=(\mathfrak{m}_1, \mathfrak{m}_2, \mathfrak{m}_3)$. Let $G\times H$ act on $M$ via the diagonal embedding $(g,h) \mapsto (g,g,h)$. Then $M$ with the 2-form \[ \omega' = \omega + {1\over 2} (\mathfrak{m}_1^\theta^L, \mathfrak{m}_2^\theta^R) \] and the moment map \[ \mathfrak{m}' \,\,=\,\, (\mathfrak{m}_1\cdot \mathfrak{m}_2, \mathfrak{m}_3): M\longrightarrow G\to H \] is a quasi-Hamiltonian $G\times H$-space, called the (intrinsic) {\em fusion} of the $G\times G\times H$-space $M$. \end{thm} \begin{rem} The geometric meaning of the fusion is that the two copies of $G$ from $G\times G\times H$ are ``attached" to the two of the tree boundary components of a 3-holed sphere, and the new diagonal copy of $G$ from $G\times H$ is then ``read off" on the remaining component, see \cite{AMM}, Ex. 9.2 and \cite{Boa} \S 2.2. Thus, in the case of smooth curves, fusion directly corresponds to gluing Riemann surfaces out of simple pieces. We will extend this to nodal curves. \end{rem} The {\em extrinsic fusion} of a quasi-Hamiltonian $G\times H_1$-space $M_1$ and a $G\times H_2$-space $M_2$ is the $G\times H_1\times H_2$-space $M_1 \circledast M_2$ which is the fusion of the $G\times H_1\times G\times H_2$-space $M_1\times M_2$ along the embedding $G\to G\times G$. We will use the following three building blocks. \begin{exas} (a) {\bf (Double of $G$: annulus).} Given $G$ as before, its {\em double} is the quasi-Hamiltonian $G\times G$-space $D(G)=G\times G$ with coordinates $a,b\in G$, the $G\times G$-action given by \[ (g_1, g_2) (a,b) = (g_1 a g_2^{-1}, g_2 b g_1^{-1}), \] the moment map given by \[ \mathfrak{m}_D: D(G) = G\times G \longrightarrow G\times G, \quad (a,b) \mapsto (ab, a^{-1}, b^{-1}) \] and the 2-form given by \[ \omega_D = {1\over 2} (a^\theta^L, b^\theta^R) + {1\over 2} (a^\theta^R, b^\theta^L). \] For a vector space $V$ and $G=GL(V)$, this space is identified with $\mathfrak{M}(Y)_W$, where $Y$ is a 2-punctured sphere and $W=(V,V)$ associates $V$ to each puncture. The surface with boundary $Y^\partial$ is an annulus. \vskip .2cm (b) {\bf Intrinsically fused double: holed torus.} With $G$ as before, its {\em intrinsically fused double} ${\bf D}(G)$ is the quasi-Hamiltonian $G$-space $G\times G$ obtained as the fusion of the $G\times G$-space $D(G)$. For a vector space $V$ and $G=GL(V)$, this space is identified with $\mathfrak{M}(Y)_V$ where $Y$ is a 1-punctured elliptic curve. The surface $Y^\partial$ is a 1-holed torus. \vskip .2cm (c) {\bf The space $\mathcal{B}(W_1,W_2)$: cross.} To treat nodal curves, we add the third type of building blocks: the varieties $\mathcal{B}(W_1,W_2)$, see Example \ref{ex:cross}. Again, this is a known quasi-Hamiltonian $GL(W_2)\times GL(W_2)$-space \cite{vdB} \cite{vdB2} with moment map \[ (a,b) \,\,\mapsto\,\, \bigl((1+ab)^{-1}, 1+ba\bigr) \,\,\in \,\, GL(W_2)\times GL(W_1) \] and the 2-form \[ \omega \,\,=\,\, {1\over 2} \bigl( \operatorname{tr}_{W_2}(1+ab)^{-1}da \wedge db- \operatorname{tr}_{W_1}(1+ba)^{-1} db\wedge da\bigr). \] As we saw in Example \ref{ex:cross}, it is identified with $\mathfrak{M}(Y)_{W_1, W_2}$, where $Y$ is a coordinate cross. The topological space $Y^\partial$ is the union of two disks meeting at one point. \end{exas} Let now $Y$ be an affine nodal curve. The topological space $Y^\partial$ can then be decomposed into elementary pieces of types (a)-(c) in the above examples, joined together by several 3-holed spheres. Let $W=(W_j)_{j\in J}$ be given. Note that for $\mathfrak{M}(Y)_W$ to be non-empty, the numbers $N_j = {\rm{dim}} W_j$ should depend only on the irreducible component of $Y$ containing $\infty_j$. This means that to each boundary component of each elementary piece we can unambiguously associate a group $GL(N_j)$ and so form the corresponding quasi-Hamiltonian space of type (a)-(c) above. Taking the product of these corresponding quasi-Hamiltonian spaces and performing the fusion along the 3-holed spheres, we get a quasi-Hamiltonian space which is identified with $\mathfrak{M}(Y)_W$. This proves Theorem \ref{thm:MW-quasi}. \begin{rem} It would be interesting to construct the 2-form on $\mathfrak{M}(Y)_W$ more intrinsically, in terms of a cohomological pairing, using some version of Poincar\'e-Verdier duality for cohomology with support on the ``nodal surface with boundary" $Y^\partial$. This does not seem to be known even for smooth $Y$. \end{rem} \eject \section{Further directions} \sparagraph{(Geometric) Langlands correspondence for nodal curves.} Since microlocal sheaves without singularities form a natural analog of local systems for nodal curves, it would be interesting to put them into the framework of the Langlands correspondence. In particular, for a compact nodal curve $X$ it would be interesting to have a derived equivalence between the de Rham version (cf. \S \ref{sec:dr}) of the ``Betti-style" derived stack $\mathfrak{M}(X,\emptyset)$ and some other moduli stack $\mathfrak B$ of ``coherent" nature, generalizing the moduli stack of vector bundles on a smooth curve. A potential candidate for $\mathfrak B$ is provided by the moduli stack of Riemann surface quiver representations in the sense of \cite{CB}. Note that the concept of microlocal sheaves makes sense for nodal curves $X$ over $\mathbb{F}_q$. So one can expect that their $L$-functions (appropriately defined) have, for projective nodal curves $X$, properties similar to those of $L$-functions of local systems on smooth projective curves over $\mathbb{F}_q$. One can even consider arithmetic analogs of nodal curves, obtained by gluing the spectra of rings of integers in number fields along closed points. An example is provided by the spectrum of the group ring $\mathbb{Z}[\mathbb{Z}/p]$, where $p$ is a prime. This scheme is the union of ${\on{Spec}}(\mathbb{Z})$ and ${\on{Spec}} (\mathbb{Z}[\sqrt[p] {1} ])$ meeting transversely at the point $(p)$, cf. \cite{Mi}, \S 2. \sparagraph {Multiplicative quiver varieties and mirror symmetry.} Let $\Gamma$ be a finite graph, possibly with loops, and $\mathbb{M}_{V,W}(X'_\Gamma)^q$ be the corresponding {\em higher genus multiplicative quiver varieties}, see Example \ref{ex:hi-MQV}. Here $q=(q_i)\in(\mathbb{C}^)^I$ is a vector of twisting parameters. We expect that the varieties $\mathbb{M}_{V,W}(X'_\Gamma)^q$ are mirror dual to the ordinary (``additive") Nakajima quiver varieties for $\Gamma$. In particular, we expect that $\mathbb{M} _\Gamma(V,W)^q$ is singular if and only if the point $q$ lies in the singular locus of the equivariant quantum connection for the ordinary quiver variety. Here, equivariance is in reference to the action of an algebraic torus which acts on the quiver variety scaling the symplectic form by a nontrivial character. See \cite{MO}, where this connection as well as its singularities, have been computed. \sparagraph {Borel/unipotent reduction and cluster varieties.} It would be interesting to extend the approach of \cite{FG} from local systems on smooth curves to microlocal sheaves on nodal curves. That is, in the situation of \S \ref{sec:framed}B we can choose any number of marked points $p_{j,\nu}$ on each boundary component $C_j$ of $Y^\partial$. After this we can consider microlocal sheaves $\mathcal{F}$ together with a Borel or unipotent reduction of the structure group at each $p_{j,\nu}$ (recall that each restriction $\mathcal{F}\|_{C_j}$ is a local system). This can lead to interesting cluster varieties. These varieties may be related to the classification of irregular DQ-modules on a symplectic surface with support in a nodal curve. \sparagraph{3-dimensional generalization.} The datum of a smooth compact curve over $\mathbb{C}$ (topologically, an oriented surface) $X$ and a finite set of points $A\subset X$ has the following 3-dimensional analog. We consider a compact oriented $C^\infty$ 3-manifold $M$ and a {\em link} in $M$, i.e., a collection $L=\{C_a\}_{a\in A}$ of disjoint embedded circles ({\em knots}). We have then a stratification of $M$ into the $C_a$ and the complement of their union. Denote the $D^b_L(M)$ the category of complexes of sheaves on $M$, constructible with respect to this stratification. For $L=\emptyset$, it is a 3-Calabi-Yau category by Poincar\'e duality. For arbitrary $L$, it has a natural abelian subcategory $\on{Perv}(M, L)$ of ``perverse sheaves". Given any surface $X\subset M$ meeting $L$ transversely, an object $\mathcal{F}\in\on{Perv}(M,L)$ gives a perverse sheaf on $X$, smooth outside $X\cap L$. One can obtain 3d analogs of compact nodal curves (``nodal 3-manifolds") by identifying several compact 3-manifolds pairwise along some knots. For example, we can glue two such manifolds $M'$ and $M''$ (say, two copies of the sphere $S^3$) by identfying a knot $C'\subset M'$ with a knot $C''\subset M''$. As the normal bundle $T_C M$ of a knot $C$ in an oriented 3-manifold $M$ is trivial, we can choose a duality structure, i.e., an identification of $T_{C'}M'$ with $T^_{C''} M''$, and then set up the formalism of microlocal sheaves and complexes. This should lead to interesting 3-Calabi-Yau categories and to $(-1)$-shifted symplectic stacks parametrizing their objects. 3-Calabi-Yau categories of the form $D\mathcal{M}^\mathcal{L}(X,\emptyset)$, see Theorem \ref{thm:CY-twist}(a), correspond to a particular type of nodal 3-manifolds: circle bundles over nodal curves over $\mathbb{C}$. \eject \appendix \section { Notations and conventions.} We fix a base field $\k$. All sheaves will be understood as sheaves of $\k$-vector spaces, similarly for complexes of sheaves. All topological spaces we consider will be understood to be homeomorphic to open sets in finite CW-complexes, in particular, they are locally compact and of finite dimension. For a space $X$ we denote by $\on{Sh}(X)$ the category of sheaves of $\k$-vector spaces on $X$. We denote by $D^b\on{Sh} (X)$ the bounded derived category of $\on{Sh}(X)$. We will consider it as a pre-triangulated category \cite{BK}, i.e., as a dg-category enriched by the complexes $R\operatorname{Hom}(\mathcal{F}, \mathcal{G})$, so that $H^0R\operatorname{Hom}(\mathcal{F},\mathcal{G})$ is the ``usual" space of morphisms from $\mathcal{F}$ to $\mathcal{G}$ in the derived category. Alternatively, we can view it as a stable $\infty$--category by passing to the dg-nerve \cite{lurie-stable} \cite{lurie-algebra} \cite{faonte}. We denote by $D^b_{\operatorname{cc}}(X)\subset D^b\on{Sh} (X)$ the full subcategory of cohomologically constructible complexes \cite{KS} and by $\mathbb{D}=\mathbb{D}_X$ the Verdier duality functor on this subcategory \cite[\S 3.4]{KS}. Thus, if $X$ is an oriented $C^\infty$-manifold of real dimension $d$, and $\mathcal{F}$ is a local system on $X$ (put in degree 0), then $\mathbb{D}(\mathcal{F}) = \mathcal{F}^\bigstar[d]$, where $\mathcal{F}^\bigstar$ is the dual local system. In general, for any compact space $X$ and any $\mathcal{F}\inD^b_{\operatorname{cc}}(X)$ we have {\em Poincar\'e-Verdier duality}, which is the canonical identification of complexes of $\k$-vector spaces with finite-dimensional cohomology, and consequently, of their cohomology spaces: \begin{equation}\label{eq:PVD} \begin{gathered} R\Gamma(X,\mathcal{F})^* \,\, \simeq\,\, R\Gamma(X, \mathbb{D}_X(\mathcal{F})); \\ \mathbb{H}^i(X, \mathcal{F})^* \,\, \simeq\,\, \mathbb{H}^{-i}(X, \mathbb{D}_X(\mathcal{F})). \end{gathered} \end{equation} Let $X$ be a complex manifold. We denote by $D^b_{\operatorname{constr}}(X)\subsetD^b_{\operatorname{cc}}(X)$ the derived category of bounded complexes of sheaves on $X$ with $\mathbb{C}$-constructible cohomology sheaves. The functor $\mathbb{D}_X$ preserves this subcategory. We denote by $\on{Perv}(X)\subsetD^b_{\operatorname{constr}}(X)$ the subcategory of perverse sheaves. The conditions of perversity are normalized so that a local system on $X$ put in degree 0, is perverse. Thus $\on{Perv}(X)$ has the perfect duality given by \[ \mathcal{F}\mapsto \mathcal{F}^\bigstar \,\, := \,\, \mathbb{D}(\mathcal{F})[-2\operatorname{dim}_\mathbb{C}(X)]. \] \eject \vskip .4cm R. B.: Department of Mathematics, MIT, Cambridge MA 02139 USA, {\tt [email protected]} \vskip .2cm M.K.: Kavli IPMU, University of Tokyo, 5-1-5 Kashiwanoha, Kashiwa, Chiba, 277-8583 Japan, {\tt [email protected] } \end{document}	arXiv
\begin{document} \begin{abstract} We define normalized versions of Berkovich spaces over a trivially valued field $k$, obtained as quotients by the action of $\mathbb R_{>0}$ defined by rescaling semivaluations. We associate such a normalized space to any special formal $k$-scheme and prove an analogue of Raynaud's theorem, characterizing categorically the spaces obtained in this way. This construction yields a locally ringed $G$-topological space, which we prove to be $G$-locally isomorphic to a Berkovich space over the field $k((t))$ with a $t$-adic valuation. These spaces can be interpreted as non-archimedean models for the links of the singularities of $k$-varieties, and allow to study the birational geometry of $k$-varieties using techniques of non-archimedean geometry available only when working over a field with non-trivial valuation. In particular, we prove that the structure of the normalized non-archimedean links of surface singularities over an algebraically closed field $k$ is analogous to the structure of non-archimedean analytic curves over $k((t))$, and deduce characterizations of the essential and of the log essential valuations, i.e. those valuations whose center on every resolution (respectively log resolution) of the given surface is a divisor. \end{abstract} \title{Normalized Berkovich spaces and surface singularities} \section{Introduction} Berkovich's geometry is an approach to non-archimedean analytic geometry developed in the late nineteen-eighties and early nineteen-nineties by Berkovich in \cite{Ber90} and \cite{Ber93}. To overcome the problems given by the fact that the metric topology of any valued field is totally disconnected, Berkovich adds many points to the usual points of a variety $X$ (not unlike what happens in algebraic geometry with generic points), to obtain an \emph{analytic space} $X^{\mathrm{an}}$, which is a locally ringed space with very nice topological properties and whose points can be seen as \emph{real semivaluations}. One important feature of Berkovich's theory is that it works also over a trivially valued base field, for example $\mathbb{C}$. This gives rise to objects that are far from being trivial, resembling some spaces studied in valuation theory, but carrying in addition an analytic structure, and containing a lot of information about the singularities of $X$. For example, Thuillier \cite{Thu07} obtained the following result (generalizing a theorem by Stepanov): if $X$ is a variety over a perfect field $k$, then the homotopy type of the dual intersection complex of the exceptional divisor of a log resolution of $X$ does not depend on the choice of the log resolution. To prove this, he associates to a subvariety $Z$ of a $k$-variety $X$ a $k$-analytic space that can be called the \emph{punctured tubular neighborhood} of $Z^{\mathrm{an}}$ in $X^{\mathrm{an}}$. It is a subspace of $X^{\mathrm{an}}$, invariant under modifications of the pair $(X,Z)$, consisting of all the semivaluations on $X$ that have center on $Z$ but are not semivaluations on $Z$. In this paper we define a normalized version $T_{X,Z}$ of this punctured tubular neighborhood, by taking the quotient of the latter by the group action of $\mathbb R_{>0}$ that corresponds to rescaling semivaluations. The space $T_{X,Z}$ can be thought of as a \emph{non-archimedean model} of the \emph{link} of $Z$ in $X$. It is a locally ringed space in $k$-algebras, endowed with the pushforward of the Grothendieck topology and structure sheaf from the punctured tubular neighborhood, and it can be seen as a wide generalization of Favre and Jonsson's \emph{valuative tree}, an object that has important applications to the dynamics of complex polynomials in two variables. Indeed, the valuative tree is homeomorphic to the topological space underlying $T_{\mathbb A^2_\mathbb{C},\{0\}}$, but the latter has much more structure. Moreover, $T_{X,Z}$ can also be thought of as a compactification of the normalized valuation space considered in \cite{JonssonMustata12} and \cite{BoucksomdeFernexFavreUrbinati13}, as the latter is homeomorphic to the subset of $T_{X,Z}$ consisting of all the points that are actual valuations on the function field of $X$. Valuation spaces homeomorphic to $T_{\mathbb A^n_\mathbb{C},\{0\}}$ appear also in \cite{BoucksomFavreJonsson08} ; there the authors develop the basics of pluripotential theory on those spaces, building on previous work of Favre and Jonsson in dimension 2. More generally, we associate a \emph{normalized space} ${T_\X}$ to any \emph{special formal $k$-scheme} $\mathscr X$. If $X$ is a $k$-variety and $Z$ is a closed subvariety of $X$, then the formal completion $\widehat{X/Z}$ of $X$ along $Z$ is a special formal $k$-scheme, and we have $T_{\widehat{X/Z}}\congT_{X,Z}$. The crucial property of ${T_\X}$ is the following: while not an analytic space itself, as a locally ringed $G$-topological space in $k$-algebras the normalized space ${T_\X}$ is $G$-locally isomorphic to an analytic space over the field $k((t))$ with a $t$-adic absolute value. Attention should be paid to the fact that these local isomorphisms are not canonical, and in general they do not induce a global $k((t))$-analytic structure on ${T_\X}$. In particular, this result explains why the valuative tree looks so much like a Berkovich curve defined over $\mathbb{C}((t))$. This interpretation permits to study ${T_\X}$, and thus deduce information about $\mathscr X$, with various tools of non-archimedean analytic geometry, including the ones that work only over non-trivially valued fields. We have only recently learned about the article \cite{Ben-BassatTemkin2013}. There the authors use the punctured tubular neighborhood of $Z^{\mathrm{an}}$ in $X^{\mathrm{an}}$ to encode the descent data necessary to glue a coherent sheaf on a formal neighborhood of $Z$ to a coherent sheaf on $X\setminus Z$. The result we just described on the structure of ${T_\X}$ is conceptually similar to the results of Sections 4.2 to 4.6 of {\it loc. cit.}. We define an affinoid domain of ${T_\X}$ as a $G$-admissible subspace $V$ of ${T_\X}$ that is isomorphic to a strict $k((t))$-affinoid space, and we show that this definition does not depend on the choice of a $k((t))$-analytic structure on $V$. This is done by showing, following \cite{Liu90}, that a reduced $k((t))$-analytic space is strictly affinoid if and only if it is Stein, compact and its ring of analytic functions bounded by one is a special $k$-algebra. Every normalized space ${T_\X}$ is compact and $G$-covered by finitely many affinoid domains, and this allows us to characterize the category of all the locally ringed $G$-topological spaces of the form ${T_\X}$ as the localization of the category of special formal $k$-schemes by the class of admissible formal blowups. This is a ``normalized spaces version'' of a classical theorem of Raynaud for non-archimedean analytic spaces (see \cite[4.1]{BosLut85}). We then apply normalized spaces to the study of surface singularities. While the importance of valuations in the study of resolutions of surface singularities was emphasized already in the work of Zariski and Abhyankar (see \cite{Zariski39} and \cite{Abhyankar56}), in our work also the additional structure given by the sheaf of analytic functions plays an important role. If $k$ is an algebraically closed field, $X$ is a $k$-surface and $Z$ is a subspace of $X$ containing its singular locus, by the structure theorem discussed above the normalized space $T_{X,Z}$ behaves like a non-archimedean analytic curve over $k((t))$. The theory of such curves is well understood, thanks to work of Bosch and L\"utkebohmert \cite{BosLut85} (\cite[Chapter 4]{Ber90} for Berkovich spaces). In particular, there is a correspondence between (semistable) models and (semistable) vertex sets (see \cite{Duc}, \cite[Chapter 6]{Temkin15} and \cite{BakerPayneRabinoff14}). We prove an analogue of this result for the normalized space $T_{X,Z}$. After showing how to construct formal log modifications of the pair $(X,Z)$ with prescribed exceptional divisors, we characterize among those modifications the ones that correspond to a log resolution of $(X,Z)$ by performing a careful study (analogous to \cite[2.2 and 2.3]{BosLut85} and \cite[4.3.1]{Ber90}) of the fibers of the map sending every semivaluation to its center on the modification. Our main source of inspiration in developing this approach was Ducros's work \cite{Duc}. The strategy described above leads to two characterizations in terms of the local structure of $T_{X,Z}$ of the \defi{essential} and \defi{log essential valuations} on $(X,Z)$, i.e. those valuations whose center on every resolution (respectively log resolution) of $(X,Z)$ is a divisor. Whenever $k$ is the field of complex numbers and $Z$ is the singular locus of $X$, this is related to a famous conjecture of Nash from the nineteen-seventies (but published only in 1995 in \cite{Nash95}) involving the \emph{arc space} $X_\infty$ of a complex variety $X$. Nash constructed an injective map from the set of irreducible components of the subspace of $X_\infty$ consisting of the arcs centered in the singular locus of $X$ to the set of essential valuations on $X$, and asked whether this map is surjective. While this is known to be false if $\mathrm{dim}(X)\geq3$ (see \cite{deFernex}), for complex surfaces a proof was given by Fern\'andez de Bobadilla and Pe Pereira in \cite{deBobadillaPereira12}. More recently de Fernex and Docampo \cite{deFernexDocampo14} proved that in arbitrary dimension every valuation that is terminal with respect to the minimal model program over $X$ is in the image of the Nash map, deducing a new proof of de Bobadilla--Pereira's theorem. The class of log essential valuations can be larger than the set of Nash's essential valuations, since in some cases the exceptional locus of the minimal resolution of $X$ may not be a divisor with normal crossings. However, for many classes of singularities (e.g. rational singularities) these two notions coincide. We now give a short overview of the content of the paper. In Section~\ref{section_preliminaries} we recall some basic constructions of the theories of formal schemes and Berkovich spaces. In Section~\ref{section_3.1} we define the normalized space of a special formal $k$-scheme, while in Section~\ref{section_3.2} we prove the structure theorem of normalized spaces and deduce some interesting consequences. In Section~\ref{section_3.3} we define affinoid domains in a normalized space, and show that this notion is independent of the choice of a $k((t))$-analytic structure. This leads to the normalized version of Raynaud's theorem in Section~\ref{section_3.4}. We then move to the study of pairs $(X,Z)$, where $X$ is a $k$-surface and $Z$ is a closed subvariety of $X$ containing its singular locus. Section~\ref{section_4.1} contains the correspondence theorem between formal modifications of $(X,Z)$ and vertex sets. In Section~\ref{section_4.2} we study discs and annuli in $T_{X,Z}$; they are used in Section~\ref{section_4.3}, where we describe the formal fibers of the specialization map. In Section~\ref{section_4.4} we show how these techniques lead to the characterization of log essential and essential valuations. Several examples have been given for the reader who might want to quickly reach a basic understanding of the applications of normalized spaces to the study of surface singularities, without spending much time learning formal and Berkovich geometry. This reader should pay attention to the examples \ref{analytification_functor}, \ref{R: interpretation algebraic case}, \ref{example_explicit_normalization}, \ref{analytic_structure_valuative_tree}, and might benefit from reading the short note \cite{Fantini14}, where some of the results of this paper were announced. \subsection{Acknowledgments} The results of this paper were part of my PhD thesis at KU Leuven. I am very thankful to my advisor, Johannes Nicaise. I am also grateful to Nero Budur, Antoine Ducros, Charles Favre, Mircea Musta{\c{t}}{\u{a}}, Sam Payne, C\'edric P\'epin, Michael Temkin, Amaury Thuillier, and Wim Veys, for helpful discussions and comments. I am extremely grateful to an anonymous referee for his/her thorough reading of the manuscript and numerous helpful comments. I acknowledge the support of the Fund for Scientific Research - Flanders (grant G.0415.10) and of the European Research Council (Starting Grant project ``Nonarcomp'' no.307856). \section{Special formal schemes and their Berkovich spaces}\label{section_preliminaries} In the section we recall the notions of special formal schemes and the associated Berkovich spaces. For a detailed study of noetherian formal schemes we refer the reader to \cite{Illusie} or \cite{Bosch14}; a quick introduction is \cite{Nicaise08}. Special formal schemes are treated for example in \cite{deJ95} and \cite{Ber96}. \pa{ Let $R$ be a complete discrete valuation ring, $K$ the fraction field of $R$ and $k$ its residue field. By definition we allow $R$ to be a trivially valued field $k$. A \defi{formal $R$-scheme} is a noetherian formal scheme endowed with a (not necessarily adic) morphism of noetherian formal schemes to $\Spf R$. Recall that a morphism of noetherian formal schemes $f:\mathscr Y\to\mathscr X$ is said to be \defi{adic} if $f^({\mathcal J}){\mathcal O}_\mathscr Y$ is an ideal of definition of $\mathscr Y$ for some (and thus for all) ideal of definition ${\mathcal J}$ of $\mathscr X$. } \pa{ A topological $R$-algebra $A$ is said to be a \defi{special $R$-algebra} if it is a noetherian adic ring and the quotient $A/J$ is a finitely generated $R$-algebra for some ideal of definition $J$ of $A$. A formal $R$-scheme $\mathscr X$ is said to be a \defi{special formal $R$-scheme} if it is separated and locally isomorphic to the formal spectrum of a special $R$-algebra. In particular the reduction $\mathscr X_0$ of $\mathscr X$ is a reduced and separated scheme locally of finite type over $k$. Observe that $\mathscr X_0$ is generally different the special fiber of $\mathscr X$, which is by definition the special formal $k$-scheme $\mathscr X_s=\mathscr X\times_R k$. } \pa{ By \cite[1.2]{Ber96}, special $R$-algebras are exactly the adic $R$-algebras of the form \[ R\{X_1,\ldots,X_n\}\lbrack\lbrack Y_1,\ldots,Y_m \rbrack\rbrack/I\cong R[[Y_1,\ldots,Y_m]]\{X_1,\ldots,X_n\}/I, \] with ideal of definition generated by an ideal of definition of $R$ and by the $Y_i$'s. Recall that if $A$ is a $I$-adic topological ring, then $A\{X_1,\ldots,X_n\} \coloneqq \varprojlim_{\ell\geq1}\big(A/I^\ell\big)[X_1,\ldots,X_n]$ is the algebra of convergent power series over $A$ in the variables $(X_1,\ldots,X_n)$. Since every $R$-algebra topologically of finite type is special (we can take $m=0$ above), every formal $R$-scheme of finite type is a special formal $R$-scheme. On the other hand, a special formal $R$-scheme is of finite type if and only if its structure morphism to $\Spf(R)$ is adic. } \begin{ex} When working over a trivially valued field $k$, we have an isomorphism of $k$-algebras $k\{X\}\lbrack\lbrack Y \rbrack\rbrack \cong k \lbrack X\rbrack \lbrack\lbrack Y \rbrack\rbrack$. The latter is not isomorphic to $k[[Y]][X]$ but to the bigger $k$-algebra $k[[Y]]\{X\}$, which consists of the $Y$-adically convergent power series in $X$ with coefficients in $k[[Y]]$. \end{ex} \Pa{Example: the algebraic case}{\label{example_algebraic} If $X$ is a separated $R$-scheme locally of finite type and $Z$ is a subscheme of the special fiber $X\otimes_R k$ of $X$, then the formal completion $\mathscr X=\widehat{X/Z}$ of $X$ along $Z$ is a special formal $R$-scheme. In this case, $\mathscr X_0=Z_{\mathrm{red}}$. For example, if $X= \mathbb A^2_R =\Spec\big(R[X_1,X_2]\big)$ and $Z$ is the origin of the special fiber of $X$, then $\widehat{X/Z}\cong\Spf\big(R\lbrack\lbrack X_1,X_2\rbrack\rbrack\big)$; the special fiber of $\widehat{X/Z}$ is $\Spf k[[X_1,X_2]]$ and its reduction is $\Spec k$. Similarly, the formal completion of a special formal $k$-scheme along a closed subscheme of its special fiber is again a special formal $k$-scheme. } \pa{ All special $R$-algebras are excellent. This follows from \cite[7]{Valabrega75} when the characteristic of $K$ is positive and from \cite[9]{Valabrega76} when it is zero. A special formal $R$-scheme $\mathscr X$ is said to be \defi{normal} if it can be covered by affine subschemes $\Spf(A)$ with $A$ normal. Since the rings $A$ are excellent, this is equivalent to the normality of all completed local rings of $\mathscr X$. } \pa{ If $A$ is a special $k$-algebra and $T$ is an element of an ideal of definition of $A$, then $T$ is topologically nilpotent and therefore it induces a homomorphism $k[[t]]\to A$ that canonically makes $A$ into a special $k[[t]]$-algebra. Conversely, any special $k[[t]]$-algebra is canonically a special $k$-algebra. We will sometimes denote a special formal $k[[t]]$-scheme by $\mathscr X_t$; and $\mathscr X$ will then denote $\mathscr X_t$ seen as a special formal $k$-scheme. } \pa{ Let $\mathscr X$ be a noetherian formal scheme with largest ideal of definition ${\mathcal J}$ and let ${\mathcal I}$ be a coherent ideal sheaf on $\mathscr X$. The \defi{formal blowup of $\mathscr X$ along ${\mathcal I}$} is the $\mathscr X$-formal scheme $$ \mathscr X':=\lim_{\stackrel{\longrightarrow}{n\geq 1}}\mathrm{Proj}\left(\oplus_{d=0}^{\infty}\mathcal{I}^d\otimes_{\mathcal{O}_{\mathscr X}}(\mathcal{O}_{\mathscr X}/\mathcal{J}^n)\right). $$ We call the closed formal subscheme of $\mathscr X$ defined by ${\mathcal I}$ the \defi{center} of the blowup. The formal blowup $\mathscr X'\to\mathscr X$ of $\mathscr X$ along ${\mathcal I}$ is characterized by the following universal property (see \cite[8.2.9]{Bosch14}): $\mathscr X'$ is a noetherian formal scheme such that the ideal $f^{-1}{\mathcal I}\mathcal O_{\mathscr X'}$ is invertible on $\mathscr X'$, and every morphism of noetherian formal schemes $\mathscr Y\to\mathscr X$ such that $f^{-1}{\mathcal I}\mathcal O_{\mathscr Y}$ is invertible on $\mathscr Y$ factors uniquely through a morphism of noetherian formal schemes $\mathscr Y\to\mathscr X'$. We say that the blowup $\mathscr X'\to\mathscr X$ of $\mathscr X$ along ${\mathcal I}$ is \defi{admissible} if the ideal ${\mathcal I}$ is ${\mathcal J}$-open, i.e. contains a power of ${\mathcal J}$. } \begin{ex} Let $X$ be a noetherian scheme, let $Z$ be a closed subscheme of $X$ defined by a coherent ideal sheaf ${\mathcal J}$ and denote by $\mathscr X=\widehat{X/Z}$ the formal completion of $X$ along $Z$. Let ${\mathcal I}$ be a ${\mathcal J}$-open coherent ideal sheaf on $X$, and $\hat{\mathcal I}$ the induced ideal sheaf on $\mathscr X$. Then the formal blowup of $\mathscr X$ along $\hat{\mathcal I}$ is isomorphic to the formal completion along $f^{\mathcal J}\mathcal O_{\Bl_{\mathcal I}(X)}$ of the blowup $\Bl_{\mathcal I}(X)$ of $X$ along ${\mathcal I}$, where $f:\Bl_{\mathcal I}(X)\to X$ is the blowup. This is \cite[2.16.(5)]{Nicaise09}. \end{ex} \pa{\label{basic properties blowup} Admissible formal blowups share many properties with blowups of ordinary schemes. In particular, the following facts are proved as for schemes: \begin{enumerate}[ref=\ref{basic properties blowup}.\roman{enumi}] \item \label{blowup: composition} a composition of admissible blowups is an admissible blowup (\cite[8.2.11]{Bosch14}); \item \label{blowup: domination} two admissible blowups can be dominated by a third one (\cite[8.2.16]{Bosch14} and the previous point); \item \label{blowup: extension} an admissible blowup of an open formal subscheme of $\mathscr X$ can be extended to an admissible blowup of $\mathscr X$ (\cite[8.2.13]{Bosch14}). \end{enumerate} } \pa{ An admissible blowup of a special formal $R$-scheme is a special formal $R$-scheme by~\cite[2.17]{Nicaise09}. Similarly, an admissible blowup of a formal $R$-scheme of finite type is of finite type. } \Pa{Berkovich theory}{ Berkovich's approach to non-archimedean analytic geometry was developed in \cite{Ber90} and \cite{Ber93}; a good introduction to the theory is \cite{Temkin15}. Since the general definition of a $K$-analytic space is quite technical, we will content ourselves with listing some properties of $K$-analytic spaces and introducing via examples those spaces that appear in the rest of the paper. In particular, we will recall how to associate a $K$-analytic space $\mathscr X^\beth$ to a special formal $R$-scheme $\mathscr X$ and define the specialization map. This construction was introduced for rigid spaces by Berthelot in \cite{Berthelot} (see also \cite[\S7]{deJ95} for a detailed exposition), while in the context of Berkovich spaces it was studied in \cite{Berkovich94} and \cite{Ber96}. If $\mathscr X$ is special over a trivially valued field $k$, we will also study a subspace of $\mathscr X^\beth$, introduced by Thuillier in \cite{Thu07}, that behaves more like a generic fiber for $\mathscr X$ (see also \cite{Ben-BassatTemkin2013}). } \pa{ A \emph{$K$-analytic space} is a locally compact and locally path connected topological space $X$ with the following additional structures: \begin{enumerate} \item For every point $x$ of $X$, a completed valued field extension $\mathscr H(x)$ of $K$, called the \defi{completed residue field} of $X$ at $x$. \item A $G$-topology on $X$, whose $G$-admissible subspaces are called \defi{analytic domains} of $X$. \item A local $G$-sheaf in $K$-algebras ${\mathcal O}_X$ on $X$, the \defi{sheaf of analytic functions}. \end{enumerate} A $G$-topology is a simple kind of Grothendieck topology, we refer to \cite[\S9.1]{BGR} for the definitions. The $G$-topology is finer than the usual topology of $X$, i.e. every open subset of $X$ is an analytic domain and every open cover of an analytic domain is a $G$-cover. If $V$ is an analytic domain of $X$, $x\in V$, $f\in{\mathcal O}_X(V)$, then $f$ can be evaluated in $x$, yielding an element $f(x)$ of $\mathscr H(x)$. Therefore, also $\|f(x)\|\in\mathbb{R}_+$ makes sense. We refer to \cite{Temkin15} for the general definition of the category $(An_K)$. } \begin{ex}\label{analytification_functor} A fundamental example of $K$-analytic space is the analytification $X^{\mathrm{an}}$ of a $K$-scheme of finite type $X$. As a topological space, \[ X^{\mathrm{an}}=\big\{(\xi_x,\|\cdot\|_x)\big\|\xi_x\in X, \|\cdot\|_x\text{ abs. value on }\kappa(\xi_x)\text{ extending the one of }K\big\}, \] with the weakest topology such that the map $\rho\colon X^{\mathrm{an}}\to X$ sending a point $x=(\xi_x,\|\cdot\|_x)$ to $\xi_x$ is continuous, and for each open $U$ of $X$ and each element $f$ of $\mathcal O_X(U)$ the induced map $\rho^{-1}(U)\to\mathbb{R}$ sending $x$ to $\|f(x)\|=\|f\|_x$ is continuous. The field $\rescompl{x}$ is the completion of $\kappa(\xi_x)$ with respect to $\|\cdot\|_x$. A morphism of $K$-schemes of finite type $Y\to X$ induces a map $Y^{\mathrm{an}}\to X^{\mathrm{an}}$. The space $X^{\mathrm{an}}$ is connected (respectively Hausdorff, compact) if and only if $X$ is connected (respectively separated, proper). Moreover, whenever $X$ is proper then GAGA type theorems hold (see \cite[\S3.4, \S3.5]{Ber90}). The sheaf $\mathcal O_{X^{\mathrm{an}}}$ can be thought of as a completion of the sheaf $\mathcal O_{X}$ with respect to some seminorm. For example, the analytic functions on an open $U$ of $\mathbb A^{n,\mathrm{an}}_K$ are the maps $f:U\to\coprod_{x\in U}\rescompl{x}$ that are locally uniform limits of rational functions without poles. More generally, every locally closed subspace of $X^{\mathrm{an}}$ can be canonically given the structure of a reduced $K$-analytic space. \end{ex} \pa{ The $G$-topology of a $K$-analytic space $X$ is constructed from an important class of distinguished compact analytic domains of $X$, that of \defi{affinoid domains}. Recall that an affinoid $K$-algebra is a quotient of a Banach $K$-algebra of the form $K\big\{r_1^{-1}T_1,\ldots,r_n^{-1}T_n\big\}= \big\{\sum_{\underline i\in{\mathbb{N}}^n} a_{\underline i}\underline T^{\underline i} \,\big\|\, a_{\underline i}\in K, \, \lim_{\|\underline i\|\to\infty}\|a_{\underline i}\|\underline r^{\underline i}=0\big\}$ (where $r_i>0$, and the Banach norm is given by $\|\|\sum a_{\underline i}\underline T^{\underline i}\|\|=\max{\|a_{\underline i}\|\underline r^{\underline i}}$), and that the affinoid spectrum $\m{{\mathcal A}}$ of ${\mathcal A}$ is the set of bounded multiplicative seminorms on ${\mathcal A}$, with the topology of pointwise convergence. Affinoid domains are then some distinguished subsets of $X$ homeomorpic to the affinoid spectrum $\m{{\mathcal A}}$ of an affinoid $K$-algebra ${\mathcal A}$, and an affinoid domain is said to be strict if we can take all $r_i$ above to be equal to 1. If $V\cong\m{{\mathcal A}}$ is an affinoid domain of $X$, then ${\mathcal O}_X(V)\cong{\mathcal A}$. A subset $U$ of $X$ is then an analytic domain if and only of for every element $u$ of $U$ there exist finitely many affinoid domains $U_1,\ldots,U_n$ of $X$ contained in $U$ and such that $u\in\cap_iU_i$ and $\cup_iU_i$ is a neighborhood of $u$ in $U$. In particular, any analytic domain of $X$ is $G$-covered by the affinoid domains it contains. } \begin{ex}\label{example_berkovich_affinoid} The \defi{analytic affine $n$-space} $\mathbb A^{n,\mathrm{an}}_K=\Spec(K[T_1,\ldots,T_n])^{\mathrm{an}}$ can be written as the union of the closed polydiscs $D^n(\underline r)=\{\|T_i\|\leq r_i\text{ for all }i\}$ of center $0$ and polyradius $r=(r_1,\ldots,r_n)\in(\mathbb{R}_+^)^n$. The polydisc $D^n(\underline r)$ is an affinoid domain, with associated affinoid $K$-algebra $K\big\{r_1^{-1}T_1,\ldots,r_n^{-1}T_n\big\}$. Explicit descriptions of the topological spaces underlying $\mathbb A^{1,\mathrm{an}}_K$, $\mathbb A^{1,\mathrm{an}}_k$ and $\mathbb A^{2,\mathrm{an}}_k$ can be found in \cite{Payne15}. \end{ex} We will now see how to associate a $K$-analytic space to a special formal $R$-scheme. \pa{\label{definition_beth_space} If $\mathscr X$ is an affine special formal $R$-scheme of the form $$\mathscr X=\Spf\left(\frac{R\{X_1,\ldots,X_n\}\lbrack\lbrack Y_1,\ldots,Y_m\rbrack\rbrack}{(f_1,\ldots,f_r)}\right),$$ then the associated Berkovich space is $$\mathscr X^\beth = V(f_1,\ldots,f_r)\subset D^n\times_K D^{m}_-\subset\mathbb A^{n+m,\mathrm{an}}_K,$$ where $D^n=D^n(\underline 1)$ is the $n$-dimensional closed unit disc in $\mathbb A^{n,\mathrm{an}}_K$ (as in Example~\ref{example_berkovich_affinoid}), $D^{m}_-=\big\{x\in\mathbb A^{m,\mathrm{an}}_K\;\big\|\;\|T_i(x)\|<1\text{ for all }i\big\}$ is the $m$-dimensional open unit disc in $\mathbb A^{m,\mathrm{an}}_K$ and $V(f_1,\ldots,f_r)$ denotes the zero locus of the $f_i$. This construction is functorial, sending an open immersion to an embedding of a closed subdomain, therefore it globalizes to general special formal $R$-schemes by gluing. If $\mathscr X$ is of finite type over $R$, this construction coincides with the one by Raynaud (see \cite{Raynaud74} or \cite[\S4]{BosLut93}) and $\mathscr X^\beth$ is compact. } \begin{ex} If $\mathscr X=\Spf\big(R\{T\}\big)$, then $\mathscr X^\beth$ is the closed unit disc in $\mathbb A^{1,\mathrm{an}}_K$. If $\mathscr X=\Spf\big(R[[T]]\big)$, then $\mathscr X^\beth$ is the open unit disc in $\mathbb A^{1,\mathrm{an}}_K$. Note that if $K=k$ is trivially valued, the latter is homeomorphic to the interval $\left[0,1\right[$. \end{ex} \pa{\label{example_affine_beth_space} If $\mathscr X=\Spf\big(R\{X_1,\ldots,X_n\}\lbrack\lbrack Y_1,\ldots,Y_m\rbrack\rbrack/(f_1,\ldots,f_r)\big)$ is an affine special formal $R$-scheme, then its associated Berkovich space $\mathscr X^\beth$ is the increasing union $\mathscr X^\beth=\bigcup_{0<\varepsilon<1}W_\varepsilon$, where $W_\varepsilon$ is the subspace of $\mathscr X^\beth$ cut out by $\|Y_i\|\leq1-\varepsilon$. Moreover, $W_\varepsilon$ is an affinoid domain of $\mathscr X^\beth$, with associated affinoid \mbox{$K$-algebra} ${K\big\{X_1,\ldots,X_n,(1-\varepsilon)^{-1}Y_1,\ldots,(1-\varepsilon)^{-1}Y_m\big\}}/{(f_1,\ldots,f_r)}.$ } \pa{\label{canonical_injection_special_algebras} Let $A$ be a special $R$-algebra and set $X=(\Spf A)^\beth$. Then the canonical homomorphism $A \otimes_R K \to {\mathcal O}_X(X)$ is injective. Indeed, let $f$ be an element of $A \otimes_R K$ which vanishes in ${\mathcal O}_X(X)$, and let ${\mathfrak M}$ be a maximal ideal of $A \otimes_R K$. By \cite[Lemma~7.1.9]{deJ95} ${\mathfrak M}$ corresponds to a point $x$ of $X$, and the image $f(x)$ of $f$ in the completed local ring of $X$ at $x$ coincides with the image $\alpha(f_{\mathfrak M})$ via the completion morphism $\alpha\colon (A \otimes_R K)_{\mathfrak M} \to (A \otimes_R K)_{\mathfrak M}^{\phantom{{\mathfrak M}}\wedge}$ of the image $f_{\mathfrak M}$ of $f$ in the localization of $A \otimes_R K$ at ${\mathfrak M}$. It follows that $\alpha(f_{\mathfrak M})=0$, hence $f_{\mathfrak M}=0$ because $(A \otimes_R K)_{\mathfrak M}$, being a localization of the commutative noetherian ring $A$, is a local noetherian ring. Since this is true for every maximal ideal of $A \otimes_R K$, it follows that $f=0$. } \pa{\label{definition_specialization_map} There is a natural \defi{specialization map} $\Sp_\mathscr X:\mathscr X^\beth\longrightarrow\mathscr X_0$ that is defined as follows. If $\mathscr X=\Spf(A)$ is affine, a point of $\mathscr X^\beth$ gives rise to a continuous character $\chi_x\colon A\to\rescompl{x}^\circ$, where we have denoted by $\rescompl{x}^\circ$ the valuation ring of $\rescompl{x}$, which in turn gives rise to a character $\widetilde\chi_x\colon A/{I}\to\widetilde{\rescompl{x}}$, where $I$ is the largest ideal of definition of $A$. The kernel of $\widetilde\chi_x$ is by definition the point $\Sp_\mathscr X(x)\in\mathscr X_0=\Spec(A/I)$. If ${\mathcal U}$ is an open formal subscheme of $\mathscr X$ then ${\mathcal U}^\beth\cong\Sp_\mathscr X^{-1}({\mathcal U}_0)$, and the restriction of $\Sp_\mathscr X$ to the latter coincides with $\Sp_{\mathcal U}$. Therefore, the definition we gave extends to general special formal $R$-schemes. The map $\Sp_\mathscr X$ is anticontinuous, i.e. the inverse image of an open subset of $\mathscr X$ is closed. For example, if $\mathscr X=\Spf(A)$ and $Z$ is a closed subset of $\mathscr X$ defined by an ideal $(f_1,\ldots,f_r)$, then \[ \Sp_\mathscr X^{-1}(Z)=\big\{ x\in\mathscr X^\beth \,\big\|\, \|f_i(x)\|<1 \text{ for all }i=1,\ldots,r \big\}. \] As in \cite[0.2.6]{Berthelot}, $\Sp_\mathscr X$ can be viewed as a morphism of locally ringed sites $\Sp_\mathscr X:\mathscr X^\beth\to\mathscr X$. Note that this map is often called also \defi{reduction map}. } \pa{\label{lemma fiber reduction} If $f:\mathscr Y\to\mathscr X$ is a morphism of special formal $k$-schemes, then $\Sp_{\mathscr X}\circ f^\beth=f\circ\Sp_\mathscr Y$. Moreover, if $Z$ is a subscheme of $\mathscr X_0$, then by \cite[0.2.7]{Berthelot} (or \cite[1.3]{Ber96}) the canonical morphism of formal $k$-schemes $\widehat{\mathscr X/Z}\to\mathscr X$ induces an isomorphism of $k$-analytic spaces $(\widehat{\mathscr X/Z})^\beth\cong\Sp_\mathscr X^{-1}(Z)$. } \pa{ Assume from now on that we are working over a trivially valued field $k$, and let $\mathscr X$ be a special formal $k$-scheme. The closed immersion $\mathscr X_0\to\mathscr X$ gives rise to an immersion $\big(\mathscr X_0\big)^\beth \to \mathscr X^\beth$, and we define the \defi{punctured Berkovich space} ${\X^}$ of $\mathscr X$ as the subspace \[ {\X^}=\mathscr X^\beth\setminus\mathscr X_0^\beth \] of $\mathscr X^\beth$. It's a $k$-analytic space, introduced by Thuillier in \cite[1.7]{Thu07} (where it is called the generic fiber of $\mathscr X$). Any adic morphism of special formal $k$-schemes $f:\mathscr Y\to\mathscr X$ induces a morphism $f^\colon\mathscr Y^\to{\X^}$, since we have $\big(f^\beth\big)^{-1}\big(\mathscr X_0^\beth\big)=\mathscr Y_0^\beth$. We will denote again by $\Sp_\mathscr X\colon{\X^}\to\mathscr X$ the restriction of the specialization map to ${\X^}$. } \Pa{Examples}{ If $\mathscr X$ is of finite type over $k$, then $\mathscr X_0=\mathscr X$, and therefore ${\X^}$ is empty. If $\mathscr X=\Spf\big(k\lbrack\lbrack t\rbrack\rbrack\big)$, then ${\X^}$ is the punctured open unit disc in $\mathbb A^{1,\mathrm{an}}_k$, which is homeomorphic to the open interval $\left]0,1\right[$. } \pa{\label{example_affine_punctured_space} If $\mathscr X=\Spf\big(k\{X_1,\ldots,X_n\}\lbrack\lbrack Y_1,\ldots,Y_m\rbrack\rbrack/(f_1,\ldots,f_r)\big)$ is an affine special formal $k$-scheme, we can describe ${\X^}$ along the lines of~\ref{example_affine_beth_space}. The complement in ${\X^}$ of the zero locus $V(Y_i)$ of one of the $Y_i$'s is the increasing union ${\X^}\setminus V(Y_i)=\bigcup_{0<\varepsilon\leq1/2}W_{i,\varepsilon}$, where $W_{i,\varepsilon}$ is the subspace of $\mathscr X^\beth$ cut out by the inequalities $\|Y_j\|\leq1-\varepsilon$ for every $j$ and $\varepsilon\leq\|Y_i\|$. The subspace $W_{i,\varepsilon}$ is an affinoid domain of ${\X^}$, with associated affinoid $k$-algebra \[ \frac{k\big\{X_1,\ldots,X_n\big\}\big\{\varepsilon Y_i^{-1},(1-\varepsilon)^{-1}Y_1,(1-\varepsilon)^{-1}Y_2,\ldots,(1-\varepsilon)^{-1}Y_m\big\}}{(f_1,\ldots,f_r)}. \] We then have ${\X^}=\bigcup_{i=1}^m\big(\bigcup_{0<\varepsilon\leq1/2}W_{i,\varepsilon}\big)$. Moreover, if we denote by $W_{i,\varepsilon}^\circ$ the open subspace of $\mathscr X^\beth$ cut out by $\|Y_j\|<1-\varepsilon$ for every $j$ and $\varepsilon>\|Y_i\|$, then the family $\big\{W_{i,\varepsilon}^\circ\big\}_{i=1,\ldots,m,\,0<\varepsilon\leq1/2}$ is an open cover of ${\X^}$. Note that if $t$ is a nonzero element of an ideal of definition of $\mathscr X$, we can analogously write ${\X^}\setminus V(t)=\mathscr X^\beth\setminus V(t)$ as an increasing union of affinoid domains $\{W_{t,\varepsilon}\}_\varepsilon$. } \Pa{Fundamental example}{\label{R: interpretation algebraic case} We now discuss in detail what happens in the algebraic case of \ref{example_algebraic}, when working over $k$. Let $X$ be a separated $k$-scheme of finite type. Then with every point $x$ of $X^{\mathrm{an}}$ we can associate a morphism $\varphi_x\colon\Spec\big(\rescompl{x}\big)\to X$, which sits in the following commutative diagram: \begin{displaymath} \xymatrix@R=1.5pc@C=3pc@M=3pt@L=3pt{ \Spec\big(\rescompl{x}\big) \ar[d] \ar[r]^(.61){\varphi_x} & X\ar[d] \\ \Spec\big(\rescompl{x}^\circ\big) \ar[r] & \Spec(k) } \end{displaymath} where $\rescompl{x}^\circ$ is the valuation ring of $\rescompl{x}$. We say that $x$ \defi{has center} on $X$ if we can fit in the diagram above a morphism $\overline{\varphi_x}\colon\Spec\big(\rescompl{x}^\circ\big)\longrightarrow X$ that extends $\varphi_x$. By the valuative criterion of separatedness if such an extension exists then it is unique. The \defi{center} of $x$ on $X$ is then by definition the image in $X$ of the closed point of $\Spec(\rescompl{x}^\circ)$ via $\overline{\varphi_x}$. We denote this point by $\Sp_X(x)$, and we write $X^\beth$ for the subset of $X^{\mathrm{an}}$ consisting of the points that have center on $X$. The space $X^\beth$ is a compact analytic domain of $X^\mathrm{an}$ which can be thought of as a bounded version of $X^{\mathrm{an}}$, and it coincides with the space defined in \ref{definition_beth_space} if $X$ is seen as a formal $k$-scheme of finite type. For example, $\big(\mathbb A^n_k\big)^\beth$ is the closed unit polydisc in $\mathbb A^{n,\mathrm{an}}_k$. If $X$ is proper, then by the valuative criterion of properness we have $X^\beth\cong X^{\mathrm{an}}$. Now let $Z$ be a closed subvariety of $X$ and set $\mathscr X=\widehat{X/Z}$. Then we have $\mathscr X^\beth=\Sp_X^{-1}(Z)$. Moreover, the restriction of $\Sp_X$ to $\mathscr X^\beth$ is the specialization map $\Sp_\mathscr X$ defined in \ref{definition_specialization_map}. The space $\mathscr X^\beth$ can be thought of as an (infinitesimal) \defi{tubular neighborhood} of $Z^\beth$ in $X^\beth$. Note that $Z^{\mathrm{an}}$ is canonically isomorphic to the subspace $\rho^{-1}(Z)$ of $X^{\mathrm{an}}$, where $\rho\colon X^{\mathrm{an}}\to X$ is the structure morphism defined in Example~\ref{analytification_functor}. Since $Z$ is closed in $X$, by the valuative criterion of properness we have $Z^\beth=Z^{\mathrm{an}}\cap X^\beth\subset X^{\mathrm{an}}$. Therefore, we have ${\X^}=\mathscr X^\beth\setminus Z^\beth = \Sp_X^{-1}(Z)\setminus\rho^{-1}(Z)$. In words, ${\X^}$ is the set of semivaluations on $X$ that have center in $Z$ but are not semivaluations on $Z$. It can be thought of as a \defi{punctured tubular neighborhood} (or \defi{link}) of $Z^\beth$ in $X^\beth$. } \pa{\label{invariance_generic_fiber} Let $f:\mathscr Y\to\mathscr X$ be an admissible blowup of special formal $k$-schemes. Then $f$ induces an isomorphism of punctured spaces $f^\colon\mathscr Y^\stackrel{\sim}{\longrightarrow}{\X^}$. In the algebraic case of Example~\ref{R: interpretation algebraic case} this follows from the valuative criterion of properness (see \cite[1.11]{Thu07}); the general case is \cite[4.5.1]{Ben-BassatTemkin2013}. } We conclude the section by giving definitions of admissibility for special formal $k$-schemes and for special $k$-algebras. \pa{\label{definition_admissible_scheme} Let $\mathscr X$ be a special formal $k$-scheme. We say that $\mathscr X$ is \defi{admissible} if the canonical morphism of sheaves $ \mathcal O_\mathscr X\to (\Sp_\mathscr X)_\mathcal O_{\X^} $ is a monomorphism. This is equivalent to the fact that $\mathcal O_\mathscr X(U)\to\mathcal O_{\X^}\big(sp_\mathscr X^{-1}(U)\big)$ is injective for every open $U$ of $\mathscr X$, so the property of being admissible can be thought of as having schematically dense generic fiber. } \pa{\label{definition_admissible_algebra} If $A$ is a special $k$-algebra and $J$ is the largest ideal of definition of $A$, we define the \defi{torsion ideal} of $A$ as $A_{\tors}=\big\{a\in A \,\,\,\big\|\,\,\,a\in A_{t-\tors}\,\, \forall t\in J\big\}$, where $A_{t-\tors}$ denotes the $t$-torsion of $A$; then $A_{\tors}$ is an ideal of $A$. We say that $A$ is \defi{admissible} if $A_{\tors}=0$. } \Pa{Remarks}{\label{admissible_equivalence} If $A$ is a nonzero admissible special $k$-algebra, then the largest ideal of definition of $A$ is nonzero, so that $A$ is not topologically of finite type over $k$. If moreover $A$ is a domain, the converse holds as well: $A$ is admissible if and only if it is not topologically of finite type over $k$. If $\{g_1,\ldots,g_s\}$ is a set of generators of $J$, then $A_{\tors}=\cap_{i=1}^sA_{g_i-\tors}$, hence $A$ is admissible if and only if the canonical morphism $A\to\prod_{i=1}^s A[g_i^{-1}]$ is injective. As is done in the finite type case in \cite[7.3.13]{Bosch14}, one can use this injectivity to deduce that, if $A$ is an admissible special $k$-algebra, then for every element $f$ of $A$ the complete localization $A\big\{f^{-1}\big\}$ is admissible. If $A$ is an algebra topologically of finite type over $k[[t]]$, seen as a special $k$-algebra, then $A$ is admissible if and only if it has no $t$-torsion. This shows that our definition of admissible algebra coincides with the usual one in this case. We will prove in Proposition~$\ref{admissible}$ that an affine special formal $k$-scheme $\Spf A$ is admissible if and only if $A$ is an admissible special $k$-algebra. It will then be clear that our definition is analogous to the usual one for formal $R$-schemes of finite type. } \section{Normalized Berkovich spaces of special formal \texorpdfstring{$k$-} -schemes} \label{section_3.1} In this section we start by defining an $\mathbb R_{>0}$-action on the punctured Berkovich space ${\X^}$ of a special formal $k$-scheme $\mathscr X$. We then introduce our primary object of study, the Normalized Berkovich space ${T_\X}$ of $\mathscr X$, as the quotient of ${\X^}$ by this action. \pa{\label{ex_action_closed_disc} One important feature of Berkovich analytic spaces is that they distinguish between equivalent but not equal seminorms. For example, if $k$ is a trivially valued field, $\gamma$ is an element of $\mathbb R_{>0}$, and $\|\cdot\|_x$ is an element of the closed unit disc $\Spf(k\{T\})^\beth=\m{k\{T\}}$ in the analytic affine line $\mathbb A^{1,\mathrm{an}}_k$, then also $\|\cdot\|_x^\gamma$ is an element of $\m{k\{T\}}$. Indeed, the Banach norm of $k\{T\}=k[T]$ is the $T$-adic one with $\|T\|=1$, so it is the trivial norm; it follows that the elements of $\m{k\{T\}}$ are the seminorms $\|\cdot\|_x$ on $k\{T\}$ satisfying $\|f\|_x\leq 1$ whenever $f\in k\{T\}$. Then $\|\cdot\|_x^\gamma$ is multiplicative, trivial on $k$, it satisfies both the ultrametric inequality and $\|f\|_x^\gamma\leq 1$ for $f\in k\{T\}\setminus\{0\}$. Similarly, if $\mathscr X=\Spf(k\lbrack\lbrack T\rbrack\rbrack)$ then $\mathscr X^{\beth}$ is the open unit disc $D_-$ in the analytic affine line $\mathbb A^{1,\mathrm{an}}_k$ and ${\X^}$ is the punctured open unit disc $D_-\setminus\{0\}$. The latter is homeomorphic to the open segment $\left]0,1\right[$, and under this identification $\mathbb R_{>0}$ acts freely on it by exponentiation. Observe that the fact that the absolute value of $k$ is trivial, and thus invariant under exponentiation by elements of $\mathbb R_{>0}$, is crucial. } \pa{ More generally, let $k$ be a trivially valued field and consider an affine special formal $k$-scheme $\mathscr X=\Spf\big(k\{X_1,\ldots,X_n\}\lbrack\lbrack Y_1,\ldots,Y_m\rbrack\rbrack/(f_1,\ldots,f_r)\big)$. It follows from the definition given in \ref{definition_beth_space} that, by seeing it as a subset of the analytic affine space $\mathbb A_k^{n+m,\mathrm{an}}$, the set $\mathscr X^\beth$ is the set of multiplicative seminorms $\|\cdot\|_x \colon k[X_1,\ldots,X_n] \lbrack\lbrack Y_1,\ldots,Y_m \rbrack\rbrack /(f_1,\ldots,f_r)$ which are trivial on $k$ and such that $\|X_i\|_x\leq1$ and $\|Y_j\|_x<1$ for every $i$ and $j$. Therefore, for every element $\gamma$ of $\mathbb R_{>0}$ the seminorm $\|\cdot\|_x^\gamma$ is itself an element of $\mathscr X^\beth$. Moreover, $\mathscr X_0^\beth$ is defined in $\mathscr X^\beth$ by the equalities $Y_1=\ldots=Y_m=0$, therefore the $\mathbb R_{>0}$-action restricts to an action on $\mathscr X^=\mathscr X^\beth\setminus \mathscr X_0^\beth$. } \pa{ If $\Spf B$ is a subscheme of the affine special formal scheme $\Spf A$, the induced map $(\Spf B)^\to(\Spf A)^$, being induced by the composition of a seminorm with the morphism $B\to A$, is equivariant with respect to the $\mathbb R_{>0}$-actions. This allows to extend the $\mathbb R_{>0}$ action to the punctured Berkovich space of a general special formal $k$-scheme $\mathscr X=\bigcup_i\Spf(A_i)$, by covering $(\Spf A_i)\cap(\Spf A_j)$ with affine subschemes. } \begin{rem}The $\mathbb R_{>0}$-action on ${\X^}$ is free (i.e. the orbits $\mathbb R_{>0}\cdot x$ are in bijection with $\mathbb R_{>0}$). Indeed, either $\mathbb R_{>0}\cdot x\cong \mathbb R_{>0}$ or $\mathbb R_{>0}\cdot x=\{x\}$; the latter is equivalent to $x$ being a trivial absolute value, but all trivial absolute values of $\mathscr X^{\beth}$ lie in $\mathscr X_0^{\mathrm{an}}$. \end{rem} \pa{ We deduce from the discussion above that the association $\mathscr X\mapsto{\X^}$ gives a functor from the category of special formal $k$-schemes with adic morphisms to the category of $k$-analytic spaces with a free $\mathbb R_{>0}$-action on the underlying topological space and equivariant analytic morphisms. } \begin{comment} \pa{\label{begin_section_local_structure} More precisely, following \ref{example_affine_punctured_space} we can write ${\X^}$ as the increasing union of the annuli $A_\varepsilon=\{\varepsilon\leq\|T\|\leq1-\varepsilon\}=\m{{\mathcal A}_\varepsilon}\subset\mathbb A^{1,\mathrm{an}}_k$ for $0<\varepsilon\leq 1/2$, where ${\mathcal A}_\varepsilon=k\{\varepsilon T^{-1},(1-\varepsilon)^{-1}T\}$. Given $\gamma$ in $\mathbb R_{>0}$ and an element $\|\cdot\|_x$ of ${\X^}$, a similar computation as in \ref{ex_action_closed_disc} shows that, for $\varepsilon$ small enough, $\|\cdot\|_x^\gamma$ is a bounded seminorm on ${\mathcal A}_\varepsilon$. This gives a well defined action of $\mathbb R_{>0}$ on ${\X^}$ since for $\varepsilon\geq\delta$ the inclusion $A_\varepsilon\hookrightarrow A_\delta$ comes from the identity map ${\mathcal A}_\delta \cong k[[T^{-1},T]]\to k[[T^{-1},T]]\cong {\mathcal A}_\varepsilon$, only the Banach norms change. } The definition of an action of $\mathbb R_{>0}$ as left composition with $\exp_\gamma$ can be extended to the punctured Berkovich space of any affine special formal $k$-scheme thanks to the following lemma. \begin{lem}\label{definition_action} Let $\mathscr X$ be an affine special formal $k$-scheme. Then there is a unique way to define an action of $\mathbb R_{>0}$ on the topological space underlying ${\X^}$ so that the following property is satisfied: if $V$ is an affinoid domain of ${\X^}$ with associated affinoid algebra ${\mathcal V}$, then $x\colon {\mathcal V}\to\mathbb R_{\geq0}$ is a point of $V$ and $\gamma$ is an element of $\mathbb R_{>0}$ such that the seminorm $x^\gamma$ is a point of $V$, then $\gamma\cdot x=x^\gamma$. \end{lem} \begin{proof} We begin by showing that for every $x\in{\X^}$ and $\gamma\in\mathbb R_{>0}$ there exists an affinoid domain $V\subset{\X^}$ such that both $x$ and $x^\gamma$ belong to $V$. Let $Y_1,\ldots,Y_m$ be a set of generators of an ideal of definition of $\mathscr X$. One of those elements does not vanish on $x$ and without loss of generality we can assume that this element is $Y_1$. In the notation of \ref{example_affine_punctured_space}, there exists $\varepsilon>0$ such that $x\in W_{1,\varepsilon}$, and by further shrinking $\varepsilon$ as before we can find a bigger affinoid domain $W_{1,\delta}$ containing both $x$ and $x^\gamma$, i.e. such that $x^\gamma$ is bounded on ${\mathcal W}_{1,\delta}$. If $V=\m{{\mathcal V}}$ is an affinoid domain of ${\X^}$ such that $x$ and $x^\gamma$ are bounded seminorms on ${\mathcal V}$, we write $\gamma\cdot_V x$ for the point of ${\X^}$ corresponding to the seminorm $x^\gamma$. We then have to show that the point $\gamma\cdot_V x$ does not depend on the choice of the affinoid domain $V$. Let us first assume that $V$ and $W$ are two such affinoid domains, with associated affinoid algebras ${\mathcal V}$ and ${\mathcal W}$ respectively, and that we have an inclusion $\iota\colon W\hookrightarrow V$. Denote by $\varphi:{\mathcal V}\to{\mathcal W}$ the corresponding morphism of affinoid algebras. Then the fact that $x$ belongs to both $V$ and $W$ amounts to the commutativity of the following diagram: \begin{displaymath} \xymatrix@C=2.5pc@R=1.5pc@M=3pt@L=3pt{ {\mathcal V} \ar[d]_\varphi \ar[r]^x & \mathbb R_{\geq0} \ar@{=}[d] \\ {\mathcal W} \ar[r]^x & \mathbb R_{\geq0} } \end{displaymath} By composing with $\exp_\gamma$ we get the following diagram, \begin{displaymath} \xymatrix@C=2.5pc@R=1.5pc@M=3pt@L=3pt{ {\mathcal V} \ar[d]_\varphi \ar@/^1.5pc/[rr]^{\gamma\cdot_V x} \ar[r]^x & \mathbb R_{\geq0} \ar[r]^{\exp_\gamma} & \mathbb R_{\geq0} \ar@{=}[d] \\ {\mathcal W} \ar[r]^x \ar@/_1.5pc/[rr]^{\gamma\cdot_W x} & \mathbb R_{\geq0} \ar[r]^{\exp_\gamma} & \mathbb R_{\geq0} } \end{displaymath} whose commutativity implies that $\iota(\gamma\cdot_W x)=\gamma\cdot_V x$, i.e. $\gamma\cdot_W x$ and $\gamma\cdot_V x$ are the same point of ${\X^}$. It is now enough to show that if $\delta$ and $V$ are as above, i.e. if both $\gamma\cdot_{W_{1,\delta}} x$ and $\gamma\cdot_V x$ can be defined, then there is an affinoid domain $W$ of ${\X^}$ that is contained both in $W_{1,\delta}$ and in $V$ and such that $\gamma\cdot_W x$ can be defined. Since $\mathscr X^\beth$ is covered by the open subspaces defined by the inequalities $\|Y_i\|<1-\varepsilon$, then $V$ being compact is contained in one of those and so it is an affinoid subdomain of the affinoid domain of $\mathscr X^\beth$ defined by the inequalities $\|Y_i\|\leq 1-\varepsilon$. By further shrinking $\delta$, we can assume that $W=V\cap W_{1,\delta}$ is the subspace of $V$ defined by $\|Y_1\|\geq\delta$. Therefore $W$ is an affinoid subdomain of $V$, with associated affinoid algebra ${\mathcal W}={\mathcal V}\{\delta Y_1^{-1}\}$. It remains to show that if we see $x$ as a bounded seminorm on ${\mathcal W}$, then $x^\gamma$ is again bounded on ${\mathcal W}$. A general element of ${\mathcal W}$ can be written as $a/Y_1^n$ for some $a$ in ${\mathcal V}$ and some $n\in\mathbb N$, and its norm $\\|a/Y_1^n\\|_{\mathcal W}$ is equal to $\\|a\\|_{\mathcal V}\delta^{-n}$, where $\\|a\\|_{\mathcal V}$ is the norm of $a$ in ${\mathcal V}$. We then have $x^\gamma(a/Y_1^n)=\frac{x(a)^\gamma}{x(Y_1)^{n\gamma}}\leq \frac{C\\|a\\|_{\mathcal V}}{\delta^n}$, where we used both the fact that $x^\gamma$ is bounded on ${\mathcal V}$ (with $C>0$ a constant such that $x(b)\leq C\\|b\\|_{\mathcal V}$ for every $b$ in ${\mathcal V}$) and the fact that $x^\gamma(Y_1)\geq\delta$ on ${\mathcal W}_{Y_1,\delta}$. This proves that $x^\gamma$ is bounded on ${\mathcal V}$ (with same constant $C$ realizing the bound), concluding the proof. \end{proof} This result allows us to extend the definition of the $\mathbb R_{>0}$-action to the Berkovich space of any special formal $k$-scheme. If $\mathscr X=\bigcup_i\Spf(A_i)$, by covering $(\Spf A_i)^\cap(\Spf A_j)^$ with affinoid domains we see that the $\mathbb R_{>0}$-actions on $(\Spf A_i)^$ and on $(\Spf A_j)^$ coincide on the intersection, and that the action we defined does not depend on a chosen presentation. We summarize the results obtained above in the following proposition. \begin{prop} Let $\mathscr X$ be a special formal scheme over $k$. Then there is a unique way to define an action of $\mathbb R_{>0}$ on the topological space underlying ${\X^}$ in such a way that it restricts on ${\mathcal U}^$ to the $\mathbb R_{>0}$-action described in Lemma~\ref{definition_action} whenever ${\mathcal U}$ is an affine open formal subscheme of $\mathscr X$. \end{prop} \begin{cor} Let $f:\mathscr Y\to\mathscr X$ be an adic morphism of special formal $k$-schemes. Then the induced morphism $f^\colon\mathscr Y^\to\mathscr{\X^}$ is equivariant. \end{cor} \begin{proof} Covering $\mathscr Y^$ and ${\X^}$ by suitable affinoid domains we can see $f^$ locally as a morphism of affinoid domains; Lemma~\ref{definition_action} allows us to conclude. \end{proof} \pa{ Putting together all the previous results, we deduce that the association $\mathscr X\mapsto{\X^}$ gives a functor from the category of special formal $k$-schemes with adic morphisms to the category of $k$-analytic spaces with a free $\mathbb R_{>0}$-action on the underlying topological space and equivariant analytic morphisms. } \end{comment} We now consider the quotient of ${\X^}$ by the $\mathbb R_{>0}$-action. \pa{\label{def_quotient_map} Let $\mathscr X$ be a special formal scheme over $k$. Denote by ${T_\X}$ the (set-theoretic) quotient of the space ${\X^}$ by the action of $\mathbb R_{>0}$, and by $\pi:{\X^}\to{T_\X}$ the quotient map. We endow ${T_\X}$ with both the quotient topology and the quotient $G$-topology. The latter is defined as follows: we declare that a subspace $U$ of ${T_\X}$ is $G$-admissible if $\pi^{-1}(U)$ is an analytic domain of ${\X^}$, and that a family $\{U_i\}_i$ of subspaces of $U$ is a $G$-cover of $U$ if $\{\pi^{-1}(U_i)\}_i$ is a $G$-cover of $\pi^{-1}(U)$. It is easy to verify that this defines a $G$-topology on ${T_\X}$, which is finer than the quotient topology because the $G$-topology on ${\X^}$ is finer than the Berkovich topology. As is the case for usual Berkovich spaces, the $G$-admissible subsets of ${T_\X}$, which should not be thought of as open subsets, will be called \defi{analytic domains} of ${T_\X}$. To provide ${T_\X}$ with the structure of a ringed $G$-topological space in $k$-algebras, we endow it with the sheaf $\mathcal O_{T_\X}=\pi_\mathcal O_{\X^}$, the push-forward of the sheaf of analytic functions on ${\X^}$ via the projection map. We will often denote by $\mathcal O_{T_\X}$ also the restriction of the previous sheaf to the usual topology of ${T_\X}$, and when talking about stalks of $\mathcal O_{T_\X}$ we will always consider the stalk with respect to the usual topology. } \begin{rem} The $G$-covers of ${T_\X}$ can be described explicitly as follows. If $U$ is an analytic domain of ${T_\X}$ and $\{U_i\}_{i\in I}$ is the family consisting of the analytic domains of ${T_\X}$ contained in $U$, then the $U_i$ form a $G$-cover of $U$ if and only if for every point $x$ of $U$ there exists a finite subset $I_x$ of $I$ such that $\bigcup_{i\in I_x}U_i$ is a neighborhood of $x$ and $x\in\bigcap_{i\in I_x}U_i$. This fact follows from the corresponding statement for ${\X^}$ and from the openness of $\pi$, which will be proven in Lemma~\ref{L: normalized hausdorff}. \end{rem} \pa{ \label{bounded_functions_on_T} If $f\in\mathcal O_T(V)$ is a function on $V\subset{T_\X}$, we do not obtain a real value by evaluating $f$ in a point of $V$. Nevertheless, it makes sense to ask whether this value lies in $\{0\}$, $\{1\}$, $\left]0,1\right[$ or $]1,\infty[$, since these sets are the orbits of the action of $\mathbb R_{>0}$ on $\mathbb R_{\geq0}$ by exponentiation. In particular, the sheaf $\pi_\mathcal O^\circ_{\X^}$, which is a subsheaf of $\mathcal O_{T_\X}$, can really be thought of as the \defi{sheaf of analytic functions bounded by $1$} on $T_\mathscr X$, and the sheaf $\pi_\mathcal O^{\circ\circ}_{\X^}$ can be thought of as the \defi{sheaf of analytic functions strictly bounded by $1$} on $T_\mathscr X$. We denote these sheaves by $\mathcal O^\circ_{T_\X}$ and $\mathcal O^{\circ\circ}_{T_\X}$ respectively. } \pa{\label{abstract_valuation_ring} A more intrinsic way to see this is to observe that with any point $x$ of ${T_\X}$ is associated an abstract valued field, the field $\mathscr H(y)$ for any point $y$ of $\pi^{-1}(x)$, endowed with an abstract valuation but not with an absolute value. If $f$ is a function on ${T_\X}$ then $f(x)$ makes sense as an element of this valued field, and $f$ is bounded by $1$ at $x$ if $f(x)$ belongs to the corresponding abstract valuation ring. Moreover, $\|f(x)\|$ makes sense as an element of the corresponding value group, while choosing a preimage $y$ for $x$ corresponds to choosing an embedding of this value group into $\mathbb{R}_+^\times$, which yields a real value for $\|f(x)\|$. } \begin{rem} \label{functions_on_T} It also makes sense to evaluate at points of ${T_\X}$ every function that is constant on the orbits of points for the $\mathbb R_{>0}$-action. For example, if $f$ and $g$ are functions on $V\subset{T_\X}$ and $x$ is a point of $V$ where $f$ and $g$ do not vanish, then we can evaluate $\log\|f\|/\log\|g\|$ at $x$. Note that this value is encoded in the structure of the normalized space. For example, if both $f$ and $g$ take values in $\left]0,1\right[$, then $(\log\|f\|/\log\|g\|)(x)$ can be defined as $\sup\{a/b \,\,\|\,\, a,b\in\mathbb N,b\neq0 \mbox{ and } \|f^b(x)\|\leq\|g^a(x)\|\}$; the other cases are similar. \end{rem} \begin{lem}\label{L: normalized hausdorff} The projection $\pi:{\X^}\to{T_\X}$ is an open map, and the topological space underlying ${T_\X}$ is Hausdorff. \end{lem} \begin{proof} The projection $\pi:{\X^}\to{T_\X}$ is an open map because $I$ acts on ${\X^}$ by homeomorphisms (this follows from the definition of the Berkovich topology on ${\X^}$ and the continuity of $x\mapsto x^\lambda$). Since ${\X^}$ is Hausdorff, to prove that the quotient ${T_\X}$ is Hausdorff as well it is sufficient to show that the orbit equivalence relation is closed in ${\X^}\times{\X^}$. Moreover, without loss of generality we can assume that $\mathscr X=\Spf A$ is affine. Now, if $x$ and $y$ be two points of ${\X^}$ that are not in the same $I$-orbit, we can find two functions $f$ and $g$ in $A$, not vanishing on $x$ and $y$, and such that $\log\|f(x)\|/\log\|g(x)\|\neq \log\|f(y)\|/\log\|g(y)\|$. Since the quotients of the logarithms are continuous and constant on orbits, the orbit equivalence relation is closed in ${\X^}\times{\X^}$, therefore ${T_\X}$ is Hausdorff. \end{proof} \begin{lem} The restriction of the $G$-sheaf $\mathcal O_{T_\X}$ to the usual topology of ${T_\X}$ is a local sheaf. \end{lem} \begin{proof} Let $x$ be a point of ${T_\X}$. Then ${\mathcal I}=\big\{f\in\mathcal O_{{T_\X},x} \mbox{ s.t. }\|f(x)\|=0\big\}$ is an ideal of $\mathcal O_{{T_\X},x}$, where we write $\mathcal O_{{T_\X},x}$ for the stalk of $\mathcal O_{{T_\X}}$ in $x$ for the usual topology. If $\|f(x)\|\neq 0$ then $f$, seen as a function on a neighborhood of $\pi^{-1}(x)$ in ${\X^}$, does not vanish in any point of some open neighborhood $U$ of some point of $\pi^{-1}(x)$. Therefore $f$ has no zero, and is hence invertible, on the $\mathbb R_{>0}$-invariant subspace $\pi^{-1}(\pi(U))$, which is open by Lemma~\ref{L: normalized hausdorff}. This proves that ${\mathcal I}$ is the unique maximal ideal of $\mathcal O_{{T_\X},x}$, and so the restriction of $\mathcal O_{{T_\X}}$ to the usual topology is local. \end{proof} \begin{ex} \label{valtree} If $\mathscr X$ is $\Spf\big(\mathbb C[[X,Y]]\big)$, the completion of the complex affine plane $\mathbb A^2_{\mathbb C}$ at the origin, the topological space ${T_\X}$ is canonically homeomorphic to the \defi{valuative tree} $T$ introduced by Favre and Jonsson in \cite{valtree}. The valuative tree is defined as the set of (semi-)valuations $v$ on $\mathbb C[[X,Y]]$ extending the trivial valuation on $\mathbb C$ and such that $\min\{v(X),v(Y)\}=1$, endowed with the topology it inherits from the Berkovich space ${\X^}$ via the inclusion that sends a valuation $v\in T$ to the seminorm $e^{-v}$. The restriction of the projection $\pi:{\X^}\to{T_\X}$ to $T$ is then a continuous bijection, therefore it is a homeomorphism because $T$ is compact by \cite[5.2]{valtree} and ${T_\X}$ is Hausdorff by Lemma~\ref{L: normalized hausdorff}. \end{ex} \begin{ex}\label{example_explicit_normalization} More generally, in the algebraic case discussed in Examples~\ref{example_algebraic} and \ref{R: interpretation algebraic case}, i.e. when $\mathscr X=\widehat{X/Z}$ is the formal completion of a $k$-variety $X$ along a closed subvariety $Z$, the normalized Berkovich space ${T_\X}$ can be thought of as the {\it normalized non-archimedean link} of $Z$ in $X$. The topological space underlying ${T_\X}$ can be described explicitly as the space of normalized valuations on $X$ that are centered on $Z$ but are not valuations on $Z$, i.e. ${T_\X}=\left(\Sp_X^{-1}(Z)\setminus Z^\beth\right)\big/\mathbb R_{>0}$. An explicit normalization can be given as follows. Let ${\mathcal I}$ be the coherent ideal sheaf of $X$ defining $Z$, and for each element $x$ of ${\X^}$ set $x({\mathcal I})=\max\big\{\|f(x)\|\;\big\|\;f\in{\mathcal I}_{\Sp_\mathscr X(x)}\big\}>0$, where ${\mathcal I}_{\Sp_\mathscr X(x)}$ denotes the stalk of ${\mathcal I}$ at ${\Sp_\mathscr X(x)}$. Then, as in Example~\ref{valtree}, since for every $\gamma$ in $\mathbb R_{>0}$ and $x$ in ${\X^}$ we have $\gamma\cdot x({\mathcal I})=x({\mathcal I})^\gamma$ the restriction of $\pi$ to the subspace $\{x\in X^\beth \;\|\; x({\mathcal I})=1/e\}$ of ${\X^}$ is a homeomorphism onto ${T_\X}$. Therefore in this case the topological space we consider is similar to the \defi{normalized valuation space} considered in \cite[\S2.2]{BoucksomdeFernexFavreUrbinati13}, but they consider only valuations with trivial kernel. Moreover, our definition as a quotient is more intrinsic as it does not depend on the choice of the real number $1/e$. \end{ex} \pa{ Let $\mathscr X$ be a special formal $k$-scheme. Then the specialization map on ${\X^}$ induces an anticontinuous map $\Sp_\mathscr X:{T_\X}\to\mathscr X$, which we call again \emph{specialization}. Indeed, as we observed in \ref{abstract_valuation_ring} the valuation ring of two elements of ${\X^}$ in the same $\mathbb R_{>0}$-orbit is the same, therefore the specialization map on ${\X^}$ passes to the quotient, inducing a map of sets $\Sp_\mathscr X:{T_\X}\to\mathscr X$ such $\Sp_\mathscr X\circ\pi=\Sp_\mathscr X$. Anticontinuity follows from the fact that the specialization on ${\X^}$ is anticontinuous and $\pi$ is open by Lemma~\ref{L: normalized hausdorff}. We will prove in Proposition~\ref{sp_surjective} that if $\mathscr X$ is admissible then the specialization map is surjective. } \pa{\label{definition_category_C} To study the spaces we have been considering so far, it is convenient to temporarily introduce a suitable category. We denote by ${\mathcal C}$ the category whose objects are the triples $\big(T,\mathcal O_T,\mathcal O^\circ_T\big)$, where $T$ is a topological space endowed with an additional $G$-topology that is finer than its given usual topology (which means that every open subspace of $T$ is also a $G$-admissible open and every open cover is a $G$-cover), $\mathcal O_T$ is a sheaf of $k$-algebras on $T$ for the $G$-topology, and $\mathcal O^\circ_T$ is a subsheaf in $k$-algebras of $\mathcal O_T$; and such that a morphisms $\big(T,\mathcal O_T,\mathcal O^\circ_T\big)\to \big(T',\mathcal O_{T'},\mathcal O^\circ_{T'}\big)$ is given by a continuous and $G$-continuous map $f:T\to T'$ and a morphism of sheaves $f_\#:\mathcal O_{T'}\to f_\mathcal O_{T}$ such that $f\big(\mathcal O^\circ_{T'}\big)\subset f_\mathcal O^\circ_T$ and inducing a local morphism of local sheaves once restricted to the usual topology. We will always write simply $\mathcal O_T$ (respectively $\mathcal O^\circ_T$) also for the restriction of $\mathcal O_T$ (resp. $\mathcal O^\circ_T$) to the topology of $T$ and, when no risk of confusion will arise, we will write $T$ for an object $\big(T,\mathcal O_T,\mathcal O^\circ_T\big)$ of ${\mathcal C}$. } \pa{ Let $\mathscr X$ be a special formal scheme over $k$. We define the \defi{normalized Berkovich space} of $\mathscr X$ as the object ${T_\X}=\big({T_\X},\mathcal O_{T_\X},\mathcal O^\circ_{T_\X}\big)$ of ${\mathcal C}$. This gives a functor $T:\big(SFor_k\big)\to{\mathcal C}$ from the category of special formal $k$-schemes with adic morphisms to ${\mathcal C}$. In Section~\ref{section_3.4} we will investigate the properties of the functor $T$ and determine its essential image. } \begin{rem}\label{homotopy_normalized} Thuillier proved in \cite{Thu07} that whenever $k$ is perfect, $X$ is a $k$-variety with singular locus $Z$ and $\mathscr X=\widehat{X/Z}$, the homotopy type of $\mathscr X^$ is the same as the homotopy type of the dual complex $\mathrm{Dual}(D)$ of the exceptional divisor $D$ of a log resolution $Y$ of $X$. Using toroidal methods, he constructs an embedding of $\mathrm{Dual}(D)\times\mathbb R_{>0}$ into $\mathscr Y^$ and a deformation retraction of the latter onto the former, where $\mathscr Y=(\widehat{Y/D})$. As both the embedding and the retraction are $\mathbb R_{>0}$-equivariant, quotienting by the action of $\mathbb R_{>0}$ we obtain a deformation retraction of $T_\mathscr Y$ onto a closed subspace homeomorphic to $\mathrm{Dual}(D)$. Since $T_\mathscr Y\congT_{X,Z}$, we deduce that the homotopy type of $T_{X,Z}$ is the same as the homotopy type of $\mathrm{Dual}(D)$. Note that by \cite{Kollar13} the homotopy type of $\mathrm{Dual}(D)$ can be almost arbitrary. However, by \cite{deFernexKollarXu12} $\mathrm{Dual}(D)$ is contractible for a wide class of singularities, namely isolated log terminal singularities (in particular, for all toric or finite quotient singularities). \end{rem} \pa{\label{def_forgetful_functor} We also have a forgetful functor $\forg:\big(An_{k((t))}\big)\to{\mathcal C}$ sending a $k((t))$-analytic space $X$ to the triple $\big(X,\mathcal O_X,\mathcal O^\circ_X\big)$. } \section{Local analytic structure} \label{section_3.2} In this section we prove one of the main properties of normalized spaces of special formal $k$-schemes. Although those spaces are not analytic spaces themselves, as locally ringed spaces in $k$-algebras they are $G$-locally isomorphic to analytic spaces defined over some Laurent series field $k((t))$. This is the content of Corollary~\ref{locally_analytic}. This result is conceptually similar to those discussed in \cite[\S4.2, 4.3, 4.4, 4.6]{Ben-BassatTemkin2013}. We deduce an analogue for normalized spaces of a theorem of de Jong (Corollary~\ref{dejong_normalized}), and a characterization of admissible special formal $k$-schemes (Proposition~\ref{admissible}). We also prove that the specialization map is surjective for the normalized space of an admissible special formal $k$-scheme (Theorem~\ref{sp_surjective}). We first need some results about the Berkovich spaces associated with affine special formal $k$-schemes. \begin{comment} \begin{lem}\label{orbit_affinoid} Let $\mathscr X$ be an affine special formal scheme over $k$, let $t$ be a nonzero element of an ideal of definition of $\mathscr X$ and let $V$ be an affinoid domain of ${\X^}$ such that $t$ has no zero on $V$. Denote by $\mathbb R_{>0}\cdot V$ the set of translates of $V$ in ${\X^}$ under the $\mathbb R_{>0}$-action. Then we can write $\mathbb R_{>0}\cdot V$ as a finite union $\cup_{i}V_i$, with each $V_i$ stable under the action of $\mathbb R_{>0}$ and such that $V_{i,\varepsilon}:=V_i\cap W_{t,\varepsilon}$ is a strict affinoid subdomain of $W_{t,\varepsilon}$ for $\varepsilon$ small enough. In particular, each $V_i$ is an increasing union of affinoid domains of ${\X^}$, and $\mathbb R_{>0}\cdot V$ is an analytic domain of ${\X^}$. \end{lem} \begin{proof} As in \ref{example_affine_punctured_space}, the increasing family $\big\{W_{t,\varepsilon}^\circ\big\}_\varepsilon$ is an open cover of ${\X^}\setminus V(t)$, so since $V$ is compact $V$ is contained in $W_{t,\delta}^\circ$ for some $\delta$. It follows that $V$ is an affinoid subdomain of $W_{t,\delta}$. Then, by Gerritzen-Grauert theorem (proofs valid in the case of a trivially valued field are given in \cite{Ducros03} and \cite{Temkin05}), $V$ is a finite union $V=\cup V'_i$ of rational domains of $W_{t,\delta}$. Each $V'_i$ is by definition determined in $W_{t,\delta}$ by finitely many inequalities $\|f_j\|\leq r_j\|g_j\|$, for some $f_j,g_j\in{\mathcal W}_{t,\delta}$ and $r_j> 0$. Then for every $\varepsilon\leq\delta$ we have \begin{align} & (\mathbb R_{>0}\cdot V'_i)\cap W_{t,\varepsilon} = \big\{x\in W_{t,\varepsilon} \mbox{ s.t. }x^\lambda\in V_i \mbox{ for some }\lambda\in\mathbb R_{>0}\big\} \\ = & \big\{x\in W_{t,\varepsilon} \mbox{ s.t. }\|f_j(x)\|\leq r_j^\lambda\|g_j(x)\|\mbox{ for some }\lambda\in\mathbb R_{>0}\mbox{ and all }j\big\} \\ = & \big\{x\in W_{t,\varepsilon} \mbox{ s.t. }\|f_j(x)\|\leq \|g_j(x)\|\mbox{ for all } j \mbox{ such that }r_j=1\big\}, \end{align} which is a strict affinoid subdomain of $W_{t,\varepsilon}$. Note that we have implicitly used the fact that the functions $f_j$ and $g_j$ extend to $W_{t,\varepsilon}$ because the map of sets underlying the Banach algebra morphism ${\mathcal W}_{t,\varepsilon}\to{\mathcal W}_{t,\delta}$ is bijective. \\ FINISH the formula above. \\ Therefore, if we set $V_i:=\mathbb R_{>0}\cdot V'_i$ and $V_{i,\varepsilon}=V_i\cap W_{t,\varepsilon}$ then the $V_i$ and $V_{i,\varepsilon}$ satisfy our requirements. Finally, $\mathbb R_{>0}\cdot V$ is an analytic domain of ${\X^}$, which is $G$-covered by the affinoid domains $V_i\cap W_{t,\varepsilon}$. Indeed, every point of $\mathbb R_{>0}\cdot V$ is contained in the interior of one set of the form $\mathbb R_{>0}\cdot V\cap W_{t,\varepsilon}$, and the latter is the finite union of the affinoid domains $V_i\cap W_{t,\varepsilon}$. \end{proof} \begin{cor}\label{G-admissible_strip} Let $\mathscr X$ be an affine special formal scheme over $k$, let $t$ be a nonzero element of an ideal of definition of $\mathscr X$ and let $U$ be a subset of ${\X^}\setminus V(t)$ stable under the action of $\mathbb R_{>0}$. Then $U$ is an analytic domain of ${\X^}$ if and only if we can write it as a union $U=\cup_i U_i$ in such a way that each $U_i$ is stable under the action of $\mathbb R_{>0}$ and is an increasing union $U_i=\cup U_{i,\varepsilon}$ for $\varepsilon$ small enough, with $U_{i,\varepsilon}$ a strict affinoid subdomain of $W_{t,\varepsilon}$, and $\{U_{i,\varepsilon}\}_{i,\varepsilon}$ is a $G$-cover of $U$. \end{cor} \begin{proof} If $U$ is an analytic domain of ${\X^}$ then it is $G$-covered by the affinoid domains that it contains, and applying Lemma~\ref{orbit_affinoid} to each of them we get the decomposition that we want. The converse implication is obvious, since $U$ is by definition $G$-covered by the affinoid domains $U_{i,\varepsilon}$. \end{proof} \end{comment} \begin{prop}\label{G-admissible_strip} Let $\mathscr X$ be an affine special formal scheme over $k$, let $t$ be a nonzero element of an ideal of definition of $\mathscr X$ and let $U$ be a subset of ${\X^}\setminus V(t)$ stable under the action of $\mathbb R_{>0}$. Then $U$ is an analytic domain of ${\X^}$ if and only if we can write it as a union $U=\cup_i U_i$ in such a way that each $U_i$ is stable under the action of $\mathbb R_{>0}$ and is an increasing union $U_i=\cup U_{i,\varepsilon}$ for $\varepsilon$ small enough, with $U_{i,\varepsilon}$ a strict affinoid subdomain of $W_{t,\varepsilon}$, and $\{U_{i,\varepsilon}\}_{i,\varepsilon}$ is a $G$-cover of $U$. \end{prop} Before proving this proposition we will establish a simple lemma. \begin{lem}\label{orbit_affinoid} Let $\mathscr X$ be an affine special formal scheme over $k$, let $t$ be a nonzero element of an ideal of definition of $\mathscr X$ and let $V$ be an affinoid domain of ${\X^}$ such that $\|t\|=1/2$ on $V$. Denote by $\mathbb R_{>0}\cdot V$ the set of translates of $V$ in ${\X^}$ under the $\mathbb R_{>0}$-action. Then we can write $\mathbb R_{>0}\cdot V$ as a finite union $\cup_{i}V_i$, with each $V_i$ stable under the action of $\mathbb R_{>0}$ and such that $V_{i,\varepsilon}:=V_i\cap W_{t,\varepsilon}$ is a strict affinoid subdomain of $W_{t,\varepsilon}$ for $\varepsilon$ small enough. In particular, each $V_i$ is an increasing union of affinoid domains of ${\X^}$, and $\mathbb R_{>0}\cdot V$ is an analytic domain of ${\X^}$. \end{lem} \begin{proof} For every $\varepsilon\leq 1/2$, $V$ is an affinoid subdomain of $W_{t,\varepsilon}$. Therefore, by Gerritzen-Grauert theorem (proofs valid in the case of a trivially valued field are given in \cite{Ducros03} and \cite{Temkin05}), $V$ is a finite union $V=\cup V'_i$ of rational domains of $W_{t,\varepsilon}$. Each $V'_i$ is by definition determined in $W_{t,\varepsilon}$ by finitely many inequalities $\|f_j\|\leq r_j\|g_j\|$, for some analytic functions $f_j,g_j$ on $W_{t,\varepsilon}$ and $r_j> 0$. Then we have \begin{align} & (\mathbb R_{>0}\cdot V'_i)\cap W_{t,\varepsilon} = \big\{x\in W_{t,\varepsilon} \mbox{ s.t. }x^\lambda\in V'_i \mbox{ for some }\lambda\in\mathbb R_{>0}\big\} \\ = & \big\{x\in W_{t,\varepsilon} \mbox{ s.t. there exists }\lambda \mbox{ s.t. } \|f_j(x)\|\leq r_j^{\frac{1}{\lambda}}\|g_j(x)\|\mbox{ for all }j\big\} \\ = & \big\{x\in W_{t,\varepsilon} \mbox{ s.t. }\|f_j(x)\|\leq \|t(x)\|^{a_j}\|g_j(x)\|\mbox{ for all } j\big\}, \end{align} where $a_j= - \log r_j / \log 2$, yielding a strict affinoid subdomain of $W_{t,\varepsilon}$. Observe that in the last equality we used the fact that $\|t\|=1/2$ on $V'_i$, hence $\|t(x)\|^{a_j}=(1/2)^{\frac{a_j}{\lambda}}=r_j^{\frac{1}{\lambda}}$ for every $x$ in $(\mathbb R_{>0}\cdot V'_i)\cap W_{t,\varepsilon}$ and every $j$. Now, if we set $V_i=\mathbb R_{>0}\cdot V'_i$ and $V_{i,\varepsilon}=V_i\cap W_{t,\varepsilon}$ for every $\varepsilon\leq 1/2$, then the $V_i$ and $V_{i,\varepsilon}$ satisfy our requirements. Finally, $\mathbb R_{>0}\cdot V$ is an analytic domain of ${\X^}$, which is $G$-covered by the affinoid domains $V_i\cap W_{t,\varepsilon}$. Indeed, every point of $\mathbb R_{>0}\cdot V$ is contained in the interior of one set of the form $\mathbb R_{>0}\cdot V\cap W_{t,\varepsilon}$, and the latter is the finite union of the affinoid domains $V_i\cap W_{t,\varepsilon}$. \end{proof} \begin{proof}[Proof of Proposition~\ref{G-admissible_strip}] If $U$ is an analytic domain of ${\X^}$ then it is $G$-covered by the affinoid domains that it contains. Then the intersection of each of them with $W_{t,1/2}$ is an affinoid domain on which $\|t\|=1/2$, and applying Lemma~\ref{orbit_affinoid} to all such intersections we get the decomposition that we want. The converse implication is obvious, since $U$ is by definition $G$-covered by the affinoid domains $U_{i,\varepsilon}$. \end{proof} \pa{ For $r\in\left]0,1\right[$, we denote by $K_r$ the affinoid $k$-algebra $k\{rt^{-1},r^{-1}t\}$; it is the completed residue field $\rescompl{r}$ of the point $r$ of $\Spf(k\lbrack\lbrack t\rbrack\rbrack)^\cong\left]0,1\right[$, and an easy computation shows that it is the field $k((t))$ with the $t$-adic absolute value such that $\|t\|=r$. For $0<\varepsilon<1/2$, the $k$-algebras ${\mathcal A}_\varepsilon=k\{\varepsilon T^{-1},(1-\varepsilon)^{-1}T\}$ are also isomorphic to the field $k((t))$, but their Banach norms are not $t$-adic (they are not even absolute values). If $r\in[\varepsilon,1-\varepsilon]$, the identity map ${\mathcal A}_{\varepsilon}\to K_r$ is a bounded morphism of Banach $k$-algebras. Its boundedness is easy to check algebraically; geometrically this corresponds to the inclusion of the point $r$, into ${\mathcal M}({\mathcal A}_\varepsilon)$, seen as the annulus $\left[\varepsilon,1-\varepsilon\right]$ in $\Spf(k\lbrack\lbrack t\rbrack\rbrack)^\cong\left]0,1\right[$. Nevertheless, note that despite having different Banach norms $K_r$ and ${\mathcal A}_{\varepsilon}$ are isomorphic not only as $k$-algebras, but also as topological $k$-algebras, since the neighborhoods of zero in both algebras coincide. This has as a very important consequence the following result. } \begin{lem} \label{lemma_tensor} Let ${\mathcal B}$ be a strict affinoid algebra over ${\mathcal A}_{\varepsilon}$ and let $r$ be an element of $\left[\varepsilon,1-\varepsilon\right]$. Then the canonical morphism ${\mathcal B}\to {\mathcal B}\hat\otimes_{{\mathcal A}_{\varepsilon}}K_r$ is an isomorphism of $k$-algebras. \end{lem} \begin{proof} If ${\mathcal B}={\mathcal A}_{\varepsilon}\{X_1,\ldots,X_n\}$, an elementary computation shows that the convergence conditions for a series of the form $\sum_{i_0,I}a_{i_0,I}t^{i_0}\underline X^I$, for $a_{i_0,I}\in k$, to belong to either ${\mathcal B}$ or ${\mathcal B}\hat\otimes_{{\mathcal A}_{\varepsilon}}K_r$ are the same (namely, the coefficients $a_{i_0,I}$ have tend to zero for the $t$-adic topology, which is the same on ${\mathcal A}_\epsilon$ and on $K_r$). Therefore ${\mathcal B}\otimes_{{\mathcal A}_{\varepsilon}}K_r$, which is isomorphic to ${\mathcal B}$ as a $k$-algebra via the canonical morphism ${\mathcal B}\to {\mathcal B}\otimes_{{\mathcal A}_{\varepsilon}}K_r$, is already complete with respect to the tensor product seminorm, and hence coincides with ${\mathcal B}\hat\otimes_{{\mathcal A}_{\varepsilon}}K_r$. In the general case, that is ${\mathcal B}={\mathcal A}_{\varepsilon}\{X_1,\ldots,X_n\}/I$, to show that ${\mathcal B}\otimes_{{\mathcal A}_{\varepsilon}}K_r$ is complete observe that ${\mathcal B}\otimes_{{\mathcal A}_{\varepsilon}}K_r\cong \big({{\mathcal A}_{\varepsilon}\{X_1,\ldots,X_n\}\otimes_{{\mathcal A}_{\varepsilon}}K_r}\big)\big/{I}$ as normed algebras. Indeed, they are canonically isomorphic as $k$-algebras, and the fact that the norms are the same follows from the fact that under the isomorphism we have $(a+I)\otimes t=(a\otimes t)+I$, and on both sides the norm of such an element is $\|a\|\cdot\|r\|$. The algebra on the right hand side is complete since it is an admissible quotient of a Banach algebra by a closed ideal, hence ${\mathcal B}\otimes_{{\mathcal A}_{\varepsilon}}K_r$ is complete. \end{proof} \begin{rem} The result above can fail if the affinoid ${\mathcal A}_\varepsilon$-algebra ${\mathcal B}$ is not strict because the first computation in the proof (that is the case $I=0$) breaks down. For example, if $0<r<s<\varepsilon\leq1/2$, then $K_r\hat\otimes_{{\mathcal A}_\varepsilon}K_s=0$. Indeed, the tensor product seminorm on $K_r\otimes_{{\mathcal A}_\varepsilon}K_s$ is the zero seminorm since the element $1\otimes 1$ of the tensor product is equal to $t^n\otimes t^{-n}$ for every $n\in{\mathbb{N}}$, so $\|1\otimes 1\|\leq r^ns^{-n}$, and the latter goes to zero as $n$ goes to infinity. \end{rem} \pa{\label{projection_structure} If $\mathscr X_t$ is a special formal scheme over $k[[t]]$ then it can be seen as a special formal scheme $\mathscr X$ over $k$, so we get a morphism of formal $k$-schemes $\mathscr X\to\Spf{(k[[t]])}$ and therefore a morphism of $k$-analytic spaces $f:\mathscr X^\beth\to\Spf{(k[[t]])}^\beth$. If $r$ is any point of $\Spf(k\lbrack\lbrack t\rbrack\rbrack)^\beth\cong\left[0,1\right[$, then we can consider the fiber product of $\mathscr X^\beth$ with the point $r$ in the category of analytic spaces over $\Spf(k\lbrack\lbrack t\rbrack\rbrack)^\beth$ (see \cite{Ber93}). This analytic space is defined over the non-archimedean field $\rescompl{r}$, which coincides with $k$ whenever $r=0$ and is otherwise isomorphic to the field $k((t))$ with the $t$-adic absolute value such that $\|t\|=r$. The topological space underlying this fiber product is canonically homeomorphic to the topological fiber: $ f^{-1}(r)\cong\mathscr X^\beth\times_{\Spf(k\lbrack\lbrack t\rbrack\rbrack)^\beth}\m{\rescompl{r}}$. If $\mathscr X_t=\Spf A$ is affine, by seing points of both $\mathscr X_t^\beth$ (considered as an analytic space over the field $k((t))$ with the $t$-adic absolute value such that $\|t\|=r$) and $\mathscr X^\beth$ as seminorms on $A$, the points of $\mathscr X_t^\beth$ satisfying the additional condition $\|t\|=r$, we deduce the existence of a map $\mathscr X_t^\beth\to\mathscr X^\beth$, which factors through $f^{-1}(r)$. In the general case, those maps glue to a map $\mathscr X_t^\beth\to f^{-1}(r)$. } \begin{lem}\label{lemma_omeo} Let $\mathscr X_t$ and $f:\mathscr X^\beth\to\Spf{(k[[t]])}^\beth$ be as above, choose $0<r<1$ and endow $k((t))$ with the $t$-adic absolute value such that $\|t\|=r$. Then the map $\mathscr X_t^\beth\to f^{-1}(r)$ constructed above induces an isomorphism of $k((t))$-analytic spaces between $\mathscr X^\beth_t$ and $f^{-1}(r)$. \end{lem} \begin{proof} We follow the lines of \cite[4.3]{Nicaise11}. Since $\mathscr X_t^\beth$ is $G$-covered by the Berkovich spaces of the affine formal subschemes of $\mathscr X_t$, we can assume that $\mathscr X_t$ is affine, with associated $k[[t]]$-algebra $k[[t]]\{X_1,\ldots,X_n\}[[Y_1,\ldots,Y_m]]/I$. Then following \ref{example_affine_beth_space} we write $\mathscr X_t^\beth$ as an increasing union $\mathscr X_t^\beth=\bigcup_{0<\varepsilon\leq1/2}U_\varepsilon$, where $U_\varepsilon$ is the subspace of $\mathscr X_t^\beth$ cut out by the inequalities $\|Y_i\|\leq1-\varepsilon$; it is an affinoid domain with associated affinoid $k((t))$-algebra $$ \frac{k((t))\{X_1,\ldots,X_n\}\{(1-\varepsilon)^{-1}Y_1,\ldots,(1-\varepsilon)^{-1}Y_m\}}{I}. $$ Similarly, by \ref{example_affine_punctured_space} we can write ${\X^}\setminus V(t)$ as an increasing union ${\X^}\setminus V(t)=\bigcup_{0<\varepsilon\leq1/2}W_{t,\varepsilon}$, where $W_{t,\varepsilon}$ is the subspace of ${\X^}$ cut out by $\varepsilon\leq\|t\|\leq1-\varepsilon$ and $\|Y_i\|\leq1-\varepsilon$; it is an affinoid domain of ${\X^}$ with associated affinoid $k$-algebra \[ \frac{{k\{\varepsilon t^{-1},(1-\varepsilon)^{-1}t\}\{X_1,\ldots,X_n\}\{(1-\varepsilon)^{-1}Y_1,\ldots,(1-\varepsilon)^{-1}Y_m\}}}{I}. \] For every $\varepsilon$ such that $\varepsilon<r<1-\varepsilon$, we get a map $U_\varepsilon\to W_{t,\varepsilon}$ that is a homeomorphism onto $f^{-1}(r)\cap W_{t,\varepsilon}$ and that coincides with the restriction to $U_\varepsilon$ of the map $\mathscr X_t^\beth\to f^{-1}(r)$ defined in \ref{projection_structure}. We deduce that $\mathscr X_t^\beth\to f^{-1}(r)$ is a homeomorphism , and it induces an isomorphism of $k((t))$-analytic spaces because, since $f^{-1}(r)\cap W_{t,\varepsilon}$ can be identified with the subspace of $f^{-1}(r)$ defined by the inequalities $\|Y_i\|\leq1-\varepsilon$, which is an affinoid domain with associated affinoid $k((t))$-algebra ${\mathcal O}_{W_{t,\varepsilon}}(W_{t,\varepsilon})\hat\otimes_{{\mathcal A}_\varepsilon}\rescompl{r}\cong{\mathcal O}_{U_\varepsilon}(U_\varepsilon)$, it identifies the affinoid domains of $\mathscr X_t^\beth$ with the ones of $f^{-1}(r)$. \end{proof} \pa{ \label{projection_structure2} The morphism $\mathscr X\to\Spf{(k[[t]])}$ of \ref{projection_structure} is adic if and only if $\mathscr X_t$ is locally of finite type over $k[[t]]$, so in general it does not induce a morphism between punctured Berkovich spaces. However, we get an $\mathbb R_{>0}$-equivariant morphism of $k$-analytic spaces $f\|_{\mathscr X^\beth\setminus V(t)}:{\mathscr X^\beth\setminus V(t)}\to\Spf(k[[t]])^$. Since $f\|_{\mathscr X^\beth\setminus V(t)}$ is equivariant and $\Spf\big(k[[t]]\big)^\cong\left]0,1\right[$ consists of a unique $\mathbb R_{>0}$-orbit, for any $r\in]0,1[$ we have a homeomorphism $f^{-1}(r)\cong \big(\mathscr X^\beth\setminus V(t)\big)/(\mathbb R_{>0})$, and the latter is homeomorphic to ${{T_\X}}\setminus V(t)$. As a consequence of the results above, we obtain the following important theorem, which is purely formal and relies essentially on the structure of analytic domains obtained in Proposition~\ref{G-admissible_strip} and on the computation of Lemma~\ref{lemma_tensor}. Recall that we have a quotient map $\pi:{\X^}\to{T_\X}$, defined in \ref{def_quotient_map}, and a forgetful functor $\forg:\big(An_{k((t))}\big)\to{\mathcal C}$, discussed in \ref{def_forgetful_functor}. } \begin{thm} \label{thm_structure_arboretum} Let $\mathscr X_t$ be a special formal scheme over $k[[t]]$ and choose $0<r<1$. Then, via the isomorphism of Lemma~\ref{lemma_omeo}, $\pi\|_{f^{-1}(r)}:f^{-1}(r)\to{T_\X}\setminus V(t)$ induces an isomorphism between $\forg(\mathscr X^\beth_t)$ and ${T_\X}\setminus V(t)$ in ${\mathcal C}$. \end{thm} \begin{proof} Without loss of generality we can assume that $\mathscr X_t$ is affine and $\mathscr X_t^\beth$ is nonempty (if $\mathscr X_t^\beth$ is empty then $t$ is identically zero on ${\X^}$, hence ${T_\X}\setminus V(t)$ is empty as well). We endow $k((t))$ with the $t$-adic absolute value such that $\|t\|=r$ and identify $\mathscr X_t^\beth$ with $f^{-1}(r)$ via Lemma~\ref{lemma_omeo} (although it will follow from the theorem that $\forg(\mathscr X_t^\beth)$ will not depend on the choice of $r$). We will prove that the continuous map $\varphi:\mathscr X^\beth_t\stackrel{\sim}{\longrightarrow} {T_\X}\setminus V(t)$ obtained from Lemma~\ref{lemma_omeo} and \ref{projection_structure2} can be upgraded to an isomorphism in ${\mathcal C}$. Observe that, once $\mathscr X_t^\beth$ is identified with $f^{-1}(r)$, the map $\varphi$ is induced by the projection $\pi$, and therefore it is a morphism of locally ringed $G$-topological spaces. The map $f:\mathscr X^\beth\to\Spf\big(k[[t]]\big)^\beth$ of~\ref{projection_structure} is equivariant and therefore, since $\mathscr X_t^\beth$, and hence ${\X^}\setminus V(t)$, is nonempty, for every $0<\varepsilon<1/2$ it induces a surjective morphism from $W_{t,\varepsilon}$ to the affinoid domain $A_\varepsilon=\m{{\mathcal A}_\varepsilon}\cong\left[\varepsilon,1-\varepsilon\right]$ of $\big(\Spf(k[[t]])\big)^\beth\cong\left[0,1\right[$. The corresponding morphism ${\mathcal A}_\varepsilon\to{\mathcal W}_{t,\varepsilon}$ is the unique $k$-morphism sending $t$ to $t$, and it endows ${\mathcal W}_{t,\varepsilon}$ with the structure of a strict affinoid algebra over ${\mathcal A}_\varepsilon$. To see that $\varphi$ is $G$-continuous we have to show that $\pi^{-1}(U)\cap \mathscr X_t^\beth$ is an analytic domain of $\mathscr X^\beth_t$ whenever $U\subset{T_\X}\setminus V(t)$ is such that $\pi^{-1}(U)$ is an analytic domain of ${\X^}$. Using Proposition~\ref{G-admissible_strip}, we write $\pi^{-1}(U)$ as $\cup U_i$, with $U_{i,\varepsilon}=U_i\cap W_{t,\varepsilon}$ strict affinoid subdomain of $W_{t,\varepsilon}$ for every $i$ and every $\varepsilon$ small enough. Following the isomorphism of Lemma~\ref{lemma_omeo}, we deduce that $\pi^{-1}(U)\cap \mathscr X_t^\beth$ is $G$-covered by the affinoid domains $U_{i,\varepsilon}\times_{A_\varepsilon}\m{K_r}$ of $\mathscr X_t^\beth$, and is therefore an analytic domain of $\mathscr X^\beth_t$. Moreover, if we choose $\varepsilon$ small enough, each $U_{i,\varepsilon}$, being strict over $W_{t,\varepsilon}$ that is strict over $A_\varepsilon$, is itself strict over $A_\varepsilon$. We deduce that, as $k$-algebras, \begin{align} \mathcal O_{\X^}(U_i)& = \varprojlim_\varepsilon\mathcal O_{\X^}(U_{i,\varepsilon}) = \varprojlim_\varepsilon \big(\mathcal O_{\X^}(U_{i,\varepsilon})\hat\otimes_{{\mathcal A}_\varepsilon}K_r\big) \\ & = \varprojlim_\varepsilon\mathcal O_{\mathscr X_t^\beth}\big(U_{i,\varepsilon}\cap \mathscr X_t^\beth\big) \\ & = \mathcal O_{\mathscr X_t^\beth}\big(U_i\cap \mathscr X_t^\beth\big), \end{align} where the second equality is given by Lemma~\ref{lemma_tensor}. Since the $U_i$ form a $G$-cover of $U$, it follows that we have an isomorphism of $k$-algebras $$ \mathcal O_{{T_\X}\setminus V(t)}(U)\cong\mathcal O_{{\X^}}\big(\pi^{-1}(U)\big)\cong\mathcal O_{\mathscr X^\beth_t}\big(\pi^{-1}(U)\cap \mathscr X_t^\beth\big). $$ Moreover, we have also $\mathcal O^\circ_{{T_\X}\setminus V(t)}(U)\cong\mathcal O^\circ_{\mathscr X^\beth_t}\big(\pi^{-1}(U)\cap \mathscr X_t^\beth\big)$, because, as noted in~\ref{bounded_functions_on_T}, an analytic function is bounded by $1$ at a point $x$ of $\mathscr X_t^\beth$ if and only if it is bounded by $1$ at all the points of the orbit $\mathbb R_{>0}\cdot x$. It remains to show that $\varphi$ is a homeomorphism of $G$-sites, i.e. that whenever $U$ is an analytic domain of $\mathscr X_t^\beth$ then $\varphi(U)$ is an analytic domain of ${T_\X}$. If $V$ is an affinoid domain of $\mathscr X_t^\beth$, then it is also an affinoid domain of ${\X^}$, hence the same reasoning as in the proof of Proposition~\ref{G-admissible_strip} shows that $\pi^{-1}\big(\varphi(U)\big)=\mathbb R_{>0}\cdot U$ is an analytic domain of ${\X^}$, therefore $\varphi(U)$ is an analytic domain of ${T_\X}$. \end{proof} \begin{cor}\label{locally_analytic} Let $\mathscr X$ be a special formal scheme over $k$. Then the normalized Berkovich space ${T_\X}$ of $\mathscr X$ is $G$-locally isomorphic in the category ${\mathcal C}$ to an object in the image of $\forg:\big(An_{k((t))}\big)\to{\mathcal C}$. \end{cor} \begin{proof} Without loss of generality we can suppose that $\mathscr X=\Spf(A)$ is affine. Choose generators $(f_1,\ldots,f_s)$ for an ideal of definition of $\mathscr X$. Each $f_i$ is topologically nilpotent in $A$ and so induces a morphism $\mathscr X\to\Spf\big(k[[t]]\big)$. Then the ${\mathscr X^\beth\setminus V({f_i})}$ cover ${\X^}$, hence ${T_\X}$ is covered by the ${\big(\mathscr X^\beth\setminus V({f_i})\big)}/(\mathbb R_{>0})$ and by~\ref{thm_structure_arboretum} ${\big(\mathscr X^\beth\setminus V({f_i})\big)}/(\mathbb R_{>0})\cong {{T_\X}\setminus V({f_i})}\cong \forg\big(\mathscr X^\beth_{f_i}\big)$. \end{proof} \Pa{Remarks}{ If $\mathscr X_t$ is a formal scheme of finite type over $k[[t]]$ then $t$ does not vanish on ${T_\X}$, and so once we have chosen a $t$-adic absolute value on $k((t))$ Theorem~\ref{thm_structure_arboretum} identifies ${T_\X}$ with the image in ${\mathcal C}$ of the $k((t))$-analytic space $\mathscr X_t^\beth$. Note that the local $k((t))$-analytic structures that we obtain in Corollary~\ref{locally_analytic} are far from being unique. Indeed, not only we have to make choices of $\|t\|\in\left]0,1\right[$, but we have to choose an affine cover of $\mathscr X$ and generators of the ideals of definition of the elements of this covers. Nonetheless, for the sake of simplicity we will ofter refer to this result by saying that normalized $k$-spaces are $G$-locally $k((t))$-analytic spaces. } \Pa{Example}{\label{analytic_structure_valuative_tree} If $\mathscr X$ is the formal scheme $\Spf\big(\mathbb C[[X,Y]]\big)$, then its normalized Berkovich space ${T_\X}$ is the valuative tree of \cite{valtree}, as observed in Example~\ref{valtree}. The largest ideal of definition of $\mathbb C[[X,Y]]$ is $(X,Y)$, so ${T_\X}$ is the union of the two $k((t))$-analytic curves $\mathscr X^\beth_X$ and $\mathscr X^\beth_Y$, both isomorphic to the $1$-dimensional open analytic disc over $k((t))$. It's important to remark that on their intersection, which is ${T_\X}\setminus V(XY)$, the two $k((t))$-analytic structures do not agree, because $t$ is sent to $X$ for one of them and to $Y$ for the other. Actually, more is true: we will show in Example~\ref{analytic_structure_valuative_tree_continued} that there is no $k((t))$-analytic space $C$ such that ${T_\X}\cong\forg(C)$. Observe that the complement of $\mathscr X^\beth_X$ in ${T_\X}$, which is the zero locus of $X$, consists of exactly one endpoint of the valuative tree, namely the order of vanishing at the origin along $X$ (it is a \defi{curve valuation} in the terminology of \cite[1.5.5]{valtree}). } \pa{\label{dejong_original} Corollary~\ref{locally_analytic} gives us a way of proving some assertions about analytic spaces over trivially valued fields by reducing to the non-trivially valued case. We prove in this way the analogue for normalized spaces of a result of A.J. de Jong \cite[7.3.6]{deJ95}. De Jong shows that if $\mathscr X=\Spf A$ is a normal and affine special formal scheme flat over a complete discrete valuation ring $R$, then the formal functions on $\mathscr X$ coincide with the analytic functions bounded by $1$ on $\mathscr X^{\beth}$. More precisely, this result holds under weaker assumptions: it is enough for $A$ to be $R$-flat, reduced, and integrally closed in the ring $A\otimes_R\mbox{Frac}(R)$ (this was remarked in \cite[7.4.2]{deJ95} and proven in \cite[2.1]{MartinKappen15}). We can deduce that the same is true in our setting: } \begin{cor}\label{dejong_normalized} Let $A$ be a special $k$-algebra and assume that $A$ is admissible and normal. If we denote by $\mathscr X$ the formal scheme $\Spf A$, then the canonical morphism $A\to\mathcal O^\circ_{T_\X}({T_\X})$ is an isomorphism. \end{cor} \begin{proof} Since $A$ is admissible, it has a nonzero ideal of definition and ${T_\X}$ is non-empty. By working separately on the connected components of $\mathscr X$ we can assume that it is connected and so $A$, being normal, is a domain. Since $A$ is normal we have $A=\cap R$ for $R$ ranging among the valuation rings of rank one of the total ring of fractions of $A$, so the canonical morphism $A\to\mathcal O^\circ_{T_\X}({T_\X})$ is an inclusion. If $t$ is a nonzero element of an ideal of definition of $A$, then $A$ is flat over $k[[t]]$ and, since $A$ is normal, it is integrally closed in $A[t^{-1}]$. Then de Jong's theorem applies to $\mathscr X_t=\Spf{A}$, seen as a special formal scheme over $k[[t]]$, yielding $A\stackrel{\sim}{\to}\mathcal O^\circ_{\mathscr X_t^\beth}\big(\mathscr X_t^\beth\big)=\mathcal O^\circ_{T_\X}(U)$, where $U=\pi\big({\X^}\setminus V(t)\big)$ is the subspace of ${T_\X}$ isomorphic to $\mathscr X^\beth_t$. Since $t$ is not a zero divisor in $A$, then $V(t)$ is a thin subset of ${\X^}$ in the sense of \cite[\S3.3]{Ber90}, so the restriction $\mathcal O^\circ_{\X^}\left({{\X^}}\right)\to\mathcal O^\circ_{\X^}\big({\X^}\setminus V(t)\big)$ is injective by \cite[3.3.14]{Ber90}. From the chain of inclusions $A\hookrightarrow\mathcal O^\circ_{T_\X}({T_\X})\hookrightarrow\mathcal O^\circ_{T_{\mathscr X}}(U)=A$ we deduce that $A\cong\mathcal O^\circ_{T_\X}({T_\X})$. \begin{comment} Since $A$ is admissible, its largest ideal of definition $J$ is non-zero and ${T_\X}$ is non-empty. Let $\{g_1,\ldots,g_s\}$ be a set of generators of $J$, so that ${T_\X}$ is covered by the subsets $U_i={T_\X}\setminus V(g_i)$. Since $\mathcal O^\circ_{T_\X}$ is a sheaf, $\mathcal O^\circ_{T_\X}({T_\X})$ is the equalizer of the diagram \begin{equation} \prod_{i} \mathcal O^\circ_{T_\X}(U_i) {{{} \atop \longrightarrow}\atop{\longrightarrow \atop {}}} \prod_{i,j} \mathcal O^\circ_{T_\X}(U_{ij}), \tag{$\ast$}\label{eq:} \end{equation} where $U_{ij}=U_i\cap U_j={T_\X}\setminus V(g_ig_j)$. Now consider $\mathscr X_i=\Spf (A/A_{g_i\mathrm{-tors}})$, where $A_{g_i-\mathrm{tors}}$ denotes the $g_i$-torsion of $A$, seen as a special formal scheme over $k[[t]]$ by sending $t$ to $g_i$. Observe that we have $\mathscr X_i^\beth\cong U_i$, because every element of $A_{g_i-\mathrm{tors}}$ is of $g_i$-torsion. Since $A$ is normal, de Jong's theorem applies to $\mathscr X_i$, yielding $A/A_{g_i\mathrm{-tors}} \stackrel{\sim}{\to}\mathcal O^\circ_{\mathscr X_i^\beth}\big(\mathscr X_i^\beth\big)\cong\mathcal O^\circ_{T_\X}(U_i)$. Similarly, we obtain $\mathcal O^\circ_{T_\X}(U_{ij})\cong A/A_{g_ig_j\mathrm{-tors}}$, and the restriction maps $\mathcal O^\circ_{T_\X}(U_i) \to \mathcal O^\circ_{T_\X}(U_{ij})$ are given by the quotient by $A_{g_ig_j\mathrm{-tors}}$, observing that $A_{g_i\mathrm{-tors}}\subset A_{g_ig_j\mathrm{-tors}}$. The canonical morphism $A\to\mathcal O^\circ_{T_\X}({T_\X})$ is then induced from \eqref{eq:} by the quotient maps $A\to A_{g_i\mathrm{-tors}}$. It follows that it is an isomorphism, because the equalizer of $\prod_{i} A_{g_i\mathrm{-tors}} {{{} \atop \longrightarrow}\atop{\longrightarrow \atop {}}} \prod_{i,j} A/(A_{g_i\mathrm{-tors}},A_{g_j\mathrm{-tors}})$ is $A/(\cap_{i=1}^sA_{g_i\mathrm{-tors}})=A/A_{\mathrm{tors}}$, which coincides with $A$ because $A$ is admissible. \end{comment} \end{proof} \pa{It $T$ is an object of ${\mathcal C}$, we define the sheaf $\mathcal O^{\circ\circ}_T$ on $T$ as the subsheaf of $\mathcal O^\circ_T$ consisting of the sections that are non-invertible in every stalk. If $T={T_\X}$ is the normalized space of a special formal scheme $\mathscr X$ over $k$ that is covered by the formal spectra of normal and admissible special $k$-algebras, then $\mathcal O^{\circ\circ}_T$ coincides with the sheaf $\mathcal O^{\circ\circ}_{T_\X}$ defined earlier. Moreover, in this case the largest ideal of definition of $\mathscr X$ coincides with $(\Sp_{\mathscr X})^\mathcal O^{\circ\circ}_{T_\X}$. Indeed, the elements of the largest ideal of definition of $\mathscr X$ are precisely the ones that are topologically nilpotent, and this property can be verified by looking at the absolute values at every point. } \Pa{Example (continued)}{\label{analytic_structure_valuative_tree_continued} We have discussed in \ref{analytic_structure_valuative_tree} a cover of the valuative tree ${T_\X}=T_{\Spf(\mathbb C[[X,Y]])}$ by $\mathbb C((t))$-analytic curves. Using Corollary~\ref{dejong_normalized} we now show that there is no global $\mathbb C((t))$-analytic structure on ${T_\X}$. To see this, assume that ${T_\X}\cong\forg(C)$ for some $\mathbb C((t))$-analytic space $C$, where $k((t))$ is endowed with the $t$-adic absolute value such that $\|t\|=r$. Then the image of $t$ in $\mathcal O_C(C)\cong\mathcal O_{T_\X}({T_\X})$, which by abuse of notation we still denote by $t$, has to be a nowhere vanishing function that is strictly bounded by 1 (because it takes the constant value $r<1$ on $C$), hence in particular an element of $\mathcal O^{\circ\circ}_{T_\X}({T_\X})$. Since $\mathcal O^{\circ\circ}_{T_\X}({T_\X})$ coincides with the largest ideal of definition of $\mathcal O^\circ_{T_\X}({T_\X})\cong$, which by Corollary~\ref{dejong_normalized} is isomorphic to $\mathbb C[[X,Y]]$, the element $t$ is a complex power series in $X$ and $Y$ with no constant term and therefore defines the germ of a curve at the origin of the affine plane $\mathbb A^2_\mathbb{C}$. Then the order of vanishing at the origin along this germ defines a point of ${T_\X}$ on which $t$ vanishes, giving a contradiction. } \begin{rem} There are other examples of subspaces of analytic spaces that are naturally analytic spaces locally but do not have a canonical field of definition. This is the case for the analytic boundaries of affinoid domains. This kind of behavior appears for example in \cite[Lemme 3.1]{Ducros12}. \end{rem} We conclude the section by applying the results we have obtained to the study of admissible formal $k$-schemes. We start with an easy lemma. \begin{lem}\label{lem_admissible_1} Let $A$ be a special $k$-algebra. Then the morphism of formal schemes $\Spf(A/A_{\tors})\to\Spf A$ induced by the quotient $\pi:A\to A/A_{\tors}$ gives an isomorphism on the level of normalized Berkovich spaces. \end{lem} \begin{proof} Let $\{g_1,\ldots,g_s\}$ be a set of generators of an ideal of definition of $A$ and denote by $\mathscr X$ and $\mathscr X'$ the formal spectra of $A$ and $A/A_{\tors}$ respectively. Then ${T_\X}$ is covered by the Berkovich spaces $\mathscr X^\beth_{g_i}$, the space $T_{\mathscr X'}$ is covered by the $(\mathscr X')^\beth_{g_i}$ and the morphism $f:T_{\mathscr X'}\to{T_\X}$ induced by $\pi$ is locally the morphism of Berkovich spaces induced by the morphism of special formal $k[[t]]$-schemes $(\mathscr X')_{g_i}\to\mathscr X_{g_i}$ coming from $\pi$. The latter is an isomorphism since every element of $A_{\tors}$ is $g_i$-torsion, hence $f$ is an isomorphism. \end{proof} We deduce the following result, which in turn implies that a formal $k[[t]]$-scheme of finite type is an admissible special formal $k$-scheme if and only if it is admissible in the classical sense. \begin{prop}\label{admissible} Let $\mathscr X=\Spf A$ be an affine special formal scheme over $k$. Then $\mathscr X$ is admissible if and only if the special $k$-algebra $A$ is admissible. \end{prop} \begin{proof} To prove the ``if'' part, since the topology on $\mathscr X$ is generated by affine open formal subschemes, it is enough to show that the map $\varphi:A\to\mathcal O_{{\X^}}({\X^})$ is injective whenever $A$ is admissible; \ref{admissible_equivalence} will then allow to conclude. So let $a$ be an element of $A\setminus\{0\}$ such that $\varphi(a)=0$. Choose $t$ in the largest ideal of definition of $A$ such that $a\notin A_{t-\tors}$ and consider the following commutative diagram: \begin{displaymath} \xymatrix@R=1.5pc@M=3pt@L=3pt{ A \ar[d]_\varphi \ar[r]^(.33){\pi} & A\otimes_{k[[t]]}k((t)) \ar@<-4ex>@{^(->}[d] \\ \mathcal O_{{\X^}}\big({\X^}\big) \ar[r] & \mathcal O_{\mathscr X_t^\beth}\big(\mathscr X_t^\beth\big)} \end{displaymath} where the bottom map is the restriction map under the identification of $\mathscr X_t^\beth$ with the subspace $\pi\big({\X^}\setminus V(t)\big)$ of ${T_\X}$. The right vertical arrow is injective, as shown in \ref{canonical_injection_special_algebras}. Since $a\notin A_{t-\tors}$ then $a$ is sent to a nonzero element by the top map, hence also $\varphi(a)$ is different from zero, which is what we had to prove. Now, to prove the ``only if'' part, denote by $\pi$ the quotient map $A\to A/A_{\tors}$, by $\mathscr X'$ the formal spectrum of $A/A_{\tors}$, and consider the following commutative diagram: \begin{displaymath} \xymatrix@C=2.5pc@R=1.5pc@M=3pt@L=3pt{ A \ar[d]_\varphi \ar[r]^(.4)\pi & A/A_{\tors} \ar[d]_\psi\\ \mathcal O_{{\X^}}\big({\X^}\big) \ar[r]^(.48)\simeq & \mathcal O_{\mathscr X'^}\big(\mathscr X'^\big) } \end{displaymath} where the bottom arrow is induced by the morphism induced on formal spectra by $\pi$; it is an isomorphism thanks to Lemma~\ref{lem_admissible_1}. Since $\psi$ is injective from the previous part, it follows that $\varphi$ is injective only if $\pi$ is injective, that is only if $A_{\tors}=0$. \end{proof} \pa{\label{associated_admissible} Let $\mathscr X$ be a special formal scheme over $k$, and let $\mathscr{T}_\mathscr X$ be the subsheaf of $\mathcal O_\mathscr X$ such that $\mathscr T_\mathscr X(\Spf A)=A_{\tors}$ for every affine subscheme $\Spf A$ of $\mathscr X$. It's a coherent ideal subsheaf of $\mathcal O_\mathscr X$, so we can consider the quotient $\mathcal O_\mathscr X/\mathscr T_\mathscr X$. The special formal scheme $\mathscr X_{\mathrm{adm}}$ is defined as the closed formal subscheme of $\mathscr X$ defined by $\mathscr T$. It is an admissible special formal scheme over $k$ that we call the \defi{admissible special formal scheme associated with} $\mathscr X$. Its normalized Berkovich space coincides with the one of $\mathscr X$. } The following proposition is the analogue of an important classical result for formal $R$-schemes of finite type. \begin{prop}\label{sp_surjective} Let $\mathscr X$ be an admissible special formal $k$-scheme. Then the maps $\Sp_\mathscr X\colon{\X^}\to\mathscr X$ and $\Sp_\mathscr X\colon{T_\X}\to\mathscr X$ are surjective. \end{prop} \begin{proof} We can replace $\mathscr X$ by its formal blowup along $\mathscr X_0$, since the blowup morphism induces a surjective map between the reductions. Indeed, without loss of generality we can assume that $\mathscr X=\Spf(A)$ is affine and since it is admissible then no component of $\Spec(A)$ is contained in $\mathscr X_0$. Therefore the image of the scheme theoretic blowup of $\Spec(A)$ along $\mathscr X_0$, which is closed since the blowup is a proper map and has to contain the complement $\Spec(A)\setminus\mathscr X_0$, is all of $\Spec(A)$. Note that the blown up formal scheme is still admissible, because its ideal of definition is locally principal, generated by a regular element. By replacing $\mathscr X$ with an affine open formal subscheme whose ideal of definition is principal, generated by an element $t$, we can assume that $\mathscr X_t=\mathscr X=\Spf(A)$ is an affine formal scheme, flat and of finite type over $k[[t]]$. Furthermore, we can assume that $A$ is integrally closed in $A[t^{-1}]$, since the morphism of formal schemes induced by taking the integral closure is surjective by \cite[V.2.1, Theorem 1]{Bourbaki-CommutativeAlgebra1-7}. By \cite[2.1]{MartinKappen15} we have $A\cong\mathcal O^\circ_{\mathscr X_t^\beth}\big({\mathscr X_t^\beth}\big)$, therefore $(\mathscr X_t)_0=\mathscr X_0$ is the canonical reduction of the affinoid space $\mathscr X_t^\beth$, and the map $\Sp_{\mathscr X_t}\colon\mathscr X_t^\beth\to\mathscr X_0$ coincides with the reduction map of \cite[\S2.4]{Ber90}. Therefore $\Sp_{\mathscr X_t}$ is surjective by \cite[2.4.4(i)]{Ber90}. The surjectivity of $\Sp_\mathscr X\colon{\X^}\to\mathscr X$ follows from the fact that $\iota\circ\Sp_\mathscr X=\Sp_{\mathscr X_t}$, where $\iota\colon\mathscr X^\beth_t\to{\X^}$ is the natural inclusion of Lemma~\ref{lemma_omeo}. \end{proof} \section{Affinoid domains and atlases} \label{section_3.3} In this section we develop more thoroughly the analogy between normalized spaces over $k$ and analytic spaces over $k((t))$, by defining the class of affinoid domains of a normalized space and showing that they behave like the affinoid domains of analytic spaces. In particular, in Proposition~\ref{projection_of_affinoids} we show that the $G$-topology of a normalized space can be described in terms of its affinoid domains, and in Proposition~\ref{existence_atlas} we prove that normalized spaces are $G$-covered by finitely many affinoid domains. Theorem~\ref{theorem_affinoids_intrinsic} shows that the property of being an affinoid domain of a normalized space is intrinsic, not depending on the choice of a $k((t))$-analytic structure. \pa{ Let $V$ be an object of ${\mathcal C}$. We say that $V$ is {\it affinoid} if it is isomorphic to $\forg(X)$ for some strictly affinoid $k((t))$-analytic space $X$. A $G$-admissible subspace $V$ of an object $T$ of ${\mathcal C}$ that is affinoid is said to be an {\it affinoid domain} of $T$. } \pa{ Equivalently, $V$ is affinoid if and only if it is isomorphic to the normalized space ${T_\X}$ of some affine formal $k[[t]]$-scheme of finite type $\mathscr X_t$, since in this case ${T_\X}\cong\forg\big(\mathscr X_t^\beth\big)$. If we want to remember the element $t\in \mathcal O^\circ_V(V)$ that is the image of $t$ under the canonical homomorphism $k[[t]]\to \mathcal O^\circ_V(V)$, we say that $V$ is {\it affinoid with respect to the parameter} $t$, and by abuse of notation we will denote by $V_t$ both a strictly affinoid $k((t))$-analytic space whose image in ${\mathcal C}$ is isomorphic to $V$ (a choice of $\|t\|\in\left]0,1\right[$ is implicit here) and the affinoid space $V$ itself. Observe that the parameter $t$ is actually an element of $\mathcal O^{\circ\circ}_V(V)$. } \begin{rem}\label{remark_affinoid_not_canonical} If $V$ is an object of ${\mathcal C}$ that is affinoid with respect to two different parameters $t_1$ and $t_2$, it is not true in general that $V_{t_1}$ and $V_{t_2}$ are isomorphic as $k((t))$-analytic spaces. For example, the strictly affinoid $k((t))$-analytic spaces $\mathcal M\big({k((t))}\big)$ and $\mathcal M\big({k((t))\{X\}/(X^2-t)}\big)$ are not isomorphic since $k((t))$ and $k((t))\{X\}/(X^2-t)$ are not isomorphic as $k((t))$-algebras, but the associated normalized spaces are isomorphic because $k((t))$ and $k((t))\{X\}/(X^2-t)\cong k((X))$ are isomorphic as special $k$-algebras. \end{rem} The following proposition pushes further the analogies between usual Berkovich spaces and normalized spaces, showing that the $G$-topology of a normalized space can be described in terms of its affinoid domains. \begin{prop}\label{projection_of_affinoids} Let $\mathscr X$ be a special formal scheme over $k$ and let $U$ be an analytic domain of the normalized space ${T_\X}$ of $\mathscr X$. Then $U$ is $G$-covered by affinoid domains of ${T_\X}$. \end{prop} \begin{proof} Since ${T_\X}$ is $G$-covered by the normalized spaces of the affine open formal subschemes of $\mathscr X$, we can assume that $\mathscr X$ is itself affine. Assume that $U$ is an analytic domain of ${T_\X}$, i.e. that $\pi^{-1}(U)$ is an analytic domain of ${\X^}$. Cover ${T_\X}$ by the $k((t))$-analytic spaces $\mathscr X_{t_i}^\beth$, for $t_i$ ranging over a finite set of generators of an ideal of definition of $\mathscr X$, and set $U_{i}=\pi^{-1}(U)\cap\mathscr X_{t_i}^\beth$. Now, each $U_{i}$ is an analytic domain of $\mathscr X_{t_i}^\beth$, so it is $G$-covered by affinoid domains $V_{i,j}$ of $\mathscr X_{t_i}^\beth$. Therefore $\big\{\pi(V_{i,j})\big\}_j$ is a $G$-cover of $\pi(U_i)$ by affinoid domains of ${T_\X}$. Since $\big\{\pi\big(\mathscr X_{t_i}^\beth\big)\big\}_i$ is a finite open cover of ${T_\X}$, the $\pi(U_{i})$ form a finite open cover of $U$, hence $\big\{\pi(V_{i,j})\big\}_{i,j}$ is a $G$-cover of $U$ by affinoid domains of ${T_\X}$. \end{proof} \begin{rem} Proposition~\ref{projection_of_affinoids} tells us that we can think about the $G$-topology of ${T_\X}$ the same way we think about the one of a Berkovich space. If $U$ is a subset of ${T_\X}$, then $U$ is an analytic domain if and only if there exists a family $\{V_i\}_{i\in I}$ of affinoid domains of ${T_\X}$ contained in $U$ such that the following property holds: for every point $x$ of $U$, there exists a finite subset $I_x$ of $I$ such that $\bigcup_{i\in I_x}V_i$ contains an open neighborhood of $x$ and $x\in\bigcap_{i\in I_x}V_i$. \end{rem} \pa{ We showed in Corollary~\ref{locally_analytic} that the normalized Berkovich space ${T_\X}$ of a special formal $k$-scheme $\mathscr X$ is $G$-locally a $k((t))$-analytic space. We will now describe a second way of covering ${T_\X}$ by $k((t))$-analytic spaces that will be very useful later; the price to pay is that we are obliged to change the formal scheme $\mathscr X$. If $T$ is an object of ${\mathcal C}$, we define an \defi{atlas} of $T$ to be a $G$-cover of $T$ by affinoid domains. } \begin{prop}\label{existence_atlas} Let $\mathscr X$ be a special formal $k$-scheme. Then the normalized Berkovich space ${T_\X}$ of $\mathscr X$ admits a finite atlas. \end{prop} \begin{proof} By replacing $\mathscr X$ with the associated admissible special formal $k$-scheme, as defined in \ref{associated_admissible}, we can assume that $\mathscr X$ is admissible. Performing an admissible blowup $\mathscr X'\to\mathscr X$ we can assume that the largest ideal of definition of $\mathscr X$ is locally principal. Now, by locally sending $t$ to a generator of this ideal, we cover $\mathscr X$ by finitely many affine open formal schemes of finite type over $k[[t]]$. Their normalized spaces then form an atlas of ${T_\X}$. \end{proof} \begin{ex} In the case of the valuative tree of Example~\ref{valtree}, we get an atlas by blowing up the origin of $\mathbb A^2_\mathbb{C}$ and using the two charts of the blowup. \end{ex} \pa{ If $X$ is a strict $k((t))$-affinoid space and $\mathscr X_t=\Spf(A)$ is a flat formal $k[[t]]$-scheme of finite type such that $\mathscr X_t^\beth\cong X$, then by flatness $A$ injects into ${\mathcal O}_X(X)\cong A \otimes_{k[[t]]} k((t))$, and so $X$ is reduced if and only if $A$ is reduced. } \pa{ If $X$ is a reduced strict $k((t))$-affinoid space, then the algebra $\mathcal O^\circ_X(X)$ is topologically of finite type over $k[[t]]$ by the Grauert-Remmert finiteness theorem \cite[\S4, Endlichkeitssatz]{GrauertRemmert1966}, and we have isomorphisms of $k((t))$-analytic spaces $X \cong \mathcal M\big({\mathcal O^\circ_X(X)\hat\otimes_{k[[t]]}k((t))}\big) \cong \Spf\big(\mathcal O^\circ_X(X)\big)_t^\beth$. If $\forg(X)$ is isomorphic to an object $V$ of ${\mathcal C}$ then obviously $\mathcal O^\circ_X(X)\cong\mathcal O^\circ_V(V)$ as $k$-algebras. Therefore, $V$ is reduced affinoid if and only if the $k$-algebra $\mathcal O^\circ_V(V)$ can be endowed with a structure of a reduced $k[[t]]$-algebra topologically of finite type and $V$ is isomorphic to $\forg\big(\Spf(\mathcal O^\circ_V(V))_t^\beth\big) \cong T_{\Spf(\mathcal O^\circ_V(V))}$ in ${\mathcal C}$. The special formal $k$-scheme $\Spf(\mathcal O^\circ_V(V))$ can then be seen as a \emph{canonical formal model} for $V$. } \pa{ Let $V$ be an affinoid object ${\mathcal C}$ with respect to the parameter $t\in\mathcal O^{\circ\circ}_V(V)$. We say that $V$ is \emph{distinguished} (with respect to the parameter $t$) if it is reduced and $\mathcal O^\circ_V(V)\otimes_{k[[t]]}k$ is reduced as well. } \pa{\label{distinguished_affinoid_equivalence} Let $X={\mathcal M}({\mathcal A})$ be a strictly affinoid $k((t))$-analytic space. Then $\forg(X)$ is distinguished with respect to $t$ if and only if the affinoid algebra ${\mathcal A}$ is distinguished in the classical sense, see \cite[\S6.4.3]{BGR}. Indeed, ${\mathcal A}$ is distinguished if and only if it is reduced and its spectral norm $\|\cdot\|_{\sup}$ takes values in $\|k((t))\|$ (this is \cite[6.4.3/1]{BGR}, since by \cite[3.6]{BGR} $k((t))$ is a stable field). In particular, to show both implication we can assume that $X$ is reduced, and therefore its canonical model is $\mathscr X_t=\Spf\mathcal O^\circ_X(X)$. Observe that $\|{\mathcal A}^\times\|_{\sup}=\|k((t))^\times\|$ if and only if $t\mathcal O^\circ_X(X)=\big\{f\in{\mathcal A}\,\big\|\,\|f\|_{\sup}\leq\|t\|\big\}$ coincides with $\mathcal O^{\circ\circ}_X(X)=\big\{f\in{\mathcal A}\,\big\|\,\|f\|_{\sup}<1\big\}$. It follows that if ${\mathcal A}$ is distinguished then the special fiber $\mathscr X_{t,s}=\Spec\big(\mathcal O^\circ_X(X)/ t\mathcal O^\circ_X(X)\big)$ of $\mathscr X$ coincides with the usual reduction $\Spec\big(\mathcal O^\circ_X(X)/\mathcal O^{\circ\circ}_X(X)\big)$ of the affinoid $X$, which in particular proves that $\forg(X)$ is distinguished. Conversely, assume that $\|{\mathcal A}\|_{\sup}\supsetneq\|k((t))\|$, so that ${\mathcal A}$ is not distinguished. Then there exists an element $f\in {\mathcal A}$ such that $\|t\|<\|f\|_{\sup}<1$. This implies that there exists some $n>0$ such that $f^n\in t\mathcal O^\circ_X(X)$ while $f\in \mathcal O^\circ_X(X)\setminus t\mathcal O^\circ_X(X)$, hence $\mathscr X_{t,s}$ is not reduced, which proves that $\forg(X)$ is not distinguished. This also shows that an affinoid object $V$ with respect to the parameter $t\in\mathcal O^{\circ\circ}_V(V)$ is distinguished with respect to $t$ if and only if $t$ is a generator of $\mathcal O^{\circ\circ}_V(V)$. } In the remaining part of the section we give a criterion for a reduced object of ${\mathcal C}$ to be affinoid that does not require checking the existence of a parameter $t$, following the approach of \cite{Liu90} for rigid analytic spaces. \pa{\label{definition_pseudo-affinoid} Let $X$ be a locally ringed $G$-topological space. Following \cite{Liu90}, we say that $X$ is a \defi{Stein space} if $\mathcal O_X$ is coherent (considered as a sheaf of modules over itself) and we have $H^n(X,\mathscr F)=0$ for every coherent sheaf of $\mathcal O_X$-modules $\mathscr F$ and every $n\geq1$. \\ Let $T$ be an object of ${\mathcal C}$. We say that $T$ is \defi{compact} if it is Hausdorff and every $G$-cover of $T$ has a finite $G$-subcover. Finally, we say that $T$ is \defi{pseudo-affinoid} if it is isomorphic to $\forg\big(\mathscr X_t^\beth\big)$ for some affine special formal scheme $\mathscr X_t$ over $k[[t]]$. } \begin{prop}\label{pseudo-affinoid_stein} Every pseudo-affinoid object of ${\mathcal C}$ is a Stein space. If $X$ is a $k((t))$-analytic space, then $X$ is a $k((t))$-affinoid space if and only if $\forg(X)$ is both compact and pseudo-affinoid. \end{prop} \begin{proof} If an object $T$ of ${\mathcal C}$ is pseudo-affinoid then as discussed in \ref{example_affine_beth_space} it is the increasing union of the affinoid domains $W_\varepsilon$, and we have surjective restriction morphisms $\mathcal O_T(W_\varepsilon)\to\mathcal O_T(W_{\varepsilon'})$ when $\varepsilon>\varepsilon'$, so $T$ is quasi-Stein in the sense of Kiehl. It follows that pseudo-affinoid objects are Stein spaces since Kiehl's Theorem B \cite[2.4]{Kie67} applies (see also \cite[\S2.1]{Nicaise09} for a definition of quasi-Stein and the statement of Kiehl's theorem). To prove the second claim, note that a $k((t))$-affinoid space $X$ is compact, so $\forg(X)$ is compact. The fact that it is pseudo-affinoid is standard: a model of $X$ is obtained by taking the formal spectrum of the image of $k[[t]]\{X_1,\ldots,X_n\}$ via an admissible epimorphism $k((t))\{X_1,\ldots,X_n\}\to\mathcal O_X(X)$. Conversely, if $\forg(X)$ is pseudo-affinoid then the $W_\varepsilon$ above form a $G$-cover of it and so, since this family is increasing, by compactness $\forg(X)$ coincides with one of the $W_\varepsilon$, which is an affinoid domain with respect to the parameter $t$. Therefore $X$ is itself affinoid. \end{proof} \pa{\label{recall_liu} Liu has proven in \cite[3.2]{Liu90} that if $X$ and $Y$ are two rigid spaces over a non-trivially valued non-archimedean field $K$ and $X$ is Stein and quasi-compact, then the canonical map \[ \Hom_{K\mathrm{-an}}(Y,X)\to\Hom_{K\mathrm{-alg}}\big(\mathcal O_X(X),\mathcal O_Y(Y)\big) \] is a bijection. He deduced that a rigid space $X$ over $K$ is strictly affinoid if and only if it is Stein, $\mathcal O_X(X)$ is a strict affinoid $K$-algebra and $X$ is quasi-compact \cite[3.2.1]{Liu90}. We will prove a similar result for the reduced objects of the category ${\mathcal C}$. } \begin{thm}\label{theorem_affinoids_intrinsic} Let $X$ be a reduced $k((t))$-analytic space. Then $X$ is strictly $k((t))$-affinoid if and only if $\forg(X)$ is Stein and compact and $\mathcal O^\circ_X(X)$, with its $\mathcal O^{\circ\circ}_X(X)$-adic topology, is a special $k$-algebra. \end{thm} \begin{proof} If $A$ is a reduced strictly affinoid $k((t))$-algebra then by \cite[\S4, Endlichkeitssatz]{GrauertRemmert1966} $A^\circ$ is a $k[[t]]$-algebra topologically of finite type. In particular it is special over $k$, so the ``only if'' implication is clear. For the converse implication, define $\mathscr X=\Spf\big(\mathcal O^\circ_X(X)\big)$. Since $X$ is a $k((t))$-analytic space, the image of $t$ in $\mathcal O_X(X)$ is strictly bounded by $1$, hence it's an element of the largest ideal of definition of $\mathcal O_\mathscr X(\mathscr X)$, so $\mathscr X_t$ is an affine special formal scheme over $k[[t]]$. It follows that $\mathscr X_t^\beth$ is a pseudo-affinoid space. Since $X$ is compact and $\|t\|<1$, we have $\mathcal O^\circ_X(X)[t^{-1}]\cong\mathcal O_X(X)$. Indeed, any $f$ in $\mathcal O_X(X)$ is bounded on $X$, hence $ft^n$ is bounded by $1$ for $n$ big enough. Therefore, by \cite[3.2]{Liu90}, as recalled in~\ref{recall_liu}, the canonical homomorphism of $k((t))$-algebras $\mathcal O^\circ_X(X)[t^{-1}]\to\mathcal O_{\mathscr X_t^\beth}\big(\mathscr X_t^\beth\big)$ induces a morphism of rigid $k((t))$-analytic spaces $\mathscr X_t^{\mathrm{rig}}\to X^{\mathrm{rig}}$. Observe that we have $X^{\mathrm{rig}}=\mathrm{Specmax}\big(\mathcal O_X(X)\big)$ by \cite[1.3]{Liu90}, and similarly $\mathscr X_t^{\mathrm{rig}}=\mathrm{Specmax}\big(\mathcal O_{\mathscr X_t^\beth}\big(\mathscr X_t^\beth\big)\big)$ by \cite[7.1.9]{deJ95}. On points, the morphism $\mathscr X_t^{\mathrm{rig}}\to X^{\mathrm{rig}}$ is then obtained by sending a maximal ideal of $\mathcal O_{\mathscr X_t^\beth}\big(\mathscr X_t^\beth\big)$ to the inverse image of this ideal under the composition $\mathcal O_X(X)\to\mathcal O_{\mathscr X_t^\beth}\big(\mathscr X_t^\beth\big)$. It follows from \cite[7.1.9]{deJ95} and \cite[1.3]{Liu90} that this morphism is a bijection and that moreover it induces isomorphisms at the level of completed local rings. Since $X^{\mathrm{rig}}$ is quasi-compact, it follows from \cite[\S7.3.3 Proposition 5]{BGR} that $X^{\mathrm{rig}}\to\mathscr X_t^{\mathrm{rig}}$ is an isomorphism of rigid spaces. Therefore by \cite{Ber93} the morphism of analytic spaces $X\to\mathscr X_t^{\beth}$ is an isomorphism as well. Hence, being both pseudo-affinoid and compact, $X$ is affinoid by Proposition~\ref{pseudo-affinoid_stein}. Since $\mathcal O^\circ_X(X)$ is a special $k$-algebra, $X$ is moreover strict. \end{proof} A $k((t))$-analytic space $X$ is Stein if and only if $\forg(X)$ is, because the groups $\mathrm H^n(X,\mathscr F)$ depend only on the locally ringed site $(X,\mathcal O_X)$ and not on the $k((t))$-algebra structure on $\mathcal O_X$. Since the same is true for the two other properties in the statement above, we obtain the following corollary. \begin{cor}\label{affinoids} Let $X$ and $X'$ be reduced $k((t))$-analytic spaces such that $\forg(X)\cong \forg(X')$. Then $X$ is strictly affinoid if and only if $X'$ is strictly affinoid. \end{cor} If $V$ is a subspace of an object of ${\mathcal C}$ of the form $\forg(X)$ for some $k((t))$-analytic space $X$, then $V$ inherits the a structure of a $k((t))$-analytic space. Therefore, the last corollary has the following useful consequence. \begin{cor}\label{affinoids2} Let $X$ be a $k((t))$-analytic space and let $V$ be a reduced analytic domain of $\forg(X)$. Then $V$ is an affinoid domain of $\forg(X)$ if and only if it is of the form $\forg(W)$ for some strict affinoid domain $W$ of $X$. Moreover, $V$ is distinguished with respect to $t\|_V$ if and only if $W$ is a distinguished strictly affinoid $k((t))$-analytic space. \end{cor} \section{Functoriality} \label{section_3.4} In this section we introduce the category of normalized spaces over $k$. The main result, Theorem~\ref{thm_functoriality}, is the analogue for normalized spaces of the fundamental result of Raynaud (\cite{Raynaud74}, a detailed proof is in \cite[4.1]{BosLut93}) that states that the functor $\mathscr X\mapsto\mathscr X^\beth$ induces an equivalence between the category of admissible formal schemes of finite type over a complete valuation ring of height one $R$, localized by the class of admissible blowups, and the category of compact and quasi-separated strictly analytic spaces over $\mathrm{Frac}(R)$. \pa{\label{definition_normalized_spaces} We say that an object $T$ of ${\mathcal C}$ is a \defi{normalized space} if it is compact and it has an atlas. Observe that such a $T$ is then \emph{quasi-separated}, which means that the intersection of any two affinoid domains of $T$, being a compact analytic domain of each of them, is a finite union of affinoid domains of $T$. A morphism $f\in \mbox{Hom}_{{\mathcal C}}(T',T)$ is said to be a \defi{morphism of normalized spaces} if there exist finite covers $\{V_i\}_{i\in I}$ of $T$ and $\{W_j\}_{j\in J}$ of $T'$ by affinoid domains such that for every $i$ in $I$ there is a subset $J_i$ of $J$ with $f^{-1}(V_i)=\cup_{j\in J_i}W_j$ and, for every $j$ in $J_i$, the restriction $f\|_{W_j}:W_j\to V_i$ of $f$ to $W_j$ is induced by a morphism of $k((t))$-analytic spaces. This means that $f\|_{W_j}$ is associated with a $k((t))$-homomorphism ${\mathcal O}_T(V_i)\to{\mathcal O}_{T'}(W_j)$, where ${\mathcal O}_T(V_i)$ and ${\mathcal O}_{T'}(W_j)$ are seen as $k((t))$-affinoid algebras. A more intrinsic way to see morphisms of normalized spaces is as those morphisms in ${\mathcal C}$ that are induced by adic morphisms between formal models, as will be clear from Theorem~\ref{thm_functoriality}. The \defi{category of normalized spaces} $(NAn_k)$ is the subcategory of ${\mathcal C}$ whose objects are normalized spaces and whose morphisms are morphisms of normalized spaces. } \pa{ We want to prove an analogue of Raynaud's theorem for the functor $T\colon \mathscr X\mapsto T_\mathscr X$ going from the category of admissible special formal $k$-schemes with adic morphisms, which we denote by $(SFor_k)$, to $(NAn_k)$. We will show that the functor $T$ is the localization of the category $(SFor_k)$ by the class $B$ of admissible formal blowups and we will characterize its essential image. The fact that $T$ sends admissible blowups to isomorphisms is~\ref{invariance_generic_fiber}. } \pa{ The category $(SFor_k)$ admits calculus of (right) fractions with respect to the class of morphisms $B$, in the sense of \cite[Ch. I]{GabrielZisman67}. This follows easily from the universal property of blowups and the results of \ref{basic properties blowup}. Therefore, the localized category $(SFor_k)_B$ can be described in a simple way: its objects are the objects of $(SFor_k)$, and a morphism $\mathscr Y\to\mathscr X$ is a two-step zigzag $$ \xymatrix@R=.3pc@M=3pt@L=3pt{ & \mathscr Y' \ar[dl]_w\ar[dr]^f\\ \mathscr Y && \mathscr X }$$ where $f$ is a morphism in $(SFor_k)$ and $w$ is an admissible blowup, modulo the equivalence relation given by further blowing up $\mathscr Y'$. Such a morphism can be thought of as a fraction $fw^{-1}$. Moreover, the localization functor $(SFor_k)\to(SFor_k)_B$ is left exact, and therefore preserves finite limits. } \begin{thm}\label{thm_functoriality} The functor $T:\mathscr X\mapsto{T_\X}$ induces an equivalence between the category $(SFor_k)_B$, the localization of the category $(SFor_k)$ of admissible special formal $k$-schemes with adic morphisms by the class $B$ of admissible blowups, and the category $(NAn_k)$. \end{thm} The remaining part of this section will be devoted to the proof of this result. \begin{lem}\label{L: model of affinoid} Let ${T_\X}$ be the normalized space of an admissible special formal $k$-scheme $\mathscr X$. Then: \begin{enumerate} \item If $V$ is an affinoid domain of ${T_\X}$, then there exist an admissible formal blowup $\mathscr X'\to\mathscr X$, a formal $k[[t]]$-scheme of finite type ${\mathcal V}$ and an open immersion of formal $k$-schemes ${\mathcal V}\hookrightarrow\mathscr X'$ inducing an isomorphism $T_{{\mathcal V}}\cong V$ in ${\mathcal C}$. \item If $\{V_j\}_{j\in J}$ is a finite atlas of ${T_\X}$, then there exist an admissible formal blowup $\mathscr X'\to\mathscr X$ and a cover $\{{\mathcal V}_j\}_{j\in J}$ of $\mathscr X$ by open formal subschemes such that $T_{{\mathcal V}_j}\cong V_j$ for every $j$. \end{enumerate} \end{lem} \begin{proof} We prove $(i)$ by reducing to the classical case of formal $k[[t]]$-schemes of finite type, where it is \cite[8.4.5]{Bosch14}. After an admissible blowup we can assume that $\mathscr X$ is covered by open formal $k[[t]]$-schemes of finite type $\mathscr X_i$, as in the proof of Proposition~\ref{existence_atlas}. Then ${T_\X}=\bigcup_i T_{\mathscr X_i}$, and so $V=\bigcup_i (V\cap T_{\mathscr X_i})$. Inducting on $i$, \cite[8.4.5]{Bosch14} tells us that after an admissible blowup $\mathscr X'_i$ of $\mathscr X_i$, which extends to an admissible blowup $\mathscr X'$ of $\mathscr X$, we can find an open formal $k[[t]]$-subscheme ${\mathcal V}_i$ of $\mathscr X'_i$, which is also an open formal $k$-subscheme of $\mathscr X'$, such that $T_{{\mathcal V}_i}=V_i$. We obtain an open formal $k$-subscheme ${\mathcal V}=\bigcup_i{\mathcal V}_i$ of an admissible blowup of $\mathscr X$ such that $T_{\mathcal V}=V$, which is what we wanted. To prove $(ii)$, we apply $(i)$ to get for every $j$ an admissible blowup $\mathscr X_j$ of $\mathscr X$ with an open formal subscheme ${\mathcal W}_j\subset\mathscr X_j$ such that $T_{{\mathcal W}_j}\cong V_j$. Using \ref{blowup: domination} we take an admissible blowup $\mathscr X'$ of $\mathscr X$ dominating all $\mathscr X_j$ via maps $f_j:\mathscr X'\to\mathscr X_j$, and we set ${\mathcal V}_j\coloneqq f_j^{-1}({\mathcal W}_j)$. The ${\mathcal V}_j$ form a cover of $\mathscr X'$ that satisfies the requirements since $T_{{\mathcal V}_j}\cong T_{{\mathcal W}_j}\cong V_j$. \end{proof} The rest of the proof of Theorem~\ref{thm_functoriality} will be divided in six steps. While the result could be proven in an analogous way as Raynaud did, we decided to deduce it from his result, to give an idea of how to apply to normalized spaces standard techniques over $k[[t]]$, in a similar way as what we did in Lemma~\ref{L: model of affinoid}. \begin{step} {\it The functor $T$ factors through $(NAn_k)$.} If $\mathscr X$ is a special formal scheme over $k$, then $T_\mathscr X$ is a normalized space, since it admits a finite atlas by Proposition~\ref{existence_atlas}. Now let $f:\mathscr Y\to\mathscr X$ be an adic morphism of special formal $k$-schemes. By replacing $\mathscr X$ and $\mathscr Y$ by admissible blowups we obtain an adic morphism $f':\mathscr Y'\to\mathscr X'$ and we can assume that $\mathscr X'$ is covered by affine open formal subschemes $\mathscr X_i$ of finite type over $k[[t]]$. The open formal $k$-subschemes $(f')^{-1}(\mathscr X_i)$ of $\mathscr Y'$ are themselves covered by affine open formal $k$-subschemes $\mathscr Y_{i,j}$, and since $f'$ is adic, the morphisms of formal $k$-schemes $f'\|_{\mathscr Y_{i,j}}:\mathscr Y_{i,j}\to\mathscr X_i$ can be upgraded to a morphism of formal $k[[t]]$-schemes, by choosing the $k[[t]]$-structure on $\mathscr Y_{i,j}$ given by the parameter $f'_\#t$, and therefore it induces a morphism of $k((t))$-analytic spaces $(\mathscr Y_{i,j})_t^\beth\to(\mathscr X_i)_t^\beth$. This shows that $T_f$ is a morphism in $(NAn_k)$. \end{step} \begin{step} {\it Faithfulness.} Let $f,g:\mathscr Y\to\mathscr X$ be two morphisms in $(SFor_k)$ such that the induced morphisms of normalized spaces $f_T,g_T:T_\mathscr Y\to{T_\X}$ coincide. Since $\mathscr X$ and $\mathscr Y$ are admissible, the specialization maps $T_\mathscr X\to\mathscr X$ and $T_\mathscr Y\to\mathscr Y$ are surjective by Proposition~\ref{sp_surjective}. Consider the following diagram: \begin{displaymath} \xymatrix@C=3.5pc@R=1.5pc@M=3pt@L=3pt{ T_\mathscr Y \ar@{>>}[d]_{\Sp_\mathscr Y} \ar[r]^{f_T=g_T} & T_\mathscr X \ar@{>>}[d]^-{\Sp_{\mathscr X}} \\ \mathscr Y \ar@<.7ex>[r]^f\ar@<-.7ex>[r]_g & \mathscr X } \end{displaymath} \noindent It commutes for both choices of the map on bottom, so it follows that $f$ and $g$ coincide as maps between the topological spaces underlying $\mathscr Y$ and $\mathscr X$. Therefore we can assume that $\mathscr X$ and $\mathscr Y$ are affine, $\mathscr X=\Spf(A)$ and $\mathscr Y=\Spf(B)$, so that $f$ and $g$ correspond to two k-algebra maps $f_\sharp$ and $g_\sharp:A\to B$ respectively. Consider then the following diagram \begin{displaymath} \xymatrix@R=1.5pc@C=3.5pc@M=4pt@L=3pt{ A \ar@{^{(}->}[d] \ar@<.7ex>[r]^{f_\sharp}\ar@<-.7ex>[r]_{g_\sharp} & B \ar@{^{(}->}[d] \\ \mathcal O({\X^}) \ar[r]_{(f^)_\sharp=(g^)_\sharp} & \mathcal O(\mathscr Y^) } \end{displaymath} that commutes for both choices of the map on top. The vertical arrows are injective since $\mathscr X$ and $\mathscr Y$ are admissible, so $f_\sharp=g_\sharp$, and hence $f=g$. \end{step} \begin{step} {\it Fullness (modulo admissible blowup).} Let $f:T'\to T$ be a morphism in $(NAn_k)$ and let $\mathscr X$ and $\mathscr Y$ be models of $T$ and $T'$ respectively. Given two finite affine covers of $\mathscr X$ and of $\mathscr Y$, using the fact that admissible blowups open formal subschemes can be extended by~\ref{blowup: extension}, after blowing up $\mathscr Y$ we can refine them to finite covers $\{(\mathscr V_i)_t\}$ of $\mathscr X$ and $\{(\mathscr W_j)_t\}$ of $\mathscr Y$ by affine formal schemes of finite type over $k[[t]]$ in such a way that, if we define $k((t))$-analytic spaces $V_i=(\mathscr V_i)_t^\beth$ and $W_j=(\mathscr W_j)_t^\beth$, then $\{\forg(V_i)\}$ and $\{\forg(W_i)\}$ are covers of $T$ and $T'$ respectively as the ones in the definition of a morphism of normalized spaces given in \ref{definition_normalized_spaces}; in particular for every $j$ there exist an $i$ and a morphism of $k((t))$-analytic spaces $W_j\to V_i$ that lifts $f\|_{\forg(W_j)}$. For every $j$ we use Raynaud's theorem (see in particular assertion (c) after Theorem 4.1 in \cite{BosLut93}): after blowing up $(\mathscr W_j)_t$ to $(\mathscr W_j')_t$, the $k((t))$-analytic morphism $W_j\to V_i$ lifts to a morphism $F:(\mathscr W_j')_t\to(\mathscr V_i)_t$ of formal schemes of finite type over $k[[t]]$. These morphisms glue to a morphism $F:\mathscr Y'\to\mathscr X$ of formal schemes over $k$ from a blowup $\mathscr Y'$ of $\mathscr Y$ to $\mathscr X$, and $T(F)=f$ since this is the case locally. The morphism $F$ is adic since it is locally a morphism of $k[[t]]$-formal schemes of finite type and such morphisms are always adic. \end{step} \begin{step}\label{proof raynaud isomorphism} {\it Isomorphisms come from admissible blowups.} If in the previous step we take for $f$ an isomorphism in ${\mathcal C}$, then it can be lifted to an admissible blowup $F:\mathscr Y'\to\mathscr X$. Indeed, if $f$ is an isomorphism, then the analytic morphisms $f\|_{W_j}:W_j\to V_i$ of the previous step are all immersions of an affinoid domain in a $k((t))$-analytic space, so we can use the analogous result in Raynaud theory. \end{step} \begin{step} {\it Existence of a model.} Let $T$ be an element of $(NAn_k)$ and let $\{X_i\}_{i\in I}$ be a finite atlas of $T$. We will prove the existence of a model of $T$, i.e. a special formal $k$-scheme $\mathscr X$ such that ${T_\X}\cong T$, by induction on the cardinality of $I$. If $I$ consists of only one element then $T$ is affinoid, and therefore has a model. If $I=\{1,2,\ldots,n\}$, then $X_n$ being affinoid has a model ${\mathcal U}$, and by induction we can find a model ${\mathcal V}$ of $V\coloneqq X_1\cup\cdots\cup X_{n-1}$. Set $W=V\cap X_n$. Since $T$ is quasi-separated, $W$ admits a finite cover by affinoid domains, and this cover can be enlarged to a cover of $V$ by affinoid domains. By Lemma~\ref{L: model of affinoid}, there exist an admissible blowup ${\mathcal V}'\to{\mathcal V}$ and an open immerson ${\mathcal W}_1\hookrightarrow{\mathcal V}'$ inducing an isomorphism $T_{{\mathcal W}_1}\cong W$. Similarly, there exist an admissible blowup ${\mathcal U}'\to{\mathcal U}$ and an open immerson ${\mathcal W}_2\hookrightarrow{\mathcal U}'$ inducing an isomorphism $T_{{\mathcal W}_2}\cong W$. Since ${\mathcal W}_1$ and ${\mathcal W}_2$ are both models of $W$, using {\it Step \ref{proof raynaud isomorphism}} we can find an admissible blowup ${\mathcal W}$ of both ${\mathcal W}_1$ and ${\mathcal W}_2$. These blowups can be extended using~\ref{blowup: extension} to admissible blowups ${\mathcal V}''\to{\mathcal V}$ and ${\mathcal U}''\to{\mathcal U}'$, so we can glue ${\mathcal V}''$ and ${\mathcal U}''$ along ${\mathcal W}$, obtaining a model of $T$. \end{step} \begin{step} {\it End of proof.} It remains to prove that the functor $T$ satisfies the universal property of the localization of categories. The fact that $T$ sends admissible blowups to isomorphisms is~\ref{invariance_generic_fiber}. Given a category ${\mathcal C}$ and a functor $F:(SFor_k)\to{\mathcal C}$ such that $F(b)$ is an isomorphism in ${\mathcal C}$ for every admissible blowup $b$, we need to show that $F$ factors as $G\circ T$ for a unique functor $G:(NAn_k)\to{\mathcal C}$. This is done in the exact same way as in the proof of Raynaud's theorem, using the basic results on blowups of~\ref{basic properties blowup}. \end{step} \noindent This completes the proof of Theorem~\ref{thm_functoriality}. \qedsymbol \begin{rem} In particular, an analytic space of the form ${\X^}$ for some special formal $k$-scheme $\mathscr X$ is uniquely determined by the associated normalized space. An explicit way of retrieving the topological space underlying ${\X^}$ from $T_\mathscr X$ is the following. If we cover ${T_\X}$ by affinoid subspaces $X_i$ with respect to parameters $t_i$, we need to glue the topological spaces $X_i\times\mathbb R_{>0}$ along subspaces homeomorphic to $X_{ij}\times\mathbb R_{>0}$. The gluing data is encoded in the $t_i$: if $x$ is a point of $X_{ij}$, we identify $(x,\gamma)\in X_i\times\mathbb R_{>0}$ to $(x,\lambda_{ij}(x)\gamma)\in X_j\times\mathbb R_{>0}$, where $\lambda_{ij}(x)\coloneqq \log\|t_j\|/\log\|t_i\|(x)$ is defined as in Remark~\ref{functions_on_T}. \end{rem} \section{Modifications of surfaces and vertex sets} \label{section_4.1} Starting from this section we move to the study of pairs $(X,Z)$, where $X$ is a surface over an arbitrary (trivially valued) field $k$ and $Z$ is a closed subvariety of $X$ containing its singular locus. After giving some general definitions, in Theorem~\ref{vertex_sets} we use normalized spaces to produce formal modifications of $(X,Z)$ with prescribed exceptional divisors. \pa{\label{setting_surfaces} Let $X$ be a surface over $k$, that is a geometrically integral and generically smooth $k$-scheme of dimension 2, and let $Z\subsetneq X$ be a nonempty closed subscheme whose support contains the singular locus of $X$. We denote by $\mathscr X$ the formal completion of $X$ along $Z$, and by $T_{X,Z}$ the normalized space ${T_\X}$ of $\mathscr X$. We also call $T_{X,Z}$ the \defi{normalized space of the pair $(X,Z)$}. As discussed in Example~\ref{example_explicit_normalization}, $T_{X,Z}$ can be viewed as a non-archimedean model for the link of $Z$ in $X$. We denote by $\widetilde{\Bl_{Z}X}$ the normalization of the blowup of $X$ along $Z$, and we write $\widetilde \mathscr X$ for the formal completion of $\widetilde{\Bl_{Z}X}$ along $\widetilde{\Bl_{Z}X}\times_{X}Z$. Then $\widetilde \mathscr X$ is another special formal $k$-scheme that is a model of $T_{X,Z}$. By~\cite[2.16.5]{Nicaise09}, $\widetilde{\mathscr X}$ is isomorphic to the normalization of the formal blowup of $\mathscr X$ along $Z$. } \pa{ A \defi{log modification} of the pair $(X,Z)$ is a pair $(Y,D)$ consisting of a normal $k$-variety $Y$ and a Cartier divisor $D$ of $Y$, together with a proper morphism of $k$-varieties $f:Y\to X$ such that $D=Y\times_XZ$ as subschemes of $Y$ and $f$ is an isomorphism outside of $D$. A log modification $(Y,D)$ of $(X,Z)$ is said to be a \defi{log resolution} of $(X,Z)$ if $Y$ is regular and $D$ has normal crossings (by which we mean that $D_{\mathrm{red}}$ has normal crossings, but not necessarily strict normal crossings, in the usual sense). Note that $D$ being Cartier is not equivalent to the set-theoretic inverse image $f^{-1}(Z)=D_{\mathrm{red}}$ being Cartier, therefore our notion of log resolution is different from the notion of good resolution that is sometimes found in the literature. } \pa{ A \defi{formal log modification} of the pair $(X,Z)$ is a normal special formal $k$-scheme $\mathscr Y$ together with an adic morphism $f:\mathscr Y\to\mathscr X$ that induces an isomorphism of normalized spaces $T_\mathscr Y\stackrel{\sim}{\longrightarrow}T_{X,Z}$, and such that $\mathscr Y\times_\mathscr X Z$ is a Cartier divisor of $\mathscr Y$. If moreover $\mathscr Y$ is regular and $\mathscr Y\times_\mathscr X Z$ has normal crossings in $\mathscr Y$, then $\mathscr Y$ is said to be a \defi{formal log resolution} of $(X,Z)$. } \begin{lem}\label{lemma_properness_formal_modifications} Let $f:\mathscr Y\to\mathscr X$ be a formal log modification of $(X,Z)$. Then $f$ is proper. \end{lem} \begin{proof} Since $f$ is adic by definition, it is enough to show that the induced morphism $f_0:\mathscr Y_0\to\mathscr X_0$ is a proper morphism of schemes. Consider the morphism $\alpha:T_\mathscr X\to T_\mathscr Y$, inverse of the isomorphism induced by $f$. Theorem~\ref{thm_functoriality} states that there exists an admissible blowup $\tau:\mathscr X'\to\mathscr X$ and a morphism $g:\mathscr X'\to\mathscr Y$ such that $\alpha$ is induced by $g$. Since the composition $\mathscr X'\stackrel{g}{\longrightarrow}\mathscr Y\stackrel{f}{\longrightarrow}\mathscr X$ is $\tau$, the induced map $f_0\circ g_0\colon\mathscr X'_0\to\mathscr X_0$ is proper. The map $g_0$ is surjective because $\Sp_{\mathscr Y}$ is surjective and the following diagram \begin{displaymath} \xymatrix@C=2.5pc@R=1.5pc@M=3pt@L=3pt{ T_{\mathscr X'} \ar[d]_{\Sp_{\mathscr X'}} \ar[r]^\simeq & T_\mathscr Y \ar[d]^-{\Sp_{\mathscr Y}} \\ \mathscr X'_0 \ar[r]_{g_0} & \mathscr Y_0 } \end{displaymath} is commutative. Since surjectivity is stable under base change by~\cite[3.5.2]{EGA1}, it follows that $f_0$ is universally closed and therefore proper. \end{proof} \pa{ If $(Y,D)\to(X,Z)$ is a log modification, then the formal completion $\mathscr Y=\widehat{Y/D}\to\mathscr X$ of $Y$ along $D$ is a formal log modification of $(X,Z)$. Such a formal log modification $\mathscr Y$ of $(X,Z)$ is said to be \defi{algebraizable}, and a log modification $(Y,D)$ of $(X,Z)$ such that $\mathscr Y\to\mathscr X$ is isomorphic to $\widehat{Y/D}\to\mathscr X$ is called an \defi{algebraization} of $\mathscr Y$. By Grothendieck's formal GAGA theorem \cite[5.1.4]{EGA3.1}, a log modification $(Y,D)\to(X,Z)$ is uniquely determined by the formal log modification $\widehat{Y/D}\to\mathscr X$ it algebraizes, and if $\mathscr Y\cong\widehat{Y/D}$ and $\mathscr Y'\cong\widehat{Y'/D'}$ are two formal log modifications that are algebraizable then $\Hom_\mathscr X(\mathscr Y',\mathscr Y)\cong\Hom_X(Y',Y)$. If $\mathscr Y$ is a formal log modification of $(X,Z)$ algebraized by the log modification $(Y,D)$, since both the properties of being regular and of having normal crossings are local properties and, by excellence, can be checked on completed local rings, then $\mathscr Y$ is a formal log resolution of $(X,Z)$ if and only if $(Y,D)$ is a log resolution of $(X,Z)$. In the following proposition we will prove that every formal log resolution if algebraized by a log resolution. Since the normal crossing condition on the exceptional divisor is not needed, we give a slightly more general statement. } \begin{prop}\label{proposition_algebraization_resolutions} Let $\mathscr Y$ be a formal log modification of $(X,Z)$ and assume that $\mathscr Y$ is regular. Then $\mathscr Y$ is algebraizable. If moreover $\mathscr Y$ is a formal log resolution, then it is algebraizable by a log resolution of $(X,Z)$. \end{prop} \begin{proof} Let $f\colon\mathscr Y\to\mathscr X$ be a formal log modification of $(X,Z)$ and assume that $\mathscr Y$ is regular. The map $f$ factors through a morphism $g:\mathscr Y\to\widetilde{\mathscr X}$. It follows from Lemma~\ref{lemma_properness_formal_modifications} that $g$ makes $\mathscr Y$ into a proper, adic formal $\widetilde{\mathscr X}$-scheme. Since $\widetilde\mathscr X$ is normal, $g$ is an isomorphism outside of the inverse image of a finite set of closed points of $\widetilde\mathscr X$. Let $\mathscr U$ be an open and affine formal subscheme of $\widetilde \mathscr X$ such that there is exactly one point $x$ in $\mathscr U$ such that $g\|_{g^{-1}(\mathscr U)}$ is an isomorphism outside of $g^{-1}(x)$, and denote by $E_1,\ldots,E_r$ the irreducible components of $g^{-1}(x)$. Since $g^{-1}(\mathscr U)\subset\mathscr Y$ is regular, each $E_i$ is a Cartier divisor on $g^{-1}(\mathscr U)$ (the theory of Cartier divisors is developed over any ringed space, see for example \cite[\S21]{EGA4.4}). The intersection matrix $(E_i\cdot E_j)_{1\leq i,j\leq r}$ is negative definite because the whole of $g^{-1}(x)$ gets contracted to $x$ by $g$, so by the elementary (albeit long) linear algebra computation in \cite[page 138, $(ii)$]{Lipman69} we can find integers $a_i\leq0$ such that if we set $E=\sum_i a_iE_i$ we have $E\cdot E_i<0$ for every $i=1,\ldots, r$. Consider the invertible sheaf ${\mathcal L}=\mathcal O_{g^{-1}(\mathscr U)}(E)\subset\mathcal O_{g^{-1}(\mathscr U)}$ on ${g^{-1}(\mathscr U)}$ associated with $E$, and denote by ${\mathcal L}_0$ the base change of ${\mathcal L}$ to ${g^{-1}(\mathscr U)}\times_{\mathscr X}Z$. Let $y$ be a point of $\widetilde \mathscr X_0$. If $y\neq x$ then $g_0^{-1}(y)$ is a point and ${\mathcal L}_0\|_{g_0^{-1}(y)}$ is therefore ample. On the other hand, if $y=x$ then ${\mathcal L}_0\|_{g_0^{-1}(y)}$ is ample by Kleiman's criterion \cite[\S III.1]{Kleiman66} because of the inequalities $E\cdot E_i<0$. Since $g_0$ is proper, by~\cite[4.7.1]{EGA3.1} this implies that the invertible sheaf ${\mathcal L}_0$ is relatively ample with respect to $g\|_{g^{-1}(\mathscr U)}$, and therefore it is ample since $\mathscr U$ is affine. Since $\mathscr U$ is algebraized by an open subscheme $U$ of $\widetilde{\Bl_{Z}X}$, Grothendieck's existence theorem~\cite[5.4.5]{EGA3.1} guarantees that $g^{-1}(\mathscr U)$ is algebraized by a proper $U$-scheme. Since those algebraizations are unique, we can glue them and so we deduce that $\mathscr Y$ is algebraized by a $k$-scheme $Y$, endowed with a proper morphism $g:Y\to\widetilde{\Bl_{Z}X}$. Set $D=Y\times_X Z$; then $D$ is Cartier in $Y$ by the universal property of ${\Bl_{Z}X}$. Since $f\colon\mathscr Y\to\mathscr X$, hence the morphism $\mathscr Y\to\widetilde{\mathscr X}$, induces an isomorphism at the level of normalized spaces, by Theorem~\ref{thm_functoriality} it is an isomorphism modulo admissible blowups, therefore its algebraization $g$ induces an isomorphism outside of $D$, and so $(Y,D)$ is a log modification of $(X,Z)$ algebraizing $\mathscr Y$. \end{proof} \pa{If $f\colon\mathscr Y\to\mathscr X$ and $f'\colon\mathscr Y'\to\mathscr X$ are two formal log modifications of $(X,Z)$, we say that $\mathscr Y'$ \defi{dominates} $\mathscr Y$ if there is a morphism of formal schemes $g\colon\mathscr Y'\to \mathscr Y$ such that $f\circ g=f'$; we denote this by $\mathscr Y'\geq \mathscr Y$. Note that if such a morphism $g$ exists, then it is unique. This follows from the fact that $f$ is uniquely determined by the fact that it induces an isomorphism of normalized spaces: to prove this we can assume that $\mathscr X$ and $\mathscr Y$ are both affine, and conclude by observing that the image of an element of ${\mathcal O}_\mathscr X(\mathscr X)$ in ${\mathcal O}_\mathscr Y(\mathscr Y)$ only depends on its image in ${\mathcal O}_{T_\mathscr Y}(T_\mathscr Y)$ because the natural map ${\mathcal O}_\mathscr Y(\mathscr Y)\to{\mathcal O}_{T_\mathscr Y}(T_\mathscr Y)$ is injective. Two formal log modifications $\mathscr Y'$ and $\mathscr Y$ are \defi{isomorphic} if $\mathscr Y \geq \mathscr Y'\geq \mathscr Y$, i.e. if there is an isomorphism $\mathscr Y' \cong \mathscr Y$ commuting with the morphisms to $\mathscr X$. The domination relation is a filtered partial order on the set of isomorphism classes of formal log modifications of $X$. By the universal properties of blowup and normalization, this partially ordered set has $\widetilde{\mathscr X}$ as unique smallest element. } \pa{ If $\mathscr Y$ is a formal log modification of $(X,Z)$, we denote by $\Div_{X,Z}(\mathscr Y)$ the finite non-empty subset of $T_{X,Z}$ consisting of the $\mathbb R_{>0}$-orbits of the divisorial valuations associated with the components of $\mathscr Y\times_\mathscr X Z$. If $\mathscr Y$ is algebraized by a log modification $(Y,D)$, we will also denote $\Div_{X,Z}(\mathscr Y)$ by $\Div_{X,Z}(Y)$. We write $\Div_{X,Z}$ for the union of the sets $\Div_{X,Z}(\mathscr Y)$, for $\mathscr Y$ ranging over all the formal log modifications of $(X,Z)$; it is the set of the $\mathbb R_{>0}$-orbits of the divisorial valuations on $X$ whose centers lie in $Z$. We call the elements of $\Div_{X,Z}$ the \defi{divisorial points} of $T_{X,Z}$. We call the finite set $\Div_{X,Z}\big(\widetilde{\mathscr X}\big)$ of divisorial points of $T_{X,Z}$ the \defi{analytic boundary} of $T_{X,Z}$, and we denote it by $\partial^{\mathrm{an}}\Txz$. Since any formal log modification $\mathscr Y$ of $(X,Z)$ dominates $\widetilde{\mathscr X}$, we always have $\partial^{\mathrm{an}}\Txz\subset\Div_{X,Z}(\mathscr Y)$. } \begin{ex} If $X=\mathbb A^2_\mathbb{C}$ and $Z=\{0\}$, so that $T_{X,Z}$ is the valuative tree as in Example~\ref{valtree}, then $\partial^{\mathrm{an}}\Txz$ consists of one point, corresponding to the exceptional divisor of the blow-up of the plane at the origin. This is the $\mathbb R_{>0}$-orbit of the order of vanishing at the origin of the complex plane, that is what Favre and Jonsson call the \defi{multiplicity valuation}. \end{ex} \begin{lem}\label{L: divisorial valuation specialization} Let $\mathscr Y$ be a formal log modification of $(X,Z)$, and let $x$ be a closed point of $\mathscr Y\times_\mathscr X Z$. Then: \begin{enumerate} \item $\Div_{X,Z}(\mathscr Y)$ is the inverse image via the specialization morphism $\Sp_\mathscr Y:T_{X,Z}\to \mathscr Y$ of the set of generic points of the irreducible components of $\mathscr Y\times_\mathscr X Z$; \item the open subset $\Sp_\mathscr Y^{-1}(x)$ of $T_{X,Z}$ can be given the structure of a pseudo-affinoid space. \end{enumerate} \end{lem} \begin{proof} If $\eta$ is the generic point of an irreducible component of $\mathscr Y\times_\mathscr X Z$, then the associated divisorial point specializes to $\eta$. Moreover, being a discrete valuation ring, $\mathcal O_{\mathscr Y,\eta}$ is the only valuation ring dominating $\mathcal O_{\mathscr Y,\eta}$, which means that there is only one point of $T_{X,Z}$ specializing to $\eta$, proving $(i)$. To show $(ii)$, set $\mathscr U=\Spf\big(\widehat{\mathcal O_{\mathscr Y,x}}\big)$. By~\ref{lemma fiber reduction}, the inverse image of $x$ in $\mathscr Y^{\beth}$ via the specialization morphism is isomorphic to $\mathscr U^{\beth}$. Let $f_x$ be a local equation for $\mathscr Y\times_\mathscr X Z$ at $x$; we then have $\Sp_\mathscr Y^{-1}(x)\cong \big(\mathscr U^\beth\setminus V(f_x)\big)/\mathbb R_{>0} \cong {\mathscr U}_{f_x}^\beth$, and the latter is pseudo-affinoid. \end{proof} \pa{ We define a family ${\mathcal W}$ of subsets of $T_{X,Z}$ as follows. Denote by $D=\widetilde{\Bl_{Z}X}\times_{X}Z$ the exceptional divisor in $\widetilde{\Bl_{Z}X}$. A nonempty subset of $T_{X,Z}$ belongs to ${\mathcal W}$ if and only if it is of the form $\Sp_{\widetilde{\mathscr X}}^{-1}(D\cap U)$, for some affine open $U$ of $\widetilde{\Bl_{Z}X}$ such that $D\cap U$ is a principal divisor of $U$. } \begin{lem} The family ${\mathcal W}$ is an atlas of $T_{X,Z}$. \end{lem} \begin{proof} Let $W=\Sp_{\widetilde{\mathscr X}}^{-1}(D\cap U)$ be an element of ${\mathcal W}$ corresponding to some affine open $U$ of $\widetilde{\Bl_{Z}X}$. Then $W=T_{\mathcal U}$, where ${\mathcal U}$ is the formal completion of $U$ along $D\cap U$. The formal scheme ${\mathcal U}$ is affine since $U$ is affine, and it can be seen as a formal scheme ${\mathcal U}_t$ of finite type over $k[[t]]$, where the $k[[t]]$-structure is defined by sending $t$ to an equation for $D\cap U$. Therefore $T_{\mathcal U}\cong\forg\big({\mathcal U}_t^\beth\big)$ is affinoid. Moreover, since $\Sp_{\widetilde{\mathscr X}}^{-1}(D)=T_{X,Z}$ and $D$ is Cartier in $\widetilde{\mathscr X}$, the elements of ${\mathcal W}$ cover $T_{X,Z}$ and therefore ${\mathcal W}$ is an atlas of $T_{X,Z}$. \end{proof} \pa{\label{facts_canonical_atlas} We call ${\mathcal W}$ the \defi{canonical atlas} of $T_{X,Z}$. The reason this is relevant is the following. Let $V\subset T_{X,Z}$ be a $k((t))$-analytic space that is a union of elements of ${\mathcal W}$. Then $V$ can be written as a finite union of elements of ${\mathcal W}$ since $D$ is quasi-compact, and the family of analytic subspaces ${\mathcal W}\|_V=\{W\in{\mathcal W} \,\,\|\,\,W\subset V\}$ is a distinguished formal atlas of $V$ (in the sense of \cite[\S1]{BosLut85}) by strict affinoid domains. This means in particular that a formal model of $V$ can be reconstructed by gluing the affine formal $k[[t]]$-schemes of finite type $\Spf\big(\mathcal O^\circ_{W}(W)\big)$, for $W$ in ${\mathcal W}\|_V$. Moreover, in our situation we can forget the $k[[t]]$-structures and glue all the affine special formal $k$-schemes $\Spf\big(\mathcal O^\circ_{W}(W)\big)$, retrieving the special formal $k$-scheme $\widetilde{\mathscr X}$. A similar idea will be used to construct formal log modifications in Theorem~\ref{vertex_sets}. Note that the union of the Shilov boundary points of elements of ${\mathcal W}$ is $\partial^{\mathrm{an}}\Txz$ (see~\cite[\S2.4]{Ber90} for the definition of the Shilov boundary of an affinoid space). In particular, if $V\subset T_{X,Z}$ is a $k((t))$-analytic space that is a union of elements of ${\mathcal W}$, then $V\cap\partial^{\mathrm{an}}\Txz$ is the analytic boundary of $V$ in the sense of~\cite[\S3.1]{Ber90}. In the following we will often implicitly chose a structure of $k((t))$-analytic curve for the elements of ${\mathcal W}$. } \begin{lem}\label{fibers_are_connected_components} If $\mathscr Y$ is a formal log modification of $(X,Z)$, then the set of the connected components of $T_{X,Z}\setminus \Div_{X,Z}(\mathscr Y)$ coincides with the family $$ \{\Sp_\mathscr Y^{-1}(x)\;\|\;x\in \mathscr Y\times_\mathscr X Z\text{ closed point}\}. $$ \end{lem} \begin{proof} Lemma~\ref{L: divisorial valuation specialization} implies that $T_{X,Z}\setminus \Div_{X,Z}(\mathscr Y)=\bigcup_x \Sp_\mathscr Y^{-1}(x)$, where this union, taken over the closed points of $\mathscr Y\times_\mathscr X Z$, is disjoint. Each $\Sp_\mathscr Y^{-1}(x)$ is open by anticointinuity of $\Sp_\mathscr Y$, so it is a union of connected components of $T_{X,Z}\setminus \Div_{X,Z}(\mathscr Y)$. The fact that $\Sp_\mathscr Y^{-1}(x)$ is connected is \cite[6.1]{Bosch77} applied to any $k((t))$-analytic curve $W\in{\mathcal W}$ such that $x\in\Sp_{\widetilde{\mathscr X}}(W)$. \end{proof} \pa{ If $C$ is a $k((t))$-analytic curve, its points can be divided into four types, according to the valuative invariants of their completed residue field (see e.g. \cite[3.3.2]{Duc}, although these ideas essentially go back to \cite{Ber90}). In particular a point $x$ of $C$ is said to be of \emph{type 2} if $\trdeg_{k} \widetilde{\rescompl{x}}=1$, where $\widetilde{\rescompl{x}}$ denotes the residue field of ${\rescompl{x}}$. The points of type 2 are precisely the points of infinite branching of $C$, i.e. a point $x$ of $C$ is of type 2 if and only if $C\setminus\{x\}$ has infinitely many connected components (if $C$ is regular, this is equivalent to $C\setminus\{x\}$ having at least three connected components). We refer to \cite[\S6]{Temkin15} or \cite{BakerPayneRabinoff14} for a description of the structure of non-archimedean analytic curves. } \begin{lem}\label{lemma_divisorial_iff_type2} If $x$ is a point of $T_{X,Z}$, $V$ is an analytic domain of $T_{X,Z}$ that contains $x$, and $C$ is a $k((t))$-analytic curve such that $\forg(C)\cong V$, then $x$ is a divisorial point of $T_{X,Z}$ if and only if it is a point of type 2 of $C$. \end{lem} \begin{proof} By abuse of notation we denote by $x$ also a point of ${\X^}$ whose image in $T_{X,Z}$ is the given point $x$. The completed residue field $\rescompl{x}$ of ${\X^}$ at $x$ can be computed also as the completed residue field of $C$ at $x$. Therefore, we deduce that it is a valued extension of $k((t))$ (for some non-trivial $t$-adic absolute value that we do not need to specify), and in particular \[ \rank_{\mathbb{Q}}\|\rescompl{x}^\times\|/\|k^\times\|\otimes_Z{\mathbb{Q}}\geq\rank_{\mathbb{Q}}\|\rescompl{x}^\times\|/\|k((t))^\times\|\otimes_Z{\mathbb{Q}}+1\geq1. \] Moreover, by Abhyankar's inequality (see \cite[Corollaire to 5.5]{Vaquie00}) we have \[ \rank_{\mathbb{Q}}\|\rescompl{x}^\times\|/\|k^\times\|\otimes_Z{\mathbb{Q}}+\trdeg_{k}\widetilde{\rescompl{x}}\leq 2. \] We said that $x$ is a type 2 point of $C$ if and only if $\trdeg_{k} \widetilde{\rescompl{x}}=1$, and by the two inequalities above this is equivalent to \[ \begin{cases} \rank_{\mathbb{Q}}\|\rescompl{x}^\times\|/\|k^\times\|\otimes_Z{\mathbb{Q}}=1 \\ \trdeg_{k}\widetilde{\rescompl{x}}=1 \end{cases} \] By \cite[Example 7, Proposition 10.1]{Vaquie00}, this is equivalent to $x$ being a divisorial point of $T_{X,Z}$. \end{proof} \pa{A \defi{vertex set} of $T_{X,Z}$ is any finite subset of $\Div_{X,Z}$ containing $\partial^{\mathrm{an}}\Txz$.} The following theorem is the main result of this section. \begin{thm}\label{vertex_sets} Let $(X,Z)$ be as in \ref{setting_surfaces}. Then the map $\mathscr Y\mapsto \Div_{X,Z}(\mathscr Y)$ induces an isomorphism between the following partially ordered sets: \begin{enumerate} \item the set of isomorphism classes of formal log modifications of $(X,Z)$, ordered by domination; \item the set of vertex sets of $T_{X,Z}$, ordered by inclusion. \end{enumerate} \end{thm} \begin{proof} We follow the lines of \cite[6.3.15]{Duc}, but the general ideas (over an algebraically closed field) go back to \cite{BosLut85} and can be found also elsewhere, for example in \cite[\S4]{BakerPayneRabinoff14}. The proof will be divided in several steps. In the first step, which is the hardest one, we rely on the proof of \cite[6.3.15]{Duc}, which is the corresponding statement for $k((t))$-analytic curves, by applying it to the elements of the canonical atlas ${\mathcal W}$ of $T_{X,Z}$. Given a vertex set of $\Div_{X,Z}$, we will first construct a special formal $k$-scheme by producing a suitable atlas of $T_{X,Z}$ and gluing the formal models of its elements. The reader may check, using the definition of ${\mathcal W}$, the facts discussed in \ref{facts_canonical_atlas}, and Lemma~\ref{fibers_are_connected_components}, that if we take $S=\partial^{\mathrm{an}}\Txz$ then the atlas that we obtain will coincide with ${\mathcal W}$. This might be a useful example to keep in mind, noting that applying what follows in this case will yield the formal log modification $\widetilde\mathscr X$. In the second step we will show that the formal scheme we obtained is a formal modification of $(X,Z)$. We will then conclude the proof by showing that this association defines a bijection, and finally that it respects the given orderings. \begin{step} {\it Construction of the formal scheme $\mathscr Y$.} Let $S$ be a vertex set of $\Div_{X,Z}$, and let ${\mathcal V}$ be the family of subsets of $T_{X,Z}$ defined as follows. A compact subset $V$ of $T_{X,Z}$ belongs to ${\mathcal V}$ if and only if there exist a subset $S'$ of $S$ and a finite family $\{U_i\}$ of connected components of $T_{X,Z}\setminus S'$ such that the following conditions are satisfied: \begin{enumerate} \item $V\subset W$ for some element $W$ of the canonical atlas ${\mathcal W}$ of $T_{X,Z}$; \item $V=T_{X,Z}\setminus\coprod U_i$; \item $V\cap S=S'$; \item for every $x\in S'\setminus \partial^{\mathrm{an}}\Txz$, there exists at least one index $i$ such that $x$ belongs to the topological boundary $\partial U_i\coloneqq \overline{U_i}\setminus U_i$ of $U_i$; \item every connected component of $T_{X,Z}$ that does not meet $S'$ is one of the $U_i$. \end{enumerate} \noindent Observe that the empty set is an element of ${\mathcal V}$, since it can be obtained by taking for $S'$ the empty set and for the family $\{U_i\}$ the set of connected components of $T_{X,Z}$. To be able to use the elements of ${\mathcal V}$ as building blocks for the formal scheme $\mathscr Y$, we will now prove that ${\mathcal V}$ is closed under intersection. Indeed, if $V_1$ and $V_2$ are elements of $V$ corresponding respectively to families $\{U_{1,i}\}$ and $\{U_{2,j}\}$ of subspaces of $T_{X,Z}$, then $V_3=V_1\cap V_2$ is the element of ${\mathcal V}$ corresponding to the subset $S_3=S\cap V_3$ of $S$ and the family of those connected components of $T_{X,Z}\setminus S_3$ that can be written as unions of sets of the form $U_{1,i}$ or $U_{2,j}$. The only part which is non-trivial to verify is the fact that these data satisfy the condition (v) above. For this, assume that $U$ is a connected component of $T_{X,Z}$ which contains no point of $S_3$. Then $U$ can not be entirely contained in $V_3$, or otherwise it would be contained in $V_1$ as well, contradicting condition (v) for $V_1$. Therefore if $U$ intersects $V_3$ non-trivially there exists a point $x$ contained both in $U$ and in the topological boundary $\partial V_3=V_3\setminus\mathrm{Int}(V_3)$ of $V_3$. But then $x$ would also belong to the topological boundary of one of the first two $V_i$, hence to $S_i$, and so as it belongs to $V_3$ it would be an element of $S_3$ as well. Since this is not possible, $U$ does not intersect $V_3$, which proves (v) because $U$ is also a connected component of $T_{X,Z}\setminus S_3$. Now let $W$ be a connected element of ${\mathcal W}$, seen as a $k((t))$-analytic space, and consider the family ${\mathcal V}\|_W=\{V\cap W \,\,\|\,\,V\in{\mathcal V}\}$ of subspaces of $W$, seen as $k((t))$-analytic subspaces themselves. Oberve that, if $V$ is an element of ${\mathcal V}$ associated as above with a family $\{U_i\}$ of subsets of $T_{X,Z}$, then the element $V'=V\cap W$ of ${\mathcal V}\|_W$ satisfies itself the conditions (i--v) above with respect to the ambient space $W$ and the family of subsets of $W$ that are connected components of subsets of the form $U_i\cap W$, where in condition (v) $\partial^{\mathrm{an}}\Txz$ is replaced by $\partial^{\mathrm{an}}\Txz\cap W$. Then, since as we observed in \ref{facts_canonical_atlas} the Shilov boundary of $W$ coincides with $\partial^{\mathrm{an}}\Txz\cap W$, we can apply \cite[6.3.15.2]{Duc} (whose hypotheses (a--d) now follow directly from our (i--v) above) to $W$ and to the family ${\mathcal V}\|_W$, deducing that ${\mathcal V}_W$ is a strict formal affinoid atlas of $W$, and moreover the associated vertex set (that is by definition the union of the Shilov boundaries of the elements of ${\mathcal V}_W$) is $S\cap W$. The associated formal $k[[t]]$-scheme $\mathscr Y_W$ is therefore a formal model of $W$ with vertex set $S\cap W$. Now, observe that the canonical model of an affinoid domain $V$ of $T_{X,Z}$, being $\Spf\big(\mathcal O^\circ_{T_{X,Z}}(V)\big)$, does not depend on the choice of a $k((t))$-analytic structure on $V$. This guarantees that we can glue all the $\mathscr Y_W$, seen as affine special formal $k$-schemes, along their intersections, obtaining a special formal $k$-scheme $\mathscr Y$. \end{step} \begin{step} {\it $\mathscr Y$ is a formal log modification of $(X,Z)$.} We defined the formal scheme $\mathscr Y$ by gluing affine special formal $k$-schemes of the form $\Spf\big(\mathcal O^\circ_{T_{X,Z}}(V)\big)$, for $V$ ranging among the elements of ${\mathcal V}$. By \cite[2.1]{MartinKappen15} (as recalled in~\ref{dejong_original}) each $\mathcal O^\circ_{T_{X,Z}}(V)$ is integrally closed in its ring of fractions, and therefore $\mathscr Y$ is normal. For each $W\in{\mathcal W}$, the inclusions of $k((t))$-analytic spaces $V\to W$, for $V\in{\mathcal V}\|_W$, induce morphisms of special $k$-algebras $\mathcal O^\circ_{T_{X,Z}}(W)\to\mathcal O^\circ_{T_{X,Z}}(V)$, and therefore a morphism of special formal $k$-schemes $\mathscr Y_W\to \widetilde \mathscr X$. These morphisms glue to an adic morphism $\mathscr Y\to\widetilde \mathscr X$, so we obtain an adic morphism $f\colon\mathscr Y\to\mathscr X$, and $\mathscr Y\times_\mathscr X Z$ is Cartier in $\mathscr Y$ by the universal property of the blowup. Moreover, since for every $W$ in the covering ${\mathcal W}$ the morphism $\mathscr Y_W\to \widetilde \mathscr X$ induces an isomorphism at the level of normalized spaces $T_{\mathscr Y_W}\cong W$, the morphism $f$ induces an isomorphism $T_\mathscr Y\congT_{X,Z}$. Therefore, $\mathscr Y$ is a formal log modification of $(X,Z)$. \end{step} \begin{step} {\it Bijectivity of the correspondence.} It follows from our construction in the first step that each element $V$ of ${\mathcal V}$ is an affinoid domain of $T_{X,Z}$ and coincides with the inverse image under the specialization morphism $\Sp_\mathscr Y$ of an affine open subset of $\mathscr Y$ in which $\mathscr Y\times_\mathscr X Z$ is principal. It follows from \cite[2.4.4]{Ber90}, applied after choosing a $k((t))$-analytic structure on $V$, that $V\cap S$, being the Shilov boundary of $V$, coincides with $V\cap\Div_{X,Z}(\mathscr Y)$. Therefore $S$, being the union of the Shilov boundaries of the elements of ${\mathcal V}$, coincides with $\Div_{X,Z}(\mathscr Y)$. This shows that the map $\mathscr Y\mapsto \Div_{X,Z}(\mathscr Y)$ is surjective. To prove its injectivity, we need to show that a formal log modification of $(X,Z)$ is determined by its divisorial set. This can be done locally, again using Ducros's results, as follows. Assume that $\mathscr Y'$ is another formal log modification of $(X,Z)$ such that $\Div_{X,Z}(\mathscr Y')=S$. Let $W$ be the element of the canonical atlas ${\mathcal W}$ associated with an open affine subspace $U$ of $\widetilde{\mathscr X}$. Then by~\cite[6.3.15]{Duc} ${\mathcal V}\|_W$ is the unique formal atlas on $W$ whose vertex set is $S\cap W$, therefore the $k[[t]]$-subspace $\tau^{-1}(U)$ of $\mathscr Y'$, where we denote by $\tau$ the composition of the canonical map $\mathscr Y'\to\widetilde{\mathscr X}$ with $\Sp_{\mathscr Y'}$, is isomorphic to the open $\mathscr Y_W$ of $\mathscr Y$, and hence $\mathscr Y'$ is isomorphic to $\mathscr Y$. \end{step} \begin{step} {\it Functoriality.} It is clear that if $\mathscr Y$ and $\mathscr Y'$ are two formal log modifications of $(X,Z)$ such that $\mathscr Y'$ dominates $\mathscr Y$, then $\Div_{X,Z}(\mathscr Y)\subset \Div_{X,Z}(\mathscr Y')$. To show that the bijective correspondence that we have constructed respects the partial orders it is then enough to note the following. Let $S_1\subset S_2$ be finite nonempty subsets of $\Div_{X,Z}$, and let $\mathscr Y_1$ and $\mathscr Y_2$ be the corresponding formal models, defined using formal atlases $\mathscr V_1$ and $\mathscr V_2$. Then from the definition of the atlases $\mathscr V_i$ it follows that we can cover $T_{X,Z}$ by $V_{1,1},\ldots,V_{1,r}\in\mathscr V_1$ and also by $V_{2,1},\ldots,V_{2,s}\in\mathscr V_2$ in such a way that each $V_{2,i}$ is a subspace of some $V_{1,i}$, and each $V_{1,i}$ is covered by the $V_{2,i}$'s that it contains. These inclusions give a morphism $\mathscr Y_2\to\mathscr Y_1$ commuting with the two morphisms $\mathscr Y_i\to\mathscr X$, hence a morphism of formal log modifications. \end{step} This completes the proof of Theorem~\ref{vertex_sets}. \end{proof} \Pa{Remarks}{ If $X$ has only rational singularities or if $k$ is an algebraic closure of the field $\mathbb F_p$ for some prime number $p$, then Theorem~\ref{vertex_sets} can also be proved using resolution of singularities to find a suitable log modification and then contractibility results \cite[2.3, 2.9]{Artin62} to contract all unnecessary divisors, and every formal log modification of $(X,Z)$ is algebraizable. In general not all of the formal log modifications given by Theorem~\ref{vertex_sets} are algebraizable, but the contractibility criterion of Grauert-Artin \cite{Artin70} guarantees that they can always be algebraized in the category of algebraic spaces over $k$. Moreover, since Artin proved that a smooth algebraic space in dimension 2 is a scheme, we retrieve Proposition~\ref{proposition_algebraization_resolutions}. } \section{Discs and annuli} \label{section_discs}\label{section_4.2} In this section we will study one-dimensional open discs and open annuli in normalized spaces. The main result, Proposition~\ref{prop_discs}, explains in which sense those discs and annuli are determined by their canonical reduction. \pa{ We say that a $k((t))$-analytic space $X$ is \defi{pseudo-affinoid} if it is the Berkovich space associated with an affine special formal $k[[t]]$-scheme. When this is the case and moreover $X$ is reduced, \cite[7.4.2]{deJ95} (recalled in~\ref{dejong_original}) tells us that $X\cong\mathscr X_t^\beth$, where $\mathscr X_t=\Spf\big({\mathcal O^\circ_X(X)}\big)$ is called the \emph{canonical formal model} of $X$, and moreover $\mathscr X_t$ is integrally closed in its generic fiber. The reduced affine special formal $k$-scheme $\big((\mathscr X_t)_s\big)_{\mathrm{red}}$ associated with the special fiber $(\mathscr X_t)_s=\mathscr X_t\otimes_{k[[t]]} k$ of $\mathscr X_t$ will then be called \emph{canonical reduction} of $X$, and will be denoted by $X_0$. \\ We say that a pseudo-affinoid $k((t))$-analytic space $X$ is \defi{distinguished} if the special fiber $(\mathscr X_t)_s$ of its canonical formal model is already reduced, i.e. if it coincides with $X_0$. We have shown in \ref{distinguished_affinoid_equivalence} that for strictly affinoid $k((t))$-analytic spaces this definition coincides with the classical one. } \pa{ We say that a normalized $k$-space $Y$ is \emph{pseudo-affinoid} (respectively, \emph{distinguished pseudo-affinoid}) if it is of the form $\forg(X)$, with $X$ a pseudo-affinoid (resp. a distinguished pseudo-affinoid) $k((t))$-analytic space. This coincides with the definition in~\ref{definition_pseudo-affinoid}. Whenever $Y$ is reduced, the affine special formal $k$-scheme $\mathscr Y=\Spf\big({\mathcal O^\circ_Y(Y)}\big)$ is called the \emph{canonical formal model} of $Y$. Note that this is an abuse of notation since $\mathscr Y$ is not a formal model of the normalized $k$-space $Y$. \\ We define the \emph{canonical reduction} of $Y$ as the closed formal subscheme $Y_0$ of $\mathscr Y$ defined by the ideal $I=\bigcap\sqrt{(f)}$, where the intersection is taken over all elements $f$ of $\mathcal O^{\circ\circ}_Y(Y)$ that do not vanish on $Y$. Being an intersection of radical ideals, $I$ is radical itself, so $Y_0$ is a reduced special formal $k$- scheme. This definition is consistent with the previous one, since $Y_0$ is isomorphic to the canonical reduction $X_0$ of $X$, and therefore the canonical reduction of a pseudo affinoid $k((t))$-analytic space only depends on its normalized space structure. To prove this, we need to show that $I=\sqrt{(t)}$. Clearly $I\subset\sqrt{(t)}$ since $t$ does not vanish on $Y$. By Theorem~\ref{thm_structure_arboretum} we have $Y=\forg(X)=T_\mathscr X\setminus V(t)$, where $\mathscr X_t$ is the canonical $k[[t]]$-formal model of $X$, as usual $\mathscr X$ is the underlying special formal $k$-scheme, and by abuse of notation we denoted by $t$ the image of $t$ in $\mathcal O_\mathscr X(\mathscr X)$. Therefore, if $f$ does not vanish on $Y$ we must have $V(f)\subset V(t)$, and so $\sqrt{(t)}\subset\sqrt{(f)}$, which implies that $\sqrt{(t)}\subset I$. Moreover, remark that $Y$ is distinguished if and only if the ideal $I$ defined above is a principal ideal. } \pa{ A $k((t))$-analytic space is called an \defi{open $k((t))$-disc}, or simply a \emph{disc}, if it is isomorphic to $\Spf\big(k[[t]][[T]]\big)^{\beth}_t$. \\ Equivalently, a disc is a $k((t))$-analytic space isomorphic to the subspace of $\mathbb A^{1,\mathrm{an}}_{k((t))}$ defined by the inequality $\|T\|<1$, where $\mathbb A^1_{k((t))}=\Spec(k((t))[T])$. } \pa{ A $k((t))$-analytic space is called an \defi{open $k((t))$-annulus of modulus $n$}, or simply an \emph{annulus of modulus $n$}, if it is of the form $A_n\coloneqq \Spf\big(k[[t]][[T_1,T_2]]/(T_1T_2-t^n)\big)_t^{\beth}$, for some $n>0$. \\ Equivalently, an annulus of modulus $n$ is a $k((t))$-analytic space isomorphic to the subspace of $\mathbb A^{1,\mathrm{an}}_{k((t))}$ defined by the inequality $\|t^n\|<\|T_1\|<1$. \\ We define a \defi{standard annulus} as an annulus of modulus one. Remark that an annulus is standard if and only if it has no $k((t))$-point. } \pa{\label{modulus_intrinsic} The modulus of an annulus $X$ is well defined, and depends only on the algebra $\mathcal O^\circ_X(X)$, hence only on the normalized space $\forg(X)$. Indeed, if $X$ has modulus $n$ then $\mathcal O^\circ_X(X)$ is the completed local ring of a $k$-surface at a du Val singularity of type $A_{n-1}$, and there is exacly one formal isomorphism class of surfaces with singularity of type $A_{n-1}$ (see for example \cite[\S2]{Artin77}). } \pa{ Discs and annuli are distinguished pseudo-affinoid $k((t))$-analytic spaces. Indeed, both are reduced and the canonical formal model of a disc is the affine formal $k[[t]]$-scheme $\Spf\big(k[[t]][[T]]\big)_t$, whose special fiber is $\Spf\big(k[[T]]\big)$; while the canonical formal model of an annulus of modulus $n$ is the affine formal $k[[t]]$-scheme $\Spf\big(k[[t]][[T_1,T_2]]/(T_1T_2-t^n)\big)_t$, whose special fiber is $\Spf\big(k[[T_1,T_2]]/(T_1T_2)\big)$. } \begin{rem} The canonical model of an annulus is regular if and only if the annulus is standard. Indeed, the maximal ideal of $k[[t,T_1,T_2]]/(T_1T_2-t^n)$ is ${\mathfrak M}=(t,T_1,T_2)$, hence ${\mathfrak M}^2=(t^2,T_1^2,T_2^2,tT_1,tT_2,t^n)$ so the $k$-vector space ${\mathfrak M}/{\mathfrak M}^2$ has dimension 2, with basis $\{T_1,T_2\}$, if and only if $n=1$. \end{rem} It is clear that any two $k((t))$-discs are always isomorphic as $k((t))$-analytic spaces, and that two $k((t))$-annuli are isomorphic if and only if they have the same modulus. In our setting we need something stronger, which will be the content of the next proposition and of the corollary following it. \begin{prop}\label{prop_discs} Let $X$ be a distinguished pseudo-affinoid $k((t))$-analytic space, and denote by $X_0$ its canonical reduction. Then: \begin{enumerate} \item if $X_0\cong\Spf\big(k[[T]]\big)$, then $X$ is a $k((t))$-disc; \item if $X_0\cong\Spf\big(k[[T_1,T_2]]/(T_1T_2)\big)$ and $X$ is irreducible, then $X$ is a $k((t))$-annulus. \end{enumerate} \end{prop} \begin{proof} Part $(i)$ follows easily from the uniqueness of deformations of smooth affine formal schemes, see \cite{PerezRodriguez08}. Indeed, up to isomorphism there is only one affine and flat special formal $k[[t]]$-scheme whose special fiber is $\Spf\big(k[[T]]\big)$, so the canonical formal model of $X$ is isomorphic to $\Spf\big(k[[t]][[T]]\big)$, hence $X$ is a $k((t))$-disc. To prove $(ii)$ we make use of the fact that the {miniversal} deformation of the formal node $\Spf\big(k[[T_1,T_2]]/(T_1T_2)\big)$ is $\Spf\big(k[[t,T_1,T_2]]/(T_1T_2-t)\big)$, which is \cite[14.0.1]{Hartshorne10}. This means that if $\mathscr X=\Spf(A)$ is an affine and flat special formal $k[[t']]$-scheme whose special fiber is isomorphic to $\Spf\big(k[[T_1,T_2]]/(T_1T_2)\big)$, then there is a local $k$-algebra morphism $\varphi\colon k[[t]]\to k[[t']]$ such that $\mathscr X \cong \Spf\big(k[[t,T_1,T_2]]/(T_1T_2-t)\big)\otimes_{k[[t]]}k[[t']]$. The morphism $\varphi$ is determined by a power series $\varphi(t)=F(t')\in k[[t']]$ such that $F(0)=0$, so we have $\mathscr X\cong\Spf\big(k[[t',T_1,T_2]]/(T_1T_2-F(t'))\big)$. Moreover, since $X$ is irreducible then $F(t')$ cannot be zero. The power series $F(t')$ can be written as $u(t')^n$ for some unit $u$ of $k[[t']]$, hence by further sending $T_1$ to $uT_1$ we obtain an isomorphism $\mathscr X\cong\Spf\big(k[[t',T_1,T_2]]/(T_1T_2-(t')^n)\big)$. By applying this to the canonical $k[[t]]$-formal model of $X$, we deduce that $X$ is a $k((t))$-annulus. \end{proof} \begin{cor}\label{cor_discs} Let $X$ be a $k((t))$-disc or a $k((t))$-annulus and let $Y$ be a distinguished pseudo-affinoid $k((t))$-analytic space such that $\forg(X)\cong \forg(Y)$. Then $X$ and $Y$ are isomorphic as $k((t))$-analytic spaces. \end{cor} \begin{proof} This follows from Proposition~\ref{prop_discs} since the special fiber of the canonical formal model of a distinguished pseudo-affinoid $k((t))$-analytic space does not depend on the chosen distinguished pseudo-affinoid $k((t))$-analytic structure. For annuli, note that the modulus of $Y$ is the same as the modulus of $X$ by~\ref{modulus_intrinsic}. \end{proof} \pa{ The last result allows us to define more intrinsically discs and annuli in normalized spaces: we say that a distinguished pseudo-affinoid analytic domain $V$ of a normalized $k$-space $T$ is a \defi{disc} (respectively an \defi{annulus of modulus $n$}) if there exists a $k((t))$-analytic space $X$ such that $V\cong\forg(X)$ and $X$ is a disc (resp. an annulus of modulus $n$). Corollary~\ref{cor_discs} tells us that this property is independent of the choice of a distinguished pseudo-affinoid structure on $V$. } \section{Formal fibers} \label{section_formal_fibers}\label{section_4.3} In this section we move to the study of the fibers of the specialization morphism. For normal surfaces we will get in Proposition~\ref{formal_fibers} very explicit results involving discs and annuli, analogous to \cite[2.2 and 2.3]{BosLut85}. For simplicity, from now on we assume that $k$ is algebraically closed. \pa{Let $\mathscr X$ be a special formal $k$-scheme and let $x$ be a point of $\mathscr X$. We define the \defi{formal fiber} of $x$ as the inverse image ${\mathcal F}_x=\Sp_\mathscr X^{-1}(x)$ of $x$ in ${T_\X}$ under the specialization morphism. It is a subspace of ${T_\X}$, open if $x$ is closed in $\mathscr X$. } \pa{ Let $X$ be a normal scheme of finite type over $k$, let $Z$ be a divisor of $X$, and let $\mathscr X=\widehat{X/Z}$ the formal completion of $X$ along $Z$. Then the argument of Lemma~\ref{L: divisorial valuation specialization} tells us that, if $\eta$ is a generic point of an irreducible component of $Z$, its formal fiber ${\mathcal F}_\eta$ is a single point of ${T_\X}$, precisely the point corresponding to the $\mathbb R_{>0}$-orbit of divisorial valuations associated with the component $\overline{\{\eta\}}$. If $x$ is a closed point of $\mathscr X$, its formal fiber ${\mathcal F}_x$ can be given the structure of a pseudo-affinoid space. } \begin{lem}\label{L: regular formal fiber} Let $\mathscr X$ be a normal special formal $k$-scheme of dimension $n$ and let $x$ be a closed point of $\mathscr X$ such that there exists an ideal of definition of $\mathscr X$ that is principal at $x$. Then $\mathscr X$ is regular at $x$ if and only if $\mathcal O^\circ_{T_\X}({\mathcal F}_x)\cong k[[X_1,\ldots,X_n]]$. \end{lem} \begin{proof} Consider the normal special formal $k$-scheme $\mathscr U=\Spf\big(\widehat{\mathcal O_{\mathscr X,x}}\big)$. By \ref{lemma fiber reduction}, the inverse image of $x$ in $\mathscr X^{\beth}$ via the specialization morphism is isomorphic to $\mathscr U^{\beth}$, so ${\mathcal F}_x$ is isomorphic to $(\mathscr U^{\beth}\setminus V(t))/\mathbb R_{>0}\cong\mathscr U_t^\beth$, where we have denoted by $t$ a local generator of an ideal of definition of $\mathscr X$ at $x$. It follows that $\mathcal O^\circ_{T_\X}({\mathcal F}_x)\cong \mathcal O^\circ_{\mathscr U_t^\beth}(\mathscr U_t^\beth)$, and since $\mathscr U_t$ is normal this is also equal to $\mathcal O_{\mathscr U_t}(\mathscr U_t)\cong \widehat{\mathcal O_{\mathscr X,x}}$ by \cite[7.3.6]{deJ95}. To conclude, by Cohen's structure theorem~\cite{Cohen46} $x$ is regular in $\mathscr X$ if and only if $\widehat{\mathcal O_{\mathscr X,x}}\cong k[[X_1,\ldots,X_n]]$. \end{proof} \begin{rem} Note that in the lemma above, while the $k$-algebra $\mathcal O^\circ_{T_\X}({\mathcal F}_x)$ does not depend on the geometry of $\mathscr X_0$ around $x$, its largest ideal of definition, and therefore also the space ${\mathcal F}_x$, strongly depends on it. Focusing now on the case of surfaces, an example of this behavior is detailed in the following proposition, which is the analogue for normalized spaces of surfaces of a classical result of Bosch and L\"utkebohmert \cite[2.2 and 2.3]{BosLut85} (see also \cite[4.3.1]{Ber90} for a formulation in the language of Berkovich curves). \end{rem} \begin{prop}\label{formal_fibers} Let $\mathscr X$ be a normal special formal $k$-scheme of dimension $2$ and let $x$ be a closed point of $\mathscr X$ such that there exists an ideal of definition of $\mathscr X$ that is principal at $x$. Then: \begin{enumerate} \item $x$ is regular both in $\mathscr X$ and in $\mathscr X_0$ if and only if its formal fiber ${\mathcal F}_x$ is a disc; \item $x$ is regular in $\mathscr X$ and an ordinary double point in $\mathscr X_0$ if and only if ${\mathcal F}_x$ is a standard annulus. \end{enumerate} \end{prop} \begin{proof} We endow ${\mathcal F}_x$ with the pseudo-affinoid $k((t))$-analytic structure from Lemma~\ref{L: divisorial valuation specialization}. Assume that $x$ is regular both in $\mathscr X$ and in $\mathscr X_0$. Then $\mathscr X_0$ is itself principal locally at $x$, and so we can choose an element $t$ of $\mathcal O_{\mathscr X,x}$ that locally defines $\mathscr X_0$. Then the image $\bar t$ of $t$ in ${\mathfrak M}/{\mathfrak M}^2$ does not vanish, where we have denoted by ${\mathfrak M}$ the maximal ideal of $\mathcal O_{\mathscr X,x}$. We then pick another element $X$ of ${\mathfrak M}$ whose image in ${\mathfrak M}/{\mathfrak M}^2$ generates it together with $\bar t$. Cohen Theorem gives us an isomorphism $\widehat{\mathcal O_{\mathscr X,x}}\cong k[[t,X]]$, therefore the formal fiber ${\mathcal F}_x$ is the disc $\mathcal{F}_x=\Spf\big(k[[t,X]]\big)_{t}^\beth$. Conversely, if ${\mathcal F}_x$ is a disc, isomorphic to $\Spf\big(k[[t,X]]\big)^{\beth}_t$, then $\mathcal O^\circ_{T_{X,Z}}({\mathcal F}_x) \cong k[[t,X]]$, so $\mathscr X$ is regular at $x$ by Lemma~\ref{L: regular formal fiber}, and $\mathscr X_0$ is locally defined by the equation $t=0$, so it is itself regular, proving $(i)$. The proof of $(ii)$ is similar. If $x$ is an ordinary double point of $\mathscr X_0$ then we can find elements $X$ and $Y$ of ${\mathfrak M}$ whose images in ${\mathfrak M}/{\mathfrak M}^2$ generate it and such that $\mathscr X_0$ is defined at $x$ by the equation $XY=0$. Then we obtain ${\mathcal F}_x=\Spf\big(k[[X,Y]]\big)_{XY}^\beth=\Spf\big(k[[t]][[X,Y]]/(XY-t)\big)_t^\beth$, which is a standard annulus. The converse implication is proved as in $(i)$, as if ${\mathcal F}_x$ is a standard annulus then $\mathcal O^\circ_{T_{X,Z}}({\mathcal F}_x) \cong k[[t]][[X,Y]]/(XY-t) \cong k[[X,Y]]$. \end{proof} \section{Log essential and essential valuations} \label{section_4.4} In this section we make use of the previous results to characterize in terms of the structure of $T_{X,Z}$ the finite sets of divisorial points of $T_{X,Z}$ that correspond via Theorem~\ref{vertex_sets} to the log resolutions of a pair $(X,Z)$. We will then be able to describe the divisorial points corresponding to the minimal log resolution of $(X,Z)$, and in Theorem~\ref{thm_log_essential_valuations} we will deduce a local characterization of the log essential valuations of $(X,Z)$. Finally, we will show that similar arguments also yield a characterization of the slightly smaller class of essential valuations which has been introduced by Nash in \cite{Nash95} and studied extensively since. This is the content of Theorem~\ref{thm_essential_valuations}. Note that in this section we use the existence of resolution of singularities for surfaces. More precisely, we admit the fact that given any set $S$ of divisorial points of $T_{X,Z}$ there exists a log resolution $Y$ of $(X,Z)$ such that $S\subset \Div_{X,Z}(Y)$. On the other hand, we do not need to assume the existence of a minimal (log) resolution of $(X,Z)$. \pa{Let $(X,Z)$ be as in \ref{setting_surfaces}. We say that a vertex set $S$ of $T_{X,Z}$ is \defi{regular} if the connected components of $T_{X,Z}\setminus S$ are discs and a finite number of standard annuli. } Putting together Lemma~\ref{fibers_are_connected_components}, Proposition~\ref{formal_fibers} and Proposition~\ref{proposition_algebraization_resolutions} we obtain the following important result. \begin{prop}\label{prop_resolution_iff_vertex_set} Let $k$ be an algebraically closed field and let $(X,Z)$ be as in \ref{setting_surfaces}. Then a formal log modification $\mathscr Y$ of $(X,Z)$ is a log resolution if and only if $\Div_{X,Z}(\mathscr Y)$ is a regular vertex set of $T_{X,Z}$. \end{prop} \pa{\label{observation_fibers_are_simple} We define the set of \defi{log essential valuations} of the pair $(X,Z)$ as the intersection $\bigcap_Y\Div_{X,Z}(Y)$, where $Y$ ranges among the log resolutions of $(X,Z)$. In words, it is the set of divisorial points of $T_{X,Z}$ whose center on every log resolution of $(X,Z)$ is a divisor; if we assume the existence of a minimal log resolution $Y_\mathrm{min}$ of $(X,Z)$ then it coincides with the set $\Div_{X,Z}(Y_\mathrm{min})$. We call an open subset $U$ of $T_{X,Z}$ \defi{simple} if it is isomorphic to either a disc or a standard annulus, $U\cap\partial^{\mathrm{an}}\Txz=\emptyset$ and the topological boundary $\partial U=\overline U\setminus U$ is contained in $\Div_{X,Z}$. Then a finite subset $S$ of $T_{X,Z}$ is a regular vertex set if and only if all the connected components of $T_{X,Z}\setminus S$ are simple subspaces of $T_{X,Z}$. } \begin{thm}\label{thm_log_essential_valuations} Let $k$ be an algebraically closed field, let $(X,Z)$ be as in \ref{setting_surfaces} and let $v$ be an element of $\Div_{X,Z}$. Then $v$ is a log essential valuation of $(X,Z)$ if and only if it has no simple neighborhood in $T_{X,Z}$. \end{thm} \begin{proof} Proposition~\ref{prop_resolution_iff_vertex_set} implies that a divisorial point which has no simple neighborhood in $T_{X,Z}$ is log essential. To prove the reverse implication, assume that $U$ is a simple subspace of $T_{X,Z}$ and that $v$ is a divisorial point contained in $U$. Since $\partial U$ is a finite set of divisorial points, there exists a log resolution $Y$ of $(X,Z)$ such that $\Div_{X,Z}(Y)$ contains $\partial U$. Set $S=\Div_{X,Z}(Y)\setminus U$. Then $S$ is a finite subset of $\Div_{X,Z}$, nonempty because it contains $\partial U$, so by Theorem~\ref{vertex_sets} it is of the form $\Div_{X,Z}(\mathscr Y')$ for some formal log modification $\mathscr Y'$ of $(X,Z)$. The connected components of $T_{X,Z}\setminus \Div_{X,Z}(\mathscr Y')$ are either $U$ or connected components of $T_{X,Z}\setminus \Div_{X,Z}(Y)$, so they are all simple. By Proposition~\ref{prop_resolution_iff_vertex_set} the formal log modification $\mathscr Y'$ can therefore be algebraized to a log resolution $Y'$ of $(X,Z)$. Since $v$ doesn't belong to $\Div_{X,Z}(Y')$, this contradicts the fact that $v$ is log essential. \end{proof} \begin{rem}\label{J4_claim_boundary} Any element of the cover discussed in Example~\ref{analytic_structure_valuative_tree} provides an example in the valuative tree $T_{\mathbb A^2_\mathbb{C},0}$ of a disc whose complement is a nondivisorial point. This shows that it is really necessary to impose the condition on the topological boundary in the definition of simple subspace. \end{rem} We will now move to studying the more classical concept of essential valuations. \pa{ Let us assume that $k=\mathbb{C}$ and that $Z=X_{sing}$ is the singular locus of $X$. Then the set of log essential valuations contains the set of essential valuations studied by Nash in~\cite{Nash95}. More generally, over an arbitrary algebraically closed field the set of \defi{essential valuations} of a pair $(X,Z)$ is defined as the intersection $\bigcap_Y\Div_{X,Z}(Y)$, where $Y$ ranges among the (not necessarily log) resolutions of $(X,Z)$ and $\Div_{X,Z}(Y)$ denotes the finite set of divisorial points associated to those components of the exceptional locus of $Y\to X$ which are divisors. The sets of essential and log essential valuations differ when the minimal resolution of the pair $(X,Z)$ is not a log resolution, i.e. when its exceptional locus is not a normal crossings divisor. An example is given in \ref{example_nonlogessential2}. However, essential and log essential valuations coincide for big classes of singularities, for example for rational singularities. } \pa{ We call an open subset $U$ of $T_{X,Z}$ \emph{elementary} if its ring of bounded analytic functions ${\mathcal O}^\circ_{T_{X,Z}}(U)$ is isomorphic to $k[[t,u]]$ and its topological boundary $\partial U=\overline{U} \setminus U$ is a finite subset of $\Div_{X,Z}$. Observe that every simple subset of $T_{X,Z}$ is elementary; a crucial difference is that elementary subsets can contain points of $\partial^{\mathrm{an}}\Txz$. } We can now give a local criterion for essential valuations analogous to Theorem~\ref{thm_log_essential_valuations}. For technical reasons we will restrict ourselves to the case of a normal surface singularity $x\in X$. This is the case that is usually considered in the literature on the Nash problem. \begin{thm}\label{thm_essential_valuations} Let $k$ be an algebraically closed field, let $X$ be a normal surface over $k$, let $x$ be a point of $X$, and let $v$ be an element of $\Div_{X,x}$. Then $v$ is an essential valuation of $(X,x)$ if and only if it has no elementary neighborhood in $T_{X,x}$. \end{thm} The proof of this result is analogous to the one of Theorem~\ref{thm_log_essential_valuations}. Before we give it, we need a proposition which combines some results similar to Lemma~\ref{fibers_are_connected_components}, Theorem~\ref{vertex_sets}, Proposition~\ref{formal_fibers} and Proposition~\ref{prop_resolution_iff_vertex_set}. \begin{prop}\label{prop_resolution_elementary_components} Let $k$ be an algebraically closed field, let $X$ be a normal surface over $k$, let $x$ be a point of $X$, and let $S$ be a finite nonempty subset of $\Div_{X,x}$. Then there exist a normal special formal $k$-scheme $\mathscr Y$ and an adic morphism $\mathscr Y \to \widehat{X/x}$ inducing an isomorphism of normalized spaces such that the associated set of divisorial points $\Div_{X,x}(\mathscr Y)$ is $S$. If moreover every connected component of $\Div_{X,x}\setminus S$ is elementary, then $\mathscr Y$ can be algebraized by a resolution $Y$ of $(X,x)$. \end{prop} \begin{proof} Choose a resolution $(Y,D)$ of $(X,x)$ such that $S\subset \Div_{X,x}(Y)$. By the contractibility criterion of Grauert-Artin \cite{Artin70} we can contract every component of $D$ which does not correspond to an element of $S$, yielding a normal algebraic spaces ${\mathcal Y}$ over $k$ with a proper morphism $f$ to $X$ (for a systematic treatment of algebraic spaces we refer the reader to \cite{Knutson1971}). Indeed, the intersection matrix of the divisor that we want to contract is negative definite because the entire exceptional divisor of $Y$ can be contracted to $x$ in $X$. By taking the formal completion of this algebraic space along $f^{-1}(x)$ we obtain the formal $k$-scheme $\mathscr Y$ that we want. The connected components of $T_{X,x}\setminus \Div_{X,x}(\mathscr Y)$ are the inverse images through the center map of the closed points of $\mathscr Y_0$ (this can be proven as in Lemma~\ref{fibers_are_connected_components}). Let $y$ be a closed point of $\mathscr Y_0$ and let $\mathcal I_y$ be the image of the ideal defining $\mathscr Y_0$ in $\mathcal O_{\mathscr Y,y}$. As in Lemma~\ref{L: divisorial valuation specialization}, it follows from \ref{lemma fiber reduction} that we have a canonical isomorphism $\Sp_\mathscr Y^{-1}(y)\cong T_{\Spf\big(\widehat{\mathcal O_{\mathscr Y,y}}\big)}\setminus V(\mathcal I_y)$. Therefore we have \begin{align} \mathcal O^\circ_{T_{X,x}}\big({\Sp_\mathscr Y^{-1}(y)}\big) & \cong \mathcal O^\circ_{T_{\Spf(\widehat{\mathcal O_{\mathscr Y,y}})}}\Big(T_{\Spf(\widehat{\mathcal O_{\mathscr Y,y}})}\setminus V(\mathcal I_y)\Big) \\ & \cong \mathcal O^\circ_{T_{\Spf(\widehat{\mathcal O_{\mathscr Y,y}})}}\Big(T_{\Spf(\widehat{\mathcal O_{\mathscr Y,y}})}\Big) \cong \widehat{\mathcal O_{\mathscr Y,y}}, \end{align} where the second isomorphism follows from the extension theorem~\cite[Proposition 3.3.14]{Ber90} and the third one holds because $\mathscr Y$ is normal at $y$. If $\Sp_\mathscr Y^{-1}(y)$ is elementary, it follows then from Cohen's theorem that $\mathscr Y$ is smooth at $y$. Since this holds for every $y$, $\mathscr Y$ is non-singular, so the algebraic space ${\mathcal Y}$ is a non-singular, separated two-dimensional algebraic space over a field, hence it can be algebraized by a scheme (see \cite[V.4.9,10]{Knutson1971}), yielding the resolution $Y$ that we are looking for. \end{proof} \Pa{Remarks}{ If we are working over the field of complex numbers, we can apply Grauert contractibility criterion \cite{Grauert62} instead of Artin's and obtain $\mathscr Y$ as a complex analytic space. Of course this is the same as analytifying the complex algebraic space given by Artin's criterion. Observe that the surface $Y$ above is not necessarily a log resolution of $(X,x)$, as the exceptional locus of the morphism $Y\to X$ may not be a divisor with normal crossings. } \begin{proof}[Proof of Theorem~\ref{thm_essential_valuations}] As before, the reasoning of Proposition~\ref{prop_resolution_elementary_components} implies that a divisorial valuation which has no elementary neighborhood in $T_{X,x}$ is essential. To prove the reverse implication, let $U$ be an elementary subspace of $T_{X,x}$ and let $v$ be a divisorial point contained in $U$. Let $Y$ be a log resolution of $(X,x)$ such that $\Div_{X,x}(Y)$ contains $\partial U$, and set $S=\Div_{X,x}(Y)\setminus U$. Then $S$ is finite and nonempty, and the connected components of $\Div_{X,x}\setminus S$, being either $U$ or connected components of $T_{X,x}\setminus \Div_{X,x}(Y)$, are all elementary. Therefore, Proposition~\ref{prop_resolution_elementary_components} tells us that there exists a resolution $Y'$ of $(X,x)$ such that $\Div_{X,x}(Y')=S$. This proves that $v$ is not an essential valuation, since it doesn't belong to $\Div_{X,x}(Y')$. \end{proof} \begin{ex}\label{example_nonlogessential2} Let us give an example of a surface for which the sets of essential and log essential valuations do not coincide. Let $X$ be the hypersurface in $\mathbb{C}^3$ defined by the equation $f=z^2+(x^3+y^3)(y^3+x^4)$. Consider the projection $X\to {\mathbb C}^2$ defined by the coordinates $x$ and $y$: it is a double cover branched on the curve $C\coloneqq \{(x^3+y^3)(y^3+x^4)=0\}$. Blow up the origin in ${\mathbb C}^2$, and let $Y$ be the surface obtained by base change and normalization: $Y$ is a double cover branched on the strict transform $C'$ of $C$ and it is smooth since $C'$ is smooth. The exceptional locus of the resolution $Y\to X$ is the inverse image $D\subset Y$ of the exceptional curve of the blow up. A standard computation shows that $D$ is irreducible and $D^2=-2$, so the resolution is minimal. Moreover, $D$ has a simple cusp as singularity, therefore $(Y,D)$ is not a log resolution of $(X,X_{sing})$. \end{ex} \begin{rem}\label{remark_existence_resolutions} We expect the approach used in this section to lead to a new proof of the existence of resolutions of surfaces, at least in characteristic 0, in a similar way as one can prove the semistable reduction theorem for curves using non-archimedean analytic spaces. A proof would go roughly as follows. The normalized space $T_{X,Z}$ can be covered by finitely many smooth affinoid $k((t))$-analytic curves, since all the points of ${\X^}$ are regular. Then \cite[5.1.14]{Duc} applied to those $k((t))$-analytic curves gives us a vertex set $S\subset\Div_{X,Z}$ such that all connected components of $T_{X,Z}\setminus S$ are \defi{virtual discs} or \defi{virtual annuli}, i.e. $k((t))$-analytic spaces that become a $k((t))$-disc or a $k((t))$-annulus after a finite separable extension of $k((t))$. If we could prove that all those virtual discs and annuli are actual discs and annuli, we would obtain a log resolution of $(X,Z)$, since by enlarging $S$ we can cut an annulus of modulus $n$ into $n$ standard annuli. If the characteristic of $k$ is zero, by a special case of \cite{Ducros13} every virtual disc is a disc. A virtual annulus is a pseudo-affinoid $k((t))$-analytic space, and to prove that it is an annulus it would be enough, by a slight generalization of Proposition~\ref{prop_discs}, to show that it is a distinguished pseudo-affinoid. We believe that it is always possible to enlarge $S$ further and break a given virtual annulus in discs and finitely many annuli. \end{rem} \end{document}	arXiv
Visualising $\mathbb CP^2$: a problem of attaching cells with a dimension gap >1 For the uninitiated Morse theory, as many other early algebraic-topology widgets, leads to a picture of smooth manifolds as being built up from 'cells', copies of $\mathbb{D}^n$ for varying $n$, 'glued' to each other by the usual topological tools; giving rise to (in some sense) a more natural picture of homology as 'coming from' cellular homology. As an example, consider the torus $\mathbb{T}^2$: we begin with empty space, attach a 0-cell ($\mathbb{D}^0=$ a point), attach a 1-cell ($\mathbb{D}^1=$ a line) to your point (both ends of the line are attached to the point, creating a circle), attach another 1-cell (in the same way, to the same point, creating a sort of figure 8). The hardest bit to visualise is next: attaching a 2-cell ($\mathbb{D}^2=$ a disk, which we will think of as its homeomorphism equivalent, a square). Begin by twisting your figure 8 so that one circle is in the xy-plane, the other in the xz-plane, now attach the top and bottom of your square (coloured red in picture below) to the xy circle (creating a 'curling round' tube) and the left and right edges (coloured blue below) of your square (now a tube) either side of the xz circle, completing the torus. The above takes some thinking, but a little reading around shows that this is fairly easy to see. What makes it so easy is that the cells we are attaching are of adjacent dimensions, that is; we may easily identify the boundary of one with the entirety of another. Where it gets harder to visualise is when the dimensions of the cells we are attaching to one another differ by >1- the canonical example of this is the complex projective plane $\mathbb{CP}^2$, a 4-manifold built by attaching a disk to a point (making a sphere) and then attaching a 4-ball to that sphere. The latter attaching map (wherein points are identified with their images), I know, may be thought of as the Hopf fibration $\partial \mathbb{D}^4=S^3 \to \mathbb{CP}^1=S^2 $, but I have no way of visualising this, particularly with regard to the interior of the 4-disk. How does a 4 cell wrap around a 2 cell without producing a singularity of some kind? Is this analagous in some sense to Dehn surgery in which one uses a thickening? Is there a right way to think about this or can it only really be thought of 'intellectually'? intuition general-topology algebraic-topology Tom BoardmanTom Boardman $\begingroup$ What do you mean by «it can only be thought of 'intellectually'»? Are you using the word intellectually to mean incorrectly? $\endgroup$ – Mariano Suárez-Álvarez Aug 4 '10 at 16:05 $\begingroup$ I guess he means as opposed to 'viscerally,' e.g. in a way that someone with no mathematical training would understand. $\endgroup$ – Qiaochu Yuan Aug 4 '10 at 17:05 $\begingroup$ @Quiaochu- correct. @Mariano is right though; it was a stupid word to use. What I meant was sort of 'in terms of higher mathematical abstractions', but even that is too mealy and doesn't quite explain what I'm after- (CP^2 is a fairly high mathematical abstraction in itself after all). $\endgroup$ – Tom Boardman Aug 4 '10 at 17:50 1) Near a point of $S^2$ the picture looks like $\mathbb{C}$ sitting inside, naturally $\mathbb{C}^2$ (because $\mathbb{C}P^2\cong(\mathbb{C}^3\setminus{0})/\mathbb{C}^{\times}$ and $S^2=\mathbb{C}P^1$ is the image of a hyperplane in $\mathbb{C}^3$). 2) Think first about $S^2$: one attaches a disk to a point, contracting the circle on the disk's boundary. Now, to make $\mathbb{C}P^2$ one takes 4-disk and do pretty much the same but not for one circle but for all, well, fibers of the Hopf fibration at once — or, in other words, for all points of $S^2$ at once. So, since there were no singularities after gluing $S^2$, there will be no singularities here either. (Not sure if it's an answer, but only hope it helps.) Grigory MGrigory M $\begingroup$ +1- not quite the whole way there, but certainly helpful! $\endgroup$ – Tom Boardman Aug 4 '10 at 12:29 Tom, I'm not sure I see how it's getting any harder in passing from a torus to a projective space. In your $\mathbb CP^2$ case, you have $\mathbb CP^1$ sitting inside of it, and the boundary of a regular (tubular) neighbourhood of the $\mathbb CP^1$ is $S^3$. And $D^4$ has $S^3$ as its boundary, so the attaching map is tautological. The normal bundle is the missing data and that's what your CW-decomposition is ignoring. This is essentially what always happens. Perhaps the conceptual hump you're dealing with is that you're asking for CW-decompositions of manifolds. Morse functions generically only build homotopy-equivalences to CW-complexes, they do not put CW-structures on the manifold without some significant work. Moreover, CW-decompositions ignore some of the most essential properties of the manifold, like smooth structures. If instead you work with handle decompositions, what I state in my first paragraph is basically a generality -- critical points amount to handle attachments and the gluing instructions are always given in a direct way from the flow lines of the Morse function's (suitably normalized) gradient. So the handle decomposition is on the given manifold -- unlike the CW-case where you only have a homotopy-equivalence to a CW-complex. Ryan BudneyRyan Budney Note that you're only attaching the boundary of the 4-cell. This is a map $S^3 \rightarrow S^2$. The resulting topological space has a completely intact copy of the interior of the 4-cell. This is similar to making $\mathbb{RP}^2$ by gluing a disk to a circle by the degree-2 map $\partial D^2 = S^1 \rightarrow S^1$ (given, say, in complex coordinates by $z\mapsto z^2$). As for the issue of having high-dimensional spheres "wrap around" low-dimensional cells: up to homotopy, this can be phrased as a question about homotopy groups, that is, maps from spheres into some space $X$. (This doesn't address the question of singularities, but I think it is still helpful for trying to think about these gluing constructions.) These are groups $\pi_k(X)$ for all $k\geq 1$; when $k=1$, this just the fundamental group. In general, they are quite poorly understood. Even for the simplest possible CW complexes $X=S^n$ (except $n=1$), we don't know all the homotopy groups! This might sound like an easy problem, but it's really....REALLY...nontrivial. Now, when you make CW complexes, you're gluing on cells $e^k$ one by one; these are really just maps from $\partial e^k = S^{k-1}$ to the stuff you already had (the "$(k-1)$-skeleton"). People often only care about the homotopy types of these maps. In summary, while it's certainly a good idea to try and visualize these things, it's also good to be able to deal abstractly with gluing constructions. The wikipedia article on homotopy groups of spheres is really interesting, and contains a sort of picture of the Hopf fibration. Basically, you want to think of it as a bunch of circles that somehow fill out $S^3$, which is just $\mathbb{R}^3$ with one extra point at $\infty$. Interestingly, any two of these circles are linked. So you can kind of think of it locally as the unit circle in the xy-plane, and then you can move that circle in 2 dimension's worth of directions, and in every one of those directions that you push the circle off itself you're going to get a new, nearby circle that's linked to the original one. (Of course, there is precisely an $S^2$'s worth of circles!) You can take the circle going through $\infty$ to be the z-axis. So by what I just said, all of these other circles need to wind once around the z-axis too. Aaron Mazel-GeeAaron Mazel-Gee Concerning the Hopf fibration, the film "Dimensions" does a superb job of visualizing it (chapters 7-8 in the table of contents). Concerning the visualization of the glueing, maybe it helps to visualize other, similar constructions to get at least some intuition. For instance, you could try to get a picture of how the torus looks like after every "meridian" (blue circle in your picture) has been squashed to a point. Greg GravitonGreg Graviton I think you understand attaching cells that have dimension gap better than you think! Consider any $S^n$ for $n>>0$, we can think of this as being the attachment of $D^n$ to a point which is 0-dimensional. Also, I would recommend that you think of this attaching cells thing a bit more carefully. if you understand the hopf fibration then you do understand the attaching of the cell. The hopf fibration is not easy to picture (as far as I know). I would recommend thinking about the map in terms of what happens when you work with coordinates. I will be back later to add more... I promise. Sean TilsonSean Tilson Not the answer you're looking for? Browse other questions tagged intuition general-topology algebraic-topology or ask your own question. Fundamental group of a certain torus-like surface Cellular Homology of the 3-Torus Fundamental group computation Degree of this attaching map — or how to define this attaching map? What to do when this theorem can't be applied: How to calculate $H_1$? CW complex structure on standard sphere identifying the south pole and north pole How is the following a CW complex Visualizing products of $CW$ complexes How exactly does a constant identification map attach a $n$-cell?	CommonCrawl
\begin{document} \centerline{\bf On a functional equation appearing in characterization of} \centerline{\bf distributions by the optimality of an estimate\footnote {This research was supported in part by the grant “Network of Mathematical Research 2013--2015”}} \vskip 1 cm \centerline{G.M. FELDMAN} \centerline{B. Verkin Institute for Low Temperature Physics and Engineering} \centerline{ Kharkiv, Ukraine} \centerline{and} \centerline{P. Graczyk} \centerline{ Laboratoire LAREMA UMR 6093 - CNRS} \centerline{Universite a'Angers, France} \vskip 1 cm \makebox[20mm]{ }\parbox{125mm}{ \small Let $X$ be a second countable locally compact Abelian group containing no subgroup topologically isomorphic to the circle group $\mathbb{T}$. Let $\mu$ be a probability distribution on $X$ such that its characteristic function $\widehat\mu(y)$ does not vanish and $\widehat\mu(y)$ for some $n \geq 3$ satisfies the equation $$ \prod_{j=1}^{n} \hat\mu(y_j + y) = \prod_{j=1}^{n} \hat\mu(y_j - y), \quad \sum_{j=1}^{n} y_j = 0, \quad y_1,\dots,y_n,y \in Y. $$ Then $\mu$ is a convolution of a Gaussian distribution and a distribution supported in the subgroup of $X$ generated by elements of order 2.} \vskip 1 cm The present note is devoted to study a functional equation on a locally compact Abelian group group which appears in characterization of probability distributions by the optimality of an estimate. Let $X$ be a second countable locally compact Abelian group, $Y=X^$ be its character group, $(x,y)$ be the value of a character $y\in Y$ at an element $x\in X$. Denote by ${\rm M}^1(X)$ the convolution semigroup of probability distribution on the group $X$, and denote by $$\widehat\mu(y) = \int_{X}(x, y)d \mu(x)$$ the characteristic function of a distribution $\mu\in {\rm M}^1(X)$. For $\mu \in {\rm M}^1(X)$ define the distribution $\bar \mu \in {\rm M}^1(X)$ by the formula $\bar \mu(B) = \mu(-B)$ for all Borel sets of $X$. Then $\widehat{\bar\mu}(y)=\overline{\widehat\mu(y)}.$ A distribution $\gamma\in {\rm M}^1(X)$ is called Gaussian if its characteristic function is represented in the form \begin{equation}\label{0} \widehat\gamma(y)= (x,y)\exp\{-\varphi(y)\}, \end{equation} where $x \in X$, and $\varphi(y)$ is a continuous nonnegative function on the group $Y$ satisfying the equation \begin{equation}\label{1} \varphi(y_1 + y_2) + \varphi(y_1 - y_2) = 2[\varphi(y_1) + \varphi(y_2)], \quad y_1, \ y_2 \in Y. \end{equation} A Gaussian distribution is called symmetric if in (\ref{0}) $x=0$. Denote by $\Gamma(X)$ the set of Gaussian distributions on the group $X$. Consider a probability space $(X, {\cal B}, \mu)$, where ${\cal B}$ is a $\sigma$-algebra of Borel subsets of $X$, and $\mu \in {\rm M}^1(X)$. Form a family of distributions $\mu_\theta(A) = \mu(A - \theta), \ A \in {\cal B}, \ \theta \in X$. Denote by $\Pi$ a class of estimates $f : X^n \mapsto X$ satisfying the condition $f(x_1 + c, \dots , x_n + c) = f(x_1, \dots, x_n) + nc$ for all $x_1,\dots ,x_n, c \in X$. According to \cite{Ru1} (see also \cite{Ru2}, \cite[Ch. 7, \S 7.10]{KaLiRa}), an estimate $f_0\in\Pi$ of a parameter $n\theta$ is called an optimal estimate in the class $\Pi$ for a sample volume $n$ if for any estimate $f\in\Pi$ and for all $y \in Y$ the inequality $$ {\bf E}_\theta\|(f_0({\bf x}), y) - (n\theta, y)\|^2 \leq {\bf E}_\theta\|(f({\bf x}), y) - (n\theta, y)\|^2 $$ holds. It turns out that the existence of an optimal estimate of the papameter $n\theta$ gives the possibility in some cases to describe completely the possible distributions $\mu$. As has been proved in (\cite{Ru1}), if an estimate $f_0$ is represented in the form \begin{equation}\label{2} (f_0({\bf x}), y) = (f({\bf x}), y) \frac{{\bf E}_0[(f({\bf x}), -y)\| {\bf z}]} {\|{\bf E}_0[(f({\bf x}), -y)\| {\bf z}]\|}, \quad y \in Y, \end{equation} where $f\in \Pi$, and ${\bf z} = (x_2 - x_1,\dots ,x_n - x_1)$, then $f_0\in \Pi$, $f_0$ does not depend on the choice of $f$ and $f_0$ is an optimal estimate. It follows from (\ref{2}) that $f_0$ is an optimal estimate if and only if $\arg{\bf E}_0[(f_0({\bf x}), y)\| {\bf z}] = 0$ . When $f_0({\bf x}) = \sum_{j=1}^{n}x_j$ it follows from this that the characteristic function $\widehat\mu(y)$ satisfies the equation \begin{equation}\label{3} \prod_{j=1}^{n} \hat\mu(y_j + y) = \prod_{j=1}^{n} \hat\mu(y_j - y), \quad \sum_{j=1}^{n} y_j = 0, \quad y_1,\dots,y_n,y \in Y, \end{equation} and $\widehat\mu^{n}(y) > 0$. When $\ n \geq 3 $ this implies that if a group $X$ contains no elements of order 2, then $\mu \in \Gamma(X)$ (see \cite{Ru1}). This note is devoted to solving of equation (\ref{3}) in a general case when $X$ is a locally compact Abelian group. Let us fix the notation. Denote by $f_2 : X \mapsto X$ the endomorphism of $X$ defined by the formula $f_2(x)=2x$. Put $X_{(2)} = {\rm Ker} \ f_2,$ $X^{(2)} = {\rm Im} \ f_2.$ Denote by $\mathbb{T}$ the circle group, and by $\mathbb{Z}$ the group of integers. Let $\psi(y)$ be an arbitrary function on the group $Y$ and $h \in Y$. Denote by $\Delta_h$ the finite difference operator $$ \Delta_h \psi(y) = \psi(y + h) - \psi(y), \quad y \in Y. $$ A continuous function $\psi(y)$ on the group $Y$ is called a polynomial if \begin{equation}\label{4} \Delta_h^{m+1} \psi(y) = 0 \end{equation} for some $m$ and for all $y, h \in Y$. The minimal $m$ for which (\ref{4}) holds is called the degree of the polynomial $\psi(y)$. From analytical point of view the result proved in \cite{Ru1} can be reformulated in the following way. Let $\mu\in {\rm M}^1(X)$, the characteristic function $\widehat\mu(y)$ satisfy equation (\ref{3}) for some $n \geq 3$ and $\widehat\mu^n(y)>0$. Then if the group $X$ contains no elements of order 2, then $\mu \in \Gamma(X)$. It is easy to see that if $\gamma$ is a symmetric Gaussian distribution on the group $X$ and $\pi \in {\rm M}^1(X_{(2)})$, then the characteristic functions $\widehat\gamma(y)$ and $\widehat\pi(y)$ satisfy equation (\ref{3}), and hence the characteristic function of the distribution $\mu = \gamma\pi$ also satisfies equation (\ref{3}). Describe first the groups $X$ for which the converse statement is true. {\bf Theorem 1}. {\it Let $X$ be a second countable locally compact Abelian group, $\mu \in {\rm M}^1(X)$. Let the characteristic function $\widehat\mu(y)$ satisfy equation $(\ref{3})$ for some $n \geq 3$ and $\widehat\mu(y) \ne 0$. Assume that the following condition holds: $(i)$ the group $X$ contains no subgroup topologically isomorphic to the circle group $\mathbb{T}$. Then $\mu=\gamma\pi$, where $\gamma\in \Gamma(X)$ and $\pi\in{\rm M}^1(X_{(2)})$}. {\bf Proof}. Set $\nu = \mu\bar\mu$. Then $\widehat\nu(y) = \|\widehat\mu(y)\|^2 > 0$. Put $\psi(y) = - \ln\widehat\nu(y)$. Equation (\ref{3}) is equivalent to the equation \begin{equation}\label{5} \sum_{j=1}^{n}\psi(y_j + y) = \sum_{j=1}^{n}\psi(y_j - y), \quad \sum_{j=1}^{n}y_j = 0, \quad y_1,\dots,y_n, y \in Y. \end{equation} We also note that \begin{equation}\label{6} \psi(-y) = \psi(y), \quad y \in Y. \end{equation} Substituting in (\ref{5}) $y_3=-y_1-y_2, \quad y_4=\dots=y_n=0$ and taking into account (\ref{6}), we get \begin{equation}\label{7} \psi(y_1 + y_2 + y) - \psi(y_1 + y_2 - y) = \psi(y_1 + y) - \psi(y_1 - y) $$ $$ + \psi(y_2 + y) - \psi(y_2 - y), \quad y_1, y_2, \ y \in Y. \end{equation} Setting successively $y = y_1 + y_2$, $y = y_1$, $y = y_2$, we find from (\ref{7}) that $$ \psi(2y_1 + 2y_2) = \psi(2y_1) + 2\psi(y_1 + y_2) - 2\psi(y_1 - y_2) + \psi(2y_2), \quad y_1, y_2 \in Y. $$ This implies that $$ \psi(2y_1 + 2y_2) + \psi(2y_1 - 2y_2) = 2[\psi(2y_1) + \psi(2y_2)], \quad y_1, y_2 \in Y, $$ i.e. the function $\psi(y)$ satisfies equation (\ref{1}) on the subgroup $Y^{(2)}$, and hence, the function $\psi(y)$ satisfies equation (\ref{1}) on the subgroup $\overline{Y^{(2)}}$. Denote by $\varphi_0(y)$ the restriction of the function $\psi(y)$ to the subgroup $\overline{Y^{(2)}}$. It is well known that we can associate to each function $\varphi(y)$ satisfying equation (\ref{1}) a symmetric 2-additive function $$ \Phi(u, v) = \frac{1}{2} [ \varphi(u + v) - \varphi(u) - \varphi(v)], \quad u, v \in Y. $$ Then $\varphi(y) = \Phi(y, y)$. Using this representation it is not difficult to verify that the function $\psi(y)$ on the subgroup $\overline{Y^{(2)}}$ satisfies the equation \begin{equation}\label{8} \Delta_k^2\Delta_{2h} \ \psi(y) = 0, \quad k, \ y \in \overline{Y^{(2)}}, \quad h \in Y. \end{equation} Return to equation (\ref{7}) and apply the finite difference method to solve it. Let $h_1$ be an arbitrary element of the group $Y$. Put $k_1 = h_1$. Substitute $y_2+h_1$ for $y_2$ and $y+k_1$ for $y$ in equation (\ref{7}). Subtracting equation (\ref{7}) from the resulting equation we obtain \begin{equation}\label{9} \Delta_{2h_1} \ \psi(y_1 + y_2 + y ) = \Delta_{h_1} \ \psi(y_1 + y) - \Delta_{-h_1} \ \psi(y_1 - y) + \Delta_{2h_1} \ \psi(y_2 + y). \end{equation} Next, let $h_2$ be an arbitrary element of the group $Y$. Put $k_2 = - h_2$. Substitute $y_2+h_2$ for $y_2$ and $y+k_2$ for $y$ in equation (\ref{9}). Subtracting equation (\ref{9}) from the resulting equation we get $$ \Delta_{-h_2}\Delta_{h_1} \ \psi(y_1 + y ) - \Delta_{h_2}\Delta_{-h_1} \ \psi(y_1 - y ) = 0. $$ Reasoning similarly we find from this $$ \Delta_{2h_3}\Delta_{-h_2}\Delta_{h_1} \ \psi(y_1 + y ) = 0, $$ and finally \begin{equation}\label{a1} \Delta_{2h_3}\Delta_{-h_2}\Delta_{h_1}\Delta_{y_1} \ \psi(y) = 0. \end{equation} Note that $h_j, y_1, y$ are arbitrary elements of the group $Y$. Setting in (\ref{a1}) $h_3=h$, $-h_2=h_1=y_1=k$, we find \begin{equation}\label{10} \Delta_k^3\Delta_{2h} \ \psi(y) = 0, \quad k, \ h, \ y \in Y. \end{equation} Fix $h \in Y$. On the one hand, it follows from (\ref{10}) that the function $\Delta_{2h}\psi(y)$ is a polynomial of degree $\leq 2$ on the group $Y$. On the other hand, as follows from (\ref{8}) the function $\Delta_{2h}\psi(y)$ is a polynomial of degree $\leq 1$ on the subgroup $\overline{Y^{(2)}}$. Then as not difficult to verify, the function $\Delta_{2h}\psi(y)$ must be a polynomial of degree $\leq 1$ on the group $Y$, i.e. \begin{equation}\label{11} \Delta_k^2\Delta_{2h} \ \psi(y) = 0, \quad k, \ h, \ y \in Y. \end{equation} Theorem 1 follows now from the following lemma. {\bf Lemma 1} (\cite[Prop. 1]{Fe2}). {\it Let $X$ be a second countable locally compact Abelian group containing no subgroup topologically isopmorphic to the circle group $\mathbb{T}$. Let $\mu \in {\rm M}^1(X)$, $\nu=\mu\bar\mu$ and $$ \hat\nu(y)=\exp\{-\psi(y)\}, $$ where the function $\psi(y)$ satisfies equation $(\ref{11})$. Then $\mu=\gamma\pi$, where $\gamma\in \Gamma(X)$ and $\pi\in{\rm M}^1(X_{(2)})$}. {\bf Remark 1}. Obviously, the above mentioned Rukhin's theorem follows directly from Theorem 1. {\bf Remark 2}. Let $X$ be a second countable locally compact Abelian group containing a subgroup topologically isomorphic to the circle group $\mathbb{T}$. Then we can consider any distribution $\mu$ on the circle group $\mathbb{T}$ as a distribution on $X$. Note that $\mathbb{Z}$ is the character group of $\mathbb{T}$. Following to \cite{Ru1} consider on the group $\mathbb{Z}$ the function $$f(m)=\begin{cases} \exp\{-m^2\}, & \text{\ if\ }\ \ m \in \mathbb{Z}^{(2)}, \\ \exp\{-m^2 + \varepsilon\}, & \text{\ if\ }\ m \not\in \mathbb{Z}^{(2)}, \end{cases} $$ where $\varepsilon > 0$ is small enough. Then $$ \rho(t) = \sum_{m=-\infty}^\infty f(m)e^{-imt} > 0.$$ Let $\mu$ be a distribution on $\mathbb{T}$ with density $ \rho(t)$ with respect to the Lebesque measure. Then $f(m)$ is the characteristic function of a distribution $\mu$ on the circle group $\mathbb{T}$. Considering $\mu$ as a distribution on the group $X$, we see that $\widehat\mu(y) >0$ and the characteristic function $\widehat\mu(y)$ satisfies equation (\ref{3}), but as easily seen, $\mu\not\in \Gamma(X){\rm M}^1(X_{(2)})$. This example shows that condition $(i)$ in Theorem 1 is sharp. {\bf Remark 3}. Let $X$ be a second countable locally compact Abelian group. In the articles \cite{Fe1} and \cite{Fe2} (see also \cite[\S 16]{Fe2}) were studied group analogs of the well-known Heyde theorem, where a Gaussian distribution is characterized by the symmetry of the conditional distribution of a linear form $L_2 =\beta_1\xi_1 + \cdots + \beta_n\xi_n$ of independent random variables $\xi_j$ given $L_1 =\alpha_1\xi_1 + \cdots + \alpha_n\xi_n$ (coefficients of the forms are topological automorphisms of the group $X$). Let $\widehat\mu_j(y)$ be the characteristic function of the random variable $\xi_j.$ It is interesting to remark that if the number of independent random variables $n=2$, then the functions $\psi_j(y)=-\ln\|\widehat\mu_j(y)\|^2$ also satisfy equation (\ref{11}). For the groups $X$ containing no subgroup topologically isomorphic to the circle group $\mathbb{T}$, and also for the two-dimensional torus $X=\mathbb{T}^2$ this implies that all $\mu_j \in \Gamma(X){\rm M}^1(X_{(2)})$. We use Theorem 1 to prove the following statement, a significant part of which refers to the case when the group $X$ contains a subgroup topologically isomorphic to the circle group $\mathbb{T}$. {\bf Theorem 2}. {\it Let $X$ be a second countable locally compact Abelian group. Let $\mu \in {\rm M}^1(X)$, let the characteristic function $\widehat\mu(y)$ satify equation $(\ref{3})$ for some odd $n$, and $\widehat\mu^n(y) > 0$. Assume that the group $X$ satisfies the condition: $(i)$ the subgroup $X_{(2)}$ is finite. Then $\mu = \gamma_0\pi$, where $\gamma_0 \in \Gamma(X)$, and $\pi$ is a signed measure on $X_{(2)}$.} {\bf Proof.} Put $\psi(y) = -\ln\|\widehat\mu(y)\|$. Then the function $\psi(y)$ satisfies equation (\ref{11}). As has been proved in \cite{Fe2} in this case the function $\psi(y)$ is represented in the form $$ \psi(y) = \varphi(y) + r_\alpha, \quad y \in y_\alpha + \overline{Y^{(2)}}, $$ where $\varphi(y)$ is a continuous function satisfying equation (\ref{1}), and $Y = \bigcup_\alpha {(y_\alpha + \overline{Y^{(2)}})}$ is a decomposition of the group $Y$ with respect to the subgroup $\overline{Y^{(2)}}$. Since $X_{(2)}$ is a finite subgroup, it is easy to see that the function $g(y) =\exp\{ - r_\alpha\}, \quad y \in y_\alpha + \overline{Y^{(2)}},$ is the chracteristic function of a signed measure $\pi$ on the subgroup $X_{(2)}$. It follows from this that $$ \|\hat\mu(y)\| = \hat\gamma(y) \hat\pi(y), $$ where $\gamma \in \Gamma(X)$ and $\widehat\gamma (y)= \exp\{-\varphi(y)\}$. Set $l(y) = \widehat\mu(y) / \|\hat\mu(y)\|$ and check that the function $l(y)$ is a character of the group $Y$. Hence, Theorem 2 will be proved. Note that the function $l(y)$ satisfies equation (\ref{3}) and \begin{equation}\label{12} l(-y) = \overline {l(y)}, \quad l^n(y) = 1, \quad y \in Y. \end{equation} Put in (\ref{3}) $y_2 = -y_1$, $y_3 =\dots= y_n = 0$. We get $$ l^{n-2}(y) l(y_1 + y) l(-y_1 + y) = l^{n-2}(-y) l(y_1 - y) l(-y_1 - y), \quad y, \ y_1, \ y_2 \in Y. $$ Taking into account (\ref{12}), it follows from this that $$ l^{2}(y + y_1) l^{2}(y - y_1) = l^{4}(y), \quad y, \ y_1 \in Y. $$ Set $m(y) = l^2(y)$. Then the function $m(y)$ satisfies the equation \begin{equation}\label{13} m(u + v) m(u - v) = m^2(u), \quad u, \ v \in Y. \end{equation} We find by induction from (\ref{13}) that \begin{equation}\label{14} m(py) = m^p(y), \quad p \in \mathbb{Z}, \ y \in Y. \end{equation} Now we formulate as a lemma the following statement. {\bf Lemma 2.} {\it Let $Y = Y_1 + Y_2$, let a continuous function $m(y)$ on $Y$ satisfy equation $(\ref{13})$ and $m^{n}(y) = 1$ for some odd $n$. Then, if the restriction of the function $m(y)$ to $Y_j$ is a character of the group $Y_j$, $j = 1, 2$, then $m(y)$ is a character of the group $Y$.} {\bf Proof}. Denote by $y = (y_1, y_2)$, $y_1\in Y_1, y_2\in Y_2$ elements of the group $Y$. Put $a(y_1, y_2) = m(y_1, 0) m(0, y_2)$, $b(y_1, y_2) = m(y_1, y_2)/a(y_1, y_2)$. Then $b(y_1, 0) = b(0, y_2) = 1$, $y_1\in Y_1, y_2\in Y_2$. It is obvious that the function $b(y_1, y_2)$ also satisfies equation (\ref{13}). Substitute in (\ref{13}) $u = (y_1, 0)$, $v = (y_1, y_2)$. We have $$ b(2y_1, y_2) b(0, -y_2) = b^2(y_1, 0), \quad y_1\in Y_1, y_2\in Y_2. $$ This implies that $b(2y_1, y_2) = 1$ for $y_1\in Y_1, y_2\in Y_2$. In particular, $b(2y_1, 2y_2) = 1$. But it follows from (\ref{14}) that $b(2y_1, 2y_2) = b^2(y_1, y_2)$. Hence, $b(y_1, y_2) = \pm 1$. Since $b^n(y_1, y_2) = 1$ and $n$ is odd, we have $b(y_1, y_2)=1$ for $y_1\in Y_1, y_2\in Y_2$, i.e. $m(y_1, y_2) = a(y_1, y_2)$ is a character of the group $Y$. Continue the proof of Theorem 2. Since, by the assumption, $X_{(2)}$ is a finite subgroup, there exist $q \ge 0$ such that the group $X$ contains a subgroup topologically isomorphic to the group $\mathbb{T}^q$, but $X$ does not contain a subgroup topologically isomorphic to the group $\mathbb{T}^{q+1}$. It is well known that a subgroup of $X$ topologically isomorphic to a group of the form $\mathbb{T}^k$ is a topologically direct summand in $X$. For this reason the group $X$ is represented in the form $X = \mathbb{T}^q + G$, where the group $G$ contains no subgroup topologically isomorphic to the circle group $\mathbb{T}$. We have $Y \cong \mathbb{Z}^q + H$, $H = G^$. It follows from Lemma 2 and (\ref{14}) by induction that the function $m(y)$ on the group $\mathbb{Z}^q$, satisfying equation (\ref{13}) and the condition $m^n(y) = 1$ is a character of the group $\mathbb{Z}^q$. By Theorem 1 the restriction of the function $\widehat\mu(y)$ to $H$ is a product of the characteristic function of a Gaussian distribution on the group $G$ and the characteristic function of a distribution on the subgroup $G_{(2)}$. Taking into account that the characteristic function of any distribution on $G_{(2)}$ takes only real values, it follows from the equality $$\widehat\mu^2(y) = \|\widehat\mu(y)\|^2 m(y), \quad y\in Y,$$ that the restriction of the function $m(y)$ to $H$ is a character of the subgroup $H$. Applying again Lemma 2 to the group $Y$, we obtain that $m(y)$ is a character of the group $Y$. Since $n$ is odd, we have $2 r + n s = 1$ for some integers $r$ and $s$. Taking into account (\ref{12}) this implies that $l(y) = (l(y))^{2r + ns} = (m(y))^r$ is a character of the group $Y$. Theorem 2 is completely proved. We note that the example given in Remark 2 shows that a signed measure $\pi$ needs not be a measure. {\bf Remark 4}. Consider the infinite-dimensional torus $X=\mathbb{T}^{\aleph_0}$. Then $Y \cong {\mathbb{Z}}^{{\aleph_0}}$, where ${\mathbb{Z}}^{{\aleph_0}}$ is the group of all sequences of integers such that in each sequence only finite number of members are not equal to zero. Consider on the group $\mathbb{Z}$ the sequence of the functions $$f_k(m) =\begin{cases} \exp\{-a_k m^2\}, & \text{\ if\ }\ \ m \in \mathbb{Z}^{(2)}, \\ \exp\{-a_k m^2 + k\}, & \text{\ if\ }\ m \not\in \mathbb{Z}^{(2)}, \end{cases} $$ where $k = 1, 2, \dots$. Put $$ f(m_1, \dots, m_l, 0, \dots) = \prod _{k=1}^{l}f_k(m_k), \quad (m_1, \dots, m_l, 0, \dots) \in {\mathbb{Z}}^{{\aleph_0}}. $$ Take $a_k > 0$ such that $$ \sum_{(m_1,\dots, m_l, 0, \dots) \in {\mathbb{Z}}^{{\aleph_0}}}f(m_1, \dots, m_l, 0, \dots) < 2. $$ Then $$ \rho(t_1, \dots, t_l, \dots)=\sum_{(m_1,\dots, m_l, 0, \dots) \in {\mathbb{Z}}^{{\aleph_0}}}f(m_1, \dots, m_l, 0, \dots)e^{-i(m_1t_1+\dots+m_lt_l+\dots)}>0, \quad t_j\in {\mathbb{R}}.$$ It follows from this that $f(m_1, \dots, m_l, 0, \dots)$ is the characteristic function of a distribution $\mu \in {\rm M}^1(\mathbb{T}^{\aleph_0})$ such that $\widehat\mu(y) >0$ and $\widehat\mu(y)$ satisfies equation (\ref{3}), but $\mu$ can not be represented as a convolution $\mu = \gamma \pi$, where $\gamma \in \Gamma(\mathbb{T}^{\aleph_0})$, and $\pi$ a signed measure on the group $\mathbb{T}^{\aleph_0}_{(2)}$. The subgroup $\mathbb{T}^{\aleph_0}_{(2)}$ is infinite in this case. This example shows that condition $(i)$ in Theorem 2 is sharp. {\bf Remark 5}. We assumed in Theorem 2 that $n$ is odd. This condition can not be omitted even for the circle group $X = \mathbb{T}$. Indeed, let $n = 4$. Take $a$ in such a way that the function $$ f(m) = \exp\{- a m^2 + i \frac{\pi}{2} m^3\}, \quad m \in \mathbb{Z} $$ be the characteristic function of a distribution $\mu \in {\rm M}^1(\mathbb{T})$. On the one hand, it is ibvious that the function $f(m)$ satisfies equation (\ref{3}) and $f^4(m) > 0$, $m \in \mathbb{Z}$. On the other hand, the distribution $\mu$ can not be represented in the form $\mu = \gamma * \pi$, where $\gamma \in \Gamma(\mathbb{T})$, and $\pi$ is a signed measure on $\mathbb{T}_{(2)}$. This example also shows that a function $f(y)$ satisfying equation (\ref{3}), generally speaking, needs not be real. \end{document}	arXiv
\begin{document} \title[Exist. and Nonexist. Warped Product Subman. of Almost Contact Man. \; \; ]{Existence and Nonexistence of Warped Product Submanifolds of Almost Contact Manifolds} \author[A. Mustafa]{Abdulqader Mustafa} \address{Department of Mathematics, Faculty of Arts and Science, Palestine Technical University, Kadoorei, Tulkarm, Palestine} \email{[email protected]} \author[C. $\ddot O$zel]{Cenap $\ddot O$zel} \address{Department of Mathematics, Faculty of Science, King Abdulaziz University, 21589 Jeddah, Saudi Arabia} \email{[email protected]} \; \author[A. Pigazzini]{Alexander{\;}Pigazzini} \address{Mathematical and Physical Science Foundation, 4200 Slagelse, Denmark} \email{[email protected]} \author[R. Pincak]{Richard Pincak} \address{Institute of Experimental Physics, Slovak Academy of Sciences, Kosice, Slovak Republic} \email{[email protected]} \maketitle \begin{abstract} This paper has two goals; the first is to generalize results for the existence and nonexistence of warped product submanifolds of almost contact manifolds, accordingly a self-contained reference of such submanifolds is offered to save efforts of potential research. Most of the results of this paper are general and decisive enough to generalize both discovered and not discovered results. Moreover, a discrete example of contact $CR$-warped product submanifold in Kenmotsu manifold is constructed. For further research direction, we addressed a couple of open problems arose from the results of this paper. \noindent{\it{AMS Subject Classification (2010)}}: {53C15; 53C40; 53C42; 53B25} \noindent{\it{Keywords}}: { Contact $CR$-warped products; Sasakian manifolds; Kenmotsu manifolds; cosymplectic manifolds; nearlt trans-Sasakian manifolds; general warped product; doubly warped product, second fundamental form; totally geodesic} \end{abstract} \sloppy \section{Introduction} Warped products have been playing some important roles in the theory of general relativity as they have been providing the best mathematical models of our universe for now; that is, the warped product scheme was successfully applied in general relativity and semi-Riemannian geometry in order to build basic cosmological models for the universe. For instance, the Robertson-Walker spacetime, the Friedmann cosmological models and the standard static spacetime are given as warped product manifolds. For more cosmological applications, warped product manifolds provide excellent setting to model spacetime near black holes or bodies with large gravitational force. For example, the relativistic model of the Schwarzschild spacetime that describes the outer space around a massive star or a black hole admits a warped product construction \cite{iijj77}. In an attempt to construct manifolds of negative curvatures, R.L. Bishop and O'Neill \cite{ddyy7} introduced the notion of {\it warped product manifolds} as follows: Let $N_1$ and $N_2$ be two Riemannian manifolds with Riemannian metrics $g_{N_1}$ and $g_{N_2}$, respectively, and $f>0$ a $C^\infty$ function on $N_1$. Consider the product manifold $N_1\times N_2$ with its projections $\pi_1:N_1\times N_2\mapsto N_1$ and $\pi_2:N_1\times N_2\mapsto N_2$. Then, the {\it warped product} $\tilde M^m= N_1\times _fN_2$ is the Riemannian manifold $N_1\times N_2=(N_1\times N_2, \tilde g)$ equipped with a Riemannian structure such that $\tilde g=g_{N_1} + f^2 g_{N_2}$. A warped product manifold $\tilde M^m=N_1\times _fN_2$ is said to be {\it trivial} if the warping function $f$ is constant. For a nontrivial warped product $N_1\times _fN_2$, we denote by $\mathfrak{D}_1$ and $\mathfrak{D}_2$ the distributions given by the vectors tangent to leaves and fibers, respectively. Thus, $\mathfrak{D}_1$ is obtained from tangent vectors of $N_1$ via the horizontal lift and $\mathfrak{D}_2$ is obtained by tangent vectors of $N_2$ via the vertical lift. Since our goal to search about existence and nonexistence of warped product submanifolds in almost contact manifolds, we hypothesize the following two problems. The first is for single warped products \begin{problem}\label{prob9} Prove existence or nonexistence of single warped product submanifolds of almost contact manifolds. \end{problem} The second problem is for doubly warped products \begin{problem}\label{prob10} Prove existence or nonexistence of doubly warped product submanifolds of almost contact manifolds. \end{problem} The present paper is organized as follows: After the introduction, we present in Section 2, the preliminaries, basic definitions and formulas. In Section 3, we provide basic results, which are necessary and useful to the next section. In Section 4, which is the main section, we generalize theorems for existence and nonexistence warped product submanifolds for single and doubly warped product submanifolds in almost contact manifolds. Moreover, in the current section we discuss the contact $CR$-warped product submaifolds in almost contact manifolds and construct an example of both types of contact $CR$-warped product submanifolds of Kenmotsu manifolds. In the final section, we address two open problems related to the obtained results in this paper. \section{Preliminaries} At first, let us recall the following important two facts regarding Riemannian submanifolds, \cite{was}. \begin{definition}\label{dfffff5555} Let $M^n$ and $\tilde M^m$ be differentiable manifolds. A differentiable mapping $\varphi: M^n\longrightarrow \tilde M^m$ is said to be an {\it immersion} if $d\varphi_x: T_xM^n\rightarrow T_{\varphi (x)} \tilde M^m$ is injective for all $x\in M^n$. If, in addition, $\varphi$ is a homeomorphism onto $\varphi(M^n)\subset \tilde M^m$, where $\varphi(M^n)$ has the subspace topology induced from $\tilde M^m$, we say that $\varphi$ is an {\it embedding}. If $M^n\subset \tilde M^m$ and the inclusion $\boldsymbol{i}: M^n\subset \tilde M^m$ is an embedding, we say that $M^n$ is a submanifold of $\tilde M^m$. \end{definition} It can be seen that if $\varphi: M^n\rightarrow \tilde M^m$ is an immersion, then $n\le m$; the difference $m-n$ is called the {\it codimension} of the immersion $\varphi$. For most local questions of geometry, it is the same to work with either immersions or embeddings. This comes from the following proposition which shows that every immersion is locally (in a certain sense) an embedding. \begin{proposition}\label {3g9stg} Let $\varphi: M^n\longrightarrow \tilde M^m$, $n\le m$, be an immersion of the differentiable manifold $M^n$ into the differentiable manifold $\tilde M^m$. For every point $x\in M^n$, there exists a neighborhood ${\mathfrak u}$ of $x$ such that the restriction $\varphi \| {\mathfrak u}\rightarrow \tilde M^m$ is an embedding. \end{proposition} Now, we turn our attention to the differential geometry of the submanifold theory. First, let $M^n$ be $n$-dimensional Riemannian manifold isometrically immersed in an $m$-dimensional Riemannian manifold $\tilde M^m$. Since we are dealing with a local study, then, by Proposition \ref{3g9stg}, we may assume that $M^n$ is embedded in $\tilde M^m$. On this infinitesimal scale, Definition \ref{dfffff5555} guarantees that $M^n$ is a {\it Riemannian submanifold} of some nearby points in $\tilde M^m$ with induced Riemannian metric $g$. Then, {\it Gauss} and {\it Weingarten formulas} are, respectively, given by \begin{equation}\label{3} \tilde \nabla_X Y=\nabla_X Y+h(X,Y) \end{equation} and \begin{equation}\label{4} \tilde\nabla_X\zeta=-A_\zeta X+\nabla^\perp_X\zeta \end{equation} for all $X,Y\in \Gamma(TM^n)$ and $\zeta\in \Gamma (T^\perp M^n)$, where $\tilde\nabla$ and $\nabla$ denote respectively the Levi-Civita and the {\it induced} Levi-Civita connections on $\tilde M^m$ and $M^n$, and $\Gamma(TM^n)$ is the module of differentiable sections of the vector bundle $TM^n$. $\nabla^\perp$ is the {\it normal connection} acting on the normal bundle $T^\perp M^n$. Here, $g$ denotes the {\it induced Riemannian metric} from $\tilde g$ on $M^n$. For simplicity's sake, the inner products which are carried by $g$, $\tilde g$ or any other induced Riemannian metric are performed via $g$. However, most of the inner products which will be applied in this thesis are equipped with $g$, other situations are rarely considered. Here, it is well-known that the {\it second fundamental form} $h$ and the {\it shape operator} $A_\zeta$ of $M^n$ are related by \begin{equation}\label{5} g(A_\zeta X,Y)=g(h(X,Y),\zeta) \end{equation} for all $X,Y\in \Gamma(TM^n)$ and $\zeta\in \Gamma(T^\perp M^n)$, \cite{pom},~\cite{iijj77}. Geometrically, $M^n$ is called a {\it totally geodesic} submanifold in $\tilde M^m$ if $h$ vanishes identically. Particularly, the {\it relative null space}, ${\mathcal N}_x$, of the submanifold $M^n$ in the Riemannian manifold $\tilde M^m$ is defined at a point $x\in M^n$ by \cite{aallr4} as \begin{equation}\label{19} {\mathcal N}_x=\{ X\in T_xM^n: h(X, Y)=0~~~ \forall~ Y\in T_xM^n\}. \end{equation} In a different line of thought, and for any $X\in \Gamma (TM^n)$, $\zeta\in \Gamma (T^\perp M^n)$ and a $(1,1)$ tensor field $\psi$ on $\tilde M^m$, we write \begin{equation}\label{6} \psi X=PX+FX, \end{equation} and \begin{equation}\label{7} \psi N=t\zeta+f\zeta, \end{equation} where $PX$, $t\zeta$ are the tangential components and $FX$, $f\zeta$ are the normal components of $\psi X$ and $\psi \zeta$, respectively, \cite{saw}. In the sake of following the common terminology, the tensor field $\psi$ is replaced by $J$ in almost Hermitian manifolds. However, the covariant derivatives of the tensor fields $\psi$, $P$ and $F$ are respectively defined as \cite{pom} \begin{equation}\label{42} (\tilde\nabla_X\psi)Y=\tilde\nabla_X\psi Y-\psi\tilde\nabla_XY, \end{equation} \begin{equation}\label{38} (\tilde\nabla_XP)Y=\tilde\nabla_XPY-P\tilde\nabla_XY \end{equation} and \begin{equation}\label{39} (\tilde\nabla_XF)Y=\nabla_{X}^{\perp}FY-F\tilde\nabla_XY. \end{equation} Likewise, we consider a local field of orthonormal frames $\{e_1, \cdots , e_n, e_{n+1}, \cdots, e_m\}$ on $\tilde M^m$, such that, restricted to $M^n$, $\{e_1, \cdots , e_n\}$ are tangent to $M^n$ and $\{e_{n+1}, \cdots, e_m\}$ are normal to $M^n$. Then, the {\it mean curvature vector} $\vec H(x)$ is introduced as \cite{pom}, \cite{iijj77} \begin{equation}\label{8} \vec H(x)=\frac{1}{n} \sum_{i=1}^{n} h(e_i, e_i), \end{equation} On one hand, we say that $M^n$ is a {\it minimal submanifold} of $\tilde M^m$ if $\vec H=0$. On the other hand, one may deduce that $M^n$ is totally umbilical in $\tilde M^m$ if and only if $h(X,Y)=g(X,Y) \vec H$, for any $X,~Y\in \Gamma (TM^n)$ \cite{yyhh88}, where $H$ and $h$ are the mean curvature vector and the second fundamental form, respectively \cite{2233ee}. For an odd dimensional real $C^\infty$ manifold $\tilde M^{2l+1}$, let $\phi$, $\xi$, $\eta$ and $\tilde g$ be respectively a $(1,~1)$ tensor field, a vector field, a $1$-form and a Riemannian metric on $\tilde M^{2l+1}$ satisfying \begin{equation}\label{151} \left. \begin{aligned} \phi^2=-I+\eta\otimes\xi,~~~~~\phi\xi=0,~~~~~\eta\circ\phi=0,~~~~~\eta(\xi) = 1,\\ \eta(X)=\tilde g(X, \xi),~~~~~\tilde g(\phi X, \phi Y)=\tilde g(X, Y)-\eta(X)\eta(Y), \end{aligned} \right\} \end{equation} for any $X,~Y\in \Gamma (T\tilde M^{2l+1})$. Then we call $(\tilde M^{2l+1}, \phi, \xi, \eta, \tilde g)$ an {\it almost contact metric manifold} and $(\phi, \xi, \eta, \tilde g)$ an {\it almost contact metric structure} on $\tilde M^{2l+1}$, see \cite{pom},~\cite{rash} and \cite{fottt}. A fundamental $2$-form $\Phi$ is defined on $\tilde M^{2l+1}$ by $\Phi(X,Y)=\tilde g(\phi X,Y)$. An almost contact metric manifold $\tilde M^{2l+1}$ is called a contact metric manifold if $\Phi =\frac{1}{2} d\eta$. If the almost contact metric manifold $(\tilde M^{2l+1},\phi, \xi, \eta, \tilde g)$ satisfies $[\phi, \phi]+2d\eta\otimes \xi=0$, then $(\tilde M^{2l+1},\phi, \xi, \eta, \tilde g)$ turns out to be a {\it normal almost contact manifold}, where the Nijenhuis tensor is defined as $$[\phi,\phi](X,Y)=[\phi X, \phi Y]+\phi^2[X,Y]-\phi [X,\phi Y]-\phi[\phi X, Y]~~~~~\forall~ X,~Y\in \Gamma(T\tilde M^{2l+1}).$$ For our purpose, we will distinguish four classes of almost contact metric structures; namely, Sasakian, Kenmotsu, cosymplectic and nearly trans-Sasakian structures. At first, an almost contact metric structure is is said to be {\it Sasakian} whenever it is both contact metric and normal, equivalently \cite{wenn} \begin{equation}\label{157} (\tilde\nabla_X\phi)Y=-\tilde g(X,Y)\xi+\eta(Y)X. \end{equation} An almost contact metric manifold $\tilde M^{2l+1}$ is called {\it Kenmotsu manifold} \cite{foss} if \begin{equation}\label{155} (\tilde\nabla_X\phi)Y=\tilde g(\phi X, Y)\xi -\eta(Y)\phi X. \end{equation} In the case of killing almost contact structure tensors, consider a normal almost contact metric structure $(\phi, \xi,\eta,\tilde g)$ with both $\Phi$ and $\eta$ are closed. Then, such $(\phi, \xi,\eta,\tilde g)$ is called {\it cosymplectic} \cite{nasso}. Explicitly, cosymplectic manifolds are characterized by normality and the vanishing of Riemannian covariant derivative of $\phi$, i.e., \begin{equation}\label{152} (\tilde\nabla_X\phi)Y=0. \end{equation} Hereafter, we call the almost contact manifold $\tilde M^{2l+1}$ a {\it nearly cosymplectic } manifold if \begin{equation}\label{154} (\tilde \nabla_X\phi)Y+(\tilde \nabla_Y\phi)X=0. \end{equation} Based on Gray-Hervella classification of almost Hermitian manifolds \cite{ogma}, an almost contact metric structure $(\phi , \xi , \eta , \tilde g)$ on $\tilde M^{2l+1}$ is called a trans-Sasakian structure \cite{zeedo} if $(\tilde M^{2l+1}\times \mathbb{R}, J, \tilde G)$ belongs to the class $W_4$ of their classification, where $J$ is the almost complex structure on $\tilde M^{2l+1}\times \mathbb{R}$ defined by $$J(X, ad/dt)= \biggl(\phi X- a\xi, \eta (X)d/dt\biggr)$$ for all vector fields $X$ on $\tilde M^{2l+1}$ and smooth functions $a$ on $\tilde M^{2l+1}\times \mathbb{R}$, where $\tilde G$ is the product metric on $\tilde M^{2l+1}\times \mathbb{R}$. This may be expressed by the condition \begin{equation}\label{274} (\tilde \nabla_X\phi)Y= \alpha \biggl(\tilde g(X,Y)\xi - \eta (Y) X\biggr) + \beta \biggl(\tilde g(\phi X, Y)\xi - \eta (Y) \phi X\biggr), \end{equation} for some smooth functions $\alpha$ and $\beta$ on $\tilde M^{2l+1}$, and we say that the trans-Sasakian structure is of type $(\alpha, \beta)$. From the above formula it follows that $$\tilde \nabla_X\xi= - \alpha \phi X + \beta \biggl(X- \eta (X) \xi \biggr).$$ Up to D. Chinea and C. Gonzalez classification of almost contact structures \cite{ches}, the class $C_6\otimes C_5$ coincides with the class of trans-Sasakian structure of type $(\alpha , \beta)$. Recently, J. C. Marrero proved that a trans-Sasakian manifold of dimension $\geq$5 is either $\alpha$-Sasakian, $\beta$-Kemnotsu or a cosymplectic manifold, \cite{massa}. In \cite{zeedo}, C. Gherghe introduced nearly trans-Sasakian structure of type $(\alpha , \beta )$. An almost contact metric structure $(\phi , \xi , \eta ,\tilde g)$ on $\tilde M^{2l+1}$ is called a {\it nearly trans-Sasakian} structure (Mustafa et al., 2014 $\&$ 2015) if \begin{equation}\label{261} (\tilde \nabla _X \phi ) Y + (\tilde \nabla _Y \phi ) X= \alpha \biggl(2\tilde g(X, Y) \xi - \eta (Y) X - \eta (X) Y\biggr)$$$$-\beta \biggl( \eta (Y) \phi X + \eta (X) \phi Y\biggr). \end{equation} Evidently, a nearly trans-Sasakian of type $(\alpha , \beta)$ is nearly-Sasakian, nearly Kenmotsu or nearly cosymplectic according as $\beta$ = 0, $\alpha$=1; or $\alpha$ = 0, $\beta$=1; or $\alpha$ = $\beta$ = 0, respectively. \section{Basic Lemmas} To relate the calculus of $N_1\times N_2$ to that of its factors the crucial notion of {\it lifting} is introduced as follows. If $f\in {\mathfrak F}(N_1)$, the {\it lift} of $f$ to $N_1\times N_2$ is $\tilde f=f\circ \pi_1\in {\mathfrak F}(N_1\times N_2)$. If $X_p\in T_p(N_1)$ and $q\in N_2$, then the {\it lift} $ X_{(p,q)}$ of $X_p$ to $(p,q)$ is the unique vector in $T_{(p,q)}(N_1)$ such that $d\pi_1( X_{(p,q)})=X_p$. If $X\in \Gamma(TN_1)$ the {\it lift} of $X$ to $N_1\times N_2$ is the vector field $X$ whose value at each $(p,q)$ is the lift of $X_p$ to $(p,q)$. The set of all such {\it horizontal lifts} $ X$ is denoted by ${\mathcal L} (N_1)$. Functions, tangent vectors and vector fields on $N_2$ are lifted to $N_1\times N_2$ in the same way using the projection $\pi_2$. Note that ${\mathcal L} (N_1)$ and symmetrically the {\it vertical lifts} ${\mathcal L} (N_2)$ are vector subspaces of $\Gamma \bigl(T(N_1\times N_2)\bigr)$, \cite{iijj77}. We recall the following two general results for warped products \cite{iijj77}. \begin{proposition}\label{1}On $\tilde M^m = N_1\times_{f}N_2$, if $X,~Y\in {\mathcal L}(N_1)$ and $Z,~W\in {\mathcal L} (N_2)$, then \begin{itemize} \item[(i)] $\tilde\nabla_XY\in {\mathcal L}(N_1)$ is the lift of $\tilde \nabla_XY$ on $N_1$. \item[(ii)]$\tilde \nabla_XZ=\tilde \nabla_ZX=(Xf/f)Z.$ \item[(iii)] $(\tilde\nabla_ZW)^\perp = h_{N_2}(Z,W)= - \bigl(g_{N_2}(Z, W)/f\bigr) \nabla (f).$ \item[(iv)] $(\tilde \nabla_ZW)^T\in {\mathcal L}(N_2)$ is the lift of $\nabla^{N_2}_ZW$ on $N_2$, \end{itemize} where $g_{N_2}$, $h_{N_2}$ and $\nabla^{N_2}$ are, respectively, the induced Riemannian metric on $N_2$, the second fundamental form of $N_2$ as a submanifold of $\tilde M^m$ and the induced Levi-Civita connection on $N_2$. \footnotemark[\value{footnote}]\footnotetext{The operators $\perp$, $T$ and $\nabla (f)$ refer to the normal projection, the tangential projection and the gradient of $f$, respectively.} \end{proposition} It is obvious that, the above proposition leads to the following geometric conclusion. \begin{corollary}\label{2} The leaves $N_1\times q$ of a warped product are totally geodesic; the fibers $p\times N_2$ are totally umbilical. \end{corollary} Clearly, the totally geodesy of the leaves follows from $(i)$, while $(iii)$ implies that the fibers are totally umbilical in $\tilde M^m$. It is significant to say that, this corollary is one of the key ingredients of this work. Since all our considered submanifolds are warped products. Here, it is well-known that the {\it second fundamental form} $\sigma$ and the {\it shape operator} $A_\xi$ of $M^n$ are related by \begin{equation}\label{5} g(A_\xi X,Y)=g(\sigma(X,Y),\xi) \end{equation} for all $X,Y\in \Gamma(TM^n)$ and $\xi\in \Gamma(T^\perp M^n)$ (for instance, see \cite{pom}, \cite{iijj77}). \section{Existence and Nonexistence of Warped Product Submanifolds in Almost Contact Manifolds } This section has two significant purposes. The first one is to provide special case solutions for Problems \ref{prob9} and \ref{prob10}, that is to see whether a warped product exists or not in almost contact manifolds. In the existence case, we prove some preparatory characteristic results which are necessary for subsequent sections, and this is the second purpose. Some new example is given to assert the existence of some important warped product manifolds. For a submanifold $M^n$ in an almost contact manifold $\tilde M^{2m+1}$ let ${\mathcal P}_XY$ denote the tangential component and ${\mathcal Q}_XY$ the normal one of $(\tilde\nabla_X\phi)Y$ in $\tilde M^{2m+1}$, where $X,~Y\in \Gamma (TM^n)$. In order to make it a self-contained reference of warped product submanifolds for immersibility and nonimmersibility problems, we hypothesize most of our statements in the current and the next section for almost contact manifolds, and for warped product submanifolds of type $N_T\times _fN_2$, where $N_T$ and $N$ are holomorphic and Riemannian submanifolds. Meaning that, a lot of particular case results are included in the theorems of the next section. It is still an open question whether or not a warped product admits isometric immersions into certain Riemannian manifolds of interest. For instance, many articles have been recently published in almost contact manifolds (see, for example \cite{xos} and \cite{wenal}). In fact, these papers and a lot others (see references in \cite{ssee44}) provide special case answers for Problems \ref{prob9} and \ref{prob10}. The following theorem generalizes all such nonexistence results as a final answer for doubly warped product submanifolds in almost contact manifolds. \begin{theorem}\label{ndo} In almost contact manifolds, there does not exist a proper doubly warped product submanifold $M^n=_{f_2}{N_1}\times _{f_1}{N_2}$ such that the characteristic vector field $\xi$ is either tangent to $N_1$ or $N_2$. \end{theorem} \begin{proof} Suppose $\xi$ in $\Gamma(TN_2)$. Then for any $X\in \Gamma(TN_1)$, we directly calculate $$2X\ln f_1=2X\ln f_1 g(\xi, \xi)=2g(\tilde\nabla_X\xi, \xi)=Xg(\xi, \xi)=X(1)=0.$$ This means that $f_1$ is constant. Similarly, it can be shown that $f_2$ is constant when $\xi$ is tangent to the first factor. Hence, we conclude that a doubly warped product submanifold of almost contact manifolds, in the sense of our hypothesis, is trivial, which completes the proof. \end{proof} Considering $\xi$ as in the above hypothesis, this theorem can be simply paraphrased by saying that: doubly warped product submanifolds in almost contact manifolds are but trivial. With this fact, some results concerning inequalities for doubly warped product submanifolds in Kenmotsu manifolds become trivial (see references in \cite{ssee44}). As a special case of Theorem \ref{ndo}, we have the following theorem for (singly) warped product submanifolds \begin{theorem}\label{fund} There is no warped product submanifolds in almost contact manifolds such that the characteristic vector field $\xi$ is tangent to the second factor. \end{theorem} The above theorem answers some special cases of Problems \ref{prob9} and \ref{prob10}. On one hand, it generalizes all related nonexistence results of this topic (see, for example \cite{xos},\cite{wenal}, \cite{zolo}, \cite{fonm} and \cite{hasss}). On the other hand, it guides us to restrict the choice of the factor that $\xi$ should be tangent to in warped product submanifolds of almost contact manifolds. From now on, the characteristic vector field $\xi$ is supposed to be tangent to the first factor of all warped product submanifolds in almost contact manifolds. Henceforth, it s strightforward to get \begin{theorem}\label{jop} For each warped product submanifold $N\times _fN_T$ of almost contact manifolds such that $\xi$ is tangent to the first factor, the following are true \begin{enumerate} \item[(i)] $g({\mathcal P}_XZ,W)=0;$ \item[(ii)] $g({\mathcal P}_ZX,JZ)-g({\mathcal P}_{JZ}X,Z)=-2 (X\ln f) \|\|Z\|\|^2,$ \end{enumerate} for every vector field $X\in \Gamma(TN)$, and $Z,~W\in \Gamma (TN_T)$. \end{theorem} As a direct application of the preceding theorem, and by using $(\ref{261})$, we state the following remark, which generalizes a lot of nonexistence results in almost contact manifolds (see, for example \cite{wsssa} and \cite{wenal}). First, by putting $\beta=0$ in $(\ref{261})$, we get the structural formula for nearly $\alpha$-Sasakian manifolds; that is, \begin{equation}\label{2ms1} (\tilde \nabla _X \phi ) Y + (\tilde \nabla _Y \phi ) X= \alpha \biggl(2\tilde g(X, Y) \xi - \eta (Y) X - \eta (X) Y\biggr). \end{equation} Now, we show that the first term on the left hand side of statement $(ii)$ above is zero. From the above equation, we directly get $$g({\mathcal P}_ZX,JZ)=-g({\mathcal P}_XZ,JZ)-\alpha\eta(X)g(Z,JZ).$$ In view of statement $(i)$ of the above theorem, the right hand side of the above equation vanishes identically. Similarly, we can show that $g({\mathcal P}_{JZ}X,Z)=0$. Hence, statement $(ii)$ implies that $X\ln f=0$. This also holds for nearly cosyplectic manifolds, one can prove that using similar analogy like above. Thus, we have the following \begin{remark}\label{fat} Warped products of the type $N\times _fN_T$ do not exist in nearly Sasakian and nearly cosymplectic manifolds if $\xi$ is tangent to the first factor, and so for Sasakian and cosymplectic manifolds. However, the situation is different in Kenmotsu manifolds as we will see in the following example and in the next chapter also. \end{remark} A submanifold $M^n$ of an almost contact metric manifold $\tilde M^{2l+1}$ is said to be a {\it contact CR-submanifold }if there exist on $M^n$ differentiable distributions $\mathfrak{D}_T$ and $\mathfrak{D}_\perp$, satisfying the following \begin{enumerate} \item [(i)] $TM^n=\mathfrak{D}_T\oplus \mathfrak{D}_\perp\oplus \langle\xi\rangle$, \item [(ii)] $\mathfrak{D}_T$ is an invariant distribution, i.e., $ \phi (\mathfrak{D}_T) \subseteq \mathfrak{D}_T$, \item [(iii)] $\mathfrak{D}_\perp$ is an anti-invariant distribution, i.e., $\phi (\mathfrak{D}_\perp) \subseteq T^\perp M^n$. \end{enumerate} In Sasakian manifolds, a concrete example of contact $CR$-warped product submanifolds of the type $N_T\times _fN_\perp$ can be found in \cite{wenal}. On the contrary, and in view of Remark \ref{fat}, we conclude that warped product submanifolds with second invariant factor are trivial in both Sasakian and cosymplectic manifolds when $\xi$ is tangent to the first factor. In particular, this implies that contact $CR$-warped product submanifolds of the type $N_\perp\times _fN_T$ reduces to be contact $CR$-products in Sasakian and cosymplectic manifolds. By contrast, such warped product submanifolds do exist in Kenmotsu manifolds. To assert the above claim, we provide a counter example that ensures such existence of warped product submanifolds in Kenmotsu manifolds when the second factor is invariant. Besides, we can get an insurance for the existence of contact $CR$-warped product submanifolds in Kenmotsu manifolds, for both types; $M^n=N_T\times _fN_\perp$ and $M^n=N_\perp\times _fN_T$, when $\xi$ is tangent to the first factor. \begin{example}\label{S1} Let $\tilde M^9=\mathbb{R}\times _{e^t}\mathbb{C}^4$ be a Kenmotsu manifold, where $\mathbb{R}$ is the real line, and $\mathbb{C}^4$ is a Kaehler manifold with Kaehlerian structure $(G,J)$. Here, $G$ and $J$ are the restrictions of $g$ and $\phi$ to $\tilde M^9(p)$, respectively, for every $p\in \tilde M^9$. Let $(t, x_1,\cdots, x_8)$ be a local coordinates frame of $\tilde M^9$ where $t$ and $(x_1, \cdots, x_8)$ denote the local coordinates of $\mathbb{R}$ and $\mathbb{C}^4$, respectively. It is well-known that the Riemannian metric tensor $g$ and the vector field $\xi$ are defined on $\tilde M^9$ as follows \cite{foss}: \[ g_{(t,x)}=\begin{pmatrix} 1 & 0 \\ 0 & e^{2t}G_(x) \\ \end{pmatrix},~~~~~~~\xi=\left(\frac{d}{dt}\right). \] Now, consider the three-dimensional submanifold $M^3$ of $\mathbb{C}^4$ given by the equations $$x_1=e^t v,~~~x_2=e^tu,~~~x_3=e^tv,~~~x_4=e^tu,~~~x_5=e^ts,~~~x_7=e^ts,~~~x_6=x_8=0.$$ Observe that the tangent bundle $TM^3$ is spanned by $Z_1$, $Z_2$ and $Z_3$, where $$Z_1=e^t\frac{\partial}{\partial x_1}+e^t\frac{\partial}{\partial x_3},~~~ Z_2=e^t\frac{\partial}{\partial x_2}+e^t\frac{\partial}{\partial x_4},~~~ Z_3=e^t\frac{\partial}{\partial x_5}+e^t\frac{\partial}{\partial x_7}.$$ Further, we define the distributions $\mathfrak{D}_T=$span$\{Z_1,~Z_2\}$, and $\mathfrak{D}_\perp=$span$\{Z_3\}$. It is obvious that $\mathfrak{D}_T$ and $\mathfrak{D}_\perp$ are holomorphic and totally real distributions on $\mathbb{C}^4$, respectively. Hence, and taking into consideration $\phi (\xi)=0$, the distributions $\mathfrak{D}_\perp \oplus\langle\xi\rangle$ and $\mathfrak{D}_T$ are respectively anti-invariant and invariant distributions on $\tilde M^9$. Thus, $N^4=\mathfrak{D}_\perp \oplus\langle\xi\rangle\oplus \mathfrak{D}_T$ is a contact $CR$-submanifold in $\tilde M^9$. In addition, it is easy to see that both $\mathfrak{D}_\perp \oplus\langle\xi\rangle$ and $\mathfrak{D}_T$ are integrable. If we denote by $N_\perp$ and $N_T$ the integral manifolds of $\mathfrak{D}_\perp \oplus\langle\xi\rangle$ and $\mathfrak{D}_T$, respectively, then the metric tensor $g$ of $N^4$ is $$g=dt^2+e^{2t} ds^2+e^{2t}(dv^2+du^2) =g_{N_\perp}+e^{2t}g_{N_T}.$$ Therefor, $N^4$ is a contact $CR$-warped product submanifold of $\tilde M^9$ of the type $N_\perp\times _fN_T$ with warping function $f=e^t$. Moreover, it straight forward to figure out that $$h(Z_1, Z_1)=h(Z_2, Z_2)=0.$$ Hence, $N^4$ is a $\mathfrak{D}_2$-minimal warped product submanifold as expected, where $\mathfrak{D}_2=\mathfrak{D}_T$.. Likewise, by an analogous procedure to the above we can deduce that $\mathfrak{D}_T\oplus \langle\xi\rangle$ is an invariant distribution on $\tilde M^9$, and $\mathfrak{D}_\perp$ is an anti-invariant. Also, it is not difficult to show integrability of $\mathfrak{D}_T\oplus \langle\xi\rangle$. Denoting the integral manifolds of $\mathfrak{D}_T\oplus \langle\xi\rangle$ and $\mathfrak{D}_\perp$ by $N_T$ and $N_\perp$, respectively, we find that $N^4=N_T\times _{e^t}N_\perp$ is a non-trivial contact $CR$-warped product in $\tilde M^9$. By calculating the coefficients of $h$ restricted to $N_T$, we deduce that $N^4=N_T\times _{e^t}N_\perp$ is a $\mathfrak{D}_1$-minimal warped product submanifold as it should be, where $\mathfrak{D}_1=\mathfrak{D}_T$. \end{example} In this sequel, proper warped product submanifolds of types $N_\theta\times _fN_T$ and $N_T\times _fN_\theta$ do exist in Kenmotsu manifolds, when $\xi$ is tangent to the first factor. Whereas, Remark \ref{fat} informs us that proper warped product submanifolds of type $N_\theta\times _fN_T$ do not exist in both Sasakian and cosymplectic manifolds. Soon we show the nonexistence of $N_T\times _fN_\theta$ in Sasakian and cosymplectic manifolds such that $N_\theta$ is proper slant. The following theorem is so significant because it will be used in the rest of this work. \begin{theorem}\label{T1} Let $M^n=N_T\times _fN$ be a warped product submanifold isometrically immersed in an almost contact manifold $\tilde M^{2l+1}$ such that $\xi$ is tangent to the first factor. Then, we have the following \begin{enumerate} \item[(i)] $g({\mathcal P}_XZ, Y)=- g(h(X,Y), FZ);$ \item[(ii)] $g({\mathcal P}_ZX, Z)=(\phi X\ln f)\|\|Z\|\|^2+g(h(X,Z), FZ);$ \item[(iii)] $g({\mathcal P}_ZX, Y)=0;$ \item[(iv)] $g({\mathcal P}_ZX, W)+g({\mathcal P}_WX, Z)=2(\phi X\ln f)g(Z,W)\newline~~~~~~~~~~~~~~~~~~~~+g(h(X,Z), FW)+g(h(X,W), FZ);$ \item[(v)] $ g({\mathcal P}_ZX-{\mathcal P}_XZ,W)-g({\mathcal P}_WX, Z)=2 (X\ln f) g(Z, PW);$ \item[(vi)] $g({\mathcal P}_XZ, W)+g({\mathcal P}_XW, Z)=0;$ \item[(vii)] $g({\mathcal Q}_XX, \phi \zeta)+g({\mathcal Q}_{\phi X}\phi X, \phi \zeta)=-g(h(X,X), \zeta)-g(h(\phi X, \phi X), \zeta),$ \end{enumerate} for arbitrary vector fields $X,~Y\in \Gamma(TN_T)$, $Z,~W\in \Gamma (TN)$ and $\zeta \in \Gamma (\nu)$. \end{theorem} \begin{proof} The assertion of statements $(i),~(ii),~(iv),~(v)$ and $(vi)$ are trivial. For statement $(iii)$, suppose that $X$ and $Z$ are taken as hypothesis. Then it is obvious that \begin{equation}\label{bil}(\tilde \nabla_X\phi)Z=\tilde \nabla_XPZ+\tilde \nabla_XFZ-\phi\tilde \nabla_XZ.\end{equation} Also, for $X$ and $Z$ we have \begin{equation}\label{bel}(\tilde \nabla_Z\phi)X=\tilde \nabla_Z\phi X-\phi\tilde \nabla_ZX.\end{equation} By subtracting $(\ref{bel})$ from $(\ref{bil})$, we obtain $$(\tilde \nabla_X\phi)Z-(\tilde \nabla_Z\phi)X=\tilde \nabla_XPZ+\tilde \nabla_XFZ-\tilde \nabla_Z\phi X.$$ Taking the inner product by $\phi Y$ in the above equation, gives $$g({\mathcal P}_XZ, \phi Y)-g({\mathcal P}_ZX, \phi Y)=-g(h(X,\phi Y), FZ).$$ Replacing $\phi Y$ by $Y$ yields $$g({\mathcal P}_ZX, Y)-\eta(Y)g({\mathcal P}_ZX, \xi)-g({\mathcal P}_XZ, Y)+\eta(Y)g({\mathcal P}_XZ,\xi)=$$$$g(h(X,Y), FZ)-\eta(Y)g(h(X,\xi),FZ).$$ By using $(i)$ in the above equation we derive $$g({\mathcal P}_ZX, Y)=\eta(Y)g({\mathcal P}_ZX, \xi).$$ Since the right hand side of the above equation vanishes identically, we obtain $(iii)$. For $(vii)$, if we take $X=\xi$ in the above theorem, then statement $(vii)$ holds directly. Now, for an arbitrary vector field tangential to the first factor and perpendicular to $\xi$, say $X$, we have $$(\tilde \nabla_X\phi)X=\tilde \nabla_X\phi X-\phi\tilde \nabla_XX.$$ First, take the inner product in the above equation with $\phi\zeta$ to get $$g({\mathcal Q}_XX, \phi\zeta)=g(h(\phi X,X), \phi\zeta)-g(h(X, X), \zeta).$$ \noindent After that, we replace $\phi X$ by $X$ in the above equation to derive $$g({\mathcal Q}_{\phi X}\phi X, \phi \zeta)=-g(h(\phi X,X), \phi\zeta)-g(h(\phi X, \phi X), \zeta).$$ Hence $(vii)$ can be obtained by adding the above two equations. \end{proof} In virtue of Theorem \ref{T1} $(v)$, we get the following decisive nonexistence result in the setting of almost contact structures, which generalizes several nonexistence results in this field (see references in \cite{ssee44}. \begin{corollary}\label{Y1} In both of Sasakian and cosymplectic manifolds, there is no warped product submanifolds with invariant first factor tangential to $\xi$, other than contact $CR$-warped products. \end{corollary} In particular, this corollary implies the nonexistence of warped product submanifolds of type $N_T\times _fN_\theta$ in Sasakian and cosymplectic manifolds such that $N_\theta$ is a proper slant. On the contrary, this is not true for Kenmotsu manifolds. Now, we prepare the following results for later use. \begin{theorem}\label{U1} Let $M^n=N_1\times _fN_2$ be a warped product submanifold isometrically immersed in a nearly trans-Sasakian manifold $\tilde M^{2l+1}$ such that $\xi$ is tangent to $N_1$. Then, the following hold \begin{enumerate} \item[(i)] $\xi \ln f=\beta;$ \item[(ii)] $g(h(\xi, Z), FZ)=-\alpha \|\|Z\|\|^2,$ \end{enumerate} for each vector field $Z$ tangent to $N_2$. \end{theorem} \begin{proof} By $(\ref{261})$, it is straightforward that \begin{equation}\label{psb} -\phi \tilde \nabla_Z\xi+\tilde \nabla_\xi \phi Z-\phi \tilde \nabla_\xi Z=-\alpha Z-\beta \phi Z. \end{equation} For $(i)$, taking the inner product with $\phi Z$ in the above equation, gives $$-2~\xi \ln f \|\|Z\|\|^2+g(\tilde \nabla_\xi\phi Z,\phi Z)=-\beta \|\|Z\|\|^2,$$ Equivalently, $$-2~\xi \ln f \|\|Z\|\|^2+\frac{1}{2}~\xi \|\|Z\|\|^2=-\beta \|\|Z\|\|^2,$$ which implies $$-2~\xi \ln f \|\|Z\|\|^2+g(\tilde \nabla_\xi Z, Z)=-\beta \|\|Z\|\|^2.$$ Hence, statement $(i)$ follows from the above equation. Now, we take the inner product with $Z$ in $(\ref{psb})$ to derive $$g(\tilde \nabla_\xi \phi Z,Z)+2~ g(\tilde \nabla_\xi Z,\phi Z)=-\alpha \|\|Z\|\|^2.$$ This can be written as $$g(\tilde \nabla_\xi PZ,Z)+g(\tilde \nabla_\xi FZ,Z)+2~g(\tilde \nabla_\xi Z,PZ)+2~g(\tilde \nabla_\xi Z,FZ)=-\alpha \|\|Z\|\|^2.$$ Hence, by the Gauss formula and part $(ii)$ of Proposition \ref{1}, we get $$g(\tilde \nabla_\xi FZ,Z)+2~g(\tilde \nabla_\xi Z,FZ)=-\alpha \|\|Z\|\|^2.$$ Consequently, $$g(\tilde \nabla_\xi Z,FZ)=-\alpha \|\|Z\|\|^2.$$ Statement $(ii)$ follows from the above equation by virtue of Gauss formula. This completes the proof. \end{proof} In the spirit of the preceding theorem, it is easy, but important, to distinguish other particular case structures. For this, we present the following table: \begin{center} \begin{table}[ht] \centering \begin{tabular}{\|c c c c c c\| } \hline $\tilde M^{2l+1}$&$\vline$ & $\xi \ln f=$ &\vline &$g(h(\xi, Z), FZ)=$&\\ [0.9ex] \hline\hline $$ Nearly trans-Sasakian& \vline &$\beta$&\vline&$-\alpha \|\|Z\|\|^2$& \\ \hline Nearly $\alpha$-Sasakian & $\vline$ &0 &\vline&$-\alpha \|\|Z\|\|^2$&\\ \hline Sasakian & $\vline$ &0&\vline&$- \|\|Z\|\|^2$& \\ \hline Nearly $\beta$-Kenmotsu& $\vline$ & $\beta$&\vline&0&\\ \hline Kenmotsu &$\vline$ & 1&\vline&0&\\ \hline Nearly cosymplectic &$\vline$ &0&\vline&0&\\ \hline Cosymplectic &$\vline$ &0&\vline&0&\\ \hline\hline \end{tabular} \caption{$\xi \ln f$ and $g(h(\xi, Z), FZ)$ for $N_1\times _fN_2$ in $\tilde M^{2l+1}$, such that $\xi$ is tangent to $N_1$ and $Z$ is tangent to $N_2$.} \end{table} \end{center} Now, assume that the warped product submanifold $N_1\times _fN_2$ in Theorem \ref{U1} is mixed totally geodesic. Thus, from statement $(ii)$ of the same theorem, we have $$\alpha \|\|Z\|\|^2=0.$$ This implies that, either $N_2$ is null, or $\alpha=0$; i.e., $\tilde M^{2l+1}$ is not $\alpha$-Sasakian. Therefore, we get the following significant nonexistence result, which will be useful in inequalities of mixed totally geodesic submanifolds in almost contact manifolds. \begin{proposition}\label{mixtgs} There is no mixed totally geodesic warped product submanifold in nearly $\alpha$-Sasakian manifolds. \end{proposition} In another line of thought, one can easily verify the following lemma. \begin{lemma}\label{conn} Let $N_1\times _fN_2$ be a warped product submanifold in almost contact manifolds $\tilde M^{2l+1}$ such that $\xi$ is tangent to the first factor. Then, $g((\tilde \nabla_\xi \phi)Z, \phi Z)=0$ for every $Z\in \Gamma (TN_2).$ \end{lemma} As another important consequence of Theorem \ref{T1}, we have the following proposition: \begin{proposition}\label{X1} For any warped product submanifold $M^n=N_T\times _fN$ of nearly trans-Sasakian manifolds with $\xi$ tangent to the first factor, the followings are true \begin{enumerate} \item[(1)] $g(h(X,Y), FZ)=0;$ \item[(2)] $g(h(X,X), \zeta)+ g(h(\phi X,\phi X), \zeta)=0;$ \item [(3)] $ g(h(X,Z), FZ)+ \alpha \eta(X) \|\|Z\|\|^2=-(\phi X\ln f) \|\|Z\|\|^2,$ \end{enumerate} where the vector fields $X,~Y$ are tangent to the first factor, $Z$ is tangent to the second and $\zeta$ is tangent to the normal subbundle $\nu$. \end{proposition} \begin{proof} Statement $(1)$ follows from $(i)$ and $(iii)$ of Theorem \ref{T1}, while $(3)$ is a consequence of $(vi)$ and $(ii)$ of the same theorem. For statement $(2)$ we apply the nearly trans-Sasakian structure for the vector fields $X$ and $\xi$ to obtain the following $$2\tilde \nabla_X\phi X= \alpha \biggl(2g(X,X)\xi -2 \eta(X)X\biggr)-2\beta \eta(X)\phi X.$$ Taking the inner product with $\zeta$ gives $$g(h(X, \phi X), \zeta)+g(\phi h(X,X)\zeta)=0.$$ Replacing $X$ by $\phi X$ gives $$g(h(-X+\eta(X)\xi, \phi X), \zeta)+g(\phi h(\phi X,\phi X)\zeta)=0.$$ By these two equations and the fact $h(X, \xi)=0$, we obtain the result. \end{proof} By means of Propositions \ref{mixtgs} and \ref{X1}, one can easily show that a mixed totally geodesic contact $CR$-warped product submanifold is indeed trivial in both Sasakian and cosymplectic manifolds. Whereas such submanifolds do exist in Kenmotsu manifolds, this is due to the fact $\xi \ln f=1$ for all warped product submanifolds of Kenmotsu manifolds when $\xi$ is tangent to the first factor. In the sequel, we prove necessary and sufficient conditions for a contact $CR$-submanifold to be locally contact $CR$-warped product in nearly trans-Sasakian manifolds. For long time, mathematicians had have interest to find an analogue of the classical de Rham theorem to warped products, a result was proved by S. Hiepko \cite{nfds}. First, let us recall this result: Let ${\mathcal H}$ be a distribution in the tangent bundle of a Riemannian manifold $M^n$ and let ${\mathcal H}^\perp$ be its orthogonal complementary distribution. Assume that the two distributions are both involutive and the integral manifolds of ${\mathcal H}$ (resp. ${\mathcal H}^\perp$) are extrinsic spheres (resp. totally geodesic). Then, $M^n$ is locally isometric to a warped product $N_1\times _fN_2$. Moreover, if $M^n$ is simply connected and complete, there exists a global isometry of $M^n$ with a warped product. Using this fundamental method we present the following characterization theorem which has been recently published in \cite{zolo}. \begin{theorem}\label{267} Every contact $CR$-submanifold $M^n$ of a nearly trans-Sasakian manifold $\tilde M^{2l+1}$ with an involutive distribution $\mathfrak{D}_\perp$ is locally a contact CR-warped product, if and only if the shape operator of $M^n$ satisfies \begin{equation}\label{1065} A_{\phi W}X=- (\phi X \mu) W-\alpha \eta (X) W,~~~~X\in \mathfrak{D}_T\oplus \langle \xi \rangle,~~~W\in \mathfrak{D}_\perp, \end{equation} for a smooth function $\mu$ on $M^n$, satisfying $V(\mu) =0$ for each $V\in \mathfrak{D}_\perp.$ \end{theorem} Observe that the above theorem generalizes many related recent results, for example contact $CR$-warped product of cosymplectic, Sasakian and Kenmotsu manifolds can be characterized in a similar way as above (see, for example \cite{wenal}). \section{Research problems based on The Results of Previous Sections} Due to the results of this paper, we hypothesize a pair of open problems. Firstly, \begin{problem}\label{ama1} Construct discrete examples of contact $CR$-warped product submanifolds of Sasakian manifolds. \end{problem} Secondly, we ask: \begin{problem}\label{pqm2} Construct a solid examples of nearly trans-Sasakian manifolds and contact $CR$-warped product submanifolds of such manifolds. \end{problem} \vskip.15in \end{document}	arXiv
\begin{document} \title{Inequalities witnessing coherence, nonlocality, and contextuality} \newcommand{INL -- International Iberian Nanotechnology Laboratory, Av. Mestre Jos\'{e} Veiga s/n, 4715-330 Braga, Portugal}{INL -- International Iberian Nanotechnology Laboratory, Av. Mestre Jos\'{e} Veiga s/n, 4715-330 Braga, Portugal} \newcommand{INL -- International Iberian Nanotechnology Laboratory, Braga, Portugal}{INL -- International Iberian Nanotechnology Laboratory, Braga, Portugal} \newcommand{Instituto de F\'{i}sica, Universidade Federal Fluminense, Av. Gal. Milton Tavares de Souza s/n, Niter\'{o}i -- RJ, 24210-340, Brazil}{Instituto de F\'{i}sica, Universidade Federal Fluminense, Av. Gal. Milton Tavares de Souza s/n, Niter\'{o}i -- RJ, 24210-340, Brazil} \newcommand{Instituto de F\'{i}sica, Universidade Federal Fluminense, Niter\'{o}i -- RJ, Brazil}{Instituto de F\'{i}sica, Universidade Federal Fluminense, Niter\'{o}i -- RJ, Brazil} \newcommand{Centro de F\'{i}sica, Universidade do Minho, Campus de Gualtar, 4710-057 Braga, Portugal}{Centro de F\'{i}sica, Universidade do Minho, Campus de Gualtar, 4710-057 Braga, Portugal} \newcommand{Centro de F\'{i}sica, Universidade do Minho, Braga, Portugal}{Centro de F\'{i}sica, Universidade do Minho, Braga, Portugal} \author{Rafael Wagner} \email{[email protected]} \affiliation{INL -- International Iberian Nanotechnology Laboratory, Braga, Portugal} \affiliation{Centro de F\'{i}sica, Universidade do Minho, Braga, Portugal} \author{Rui Soares Barbosa} \email{[email protected]} \affiliation{INL -- International Iberian Nanotechnology Laboratory, Braga, Portugal} \author{Ernesto F.~Galvão} \email{[email protected]} \affiliation{INL -- International Iberian Nanotechnology Laboratory, Braga, Portugal} \affiliation{Instituto de F\'{i}sica, Universidade Federal Fluminense, Niter\'{o}i -- RJ, Brazil} \date{\today} \begin{abstract} Quantum coherence, nonlocality, and contextuality are key resources for quantum advantage in metrology, communication, and computation. We introduce a graph-based approach to derive classicality inequalities that bound local, noncontextual, and coherence-free models, offering a unified description of these seemingly disparate quantum resources. Our approach generalizes recently proposed basis-independent coherence witnesses, and recovers all noncontextuality inequalities of the exclusivity graph approach. Moreover, violations of certain classicality inequalities witness preparation contextuality. We describe an algorithm to find all such classicality inequalities, and use it to analyze some of the simplest scenarios. \end{abstract} \newcommand\redsout{\bgroup\markoverwith{\textcolor{magenta}{\rule[0.5ex]{2pt}{0.4pt}}}\ULon} \addtocounter{page}{-1} \let\reusablemaketitle\maketitle \let\reusableauthor\author \let\reusableaffiliation\affiliation \maketitle \psection{Introduction} Non-classical resources provided by quantum theory are key to quantum advantage for information processing \cite{HowardWVE14,Abbot12,Raussendorf13,Abramsky17,Mansfield18,Saha19,lostaglio2020certifying,Kirby20}; see \cite{Budroni21,Brunner14,Streltsov17} for comprehensive reviews of applications. Many different quantum resources have been studied and quantified using the framework of resource theory \cite{Winter16,Selby20,Amaral19,Duarte18,Wolfe20,Schmid20resource,KangDaWu21,Chitambar19,Abramsky17,Abramsky19,Barbosa23,wagner2021using,regula2017convex,regula2022tight,theurer2017resource,designolle2021set,regula2022probabilistic}. It is natural to wonder to what extent different quantum resources can be characterized in a unified way. Here we address this question by proposing a single formalism that yields inequalities bounding three different notions of classicality: noncontextual, local, and coherence-free models. A number of modern approaches to contextuality have successfully incorporated nonlocality as a special case \cite{cabello2014graph,AbramskyB11,acin2015combinatorial,amaral2018graph}. The relationship between this unified notion of non-classical correlations and coherence, however, has been harder to establish. One roadblock is that most approaches to characterize coherence presuppose the choice of a fixed reference basis \cite{Streltsov17}. Recently, different approaches have been proposed to define basis-independent notions of coherence. One example is the introduction of the concept of set coherence \cite{designolle2021set}. Another recent approach, on which the present work builds, defines basis-independent coherence witnesses using only relational information between states in the form of two-state overlaps \cite{GalvaoB20}. Still, so far there has been no clear identification between non-locality and contextuality on one hand, and coherence on the other. There are examples of models that mimic quantum coherence \textit{without} displaying contextuality or nonlocality, such as the toy models from Refs.~\cite{spekkens2007evidence,catani2021interference}, while on the other hand incoherent states -- even maximally mixed states -- can of course be used to witness state-independent quantum contextuality~\cite{cabello2008experimentally,amselem2009state}. Theory-independent approaches have been used to compare relevant types of nonclassical resources~\cite{takagi2019general,regula2017convex,regula2022probabilistic}, but an understanding of the special case of coherence and contextuality is still lacking. Understanding which elements of quantum coherence cannot be described by noncontextual models has both important foundational impact and potential technological applications. Building on the study of coherence using two-state overlaps \cite{GalvaoB20}, we propose a framework that associates to any (simple) graph $G$ a probability polytope $C_G$ of edge weightings. Vertices of the graph $G$ represent probabilistic processes, while edges of $G$ correspond to correlations between neighbouring processes. We show that the faces of the polytope $C_G$ describe bounds on noncontextual, local, and coherence-free models, depending on the interpretation of vertices of the graph $G$ as preparations and measurements. The description of three notions of classicality under a single framework represents a significant conceptual advance towards clarifying the source of quantum computational advantage. \psection{The classical polytope $C_G$} Let $G = (V(G), E(G))$ be an undirected graph, which we call the \stress{event graph}. We consider edge weightings $r \colon E(G) \to [0,1]$, which assign a weight $r_e = r_{ij}$ to each edge $e=\{i,j\}$ of $G$. We regard these weightings as points forming a polytope, the unit hypercube, $r \in [0,1]^{E(G)}$. To define the \stress{classical polytope} $C_G \subseteq [0,1]^{E(G)}$, take each vertex $i \in V(G)$ to represent a random variable $A_i$ with values belonging to an alphabet $\Lambda$, and suppose these are jointly distributed. This determines an edge weighting $r$ where each weight $r_{ij}$ is the probability that the processes corresponding to vertices $i$ and $j$ output equal values, \ie $r_{ij} = P(A_i = A_j)$. An edge weighting $r$ is in the classical polytope $C_G$ if it arises in this fashion from jointly distributed random variables $(A_i)_{i \in V(G)}$. Each weight $r_{ij}$ is then a measure of the correlation between the output values of $A_i$ and of $A_j$. In the case of dichotomic values $\Lambda = \{ +1, -1\}$, this quantity is related to the expected value of the product by $\left\langle A_i A_j\right\rangle = 2r_{ij}-1$ \footnote{Note that we do not assume a fixed finite outcome set $\Lambda$, or a bound on its size. The classical polytope consists of the edge weightings that arise from jointly distributed random variables with outcomes in \stress{some} set $\Lambda$. We could fix a single $\Lambda$ as long as it is countably infinite. But in practice, for a fixed graph $G$ with $n$ vertices, it suffices to consider $\Lambda = \enset{1, \ldots, n}$.}. An (alternative) formal description of $C_G$ is given in detail in \Cref{app:algo} \if\forprl1 in the supplemental material \footnote{See Supplemental Material at [URL will be inserted by publisher] for further details including proofs and computational explorations.} \fi. \psection{Inequalities defining $C_G$} The inequalities defining the polytope $C_G$ impose logical conditions determining the set of classical edge weightings. The existence of non-trivial facets of $C_G$ can be illustrated with the example of \cref{fig:all_graphs_together}--(a), the 3-vertex complete graph $K_3$, with edge weights $r_{12}, r_{23}, r_{13}$. We cannot have e.g.\ $r_{12}=r_{23}=1$ and $r_{13}=0$, as this would contradict transitivity of equality on the deterministic values corresponding to each of the three vertices: $A_1 = A_2 = A_3 \neq A_1$. In Ref.~\cite{GalvaoB20} it was shown that the only non-trivial inequalities for the $n$-cycle event graph $C_n$ are \begin{equation} -r_{e} +\sum_{{e'} \neq {e}} r_{e'} \leq n-2,\;\; \text{for each $e \in E(C_n)$.} \label{eq:ncycle} \end{equation} Incidentally, these inequalities have been known at least since the work of Boole \cite{Boole1854, Pitowsky94}. We now give a high-level description of an algorithm to completely characterize $C_G$ for general event graphs $G$. We start by enumerating the vertices of $C_G$. These are all the `deterministic' labellings of the edges of $G$ with values in $\{0,1\}$ that are logically consistent with transitivity of equality. The facets of $C_G$ can then be found using standard convex geometry tools~\footnote{The inequalities found in this work were obtained using the \textbf{traf} option from the \textbf{PORTA} program, which converts a V-representation of a polytope into an H-representation.}. Whether a given deterministic edge labelling is consistent -- and therefore a vertex of $C_G$ -- can be checked in linear time on the size of $G$ by a graph traversal. However, it is unnecessary to generate all $2^{\lvert E(G) \rvert}$-many labellings and discard the inconsistent ones. Instead, one can directly generate only the consistent ones by searching through underlying value assignments to the vertices of $G$. Despite being much more efficient for most graphs, this also quickly becomes unavoidably intractable due to the exponentially-increasing number of vertices of the polytope $C_G$. We deepen this discussion in \Cref{app:algo}. Using the method just outlined, we find all facets of $C_G$ for some small graphs, including all graphs shown in \cref{fig:all_graphs_together}. Interestingly, already for $K_4$ (\cref{fig:all_graphs_together}--(c)) a new type of facet appears which is different from the cycle inequalities of \cref{eq:ncycle}. These new facets of $K_4$ are described by the inequalities of the form \begin{equation} (r_{12}+r_{13}+r_{14}) - (r_{23}+r_{34}+r_{24}) \le 1, \label{eq:k4nontrivial} \end{equation} (and others obtained by label permutations). In \Cref{app:buildgraphs}, we prove that some constructions of graphs by combining smaller graphs do not give rise to new facet inequalities, trimming the class of graphs worth analyzing. In \Cref{app:smallgraphs}, we list all facet inequalities of the classical polytope for the complete graphs $K_4$, $K_5$, and $K_6$. We also give numerically-found examples of quantum violations -- in the sense described in the next section -- of all non-trivial facets of $K_4$ and $K_5$. In \Cref{app:family}, we generalize the inequalities of \cref{eq:k4nontrivial} to complete graphs of $n \ge 4$ vertices, and prove that these define facets of the classical polytopes $C_{K_n}$ for all such $n$. We now proceed to describe how the inequalities obtained for the abstract scenarios considered above establish bounds both on coherence-free models and on noncontextual/local models. Each type of operational scenario suggests an interpretation for edge weights, and naturally imposes further constraints on them, resulting in cross-sections of the polytope $C_G$. These cross-sections recover known noncontextuality/locality polytopes, as well as basis-independent coherence witnesses. \psection{$C_G$ bounds coherence-free models} In Ref.~\cite{GalvaoB20}, basis-independent coherence witnesses were described using only pairwise overlaps $r_{ij}=\mathrm{tr}(\rho_i \rho_j)$ among a set of quantum states. We review the correspondence between the facet inequalities of $C_G$ and the coherence witnesses of Ref.~\cite{GalvaoB20}. Let $G$ be any graph with $n$ vertices. We start by preparing a general separable state of $n$ quantum systems of the same type (e.g. qudits), each associated to a vertex of the graph. Each edge of $G$ is given a weight equal to the overlap between the two states of its incident vertices. These overlaps can be estimated using the well-known SWAP test \cite{BuhrmanCWdW01}. In Ref.~\cite{GalvaoB20} it was shown that the facet-inequalities of $C_G$ describe necessary conditions on the set of overlaps, \ie on edge weightings of $G$, for the set of single-system states to be coherence-free, that is, all of them diagonal in a common single-system basis. This is so because for such coherence-free states, the overlap $r_{ij}$ equals the probability of obtaining equal outcomes in independent measurements of states $i$ and $j$ using the observable that projects onto the reference basis. \psection{$C_G$ bounds local and noncontextual models} The faces of $C_G$ can also be understood as bounds on noncontextual models~\cite{KochenS67,Bell64}. A simple first approach consists in having vertices of $G$ represent measurements, while edges identify two-measurement contexts, i.e. pairs of observables that can be measured simultaneously. The weight of an edge corresponds to the probability, with respect to a given global state, that the two incident measurements yield equal outcomes. A necessary and sufficient condition for the existence of a noncontextual model whose behaviour is consistent with a given edge weighting is the existence of a global probability distribution (on outcome assignments to all measurements) whose marginals recover the correct outcome probabilities. This is the content of the Fine--Abramsky--Brandenburger theorem \cite{Fine82PRL,Fine82JMP,AbramskyB11}. Such a global distribution, when it exists, can also be interpreted as a classical coherence-free model. This dual role of global probability distributions is the link connecting coherence-free models and noncontextual models, and allowing violations of facet inequalities of $C_G$ to witness either property, depending on the interpretation of the scenario at hand. In general, this simple approach, interpreting vertices as measurements and edges as equality of outcome in two-measurement contexts, is not sufficient to capture contextuality in full generality~\cite[Section 2.5.3]{amaral2018graph}. Even restricting to contextuality scenarios whose maximal contexts have size two, the facets of $C_G$ are not necessarily facet, or even tight, noncontextuality inequalities, except in the case of dichotomic measurements~\cite[Theorem 38]{araujo2014quantum}, where equality of outcomes fully determines the measurement statistics. An important example is the Clauser--Horne--Shimony--Holt (CHSH) inequality. \begin{figure} \caption{\textbf{Event graphs corresponding to bounds on classical models.} Each of these graphs can be used to obtain the following nonclassicality inequalities: (a) constrained CHSH inequality; (b),(d) CHSH Bell locality inequality; (c) new $K_4$ classicality inequality from \cref{eq:k4nontrivial}, and (e) Klyachko, Can, Binicio\u{g}lu, and Schumovsky (KCBS) noncontextuality inequality.} \label{fig:all_graphs_together} \end{figure} Some contextuality scenarios require the imposition of further constraints, which geometrically determine cross-sections on the classical polytope $C_G$. These constraints may, for example, represent operational symmetries of the measurement scenario, e.g. making two vertices equal, or may encode given conditions on the compatibility of observables. One example is the exclusivity constraint present in the Cabello--Severini--Winter (CSW) graph approach~\cite{cabello2014graph}. We now show how both CHSH and the original 3-setting Bell inequality can be obtained from cycle inequalities, before describing a more systematic approach that recovers all noncontextuality inequalities obtainable from the exclusivity graph approach~\cite{cabello2014graph,amaral2018graph}. \psection{Example: CHSH inequality from the 4-cycle graph $C_4$} It is easy to check from \cref{eq:ncycle} that the 4-cycle graph $C_4$ with edges $r_{12}, r_{23}, r_{34}, r_{14}$ (see \cref{fig:all_graphs_together}--(b)) has $4$ non-trivial facets given by the inequality \begin{equation} r_{12}+r_{23}+r_{34}-r_{14} \le 2, \label{eq:4cycle} \end{equation} and label permutations thereof. We translate this into the CHSH~\cite{ClauserHSH69} Bell scenario, with Alice locally measuring one of two rank-1 projectors $A_1$ or $A_2$, and Bob locally measuring either $B_1$ or $B_2$, on the singlet state $\ket{\psi}=\frac{1}{\sqrt{2}} (\ket{01}-\ket{10})$. As a contextuality scenario, the CHSH graph $C_4$ is a graph with no clique with more than two vertices, and the only non-trivial noncontextuality inequality is given in terms of correlations. From the event graph perspective, each vertex can be understood as a two-outcome measurement at either Alice or Bob. It is easy to check that the overlap between two single-qubit projectors $A$, $B$ is the probability of obtaining different outcomes~\footnote{Equal outcomes and different outcomes are described in a dual way. Without loss of generality, we can use either the probability that the two measurements return equal outcomes or the probability that they return different outcomes. The forbidden deterministic edge labellings are essentially the same up to permuting $0$ and $1$.} when measuring those projectors on each part of the singlet state: $r_{AB}=p_{\neq}^{AB}=1-p_{=}^{AB}$. Using this interpretation, the facet of $C_{C_4}$ given by \cref{eq:4cycle} can be rewritten as \begin{equation} \label{eq:eqchsh} p_{\neq}^{{A_1}{B_1}}+p_{\neq}^{{A_2}{B_1}}+p_{\neq}^{{A_2}{B_2}}-p_{\neq}^{{A_1}{B_2}} \le 2, \end{equation} which is a well-known way to write the CHSH inequality~\cite{collins2002bell}. This same procedure can be used to obtain chained Bell inequalities \cite{braunstein1990wringing,araujo2013all} from cycle inequalities. \psection{Example: Original Bell inequality from the 3-cycle graph $C_3$} If on the $C_4$ graph we have just analyzed we impose the constraint that one of the edge weights equal 1, we recover the non-trivial facets for the 3-cycle $C_3$, namely $r_{12}+r_{23}-r_{13} \le 1$ and label permutations. The embedded tetrahedron with these 3 facets delimits the local correlations in the original two-party Bell inequality \cite{Bell64}, featuring three settings at each party, and assuming perfect anticorrelation for pairs of aligned settings. For a geometrical description of the elliptope of quantum correlations, see Ref.~\cite{Le2023quantumcorrelations}. \psection{Example: CHSH inequality from the 5-vertex wheel graph $W_5$} An alternative way of interpreting an event graph as a contextuality scenario involves having a single vertex, the handle, represent a quantum state, and all the others represent measurement operators. Take the 5-vertex wheel graph $W_5$ of \cref{fig:all_graphs_together}--(d) as an instructive example. A simple calculation shows that if we impose $r_{12}=r_{34}$ and $r_{23}=r_{14}$, then adding together four $3$-cycle inequalities for this graph recovers the CHSH inequality in the form of \cref{eq:eqchsh}. The quantum realization of this graph scenario has the central vertex $5$ representing a singlet state, with the other vertices representing the four projectors measured jointly by Alice and Bob. The imposed constraints reflect the fact that opposing edges represent the same quantity, the overlap between the two projectors locally measured by one of the parties. \psection{Recovering all noncontextuality inequalities of the exclusivity graph formalism} The second approach to obtaining the CHSH inequality does not rely on particular properties of the singlet state. The use of a handle vertex to represent a state can be generalized to other scenarios, as we now describe. In the \textit{exclusivity graph approach} to contextuality we have vertices in a graph $H$ representing measurement events (in a quantum realization, projection operators), with edges connecting mutually exclusive events (in the quantum setting, orthogonal projectors). In this formalism, the noncontextual behaviours are described as a well-known convex set, the stable polytope of the graph, denoted $\STAB(H)$~\cite{amaral2018graph}. We can understand this setup in terms of our formalism as follows. We define an event graph $H_\star$ obtained from the exclusivity graph $H$ by adding a new vertex connected to all other vertices. This new vertex is used to represent a handle state $\psi$. Formally, $H_\star$ is given by $V(H_\star) \defeq V(H) \sqcup \{\psi\}$ and $E(H_\star) \defeq E(H)\cup \setdef{\{v,\psi\}}{v\in V(H)}$. The structure of the exclusivity graph $H$ is then used to force a constraint on edge weightings of $H_\star$, namely that all edges already present in $H$ be assigned zero weight. The resulting cross-section $C_{H_\star}^0 \defeq \setdef{r \in C_{H_\star}}{\Forall{e\in E(H)} r_e=0}$ of the polytope $C_{H_\star}$, which moreover is a subpolytope, then carries information about the noncontextual behaviours in $\STAB(H)$. Formally, in \Cref{app:contextuality}, we exhibit an isomorphism between the polytopes $\STAB(H)$ and $C_{H_{\star}}^0$ for any exclusivity graph $H$. As a consequence, we show that the \textit{facet-defining noncontextuality inequalities bounding noncontextual behaviours for $H$ are precisely the facet-defining inequalities of $C_{H_\star}^0$}. Moreover, these inequalities can be obtained from the inequalities defining facets of the whole classical polytope $C_{H_\star}$ by removing (\ie setting to zero) the variables $r_e$ with $e \in E(H)$. \psection{Example: KCBS noncontextuality inequality} We illustrate this correspondence between formalisms with the noncontextuality inequality obtained by Klyachko, Can, Binicio\u{g}lu, and Schumovsky (KCBS) \cite{KlyachkoCBS08}, and expressed in the CSW formalism in Ref.~\cite{cabello2014graph}. Consider the 6-vertex wheel graph $W_6$ of \cref{fig:all_graphs_together}--(e). Let the central vertex represent a quantum state, and let neighbouring vertices in the outer 5-cycle represent mutually exclusive measurement events (quantum mechanically: orthogonal projectors) so as to impose $r_{vw}=0$ for neighbouring $v$ and $w$ in this outer subgraph. The KCBS noncontextuality inequality is a bound on weightings of the edges connected to the central vertex: \begin{equation}\label{eq: KCBS event graph} \sum_{v=1}^{5} r_{v6} \le 2 . \end{equation} Note that each edge weight $r_{v6}$ in \cref{eq: KCBS event graph} is the probability of successful projection of the central vertex state onto the projector associated with vertex $v$. In our framework, this inequality is obtained from a facet-defining inequality of $C_{W_6}$, \[ -r_{12}-r_{23}-r_{34}-r_{45}-r_{15}+r_{16}+r_{26}+r_{36}+r_{46}+r_{56} \leq 2, \] by imposing the exclusivity (or orthogonality) condition of null edge weights on the 5-cycle outer subgraph. \psection{Cycle inequalities witness preparation contextuality} Besides considering different approaches to Kochen--Specker noncontextuality, one can also consider different \textit{notions} of noncontextuality. One such proposal, put forth by Spekkens in Ref.~\cite{spekkens2005contextuality}, is that of preparation (generalized) noncontextuality~\cite{Lostaglio2020contextualadvantage,schmid2018discrimination,Spekkens08,baldijao_emergence_2021,lostaglio2020certifying}. We consider once more a quantum realization of the event graph representing vertices as states and edges as two-state overlaps. In \Cref{app:preparationcontextuality} we prove that \textit{violations of the inequalities for the classical polytope of the cycle event graph $C_n$ are witnesses of preparation contextuality}. This result is shown for a class of prepare-and-measure operational scenarios~\cite{Lostaglio2020contextualadvantage,schmid2018discrimination}, which includes quantum theory viewed as an operational theory. In contrast to quantum theory, the well-known noncontextual toy theory of Ref.~\cite{spekkens2007evidence} does not violate these event graph inequalities, if vertices of the event graph are taken to represent toy theory states. \psection{Discussion and future directions} We proposed a new graph-theoretic approach that unifies the study of three different quantum resources, namely contextuality, nonlocality, and coherence. Non-classicality inequalities are obtained as facets of a polytope $C_G$ of edge weightings associated with an \textit{event graph} $G$, with suitable constraints that depend on the chosen interpretation of vertices as quantum states or measurements, as required by each scenario. Connections with the theory of contextuality were presented with respect to different approaches and definitions. In particular, we recovered all inequalities of the CSW exclusivity graph approach \cite{cabello2014graph}, and we explicitly derived CHSH and KCBS inequalities as examples. We also showed that for cycle graphs the classical polytope bounds Spekkens preparation noncontextuality. It would be interesting to understand whether these results can be made more robust. In particular, we observed that the noncontextuality inequalities for exclusivity graphs $H$ are obtained from the inequalities of a classical polytope $C_{H_\star}$ by assigning weight zero to some edges. But many of these inequalities of $C_{H_\star}$ \textit{allow} for deviations from such null weights without leaving the classical polytope $C_{H_\star}$. This suggests that perhaps those inequalities could still be interpreted as a robust form of noncontextuality inequalities, where exclusivity is relaxed. Future research directions include characterizing this framework in the landscape of general probabilistic theories (GPTs) and understanding how this approach bounds relational unitary invariants involving three or more states, such as Bargmann invariants \cite{Oszmaniec21}. It would also be interesting to relate violation of our inequalities with advantage in quantum protocols, as recently done by some of us in \cite{WagnerCG22} for the task of quantum interrogation. \psection{Acknowledgements} We would like to thank Marcelo Terra Cunha, John Selby, David Schmid, and Raman Choudhary for helpful discussions. We also thank Roberto D.~Baldijão for critically reviewing an early version of this work. We acknowledge financial support from FCT -- Fundação para a Ciência e a Tecnologia (Portugal) through PhD Grant SFRH/BD/151199/2021 (RW) and through CEECINST/00062/2018 (RSB and EFG). This work was supported by the ERC Advanced Grant QU-BOSS, GA no. 884676. \if\forprl1 \setcounter{affil}{0} \title{{Supplemental Material \\ Inequalities witnessing coherence, nonlocality, and contextuality}} \reusableauthor{Rafael Wagner} \reusableaffiliation{INL -- International Iberian Nanotechnology Laboratory, Braga, Portugal} \reusableaffiliation{Centro de F\'{i}sica, Universidade do Minho, Braga, Portugal} \reusableauthor{Rui Soares Barbosa} \reusableaffiliation{INL -- International Iberian Nanotechnology Laboratory, Braga, Portugal} \reusableauthor{Ernesto F.~Galvão} \reusableaffiliation{INL -- International Iberian Nanotechnology Laboratory, Braga, Portugal} \reusableaffiliation{Instituto de F\'{i}sica, Universidade Federal Fluminense, Niter\'{o}i -- RJ, Brazil} \date{\today} \reusablemaketitle \fi \appendix \setcounter{secnumdepth}{2} \section{Characterizing the classical polytope}\label{app:algo} In Ref.~\cite{GalvaoB20}, Galv\~{a}o and Brod derived the facet-defining inequalities of the classical polytope $C_{C_n}$ of the $n$-cycle event graph $C_n$, as discussed in the text. The construction uses an argument based on Boole's inequalities for logically consistent processes~\cite{Boole1854}. In the main text we discuss that, in fact, \textit{any} event graph, and not only cycle graphs, can be used to bound classicality of different forms. In this section, we consider the computational problem of characterizing the classical polytope $C_G$ for any event graph $G$. We propose a simple algorithm for computing all its vertices and facets. This proceeds by first calculating the list of vertices of $C_G$, \ie its V-representation, and then finding its facet-defining inequalities, \ie its H-representation, using standard convex geometry tools. As discussed in the main text, this last step is computationally efficient on the size of the polytope. However, the overall efficiency of the procedure is intrinsically limited by the fact that the number of vertices and facets of $C_G$ grows exponentially on the size of $G$. The brunt of this section is dedicated to computing the set of vertices of $C_G$. After setting out the formal definitions, we characterize the edge $\enset{0,1}$-labellings $E(G) \to \enset{0,1}$ that respect logical consistency conditions and thus correspond to the vertices of $C_G$. This characterization yields an efficient procedure for checking whether such an edge labelling is a vertex of $C_G$, whose complexity we analyze. However, when the goal is to generate all vertices of $C_G$, it is needlessly wasteful to generate all the $2^{\|E(G)\|}$-many edge $\{0,1\}$-labellings and then filter them one by one. Instead, we present a procedure that generates the edge labellings that are vertices of $C_G$ by generating vertex labellings underlying them, thus limiting the search through the space $\enset{0,1}^{E(G)}$ of edge labellings. Even though it might output the same vertex more than once, the method works well, especially for dense graphs. It is optimal for the complete graphs $K_n$, which as we will see in \Cref{app:smallgraphs} are our main examples of interest. We observe that the number of vertices of the polytope $C_{K_n}$ is given by a well-known combinatorial sequence, known as the Bell numbers \cite{OEISBell}, which count the number of partitions of a set, precisely the space that is searched by this procedure. Finally, we discuss an alternative method that might be more efficient for sparse graphs. \psection{Basic definitions} We start with the relevant definitions. \begin{definition}[Graph]\label{def: graph} A \textit{graph} $G = (V(G), E(G))$ consists of a finite set $V(G)$ of vertices and a set $E(G)$ of edges, which are two-element subsets of $V(G)$, \ie sets of the form $\enset{v,w}$ where $v, w \in V(G)$ are distinct vertices. \end{definition} Note that the graphs we consider in this text are so-called \textit{simple} graphs: they are undirected (since $\enset{v,w} = \enset{w,v}$), have at most one edge between any two vertices $v$ and $w$, and have no loops (i.e. have no edges from a vertex to itself). In one well-delimited passage, however, we will need to consider \textit{possibly loopy graphs}, which may have loops. This corresponds to dropping the requirement that $v$ and $w$ be distinct in the definition above. A possibly loopy graph is said to be \textit{loop-free} if it has no loops, \ie if is is a bona fide (simple) graph. \begin{definition}[Labellings and colouring]\label{def: vertex labelling and colouring} A \textit{vertex labelling} by a set $\Lambda$, or a \textit{vertex $\Lambda$-labelling} for short, is a function $\fdec{\lambda}{V(G)}{\Lambda}$ assigning to each vertex a label from $\Lambda$. It is called a \textit{colouring} if $\{v,w\} \in E(G)$ implies $\lambda(v) \neq \lambda(w)$. The graph $G$ is said to be $k$-colourable for $k \in \NN$ when it admits a colouring by a set of size $k$. Similarly, an \textit{edge labelling} by a set $\Lambda$, or an \textit{edge $\Lambda$-labelling} for short, is a function $\fdec{\alpha}{E(G)}{\Lambda}$ assigning a label from $\Lambda$ to each edge. When $\Lambda=[0,1]$, we call this an \textit{edge weighting}. \end{definition} \begin{definition}[Chromatic number] The \textit{chromatic number} of a graph $G$, written $\chi(G)$, is the smallest $k \in \NN$ such that $G$ is $k$-colourable. \end{definition} In the classical, deterministic situation modelled by our framework, we consider a vertex labelling of a graph $G$ by an arbitrary labelling set $\Lambda$. However, operationally, we do not have access to the vertex labels, but only to the information of whether the labels of neighbouring edges are equal or different. \begin{definition} Given any vertex labelling $\fdec{\lambda}{V(G)}{\Lambda}$, its \textit{equality labelling} $\epsilon_\lambda$ is the edge $\enset{0,1}$-labelling given by: \begin{align} \fdec{\epsilon_\lambda&}{E(G)}{\enset{0,1}} \\ \epsilon_\lambda& \,\{v,w\} \,\defeq\, \delta_{\lambda(v),\lambda(w)} = \begin{cases} 1 & \text{if $\lambda(v)=\lambda(w)$} \\ 0 & \text{if $\lambda(u) \neq \lambda(v)$}\end{cases} \Mdot \end{align} \end{definition} We are interested in chracterizing the edge $\enset{0,1}$-labellings that arise as equality labellings of vertex labellings. \begin{definition} An edge $\enset{0,1}$-labelling $\fdec{\alpha}{E(G)}{\enset{0,1}}$ is said to be \textit{$\Lambda$-realizable} if it is the equality labelling of some vertex $\Lambda$-labelling, \ie if $\alpha = \epsilon_\lambda$ for some $\fdec{\lambda}{V(G)}{\Lambda}$. If $\Lambda$ has size $k \in \NN$, we say that $\alpha$ is $k$-realizable. \end{definition} We write $\EqG$ for the set of realizable edge labellings of $G$ (with any $\Lambda$), and $\EqGk$ for the set of $k$-realizable ones. We have that $\EqGk \subseteq \Eqk{G}{k'}$ whenever $k \leq k'$, and $\EqG = \cup_{k \in\NN}\EqGk$. Moreover, $\EqG = \Eqk{G}{\|V(G)\|}$ because a vertex labelling uses at most one distinct label per vertex of the graph. We often refer to these realisable edge $\enset{0,1}$-labellings as the \textit{classical} edge labellings. By the inclusion $\{0,1\}\subseteq[0,1]$, we can think of any edge $\enset{0,1}$-labelling as a (deterministic) edge weighting. This gives an alternative description of the classical polytope $C_G$ in the main text. \begin{definition} Given a graph $G$, its \textit{classical polytope} $C_G \subseteq [0,1]^{E(G)}$ is the convex hull of the set $\EqG$ seen as a set of points in $[0,1]^{E(G)}$. \end{definition} \psection{Characterizing the vertices of $C_G$} We now consider the question of determining whether a given edge $\enset{0,1}$-labelling is realizable (as the equality labelling of some vertex labelling). Given $\fdec{\alpha}{E(G)}{\enset{0,1}}$, define a relation $\sim_\alpha$ on the set of vertices of $G$ whereby $v \sim_\alpha w$ if and only if there is a path from $v$ to $w$ through edges labelled by $1$, \ie there is a sequence $e_1, \ldots, e_n \in E(G)$ such that $v \in e_1$, $w \in e_n$, $e_i \cap e_{i+1} \neq \es$, and $\alpha(e_i)=1$. This is easily seen to be an equivalence relation. It yields the following characterization of the classical edge labellings. \begin{proposition}\label{prop:charclassicalvertices} An edge labelling $\fdec{\alpha}{E(G)}{\enset{0,1}}$ is realizable (\ie classical) if and only if for all edges $\enset{v,w} \in E(G)$, $v \sim_\alpha w$ implies $\alpha(\enset{v,w})=1$. \end{proposition} In other words, an edge labelling $\fdec{\alpha}{E(G)}{\enset{0,1}}$ fails to be realizable precisely when there is an edge $\enset{v,w} \in E(G)$ such that $v \sim_\alpha w$ and $\alpha(\enset{v,w})=0$. In terms of the underlying vertex labellings, such a situation would violate the transitivity of equality. A slightly different perspective is given by using $\alpha$ to construct a new graph that `collapses' $G$ through paths labelled by $1$. Note that this construction yields a possibly loopy graph. An edge $\enset{0,1}$-labelling $\alpha$ partitions the edges of $G$ into two sets. This determines two graphs $G_{\alpha = 0}$ and $G_{\alpha = 1}$, both with the same vertex set as $G$, but each retaining only the edges of $G$ with the corresponding label, i.e. for each $\lambda \in \enset{0,1}$, \begin{align} V(G_{\alpha = \lambda}) &\defeq V(G)\\ E(G_{\alpha = \lambda}) &\defeq \setdef{e \in E(G)}{\alpha(e) = \lambda} \end{align} A possibly loopy graph $G/\alpha$ is then defined as follows: \begin{itemize} \item its vertices are connected components of $G_{\alpha = 1}$, or equivalently, the equivalence classes of $\sim_\alpha$; \item there is an edge between two connected components $A$ and $B$ of $G_{\alpha = 1}$ whenever there exist vertices $v \in A$, $w \in B$, such that $\{v,w\} \in E(G_{\alpha=0})$. \end{itemize} \begin{lemma} Let $\fdec{\alpha}{E(G)}{\enset{0,1}}$ and $\Lambda$ be any set. There is a one-to-one correspondence between $\Lambda$-realizations of $\alpha$ and $\Lambda$-colourings of ${G / \alpha}$. \end{lemma} \begin{proof} Let $\fdec{\lambda}{V(G)}{\Lambda}$ such that $\alpha = \epsilon_\lambda$. If $v \sim_\alpha w$, then $\lambda(v)=\lambda(w)$, by propagating equality along the path labelled by $1$. Hence, the map $\fdec{\kappa}{V({G / \alpha})}{\Lambda}$ given by $\kappa([v])\defeq\lambda(v)$ is well defined. Now, an edge $e \in E_{G / \alpha}$ is of the form $e=\enset{[v],[w]}$ for some $v, w \in V(G)$ such that $\alpha(\enset{v,w}) = 0$. Since $\alpha = \epsilon_\lambda$, this means that $\lambda(v) \neq \lambda(w)$, hence $\kappa([v]) \neq \kappa([w])$. Thus, $\kappa$ is a colouring. Conversely, given a colouring $\fdec{\kappa}{V_{G / \alpha}}{\Lambda}$, set $\lambda(v) \defeq \kappa([v])$. Let $e = \enset{v, w} \in E(G)$. If $\alpha(e)=1$, then $[v]=[w]$, hence $\lambda(v)=\lambda(w)$ because $\kappa$ is a colouring. If $\alpha(e)=0$, then $\enset{[v],[w]}\in E_{G / \alpha}$, hence $\lambda(v)\neq\lambda(w)$. In either case, $\alpha(e) = \epsilon_\lambda(e)$. The two processes just described are inverses of one another. \end{proof} \begin{corollary}\label{cor:realizablecolourable} An edge $\enset{0,1}$-labelling is $\Lambda$-realizable if and only if the possibly loopy graph $G/\alpha$ is $\Lambda$-colourable. In particular, it is realizable (\ie classical) if and only if $G/\alpha$ is loop-free. \end{corollary} \begin{proposition} Checking whether an edge $\enset{0,1}$-labelling for a graph $G$ is realizable can be done in time $O(n+m)$ where $n = \|V(G)\|$ and $m = \|E(G)\|$. Checking $k$-realizability in a given $k\geq 3$ is $NP$-complete. \end{proposition} \begin{proof} For the first part, transverse the graph $G_{\alpha=1}$ using a depth-first search (DFS). When visting each vertex, run through all the departing edges of $G_{\alpha=0}$ to see if any is linked to an already visited vertex in the connected component of $G_{\alpha=1}$ currently being traversed. If any is found, reject $\alpha$. For the second part, use \cref{cor:realizablecolourable} to reduce to graph colouring: a graph $G$ is $k$-colourable if and only if the constant $0$ edge labelling is realizable. \end{proof} The procedure outlined in the proof above is described below in more detail using pseudo-code. {\scriptsize \begin{align} &\rule{\columnwidth}{.7pt} \\ &\textbf{Input: } \text{graph $G$ with } V(G) = \enset{1, \ldots, N}. \hspace{100cm}\\[-2pt] &\hphantom{\textbf{Input: }} \text{edge-labelling } \fdec{\alpha}{E(G)}{\enset{0,1}} \\[-2pt] &\textbf{Output: } \text{whether $\alpha$ is realizable, hence a vertex of the polytope $C_G$}. \\[-2pt] &\\[-2pt] &\textbf{global variable } d_i \textbf{ for each } i \in V(G) \\[-2pt] &\textbf{global variable } c_i \textbf{ for each } i \in V(G) \\[-2pt] & \\[-2pt] &\textbf{procedure } \textsc{Main} ()\\[-2pt] &\algtab d_i \leftarrow \textrm{false} \textbf{ for all } i \in V(G)\\[-2pt] &\algtab \textbf{for } i \in V(G) \textbf{ do} \\[-2pt] &\algtab\algtab \textbf{if } \lnot d_i \textbf{ then} \\[-2pt] &\algtab\algtab\algtab c_j \leftarrow \textrm{false} \textbf{ for all } j \in V(G) \\[-2pt] &\algtab\algtab\algtab \textsc{Search }(i) \\[-2pt] &\algtab\algtab \textbf{end if} \\[-2pt] &\algtab \textbf{end for} \\[-2pt] &\algtab \textbf{terminate with output } \textrm{true}\\[-2pt] &\\[-2pt] &\textbf{procedure } \textsc{Search}(i) \\[-2pt] &\algtab d_i, c_i \leftarrow \textrm{true} \\[-2pt] &\algtab \textbf{for } j \in \textsc{Neighbours }(i) \textbf{ do} \\[-2pt] &\algtab\algtab \textbf{if } \alpha(\enset{i,j})=0 \;\land\; c_j \textbf{ then}\\[-2pt] &\algtab\algtab\algtab \textbf{terminate with output } \textrm{false}\\[-2pt] &\algtab\algtab \textbf{else if } \alpha(\enset{i,j}=1) \;\land\; \lnot d_j \textbf{ then}\\[-2pt] &\algtab\algtab\algtab \textsc{Search }(j) \\[-2pt] &\algtab\algtab \textbf{end if} \\[-2pt] &\algtab \textbf{end for} \\[-2pt] & \rule{\columnwidth}{.7pt} \end{align} } \psection{Computing all the vertices of $C_G$} We conclude that it is computationally easy to check whether a given edge $\{0,1\}$-labelling, \ie a given deterministic edge weighting, is classical. Nevertheless, determining the whole set of vertices of the classical polytope is computationally hard since the number of edge labelling to be tested grows exponentially with the number of edges of the graph. It is interesting to note that for the complete event graph $K_n$ of $n$ vertices the number of classical edge labellings, \ie vertices of the classical polytope $C_{K_n}$, is given by a well-known sequence, the Bell or exponential numbers \cite{OEISBell,bell1934exponential}. The $n$-th Bell number is the number of partitions, or equivalence relations, of a set of size $n$. It is clear that edge $\enset{0,1}$-labellings of $K_n$ are in one-to-one correspondence with symmetric reflexive relations on the set of vertices $\enset{1, \ldots, n}$, where the label of an edge $\enset{v,w}$ determines whether the pairs $(v,w)$ and $(w,v)$ are in the relation. Among these, the classical edge labellings correspond to the equivalence relations (which additionally satisfy transitivity), with the underlying vertex labelling determining a partition of the vertices. For a general graph $G$, it is still true that the classical edge labellings arise from partitions, or equivalence relations, on the set of vertices, determined by the underlying vertex labelling. The difference is that an edge labelling does not carry enough information to characterize a relation fully. So, in particular, different vertex partitions may give rise to the same classical edge labelling. We can use this observation to propose a different method for generating all vertices of $C_G$ by constructing vertex-labellings of $G$. The procedure is given below in pseudo-code. {\scriptsize \begin{align} &\rule{\columnwidth}{.7pt} \\ &\textbf{Input: } \text{graph $G$ with } V(G) = \enset{1, \ldots, N}. \hspace{100cm}\\[-2pt] &\textbf{Output: } \text{vertices of the polytope $C_G$}. \\[-2pt] &\\[-2pt] &\textbf{global variable } \lambda_i \textbf{ for each } i \in V(G) \\[-2pt] &\textbf{global variable } \alpha_e \textbf{ for each } e \in E(G) \\[-2pt] & \\[-2pt] &\textbf{procedure } \textsc{Main} ()\\[-2pt] &\algtab\textsc{Generate }(1,1)\\[-2pt] &\textbf{end procedure} \\[-2pt] &\\[-2pt] &\textbf{procedure } \textsc{Generate }(i,next) \\[-2pt] &\algtab\textbf{if } i=N+1 \textbf{ then} \\[-2pt] &\algtab\algtab \textbf{output } (\alpha_e)_{e \in E(G)} \\[-2pt] &\algtab\textbf{else} \\[-2pt] &\algtab\algtab \textbf{for } x < next \textbf{ do} \\[-2pt] &\algtab\algtab\algtab \textsc{Update } (i,x) \\[-2pt] &\algtab\algtab\algtab \textsc{Generate } (i+1,next) \\[-2pt] &\algtab\algtab \textbf{end for} \\[-2pt] &\algtab\algtab \textsc{Update } (i,next) \\[-2pt] &\algtab\algtab \textsc{Generate } (i+1,next+1) \\[-2pt] &\algtab\textbf{end if} \\[-2pt] &\textbf{end procedure} \\[-2pt] &\\[-2pt] &\textbf{procedure } \textsc{Update }(i,x) \\[-2pt] &\algtab \lambda_i \leftarrow x \\[-2pt] &\algtab \textbf{for } j < i \textbf{ with } \enset{i,j} \in E(G) \textbf{ do} \\[-2pt] &\algtab \algtab \alpha_{\enset{i,j}} \leftarrow \textbf{if } \lambda_j = x \textbf{ then } 1 \textbf{ else } 0 \\[-2pt] &\algtab \textbf{end for} \\[-2pt] &\textbf{end procedure}\\[-2pt] & \rule{\columnwidth}{.7pt} \end{align} } The procedure above has the disadvantage that it might output the same vertex of the polytope multiple times. This is because, as already discussed, different partitions of the vertices of $G$ can give rise to the same edge labelling. The problem is especially noticeable for sparse graphs. An alternative method for generating the vertices of $C_G$, which might be more efficient in the case of sparser graphs, is to directly search through $\enset{0,1}^{E(G)}$ while checking for consistency on the fly, in order to trim the search space so that only the realizable edge labellings are constructed. This can be done by keeping a representation of the current vertex partition (induced by the edges labelled $1$ in the edge labelling being constructed), for example using a union-find data structure, together with a record of forbidden merges between partition components (induced by the edges labelled $0$s in the edge labelling being constructed). The disadvantage is that the upkeep of this representation, necessary for checking consistency on the fly, cannot be done in constant time. This incurs an overhead at each step in the search. \section{Characterizing classical polytopes \\ by graph decompositions}\label{app:buildgraphs} In this section, we prove some general facts that relate the classical polytopes of different graphs. In particular, we show that some methods of combining graphs to build larger graphs do not give rise to new classicality inequalities. Or, seen analytically rather than synthetically, that some graphs $G$ can be decomposed into smaller component graphs in a way that reduces the question of characterizing $C_G$ to that of characterizing the polytopes of these components. These observations help trim down the class of graphs that is worth analyzing in the search for new classicality inequalities. As a by-product, we characterize the class of graphs for which all edge weightings are classical as being that of trees, an analogue of Vorob{\textquotesingle}ev's theorem \cite{vorobev1962consistent} in this framework. \begin{proposition}\label{prop:disjointunion} Let $G_1$ and $G_2$ be graphs, and write $G_1 + G_2$ for their disjoint union. Then \[C_{G_1 + G_2} = C_{G_1} \times C_{G_2} = \setdef{(r_1,r_2)}{r_1 \in G_1, r_2 \in G_2}.\] \end{proposition} \begin{proof} Given vertex labellings $\fdec{\lambda_i}{V(G_i)}{\Lambda_i}$ for each $i=1,2$, one obtains a function \[\fdec{\lambda_1+\lambda_2}{V(G_1) \sqcup V(G_2)}{\Lambda_1 \sqcup \Lambda_2}\] which is a vertex labelling of $G_1+G_2$ since $V(G_1+G_2) = V(G_1)\sqcup V(G_2)$. The corresponding equality edge labelling, $\fdec{\epsilon_{\lambda_1+\lambda_2}}{E(G_1+G_2)}{\enset{0,1}}$, is precisely the function \[ \fdec{[\epsilon_{\lambda_1},\epsilon_{\lambda_2}]}{E(G_1) \sqcup E(G_2)}{\enset{0,1}} \] given by \[ e \longmapsto \begin{cases} \epsilon_{\lambda_1}(e) & \text{ if $e \in E(G_1)$} \\ \epsilon_{\lambda_2}(e) & \text{ if $e \in E(G_2)$} \end{cases}, \] implying the result. \end{proof} In particular, vertices of the polytope $C_{G_1 + G_2}$ are in bijective correspondence with pairs consisting of one vertex from each of the polytopes $C_{G_i}$, while the facets of $C_{G_1 + G_2}$ are in bijective correspondence with the union of the facets of $C_{G_1}$ and the facets of $C_{G_2}$. That is, the inequalities defining $C_{G_1+G_2}$ are those defining $C_{G_1}$ plus those defining $C_{G_2}$. Taking the disjoint union of event graphs thus creates no new classicality inequalities. As a consequence, we might as well focus solely on studying the classical polytopes of connected graphs. The result above considers the construction of a new graph by placing two graphs side by side. But similar results can be obtained for more complicated ways of combining graphs, namely gluing along a vertex or along an edge. \begin{definition}[Gluing] Given graphs $G_1$ and $G_2$, and tuples of vertices \begin{align} \vc{v}_1&=(v_1^1, \ldots, v_1^k) \in V(G_1)^k,\\ \vc{v}_2&=(v_2^1, \ldots, v_2^k) \in V(G_2)^k, \end{align} the \stress{gluing of $G_1$ and $G_2$ along $\vc{v}_1$ and $\vc{v}_2$}, written $G_1 +_{\vc{v}_1=\vc{v}_2} G_2$, is the graph obtained by taking the disjoint union $G_1+G_2$ and identifying the vertices $v_1^j$ and $v_2^j$ for $j = 1,\ldots, k$. Explicitly: its vertices are \[ V(G_1 +_{\vc{v}_1=\vc{v}_2} G_2) \;\defeq\; O_1 \sqcup O_2 \sqcup N , \] where $O_i \defeq V(G_i) \setminus \{v_i^1, \ldots, v_i^k\}$ is the set of vertices of $G_i$ not being identified and $N = \enset{v^1, \ldots v^k}$ is a set of `new' vertices (\ie not in either $G_i$); its edges are \[ E(G_1 +_{\vc{v}_1=\vc{v}_2} G_2) \;\defeq\; E_1 \cup E_2 , \] where $E_i$ is equal to $E(G_i)$ but with occurrences of $v_i^j$ replaced by the new $v^j$. \end{definition} \begin{proposition}\label{prop:glue_vertex} Let $G_1$ and $G_2$ be graphs, $v_1 \in V(G_1)$ and $v_2 \in V(G_2)$, then $C_{G_1 +_{v_1=v_2} G_2} = C_{G_1} \times C_{G_2}$. \end{proposition} \begin{proof} We proceed as in the proof of \cref{prop:disjointunion}, using the same notation, but then take a quotient of the merged alphabet $\Lambda_1 \sqcup \Lambda_2$ identifying two labels, one from each component: $\lambda_1(v_1) \in \Lambda_1$ with $\lambda_2(v_2) \in \Lambda_2$. This yields a well-defined labelling for $G_1 +_{v_1=v_2} G_2$ where the new vertex $v$ is labelled by the element resulting from this identification. This does not affect the equality edge-labellings, and so we obtain the same result. \end{proof} Read analytically, if $G$ is a graph with a cut vertex $V$, \ie a vertex whose removal disconnects the graph into two components with vertex sets $V_1$ and $V_2$, then its polytope can be characterized in terms of the polytopes of the induced subgraph on $V_1 \cup \enset{v}$ and $V_2 \cup \enset{v}$. In particular, the facet-defining inequalities of $C_G$ are those of each of these two components. As an aside, this result is the missing ingredient for fully characterizing the event graphs that cannot display any nonclassicality, \ie for which all edge weightings $E(G) \to [0,1]$ are classical. This could be seen as an analogue of Vorob{\textquotesingle}ev's \cite{vorobev1962consistent} theorem in our framework. \begin{corollary} A graph $G$ is such that $C_G = [0,1]^{E(G)}$ if and only if it is a tree. \end{corollary} \begin{proof} For `only if' part, if $G$ has a cycle then any edge labelling $E(G) \to \enset{0,1}$ that restricts to $(1,\ldots,1,0)$ on said cycle is not in $C_G$. For the `if' part, apply \cref{prop:glue_vertex} multiple times, following the construction of a tree as a sequence of gluings along a vertex of copies of $K_2$, whose classical polytope is $[0,1]$. \end{proof} We now move to consider gluing along an edge. \begin{proposition} Let $G_1$ and $G_2$ be graphs, $v_1,w_1 \in V(G_1)$ and $v_2,w_2 \in V(G_2)$ such that $e_i \defeq \enset{v_i,w_i} \in E(G_i)$. Writing \[G \defeq G_1 +_{(v_1,w_1)=(v_2,w_2)} G_2,\] we have \begin{align} C_{G} =& \setdef{r \in [0,1]^{E(G)}}{r\|_{E(G_1)} \in C_{G_1}, r\|_{E(G_2)} \in C_{G_2}} \\ \cong& \setdef{(r,s)}{r \in C_{G_1}, s \in C_{G_2}, r_{e_1} = s_{e_2}} \\ \cong& (C_{G_1} \times [0,1]^{E(G_2)\setminus\enset{e_2}}) \cap ([0,1]^{E(G_2)\setminus\enset{e_1}} \times C_{G_2}) , \end{align} where for the last line we assume that $C_{G_1}$ is written with $e_1$ as its last coordinate and $C_{G_2}$ with $e_2$ as its first coordinate. \end{proposition} \begin{proof} The proof is similar to that of \cref{prop:glue_vertex}, but now we are forced to make two identifications between elements of $\Lambda_1$ and of $\Lambda_2$ in $\Lambda_1 \sqcup \Lambda_2$. When $\lambda_1$ and $\lambda_2$ are such that $\epsilon_{\lambda_1}(e_1) = \epsilon_{\lambda_2}(e_2)$, \ie such that \[ \lambda_1(v_1) = \lambda_1(w_1) \;\;\Leftrightarrow\;\; \lambda_2(v_2) = \lambda_2(w_2),\] then this yields a well-defined vertex labelling of $G$ and the result follows. \end{proof} Note that the result is not quite as strong as \cref{prop:disjointunion,prop:glue_vertex}. While the inequalities of $C_{G_1}$ plus those of $C_{G_2}$ form a complete set of inequalities for the classical polytope of the resulting graph $G_1 +_{(v_1,w_1)=(v_2,w_2)} G_2$, this is not necessarily a minimal set. \begingroup \squeezetable \begin{table}[thb] \caption{Quantum violations for facet inequalities of $C_{K_5}$\label{tab: inequalities K5}} \begin{ruledtabular} \begin{tabular}{ccccc} Class & Violation &\multicolumn{2}{l}{Inequality Representative for the Class}& Dim. \\ \hline 11--40 & 1/4 &\multicolumn{2}{l}{$-r_{12}+r_{15}+r_{25}\leq 1$}&2\\ 41--60 & 1/3 &\multicolumn{2}{l}{$+r_{15}+r_{25}+r_{35}-(r_{12}+r_{13}+r_{23})\leq 1$}&3\\ 61--65 & 0.243 &\multicolumn{2}{l}{$+r_{12}+r_{13}+r_{14}+r_{15}-(r_{23}+r_{24}+r_{25}+r_{34}+r_{35}+r_{45})\leq 1$}&4\\ 66--75 & 0.312 &\multicolumn{2}{l}{$+r_{12}+r_{14}+r_{15}+r_{23}+r_{34}+r_{35}-(r_{13}+r_{24}+r_{25}+r_{45})\leq 2$}&3\\ 76--87 & $0.795$ &\multicolumn{2}{l}{$+r_{12}+r_{15}+r_{23}+r_{34}+r_{45}-(r_{13}+r_{14}+r_{24}+r_{25}+r_{35})\leq 2$}&2\\ 88--92 & 0.344 &\multicolumn{2}{l}{$+2r_{12}+2r_{23}+2r_{24}+2r_{25}-(r_{13}+r_{14}+r_{15}+r_{34}+r_{35}+r_{45})\leq 3$}&4\\ 93--152 & 0.688 &\multicolumn{2}{l}{$+r_{13}+r_{14}+2r_{24}+r_{34}+2r_{45}-(2r_{12}+2r_{25}+2r_{35})\leq 3$}&3\\ 153--212 & 0.7306 &\multicolumn{2}{l}{$+2r_{12}+2r_{14}+2r_{15}+r_{23}+r_{35}-(2r_{13}+2r_{24}+r_{25}+2r_{45})\leq 3$}&2\\ 213--242 & 0.855 &\multicolumn{2}{l}{$+2r_{13}+2r_{14}+2r_{23}+2r_{24}+3r_{35}+3r_{45}-(2r_{12}+4r_{15}+4r_{25}+r_{34})\leq 5$}&3\\ \end{tabular} \end{ruledtabular} \end{table} \endgroup \begin{proposition}\label{prop:subgraph} Let $G$ be a graph and $G'$ be a subgraph of $G$ on the same set of vertices, \ie $V(G') = V(G)$ and $E(G') \subseteq E(G)$. Then $C_{G}$ is a subpolytope of $C_{G'} \times [0,1]^{E(G)\setminus E(G')}$. \end{proposition} \begin{proof} We need to show that the vertices of $C_{G}$ constitute a subset of the vertices of $C_{G'} \times [0,1]^{E(G)\setminus E(G')}$, \ie that $\EqG \subseteq \EqGprime \times \enset{0,1}^{E(G)\setminus E(G')}$. Given a classical edge labelling of $G$, \ie an edge labelling of the form $\epsilon_\lambda$ for some vertex labelling $\fdec{\lambda}{V(G)}{\Lambda}$, we can regard $\lambda$ as a vertex labelling of $G'$ and conclude that its equality labelling is simply the restriction of $\fdec{\epsilon_\lambda}{E(G)}{\enset{0,1}}$ to the subset $E(G')$ of its domain. \end{proof} In particular, $C_{K_n}$ is a subpolytope of $C_G$ for any event graph $G$ with $n$ vertices. \section{Classical polytope facets and quantum violations for small graphs}\label{app:smallgraphs} In this section, we study the facet-defining inequalities of some small graphs. In particular, we analyze and classify the facet-defining inequalities for the classical polytopes $C_G$ corresponding to complete event graphs of 4 and 5 vertices ($G=K_4$ and $G=K_5$, respectively). We also find quantum violations of these inequalities with pure states that are sampled from the set of quantum states. For sampling we used the Python library \textrm{QuTip}~\cite{johansson2012qutip}. Ref.~\cite{GalvaoB20} gave a complete characterization of the classical polytope of the graph $K_3 = C_3$, the smallest graph with non-trivial inequalities, together with a characterization of its maximal quantum violations, as well as applications. More generally, Ref.~\cite{GalvaoB20} gave the complete set of inequalities for the classical polytope of the cycle graphs $C_n$, which take the very simple form in \cref{eq:ncycle}. Here, we move to consider graphs with more than three edges and which are not cycles. \psection{Facet-defining inequalities for small complete graphs} The facet-defining inequalities of the classical polytope of the graph $C_4$ (the $4$-cycle) are those of the form given by the CHSH inequality mentioned in the main text. If we add one more edge to this graph, the corresponding polytope ends up being described by $3$-cycle inequalities alone. Therefore, the first interesting graph yielding non-trivial and non-cycle inequalities is $K_4$, the complete graph of 4 vertices. The classical polytope of this graph has facets defined by $3$- and $4$-cycle inequalities, together with facets defined by the new inequalities described in \cref{eq:k4nontrivial} in the main text, \ie those of the form \[ (r_{12}+r_{13}+r_{14})-(r_{23}+r_{34}+r_{24}) \le 1 . \] This inequality has a structure that is present for all $K_n$ graphs, as will be discussed in \Cref{app:family}. Since complete graphs have all possible edges, these are the graphs that impose the largest number of non-trivial constraints on edge assignments, as per \cref{prop:subgraph}. Therefore, it is natural to look at those graphs first. We addressed the complete characterization of the classical polytopes of complete graphs, proceeding as far as the computational complexity of the problem allowed. In particular, we found complete sets of facet-defining inequalities for $C_{K_5}$ and $C_{K_6}$. The polytope $C_{K_5}$ has 52 vertices and 242 facets. These facets fall are divided into 9 symmetry classes. Representative inequalities from each of these classes are shown in \cref{tab: inequalities K5}. The polytope $C_{K_6}$ has 203 vertices and requires 50,652 inequalities. A list of inequalities and Python code used to obtain them can be found in Ref.~\cite{wagner2022github}. \psection{Quantum violations} We looked for quantum violations of each inequality class of $C_{K_5}$ obtained by pure states in Hilbert spaces of dimensions 2, 3, and 4. The violations found are included in \cref{tab: inequalities K5}. The inequality in the third row is apparently not violated by either qubit or qutrit states. The largest violation found among all the inequalities was $0.855$, for the inequality in the last row of the table. The sets of quantum states yielding the violations found are presented in Ref.~\cite{wagner2022github}. \begin{figure}\label{fig: qubit_violations} \end{figure} For some classes of inequalities, we also found violations using pure qubit states that display interesting symmetries in the Bloch sphere. We present those violations in \cref{fig: qubit_violations}. For instance, consider the inequality in the fifth row of \cref{tab: inequalities K5}. It can be violated with the quantum states \begin{align} \vert \psi_k \rangle = \frac{1}{\sqrt{2}}\left(\vert 0 \rangle + e^{2\pi i k/5}\vert 1 \rangle\right) \end{align} with $k=0,\dots,4$. This quantum realization attains a value of $\sfrac{5\sqrt{5}}{4}$ and hence a violation of $\sfrac{5\sqrt{5}}{4}-2\approx 0.79508$. Another interesting violation with qubits is for the inequality in the fourth row of the table. There, a maximal qubit violation is achieved by the states depicted in \cref{fig: qubit_violations}: choosing $\vert \psi_2 \rangle, \vert \psi_4 \rangle, \vert \psi_5 \rangle $ equally distributed on the equator of the Bloch sphere, \ie separated by angles of $\sfrac{2\pi}{3}$, implying that $r_{24}=r_{25}=r_{45}=\sfrac{1}{4}$, and choosing $\vert \psi_1 \rangle = \vert 0 \rangle ,\vert \psi_3 \rangle = \vert 1 \rangle$, implying that $r_{13}=0$ and all remaining overlaps are equal to $\sfrac{1}{2}$. This set of vectors attains the value $\sfrac{6}{2}-\sfrac{3}{4}=\sfrac{9}{4}$ and hence a violation of $\sfrac{9}{4}-2=\sfrac{1}{4}$. These symmetrically arranged qubit states are also the states used in the construction of the elegant joint measurement of Ref.~\cite{gisin2019entanglement}. However, we could find a higher violation of the same inequality using qutrits, as shown in the table. We will see in \cref{app:family} that the inequality in \cref{eq:k4nontrivial} generalizes to an infinite family of inequalities for the polytope of $K_n$. The quantum violation found for this non-cycle $K_4$ inequality used the following four qutrit states: \begin{align} \ket{\psi_1} &= \ket{0} \\ \ket{\psi_2} &= \sqrt{\frac{5}{9}}\ket{0} + \sqrt{\frac{4}{9}}\ket{1} \\ \ket{\psi_3} &= \sqrt{\frac{5}{9}}\ket{0} - \sqrt{\frac{1}{9}}\ket{1} + i\sqrt{\frac{1}{3}}\ket{2} \\ \ket{\psi_4} &= \sqrt{\frac{5}{9}}\ket{0} - \sqrt{\frac{1}{9}}\ket{1} - i\sqrt{\frac{1}{3}}\ket{2}\\ \end{align} This set of states attains a value of $\sfrac{4}{3}$ and hence violation of $\sfrac{4}{3}-1=\sfrac{1}{3}$. This corresponds to the second class of inequalities of $C_{K_5}$ in \cref{tab: inequalities K5}. We remark once more that the above violations are \stress{not} necessarily optimal. They were not found using \eg techniques of semidefinite programming over the quantum set. We found this landscape of violations by sampling quantum states and calculating the value of the left-hand side of the inequality, which is suboptimal. An important remark is that the quantum violation for the $3$-cycle inequality class (first row in \cref{tab: inequalities K5}) is \stress{provably maximal}, as shown in Ref.~\cite{GalvaoB20}. \section{Infinite family of \\ classical polytope facets}\label{app:family} \Cref{eq:k4nontrivial} in the main text shows a facet-defining inequality of the polytope $C_{K_4}$ that is not of the previously known form of inequalities derived from cycles in Ref.~\cite{GalvaoB20} (which where enough, incidentally, to characterise the classical polytope of the graph $K_3 = C_3$). In this section, we generalize it to an infinite family of new classicality inequalities. More concretely, we present a construction of a facet-defining inequality of the classical polytope $C_{K_n}$ for any $n \geq 2$. Moreover, each inequality on this family cannot be obtained from combining prior members of the family. For $n = 4$, this recovers the just-mentioned inequality from \cref{eq:k4nontrivial}, while for $n=3$ it naturally reduces to the 3-cycle inequality. Fix a natural number $n \geq 2$. Write $V_n = \enset{1, \ldots, n}$ for the vertices of $K_n$, and let $E_n$ denote the set of edges of $K_n$, \ie all two-element subsets of $V_n$. Consider a partition of $E_n$ into the subsets $G_n, R_n \subseteq E_n$ given as \begin{align} G_n &\defeq \setdef{\enset{1,i}}{i=2,\ldots,n} \\ R_n &\defeq E_{n} \setminus G_n. \end{align} The edges in $R_n$ determine a complete subgraph of $K_n$ with one fewer vertex, \ie a subgraph isomorphic to $K_{n-1}$. In turn, the edges in $G_n$ form a subgraph isomorphic to $K_{1,n-1}$, a star graph with $n$ vertices. We use this specific partition of $E_n$ to define a generalized version of the inequality from \cref{eq:k4nontrivial}: \begin{equation}\label{eq: new K4 generalized} \sum_{e\in G_n}r_e - \sum_{e\in R_n} r_e \leq 1. \end{equation} \begin{figure} \caption{\textbf{Depiction of the sets $R_n$ and $G_n$ for a given complete graph $K_n$.} The set $R_n$ is always a complete subgraph (isomorphic to) $K_{n-1}$ of $K_n$. Here we considered $n=4$ as an example.} \label{fig: R_and_G_figure_cut_Kn} \end{figure} We first show that this is indeed a classicality inequality for the complete graph $K_n$. \begin{proposition} For any $n \geq 2$, the classical polytope $C_{K_n}$ of the complete event graph $K_n$ is contained in the half-space defined by the inequality in \cref{eq: new K4 generalized}, \ie all classical edge weightings of $K_n$ satisfy the inequality. \end{proposition} \begin{proof} It suffices to check that the inequality is satisfied by any vertex of the polytope $C_{K_n}$. Recall that the vertices of this polytope correspond to classical edge $\enset{0,1}$-labellings of the graph $K_n$, that is, those realisable as the equality labelling of some vertex labelling. So, let $\fdec{\lambda}{V_n}{\Lambda}$ be any vertex labelling and $r \in [0,1]^{E_n}$ be the vertex of the classical polytope corresponding to its equality edge labelling. That is, for all $e=\enset{i,j} \in E_n$, \[r_{ij} = \epsilon_\lambda(\enset{i,j})=\begin{cases}1 & \text{ if $\lambda(i)=\lambda(j)$} \\ 0 & \text{ if $\lambda(i)\neq\lambda(j)$}\end{cases}.\] Consider the set of vertices in $\enset{2, \ldots, n}$ that are labelled the same as vertex $1$, \begin{align} S_\lambda &= \setdef{i \in \enset{2, \ldots, n}}{\lambda(i)=\lambda(1)} \\&= \setdef{i \in \enset{2, \ldots, n}}{r_{1 i} = 1} \end{align} By construction, an edge in $G_n$, which is of the form $\enset{1,i}$, is labelled $1$ or $0$ depending on whether $i$ is in $S_\lambda$ or not. Moreover, by transitivity of equality, if $i,j \in S_\lambda$ then $\lambda(i)=\lambda(j)$, meaning that the edge $\enset{i,j}$ is also labelled $1$. Writing $k \defeq \|S_\lambda\|$, one can therefore bound the left-hand side of \cref{eq: new K4 generalized}: \begin{align} \sum_{e\in G_n}r_e - \sum_{e\in R_n} r_e &= \sum_{i \in S_\lambda}r_{1i} - \sum_{e\in R_n} r_e \\ &= k - \sum_{e\in R_n} r_e \\ &\leq k - \sum_{i,j \in S_\lambda} r_{ij} \\ &= k - \binom{k}{2} \\&= 1-\binom{k-1}{2} \\&\leq 1 \end{align} where for the corner case $k=0$ this still holds putting $\binom{-1}{2}=1$ \end{proof} We now state the central result of this section. \begin{theorem} The inequality in \cref{eq: new K4 generalized} defines a facet of the classical polytope $C_{K_n}$ of the complete event graph $K_n$ for any $n \geq 2$. \end{theorem} \begin{proof} We establish this result by finding the set of vertices of the polytope $C_{K_n}$ that belongs to -- and therefore determines -- this facet. In fact, it suffices to find a set of points $F$ in the space (of edge weightings) such that: (i) all the points in $F$ belong to the polytope $C_{K_n}$, (ii) all the points in $F$ saturate the inequality, \ie belong to the hyperplane determined by it, (iii) the set $F$ is affinely independent, and (iv) $F$ contains as many points as the dimension $D$ of the polytope, so that it generates an affine subspace of dimension $D-1$. In our proof, the chosen points are moreover vertices of the polytope, as they are edge $\enset{0,1}$-labellings. We construct a set $F$ of polytope vertices. This consists of two kinds of edge labellings: those that assign $1$ to exactly one edge of $G_n$ (and $0$ to all other edges of $E_n$) and those that assign $1$ precisely to a triangle consisting of two edges from $G_n$ and another from $R_n$. More formally, we define a family of edge $\enset{0,1}$-labellings indexed by subsets of size $1$ or $2$ of the vertex set $\enset{2,\ldots,n}$, as follows: for each $i = 2, \ldots, n$, define the edge $\enset{0,1}$-labelling $r^{(i)}$ with $r^{(i)}_{1i} = 1$ and $r^{(i)}_e=0$ for all other edges $e$; for each pair $i,j = 2, \ldots, n$ with $i \neq j$ define the edge $\enset{0,1}$-labelling $r^{(i,j)}$ with $r^{(i,j)}_{1i} = r^{(i,j)}_{1j} = r^{(i,j)}_{ij} = 1$ and $r^{(i,j)}_e=0$ for all other edges $e$. The set $F$ is then given by \[ F \defeq \setdef{r^{(i)}}{i = 2, \ldots, n} \cup \setdef{r^{(i,j)}}{i,j = 2, \ldots ., n, i \neq j}. \] \Cref{fig:principle_argument_newk4_infinite} depicts the construction of the set $F$ for the case of $n=5$. We now check conditions (i)--(iv) to establish the desired result. \begin{figure} \caption{\textbf{The construction of the set $F$ for the $K_5$ graph.} Each edge is labelled $1$ where explicitly noted, otherwise it is labelled $0$ (to keep the figures easy to read). The first row shows the four labellings of the form $r^{(i)}$ with only one edge labelled $1$ from $G_5$. The remaining rows show the labellings of the form $r^{(i,j)}$, which assign label $1$ to exactly one triangle consisting of two edges from $G_5$ and the connecting edge from $R_5$.} \label{fig:principle_argument_newk4_infinite} \end{figure} For condition (i), we use \cref{prop:charclassicalvertices} to show that all the edge labellings in the set $S$ are classical and thus vertices of the polytope $C_{K_n}$. Indeed, no cycle can have exactly one edge with label $0$. In the case of the labellings of the form $r^{(i)}$, this is immediate as there is only one edge not labelled $0$. For the labellings of the form $r^{(i,j)}$, no triangle (\ie subgraph isomorphic to $C_3$) has exactly one edge labelled $0$: if one chooses two edges labelled $1$ then the remaining edge that completes the 3-cycle also has label $1$. Moreover, any larger cycle can have at most two edges labelled $1$. Alternatively, we can show this by constructing an underlying vertex labelling: for $r^{(i)}$ pick $\fdec{\lambda}{V_n}{\Lambda}$ with $\lambda(1)=\lambda(i)$ and all other vertex labelled differently; for $r^{(i,j)}$ pick $\lambda$ with $\lambda(1)=\lambda(i)=\lambda(j)$ and the other vertices labelled differently. Condition (ii) is directly checked: for each $i = 2, \ldots, n$ we have \[ \sum_{e\in G_n}r^{(i)}_e - \sum_{e\in R_n} r^{(i)}_e = r^{(i)}_{1i} - 0 = 1 - 0 = 1 , \] and for each pair $i, j = 2, \ldots, n$ with $i \neq j$, \[ \sum_{e\in G_n}r^{(i,j)}_e - \sum_{e\in R_n} r^{(i,j)}_e = r^{(i,j)}_{1i} + r^{(i,j)}_{1j} - r^{(i,j)}_{ij} = 2 - 1 = 1. \] For condition (iii), affine independence can be verified by inspecting the matrix whose columns are the vectors corresponding to the edge-labellings in $F$. Ordering the components of each vector (corresponding to the edges of $K_n$) in lexicographic order and listing $r^{(i)}$ followed by $r^{(i,j)}$ also in that order, the matrix is arranged to be triangular with diagonal entries all equal to 1, hence its determinant is equal to $1$, implying linear independence of the vectors. Finally, for condition (iv), as all these labellings are distinct, one can count the number of elements of $S$ from the way they were constructed: \[\|F\| = \binom{n-1}{1} + \binom{n-1}{2} = \binom{n}{2} = \frac{n(n-1)}{2}.\] We conclude that it is the same as the dimension of the ambient space (of edge labellings) where the polytope lives, and thus also of the polytope itself. \end{proof} \section{Event graphs and \\ Kochen--Specker contextuality}\label{app:contextuality} In this section, we establish a formal connection between our framework and (Kochen--Specker) contextuality. The central result (\cref{theorem: STAB = C}) shows how our event graph formalism recovers all noncontextuality inequalities obtainable from the Cabello--Severini--Winter (CSW) exclusivity graph approach~\cite{cabello2014graph}. To achieve this, we encode a contextuality setup, represented in CSW by an exclusivity graph $H$, by imposing exclusivity constraints on a related event graph $H_\star$. This process amounts to taking a cross-section yielding a subpolytope of the classical polytope $C_{H_\star}$. We show that the resulting facet inequalities bound noncontextual models for $H$. In fact, we prove something \stress{stronger}. We describe an explicit isomorphism between the noncontextual polytope associated by CSW to the exclusivity graph $H$ and this cross-section subpolytope of the classical polytope $C_{H_\star}$ associated by our approach to the event graph $H_\star$. In particular, these polytopes have the same non-trivial facet-defining inequalities. These are obtainable from the inequalities that define the full (unconstrained) classical polytope of the event graph $H_\star$ by setting some coefficients to zero. \Cref{theorem: STAB = C} thus establishes a tight correspondence between our event graph approach and a broad, well-established framework for contextuality. In what follows, we introduce the relevant definitions regarding the exclusivity graph approach, the associated event graphs, and the constraints to be imposed on them, before proving the new results. \psection{The exclusivity graph approach} In the CSW framework from Ref. \cite{cabello2014graph}, contextuality scenarios are described by so-called exclusivity graphs. Hence this formalism is also known as the exclusivity graph approach; see also~\cite[Chapter 3]{amaral2018graph} for a recent and comprehensive discussion. The vertices of an exclusivity graph $H$ represent measurement events, and its edges indicate exclusivity between events, where two events are exclusive that can be distinguished by a measurement procedure. Even though the CSW framework is theory-independent, it is helpful for clarity of exposition to consider its instantiation in quantum theory, in order to better convey the underlying intuitions. In quantum theory, measurement events are represented by projectors (PVM elements) on a Hilbert space, or equivalently, by closed subspaces of the Hilbert space. Exclusivity is captured by orthogonality, which characterizes when two projectors may appear as elements of the same PVM, \ie events from \stress{the same} measurement procedure. Given a set of projectors $\{\Pi_v\}_{v \in V}$ on a fixed Hilbert space, the corresponding contextuality scenario is thus described by its orthogonality graph. This graph has set of vertices $V$ and has an edge $\{u,v\}$ if and only if the projectors $\Pi_u$ and $\Pi_v$ are orthogonal to each other, \ie when $\Pi_u\Pi_v=0$. In this approach, a non-negative vertex weighting $\fdec{\gamma}{V(H)}{\RR_{\geq 0}}$ on the exclusivity graph $H$ determines a noncontextuality inequality on the probabilities $P(v)$ of measurement events $v \in V(H)$: \[\sum_{v \in V(H)} \gamma(v) P(v) \;\leq\; \alpha(H,\gamma),\] where $\alpha(H,\gamma)$ is the independence number of the vertex-weighted graph. In the quantum case, this yields a noncontextuality condition on the statistics predicted by a given quantum state $\psi$: \[\sum_{v \in V(H)} \gamma(v) \langle \psi \vert \Pi_v \vert \psi \rangle \;\leq\; \alpha(H,\gamma) .\] Such noncontextuality inequalities above determine the polytope of noncontextual behaviours for any exclusivity graph $H$. This polytope, known as the stable set polytope of $H$, $\STAB(H)$, is most readily defined by its V-representation, which we now present, following \cite[Chapter 3]{amaral2018graph}. \begin{definition}\label{def: stable set} Let $H$ be a graph. A subset $S \subseteq V(H)$ of vertices is called a \stress{stable set} if no two vertices of $S$ are adjacent in $H$, \ie for all $v, w \in S$, $\{v,w\} \not\in E(H)$. Write $\mathcal{S}(H)$ for the set of stable sets of $H$. \end{definition} To any subset of vertices $W\subseteq V(H)$ corresponds its characteristic map, the vertex $\{0,1\}$-labelling $\fdec{\chiW}{V(H)}{\enset{0,1}}$ given by: \[ \chiW(v) \defeq \begin{cases} 1 & \text{ if $v \in W$,}\\ 0 & \text{ if $v \notin W$.} \end{cases} \] Through the inclusion $\enset{0,1} \subseteq [0,1]$, one regards a vertex $\{0,1\}$-labelling (equivalently, a subset of vertices) as a point in $[0,1]^{V(H)} \subseteq \RR^{V(H)}$. Those arising from stable sets $S \in \mathcal{S}(H)$ correspond to the deterministic noncontextual models, which determine the whole convex set of noncontextual behaviours. \begin{definition} \label{def: STAB} The \stress{stable set polytope} of a graph $H$, denoted $\STAB(H)$, is the convex hull of the points $\chiS \in [0,1]^{V(H)}$ with $S$ ranging over all stable sets of $H$, \[ \STAB(H) \defeq \textrm{ConvHull}\setdef{\chiS}{S \in \mathcal{S}(H)}. \] \end{definition} To get the intuition underlying this description, one may think of a vertex $\{0,1\}$-labelling $\fdec{\chi_W}{V(H)}{\enset{0,1}}$ as a deterministic assignment of truth values to all measurement events (vertices of the exclusivity graph). In this interpretation, the subset of vertices $W \subseteq V(H)$ is the set of measurement events that are assigned \textit{true}. The stability condition indicates that no two adjacent vertices of the exclusivity graph $H$ are labelled with $1$, that is, two exclusive measurement events cannot be simultaneously true. This captures the exclusivity condition at the deterministic level, thus yielding the deterministic noncontextual models. \psection{From exclusivity graphs to constrained event graphs} We relate this approach to our framework by constructing a new (event) graph $H_\star$ from any (exclusivity) graph $H$. This is obtained by adding a new vertex $\psi$ with an edge connecting it to all the vertices of $H$. See \cref{fig:orthogonality_to_event} for an instance of this construction for the KCBS scenario and \cref{fig:ProofAppD} for a more generic description. The construction is formally described in \cref{def: G out of G prime} below. The relevance of the new vertex $\psi$ is well known; it is usually called the `handle' and it appears in the literature on the graph approaches~\cite{amaral2018graph,baldijao2020classical,vandre2022quantum}. Its name comes from the geometric arrangement of the vectors providing the maximal quantum violation of the KCBS inequality of \cref{eq: KCBS event graph}: the quantum state resembles the handle of an umbrella made of the vectors that describe measurement events. \begin{definition}\label{def: G out of G prime} Let $H$ be a graph. Define a new graph $H_\star$ by \begin{align} V(H_\star) &\defeq V(H) \sqcup \enset{\psi} \\ E(H_\star) &\defeq E(H) \cup \setdef{\enset{\psi,v}}{v \in V(H)}. \end{align} Moreover, define $C_{H_\star}^0$ to consist of the classical edge weightings of $H_\star$ that assign value $0$ to all edges in $H$, \[ C_{H_\star}^0 \defeq \setdef{r \in C_{H_\star}}{\Forall{e\in E(H)} r_e=0} . \] \end{definition} The set $C_{H_\star}^0$ is, by construction, a cross-section of the classical polytope $C_{H_\star}$ of the event graph $H_\star$, being its intersection with the $\|V(H)\|$-dimensional subspace defined by the equations $\bigwedge_{e \in E(H)}r_e=0$. Moreover, it is a subpolytope of $C_{H_\star}$, \ie the convex hull of a subset of its vertices. These vertices are the classical edge $\enset{0,1}$-labellings that assign label $0$ to edges in $H$. In terms of the underlying vertex labellings (from which classical edge labellings arise as equality labellings), the requirement is that any two vertices adjacent in $H$ must be labelled differently. \begin{figure}\label{fig:orthogonality_to_event} \end{figure} \psection{Recovering the noncontextual polytope} The edge set of the graph $H_\star$ can be partitioned into two sets: the edges already present in $H$ and the new edges of the form $\{\psi,v\}$ for $v \in V(H)$. The latter are in one-to-one correspondence with vertices of $H$. So, there is a bijection $E(H_\star) \cong E(H) \sqcup V(H)$. When considering the polytope $C_{H_\star} \subseteq [0,1]^{E(H_\star)}$ we adopt the convention of ordering the coordinates with the edges already in $H$ listed first, so that \[\RR^{E(H_\star)} \cong \RR^{E(H) \sqcup V(H)} \cong \RR^{E(H)} \times \RR^{V(H)}.\] The subpolytope $C_{H_\star}^0$ is thus written as the set of points of $C_{H_\star}$ of the form $(\mathbf{0}_{H},r)$ where $\mathbf{0}_{H}$ is the zero vector in $\RR^{E(H)}$ (corresponding to the edges inherited from $H$) and $r$ is a weighting of the remaining (new) edges. In particular, the vertices of $C_{H_\star}^0$ are precisely the classical $\{0,1\}$-labellings of $H_\star$ that assign the label $0$ to all the edges in $H$. We can now prove our main result, showing that $C_{H_\star}$ is indeed (isomorphic to) the polytope of noncontextual behaviours for $H$. \begin{theorem}\label{theorem: STAB = C} For any (exclusivity) graph $H$, there is an isomorphism of polytopes \[C_{H_\star}^0 \; \cong \; \STAB(H)\] between the stable set polytope (of noncontextual models) of $H$ and the subpolytope of the classical polytope of event graph $H_\star$ constrained by the exclusivity conditions. More explicitly, this is given by the identification \[C_{H_\star}^0 \; = \; \{\mathbf{0}_H\} \times \STAB(H)\] where $\mathbf{0}_H$ is the zero vector in $\mathbb{R}^{E(H)}$. \end{theorem} \begin{proof} To establish the result, we consider the vertices of these polytopes. Per the above discussion, we have $E(H_\star) \cong E(H) \sqcup V(H)$. Consequently, there is a bijection between vertex $\{0,1\}$-labellings of $H$ (equivalently, subsets of $V(H)$), on the one hand, and edge $\{0,1\}$-labellings of $H_\star$ that assign label $0$ to all the edges in $E(H)$, on the other. Explicitly, to each subset of vertices $W \subseteq V(H)$ corresponds the edge-labelling of $H_\star$ \[\fdec{[\mathbf{0}_H,\chi_{W}]}{E(H_\star) \cong E(H) \sqcup V(H)}{\{0,1\}},\] as depicted in \cref{fig:ProofAppD}. We show that this bijection restricts to a bijection between the \stress{classical} assignments in both cases. Concretely, a subset of vertices $S \subseteq V(H)$ is \stress{stable}, hence (its characteristic map $\fdec{\chi_{S}}{V(H)}{\{0,1\}}$ is) a vertex of the polytope $\STAB(H)$, if and only if the corresponding edge labelling $[\mathbf{0}_H,\chi_{S}]$ of $H_\star$ is classical, hence a vertex of the polytope $C_{H_\star}$ and thus of $C_{H_\star}^0$. We establish the two directions of this equivalence simultaneously, recalling the characterisation of classical edge labellings from \cref{prop:charclassicalvertices}. Consider $H_\star$ with edge labelling $[\mathbf{0}_H,\chi_{S}]$. The labelling fails to be classical if and only if there is an edge with label $0$ between two vertices linked by a path consisting of edges with label $1$. Since all the edges between vertices in $H$ have label $0$, the only way to build such a path of $1$-labelled edges is via the handle $\psi$: \eg $\{u,\psi\},\, \{\psi, v\}$ where both $u$ and $v$ must belong to $S$. So, two vertices $u$ and $v$ of $H_\star$ are linked by a $1$-labelled path if and only if they both belong to $S \cup \{\psi\}$. Therefore, the labelling is classical if and only if there is no edge with label $0$ between vertices in this set $S \cup \{\psi\}$. To further simplify this condition, note that edges between $\psi$ and a vertex from $S$ have label $1$ by construction of the second component of $[\mathbf{0}_H,\chi_{S}]$, while from the first component, all edges between vertices in $H$ have label $0$. The classicality condition is thus equivalent to there being no edges in $H$ between vertices in $S$, which is precisely to say that $S$ is stable. \end{proof} \begin{figure}\label{fig:ProofAppD} \end{figure} \psection{Recovering all noncontextuality inequalities} We established \cref{theorem: STAB = C} in terms of the vertices of the polytopes, \ie by working with their V-representations. We now consider the relationship between their H-representations, \ie their facet-defining inequalities.\footnote{H-representation is standard terminology referring to the description of a polytope as an intersection of half-spaces, \ie in terms of (facet-defining) inequalities. The `H' in `H-representation' is not to be confused with the symbol `$H$' that we use to denote an exclusivity graph.} Of course, there is also a bijection between the facets of $\STAB(H)$ and those of $C_{H_\star}^0$. Given the particularly simple description of the isomorphism, whereby $C_{H_\star}^0$ is written as a product of polytopes, we can write this correspondence explicitly. It turns out that the facet-defining inequalities of the subpolytope $C_{H_\star}^0$ are precisely the same as the facet-defining inequalities of the stable polytope of $H$. Moreover, these can be obtained from the inequalities defining the (unconstrained) polytope $C_{H_\star}$ of the event graph $H_\star$ by setting some coefficients to zero. We thus recover the full set of noncontextuality inequalities from our event graph formalism. To see this, recall that if $P$ and $Q$ are two convex polytopes with H-representations $P = \setdef{x}{A_1 \, x \preccurlyeq b_1}$ and $Q = \setdef{y}{A_2 \, y \preccurlyeq b_2}$ then their product has H-representation \[ P \times Q = \setdef{(x,y)}{A_1 \, x \preccurlyeq b_1 \text{ and } A_2 \, y \preccurlyeq b_2}. \] Here, the notation $A \, z \preccurlyeq b$ describes a set of linear inequalities on $z$ in matrix form, with the symbol $\preccurlyeq$ standing for component-wise inequality $\leq$ between real numbers. Applying this to \begin{align} C_{H_\star}^0 &= \{\mathbf{0}_{H}\} \times \STAB(H)\\ &= \setdef{(x,y)}{x \in \{\mathbf{0}_{H}\}, y \in \STAB(H)}. \end{align} we obtain that the H-representation of $C_{H_\star}^0$ is the conjunction of the H-representations of $\{\mathbf{0}_{H}\}$ and of $\STAB(H)$. The former consists simply of the equations $r_e = 0$ for each $e \in E(H)$, zeroing out the first components, which corresponds to the weights of edges already in $H$. Thus the non-trivial inequalities bounding $C_{H_\star}^0$ are thus the same as the inequalities bounding $\STAB(H)$. Since $C_{H_\star}^0$ is obtained from $C_{H_\star}$ by intersecting with the subspace that zeroes the components corresponding to edges in $E(H)$, a complete set of inequalities for $C_{H_\star}^0$ can be obtained from the facet-defining inequalities of $C_{H_\star}$ by disregarding those components, \ie setting the corresponding coefficients to zero. This process is illustrated by the derivation of the KCBS inequality presented in the main text. There, the exclusivity graph is the 5-cycle, with neighbouring vertices representing orthogonal projectors. The graph $H_\star$ is then the 6-vertex wheel graph $W_6$ of \cref{fig:all_graphs_together}--(e). As shown in the main text, the KCBS noncontextuality inequality $\sum_a\gamma_a\vert\langle \psi \vert a \rangle \vert^2 \leq \alpha(H,\gamma)$ arises as a $C_{H_\star}^0$ inequality, being obtained from a classicality inequality for the event graph $W_6$ (a facet-defining inequality of $C_{H_\star}$) by setting to zero the coefficients relating to edges already in $H$. \section{Event graphs and \\ preparation contextuality}\label{app:preparationcontextuality} In this section, we relate our approach to Spekkens's notion of preparation contextuality. This may be understood as providing a \stress{theory-independent} perspective on the use of our formalism to witness quantum coherence. There, the vertices of event graphs were interpreted as quantum states and the edges as two-state overlaps. A similar treatment can be carried out for a certain class of operational theories which support a notion of confusability, with vertices interpreted as (abstract) preparation procedures. \psection{Operational probabilistic theories} Spekkens's notion of generalized contextuality is associated to operational probabilistic theories~\cite{dariano2017quantum,schmid2020structure,kunjwal2019beyondcabello}. The description of an operational theory starts with a set of basic (operational) physical processes: in the simplest scenarios, one considers preparations and measurements. One considers experiments consisting of a preparation $P$ followed by a measurement $M$ that returns an outcome $k$. A probability rule associates a probability $p(k \mid M, P)$ of obtaining outcome $k$ when performing measurement $M$ after the preparation $P$. More precisely, it associates a probability distribution over outcomes $k$ to each choice of preparation $P$ and measurement $M$. For a dichotomic measurement $M$, \ie one with only two possible outcomes $0$ and $1$, we simplify notation and write $p(M \mid P)$ for $p(1 \mid M, P)$. A crucial -- if sometimes overlooked -- aspect is that the full set of procedures includes also classical probabilistic mixtures (\ie convex combinations) of basic procedures, with the probability rule extended accordingly (\ie linearly). Given an operational theory, one defines an equivalence relation identifying indistinguishable procedures. Following Ref.~\cite{spekkens2005contextuality}, two preparation procedures are \textit{operationally equivalent}, written $P \simeq P'$, if and only if for all measurements $M$ and possible outcomes $k$, \[ p(k\|M,P) = p(k\|M,P') . \] A similar definition applies to measurement procedures, but this will not be needed in what follows. When one treats quantum theory as an operational theory, quantum states $\vert \phi \rangle$ correspond to equivalence classes of operational procedures. For instance, a state $\ket{0}$ may represent preparing a ground state of a nitrogen atom, or preparing the horizontal polarization in photonic qubits. We relax this terminology and refer to `the preparation $P$ associated with a state $\ket{\phi}$', even though strictly speaking $P$ is only an instance of an equivalence class of procedures. Such relaxation is safe for our purposes. In effect, it corresponds to treating pure quantum states as the basic procedures. The interesting operational equivalences relevant for preparation contextuality go beyond these, holding between classical mixtures of basic procedures. For example, in quantum theory, the preparation procedure corresponding of an equal mixture of pure qubit states $\ket{0}$ and $\ket{1}$ is operationally equivalent to that corresponding to an equal mixture of states $\ket{+}$ and $\ket{-}$. Indeed, both these classical mixtures define the same qubit mixed state, the totally mixed state. \psection{LSSS operational constraints} We wish to generalize the situation in which our graph-theoretic framework is used to witness quantum coherence. There, vertices of an event graph $G$ are interpreted as representing vectors $\{\ket{\phi_i}\}_{i \in V(G)}$ in some Hilbert space $\mathcal{H}$, \ie pure quantum states. Edge weights then correspond to two-state quantum overlaps, $\vert \langle \phi_i \vert \phi_j \rangle \vert^2$. Such overlaps can be accessed empirically by \eg measuring one of the states on a measurement basis that includes the other. Abstracting from this, we consider a situation in which we associate a preparation procedure $P_i$ to each vertex $i \in V(G)$ of a given graph $G$. But in order to emulate the setup above for more general operational theories, it is necessary to impose some additional operational constraints. These constraints distil the aspects of quantum theory that make this work, allowing (a theory-independent version of) two-state overlaps. We shall refer to them as the Lostaglio--Senno--Schmid--Spekkens (LSSS) operational constraints, after Refs.~\cite{Lostaglio2020contextualadvantage,schmid2018discrimination}. Note that these constraints apply to preparation procedures; we need not assume any operational equivalences for measurement procedures. Therefore, the scenarios under consideration aim to probe preparation contextuality only. First, for any preparation $P_i$, we assume that there is a corresponding `test measurement' $M_i$ with outcomes $\{0,1\}$ satisfying the operational statistics $p(M_i \mid P_i) = 1$. In quantum theory, if $P_i$ is the preparation associated with state $\ket{\phi_i}$ then $M_i$ is realised by the projective measurement $\{\ket{\phi_i}\!\bra{\phi_i}, \mathbb{1} - \ket{\phi_i}\!\bra{\phi_i}\}$ where the first projector corresponds to the outcome $k=1$. Moreover, for any edge $\{i,j\} \in E(G)$, whose incident vertices have preparations $P_i$ and $P_j$, we assume that there exists another pair of preparations $P_{i^\perp}$ and $P_{j^\perp}$ satisfying $p(M_i \mid P_{i^\perp}) = 0$, $p(M_j \mid P_{j^\perp}) = 0$, and the operational equivalence $\frac{1}{2}P_i + \frac{1}{2}P_{i^\perp} \simeq \frac{1}{2}P_j + \frac{1}{2}P_{j^\perp}$. In quantum theory, this is always satisfied: given a pair of pure states $\ket{\phi_i}$ and $\ket{\phi_j}$, one picks $\ket{\phi_{i^\perp}}$ to be the vector orthogonal to $\ket{\phi_i}$ living in the two-dimensional space spanned by $\{\ket{\phi_i},\ket{\phi_j}\}$, and similarly for $\ket{\phi_{j^\perp}}$. The probabilities $p(M_i\mid P_j)$ are usually called the \stress{confusability}~\cite{lostaglio2020certifying,schmid2018discrimination}, because they may be interpreted as the probability of guessing incorrectly that the preparation performed had been $P_i$ instead of $P_j$. These probabilities provide a theory-independent, operational treatment of two-state overlaps, which reduces to the familiar notion in the case of quantum theory viewed as an operational theory: \[p(M_i \mid P_j) \stackrel{QT}{=} \Tr\left(\ket{\phi_i}\!\bra{\phi_i}\ket{\phi_j}\!\bra{\phi_j}\right) = \vert \langle \phi_i \vert \phi_j \rangle \vert^2 .\] Therefore, we use these confusability probabilities to provide edge weights $r_{ij} = p(M_i \mid P_j)$ in our framework. In summary, an assignment of preparation procedures to the vertices of $G$ such that the LSSS operational constraints are satisfied for the pairs of preparations associated to each edge determines an edge weighting $\fdec{r}{E(G)}{[0,1]}$. \psection{Preparation noncontextuality} When faced with an operational theory, a natural question is whether it admits a (noncontextual) hidden-variable explanation, that is, whether it can be realised by a noncontextual ontological model. In general, an ontological model consists of a measurable space $(\Lambda, \mathcal{F}_\Lambda)$ of \stress{ontic} states equipped with ontological interpretations for preparation and measurement procedures: preparation procedures $P$ determine probability measures $\mu_P$ on $\Lambda$, whereas measurement procedures $M$ determine measurable functions $\xi_M$ mapping each ontic state $\lambda \in \Lambda$ to (a distribution on) outcomes. Note that the interpretation of classical mixtures of procedures must be determined linearly from that of basic procedures, \eg $\mu_{\frac{1}{2}P+\frac{1}{2}Q} = \frac{1}{2}\mu_P+\frac{1}{2}\mu_Q$. The composition of the interpretations of a preparation and a measurement (going via the ontic space $\Lambda$) is required to recover the empirical or operational predictions, \ie \[p(\cdot \mid M, P) \;=\; \int_\Lambda \xi_M \, \mathrm{d} \mu_P ,\] or with variables, \[p( k \mid M, P) \;=\; \int_\Lambda \xi_M(k \mid \lambda) \, \mathrm{d} \mu_P(\lambda) .\] Such a realization by an ontological model is said to be noncontextual if operationally equivalent procedures are given the same interpretation. For preparations, the requirement is that two operationally equivalent preparation procedures determine the same probability measure on $\Lambda$. We refrain from going into detail on the general definition, as the characterization that follows suffices. In Refs.~\cite{Lostaglio2020contextualadvantage,schmid2018discrimination}, it was shown that the LSSS constraints imply that any preparation noncontextual model explaining preparation procedures $P_i$ as probability measures $\mu_i$ on $\Lambda$ must satisfy \begin{equation}\label{equation: noncontextual overlaps} p(M_i\mid P_j) = 1 - \\|\mu_i - \mu_j\\|_{_{\mathsf{TV}}} , \end{equation} where $\\|\cdot - \cdot\\|_{_{\mathsf{TV}}}$ denotes the total variation distance between probability measures, given for an arbitrary measurable space $(\Lambda, \mathcal{F}_\Lambda)$ by \[\\|\mu_i - \mu_j\\|_{_{\mathsf{TV}}} = \sup_{E \in \mathcal{F}_\Lambda}\|\mu_i(E) - \mu_j(E)\|.\] In the case when $\Lambda$ is discrete (which is effectively all we actually need), this distance is related to the $l_1$ norm\footnote{{In the continuous case, it is often rendered as $\\|\mu_i - \mu_j\\|_{_{\mathsf{TV}}} = \int_\Lambda \vert \mu_i(\lambda) - \mu_j(\lambda) \vert \,\mathrm{d}\lambda$ in terms of a reference measure such as the Lebesgue measure on the real line.}}: \begin{align} \\|\mu_i - \mu_j\\|_{_{\mathsf{TV}}} = \frac{1}{2}\\|\mu_i - \mu_j\\|_{_1} = \frac{1}{2}\sum_{\lambda \in \Lambda}\|\mu_i(\lambda)-\mu_j(\lambda)\|. \end{align} We can take that as a \textit{definition} of preparation noncontextual edge weightings. \begin{definition}\label{def:PNCedgeweighting} Let $G$ be an event graph. An edge weighting $\fdec{r}{E(G)}{[0,1]}$ is said to be \stress{preparation noncontextual} if the edge weights are of the form in the right-hand side of \cref{equation: noncontextual overlaps}, \ie $r_{ij} = 1 - \\|\mu_i - \mu_j\\|_{_{\mathsf{TV}}}$, for some choice of an (ontic) measurable space $\Lambda$ and of probability measures $\mu_i$ on $\Lambda$ for each vertex $i\in V(G)$. \end{definition} \psection{Cycle inequalities witness preparation contextuality} We now show how in the case of cycle graphs the inequalities derived from our framework serve as witnesses of preparation contextuality for operational theories satisfying the LSSS constraints. The technical result is stated in the following proposition; it follows from the triangle inequality. \begin{proposition} Any inequality bounding the set $C_{C_n}$ cannot be violated by a preparation noncontextual edge weighting (\cref{def:PNCedgeweighting}). \end{proposition} \begin{proof} For simplicity, we use addition modulo $n$ when labelling the vertices of the cycle graph $C_n$, meaning that $i = i+n$. From the triangle inequality of the norm $\\| \cdot \\|_{_{\mathsf{TV}}}$ it follows that \begin{align} & \\| \mu_i - \mu_{i+n-1}\\|_{_{\mathsf{TV}}} \\ = \; & \\| \mu_i \underbrace{-\mu_{i+1}+\mu_{i+1}-\dots -\mu_{i+n-2}+\mu_{i+n-2}}_{n-2\text{ zeros}}-\mu_{i+n-1}\\|_{_{\mathsf{TV}}}\\ \leq \; & \\| \mu_i - \mu_{i+1}\Vert_{_{\mathsf{TV}}} + \dots + \Vert \mu_{i+n-2}-\mu_{i+n-1}\\|_{_{\mathsf{TV}}} . \end{align} Therefore, writing $\\|\mu_{i,j}\\|_{_{\mathsf{TV}}} \defeq \\|\mu_{i}-\mu_j\\|_{_{\mathsf{TV}}}$ for clarity, \[ \Vert \mu_{i,i+n-1}\Vert_{_{\mathsf{TV}}} - \Vert \mu_{i,i+1}\Vert_{_{\mathsf{TV}}} - \dots - \Vert \mu_{i+n-2,i+n-1}\Vert_{_{\mathsf{TV}}} \leq 0 . \] We must now add $1$ to each term to recover the noncontextual overlaps of \cref{equation: noncontextual overlaps}. We have $n$ terms, but since the first term has a different sign, two of these $1$s will cancel, leaving $n-2$ added to both sides of the inequality: \begin{align} -1+\Vert \mu_{i,i+n-1}\Vert_{_{\mathsf{TV}}} +1-\Vert \mu_{i,i+1}\Vert_{_{\mathsf{TV}}} &\\ +\;\cdots\; + 1 - \Vert \mu_{i+n-2,i+n-1} \Vert_{_{\mathsf{TV}}} & \;\;\leq\;\; n-2. \end{align} Recalling that $r_{ij} = 1-\Vert \mu_{i,j} \Vert_{_{\mathsf{TV}}}$, we recover a cycle inequality for any chosen vertex $i$: \[ -r_{i,i+n-1}+r_{i,i+1}+\dots+r_{i+n-2,i+n-1}\leq n-2 . \] \end{proof} We may see this result from two perspectives. We can take a \textit{theory-dependent perspective} and look for what information we can extract assuming quantum theory as the relevant operational theory; this proposition then shows that pure quantum states that violate the $n$-cycle inequalities can be used to construct a proof of quantum preparation contextuality. The construction is done by constructing states and measurements that represent a realization of the prepare-and-measure scenario described by the LSSS constraints. In summary, violations of these inequalities serve as \textit{witnesses of quantum preparation contextuality}. In light of this result, the experiment of Ref.~\cite{Giordani21} can be understood as an experimental test that witnessed preparation contextuality of quantum theory; however since the purpose was not to witness preparation contextuality the authors have not experimentally probed the relevant operational equivalences, and have not ruled out loopholes for such a test. We can also take a \textit{theory-independent perspective}. If a given operational theory satisfying the LSSS constraints for some cycle graph admits a preparation noncontextual ontological model, then the confusabilities $r_{ij} = p(M_i\vert P_j)$ are bounded by the cycle inequalities. For instance, the Spekkens Toy Theory~\cite{spekkens2007evidence} satisfies the LSSS constraints for any pair of preparation procedures. Since it admits a noncontextual ontological model, it cannot violate the cycle inequalities. \end{document}	arXiv
\begin{definition}[Definition:Product Sigma-Algebra/Binary Case] Let $\struct {X_1, \Sigma_1}$ and $\struct {X_2, \Sigma_2}$ be measurable spaces. The '''product $\sigma$-algebra''' of $\Sigma_1$ and $\Sigma_2$ is denoted $\Sigma_1 \otimes \Sigma_2$, and defined as: :$\Sigma_1 \otimes \Sigma_2 := \map \sigma {\set {S_1 \times S_2: S_1 \in \Sigma_1 \text { and } S_2 \in \Sigma_2} }$ where: :$\sigma$ denotes generated $\sigma$-algebra :$\times$ denotes Cartesian product. This is a $\sigma$-algebra on the Cartesian product $X \times Y$. \end{definition}	ProofWiki
Analysis and classification of heart diseases using heartbeat features and machine learning algorithms Fajr Ibrahem Alarsan ORCID: orcid.org/0000-0001-6057-83391 & Mamoon Younes2 This study proposed an ECG (Electrocardiogram) classification approach using machine learning based on several ECG features. An electrocardiogram (ECG) is a signal that measures the electric activity of the heart. The proposed approach is implemented using ML-libs and Scala language on Apache Spark framework; MLlib is Apache Spark's scalable machine learning library. The key challenge in ECG classification is to handle the irregularities in the ECG signals which is very important to detect the patient status. Therefore, we have proposed an efficient approach to classify ECG signals with high accuracy Each heartbeat is a combination of action impulse waveforms produced by different specialized cardiac heart tissues. Heartbeats classification faces some difficulties because these waveforms differ from person to another, they are described by some features. These features are the inputs of machine learning algorithm. In general, using Spark–Scala tools simplifies the usage of many algorithms such as machine-learning (ML) algorithms. On other hand, Spark–Scala is preferred to be used more than other tools when size of processing data is too large. In our case, we have used a dataset with 205,146 records to evaluate the performance of our approach. Machine learning libraries in Spark–Scala provide easy ways to implement many classification algorithms (Decision Tree, Random Forests, Gradient-Boosted Trees (GDB), etc.). The proposed method is evaluated and validated on baseline MIT-BIH Arrhythmia and MIT-BIH Supraventricular Arrhythmia database. The results show that our approach achieved an overall accuracy of 96.75% using GDB Tree algorithm and 97.98% using random Forest for binary classification. For multi class classification, it achieved to 98.03% accuracy using Random Forest, Gradient Boosting tree supports only binary classification. An electrocardiogram (ECG) is a complete representation of the electrical activity of the heart on the surface of the human body, and it is extensively applied in the clinical diagnosis of heart diseases [1], it can be reliably used as a measure to monitor the functionality of the cardiovascular system. ECG signals have been widely used for detecting heart diseases due to its simplicity and non-invasive nature. Features of ECG signals can be computed from ECG samples and extracted using some softwares (ex: Matlab). For instance, millions of people suffer from irregular heartbeats which can be lethal in some cases. Therefore, accurate and low-cost diagnosis of arrhythmic heartbeats is highly desirable [2]. Many studies have developed arrhythmia classification approaches that use automatic analysis and diagnosis systems based on ECG signals. The most important factors for the analysis and diagnosis of cardiac diseases are features extraction and beats classification. Numerous techniques for classifying ECG signals were proposed in recent years and good results achieved [3,4,5]. The performance of ECG pattern classification strongly depends on the characterization power of the features that are extracted from the ECG signal and the design of the classifier (classification model). Automated classification of heartbeats has been previously reported by many investigators using a variety of features to represent the ECG and a number of classification methods. In general, heartbeat features include ECG morphology, heartbeat interval features (temporal features), beats correlations and summits values [6]. The target of classification process is obtaining an intelligent model, that is capable to class any heartbeat signal to specific type of heartbeats. Experiments have been conducted on the well-known MIT-BIH Arrhythmia database using obtained model, and results have been compared with the previous scientific literature. The final results show that our model is not only more efficient than related works in terms of accuracy, but also competitive in terms of sensitivity and specificity. Big data analytic plays a vital role in managing the huge amount of health-care data and improving the quality of health-care services offered to patients. In this context, one of the challenges lies in the classification of data, which relies on effectively distributed processing platforms, advanced data mining and machine learning techniques. Therefore, a Big data technique is introduced in this work to meet the challenges faced by classify the ECG beats. Recently, deep learning techniques have been used by many companies, including Facebook, Google, IBM, Microsoft, NEC, Netflix, and NVIDIA [7, 8], and in a very large set of application domains such as customer churn prediction in telecom company [9]. In this paper, a novel deep learning approach for ECG beats classification is presented. Background and related work There are many works related to ECG classification without using big data tools when size of dataset is not large. On other hand, there are several studies that depend on big data techniques. In [10], Indonesia has high mortality caused by cardiovascular diseases. To minimize the mortality, a tele-ecg system was built for heart diseases early detection and monitoring using Hadoop framework, in order to deal with big data processing. The system can classify the ECG data using decision tree (DT) and random forest (RF), it was the first real system for heartbeats classification using big data tools. The system was build on cluster computer with 4 nodes. The server was able to handle 60 requests at the same time. The accuracy was 97.14% and 98,92% for decision tree and random forest respectively. In [11], Neural networks and dimensionality reduction technique was used and the approach was tested on the Massachusetts Institute of Technology arrhythmia database. The classification performance on a test set of only 18 ECG records of 30 min each achieved an accuracy of 96.97%. In [1], Many types of heartbeat were extracted and used for classification, classification method is used to classify independent type (3 records for each type); each type of heartbeats has its own model (model to classify normal heartbeats, model to classify type 1 of heartbeats and so forth). Neural network and SVM were applied and the accuracy of results was high good (more than 90%), but there was not unified model to classify all multi-types together at once. In [12], Feed-forward and fully connected artificial neural networks aided by particle swarm optimization technique are employed to recognize two patterns of heartbeats [Ventricular ectopic beats (VEBs) and supra-VEBs (SVEBs)]. Tuning parameters of proposed method has improved accuracy of classification to 96% comparing to the same method with default value of parameters. Not all features were used (only morphological and temporal) and a total of 83,648 beats were selected for training and testing. In [13], Hidden Markov Models and Spark were used to mine ECG data, combining accurate Hidden Markov Model (HMM) techniques with Apache Spark to improve the speed of ECG analysis. The paper has proven that there is potential for developing a fast classifier for heartbeats classification. In [8], DNN (Deep Neural Network) has been used for deep learning to classify heartbeats. The author has compared results with many studies, accuracy of classifying has reached 99%, but the classification was only two types (Normal and Abnormal) and dataset size was almost 85,000 records. In [2], multiply types of heartbeats have been studied and the author has reached accuracy 93.4%. Convolutional neural network for classification of ECG beat types has been developed by the author. All studies have proven that machine learning algorithms are very effective in heartbeats classification. Objective of the paper In this paper, multiples classifiers are proposed for ECG classification, these classifiers are used mostly in Big Data and Machine Learning fields by the weighted voting principle. Each classifier influences the final decision according to its performance on the training data. Parameters of each classifier are adjusted on the basis of an individual classifier's performance on the training data by applying the pseudoinverse technique. The proposed approach is validated in the MIT BIH Arrhythmia Database. The classification performance was validated on a set of 51 ECG records with different temporal length. So our work is distinguished by: Number of tested records (205,146 records of 51 patients). Complexity of heartbeat types in training and testing (training records contains Normal and Abnormal beats). Using Machine learning algorithms for classification. Using big data tool (Spark–Scala). Using local host pc (according to the lack of requirements). Binary and Multi Classification. In general, previous studies are using known methods (SVM, NN, PCA, Adaptive methods, etc.) and limited number of records for testing and training. Experimental study for whole work will be introduced in the following points: Heartbeat dataset Data set preparing The development of this work requires a database with digital ECG records for computational analysis of many different patients with different pathologies. Accordingly, we employed the widely known Massachusetts Institute of Technology (MIT) arrhythmia database. The original dataset is the MIT-BIH Arrhythmia Dataset. Using physionet ATM Bank [14], to get records' annotations with specific configuration as shown in Fig. 1. Physionet configuration As shown in Fig. 1, each record was extracted to its end (ex: record 100 is 1805 s). In general, completed dataset contains information of 51 patients (totally 205,146 rows for all patients). Each row has 16 columns of features. Each record of data has three main files with three different extentions .atr, .hea and .dat, atr file contains annotations, dat file is the digitalized signals and hea file is header file. The recordings were digitized at 360 samples per second per channel with 11-bit resolution over a 10 mV range. Two or more cardiologists independently annotated each record; disagreements were resolved to obtain the computer-readable reference annotations for each beat included with the database. More details about dataset in [15]. Annotations was saved as text files from Physionet website. Many of the considered features based on the Discrete Wavelet Transform (DWT) of the continuous ECG signal, final dataset was saved as csv file. In addition to the ECG signal, annotations contain the beat localization and the beat class [16]. Normal beat pattern is shown in Fig. 2. Firstly, two classes (Normal and Abnormal) were classified, then multi classes (4 classes) were classified. Normal beat pattern The selected features are beat properties, Fig. 3 shows 3 summits (Q, R and S) of heartbeat Beat properties (not normalized) Features can be divided into three classes: Summits features, Temporal features and Morphological features. Feature extraction and selection Discrete Wavelet Transform (DWT) was used to get features from downloaded annotations. All chosen features were used in classification model. Features can be divided into three types as described in the following. Summits features Three main summits were considered in this paper as features, these features are related to the amplitude of three summits QRS as shown in Fig. 3. Temporal features Nine temporal features were calculated and used, one of them is the RR interval, defined as the time delay between two QRS peaks. Two other features are the interval between the current and previous beat and the one between the current and subsequent beat, which are called RR1 and RR2 respectively. Another interval is defined as the distance between the previous beat and its predecessor, called RR0. Figure 4 shows all RR intervals: RR intervals Three other features were extracted based on previous intervals. These features are called Ratio1, Ratio2 and Ratio3. They are defined as below: $$\begin{aligned} Ratio1= \frac{R R_{0}}{R R_{1}} \quad Ratio2= \frac{R R_{2}}{R R_{1}} \quad Ratio3= \frac{R R_{m}}{R R_{1}} \end{aligned}$$ $$\begin{aligned} R R_{m}=mean(R R_{0} \mathbf , R R_{1} \mathbf , R R_{2}) \end{aligned}$$ Three other features are chosen and selected to define each summit period as shown in Fig. 5. QRS summits periods Morphological features Using the QRS summits (after normalization), the maximum of cross-correlation function between each detected beat and the following beat was calculated, as well as the maximum of cross-correlation between the current beat and the previous beat detected, called respectively Corr1 and Corr2 [17]. Another feature was the maximum of cross-correlation between a template of normal beat, with each QRS complex detected, called $C_{xy}$, was computed. For each record, the template was calculated as the averaged beat of a sequence of many normal sinus beats. Finally, a feature was defined as the QRS duration when QRS beat equals to 0.5 in the normalized QRS complex, as shown in Fig. 5. Morphological features are 4 features. The total features are 16 features (3 QRS amplitude, 9 temporal, 4 morphological) as shown in Fig. 6. Total features for classification For features selection, many tests have been done to get best features for final model. Basically, all extracted features were selected for classification process. Heartbeat classification using machine learning Features acquisition and storing Dataset was used after downloading it from [18], there are a lot of patients records in this website and many types of databases; they belongs to real patients. As mentioned before, MIT-BIH Arrhythmia and MIT-BIH Supraventricular Arrhythmia databases were chosen. Figure 7 shows stages from dataset to get model to classify heartbeats. Dataset to model stages The records are described in Table 1. Table 1 Used records Features were obtained using Matlab software and stored in csv file with known columns types; some columns are integer type and others are double types, columns types is needed when reading csv file in Spark–Scala. Processing was implemented using Spark–Scala firmware; most of papers used Matlab software for classification, Matlab software is very helpful tool in classification problems but when size of dataset is too large, other techniques are preferred. On other hand, machine learning algorithms are not implemented easily in Matlab. This case can be summarized as follows: Dataset size is too large. Need to implement algorithm like: Decision Tree, Random Forests, Gradient-Boosted Trees. Processing speed. So, using big data tools would be helpful in this case. There are a lot of big data tools. For our case, Spark-shell and Scala were used in local host PC (Fig. 8: Local host PC Spark). Local host PC Spark Figure 9 shows stages frame sampling data to get final model Classification model CSV file can be read by Spark–Scala easily, we just built a schema for its columns. The schema defines type of each column in the csv file as shown in Table 2. Table 2 Dataset schema All columns are double values except the last column; it is label column and its type is integer. Figure 10 shows sample of dataset, it contains 16 columns as features and the column 17 contains beat type. Sample of dataset Every generic model of machine learning consists of some components independent of the algorithm adopted [19]. In our case, they are: Sampling dataset: divide dataset into two groups, one for training the model and the other for testing the model. Random sampling is used Pre-processing: all needed operations to get the data ready for classification model and it depends on dataset structure: Fill null values: null values in columns might yield to mis-classification, so replacing all null values in columns with other value is very important in classification mode. Null values might replaced with static value (such as 0 value) or with values like mean value of all column or max value of column values. In our case, null values are filled with 0 value. Process column 17 (heartbeat type): labeling this column with two classes (Normal and Abnormal) or multi classes (Normal and specified types of irregular heartbeat types). Over fitting handling: it means when training data has many rows with type 1 and few rows with type 2. In our case more than 10000 rows have type "Normal" , while type "Abnormal" are less than that. Mapping dataset with fractions is done for fitting data. Separate columns according to their type (Integer, Double) Using String-indexer for labeling beat type. Features Selection: Select columns from data columns as features. Using algorithm GBT: Gradient-Boosted Trees with parameters MaxDepth and MaxIter, Or RF: Random Forest with parameters MaxDepth and NumTrees. Training model: after this step, a trained model is generated and is ready for testing on testing data. Evaluation trained model: evaluators are needed to calculate accuracy for each trained model, each algorithm has its own evaluator. In this paper, Gradient-Boosted Trees model (GBT) and Random Forest model (RF) are implemented and tested. Gradient-Boosted Trees model Gradient-Boosted (GDB) Tree is a machine learning technique for regression and classification issues, which produces a prediction model in the form of an ensemble of weak prediction models. The idea of gradient boosting originated in the observation that boosting can be interpreted as an optimization algorithm on a suitable cost function [20]. The built model basically depends on two parameters of gradient boosted tree; these two parameters are most important parameters of GBT. The GBT model is in Table 3. Table 3 GBT tree GBT trained model was built according to many values of Max iteration and Max Depth; values were changed manually as in Table 4. Table 4 values of iteration and depth Random Forest model (RF model) Random forests or random decision forests are an ensemble learning method for classification, regression and other tasks that operates by constructing a multitude of decision trees at training time and outputting the class that is the mode of the classes (classification) or mean prediction (regression) of the individual trees [16, 21]. The built model depended basically on two parameters of random forest; these two parameters are most important parameters of RF. The RF model is in Table 5. Table 5 RF model RF trained model was built according to many values of Number of Trees and Max Depth; values were changed manually as in Table 6. Table 6 RF models The dataset contains 205,146 rows, they were randomly split into two parts: training and testing. After that, the built model was tested validated on different dataset (32,168 rows). To validate this work, accuracy of model was calculated using binary evaluator BinaryClassificationEvaluator for GBT model and MulticlassClassificationEvaluator for RF model. In addition, Sensitivity and specificity were calculated based on Eq. (3). Where [22, 23]: SE (Sensitivity): The sensitivity of a clinical test refers to the ability of the test to correctly identify those patients with the disease. SP (Specificity): The specificity of a clinical test refers to the ability of the test to correctly identify those patients without the disease. To calculate SE and SP, these terms should be defined as follows: TP True positive: the patient has the disease and the test is positive. FP False positive: the patient does not have the disease but the test is positive. TN True negative: the patient does not have the disease and the test is negative. FN False negative: the patient has the disease but the test is negative. And equations of SE and SP: $$\begin{aligned} SE= \frac{TP }{TP+FN} * 100 \quad SP= \frac{TN }{TN+FP} 100 \end{aligned}$$ CC: Correct Classification is computed as below: $$\begin{aligned} CC=\frac{TP+TN }{TP+TN+FP+FN}100 \end{aligned}$$ Tables 7 and 8 summarized results of GBT and RF algorithms. Table 7 GBT results Table 8 RF results Training process in random forest is faster than decision tree, while testing process in decision tree is faster than in random forest. Parameters of both algorithms were changed manually. The optimal values for tuned parameters can be obtained by running methods with cross validation, but they need too much time. For production system, cross validation can be used and the resulted optimal values can be used instead. Tables above show that built models of both algorithms are capable to predict types of heartbeats with accuracy 96.75% and 97.98 for GBT model and RF model respectively. 4-Classes classification After two classes classification, multi classes classification was validated using RF Algorithm. RF Algorithm supports multi classes classification, while GBT supports only binary classification. Originally, the dataset has a column named label, it has many different integer values such as 1, 5, 9, 2 and others. Each value labels a class such as 1 labels Normal beat and all other values labels Abnormal beat (5 labels PVC and 9 labels PAC).In binary classification, this column is handled to be just two classes in the pre-processing stage (Normal and Abnormal). In multi classification, this column is handled to be 4 classes in the pre-processing stage ( Normal, PVC, PAC and Other) as explained in Table 9. Table 9 4 classes classification Table 10 shows results using random forest algorithm: Table 10 RF results for multi classification Table above shows that built model for multi classification is able to predict multi types of heartbeats with accuracy 98.03%, this contribution is very useful; it predicts 4 classes of heartbeats at once. Max accuracy of binary classification in our case was 97%, it is better the results in [11, 12]. In [11], the accuracy of classification was 96.97% and 96% in [12]. Max accuracy of multi classification in our case was 98.03%, it is better comparing to [7], where multiply types of heartbeats have been studied and classified using convolutional neural network and the accuracy of classification was 93.4%. Conclusion and future scope In summary, This work has validated an ability to classify heartbeats. Classification process is using some features of heartbeats and machine learning classification algorithms with local host pc working using only one node, which are crucial for diagnosis of cardiac arrhythmia. The developed GBT and RF models can classify different ECG heartbeat types and thus, can be implemented into a CAD ECG system to perform a quick and reliable diagnosis. The proposed model has the potential to be introduced into clinical settings as a helpful tool to aid the cardiologists in the reading of ECG heartbeat signals and to understand more about them. The occurrence, sequential patterns and persistence of the classes of ECG heartbeats considered in this work can be grouped under three main categories which represents normal, PVC, PAC, and other. As a future work, implemented methods can be rebuilt to work with many classes (Ex: more than 5 types of heartbeats), the work can be developed to be used in real time and be trained continuously to enhance it and increase its accuracy. Moreover, the whole process of classification can be used with other types of datasets such as stress and clinical datasets. The data set is available to public and can be found in https://physionet.org/physiobank/, it is simple to extract the data in many formats using physionet ATM Bank https://physionet.org/cgi-bin/atm/ATM. ECG: NN: PCA: Discrete Wavelet Transform GBT: Gradient-Boosted Trees RF: Li H, Yuan D, Ma X, Cui D, Cao L. Genetic algorithm for the optimization of features and neural networks in ECG signals classification. Sci Rep. 2017;7:41011. Kachuee M, Fazeli S, Sarrafzadeh M. ECG heartbeat classification: a deep transferable representation. In: 2018 IEEE international conference on healthcare informatics (ICHI). New York: IEEE; 2018. p. 443–4. Zhao Z, Yang L, Chen D, Luo Y. A human ECG identification system based on ensemble empirical mode decomposition. Sensors. 2013;13(5):6832–64. Valenza G, Citi L, Lanatá A, Scilingo EP, Barbieri R. Revealing real-time emotional responses: a personalized assessment based on heartbeat dynamics. Sci Rep. 2014;4:4998. Valenza G, Greco A, Citi L, Bianchi M, Barbieri R, Scilingo E. Inhomogeneous point-processes to instantaneously assess affective haptic perception through heartbeat dynamics information. Sci Rep. 2016;6:28567. Christov I, Jekova I, Bortolan G. Premature ventricular contraction classification by the kth nearest-neighbours rule. Physiol Meas. 2005;26(1):123. Sannino G, De Pietro G. A deep learning approach for ECG-based heartbeat classification for arrhythmia detection. Fut Gener Comput Syst. 2018;86:446–55. Celesti F, Celesti A, Carnevale L, Galletta A, Campo S, Romano A, Bramanti P, Villari M. Big data analytics in genomics: the point on deep learning solutions. In: 2017 IEEE symposium on computers and communications (ISCC). New York: IEEE; 2017. p. 306–9. Ahmad AK, Jafar A, Aljoumaa K. Customer churn prediction in telecom using machine learning in big data platform. J Big Data. 2019;6(1):28. Ma'sum M.A, Jatmiko W, Suhartanto H. Enhanced tele ECG system using hadoop framework to deal with big data processing. In: 2016 international workshop on Big Data and information security (IWBIS). New York: IEEE; 2016. p. 121–6. Dalvi RdF, Zago GT, Andreão RV. Heartbeat classification system based on neural networks and dimensionality reduction. Res Biomed Eng. 2016;32(4):318–26. Ince T, Kiranyaz S, Gabbouj M. A generic and robust system for automated patient-specific classification of ECG signals. IEEE Trans Biomed Eng. 2009;56(5):1415–26. O'Brien J. Using hidden Markov models and spark to mine ECG data. PhysioBank ATM. 2019. https://physionet.org/cgi-bin/atm/ATM. Accessed 21 June 2019. Moody GB, Mark RG. The impact of the mit-bih arrhythmia database. IEEE Eng Med Biol Mag. 2001;20(3):45–50. Ho TK. Random decision forests. In: Proceedings of 3rd international conference on document analysis and recognition, vol. 1. New York: IEEE; 1995. p. 278–82. Gharaviri A, Dehghan F, Teshnelab M, Moghaddam HA. Comparison of neural network, ANFIS, and SVM classifiers for PVC arrhythmia detection. In: 2008 international conference on machine learning and cybernetics, vol. 2. New York: IEEE; 2008. p. 750–5. PhysioBank ATM Dataset. 2019. https://physionet.org/physiobank/. Accessed 21 June 2019. Alzubi J, Nayyar A, Kumar A. Machine learning from theory to algorithms: an overview. In: Journal of physics: conference series, vol. 1142. Bristol: IOP Publishing; 2018. p. 012012. Friedman JH. Stochastic gradient boosting. Comput Stat Data Anal. 2002;38(4):367–78. Barandiaran I. The random subspace method for constructing decision forests. IEEE Trans Pattern Anal Mach Intell. 1998;20(8):122. Parikh R, Mathai A, Parikh S, Sekhar GC, Thomas R. Understanding and using sensitivity, specificity and predictive values. Indian J Ophthalmol. 2008;56(1):45. Lalkhen AG, McCluskey A. Clinical tests: sensitivity and specificity. Continuing Educ Anaesth Crit Care Pain. 2008;8(6):221–3. Thanks for Mr. Ammar Asaad, and Mr. Ahmad Ali for their co-operation and help. The authors declare that they have no funding. Informatics and Decision Supporting Systems, Higher Institute for Applied Sciences and Technology, Damascus, Syria Fajr Ibrahem Alarsan Faculty of Computer and Automation Engineering, Damascus University, Damascus, Syria Mamoon Younes Search for Fajr Ibrahem Alarsan in: Search for Mamoon Younes in: FIA took the role of performing the literature review, finding best model for heartbeat classification. MY took on a supervisory role and oversaw the completion of the work, Proposed data set for implementation and validation. Both authors read and approved the final manuscript. Correspondence to Fajr Ibrahem Alarsan. The authors declare that this study contain real data for real patients obtained from famous dataset on internet and this data is available for all. The database is free to use by all users. Alarsan, F.I., Younes, M. Analysis and classification of heart diseases using heartbeat features and machine learning algorithms. J Big Data 6, 81 (2019) doi:10.1186/s40537-019-0244-x DOI: https://doi.org/10.1186/s40537-019-0244-x Heartbeats classification Machine-learning libraries (MLlib) Spark–Scala	CommonCrawl
eMathZone From basic to higher mathematics Everyday Math Higher Mathematics General Topology Math Results And Formulas The Area Between a Curve and the X-axis Let us consider an example to illustrate the application of the definite integral to find the area function $$A\left( x \right)$$ of a shaded region under a curve $$y = {x^2}$$ as shown in the given diagram. A small change $$\delta x$$ in $$x$$ corresponds to a change in $$A$$ is $$\delta A$$, as shown in the given diagram. It is clear from the diagram that \[\begin{gathered} {\text{area}}\,{\text{of}}\,PQML < \delta A < {\text{area}}\,{\text{of}}\,RSML \\ \Rightarrow PL \times LM < \delta A < SM \times LM\,\,\,\,{\text{ – – – }}\left( {\text{i}} \right) \\ \end{gathered} \] Since $$PL = y$$, $$LM = \delta x$$, $$SM = y + \delta y$$ as shown in the figure, so equation (i) takes the form \[\begin{gathered} y\delta x < \delta A < \left( {y + \delta y} \right)\delta x \\ \Rightarrow y < \frac{{\delta A}}{{\delta x}} < y + \delta y \\ \end{gathered} \] Since $$\delta y \to 0$$ as $$\delta x \to 0$$, so taking the limit, we have \[\begin{gathered} \mathop {\lim }\limits_{\delta x \to 0} y < \mathop {\lim }\limits_{\delta x \to 0} \frac{{\delta A}}{{\delta x}} < \mathop {\lim }\limits_{\delta x \to 0} \left( {y + \delta y} \right) \\ \Rightarrow y < \frac{{dA}}{{dx}} < y + 0 \\ \Rightarrow y < \frac{{dA}}{{dx}} < y \\ \Rightarrow \frac{{dA}}{{dx}} = y \\ \Rightarrow \frac{{dA}}{{dx}} = {x^2} \\ \end{gathered} \] Since this shows that for the curve $$y = {x^2}$$, the derivative of the corresponding area function $$A$$ is \[\frac{{dA}}{{dx}} = {x^2}\,\,\,\,{\text{ – – – }}\left( {{\text{ii}}} \right)\] The definition of a definite integral is \[\frac{d}{{dx}}\left( {F\left( x \right)} \right) = f\left( x \right) \Rightarrow F\left( b \right) – F\left( a \right) = \int\limits_a^b {f\left( x \right)dx} \,\,\,\,{\text{ – – – }}\left( {{\text{iii}}} \right)\] Using this definition, we have \[\frac{d}{{dx}}\left( {A\left( x \right)} \right) = {x^2} \Rightarrow A\left( b \right) – A\left( a \right) = \int\limits_a^b {{x^2}dx} \,\,\,\,{\text{ – – – }}\left( {{\text{iv}}} \right)\] Equation (iv) shows that the area under the curve $$y = {x^2}$$ and between $$x = a$$ and $$x = b$$ is as shown in the diagram \[A = \int\limits_a^b {{x^2}dx} \] ⇐ Example of an Area Under a Curve ⇒ The Area Bounded by the Curve y=x^3-x and the x-axis ⇒ © emathzone.com - All rights reserved	CommonCrawl
molecular-clouds Zucker et al. (2018) – Mapping Distances across the Perseus Molecular Cloud Using CO Observations, Stellar Photometry, and Gaia DR2 Parallax Measurements We present a new technique to determine distances to major star-forming regions across the Perseus Molecular Cloud, using a combination of stellar photometry, astrometric data, and 12CO spectral-line maps. Incorporating Tags distances, extinction, molecular-clouds, Perseus Tahani et al. (2018) – Helical magnetic fields in molecular clouds?. A new method to determine the line-of-sight magnetic field structure in molecular clouds Context. Magnetic fields pervade in the interstellar medium (ISM) and are believed to be important in the process of star formation, yet probing magnetic fields in star formation regions is Tags Faraday, magnetic-field, molecular-clouds, Orion, Zeeman Soler (2019) – Using Herschel and Planck observations to delineate the role of magnetic fields in molecular cloud structure We present a study of the relative orientation between the magnetic field projected onto the plane of sky (B⊥) on scales down to 0.4 pc, inferred from the polarized thermal Tags Herschel, magnetic-field, molecular-clouds, Planck Zamora-Avilés et al. (2019) – Structure and Expansion Law of HII Regions in structured Molecular Clouds We present radiation-magnetohydrodynamic simulations aimed at studying evolutionary properties of HII regions in turbulent, magnetised, and collapsing molecular clouds formed by converging flows in the warm neutral medium. We focus Tags feedback, HII-region, molecular-clouds Corbelli, Braine & Giovanardi (2019) – Rise and fall of molecular clouds across the M33 disk We carried out deep searches for CO line emission in the outer disk of M33, at R>7 kpc, and examined the dynamical conditions that can explain variations in the mass Tags CO, galaxies, general-ISM, kinematics, molecular-clouds, star-formation Zucker et al. (2019) – A Large Catalog of Accurate Distances to Local Molecular Clouds: The Gaia DR2 Edition We present a uniform catalog of accurate distances to local molecular clouds informed by the Gaia DR2 data release. Our methodology builds on that of Schlafly et al. (2014). First, Tags 3D, distances, dust, extinction, Gaia, interstellar, molecular-clouds Körtgen, Federrath & Banerjee (2019) – On the shape and completeness of the column density probability distribution function of molecular clouds Both observational and theoretical research over the past decade has demonstrated that the probability distribution function (PDF) of the gas density in turbulent molecular clouds is a key ingredient for Tags interstellar, molecular-clouds, statistical-analysis, turbulence Li & Klein (2019) – Magnetized interstellar molecular clouds: II. The Large-Scale Structure and Dynamics of Filamentary Molecular Clouds Ideal MHD high resolution AMR simulations with driven turbulence and self-gravity have been performed that demonstrate the formation of long filamentary molecular clouds at the converging location of large-scale turbulence Tags dynamics, filament, magnetic-field, MHD, molecular-clouds, structure, turbulence Ortiz-León et al. (2018) – Gaia-DR2 confirms VLBA parallaxes in Ophiuchus, Serpens and Aquila Abstract We present Gaia-DR2 astrometry of a sample of YSO candidates in Ophiuchus, Serpens Main and Serpens South/W40 in the Aquila Rift, which had been mainly identified by their infrared Tags distances, Gaia, molecular-clouds Utomo, Blitz & Falgarone (2018) – The Origin of Interstellar Turbulence in M33 We utilize the multi-wavelength data of M33 to study the origin of turbulence in its interstellar medium. We find that the HI turbulent energy surface density inside 8 kpc is Tags 2MASS, CO, GALEX, HI, M33, molecular-clouds Vázquez-Semadeni et al. (2018) – Molecular cloud evolution – VI. Measuring cloud ages In previous contributions, we have presented an analytical model describing the evolution and star formation rate (SFR) of molecular clouds (MCs) undergoing hierarchical gravitational contraction. The cloud's evolution is characterized Tags Cloud Evolution, Comparison of Models with Observations, molecular-clouds, star-formation Forgan & Bonnell (2018) – Clumpy shocks as the driver of velocity dispersion in molecular clouds: the effects of self-gravity and magnetic fields We revisit an alternate explanation for the turbulent nature of molecular clouds – namely, that velocity dispersions matching classical predictions of driven turbulence can be generated by the passage of Tags gravitational-instability, Larson, molecular-clouds, shock, spiral-arms, turbulence Clark et al. (2018) – Tracing the formation of molecular clouds via [CII], [CI] and CO emission Our understanding of how molecular clouds form in the interstellar medium (ISM) would be greatly helped if we had a reliable observational tracer of the gas flows responsible for forming Tags CII, CO, molecular-clouds, synthetic-observations Bertram et al. (2015) – Centroid velocity statistics of molecular clouds We compute structure functions and Fourier spectra of 2D centroid velocity maps in order to study the gas dynamics of typical molecular clouds in numerical simulations. We account for a Tags centroid-velocity, intermittency, molecular-clouds, numerical-simulation, structure-function, turbulence Henshaw et al. (2016) – Molecular gas kinematics within the central 250 pc of the Milky Way Using spectral line observations of HNCO, N2H+, and HNC, we investigate the kinematics of dense gas in the central ˜250 pc of the Galaxy. We present SCOUSE (Semi-automated multi-COmponent Universal Tags CMZ, CO, data-science, hyperspectral, kinematics, molecular-clouds, turbulence Iwasaki et al. (2018) – The Early Stage of Molecular Cloud Formation by Compression of Two-phase Atomic Gases We investigate the formation of molecular clouds from atomic gas by using three-dimensional magnetohydrodynamical simulations including chemical reactions and heating/cooling processes. We consider super-Alfv\'enic head-on colliding flows of atomic gas Tags cooling, heating, magnetic-field, MHD, molecular-clouds, thermal-instability, turbulence Geen et al. (2018) – The (Un)predictability of Star Formation on a Cloud Scale Molecular clouds are turbulent structures whose star formation efficiency (SFE) is strongly affected by internal stellar feedback processes. In this paper we determine how sensitive the SFE of molecular clouds Tags HII-region, molecular-clouds, numerical-simulation, star-formation Punanova et al. (2018) – Kinematics of dense gas in the L1495 filament We study the kinematics of the dense gas of starless and protostellar cores traced by the N2D+(2-1), N2H+(1-0), DCO+(2-1), and H13CO+(1-0) transitions along the L1495 filament and the kinematic links Tags CO, filament, kinematics, molecular-clouds Grossschedl et al. (2018) – 3D shape of Orion A from Gaia DR2 We use the $\mathit{Gaia}$ DR2 distances of about 700 mid-infrared selected young stellar objects in the benchmark giant molecular cloud Orion A to infer its 3D shape and orientation. We Tags filament, Gaia, molecular-clouds, Orion, structure Mattern et al. (2018) – SEDIGISM: The kinematics of ATLASGAL filaments Analysing the kinematics of filamentary molecular clouds is a crucial step towards understanding their role in the star formation process. Therefore, we study the kinematics of 283 filament candidates in Tags CO, filament, Milky-Way, molecular-clouds Körtgen et al. (2018) – The Origin of Filamentary Star Forming Clouds in Magnetised Galaxies Observations show that galaxies and their interstellar media are pervaded by strong magnetic fields with energies in the diffuse component being at least comparable to the thermal and even as Tags filament, galaxies, interstellar, magnetic-field, molecular-clouds, star-formation Zucker, Battersby & Goodman (2018) – The Physical Properties of Large-Scale Galactic Filaments The characterization of our Galaxy's longest filamentary gas features has been the subject of several studies in recent years, producing not only a sizeable sample of large-scale filaments, but also Tags dust, filament, hyperspectral, kinematics, molecular-clouds Palouš & Ehlerová (2017) – Gould's Belt: Local Large-Scale Structure in the Milky Way Gould's Belt is a flat local system composed of young massive stars of spectral types O or early-type B (OB stars), molecular clouds, and neutral hydrogen within 500 pc of Tags local-bubble, Milky-Way, molecular-clouds, solar-neighborhood	CommonCrawl
Janko group J2 In the area of modern algebra known as group theory, the Janko group J2 or the Hall-Janko group HJ is a sporadic simple group of order 27 · 33 · 52 · 7 = 604800 ≈ 6×105. For general background and history of the Janko sporadic groups, see Janko group. Algebraic structure → Group theory Group theory Basic notions • Subgroup • Normal subgroup • Quotient group • (Semi-)direct product Group homomorphisms • kernel • image • direct sum • wreath product • simple • finite • infinite • continuous • multiplicative • additive • cyclic • abelian • dihedral • nilpotent • solvable • action • Glossary of group theory • List of group theory topics Finite groups • Cyclic group Zn • Symmetric group Sn • Alternating group An • Dihedral group Dn • Quaternion group Q • Cauchy's theorem • Lagrange's theorem • Sylow theorems • Hall's theorem • p-group • Elementary abelian group • Frobenius group • Schur multiplier Classification of finite simple groups • cyclic • alternating • Lie type • sporadic • Discrete groups • Lattices • Integers ($\mathbb {Z} $) • Free group Modular groups • PSL(2, $\mathbb {Z} $) • SL(2, $\mathbb {Z} $) • Arithmetic group • Lattice • Hyperbolic group Topological and Lie groups • Solenoid • Circle • General linear GL(n) • Special linear SL(n) • Orthogonal O(n) • Euclidean E(n) • Special orthogonal SO(n) • Unitary U(n) • Special unitary SU(n) • Symplectic Sp(n) • G2 • F4 • E6 • E7 • E8 • Lorentz • Poincaré • Conformal • Diffeomorphism • Loop Infinite dimensional Lie group • O(∞) • SU(∞) • Sp(∞) Algebraic groups • Linear algebraic group • Reductive group • Abelian variety • Elliptic curve History and properties J2 is one of the 26 Sporadic groups and is also called Hall–Janko–Wales group. In 1969 Zvonimir Janko predicted J2 as one of two new simple groups having 21+4:A5 as a centralizer of an involution (the other is the Janko group J3). It was constructed by Marshall Hall and David Wales (1968) as a rank 3 permutation group on 100 points. Both the Schur multiplier and the outer automorphism group have order 2. As a permutation group on 100 points J2 has involutions moving all 100 points and involutions moving just 80 points. The former involutions are products of 25 double transportions, an odd number, and hence lift to 4-elements in the double cover 2.A100. The double cover 2.J2 occurs as a subgroup of the Conway group Co0. J2 is the only one of the 4 Janko groups that is a subquotient of the monster group; it is thus part of what Robert Griess calls the Happy Family. Since it is also found in the Conway group Co1, it is therefore part of the second generation of the Happy Family. Representations It is a subgroup of index two of the group of automorphisms of the Hall–Janko graph, leading to a permutation representation of degree 100. It is also a subgroup of index two of the group of automorphisms of the Hall–Janko Near Octagon,[1] leading to a permutation representation of degree 315. It has a modular representation of dimension six over the field of four elements; if in characteristic two we have w2 + w + 1 = 0, then J2 is generated by the two matrices ${\mathbf {A} }={\begin{pmatrix}w^{2}&w^{2}&0&0&0&0\\1&w^{2}&0&0&0&0\\1&1&w^{2}&w^{2}&0&0\\w&1&1&w^{2}&0&0\\0&w^{2}&w^{2}&w^{2}&0&w\\w^{2}&1&w^{2}&0&w^{2}&0\end{pmatrix}}$ and ${\mathbf {B} }={\begin{pmatrix}w&1&w^{2}&1&w^{2}&w^{2}\\w&1&w&1&1&w\\w&w&w^{2}&w^{2}&1&0\\0&0&0&0&1&1\\w^{2}&1&w^{2}&w^{2}&w&w^{2}\\w^{2}&1&w^{2}&w&w^{2}&w\end{pmatrix}}.$ These matrices satisfy the equations ${\mathbf {A} }^{2}={\mathbf {B} }^{3}=({\mathbf {A} }{\mathbf {B} })^{7}=({\mathbf {A} }{\mathbf {B} }{\mathbf {A} }{\mathbf {B} }{\mathbf {B} })^{12}=1.$ (Note that matrix multiplication on a finite field of order 4 is defined slightly differently from ordinary matrix multiplication. See Finite field § Field with four elements for the specific addition and multiplication tables, with w the same as a and w2 the same as 1 + a.) J2 is thus a Hurwitz group, a finite homomorphic image of the (2,3,7) triangle group. The matrix representation given above constitutes an embedding into Dickson's group G2(4). There is only one conjugacy class of J2 in G2(4). Every subgroup J2 contained in G2(4) extends to a subgroup J2:2 = Aut(J2) in G2(4):2 = Aut(G2(4)) (G2(4) extended by the field automorphisms of F4). G2(4) is in turn isomorphic to a subgroup of the Conway group Co1. Maximal subgroups There are 9 conjugacy classes of maximal subgroups of J2. Some are here described in terms of action on the Hall–Janko graph. • U3(3) order 6048 – one-point stabilizer, with orbits of 36 and 63 Simple, containing 36 simple subgroups of order 168 and 63 involutions, all conjugate, each moving 80 points. A given involution is found in 12 168-subgroups, thus fixes them under conjugacy. Its centralizer has structure 4.S4, which contains 6 additional involutions. • 3.PGL(2,9) order 2160 – has a subquotient A6 • 21+4:A5 order 1920 – centralizer of involution moving 80 points • 22+4:(3 × S3) order 1152 • A4 × A5 order 720 Containing 22 × A5 (order 240), centralizer of 3 involutions each moving 100 points • A5 × D10 order 600 • PGL(2,7) order 336 • 52:D12 order 300 • A5 order 60 Conjugacy classes The maximum order of any element is 15. As permutations, elements act on the 100 vertices of the Hall–Janko graph. OrderNo. elementsCycle structure and conjugacy 1 = 11 = 11 class 2 = 2315 = 32 · 5 · 7240, 1 class 2520 = 23 · 32 · 5 · 7250, 1 class 3 = 3560 = 24 · 5 · 7330, 1 class 16800 = 25 · 3 · 52 · 7332, 1 class 4 = 226300 = 22 · 32 · 52 · 726420, 1 class 5 = 54032 = 26 · 32 · 7520, 2 classes, power equivalent 24192 = 27 · 33 · 7520, 2 classes, power equivalent 6 = 2 · 325200 = 24 · 32 · 52 · 72436612, 1 class 50400 = 25 · 32 · 52 · 722616, 1 class 7 = 786400 = 27 · 33 · 52714, 1 class 8 = 2375600 = 24 · 33 · 52 · 72343810, 1 class 10 = 2 · 560480 = 26 · 33 · 5 · 71010, 2 classes, power equivalent 120960 = 27 · 33 · 5 · 754108, 2 classes, power equivalent 12 = 22 · 350400 = 25 · 32 · 52 · 7324262126, 1 class 15 = 3 · 580640 = 28 · 32 · 5 · 752156, 2 classes, power equivalent References 1. "The near octagon on 315 points". • Robert L. Griess, Jr., "Twelve Sporadic Groups", Springer-Verlag, 1998. • Hall, Marshall; Wales, David (1968), "The simple group of order 604,800", Journal of Algebra, 9 (4): 417–450, doi:10.1016/0021-8693(68)90014-8, ISSN 0021-8693, MR 0240192 (Griess relates [p. 123] how Marshall Hall, as editor of The Journal of Algebra, received a very short paper entitled "A simple group of order 604801." Yes, 604801 is prime.) • Janko, Zvonimir (1969), "Some new simple groups of finite order. I", Symposia Mathematica (INDAM, Rome, 1967/68), Vol. 1, Boston, MA: Academic Press, pp. 25–64, MR 0244371 • Wales, David B., "The uniqueness of the simple group of order 604800 as a subgroup of SL(6,4)", Journal of Algebra 11 (1969), 455–460. • Wales, David B., "Generators of the Hall–Janko group as a subgroup of G2(4)", Journal of Algebra 13 (1969), 513–516, doi:10.1016/0021-8693(69)90113-6, MR0251133, ISSN 0021-8693 External links • MathWorld: Janko Groups • Atlas of Finite Group Representations: J2 • The subgroup lattice of J2 Groups Basic notions • Subgroup • Normal subgroup • Commutator subgroup • Quotient group • Group homomorphism • (Semi-) direct product • direct sum Types of groups • Finite groups • Abelian groups • Cyclic groups • Infinite group • Simple groups • Solvable groups • Symmetry group • Space group • Point group • Wallpaper group • Trivial group Discrete groups Classification of finite simple groups Cyclic group Zn Alternating group An Sporadic groups Mathieu group M11..12,M22..24 Conway group Co1..3 Janko groups J1, J2, J3, J4 Fischer group F22..24 Baby monster group B Monster group M Other finite groups Symmetric group Sn Dihedral group Dn Rubik's Cube group Lie groups • General linear group GL(n) • Special linear group SL(n) • Orthogonal group O(n) • Special orthogonal group SO(n) • Unitary group U(n) • Special unitary group SU(n) • Symplectic group Sp(n) Exceptional Lie groups G2 F4 E6 E7 E8 • Circle group • Lorentz group • Poincaré group • Quaternion group Infinite dimensional groups • Conformal group • Diffeomorphism group • Loop group • Quantum group • O(∞) • SU(∞) • Sp(∞) • History • Applications • Abstract algebra	Wikipedia
\begin{document} \sffamily \sffamily \title{Complex flows, escape to infinity\\ and a question of Rubel } \author{J.K. Langley} \maketitle \begin{abstract} Let $f$ be a transcendental entire function. It was shown in a previous paper \cite{Latraj} that the holomorphic flow $\dot z = f(z)$ always has infinitely many trajectories tending to infinity in finite time. It will be proved here that such trajectories are in a certain sense rare, although an example will be given to show that there can be uncountably many. In contrast, for the classical antiholomorphic flow $\dot z = \bar f(z)$, such trajectories need not exist at all, although they must if $f$ belongs to the Eremenko-Lyubich class $\mathcal{B}$. It is also shown that for transcendental entire $f$ in $\mathcal{B}$ there exists a path tending to infinity on which $f$ and all its derivatives tend to infinity, thus affirming a conjecture of Rubel for this class. \\ MSC 2000: 30D35. \end{abstract} \section{Introduction} The starting point of this note is the flow \begin{equation} \label{H1} \dot z = f(z), \end{equation} in which $f $ or its conjugate $\bar f$ is an entire function. A trajectory for (\ref{H1}) is a path $z(t)$ in the plane with $z'(t) = f(z(t)) \in \mathbb C$ for $t$ in some maximal interval $(\alpha, \beta) \subseteq \mathbb R$. By the existence-uniqueness theorem, such trajectories are either constant (with $z(t)$ a zero of $f$), periodic or injective. It was shown in \cite[Theorem 5]{kingneedham} that if $f$ is a polynomial in $z$ of degree $n \geq 2$ then there exist $n-1$ disjoint trajectories for (\ref{H1}) which tend to infinity in finite increasing time, that is, which satisfy $\beta \in \mathbb R$ and $\lim_{t \to \beta - } z(t) = \infty$.The following theorem for holomorphic flows with transcendental entire $f$ was proved in \cite[Theorem 1.1]{Latraj}. \begin{thm}[\cite{Latraj}] \label{thm0} Let the function $f$ be transcendental entire: then (\ref{H1}) has infinitely many pairwise disjoint trajectories which tend to infinity in finite increasing time. \end{thm} For meromorphic functions in general, such trajectories need not exist at all \cite{Latraj}, but a result was also proved in \cite{Latraj} for the case where $f$ is transcendental and meromorphic in the plane and the inverse function $f^{-1}$ has a logarithmic singularity over $\infty$: this means that there exist $M > 0$ and a component $U$ of the set $\{ z \in \mathbb C : \, \|f(z)\| > M \}$ such that $U$ contains no poles of $f$ and $\log f$ maps $U$ conformally onto the half-plane $H = \{ v \in \mathbb C : \, {\rm Re } \, v > \log M \}$ \cite{BE,Nev}. In this case \cite[Theorem 1.2]{Latraj}, (\ref{H1}) has infinitely many pairwise disjoint trajectories tending to infinity in finite increasing time from within a neighbourhood $\{ z \in U : \|f(z)\| > M' \geq M \}$ of the singularity. On the other hand, for entire $f$ in (\ref{H1}), it seems that trajectories which tend to infinity in finite increasing time are somewhat exceptional. For the simple example $\dot z = - \exp( -z)$, it is easy to check that all trajectories satisfy $\exp(z(t)) = \exp(z(0)) -t$ and so tend to infinity as $t$ increases, but take infinite time to do so unless $\exp( z(0))$ is real and positive. It will be shown that for transcendental entire $f$ there is, in a certain sense, zero probability of landing on a trajectory of (\ref{H1}) which tends to infinity in finite time. To state the theorem, let $f$ be transcendental entire and let \begin{equation} \label{Fdef1} z_0 \in \mathbb C, \quad f(z_0) \neq 0, \quad F(z) = \int_{z_0}^z \frac{du}{f(u)} . \end{equation} Then $F(z)$ is defined near $z_0$ and is real and increasing as $z$ follows the trajectory $\zeta_{z_0} (t)$ of (\ref{H1}) starting at $z_0$. Let $\delta $ be small and positive and take the pre-image $ L_\delta(z_0)$ of the real interval $(- \delta, \delta)$ under the function $- i F(z) $; then $ L_\delta(z_0)$ is perpendicular to $\zeta_{z_0} (t)$ at $z_0$. The proof of the following result is adapted from that of the Gross star theorem \cite[p.292]{Nev}. \begin{thm} \label{thmhol} Let $f$ be a transcendental entire function and let $z_0$ and $F$ be as in (\ref{Fdef1}). For small positive $\delta$ let $Y_\delta$ be the set of $y \in (- \delta, \delta)$ such that the trajectory of (\ref{H1}) starting at $F^{-1}(iy)$ tends to infinity in finite increasing time. Then $Y_\delta$ has Lebesgue measure $0$. \end{thm} Theorem \ref{thmhol} seems unlikely to be best possible, but an example from \cite{Volk} (see \S \ref{uncountable}) shows that there exists a transcendental entire $f$ for which (\ref{H1}) has uncountably many trajectories tending to infinity in finite increasing time. It seems natural to ask similar questions in respect of the antiholomorphic flow \begin{equation} \label{AH} \dot z = \frac{dz}{dt} = \bar g(z), \end{equation} where $g$ is a non-constant entire function. Equation (\ref{AH}) appears widely in textbooks as a model for incompressible irrotational plane fluid flow, and is linked to (\ref{H1}) insofar as if $f = 1/g$ then (\ref{AH}) has the same trajectories as (\ref{H1}), since $\bar g = f/\|f\|^2$, although zeros of one of $f$ and $g$ are of course poles of the other and in general the speeds of travel differ. The trajectories of (\ref{AH}) are determined by choosing $G$ with $G'(z) = g(z)$ and writing \begin{equation} \label{transform1} v = G(z), \quad \dot v = g(z) \dot z = \|g(z)\|^2 \geq 0 , \end{equation} which leads to the classical fact that trajectories for (\ref{AH}) are level curves of ${\rm Im} \, G(z)$ on which ${\rm Re} \, G(z)$ increases with $t$. By the maximum principle, ${\rm Im} \, G(z)$ cannot be constant on a closed curve. Thus, apart from the countably many which tend to a zero of $G' = g$, all trajectories for (\ref{AH}) go to infinity, but this leaves open the question as to how long they take to do so. If a non-constant trajectory $\Gamma$ of (\ref{AH}) passes from $z_1 $ to $ z_2$ along an arc meeting no zeros of $g$, then ${\rm Im} \, v = \beta$ is constant on $\Gamma$ and $X = {\rm Re} \, v$ increases from $X_1 = {\rm Re} \, G(z_1) $ to $X_2 = {\rm Re} \, G(z_2)$. Thus (\ref{transform1}) implies that the transit time is \begin{equation} \int_{X_1+i\beta}^{X_2+i \beta } \frac1{\|g(z)\|^2} \, dv = \int_{X_1+i \beta}^{X_2+i \beta} \left\| \frac{dz}{dv} \right\|^2 \, dv = \int_{X_1}^{X_2} \left\| \frac{dz}{dX} \right\|^2 \, dX . \label{transit} \end{equation} This formula shows that a zero of $g$ cannot be reached in finite time, because if $z$ tends to a zero $z_3$ of $g$ of multiplicity $m$ as $X \to X_3$ then, with $c_j$ denoting non-zero constants, \begin{eqnarray} X - X_3 &=& G(z)-G(z_3) \sim c_1 (z-z_3)^{m+1}, \\ \left\| \frac{dz}{dX} \right\|^2 &=& \frac1{ \|g(z)\|^2 } \sim \frac{c_2}{ \|X-X_3\|^{2m/(m+1)} } \geq \frac{ c_2 }{\|X - X_3\|} . \end{eqnarray} Suppose now that $G' = g$ is a polynomial of degree $n \geq 1$ in (\ref{AH}), (\ref{transform1}) and (\ref{transit}). If $S \in \mathbb R$ and $R$ is sufficiently large and positive then each pre-image under $v = G(z)$ of the half-line $v = r + iS, r \geq R,$ gives a trajectory of (\ref{AH}) which tends to infinity, on which (\ref{transform1}) delivers $$\frac{dt}{dv} = \frac1{\|g(z)\|^2} \sim \frac{c_3 }{ \|z\|^{2n}} \sim \frac{c_4}{ \|v\|^{2n/(n+1)}} .$$ Hence (\ref{transit}) implies that the transit time to infinity is finite for $n \geq 2$ and infinite for $n=1$. Thus, if $g$ is a non-linear polynomial, (\ref{AH}) always has uncountably many trajectories tending to infinity in finite increasing time, but this need not be the case for transcendental entire $g$. \begin{thm} \label{thmbbh} There exists a transcendental entire function $g$ such that (\ref{AH}) has no trajectories tending to infinity in finite increasing time. \end{thm} Theorem \ref{thmbbh} also marks a sharp contrast with Theorem~\ref{thm0}, and its proof rests on the following immediate consequence of a result of Barth, Brannan and Hayman \cite[Theorem 2]{BBH}. \begin{thm}[\cite{BBH}] \label{BBHthm} There exists a transcendental entire function $G$ such that any unbounded connected plane set contains a sequence $(w_n)$ tending to infinity on which $U = {\rm Re} \, G$ satisfies $(-1)^n U(w_n) \leq \|w_n\|^{1/2 } $. \end{thm} To establish Theorem \ref{BBHthm}, it is only necessary to take the plane harmonic function $v$ constructed in \cite[Theorem 2]{BBH}, with the choice of $\psi(r)$ given by \cite[p.364]{BBH}. With $U = v$, and $V$ a harmonic conjugate of $U$, elementary considerations show that the resulting entire function $G = U+iV$ cannot be a polynomial. On the other hand, in the presence of a logarithmic singularity of the inverse function over infinity, trajectories of (\ref{AH}) tending to infinity in finite increasing time exist in abundance. \begin{thm} \label{thm2} Let $g$ and $G$ be transcendental meromorphic functions in the plane such that $G'=g$ and either $G^{-1}$ or $g^{-1}$ has a logarithmic singularity over $\infty$. Then in each neighbourhood of the singularity the flow (\ref{AH}) has a family of pairwise disjoint trajectories $\gamma_Y, Y \in \mathbb R$, each of which tends to infinity in finite increasing time. \end{thm} Theorem \ref{thm2} applies in particular if $g$ or its antiderivative $G$ is a transcendental entire function and belongs to the Eremenko-Lyubich class $\mathcal{B}$, which plays a salient role in complex dynamics \cite{Ber4,EL,sixsmithEL} and is defined by the property that $F \in \mathcal{B}$ if the finite critical and asymptotic values of $F$ form a bounded set, from which it follows that if $F \in \mathcal{B}$ is transcendental entire then $F^{-1}$ automatically has a logarithmic singularity over $\infty$. A specific function to which Theorem \ref{thm2} may be applied is $g(z) = e^{-z} + 1$; here $g$ is in $\mathcal{B}$, but its antiderivative $G$ is not, and this example also gives uncountably many trajectories of (\ref{AH}) taking infinite time to reach infinity through the right half-plane. Theorem \ref{thm2} is quite straightforward to prove when the inverse of $G$ has a logarithmic singularity over infinity, but the method turns out to have a bearing on the following question of Rubel \cite[pp.595-6]{Linear}: if $f$ is a transcendental entire function, must there exist a path tending to infinity on which $f$ and its derivative $f'$ both have asymptotic value $\infty$? This problem was motivated by the classical theorem of Iversen \cite{Nev}, which states that $\infty$ is an asymptotic value of every non-constant entire function. For transcendental entire $f$ of finite order, a strongly affirmative answer to Rubel's question was provided by the following result \cite[Theorem 1.5]{Larubel}. \begin{thm}[\cite{Larubel}] \label{rubelthm} Let the function $f$ be transcendental and meromorphic in the plane, of finite order of growth, and with finitely many poles. Then there exists a path $\gamma$ tending to infinity such that, for each non-negative integer $m$ and each positive real number $c$, \begin{equation} \lim_{z \to \infty, z \in \gamma } \frac{ \log \|f^{(m)}(z)\|}{ \log \|z\|} = + \infty \quad \hbox{and} \quad \int_\gamma \|f^{(m)}(z)\|^{-c} \|dz\| < + \infty . \label{rr3} \end{equation} \end{thm} For functions of infinite order, Rubel's question appears to be difficult, although a path satisfying (\ref{rr3}) for $m=0$ is known to exist for any transcendental entire function $f$ \cite{LRW}. However, a direct analogue of Theorem \ref{rubelthm} goes through relatively straightforwardly for transcendental entire functions $f$ in the Eremenko-Lyubich class $\mathcal{B}$. \begin{thm} \label{thm1} Let $f$ be a transcendental meromorphic function in the plane such that $f^{-1}$ has a logarithmic singularity over $\infty$, and let $D \in \mathbb R$. Then there exists a path $\gamma$ tending to infinity in a neighbourhood of the singularity, such that $f(z) -iD$ is real, positive and increasing on $\gamma$ and (\ref{rr3}) holds for each integer $m \geq 0 $ and real $c > 0$. \end{thm} This paper is organised as follows: Theorem \ref{thmhol} is proved in \S\ref{pfthmhol}, followed by an example in \S\ref{uncountable} and the proof of Theorem \ref{thmbbh} in \S\ref{pfthmbbh}. It is then convenient to give the proof of Theorem \ref{thm1} in \S\ref{pfthm1}, prior to that of Theorem \ref{thm2} in \S\ref{pfthm2}. \section{Proof of Theorem \ref{thmhol}}\label{pfthmhol} Let $f$, $F$, $z_0$ and $\delta$ be as in the statement of Theorem \ref{thmhol}. For $y \in (- \delta, \delta)$ let $g(y) = F^{-1}(iy)$ and let $T(y)$ be the supremum of $s > 0$ such that the trajectory $\zeta_{g(y)}(t)$ of (\ref{H1}) with $\zeta_{g(y)}(0) = g(y)$ is defined and injective for $0 \leq t < s$. If the trajectory $\zeta_{g(y)}(t)$ is periodic with minimal period $S_y$ then $T(y) = S_y$ and $\zeta_{g(y')}(t)$ has the same period for $y'$ close to $y$ \cite{brickman}. Furthermore, if $\zeta_{g(y)}(t)$ tends to infinity in finite time then $T(y) < + \infty$, while if $T(y)$ is finite but $\zeta_{g(y)}(t)$ is not periodic then $\lim_{t \uparrow T(y)} \zeta_{g(y)}(t) = \infty$ \cite[Lemma 2.1]{Latraj}. Set $$ A = \{ iy + t: \, \, y \in (- \delta, \delta), \, 0 < t < T(y) \} , \quad B = \{ \zeta_{g(y)} (t) : \, y \in (- \delta, \delta), \, 0 < t < T(y) \}. $$ Then $G( iy + t ) = \zeta_{g(y)} (t) $ is a bijection from $A$ to $B$. For $u = \zeta_{g(y)} (t)$, where $y \in (- \delta, \delta)$ and $0 < t < T(y)$, let $\sigma_u$ be the subarc of $ L_\delta(z_0)$ from $z_0$ to $g(y)$ followed by the sub-trajectory of (\ref{H1}) from $g(y)$ to $u$, and define $F$ by (\ref{Fdef1}) on a simply connected neighbourhood $D_u$ of $\sigma_u$. Then $F$ maps $\sigma_u$ bijectively to the line segment $[0, iy]$ followed by the line segment $[iy, iy+t]$, and taking a sub-domain if necessary makes it possible to assume that $F$ is univalent on $D_u$, with inverse function defined on a neighbourhood of $[iy, iy+t]$. Let $y'$ and $t'$ be real and close to $y$ and $t$ respectively. Then the image under $F^{-1}$ of the line segment $[iy', iy' + t']$ is an injective sub-trajectory of (\ref{H1}) joining $g(y') \in L_\delta(z_0)$ to $F^{-1}(iy' +t') = \zeta_{g(y')} (t') = G(iy'+t')$, and so $T(y') \geq t'$. Thus $y \rightarrow T(y)$ is lower semi-continuous and $A$ is a domain, while $G: A \to B$ is analytic. Moreover, $A$ is simply connected, because its complement in $\mathbb C \cup \{ \infty \}$ is connected, and so is $B$. Furthermore, $F$ extends to be analytic on $B$, by (\ref{Fdef1}) and the fact that $f \neq 0$ on $B$, and $F \circ G$ is the identity on $A$ because $F(G(t)) = t$ for small positive $t$. For $N \in (0, + \infty ) $, let $M_N $ be the set of all $ y$ in $ (- \delta, \delta)$ such that $\zeta_{g(y)} (t)$ tends to infinity and $T(y) < N $. To prove Theorem \ref{thmhol}, it suffices to show that each such $M_N$ has measure $0$, and the subsequent steps will be adapted from the proof of the Gross star theorem \cite[p.292]{Nev} and its extensions due to Kaplan \cite{Kaplan}. Let $\Lambda_N \subseteq B$ be the image of $\Omega_N = \{ w \in A : \, {\rm Re} \, w < N \}$ under $G$, let $r$ be large and positive and denote the circle $\|z\| = r$ by $S(0, r)$. Then $S(0, r) \cap \Lambda_N$ is a union of countably many open arcs $\Sigma_r$. If $y \in M_N$ then $T(y) < N$ and as $t \to T(y)$ the image $z = G(iy+t) $ tends to infinity in $\Lambda_N$ and so crosses $S(0, r)$, and hence there exists $\zeta $ in some $ \Sigma_r$ with ${\rm Im} \, F(\zeta) = y$, since $F: B \to A$ is the inverse of $G$. Thus the measure $\mu_N$ of $M_N$ is at most the total length $s(r)$ of the arcs $F(\Sigma_r)$. It follows from the Cauchy-Schwarz inequality that, as $t \to + \infty$, \begin{eqnarray} \mu_N^2 &\leq& s(t)^2 = \left( \int_{t e^{i \phi } \in \Lambda_N } \|F'(t e^{i \phi } )\| t \, d \phi \, \right)^2 \\ &\leq& \left( \int_{t e^{i \phi } \in \Lambda_N } \|F'(t e^{i \phi } )\|^2 t \, d \phi \, \right) \left( \int_{t e^{i \phi } \in \Lambda_N } t \, d \phi \, \right) \leq 2 \pi t \left( \int_{t e^{i \phi } \in \Lambda_N } \|F'(t e^{i \phi } )\|^2 t \, d \phi \, \right) . \end{eqnarray} Thus $\mu_N = 0$, since dividing by $2 \pi t$ and integrating from $r$ to $r^2$ yields, as $r \to + \infty$, \begin{eqnarray} \frac{ \mu_N^2 \log r }{2 \pi} &\leq& \int_r^{r^2} \int_{t e^{i \phi } \in \Lambda_N } \|F'(t e^{i \phi } )\|^2 \, t \, d \phi \, dt \leq \int_{\Lambda_N} \|F'(t e^{i \phi } )\|^2 \, t \, d \phi \, dt = \hbox{area $(\Omega_N)$} \leq 2 \delta N . \end{eqnarray} $\Box$ \section{An example}\label{uncountable} Suppose that $G$ is a locally univalent meromorphic function in the plane, whose set of asymptotic values is an uncountable subset $E$ of the unit circle $\mathbb T$. Suppose further that there exists a simply connected plane domain $D$, mapped univalently onto the unit disc $\Delta$ by $G$, such that the branch $\phi$ of $G^{-1}$ mapping $\Delta$ to $D$ has no analytic extension to a neighbourhood of any $\beta \in E$. Let $F = S(G)$, where $S$ is a M\"obius transformation mapping $\Delta$ onto $\{ w \in \mathbb C : \, {\rm Re} \, w < 0 \}$, and for $\beta \in E$ let $\alpha = S(\beta)$ and let $L$ be the half-open line segment $[\alpha -1, \alpha)$. Then $M = S^{-1}(L)$ is a line segment or circular arc in $\Delta$ which meets $\mathbb T$ orthogonally at $\beta$. Moreover, $\phi(M)$ is a level curve of ${\rm Im} \, F$ in $D$, which cannot tend to a simple $\beta$-point of $G$ in $\mathbb C$ because this would imply that $\phi$ extends to a neighbourhood of $\beta$. Hence $\phi(M)$ is a path tending to infinity in $D$, on which ${\rm Im} \, F(z)$ is constant and $F(z)$ tends to $\alpha$. Since $G$ and $F$ are locally univalent, $f = 1/F'$ is entire. As $t \to 0-$ write, on $\phi(M)$, $$ F(z) = \alpha + t, \quad \quad \frac{dt}{dz} = F'(z) = \frac1{f(z)}, \quad \frac{dz}{dt} = f(z), $$ so that $\phi(M)$ is a trajectory of (\ref{H1}) which tends to infinity in finite increasing time, and there exists one of these for every $\beta$ in the uncountable set $E$. A suitable $G$ is furnished by a construction of Volkovyskii \cite{Ermich,Volk}, in which $\mathbb T \setminus E$ is a union of disjoint open circular arcs $I_k = (a_k, b_k)$, oriented counter-clockwise. For each $k$, take the multi-sheeted Riemann surface onto which $(a_k - b_k e^z)/(1-e^z)$ maps the plane, cut it along a curve which projects to $I_k$, and glue to $\Delta$ that half which lies to the right as $I_k$ is followed counter-clockwise. This forms a simply connected Riemann surface $R$ with no algebraic branch points. By \cite[Theorem 17, p.71]{Volk} (see also \cite[p.6]{Ermich}), the $I_k$ can be chosen so that $R$ is parabolic and is thereby the image surface of a locally univalent meromorphic function $G$ in the plane. $\Box$ \section{Proof of Theorem \ref{thmbbh}}\label{pfthmbbh} Following the notation of the introduction, suppose that $v=G(z)$ is a transcendental entire function with derivative $g$ in (\ref{AH}), (\ref{transform1}) and (\ref{transit}). \begin{prop} \label{propbbh} Let $\Gamma$ be a level curve tending to infinity on which $Y = {\rm Im} \, G(z) = \beta \in \mathbb R $ and $X = {\rm Re} \, G(z) $ increases, with $X \geq \alpha \in \mathbb R $, and assume that $\Gamma$ meets no zero of $g$. Suppose that $(z_n)$ is a sequence tending to infinity on $\Gamma$ such that $v_n = G(z_n ) = X_n + i \beta $ satisfies $v_n = o( \|z_n\|)^2 $. Then the trajectory of (\ref{AH}) which follows $\Gamma$ takes infinite time in tending to infinity. \end{prop} Here it is not assumed or required that $X \to + \infty$ as $z \to \infty$ on $\Gamma$. \\ \\ \textit{Proof of Proposition \ref{propbbh}.} It may be assumed that $\Gamma$ starts at $z^$ and $G(z^) = \alpha + i \beta$. Denote positive constants, independent of $n$, by $C_j$. Then the Cauchy-Schwarz inequality gives, as $n $ and $z_n$ tend to infinity, \begin{eqnarray} \|z_n\|^2 &\leq& \left( C_1 + \int_\alpha^{X_n} \left\| \frac{dz}{dX} \right\| \, dX \right)^2 \\ &\leq& 2 \left( \int_\alpha^{X_n} \left\| \frac{dz}{dX} \right\| \, dX \right)^2 \\ &\leq& 2 \left( \int_\alpha^{X_n} \, dX \right) \left( \int_\alpha^{X_n} \left\| \frac{dz}{dX} \right\|^2 \, dX \right) \\ &\leq& 2 \left( \|v_n\| + C_2 \right) \left( \int_\alpha^{X_n} \left\| \frac{dz}{dX} \right\|^2 \, dX \right) \\ &\leq& o \left( \|z_n\|^2 \right) \left( \int_\alpha^{X_n} \left\| \frac{dz}{dX} \right\|^2 \, dX \right) . \end{eqnarray} Thus (\ref{transit}) shows that the transit time from $z^$ to $z_n$ tends to infinity with $n$. $\Box$ \textit{Proof of Theorem \ref{thmbbh}.} Let $G$ be the entire function given by Theorem \ref{BBHthm}, and set $g = G'$. As noted in the introduction, no trajectory of (\ref{AH}) can pass through a zero of $g$, and in any case it takes infinite time for a trajectory to approach a zero of $g$. Furthermore, if $\Gamma$ is a level curve, starting at $z^$ say, on which ${\rm Im} \, G(z)$ is constant and $U(z) = {\rm Re} \, G(z)$ increases, and on which $g$ has no zeros, then there exists a sequence $z_n = w_{2n}$ which tends to infinity on $\Gamma$ and satisfies $$U(z^) \leq U(z_n) \leq \|z_n\|^{1/2} , \quad \|G(z_n)\| \leq \|U(z_n)\| + O(1) \leq \|z_n\|^{1/2} + O(1).$$ Hence $\Gamma$ satisfies the hypotheses of Proposition \ref{propbbh}. It now follows that (\ref{AH}) has no trajectories tending to infinity in finite increasing time. Since time can be reversed for these flows by setting $s = -t$ and $dz/ds = - \bar g(z)$, the same example has no trajectories tending to infinity in finite decreasing time either. $\Box$ \section{Proof of Theorem \ref{thm1}}\label{pfthm1} Let $f$ be as in the hypotheses. Then there exist $M > 0$ and a component $U$ of $\{ z \in \mathbb C : \, \|f(z)\| > M \} $ such that $v = \log f(z)$ is a conformal bijection from $U$ to the half-plane $H$ given by ${\rm Re} \, v > N = \log M$; it may be assumed that $0 \not \in U$. Let $\phi: H \to U$ be the inverse function. If $u \in H$ then $\phi$ and $\log \phi$ are univalent on the disc $\|w-u\| < {\rm Re} \, u - N$ and so Bieberbach's theorem and Koebe's quarter theorem \cite[Chapter 1]{Hay9} imply that \begin{equation} \label{h3} \left\| \frac{\phi''(u)}{\phi'(u)} \right\| \leq \frac4{{\rm Re} \, u -N } , \quad \left\| \frac{\phi'(u)}{\phi(u)} \right\| \leq \frac{4 \pi}{{\rm Re} \, u -N } . \end{equation} \begin{lem} \label{lem1} Let $v_0 $ be large and positive and for $0 \leq k \in \mathbb Z$ write \begin{equation} \label{rub1} V_k = \left\{ v_0 + t e^{i \theta} : \, t \geq 0, \, - \, \frac{\pi}{2^{k+2}} \leq \theta \leq \frac{\pi}{2^{k+2}} \right\}, \quad G_k(v) = \frac{f^{(k)}(z)}{f(z)}, \quad z = \phi(v). \end{equation} Then there exist positive constants $d$ and $c_k $ such that $\| \log \phi'(v) \| \leq d \log ( {\rm Re} \, v )$ as $v \to \infty$ in $V_1$ and $\| \log \|G_k(v)\| \| \leq c_k \log ( {\rm Re} \, v )$ as $v \to \infty$ in $V_k$. \end{lem} \textit{Proof.} For $v \in V_1$, parametrise the straight line segment from $v_0$ to $v$ with respect to $s = {\rm Re} \, u$. Then (\ref{h3}) and the simple estimate $\|du\| \leq \sqrt{2} ds $ yield $\| \log \phi'(v) \| = O( \log ( {\rm Re} \, v ))$ as $v \to \infty$ in $V_1$. Next, the assertion for $G_k$ is trivially true for $k=0$, so assume that it holds for some $k \geq 0$ and write \begin{eqnarray} G_{k+1}(v) &=& \frac{f^{(k+1)}(z)}{f(z)} = \frac{f^{(k)}(z)}{f(z)} \cdot \frac{f'(z)}{f(z)} +\frac{d}{dz} \left( \frac{f^{(k)}(z)}{f(z)} \right) \nonumber \\ &=& G_k(v) G_1(v) + \frac{G_k'(v)}{\phi'(v)} = \frac{G_k(v)}{\phi'(v)} \left( 1 + \frac{G_k'(v)}{G_k(v)} \right). \end{eqnarray} Thus it suffices to show that $G_k'(v)/G_k(v) \to 0$ as as $v \to \infty$ in $V_{k+1}$. By (\ref{rub1}) there exists a small positive $d_1$ such that if $v \in V_{k+1}$ is large then the circle $\|u - v\| = r_v = d_1 {\rm Re} \, v$ lies in $V_k$, and the differentiated Poisson-Jensen formula \cite[p.22]{Hay2} delivers $$ \frac{G_k'(v)}{G_k(v)} = \frac1{ \pi} \int_0^{2 \pi} \, \frac{ \log \| G_k(v + r_v e^{i \theta } )\|}{r_v e^{i \theta}} \, d \theta = O \left( \frac{ \log ( {\rm Re} \, v ) }{ {\rm Re} \, v } \right) \to 0 $$ as $v \to \infty$ in $V_{k+1}$. This proves the lemma. $\Box$ To establish Theorem \ref{thm1}, take any $D \in \mathbb R$. Then there exist $v_1 \in [1, + \infty) $ and a path $$\Gamma \subseteq \{ v \in \mathbb C : \, {\rm Re} \, v > N , \, \| {\rm Im} \, v \| < \pi/4 \} \subseteq H$$ which is mapped by $e^v$ to the half-line $\{ t + iD : \, t \geq v_1 \}$. Thus $f(z) -i D = e^v -i D $ is real and positive for $z$ on $\gamma = \phi (\Gamma)$, and $\Gamma \setminus V_k $ is bounded for each $k \geq 0$. Now write, on $\Gamma$, $$ e^v = t+iD, \quad \frac{dv}{dt} = \frac1{t+iD}, \quad s = {\rm Re} \, v = \frac12 \ln (t^2+D^2) .$$ Hence, for any non-negative integers $k, m$, Lemma \ref{lem1} gives, as $v \to \infty$ on $\Gamma $, $$ \left\| \frac{f^{(k)}(z)}{z^m} \right\| = \left\| \frac{f(z) G_k(v)}{z^m} \right\| = \left\| \frac{e^v G_k(v)}{\phi(v)^m} \right\| \geq \frac{e^s }{ s^{c_k+md} } \geq e^{s/2} \to \infty . $$ It then follows that, for $c > 0$, \begin{eqnarray} \int_\gamma \|f^{(k)}(z)\|^{-c} \, \|dz\| &\leq & O(1) + \int_\Gamma e^{- cs/2} \|\phi'(v)\| \, \|dv\| \\ &\leq& O(1) + \int_\Gamma e^{- cs/4} \, \|dv\| \\ &=& O(1) + \int_{v_1}^{+\infty} \frac1{(t^2+D^2)^{1/2+c/8}} \, dt < + \infty . \end{eqnarray} $\Box$ \section{Proof of Theorem \ref{thm2}}\label{pfthm2} Suppose first that the inverse function of the antiderivative $G$ of $g$ has a logarithmic singularity over infinity, and take $D \in \mathbb R$. Then Theorem \ref{thm1} may be applied with $f = G$ and $m= c = 1$, giving a level curve $\gamma = \gamma_D$, lying in a neighbourhood of the singularity, on which ${\rm Im} \, G(z) = D$ and ${\rm Re} \, G(z) $ increases. This curve is a trajectory for (\ref{AH}), traversed in time $$ \int_\gamma \frac1{\bar g(z)} \, dz \leq \int_\gamma \|G'(z)\|^{-1} \, \|dz\| < + \infty ,$$ which completes the proof in this case. For the proof of the following lemma the reader is referred to the statement and proof of \cite[Lemma 3.1]{blnewqc}. \begin{lem}[\cite{blnewqc}] \label{lemfirstest} Let the function $\phi : H \to \mathbb C \setminus \{ 0 \}$ be analytic and univalent, where $H = \{ v \in \mathbb C : \, {\rm Re} \, v > 0 \}$, and for $v, v_1 \in H$ define $Z(v) = Z(v, v_1)$ by \begin{equation} \label{h1} Z(v, v_1) = \int_{v_1}^v e^{u/2} \phi'(u) \, du = 2 e^{v/2} \phi'(v) - 2 e^{v_1/2} \phi'(v_1) - 2 \int_{v_1}^v e^{u/2} \phi''(u) \, du . \end{equation} Let $\varepsilon $ be a small positive real number. Then there exists a large positive real number $N_0$, depending on $\varepsilon$ but not on $\phi$, with the following property. Let $v_0 \in H$ be such that $S_0 = {\rm Re} \, v_0 \geq N_0$, and define $v_1, v_2, v_3, K_2$ and $ K_3$ by \begin{equation} \label{vjdef} v_j = \frac{2^j S_0}{128} + i T_0, \quad T_0 = {\rm Im} \, v_0, \quad K_j = \left\{ v_j + r e^{i \theta} : \, r \geq 0, \, - \frac{\pi}{2^j} \leq \theta \leq \frac{\pi}{2^j} \right\}. \end{equation*} Then the following two conclusions both hold:\\ (i) $Z = Z(v, v_1)$ satisfies, for $v \in K_2$, \begin{equation} \label{h2} Z(v, v_1 ) = \int_{v_1}^v e^{u/2} \phi'(u) \, du = 2 e^{v/2} \phi'(v) (1 + \delta (v) ), \quad \| \delta (v) \| < \varepsilon . \end{equation} (ii) $\psi = \psi (v, v_1) = \log Z(v, v_1) $ is univalent on a domain $H_1$, with $v_0 \in H_1 \subseteq K_3$, and $\psi(H_1)$ contains the strip \begin{equation} \label{Omegaimage} \left\{ \psi (v_0) + \sigma+ i \tau : \, \sigma \geq \log \frac18 \, , \, - 2 \pi \leq \tau \leq 2 \pi \right\} . \end{equation} \end{lem} $\Box$ Assume henceforth that $g$ is as in the hypotheses of Theorem \ref{thm2} and the inverse function of $g$ has a logarithmic singularity over infinity. This time there exist $M > 0$ and a component $C$ of $\{ z \in \mathbb C : \, \|g(z)\| > M \} $ such that $\zeta = \log g(z)$ is a conformal mapping of $C$ onto the half-plane given by ${\rm Re} \, \zeta > \log M$. Since (\ref{AH}) may be re-scaled via $z = Mw$ and $g(z) = Mh(w) $, it may be assumed that $M = 1$ and $0 \not \in C$. In order to apply Lemma \ref{lemfirstest}, let $\phi: H \to C$ be the inverse function $z = \phi(v)$ of the mapping from $C$ onto $H$ given by $$ v = 2 \zeta = 2 \log g(z), \quad g(z) = e^{v/2} , $$ As in the proof of Theorem \ref{thm1}, (\ref{h3}) holds for $u \in H$, with $N = 0$. By (\ref{Omegaimage}) there exists $X_0 > 0$ such that $ Z(v, v_1)$ maps a domain $H_2 \subseteq H_1 \subseteq K_3 \subseteq H$ univalently onto a half-plane ${\rm Re} \, Z > X_0$. Hence, for any $Y_0 \in \mathbb R$, there exists a path $\Gamma $ which tends to infinity in $ H_1 \subseteq K_3$ and is mapped by $ Z(v, v_1)$ onto the half-line $L_0 = \{ X + i Y_0, \, X \geq X_0 + 1 \}$. Consider the flow in $H_2$ given by \begin{equation} \label{vflow} \phi'(v) \dot v = \overline{e^{v/2}} ; \end{equation} by (\ref{h2}) this transforms under $Z = Z(v, v_1)$ to \begin{equation} \label{wflow} \dot Z = \frac{dZ}{dv} \, \dot v = e^{v/2} \phi'(v) \dot v = \| e^{v} \| . \end{equation} Combining (\ref{h3}) and (\ref{h2}) shows that $ \| e^{v} \| \geq \|Z(v)\|^{3/2} $ for large $v$ on $\Gamma$. Hence there exists a trajectory of (\ref{wflow}) which starts at $X_0+1 + iY_0$ and tends to infinity along $L_0$ in time $$ T_0 \leq \int_{X_0+1}^\infty \left\| \frac{dt}{dX} \right\| \, dX \leq O(1) + \int_{X_0+1}^\infty (X^2 + Y_0^2)^{-3/4} \, dX < + \infty . $$ This gives a trajectory of (\ref{vflow}) tending to infinity along $\Gamma$ and taking finite time to do so, and hence a trajectory $\gamma$ of (\ref{AH}) in $C$, tending to infinity in finite increasing time. Since $Y_0 \in \mathbb R$ may be chosen at will, this proves Theorem \ref{thm2}. $\Box$ {\footnotesize } \noindent School of Mathematical Sciences, University of Nottingham, NG7 2RD.\\ [email protected] \end{document}	arXiv
Definition of definition I was wondering if there is a good way to "define" what definition means exactly in mathematics. Since the answers may be subjective or philosophical, I want to ask only for references on this topic. So I am looking for references which answer the question "What does definition mean in mathematics" in a concise and commonly accepted way. and also for references which discuss philosophical problems connected with this question (if there are any). reference-request education definition philosophy meta-math $\begingroup$ You mean you don't like this: A line is breadthless length. ... or ... A straight line is a line which lies evenly with the points on itself $\endgroup$ – GEdgar Jul 10 '11 at 16:53 I would summarize my personal views about what "definition" means in mathematics as follows: "[M]eaning is use — words are not defined by reference to the objects they designate, nor by the mental representations one might associate with them, but by how they are used. (source) "Mathematicians do not study objects, but relations between objects. Thus, they are free to replace some objects by others so long as the relations remain unchanged. Content to them is irrelevant: they are interested in form only." - Henri Poincaré For example consider the symbol "$\mathbb{Z}$". If I wanted to tell someone what I meant by it, I might write $$\mathbb{Z}=\{\ldots,-2,-1,0,1,2,\ldots\}$$ or if were being highbrow, perhaps $$\mathbb{Z}\text{ is the infinite cyclic group.}$$ or $$\mathbb{Z}\text{ is the initial object in }\mathsf{Ring}.$$ If I spoke another language, or used a different method of writing integers, or used a different notation for sets, these would appear quite different. But such differences are irrelevant; it doesn't even matter if someone else's mental conception of the integers is radically different from my own. What matters is that our usages agree - if it is the case that, any time I write a statement about $\mathbb{Z}$ that I consider true, anyone else agrees that (modulo differences of language / notation) that is a true statement about whatever it is they think of when they see "$\mathbb{Z}$", then functionally, our "definitions" agree. So, I don't think of "definition" as a formal concept in math (I know almost nothing about logic / set theory / metamathematics - I am just expressing my opinions). Even in formal logic, how can we hope to define parentheses? Or "$\in$"? We just start using them, and if people agree what we've written makes sense to them, that's the best we can hope for - we can try to use natural language to convey our mental conceptions to other people, but we can't dive into their heads and check that their mental conception is actually the same. (Obviously, intuitions / mental conceptions are of the utmost importance in doing mathematics - we won't get anywhere with blind manipulation of symbols. I'm just saying that all we can check our agreement on are external expressions such as equations or sentences.) Finally, I'd just like to add this comic from SMBC: Zev Chonoles $\begingroup$ Minor note: of course in your example, the three meanings are different: the first is $\mathbb{Z}$ as a set, the second as a group, the third as a ring. $\endgroup$ – wildildildlife Jul 10 '11 at 10:52 $\begingroup$ That's a good point, I was glossing over that detail. To be more precise, if I wrote down the relevant definitions in group theory, ring theory, and category theory necessary to parse those statements, then if someone's definition of $\mathbb{Z}$-as-a-ring (and all the previous definitions) agreed with mine, they would also agree that (whatever they thought of when they saw the phrase) "the additive group of $\mathbb{Z}$" was (whatever they thought of when they saw the phrase) "the infinite cyclic group". $\endgroup$ – Zev Chonoles Jul 10 '11 at 11:13 $\begingroup$ Wrt the three definitions of $\mathbb Z$, the first is referred to as Extensional and the last two are called Intensional. In philosophy, this distinction cuts through to the differences between Realism and Nominalism. $\endgroup$ – Dactyl Jul 10 '11 at 12:11 To define a word, even the word "define", you need a language with which to define it. Trying to do so in English is difficult, because English is not what we call a formal language. A formal language is a list of symbols and an acceptable grammar for these symbols to follow. In mathematics, we generally use the formal language of Zermelo-Frankel set theory (or ZFC) to talk to each other (although many alternative ways have been studied). In this language, I would define a definition to be a finitely generated formula (would you accept an infinitely long definition of something?) of set theory that is legitimate according to the grammar whose quantifiers range over previously known results. For example, in ZFC the definition "A number is an even number if it is a multiple of 2," can be written as "If x is a natural number and there exists another natural number y so that x=2y, then we define x as an even number," in ZFC which, in the scope of set theory, is a legitimate sentence whose quantifier ("all") ranges over the set of natural numbers. A sentence that isn't definable would be something like "Call a set U universal if it contains all possible sets," because to define it in ZFC, you would need a formula "If U is a set so that for any set X, X is in U, then we call U universal," this formula quantifies over the set of all sets, which is not a set by Russell's Paradox, so this is not a legitimate definition. Kurt Gödel studied "definable" structures in set theory and came up with the constructible universe, called L, which is a very useful concept in studying models of set theory. L is basically the "set of things definable by a formula of ZFC". Notably, under the assumption that all of the universe of set theory is actually equal to L, one can prove the generalized form of continuum hypothesis, one of the biggest problems in set theory during the 20th century. tomcuchta $\begingroup$ Your remarks about $L$ at the end are not quite accurate, since you are conflating the constructible sets with the definable sets. The constructible sets are those that appear in the constructible hierarchy, which is defined by a very restricted concept of definability. But it is fully consistent with ZFC that there are definable objects not in $L$. Indeed, the set $\mathbb{R}$ is definable, but it is consistent that $\mathbb{R}$ is not in $L$, and it cannot be in $L$ in any model in which CH fails. $\endgroup$ – JDH Jul 10 '11 at 17:14 How can there be 6 other answers, yet no one has so far mentioned conservative extensions and/or extensions by definition? This is the correct framework in which to view definitions. References (of a philosophy-of-mathematics nature): Bertrand Russell's On Denoting. Ludwig Wittgenstein's Tracticus Logico-Philosophicus. Wittgenstein's later work, Philosophische Untersuchungen/Philosophical Investigations in which Wittgenstein refutes his earlier work (and deals with the problem you've brought up more or less directly). (Note that Wittgenstein studied under Russell's direction at Cambridge, at Frege's suggestion). Lastly, for context (and a deeper understanding of the strengths/weaknesses of formal languages), you'll probably want to study a formal language or two, and perhaps additionally study Godel's Incompleteness Theorems. Raeez $\begingroup$ Saying that Wittgenstein was "influenced" by Russell (btw, two 'l's) is an understatement, considering that Wittgenstein travelled to Cambridge expressly to study with Russell (by suggestion of Frege). $\endgroup$ – Willie Wong Jul 10 '11 at 11:56 $\begingroup$ @Willie Wong: Thanks—suggestions have been incorporated. $\endgroup$ – Raeez Jul 10 '11 at 16:25 $\begingroup$ @WillieWong - The Lord gave, and the Lord hath taken away. " I have not found in Wittgenstein's Philosophical Investigations anything that seemed to me interesting and I do not understand why a whole school finds important wisdom in its pages." -Russell, Bertrand. "Some Replies to Criticism." My Philosophical Development. New York, NY: Simon and Schuster. 1959 $\endgroup$ – George Chen Dec 26 '14 at 19:17 $\begingroup$ In other words, Wittgenstein mooned the world with someone else's glory, and went back to where he came from buttnaked. $\endgroup$ – George Chen Dec 28 '14 at 18:40 I was wondering if there is a good way to "define" what definition means exactly in mathematics. To make the definition of definition exact IMHO you need to make everything exact. This is what tomcuchta is talking about. Use some computer language to do mathematics, look for proof checkers and proof assistants. (If a language is computerized, this guarantees that the language is completely formal.) Then definition is a syntactic construction which binds a name. beroal No references from me, sorry, but allow me a quick and handwaving answer. The way I understand it, a definition of some object within a given theory is a meaningful shortcut. "Shortcut", because it provides a name for a bunch of certain properties that an object in the theory may or may not have; then every time the properties have to be refered to, the name is used. And "meaningful", in the sense that the properties being grouped under the name, either already have an important counterpart in the intended model of the theory, or else, they prove to be mathematically (read "technically") useful in the elaboration of the theory. Of course, this is still a narrow understanding, even mathematically (as opposed to "philosophically" I suppose), at least in that definitions are set forth in multiple levels of everyday mathematical practice--not just within a given theory, but on various meta-levels as well. In fact, as an aside, I believe this might be a key question to ask before trying out questions of "invention vs discovery" (see Is there any difference between a math invention and a math discovery? for example): do mathematicians choose their definitions, or are these forced upon them? Is there a uniform answer to be applied to all definitions (within a given theory)? Et cetera. A definition is just an abbreviation for typographical conveniences. In Whitehead & Russell's words: Theoretically, it is unnecessary ever to give a definition ... the definitions are no part of our subject, but are, strictly speaking, mere typographical conveniences. Practically, of course, if we introduced no definitions, our formulae would very soon become so lengthy as to be unmanageable; but theoretically, all definitions are superfluous. * Pardon me for being petty, but definition itself cannot be defined; it can only be explained, because defining definition leads to endless regression. For in-depth illustration, please see What is the difference between ❋3.01 and ❋4.5 in Whitehead and Russell's PM? *Source: Whitehead & Russell. Principia Mathematica. Chapter 1. Page 12, Merchant Books, 1910 George Chen $\begingroup$ The above definition is called dictionary definition. Ultimately ostensive definition, e.g., putting the word next to the thing it stands for, is needed in order to assign meanings to words in minimum vocabulary. $\endgroup$ – George Chen Jan 6 '16 at 20:07 [The Role of Mathematical Definitions in Mathematics and in Undergraduate Mathematics Courses][1]: http://pdfs.semanticscholar.org/2fca/dfddcc2e87d92ce0e80079863050aaa887a4.pdf yavuz Not the answer you're looking for? Browse other questions tagged reference-request education definition philosophy meta-math or ask your own question. Book on the Rigorous Foundations of Mathematics- Logic and Set Theory How is "point" in geometry undefined? And What is a "mathematical definition"? 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3E Chapter Exercises [ "stage:draft", "article:topic", "calcplot:yes", "license:ccbyncsa", "showtoc:yes", "transcluded:yes" ] 3: Applications of Derivatives Chapter Review Exercises True or False? Justify your answer with a proof or a counterexample. Assume that $f(x)$ is continuous and differentiable unless stated otherwise. 1) If $f(−1)=−6$ and $f(1)=2$, then there exists at least one point $x∈[−1,1]$ such that $f′(x)=4.$ True, by Mean Value Theorem 2) If $f′(c)=0,$ there is a maximum or minimum at $x=c.$ 3) There is a function such that $f(x)<0,f′(x)>0,$ and $f''(x)<0.$ (A graphical "proof" is acceptable for this answer.) 4) There is a function such that there is both an inflection point and a critical point for some value $x=a.$ Given the graph of $f′$, determine where $f$ is increasing or decreasing. Increasing: $(−2,0)∪(4,∞)$, decreasing: $(−∞,−2)∪(0,4)$ The graph of $f$ is given below. Draw $f′$. Find the linear approximation $L(x)$ to $y=x^2+tan(πx)$ near $x=\frac{1}{4}.$ $L(x)=\frac{17}{16}+\frac{1}{2}(1+4π)(x−\frac{1}{4})$ Find the differential of $y=x^2−5x−6$ and evaluate for $x=2$ with $dx=0.1.$ Find the critical points and the local and absolute extrema of the following functions on the given interval. 1) $f(x)=x+sin^2(x)$ over $[0,π]$ Critical point: $x=\frac{3π}{4},$ absolute minimum: $x=0,$ absolute maximum: $x=π$ 2) $f(x)=3x^4−4x^3−12x^2+6$ over $[−3,3]$ Determine over which intervals the following functions are increasing, decreasing, concave up, and concave down. 1) $x(t)=3t^4−8t^3−18t^2$ Increasing: $(−1,0)∪(3,∞)$ Decreasing: $(−∞,−1)∪(0,3)$ Concave up: $(−∞,\frac{1}{3}(2−\sqrt{13}))∪(\frac{1}{3}(2+\sqrt{13}),∞)$ Concave down: $(\frac{1}{3}(2−\sqrt{13}),\frac{1}{3}(2+\sqrt{13}))$ 2) $y=x+sin(πx)$ 3) $g(x)=x−\sqrt{x}$ Increasing: $(\frac{1}{4}∞),$ Decreasing: $(0,\frac{1}{4})$ Concave up: $(0,∞)$ Concave down: nowhere 4) $f(θ)=sin(3θ)\ Evaluate the following limits. 1) \(lim_{x→∞}\frac{3x\sqrt{x^2+1}}{\sqrt{x4−1}}$ $3$ 2) $lim_{x→∞}cos(\frac{1}{x})$ 3) $lim_{x→1}\frac{x−1}{sin(πx)}$ $−\frac{1}{π}$ 4) $lim_{x→∞}(3x)^{1/x}$ Use Newton's method to find the first two iterations, given the starting point. 1) $y=x^3+1,x_0=0.5$ $x_1=−1,x_2=−1$ 2) $\frac{1}{x+1}=\frac{1}{2},x_0=0$ Find the antiderivatives $F(x)$ of the following functions. 1) $g(x)=\sqrt{x}−\frac{1}{x^2}$ $F(x)=\frac{2x^{3/2}}{3}+\frac{1}{x}+C$ 2) $f(x)=2x+6cosx,F(π)=π^2+2\ Graph the following functions by hand. Make sure to label the inflection points, critical points, zeros, and asymptotes. 1) \(y=\frac{1}{x(x+1)^2}$ Inflection points: none Critical points: $x=−\frac{1}{3}$ Zeros: none; vertical asymptotes: $x=−1, x=0$ 2) $y=x−\sqrt{4−x^2}$ A car is being compacted into a rectangular solid. The volume is decreasing at a rate of $2 m^3/sec$. The length and width of the compactor are square, but the height is not the same length as the length and width. If the length and width walls move toward each other at a rate of $0.25$ m/sec, find the rate at which the height is changing when the length and width are $2$ m and the height is $1.5$ m. The height is decreasing at a rate of $0.125$ m/sec A rocket is launched into space; its kinetic energy is given by $K(t)=(\frac{1}{2})m(t)v(t)^2$, where $K$ is the kinetic energy in joules, $m$ is the mass of the rocket in kilograms, and $v$ is the velocity of the rocket in meters/second. Assume the velocity is increasing at a rate of $15 m/sec^2$ and the mass is decreasing at a rate of $10$ kg/sec because the fuel is being burned. At what rate is the rocket's kinetic energy changing when the mass is $2000$ kg and the velocity is $5000$ m/sec? Give your answer in mega-Joules (MJ), which is equivalent to $10^6$ J. The famous Regiomontanus' problem for angle maximization was proposed during the $15$ th century. A painting hangs on a wall with the bottom of the painting a distance $a$ feet above eye level, and the top $b$ feet above eye level. What distance x (in feet) from the wall should the viewer stand to maximize the angle subtended by the painting, $θ$? $x=\sqrt{ab}$ feet An airline sells tickets from Tokyo to Detroit for $$1200.$ There are $500$ seats available and a typical flight books $350$ seats. For every $$10$ decrease in price, the airline observes an additional five seats sold. What should the fare be to maximize profit? How many passengers would be onboard? For the following functions: a. Find the interval(s) on which $f(x)$ is increasing b. Find the interval(s) on which $f(x)$ is decreasing c. Find the local (relative) maximum and local (relative) minimum (if any). 1) $f(x)=x^2 e^{-x}$ 2) $f(x)=3(x^2-4)^{2/3}$ 3) $f(x)=x+\frac{1}{x}$ a. Find the interval(s) on which $g(x)$ is concave up b. Find the interval(s) on which $g(x)$ is concave down c. Find the inflection points (if any) 1) $g(x)=10x^3+ 3x^5$ 2) $g(x)=e^{-x^2/2}$ 3) $g(x)=5x^3+ 2x^5$ 4) $g(x)=\ln(1+x^2)$ 5) $g(x)=(3-x^2)^2$ Find the horizontal asymptote $y=f(x)= x- \sqrt{x^2+2x-6}$ $f(t)=\frac{1}{1-t^2}$ a. Find vertical and horizontal asymptotes (if any) b. Find the intervals when $f(t)$ is concave up and concave down $f(t)=e^{-t^2}$ a. Find the local maxima and minima for the function $f(t)$ b. Find the intervals when $f(t)$ is increasing and decreasing c. Find the intervals when $f(t)$ is concave up and concave down d. Find the inflection points for $f(t)$ Find all vertical and horizontal asymptotes for the function and justify your work using limits. $f(x)=\frac{x^2-5x+6}{x^2-4x+3}$ Using the following information: $y=\frac{x}{x^2-1},\qquad y^\prime=-\frac{x^2+1}{\left(x^2-1\right)^2},\mbox{ and }y^{\prime\prime}=\frac{2x\left(x^2+3\right)}{\left(x^2-1\right)^3}$ a) Identify any asymptotes for the function $y$ using limits. b) Find where the function $y$ is increasing/decreasing and any relative extrema. c) Find where the function $y$ is concave up/down and any points of inflection. d) Graph the function $y$ Let $f(x) = x^{5/4}-x^{1/4}$ Determine the intervals on which $f$ is increasing and decreasing and find any local extrema. Let $g(x) = \ln (x^2+1)$ Determine the intervals of concavity of $g(x)$ and locate any inflection points. Determine the intervals of concavity of $g(x)$ and locate any inflection points: 1) $g(x) = (\ln x)^2$ 2) $g(x) = x-2 \tan^{-1}(x)$ Find interval(s) on which the function is concave up and the interval(s) on which the function is concave down, and inflection point(s): $y=f(x)= x-2tan^{-1}(x)$ Let $f(x) = x^2e^{-16x^2}$ 3.9 E Exercises 4 Integral Calculus	CommonCrawl
Chemical and structural properties of reduced graphene oxide—dependence on the reducing agent Composites & nanocomposites B. Lesiak1, G. Trykowski2, J. Tóth3, S. Biniak2, L. Kövér3, N. Rangam1, L. Stobinski4,5,6 & A. Malolepszy4 Journal of Materials Science volume 56, pages 3738–3754 (2021)Cite this article Graphene oxide (GO) prepared from graphite powder using a modified Hummers method and reduced graphene oxide (rGO) obtained from GO using different reductants, i.e., sodium borohydride, hydrazine, formaldehyde, sodium hydroxide and L-ascorbic acid, were investigated using transmission electron microscopy, X-ray diffraction, Raman, infrared and electron spectroscopic methods. The GO and rGOs' stacking nanostructure (flake) size (height x diameter), interlayer distance, average number of layers, distance between defects, elementary composition, content of oxygen groups, C sp3 and vacancy defects were determined. Different reductants applied to GO led to modification of carbon to oxygen ratio, carbon lattice (vacancy) and C sp3 defects with various in-depth distribution of C sp3 due to oxygen group reduction proceeding as competing processes at different rates between interstitial layers and in planes. The reduction using sodium borohydride and hydrazine in contrary to other reductants results in a larger content of vacancy defects than in GO. The thinnest flakes rGO obtained using sodium borohydride reductant exhibits the largest content of vacancy, C sp3 defects and hydroxyl group accompanied by the smallest content of epoxy, carboxyl and carbonyl groups due to a mechanism of carbonyl and carboxyl group reduction to hydroxyl groups. This rGO similar diameter to GO seems to result from a predominant reduction rate between the interstitial layers. The thicker flakes of a smaller diameter than in GO are obtained in rGOs prepared using remaining reductants and result from a higher rate of reduction of in plane defects. Avoid the most common mistakes and prepare your manuscript for journal editors. Wide varieties of methods for graphene oxide (GO) reduction have been recently utilized [1,2,3,4,5]. These methods include thermal, microwave, photo, chemical, electrochemical and solvothermal reduction. The most commonly applied chemical reduction routes use a wide variety of chemical compounds. Strong reducing agents are hydrazine and its derivatives (hydrazine hydrate and dimethylhydrazine), metal hydrides (sodium hydride, sodium borohydride), hydroiodic acid, etc. Other, weak reducing agents include ascorbic acid, hydroquinone, pyrogallol, hot strong alkaline solutions (KOH, NaOH), hydroxylamine and urea [1 and references included]. Reduction procedures of GO using different conditions, i.e. vapor, aqueous solutions of different concentrations and time, result in preparation of reduced graphene oxide (rGO) of variety of chemical and structural properties. The main criteria for estimating the effect of reduction are C/O ratio, optical properties, structural defects and electrical conductivity. The electrical conductivity of a monolayer graphene depends on the carrier transport and therefore rGO chemical properties, where the oxygen groups attached to the charge carrying plane are more important in contrary to oxygen groups attached to the edges of the charge carrying planes. Reduction of chemically prepared GO results in rGO arranged in graphene stacking nanostructures (flakes) of various average height and diameter with graphene interlayers separated by a distance depending on the degree of the reduction, which implicates the average number of layers in graphene flakes [5]. Therefore, the conductivity of rGO depends not only on C/O ratios, the number of still existing in rGO different oxygen groups and their ratios, but also on structural properties of the obtained rGO flakes and layers like size of flakes, type and content of defects in graphene layers. Description of structural and chemical properties of graphene materials basing on results obtained using X-ray photoelectron spectroscopy (XPS), X-ray excited Auger electron spectroscopy (XAES), reflection electron energy loss spectroscopy (REELS), Raman spectroscopy, X-ray diffraction (XRD), transmission electron microscopy (TEM), Fourier transformed infrared spectroscopy (FTIR), atomic force microscopy (AFM), etc., has been often reported [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19]. The FTIR and XPS analyses are usually applied for probing the chemical properties of atoms, carbon hybridizations and oxygen groups, where FTIR analysis information is related to the bulk, and XPS information depth is limited to the surface, i.e. few nanometers. By the recently developed complementary approaches of Raman spectroscopy [8,9,10,11,12,13,14,15,16,17,18,19] of few micrometers information depth, the surface AFM [8] and REELS [5] methods provide the information on the structural and chemical properties like level of disorder, dopants, the number of layers, the linear dispersion of electronic energy, the crystal orientation, the defects and strain [8,9,10,11,12,13,14,15,16,17,18,19]. Although, numerous procedures of reduction of GO using various reducing agents have been proposed [1 and references within], there are still many points to clarify like how oxygen groups are reduced, how lattice defects are being formed and reconstructed and what are the mechanisms of reduction and carbon lattice reconstruction using various chemical preparations and compounds. An attempt to reveal these differences was made using FTIR, Raman and surface XPS, XAES and REELS studies utilizing different in-depth sensitivity. The particular aim of the present work was to obtain information on differences in effectivity of the chemical reduction of GO flakes performed in aqueous solutions of various chemical compounds, i.e. strong and weak reductants like sodium borohydride (NaBH4), hydrazine (N2H4), formaldehyde (CH2O), sodium hydroxide (NaOH), L-ascorbic acid (C6H8O6), as well as L-ascorbic acid (C6H8O6) prepared at room temperature (RT). The study presents the detailed comparison of the structural and chemical properties of the reduced GO samples, however is not concerned with investigating the kinetics of the reduction reaction. Information depth of electron spectroscopic methods A convenient measure of surface sensitivity in electron spectroscopic methods (XPS, XAES, REELS) is the information depth, ID, defined as the maximum depth, normal to the surface, from which useful signal information of a specified percentage, P, of signal originates [20, 21]. Therefore, ID can describe the sampling depth of this specified percentage of signal. The averaged information depth called a mean escape depth, MED, in XPS and XAES and mean penetration depth, MPD, in REELS is defined as an average depth normal to the surface from which the specified particles escape [22, 23]. Assuming single scattering of electrons, these values can be evaluated from the equations given as follows [20, 21]. $$ID(P)_{XPS} = \lambda \cos \alpha_{out} \ln \left( {\frac{1}{{1 - {\raise0.7ex\hbox{$P$} \!\mathord{\left/ {\vphantom {P {100}}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{${100}$}}}}} \right),\,ID(P)_{{{\varvec{REELS}}}} = \lambda \frac{{\cos \alpha_{{{\varvec{in}}}} \cos \alpha_{{{\varvec{out}}}} }}{{\cos \alpha_{{{\varvec{in}}}} + \cos \alpha_{{{\varvec{out}}}} }}\ln \left( {\frac{1}{{1 - {\raise0.7ex\hbox{$P$} \!\mathord{\left/ {\vphantom {P {100}}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{${100}$}}}}} \right),$$ $$MED_{{{\varvec{XPS}}}} = \lambda \cos \alpha_{{{\varvec{out}}}} ,\,MPD{}_{{{\varvec{REELS}}}} = \lambda \frac{{\cos \alpha_{{{\varvec{in}}}} \cos \alpha_{{{\varvec{out}}}} }}{{\cos \alpha_{{{\varvec{in}}}} + \cos \alpha_{{{\varvec{out}}}} }}$$ (1b) where λ is the inelastic mean free path, IMFP, of electrons scattered inelastically dependent on the kinetic energy, KE, of the electrons and on the material, cosαin and cosαout are angles of incidence and emission of electrons with respect to the surface normal of the specimen and P is a specified percentage of the signal considered. The values of ID(P = 99%), MED, MPD for the applied geometry of analysis and values of IMFPs in graphite published by Shinotsuka et al. [24] evaluated for the investigated transitions are listed in Table 1. The considered theoretical values of IMFPs [24] are in agreement with the experimentally determined values of IMFPs for bi-, tri- and multilayer graphene as found by Xu et al. [25]. Table 1 The MED, MPD and (ID(P = 99%)) (Eq. (1)) of the investigated signal evaluated using the IMFPs, λ, in carbon from Shinotsuka [24] and a number of layers evaluated assuming an interlayer distance in GO of 0.721 nm, rGOs of 0.4 nm and graphite of 0.335 nm The in-depth profile of graphene layers from REELS spectra is evaluated using an exponential interpolation of intensity of reflected inelastically scattered electrons, where the signal can be attenuated according to equation [26]: $$I = I^{\infty} ({1} - {\exp}( - z/l{\cos}a_{{{\text{out}}}} ),$$ where I is the signal intensity from the overlayer of thickness z, I∞ is the signal intensity from the overlayer material of infinite thickness [5], λ is the IMFP [24] and αout is the electron emission angle with respect to the surface normal. The following samples were investigated: a commercially available graphite (denoted as Gr) of 99.0% purity (AcrosOrganics, USA, 325 mesh) and expanded graphite (denoted as Gr-exp), graphene oxide (denoted as GO) and reduced graphite oxide samples prepared from GO using different reductants, i.e. sodium borohydride (NaBH4), hydrazine (N2H4) [5], sodium hydroxide (NaOH), formaldehyde (CH2O) and L-ascorbic acid (C6H8O6). These reduced graphene oxide samples were denoted as rGO-NaBH4, rGO-N2H4, rGO-NaOH, rGO-CH2O and rGO-C6H8O6. The GO sample was prepared from commercially available graphite using a modified Hummers method [27]. The reduction using NaBH4, NaOH, C6H8O6 and CH2O reductants proceeded in 100 ml water suspension of GO (0.6 wt.%) mixed with 50 ml 1 M water solution of NaBH4, NaOH, C6H8O6 reducing agents, whereas reduction of GO using N2H4 proceeded using 50 ml of 50% solution of hydrazine hydrate. The GO and reducing agent solutions were boiled under a reflux for 30 min. Then, all the reaction mixtures were cooled to RT, washed in deionized water till the pH of filtrate stabilized to ca. 7–8, and then they were dried. Additional reduction procedure using L-ascorbic acid (C6H8O6) at RT was proceeded under stirring of GO and C6H8O6 in water solution for 30 min and then washing in deionized water till pH filtrate stabilized to ca. 7–8 and drying. HR-TEM, EDX, PEELS and XRD apparatus A high resolution transmission electron microscope (HR-TEM), Tecnai 20F X-Twin, equipped with an electron source, a cathode with field emission gun (FEG), EHT = 200 keV, a camera Orius and a high-angle annular dark-field (HAADF) detector, was used. This TEM was equipped with energy-dispersive X-ray spectrometer (EDX) with the energy resolution of 134 eV (EDAX RTEM SN9577+) and parallel electron energy loss spectrometer (PEELS) with the energy resolution of 0.8 eV. The quantification was performed according to the modified standardless/thin foil method. The preparation of samples proceeded in the following steps: sonication for 5 s of a few milligrams of sample in ethanol (99.8% anhydrous) using ultrasounds, applying a drop of the solution of 5 μl on a carbon coated copper mesh with holes (Lacey type Cu 400 mesh, Plano), evaporating the solvent at room temperature and then investigating the remaining dried powder stuck on the copper mesh. Powder X-ray diffraction (pXRD) patterns were obtained by using an X'Pert Pro diffractometer, with Cu Kα radiation (λ = 1.5406 Å) and with a X'Celerator detector. Raman apparatus A Raman micro-spectrometer, Senterra, Bruker Optik with a 532 nm laser was used. The measurement parameters are the following: laser power of 2 mW, acquisition time of 360 s, number of scans 2 and analyzed range of 800–3000 cm−1. FTIR apparatus The FTIR spectra in the 70–4000 cm−1 range were recorded in vacuum spectrometer with Fourier transformation, Vertex 70 V, Bruker Optik, at T = 22 °C, p = 10–1 Pa, resolution 4 cm−1 and number of scans 100. Before the sample measurement, a "vacuum spectrum" was recorded and subtracted automatically as background during registration of spectra for the investigated samples. The sample was mixed with KBr at a ratio of 1/300 mg and then compressed at 7 MPa cm−2 to form a pellet, and the transmission spectrum was recorded in the range of 750–1900 cm−1. XPS apparatus The XPS measurements were carried out in an ultra-high-vacuum (UHV) using the ESA-31 electron spectrometer (homemade) [28]. The spectrometer is equipped with a hemispherical electron energy analyzer of a high relative energy resolution of 0.5% without retardation (the retarding ratio, k, can be changed from 2 to 100), an electron gun (LEG62-VG Microtech), a homemade X-ray excitation source (Al Kα X-rays hν = 1486.67 eV) and an Ar+ ion source of AG21 (VG Scientific). The XPS spectra were measured in the fixed retarding ratio (FRR) mode (k = 4, 8, 16) at a photon incidence and electron emission angles of 70° and 0°, respectively, with respect to the surface normal of the specimen. The REELS spectra were measured using a primary electron energy of 4 keV (k = 41, generally keeping the analyzer pass energy at about 50–100 eV) and the electron beam current intensity of a few nA at electron incidence and emission angles of 50° and 0o with respect to the surface normal of the specimen, respectively. The prepared powder of graphite, GO and rGO samples was placed on a holder using an UHV tape and investigated without any thermal and/or chemical pretreatment. Structural properties of graphene flakes by HR-TEM, EDX and PEELS The EDX spectra of exemplary rGO-NaBH4 and rGO-CH2O samples (Fig. S1) reveal in the bulk the presence of C, O and Na. The PEELS spectra show differences in the intensity of elastic peak and inelastic loss peaks in the energy loss region of about 50–400 eV, as well as inelastic energy loss values indicating structural variations between the presented samples. The TEM images of exemplary rGOs (Fig. 1) show transparent parts confirming graphene layers structure with dark parts due to the overlapping layers. The HR-TEM images of a rGO-NaBH4 and b rGO-CH2O Graphene interlayer distance and size of flakes by XRD The XRD (002) and (10) patterns of the investigated samples (Fig. 2) were fitted using Pearson7 function (Fig. S2). The resulting fitting parameters, i.e. Bragg's angle of (002) and (10) patterns and the respective full-width at half maximum (FWHM) values, are listed in Table S1. The structural parameters, i.e. the average interlayer distance, d, height of the stacking nanostructures, H, number of layers in the flakes, n, and diameter of the flakes, D, evaluated using Bragg's and Scherrer's equations are listed in Table 2. For evaluating the value of H from (002) pattern and D from (10) pattern, the constants of 0.9 and 1.84 were considered, respectively. The XRD diffractograms recorded from the investigated samples Table 2 Comparison of structural parameters of GOs and rGOs from XRD patterns. Symbols denote: d-average distance between graphene layers, H-average height of flakes, n-average number of graphene layers and D-average diameter of flakes The structural parameters of GO, rGOs vary depending on the reductant (Table 2). The interlayer distance is 0.721 nm (GO) and 0.446–0.345 nm (rGOs). These values confirmed elsewhere, i.e. 0.335 nm (graphite), 0.335–0.4 nm (rGO) and 0.77–0.9 nm (GO) increase with humidity [5, 29,30,31]. The rGO reduced using NaBH4 shows considerably large flake diameter of the smallest thickness and average number of graphene layers. These average number of layers in different rGO flakes increases in the following order: rGO-NaBH4 < rGO-N2H4 < rGO-C6H8O6–RT < rGO-C6H8O6 < rGO-CH2O < rGO-NaOH. Structural properties of graphene flakes by Raman spectroscopy The Raman spectrum is considered to depend on: (i) clustering of the sp2 phase, (ii) bond disorder, (iii) the presence of sp2 rings or chains and (iv) the sp2/sp3 ratio, where these factors act as competing forces influencing the shape of the Raman spectra. Classification according to these features was proposed by Ferrari and Robertson [12] as an amorphization trajectory consisting of three stages ranging from graphite to tetrahedral amorphous carbon (ta-C) or defected diamond, i.e. (1) graphite to nanocrystalline graphite (nc-Gr), (2) nanocrystalline graphite to amorphous carbon (a-C) and (3) a-C to 100% sp3 ta-C or defected diamond. The classification and interpretation of the Raman spectra require consideration of intensity and width (FWHM) of the Raman modes. The main features of Raman spectra of graphene show characteristic major bands assigned to the first-order D (~ 1320–1350 cm−1), G (~ 1570–1605 cm−1), D' (~ 1620 cm−1), D + D' (~ 2900 cm−1) modes and the second-order 2D mode (2640- 2680 cm−1) (Fig. 3). The D mode is an in-plane breathing vibration of the six-membered carbon rings and is considered as a disorder-induced band. Therefore, the D band reflects defects like destroyed carbon hexagons, and it is not present for a perfect graphite structure in contrary to amorphous structures, for which its intensity is high. The intensity and FWHM of D band depend on disorder and the type of edges, where the intensity is higher for armchair than for zigzag edges. The G mode is due to the in-plane stretching vibration of carbon atom pairs and is observed for all carbon structures containing sp2 bonds, both aromatic carbon and other sp2 structures. The 2D allowed mode is the most intense for an ideal single-layer graphene. A low intensity D band accompanied by a presence of 2D band indicates a high-quality graphene. The main features of the evolution of Raman spectrum in stage 1 are appearance of D mode, increasing of I(D)/I(G) ratio following Tuinstra and Koenig (TK) relations [13], appearance of D' mode, disappearance of D and 2D mode doublet structure, appearance of D + D' mode and at the end of stage 1, overlapping of G and D' modes. The ratio of intensity of D to G mode is inversely proportional to the size of effective crystallite size in the direction of the graphite plane, La, or graphitic cluster, i.e.: $$I\left( D \right)/I\left( G \right) = C(\lambda)/L_{a}$$ where λ is Raman wavelength and C(515.5 nm) = 4.4 nm [12]. In the stage 2, the position of G mode decreases, the TK relations fails and I(D)/I(G) decreases. The broad feature from about 2300 cm−1 to 3200 cm−1 modulated by 2D, D + D' and 2D' modes appears instead of a well-defined second-order peaks. For stage 2, the new relation was proposed by Ferrari and Robertson [12]: $$I\left( D \right)/I\left( G \right) = C^{\prime}(\lambda)L_{a}^{2}$$ with C'(514) ~ 0.55 nm−2. The Raman spectra recorded from the investigated samples The stage 3 is beyond the data presented and therefore is omitted in discussion. In all stages of amorphization trajectory, the structural defects exist lowering the symmetry of the infinite crystal and they include: (i) the point-like defects, (ii) cluster defects and (iii) boundary or edge defects. The experimental investigation of defects [17] in Ar bombarded graphene and graphite with different grain size like: (i) vacancy defects produced by the deformation of the carbon lattice bond, (ii) on-site defects, which describe out-of-plane atoms bonded to carbon atoms (namely sp3 hybridization) and (iii) boundary or edge defects, provided detectable D and D' modes modification. The experimental data showed that I(D)/I(D′) is maximum (∼ 13) for defects associated with sp3 hybridization, it decreases for vacancy-like defects (∼ 7) and reaches a minimum for boundary-like defects in graphite (∼ 3.5) [17]. Other work by Lucchese et al. [18] studying disorder and defects in graphene caused by Ar ion bombardment provided quantification of defects, i.e. the average distance between point-like defects, LD, and density of defects, nD (LD~ 1/$\sqrt{{n}_{D}}$) by fitting the observed I(D)/I(G) versus LD using a phenomenological model. The values of I(D)/I(G) have a non-monotonic dependence on LD, increasing with increasing LD up to LD ~ 4 nm in stage 2, and then decreasing for LD > 4 nm (stage 1) and being in agreement with a well-established graphitization trajectory for carbon materials proposed by Ferrari and Robertson [12]. Such behavior suggests the existence of two disorder-induced competing mechanisms contributing to the Raman D band. The phenomenological model considers that the impact of a single ion in the graphene sheet causing modifications on two length scales, here denoted by rA and rS (with rA > rS), which are the radii of two circular areas measured from the impact point and subscript A stands for "activated," whereas the subscript S stands for "structurally-defective," with values of rS = 1 nm and rA = 3 nm [19]. This model extended by Concado et al. [19] to other Raman lines is valid for LD > 10, i.e. for a low-density defects materials. The measured shapes, positions, relative intensities of D, G, 2D, D + D' modes and FWHM of Raman spectra are characteristic for carbon materials (Fig. 3, Table 3, Fig. S3–S5). According to the above, both graphite samples can be classified to stage 1, whereas GO and rGOs to stage 2. The values of La evaluated according to Eqs. (3) are listed in Table 3. Therefore, graphite samples exhibit a low defect density, whereas GO and rGO exhibit a high defect density. The stage 1 Gr and Gr-exp samples show spectra of a high intensity and small FWHM of G mode indicating dominating graphite structure of low defect density. These samples are characterized by the smallest ratio of I(D)/I(G) (Table 3) at maximum of G mode at 1572–1575 cm−1. The stage 2 of amorphization trajectory includes GO and rGO samples, which characteristic features are high intensities and width of FWHM of D and G modes, very weak intensity of 2D mode and values of I(D)/I(G) close to 1. The variation of positions and FWHM of D, G and 2D modes (Fig. S3-S5) confirms that the investigated samples have different thickness, C sp3 content, graphitic clusters size, distance between defects and their density (Fig. S5), crystallinity (Fig. S4, Table 3) [8,9,10,11,12,13,14,15,16,17,18,19]. The FWHM and intensity values of D and G mode indicate the distance between point-like defects smaller than 3 nm [19], for which phenomenological evaluation of distance between defects proposed by Concado et al. [19] is not valid. The rGOs exhibit higher crystallinity than GO, whereas the highest crystallinity is shown by graphite and expanded graphite (Fig. S4). As reported elsewhere for rGOs [12], increasing frequency of D band position and decreasing frequency of G band position accompanied by decreasing FWHM of D and G bands (Fig. S3) is related to decreasing C sp3 content and increasing La. While increasing La value provides decreasing position and FWHM of 2D mode [20], the increasing number of layers leads to an opposite effect [10]. Therefore, the 2D band in Raman spectra exhibits these two effects (Fig. S3). Table 3 The parameters of D, G and 2D bands of Raman spectra and the effective crystallite size in the direction of the graphite plane, La, for the investigated samples Chemical groups in graphene flakes by FTIR spectroscopy The FTIR measurement procedures (transmittance of KBr pellets) do not provide quantitative determination of surface (or volumetric) concentration of chemical moieties (Fig. 4). The comparison of band intensity for various samples cannot provide clear information about their chemical structure because of the arbitrary background subtraction and band extension. A stoichiometric structure of GO, heterogeneity of the oxygenated functional groups and their interaction lead to overlapping of the characteristic bands and affect their position and intensity. However, the recorded spectra indicate: (i) the presence (or absence) of the chemical structures and (ii) changes in relative intensities of the respective bands caused by reduction procedures informing on chemical structures transformation. The FTIR spectra recorded from the investigated samples In all FTIR spectra (Fig. 4), two main range of absorption bands 1750–1450 cm−1 and 1300–950 cm−1 characteristic for C = O and C-O moieties, respectively, can be noticed [6, 7]. The left side of the spectra starts with more or less shaped band near 1720 cm−1 attributed to the stretching vibration of C = O moiety in carboxylic (conjugated and/or non-conjugated) or carbonate systems (acid, ester, anhydride, dioxolan). The bands at 1635 cm−1 and 1565 cm−1 can be assigned to aromatic carbon–carbon bonds, carbonyl moieties in various chemical surroundings (quinone-, ketone-, aldehyde-like, amide-like), carbon–oxygen ion-radical structures and conjugated systems (diketone, keto-esters, keto-enol and quinone-hydroquinone structures). The adsorbed/intercalated water provides deformation vibration band δ(HOH) located near 1630 cm−1 overlapping with C–OH mode at 1620 cm−1. The next set of the overlapping peaks, which form an absorption band in 1300–950 cm−1 region, can be attributed to C-O moieties existing in a different structural environment. In this spectral region, the presence of C–O–C symmetric stretching vibration bands in ether-, ester- lactone-, pyrone-, furane-like structures should be taken into consideration. Also, phenol and hydroxyl molecular groups exhibit characteristic mode located near 1070 cm−1 (C–OH vibration), while the epoxide group provides a mode near 1290 cm−1 (asymmetric C–O–C stretching). The graphite (Gr) spectrum was arbitrary extended and small amount of oxygen bonded with carbon in the form of carboxylic and ether groups, as well as strongly adsorbed water can be observed. The peak assigned to in-plane stretching of aromatic rings (v(C = C)) with frequency 1580 cm−1 is visible (only partially overlapped). The oxidative treatment of graphite (GO samples) gives large enhancement of C = O and C-O bands described by the above mentioned oxygen-containing functional groups—the increase in relative intensity of these peaks and covering the aromatic peaks can be observed. The difference in the shape of the spectra recorded for Gr-exp and GO-foil—markedly the increase in relative intensity of the band in C–O–C region (near 1200 cm−1)—can be explained by stronger dehydration of GO-foil with creation of anhydride structures (lactone). The GO reduction using different procedures modifies the FTIR spectra. After reduction with sodium hydroxide in rGO-NaOH, a relative decrease in the peak ascribed to carboxylic moieties (1720 cm−1) and a relative increase of the band associated with the presence of hydroxyl groups (near 1620 cm−1) take place. Similar changes can be observed for GO reduced with formaldehydes (rGO-CH2O). The GO reduction with other weak reduction agent—L-ascorbic acid at RT (rGO-C6H8O6-RT) and (rGO-C6H8O6) changes the FTIR spectra due to extensive adsorption of organic molecules. Mainly the presence of the overlapped bands of aliphatic chains and hydroxyl functional groups can be observed. The reduction of GO with sodium borohydride (sample rGO-NaBH4) completely dehydrates carbon materials (disappearance of a peak near 1630 cm−1) creating anhydride structures (lactone, cyclic esters) without removing carbon–oxygen functionalities. In samples reduced using sodium borohydride and L-ascorbic acid, a decrease of alkene stretching vibration (1535 cm−1) and a presence of a strong amide stretching vibration (1565 cm−1) in contrary to GO and rGOs reduced using formaldehyde and sodium hydroxide are observed. The smallest content of water is observed in rGO-NaBH4, rGO-C6H8O6-RT and rGO-C6H8O6 in contrary to GO, rGO-NaOH and rGO-CH2O. Average number of layers in graphene flakes by REELS The REELS spectrum of carbon nanomaterials reflects the structural features in intensity and energy loss values of reflected electrons losing their kinetic energy on valence electrons forming π and π + σ bonds [32, 33]. The values of intensity and energy loss are characteristic for the type of bonds (C sp2 and C sp3 hybridizations), and content of these hybridizations at the surface (S) and in the bulk (B) varies with chemical and structural properties of carbon nanomaterials, i.e. graphite and diamond [32], single, bi-, triple-, multilayer graphene and graphene oxide [34,35,36,37]. For fitting the spectroscopic data reflecting the interaction of matter with electromagnetic radiation, the most common pseudo-Voigt profile, i.e. convolution of Lorentzian and Gaussian symmetric components, is used since Lorentzian refers to distribution of decaying oscillations and Gaussian to distribution of velocities leading to Doppler broadening. The REELS spectra recorded for the investigated carbon nanomaterials resulted as fitted Gaussian functions of surface and bulk C sp2/sp3 components (Fig. S6) as reported elsewhere [5]. Results of this fitting presenting the electron kinetic energy loss values typical for surface (S) and bulk (B) C sp2 and C sp3 components, these components' percentage contributions and the ratio of C sp2B/C sp2S are listed in Table 4. For approximating the intensity ratio of π + σ energy loss peak components from C sp2B bulk to C sp2S surface as a function of graphene layer in-depth profile, z, Eq. (2) is applied. For a single layer graphene, the ratio of C sp2B/C sp2S is zero, whereas for graphite, it results from the experimental REELS spectra fitting (Fig. S6). The average number of layers, n, within the in-depth profile, z, of graphene flakes (Fig. 5) was evaluated using interlayer distances from XRD (Table 2). The values of n resulting from REELS (Table 4) and XRD (Table 2) for GO and rGOs flakes are consistent, as also reported previously [5]. Table 4 Comparison of REELS spectral parameters for the investigated GOs and rGOs resulting from the fitting of the respective spectra to Gaussian functions Dependence of C sp2B/C sp2S ratio on the penetration depth for electron signal information depth ID (P = 99%), IMFP in graphite from Shinotsuka [24] for evaluating the average number of layers resulting from REELS and Raman spectra Surface elemental content, C sp 2 /sp 3 hybridizations and oxygen groups in graphene flakes The quantification of the surface atomic content of GO and rGOs was carried out from the area under C 1 s, O 1 s, N 1 s, Na 1 s, Si 2p and S 2p photoelectron peaks after using Tougaard background subtraction [38]. The atomic content was evaluated from the XPS MultiQuant software [39, 40] accounting for Scofield photoionization cross sections [41], electron elastic scattering and analyzer transmission function. The results are presented in Table S2 and Fig. 6. a Surface atomic content determined from XPS in carbon nanomaterials (Table S2) the samples are indicated in the order of increasing number of layers in graphene flakes and b C/O (Table S2) ratio as a function of the average number of graphene layers (Table 4). Notation "" – sample from Ref [5] Samples of GO and rGO show contaminations of N, Na, S and Si (Table S2, Fig. 6a). For rGOs, the increasing content of C and decreasing content of O are observed with increasing number of layers in flakes (Fig. 6). The C 1 s and O 1 s spectra after Tougaard inelastic background subtraction [38] were fitted as suggested by Kovtun et al. [42] to pseudo-Voigt functions (combination of Gauss and Lorentz asymmetric and symmetric components) using the XPSPeakfit software [43]. The C1s spectra components binding energy (BE) values for different chemical forms, i.e. C sp2/ C sp3 hybridizations and oxygen groups (hydroxyl – C–OH, epoxy – C–O–C, carbonyl – C = O and carboxyl – C-OOH), were applied as reported elsewhere for different carbon nanomaterials like nanodiamonds, graphite, carbon nanotubes, graphene oxide and graphene [42,44–48 and references within]. The vacancy defects were considered at about 283.9 eV as suggested by Barinow et al. [48]. The O 1 s BE of carbon–oxygen groups was selected in the range of experimental [5, 49, 50] and calculated values [50]. The resulting fitted spectra are shown in Fig. S7, S8. The values of C 1 s and O 1 s BEs for carbon lattice vacancies, C sp2/sp3 hybridizations and carbon–oxygen groups resulting from the above fitting procedure shown in Table S3 and S4 are in agreement with a literature data [5, 42, 44,45,46,47,48,49,50]. The evaluated oxygen content resulting from C 1 s and O 1 s spectra fitting is in a reasonable agreement (Fig. S9). Comparison of carbon–oxygen groups content is provided in Fig. 7a–b. For all carbon nanomaterials the C 1 s BE of C sp2 component is 284.5 ± 0.1 eV (Table S3). Samples of graphite show at the surface C–OH and C = O groups and traces of an adsorbed water. Samples of GO and rGO show additionally C–O–C and C-OOH groups and a larger content of water at O 1 s BE of 535.2 ± 0.1 eV, 536.4 ± 0.3 eV, 538.3 ± 0.3 eV, attributed to adsorbed water, liquid water and liquid and gas phase water (overlapping peaks), respectively, as reported elsewhere [51,52,53,54]. Reduction of GO decreases the amount of carbon–oxygen groups C–OH > C–O–C > C = O > C-OOH and vacancy and C sp3 defects (Table S3). This is observed with increasing number of graphene layers in flakes (Fig. 7a). The ratio of rGO to GO carbon–oxygen groups decreasing for C–OH and increasing for C-OOH > C = O > C–O–C accompanied by dehydration (Figs. 4, 7a, Table S4) with number of layers suggests different rate of oxygen groups reduction influencing the thickness of rGO flakes (Fig. 7b). a Surface atomic content of C sp3 hybridizations, carbon–oxygen groups with respect to C sp2, b the ratio of oxygen group content in rGOs to GO (Table S3); in a–b the rGOs are indicated in the order of increasing number of layers in graphene flakes The dependence of number of graphene layers on the in-depth distribution of C sp3 defects resulting from evaluation using: (i) the first derivative of Auger C KLL spectrum [55] from which the parameter D is calculated [46] (C sp3C KLL) (Fig. S10), (ii) C 1 s spectra fitting (C sp3C 1 s) (Table S5), REELS spectra fitting of (iii) surface (C sp3REELS-surface) and (iv) bulk (C sp3REELS-bulk) components (Table 4) is provided in Fig. 8a. The GO-rGOs show surface C sp3 enrichment. The content of C sp3 decreases with a depth since an average information depth increases as following: REELS-surface (outer surface layer) < C KLL (1–2 layers) < REELS-bulk ~ C 1 s (2–5 layers) (Table 1). The smallest difference between the outer layer and 1–2 layers below obtained for rGOs of the thinnest flakes indicates the most efficient penetration of a reducing agent between the layers, i.e. case of rGO-NaBH4 and rGO-N2H4 (Fig. 8b). Comparison of a C sp3 content resulting from Auger C KLL spectra parameter D evaluation (Table S5), C 1 s spectra fitting (Table S3) and REELS spectra fitting (Table 4) the samples are indicated in the order of increasing number of layers and b difference in C sp3 percent resulting from C KLL, C 1 s and REELS-surface and REELS-bulk components. Notation ""—sample from Ref [5] Influence of reducing agents on structural and chemical properties of graphene flakes Dependence of different physicochemical properties of rGO flakes on the number of layers is presented in Figs. 6–9. The REELS indicates that the thickness of graphene flakes in GO and rGOs increases in the order of GO-foil-fresh < rGO-NaBH4 < rGO-N2H4 < rGO-C6H8O6 < rGO-CH2O < rGO-NaOH. This is accompanied by increasing graphitic cluster size, La, from 1.34 nm to 1.5 nm (Table 2) and C/O ratio (Fig. 6), distance between defects and decreasing defect density, decreasing interlayer distance, content of C sp3 hybridizations and vacancy defects (Fig. 9), carbon–oxygen groups and water situated between the interstitial layers (Fig. 7, Table S3, S4). The reduction using various reductants proceeds with different rate, where oxygen group content decreases in the following order: C–OH > C–O–C > C = O > C-OOH (Fig. 7a). The intensity and width of D and G modes in Raman spectra provide evidence on average distance between defects LD < 3 nm [19]. For rGOs, the XPS results indicate decreasing density of C sp3 and vacancy defects with increasing number of layers in graphene flakes (Fig. 9a). The estimate of LD from C sp3 and vacancy defects (Table S3) provides for GO the value of LD < 1 nm (C sp3 defects) and LD~ 15 nm (vacancy defects), whereas for rGOs, LD ~ 1–2 nm (C sp3 defects) and 4–15 nm (vacancy defects). The oxygen content obtained from XPS (Table S2) in a range of 26.1 at.% (GO-foil-fresh) to 8.6 at.% (rGO-N2H4) provides an estimate of distance between defects of LD ~ 1 nm and LD ~ 3 nm, respectively, which remains in agreement with the Raman spectroscopy data. Dependence of a the atomic percent of vacancy and C sp3 defects (Table S3) obtained from XPS spectra and b ratio of height and diameter values of rGOs to GO flakes (Table 2) on average number of layers in graphene flakes obtained from XRD and REELS (Table 4) The modification of the flake size of rGOs, i.e. height and diameter, depends on the applied reductants (Fig. 9b). The rGO-NaBH4 of the thinnest flakes (Table 4) and flake diameter similar to GO (Table 2) shows the largest number of surface carbon–oxygen groups and vacancy defects of larger content than in GO (Table S3, Fig. 7a, Fig. 9a). Also, no water is present between the interstitial sites (Fig. 4) and the ratio of rGO to GO group content decreases in the following order: C–OH > C-OOH > C = O > C–O–C (Fig. 7b) due to a mechanism of NaBH4 reduction converting C = O and C-OOH to C–OH groups as proposed by Samulski [56]. The rGO-N2H4, rGO-C6H8O6, rGO-NaOH, rGO-CH2O of thicker flakes and smaller diameters than GO exhibit a similar content of vacancy defects than GO, a smaller content of C sp3 defects and a content of oxygen groups decreasing in an order: C-OOH > C = O > C–OH > C–O–C. The binding energies for carbon–oxygen groups in graphene and defected graphene, where edges may be considered as defected graphene, do not vary distinctly [57]. However, a predominant rate of reduction between the interstitial site over that in plane results in thin flakes of a large diameter, whereas a higher rate of reduction in plane leads to smaller diameter flakes of larger number of layers. The reduction of GO at RT conditions in contrary to boiling conditions (e.g. exemplary rGO-C6H8O6-RT sample) results in a larger number of C sp3 and vacancy defects, carbon–oxygen groups, interstitial water and smaller average number of layer in flakes. The boiling reduction conditions lead to reparation of the vacancy defects and provide a higher reduction rate between the interstitial sites. Aging of GO (sample GO-foil-fresh and GO-foil) provides a larger number of layers in flakes, which could be influenced by additional oxidation and adsorption of water leading to graphene layers stacking. Summary of results and conclusions The bulk and surface properties of GO prepared from graphite using the modified Hummers method and rGOs obtained from this initial GO using a range of reducing agents were characterized by applying a set of complementary methods. The dependence of rGOs chemical and structural properties on the preparation procedures applying different reductants was investigated. Novelty and significance of these study rely on: (i) investigating the effect of few reducing agents like conventional and "green" reductants in similar conditions, (ii) characterizing GO and rGOs structural and chemical bulk and surface properties, (iii) providing correlation between rGOs bulk and surface properties due to the applied reductants, (iv) applying various complementary methods of analysis such as XRD, EDX, HR-TEM, PEELS, FTIR, Raman spectroscopy, REELS, XPS and XAES, (v) confirming reliability of different methods for evaluating the selected physicochemical properties. The GO and rGOs showed different interlayer distances in the range of ca. 0.446–0.345 nm with the largest interlayer distance for GO (0.721 nm). The GO and GO flakes of different size, i.e. diameter and height containing various average number of layers, were obtained. The number of layers in flakes resulting from XRD and REELS analyses was in agreement. For rGOs, the number of layers in flakes increased in the following order: NaBH4 < N2H4 < C6H8O6–RT < C6H8O6 < CH2O < NaOH from 3.4 ± 2 nm to 12.4 ± 2.1 nm. This was accompanied by increasing graphitic cluster size from 1.34 nm to 1.5 nm, distance between defects and decreasing defect density. The Raman and XPS data provided in GO-rGOs a consistent value of distance between defects, LD, from 1 to 3 nm. The content of oxygen groups decreased in the order of C–OH > C–O–C > C = O > C-OOH and the ratio of these oxygen groups in rGO to GO showed decrease for C–OH groups and increase for C-OOH > C = O > C–O–C groups. This provides an evidence on the importance of C–OH groups present in planes and edges to obtain thin graphene flakes. The GO showed the most inhomogeneous distribution of oxygen groups and C sp3 hybridizations at the surface and in the bulk. 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The authors (B.L., N.R., L.K.) acknowledge the support of the Horizon 2020 MSCA-COFUND agreement No. 711859 and the financial resources for science in the years 2017–2021 awarded for the implementation of an international co-financed project (3549/H2020/COFUND/2016/2). The author (J.T.) acknowledges the support by the European Regional Development Fund and Hungary in the frame of the project GINOP-2.2.1-15-2016-00012. The author (G.T.) acknowledges the support by the National Science Centre Poland for financial support (Grant "MINIATURA" No. 2017/01/X/ST5/01248). The author A. M. thanks the National Centre for Research and Development for support of the project LIDER/33/0117/L-9/17/NCBR/2018. The author L. S. would like to thank for the support of the project NCBiR (2018-2012) POIR.01.01.01-00-0802/17-00 (Polski Bazalt S.A.). Institute of Physical Chemistry, Polish Academy of Sciences, Kasprzaka 44/52, 01-224, Warsaw, Poland B. Lesiak & N. Rangam Faculty of Chemistry, Nicolaus Copernicus University in Torun, Gagarina 7, 87-100, Torun, Poland G. Trykowski & S. Biniak Section of Surface Physics, Laboratory of Materials Science, Institute for Nuclear Research, P.O. Box 51, 4001, Debrecen, Hungary J. Tóth & L. Kövér Faculty of Chemical and Process Engineering, Warsaw University of Technology, Waryńskiego 1, 00-645, Warsaw, Poland L. Stobinski & A. Malolepszy Polski Bazalt S.A, 157 Weteranów Str., 05-250, Radzymin, Poland L. Stobinski NANOMATERIALS LS, 14/38 Wyszogrodzka Str., 03-337, Warsaw, Poland B. Lesiak G. Trykowski J. Tóth S. Biniak L. Kövér N. Rangam A. Malolepszy Correspondence to B. Lesiak or G. Trykowski. The authors declare that they have no conflict of interest. Handling Editor: Yaroslava Yingling. Electronic supplementary material Below is the link to the electronic supplementary material. Supplementary file1 (DOCX 2540 kb) Lesiak, B., Trykowski, G., Tóth, J. et al. Chemical and structural properties of reduced graphene oxide—dependence on the reducing agent. J Mater Sci 56, 3738–3754 (2021). https://doi.org/10.1007/s10853-020-05461-1 Issue Date: February 2021	CommonCrawl
\begin{document} \baselineskip=17pt \title{On Qian's problem for $\mathcal{L}_{\infty}$-spaces} \author{Duanxu Dai\\ College of Mathematics and Computer Science\\ Quanzhou Normal University\\ Quanzhou 362000, China\\ E-mail: [email protected] \and} \date{} \maketitle \renewcommand{\arabic{footnote}}{} \footnote{2010 \emph{Mathematics Subject Classification}: Primary 46B04, 46B20; Secondary 46A22, 54C60.} \footnote{\emph{Key words and phrases}: $\varepsilon$-Isometry, Stability, Figiel theorem, Banach space.} \footnote{ Supported by the Natural Science Foundation of China (Grant No. 11601264) and the Outstanding Youth Scientific Research Personnel Training Program of Fujian Province and the High level Talents Innovation and Entrepreneurship Project of Quanzhou City and the Research Foundation of Quanzhou Normal University(Grant No. 2016YYKJ12). } \footnote{} \renewcommand{\arabic{footnote}}{\arabic{footnote}} \setcounter{footnote}{0} \begin{abstract} In this paper we devote to study Qian's problem for $\mathcal{L}_{\infty}$-spaces. Firstly, a positive answer to Qian's problem for $C(K)$-spaces is given by the assumption that $K$ has the C$\check{e}$ch-Stone property. Secondly, we obtain quantitative characterizations of separably injective spaces that turn out to give a positive answer to Qian's problem of 1995 in the setting of separable universality. Thirdly, we prove a sharpen quantitative and generalized Sobczyk theorem, which gives sharpen constants ($\alpha,\gamma$) for Qian's Problem. Finally, we give a more generalized Figiel theorem for $\mathcal{L}_{\infty}$-spaces. \end{abstract} \section{Introduction} Mazur and Ulam \cite{ma} in 1932 proved that every surjective isometry between two Banach spaces $X$ and $Y$ is necessarily affine. Since then, properties of isometries and generalizations there of between Banach spaces has continued for 86 years. On this period, many significant problems about perturbation properties of surjective $\varepsilon$-isometries were proposed and solved by numerous mathematicians. In particular, we mention the Hyers-Ulam problem \cite{hy} (see, for instance, \cite{ge}, \cite{gru}, and \cite{om}). In 1968, Figiel \cite{figiel} showed the following remarkable result(Figiel theorem): For every standard isometry $f:X\rightarrow Y$ there is a linear operator $T:L(f)\rightarrow X$ with $\\|T\\|=1$ so that $Tf=Id$ on $X$, where $L(f)$ is the closure of span $f(X)$ in $Y$ (see also \cite{ben} and \cite{du}). \begin{defin} Let $X,Y$ be two Banach spaces, $\varepsilon\geq0$, and let $f:X\rightarrow Y$ be a mapping. (1) $f$ is said to be an $\varepsilon$-isometry if \begin{align} \|\\|f(x)-f(y)\\|-\\|x-y\\|\|\leq\varepsilon\;\;{\rm for\; all}\; x,y\in X. \end{align} In particular, a $0$-isometry $f$ is simply called an isometry. (2) We say an $\varepsilon$-isometry $f$ is standard if $f(0)=0$. (3) A standard $\varepsilon$-isometry $f$ is $(\alpha,\gamma)$-stable if there exist $\alpha, \gamma>0$ and a bounded linear operator $T:L(f)\rightarrow X$ with $\\|T\\|\leq\alpha$ such that \begin{align}\label{E:1.2}\\|Tf(x)-x\\|\leq\gamma\varepsilon,\;\;{\rm for\;all\;}x\in X. \end{align} In this case, we also simply say $f$ is stable, if no confusion arises. (4) A pair $(X,Y)$ of Banach spaces $X$ and $Y$ is said to be stable if every standard $\varepsilon$-isometry $f:X\rightarrow Y$ is $(\alpha,\gamma)$-stable for some $\alpha,\gamma>0$. (5) A pair $(X,Y)$ of Banach spaces $X$ and $Y$ is called $(\alpha,\gamma)$-stable for some $\alpha,\gamma>0$ if every standard $\varepsilon$-isometry $f:X\rightarrow Y$ is $(\alpha,\gamma)$-stable. \end{defin} The study of non-surjective $\varepsilon$-isometries has also been considered (see, for instance, \cite{bao}, \cite{cheng}, \cite{cheng2}, \cite{Dai}, \cite{dil}, \cite{om}, \cite{qian}, \cite{sm} and \cite{ta}). Qian\cite{qian} proposed the following problem in 1995. \begin{prob} \label{P1} Is it true that for every pair $(X,Y)$ of Banach spaces $X$ and $Y$ there exists $\gamma>0$ such that every standard $\varepsilon$-isometry $f:X\rightarrow Y$ is $(\alpha,\gamma)$-stable for some $\alpha>0$? \end{prob} \noindent However, Qian \cite{qian} presented a counterexample showing that if a separable Banach space $Y$ contains an uncomplemented closed subspace $X$ then for every $\varepsilon>0$ there is a standard $\varepsilon$-isometry $f:X\rightarrow Y$ which is not stable. Recently, Cheng et al \cite{cheng3} showed the following sharp weak stability version. \begin{thm}[Cheng et al]\label{T:1.3} Let $X$ and $Y$ be Banach spaces, and let $f:X\rightarrow Y$ be a standard $\varepsilon$-isometry for some $\varepsilon\geq 0$. Then for every $x^\in X^$, there exists $\phi\in Y^$ with $\\|\phi\\|=\\|x^\\|\equiv r$ such that \begin{align} \|\langle\phi,f(x)\rangle-\langle x^,x\rangle\|\leq2\varepsilon r, \; for\;all\;x\in X.\end{align} \end{thm} For study of the stability of $\varepsilon$-isometries of Banach spaces, the following question was proposed in \cite{cheng2}. \begin{prob} Is there a characterization for the class of Banach spaces $\mathcal{X}$ satisfying given any $X\in \mathcal{X}$ and Banach space $Y$, the pair $(X,Y)$ is $($$(\alpha,\gamma)$-, resp.$)$ stable? \end{prob} Every space $X$ of this class is said to be a universally ($(\alpha,\gamma)$-, resp.) left-stable space. On one hand, Cheng, Dai, Dong et.al. \cite{cheng2} proved that every injective Banach space is a universally left-stable space. On the other hand, the first two authors Cheng and Dai, together with others \cite{bao} showed that every universally left-stable space is just a cardinality injective Banach space (i.e., a Banach space which is complemented in every superspace with the same cardinality) and they also showed that a dual space is injective (i.e., $X$ is complemented in $\ell_\infty(B_{X^})$)if and only if it is a universally left-stable space. This paper devotes to study Qian's problem for $\mathcal{L}_{\infty}$-spaces as follows. In Section 3, we obtain a weak positive answer to Qian's problem for $C(K)$-spaces (see Corollary \ref{main3}). Then by assuming that $K$ has the C$\check{e}$ch-Stone property, a positive answer for such a $C(K)$-space is given. The following Problem \ref{P} is also very natural. \begin{prob}\label{P} Is there a characterization for the class of Banach spaces $\mathcal{S}$ satisfying given any $X\in \mathcal{S}$ and separable Banach space $Y$, the pair $(X,Y)$ is $((\alpha,\gamma)$-, resp.$)$ stable? \end{prob} Every space $X$ of this class is said to be a separably universally (resp. $(\alpha,\gamma)$) left-stable space. In Section 4, we will show that all of these spaces of the class $\mathcal{S}$ coincide with separably injective Banach spaces. We here refer the reader to a very excellent paper \cite{ASC} by Avil$\acute{e}$s-S$\acute{a}$nchez-Castillo-Gonz$\acute{a}$lez-Moreno for further information about injective Banach spaces and separably injective Banach spaces where they resolved (under an additional assumption) a long standing problem proposed by Lindenstrauss in the middle sixties. The following theorem was proved by Sobczyk \cite{sob} which says that $c_0$ is separable separably injective space( A Banach space $X$ is said to be $\lambda$-separably injective if it has the following extension property: Every bounded linear operator $T$ from a closed subspace of a separable Banach space into $X$ can be extended to be a bounded operator on the whole space with its norm at most $\lambda\\|T\\|$. In this case, $X$ is said to be separably injective if it is $\lambda$-separably injective for some $\lambda\geq 1$ \cite{zip}) while Zippin \cite{zip} showed that $c_0$ is the unique separable separably injective space, up to an isomorphism. In 2014, Cheng, Dai et al \cite{cheng2} proved that $c_0$, up to an isomorphism, is the unique separable space such that the couple $(c_0, Y)$ is stable for every separable space $Y$. \begin{thm}[Sobczyk theorem \cite{sob}]\label{T:1.4} Let $X$ be a separable Banach space. If $E$ is a closed subspace of $X$ and $T: E\rightarrow c_0$ is a bounded operator then there exists an operator $\widetilde{T}: X \rightarrow c_0$ such that $\widetilde{T}\|_E=T$ and $\\|\widetilde{T}\\|\leq 2\\|T\\|$. \end{thm} In section 5, we also prove a sharpen quantitative and generalized Sobczyk theorem (see Theorem \ref{T:1.4}), that is, Theorem \ref{T:3.7}, which gives examples of nonseparable separably injective spaces $X$ (but not injective, i.e., $X$ is not complemented in $\ell_\infty(B_{X^})$) such that for some sharpen constants $\alpha,\gamma>0$, the couple $(X,Y)$ is $(\alpha,\gamma)$-stable for every separable space $Y$. In Section 6, we prove a more generalized Figiel theorem for $\mathcal{L}_{\infty,\lambda}$-spaces ( see \cite{AH}, \cite {ASC}, \cite{Bou}). All symbols and notations in this paper are standard. We use $X$ to denote a real Banach space and $X^$ its dual. $B_X$, ext $(B_{X^})$ and $S_X$ denote the closed unit ball of $X$, the set of all extremal points of $B_{X^}$ and the unit sphere of $X$, respectively. For a subset $A\subset X$, $\overline{A}$ and card $(A)$ stand respectively for the closure of $A$, the cardinality of $A$. Given a bounded linear operator $T:X\rightarrow Y$, $T^:Y^\rightarrow X^$ stands for its conjugate operator. We denote by $d(X,Y)=\inf\{\\|T\\|\cdot\\|T^{-1}\\|: T\;\text {is\;an\;isomorphism\; between}\;X\;\text{and} \;Y \}$ the Banach-Mazur distance between $X$ and $Y$. \section{Preliminaries } Recall that a Banach space $X$ is said to be $\lambda$-(resp. separably injective) injective if it has the following extension property: Every bounded linear operator $T$ from a closed subspace of a (resp. separable) Banach space into $X$ can be extended to be a bounded operator on the whole space with its norm at most $\lambda\\|T\\|$ (see, for instance, \cite{Alb}, \cite{ASC}, \cite{fa}, \cite{Wo}, \cite{zip}). In this case, $X$ is said to be injective (resp. separably injective) if it is $\lambda$-(resp. separably injective) injective for some $\lambda\geq 1$. The following Proposition \ref{P:3.1} follows easily from Remark \ref{R:3.2}. \begin{prop}\label{P:3.1} A (resp. separable) Banach space $X$ is $\lambda$-(resp. separably injective) injective if and only if it is $\lambda$-complemented in every (resp. separable) superspace (i.e., a normed linear space which contains $X$). \end{prop} The following Proposition \ref{P:3.2} was proved by Avil$\acute{e}$s, S$\acute{a}$nchez, Castillo, Gonz$\acute{a}$lez and Moreno (see \cite[Prop. 3.2]{ASC}). \begin{prop}\label{P:3.2}\rm (1) If a Banach space $X$ is $\lambda$-separably injective, then it is $3\lambda$-complemented in every superspace $Y$ such that $Y/X$ is separable. (2) If a Banach space $X$ is $\lambda$-complemented in every superspace $Y$ such that $Y/X$ is separable, then $X$ is $\lambda$-separably injective. \end{prop} \begin{rem}\label{R:3.2} For any set $\Gamma$, that $\ell_\infty(\Gamma)$ is $1$-injective follows from the Hahn-Banach theorem. \end{rem} Recall that $\mathcal{S}$ is the class of Banach spaces satisfying given any $X\in \mathcal{S}$ and separable Banach space $Y$, the pair $(X,Y)$ is ($(\alpha,\gamma)$-, resp.) stable. Every space $X$ of this class is said to be a separably universally ($(\alpha,\gamma)$-, resp.) left-stable space. In section 3, we completely solve Problem \ref{P}. That is, we prove that all of these spaces of the class $\mathcal{S}$ coincide with separably injective Banach spaces. \begin{lem}\label{main2} Suppose that $X$, $Y$ are Banach spaces. Let $\varepsilon\geq 0$. Assume that $f$ is a $\varepsilon-$ isometry from $X$ into $Y$ with $f(0)=0$. Then for every $w^$-dense subset $\Omega\subset$ ext $(B_{X^})$ there is a bounded linear operator $T: Y\rightarrow \ell_\infty(\Omega)$ such that $$\\|T f(x)-x\\|\leq 2\varepsilon,\;\;\text{for\;all}\;x\in X .$$ \end{lem} \begin{proof} By Theorem \ref{T:1.3}, for every $x^\in \Omega$, there exists a functional $Q(x^)\in S_{Y^}$ such that \begin{align} \|\langle Q(x^),f(x)\rangle-\langle x^,x\rangle\|\leq2\varepsilon , \; for\;all\;x\in X.\end{align} We now define a mapping $T:Y\rightarrow \ell_\infty(\Omega)$ by $$T(y)=\{Q(x^)(y)\}_{x^\in\Omega}.$$ It is clear that $T$ is a bounded linear operator with norm one and $$\\|T f(x)-x\\|=\sup_{x^\in\Omega}\|Q(x^)f(x)-x^(x)\|\leq 2 \varepsilon,\;\;\text{for\;all}\;x\in X .$$ \end{proof} The following Lemma \ref{L:1.1} follows from Qian's counterexample in \cite{qian} (see also \cite{cheng2}). \begin{lem}\label{L:1.1} Let $X$ be a closed subspace of Banach space $Y$. If $\text{card}\;(X)=\text{card}\;(Y)$, then for every $\varepsilon>0$ and every bijective mapping $g: X\rightarrow B_Y$ with $g(0)=0$, there is a standard $\varepsilon$-isometry $f:X\rightarrow Y$ defined for all $x\in X$ by $f(x)=x+\frac{\varepsilon}{2}g(x)$ such that (1) $L(f)\equiv\overline{\rm {span}}\;f(X)=Y$; (2) $X$ is $\lambda$ complemented in $Y$ whenever $f$ is $(\lambda,\gamma)$ stable for some $\lambda$, $\gamma>0$. \end{lem} \section{On Qian's problem for $C(K)$-spaces } Recall that a dual Banach space $Y^$ is said to have the point of weak star to norm continuity property (in short, $w^$-PCP) if every nonempty bounded subset of $Y^$ admits relative weak star neighborhoods of arbitrarily small norm diameter. For example, if $Y$ is an Asplund space, then $Y^$ has the $w^$-PCP (see, for instance, \cite{Phe}). Recall that a set valued mapping $F:X \rightarrow 2^{Y}$ is said to be usco provided it is nonempty compact valued and upper semicontinuous, i.e., $F(x)$ is nonempty compact for each $x\in X$ and $\{x\in X: F(x)\subset U\}$ is open in $X$ whenever $U$ is open in $Y$. We say that $F$ is usco at $x\in X$ if $F$ is nonempty compact valued and upper semicontinuous at $x$, i.e., for every open set $V$ of $Y$ containing $F(x)$ there exists a open neighborhood $U$ of $X$ such that $F(U)\subset V$. Therefore, $F$ is usco if and only if $F$ is usco at each $x\in X$. Recall that a mapping $\varphi: X \rightarrow Y$ is called a selection of $F$ if $\varphi(x)\in F(x)$ for each $x\in X$, moreover, we say $\varphi$ is a continuous (linear) selection of $F$ if $\varphi$ is a continuous (linear) mapping. We denote the graph of $F$ by $G(F)\equiv\{(x,y)\in X\times Y:y\in F(x)\}$, we write $F_1\subset F_2$ if $G(F_1)\subset G(F_2)$. A usco mapping $F$ is said to be minimal if $E=F$ whenever $E$ is a usco mapping and $E \subset F$ (see, for instance, \cite {Dai}, \cite[page 19, 102-109]{Phe}). The following Problem \ref{P2} is equivalent to Problem \ref{P1}. \begin{prob}\label{P2}Does there exist a constant $\gamma>0$ depending only on $X$ and $Y$ with the following property: For each $\varepsilon$-isometry $f:$ $X\rightarrow Y$ with $f(0)=0$ there is a $w^-w^$ continuous linear selection $Q$ of the set-valued mapping $\Phi$ from $X^$ into $2^{L(f)^}$ defined by $$\Phi(x^)=\{\phi\in L(f)^:\|\langle \phi,f(x)\rangle-\langle x^,x\rangle\|\leq\gamma\\|x^\\|\varepsilon ,\;\;\text{for\;all}\;x\in X \},$$ where $L(f)$ $=$ $\overline{\text{span}}\;f(X)$? \end{prob} The following Lemma \ref{main1} was motivated by Dai et.al. in \cite[Lemma 4.2]{Dai}. By an analogous argument we conclude the result on $w^-w^$ usco mappings, which will be used to prove Corollary \ref{main3}. \begin{lem}\label{main1} Suppose that $X$, $Y$ are Banach spaces. Let $\varepsilon\geq 0$. Assume that $f$ is a $\varepsilon-$ isometry from $X$ into $Y$ with $f(0)=0$, $H$ itself is a Baire subspace contained in $S_{X^}$ with respect to $w^$-topology. If we define a set-valued mapping $\Phi_1:S_{X^}\rightarrow 2^{S_{L(f)^}}$ by $$\Phi_1(x^)=\{\phi\in S_{L(f)^}:\|\langle \phi,f(x)\rangle-\langle x^,x\rangle\|\leq 4 \varepsilon ,\;\;\text{for\;all}\;x\in X \},$$ where $L(f)$ $=$ $\overline{\rm{span}}\,f(X)$, then (i) $\Phi_1$ is convex $w^$-usco at each point of $S_{X^}$. (ii) There exists a minimal convex $w^-w^$ usco mapping contained in $\Phi_1$. (iii) If, in addition, $Y^$ has the $w^$-PCP (especially, if $Y$ is an Asplund space) or $Y$ is separable, then there exists a selection $Q$ of $\Phi_1$ such that $Q$ is $w^-w^$ continuous on a $w^$-dense $G_\delta$ subset of $H$. \end{lem} \begin{proof} (i) It follows easily from \cite[Lemma 4.2 (i)]{Dai}. (ii)By Zorn Lemma (see \cite[Lemma 4.2 (ii)]{Dai} or \cite[Prop.7.3, p.103]{Phe}) there exists a minimal convex $w^-w^$ usco mapping contained in $\Phi_1$. (iii) By (ii) there is a minimal convex $w^-w^$ usco mapping $F\subset\Phi_1$, and $H$ itself is a Baire space with respect to $w^$-topology, and $Y^$ has the $w^$-PCP (especially, if $Y$ is an Asplund space) or $Y$ is separable, which follows easily from \cite[Lemma 7.14, p.106-107]{Phe} and \cite[Lemma 4.2 (iii)]{Dai}. \end{proof} \begin{rem} The above Lemma \ref{main1} also holds if we substitute $Y^$ and $S_{Y^}$ for $L(f)^$ and $S_{L(f)^}$, respectively. \end{rem} \begin{lem} \label{main2} Suppose that $X$, $Y$ are Banach spaces. Let $\varepsilon\geq 0$. Assume that $f$ is a $\varepsilon-$ isometry from $X$ into $Y$ with $f(0)=0$. Then (1) for every $w^$-dense subset $\Omega\subset$ ext $(B_{X^})$ there is a bounded linear operator $T: Y\rightarrow \ell_\infty(\Omega)$ such that $$\\|T f(x)-x\\|\leq 2\varepsilon,\;\;\text{for\;all}\;x\in X .$$ (2) If $Y^$ has the $w^$-PCP or $Y$ is separable, then there exists a $w^$-dense $G_\delta$ subset $\Omega\subset$ ext $B_{X^}$ such that there is a bounded linear operator $T: Y\rightarrow C(\Omega)$ such that $$\\|T f(x)-x\\|\leq 2 \varepsilon,\;\;\text{for\;all}\;x\in X .$$ \end{lem} \begin{proof} (1) By Theorem \ref{T:1.3}, for every $x^\in \Omega$, there exists a functional $Q(x^)\in S_{Y^}$ such that \begin{align} \|\langle Q(x^),f(x)\rangle-\langle x^,x\rangle\|\leq2\varepsilon , \; for\;all\;x\in X.\end{align} We now define a mapping $T:Y\rightarrow \ell_\infty(\Omega)$ by $$T(y)=\{Q(x^)(y)\}_{x^\in\Omega}.$$ It is clear that $T$ is a bounded linear operator with norm one and $$\\|T f(x)-x\\|=\sup_{x^\in\Omega}\|Q(x^)f(x)-x^(x)\|\leq 2 \varepsilon,\;\;\text{for\;all}\;x\in X .$$ (2) Since ext $(B_{X^})$ itself is a Baire space in its relative $w^$-topology (see \cite[p.217, line 17-19 ]{Hol}), it follows from Lemma \ref{main1} that there is a $w^-$ dense $G_\delta$ subset $\Omega$ in ext $(B_{X^})$ such that there is a $w^-w^$ continuous selection $Q$ of $\Phi_1$ on $\Omega$ satisfying that for every $x\in X$ and $x^\in \Omega$, the following inequality holds : \begin{align} \|\langle Q(x^),f(x)\rangle-\langle x^,x\rangle\|\leq2\varepsilon.\end{align} Let $T:Y\rightarrow \ell_\infty(\Omega)$ be defined as in (i). Therefore, $ T(y)\in C(\Omega)$ and $$\\|T f(x)-x\\|\leq 2\varepsilon,\;\;\text{for\;all}\;x\in X .$$ \end{proof} \begin{cor}\label{main3} Suppose that $X=C(K)$ for a compact Hausdorff space $K$ and $Y^$ has the $w^$-PCP (especially, if $Y$ is an Asplund space) or $Y$ is separable. Let $\varepsilon\geq 0$. Assume that $f$ is a standard $\varepsilon$-isometry from $X$ into $Y$. Then there exists a dense $G_\delta$ subset $\Omega$ of $K$ such that there is a bounded linear operator $T: Y\rightarrow C(\Omega)$ such that $T f-Id$ is uniformly bounded by $2\varepsilon$ on $X$. \end{cor} \begin{proof} It suffices to note that ext $(B_{X^})=\{\pm\delta_t:t\in K\}$ and $\{\delta_t:t\in K\}$ is a compact Baire space norming for $X$, and then apply Lemma \ref{main1} and Lemma \ref{main2} to conclude the results we desired by substituting $\{\delta_t:t\in K\}$ respectively for $H$ and ext $(B_{X^})$ everywhere. \end{proof} Now we introduce a new notion the so called the C$\check{e}$ch-Stone property that a topological space $K$ is said to have the C$\check{e}$ch-Stone property provided that for every $G_\delta$ dense subset $S\subset K$ there is a dense subset $\Omega\subset S$ such that $K$ is the C$\check{e}$ch-Stone compactification of $\Omega$. For example, $\beta\mathbb{N}$, the C$\check{e}$ch-Stone compactification of $\mathbb{N}$ has the C$\check{e}$ch-Stone property since every $G_\delta$ dense subset of it must contain the discrete space $\mathbb{N}$. As a consequence of Corollary \ref{main3} we have \begin{thm}Suppose that $X=C(K)$ where $K$ is a compact Hausdorff space and admitting the C$\check{e}$ch-Stone property. If either $Y^$ has the $w^$-PCP or $Y$ is separable, then the pair $(X,Y)$ is $(1,2)$-stable. \end{thm} \section{A quantitative characterization of separably injective Banach spaces} In this section, we combine Lemma \ref{main2} with some results from \cite{John} by Johnson-Oikhberg (Lindenstrass\cite{lin}, Rosenthal \cite{Ros}, S$\acute{a}$nchez \cite{San} and Castillo-Moreno \cite{Cas}) and from \cite{ASC} by Avil$\acute{e}$s-S$\acute{a}$nchez-Castillo-Gonz$\acute{a}$lez-Moreno to conclude a quantitative characterization of separably injective Banach spaces which completely solves Problem \ref{P}. \begin{thm}\label{T} \rm (i) If $X$ is a $\lambda$-separably injective Banach space, then the pair $(X, Y)$ is $(3\lambda,6\lambda)$ stable for every separable Banach space $Y$. (ii) If the pair $(X, Y)$ is $(\lambda,\gamma)$ stable for every separable Banach space $Y$, then $X$ is a $\lambda$-separably injective Banach space. \end{thm} \begin{proof} (i) Since $Y$ is separable, it follows from Lemma \ref{main2} that for every $w^$-dense subset $\Omega\subset$ ext $(B_{X^})$, there is a bounded linear operator $T: Y\rightarrow \ell_{\infty}(\Omega)$ such that $$\\| T f(x)-x\\|\leq 2 \varepsilon,\;\;\text{for\;all}\;x\in X .$$ Let $Z=\rm {\overline{span}}\;\{Tf(X)\cup X\}$. It follows from the continuity of $T$ that $Z/X$ is separable quotient space since $Y$ is separable. Since $X$ is $\lambda$- separable injective, it follows from Proposition \ref{P:3.2} that $X$ is $3\lambda$-complemented in $Z$. Therefore, there is a bounded linear operator $P:Z\rightarrow X$ with $\\|P\\|\leq 3\lambda$ such that $$\\| PT f(x)-x\|=\\|PT f(x)-Px\\| \leq 6\lambda \varepsilon,\;\;\text{for\;all}\;x\in X ,$$ where $PT: L(f)\rightarrow X$ satisfies that $\\|PT\\|\leq 3\lambda$. (ii) By Proposition \ref{P:3.2}, it suffices to show that $X$ is $\lambda$-complemented in every superspace $Y$ such that $Y/X$ is separable. Let $Y=X+Y/X$ be the algebraic direct sum. Since $Y/X$ is separable, card $(X)$ $=$ card $(Y)$. It follows from Qian's counterexample (i.e., Lemma \ref{L:1.1}) that there is an $\varepsilon$-isometry $f: X \rightarrow Y$ such that $Y=L(f)$ and $f(0)=0$. Hence by the assumption, there is a projection $P:Y\rightarrow X$ with $\\|P\\|\leq \lambda$ and we complete the proof. \end{proof} Recall that a compact Hausdorff space $K$ is said to be an $F$-space if disjoint open $F_\sigma$ sets have disjoint closures. For example, $\beta\mathbb{N}$, the C$\check{e}$ch-Stone compactification of $\mathbb{N}$ and $\beta\mathbb{N}\backslash\mathbb{N}$ are $F$-spaces. Since $C(K)$ is 1-separably injective for every $F$-space $K$ (see, for instance, \cite[p.202-203]{ASC}, \cite{lin}), we have \begin{cor}\label{C1} For every compact $F$-space $K$ (for example, $K=\beta\mathbb{N}\backslash\mathbb{N}$), the pair $(C(K), Y)$ $($resp. $(\ell_\infty/c_0, Y)$ $)$ is $(3,6)$ stable for every separable Banach space $Y$. \end{cor} \begin{proof} It is sufficient to note that $\ell_\infty/c_0$ is linearly isometric to $C(\beta\mathbb{N}\backslash\mathbb{N})$. \end{proof} Recall that a compact space $K$ has height n if $K^{(n)}=\emptyset$, where we write $K'$ for the derived set of $K$ and $K^{(n+1)}=(K^{(n)})'$. Since $C(K)$ is $(2n-1)$-separably injective for every $K$ of height $n$ (see, for instance, \cite[p.203]{ASC}), we have \begin{cor}\label{C1.1} For every compact space $K$ of height $n$, the pair $(C(K), Y)$ is $(6n-3,12n-6)$ stable for every separable Banach space $Y$. \end{cor} To combine Theorem \ref{T} with the results of Johnson-Oikhberg \cite{John} that for every family of $\lambda$-separably injective spaces $\{E_i\}_{i\in\Lambda}$, $(\sum_{i\in\Lambda} E_i)_{\ell_\infty}$ and $(\sum_{i\in\Lambda} E_i)c_0)$ are respectively $\lambda$-separably injective and $2\lambda^2$-separably injective, which was also proved by Rosenthal \cite{Ros}, S$\acute{a}$nchez \cite{San} and Castillo-Moreno \cite{Cas} with the estimates for the constant, respectively $\lambda(1+\lambda)^{+}$, $(3\lambda^2)^+$ and $6(1+\lambda)$, we have the following corollaries. \begin{cor}\label{C2} The pair $((\sum_{i\in\Lambda} E_i)_{\ell_\infty}, Y)$ is $(3\lambda,6\lambda)$ stable for every separable Banach space $Y$, where $\{E_i\}_{i\in\Lambda}$ is a family of $\lambda$-separably injective spaces. \end{cor} \begin{cor}\label{C3} The pair $((\sum_{i\in\Lambda} E_i)c_0), Y)$ is $(6\lambda^2, 12\lambda^2)$ (resp. $(3\lambda (1+\lambda)^{+}, 6\lambda (1+\lambda)^{+})$, $((9\lambda^2)^{+}, (18\lambda^2)^+)$ and $(18(1+\lambda),36(1+\lambda))$ stable for every separable Banach space $Y$, where $\{E_i\}_{i\in\Lambda}$ is a family of $\lambda$-separably injective spaces. \end{cor} \begin{rem} There are many other examples for separably injective Banach spaces, such as the Johnson-Lindenstrauss spaces \cite{John2}, Benyamini-space which is an M-space nonisomorphic to a $C(K)$-space \cite{ben1} and the WCG nontrivial twisted sums of $c_0(\Gamma)$ constructed by Argyros, Castillo, Granero, Jimenez and Moreno \cite{Arg} (see, for instance, \cite{ASC}). \end{rem} Qian \cite{qian} proved that the pair $(L_p,L_p)$ is stable for $1< p<\infty$. \v{S}emrl and V\"{a}is\"{a}l\"{a} \cite{sm} gave a sharp estimate for the constant pair $(\alpha,\gamma)$ with $\gamma=2$. Therefore, it is very natural to ask: \begin{prob}\label{P3} Is it true that the following pairs are stable for $1\leq p\leq\infty$ and $p\neq q<\infty$? \rm (1) $((\sum_{n=1}^{\infty} l_p^n)_{c_0}, (\sum_{n=1}^{\infty} l_p^n)_{c_0})$; (2)$((\sum_{n=1}^{\infty} l_p^n)_{\ell_\infty}, (\sum_{n=1}^{\infty} l_p^n)_{\ell_\infty})$; (3) $((\sum_{n=1}^{\infty} \ell_\infty)_{l_p}, (\sum_{n=1}^{\infty} \ell_\infty)_{l_p})$; (4) $((\sum_{n=1}^{\infty} l_p)_{\ell_\infty}, (\sum_{n=1}^{\infty} l_p)_{\ell_\infty})$; (5) $((\sum_{n=1}^{\infty} L_p)_{\ell_\infty}, (\sum_{n=1}^{\infty} L_p)_{\ell_\infty})$; (6) $((\sum_{n=1}^{\infty} c_0)_{l_p}, (\sum_{n=1}^{\infty} c_0)_{l_p})$; (7) $((\sum_{n=1}^{\infty} L_p)_{c_0}, (\sum_{n=1}^{\infty} L_p)_{c_0})$; (8) $((\sum_{n=1}^{\infty} \ell_p)_{c_0}, (\sum_{n=1}^{\infty} l_p)_{c_0})$. (9) $((\sum_{n=1}^{\infty} l_p)_{\ell_q}, (\sum_{n=1}^{\infty} l_p)_{\ell_q})$; (10) $((\sum_{n=1}^{\infty} L_p)_{\ell_q}, (\sum_{n=1}^{\infty} L_p)_{\ell_q})$. \end{prob} It is true for (1), (2), (3), (4) and (5) if $p=\infty$ as we have proved. In this case, it is not true for (6), (7) and (8) since $(\sum_{n=1}^{\infty} c_0)_{\ell_\infty}$, $(\sum_{n=1}^{\infty} L_\infty)_{c_0}$ and $(\sum_{n=1}^{\infty} \ell_\infty)_{c_0}$ are not complemented in $\ell_\infty$. If $1\leq p<\infty$, then it is also not true for (3), (4) and (5) since $(\sum_{n=1}^{\infty} \ell_\infty)_{l_p}$, $(\sum_{n=1}^{\infty} l_p)_{\ell_\infty}$ and $(\sum_{n=1}^{\infty} L_p)_{\ell_\infty}$ are not complemented in $\ell_\infty$. However, we do not know if it is true or not for the above problem \ref{P3} in general case. \section{A quantitative and generalized Sobczyk theorem } If $E_i$ is a $\lambda$-injective Banach spaces for each $i\in\Lambda$ (A Banach space $X$ is said to be $\lambda$-injective if it is $\lambda$-complemented in every superspace i.e., a Banach space which contains $X$), then by Theorem \ref{T:1.3} we have the following Theorem \ref{T:3.7} which gives sharpen constants ($\alpha,\gamma$) for Qian's Problem. In some sense, it could be seen as a quantitative and generalized Sobczyk theorem (See Theorem \ref{T:1.4}). \begin{thm}\label{T:3.7} Let $\Lambda$ and $\Gamma_i$ for each $i \in \Lambda$ are index sets. Suppose that one of the following three statements holds i) $X$ is isomorphic to $Z=$ $ (\sum_{i\in\Lambda} c_0(\Gamma_i))_{\ell_\infty}$ and $\lambda>d(X,Z)$; ii) $X$ is isomorphic to $Z=$ $(\sum_{i\in\Lambda} \ell_\infty(\Gamma_i))_{c_0}$ and $\lambda>d(X,Z)$; iii) $X=(\sum_{i\in\Lambda} E_i)c_0$ and $\{E_i\}_{i\in\Lambda}$ is a family of $\lambda$-injective Banach spaces. Then $(X,Y)$ is $(2\lambda,4\lambda)$-stable for every separable Banach space $Y$. \end{thm} \begin{proof} i) Let $X$ be a Banach space isomorphic to $ (\sum_{i\in\Lambda} c_0(\Gamma_i))_{\ell_\infty}$ and $T: X\rightarrow (\sum_{i\in\Lambda} c_0(\Gamma_i))_{\ell_\infty}$ be an isomorphism such that $\\|T\\|\cdot\\|T^{-1}\\|<\lambda$. For each $n\in\Lambda$ and $m\in\Gamma_n$, let $e_{nm}\in(\sum_{i\in\Lambda} c_0(\Gamma_i))_{\ell_\infty}$ with the standard biorthogonal functionals $e_{nm}^\in(\sum_{i\in\Lambda} c_0(\Gamma_i))_{\ell_\infty}^$ such that $e_{ij}^(e_{nm})=\delta_{in}\delta_{jm}$. For all $n\in\Lambda$ and $m\in\Gamma_n$, let $x_{nm}\in X$ be such that $T(x_{nm})=e_{nm}$. Let $T^:Z^\rightarrow X^$ be the conjugate operator of $T$. Then $$T(x)=\{\sum_{m\in\Gamma_n} (T^e_{nm}^)(x)e_{nm}\}_{n\in\Lambda}$$ and $$x=T^{-1}\{\sum_{m\in\Gamma_n} (T^e_{nm}^)(x)e_{nm}\}_{n\in\Lambda},\;\;{\rm for\;all}\;x\in X.$$ For all $n\in\Lambda$ and $m\in\Gamma_n$, let $x_{nm}^= T^e_{nm}^\in \\|T\\|B_{X^}$. It follows from Theorem \ref{T:1.3} that for every $n\in\Lambda$ and $m\in\Gamma_n$, there exists a functional $\phi_{nm}\in \\|T\\|B_{Y^}$ with $\\|\phi_{nm}\\|=\\|x_{nm}^\\|$ such that \begin{align}\label{E:4.3} \|\langle\phi_{nm},f(x)\rangle-\langle x_{nm}^,x\rangle\| \leq2 \varepsilon\\|T\\|,\; {\rm for \;all}\; x\in X.\end{align} It follows from the $w^-w^$ continuity of $T^$ that for each $n\in\Lambda$, $x^_{nm}\rightarrow0$ in the $w^$-topology of $X^$ Since $e^_{nm}\rightarrow0$ in the $w^$-topology of $Z^$. Let \begin{align} K=\{\psi\in \\|T\\|B(Y^):\|\langle\psi, f(x)\rangle\|\leq 2\varepsilon\\|T\\|, \; {\rm for \;all}\; x\in X\}.\end{align} Then $K$ is a nonempty $w^$-compact subset of $Y^$. Since $Y$ is separable, $(\\|T\\|B_{Y^},w^)$ is metrizable. Let $d$ be a metric such that $(\\|T\\|B_{Y^}, d)$ is homeomorphic to $(\\|T\\|B_{Y^},w^)$. Since for each $n\in\Lambda$, $(x^_{nm})$ is a $w^$-null net in $X^$, inequality \eqref{E:4.3} implies that for each $n\in\Lambda$, every $w^$-cluster point $\phi$ of $(\phi_{nm})$ is in $K$ such that $\\|\phi\\|\leq\\|T\\|$, which yields that ${\rm d}(\phi_{nm},K)\rightarrow0$ for each $n\in\Lambda$. Hence, for each $n\in\Lambda$, there is a net $(\psi_{nm})\subset K$ such that d$(\phi_{nm},\psi_{nm})\rightarrow0$, or equivalently, $\phi_{nm}-\psi_{nm}\rightarrow0$ in the $w^$-topology of $Y^$. Let $S:Y\rightarrow X$ be defined for every $y\in Y$ by \begin{align}S(y)= T^{-1}\{\sum_{m\in\Gamma_n}\langle \phi_{nm}-\psi_{nm},y\rangle e_{nm}\}_{n\in\Lambda}\in X.\end{align} Hence $$\\|S\\|\leq2\\|T\\|\cdot\\|T^{-1}\\|<2\lambda$$ and \begin{align}&\\|Sf(x)-x\\|=\\|T^{-1}\{\sum_{m\in\Gamma_n} \langle\phi_{nm}-\psi_{nm},f(x)\rangle e_{nm}\}_{n\in\Lambda}-T^{-1}\{\sum_{m\in\Gamma_n} \langle x_{nm}^,x\rangle e_{nm}\}_{n\in\Lambda}\\|\notag\\ &\leq\\|T^{-1}\\|\sup_{n\in\Lambda}(\\|\sum_{m\in\Gamma_n} \langle\phi_{nm}-\psi_{nm},f(x)\rangle e_{nm}-\sum_{m\in\Gamma_n} \langle x_{nm}^,x\rangle e_{nm}\\|)\notag\\ &\leq\\|T^{-1}\\|\cdot \sup_{n\in\Lambda}\sup_{m\in\Gamma_n}\| \langle\phi_{nm},f(x)\rangle-\langle x_{nm}^,x\rangle -\langle\psi_{nm},f(x)\rangle \|\notag\\ &\leq\\|T^{-1}\\|(\sup_{n\in\Lambda}\sup_{m\in\Gamma_n} \|\langle\phi_{nm},f(x)\rangle-\langle x_{nm}^,x\rangle\| +\sup_{n\in\Lambda}\sup_{m\in\Gamma_n}\|\langle\psi_{nm},f(x)\rangle\|)\notag\\ &\leq 4 \varepsilon\\|T\\|\cdot\\|T^{-1}\\|<4\varepsilon\lambda.\notag\end{align} ii-iii) For each $i\in\Lambda$, $\Gamma_i$ denotes by $B_{E_i^}$. It suffices to show this case that $X=(\sum_{i\in\Lambda} E_i)c_0$. Let $J: X=(\sum_{i\in\Lambda} E_i)c_0\rightarrow (\sum_{i\in\Lambda}\ell_\infty (B_{E_i^}))c_0=(\sum_{i\in\Lambda}\ell_\infty (\Gamma_i))c_0$ be the canonical embedding. For each $n\in\Lambda$, let $Q_n:(\sum_{i\in\Lambda}\ell_\infty (\Gamma_i))c_0\rightarrow\ell_\infty (\Gamma_n)$ be the canonical projection. Let $P_n:\ell_\infty (\Gamma_n) \rightarrow E_n$ be a family of projections with $\\|P_n\\|\leq \lambda$. For each $n\in\Lambda$ and $m\in\Gamma_n$, let $e_{nm}\in(\sum_{i\in\Lambda}\ell_\infty(\Gamma_i))_{c_0}$ with the standard biorthogonal functionals $e_{nm}^\in((\sum_{i\in\Lambda}\ell_\infty(\Gamma_i))_{c_0})^$ such that $e_{ij}^(e_{nm})=\delta_{in}\delta_{jm}$. Then $$x=\sum_{n\in\Lambda} \{(e_{nm}^)(x)\}_{m\in\Gamma_n} \;\;{\rm for\;all}\;x\in X.$$ By Theorem \ref{T:1.3}, for each $n\in\Lambda$ and $m\in\Gamma_n$, there exists $\phi_{nm}\in B_{Y^}$ with $\\|\phi_{nm}\\|=\\|e_{nm}^\\|$ such that \begin{align} \|\langle\phi_{nm},f(x)\rangle-\langle e_{nm}^,x\rangle\| \leq2 \varepsilon,\; {\rm for \;all}\; x\in X.\end{align} Clearly, $e^_{nm}\rightarrow0$ uniformly for each $m\in\Gamma_n$ in the $w^$-topology of $Z^$. Let \begin{align}K=\{\psi\in B(Y^):\|\langle\psi, f(x)\rangle\|\leq 2\varepsilon, \; {\rm for \;all}\; x\in X\}.\end{align} Since $\Gamma_n$ can be well ordered for every $n\in \Lambda$, we write $$\Gamma_n=\{0,1,2,\cdots, w_0, w_0+1, \cdots, w_1,\cdots \prec\Gamma_n \},$$ where $\Gamma_n$ also denotes by its ordinal number. It follows from i) that for each $n\in \Lambda$, there is a net $(\psi_{n0})\subset K$ such that d$(\phi_{n0},\psi_{n0})\rightarrow0$. We can choose $(\psi_{nm})\subset K$ such that for every $n\in \Lambda$ and $m\in\Gamma_n$, d$(\phi_{nm}, \psi_{nm})\leq$ d$(\phi_{n0},\psi_{n0})$ or equivalently, $(\phi_{nm}-\psi_{nm})\rightarrow0$ uniformly for each $m\in\Gamma_n$ in the $w^$-topology of $Y^$. Let $Q:Y \rightarrow (\sum_{i\in\Lambda}\ell_\infty (\Gamma_i))c_0$ be defined for all $y\in Y$ by \begin{align}Q(y)=\sum_{n\in \Lambda}\{\langle\phi_{nm}-\psi_{nm},y\rangle \}_{m\in\Gamma_n} \in (\sum_{i\in\Lambda}\ell_\infty (\Gamma_i))c_0, \end{align} which yields that \begin{align}\\|Q(y)\\|\leq(\sup_{n\in\Lambda, m\in\Gamma_n}\\|\phi_{nm}-\psi_{nm}\\|)\\|y\\|\leq2\\|y\\|.\end{align} Thus $$\\|Q\\|\leq2.$$ Let $S:Y\rightarrow X$ be defined for all $y\in Y$ by \begin{align}S(y)=\sum_{n\in\Lambda} P_nQ_nQ(y) =\sum_{n\in\Lambda}P_n\{\langle \phi_{nm}-\psi_{nm},y\rangle\}_{m\in\Gamma_n}. \end{align} Hence $$\\|S\\|=\sup_{n\in\Lambda} \\|P_nQ_nQ\\|\leq2\lambda$$ and \begin{align}&\\|Sf(x)-x\\|=\\|\sum_{n\in\Lambda} P_n\{\langle\phi_{nm}-\psi_{nm},f(x)\rangle \}_{m\in\Gamma_n}-\sum_{n\in\Lambda} P_n\{\langle e_{nm}^,x\rangle \}_{m\in\Gamma_n}\\|\notag\\ &\leq\lambda\sup_{n\in\Lambda}\sup_{m\in\Gamma_n}\|\langle\phi_{nm},f(x)\rangle-\langle e_{nm}^,x\rangle-\langle\psi_{nm},f(x)\rangle\|\notag\\ &\leq\lambda(\sup_{n\in\Lambda}\sup_{m\in\Gamma_n} \|\langle\phi_{nm},f(x)\rangle-\langle e_{nm}^,x\rangle\| +\sup_{n\in\Lambda}\sup_{m\in\Gamma_n}\|\langle\psi_{nm},f(x)\rangle\|)\notag\\ &\leq 4\varepsilon\lambda.\notag\end{align} Thus, our proof is completed. \end{proof} \section{A generalized Figiel theorem for $\mathcal{L}_{\infty,\lambda}$-spaces} Recall that a Banach space $X$ is said to be a $\mathcal{L}_{\infty,\lambda}$-space if every finite dimensional subspace $F$ of $X$ is contained in another finite dimensional subspace $E$ of $X$ such that $d(E, \ell_\infty^{\dim E})\leq\lambda$. In this case, a Banach space $X$ is said to be a $\mathcal{L}_{\infty}$-space if for some $\lambda>0$, $X$ is a $\mathcal{L}_{\infty,\lambda}$-space(see, for instance, \cite{AH}, \cite {ASC}, \cite{Bou}). For example, a $\lambda$-separably injective Banach space is a $\mathcal{L}_{\infty,9\lambda^+}$-space (see \cite [ p.199, Prop.3.5 (a)]{ASC})and every $C(K)$-space is also a $\mathcal{L}_{\infty}$-space. \begin{thm}\label{T:1.2} Suppose that $X$ is a $\mathcal{L}_{\infty,\lambda}$-space and $Y$ is a Banach space. Then for every standard $\varepsilon$-isometry $f:X \rightarrow Y$, there is a bounded linear operator $T: Y\rightarrow X^{*}$ such that $Tf-Id$ is uniformly bounded by $ 2 \lambda\varepsilon$ on $X$. \end{thm} \begin{proof} By Lemma \ref{main2}, for every $w^$-dense subset $\Omega\subset$ ext $(B_{X^})$ there is a bounded linear operator $S: Y\rightarrow \ell_\infty(\Omega)$ with norm one such that $$\\|S f(x)-x\\|\leq 2\varepsilon,\;\;\text{for\;all}\;x\in X .$$ Let $X=\cup_{i\in I} E_i$ such that for every $i, j\in (I,\succeq)$, $i\succeq j$ if and only if $E_i\supseteq E_j$ satisfying that for each $i\in I$, $\dim E_i<\infty$ and $d(E_i, \ell_\infty^{\dim E_i})\leq\lambda$. Hence for each $i\in I$, there exists a projection $P_i:\ell_\infty(\Omega)\rightarrow E_i$ such that $\\|P_i\\|<\lambda+\frac{1}{1+\dim E_i}$. Since $\{P_i\}_{i\in I}$ is uniformly bounded on $B_{\ell_\infty(\Omega)}$, it follows from the Arzel$\grave{a}$-Ascoli theorem that there is a subnet $\{\delta_i\}_{i\in\Lambda}$ of $I$ for an partial order set $\Lambda$ such that $P: \ell_\infty(\Omega)\rightarrow X^{}$ is well defined by $$P(y)=w^-\lim_{i\in\Lambda}P_{\delta_i}(y),\;\text {for\;all}\; y\in \ell_\infty(\Omega),$$ which yields that $$\\|P\\|\leq \lambda \;\text {and}\; P\|_X=Id.$$ Hence $$\\| Tf(x)-x\\|\leq 2\varepsilon\lambda,\;\;\text{for\;all}\;x\in X ,$$ where $T=PS: Y\rightarrow X^{**}$ with $\\|T \\|\leq\lambda$. \end{proof} \end{document}	arXiv
# The concept of force and its relationship to shapeless objects In physics and engineering, the concept of force is fundamental. Force is what causes an object to accelerate or change its state of motion. It can be thought of as a push or a pull on an object. When it comes to shapeless objects, force plays a crucial role in determining their behavior. Shapeless objects, such as fluids or gases, do not have a fixed shape or volume. Instead, they take the shape of their container and can flow or deform under the influence of external forces. The relationship between force and shapeless objects can be understood through the concept of pressure. Pressure is defined as the force exerted per unit area. It is the force distributed over the surface of an object. To illustrate this concept, let's consider a simple example. Imagine you have a balloon filled with air. When you squeeze the balloon, you are applying a force to it. This force causes the air molecules inside the balloon to move closer together, increasing the pressure inside the balloon. As a result, the balloon may deform or even burst if the force applied is too great. This example demonstrates how force can affect the shape and behavior of a shapeless object like air. Understanding the relationship between force and shapeless objects is crucial in various fields, including fluid mechanics, aerodynamics, and structural engineering. Another example of the relationship between force and shapeless objects is the flow of water through a pipe. When water flows through a pipe, it experiences resistance due to friction between the water molecules and the walls of the pipe. This resistance is caused by the force exerted by the pipe walls on the water. The pressure difference between the two ends of the pipe creates a force that pushes the water through the pipe. This force is what allows water to flow from a higher pressure region to a lower pressure region. ## Exercise Think of a real-life situation where force is applied to a shapeless object. Describe the object, the force applied, and the resulting behavior of the object. ### Solution One example could be squeezing toothpaste out of a tube. The toothpaste inside the tube is a shapeless object that takes the shape of the tube. When you apply force by squeezing the tube, the toothpaste is pushed out due to the pressure exerted on it. The behavior of the toothpaste is to flow out of the tube. # Understanding mass and weight and their role in shaping objects In physics and engineering, mass and weight are important concepts that play a role in shaping objects. Mass refers to the amount of matter an object contains, while weight is the force exerted on an object due to gravity. Mass is a fundamental property of an object and is measured in kilograms (kg). It is a scalar quantity, meaning it only has magnitude and no direction. The mass of an object remains constant regardless of its location. Weight, on the other hand, is a force and is measured in newtons (N). It is a vector quantity, meaning it has both magnitude and direction. The weight of an object depends on its mass and the acceleration due to gravity. The relationship between mass and weight can be understood through the equation: $$weight = mass \times acceleration\ due\ to\ gravity$$ The acceleration due to gravity on Earth is approximately 9.8 m/s^2. This means that for every kilogram of mass, an object experiences a weight of approximately 9.8 newtons. It is important to note that mass and weight are not the same thing. Mass is a measure of the amount of matter in an object, while weight is the force exerted on an object due to gravity. The mass of an object remains constant, while its weight can vary depending on the gravitational field it is in. To illustrate the concept of mass and weight, let's consider a simple example. Imagine you have a box with a mass of 5 kg. On Earth, where the acceleration due to gravity is 9.8 m/s^2, the weight of the box can be calculated as: $$weight = 5\ kg \times 9.8\ m/s^2 = 49\ N$$ This means that the box experiences a weight of 49 newtons on Earth. ## Exercise Calculate the weight of an object with a mass of 2.5 kg on Earth. ### Solution The weight of the object can be calculated as: $$weight = 2.5\ kg \times 9.8\ m/s^2 = 24.5\ N$$ # Exploring motion and its connection to shapeless objects Motion is a fundamental concept in physics that is closely connected to shapeless objects. Motion refers to the change in position of an object over time. It can be described in terms of displacement, velocity, and acceleration. Displacement is a vector quantity that represents the change in position of an object. It is defined as the difference between the final and initial positions of an object. Displacement has both magnitude and direction. Velocity is a vector quantity that describes the rate of change of displacement. It is defined as the change in displacement divided by the change in time. Velocity has both magnitude and direction. Acceleration is a vector quantity that describes the rate of change of velocity. It is defined as the change in velocity divided by the change in time. Acceleration has both magnitude and direction. To understand the connection between motion and shapeless objects, let's consider an example. Imagine a car traveling along a straight road. The car's motion can be described in terms of its displacement, velocity, and acceleration. If the car starts at position 0 and moves to position 100 meters in 10 seconds, its displacement would be 100 meters. The displacement has a magnitude of 100 meters and a direction of the positive x-axis. The car's velocity can be calculated by dividing the displacement by the time taken: $$velocity = \frac{displacement}{time} = \frac{100\ m}{10\ s} = 10\ m/s$$ The velocity has a magnitude of 10 m/s and a direction of the positive x-axis. If the car's velocity changes over time, it is experiencing acceleration. For example, if the car's velocity changes from 10 m/s to 20 m/s in 5 seconds, the acceleration can be calculated by dividing the change in velocity by the change in time: $$acceleration = \frac{change\ in\ velocity}{change\ in\ time} = \frac{20\ m/s - 10\ m/s}{5\ s} = 2\ m/s^2$$ The acceleration has a magnitude of 2 m/s^2 and a direction of the positive x-axis. To further illustrate the concept of motion and its connection to shapeless objects, let's consider the motion of a ball thrown into the air. When the ball is thrown upward, it experiences a positive acceleration due to gravity. As it reaches its highest point and starts to fall back down, it experiences a negative acceleration due to gravity. The motion of the ball can be described in terms of its displacement, velocity, and acceleration at different points in time. By analyzing the motion of the ball, we can gain a better understanding of how shapeless objects move in different scenarios. ## Exercise A car travels a distance of 200 meters in 20 seconds. Calculate the car's velocity. ### Solution The car's velocity can be calculated by dividing the displacement by the time taken: $$velocity = \frac{displacement}{time} = \frac{200\ m}{20\ s} = 10\ m/s$$ # The properties of matter and how they affect shapeless objects The properties of matter play a crucial role in shaping and influencing the behavior of shapeless objects. Matter refers to anything that has mass and occupies space. It can exist in different states, such as solid, liquid, or gas, and can undergo physical and chemical changes. One important property of matter is its mass. Mass is a measure of the amount of matter in an object and is typically measured in kilograms (kg). It is a scalar quantity, meaning it has magnitude but no direction. Another property of matter is its density. Density is defined as the mass of an object divided by its volume. It represents how tightly packed the particles of matter are within an object. Density is typically measured in kilograms per cubic meter (kg/m^3) or grams per cubic centimeter (g/cm^3). The shape of an object is also a property of matter. Objects can have different shapes, such as rectangular, cylindrical, or irregular. The shape of an object can affect its stability, strength, and ability to withstand external forces. To understand how the properties of matter affect shapeless objects, let's consider an example. Imagine two objects: a solid iron ball and a hollow plastic ball. Both balls have the same mass, but their densities and shapes are different. The solid iron ball has a higher density than the hollow plastic ball. This means that the particles of matter in the iron ball are more tightly packed than in the plastic ball. As a result, the iron ball is heavier and more compact. The shape of the objects also affects their behavior when subjected to external forces. The solid iron ball, with its compact and dense structure, is more resistant to deformation and can withstand higher forces without breaking. On the other hand, the hollow plastic ball, with its less dense and less compact structure, is more flexible and can deform under lower forces. Understanding the properties of matter and how they affect shapeless objects is essential in fields such as engineering and materials science. Engineers need to consider the properties of different materials when designing structures or selecting materials for specific applications. By understanding how matter behaves and interacts with external forces, engineers can create safer and more efficient structures. To further illustrate the concept of how the properties of matter affect shapeless objects, let's consider the example of a bridge. Bridges are designed to withstand the forces exerted on them by the weight of vehicles, wind, and other external factors. The choice of materials for building a bridge is crucial. Different materials have different properties, such as strength, density, and durability. Engineers need to select materials that can withstand the expected forces and environmental conditions. For example, steel is a commonly used material for bridge construction due to its high strength and durability. It can withstand heavy loads and resist deformation. Concrete is another commonly used material for bridge construction, as it has high compressive strength and can be molded into different shapes. By understanding the properties of matter and how they affect shapeless objects, engineers can design structures that are safe, efficient, and long-lasting. ## Exercise Why is the density of an object an important property to consider when designing structures? ### Solution The density of an object is an important property to consider when designing structures because it affects the weight and stability of the structure. Objects with higher densities are heavier and may require stronger supports or foundations. Additionally, the density of materials used in the structure can affect its overall strength and ability to withstand external forces. # Using vectors to represent and manipulate shapeless objects Vectors are mathematical objects that are used to represent and manipulate shapeless objects. A vector is defined by both magnitude and direction, and it can be represented as an arrow in a coordinate system. In physics and engineering, vectors are often used to describe the motion, forces, and other physical quantities associated with shapeless objects. Vectors can be added, subtracted, multiplied, and divided to perform various operations and calculations. One important concept in vector manipulation is vector addition. When two vectors are added together, their magnitudes are added, and their directions are combined. This operation is often used to find the resultant vector of multiple forces acting on an object. Another important operation is vector subtraction. When two vectors are subtracted, their magnitudes are subtracted, and their directions are combined. This operation is often used to find the net force acting on an object when multiple forces are present. To better understand how vectors are used to represent and manipulate shapeless objects, let's consider an example. Imagine a car traveling on a curved road. The car's motion can be described by a velocity vector, which has both magnitude (speed) and direction. If the car is traveling at a constant speed of 60 miles per hour in a northward direction, the velocity vector can be represented as $\vec{v} = 60 \, \text{mph} \, \hat{i}$, where $\hat{i}$ represents the northward direction. Now, let's say the car encounters a strong crosswind blowing from the west. The crosswind can be represented by a wind vector, $\vec{w} = 20 \, \text{mph} \, \hat{j}$, where $\hat{j}$ represents the westward direction. To find the resultant velocity of the car, we can add the velocity vector and the wind vector. The resultant vector, $\vec{r}$, is given by $\vec{r} = \vec{v} + \vec{w}$. $\vec{r} = 60 \, \text{mph} \, \hat{i} + 20 \, \text{mph} \, \hat{j}$ By adding the magnitudes and combining the directions, we can find the resultant velocity vector of the car. Let's calculate the resultant velocity vector of the car using the given values. $\vec{r} = 60 \, \text{mph} \, \hat{i} + 20 \, \text{mph} \, \hat{j}$ To add the magnitudes, we simply add 60 mph and 20 mph: $\vec{r} = 80 \, \text{mph} \, \hat{i} + \hat{j}$ So, the resultant velocity vector of the car is 80 mph in the northward direction with a slight westward component. ## Exercise Consider a boat traveling at a speed of 30 knots in a southeast direction. The boat encounters a current flowing at a speed of 10 knots in a northwest direction. Calculate the resultant velocity vector of the boat. ### Solution The velocity vector of the boat, $\vec{v}$, is 30 knots in the southeast direction, which can be represented as $\vec{v} = 30 \, \text{knots} \, \hat{i} + 30 \, \text{knots} \, \hat{j}$. The current vector, $\vec{c}$, is 10 knots in the northwest direction, which can be represented as $\vec{c} = -10 \, \text{knots} \, \hat{i} - 10 \, \text{knots} \, \hat{j}$. To find the resultant velocity vector, we can add the velocity vector and the current vector: $\vec{r} = \vec{v} + \vec{c}$ $\vec{r} = (30 \, \text{knots} \, \hat{i} + 30 \, \text{knots} \, \hat{j}) + (-10 \, \text{knots} \, \hat{i} - 10 \, \text{knots} \, \hat{j})$ Simplifying the expression, we get: $\vec{r} = 20 \, \text{knots} \, \hat{i} + 20 \, \text{knots} \, \hat{j}$ So, the resultant velocity vector of the boat is 20 knots in the southeast direction. # Applications of shapeless objects in engineering Shapeless objects, such as fluids and gases, play a crucial role in engineering. They are used in various applications, from designing hydraulic systems to analyzing fluid flow in pipes. Understanding how shapeless objects behave and interact with their surroundings is essential for engineers. One common application of shapeless objects in engineering is in fluid mechanics. Fluids, such as water and air, are shapeless and can flow freely. Engineers use the principles of fluid mechanics to design and analyze systems that involve the movement of fluids, such as pumps, turbines, and pipes. For example, in designing a water distribution system for a city, engineers need to consider the flow of water through pipes of different sizes and shapes. They use the principles of fluid mechanics to determine the pressure, velocity, and flow rate of the water in the pipes. This information is crucial for ensuring that water reaches every household with adequate pressure and volume. Another application of shapeless objects in engineering is in heat transfer. Heat can be transferred through conduction, convection, and radiation. Engineers use the principles of heat transfer to design systems that efficiently transfer heat, such as heat exchangers and cooling systems. For instance, in designing a car engine, engineers need to ensure that heat generated by the combustion process is effectively dissipated to prevent overheating. They use the principles of heat transfer to design cooling systems that remove excess heat from the engine and maintain its operating temperature within a safe range. An example of the application of shapeless objects in engineering is the design of an airplane wing. The shape of the wing is carefully designed to generate lift, which allows the airplane to overcome gravity and stay airborne. The shape of the wing is designed to create a pressure difference between the upper and lower surfaces. This pressure difference, combined with the shape of the wing, creates lift. Engineers use the principles of fluid mechanics to analyze the flow of air over the wing and optimize its shape for maximum lift and efficiency. ## Exercise Consider a hydraulic system that uses a pump to generate pressure and move fluid through pipes. How would you apply the principles of fluid mechanics to design and analyze this system? ### Solution To design and analyze a hydraulic system, you would apply the principles of fluid mechanics. This includes determining the pressure, velocity, and flow rate of the fluid in the pipes, as well as considering factors such as pipe size, shape, and material. By applying these principles, you can ensure that the hydraulic system operates efficiently and effectively. # The role of shapeless objects in physics Shapeless objects, such as fluids and gases, play a significant role in physics. They are used to study and understand various phenomena, from the behavior of gases under different conditions to the motion of fluids in pipes. Understanding the properties and behavior of shapeless objects is crucial for physicists. One important concept in physics is the study of gases. Gases are shapeless and can expand and contract to fill the space available to them. Physicists use the principles of gas laws, such as Boyle's law and Charles's law, to describe and predict the behavior of gases under different conditions, such as changes in pressure and temperature. For example, Boyle's law states that the pressure of a gas is inversely proportional to its volume, assuming constant temperature. This law helps physicists understand how gases behave when compressed or expanded, and it has practical applications in fields such as chemistry and engineering. Another important concept in physics is the study of fluid dynamics. Fluids, such as liquids and gases, are shapeless and can flow freely. Physicists use the principles of fluid dynamics to study and analyze the motion of fluids, such as the flow of water in rivers and the behavior of air around objects. For instance, physicists use the principles of fluid dynamics to study the flow of blood in the human circulatory system. By understanding how blood flows through the arteries and veins, physicists can gain insights into the functioning of the cardiovascular system and develop treatments for various cardiovascular diseases. An example of the role of shapeless objects in physics is the study of atmospheric pressure. Atmospheric pressure is the force exerted by the weight of the air above a given point on the Earth's surface. Physicists study atmospheric pressure to understand its effects on weather patterns, air density, and the behavior of gases in the atmosphere. Atmospheric pressure is responsible for various phenomena, such as the formation of weather systems, the movement of air masses, and the behavior of gases in the atmosphere. By studying atmospheric pressure, physicists can gain insights into the dynamics of the Earth's atmosphere and develop models to predict and understand weather patterns. ## Exercise Consider the behavior of gases under different conditions, such as changes in pressure and temperature. How would you apply the principles of gas laws to describe and predict the behavior of gases? ### Solution To describe and predict the behavior of gases under different conditions, you would apply the principles of gas laws. For example, Boyle's law states that the pressure of a gas is inversely proportional to its volume, assuming constant temperature. This law can be used to predict how the volume of a gas changes when its pressure is changed, or vice versa. Another gas law, Charles's law, states that the volume of a gas is directly proportional to its temperature, assuming constant pressure. This law can be used to predict how the volume of a gas changes when its temperature is changed, or vice versa. By applying these gas laws and other principles of gas behavior, physicists can describe and predict the behavior of gases under different conditions, such as changes in pressure and temperature. This knowledge is crucial for understanding various phenomena, from the behavior of gases in chemical reactions to the properties of air in the atmosphere. # Analyzing and calculating forces on shapeless objects Analyzing and calculating forces on shapeless objects is an essential part of physics and engineering. Forces can cause shapeless objects to accelerate, deform, or change their state of motion. Understanding the forces acting on shapeless objects is crucial for predicting and explaining their behavior. One important concept in analyzing forces on shapeless objects is Newton's second law of motion. This law states that the acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass. Mathematically, this can be expressed as $F = ma$, where $F$ is the net force acting on the object, $m$ is its mass, and $a$ is its acceleration. This equation allows physicists and engineers to calculate the net force acting on a shapeless object based on its mass and acceleration. For example, if a car with a mass of 1000 kg accelerates at a rate of 5 m/s^2, the net force acting on the car can be calculated using Newton's second law: $F = ma = 1000 \, \text{kg} \times 5 \, \text{m/s}^2 = 5000 \, \text{N}$ This means that a net force of 5000 Newtons is required to accelerate the car with a mass of 1000 kg at a rate of 5 m/s^2. Another important concept in analyzing forces on shapeless objects is the concept of equilibrium. An object is in equilibrium when the net force acting on it is zero. This means that the forces acting on the object are balanced, and it is either at rest or moving with constant velocity. An example of analyzing forces on shapeless objects is the study of fluid flow in pipes. When fluid flows through a pipe, various forces act on the fluid, such as pressure forces, gravity forces, and frictional forces. To analyze and calculate these forces, physicists and engineers use principles such as Bernoulli's equation and the Navier-Stokes equations. These equations allow them to determine the pressure, velocity, and flow rate of the fluid in the pipe based on various factors, such as the pipe's geometry, the fluid's properties, and the forces acting on the fluid. By analyzing and calculating these forces, physicists and engineers can design and optimize pipe systems for various applications, such as water distribution, oil and gas transportation, and air conditioning. ## Exercise Consider a block of ice sliding down a frictionless inclined plane. How would you analyze and calculate the forces acting on the block of ice? ### Solution To analyze and calculate the forces acting on the block of ice sliding down a frictionless inclined plane, you would consider the forces of gravity and normal force. The force of gravity, or weight, acts vertically downward and can be calculated using the equation $F = mg$, where $m$ is the mass of the block and $g$ is the acceleration due to gravity. The normal force acts perpendicular to the inclined plane and counteracts the force of gravity. It can be calculated using the equation $F_{\text{normal}} = mg \cos(\theta)$, where $\theta$ is the angle of the inclined plane. Since the inclined plane is frictionless, there is no frictional force acting on the block of ice. By analyzing and calculating these forces, physicists and engineers can predict and explain the motion of the block of ice sliding down the inclined plane. # Designing structures using shapeless objects Designing structures using shapeless objects is a crucial aspect of engineering. Shapeless objects, such as beams, columns, and trusses, are used to create stable and functional structures that can withstand various forces and loads. One important consideration in designing structures is the concept of load-bearing capacity. The load-bearing capacity of a structure refers to its ability to support and distribute loads without experiencing failure or collapse. Engineers must carefully analyze and calculate the forces acting on the structure to ensure that it can safely bear the intended loads. To design structures using shapeless objects, engineers must consider factors such as material properties, structural geometry, and the types of loads that the structure will be subjected to. They use principles of mechanics and structural analysis to determine the appropriate size, shape, and arrangement of shapeless objects in the structure. An example of designing structures using shapeless objects is the design of a bridge. Bridges are complex structures that must be able to support the weight of vehicles, pedestrians, and other loads while spanning a gap, such as a river or a valley. Engineers use shapeless objects, such as beams and trusses, to create the main structural elements of the bridge. They analyze and calculate the forces acting on these shapeless objects, such as the weight of the bridge itself, the weight of the vehicles crossing the bridge, and the forces caused by wind and earthquakes. By carefully designing and arranging the shapeless objects in the bridge, engineers can ensure that the bridge has the necessary load-bearing capacity to safely support the intended loads. They also consider factors such as aesthetics, cost, and environmental impact in the design process. ## Exercise Consider a building that needs to support a heavy rooftop garden. How would you design the structure using shapeless objects? ### Solution To design a structure that can support a heavy rooftop garden, engineers would need to consider factors such as the weight of the garden, the weight of the building itself, and the forces caused by wind and earthquakes. They would use shapeless objects, such as columns and beams, to create the main structural elements of the building. By analyzing and calculating the forces acting on these shapeless objects, engineers can determine the appropriate size, shape, and arrangement of the columns and beams to ensure that the structure has the necessary load-bearing capacity. Engineers would also consider other factors, such as the material properties of the shapeless objects, the structural geometry of the building, and any additional design requirements or constraints. By carefully designing the structure using shapeless objects, engineers can create a stable and functional building that can safely support the heavy rooftop garden. # Using shapeless objects to solve real-world problems Shapeless objects, such as springs, ropes, and pulleys, can be used to solve a wide range of real-world problems. These objects have unique properties that allow them to transfer and transform forces, making them valuable tools in engineering and physics. One example of using shapeless objects to solve real-world problems is in the design of suspension systems for vehicles. Suspension systems use springs and shock absorbers to provide a smooth and comfortable ride by absorbing and dampening the forces caused by bumps and uneven surfaces. In a typical suspension system, shapeless objects such as coil springs are used to support the weight of the vehicle and provide flexibility to absorb shocks. When the vehicle encounters a bump, the spring compresses, storing potential energy. This energy is then released, pushing the vehicle back up and reducing the impact of the bump. Another example is the use of ropes and pulleys in lifting and hoisting systems. Shapeless objects such as ropes and pulleys allow us to lift heavy objects with less effort by distributing the force over multiple ropes and changing the direction of the force. For instance, a block and tackle system consists of multiple pulleys and ropes. By pulling the ropes, we can lift heavy loads with less force. The shapeless objects in the system distribute the force evenly, making it easier to lift the load. Shapeless objects can also be used in engineering structures to distribute and redirect forces. For example, in a suspension bridge, shapeless objects such as cables and suspension rods are used to support the weight of the bridge deck and transfer it to the bridge towers and anchorages. The cables in a suspension bridge are under tension and can withstand large forces. They distribute the weight of the bridge deck along their length, allowing the bridge to span long distances without the need for additional support columns. ## Exercise Think of a real-world problem that can be solved using shapeless objects. Describe the problem and explain how shapeless objects can be used to solve it. ### Solution One real-world problem that can be solved using shapeless objects is the construction of a zip line. A zip line is a recreational activity where a person slides down a cable suspended between two points, usually at a high elevation. The problem is how to create a safe and enjoyable zip line experience. Shapeless objects, such as the cable and the pulleys, are used to solve this problem. The cable provides a strong and flexible structure that can support the weight of the person and the forces caused by the sliding motion. The pulleys allow for smooth movement along the cable and help to distribute the forces evenly. By using shapeless objects in the design of the zip line, engineers can create a thrilling and safe experience for participants, allowing them to enjoy the sensation of flying through the air while minimizing the risk of accidents. # Future advancements and possibilities in shapeless objects The field of shapeless objects is constantly evolving, and there are many exciting advancements and possibilities on the horizon. As technology continues to advance, we can expect to see new applications and uses for shapeless objects in various industries, including physics and engineering. One area of future advancement is the development of shape-memory materials. These materials have the ability to change their shape in response to external stimuli, such as temperature or pressure. This opens up possibilities for creating self-adjusting structures and devices that can adapt to different conditions. For example, imagine a building that can change its shape based on the weather. On a hot day, the building could expand to provide more ventilation and cooling, while on a cold day, it could contract to conserve heat. This would not only improve energy efficiency but also enhance the comfort and functionality of the building. Another area of future advancement is the use of shapeless objects in robotics and automation. Shapeless objects can be used to create flexible and adaptable robotic systems that can perform a wide range of tasks. These robots could be used in various industries, such as manufacturing, healthcare, and exploration. For instance, imagine a robot that can change its shape to fit through narrow spaces or manipulate objects of different sizes and shapes. This would greatly expand the capabilities of robots and enable them to perform tasks that were previously impossible or difficult. In addition, advancements in materials science and nanotechnology may lead to the development of shapeless objects with even more unique and extraordinary properties. For example, researchers are exploring the use of shapeless objects with programmable properties, such as the ability to change their stiffness or conductivity on demand. Imagine a material that can change from being soft and flexible to rigid and strong with the flip of a switch. This could have applications in various fields, such as aerospace, where lightweight and adaptable materials are highly desirable. Overall, the future of shapeless objects holds great promise. As we continue to push the boundaries of science and engineering, we can expect to see new and innovative uses for shapeless objects that will revolutionize various industries and improve our lives in countless ways. ## Exercise Brainstorm and describe one potential future advancement or possibility in the field of shapeless objects. Explain how this advancement or possibility could impact a specific industry or application. ### Solution One potential future advancement in the field of shapeless objects is the development of shapeless objects with self-healing properties. Imagine a material that can repair itself when damaged, similar to how our skin heals when we get a cut. This advancement could have significant implications in the automotive industry. Currently, when a car gets damaged, it often requires expensive repairs or replacement parts. However, with shapeless objects that can self-heal, cars could potentially repair themselves, reducing the need for costly repairs and improving the longevity and durability of vehicles. This advancement could also have environmental benefits, as it would reduce the amount of waste generated from damaged or worn-out parts. Additionally, self-healing shapeless objects could be used in other industries, such as aerospace and construction, where the ability to repair and maintain structures and equipment is crucial. Overall, the development of shapeless objects with self-healing properties has the potential to revolutionize various industries by improving the efficiency, durability, and sustainability of products and structures.	Textbooks
Optimal voltage control of non-stationary eddy current problems MCRF Home Preface: A tribute to professor Eduardo Casas on his 60th birthday March 2018, 8(1): 1-34. doi: 10.3934/mcrf.2018001 Second order optimality conditions for optimal control of quasilinear parabolic equations Lucas Bonifacius 1,, and Ira Neitzel 2, Technische Universität München, Fakultät für Mathematik, Boltzmannstr. 3, 85748 Garching, Germany Rheinische Friedrich-Wilhelms-Universität Bonn, Institut für Numerische Simulation, Wegelerstr. 6, 53115 Bonn, Germany * Corresponding author: Ira Neitzel Received March 2017 Revised September 2017 Published January 2018 Fund Project: The first author is supported by the International Research Training Group IGDK, funded by the German Science Foundation (DFG) and the Austrian Science Fund (FWF) Full Text(HTML) We discuss an optimal control problem governed by a quasilinear parabolic PDE including mixed boundary conditions and Neumann boundary control, as well as distributed control. Second order necessary and sufficient optimality conditions are derived. The latter leads to a quadratic growth condition without two-norm discrepancy. Furthermore, maximal parabolic regularity of the state equation in Bessel-potential spaces $H_D^{-\zeta,p}$ with uniform bound on the norm of the solution operator is proved and used to derive stability results with respect to perturbations of the nonlinear differential operator. Keywords: Optimal control, second order optimality conditions, quasilinear parabolic partial differential equation, nonautonomous equation, maximal parabolic regularity. Mathematics Subject Classification: 35K59, 49K20, 90C48. Citation: Lucas Bonifacius, Ira Neitzel. Second order optimality conditions for optimal control of quasilinear parabolic equations. Mathematical Control & Related Fields, 2018, 8 (1) : 1-34. doi: 10.3934/mcrf.2018001 P. 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Summary of differentiability and integrability exponents $p$ Given by isomorphism $-\nabla\cdot\mu\nabla + 1 \colon W_D^{1,p}\rightarrow W_D^{-1,p}$ and close to the spatial dimension $d$ ; see Assumption 3. $\zeta$ Differentiability exponent close to one defining $H_D^{-\zeta,p}$ . $s$ Integrability exponent for the controls determined by $p$ , $\zeta$ , and $d$ , possibly large; see Assumption 4. $r$ , $r'$ Integrability exponents for linearized and adjoint state equation introduced in Section 4. J.-P. Raymond, F. Tröltzsch. Second order sufficient optimality conditions for nonlinear parabolic control problems with state constraints. Discrete & Continuous Dynamical Systems - A, 2000, 6 (2) : 431-450. doi: 10.3934/dcds.2000.6.431 Jeremy LeCrone, Gieri Simonett. On quasilinear parabolic equations and continuous maximal regularity. Evolution Equations & Control Theory, 2020, 9 (1) : 61-86. doi: 10.3934/eect.2020017 William G. Litvinov. 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Communications on Pure & Applied Analysis, 2017, 16 (5) : 1697-1706. doi: 10.3934/cpaa.2017081 Ugur G. Abdulla. On the optimal control of the free boundary problems for the second order parabolic equations. II. Convergence of the method of finite differences. Inverse Problems & Imaging, 2016, 10 (4) : 869-898. doi: 10.3934/ipi.2016025 Thierry Horsin, Peter I. Kogut, Olivier Wilk. Optimal $L^2$-control problem in coefficients for a linear elliptic equation. II. Approximation of solutions and optimality conditions. Mathematical Control & Related Fields, 2016, 6 (4) : 595-628. doi: 10.3934/mcrf.2016017 Sachiko Ishida, Tomomi Yokota. Boundedness in a quasilinear fully parabolic Keller-Segel system via maximal Sobolev regularity. Discrete & Continuous Dynamical Systems - S, 2020, 13 (2) : 211-232. doi: 10.3934/dcdss.2020012 Mehdi Badra, Kaushik Bal, Jacques Giacomoni. Existence results to a quasilinear and singular parabolic equation. Conference Publications, 2011, 2011 (Special) : 117-125. doi: 10.3934/proc.2011.2011.117 Peter Weidemaier. Maximal regularity for parabolic equations with inhomogeneous boundary conditions in Sobolev spaces with mixed $L_p$-norm. Electronic Research Announcements, 2002, 8: 47-51. Serge Nicaise, Fredi Tröltzsch. Optimal control of some quasilinear Maxwell equations of parabolic type. Discrete & Continuous Dynamical Systems - S, 2017, 10 (6) : 1375-1391. doi: 10.3934/dcdss.2017073 Giovanni Bellettini, Matteo Novaga, Giandomenico Orlandi. Eventual regularity for the parabolic minimal surface equation. Discrete & Continuous Dynamical Systems - A, 2015, 35 (12) : 5711-5723. doi: 10.3934/dcds.2015.35.5711 M. Soledad Aronna. Second order necessary and sufficient optimality conditions for singular solutions of partially-affine control problems. Discrete & Continuous Dynamical Systems - S, 2018, 11 (6) : 1233-1258. doi: 10.3934/dcdss.2018070 Ugur G. Abdulla. On the optimal control of the free boundary problems for the second order parabolic equations. I. Well-posedness and convergence of the method of lines. Inverse Problems & Imaging, 2013, 7 (2) : 307-340. doi: 10.3934/ipi.2013.7.307 PDF downloads (127) HTML views (347) Lucas Bonifacius Ira Neitzel	CommonCrawl
Generalized flag variety In mathematics, a generalized flag variety (or simply flag variety) is a homogeneous space whose points are flags in a finite-dimensional vector space V over a field F. When F is the real or complex numbers, a generalized flag variety is a smooth or complex manifold, called a real or complex flag manifold. Flag varieties are naturally projective varieties. Flag varieties can be defined in various degrees of generality. A prototype is the variety of complete flags in a vector space V over a field F, which is a flag variety for the special linear group over F. Other flag varieties arise by considering partial flags, or by restriction from the special linear group to subgroups such as the symplectic group. For partial flags, one needs to specify the sequence of dimensions of the flags under consideration. For subgroups of the linear group, additional conditions must be imposed on the flags. In the most general sense, a generalized flag variety is defined to mean a projective homogeneous variety, that is, a smooth projective variety X over a field F with a transitive action of a reductive group G (and smooth stabilizer subgroup; that is no restriction for F of characteristic zero). If X has an F-rational point, then it is isomorphic to G/P for some parabolic subgroup P of G. A projective homogeneous variety may also be realised as the orbit of a highest weight vector in a projectivized representation of G. The complex projective homogeneous varieties are the compact flat model spaces for Cartan geometries of parabolic type. They are homogeneous Riemannian manifolds under any maximal compact subgroup of G, and they are precisely the coadjoint orbits of compact Lie groups. Flag manifolds can be symmetric spaces. Over the complex numbers, the corresponding flag manifolds are the Hermitian symmetric spaces. Over the real numbers, an R-space is a synonym for a real flag manifold and the corresponding symmetric spaces are called symmetric R-spaces. Flags in a vector space Main article: flag (linear algebra) A flag in a finite dimensional vector space V over a field F is an increasing sequence of subspaces, where "increasing" means each is a proper subspace of the next (see filtration): $\{0\}=V_{0}\subset V_{1}\subset V_{2}\subset \cdots \subset V_{k}=V.$ If we write the dim Vi = di then we have $0=d_{0}<d_{1}<d_{2}<\cdots <d_{k}=n,$ where n is the dimension of V. Hence, we must have k ≤ n. A flag is called a complete flag if di = i for all i, otherwise it is called a partial flag. The signature of the flag is the sequence (d1, ..., dk). A partial flag can be obtained from a complete flag by deleting some of the subspaces. Conversely, any partial flag can be completed (in many different ways) by inserting suitable subspaces. Prototype: the complete flag variety According to basic results of linear algebra, any two complete flags in an n-dimensional vector space V over a field F are no different from each other from a geometric point of view. That is to say, the general linear group acts transitively on the set of all complete flags. Fix an ordered basis for V, identifying it with Fn, whose general linear group is the group GL(n,F) of n × n invertible matrices. The standard flag associated with this basis is the one where the ith subspace is spanned by the first i vectors of the basis. Relative to this basis, the stabilizer of the standard flag is the group of nonsingular lower triangular matrices, which we denote by Bn. The complete flag variety can therefore be written as a homogeneous space GL(n,F) / Bn, which shows in particular that it has dimension n(n−1)/2 over F. Note that the multiples of the identity act trivially on all flags, and so one can restrict attention to the special linear group SL(n,F) of matrices with determinant one, which is a semisimple algebraic group; the set of lower triangular matrices of determinant one is a Borel subgroup. If the field F is the real or complex numbers we can introduce an inner product on V such that the chosen basis is orthonormal. Any complete flag then splits into a direct sum of one-dimensional subspaces by taking orthogonal complements. It follows that the complete flag manifold over the complex numbers is the homogeneous space $U(n)/T^{n}$ where U(n) is the unitary group and Tn is the n-torus of diagonal unitary matrices. There is a similar description over the real numbers with U(n) replaced by the orthogonal group O(n), and Tn by the diagonal orthogonal matrices (which have diagonal entries ±1). Partial flag varieties The partial flag variety $F(d_{1},d_{2},\ldots d_{k},\mathbb {F} )$ is the space of all flags of signature (d1, d2, ... dk) in a vector space V of dimension n = dk over F. The complete flag variety is the special case that di = i for all i. When k=2, this is a Grassmannian of d1-dimensional subspaces of V. This is a homogeneous space for the general linear group G of V over F. To be explicit, take V = Fn so that G = GL(n,F). The stabilizer of a flag of nested subspaces Vi of dimension di can be taken to be the group of nonsingular block lower triangular matrices, where the dimensions of the blocks are ni := di − di−1 (with d0 = 0). Restricting to matrices of determinant one, this is a parabolic subgroup P of SL(n,F), and thus the partial flag variety is isomorphic to the homogeneous space SL(n,F)/P. If F is the real or complex numbers, then an inner product can be used to split any flag into a direct sum, and so the partial flag variety is also isomorphic to the homogeneous space $U(n)/U(n_{1})\times \cdots \times U(n_{k})$ in the complex case, or $O(n)/O(n_{1})\times \cdots \times O(n_{k})$ in the real case. Generalization to semisimple groups The upper triangular matrices of determinant one are a Borel subgroup of SL(n,F), and hence the stabilizers of partial flags are parabolic subgroups. Furthermore, a partial flag is determined by the parabolic subgroup which stabilizes it. Hence, more generally, if G is a semisimple algebraic or Lie group, then the (generalized) flag variety for G is G/P where P is a parabolic subgroup of G. The correspondence between parabolic subgroups and generalized flag varieties allows each to be understood in terms of the other. The extension of the terminology "flag variety" is reasonable, because points of G/P can still be described using flags. When G is a classical group, such as a symplectic group or orthogonal group, this is particularly transparent. If (V, ω) is a symplectic vector space then a partial flag in V is isotropic if the symplectic form vanishes on proper subspaces of V in the flag. The stabilizer of an isotropic flag is a parabolic subgroup of the symplectic group Sp(V,ω). For orthogonal groups there is a similar picture, with a couple of complications. First, if F is not algebraically closed, then isotropic subspaces may not exist: for a general theory, one needs to use the split orthogonal groups. Second, for vector spaces of even dimension 2m, isotropic subspaces of dimension m come in two flavours ("self-dual" and "anti-self-dual") and one needs to distinguish these to obtain a homogeneous space. Cohomology If G is a compact, connected Lie group, it contains a maximal torus T and the space G/T of left cosets with the quotient topology is a compact real manifold. If H is any other closed, connected subgroup of G containing T, then G/H is another compact real manifold. (Both are actually complex homogeneous spaces in a canonical way through complexification.) The presence of a complex structure and cellular (co)homology make it easy to see that the cohomology ring of G/H is concentrated in even degrees, but in fact, something much stronger can be said. Because G → G/H is a principal H-bundle, there exists a classifying map G/H → BH with target the classifying space BH. If we replace G/H with the homotopy quotient GH in the sequence G → G/H → BH, we obtain a principal G-bundle called the Borel fibration of the right multiplication action of H on G, and we can use the cohomological Serre spectral sequence of this bundle to understand the fiber-restriction homomorphism H(G/H) → H(G) and the characteristic map H(BH) → H(G/H), so called because its image, the characteristic subring of H(G/H), carries the characteristic classes of the original bundle H → G → G/H. Let us now restrict our coefficient ring to be a field k of characteristic zero, so that, by Hopf's theorem, H(G) is an exterior algebra on generators of odd degree (the subspace of primitive elements). It follows that the edge homomorphisms $E_{r+1}^{0,r}\to E_{r+1}^{r+1,0}$ of the spectral sequence must eventually take the space of primitive elements in the left column H(G) of the page E2 bijectively into the bottom row H(BH): we know G and H have the same rank, so if the collection of edge homomorphisms were not full rank on the primitive subspace, then the image of the bottom row H(BH) in the final page H(G/H) of the sequence would be infinite-dimensional as a k-vector space, which is impossible, for instance by cellular cohomology again, because a compact homogeneous space admits a finite CW structure. Thus the ring map H(G/H) → H(G) is trivial in this case, and the characteristic map is surjective, so that H(G/H) is a quotient of H(BH). The kernel of the map is the ideal generated by the images of primitive elements under the edge homomorphisms, which is also the ideal generated by positive-degree elements in the image of the canonical map H(BG) → H(BH) induced by the inclusion of H in G. The map H(BG) → H(BT) is injective, and likewise for H, with image the subring H(BT)W(G) of elements invariant under the action of the Weyl group, so one finally obtains the concise description $H^{}(G/H)\cong H^{}(BT)^{W(H)}/{\big (}{\widetilde {H}}^{}(BT)^{W(G)}{\big )},$ where ${\widetilde {H}}^{}$ denotes positive-degree elements and the parentheses the generation of an ideal. For example, for the complete complex flag manifold U(n)/Tn, one has $H^{}{\big (}U(n)/T^{n}{\big )}\cong \mathbb {Q} [t_{1},\ldots ,t_{n}]/(\sigma _{1},\ldots ,\sigma _{n}),$ where the tj are of degree 2 and the σj are the first n elementary symmetric polynomials in the variables tj. For a more concrete example, take n = 2, so that U(2)/[U(1) × U(1)] is the complex Grassmannian Gr(1,$\mathbb {C} $2) ≈ $\mathbb {C} $P1 ≈ S2. Then we expect the cohomology ring to be an exterior algebra on a generator of degree two (the fundamental class), and indeed, $H^{*}{\big (}U(2)/T^{2}{\big )}\cong \mathbb {Q} [t_{1},t_{2}]/(t_{1}+t_{2},t_{1}t_{2})\cong \mathbb {Q} [t_{1}]/(t_{1}^{2}),$ as hoped. Highest weight orbits and projective homogeneous varieties If G is a semisimple algebraic group (or Lie group) and V is a (finite dimensional) highest weight representation of G, then the highest weight space is a point in the projective space P(V) and its orbit under the action of G is a projective algebraic variety. This variety is a (generalized) flag variety, and furthermore, every (generalized) flag variety for G arises in this way. Armand Borel showed that this characterizes the flag varieties of a general semisimple algebraic group G: they are precisely the complete homogeneous spaces of G, or equivalently (in this context), the projective homogeneous G-varieties. Symmetric spaces Main article: Symmetric space Let G be a semisimple Lie group with maximal compact subgroup K. Then K acts transitively on any conjugacy class of parabolic subgroups, and hence the generalized flag variety G/P is a compact homogeneous Riemannian manifold K/(K∩P) with isometry group K. Furthermore, if G is a complex Lie group, G/P is a homogeneous Kähler manifold. Turning this around, the Riemannian homogeneous spaces M = K/(K∩P) admit a strictly larger Lie group of transformations, namely G. Specializing to the case that M is a symmetric space, this observation yields all symmetric spaces admitting such a larger symmetry group, and these spaces have been classified by Kobayashi and Nagano. If G is a complex Lie group, the symmetric spaces M arising in this way are the compact Hermitian symmetric spaces: K is the isometry group, and G is the biholomorphism group of M. Over the real numbers, a real flag manifold is also called an R-space, and the R-spaces which are Riemannian symmetric spaces under K are known as symmetric R-spaces. The symmetric R-spaces which are not Hermitian symmetric are obtained by taking G to be a real form of the biholomorphism group Gc of a Hermitian symmetric space Gc/Pc such that P := Pc∩G is a parabolic subgroup of G. Examples include projective spaces (with G the group of projective transformations) and spheres (with G the group of conformal transformations). See also • Parabolic Lie algebra • Bruhat decomposition References • Robert J. Baston and Michael G. Eastwood, The Penrose Transform: its Interaction with Representation Theory, Oxford University Press, 1989. • Jürgen Berndt, Lie group actions on manifolds, Lecture notes, Tokyo, 2002. • Jürgen Berndt, Sergio Console and Carlos Olmos, Submanifolds and Holonomy, Chapman & Hall/CRC Press, 2003. • Michel Brion, Lectures on the geometry of flag varieties, Lecture notes, Varsovie, 2003. • James E. Humphreys, Linear Algebraic Groups, Graduate Texts in Mathematics, 21, Springer-Verlag, 1972. • S. Kobayashi and T. Nagano, On filtered Lie algebras and geometric structures I, II, J. Math. Mech. 13 (1964), 875–907, 14 (1965) 513–521. Authority control: National • Germany	Wikipedia
Results for 'Sre��ko Kova��' Generalist Genes: Implications for the Cognitive Sciences.Yulia Kovas & Robert Plomin - 2006 - Trends in Cognitive Sciences 10 (5):198-203.details Philosophy of Cognitive Science, Misc in Philosophy of Cognitive Science Philosophy of Cognitive Science, Miscellaneous in Philosophy of Cognitive Science Matthias Koßler, Substantielles Wissen und subjektives Handeln. Dargestellt in einem Vergleich von Hegel und Schopenhauer. [REVIEW]Matthias Koßler - 1992 - Schopenhauer Jahrbuch:167-172.details Genes and Gini: What Inequality Means for Heritability.Fatos Selita & Yulia Kovas - 2019 - Journal of Biosocial Science 51 (1):18-47.details Introducing the Ko Corpus of Korean Mother–Child Interaction.Eon-Suk Ko, Jinyoung Jo, Kyung-Woon On & Byoung-Tak Zhang - 2020 - Frontiers in Psychology 11.details We describe a corpus of speech taking place between 30 Korean mother–child pairs, divided in three groups of Prelexical, Early-Lexical, and Advanced-Lexical. In addition to the child-directed speech, this corpus includes two different formalities of adult-directed speech, i.e., family-directed ADS and experimenter-directed ADS. Our analysis of the MLU in CDS, family-, and experimenter-directed ADS found significant differences between CDS and ADS_Fam, and between ADS_Fam and ADS_Exp, but not between CDS and ADS_Exp. Our finding suggests that researchers should pay more attention (...) to controlling the level of formality in CDS and ADS when comparing the two registers for their speech characteristics. The corpus was transcribed in the CHAT format of the CHILDES system, so users can easily extract data related to verbal behavior in the mother–child interaction using the CLAN program of CHILDES. (shrink) Beyond linear conciliation.Ko-Hung Kuan - 2020 - Synthese 198 (12):11483-11504.details Formal epistemologists criticise the Conciliatory View of peer disagreement for being non-commutative with conditionalisation, path dependent and does not preserve the independence between propositions. Failing to commute with conditionalisation, one may switch the order between conciliating and conditionalising and obtain different outcomes. Failing to be path independent, the outcome of conciliation varies with the order of the acquisition of new testimonies. Failing to preserve the independence between propositions, one may suffer from a sure-loss and hence be deemed irrational. The three (...) formal deficiencies urge people to abandon the Conciliatory View. This paper aims to show that one may save the Conciliatory View by conciliating with nonlinear functions. Research in the study of opinion pooling shows that the three deficiencies are not problems of the Conciliatory View, but problems of linear averaging. Hence, one can get rid of these formal deficiencies by making conciliation with nonlinear averaging functions. After showing how the three deficiencies can be avoided, I will explore the features of nonlinear averaging functions and argue that they have properties that correctly capture people's intuition concerning disagreement. The conclusion, therefore, is to suggest epistemologists develop a more fine-grained taxonomy for cases of disagreement. With a deliberate categorisation of different kinds of disagreement, epistemologists can pick the proper averaging rule to apply in each specific case, and get rid of possible formal deficiencies. (shrink) Epistemology of Disagreement in Epistemology Hybrid Firefly Model in Routing Heterogeneous Fleet of Vehicles in Logistics Distribution.D. Simi, I. Kova evi, V. Svir evi & S. Simi - 2015 - Logic Journal of the IGPL 23 (3):521-532.details Structural and Functional Genetic Neuroimaging.Yulia Kovas & Robert Plomin - 2006 - Trends in Cognitive Sciences 10 (5):198-203.details Brain Imaging and Localization in Philosophy of Cognitive Science Response to Marcus and Rabagliati 'Genes and Domain Specificity'.Yulia Kovas & Robert Plomin - 2006 - Trends in Cognitive Sciences 10 (9):398.details Modularity in Cognitive Science in Philosophy of Cognitive Science Perceptual Representation and Effectiveness of Local Figure–Ground Cues in Natural Contours.Ko Sakai, Shouhei Matsuoka, Ken Kurematsu & Yasuhiro Hatori - 2015 - Frontiers in Psychology 6.details Complexity and the Paradigm of Wolfram's A New Kind of Science: From the Computational Sciences to the Science of Computation.Kovas Boguta - 2005 - Complexity 10 (4):15-21.details Complexity in Philosophy of Physical Science Computationalism in Philosophy of Cognitive Science Chapter Four Divine Paradox and Harmony: Whitehead and Jung on the Love of God Young Woon Ko.Young Woon Ko - 2007 - In Thomas Jay Oord (ed.), The Many Facets of Love: Philosophical Explorations. Cambridge Scholars Press.details $60.62 new (collection) Amazon page Osvajanja Sre Ce.Bertrand Russell & C. Ljuba Popovi - 1982 - Minerva.details Is That the Same Person? Case Studies in Neurosurgery.Nancy S. Jecker & Andrew L. Ko - 2017 - American Journal of Bioethics Neuroscience 8 (3):160-170.details Probability Description and Entropy of Classical and Quantum Systems.Margarita A. Man'ko & Vladimir I. Man'ko - 2011 - Foundations of Physics 41 (3):330-344.details Tomographic approach to describing both the states in classical statistical mechanics and the states in quantum mechanics using the fair probability distributions is reviewed. The entropy associated with the probability distribution (tomographic entropy) for classical and quantum systems is studied. The experimental possibility to check the inequalities like the position–momentum uncertainty relations and entropic uncertainty relations are considered. Classical Mechanics in Philosophy of Physical Science Probabilities in Quantum Mechanics in Philosophy of Physical Science Thermodynamics and Statistical Mechanics in Philosophy of Physical Science Uncertainty Principle in Philosophy of Physical Science Ethical Leadership: An Integrative Review and Future Research Agenda.Changsuk Ko, Jianhong Ma, Roman Bartnik, Mark H. Haney & Mingu Kang - 2018 - Ethics and Behavior 28 (2):104-132.details Over the past decade, ethical leadership has increasingly become one of the most popular topics in the areas of leadership and business ethics. As a result, there now exists a substantial body of empirical research addressing ethical leadership issues, but the findings reported by this body of research are highly fragmented. The topic has advanced to the stage where a review and synthesis of existing literature can provide great value and help move the scholarly conversation forward. The primary purposes of (...) this article are to review empirical findings from the ethical leadership literature utilizing a framework consisting of the antecedents, mediators, moderators and outcomes of ethical leadership, and suggest a set of interesting research opportunities, thereby facilitating future investigation. We base our synthesis on a review of 62 empirical studies on ethical leadership that were published between 2005 and mid-2015. (shrink) Srê Phonemes and SyllablesSre Phonemes and Syllables.William A. Smalley - 1954 - Journal of the American Oriental Society 74 (4):217.details Philosophy of Psychology in Philosophy of Cognitive Science Truth, KoΣmoΣ, and Apeth in the Homeric Poems.A. W. H. Adkins - 1972 - Classical Quarterly 22 (01):5-.details A number of scholars have discussed the difficulty of preserving accurately—or at all—information about the past1 in the Greek Dark Ages when the literacy of Minoan/Mycenean Greece had been lost. Such preservation necessarily depended on the memories of the members of the society, especially those of the professional 'rememberers', the bards of the oral tradition: in such a society, if knowledge of an event is to be available to future generations, it must not be forgotten. Classics in Arts and Humanities Kova Dėl Tuščios Vietos.Jonas Dagys - 2013 - Problemos 84:184-186.details Visual and Linguistic Stimuli in the Remote Associates Test: A Cross-Cultural Investigation.Teemu Toivainen, Ana-Maria Olteteanu, Vlada Repeykova, Maxim Likhanov & Yulia Kovas - 2019 - Frontiers in Psychology 10.details Controls on Pore Types and Pore-Size Distribution in the Upper Triassic Yanchang Formation, Ordos Basin, China: Implications for Pore-Evolution Models of Lacustrine Mudrocks.Lucy T. Ko, Robert G. Loucks, Kitty L. Milliken, Quansheng Liang, Tongwei Zhang, Xun Sun, Paul C. Hackley, Stephen C. Ruppel & Sheng Peng - 2017 - Interpretation: SEG 5 (2):SF127-SF148.details Our main objectives are to learn if pore-evolution models developed from marine mudrocks can be directly applied to lacustrine mudrocks, investigate what controls the different pore types and sizes of Chang 7 organic matter -rich argillaceous mudstones of the Upper Triassic Yanchang Formation, and describe the texture, fabric, mineralogy, and thermal maturity variation in the Chang 7 mudstones. Lacustrine mudstones from nine cored wells along a depositional dip in the southeastern Ordos Basin, China, were investigated. Helium porosimetry, nitrogen adsorption, and (...) field-emission scanning electron microscopy of Ar-ion milled samples were applied. Measured average total porosity of samples from a proximal to distal transect is higher than those from the two adjacent cored wells. This difference in porosity partly caused by differences in the clay mineral content implies that in the fluvial-deltaic-lacustrine depositional environment, reservoir quality can vary significantly in a short distance. Owing to the uneven distribution of the sample set from proximal to distal area, we mainly evaluate variations in the proximal setting. Results from nitrogen-gas adsorption experiments show that there are four distinct patterns of pore-size distribution within the Chang 7 member of the Yanchang Formation with no particular correlation with mineralogical composition and thermal maturity. The pore network within Chang 7 mudstones is dominated by OM-hosted pores, with a lesser abundance of interparticle and intraparticle pores. The size distribution of mineral-hosted pores within these mudstones is found to be closely related to the rock texture and fabric. Mudstones with well-sorted grains and a higher percentage of coarser grains have more abundant mineral pores. The sizes of OM-hosted pores in these compaction-dominated lacustrine mudstones were one to two orders of magnitude smaller than those in the marine mudstones that display abundant early cementation. (shrink) Co‐Prints and Translation.Cay Dollerup & Silvana Orel‐Kos - 2001 - Perspectives 9 (2):87-108.details Is Discharge Knee Range of Motion a Useful and Relevant Clinical Indicator After Total Knee Replacement? Part 2.Justine M. Naylor, Victoria Ko, Steve Rougellis, Nick Green, Rajat Mittal, Rob Heard, Anthony E. T. Yeo, Anne Barnett, Danella Hackett, Chris Saliba, Nicole Smith, Martin Mackey, Alison Harmer, Ian A. Harris, Sam Adie & Lynette McEvoy - 2012 - Journal of Evaluation in Clinical Practice 18 (3):652-658.details Philosophy of Medicine in Philosophy of Science, Misc Development and Pilot Testing of an Informed Consent Video for Patients with Limb Trauma Prior to Debridement Surgery Using a Modified Delphi Technique.Yen-Ko Lin, Chao-Wen Chen, Wei-Che Lee, Tsung-Ying Lin, Liang-Chi Kuo, Chia-Ju Lin, Leiyu Shi, Yin-Chun Tien & Yuan-Chia Cheng - 2017 - BMC Medical Ethics 18 (1):1-12.details Background Ensuring adequate informed consent for surgery in a trauma setting is challenging. We developed and pilot tested an educational video containing information regarding the informed consent process for surgery in trauma patients and a knowledge measure instrument and evaluated whether the audiovisual presentation improved the patients' knowledge regarding their procedure and aftercare and their satisfaction with the informed consent process. Methods A modified Delphi technique in which a panel of experts participated in successive rounds of shared scoring of items (...) to forecast outcomes was applied to reach a consensus among the experts. The resulting consensus was used to develop the video content and questions for measuring the understanding of the informed consent for debridement surgery in limb trauma patients. The expert panel included experienced patients. The participants in this pilot study were enrolled as a convenience sample of adult trauma patients scheduled to receive surgery. Results The modified Delphi technique comprised three rounds over a 4-month period. The items given higher scores by the experts in several categories were chosen for the subsequent rounds until consensus was reached. The experts reached a consensus on each item after the three-round process. The final knowledge measure comprising 10 questions was developed and validated. Thirty eligible trauma patients presenting to the Emergency Department were approached and completed the questionnaires in this pilot study. The participants exhibited significantly higher mean knowledge and satisfaction scores after watching the educational video than before watching the video. Conclusions Our process is promising for developing procedure-specific informed consent and audiovisual aids in medical and surgical specialties. The educational video was developed using a scientific method that integrated the opinions of different stakeholders, particularly patients. This video is a useful tool for improving the knowledge and satisfaction of trauma patients in the ED. The modified Delphi technique is an effective method for collecting experts' opinions and reaching a consensus on the content of educational materials for informed consent. Institutions should prioritize patient-centered health care and develop a structured informed consent process to improve the quality of care. Trial registration The ClinicalTrials.gov Identifier is NCT01338480. The date of registration was April 18, 2011. (shrink) Phantasie und Einbildungskraft: Zur Rolle der Einbildungskraft bei Fichte und Solger.Matthias Koßler - 2003 - Fichte-Studien 21:163-181.details Fichtes Konzeption vom »Schweben der Einbildungskraft« ist in der Ästhetik der Romantik aufgegriffen und zu einer zentralen Denkfigur geworden. Auch die Philosophie Karl Wilhelm Ferdinand Solgers, des Nachfolgers Fichtes auf dem Lehrstuhl in Berlin, pflegt in die Romantische Ästhetik im Gefolge Friedrich Schlegels eingeordnet zu werden, gerade auch im Hinblick auf die Bedeutung der Einbildungskraft. Um so überraschender ist dann aber die Tatsache, dass die Einbildungskraft für Solgers Ästhetik, anders als etwa bei Schlegel und Novalis, gar keine Rolle spielt. Sie (...) gehört zu den »gemeinen Erkenntnisarten« und wird der Phantasie, die die Form der »höheren Erkenntnis« ist, entgegengesetzt. Die »Einbildungskraft [ist] an die sinnliche Wahrnehmung geheftet und vom Triebe ganz bestimmt, die Phantasie aber führt das göttliche Wesen in die Erscheinung über«. Nun spricht Solger häufig in diesem Sinne von der »gemeinen Einbildungskraft«, und man könnte meinen, dass es hier nur um eine Frage der Benennung gehe, indem die produktive Einbildungskraft bei ihm den Namen Phantasie erhält, während er die Bezeichnung,Einbildungskraft' für die reproduktive Einbildungskraft reserviert, wofür der Zusatz »gemeine« steht. Auch wenn dem so wäre - die Frage, ob es sich so verhält, soll noch untersucht werden –, so muss es doch einen Grund geben, warum Solger sich von der Tradition abkehrt und einen in ihr bewährten Begriff ersetzt. (shrink) ΣϒPIΣKOΣ EΓPΦΣEN: Loaded Names, Artistic Identity, and Reading an Athenian Vase.Seth D. Pevnick - 2010 - Classical Antiquity 29 (2):222-253.details This paper examines the importance of artist names and artistic identity, especially as expressed in artist signatures, to the interpretation of ancient Greek pottery. Attention is focused on a calyx krater signed ΣϒPIΣKOΣ EΓPΦΣEN [sic], and it is argued that the non-Greek ethnikon used as artist name encourages a non-Athenian reading of the iconography. The painted labels for all six figures on this vase, together with parallels from other Athenian red-figure vases—including others from the Syriskos workshop—all suggest the presentation of (...) an alternative, un-Athenian world view. Okeanos, Dionysos, and Epaphos are read as representing faraway lands at the edges of the Ge Panteleia, or "entire earth," while the central figure of Themis, Greek personification of divine right, is depicted pouring a libation to Balos, the Hellenized form of the Syrian supreme god Baal, thereby recognizing his status as a supreme deity. Other overtly political messages have been read elsewhere in the oeuvre of the Syriskos Workshop, where it seems that at least two distinct artistic identities were at play—the explicitly foreign "little Syrian," and the more conventional Pistoxenos, or "trustworthy foreigner." When explicitly signed on vessels, these artistic identities necessarily sway interpretation, whereas on the many unsigned pieces, the viewer is left to consider which identity is at play. (shrink) Development and pilot testing of an informed consent video for patients with limb trauma prior to debridement surgery using a modified Delphi technique.Yen-Ko Lin, Chao-Wen Chen, Wei-Che Lee, Tsung-Ying Lin, Liang-Chi Kuo, Chia-Ju Lin, Leiyu Shi, Yin-Chun Tien & Yuan-Chia Cheng - 2017 - BMC Medical Ethics 18 (1):67.details Ensuring adequate informed consent for surgery in a trauma setting is challenging. We developed and pilot tested an educational video containing information regarding the informed consent process for surgery in trauma patients and a knowledge measure instrument and evaluated whether the audiovisual presentation improved the patients' knowledge regarding their procedure and aftercare and their satisfaction with the informed consent process. A modified Delphi technique in which a panel of experts participated in successive rounds of shared scoring of items to forecast (...) outcomes was applied to reach a consensus among the experts. The resulting consensus was used to develop the video content and questions for measuring the understanding of the informed consent for debridement surgery in limb trauma patients. The expert panel included experienced patients. The participants in this pilot study were enrolled as a convenience sample of adult trauma patients scheduled to receive surgery. The modified Delphi technique comprised three rounds over a 4-month period. The items given higher scores by the experts in several categories were chosen for the subsequent rounds until consensus was reached. The experts reached a consensus on each item after the three-round process. The final knowledge measure comprising 10 questions was developed and validated. Thirty eligible trauma patients presenting to the Emergency Department were approached and completed the questionnaires in this pilot study. The participants exhibited significantly higher mean knowledge and satisfaction scores after watching the educational video than before watching the video. Our process is promising for developing procedure-specific informed consent and audiovisual aids in medical and surgical specialties. The educational video was developed using a scientific method that integrated the opinions of different stakeholders, particularly patients. This video is a useful tool for improving the knowledge and satisfaction of trauma patients in the ED. The modified Delphi technique is an effective method for collecting experts' opinions and reaching a consensus on the content of educational materials for informed consent. Institutions should prioritize patient-centered health care and develop a structured informed consent process to improve the quality of care. The ClinicalTrials.gov Identifier is NCT01338480. The date of registration was April 18, 2011. (shrink) Building an Ethical Environment Improves Patient Privacy and Satisfaction in the Crowded Emergency Department: A Quasi-Experimental Study. [REVIEW]Yen-Ko Lin, Wei-Che Lee, Liang-Chi Kuo, Yuan-Chia Cheng, Chia-Ju Lin, Hsing-Lin Lin, Chao-Wen Chen & Tsung-Ying Lin - 2013 - BMC Medical Ethics 14 (1):8-.details Background: To evaluate the effectiveness of a multifaceted intervention in improving emergency department (ED) patient privacy and satisfaction in the crowded ED setting. Methods: A pre- and post-intervention study was conducted. A multifaceted intervention was implemented in a university-affiliated hospital ED. The intervention developed strategies to improve ED patient privacy and satisfaction, including redesigning the ED environment, process management, access control, and staff education and training, and encouraging ethics consultation. The effectiveness of the intervention was evaluated using patient surveys. Eligibility (...) data were collected after the intervention and compared to data collected before the intervention. Differences in patient satisfaction and patient perception of privacy were adjusted for predefined covariates using multivariable ordinal logistic regression. Results: Structured questionnaires were collected with 313 ED patients before the intervention and 341 ED patients after the intervention. There were no important covariate differences, except for treatment area, between the two groups. Significant improvements were observed in patient perception of "personal information overheard by others", being "seen by irrelevant persons", having "unintentionally heard inappropriate conversations from healthcare providers", and experiencing "providers' respect for my privacy". There was significant improvement in patient overall perception of privacy and satisfaction. There were statistically significant correlations between the intervention and patient overall perception of privacy and satisfaction on multivariable analysis. Conclusions: Significant improvements were achieved with an intervention. Patients perceived significantly more privacy and satisfaction in ED care after the intervention. We believe that these improvements were the result of major philosophical, administrative, and operational changes aimed at respecting both patient privacy and satisfaction. (shrink) Direct download (16 more) Growth and Physical Properties of the Decagonal Al–Cu–Co Quasicrystal Grown From the Ternary Melt.R. A. Ribeiro, S. L. Bud'ko, F. C. Laabs, M. J. Kramer & P. C. Canfield - 2004 - Philosophical Magazine 84 (12):1291-1302.details Metaphysics of Mind in Philosophy of Mind Matthias Koßler (Hg.): Musik als Wille und Welt. Schopenhauers Philosophie der Musik.Regina Schidel & Matthias Neuber - 2011 - Philosophischer Literaturanzeiger 64 (4):349.details Effects of Substitution on Low-Temperature Physical Properties of LuFe2Ge2.Sheng Ran, Sergey L. Bud'ko & Paul C. Canfield - 2011 - Philosophical Magazine 91 (34):4388-4400.details Between Technical Features and Analytic Capabilities: Charting a Relational Affordance Space for Digital Social Analytics.Anders Koed Madsen - 2015 - Big Data and Society 2 (1).details Digital social analytics is a subset of Big Data methods that is used to understand the social environment in which people and organizations have to act. This paper presents an analysis of eight projects that are experimenting with the use of these methods for various purposes. It shows that two specific technological features influence the work with such methods in all the cases. The first concerns the need to distribute choices about the structure of data to third-party actors and the (...) second concerns the need to balance machine intelligence and human intuition when automating the analysis. These features set specific conditions for knowledge production, and the paper identifies two opposite approaches for engaging with each of these conditions. These features and approaches are finally combined into a two-dimensional affordance space that illustrates how there is flexibility in the way project leaders interact with the features of the data environment. It thereby also shows how digital social analytics come to have different affordances for different projects. (shrink) Sculpture and the Sculptural.Erik Koed - 2005 - Journal of Aesthetics and Art Criticism 63 (2):147–154.details Sculpture in Aesthetics Sculpture and the Sculptural.Erik Koed - 2005 - Journal of Aesthetics and Art Criticism 63 (2):147 - 154.details Journalism Ethics in Multinational Family: "When in the EU, Should One Do as the EU Journalists Do?".Melita Poler Kova - 2008 - Journal of Mass Media Ethics 23 (2):141 – 157.details This essay reviews a number of issues regarding self-regulation and professional ethics which journalists across Europe might face in the scaling down of national borders. The dilemma of whether a pan-European ideal standards code of ethics can help journalists when working across borders and encountering other traditions is explored by referring to Slovenia, one of the new European Union (EU) members. Presenting a critique of the traditional professionalization concept, cogent arguments are found for rejecting a universal code of ethics. By (...) acknowledging the limitations and even deficiencies of such codified morality, a journalist's responsibility is emphasized and a different concept of ethics is indicated. Ethical journalists in this international context must focus on responsibility, positive tolerance, and empathy that transcends mere obedience to a code. The EU citizen's ethics rather than EU professional ethics should be advanced, based on universal principles and grounded in personal responsibility. (shrink) Toleration in Applied Ethics in Social and Political Philosophy ST Sur Quelques Questions de la Théorie de l'Art Et de l'Esthétique Marxiste-Léniniste.M. Kova, M. Pokorna & M. Svoboda - 1985 - Estetika 22 (1):33-40.details Social Alliance and Employee Voluntary Activities: A Resource-Based Perspective. [REVIEW]Gordon Liu & Wai-Wai Ko - 2011 - Journal of Business Ethics 104 (2):251-268.details The corporate social responsibility literature devotes relatively little attention to the strategic role played by employee voluntary activities (EVAs) in social alliances. Using the resource-based perspective of the organization to frame the data collection and the analyses, this article investigates: (1) the role of EVAs in the development of corporate and non-profit organizations (NPOs) competitive assets and (2) the management approaches to how both parties can develop their own resources by combining them with the shared resources with the purpose of (...) enhancing its competitive advantage in its own sector. The database is composed of 70 specifically designed interviews with managers of UK-based firms and NPOs. The analyses suggest, among other things, that the majority of corporate and non-profit managers find that EVAs generate substantial tangible and intangible benefits for their respective organisations, creating genuine synergies. We also find evidence of a general preference for the management approaches of such programmes in both types of organisation. (shrink) Business Ethics in Applied Ethics Data in the Smart City: How Incongruent Frames Challenge the Transition From Ideal to Practice.Anders Koed Madsen - 2018 - Big Data and Society 5 (2).details This paper presents an analysis of interviews, focus groups and workshops with employees in the technical administration in the municipality of Copenhagen in the year after it won a prestigious Smart City award. The administration is interpreted as a 'most likely' to succeed in translating the idealised version of the smart city into a workable bureaucratic practice. Drawing on the work of Orlikowski and Gash, the empirical analysis identifies and describes two incongruent 'technological frames' that illustrates different ways of making (...) sense of data and the smart city within this single organisational unit. One is called the experimentalist's credo and it is characterised by inspiration from the development of an Internet of Things as well as a readiness to learn from the open source community in software development. The other is called the data-owners vocation and it is characterised by a more situated approach that interprets data as strategic and political. It is argued that the existence of these frames provides two insights relevant for the literature on smart cities. First, they illustrate that one should be careful not to reify the smart city as a phenomenon that can be criticised in generic terms. Second, they suggest that even if there exists a transition toward the implementation of a technocratic smart city paradigm across public administrations, this paradigm is not unique in its focus on markets and evidence in governance. (shrink) How and When Socially Entrepreneurial Nonprofit Organizations Benefit From Adopting Social Alliance Management Routines to Manage Social Alliances?Gordon Liu, Wai Wai Ko & Chris Chapleo - 2018 - Journal of Business Ethics 151 (2):497-516.details Social alliance is defined as the collaboration between for-profit and nonprofit organizations. Building on the insights derived from the resource-based theory, we develop a conceptual framework to explain how socially entrepreneurial nonprofit organizations can improve their social alliance performance by adopting strategic alliance management routines. We test our framework using the data collected from 203 UK-based SENPOs in the context of cause-related marketing campaign-derived social alliances. Our results confirm a positive relationship between social alliance management routines and social alliance performance. (...) We also find that relational mechanisms, such as mutual trust, relational embeddedness, and relational commitment, mediate the relationship between social alliance management routines and social alliance performance. Moreover, our findings suggest that different types of social alliance motivation can influence the impact of social alliance management routines on different types of the relational mechanisms. In general, we demonstrate that SENPOs can benefit from adopting social alliance management routines and, in addition, highlight how and when the social alliance management routines–social alliance performance relationship might be shaped. Our study offers important academic and managerial implications, and points out future research directions. (shrink) Evolution of Cognition: Towards the Theory of Origin of Human Logic. [REVIEW]Vladimir G. Red'ko - 2000 - Foundations of Science 5 (3):323-338.details The main problem discussed in this paper is: Why and how did animal cognition abilities arise? It is argued that investigations of the evolution of animal cognition abilities are very important from an epistemological point of view. A new direction for interdisciplinary researches – the creation and development of the theory of human logic origin – is proposed. The approaches to the origination of such a theory (mathematical models of ``intelligent invention'' of biological evolution, the cybernetic schemes of evolutionary progress (...) and purposeful adaptive behavior) as well as potential interdisciplinary links of the theory are described and analyzed. (shrink) Epistemology of Logic in Logic and Philosophy of Logic Misfit Dislocation Loops in Composite Nanowires.I. A. Ovid'ko & A. G. Sheinerman - 2004 - Philosophical Magazine 84 (20):2103-2118.details Let's Celebrate Father Ivan MUSICHKA!Ivan Datsʹko - 2016 - Ukrainian Religious Studies 77:144-149.details Every Christian who carefully reads the message of St. Apostle Paul, can not do it pay attention to the number of times he uses, so to speak, military terminology. Suffice it to read the sixth chapter of the epistle to the Ephesians - the words, which is also St. Mr. Patriarch Joseph finished his Testament: "Fix in the Lord and in the power of his power. Put on a full armor of God so that you can resist the tricks devilish (...) For we have to fight not against the body and blood, but against the beginning, against the authorities, against the rulers of this world of darkness, against the spirits of malice Therefore, take a full weapon God so that you can resist and... stand firmly. Stand, then, girth Your hips are right, putting on the armor of justice, and putting your legs ready preaching the gospel of peace. (shrink) Kóρος.Julio César Díaz - 2010 - International Studies in Philosophy Monograph Series:11-19.details Ko Hung, der Philosoph und Alchimist.Alfred Forke - 1932 - Archiv für Geschichte der Philosophie 41:115.details Three-Level Mechanism of Consumer Digital Piracy: Development and Cross-Cultural Validation.Mateja Kos Koklic, Monika Kukar-Kinney & Irena Vida - 2016 - Journal of Business Ethics 134 (1):15-27.details Digital piracy as a continuing problem significantly impacts various stakeholders, including consumers, enterprises, and countries. This study develops a three-level mechanism of determinants of consumer digital piracy behavior, with personal risk as an individual factor, susceptibility to interpersonal influence as an inter-personal factor, and moral intensity as a broad societal factor. Further, it explores the role of rationalization and future piracy intent as outcomes of past piracy behaviors. The authors use survey data from four countries in the European Union to (...) test the system of structural relationships. With an exception of the effect of consumers' susceptibility to interpersonal influence on piracy behavior, the conceptual model receives remarkably consistent support across the four countries. Specifically, perception of personal risk and moral intensity negatively affected the reported piracy behavior in all four countries. The results further support the negative influence of moral intensity and the positive influence of past digital piracy behavior on consumers' use of rationalization. Lastly, personal risk, rationalization, and past digital piracy behavior directly influenced consumers' intention to engage in digital piracy in the future. The study also discusses implications of the findings and identifies areas of future research. (shrink) Physical Properties of Single Crystalline BaSn5.Xiao Lin, Sergey L. Bud'ko & Paul C. Canfield - 2012 - Philosophical Magazine 92 (24):3006-3014.details Philosophy of Physical Science, Misc in Philosophy of Physical Science On the Structure of Paradoxes.Du?ko Pavlovi? - 1992 - Archive for Mathematical Logic 31 (6):397-406.details Paradox is a logical phenomenon. Usually, it is produced in type theory, on a type Ω of "truth values". A formula Ψ (i.e., a term of type Ω) is presented, such that Ψ↔¬Ψ (with negation as a term¬∶Ω→Ω)-whereupon everything can be proved: In Sect. 1 we describe a general pattern which many constructions of the formula Ψ follow: for example, the well known arguments of Cantor, Russell, and Gödel. The structure uncovered behind these paradoxes is generalized in Sect. 2. This (...) allows us to show that Reynolds' [R] construction of a typeA ≃℘℘A in polymorphic λ-calculus cannot be extended, as conjectured, to give a fixed point ofevery variable type derived from the exponentiation: for some (contravariant) types, such a fixed point causes a paradox.Pursueing the idea that $$\frac{{{\text{type theory}}}}{{{\text{categorical interpretation}}}} = \frac{{{\text{(propositional) logic}}}}{{{\text{Lindebaum algebra}}}}$$ the language of categories appears here as a natural medium for logical structures. It allows us to abstract from the specific predicates that appear in particular paradoxes, and to display the underlying constructions in "pure state". The essential role of cartesian closed categories in this context has been pointed out in [L]. The paradoxes studied here remain within the limits of the cartesian closed structure of types, as sketched in this Lawvere's seminal paper — and do not depend on any logical operations on the type Ω. Our results can be translated in simply typed λ-calculus in a straightforward way (although some of them do become a bit messy). (shrink) Liar Paradox in Logic and Philosophy of Logic Neural Mechanisms of Inhibitory Response in a Battlefield Scenario: A Simultaneous fMRI-EEG Study.Li-Wei Ko, Yi-Cheng Shih, Rupesh Kumar Chikara, Ya-Ting Chuang & Erik C. Chang - 2016 - Frontiers in Human Neuroscience 10.details Philosophy of Neuroscience in Philosophy of Cognitive Science Electoral Incentives, Policy Compromise, and Coalition Durability: Japan's LDP–Komeito Government in a Mixed Electoral System.Adam P. Liff & Ko Maeda - 2019 - Japanese Journal of Political Science 20 (1):53-73.details Generation of Cracks at Triple Junctions of Grain Boundaries in Mechanically Loaded Polysilicon.I. A. Ovid'ko & A. G. Sheinerman - 2007 - Philosophical Magazine 87 (27):4181-4195.details 1 — 50 / 438	CommonCrawl
assume a poisson distribution The Poisson distribution. the probability that n falls within the range of 0 and n. For instance, we might be interested in the number of phone calls EXACTLY n successes in a Poisson getting AT MOST 1 phone call in the next hour would be an example of a cumulative Source: National Vital Statistics Report. Solution for Assume the Poisson distribution applies. Frequently-Asked Questions or review the The number of trials is large and the probability of success on any trial is small, so we assume $X$ has an approximate Poisson … tutorial So X~Po(0.7) P(X =0) e −0.7 0.4966 (4 d.p.) Assume arrivals occur according to a Poisson process with average 7 per hour. Find the probability that in a year, there will be 7 hurricanes.b. P(x)<1. average. distribution is a assume a poisson distribution with λ 5.6 find the following probabilities? What is the probability that schools in Dekalb County will close for 4 days Assume a Poisson distribution with? For example, suppose we know that a receptionist receives an For the Poission λ = μ. 🎉 The Study-to-Win Winning Ticket number has been announced! Find the probability that there will be 4 … Normal: It really depends on how you are going to use n since NORMDIST doesn't directly use n. assume poisson distribution. (a) The probability that a randomly selected 55-year-old African American female will live beyond 80 years of age (at least 25 more years)(b) The probability that a randomly selected 55-year-old African American female will live less than 20 more years, Life Expectancy According to the National Center for Health Statistics, the life expectancy for a 55 -year-old African American female is 26.1 years. The Poisson probability distribution gives the probability of a number of events occurring in a fixed interval of time or space if these events happen with a known average rate and independently of the time since the last event. Poisson distribution. What is the probability of at least 39 absences in 5 days? So we're going to do f sub two of two. Suppose she received 1 phone call per hour on The average rate of success refers to the average number of Since the schools have closed historically 3 days each year due to What is the probability that South Florida will be hit by a major hurricane at least once in the next ten years? Poisson experiment. The Poisson Distribution. In a 55 -year period, how many years are expected to have 4 hurricanes?c. experiment. The Poisson Distribution, on the other hand, doesn't require you to know n or p. We are assuming n is infinitely large and p is infinitesimal. B) If λ = 8.0 , find P (X ≥ 3). phone call per hour on average. distribution. a. assume a Poisson distribution with (upside down looking y symbol) = 5.2. Use the given mean to find the indicated probability Find P(5) when u= 9. ; The average rate at which events occur is constant; The occurrence of one event … will get 0, 1, 2, 3, or 4 calls next hour. The Poisson distribution is a probability distribution that does not predict the probability of an event occurring. All right, party. calculated, as shown in the table below. For instance, we might be interested in the number of phone calls The probability of a success during a small time interval is proportional to the … Poisson distribution calculator calculates the probability of given number of events that occurred in a fixed interval of time with respect to the known average rate of events occurred. =0.2700 (4 d.p.) So it's gonna be 6 to 6. In a 55 -year period, how many years are expected to have 7 hurricanes? The expected value of 8.7 years is close to the actual value of 8 years, so the Poisson distribution works well here. help_outline. Assuming that from age 55, the survival of African American females follows an exponential distribution, determine the following probabilities. So when x = 5 and mu = 7. If A = 8.0, find P(X = 8). snow, the average rate of success is 3. Assume a Poisson distribution with A = 5.0. A) If λ = 2.0 , find P (X ≥ 2 ). b. So six is six is 46,656. Similarly, if we focused on a 2-hour We also … The probability of getting EXACTLY This hotline receives an average of 3 calls per day that deal with sexual harassment. Asked Oct 4, 2020. a. C) If λ = 0.5 , find P (X≤ 1). For help in using the calculator, read the I don't have an account. So Y~Po(2.1) P(Y=2)= e−2.1×2.1 2 2! Find the probability that in a year, there will be 7 hurricanes. Assume that the Poisson distribution applies; assume that the mean number of Atlantic hurricanes in the United States is 6.1 per year, as in Example $I$; and proceed to find the indicated probability.Hurricanesa. Favorite Answer. Consider a Poisson distribution with $\mu=3$ .a. The Poisson approximation seems to fit the simulation results fairly well. b. X 1? Assume that a large Fortune 500 company has set up a hotline as part of a policy to eliminate sexual harassment among their employees and to protect themselves from future suits.) Does the Poisson distribution work well here? Does this data follow a Poisson distribution? to the probability of getting zero phone calls PLUS the probability of getting Let Lambda = 5.0, find P(x greaterthanorequalto 3) b. (Source: National Hurricane Center)a. What is the probability that between 6 and 10 processors fail "Looking for a Similar Assignment? = 8.0, find P(X ? But it's neat to know that it really is just the binomial distribution and the binomial distribution really did come from kind of the common sense of flipping coins. The Poisson distribution is a discrete probability distribution for the counts of events that occur randomly in a given interval of time (or space). Assume the Poisson distribution applies. We might ask: What is the likelihood that she Assume the Poisson distribution applies. Compute the probability of six occurrences in three time periods.f. A Poisson experiment has the following characteristics: The number of successes in a Poisson experiment is referred to as What is the probability that South Florida will be hit by a major hurricane at least once in the next ten years? Assume that a large Fortune 500 company has set up a hotline as part of a policy to eliminate sexual harassment among their employees and to protect themselves from future suits.) The Poisson distribution and the binomial distribution have some similarities, but also several differences. Write the appropriate Poisson probability function.b. Now we're for part C. We're gonna make another probability distribution function, And this time it's gonna be x number of occurrences over three time periods. Use the following information to answer the next seven exercises: A ballet instructor is interested in knowing what percent of each year's class will continue on to the next, so that she can plan what classes to offer. E) If λ = 5.0 , find P (X ≤ 3). What is a Poisson distribution. Compute $P(x \geq 2)$. Does the Poisson distribution work well here? The only parameter of the Poisson distribution is the rate λ (the expected value of x). probability distribution of a Poisson random variable. Here, n would be a Poisson Find the probability that in a year, there will be no hurricanes. Rather, it predicts the probability of how many times an event will occur. Instructions: To find the answer to a frequently-asked Here, we define a "success" as a school Multiply that by eating Lego six and divided by six Factorial, which is 720 you'll get 0.1606 All right, Finally, we need to compute the probability of five occurrences in two time periods. Assume a Poisson distribution. Click 'Join' if it's correct, By clicking Sign up you accept Numerade's Terms of Service and Privacy Policy. assume a poisson distribution. Compute $f(1)$d. I discuss the conditions required for a random variable to have a Poisson distribution. If none of the questions addresses your The probability that a success will occur within a short interval is We will later look at Poisson regression: we assume the response variable has a Poisson distribution (as an alternative to the normal Lv 7. And this is really interesting because a lot of times people give you the formula for the Poisson distribution and you can kind of just plug in the numbers and use it. 12 views. We need the Poisson Distribution to do interesting things like finding the probability of a number of events in a time period or finding the probability of waiting some time until the next event.. Our educator team will work on creating an answer for you in the next 6 hours. Use the given mean to find the indicated probability. experiment. Over the years, she has established the following probability distribution.$\bullet$ Let $X=$ the number of years a student will study ballet with the teacher.$\bullet$ Let $P(x)=$ the probability that a student will study ballet $x$ years.On average, how many years would you expect a child to study ballet with this teacher? Sample Problems. (Note: The Poisson probability in this example is equal to 0.061. We need to assume that the probability of getting an infection over a short time period is proportional to the length of the time period. So we're giving a possum distribution with unexpected number of occurrences in one time period of two and were asked to find the probability function for part A for this distribution. Relevance. P(x)=1. getting AT MOST n successes in a Poisson The Poisson Distribution … 3 phone calls in the next hour would be an example of a Poisson probability. The Poisson Process is the model we use for describing randomly occurring events and by itself, isn't that useful. an hour by a receptionist. b. The discrete compound Poisson distribution is also widely used in actuarial science for modelling the distribution of the total claim amount. Assume that, we conduct a Poisson experiment, in which the average number of successes within a given range is taken as λ. In probability theory and statistics, the Poisson distribution (/ ˈ p w ɑː s ɒ n /; French pronunciation: ), named after French mathematician Siméon Denis Poisson, is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time or space if these events occur with a … So from our formula, you know that this couldn't be are means the X power said to the X times eat the negative, mean so eat an egg to all over X factorial. Find the following probabilities.. a) X= 1 b) X< 1 c) X> 1 d) X < or equal to 1 A. Let Lambda = 0.4, find P(x lessthanorequalto 1) c. Let Lambda = 6.0, find P( x lessthanorequalto 2) Statisticians (especially in textbooks and classes) assume things fit a given distribution for the same reason that physics teachers start off problems with "Assume … For example, A Poisson probability refers to the probability of getting The count of events that will occur during the interval k being usually interval of time, a distance, volume or area. random variable and their associated probabilities represent a Poisson P(5) -U (Round to the nearest thousandth as needed.) The actual amount can vary. 1.). We might ask: What is the likelihood next hour that she will Assume a Poisson distribution. If we treated this as a Poisson experiment, then the average rate Obviously some days … Find the mean number of murders per day, then use that result to find the probability that in a day, there are no murders. = 2.0, find P(X ? average of 1 phone call per hour. The Calculator will compute the Poisson and help_outline. Poisson Distribution Mean and Variance. A Poisson experiment examines the number of times an event occurs What is the probability that South Florida will not be hit by a major hurricane in the next ten years?d. Poisson: If you assume that the mean of the distribution = np, then the cumulative distribution values decrease (e.g. What is the probability that South Florida will be hit by a major hurricane two years in a row?b. Here, n would be a Poisson Find the probability that in a year, there will be 5 hurricanes.b. 1 Answer VSH Mar 13, 2018 Answer link. Hurricanes a. How does the result from part (b) compare to the recent period of 55 years in which 8 years had 5 hurricanes? Find P(5 ) when μ=8 - Answered by a verified Tutor If we treat the number of phone Poisson sampling assumes that the random mechanism to generate the data can be described by a Poisson distribution. Suppose #X# has a Poisson distribution with a mean of .4. Asked Oct 4, 2020. an hour by a receptionist. on the Poisson distribution or visit the P(x)>1. 🎉View Winning Ticket. The interval could be anything - a unit of time, On average 4 of every 1000 processors Fails. (And the average rate of success would be 2 Answers. proportional to the size of the interval. Assume the Poisson distribution applies. The Poisson distribution The Poisson distribution is a discrete probability distribution for the counts of events that occur randomly in a given interval of time (or space). ? calls as a Poisson random variable, the various probabilities can be The probability that a randomly selected 55 -year-old African American female will live less than 20 more years, Let $X,$ the number of flaws on the surface of a randomly selected carpet of a particular type, have a Poisson distribution with parameter $\mu=5 .$ Use software or Appendix Table A.2 to compute the following probabilities:(a) $P(X \leq 8)$(b) $P(X=8)$(c) $P(9 \leq X)$(d) $P(5 \leq X \leq 8)$(e) $P(5<8)$. Poisson distributions Mixed exercise 2 1 a Let X be the number of accidents in a one-month period. The Poisson distribution is discussed in Appendix 5–D at the end of this chapter. A random variable X that obeys a Poisson distribution takes on only nonnegative values; the probability that X = k is. All right. x=1 B.X<1 C. X>1 D. View the step-by-step solution to: Question Assume a Poisson distribution with λequals=4.2 Find the following probabilities. It is named after Simeon-Denis Poisson (1781-1840), a French mathematician, who published its essentials in a paper in 1837. Find the probability that in a year, there will be no hurricanes.b. b Let Y be the number of accidents in a three-month period. during a specified interval. The average rate of success 6. find the following probabilities. Question. Ah, we have two occurrences in one time, period. that the average rate of success is 2 errors for every five pages. In general, assume that X 1, …, X p are p regression variables observed jointly with a count response variable Y that follows the Poisson distribution. The Poisson distribution was discovered by a French Mathematician-cum- Physicist, Simeon Denis Poisson in 1837. A Poisson distribution is a probability distribution of a Poisson random variable. c. X > 1? You da real mvps! All right, now we're asked to find the probability of two occurrences over one time period, and that corresponds Sid F sub two over here. A Poisson random variable refers to the number of successes in a Poisson random variable would be 4. ... To intuitively understand the Poisson distribution, assume we have a collection of … Click to sign up. Then, the average rate of Whoops, there might be a typo in your email. Go to your Tickets dashboard to see if you won! In general, assume that X 1, …, X p are p regression variables observed jointly with a count response variable Y that follows the Poisson distribution. In Poisson distribution, the mean of the distribution is represented by λ and e is constant, which is approximately equal to 2.71828. Suppose we knew that she received 1 How does the result from part (b) compare to the recent period of 55 years in which 10 years had 4 hurricanes? Six all over X factorial. The probability of a success during a small time interval is proportional to the entire length of the time interval. See the answer. Does the Poisson distribution work well here? For example, at any particular time, there is a certain probability that a particular cell within a large population of cells will acquire a mutation. next year? But it's neat to know that it really is just the binomial distribution and the binomial distribution really did come from kind of the common sense of flipping coins. = 5.0. For a Poisson Distribution you have. View Answer. In Exercises 5–8, assume that the Poisson distribution applies; assume that the mean number of Atlantic hurricanes in the United States is 6.1 per year, as in Example 1; and proceed to find the indicated probability. question, simply click on the question. is small. Properties of the Poisson distribution. And you should get zero point one 563 and those were your answers. In a 55 -year period, how many years are expected to have 7 hurricanes?c. And we will get that that probability is equal to 0.2 707 rounded to four decimal places. It will calculate all the poisson probabilities from 0 to x. Thanks to all of you who support me on Patreon. We might, for example, ask how many customers visit a Source:National Vital Statistics Report. Write the appropriate Poisson probability function.b. Three time periods. A. x=1 B.X<1 C. X>1 D.X ≤ 1. We will use the term "interval" to refer to either a time interval or an area, depending on the context of the problem. It's gonna become to square times eat in anger to over two factorial, and we're gonna punch them to a calculator real quick. of success over a 1-hour period would be 1 phone call. d. If A = 3.7, find P(X Attributes of a Poisson Experiment A Poisson experiment is a statistical experiment that has the following properties: The experiment results in outcomes that can be classified as successes or failures. Normal: It really depends on how you are going to use n since NORMDIST doesn't directly use n. However, The Poisson Distribution was developed by the French mathematician Simeon Denis Poisson in 1837. Answer Save. At least 6$?$ At least 10$?$(b) What are the expected value and standard deviation of the number of small aircraft that arrive during a 90 -min period? success would be 0.5 calls per half hour. The average rate of success is 3. The probability that a randomly selected 55 -year-old African American female will live beyond 80 years of age (at least 25 more years)b. Use the Poisson distribution to find the indicated probabilities. The shift geometric distribution is discrete compound Poisson distribution since it is a trivial case of negative binomial distribution. The Poisson distribution refers to a discrete probability distribution that expresses the probability of a specific number of events to take place in a fixed interval of time and/or space assuming that these events take place with a given average rate and independently of … For example, a book editor might be interested in the number of words spelled incorrectly in a particular book. The Poisson Calculator makes it easy to compute individual and cumulative a. An introduction to the Poisson distribution. The properties of the Poisson distribution have relation to those of the binomial distribution:. If we treated this as a Poisson experiment, then the value of the View Answer. Assume that the Poisson distribution applies; assume that the mean number of Atlantic hurricanes in the United States is 6.1 per year, as in Example $I$; and proceed to find the indicated probability.Hurricanesa. Statistics Random Variables Probability Distribution. Use the Poisson distribution to find the indicated probabilities.In a recent year, NYU-Langone Medical Center had 4221 birhs. Question. need, refer to Stat Trek's tutorial Enter a value in BOTH of the first two text boxes. Poisson Distribution. Poisson distribution. A Poisson distribution is the probability distribution that results from a Poisson experiment. We're supposed to find the probability of six occurrences and three time periods. The probability that a single success will occur during a short interval is Suppose we knew that she received 1 What is the Applications of the Poisson distribution can be found in many fields including: The Poisson Distribution is a discrete distribution. And the cumulative Poisson probability would be Write the appropriate Poisson probability function to determine the probability of $x$ occurrences in three time periods.d. hour on average. Taken together, the values for the Poisson What is the probability that South Florida will not be hit by a major hurricane in the next ten years?d. The probability that South Florida will be hit by a major hurricane (category 4 or 5 ) any single year is $\frac{1}{16}$ . Poisson probability. 60 accents eat in the negative. Use the given mean to find the indicated probability. What is the probability that a. X = 1? Suppose we knew that she received 1 phone call per Hurricanes 2010 The data below give the number of hurricanes classified as major hurricanes in the Atlantic Ocean each year from 1944 through 2006, as reported by NOAA (www.nhc.noaa.gov):3, 3, 1, 2, 4, 3, 8, 5, 3, 4, 2, 6, 2, 2, 5, 2, 2, 7, 1, 2, 6, 1, 3,1, 0, 5, 2, 1, 0, 1, 2, 3, 2, 1, 2, 2, 2, 3, 1, 1, 1, 3, 0, 1, 3, 2,1, 2, 1, 1, 0, 5, 6, 1, 3, 5, 3, 4, 2, 3, 6, 7, 2, 2, 5, 2, 5a) Create a dotplot of these data.b) Describe the distribution. a Poisson random variable. The Poisson distribution became useful as it models events, particularly uncommon events. The Poisson distribution has mean (expected value) λ = 0.5 = μ and variance σ 2 = λ = 0.5, that is, the mean and variance are the same. The Poisson distribution … table.). experiment might involve a different unit of time. on the Poisson distribution. The Poisson Process is the model we use for describing randomly occurring events and by itself, isn't that useful. Denote a Poisson process as a random experiment that consist on observe the occurrence of specific events over a continuous support (generally the space or the time), such that the process is stable (the number of occurrences, \lambda is constant in the long run) and the events occur randomly and independently.. The Poisson distribution refers to a discrete probability distribution that expresses the probability of a specific number of events to take place in a fixed interval of time and/or space assuming that these events take place with a given average rate and independently of the time since the occurrence of the last event. random variable. For this, we're gonna need to make yet another probability function. Related questions. two main characteristics of a Poisson experiment. To learn more about the Poisson distribution, read Stat Trek's Denote a Poisson process as a random experiment that consist on observe the occurrence of specific events over a continuous support (generally the space or the time), such that the process is stable (the number of occurrences, \lambda is constant in the long run) and the events occur randomly and independently.. Image Transcriptionclose. You must be logged in to bookmark a video. A Poisson distribution is often used to model data which arises from counting the number of occurrences of an outcome within a specified time period or area. This distribution represents the probability of an amount of time passing before an event occurs. In this article, we will discuss the Poisson distribution formula with examples. 2). Let's make this you thio. The Poisson distribution is the discrete probability distribution of the number of events occurring in a given time period, given the average number of times the event occurs over that time period. For example, suppose we know that a receptionist receives an average of 1 phone call per hour. Assume the variable follows a Poisson distribution. pages? b. M3. The Poisson distribution has mean (expected value) λ = 0.5 = μ and variance σ 2 = λ = 0.5, that is, the mean and variance are the same. 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\begin{definition}[Definition:Smallest/Ordered Set/Subset] Let $\struct {S, \preceq}$ be an ordered set. Let $T \subseteq S$ be a subset of $S$. An element $x \in T$ is '''the smallest element of $T$''' {{iff}}: :$\forall y \in T: x \preceq \restriction_T y$ where $\preceq \restriction_T$ denotes the restriction of $\preceq$ to $T$. \end{definition}	ProofWiki
Decoherence effects in the quantum qubit flip game using Markovian approximation Piotr Gawron1, Dariusz Kurzyk2,1 & Łukasz Pawela1 Quantum Information Processing volume 13, pages 665–682 (2014)Cite this article We are considering a quantum version of the penny flip game, whose implementation is influenced by the environment that causes decoherence of the system. In order to model the decoherence, we assume Markovian approximation of open quantum system dynamics. We focus our attention on the phase damping, amplitude damping and amplitude raising channels. Our results show that the Pauli strategy is no longer a Nash equilibrium under decoherence. We attempt to optimize the players' control pulses in the aforementioned setup to allow them to achieve higher probability of winning the game compared with the Pauli strategy. Quantum information experiments can be described as a sequence of three operations: state preparation, evolution and measurement [1]. In most cases, one cannot assume that experiments are conducted perfectly; therefore, imperfections have to be taken into account while modeling them. In this work, we are interested in how the knowledge about imperfect evolution of a quantum system can be exploited by players engaged in a quantum game. We assume that one of the players possesses the knowledge about imperfections in the system, while the other is ignorant of their existence. We ask a question of how much the player's knowledge about those imperfections can be exploited by him/her for their advantage. We consider implementation of the quantum version of the penny flip game, which is influenced by the environment that causes decoherence of the system. In order to model the decoherence, we assume Markovian approximation of open quantum system dynamics. This assumption is valid, for example, in the case of two-level atom coupled to the vacuum, undergoing spontaneous emission (amplitude damping). The coherent part of the atom's evolution is described by one-qubit Hamiltonian. Spontaneous emission causes an atom in the excited state to drop down into the ground state, emitting a photon in the process. Similarly, phase damping channel can be considered. This channel causes a continuous decay of coherence without energy dissipation in a quantum system [2]. The paper is organized as follows: in the two following subsections, we discuss related work and present our motivation to undertake this task. In Sect. 2, we recall the penny flip game and its quantum version; in Sect. 3, we present the noise model; in Sect. 4, we discuss the strategies applied in the presence of noise and finally in Sect. 5, we conclude the obtained results. Imperfect realizations of quantum games have been discussed in literature since the beginning of the century. Johnson [3] discusses a three-player quantum game played with a corrupted source of entangled qubits. The author implicitly assumes that the initial state of the game had passed through a bit-flip noisy channel before the game began. The corruption of quantum states in schemes implementing quantum games was studied by various authors, e.g., in [4], the authors study the general treatment of decoherence in two-player, two-strategy quantum games; in [5], the authors perform an analysis of the two-player prisoners' dilemma game; in [6], the multiplayer quantum minority game with decoherence is studied; in [7, 8], the authors analyze the influence of the local noisy channels on quantum Magic Squares games, while the quantum Monty Hall problem under decoherence is studied first in [9] and subsequently in [10]. In [11], the authors study the influence of the interaction of qubits forming a spin chain on the qubit flip game. An analysis of trembling hand-perfect equilibria in quantum games was done in [12]. Prisoners' dilemma in the presence of collective dephasing modeled by using the Markovian approximation of open quantum systems dynamics is studied in [13]. Unfortunately, the model applied in this work assumes that decoherence acts only after the initial state has been prepared and ceases to act before unitary strategies are applied. Another interesting approach to quantum games is the study of relativistic quantum games [14, 15]. This setup has also been studied in a noisy setup [16]. In the quantum game, theoretic literature decoherence is typically applied to a quantum game in the following way: The entangled state is prepared, It is transferred through a noisy channel, Players' strategies are applied, The resulting state is transferred once again through a noisy channel, The state is disentangled, Quantum local measurements are performed, and the outcomes of the games are calculated. In some cases, where it is appropriate, steps 4 and 5 are omitted. The problem with the above procedure is that it separates unitary evolution from the decoherent evolution. In Miszczak et al. [11], it was proposed to observe the behavior of the quantum version of the penny flip game under more physically realistic assumptions where decoherence due to coupling with the environment and unitary evolution happen simultaneously. In the papers, the authors study an implementation of the qubit flip game on quantum spin chains. First, a design, expressed in the form of quantum control problem, of the game on the trivial, one-qubit spin chain is proposed. Then the environment in the form of an additional qubit is added, and spin-spin coupling is adjusted, so one of the players, under some assumptions, can not detect that the system is implemented on two qubits rather than on one qubit. In the paper, it is shown that if one of the players posses the knowledge about the spin coupling, he or she can exploit it for augmenting his or hers winning probability. Game as a quantum experiment In this work, our goal is to follow the work done in [11] and to discuss the quantum penny flip game as a physical experiment consisting in preparation, evolution and measurement of the system. For the purpose of this paper, we assume that preparation and measurement, contrary to noisy evolution of the system are perfect. We investigate the influence of the noise on the players' odds and how the noisiness of the system can be exploited by them. The noise model we use is described by the Lindblad master equation, and the dynamics of the system is expressed in the language of quantum systems control. Penny flip game In order to provide classical background for our problem, let us consider a classical two-player game, consisting in flipping over a coin by the players in three consecutive rounds. As usual, the players are called Alice and Bob. In each round, Alice and Bob performs one of two operations on the coin: flips it over or retains it unchanged. At the beginning of the game, the coin is turned heads up. During the course of the game the coin is hidden and the players do not know the opponents actions. If after the last round, the coin tails up, then Alice wins, otherwise the winner is Bob. The game consists of three rounds: Alice performs her action in the first and the third round, while Bob performs his in the second round of the game. Therefore, the set of allowed strategies consists of eight sequences $(N,N,N), (N,N,F),$ $ \ldots , (F,F,F)$, where $N$ corresponds to the non-flipping strategy and $F$ to the flipping strategy. Bob's pay-off table for this game is presented in Table 1. Looking at the pay-off tables, it can be seen that utility function of players in the game is balanced; thus, the penny flip game is a zero-sum game. Table 1 Bob's pay-off table for the penny flip game A detailed analysis of this game and its asymmetrical quantization can be found in [17]. In this work it was shown that there is no winning strategy for any player in the penny flip game. It was also shown, that if Alice was allowed to extend her set of strategies to quantum strategies she could always win. In Miszczak et al. [11] it was shown that when both players have access to quantum strategies the game becomes fair and it has the Nash equilibrium. Qubit flip game The quantum version of the qubit flip game was studied for the first time by Meyer [18]. In our study, we wish to follow the work done in the aforementioned paper [11]. Hence, we consider a quantum version of the penny flip game. In this case, we treat a qubit as a quantum coin. As in the classical case the game is divided into three rounds. Starting with Alice, in each round, one player performs a unitary operation on the quantum coin. The rules of the game are constrained by its physical implementation. In order to obtain an arbitrary one-qubit unitary operation it is sufficient to use a control Hamiltonian built using only two traceless Pauli operators [19]. Therefore, we assume that in each round each of the players can choose three control parameters $\alpha _1,\alpha _2,\alpha _3$ in order to realize his/hers strategy. The resulting unitary gate is given by the equation: $$\begin{aligned} U(\alpha _1,\alpha _2,\alpha _3)=\hbox {e}^{-\mathrm{i}\alpha _3\sigma _z \Delta t} \hbox {e}^{-\mathrm{i}\alpha _2\sigma _y \Delta t} \hbox {e}^{-\mathrm{i}\alpha _1\sigma _z \Delta t}, \end{aligned}$$ where $\Delta t$ is an arbitrarily chosen constant time interval. Therefore, the system defined above forms a single qubit system driven by time-dependent Hamiltonian $H(t)$, which is a piecewise constant and can be expressed in the following form $$\begin{aligned} H(t)= {\left\{ \begin{array}{ll} \alpha _1^{A_1}\sigma _z &{} \text { for } 0\le t < \Delta t,\\ \alpha _2^{A_1}\sigma _y &{} \text { for } \Delta t\le t < 2\Delta t,\\ \alpha _3^{A_1}\sigma _z &{} \text { for } 2\Delta t\le t < 3\Delta t,\\ \alpha _1^{B}\sigma _z &{} \text { for } 3\Delta t\le t < 4\Delta t,\\ \alpha _2^{B}\sigma _y &{} \text { for } 4\Delta t\le t < 5\Delta t,\\ \alpha _3^{B}\sigma _z &{} \text { for } 5\Delta t\le t < 6\Delta t,\\ \alpha _1^{A_2}\sigma _z &{} \text { for } 6\Delta t\le t < 7\Delta t,\\ \alpha _2^{A_2}\sigma _y &{} \text { for } 7\Delta t\le t < 8\Delta t,\\ \alpha _3^{A_2}\sigma _z &{} \text { for } 8\Delta t\le t \le 9\Delta t. \end{array}\right. } \end{aligned}$$ Control parameters in the Hamiltonian $H(t)$ will be referred to vector $\mathrm {\alpha }=(\alpha _1^{A_1}, \alpha _2^{A_1}, \alpha _3^{A_1}, \alpha _1^{B}, \alpha _2^{B}, \alpha _3^{B}, \alpha _1^{A_2}, \alpha _2^{A_2}, \alpha _3^{A_2})$, where $\alpha _i^{A_1},\alpha _i^{A_2}$ are determined by Alice and $\alpha _i^{B}$ are selected by Bob. Suppose that players are allowed to play the game by manipulating the control parameters in the Hamiltonian $H(t)$ representing the coherent part of the dynamics, but they are not aware of the action of the environment on the system. Hence, the time evolution of the system is non-unitary and is described by a master equation, which can be written generally in the Lindblad form as $$\begin{aligned} \frac{\mathrm{d}\rho }{\mathrm{d}t}=-\mathrm{i}[H(t),\rho ] + \sum _j \gamma _j(L_j\rho L_j^\dagger - \frac{1}{2}\{L_j^\dagger L_j,\rho \}), \end{aligned}$$ where $H(t)$ is the system Hamiltonian, $L_j$ are the Lindblad operators, representing the environment influence on the system [2] and $\rho $ is the state of the system. For the purpose of this paper we chose three classes of decoherence: amplitude damping, amplitude raising and phase damping which correspond to noisy operators $\sigma _{-}=\| 0 \rangle \langle 1 \|$, $\sigma _{+}=\| 1 \rangle \langle 0 \|$ and $\sigma _z$, respectively. Let us suppose that initially the quantum coin is in the state $\| 0 \rangle \langle 0 \|$. Next, in each round, Alice and Bob perform their sequences of controls on the qubit, where each control pulse is applied according to Eq. (3). After applying all of the nine pulses, we measure the expected value of the $\sigma _z$ operator. If $\mathrm{tr}(\sigma _z\rho (T))=-1$ Alice wins, if $\mathrm{tr}(\sigma _z\rho (T))=1$ Bob wins. Here, $\rho (T)$ denotes the state of the system at time $T=9\Delta t$. Alternatively we can say that the final step of the procedures consists in performing orthogonal measurement $\{O_\mathrm{tails}\rightarrow \| 1 \rangle \langle 1 \|,O_\mathrm{heads}\rightarrow \| 0 \rangle \langle 0 \|\}$ on state $\rho (T)$. The probability of measuring $O_\mathrm{tails}$ and $O_\mathrm{heads}$ determines pay-off functions for Alice and Bob, respectively. These probabilities can be obtained from relations $p(\mathrm{tails})=\langle 1 \|\rho (T)\| 1 \rangle $ and $p(\mathrm{heads})=\langle 0 \|\rho (T)\| 0 \rangle $. Nash equilibrium In this game, pure strategies cannot be in Nash equilibrium [18]. Hence, the players choose mixed strategies, which are better than the pure ones. We assume that Alice and Bob use the Pauli strategy, which is mixed and gives Nash equilibrium [11]; therefore, this strategy is a reasonable choice for the players. According to the Pauli strategy, each player chooses one of the four unitary operations $\{{1\!\!1}, \mathrm{i}\sigma _{x}, \mathrm{i}\sigma _{y}, \mathrm{i}\sigma _{z}\}$ with equal probability. Thus, to obtain the Pauli strategy, each player chooses a sequence of control parameters $(\alpha _1^\square , \alpha _2^\square , \alpha _3^\square )$ listed in Table 2. The symbol $\square $ can be substituted by $A_1,B,A_2$. It means that in each round, one player performs a unitary operation chosen randomly with a uniform probability distribution from the set $\{ {1\!\!1}, \mathrm{i}\sigma _x, \mathrm{i}\sigma _y, \mathrm{i}\sigma _z \}$. Table 2 Control parameters for realizing the Pauli strategy Influence of decoherence on the game In this section, we perform an analytical investigation which shows the influence of decoherence on the game result. In accordance with the Lindblad master equation, the environment influence on the system is represented by Lindblad operators $L_j$, while the rate of decoherence is described by parameters $\gamma _j$. In our game, players use the Pauli strategy; hence, the quantum system evolves depending on the Hamiltonians expressed as $H(t)=\alpha _i^\square \sigma _y$ or $H(t)=\alpha _i^\square \sigma _z$. To simplify the discussion, we consider Hamiltonians represented by diagonal matrices. In our case, $H=\alpha _i^\square \sigma _z$ is diagonal, but Hamiltonian $\alpha _i^\square \sigma _y$ requires diagonalization. Therefore, we will consider solutions of Lindblad equations for the Hamiltonians given by $H_z = \alpha _i^\square \sigma _z$ and $H_y = \alpha _i^\square U^\dagger \sigma _y U=\alpha _i^\square \left( \begin{array}{ll} -1 &{} 0\\ 0 &{} 1 \end{array} \right) $, where $U=\left( \begin{array}{ll} -\frac{\sqrt{2}}{2} &{} -\frac{\sqrt{2}}{2} \\ \mathrm{i}\frac{\sqrt{2}}{2} &{} -\mathrm{i}\frac{\sqrt{2}}{2} \end{array} \right) $ is unitary matrix, whose columns are the eigenvectors of $\sigma _y$. Thus, we consider the solutions of the Lindblad equation for the Hamiltonian of the form $$\begin{aligned} H=\beta _1\| 0 \rangle \langle 0 \| + \beta _2\| 1 \rangle \langle 1 \|. \end{aligned}$$ Amplitude damping and amplitude raising First we consider the amplitude damping decoherence, which corresponds to the Lindblad operator $\sigma _{-}$. Thus, the master Eq. (3) is expressed as $$\begin{aligned} \frac{\mathrm{d}\rho }{\mathrm{d}t}=-\mathrm{i}[H,\rho (t)] + \gamma (\sigma _{-}\rho (t)\sigma _{+}-\frac{1}{2}\sigma _{+}\sigma _{-}\rho (t)-\frac{1}{2}\ \rho (t)\sigma _{+}\sigma _{-}), \end{aligned}$$ where $\sigma _{+}=\sigma _{-}^\dagger =\| 1 \rangle \langle 0 \|$. The equation can be rewritten in the following form $$\begin{aligned} \frac{\mathrm{d}\rho }{\mathrm{d}t}=A\rho (t)+\rho (t) A^\dagger + \gamma \sigma _{-}\rho (t)\sigma _{+}, \end{aligned}$$ where $A=-\mathrm{i}H(t)-\frac{1}{2}\gamma \sigma _{+}\sigma _{-}$. In solving this equation it is helpful to make a change of variables $\rho (t)=\hbox {e}^{At}\hat{\rho }(t)\hbox {e}^{A^\dagger t}$. Hence, we obtain $$\begin{aligned} \frac{\mathrm{d}\hat{\rho }}{\mathrm{d}t}=\gamma B(t)\hat{\rho }(t) B^{\dagger }(t), \end{aligned}$$ where $B(t)=\hbox {e}^{-At}\sigma _{-}\hbox {e}^{At}=\hbox {e}^{-\mathrm{i}(\beta _2-\beta _1)t-\frac{\gamma }{2}t}\sigma _{-}$. It follows that $$\begin{aligned} \frac{\mathrm{d}\hat{\rho }}{\mathrm{d}t}=\gamma \hbox {e}^{-\gamma t} \sigma _{-}\hat{\rho }(t) \sigma _{+}. \end{aligned}$$ Due to the fact that $\sigma _{-}\sigma _{-}=\sigma _{+}\sigma _{+}=0$ and $\sigma _{-}\frac{\mathrm{d}\hat{\rho }}{\mathrm{d}t}\sigma _{+}=0$ it is possible to write $\hat{\rho }(t)$ as $$\begin{aligned} \hat{\rho }(t)=\hat{\rho }(0) -\hbox {e}^{-\gamma t}\sigma _{-}\hat{\rho }(0)\sigma _{+}. \end{aligned}$$ Coming back to the original variables we get the expression $$\begin{aligned} \rho (t)=\hbox {e}^{At}\rho (0)\hbox {e}^{A^\dagger t}-\hbox {e}^{-\gamma t}\sigma _{-}\rho (0)\sigma _{+}. \end{aligned}$$ In order to study the asymptotic effects of decoherence on the results of the game, we consider the following limit $$\begin{aligned} \lim _{\gamma \rightarrow \infty } \hbox {e}^{At}\rho (0)\hbox {e}^{A^\dagger t}-\hbox {e}^{-\gamma t}\sigma _{+}\rho (0)\sigma _{-} = \| 0 \rangle \langle 0 \|\rho (0)\| 0 \rangle \langle 0 \|. \end{aligned}$$ Let $\rho (0)=\| 0 \rangle \langle 0 \|$; thus, the above limit is equal to $\| 0 \rangle \langle 0 \|$. This result shows that for high values of $\gamma $, chances of winning the game by Bob increase to 1 as $\gamma $ increases. Figure 1 shows an example of the evolution of a quantum system with amplitude damping decoherence for two values of the parameter $\gamma $. Figure 1a, b show the player's control pulses. In this case they are the ones implementing the Pauli strategy. Figure 1c, d show the time evolution of the state expressed as the expectation values of the observables $\sigma _x$, $\sigma _y$ and $\sigma _z$ for both cases. Finally, Fig. 1e, f show the evolution of the qubit's state in the Bloch sphere. This shows how a little amount of noise influences the evolution of the system and changes the probability of winning the game. Example of the time evolution of a quantum system with the amplitude damping decoherence for a sequence of control parameters $\alpha $ and fixed $\gamma =0.1$ (left side), $\gamma =0.7$ (right side). a Control parameters $\alpha =(-\frac{\pi }{4},-\frac{\pi }{2},\frac{\pi }{4},0,-\frac{\pi }{2},0,-\frac{\pi }{4},-\frac{\pi }{2},\frac{\pi }{4})$. b Control parameters $\alpha =(0,-\frac{\pi }{2},0,-\frac{\pi }{4},-\frac{\pi }{2},\frac{\pi }{4},-\frac{\pi }{4},0,-\frac{\pi }{4})$. c Mean values of $\sigma _x,\sigma _y$ and $\sigma _z$. d Mean values of $\sigma _x,\sigma _y$ and $\sigma _z$. e Time evolution of a quantum coin. f Time evolution of a quantum coin The noisy operator $\sigma _{+}$ is related to amplitude raising decoherence, and the solution of the master equation has the following form $$\begin{aligned} \rho (t)=\hbox {e}^{At}\rho (0)\hbox {e}^{A^\dagger t}-\hbox {e}^{-\gamma t}\sigma _{+}\rho (0)\sigma _{-}, \end{aligned}$$ where $A=-\mathrm{i}H(t) -\frac{1}{2}\gamma \sigma _{-}\sigma _{+}$. It is easy to check that as $\gamma \rightarrow \infty $ the state $\| 1 \rangle \langle 1 \|$ is the solution of the above equation, in which case Alice wins. Phase damping Now, we consider the impact of the phase damping decoherence on the outcome of the game. In this case, the Lindblad operator is given by $\sigma _z$. Hence, the Lindblad equation has the following form $$\begin{aligned} \frac{\mathrm{d}\rho }{\mathrm{d}t}&= -\mathrm{i}[H,\rho (t)] + \gamma (\sigma _z\rho (t)\sigma _z - \frac{1}{2}\sigma _z\sigma _z\rho (t) -\frac{1}{2}\rho (t)\sigma _z\sigma _z)\nonumber \\&= -\mathrm{i}[H,\rho (t)] + \gamma (\sigma _z\rho (t)\sigma _z - \rho (t)). \end{aligned}$$ Next, we make a change of variables $\hat{\rho }(t)=\hbox {e}^{\mathrm{i}Ht}\rho (t)\hbox {e}^{-\mathrm{i}Ht}$, which is helpful to solve the equation. We obtain $$\begin{aligned} \frac{\mathrm{d}\hat{\rho }}{\mathrm{d}t}&= \frac{\mathrm{d}\hbox {e}^{\mathrm{i}Ht}}{\mathrm{d}t}\rho (t)\hbox {e}^{-\mathrm{i}Ht}+ \hbox {e}^{\mathrm{i}Ht}\frac{\mathrm{d}\rho }{\mathrm{d}t}\hbox {e}^{-\mathrm{i}Ht}+ \hbox {e}^{\mathrm{i}Ht}\rho (t)\frac{\mathrm{d}\hbox {e}^{-\mathrm{i}Ht}}{\mathrm{d}t}\nonumber \\&= \mathrm{i}H \hbox {e}^{\mathrm{i}Ht}\hbox {e}^{-\mathrm{i}Ht}\hat{\rho }(t)\hbox {e}^{\mathrm{i}Ht}\hbox {e}^{-\mathrm{i}Ht} - \mathrm{i}\hbox {e}^{\mathrm{i}Ht}H\hbox {e}^{-\mathrm{i}Ht}\hat{\rho }(t)\hbox {e}^{\mathrm{i}Ht}\hbox {e}^{-\mathrm{i}Ht}\nonumber \\&+\, \mathrm{i}\hbox {e}^{\mathrm{i}Ht}\hbox {e}^{-\mathrm{i}Ht}\hat{\rho }(t)\hbox {e}^{\mathrm{i}Ht}H\hbox {e}^{-\mathrm{i}Ht}+ \gamma \hbox {e}^{\mathrm{i}Ht}\sigma _z\hbox {e}^{-\mathrm{i}Ht}\hat{\rho }(t) \hbox {e}^{\mathrm{i}Ht}\sigma _z\hbox {e}^{-\mathrm{i}Ht}\nonumber \\&-\,\hbox {e}^{\mathrm{i}Ht}\hbox {e}^{-\mathrm{i}Ht}\hat{\rho }\hbox {e}^{\mathrm{i}Ht}\hbox {e}^{-\mathrm{i}Ht} -\mathrm{i}\hbox {e}^{\mathrm{i}Ht}\hbox {e}^{-\mathrm{i}Ht}\hat{\rho }\hbox {e}^{\mathrm{i}Ht}\hbox {e}^{-\mathrm{i}Ht}H \nonumber \\&= \gamma (\sigma _z\hat{\rho (t)}\sigma _z - \hat{\rho (t)}). \end{aligned}$$ It follows that the solution of the above equation is given by $$\begin{aligned} \hat{\rho }(t)&= \| 0 \rangle \langle 0 \|\rho (0)\| 0 \rangle \langle 0 \| + \| 1 \rangle \langle 1 \|\rho (0)\| 1 \rangle \langle 1 \| + \nonumber \\&+\, \mathrm{e}^{-2\gamma t} (\| 0 \rangle \langle 0 \|\rho (0)\| 1 \rangle \langle 1 \|+\| 1 \rangle \langle 1 \|\rho (0)\| 0 \rangle \langle 0 \|). \end{aligned}$$ $$\begin{aligned} \rho (t)&= \| 0 \rangle \langle 0 \|\rho (0)\| 0 \rangle \langle 0 \| + \| 1 \rangle \langle 1 \|\rho (0)\| 1 \rangle \langle 1 \| + \nonumber \\&+\, \mathrm{e}^{-2\gamma t}\mathrm{e}^{-\mathrm{i}H t}(\| 0 \rangle \langle 0 \|\rho (0)\| 1 \rangle \langle 1 \|+\| 1 \rangle \langle 1 \|\rho (0)\| 0 \rangle \langle 0 \|) \mathrm{e}^{\mathrm{i}H t}. \end{aligned}$$ Consider the following limit $$\begin{aligned} \lim _{\gamma \rightarrow \infty } \rho (t)= \| 0 \rangle \langle 0 \|\rho (0)\| 0 \rangle \langle 0 \| + \| 1 \rangle \langle 1 \|\rho (0)\| 1 \rangle \langle 1 \|. \end{aligned}$$ The above result is a diagonal matrix dependent on the initial state. For high values of $\gamma $, the initial state $\rho (0)$ has a significant impact on the game. If $\rho (0)=\| 0 \rangle \langle 0 \|$ then $\lim _{\gamma \rightarrow \infty } \rho (t)=\| 0 \rangle \langle 0 \|$. This kind of decoherence is conducive to Bob. Similarly, if $\rho (0) = \| 1 \rangle \langle 1 \|$, then Alice wins. The evolution of a quantum system with the phase damping decoherence and fixed Hamiltonian is shown in Fig. 2. Figures 2a,b show the player's control pulses. In this case they are the ones implementing the Pauli strategy. Figure 2c,d show the time evolution of the state expressed as the expectation values of the observables $\sigma _x$, $\sigma _y$ and $\sigma _z$ for both cases. Finally, Fig. 2e,f show the evolution of the qubit's state in the Bloch sphere. In this case, we can see that a low amount of phase damping noise does not have a significant impact on the outcome of the game. On the other hand, for higher values of $\gamma $ we can see mainly the effect of the decoherence rather than the effect of player's actions, i.e., the state evolves almost directly toward the maximally mixed state. Example of the time evolution of a quantum system with the phase damping decoherence for fixed $\gamma =0.5$ (left side), $\gamma =5$ (right side) and a sequence of control parameters $\alpha $. a Control parameters $\alpha =(-\frac{\pi }{4},-\frac{\pi }{2},\frac{\pi }{4},-\frac{\pi }{4},-\frac{\pi }{2},\frac{\pi }{4},-\frac{\pi }{4},-\frac{\pi }{2},\frac{\pi }{4})$. b Control parameters $\alpha =(0,-\frac{\pi }{2},0,0,-\frac{\pi }{2},0,0,-\frac{\pi }{2},0)$. c Mean values of $\sigma _x,\sigma _y$ and $\sigma _z$. d Mean values of $\sigma _x,\sigma _y$ and $\sigma _z$. e Time evolution of a quantum coin. f Time evolution of a quantum coin Optimal strategy for the players Due to the noisy evolution of the underlying qubit, the strategy given by Table 2 is no longer a Nash equilibrium. We study the possibility of optimizing one player's strategy, while the other one uses the Pauli strategy. It turns out that this optimization is not always possible. If the rate of decoherence is high enough, then the players' strategies have little impact on the game outcome. In the low noise scenario, it is possible to optimize the strategy of both players. In each round, one player performs a series of unitary operations, which are chosen randomly from a uniform distribution. Therefore, the strategy of a player can be seen as a random unitary channel. In this section $\Phi _{A_1},\Phi _{A_2}$ denote mixed unitary channels used by Alice who implements the Pauli strategy. Similarly, $\Phi _B$ denotes channels used by Bob. Optimization method In order to find optimal strategies for the players, we assume the Hamiltonian in (3) to have the form $$\begin{aligned} H = H(\varepsilon (t)), \end{aligned}$$ where $\varepsilon (t)$ are the control pulses. As the optimization target, we introduce the cost functional $$\begin{aligned} J(\varepsilon )=\mathrm{tr}\{ F_0(\rho (T)) \}, \end{aligned}$$ where $F_0(\rho (T))$ is a functional that is bounded from below and differentiable with respect to $\rho (T)$. A sequence of control pulses that minimizes the functional (19) is said to be optimal. In our case we assume that $$\begin{aligned} \mathrm{tr}\{ F_0(\rho (T)) \} = \frac{1}{2} \|\| \rho (T) - \rho _\mathrm{T} \|\|_\mathrm{F}^2, \end{aligned}$$ where $\rho _\mathrm{T}$ is the target density matrix of the system. In order to solve this optimization problem, we need to find an analytical formula for the derivative of the cost functional (19) with respect to control pulses $\varepsilon (t)$. Using the Pontryagin principle [20], it is possible to show that we need to solve the following equations to obtain the analytical formula for the derivative $$\begin{aligned} \frac{\mathrm{d}\rho (t)}{\mathrm{d}t}&= -\mathrm{i}[H(\varepsilon (t)) ,\rho (t)] - \mathrm{i}L_\mathrm{D} [\rho (t)],\; t\in [0, T],\end{aligned}$$ $$\begin{aligned} \frac{\mathrm{d}\lambda (t)}{\mathrm{d}t}&= -\mathrm{i}[H(\varepsilon (t)) ,\lambda (t)] - \mathrm{i}L_\mathrm{D}^\dagger [\lambda (t)],\; t\in [0, T],\end{aligned}$$ $$\begin{aligned} L_\mathrm{D}[A]&= \mathrm{i}\sum _j \gamma _j(L_j A L_j^\dagger - \frac{1}{2}\{L_j^\dagger L_j,A\}),\end{aligned}$$ $$\begin{aligned} \rho (0)&= \rho _\mathrm{s}, \end{aligned}$$ $$\begin{aligned} \lambda (T)&= F'_0(\rho (T)), \end{aligned}$$ where $\rho _\mathrm{s}$ denotes the initial density matrix, $\lambda (t)$ is called the adjoint state and $$\begin{aligned} F'_0(\rho (T)) = \rho (T) - \rho _\mathrm{T}. \end{aligned}$$ The derivation of these equations can be found in [21]. In order to optimize the control pulses using a gradient method, we convert the problem from an infinite dimensional (continuous time) to a finite dimensional (discrete time) one. For this purpose, we discretize the time interval $[0, T]$ into $M$ equal sized subintervals $\Delta t_k$. Thus, the problem becomes that of finding $\varepsilon =[\varepsilon _1,\ldots ,\varepsilon _M]^\mathrm{T}$ such that $$\begin{aligned} J(\varepsilon ) = \inf _{\zeta \in \mathbb {R}^M}J(\zeta ). \end{aligned}$$ The gradient of the cost functional is $$\begin{aligned} G = \left[ \frac{\partial J}{\partial \varepsilon _1}, \ldots , \frac{\partial J}{\partial \varepsilon _M} \right] ^\mathrm{T}. \end{aligned}$$ It can be shown [21] that elements of vector (28) are given by $$\begin{aligned} \frac{\partial J}{\partial \varepsilon _k} = \mathrm{tr}\left\{ -\mathrm{i}\lambda _k \left[ \frac{\partial H(\varepsilon _k)}{\partial \varepsilon _k}, \rho _k \right] \right\} \Delta t_k, \end{aligned}$$ where $\rho _k$ and $\lambda _k$ are solutions of the Lindblad equation and the adjoint system corresponding to time subinterval $\Delta t_k$, respectively. To minimize the gradient given in Eq. (28) we use the BFGS algorithm [22]. Optimization setup Our goal is to find control strategies for players, which maximize their respective chances of winning the game. We study three noise channels: the amplitude damping, the phase damping and the amplitude raising channel. They are given by the Lindblad operators $\sigma _-$, $\sigma _z$ and $\sigma _+ = \sigma _-^\dagger $, respectively. In all cases, we assume that one of the players uses the Pauli strategy, while for the other player we try to optimize a control strategy that maximizes that player's probability of winning. However, in our setup it is convenient to use the value of the observable $\sigma _z$ rather than probabilities. Value 0 means that each player has a probability of $\frac{1}{2}$ of winning the game. Values closer to 1 mean higher probability of winning for Bob, while values closer to -1 mean higher probability of winning for Alice. Optimization results The results for the phase damping channel are shown in Fig. 3. As it can be seen, in this case, both players are able to optimize their strategies, and so Alice can optimize her strategy for low values of $\gamma $ to obtain the probability of winning grater than $\frac{1}{2}$. The region where this occurs is shown in the inset. For high noise values, she is able to achieve the probability of winning equal to $\frac{1}{2}$. In the case of high values of $\gamma $, the best strategy for Alice is to drive the state as close as possible to the maximally mixed state on her first move. This state can not be changed neither by Bob's actions, nor by the phase damping channel. On the other hand, optimization of Bob's strategy shows that he is able to achieve high probabilities of winning for relatively low values of $\gamma $. This is consistent with the limit shown in Eq. (17) as our initial state is $\rho =\| 0 \rangle \langle 0 \|$. Figure 4 presents optimal game strategies for both players. For Alice we chose $\gamma =1.172$ which corresponds to her maximal probability of winning the game. In the case of Bob's strategies we arbitrarily choose the value $\gamma =1.610$. In these cases the evolution of the qubit is much more complex. This is due to the fact that the players are not restricted to the Pauli strategy. Mean value of the pay-off for the phase damping channel with and without optimization of the player's strategies. The inset shows the region where Alice is able to increase her probability of winning to exceed $\frac{1}{2}$ Game results for the phase damping channel. Optimal Alice's strategy when $\gamma = 1.172$ (left side), and optimal Bob's strategy when $\gamma = 1.610$ (right side). a Optimal controls for Alice, b Optimal controls for Bob, c Mean values of $\sigma _x,\sigma _y$ and $\sigma _z$, d Mean values of $\sigma _x,\sigma _y$ and $\sigma _z$, e Time evolution of a quantum coin, f Time evolution of a quantum coin Amplitude damping Next, we present the results obtained for the amplitude damping channel. They are shown in Fig. 5. Unfortunately, for Alice, for high values of $\gamma $ Bob always wins. This is due to the fact that in this case the state quickly decays to state $\| 0 \rangle \langle 0 \|$. Additionally, Bob is also able to optimize his strategies. He is able to achieve probability of winning equal to 1 for relatively low values of $\gamma $. However, for low values of $\gamma $, the interaction allows Alice to achieve higher than $\frac{1}{2}$ probability of winning. The region where this happens is magnified in the inset. Interestingly, for very low values of $\gamma $, Alice can increase her probability of winning. This is due to the fact that low noise values are sufficient to distort Bob's attempts to perform the Pauli strategy. On the other hand, they are not high enough to drive the system toward state $\| 0 \rangle \langle 0 \|$. Optimal game results for both players are shown in Fig. 6. For both players, we chose $\gamma =0.621$ which corresponds to Alice's maximal probability of winning the game. As can be seen, in this case, the evolutions of the observables $\sigma _x$, $\sigma _y$ and $\sigma _z$ show rapid oscillations. This behavior is turned on by applying control pulses associated with the $\sigma _y$ Hamiltonian. Mean value of the pay-off for the amplitude damping channel with and without optimization of the player's strategies. The inset shows the region where Alice is able to increase her probability of winning to exceed $\frac{1}{2}$ Game results obtained for the amplitude damping channel with $\gamma $ equal to $0.621$. Optimal Alice's strategy (left side), and optimal Bob's strategy (right side). a Optimal controls for Alice, b Optimal controls for Bob, c Mean values of $\sigma _x,\sigma _y$ and $\sigma _z$, d Mean values of $\sigma _x,\sigma _y$ and $\sigma _z$, e Time evolution of a quantum coin, f Time evolution of a quantum coin Amplitude raising Finally, we present optimization results for the amplitude raising channel. The optimization results, shown in Fig. 7, indicate that Alice can achieve probability of winning equal to 1 for lower values of $\gamma $ compared with the unoptimized case. In this case, Bob cannot do any better than in the unoptimized case due to a limited number of available control pulses. Mean value of the pay-off for the amplitude raising channel with and without optimization of the player's strategies We studied the quantum version of the coin flip game under decoherence. To model the interaction with external environment, we used the Markovian approximation in the form of the Lindblad equation. Because of the fact that Pauli strategy is a known Nash equilibrium of the game, therefore, it was natural to investigate this strategy in the presence noise. Our results show that in the presence of noise, the Pauli strategy is no longer a Nash equilibrium. One of the players, Bob in our case, is always favoured by amplitude and phase damping noise. If we had considered a game with another initial state i.e.,, $\rho _0=\| 1 \rangle \langle 1 \|$, Alice would have been favoured in this case. Our next step was to check if the players were able to do better than the Pauli strategy. For this, we used the BFGS gradient method to optimize the players' strategies. 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Institute of Theoretical and Applied Informatics, Polish Academy of Sciences, Bałtycka 5, 44-100 , Gliwice, Poland Piotr Gawron, Dariusz Kurzyk & Łukasz Pawela Institute of Mathematics, Silesian University of Technology, Kaszubska 23, 44-100 , Gliwice, Poland Dariusz Kurzyk Piotr Gawron Łukasz Pawela Correspondence to Dariusz Kurzyk. Open Access This article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution, and reproduction in any medium, provided the original author(s) and the source are credited. Gawron, P., Kurzyk, D. & Pawela, Ł. Decoherence effects in the quantum qubit flip game using Markovian approximation. Quantum Inf Process 13, 665–682 (2014). https://doi.org/10.1007/s11128-013-0681-y Issue Date: March 2014 Lindblad master equation Decoherence effects Quantum games Open quantum systems	CommonCrawl
Lindström quantifier In mathematical logic, a Lindström quantifier is a generalized polyadic quantifier. Lindström quantifiers generalize first-order quantifiers, such as the existential quantifier, the universal quantifier, and the counting quantifiers. They were introduced by Per Lindström in 1966. They were later studied for their applications in logic in computer science and database query languages. Generalization of first-order quantifiers In order to facilitate discussion, some notational conventions need explaining. The expression $\phi ^{A,x,{\bar {a}}}=\{x\in A\colon A\models \phi [x,{\bar {a}}]\}$ for A an L-structure (or L-model) in a language L, φ an L-formula, and ${\bar {a}}$ a tuple of elements of the domain dom(A) of A. In other words, $\phi ^{A,x,{\bar {a}}}$ denotes a (monadic) property defined on dom(A). In general, where x is replaced by an n-tuple ${\bar {x}}$ of free variables, $\phi ^{A,{\bar {x}},{\bar {a}}}$ denotes an n-ary relation defined on dom(A). Each quantifier $Q_{A}$ is relativized to a structure, since each quantifier is viewed as a family of relations (between relations) on that structure. For a concrete example, take the universal and existential quantifiers ∀ and ∃, respectively. Their truth conditions can be specified as $A\models \forall x\phi [x,{\bar {a}}]\iff \phi ^{A,x,{\bar {a}}}\in \forall _{A}$ $A\models \exists x\phi [x,{\bar {a}}]\iff \phi ^{A,x,{\bar {a}}}\in \exists _{A},$ where $\forall _{A}$ is the singleton whose sole member is dom(A), and $\exists _{A}$ is the set of all non-empty subsets of dom(A) (i.e. the power set of dom(A) minus the empty set). In other words, each quantifier is a family of properties on dom(A), so each is called a monadic quantifier. Any quantifier defined as an n > 0-ary relation between properties on dom(A) is called monadic. Lindström introduced polyadic ones that are n > 0-ary relations between relations on domains of structures. Before we go on to Lindström's generalization, notice that any family of properties on dom(A) can be regarded as a monadic generalized quantifier. For example, the quantifier "there are exactly n things such that..." is a family of subsets of the domain of a structure, each of which has a cardinality of size n. Then, "there are exactly 2 things such that φ" is true in A iff the set of things that are such that φ is a member of the set of all subsets of dom(A) of size 2. A Lindström quantifier is a polyadic generalized quantifier, so instead being a relation between subsets of the domain, it is a relation between relations defined on the domain. For example, the quantifier $Q_{A}x_{1}x_{2}y_{1}z_{1}z_{2}z_{3}(\phi (x_{1}x_{2}),\psi (y_{1}),\theta (z_{1}z_{2}z_{3}))$ is defined semantically as $A\models Q_{A}x_{1}x_{2}y_{1}z_{1}z_{2}z_{3}(\phi ,\psi ,\theta )[a]\iff (\phi ^{A,x_{1}x_{2},{\bar {a}}},\psi ^{A,y_{1},{\bar {a}}},\theta ^{A,z_{1}z_{2}z_{3},{\bar {a}}})\in Q_{A}$ where $\phi ^{A,{\bar {x}},{\bar {a}}}=\{(x_{1},\dots ,x_{n})\in A^{n}\colon A\models \phi [{\bar {x}},{\bar {a}}]\}$ for an n-tuple ${\bar {x}}$ of variables. Lindström quantifiers are classified according to the number structure of their parameters. For example $Qxy\phi (x)\psi (y)$ is a type (1,1) quantifier, whereas $Qxy\phi (x,y)$ is a type (2) quantifier. An example of type (1,1) quantifier is Hartig's quantifier testing equicardinality, i.e. the extension of {A, B ⊆ M: \|A\| = \|B\|}. An example of a type (4) quantifier is the Henkin quantifier. Expressiveness hierarchy The first result in this direction was obtained by Lindström (1966) who showed that a type (1,1) quantifier was not definable in terms of a type (1) quantifier. After Lauri Hella (1989) developed a general technique for proving the relative expressiveness of quantifiers, the resulting hierarchy turned out to be lexicographically ordered by quantifier type: (1) < (1, 1) < . . . < (2) < (2, 1) < (2, 1, 1) < . . . < (2, 2) < . . . (3) < . . . For every type t, there is a quantifier of that type that is not definable in first-order logic extended with quantifiers that are of types less than t. As precursors to Lindström's theorem Although Lindström had only partially developed the hierarchy of quantifiers which now bear his name, it was enough for him to observe that some nice properties of first-order logic are lost when it is extended with certain generalized quantifiers. For example, adding a "there exist finitely many" quantifier results in a loss of compactness, whereas adding a "there exist uncountably many" quantifier to first-order logic results in a logic no longer satisfying the Löwenheim–Skolem theorem. In 1969 Lindström proved a much stronger result now known as Lindström's theorem, which intuitively states that first-order logic is the "strongest" logic having both properties. Algorithmic characterization References • Lindstrom, P. (1966). "First order predicate logic with generalized quantifiers". Theoria. 32 (3): 186–195. doi:10.1111/j.1755-2567.1966.tb00600.x. • L. Hella. "Definability hierarchies of generalized quantifiers", Annals of Pure and Applied Logic, 43(3):235–271, 1989, doi:10.1016/0168-0072(89)90070-5. • L. Hella. "Logical hierarchies in PTIME". In Proceedings of the 7th IEEE Symposium on Logic in Computer Science, 1992. • L. Hella, K. Luosto, and J. Vaananen. "The hierarchy theorem for generalized quantifiers". Journal of Symbolic Logic, 61(3):802–817, 1996. • Burtschick, Hans-Jörg; Vollmer, Heribert (1999), Lindström Quantifiers and Leaf Language Definability, ECCC TR96-005 • Westerståhl, Dag (2001), "Quantifiers", in Goble, Lou (ed.), The Blackwell Guide to Philosophical Logic, Blackwell Publishing, pp. 437–460. • Antonio Badia (2009). Quantifiers in Action: Generalized Quantification in Query, Logical and Natural Languages. Springer. ISBN 978-0-387-09563-9. Further reading • Jouko Väänanen (ed.), Generalized Quantifiers and Computation. 9th European Summer School in Logic, Language, and Information. ESSLLI’97 Workshop. Aix-en-Provence, France, August 11–22, 1997. Revised Lectures, Springer Lecture Notes in Computer Science 1754, ISBN 3-540-66993-0 External links • Dag Westerståhl, 2011. 'Generalized Quantifiers'. Stanford Encyclopedia of Philosophy.	Wikipedia
Research \| Open \| Published: 19 July 2017 Synergy between dinotefuran and fipronil against the cat flea (Ctenocephalides felis): improved onset of action and residual speed of kill in adult cats Romain Delcombel1, Hamadi Karembe1, Bakela Nare3, Audrey Burton3, Julian Liebenberg2, Josephus Fourie2 & Marie Varloud1 Parasites & Vectorsvolume 10, Article number: 341 (2017) \| Download Citation The cat flea, Ctenocephalides felis felis (C. felis), is a cosmopolitan hematophagous ectoparasite, and is considered to be the most prevalent flea species in both Europe and the USA. Clinical signs frequently associated with flea bites include pruritus, dermatitis and in severe cases even pyodermatitis and alopecia. Ctenocephalides felis is also a vector for several pathogens and is an intermediate host for the cestode Dipylidium caninum. Treatment of cats with a fast-acting pulicide, that is persistently effective in protecting the animal against re-infestation, is therefore imperative to their health. In addition, a rapid onset of activity ("speed of kill") may also reduce the risks of disease transmission and flea allergic dermatitis. The aim of this study was to evaluate the in vitro insecticidal activity and potential synergism between dinotefuran and fipronil against C. felis. A further aim was to evaluate the onset of activity and residual speed of kill of the combination in vivo on cats artificially infested with C. felis. In the first study, the insecticidal activity of dinotefuran and fipronil separately and dinotefuran/fipronil (DF) in combination, at a fixed ratio (2:1), was evaluated using an in vitro coated-vial bioassay. In the second study, the onset of activity against existing flea infestations and residual speed of kill of DF against artificial flea infestations on cats was assessed in vivo. Onset of activity against existing flea infestations was assessed in terms of knock-down effect within 2 h post-treatment and onset of speed of kill assessed at 3 h, 6 h and 12 h post-treatment. Residual speed of kill was evaluated 6 h and 48 h after infestation, over a period of six weeks post-treatment. In vitro results revealed that the DF combination was synergistic and more potent against fleas than either compound alone. The combination also proved effective when tested in vivo. Efficacy was > 97% [geometric mean (GM) and arithmetic mean (AM)] at 3 h after treatment, and ≥ 99.8% (GM and AM) at 6 h and 12 h post-treatment. At 6 h after flea re-infestations, the efficacy of DF remained ≥ 90.8% (GM and AM) for up to 28 days, and at 42 days post-treatment persistent efficacy was still ≥ 54.3% (GM and AM). At 48 h after flea re-infestations, DF remained almost fully effective for up to 28 days, with efficacies ≥ 99.4% (GM and AM) and was persistently ≥ 93.0% (GM and AM) effective for up to 42 days post-treatment. The combination of dinotefuran and fipronil in a single formulation exhibited strong synergistic insecticidal activity against C. felis in vitro, and also proved effective on artificially infested cats. This activity had a rapid onset that persisted for 6 weeks against re-infestations of C. felis on cats. The rapid curative insecticidal effect was observed as early as 3 h after treatment, and as early as 6 h after re-infestations for up to 6 weeks post-treatment. The insecticidal activity profile of DF makes it an optimal candidate for the protection of cats against flea infestations, and possibly also associated diseases. The cat flea, Ctenocephalides felis felis (C. felis) is a hematophagous ectoparasite that infests cats and other domestic animals worldwide [1]. Both outdoor and indoor domestic animals are affected. A recent epidemiological study conducted in several European countries, indicated that 15.5% of cats were infested with fleas [2]. Moreover, C. felis was the most prevalent flea species in this study. Similarly, C. felis has recently been reported to be the most prevalent flea species in the USA [3]. In this epidemiological survey conducted by Blagburn and collaborators in 2016, 96% of the fleas collected countrywide were identified as C. felis. Fleas are highly efficient in acquiring a host by leaping from their immediate environment onto such an animal [4]. In addition, when more than one animal is accommodated in a household, an infested host can transmit live and prolific fleas to its congeners, and this inter-host transfer can occur within an hour after contact [5]. Immediately after a host has been acquired, fleas start feeding and continue to take numerous blood meals daily [6,7,8,9]. Consequently, symptoms associated with flea infestations can soon be observed. For instance, allergenic proteins contained in C. felis saliva, may result in immediate hypersensitivity resulting in flea allergic dermatitis (FAD) [10]. Cats with FAD suffer from intense discomfort caused by severe pruritus and dermatitis characterized by excoriation, scaling, crusting, miliary lesions and papules. If untreated, the disease can lead to pyodermitis and alopecia. Although the threshold in unknown, a small number of flea bites are expected to induce a resurgence of the symptoms in cats already sensitized to flea allergens [11]. The detrimental effects of flea infestations are, however, not limited to sensitized animals, but can be responsible for skin irritation in non-flea allergen sensitized hosts, resulting in intense grooming and itching, and can even induce anemia in susceptible cats [11]. Ctenocephalides felis is also a vector of several pathogens, namely viruses such as feline leukemia virus [12], bacteria such as Rickettsia felis [13], Haemoplasma species [14] and Bartonella species [15]. Fleas also act as intermediate host for intestinal helminths such as the cestode Dipylidium caninum [16]. Effective treatment of cats with a fast-acting pulicide, with a persistent efficacy against re-infestation, is therefore imperative to their health. Whilst the efficacy of pulicidal products has generally been evaluated 48 h after treatment against existing flea infestations or re-infestation [17], some of them, especially spot-on formulations, are expected to act much sooner. This property, referred to as speed of kill, represents the rapid onset of activity, thus freeing cats from their fleas. Rapid speed of kill may also reduce the risks of disease transmission and FAD [18]. Dinotefuran is a furanicotinyl insecticide belonging to the third generation of neonicotinoids [19]. It acts specifically on the nervous system of fleas by inhibiting a nicotinic subclass of acetylcholine receptors. Dinotefuran is a contact pulicide and has a rapid speed of kill against fleas, as early as 2 h [20] to 6 h post-treatment on infested cats [21]. Moreover, it has been demonstrated that dinotefuran has a residual efficacy lasting for 30 days, when evaluated against 4 consecutive weekly infestations with fleas [22]. The activity of this compound is, however, limited to insects and no acaricidal activity has been demonstrated, except at high concentrations and after a week of exposure [23, 24]. Fipronil on the other hand, is a proven insecticide and acaricide. This compound acts on insects and acarines by blocking the action of gamma-aminobutyric acid. It also acts by contact and because it accumulates in the skin and sebaceous glands, remains active for at least 4 weeks against fleas and ticks [25]. In order to strengthen and extend insecticidal activity, a combination of active ingredients with different modes of action and potency in a single spot-on formulation has been proposed [26, 27]. Moreover, if compounds in combination are synergistic, the same insecticidal efficacy can be achieved as when they are administered separately, but at lower active ingredient concentrations. This is likely to improve the safety profile of such products. In this study, dinotefuran and fipronil were combined in a single solution, taking advantage of their different mechanisms of action. It was anticipated that this novel combination would provide a more complete topical protection of cats against ectoparasites, a rapid onset of efficacy to alleviate flea bite dermatitis and a long lasting residual speed of kill effect, aiding in the protection of cats against FAD and flea-borne pathogens. The aim of this study was to evaluate the insecticidal activity and potential synergism between dinotefuran and fipronil against C. felis in vitro, when tested separately or in combination, and also to evaluate the onset of activity and residual speed of kill in vivo on artificially infested cats. In vitro insecticidal activity and interaction between dinotefuran and fipronil The insecticidal activity of dinotefuran, fipronil and dinotefuran/fipronil (DF) in a 2:1 fixed ratio (2:1) was evaluated using an in vitro coated-vial bioassay. Test compounds were dissolved in DMSO to a final stock concentrations. An aliquot was taken from each compound stock and added to an acetone/triton solution to achieve the desired top concentrations for the study. The top concentrations of test compound, individually or in combination, were serially diluted with the same diluent to achieve the desired titration range. Vials were treated with dinotefuran, fipronil, DF or solvent alone. The final DMSO concentration in each vial was less than 0.5%. Vials were capped and allowed to dry for at least 4 h before adding 10 newly emerged (0 to 7 days old) unfed adult fleas in each vial. Flea susceptibility was assessed at 48 h post-exposure by evaluating mortality. Those showing normal movement and/or jumping ability were considered live, and those showing no movement after tapping the vials were scored as dead. In vivo efficacy study on cats An in vivo efficacy study was designed to assess the onset of activity against existing flea infestations, as well as residual speed of kill of DF (2:1 ratio) against artificial flea infestations on cats. This study was an unblinded, randomized, 3-arms study, comparing the results obtained on 2 groups of cats treated with the DF combination with the untreated control group. Group 1 was untreated, Groups 2 and 3 were treated with DF. Groups 1 and 2 were used to evaluate the onset of activity of DF against existing flea infestations, while Groups 1 and 3 were used to evaluate its residual speed of kill. The study was conducted in accordance with the appropriate European and International guidelines at the time [17, 28]. Animals, animal housing and environmental monitoring Adult domestic short hair type cats originating from a purpose-bred colony were used in this study (Tables 1, 2). Before enrollment, cats had not been treated with an acaricide, insecticide or an insect growth regulator for at least 12 weeks, were free of fleas and were dewormed. A total of 34 cats were enrolled in the study of which 24 were included in the experimental phase after an acclimatization period of 7 days. Table 1 Details on age and gender of cats included in the study Table 2 Details on body weight and hair lengths of cats included in the study Cats were included in the study if they were considered healthy based on a veterinary examination, and if their pre-treatment flea retention was > 60%. The animals were housed individually in a temperature controlled animal unit where a photoperiod of 12 h light - 12 h darkness was maintained. The temperature in the housing unit fluctuated between 17.3 °C and 24.9 °C. All cats were observed daily for general health and if required examined by a veterinarian. The cats were fed daily with dry food at the recommended rate and water was available ad libitum and renewed at least twice daily. Allocation and treatment Random allocation to the three study groups was performed within gender, based on individual pre-treatment flea counts. Group 1 was untreated while Groups 2 and 3 were treated topically with 0.5 ml of DF containing dinotefuran (22% w/w, 252.2 mg/ml) and fipronil (8.92% w/w, 98.9 mg/ml) on study day (day) 0 (Table 3). Actual doses administered to each cat in Group 2 ranged between 34.4 mg/kg and 49.8 mg/kg for dinotefuran, and between 13.5 mg/kg and 19.5 mg/kg for fipronil. Actual doses administered to each cat in Group 3 ranged between 27.2 mg/kg and 48.1 mg/kg for dinotefuran, and between 10.7 mg/kg and 18.9 mg/kg for fipronil. Using a 1 ml syringe, DF was applied topically. The hair was parted at the base of the neck in front of the shoulder blades, until the skin was visible. The content of the syringe was then administered directly on the skin at a single spot. The cats were observed for possible adverse events hourly for 4 h after the last animal had been treated. Local tolerance observations were conducted prior to treatment and at 4 h, 8 h, 1 day, 2 days and 3 days post-treatment. Table 3 Details on flea retention as evaluated on day −5, as well as Investigational Veterinary Product (IVP) treatment dose (ml/kg) administered on day 0 Flea infestation and counts Laboratory reared fleas, originally isolated in the USA, were used for both in vitro and in vivo artificial infestations. At each infestation, every cat was infested with approximately 100 unfed mixed gender fleas. During acclimatization, all enrolled cats were infested with fleas on day -6 and the fleas were removed and counted on day -5, to determine the cat's suitability for inclusion (Table 3). These counts were also used for subsequent group allocation purposes. Following allocation to the three study groups, flea infestations were performed on days -1, 7, 14, 21, 28, 35 and 42 for the control Group 1, only on the day prior to day 0 treatment (i.e. on day -1) for the DF treated Group 2, and only on the post-treatment days (i.e. on days 7, 14, 21, 28, 35 and 42) for the DF treated Group 3 (Figs. 1 and 2). Flow diagram summarizing experimental design. Knock down and curative efficacy flea count assessments Flow diagram summarizing experimental design. Preventive efficacy flea count assessments Flea counts were conducted by combing. Counts were performed on day 0 in Groups 1 and 2 at 3 h, 6 h and 12 h after treatment. On days 7, 14, 21, 28, 35 and 42, counts were performed at 6 h and 48 h after infestation in Groups 1 and 3. At the 3 h and 6 h time points, the fleas were returned to the animals, while they were removed at the final 12 h or 48 h assessment time points (Figs. 1 and 2). Additionally, all fleas dropping from cats in Groups 1 and 2 were collected in collection pans placed underneath the cages after 5, 15, 30 and 120 min post-treatment, and counted (Fig. 1). Calculations and statistics The mortality data derived from at least 2 independent replicates of duplicate testing, were analyzed using the CalcuSyn Version 2.0 software (Biosoft) and Sigmaplot version 12.5 (Systat software). Half maximal effective concentration (EC50) and combination index (CI) values were calculated to assess the potential for synergistic activity [29]. The CI was computed by CalcuSyn according to Chou [29]: $$ CI=\frac{EC_{50}\kern0.5em Dinotefuran\kern0.5em in\kern0.5em combination}{EC_{50}\kern0.5em Dinotefuran\kern0.5em alone}+\frac{EC_{50}\kern0.5em Fipronil\kern0.5em in\kern0.5em combination}{EC_{50}\kern0.5em Fipronil\kern0.5em alone} $$ A CI value of approximately 1 indicated that the efficacy of the compounds was simply additive, a CI < 1 was interpreted as synergistic and a CI > 1 as antagonistic. In vivo efficacy study in cats Efficacies were calculated by comparing control vs DF treated flea counts using Abbot's formula. Efficacy values were calculated using geometric (GM) and arithmetic mean (AM) flea counts. For GM, the calculations were based on the flea (count +1) data, and one (1) was subsequently subtracted from the result to obtain a meaningful value for the geometric mean of the study groups. Groups were compared using a one-way ANOVA. The proportion of dead/moribund/live dislodged fleas collected at each time collection point was calculated as follows: Cumulative falling-off (%) = $ \frac{\left(\frac{N_t}{T_t}-\frac{N_c}{T_c}\right)}{\left(1-\frac{N_c}{T_c}\right)}\times 100 $ where Nt is the mean of the cumulative total number of dead/moribund/live fleas in the treated group (Group 2); Nc is the mean of the cumulative total number of dead/moribund/live fleas in the untreated control group (Group 1); Tt is the mean number of fleas infested to cats in the treated group (Group 2) = 100; and Tc is the mean number of fleas infested to cats in the untreated control group (group 1) = 100. No significant mortality was observed in the control treatments (solvent only and untreated vials), consequently mortality correction was not required to calculate EC50 values. Activity against C. felis was dose-dependent for both chemicals (Table 4). Dinotefuran (EC50 = 2.74 ppm) was more potent than fipronil (EC50 = 10.8 ppm) against fleas. Table 4 EC50 values and combination indexes of single drugs and drug combination - Ctenocephalides felis, coated glass assay Tests to determine the effect of dinotefuran and fipronil in combination on adult fleas were conducted using a fixed ratio design [29]. Given that the efficacy against adult fleas of each compound separately was dose-dependent, the results were progressed into a synergy analysis. This analysis revealed that the combination was more effective against fleas than either compound alone, with a CI (combination index) of 0.44, indicating strong synergy (Table 4). Combination of dinotefuran with fipronil significantly shifted the dose response curve towards the left and significantly reduced the IC50 values of dinotefuran and fipronil, as indicated by the non-overlap of 95% confidence intervals (Table 4). Topical DF administration was well-tolerated. Except for wet hair (cosmetic effects), no abnormal signs that could be attributed to treatment were observed. Crusts, associated with flea combing or excessive grooming of the cats were noticed on some animals, especially in the control group (4 out of 6), and were detected prior to treatment in some cats allocated to treated groups (1 and 2 out of 6, respectively). Flea-bite dermatitis was detected in one cat from the control group. The flea retention rate on day -6 was > 60% (65 ± 5%). Moreover, at all post-treatment assessment time points the GM flea counts of the untreated control group were > 55.5 (AM > 49.5), indicating a vigorous flea challenge at each occasion (Tables 5, 6). Table 5 The curative efficacy of a combination of dinotefuran/fipronil administered topically on Day 0 following infestation of cats with Ctenocephalides felis on Day -1 Table 6 Preventative efficacy of dinotefuran/fipronil administered topically to cats on day 0, against re-infestation with Ctenocephalides felis on days 7, 14, 21, 28, 35 and 42 Onset of activity and curative efficacy (Group 1 vs Group 2) Significantly fewer fleas (Table 7) were recorded on cats in the DF treated group (Group 2) than on cats in the untreated control group (Group 1) at all post-treatment assessment time points (3 h, 6 h and 12 h). Table 7 Results of ANOVA comparisons between groups The efficacy of DF against an existing population of fleas was 97.4% (97.2%, AM) at 3 h after treatment and at 12 h it had increased to ≥ 99.8% (GM and AM) (Table 5). Preventative efficacy (Group 1 vs Group 3) Significantly fewer fleas (Table 7) were recorded on cats in the treated group (Group 3) than on cats in the untreated control group (Group 1) at all post-treatment assessment time points and days. At 6 h after re-infestations, the efficacy of DF remained ≥ 90.8% (GM and AM) for up to 28 days and efficacy was maintained at ≥ 54.3% (GM and AM) for up to 42 days post-treatment (Table 6). At 48 h after re-infestations, DF remained almost fully effective for up to 28 days with efficacies ≥ 99.4% (GM and AM) and remained ≥ 93.0% (GM and AM) effective for up to 42 days post treatment (Table 6). Knock-down effect (Group 1 vs Group 2) Very few fleas (mean ≤ 1.0) were collected from the pans placed under the control cats, indicating vigorous on-host infestations. In contrast, the cumulative number of fleas dislodged and collected from the treated cats increased gradually from 5 min after treatment, and was greater at 2 h post-treatment, compared to the number collected from control cats (Fig. 3). Cumulative number of fleas (arithmetic mean + standard deviation) that dropped from cats in Group 1 (untreated animals) and Group 2 (dinotefuran/fipronil treated animals) The successful protection of cats against the detrimental effects of flea infestations, which may include FAD and possibly also flea borne diseases, is dependent on the use of a pulicidal product with rapid and persistently maintained activity. To achieve these performances, the combination of active ingredients with complementary modes of action and potencies in a single spot-on formulation has been developed. Dinotefuran, a third generation neonicotinoid and fipronil, a phenylpyrazole were selected. As a first step, the insecticidal efficacy of the two actives separately and in combination were tested against C. felis in vitro. This test indicated that dinotefuran was more potent than fipronil, and when they were administered in a 2:1 combination, a strong synergistic effect was observed in vitro. It was assumed that this synergistic effect would enhance not only the in vivo duration of activity of the combination against C. felis, but would also promote and maintain a quick speed of kill over this period. In the subsequent in vivo study, the persistent efficacy of DF remained ≥ 95.2% (GM counts) for 42 days. This is 12 days longer than previously reported for dinotefuran in combination with an insect growth regulator without adulticidal activity (pyriproxyfen) [22, 30]. Not only did the duration of efficacy improve, but the dose volume of a 22% w/w (252.2 mg/ml) DF formulation decreased to 0.5 ml/cat, where it was previously applied at 0.8 ml on cats that weighed between 2.4 kg and 3.9 kg [22]. In the current study, the dose rate of dinotefuran administered to individual animals thus ranged between 27.2 mg/kg and 49.8 mg/kg, compared to a range of between 51.7 mg/kg and 84.1 mg/kg administered by Murphy et al. [22]. Moreover, the speed of kill of the combination was also markedly improved in comparison to fipronil-based formulations [31]. In the latter study, the authors demonstrated that a fipronil/(s)-methoprene formulation administered to cats at a dose of 7.5 to 15 mg/kg was only 41.1% effective against infestation with a Kansas 1 (KS1) C. felis strain at 6 h post-infestation, assessed 28 days post-treatment [31]. The efficacy demonstrated in that study was markedly lower than that observed for the DF formulation in the current study, where a 94.2% efficacy (GM) was observed at the same assessment time point and remained ≥ 63.0% up to 42 days post-treatment. Such difference can be explained not only by the flea strain differences, but also by an improved activity related to the synergy between the two active ingredients tested in the present study. The rapid and maintained action of DF is advantageous for protecting cats against the detrimental health effects of flea infestations. By treating cats with DF, they can rapidly be rid of fleas even when repeatedly exposed under high environmental flea challenges. This will aid in preventing or rapidly relieving cats suffering from dermatological symptoms associated with severe flea infestations. Moreover, the rapid and maintained action of DF can assist in the protection against flea-borne pathogens. For instance, Bartonella henselae, a Gram negative bacteria responsible for a zoonotic disease, is excreted in flea feces within 24 h after a blood meal [32] and fleas can start to excrete faeces about 30 min after infestation. Consequently, killing fleas as fast as possible before they excrete infective faeces, will contribute to the protection of cats and their owners against infection. The combination of actives with different modes of activity can be beneficial in controlling infestations with specific field isolates that may be less susceptible to a specific active ingredient. For example, fipronil has a modest killing effect on the KS1 flea strain while dinotefuran is highly potent [21, 33]. The combination of these actives can therefore reduce the risk of potential efficacy failures in the field due to the presence of less susceptible flea populations. An obvious advantage of the DF formulation is the well-known acaricidal activity of fipronil, affecting not only the central nervous system of the parasite, but also important organs of ticks such as the saliva glands and ovaries. This attribute aids in preventing disease transmission as well as parasite reproduction [18, 34]. Although the experimental DF formulation used in this study exhibited great benefits and promise, it can potentially be improved by including a potent insect growth regulator (IGR). This will not only result in the on host-control of fleas, but the effective control of fleas in the animal's immediate environment. Pyriproxyfen, a potent juvenile hormone analog effectively inhibiting the development of fleas in the environment, would be the appropriate choice. This molecule prevents eggs from hatching, and the developmental stages from molting and ultimately checks C. felis proliferation in the environment [35, 36]. 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Murphy M, Ball CA, Gross S. Comparative in vivo adulticidal activity of a topical dinotefuran versus an imidacloprid-based formulation against cat fleas (Ctenocephalides felis) on cats. Vet Ther. 2009;10:9–16. de Oliveira PR, Pizano MA, Remédio RN, Bechara GH, de Abreu RMM, Camargo Mathias MI. Potential of the chemical dinotefuran in the control of Rhipicephalus sanguineus (Latreille, 1806) (Acari: Ixodidae) semi-engorged female ticks. Exp Parasitol. 2015;155:82–8. Blair J, Fourie JJ, Varloud M, Horak IG. Efficacy and speed of kill of a topically applied formulation of dinotefuran-permethrin-pyriproxyfen against weekly tick infestations with Rhipicephalus sanguineus (sensu lato) on dogs. Parasit Vectors. 2016;9(1):283. Beugnet F, Franc M. Insecticide and acaricide molecules and/or combinations to prevent pet infestation by ectoparasites. Trends Parasitol. 2012;28:267–79. Baker C, Tielemans E, Prullage JB, Chester ST, Knaus M, Rehbein S, et al. Efficacy of a novel topical combination of fipronil, (S)-methoprene, eprinomectin and praziquantel against adult and immature stages of the cat flea (Ctenocephalides felis) on cats. Vet Parasitol. 2014;202:54–8. Dryden MW, Payne PA, Vicki S, Riggs B, Davenport J, Kobuszewski D. Efficacy of dinotefuran-pyriproxyfen, dinotefuran-pyriproxyfen-permethrin and fipronil-(S)-methoprene topical spot-on formulations to control flea populations in naturally infested pets and private residences in Tampa. FL Vet Parasitol. 2011;182:281–6. "Guidelines for the Testing and Evaluation of the Efficacy of Antiparasitic Substances for the Treatment and Prevention of Tick and Flea infestation in Dogs and Cats" adopted on 7 November 2007 by the Committee for Veterinary Medicinal Product of the European Agency for the Evaluation of Medicinal Products (EMEA/CVMP/005/2000-Rev.2). http://www.ema.europa.eu/docs/en_GB/document_library/Scientific_guideline/2016/07/WC500210927.pdf. Accessed July 2017. Chou TC. Theoretical basis, experimental design, and computerized simulation of synergism and antagonism in drug combination studies. Pharmacol Rev. 2006;58(3):621–81. European Medicines Agency. Committee for Medicinal Products for Veterinary Use (CVMP) assessment report for Vectra Felis (EMEA/V/C/002746/0000). Revised 2015. http://www.ema.europa.eu/docs/en_GB/document_library/EPAR_-_Public_assessment_report/veterinary/002746/WC500169376.pdf. Accessed July 2017. Dryden MW, Smith V, Payne PA, McTier TL. Comparative speed of kill of selamectin, imidacloprid, and fipronil-(S)-methoprene spot-on formulations against fleas on cats. Vet Therapeut. 2005;6(3):228–36. Boushira E, Franc M, Boulouis HJ, Jacquiet P, Raymond-Letron I, Liénard E. Assessment of persistence of Bartonella henselae in Ctenocephalides felis. Appl Environ Microbiol. 2013;79:7439–44. Dryden M, Payne P, Smith V. Efficacy of selamectin and fipronil-(S)-methoprene spot-on formulations applied to cats against adult cat fleas (Ctenocephalides felis), flea eggs, and adult flea emergence. Vet Ther. 2007;8:255–62. Oliveira PR, Bechara HG, Morales MA, Mathias MI. Action of the chemical agent fipronil on the reproductive process of semi-engorged females of the tick Rhipicephalus sanguineus (Latreille, 1806) (Acari: Ixodidae). Ultrastructural evaluation of ovary cells. Food Chem Toxicol. 2009;47(6):1255–64. Stanneck et al. 2002. Stanneck D, Larsen KS, Mencke N. Pyriproxyfen concentration in the coat of cats and dogs after topical treatment with a 1.0% w/v spot-on formulation. J Vet Pharmacol Ther. 2003;26(3):233-235. Maynard L, Houffschmitt P, Lebreux B. Field efficacy of a 10 per cent pyriproxyfen spot-on for the prevention of flea infestations on cats. J Small Anim Pract. 2001;42(10):491–4. The authors would like to acknowledge and thank JL and his team at Clinvet for conduct of the in vivo study. The authors would also like to thank Prof Ivan Horak and Dr. Dionne Crafford whom assisted with review and formatting of the manuscript, respectively. This study was fully funded by Ceva Santé Animale, Libourne, France, of which RD, HK and MV are employees. Please refer to "Authors' contributions" below for details on the role of the funding body in the study design, collection and analysis of data and writing of this manuscript. Data will not be shared as study documentation is protected by confidentiality agreements. Ceva Santé Animale, 10 avenue de la Ballastière, 33500, Libourne, France Romain Delcombel , Hamadi Karembe & Marie Varloud Clinvet International (Pty) Ltd, Uitzich Road, Bainsvlei, Bloemfontein, South Africa Julian Liebenberg & Josephus Fourie Avista Pharma Solutions, 3501-C TriCenter Blvd, Durham, NC, 27713, USA Bakela Nare & Audrey Burton Search for Romain Delcombel in: Search for Hamadi Karembe in: Search for Bakela Nare in: Search for Audrey Burton in: Search for Julian Liebenberg in: Search for Josephus Fourie in: Search for Marie Varloud in: The study was designed by AB, RD, HK, BN, AB and MV with input from JL. The study was carried out by JL and AB. RD, HK, BN, AB and MV compiled the first draft report. The draft was revised by JF and JL. The second draft of the manuscript was revised and improved by all authors. All authors read and approved the final manuscript. Correspondence to Marie Varloud. Ethics declarations Ethics approval Prior to the commencement of the study its conduct was approved by the "Clinvet Committee for Animal Ethics and Welfare", with ethics approval reference CV16/056. This study was fully funded by the Sponsor, Ceva Santé Animale, Libourne, France, of which RD, HK and MV are employees. The in vitro study was conducted by a Contract Research Organisation (CRO), Avista Pharma solutions, of which BN and BA are employees. The in vivo study was conducted by a CRO, Clinvet International (Pty) Ltd., of which JF and JL are employees. The authors confirm that there are no financial or personal interest or belief that could affect their objectivity in reporting on the results obtained in this study. Dinotefuran Fipronil Ctenocephalides felis	CommonCrawl
What makes a theory "Quantum"? Say you cook up a model about a physical system. Such a model consists of, say, a system of differential equations. What criterion decides whether the model is classical or quantum-mechanical? None of the following criteria are valid: Partial differential equations: Both the Maxwell equations and the Schrödinger equation are PDE's, but the first model is clearly classical and the second one is not. Conversely, finite-dimensional quantum systems have as equations of motion ordinary differential equations, so the latter are not restricted to classical systems only. Complex numbers: You can use those to analyse electric circuits, so that's not enough. Conversely, you don't need complex numbers to formulate standard QM (cf. this PSE post). Operators and Hilbert spaces: You can formulate classical mechanics à la Koopman-von Neumann. In the same vein: Dirac-von Neumann axioms: These are too restrictive (e.g., they do not accommodate topological quantum field theories). Also, a certain model may be formulated in such a way that it's very hard to tell whether it satisfies these axioms or not. For example, the Schrödinger equation corresponds to a model that does not explicitly satisfy these axioms; and only when formulated in abstract terms this becomes obvious. It's not clear whether the same thing could be done with e.g. the Maxwell equations. In fact, one can formulate these equations as a Dirac-like equation $(\Gamma^\mu\partial_\mu+\Gamma^0)\Psi=0$ (see e.g. 1804.00556), which can be recast in abstract terms as $i\dot\Psi=H\Psi$ for a certain $H$. Probabilities: Classical statistical mechanics does also deal with probabilistic concepts. Also, one could argue that standard QM is not inherently probabilistic, but that probabilities are an emergent property due to the measurement process and our choice of observable degrees of freedom. Planck's constant: It's just a matter of units. You can eliminate this constant by means of the redefinition $t\to \hbar t$. One could even argue that this would be a natural definition from an experimental point of view, if we agree to measure frequencies instead of energies. Conversely, you may introduce this constant in classical mechanics by a similar change of variables (say, $F=\hbar\tilde F$ in the Newton equation). Needless to say, such a change of variables would be unnatural, but naturalness is not a well-defined criterion for classical vs. quantum. Realism/determinism: This seems to depend on interpretations. But whether a theory is classical or quantum mechanical should not depend on how we interpret the theory; it should be intrinsic to the formalism. People are after a quantum theory of gravity. What prevents me from saying that General Relativity is already quantum mechanical? It seems intuitively obvious that it is a classical theory, but I'm not sure how to put that intuition into words. None of the criteria above is conclusive. quantum-mechanics classical-mechanics foundations AccidentalFourierTransformAccidentalFourierTransform $\begingroup$ I've removed some comments which didn't seem to be intended to request clarifications or suggest improvements. $\endgroup$ – David Z♦ Apr 4 '18 at 21:44 $\begingroup$ Note that the appropriate answer to this question depends quite heavily on whether you mean "what distinguishes quantum theories from classical theories specifically", or "what distinguishes quantum theories from other theories in general" - for example, the class of what are often referred to as generalized probabilistic theories, which include classical, quantum and many other theories besides. In this latter class, classical theories are distinguished by many properties, and so the lack of any of these tells us we are dealing with a non-classical theory - but not necessarily a quantum one $\endgroup$ – Robin Saunders Apr 5 '18 at 0:19 $\begingroup$ @RobinSaunders Hmm that's actually a very good point, I like the way you put it. If you ever have some free time, please consider making that comment into an answer. Cheers! $\endgroup$ – AccidentalFourierTransform Apr 8 '18 at 2:11 $\begingroup$ I'm interested in why you say TQFT does not fit within the Dirac-von Neumann axioms. It's true that those axioms don't tell you much about the structure of the theory, but it's not really different for any QFT, for which there is a Hilbert space associated to any spatial manifold. I'd say those axioms are insufficiently strong, rather than being too restrictive. $\endgroup$ – Holographer Apr 8 '18 at 17:07 I think this is a subtle question and I think it depends somewhat on how you choose to represent quantum mechanics. To see one extreme of this, consider the viewpoint put forth by Kibble in [1]. For simplicity I will be thinking of finite-dimensional quantum systems here; there are some subtleties in infinite dimensions but as far as I know the basic picture still holds. In this, he shows that if we describe the theory in terms of physical states (rays in the Hilbert space), then the dynamics of Schrödinger evolution correspond exactly to Hamiltonian evolution via the symplectic form from the Kähler structure on the projective Hilbert space (which is to say, the evolution is that of a classical system). However there are two distinctions which make quantum mechanics different from classical mechanics: The phase space must be a projective Hilbert space (as opposed to just a symplectic manifold), and the Hamiltonian is restricted to being a quadratic form in the homogeneous coordinates on projective space. In classical mechanics any (sufficiently smooth) function is admissible as a Hamiltonian. Composite systems are described differently. In classical mechanics the phase space of a composite system is the Cartesian product of the phase spaces. In quantum mechanics, it is the Segre embedding (which descends from the tensor product of Hilbert spaces). This is parametrically different; if the phase spaces of the two subsystems are $2m$ and $2n$, then in classical mechanics the composite system has dimension $2m+2n$, whereas in quantum mechanics it has dimension $2(n+1)(m+1)-2$. The extra states are the entangled states. Virtually all the observable consequences of QM come here, e.g. Bell inequalities. Of course if we consider identical particles things get even a bit more complicated. If you ignore the second point, and focus only on a single quantum system, the surprising conclusion is that every quantum mechanical system is a special case of classical mechanics (with the provision that again I haven't checked the details in infinite dimensions but it is at least morally true). However, part of the structure of quantum mechanics is how it describes composite systems so you can't just ignore this second point. A mathematician would say that this gives an injective functor from the category of quantum mechanical theories to the category of classical theories which is not compatible with the symmetric monoidal structures on the two. I want to point out that this is emphatically not how we typically think of the correspondence principle in quantum mechanics. That is, it is a mapping from a finite-dimensional quantum mechanical system to a finite-dimensional classical system (of the same dimension). Normally, if we think about e.g. a free particle in one dimension, the Hilbert space for that quantum system is infinite dimensional, yet it corresponds to a 2-dimensional classical phase space. But the point is that, at least in this question, we can't restrict to the ordinary notion of correspondence since we don't have a physical interpretation for the system of equations describing the theory. Additionally, despite the above example, whether a theory is classical or quantum has essentially nothing to do with where the states live. Indeed, if we just want to consider a free particle in one dimension again, we would typically describe its state as a self-adjoint trace class unit trace operator $\hat \rho$ on the Hilbert space $L^2(\mathbb R)$. In contrast, in classical mechanics we would describe a state as a probability distribution $\rho$ on the phase space $\mathbb R^2$ (note that in the above example we had only pure classical states i.e. only those described by a $\delta$ function on the phase space whereas now we have mixed states). However we could just as easily describe the quantum state by its Wigner function, in which case it lives in exactly the same affine space as the classical distribution. However the Wigner function satisfies slightly different inequalities than the classical probability distribution; in particular it can be slightly negative and cannot be too positive. The details of this were first worked out in [2]. In this case, it is the dynamics that give away the quantum nature. Specifically, to go from classical to quantum mechanics, we must replace the Poisson bracket by the Moyal bracket (which has $O(\hbar^2)$ corrections), indicating the failure of Liouville's theorem in the phase space formulation of quantum mechanics: (quasi)probability density is not conserved along trajectories of the system. All of this is to say that it seems difficult (and maybe impossible) to try to find a single distinguishing feature between classical and quantum mechanics without considering composite systems, so if that is what you want, I'm not sure I have an answer. If you do allow for composite systems though, it is a pretty unambiguous distinction. Given this, it is perhaps not surprising that all the experimental tests we have which demonstrate that the world is quantum and not classical are based on entanglement. [1]: Kibble, T. W. B. "Geometrization of quantum mechanics". Comm. Math. Phys. 65 (1979), no. 2, 189--201. [2]: H.J. Groenewold (1946), "On the Principles of elementary quantum mechanics", Physica 12, pp. 405-460. Logan MLogan M As far as I know, the commutator relations make a theory quantum. If all observables commute, the theory is classical. If some observables have non-zero commutators (no matter if they are proportional to $\hbar$ or not), the theory is quantum. Intuitively, what makes a theory quantum is the fact that observations affect the state of the system. In some sense, this is encoded in the commutator relations: The order of the measurements affects their outcome, the first measurement affects the result of the second one. PhotonPhoton $\begingroup$ I think this answer is on the right track. In quantum mechanics, the transfer of information is intrinsically tied to the dynamics of the system, whereas in classical physics that is not the case. $\endgroup$ – DanielSank Apr 4 '18 at 20:16 $\begingroup$ I would agree with this. It was my answer also but I came too late. So, in any situation, what exactly is quantum is best shown in experiments such as Stern-Gerlach type. If you measure for x dirrection you get + and - or spin up or down, but if you measure in y, you get spins in that direction. If you measure first in x, thaen in y, you get as a rersult a y direction, but if you measure in x, then again in x, you get only x..... $\endgroup$ – Žarko Tomičić Apr 4 '18 at 20:17 $\begingroup$ I would say on the contrary that observations affect the state of a classical systems where everything is physical. $\endgroup$ – Bill Alsept Apr 4 '18 at 20:55 $\begingroup$ In MWI, observations don't affect the state of the system in some mysterious way. Rather, you should consider the composite Hilbert space describing both the system and the measuring device (large-dimensional Hilbert space). A measurement is a time-dependent interaction and in the measurement limit you produce a fully entangled state between the two. If you compute the reduced density matrix for the system of interest, you get a diagonal matrix of the probabilities. The point being that "observations affect the state of the system" is arguably really a statement about composite systems. $\endgroup$ – Logan M Apr 4 '18 at 21:07 $\begingroup$ @AccidentalFourierTransform I think that the existence of Poisson brackets is a semantic point. As long as you take the specific definition of a commutator as $[A, B] = A B - B A$ then this answer holds, IMO. After all, non-commuting operators, under this specific definition, are what lead to all the "quantum weirdness" such as Bell's theorem and the Uncertainty principles. All operators in KvN commute under this definition. $\endgroup$ – Bridgeburners Apr 5 '18 at 20:06 Frame challenge: I think the question is based on a misleading premise. While there are a number of characteristics typical of quantum theories as opposed to classical theories - some you've already listed in the question, and others have been suggested in the existing answers - there's no particular reason to expect there to be a single unambiguous rule that categorizes any arbitrary theory as either quantum or classical. Nor is there any particular need for such a rule. You give the example of quantum gravity. However, the reason we want a quantum theory of gravity is not because it has the tag "quantum" attached to it, as if it were a handbag that would not be adequately fashionable without the correct label, but because we want it to be able to answer certain questions about reality which we already know General Relativity can't answer. In short, don't worry about whether the theory is "quantum" or not - worry about whether it answers the questions you want answered or not. Also relevant. Addendum: the same goes for the existing theories, of course. We don't like the Standard Model because it is quantum. We like it because it works. Harry JohnstonHarry Johnston $\begingroup$ @JerrySchirmer, that's not really what this question asks, though. $\endgroup$ – Harry Johnston Apr 5 '18 at 19:37 $\begingroup$ It asks "what is it about a theory that makes it 'quantum'". And the answer would be "we apply quantization to some classical theory" $\endgroup$ – Jerry Schirmer Apr 5 '18 at 20:43 $\begingroup$ @JerrySchirmer, that's one possible answer, certainly. But I think the OP is asking for criteria that are based directly on the mathematical characteristics of a particular model, rather than on how the model was developed. (And I think in practice that, if presented with a theory with characteristics similar to other quantum theories, most physicists would call it a quantum theory regardless of whether it was derived from a classical model or not.) $\endgroup$ – Harry Johnston Apr 5 '18 at 21:20 $\begingroup$ ... incidentally, unless I've overlooked something, none of the existing answers mention quantization as a possible criteria so you might want to post that as an answer @JerrySchirmer $\endgroup$ – Harry Johnston Apr 5 '18 at 21:23 $\begingroup$ All that said, if I had to choose one feature that was the most important characteristic of quantum theories, I'd have to endorse Photon's answer. $\endgroup$ – Harry Johnston Apr 6 '18 at 23:06 TL;DR: Correlations. First things first: since the OP asks for a criterion to tell whether a model is quantum mechanical, the answer has to involve observables. After all if you could rewrite your "quantum" model as a "classical" model, those labels would not be worth much after all. Furthermore all quantum theories (that I know of) are probabilistic, therefore this answer focuses on probabilistic observables, i.e. correlation functions. The fundamental difference between a quantum theory and a classical theory is their correlation structure. That is, quantum theories can show correlations that classical theories cannot. The historically first and simplest example of this is Bell's inequality. By now there are many such inequalities for all kinds of observables, a frequently used one being the CHSH inequality. In general these inequalities set bounds on correlation functions that cannot be violated by a classical probability theory, where the latter can be made precisely (see below). Quantum probability theories can violate some of these inequalities, which makes them intrinsically different. Interestingly, there are also theories that have correlations that are even stronger than in quantum theory. These are known as Popescu-Rohrlich boxes and they have been shown to allow maximal violation of the so called Tsirelson bound, another inequality which is however fulfilled by quantum theory. Making these statements (which all work on the level of probability distributions on a space of observables) is a whole field. Some references (I'll try to put some more tomorrow, too tired now): One can try to uniquely single out quantum theory as a 'special' probability theory by starting from certain information theoretic postulates: https://arxiv.org/abs/1203.4516 So called 'loophole free' Bell tests have shown that we live in a world that violates classical probability theory (even though some people will argue against that): https://www.nature.com/articles/nature15759 A nice presentation about the ideas mentioned above of a guy who (unlike me) actually knows what he is talking about: http://www.math.umd.edu/~diom/RIT/QI-Spring10/ClassvsQuantInfo.pdf Thomas Lee Abshier ND WolpertingerWolpertinger Here is an experimentalist's answer: A mathematical system, either algebraic or differential equations, has axioms and theorems and is self contained and self consistent. A physics theory is a subset of a mathematical system that is defined by imposing extra axioms, called laws or postulates, which are necessary by construction, in order to pickup from the overall mathematical set, those solutions which fit data, i.e. measurements and observations. Classical theories are those that use classical laws, such as: Newton's laws for mechanics, the set of laws of electricity and magnetism unified in Maxwell's equations, the thermodynamic laws (and maybe etc). Quantum theories are the ones obeying quantum mechanical laws, i.e. the postulates of quantum mechanics, no matter the mathematical formulation. In order to fit the data and observations, quantum mechanics postulates were necessary, and this is what distinguishes classical from quantum, IMO. Edit after comments: In your list: Dirac-von Neumann axioms: These are too restrictive (e.g., they do not accommodate topological quantum field theories). This was the first time I met Topological Quantum Field Theories (TQFT). (Such introductions are one of the reasons I follow this site - to get whiffs of new-to-me physics.) The gauge is, if this set of theories fit data and predict measurements. In axiomatic mathematical theories, theorems can be set up as axioms, and then the former axioms have to be proven as theorems, for a self consistent theory. Usually the axioms are chosen as the simplest expression from a set of consistent theorems. Since TQFTs fit data and are predictive of quantum states, it is necessary that from the axiomatic postulates for TQFT one should be able to derive the postulates of quantum mechanics (possibly in a very complicated mathematical method). The wikipedia article on TQFT seems to indicate this. This is necessary for a theory to be quantum IMO. I.e. it is the postulates that connect measurements to the mathematical formulas, by construction. anna vanna v $\begingroup$ +1 Thank you for the answer, but I'm not convinced. As I said in the OP, the postulates of QM are too restrictive. There are systems that we deem quantum-mechanical, yet they fail to satisfy these axioms. For example, topological quantum field theories (which have their own set of axioms). $\endgroup$ – AccidentalFourierTransform Apr 7 '18 at 19:47 $\begingroup$ These topological theories, do they fit any data? ? If they fit the data, then this just means that some of the postulates (linked above) of quantum mechanics can be relaxed/ignored. Otherwise , as when theorems in axiomatic mathematics can be turned into axioms, they become theorems.Or are they just a science fiction game with mathematics $\endgroup$ – anna v Apr 8 '18 at 3:12 $\begingroup$ Wow, that's a very condescending comment. Just because you don't find them useful does not make them "science fiction games". Wow, just wow. I really didn't expect that attitude from you... $\endgroup$ – AccidentalFourierTransform Apr 8 '18 at 3:13 $\begingroup$ And of course they fit data; TQFT's are essential to study the low-energy behaviour of some condensed matter systems. $\endgroup$ – AccidentalFourierTransform Apr 8 '18 at 3:16 $\begingroup$ +1 for a very good point: "Quantum theories are the ones obeying quantum mechanical laws, i.e. the postulates of quantum mechanics, no matter the mathematical formulation." $\endgroup$ – AlQuemist Apr 9 '18 at 8:24 I would say that something intrinsically quantum is the way in which probabilities and the function which obeys the partial differential equation are related. As you note, both interference and probabilities are present in classical theories. What's new are probability amplitudes where interference leads to a supression of probabilities which is not possible in classical theories. For the finite-dimensional case, there's also Lucien Hardy's proposal "Quantum Theory From Five Reasonable Axioms" (https://arxiv.org/abs/quant-ph/0101012). There, the distinguishing factor between quantum theory and classical probability theory is that "there exists a continuous reversible transformation on a system between any two pure states of that system." Another reference along similar lines is Chapter 9 of Scott Aaronson's book "Quantum Computing since Democritus". MarcMarc $\begingroup$ Isn't interference of probabilities basically how we express wave-particle duality mathematically? $\endgroup$ – asmaier Apr 24 '18 at 13:06 $\begingroup$ I am not sure what you are getting at. Frist, there is no interference of probabilities but only probability amplitudes and second, sure, the physical phenomenon of wave-particle duality is related to this mathematical mechanism. $\endgroup$ – Marc Apr 25 '18 at 14:15 tl; dr Erm... You do. Say you cook up a model about a physical system ... Equations do not exists by themselves, they always have a surrounding. The head is assumptions and the tail usually describes limitations of said mathematical model. So really, it is up to your interpretation of question at hand OR the data available to you, that can consistently (deterministically?) predict if a theory is "Quantum". Conversely, if you do not have a head and tail, you can make a lot of cases about what an equations is talking about but can't say anything concretely. All the answers here are inspiring, and frankly sexy, but take time to consider my rudimentary examples below This way of thinking "what characteristic of equation predicts its applicability in <name of physics branch>" is a misuse of mathematics. Maths is, perhaps, the ultimate but we must remember that in physics we use it as a tool. My illustration below might seem childish but please consider the following equations Equation 1: $$ x^2 + x - 6 = 0 $$ $$ 2x + 5y = 20 $$ Just looking at these, a mathematician can happily say that Equation 1 has two solutions +2 and -3, and the curve is upward facing, with maxima at x = -0.5 has a slope of -0.4 has intercepts 4 and 10 has infinite ordered pairs (x, y) satisfying the equation describes a curve that encloses the origin And we would all agree with the above points. But the wise physicist stays mum, because s/he knows that these equations aren't just scribblings of some dyslexic Vulcan but are models of something, they represent something or some phenomena. So a physicist agrees with the mathematician but doesn't come to a conclusion. Let us look at the questions which lead us to these equations Question 1: The product of a quantity and one more than itself is 6, find the value of this quantity if a. the quantity is money lent b. the quantity is time Two times the number of my sons and five times the number of my daughters always equals two times the number of appendages a normal person has on his hands. How many sons and daughters do I have? Now, I hope you have an aha! moment. The answer of Q1 b is just +2 because time can not be negative (we've all solved such questions as kids) and the answer to Q2 can be quite surprising - 5 sons and 2 daughters - because physicists are good people and don't make fractional children or negative children. Did you see that -- one equation, two variables, and we still get a unique answer - constraints. So mathematician (the equation) and physicist (the big picture) are both correct where they stand. But the physicists wins, because we are at physics.stackexchange.com math in itself is very strong, pure, almost unpalatable; we need both the background information and the constraints to understand what this wonderful tool is trying to tell us through equations. On a serious note, I'd like to point out that there's probably no (respectable) book on classical physics which teaches F = ma without first explicitly-and-clearly stating the following: Assumptions required e.g. frictionless surfaces and perfectly rigid bodies Newton's Three Laws of Motion (word-by-word) That dF = d(m.v), which can be simplified if mass is (almost) constant and most importantly, the fact that objects we are dealing with are not of super-tiny scale, i.e. larger than 10-9m in diameter. Authors don't do this for pedagogy, most 9th grade students wouldn't give a damn about rigidity, but in fact they do it because these statements are necessary for the equation/theory to work. Trying to predict if an equation describes a Quantum thingy is a discussion-based question at best, or meta-math. To the OP specifically, If you are an inventor, working on something like GUT (why else would you have a equation whose origin you do not know) and you are curious if it applies equally well to big and small bodies - apply constraints. I do not have the mathematical foresight but logically I can say that variations in constraints will define the way system behaves for Quantum and Classical bodies. In Thinking Fast and Slow there's a chapter which illustrates that we have a tendency to support what is popular/fancy rather than what is correct/plausible. I think the question is primarily opinion based. edited Apr 8 '18 at 5:11 RinkyPinkuRinkyPinku $\begingroup$ Apropos of equation 1, a mathematician would perhaps say minimum rather than maxima (sic). $\endgroup$ – Deepak Apr 9 '18 at 15:50 Physical models are determined by their lattice of events. The set of physical events form an algebraic lattice with the two binary operators that serve as the OR and AND between events. We assume the lattice of events to be sigma-additive and orthomodular. We call this lattice the logic of the model. In this sense events are the elements of logic. System states are probability measures over this algebra. Physical quantities are mappings between statements on measurements of a quantity (think of Borel-sets of the reals) and the logic. The logic of a classical model is isomorphic to a set algebra so it is distributive ( a ∨ (b ∧ c) = (a ∨ b) ∧ (a ∨ c) and vice versa) and fully atomic. The logic of a quantum model isomorphic to the lattice of the subspaces of a Hilbert space and therefore it is not distributive but also fully atomic. The above alone is sufficient to explain many features associated with quantum models, including real valued physical quantities can be represented as self-adjoint operators commutation relations superposition of states the Schrödinger equation g.kerteszg.kertesz $\begingroup$ Can you please add some references? I think the answer could benefit from that. $\endgroup$ – Kiro Apr 7 '18 at 7:35 TLDR: Wave-Particle duality I want to answer this question from a historical perspective: According to our current understanding a quantum theory shows features of both classical mechanics and electrodynamics (e.g. light) at the same time. The first person to notice such a connection between mechanics and the theory of light was Hamilton. He developed Hamiltonian Optics, which described light as a particle (aka corpuscle). Theorists soon recognized that Hamiltonian Optics cannot account for light phenomena like interference, diffraction, and polarisation. They realized that Hamiltonian Optics is only an approximation, which works well as long as the wavelength of light is much smaller than the measurement apparatus (e.g. for geometrical optics based on light rays and lenses). Nevertheless, the language of Hamiltonian Optics worked perfectly to describe classical mechanics, which is now commonly known as Hamiltonian mechanics. Maxwell's field theory of Electrodynamics was a more correct description of light, but then came Planck and Einstein. They showed that to describe Black Body Radiation and the photoelectric effect it was necessary to assume that light cannot be a field with infinite divisibility (i.e. continuity) as assumed in Maxwell's Wave theory of light. Rather, light must consist of countable entities they called "quanta". But, this theory was ad hoc and not consistent with special relativity. (Note: the consistent version is Quantum Electrodynamics.) Although immature, the Planck and Einstein explanation of these phenomena was the first quantum theory because it showed (or better, assumed) wave-particle duality. (Note: Quantisation doesn't mean going from a wave theory of light back to a corpuscle theory like Hamiltonian Optics. Rather it combines features of waves and particles.) The crazy genius of deBroglie and Schrödinger was needed to apply this theory in the opposite direction - to particles. They noticed that if Maxwell's wave theory of light must be extended to contain quanta/particles, classical theory (which consists only of particles) must be extended to produce the features of waves. They saw classical theory could be an approximation like Hamiltonian Optics, which is valid only for short wavelengths. Thus, Schrödinger developed wave mechanics not by postulating quanta, but by reversing the approximations necessary to go from Maxwell's theory of light to Hamiltonian Optics. In opposition to Electrodynamics, Classical Mechanics needed to be "wavized" to become a complete theory showing wave-particle duality. (Note: here again, quantisation is not going from a particle theory to a complete wave theory of infinite divisibility, rather, it combines features of both worlds.) So, a theory is "Quantum" when it integrates/combines the features of both waves and particles. A classical theory is either only waves/fields or only particles. Regarding the quantisation of General Relativity, it is instructive to compare this classical field theory with another classical field theory, namely fluid dynamics. What both theories have in common is their high non-linearity. Both can only be quantised if they get linearized first. If one linearizes fluid dynamics, one gets the equation for sound waves. If one linearizes the equations of GR, one gets the equations of gravitational waves. If one quantizes the equation of sound waves, one gets phonons. If one quantizes gravitational waves, one gets gravitons. Again, both Gravitons and Phonons show wave-particle duality. But in both cases, we need to linearize our theory first to be able to quantize it. (Note: Phonons only exist in solids. Gravitons might also only exist in "solid" space-time.) asmaierasmaier I'm astonished that nobody appears to mention that a quantum theory describes quantities which have discrete values. All quantities which appear continuous on the macroscopic level can only take on discrete values in a quantum theory. The differences are "communicated" by "particles" (photons etc.). That's the heart of a quantum theory. Describing the states and interacting particles has not been achieved, or has only been tentatively achieved, for gravitation. Peter A. SchneiderPeter A. Schneider $\begingroup$ -1 This answer is basically incorrect; in particular, "All quantities which appear continuous on the macroscopic level can only take on discrete values in a quantum theory". $\endgroup$ – AlQuemist Apr 9 '18 at 14:32 $\begingroup$ @PeterA.Schneider No, that's a very simplistic view of classical mechanics (and physics in general): a single system always has an infinite number of different descriptions, some of which are typically more accurate than others. It's turtles all the way down: you can always add more levels of sophistication to a certain model. In this sense, speaking of a "coin" is not meaningful: you have to decide which degrees of freedom you want to study (only heads/tails? or also it's final temperature? what about any possible deformation due to the impact?) (1/2) $\endgroup$ – AccidentalFourierTransform Apr 9 '18 at 16:52 $\begingroup$ (2/2) At some point you truncate the problem, and pick a certain finite set of degrees of freedom. Once you do this, you should be able to decide whether the model is classical or quantum-mechanical independently of other "more sophisticated" models. The binary model is consistent in and of itself, independently of more accurate descriptions. It is a valid model, and complete as far as the degrees of freedom we chose to describe is concerned. Whether there is a Newtonian description that is more accurate is completely irrelevant. FWIW, I appreciate your answer anyway, and I upvoted it. $\endgroup$ – AccidentalFourierTransform Apr 9 '18 at 16:54 $\begingroup$ @PeterA.Schneider Take guitar string or some other resonating system - you get discrete results. $\endgroup$ – Arvo Apr 10 '18 at 12:03 $\begingroup$ As with @Arvo I immediately glanced at classical standing waves. As with quantum systems they discreteness comes from the application of boundary conditions. As with quantum systems they are a steady-state effect and you can observe results that don't meet the quantization condition in the immediate aftermath of disturbing the system. $\endgroup$ – dmckee♦ Apr 11 '18 at 19:01 protected by Qmechanic♦ Apr 4 '18 at 22:19 Not the answer you're looking for? Browse other questions tagged quantum-mechanics classical-mechanics foundations or ask your own question. QM without complex numbers Can we derive Schrödinger equation from classical wave equation? Where is the line between Quantum and Relativity? How does a vanishing $[x, p]$ work with the group theoretical definition of $p \propto \frac{\partial}{\partial x}$? Does the electro-dynamical lagrangian contain a (Dirac) wave-function? Differentiating quantum mechanics and string theory Quantum mechanics as classical field theory States versus ensembles in quantum mechanics "Consciousness causes collapse"-interpretation or free will excluded by quantum zeno effect? The implications of Gödel's Second Incompleteness Theorem on Theoretical Physics models Does quantum mechanics imply that particles have no trajectories? What is the program of quantum field theory? What is its derivation? Where is quantum plasticity theory? Modern axioms of quantum mechanics Normalisation of quantum states: why?	CommonCrawl
\begin{document} \title{Convexity Estimates for High Codimension Mean Curvature Flow} \maketitle \begin{abstract} We consider the evolution by mean curvature of smooth $n$-dimensional submanifolds in $\mathbb{R}^{n+k}$ which are compact and quadratically pinched. We will be primarily interested in flows of high codimension, the case $k\geq 2$. We prove that our submanifold is asymptotically convex, that is the first eigenvalue of the second fundamental form in the principal mean curvature direction blows up at a strictly slower rate than the mean curvature vector. We use this convexity estimate to show that at a singular time of the flow, there exists a rescaling that converges to a smooth codimension-one limiting flow which is convex and moves by translation. \end{abstract} \section{Introduction} Let us consider a compact smooth $n$-manifold $M$, and a family of immersions \[F : M\times [0,T) \to \mathbb{R}^{n+k}\] which move by mean curvature flow, that is \[\partial_t F(x,t) = H(x,t)\] for each $(x,t) \in M\times [0,T)$ where $H$ is the mean curvature vector. The mean curvature flow constitutes a system of quasilinear weakly parabolic partial differential equations for $F$, and since $M$ is compact the flow must form a singularity in finite time. Singularity formation may be characterised analytically as follows: if we let $A$ denote the second fundamental form and take $T$ to be the maximal time then there holds \[\limsup_{t\nearrow T} \sup_M \|A\|(\cdot,t) = \infty.\] A profound and challenging problem is characterising and classifying the geometry of singularities, whose formation depends on the initial submanifold $M_0$ where $M_t:=F(M,t)$. In the seminal work of Huisken-Sinestrari \cite{Huisk-Sin99a}, the singularities formed by mean convex codimension one solutions were shown to be weakly convex (White obtained a similar result for embedded mean convex solutions in \cite{White}). It is natural to seek a corresponding theorem for solutions of higher codimension. However, we encounter a number of new difficulties, the foremost being that the second fundamental form and the mean curvature are vector-valued and consequently there is no direct corresponding notion of mean convexity. We instead use a different but related condition introduced by Andrews-Baker\cite{Andrews2010}. They showed that when $n \geq 2$, the quadratic pinching condition \begin{equation}\label{eqn_quadratic} \|A\|^2 - c \|H\|^2 + a\leq 0 \end{equation} is preserved for each $c < \frac{4}{3n}$ and $a > 0$. That is, if the condition is satisfied at the initial time, then it is satisfied by $M_t$ for every $t\in[0,T)$. We note that for compact hypersurfaces, positive mean curvature implies this condition but for all sufficiently large $c$. We will refer to submanifolds satisfying \eqref{eqn_quadratic} as being quadratically pinched. Andrews-Baker showed that if $c < \{\frac{4}{3n}, \frac{1}{n-1}\}$ the flow contracts quadratically pinched solutions to round points. This result is a high codimension generalisation of Huisken's work on convex solutions of codimension one \cite{Huisken84}. Recently the second-named author has constructed a flow with surgeries for solutions of dimension $n \geq 5$ which are quadratically pinched with \[c< c_n : =\begin{cases} \frac{3(n+1)}{2n(n+2)} & n = 5, 6, 7, \\ \frac{4}{3n} & n \geq 8. \end{cases} \] This generalises the surgery construction for two-convex hypersurface flows due to Huisken-Sinestrari \cite{Huisk-Sin09}. An important ingredient in \cite{Nguyen20} (and in the present work) is the codimension estimate due to Naff, which implies the singularities formed by a quadratically pinched solution are codimension one if $c < c_n$. In this paper we show that a quadratically pinched mean curvature flow with $c < c_n$ is asymptotically convex in a quantifiable manner. As the second fundamental form is vector-valued, we denote by $\lambda_1 \leq \dots \leq \lambda_n$ the eigenvalues of the second fundamental form in the principal normal direction, that is $\nu_1= \frac{H}{\|H\|}$ (see Section 2 for precise definitions). \begin{theorem} \label{thm:convex} Let $F:M\times[0,T) \to \mathbb{R}^{n+k}$ be a mean curvature flow of dimension $n \geq 5$ which is quadratically pinched with $c < c_n$. Then for any $\varepsilon >0$ there exists a constant $C_\varepsilon>0$ which depends only on $n$, $\varepsilon$ and $M_0$ such that \begin{align} \lambda_1 \geq -\varepsilon \|H\| - C_\varepsilon \end{align} on $M_t$ for each $t \in [0,T)$. \end{theorem} As $\varepsilon>0$ is arbitrary, this shows that the negative part of the first eigenvalue in the principal normal direction does not grow as fast as $\|H\|$. We then use this estimate to characterise type II singularities of the quadratically pinched mean curvature flow near the maximum of the curvature: \begin{theorem} \label{thm:sing} Suppose a type II singularity forms at time $T$, that is \[\limsup_{t \nearrow T} \Big[(T-t)\max_{M_t} \|A\|^2 \Big] =\infty.\] Then there exists a sequence of rescalings of $F$ that subconverges smoothly to a codimension one limiting flow which is either: a strictly convex translating solution; or the isometric product of $\mathbb{R}^m$ with a strictly convex translating solution of dimension $n -m$. \end{theorem} The paper is set out as follows. In Section \ref{sec:evolution_equations} we gather together the necessary evolution equations and technical tools. In particular, $\lambda_1$ is not smooth but is locally Lipschitz and semiconvex, and its evolution equation must be understood in a distributional sense. In Section \ref{sec:Poinc} we obtain a Poincar\'e-type inequality which requires the Simons' identity for high codimension submanifolds. In Section \ref{sec:Stamp}, we complete the proof of the convexity estimate by applying Huisken's Stampacchia iteration. Finally, in Section \ref{sec:singularity_formation} we study singularity formation and prove Theorem \ref{thm:sing}. \textbf{Acknowledgements.} The second named author was supported by the EPSRC grant EP/S012907/1. \section{Evolution equations} \label{sec:evolution_equations} Let $F:M\times [0,T) \to \mathbb{R}^{n+k}$ solve mean curvature flow and write $M_t := F(M,t)$. We recall from the work of Andrews-Baker \cite{Andrews2010} the following evolution equations for the second fundamental form and mean curvature vector. With respect to local orthonormal frames $\{e_i\}$ and $\{\nu_\alpha\}$ for the tangent and normal bundles, \begin{align} \nabla_{\partial_t} A_{ij\alpha} & = \Delta A_{ij\alpha} + A_{ij\beta}A_{pq\beta} A_{pq\alpha} \\ &+A_{iq\beta}A_{qp\beta} A_{pj\alpha}+A_{jq\beta}A_{qp\beta} A_{pi\alpha} - 2 A_{ip\beta}A_{jq\beta}A_{pq\alpha} , \end{align} and \begin{align} \nabla_{\partial_t} H_{\alpha} & = \Delta H_{\alpha} + H_\beta A_{pq\beta} A_{pq \alpha}. \end{align} From these equations we can compute that \begin{align} (\partial_t -\Delta ) \|A\|^2 &= - 2 \|\nabla A\|^2 + 2 \| \langle A, A\rangle \|^2 + 2 \|R^{\perp}\|^2 \\ (\partial_t - \Delta) \|H\|^2 & = - 2 \|\nabla H\|^2 + 2 \|\langle A, H\rangle \|^2 . \end{align} We use $R^{\perp}$ to denote the normal curvature, which is given by \begin{align} R^\perp_{ij \alpha \beta} = A_{ip\alpha} A_{jp\beta}- A_{jp \alpha} A_{ip\beta}. \end{align} Under the quadratic pinching assumption we have $\|H\|>0$, so at any point in $M_t$ we can choose a local orthonormal frame for the normal bundle which is such that \[\nu_1 = \frac{H}{\|H\|}.\] We also use the notation \[\hat A = A - \frac{1}{\|H\|} \langle A, H\rangle \frac{H}{\|H\|} = A - A_1 \nu_1\] to denote the components of the second fundamental form orthogonal to the mean curvature vector, and write \[h = \frac{1}{\|H\|}\langle A, H\rangle = A_1\] for the scalar part of the mean curvature component of $A$. Hence $A$ admits the decomposition \[A = h \nu_1 + \hat A.\] \subsection{Pinching is preserved} With this notation in place we can state the estimate proven by Andrews-Baker showing that quadratic pinching is preserved by the flow. \begin{lemma}[\cite{Andrews2010}, Section 3] \label{lem:pinch_pres} Fix constants $0 <c < \frac{4}{3n}$ and $a >0$ and let \[\mathcal Q := \|A\|^2 - c\|H\|^2 + a.\] At every point in $M\times[0,T)$ where $\mathcal Q \leq 0$ there holds \begin{align} \label{eqn_pinchpres2} \nonumber(\partial_t-\Delta ) \mathcal Q & \leq -2 ( \|\nabla A \|^2-c \|\nabla H\|^2 )+2 \|h\|^2 \mathcal Q-2 a \|h\|^2-\frac{2a}{n} \frac{1}{c-\nicefrac{1}{n}}\|\hat A\|^2 \\ &+\frac{2}{n}\frac{1}{c-\nicefrac{1}{n}}\| \hat A\|^2 \mathcal Q+ \left(6-\frac{2}{n (c-\nicefrac{1}{n})} \right) \|\accentset{\circ} h\|^2 \| \hat A\|^2+\left(3-\frac{2}{n (c-\nicefrac{1}{n})} \right)\|\hat A\|^4. \end{align} \end{lemma} Note that by Proposition 6 in \cite{Andrews2010} we have \begin{align} \label{eq:grad_trace} \|\nabla A\|^2 \geq \frac{3}{n+2} \|\nabla H\|^2 \end{align} so the gradient term on the right-hand side is nonpositive. At points where $\mathcal Q\leq 0$, each of the zeroth-order reaction terms is also nonpositive. From now on we suppose the initial submanifold $M_0$, and hence $M_t$ for all $t \in [0,T)$, is quadratically pinched with \[c \leq \frac{4}{3n} - \varepsilon_0, \qquad \varepsilon >0.\] For ease of notation let us define \[W := \bigg( \frac{4}{3n} - \frac{\varepsilon_0}{2}\bigg) \|H\|^2 - \|A\|^2, \qquad w:= W^\frac{1}{2},\] and observe that by the quadratic pinching $W \geq \frac{\varepsilon_0}{2} \|H\|^2$ on $M_t$. \begin{lemma} \label{lem:w_evol} At each point in $M \times [0,T)$ we have the inequalities \begin{align} (\partial_t - \Delta) W \geq 2 \|h\|^2 W + \frac{(n+2)}{3} \varepsilon_0 \|\nabla A\|^2. \end{align} and \begin{align} (\partial_t - \Delta)w \geq \|h\|^2 w + \delta_0 \frac{\|\nabla A\|^2}{\|H\|}, \end{align} where $\delta_0 >0$ depends only on $n$ and $\varepsilon_0$. \end{lemma} \begin{proof} From Lemma \ref{lem:pinch_pres} we obtain \begin{align} (\partial_t - \Delta) W\geq 2 \|h\|^2 W + 2\|\nabla A\|^2 -2 \bigg( \frac{4}{3n} - \frac{\varepsilon_0}{2}\bigg)\|\nabla H\|^2. \end{align} Using \eqref{eq:grad_trace} we estimate \begin{align} \|\nabla A\|^2 - \bigg( \frac{4}{3n} - \frac{\varepsilon_0}{2}\bigg)\|\nabla H\|^2 &= \bigg(1 - \frac{(n+2)}{3} \bigg( \frac{4}{3n} - \frac{\varepsilon_0}{2}\bigg)\bigg) \|\nabla A\|^2\\ &+\bigg( \frac{4}{3n} - \frac{\varepsilon_0}{2}\bigg) \bigg( \frac{n+2}{3} \|\nabla A\|^2 - \|\nabla H\|^2\bigg)\\ & \geq \bigg(1 - \frac{4(n+2)}{9n} \bigg) \|\nabla A\|^2 + \frac{(n+2)}{6} \varepsilon_0\|\nabla A\|^2\\ &\geq \frac{(n+2)}{6} \varepsilon_0\|\nabla A\|^2, \end{align} which gives the desired inequality for $W$. It follows that \begin{align} (\partial_t - \Delta) W^\frac{1}{2} &= \frac{1}{4 W^{3/2} } \|\nabla W\|^2 + \frac{1}{2 W^\frac{1}{2} } (\partial_t - \Delta) W\\ &\geq \|h\|^2 W^\frac{1}{2} + \frac{(n+2)}{6} \varepsilon_0 \frac{\|\nabla A\|^2}{W^\frac{1}{2}}, \end{align} and since $W \leq \frac{4}{3n} \|H\|^2$ we have \begin{align} (\partial_t - \Delta) w &\geq \|h\|^2 w + \frac{(3n)^\frac{1}{2}}{2}\frac{(n+2)}{6} \varepsilon_0 \frac{\|\nabla A\|^2}{\|H\|}. \end{align} Thus it suffices to take \[\delta_0 = \frac{(3n)^\frac{1}{2}}{2}\frac{(n+2)}{6} \varepsilon_0 .\] \end{proof} \subsection{The evolution of $h$} From the equations for $A$ and $H$, we readily compute the projection $\langle A, H\rangle $ satisfies \begin{align} (\partial_t - \Delta) A_{ij\alpha} H_\alpha &= -2\nabla_p A_{ij\alpha} \nabla_p H_\alpha + 2 H_\alpha A_{ij\beta}A_{pq\beta} A_{pq\alpha} \\ &+H_\alpha(A_{iq\beta}A_{qp\beta} A_{pj\alpha}+A_{jq\beta}A_{qp\beta} A_{pi\alpha} - 2 A_{ip\beta}A_{jq\beta}A_{pq\alpha} ). \end{align} The first of the reaction terms can be split into a hypersurface and a codimension component, as follows: \begin{align} 2 H_\alpha A_{ij\beta}A_{pq\beta} A_{pq\alpha} &= 2 A_{ij\beta} A_{kl\beta}h_{kl} H_1 \\ &= 2 h_{ij}h_{pq}h_{pq}H_1 + 2 \sum_{\beta \neq 1} A_{ij\beta}A_{pq\beta} h_{pq} H_1 \\ &= 2 h_{ij} H_1 \|h\|^2 + 2 \sum_{\beta\neq1}^n \hat A_{ij\beta} \hat A_{pq\beta} h_{pq}H_1. \end{align} Similarly, the remaining reaction terms can be written as \begin{align} H_\alpha(A_{iq\beta}A_{qp\beta} A_{pj\alpha}&+A_{jq\beta}A_{qp\beta} A_{pi\alpha} - 2 A_{ip\beta}A_{jq\beta}A_{pq\alpha} )\\ & = h_{ip}h_{pq}h_{qj}H_1 + h_{jp}h_{pq}h_{qi}H_1 - 2 h_{ip}h_{jq}h_{pq}H_1\\ &+ \sum_{\beta \neq 1} A_{ip\beta} A_{pq \beta} h_{qj} H_1 + \sum_{\beta \neq 1}A_{jp\beta}A_{pq\beta}h_{qi}H_1 - 2 \sum_{\beta \neq 1} A_{ip\beta}A_{jq\beta}h_{pq}H_1 \\ &= \sum_{\beta \neq 1} \hat A_{ip\beta} \hat A_{pq \beta} h_{qj} H_1 + \sum_{\beta \neq 1}\hat A_{jp\beta}\hat A_{pq\beta}h_{qi}H_1 - 2 \sum_{\beta \neq 1} \hat A_{ip\beta}\hat A_{jq\beta}h_{pq}H_1. \end{align} Therefore, \begin{align} (\partial_t - \Delta) A_{ij\alpha} H_\alpha &= -2\nabla_p A_{ij\alpha} \nabla_p H_\alpha +2\|h\|^2 h_{ij} H_1 + 2 \sum_{\beta\neq1}^n \hat A_{ij\beta} \hat A_{pq\beta} h_{pq}H_1\\ &+\sum_{\beta \neq 1} \hat A_{ip\beta} \hat A_{pq \beta} h_{qj} H_1 + \sum_{\beta \neq 1}\hat A_{jp\beta}\hat A_{pq\beta}h_{qi}H_1 - 2 \sum_{\beta \neq 1} \hat A_{ip\beta}\hat A_{jq\beta}h_{pq}H_1. \end{align} For a positive function $f$, we have \begin{align} (\partial_t-\Delta) \sqrt{f} & =\frac{1}{4 f^{3/2} } \|\nabla f\|^2 + \frac{1}{2 \sqrt{f} } (\partial_t - \Delta) f, \end{align} hence the quantity $\|H\|$ satisfies \begin{align} (\partial_t - \Delta) \|H\| & =\frac{1}{4 \|H\|^3} \|\nabla \|H\|^2\|^2 + \frac{ 1}{2 \|H\| }(- 2 \|\nabla H\|^2 + 2 \|\langle A, H\rangle \|^2)\\ &= \frac{\|\langle A, H\rangle \|^2 }{\|H\|} - \frac{\|\nabla H\|^2}{\|H\|}+ \frac{1}{\|H\|^3} \langle H, \nabla_i H\rangle \langle H, \nabla_i H\rangle. \end{align} There holds \[\frac{\|\langle A, H\rangle \|^2 }{\|H\|} = \|\langle A, \|H\|^{-1} H\rangle\|^2 \|H\|= \|h\|^2 H_1\] and \begin{align} - \frac{\|\nabla H\|^2}{\|H\|}+ \frac{1}{\|H\|^3}& \langle H, \nabla_i H\rangle \langle H, \nabla_i H\rangle \\ &= -\frac{1}{H_1} ( H_1^2 \|\nabla \nu_1\|^2 + \|\nabla H_1\|^2) + \frac{1}{H_1} \langle \nu_1 , H_1 \nabla \nu_1 + \nabla H_1 \nu_1\rangle \\ & = - H_1 \|\nabla \nu_1\|^2 \end{align} so we have \begin{align} \label{eq:H_evol} (\partial_t - \Delta) \|H\| & = \|h\|^2 H_1 - H_1\|\nabla \nu_1\|^2. \end{align} For a tensor $B_{ij}$ divided by a positive scalar function $f$ there holds \begin{align} (\nabla_{\partial_t} - \Delta) \frac{B_{ij}}{f} = \frac{1}{f} (\nabla_{\partial_t} - \Delta) B_{ij} - \frac{B_{ij}}{f^2} (\partial_t - \Delta) f + \frac{2}{f} \bigg\langle \nabla \frac{B_{ij}}{f}, \nabla f \bigg\rangle, \end{align} Therefore, dividing $\langle A, H\rangle$ by $\|H\|$, we obtain \begin{align} (\nabla_{\partial_t} - \Delta) h_{ij} &= \|h\|^2 h_{ij} +2 H_1^{-1} \sum_{\beta\neq1}^n \hat A_{ij\beta} \hat A_{pq\beta} h_{pq}H_1 +H_1^{-1}\sum_{\beta \neq 1} \hat A_{ip\beta} \hat A_{pq \beta} h_{qj} H_1 \\ &+ H_1^{-1}\sum_{\beta \neq 1}\hat A_{jp\beta}\hat A_{pq\beta}h_{qi}H_1 - 2 H_1^{-1}\sum_{\beta \neq 1} \hat A_{ip\beta}\hat A_{jq\beta}h_{pq}H_1\\ &-2H_1^{-1} \langle \nabla A_{ij} ,\nabla H \rangle + h_{ij} \|\nabla \nu_1\|^2 +2 H_1^{-1} \langle \nabla h_{ij}, \nabla H_1 \rangle. \end{align} Let us introduce the abbreviation \begin{align} \label{eq:T_def} T_{ij} &:= 2 \sum_{\beta\neq1}^n \hat A_{ij\beta} \hat A_{pq\beta} h_{pq}+\sum_{\beta \neq 1} \hat A_{ip\beta} \hat A_{pq \beta} h_{qj} \notag \\ &+ \sum_{\beta \neq 1}\hat A_{jp\beta}\hat A_{pq\beta}h_{qi} - 2 \sum_{\beta \neq 1} \hat A_{ip\beta}\hat A_{jq\beta}h_{pq}, \end{align} so that we may write \begin{align} (\nabla_{\partial_t} - \Delta) h_{ij} &= \|h\|^2 h_{ij} +T_{ij} -2H_1^{-1} \langle \nabla A_{ij}, \nabla H \rangle \\ &+ h_{ij} \|\nabla \nu_1\|^2 + 2 H_1^{-1} \langle \nabla h_{ij}, \nabla H_1 \rangle. \end{align} We simplify the gradient terms by decomposing \begin{align} - 2\langle \nabla A_{ij} , \nabla H\rangle & = - 2\langle \nabla h_{ij} \nu_1 + h_{ij} \nabla \nu_1 + \nabla \hat A_{ij}, \nabla H_1 \nu_1 + 2H_1 \nabla \nu_1 \rangle\\ &= - 2\langle \nabla h_{ij} , \nabla H_1\rangle - 2H_1 h_{ij} \|\nabla \nu_1\|^2 - 2\langle \nabla \hat A_{ij}, \nabla H_1 \nu_1 \rangle \\ &- 2H_1 \langle \nabla \hat A_{ij}, \nabla \nu_1 \rangle, \end{align} and so obtain: \begin{lemma} At each point in $M\times [0,T)$ there holds \begin{align} (\nabla_{\partial_t} - \Delta) h_{ij} &= \|h\|^2 h_{ij} +T_{ij}- h_{ij} \|\nabla \nu_1\|^2 - 2H_1^{-1} \langle \nabla \hat A_{ij}, \nabla H_1 \nu_1 \rangle\\ &- 2\langle \nabla \hat A_{ij}, \nabla \nu_1 \rangle , \end{align} where $T_{ij}$ is the quantity defined in \eqref{eq:T_def}. \end{lemma} Since $h$ is a symmetric bilinear form it has $n$ real eigenvalues, which we denote by \[\lambda_1 \leq \dots \leq \lambda_n.\] The smallest eigenvalue can be written as \[\lambda_1(x,t) = \min_{\|v\| = 1} h(x,t) (v,v),\] and is therefore a Lipschitz continuous function on $M \times [0,T)$. We will use the evolution equation for $h$ to estimate $(\partial_t -\Delta)\lambda_1$, interpreted in an appropriate weak sense (cf. \cite{White} and \cite{Langford17}). \begin{definition} Let $f:M\times [0,T) \to \mathbb{R}$ be locally Lipschitz continuous and fix a point $(x_0,t_0) \in M \times (0,T)$. We say that a function $\varphi$ is a lower support for $f$ at $(x_0,t_0)$ if $\varphi$ is $C^2$ on the set $B_{g(t_0)}(x_0,r) \times [-r^2 +t_0,t_0]$ for some $r >0$ and there holds \[f(x,t) \geq \varphi(x,t),\] with equality at $(x_0,t_0)$. If the inequality is reversed then $\varphi$ is an upper support for $f$ at $(x_0,t_0)$. \end{definition} With this definition in place we have the following estimate: \begin{lemma} \label{lem:lambda_1_evol_1} Fix $(x_0,t_0) \in M \times (0,T)$ and suppose $\varphi$ is a lower support for $\lambda_1$ at $(x_0,t_0)$. Then at $(x_0,t_0)$ there holds \begin{align} (\partial_t - \Delta) \varphi &\geq \|h\|^2 \varphi +T_{11}- \varphi \|\nabla \nu_1\|^2 - 2H_1^{-1} \langle \nabla \hat A_{11}, \nabla H_1 \nu_1 \rangle\\ &- 2\langle \nabla \hat A_{11}, \nabla \nu_1 \rangle \end{align} \end{lemma} \begin{proof} We choose at the point $(x_0,t_0)$ an orthonormal basis of tangent vectors $\{e_i\}$ which are such that \[h(x_0,t_0) (e_i,e_i) = \lambda_i,\] and extend the $\{e_i\}$ to a spatial neighbourhood of $x_0$ by parallel transport with respect to $g(t_0)$. We then extend to an orthonormal frame on a backward spacetime neighbourhood of $(x_0,t_0)$ by parallel transport with respect to the connection $\nabla_{\partial_t}$. On this neighbourhood we can define a smooth function \[\eta(x,t) := h(x,t)(e_1(x,t) ,e_1(x,t)).\] Observe that by the definition of $\lambda_1$ there holds \[\eta(x,t) \geq \lambda_1(x,t) \geq \varphi(x,t),\] with equality at $(x_0,t_0)$. It follows that at the point $(x_0,t_0)$ we have \[\partial_t \eta \leq \partial_t \varphi, \qquad \Delta \eta \geq \Delta \varphi,\] hence \[(\partial_t - \Delta) \varphi \geq (\partial_t - \Delta) \eta.\] At $(x_0,t_0)$ we compute \begin{align} \partial_t \eta & = \nabla_{\partial_t} h(e_1, e_1) + 2 h(e_1, \nabla_{\partial_t} e_1)\\ & = \Delta h_{11} +\|h\|^2 h_{11} +T_{11}- h_{11} \|\nabla \nu_1\|^2 - 2H_1^{-1} \langle \nabla \hat A_{11}, \nabla H_1 \nu_1 \rangle\\ &- 2\langle \nabla \hat A_{11}, \nabla \nu_1 \rangle \\ & = \Delta h_{11} +\|h\|^2 \varphi +T_{11}- \varphi \|\nabla \nu_1\|^2 - 2H_1^{-1} \langle \nabla \hat A_{11}, \nabla H_1 \nu_1 \rangle\\ &- 2\langle \nabla \hat A_{11}, \nabla \nu_1 \rangle, \end{align} and \begin{align} \Delta h_{11} &= \nabla_i (\nabla_i (h_{11}) - 2 h(e_1, \nabla_i e_1))\\ &=\Delta (h_{11}) - 2 h(e_1, \Delta e_1)\\ & = \Delta \eta - 2 h (e_1, \Delta e_1). \end{align} On the other hand, at $(x_0,t_0)$ there holds \begin{align} \langle e_1, \Delta e_1\rangle = \nabla_k \langle e_1, \nabla_k e_1 \rangle =0, \end{align} which shows that $\Delta e_1$ is orthogonal to $e_1$, so since $h$ is diagonal at $(x_0,t_0)$ we obtain \begin{align} \Delta h_{11} = \Delta \eta, \end{align} and consequently \begin{align} \partial_t \eta & = \Delta \eta +\|h\|^2 \varphi +T_{11}- \varphi \|\nabla \nu_1\|^2 - 2H_1^{-1} \langle \nabla \hat A_{11}, \nabla H_1 \nu_1 \rangle\\ &- 2\langle \nabla \hat A_{11}, \nabla \nu_1 \rangle. \end{align} It follows then \begin{align} (\partial_t - \Delta) \varphi &\geq (\partial_t -\Delta)\eta\\ & = \|h\|^2 \varphi +T_{11}- \varphi \|\nabla \nu_1\|^2 - 2H_1^{-1} \langle \nabla \hat A_{11}, \nabla H_1 \nu_1 \rangle- 2\langle \nabla \hat A_{11}, \nabla \nu_1 \rangle \end{align} at $(x_0,t_0)$ as required. \end{proof} Eventually we will want to prove integral estimates for the function $\lambda_1$. To do so we appeal to Alexandrov's theorem, following Brendle \cite{Brendle15} (see also \cite{Langford17}). We call a function $f:M\times[0,T) \to \mathbb{R}$ locally semiconvex (resp. semiconcave) if about every $(x_0,t_0)$ there is a small open neighbourhood on which $f$ can be expressed as the sum of a smooth and a convex (resp. concave) function. \begin{lemma} \label{lem:alex} Let $f:M\times[0,T) \to \mathbb{R}$ be locally semiconvex. Then $f$ is twice differentiable almost everywhere in $M\times[0,T)$, and if $\varphi$ is a nonnegative Lipschitz function on $M$ then for each $t \in [0,T)$ there holds \[\int_M \Delta f \cdot \varphi \, d\mu_t \leq -\int_M \langle \nabla f, \nabla \varphi \rangle \,d\mu_t.\] Here $\mu_t$ is the measure induced by the immersion $F(\cdot,t)$. \end{lemma} \begin{proof} Choosing local coordinates and applying Alexandrov's theorem \cite[Section 6.4]{Ev-Gar}, we see that $f$ has two derivatives at a.e. point in $M\times[0,T)$. Furthermore, by \cite[Section 6.3]{Ev-Gar}, for each $t \in [0,T)$ there is a singular Radon measure $\chi$ on $M$ with the property that \[\int_M \Delta f \cdot \varphi \,d\mu_t + \int_M \varphi \,d\chi = - \int_M \langle \nabla f, \nabla \varphi \rangle\,d\mu_t\] for every $\varphi \in C^2(M)$ . Hence if $\varphi \geq 0$ there holds \[\int_M \Delta f \cdot \varphi \,d\mu_t \leq - \int_M \langle \nabla f, \nabla \varphi \rangle\,d\mu_t.\] By approximation, the same inequality also holds if $\varphi$ is only Lipschitz continuous. \end{proof} Since $h$ is smooth, on every small enough set in spacetime, $\lambda_1$ can be expressed as the minimum over a set of smooth functions which is compact in $C^2$. This is sufficient to ensure that $\lambda_1$ is locally semiconcave on $M\times[0,T)$, so by the lemma we conclude that there is a set of full measure $Q \subset M\times [0,T)$ where $\lambda_1$ is twice differentiable. \begin{lemma} \label{lem:lambda_1_evol_2} At each point in $Q$ there holds \begin{align} (\partial_t - \Delta) \lambda_1 &\geq \|h\|^2 \lambda_1 +T_{11}- \lambda_1 \|\nabla \nu_1\|^2 - 2H_1^{-1} \langle \nabla \hat A_{11}, \nabla H_1 \nu_1 \rangle\\ &- 2\langle \nabla \hat A_{11}, \nabla \nu_1 \rangle. \end{align} \end{lemma} \begin{proof} Fix a point $(x_0,t_0)\in Q$. Then $\lambda_1$ admits a lower support $\varphi$ at $(x_0,t_0)$, to which we can apply Lemma \ref{lem:lambda_1_evol_1}. Since $\varphi(x_0,t_0) = \lambda_1(x_0,t_0)$, this gives the desired inequality. \end{proof} \begin{remark} Notice that the first of the gradient terms is nonnegative whenever $\lambda_1 \leq 0$, whereas the remaining gradient terms both contain $\nabla \hat A$ as a factor. It is this structure of the gradient terms which allows us to prove the convexity estimate. \end{remark} \subsection{The evolution of $\|\hat A\|^2$} The following evolution equation for $\|\hat A\|^2$ was derived by Naff \cite{Naff2019}: \begin{align} (\partial_t - \Delta) \|\hat A\|^2 &= 2 \|\langle \hat A , \hat A\rangle \|^2 + 2 \sum_{i,j} \bigg\| \sum_k (\hat A_{ik} \otimes \hat A_{jk} - \hat A_{jk} \otimes \hat A_{ik} )\bigg\|^2 + 2 \sum_\alpha \|R^{\perp}_{ij1 \alpha}\|^2\\ &- 2\|\nabla \hat A\|^2 + 4 \sum_{i,j,k} (\langle \nabla_k \accentset{\circ} h_{ij}, \nu_1 \rangle - H_1^{-1} \accentset{\circ} h_{ij} \nabla_k H_1 ) \langle \hat A_{ij}, \nabla_k \nu_1\rangle. \end{align} We make use of the quantity \[v := \frac{\|\hat A\|^2}{\|H\|}.\] \begin{lemma} \label{lem:v_evol} There is a positive constant $C = C(n)$ such that \begin{align} (\partial_t - \Delta) v &\leq C\|A\|^2 \|\hat A\| + C \frac{\|\hat A\|}{H_1} \frac{\|\nabla A\|^2}{H_1} - 2\frac{\|\nabla \hat A\|^2}{H_1} \end{align} holds on $M\times [0,T)$. \end{lemma} \begin{proof} We use the formula \begin{align} (\partial_t - \Delta) \frac{f_1}{f_2} = \frac{1}{f_2} (\partial_t - \Delta) f_1 - \frac{f_1}{f_2^2} (\partial_t - \Delta) f_2 + \frac{2}{f_2} \bigg\langle \nabla \frac{f_1}{f_2}, \nabla f_2 \bigg\rangle \end{align} to derive \begin{align} (\partial_t - \Delta) v &= \frac{1}{\|H\|} (\partial_t - \Delta) \|\hat A\|^2 - \frac{\|\hat A\|^2}{\|H\|^2} (\partial_t - \Delta) \|H\| + \frac{2}{\|H\|} \bigg\langle \nabla \frac{\|\hat A\|^2}{\|H\|}, \nabla \|H\|\bigg\rangle. \end{align} Let us estimate \begin{align} \frac{2}{\|H\|} \bigg\langle \nabla \frac{\|\hat A\|^2}{\|H\|}, \nabla \|H\|\bigg\rangle & = \frac{2}{H_1} \bigg\langle \frac{1}{H_1} \nabla \|\hat A\|^2 - \frac{\|\hat A\|^2}{H_1^2} \nabla H_1, \nabla H_1\bigg\rangle \\ & = \frac{4}{H_1^2} \hat A_{ij}\langle \nabla \hat A_{ij}, \nabla H_1\rangle - 2 \frac{\|\hat A\|^2}{H_1^2} \frac{\|\nabla H_1\|^2}{H_1}\\ & \leq C(n) \frac{\|\hat A\|}{H_1} \frac{\|\nabla A\|^2}{H_1} , \end{align} where in the last line we have used $\|A\|^2 \leq \frac{4}{3n} \|H\|^2$. By \eqref{eq:H_evol} we have \begin{align} - \frac{\|\hat A\|^2}{\|H\|^2}(\partial_t - \Delta) \|H\| & = - \frac{\|\hat A\|^2}{H_1^2}( \|h\|^2 H_1 - H_1\|\nabla \nu_1\|^2) \\ &\leq \frac{\|\hat A\|^2}{H_1} \bigg\| \nabla \frac{H}{H_1} \bigg\|^2\\ &\leq C(n) \frac{\|\hat A\|}{H_1} \frac{\|\nabla A\|^2}{H_1}, \end{align} so there holds \begin{align} \label{eq:v_evol_1} (\partial_t - \Delta) v &\leq \frac{1}{H_1} (\partial_t - \Delta) \|\hat A\|^2 + C(n) \frac{\|\hat A\|}{H_1} \frac{\|\nabla A\|^2}{H_1}. \end{align} We recall \begin{align} (\partial_t - \Delta) \|\hat A\|^2 &= 2 \|\langle \hat A , \hat A\rangle \|^2 + 2 \sum_{i,j} \bigg\| \sum_k (\hat A_{ik} \otimes \hat A_{jk} - \hat A_{jk} \otimes \hat A_{ik} )\bigg\|^2 + 2 \sum_\alpha \|R^{\perp}_{ij1 \alpha}\|^2\\ &- 2\|\nabla \hat A\|^2 + 4 \sum_{i,j,k} (\langle \nabla_k \accentset{\circ} h_{ij}, \nu_1 \rangle - H_1^{-1} \accentset{\circ} h_{ij} \nabla_k H_1 ) \langle \hat A_{ij}, \nabla_k \nu_1\rangle, \end{align} and estimate \begin{align} (\partial_t - \Delta) \|\hat A\|^2 & \leq C(n)\|\hat A\|^4 + 2 \sum_\alpha \|R^{\perp}_{ij1 \alpha}\|^2 - 2\|\nabla \hat A\|^2 + C(n) \frac{\|\hat A\|}{H_1} \|\nabla A\|^2. \end{align} Then since \begin{align} R^\perp_{ij\alpha \beta} = A_{ip\alpha} A_{jp\beta} - A_{ip\beta}A_{jp\alpha} \end{align} we can write \begin{align} 2 \sum_\alpha \|R^{\perp}_{ij1 \alpha}\|^2 = 2 \sum_{\alpha \geq 2} \|h_{ip} \hat A_{jp\alpha} - \hat A_{ip\alpha} h_{jp}\|^2 \end{align} and use this to bound \begin{align} (\partial_t - \Delta) \|\hat A\|^2 & \leq C(n)\|A\|^2 \|\hat A\|^2 + C(n) \frac{\|\hat A\|}{H_1} \|\nabla A\|^2 - 2\|\nabla \hat A\|^2 . \end{align} Substituting this inequality into \eqref{eq:v_evol_1} and using the quadratic pinching gives the desired estimate. \end{proof} \subsection{Modifying $\lambda_1$} We now form the quantity \[f(x,t) := -\lambda_1(x,t) - \varepsilon w(x,t) + \Lambda v(x,t),\] where $\varepsilon$ and $\Lambda$ are positive constants to be chosen later. Combining the evolution equations for the three components we obtain: \begin{lemma} \label{lem:f_evol} At each point in $Q$ there holds \begin{align} (\partial_t - \Delta )f &\leq \|h\|^2 f + C(1+ \Lambda) \|A\|^2 \|\hat A\| - ( f + \varepsilon w )\|\nabla \nu_1\|^2 \\ & - \bigg( \frac{\varepsilon \delta_0}{2}- C\Lambda \frac{\|\hat A\|}{H_1} \bigg) \frac{\|\nabla A\|^2}{H_1} - \bigg(2\Lambda - \frac{C}{\varepsilon \delta_0}\bigg)\frac{\|\nabla \hat A\|^2}{H_1}, \end{align} where $C = C(n)$. \end{lemma} \begin{proof} At any point in $Q$ we compute \begin{align} (\partial_t - \Delta ) f = - (\partial_t - \Delta )\lambda_1 -\varepsilon(\partial_t - \Delta ) w+\Lambda (\partial_t - \Delta )v, \end{align} so by Lemma \ref{lem:lambda_1_evol_2}, \begin{align} (\partial_t - \Delta )f &\leq -\|h\|^2 \lambda_1 -T_{11}+ \lambda_1 \|\nabla \nu_1\|^2 +2H_1^{-1} \langle \nabla \hat A_{11}, \nabla H_1 \nu_1 \rangle\\ &+ 2\langle \nabla \hat A_{11}, \nabla \nu_1 \rangle-\varepsilon(\partial_t - \Delta ) w+ \Lambda (\partial_t - \Delta )v. \end{align} Inserting the estimates from Lemma \ref{lem:w_evol} and Lemma \ref{lem:v_evol} we find \begin{align} (\partial_t - \Delta )f &\leq \|h\|^2 (-\lambda_1 - \varepsilon w) -T_{11}+ \lambda_1 \|\nabla \nu_1\|^2 + 2H_1^{-1} \langle \nabla \hat A_{11}, \nabla H_1 \nu_1 \rangle\\ &+ 2\langle \nabla \hat A_{11}, \nabla \nu_1 \rangle-\varepsilon \delta_0 \frac{\|\nabla A\|^2}{H_1}+ \Lambda \bigg( C\|A\|^2 \|\hat A\| + C \frac{\|\hat A\|}{H_1} \frac{\|\nabla A\|^2}{H_1} - 2\frac{\|\nabla \hat A\|^2}{H_1}\bigg), \end{align} where $C = C(n)$. Using the definition of $f$ and rearranging we obtain \begin{align} (\partial_t - \Delta )f &\leq \|h\|^2 (f - \Lambda v) -T_{11}+ C \Lambda \|A\|^2 \|\hat A\| + (- f - \varepsilon w + \Lambda v)\|\nabla \nu_1\|^2 \\ & + 2H_1^{-1} \langle \nabla \hat A_{11}, \nabla H_1 \nu_1 \rangle+ 2\langle \nabla \hat A_{11}, \nabla \nu_1 \rangle + C\Lambda \frac{\|\hat A\|}{H_1} \frac{\|\nabla A\|^2}{H_1}\\ &-\varepsilon \delta_0 \frac{\|\nabla A\|^2}{H_1} - 2\Lambda\frac{\|\nabla \hat A\|^2}{H_1}. \end{align} Next we estimate \begin{align} -T_{11} &= 2 \sum_{\beta\neq1}^n \hat A_{ij\beta} \hat A_{pq\beta} h_{pq} +\sum_{\beta \neq 1} \hat A_{ip\beta} \hat A_{pq \beta} h_{qj} \\ &+\sum_{\beta \neq 1}\hat A_{jp\beta}\hat A_{pq\beta}h_{qi} - 2 \sum_{\beta \neq 1} \hat A_{ip\beta}\hat A_{jq\beta}h_{pq}\\ &\leq C(n) \|A\|^2 \|\hat A\| \end{align} and \begin{align} 2H_1^{-1} \langle \nabla \hat A_{11}, \nabla H_1 \nu_1 \rangle + 2\langle \nabla \hat A_{11}, \nabla \nu_1 \rangle &\leq C(n) H_1^{-1} \|\nabla \hat A\| \|\nabla A\|\\ &\leq \frac{\varepsilon \delta_0}{2} \frac{\|\nabla A\|^2}{H_1} + \frac{C(n)}{\varepsilon \delta_0} \frac{\|\nabla \hat A\|^2}{H_1} \end{align} in order to obtain \begin{align} (\partial_t - \Delta )f &\leq \|h\|^2 (f - \Lambda v) + C(1+ \Lambda) \|A\|^2 \|\hat A\| - ( f + \varepsilon w -\Lambda v)\|\nabla \nu_1\|^2 \\ & -\frac{\varepsilon \delta_0}{2} \frac{\|\nabla A\|^2}{H_1} + C\Lambda \frac{\|\hat A\|}{H_1} \frac{\|\nabla A\|^2}{H_1} + \bigg(\frac{C}{\varepsilon \delta_0} - 2\Lambda \bigg)\frac{\|\nabla \hat A\|^2}{H_1}. \end{align} Finally, by bounding \begin{align} \Lambda v \|\nabla \nu_1\|^2 \leq C(n) \Lambda \frac{\|\hat A\|}{H_1} \frac{\|\nabla A\|^2}{H_1} , \end{align} we find \begin{align} (\partial_t - \Delta ) f &\leq \|h\|^2 (f-\Lambda v) + C(1+ \Lambda) \|A\|^2 \|\hat A\| - ( f + \varepsilon w )\|\nabla \nu_1\|^2 \\ & - \bigg( \frac{\varepsilon \delta_0}{2}- C\Lambda \frac{\|\hat A\|}{H_1} \bigg) \frac{\|\nabla A\|^2}{H_1} - \bigg(2\Lambda - \frac{C}{\varepsilon \delta_0}\bigg)\frac{\|\nabla \hat A\|^2}{H_1}. \end{align} \end{proof} \section{A Poincar\'{e} inequality} \label{sec:Poinc} In this section we establish a Poincar\'{e}-type inequality for the high codimension solution $M_t$. The proof loosely follows \cite[Lemma 5.4]{Huisken84}, in that we combine Simons' identity with an integration by parts argument. We also incorporate an idea from \cite[Proposition 3.1]{Bren-Huisk17}, where the authors symmetrise and then take the square of Simons' identity to fully exploit the structure of the cubic zeroth-order terms. Simons' identity for high codimension submanifolds states that \begin{align} \nabla _k \nabla_ l A _{ij\alpha } & = \nabla _i \nabla_j A_{kl \alpha } + A_{ kl \beta} A _{ ip \beta} A _{ jp \alpha} - A_{ ij \beta} A _{ kp \beta} A _{ lp \alpha}\\ &+ A_{ jl \beta} A_{ ip \beta} A _{ kp \alpha} + A_{ jk \beta} A _{ ip \beta} A _{ lp \alpha}-A _{ il \beta} A _{ kp \beta} A _{ jp \alpha}- A_{ jl \beta}A _{ kp \beta}A _{ ip \alpha}. \end{align} We symmetrise to get \begin{align} \nabla _k \nabla_ l A _{ij\alpha } +\nabla_l\nabla_k A_{ji\alpha} =& \ \nabla _i \nabla_j A_{kl \alpha } + \nabla_j\nabla_i A_{lk\alpha} + E_{klij\alpha}. \end{align} where \begin{align} E_{klij\alpha} &= A_{ kl \beta} A _{ ip \beta} A _{ jp \alpha} + A_{lk\beta} A_{jp\beta} A_{ip\alpha}\\ &- A_{ ij \beta} A _{ kp \beta} A _{ lp \alpha}-A_{ji\beta} A_{lp\beta}A_{kp\alpha}\\ &+ A_{ jl \beta} A_{ ip \beta} A _{ kp \alpha} +A_{ik\beta} A_{jp\beta}A_{lp\alpha}\\ &+ A_{ jk \beta} A _{ ip \beta} A _{ lp \alpha}+ A_{il\beta} A_{jp\beta}A_{kp\alpha}\\ &-A _{ il \beta} A _{ kp \beta} A _{ jp \alpha} -A_{jk\beta}A_{lp\beta}A_{ip\alpha}\\ &- A_{ jl \beta} A _{ kp \beta}A _{ ip \alpha}-A_{ik\beta} A_{lp\beta}A_{jp\alpha}. \end{align} Using the relation \begin{align} R^\perp_{ij \alpha \beta} = A_{ip\alpha} A_{jp \beta} - A_{ip\beta} A_{jp\alpha} \end{align} we can rewrite the components of $E$ as \begin{align} E_{klij\alpha}=& \ A_{kl\beta} ( A_{ip\beta} A_{jp \alpha} + A_{jp\beta} A_{ip\alpha} ) - A_{ij \beta} (A_{kp\beta}A_{lp \alpha}+A_{lp\beta} A_{kp\alpha} ) \\ &+A_{jl\beta} (A_{ip\beta} A_{kp\alpha} - A_{kp\beta}A_{ip\alpha} ) + A_{jk\beta} ( A_{ip\beta}A_{lp\alpha} - A_{lp\beta}A_{ip\alpha } ) \\ &+A_{ik\beta} (A_{lp\alpha}A_{jp\beta} - A_{lp\beta} A_{jp\alpha} ) + A_{il\beta}( A_{kp\alpha} A_{jp\beta} - A_{kp\beta} A_{jp \alpha})\\ =& \ A_{kl\beta} ( A_{ip\beta} A_{jp \alpha} + A_{jp\beta} A_{ip\alpha} ) - A_{ij \beta} (A_{kp\beta}A_{lp \alpha}+A_{lp\beta} A_{kp\alpha} ) \\ & + A_{jl\beta} R^\perp_{ki\alpha\beta} + A _{ jk\beta} R^\perp _{li\alpha\beta} + A_{ik\beta} R^\perp_{lj\alpha\beta} + A _{il\beta}R^\perp _{ kj\alpha\beta}\\ = & 2 A_{kl\beta} A_{ip\beta} A_{jp \alpha} - 2A_{ij \beta} A_{kp\beta}A_{lp \alpha} + A_{kl\beta} R^\perp_{ij\alpha \beta} - A_{ij\beta} R^\perp_{kl\alpha\beta}\\ & + A_{jl\beta} R^\perp_{ki\alpha\beta} + A _{ jk\beta} R^\perp _{li\alpha\beta} + A_{ik\beta} R^\perp_{lj\alpha\beta} + A _{il\beta}R^\perp _{ kj\alpha\beta}. \end{align} \begin{lemma} There is a positive constant $C=C(n)$ such that \[\|E\|^2 \geq 8\|h\|^2 \tr (h^4) - 8 \tr(h^3)^2 - C\|A\|^5\|\hat A\|.\] \end{lemma} \begin{proof} Let us decompose $E$ as \[E_{klij\alpha} = U_{klij\alpha} + V_{klij\alpha}\] where \begin{align} U_{klij\alpha} &:= 2 A_{kl\beta} A_{ip\beta} A_{jp \alpha} - 2A_{ij \beta} A_{kp\beta}A_{lp \alpha} ,\\ V_{klij\alpha} &:= A_{kl\beta} R^\perp_{ij\alpha \beta} - A_{ij\beta} R^\perp_{kl\alpha\beta} + A_{jl\beta} R^\perp_{ki\alpha\beta}\\ & + A _{ jk\beta} R^\perp _{li\alpha\beta} + A_{ik\beta} R^\perp_{lj\alpha\beta} + A _{il\beta}R^\perp _{ kj\alpha\beta}. \end{align} There then holds \[\|E\|^2 = \|U\|^2 + 2 \langle U, V\rangle + \|V\|^2.\] Breaking $U$ into components parallel and orthogonal to the mean curvature vector we obtain \begin{align} U_{klij} &= 2 \langle A_{kl} ,A_{ip} \rangle A_{jp} - 2\langle A_{ij}, A_{kp}\rangle A_{lp}\\ & = 2 h_{kl} h_{ip} A_{jp} - 2 h_{ij}h_{kp} A_{lp} + 2 \langle \hat A_{kl}, \hat A_{ip} \rangle A_{jp} - 2 \langle \hat A_{ij}, \hat A_{lp} \rangle A_{lp}\\ & = 2 h_{kl} h_{ip} h_{jp} \nu_1 - 2 h_{ij}h_{kp} h_{lp} \nu_1+ 2 h_{kl} h_{ip} \hat A_{jp} - 2 h_{ij}h_{kp} \hat A_{lp}\\ &+ 2 \langle \hat A_{kl}, \hat A_{ip} \rangle A_{jp} - 2 \langle \hat A_{ij}, \hat A_{lp} \rangle A_{lp}, \end{align} hence \begin{align} \|U\|^2 &\geq 4 \| h_{kl} h_{ip} h_{jp} - h_{ij}h_{kp} h_{lp} \|^2 - C(n) \|\hat A\| \|A\|^5+ 4 \|\langle \hat A_{kl}, \hat A_{ip} \rangle A_{jp} - \langle \hat A_{ij}, \hat A_{lp} \rangle A_{lp}\|^2 \\ & \geq 8 \|h\|^2 \tr(h^4)-8\tr(h^3)^2 - C(n) \|\hat A\|\|A\|^5. \end{align} Substituting this back in we arrive at \begin{align} \|E\|^2 \geq 8 \|h\|^2 \tr(h^4)-8\tr(h^3)^2 - 2 \|U\|\|V\| - C(n) \|\hat A\|\|A\|^5. \end{align} There is a purely dimensional constant $C$ such that \[\|V\| \leq C\|A\|\|R^\perp\|,\] and we have \[R^{\perp}_{ij1\beta} = h_{ip} \hat A_{jp\beta} - \hat A_{ip\beta} h_{jp}\] and \[R^{\perp}_{ij\alpha\beta} = \hat A_{ip\alpha} A_{jp\beta} - A_{ip\beta} \hat A_{jp\alpha},\qquad \alpha \geq 2,\] so for a larger constant $C$ there holds \[\|V\| \leq C \|A\|^2 \|\hat A\|.\] Since $\|U\| \leq C\|A\|^3$ we have \[\|E\|^2 \geq 8 \|h\|^2 \tr(h^4)-8\tr(h^3)^2 - C\|A\|^5 \|\hat A\|.\] \end{proof} We are now ready to prove the Poincar\'{e} inequality. The proof does not actually use the fact that $M_t$ moves by mean curvature, so this result can be viewed as a general statement about high codimension submanifolds. \begin{proposition} Fix $t \in [0,T)$ and let $u:M\to \mathbb{R}$ be a nonnegative Lipschitz function which is supported in $\supp(f(\cdot,t))$. Then there is a positive constant $C = C(n,\varepsilon_0,\varepsilon,\Lambda)$ such that \begin{align} \int_M \|h\|^2 u^2 \, d\mu_t & \leq C \int_M u^2 \frac{\|\nabla A\|^2}{\|A\|^2} \, d\mu_t +C\int_M u \|\nabla u\| \frac{\|\nabla A\|}{\|A\|} \,d \mu_t + C \int_M \|A\| \|\hat A\| u^2\,d\mu_t. \end{align} \end{proposition} \begin{proof} For a symmetric matrix $B$ with eigenvalues $\mu_i$ there holds \begin{align} \|B\|^2 \tr (B^4) - \tr(B^3)^2 &= \frac{1}{2} \sum_{i,j}( \mu_i^2 \mu_j^4 - \mu_i^3 \mu_j^3 )+\frac{1}{2} \sum_{i,j}( \mu_j^2 \mu_i^4 - \mu_i^3 \mu_j^3 )\\ &=\frac{1}{2}\sum_{i,j} \mu_i^2 \mu_j^2 (\mu_i^2 + \mu_j^2 - 2 \mu_i \mu_j)\\ &= \frac{1}{2}\sum_{i,j}\mu_i^2 \mu_j^2(\mu_i - \mu_j)^2. \end{align} Observe that the right-hand side vanishes if and only if $B$ is the second fundamental form of a codimension-one cylinder. Let us define \[B(x,t) = h(x,t) - \Lambda v(x,t) g(x,t),\] which has as eigenvalues $\mu_i = \lambda_i - \Lambda v$. In particular, the computation above shows that \begin{align} \|B\|^2 \tr (B^4) - \tr(B^3)^2 &\geq \mu_n^2 \mu_1^2 (\mu_n-\mu_1)^2\\ &=\lambda_n^2 \mu_1^2 (\mu_n - \mu_1)^2 - 2 \Lambda \lambda_n \mu_1^2 (\mu_n - \mu_1)^2 v + \Lambda^2 \mu_1^2 (\mu_n - \mu_1)^2 v^2\\ &\geq \frac{1}{C} \|h\|^2 \mu_1^2 (\mu_n - \mu_1)^2 - C \|A\|^5 \|\hat A\| \end{align} where $C = C(n,\Lambda)$. At any point where $f(x,t) >0$ we have \[\lambda_1(x,t) < - \varepsilon w(x,t) + \Lambda v(x,t),\] which is to say that $\mu_1(x,t) \leq - \varepsilon w(x,t)$. Furthermore, since \[0< H_1(x,t) = \lambda_1(x,t) + \dots + \lambda_n(x,t) \leq \lambda_1(x,t) + (n-1)\lambda_n(x,t)\] there holds \begin{align} \mu_n(x,t) - \mu_1(x,t) &= \lambda_n(x,t) - \lambda_1(x,t) \\ &\geq - \bigg(1+\frac{1}{n-1}\bigg) \lambda_1(x,t)\\ &\geq \frac{n}{n-1} \varepsilon w(x,t) - \frac{n}{n-1} \Lambda v(x,t). \end{align} If the right-hand side is nonnegative then we can square both sides to get an estimate of the form \begin{align} (\mu_n(x,t) - \mu_1(x,t))^2 & \geq \frac{\varepsilon^2}{C} w(x,t)^2 - C\varepsilon \Lambda \|A\|\|\hat A\| \end{align} where $C = C(n)$. On the other hand if \[\frac{n}{n-1} \varepsilon w(x,t) - \frac{n}{n-1} \Lambda v(x,t)< 0 \] then trivially there holds \[(\mu_n(x,t) - \mu_1(x,t))^2 \geq 0 \geq \frac{n^2}{(n-1)^2} \varepsilon^2 w(x,t)^2 - \frac{n^2}{(n-1)^2} \Lambda^2 v(x,t)^2,\] so in either case we can bound \begin{align} (\mu_n(x,t) - \mu_1(x,t))^2 & \geq \frac{\varepsilon^2}{C} w(x,t)^2 - C \|A\|\|\hat A\| \end{align} with $C = C(n,\varepsilon, \Lambda)$. Putting these estimates together we find on the support of $f$ there holds \begin{align} \|B\|^2 \tr (B^4) - \tr(B^3)^2 &\geq \mu_n^2 \mu_1^2 (\mu_n-\mu_1)^2\\ &\geq \frac{\varepsilon^4}{C} \|h\|^2 w^4 - C \|A\|^5 \|\hat A\|, \end{align} and since \[\|B\|^2 \tr (B^4) - \tr(B^3)^2 \leq \|h\|^2 \tr(h^4) - \tr(h^3)^2 + C \|A\|^5 \|\hat A\|\] we finally get \[\|h\|^2 \tr(h^4) - \tr(h^3)^2 \geq C^{-1} \|h\|^2 \|A\|^4 - C \| A\|^5\|\|\hat A\| \] where $C = C(n, \varepsilon_0, \varepsilon, \Lambda)$. Combining this inequality with the result of the last lemma we find on the support of $f$ there holds \begin{align} \|h\|^2 \|A\|^4 &\leq C(\|h\|^2 \tr(h^4) - \tr(h^3)^2 )+ C\| A\|^5\|\|\hat A\|\\ & \leq C\|E\|^2 + C\|A\|^5 \|\hat A\|. \end{align} Let $u$ be a nonnegative Lipschitz function supported in $\supp(f)$. Then we can multiply this inequality by $\|A\|^{-4} u^2$ and integrate over $M$ to get \begin{align} \int_M & \|h\|^2 u^2 \,d\mu_t \\ &\leq C \int_M \|A\|^{-4} u^2 \|E\|^2 + \|A\| \|\hat A\| u^2 \, d\mu_t\\ & = C \int_M \|A\|^{-4} u^2 E_{klij\alpha} (\nabla _k \nabla_ l A _{ij\alpha } +\nabla_l\nabla_k A_{ji\alpha} -\nabla _i \nabla_j A_{kl \alpha } - \nabla_j\nabla_i A_{lk\alpha}) \, d\mu_t\\ & + C \int_M \|A\| \|\hat A\| u^2 \,d\mu_t \end{align} We are going to estimate each of the four Hessian terms on the right. Since each of these is handled in the same way, we only give the argument for the first one. Defining \[T_k := \|A\|^{-4} u^2 E_{ijkl\alpha} \nabla_ l A _{ij\alpha },\] we can write \begin{align} \|A\|^{-4} u^2 E_{ijkl\alpha} \nabla _k \nabla_ l A _{ij\alpha } & = \nabla_k T_k + 4 \|A\|^{-5} u^2 E_{ijkl\alpha} \nabla_ l A _{ij\alpha } \nabla_k \|A\| \\ &-2 \|A\|^{-4} u E_{ijkl\alpha}\nabla_ l A _{ij\alpha } \nabla_k u - \|A\|^{-4} u^2 \nabla_k E_{ijkl\alpha} \nabla_ l A _{ij\alpha }. \end{align} The divergence term vanishes upon integration, and there is a purely dimensional constant $C$ such that \[\|E\| \leq C\|A\|^3 , \qquad \|\nabla E\| \leq C \|A\|^2 \|\nabla A\|, \qquad \|\nabla \|A\|\| \leq C \|\nabla A\|,\] so making $C$ a bit larger, we have \begin{align} \int_M \|A\|^{-4} u^2 E_{ijkl\alpha}\nabla _k \nabla_ l A _{ij\alpha } \, d\mu_t &\leq C \int_M u^2 \|A\|^{-5} \|A\|^3 \|\nabla A\|^2 \, d\mu_t\\ & + C \int_M u \|\nabla u\| \|A\|^{-4} \|A\|^3 \|\nabla A\| \,d \mu_t\\ & + C\int_M u^2 \|A\|^{-4} \|A\|^2 \|\nabla A\|^2 \,d\mu_t. \end{align} Estimating the remaining Hessian terms in the same way and substituting back in we arrive at \begin{align} \int_M \|h\|^2 u^2 \, d\mu_t & \leq C \int_M u^2 \frac{\|\nabla A\|^2}{\|A\|^2} \, d\mu_t +C\int_M u \|\nabla u\| \frac{\|\nabla A\|}{\|A\|} \,d \mu_t + C \int_M \|A\| \|\hat A\| u^2\,d\mu_t. \end{align} \end{proof} \section{Stampacchia iteration} \label{sec:Stamp} In this section we establish the convexity estimate by proving an a priori supremum estimate for the function \[f_\sigma := \frac{f}{\|H\|^{1-\sigma}}\] where $\sigma \in (0,1)$ is chosen small depending on $n$ and $M_0$. Recall from Lemma \ref{lem:f_evol} that at each point in $Q$ there holds \begin{align} (\partial_t - \Delta )f &\leq \|h\|^2 f + C(1+ \Lambda) \|A\|^2 \|\hat A\| - ( f + \varepsilon w )\|\nabla \nu_1\|^2 \\ & - \bigg( \frac{\varepsilon \delta_0}{2}- C\Lambda \frac{\|\hat A\|}{H_1} \bigg) \frac{\|\nabla A\|^2}{H_1} - \bigg(2\Lambda - \frac{C}{\varepsilon \delta_0}\bigg)\frac{\|\nabla \hat A\|^2}{H_1}, \end{align} where $C = C(n)$. Let us fix \[\Lambda = \frac{C}{2\varepsilon \delta_0}\] so that the last term vanishes. Then using \begin{align} (\partial_t - \Delta) \|H\|^{1-\sigma} = (1-\sigma) \|h\|^2 H_1^{1-\sigma} - (1-\sigma)H_1^{1-\sigma}\|\nabla \nu_1\|^2 + \sigma (1-\sigma) H_1^{-\sigma - 1} \|\nabla H_1\|^2 \end{align} we compute that \begin{align} (\partial_t - \Delta) f_\sigma &\leq \sigma \|h\|^2 f_\sigma + C(1+ \Lambda) \|A\|^2 \frac{\|\hat A\|}{H_1^{1-\sigma}} - \bigg(\sigma f_\sigma + \varepsilon \frac{w}{H_1^{1-\sigma} }\bigg)\|\nabla \nu_1\|^2 \\ & - \bigg( \frac{\varepsilon \delta_0}{2}- C\Lambda \frac{\|\hat A\|}{H_1} \bigg) H^\sigma \frac{\|\nabla A\|^2}{H_1^2} - \sigma(1-\sigma) f_\sigma \frac{ \|\nabla H_1\|^2}{H_1^2}\\ & + 2(1-\sigma) \bigg\langle \nabla f_\sigma, \frac{\nabla H_1}{H_1} \bigg\rangle. \end{align} Hence at points in $Q \cap \supp(f_\sigma)$ we have \begin{align} \label{eq:f_sigma_evol} (\partial_t - \Delta) f_\sigma &\leq \sigma \|h\|^2 f_\sigma+C(1+ \Lambda) \|A\|^2 \frac{\|\hat A\|}{H_1^{1-\sigma}}- \bigg( \frac{\varepsilon \delta_0}{2}- C\Lambda \frac{\|\hat A\|}{H_1} \bigg) H_1^\sigma\frac{\|\nabla A\|^2}{H_1^{2}} \notag\\ & + 2(1-\sigma) \bigg\langle \nabla f_\sigma, \frac{\nabla H_1}{H_1} \bigg\rangle, \end{align} where $C=C(n)$. All of the computations until now were for a quadratically pinched solution with \[c \leq \frac{4}{3n} - \varepsilon_0.\] From here on we assume $n\geq 5$ and the more restrictive condition $c \leq c_n - \varepsilon_0$ where \[c_n : =\begin{cases} \frac{3(n+1)}{2n(n+2)} & n = 5, 6, 7, \\ \frac{4}{3n} & n \geq 8. \end{cases} \] This is the range of pinching constants for which Naff's codimension estimate is valid. \begin{theorem}[\cite{Naff2019}] \label{thm:codim} Let $F:M\times[0,T) \to \mathbb{R}^{n+k}$, $n \geq 5$, be a quadratically pinched mean curvature flow with $c \leq c_n - \varepsilon_0$. Then there is a constant $\eta = \eta(n,\varepsilon_0)$ in $(0,1)$ such that \[\max_{M_t} \frac{\|\hat A\|^2}{\|H\|^{2-2\eta}} \leq \max_{M_0} \frac{\|\hat A\|^2}{\|H\|^{2-2\eta}}\] for each $t \in [0,T)$. \end{theorem} Hence if we set \[L := \max_{M_0} \|H\|\] then the inequality \[\frac{\|\hat A\|}{\|H\|} \leq C(n)L^\eta \|H\|^{-\eta}\] holds on $M_t$ for every $t \in [0,T)$. Inserting this estimate into \eqref{eq:f_sigma_evol} we find \begin{align} \label{eq:f_sigma_est_1} (\partial_t - \Delta) f_\sigma &\leq \sigma \|h\|^2 f_\sigma+C(1+ \Lambda)L^\eta \|A\|^2 H_1^{\sigma-\eta}- \bigg( \frac{\varepsilon \delta_0}{2}- C\Lambda L^\eta H_1^{-\eta} \bigg) H_1^\sigma\frac{\|\nabla A\|^2}{H_1^{2}}\notag\\ & + 2(1-\sigma) \bigg\langle \nabla f_\sigma, \frac{\nabla H_1}{H_1} \bigg\rangle \end{align} on $Q \cap \supp(f_\sigma)$, where $C=C(n)$. \subsection{$L^p$-estimates} For each $k >0$ let us define \[ f_{\sigma,k}(x,t) := \max\{ f_\sigma(x,t) -k,0\}.\] Using the Poincar\'{e} inequality we now establish an $L^p$-estimate for $f_{\sigma,k}$. In the codimension one case similar estimates have appeared in \cite{Huisken84} and \cite{Huisk-Sin99a}. \begin{proposition} \label{prop:Lp} There are positive constants $p_0$ and $\ell_0$ depending on $n$, $\varepsilon_0$, $\eta$, $\varepsilon$ and $\Lambda$, and a positive constant $k_0 = k_0(n, \varepsilon_0, \eta, \varepsilon, \Lambda, L)$, with the following property. For every \[p \geq p_0, \qquad \sigma \leq \ell_0 p^{-\frac{1}{2}}, \qquad k \geq k_0,\] we have \[\sup_{t \in [0,T)} \bigg( \int_M f_{\sigma,k}^p \,d\mu_t \bigg)\leq C,\] where $C = C(n, \varepsilon_0,\eta,\varepsilon, \Lambda, L, \mu_0(M), T, k, \sigma, p)$. \end{proposition} \begin{proof} Suppose for now that $p_0 \geq 4$ and $\ell_0 \leq \eta$. Then the condition $\sigma \leq \ell_0 p^{-\frac{1}{2}}$ ensures that $\sigma \leq \eta /2$. On $\supp(f_{\sigma,k})$ we have \[k < \frac{f}{H_1} H_1^\sigma \leq C_0(n,\Lambda) H_1^\sigma,\] so if we take $k_0 \geq C_0$ and impose $k \geq k_0$ then on $\supp(f_{\sigma,k})$ there holds \[H_1 \geq (k/C_0)^\frac{1}{\sigma} \geq \max\{k/C_0,1\}.\] Substituting this into \eqref{eq:f_sigma_est_1} we find \begin{align} (\partial_t - \Delta) f_\sigma &\leq \sigma \|h\|^2 f_\sigma+C(1+ \Lambda) L^\eta \|A\|^2 H_1^{-\frac{\eta}{2}}- \bigg( \frac{\varepsilon \delta_0}{2}- C_1 k^{-\eta} \bigg) H_1^\sigma\frac{\|\nabla A\|^2}{H_1^{2}}\notag\\ & + 2(1-\sigma) \bigg\langle \nabla f_\sigma, \frac{\nabla H_1}{H_1} \bigg\rangle \end{align} on $Q\cap \supp(f_{\sigma, k})$, where $C = C(n)$ and $C_1 = C_1(n,\eta,\Lambda,L)$. Choosing $k_0$ a bit larger so that \[k_0 \geq \max\bigg\{1,C_0, \bigg( \frac{4C_1}{\varepsilon \delta_0}\bigg)^{1/\eta} \bigg\}\] and using $f/H_1 \leq C_0$, we find on $Q \cap \supp(f_{\sigma,k})$, \begin{align} (\partial_t - \Delta) f_\sigma &\leq \sigma \|h\|^2 f_\sigma+C(1+ \Lambda)L^\eta \|A\|^2 H_1^{-\frac{\eta}{2}}- \frac{\varepsilon \delta_0}{4 C_0} f_\sigma \frac{\|\nabla A\|^2}{H_1^{2}}\notag\\ & + 2(1-\sigma) \bigg\langle \nabla f_\sigma, \frac{\nabla H_1}{H_1} \bigg\rangle. \end{align} By Young's inequality we have \[2(1-\sigma) \bigg\langle \nabla f_\sigma, \frac{\nabla H_1}{H_1} \bigg\rangle \leq C_2 \frac{\|\nabla f_\sigma\|^2}{f_\sigma} + \frac{\varepsilon \delta_0}{8 C_0} f_\sigma \frac{\|\nabla A\|^2}{H_1^2}\] on $\supp(f_\sigma)$, where $C_2 = C_2(n,\varepsilon_0,\varepsilon, C_0)$. Hence on $Q \cap \supp(f_{\sigma,k})$, \begin{align} (\partial_t - \Delta) f_\sigma &\leq \sigma \|h\|^2 f_\sigma+C(1+ \Lambda) L^\eta \|A\|^2 H_1^{-\frac{\eta}{2}}- \frac{\varepsilon \delta_0}{8 C_0} f_\sigma \frac{\|\nabla A\|^2}{H_1^{2}} + C_2 \frac{\|\nabla f_\sigma\|^2}{f_\sigma}. \end{align} Applying the pinching we can bound \[C(1+ \Lambda) L^\eta \|A\|^2 H_1^{-\eta/2} \leq C_3(n,\eta,\Lambda,L) \|h\|^2 H_1^{-\eta/2},\] and by Young's inequality \[H_1^{-\eta/2} \leq \frac{4 - \eta}{4}s^{4/(4-\eta)} + \frac{\eta}{4} \frac{1}{s^{4/\eta} } \frac{1}{H_1^2} \leq s^{4/(4-\eta)} + \frac{1}{s^{4/\eta} } \frac{1}{H_1^2} \] for every positive $s$. Setting $s = \sigma^{(4-\eta)/4}$ gives \[H_1^{-\eta/2} \leq \sigma + \frac{1}{\sigma^{(4-\eta)/\eta}} \frac{1}{H_1^2} \leq \sigma + \sigma^{-4/\eta} H_1^{-2},\] so using the pinching we get \[C(1+ \Lambda) L^\eta\|A\|^2 H_1^{-\frac{\eta}{2}} \leq C_3 \sigma \|h\|^2 + C_4 \sigma^{-4/\eta}\] for some $C_4 = C_4(n,\eta,\Lambda,L)$. Substituting back in, we have \begin{align} (\partial_t - \Delta) f_\sigma &\leq \sigma \|h\|^2 f_\sigma + C_3\sigma \|h\|^2- c_0 f_\sigma \frac{\|\nabla A\|^2}{H_1^{2}} + C_2 \frac{\|\nabla f_\sigma\|^2}{f_\sigma} + C_4 \sigma^{-4/\eta} \end{align} on $Q \cap \supp (f_{\sigma,k})$, where \[c_0 := \frac{\varepsilon \delta_0}{8 C_0}.\] If $\varphi$ is any nonnegative Lipschitz function supported in $\supp(f_{\sigma,k})$, then on almost every timeslice we can multiply the last inequality by $\varphi$ and integrate to get \begin{align} \int_M \partial_t f_\sigma \cdot \varphi \,d\mu_t &\leq \int_M \Delta f_\sigma \cdot \varphi\,d\mu_t + \sigma \int_M \|h\|^2 f_\sigma \varphi \,d\mu_t + C_3 \sigma \int_M \|h\|^2 \varphi\, d\mu_t \\ &- c_0 \int_M f_\sigma \varphi \frac{\|\nabla A\|^2}{\|H\|^2} \,d\mu_t + C_2 \int_M \varphi \frac{\|\nabla f_\sigma\|^2}{f_\sigma}\,d\mu_t + C_4 \sigma^{-4/\eta}\int_M \varphi\,d\mu_t. \end{align} Since $f_\sigma$ is a locally semiconvex function we can use Lemma \ref{lem:alex} to integrate by parts, and so obtain \begin{align} \int_M \partial_t f_\sigma \cdot \varphi \,d\mu_t & \leq -\int_M \langle \nabla f_\sigma, \nabla \varphi \rangle\,d\mu_t + \sigma \int_M \|h\|^2 f_\sigma \varphi \,d\mu_t + C_3 \sigma \int_M \|h\|^2 \varphi\, d\mu_t \\ &- c_0 \int_M f_\sigma \varphi \frac{\|\nabla A\|^2}{\|H\|^2} \,d\mu_t + C_2 \int_M \varphi \frac{\|\nabla f_\sigma\|^2}{f_\sigma}\,d\mu_t + C_4 \sigma^{-4/\eta}\int_M \varphi\,d\mu_t. \end{align} We set $\varphi = p f_{\sigma, k}^{p-1}$ in this inequality and use \[\frac{d}{dt} \int_M f_{\sigma, k}^p \,d\mu_t = p \int_M \partial_t f_\sigma \cdot f_{\sigma,k}^{p-1} \,d\mu_t - \int_M \|H\|^2 f_{\sigma,k}^p\,d\mu_t\] to estimate \begin{align} \frac{d}{dt} &\int_M f_{\sigma,k}^p \,d\mu_t\\ &\leq - p(p-1) \int_M f_{\sigma,k}^{p-2}\|\nabla f_\sigma\|^2\,d\mu_t + \sigma p\int_M \|h\|^2 f_\sigma f_{\sigma,k}^{p-1} \,d\mu_t + C_3 \sigma p \int_M \|h\|^2 f_{\sigma,k}^{p-1}\, d\mu_t \\ &- c_0 p\int_M f_\sigma f_{\sigma,k}^{p-1} \frac{\|\nabla A\|^2}{\|H\|^2} \,d\mu_t + C_2 p\int_M f_{\sigma,k}^{p-1} \frac{\|\nabla f_\sigma\|^2}{f_\sigma}\,d\mu_t + C_4 \sigma^{-4/\eta} p \int_M f_{\sigma,k}^{p-1} \,d\mu_t \end{align} for almost every $t \in [0,T)$. Using that $f_{\sigma,k} = f_\sigma - k$ on $\supp(f_{\sigma,k})$ and rearranging slightly, this gives \begin{align} \frac{d}{dt} \int_M f_{\sigma,k}^p \,d\mu_t &\leq - (p(p-1) - C_2p)\int_M f_{\sigma,k}^{p-2}\|\nabla f_\sigma\|^2\,d\mu_t - c_0 p\int_M f_{\sigma,k}^{p} \frac{\|\nabla A\|^2}{\|H\|^2} \,d\mu_t \\ & + \sigma p\int_M \|h\|^2 f_{\sigma,k}^{p} \,d\mu_t + (C_3+k) \sigma p \int_M \|h\|^2 f_{\sigma,k}^{p-1}\, d\mu_t \\ &+ C_4 \sigma^{-4/\eta} p \int_M f_{\sigma,k}^{p-1} \,d\mu_t. \end{align} Using Young's inequality we estimate \begin{align} (C_3 +k)\sigma p\int_M \|h\|^2 f_{\sigma,k}^{p-1}\, d\mu_t & \leq \sigma (p-1) \int_M \|h\|^2 f_{\sigma,k}^p \,d\mu_t + (C_3+k)^p \sigma \int_M \|h\|^2 \, d\mu_t \end{align} and \begin{align} C_4 \sigma^{-4/\eta} p \int_M f_{\sigma,k}^{p-1} \,d\mu_t \leq C_4 \sigma^{-4/\eta} (p-1) \int_M f_{\sigma,k}^{p} \,d\mu_t + C_4 \sigma^{-4/\eta} \mu_t(M). \end{align} Inserting these inequalities we arrive at \begin{align} \label{eq:Lp_step} \frac{d}{dt} \int_M f_{\sigma,k}^p \,d\mu_t &\leq - (p(p-1) - C_2p)\int_M f_{\sigma,k}^{p-2}\|\nabla f_\sigma\|^2\,d\mu_t - c_0 p\int_M f_{\sigma,k}^{p} \frac{\|\nabla A\|^2}{\|H\|^2} \,d\mu_t \notag\\ & + 2\sigma p\int_M \|h\|^2 f_{\sigma,k}^{p} \,d\mu_t + (C_3+k)^p \sigma \int_M \|h\|^2 \, d\mu_t \notag\\ &+C_4 \sigma^{-4/\eta} p \int_M f_{\sigma,k}^{p} \,d\mu_t + C_4 \sigma^{-4/\eta} \mu_t(M). \end{align} Since $ f_{\sigma, k}$ is supported in $\supp(f)$, we can apply the Poincar\'{e} inequality with $u = f_{\sigma, k}^{\frac{p}{2}}$ to obtain \begin{align} \int_M \|h\|^2 f_{\sigma,k}^p \, d\mu_t & \leq C_5 \int_M f_{\sigma,k}^p \frac{\|\nabla A\|^2}{\|H\|^2} \, d\mu_t +C_5 p \int_M f_{\sigma,k}^{p-1} \|\nabla f_\sigma\| \frac{\|\nabla A\|}{\|H\|} \,d \mu_t\\ & + C_5 \int_M \|A\| \|\hat A\| f_{\sigma,k}^p\,d\mu_t, \end{align} where the constant $C_5$ depends on $n$, $\varepsilon_0$, $\varepsilon$ and $\Lambda$. Applying Young's inequality we obtain \begin{align} \int_M \|h\|^2 f_{\sigma,k}^p \, d\mu_t & \leq C_5(1+p^\frac{1}{2}) \int_M f_{\sigma,k}^p \frac{\|\nabla A\|^2}{\|H\|^2} \, d\mu_t +C_5 p^\frac{3}{2} \int_M f_{\sigma,k}^{p-2} \|\nabla f_\sigma\|^2 \,d \mu_t\\ & + C_5 \int_M \|A\| \|\hat A\| f_{\sigma,k}^p\,d\mu_t. \end{align} Inserting the codimension estimate and quadratic pinching we get \begin{align} C_5 \int_M \|A\| \|\hat A\| f_{\sigma,k}^p\,d\mu_t \leq C_6(n, L, C_5) \int_M \|h\|^2 \|H\|^{-\eta} f_{\sigma,k}^p\,d\mu_t, \end{align} and we know that $\|H\| \geq k/C_0$ on $\supp(f_{\sigma,k})$, so if we take \[ k_0 \geq \max\bigg\{1,C_0, \bigg( \frac{4C_1}{\varepsilon \delta_0}\bigg)^{1/\eta} , C_0(2C_6)^{1/\eta}\bigg\}\] then \begin{align} C_5 \int_M \|A\| \|\hat A\| f_{\sigma,k}^p\,d\mu_t \leq \frac{1}{2} \int_M \|h\|^2 f_{\sigma,k}^p\,d\mu_t. \end{align} In this case \begin{align} \frac{1}{2} \int_M \|h\|^2 f_{\sigma,k}^p \, d\mu_t & \leq C_5(1+p^\frac{1}{2}) \int_M f_{\sigma,k}^p \frac{\|\nabla A\|^2}{\|H\|^2} \, d\mu_t +C_5 p^\frac{3}{2} \int_M f_{\sigma,k}^{p-2} \|\nabla f_\sigma\|^2 \,d \mu_t. \end{align} Multiplying this inequality through by $4\sigma p$ and substituting back into \eqref{eq:Lp_step} gives \begin{align} \frac{d}{dt} \int_M f_{\sigma,k}^{p} \,d\mu_t &\leq- (p(p-1)-C_2p - 4 C_5 \sigma p^\frac{5}{2}) \int_M f_{\sigma, k}^{p-2} \|\nabla f_\sigma\|^2 \,d\mu_t \\ &- (c_0 p - 4 C_5\sigma p -4 C_5\sigma p^\frac{3}{2} ) \int_M f_{\sigma,k}^{p} \frac{\|\nabla A\|^2}{\|H\|^{2}}\,d\mu_t\\ &+ (C_3+k)^p \sigma \int_M \|h\|^2 \, d\mu_t +C_4 \sigma^{-4/\eta} p \int_M f_{\sigma,k}^{p} \,d\mu_t \\ &+ C_4 \sigma^{-4/\eta} \mu_t(M). \end{align} Now we insert the assumption $\sigma \leq \ell_0 p^{-\frac{1}{2}}$ and thus obtain \begin{align} \frac{d}{dt} \int_M f_{\sigma,k}^{p} \,d\mu_t &\leq- (p(p-1)-C_2p -4 C_5 \ell_0 p^2) \int_M f_{\sigma, k}^{p-2} \|\nabla f_\sigma\|^2 \,d\mu_t \\ &- (c_0 p - 4 C_5\ell_0 p^\frac{1}{2} -4 C_5 \ell_0 p ) \int_M f_{\sigma,k}^{p} \frac{\|\nabla A\|^2}{\|H\|^{2}}\,d\mu_t\\ &+ (C_3+k)^p \sigma \int_M \|h\|^2 \, d\mu_t +C_4 \sigma^{-4/\eta} p \int_M f_{\sigma,k}^{p} \,d\mu_t \\ &+ C_4 \sigma^{-4/\eta} \mu_t(M). \end{align} Decreasing $\ell_0$ so that \[\ell_0 \leq \min\bigg\{\eta, \frac{c_0}{8C_5}, \frac{1}{8C_5} \bigg\}\] now gives \begin{align} \frac{d}{dt} \int_M f_{\sigma,k}^{p} \,d\mu_t &\leq- (p^2/2 -p-C_2p ) \int_M f_{\sigma, k}^{p-2} \|\nabla f_\sigma\|^2 \,d\mu_t \\ &- (c_0 p /2 - 2 C_5\ell_0 p^\frac{1}{2} ) \int_M f_{\sigma,k}^{p} \frac{\|\nabla A\|^2}{\|H\|^{2}}\,d\mu_t\\ &+ (C_3+k)^p \sigma \int_M \|h\|^2 \, d\mu_t +C_4 \sigma^{-4/\eta} p \int_M f_{\sigma,k}^{p} \,d\mu_t \\ &+ C_4 \sigma^{-4/\eta} \mu_t(M). \end{align} We can now take $p_0$ large depending only on $c_0$ and $C_5$ to ensure that for $p \geq p_0$ the inequality \begin{align} \frac{d}{dt} \int_M f_{\sigma,k}^{p} \,d\mu_t &\leq (C_3+k)^p \sigma \int_M \|h\|^2 \, d\mu_t +C_4 \sigma^{-4/\eta} p \int_M f_{\sigma,k}^{p} \,d\mu_t \\ &+ C_4 \sigma^{-4/\eta} \mu_t(M). \end{align} holds for almost every $t \in [0,T)$. Taking $k_0$ a bit larger depending on $n$ and $C_3$, using $k \geq k_0$ we can bound \begin{align} \frac{d}{dt} \int_M f_{\sigma,k}^{p} \,d\mu_t &\leq 2^p k^p \sigma \int_M \|H\|^2 \, d\mu_t +C_4 \sigma^{-4/\eta} p \int_M f_{\sigma,k}^{p} \,d\mu_t \\ &+ C_4 \sigma^{-4/\eta} \mu_t(M). \end{align} Since \[\frac{d}{dt} \int_M 2^p k^p \sigma \, d\mu_t = - 2^p k^p \sigma \int_M \|H\|^2 \,d\mu_t\] this implies \begin{align} \frac{d}{dt} \int_M f_{\sigma,k}^{p} + 2^p k^p \sigma \,d\mu_t &\leq C_4 \sigma^{-4/\eta} p \int_M f_{\sigma,k}^{p} \,d\mu_t + C_4 \sigma^{-4/\eta} \mu_t(M)\\ &= C_4 \sigma^{-4/\eta} p \int_M f_{\sigma,k}^{p} + p^{-1} \,d\mu_t . \end{align} Hence the function \[\varphi(t):= \int_M f_{\sigma,k}^p +2^pk^p\sigma + p^{-1} \,d\mu_t \] satisfies \[\varphi'(t) \leq C_4 \sigma^{-4/\eta} p \varphi(t)\] for almost every $t \in [0,T)$. Since $\varphi$ is Lipschitz continuous in time it follows that \[\varphi(t) \leq \varphi(0) \exp( C_4 \sigma^{-4/\eta} p t).\] In particular, $\varphi$ can be bounded from above in terms of its value at the initial time, and the constants $C_4$, $\eta$, $\sigma$, $p$ and $T$. Recall that $C_4$ depends only on $n$, $\eta$, $\Lambda$ and $L$. Also, \[f_{\sigma, k}^{p} \leq C_0^p \|H\|^{\sigma p},\] so $\varphi(0)$ can be bounded purely in terms of $n$, $\Lambda$, $\sigma$, $p$, $L$ and $\mu_0(M)$. This completes the proof. \end{proof} \subsection{The supremum estimate} Combining the $L^p$-estimates just established with the Michael-Simon Sobolev inequality \cite{Michael-Simon73} we obtain the following iteration inequality. The proof is very similar to that of Theorem 5.1 in \cite{Huisken84}, so we omit the details. \begin{proposition} There are positive constants $p_1 \geq p_0$ and $\ell_1 \leq \ell_0$ depending on $n$, $\varepsilon_0$, $\eta$, $\varepsilon$ and $\Lambda$, and a positive constant $k_1 \geq k_0$ depending on $n$, $\varepsilon_0$, $\eta$, $\varepsilon$, $\Lambda$ and $L$, with the following property. Suppose $p \geq p_1$ and $\sigma \leq \ell_1 p^{-\frac{1}{2}}$ and set \[A(k) := \int_0^T \int_{\supp(f_{\sigma,k}(\cdot,t))} \,d\mu_t dt.\] Then for every $h>k\geq k_1$ we have \begin{align} A(h) \leq \frac{C}{(h-k)^p} A(k)^{\gamma}. \end{align} where $\gamma >1$ depends on $n$ and $C = C(n, \varepsilon_0, \eta, \varepsilon, \Lambda, L, \mu_0(M), T, \sigma, p)$. \end{proposition} Appealing to Stampacchia's lemma (see for example Lemma B.1 in \cite{Kind-Stamp}) we obtain: \begin{corollary} There is a constant $k_2 = k_2 (n,\varepsilon_0, \eta, \varepsilon, \Lambda, L, \mu_0(M), T)$ such that \[f_{\sigma_0, k_2} \equiv 0\] on $M\times [0,T)$, where $\sigma_0 := \ell_1 p_1^{-\frac{1}{2}}$ depends only on $n$, $\varepsilon_0$, $\eta$, $\varepsilon$ and $\Lambda$. \end{corollary} Recall that $\eta$ depends only on $n$ and $\varepsilon_0$, and we chose $\Lambda$ depending only on $n$, $\varepsilon_0$ and $\varepsilon$. Therefore, by the corollary we have an estimate of the form \begin{align} \frac{\lambda_1 + \varepsilon w}{\|H\|} \geq - C\|H\|^{-\sigma_0} -\Lambda \frac{\|\hat A\|^2}{\|H\|^2} \end{align} on $M\times [0,T)$, where $C = C(n,\varepsilon_0, \varepsilon, L, \mu_0(M), T)$. Appealing to the codimension estimate of Theorem \ref{thm:codim}, we finally obtain \begin{equation} \label{eq:conv_est} \frac{\lambda_1 + \varepsilon w}{\|H\|} \geq - C\|H\|^{-\sigma_0} -C\|H\|^{-2\eta}, \end{equation} where $C$ has the same dependencies as before. From here, since $\varepsilon$ can be made arbitrarily small, an application of Young's inequality to the two lower-order terms on the right-hand side gives the convexity estimate of Theorem \ref{thm:convex} (note that $T$ can be bounded in terms of $M_0$ by applying the maximum principle to the evolution equation of $W$). \begin{remark} There is another way to prove Theorem \ref{thm:convex} using compactness and the strong maximum principle, which we now sketch. Let $F:M \times [0,T) \to \mathbb{R}^{n+k}$ be a quadratically pinched mean curvature flow with $n \geq 5$ and $c < c_n$, fix a constant $\varepsilon >0$, and set \[f_\varepsilon := \frac{\lambda_1 + \varepsilon w}{\|H\|}.\] For each $j \in \mathbb{N}$ set \[\delta_{j} := \inf \{f_\varepsilon(x,t): \|H\|(x,t) \geq j\}.\] Then the $\delta_j$ form a bounded nondecreasing sequence and therefore converge to some $\delta$. Choose a sequence $(x_j, t_j) \in P_j$ such that $f_\varepsilon(x_j,t_j) \to \delta$ and form a sequence of solutions by shifting $F(x_j,t_j)$ to the spacetime origin in $\mathbb{R}^{n+k} \times \mathbb{R}$ and parabolically rescaling by $\|H\|(x_j,t_j)$. Then the gradient estimates established in Section 3 of \cite{Nguyen2018a} ensure that this sequence converges smoothly in a small spacetime neighbourhood about the origin. The limit lies in $\mathbb{R}^{n+1}$ by Naff's codimension estimate, and the quantity $f_\varepsilon$ attains its minimum $\delta$ at the spacetime origin. The strong maximum principle applied to the evolution equation for $f_\varepsilon$ shows that $\|\nabla h\|\equiv 0$ on the limiting solution, which is consequently either a piece of shrinking sphere or cylinder. In either case we have $\lambda_1 \geq 0$, so $\delta >0$. In other words, on the original solution, there exists a threshold $C_\varepsilon$ depending on $\varepsilon$ and $M_0$ such that whenever $\|H\|(x,t) \geq C_\varepsilon$ there holds \[ \frac{\lambda_1(x,t)}{\|H\|(x,t)} \geq -\varepsilon \frac{w(x,t)}{\|H\|(x,t)} \geq - C(n) \varepsilon.\] Note that this argument does not yield a quantitative blow-up rate for the negative part of the second fundamental form, in contrast to the estimate \eqref{eq:conv_est} proven above. Moreover, the gradient estimates in \cite{Nguyen2018a} are difficult to establish and will not be available in other situations where the Stampacchia iteration goes through. Indeed, an estimate showing asymptotic positivity of curvature is often needed to establish gradient estimates; this is the case for the fully nonlinear flow studied in \cite{Bren-Huisk17} and for three-dimensional Ricci flow \cite{Perelman_a}. \end{remark} \section{Singularity formation} \label{sec:singularity_formation} In the study of parabolic evolution equations it is natural to distinguish between singularities which form at different rates. For a solution of mean curvature flow $F:M\times[0,T) \to \mathbb{R}^{n+k}$ where $T$ is the maximal time we say that a type I singularity forms as $t \to T$ if there is a positive constant $C$ such that \[\max_{M_t} \|A\|^2 \leq \frac{C}{T-t}.\] Note that this is the blow-up rate for solutions which shrink homothetically (such as shrinking spheres and cylinders). If on the other hand \[\limsup_{t \nearrow T} \Big[(T-t)\max_{M_t} \|A\|^2 \Big] =\infty\] then the singularity forming at time $T$ is said to be of type II. For certain type I singularities, Baker used Huisken's monotonicity formula to show that appropriate rescalings about the singularity converge to a homothetically shrinking solution \cite{Baker_thesis}. Moreover, Baker could show that the only such solutions satisfying the quadratic pinching condition are shrinking spheres and (generalised) cylinders. The analogous result for mean-convex solutions of codimension one was proven earlier by Huisken \cite{Huisken90}. In \cite{Huisk-Sin99a} Huisken and Sinestrari used their convexity estimate to show that at a type II singularity, appropriate rescalings about the maximum of the curvature converge to a convex translating solution. In this section we use our convexity estimate to generalise their result to higher codimensions. Fix a smooth mean curvature flow $F:M\times[0,T) \to \mathbb{R}^{n+k}$ of dimension $n \geq 5$ which is quadratically pinched with $c < c_n$, and suppose a type II singularity is forming as $t \to T$. Consider a sequence of times $\tilde t_j \to T$ and let $(x_j,t_j)$ be such that \[(\tilde t_j - t_j) \|H\|^2(x_j,t_j) := \max_{M \times [0,\tilde t_j]} (\tilde t_j - t) \|H\|^2(x,t).\] Then we have \[\|H\|^2(x_j, t_j) = \max_{M} \|H\|^2(x,t_j) \] By the type II assumption, for each $K > 0$ there is a point $(y,\tau) \in M\times [0,T)$ such that \[(T-\tau) \|H\|^2(y,\tau) \geq K.\] If $j$ is large enough so that $\tilde t_j > \tau$ then we have \begin{align} (\tilde t_j - t_j) \|H\|^2(x_j,t_j) = (\tilde t_j - \tau)\|H\|^2(y,\tau) \geq K - (T - \tilde t_j) \|H\|^2(y,\tau). \end{align} Hence if $j$ is sufficiently large there holds \[(\tilde t_j - t_j)\|H\|^2(x_j, t_j) \geq K/2,\] and since $K$ can be made arbitrarily large this shows that \[(\tilde t_j - t_j)\|H\|^2(x_j, t_j) \to \infty.\] It follows that $t_j \to T$. Let $L_j^2 := \|H\|^2(x_j,t_j)$ and consider the sequence of rescaled solutions defined by \[F_j(x,t) := L_j (F(x,L_j^{-2}t + t_j ) - F(x_j,t_j)), \qquad (x,t) \in M\times [- L_j^2t_j, L_j^2(T- t_j)),\] which satisfy the conditions \[F_j(0,0) = 0, \qquad \|H_j\|^2(0,0) = 1,\] where $H_j$ is the mean curvature vector of $F_j$. More generally, for $t \leq L_j^2 (\tilde t_j - t_j)$ there holds \begin{align} \|H_j\|^2(x,t) &= L_j^{-2} \|H\|^2(x, L_j^{-2} t + t_j) \\ &=L_j^{-2} \frac{(\tilde t_j - L_j^{-2} t -t_j ) \|H\|^2(x, L_j^{-2} t + t_j)}{\tilde t_j - L_j^{-2} t - t_j} \\ & \leq L_j^{-2} \frac{(\tilde t_j - t_j) \|H\|^2(x_j, t_j) }{\tilde t_j - L_j^{-2} t - t_j}\\ & = \frac{\tilde t_j - t_j}{\tilde t_j - t_j - L_j^{-2} t }. \end{align} Therefore, for times $t \leq \delta L_j^2 (\tilde t_j - t_j)$ with $\delta <1$ we have \[\max_{M} \|H_j\|^2(\cdot,t) \leq \frac{1}{1-\delta}.\] Passing to a subsequence in $j$, we can guarantee that there is a sequence $\tau_j \to \infty$ such that \[\max_{M} \|H_j\|^2(\cdot, t) \leq 1 + \frac{1}{j}, \qquad \forall \; t \in [-\tau_j, \tau_j]. \] It is well known that for a compact solution of mean curvature flow, a global upper bound for $\|A\|$ implies bounds on all of the higher derivatives of $A$. This follows from the estimates in \cite{Ecker-Huisken} in the codimension-one case, and similar arguments work in higher codimensions (the details can be found in Section 4.3 of \cite{Baker_thesis}). Standard compactness theorems therefore imply that there is a smooth solution \[\tilde F : \tilde M \times (-\infty, \infty ) \to \mathbb{R}^{n+k}\] such that the sequence $F_j$ subconverges smoothly to $\tilde F$ in the following local sense. There is a sequence of nested open sets $U_l \subset \tilde M$ such that \[\tilde M = \bigcup_{l \in \mathbb N} U_l\] and local diffeomorphisms $\varphi_l : U_l \to M$ such that the sequence \[(x,t) \mapsto F_j(\varphi_l(x), t), \qquad (x,t) \in U_l \times [-l, l] \] converges smoothly to \[(x,t) \mapsto \tilde F(x,t), \qquad (x,t) \in U_l \times [-l, l]\] as $j \to \infty$ for every $l \in \mathbb{N}$. This follows for example from Hamilton's compactness theorem \cite{Ham_compactness}, as is illustrated in Section 6.1 of \cite{Baker_thesis}. \begin{theorem} The smooth limiting solution $\tilde F : \tilde M \times (-\infty, \infty) \to \mathbb{R}^{n+k}$ obtained by the above rescaling procedure lies in an $(n+1)$-dimensional affine subspace and is either: a strictly convex translating solution; or the isometric product of $\mathbb{R}^m$ with a strictly convex translating solution of dimension $n -m$. \end{theorem} \begin{proof} We denote the second fundamental form of $\tilde F$ by $\tilde A$, and use this convention for other curvature quantities as well. We know that the mean curvature vector of $\tilde F$ satisfies $\|\tilde H\|(0,0) = 1$, but a priori, there could be a point on the limiting solution where $\|\tilde H\| = 0$. However the evolution equation \[(\partial_t - \Delta ) \tilde W \geq 2 \|\tilde h\|^2 \tilde W \geq \frac{3}{2} \tilde W^3\] is still valid, so by the strong maximum principle we either have $\tilde W >0$ or $\tilde W \equiv 0$. In the latter situation $\|\tilde H\|^2(0,0) = 0$, which is a contradiction, so appealing to the pinching we conclude that \[\frac{4}{3n} \|\tilde H\|^2 > \tilde W >0 \] on $\tilde M \times (-\infty, \infty)$. Hence Naff's codimension estimate implies $\tilde h \equiv \tilde A$, and consequently, the image of $\tilde F$ lies in an $(n+1)$-dimensional subspace of $\mathbb{R}^{n+1}$. By the convexity estimate, $\tilde h \geq 0$ on $\tilde M \times (-\infty,\infty)$. To recap, the blow-up limit $\tilde F$ is codimension one, has nonnegative second fundamental form, its scalar mean curvature $\|\tilde H\|$ is globally bounded from above by one, and this global upper bound is attained at the spacetime origin. This is exactly the situation considered in Section 4 of \cite{Huisk-Sin99a}. Applying Hamilton's strong maximum principle for tensors to the evolution of the second fundamental form, \[(\partial_t - \Delta) \tilde h_j^i = \|\tilde h\|^2 \tilde h^i_j,\] we conclude that the solution $\tilde M_t := \tilde F(\tilde M, t)$ splits as an isometric product $\mathbb{R}^{m} \times N_t$, where $N_t$ is a strictly convex solution of dimension $n -m$ which exists for all $ t \in (-\infty, \infty)$. Since the spacetime maximum of the mean curvature of $N_t$ is attained at the spacetime origin, the rigidity case of Hamilton's Harnack inequality \cite{Ham_harnack} implies the family $N_t$ moves by translation. \end{proof} \begin{remark} By the gradient estimate in \cite{Nguyen2018a}, the limiting flow $\tilde F$ cannot be the product of $\mathbb{R}^{n-1}$ with a grim reaper. On the other hand, the grim reaper is the only strictly convex translator in $\mathbb{R}^2$, so we conclude that $m \leq n-2$. \end{remark} \begin{remark} If $N_t$ is uniformly two-convex in the sense that the smallest two principal curvatures satisfy \[\lambda_1 + \lambda_2 \geq \alpha H\] globally for some $\alpha >0$, then by the gradient estimate in \cite{Nguyen2018a} and work of Bourni-Langford \cite{Bourni-Langford}, $N_t$ must be rotationally symmetric and hence a bowl soliton of dimension $m$. See also the paper \cite{Naff2019a}. \end{remark} \end{document}	arXiv
Preordered class In mathematics, a preordered class is a class equipped with a preorder. Definition When dealing with a class C, it is possible to define a class relation on C as a subclass of the power class C $\times $ C . Then, it is convenient to use the language of relations on a set. A preordered class is a class with a preorder on it. Partially ordered class and totally ordered class are defined in a similar way. These concepts generalize respectively those of preordered set, partially ordered set and totally ordered set. However, it is difficult to work with them as in the small case because many constructions common in a set theory are no longer possible in this framework. Equivalently, a preordered class is a thin category, that is, a category with at most one morphism from an object to another. Examples • In any category C, when D is a class of morphisms of C containing identities and closed under composition, the relation 'there exists a D-morphism from X to Y' is a preorder on the class of objects of C. • The class Ord of all ordinals is a totally ordered class with the classical ordering of ordinals. References • Nicola Gambino and Peter Schuster, Spatiality for formal topologies • Adámek, Jiří; Horst Herrlich; George E. Strecker (1990). Abstract and Concrete Categories (PDF). John Wiley & Sons. ISBN 0-471-60922-6.	Wikipedia
Dessin d'enfant In mathematics, a dessin d'enfant is a type of graph embedding used to study Riemann surfaces and to provide combinatorial invariants for the action of the absolute Galois group of the rational numbers. The name of these embeddings is French for a "child's drawing"; its plural is either dessins d'enfant, "child's drawings", or dessins d'enfants, "children's drawings". A dessin d'enfant is a graph, with its vertices colored alternately black and white, embedded in an oriented surface that, in many cases, is simply a plane. For the coloring to exist, the graph must be bipartite. The faces of the embedding are required be topological disks. The surface and the embedding may be described combinatorially using a rotation system, a cyclic order of the edges surrounding each vertex of the graph that describes the order in which the edges would be crossed by a path that travels clockwise on the surface in a small loop around the vertex. Any dessin can provide the surface it is embedded in with a structure as a Riemann surface. It is natural to ask which Riemann surfaces arise in this way. The answer is provided by Belyi's theorem, which states that the Riemann surfaces that can be described by dessins are precisely those that can be defined as algebraic curves over the field of algebraic numbers. The absolute Galois group transforms these particular curves into each other, and thereby also transforms the underlying dessins. For a more detailed treatment of this subject, see Schneps (1994) or Lando & Zvonkin (2004). History 19th century Early proto-forms of dessins d'enfants appeared as early as 1856 in the icosian calculus of William Rowan Hamilton;[1] in modern terms, these are Hamiltonian paths on the icosahedral graph. Recognizable modern dessins d'enfants and Belyi functions were used by Felix Klein.[2] Klein called these diagrams Linienzüge (German, plural of Linienzug "line-track", also used as a term for polygon); he used a white circle for the preimage of 0 and a '+' for the preimage of 1, rather than a black circle for 0 and white circle for 1 as in modern notation.[3] He used these diagrams to construct an 11-fold cover of the Riemann sphere by itself, with monodromy group $PSL(2,11)$, following earlier constructions of a 7-fold cover with monodromy $PSL(2,7)$ connected to the Klein quartic.[4] These were all related to his investigations of the geometry of the quintic equation and the group $A_{5}=PSL(2,5)$, collected in his famous 1884/88 Lectures on the Icosahedron. The three surfaces constructed in this way from these three groups were much later shown to be closely related through the phenomenon of trinity. 20th century Dessins d'enfant in their modern form were then rediscovered over a century later and named by Alexander Grothendieck in 1984 in his Esquisse d'un Programme.[5] Zapponi (2003) quotes Grothendieck regarding his discovery of the Galois action on dessins d'enfants: This discovery, which is technically so simple, made a very strong impression on me, and it represents a decisive turning point in the course of my reflections, a shift in particular of my centre of interest in mathematics, which suddenly found itself strongly focused. I do not believe that a mathematical fact has ever struck me quite so strongly as this one, nor had a comparable psychological impact. This is surely because of the very familiar, non-technical nature of the objects considered, of which any child’s drawing scrawled on a bit of paper (at least if the drawing is made without lifting the pencil) gives a perfectly explicit example. To such a dessin we find associated subtle arithmetic invariants, which are completely turned topsy-turvy as soon as we add one more stroke. Part of the theory had already been developed independently by Jones & Singerman (1978) some time before Grothendieck. They outline the correspondence between maps on topological surfaces, maps on Riemann surfaces, and groups with certain distinguished generators, but do not consider the Galois action. Their notion of a map corresponds to a particular instance of a dessin d'enfant. Later work by Bryant & Singerman (1985) extends the treatment to surfaces with a boundary. Riemann surfaces and Belyi pairs The complex numbers, together with a special point designated as $\infty $, form a topological space known as the Riemann sphere. Any polynomial, and more generally any rational function $p(x)/q(x)$ where $p$ and $q$ are polynomials, transforms the Riemann sphere by mapping it to itself. Consider, for example,[6] the rational function $f(x)=-{\frac {(x-1)^{3}(x-9)}{64x}}=1-{\frac {(x^{2}-6x-3)^{2}}{64x}}.$ At most points of the Riemann sphere, this transformation is a local homeomorphism: it maps a small disk centered at any point in a one-to-one way into another disk. However, at certain critical points, the mapping is more complicated, and maps a disk centered at the point in a $k$-to-one way onto its image. The number $k$ is known as the degree of the critical point and the transformed image of a critical point is known as a critical value. The example given above, $f$, has the following critical points and critical values. (Some points of the Riemann sphere that, while not themselves critical, map to one of the critical values, are also included; these are indicated by having degree one.) critical point x critical value f(x) degree $0$ $\infty $ $1$ $1$ $0$ $3$ $9$ $0$ $1$ $3+2{\sqrt {3}}\approx 6.464$ $1$ $2$ $3-2{\sqrt {3}}\approx -0.464$ $1$ $2$ $\infty $ $\infty $ $3$ One may form a dessin d'enfant from $f$ by placing black points at the preimages of 0 (that is, at 1 and 9), white points at the preimages of 1 (that is, at $3\pm 2{\sqrt {3}}$), and arcs at the preimages of the line segment [0, 1]. This line segment has four preimages, two along the line segment from 1 to 9 and two forming a simple closed curve that loops from 1 to itself, surrounding 0; the resulting dessin is shown in the figure. In the other direction, from this dessin, described as a combinatorial object without specifying the locations of the critical points, one may form a compact Riemann surface, and a map from that surface to the Riemann sphere, equivalent to the map from which the dessin was originally constructed. To do so, place a point labeled $\infty $ within each region of the dessin (shown as the red points in the second figure), and triangulate each region by connecting this point to the black and white points forming the boundary of the region, connecting multiple times to the same black or white point if it appears multiple times on the boundary of the region. Each triangle in the triangulation has three vertices labeled 0 (for the black points), 1 (for the white points), or $\infty $. For each triangle, substitute a half-plane, either the upper half-plane for a triangle that has 0, 1, and $\infty $ in counterclockwise order or the lower half-plane for a triangle that has them in clockwise order, and for every adjacent pair of triangles glue the corresponding half-planes together along the portion of their boundaries indicated by the vertex labels. The resulting Riemann surface can be mapped to the Riemann sphere by using the identity map within each half-plane. Thus, the dessin d'enfant formed from $f$ is sufficient to describe $f$ itself up to biholomorphism. However, this construction identifies the Riemann surface only as a manifold with complex structure; it does not construct an embedding of this manifold as an algebraic curve in the complex projective plane, although such an embedding always exists. The same construction applies more generally when $X$ is any Riemann surface and $f$ is a Belyi function; that is, a holomorphic function $f$ from $X$ to the Riemann sphere having only 0, 1, and $\infty $ as critical values. A pair $(X,f)$ of this type is known as a Belyi pair. From any Belyi pair $(X,f)$ one can form a dessin d'enfant, drawn on the surface $X$, that has its black points at the preimages $f^{-1}(0)$ of 0, its white points at the preimages $f^{-1}(1)$ of 1, and its edges placed along the preimages $f^{-1}([0,1])$ of the line segment $[0,1]$. Conversely, any dessin d'enfant on any surface $X$ can be used to define gluing instructions for a collection of halfspaces that together form a Riemann surface homeomorphic to $X$; mapping each halfspace by the identity to the Riemann sphere produces a Belyi function $f$ on $X$, and therefore leads to a Belyi pair $(X,f)$. Any two Belyi pairs $(X,f)$ that lead to combinatorially equivalent dessins d'enfants are biholomorphic, and Belyi's theorem implies that, for any compact Riemann surface $X$ defined over the algebraic numbers, there are a Belyi function $f$ and a dessin d'enfant that provides a combinatorial description of both $X$ and $f$. Maps and hypermaps A vertex in a dessin has a graph-theoretic degree, the number of incident edges, that equals its degree as a critical point of the Belyi function. In the example above, all white points have degree two; dessins with the property that each white point has two edges are known as clean, and their corresponding Belyi functions are called pure. When this happens, one can describe the dessin by a simpler embedded graph, one that has only the black points as its vertices and that has an edge for each white point with endpoints at the white point's two black neighbors. For instance, the dessin shown in the figure could be drawn more simply in this way as a pair of black points with an edge between them and a self-loop on one of the points. It is common to draw only the black points of a clean dessin and to leave the white points unmarked; one can recover the full dessin by adding a white point at the midpoint of each edge of the map. Thus, any embedding of a graph in a surface in which each face is a disk (that is, a topological map) gives rise to a dessin by treating the graph vertices as black points of a dessin, and placing white points at the midpoint of each embedded graph edge. If a map corresponds to a Belyi function $f$, its dual map (the dessin formed from the preimages of the line segment $[1,\infty ]$) corresponds to the multiplicative inverse $1/f$.[7] A dessin that is not clean can be transformed into a clean dessin in the same surface, by recoloring all of its points as black and adding new white points on each of its edges. The corresponding transformation of Belyi pairs is to replace a Belyi function $\beta $ by the pure Belyi function $\gamma =4\beta (1-\beta )$. One may calculate the critical points of $\gamma $ directly from this formula: $\gamma ^{-1}(0)=\beta ^{-1}(0)\cup \beta ^{-1}(1)$, $\gamma ^{-1}(\infty )=\beta ^{-1}(\infty )$, and $\gamma ^{-1}(1)=\beta ^{-1}({\tfrac {1}{2}})$. Thus, $\gamma ^{-1}(1)$ is the preimage under $\beta $ of the midpoint of the line segment $[0,1]$, and the edges of the dessin formed from $\gamma $ subdivide the edges of the dessin formed from $\beta $. Under the interpretation of a clean dessin as a map, an arbitrary dessin is a hypermap: that is, a drawing of a hypergraph in which the black points represent vertices and the white points represent hyperedges. Regular maps and triangle groups The five Platonic solids – the regular tetrahedron, cube, octahedron, dodecahedron, and icosahedron – viewed as two-dimensional surfaces, have the property that any flag (a triple of a vertex, edge, and face that all meet each other) can be taken to any other flag by a symmetry of the surface. More generally, a map embedded in a surface with the same property, that any flag can be transformed to any other flag by a symmetry, is called a regular map. If a regular map is used to generate a clean dessin, and the resulting dessin is used to generate a triangulated Riemann surface, then the edges of the triangles lie along lines of symmetry of the surface, and the reflections across those lines generate a symmetry group called a triangle group, for which the triangles form the fundamental domains. For example, the figure shows the set of triangles generated in this way starting from a regular dodecahedron. When the regular map lies in a surface whose genus is greater than one, the universal cover of the surface is the hyperbolic plane, and the triangle group in the hyperbolic plane formed from the lifted triangulation is a (cocompact) Fuchsian group representing a discrete set of isometries of the hyperbolic plane. In this case, the starting surface is the quotient of the hyperbolic plane by a finite index subgroup Γ in this group. Conversely, given a Riemann surface that is a quotient of a $(2,3,n)$ tiling (a tiling of the sphere, Euclidean plane, or hyperbolic plane by triangles with angles ${\tfrac {\pi }{2}}$, ${\tfrac {\pi }{3}}$, and ${\tfrac {\pi }{n}}$, the associated dessin is the Cayley graph given by the order two and order three generators of the group, or equivalently, the tiling of the same surface by $n$-gons meeting three per vertex. Vertices of this tiling give black dots of the dessin, centers of edges give white dots, and centers of faces give the points over infinity. Trees and Shabat polynomials The simplest bipartite graphs are the trees. Any embedding of a tree has a single region, and therefore by Euler's formula lies in a spherical surface. The corresponding Belyi pair forms a transformation of the Riemann sphere that, if one places the pole at $\infty $, can be represented as a polynomial. Conversely, any polynomial with 0 and 1 as its finite critical values forms a Belyi function from the Riemann sphere to itself, having a single infinite-valued critical point, and corresponding to a dessin d'enfant that is a tree. The degree of the polynomial equals the number of edges in the corresponding tree. Such a polynomial Belyi function is known as a Shabat polynomial,[8] after George Shabat. For example, take $p$ to be the monomial $p(x)=x^{d}$ having only one finite critical point and critical value, both zero. Although 1 is not a critical value for $p$, it is still possible to interpret $p$ as a Belyi function from the Riemann sphere to itself because its critical values all lie in the set $\{0,1,\infty \}$. The corresponding dessin d'enfant is a star having one central black vertex connected to $d$ white leaves (a complete bipartite graph $K_{1,d}$). More generally, a polynomial $p(x)$ having two critical values $y_{1}$ and $y_{2}$ may be termed a Shabat polynomial. Such a polynomial may be normalized into a Belyi function, with its critical values at 0 and 1, by the formula $q(x)={\frac {p(x)-y_{1}}{y_{2}-y_{1}}},$ but it may be more convenient to leave $p$ in its un-normalized form.[9] An important family of examples of Shabat polynomials are given by the Chebyshev polynomials of the first kind, $T_{n}(x)$, which have −1 and 1 as critical values. The corresponding dessins take the form of path graphs, alternating between black and white vertices, with $n$ edges in the path. Due to the connection between Shabat polynomials and Chebyshev polynomials, Shabat polynomials themselves are sometimes called generalized Chebyshev polynomials.[9][10] Different trees will, in general, correspond to different Shabat polynomials, as will different embeddings or colorings of the same tree. Up to normalization and linear transformations of its argument, the Shabat polynomial is uniquely determined from a coloring of an embedded tree, but it is not always straightforward to find a Shabat polynomial that has a given embedded tree as its dessin d'enfant. The absolute Galois group and its invariants The polynomial $p(x)=x^{3}(x^{2}-2x+a)^{2}\,$ may be made into a Shabat polynomial by choosing[11] $a={\frac {34\pm 6{\sqrt {21}}}{7}}.$ The two choices of $a$ lead to two Belyi functions $f_{1}$ and $f_{2}$. These functions, though closely related to each other, are not equivalent, as they are described by the two nonisomorphic trees shown in the figure. However, as these polynomials are defined over the algebraic number field $\mathbb {Q} ({\sqrt {21}})$, they may be transformed by the action of the absolute Galois group $\Gamma $ of the rational numbers. An element of $\Gamma $ that transforms ${\sqrt {21}}$ to $-{\sqrt {21}}$ will transform $f_{1}$ into $f_{2}$ and vice versa, and thus can also be said to transform each of the two trees shown in the figure into the other tree. More generally, due to the fact that the critical values of any Belyi function are the pure rationals 0, 1, and $\infty $, these critical values are unchanged by the Galois action, so this action takes Belyi pairs to other Belyi pairs. One may define an action of $\Gamma $ on any dessin d'enfant by the corresponding action on Belyi pairs; this action, for instance, permutes the two trees shown in the figure. Due to Belyi's theorem, the action of $\Gamma $ on dessins is faithful (that is, every two elements of $\Gamma $ define different permutations on the set of dessins),[12] so the study of dessins d'enfants can tell us much about $\Gamma $ itself. In this light, it is of great interest to understand which dessins may be transformed into each other by the action of $\Gamma $ and which may not. For instance, one may observe that the two trees shown have the same degree sequences for their black nodes and white nodes: both have a black node with degree three, two black nodes with degree two, two white nodes with degree two, and three white nodes with degree one. This equality is not a coincidence: whenever $\Gamma $ transforms one dessin into another, both will have the same degree sequence. The degree sequence is one known invariant of the Galois action, but not the only invariant. The stabilizer of a dessin is the subgroup of $\Gamma $ consisting of group elements that leave the dessin unchanged. Due to the Galois correspondence between subgroups of $\Gamma $ and algebraic number fields, the stabilizer corresponds to a field, the field of moduli of the dessin. An orbit of a dessin is the set of all other dessins into which it may be transformed; due to the degree invariant, orbits are necessarily finite and stabilizers are of finite index. One may similarly define the stabilizer of an orbit (the subgroup that fixes all elements of the orbit) and the corresponding field of moduli of the orbit, another invariant of the dessin. The stabilizer of the orbit is the maximal normal subgroup of $\Gamma $ contained in the stabilizer of the dessin, and the field of moduli of the orbit corresponds to the smallest normal extension of $\mathbb {Q} $ that contains the field of moduli of the dessin. For instance, for the two conjugate dessins considered in this section, the field of moduli of the orbit is $\mathbb {Q} ({\sqrt {21}})$. The two Belyi functions $f_{1}$ and $f_{2}$ of this example are defined over the field of moduli, but there exist dessins for which the field of definition of the Belyi function must be larger than the field of moduli.[13] Notes 1. Hamilton (1856). See also Jones (1995). 2. Klein (1879). 3. le Bruyn (2008). 4. Klein (1878–1879a); Klein (1878–1879b). 5. Grothendieck (1984) 6. This example was suggested by Lando & Zvonkin (2004), pp. 109–110. 7. Lando & Zvonkin (2004), pp. 120–121. 8. Girondo & González-Diez (2012) p. 252 9. Lando & Zvonkin (2004), p. 82. 10. Jones, G. and Streit, M. "Galois groups, monodromy groups and cartographic groups", p. 43 in Schneps & Lochak (2007) pp. 25–66. Zbl 0898.14012 11. Lando & Zvonkin (2004), pp. 90–91. For the purposes of this example, ignore the parasitic solution $a={\tfrac {25}{21}}$. 12. $\Gamma $ acts faithfully even when restricted to dessins that are trees; see Lando & Zvonkin (2004), Theorem 2.4.15, pp. 125–126. 13. Lando & Zvonkin (2004), pp. 122–123. References • le Bruyn, Lieven (2008), Klein's dessins d'enfant and the buckyball. • Bryant, Robin P.; Singerman, David (1985), "Foundations of the theory of maps on surfaces with boundary", Quarterly Journal of Mathematics, Second Series, 36 (141): 17–41, doi:10.1093/qmath/36.1.17, MR 0780347. • Girondo, Ernesto; González-Diez, Gabino (2012), Introduction to compact Riemann surfaces and dessins d'enfants, London Mathematical Society Student Texts, vol. 79, Cambridge: Cambridge University Press, ISBN 978-0-521-74022-7, Zbl 1253.30001. • Grothendieck, A. (1984), Esquisse d'un programme • Hamilton, W. R. (17 October 1856), Letter to John T. Graves "On the Icosian". Collected in Halberstam, H.; Ingram, R. E., eds. (1967), Mathematical papers, Vol. III, Algebra, Cambridge: Cambridge University Press, pp. 612–625. • Jones, Gareth (1995), "Dessins d'enfants: bipartite maps and Galois groups", Séminaire Lotharingien de Combinatoire, B35d: 4, archived from the original on 8 April 2017, retrieved 2 June 2010. • Jones, Gareth; Singerman, David (1978), "Theory of maps on orientable surfaces", Proceedings of the London Mathematical Society, 37 (2): 273–307, doi:10.1112/plms/s3-37.2.273. • Klein, Felix (1878–79), "Über die Transformation der elliptischen Funktionen und die Auflösung der Gleichungen fünften Grades (On the transformation of elliptic functions and ...)", Mathematische Annalen, 14: 13–75 (in Oeuvres, Tome 3), doi:10.1007/BF02297507, S2CID 121056952, archived from the original on 19 July 2011, retrieved 2 June 2010. • Klein, Felix (1878–79), "Über die Transformation siebenter Ordnung der elliptischen Funktionen (On the seventh order transformation of elliptic functions)", Mathematische Annalen, 14: 90–135 (in Oeuvres, Tome 3), doi:10.1007/BF01677143, S2CID 121407539. • Klein, Felix (1879), "Ueber die Transformation elfter Ordnung der elliptischen Functionen (On the eleventh order transformation of elliptic functions)", Mathematische Annalen, 15 (3–4): 533–555, doi:10.1007/BF02086276, S2CID 120316938, collected as pp. 140–165 in Oeuvres, Tome 3 Archived 19 July 2011 at the Wayback Machine. • Lando, Sergei K.; Zvonkin, Alexander K. (2004), Graphs on Surfaces and Their Applications, Encyclopaedia of Mathematical Sciences: Lower-Dimensional Topology II, vol. 141, Berlin, New York: Springer-Verlag, ISBN 978-3-540-00203-1, Zbl 1040.05001. See especially chapter 2, "Dessins d'Enfants", pp. 79–153. • Schneps, Leila, ed. (1994), The Grothendieck Theory of Dessins d'Enfants, London Mathematical Society Lecture Note Series, Cambridge: Cambridge University Press, ISBN 978-0-521-47821-2. • Schneps, Leila; Lochak, Pierre, eds. (1997), Geometric Galois actions II. The inverse Galois problem, moduli spaces and mapping class groups. Proceedings of the conference on geometry and arithmetic of moduli spaces, Luminy, France, August 1995, London Mathematical Society Lecture Note Series, vol. 243, Cambridge University Press, ISBN 0-521-59641-6, Zbl 0868.00040. • Shabat, G.B.; Voedvodsky, V.A. (2007) [1990], "Drawing curves over number fields", in Cartier, P.; Illusie, L.; Katz, N.M.; Laumon, G.; Manin, Yu.I.; Ribet, K.A. (eds.), The Grothendieck Festschrift Volume III, Modern Birkhäuser Classics, Birkhäuser, pp. 199–227, ISBN 978-0-8176-4568-7, Zbl 0790.14026. • Singerman, David; Syddall, Robert I. (2003), "The Riemann Surface of a Uniform Dessin", Beiträge zur Algebra und Geometrie, 44 (2): 413–430, MR 2017042, Zbl 1064.14030. • Zapponi, Leonardo (August 2003), "What is a Dessin d'Enfant" (PDF), Notices of the American Mathematical Society, 50 (7): 788–789, Zbl 1211.14001. Topics in algebraic curves Rational curves • Five points determine a conic • Projective line • Rational normal curve • Riemann sphere • Twisted cubic Elliptic curves Analytic theory • Elliptic function • Elliptic integral • Fundamental pair of periods • Modular form Arithmetic theory • Counting points on elliptic curves • Division polynomials • Hasse's theorem on elliptic curves • Mazur's torsion theorem • Modular elliptic curve • Modularity theorem • Mordell–Weil theorem • Nagell–Lutz theorem • Supersingular elliptic curve • Schoof's algorithm • Schoof–Elkies–Atkin algorithm Applications • Elliptic curve cryptography • Elliptic curve primality Higher genus • De Franchis theorem • Faltings's theorem • Hurwitz's automorphisms theorem • Hurwitz surface • Hyperelliptic curve Plane curves • AF+BG theorem • Bézout's theorem • Bitangent • Cayley–Bacharach theorem • Conic section • Cramer's paradox • Cubic plane curve • Fermat curve • Genus–degree formula • Hilbert's sixteenth problem • Nagata's conjecture on curves • Plücker formula • Quartic plane curve • Real plane curve Riemann surfaces • Belyi's theorem • Bring's curve • Bolza surface • Compact Riemann surface • Dessin d'enfant • Differential of the first kind • Klein quartic • Riemann's existence theorem • Riemann–Roch theorem • Teichmüller space • Torelli theorem Constructions • Dual curve • Polar curve • Smooth completion Structure of curves Divisors on curves • Abel–Jacobi map • Brill–Noether theory • Clifford's theorem on special divisors • Gonality of an algebraic curve • Jacobian variety • Riemann–Roch theorem • Weierstrass point • Weil reciprocity law Moduli • ELSV formula • Gromov–Witten invariant • Hodge bundle • Moduli of algebraic curves • Stable curve Morphisms • Hasse–Witt matrix • Riemann–Hurwitz formula • Prym variety • Weber's theorem (Algebraic curves) Singularities • Acnode • Crunode • Cusp • Delta invariant • Tacnode Vector bundles • Birkhoff–Grothendieck theorem • Stable vector bundle • Vector bundles on algebraic curves	Wikipedia
\begin{definition}[Definition:Summation/Infinite] Let $\struct {S, +}$ be an algebraic structure where the operation $+$ is an operation derived from, or arising from, the addition operation on the natural numbers. Let an infinite number of values of $j$ satisfy the propositional function $\map R j$. Then the precise meaning of $\ds \sum_{\map R j} a_j$ is: :$\ds \sum_{\map R j} a_j = \paren {\lim_{n \mathop \to \infty} \sum_{\substack {\map R j \\ -n \mathop \le j \mathop < 0}} a_j} + \paren {\lim_{n \mathop \to \infty} \sum_{\substack {\map R j \\ 0 \mathop \le j \mathop \le n} } a_j}$ provided that both limits exist. If either limit ''does'' fail to exist, then the '''infinite summation''' does not exist. \end{definition}	ProofWiki
Double-negation translation In proof theory, a discipline within mathematical logic, double-negation translation, sometimes called negative translation, is a general approach for embedding classical logic into intuitionistic logic. Typically it is done by translating formulas to formulas which are classically equivalent but intuitionistically inequivalent. Particular instances of double-negation translations include Glivenko's translation for propositional logic, and the Gödel–Gentzen translation and Kuroda's translation for first-order logic. Propositional logic The easiest double-negation translation to describe comes from Glivenko's theorem, proved by Valery Glivenko in 1929. It maps each classical formula φ to its double negation ¬¬φ. Results Glivenko's theorem states: If φ is a propositional formula, then φ is a classical tautology if and only if ¬¬φ is an intuitionistic tautology. Glivenko's theorem implies the more general statement: If T is a set of propositional formulas and φ a propositional formula, then T ⊢ φ in classical logic if and only if T ⊢ ¬¬φ in intuitionistic logic. In particular, a set of propositional formulas is intuitionistically consistent if and only if it is classically satisfiable. First-order logic The Gödel–Gentzen translation (named after Kurt Gödel and Gerhard Gentzen) associates with each formula φ in a first-order language another formula φN, which is defined inductively: • If φ is atomic, then φN is ¬¬φ as above, but furthermore • (φ ∨ θ)N is ¬(¬φN ∧ ¬θN) • (∃x φ)N is ¬(∀x ¬φN) and otherwise • (φ ∧ θ)N is φN ∧ θN • (φ → θ)N is φN → θN • (¬φ)N is ¬φN • (∀x φ)N is ∀x φN This translation has the property that φN is classically equivalent to φ. But in intuitionistic first-order logic, neither direction is necessarily provable. Troelstra and van Dalen (1988, Ch. 2, Sec. 3) give a description, due to Leivant, of formulas that do imply their Gödel–Gentzen translation. Equivalent variants Due to constructive equivalences, there are several alternative definitions of the translation. For example, a valid De Morgan's law allows one to rewrite a negated disjunction. One possibility can thus succinctly be described as follows: Prefix "¬¬" before every atomic formula, but also to every disjunction and existential quantifier, • (φ ∨ θ)N is ¬¬(φN ∨ θN) • (∃x φ)N is ¬¬∃x φN Another procedure, known as Kuroda's translation, is to construct a translated φ by putting "¬¬" before the whole formula and after every universal quantifier. This procedure exactly reduces to the propositional translation whenever φ is propositional. Thirdly, one may instead prefix "¬¬" before every subformula of φ, as done by Kolmogorov. Such a translation is the logical counterpart to the call-by-name continuation-passing style translation of functional programming languages along the lines of the Curry–Howard correspondence between proofs and programs. The Gödel-Gentzen- and Kuroda-translated formulas of each φ are provenly equivalent, and this result holds already in minimal propositional logic. And further, in intuitionistic propositional logic, the Kuroda- and Kolmogorov-translated formulas are equivalent also. The mere propositional mapping of φ to ¬¬φ does not extend to a sound translation of first-order logic, as the so called double negation shift ∀x ¬¬φ(x) → ¬¬∀x φ(x) is not a theorem of intuitionistic predicate logic. So the negations in φN have to be placed in a more particular way. Results Let TN consist of the double-negation translations of the formulas in T. The fundamental soundness theorem (Avigad and Feferman 1998, p. 342; Buss 1998 p. 66) states: If T is a set of axioms and φ is a formula, then T proves φ using classical logic if and only if TN proves φN using intuitionistic logic. Arithmetic The double-negation translation was used by Gödel (1933) to study the relationship between classical and intuitionistic theories of the natural numbers ("arithmetic"). He obtains the following result: If a formula φ is provable from the axioms of Peano arithmetic then φN is provable from the axioms of Heyting arithmetic. This result shows that if Heyting arithmetic is consistent then so is Peano arithmetic. This is because a contradictory formula θ ∧ ¬θ is interpreted as θN ∧ ¬θN, which is still contradictory. Moreover, the proof of the relationship is entirely constructive, giving a way to transform a proof of θ ∧ ¬θ in Peano arithmetic into a proof of θN ∧ ¬θN in Heyting arithmetic. By combining the double-negation translation with the Friedman translation, it is in fact possible to prove that Peano arithmetic is Π02-conservative over Heyting arithmetic. See also • Dialectica interpretation References • J. Avigad and S. Feferman (1998), "Gödel's Functional ("Dialectica") Interpretation", Handbook of Proof Theory, S. Buss, ed., Elsevier. ISBN 0-444-89840-9 • S. Buss (1998), "Introduction to Proof Theory", Handbook of Proof Theory, S. Buss, ed., Elsevier. ISBN 0-444-89840-9 • G. Gentzen (1936), "Die Widerspruchfreiheit der reinen Zahlentheorie", Mathematische Annalen, v. 112, pp. 493–565 (German). Reprinted in English translation as "The consistency of arithmetic" in The collected papers of Gerhard Gentzen, M. E. Szabo, ed. • V. Glivenko (1929), Sur quelques points de la logique de M. Brouwer, Bull. Soc. Math. Belg. 15, 183-188 • K. Gödel (1933), "Zur intuitionistischen Arithmetik und Zahlentheorie", Ergebnisse eines mathematischen Kolloquiums, v. 4, pp. 34–38 (German). Reprinted in English translation as "On intuitionistic arithmetic and number theory" in The Undecidable, M. Davis, ed., pp. 75–81. • A. N. Kolmogorov (1925), "O principe tertium non datur" (Russian). Reprinted in English translation as "On the principle of the excluded middle" in From Frege to Gödel, van Heijenoort, ed., pp. 414–447. • A. S. Troelstra (1977), "Aspects of Constructive Mathematics", Handbook of Mathematical Logic, J. Barwise, ed., North-Holland. ISBN 0-7204-2285-X • A. S. Troelstra and D. van Dalen (1988), Constructivism in Mathematics. An Introduction, volumes 121, 123 of Studies in Logic and the Foundations of Mathematics, North–Holland. External links • "Intuitionistic logic", Stanford Encyclopedia of Philosophy.	Wikipedia
Irreducible holomorphic symplectic (IHS) manifolds are a higher dimensional analogue of K3 surfaces; if $X$ is such a manifold, we can define a quadratic form on $H^2(X,\mathbb Z)$ that bears a formal resemblance to the intersection product on a surface. A birational transformation $f$ of a manifold $X$ is said "imprimitive" if it preserves the fibres of a non-trivial fibration $\pi\colon X\dashrightarrow B$. Analogously to the surface case (Gizatullin), I will show that, if $X$ is IHS and $f$ induces a linear automorphism of $H^2(X,\mathbb Z)$ with at least an eigenvalue with modulus different then $1$, then it is primitive.	CommonCrawl
Smooth structure In mathematics, a smooth structure on a manifold allows for an unambiguous notion of smooth function. In particular, a smooth structure allows one to perform mathematical analysis on the manifold.[1] Definition A smooth structure on a manifold $M$ is a collection of smoothly equivalent smooth atlases. Here, a smooth atlas for a topological manifold $M$ is an atlas for $M$ such that each transition function is a smooth map, and two smooth atlases for $M$ are smoothly equivalent provided their union is again a smooth atlas for $M.$ This gives a natural equivalence relation on the set of smooth atlases. A smooth manifold is a topological manifold $M$ together with a smooth structure on $M.$ Maximal smooth atlases By taking the union of all atlases belonging to a smooth structure, we obtain a maximal smooth atlas. This atlas contains every chart that is compatible with the smooth structure. There is a natural one-to-one correspondence between smooth structures and maximal smooth atlases. Thus, we may regard a smooth structure as a maximal smooth atlas and vice versa. In general, computations with the maximal atlas of a manifold are rather unwieldy. For most applications, it suffices to choose a smaller atlas. For example, if the manifold is compact, then one can find an atlas with only finitely many charts. Equivalence of smooth structures Let $\mu $ and $\nu $ be two maximal atlases on $M.$ The two smooth structures associated to $\mu $ and $\nu $ are said to be equivalent if there is a diffeomorphism $f:M\to M$ such that $\mu \circ f=\nu .$ Exotic spheres John Milnor showed in 1956 that the 7-dimensional sphere admits a smooth structure that is not equivalent to the standard smooth structure. A sphere equipped with a nonstandard smooth structure is called an exotic sphere. E8 manifold The E8 manifold is an example of a topological manifold that does not admit a smooth structure. This essentially demonstrates that Rokhlin's theorem holds only for smooth structures, and not topological manifolds in general. Related structures The smoothness requirements on the transition functions can be weakened, so that we only require the transition maps to be $k$-times continuously differentiable; or strengthened, so that we require the transition maps to be real-analytic. Accordingly, this gives a $C^{k}$ or (real-)analytic structure on the manifold rather than a smooth one. Similarly, we can define a complex structure by requiring the transition maps to be holomorphic. See also • Smooth frame – Generalization of an ordered basis of a vector spacePages displaying short descriptions of redirect targets • Atlas (topology) – Set of charts that describes a manifold References 1. Callahan, James J. (1974). "Singularities and plane maps". Amer. Math. Monthly. 81: 211–240. doi:10.2307/2319521. • Hirsch, Morris (1976). Differential Topology. Springer-Verlag. ISBN 3-540-90148-5. • Lee, John M. (2006). Introduction to Smooth Manifolds. Springer-Verlag. ISBN 978-0-387-95448-6. • Sepanski, Mark R. (2007). Compact Lie Groups. Springer-Verlag. ISBN 978-0-387-30263-8. Manifolds (Glossary) Basic concepts • Topological manifold • Atlas • Differentiable/Smooth manifold • Differential structure • Smooth atlas • Submanifold • Riemannian manifold • Smooth map • Submersion • Pushforward • Tangent space • Differential form • Vector field Main results (list) • Atiyah–Singer index • Darboux's • De Rham's • Frobenius • Generalized Stokes • Hopf–Rinow • Noether's • Sard's • Whitney embedding Maps • Curve • Diffeomorphism • Local • Geodesic • Exponential map • in Lie theory • Foliation • Immersion • Integral curve • Lie derivative • Section • Submersion Types of manifolds • Closed • (Almost) Complex • (Almost) Contact • Fibered • Finsler • Flat • G-structure • Hadamard • Hermitian • Hyperbolic • Kähler • Kenmotsu • Lie group • Lie algebra • Manifold with boundary • Oriented • Parallelizable • Poisson • Prime • Quaternionic • Hypercomplex • (Pseudo−, Sub−) Riemannian • Rizza • (Almost) Symplectic • Tame Tensors Vectors • Distribution • Lie bracket • Pushforward • Tangent space • bundle • Torsion • Vector field • Vector flow Covectors • Closed/Exact • Covariant derivative • Cotangent space • bundle • De Rham cohomology • Differential form • Vector-valued • Exterior derivative • Interior product • Pullback • Ricci curvature • flow • Riemann curvature tensor • Tensor field • density • Volume form • Wedge product Bundles • Adjoint • Affine • Associated • Cotangent • Dual • Fiber • (Co) Fibration • Jet • Lie algebra • (Stable) Normal • Principal • Spinor • Subbundle • Tangent • Tensor • Vector Connections • Affine • Cartan • Ehresmann • Form • Generalized • Koszul • Levi-Civita • Principal • Vector • Parallel transport Related • Classification of manifolds • Gauge theory • History • Morse theory • Moving frame • Singularity theory Generalizations • Banach manifold • Diffeology • Diffiety • Fréchet manifold • K-theory • Orbifold • Secondary calculus • over commutative algebras • Sheaf • Stratifold • Supermanifold • Stratified space	Wikipedia
\begin{document} \title{Descents on quasi-Stirling permutations} \begin{abstract} Stirling permutations were introduced by Gessel and Stanley~\cite{gessel_stirling_1978}, who used their enumeration by the number of descents to give a combinatorial interpretation of certain polynomials related to Stirling numbers. Quasi-Stirling permutations, which can be viewed as labeled noncrossing matchings, were introduced by Archer et al.\ \cite{archer_pattern_2019} as a natural extension of Stirling permutations. Janson's correspondence~\cite{janson_plane_2008} between Stirling permutations and labeled increasing plane trees extends to a bijection between quasi-Stirling permutations and the same set of trees without the increasing restriction. Archer et al.~\cite{archer_pattern_2019} posed the problem of enumerating quasi-Stirling permutations by the number of descents, and conjectured that there are $(n+1)^{n-1}$ such permutations of size $n$ having the maximum number of descents. In this paper we prove their conjecture, and we give the generating function for quasi-Stirling permutations by the number of descents, expressed as a compositional inverse of the generating function of Eulerian polynomials. We also find the analogue for quasi-Stirling permutations of the main result from~\cite{gessel_stirling_1978}. We prove that the distribution of descents on these permutations is asymptotically normal, and that the roots of the corresponding quasi-Stirling polynomials are all real, in analogy to B\'ona's results for Stirling permutations~\cite{bona_real_2008}. Finally, we generalize our results to a one-parameter family of permutations that extends $k$-Stirling permutations, and we refine them by also keeping track of the number of ascents and the number of plateaus. \end{abstract} \section{Introduction} \subsection{Stirling permutations} In 1978, Gessel and Stanley~\cite{gessel_stirling_1978} introduced the set $\mathcal{Q}_n$ of Stirling permutations. They are defined as those permutations $\pi_1\pi_2\dots\pi_{2n}$ of the multiset $\{1,1,2,2,\dots,n,n\}$ satisfying that, if $i<j<k$ and $\pi_i=\pi_k$, then $\pi_j>\pi_i$. In pattern avoidance terminology, we can describe this condition as avoiding the pattern $212$. In general, given two sequences of positive integers $\pi=\pi_1\pi_2\dots\pi_r$ and $\sigma=\sigma_1\sigma_2\dots\sigma_s$, we say that $\pi$ avoids $\sigma$ if there is no subsequence $\pi_{i_1}\pi_{i_2}\dots \pi_{i_s}$ (with $i_1<i_2<\dots<i_s$) whose entries are in the same relative order as $\sigma_1\sigma_2\dots\sigma_s$. Using the notation $[r]=\{1,2,\dots,r\}$, define $i\in[r]$ to be a {\em descent} of $\pi=\pi_1\pi_2\dots\pi_r$ if $\pi_i>\pi_{i+1}$ or $i=r$, and let $\des(\pi)$ denote the number of descents of $\pi$. This is the same definition used in~\cite{gessel_stirling_1978,bona_real_2008,janson_plane_2008,janson_generalized_2011}, even though other papers, such as~\cite{archer_pattern_2019}, do not consider $i=r$ to be a descent. Descents are closely related to {\em ascents}, which are indices $i\in\{0,\dots,r-1\}$ such that $\pi_i<\pi_{i+1}$ or $i=0$, and to {\em plateaus}, which are indices $i\in[r-1]$ such that $\pi_i=\pi_{i+1}$. Let $\asc(\pi)$ and $\plat(\pi)$ denote the number of ascents and the number of plateaus of $\pi$, respectively. Denoting by $\mathcal{S}_n$ the set of permutations of $[n]$, the polynomials \begin{equation}\label{eq:Eulerian_def} A_n(t)=\sum_{\pi\in\mathcal{S}_n} t^{\des(\pi)} \end{equation} are called {\em Eulerian polynomials}. It is well known (see for example \cite[Prop.\ 1.4.4]{stanley_enumerative_2012}) that \begin{equation}\label{eq:Eulerian} \sum_{m\ge0} m^n t^m=\frac{A_n(t)}{(1-t)^{n+1}}, \end{equation} and in fact this formula is often used as the definition of Eulerian polynomials. Gessel and Stanley~\cite{gessel_stirling_1978} show that, when replacing the coefficients in the left-hand side of Equation~\eqref{eq:Eulerian} by Stirling numbers of the second kind, then the role of the Eulerian polynomials is played by the {\em Stirling polynomials} $$Q_n(t)=\sum_{\pi\in\mathcal{Q}_n} t^{\des(\pi)},$$ which count Stirling permutations by the number of descents. Specifically, denoting by $S(n,m)$ the number of partitions of an $n$-element set into $m$ blocks, they prove the following. \begin{theorem}[\cite{gessel_stirling_1978}]\label{thm:GS} $$\sum_{m\ge0} S(m+n,m)\, t^m=\frac{Q_n(t)}{(1-t)^{2n+1}}.$$ \end{theorem} It follows, in particular, that $\|\mathcal{Q}_n\|=(2n-1)!!=(2n-1)\cdot(2n-3)\cdot\dots\cdot 3\cdot 1$. There is an extensive literature on Stirling permutations and their generalizations. B\'ona~\cite{bona_real_2008} showed that the distribution of plateaus on $\mathcal{Q}_n$ is also given by the polynomial $Q_n(t)$, that this polynomial has only real roots (this had also been proved by Brenti~\cite[Thm.\ 6.6.3]{brenti_unimodal_1989}), and that this distribution converges to a normal distribution. More generally, Janson~\cite{janson_plane_2008} showed that the joint distribution ascents, descents and plateaus is asymptotically normal, and Haglund and Visontai~\cite{haglund_stable_2012} proved the stability of the corresponding multivariate polynomials. Gessel and Stanley \cite{gessel_stirling_1978} proposed an extension of Stirling permutations by allowing $k$ copies of each element in $[n]$. These permutations were studied by Brenti~\cite{brenti_unimodal_1989} in an even more general setting, proving real-rootedness of their descent polynomials; by Park~\cite{park_r-multipermutations_1994,park_inverse_1994}, who studied the distribution of various statistics on them; and by Janson, Kuba and Panholzer~\cite{janson_generalized_2011}, who proved a joint normal law for ascents, descents and plateaus. Other generalizations have been studied by Barbero et al.~\cite{barbero_g._generalized_2015}. \subsection{Quasi-Stirling permutations} In~\cite{archer_pattern_2019}, Archer, Gregory, Pennington and Slayden introduce the set $\overline{\mathcal{Q}}_n$ of {\em quasi-Stirling} permutations. These are permutations $\pi_1\pi_2\dots\pi_{2n}$ of the multiset $\{1,1,2,2,\dots,n,n\}$ avoiding $1212$ and $2121$; that is, those for which there do not exist $i<j<k<\ell$ such that $\pi_i=\pi_k$ and $\pi_j=\pi_\ell$. Thinking of $\pi$ as a labeled matching of $[2n]$, by placing an arc between with label $k$ between $i$ with $j$ if $\pi_i=\pi_j=k$, the avoidance requirement is equivalent to the matching being {\em noncrossing} (see~\cite[Exercise 6.19(o)]{stanley_enumerative_1999}). By definition, $\mathcal{Q}_n\subseteq\overline{\mathcal{Q}}_n$. Archer et al.~\cite{archer_pattern_2019} note that \begin{equation}\label{eq:nCat} \|\overline{\mathcal{Q}}_n\|=n!\,C_n=\frac{(2n)!}{(n+1)!},\quad \text{where }C_n=\frac{1}{n+1}\binom{2n}{n} \end{equation} is the $n$th Catalan number. They also compute the number of permutations in $\overline{\mathcal{Q}}_n$ avoiding some sets of patterns of length 3, and they enumerate quasi-Stirling permutations by the number of plateaus. They pose the open problem of enumerating quasi-Stirling permutations by the number of descents, and they conjecture the following intriguing formula\footnote{The statement of the conjecture in~\cite{archer_pattern_2019} mentions permutations with $n-1$ descents, since their definition does not consider the last position $2n$ to be a descent.}. \begin{conjecture}[\cite{archer_pattern_2019}]\label{conj:archer} The number of $\pi\in\overline{\mathcal{Q}}_n$ with $\des(\pi)=n$ is equal to $(n+1)^{n-1}$. \end{conjecture} \subsection{Structure of the paper} In Section~\ref{sec:des} we prove Conjecture~\ref{conj:archer}, stated as Theorem~\ref{thm:desn}. More generally, in Theorem~\ref{thm:main}, we describe the generating function enumerating quasi-Stirling permutations by the number of descents. An analogue of Theorem~\ref{thm:GS} for quasi-Stirling permutations is given in Theorem~\ref{thm:QQn}. In Section~\ref{sec:properties} we study some properties of the distribution of descents on quasi-Stirling permutations, analogous to those studied by B\'ona~\cite{bona_real_2008} for Stirling permutations. We show that the corresponding polynomials have real roots only, and that the distribution of descents is asymptotically normal. In Section~\ref{sec:k} we consider an extension of quasi-Stirling permutations by allowing $k$ copies of each element in $[n]$, in analogy to $k$-Stirling permutations. We generalize the results from Section~\ref{sec:des} to this setting, and we refine them by considering the joint distribution of the number of ascents, the number of descents, and the number of plateaus on our generalized quasi-Stirling permutations. Finally, we give a simple description of the joint distribution of the same statistics on $k$-Stirling permutations. It is worth pointing out that Stirling permutations (and, more generally, $k$-Stirling permutations) have a simple recursive description, since elements in $\mathcal{Q}_n$ can be obtained by inserting the adjacent pair $nn$ into elements in $\mathcal{Q}_{n-1}$. This fact is repeatedly used in~\cite{gessel_stirling_1978,bona_real_2008,janson_plane_2008}, and it greatly simplifies the enumeration of Stirling permutations by certain statistics. For example, it yields a recurrence for the number of Stirling permutations with a given number of descents or plateaus~\cite{bona_real_2008}, and it enables a probabilistic proof of the symmetry of the numbers of ascents, descents and plateaus~\cite{janson_plane_2008}. Unfortunately, quasi-Stirling permutations do not have such a simple recursive description (in fact, the quotient $\|\overline{\mathcal{Q}}_{n}\|/\|\overline{\mathcal{Q}}_{n-1}\|=2n(2n-1)/(n+1)$ is not an integer in general), which makes their enumeration with respect to the number of descents more challenging. Nevertheless, we are still able to find analogues for quasi-Stirling permutations of most of the results from~\cite{gessel_stirling_1978} and \cite{bona_real_2008}. \subsection{A bijection to plane trees}\label{sec:bij} The original motivation for considering quasi-Stirling permutations in~\cite{archer_pattern_2019} is that they are in bijection with labeled plane rooted trees, in much the same way that Stirling permutations are in bijection with increasing trees. Next we describe these bijections. Denote by $\mathcal{T}_n$ the set of edge-labeled plane (i.e., ordered) rooted trees with $n$ edges. Each edge of such a tree receives a unique label from $[n]$. The {\em root} is a distinguished vertex of the tree, which we place at the top. The {\em children} of a vertex $i$ are the neighbors of $i$ that are not in the path from $i$ to the root; the neighbor of $i$ in the path to the root (if $i$ is not the root) is called the {\em parent} of $i$. The children of $i$ are placed below $i$, and the left-to-right order in which they are placed matters. Vertices with no children are called {\em leaves}. Disregarding the labels, it is well known that the number of unlabeled plane rooted trees with $n$ edges is $C_n$. Since there are $n!$ ways to label the edges of a particular tree, it follows that $\|\mathcal{T}_n\|=n!\,C_n$. Denote by $\mathcal{I}_n\subseteq\mathcal{T}_n$ be the subset of those trees whose labels along any path from the root to a leaf are increasing. Elements of $\mathcal{I}_n$ are called edge-labeled increasing plane trees, or simply increasing trees when there is no confusion. A simple bijection between $\mathcal{I}_n$ and $\mathcal{Q}_n$ was given by Janson in~\cite{janson_plane_2008}. Archer at al.~\cite{archer_pattern_2019} showed that this bijection naturally extends to a bijection $\varphi$ between $\mathcal{T}_n$ and $\overline{\mathcal{Q}}_n$. Both bijections can be described as follows. Given a tree $T\in\mathcal{T}_n$, traverse its edges by following a depth-first walk from left to right (i.e., counterclockwise); that is, start at the root, go to the leftmost child and explore that branch recursively, return to the root, then continue to the next child, and so on (see \cite[Fig.\ 5-14]{stanley_enumerative_1999} for a visual description). Recording the labels of the edges as they are traversed gives a permutation $\varphi(T)\in\overline{\mathcal{Q}}_n$; see Figure~\ref{fig:varphi} for an example. Note that each edge is traversed twice, once in each direction. As shown in~\cite{archer_pattern_2019}, the map $\varphi:\mathcal{T}_n\to\overline{\mathcal{Q}}_n$ is a bijection. Additionally, the image of the subset of increasing trees is precisely the set of Stirling permutations, and so $\varphi$ induces a bijection between $\mathcal{I}_n$ and $\mathcal{Q}_n$, which is the map described in~\cite{janson_plane_2008}. \begin{figure}\label{fig:varphi} \end{figure} \section{Descents on quasi-Stirling permutations}\label{sec:des} In order to enumerate quasi-Stirling permutations by the number of descents, let us first analyze how descents are transformed by the bijection $\varphi$. Define the number of {\em cyclic descents} of a sequence of positive integers $\pi=\pi_1\pi_2\dots\pi_r$ to be \begin{equation}\label{eq:cdes_def} \cdes(\pi)=\|\{i\in[r]: \pi_i>\pi_{i+1}\}\|, \end{equation} with the convention $\pi_{r+1}:=\pi_1$. Note that rotating the entries of $\pi$ does not change the number of cyclic descents, that is, $\cdes(\pi_{i+1}\dots\pi_r\pi_1\dots\pi_i)=\cdes(\pi)$ for all $i\in[r]$. Let $T\in\mathcal{T}_n$, and let $v$ a vertex of $T$. If $v$ is not the root, define $\cdes(v)$ to be the number of cyclic descents of the sequence obtained by listing the labels of the edges incident to $v$ in counterclockwise order (note that the starting point is irrelevant). Equivalently, if the label of the edge between $v$ and its parent is $\ell$, and the labels of the edges between $v$ and its children are $a_1,a_2,\dots,a_{d}$ from left to right, then $\cdes(v)=\cdes(\ell a_1\dots a_d)$. If $v$ is the root of $T$, define $\cdes(v)$ to be the number of descents of the sequence obtained by listing the labels of the edges incident to $v$ from left to right, that is, $\cdes(v)=\des(a_1\dots a_d)$ with the above notation. Finally, define the number of cyclic descents of $T$ to be $$\cdes(T)=\sum_v \cdes(v),$$ where the sum ranges over all the vertices $v$ of $T$. For example, if $T$ is the tree on the left of Figure~\ref{fig:varphi}, then $\cdes(T)=2+1+0+0+2+0+1+0+0=6$, where the two vertices with $\cdes(v)=2$ are the root and the other vertex with 3 children. \begin{lemma}\label{lem:cdes} The bijection $\varphi:\mathcal{T}_n\to\overline{\mathcal{Q}}_n$ has the following property: if $T\in\mathcal{T}_n$ and $\pi=\varphi(T)\in\overline{\mathcal{Q}}_n$, then $$\des(\pi)=\cdes(T).$$ \end{lemma} \begin{proof} In the counterclockwise depth-first walk performed on $T$ to obtain $\pi$, suppose that the $i$th step of the walk traverses edge $e$ in the direction from vertex $u$ to vertex $v$, and let $\ell$ be its label. Then $i$ is a descent of $\pi$ if and only if one of the following holds: \begin{itemize} \item $u$ is the parent of $v$, and $\ell$ is larger than the label of the edge between $v$ and its leftmost child; \item $u$ is a child of $v$ (but not its rightmost child), and $\ell$ is larger than the label of the edge between $v$ and its next child in the left-right order; \item $u$ is the rightmost child of $v$, and either $v$ is the root or $T$, or $\ell$ is larger than the label of the edge between $v$ and its parent. \end{itemize} By definition, $\cdes(v)$ counts the number of times that $v$ is involved in one of these scenarios at some point in the depth-first walk on $T$, and so $\cdes(T)$ equals the total number of descents of~$\pi$. \end{proof} Another property of $\varphi$ that follows easily from its description, as noted in~\cite{archer_pattern_2019}, is that plateaus of quasi-Stirling permutations correspond to leaves of edge-labeled plane rooted trees. Specifically, denoting by $\lea(T)$ the number of leaves of $T\in\mathcal{T}_n$, and letting $\pi=\varphi(T)\in\overline{\mathcal{Q}}_n$, we have that \begin{equation}\label{eq:lea} \plat(\pi)=\lea(T). \end{equation} \subsection{Quasi-Stirling permutations with most descents} It follows from Lemma~\ref{lem:cdes} that the maximum value that $\des(\pi)$ can attain for $\pi\in\overline{\mathcal{Q}}_n$ is~$n$. To see this, note that if $\pi=\varphi(T)$ and $v$ is a vertex of $T$, then $\cdes(v)$ is bounded from above by the number of children of $v$, which we denote by $d(v)$. Summing over all the vertices $v$ of $T$, we get $\des(\pi)=\cdes(T)=\sum_{v} \cdes(v) \le \sum_{v} d(v)=n$, the number of edges of $T$. This bound is attained, for example, by the permutation $\pi=12\dots nn\dots 21\in\overline{\mathcal{Q}}_n$. Next we count how many permutations attain this upper bound, proving Conjecture~\ref{conj:archer}. \begin{theorem}\label{thm:desn} The number of $\pi\in\overline{\mathcal{Q}}_n$ with $\des(\pi)=n$ is equal to $(n+1)^{n-1}$. \end{theorem} \begin{proof} By Lemma~\ref{lem:cdes}, the problem is equivalent to counting the number of trees $T\in\mathcal{T}_n$ such that $\cdes(T)=n$. As discussed above, $\cdes(T)=n$ if and only if $\cdes(v)=d(v)$ for every vertex $v$ of $T$. Let $\mathcal{T}^{\max}_n\subseteq\mathcal{T}_n$ be the subset of trees satisfying this condition. If $v$ is the root of $T\in\mathcal{T}_n$, then $\cdes(v)=d(v)$ precisely when the labels of the edges incident to $v$ decrease from left to right. If $v$ is a non-root vertex, then $\cdes(v)=d(v)$ if and only if the labels of the edges incident to $v$ decrease when read counterclockwise starting from the largest label. Let $\mathcal{U}_n$ be the set of edge-labeled {\em unordered} rooted trees with $n$ edges. The difference with $\mathcal{T}_n$ is that, for trees in $\mathcal{U}_n$, the order of the children of a vertex is irrelevant; that is, trees are determined by their combinatorial structure and not by their particular embedding on the plane. It is known that $\|\mathcal{U}_n\|=(n+1)^{n-1}$. Indeed, transferring edge labels to vertex labels by moving each label to the endpoint away from the root, and labeling the root with $n+1$, trees in $\mathcal{U}_n$ are in bijection with vertex-labeled unordered unrooted trees on $n+1$ vertices (an example of this bijection appears on the left of Figure~\ref{fig:unordered}). By Cayley's formula, the number of such trees is $(n+1)^{n-1}$. Thus, it suffices to describe a bijection between $\mathcal{T}^{\max}_n$ and $\mathcal{U}_n$. Given a tree in $\mathcal{T}^{\max}_n$, the corresponding unordered tree is obtained simply by forgetting the order of the children of each vertex. Conversely, given a tree in $\mathcal{U}_n$, the unique tree in $\mathcal{T}^{\max}_n$ with the same combinatorial structure is obtained as follows. First, place the root and its children so that the labels of the corresponding edges decrease from left to right. Then, for each placed vertex $v$, recursively place its children in the only possible order that yields $\cdes(v)=d(v)$. Specifically, writing $d=d(v)$, suppose that the labels between $v$ and its children are $a_1,a_2,\dots,a_d$ in increasing order, that the label between $v$ and its parent is $a$, and that $0\le i\le d$ is the index such that $$a_1<a_2<\dots<a_i<a<a_{i+1}<\dots<a_d.$$ Then place the children of $v$ so that the edge labels are $a_i,a_{i-1},\dots,a_1,a_d,a_{d-1},\dots,a_{i+1}$ from left to right. This guarantees that $\cdes(aa_ia_{i-1}\dots a_1a_da_{d-1}\dots a_{i+1})=d$. An example of this bijection is shown on the right of Figure~\ref{fig:unordered}. \end{proof} \begin{figure} \caption{The bijections in the proof of Theorem~\ref{thm:desn}: a vertex-labeled unordered unrooted tree (left), its corresponding edge-labeled unordered rooted tree (center), and its corresponding edge-labeled plane rooted tree with maximum number of descents (right).} \label{fig:unordered} \end{figure} \subsection{A generating function for the number of descents} Denote by $$A(t,z)=\sum_{n\ge0} A_n(t) \frac{z^n}{n!}$$ the exponential generating function (EGF for short) of the Eulerian polynomials, defined in Equation~\eqref{eq:Eulerian}. It is well known \cite[Prop.\ 1.4.5]{stanley_enumerative_2012} that \begin{equation}\label{eq:A_GF} A(t,z)=\frac{1-t}{1-te^{(1-t)z}}. \end{equation} In analogy to $A_n(t)$ and $Q_n(t)$, define the {\em quasi-Stirling polynomials} \begin{equation}\label{eq:defoQn} \overline{Q}_n(t)=\sum_{\pi\in\overline{\mathcal{Q}}_n} t^{\des(\pi)}, \end{equation} and their EGF $$\overline{Q}(t,z)=\sum_{n\ge0}\overline{Q}_n(t)\frac{z^n}{n!}.$$ The first few quasi-Stirling polynomials are \begin{align} &\overline{Q}_1(t)=t,\\ &\overline{Q}_2(t)=t+3\,{t}^{2},\\ &\overline{Q}_3(t)=t+13\,{t}^{2}+16\,{t}^{3}, \\ &\overline{Q}_4(t)=t+39\,{t}^{2}+171\,{t}^{3}+125\,{t}^{4},\\ &\overline{Q}_5(t)=t+101\,{t}^{2}+1091\,{t}^{3}+2551\,{t}^{4}+1296\,{t}^{5},\\ &\overline{Q}_6(t)=t+243\,{t}^{2}+5498\,{t}^{3}+28838\,{t}^{4}+43653\,{t}^{5}+16807\,{t}^{6},\\ &\overline{Q}_7(t)=t+561\,{t}^{2}+24270\,{t}^{3}+243790\,{t}^{4}+780585\,{t}^{5}+850809\,{t}^{6}+262144\,{t}^{7}. \end{align} The main result in this section is an equation that describes $\overline{Q}(t,z)$, and allows us to compute the quasi-Stirling polynomials. The notation $[z^n]F(z)$ refers to the coefficient of $z^n$ in the generating function $F(z)$. \begin{theorem}\label{thm:main} The EGF of quasi-Stirling permutations by the number of descents satisfies the implicit equation $\overline{Q}(t,z)=A(t,z\overline{Q}(t,z))$, that is, \begin{equation}\label{eq:oQ} \overline{Q}(t,z)=\frac{1-t}{1-te^{(1-t)z\overline{Q}(t,z)}}. \end{equation} In particular, its coefficients satisfy \begin{equation}\label{eq:Lagrange} \overline{Q}_n(t)=\frac{n!}{n+1}\,[z^n]A(t,z)^{n+1}. \end{equation} \end{theorem} Before proving this theorem, note that if we ignore descents by setting $t=1$, then $A(1,t)=\frac{1}{1-z}$. In this case, Theorem~\ref{thm:main} simply states that $$\overline{Q}(1,z)=A(1,z\overline{Q}(t,z))=\frac{1}{1-z\overline{Q}(1,z)}.$$ It follows that \begin{equation}\label{eq:Catalan} \overline{Q}(1,z)=\frac{1-\sqrt{1-4z}}{2z}, \end{equation} the ordinary generating function for the Catalan numbers, and so $\|\overline{\mathcal{Q}}_n\|=n!C_n$, which agrees with Equation~\eqref{eq:nCat}. Equation~\eqref{eq:Lagrange} in this case states that $$\overline{Q}_n(1)=\frac{n!}{n+1}\,[x^{n}]\frac{1}{(1-x)^{n+1}}=\frac{n!}{n+1}\binom{2n}{n}=n!C_n.$$ It is also worth noting that a non-bijective proof of Theorem~\ref{thm:desn} can be deduced from Theorem~\ref{thm:main}, by noting that the number of $\pi\in\overline{\mathcal{Q}}_n$ with $n$ descents is $$[t^n]\overline{Q}_n(t)=\frac{n!}{n+1}\,[t^nz^n]A(t,z)^{n+1}=\frac{n!}{n+1}\,[t^nz^n]\left(\sum_{n\ge0}\frac{t^nz^n}{n!}\right)^{n+1}=\frac{n!}{n+1}\,[t^nz^n]e^{(n+1)tz}=(n+1)^{n-1}.$$ By Lemma~\ref{lem:cdes}, $\overline{Q}(t,z)$ is also the EGF for edge-labeled plane rooted trees by the number of cyclic descents, that is, \begin{equation}\label{eq:oQT} \overline{Q}_n(t)=\sum_{T\in\mathcal{T}_n} t^{\cdes(T)} \quad \text{and}\quad \overline{Q}(t,z)=\sum_{n\ge0} \sum_{T\in\mathcal{T}_n} t^{\cdes(T)} \frac{z^n}{n!}. \end{equation} For $n\ge1$, let $\mathcal{T}'_n\subseteq\mathcal{T}_n$ be the subset of trees whose root has exactly one child, and let \begin{equation} \label{eq:Rn} R_n(t)=\sum_{T'\in\mathcal{T}'_n} t^{\cdes(T')-1}. \end{equation} The next lemma will be used in the proof of Theorem~\ref{thm:main}. \begin{lemma}\label{lem:T'} We have $$\sum_{n\ge1} R_n(t) \frac{z^n}{n!}=z\,\overline{Q}(t,z).$$ \end{lemma} \begin{proof} We define two operations on trees. For $T\in\mathcal{T}_{n-1}$, let $T^{\|n}\in\mathcal{T}'_n$ be the tree obtained from $T$ by attaching an edge with label $n$ from the root of $T$ to a new root vertex. For $T'\in\mathcal{T}'_n$ and $i\in\{0,1,\dots,n-1\}$, let $T'+i\in\mathcal{T}'_n$ be tree obtained by adding $i$ modulo $n$ to each label of $T'$, so that the resulting labels are again the numbers $1,2,\dots,n$. Next we analyze how the statistic $\cdes$ behaves under these two operations. We have that $\cdes(T^{\|n})=\cdes(T)+1$ for all $T\in\mathcal{T}_{n-1}$, since $\cdes(v)$ stays the same for each vertex $v$ of $T$, and the new root contributes one cyclic descent. On the other hand, $\cdes(T'+i)=\cdes(T')$ for all $T'\in\mathcal{T}'_n$ because, when adding $1$ modulo $n$ to each label, the relative order of the labels around a vertex $v$ does not change unless $v$ is an endpoint of the edge with label $n$ in $T'$, which has label $1$ in $T'+1$. But, even for such $v$, the value of $\cdes(v)$ does not change, because when reading the labels around $v$ in counterclockwise order, the old label $n$ was bigger than the next label, whereas the new label $1$ is smaller than the previous label. Since every tree in $T'\in\mathcal{T}'_n$ can be obtained as $T'=T^{\|n}+i$ for a unique $i\in\{0,1,\dots,n-1\}$ and a unique $T\in\mathcal{T}_{n-1}$, we have \begin{align} R_n(t)&=\sum_{T'\in\mathcal{T}'_n} t^{\cdes(T')-1}=\sum_{T\in\mathcal{T}_{n-1}} \sum_{i=0}^{n-1} t^{\cdes(T^{\|n}+i)-1}=\sum_{T\in\mathcal{T}_{n-1}} n t^{\cdes(T^{\|n})-1} =n \sum_{T\in\mathcal{T}_{n-1}} t^{\cdes(T)}\\ &=n\overline{Q}_{n-1}(t) \end{align} for every $n\ge1$, where we used Equation~\eqref{eq:oQT} in the last equality. Multiplying both sides by $z^n/n!$ and summing over $n$, we get $$\sum_{n\ge1} R_n(t) \frac{z^n}{n!}=\sum_{n\ge1} \overline{Q}_{n-1}(t) \frac{z^n}{(n-1)!}=z\,\overline{Q}(t,z).$$ \end{proof} \begin{proof}[Proof of Theorem~\ref{thm:main}] We consider a recursive description of edge-labeled plane rooted trees. Let $\mathcal{T}=\bigcup_{n\ge0}\mathcal{T}_n$ and $\mathcal{T}'=\bigcup_{n\ge1}\mathcal{T}'_n$. We think of these sets as labeled combinatorial classes in the sense of Flajolet and Sedgewick~\cite{flajolet_analytic_2009}. The EGFs for these classes with a variable $t$ keeping track of the number of cyclic descents are $\overline{Q}(t,z)$ for $\mathcal{T}$, by Equation~\eqref{eq:oQT}, and $tz\overline{Q}(t,z)$ for $\mathcal{T}'$, by Lemma~\ref{lem:T'}. Every tree in $\mathcal{T}$ can be decomposed as a sequence of trees in $\mathcal{T}'$ with a common root. Let us first consider the simplified version where we momentarily disregard the parameter $\cdes$. Recall from~\cite{flajolet_analytic_2009} that if $F(z)$ is the EGF for a labeled class, then the EGF for sequences of objects in that class is $\frac{1}{1-F(z)}$. Thus, the decomposition of trees in $\mathcal{T}$ as sequences of trees in $\mathcal{T}'$ yields the equation $$\overline{Q}(1,z)=\frac{1}{1-z\overline{Q}(1,z)}.$$ Next we incorporate the parameter $\cdes$, by analyzing how it behaves under this decomposition. Suppose that $T\in\mathcal{T}$ is obtained by attaching $r$ non-empty trees $T'_1,T'_2,\dots,T'_r\in\mathcal{T}'$ to a common root, denoted by $v_0$, and relabeling their edges with distinct labels from $1$ up to the total number of edges, so that the relative order of the labels within each tree is preserved. This relabeling does not change the value of $\cdes(v)$ at any vertex $v$ other than $v_0$. However, whereas the root of each $T'_i$ contributed $1$ to its number of cyclic descents, the contribution to $\cdes(T)$ of the common root $v_0$ equals the number of descents of the sequence of labels in $T$ of the edges incident to this vertex. Specifically, if the edges incident to $v_0$ in $T$ have labels $a_1,a_2,\dots,a_r$ from left to right, then $$\cdes(T)=\sum_{i=1}^r(\cdes(T'_i)-1)+\des(a_1a_2\dots a_r).$$ Since the labels of these $r$ subtrees of $T$ form a partition of $[n]$, and the order in which these trees are attached to $v_0$ can be any of the $r!$ permutations, it follows that $$\sum_{T\in\mathcal{T}_n} t^{\cdes(T)} = \sum_{\substack{\{B_1,B_2,\dots,B_r\}\\ \text{partition of $[n]$}}} \left(\sum_{T'_1\in\mathcal{T}'_{\|B_1\|}} t^{\cdes(T'_1)-1}\right)\cdots\left(\sum_{T'_r\in\mathcal{T}'_{\|B_r\|}} t^{\cdes(T'_r)-1}\right)\left(\sum_{\pi\in\mathcal{S}_r} t^{\des(\pi)}\right),$$ where the first sum on the right-hand side is over all partitions of $[n]$, not just those with a fixed number of blocks. Using the notation $R_n(t)$ from Equation~\eqref{eq:Rn}, we can rewrite this equation as \begin{align}\overline{Q}_n(t)&=\sum_{\substack{\{B_1,B_2,\dots,B_r\}\\ \text{partition of $[n]$}}} R_{\|B_1\|}(t) \cdots R_{\|B_r\|}(t) A_r(t)\\ &=\sum_{r=1}^n\frac{1}{r!}\sum_{\substack{b_1+\dots+b_r=n \\ b_1,\dots,b_r\ge1}}\binom{n}{b_1,\dots,b_r} R_{b_1}(t) \cdots R_{b_r}(t) A_r(t). \end{align} To turn this equality into an equation for EGFs, we use a bivariate version of the Compositional Formula (see \cite[Thm.\ 5.1.4]{stanley_enumerative_1999}). Explicitly, we multiply both sides by $z^n/n!$, sum over $n\ge0$, and apply Lemma~\ref{lem:T'}: \begin{align}\overline{Q}(t,z)=\sum_{n\ge0}\overline{Q}_n(t)\frac{z^n}{n!}&=1+\sum_{n\ge1}\sum_{r=1}^n\frac{1}{r!}\sum_{\substack{b_1+\dots+b_r=n\\ b_1,\dots,b_r\ge1}} \frac{1}{b_1!\cdots b_r!} R_{b_1}(t) \cdots R_{b_r}(t) A_r(t) z^n\\ &=1+\sum_{r\ge1}\frac{1}{r!}\sum_{b_1,\dots,b_r\ge1} R_{b_1}(t)\frac{z^{b_1}}{b_1!} \cdots R_{b_r}(t)\frac{z^{b_r}}{b_r!} A_r(t)\\ &=1+\sum_{r\ge1} \frac{1}{r!}(z\overline{Q}(t,z))^r A_r(t) \\ &=A(t,z\overline{Q}(t,z)), \end{align} proving Equation~\eqref{eq:oQ}. Next, we extract the coefficients of $\overline{Q}(t,z)$. The generating function $F(t,z):=z\overline{Q}(t,z)$ satisfies the equation $$F(t,z)=zA(t,F(t,z)).$$ Thus, by the Lagrange's inversion formula (see for example \cite[Thm.\ 5.4.2]{stanley_enumerative_1999}), we have $$[z^n]F(t,z)=\frac{1}{n}\,[z^{n-1}]A(t,z)^n,$$ so $$\overline{Q}_n(t)=n!\,[z^n]\overline{Q}(t,z)=n!\,[z^{n+1}]F(t,z)=\frac{n!}{n+1}\,[z^{n}]A(t,z)^{n+1}.$$ \end{proof} Gessel and Stanley's main result from~\cite{gessel_stirling_1978} (stated above as Theorem~\ref{thm:GS}) is the analogue for Stirling polynomials of Equation~\eqref{eq:Eulerian} for Eulerian polynomials. As a consequence of Theorem~\ref{thm:main}, we obtain the following analogue for quasi-Stirling polynomials of these two results. \begin{theorem}\label{thm:QQn} $$\sum_{m\ge 0} \frac{m^n}{n+1}\binom{m+n}{m}\, t^m=\frac{\overline{Q}_n(t)}{(1-t)^{2n+1}}.$$ \end{theorem} \begin{proof} We use Equation~\eqref{eq:Lagrange} and extract the coefficient of $z^n$ in $A(t,z)^{n+1}$. By Equation~\eqref{eq:A_GF}, this expression equals $$\left(\frac{1}{1-te^{(1-t)z}}\right)^{n+1}=\sum_{m\ge0} \binom{m+n}{m}t^m e^{m(1-t)z},$$ and so $$[z^n]\left(\frac{1}{1-te^{(1-t)z}}\right)^{n+1}=\sum_{m\ge0} \binom{m+n}{m} \frac{t^m m^n(1-t)^n}{n!}.$$ Thus, by Equation~\eqref{eq:Lagrange}, $$\overline{Q}_n(t)=\frac{n!}{n+1}[z^n]\left(\frac{1-t}{1-te^{(1-t)z}}\right)^{n+1}=\frac{(1-t)^{2n+1}}{n+1}\sum_{m\ge0} \binom{m+n}{m} m^n t^m,$$ which is equivalent to the stated formula. \end{proof} \section{Properties of quasi-Stirling polynomials}\label{sec:properties} In the section, we prove some properties of the distribution of the number of descents ---as well as the number of ascents and the number of plateaus--- on quasi-Stirling permutations, in analogy with B\'ona's results for Stirling permutations~\cite{bona_real_2008}. By symmetry, the number of ascents and the number of descents are equidistributed on $\overline{\mathcal{Q}}_n$, since reversing a permutation in $\overline{\mathcal{Q}}_n$ turns ascents into descents and vice versa. Using Equation~\eqref{eq:lea}, it is shown by Archer et al.~\cite{archer_pattern_2019} that the number of elements in $\overline{\mathcal{Q}}_n$ with $m$ plateaus is $$n!N(n,m)=(n-1)!\binom{n}{m}\binom{n}{m-1},$$ where the Narayana number $N(n,m)$ is the number of unlabeled plane rooted trees with $n$ edges and $m$ leaves. The corresponding generating function is \begin{equation}\label{eq:Narayana} \sum_{n\ge0} \sum_{\pi\in\overline{\mathcal{Q}}_n} u^{\plat(\pi)} \frac{z^n}{n!} =1+\sum_{n,m\ge1} N(n,m) u^m z^n=\frac{1-(u-1)z-\sqrt{1-2(1+u)z+(1-u)^2z^2}}{2z}. \end{equation} \subsection{Average number of ascents, descents, and plateaus} B\'ona proves in~\cite[Cor.\ 1]{bona_real_2008} that Stirling permutations in $\mathcal{Q}_n$ have, on average, $(2n+1)/3$ ascents, $(2n+1)/3$ descents, and $(2n+1)/3$ plateaus. From Theorem~\ref{thm:main}, we can derive the following analogue for quasi-Stirling permutations. \begin{corollary}\label{cor:average} Let $n\ge1$. On average, elements of $\overline{\mathcal{Q}}_n$ have $(3n+1)/4$ ascents, $(3n+1)/4$ descents, and $(n+1)/2$ plateaus. \end{corollary} \begin{proof} By Equation~\eqref{eq:defoQn}, $$\frac{\overline{Q}_n'(1)}{\|\overline{\mathcal{Q}}_n\|}=\frac{\sum_{\pi\in\overline{\mathcal{Q}}_n} \des(\pi)}{\|\overline{\mathcal{Q}}_n\|}$$ is the average number of descents in elements of $\overline{\mathcal{Q}}_n$. Differentiating Equation~\eqref{eq:oQ} with respect to $t$, setting $t=1$, and solving for $\frac{\partial \overline{Q}}{\partial t}(1,z)$, we get \begin{equation}\label{eq:oQ'} \frac{\partial \overline{Q}}{\partial t}(1,z)=\frac{z\overline{Q}(1,z)(2-z\overline{Q}(1,z))}{2(1-z-2z\overline{Q}(1,z)+z^2\overline{Q}(1,z)^2)}=\frac{1}{4z}\left(\frac{1-z}{\sqrt{1-4z}}-1+z\right), \end{equation} where we have used Equation~\eqref{eq:Catalan} in the last equality. Extracting the coefficient of $z^n$ for $n\ge1$ on both sides of~\eqref{eq:oQ'}, $$\frac{\overline{Q}_n'(1)}{n!}=\frac{1}{4}\,[z^{n+1}]\frac{1-z}{\sqrt{1-4z}}=\frac{1}{4}\left(\binom{2n+2}{n+1}-\binom{2n}{n}\right)=\frac{3n+1}{4(n+1)}\binom{2n}{n}.$$ Dividing by $\|\overline{\mathcal{Q}}_n\|/n!=C_n$, we conclude that $$\frac{\overline{Q}_n'(1)}{\|\overline{\mathcal{Q}}_n\|}=\frac{3n+1}{4}.$$ The average number of descents in elements of $\overline{\mathcal{Q}}_n$ equals the average number of ascents, by symmetry. Additionally, the sum of the numbers of ascents, descents and plateaus of any given $\pi\in\overline{\mathcal{Q}}_n$ is $2n+1$, since every $i\in\{0,1,\dots,2n\}$ is an ascent, a descent or a plateau of $\pi$. It follows that the average number of plateaus is $$2n+1-2\cdot\frac{3n+1}{4}=\frac{n+1}{2}.$$ Alternatively, this average can be deduced directly from the generating function in Equation~\eqref{eq:Narayana}. \end{proof} \subsection{Real roots of quasi-Stirling polynomials and $r$-Eulerian polynomials} It is well-known result of Frobenius that the roots of the Eulerian polynomials $A_n(t)$ are real, distinct, and nonpositive. In~\cite[Thm.\ 1]{bona_real_2008}, B\'ona proves the analogous result for the Stirling polynomials $Q_n(t)$, although their real-rootedness had already been shown by Brenti~\cite[Thm.\ 6.6.3]{brenti_unimodal_1989} in more generality. In this subsection, we prove that quasi-Stirling polynomials $\overline{Q}_n(t)$ also have this property. Unlike the proofs for $A_n(t)$ and $Q_n(t)$ that give direct recurrences for these polynomials, our proof relates $\overline{Q}_n(t)$ to the so-called $r$-Eulerian polynomials. For $r\ge1$, define the number of $r$-excedances of a sequence $\pi=\pi_1\pi_2\dots\pi_s$ to be $$\exc_r(\pi)=\{i\in[s]:\pi_i\ge i+r\}.$$ In particular, we write $\exc(\pi)=\exc_1(\pi)$ to denote the number of excedances in the usual sense. Riordan~\cite{riordan_introduction_1958}, and later Foata and Sch\"utzenberger~\cite{foata_theorie_1970}, defined the polynomials $$A_{n,r}(t)=\sum_{\pi\in\mathcal{S}_n} t^{\exc_r(\pi)}.$$ For $r=1$, we have $A_{n,1}(t)=A_n(t)/t$, by the well-known fact (see \cite{foata_theorie_1970} or \cite[Prop.\ 1.4.3]{stanley_enumerative_2012}) that the number of excedances in $\mathcal{S}_n$ is equidistributed with the number of descents, if we do not consider the last position $n$ to be a descent. Let $\mathcal{J}_{n,r}$ denote the set of injections $\pi:[n-r]\to[n]$. Identifying such an injection with the sequence $\pi=\pi_1\pi_2\dots\pi_{n-r}$ of its images, define the polynomials $$J_{n,r}(t)=\sum_{\pi\in\mathcal{J}_{n,r}} t^{\exc(\pi)}.$$ For small values of $r$, these polynomials appear in \cite[A144696--A144699]{sloane_-line_nodate}. Adapting the notation, it is shown in~\cite{riordan_introduction_1958,foata_theorie_1970} that, for $r\ge1$, \begin{equation}\label{eq:IA} J_{n,r}(t)=\frac{t^{n-r}\,A_{n,r}(1/t)}{r!} \end{equation} (in particular, $J_{n,1}(t)=A_{n,1}(t)=A_n(t)/t$) and that $$\sum_{m\ge1} t\,J_{m+r-1,r}(t)\,\frac{z^m}{m!}=\frac{A(t,z)^r}{r}.$$ Setting $r=n+1$ and taking the coefficient of $z^n$, it follows from Theorem~\ref{thm:main} that \begin{equation}\label{eq:QI} \overline{Q}_n(t)=t\,J_{2n,n+1}(t). \end{equation} We are now ready to prove the real-rootedness of quasi-Stirling and $r$-Eulerian polynomials. \begin{theorem}\label{thm:roots} For every $1\le r\le n$, each of the polynomials $A_{n,r}(t)$, $J_{n,r}(t)$ and $\overline{Q}_n(t)$ has real, distinct, and nonpositive roots. \end{theorem} \begin{proof} We will prove that the polynomials \begin{equation}\label{eq:def_pnr} p_{n,r}(t):=t\,J_{n,r}(t) \end{equation} have real, distinct, and nonpositive roots. The statement for $J_{n,r}(t)$ will then follow immediately, as well as for $A_{n,r}(t)$ because, by Equation~\eqref{eq:IA}, its roots are the reciprocals of the roots of $J_{n,r}(t)$. The statement for the polynomials $\overline{Q}_n(t)$ will follow from Equation~\eqref{eq:QI}. First we claim that, for any fixed $r\ge1$, the polynomials $p_{n,r}(t)$ satisfy the recurrence \begin{equation}\label{eq:recp} p_{n,r}(t)=n\,t\,p_{{n-1},r}(t)+t(1-t)\,p_{n-1,r}'(t) \end{equation} for $n>r$, with initial condition $p_{r,r}(t)=t$. Indeed, this is a direct translation, using Equations~\eqref{eq:IA} and~\eqref{eq:def_pnr}, of the recurrence for $A_{n,r}(t)$ proved in \cite[p.\ 214]{riordan_introduction_1958}: $$A_{n,r}(t)=(r+(n-r)t)A_{n-1,r}(t)+t(1-t)A'_{n-1,r}(t).$$ Note that, aside from the initial condition, recurrence~\eqref{eq:recp} does not depend on $r$, and it extends the well-known recurrence satisfied by the Eulerian polynomials $A_n(t)=p_{n,1}(t)$. By definition, $p_{n,r}(t)$ is a polynomial of degree $n-r+1$ with a positive leading coefficient, and $0$ is one of its roots. Next we show by induction on $n$ that $p_{n,r}(t)$ has $n-r+1$ real, distinct, and nonpositive roots. This is trivially true for base case $n=r$, since $p_{r,r}(t)=t$. Let $n>r$, and suppose that $p_{n-1,r}(t)$ has $n-r$ real, distinct roots $x_1<x_2<\dots<x_{n-r}=0$. The sign of the derivative $p'_{n-1,r}(t)$ alternates on these roots; specifically, $p'_{n-1,r}(x_i)$ is positive if $i$ and $n-r$ have the same parity, and negative otherwise. Since $p_{{n-1},r}(x_i)=0$ and $1-x_i>0$ for all $i$, the same assertion applies to the sign of $n\,p_{{n-1},r}(x_i)+(1-x_i)\,p_{n-1,r}'(x_i)$. It follows that the polynomial $n\,p_{{n-1},r}(t)+(1-t)\,p_{n-1,r}'(t)$ must have a root between any pair of consecutive roots of $p_{n-1,r}(t)$, let us denote these roots by $y_1,\dots,y_{n-r-1}$ where $$x_1<y_1<x_2<y_2<\dots<y_{n-r-1}<x_{n-r}=0.$$ Using that $$p_{n,r}(t)=t\left(n\,p_{{n-1},r}(t)+(1-t)\,p_{n-1,r}'(t)\right)$$ by Equation~\eqref{eq:recp}, the polynomial $p_{n,r}(t)$ has the roots $y_1,\dots,y_{n-r-1}$, plus a root $y_{n-r}=0$. Note also that, if $n-r$ is even, then $p_{n,r}(x_1)>0$ and $\lim_{n\to-\infty}p_{n,r}(t)=-\infty$; if $n-r$ is odd, then $p_{n,r}(x_1)<0$ and $\lim_{n\to-\infty}p_{n,r}(t)=+\infty$. In both cases, $p_{n,r}(t)$ has an additional root $y_0<x_1$, for a total of $n-r+1$ roots $y_0<y_1<\dots<y_{n-r-1}<y_{n-r}=0$. \end{proof} In analogy to B\'ona's results for Stirling permutations \cite[Thm.\ 3]{bona_real_2008}, we can infer the modal number of descents in $\overline{\mathcal{Q}}_n$ from Theorem~\ref{thm:roots}. \begin{corollary} Fix $n\ge1$, and let $m$ be an index that maximizes $\|\{\pi\in\overline{\mathcal{Q}}_n:\des(\pi)=m\}\|$ (equivalently, $\|\{\pi\in\overline{\mathcal{Q}}_n:\asc(\pi)=m\}\|$). Then $$\left\|m-\frac{3n+1}{4}\right\|<1.$$ Similarly, let $m'$ be an index that maximizes $\|\{\pi\in\overline{\mathcal{Q}}_n: \plat(\pi)=m'\}\|$. Then $$\left\|m'-\frac{n+1}{2}\right\|<1.$$ \end{corollary} \begin{proof} Like B\'ona's proof of \cite[Thm.\ 3]{bona_real_2008}, our proof relies on a theorem of Darroch~\cite{darroch_distribution_1964} (see also \cite[Thm.\ 3.25]{bona_combinatorics_2012} and \cite[Prop.\ 1]{pitman_probabilistic_1997}), which implies that if $p(t)=\sum_{m=0}^n p_m t^m$ is a polynomial that has only real roots and satisfies $p(1)>0$, then an index $m$ that maximizes $p_m$ must satisfy $\|m-p'(1)/p(1)\|<1$. By Theorem~\ref{thm:roots}, the polynomial $\overline{Q}_n(t)$ has only real roots, and by Corollary~\ref{cor:average}, $$\frac{\overline{Q}_n'(1)}{\overline{Q}_n(1)} =\frac{\overline{Q}_n'(1)}{\|\overline{\mathcal{Q}}_n\|}=\frac{3n+1}{4},$$ so the first statement follows. On the other hand, it is well-known (see \cite[Thm.\ 5.3.1]{brenti_unimodal_1989} and \cite{bona_combinatorics_2012}) that the Narayana polynomials $\sum_{m=1}^n N(n,m) u^m$, which give the distribution of the number of plateaus by Equation~\eqref{eq:Narayana}, have only real roots. Thus, the statement regarding plateaus follows similarly Corollary~\ref{cor:average}. \end{proof} \subsection{Asymptotically normal distribution} Here we prove that the distribution of each of the statistics $\asc$, $\des$ and $\plat$ on quasi-Stirling permutations is asymptotically normal. We use a result of Bender, that can be stated as follows. \begin{theorem}[{\cite{bender_central_1973}, see also \cite{canfield_central_1977,bona_real_2008,pitman_probabilistic_1997}}]\label{thm:bender} Let $\{X_n\}_n$ be a sequence of random variables, where $X_n$ takes values in $[n]$. Suppose that the polynomials $g_n(t)=\sum_{m=1}^n P(X_n=m)\, t^m$ have only real roots, and that \begin{equation}\label{eq:var} \sigma_n=\sqrt{\Var(X_n)}\to\infty \end{equation} as $n\to\infty$. Then $$\frac{X_n-E(X_n)}{\sigma_n}\rightarrow N(0,1),$$ which denotes convergence in distribution to the standard normal distribution. \end{theorem} \begin{theorem} The distribution of the number of descents (resp.\ ascents, plateaus) on elements of $\overline{\mathcal{Q}}_n$ converges to a normal distribution as $n\to\infty$. \end{theorem} \begin{proof} In order to apply Theorem~\ref{thm:bender} to descents (equivalently, ascents) on quasi-Stirling permutations, we let $X_n$ be the number of descents of a random element of $\overline{\mathcal{Q}}_n$. Then the polynomials $$g_n(t)=\sum_{m=1}^n P(X_n=m)\, t^m=\frac{\overline{Q}_n(t)}{\|\overline{\mathcal{Q}}_n\|}$$ have only real roots by Theorem~\ref{thm:roots}. It remains to show that Equation~\eqref{eq:var} holds. Using that $\overline{Q}'_n(1)+\overline{Q}''_n(1)=\sum_{\pi\in\overline{\mathcal{Q}}_n} \des(\pi)^2$, we have \begin{equation}\label{eq:varformula} \Var(X_n)=E(X_n^2)-E(X_n)^2=\frac{\overline{Q}'_n(1)+\overline{Q}''_n(1)}{\|\overline{\mathcal{Q}}_n\|}-\left(\frac{3n+1}{4}\right)^2, \end{equation} since $E(X_n)=\overline{Q}'_n(1)/\|\overline{\mathcal{Q}}_n\|=(3n+1)/4$ by Corollary~\ref{cor:average}. Differentiating Equation~\eqref{eq:oQ} twice with respect to $t$, setting $t=1$, and solving for $\frac{\partial^2 \overline{Q}}{\partial t^2}(1,z)$, we get \begin{align} \frac{\partial^2 \overline{Q}}{\partial t^2}(1,z) &=\frac{z}{6}\,\frac{z^3\overline{Q}(1,z)^4-4z^2\overline{Q}(1,z)^3+6z\overline{Q}(1,z)^2+12 z\left(\frac{\partial \overline{Q}}{\partial t}(1,z)\right)^2+12\frac{\partial \overline{Q}}{\partial t}(1,z)}{1-z-3z\overline{Q}(1,z)+z^2\overline{Q}(1,z)+3z^2\overline{Q}(1,z)^2-z^3\overline{Q}(1,z)^3}\\ &=\frac{1}{12z}\left(\frac{14{z}^{3}-18{z}^{2}+23z-4}{(1-4z)^{3/2}}+z+4\right), \end{align} where we have used Equations~\eqref{eq:Catalan} and~\eqref{eq:oQ'} in the last equality. Extracting the coefficient of $z^n$ in $$\frac{\partial^2 \overline{Q}}{\partial t^2}(1,z)+\frac{\partial \overline{Q}}{\partial t^2}(1,z)=\frac{1}{12z}\left(\frac{14{z}^{3}-6{z}^{2}+8z-1}{(1-4z)^{3/2}}-2z+1\right),$$ we get that, for $n\ge1$, \begin{align}\frac{\overline{Q}'_n(1)+\overline{Q}''_n(1)}{n!}&=\frac{1}{12}[z^{n+1}]\frac{14{z}^{3}-6{z}^{2}+8z-1}{(1-4z)^{3/2}}\\ &=14(2n-3)\binom{2n-4}{n-2}-6(2n-1)\binom{2n-2}{n-1}+8(2n+1)\binom{2n}{n}-(2n+3)\binom{2n+2}{n+1}\\ &=\frac{27n^3 + 10n^2 - 9n - 4}{12n(n + 1)}\binom{2n-2}{n-1}. \end{align} Plugging this expression and the equality $\|\overline{\mathcal{Q}}_n\|=n!C_n$ into Equation~\eqref{eq:varformula}, we obtain $$\Var(X_n)=\frac {11{n}^{2}-6n-5}{48(2n-1)},$$ and so Equation~\eqref{eq:var} holds. By Theorem~\ref{thm:bender}, we conclude that $$\frac{X_n-\frac{3n+1}{4}}{\sqrt{\frac {11{n}^{2}-6n-5}{48(2n-1)}}}\rightarrow N(0,1).$$ Asymptotic normality of the distribution of the number of plateaus can be proved similarly using Equation~\eqref{eq:Narayana}; in fact, it is already known that the Narayana distribution is asymtotically normal, see for example~\cite{fulman_steins_2018}. \end{proof} \section{Generalization to $k$-quasi-Stirling permutations} \label{sec:k} In the rest of the paper, we significantly extend the results from Section~\ref{sec:des}. On the one hand, we refine them to track the joint distribution of the number of ascents, the number of descents, and the number of plateaus. On the other hand, we generalize them to a one-parameter family of permutations, by allowing $k$ copies of each element. Gessel and Stanley proposed in~\cite{gessel_stirling_1978} a generalization of Stirling permutations by considering, for a fixed positive integer~$k$, permutations of the multiset $\{1^k,2^k,\dots,n^k\}$ (where the notation $i^k$ indicates $k$ copies of $i$) that avoid the pattern $212$. Let $\mathcal{Q}^k_n$ denote the set of these permutations, which we call {\em $k$-Stirling permutations}, following~\cite{janson_generalized_2011,kuba_analysis_2011}, although they are called $r$-multipermutations in~\cite{park_r-multipermutations_1994,park_inverse_1994}. Note that $\mathcal{Q}^1_n=\mathcal{S}_n$ and $\mathcal{Q}^2_n=\mathcal{Q}_n$. For this reason, the coefficients of the Stirling polynomials $Q_n(t)$ are called {\em second-order Eulerian numbers} in \cite{graham_concrete_1994}. An even more general version where each element $i$ appears an arbitrary number of times, which may be different for each $i$, was considered by Brenti~\cite{brenti_unimodal_1989}. Generalizing the definition of quasi-Stirling permutations in an analogous manner, we define {\em $k$-quasi-Stirling permutations} as those permutations of $\{1^k,2^k,\dots,n^k\}$ that avoid the patterns $1212$ and $2121$, and denote this set by $\overline{\mathcal{Q}}^k_n$. Note that $\overline{\mathcal{Q}}^1_n=\mathcal{S}_n$ and $\overline{\mathcal{Q}}^2_n=\overline{\mathcal{Q}}_n$. Viewing permutations of $\{1^k,2^k,\dots,n^k\}$ as ordered set partitions into blocks of size $k$, the avoidance requirement is equivalent to the partition being {\em noncrossing }. In this section, we enumerate $k$-quasi-Stirling permutations by the number of ascents, the number of descents, and the number of plateaus. \subsection{Bijections to trees}\label{sec:bij_k} We present two bijections between $k$-quasi-Stirling permutations and different kinds of trees, each one extending a bijection in the literature between $k$-Stirling permutations and certain increasing trees. In~\cite[Thm.\ 1]{janson_generalized_2011}, Janson, Kuba and Panholzer describe a bijection between $k$-Stirling permutations and {\em $(k+1)$-ary increasing trees} ---this bijection is attributed to Gessel in~\cite{park_r-multipermutations_1994}---, and they use it to study the distribution of ascents, descents and plateaus on $k$-Stirling permutations. A (vertex-labeled) $k$-ary tree is a plane rooted tree where each vertex has either $0$ or $k$ children, and each of the internal (i.e. non-leaf) vertices receives a distinct label between 1 and the number of internal vertices. Let $\mathcal{A}^k_n$ denote the set of $k$-ary trees with $n$ internal vertices. Such a tree is {\em increasing} if the label of each vertex is smaller than the labels of its children, disregarding unlabeled leaves. Inspired by this bijection, we can construct a bijection $\psi$ between $k$-ary trees without the increasing condition, and $k$-quasi-Stirling permutations. \begin{theorem}\label{thm:psi} There is a natural bijection $\psi:\mathcal{A}^k_n\to\overline{\mathcal{Q}}^k_n$. \end{theorem} \begin{proof} Given a tree in $\mathcal{A}^k_n$, traverse its edges by following a depth-first walk from left to right, and record the label of each vertex that the path returns to; in other words, record every time that a vertex is visited except for the first time. See Figure~\ref{fig:kary} for examples with $k=2$ and $k=3$. Let us show that the resulting sequence belongs to $\overline{\mathcal{Q}}^k_n$. First, it contains $k$ copies of each element in $[n]$, since the path returns to each internal vertex once coming from each of its $k$ children. Second, it avoids the patterns $1212$ and $2121$ because, if a label $a$ appears between two readings of a label $b$ in the depth-first walk, then vertex $b$ is in the path between vertex $a$ and the root, and so all occurrences of $a$ in the sequence appear between those two occurrences of~$b$. To see that $\psi$ is a bijection, let us describe its inverse. Given $\pi\in\overline{\mathcal{Q}}^k_n$, let $b=\pi_n$, and decompose $\pi$ as $\pi=\sigma_1 b \sigma_2 b \dots \sigma_{k-1} b \sigma_k b$. The subsequences $\sigma_1,\sigma_2,\dots,\sigma_k$ must have pairwise disjoint entries, because $\pi$ avoids $1212$ and $2121$, and so each $\sigma_i$ is a $k$-quasi-Stirling permutation whose entries have been relabeled by an order-preserving function. Then $\psi^{-1}(\pi)$ is the $k$-ary tree consisting of a root labeled $b$ with subtrees $\psi^{-1}(\sigma_1),\psi^{-1}(\sigma_2),\dots,\psi^{-1}(\sigma_k)$ from left to right, constructed recursively. \end{proof} For $k=2$, $\psi$ is a bijection between (vertex-labeled) binary trees and quasi-Stirling permutations. Interestingly, the shift of the parameter $k$ in the bijection from~\cite{janson_generalized_2011} between $(k+1)$-ary increasing trees and $k$-Stirling permutations disappears in our bijection $\psi$. \begin{figure}\label{fig:kary} \end{figure} In order to study ascents, descents and plateaus, we introduce a second bijection $\phi$ between trees and $k$-quasi-Stirling permutations that will be more suitable to track these statistics. It extends a construction of Kuba and Panholzer \cite[Thm.\ 2.2]{kuba_analysis_2011} that, with a slight modification, yields a bijection between $k$-Stirling permutations and certain modified increasing trees that we describe next. For $k\ge2$, let $\mathcal{T}^k_n$ be the set of edge-labeled plane rooted trees with $n$ edges, where every node other than the root has $k-2$ unlabeled half-edges which serve as walls, separating its children into $k-1$ (possibly empty) compartments. By definition, $\mathcal{T}^2_n=\mathcal{T}_n$. A tree in $\mathcal{T}^3_n$ is drawn on the left of Figure~\ref{fig:phi}. We call trees in $\mathcal{T}^k_n$ {\em compartmented trees}. Denote by $\mathcal{I}^k_n\subseteq\mathcal{T}^k_n$ the subset of those trees whose labels along any path from the root to a leaf are increasing. \begin{figure}\label{fig:phi} \end{figure} We are now ready to describe the bijection $\phi$ between $\mathcal{T}^k_n$ and $\overline{\mathcal{Q}}^k_n$. For $k=2$, $\phi$ coincides with $\varphi:\mathcal{T}_n\to\overline{\mathcal{Q}}_n$ described in Section~\ref{sec:bij}. When restricted to increasing trees, $\phi$ becomes a bijection between $\mathcal{I}^k_n$ and $\mathcal{Q}^k_n$, which is a version of~\cite[Thm.\ 2.2]{kuba_analysis_2011}. \begin{theorem}\label{thm:phi} There is a natural bijection $\phi:\mathcal{T}^k_n\to\overline{\mathcal{Q}}^k_n$. \end{theorem} \begin{proof} Given a tree $T\in\mathcal{T}^k_n$, first label the half-edges at each node $v$ with the label of the edge between $v$ and its parent. Then traverse the edges and half-edges of $T$ following a depth-first walk from left to right, and record their labels as they are traversed, with the rule that the label of each half-edge only contributes once. Let $\phi(T)$ be the resulting sequence of recorded labels; see Figure~\ref{fig:phi} for an example. We claim that $\phi(T)\in\overline{\mathcal{Q}}^k_n$. Indeed, each element in $[n]$ appears $k$ times: twice from traversing the edge with that label, and $k-2$ times from traversing the half-edges with that label. Additionally, the sequence does not contain the patterns $1212$ and $2121$ because, if a label $a$ appears between two readings of a label $b$ in the depth-first walk, then the whole subtree containing the edge labeled $a$ (which includes all occurrences of $a$) must be read between those two occurrences of~$b$. Next we show that $\phi$ is a bijection by describing its inverse. Given $\pi\in\overline{\mathcal{Q}}^k_n$, let $a=\pi_1$, and decompose $\pi$ according to the occurrences of $a$ as $\pi=a\sigma_1 a \sigma_2 a \dots a \sigma_{k-1} a \sigma_k$. The subsequences $\sigma_1,\sigma_2,\dots,\sigma_k$ must have pairwise disjoint entries, because $\pi$ avoids $1212$ and $2121$, and so each $\sigma_i$ is a $k$-quasi-Stirling permutation with relabeled entries. We obtain $\phi^{-1}(\pi)$ as follows. First, place an edge with label $a$ from the root to its leftmost child. In the $k-1$ compartments hanging from that child, place the subtrees $\phi^{-1}(\sigma_1),\dots,\phi^{-1}(\sigma_{k-1})$ from left to right, constructed recursively. Finally, place the subtree $\phi^{-1}(\sigma_k)$ hanging from the root of $\phi^{-1}(\pi)$, to the right of the initially placed edge. \end{proof} Either of the bijections $\psi$ or $\phi$ allows us to easily count the number of $k$-quasi-Stirling permutations. We denote the {\em $k$-Catalan numbers} (see~\cite[pp.\ 168--173]{stanley_enumerative_1999}) by $$C_{n,k}=\frac{1}{(k-1)n+1}\binom{kn}{n}.$$ \begin{theorem}\label{thm:QQkn} For $n\ge1$ and $k\ge1$, $$\|\overline{\mathcal{Q}}^k_n\|=\frac{(kn)!}{((k-1)n+1)!}=n!\,C_{n,k}.$$ \end{theorem} \begin{proof} Consider the EGF $G(z)=\sum_{n\ge0}\|\overline{\mathcal{Q}}^k_n\|z^n/n!$. By Theorem~\ref{thm:phi}, $\|\overline{\mathcal{Q}}^k_n\|=\|\mathcal{T}^k_n\|$. We enumerate compartmented trees by giving a recursive description of the class $\mathcal{T}^k=\bigcup_{n\ge0} \mathcal{T}^k_n$. Every non-empty tree in $\mathcal{T}^k$ consists of a root having a sequence of children, where each of these children plays itself the role of a root of a sequence of $k-1$ trees from $\mathcal{T}^k$, one in each of the compartments. This description translates into the equation $$G(z)=\frac{1}{1-zG(z)^{k-1}},$$ which is equivalent to $G(z)-zG(z)^k=1$. Alternatively, using Theorem~\ref{thm:psi}, $\|\overline{\mathcal{Q}}^k_n\|=\|\mathcal{A}^k_n\|$, so one can enumerate $k$-ary trees instead. The obvious recursive description of such trees yields the equation $G(z)=1+zG(z)^k$ for their EGF. Applying the Lagrange inversion formula to $F(z):=G(z)-1$, which satisfies $F(z)=z(1+F(z))^k$, we get, for $n\ge1$, $$\frac{\|\overline{\mathcal{Q}}^k_n\|}{n!}=[z^n]F(z)=\frac{1}{n}[z^{n-1}](1+z)^{kn}=\frac{1}{n}\binom{kn}{n-1}=C_{n,k}.$$ \end{proof} \subsection{Ascents, descents and plateaus on $k$-quasi-Stirling permutations} We are now ready to generalize Theorem~\ref{thm:main} to give an equation for the refined $k$-quasi-Stirling polynomials $$\overline{P}^{(k)}_n(q,t,u)=\sum_{\pi\in\overline{\mathcal{Q}}^k_n} q^{\asc(\pi)}t^{\des(\pi)} u^{\plat(\pi)},$$ and their EGF $$\overline{P}^{(k)}(q,t,u;z)=\sum_{n\ge0}\overline{P}^{(k)}_n(q,t,u)\frac{z^n}{n!}.$$ For $k=2$, we have $\overline{P}^{(2)}(1,t,1;z)=\overline{Q}(t,z)$ by definition, which we computed in Theorem~\ref{thm:main}. On the other hand, an expression for $\overline{P}^{(2)}(1,1,u;z)$ is given by Equation~\eqref{eq:Narayana}. In order to track ascents, it will be useful to consider the homogenization of the Eulerian polynomials, $$\hat{A}_n(q,t)=\sum_{\pi\in\mathcal{S}_n} q^{\asc(\pi)} t^{\des(\pi)},$$ and their EGF \begin{align}\hat{A}(q,t;z)&=\sum_{n\ge0} \hat{A}_n(q,t) \frac{z^n}{n!}= 1+\sum_{n\ge1} A_n(t/q) q^{n+1} \frac{z^n}{n!} = 1+q(A(t/q,qz)-1)\\ &=1-q+\frac{q(q-t)}{q-te^{(q-t)z}}, \end{align} using Equation~\eqref{eq:A_GF}. \begin{theorem}\label{thm:main_plat_k} Fix $k\ge1$. The EGF of $k$-quasi-Stirling permutations by the number of ascents, the number of descents, and the number of plateaus satisfies the implicit equation $\overline{P}^{(k)}(q,t,u;z)=\hat{A}(q,t,z(\overline{P}^{(k)}(q,t,u;z)-1+u)^{k-1})$, that is, \begin{equation}\label{eq:oP} \overline{P}^{(k)}(q,t,u;z)=1-q+\frac{q(q-t)}{q-t\,\exp\left((q-t)z(\overline{P}^{(k)}(q,t,u;z)-1+u)^{k-1}\right)}. \end{equation} In particular, for $n\ge1$, its coefficients satisfy \begin{equation} \overline{P}^{(k)}_n(q,t,u)=\frac{n!}{(k-1)n+1}\,[z^n]\left(\hat{A}(q,t;z)-1+u\right)^{(k-1)n+1}. \end{equation} \end{theorem} Our proof relies on the bijection $\phi$. The first step is to analyze how the statistics $\asc$, $\des$ and $\plat$ are transformed by this bijection. To this end, we extend the notion of cyclic descents to compartmented trees. Let $T\in\mathcal{T}^k_n$, and let $v$ a vertex of $T$. If $v$ is not the root, define $\cdes(v)$ to be the number of cyclic descents of the sequence obtained by listing the labels of the edges and half-edges incident to $v$ in counterclockwise order, where the half-edges are given the label of the edge between $v$ and its parent. Equivalently, if this label is $\ell$, and the labels of the edges between $v$ and its children in the $j$-th compartment from the left are $a_{j,1},a_{j,2},\dots,a_{j,d_j}$ from left to right, for $1\le j\le k-1$, then \begin{equation}\label{eq:cdes_k} \cdes(v)=\cdes(\ell a_{1,1}\dots a_{1,d_1}\ell a_{2,1}\dots a_{2,d_2}\ell \dots \ell a_{k-1,1}\dots a_{k-1,d_{k-1}}). \end{equation} If $v$ is the root of $T$, define $\cdes(v)$ to be the number of descents of the sequence obtained by listing the labels of the edges incident to $v$ from left to right. Finally, define the number of cyclic descents of $T$ to be $$\cdes(T)=\sum_v \cdes(v),$$ where the sum ranges over all the vertices $v$ of $T$. For example, if $T$ is the tree on the left of Figure~\ref{fig:phi}, then $\cdes(T)=2+1+0+2+0+1+0+0=6$, where the two vertices with $\cdes(v)=2$ are the root and the vertex with 3 children. Switching the direction of the inequalities, we define the number of {\em cyclic ascents} of a sequence of positive integers as $\casc(\pi_1\pi_2\dots\pi_r)=\|\{i\in[r]: \pi_i<\pi_{i+1}\}\|$, with the convention $\pi_{r+1}:=\pi_1$. For a non-root vertex $v$ of $T\in\mathcal{T}^k_n$, define $\casc(v)$ in analogy to Equation~\eqref{eq:cdes_k}, replacing $\cdes$ with $\casc$. If $v$ is the root of $T$, define $\casc(v)$ to be the number of ascents of the sequence obtained by listing the labels of the edges incident to $v$ from left to right. The number of cyclic ascents of $T$ is then defined as $\casc(T)=\sum_v \casc(v)$, summing over all the vertices of $T$. Recall that the children of each non-root vertex of $T\in\mathcal{T}^k_n$ are separated into $k-1$ (possibly empty) compartments. Denote by $\emp(T)$ the total number of empty compartments of $T$. For example, if $T$ is the tree on the left of Figure~\ref{fig:phi}, then $\emp(T)=10$. \begin{lemma}\label{lem:cdes_k} The bijection $\phi:\mathcal{T}^k_n\to\overline{\mathcal{Q}}^k_n$ has the following property: if $T\in\mathcal{T}^k_n$ and $\pi=\phi(T)\in\overline{\mathcal{Q}}^k_n$, then $$\asc(\pi)=\casc(T), \quad \des(\pi)=\cdes(T) \quad \text{and} \quad \plat(\pi)=\emp(T).$$ \end{lemma} \begin{proof} The proof of the equality $\des(\pi)=\cdes(T)$ is similar to the proof of Lemma~\ref{lem:cdes}, the only difference being the contribution to $\pi=\phi(T)$ of the half-edges at each non-root vertex $v$ of $T$. These half-edges are labeled with the label $\ell$ of the edge $e$ between $v$ and its parent, and then traversed by the depth-first walk in the definition of $\phi$. Each such half-edge $h$ creates an entry $\ell$ in $\pi$, which is inserted between the label of the edge or half-edge incident to $v$ immediately to the left of $h$ (or, in its absence, the edge $e$), and the label of the edge or half-edge immediately to the right of $h$ (or, in its absence, the edge $e$). With these additional entries in $\pi=\phi(T)$, the contribution to $\des(\pi)$ of all the visits to $v$ of the depth-first walk around $T$ equals $\cdes(v)$, as defined in Equation~\eqref{eq:cdes_k}. If $v$ is the root of $T$, its contribution to $\des(\pi)$ is also $\cdes(v)$, like in the proof of Lemma~\ref{lem:cdes}. Adding the contributions of all the vertices of $T$, we see that $\cdes(T)=\sum_v \cdes(v)=\des(\pi)$. The equality $\asc(\pi)=\casc(T)$ is proved analogously by symmetry. Finally, to show that $\plat(\pi)=\emp(T)$, note that a plateau in $\pi=\phi(T)$ occurs when two edges or half-edges (or one of each) of $T$ with the same label are traversed one immediately after the other by the depth-first walk on $T$, which happens precisely at empty compartments of $T$. \end{proof} By Lemma~\ref{lem:cdes_k}, $\overline{P}^{(k)}(q,t,u;z)$ is also the EGF for compartmented trees by the number of cyclic ascents, the number of cyclic descents, and the number of empty compartments: \begin{equation}\label{eq:oPT} \overline{P}^{(k)}(q,t,u;z)=\sum_{n\ge0} \sum_{T\in\mathcal{T}^k_n} q^{\casc(T)}t^{\cdes(T)}u^{\emp(T)} \frac{z^n}{n!}. \end{equation} Paralleling the proof of Theorem~\ref{thm:main}, we will consider a recursive description of compartmented trees. Let ${\mathcal{T}^k_n}'\subseteq\mathcal{T}^k_n$ be the subset of trees whose root has exactly one child, and let \begin{equation}\label{eq:Rkn} R^{(k)}_n(q,t,u)=\sum_{T'\in{\mathcal{T}^k_n}'} q^{\casc(T')-1}t^{\cdes(T')-1}u^{\emp(T')}. \end{equation} Let $\mathcal{T}^k=\bigcup_{n\ge0}\mathcal{T}^k_n$ and ${\mathcal{T}^k}'=\bigcup_{n\ge1}{\mathcal{T}^k_n}'$. The next lemma generalizes Lemma~\ref{lem:T'}. \begin{lemma}\label{lem:T'_k} We have $$\sum_{n\ge1} R^{(k)}_n(q,t,u) \frac{z^n}{n!}=z\,(\overline{P}^{(k)}(q,t,u;z)-1+u)^{k-1}.$$ \end{lemma} \begin{proof} In the proof of Lemma~\ref{lem:T'}, we showed that every tree $T'\in\mathcal{T}'_n$ can be obtained as $T'=T^{\|n}+j$ for a unique $j\in\{0,1,\dots,n-1\}$ and a unique $T\in\mathcal{T}_{n-1}$. Here we generalize this construction to compartmented trees. Using the notation from~\cite{flajolet_analytic_2009}, consider the class $\Seq_{k-1}(\mathcal{T}^{k})$, which consists of sequences of $k-1$ trees from $\mathcal{T}^{k}$ whose edges have been relabeled with distinct labels from $1$ up to the total number of edges, so that the relative order of labels within each tree is preserved. If $\vec{T}\in \Seq_{k-1}(\mathcal{T}^{k})$ is a relabeling of the tuple $(T_1,\dots,T_{k-1})$, where $T_j\in\mathcal{T}^{k}$ for all $j$, define \begin{equation}\label{eq:cdesvecT} \cdes(\vec{T})=\sum_{j=1}^{k-1}\cdes(T_j),\quad \casc(\vec{T})=\sum_{j=1}^{k-1}\casc(T_j),\quad \emp(\vec{T})=\sum_{j=1}^{k-1}\emp(T_j)+\|\{j:T_j \text{ is empty}\}\|, \end{equation} and let $\|\vec{T}\|$ denote the total number of edges. Using Equation~\eqref{eq:oPT}, the corresponding multivariate EGF is \begin{equation}\label{eq:EGFSeq} \sum_{\vec{T}\in\Seq_{k-1}(\mathcal{T}^{k})} q^{\casc(\vec{T})}t^{\cdes(\vec{T})}u^{\emp(\vec{T})}\frac{z^{\|\vec{T}\|}}{\|\vec{T}\|!} = (\overline{P}^{(k)}(q,t,u;z)-1+u)^{k-1}. \end{equation} Given $\vec{T}\in \Seq_{k-1}(\mathcal{T}^{k})$ with $\|\vec{T}\|=n-1$, construct a new tree $\vec{T}^{\|n}\in{\mathcal{T}^k_n}'$ as follows: combine the $k-1$ trees in $\vec{T}$ by identifying their roots into a common vertex $v_0$, placing the trees from left to right with $k-2$ half-edges at $v_0$ separating them, and attach an edge with label $n$ from $v_0$ to a new root vertex. Using the definitions from Equation~\eqref{eq:cdesvecT}, we claim that \begin{equation}\label{eq:cdesvecTplus1} \cdes(\vec{T}^{\|n})=\cdes(\vec{T})+1, \quad \casc(\vec{T}^{\|n})=\casc(\vec{T})+1, \quad \emp(\vec{T}^{\|n})=\emp(\vec{T}). \end{equation} To prove the first equality, suppose that $\vec{T}$ is a relabeling of the tuple $(T_1,\dots,T_{k-1})$. If $v$ is a non-root vertex of $T_j$ for some $j$, then $\cdes(v)$ is the same in $T_j$ as it is in $\vec{T}^{\|n}$. On the other hand, assuming that the edges incident to the root of $T_j$ have labels $a_{j,1},a_{j,2},\dots,a_{j,d_j}$ from left to right, the root of $T_j$ contributes $\des(a_{j,1}a_{j,2}\dots a_{j,d_j})$ to $\cdes(\vec{T})$ for each $j$, whereas the corresponding vertex $v_0$ in $\vec{T}^{\|n}$ contributes \begin{align} \cdes(v_0)&=\cdes(n a_{1,1}\dots a_{1,d_1}n a_{2,1}\dots a_{2,d_2}n \dots n a_{k-1,1}\dots a_{k-1,d_{k-1}})= \sum_{j=1}^{k-1} \cdes(n a_{j,1}a_{j,2}\dots a_{j,d_j})\\ &=\sum_{j=1}^{k-1} \des(a_{j,1}a_{j,2}\dots a_{j,d_j}) \end{align} to $\cdes(\vec{T}^{\|n})$. Finally, the new root of $\vec{T}^{\|n}$ contributes an additional cyclic descent, proving the first equality in Equation~\eqref{eq:cdesvecTplus1}. A symmetric argument proves the analogous statement for $\casc$. The third equality in Equation~\eqref{eq:cdesvecTplus1} is clear by construction, since each empty $T_j$ contributes an additional empty compartment in $\vec{T}^{\|n}$. For $T'\in{\mathcal{T}^k_n}'$ and $i\in\{0,1,\dots,n-1\}$, let $T'+i\in{\mathcal{T}^k_n}'$ be the tree obtained by adding $i$ modulo $n$ to each label of $T'$. As in the proof of Lemma~\ref{lem:T'}, this operation preserves the number of cyclic descents, that is, $\cdes(T'+i)=\cdes(T')$ for all $T'\in{\mathcal{T}^k_n}'$. In addition, $\casc(T'+i)=\casc(T')$ by symmetry, and $\emp(T'+i)=\emp(T')$ because adding $i$ does not change the underlying unlabeled tree, and in particular its number of empty compartments. Noting that every tree $T'\in{\mathcal{T}^k_n}'$ can be obtained as $T'=\vec{T}^{\|n}+i$ for a unique $\vec{T}\in \Seq_{k-1}(\mathcal{T}^{k})$ with $\|\vec{T}\|=n-1$, and a unique $i\in\{0,1,\dots,n-1\}$, we have \begin{align} R^{(k)}_n(q,t,u)&=\sum_{T'\in{\mathcal{T}^k_n}'} q^{\casc(T')-1}t^{\cdes(T')-1}u^{\emp(T')}\\ &=\sum_{\substack{\vec{T}\in\Seq_{k-1}(\mathcal{T}^{k}) \\ \|\vec{T}\|=n-1}} \sum_{i=0}^{n-1} q^{\casc(\vec{T}^{\|n}+i)-1}t^{\cdes(\vec{T}^{\|n}+i)-1}u^{\emp(\vec{T}^{\|n}+ )}\\ &=\sum_{\substack{\vec{T}\in\Seq_{k-1}(\mathcal{T}^{k}) \\ \|\vec{T}\|=n-1}} n\, q^{\casc(\vec{T}^{\|n})-1}t^{\cdes(\vec{T}^{\|n})-1}u^{\emp(\vec{T}^{\|n})}\\ &=n\sum_{\substack{\vec{T}\in\Seq_{k-1}(\mathcal{T}^{k}) \\ \|\vec{T}\|=n-1}} q^{\casc(\vec{T})}t^{\cdes(\vec{T})}u^{\emp(\vec{T})} \end{align} for every $n\ge1$. Multiplying both sides by $z^n/n!$, summing over $n$, and using Equation~\eqref{eq:EGFSeq}, we get $$\sum_{n\ge1} R^{(k)}_n(q,t,u) \frac{z^n}{n!}= \sum_{n\ge1} \sum_{\substack{\vec{T}\in\Seq_{k-1}(\mathcal{T}^{k}) \\ \|\vec{T}\|=n-1}} q^{\casc(\vec{T})}t^{\cdes(\vec{T})}u^{\emp(\vec{T})} \frac{z^n}{(n-1)!} =z\,(\overline{P}^{(k)}(q,t,u;z)-1+u)^{k-1}.$$ \end{proof} \begin{proof}[Proof of Theorem~\ref{thm:main_plat_k}] Recall that, by Equation~\eqref{eq:oPT}, $\overline{P}^{(k)}(q,t,u;z)$ is the EGF for the class $\mathcal{T}^k$ with respect to $\casc$, $\cdes$ and $\emp$. Every $T\in\mathcal{T}^k$ can be decomposed as a sequence of non-empty trees $T'_1,T'_2,\dots,T'_r\in{\mathcal{T}^k}'$ with a common root, denoted by $v_0$, with edges relabeled from $1$ up to the number of edges of $T$ as usual. In this decomposition, $\emp(T)=\sum_{i=1}^r \emp(T'_j)$. As in the proof of Theorem~\ref{thm:main}, if the edges incident to $v_0$ in $T$ have labels $a_1,a_2,\dots,a_r$ from left to right, then $$\cdes(T)=\sum_{i=1}^r(\cdes(T'_i)-1)+\des(a_1a_2\dots a_r),$$ and similarly $$\casc(T)=\sum_{i=1}^r(\casc(T'_i)-1)+\asc(a_1a_2\dots a_r).$$ It follows that \begin{multline} \sum_{T\in\mathcal{T}^k_n} q^{\casc(T)}t^{\cdes(T)}u^{\emp(T)} \\ = \sum_{\substack{\{B_1,B_2,\dots,B_r\}\\ \text{partition of $[n]$}}} \left(\sum_{\pi\in\mathcal{S}_r} q^{\asc(\pi)} t^{\des(\pi)}\right)\cdot\prod_{i=1}^r \left(\sum_{T'_i\in\mathcal{T}'_{\|B_i\|}} q^{\casc(T'_i)-1}t^{\cdes(T'_i)-1}u^{\emp(T'_i)} \right). \end{multline} Using the notation $R^{(k)}_n(q,t,u)$ from Equation~\eqref{eq:Rkn}, we can rewrite this equation as $$\overline{P}^{(k)}_n(q,t,u)=\sum_{\substack{\{B_1,B_2,\dots,B_r\}\\ \text{partition of $[n]$}}} \hat{A}_r(q,t)\cdot \prod_{i=1}^r R^{(k)}_{\|B_i\|}(q,t,u).$$ By the Compositional Formula, and using Lemma~\ref{lem:T'_k}, \begin{align} \overline{P}^{(k)}(q,t,u;z)&=\sum_{n\ge0}\overline{P}^{(k)}_n(q,t,u)\frac{z^n}{n!}=1+\sum_{r\ge1}\hat{A}_r(q,t)\frac{1}{r!} \left(\sum_{b\ge1} R^{(k)}_{b}(q,t,u)\frac{z^{b}}{b!}\right)^r\\ &=1+\sum_{r\ge1} \hat{A}_r(q,t) \frac{1}{r!} \left(z\,(\overline{P}^{(k)}(q,t,u;z)-1+u)^{k-1}\right)^r \\ &=\hat{A}(q,t,z(\overline{P}^{(k)}(q,t,u;z)-1+u)^{k-1}), \end{align} proving Equation~\eqref{eq:oP}. To extract the coefficient of $z^n$ in $\overline{P}^{(k)}(q,t,u;z)$, first write $z=y^{k-1}$, so that $$\overline{P}^{(k)}(q,t,u;y^{k-1})=\hat{A}(q,t,y^{k-1}(\overline{P}^{(k)}(q,t,u;y^{k-1})-1+u)^{k-1}).$$ Then the generating function $F(y):=F(q,t,u;y):=y\left(\overline{P}^{(k)}(q,t,u;y^{k-1})-1+u\right)$ satisfies the equation $$F(y)=y\left(\hat{A}(q,t,F(y)^{k-1})-1+u\right).$$ Applying Lagrange's inversion formula to $F(y)$ yields, for $n\ge1$, \begin{align} \overline{P}^{(k)}_n(q,t,u)&=n!\,[z^n]\overline{P}^{(k)}(q,t,u;z)=n!\,[y^{(k-1)n+1}]F(y)\\ &=\frac{n!}{(k-1)n+1}\,[y^{(k-1)n}]\left(\hat{A}(q,t;y^{k-1})-1+u\right)^{(k-1)n+1}\\ &=\frac{n!}{(k-1)n+1}\,[z^n]\left(\hat{A}(q,t;z)-1+u\right)^{(k-1)n+1}. \end{align} \end{proof} \subsection{Ascents, descents and plateaus on $k$-Stirling permutations} B\'ona proves in \cite[Prop.\ 1]{bona_real_2008} that, on the set $\mathcal{Q}_n$ of Stirling permutations, the statistics $\asc$, $\des$ and $\plat$ are equidistributed. Janson generalizes this result in \cite[Thm.\ 2.1]{janson_plane_2008}, proving, using a probabilistic argument, that the polynomials \begin{equation}\label{eq:Pn} P_n(q,t,u)=\sum_{\pi\in\mathcal{Q}_n} q^{\asc(\pi)} t^{\des(\pi)} u^{\plat(\pi)} \end{equation} are symmetric in the three variables, and he suggests in \cite[Sec.\ 3]{janson_plane_2008} that the study of these polynomials would be interesting. In~\cite{haglund_stable_2012}, Haglund and Visontai prove that these polynomials are {\em stable}, meaning that they do not vanish when all the variables have a positive imaginary part. In this subsection we present a nice differential equation satisfied by the EGF of these polynomials, which we denote by $$P(q,t,u;z)=\sum_{n\ge0}P_n(q,t,u)\frac{z^n}{n!},$$ and more generally, by the analogous multivariate EGF for $k$-Stirling permutations: $$P^{(k)}(q,t,u;z)=\sum_{n\ge0}\sum_{\pi\in\mathcal{Q}^k_n} q^{\asc(\pi)} t^{\des(\pi)} u^{\plat(\pi)}\frac{z^n}{n!}.$$ We give two derivations of the differential equation. The first one uses the same techniques as in the proofs of Lemma~\ref{lem:T'_k} and Theorem~\ref{thm:main_plat_k}, including our interpretation of ascents, descents and plateaus in permutations as cyclic ascents, cyclic descents and empty compartments in compartmented trees. The second one is a more direct, conceptual proof. \begin{theorem}\label{thm:Qplat_k} The EGF $P(z):=P^{(k)}(q,t,u;z)$ of $k$-Stirling permutations by the number of ascents, the number of descents, and the number of plateaus satisfies the differential equation $$P'(z)=(P(z)-1+q)(P(z)-1+t)(P(z)-1+u)^{k-1},$$ with initial condition $P(0)=1$. \end{theorem} \begin{proof}[First proof] As discussed in Section~\ref{sec:bij_k}, the map $\phi$ restricts to a bijection between increasing compartmented trees $\mathcal{I}^k_n$ and $k$-Stirling permutations $\mathcal{Q}^k_n$. Thus, by Lemma~\ref{lem:cdes_k}, $P^{(k)}(q,t,u;z)$ is the EGF of increasing compartmented trees where $q$, $t$ and $u$ mark the number of cyclic ascents, the number of cyclic descents, and the number of empty compartments, respectively. Next we adapt the construction from the proof of Lemma~\ref{lem:T'_k} to increasing trees. Let $\mathcal{I}^k=\bigcup_{n\ge0}\mathcal{I}^k_n$ be the class of increasing compartmented trees, and let ${\mathcal{I}^k}'\subseteq\mathcal{I}^k$ be the subclass of those trees whose root has exactly one child. Every tree in ${\mathcal{I}^k}'$ can be obtained from a sequence $\vec{T}\in\Seq_{k-1}(\mathcal{I}^k)$, consisting of $k-1$ trees $T'_1,\dots,T'_{k-1}\in\mathcal{I}^k$ with edges relabeled as usual, by identifying the $k-1$ roots into a common vertex $v_0$, placing $k-2$ half-edges at $v_0$ to separate these trees into $k-1$ compartments, adding $1$ to every edge label, and attaching a new edge with label $1$ from $v_0$ to a new root vertex. Denote the resulting tree by $\vec{T}^{\|1}\in{\mathcal{I}^k}'$, and observe that, with the notation from the proof of Lemma~\ref{lem:T'_k} and letting $n=\|\vec{T}\|+1$, we have $\vec{T}^{\|1}=\vec{T}^{\|n}+1$. Since adding $1$ modulo $n$ to the labels preserves the statistics $\cdes$, $\casc$ and $\emp$, it follows from Equation~\eqref{eq:cdesvecTplus1} that $\cdes(\vec{T}^{\|1})=\cdes(\vec{T}^{\|n})=\cdes(\vec{T})+1$, $\casc(\vec{T}^{\|1})=\casc(\vec{T}^{\|n})=\casc(\vec{T})+1$, and $\emp(\vec{T}^{\|1})=\emp(\vec{T}^{\|n})=\emp(\vec{T})$. The number of edges of $\vec{T}^{\|1}$ is $\|\vec{T}\|+1$. Noting that every $T'\in{\mathcal{I}^k}'$ is obtained as $T'=\vec{T}^{\|1}$ for a unique $\vec{T}\in\Seq_{k-1}(\mathcal{I}^k)$, and using Equation~\eqref{eq:EGFSeq}, we deduce that the refined EGF for the class ${\mathcal{I}^k}'$ is \begin{align}\nonumber \sum_{T'\in{\mathcal{I}^k}'} q^{\casc(\vec{T'})-1}t^{\cdes(\vec{T'})-1}u^{\emp(\vec{T'})}\frac{z^{\|T'\|}}{\|T'\|!}& =\sum_{\vec{T}\in\Seq_{k-1}(\mathcal{T}^{k})} q^{\casc(\vec{T})}t^{\cdes(\vec{T})}u^{\emp(\vec{T})}\frac{z^{\|\vec{T}\|+1}}{(\|\vec{T}\|+1)!}\\ &=\int_0^z(P(y)-1+u)^{k-1}\,dy. \label{eq:Pplat} \end{align} The decomposition from the proof of Theorem~\ref{thm:main_plat_k}, expressing trees in $\mathcal{T}^k$ as sequences of trees in ${\mathcal{T}^k}'$, restricts straightforwardly to increasing trees: every tree in $ \mathcal{I}^k$ can be decomposed as a sequence of trees in ${\mathcal{I}^k}'$ with a common root, relabeling as usual. The same argument, using now Equation~\eqref{eq:Pplat}, implies that $$P(z)=\hat{A}\left(q,t,\int_0^z(P(y)-1+u)^{k-1}\,dy\right)=1-q+\frac{q(q-t)}{q-te^{(q-t)\int_0^z(P(y)-1+u)^{k-1}\,dy}}.$$ Rewriting this equation as $$(q-t)\int_0^z(P(y)-1+u)^{k-1}\,dy=\ln(P(z)-1+t)-\ln(P(z)-1+q)+\ln(q)-\ln(t)$$ and differentiating with respect to $z$ gives the stated differential equation. \end{proof} \begin{proof}[Second proof] By looking at the positions of the $1$s, every non-empty $k$-Stirling permutation $\pi\in\mathcal{Q}^k_n$ can be decomposed uniquely as $\pi=\sigma_0 1\sigma_1 1\dots 1\sigma_{k}$. The entries in each $\sigma_i$ must be pairwise disjoint, since otherwise the repeated entry together with a $1$ in between would form an occurrence of $212$ in $\pi$. Thus, each $\sigma_i$ is itself a $k$-Stirling permutation with relabeled entries. Conversely, any $(k+1)$-tuple of $k$-Stirling permutations can be used to construct a new Stirling permutation by relabeling the entries with $\{2^k,\dots,n^k\}$ so that their relative order is preserved, and using the relabeled permutations as the subsequences $\sigma_0,\sigma_1,\dots,\sigma_{k}$ above. Next, we examine how ascents, descents, and plateaus behave under this decomposition. Note that $\des(\pi)=\sum_{i=0}^{k} \des(\sigma_i)$, unless $\sigma_{k}$ is empty, in which case $\pi$ has an extra descent. Similarly, $\asc(\pi)=\sum_{i=0}^{k} \asc(\sigma_i)$, unless $\sigma_0$ is empty, in which case $\pi$ has an extra ascent. On the other hand, $$\plat(\pi)=\sum_{i=0}^{k} \plat(\sigma_i)+\|\{i\in[k-1]:\sigma_i\text{ is empty}\}\|.$$ It follows that $$P'(z)=(P(z)-1+q)(P(z)-1+t)(P(z)-1+u)^{k-1}.$$ The initial condition $P(0)=1$ comes from the empty permutation, completing the proof. We remark that, through the bijection $\phi$, this decomposition of $k$-Stirling permutations is equivalent to the decomposition of non-empty increasing compartmented trees $T\in\mathcal{I}^k$ into $k+1$ trees $T_0,T_1,\dots,T_k\in\mathcal{I}^k$, obtained by splitting $T$ at the edge with label $1$ and at the half-edges of its lower endpoint, as shown in Figure~\ref{fig:tree_decomp}. \end{proof} \begin{figure} \caption{The decomposition of increasing compartmented trees from the second proof of Theorem~\ref{thm:Qplat_k}.} \label{fig:tree_decomp} \end{figure} For $k=2$, the differential equation in Theorem~\ref{thm:Qplat_k} becomes simply \begin{equation}\label{eq:Qplat} P'(z)=(P(z)-1+q)(P(z)-1+t)(P(z)-1+u). \end{equation} The symmetry among the variables $q,t,u$ in this equation immediately implies that the joint distribution of the statistics $\asc$, $\des$ and $\plat$ on $\mathcal{Q}_n$ is symmetric, in the sense that it is invariant under any permutation of the three statistics, agreeing with \cite[Thm.\ 2.1]{janson_plane_2008}. Setting $q=u=1$ and making the change of variable $z=\frac{z}{(1-t)^2}$ in Equation~\eqref{eq:Qplat} yields a different proof of \cite[Thm.\ 2.4]{gessel_stirling_1978}. \section{Further directions} \label{sec:further} In this section we discuss some open problems and potential directions of further research. \begin{problem} Give a combinatorial proof of Theorem~\ref{thm:QQn}, in analogy to Gessel and Stanley's second proof of \cite[Thm.\ 2.1]{gessel_stirling_1978} for Stirling polynomials. \end{problem} In the statement of Theorem~\ref{thm:QQn}, the coefficient of $t^m$ in $\overline{Q}_n(t)/(1-t)^{2n+1}$ can be interpreted as the number of permutations $\pi\in\overline{\mathcal{Q}}_n$ with $m$ bars inserted in some of the $2n+1$ spaces between the entries of $\pi$ (allowing bars to be inserted before the first entry and after the last entry), satisfying that there is some bar in each descent of $\pi$. A combinatorial proof would be obtained by showing that the number of such barred permutations equals $\frac{m^n}{n+1}\binom{m+n}{m}$. Our second question asks for a bijective proof of Equation~\eqref{eq:QI}, which states that the number of quasi-Stirling permutations $\pi\in\overline{\mathcal{Q}}_n$ with $\des(\pi)=m+1$ equals the number of injections $[n-1]\to[2n]$ with $m$ excedances, for all $n$ and $m$ . \begin{problem} Describe a bijection $f:\overline{\mathcal{Q}}_n\to\mathcal{J}_{2n,n+1}$ such that $\des(\pi)-1=\exc(f(\pi))$ for all $\pi\in\overline{\mathcal{Q}}_n$. \end{problem} Viewing quasi-Stirling permutations as noncrossing matchings of $[2n]$ whose arcs are labeled with the labels $1,2,\dots,n$, it is reasonable to consider the closely related set of labeled {\em nonnesting} matchings. These correspond to the set $\widetilde{\mathcal{Q}}_n$ of permutations of $\{1,1,2,2,\dots,n,n\}$ that avoid $1221$ and $2112$. By definition, $\mathcal{Q}_n\subseteq\widetilde{\mathcal{Q}}_n$. It is easy to see that $\|\widetilde{\mathcal{Q}}_n\|=n!C_n=\|\overline{\mathcal{Q}}_n\|$, since unlabeled nonnesting matchings, just like their noncrossing counterparts, are also counted by the Catalan numbers (see \cite{chen_crossings_2007} for more information on crossings and nestings in matchings), and there are $n!$ ways to assign labels to the arcs. The distribution of the number of descents on $\widetilde{\mathcal{Q}}_n$, which is different from its ditribution on $\overline{\mathcal{Q}}_n$, is the topic of a forthcoming preprint~\cite{archer_descents_nodate}. Finally, it may be interesting to study the distribution on $k$-quasi-Stirling permutations of other statistics such as the number of inversions, the major index, the number of left-to-right minima, the number of inverse descents, the number of blocks of specified sizes, or the distance between occurrences of elements. The distribution of these statistics on Stirling and $k$-Stirling permutations has been studied in~\cite{park_r-multipermutations_1994,park_inverse_1994,janson_generalized_2011,kuba_analysis_2011}. \subsection*{Acknlowledgements} The author thanks Kassie Archer and Adam Gregory for making him aware of Conjecture~\ref{conj:archer} and the notion of quasi-Stirling permutations, as well as for helpful discussions. \end{document}	arXiv
Hostname: page-component-7ccbd9845f-mpxzb Total loading time: 1.836 Render date: 2023-01-29T20:10:14.344Z Has data issue: true Feature Flags: { "useRatesEcommerce": false } hasContentIssue true >Journals >British Journal of Political Science >Volume 46 Issue 3 >Places and Preferences: A Longitudinal Analysis of... THE SOURCE OF THE SPATIAL CLUSTERING OF POLITICAL PREFERENCES: CONTEXTUAL EFFECTS OR SELECTION? DATA AND METHODOLOGY DISCUSSION AND CONCLUSIONS Places and Preferences: A Longitudinal Analysis of Self-Selection and Contextual Effects Published online by Cambridge University Press: 21 October 2014 Aina Gallego , Franz Buscha , Patrick Sturgis and Daniel Oberski Save PDF (0.41 mb) View PDF[Opens in a new window] Rights & Permissions[Opens in a new window] Contextual theories of political behaviour assert that the contexts in which people live influence their political beliefs and vote choices. Most studies, however, fail to distinguish contextual influence from self-selection of individuals into areas. This article advances understanding of this controversy by tracking the left–right position and party identification of thousands of individuals over an eighteen-year period in England before and after residential moves across areas with different political orientations. There is evidence of both non-random selection into areas and assimilation of new entrants to the majority political orientation. These effects are contingent on the type of area an individual moves into and contextual effects are weak and dominated by the larger effect of self-selection into areas. British Journal of Political Science , Volume 46 , Issue 3 , July 2016 , pp. 529 - 550 DOI: https://doi.org/10.1017/S0007123414000337[Opens in a new window] Copyright © Cambridge University Press 2014 It is unexceptional to remark that the political preferences of a national population are not randomly distributed across geographical areas. We know considerably less, however, about how this spatial clustering comes about. Contextual theories of political behaviour assert that elements of the environment in which individuals are situated exert a causal influence on the political parties and policies they prefer.Footnote 1 People, the argument goes, progressively assimilate through a variety of social-psychological mechanisms to the dominant political orientation of the environments in which they live. Consistent with these theories, a long tradition of research in electoral geography has examined the cross-sectional correlation between individual political preferences and social and political characteristics of local contexts.Footnote 2 Despite the large body of existing research in this area, however, the question of whether local contexts actually cause political preferences to change remains contested. Scholars have long argued that the correlation between individual political preferences and contextual characteristics is driven, in whole or in part, by self-selection of people into areas with congruent political beliefs.Footnote 3 It is our contention in this article that this longstanding debate remains unresolved because previous studies based on cross-sectional data have not been able to separate self-selection and contextual effects from one another convincingly. Although falling some way short of the gold standard of random assignment of individuals to areas, a longitudinal research design provides a considerably more satisfactory means of identifying the independent effects of assimilation and self-selection. This is because panel data enable the tracking of changes in political preferences before and after individuals move to contexts with different political majorities. Yet, because of the strong data requirements that longitudinal approaches impose, few examples of this type of strategy can be found in the existing literature. This article advances understanding in this area by assessing the causal effect of contexts on individual political orientations by tracking the preferences of individuals before and after residential moves, over an eighteen-year period. We do not seek to address the full range of causes and political consequences of internal migration, but focus our attention on two narrow yet fundamental research questions: Do people select into areas that exhibit majority political beliefs congruent with their own? And are individual political preferences influenced by the political orientation of the area into which an individual moves? To foreshadow our later results, our findings show that people are more likely to choose areas in which to live that are congruent with their pre-existing political preferences. Yet political orientation plays only a very limited, if any, role in location choice. Rather it is the socio-economic characteristics of individuals that are correlated with political preference, such as work and parental status, which are consequential in this regard. Self-selection into areas occurs, in short, for non-political reasons. We also find that in the years following a residential move, an individual's political preferences become more aligned with the majority political orientation of the area into which they moved. However, this process of assimilation is both weak and contingent upon area type; in England only those moving into strongly Conservative areas from other types of area exhibit evidence of contextual effects. Establishing that both self-selection and, albeit limited, contextual effects exist is important for several reasons. In a fundamental sense, it provides political scientists with a better understanding of the origins of political preferences. From a more practical perspective, both processes are likely to have important political consequences because they produce, over time, more spatially polarized political preferences. In politically homogeneous communities, residents are less likely to be exposed to diverse opinions, with negative implications for social and political tolerance, and the majority party will have larger margins of victory, resulting in less competitive elections. Geographical polarization of political preferences has also been shown to generate electoral biases in the translation from votes to seats in majoritarian systems and affects the incentives of parties to modify their policy platforms.Footnote 4 The remainder of this article is structured as follows. First, we review the existing literature on the geographical clustering of political preferences. A discussion of the limitations of existing methodological approaches to untangling contextual effects and selection mechanisms leads us to conclude that a longitudinal research strategy is required. We then describe the dataset and key measures to be used in our analysis and detail our model-fitting strategy, before presenting our empirical results. We conclude with a discussion of the limitations of our own approach and a consideration of the substantive implications of our findings. Scholars have argued that contextual influence occurs through a variety of social-psychological mechanisms,Footnote 5 including interpersonal contact and persuasion,Footnote 6 party mobilization,Footnote 7 exposure to shared local socio-economic conditions,Footnote 8 common local interests,Footnote 9 and exposure to low-intensity information cues.Footnote 10 For instance, theories of interpersonal contact argue that members of the political majority will tend to have their views reinforced, whereas those in the minority will change their views through processes of persuasion and conformity, generating a more homogeneous community outlook over time. Strong local party branches can also persuade local residents to change their political views through outreach activities. A large body of empirical evidence supports the main prediction of contextual theories in showing that there is a robust correlation between individual political preferences and area-level characteristics. For instance, Butler and Stokes showed that residents of British parliamentary constituencies vote for the local majority more frequently than would be expected for the population as a whole.Footnote 11 Similar correlations between measures of aggregate political orientations and individual political preferences have been replicated many times since.Footnote 12 Other studies have shown that socio-economic characteristics of neighbourhoods, variously defined and measured, predict vote choice above and beyond individual level characteristics.Footnote 13 Many studies have also found evidence consistent with the mechanisms hypothesized to underpin assimilation effects, such as interpersonal discussion or political mobilization.Footnote 14 While the large majority of existing studies rely on cross-sectional evidence, some have used short-run panels. Pattie and Johnston examined the characteristics of individuals who switched parties between the 1987 and 1992 general elections in Britain and found that people reporting discussion partners who supported a different party were more likely to have changed allegiance.Footnote 15 Johnston et al. showed with data from the British Household Panel Survey (BHPS) that people in the same constituency had changed their votes in similar directions across elections in 1992 and 1997, leading them to conclude that 'place matters'.Footnote 16 However, these two-wave studies remain somewhat inconclusive about causal order. That is to say, it may be the case that individuals who do not identify strongly with a party are both more likely to select discussion partners with different opinions and to switch parties. Hence, with only a few exceptions, it is still true that 'the standard approach in studies of context is to examine the effect that an aggregate-level compositional measure has on an individual behavior or attitude … covariation between the individual variable and aggregate variable is taken as evidence of a contextual effect'.Footnote 17 There are good reasons to assume that an individual's choice of a residential location will be correlated with prior political preferences. While it is unlikely that political preferences have a strong direct causal effect on residential choices, an indirect influence seems more plausible. This is because, when deciding where to relocate, movers must balance a broad range of preferences and constraints, including local housing prices, distance to work, quality of services, and tastes over the type of neighbourhood and the features of the house.Footnote 18 Political preferences are correlated with the socio-economic characteristics that constrain the choice of destination, such as income and family situation. Preferences are also correlated with tastes for many types of public goods (safety, scenic landscapes, cultural events, nightlife) which different localities provide.Footnote 19 Because of differences in both socio-economic constraints and location preferences we expect that those on the left and on the right will exhibit different location choices when they move and, specifically, that they will be more likely to move to politically like-minded communities. Recent research on the dynamics of partisan support have shown that, in the United States at least, movers tend to select areas that have majority political preferences more similar to their own than their original location.Footnote 20 While these studies provide evidence of non-random selection into areas, they tell us nothing about contextual effects and do not examine whether self-selection is political in nature. Consistent with the view that self-selection trumps contextual effects, critics of contextual effect theories have pointed out that the correlation between contextual characteristics and individual preferences generally becomes negligible, or zero, when controlling for individual level characteristics.Footnote 21 Hence, summarizing the critics' position in the debate, King concludes: 'The geographical variation is usually quite large to begin with, but after we control for what we have learned about voters, there isn't much left for contextual effects'.Footnote 22 Concerns about selection bias can, of course, be mitigated by controlling for socio-economic characteristics, which influence both the propensity to move and political preferences. A conditioning strategy, though, requires that all the requisite control variables are known and measurable, which seems unrealistic given that many of the candidate variables are notoriously difficult to measure in surveys (for example, housing tastes, early socialization experiences, personality). Thus, statistical control using cross-sectional data is unlikely to be a wholly effective strategy for dealing with selection bias. In summary, despite advances in both data and method, doubts remain over the core claim of theories of contextual effects: that contexts cause changes in political attitudes and behaviour. A more robust strategy for estimating the effects of contexts on political preferences is to make use of data containing longitudinal information about the same individuals over time. The primary advantage of repeated measurements or panel data is that, under certain model specifications, it is possible to partial out all observed and unobserved time-invariant characteristics of individual units.Footnote 23 As Halaby puts it, 'the problem of causal inference is fundamentally one of unobservables, and unobservables are at the heart of the contribution of panel data to solving problems of causal inference'.Footnote 24 The incorporation of a longitudinal dimension yields crucial additional leverage on questions of causal order, making it possible to model within-individual change as a function of preceding events.Footnote 25 Because this approach is based on the analysis of change in both dependent and independent variables within individuals over time, the estimated model coefficients are purged of the effects of all fixed (or 'time-invariant') respondent characteristics. Such fixed characteristics comprise both the 'usual suspects' such as gender, age cohort, and ethnicity, as well as less easily measurable variables such as personality traits and pre-adult socialization experiences. We are unaware of any existing study which has used this type of design to evaluate the extent and magnitude of self-selection and political assimilation to areas. This article has been motivated by the need to address this longstanding lacuna. To estimate the effect of area-level political orientation on individual political preferences, we have used the BHPS. We tracked individuals who moved across electoral constituencies over an eighteen-year period and observed whether movers were more likely to choose constituencies where their pre-existing views were closer to the views of existing residents than other potential choices of location (self-selection). Additionally, we evaluated whether the self-selection effects observed were political in nature, which is to say that individuals chose 'like-minded' areas because of their political orientation. We also assessed whether individuals adopted the political preferences prevalent in their new contexts over time. The BHPS is a large, high-quality repeated measures survey in which a stratified, multi-stage, random sample of British households had been interviewed annually since 1991. Computer-assisted face-to-face interviews were attempted with all household members aged 16 years or older. The initial Wave 1 household response rate was 74 per cent. Extensive efforts were made to track responding individuals across waves when a household had moved address, or when an individual moved from an existing household to a new one, such as when adult children had left home, or when a cohabiting couple separated. The study achieved a tracking rate averaging 95 per cent across all waves. The BHPS was thus ideally suited to our objectives because it contains a large number of residentially mobile individuals for whom self-reports of political preferences are observed before and after a move. Our analysis uses eighteen waves of data from 1991 to 2008 inclusive, with almost 10,000 individuals clustered within over 5,000 households in the first wave (1991). We restrict our focus to England only, excluding households in Wales, Scotland and Northern Ireland because the party systems in these countries are sufficiently different from England to make combined analyses difficult to interpret. We also exclude observations of those aged under 18 in order to match our analysis sample with the voting age population in England. We include 'new sample members' who join the BHPS through the formation of new households with 'original sample members' as well as 're-entrants' (i.e. those who had been non-respondents in the previous wave). These inclusion criteria yield an analysis sample of 17,373 individuals, who provide a combined total of 158,000 unique observations over the eighteen waves. The average number of waves completed by individuals is 9.14 and 4,100 individuals responded in all eighteen waves. To deal with the issue of non-random attrition, we include a range of control variables that predict drop-out from the study. Our estimates are, therefore, unbiased under the 'missing at random (MAR)' assumption, which we consider to be plausible in the current context.Footnote 26 Individual Political Preferences The BHPS offers two options for specifying individual political preferences. The first is a standard measure of party identification, which was administered in all eighteen waves; the second is a multi-item scale designed to measure people's 'left–right' economic value orientation, which was administered in Waves 1, 3, 5, 7, 10, 14 and 17.Footnote 27 Each measure has contrasting advantages and disadvantages. Party identification was measured in every wave and enables us to detect potential assimilation effects which are not based on changes in an individual's underlying preferences and beliefs, for example as a result of differences in the quality of candidates. By contrast, the left–right scale provides a finer-grained measure of political orientation, which enables detection of smaller changes across and within individuals over time and is not subject to the potentially distorting influence of tactical voting. Because of their differing theoretical and empirical properties we undertook all analyses using both measures of political preference. For the left–right scale, we took the first principal component of the six items, which is appropriate for these items.Footnote 28 For party identification, we considered only supporters of the two main parties, Labour and the Conservatives. This yields a binary variable for party support which is considerably more straightforward to handle in a panel data regression framework than a nominal variable with more than two categories. Areal Units An important question in the study of contextual effects is how the areal units defining spatial location should be defined. The mechanisms through which contextual influence operates can manifest themselves at small (e.g. interaction with neighbours), intermediate (e.g. party mobilization in a constituency), or large (e.g. regional media) spatial scales. Studies of the influence of spatial scales on political behaviour have found, as in other substantive contexts, that choice of scale is consequential for the estimates obtained.Footnote 29 In this study, we have used electoral constituencies as our areal units. Constituencies are the key electoral boundary in first-order, parliamentary elections in the United Kingdom. In England, parliamentary constituencies contain an average of 70,000 voters, having approximately the population size of a small town. While interpersonal interactions with neighbours that might result in political assimilation happen at finer-grained levels of geography, the constituency level should still be capable of capturing local conditions and interactions in school and work-place settings that require some short-range mobility. Additionally, since party organizations work to win a majority of the vote within the constituency boundaries and constituencies share the same MP, the political environment within a constituency will probably be more internally homogeneous than the country as a whole. Of more practical importance, however, is the fact that constituencies are the lowest geographical level at which it is possible to derive a useable measure of aggregate political orientation for the period in question. Electoral wards, which would be preferable with regard to size (they are smaller) are problematic because of the limited nature of information that can be attached to them and because their boundaries changed substantially between 1991 and 2008, rendering longitudinal analysis difficult. Other boundaries, such as census output areas,Footnote 30 have no feasible way of being linked to electoral results or to other measures of aggregate political orientation. Thus, while our choice of areal unit is not perfect, we believe it to be the best amongst the available alternatives. We return to the implications of our use of constituencies as the areal unit for the interpretation of our findings in the discussion section. While English constituency boundaries were quite stable in the period between 1991 and 2008, the redistricting for the 1997 general election affected a non-trivial number of constituency boundaries. However, most of the boundary changes affected only a small number of electors and the results we present here are robust to excluding observations located in constituencies which were subject to boundary changes during the reference period.Footnote 31 Area-Level Political Orientation We focus on the political orientation of the constituency as the main independent variable at the contextual level. Our measure is based on the electoral results for the four parliamentary elections held in 1992, 1997, 2001 and 2005. An intuitively appealing strategy would be to define constituency-level political orientation as the ratio (or similar function) of the vote share of the two main parties. However, this is challenging because of the sometimes significant role of tactical voting and of minor parties, which vary across constituencies and elections. Neither would it be clear how to apply vote shares to constituencies in non-election years. Therefore, we applied a typology to constituencies, placing them into one of six mutually exclusive categories:Footnote 32 — Safe Conservative constituencies (N=154): The Conservative party won a parliamentary seat in all four elections. — Safe Labour constituencies (N=211): The Labour party won a parliamentary seat in all four elections. — Marginal Conservative constituencies (N=12): The Conservative party won a parliamentary seat in three of the four elections. — Marginal Labour constituencies (N=111): The Labour party won a parliamentary seat in three of the four elections. — Safe or marginal Liberal-Democrat constituencies (N=31): The Liberal-Democratic party won a parliamentary seat in three or four elections. — Mixed constituencies (N=47): None of the three main parties won a seat in three or four of the elections. During the period of analysis, parties other than the main three won a parliamentary seat once in eight constituencies,Footnote 33 and only in one constituency did a different party win two elections. The mixed category is mostly made up of constituencies in which one of the three large parties won two elections and another of the large parties won the other two elections. Thus, these are the most competitive constituencies, where no party has a clear dominance. This classification of constituencies captures large differences in voting patterns. For instance, in the 1992 election the Conservative party won on average 57 per cent of the vote in safe Conservative seats but only 33 per cent in safe Labour seats. The Labour party received 18 per cent of the vote in safe Conservative seats and 53 per cent in safe Labour seats. Residential Mobility In our analysis sample, we observed a total of 14,500 residential moves from one wave to the next, which represents an average annual move rate of just over 9 per cent across the sample as a whole. Many of these relocations are, however, over a small distance within the same constituency and so would not be expected to result in discernible change in the external political environment. Therefore, we further restricted our definition of 'movers' to individuals who relocated to a different parliamentary constituency. This reduced the number of moves we observe by approximately half, yielding a total of 7,437. Using this definition, 69 per cent of respondents did not move at all, 17.5 per cent moved once, 7 per cent moved twice and 6.5 per cent moved three times or more during the period of observation.Footnote 34 Table 1 shows one-year transition probabilities (as percentages) for moves between different constituency types. Only individuals who were observed in at least two consecutive waves can be included in transition tables, which results in a reduction of the sample size from 142,000 to 125,000 observations.Footnote 35 The diagonal row in Table 2 comprises observations which remained in the same constituency type in any two-year period. The majority of observations did not move into different constituency types. However, of the 36,900 observations in safe Conservative constituencies, 315 moved into safe Labour constituencies in a subsequent year. Similarly, of the 41,100 observations in safe Labour constituencies, 417 moved to a safe Conservative constituency in a subsequent year. Of the 28,000 observations in marginal Labour constituencies, approximately 400 moved to safe Labour and safe Conservative constituencies respectively. Table 1 BHP3 Constituency Trasition Probabiilities Source: BHPS 1991–2008. Note: The table shows the total number of moves over all transition pair-years and, below these, the average percentage of respondents transitioning in such pair-years. Table 2 Panel Regression Models for Political Assimilation and Selection Effects Notes: Cluster standard errors in parentheses. Additional move types include Safe Con to Safe Con, Safe Lab to Safe Lab and Other Mover Types. Individual controls include: age, gender, education, marital status, children, income, class status, employment status, part/full-time, health status and time dummies. Coefficients in Models 1 to 5 are ordinary least squares, in Models 6 to 10 coefficients are odds ratios. Note that the long-run multiplier for Models 6 to 10 is derived by summing the exponentiated logit coefficients and converting this to odds ratios rather than summing the individual odds ratios. p<0.05, p<0.01, *p<0.001. † Reference group consists of those who have not moved. Because the inclusion of the full set of transition probabilities results in categories with small cell sizes we collapsed the full set of transitions into the following six categories: 1. No move 2. Moves from any constituency type (apart from safe Conservative) into safe Conservative 3. Moves from any constituency type (apart from safe Labour) into safe Labour 4. Moves from safe Conservative into safe Conservative 5. Moves from safe Labour into safe Labour 6. All other move types. It is moves of Types 2 and 3 which are of greatest analytical interest because they represent a clear change in the political orientation of the area in which an individual lives.Footnote 36 They can be considered, therefore, as ideal test-cases for theories of contextual effects. We focus our attention on these two move types in the analyses that follow, although analyses have been undertaken for all move types and will be made available upon request. To estimate the effect on individual political preferences of people moving to an area with a different political context, we use a panel data model with fixed effects and distributed lags and leads.Footnote 37 We include lagged effects because the influence of a new area on a mover is unlikely to occur immediately, potentially taking several years or more to materialize. The baseline model has the following form: (1) $$y_{{it}} =\mathop{\sum}\nolimits_{k={\rm 0}}^{\rm 5} {{\bf MovCon'}_{{i,t{-}k}} \beta _{{{-}k}} {\plus}x_{{it}} '\lambda } {\plus}e_{{it}} ,$$ where the political preferences of the i-th person in the t-th year, y it , are modelled as depending on the type of move in the preceding five years, MovCon′ i,t−k , a design vector corresponding to the six categories of move type described in the previous section, and 'no move' as the reference category. Time-varying covariates are collected in the vector x it . At the individual level we control for sex, age, age squared, educational level, income, social class, employment status, health status, marital status, and parental status. Where models are estimated with fixed effects, time-invariant characteristics such as sex are excluded. To control for spurious effects caused by unobserved differences between individuals that are time-invariant, each person's observations are centred on the within-person mean. That is, we use time-demeaned data such that an individual's score at time t is subtracted from their person specific mean over all observations across all time points $$(y_{{it}} =\tilde{y}_{{it}} {\minus}\mathop{y}^\limits{{{\vskip-1.5pt\hskip-5pt\tf="Els-ent4" \char 126}}} _{{i.}} )$$ . The well-known consequence of using time-demeaned data is that all time-invariant characteristics of sample units are 'differenced out', yielding the fixed effects model. The coefficients of primary interest in Equation 1 are the lagged coefficient vectors β −k where k is set to a maximum of 5. The choice of a five-year maximum lag is a trade-off between our theoretical expectation that it may take several years for a contextual effect to be manifested and the fact that extending lags beyond five years results in small cell sizes and, therefore, imprecise estimates. Moreover, as we shall show later, increasing the number of lags to nine years does not alter our key substantive findings. One way of thinking about how to interpret the lagged coefficients is to consider a hypothetical person who moves to a safe Labour constituency one year but then moves back the next year: β −k, labour will then be the effect on political preferences of that one-time move after k years, controlling for time-invariant unobserved variables and time-varying covariates. However, people typically stay in their new place of residence for longer than one year, so it is also of interest to know what the effect of the move on preferences will be when the effect is aggregated over ensuing years. This 'long-run effect' can be obtained by summing the coefficients over the lag vector: (2) $$\bibeta _{{{\rm long{\hbox-}run}}} =\mathop{\sum}\nolimits_{k={\rm 0}}^{\rm 5} {\bibeta _{{{-}k}} } $$ Hypothesis tests of zero long-run effects can be performed by obtaining the sampling variance of $\hat{\bibeta }_{{{\rm long{\hbox-}run}}} $ , which by standard methods, can be shown to be: (3) $${\mathop{\rm var}} (\hat{\bibeta }_{{{\rm long{\hbox-}run}}} )=\mathop{\sum}\nolimits_{k{\rm =0}}^{\rm 5} {{\rm var(}\hat{\bibeta }_{{{-}k}} {\rm ){\plus}2}\mathop{\sum}\nolimits_{k\,\lt\, l} {{\mathop{\rm cov}} } (\hat{\bibeta }_{{{\minus}k}} ,\hat{\bibeta }_{{{\minus}l}} )} $$ Note that this means that individual hypothesis tests performed on the lagged effects β −k may be non-significant while the overall hypothesis test on β long-run is significantly greater than 0. By using a fixed effects model with time-varying covariates, we control for possible confounding due to unobserved between-person differences, as well as observed differences due to the covariates. Although this is already a strong research design, confounding could conceivably still occur if the probability of moving is correlated with the propensity to change political preferences. For example, becoming a parent is an event that can cause both a residential move and a change in political preferences. Blanden et al. propose controlling for the effect of such 'pre-programme trends' by including 'lead' dummies in the model.Footnote 38 The coefficients for the effect of the future on the present, γ +k , should not be interpreted causally, but as evidence for selection effects on the change in preferences. By including leads in the model, our final specification becomes: (4) $$y_{{it}} =\mathop{\sum}\nolimits_{k{\rm =0}}^{\rm 5} {{\bf MovCon'}_{{i,t{-}k}} \bibeta _{{{\rm {-}}k}} {\plus}} \mathop{\sum}\nolimits_{k{\rm =0}}^{\rm 5} {{\bf MovCon'}_{{i,t{\plus}k}} \bigamma _{{{\rm {\plus}}k}} {\plus}x_{{it}} '\lambda {\plus}e_{{it}} } $$ In addition to their substantive interpretation as indicators of political selection into areas, the inclusion of leads also enables us to obtain estimates of the lagged effects, adjusted for non-random selection into areas. Thus, our control for time-constant, time-varying, as well as trend selection effects motivates the interpretation of the lagged coefficients β − k and their long-run versions β long-run as the effect of the constituency on an individual's political preferences. Finally, it should be noted that the use of lags and leads in their raw form results in reduced sample size. This is because for some observations, we do not observe political preferences five years before and after a move. Rather than dropping such observations we set the unobserved prior and subsequent moves to 0 and include a vector of dummies representing 'missingness' in the model, although we do not report the coefficient estimates in our results. Specification tests show that our findings are substantively unaffected by the inclusion or exclusion of these cases. We begin by presenting some descriptive statistics before moving to more causally-focused analyses. An implication of contextual theories of political behaviour is that the magnitude of an assimilation effect should (initially at least) increase over time, because the opportunity for and experience of the various influence mechanisms will grow as a function of time spent in a locale.Footnote 39 Thus, we should expect the association between constituency type and individual policy preferences to increase over the number of years an individual has lived in the area. Figure 1 shows percentage support for the Conservative party and the mean of the left–right scale by type of constituency and the number of years the individual has lived at their current address.Footnote 40 Individuals who live in safe Conservative constituencies are likely to support the Conservative party even immediately after moving to their new place of residence. Additionally, the longer the period of residence within a safe Conservative constituency, the more likely an individual is to support the Conservatives. Conversely, individuals in safe Labour constituencies, while considerably more likely to support the Labour party, show no trend towards increasing support for Labour the longer they have lived in a safe Labour constituency. This may be due, partially at least, to a ceiling effect because the level of support for Labour in safe Labour constituencies is already close to 75 per cent at year zero. The pattern for the left–right scale is clearer and more consistent, with individuals in safe Conservative constituencies expressing more right-wing views the longer they have lived in that area and the opposite being the case for individuals in safe Labour constituencies. Thus, the BHPS provides quite strong preliminary evidence of contextual effects; people are closer to the aggregate political orientation of their constituency the longer they have lived in it. However, although Figure 1 shows an apparent trend over time, the data is analysed cross-sectionally and the variation is, therefore, between rather than within individuals. Thus, the patterns we observe may have emerged due to non-random selection of individuals into (and out of) areas rather than to the effect of areas on individuals. It is to this possibility that we now turn via regression analysis. Table 2 presents the results of our regression models for the left–right value scale and party support over ten columns, with each column representing a different model specification. In Model 1 we include only lags, no individual-level controls and no fixed effects. Model 2 adds leads to this specification. Model 3 contains lags and individual level controls but no leads or fixed effects, whilst Model 4 introduces individual fixed effects to Model 3. Finally, Model 5 reintroduces lead indicators. The same pattern is repeated for party support in Models 6 to 10, though now we use a logistic link function because the outcome is binary. In Table 2 we suppress the results for other move types (such as moving from a safe Conservative constituency to another safe Conservative constituency) and for the control variables.Footnote 41 The results of these additional contrasts make no material difference to our substantive conclusions. In all models the reference category for the different move types is non-movers. Thus the coefficients should be interpreted as the effect of making the move type in question on political preferences, compared to (covariate adjusted) non-movers. Model 1 in Table 2 shows that moving into a safe Conservative constituency from any other type of constituency is associated with a significant move to the right on the left–right scale (higher scores indicate more right-wing preferences). The long-run multiplier for this move type is 0.946 (p<0.001), indicating significant and quite substantial contextual effects over the five years following the move. The magnitude of this effect is approximately equivalent to the average cross-sectional difference in left–right scores between a Liberal Democrat and a Conservative party identifier. The negative coefficients for the effect of moving into a safe Labour constituency from other constituency types provides some evidence of assimilation for this type of move, though none of these is statistically significant, either singly or in combination. This pattern corresponds to that which was observed for the cross-sectional analysis in Figure 1, where the trend for Conservative support was stronger than for Labour support. Model 2 introduces lead indicators. Significant coefficients for the lead dummies suggest that, before controlling for other characteristics, individuals behave as if they choose their new areas, at least in part, on the basis of their prior political preferences. Our results suggest that those moving to safe Conservative areas become significantly more right-wing prior to a move. The coefficient estimates for those moving to safe Labour constituencies are not statistically significant, although they are of the correct sign (they become more left-wing). The long-run multiplier for those moving to safe Conservative areas between time period 0 and time period 5 has been reduced slightly to 0.845 (p<0.003), indicating smaller, but still substantial, assimilation effects after controlling for non-random selection into constituency types. However, these estimates cannot be considered causal as we have not yet controlled for selection on observed time-varying and time-invariant characteristics. In Model 3, we remove the lead dummies and introduce individual level controls for characteristics which might lead people to relocate to a different constituency and also to change their political orientation. These are: age, gender, marital status, parental status, household income, social class, employment status and health status. We also include year dummies to control for macro-level events in the external environment. Introduction of these controls results in the lagged coefficients becoming somewhat reduced in magnitude, which suggests that the assimilation effects observed in Model 1 are at least partly due to non-random selection of individuals into constituencies. Controlling for these individual characteristics and survey year reduces the long-run multiplier to 0.639, which although statistically significant (p<0.018), represents a 30 per cent reduction in magnitude compared to Model 1. Estimates for those moving into safe Labour constituencies change only marginally with the introduction of controls in Model 3; these individuals become more left-wing over time, although the effect cannot be distinguished from zero when inference is made to the broader population (five-year cumulative estimate=−0.865 (p<0.772)). The introduction of individual fixed effects, to control for time invariant characteristics, in Model 4 reduces the magnitude of the assimilation effects quite substantially, with the contemporaneous coefficient for Conservative constituency moves, in particular, reduced from 0.206 to a statistically non-significant 0.047. However, while the majority of coefficients are reduced in magnitude, there are exceptions at the three-year and four-year lags for safe Conservative (0.139; p<0.05) and safe Labour (−0.201; p<0.01) constituency moves, respectively. This suggests that assimilation, rather than occurring immediately after a move, takes place after some years in the new location. For safe Conservative constituency moves, the long-run effect is reduced to 0.428, but this is still statistically different from zero (p<0.011). For Labour, the long-run multiplier is −0.200 (p<0.310) which suggests that the small apparent assimilation effect observed after four years is removed when combined with the effects in the other years. Finally, Model 5 re-introduces lead dummies in order to control for all forms of self-selection in one model. Results suggest that re-introducing the leads has only a small effect on the estimates. The leads are statistically non-significant for those moving to safe Conservative constituencies and have little effect on the coefficient estimates of Model 4. The long-run multiplier becomes marginally non-significant in Model 5. However, none of the lead coefficients in Model 5 are themselves significantly different from zero; hence our preferred estimates are those in Model 4. For those moving to safe Labour constituencies, however, there is some limited evidence of political self-selection; two years prior to a move, individuals moving to a safe Labour constituency become somewhat more left-wing, suggesting that move choice is related to prior shifts in political orientation, though the magnitude of the effect is weak. Moreover, political self-selection has no material effect on our conclusions regarding assimilation for those moving to safe Labour constituencies, with the magnitude and significance of the coefficients unaltered by the introduction of the leads. We now turn to the results for party identification, which are presented in Models 6 to 10 in Table 2. It should be noted that the sample size for some of these models is reduced substantially compared to the linear specifications in Models 1 to 5. This is because relatively few individuals changed their party identification between Labour and the Conservatives during the period of observation and only individuals who change on the outcome contribute to the parameter estimates in a fixed effects model. We must, therefore, be cautious in our interpretation of these models, because our power of inference is weak. The pattern of coefficients across Models 6 to 10 is quite similar to that found for the left–right scale models. There is evidence in the 'naïve' Models 6, 7 and 8 of political assimilation for individuals moving into safe Conservative constituencies but only weak and inconsistent support for an effect of moves into safe Labour constituencies. Once individual fixed effects are introduced in Models 9 and 10, there is no evidence of assimilation occurring for either type of move. Indeed, these results suggest that individuals moving to safe Conservative constituencies become less likely to support the Conservative party over time, although these estimates are not statistically significant. Long-run multiplier coefficients for those moving into safe Conservative and safe Labour constituencies are also not statistically distinguishable from zero. Because the BHPS has eighteen waves of data, it is possible to extend the annual lagged and cumulative effects beyond five years after a move. Presentation of these models in tabular form is cumbersome, so we show them in graphical summary form in Figures 2 and 3. We present estimates corresponding to the left–right scale Models 1 and 4 and party support Models 6 and 9 with nine lags instead of five. This provides a contrast between naïve estimation (lags only model) and estimation which is more robust to potential confounders.Footnote 42 Figure 2 shows the effect of moving to safe Conservative and Labour constituencies on left–right attitudes up to nine years after a move. Fig. 1 Party identification and left–right economic values over time lived in constituency Fig. 2 Assimilation to left–right values by type of moves over time with extended lags These longer-run estimates produce similar results to those presented in Table 2; a long-run effect is apparent for left–right attitudes for those moving to a safe Conservative constituency in the model with individual-level controls only. However, once individual fixed effects and time-varying individual level characteristics are controlled for, this long-run effect is approximately halved. There is no commensurate effect on left–right attitudes for individuals moving to safe Labour constituencies, with or without fixed effects. The models for party support in Figure 3 also suggest the presence of an assimilation effect for individuals moving to safe Conservative constituencies, but these are statistically non-significant when individual fixed effects are added to the model. No evidence of assimilation effects is evident for individuals who move to safe Labour constituencies, even before the inclusion of fixed effects. Fig. 3 Assimilation to Conservative party support by type of moves over time with extended lags In this article we have taken a new approach to addressing an enduring controversy in the study of political behaviour. While theories which posit a causal effect of geographical context on individual political preferences have a long tradition in political science, existing studies have yet to provide convincing evidence that individuals do indeed assimilate, over time, to the majority preferences of the areas in which they live. Our analyses advance the existing state-of-the art in this field by tracking the political preferences of a large sample of individuals over an eighteen-year period. Our analysis used panel data models with fixed effects and controls for time-varying individual characteristics. This longitudinal approach yields a considerably stronger protection against the primary threat to valid causal inference in standard cross-sectional designs, namely that individuals choose which areas they wish to move to (and remain in) and that these choices are themselves correlated with political preferences. Our results suggest that political assimilation effects were evident in England during the period 1991–2008 but that these were weak and differential across different types of areas. On the one hand, movers to safe Conservative seats became more economically right-wing and more likely to vote Conservative following the move. This suggests that, consistent with the predictions of contextual theories of political behaviour, moving to a more conservative area leads individuals to become more aligned in their political preferences with the local majority. On the other hand, we found no change in left–right attitudes and only very weak evidence of change in party identification amongst movers to safe Labour constituencies. Several factors may account for the differential assimilation effects across constituency types. First, people who move to a safe Labour area already have a high probability of voting for the Labour party and of having economically left-wing attitudes. Thus, the contingent nature of our findings may be due to a ceiling effect; there is little scope for movers into Labour areas to become more left-wing than they already are immediately prior to moving. An alternative possibility is that the mechanisms through which contextual effects are manifested are less powerful in safe Labour seats. Safe Labour seats are mostly located in urban areas such as London, Birmingham, Manchester, Newcastle upon Tyne and Liverpool. The social pressure to conform to the local majority may be less strong in socially diverse, urban areas than in more homogeneous rural or suburban areas.Footnote 43 Conservatives, who traditionally value conformity to existing social norms, may be more likely than those on the left to pressure newcomers to conform to the local majority position. These possibilities are, however, speculative and it is not possible to establish, with the data available to us, why moving to a Conservative area has an effect on political preferences, while moving to a Labour context does not. Be that as it may, the finding that contexts have heterogeneous effects is important in its own right because it suggests that it is necessary, for a complete account, to clearly specify the conditions under which we should expect areal units to affect political preferences. With regard to selection into areas, previous studies have demonstrated that American movers, on average, relocate into areas with more congruent political beliefs than the constituencies from which they moved.Footnote 44 Our results show that this kind of geographic sorting generalizes to the British context; an individual's existing political preference is a strong and significant predictor of the political orientation of the area into which he or she moves. However, the finding that citizens relocate to constituencies that are more congruent with their existing political beliefs does not imply that the choice of locale is caused by political orientation. In our analysis, when individual level controls are introduced, political preferences prior to the time of moving no longer predict the political orientation of the destination constituency. This suggests that sorting of politically like-minded individuals into areas arises indirectly, because people with different political preferences also have different socio-economic characteristics which are the actual causal determinants of residential location choices. In short, self-selection into areas appears to be almost entirely non-political in nature. While our study significantly improves on previous attempts to identify contextual effects, it has limitations of its own which should be acknowledged. In particular our choice of areal unit (Westminster constituencies) can be criticized for inadequately representing the spatial scale at which the mechanisms generally thought to underlie assimilation are likely to operate. Neighbourhood effects theories generally contend that context effects operate primarily through social-psychological processes, such as interpersonal influence and persuasion, which are likely to happen at smaller spatial scales than a parliamentary constituency. Using smaller areal units would almost certainly result in different estimates of assimilation and selection effects than those we have presented here.Footnote 45 While we acknowledge that this problem is pertinent to the interpretation of our findings, we do not believe that it invalidates our findings and conclusions. As we have argued, the constituency is a substantively important context because it is the relevant electoral scale in parliamentary elections and political parties are incentivized to target mobilization efforts strategically across constituencies. In addition, the fact that we find evidence consistent with contextual effects for movers to Conservative constituencies suggests that the null results for some mover types are not due merely to the use of a large areal unit. While we cannot conclude that our results will necessarily generalize across different spatial scales, or to different political contexts, this is not something that we should expect to be the case in any event. The modifiable areal unit problem should not, in short, be taken as a threat to the validity of our conclusions as it is equally pertinent to any spatial scale that an analyst happens, or is able, to select. A second limitation of our approach is that it overlooks other important ways in which assimilation is likely to take place. In particular, our focus on adult movers means that we cannot draw conclusions about two important groups: young people and those who reside in the same area (or same type of area) for long periods. The places where people live in their childhood, adolescence and early adulthood are likely to shape their political outlook more profoundly than at other times in their lives.Footnote 46 Because we focus only on adults aged 18 and over, we would not detect effects which occur prior to adulthood. And, although it would, in principle, be possible to break our analyses down across age groups, in practice our sample of movers is too small to be able to detect differences between age groups that might exist in the population reliably. Similarly, it is plausible that contextual effects occur for individuals who never move but are nonetheless influenced by the changing political environment within their own locale over time. However, identification of causal assimilation effects for non-movers is a considerably more challenging analytical task than for a sample of movers and consideration of this important group is beyond the scope of this article. A third limitation of our research is that it only addresses one possible source of heterogeneity in treatment effects (the political hue of the destination constituency). Both individual and contextual characteristics may influence the extent to which individuals adopt the political preferences prevalent in their local environments. For instance, the reasons why individuals move – such as getting a new job, moving to a more family-friendly neighbourhood, or attending university – may themselves shape the propensity to assimilate to the new context. Very stable local communities may affect newcomers in different ways than gentrifying communities or places that are growing fast over a specific period of time. Long-distance moves that disrupt social relationships may have different implications for political preferences than short-distance moves where it is easier to maintain pre-existing social networks. While we acknowledge, then, that our conclusions cannot be generalized without appropriate caution to other spatial levels or to other population sub-groups, we believe that our findings are important nevertheless. The evidence that we have presented suggests that self-selection into areas is considerably more important than assimilation effects in producing the spatial clustering of political preferences long observed by political geographers. This confirms the dominance of selection over assimilation that has been observed in other contexts, including those which have used experimental designs.Footnote 47 We have also shown for the first time that self-selection into areas is almost entirely non-political in nature, in the sense that individuals do not choose where to live on the basis of political preference per se, but as a result of socio-economic characteristics which are jointly correlated with choice of location and political orientation. Location affects individual political preferences, but only weakly, in some areas, and for some outcomes. Thus, while contexts are certainly relevant to our understanding of political preferences, they appear to be considerably less important than proponents of contextual theories have sometimes maintained. Gallego: Institut de Barcelona d'Estudis Internacionals (email: [email protected]); Buscha: Department of Economics and Quantitative Methods, University of Westminster (email: [email protected]); Sturgis: Department of Social Statistics and Demography, University of Southampton (email: [email protected]); Oberski: Methodology Department, Tilburg University (email: [email protected]). The authors gratefully acknowledge the support of the Economic and Social Research Council through the grant for the National Centre for Research Methods (NCRM; grant reference: RES-576-47-5001-01) and from the Marie Curie Actions of the European Union's Seventh Framework Programme under REA grant agreement no. 334054 (PCIG12-GA-2012-334054). The code utilized to produce the results is posted in the BJPS repository. Data replication sets and online appendices are available at http://dx.doi.org/doi: 10.1017/S0007123414000337. The British Household Panel Study, however, does not allow dissemination of the micro-data. 1 Agnew Reference Agnew1987; Books and Prysby Reference Books and Prysby1988; Burbank Reference Burbank1997; Cox Reference Cox1969; Ethington and McDaniel Reference Ethington and McDaniel2007; Huckfeldt and Sprague Reference Huckfeldt and Sprague1995; Johnston and Pattie Reference Johnston and Pattie2006. 2 Andersen and Heath Reference Andersen and Heath2002; Butler and Stokes Reference Butler and Stokes1974; Crewe and Payne Reference Crewe and Payne1976; Johnston et al. Reference Johnston, Jones, Sarker, Propper, Burgess and Bolster2004; Johnston, Pattie and Allsopp Reference Johnston, Pattie and Allsopp1988; McAllister et al. Reference McAllister, Johnston, Pattie, Tunstall, Dorling and Rossiter2001; Miller Reference Miller1978. 3 Dunleavy Reference Dunleavy1979; Kelley and McAllister Reference Kelley and McAllister1985; King Reference King1996; McAllister and Studlar Reference McAllister and Studlar1992. 4 Chen and Rodden Reference Chen and Rodden2009; Rodden Reference Rodden2010; Rodden Reference Rodden2012. 5 While existing work focuses mostly on social mechanisms, other contextual characteristics such as climate or geographic features can also conceivably affect political behaviour. 6 Butler and Stokes Reference Butler and Stokes1974; Huckfeldt, Ikeda and Pappi Reference Huckfeldt, Ken'ichi and Pappi2005; Huckfeldt and Sprague Reference Huckfeldt and Sprague1995. 7 Denver and Hands Reference Denver and Hands1997; Pattie, Johnston and Fieldhouse Reference Pattie, Johnston and Fieldhouse1995. 8 Books and Prysby Reference Books and Prysby1988. 9 Cutler Reference Cutler2007. 10 Cho and Rudolph Reference Cho and Rudolph2008; Huckfeldt and Sprague Reference Huckfeldt and Sprague1992. 11 Butler and Stokes Reference Butler and Stokes1974. 12 E.g. Cox Reference Cox1969; Crewe and Payne Reference Crewe and Payne1976; Taylor and Johnston Reference Taylor and Johnston1979. 13 E.g. Andersen and Heath Reference Andersen and Heath2002; Johnston, Pattie and Allsopp Reference Johnston, Pattie and Allsopp1988; Johnston et al. Reference Johnston, Propper, Burgess, Sarker, Bolster and Jones2005; McAllister et al. Reference McAllister, Johnston, Pattie, Tunstall, Dorling and Rossiter2001; Miller Reference Miller1978. 14 For a review, see Johnston and Pattie Reference Johnston and Pattie2006. 15 Pattie and Johnston Reference Pattie and Johnston2000. 16 Johnston et al. (Reference Johnston, Pattie, Dorling, MacAllister, Tunstall and Rossiter2001), p. 107. 17 Baybeck and McClurg (Reference Baybeck and McClurg2005), p. 494. 18 Rabe and Taylor 2010. 19 E.g., see Florida (2003) on cities of the creative class; and Tiebout (Reference Tiebout1956) on political orientation and preferred bundles of taxes and services. 20 Cho, Gimpel and Hui Reference Cho, Gimpel and Hui2012; McDonald Reference McDonald2011. 21 Kelley and McAllister Reference Kelley and McAllister1985; McAllister and Studlar Reference McAllister and Studlar1992. 22 King (Reference King1996), p. 160. 23 Halaby Reference Halaby2003; Halaby Reference Halaby2004; Wooldridge Reference Wooldridge2002. 24 Halaby (Reference Halaby2003), p. 2. 25 Allison Reference Allison1994. 26 That is, we consider it unlikely that political orientation is a strong determinant of drop-out from the study. Although the BHPS contains a longitudinal weight, using this to correct for differential attrition is not attractive because any unit with a single missing wave of data over the eighteen years of observation is dropped from the weighted estimator. 27 Evans, Heath and Lalljee 1996; Heath, Evans and Martin Reference Heath, Evans and Martin1994. 28 Sturgis Reference Sturgis2002. 29 Cutts and Fieldhouse Reference Cutts and Fieldhouse2009; Johnston et al. Reference Johnston, Jones, Propper and Burgess2007. 30 Martin 2008. 31 Constituency identifiers in the BHPS are over-written with the new constituency code when boundaries change. This means that it is not possible to identify which respondents in our analysis sample changed to a different constituency without moving house due to the 1997 boundary revisions. However, we are able to identify respondents who are resident in constituencies which were created in 1997 and therefore we can estimate models including and excluding this group. We find no significant differences in the patterns reported in the analyses. 32 In case of boundary revision, the coding matches any constituency revised in 1997 to the 1992 constituency with which it has the largest overlap (see Norris Reference Norris2005). 33 These are: Bethnal Green and Bow, Birmingham Sparkbrook and Small Heath, Brentwood and Ongar, East Ham, Staffordshire South, Tatton, West Bromwich West, West Ham. 34 While the United States is widely considered to have high rates of residential mobility, one-year mobility rates are similar to those observed in Britain; 12 per cent of the US population moved to a different address in 2005, while 11 per cent of the population in Britain did so (Molloy, Smith and Wozniak Reference Molloy, Smith and Wozniak2011). Data from the 2001 Census and administrative records suggest that about 6.7 million UK residents, or 11.4 per cent of the population, moved from one address to another in the previous twelve months (Champion Reference Champion2005). Most of these moves are over short distances, with approximately two fifths moving less than 2 km away and only one third moving more than 10 km away. 35 All observations in the first wave (1991) of the BHPS do not have earlier information and can, therefore, not be used. Individuals who drop out of waves cannot provide transition information. 36 In principle moves from safe Conservative to safe Labour constituencies and vice versa are of greatest theoretical interest, although our sample size is insufficient to go to this level of granularity. 37 Blanden et al. Reference Blanden, Buscha, Sturgis and Urwin2012; Laporte and Windmeijer Reference Laporte and Windmeijer2005. 38 Blanden et al. Reference Blanden, Buscha, Sturgis and Urwin2012. 39 A corollary example is the long-established association between the political preferences of married and co-habiting couples. Alford et al. (Reference Alford, Hatemi, Hibbing, Martin and Eaves2011) argue that, if the intra-spousal correlation is due to influence rather than self-selection, we should observe partners becoming more similar to each other the longer they are together. 40 The 'time lived in constituency' variable is derived from the address record rather than by self-report. We limit the upper bound of time at address to ten years because 'time at address' is increasingly confounded with age as 'time at residence' increases. 41 These results are available from the corresponding author upon request. 42 The overall pattern is very similar when using different specifications, e.g. when plotting estimates from Table 2 with five lags or from lag and lead models. 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\begin{document} \title{Scopes and limits of modality in quantum mechanics} \author{{\sc Graciela Domenech}\thanks{ Fellow of the Consejo Nacional de Investigaciones Cient\'{\i}ficas y T\'ecnicas (CONICET)} $^{1, 3}$,\ \ {\sc Hector Freytes} $^{2}$\ {\sc and} \ {\sc Christian de Ronde} $^{3, 4}$} \maketitle \begin{center} \begin{small} 1. Instituto de Astronom\'{\i}a y F\'{\i}sica del Espacio (IAFE)\\ Casilla de Correo 67, Sucursal 28, 1428 Buenos Aires, Argentina\\ 2. Dipartimento di Scienze e Pedagogiche e Filosofiche - Universita degli Studi di Cagliari \\ Via Is Mirrionis 1, 09123, Cagliari - Italia\\ 3. Center Leo Apostel (CLEA)\\ 4. Foundations of the Exact Sciences (FUND) \\ Brussels Free University - Krijgskundestraat 33, 1160 Brussels - Belgium \end{small} \end{center} \begin{abstract} We develop an algebraic frame for the simultaneous treatment of actual and possible properties of quantum systems. We show that, in spite of the fact that the language is enriched with the addition of a modal operator to the orthomodular structure, contextuality remains a central feature of quantum systems. \end{abstract} \maketitle \newtheorem{theo}{Theorem}[section] \newtheorem{definition}[theo]{Definition} \newtheorem{lem}[theo]{Lemma} \newtheorem{prop}[theo]{Proposition} \newtheorem{coro}[theo]{Corollary} \newtheorem{exam}[theo]{Example} \newtheorem{rema}[theo]{Remark}{\hspace{4mm}} \newtheorem{example}[theo]{Example} \newcommand{\proof}{\noindent {\em Proof:\/}{\hspace{4mm}}} \newcommand{ $\Box$}{ $\Box$} \section{Introduction} Contextuality is one of the main features of the discourse about quantum systems and has been studied from different approaches. We are interested here in algebraic versions related to partial valuations of the orthomodular lattice of closed subspaces of Hilbert space. This proposal allows to identify the constraints imposed by the structure to the relation between actuality and possibility and the discourse that includes both type of propositions. The paper is organized as follows: in Sec. 2 we discuss the contextual character of quantum systems from an algebraic perspective. Sec. 3 introduces the desiderata of modal interpretations of quantum mechanics involved in our treatment. We devote Sec. 4 to expose the algebraic structure which represents the orthomodular lattice enriched with modal operators. In Sec. 5 we show how the discourse about properties is genuinely enlarged giving an adequate framework to represent the Born rule. In Sec. 6 we use Kochen-Specker theorem to prove that the contextual character of quantum mechanics is maintained even when the discourse is enriched with modalities. Finally, we outline our conclusions. \section{Contextuality in quantum systems} In paper \cite{DF}, we dealt with the problem of the limits imposed by the orthomodular structure of projection operators to the possibility of thinking of properties possessed by an isolated quantum system. This is an important point in the discussion about quantum systems because almost every problem in the relation between the mathematical formalism and what may be called our experience about the behavior of physical objects can be encoded in the question about the possible meaning of the proposition {\it the physical magnitude ${\cal A}$ has a value and the value is this or that real number}. Already from the first formalizations, this point was recognized. For example, P.A.M. Dirac stated in his famous book: ``The expression that an observable `has a particular value' for a particular state is permissible in quantum mechanics in the special case when a measurement of the observable is certain to lead to the particular value, so that the state is an eigenstate of the observable. It may easily be verified from the algebra that, with this restricted meaning for an observable `having a value', if two observables have values for a particular state, then for this state the sum of the two observables (if the sum is an observable) has a value equal to the sum of the values of the two observables separately and the product of the two observables (if this product is an observable) has a value equal to the product of the values of the two observables separately'' \cite{Dirac}. This last point is the requirement of the functional compatibility condition (FUNC), to which we will return later. As long as we limit ourselves to speaking about measuring results and avoid being concerned with what happens to nature when she is not measured, quantum mechanics carries out predictions with great accuracy. But if we naively try to interpret eigenvalues as actual values of the physical properties of a system, we are faced with all kind of no-go theorems that preclude this possibility. Most remarkable is the Kochen-Specker (KS) theorem that rules out the non-contextual assignment of values to physical magnitudes \cite{ks}. In \cite{DF} we gave algebraic, topological and categorial versions of the KS theorem. To include modalities in our analysis we will deal with the algebraic form in terms of valuations. So we recall the main features of our discussion there. Let ${\mathcal H}$ be the Hilbert space associated with the physical system and $L({\mathcal H})$ be the set of closed subspaces on ${\mathcal H}$. If we consider the set of these subspaces ordered by inclusion, then $L({\mathcal H})$ is a complete orthomodular lattice \cite{MM}. It is well known that each self-adjoint operator $\bf A$ representing a physical magnitude ${\cal A}$ has associated a Boolean sublattice $W_A$ of $L({\mathcal H})$. More precisely, $W_A$ is the Boolean algebra of projectors ${\bf P}_i$ of the spectral decomposition ${\bf A}=\sum_{i} a_i {\bf P}_i$. We will refer to $W_A$ as the spectral algebra of the operator $\bf A$. Any proposition about the system is represented by an element of $L({\mathcal H})$ which is the algebra of quantum logic introduced by G. Birkhoff and J. von Neumann \cite{BV}. Assigning values to a physical quantity ${\cal A}$ is equivalent to establishing a Boolean homomorphism $v: W_A \rightarrow {\bf 2}$ \cite{IB}, being ${\bf 2}$ the two elements Boolean algebra. So it is natural to consider the following definition: \begin{definition} Let $(W_i)_{i\in I}$ be the family of Boolean sublattices of $L({\mathcal H})$. A global valuation over $L({\mathcal H})$ is a family of Boolean homomorphisms $(v_i: W_i \rightarrow {\bf 2})_{i\in I}$ such that $v_i\mid W_i \cap W_j = v_j\mid W_i \cap W_j$ for each $i,j \in I$ \end{definition} Were it possible, this global valuation would give the values of all magnitudes at the same time maintaining a {\it compatibility condition} in the sense that whenever two magnitudes share one or more projectors, the values assigned to those projectors are the same from every context. But the KS theorem assures that we cannot assign real numbers pertaining to their spectra to operators ${\bf A}$ in such a way to satisfy the functional composition principle (FUNC) which is the expression of the ``natural'' requirement mentioned by Dirac that, for any operator ${\bf A}$ representing a dynamical variable and any real-valued function $f({\bf A})$, the value of $f({\bf A})$ is the corresponding function of the value of ${\bf A}$. In the algebraic terms of the previous definition, the KS theorem reads: \begin{theo}\label{CS2} If $\mathcal{H}$ is a Hilbert space such that $dim({\cal H}) > 2$, then a global valuation over $L({\mathcal H})$ is not possible. $\Box$ \\ \end{theo} Of course contextual valuations allow us to refer to different sets of actual properties of the system which define its state in each case. Algebraically, a {\it contextual valuation} is a Boolean valuation over one chosen spectral algebra. In classical particle mechanics it is possible to define a Boolean valuation of all propositions, that is to say, it is possible to give a value to all the properties in such a way of satisfying FUNC. This possibility is lost in the quantum case. Now we intend to show how this discussion is able to include {\it modalities}, i.e. to consider also possibility and necessity of propositions about the properties of physical systems: what {\it may be the case} (what the possible physical situations are) and what {\it is necessarily the case}. The consideration of {\it possibility} is of course always present in quantum theories. What we propose here is an algebraic consideration of this type of sentences and their articulation with those about actual properties. Several attempts to obtain modal extensions of the orthomodular systems are found in the literature. One possibility developed in \cite{Dish} allows to embed orthomodular propositional systems in modal systems. Another extension is provided by adding quantifiers to the orthomodular structure \cite{JAN1, JAN2}, so generalizing the monadic extension of the Boolean algebras \cite{HAL}. In our case, we enrich the orthomodular structure with a modal operator thus obtaining an algebraic variety such that each orthomodular lattice can be represented by an algebra of this variety. On the other hand, this operator acts as a quantifier in the sense of \cite{JAN1, JAN2}. The physical motivation for this construction is the purpose to link consistently the propositions about actual and possible properties of the system in a single structure. \section{Modal interpretations} Modal interpretations of quantum mechanics \cite{vF91, D88, D89, DD} face the problem of finding an objective reading of the accepted mathematical formalism of the theory, a reading ``in terms of properties possessed by physical systems, independently of consciousness and measurements (in the sense of human interventions)'' \cite{DD}. These interpretations intend to consistently include the possible properties of the system in the discourse and so find a new link between the state of the system and the probabilistic character of its properties, namely, sustaining that the interpretation of the quantum state must contain a modal aspect. The name {\it modal interpretation} was for the first time used by B. van Fraassen \cite{BvF} following {\it modal logic}, precisely the logic that deals with possibility and necessity. The fundamental point is the purpose of interpreting ``the formalism as providing information about properties of physical systems'' \cite{DD}. In this context, a physical property of a system is ``a definite value of a physical quantity belonging to this system; i.e., a feature of physical reality'' \cite{DD}. As usual, definite values of physical magnitudes correspond to yes/no propositions represented by orthogonal projection operators acting on vectors belonging to the Hilbert space of the (pure) states of the system \cite{jauch}. Modal interpretations may be thought to be a study of the constraints under which one is able to talk a consistent classical discourse without contradiction with the quantum formalism. To study this issue and in order to avoid inconsistencies, we face the problem of modalities in the frame of algebraic logic. To do so, we build a variety that is an expansion of the orthomodular lattices by adding an operator, {\it the possibility operator}, to these structures. It will represent the possibility of occurrence of a property, measurable in terms of the Born rule. The analysis of the changes introduced by allowing modalities will be performed, as in \cite{DF}, for the case of pure states. In spite of the restrictions this imposes to the comparison with the general case of modal interpretations, we think it contributes to enlighten the discussion all the same. In a following step, we will extend the treatment to the factorized space of subsystems. \section{An algebraic study of modality} First we recall from \cite{Bur, MM, BD} some notions of the universal algebra and lattice theory that will play an important role in what follows. For each algebra $A$, we denote by $Con(A)$, the congruence lattice of $A$, the diagonal congruence is denoted by $\Delta$ and the largest congruence $A^2$ is denoted by $\nabla$. $\theta$ is called {\it factor congruence} iff there is a congruence $\theta^$ on $A$ such that, $\theta \land \theta^ = \Delta$, $\theta \lor \theta^* = \nabla$ and $\theta$ permutes with $\theta^$. If $\theta$ and $\theta^$ is a pair of factor congruences on $A$ then $A \cong A/\theta \times A/\theta^$. $A$ is {\it directly indecomposable} if $A$ is not isomorphic to a product of two non trivial algebras or, equivalently $\Delta,\nabla$ are the only factor congruences in $A$. We say that $A$ is {\it subdirect product} of a family of $(A_i)_{i\in I}$ of algebras if there exists an embedding $f: A \rightarrow \prod_{i\in I} A_i$ such that $\pi_i f : A\! \rightarrow A_i$ is a surjective homomorphism for each $i\in I$ where $\pi_i$ is the projection onto $A_i$. $A$ is {\it subdirectly irreducible} iff $A$ is trivial or there is a minimum congruence in $Con(A) - \Delta$. It is clear that a subdirectly irreducible algebra is directly indecomposable. An important result due to Birkhoff is that every algebra $A$ is subdirect product of subdirectly irreducibles algebras. In a Boolean algebra $A$, congruences are identifiable to certain subsets called {\it filters}. $F \subset A$ is a filter iff it satisfies: if $a\in F$ and $a\leq x$ then $x\in F$ and if $a,b\in F$ then $a\land b \in F$. $F$ is a proper filter iff $F\not = A$ or, equivalently, $0\not \in F$. If $X\subseteq A$, the filter $F_X$ generated by $X$ is the minimum filter containing $X$. A proper filter $F$ is maximal iff the quotient algebra $A/F$ is isomorphic to $\bf 2$. It is well known that each proper filter can be extended to a maximal one. We denote by ${\cal OML}$ the variety of orthomodular lattices. Let $L=\langle L,\lor,\land, \neg, 0, 1\rangle$ be an orthomodular lattice. Given $a, b, c$ in $L$, we write: $(a,b,c)D$\ \ iff $(a\lor b)\land c = (a\land c)\lor (b\land c)$; $(a,b,c)D^{}$ iff $(a\land b)\lor c = (a\lor c)\land (b\lor c)$ and $(a,b,c)T$\ \ iff $(a,b,c)D$, (a,b,c)$D^{}$ hold for all permutations of $a, b, c$. An element $z$ of a lattice $L$ is called a {\it central} iff for all elements $a,b\in L$ we have\ $(a,b,z)T$. We denote by $Z(L)$ the set of all central elements of $L$ and it is called the {\it center} of $L$. $Z(L)$ is a Boolean sublattice of $L$ {\rm \cite[Theorem 4.15]{MM}}. \begin{prop}\label{eqcentro} {\rm \cite[Lemma 29.9 and Lemma 29.16]{MM}} Let $L$ be an orthomodular lattice then we have \begin{enumerate} \item $z \in Z(L)$ if and only if $a = (a\land z) \lor (a \land \neg z)$ for each $a\in L$ \item If $L$ is complete then $Z(L)$ is a complete lattice and for each family $(z_i)_i$ in Z(L) and $a\in L$, $a \land \bigvee z_i = \bigvee (a \land Z_i)$. \end{enumerate} $\Box$ \end{prop} Factor congruences in $L$ are identifiable to the elements of the center $Z(L)$. More precisely if $z\in Z(L)$, the binary relation ${\Theta}_z$ on $A$ defined by $a \Theta_z b$ iff $a\land z = b\land z$ is a congruence on $L$, such that $L\cong L/{\Theta}_z\times L/{\Theta}_{\neg z}$. Now we build up a framework to include modal propositions in the same structure as actual ones. To do so, we enrich the orthomodular lattice with a modal operator taking into account the following considerations: \begin{enumerate} \item Propositions about the properties of the physical system will be interpreted in the orthomodular lattice of subspaces of the Hilbert space of the (pure) states of the system. Thus we will retain this structure in our extension. \item Given a proposition about the system, it is possible to define a context from which one can predicate with certainty about it (and about a set of propositions that are compatible with it) and predicate probabilities about the other ones. This is to say that one may predicate truth or falsity of all possibilities at the same time, i.e. possibilities allow an interpretation in a Boolean algebra. In rigorous terms, for each proposition $P$, if we refer with $\Diamond P$ to the possibility of $P$, then $\Diamond P$ will be a central element of the orthomodular structure. \item If $P$ is a proposition about the system and $P$ occurs, then it is trivially possible that $P$ occurs. This is expressed as $P \leq \Diamond P$. \item To assume an actual property and a complete set of properties that are compatible with it determines a context in which the classical discourse holds. Classical consequences that are compatible with it, for example probability asignements to the actuality of other propositions, share the classical frame. These consequences are the same ones as those which would be obtained by considering the original actual property as a possible one. This is interpreted as, if $P$ is a property of the system, $\Diamond P$ is the smallest central element greater than $P$. \end{enumerate} The algebraic study will be performed using the necessity operator $\Box$ instead of the possibility one $\Diamond$ because of technical reasons. Then it will be possible to define the possibility operator from the necessity one. \begin{definition} {\rm Let $A$ be an orthomodular lattice. We say that $A$ is {\it Boolean saturated} iff for each $a\in A$ the set $\{z\in Z(A): z\leq a \}$ has a maximum. In this case such maximum is denoted by $\Box (a)$. } \end{definition} \begin{example} {\rm In view of Proposition \ref{eqcentro}, orthomodular complete lattices with $\Box(a) = \bigvee \{z \in Z(L) : z \leq a \}$ as an operator, are examples of Boolean saturated orthomodular lattices.} \end{example} \begin{prop}\label{PROST} Let $A$ be an orthomodular lattice. Then $A$ is Boolean saturated iff there exists an unary operator $\Box$ acting on the elements of $A$ satisfying \begin{enumerate} \item[S1] $\Box x \leq x$ \item[S2] $\Box 1 = 1$ \item[S3] $\Box \Box x = \Box x$ \item[S4] $\Box(x \land y) = \Box(x) \land \Box(y)$ \item[S5] $y = (y\land \Box x) \lor (y \land \neg \Box x)$ \item[S6] $\Box (x \lor \Box y ) = \Box x \lor \Box y $ \item[S7] $\Box (\neg x \lor (y \land x)) \leq \neg \Box x \lor \Box y $ \end{enumerate} \end{prop} \begin{proof} Suppose that $A$ is Boolean saturated. S1), S2) and S3) are trivial. \hspace{0.2cm} S4) Since $x\land y \leq x$ and $x\land y \leq y$ then $\Box(x\land y) \leq \Box(x) \land \Box(y)$. For the converse, $\Box(x) \leq x$ and $\Box(y) \leq y$, thus $\Box(x) \land \Box(y) \leq \Box(x\land y)$. \hspace{0.2cm} S5) Follows from Proposition \ref{eqcentro} since $\Box(x) \in Z(A)$. \hspace{0.2cm} S6) For simplicity let $z = \Box y$. From the precedent item and taking into account that $z \in Z(L)$ we have that $\Box(z\lor x) \land \Box(\neg z \lor x) = \Box ((z\lor x)\land (\neg z \lor x )) = \Box(x)$. Since $\neg z \leq \Box(\neg z \lor x)$ then we have that $1 = z \lor \neg z \leq z \lor \Box(\neg z \lor x)$. Also we have $z\leq \Box(z\lor x)$. Finally $z\lor \Box(x) = (z\lor \Box(z\lor x)) \land (z\lor\Box(\neg z\lor x)) = (z\lor \Box(z\lor x)) \land 1 = \Box(z\lor x)$ i.e. $\Box (x \lor \Box y ) = \Box x \lor \Box y $. \hspace{0.2cm} S7) Since $\Box(x) \leq x$ then $\neg x \leq \neg \Box x$, we have that $\neg x \lor (y \land x) \leq \neg \Box x \lor y$. Using the precedent item $\Box (\neg x \lor (y \land x)) \leq \Box (\neg \Box x \lor y) = \neg \Box x \lor \Box y$ since $\neg \Box x \in Z(A)$. \\ For the converse, let $a\in A$ and $\{z \in Z(A) : z\leq a \}$. By $S1$ and $S5$ it is clear that $\Box a \in \{z \in Z(A) : z\leq a \}$. We see that $\Box a$ is the upper bound of the set. Let $z\in Z(A)$ such that $z \leq a$ then $1 = \neg z \lor (a \land z)$. Using $S2$ and $S7$ we have $1 = \Box 1 = \Box (\neg z \lor (a\land z)) \leq \neg \Box z \lor a = \neg z \lor a $. Therefore $z = z \land (\neg z \lor \Box a )$ and since $z$ is central $z = z \land \Box a$ resulting $z \leq \Box a$. Finally $ \Box a = Max\{z \in Z(A) : z\leq a \} $. \end{proof} \begin{theo} The class of Boolean saturated orthomodular lattices constitutes a variety which is axiomatized by \begin{enumerate} \item Axioms of ${\cal OML}$ \item $S1,...,S7$ \end{enumerate} \end{theo} \begin{proof} Obvious by Proposition \ref{PROST} \end{proof} Boolean saturated orthomodular lattices are algebras $ \langle A, \land, \lor, \neg, \Box, 0, 1 \rangle$ of type $ \langle 2, 2, 1, 1, 0, 0 \rangle$ and the variety they constitute will be noted as ${\cal OML}^\Box$. On each algebra of ${\cal OML}^\Box$ we can define the possibility operator as unary operation $\Diamond$ given by $$\Diamond x = \neg \Box \neg x$$ \begin{prop}\label{POS} Let $A$ be a Boolean saturated orthomodular lattice and $a, b \in A$. Then we have \begin{enumerate} \item $a\leq \Diamond a$ \item $\Diamond a = Min \{z\in Z(A): a\leq z \}$ \end{enumerate} \end{prop} \begin{proof} We first note that $\Diamond a \in Z(A)$ since $\Box \neg a \in Z(A)$. On the other hand $\Box \neg a \leq \neg a$ and then $a = \neg \neg a \leq \neg \Box \neg a = \Diamond a$. If $z \in Z(A)$ such that $a \leq z$ then $\neg z \leq \neg a$ resulting $\neg z \leq \Box \neg a$. Thus $\Diamond a = \neg \Box \neg a \leq z$. \end{proof} \begin{rema} {\rm From \ref{POS} it may be seen that Boolean saturated orthomodular lattices satisfy the four items which motivate our approach. } \end{rema} \begin{theo}\label{COMPZ} Let $L$ be an orthomodular lattice. Then there exists an orthomodular monomorphism $f:L \rightarrow L^\Box$ such that $L^\Box \in {\cal OML}^\Box$. \end{theo} \begin{proof} Let $f: L \rightarrow \prod_{i\in I} L_i$ be a subdirect embedding of $L$. Since $L_i$ is subdirectly irreducible then $Z(L_i) = \{0,1\}$ for each $i\in I$ resulting $Z(\prod_{i\in I} L_i)$ a complete Boolean algebra and so $\prod_{i\in I} L_i$ is Boolean saturated. \\ \end{proof} In this case we may say that each orthomodular lattice can be represented by a Boolean saturated one. In this sense, the embedding of orthomodular systems in modal systems proposed in \cite{Dish} is maintained. On the other hand, we see that the defined modal operators are quantifiers in the sense of \cite{JAN1, JAN2}. \vskip 0.5truecm For each orthomodular lattice $L$, if $f:L \rightarrow L^\Box$ such that $L^\Box \in {\cal OML}^\Box$ is an orthomodular monomorphism, we refer to $L^\Box$ as a {\it modal extension} of $L$. In this case, we may see the lattice $L$ as a subset of $L^\Box$. \section{Modalities: enlargement of the expressivity of the discourse} It is clear that the addition of modalities gives by itself greater expressive power to the language of propositions about the system. But what we want to emphasize is that it gives an adequate framework to represent, for example, the Born rule for the probability of actualization of a property, something that has no place in the orthomodular lattice alone. In order to develop these ideas, we need to prove which conditions on elements of a subset $A$ are necessary to make $\langle A \rangle_L $, the sublattice generated by $A$, a Boolean sublattice. \begin{definition} {\rm Let $L$ be an orthomodular lattice and $a,b \in L$. Then $a$ commutes with $b$ if and only if $a = (a\land b) \lor (a \land \neg b)$. A non-empty subset $A$ is called a Greechie set iff for any three different elements of $A$, at least one commutes with the other two. } \end{definition} \begin{prop}\label{GREE} Let $L$ be an orthomodular lattice. If $A$ is a Greechie set in $L$ such that for each $a\in A, \neg a \in A$ then, $\langle A \rangle_L $ is Boolean sublattice. \end{prop} \begin{proof} It is well known from {\rm \cite{GREE}} that $\langle A \rangle_L $ is a distributive sublattice of $L$. Since distributive orthomodular lattices are Boolean algebras, we only need to see that $\langle A \rangle_L $ is closed by $\neg$. To do that we use induction on the complexity of terms of the subuniverse generated by $A$. For $comp(a) = 0$, it follows from the fact that $A$ is closed by negation. Assume validity for terms of the complexity less than $n$. Let $\tau$ be a term such that $comp(\tau)= n$. If $\tau = \neg \tau_1$ then $\neg \tau \in \langle A \rangle_L$ since $\neg \tau = \neg \neg \tau_1 = \tau_1$ and $\tau_1 \in \langle A \rangle_L$. If $\tau = \tau_1 \land \tau_2$, $\neg \tau = \neg \tau_1 \lor \neg \tau_2$. Since $comp(\tau_i) < n$, $\neg \tau_i \in \langle A \rangle_L$ for $i= 1,2$ resulting $\neg \tau \in \langle A \rangle_L$. We use the same argument in the case $\tau = \tau_1 \lor \tau_2$. Finally $\langle A \rangle_L$ is a Boolean sublattice. \end{proof} \begin{definition} {\rm Let $L$ be an orthomodular lattice and $L^\Box \in {\cal OML}^\Box$ be a modal extension of $L$. We define the {\it possibility space} of $L$ in $L^\Box$ as $$\Diamond L = \langle \{\Diamond p : p \in L \} \rangle_{L^\Box} $$ } \end{definition} The {\it possibility space} represents the modal content added to the discourse about properties of the system. \begin{prop}\label{POSSPACE} Let $L$ be an orthomodular lattice, $W$ a Boolean sublattice of $L$ and $L^\Box \in {\cal OML}^\Box$ a modal extension of $L$. Then $\langle W \cup \Diamond L \rangle_{L^\Box}$ is a Boolean sublattice of $L^\Box$. In particular $\Diamond L$ is a Boolean sublattice of $Z(L^\Box)$. \end{prop} \begin{proof} Follows from Proposition \ref{GREE} since $W \cup \Diamond L$ is a Greechie set closed by $\neg$. \\ \end{proof} We know that, in the orthomodular lattice of the properties of the system, it is always possible to choose a context in which any possible property pertaining to this context can be considered as an actual property. We formalize this fact in the following definition and then we prove that this is always possible in our modal structure. \begin{definition} {\rm Let $L$ be an orthomodular lattice, $W$ a Boolean sublattice of $L$, $p\in W$ and $L^\Box$ be a modal extension of $L$. If $f: \Diamond L \rightarrow {\bf 2}$ is a Boolean homomorphism such that $f(\Diamond p) = 1$ then an actualization of $p$ compatible with $f$ is a Boolean homomorphism $f_p: W \rightarrow {\bf 2}$ such that \begin{enumerate} \item $f_p(p) = 1$ \item There exists a Boolean homomorphism $g : \langle W \cup \Diamond L \rangle_{L^\Box} \rightarrow {\bf 2}$ such that $g\mid W = f_p$ and $g \mid \Diamond L = f $ \end{enumerate} } \end{definition} \begin{theo} \label{ACT} Let $L$ be an orthomodular lattice, $W$ a Boolean sublattice of $L$, $p\in W$ and $L^\Box$ be a modal extension of $L$. If $f: \Diamond L \rightarrow {\bf 2}$ is a Boolean homomorphism such that $f(\Diamond p) = 1$ then there exists an actualization of $p$ compatible with $f$. \end{theo} \begin{proof} Let $F$ be the filter associated with the Boolean homomorphism $f$. We consider the $\langle W \cup \Diamond L \rangle_{L^\Box}$-filter $F_p$ generated by $ F \cup \{p \}$. We want to see that $F_p$ is a proper filter. If $F_p$ is not proper, then there exists $a\in F$ such that $a \land p \leq 0$. Thus $p\leq \neg a$ being $\neg a$ a central element. But $\Diamond p$ is the smallest Boolean element greater than $p$, then $\Diamond p \leq \neg a$ or equivalently $\Diamond p \land a = 0$ And this is a contradiction since $\Diamond p, a \in F $ being $F$ a proper filter. Thus we may extend $F_p$ to be a maximal filter $F_M$ in $\langle W \cup \Diamond L \rangle_{L^\Box}$, resulting the natural projection $\langle W \cup \Diamond L \rangle_{L^\Box} \rightarrow \langle W \cup \Diamond L \rangle_{L^\Box} /F_M \approx {\bf 2} $ an actualization of $p$ compatible with $f$. $\Box$ \\ \end{proof} The next theorem allows an algebraic representation of the Born rule which quantifies possibilities from a chosen spectral algebra. \begin{theo} \label{BORN} Let $L$ be an orthomodular lattice, $W$ a Boolean sublattice of $L$ and $f: W \rightarrow {\bf 2}$ a Boolean homomorphism. If we consider a modal extension $L^\Box$ of $L$ then there exists a Boolean homomorphism $f^: \langle W \cup \Diamond L \rangle_{L^\Box} \rightarrow {\bf 2} $ such that $f^* \mid W = f$. \end{theo} \begin{proof} Let $i: W \rightarrow \langle W \cup \Diamond L \rangle_{L^\Box}$ be the Boolean canonical embedding. If we consider the following diagram: \begin{center} \unitlength=1mm \begin{picture}(60,20)(0,0) \put(8,16){\vector(3,0){5}} \put(2,10){\vector(0,-2){5}} \put(2,16){\makebox(0,0){$W$}} \put(20,16){\makebox(0,0){$\bf 2$}} \put(2,0){\makebox(0,0){$ \langle W \cup \Diamond L \rangle_{L^\Box} $}} \put(2,20){\makebox(17,0){$f$}} \put(2,8){\makebox(-5,0){$i$}} \end{picture} \end{center} \noindent we see that there exists a Boolean homomorphism $f^: \langle W \cup \Diamond L \rangle_{L^\Box} \rightarrow {\bf 2}$ such that $f^ \mid W_A = f$ because ${\bf 2}$ is injective in the variety of Boolean algebras \cite{SIK}. \\ \end{proof} \noindent We note that this reading of the Born rule is a kind of converse of the possibility of actualizing properties given by Theorem \ref{ACT}. \section{Kochen-Specker theorem: a limit also for modalities} The addition of modalities to the discourse about the properties of a quantum system enlarges its expressive power. At first sight it may be thought that this could help to circumvent contextuality, allowing to refer to physical properties belonging to the system in an objective way that resembles the classical picture. But this is not the case as we have announced in \cite{DFR}. To prove it here, we introduce an algebraic representation of the notion of global actualization: \begin{definition} {\rm Let $L$ be an orthomodular lattice, $(W_i)_{i \in I}$ the family of Boolean sublattices of $L$ and $L^\Box$ a modal extension of $L$. If $f: \Diamond L \rightarrow {\bf 2}$ is a Boolean homomorphism, an {\it actualization compatible with } $f$ is a global valuation $(v_i: W_i \rightarrow {\bf 2})_{i\in I}$ such that $v_i\mid W_i \cap \Diamond L = f\mid W_i \cap \Diamond L $ for each $i\in I$.} \end{definition} Compatible actualizations represent the passage from possibility to actuality. \begin{theo} Let $L$ be an orthomodular lattice. Then $L$ admits a global valuation iff for each possibility space there exists a Boolean homomorphism $f: \Diamond L \rightarrow {\bf 2}$ that admits a compatible actualization. \end{theo} \begin{proof} Suppose that $L$ admits a global valuation $(v_i: W_i \rightarrow {\bf 2})_{i\in I}$. Let ${L^\Box}$ be a modal extension of $L$ and consider $A_i = W_i \cap \Diamond L$. Let $f_0 = \bigcup_i A_i \rightarrow {\bf 2}$ such that $f_0(x) = v_i(x)$ if $x\in W_i$. $f_0$ is well defined since $(v_i)_i$ is a global valuation. If we consider $\langle \bigcup_i A_i \rangle_{L^\Box}$, the subalgebra of $L^{\Box}$ generated by the join of the family $(A_{i})$, it may be proved that it is a Boolean subalgebra of the possibility space $\Diamond L$. We can extended $f_0$ to a Boolean homomorphism $f_0^:\langle \bigcup_i A_i \rangle_{L^\Box} \rightarrow {\bf 2}$. Since ${\bf 2}$ is injective in the variety of Boolean algebras \cite{SIK}, there exits a Boolean homomorphism $f: \Diamond L \rightarrow {\bf 2}$ such that the following diagram is commutative \begin{center} \unitlength=1mm \begin{picture}(20,20)(0,0) \put(12,16){\vector(3,0){8}} \put(2,10){\vector(0,-2){5}} \put(10,4){\vector(1,1){7}} \put(2,10){\makebox(13,0){$\equiv$}} \put(2,16){\makebox(0,0){$\langle \bigcup_i A_i \rangle_{L^\Box} $}} \put(23,16){\makebox(0,0){${\bf 2}$}} \put(2,0){\makebox(0,0){$\Diamond L$}} \put(7,20){\makebox(17,0){$f_0^$}} \put(2,8){\makebox(-6,0){$i$}} \put(18,2){\makebox(-4,3){$f$}} \end{picture} \end{center} Thus $f: \Diamond L \rightarrow {\bf 2}$ admits a compatible actualization. The converse is immediate. \\ \end{proof} Since the possibility space is a Boolean algebra, there exists a Boolean valuation of the possible properties. But in view of the last theorem, an actualization that would correspond to a family of compatible global valuations is prohibited. Thus the theorem states that the contextual character of quantum mechanics is maintained even when the discourse is enriched with modalities. \section{Conclusions} From the algebraic characterization of contextuality given by the non existence of compatible global valuations over the orthomodular structure, we show that, if this structure is enriched with modal operators, the discourse about properties is genuinely enlarged. However, the contextual character of the complete language is maintained. Thus contextuality remains a main feature of quantum systems even when modalities are taken into account. \end{document}	arXiv
Digital root The digital root (also repeated digital sum) of a natural number in a given radix is the (single digit) value obtained by an iterative process of summing digits, on each iteration using the result from the previous iteration to compute a digit sum. The process continues until a single-digit number is reached. For example, in base 10, the digital root of the number 12345 is 6 because the sum of the digits in the number is 1 + 2 + 3 + 4 + 5 = 15, then the addition process is repeated again for the resulting number 15, so that the sum of 1 + 5 equals 6, which is the digital root of that number. In base 10, this is equivalent to taking the remainder upon division by 9 (except when the digital root is 9, where the remainder upon division by 9 will be 0), which allows it to be used as a divisibility rule. Formal definition Let $n$ be a natural number. For base $b>1$, we define the digit sum $F_{b}:\mathbb {N} \rightarrow \mathbb {N} $ to be the following: $F_{b}(n)=\sum _{i=0}^{k-1}d_{i}$ where $k=\lfloor \log _{b}{n}\rfloor +1$ is the number of digits in the number in base $b$, and $d_{i}={\frac {n{\bmod {b^{i+1}}}-n{\bmod {b}}^{i}}{b^{i}}}$ is the value of each digit of the number. A natural number $n$ is a digital root if it is a fixed point for $F_{b}$, which occurs if $F_{b}(n)=n$. All natural numbers $n$ are preperiodic points for $F_{b}$, regardless of the base. This is because if $n\geq b$, then $n=\sum _{i=0}^{k-1}d_{i}b^{i}$ and therefore $F_{b}(n)=\sum _{i=0}^{k-1}d_{i}<\sum _{i=0}^{k-1}d_{i}b^{i}=n$ because $b>1$. If $n<b$, then trivially $F_{b}(n)=n$ Therefore, the only possible digital roots are the natural numbers $0\leq n<b$, and there are no cycles other than the fixed points of $0\leq n<b$. Example In base 12, 8 is the additive digital root of the base 10 number 3110, as for $n=3110$ $d_{0}={\frac {3110{\bmod {12^{0+1}}}-3110{\bmod {1}}2^{0}}{12^{0}}}={\frac {3110{\bmod {12}}-3110{\bmod {1}}}{1}}={\frac {2-0}{1}}={\frac {2}{1}}=2$ $d_{1}={\frac {3110{\bmod {12^{1+1}}}-3110{\bmod {1}}2^{1}}{12^{1}}}={\frac {3110{\bmod {144}}-3110{\bmod {1}}2}{12}}={\frac {86-2}{12}}={\frac {84}{12}}=7$ $d_{2}={\frac {3110{\bmod {12^{2+1}}}-3110{\bmod {1}}2^{2}}{12^{2}}}={\frac {3110{\bmod {1728}}-3110{\bmod {1}}44}{144}}={\frac {1382-86}{144}}={\frac {1296}{144}}=9$ $d_{3}={\frac {3110{\bmod {12^{3+1}}}-3110{\bmod {1}}2^{3}}{12^{3}}}={\frac {3110{\bmod {20736}}-3110{\bmod {1}}728}{1728}}={\frac {3110-1382}{1728}}={\frac {1728}{1728}}=1$ $F_{12}(3110)=\sum _{i=0}^{4-1}d_{i}=2+7+9+1=19$ This process shows that 3110 is 1972 in base 12. Now for $F_{12}(3110)=19$ $d_{0}={\frac {19{\bmod {12^{0+1}}}-19{\bmod {1}}2^{0}}{12^{0}}}={\frac {19{\bmod {12}}-19{\bmod {1}}}{1}}={\frac {7-0}{1}}={\frac {7}{1}}=7$ $d_{1}={\frac {19{\bmod {12^{1+1}}}-19{\bmod {1}}2^{1}}{12^{1}}}={\frac {19{\bmod {144}}-19{\bmod {1}}2}{12}}={\frac {19-7}{12}}={\frac {12}{12}}=1$ $F_{12}(19)=\sum _{i=0}^{2-1}d_{i}=1+7=8$ shows that 19 is 17 in base 12. And as 8 is a 1-digit number in base 12, $F_{12}(8)=8$. Direct formulas We can define the digit root directly for base $b>1$ $\operatorname {dr} _{b}:\mathbb {N} \rightarrow \mathbb {N} $ in the following ways: Congruence formula The formula in base $b$ is: $\operatorname {dr} _{b}(n)={\begin{cases}0&{\mbox{if}}\ n=0,\\b-1&{\mbox{if}}\ n\neq 0,\ n\ \equiv 0{\bmod {b-1}},\\n\ {\rm {mod}}\ (b-1)&{\mbox{if}}\ n\not \equiv 0{\bmod {b-1}}\end{cases}}$ or, $\operatorname {dr} _{b}(n)={\begin{cases}0&{\mbox{if}}\ n=0,\\1\ +\ ((n-1)\ {\rm {mod}}\ (b-1))&{\mbox{if}}\ n\neq 0.\end{cases}}$ In base 10, the corresponding sequence is (sequence A010888 in the OEIS). The digital root is the value modulo $b-1$ because $b\equiv 1{\bmod {b-1}},$ and thus $b^{k}\equiv 1^{k}\equiv 1{\bmod {b-1}},$ so regardless of position, the value $n{\bmod {b}}-1$ is the same – $ab^{2}\equiv ab\equiv a{\bmod {b-1}}$ – which is why digits can be meaningfully added. Concretely, for a three-digit number $n=a_{1}b^{2}+a_{2}b^{1}+a_{3}b^{0}$ $\operatorname {dr} _{b}(n)\equiv a_{1}b^{2}+a_{2}b^{1}+a_{3}b^{0}\equiv a_{1}(1)+a_{2}(1)+a_{3}(1)\equiv (a_{1}+a_{2}+a_{3}){\bmod {b-1}}$. To obtain the modular value with respect to other numbers $n$, one can take weighted sums, where the weight on the $k$-th digit corresponds to the value of $b^{k}$ modulo $n$. In base 10, this is simplest for 2, 5, and 10, where higher digits vanish (since 2 and 5 divide 10), which corresponds to the familiar fact that the divisibility of a decimal number with respect to 2, 5, and 10 can be checked by the last digit (even numbers end in 0, 2, 4, 6, or 8). Also of note is the modulus $n=b+1$: since $b\equiv -1{\bmod {b+1}},$ and thus $b^{2}\equiv (-1)^{2}\equiv 1{\pmod {b+1}},$ taking the alternating sum of digits yields the value modulo $b+1$. Using the floor function It helps to see the digital root of a positive integer as the position it holds with respect to the largest multiple of $b-1$ less than the number itself. For example, in base 6 the digital root of 11 is 2, which means that 11 is the second number after $6-1=5$. Likewise, in base 10 the digital root of 2035 is 1, which means that $2035-1=2034\|9$. If a number produces a digital root of exactly $b-1$, then the number is a multiple of $b-1$. With this in mind the digital root of a positive integer $n$ may be defined by using floor function $\lfloor x\rfloor $, as $\operatorname {dr} _{b}(n)=n-(b-1)\left\lfloor {\frac {n-1}{b-1}}\right\rfloor .$ Properties • The digital root of $a_{1}+a_{2}$ in base $b$ is the digital root of the sum of the digital root of $a_{1}$ and the digital root of $a_{2}$. This property can be used as a sort of checksum, to check that a sum has been performed correctly. $\operatorname {dr} _{b}(a_{1}+a_{2})=\operatorname {dr} _{b}(\operatorname {dr} _{b}(a_{1})+\operatorname {dr} _{b}(a_{2})).$ • The digital root of $a_{1}-a_{2}$ in base $b$ is congruent to the difference of the digital root of $a_{1}$ and the digital root of $a_{2}$ modulo $b-1$. $\operatorname {dr} _{b}(a_{1}-a_{2})\equiv (\operatorname {dr} _{b}(a_{1})-\operatorname {dr} _{b}(a_{2})){\bmod {b-1}}.$ • The digital root of $-n$ in base $b$ as follows: $\operatorname {dr} _{b}(-n)\equiv -\operatorname {dr} _{b}(n){\bmod {b-1}}.$ • The digital root of the product of nonzero single digit numbers $a_{1}\cdot a_{2}$ in base $b$ is given by the Vedic Square in base $b$. • The digital root of $a_{1}\cdot a_{2}$ in base $b$ is the digital root of the product of the digital root of $a_{1}$ and the digital root of $a_{2}$. $\operatorname {dr} _{b}(a_{1}a_{2})=\operatorname {dr} _{b}(\operatorname {dr} _{b}(a_{1})\cdot \operatorname {dr} _{b}(a_{2})).$ Additive persistence The additive persistence counts how many times we must sum its digits to arrive at its digital root. For example, the additive persistence of 2718 in base 10 is 2: first we find that 2 + 7 + 1 + 8 = 18, then that 1 + 8 = 9. There is no limit to the additive persistence of a number in a number base $b$. Proof: For a given number $n$, the persistence of the number consisting of $n$ repetitions of the digit 1 is 1 higher than that of $n$. The smallest numbers of additive persistence 0, 1, ... in base 10 are: 0, 10, 19, 199, 19 999 999 999 999 999 999 999, ... (sequence A006050 in the OEIS) The next number in the sequence (the smallest number of additive persistence 5) is 2 × 102×(1022 − 1)/9 − 1 (that is, 1 followed by 2 222 222 222 222 222 222 222 nines). For any fixed base, the sum of the digits of a number is proportional to its logarithm; therefore, the additive persistence is proportional to the iterated logarithm.[1] Programming example The example below implements the digit sum described in the definition above to search for digital roots and additive persistences in Python. def digit_sum(x: int, b: int) -> int: total = 0 while x > 0: total = total + (x % b) x = x // b return total def digital_root(x: int, b: int) -> int: seen = set() while x not in seen: seen.add(x) x = digit_sum(x, b) return x def additive_persistence(x: int, b: int) -> int: seen = set() while x not in seen: seen.add(x) x = digit_sum(x, b) return len(seen) - 1 In popular culture Digital roots are used in Western numerology, but certain numbers deemed to have occult significance (such as 11 and 22) are not always completely reduced to a single digit. Digital roots form an important mechanic in the visual novel adventure game Nine Hours, Nine Persons, Nine Doors. See also • Arithmetic dynamics • Base 9 • Casting out nines • Digit sum • Divisibility rule • Hamming weight • Multiplicative digital root References 1. Meimaris, Antonios (2015), On the additive persistence of a number in base p, Preprint • Averbach, Bonnie; Chein, Orin (27 May 1999), Problem Solving Through Recreational Mathematics, Dover Books on Mathematics (reprinted ed.), Mineola, NY: Courier Dover Publications, pp. 125–127, ISBN 0-486-40917-1 (online copy, p. 125, at Google Books) • Ghannam, Talal (4 January 2011), The Mystery of Numbers: Revealed Through Their Digital Root, CreateSpace Publications, pp. 68–73, ISBN 978-1-4776-7841-1, archived from the original on 29 March 2016, retrieved 11 February 2016 (online copy, p. 68, at Google Books) • Hall, F. M. (1980), An Introduction into Abstract Algebra, vol. 1 (2nd ed.), Cambridge, U.K.: CUP Archive, p. 101, ISBN 978-0-521-29861-2 (online copy, p. 101, at Google Books) • O'Beirne, T. H. (13 March 1961), "Puzzles and Paradoxes", New Scientist, Reed Business Information, 10 (230): 53–54, ISSN 0262-4079 (online copy, p. 53, at Google Books) • Rouse Ball, W. W.; Coxeter, H. S. M. (6 May 2010), Mathematical Recreations and Essays, Dover Recreational Mathematics (13th ed.), NY: Dover Publications, ISBN 978-0-486-25357-2 (online copy at Google Books) External links • Patterns of digital roots using MS Excel • Weisstein, Eric W. "Digital Root". MathWorld. 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Let $D\subset T\times X$, where $T$ is a measurable space, and $X$ a topological space. We study inclusions between three classes of extended real-valued functions on $D$ which are upper semicontinuous in $x$ and satisfy some measurability conditions. Kucia A.: Some counterexamples for Carathéodory functions and multifunctions. submitted to Fund. Math. Zygmunt W.: Scorza-Dragoni property (in Polish). UMCS, Lublin, 1990.	CommonCrawl
Sorgenfrey plane In topology, the Sorgenfrey plane is a frequently-cited counterexample to many otherwise plausible-sounding conjectures. It consists of the product of two copies of the Sorgenfrey line, which is the real line $\mathbb {R} $ under the half-open interval topology. The Sorgenfrey line and plane are named for the American mathematician Robert Sorgenfrey. A basis for the Sorgenfrey plane, denoted $\mathbb {S} $ from now on, is therefore the set of rectangles that include the west edge, southwest corner, and south edge, and omit the southeast corner, east edge, northeast corner, north edge, and northwest corner. Open sets in $\mathbb {S} $ are unions of such rectangles. $\mathbb {S} $ is an example of a space that is a product of Lindelöf spaces that is not itself a Lindelöf space. The so-called anti-diagonal $\Delta =\{(x,-x)\mid x\in \mathbb {R} \}$ is an uncountable discrete subset of this space, and this is a non-separable subset of the separable space $\mathbb {S} $. It shows that separability does not inherit to closed subspaces. Note that $K=\{(x,-x)\mid x\in \mathbb {Q} \}$ and $\Delta \setminus K$ are closed sets; it can be proved that they cannot be separated by open sets, showing that $\mathbb {S} $ is not normal. Thus it serves as a counterexample to the notion that the product of normal spaces is normal; in fact, it shows that even the finite product of perfectly normal spaces need not be normal. See also • List of topologies • Moore plane References • Kelley, John L. (1955). General Topology. van Nostrand. Reprinted as Kelley, John L. (1975). General Topology. Springer-Verlag. ISBN 0-387-90125-6. • Robert Sorgenfrey, "On the topological product of paracompact spaces", Bull. Amer. Math. Soc. 53 (1947) 631–632. • Steen, Lynn Arthur; Seebach, J. Arthur Jr. (1995) [1978]. Counterexamples in Topology (Dover reprint of 1978 ed.). Berlin, New York: Springer-Verlag. ISBN 978-0-486-68735-3. MR 0507446.	Wikipedia
If the least common multiple of $A$ and $B$ is $1575$, and the ratio of $A$ to $B$ is $3:7$, then what is their greatest common divisor? Since the ratio of $A$ to $B$ is $3:7$, there is an integer $k$ for which $A=3k$ and $B=7k$. Moveover, $k$ is the greatest common divisor of $A$ and $B$, since 3 and 7 are relatively prime. Recalling the identity $\mathop{\text{lcm}}[A,B]\cdot\gcd(A,B)=AB$, we find that $1575k=(3k)(7k),$ which implies $k=1575/21=\boxed{75}$.	Math Dataset
Graciela Boente Graciela Lina Boente Boente is an Argentine mathematical statistician at the University of Buenos Aires.[1] She is known for her research in robust statistics, and particularly for robust methods for principal component analysis and regression analysis. Graciela Lina Boente Boente NationalityArgentine Alma materUniversity of Buenos Aires OccupationMathematical statistician AwardsGuggenheim Fellow Education Boente earned her Ph.D. in 1983 from the University of Buenos Aires. Her dissertation, Robust Principal Components, was supervised by Victor J. Yohai.[2] Awards and honors Boente became a Guggenheim Fellow in 2001.[3] In 2008, the Argentine National Academy of Exact, Physical and Natural Sciences gave her their Consecration Prize in recognition of her contributions and teaching.[4][5] She became an honored fellow of the Institute of Mathematical Statistics in 2013, "for her research in robust statistics and estimation, and for outstanding service to the statistical community".[6] References 1. Investigator profile, University of Buenos Aires, Institute of Mathematical Investigations, retrieved 2018-08-04 2. Graciela Boente at the Mathematics Genealogy Project 3. Graciela Lina Boente Boente, John Simon Guggenheim Memorial Foundation, retrieved 2018-08-04 4. "Premian a científicos destacados", La Nacion, December 9, 2008 5. Memoria 2008, Argentine National Academy of Exact, Physical and Natural Sciences, retrieved 2018-08-04 6. IMS Fellows 2013 (PDF), Institute of Mathematical Statistics, retrieved 2018-08-04 Authority control International • VIAF Academics • MathSciNet • Mathematics Genealogy Project • zbMATH	Wikipedia
Inline formulas are surrounded by single dollar signs. For example, $f(x) = ax^2 + bx + c$ renders as $f(x)=ax^2+bx+c$. Greek letters are mostly written by simply spelling them after a backslash with capitalization indicated by first letter. For example, $\alpha \beta \Gamma \Delta$ renders as $\alpha \beta \Gamma \Delta$. For a quick tutorial and command reference, please see our MathJax quick reference page. See also our additional pointers and references.	CommonCrawl
\begin{document} \title{Hypercyclicity, existence and approximation results for convolution operators on spaces of entire functions} \author{Vin\'icius V. F\'avaro\thanks{The first named author is supported by FAPESP Grant 2014/50536-7; FAPEMIG Grant PPM-00086-14; and CNPq Grants 482515/2013-9, 307517/2014-4.} and Ariosvaldo M. Jatob\'a} \date{} \maketitle \vspace{-1.0em} \begin{abstract} \noindent In this work we shall prove new results on the theory of convolution operators on spaces of entire functions. The focus is on hypercyclicity results for convolution operators on spaces of entire functions of a given type and order; and existence and approximation results for convolution equations on spaces of entire functions of a given type and order. In both cases we give a general method to prove new results that recover, as particular cases, several results of the literature. Applications of these more general results are given, including new hypercyclicity results for convolution operators on spaces on entire functions on $\mathbb{C}^n.$ \end{abstract} \noindent\textbf{Mathematics Subject Classifications (2010):} 47A16, 46G20, 46E10, 46E50. \newline \textbf{Key words:} Convolution operators, holomorphic functions, hypercyclicity, existence and approximation results. \section{Introduction} \qquad For a topological vector space $X$, a continuous linear operator $T \colon X \longrightarrow X$ is {\it hypercyclic} if the \emph{orbit of} $x$, given by $\{x, T(x), T^2(x), \ldots\}$ is dense in $X$ for some $x \in X$. In this case, $x$ is said to be a \emph{hypercyclic vector for $T$}. Hypercyclic translation and differentiation operators on spaces of entire functions of one complex variable were first investigated by Birkhoff \cite{birkhoff} and MacLane \cite{maclane}, respectively. Godefroy and Shapiro \cite{godefroy} pushed these results quite further by proving that every nontrivial convolution operator on spaces of entire functions of several complex variables is hypercyclic. By a \emph{nontrivial convolution operator} we mean a convolution operator which is not a scalar multiple of the identity. For the theory of hypercyclic operators and its ramifications we refer to \cite{bay, goswinBAMS, grosse_peris}. We remark that several results on the hypercyclicity of operators on spaces of entire functions in infinitely many complex variables appeared later (see, e.g., \cite{aron, bay2, favaro3, bes2012, CDSjmaa, chan, FM, GS, goswinBAMS, MPS, peterssonjmaa}). In 2007, Carando, Dimant and Muro \cite{CDSjmaa} proved some general results that encompass as particular cases several of the above mentioned results. In \cite{favaro3}, using the theory of holomorphy types, Bertoloto, Botelho, F\'{a}varo and Jatob\'{a} generalized strictly the results of \cite{CDSjmaa} to a more general setting. For instance, \cite[Theorem 2.7]{favaro3} recovers, as a very particular case, the famous result of Godefroy and Shapiro \cite{godefroy} on the hypercyclicity of convolution operators on $\mathcal{H}(\mathbb{C}^{n})$. The techniques of \cite{favaro3} are a refinement of a general method introduced in \cite{favaro1} to prove existence and approximation results for convolution equations defined on the space $\mathcal{H}_{\Theta b}(E)$ of all entire functions of $\Theta$-bounded type defined on a complex Banach space $E$. The investigation of existence and approximation results for convolution equations was initiated by Malgrange \cite{malgrange} and developed by several authors (see, for instance \cite{Col-Mat, cgp, cp, DW, DW2, favaro, FaBelg, favaro1, favaro2, gupta, G, martineau, Matos-F, Matos-Z, Matos-Z2, Matos-livro, MN, Nach-B}). In this work we give contributions in two directions. In the first, we explore hypercyclicity results for convolution operators on the space $Exp_{\Theta,0,A}^{k}\left( E\right) $, introduced in \cite{favaro2}, of $\Theta$ entire functions of a given type $A$ and order $k$ on a complex Banach space $E$, where $\Theta$ is a given holomorphy type. These results generalize the hypercyclicity results obtained in \cite{favaro3, birkhoff, CDSjmaa, godefroy, maclane}. In the second direction, we obtain a general method to prove existence and approximation results for convolution equations on $Exp_{\Theta,0,A}^{k}\left( E\right) $. These results generalize results of the same type obtained in \cite{favaro3, FaBelg, gupta, G, malgrange, martineau, matos3}. Both in the hypercyclicity results and in the existence and approximation results, the duality result via Fourier-Borel transform proved in \cite{favaro2} plays a central role. In the fashion of Dineen \cite{dineen}, we identify the properties a holomorphy type must enjoy for the results to hold true (more precisely, the $\pi_1$-$\pi_2$-holomorphy types introduced in \cite{favaro1} and refined in \cite{favaro3}, and the notion of $\pi_{2,k}$-holomorphy type introduced in Definition \ref{pi2k}). Moreover, our proofs of the hypercyclicity results rest on a classical hypercyclicity criterion, first obtained by Kitai \cite{kitai} and later on rediscovered by Gethner and Shapiro \cite{GS}. Nowadays it is known, by a result of Costakis and Sambarino \cite{costakis}, that the classical hypercyclicity criterion of Kitai ensures that the operator is mixing, a property stronger than hypercyclicity. We recall that if $X$ is a topological vector space, then a continuous linear operator $T\colon X\rightarrow X$ is said to be \emph{mixing} if for any two non-empty open sets $U,V\subset X,$ there is $n_0\in \mathbb{N}$ such that $T^n(U)\cap V\neq\emptyset$ for all $n\geq n_0.$ This paper is organized as follows. In Section 2 we collect some general results which are often used in subsequent sections. In Section 3 we prove some preparatory results about convolution operators. In Section 4 we prove hypercyclicity results for convolution operators. Section 5 is devoted to the study of existence and approximation results for convolution equations. Finally, in Section 6 we provide new examples and show that well known examples are recovered by our results. Throughout this paper $\mathbb{N}$ denotes the set of positive integers and $\mathbb{N}_{0}$ denotes the set $\mathbb{N}\cup\{0\}$. By $\Delta$ we mean the open unit disk in the complex field $\mathbb{C}$. As usual, for $k\in\left( 1,+\infty\right) ,$ we denote by $k^{\prime}$ its conjugate, that is, $\frac{1}{k}+\frac{1}{k^{\prime}}=1.$ For $k=1,$ we set $k^{\prime }=+\infty.$ $E$ and $F$ are always complex Banach spaces and $E^{\prime}$ denotes the topological dual of $E$. The Banach space of all continuous $m$-homogeneous polynomials from $E$ into $F$ endowed with its usual sup norm is denoted by $\mathcal{P}(^{m}E;F)$. The subspace of $\mathcal{P}(^{m}E;F)$ of all polynomials of finite type is represented by $\mathcal{P}_f(^{m}E;F)$. $\mathcal{H}(E;F)$ denotes the vector space of all holomorphic mappings from $E$ into $F$. In all these cases, when $F = \mathbb{C}$ we write $\mathcal{P}(^{m}E)$, $\mathcal{P}_f(^{m}E)$ and $\mathcal{H}(E)$ instead of $\mathcal{P}(^{m}E;\mathbb{C})$, $\mathcal{P}_f(^{m}E;\mathbb{C})$ and $\mathcal{H}(E;\mathbb{C})$, respectively. For the general theory of homogeneous polynomials and holomorphic functions or any unexplained notation we refer to Dineen \cite{dineenlivro}, Mujica \cite{mujica} and Nachbin \cite{nachbin2}. \section{Preliminaires} \qquad We start recalling several concepts and results involving holomorphy on infinite dimensional spaces. \begin{definition}\label{def_holomorfia} \rm Let $U$ be an open subset of $E$. A mapping $f\colon U\longrightarrow F$ is said to be \textit{holomorphic on $U$} if for every $a\in U$ there exists a sequence $(P_{m})_{m=0}^{\infty}$, where each $P_{m}\in \mathcal{P}(^{m}E;F)$ ($\mathcal{P}(^{0}E;F)=F$), such that $f(x)=\sum\limits _{m=0}^{\infty}P_{m}(x-a)$ uniformly on some open ball with center $a$. The $m$-homogeneous polynomial $m!P_{m}$ is called the \textit{$m$-th derivative of $f$ at $a$} and is denoted by $\hat{d}^{m}f(a)$. In particular, if $P\in\mathcal{P}(^{m}E;F)$, $a\in E$ and $k\in\{0,1,\ldots,m\}$, then \[ \hat{d}^{k}P(a)(x)=\frac{m!}{(m-k)!}\check{P}(\underbrace{x,\ldots ,x}_{k\,{times}},a,\ldots,a) \] for every $x\in E$, where $\check{P}$ is the unique symmetric $m$-linear mapping associated to $P$. \end{definition} \begin{definition} \rm\label{holomorphy type} (Nachbin \cite{nachbin2}) A \emph{holomorphy type $\Theta$ from $E$ to $F$} is a sequence of Banach spaces $(P_{\Theta}(^{j} E;F))_{j=0}^{\infty}$, the norm on each of them being denoted by $\\|\cdot\\|_{\Theta}$, such that the following conditions hold true: \begin{enumerate} \item[$(1)$] \textrm{Each $P_{\Theta}(^{j} E;F)$ is a vector subspace of $P(^{j} E; F)$ and $P_{\Theta}(^{0} E;F)$ coincides with $F$ as a normed vector space;} \item[$(2)$] \textrm{There is a real number $\sigma\geq1$ for which the following is true: given any $k\in\mathbb{N}_{0}$, $j\in\mathbb{N}_{0}$, $k\leq j $, $a\in E$, and $P\in\mathcal{P}_{\Theta}(^{j}E;F)$, we have $\hat{d}^{k}P(a)\in\mathcal{P}_{\Theta}(^{k}E;F)$ and \[ \left\\| \frac{1}{k!}\hat{d}^{k}P(a)\right\\| _{\Theta}\leq\sigma ^{j}\\|P\\|_{\Theta}\\|a\\|^{j-k}. \] } \end{enumerate} A holomorphy type from $E$ to $F$ shall be denoted by either $\Theta$ or $( \mathcal{P}_{\Theta}(^{j}E;F))_{j=0}^\infty$. When $F=\mathbb{C}$ we write $P_\Theta(^{j}E)$ instead of $P_{\Theta}(^{j} E;\mathbb{C})$, for every $j\in\mathbb{N}_0.$ \end{definition} \noindent It is obvious that each inclusion $\mathcal{P}_{\Theta}(^{j}E;F)\subset \mathcal{P}(^{j}E;F)$ is continuous and $\\|P\\|\leq\sigma^{j}\\|P\\|_{\Theta}$. \begin{definition}\rm(\cite[Definition 2.2]{favaro2}) \textrm{\label{bk} Let $(\mathcal{P}_{\Theta}(^{j}E))_{j=0}^{\infty}$ be a holomorphy type from $E$ to $\mathbb{C}$. For $\ \rho>0$ and $k\geq1,$ we denote by $\mathcal{B}_{\Theta,\rho}^{k}\left( E\right) $ the complex Banach space of all $f\in\mathcal{H}\left( E\right) $ such that $\widehat{d} ^{j}f\left( 0\right) \in\mathcal{P}_{\Theta}\left( ^{j}E\right) ,$ for all $j\in\mathbb{N}_{0}$\ and \[ \left\Vert f\right\Vert _{\Theta,k,\rho}={\displaystyle\sum\limits_{j=0} ^{\infty}} \rho^{-j}\left( \frac{j}{ke}\right) ^{\frac{j}{k}}\left\Vert \frac{1}{j!}\widehat{d}^{j}f\left( 0\right) \right\Vert _{\Theta}<+\infty, \] with the norm given by $\left\Vert \cdot\right\Vert _{\Theta,k,\rho}.$ } \end{definition} Now we recall the definition of the spaces of entire functions of a given type $A$ and finite order $k$. \begin{definition}\rm(\cite[Definition 2.4]{favaro2}) \label{expk}Let $\left( \mathcal{P}_{\Theta}(^{j}E)\right) _{j=0}^{\infty}$ be a holomorphy type from $E$ to $\mathbb{C}$, $A\in\left[ 0,+\infty\right) $ and $k\geq1.$ We denote by $Exp_{\Theta,0 ,A}^{k}\left( E\right) $ the complex vector space ${\displaystyle\bigcap \limits_{\rho>A}} \mathcal{B}_{\Theta,\rho}^{k}\left( E\right) $ with the locally convex projective limit topology. In case $A=0$ we denote $Exp_{\Theta,0}^{k}\left( E\right) :=Exp_{\Theta ,0,0}^{k}\left( E\right) = {\displaystyle\bigcap\limits_{\rho>0}} \mathcal{B}_{\Theta,\rho}^{k}\left( E\right) $. By \cite[Proposition 2.7]{favaro2} $Exp_{\Theta,0 ,A}^{k}\left( E\right) $ is a Fr\'echet space. \end{definition} \begin{proposition}(\cite[Proposition 2.5]{favaro2}) \label{caracterizacao sm(r,q)}Let $\left( \mathcal{P}_{\Theta}(^{j}E)\right) _{j=0}^{\infty}$ be a holomorphy type from $E$ to $\mathbb{C}$ and $k\in[1,+\infty)$. If $f\in$ $\mathcal{H}\left( E\right) $ is such that $\widehat{d}^{j}f\left( 0\right) \in\mathcal{P}_{\Theta}\left( ^{j}E\right) ,$ $\forall j\in\mathbb{N}_{0}$, then for each $A\in\left[ 0,+\infty\right), f\in Exp_{\Theta,0,A}^{k}\left( E\right) $ if, and only if, $$\limsup \limits_{j\rightarrow\infty}\left( \frac{j}{ke}\right) ^{\frac{1}{k} }\left\Vert \frac{1}{j!}\widehat{d}^{j}f\left( 0\right) \right\Vert _{\Theta}^{\frac{1}{j}}\leq A.$$ \end{proposition} Now we recall the definition of the spaces of holomorphic functions of a given type $A$ and infinite order. \begin{definition}\rm(\cite[Definition 2.8]{favaro2}) \rm{\label{expInfinito}Let $\left( \mathcal{P}_{\Theta}(^{j}E)\right) _{j=0}^{\infty}$ be a holomorphy type from $E$ to $\mathbb{C}$. If $A\in\left[ 0,+\infty\right) ,$ we denote by $\mathcal{H}_{\Theta b}\left( B_{\frac{1} {A}}\left( 0\right) \right) $ the Fr\'echet space of all $f\in\mathcal{H}\left( B_{\frac{1}{A}}\left( 0\right) \right) $ such that $\widehat{d}^{j}f\left( 0\right) \in\mathcal{P}_{\Theta}\left( ^{j}E\right) ,$ for all $j\in\mathbb{N}_{0}$ and} \[ \limsup\limits_{j\rightarrow\infty}\left\Vert \frac{1}{j!}\widehat{d} ^{j}f\left( 0\right) \right\Vert _{\Theta}^{\frac{1}{j}}\leq A, \] \textrm{endowed with the locally convex topology generated by the family of seminorms $\left( p_{\Theta,\rho}^{\infty}\right) _{\rho>A},$ where \[ p_{\Theta,\rho}^{\infty}\left( f\right) = {\displaystyle\sum\limits_{j=0} ^{\infty}} \rho^{-j}\left\Vert \frac{1}{j!}\widehat{d}^{j}f\left( 0\right) \right\Vert _{\Theta}. \] } \end{definition} We also denote $\mathcal{H}_{\Theta b}\left( B_{\frac{1}{A}}\left( 0\right) \right) $ by $Exp_{\Theta,0,A}^{\infty}\left( E\right) $ and we also write $Exp_{\Theta,0}^{\infty}\left( E\right) =Exp_{\Theta,0,0}^{\infty}\left( E\right) $. \begin{remark} \rm Note that the space $Exp_{\Theta,0}^{\infty}\left( E\right)$ coincides with the space $\mathcal{H}_{\Theta b}(E)$ introduced by Nachbin \cite{nachbin2}. \end{remark} We need to recall the algebraic isomorphism of the Fourier-Borel transform proved in \cite[Theorems 4.6 and 4.9]{favaro2}. This isomorphism plays a key role in the proofs of the hypercyclicity results for convolution operators and existence and approximation results for convolution equations. To introduce the Fourier-Borel transform it is necessary to recall the definition of the Borel transform and to use that $\Theta$ is a $\pi_{1}$-holomorphy type. The concept of $\pi_{1}$-holomorphy type was originally introduced in \cite[Definitions 2.3]{favaro1} and nowadays we use a slight variation of this concept which can be found in \cite[Definition 2.5]{favaro3}. This concept and related notions are very useful to prove results of this type, see e.g. \cite{favaro3, CDSjmaa, favaro1, favaro2,FM, gupta, G, MPS}. \begin{definition}\label{pi-tipo de holomorfia} \rm Let $(\mathcal{P}_{\Theta}(^{j}E;F))_{j=0}^{\infty}$ be a holomorphy type from $E$ to $F$. We say that this holomorphy type is a \emph{$\pi_1$-holomorphy type} if the following conditions hold: \begin{enumerate} \item[($1$)] $P_f(^{j}E;F) \subset \mathcal{P}_{\Theta}(^{j}E;F)$ and there exists $K>0$ such that $\\|\phi^j\cdot b\\|_{\Theta}\leq K^ j\\|\phi\\|^j\\|b\\|,$ for all $\phi\in E^{\prime}$, $b\in F$ and $j\in \mathbb{N}_0$; \item[($2$)] For $j\in \mathbb{N}_0$, $P_f(^{j}E;F)$ is dense in $\left(\mathcal{P}_{\Theta}(^{j}E;F), \\|\cdot \\|_{\Theta}\right)$. \end{enumerate} \end{definition} Now we recall the isomorphism given by Borel transform (see \cite[Definition 4.1]{favaro3} or \cite[p. 915]{favaro1}). \begin{definition}\rm Let $\Theta$ be a $\pi_{1}$-holomorphy type from $E$ to $F$. It is clear that the {\it Borel transform} \[ \mathcal{B}_{\Theta}\colon\left[ \mathcal{P}_{\Theta}(^{m}E;F)\right] ^{\prime}\longrightarrow\mathcal{P}(^{m}E^{\prime};F^{\prime})~, ~ \mathcal{B}_{\Theta} T(\phi)(y)=T(\phi^{m}y),\] for $T\in\left[ \mathcal{P}_{\Theta}(^{m}E;F)\right] ^{\prime}$, $\phi\in E^{\prime}$ and $y\in F$, is well defined and linear. Moreover, ${\cal B}_\Theta$ is continuous and injective by conditions (a1) and (a2) of Definition \ref{pi-tipo de holomorfia}. So, denoting the range of $\mathcal{B}_{\Theta}$ in $\mathcal{P} (^{m}E^{\prime};F^{\prime})$ by $\mathcal{P}_{\Theta^{\prime}}(^{m}E^{\prime };F^{\prime})$, the correspondence $$\mathcal{B}_{\Theta} T \in \mathcal{P} _{\Theta^{\prime}}(^{m}E^{\prime};F^{\prime})\mapsto \\|\mathcal{B}_{\Theta} T\\|_{\Theta^{\prime}}:=\\|T\\|, $$ defines a norm on $\mathcal{P} _{\Theta^{\prime}}(^{m}E^{\prime};F^{\prime})$. In this fashion the spaces $\left( \left[ \mathcal{P}_{\Theta} (^{m}E;F)\right] ^{\prime}\;,\\|\cdot\\|\right) $ and $(\mathcal{P}_{\Theta^{\prime}}(^{m}E^{\prime};F^{\prime}),\;\\|\cdot \\|_{\Theta^{\prime}})$ are isometrically isomorphic. \end{definition} \begin{definition}\rm(\cite[Definition 4.3]{favaro2}) \label{bklinha} Let $(\mathcal{P}_{\Theta}(^{j}E))_{j=0}^{\infty}$ be a $\pi_{1}$-holomorphy type from $E$ to $\mathbb{C}$. If $\ \rho>0$ and $k\geq1,$ we denote by $\mathcal{B}_{\Theta^{\prime},\rho}^{k}\left( E^{\prime}\right) $ the complex Banach space of all $f\in\mathcal{H}\left( E^{\prime}\right) $ such that $\widehat{d} ^{j}f\left( 0\right) \in\mathcal{P}_{\Theta^{\prime}}\left( ^{j}E^{\prime}\right) ,$ for all $j\in\mathbb{N}_{0}$\ and \[ \left\Vert f\right\Vert _{\Theta^{\prime},k,\rho}={\displaystyle\sum \limits_{j=0} ^{\infty}} \rho^{-j}\left( \frac{j}{ke}\right) ^{\frac{j}{k} }\left\Vert \frac{1}{j!}\widehat{d}^{j}f\left( 0\right) \right\Vert _{\Theta^{\prime}}<+\infty. \] \end{definition} As done just below of \cite[Definition 4.3]{favaro2} we may consider the space $Exp_{\Theta^\prime ,A}^{k}\left( E\right) $, for every $k\in[1,+\infty]$ and $A\in\left( 0,+\infty\right]$, but in this paper we are particularly interest in case $A=+\infty$. We recall the definition now. \begin{definition} \rm Let $\left( \mathcal{P}_{\Theta}(^{j}E)\right) _{j=0}^{\infty}$ be a $\pi_1$-holomorphy type from $E$ to $\mathbb{C}$ and $A\in\left( 0,+\infty\right]$. \item[(a)] For $k\geq1,$ we denote by $Exp_{\Theta^\prime ,A}^{k}\left( E\right) $ the complex vector space ${\displaystyle\bigcup \limits_{\rho<A}} \mathcal{B}_{\Theta^\prime,\rho}^{k}\left( E\right) $ with the locally convex inductive limit topology. \item[(b)] For $k=+\infty,$ $Exp_{\Theta^\prime ,\infty}^{\infty}\left( E^\prime\right) $ is a space of germs of holomorphic functions described in the following way: For $\rho>0,$ we define the complex vector space $\mathcal{H}_{\Theta^\prime}^{\infty}\left( B_{\frac{1}{\rho}}\left( 0\right) \right) $ of all $f\in\mathcal{H}\left( B_{\frac{1}{\rho}}\left( 0\right) \right) ,$ where $B_{\rho}\left( 0\right) $ denotes the open ball centered in $0$ and radius $\rho$ in $E^{\prime },$ such that $\widehat{d} ^{j}f\left( 0\right) \in\mathcal{P}_{\Theta^\prime}\left( ^{j}E\right) ,$ for all $j\in\mathbb{N}_{0}$\ and \[ p_{\Theta^\prime,\rho}^{\infty}={\displaystyle\sum\limits_{j=0}^{\infty}} \rho^{-j}\left\Vert \frac{1} {j!}\widehat{d}^{j}f\left( 0\right) \right\Vert _{\Theta^\prime}<+\infty, \] which is a Banach space with the norm $p_{\Theta^\prime,\rho}^{\infty}.$ Let $S= {\displaystyle\bigcup\limits_{0<\rho< A}} \mathcal{H}_{\Theta^\prime }^\infty\left( B_{\frac{1}{\rho}}\left( 0\right) \right) $ and define the following equivalence relation: \[ f\sim g\Longleftrightarrow\text{ there is }\rho>0 \text{ such that }f\|_{B_{\frac{1}{\rho}}\left( 0\right) }=g\|_{B_{\frac{1}{\rho} }\left( 0\right) .} \] We denote by $S\left/\sim\right. $ the set of all equivalence classes of elements of $S$ and by $\left[ f\right] $ the equivalence class which has $f$ as one representative. If we define the operations \[ \left[ f\right] +\left[ g\right] =\left[ f\|_{B_{\frac{1}{\rho}}\left( 0\right) }+g\|_{B_{\frac{1}{\rho}}\left( 0\right) }\right] , \] where $\rho\in (0,A) $ is such that $f\|_{B_{\frac{1}{\rho} }\left( 0\right) },g\|_{B_{\frac{1}{\rho}}\left( 0\right) }\in \mathcal{H}_{\Theta^\prime }^\infty\left( B_{\frac{1}{\rho}}\left( 0\right) \right) ,$ and \[ \lambda\left[ f\right] =\left[ \lambda f\right] ,\text{\qquad}\lambda \in\mathbb{C}, \] then $S\left/ \sim\right. $ becomes a vector space. For each $\rho\in (0,A) $, let $i_{\rho}\colon\mathcal{H}_{\Theta^\prime }^\infty\left( B_{\frac{1}{\rho}}\left( 0\right) \right) \longrightarrow S\left/ \sim\right. $ be given by $i_{\rho}\left( f\right) =\left[ f\right] .$ So we define $Exp_{\Theta^\prime ,A}^{\infty}\left( E\right) $ being the space $S\left/ \sim\right. $ with the locally convex inductive limit topology generated by the family $\left( i_{\rho}\right) _{\rho\in (0,A)}$. In both cases, $Exp_{\Theta^\prime ,A}^{k}\left( E\right) $ becomes a $DF$-space. \end{definition} Now we are able to recall the algebraic isomorphism given by the Fourier-Borel transform $\mathcal{F}$ in \cite[Theorem 4.6]{favaro2}. We define $\lambda\left( k\right) =\frac{k}{\left( k-1\right) ^{\frac{k-1}{k}}},$ for$\ k\in\left( 1,+\infty\right) .$ Since $\lim\limits_{k\rightarrow\infty}\lambda\left( k\right) =1 ,$ we set $\lambda\left( \infty\right) =1.$ When $A=0$ we write $A^{-1}=+\infty.$ \begin{theorem}\label{fourier_borel} Let $k\in\left( 1,+\infty\right] $, $A\in\left[ 0,+\infty\right) $ and $(\mathcal{P}_{\Theta}(^{j}E))_{j=0}^{\infty}$ be a $\pi_{1}$-holomorphy type from $E$ to $\mathbb{C}$. Then the Fourier-Borel transform \[ \mathcal{F}\colon\left[ Exp_{\Theta ,0, A} ^{k}\left( E\right) \right] ^{\prime}\longrightarrow Exp_{\Theta^{\prime} , (\lambda(k)A)^{-1}}^{k^{\prime}}\left( E^{\prime }\right) , \] given by $\mathcal{F}T\left( \varphi\right) =T\left( e^{\varphi}\right) ,$ for all $T\in\left[ Exp_{\Theta ,0,A} ^{k}\left( E\right) \right] ^{\prime}$\ and $\varphi\in E^{\prime},$ establishes an algebraic isomorphism. Note that, when $A=0$, we have $\mathcal{F}$ from $\left[ Exp_{\Theta ,0} ^{k}\left( E\right) \right] ^{\prime}$ to $Exp_{\Theta^{\prime} , \infty}^{k^{\prime}}\left( E^{\prime }\right) .$ \end{theorem} \section{Convolution operators} We start with a preliminary result we need to introduce convolution operators on $Exp_{\Theta ,0}^{k}\left( E\right)$. \begin{proposition}\label{derivadas} \label{dnfa}Let $\left( \mathcal{P}_{\Theta}(^{j}E)\right) _{j=0}^{\infty}$ be a holomorphy type from $E$ to $\mathbb{C}$, $a\in E,$ $k\in\left[ 1,+\infty\right] ,$ $A\in\left[ 0,+\infty\right) $ and $f\in Exp_{\Theta ,0,A}^{k}\left( E\right) .$ Then $\widehat{d}^{n}f\left( \cdot\right) a\in Exp_{\Theta ,0,\sigma A}^{k}\left( E\right) ,$ for any constant $\sigma$ satisfying condition (3) of Definition \ref{holomorphy type}. Besides {\small \[ \widehat{d}^{n}f\left( \cdot\right) a= {\displaystyle\sum\limits_{j=0}^{\infty}} \frac{1}{ j!}\overset {\begin{picture}(60,5)\put(0,0){\line(6,1){30}}\put(30,5){\line(6,-1){30}}\end{picture}} {d^{j+n}f\left( 0\right) \cdot^{j}}\left( a\right) , \]} in the topology of $Exp_{\Theta ,0,\sigma A}^{k}\left( E\right) .$ \end{proposition} \begin{proof} It is known (see Nachbin \cite[p. 29]{nachbin2}) that, for a fixed $j\in\mathbb{N}_0,$ {\small \begin{equation} \widehat{d}^{j}f\left( x\right) a= {\displaystyle\sum\limits_{n=0}^{\infty}} \frac{1}{ n!}\overset {\begin{picture}(60,5)\put(0,0){\line(6,1){30}}\put(30,5){\line(6,-1){30}}\end{picture}} {d^{j+n}f\left( 0\right) x^{n}}\left( a\right) = {\displaystyle\sum\limits_{n=0}^{\infty}} \frac{1}{ n!}\overset {\begin{picture}(60,5)\put(0,0){\line(6,1){30}}\put(30,5){\line(6,-1){30}}\end{picture}} {d^{j+n}f\left( 0\right) a^{j}}\left( x\right) , \label{serieConvPontual} \end{equation} } for all $x\in E$. Since $\hat{d}^{m}f(0)\in\mathcal{P}_{\Theta}(^{m}E),$ for all $m\in \mathbb{N}_{0},$ then $\Hhat{d^{j+n}f(0)a^j}\in \mathcal{P}_{\Theta}(^{n}E)$ and {\small \begin{equation} \label{Teo1_desig1}\left\Vert\Hhat{d^{j+n}f(0)a^j}\right\Vert_{\Theta}\leq\frac{n!j!\sigma^{j+n}} {(j+n)!}\left\Vert\hat{d}^{j+n}f(0)\right\Vert_{\Theta}\Vert a\Vert^{j}, \end{equation} } \noindent for all $n\in\mathbb{N}_0.$ In fact, let $P$ being the $(n+j)$-homogeneous polynomial $\hat{d}^{j+n}f(0)$. Thus $\check{P}=d^{j+n}f(0)$ and it follows from Definition \ref{def_holomorfia} that \small{\[ \Hhat{d^{j+n}f(0)a^j}=\widehat{\check{P}a^j}=\frac{j!}{(n+j)!}\hat{d}^n P(a). \]} Using condition $(3)$ of Definition $\ref{holomorphy type}$ we have \small{\[ \left\Vert\Hhat{d^{j+n}f(0)a^j}\right\Vert_{\Theta}=\frac{j!}{(n+j)!}\left\Vert \hat{d}^n P(a)\right\Vert_{\Theta}\leq\frac{n!j!\sigma^{j+n}} {(j+n)!}\left\Vert P\right\Vert_{\Theta}\Vert a\Vert^{j}=\frac{n!j!\sigma^{j+n}} {(j+n)!}\left\Vert\hat{d}^{j+n}f(0)\right\Vert_{\Theta}\Vert a\Vert^{j}. \]} For $k\in\left[ 1,+\infty\right) ,$ let {\small \[ N=\limsup\limits_{n\rightarrow\infty}\left( \frac{n+j}{ke}\right) ^{\frac {1}{k}}\left\Vert \frac{\widehat{d}^{n+j}f\left( 0\right) }{\left( n+j\right) !}\right\Vert _{\Theta }^{\frac{1}{n+j}}. \]} By Proposition \ref{caracterizacao sm(r,q)} we have $N\leq A<+\infty$. Then for every $\varepsilon>0,$ there is $C\left( \varepsilon\right) >0$ such that {\small \begin{equation} \label{Teo1_desig2}\left( \frac{n+j}{ke}\right) ^{\frac{n+j}{k}}\left\Vert \frac{\widehat {d}^{n+j}f\left( 0\right) }{\left( n+j\right) !}\right\Vert _{\Theta }\leq C\left( \varepsilon\right) \left( N+\varepsilon\right) ^{n+j}, \end{equation}} for all $n\in\mathbb{N}_0$. Hence {\small \begin{gather} \left( \frac{n}{ke}\right) ^{\frac{n}{k}}\frac{1}{n!}\left\Vert \overset {\begin{picture}(60,5)\put(0,0){\line(6,1){30}}\put(30,5){\line(6,-1){30}}\end{picture}} {d^{j+n}f\left( 0\right) a^{j}}\right\Vert _{\Theta}\overset {(\ref{Teo1_desig1})}{\leq}\left( \frac{n}{ke}\right) ^{\frac{n}{k}}\frac{j!\sigma^{j+n}}{(j+n)!}\left\Vert \widehat{d} ^{n+j}f\left( 0\right) \right\Vert _{\Theta}\left\Vert a\right\Vert ^{j}\nonumber\\ =j!\sigma^{j+n}\left( \frac{n}{ke}\right) ^{\frac{n}{k}}\left\Vert \frac{\widehat{d}^{n+j}f\left( 0\right) }{\left( n+j\right) !}\right\Vert _{\Theta}\left\Vert a\right\Vert ^{j}\overset{(\ref{Teo1_desig2})}{\leq }j!\sigma^{j+n}\left( \frac{n}{ke}\right) ^{\frac{n}{k}}\left( \frac {ke}{n+j}\right) ^{\frac{n+j}{k}}C\left( \varepsilon\right) \left( N+\varepsilon\right) ^{n+j}\left\Vert a\right\Vert ^{j}\nonumber\\ =j!\sigma^{j+n}\left( \frac{n}{n+j}\right) ^{\frac{n}{k}}\left( \frac {ke}{n+j}\right) ^{\frac{j}{k}}C\left( \varepsilon\right) \left( N+\varepsilon\right) ^{n+j}\left\Vert a\right\Vert ^{j}.\label{des1} \end{gather} } Since {\small \[ \lim_{n\rightarrow\infty}\left( j!\right) ^{\frac{1}{n}}\sigma^{\frac{j} {n}+1}\left( \frac{n}{n+j}\right) ^{\frac{1}{k}}\left( \frac{ke} {n+j}\right) ^{\frac{j}{kn}}=\sigma, \] } there is $D\left( \varepsilon\right) >0$ such that {\small \begin{equation} j!\sigma^{j+n}\left( \frac{n}{n+j}\right) ^{\frac{n}{k}}\left( \frac {ke}{n+j}\right) ^{\frac{j}{k}}\leq D\left( \varepsilon\right) \left( \sigma+\varepsilon\right) ^{n}, \label{des2} \end{equation}} \noindent for all $n\in\mathbb{N}_0$. From (\ref{des1}) and (\ref{des2}) we obtain {\small \[ \left( \frac{n}{ke}\right) ^{\frac{n}{k}}\frac{1}{n!}\left\Vert \overset {\begin{picture}(60,5)\put(0,0){\line(6,1){30}}\put(30,5){\line(6,-1){30}}\end{picture}} {d^{j+n}f\left( 0\right) a^{j}}\right\Vert _{\Theta}\leq C\left( \varepsilon\right) D\left( \varepsilon\right) \left\Vert a\right\Vert ^{j}\left( N+\varepsilon\right) ^{j}\left[ \left( \sigma +\varepsilon\right) \left( N+\varepsilon\right) \right] ^{n}, \]} for all $n\in\mathbb{N}_0$ and $\varepsilon>0.$ Therefore {\small \[ \limsup\limits_{n\rightarrow\infty}\left( \frac{n}{ke}\right) ^{\frac{1}{k} }\left\Vert \frac{\overset {\begin{picture}(60,5)\put(0,0){\line(6,1){30}}\put(30,5){\line(6,-1){30}}\end{picture}} {d^{j+n}f\left( 0\right) a^{j}}}{n!}\right\Vert _{\Theta }^{\frac{1}{n}}\leq\left( \sigma+\varepsilon \right) \left( N+\varepsilon\right) , \]} for all $\varepsilon>0$, which implies {\small \[ \limsup\limits_{n\rightarrow\infty}\left( \frac{n}{ke}\right) ^{\frac{1}{k} }\left\Vert \frac{\overset {\begin{picture}(60,5)\put(0,0){\line(6,1){30}}\put(30,5){\line(6,-1){30}}\end{picture}} {d^{j+n}f\left( 0\right) a^{j}}}{n!}\right\Vert _{\Theta }^{\frac{1}{n}}\leq\sigma N. \]} \noindent Since $N\leq A$, then $\sigma N\leq\sigma A$ and so $\widehat{d}^{n}f\left( \cdot\right) a\in Exp_{\Theta ,0,\sigma A}^{k}\left( E\right) $. \newline Now we consider $k=+\infty.$ If $A=0$, then $\mathcal{H}_{\Theta b }\left( E\right) =Exp_{\Theta ,0}^{\infty}\left( E\right) $ and this case was proved in \cite[Proposition 3.1 (i)]{favaro1}. If $A\neq 0$, then for $f\in Exp_{\Theta ,0,A}^{\infty}\left( E\right) =\mathcal{H}_{\Theta b }\left( B_{\frac{1}{A}}\left( 0\right) \right) $ we have {\small \[ \limsup\limits_{n\rightarrow\infty}\left\Vert \frac{\widehat{d}^{n+j}f\left( 0\right) }{\left( n+j\right) !}\right\Vert _{\Theta }^{\frac{1}{n+j}}\leq A \]} and as above we obtain {\small \[ \limsup\limits_{n\rightarrow\infty}\left\Vert \frac{\overset {\begin{picture}(60,5)\put(0,0){\line(6,1){30}}\put(30,5){\line(6,-1){30}}\end{picture}} {d^{j+n}f\left( 0\right) a^{j}}}{n!}\right\Vert _{\Theta }^{\frac{1}{n}}\leq \sigma A. \]} Thus $\widehat{d}^{n}f\left( \cdot\right) a\in\mathcal{H}_{\Theta b }\left( B_{\frac{1}{\sigma A}}\left( 0\right) \right) =Exp_{\Theta ,0,\sigma A}^{\infty}\left( E\right) .$ Now we only have to prove the convergence of the series in the topology of $Exp_{\Theta ,0,\sigma A}^{k}\left( E\right) $. If $f\in\mathcal{B}_{\Theta ,\rho}^{k}\left( E\right) $ for some $\rho>0$ with $k\in\left[ 1,+\infty \right) ,$ repeating the argument above with $\rho$ instead of $L$ we obtain constants $C_{1}\left( \varepsilon\right)>0$ and $D_{1}\left( \varepsilon\right)>0$ such that {\small \begin{gather} \left\Vert \widehat{d}^{j}f\left( \cdot\right) a- {\displaystyle\sum\limits_{n=0}^{v}} \left( n!\right) ^{-1}\overset {\begin{picture}(60,5)\put(0,0){\line(6,1){30}}\put(30,5){\line(6,-1){30}}\end{picture}} {d^{j+n}f\left( 0\right) \cdot^{n}}\left( a\right) \right\Vert _{\Theta ,k,\rho_{0}} \leq {\displaystyle\sum\limits_{n=v+1}^{\infty}} \rho_{0}^{-n}\left( \frac{n}{ke}\right) ^{\frac{1}{k}}\left\Vert \left( n!\right) ^{-1}\widehat{d}^{j+n}f\left( 0\right) \right\Vert _{\Theta }\left\Vert a\right\Vert ^{j}\sigma^{ n+j}\\ \leq C_{1}\left( \varepsilon\right) D_{1}\left( \varepsilon\right) \left\Vert a\right\Vert ^{j}\left( \rho+\varepsilon\right) ^{j} {\displaystyle\sum\limits_{n=v+1}^{\infty}} \left[ \rho_{0}^{-1}\left( \rho+\varepsilon\right) \left( \sigma+\varepsilon \right) \right] ^{n}, \end{gather} } and this tends to zero when $v\rightarrow\infty,$ for $\rho_{0}>\rho$ and $\varepsilon>0$ such that $\left( \rho+\varepsilon\right) \left( \sigma+\varepsilon\right) <\rho_{0}$. Hence $ {\displaystyle\sum\limits_{j=0}^{\infty}} \frac{1}{ j!}\overset {\begin{picture}(60,5)\put(0,0){\line(6,1){30}}\put(30,5){\line(6,-1){30}}\end{picture}} {d^{j+n}f\left( 0\right) \cdot^{j}}\left( a\right) $ converges to $\widehat{d}^{n}f\left( \cdot\right) a$ in the topology of $Exp_{\Theta ,0,\sigma A}^{k}\left( E\right) .$ The case $k=+\infty$ is analogous. \end{proof} Now we restrict ourselves to the case $A=0.$ The case for an arbitrary $A$ shall be treated later (cf. Theorem \ref{main3}). The following concept is well defined due to Proposition \ref{derivadas}. \begin{definition} \rm\textrm{\label{defOperConv} For $k\in\left[ 1,+\infty\right] ,$ a \emph{convolution operator on }$Exp_{\Theta ,0}^{k}\left( E\right) $ is a continuous linear mapping {\small \[ L\colon Exp_{\Theta ,0}^{k}\left( E\right) \longrightarrow Exp_{\Theta ,0}^{k}\left( E\right) \]} such that $d\left( Lf\right) \left( \cdot\right) a=L\left( df\left( \cdot\right) a\right) $ for all $a\in E$ and $f\in Exp_{\Theta ,0}^{k}\left( E\right) .$ We denote the set of all convolution operators on $Exp_{\Theta ,0} ^{k}\left( E\right) $ by $\mathcal{A}_{\Theta ,0}^{k}.$} \end{definition} Using induction it is easy to check that convolution operators commute with all the directional derivatives of all orders, that is, for all $a\in E, n\in\mathbb{N} $ and $L\in\mathcal{A}_{\Theta ,0}^{k}$, $L\left( \widehat{d}^{n}f\left( \cdot\right) a\right) =\widehat{d}^{n}\left( Lf\right) \left( \cdot\right) a.$ In Theorem \ref{teorema_translacao} we shall prove that convolution operators could have been defined replacing the condition $d\left( Lf\right) \left( \cdot\right) a=L\left( df\left( \cdot\right) a\right) $ by $\tau_{-a}\left( L\left( f\right) \right) =L\left( \tau_{-a}f\right) $ for all $a\in E,$ where $\tau_{-a}f\left( x\right) =f\left( x+a\right) ,$ for all $x\in E.$ So, commutativity with the directional derivatives or translations are equivalent concepts. First we need to prove that the translations are well-defined. \begin{proposition} \label{traslacao} Let $\left( \mathcal{P}_{\Theta}(^{j}E)\right) _{j=0}^{\infty}$ be a holomorphy type from $E$ to $\mathbb{C}$ and $k\in\left[ 1,+\infty\right] .$ If $f\in Exp_{\Theta ,0}^{k}\left( E\right) $ and $a\in E,$ then $\tau_{-a}f\in Exp_{\Theta ,0}^{k}\left( E\right) $ and {\small \[ \tau_{-a}f= {\displaystyle\sum\limits_{n=0}^{\infty}} \frac{1}{n!}\widehat{d}^{n}f\left( \cdot\right) a, \]} in the topology of $Exp_{\Theta ,0}^{k}\left( E\right) .$ \end{proposition} \begin{proof} The case $k=+\infty$ was proved in \cite[Proposition 3.1 (ii)]{favaro1}. For $k\in\left[ 1,+\infty\right) $ and $f\in Exp_{\Theta,0} ^{k}\left( E\right) $ we have that {\small \begin{equation} \limsup\limits_{j\rightarrow\infty}\left( \frac{j}{ke}\right) ^{\frac{1}{k} }\left\Vert \frac{\widehat{d}^{j}f\left( 0\right) }{j!}\right\Vert _{\Theta }^{\frac{1}{j}}= 0. \label{limsup3} \end{equation} } Thus for all $\varepsilon>0$ there is $C\left( \varepsilon\right) >0$ such that {\small \begin{equation} \left( \frac{j}{ke}\right) ^{\frac{j}{k}}\left\Vert \frac{\widehat{d} ^{j}f\left( 0\right) }{j!}\right\Vert _{\Theta }\leq C\left( \varepsilon\right) \varepsilon^{j}, \label{epsilon+limsup} \end{equation} } for all $j\in\mathbb{N}$. Since, for each $n\in\mathbb{N}$, $\widehat{d}^{n}\left( \tau_{-a}f\right) \left( 0\right) =\widehat{d}^{n}f\left( a\right) $ then we have {\small \[ \left\Vert \widehat{d}^{n}\left( \tau_{-a}f\right) \left( 0\right) \right\Vert _{\Theta }=\left\Vert \widehat{d}^{n}f\left( a\right) \right\Vert _{\Theta }\leq {\displaystyle\sum\limits_{j=0}^{\infty}} \frac{1}{j!}\left\Vert\overset {\begin{picture}(60,5)\put(0,0){\line(6,1){30}}\put(30,5){\line(6,-1){30}}\end{picture}} {d^{n+j}f\left( 0\right) a^{j}}\right \Vert_{\Theta} \leq {\displaystyle\sum\limits_{j=0}^{\infty}} \dfrac{n!\cdot \sigma^{n+j}}{(n+j)!}\left\Vert \widehat{d}^{n+j}f\left( 0\right) \right\Vert _{\Theta }\left\Vert a\right\Vert ^{j} \]} and {\small \begin{gather} \left( \frac{n}{ke}\right) ^{\frac{n}{k}}\frac{1}{n!}\left\Vert \widehat {d}^{n}\left( \tau_{-a}f\right) \left( 0\right) \right\Vert _{\Theta }\leq {\displaystyle\sum\limits_{j=0}^{\infty}} \left( \frac{n}{ke}\right) ^{\frac{n}{k}}\dfrac{ \sigma^{n+j}}{(n+j)!}\left\Vert \widehat{d}^{n+j}f\left( 0\right) \right\Vert _{\Theta }\left\Vert a\right\Vert ^{j}\\ = {\displaystyle\sum\limits_{j=0}^{\infty}} \left( \frac{n}{ke}\right) ^{\frac{n}{k}}\left( \frac{ke}{n+j}\right) ^{\frac{n+j}{k}}\left( \frac{n+j} {ke}\right) ^{\frac{n+j}{k}}\frac{\sigma^{n+j}}{\left( n+j\right) !}\left\Vert \widehat{d}^{n+j}f\left( 0\right) \right\Vert _{\Theta }\left\Vert a\right\Vert ^{j}\\ \leq {\displaystyle\sum\limits_{j=0}^{\infty}} \left( \frac{ke}{j}\right) ^{\frac{j}{k}}\sigma^{n+j}\left\Vert a\right\Vert ^{j}\left( \frac{n+j}{ke}\right) ^{\frac{n+j}{k}}\frac{1}{\left( n+j\right) !}\left\Vert \widehat{d}^{n+j}f\left( 0\right) \right\Vert _{\Theta }. \end{gather} } \newline Since $\lim\limits_{j\rightarrow\infty}\left( \frac{ke}{j}\right) ^{\frac{1}{k} }=0,$ for each $\varepsilon>0$ there is $D\left( \varepsilon\right) >0$ such that {\small \begin{equation} \left( \frac{ke}{j}\right) ^{\frac{j}{k}}\leq D\left( \varepsilon\right) \varepsilon^{j}, \label{depsilon} \end{equation} } for all $j\in\mathbb{N}$. Considering $\varepsilon>0$ such that $\sigma\varepsilon^2\left\Vert a\right\Vert <1$ and using (\ref{epsilon+limsup}), we obtain {\small \[\left( \frac{n}{ke}\right) ^{\frac{n}{k}}\frac{1}{n!}\left\Vert \widehat {d}^{n}\left( \tau_{-a}f\right) \left( 0\right) \right\Vert _{\Theta}\leq C\left( \varepsilon\right) D\left( \varepsilon\right) \sigma^{n}\varepsilon^{n} {\displaystyle\sum\limits_{j=0}^{\infty}} \sigma^{j}\varepsilon^{2j}\left\Vert a\right\Vert ^{j} =C\left( \varepsilon\right) D\left( \varepsilon\right) \sigma^{n}\varepsilon^{n}\frac{1}{1-\sigma\varepsilon ^2\left\Vert a\right\Vert}. \] } Hence {\small \begin{equation} \limsup\limits_{n\rightarrow\infty}\left( \frac{n}{ke}\right) ^{\frac{1}{k} }\left\Vert \frac{\widehat{d}^{n}\left( \tau_{-a}f\right) \left( 0\right) }{n!}\right\Vert _{\Theta } ^{\frac{1}{n}}=0, \label{limsuptrasl} \end{equation} } and so $\tau_{-a}f\in Exp_{\Theta,0}^{k}\left( E\right)$. To prove the convergence, let $f\in Exp_{\Theta ,0}^{k}\left( E\right) $ and $\rho>0$. Then {\small \begin{gather} \left\Vert \tau_{-a}f- {\displaystyle\sum\limits_{n=0}^{v}} \frac{1}{n!}\widehat{d}^{n}f\left( \cdot\right) a\right\Vert _{\Theta ,k,\rho}\leq {\displaystyle\sum\limits_{j=0}^{\infty}} \rho^{-j}\left( \frac{j}{ke}\right) ^{\frac{j}{k}} {\displaystyle\sum\limits_{n=v+1}^{\infty}} \frac{1}{j!n!}\left\Vert \widehat{d}^{j}\left( \widehat{d}^{n}f\left( \cdot\right) a\right) \left( 0\right) \right\Vert _{\Theta }\\ \leq {\displaystyle\sum\limits_{j=0}^{\infty}} \rho^{-j}\left( \frac{j}{ke}\right) ^{\frac{j}{k}} {\displaystyle\sum\limits_{n=v+1}^{\infty}} \frac{\sigma^{n+j}}{(n+j)!}\left\Vert \widehat{d}^{n+j}f\left( 0\right) \right\Vert _{\Theta }\left\Vert a\right\Vert ^{n}\\ ={\displaystyle\sum\limits_{j=0}^{\infty}} {\displaystyle\sum\limits_{n=v+1}^{\infty}} \rho^{-j}\left( \frac{ke}{n+j}\right) ^{\frac{n}{k}}\sigma^{n+j}\left( \frac{n+j}{ke}\right) ^{\frac{n+j}{k}}\left\Vert \frac{\widehat{d}^{n+j}f\left( 0\right) }{\left( n+j\right) !}\right\Vert _{\Theta }\left\Vert a\right\Vert ^{n}\\ \leq C\left( \varepsilon\right) D\left( \varepsilon\right) {\displaystyle\sum\limits_{j=0}^{\infty}} {\displaystyle\sum\limits_{n=v+1}^{\infty}} \rho^{-j}\varepsilon^{n}\varepsilon^{n+j}\sigma^{n+j}\left\Vert a\right\Vert ^{n}\\ \leq C\left( \varepsilon\right) D\left( \varepsilon\right) {\displaystyle\sum\limits_{j=0}^{\infty}}\rho^{-j}\varepsilon^j\sigma^j {\displaystyle\sum\limits_{n=v+1}^{\infty}} \varepsilon^{2n}\sigma^{n}\left\Vert a\right\Vert ^{n}. \end{gather} } Now, if for each $\rho>0$ we choose $\varepsilon>0$ such that $\varepsilon < \frac{\rho}{\sigma}$ and $\varepsilon^ 2\sigma\left\Vert a\right\Vert <1,$ then we obtain {\small \[ \lim\limits_{v\rightarrow\infty}\left\Vert \tau_{-a}f- {\displaystyle\sum\limits_{n=0}^{v}} \frac{1}{n!}\widehat{d}^{n}f\left( \cdot\right) a\right\Vert _{\Theta ,\rho}=0, \]} and the convergence follows from the definition of the topology. \end{proof} Using Proposition \ref{traslacao} it is not difficult to show the following result. \begin{proposition} \label{limLambdatende a zero}Let $\left( \mathcal{P}_{\Theta}(^{j}E)\right) _{j=0}^{\infty}$ be a holomorphy type from $E$ to $\mathbb{C}$, $k\in\left[ 1,+\infty\right] $, $f\in Exp_{\Theta ,0}^{k}\left( E\right) $ and $a\in E.$ Then {\small \[ \lim_{\lambda\rightarrow0}\lambda^{-1}\left( \tau_{-\lambda a}f-f\right) =\widehat{d}^{1}f\left( \cdot\right) a, \]} in the topology of $Exp_{\Theta ,0}^{k}\left( E\right) .$ \end{proposition} \begin{theorem}\label{teorema_translacao}Let $\left( \mathcal{P}_{\Theta}(^{j}E)\right) _{j=0}^{\infty}$ be a holomorphy type from $E$ to $\mathbb{C}$, $k\in\left[ 1,+\infty\right] $ and $L$ be a continuous linear mapping from $Exp_{\Theta,0}^{k}\left( E\right) $ into itself. Then $L$ is a convolution operator if, and only if, $L\left( \tau_{a}f\right) =\tau_{a}\left( Lf\right) $ for all $a\in E$ and $f\in Exp_{\Theta ,0}^{k}\left( E\right) .$ \end{theorem} \begin{proof} We saw that $L\left( \widehat{d}^{n}f\left( \cdot\right) a\right) =\widehat{d}^{n}\left( Lf\right) \left( \cdot\right) a$ for all $n\in\mathbb{N}$ and $a\in E.$ Using this fact and Proposition \ref{traslacao} we have {\small \[ L\left( \tau_{-a}f\right) = {\displaystyle\sum\limits_{n=0}^{\infty}} \frac{1}{n!}L\left( \widehat{d}^{n}f\left( \cdot\right) \left( a\right) \right) = {\displaystyle\sum\limits_{n=0}^{\infty}} \frac{1}{n!}\widehat{d}^{n}\left( Lf\right) \left( \cdot\right) a=\tau_{-a}\left( Lf\right) , \]} which implies $L\left( \tau_{a}f\right) =\tau_{a}\left( Lf\right) .$ Now suppose that $L$ satisfies $L\left( \tau_{a}f\right) =\tau_{a}\left( Lf\right) $ for all $a\in E.$ Thus it follows from Proposition \ref{limLambdatende a zero} that {\small \begin{gather} \widehat{d}^{1}\left( Lf\right) \left( \cdot\right) a=\lim_{\lambda\rightarrow0}\lambda^{-1}\left( \tau_{-\lambda a}\left( Lf\right) -Lf\right) =\lim_{\lambda\rightarrow0} \lambda^{-1}\left( L\left( \tau_{-\lambda a}f\right) -Lf\right) \\ =\lim_{\lambda\rightarrow0}L\left( \lambda^{-1}\left( \tau_{-\lambda a}f-f\right) \right) =L\left( \lim_{\lambda \rightarrow0}\lambda^{-1}\left( \tau_{-\lambda a}f-f\right) \right) =L\left( \widehat{d}^{1}f\left( \cdot\right) a\right) . \end{gather} } Hence $L$ is a convolution operator. \end{proof} Now we are interested to provide a description of all convolution operators on $Exp_{\Theta ,0}^{k}\left( E\right) .$ To do this, we need to introduce the convolution product. \begin{definition}\rm Let $\left( \mathcal{P}_{\Theta}(^{j}E)\right) _{j=0}^{\infty}$ be a holomorphy type from $E$ to $\mathbb{C}$, $k\in\left[ 1,+\infty\right] ,$ $T\in\left[ Exp_{\Theta ,0}^{k}\left( E\right) \right] ^{\prime}$ and $f\in Exp_{\Theta ,0}^{k}\left( E\right) $. We define \emph{the convolution product of }$T$ \emph{and} $f$ by $\left( T\ast f\right) \left( x\right) =T\left( \tau_{-x}f\right) ,$ for all $x\in E.$ \end{definition} We will prove that all convolution operators are of the form $T\ast,$ but to prove that $T\ast$ defines a convolution operator on $Exp_{\Theta ,0}^{k}\left( E\right) ,$ for $k\in\left[ 1,+\infty\right] ,$ we need the following definition, which is a generalization of the concept of $\pi_2$-holomorphy type \cite[Definition 2.5]{favaro3}. The concept of $\pi_2$-holomorphy type was used in \cite{favaro3} to describe convolution operators on $\mathcal{H}_{\Theta b}(E)$. \begin{definition}\rm\label{pi2k}Let $k\in\left[ 1,+\infty\right] $ and $A\in\left[ 0,+\infty\right) $. A holomorphy type $(\mathcal{P}_{\Theta}(^{j}E))_{j=0}^{\infty}$ from $E$ to $\mathbb{C}$ is said to be a \emph{$\pi_{2,k}$-holomorphy type} if, for each $T\in\left[ Exp_{\Theta, 0,A}^{k}\left( E\right) \right] ^{\prime}$, the following conditions hold: \begin{enumerate} \item[(1)] For $j\in \mathbb{N}_{0}$ and $m \in \mathbb{N}_{0}$, $m\leq j,$ if $P\in\mathcal{P}_{\Theta }\left( ^{j}E\right) $ with $B\in\mathcal{L}\left( ^{j}E\right) $ such that $P=\hat{B},$ then the polynomial {\small \begin{align} T\left( \widehat{B\cdot^{m}}\right) \colon E & \longrightarrow\mathbb{C}\\ y & \longmapsto T\left( B\cdot^{m}y^{j-m}\right) \end{align} } belongs to $\mathcal{P}_{\Theta }\left( ^{j-m}E\right)$; \item[(2)]For constants $C> 0$ and $\rho>A$ such that {\small \[ \left\vert T\left( f\right) \right\vert \leq C \left\Vert f\right\Vert _{\Theta ,k,\rho}, \quad\text{if}\; k\in [1,+\infty), \]} {\small \[ \left\vert T\left( f\right) \right\vert \leq Cp_{\Theta ,\rho}^{\infty}\left( f\right) ,\quad\text{if }k=+\infty, \]} for all $f\in Exp_{\Theta,0,A}^{k}\left( E\right)$, there is a constant $M > 0$ such that {\small \[ \left\Vert T\left( \widehat{A\cdot^{m}}\right) \right\Vert _{\Theta }\leq C M^ j\rho^{-m}\left( \frac{m}{ke}\right) ^{\frac{m}{k}}\left\Vert P\right\Vert _{\Theta },\quad\text{if}\; k\in [1,+\infty), \]} {\small \[ \left\Vert T\left( \widehat{A\cdot^{m}}\right) \right\Vert _{\Theta }\leq CM^j\rho^{-m}\left\Vert P\right\Vert _{\Theta },\quad\text{if}\; k=+\infty. \]} for every $P\in\mathcal{P}_{\Theta }\left( ^{j}E\right) $, $j\in \mathbb{N}_{0}$ and $m \in \mathbb{N}_{0}$, $m\leq j$. \end{enumerate} \end{definition} \begin{remark}\rm \begin{enumerate} \item[(i)] Note that the constants $C$ and $\rho$ of Definition \ref{pi2k} (2) exist because $T\in\left[ Exp_{\Theta, 0,A}^{k}\left( E\right) \right] ^{\prime}$. \item[(ii)] When $k=+\infty$ and $A=0$ the concepts of $\pi_{2,\infty}$-holomorphy type and $\pi_2$-holomorphy (see \cite[Definition 2.5]{favaro3}) type coincide. So in this case we write $\pi_{2,\infty}=\pi_{2}$. \end{enumerate} \end{remark} \begin{theorem}\label{T}Let $k\in\left[ 1,+\infty\right] $ and $(\mathcal{P}_{\Theta}(^{j}E))_{j=0}^{\infty}$ be a $\pi_{2,k}$-holomorphy type from $E$ to $\mathbb{C}$. If $T\in\left[ Exp_{\Theta ,0}^{k}\left( E\right) \right] ^{\prime}$ and $f\in Exp_{\Theta ,0}^{k}\left( E\right) ,$ then $T\ast f\in Exp_{\Theta ,0}^{k}\left( E\right) $ and $T\ast \in\mathcal{A}_{\Theta ,0}^{k} .$ \end{theorem} \begin{proof} The linearity of $T\ast$ is clear. By Propositions \ref{dnfa} and \ref{traslacao} we have that {\small \[ \left( T\ast f\right) \left( x\right) =T\left( \tau_{-x}f\right) = {\displaystyle\sum\limits_{n=0}^{\infty}} \frac{1}{n!}T\left( \widehat{d}^{n}f\left( \cdot\right) \left( x\right) \right) = {\displaystyle\sum\limits_{n=0}^{\infty}} \frac{1}{n!} {\displaystyle\sum\limits_{j=0}^{\infty}} \frac{1}{j!}T\left( \overset {\begin{picture}(60,5)\put(0,0){\line(6,1){30}}\put(30,5){\line(6,-1){30}}\end{picture}} {d^{j+n}f\left( 0\right) \cdot^{j}}\left( x\right) \right) . \]} Let $f\in Exp_{\Theta ,0}^{k}\left( E\right)$ and $T\in\left[ Exp_{\Theta ,0}^{k}\left( E\right) \right] ^{\prime}$. Since $\Theta$ is a $\pi_{2,k}$-holomorphy type we have that $T\left( \overset {\begin{picture}(60,5)\put(0,0){\line(6,1){30}}\put(30,5){\line(6,-1){30}}\end{picture}} {d^{j+n}f\left( 0\right) \cdot^{j}}\right)\in\mathcal{P} _{\Theta }\left( ^{n}E\right) $ and {\small \[ \left\Vert T\left( \overset {\begin{picture}(60,5)\put(0,0){\line(6,1){30}}\put(30,5){\line(6,-1){30}}\end{picture}} {d^{j+n}f\left( 0\right) \cdot^{j}}\right) \right\Vert _{\Theta }\leq CM^{j+n}\rho^{-j}\left( \frac{j}{ke}\right) ^{\frac{j}{k}}\left\Vert \widehat{d}^{j+n}f\left( 0\right) \right\Vert _{\Theta }, \]} for $k\in\left[ 1,+\infty\right) ,$ and {\small \[ \left\Vert T\left( \overset {\begin{picture}(60,5)\put(0,0){\line(6,1){30}}\put(30,5){\line(6,-1){30}}\end{picture}} {d^{j+n}f\left( 0\right) \cdot^{j}}\right) \right\Vert _{\Theta }\leq CM^{j+n}\rho^{-j}\left\Vert \widehat{d} ^{j+n}f\left( 0\right) \right\Vert _{\Theta }, \]} for $k=+\infty,$ where $C$, $\rho$ and $M$ are as in Definition \ref{pi2k}. If $k\in\left[ 1,+\infty\right) $ and $0<\rho^{\prime}<\rho,$ then {\small \begin{gather} {\displaystyle\sum\limits_{j=0}^{\infty}} \frac{1}{j!}\left\Vert T\left( \overset {\begin{picture}(60,5)\put(0,0){\line(6,1){30}}\put(30,5){\line(6,-1){30}}\end{picture}} {d^{j+n}f\left( 0\right) \cdot^{j}}\right) \right\Vert _{\Theta }\leq C {\displaystyle\sum\limits_{j=0}^{\infty}} \frac{1}{j!}M^{j+n}(\rho^{\prime})^{-j}\left( \frac{j}{ke}\right) ^{\frac{j}{k}}\left\Vert \widehat{d}^{j+n}f\left( 0\right) \right\Vert _{\Theta }\nonumber\\ =(\rho^{\prime})^{n}C n! {\displaystyle\sum\limits_{j=0}^{\infty}} M^{j+n}\frac{\left( j+n\right) !}{j!n!}\left( \frac{j}{j+n}\right) ^{\frac{j}{k} }\left( \frac{ke}{j+n}\right) ^{\frac{n}{k}}\left( \frac{j+n}{ke}\right) ^{\frac{j+n}{k}}(\rho^{\prime})^{-\left( j+n\right) }\left\Vert \frac{\widehat{d} ^{j+n}f\left( 0\right) }{\left( j+n\right) !}\right\Vert _{\Theta }. \label{desig3.11} \end{gather} } Since {\small \[ \limsup\limits_{j\rightarrow\infty}\binom{j+n}{n}^{\frac{1}{j+n}} =1, \]} then for every $\varepsilon>0$ there is $D\left( \varepsilon\right) >0$ such that {\small \[ \binom{j+n}{n}\leq D\left( \varepsilon\right) \left( 1+\varepsilon\right) ^{j+n}, \]} for all $j\in\mathbb{N}$. Hence {\small \begin{gather} {\displaystyle\sum\limits_{j=0}^{\infty}} \frac{1}{j!}\left\Vert T\left( \overset {\begin{picture}(60,5)\put(0,0){\line(6,1){30}}\put(30,5){\line(6,-1){30}}\end{picture}} {d^{j+n}f\left( 0\right) \cdot^{j}}\right) \right\Vert _{\Theta } \\ \leq C\cdot D\left( \varepsilon\right) (\rho^{\prime})^{n}n!\left( \frac{ke}{n}\right) ^{\frac{n}{k} }\sum\limits_{j=0}^{\infty}\left(\frac{\rho^{\prime}}{M(1+\varepsilon)}\right)^{-(j+n)}\left( \frac{j+n}{ke}\right) ^{\frac{j+n}{k}}\left\Vert \frac{\widehat{d} ^{j+n}f\left( 0\right) }{\left( j+n\right) !}\right\Vert _{\Theta }. \\ \leq C\cdot D\left( \varepsilon\right) (\rho^{\prime})^{n}n!\left( \frac{ke}{n}\right) ^{\frac{n}{k} }\left\Vert f\right\Vert _{\Theta ,k,\frac{\rho^{\prime}}{M(1+\varepsilon)}} \end{gather} } and so {\small \[ P_{n}= {\displaystyle\sum\limits_{j=0}^{\infty}} \frac{1}{j!}T\left( \overset {\begin{picture}(60,5)\put(0,0){\line(6,1){30}}\put(30,5){\line(6,-1){30}}\end{picture}} {d^{j+n}f\left( 0\right) \cdot^{j}}\right) \in\mathcal{P}_{\Theta }\left( ^{n}E\right) , \]} for each $n\in\mathbb{N}$ and {\small \[ \left\Vert P_{n}\right\Vert _{\Theta }\leq C\cdot D\left( \varepsilon\right) (\rho^{\prime})^{n}n!\left( \frac{ke}{n}\right) ^{\frac{n}{k}}\left\Vert f\right\Vert _{\Theta ,k,\frac{\rho^{\prime}}{M(1+\varepsilon)}}. \]} Hence {\small \[ \limsup\limits_{n\rightarrow\infty}\left( \frac{n}{ke}\right) ^{\frac{1}{k} }\left\Vert \frac{P_{n}}{n!}\right\Vert _{\Theta }^{\frac{1}{n}}\leq\limsup\limits_{n\rightarrow\infty }(C\cdot D(\varepsilon))^{\frac{1}{n}}\rho^{\prime}\left\Vert f\right\Vert _{\Theta ,k,\frac{\rho^{\prime}}{M(1+\varepsilon)}}^{\frac{1}{n}} =\rho^{\prime}. \]} Since $0<\rho^{\prime}<\rho$ is arbitrary we get {\small \[ \limsup\limits_{n\rightarrow\infty}\left( \frac{n}{ke}\right) ^{\frac{1}{k} }\left\Vert \frac{P_{n}}{n!}\right\Vert _{\Theta }^{\frac{1}{n}}=0. \]} Thus $T\ast f\in Exp_{\Theta ,0}^{k}\left( E\right) .$ For $\rho_{1}>0,$ if we choose $0<\rho^{\prime}<\rho$ and $\rho^{\prime}<\rho_{1},$ then we have {\small \begin{gather} \left\Vert T\ast f\right\Vert _{\Theta ,k,\rho_{1}}= {\displaystyle\sum\limits_{n=0}^{\infty}} \frac{1}{n!}\left( \frac{n}{ke}\right) ^{\frac{1}{k}}\rho_{1}^{-n}\left\Vert P_{n}\right\Vert _{\Theta }\\ \leq C\left\Vert f\right\Vert _{\Theta ,k,\frac{\rho^{\prime}}{M(1+\varepsilon)}} {\displaystyle\sum\limits_{n=0}^{\infty}} \left( \frac{\rho^{\prime}}{\rho_{1}}\right) ^{n}=C\left( 1-\frac {\rho^{\prime}}{\rho_{1}}\right) ^{-1}\left\Vert f\right\Vert _{\Theta ,k,\frac{\rho^{\prime}}{M(1+\varepsilon)}}. \end{gather} } and since the topology of $Exp_{\Theta ,0}^{k}\left( E\right)$ is generated by the family of norms $(\\|\cdot \\|_{\Theta,k,\rho})_{\rho>0}$, we have that $T\ast$ is continuous. The case $k=+\infty,$ was proved in \cite[Proposition 4.7]{favaro3}. The fact that $T$ commutes with translations is clear. \end{proof} \begin{definition}\rm\label{gamma_k} If $k\in\left[ 1,+\infty\right] $ and $(\mathcal{P}_{\Theta}(^{j}E))_{j=0}^{\infty}$ is a $\pi_{2,k}$-holomorphy type from $E$ to $\mathbb{C}$, we define {\small \[ \gamma_{\Theta ,0}^{k} \colon\mathcal{A}_{\Theta ,0} ^{k}\longrightarrow\left[ Exp_{\Theta ,0}^{k}\left( E\right) \right] ^{\prime} \]} by $\gamma_{\Theta ,0}^{k}\left( L\right) \left( f\right) =\left( Lf\right) \left( 0\right) ,$ for $f\in Exp_{\Theta ,0}^{k}\left( E\right) $ and $L\in\mathcal{A}_{\Theta ,0}^{k}.$ \end{definition} \begin{theorem} \label{bijLinear}If $k\in\left[ 1,+\infty\right] $ and $(\mathcal{P}_{\Theta}(^{j}E))_{j=0}^{\infty}$ is a $\pi_{2,k}$-holomorphy type from $E$ to $\mathbb{C}$, then the mapping $\gamma_{\Theta ,0}^{k}$ is a linear bijection and its inverse is the mapping {\small \[\Gamma_{\Theta ,0}^{k}\colon\left[ Exp_{\Theta ,0}^{k}\left( E\right) \right] ^{\prime}\longrightarrow\mathcal{A}_{\Theta ,0}^{k} \]} given by $\Gamma_{\Theta ,0} ^{k}\left( T\right) \left( f\right) =T\ast f,$ for $T\in\left[ Exp_{\Theta ,0}^{k}\left( E\right) \right] ^{\prime},$ $f\in Exp_{\Theta ,0}^{k}\left( E\right) $ and $k\in\left[ 1,+\infty\right] .$ \end{theorem} As a last result presented on this section we will prove that the Fourier-Borel transform becomes an isomorphism of algebras. We introduce the following product on $\left[ Exp_{\Theta ,0}^{k}\left( E\right) \right] ^{\prime}$. \begin{definition} \rm\label{produtoConvolucao} Let $k\in\left[ 1,+\infty\right] $, $(\mathcal{P}_{\Theta}(^{j}E))_{j=0}^{\infty}$ a $\pi_{2,k}$-holomorphy type from $E$ to $\mathbb{C}$ and $T_{1},T_{2}\in\left[ Exp_{\Theta ,0}^{k}\left( E\right) \right] ^{\prime}$. We define the \emph{convolution product} $T_{1} \ast T_{2}\in \left[ Exp_{\Theta ,0}^{k}\left( E\right) \right] ^{\prime}$ by {\small \[ T_{1}\ast T_{2}=\gamma_{\Theta ,0}^{k}\left( L_{1}\circ L_{2}\right) \in\left[ Exp_{\Theta ,0}^{k}\left( E\right) \right] ^{\prime}, \]} where $L_{1}=T_{1}\ast$ and $L_{2}=T_{2}\ast.$ \end{definition} It is easy to check that $\gamma_{\Theta ,0}^{k}$ preserves product, that is, $$\gamma_{\Theta ,0}^{k}\left( L_{1}\circ L_{2}\right) =\left( \gamma_{\Theta ,0}^{k}L_{1}\right) \ast\left( \gamma _{\Theta ,0}^{k}L _{2}\right) .$$ With this product $\left[ Exp_{\Theta ,0}^{k}\left( E\right) \right] ^{\prime}$ becomes an algebra with unit element $\delta\colon Exp_{\Theta ,0}^{k}\left( E\right) \rightarrow \mathbb{C}$ given by $\delta\left( f\right) =f\left( 0\right) $, for all $f\in Exp_{\Theta ,0}^{k}\left( E\right)$. \begin{theorem} \label{isomorfismoDeAlgebras}If $k\in\left( 1,+\infty\right] $ and $(\mathcal{P}_{\Theta}(^{j}E))_{j=0}^{\infty}$ is a $\pi_{1}$-$\pi_{2,k}$-holomorphy type from $E$ to $\mathbb{C}$, then the Fourier-Borel transform $\mathcal{F}$ is an isomorphism between the algebras $\left[ Exp_{\Theta ,0}^{k}\left( E\right) \right] ^{\prime}$ and $Exp_{\Theta^{\prime},\infty}^{k^{\prime}}\left( E^{\prime}\right) =Exp_{\Theta^{\prime}}^{k^{\prime}}\left( E^{\prime}\right) .$ \end{theorem} \begin{proof} Since $\Theta$ is a $\pi_{1}$-holomorphy type, then it follows from Theorem \ref{fourier_borel} that $\mathcal{F}$ is an algebraic isomorphism between these spaces. Since $\Theta$ is also a $\pi_{2,k}$-holomorphy type, then it is easy to see that $\mathcal{F}\left( T_{1}\ast T_{2}\right) =\left( \mathcal{F}T_{1}\right) \cdot\left( \mathcal{F}T_{2}\right) .$ \end{proof} \section{Hypercyclicity results} The main result of this section is the following theorem: \begin{theorem}\label{main1} Let $k\in\left( 1,+\infty\right] $, $E^{\prime}$ be separable and $(\mathcal{P}_{\Theta}(^{m}E))_{m=0}^{\infty}$ be a $\pi_1$-holomorphy type from $E$ to $\mathbb{C}$. Then every nontrivial convolution operator on $Exp_{\Theta ,0}^{k}\left( E\right)$ is mixing and thus in particular hypercyclic. \end{theorem} The proof of this result rests on the following theorem, which, as mentioned in the Introduction, is due to Costakis and Sambarino \cite{costakis} and sharpens an earlier result of Kitai \cite{kitai} and Gethner and Shapiro \cite{GS}. \begin{theorem} \label{Hypercyclity criterion}Let $X$ be a separable Fr\'echet space. Then a continuous linear mapping $T\colon X \rightarrow X$ is mixing if there are dense subsets $Z,Y$ of $X$ and a mapping $S\colon Y \rightarrow Y$ satisfying the following three conditions: \textit{(a) $T^{n}(z) \rightarrow 0$ when $n \rightarrow \infty$ for every $z \in Z$.} \textit{(b) $S^{n}(y) \rightarrow 0$ when $n \rightarrow \infty$ for every $y \in Y$.} \textit{(c) $T \circ S (y) = y$ for every $y \in Y$.} \end{theorem} Before proving Theorem \ref{main1} we need some auxiliary results. \begin{proposition}\label{densidade de ephi b} Let $k\in\left(1,+\infty\right] $, $(\mathcal{P}_{\Theta}(^{j}E))_{j=0}^{\infty}$ be a $\pi_{1}$-holomorphy type from $E$ to $\mathbb{C}$ and $U$ be a non-empty open subset of $E^{\prime}$. Then the set $S={\rm span}\{e^{\phi}:\phi\in U \}$ is dense in $Exp_{\Theta ,0}^{k}\left( E\right) $. \end{proposition} \begin{proof} Assume that $S$ is not dense in $Exp_{\Theta ,0}^{k}\left( E\right) $. Since $Exp_{\Theta ,0}^{k}\left( E\right) $ is a locally convex space, it follows as a consequence of Hahn-Banach Theorem that there exists a nonzero functional $T\in [Exp_{\Theta ,0}^{k}\left( E\right) ]^{\prime}$ that vanishes on $\overline{S}$. In particular $T(e^{\phi})=0$ for each $\phi\in U$ and so $\mathcal{F}T(\phi)=T(e^{\phi})=0$ for every $\phi \in U$ Thus $\mathcal{F} T$ is a holomorphic function that vanishes on the open non-empty set $U$ and this implies that $\mathcal{F} T\equiv 0$ on $E^\prime$. Since $\mathcal{F}$ is injective we have $T\equiv 0,$ a contradiction. Hence $S$ is dense in $Exp_{\Theta ,0}^{k}\left( E\right) $. \end{proof} Now we will prove that the exponential functions are eigenvectors for the convolution operators $L\in\mathcal{A}_{\Theta ,0}^{k}.$ Moreover, for each $L$ we will describe the eigenvalues associated to the exponential functions. This result is the key of the proof of Theorem \ref{main1}. \begin{lemma}\label{lemma scalar} Let $k\in\left( 1,+\infty\right] ,$ $(\mathcal{P}_{\Theta}(^{j}E))_{j=0}^{\infty}$ be a $\pi_{1}$-holomorphy type from $E$ to $\mathbb{C}$ and $L\in\mathcal{A}_{\Theta ,0}^{k}.$ Then: \begin{enumerate} \item[{\rm (a)}] $L(e^{\phi})=\mathcal{F}[\gamma_{\Theta ,0}^{k}\left( L\right)](\phi)\cdot e^{\phi }$ for every $\phi\in E^{\prime}.$ \item[{\rm(b)}] $L$ is a scalar multiple of the identity if and only if $\mathcal{F}\left[\gamma_{\Theta ,0}^{k}\left( L\right)\right]$ is constant. \end{enumerate} \end{lemma} \begin{proof} (a) By Definition \ref{gamma_k} and Theorem \ref{fourier_borel} we have $\gamma_{\Theta ,0}^{k}\left( L\right)\in\lbrack Exp_{\Theta ,0}^{k}\left( E\right)]^{\prime}$ and \[ \mathcal{F}[\gamma_{\Theta,0}^{k}(L)](\phi)=\gamma_{\Theta,0}^{k}(L)(e^{\phi })=L(e^{\phi})(0) \] for each $\phi\in E^{\prime}.$ Therefore \begin{align} L(e^{\phi})(y) & =[\tau_{-y}(L(e^{\phi}))](0) =[L\left( \tau_{-y}(e^{\phi})\right) ](0) =[L\left( e^{\phi(y)}\cdot e^{\phi}\right) ](0)\\ & =e^{\phi(y)}\cdot L(e^{\phi})(0) =e^{\phi(y)}\cdot \mathcal{F}[\gamma_{\Theta,0}^{k}(L)](\phi) = \left(\mathcal{F}[\gamma_{\Theta,0}^{k}(L)](\phi) \cdot e^\phi \right)(y), \end{align} for all $y\in E.$\newline(b) Let $\lambda\in\mathbb{C}$ be such that $\mathcal{F}(\gamma_{\Theta,0}^{k}(L)(\phi)=\lambda$ for every $\phi\in E^{\prime}.$ By (a) it follows that \[ L(e^{\phi})=\mathcal{F}[\gamma_{\Theta,0}^{k}(L)](\phi)\cdot e^{\phi}=\lambda e^{\phi} \] for every $\phi\in E^{\prime}.$ The continuity of $L$ and the denseness of $\{e^\phi : \phi \in E'\}$ in $Exp_{\Theta ,0}^{k}\left( E\right)$ (Proposition \ref{densidade de ephi b}) yield that $L(f)=\lambda f$ for every $f\in Exp_{\Theta ,0}^{k}\left( E\right)$. Conversely, let $\lambda\in\mathbb{C}$ be such that $L(f)=\lambda f,$ for every $f\in Exp_{\Theta ,0}^{k}\left( E\right)$. Using part (a) again we get \[ \lambda e^{\phi}=L(e^{\phi})=\mathcal{F}[\gamma_{\Theta,0}^{k} (L)](\phi)\cdot e^{\phi} \] and so $\mathcal{F}[\gamma_{\Theta,0}^{k} (L)](\phi)=\lambda,$ for every $\phi\in E^{\prime}$. \end{proof} \begin{proposition}\label{Linearmente indep} Let $k\in\left( 1,+\infty\right] $ and $(\mathcal{P}_{\Theta}(^{j}E))_{j=0}^{\infty}$ be a $\pi_{1}$-holomorphy type from $E$ to $\mathbb{C}$. Then the set $$B=\{e^\phi : \phi\in E^{\prime}\}$$ \noindent is a linearly independent subset of $Exp_{\Theta ,0}^{k}\left( E\right) $. \end{proposition} \begin{proof} We know that $B \subseteq Exp_{\Theta ,0}^{k}\left( E\right)$. Let $\{e^{\phi_i}\}_{i\in I}$ be a maximal linearly independent subset of $B$. Fix $\phi\in E'$ and assume that there exist non-zero constants $ \alpha_{i_1}, \ldots, \alpha_{i_n}\in \mathbb{C}$ such that \begin{equation} \alpha_{i_1} e^{\phi_{i_1}} + \cdots + \alpha_{i_n} e^{\phi_{i_n}} = e^{\phi}\label{linear} \end{equation} Given $a \in E$, it follows from Proposition \ref{derivadas} that the differentiation operator $$D_a\colon Exp_{\Theta ,0}^{k}\left( E\right)\longrightarrow Exp_{\Theta ,0}^{k}\left( E\right)~,~D_a\left( f\right) =\hat{d}^1f\left( \cdot\right) \left( a\right)$$ is well defined. Applying the operator $D_{a}$ in $(\ref{linear})$, it follows that \begin{equation} \alpha_{i_1} \cdot \phi_{i_1}(a)\cdot e^{\phi_{i_1}} + \cdots + \alpha_{i_n}\cdot \phi_{i_n}(a)\cdot e^{\phi_{i_n}} = \phi(a)\cdot e^{\phi}\label{linear1} \end{equation} Since $\{e^{\phi_i}\}_{i\in I}$ is linearly independent and $ \alpha_{i_1}, \ldots, \alpha_{i_n}$ are non-zero, by $(\ref{linear})$ and $(\ref{linear1})$ we have $$ \phi_{i_1}(a) = \cdots = \phi_{i_n}(a)=\phi(a).$$ Since $a\in E$ is arbitrary we obtain $$ \phi_{i_1} = \cdots = \phi_{i_n}=\phi.$$ Hence $\{\phi_i\}_{i\in I}=E'$ and so $B$ is linearly independent. \end{proof} \textit{Proof of Theorem \ref{main1}.} Let $L \colon Exp_{\Theta ,0}^{k}\left( E\right) \longrightarrow Exp_{\Theta ,0}^{k}\left( E\right)$ be a nontrivial convolution operator. We shall show that $L$ satisfies the Hypercyclicity Criterion of Theorem \ref{Hypercyclity criterion}. First of all, since $E'$ is separable and $\Theta$ is a $\pi_1$-holomorphy type, we have that $Exp_{\Theta ,0}^{k}\left( E\right) $ is separable as well. By \cite[Propositions 2.7]{favaro2}, $Exp_{\Theta ,0}^{k}\left( E\right) $ is a Fr\'{e}chet space. By $\Delta$ we mean the open unit disk in $\mathbb{C}$. Consider the sets \[ V=\{\phi\in E^{\prime}: \|\mathcal{F} [\gamma_{\Theta,0}^{k}(L)](\phi)\| <1 \}=\mathcal{F}[\gamma_{\Theta,0}^{k}(L)]^{-1}(\Delta) \] \noindent and \[ W=\{\phi\in E^{\prime}: \|\mathcal{F}[\gamma_{\Theta,0}^{k}(L)](\phi)\| >1 \}=\mathcal{F}[\gamma_{\Theta,0}^{k}(L)]^{-1}(\mathbb{C}-\overline{\Delta}). \] Since $L$ is not a scalar multiple of the identity, Lemma \ref{lemma scalar} (b) yields that $\mathcal{F}[\gamma_{\Theta,0}^{k}(L)]$ is non constant. Therefore, it follows from Liouville's Theorem that $V$ and $W$ are non-empty open subsets of $E^{\prime}$. Consider now the following subspaces of $Exp_{\Theta ,0}^{k}\left( E\right)$: \[ H_V = {\rm span}\{e^{\phi}: \phi\in V\} {\rm ~~and~~}H_W ={\rm span}\{e^{\phi}: \phi\in W\}. \] By Proposition \ref{densidade de ephi b} we know that both $H_V$ and $H_W$ are dense in $Exp_{\Theta ,0}^{k}\left( E\right)$. Let us deal with $H_V$ first. Given $\phi \in V$, from Lemma \ref{lemma scalar}(a) we have \[ L(e^{\phi})=\mathcal{F}[\gamma_{\Theta,0}^{k}(L)](\phi)\cdot e^{\phi} \in H_V. \] So $L(H_V) \subseteq H_V$ because $L$ is linear. Applying Lemma \ref{lemma scalar}(a) and the linearity of $L$ once again we get \[ L^{n}(e^{\phi})=\left (\mathcal{F}[\gamma_{\Theta,0}^{k}(L)](\phi)\right)^{n}\cdot e^{\phi} \] for all $n\in\mathbb{N}$ and $\phi \in V$. Now let $f \in H_{V}$, that is $f = \sum\limits_{j=1}^{m} \alpha_{j} e^{\phi_{j}}$, with $\alpha_{j} \in \mathbb{C}$ and $\phi_{j} \in V$. It follows that $$ L^{n}(f) = \sum_{j=1}^{m} \alpha_j L^{n}(e^{\phi_{j}}) = \sum_{j=1}^{m} \alpha_{j} \left(\mathcal{F}[\gamma_{\Theta,0}^{k}(L)](\phi_j)\right)^{n}e^{\phi_{j}}. $$ Since $\left\vert\mathcal{F}[\gamma_{\Theta,0}^{k}(L)](\phi_j)\right\vert<1$ for every $j=1,\ldots,m$, it follows that $L^{n}f \rightarrow 0$ when $n \rightarrow \infty$. Now we handle $H_W$. For each $\phi\in W$, $\mathcal{F}[\gamma_{\Theta,0}^k(L)](\phi) \neq 0$, so we can define \[ S(e^{\phi}):=\dfrac{e^{\phi}}{\mathcal{F}[\gamma_{\Theta,0}^k(L)](\phi)} \in Exp_{\Theta ,0}^{k}\left( E\right). \] By Proposition \ref{Linearmente indep}, $\{e^{\phi}: \phi\in W\}$ is a linearly independent set. Hence $S$ admits a unique extension to a linear mapping $S\colon H_{W} \rightarrow H_{W}$. Now let $f \in H_{W}$, that is $f = \sum\limits_{j=1}^{m} \alpha_{j} e^{\phi_{j}}$, with $\alpha_{j} \in \mathbb{C}$ and $\phi_{j} \in W$. It follows that $$ S^{n}f = \sum_{j=1}^{m} \frac{ \alpha_{j} e^{\phi_{j}} }{\left (\mathcal{F}[\gamma_{\Theta,0}^{k}(L)](\phi_j)\right)^{n}}. $$ Since $\left\vert\mathcal{F}[\gamma_{\Theta,0}^{k}(L)](\phi_j)\right\vert > 1$ for every $j$, it follows that $S^{n}f \rightarrow 0$ when $n \rightarrow \infty$. Finally, $L\circ S(f)=f$ for every $f\in H_W$, so $L$ is mixing. $\Box$ \begin{theorem}\label{main2} Let $k\in\left( 1,+\infty\right] $, $E^{\prime}$ be separable, $(\mathcal{P}_{\Theta}(^{m}E))_{m=0}^{\infty}$ be a $\pi_1$-$\pi_{2,k}$-holomorphy type and $T\in[Exp_{\Theta ,0}^{k}\left( E\right)]^{\prime}$ be a linear functional which is not a scalar multiple of $\delta_0$, where $\delta_0$ is defined by $\delta_0(f)=f(0)$. Then $\Gamma_{\Theta,0}^{k}(T)$ is a convolution operator that is not a scalar multiple of the identity, hence mixing. \end{theorem} \begin{proof} By Theorem \ref{bijLinear}, for each $T\in[Exp_{\Theta ,0}^{k}\left( E\right)]^{\prime}$, the mapping $\Gamma_{\Theta,0}^{k}(T)$ is a convolution operator. Suppose that there is $\lambda\in\mathbb{C}$ such that $\Gamma_{\Theta,0}^{k}(T)(f)=\lambda\cdot f$ for all $f\in Exp_{\Theta ,0}^{k}\left( E\right)$. Then \[ \lambda\cdot f(x)=\Gamma_{\Theta,0}^{k}(T)(f)(x)=(T\ast f)(x)=T(\tau _{-x}f) \] for every $x \in E$. In particular, \[ \lambda\cdot\delta_{0}(f)=\lambda\cdot f(0)=T(\tau_{0}f)=T(f) \] for every $f\in Exp_{\Theta ,0}^{k}\left( E\right)$. Hence $T=\lambda\cdot\delta_{0}$. This contradiction shows that $\Gamma_{\Theta,0}^{k}(T)$ is not a scalar multiple of the identity, hence mixing by Theorem \ref{main1}. \end{proof} Now we have an easy but interesting application of Lemma \ref{lemma scalar}. \begin{proposition} \label{dense_range}Let $k\in\left( 1,+\infty\right] $, $E^{\prime}$ be separable and $(\mathcal{P}_{\Theta}(^{n}E))_{n=0}^{\infty }$ be a $\pi_{1}$-holomorphy type from $E$ to $\mathbb{C}$. Then every nonzero convolution operator on $Exp_{\Theta ,0}^{k}\left( E\right)$ has dense range. \end{proposition} \begin{proof} Let $L\neq0$ be a convolution operator. If $L$ is a scalar multiple of the identity, then clearly $L$ is surjective. Suppose now that $L$ is not a scalar multiple of the identity. By Proposition \ref{densidade de ephi b}, ${\rm span}\{e^{\phi}: \phi\in E^{\prime }\}$ is dense in $Exp_{\Theta ,0}^{k}\left( E\right)$. By Lemma \ref{lemma scalar} (a), $L(e^{\phi})=\mathcal{F}[\gamma_{\Theta,0}^k(L))(\phi)\cdot e^{\phi}$ for every $\phi \in E'$, and this implies that each $e^{\phi}$ belongs to the range of $L$. Therefore, \[ Exp_{\Theta ,0}^{k}\left( E\right)=\overline{{\rm span}\{e^{\phi}: \phi\in E^{\prime} \}}=\overline{L(Exp_{\Theta ,0}^{k}\left( E\right))}. \] \end{proof} \begin{remark} \label{canonical_cte}\rm Note that, if $\sigma\neq 1$ and $A\neq 0$ then the concept of convolution operator on $Exp_{\Theta ,0,A}^{k}\left( E\right) $ is senseless because in this case Proposition \ref{derivadas} does not assure that $ d^{n}f\left( \cdot\right) a$ belongs to $Exp_{\Theta ,0,A}^{k}\left( E\right) $. In the most usual examples the inequality (2) of definition of holomorphy type (Definition \ref{holomorphy type}) works with the constant $\frac{j!}{k!(j-k)!}$ instead of $\sigma^{j}$. Since $$\frac{j!}{k!(j-k)!}\leq 2^{j},$$ it follows that, in this case, inequality (2) of Definition \ref{holomorphy type} is valid with $\sigma=2$. So, if $\Theta$ is a holomorphy type such that condition (2) of Definition \ref{holomorphy type} is valid with $\frac{j!}{k!(j-k)!}$ instead of $\sigma^{j}$, for every $k,j\in\mathbb{N}_{0}$, $k\leq j $, then in Proposition \ref{derivadas} we obtain that $\widehat{d}^{n}f\left( \cdot\right) a\in Exp_{\Theta ,0, A}^{k}\left( E\right) ,$ instead of $\widehat{d}^{n}f\left( \cdot\right) a\in Exp_{\Theta ,0, \sigma A}^{k}\left( E\right) .$ Consequently, in this case, we also may define convolution operators from $Exp_{\Theta ,0, A}^{k}\left( E\right)$ to itself as in Definition \ref{defOperConv}. We denote the set of all convolution operators on $Exp_{\Theta ,0,A} ^{k}\left( E\right) $ by $\mathcal{A}_{\Theta ,0,A}^{k}.$ In this case we say that $\Theta$ is a holomorphy type with \emph{canonical constants}. \end{remark} We finish this section stating the analogous of Theorem \ref{main1} in the case that $\Theta$ is a holomorphy type with canonical constants, which encompasses most known examples. The proof is a straightforward adaptation of the previous results. \begin{theorem}\label{main3} Let $k\in\left( 1,+\infty\right] $, $A\in\left[ 0,+\infty\right) $, $E^{\prime}$ be separable and $(\mathcal{P}_{\Theta}(^{m}E))_{m=0}^{\infty}$ be a $\pi_1$-holomorphy type from $E$ to $\mathbb{C}$, with canonical constants. Then every nontrivial convolution operator on $Exp_{\Theta ,0,A}^{k}\left( E\right)$ is mixing and thus in particular hypercyclic. \end{theorem} \section{Existence and Approximation Theorems for Convolution Equations} \begin{definition} \label{division space}\rm Let $U$ be an open subset of $E$ and $\mathcal{F}(U)$ be a collection of holomorphic functions from $U$ into $\mathbb{C}$. We say that $\mathcal{F}(U)$ is \emph{closed under division } if, for each $f$ and $g$ in $\mathcal{F}(U)$, with $g\neq 0$ and $h=f/g$ a holomorphic function on $U$, we have $h$ in $\mathcal{F}(U)$.\newline The quotient notation $h=f/g$ means that $f(x)=h(x)\cdot g(x)$, for all $x\in U$. \end{definition} The next useful result was proved by Gupta in \cite{gupta}. \begin{lemma} \label{lema gupta} Let $U$ be an open connected subset of $E.$ Let $f$ and $g$ be holomorphic functions on $U,$ with $g$ non identically zero, such that, for any affine subspace $S$ of $E$ of dimension one, and for any connected component $S^{\prime}$ of $S\cap U$ on which $g$ is not identically zero, the restriction $f\|_{S^{\prime}}$ is divisible by the restriction $g\|_{S^{\prime}},$ with the quotient being holomorphic in $S^{\prime}.$ Then $f$ is divisible by $g$ and the quotient is holomorphic on $U.$ \end{lemma} \begin{theorem} \label{divisao} Let $k\in\left( 1,+\infty\right] $ and $(\mathcal{P}_{\Theta}(^{j}E))_{j=0}^{\infty}$ be a $\pi_{1}$-$\pi_{2,k}$-holomorphy type from $E$ to $\mathbb{C}$. If\linebreak $Exp^{k'}_{\Theta^{\prime}}(E^{\prime})$ is closed under division and $T_{1},T_{2}\in[Exp_{\Theta ,0}^{k}\left( E\right)]^{\prime}$ are such that\ $T_{2}\neq0$ and $T_{1}\left( P\cdot e^\phi\right) =0$ whenever $T_{2}\ast( P\cdot e^\phi)=0$ with $\phi\in E^{\prime}$ and $P\in\mathcal{P}_{\Theta}\left( ^{m}E\right) ,$ $m\in\mathbb{N}_{0},$ then $\mathcal{F}T_{1}$ is divisible by $\mathcal{F}T_{2}$ with the quotient being an element of $Exp^{k'}_{\Theta^{\prime}}(E^{\prime}).$ \end{theorem} \begin{proof} Let $S$ be an one dimensional affine subspace of $E^{\prime}.$ It is clear that $S$ is of the form $\left\{ \phi_{1}+t\phi_{2};t\in\mathbb{C}\right\} ,$ where $\phi_{1},\phi_{2}\in E^{\prime}$ are fixed. We suppose that $t_{0}$ is a zero of order $k$ of \[ g_{2}\left( t\right) =\mathcal{F}T_{2} \left( \phi _{1}+t\phi_{2}\right) =T_{2}\left( e^{ \phi_{1}+t\phi_{2}} \right) . \] Then we have \[ T_{2}\left( \phi_{2}^{j}\cdot e^{ \phi_{1}+t_{0}\phi_{2}} \right) =0, \] for each $j<k,$ and this implies $$ T_{2}\ast\left(\phi_{2}^{j}\cdot e^{ \phi_{1}+t_{0}\phi_{2}}\right) = {\displaystyle\sum\limits_{m=0}^{j}} \binom{j}{m}\phi_{2}^{j-m}\cdot e^{ \phi_{1}+t_{0}\phi_{2}}\cdot T_{2}\left( \phi_{2}^{m}\cdot e^{ \phi_{1}+t_{0}\phi_{2}} \right) =0, $$ for each $j<k.$ Hence, it follows from the hypothesis that $T_{1}\left( \phi_{2}^{j}\cdot e^{ \phi_{1}+t_{0}\phi_{2}} \right) =0,$ for all $j<k,$ and this implies that $t_{0}$ is a zero of order at least $k$ of $g_{1}\left( t\right) =\mathcal{F}T_{1}\left( \phi_{1}+t\phi_{2}\right) .$ Therefore $\mathcal{F}T_{1}\|_{S}$ is divisible by $\mathcal{F}T_{2}\|_{S}$ and the quotient is holomorphic on $S.$ By Lemma \ref{lema gupta}, we have that $\mathcal{F}T_{1}$ is divisible by $\mathcal{F}T_{2}$ on $E^{\prime}$ and the quotient is an entire function. Since $Exp^{k'}_{\Theta^{\prime}}(E^{\prime})$ is closed under division, then the quotient $\mathcal{F}T_{1}/\mathcal{F}T_{2}$ belongs to $Exp^{k'}_{\Theta^{\prime}}(E^{\prime}).$ \end{proof} \begin{theorem} \label{teoremaAproximacao1}Let $k\in\left( 1,+\infty\right] $ and $(\mathcal{P}_{\Theta}(^{j}E))_{j=0}^{\infty}$ be a $\pi_{1}$-$\pi_{2,k}$-holomorphy type from $E$ to $\mathbb{C}$. If $Exp^{k'}_{\Theta^{\prime}}(E^{\prime})$ is closed under division and $L\in\mathcal{A}_{\Theta ,0}^{k},$ then the vector subspace of $Exp_{\Theta ,0} ^{k}\left( E\right) $ ge\-ne\-ra\-ted by {\small \[ \mathcal{L=}\left\{ P\cdot e^\varphi\colon P\in\mathcal{P}_{\Theta }\left( ^{n}E\right) ,n\in\mathbb{N}_0,\varphi\in E^{\prime} \textrm{ and }L\left( P\cdot e^\varphi\right) =0\right\} \]} is dense in {\small \[ \ker L=\left\{ f\in Exp_{\Theta ,0}^{k}\left( E\right) \colon Lf=0\right\} . \]} \end{theorem} \begin{proof} First let us consider $L$ equal to $0.$ In this case $\ker L= Exp_{\Theta ,0}^{k}\left( E\right)$ and the result follows from Proposition \ref{densidade de ephi b}. Now consider $L\neq 0$. By Theorem \ref{bijLinear} there is $T\in [Exp_{\Theta ,0}^{k}\left( E\right) ]'$ such that $L=T\ast.$ Suppose that $R\in [Exp_{\Theta ,0}^{k}\left( E\right) ]'$ is such that $R\|_{\mathcal{L}}=0.$ Thus by Theorem \ref{divisao}, there is $H\in Exp_{\Theta^{\prime}}^{k'}\left( E^{\prime}\right) $ such that $\mathcal{F}\left( R\right) =H\cdot\mathcal{F}\left( T\right) .$ By the isomorphism of the Fourier-Borel transform (see Theorem \ref{fourier_borel}) there is $S\in[Exp_{\Theta ,0}^{k}\left( E\right) ]'$ such that $H=\mathcal{F}\left( S\right) $ and $\mathcal{F}\left( R\right) =\mathcal{F}\left( S\right)\cdot \mathcal{F}\left( T\right) =\mathcal{F}\left( S\ast T\right) .$ Hence $R=S\ast T$ and for each $f\in {\rm ker }L,$ we have $R\ast f=S\ast\left( T\ast f\right)=S\ast (Lf) =0$ and $R\left( f\right) =\left( R\ast f\right) \left( 0\right) =0.$ We showed that every $R\in [Exp_{\Theta ,0}^{k}\left( E\right) ]'$ vanishing on the vector subspace of $Exp_{\Theta ,0}^{k}\left( E\right) $ generated by $\mathcal{L}$ vanishes on $\ker L$. Now the result follows from the Hahn-Banach Theorem. \end{proof} \begin{theorem} \label{teorema Ortogonal e fraco}Let $k\in\left( 1,+\infty\right] $ and $(\mathcal{P}_{\Theta}(^{j}E))_{j=0}^{\infty}$ be a $\pi_{1}$-$\pi_{2,k}$-holomorphy type from $E$ to $\mathbb{C}$. If $Exp^{k'}_{\Theta^{\prime}}(E^{\prime})$ is closed under division and $L\in\mathcal{A}_{\Theta ,0}^{k}$, then its transpose \newline$^{t}L\colon\left[ Exp_{\Theta ,0}^{k}\left( E\right) \right] ^{\prime}\longrightarrow\left[ Exp_{\Theta ,0}^{k}\left( E\right) \right] ^{\prime}$is such that\newline\emph{(a) }$^{t}L\left( \left[ Exp_{\Theta ,0}^{k}\left( E\right) \right] ^{\prime}\right) $ is the orthogonal of $\ker L$ in $\left[ Exp_{\Theta ,0} ^{k}\left( E\right) \right] ^{\prime}.$\newline\emph{(b) }$^{t} L\left( \left[ Exp_{\Theta ,0}^{k}\left( E\right) \right] ^{\prime}\right) $ is closed for the weak-star topology in $\left[ Exp_{\Theta ,0}^{k}\left( E\right) \right] ^{\prime}$ defined by $Exp_{\Theta,0}^{k}\left( E\right) .$ \end{theorem} \begin{proof} If $L$ is equal to $0,$ the result is clear. Let $L\neq 0$ and $T\in \left[ Exp_{\Theta ,0}^{k}\left( E\right) \right] ^{\prime}$ be such that $L=T\ast.$ For each $R\in \hspace{0.1 cm}^{t}L\left( \left[ Exp_{\Theta ,0}^{k}\left( E\right) \right] ^{\prime}\right) $ there is $S\in \left[ Exp_{\Theta ,0}^{k}\left( E\right) \right] ^{\prime}$ satisfying $R=\hspace{0.1 cm}^{t}L\left( S\right) .$ Hence, for each $f\in \ker L$ we have $R\left( f\right) =\hspace{0.1 cm}^{t}L\left( S\right) \left( f\right) =S\left( Lf\right) =0,$ and then $^{t}L\left( \left[ Exp_{\Theta ,0}^{k}\left( E\right) \right] ^{\prime}\right) $ is contained in the orthogonal of $\ker L.$ Conversely, if $R$ is in the orthogonal of $\ker L,$ then by Theorem \ref{divisao} there is $H\in Exp_{\Theta^{\prime}}^{k'}\left( E^{\prime }\right) $ such that $\mathcal{F}\left( R\right) =H\cdot\mathcal{F}\left( T\right) $ and by Theorem \ref{fourier_borel} there is $S\in \left[ Exp_{\Theta ,0}^{k}\left( E\right) \right] ^{\prime}$ such that $H=\mathcal{F}\left( S\right) $ and $\mathcal{F}\left( R\right) =\mathcal{F}\left( S\right)\cdot\mathcal{F}\left( T\right) =\mathcal{F}\left( S\ast T\right) .$ Hence $R=S\ast T$ and for each $f\in Exp_{\Theta ,0}^{k}\left( E\right) ,$ we have \begin{gather} R\left( f\right) =\left( S\ast T\right) \left( f\right) =\left( \left( S\ast T\right) \ast f\right) \left( 0\right) =\left( S\ast\left( T\ast f\right) \right) \left( 0\right) \\ =S\left( T\ast f\right) =S\left( Lf\right) =\hspace{0.1cm}^{t}L \left( S\right) \left( f\right) \end{gather} and this implies that $R=\hspace{0.1cm}^tL\left( S\right) $ and so $R\in\hspace{0.1cm}^{t}L\left( \left[ Exp_{\Theta ,0}^{k}\left( E\right) \right] ^{\prime}\right) $, proving $\left( a\right) $. To prove $(b)$ note that the orthogonal of $\ker L$ is equal to \[ {\displaystyle\bigcap\limits_{f\in\ker L}} \left\{ T\in\left[ Exp_{\Theta ,0}^{k}\left( E\right) \right] ^{\prime};\text{ }T\left( f\right) =0\right\} . \] Since, for each $f\in Exp_{\Theta ,0}^{k}\left( E\right)$ the set $\left\{ T\in\left[ \mathcal{H}_{\Theta b}\left( E\right) \right] ^{\prime}\colon T\left( f\right) =0\right\} $ is closed for the weak-star topology, the result follows. \end{proof} \qquad The next result of this article is a theorem about existence of solution of convolution equations. In order to prove this result we need the following Dieudonn\'{e}-Schwartz result (see \cite[Th\'eor\`eme 7]{DS} or \cite[p. 308]{horvath}). \begin{lemma} \label{lema teorema de existencia}If $E$ and $F$ are Fr\'{e}chet spaces and $u\colon E\longrightarrow F$ is a linear continuous mapping, then the following conditions are equivalent:\newline\emph{(a) }$u\left( E\right) =F;$\newline\emph{(b) }$^{t}u\colon F^{\prime}\longrightarrow E^{\prime}$ is injective and $^{t}u\left( F^{\prime}\right) $ is closed for the weak-star topology of $E^{\prime}$ defined by $E.$ \end{lemma} \begin{theorem} \label{teorema de existencia}Let $k\in\left( 1,+\infty\right] $ and $(\mathcal{P}_{\Theta}(^{j}E))_{j=0}^{\infty}$ be a $\pi_{1}$-$\pi_{2,k}$-holomorphy type from $E$ to $\mathbb{C}$. If $Exp^{k'}_{\Theta^{\prime}}(E^{\prime})$ is closed under division and $L\in\mathcal{A}_{\Theta ,0}^{k},$ then $L\left( Exp_{\Theta ,0}^{k}\left(E\right) \right) =Exp_{\Theta ,0}^{k}\left( E\right) .$ \end{theorem} \begin{proof} By \cite[Proposition 2.7]{favaro2}, $Exp_{\Theta ,0}^{k}\left( E\right) $ is a Fr\'{e}chet space. By Lemma \ref{lema teorema de existencia} (b) and Theorem \ref{teorema Ortogonal e fraco*} $(b)$, it is enough to show that $^{t}L$ is injective. Since $L=T\ast$ for some $T\in \left[ \mathcal{H}_{\Theta b}\left( E\right) \right] ^{\prime}$ then, for all $S\in \left[ Exp_{\Theta ,0}^{k}\left( E\right) \right] ^{\prime }$ and $f\in Exp_{\Theta ,0}^{k}\left( E\right) $ we have $$ ^tL (S) \left( f\right) =S\left( L f\right) =S\left( T\ast f\right) =\left( S\ast T\right) \left( f\right) .$$ Thus $^tL (S)=S\ast T$ and if $^tL(S)=0,$ then $S\ast T=0$ and $\mathcal{F}\left( S\ast T\right) =0.$ Since $L=T\ast$ is non zero, then it follows that $\mathcal{F}T$ is non zero and since $\mathcal{F}\left( S\ast T\right) =\mathcal{F}S\cdot\mathcal{F}T,$ we get $\mathcal{F}S=0.$ Hence $S=0$ and $^tL$ is injective. \end{proof} \section{Applications} We finish the paper showing the applicability of Theorems \ref{main1}, \ref{main2}, \ref{main3}, \ref{teoremaAproximacao1} and \ref{teorema de existencia}. \noindent (1) Consider the finite dimensional case $E=\mathbb{C}^n.$ Then, $\mathcal{P}_{\Theta}(^{m}E)=\mathcal{P}(^{m}\mathbb{C}^{n}),$ for all $m\in\mathbb{N}_0$. Considering $k=\infty$ we get $Exp_{\Theta ,0}^{k}\left( E\right)=\mathcal{H}(\mathbb{C}^{n})$. Thus, using Theorem \ref{main1} we recover the result of Godefroy and Shapiro \cite{godefroy} (and consequently the results of Birkhoff \cite{birkhoff} and MacLane \cite{maclane}) which states that every nontrivial convolution operator $L\colon\mathcal{H}(\mathbb{C}^{n})\rightarrow \mathcal{H}(\mathbb{C}^{n})$ is hypercyclic. Moreover, for an arbitrary $k\in(1,+\infty]$ we obtain the unknown result that every nontrivial convolution operator $L\colon Exp_{0}^{k}(\mathbb{C}^{n})\rightarrow Exp_{0}^{k}(\mathbb{C}^{n})$ is mixing. More generally, since $(\mathcal{P}(^{m}\mathbb{C}^{n}))_{m=0}^\infty$ is a holomorphy type with canonical constants according to Remark \ref{canonical_cte}, then, for $A\in\left[ 0,+\infty\right) $, we also obtain that every nontrivial convolution operator $L\colon Exp_{0, A}^{k}(\mathbb{C}^{n})\rightarrow Exp_{0, A}^{k}(\mathbb{C}^{n})$ is mixing. It is easy to check that $(\mathcal{P}(^{m}\mathbb{C}^{n}))_{m=0}^\infty$ is a $\pi_{2,k}$-holomorphy type. Hence it follows from Theorem \ref{main2} that if $T\in[Exp_{0}^{k}\left( \mathbb{C}^{n}\right)]^{\prime}$ is a linear functional which is not a scalar multiple of $\delta_0$, then the convolution operator $\Gamma_{\Theta,0}^{k}(T)$ on $Exp_{0}^{k}(\mathbb{C}^{n})$ is mixing. Finally, using \cite[Corollaire 1]{martineau} it is easy to check that $Exp^{k'}_{\Theta^{\prime}}(\mathbb{C}^{n})=Exp^{k'}(\mathbb{C}^{n})$ is closed under division. Therefore it follows from Theorem \ref{teorema de existencia} that for each $g\in Exp_{0}^{k}(\mathbb{C}^{n}),$ the convolution equation $Lf=g$ has a solution $f\in Exp_{0}^{k}(\mathbb{C}^{n})$. Moreover, it follows from Theorem \ref{teoremaAproximacao1} that each solution of the homogeneous equation $Lf=0$ can be approximated by exponential polynomials solutions. Hence we recover the existence and approximation results for convolution operators on $Exp_{0}^{k}(\mathbb{C}^{n})$ obtained by Martineau \cite{martineau}. \noindent (2) Let $E$ be a complex Banach space such that $E^\prime$ has the bounded approximation property and consider the space $\mathcal{P}_{N}(^{m}E)$ of all nuclear $m$-homogeneous polynomials on $E$. It is well known that $\Theta=N$ is a $\pi_1$-holomorphy type (see for instance Dineen \cite[Example 3]{dineen}). It is also easy to check that $\Theta=N$ is a holomorphy type with canonical constants. By Matos \cite[Proposition 3.9]{matos3} $\Theta=N$ is also a $\pi_{2,k}$-holomorphy type. Since $\Theta^\prime$ is the holomorphy type of all continuous $m$-homogeneous polynomials on $E^\prime$ we have that $Exp^{k'}_{\Theta^{\prime}}(E^\prime)=Exp^{k'}(E^\prime)$ and by Matos \cite[Theorem 4.3]{matos2} $$ \left[Exp^{k}_{N,0}(E)\right]^{\prime}=Exp^{k'}(E^\prime). $$ Moreover, since $$Exp_0^{k'}(E^\prime)\subset Exp_A^{k'}(E^\prime)\subset Exp_{0,A}^{k'}(E^\prime)\subset Exp_B^{k'}(E^\prime)\subset Exp^{k'}(E^\prime),$$ for every $k\in[1,\infty]$ and $0<A<B<\infty$, then using Matos \cite[Corollary 4.5]{matos3} it is easy to check that $Exp^{k'}(E^\prime)$ is closed under division. Thus, we have the following unknown results: \begin{itemize} \item Every nontrivial convolution operator $L\colon Exp_{N,0, A}^{k}(E)\rightarrow Exp_{N,0, A}^{k}(E)$ is mixing, for every $k\in(1,+\infty]$ and $A\in\left[ 0,+\infty\right) $. \item If $T\in[Exp_{N,0}^{k}\left( E\right)]^{\prime}$ is a linear functional which is not a scalar multiple of $\delta_0$, then the convolution operator $\Gamma_{N,0}^{k}(T)$ on $Exp_{N,0}^{k}(E^{\prime})$ is mixing. \end{itemize} If $k\in(1,+\infty]$ and $L\colon Exp_{N,0}^{k}(E)\rightarrow Exp_{N,0}^{k}(E)$ is a convolution operator, then we also recover the following results of \cite{matos3}: \begin{itemize} \item For each $g\in Exp_{N,0}^{k}(E),$ the convolution equation $Lf=g$ has a solution $f\in Exp_{N,0}^{k}(E)$. \item Each solution of the homogeneous equation $Lf=0$ can be approximated by exponential polynomials solutions in $Exp_{N,0}^{k}(E).$ \end{itemize} \noindent (3) Let $0<s\leq\infty$ and $1\leq q,r\leq\infty$ such that $q^\prime\leq r^\prime$ and $$1\leq\frac{1}{s}+\frac{m}{q^\prime},$$ for every $m\in\mathbb{N}$ (as usual $s^\prime, r^\prime, q^\prime$ denote the conjugates of $s, r, q$, respectively). Consider the space $\mathcal{P}_{\widetilde{N},\left( s;\left( r,q\right) \right)}\left(^{m}E\right)$ of all $\left( s;\left( r,q\right) \right)$-quasi-nuclear $m$-homogeneous polynomials on the complex Banach space $E$ introduced by Matos \cite[Section 7.2]{Matos-livro}. Let us consider also $E^\prime$ having the bounded approximation property. The proof that $\left(\mathcal{P}_{\widetilde{N},\left( s;\left( r,q\right) \right)}\left(^{m}E\right)\right)_{m=0}^\infty$ is a $\pi_1$-holomorphy type can be found in \cite[Sections 8.2 and 8.3]{Matos-livro} (see also \cite[Example 3.11. (a)]{favaro3}) and it is easy to check that this is a holomorphy type with canonical constants. By F\'avaro \cite[Proposition 2.14]{FaBelg} $\left(\mathcal{P}_{\widetilde{N},\left( s;\left( r,q\right) \right)}\left(^{m}E\right)\right)_{m=0}^\infty$ is a $\pi_{2,k}$-holomorphy type. Moreover, Matos proved in \cite[Section 8.2]{Matos-livro} that when $E^{\prime}$ has the bounded approximation property, then the Borel transform $\mathcal{B}_{\widetilde{N},\left( s;\left( r,q\right) \right) }$ establishes an isometric isomorphism between $\left[\mathcal{P}_{\widetilde{N},\left( s;\left( r,q\right) \right)}\left(^{m}E\right)\right]^{\prime}$ and $\mathcal{P}_{\left( s^{\prime},m\left( r^{\prime};q^{\prime}\right) \right)}\left(^{m}E^{\prime}\right)$, where $\mathcal{P}_{\left( s^{\prime},m\left( r^{\prime};q^{\prime}\right) \right)}\left(^{m}E^{\prime}\right)$ denotes the space of all absolutely $\left( s^{\prime},m\left( r^{\prime};q^{\prime}\right) \right)$-summing $m$-homogeneous polynomials on $E^\prime$ introduced by Matos \cite[Section 3]{matosjmaa}. So, in this case the role of $Exp^{k'}_{\Theta^{\prime}}(E^{\prime})$ is played by $Exp^{k'}_{\left( s^{\prime},m\left( r^{\prime};q^{\prime}\right) \right)}(E^{\prime})$ and the isomophism $$ \left[Exp^{k}_{\widetilde{N},\left( s;\left( r,q\right) \right) }(E)\right]^{\prime}=Exp^{k'}_{\left( s^{\prime},m\left( r^{\prime};q^{\prime}\right) \right)}(E^\prime) $$ given by the Fourier-Borel transform is in F\'avaro \cite[Theorem 3.5]{favaro}. Furthermore, since $$Exp_{0,\Theta^\prime}^{k'}(E^\prime)\subset Exp_{\Theta^\prime,A}^{k'}(E^\prime)\subset Exp_{\Theta^\prime,0,A}^{k'}(E^\prime)\subset Exp_{\Theta^\prime,B}^{k'}(E^\prime)\subset Exp^{k'}_{\Theta^\prime}(E^\prime),$$ for every $k\in[1,\infty]$ and $0<A<B<\infty$, then using F\'avaro \cite[Theorem 3.5 and Remark 3.6]{FaBelg} it is easy to check that $Exp^{k'}_{\left( s^{\prime},m\left( r^{\prime};q^{\prime}\right) \right)}(E^\prime)$ is closed under division. Thus, we have the following unknown results: \begin{itemize} \item Every nontrivial convolution operator $L\colon Exp_{\widetilde{N},\left( s;\left( r,q\right) \right),0, A}^{k}(E)\rightarrow Exp_{\widetilde{N},\left( s;\left( r,q\right) \right),0, A}^{k}(E)$ is mixing, for every $k\in(1,+\infty]$ and $A\in\left[ 0,+\infty\right) $. \item If $T\in[Exp_{\widetilde{N},\left( s;\left( r,q\right) \right),0}^{k}\left( E\right)]^{\prime}$ is a linear functional which is not a scalar multiple of $\delta_0$, then the convolution operator $\Gamma_{\widetilde{N},\left( s;\left( r,q\right) \right),0}^{k}(T)$ on $Exp_{\widetilde{N},\left( s;\left( r,q\right) \right),0}^{k}(E^{\prime})$ is mixing. \end{itemize} If $k\in(1,+\infty]$ and $L\colon Exp_{\widetilde{N},\left( s;\left( r,q\right) \right),0}^{k}(E)\rightarrow Exp_{\widetilde{N},\left( s;\left( r,q\right) \right),0}^{k}(E)$ is a convolution operator, then we also recover the following results of \cite{FaBelg}: \begin{itemize} \item For each $g\in Exp_{\widetilde{N},\left(s;\left( r,q\right) \right),0}^{k}(E),$ the convolution equation $Lf=g$ has a solution $f\in Exp_{\widetilde{N},\left( s;\left( r,q\right) \right),0}^{k}(E)$. \item Each solution of the homogeneous equation $Lf=0$ can be approximated by exponential polynomials solutions in $Exp_{\widetilde{N},\left( s;\left( r,q\right) \right),0}^{k}(E).$ \end{itemize} \noindent (4) We can also obtain the hypercyclicity result given in Theorem \ref{main3} for the following holomorphy types (both are holomorphy types with canonical constants): \begin{itemize} \item $\left( \mathcal{P}_{\widetilde{N},\left( \left( r,q\right) ;\left( s,p\right) \right) }\left( ^{m}E\right) \right) _{m=0}^{\infty}:$ the holomorphy type of all \emph{Lorentz $((r,q);(s,p))$ -quasi-nuclear} $m$-homogeneous polynomials from $E$ to $\mathbb{C}$, $m\in\mathbb{C},$ where $r,q,s,p\in\lbrack1,\infty\lbrack$, $r\leq q$, $s^{\prime}\leq p^{\prime}$ and \[ 1\leq\frac{1}{q}+\frac{m}{p^{\prime}}, \textrm{ for all } m\in\mathbb{C}. \] See \cite[Section 2 and Definition 4.4]{FP}. \item $\left(\mathcal{P}_{\sigma(p)}\left(^{m}E\right)\right) _{m=0}^{\infty}:$ the holomorphy type of all $\sigma(p)$\emph{-nuclear} $m$-homogeneous polynomials from $E$ to $\mathbb{C}$, $m\in\mathbb{C}$, where $p\geq 1,$ defined in the obvious way according to the multilinear case studied in \cite{BM}. \end{itemize} Consider $E^\prime$ having the bounded approximation property. Then \cite[Propositions 4.9]{FP} and \cite[p.7]{BM} ensure that $\left( \mathcal{P}_{\widetilde{N},\left( \left( r,q\right) ;\left( s,p\right) \right) }\left( ^{m}E\right) \right) _{m=0}^{\infty}$ and $\left(\mathcal{P}_{\sigma(p)}\left(^{m}E\right)\right) _{m=0}^{\infty}$ are $\pi_1$-holomorphy types, respectively. Hence, for $k\in(1,+\infty]$ and $A\in\left[ 0,+\infty\right) $, every nontrivial convolution operator $$L\colon Exp_{\widetilde{N},\left( \left( r,q\right) ;\left( s,p\right) \right),0, A}^{k}(E)\rightarrow Exp_{\widetilde{N},\left( \left( r,q\right) ;\left( s,p\right) \right),0, A}^{k}(E)$$ or $$L\colon Exp_{\sigma(p),0, A}^{k}(E)\rightarrow Exp_{\sigma(p),0, A}^{k}(E)$$ is mixing. For details about the duality results given by the Borel transform and the theory involving the Lorentz polynomials and $\sigma(p)$-nuclear polynomials we refer to \cite{FMP, FP, Matos-Pellegrino} and \cite{favaro3, BM}, respectively. {\em Authors' addresses}: Faculdade de Matem\'{a}tica, Universidade Federal de Uberl\^{a}ndia, 38.400-902 - Uberl\^{a}ndia, Brazil,\newline e-mails: \texttt{[email protected]} \qquad\texttt{[email protected]} \end{document}	arXiv
Journal of Cryptographic Engineering November 2012 , Volume 2, Issue 4, pp 221–240 \| Cite as Co-$Z$ ECC scalar multiplications for hardware, software and hardware–software co-design on embedded systems Brian Baldwin Raveen R. Goundar Mark Hamilton William P. Marnane Regular Paper First Online: 16 October 2012 Recent elliptic curve scalar multiplication algorithms are based on efficient co-$Z$ arithmetics. These arithmetics were initially introduced by Meloni in 2007 where addition of projective points share the same $Z$-coordinate. The co-$Z$ version algorithms are sufficiently fast and secure against a large variety of implementation attacks. This paper analyses the performance of these algorithms in hardware and then compares them against software and hardware–software co-design environments on FPGA, in terms of speed, memory, power and energy consumption. Specifically, this paper presents a survey and performance comparison of implementations of co-$Z$ versions of the Montgomery ladder and the Joye's double-add algorithm in an embedded system environment. Elliptic curves regular ladders FPGA Microblaze Hardware Hardware–software co-design This material is based upon works supported by the Science Foundation Ireland under Grant No. 06/MI/006. Research supported by Profs. Francisco Rodríguez-Henríquez and Çetin K. Koç through UC MEXUS Grant, administered by UCSB and CINVESTAV-IPN. Co-$Z$ Algorithms In this section we present some of the Co-$Z$ operations defined in this paper, presented as Algorithm 12–Algorithm 18. Open image in new window Point doubling formulæ with update in homogeneous coordinates A double of point ${\varvec{P}} = (X_1:Y_1:Z_1)$ on $E_\mathcal{H }$, denoted $DBL_\mathcal{H }$, is computed as ${\varvec{2P}}=(X_3:Y_3:Z_3)$ with $$\begin{aligned} X_3 = 2BD, Y_3 = A(4C-D)-8(Y_1B)^2, Z_3 = 8B^3 \end{aligned}$$ where $A=a Z_1^2+3X_1^2$, $B=Y_1Z_1$, $C=X_1(Y_1B)$, and $D=A^2-8C$. The cost of it is $\underline{6\mathsf M + 5\mathsf S + 1\mathsf c }$. We optimised $DBL_\mathcal{H }$ by trading one multiplication with one squaring which results in a cost of $\underline{5\mathsf M + 6\mathsf S + 1\mathsf c }$ and is given as $$\begin{aligned} X_3 = 4BD, Y_3 = A(4C-D)-64(Y_1B)^2, Z_3 = 64B^3 \end{aligned}$$ where $A=2(a Z_1^2+3X_1^2)$, $B=Y_1Z_1$, $C=2[(X_1+Y_1B)^2-X_1^2-(Y_1B)^2]$, and $D=A^2-8C$. If $Z_1 = 1$, the cost drops to $\underline{3\mathsf M +5\mathsf S }$, with $$\begin{aligned} X_3 = 4Y_1D, Y_3 = A(4C-D)-64B, Z_3 = 64Y_1\alpha \end{aligned}$$ where $A=2(a+3X_1^2)$, $\alpha =Y_1^2$, $B=\alpha ^2$, $C=2[(X_1+\alpha )^2-X_1^2-B]$, and $D=A^2-8C$. We notice that together with ${\varvec{2P}}$ we obtain a representation of a point ${\varvec{P}}$ having the same $Z$ coordinate at a cost of only one multiplication. $$\begin{aligned} \varvec{\tilde{P}}=(64Y_1\alpha \cdot X_1:64B:64Y_1\alpha )\sim (X_1:Y_1:Z_1)={\varvec{P}}. \end{aligned}$$ We let $(\varvec{\tilde{P}}, 2{\varvec{P}}) \leftarrow DBLU_\mathcal{H }({\varvec{P}})$ denote the corresponding operation, where $\varvec{\tilde{P}} \sim {\varvec{P}}$ and $\mathrm Z (\varvec{\tilde{P}}) = \mathrm Z (2{\varvec{P}})$. The cost of $DBLU_\mathcal{H }$ operation (doubling with update) is $\underline{4\mathsf M + 5\mathsf S }$. Furthermore, for implementation purpose of Algorithm 4 we define an $(X,Z)$-only point doubling with an update in homogeneous coordinate, denoted as $DBLU_\mathcal{H }^{}$ as $DBLU_\mathcal{H }^{}({\varvec{P}})\leftarrow (X(\varvec{\tilde{P}}):X({\varvec{2P}}):Z({\varvec{2P}}))= (X_1\cdot 64 Y_1\alpha : 4Y_1D: 64Y_1\alpha )$. The cost of $DBLU_\mathcal{H }^{*}$ operation is $\underline{3\mathsf M +5\mathsf S }$. Full coordinate recovery The formula for the recovery of the full projective coordinates of the output point ${\varvec{Q}}=k{\varvec{P}}$, from the $x$-coordinates ${\varvec{R}}_\mathbf 0 = (X_1,Z)$ and ${\varvec{R}}_\mathbf 1 ,{\varvec{Z}} = (X_2,Z)$ at the end of the Montgomery ladder is described in Algorithm 19. Note that $D=(x_D,y_D)$ represents the invariant, input point ${\varvec{P}}$, of the Montgomery ladder in affine coordinates. The cost of this formula is $\underline{8\mathsf M +2\mathsf S +1M_a+1M_{4b}+8add}$ and its implementation requires $11$ registers as detailed in algorithm 7 of [17]. The full coordinates recovery formula given by Algorithm 4 is evaluated in Algorithm 20. The cost of which is $\underline{10\mathsf M +3\mathsf S +8add}$ and its implementation requires $13$ registers as detailed in algorithm 8 of [17]. Point doubling and tripling with co-$Z$ update Algorithms 3, 7, 8, 9 and 10 require a point doubling or a point tripling operation for their initialisation. We describe here how this can be implemented. Initial Point Doubling The double of a point is computed using the DBLU operation below. $$\begin{aligned} {\left\{ \begin{array}{ll} \mathrm X (2{\varvec{P}}) = M^2 -2S,\\ \mathrm Y (2{\varvec{P}}) = M(S - \mathrm X (2{\varvec{P}})) - 8L, \\ \mathrm Z (2{\varvec{P}}) = 2Y_1 \end{array}\right.} \end{aligned}$$ with $M = 3B + a$, $S = 2((X_1+E)^2 - B - L)$, $L = E^2$, $B = {X_1}^2$, and $E = {Y_1}^2$. Since $Z(2{\varvec{P}}) = 2Y_1$, it follows that $$\begin{aligned} (S:8L:Z(2{\varvec{P}})) \sim {\varvec{P}}\quad {\text{ with}}\, S = 4X_1{Y_1}^2\, {\text{ and}}\,L={Y_1}^4 \end{aligned}$$ is an equivalent representation for point ${\varvec{P}}$. Updating point ${\varvec{P}}$ such that its $Z$-coordinate is equal to that of $2{\varvec{P}}$ comes thus for free [29]. We let $(2{\varvec{P}}, \varvec{\tilde{P}}) \leftarrow \text{ DBLU}({\varvec{P}})$ denote the corresponding operation, where $\varvec{\tilde{P}} \sim {\varvec{P}}$ and $\mathrm Z (\varvec{\tilde{P}}) = \mathrm Z (2{\varvec{P}})$. The cost of DBLU operation (doubling with update) is $\underline{1\mathsf M + 5\mathsf S }$. Initial Point Tripling The triple of ${\varvec{P}} = (X_1:Y_1:1)$ can be evaluated as $3{\varvec{P}} = {\varvec{P}} + 2{\varvec{P}}$ using co-$Z$ arithmetic [26]. From $(2{\varvec{P}}, \varvec{\tilde{P}}) \leftarrow \text{ DBLU}({\varvec{P}})$, this can be obtained as ZADDU$(\varvec{\tilde{P}}, 2{\varvec{P}})$ with $5\mathsf M + 2\mathsf S $ and no additional cost to update ${\varvec{P}}$ for its $Z$-coordinate becoming equal to that of $3{\varvec{P}}$. The corresponding operation, tripling with update, is denoted TPLU$({\varvec{P}})$ and its total cost is of $\underline{6\mathsf M + 7\mathsf S }$. 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Sci. 2162, 348–363 (2001)MathSciNetCrossRefGoogle Scholar Research, C.: Sec 2: Recommended elliptic curve domain, parameters (2000) Google Scholar Rivain, M.: Fast and regular algorithms for scalar multiplication over elliptic curves. Cryptology ePrint Archive, Report 2011/338 (2011). http://eprint.iacr.org/ Slla, A.M., Drabek, V.: An efficient list-based scheduling algorithm for high-level synthesis. In: Proceedings of the Euromicro Symposium on Digital Systems Design, pp. 316–323. IEEE Computer Society, New York (2002)Google Scholar Venelli, A., Dassance, F.: Faster side-channel resistant elliptic curve scalar multiplication. Contemp. Math. 521, 29–40 (2010)MathSciNetCrossRefGoogle Scholar Walter, C.D.: Montgomery exponentiation needs no final subtractions. Electron. Lett. 35(21), 1831–1832 (1999)CrossRefGoogle Scholar Xilinx: Microblaze soft processor core. http://www.xilinx.com/tools/microblaze.htm Yen, S.M., Joye, M.: Checking before output may not be enough against fault-based cryptanalysis. IEEE Trans. Comput. 49(9), 967–970 (2000)CrossRefGoogle Scholar Yen, S.M., Kim, S., Lim, S., Moon, S.J.: A countermeasure against one physical cryptanalysis may benefit another attack. In: Kim, K. (ed.) Information Security and Cryptology—ICISC 2001. Lecture Notes in Computer Science, vol. 2288, pp. 414–427. Springer, Berlin (2002)Google Scholar 1.Department of Electrical and Electronic Engineering, Claude Shannon Institute for Discrete Mathematics, Coding and CryptographyUniversity College CorkCorkIreland 2.Centro de investigación y de Estudios Avanzados del I.P.N.MexicoMexico Baldwin, B., Goundar, R.R., Hamilton, M. et al. J Cryptogr Eng (2012) 2: 221. https://doi.org/10.1007/s13389-012-0042-2 Received 24 January 2012 Accepted 04 September 2012 First Online 16 October 2012 Publisher Name Springer-Verlag	CommonCrawl
How Big A Table Can The Carpenter Build? Sep. 23, 2016 , at 8:00 AM Edited by Oliver Roeder Filed under The Riddler Illustration by Guillaume Kurkdjian Welcome to The Riddler. Every week, I offer up a problem related to the things we hold dear around here: math, logic and probability. These problems, puzzles and riddles come from many top-notch puzzle folks around the world — including you! Last week, we started something new: Riddler Express problems. These will be bite-size puzzles that don't take as much fancy math or computational power to solve. For those of you in the slow-puzzle movement, worry not — we still feature our classic, more challenging Riddler. You can mull both over on your commute, dissect them on your lunch break and argue about them with your friends and lovers. When you're ready, submit your answer(s) using the links below. I'll reveal the solutions next week, and a correct submission (chosen at random) will earn a shoutout in this column.1 Before we get to the new puzzles, let's reveal the winners of last week's. Congratulations to 👏 Katie Andrews 👏 of Great Falls, Virginia, and 👏 Ben Sokolowsky 👏 of Stony Brook, New York, our respective Express and Classic winners. You can find solutions to the previous Riddlers at the bottom of this post. First up, Riddler Express, inspired by Eric Beck: Suppose the NCAA College Football Playoff (a single-elimination tournament) expanded to include not just four, but all 128 Division I Football Bowl Subdivision teams. How many individual games would be played in that playoff? (Hint: You probably don't even need a piece of paper for this one.) Submit your answer If you need a hint you can try asking me nicely. Want to submit a new Riddler Express puzzle or problem? Email me. And now, for Riddler Classic, a handy puzzle from Eric Valpey: You're on a DIY kick and want to build a circular dining table which can be split in half so leaves can be added when entertaining guests. As luck would have it, on your last trip to the lumber yard, you came across the most pristine piece of exotic wood that would be perfect for the circular table top. Trouble is, the piece is rectangular. You are happy to have the leaves fashioned from one of the slightly-less-than-perfect pieces underneath it, but there's still the issue of the main circle. You devise a plan: cut two congruent semicircles from the perfect 4-by-8-foot piece and reassemble them to form the circular top of your table. What is the radius of the largest possible circular table you can make? Extra credit: What is the largest circular table that can be made from N congruent pieces? If you need a hint you can try asking me nicely. Want to submit a new Riddler? Email me. Here's the solution to last week's Riddler Express, which asked how high Count Von Count can count on Twitter, given the 140-character limit. If he spells out the numbers, without commas or "and"s, plus an exclamation point, the highest he'll get is the number 1,111,373,373,372. It fits nice and snug in a tweet! one trillion one hundred eleven billion three hundred seventy three million three hundred seventy three thousand three hundred seventy two! — Oliver Roeder (@ollie) September 20, 2016 There's no silver bullet for arriving at this answer. Some trial and error is eventually required after you recognize that spelling out "one," "two" and "ten" is relatively short compared to "three," "seven" and "eight," for example. Eventually, you'll hit the character limit. Tim Supinie graphed the length of the count's tweets as the numbers being counted crept up: @hypergeometricx @ollie My puny laptop won't make it to 1 trillion this decade. (Good call on the log scale, tho.) pic.twitter.com/BLimA9ASZo — Tim Supinie (@plustssn) September 16, 2016 But worry not, Count fans. At his current rate of about two tweets per day, he won't hit the limit for another 1.5 billion years, at which point the oceans will have evaporated and only single-celled organisms and fictional nobles will have survived. And here's the solution to last week's Riddler Classic, concerning an experimental sports league draft. Suppose a 30-team league switches from a system in which the previous year's worst teams draft first to a system in which teams are randomly assigned to two groups via a series of coin flips, with all of the teams in one group drafting first, but the order within those groups still being sorted out by which teams were worst last season. If your team's draft position was 10th in the old system, its expected draft position is 12.75 under the new system. Why? Half of the time, your team wins its coin flip, putting it in the first group. In this case, you expect to draft after 9/2 other teams. The '9' represents the nine teams with worse records than yours, and the '2' represents the 50-50 chance any of those teams ends up in your group. Therefore, in this case, you expect to draft in (9/2)+1, or 11/2, position. The other half of the time, your team loses its coin flip, putting it in the second group. In this case, you expect to draft before 20/2, or 10, teams. The 20 is the 20 teams with better records than yours, and the 2 represents the chance that their coin flips place them in the first group. Therefore, in this case, you expect to draft in 30-10=20th position. Combining these two cases, your team expects to draft in $1/2\cdot (11/2)+1/2\cdot (20)=12.75$ position. Turns out that the league's rule change does help in alleviating the incentive for tanking at the end of a season. For extra credit, I asked what a team's expected draft position would be if a league of N teams was divided randomly into T groups. Per the puzzle's submitter, Stephen Penrice, the expected draft position of the ith best team is $$\frac{i}{t}+\frac{(t-1)(N+1)}{2T}$$ The math gets a little messy there, but the ideas are just the same as in the 30-team, 10th place case. For more, as ever, Laurent Lessard provides a lucid explanation. And while the average pick becomes 12.75 rather than 10th under the new system, the distribution of that new pick is bimodal, and has a high variance, as this animated simulation from Russell Maier shows: My ans to this wknds @FiveThirtyEight riddler @ollie. Nice bimodal dist. Avg pick in new sys is 13th for 10th worst pic.twitter.com/vhfw4JSA9G — Russell Maier (@MaierRussell) September 19, 2016 Elsewhere in the puzzling world: The world's greatest detective [The New York Times] More mystery puzzles [Expii] A "September" puzzle [NPR] Slate now has a crossword puzzle [Slate] Better know a puzzle master: David Kwong [NBC News] Have a wonderful weekend! May you win all of your coin tosses. Important small print: To be eligible, I need to receive your correct answer before 11:59 p.m. EDT on Sunday. Have a great weekend! The Riddler (180 posts)	CommonCrawl
Poisson Summation Formula Created by: Jens Fischer The Poisson Summation Formula (PSF) expresses the fact that discretization (sampling) in one domain (time or frequency) means periodization in the other domain (frequency or time). In other words, discretizing a function means to periodize its spectrum and, vice versa, periodizing a function means to discretize its spectrum. Variants of the PSF are required to formally describe the transition from functions which are fully smooth, i.e., infinitely differentiable in both time and frequency domain, to fully discrete functions, i.e., functions which are discrete in both time and frequency domain such that they can be Fourier transformed via the Discrete Fourier Transform (DFT). An intermediate stage between fully smooth and fully discrete functions are Fourier series (semi-discrete). They are represented either by the Fourier series itself (periodic function) or its coefficients (discrete function). The latter duality between discrete and periodic functions is another interpretation of the Poisson Summation Formula. The Poisson Summation Formula is known in many different variants. But there are basically two different types. In one case, the sum on both sides amounts to one complex value. The equality is then true in the sense of complex numbers. In the other case, we have an equality between two (ordinary or generalized) functions and (provided both functions fulfill a certain duality condition) the equality is then true in the generalized functions sense. An evaluation of both sides of the equation, for example at t=0, leads to the first case again, an equality between two complex numbers. Two, dual versions of the Poisson Summation Formula (in the generalized functions sense) are known[1], \[ \sum_{k=-\infty}^{+\infty} f(t - k T) = \frac{1}{T} \; \sum_{m=-\infty}^{+\infty} \hat{f}(\frac{m}{T}) \, e^{\,2 \pi \, t \, \frac{m}{T}} \] \[ \sum_{k=-\infty}^{+\infty} \hat{g}(\sigma - \frac{k}{T}) = T \; \sum_{m=-\infty}^{+\infty} g(m T) \, e^{\,-2 \pi \, \sigma \, m T} \] where is the Fourier transform of and is the Fourier transform of . Without loss of generality, one may moreover think of as the Fourier transform of and of as the Fourier transform of . We refer to the so-called "unitary, ordinary frequency" Fourier transform, i.e., its exponent includes the factor of 2 π. Choosing T = 1 and t = 0 in the first or σ = 0 in the second equation, yields the more simplified version $$ \sum_{k=-\infty}^{+\infty} f(k) = \sum_{m=-\infty}^{+\infty} \hat{f}(m) $$ which is known to be true for Schwartz functions, i.e., infinitely differentiable functions which rapidly decay to zero for large arguments. Choosing t = 0 and T = 2 π in the first equation, yields the formula originally found by Poisson. Choosing f = δ, the Dirac delta function, it yields the so-called distributional formulation of the Poisson Summation Formula, i.e., the equality between two Dirac combs. If the distance between two Diracs on the left is T, then the distance between two Diracs on the right becomes 1/T where T > 0 is an arbitrary real number. The Poisson Summation Formula can be easily understood if two symbols are introduced, one for discretization and one for periodization . Using these symbols, the two PSF variants above become $${\triangle \!\! \triangle \!\! \triangle}_{T} f = \frac{1}{T} \; {\cal F}^{-1}(\, {\bot \!\! \bot \!\! \bot}_{\frac{1}{T}} ({\cal F} f) \,)$$ $${\triangle \!\! \triangle \!\! \triangle}_{\frac{1}{T}} \, ({\cal F} g) = \, T \; {\cal F}({\bot \!\! \bot \!\! \bot}_{T} g).$$ They therefore represent the rules $${\cal F}(\, {\triangle \!\! \triangle \!\! \triangle}_{T} f \,) = \frac{1}{T} \; {\bot \!\! \bot \!\! \bot}_{\frac{1}{T}} ({\cal F} f)$$ $${\cal F}({\bot \!\! \bot \!\! \bot}_{T} \, g) = \frac{1}{T} \; {\triangle \!\! \triangle \!\! \triangle}_{\frac{1}{T}} ({\cal F} g)$$ which are known as Fourier series analysis and Fourier series synthesis formula, respectively. They moreover represent the inverse Discrete-Time Fourier Transform (IDTFT) and the Discrete-Time Fourier Transform (DTFT), respectively. For T = 1 they yield $${\cal F}(\, {\triangle \!\! \triangle \!\! \triangle} f \,) = {\bot \!\! \bot \!\! \bot} ({\cal F} f)$$ $${\cal F}({\bot \!\! \bot \!\! \bot} \, g) = {\triangle \!\! \triangle \!\! \triangle} ({\cal F} g).$$ Hence, the Fourier transform turns periodization into discretization and discretization into periodization. In the tempered distributions sense, if is the Dirac impulse then is its Fourier transform and these formulas become $${\cal F}(\, {\triangle \!\! \triangle \!\! \triangle} \, \delta) \,=\, {\bot \!\! \bot \!\! \bot} \, 1$$ $${\cal F}({\bot \!\! \bot \!\! \bot} \, 1) \,=\, {\triangle \!\! \triangle \!\! \triangle} \, \delta$$ and using the Dirac comb identity it yields $${\cal F}(\, \text{III} \,) \,=\, \text{III}$$ where is the Dirac comb. The Dirac comb identity states that it can be created either via periodizing or via discretizing the function that is constantly 1. It can be shown that these formulas actually link four different Fourier transforms with one another, the Integral Fourier Transform, the Discrete-Time Fourier Transform (DTFT), the Discrete Fourier Transform (DFT) and the Integral Fourier Transform for periodic functions used to analyze Fourier Series. The DTFT is moreover the inverse of the Integral Fourier Transform for periodic functions. [2] C. Gasquet, P. Witomski. Fourier Analysis and Applications: Filtering, Numerical Computation, Wavelets; Marsden et.al., Eds.; Springer: New York, 1999; pp. 344. Jens V. Fischer; Four Particular Cases of the Fourier Transform. Mathematics 2018, 6, 335, 10.3390/math6120335. Poisson Summation Formula Fourier transform Discrete-Time Fourier Transform (DTFT) Discrete Fourier Transform (DFT) Fourier series Dirac impulse Dirac comb discrete function periodic function discretization periodization sampling Jens, Fischer. Poisson Summation Formula, Encyclopedia, 2018, v24, Available online: https://encyclopedia.pub/96	CommonCrawl
\begin{document} \title{Generation of interface for an Allen-Cahn equation \ with nonlinear diffusion} \begin{center} {\large\bf Matthieu Alfaro }\\[1ex] I3M, Universit\'e de Montpellier 2,\\ CC051, Place Eug\`ene Bataillon, 34095 Montpellier Cedex 5, France,\\[2ex] {\large\bf Danielle Hilhorst }\\[1ex] CNRS et Laboratoire de Math\'ematiques\\ Universit\'e de Paris-Sud 11, 91405 Orsay Cedex, France. \\[2ex] \end{center} \begin{abstract} In this note, we consider a nonlinear diffusion equation with a bistable reaction term arising in population dynamics. Given a rather general initial data, we investigate its behavior for small times as the reaction coefficient tends to infinity: we prove a generation of interface property. \\ \noindent{\underline{Key Words:}} degenerate diffusion, singular perturbation, motion by mean curvature, population dynamics.\footnote{AMS Subject Classifications: 35K65, 35B25, 35R35, 92D25.} \end{abstract} \section{Introduction}\label{intro-poreux} We consider the degenerate parabolic problem \[ (P^{\;\!\ep}) \quad\begin{cases} u_t=\Delta (u^m)+\edeux f(u)&\text{in }Q_T:=\Omega \times (0,T) \\ \displaystyle \frac{\partial (u^m)}{\partial \nu} = 0 &\text{on }\partial \Omega \times (0,T) \\ u(x,0)=u_0(x) &\text{in }\Omega\,, \end{cases} \] with $\varepsilon >0$ a small parameter. Here $\Omega$ is a smooth bounded domain in $\mathbb{R}^N$ ($N\geq 2$), $\nu$ is the Euclidian unit normal vector exterior to $\partial \Omega$ and $m\geq 2$. We assume that $f$ is smooth and has exactly three zeros $0<a<1$ such that \begin{equation}\label{der-f-poreux} f'(0)<0\,, \qquad f'(a)>0\,, \qquad f'(1)<0\,. \end{equation} Moreover we suppose that the initial function $u_0 \in C^2(\overline{\Omega})$ is nonnegative, with support $ Supp\, u_0:=\{x\in\Omega\|\ u_0(x)>0\} \subset\subset \Omega$. Furthermore we define the initial interface $\Gamma _0$ by $$ \Gamma _0:=\{x\in\Omega\|\ u_0(x)=a \}\,, $$ and suppose that $\Gamma _0$ is a smooth hypersurface without boundary such that \begin{equation}\label{dalltint-poreux} \Gamma _0 \subset\subset \Omega \quad \mbox { and } \quad \nabla u_0(x) \neq 0\quad\text{if $x\in\Gamma _0$\,,} \end{equation} \begin{equation}\label{initial-data-poreux} u_0>a \quad \text { in } \quad \om _0 ^{(1)}\,,\quad u_0<a \quad \text { in } \quad \om _0 ^{(0)}\,, \end{equation} where $\om _0 ^{(1)}$ denotes the region enclosed by $\Gamma _0$ and $\om _0 ^{(0)}$ the region enclosed between $\partial \Omega$ and $\Gamma _0$.\\ We prove a generation of interface property, namely that the solution $u^\ep$ quickly becomes close to $1$ or $0$, except in an $\mathcal O(\varepsilon)$ neighborhood of the initial interface $\Gamma _0$, creating a steep transition layer around $\Gamma _0$. More precisely, we are in presence of the following phenomenon: in the very early stage, the nonlinear diffusion term is negligible when compared with the reaction term $\varepsilon ^{-2}f(u)$. Hence, under the rescaling in time $\tau=t/\varepsilon^2$, the equation is well approximated by the ordinary differential equation $u_\tau=f(u)$. In view of the bistable nature of $f$, $u^\ep$ quickly approaches the stable equilibria of the ordinary differential equation, $0$ or $1$, and an interface is formed between the regions $\{u^\ep\approx 0\}$ and $\{u^\ep\approx 1\}$.\\ The organization of this note is as follows. In Section \ref{s:biology} we briefly explain how Problem $(P^\varepsilon)$ arises in population dynamics. In Section \ref{s:existence}, we recall known results about the well-posedness of Problem $(P^{\;\!\ep})$ and a comparison principle. In Section \ref{s:generation}, we prove the generation of interface property for Problem $(P^{\;\!\ep})$. To that purpose we construct sub- and super-solutions by modifying the solution of the corresponding ordinary differential equation $u_t=\varepsilon ^{-2}f(u)$. We also show the optimality of the generation time ${t}^{\,\varepsilon}:=f'(a) ^{-1}\varepsilon^2\|\ln\varepsilon\|$ and prove that the thickness of the interface is of order $\mathcal O(\varepsilon)$ at the generation time ${t}^{\,\varepsilon}$. Our method of proof follows the same lines as that of \cite{AHM} and \cite{A}. It is slightly different from those of Xinfu Chen \cite{C1} and \cite{C2}, who transforms the reaction function $f$. We postpone to future work the study of the interface motion after the generation time of the interface. \\ Finally let us mention articles involving the singular limit of reaction-diffusion equations with nonlinear diffusion. Feireisl \cite{F} studies the singular limit of a degenerate parabolic equation in the whole space $\mathbb{R} ^N$. He studies the problem in the scaling \begin{equation}\label{feireisl} u_t=\varepsilon \Delta (u^m)+\frac 1 \varepsilon f(u)\,, \end{equation} where, in the limit $\varepsilon \to 0$, the limit free boundary moves according to motion by constant speed (that of a related traveling wave). In a similar scaling, Hilhorst, Kersner, Logak and Mimura \cite{HKLM} investigate the singular limit of this equation in a bounded domain with a monostable reaction term. In both of these papers, they prove that the solution $u^\ep$ of the nonlinear diffusion equation converges to 0 or 1 on both sides of an interface moving with constant normal velocity. In this scaling, proofs about the interface motion can be performed with using only one term in the asymptotic expansion whereas we would need to use two terms in the case of Problem $(P^\varepsilon)$ as well as a suitable linearization procedure; this is far from trivial here since Problem $(P^{\;\!\ep})$ is degenerate parabolic. \section{The biological context}\label{s:biology} In this section, we discuss nonlinear diffusion in population dynamics. It is well-known that the control of a population can be achieved by introducing density dependent birth or death rates. In \cite{GN}, Gurney and Nisbet show that the introduction of a nonlinearity into the dispersal behavior of a species --- which behaves in an otherwise linear way--- can, in an inhomogeneous environment, lead to a regulatory effect. More precisely, they consider the equation $$ u_t=-\operatorname{div} {\bf j}+G(x)u\,, $$ where $u(x,t)$ denotes the population density, $G=G(x)$ the growth function only depending on the location and ${\bf j}(x,t)$ the local population current density. By using the well-known random motion model one obtains the linear equation $u_t=\Delta u+G(x)u$. Another possibility is to choose the biased random motion model where movements are largely random but slightly modified by the distribution of the fellows; the corresponding equation is then written as $u_t=\Delta u+\operatorname{div}(u\operatorname{grad} u)+G(x)u$. Nevertheless, Carl \cite{Carl} has observed that arctic ground squirrels migrate from densely populated areas into sparsely populated ones, even when the latter is less favorable (burrow sites not available, intensive predation). For such species, migration to avoid crowding, rather than random motion, is the primary cause of dispersal. To describe such movements, Gurney and Nisbet use the directed motion model where individuals can only stay put or move down the population gradient; this model yields the degenerate parabolic equation \begin{equation}\label{gurney} u_t=\Delta (u^2)+G(x)u\,. \end{equation} In \cite{GN}, the authors perform a qualitative analysis of the three different dispersal models (random motion, biased random motion and directed motion). They conclude that the introduction of density dependent diffusion is efficient to study the dynamics of a population which regulates its size below the carrying capacity set by the supply of nutrients. \\ Gurtin and Mac Camy \cite{GM} proposed the class of equations which we study here and which involves degenerate diffusion and nonlinear reaction, namely \begin{equation}\label{general} u_t=\Delta (u^m)+f(u)\,,\quad \quad m\geq 2\,. \end{equation} In absence of a reaction term, equation \eqref{general} reduces to the so-called porous medium equation \begin{equation}\label{pme} u_t=\Delta (u^m)\,, \end{equation} which describes, among others, the flow of an ideal gas in a homogeneous medium ($m\geq 2$), groundwater infiltration (Boussinesq's equation, $m=2$), the spread of a thin viscous film under gravity ($m=4$), and thermal propagation in plasma ($m \simeq 6$). The porous medium equation has been extensively investigated in literature: we refer to the book of V\'asquez \cite{V} and the references therein. The main feature of these equations is that they degenerate at the points where $u=0$. As a consequence, a loss of regularity of solutions occurs and disturbances propagate with finite speed. This phenomenon contrasts with the infinite speed of propagation of solutions of the heat equation $u_t=\Delta u$. \\ \section{Comparison principle and well-posedness}\label{s:existence} Since the diffusion term degenerates at the points where $u^\varepsilon = 0$, $u^\varepsilon$ is not smooth. This leads us to define a notion of weak solution for Problem $(P^{\;\!\ep})$, in a similar way as it is done by Aronson, Crandall and Peletier \cite{ACP} for a corresponding one-dimensional problem. \begin{defi}\label{definition-weaksol-poreux} A function $u^\varepsilon : [0,\infty)\to L^1(\Omega)$ is a weak solution of Problem $(P^\varepsilon)$ if, for all $T>0$, \begin{enumerate} \item $u^\varepsilon \in C\left([0,\infty);L^1(\Omega)\right)\cap L^\infty (\Q)$\,; \item $u^\varepsilon$ satisfies the integral equality \begin{multline}\label{deqexi-poreux} \int_ \Omega {u^\varepsilon}(T)\varphi(T)-\int\int_{\Q}({u^\varepsilon}\varphi _t +(u^\varepsilon)^m\Delta \varphi)= \int _\Omega u_0\varphi(0) \\+\int\int _{\Q} \edeux f(u^\varepsilon)\varphi\,, \end{multline} for all $\varphi \in C^2(\overline{\Q})$ such that $\varphi \geq 0$ and $\displaystyle \frac {\partial \varphi}{\partial \nu}=0$ on $\partial \Omega$. \end{enumerate} A sub-solution (respectively a super-solution) of Problem $(P^\varepsilon)$ is a function satisfying (i) and (ii) with equality replaced by $\leq$ (respectively $\geq$). \end{defi} \begin{thm}[Existence and comparison principle]\label{Existence-comparison} Let $T>0$ be arbitrary. The following properties hold: \begin{enumerate} \item Let $u^-$ and $u^+$ be a sub-solution and a super-solution of Problem $(P^\varepsilon)$ with initial data $u_0^-$ and $u_0^+$ respectively. $$ \mbox{If~}\quad u_0^- \leq u_0^+\,\quad \mbox{~then~}\quad u^-\leq u^+ \mbox{~in~} Q_T\,; $$ \item Problem $(P^\varepsilon)$ has a unique weak solution $u^\varepsilon$ which is such that \begin{equation}\label{encadrement} 0\leq u^\varepsilon \leq \max(1,\Vert u_0 \Vert _{L^\infty(\Omega)}) \text{~in~} Q_T\,; \end{equation} \item $u^\varepsilon \in C(\overline{Q_T}).$ \end{enumerate} \end{thm} The proof of Theorem \ref{Existence-comparison} is standard; it can be performed by using the same lines as that of Theorem 5 in \cite{ACP}. The continuity of $u^\varepsilon$ follows from \cite{DB}.\\ The following result turns out to be an essential tool when constructing smooth sub- and super-solutions of Problem $(P^{\;\!\ep})$. \begin{lem}\label{lemma-sup-poreux} Let $u^\varepsilon$ be a continuous nonnegative function in $\overline{\Omega} \times [0,T]$. Define $\om _t ^{supp} = \{x \in \Omega \|\ u^\varepsilon (x,t)>0\}$ and $\Gamma _t ^{supp}=\partial \om _t ^{supp}$ for all $t\in[0,T]$. Suppose that the family $\Gamma:=\cup _{0< t \leq T} \Gamma _t ^{supp} \times \{t\}$ is sufficiently smooth and let $\nu _t ^{supp}$ be the outward normal vector on $\Gamma _t ^{supp}$. Suppose moreover that \begin{enumerate} \item $\nabla (u^\varepsilon)^m \mbox{~is~continuous~in~} \overline{\Omega} \times [0,T]\,;$ \item ${\cal L}[u^\varepsilon]:=u^\varepsilon_t -\Delta ({u^\varepsilon})^m -\edeux f(u^\varepsilon) = 0 \mbox{ in } \{(x,t) \in \overline{\Omega} \times [0,T] \\ \mbox{~such~that~} {u^\varepsilon}(x,t) > 0\}\,;$ \item $\displaystyle{\frac{\partial (u^\varepsilon)^m}{\partial \nu _t ^{supp}}}=0\; \text{ on }\ \partial \om _t ^{supp},\; \text{ for all }\ t\in[0,T]$\,. \end{enumerate} Then $u^\varepsilon$ is a solution of Problem $(P^\varepsilon)$. Similarly $u$ is a sub-solution (respectively a super-solution) of Problem $(P^\varepsilon)$ if the equality in (ii) is replaced by $\leq$ (resp. $\geq$) and if the equality in (iii) is replaced by $\leq$ (resp. $\geq$). \end{lem} We refer to \cite{HKLM} for the proof. \section{Generation of interface} \label{s:generation} In this section we prove that, given a nearly arbitrary initial function $u_0$, the solution $u^\ep$ quickly becomes close to $1$ or $0$, except in an $\mathcal O(\varepsilon)$ neighborhood of the initial interface $\Gamma_0$, creating a steep transition layer around $\Gamma _0$. The time needed to develop such a transition layer is of order $\mathcal O (\varepsilon^2\|\ln\varepsilon\|)$. \begin{thm}[Generation of interface]\label{g-th-gen-poreux} Assume $m\geq 2$. Let $\gamma \in (0,\min (a,\\ 1 -a))$ be arbitrary and define $\mu$ as the derivative of $f(u)$ at the unstable equilibrium $u=a$, namely \begin{equation}\label{g-def-mu-poreux} \mu=f'(a)\,. \end{equation} Moreover, set $$ t^{\,\varepsilon}:=\mu ^{-1}\varepsilon ^2\|\ln\varepsilon\|\,. $$ Then there exist positive constants $\varepsilon_0$ and $M_0$ such that, for all $\varepsilon\in(0,\varepsilon _0)$, \begin{enumerate} \item for all $x \in \Omega$, we have that $$ 0 \leq u^\varepsilon(x,t^{\,\varepsilon}) \leq 1+\gamma\,; $$ \item for all $x\in\Omega$ such that $\|u_0(x)-a\|\geq M_0 \varepsilon$, we have that \begin{align} &\text{if}\;~~u_0(x)\geq a+M_0\varepsilon\;~~\text{then}\;~~u^\varepsilon(x,t^{\,\varepsilon}) \geq 1-\gamma\,,\label{g-part2}\\ &\text{if}\;~~u_0(x)\leq a-M_0\varepsilon\;~~\text{then}\;~~u^\varepsilon(x,t^{\,\varepsilon}) \leq \gamma \label{g-part3}\,. \end{align} \end{enumerate} \end{thm} Theorem \ref{g-th-gen-poreux} will be proved by constructing a suitable pair of sub- and super-solutions. As mentioned above, the nonlinear diffusion term is negligible in this early stage so that the behavior of the solution is governed by the ordinary differential equation $u_t=\edeux f(u)$. An immediate consequence of Theorem \ref{g-th-gen-poreux} is that $u^\ep(x,t^{\,\varepsilon})$ is close to 0 or 1, except in an $\mathcal O(\varepsilon)$ neighborhood of the initial interface $\Gamma _0$. In other words, the transition layers which have developed have an $\mathcal O(\varepsilon)$ thickness. \begin{cor}[Thickness of the transition layers at time $t^{\,\varepsilon}$]\label{cor-thi} Let $\eta \in (0,\min (a, 1 -a))$ be an arbitrary constant. Then there exist positive constants $\varepsilon _0 $ and $\mathcal C$ such that, for all $\varepsilon \in(0,\varepsilon _0)$, \begin{equation}\label{resultat} u^\ep(x,t^{\,\varepsilon}) \in \begin{cases} \,[0,1+\eta]&\quad\text{if}\quad x\in\mathcal N_{\mathcal C\varepsilon}(\Gamma_0)\\ \,[0,\eta]&\quad\text{if}\quad x\in \Omega ^{(0)} _0 \setminus\mathcal N_{\mathcal C\varepsilon}(\Gamma _0)\\ \,[1-\eta,1+\eta]&\quad\text{if}\quad x\in\Omega ^{(1)} _0\setminus\mathcal N_{\mathcal C\varepsilon}(\Gamma _0)\,, \end{cases} \end{equation} where $\Omega ^{(1)} _0$ denotes the region enclosed by $\Gamma _0$, $\Omega ^{(0)} _0$ the region enclosed between $\partial \Omega$ and $\Gamma_0$, and $$ \mathcal N _r(\Gamma _0):=\{x\in \Omega,\, dist(x,\Gamma _0)<r\} $$ denotes the $r$-neighborhood of $\Gamma _0$. \end{cor} We will also show that the generation time ${t}^{\,\varepsilon}:=\mu^{-1}\varepsilon^2\|\ln\varepsilon\|$ is optimal. In other words, the interface is not fully developed until $t$ becomes close to $t^{\,\varepsilon}$. More precisely, the following result holds. \begin{prop}\label{pr:optimal-time} Denote by $t^{\,\varepsilon}_{min}$ the smallest time such that \eqref{resultat} holds. Then there exists a constant $b=b(\mathcal C)$ such that \[ {t}^{\,\varepsilon}_{min}\geq \mu^{-1} \varepsilon^2 (\|\ln\varepsilon\| - b)\, \] for all $\,\varepsilon \in (0,\varepsilon_0).$ \end{prop} \subsection{Proof of the generation of interface property} \subsubsection{The bistable ordinary differential equation} Let us first consider the problem without diffusion, namely \begin{equation}\label{no-diffusion} \bar{u}_t=\frac{1}{\varepsilon^2}\,f(\bar{u})\,, \qquad \bar{u}(x,0)=u_0(x)\,. \end{equation} Its solution can be written in the form \[ \bar{u}(x,t)=Y\left(\frac{t}{\varepsilon^2},\,u_0(x)\right)\,, \] where $Y(\tau,\xi)$ is the solution of the ordinary differential equation \begin{equation}\label{ode-poreux} \left\{\begin{array}{ll} Y_\tau (\tau,\xi)&=f(Y(\tau,\xi)) \quad \text { for } \tau >0 \\ Y(0,\xi)&=\xi\,. \end{array}\right. \end{equation} Here $\xi$ ranges over the interval $(-C_0,C_0)$, where $C_0:=\Vert u_0 \Vert _{L^\infty(\Omega)} +1$. We claim that $Y$ has the following properties. \begin{lem}\label{properties-Y}There exists a positive constant $C$ such that the following holds \begin{enumerate} \item $\text{If }\ \xi >0 \;\text{ then }\ Y(\tau,\xi)>0\,,$\\ $\text{If }\ \xi <0 \;\text{ then }\ Y(\tau,\xi)<0\,;$ \item $\|Y(\tau,\xi)\|\leq C_0\,;$ \item $Y _ \xi (\tau,\xi) > 0\,;$ \item $\|\displaystyle{\frac{Y_{\xi\xi}}{Y_\xi}(\tau,\xi)}\|\leq C (e^{\mu \tau}-1)$\,, \end{enumerate} for all $\tau >0$ and all $\xi \in(-C_0,C_0)$. \end{lem} Properties (i) and (ii) are direct consequences of the profile of $f$ --- more precisely of the sign conditions $f>0$ in $(-\infty,0)\cup(a,1)$ and $f<0$ in $(0,a)\cup (1,\infty)$--- and of the qualitative properties of the solution of the bistable ordinary differential equation \eqref{ode-poreux}; for proofs of (iii) and (iv) we refer to \cite{AHM}, subsection 3.1.\qed \subsubsection{Construction of sub- and super-solutions} We use the notation $a^+=\max(a,0)$. The sub- and super-solutions are given by \begin{equation}\label{w+-} w_\varepsilon^\pm(x,t)=\left[Y\left(\frac{t}{\varepsilon^2},\,u_0(x)\pm\varepsilon^2C^\star(e^{\mu t/\ep ^2}-1)\right)\right]^+\,. \end{equation} \begin{lem}\label{g-w} There exist positive constants $\varepsilon_0$ and $C^\star$ such that, for all $\, \varepsilon \in (0,\varepsilon _0)$, $(w_\varepsilon^-,w_\varepsilon^+)$ is a pair of sub- and super-solutions for Problem $(P^{\;\!\ep})$, in the domain $\overline{\Omega}\times [0,\mu ^{-1} \varepsilon^2\|\ln \varepsilon\|]$. Moreover, since also $w^-_\varepsilon(x,0)=w^+ _\varepsilon(x,0)=u_0(x)$, it follows that \begin{equation}\label{g-coincee1} w_\varepsilon^-(x,t) \leq u^\varepsilon(x,t) \leq w_\varepsilon^+(x,t) \quad\ \mbox{~for~all~} (x,t) \in \overline{\Omega}\times [0,\mu ^{-1} \varepsilon^2\|\ln \varepsilon\|]\,. \end{equation} \end{lem} {\noindent \bf Proof.} In order to prove that $(w_\varepsilon ^-,w_\varepsilon ^+)$ is a pair of sub- and super-solutions for Problem $(P^{\;\!\ep})$ for a suitable choice of $\varepsilon _0$ and $C_g$, we check that the sufficient conditions in Lemma \ref{lemma-sup-poreux} are satisfied. As for the sub-solution $w_\varepsilon ^-$, we remak that property (i) in Lemma \ref{properties-Y} implies that, for all $t>0$, $$ \begin{array}{ll} \om _t ^{supp} [w_\varepsilon^-]=\{x\in\Omega\| \ u_0(x)>\varepsilon ^2 C_g (e^{\mu t/\ep ^2} -1)\} \\ \Gamma _t ^{supp} [w_\varepsilon^-]:=\partial \om _t ^{supp} [w_\varepsilon ^-]= \{x\in\Omega\| \ u_0(x)=\varepsilon ^2 C_g (e^{\mu t/\ep ^2} -1)\}\,. \end{array} $$ Choose $(x_0,t_0)$ such that $x_0 \in\Gamma _{t_0} ^{supp} [w_\varepsilon ^-]$; for $(x,t)$ such that $x\in \om _t ^{supp} [w_\varepsilon ^-]$ we have $$ \nabla (w_\varepsilon ^-)^m (x,t)= mY^{m-1}Y_\xi\left(\frac{t}{\varepsilon^2},\,u_0(x)-\varepsilon^2C^\star(e^{\mu t/\ep ^2}-1)\right)\nabla u_0(x)\,. $$ Since $Y(\tau,0)=0$ the equality above implies $$ \lim _{\substack{(x,t)\to(x_0,t_0) \\ x\in \om _t ^{supp} [w_\varepsilon ^-]}} \nabla (w_\varepsilon ^-)^m (x,t)=0. $$ Therefore conditions (i) and (iii) of Lemma \ref{lemma-sup-poreux} are satisfied by the sub-solution. As for the super-solution $w_\varepsilon ^+$, we remark that property (i) of Lemma \ref{properties-Y} implies that, for all $t>0$, $$ \begin{array}{ll} \om _t ^{supp}[w_\varepsilon^+]=\Omega \\ \Gamma _t ^{supp}[w_\varepsilon ^+]:=\partial \om _t ^{supp} [w_\varepsilon ^+]=\partial \Omega\,. \end{array} $$ Hence condition (iii) of Lemma \ref{lemma-sup-poreux} for the super-solution is a direct consequence of the fact that $ Supp\, u_0 \subset\subset \Omega$, whereas condition (i) is obviously satisfied. It remains to prove that $$ {\cal L} [w_\varepsilon ^-]:=(w_\varepsilon ^-)_t-\Delta ({w_\varepsilon ^-})^m-\edeux f(w_\varepsilon ^-)\leq 0\,, $$ in $\{(x,t)\in \overline{\Omega}\times [0,\mu ^{-1}\varepsilon ^2 \|\ln \varepsilon\|] \mbox{~such~that~} w_\varepsilon ^-(x,t)>0\}$ and that ${\cal L} [w_\varepsilon ^+] \geq 0$ in $\{(x,t)\in \overline{\Omega}\times [0,\mu ^{-1}\varepsilon ^2 \|\ln \varepsilon\|]\}$. In view of the ordinary differential equation \eqref{ode-poreux}, straightforward computations yield \begin{multline} {\cal L} [{w_\varepsilon^-}] =-Y_\xi\Big[C^\star \,\mu\, e^{\mu t/\ep ^2}+m(m-1) Y^{m-2}Y_\xi \|\nabla u_0\|^2\\ +mY^{m-1}{\frac{Y_{\xi\xi}}{Y_\xi}}\,\|\nabla u_0\|^2 +mY^{m-1}\Delta u_0\Big]\,, \end{multline} in $\om _t ^{supp} [w_\varepsilon^-]$, where the function $Y$ and its derivatives are taken at the point $(\tau,\xi)=(t /{\varepsilon ^2}, u_0(x)-\varepsilon^2C^\star(e^{\mu t/\ep ^2}-1))$. Moreover since the term $m(m-1) Y^{m-2}Y_\xi^2 \|\nabla u_0\|^2$ is nonnegative, it follows that $$ {\cal L} [w_\varepsilon^-] \leq -Y_\xi\Big[C^\star \,\mu\, e^{\mu t/\ep ^2}+mY^{m-1}{\frac{Y_{\xi\xi}}{Y_\xi}}\,\|\nabla u_0\|^2 +mY^{m-1}\Delta u_0\Big]\,. $$ We note that, in the range $0 \leq t \leq \mu ^{-1} \varepsilon ^2\|\ln \varepsilon\|$, we have, for $\varepsilon _0$ sufficiently small, $$ \xi=u_0(x)- \varepsilon ^2 C^\star(e^{\mu t/\ep ^2}-1)\in (-C_0,C_0)\,. $$ We deduce from the properties (ii)-(iv) stated in Lemma \ref{properties-Y} that there exist positive constants $C_1$ and $C_2$ --- only depending on $m$, $C_0$, $C$, $\Vert \nabla u_0\Vert _{L^\infty(\Omega)}$ and $\Vert \Delta u_0 \Vert _{L^\infty(\Omega)}$--- such that $$ {\cal L}[w_\varepsilon^-] \leq -Y_\xi \Big[(C^\star\,\mu-C_1)e^{\mu t/\ep ^2}-C_2\Big]\,, $$ which implies that ${\cal L} [w_\varepsilon ^-] \leq 0$ if $C^\star$ is chosen large enough. As for the super-solution we obtain \begin{multline} {\cal L} [w_\varepsilon^+] =Y_\xi\Big[C^\star \,\mu\, e^{\mu t/\ep ^2}-m(m-1) Y^{m-2}Y_\xi \|\nabla u_0\|^2\\ -mY^{m-1}{\frac{Y_{\xi\xi}}{Y_\xi}}\,\|\nabla u_0\|^2 -mY^{m-1}\Delta u_0\Big]\,, \end{multline} and the assumption that $m\geq 2$ gives an upper bound for $\|Y^{m-2}\|$. Following the same argument as above one can prove that ${\cal L} [w_\varepsilon ^+] \geq 0$ for $C^\star$ sufficiently large. This completes the proof of Lemma \ref{g-w}.\qed \subsubsection{Proof of Theorem \ref{g-th-gen-poreux}} In order to prove Theorem \ref{g-th-gen-poreux} we first present basic estimates of the function $Y$ after a time of order $\tau\sim \|\ln \varepsilon\|$. \begin{lem}\label{after-time} Let $\gamma \in (0,\min (a,1 -a))$ be arbitrary. There exist positive constants $\varepsilon_0$ and $C_Y$ such that, for all $\,\varepsilon\in(0,\varepsilon _0)$, \begin{enumerate} \item for all $\xi\in (-C_0,C_0)$, \begin{equation}\label{g-part11} -\gamma \leq Y(\mu ^{-1} \| \ln \varepsilon \|,\xi) \leq 1+\gamma\,; \end{equation} \item for all $\xi\in (-C_0,C_0)$ such that $\|\xi-a\|\geq C_Y \varepsilon$, we have that \begin{align} &\text{if}\;~~\xi\geq a+C_Y \varepsilon\;~~\text{then}\;~~Y(\mu ^{-1}\| \ln \varepsilon \|,\xi) \geq 1-\gamma\label{g-part22}\\ &\text{if}\;~~\xi\leq a-C_Y \varepsilon\;~~\text{then}\;~~Y(\mu ^{-1}\| \ln \varepsilon \|,\xi)\leq \gamma \label{g-part33}\,. \end{align} \end{enumerate} \end{lem} These estimates illustrate the stability of the equilibria 0 and 1 for the bistable ordinary differential equation \eqref{ode-poreux}. For more details we refer the reader to the proof of Lemma 3.9 in \cite{AHM}.\qed \vskip 8pt We are now ready to prove Theorem \ref{g-th-gen-poreux}. By setting $t=\mu ^{-1} \varepsilon ^2\|\ln \varepsilon\|$ in \eqref{g-coincee1}, we obtain \begin{multline}\label{g-gr} \left[Y\left(\mu ^{-1}\|\ln \varepsilon\|, u_0(x)-(C^\star \varepsilon -C^\star \varepsilon ^2 )\right)\right] ^+\\ \leq u^\varepsilon(x,\mu ^{-1} \varepsilon^2\|\ln \varepsilon\|) \leq \left[Y\left(\mu ^{-1}\|\ln \varepsilon\|, u_0(x)+C^\star \varepsilon -C^\star \varepsilon ^2\right)\right]^+\,. \end{multline} Since, for $\varepsilon_0$ small enough, $u_0(x)+ (C^\star \varepsilon -C^\star \varepsilon ^2) \in (0,C_0)$, the assertion $(i)$ of Theorem \ref{g-th-gen-poreux} is a direct consequence of \eqref{g-part11} and \eqref{g-gr}. Next we prove \eqref{g-part2}. We choose $M_0$ large enough so that $M_0\varepsilon- C^\star \varepsilon+C^\star \varepsilon ^2 \geq C_Y \varepsilon$. Then, for all $x\in \Omega$ such that $u_0(x)\geq a+M_0 \varepsilon$, we have that $ u_0(x)-(C^\star \varepsilon -C^\star \varepsilon ^2) \geq a+C_Y \varepsilon$, which we combine with \eqref{g-gr} and \eqref{g-part22} to deduce that \[ u^\varepsilon(x,\mu^{-1} \varepsilon ^2 \| \ln \varepsilon \|)\geq 1-\gamma. \] The inequality \eqref{g-part3} can be shown in a similar way. This completes the proof of Theorem \ref{g-th-gen-poreux}.\qed \begin{rem}\label{gpaszero} Theorem \ref{g-th-gen-poreux} remains true if we perturb the reaction function $f(u)$ by order $\varepsilon$, setting for instance ${\widetilde{f}}(u) = f(u)-\varepsilon g(x,t,u)$. To deal with this more general case, we proceed as follows. We first consider a slightly perturbed reaction function, namely ${f_\delta}(u)=f(u)+\delta$ which, for $\delta$ small enough, is still of bistable type. Define $a(\delta)$ as its unstable zero and $\mu(\delta):=f'(a(\delta))$ as the slope of $f_\delta$ in this point. We then define $Y(\tau,\xi;\delta)$ as the solution of the initial value problem \begin{equation} \left\{\begin{array}{ll} Y_\tau (\tau,\xi;\delta)&=f_\delta (Y(\tau,\xi;\delta)) \quad \text { for }\tau >0 \\ Y(0,\xi;\delta)&=\xi\,. \end{array}\right. \end{equation} Finally we construct a pair of sub- and super-solutions in the form \[ w_\varepsilon^\pm(x,t)=\left[Y\left(\frac{t}{\varepsilon^2},u_0(x)\pm\varepsilon^2r(\pm \varepsilon \mathcal G,\frac{t}{\varepsilon^2});\pm \varepsilon \mathcal G\right)\right]^+\,, \] where $r(\delta,\tau):=C^\star(e^{\mu (\delta) \tau}-1)$ and $\mathcal G:=\Vert g\Vert _{L^\infty(0,C_0)}$. For more details and proofs we refer to \cite{AHM}, more precisely to Section 4 which deals with a generation of interface property for an equation with linear diffusion and an unbalanced reaction term $f(u)$.\qed \end{rem} \subsection{Proof of the optimality of the generation time} We show below that the generation time ${t}^{\,\varepsilon}:=\mu^{-1}\varepsilon^2\|\ln\varepsilon\|$ is optimal. In other words, the interface is not fully developed before $t$ is close to $t^{\,\varepsilon}$. We will need the following lemma about the solution $Y(\tau,\xi)$ of the corresponding bistable ordinary differential equation. \begin{lem}\label{lemme-opt} Let $\eta \in (0,\min(a,1-a))$ be arbitrary. Then there exist positive constants $C_1=C_1(\eta)$ and $C_2=C_2(\eta)$ such that, if $\xi\in (a,1-\eta)$ then, for every $\tau>0$ such that $Y(\tau,\xi)$ remains in the interval $(a,1-\eta)$, we have \begin{equation}\label{g-est-Y-1} C_1e^{\mu \tau}(\xi-a)\leq Y(\tau,\xi)-a \leq C_2e^{\mu \tau}(\xi-a)\,. \end{equation} \end{lem} We refer to \cite{AHM} Corollary 3.5 for the proof of Lemma \ref{lemme-opt}.\qed \\ {\noindent \bf Proof of Proposition \ref{pr:optimal-time}.} For each $b>0$, we set \[ t{\, ^\varepsilon} (b):=\mu ^{-1} \varepsilon ^2 (\|\ln \varepsilon\|-b)\,, \] and evaluate $u^\ep(x,t^{\,\varepsilon}(b))$ at a point $x \in \Omega _0 ^{(1)}$ such that $dist(x,\Gamma _0)=\mathcal C\varepsilon$. Define $C:= \\|u_0\\|_{{C^2}(\overline{\Omega})}$. Since $u_0=a$ on $\Gamma_0$, we have that \begin{equation}\label{m-M} u_0(x) \leq a +C \mathcal C \varepsilon\,, \end{equation} which implies together with Lemma \ref{lemme-opt} that \[ \begin{array}{lll} w^+ _\varepsilon (x, t^{\,\varepsilon}(b))& =Y\Big ( \mu ^{-1} (\|\ln \varepsilon\|-b), u_0(x)+\varepsilon C^\star e^{-b}-\varepsilon ^2 C^\star \Big ) \\ & \leq a+C_2e^{\|\ln \varepsilon\|-b} \big(u_0(x)+\varepsilon C^\star e^{-b}-\varepsilon^2 C^\star -a ) \\ & \leq a+C_2 \varepsilon^{-1} e^{-b} (C\mathcal C \varepsilon+\varepsilon C^\star e^{-b}) \\ & = a+C_2 e^{-b} (C\mathcal C + C^\star e^{-b})\,. \end{array} \] Now choose $b$ large enough so that \begin{equation}\label{1-eta} a+C_2 e^{-b} (C\mathcal C + C^\star e^{-b}) <1-\eta\,; \end{equation} the inequalities \eqref{1-eta} and \eqref{g-coincee1} then yield \[ u^\ep(x,t^{\,\varepsilon}(b))\leq w^+ _\varepsilon(x, t^{\,\varepsilon}(b))<1-\eta\,. \] Therefore \eqref{resultat} does not hold at $t=t^{\,\varepsilon}(b)$, and hence $t^{\,\varepsilon}(b)<t^{\,\varepsilon}_{min}$. This completes the proof of Proposition \ref{pr:optimal-time}. \qed \end{document}	arXiv
Asian-Australasian Journal of Animal Sciences (아세아태평양축산학회지) Volume 23 Issue 11 Asian Australasian Association of Animal Production Societies (아세아태평양축산학회) Effects of Diet Complexity and Fermented Soy Protein on Growth Performance and Apparent Ileal Amino Acid Digestibility in Weanling Pigs Ao, X. (Department of Animal Resource and Science, Dankook University) ; Kim, H.J. (Department of Animal Resource and Science, Dankook University) ; Meng, Q.W. (Department of Animal Resource and Science, Dankook University) ; Yan, L. (Department of Animal Resource and Science, Dankook University) ; Cho, J.H. (Department of Animal Resource and Science, Dankook University) ; Kim, I.H. (Department of Animal Resource and Science, Dankook University) https://doi.org/10.5713/ajas.2010.10109 Two experiments were conducted to evaluate the effects of diet complexity and fermented soy protein on growth performance and amino acid digestibility. In Exp. 1, a total of 120 crossbred weanling pigs ($5.68{\pm}0.80\;kg$ BW) were randomly allocated into 4 treatments. Each treatment had 6 replicate pens comprising 5 pigs in each replicate. Experimental diets consisted of simple (soybean meal as protein source) and complex (soybean meal, rice protein concentrate, potato protein concentrate and fish meal as protein sources) diets; each diet contained 0 or 5% fermented soy protein (FSP), respectively. Dietary treatments included: i) simple diet; ii) simple diet with 5% FSP; iii) complex diet; iv) complex diet with 5% FSP. Pigs were provided each experimental diet for 20 d (phase 1) and then fed the same common diet for 10 d (phase 2). During days 0-10, pigs fed FSP diets had greater ADG than those fed non-FSP diets (p<0.05). G/F in FSP treatments was significantly higher than that in non-FSP treatments (p<0.05) from days 0 to 10. Throughout the overall period, G/F was greater in FSP treatments compared with non-FSP treatments (p<0.05). On d 10, N digestibility was higher in pigs fed FSP diets than in those fed non-FSP diets (p<0.05). Diet complexity did not affect growth performance and nutrient digestibility (p>0.05) in this experiment. In Exp 2, 12 ileal-cannulated, weanling barrows were housed in individual metabolism crates and randomly assigned to 1 of 4 treatments (same as Exp. 1) by using a $4{\times}4$ Latin square design. Among the essential amino acids, apparent ileal digestibility (AID) of Met and Val were increased in pigs fed FSP diets compared with those fed non-FSP diets (p<0.05). AID of Met, Phe and total essential amino acids were higher in pigs fed complex diets than in those fed simple diets (p<0.05). Among the non-essential amino acids, AID of Ala in FSP treatments was greater than that in non-FSP treatments (p<0.05). In addition, Asp, Cys, Glu, Pro, Ser and total non-essential amino acid digestibilities in pigs fed complex diets were higher compared with those fed simple diets (p<0.05). Interaction was observed in AID of Met, Asp and Pro. In conclusion, these results indicated that feeding of 5% FSP to nursery pigs improved feed efficiency and AID of amino acids, and diet complexity did not maximize the growth performance of pigs in the subsequent phase. Apparent Ileal Amino Acid Digestibility;Diet Complexity;Fermented Soy Protein;Growth Performance;Pigs AOAC. 1994. Official method of analysis. 16th Edition. Association of Official Analytical Chemists, Washington, DC. Anderson, R. L., J. J. Rackis and W. H. Tallent. 1979. Biologically active substances in soy products (Ed. H. L. Wilcke, D. T. Hopkins and D. H. Waggle) Soy Protein and Human Nutrition. Academic Press, New York. pp. 209-233. Bassily, J. A., K. G. Michael and A. K. Said. 1982. Blood urea content for evaluating dietary protein quality. Br. J. Nutr. 24:983. Baker, D. H. 2000. Nutritional constraints to the use of soy products by animals. In Soy in Animal Nutrition (Ed. J. K. Drackley). Federation of Animal Science Societies, Savoy, IL. pp. 1-12. Cho, J. H., B. J. Min, Y. J. Chen, J. S. Yoo, Q. Wang, J. H. Ahn, I.B. Chung and I. H. Kim. 2008. Evaluation of FSP (fermented soy protein) to replace soybean meal in weaned pigs: Growth performance, blood urea nitrogen and total protein concentrations in serum and nutrient digestibility. J. Anim. Sci. 86 (Suppl. 3):95 (Abstr.). Dritz, S. S., K. Q. Owen, J. L. Nelssen, R. D. Goodband and M. D.Tokach. 1996. Influence of weaning age and nursery diet complexity on growth performance and carcass characteristics and composition of high-health status pigs from weaning to 109 kilograms. J. Anim. Sci. 74:2975-2984. Feng, J., X. Liu, Z. R. Xu, Y. P. Lu and Y. Y. Liu. 2007. The effect of Aspergillus oryzae fermented soybean meal on growth performance, digestibility of dietary components and activities of intestinal enzymes in weaned piglets. Anim. Feed Sci. Technol. 134:295-303. https://doi.org/10.1016/j.anifeedsci.2006.10.004 Jones, C. K., J. M. DeRouchey, J. L. Nelssen, M. D. Tokach, S. S. Dritz and R. D. Goodband. 2010. Effects of fermented soybean meal and specialty animal protein sources on nursery pig performance. J. Anim. Sci. 88:1725-1732. https://doi.org/10.2527/jas.2009-2110 Kiers, J. L., A. E. A. van Laeken, F. M. Rombouts and M. J. R. Nout. 2000. In vitro digestibility of bacillus fermented soybean. Int. J. Food. Micobiol. 60:163. https://doi.org/10.1016/S0168-1605(00)00308-1 Kim, Y. C. 2004. Evaluation of availability for fermented soybean meal in weanling pigs. Ph. D. Thesis, Department of Animal Resources and Science, Seoul University, Korea. Kim, S. W., R. D. Mateo and J. Feng. 2005. Fermented soybean meal as a protein source in nursery diets replacing dried skim milk. J. Anim. Sci. 83 (Suppl. 1):116. Kim, Y. G., J. D. Lohakare, J. H. Yun, S. Heo and B. J. Chae. 2007.Effect of feeding levels of microbial fermented soy protein on the growth performance, nutrient digestibility and intestinal morphology in weaned piglets. Asian-Aust. J. Anim. Sci. 20:399-404. Kim, S. W., E. van Heugten, F. Ji, C. H. Lee and R. D. Mateo.2010. Fermented soybean meal as a vegetable protein source for nursery pigs: I. Effects on growth performance of nursery pigs. J. Anim. Sci. 88:214-224. https://doi.org/10.2527/jas.2009-1993 Lalles, J. P. 2000. Soy products as protein sources for preruminants and young pigs. Pages 106-126 in Soy in Animal Nutrition (Ed. J. K. Drackley). Federation of Animal Science Societies, Savoy, IL. Li, D. F., J. L. Nelssen, P. G. Reddy, F. Blecha, J. D. Hancock, G. L.Allee, R. D. Goodband and R. D. Klemm. 1990. Transient hypersensitivity to soybean meal in the early weaned pig. J. Anim. Sci. 68:1790-1799. Li, D. F., J. L. Nelssen, P. G. Reddy, F. Blecha, R. D. Klemm andR. D. Goodband. 1991. Interrelationship between hypersensitivity to soybean proteins and growth performance in early-weaned pigs. J. Anim. Sci. 69:4062-4069. Min, B. J., J. W. Hong, O. S. Kwon, W. B. Lee, Y. C. Kim, I. H.Kim, W. T. Cho and J. H. Kim. 2004. The effect of feeding processed soy protein on the growth performance and apparent ileal digestibility in weanling pigs. Asian-Aust. J. Anim. Sci. 17:1271. https://doi.org/10.5713/ajas.2004.1271 Min, B. J. 2006. Nutritional value of fermented soy protein (FSP) and effect of FSP on performance and meat quality of pigs. Ph. D. Thesis, Department of Animal Resources and Science, Dankook University, Korea. NRC. 1998. Nutrient requirements of swine (10 th Rev Ed.). National Academy Press, Washington, DC. Rerat, A., C. F. Simones-Nunes, P. Mendy and P. Vaugelade. 1992. Spalnchnic fluxes of amino acids after duodenal infusion of carbohydrate solutions containing free amino acids or oligopeptides in the non-anaesthetizes pig. Br. J. Nutr. 68:111-138. https://doi.org/10.1079/BJN19920071 SAS. 1996. SAS user's guide. Release 6. 12 edition. SAS Inst Inc Cary NC. USA. Sarkar, P. K., L. J. Jones, G. S. Craven, S. M. Somerset and C. Palmer. 1997. Amino acid profiles of kinema, a soybean fermented food. Food Chem. 59:69. https://doi.org/10.1016/S0308-8146(96)00118-5 Sohn, K. S., C. V. Maxwell, D. S. Buchanan and L. L. Southern.1994. Improved soybean protein for early-weaned pigs: I. Effects on performance and total tract amino acids digestibility. J. Anim. Sci. 72:622-630. Tokach, M. D., R. D. Goodband and J. L. Nelssen. 1994. Recent developments in nutrition for the early-weaned pigs. Comp. Cont. Ed. Pract. Vet. 16:406. Whang, K. Y., F. K. M. McKeith, S. W. Kim and R. A. Easter.2000. Effect of starter feeding program on growth performance and gains of body components from weaning to marker weight in swine. J. Anim. Sci. 78:2885-2895. Wolter, B. F., M. Ellis, B. P. Corrigan, J. M. DeDecker, S. E. Curtis,E. N. Parr and D. M. Webel. 2003. Impact of early postweaning growth rate as affected by diet complexity and space allocation on subsequent growth performance of pigs in a wean-to-finish production system. J. Anim. Sci. 81:353-359. Yun, J. H., I. K. Kwon, J. D. Lohakare, J. Y. Choi, J. S. Yong, J.Zheng, W. T. Cho and B. J. Chae. 2005. Comparative efficacy of plant and animal protein sources on the growth performance, nutrient digestibility, morphology and caecal microbiology of early-weaned pigs. Asian-Aust. J. Anim. Sci. 18:1285-1293. https://doi.org/10.5713/ajas.2005.1285 Effects of different fermented soy protein and apparent ileal digestible lysine levels on weaning pigs fed fermented soy protein-amended diets vol.83, pp.5, 2011, https://doi.org/10.1111/j.1740-0929.2011.00966.x Influence of source and micronization of soybean meal on nutrient digestibility and growth performance of weanling pigs1 vol.91, pp.1, 2013, https://doi.org/10.2527/jas.2011-4924	CommonCrawl
\begin{definition}[Definition:Real Number/Number Line Definition] A '''real number''' is defined as a '''number''' which is identified with a point on the '''real number line'''. \end{definition}	ProofWiki
What are the orbital mechanical consideration behind hand-launched nanosatellites from the ISS? The NASA Spaceflight article Extended Russian EVA complete – conducts satellite deployments says: The first order of the day was for Ryazanskiy to head into the manual deployment of five nanosatellites from a ladder outside the airlock, following set up tasks including the hosting of a Go-Pro camera. (emphasis added) The satellites, each of which has a mass of about 11 pounds, have a variety of purposes. One of the satellites, with casings made using 3-D printing technology, will test the effect of the low-Earth-orbit environment on the composition of 3-D printed materials. Another satellite contains recorded greetings to the people of Earth in 11 languages. It sounds like they were just tossed by hand into space. What are the orbital mechanical consideration behind "hand-launching" things from the ISS? What keeps it from coming back and hitting you in the back of the head 93 minutes later? (or from the side 46.5 minutes later for that matter) Seriously, are there rules and constraints applied to launching satellites by hand from the ISS so that there is very little risk of it intercepting the ISS at some point later in time? Certain directions and/or speeds only? More background on the nanosatellites: A third satellite commemorates the 60th anniversary of the Sputnik 1 launch and the 160th anniversary of the birth of Russian scientist Konstantin Tsiolkovsky. iss cubesat payload-deployment $\begingroup$ If you toss a satellite by hand from ISS, you modify its orbit slightly. But on a different orbit it could not hit the ISS. The effect of drag in the low orbit will be different for the large ISS and the tiny nanosatellite, therefore a hit after 93 minutes later is unlikely. The orbit height decay of the ISS is about 2 km in half a month, that is 8 m per cycle. $\endgroup$ – Uwe Aug 20 '17 at 16:32 $\begingroup$ @Uwe sounds good but remember, when hand-launching you control both the magnitude and direction of the $\Delta v$, and an 8m loss of altitude corresponds to only a few millimeters/second. Also the non-uniformity of Earth's gravity introduces another source of uncertainty. I'm not sure "unlikely" is good enough when it comes to the safety of the ISS, so I'm guessing that there are " rules and constraints" so that there will be "very little risk". If you are convinced that there are no rules, you could leave that as an answer and see what happens. $\endgroup$ – uhoh Aug 20 '17 at 23:23 $\begingroup$ Atmospheric and other effects aside: Even if the satellite encounters the ISS again at some point their relative velocities won't be much different from the initial toss. $\endgroup$ – Roman Reiner Aug 21 '17 at 9:53 $\begingroup$ @RomanReiner and all the photovoltaic panels and ammonia-filled thermal radiator panels, and high-gain antennas (some shown in youtu.be/PJzjs4EI22k) are all designed and rated to accept impacts from astronaut-thrown and returning 5kg cubesats? That's not considering that the velocity difference could change different after a few orbits - the Earth is not a perfect monopole. If you think it's safe to throw things off the ISS any way you feel like because nothing could go wrong and so there are no rules, consider posting that as an answer! $\endgroup$ – uhoh Aug 21 '17 at 10:18 $\begingroup$ @uhoh If the orbit of the ISS and the nanosatellite differ only by 1 m in height by different decay, their periods differ only by 1.2 millisecond for a circular orbit in 400 km height. But at a speed of 7.67 km/s, 1.2 millisecond means a distance of 9.2 m. As you wrote the difference in speed is only 0.56 mm/s, but the circumference of the orbit is 42543 km. If the difference in orbit height increases by one 1m for each orbit, the tangential distance increases by another 9.2 m. I think it is very unlikely that the decay of the orbits of the ISS and the nanosatellite would be equal. $\endgroup$ – Uwe Aug 21 '17 at 21:38 If you launch nadir and retrograde, you will put the object into a lower energy orbit such that, barring ISS deboosts (they do happen, but they are rare), it will never again intersect ISS's orbit. Launching prograde, zenith, or out of plane will set you up for a potential recontact scenario. These objects are also large enough that they can be independently tracked once they've gained enough separation from ISS, so any unlikely potential recontact scenario can be averted. TristanTristan $\begingroup$ Is this what's actually done in practice? $\endgroup$ – Chris Aug 21 '17 at 22:53 $\begingroup$ Arguments based on intuition assume the Earth's gravity field is a perfect monopole; $\mathbf{r}/\|r\|^3$ but this isn't the case, so hand waving arguments on the level of meters and milliseconds are not necessarily valid. Also I've asked "...are there rules and constraints applied to launching satellites by hand from the ISS...?" rather than "Do you personally feel that the chances of something bad happening are small?" Is it true that you can just do whatever you want, or are there rules and constraints? $\endgroup$ – uhoh Aug 21 '17 at 23:42 $\begingroup$ There is a pre-prescribed acceptable range of jettison trajectories that is already analyzed and acceptable to the program. There's a cone that faces down and aft, approximately 45 degrees in width, where as long as something is launched in that direction with a good velocity (at least 1 m/s or so), recontact is all but eliminated as a possibility. I promise you that these aren't hand-wavey arguments but are informed by actual analysis, which I cannot post here. $\endgroup$ – Tristan Aug 22 '17 at 13:06 $\begingroup$ @Tristan Would the answer be improved by having the information in that comment edited into it? $\endgroup$ – Wayne Conrad Aug 22 '17 at 19:37 $\begingroup$ @Tristan the wording in your comment is better than in the answer; can you consider porting some of it back up there in some form? 'there is a cone' and 'at least 1 m/s'. I think that's certainly good enough, and is easy enough to be checked with some simulations that in this particular case would not need a further link to make it verifiable. Thanks! $\endgroup$ – uhoh Sep 2 '17 at 11:49 Not the answer you're looking for? Browse other questions tagged iss cubesat payload-deployment or ask your own question. If you throw a baseball from the space station, will it return to you in 90 minutes? Why is there a need for getting rid of ISS trash using the empty ATV and similar vehicles What kinds of things have been tossed out of the ISS? Was this large pieces of "space junk" just released from the ISS in the "nadir and retrograde" direction? Are the cubesats deployed from the ISS always directed "nadir and retrograde"? Would it be possible to de-orbit a small object by throwing it with human strength? What are the major results of ISS experiments? Why would the future Nanoracks airlock be built so that it must be removed from the ISS to deploy cubesats? Observing ISS from the earth How long does trash jettisoned by hand from the ISS fall before burning up on reentry? What brand and model is the hand-held sextant tested on the ISS? What can be done in future mass-cubesat deploys to make them "less irksome" to orbital space debris experts? What is the orbital boost acceleration of the ISS? Are mechanical pencils used on the ISS?	CommonCrawl
Kim P. Huynh1, Yuri Ostrovsky2, Robert J. Petrunia3 & Marcel C. Voia4 Firm shutdown creates a turbulent situation for workers as it leads directly to layoffs for its workers. An additional consideration is whether a firm's shutdown within an industry creates turbulence for workers at other continuing firms. Using data drawn from the Longitudinal Worker File, a Canadian firm-worker matched employment database, we investigate the impact of industry shutdown rates on workers at continuing firm. This paper exploits variation in shutdown rates across industries and within an industry over time to explain the rate of permanent layoffs and the growth of workers' earnings. We find an increase in industry shutdown rates increases the probability of permanent layoffs and decreases earnings growth for workers at continuing firms. The fortunes of firms and workers are inextricably linked. Firm shutdown results in displacement of workers through layoffs. Firm turnover creates uncertainty for workers by affecting their employment status and wages. These first-round effects have negative consequences for the laid-off workers of the shutting down firms. When examining firm shutdown within an industry and its impact on workers, industry shutdown rates also provide an indication of the state of an industry. If industry shutdown rates capture industry wide shocks and fluctuations, then industry shutdown rates may also tell us something about the fortunes of workers at continuing firms. Negative shocks within an industry cause firm profits to fall, which results in rising shutdown rates. Further, the falling demand also causes layoffs at continuing firms to rise, as these firms must reduce production and shed costs. The issue becomes whether industry shutdown rates capture turbulence and fluctuations within an industry, which spill over to cause second-round layoffs at the continuing firms. This paper empirically investigates the effect of industry shutdown rates on the probability of worker layoffs at continuing firms and, by extension, earnings growth of these laid-off workers. A firm's exit or shutdown results in separations as the firm must lay off its worker. The purpose of this paper is not to consider these direct effects of firm shutdown on worker outcomes. Rather, we look at whether industry shutdown rates contain information indirectly relevant for layoff probabilities and earnings of workers at continuing firms. We focus on industry shutdown rates as its impact on workers receives little attention in the literature. This paper addresses the impact of industry shutdown rates by examining the questions: (i) does the industry shutdown rates affect workers at continuing firms; (ii) how does the industry shutdown rate affect workers at continuing firms; and (iii) what are the future earnings prospects of workers experiencing a permanent layoff from a continuing firm? Understanding the labor market interaction of firms and workers requires access to firm-worker matched datasets.1 Our study utilizes one such Canadian administrative employer-employee dataset called the Longitudinal Worker File (LWF).2 Earnings growth allows us to look at the future prospects of laid-off workers. This analysis captures the intensive margin of associated with layoffs. Using administrative data on workers in the USA, von Wachter et al. (2009) find that the annual earnings of workers in relative stable jobs experiencing a surprise layoff during the 1982 recession are still 20% lower than their nondisplaced counterparts after more than 20 years. Using Canadian data, Morissette R et al. (2007) find that mass layoffs due to firm closure have a greater impact on more senior workers. Further, Song and von Wachter (2014) show that the long-term nonemployment rate increase is similar across recessions in the past 30 years. However, they find the long-term unemployment rate increase is higher in the 2008 recession than in previous recessions. These studies demonstrate that layoffs, especially mass layoffs typically occurring during recessions, have long-term consequences for the earnings and employment prospects of displaced workers.3 Our study is similar to Quintin and Stevens (2005a); Quintin and Stevens (2005b), who investigate the impact of industry exit rates on firm-worker separation rates using cross-sectional French data. However, three additional aspects of the LWF database allow us to build on these previous studies. First, the LWF database classifies separations as (i) voluntary separation when a worker quits or an involuntary separation as a result of a layoff and (ii) permanent or temporary. The data used in Quintin and Stevens (2005a); Quintin and Stevens (2005b) only identifies worker separations with no classification on type of separation. Due to these data limitations, Quintin and Stevens (2005a); Quintin and Stevens (2005b) focus on explanations for worker separations related to the workers choice to leave the firm. In contrast, the LWF database allows us to empirically analyze the firm's decision to separate from workers through permanent layoffs. The second aspect is that the LWF is a longitudinal database, while the third aspect is that the LWF contains worker earnings information. Unlike the data in Quintin and Stevens (2005a); Quintin and Stevens (2005b), the longitudinal aspect of the LWF database allows us to exploit variation in industry shutdown rates both across industries and within an industry over time and also allows us to follow workers over time. As the empirical specifications includes industry dummy variables as controls, the analysis focuses on the within industry variation in shutdown rates. Using the longitudinal worker information, this paper provides further analysis of the growth rate of individual worker earnings following a permanent layoff. The findings of our study are4: Industry shutdown rates have a positive and significant effect on the probability of a permanent layoff at continuing firms. The impact of industry shutdown rates on the probability of a permanent layoff captures the extensive margin or the number of affected workers. For men, a 1% increase in industry shutdown rates means approximately a 0.13% increase in the probability of a worker layoff. For women, the marginal effect can be negative or positive and ranges from −0.01% at extra small-sized (less than 5 employees) firms to 0.11% at small-sized firms (5 to 19 employees). The effect of industry shutdown rates on earnings growth is generally negative for both laid-off men and women. The exceptions include men at medium-sized firms and women at small-sized firms. The impact of industry shutdown rates on individual workers through wage growth captures an intensive margin. For workers experiencing a permanent layoff, their post-layoff wage prospects vary with the size of firm at which they eventually find employment. Most laid-off workers moving to a larger firm see their wages increase, while most laid-off workers moving to a smaller firm see their wages fall. The first result extends the finding of Quintin and Stevens (2005b). Quintin and Stevens (2005a, b) are not able to distinguish between layoffs and quits. They focus on workers voluntarily leaving continuing firms to explain the positive relationship between worker separation rates and industry shutdown rates. Our first finding indicates that layoff rates at continuing firms also increase with industry shutdown rates. Therefore, models of worker turnover must capture both workers choosing to quit firms and firms choosing to lay off workers when investigating worker separations in the context of industry fluctuations. The second result also extends the previous work by demonstrating that rising industry shutdown rates also cause deterioration of the earnings prospects for laid-off workers. However, the final result shows that some workers do find "good" jobs after experiencing a layoff, which allows them to increase their earnings. Thus, a layoff need not necessarily result in a "bad" outcome for a displaced worker. These results demonstrate the necessity of the joint analysis of firm shutdown with either permanent layoff or worker wages. Industry shutdown rates provide a measure of firm turnover or churn within an industry. Exogenous conditions within an industry, such things as cyclical movements or demand decline, cause profits of firms to change. The typical view of the firm in economics is that falling profits for firms within an industry lead to firm shutdown and possible exit. Thus, increasing shutdown rates indicate falling profits within an industry. For continuing firms, direct and indirect effects on employment result when moving to a new equilibrium. With these falling profits, output falls at continuing firms, which leads directly to worker layoffs. Indirect effects occur for continuing firms for two reasons. First, they now face less competition with greater shutdown of competitors. Second, more workers are available to hire with the shutdown of competitors. Continuing firms are now better able to substitute for current workers as new hires are cheaper (see Farber (1999)). Direct effects of falling profits result in increased layoffs at continuing firms and, by extension, lower earnings for laid-off workers. Indirect effects are ambiguous. Recent research suggests that, in the case of involuntary separations, there are large differences in the income losses associated with differences in human capital. Kambourov and Manovskii (2009) argue that many skills acquired by workers during their working careers are job-specific. Job displacement is especially detrimental to those workers with job-specific skills not easily transferable. Davis and Wachter (2011) provide an extensive review of the literature on the effects of large cyclical movements in job displacement and how worker anxieties about job loss, wage cuts, and job opportunities respond to contemporaneous economic conditions. They find that the job loss as a result of mass layoffs results in a loss of earnings results in roughly 1.4 to 2.8 years of pre-displacement earnings (depending on the current unemployment rate). This macroeffect is of first-order importance. However, there are spillover effects of mass layoffs.5 Gathmann et al. (2017) exploit regional variation to find spillover effects of mass layoffs are about 35% of local employment losses stem from spillover effects in plants not directly affected by the mass layoff (55% after a decade). In our analysis, we are not able to use regional variations but rather rely on the industry shutdown rate as a proxy for industry variation. Depending on the firm size class of a worker, we compute that there is an annual earnings loss of between 10 and 60% for laid-off men and 20 to 60% for laid-off women as a result of a 1% increase in the shutdown rate. The rest of the paper is organized in the following fashion: the LWF (firm-worker matched) dataset is described in Section 2 while Section 3 provides an empirical model of permanent layoffs which discusses the issue of selection due to firm survival. Section 4 discusses the effect of firm shutdown rates on workers' earnings. Finally, Section 5 concludes. The Longitudinal Worker File Our data are from the Longitudinal Worker File (LWF). The LWF is an annual administrative dataset from 1983 onwards and contains a 10% random sample of Canadians who either filed a tax return (T1 form) or received a statement of remuneration (T4 form). Appendix A gives a brief description of the LWF data sources and its construction. The LWF has information on individuals' earnings, demographics, and occupation, as well as on the the firm of employment. LWF's matched employer-employed structure allows for examining workers' mobility, turnover, and earnings dynamics. Our sample consists of individuals living in the 10 Canadian provinces who are between 25 and 64 years of age. The source of firm-level information is the Longitudinal Employment Analysis Program (LEAP) database. Given that the LWF and LEAP databases contain common firm identifiers, firm information from the LEAP database is linkable to the worker in the LWF database. LEAP contains annual employment information on firms with at least one dollar in payroll in a given year from 1991 to 2008. The LEAP payroll information allows the identification in year t of continuing firms with a positive payroll versus temporarily or permanently (exit) firm shutdown with a zero payroll. Industry j's shutdown rate in year t, SR jt , is $$\begin{array}{@{}rcl@{}} {SR}_{{jt}}= {SD}_{j,t+1}/N_{{jt}} \end{array} $$ where SD j,t+1 gives the total number of firms in industry j with a positive payroll in year t and a zero payroll in period t+1 and N jt gives the total number of firms with in industry j positive payroll in period t. The structure of the LEAP database implies that firm shutdown is not due to merger or acquisition activity. Table 1 provides the list of the 39 industries in the data. LEAP assigns a NAICS code to each firm from 1992 onwards6. We restrict our sample of workers to the period from 1992 to 2007 since the analysis uses firm and NAICS information taken from the LEAP database. Table 1 Industry classification by NAICS A separation occurs in year t, if t is the last year of an individual's tenure in firm j (i.e., the end of a job spell). The LWF database allows for the categorization of employee-employer separations. Quits and layoffs are two such categories. Layoffs are further broken into temporary, worker subject to recall, and permanent, worker not subject to recall subcategories. These categories allow for the creation of dummy variables. The value of a given separation dummy variable is 1 for any type of the given separation, including, but not limited to, quits and layoffs. For example, the value of the layoff variable is 1 if the Record of Employment (ROE) states that the shortage of work is the reason for the separation, i.e., layoff. Table 2 provides summary statistics across industries. There is industry heterogeneity in terms of (i) workers' characteristics of age, gender, tenure, and earnings and (ii) industry characteristics of shutdown rate, permanent layoff rate, number of firms, and number of workers. The age range of average worker varies from a low of 37.8 years in the motion picture and recording industry to a high of 44.0 years in the primary metal manufacturing industry. Women dominate clothing manufacturing and leather and allied manufacturing at 76% of workers but only constitute 10% of workers in mining. Tenure ranges from 3.81 years in administrative and support services to 11.45 years in primary metal manufacturing. Average earnings are the highest in oil and gas extraction at $107,090 per year while earnings in accommodation and food services are $18,800 per year on average. The shutdown rate is the highest in utilities at 16.1% and the lowest in fabricated metal product manufacturing at 7.4%. Forestry has the highest permanent layoff rate 12.4%, while oil extractions has the lowest at 1.5%. Table 2 Summary statistics by industry Table 3 provides summary statistics on worker characteristics across five firm size classes. We define firm size groupings as (i) extra small (XS)—less than 5 employees; (ii) small (S)—5–19 employees; (iii) medium (M)—20–99 employees; (iv) large (L)—100–500 employees; and (v) extra large (XL)—greater than 500 employees. XS size class firms have workers with the lowest tenure and earnings relative to the other size classes, but these firms experience the highest shutdown rates. The permanent layoff rate is the highest for the firm size classes XS, S, and M at around 5%. L size class firms have a 3.7% layoff rate, while XL firms have a 2% layoff rate. Table 3 Summary statistics by size of firms Table 4 provides summary statistics for worker characteristics across five regions: (1) Atlantic provinces—Newfoundland, New Brunswick Nova Scotia, and Prince Edward Island; (2) Quebec; (3) Ontario; (4) Prairie provinces—Alberta, Saskatchewan, and Manitoba; and (5) British Columbia. Across the regions, average age, proportion of men versus women, and exit rate are similar. The eastern Canadian regions of the Atlantic provinces, Quebec, and Ontario tend to have longer tenure rates compared to the Prairie provinces and British Columbia. Wage rates range from an average high of $45,780 in Ontario to a low of $29,710 in the Atlantic provinces. The opposite occurs for layoff rate as the Atlantic provinces have the highest permanent layoff rate of 6.7% and Ontario has lowest at 3%. Table 4 Summary statistics by region Comparison of continuing and shutting down firms One issue to consider when investigating the impact of industry shutdown rates on worker layoff rate is that workers may choose to quit in anticipation of deteriorating industry conditions in order to avoid any negative consequences of being laid off. A worker may quit in anticipation of being laid off or firm shutdown. This may create a possible selection bias when investigating firm layoffs of workers. Given that a random sample of workers forms the basis of the LWF database, we observe separations for workers in the LWF sample but do not observe separations rates at the firm level. Therefore, we are unable to determine quit rates in the years prior to a firm's shutdown. However, the data contain a measure of firm employment which allows us to look at overall employment activity at firms. Figure 1 presents the median employment size and growth for firms in their last 3 years prior to shutdown. As a comparison, the figure also presents median employment size and growth for rival continuing firms over a similar 3-year window. Continuing firms tend to be larger and have higher growth than shutting down firms. The median employment size and growth both tend to be flat for continuing firms. Alternatively, shutting firms experience a drop in size and increasingly negative growth as shutdown approaches. Comparison of shutdown and continuing firms. Note: This graph provides a comparison between shutting down firms in year t with continuing firms. For these two groups of firms, the graph provides the median employment size and growth rate in 3 years prior to firm shutdown in the former group. For a full comparison by industry, see Tables 5 and 6 Table 5 Size comparison of shutting down and continuing firms Table 6 Growth comparison of shutting down and continuing firms Tables 5 and 6 provide these comparisons between continuing and shutting down and firms across the industries. Similar results occur at the industry level. The shedding of workers, whether through layoffs or quits, appears to occur in the years leading to firm shutdown. Permanent layoffs—extensive margin Industry shutdown rates measure the short-run performance of firms within an industry. High shutdown rates indicate firms within an industry deem that shutdown is more profitable than continuing operations. The implication of shutdown is that a firm must become profitable or eventually exit. One method to reduce costs is worker layoffs. These layoffs can be temporary or permanent depending on circumstances. Temporary layoffs may lead to permanent layoffs in the long run if the firm eventually exits or workers are not recalled. Thus, our analysis focuses on permanent layoffs by firms as a method to analyze the process of shedding workers. We consider the effects of industry shutdown rates along with the other controls to assess the qualitative and quantitative impacts of industry conditions on a firm's decision to permanently layoff workers. We identify shutdowns in year t as those firms transitioning from a positive payroll in year t to a zero payroll in year t+1. A firm's shutdown does not imply an exit, as the firm may have a positive payroll in some future period. Our focus on anticipated separations motivates the choice of shutdown rates. The absence of a positive annual payroll in year t signals at least a year-long closure. From the worker's point of view, there is little difference whether or not his/her firm reopens in some future year following shutdown. In either case, the firm's workers anticipate prolonged separations and adjust their labor market decisions. Shutdowns are also more easily identified in the data than firm exits since they only require the knowledge of the firm's payroll in two consecutive periods. For the analysis, we perform separate analysis for men and women and across firms in different size classes. We analyzed the pooled data but found the assumption of homogeneity of effects across men and women is rejected statistically and economically.7 Selection issues and identification strategy A selection issue arises as the permanent layoff decisions are only observable for continuing firms in year t. In the remainder of the paper, we will refer to continuing firms to indicate those firms not experiencing a shutdown at year t. To account for the selection bias, we consider two separate dichotomous variables and allow for correlated disturbances. For worker i at firm k in industry j at time t, we estimate a bivariate probit model. The continuing firm (FS) equation accounts for firm selection and the permanent layoff (PL) equation captures a worker's outcome or the probability of a permanent layoff, which gives the following bivariate probit worker selection (BPWS) model: $$\begin{array}{@{}rcl@{}} \text{FS}_{ikjt}^{} &=& \alpha^{\text{FS}} + \beta^{\text{FS}} \text{SR}_{jt} + \gamma^{\text{FS}} B_{it} + \sum_{j=1}^{J} \psi_{j}^{\text{FS}} I_{j} + \sum_{t=1993}^{2002} \delta_{t}^{\text{FS}} D_{t} + \lambda Z_{kjt} + v_{ikjt}, \\ \text{PL}_{ikjt}^{} &=& \alpha^{\text{PL}} + \beta^{\text{PL}} \text{SR}_{jt} + \gamma^{\text{PL}} B_{it} + \sum_{j=1}^{J} \psi_{j}^{\text{PL}} I_{j} +\sum_{t=1993}^{2002} \delta_{t}^{\text{PL}} D_{t} + u_{ikjt}. \end{array} $$ $$\begin{array}{@{}rcl@{}} v_{ikjt},u_{ikjt}\sim N(\mu,\Sigma), \mu= \left[\begin{array}{cc} 0\\0 \end{array}\right], \Sigma= \left[\begin{array}{cc} 1&\rho \\ \rho&1 \end{array}\right] \end{array} $$ The sample includes only continuing workers or workers experiencing a permanent layoff. Thus, the indicator variable, PL ikjt , equals 1 if a worker experiences a permanent layoff with $\text {PL}_{ikjt}^{} \geq 0$ and 0 if a worker continues employment. A second indicator variable, FS ikjt , equals 1 if a firm remains active with $\text {FS}_{ikjt}^{} \geq 0 $ and 0 otherwise. SR jt is the annual shutdown rate in industry j in period t. The PL equation includes individual-, firm-, and industry-specific control variables: (i) B it is a set of worker including an age categories, marital status, job tenure and tenure squared, region of residence, union membership, and earnings in period year t−1; (ii) I j is industry-specific dummy variables; and (iii) D t is a set of year-specific dummy variables. We break the sample of workers into subsamples for estimation purposes based on their firm's employment size. The FS equation includes all the relevant variables from PL equation but with Z kjt as the exclusion restrictions both at the firm (k) and industry (j) levels. For a technical discussion of this method, please refer to Maddala (1983). Identification strategy The BPWS model given in Eq. 2 identifies the impact of selection in two ways: (1) the correlation parameter (ρ) of the joint model and (2) using exclusion restrictions of variables (Z jt ). The correlation parameter achieves identification through functional form. Han and Vytlacil (2017) prove that identification is achievable in bivariate models without exclusion restrictions (i.e., instruments) if there are common exogenous regressors in both equations. They also show that having an exclusion restriction is necessary and sufficient for identification in these models without common exogenous variables but is sufficient only in models with common exogenous covariates. The second method requires at least one variable that affects whether a firm continues or not but not whether a worker experiences a permanent layoff or not, contemporaneously. There are two exclusion restrictions. The first exclusion restriction is the use of industry-level US-Canada bilateral real exchange rate: $$\begin{array}{@{}rcl@{}} \text{RER}_{jt}= P_{jt}^{\text{US}}/P_{jt}^{\text{CDN}} \times e_{t}, \end{array} $$ where $P_{jt}^{\text {US}}$ is the US industry gross output price index, $P_{jt}^{\text {CDN}}$ is the Canada industry gross output price index and e t is the nominal bilateral exchange rate between Canadian and USA in year t. The choice of RER jt as the exclusion restriction is motivated by the fact that the USA is the major trading partner of Canada. The real exchange rate affects Canadian export and import propensities with the USA. Short-run profits of Canadian firms likely fluctuate with export/import propensities. Thus, real exchange rate movements likely affect the probability of whether a Canadian firm continues to operate or temporarily shutdown; see for example Huynh et al. (2010). For employment, the impact of exchange rates differs. Huang et al. (2014) provide empirical evidence that exchange rate movements have little effect on manufacturing employment and no effect on non-manufacturing employment in Canada for the period 1994 to 2010. Commodity prices and exchange rate movements are tied together. The authors show that commodity price movements are a main driver to employment changes in manufacturing resulting from exchange rate movements. Further, Campa and Goldberg (2001) show that the real exchange rate movements for the USA have effects on wages and hours worked but have negligible effects on total employment and number of jobs. Based on these empirical findings, we argue that fluctuations of the real exchange rate is correlated with firm exit rates but are unlikely to affect the contemporaneous probability a worker experiences a permanent separation. The second exclusion restriction is a relative firm-to-industry variable. We compute the logarithm of the ratio of the wage bill of firm k at time t relative to the average wage bill of firms in industry j and size class s at time t or: $$\begin{array}{@{}rcl@{}} \log \overline{\text{wage bill}}_{kjst} = \log \bigg(\frac{{\text{wage bill}}_{kjst}} {\overline{\text{wage bill}}_{jst}} \bigg). \end{array} $$ This variable is strongly correlated with whether a firm continues operations, as it proxies for how competitive a firm is relative to its industry peers. Controlling for the employment size of a firm, the relative wage bill provides a measure of firm efficiency/productivity within an industry. More productive firms pay higher wages and, thus, have a higher wage bill as discussed in Abowd et al. (1999), Michelacci and Quadrini (2009) and Moscarini and Postel-Vinay (2012). More productive firms with higher wage bills should be more likely to continue operations. However, the contemporaneous relative wage bill of a firm is unlikely to contain information about worker layoff probabilities at continuing firms. The BPWS results provide estimates of the impact of industry shutdown rates on worker layoffs with an additional selection control for whether a firm is active or not. Table 7 presents estimation coefficients for the probability of a permanent layoff when controlling for firm shutdown selection effects for men while Table 8 provides estimation coefficients for women. Table 7 Bivariate probability of permanent layoff: men Table 8 Bivariate probability of permanent layoff: women The descriptive statistics illustrate that there is substantial variation in the shutdown rates across industry and time. Therefore, the impact of industry shutdown rates on permanent layoffs should be well-identified. A likelihood ratio test reveals that selection is statistically significant in all cases for men and three out of five cases for women. The exceptional cases are women at large and extra large firms. Therefore, selection via the impact of firm shutdown affects the probability of permanent layoff on a worker. Most of the discussion emphasizes the variable of interest, industry shutdown rates. With the exception of women at small-sized firms, the coefficient on the shutdown rate is positive for both men and women across the firm size classes. Thus, these estimates indicate that the impact of industry shutdown rates on worker layoff rates are positive. Figure 2 provides estimated marginal effects of an increase in industry shutdown rate on the probability of a worker layoff across the firm size classes. For comparison, this figure also provides the estimated marginal effect without accounting for selection.8 For both men and women, these quantitative impacts of the industry shutdown rate on permanent layoff probability change when accounting for selection. After controlling for selection, the results for men indicate that a 1% increase in industry shutdown rate causes between 0.04 and 0.14% increase in the probability of a permanent layoff. For women, the marginal effects vary across the firm size classes; a 1% increase in industry shutdown rates implies (i) a 0.01% decrease in the probability of a permanent layoff at extra small-sized firms and (ii) a 0.11, 0.03, 0.01, and 0.05% increase in the probability of a permanent layoff at small-, medium-, large-, and extra large-sized firms, respectively. Probability of permanent layoff and the effect of selection. Note: The figure provides the marginal effects of industry shutdown rates on the probability of a permanent layoff for a worker across various size classes of firms. Selection corresponds to estimates from Tables 7 and 8 for men and women, respectively. For comparison, no selection are estimates when not accounting for selection effects of continuing or shutdown of a firm Returning to Tables 7 and 8, coefficients on the other control variables remain fairly constant across the firm size classifications and qualitatively identical for men and women. The probability of a permanent layoff falls with a worker's income. Tenure effects are concave in shape. Married workers have a lower probability of permanent layoff separation, while unionized workers have a higher permanent separation probability. Across the regions, workers in the Atlantic provinces experience the highest probability of a permanent layoff, where the lowest permanent layoff separation probability occurs for workers in the Prairie provinces. Tables 7 and 8 also report the coefficients on the log of the firm's wage bill and the log of the real exchange rate, which are our exclusion restriction variables in the selection equation. The coefficient on the wage bill variable is always positive and significant. This result likely captures the effect of firm size on firm survival as larger firms tend to have higher survival rates. The coefficient on the log of the real exchange rate varies between negative and positive and is only statistically significant with a negative value for men at small-sized firms and women at large-sized firms. For men, the correlation in the error terms between the two equations is approximately −0.45 in the extra small-, small-, medium-sized firm categories and approximately 0.9 in the large- and extra large-sized firm categories. A negative correlation implies that a positive shock to a firm remaining active has a negative impact on the probability of a male worker being permanently laid off. This correlatation also varies for women across firm size classes. Earnings transitions—intensive margin The previous section discusses permanent layoffs or the extensive margin of employment. In this section, we discuss workers earnings transitions or the intensive margins of permanent layoffs by looking at the earnings growth for those workers experiencing a permanent layoff. We do not use the identification strategy found in Abowd et al. (1999), where the worker and firm fixed effects enter additively. The LWF allows us to follow the worker transitions from a separation (layoff) to possible employment to a another firm. Eeckhout and Kircher (2011) provide motivation for using transitions. They show the estimated worker and firm fixed effects from the log-linear wage equation do not directly identify the underlying worker skill and firm productivity heterogeneity. In particular, the correlation between the estimated worker and firm fixed effects does not identify sorting in the matching between worker skill and firm productivity. Earnings and selection Similar to the previous selection problem, the estimated earnings growth model must account for selection effects due to firm shutdown. To deal with this selection problem, we estimate the effect of the transitions on the change in log wage using a Heckman-selection model. Again, the selection equation describes the probability of a firm continuing $\left (\text {FS}_{kjt}^{}\right)$, while the outcome equation describes the log wage $\left (\ln w_{ikjt}^{}\right)$ of a specific transition: $$\begin{array}{@{}rcl@{}} \text{FS}_{ikjt}^{} &=& \alpha^{\text{FS}} + \beta^{\text{FS}} \text{SR}_{jt} + \gamma^{\text{FS}} B_{it} + \sum_{j=1}^{J} \psi_{j}^{\text{FS}} I_{j} + \sum_{t=1993}^{2002} \delta_{t}^{\text{FS}} D_{t} + \lambda Z_{kjt} + v_{ikjt}, \\ \Delta \log w_{ikjt}^{} &=& \alpha^{w} + \beta^{w} \text{SR}_{jt} + \gamma^{w} B_{it} + \sum_{j=1}^{J} \psi_{j}^{w} I_{j} +\sum_{t=1993}^{2002} \delta_{t}^{w} D_{t} + u_{ikjt}. \end{array} $$ where Δ lnw ikjt is wage growth of worker i from firm k in industry j at time t and the errors u ikjt and v ikjt are normally distributed with zero means and correlation ρ. The other variables are defined as in the BPWS model from Eq. 2. The analysis examines wage growth as a way to control for potentially unobservable factors. For example, there may be wages differentials due to job risk, education, or occupations with higher layoff rates. The analysis includes industry, location, and firm size variables which partially capture some of these differentials. Further, these unobservable-time invariant worker or job characteristics are unlikely to affect wage growth. Dostie (2005) and Abowd et al. (2005) show unobserved heterogeneity affects the level of lnw ikjt . However, the analysis of wage growth, Δ lnw ikjt , differences out time invariant factors and, thus, removes these unobservable variables. In contrast to the BPWS model, the exclusion restriction only includes the firm-to-industry relative wage ($\log \overline {\text {wage bill}}_{ikjt}$). The specification does not include the relative real exchange rate as an exclusion restriction. Campa and Goldberg (2001) show an impact of the real exchange rate on wages, which justifies this change from the previous worker separation analysis. Tables 9 and 10 present the coefficient estimates for the earnings regression accounting for selection effects for men and women, respectively. The selection parameter (λ) is significant for all size classes except small- and large-sized firm categories for men and small- and extra large-sized firm categories for women.9 This result is due to the small correlation (ρ) between the two equations. For comparison purposes, Fig. 3 provides coefficient estimates on the industry shutdown variable for the selection and non-selection models. For men, the coefficient on the industry shutdown rate variable becomes positive and statistically significant for workers at medium-sized firms, while the coefficients remain negative, statistically significant and increase slightly in magnitude for workers at other size classes when moving from the non-selection to the selection model. For women, there is no change in the qualitative findings and little change in the quantitative effects of the industry shutdown rate after accounting for selection. Thus, the impact of selection effects of firm shutdown is small when examining worker earnings growth. With the exception of men at medium-sized firms, the correlation between the error terms in the two equations, ρ, is positive. Positive correlation indicates that firms with unexplained increases in the probability of remaining active also have unexplained increases to wages paid. Δ logw ikjt and the effect of selection. Note: The figure provides the marginal effects of industry shutdown rates on the probability of a permanent layoff for a worker across various size classes of firms. Selection corresponds to estimates from Tables 9 and 10 for men and women, respectively. For comparison, no selection are estimates when not accounting for selection effects of continuing or shutdown of a firm Table 9 Earnings regression with selection: men Table 10 Earnings regression with selection: women The change in the logarithm of worker wages measures the wage growth for a worker. Thus, the coefficient on the industry shutdown rate variable gives the response of worker wage growth to changes in the industry shutdown rate. Equivalently, this coefficient gives an elasticity or the percentage change in worker earnings in response to a 1% change in the industry shutdown rate. The estimated coefficient values indicate economic significance in that worker wage growth is highly responsive to industry shutdown rates. For men, extra small-sized firms show the least response of wage growth to industry shutdown rates with a coefficient of −0.98, while men at large-sized firms have the most response with a coefficient of −3.05. For women, workers at the extra small-sized firms have the largest response as the coefficient estimate indicates a 1% increase in industry shutdown rate causes a 3% decrease in worker wage growth. The coefficients on the other variables indicate similar patterns across firm size classes and genders. Earnings growth falls with age and rise with being married or part of a unionized firm. The effect of job tenure is nonlinear. Wage growth initially falls with tenure but begins to rise after approximately 11 years at a job. We investigate worker earnings while controlling for the possible association of the firm size class with the worker earning changes. There are two potential reasons for a worker's firm size class to change. First, the worker moves to a different firm belonging to a different size class. Second, the worker stays at the same firm, but the firm moves to a different firm size class. Since we look at workers experiencing a permanent layoff, our analysis focuses on the group of workers moving to a different firm. This analysis demonstrates whether a layoff necessarily results in a worse situation for a worker. We examine the impact of firm size class switches on the earnings of laid-off workers since firm size provides a clear dimension for improvement in worker's earnings. Oi and Idson (1999) document that larger firms pay higher wages. Therefore, workers experiencing a layoff but moving to firms in larger size classes may actually see their wages increase.10 Figure 4 presents the probability distribution function (PDF) for Δlog(wage ikjt ) for those men and women, respectively, who experience a permanent layoff but move to a different firm. Each figure shows CDFs for three subgroups: (i) switch down—worker moves to a firm in a smaller size class; (ii) switch to same size—worker moves to a firm in the same size class; and (iii) switch up—worker moves to a firm in a larger size class. For both men and women, the wage growth PDFs for the switch down, switch to the same size, and switch up are left, middle, and right, respectively. These figures indicate that workers who transition to larger sized firms do better than workers who move to a firm in the same size class, while workers who move to smaller sized firms do worse. An asymmetry results when comparing the distributions across the three groups. For negative values of wage growth, the lower tail for the switch down group of workers is much fatter than for the other two groups, while the lower tail looks similar for the switch to same size and switch up groups. For positive values of wage growth, the opposite occurs. The distribution switch down and switch to same size groups have similar upper tails while the switch up group has a fatter upper tail. Unconditional probability distribution of Δ logw ikjt . Note: This graph illustrates the unconditional growth rate of wages (Δ logw ikjt ) for male (top graph) and female (bottom graph) workers who experienced a permanent layoff and found a new job. The following three lines are for groups of workers: (1) transition to a smaller size firm (switch down), (2) transition to a larger size firm (switch up), and (3) transition to a same size firm This unconditional analysis ignores the rich characteristics of firms and workers. So, we amend the wage model with selection (5) to include the firm size class switches. The switchers are treated as exogenous as we focus only on involuntary separations or permanent layoffs. The following specification combines workers experiencing a firm size class switch with the selection wage model: $$\begin{array}{@{}rcl@{}} \text{FS}_{ikjt}^{} &=& \alpha^{\text{FS}} \,+\, \beta^{\text{FS}} \text{SR}_{jt} \,+\, \gamma^{\text{FS}} B_{it} \,+\, \sum_{j=1}^{J} \psi_{j}^{\text{FS}} I_{j} \,+\,\! \sum_{t=1993}^{2002} \delta_{t}^{\text{FS}} D_{t} \,+\, \lambda Z_{kjt} \,+\, \sum_{i \in m}\eta^{\text{FS}} \text{SW}_{it} \,+\, v_{ikjt}, \\ \Delta \log w_{ikjt}^{} &=& \alpha^{w} + \beta^{w} \text{SR}_{jt} + \gamma^{w} B_{it} + \sum_{j=1}^{J} \psi_{j}^{w} I_{j} +\sum_{t=1993}^{2002} \delta_{t}^{w} D_{t} + \sum_{i \in m}\eta^{w} \text{SW}_{it} + u_{ikjt}. \end{array} $$ where SW it is a series of indicator variables for individuals across various firm size transitions between time t−1 and t and η w is the corresponding coefficients on the indicator variables. Firm size transition classes, m, are: (i) extra small to small (XS–S); (ii) small to extra small (S–XS); (iii) small to small (S–S); (iv) small to medium (S–M); (v) medium to small (M–S); (vi) medium to medium (M–M); (vii) medium to large (M–L); (viii) large to medium (L–M); (ix) large to large (L–L); (x) large to extra large (L–XL); (xi) extra large to large (XL–L); and (xii) extra large to extra large (XL–XL). Table 11 provides estimates for the earnings regressions controlling for firm size class changes. Industry shutdown rate continues to have a negative impact on worker earnings even with the additional control for switching firm size class. The coefficients on the switching variables have the expected sign. An increase in the firm size class of a worker sees the worker's earnings increase, while a decrease in firm size class sees the worker's earnings fall. Table 11 Earnings switcher regression with selection: pooled Switching from extra small- to small-sized firm causes wages to increase by 0.22% for men and 0.18% for women. The magnitude is not as great in the reverse direction as switching from small- to an extra small-sized firms causes earnings for men to fall by 0.20% and women earnings to fall by 0.14%. A movement from medium- to large-sized firms causes earnings of men to increase by 0.11% and earnings of women to increase by 0.09%, while a movement from large- to medium-sized firms causes the earnings of men and women to fall by 0.06 and 0.01%, respectively. Those workers not changing firm size class generally do not see changes in their earnings. The exceptions to this rule are men at medium- and large-sized firms who see a statistically significant increase in earnings of 6%. Figure 5 present the PDFs of the residuals from the regressions in Table 11 for men and women. As in Fig. 4, these workers are broken into three categories based on pre-layoff to post-layoff size class transition of their firms. The conditioning removes a significant amount of the difference between the distributions across the three categories. Further, the asymmetries at the tails of the distributions across the three categories disappear after the conditioning. A worker does not necessarily end up in a worse position with a lower earning job after being permanently laid off. However, almost 60% of those laid-off workers who move to smaller or similarly sized firms see a fall in wages. In contrast, less than 50% of laid-off workers eventually moving to a larger sized firm see their earnings fall. Thus, the type of firm a worker ends up at after being laid off explains a significant amount of the resulting wages. Conditional probability distribution of Δ logw ikjt . Note: This graph illustrates the conditional growth rate of wages (Δ logw ikjt ) for male (top graph) and female (bottom graph) workers who experienced a permanent layoff and found a new job. The following three lines are for groups of workers: (1) transition to a smaller size firm (switch down), (2) transition to a larger size firm (switch up), and (3) transition to a same size firm. The residual wage growth is generated by the Heckman selection model (6) and results in Table 11 We quantify the effect of industry shutdown rates on worker outcomes such as involuntary separations or permanent layoffs (extensive margin) and wage earnings (intensive margin). Our empirical work shows that when controlling for individual- and firm-specific characteristics, industry shutdown rates generally have a positive and significant effect on the probability of a permanent worker layoff. For wage growth, shutdown rates have a negative effect but the effects are amplified for workers in smaller firms. The unique structure of the LWF database allows us to differentiate among different industries in our analysis. We find substantial differences across industries in the roles of individual- and firm-level attributes on permanent layoff and wage growth. Our analysis controls for firm selection effects on worker outcomes due to firm shutdown. Accounting for selection effects does alter the estimated impact of industry shutdown rates on worker outcomes. Determining the relative contribution of worker, firm, industry, and time factors to the overall employment instability is an essential step in developing training programs to counter the adverse effects of employment loss. If job instability is mostly determined by differences in individual human capital, then future policies may focus on providing opportunities for workers to improve their education or skills. If, on the other hand, job instability is mostly a reflection of industry conditions or, more specifically, firm shutdown within an industry, then education and skill development programs may not be as effective. Hence, understanding the relative impact of individual and firm characteristics on worker turnover is important in determining the effectiveness of specific training and skill-development programs provided both privately and publicly. In the light of the recent economic downturn that affected many Western countries including Canada, the costs and benefits associated with such programs are likely to remain subject to intense policy discussions in the foreseeable future. Our estimates of the impact of industry shutdown rates on earnings growth is line with other papers that focus on uncertainty and variability such as Gathmann et al. (2017). These results demonstrate the necessity of the joint analysis of firm shutdown with either permanent layoff or worker wages. Industry shutdown rates provide a measure of turbulence and firm turnover within an industry. Without controlling for firm selection, the analysis ignores a major portion of workers and firms. Higher industry shutdown rates suggest more turbulence within an industry. Substantial hiring and firing costs lead to a desire by continuing firms to keep and not lay off their workers. These costs factor into a firm's choice to continue operations or shutdown. Higher hiring and firing costs within an industry also factor into a firm's choice between temporary shutdown or permanent exit. Controlling for a firm's shutdown probability allows the industry shutdown rate to fully capture industry turnover which leads to the positive correlation between industry shutdown and the permanent worker layoff rate. This finding complements the work by Moscarini and Postel-Vinay (2012) who document that the negative correlation between net job creation rates and the unemployment rate is larger for small firms versus large firms. Job turnover has a rich set of dynamics that cannot necessarily be explored with reduced-form methods. As suggested by Postel-Vinay and Robin (2006), they highlight the role for modeling job turnover using frictional models of unemployment. In these models, job turnover is a dynamic process that involves explicitly laying out the microfoundations. However, there is an important opportunity for further research on voluntary separations or a worker quitting their job to find a new one. Recent work by Lise et al. (2016) allows for matched agents to undertake on-the-job search and illustrates the complexity of labor outcomes in terms employment prospects and earnings. A fruitful extension would consider both involuntary and voluntary quits. 1 Work in this literature is driven by collection of administrative data, which usually have restricted access. For example, a recent study by Song et al. (2015) shows that rising labor earnings dispersion in the USA is driven by increasing wage dispersion across firms and not by changes to within firm wage dispersion. Haltiwanger et al. (2006) provide a broad overview. 2 Morissette (2004), Morissette R et al. (2007) and Morissette et al. (2013) use the LWF database to investigate permanent layoffs and worker reallocation. 3 Job instability has wide ranging financial and other consequences for individuals and families (Jacobson et al. (1993); Gottschalk and Moffitt (1994) Gottschalk and Moffitt (2009); Beach et al. (2003); Morissette and Ostrovsky (2005)). Often, it signals high earnings uncertainty, which may, in turn, lead to lower consumption Browning and Lusardi (1996) and alter family savings and labor supply decisions (Pistaferri (2003)). It may also affect families' schooling and occupational choices (Guiso et al. (2002)) and even their fertility behavior (Fraser (2001)). 4 We perform separate analysis on men and women as labor market decisions and outcomes are likely to differ; see Killingsworth and Heckman (1987), Loprest (1992), and Altonji and Blank (1999), inter alia. 5 We thank an anonymous referee for pointing out this salient feature. 6 This NAICS coding is partially due to retro-coding by Statistics Canada. 7 Results are available upon request. 8 The coefficients on the other variables are quite similar for the models with and without a firm selection control. A complete set of estimates for the model with no control for selection are available from the authors upon request. 9 In a full-information maximum likelihood estimation, the selection parameter is a function of correlation and variance (σ) or λ=ρ×σ. 10 Other dimensions to look at when investigating worker earnings following layoffs include workers moving to new occupations or industries. Our dataset does not include information regarding worker occupation. Further, there is no clear direction to the change in worker earnings when moving to a new industry or occupation unlike moving to larger firms. 11A T4 form closely resembles a W-2 form in the USA. Construction of Longitudinal Worker File Statistics Canada constructs the LWF database from four data sources. The first data source in the LWF is the T4 Supplementary Tax File, which is a random sample of all individuals who received a T4 supplementary tax form and filed a tax return. A T4 supplementary tax form is issued by an employer to each employee for any earnings that either exceed a certain threshold or trigger income tax, Canada/Quebec Pension Plan (C/QPP), or unemployment insurance premiums. It contains information about the earnings received from an employer in a given year, tax deducted, pension contributions, union dues, and other information. The second data source is the Record of Employment (ROE), which includes employer provided information on separations and their reasons. Canadian employers are by law required to provide such information for any separation. A detailed list of reasons for separations includes voluntary and involuntary separations such as the shortage of work, labor dispute, injury or illness, quit, pregnancy and parental leaves, retirement, and other reasons. The third data source is the Longitudinal Employment Analysis Program (LEAP). Statistics Canada constructs and maintains the LEAP database. This database includes information about the size of the employee's firm and tracks employees who move from one firm to another. The LEAP database covers the entire Canadian economy and includes firms (but not establishments) with at least one dollar in annual payroll. The key information that comes from the LEAP is the firm's employment derived from its payroll using average labor units (ALU). LEAP tracks employees who move from one firm to another. Statistics Canada constructs LEAP, and by extension the LWF database, to handle mergers and acquisitions in a retrospective manner. Suppose two firms, A and B, merge in year t to create firm C. Within the database prior to year t, a synthetic history for firm C is created by aggregating information from firms A and B, so that only firm C's information appears in the database. Thus, identification of a firm's exit or shutdown imply these are not due to merger activity. The final data source is personal income tax files (T1), which add demographic variables such as age, sex, family status, and area of residence. They also provide information about individuals' income sources other than T4 earnings. Our data was constructed by using information from the LEAP to classify firm entries and shutdowns and to compute industry-specific shutdown rates. Identification of firm entries and shutdowns is based on firm payroll transitions from 1 year to the next one. A firm's entry year is the first year; the firm has a positive payroll. We identify firm shutdown in year t when a firm has zero payroll in year t but positive payroll in year t−1. Thus, entry year is not identifiable for firms existing in 1991 or the first year of the LEAP database, while firm shutdown is not identifiable in 2008 or the last year of the database. Further, LEAP includes NAICS codes for firms from 1992 and onwards. Consequently, NAICS industry-specific shutdown rates can be computed only from 1992 to 2007. We proceed by extracting individual data from the LWF. Since NAICS codes in the LWF are available only from 1992, we used the LWF data from 1992 to 2008. We kept men and women aged 24 to 64. Total earnings in year 4t were defined as individual's total annual paid employment income (wages and salaries) computed from all T4 forms issued to the individual in year t. All earnings are adjusted to 2007 constant dollars using the Consumer Price Index for Canada. For individuals who held multiple jobs in a given year, we then retained only the characteristics of main jobs defined as jobs with the highest T4 amount in that year.11 To each individual record in the LWF, we added industry-specific shutdown rates by matching firm identifiers in the LWF to those in the LEAP. We excluded individuals who died and whose employer's industry classification was unknown. Next, individual employer-employee records from the LWF are matched to industry price information available for the period from 1987 to 2007. US industry prices are taken from Industry Economic Accounts tables available from the Bureau of Economic Analysis, US Department of Commerce (Chain-Type Price Indexes for Gross Output by Industry series). Canadian industry price indexes are computed from the information on gross output and real gross output, by industry (Statistics Canada CANSIM series 383-0022). Although both the US and Canadian industry price indexes are based on the North American Industry Classification System (NAICS) codes, there are some differences between the industries available in each series. We identified 42 industries for which a direct correspondence between the two series could be established. Excluded are primarily industries that are most likely to be represented by the public sector, such as, for instance, public administration, education and healthcare. Three industries ("petroleum and coal product manufacturing," "pipeline transportation," and "waste management") had to be excluded because of insufficient sample size. Therefore, our final sample includes 39 industry categories. The list of included industries is given in Table 1. Finally, the LWF records are also matched to annual Canada/US nominal exchange rates necessary to produce real exchange rates used in the study. The rates used in the study are from the G.5 Foreign Exchange Rates series provided by the Board of Governors of the Federal Reserve System (Series ID: EXCAUS). Abowd, J, Kramarz F, Roux S (2005) Wages, mobility and firm performance: advantages and insights from using matched worker-firm data. Econ J116(512): F245–F285. Abowd, JM, Kramarz F, Margolis DN (1999) High wage workers and high wage firms. Econometrica67(2): 251–334. Altonji, JG, Blank RM (1999) Race and gender in the labor market. In: Ashenfelter O Card D (eds)Handbook of Labor Economics, 3143–3259.. Elsevier, Amsterdam. Beach, CM, Finnie R, Gray D (2003) Earnings variability and earnings instability of women and men in Canada: how do the 1990s compare to the 1980s?. Can Public Policy29(s1): 41–64. Browning, M, Lusardi A (1996) Household saving: micro theories and micro facts. J Econ Lit34(4): 1797–1855. Campa, JM, Goldberg LS (2001) Employment versus wage adjustment and the U.S. dollar. Rev Econ Stat83(3): 477–489. Davis, SJ, Wachter TV (2011) Recessions and the costs of job los. Brook Pap Econ Act2: 1–72. Dostie, B (2005) Job turnover and the returns to seniority. J Bus Econ Stat23(2): 192–199. Eeckhout, J, Kircher P (2011) Identifying sorting–in theory. Rev Econ Stud78(3): 872–906. Fraser, CD (2001) Income risk, the tax-benefit system and the demand for children. Economica68(269): 105–25. Farber, HS (1999) Mobility and stability: the dynamics of job change in labor markets. In: Ashenfelter O Card D (eds)Handbook of Labor Economics,2439–2483.. Elsevier, Amsterdam. Gathmann, C, Helm I, Schönberg U (2017) Spillover effects of mass layoffs, working paper, University College London. Gottschalk, P, Moffitt R (1994) The growth of earnings instability in the U.S. labor market. Brook Pap Econ Act25(1994-2): 217–272. Gottschalk, P, Moffitt R (2009) The rising instability of U.S. earnings. J Econ Perspect23(4): 3–24. Guiso, L, Jappelli T, Pistaferri L (2002) An empirical analysis of earnings and employment risk. J Bus Econ Stat20(2): 241–53. Haltiwanger, J, Brown C, Lane J (2006) Economic turbulence: the impact on workers, firms and economic growth. University of Chicago Press, Chicago. Han, S, Vytlacil E (2017) Identification in a generalization of bivariate probit models with dummy endogenous regressors. mimeo. J Econ199: 63–73. Huang, H, Pang K, Tang Y (2014) Effects of exchange rates on employment in Canada. Can Public Policy40(4): 339–352. Huynh, KP, Petrunia RJ, Voia M (2010) The impact of initial financial state on firm duration across entry cohorts. J Ind Econ58(3): 661–689. Jacobson, LS, LaLonde RJ, Sullivan DG (1993) Earnings losses of displaced workers. Am Econ Rev83(4): 685–709. Kambourov, G, Manovskii I (2009) Occupational specificity of human capital. Int Econ Rev50(1): 63–115. Killingsworth, MR, Heckman JJ (1987) Female labor supply: a survey. In: Ashenfelter O Layard R (eds)Handbook of Labor Economics,103–204.. Elsevier, Amsterdam. Lise, J, Meghir C, Robin J-M (2016) Mismatch, sorting and wage dynamics. Rev Econ Dyn19(1): 63–87. Loprest, PJ (1992) Gender differences in wage growth and job mobility. Am Econ Rev82(2): 526–532. Maddala, G (1983) Limited dependent and qualitative variables in econometrics. Cambridge University Press, Amsterdam. Michelacci, C, Quadrini V (2009) Financial markets and wages. Rev Econ Stud76(2): 795–827. Morissette, R (2004) Have permanent layoff rates increased in Canada? Analytical studies branch research paper No. 218 Statistics Canada. Morissette, R, Lu Y, Qiu T (2013) Worker reallocation in Canada. Analytical Studies Branch Research Paper No. 348, Statistics Canada. Morissette, R, Ostrovsky Y (2005) The instability of family earnings and family income in Canada, 19861991 and 1992001. Can Public Policy31(3): 273–302. Morissette R, Zhang X, Frenette M (2007) Earnings losses of displaced workers: Canadian evidence from a large administrative database on firm closures and mass layoffs. Analytical Studies Branch Research Paper No. 291, Statistics Canada. Moscarini, G, Postel-Vinay F (2012) The contribution of large and small employers to job creation in times of high and low unemployment. Am Econ Rev102(6): 2509–39. Oi, W, Idson T (1999) Firm size and wages. In: Ashenfelter O Card D (eds)Handbook of Labor Economics,2165–2214.. Elsevier, Amsterdam. Pistaferri, L (2003) Anticipated and unanticipated wage changes, wage risk, and intertemporal labor supply. J Labor Econ21(3): 729–728. Postel-Vinay, F, Robin J-M (2006) Microeconometric search-matching models and matched employer-employee data. Open Access publications from University College London. http://eprints.ucl.ac.uk/. Quintin, E, Stevens J (2005a) Growing old together: firm survival and employee turnover. Top Macroecon5(1): 1319–1319. Quintin, E, Stevens JJ (2005b) Raising the bar for models of turnover. Finance and Economics Discussion Series 2005-23, Board of Governors of the Federal Reserve System (U.S.) Song, J, Price DJ, Guvenen F, Bloom N, von Wachter T (2015) Firming up inequality. NBER working paper 11199, National Bureau of Economic Research, Inc. Song, J, von Wachter T (2014) Long-term nonemployment and job displacement. mimeo, UCLA. von Wachter, T, Song J, Manchester J (2009) Long-term earnings losses due to mass layoffs during the 1982 recession: an analysis using U.S. Administrative Data from 1974 to 2004. mimeo. The assistance and hospitality of Statistics Canada is gratefully acknowledged. Comments and suggestions from the participants at various conferences and seminars are greatly appreciated. The authors thank Kathryn Shaw, Michael Veall, Gueorgui Kambourov, John Stevens, Joni Hersch, and anonymous referees for the valuable comments. The views in this paper represent those of the authors alone and are not those of the Bank of Canada or Statistics Canada. All errors and opinions are our own. No funding was received for this paper. Bank of Canada, 234 Wellington Street, Ottawa, K1A 0G9, ON, Canada Kim P. Huynh Statistics Canada, 24-J RHC, 100 Tunney's Pasture Driveway, Ottawa, K1A 0T6, ON, Canada Yuri Ostrovsky Lakehead University, 955 Oliver Road, Thunder Bay, P7B 5E1, ON, Canada Robert J. Petrunia Department of Economics, Carleton University, 1125 Colonel By Drive, Ottawa, K1S 5B6, ON, Canada Marcel C. Voia Search for Kim P. Huynh in: Search for Yuri Ostrovsky in: Search for Robert J. Petrunia in: Search for Marcel C. Voia in: Correspondence to Marcel C. Voia. 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. Huynh, K.P., Ostrovsky, Y., Petrunia, R.J. et al. Industry shutdown rates and permanent layoffs: evidence from firm-worker matched data. IZA J Labor Econ 6, 7 (2017) doi:10.1186/s40172-017-0057-0 Worker separation Firm survival	CommonCrawl
# Understanding the Discrete Fourier Transform The Discrete Fourier Transform (DFT) is a mathematical technique used to analyze and transform discrete-time signals. It is an important tool in various fields such as signal processing, image processing, and data compression. The DFT is based on the Fourier series, which is a mathematical concept that describes how any periodic function can be represented as an infinite sum of sine and cosine functions. Consider a discrete-time signal $x[n]$ sampled at a frequency of 1 Hz. The DFT of this signal can be represented as: $$X(k) = \sum_{n=0}^{N-1} x[n] e^{-j2\pi kn/N}$$ where $N$ is the number of samples, $j$ is the imaginary unit, and $k$ is the frequency index. ## Exercise Determine the DFT of the following signal: $$x[n] = \begin{cases} 1 & \text{if } n = 0 \\ 0 & \text{otherwise} \end{cases}$$ The DFT of the given signal is: $$X(k) = \begin{cases} 1 & \text{if } k = 0 \\ 0 & \text{otherwise} \end{cases}$$ # Introduction to C programming and complex numbers To implement the FFT algorithm in C, it is important to understand the basics of C programming and how to work with complex numbers. C programming is a general-purpose, procedural programming language that is widely used for system programming, embedded systems, and various applications. Complex numbers are mathematical entities that consist of a real part and an imaginary part. In C, the `complex` type is used to represent complex numbers. The `complex` type is defined in the `<complex.h>` header file. To create a complex number in C, you can use the `complex()` function. For example, to create a complex number with a real part of 3 and an imaginary part of 4, you can write: ```c #include <complex.h> complex double z = complex(3, 4); ``` In C, complex numbers can be used in arithmetic operations just like real numbers. For example, you can add two complex numbers using the `+` operator: ```c complex double z1 = complex(1, 2); complex double z2 = complex(3, 4); complex double z3 = z1 + z2; ``` ## Exercise Write a C program that creates two complex numbers and calculates their sum. # Understanding the basics of the FFT algorithm The Fast Fourier Transform (FFT) is an efficient algorithm for computing the DFT of a sequence. The FFT algorithm is based on the divide-and-conquer technique, which reduces the time complexity of the DFT from $O(N^2)$ to $O(N\log N)$. The FFT algorithm can be implemented using recursion or iterative methods. The Cooley-Tukey algorithm is one of the most widely used FFT algorithms, which is based on the radix-2 decimation-in-time (DIT) algorithm. Consider a sequence of length $N = 8$. The FFT of this sequence can be computed using the Cooley-Tukey algorithm by recursively dividing the sequence into smaller subproblems. The Cooley-Tukey algorithm involves the following steps: 1. Divide the input sequence into $N/2$ subproblems. 2. Compute the FFT of each subproblem recursively. 3. Combine the results of the subproblems to obtain the FFT of the input sequence. ## Exercise Explain how the Cooley-Tukey algorithm can be used to compute the FFT of a sequence of length $N = 8$. # Implementing the FFT algorithm in C To implement the FFT algorithm in C, you can use the `fftw3` library, which provides functions for computing the FFT of real and complex sequences. To compute the FFT of a sequence using the `fftw3` library, you can follow these steps: 1. Include the `fftw3.h` header file. 2. Allocate memory for the input and output sequences. 3. Initialize the `fftw_plan` structure, which contains the FFT plan. 4. Execute the FFT using the `fftw_execute()` function. 5. Free the memory allocated for the input and output sequences. Here is an example of a C program that computes the FFT of a sequence using the `fftw3` library: ```c #include <complex.h> #include <fftw3.h> int main() { int N = 8; double input[N] = {1, 0, 1, 0, 1, 0, 1, 0}; complex double output[N]; fftw_plan plan = fftw_plan_dft_r2c_1d(N, input, output, FFTW_ESTIMATE); fftw_execute(plan); fftw_destroy_plan(plan); return 0; } ``` ## Exercise Write a C program that computes the FFT of a sequence using the `fftw3` library. # Optimization techniques for the FFT algorithm There are several optimization techniques that can be used to improve the performance of the FFT algorithm. Some of these techniques include: - In-place FFT: The FFT algorithm can be modified to use in-place computations, which reduces the memory overhead. - Radix-2 FFT: The FFT algorithm can be optimized for sequences with lengths that are powers of 2. - Parallel FFT: The FFT algorithm can be parallelized using multicore processors or GPUs to improve the performance. The in-place FFT algorithm can be used to compute the FFT of a sequence using the `fftw3` library. The in-place FFT algorithm uses a smaller amount of memory compared to the out-of-place FFT algorithm. Here is an example of a C program that computes the in-place FFT of a sequence using the `fftw3` library: ```c #include <complex.h> #include <fftw3.h> int main() { int N = 8; complex double input[N] = {1, 0, 1, 0, 1, 0, 1, 0}; fftw_plan plan = fftw_plan_dft_1d(N, input, input, FFTW_FORWARD, FFTW_ESTIMATE); fftw_execute(plan); fftw_destroy_plan(plan); return 0; } ``` ## Exercise Write a C program that computes the in-place FFT of a sequence using the `fftw3` library. # Applications of the FFT algorithm in various fields The FFT algorithm finds applications in various fields, such as: - Signal processing: The FFT is widely used for analyzing and processing signals in communication systems, audio processing, and image processing. - Image processing: The FFT is used for image compression, filtering, and feature extraction. - Data compression: The FFT is used for lossless data compression algorithms, such as JPEG and MP3. - Cryptography: The FFT is used in cryptographic algorithms, such as RSA and elliptic curve cryptography. The FFT is used in the JPEG image compression algorithm to transform the image data from the spatial domain to the frequency domain. ## Exercise Explain how the FFT is used in the JPEG image compression algorithm. # Comparing the FFT algorithm with other algorithms for computing the Discrete Fourier Transform The FFT algorithm is not the only algorithm for computing the DFT. Other algorithms include: - Direct computation: The DFT can be computed directly using the DFT formula. - Decimation-in-frequency (DIF) algorithm: The DFT can be computed using the DIF algorithm, which involves the decimation of the input sequence. - Fast cosine transform (FCT): The FCT is a fast algorithm for computing the DFT of real-valued sequences. The decimation-in-frequency (DIF) algorithm can be used to compute the DFT of a sequence using the `fftw3` library. Here is an example of a C program that computes the DFT of a sequence using the DIF algorithm and the `fftw3` library: ```c #include <complex.h> #include <fftw3.h> int main() { int N = 8; double input[N] = {1, 0, 1, 0, 1, 0, 1, 0}; complex double output[N]; fftw_plan plan = fftw_plan_dft_r2c_1d(N, input, output, FFTW_ESTIMATE); fftw_execute(plan); fftw_destroy_plan(plan); return 0; } ``` ## Exercise Write a C program that computes the DFT of a sequence using the DIF algorithm and the `fftw3` library. # Real-world examples and case studies of implementing the FFT algorithm in C The FFT algorithm has been widely used in various real-world applications. Some examples include: - Audio processing: The FFT is used for audio compression, filtering, and feature extraction. - Image processing: The FFT is used for image compression, filtering, and feature extraction. - Communication systems: The FFT is used for signal analysis and modulation. - Cryptography: The FFT is used in cryptographic algorithms, such as RSA and elliptic curve cryptography. The FFT algorithm has been used in the development of the JPEG image compression standard, which is widely used for compressing digital images. ## Exercise Discuss the role of the FFT algorithm in the development of the JPEG image compression standard. # Troubleshooting common issues in the implementation of the FFT algorithm When implementing the FFT algorithm in C, you may encounter common issues, such as: - Memory allocation issues: Ensure that you allocate enough memory for the input and output sequences. - Incorrect usage of the `fftw3` library functions: Check that you are using the correct functions and parameters from the `fftw3` library. - Incorrect computation of the DFT: Verify that you are using the correct formula for the DFT. To troubleshoot memory allocation issues, you can use the `malloc()` function to allocate memory for the input and output sequences. Here is an example of a C program that computes the FFT of a sequence using the `fftw3` library: ```c #include <complex.h> #include <fftw3.h> #include <stdlib.h> int main() { int N = 8; double input = (double )malloc(N * sizeof(double)); complex double output = (complex double )malloc(N * sizeof(complex double)); fftw_plan plan = fftw_plan_dft_r2c_1d(N, input, output, FFTW_ESTIMATE); fftw_execute(plan); fftw_destroy_plan(plan); free(input); free(output); return 0; } ``` ## Exercise Write a C program that computes the FFT of a sequence using the `fftw3` library and allocates memory using the `malloc()` function. # Conclusion and further resources In this textbook, we have covered the fundamentals of the FFT algorithm and its implementation in C programming. The FFT algorithm is a powerful tool for analyzing and transforming discrete-time signals. For further learning, you can explore the following resources: - The `fftw3` library documentation: https://www.fftw.org/doc/ - The "Numerical Recipes in C" book by William H. Press, Saul A. Teukolsky, William T. Vetterling, and Brian P. Flannery - The "Understanding Digital Signal Processing" book by Richard G. Lyons These resources provide in-depth explanations and practical examples of the FFT algorithm and its applications in various fields.	Textbooks
Current Search: Research Repository (x) » Department of English (x) (26,451 - 26,500 of 26,500) Youth Participation in Qualitative Research: Challenges and Possibilities. Schelbe, Lisa, Chanmugam, Amy, Moses, Tally, Saltzburg, Susan, Williams, Lela Rankin, Letendre, Joan Research often excludes youth participants, omitting their social and psychological realities, undermining their rights to participate and benefit from research, and weakening the validity of research. Researchers may be discouraged from including youth due to logistical (e.g. gaining access) or ethical (e.g. coercion risks based on developmental level) concerns. Increased discussion is needed around appropriate methods to use with child and youth participants that manage challenges related... Show moreResearch often excludes youth participants, omitting their social and psychological realities, undermining their rights to participate and benefit from research, and weakening the validity of research. Researchers may be discouraged from including youth due to logistical (e.g. gaining access) or ethical (e.g. coercion risks based on developmental level) concerns. Increased discussion is needed around appropriate methods to use with child and youth participants that manage challenges related to developmental capacities, legal status, power differentials, and unpredictable aspects of qualitative research. This paper pools experiences of six researchers, describing solutions we have developed in studies employing varied qualitative methodologies with varied vulnerable youth sub-populations. We detail successful approaches to access, compensation, consent, assent, and confidentiality. Social work researchers are well suited to navigate the challenges, and we share our examples with the aim of facilitating increased youth participation in research. FSU_libsubv1_scholarship_submission_1505837771_741d9110, 10.1177/1473325014556792 YOUTHFUL OFFENDER'S AGGRESSIVE AND AUTONOMIC REACTIONS TO STRESS AS A FUNCTION OF RACE OF EXAMINER AND RACE OF SUBJECT. PERRY, AUBREY MACDONNELL., The Florida State University The yrast rotational bands of selenium-74 and krypton-77. Gross, Carl J., Florida State University New states in the positive parity yrast bands of $\sp{74}$Se and $\sp{77}$Kr have been observed with the reactions $\sp{52}$Cr($\sp{28}$Si,$\alpha$2p)$\sp{74}$Se and $\sp{52}$Cr($\sp{28}$Si,2pn)$\sp{77}$Kr at 98 MeV. The target consisted of approximately 1 mg/cm$\sp2$ natural chromium (84% $\sp{52}$Cr abundance) evaporated on a thick lead backing., The new states extend the known level scheme of $\sp{74}$Se up to I$\sp{\pi}$ = (22$\sp+$) and most of the transitions in the other previously... Show moreNew states in the positive parity yrast bands of $\sp{74}$Se and $\sp{77}$Kr have been observed with the reactions $\sp{52}$Cr($\sp{28}$Si,$\alpha$2p)$\sp{74}$Se and $\sp{52}$Cr($\sp{28}$Si,2pn)$\sp{77}$Kr at 98 MeV. The target consisted of approximately 1 mg/cm$\sp2$ natural chromium (84% $\sp{52}$Cr abundance) evaporated on a thick lead backing., The new states extend the known level scheme of $\sp{74}$Se up to I$\sp{\pi}$ = (22$\sp+$) and most of the transitions in the other previously reported bands have been seen. For the states I$\sp{\pi}$ $\geq$ 6$\sp+$ the spectrum shows a relatively constant moment of inertia parameter $(\hbar\sp2$/2$\Theta)$ = 27.8 $\pm$ 0.5 keV., Excited positive parity states up to spin (41/2) have been observed in $\sp{77}$Kr. $\Delta$I = 1 transitions have been identified throughout the positive parity band. The energies, mixing ratios and B(M1) transition rates for these transitions alternate in size as the spin increases. A cranked shell model analysis was performed along with Strutinsky-Bogolyubov cranking calculations. The observed decrease in the signature splitting of the $\nu$g$\sb{9/2}$ band has been attributed to a band crossing due to an aligning pair of g$\sb{9/2}$ protons. Prolate quadrupole deformations of $\beta\sb2$ = 0.34 for the ground band and $\beta\sb2$ = 0.26 for the first excited band are predicted. This band crossing is associated with a shape change caused by the polarization effect of aligned quasiparticles. Yukawa Unification in SO(10) Susy Guts. Auto, Daniel M., Baer, Howard, Hunter, Christopher, Reina, Laura, Prosper, Harrison, Piekarewicz, Jorge, Department of Physics, Florida State University Supersymmetric grand unified models based on the SO(10) gauge group are especially attractive in light of recent data on neutrino masses. The simplest SO(10) SUSY GUT models predict unification of third generation Yukawa couplings (t –b – Ƭ) in addition to the usual gauge coupling unification. An assessment of the viability of such Yukawa unified models is presented. For the superpotential Higgs mass parameter μ>0, it is found that unification to less than 1% is possible, but only for GUT... Show moreSupersymmetric grand unified models based on the SO(10) gauge group are especially attractive in light of recent data on neutrino masses. The simplest SO(10) SUSY GUT models predict unification of third generation Yukawa couplings (t –b – Ƭ) in addition to the usual gauge coupling unification. An assessment of the viability of such Yukawa unified models is presented. For the superpotential Higgs mass parameter μ>0, it is found that unification to less than 1% is possible, but only for GUT scale scalar mass parameter m16 ~ 8 – 20 TeV, and small values of gaugino mass m1/2 ≤ 150 GeV. Such models require tha a GUT scale mass splitting exists amongst Higgs scalars with m2Hu < m2Hd. Viable solutions lead to a radiatively generated inverted scalar mass hierarchy, with third generation and Higgs scalars being lighter than other sfermions. These models have a very heavy sfermions, so that unwanted flavor changing and CP violating SUSY processes are suppressed, but may suffer from some fine-tuning requirements. While the generated spectra satisify b → sγ and (g – 2)μ constraints, there exists tension with the dark matter relic density unless m16 ≤ 3TeV. These models offer prospects for SUSY discovery at the Fermilab Tevatron collider via the search for W1Z2 → 3l events, or via gluino pair production. If μ < 0, Yujawa coupling unification to less than 5% can occur for m16 and m 1/2≥ 1 – 2 TeV. Consistency of negative μ Yukawa unified models with b → sγ, (g – 2)μ, and relic density Ωh2 all imply very large values of m1/2 typically greater than about 2.5 TeV, in which case direct dection of sparticles may be a challenge even at the LHC. To address the tension between Yukawa unification and the excess of dark matter that the μ>0 models tend to predict, a couple of possible improvements are surveyed. One solution- lowering the GUT scale mass value of first and second generation scalars, leads to uR and cR squark masses in the 90 – 120 GeV regime, which should be accessible to Fermilab Tavatron experiments. Another possibility is relaxing gaugino mass universality which may solve the relic density problem by having neutralino annihilations via the Z or h resonances, or by having a wino-like LSP. Z-Sum Approach to Loop Integrals. Rottmann, Paulo A., Reina, Laura, Aluffi, Paolo, Berg, Bernd A., Wahl, Horst D., Rikvold, Per Arne, Department of Physics, Florida State University We study the applicability of the Z-Sum approach to multi-loop calculations with massive particles in perturbative quantum field theory. We systematically analyze the case of one-loop scalar integrals, which represent the building blocks of any higher-loop calculation. We focus in particular on triangle one-loop integrals and identify strengths and limitations of the Z-Sum approach, extending our results to the case of one-loop box integrals when appropriate. We conclude with the calculation... Show moreWe study the applicability of the Z-Sum approach to multi-loop calculations with massive particles in perturbative quantum field theory. We systematically analyze the case of one-loop scalar integrals, which represent the building blocks of any higher-loop calculation. We focus in particular on triangle one-loop integrals and identify strengths and limitations of the Z-Sum approach, extending our results to the case of one-loop box integrals when appropriate. We conclude with the calculation of a specific physical example: the calculation of heavy flavor corrections to the renormalized scattering amplitude for deep inelastic scattering. Zeta regularized products and modular constants. Heydari, Shahryar., Florida State University The purpose of this dissertation is to, first outline a theory of Zeta regularized products which will work for sequences of complex numbers, and second to use this theory to compute Zeta regularized products and modular constants for sequences which are integer combinations of a fixed set of complex numbers., The gamma function $\Gamma(z)$ is represented as the ratio of two Zeta regularized products. This relation is then extended to define multiple gamma functions as the ratio of two... Show moreThe purpose of this dissertation is to, first outline a theory of Zeta regularized products which will work for sequences of complex numbers, and second to use this theory to compute Zeta regularized products and modular constants for sequences which are integer combinations of a fixed set of complex numbers., The gamma function $\Gamma(z)$ is represented as the ratio of two Zeta regularized products. This relation is then extended to define multiple gamma functions as the ratio of two corresponding Zeta regularized products. A full account of the functional equations associated with multiple gamma functions is also given. The double gamma function is investigated in detail., Some other special functions are also discussed. Namely Jacobi's theta function $\theta\sb1$, the Weierstrass sigma function $\sigma(z),$ and $P(z\vert\tau)$ defined by, The determinant of the Laplacian on an n-dimensional flat Torus is computed for $n \geq$ 2, by computing Zika virus and neural developmental defects: building a case for a cause. Ogden, Sarah C., Hammack, Christy, Tang, Hengli FSU_libsubv1_wos_000375885300013, 10.1007/s11427-016-5053-2 Zika virus directly infects peripheral neurons and induces cell death. Oh, Yohan, Zhang, Feiran, Wang, Yaqing, Lee, Emily M, Choi, In Young, Lim, Hotae, Mirakhori, Fahimeh, Li, Ronghua, Huang, Luoxiu, Xu, Tianlei, Wu, Hao, Li, Cui, Qin, Cheng-Feng,... Show moreOh, Yohan, Zhang, Feiran, Wang, Yaqing, Lee, Emily M, Choi, In Young, Lim, Hotae, Mirakhori, Fahimeh, Li, Ronghua, Huang, Luoxiu, Xu, Tianlei, Wu, Hao, Li, Cui, Qin, Cheng-Feng, Wen, Zhexing, Wu, Qing-Feng, Tang, Hengli, Xu, Zhiheng, Jin, Peng, Song, Hongjun, Ming, Guo-Li, Lee, Gabsang Zika virus (ZIKV) infection is associated with neurological disorders of both the CNS and peripheral nervous systems (PNS), yet few studies have directly examined PNS infection. Here we show that intraperitoneally or intraventricularly injected ZIKV in the mouse can infect and impact peripheral neurons in vivo. Moreover, ZIKV productively infects stem-cell-derived human neural crest cells and peripheral neurons in vitro, leading to increased cell death, transcriptional dysregulation and cell... Show moreZika virus (ZIKV) infection is associated with neurological disorders of both the CNS and peripheral nervous systems (PNS), yet few studies have directly examined PNS infection. Here we show that intraperitoneally or intraventricularly injected ZIKV in the mouse can infect and impact peripheral neurons in vivo. Moreover, ZIKV productively infects stem-cell-derived human neural crest cells and peripheral neurons in vitro, leading to increased cell death, transcriptional dysregulation and cell-type-specific molecular pathology. FSU_pmch_28758997, 10.1038/nn.4612, PMC5575960, 28758997, 28758997, nn.4612 Zika Virus Infection Induces DNA Damage Response and S-Phase Arrest in Human Cortical Neural Progenitors. Hammack, Christy, Tang, Hengli, Megraw, Timothy L., Chadwick, Brian P., Gilbert, David M., Li, Yan, Zhu, Fanxiu, Florida State University, College of Arts and Sciences,... Show moreHammack, Christy, Tang, Hengli, Megraw, Timothy L., Chadwick, Brian P., Gilbert, David M., Li, Yan, Zhu, Fanxiu, Florida State University, College of Arts and Sciences, Department of Biological Science Zika virus (ZIKV) is a re-emerging mosquito-borne flavivirus of significant public health concern closely related to other highly pathogenic flaviviruses, such as dengue virus (DENV) and West Nile virus (WNV). With the rise of ZIKV in Brazil in 2015, its potential link to microcephaly and other severe neurological birth defects prompted the World Health Organization to declare ZIKV a Public Health Emergency of International Concern. Since this time, numerous studies have provided ample... Show moreZika virus (ZIKV) is a re-emerging mosquito-borne flavivirus of significant public health concern closely related to other highly pathogenic flaviviruses, such as dengue virus (DENV) and West Nile virus (WNV). With the rise of ZIKV in Brazil in 2015, its potential link to microcephaly and other severe neurological birth defects prompted the World Health Organization to declare ZIKV a Public Health Emergency of International Concern. Since this time, numerous studies have provided ample evidence to establish ZIKV as the causative agent of microcephaly, yet the molecular mechanisms underlying these neurodevelopmental defects are not well understood. We therefore establish a tractable experimental model system to investigate the impact of ZIKV on human neural development. We demonstrate that ZIKV efficiently infects human cortical neural progenitor cells (hNPCs) derived from induced pluripotent stem cells, but less efficiently infects other cells along the neural differentiation pathway, including immature cortical neurons. Infected hNPCs further release infectious ZIKV particles. Importantly, ZIKV infection disrupts cell cycle progression and induces cell death in hNPCs contributing to their attenuated growth. Global transcriptome analyses of ZIKV-infected hNPCs reveal transcriptional dysregulation, notably a downregulation of cell-cycle-related genes, highlighting the potential involvement of cell cycle pathways in ZIKV biology. We then study the molecular mechanisms by which ZIKV manipulates the cell cycle in hNPCs and the functional consequences of cell-cycle perturbation on the replication of ZIKV and related flaviviruses. We demonstrate that host cell-cycle disruption is unique to ZIKV among the flaviviruses tested, including DENV and WNV, however similar among the two strains of ZIKV tested, including the prototype Uganda strain and a Puerto Rican strain. ZIKV, but not DENV, infection induces DNA double-strand breaks, triggering the DNA damage response through the ATM/Chk2 signaling pathway, while suppressing activation of the ATR/Chk1 signaling pathway in hNPCs. Furthermore, ZIKV infection impedes the progression of cells through S phase thereby preventing the completion of host DNA replication. Recapitulating the S-phase arrest state with S-phase inhibitors leads to an increase in ZIKV replication, but not of WNV or DENV replication. Together, our results identify hNPCs as a direct target of ZIKV and the damaging impact of ZIKV on the growth of hNPCs. Importantly, our data demonstrate ZIKV's ability to induce host DNA damage and arrest cell cycle progression, which results in a cellular environment favorable for its replication. As hNPCs generate the cortical neurons during early fetal brain development, the ZIKV-mediated growth retardation likely contributes to the neurodevelopmental defects of the congenital Zika syndrome. 2018_Sp_Hammack_fsu_0071E_14286 Zika Virus Infects Human Cortical Neural Progenitors and Attenuates Their Growth. Tang, Hengli, Hammack, Christy, Ogden, Sarah C, Wen, Zhexing, Qian, Xuyu, Li, Yujing, Yao, Bing, Shin, Jaehoon, Zhang, Feiran, Lee, Emily M, Christian, Kimberly M, Didier, Ruth... Show moreTang, Hengli, Hammack, Christy, Ogden, Sarah C, Wen, Zhexing, Qian, Xuyu, Li, Yujing, Yao, Bing, Shin, Jaehoon, Zhang, Feiran, Lee, Emily M, Christian, Kimberly M, Didier, Ruth A, Jin, Peng, Song, Hongjun, Ming, Guo-Li The suspected link between infection by Zika virus (ZIKV), a re-emerging flavivirus, and microcephaly is an urgent global health concern. The direct target cells of ZIKV in the developing human fetus are not clear. Here we show that a strain of the ZIKV, MR766, serially passaged in monkey and mosquito cells efficiently infects human neural progenitor cells (hNPCs) derived from induced pluripotent stem cells. Infected hNPCs further release infectious ZIKV particles. Importantly, ZIKV infection... Show moreThe suspected link between infection by Zika virus (ZIKV), a re-emerging flavivirus, and microcephaly is an urgent global health concern. The direct target cells of ZIKV in the developing human fetus are not clear. Here we show that a strain of the ZIKV, MR766, serially passaged in monkey and mosquito cells efficiently infects human neural progenitor cells (hNPCs) derived from induced pluripotent stem cells. Infected hNPCs further release infectious ZIKV particles. Importantly, ZIKV infection increases cell death and dysregulates cell-cycle progression, resulting in attenuated hNPC growth. Global gene expression analysis of infected hNPCs reveals transcriptional dysregulation, notably of cell-cycle-related pathways. Our results identify hNPCs as a direct ZIKV target. In addition, we establish a tractable experimental model system to investigate the impact and mechanism of ZIKV on human brain development and provide a platform to screen therapeutic compounds. FSU_pmch_26952870, 10.1016/j.stem.2016.02.016, PMC5299540, 26952870, 26952870, S1934-5909(16)00106-5 Zika-Virus-Encoded NS2A Disrupts Mammalian Cortical Neurogenesis by Degrading Adherens Junction Proteins. Yoon, Ki-Jun, Song, Guang, Qian, Xuyu, Pan, Jianbo, Xu, Dan, Rho, Hee-Sool, Kim, Nam-Shik, Habela, Christa, Zheng, Lily, Jacob, Fadi, Zhang, Feiran, Lee, Emily M, Huang, Wei-Kai... Show moreYoon, Ki-Jun, Song, Guang, Qian, Xuyu, Pan, Jianbo, Xu, Dan, Rho, Hee-Sool, Kim, Nam-Shik, Habela, Christa, Zheng, Lily, Jacob, Fadi, Zhang, Feiran, Lee, Emily M, Huang, Wei-Kai, Ringeling, Francisca Rojas, Vissers, Caroline, Li, Cui, Yuan, Ling, Kang, Koeun, Kim, Sunghan, Yeo, Junghoon, Cheng, Yichen, Liu, Sheng, Wen, Zhexing, Qin, Cheng-Feng, Wu, Qingfeng, Christian, Kimberly M, Tang, Hengli, Jin, Peng, Xu, Zhiheng, Qian, Jiang, Zhu, Heng, Song, Hongjun, Ming, Guo-Li Zika virus (ZIKV) directly infects neural progenitors and impairs their proliferation. How ZIKV interacts with the host molecular machinery to impact neurogenesis in vivo is not well understood. Here, by systematically introducing individual proteins encoded by ZIKV into the embryonic mouse cortex, we show that expression of ZIKV-NS2A, but not Dengue virus (DENV)-NS2A, leads to reduced proliferation and premature differentiation of radial glial cells and aberrant positioning of newborn... Show moreZika virus (ZIKV) directly infects neural progenitors and impairs their proliferation. How ZIKV interacts with the host molecular machinery to impact neurogenesis in vivo is not well understood. Here, by systematically introducing individual proteins encoded by ZIKV into the embryonic mouse cortex, we show that expression of ZIKV-NS2A, but not Dengue virus (DENV)-NS2A, leads to reduced proliferation and premature differentiation of radial glial cells and aberrant positioning of newborn neurons. Mechanistically, in vitro mapping of protein-interactomes and biochemical analysis suggest interactions between ZIKA-NS2A and multiple adherens junction complex (AJ) components. Functionally, ZIKV-NS2A, but not DENV-NS2A, destabilizes the AJ complex, resulting in impaired AJ formation and aberrant radial glial fiber scaffolding in the embryonic mouse cortex. Similarly, ZIKA-NS2A, but not DENV-NS2A, reduces radial glial cell proliferation and causes AJ deficits in human forebrain organoids. Together, our results reveal pathogenic mechanisms underlying ZIKV infection in the developing mammalian brain. FSU_pmch_28826723, 10.1016/j.stem.2017.07.014, PMC5600197, 28826723, 28826723, S1934-5909(17)30293-X ZIKV AND DENV DIFFERENTIALLY PERTURB ISG15 DESPITE SIMILAR RESTRICTION BY INTERFERON. Sherman, Allaura Zinc and carbonic anhydrase in oysters. Nielsen, Stephen Ashley Zinc and neurogenesis: making new neurons from development to adulthood.. Levenson, Cathy W, Morris, Deborah Stem cell proliferation, neuronal differentiation, cell survival, and migration in the central nervous system are all important steps in the normal process of neurogenesis. These mechanisms are highly active during gestational and early neonatal brain development. Additionally, in select regions of the brain, stem cells give rise to new neurons throughout the human lifespan. Recent work has revealed key roles for the essential trace element zinc in the control of both developmental and adult... Show moreStem cell proliferation, neuronal differentiation, cell survival, and migration in the central nervous system are all important steps in the normal process of neurogenesis. These mechanisms are highly active during gestational and early neonatal brain development. Additionally, in select regions of the brain, stem cells give rise to new neurons throughout the human lifespan. Recent work has revealed key roles for the essential trace element zinc in the control of both developmental and adult neurogenesis. Given the prevalence of zinc deficiency, these findings have implications for brain development, cognition, and the regulation of mood. FSU_pmch_22332038, 10.3945/an.110.000174, PMC3065768, 22332038, 22332038, 000174 Zinc Deficiency Impairs Retinoic Acid-Induced Differentiation of Human Neurons. Gower-Winter, Shannon Dooies, Levenson, Cathy W., Ilich-Ernst, Jasminka, Eckel, Lisa, Department of Nutrition, Food, and Exercise Science, Florida State University Neurogenesis is the process of stem cell proliferation, survival, and differentiation. Recent research has confirmed the presence of ongoing neurogenesis throughout life in humans. This fact has led to vast interest in the mechanisms that underlie this process. Manipulation of adult neurogenesis has the potential to enhance the treatment of a multitude of neurodegenerative diseases including Alzheimer's disease, Parkinson's disease, and depression as well as injury and stroke. Previous work... Show moreNeurogenesis is the process of stem cell proliferation, survival, and differentiation. Recent research has confirmed the presence of ongoing neurogenesis throughout life in humans. This fact has led to vast interest in the mechanisms that underlie this process. Manipulation of adult neurogenesis has the potential to enhance the treatment of a multitude of neurodegenerative diseases including Alzheimer's disease, Parkinson's disease, and depression as well as injury and stroke. Previous work has shown that the essential trace metal zinc regulates neuronal precursor proliferation and survival. Thus, this work is based on the central hypothesis that zinc is also needed for neuronal differentiation. Furthermore we proposed that transforming growth factor signaling may be involved in the zinc regulated mechanisms of differentiation. Zinc deficiency (ZD; 0.4µM) impaired the ability of neuronal precursor cells (NT2) to differentiate into mature neurons (NT2-N) when exposed to 2 wks of 10µM retinoic acid (RA), as measured by the early neuronal marker TuJ1. Additionally, we demonstrated a differential regulation of Transforming Growth Factor Beta (TGF-β) receptor isoforms type I (RI) and II (RII) under zinc deficient (0.4µM) conditions in NT2 cells undergoing RA-induced differentiation. Measurements of TGF-β RI and RII in zinc adequate (ZA; 2.5µM) differentiated NT2-N neurons showed that neither receptor isoform was expressed in these cells. TGF-β RI was up-regulated in NT2-N cells in response to ZD (0.4µM) however, while TGF-β RII remained down-regulated under ZD (0.4µM) conditions, as demonstrated via TGF-β RI and RII immunocytochemistry. These data confirmed that ZD (0.4µM) does impair RA-induced differentiation of human NT2 neuronal cells. There is also evidence that a differential regulation of the TGF-β receptor I and II isoforms may be involved in this mechanism, as the loss of RII expression in ZD (0.4µM) NT2-N cells could be responsible for a decline in TGF-β signaling in these cells and thus an attenuated cellular response to TGF-β responsive genes. This research suggests an important role for TGF-β and the trace metal zinc in regulating neuronal differentiation, and helps to improve understanding of adult neurogenesis in the human brain. Zinc in the central nervous system: From molecules to behavior.. Gower-Winter, Shannon D, Levenson, Cathy W The trace metal zinc is a biofactor that plays essential roles in the central nervous system across the lifespan from early neonatal brain development through the maintenance of brain function in adults. At the molecular level, zinc regulates gene expression through transcription factor activity and is responsible for the activity of dozens of key enzymes in neuronal metabolism. At the cellular level, zinc is a modulator of synaptic activity and neuronal plasticity in both development and... Show moreThe trace metal zinc is a biofactor that plays essential roles in the central nervous system across the lifespan from early neonatal brain development through the maintenance of brain function in adults. At the molecular level, zinc regulates gene expression through transcription factor activity and is responsible for the activity of dozens of key enzymes in neuronal metabolism. At the cellular level, zinc is a modulator of synaptic activity and neuronal plasticity in both development and adulthood. Given these key roles, it is not surprising that alterations in brain zinc status have been implicated in a wide array of neurological disorders including impaired brain development, neurodegenerative disorders such as Alzheimer's disease, and mood disorders including depression. Zinc has also been implicated in neuronal damage associated with traumatic brain injury, stroke, and seizure. Understanding the mechanisms that control brain zinc homeostasis is thus critical to the development of preventive and treatment strategies for these and other neurological disorders. FSU_pmch_22473811, 10.1002/biof.1012, PMC3757551, 22473811, 22473811 Zinc Regulation of Bone Marrow-Derived Mesenchymal Stem Cell Neuronal Differentiation. Faye, Sari, Department of Chemistry and Biochemistry The multipotent ability of mesenchymal stem cells (MSC) to differentiate into a large variety of mature cell types gives them a high potential for use in a variety of therapeutic purposes. Recently, it was discovered that bone marrow derived MSC could be induced to take on a neuronal phenotype through the addition of cobalt chloride (CoCl2) to the growth media. It is also well known that the trace element zinc is vital for both neuronal proliferation and differentiation from neuronal... Show moreThe multipotent ability of mesenchymal stem cells (MSC) to differentiate into a large variety of mature cell types gives them a high potential for use in a variety of therapeutic purposes. Recently, it was discovered that bone marrow derived MSC could be induced to take on a neuronal phenotype through the addition of cobalt chloride (CoCl2) to the growth media. It is also well known that the trace element zinc is vital for both neuronal proliferation and differentiation from neuronal precursor cells. Thus, this work tested the hypothesis that zinc plays a role in the differentiation of MSC into neurons. Secondly, because zinc is unable to enter or exit cells without the assistance of zinc transport proteins (ZnT), this work tested the hypothesis that two transport proteins, ZnT-1 and ZnT-4, would be regulated both by zinc and by treatment with cobalt. This work used both cell morphology and markers of neuronal differentiation (TuJ1 and neuronal specific enolase) to show that zinc deficiency (ZD) combined with CoCl2 treatment appeared to induce differentiation of rat MSC. Furthermore, the zinc transporters were differentially regulated such that ZnT-4 was increased on the cell membrane by zinc deficiency, while ZnT-1 levels at the membrane were highest in the combined zinc deficiency-cobalt treatment group. These data implicate zinc in the mechanisms associated with MSC function. Zinc Regulation of Mesenchymal Stem Cell Proliferation and Survival. Hagler, Shaye, Department of Chemistry and Biochemistry Mesenchymal stem cells (MSC) have a wide variety of promising clinical applications including the treatment of brain disorders and injury, cardiovascular disease, and cancer. To fully exploit their potential, we need a better understanding of the cellular and molecular mechanisms that govern stem cell division and survival. We have hypothesized that the essential trace element zinc regulates the proliferation and survival of rat and human bone marrow-derived MSC. Proliferation of MSC is... Show moreMesenchymal stem cells (MSC) have a wide variety of promising clinical applications including the treatment of brain disorders and injury, cardiovascular disease, and cancer. To fully exploit their potential, we need a better understanding of the cellular and molecular mechanisms that govern stem cell division and survival. We have hypothesized that the essential trace element zinc regulates the proliferation and survival of rat and human bone marrow-derived MSC. Proliferation of MSC is impaired by zinc deficiency. For example, after 48h of zinc deficiency, proliferation was reduced by 50% (p Zinc Regulation of Neural Stem Cell Proliferation and Antidepressant Efficacy. Mullin, Tatyana, Levenson, Cathy W., Ilich-Ernst, Jasminka, Hurt, Myra, Department of Nutrition, Food, and Exercise Science, Florida State University Changes in zinc homeostasis are strongly associated with abnormal brain function and a variety of neurological and neuropsychiatric disorders, including depression. It is hypothesized that the neurogenic potential of chronic antidepressant administration contributes to its therapeutic effects in depression. Thus, the goal of this work was to determine the extent to which zinc is needed for antidepressant drug induction of neural stem cell proliferation and differentiation. Human NTERA-2/D1 ... Show moreChanges in zinc homeostasis are strongly associated with abnormal brain function and a variety of neurological and neuropsychiatric disorders, including depression. It is hypothesized that the neurogenic potential of chronic antidepressant administration contributes to its therapeutic effects in depression. Thus, the goal of this work was to determine the extent to which zinc is needed for antidepressant drug induction of neural stem cell proliferation and differentiation. Human NTERA-2/D1 (NT2) cell culture, an established in vitro model system to study neuronal development, was utilized. Zinc deficiency impaired NT2 cell proliferation measured by the number of Ki67-positive cells. Treatment with fluoxetine or lithium did not result in a significant increase in cell proliferation rate. However, six-day treatment with these antidepressants had a stimulatory effect on NT2 cell differentiation revealed by immunofluorecent detection of the neuron-specific marker TuJ1. Furthermore, zinc deficient cultures treated with fluoxetine or lithium appeared to have a decreased expression of this neuronal marker. Taken together, these results suggest that the essential trace element zinc is needed for neuronal stem cell proliferation and differentiation. Zinc Regulation of Neural Stem Cells and Behavior in Brain Injury Complicated by Ethanol Intake. Morris, Deborah R., Levenson, Cathy W., Zhu, Lei, Olcese, James, Zhou, Yi, Florida State University, College of Medicine, Department of Biological Science In addition to the known behavioral and cognitive impairments, including memory deficits, depression, and anxiety associated with traumatic brain injury (TBI), there is an increased risk for new onset heavy weekly drinking, binge drinking, and alcohol-related problems. Our previously published work has shown that zinc supplementation reduced TBI-associated deficits, particularly the depression-like symptom anhedonia and stress-induced anxiety. Our objective was to examine the behavioral and... Show moreIn addition to the known behavioral and cognitive impairments, including memory deficits, depression, and anxiety associated with traumatic brain injury (TBI), there is an increased risk for new onset heavy weekly drinking, binge drinking, and alcohol-related problems. Our previously published work has shown that zinc supplementation reduced TBI-associated deficits, particularly the depression-like symptom anhedonia and stress-induced anxiety. Our objective was to examine the behavioral and cellular outcomes associated with TBI that are complicated by ethanol consumption, as well as the effect of zinc supplementation on these outcomes. Adult male rats were fed a zinc supplemented (180 ppm) or zinc adequate (30 ppm) diet for 4 weeks followed by a moderate TBI using to the medial frontal cortex produced by controlled cortical impact. After injury, rats were given 3 g/kg of ethanol daily for 7 days via gavage. Ethanol intake exacerbated TBI-induced anxiety-like and depression-like behaviors as well inducing recognition memory impairments. Furthermore, zinc supplementation was unable to reduce these behavioral deficits when injury was accompanied by ethanol intake. While ethanol did not worsen learning and memory, zinc supplementation also did not improve Morris water maze performance in ethanol-treated animals. Evidence in the literature has demonstrated that both brain injury and ethanol can regulate neurogenesis. We then wanted to examine the extent to which changes in stem cells are responsible for our behavioral observations. TBI produced a trend towards increased hippocampal stem cells, and zinc supplementation with injury resulted in a significant increase in stem cells 8 days post-injury. Ethanol did not appear to impair TBI or zinc supplemented induced proliferation. There was a small trend towards a decrease in differentiation of these labelled proliferating stem cells with ethanol and TBI combined. Finally, the molecular mechanisms responsible for the role of zinc in neuronal precursor cells and neuronal differentiation were examined. Zinc regulation of transcriptional activity during retinoic acid-induced neuronal differentiation. Morris, Deborah R, Levenson, Cathy W Zinc deficiency impairs the proliferation and differentiation of stem cells in the central nervous system that participate in neurogenesis. To examine the molecular mechanisms responsible for the role of this essential nutrient in neuronal precursor cells and neuronal differentiation, we identified zinc-dependent changes in the DNA-binding activity of zinc finger proteins and other transcription factors in proliferating human Ntera-2 neuronal precursor cells undergoing retinoic acid... Show moreZinc deficiency impairs the proliferation and differentiation of stem cells in the central nervous system that participate in neurogenesis. To examine the molecular mechanisms responsible for the role of this essential nutrient in neuronal precursor cells and neuronal differentiation, we identified zinc-dependent changes in the DNA-binding activity of zinc finger proteins and other transcription factors in proliferating human Ntera-2 neuronal precursor cells undergoing retinoic acid-stimulated differentiation into a neuronal phenotype. We found that zinc deficiency altered binding activity of 28 transcription factors including retinoid X receptor (RXR) known to participate in neuronal differentiation. Alterations in zinc finger transcription factor activity were not simply the result of removal of zinc from these proteins during zinc deficiency, as the activity of other zinc-binding transcription factors such as the glucocorticoid receptor was increased by as much as twofold over zinc-adequate conditions, and nonzinc-binding transcription factors such as nuclear factor-1 and heat shock transcription factor-1 were increased by as much as fourfold over control. Western analysis did not detect significant decreases in total RXR protein abundance in neuronal precursors, suggesting that the decrease in DNA-binding activity was not simply the result of a reduction in RXR levels in neuronal precursor cells. Rather, use of a reporter gene construct containing retinoic acid response elements upstream from a luciferase coding sequence revealed that zinc deficiency results in decreased transcriptional activity of RXR and reductions in retinoic acid-mediated gene transcription during neuronal differentiation. These results show that zinc deficiency has implications for both developmental and adult neurogenesis. FSU_pmch_24029070, 10.1016/j.jnutbio.2013.06.002, PMC3832953, 24029070, 24029070, S0955-2863(13)00126-5 Zinc supplementation provides behavioral resiliency in a rat model of traumatic brain injury. Cope, Elise C, Morris, Deborah R, Scrimgeour, Angus G, VanLandingham, Jacob W, Levenson, Cathy W Depression, anxiety, and impairments in learning and memory are all associated with traumatic brain injury (TBI). Because of the strong link between zinc deficiency, depression, and anxiety, in both humans and rodent models, we hypothesized that dietary zinc supplementation prior to injury could provide behavioral resiliency to lessen the severity of these outcomes after TBI. Rats were fed a marginal zinc deficient (5 ppm), zinc adequate (30 ppm), or zinc supplemented (180 ppm) diet for 4... Show moreDepression, anxiety, and impairments in learning and memory are all associated with traumatic brain injury (TBI). Because of the strong link between zinc deficiency, depression, and anxiety, in both humans and rodent models, we hypothesized that dietary zinc supplementation prior to injury could provide behavioral resiliency to lessen the severity of these outcomes after TBI. Rats were fed a marginal zinc deficient (5 ppm), zinc adequate (30 ppm), or zinc supplemented (180 ppm) diet for 4 weeks followed by a moderately-severe TBI using the well-established model of controlled cortical impact (CCI). Following CCI, rats displayed depression-like behaviors as measured by the 2-bottle saccharin preference test for anhedonia. Injury also resulted in evidence of stress and impairments in Morris water maze (MWM) performance compared to sham-injured controls. While moderate zinc deficiency did not worsen outcomes following TBI, rats that were fed the zinc supplemented diet for 4 weeks showed significantly attenuated increases in adrenal weight (p<0.05) as well as reduced depression-like behaviors (p<0.001). Supplementation prior to injury improved resilience such that there was not only significant improvements in cognitive behavior compared to injured rats fed an adequate diet (p<0.01), there were no significant differences between supplemented and sham-operated rats in MWM performance at any point in the 10-day trial. These data suggest a role for supplemental zinc in preventing cognitive and behavioral deficits associated with TBI. FSU_pmch_21699908, 10.1016/j.physbeh.2011.06.007, PMC3506179, 21699908, 21699908, S0031-9384(11)00322-2 Zintl and Intermetallic Phases Grown from Calcium/Lithium Flux. Blankenship, Trevor, Latturner, Susan, Locke, Bruce R., Stiegman, Albert E., Alabugin, Igor V., Florida State University, College of Arts and Sciences, Department of Chemistry... Show moreBlankenship, Trevor, Latturner, Susan, Locke, Bruce R., Stiegman, Albert E., Alabugin, Igor V., Florida State University, College of Arts and Sciences, Department of Chemistry and Biochemistry Metal flux synthes is a useful alternative method to high temperature solid state synthesis; it allows easy diffusion of reactants at lower temperatures, and presents favorable conditions for crystal growth. A mixed flux of calcium and lithium in a 1:1 ratio was explored in this work; this mixture melts at 300°C and is an excellent solvent for main group elements and CaH₂. Reactions of p-block elements in a 1:1 Ca/Li flux have produced several new intermetallic and Zintl phases.... Show moreMetal flux synthes is a useful alternative method to high temperature solid state synthesis; it allows easy diffusion of reactants at lower temperatures, and presents favorable conditions for crystal growth. A mixed flux of calcium and lithium in a 1:1 ratio was explored in this work; this mixture melts at 300°C and is an excellent solvent for main group elements and CaH₂. Reactions of p-block elements in a 1:1 Ca/Li flux have produced several new intermetallic and Zintl phases. Electronegative elements from groups 14 and 15 are reduced to anions in this flux, yielding charge-balanced products. More electropositive metals from group 13 are not fully reduced; the resulting products are complex intermetallics. The reactions of tin or lead and carbon in Ca/Li flux produced the analogous phases Ca₁₁Tt₃C8 (Tt = Sn, Pb) in the monoclinic C21/c space group (a = 13.2117(8) Å, b =10.7029(7) Å, c = 14.2493(9) Å, β = 105.650(1)° for the Sn analog). These compounds are carbide Zintl phases that includes the rare combination of C₃⁴ and C₂² units as well as Sn⁴ or Pb⁴ anions. Ca/Li flux reactions of CaH2 and arsenic have produced the Zintl phases LiCa₃As₂H in orthorhombic Pnma (a = 11.4064(7), b = 4.2702(3), c = 11.8762(8) Å), and Ca13As6C0.46N1.155H6.045in tetragonal P4/mbm (a = 15.7493(15), c = 9.1062(9) Å). The complex stoichiometry of the latter phase was caused by incorporation of light element contaminants and was studied by neutron diffraction, showing mixing of anionic sites to achieve charge balance. Ca/Li flux reactions with group 13 metals have resulted in several new intermetallic phases. Reactions of indium and CaH₂ in the Ca/Li flux (with or without boron) formed Ca₅₃In₁₃B₄₋ₓH₂₃(2.4 < x < 4.0) in cubic space group Im-3 (a = 16.3608(6) Å) which features metallic indium atoms and ionic hydride sites. The electronic properties of this "subhydride" were confirmed by ¹H and ¹¹⁵In NMR spectroscopy. Attempts to replace boron with carbon yielded Ca₁₂InC₁₃₋ₓ, (Im-3, a = 9.6055(8)Å) which contains C34- units. A very similar phase, Ba12InC18H4 (Im-3,a = 11.1415(8) Å), was grown from the reaction of indium, carbon, and LiH in Ba/Li flux. This compound also includes C₃⁴ units. Preliminary Ca/Li flux reactions of aluminum with other main group elements have produced several new phases: a hydride clathrate Ca₃₁Al₂H₂₅ in cubic Fd-3m (a=18.0835(15) Å), Ca24Al2(C1-xHx)N2H16 in tetragonal P42/nmc (a=15.9069(12) Å, c=13.7323(10) Å, and Ca4Al2N5 in orthorhombic Pna2₁ (a = 11.2331(1) Å, b=9.0768(8) Å, c=6.0093(5) Å. Zn(II)-coordination modulated ligand photophysical processes – the development of fluorescent indicators for imaging biological Zn(II) ions. Zhu, Lei, Yuan, Zhao, Simmons, J., Sreenath, Kesavapillai Molecular photophysics and metal coordination chemistry are the two fundamental pillars that support the development of fluorescent cation indicators. In this article, we describe how Zn(II)-coordination alters various ligand-centered photophysical processes that are pertinent to developing Zn(II) indicators. The main aim is to show how small organic Zn(II) indicators work under the constraints of specific requirements, including Zn(II) detection range, photophysical requirements such as... Show moreMolecular photophysics and metal coordination chemistry are the two fundamental pillars that support the development of fluorescent cation indicators. In this article, we describe how Zn(II)-coordination alters various ligand-centered photophysical processes that are pertinent to developing Zn(II) indicators. The main aim is to show how small organic Zn(II) indicators work under the constraints of specific requirements, including Zn(II) detection range, photophysical requirements such as excitation energy and emission color, temporal and spatial resolutions in a heterogeneous intracellular environment, and fluorescence response selectivity between similar cations such as Zn(II) and Cd(II). In the last section, the biological questions that fluorescent Zn(II) indicators help to answer are described, which have been motivating and challenging this field of research. FSU_migr_chm_faculty_publications-0016, 10.1039/C4RA00354C Zooarchaeological Analysis of a Multicomponent Shell-Bearing Site in Davidson County, Tennessee. Peres, Tanya M., Deter-Wolf, Aaron, Myers, Gage A. Site 40DV7 is one of several large shell-bearing sites located along the Cumberland River near Nashville which were heavily impacted by catastrophic flooding and looting activity during the spring of 2010. Emergency sampling and ongoing monitoring at 40DV7 since that time have identified deeply-stratified deposits spanning the Archaic through Mississippian periods. These deposits, and particularly the temporally-distinct shell midden components, may help inform our understanding of human... Show moreSite 40DV7 is one of several large shell-bearing sites located along the Cumberland River near Nashville which were heavily impacted by catastrophic flooding and looting activity during the spring of 2010. Emergency sampling and ongoing monitoring at 40DV7 since that time have identified deeply-stratified deposits spanning the Archaic through Mississippian periods. These deposits, and particularly the temporally-distinct shell midden components, may help inform our understanding of human occupation, species interdependence, and environmental change along the Cumberland River over a period of more than 5000 years. Zooarchaeological Analysis of Faunal Remains Recovered from Sands Key #2 (8D2) (SEAC Accession #1930), Biscayne National Park, Miami-Dade County Florida. Peres, Tanya M., McLean, Emily This is a report of the zooarchaeological analysis of faunal remains recovered as part of the excavations by archeologists with the Southeastern Archeological Center of the National Park Service at the Sands Key #2 site (8DA2) (SEAC Acc #1930), located in the Biscayne National Park, Miami-Dade County, Florida. This analysis was performed under the Southern Appalachian Cooperative Ecosystems Studies Unit (SA-CESU) Task Agreement Number (P14AC01652) under Cooperative Agreement Number P14AC00882... Show moreThis is a report of the zooarchaeological analysis of faunal remains recovered as part of the excavations by archeologists with the Southeastern Archeological Center of the National Park Service at the Sands Key #2 site (8DA2) (SEAC Acc #1930), located in the Biscayne National Park, Miami-Dade County, Florida. This analysis was performed under the Southern Appalachian Cooperative Ecosystems Studies Unit (SA-CESU) Task Agreement Number (P14AC01652) under Cooperative Agreement Number P14AC00882 between the United States Department of the Interior - The National Park Service/Southeast Archeological Center and Middle Tennessee State University (PI Tanya M. Peres, September 2014). The Project title is "Documenting Subsistence Strategies in the Southeast Using the National Park Service's Archeological Resources." In October 2015, the remainder of the zooarchaeological analysis and reporting was subcontracted by Middle Tennessee State University to Tanya M. Peres at Florida State University (FSU Project# 037433 \| MTSU Award# 536858S). The Sands Key #2 faunal assemblage reported on here contains 9,812 specimens weighing 12,791.65 g. The data generated from the zooarchaeological analysis is detailed in this report. Preliminary interpretations about the use of aquatic resources by the Tequesta are offered. FSU_libsubv1_scholarship_submission_1532018990_f2e488bf, 10.17125/fsu.1532018990 Zooarchaeological Analysis of Faunal Remains Recovered from Totten Key (8DA3439) (SEAC Accession #2628), Biscayne National Park, Miami-Dade County Florida. This is a report of the zooarchaeological analysis of faunal remains recovered as part of the excavations by archeologists with the Southeastern Archeological Center (SEAC) of the National Park Service at the Totten Key Site (8DA3439) on Totten Key, Miami-Dade County, Florida. This analysis was performed under the Southern Appalachian Cooperative Ecosystems Studies Unit (SA-CESU) Task Agreement Number (P14AC01652) under Cooperative Agreement Number P14AC00882 between the United States... Show moreThis is a report of the zooarchaeological analysis of faunal remains recovered as part of the excavations by archeologists with the Southeastern Archeological Center (SEAC) of the National Park Service at the Totten Key Site (8DA3439) on Totten Key, Miami-Dade County, Florida. This analysis was performed under the Southern Appalachian Cooperative Ecosystems Studies Unit (SA-CESU) Task Agreement Number (P14AC01652) under Cooperative Agreement Number P14AC00882 between the United States Department of the Interior - The National Park Service/Southeast Archeological Center and Middle Tennessee State University (PI Tanya M. Peres, September 2014) (Appendix 1). The Project title is "Documenting Subsistence Strategies in the Southeast Using the National Park Service's Archeological Resources." In October 2015, the remainder of the zooarchaeological analysis and reporting was subcontracted by Middle Tennessee State University to Tanya M. Peres at Florida State University (FSU Project# 037433 \| MTSU Award# 536858S). FSU_libsubv1_scholarship_submission_1532019273_4d277f12, 10.17125/fsu.1532019273 Zooarchaeological Remains from the 1998 Fewkes Site Excavations, Williamson County, Tennessee. Peres, Tanya M. The Fewkes site faunal assemblage, excavated as part of a Phase III data recovery project for the Tennessee Department of Transportation in 1998, was analyzed and evaluated in light of its potential to provide significant information about Middle Mississippian subsistence practices and environmental conditions of the area during the time of occupation. Specific goals of the analysis included: (1) defining the subsistence strategies and practices of the people that inhabited the site; (2)... Show moreThe Fewkes site faunal assemblage, excavated as part of a Phase III data recovery project for the Tennessee Department of Transportation in 1998, was analyzed and evaluated in light of its potential to provide significant information about Middle Mississippian subsistence practices and environmental conditions of the area during the time of occupation. Specific goals of the analysis included: (1) defining the subsistence strategies and practices of the people that inhabited the site; (2) determining the relationship of the site to the surrounding ecological habitats; and (3) determining the seasonality of the site. Additionally, the Fewkes faunal assemblage was compared to animal exploitation practices as outlined for the Cumberland River drainage model of Mississippian period sites. The results of the analysis of selected contexts are presented here. Zora, Color Struck and Weary Blues and Tea with Zora and Marjorie (Three plays about the life of Zora Neale Hurston). Speisman, Barbara Waddell A trilogy of three plays based upon the life of Florida-born author, Zora Neale Hurston, which emphasizes Hurston's unique place in American literary history. The plays, Zora, Color Struck and Weary Blues, and Tea with Zora and Marjorie are based on not only interpretations of Zora's works, letters, and conversations with people who remember her, but also the works and letters of Marjorie Kinnan Rawlings, Carl Van Vechten, Fanny Hurst, and Langston Hughes. The three plays present Hurston... Show moreA trilogy of three plays based upon the life of Florida-born author, Zora Neale Hurston, which emphasizes Hurston's unique place in American literary history. The plays, Zora, Color Struck and Weary Blues, and Tea with Zora and Marjorie are based on not only interpretations of Zora's works, letters, and conversations with people who remember her, but also the works and letters of Marjorie Kinnan Rawlings, Carl Van Vechten, Fanny Hurst, and Langston Hughes. The three plays present Hurston first as a child in Eatonville at the turn of the century, then as a young woman during the Harlem Renaissance, and finally in her full maturity. The structure of Zora and Color Struck and Weary Blues is concentrated on two of the most important days of Zora's life, which are the day of her mother's death when she was about 12 and the night of the Opportunity Award's Banquet which launched the Harlem Renaissance. The structure of Tea with Zora and Marjorie is different from the two previous plays because it relates to the period from 1942 until 1952 in the life of not only Zora Hurston but Marjorie Kinnan Rawlings, another prominent Florida writer. Zwitteration: A Different Approach to Non Stick Surfaces. Estephan, Zaki Georges, Schlenoff, Joseph B., Ma, Teng, Roper, Michael, Strouse, Geoffrey, Ramakrishnan, Subramanian, Department of Chemistry and Biochemistry, Florida State... Show moreEstephan, Zaki Georges, Schlenoff, Joseph B., Ma, Teng, Roper, Michael, Strouse, Geoffrey, Ramakrishnan, Subramanian, Department of Chemistry and Biochemistry, Florida State University Limiting undesired interactions of proteins with surfaces is a vital task for implementation of many technologies that require direct exposure to protein media. This includes sensors, single molecule spectroscopy studies, and nanoparticles that would act as vehicles for therapeutic agents or diagnostic agents. Current technology relies on the resistive properties of poly(ethylene glycol), PEG, to protein adsorption. PEG has been therefore the subject of thorough studies to decipher the... Show moreLimiting undesired interactions of proteins with surfaces is a vital task for implementation of many technologies that require direct exposure to protein media. This includes sensors, single molecule spectroscopy studies, and nanoparticles that would act as vehicles for therapeutic agents or diagnostic agents. Current technology relies on the resistive properties of poly(ethylene glycol), PEG, to protein adsorption. PEG has been therefore the subject of thorough studies to decipher the mechanism involved in protein resistivity. The latter has been mainly attributed either to chain mobility, that would suffer from entropic penalty upon protein adsorption, or due to a hydration layer that prevents close encounter of proteins to the surface. Regardless of the mechanism, PEG has been reported to suffer from performance degradation in biological media due to oxidation, and its properties have been reported to differ with temperature. Given their biocompatibility, zwitterions have been proposed as a viable alternative mimicking the cell membrane. Polymeric zwitterions, the most commonly studied alternatives, result in an increase in the hydrodynamic size of particles upon grafting to surfaces. Control over size is essential as it controls the distribution of particles in the body. This work attempts to provide a different approach to nanoparticle stabilization against different aggregating factors to alleviate some of the above mentioned shortcomings of PEG and other polymers. A monomeric zwitterion siloxane was synthesized. The zwitterion siloxane covalently bonds to the oxide surface of nanoparticles without significantly changing their hydrodynamic size. The "zwitterated" particles remain stable even when challenged with high salt solutions or incubated with serum; two factors that are known to induce aggregation. The efficacy of the zwitterionic coating was compared head-to-head with a PEG coating for its ability to prevent protein adsorption to silica nanoparticles. The same siloxane coupling chemistry is employed to yield surfaces with similar coverages of both types of ligand on two geometrically different surfaces (nanoparticlesversusplanar). While both types of surface modification are highly effective in preventing protein adsorption and nanoparticle aggregation, the zwitterion provided monolayer-type coverage with minimal thickness whereas the PEG appeared to yield a more three-dimensional coating. A mechanism is proposed to explain the resistive properties of passivating ligands such as PEG and other neutral surfaces. The role of the passivating ligand is broken down to ion-coupled and ion-decoupled processes. The ion-decoupled process minimizes intermolecular interactions, whereas the ion-coupled mechanism prevents ion pairing between protein and surface charges which releases counterions and water molecules, an entropic driving force enough to overcome a disfavored enthalpy of adsorption. Finally, the synthesis of zwitterated iron oxide nanoparticles by co-precipitation of iron salts in presence of zwitterion siloxane as the stabilizing ligand is reported. This procedure yields superparamagnetic maghemite nanoparticles whose polydispersity varies as a function of the amount of zwitterion siloxane present during synthesis. The latter has the effect of changing the effective hydrodynamic radius of the particles from 5.4 nm to 35 nm. The presence of zwitterions on the surface is validated with thermogravimetric analysis and Diffuse Reflectance Infrared Fourier Transform. Magnetization versus applied field data shows the absence of coercive field and low magnetization values attributed to the decreasing particle size as well as the diamagnetic coating. The particles are tested for their possible use as MRI contrast agents. The calculated relaxation rates are low indicating that a high concentration of iron is needed for good contrast. Introduction of amine functionality for incorporation of targeting agents is achieved by the addition of aminopropyltriethoxysilane post-synthesis. The presence of the latter is verified by fluorescence spectroscopy. Zymancer. Barron, Justin, Wingate, Mark, Kubik, Ladislav, Spencer, Peter, College of Music, Florida State University Zymancer is an approximately 13 minute work for what is essentially a chamber orchestra, but with just one player to a part. Though it is written and performed as a single movement, there are three main formal sections of the piece that could be considered movements. These three sections are distinct from each other in tempi, meter, harmony, and mood. There is, however, a return of material from the first section at the end of the third. Zymosan Fungal Infection Induces Necleosome Distributions During the Innate Immune Response on a Time Dependent Manner. Gruder, Olivia, Department of Biological Sciences Chromatin structure plays a critical role in the regulation of the human genome. An understanding of the role of chromatin structure and its relationship to gene regulation is critical to developing new strategies to prevent and treat diseases. We chose to investigate the anti-inflammatory response of human macrophage-like cell line (THP1) to Zymosan, in order to elucidate the regulation of chromatin. Zymosan is a component of the fungal cell wall that induces an innate immune response. After... Show moreChromatin structure plays a critical role in the regulation of the human genome. An understanding of the role of chromatin structure and its relationship to gene regulation is critical to developing new strategies to prevent and treat diseases. We chose to investigate the anti-inflammatory response of human macrophage-like cell line (THP1) to Zymosan, in order to elucidate the regulation of chromatin. Zymosan is a component of the fungal cell wall that induces an innate immune response. After THP1 were treated with Zymosan, we hypothesized that the fungal infection would initiate an inflammatory response by altering nucleosome redistribution and/or altering chromatin structure in a time dependent manner. Based on previous results that showed rapid, widespread, transient changes in nucleosome distribution in the innate immune response, we chose to look at multiple time points at high temporal resolution: 0 (control), 20', 40', 60', 80', 100', 2h, 3h, 4h and 12h. We measured nucleosome distribution at each of these time points at hundreds of genes transcription start sites involved in the immune response. We saw the greatest changes in nucleosome positioning from 20 to 60 minutes, and it appeared that these changes were transient since they reverted back to their original after the 60-minute time point. These results support our prediction that all cells have the same nucleosome distributions during their resting states, but can be altered with the addition of an insult. In response to a stimulus, a biochemical "yawn" occurs to provide accessibility to genes needed to provide a response. The data indicates that widespread but transient changes occur to the entire genome upon response to an environmental stimulus. Zymosan Fungal Infection Induces Nucleosome Redistributions During the Innate Immune Response. Gruder, Olivia, Dennis, Jonathan, Department of Biological Science Chromatin structure plays a critical role in the regulation of the human genome. An understanding of the role of chromatin structure and its relationship to gene regulation is critical to developing new strategies to prevent and treat diseases. We chose to investigate the anti-inflammatory response of human macrophage like cell line (THP1) to Zymosan, in order to elucidate the regulation of chromatin. Zymosan is a component the fungal cell wall that induces an innate immune response. After... Show moreChromatin structure plays a critical role in the regulation of the human genome. An understanding of the role of chromatin structure and its relationship to gene regulation is critical to developing new strategies to prevent and treat diseases. We chose to investigate the anti-inflammatory response of human macrophage like cell line (THP1) to Zymosan, in order to elucidate the regulation of chromatin. Zymosan is a component the fungal cell wall that induces an innate immune response. After THP1 were treated with zymosan, we hypothesized that the fungal infection would initiate an inflammatory response by altering nucleosome redistribution and/or altering chromatin structure in a time dependent manner. Based on previous results that showed rapid, widespread, transient changes in nucleosome distribution in the innate immune response, we chose to look at multiple time points at high temporal resolution: 0 (control), 20', 40', 60', 80', 100', 2h, 3h, 4h and 12h. We measured nucleosome distribution at each of these time points at hundreds of genes transcription start sites involved in the immune response. nucleosome distribution changes in the innate immune response to fungal infection. ¡Casinando!: Identity, Meaning, and the Kinesthetic Language of Cuban Casino Dancing. Martinez, Brian, Gunderson, Frank, Bakan, Michael, Brewer, Charles, College of Music, Florida State University A genre of Cuban music known as timba and a genre of Cuban social dance known as casino have often been mistakenly categorized as styles of salsa music and dance. Because of this association, along with political relations between the United States and Cuba, these genres have been marginalized in favor of mainstream salsa. In this thesis, I argue that casino and timba must be understood as distinct genres from an historical perspective. Additionally, I examine casino from a linguistic... Show moreA genre of Cuban music known as timba and a genre of Cuban social dance known as casino have often been mistakenly categorized as styles of salsa music and dance. Because of this association, along with political relations between the United States and Cuba, these genres have been marginalized in favor of mainstream salsa. In this thesis, I argue that casino and timba must be understood as distinct genres from an historical perspective. Additionally, I examine casino from a linguistic perspective and apply principles of linguistic relativity to create a linguistic analogy for social partner dance. By understanding casino and timba as separate from the international salsa phenomenon, they can be studied and appreciated as the unique cultural forms that they truly are. ¡Guerra Al Metate!: The Visuality of Foodways in Postrevolutionary Mexico City (1920 1960). Wolff, Lesley Anne, Carrasco, Michael, Herrera, Robinson A., Niell, Paul B., Bearor, Karen A., Florida State University, College of Fine Arts, Department of Art History This dissertation considers foodways as a vital symbolic and material force in the arts of Mexico's volatile postrevolutionary reconstruction (1920 – 1960). Although Mexican food history has stood at the forefront of a growing food studies movement, the field has been slow to appropriate image-based methodologies. Likewise, art history has been hesitant to embrace the historical performativity and materiality of foodways. This project thus seeks to fill a gap at the margins of food studies... Show moreThis dissertation considers foodways as a vital symbolic and material force in the arts of Mexico's volatile postrevolutionary reconstruction (1920 – 1960). Although Mexican food history has stood at the forefront of a growing food studies movement, the field has been slow to appropriate image-based methodologies. Likewise, art history has been hesitant to embrace the historical performativity and materiality of foodways. This project thus seeks to fill a gap at the margins of food studies and art history, particularly at the nexus of indigeneity and urbanization. The dissertation traces the shifting relationships between art and food during a period of rampant modernization, in which the rise of modern cookery through electrical appliances and industrial foodstuffs converged and clashed with the nation's growing nostalgia for its pre-Columbian heritage. The book focuses on three case studies of artistic production and alimentary consumption—Tina Modotti and pulque, Carlos E. González and mole poblano, and Rufino Tamayo and watermelon—that highlight the various ways in which visual renderings of food were used to frame indigenous culture as both the foundation of and a threat to the modern state. Each case study engages the convergence of racial imaginaries, artistic production, and foodways to show how conflictive attitudes toward indigenous heritage and bodies were made manifest through images of food and foodways. Therefore, this project demonstrates how seemingly innocuous images of foodstuffs and consumption became implicated in a broader visual, experiential, and commercial battle over the definition of nationalist attitudes toward indigeneity. The manuscript consists of five chapters and an appendix. Chapter 1, "Introduction," surveys Mexican food and art histories and establishes my intersectional framework. Chapter 2, "Nursing the Nation: Pulque and the Indigenous Body in Tina Modotti's Baby Nursing," argues that Tina Modotti's celebrated photograph Baby Nursing (1926) invokes the problematic consumption of pulque, an indigenous fermented beverage, as a metonym for nationalist ideologies that simultaneously celebrate and rebuke indigenous lifeways. Chapter 3, "The 'Spirit of Mexico': Consuming Heritage in Café de Tacuba," demonstrates how an iconic but previously unstudied painting depicting the mythic invention of mole poblano, commissioned for Mexico City's famous Café de Tacuba (1946), negotiates modern consumption by evoking colonial production. Chapter 4, "Mister Watermelon/Señor Sandía: Fruitful Anxieties in the Work of Rufino Tamayo," argues that Rufino Tamayo's still life mural Naturaleza muerta (1954), commissioned for the Sanborns department store café, mediated the state's aggressive removal of fruteros [informal fruit vendors] by acting as both an icon of Anglophone modernity and a visual celebration of Mexican tropicalia. Chapter 5, "The Colonial in the Contemporary: On the State of Mexican Gastronomy," presents the book's conclusions while engaging in a critique of Mexico's contemporary gastronomic movement and its reliance upon colonial aesthetics to veil Mexico City's socio-economic fragmentation. The Appendix catalogues recipes for pulque, mole poblano, and watermelon-based dishes, all of which have been compiled from nineteenth- and twentieth-century cookbooks and manuscripts. 2018_Su_Wolff_fsu_0071E_14737 βTRCP: Linking Circadian Rhythms and Metabolism. Sweeney, Megan C., Department of Biomedical Sciences Shifts in circadian rhythms, like in shift work or jet lag, have been shown to increase the risk of many metabolic disorders. Therefore, it is not surprising that many genes involved in the circadian clock mechanism have demonstrated a regulatory role in metabolism. It has been shown that E3 ubiquitin ligases can influence metabolism as well. In initial studies, my lab created a knockout of two E3 ubiquitin ligases thought to be essential to the clock, βTRCP1/2, in a mouse model in order to... Show moreShifts in circadian rhythms, like in shift work or jet lag, have been shown to increase the risk of many metabolic disorders. Therefore, it is not surprising that many genes involved in the circadian clock mechanism have demonstrated a regulatory role in metabolism. It has been shown that E3 ubiquitin ligases can influence metabolism as well. In initial studies, my lab created a knockout of two E3 ubiquitin ligases thought to be essential to the clock, βTRCP1/2, in a mouse model in order to study the proteasomal degradation machinery in mammals. Upon characterizing the circadian phenotype of this mouse, we noticed an unprecedented, metabolic phenotype after deletion of these vital ligases. These novel mutant mice lose over 30% of their body weight within 5 days while still maintaining an eating and drinking regime similar to wild-type mice. In this project, in vivo and sequence analysis studies aimed to look further into the causes of this phenomenon and the molecular mechanisms underlying them. Γ-Ray Spectroscopic Study of Calcium-48,49 and Scandium-50 Focusing on Low Lying Octupole Vibration Excitations. McPherson, David M. (David Marc), Cottle, Paul D. (Paul Davis), Kercheval, Alec N., Cao, Jianming, Piekarewicz, Jorge, Riley, Mark A., Florida State University, College of Arts... Show moreMcPherson, David M. (David Marc), Cottle, Paul D. (Paul Davis), Kercheval, Alec N., Cao, Jianming, Piekarewicz, Jorge, Riley, Mark A., Florida State University, College of Arts and Sciences, Department of Physics An inverse kinematic proton scattering experiment was performed at the National Superconducting Cyclotron Laboratory (NSCL) using the GRETINA-S800 detector system in conjunction with the Ursinus College liquid hydrogen target. $\gamma$-ray yields from the experiment were determined using geant4 simulations, generating state population cross sections. These cross sections were used to extract the delta_3 deformation length for the low-lying octupole vibration excitations in Ca-48,49 using the... Show moreAn inverse kinematic proton scattering experiment was performed at the National Superconducting Cyclotron Laboratory (NSCL) using the GRETINA-S800 detector system in conjunction with the Ursinus College liquid hydrogen target. $\gamma$-ray yields from the experiment were determined using geant4 simulations, generating state population cross sections. These cross sections were used to extract the delta_3 deformation length for the low-lying octupole vibration excitations in Ca-48,49 using the coupled channels analysis code fresco. Particle-core coupling in Ca-49 was studied in comparison to Ca-48 through determination of the neutron and proton deformation lengths. The total inverse kinematic proton scattering deformation lengths were evaluated for the low-lying octupole vibration excitations in Ca-48,49 to be delta_3(Ca-48, 3^-_1) = 1.0(2)fm, delta_3(Ca-49, 9/2^+_1) = 1.2(1)fm, delta_3 (Ca-49, 9/2^+_1) = 1.5(2)fm, delta_3(Ca-49, 5/2^+_1) = 1.1(1)fm. Proton and neutron deformation lengths for two of these octupole states were also determined to be delta_p(Ca-48, 3^-_1) = 0.9(1)fm, delta_p (Ca-49, 9/2^+_1) = 1.0(1)fm, delta_n(Ca-48, 3^-_1) = 1.1(3)fm, and delta_n(Ca-49, 9/2^+_1) = 1.3(3)fm. Additionally, the ratios of the neutron to proton transition matrix elements were also determined for these two states to be M_n/M_p(Ca-48, 3^-_1) = 1.7(6) and M_n/M_p(Ca-49, 9/2^+_1) = 2.0(5). Statistically, the derived values for these two nuclei are nearly identical. δ/ω-Plectoxin-Pt1a: An Excitatory Spider Toxin with Actions on both Ca(2+) and Na(+) Channels. Zhou, Yi, Zhao, Mingli, Fields, Gregg B., Wu, Chun-Fang, Branton, W. FSU_migr_biomed_faculty_publications-0044 ΛC Semileptonic Decays in a Quark Model. Hussain, Md Mozammel, Roberts, Winston, Goldsby, Kenneth A,, Volya, Alexander, Crede, Volker, Owens, Joseph F., Florida State University, College of Arts and Sciences,... Show moreHussain, Md Mozammel, Roberts, Winston, Goldsby, Kenneth A,, Volya, Alexander, Crede, Volker, Owens, Joseph F., Florida State University, College of Arts and Sciences, Department of Physics Hadronic form factors for semileptonic decay of the Λ[subscript c] are calculated in a nonrelativistic quark model. The full quark model wave functions are employed to numerically calculate the form factors to all orders in (1/m[subscript c], 1/m[subscript s]). The form factors satisfy relationships expected from the heavy quark effective theory (HQET) form factors. No other semileptonic decays of Λ[subscript c] has been reported other than the decay to the ground state Λ that implies f = B(Λ... Show moreHadronic form factors for semileptonic decay of the Λ[subscript c] are calculated in a nonrelativistic quark model. The full quark model wave functions are employed to numerically calculate the form factors to all orders in (1/m[subscript c], 1/m[subscript s]). The form factors satisfy relationships expected from the heavy quark effective theory (HQET) form factors. No other semileptonic decays of Λ[subscript c] has been reported other than the decay to the ground state Λ that implies f = B(Λ[subscript c]⁺ → Λl⁺ν[subscript l])/B(Λ[subscript c]⁺ → X[subscript s]l⁺ν[subscript l]) = 1. In this work, the differential decay rates and branching fractions are calculated for transitions to the ground state and a number of excited states of Λ. The branching fraction of the semileptonic decay width to the total width of Λ[subscript c] has been calculated and compared with other theoretical estimates and experimental results. The branching fractions for Λ[subscript c] → Λl⁺ν[subscript l] → Σπl⁺ν[subscript l] and Λ[subscript c] → Λl⁺ν[subscript l] → NǨl⁺ν[subscript l] are also calculated. Apart from decays to the ground state Λ(1115), it is found that decays through the Λ(1405) provide a significant portion of the branching fraction Λ[subscript c] → X[subscript s]lν[subscript l]. There are various conjectures on the structure of the Λ(1405) while we treated it as a three quark state. A new estimate for f = B(Λ[subscript c]⁺ → Λl⁺ν[subscript l] is obtained. FSU_2017SP_Hussain_fsu_0071E_13920 π Berry phase and Zeeman splitting of Weyl semimetal TaP. Hu, J, Liu, J Y, Graf, D, Radmanesh, S M A, Adams, D J, Chuang, A, Wang, Y, Chiorescu, I, Wei, J, Spinu, L, Mao, Z Q The recent breakthrough in the discovery of Weyl fermions in monopnictide semimetals provides opportunities to explore the exotic properties of relativistic fermions in condensed matter. The chiral anomaly-induced negative magnetoresistance and π Berry phase are two fundamental transport properties associated with the topological characteristics of Weyl semimetals. Since monopnictide semimetals are multiple-band systems, resolving clear Berry phase for each Fermi pocket remains a challenge.... Show moreThe recent breakthrough in the discovery of Weyl fermions in monopnictide semimetals provides opportunities to explore the exotic properties of relativistic fermions in condensed matter. The chiral anomaly-induced negative magnetoresistance and π Berry phase are two fundamental transport properties associated with the topological characteristics of Weyl semimetals. Since monopnictide semimetals are multiple-band systems, resolving clear Berry phase for each Fermi pocket remains a challenge. Here we report the determination of Berry phases of multiple Fermi pockets of Weyl semimetal TaP through high field quantum transport measurements. We show our TaP single crystal has the signatures of a Weyl state, including light effective quasiparticle masses, ultrahigh carrier mobility, as well as negative longitudinal magnetoresistance. Furthermore, we have generalized the Lifshitz-Kosevich formula for multiple-band Shubnikov-de Haas (SdH) oscillations and extracted the Berry phases of π for multiple Fermi pockets in TaP through the direct fits of the modified LK formula to the SdH oscillations. In high fields, we also probed signatures of Zeeman splitting, from which the Landé g-factor is extracted. Φ-Value Analysis of Symfoil-4T. Sutherland, Mason A., Department of Biological Science A critical consideration in the process of de novo protein architecture design and protein evolution is the folding pathway and behavior a protein undertakes in transitioning to its functional tertiary structure. Of particular interest is a cryptic element within protein primary structure that enables an efficient folding pathway, and is postulated to be a heritable element in the evolution of protein architecture, the "folding nucleus" (FN). However, almost nothing is known regarding how the... Show moreA critical consideration in the process of de novo protein architecture design and protein evolution is the folding pathway and behavior a protein undertakes in transitioning to its functional tertiary structure. Of particular interest is a cryptic element within protein primary structure that enables an efficient folding pathway, and is postulated to be a heritable element in the evolution of protein architecture, the "folding nucleus" (FN). However, almost nothing is known regarding how the FN changes as simpler peptide motifs join to form more complex polypeptides. To this effect, the structure and folding properties of foldable intermediates along the evolutionary trajectory of the β-trefoil protein type were tested. This study specifically used and compared data from Symfoil-4T (an engineered β-trefoil protein) to several mutants to show that the FN is acquired during gene fusion events, incorporating novel turn structure generated by gene fusion. Furthermore, the FN of β-trefoils are adjusted by circular permutation in response to destabilizing functional mutations to allow the survival of FN (which is made possible by the intrinsic C3 cyclic symmetry of β-trefoil architecture) identifying a selective advantage that helps explain extant cyclic structural symmetry in the proteome. Picture Superiority Effect on Encoding and Reminding Recent literature has demonstrated that if someone studies one list of word pairs followed by another list of word pairs in which the second half of some of the pairs have changed (e.g., they saw "knee-bone" on the first list and "knee-bend" on the second list), this change can remind them of the pair's presence on the original list and improve memory performance for both of the pairs. Previously, this kind of design would lead researchers to observe interference, which would cause memory... Show moreRecent literature has demonstrated that if someone studies one list of word pairs followed by another list of word pairs in which the second half of some of the pairs have changed (e.g., they saw "knee-bone" on the first list and "knee-bend" on the second list), this change can remind them of the pair's presence on the original list and improve memory performance for both of the pairs. Previously, this kind of design would lead researchers to observe interference, which would cause memory performance to suffer; the discovery of the memory enhancing effects of reminders inspired further research to understand how their underlying mechanisms and what makes them more likely to occur. The presented research examines whether picture-word pairs will trigger more remindings than word-word pairs on the presumption that the picture superiority effect will play an important role. The results indicate that participants did experience more remindings for the picture-word pairs than for the word-word pairs and that better memory performance for picture-word pairs is due to a combination of this effect and an encoding effect. FSU_libsubv1_scholarship_submission_1524707531_55417f4e Liu, H. (212)	CommonCrawl
\begin{definition}[Definition:Lattice Ordering] Let $\struct {S, \preceq}$ be a lattice. Then the ordering $\preceq$ is referred to as a '''lattice ordering'''. \end{definition}	ProofWiki
Automorphisms of the symmetric and alternating groups In group theory, a branch of mathematics, the automorphisms and outer automorphisms of the symmetric groups and alternating groups are both standard examples of these automorphisms, and objects of study in their own right, particularly the exceptional outer automorphism of S6, the symmetric group on 6 elements. Summary $n$ $\operatorname {Aut} (\mathrm {S} _{n})$ $\operatorname {Out} (\mathrm {S} _{n})$ $n\neq 2,6$ $\mathrm {S} _{n}$ $\mathrm {C} _{1}$ $n=2$ $\mathrm {C} _{1}$ $\mathrm {C} _{1}$ $n=6$ $\mathrm {S} _{6}\rtimes \mathrm {C} _{2}$[1] $\mathrm {C} _{2}$ $n$ $\operatorname {Aut} (\mathrm {A} _{n})$ $\operatorname {Out} (\mathrm {A} _{n})$ $n\geq 4,n\neq 6$ $\mathrm {S} _{n}$ $\mathrm {C} _{2}$ $n=1,2$ $\mathrm {C} _{1}$ $\mathrm {C} _{1}$ $n=3$ $\mathrm {C} _{2}$ $\mathrm {C} _{2}$ $n=6$ $\mathrm {S} _{6}\rtimes \mathrm {C} _{2}$ $\mathrm {V} =\mathrm {C} _{2}\times \mathrm {C} _{2}$ Generic case • $n\neq 2,6$: $\operatorname {Aut} (\mathrm {S} _{n})=\mathrm {S} _{n}$, and thus $\operatorname {Out} (\mathrm {S} _{n})=\mathrm {C} _{1}$. Formally, $\mathrm {S} _{n}$ is complete and the natural map $\mathrm {S} _{n}\to \operatorname {Aut} (\mathrm {S} _{n})$ is an isomorphism. • $n\neq 1,2,6$: $\operatorname {Out} (\mathrm {A} _{n})=\mathrm {S} _{n}/\mathrm {A} _{n}=\mathrm {C} _{2}$, and the outer automorphism is conjugation by an odd permutation. • $n\neq 2,3,6$: $\operatorname {Aut} (\mathrm {A} _{n})=\operatorname {Aut} (\mathrm {S} _{n})=\mathrm {S} _{n}$ Indeed, the natural maps $\mathrm {S} _{n}\to \operatorname {Aut} (\mathrm {S} _{n})\to \operatorname {Aut} (\mathrm {A} _{n})$ are isomorphisms. Exceptional cases • $n=1,2$: trivial: $\operatorname {Aut} (\mathrm {S} _{1})=\operatorname {Out} (\mathrm {S} _{1})=\operatorname {Aut} (\mathrm {A} _{1})=\operatorname {Out} (\mathrm {A} _{1})=\mathrm {C} _{1}$ $\operatorname {Aut} (\mathrm {S} _{2})=\operatorname {Out} (\mathrm {S} _{2})=\operatorname {Aut} (\mathrm {A} _{2})=\operatorname {Out} (\mathrm {A} _{2})=\mathrm {C} _{1}$ • $n=3$: $\operatorname {Aut} (\mathrm {A} _{3})=\operatorname {Out} (\mathrm {A} _{3})=\mathrm {S} _{3}/\mathrm {A} _{3}=\mathrm {C} _{2}$ • $n=6$: $\operatorname {Out} (\mathrm {S} _{6})=\mathrm {C} _{2}$, and $\operatorname {Aut} (\mathrm {S} _{6})=\mathrm {S} _{6}\rtimes \mathrm {C} _{2}$ is a semidirect product. • $n=6$: $\operatorname {Out} (\mathrm {A} _{6})=\mathrm {C} _{2}\times \mathrm {C} _{2}$, and $\operatorname {Aut} (\mathrm {A} _{6})=\operatorname {Aut} (\mathrm {S} _{6})=\mathrm {S} _{6}\rtimes \mathrm {C} _{2}.$ The exceptional outer automorphism of S6 Among symmetric groups, only S6 has a non-trivial outer automorphism, which one can call exceptional (in analogy with exceptional Lie algebras) or exotic. In fact, Out(S6) = C2.[2] This was discovered by Otto Hölder in 1895.[2][3] The specific nature of the outer automorphism is as follows: • the sole identity permutation maps to itself; • a 2-cycle such as (1 2) maps to the product of three 2-cycles such as (1 2)(3 4)(5 6) and vice versa, there being 15 permutations each way; • a 3-cycle such as (1 2 3) maps to the product of two 3-cycles such as (1 4 5)(2 6 3) and vice versa, accounting for 40 permutations each way; • a 4-cycle such as (1 2 3 4) maps to another 4-cycle such as (1 6 2 4) accounting for 90 permutations; • the product of two 2-cycles such as (1 2)(3 4) maps to another product of two 2-cycles such as (3 5)(4 6), accounting for 45 permutations; • a 5-cycle such as (1 2 3 4 5) maps to other 5-cycles such as (1 3 6 5 2) accounting for 144 permutations; • the product of a 2-cycle and a 3-cycle such as (1 2 3)(4 5) maps to a 6-cycle such as (1 2 5 3 4 6) and vice versa, accounting for 120 permutations each way; • the product of a 2-cycle and a 4-cycle such as (1 2 3 4)(5 6) maps to another such permutation such as (1 4 2 6)(3 5) accounting for the 90 remaining permutations. Thus, all 720 permutations on 6 elements are accounted for. The outer automorphism does not preserve cycle structure in general, mapping single cycles to the product of two cycles and vice versa. This also yields another outer automorphism of A6, and this is the only exceptional outer automorphism of a finite simple group:[4] for the infinite families of simple groups, there are formulas for the number of outer automorphisms, and the simple group of order 360, thought of as A6, would be expected to have two outer automorphisms, not four. However, when A6 is viewed as PSL(2, 9) the outer automorphism group has the expected order. (For sporadic groups – i.e. those not falling in an infinite family – the notion of exceptional outer automorphism is ill-defined, as there is no general formula.) Construction There are numerous constructions, listed in (Janusz & Rotman 1982). Note that as an outer automorphism, it is a class of automorphisms, well-determined only up to an inner automorphism, hence there is not a natural one to write down. One method is: • Construct an exotic map (embedding) S5 → S6; see below • S6 acts by conjugation on the six conjugates of this subgroup, yielding a map S6 → SX, where X is the set of conjugates. Identifying X with the numbers 1, ..., 6 (which depends on a choice of numbering of the conjugates, i.e., up to an element of S6 (an inner automorphism)) yields an outer automorphism S6 → S6. • This map is an outer automorphism, since a transposition does not map to a transposition, but inner automorphisms preserve cycle structure. Throughout the following, one can work with the multiplication action on cosets or the conjugation action on conjugates. To see that S6 has an outer automorphism, recall that homomorphisms from a group G to a symmetric group Sn are essentially the same as actions of G on a set of n elements, and the subgroup fixing a point is then a subgroup of index at most n in G. Conversely if we have a subgroup of index n in G, the action on the cosets gives a transitive action of G on n points, and therefore a homomorphism to Sn. Construction from graph partitions Before the more mathematically rigorous constructions, it helps to understand a simple construction. Take a complete graph with 6 vertices, K6. It has 15 edges, which can be partitioned into perfect matchings in 15 different ways, each perfect matching being a set of three edges no two of which share a vertex. It is possible to find a set of 5 perfect matchings from the set of 15 such that no two matchings share an edge, and that between them include all 5 × 3 = 15 edges of the graph; this graph factorization can be done in 6 different ways. Consider a permutation of the 6 vertices, and see its effect on the 6 different factorizations. We get a map from 720 input permutations to 720 output permutations. That map is precisely the outer automorphism of S6. Being an automorphism, the map must preserve the order of elements, but it does not preserve cycle structure. For instance, a 2-cycle maps to a product of three 2-cycles; it is easy to see that a 2-cycle affects all of the 6 graph factorizations in some way, and hence has no fixed points when viewed as a permutation of factorizations. The fact that it is possible to construct this automorphism at all relies on a large number of numerical coincidences which apply only to n = 6. Exotic map S5 → S6 There is a subgroup (indeed, 6 conjugate subgroups) of S6 which is abstractly isomorphic to S5, but which acts transitively as subgroups of S6 on a set of 6 elements. (The image of the obvious map Sn → Sn+1 fixes an element and thus is not transitive.) Sylow 5-subgroups Janusz and Rotman construct it thus: • S5 acts transitively by conjugation on the set of its 6 Sylow 5-subgroups, yielding an embedding S5 → S6 as a transitive subgroup of order 120. This follows from inspection of 5-cycles: each 5-cycle generates a group of order 5 (thus a Sylow subgroup), there are 5!/5 = 120/5 = 24 5-cycles, yielding 6 subgroups (as each subgroup also includes the identity), and Sn acts transitively by conjugation on the set of cycles of a given class, hence transitively by conjugation on these subgroups. Alternately, one could use the Sylow theorems, which state generally that all Sylow p-subgroups are conjugate. PGL(2,5) The projective linear group of dimension two over the finite field with five elements, PGL(2, 5), acts on the projective line over the field with five elements, P1(F5), which has six elements. Further, this action is faithful and 3-transitive, as is always the case for the action of the projective linear group on the projective line. This yields a map PGL(2, 5) → S6 as a transitive subgroup. Identifying PGL(2, 5) with S5 and the projective special linear group PSL(2, 5) with A5 yields the desired exotic maps S5 → S6 and A5 → A6.[5] Following the same philosophy, one can realize the outer automorphism as the following two inequivalent actions of S6 on a set with six elements:[6] • the usual action as a permutation group; • the six inequivalent structures of an abstract 6-element set as the projective line P1(F5) – the line has 6 points, and the projective linear group acts 3-transitively, so fixing 3 of the points, there are 3! = 6 different ways to arrange the remaining 3 points, which yields the desired alternative action. Frobenius group Another way: To construct an outer automorphism of S6, we need to construct an "unusual" subgroup of index 6 in S6, in other words one that is not one of the six obvious S5 subgroups fixing a point (which just correspond to inner automorphisms of S6). The Frobenius group of affine transformations of F5 (maps x $\mapsto $ ax + b where a ≠ 0) has order 20 = (5 − 1) · 5 and acts on the field with 5 elements, hence is a subgroup of S5. (Indeed, it is the normalizer of a Sylow 5-group mentioned above, thought of as the order-5 group of translations of F5.) S5 acts transitively on the coset space, which is a set of 120/20 = 6 elements (or by conjugation, which yields the action above). Other constructions Ernst Witt found a copy of Aut(S6) in the Mathieu group M12 (a subgroup T isomorphic to S6 and an element σ that normalizes T and acts by outer automorphism). Similarly to S6 acting on a set of 6 elements in 2 different ways (having an outer automorphism), M12 acts on a set of 12 elements in 2 different ways (has an outer automorphism), though since M12 is itself exceptional, one does not consider this outer automorphism to be exceptional itself. The full automorphism group of A6 appears naturally as a maximal subgroup of the Mathieu group M12 in 2 ways, as either a subgroup fixing a division of the 12 points into a pair of 6-element sets, or as a subgroup fixing a subset of 2 points. Another way to see that S6 has a nontrivial outer automorphism is to use the fact that A6 is isomorphic to PSL2(9), whose automorphism group is the projective semilinear group PΓL2(9), in which PSL2(9) is of index 4, yielding an outer automorphism group of order 4. The most visual way to see this automorphism is to give an interpretation via algebraic geometry over finite fields, as follows. Consider the action of S6 on affine 6-space over the field k with 3 elements. This action preserves several things: the hyperplane H on which the coordinates sum to 0, the line L in H where all coordinates coincide, and the quadratic form q given by the sum of the squares of all 6 coordinates. The restriction of q to H has defect line L, so there is an induced quadratic form Q on the 4-dimensional H/L that one checks is non-degenerate and non-split. The zero scheme of Q in H/L defines a smooth quadric surface X in the associated projective 3-space over k. Over an algebraic closure of k, X is a product of two projective lines, so by a descent argument X is the Weil restriction to k of the projective line over a quadratic étale algebra K. Since Q is not split over k, an auxiliary argument with special orthogonal groups over k forces K to be a field (rather than a product of two copies of k). The natural S6-action on everything in sight defines a map from S6 to the k-automorphism group of X, which is the semi-direct product G of PGL2(K) = PGL2(9) against the Galois involution. This map carries the simple group A6 nontrivially into (hence onto) the subgroup PSL2(9) of index 4 in the semi-direct product G, so S6 is thereby identified as an index-2 subgroup of G (namely, the subgroup of G generated by PSL2(9) and the Galois involution). Conjugation by any element of G outside of S6 defines the nontrivial outer automorphism of S6. Structure of outer automorphism On cycles, it exchanges permutations of type (12) with (12)(34)(56) (class 21 with class 23), and of type (123) with (145)(263) (class 31 with class 32). The outer automorphism also exchanges permutations of type (12)(345) with (123456) (class 2131 with class 61). For each of the other cycle types in S6, the outer automorphism fixes the class of permutations of the cycle type. On A6, it interchanges the 3-cycles (like (123)) with elements of class 32 (like (123)(456)). No other outer automorphisms To see that none of the other symmetric groups have outer automorphisms, it is easiest to proceed in two steps: 1. First, show that any automorphism that preserves the conjugacy class of transpositions is an inner automorphism. (This also shows that the outer automorphism of S6 is unique; see below.) Note that an automorphism must send each conjugacy class (characterized by the cyclic structure that its elements share) to a (possibly different) conjugacy class. 2. Second, show that every automorphism (other than the above for S6) stabilizes the class of transpositions. The latter can be shown in two ways: • For every symmetric group other than S6, there is no other conjugacy class consisting of elements of order 2 that has the same number of elements as the class of transpositions. • Or as follows: Each permutation of order two (called an involution) is a product of k > 0 disjoint transpositions, so that it has cyclic structure 2k1n−2k. What is special about the class of transpositions (k = 1)? If one forms the product of two distinct transpositions τ1 and τ2, then one always obtains either a 3-cycle or a permutation of type 221n−4, so the order of the produced element is either 2 or 3. On the other hand, if one forms the product of two distinct involutions σ1, σ2 of type k > 1, then provided n ≥ 7, it is always possible to produce an element of order 6, 7 or 4, as follows. We can arrange that the product contains either • two 2-cycles and a 3-cycle (for k = 2 and n ≥ 7) • a 7-cycle (for k = 3 and n ≥ 7) • two 4-cycles (for k = 4 and n ≥ 8) For k ≥ 5, adjoin to the permutations σ1, σ2 of the last example redundant 2-cycles that cancel each other, and we still get two 4-cycles. Now we arrive at a contradiction, because if the class of transpositions is sent via the automorphism f to a class of involutions that has k > 1, then there exist two transpositions τ1, τ2 such that f(τ1) f(τ2) has order 6, 7 or 4, but we know that τ1τ2 has order 2 or 3. No other outer automorphisms of S6 S6 has exactly one (class) of outer automorphisms: Out(S6) = C2. To see this, observe that there are only two conjugacy classes of S6 of size 15: the transpositions and those of class 23. Each element of Aut(S6) either preserves each of these conjugacy classes, or exchanges them. Any representative of the outer automorphism constructed above exchanges the conjugacy classes, whereas an index 2 subgroup stabilizes the transpositions. But an automorphism that stabilizes the transpositions is inner, so the inner automorphisms form an index 2 subgroup of Aut(S6), so Out(S6) = C2. More pithily: an automorphism that stabilizes transpositions is inner, and there are only two conjugacy classes of order 15 (transpositions and triple transpositions), hence the outer automorphism group is at most order 2. Small n Symmetric For n = 2, S2 = C2 = Z/2 and the automorphism group is trivial (obviously, but more formally because Aut(Z/2) = GL(1, Z/2) = Z/2* = C1). The inner automorphism group is thus also trivial (also because S2 is abelian). Alternating For n = 1 and 2, A1 = A2 = C1 is trivial, so the automorphism group is also trivial. For n = 3, A3 = C3 = Z/3 is abelian (and cyclic): the automorphism group is GL(1, Z/3*) = C2, and the inner automorphism group is trivial (because it is abelian). Notes 1. Janusz & Rotman 1982. 2. Lam, T. Y., & Leep, D. B. (1993). "Combinatorial structure on the automorphism group of S6". Expositiones Mathematicae, 11(4), 289–308. 3. Otto Hölder (1895), "Bildung zusammengesetzter Gruppen", Mathematische Annalen, 46, 321–422. 4. ATLAS p. xvi 5. Carnahan, Scott (2007-10-27), "Small finite sets", Secret Blogging Seminar, notes on a talk by Jean-Pierre Serre.{{citation}}: CS1 maint: postscript (link) 6. Snyder, Noah (2007-10-28), "The Outer Automorphism of S6", Secret Blogging Seminar References • https://web.archive.org/web/20071227060045/http://polyomino.f2s.com/david/haskell/outers6.html • Some Thoughts on the Number 6, by John Baez: relates outer automorphism to the regular icosahedron • "12 points in PG(3, 5) with 95040 self-transformations" in "The Beauty of Geometry", by Coxeter: discusses outer automorphism on first 2 pages • Janusz, Gerald; Rotman, Joseph (June–July 1982). "Outer Automorphisms of S6". The American Mathematical Monthly. 89 (6): 407–410. doi:10.2307/2321657. JSTOR 2321657. • Fournelle, Thomas A. (1 January 1993). "Symmetries of the Cube and Outer Automorphisms of S6". The American Mathematical Monthly. 100 (4): 377–380. doi:10.2307/2324961. JSTOR 2324961. • Lorimer, P. J. (1 January 1966). "The Outer Automorphisms of S6". The American Mathematical Monthly. 73 (6): 642–643. doi:10.2307/2314806. JSTOR 2314806. • Miller, Donald W. (1 January 1958). "On a Theorem of Hölder". The American Mathematical Monthly. 65 (4): 252–254. doi:10.2307/2310241. JSTOR 2310241.	Wikipedia
\begin{document} \title{Grothendieck--Serre for constant reductive group schemes} \maketitle \begin{abstract} The Grothendieck--Serre conjecture asserts that on a regular local ring there is no nontrivial reductive torsor becomes trivial over the fraction field. This conjecture is answered in the affirmative in the equicharacteristic case but is still open in the mixed characteristic case. In this article, we establish its generalized version over Pr\"ufer bases for constant reductive group schemes. In particular, the Noetherian restriction of our main result settles a new case of the Grothendieck--Serre conjecture. Subsequently, we use this as a key input to resolve the variant of Bass--Quillen conjecture for torsors under constant reductive group schemes in our Pr\"uferian context. Along the way, inspired by the recent preprint of $\check{\mathrm{C}}$esnavi$\check{\mathrm{c}}$ius \cite{Ces22b}, we also prove several versions of Nisnevich conjecture in our context. \end{abstract} \hypersetup{ linktoc=page, } \renewcommand\contentsname{} \quad\\ \tableofcontents \section{Grothendieck--Serre on schemes smooth over Pr\"ufer bases} The Grothendieck--Serre conjecture predicts that every torsor under a reductive group scheme $G$ over a regular local ring $A$ is trivial if it becomes trivial over $\Frac A$. In other words, the following map \[ H^1_{\mathrm{\acute{e}t}}(A,G)\rightarrow H^1_{\mathrm{\acute{e}t}}(\Frac A,G) \] between nonabelian cohomology pointed sets, has trivial kernel. The conjecture was settled in the affirmative when $\dim A=1$, or when $A$ contains a field (that is, $A$ is of equicharacteristic). When $A$ is of mixed characteristic, the conjecture holds when $A$ is unramified and $G$ is quasi-split, whereas most of the remaining scenarios are unknown, see the recent survey \cite{Pan18}{\S 5} for a detailed review of the state of the art, as well as \S \ref{history} below for a summary. The present article aims to establish a new mixed characterstic case of the Grothendieck--Serre conjecture, in the following much more general non-Noetherian setup: \begin{thm-tweak} [\Cref{G-S for constant reductive gps}~\ref{G-S for constant reductive gps i}]\label{main thm} For a semilocal Pr\"ufer ring $R$, a reductive $R$-group scheme $G$, and an irreducible, affine, $R$-smooth scheme $X$, every generically trivial $G$-torsor on $X$ is Zariski-semilocally trivial. In other words, if $A:=\mathscr{O}_{X,\textbf{x}}$ is the semilocal ring of $X$ at a finite subset $\textbf{x}\subset X$, we have \[ \ker\,(H^1_{\mathrm{\acute{e}t}}(A,G)\rightarrow H^1_{\mathrm{\acute{e}t}}(\Frac A, G))=\{\ast\}. \] \end{thm-tweak} A ring is \emph{Pr\"ufer} if all its local rings are valuation rings, that are, integral domains $V$ such that every $x\in (\Frac V)\backslash V$ satisfies $x^{-1}\in V$. Noetherian valuation rings are exactly discrete valuation rings. Therefore, \Cref{main thm} specializes to a previously unsolved case of the Grothendieck--Serre in the sense that the aforementioned regular local ring $A$ is taken as a local ring of a scheme smooth over a discrete valuation ring. By Popescu's theorem \SP{07GC}, \Cref{main thm} still holds when $A$ is a semilocal ring that is geometrically regular over a Dedekind ring. \begin{pp}[Known cases of the Grothendieck--Serre conjecture] \label{history} Since proposed by Serre \cite{Ser58}{page~31} and Grothendieck \cite{Gro58}{pages 26--27, Remarques~3}, \cite{Gro68}{Remarques~1.11~a)}, the Grothendieck--Serre conjecture has already various known cases, as listed below. \begin{enumerate}[label={{\upshape(\roman)}}] \item The case when $G$ is a torus was proved by Colliot-Th\'el\`ene and Sansuc in \cite{CTS87}. \item The case when $\dim A=1$, namely, $A$ is a discrete valuation ring, was addressed by Nisnevich in \cite{Nis82} and \cite{Nis84}, then is improved and generalized to the semilocal Dedekind case in \cite{Guo22}. Several special cases were proved in \cite{Har67}, \cite{BB70}, \cite{BrT3} over discrete valuation rings, and in \cite{PS16}, \cite{BVG14}, \cite{BFF17}, \cite{BFFH20} for the semilocal Dedekind case. \item The case when $A$ is Henselian was settled in \cite{BB70} and \cite{CTS79}{Assertion~6.6.1} by reducing the triviality of $G$-torsors to residue fields then inducting on $\dim A$ to reach Nisnevich's resolved case. \item The equicharacteristic case, namely, when $A$ contains a field $k$, was established by Fedorov and Panin \cite{FP15} when $k$ is infinite (see also \cite{PSV15}, \cite{Pan20b} for crucial techniques) and by Panin \cite{Pan20a} when $k$ is finite, which was later simplified by \cite{Fed22a}. Before these, several equicharacteristic subcases were proved in \cite{Oja80},\cite{CTO92}, \cite{Rag94}, \cite{PS97}, \cite{Zai00}, \cite{OP01}, \cite{OPZ04}, \cite{Pan05}, \cite{Zai05}, \cite{Che10}, \cite{PSV15}. \item When $A$ is of mixed characteristic, \v{C}esnavi\v{c}ius \cite{Ces22a} settled the case when $G$ is quasi-split and $A$ is unramified (that is, for $p\colonequals \mathrm{char} (A/\mathfrak{m}_A)$, the ring $A/pA$ is regular). Priorly, Fedorov \cite{Fed22b} proved the split case under additional assumptions. Recently, \v{C}esnavi\v{c}ius \cite{Ces22b}{Theorem~1.3} settled a generalized Nisnevich conjecture under certain conditions, which specializes to the equal and mixed characteristic cases of the Grothendieck--Serre proved in \cite{FP15}, \cite{Pan20a}, \cite{Ces22a}. \item There are sporadic cases where $A$ or $G$ are speical (with possible mixed characteristic condition), see \cite{Gro68}{Remarque~1.11~a)}, \cite{Oja82}, \cite{Nis89}, \cite{Fed22b}, \cite{Fir22}, \cite{BFFP22}, \cite{Pan19b}. \end{enumerate} For arguing \Cref{main thm}, we will use our Pr\"uferian counterparts of the toral case \cite{CTS87} (see \Cref{G-S type results for mult type}~\ref{G-S for mult type gp}) and the semilocal Dedekind case \cite{Guo22} (see \Cref{G-S over semi-local prufer}) but no other known case of the Grothendieck--Serre Conjecture. \end{pp} \begin{pp}[Outline of the proof of \Cref{main thm}] \label{intro-outline-pf of main thm} As an initial geometric step, similar to \v{C}esnavi\v{c}ius's approach, one tries to use Gabber--Quillen type presentation lemma to fiber a suitable open neighbourhood $U\subset X$ of $\textbf{x}$ into smooth affine curves $U\to S$ over an open \[ S\subset \mathbb{A}_{R}^{\dim(X/R)-1} \] in such a way that a given small\footnote{Typically, it is not so small, and is only $R$-fiberwise of codimension $\ge 1$ in $X$.} closed subscheme $Y\subset X$ becomes \emph{finite} over $S$. For us, $Y$ could be any closed subscheme away from which the generically trivial torsor in question is trivial. Practically, this could be very hard (and even impossible) to achieve, partly due to the remaining mysterious of algebraic geometry in mixed characteristics; however, by the same reasoning as \cite{Ces22a}{Variant 3.7}, it is indeed possible if our $X$ admits a projective, flat compactification $\overline{X}$ over $R$ such that the boundary $\overline{Y}\backslash Y$ of $Y$ is $R$-fiberwise of codimension $\ge 2$ in $\overline{X}$. Assume for the moment that our compactification $\overline{X}$ is even $R$-smooth. (Although this looks very limited, it is strong enough for our later application in which $\overline{X}=\mathbb{P}_{R}^d$ .) Then, a key consequence of the purity for reductive torsors on scheme smooth over Pr\"ufer schemes (which takes advantage of the fact that our $G$ descends to $R$ hence is defined over the whole $\overline{X}$) is that we can enlarge the domain $X$ of our torsor so that $\overline{X}\backslash X$ is $R$-fiberwise of codimension $\ge 2$ in $\overline{X}$, see \Cref{extend generically trivial torsors} for a more precise statement. In particular, since $\overline{Y}\backslash Y \subset \overline{X}\backslash X$, we can ensure that $\overline{Y}\backslash Y$ is also $R$-fiberwise of codimension $\ge 2$ in $\overline{X}$, and it is therefore possible to run Panin--Fedorov--\v{C}esnavi\v{c}ius type arguments to finish the proof in the present case. However, our $R$-smooth $X$ from the beginning is arbitrary and need not have a smooth projective compactification. To circumvent this difficulty, we prove the following result, which is interesting in its own right: the pair $(Y,X)$ Zariski-locally on $X$ can be presented as an elementary étale neighbourhood of a similar pair $(Y',X')$, where $X'$ is an open of some projective $R$-space, see \Cref{variant of Lindel's lem} for a finer statement. Standard glueing techniques then allows us to replace $(Y,X)$ by $(Y',X')$ and to study generically trivial torsors on $X'$, but now $X'$ has smooth projective compactifications, namely, the projective $R$-spaces, we come back to the situation already settled in the previous paragraph. \end{pp} \begin{pp}[Nisnevich's purity conjecture] Now, we turn to Nisnevich's purity conjecture, where we require the total isotropicity of group schemes. A reductive group scheme $G$ defined over a scheme $S$ is \emph{totally isotropic} at $s\in S$ if every $G_i$ in the decomposition \cite{SGA3IIInew}{Exposé~XXIV, Proposition~5.10~(i)} \[ \textstyle G^{\mathrm{ad}}_{\mathscr{O}_{S,s}}\cong \prod_i\mathrm{Res}_{R_i/\mathscr{O}_{S,s}}(G_i) \] contains a $\mathbb{G}_{m,R_i}$, equivalently, every $G_i$ has a parabolic $R_i$-subgroup that is fiberwisely proper, see \cite{SGA3IIInew}{Expos\'e XXVI, Corollaire 6.12}. If this holds for all $s\in S$, then $G$ is \emph{totally isotropic}. Proposed by Nisnevich \cite{Nis89}{Conjecture~1.3} and modified due to the anisotropic counterexamples of Fedorov \cite{Fed22b}{Proposition~4.1}, the Nisnevich conjecure predicts that, for a regular semilocal ring $A$, a regular parameter $r\in A$ (that is, $r\in \mathfrak{m}\backslash \mathfrak{m}^2$ for every maximal ideal $\mathfrak{m}\subset A$), and a reductive $A$-group scheme $G$ such that $G_{A/rA}$ is totally isotropic, every generically trivial $G$-torsor on $A[\f{1}{r}]$ is trivial, that is, the following map \[ \textstyle \text{ $H^1(A[\f{1}{r}],G)\rightarrow H^1(\Frac A,G)$ \quad has trivial kernel}. \] The case when $A$ is a local ring of a regular affine variety over a field and $G=\GL_n$ was settled by Bhatwadekar--Rao in \cite{BR83} and was subsequently extended to arbitrary regular local rings containing fields by Popescu \cite{Pop02}{Theorem~1}. Nisnevich in \cite{Nis89} proved the conjecture in dimension two, assuming that $A$ is a local ring with infinite residue field and that $G$ is quasi-split. For the state of the art, the conjecure was settled in equicharacteristic case and in several mixed characteristic case by {\v{C}}esnavi{\v{c}}ius in \cite{Ces22b}{Theorem~1.3} (previously, Fedorov \cite{Fed21} proved the case when $A$ contains an infinite field). Besides, the toral case and some low dimensional cases are known and surveyed in \cite{Ces22}{Section~3.4.2~(1)} including Gabber's result \cite{Gab81}{Chapter~I, Theorem~1} for the local case $\dim A\leq 3$ when $G$ is either $\GL_n$ or $\mathrm{PGL}_n$. In this article, we prove several variants of Nisnevich conjecture over Pr\"ufer bases, see \Cref{torsors-Sm proj base}~\ref{Nis-sm-proj} and \Cref{G-S for constant reductive gps}~\ref{G-S for constant reductive gps ii}. \end{pp} \begin{pp}[Outline of the paper] \label{outline of the paper} In this \S \ref{outline of the paper}, unless stated otherwise, $R$ is a semilocal Pr\"ufer ring, $S:=\Spec R$ is its spectrum, $X$ is an irreducible, affine, $R$-smooth scheme, $A\colonequals \mathscr{O}_{X,\textbf{x}}$ is the semilocal ring of $X$ at a finite subset $\textbf{x}\subset X$, and $G$ is a reductive $X$-group scheme. \begin{itemize} \item [(1)] In \S \ref{sect-purity of reductive torsors}, we establish purity of reductive torsors on schemes smooth over Pr\"ufer rings. The global statement is \Cref{purity for rel. dim 1} : if $X$ is a smooth $S$-curve and if a closed subset $Z\subset X$ satisfies \[ \text{$Z_{\eta}= \emptyset$\quad for each generic point $\eta\in S$ \quad and\quad $\codim(Z_s,X_s)\ge 1$ for all $s\in S$,} \] then restriction induces the following equivalence of categories of $G$-torsors \[ \mathbf{Tors}(X_{\mathrm{\acute{e}t}},G) \isoto \mathbf{Tors}((X\backslash Z)_{\mathrm{\acute{e}t}},G). \] In particular, passing to isomorphism classes of objects, we have the following bijection of pointed sets \[ H^1_{\mathrm{\acute{e}t}}(X,G)\simeq H^1_{\mathrm{\acute{e}t}}(X\backslash Z,G). \] In its local variant \Cref{extends across codim-2 points}, we show that if $x\in X$ is a point such that \[ \text{either $x\in X_{\eta}$ with $\dim \mathscr{O}_{X_{\eta},x} =2$, \quad or\; $x\in X_s$ with $s \neq \eta$ and $\dim \mathscr{O}_{X_s,x} =1$,} \] then every $G$-torsor over $\Spec \mathscr{O}_{X,x} \backslash \{x\}$ extends uniquely to a $G$-torsor over $\Spec \mathscr{O}_{X,x}$. An immediate consequence is \Cref{extend generically trivial torsors}: every generically trivial $G$-torsor on $\mathscr{O}_{X,\textbf{x}}$ extends to a $G$-torsor on an open neighbourhood of $\textbf{x}$ whose complementary closed has codimension $\ge 3$ (resp., $\ge 2$) in the generic (resp., non-generic) $S$-fibers of $X$. As for their proofs, the key case to treat is that of vector bundles, that is, $G=\GL_n$, whose analysis ultimately depends on the estimate of the projective dimensions of reflexive sheaves on $X$ and the equivalence of categories of reflexive sheaves on $X$ and on $X\backslash Z$ (under suitable codimensional constraints on $Z$), both are essentially obtained by Gabber--Ramero \cite{GR18}. \item [(2)] In \S \ref{low dim cohomology of mult gp}, based on the low-dimensional vanishing of local cohomology of tori, we prove the purity for cohomology of group schemes of multiplicative type over Pr\"ufer bases, as well as the surjectivity of $H^1_{\mathrm{\acute{e}t}}(-,T) $ and the injectivity of $H^2_{\mathrm{\acute{e}t}}(-,T)$ upon restricting to opens for flasque tori $T$ over an $S$-smooth scheme, see \Cref{purity for gp of mult type} for a more precise statement. After these preliminaries, we are able to prove the Pr\"uferian counterparts of Colliot-Thélène--Sansuc's results for tori and, in particular, the Grothendieck-Serre for tori, see \Cref{G-S type results for mult type}~\ref{G-S for mult type gp}. \item [(3)] In \S \ref{sect-geom lem}, we first present the Pr\"uferian analog of \v{C}esnavi\v{c}ius's formulation of the geometric presentation lemma in the style of Gabber--Quillen, see \Cref{Ces's Variant 3.7}. The rest of \S \ref{sect-geom lem} is devoted to proving the following variant (in some aspect, a stronger form) of Lindel's lemma which should be of independent interest: for an $S$-smooth scheme $X$ and a closed subscheme $Y\subset X$ that avoids all the maximal points of the $S$-fibers of $X$, the pair $(Y,X)$ Zariski-locally on $X$ can be presented as an elementary étale neighbourhood of a similar pair $(Y',X')$, where $X'$ is an open of some projective $S$-space, see \Cref{variant of Lindel's lem} for a finer statement where one works Zariski semilocally on $X$. \item [(4)] The main result of \S \ref{section-torsors on sm aff curves} is the Section \Cref{triviality on sm rel. affine curves} on triviality of torsors on a smooth affine relative curve. The idea of the proof ultimately depends on the geometry of affine Grassmannians developed by Fedorov, who proved \Cref{triviality on sm rel. affine curves}~\ref{sec-thm-semilocal} for $C=\mathbb{A}_R^1$. \item [(5)] In \S \ref{sect-torsor on sm proj base}, we prove \Cref{main thm} under the additional assumption that $X$ is $R$-projective, only assuming that $G$ is a reductive $X$-group scheme (thus not necessarily descends to $R$). This result was proved by the second author and simultaneously by an unpublished work of Panin and the first author in the Noetherian case. As explained in \S \ref{intro-outline-pf of main thm}, the proof crucially uses the results of \S \ref{sect-purity of reductive torsors} on purity for reductive torsors to extends the domain of the relevant torsors so that the geometric presentation \Cref{Ces's Variant 3.7} could apply. It turns out that, without much extra efforts, we could simultaneously obtain a version of Nisnevich statement as in \Cref{torsors-Sm proj base}~\ref{Nis-sm-proj}. \item [(6)] In \S \ref{sect-torsor under constant redu}, we prove \Cref{main thm} as well as the corresponding Nisnevich statement \Cref{G-S for constant reductive gps}~\ref{G-S for constant reductive gps ii}. As mentioned in \S \ref{intro-outline-pf of main thm}, the proof is via reduction to the case settled in \S \ref{sect-torsor on sm proj base} where $X$ is an open of some projective $R$-space, using \Cref{variant of Lindel's lem}. \item [(7)] In \S \ref{sect-Bass-Quillen}, we prove the following Bass--Quillen statement for torsors: for a ring $A$ that is smooth over a Pr\"{u}fer ring $R$ and a totally isotropic reductive $R$-group scheme $G$, a $G$-torsor on $\mathbb{A}_A^N$ descends to $A$ if it is Zariski-locally trivial, equivalently, if it is Nisnevich-locally trivial. For the proof, the key input is the main \Cref{main thm} as well as the purity \Cref{purity for rel. dim 1} that we utilize to show that every generically trivial $G_{R(t)}$-torsor is trivial for a reductive $R$-group scheme $G$ (see \Cref{triviality over R(t)}). Granted them, we will closely follow the line of the proofs of the classical Bass--Quillen conjecture for vector bundles in the unramified case (as gradually developed by Quillen \cite{Qui76}, Lindel \cite{Lin81} and Popescu \cite{Pop89}): Quillen patching and the approximation \Cref{approxm semi-local Prufer ring} reduce us to the case of a finite-rank valuation ring $R$, inducting on $\mathrm{rank}(R)$ and the number of variables involving \Cref{triviality over R(t)} establishes the case $A$ being a polynomial ring over $R$, an application of the `inverse' to Quillen patching \Cref{inverse patching} settles the case $A$ being a localization of a polynomial ring over $R$, and finally, by inducting on the pair $(\dim R,\dim A-\dim R)$ and using \Cref{main thm}, a glueing argument involving \Cref{variant of Lindel's lem} reduces the general case to the already settled case $A$ being a localization of a polynomial ring. \item [(8)] In appendix \ref{section-G-S on semilocal prufer}, we prove the Grothendieck--Serre on semilocal Pr\"ufer rings (\Cref{G-S over semi-local prufer}), which simultaneously generalizes the main results of \cite{Guo22} and \cite{Guo20b}. In essence, one has to prove a product formula for reductive groups scheme on semilocal Pr\"ufer rings (\Cref{decomp-gp}). The toral case is immediately obtained, thanks to the already settled Grothendieck-Serre for tori (see \Cref{G-S type results for mult type}~\ref{G-S for mult type gp}). The general case would follow from the similar arguments as in \emph{loc. cit}, once Harder's weak approximation argument was carried out to establish the key \Cref{open-normal}. \end{itemize} \end{pp} \begin{pp}[Notations and conventions] All rings in this paper are commutative with units, unless stated otherwise. For a point $s$ of a scheme (resp., for a prime ideal $\mathfrak{p}$ of a ring), we let $\kappa(s)$ (resp., $\kappa(\mathfrak{p})$) denote its residue field. For a global section $s$ of a scheme $S$, we write $S[\frac{1}{s}]$ for the open locus where $s$ does not vanish. For a ring $A$, we let $\text{Frac}\, A$ denote its total ring of fractions. For a morphism of algebraic spaces $S'\to S$, we let $(-)_{S'}$ denote the base change functor from $S$ to $S'$; if $S=\text{Spec}\,R$ and $S'=\text{Spec}\,R'$ are affine schemes, we will also write $(-)_{R'}$ for $(-)_{S'}$. Let $S$ be an algebraic space, and let $G$ be an $S$-group algebraic space. For an $S$-algebraic space $T$, by a $G$-torsor over $T$ we shall mean a $G_T:=G\times_RT$-torsor. Denote by $\textbf{Tors}(S_{\mathrm{fppf}},G)$ (resp., $\textbf{Tors}(S_{\mathrm{\acute{e}t}},G)$) the groupoid of $G$-torsors on $S$ that are fppf-locally (resp., \'etale-locally) trivial; specifically, if $G$ is $S$-smooth (e.g., $G$ is $S$-reductive), then every fppf-locally trivial $G$-torsor is \'etale-locally trivial, so we have $$\textbf{Tors}(S_{\mathrm{fppf}},G)= \textbf{Tors}(S_{\mathrm{\acute{e}t}},G).$$ Let $X$ be a scheme. The category of invertible $\mathscr{O}_X$-modules is denoted $\mathbf{Pic} X$. Assume that $X$ is locally coherent, for example, $X$ could be locally Noetherian or locally finitely presented over a Pr\"ufer ring. The category of reflexive $\mathscr{O}_X$-modules is denoted $\mathscr{O}_X$-$\mathbf{Rflx}$. \end{pp} \section{Purity for reductive torsors on schemes smooth over Pr\"ufer rings} \label{sect-purity of reductive torsors} In this section, we recollect useful geometric properties on scheme over Pr\"ufer bases. \begin{lemma-tweak}\label{geom} For a Pr\"{u}fer scheme $S$, an $S$-flat, finite type, irreducible scheme $X$, and a point $s\in S$, \begin{enumerate}[label={{\upshape(\roman)}}] \item\label{geo-i} all nonempty $S$-fibers have the same dimension; \item\label{geo-iii} if $\mathscr{O}_{X_s,\xi}$ is reduced for a maximal point $\xi\in X_s$, then the local ring $\mathscr{O}_{X,\xi}$ is a valuation ring such that the extension $\mathscr{O}_{S,s}\hookrightarrow \mathscr{O}_{X,\xi}$ induces an isomorphism of value groups. \end{enumerate} \end{lemma-tweak} \begin{proof} For \ref{geo-i}, see \cite{EGAIV3}{Lemme~14.3.10}. For \ref{geo-iii}, see \cite{MB22}{Théorème~A}. \end{proof} \begin{lemma}\label{enlarge valuation rings} For a valuation ring $V$, an essentially finitely presented (resp., essentially smooth) $V$-local algebra $A$, there are an extension of valuation rings $V'/V$ with trivial extension of value groups, and an essentially finitely presented (resp., essentially smooth) $V$-map $V'\to A$ with finite residue fields extension. \end{lemma} \begin{proof} Assume $A=\mathscr{O}_{X,x}$ for an affine scheme $X$ finitely presented over $V$ and a point $x\in X$ lying over the closed point $s \in \text{Spec}(V)$. Let $t=\text{tr.deg}(\kappa(x)/\kappa(s))$. As $\kappa(x)$ is a finite extension of $l\colonequals \kappa(s)(a_1,\cdots, a_t)$ for a transcendence basis $(a_i)_1^t$ of $\kappa(x)/\kappa(s)$, we have $t=\dim_l\Omega^1_{l/\kappa(s)}\leq \dim_{\kappa(x)}\Omega_{\kappa(x)/\kappa(s)}^1$. Choose sections $b_1,\cdots,b_t\in \Gamma(X,\mathscr{O}_X)$ such that $d\overline{b_1},\cdots,d\overline{b_t} \in \Omega^1_{\kappa(x)/\kappa(s)} $ are linearly independent over $\kappa(x)$, where the bar stands for their images in $\kappa(x)$. Define $p:X\to \mathbb{A}_V^t$ by sending the standard coordinates $T_1,\cdots, T_t$ of $\mathbb{A}_V^t$ to $b_1,\cdots, b_t$, respectively. Since $d\overline{b_1},\cdots,d\overline{b_t} \in \Omega^1_{\kappa(x)/\kappa(s)} $ are linearly independent, the image $\eta\colonequals p(x)$ is the generic point of $\mathbb{A}^t_{\kappa(s)}$, so $V^{\prime}\colonequals \mathscr{O}_{\mathbb{A}^t_{V},\eta}$ is a valuation ring whose value group is $\Gamma_{V^{\prime}}\simeq \Gamma_V$. Note that $\kappa(x)/\kappa(\eta)$ is finite, the map $V^{\prime}\rightarrow A$ induces a finite residue fields extension. When $V\rightarrow A$ is essentially smooth, the images of $db_1,\cdots,db_t $ under the map $\Omega^1_{X/V}\otimes \kappa(x) \to \Omega^1_{\kappa(x)/\kappa(s)}$ are linearly independent, so are their images in $\Omega^1_{X/V}\otimes \kappa(x)$. Hence, $p$ is essentially smooth at $x$. \end{proof} In the sequel, we will use the following limit argument repeatedly. \begin{lemma-tweak} \label{approxm semi-local Prufer ring} Every semilocal Pr\"{u}fer domain $R$ is a filtered direct union of its subrings $ R_i$ such that: \begin{enumerate}[label={{\upshape(\roman)}}] \item for every $i$, $R_i$ is a semilocal Pr\"{u}fer domain of finite Krull dimension; and \item for $i$ large enough, $R_i\to R$ induces a bijection on the sets of maximal ideals hence is fpqc. \end{enumerate} \end{lemma-tweak} \begin{proof} Write $\Frac (R)=\cup_i K_i$ as the filtered direct union of the subfields of $\Frac (R)$ that are finitely generated over its prime field $\mathfrak{K}$. For $R_i\colonequals R\cap K_i$, we have $R=\cup_i R_i$. It remains to see that every $R_i$ is a semilocal Pr\"{u}fer domain whose local rings have finite ranks. Let $\{\mathfrak{p}_j\}_{1\le j \le n}$ be the set of maximal ideals of $R$. Then $R=\bigcap_{1\le j \le n} R_{\mathfrak{p}_j}$ is the intersection of the valuation rings $R_{\mathfrak{p}_j}$. Thus we have \[ \textstyle R_i=\bigcap_{1\le j \le n} \left(K_i \cap R_{\mathfrak{p}_j}\right). \] Since $K_i/\mathfrak{K}$ has finite transcendence degree, by Abhyankar's inequality, every $K_i \cap R_{\mathfrak{p}_j}$ is a valuation ring of finite rank. By \cite{Bou98}{VI, \S7, Proposition~1--2}, $R_i$ is a semilocal Pr\"{u}fer domain, and its local rings at maximal ideals are precisely the minimal elements of the set $\{K_i \cap R_{\mathfrak{p}_j}\}_{1\le j \le n}$ under inclusion. This implies (i). For (ii), it suffices to show that for $i$ large enough there are no strict inclusion relation between $K_i \cap R_{\mathfrak{p}_{j_1}}$ and $K_i \cap R_{\mathfrak{p}_{j_2}}$ for $j_1\neq j_2$. Indeed, if $\pi_j \in \mathfrak{p}_j\backslash \bigcup_{j'\neq j} \mathfrak{p}_{j'}$ for $1\le j \le n$, then (ii) holds for any $i$ for which $\{\pi_j\}_{1\le j \le n} \subset K_i$. \end{proof} \begin{pp}[Coherence and reflexive sheaves] For a ring $R$, if an $R$-module $M$ is finitely generated and its finitely generated submodules are all finitely presented, then $M$ is a \emph{coherent} $R$-module. The ring $R$ is \emph{coherent} if it is a coherent $R$-module. An $\mathscr{O}_X$-module $\mathscr{F}$ on a scheme $X$ is \emph{coherent} if, for every affine open $U\subset X$, $\mathscr{F}(U)$ is a coherent $\mathscr{O}_X(U)$-module. A scheme $X$ is \emph{locally coherent} if $\mathscr{O}_X$ is a coherent $\mathscr{O}_X$-module. For a locally coherent scheme $X$, an $\mathscr{O}_X$-module $\mathscr{F}$ is coherent if and only if it is finitely presented (\SP{05CX}), and if this is true then its dual $\mathscr{F}^{\vee}\colonequals \mathscr{H}\! om_{\mathscr{O}_X}(\mathscr{F},\mathscr{O}_X)$ is also coherent. Examples of locally coherent schemes include locally Noetherian schemes, Pr\"ufer schemes and, more generally, any scheme flat and locally of finite type over them, see \cite{GR71}{Partie~I, Théorème~3.4.6}. Let $X$ be a locally coherent scheme. A coherent $\mathscr{O}_X$-module $\mathscr{F}$ is \emph{reflexive} if taking double dual $\mathscr{F}\rightarrow \mathscr{F}^{\vee\!\vee}$ is an isomorphism. For a reflexive $\mathscr{O}_X$-module $\mathscr{F}$, locally on $X$, taking dual of a finite presentation of $\mathscr{F}^{\vee}$ yields a \emph{finite copresentation} of $\mathscr{F}\simeq \mathscr{F}^{\vee\!\vee}$ of the form $0\rightarrow \mathscr{F}\rightarrow \mathscr{O}_{X}^{\oplus m}\xrightarrow{\alpha} \mathscr{O}_{X}^{\oplus n}$. Conversely, if $\mathscr{F}$ admits such a copresentation, then $\mathscr{F}\simeq \text{coker} (\alpha^{\vee})^{\vee}$, so by \SP{01BY} is reflexive. Hence, a coherent sheaf on $X$ is reflexive if and only if it is finitely copresented. As a consequence, we see that reflexivity is compatible with limit formalism in the following sense: if $X=\lim_{i}X_i$ is the limit of an inverse system $(X_i)$ of qcqs locally coherent schemes with affine transition morphisms, then we have \begin{equation}\label{lim of Rflx} \text{colim}_i\,\mathscr{O}_{X_i}\text{-}\mathbf{Rflx}\xrightarrow{\sim}\mathscr{O}_{X} \text{-}\mathbf{Rflx}. \end{equation} \end{pp} \begin{lemma-tweak}\label{lim-codim} Let $X\rightarrow S$ be a finite type morphism with regular fibers between topologically Noetherian schemes, let $j\colon U\hookrightarrow X$ be a quasi-compact open immersion with complement $Z\colonequals X\backslash U$ satisfying \[ \quad \text{$\codim(Z_s, X_s)\geq 1$ for every $s\in S$\quad and \quad $\codim(Z_{\eta},X_{\eta})\geq 2$ for every generic point $\eta\in S$,} \] and let $\mathscr{F}$ be a reflexive $\mathscr{O}_X$-module. Assume that $S$ is a cofiltered inverse limit of integral schemes $(S_{\lambda})_{\lambda\in \Lambda}$ with generic point $\eta_{\lambda}$ and surjective transition maps. Then, there is a $\lambda_{0}\in \Lambda$, a finite type morphism $X_{\lambda_0}\rightarrow S_{\lambda_0}$ with regular fibers such that $X_{\lambda_0}\times_{S_{\lambda_0}}S\simeq X$, a closed subscheme $Z_{\lambda_0}\subset X_{\lambda_0}$ such that $Z_{\lambda_{0}}\times_{S_{\lambda_0}}S\simeq Z$, the open immersion $j_{\lambda_0}\colon X_{\lambda_0}\backslash Z_{\lambda_0}\hookrightarrow X_{\lambda_0}$ is quasi-compact, and the following \[ \quad \text{$\codim(({Z_{\lambda_0}})_s, ({X_{\lambda_0}})_s)\geq 1$ for every $s\in S_{\lambda_0}$\quad and \quad $\codim(({Z_{\lambda_0}})_{\eta_0},({X_{\lambda_0}})_{\eta_0})\geq 2$} \] is satisfied. Also, there is a reflexive $\mathscr{O}_{X_{\lambda_0}}$-module $\mathscr{F}_{\lambda_0}$ whose inverse image on $X$ is $\mathscr{F}$. \end{lemma-tweak} \begin{proof} The condition that $X$ has regular $S$-fibers descends to $X_{\lambda_0}$ by \cite{EGAIV2}{Proposition~6.5.3}. The reflexive $\mathscr{O}_X$-module $\mathscr{F}$ descends thanks to \cite{EGAIV3}{Théorème~8.5.2} and by applying \cite{EGAIV3}{Corollaire~8.5.2.5} to $\mathscr{F}\isoto \mathscr{F}^{\vee\!\vee}$. Because $Z$ is contructible closed, by \cite{EGAIV3}{Théorème~8.3.11}, it descends to $Z_{\lambda}$ such that $p_{\lambda}^{-1}(Z_{\lambda})=Z$. For $f_{\lambda}\colon X_{\lambda}\rightarrow S_{\lambda}$, by the transversity of fibers and \cite{EGAIV2}{Corollaire~4.2.6}, $Z_{\lambda}$ does not contain any irreducible components of $f_{\lambda}^{-1}(s_{\lambda})$ for any $s_{\lambda}\in S_{\lambda}$. Finally, the image of the generic point $\eta\in S$ is the generic point $\eta_{\lambda}\in S_{\lambda}$. By \cite{EGAIV2}{Corollaire~6.1.4}, we have $\codim((Z_{\lambda})_{\eta_{\lambda}}, (X_{\lambda})_{\eta_{\lambda}})=\codim(Z_{\eta}, X_{\eta})\geq 2$. Finally, $\mathscr{F}$ descends to a reflexive sheaf by (\ref{lim of Rflx}). \end{proof} \begin{prop-tweak}\label{GR18} For a semilocal Pr\"{u}fer ring $R$ and a flat, locally of finite type morphism $f\colon X\rightarrow S:=\Spec R$ of schemes with regular fibers, the following assertions hold. \begin{enumerate}[label={{\upshape(\roman)}}] \item\label{GR18-11.4.1} For every $x\in X$ and every coherent $\mathscr{O}_X$-module $\mathscr{F}$ that is reflexive at $x$, we have \[ \text{$\mathrm{proj.dim}_{\mathscr{O}_{X,x}}\mathscr{F}_x\leq \max (0,n-1)$, \quad where \quad $n= \dim \mathscr{O}_{f^{-1}(f(x)),x}$.} \] \item\label{GR18-11.4.6} For a closed subset $Z\subset X$ such that $j\colon X\backslash Z\hookrightarrow X$ is quasi-compact and satisfies the following \[ \quad \text{$\codim(Z_s, X_s)\geq 1$ for all $s\in S$\quad and \quad $\codim(Z_{\eta},X_{\eta})\geq 2$ for the generic point $\eta\in S$,} \] the restriction functors induce the following equivalences: \begin{equation}\label{restriction} \text{$\mathscr{O}_{X}\text{-}\mathbf{Rflx}\isoto \mathscr{O}_{X\backslash Z}\text{-}\mathbf{Rflx}$\quad\quad and \quad\quad $\mathbf{Pic}\,X\isoto \mathbf{Pic}\, X\backslash Z$} \end{equation} In particular, for every $X$-affine finite type scheme $Y$, we have a bijection of sets \[ \textstyle Y(X)\simeq Y(X\backslash Z). \] \end{enumerate} \end{prop-tweak} \begin{proof} The assertion \ref{GR18-11.4.1} is \cite{GR18}{Proposition~11.4.1~(iii)}. For the first assertion of \ref{GR18-11.4.6}, by glueing, the equivalence is Zariski-local on $X$, so we may assume that $X$ is affine and thus, by \cite{Nag66}{Theorem 3'}, $X\to S$ is finitely presented. By a standard limit argument involving Lemmata \ref{approxm semi-local Prufer ring} and \ref{lim-codim}, we reduce to the case of a finite-dimensional $S$. Now since $\|X\|$ is the finite disjoint union of its $S$-fibers $X_s$, which are Noetherian spaces, we see that $X$ is topologically Noetherian. In particular, every open subset of $X$ is quasi-compact. By \cite{GR18}{Proposition~11.3.8~(i)}, the functor (\ref{restriction}) is essentially surjective. For the fully faithfulness, we let $ \mathscr{F}$ and $ \mathscr{G}$ be two reflexive $\mathscr{O}_X$-modules and need to show that restriction induces $$ \text{Hom}_{\mathscr{O}_X}(\mathscr{F},\mathscr{G})\xrightarrow{\sim }\text{Hom}_{\mathscr{O}_U}(\mathscr{F}\|_U,\mathscr{G}\|_U), \quad\quad \text{ where } \quad U:=X\backslash Z. $$ Choose a finite presentation $\mathscr{O}_{X}^{\oplus m}\rightarrow \mathscr{O}_{X}^{\oplus n} \rightarrow \mathscr{F} \rightarrow 0$. Then chasing the following commutative diagram \[ \begin{tikzcd} 0 \arrow[r] & \mathrm{Hom}_{\mathscr{O}_X}(\mathscr{F},\mathscr{G}) \arrow[d] \arrow[r] & \mathrm{Hom}_{\mathscr{O}_X}(\mathscr{O}_X^{\oplus n},\mathscr{G}) \arrow[d] \arrow[r] & \mathrm{Hom}_{\mathscr{O}_X}(\mathscr{O}_X^{\oplus m},\mathscr{G}) \arrow[d] \\ 0 \arrow[r] & \mathrm{Hom}_{\mathscr{O}_U}(\mathscr{F}\|_U,\mathscr{G}\|_U) \arrow[r] & \mathrm{Hom}_{\mathscr{O}_U}(\mathscr{O}_U^{\oplus n},\mathscr{G}\|_U) \arrow[r] & \mathrm{Hom}_{\mathscr{O}_U}(\mathscr{O}_U^{\oplus m},\mathscr{G}\|_U) \end{tikzcd} \] with exact rows reduces us first to the case when $\mathscr{F}$ is free. Choose a finite copresentation $0\rightarrow \mathscr{G}\rightarrow \mathscr{O}_{X}^{\oplus m^{\prime}}\rightarrow \mathscr{O}_{X}^{\oplus n^{\prime}}$ and chasing a similar commutative diagram reduces us further to the case $\mathscr{G}$ is free. Thus we may assume that $\mathscr{F}=\mathscr{G}=\mathscr{O}_{X}$ and need to show that $\mathscr{O}_X(X)\xrightarrow{\sim} \mathscr{O}_X(U)$. Through the terminology of \cite{GR18}{10.4.19}, our assumption on the fiberwise codimension of $Z$ in $X$ implies that $\delta'(z,\mathscr{O}_X)\ge 2$, see \cite{GR18}{Corollary~10.4.46}, so we may apply \cite{GR18}{Proposition~11.3.8 (ii)} to $\mathscr{F}:=\mathscr{O}_X$ and deduce that $\mathscr{O}_X\simeq j_(\mathscr{O}_{U})$. Taking global sections yields the desired isomorphism. For the second assertion of \ref{GR18-11.4.6}, by glueing, the problem is Zariski-local on $X$, so we can assume that $X$ is affine. Choose an embedding $Y\hookrightarrow \mathbb{A}_X^n$ over $X$ for some integer $n$. Since $U$ is schematically dense in $X$, for every morphism $\phi\colon U \rightarrow Y$, if $\phi$ extends uniquely to a morphism $\widetilde{\phi}\colon X\rightarrow \mathbb{A}^n_X$, then $\widetilde{\phi}^{-1}(Y)$ is a closed subscheme of $X$ containing $U$ and, by \cite{EGAIV4}{Lemme~20.3.8.8}, must coincide with $X$. In other words, if $\widetilde{\phi}$ exists uniquely, then it factorises as $X\overset{\psi}{\rightarrow} Y\hookrightarrow \mathbb{A}^n_X$ such that $\psi$ is the unique extension of $\phi$. This reduces us to the case $Y=\mathbb{A}_X^n$. Now, by the reflexivity of $\mathscr{O}_X$ and the full faithfulness of $\mathscr{O}_{X}\text{-}\mathbf{Rflx}\isoto \mathscr{O}_{U}\text{-}\mathbf{Rflx}$, we have the desired bijection \[ \mathbb{A}_X^n(X)=\mathrm{Hom}_{\mathscr{O}_X}(\mathscr{O}_X,\mathscr{O}_X^{\oplus n}) \simeq \mathrm{Hom}_{\mathscr{O}_U}(\mathscr{O}_{U},\mathscr{O}_{U}^{\oplus n}) =\mathbb{A}_X^n(U).\qedhere \] \end{proof} \begin{cor-tweak}\label{vect-ext} For a semilocal affine Pr\"{u}fer scheme $S$, an $S$-flat, locally of finite type scheme $X$ with regular one-dimensional $S$-fibers, and its closed subset $Z$ such that $j\colon X\backslash Z\hookrightarrow X$ is quasi-compact and \[ \text{$Z_{\eta}= \emptyset$\quad for each generic point $\eta\in S$ \quad and\quad $\codim(Z_s,X_s)\ge 1$ for all $s\in S$,} \] the restriction and the pushforward $j_{\ast}(-)$ as inverse induce an equivalence of categories of vector bundles \[ \mathbf{Vect}_{X} \isoto \mathbf{Vect}_{X\backslash Z}. \] \end{cor-tweak} \begin{proof} For vector bundles $\mathscr{E}_1$ and $\mathscr{E}_2$ on $X$, the scheme $Y\colonequals \underline{\mathrm{Isom}}_X(\mathscr{E}_1,\mathscr{E}_2)$ is $X$-affine of finite type, so $Y(X\backslash Z)=Y(X)$ by \Cref{GR18}\ref{GR18-11.4.6}. This proves the full faithfulness. For the essential surjectivity, by \Cref{GR18}\ref{GR18-11.4.6}, every vector bundle $\mathscr{E}$ on $X\backslash Z$ extends to a reflexive $\mathscr{O}_X$-module $j_{\ast}\mathscr{E}$. To show that the reflexive $\mathscr{O}_X$-module $j_{\ast}\mathscr{E}$ is a vector bundle, it suffices to exploit \Cref{GR18}\ref{GR18-11.4.1}. \end{proof} As a consequence, we obtain the following purity on regular one-dimensional relative curves for torsors under reductive group schemes. \begin{thm-tweak}\label{purity for rel. dim 1} For a semilocal affine Pr\"{u}fer scheme $S$, an $S$-flat, locally of finite type scheme $X$ with regular $\mathrm{one\text{-}dimensional}$ $S$-fibers, and a closed subset $Z\subset X$ such that $X\backslash Z\hookrightarrow X$ is quasi-compact and \[ \text{$Z_{\eta}= \emptyset$\quad for each generic point $\eta\in S$ \quad and\quad $\codim(Z_s,X_s)\ge 1$ for all $s\in S$,} \] and a reductive $X$-group scheme $G$, restriction induces the following equivalence of categories of $G$-torsors \begin{equation}\label{restriction of G-torsors} \mathbf{Tors}(X_{\mathrm{\acute{e}t}},G) \isoto \mathbf{Tors}((X\backslash Z)_{\mathrm{\acute{e}t}},G). \end{equation} In particular, passing to isomorphism classes of objects, we have an isomorphism $$H^1_{\mathrm{\acute{e}t}}(X,G)\simeq H^1_{\mathrm{\acute{e}t}}(X\backslash Z,G).$$ \end{thm-tweak} \begin{proof} It suffices to show that (\ref{restriction of G-torsors}) is an equivalence. To show that (\ref{restriction of G-torsors}) is fully faithful, we let $\mathcal{P}_1$ and $\mathcal{P}_2$ be two $G$-torsors, and consider $Y\colonequals \underline{\mathrm{Isom}}_X(\mathcal{P}_1,\mathcal{P}_2)$, the functor on $X$-schemes parameterizing isomorphisms from $\mathcal{P}_1$ to $\mathcal{P}_2$. By descent theory, $Y$ is $X$-affine and of finite type, so $Y(X\backslash Z)=Y(X)$ by \Cref{GR18}~\ref{GR18-11.4.6}. This proves the full faithfulness. To show that (\ref{restriction of G-torsors}) is essentially surjective, we pick a $G$-torsor $\mathcal{P}$ on $X\backslash Z$ and need to show that $\mathcal{P}$ extends to a $G$-torsor on $X$. Since the assumption on the fiber codimension still holds when we base change to every \'etale scheme $X'$ over $X$, we obtain an equivalence $\mathbf{Tors}(X'_{\mathrm{\acute{e}t}},G) \isoto \mathbf{Tors}((X'\backslash Z')_{\mathrm{\acute{e}t}},G)$, where $Z':=Z\times_{X}X'$. Consequently, by glueing in the \'etale topology, it suffices to show that, \'etale locally on $X$, $\mathcal{P}$ extends to a $G$-torsor on $X$. To see this, we may assume that $X$ is affine and $G\subset \GL_{n,X}$, then exploit the commutative diagram with exact rows \[ \begin{tikzcd} {(\mathrm{GL}_{n,X}/G)(X)} \arrow[r] \arrow[d,"\simeq" labl, ] & {H^1_{\mathrm{\acute{e}t}}(X,G)} \arrow[r] \arrow[d] & {H^1_{\mathrm{\acute{e}t}}(X,\mathrm{GL}_{n,X})} \arrow[d] \\ {(\mathrm{GL}_{n,X}/G)(X\backslash Z)} \arrow[r] & {H^1_{\mathrm{\acute{e}t}}(X\backslash Z,G)} \arrow[r] & {H^1_{\mathrm{\acute{e}t}}(X\backslash Z,\mathrm{GL}_{n,X})}, \end{tikzcd} \] where the bijectivity of the left vertical arrow follows from \Cref{GR18}\ref{GR18-11.4.6} and, as $G$ is reductive, $\mathrm{GL}_{n,X}/G$ is affine over $X$ by \cite{Alp14}{9.4.1}. By \Cref{vect-ext}, we may replace $X$ by an affine open cover to assume that the induced $\GL_{n,X\backslash Z}$-torsor $\mathcal{P}\times ^{G_{X\backslash Z}}\GL_{n,X\backslash Z}$ is trivial. A diagram chase implies that there exists a $G$-torsor $\mathcal{Q}$ on $X$ such that $\mathcal{Q}\|_{X\backslash Z}\simeq \mathcal{P}$, as claimed. \end{proof} The following local variant of \Cref{purity for rel. dim 1} is a non-Noetherian counterpart of \cite{CTS79}{Théorème~6.13}. \begin{thm-tweak} \label{extends across codim-2 points} For a finite-rank valuation ring $R$ with spectrum $(S,\eta)$, an $S$-flat finite type scheme $X$ with regular fibers, a reductive $X$-group scheme $G$, and a point $x$ that is \begin{enumerate}[label={{\upshape(\roman)}}] \item\label{g-tor-local-gen} either $x\in X_{\eta}$ with $\dim \mathscr{O}_{X_{\eta},x} =2$, or \item\label{g-tor-local-ngen} $x\in X_s$ with $s \neq \eta$ and $\dim \mathscr{O}_{X_s,x} =1$, \end{enumerate} every $G$-torsor on $\Spec \mathscr{O}_{X,x} \backslash \{x\}$ extends uniquely to a $G$-torsor on $\Spec \mathscr{O}_{X,x}$. \end{thm-tweak} \begin{proof} It suffices to show that restriction induces an equivalence of categories of $G$-torsors on $\Spec \mathscr{O}_{X,x}$ and on $\Spec \mathscr{O}_{X,x}\backslash \{x\}$. The argument of \Cref{purity for rel. dim 1} reduces us to the case of vector bundles, namely, to the case $G=\GL_n$. Now the case \ref{g-tor-local-gen} is classical (see for instance, \cite{Gab81}{\S1, Lemma~1}). For \ref{g-tor-local-ngen}, the finite-rank assumption on $V$ guarantees $X$ and hence also $\Spec\,\mathscr{O}_{X,x} \backslash \{x\}$ to be topologically Noetherian and, in particular, quasi-compact. Therefore, by \Cref{GR18}\ref{GR18-11.4.6}, every vector bundle $\mathscr{E}$ on $\Spec \mathscr{O}_{X,x}\backslash \{x\}$, extends to a reflexive sheaf $j_{\ast}(\mathscr{E})$ on $\Spec \mathscr{O}_{X,x}$, which, by the assumption $\dim \mathscr{O}_{X_s,x}=1$ and \Cref{GR18}\ref{GR18-11.4.1}, is projective, hence the assertion follows. \end{proof} Granted the purity \Cref{extends across codim-2 points}, we extend reductive torsors outside a closed subset of higher codimension that is crucial for the proof of \Cref{G-S for constant reductive gps}. \begin{prop-tweak} \label{extension torsors} For a semilocal Pr\"{u}fer affine scheme $S$, an $S$-flat finite type quasi-separated scheme $X$ with regular $S$-fibers, a closed subset $Z\subset X$ such that $X\backslash Z\subset X$ is quasi-compact and satisfies \[ \text{$\codim(Z_{\eta}, X_{\eta})\geq 2$\,\, for each generic point $\eta\in S$ \quad and\quad $\codim(Z_s,X_s)\geq 1$\,\, for all $s\in S$,} \] a reductive $X$-group scheme $G$, and a $G$-torsor $\mathcal{P}$ on $X\backslash Z$, there is a closed subset $Z^{\prime}\subset Z$ satisfying \[ \text{$\codim(Z_{\eta}', X_{\eta})\geq 3$\,\, for each generic point $\eta\in S$ \quad and\quad $\codim(Z_s',X_s)\geq 2$\,\, for all $s\in S$,} \] and a $G$-torsor $\mathcal{Q}$ on $X\backslash Z'$ such that $\mathcal{P}\simeq \mathcal{Q}\|_{X\backslash Z}$. \end{prop-tweak} \begin{proof} By \cite{Nag66}{Theorem 3'}, $X$ is finitely presented over $S$, hence a limit argument involving Lemmata \ref{approxm semi-local Prufer ring} and \ref{lim-codim}, we are reduced to the case when all local rings of $R$ are valuation rings of finite ranks. Let $\mathcal{P}_{X\backslash Z}$ be a $G$-torsor on $X\backslash Z$. Since $\|S\|$ is finite and each fiber $X_s$ is Noetherian, there are finitely many points $x\in Z$ satisfying one of the assumptions (i)--(ii) of \Cref{extends across codim-2 points}; among these points we pick a maximal one under the generalization, say $x$. The maximality of $x$ implies that $(X\backslash Z)\cap \Spec \,\mathscr{O}_{X,x} = \Spec \,\mathscr{O}_{X,x} \backslash \{x\}$, so, by \Cref{extends across codim-2 points}, the $G$-torsor $\mathcal{P}_{X\backslash Z}\|_{(X\backslash Z) \cap \Spec (\mathscr{O}_{X,x})}$ extends to a $G$-torsor $\mathcal{P}_x$ on $\Spec \,\mathscr{O}_{X,x}$. As $X$ is topologically Noetherian, we may spread out $\mathcal{P}_x$ to obtain a $G$-torsor $\mathcal{P}_{U_x}$ over an open neighbourhood ${U}_x$ of $x$ such that $\mathcal{P}_{X\backslash Z}\|_{({X\backslash Z})\cap U_x} \simeq \mathcal{P}_{U_x}\|_{(X\backslash Z)\cap U_x}$ as $G$-torsors over $(X\backslash Z)\cap U_x$. Consequently, we may glue $\mathcal{P}_{X\backslash Z}$ with $\mathcal{P}_{U_x}$ to extend $\mathcal{P}_{X\backslash Z}$ to a $G$-torsor on $U_1\colonequals (X\backslash Z)\cup U_x$. Since $Z_1\colonequals X\backslash U_1$ contains strictly fewer points $x$ satisfying the assumptions (i) or (ii) of \Cref{extends across codim-2 points}, we extend $\mathcal{P}$ iteratively to find the desired closed subset $Z^{\prime}\subset X$ such that $\mathcal{P}_{X\backslash Z}$ extends over $X\backslash Z^{\prime}$. \end{proof} \begin{cor-tweak} \label{extend generically trivial torsors} For a semilocal Pr\"{u}fer affine scheme $S$, an $S$-flat finite type quasi-separated scheme $X$ with regular $S$-fibers, a finite subset $\textbf{x}\subset X$ contained in an single affine open, a nonzero $r\in \mathscr{O}_{X,\mathbf{x}}$, and a reductive $X$-group scheme $G$, every generically trivial $G$-torsor on $\mathscr{O}_{X,\textbf{x}}[\frac{1}{r}]$ extends to a $G$-torsor on an open neighbourhood $U$ of $\mathrm{Spec}\,\mathscr{O}_{X,\textbf{x}}[\frac{1}{r}]$ whose complementary closed $Z\colonequals X\backslash U$ satisfies the following \[ \text{$\codim(Z_{\eta}, X_{\eta})\geq 3$\,\, for each generic point $\eta\in S$ \quad and\quad $\codim(Z_s,X_s)\geq 2$\,\, for all $s\in S$.} \] \end{cor-tweak} \begin{proof} As in the proof of \Cref{extension torsors}, we may assume that $S$ has finite Krull dimension; in particular, $X$ is topologically Noetherian. Let $\mathcal{P}$ be a generically trivial $G$-torsor on $\mathscr{O}_{X,\textbf{x}}[\frac{1}{r}]$. By spreading out, $\mathcal{P}$ extends to a $G$-torsor $\mathcal{P}_U$ on $U\colonequals \Spec R[\f{1}{r}]$ for an affine open neighbourhood $\Spec R \subset X$ of $\textbf{x}$. It remains to extend $U$ and $\mathcal{P}_U$ to ensure that $Z\colonequals X\backslash U$ satisfies the assumptions of \Cref{extension torsors}. Assume that $Z$ contains a point $z$ such that either \begin{enumerate}[label={{\upshape(\roman)}}] \item $z\in X_{\eta}$ and $\dim \mathscr{O}_{X,z} =1$, in which case $\Spec (\mathscr{O}_{X,z}) \cap U$ is a maximal point of $X$, or \item $z$ is a maximal point of $X_s$ with $s \neq \eta$, in which case $\Spec \,\mathscr{O}_{X,z}$, and hence also $\Spec (\mathscr{O}_{X,z}) \cap U$, is the spectrum of a valuation ring (\Cref{geom}~\ref{geo-iii}). \end{enumerate} By the Grothendieck--Serre on valuation rings \cite{Guo20b}, the generically trivial $G$-torsor $\mathcal{P}_U\|_{\Spec (\mathscr{O}_{X,z}) \cap U}$ is trivial. Thus, as in the proof of \Cref{extension torsors}, we can glue $\mathcal{P}_U$ with the trivial $G$-torsor over a small enough open neighbourhood of $z$ to extend $\mathcal{P}_U$ across such a point $z\in Z$. Note that, due to the topologically Noetherianness of $X$, $Z$ contains at most finitely many points $z$ satisfying the above assumption (i) or (ii). Therefore, iteratively extend $U$ and $\mathcal{P}_U$, we may assume that $Z$ does not contain any point $z$ satisfying (i) or (ii), whence \Cref{extension torsors} applies. \end{proof} \section{Low dimensional cohomology of groups of multiplicative type} \label{low dim cohomology of mult gp} \begin{pp}[Vanishing of low-dimensional local cohomology of tori] \end{pp} \begin{prop-tweak}\label{local cohomology of tori} For a finite-rank valuation ring $R$ with spectrum $S$ and generic point $\eta\in S$, an $S$-flat finite type scheme $X$ with regular $S$-fibers, a point $x\in X$, and an $\mathscr{O}_{X,x}$-torus $T$, \begin{enumerate}[label={{\upshape(\roman)}}] \item \label{paraf} if either $x\in X_{\eta}$ with $\dim \mathscr{O}_{X_{\eta},x} \ge 2$, or $x\in X_s$ with $s \neq \eta$ and $\dim \mathscr{O}_{X_s,x} \ge 1$, then we have \[ \text{$H^i_{\{x\}}(\mathscr{O}_{X,x},T)=0$ \quad for \, $0\le i \le 2$; } \] \item \label{val} otherwise, $\mathscr{O}_{X,x}$ is a valuation ring, and, in case $T$ is $\mathrm{flasque}$, we have \[ H^2_{\{x\}}(\mathscr{O}_{X,x},T)=0. \] \end{enumerate} \end{prop-tweak} \begin{proof} In \ref{paraf}, by \cite{EGAIV3}{Lemme~14.3.10}, $\overline{\{x\}}$ is $R$-fiberwise of codimension $\ge 2$ (resp., $\ge 1$) in the generic fiber (resp., non-generic fiber) of $X$, so, by \Cref{GR18}~\ref{GR18-11.4.6}, for any open $x\in U \subset X$ we have \[ H^i(U,\mathbb{G}_{m})\simeq H^i(U\backslash \overline{\{x\}},\mathbb{G}_{m}) \quad \text{ for } \quad 0\le i \le 1. \] The same is true with $(U,x)$ replaced by any scheme $(W,y)$ \'etale over it, so taking colimit yields \[ H^i_{\mathrm{\acute{e}t}}(\mathscr{O}_{X,\bar{x}}^{\text{sh}},\mathbb{G}_m)\simeq \text{colim}_{(W,y)}\,H_{\mathrm{\acute{e}t}}^i(W,\mathbb{G}_{m})\simeq \text{colim}_{(W,y)}\,H_{\mathrm{\acute{e}t}}^i(W\backslash \overline{\{y\}},\mathbb{G}_{m}) \quad \text{ for } \quad 0\le i \le 1. \] By the finite-rank assumption on $R$, $\mathscr{O}_{W,y}$ and hence also $\mathscr{O}_{X,\bar{x}}^{\text{sh}}$ has quasi-compact punctured spectrum, so the rightmost term above identifies with $H^i_{\mathrm{\acute{e}t}}(\Spec(\mathscr{O}_{X,\bar{x}}^{\mathrm{sh}})\backslash \{\overline{x}\},\mathbb{G}_m)$. Looking at the local cohomology exact sequence for the pair $(\Spec(\mathscr{O}_{X,\bar{x}}^{\mathrm{sh}}),\bar{x})$ and $T\simeq \mathbb{G}_{m}^{\dim\,T}$, we deduce that $$H_{\{\bar{x}\}}^i(\mathscr{O}_{X,\bar{x}}^{\mathrm{sh}},T)=0 \quad \text{ for } \quad 0\le i\le 2.$$ By the local-to-global $E_2$-spectral sequence \cite{SGA4II}{Exposé~V, Proposition~6.4}, this implies the desired vanishing. In \ref{val}, either $x\in X_{\eta}$ with $\dim \mathscr{O}_{X_{\eta},x}\le 1,$ then $\mathscr{O}_{X,x}$ is a discrete valuation ring, or $x$ is a maximal point of some fiber of $X\to S$, then, by \Cref{geom}~\ref{geo-iii}, $\mathscr{O}_{X,x}$ is a valuation ring. The desired vanishing is proven in \cite{Guo20b}{Lemma~2.3}. \end{proof} \Cref{local cohomology of tori} has the following global consequence. \begin{thm-tweak} \label{purity for gp of mult type} For a semilocal Pr\"ufer scheme $S$, a flat, locally of finite type $S$-scheme $X$ with regular $S$-fibers, a closed $Z\subset X$ with retrocompact $X\backslash Z$, and a finite type $X$-group $M$ of multiplicative type, \begin{enumerate}[label={{\upshape(\roman)}}] \item \label{mult-paraf} if $Z$ satisfies the following condition \[ \text{$\codim(Z_{\eta}, X_{\eta})\geq 2$ for every generic point $\eta\in S$ \quad and\quad $\codim(Z_s,X_s)\geq 1$ for all $s\in S$,} \] then we have \[ H^i_{\mathrm{fppf}}(X,M)\simeq H^i_{\mathrm{fppf}}(U,M)\quad \,\text{for }0\le i\le 1 \quad \text{and} \quad\, H^2_{\mathrm{fppf}}(X,M)\hookrightarrow H^2_{\mathrm{fppf}}(U,M). \] \item \label{gl-H1-H2-flasque} if $X$ is qcqs and $M=T$ is an $X$-torus such that $T_{\mathscr{O}_{X,z}}$ is $\mathrm{flasque}$ for every $z\in Z$ for which $\mathscr{O}_{X,z}$ is a valuation ring, then we have \[ \text{$H^1_{\mathrm{\acute{e}t}}(X,T) \twoheadrightarrow H^1_{\mathrm{\acute{e}t}}(U,T)$ \quad and \quad $H^2_{\mathrm{\acute{e}t}}(X,T)\hookrightarrow H^2_{\mathrm{\acute{e}t}}(U,T)$;} \] in particular, if $K(X)$ denotes the total ring of fractions of $X$, \[ \text{$\mathrm{Pic}\, X \twoheadrightarrow \mathrm{Pic}\, U$ \quad and \quad $\mathrm{Br}(X)\hookrightarrow \mathrm{Br}(K(X))$.} \] \end{enumerate} \end{thm-tweak} \begin{proof} For \ref{mult-paraf}, by the local cohomology exact sequence for the pair $(X,Z)$ and the sheaf $M$, the statement is equivalent to the vanishings $H^i_Z(X,M)=0$ for $0\le q \le 2$. By the spectral sequence in \cite{CS21}{Lemma 7.1.1}, it suffices to show the vanishings of $\mathcal{H}^q_Z(M)$, the \'etale-sheafification of the presheaf $X^{\prime}\mapsto H^q_{Z^{\prime}}(X^{\prime}, M)$ on $X_{\mathrm{\acute{e}t}}$, where $Z^{\prime}\colonequals Z\times_X X^{\prime}$. This problem is \'etale-local on $X$, so we may assume that $X$ is affine, and $M$ splits as $\mathbb{G}_m$ or $\mu_n$, and, since $\mu_n=\ker(\mathbb{G}_m\overset{\times n}{\rightarrow}\mathbb{G}_m)$, even $M=\mathbb{G}_m$. Now, $X$ is qcqs and, by \cite{Nag66}{Theorem 3'}, is finitely presented over $S$, so a standard limit argument involving Lemmata \ref{approxm semi-local Prufer ring} and \ref{lim-codim} reduces us to the case $R$ having finite Krull dimension; in particular, $X$ is topologically Noetherian. Recall the coniveau spectral sequence \cite{Gro68c}{\S10.1} \[ \textstyle E^{pq}_2=\bigoplus_{z\in Z^{(p)}} H^{p+q}_{\{z\}}(M)\Rightarrow H^{p+q}_Z(X,M); \] the topological Noetherianness of $X$ allows us to identify \[ \text{$H^{p+q}_{\{z\}}(M)\colonequals \text{colim}\,H^{p+q}_{\overline{\{z\}}\cap U}(U,M)$ \quad as \quad $H^{p+q}_{\{z\}}(\mathscr{O}_{X,z},M)$,} \] where $U$ runs over the open neighbourhoods of $z$ in $X$. Therefore, it is enough to show $H^i_{\{z\}}(\mathscr{O}_{X,z},\mathbb{G}_m)=0$ for $z\in Z$ and $0\le i \le 2$, which was settled by \Cref{local cohomology of tori}\ref{paraf}. For \ref{gl-H1-H2-flasque}, the local cohomology exact sequence reduces us to show the vanishing $H^2_Z(X,T)=0$. Similar to the above, as $X$ is qcqs and $X\backslash Z\subset X$ is a qc open, a standard limit argument involving \Cref{approxm semi-local Prufer ring} reduces us to the case $R$ having finite Krull dimension, in particular, $X$ is topologically Noetherian. The coniveau spectral sequence reduces us to show $H^2_{\{z\}}(\mathscr{O}_{X,z},T)=0$, which was settled by \Cref{local cohomology of tori}. \end{proof} \begin{pp}[Grothendieck--Serre type results for tori] \end{pp} \begin{lemma-tweak} \label{non-Noeth-pullback line bundle} Let $\phi:X\to Y$ be a morphism of schemes. Let $\mathscr{L}$ be an invertible $\mathscr{O}_X$-module. If \begin{itemize} \item [(1)] $Y$ is quasi-compact quasi-separated, integral, and normal, \item [(2)] there exist a smooth projective morphism $\overline{\phi}:\overline{X}\to Y$, with geometrically integral fibers, and a quasi-compact open immersion $X\hookrightarrow \overline{X}$ over $Y$, and \item [(3)] $\mathscr{L}$ is trivial when restricted to the generic fiber of $\phi$, \end{itemize} then $\mathscr{L}\simeq \phi^\mathscr{N}$ for some invertible $\mathscr{O}_Y$-module $\mathscr{N}$. \end{lemma-tweak} \begin{proof} When $Y$ is Noetherian, this follows from a much more general result \SP{0BD6}, where (2) can be replaced by the weaker assumption that $X\to Y$ is faithfully flat of finite presentation, with integral fibers. The general case follows from this via Noetherian approximations. Precisely, as $Y$ is qcqs, we can use \SP{01ZA} to write $Y=\lim_i Y_i$ for a filtered inverse system $\{Y_i\}$ of finite type integral $\mathbf{Z}$-schemes with affine transition morphisms. Since $Y$ is normal and the normalization of a finite type integral $\mathbf{Z}$-scheme is again of finite type over $\mathbf{Z}$, we may assume that each $Y_i$ is normal. Next, as $\phi$ is qcqs and of finite presentation, by \SPD{01ZM}{0C0C}, for some $i_0$ there exist a finite type smooth morphism $\overline{\phi}_{i_0}:\overline{X}_{i_0}\to Y_{i_0}$ such that $\overline{X}\simeq \overline{X}_{i_0}\times_{Y_{i_0}}Y$ as $Y$-schemes, an open subscheme $X_{i_0}\subset \overline{X}_{i_0}$ whose pullback to $\overline{X}$ identifies with $X$, and, by \SP{{0B8W}}, there is an invertible $\mathscr{O}_{X_{i_0}}$-module $\mathscr{L}_{i_0}$ whose pullback to $X$ is isomorphic to $\mathscr{L}$. For any $i\ge i_0$ denote by $\phi_i:X_i\colonequals X_{i_0}\times_{Y_{i_0}}Y_i\to Y_i$ the base change of $\overline{\phi}_{i_0}\|_{X_{i_0}}$ to $Y_i$, and denote by $\mathscr{L}_i$ the pullback of $\mathscr{L}_{i_0}$ to $X_i$. By \SPD{01ZM}{01ZP}, any projective embedding of $\overline{X}$ over $Y$ descends to a projective embedding of $\overline{X}_{i}$ over $Y_i$ for large $i$; in particular, $\overline{\phi}_i$ is projective for large $i$. Since $Y$ is normal, the assumption (2) implies that the Stein factorization of $\overline{\phi}$ is itself; in particular, $\mathscr{O}_Y\isoto \overline{\phi}_\mathscr{O}_{\overline{X}}$. This implies that the finite extension $\mathscr{O}_{Y_{i_0}} \hookrightarrow \overline{\phi}_{i_0,}\mathscr{O}_{\overline{X}_{i_0}}$ is an isomorphism, because its base change to the function field of $Y$ is so and $Y_{i_0}$ is normal. In particular, by Zariski's main theorem, $\overline{\phi}_{i_0}$ has connected geometric fibers; as it is also smooth, all its fibers are even geometrically integral. By limit formalism, for large $i$, $\mathscr{L}_i$ is trivial when restricted to the generic fiber of $\phi_i$. Consequently, for large $i$, the morphism $\phi_i:X_i\to Y_i$ and the invertible $\mathscr{O}_{X_i}$-module $\mathscr{L}_i$ satisfy all the assumptions of the Lemma, so $\mathscr{L}_i\simeq \phi_i^\mathscr{N}_i$ for some invertible $\mathscr{O}_{Y_i}$-module $\mathscr{N}_i$. Then $\mathscr{L}\simeq \phi^\mathscr{N}$ for $\mathscr{N}$ the pullback of $\mathscr{N}_i$ to $Y$. \end{proof} \begin{prop-tweak} [\emph{cf.}~\cite{CTS87}{4.1--4.3}]\label{G-S type results for mult type} For a Pr\"ufer domain $R$, an irreducible scheme $X$ essentially smooth over $R$ with function field $K(X)$, an $X$-group scheme $M$ of multiplicative type, and a connected finite \'etale Galois covering $X'\to X$ splitting $M$, the restriction maps \[ H^1_{\mathrm{fppf}}(X,M) \rightarrow H^1_{\mathrm{fppf}}(K(X),M)\quad \text{ and } \quad H^2_{\mathrm{fppf}}(X,M) \rightarrow H^2_{\mathrm{fppf}}(K(X),M) \] are injective in each of the following cases: \begin{enumerate}[label={{\upshape(\roman)}}] \item \label{G-S for mult type gp}$X=\Spec A$ for $A$ a semilocal ring essentially smooth over $R$; \item \label{B-Q for mult type} For some essentially smooth semilocal $R$-algebra $A$, there exists a quasi-compact open immersion $X\hookrightarrow \overline{X}$, where $\overline{X}$ is a smooth projective $A$-scheme, with geometrically integral fibers, such that $\Pic X_L=0$ for any finite separable fields extension $L/\Frac A$, and $M=N_X$ for $N$ an $A$-group of multiplicative type (for instance, $X$ could be any quasi-compact open subscheme of $\mathbb{P}_A^N$); \item any subcovering $X''\to X$ of $X'\to X$ satisfies $\mathrm{Pic}\,X''=0$. \end{enumerate} Further, if $M$ is a $\mathrm{flasque}$ $X$-torus, then in all cases $\mathrm{(i)}$--$\mathrm{(iii)}$ the restriction map \[ H^1_{\mathrm{\acute{e}t}}(X,M) \isoto H^1_{\mathrm{\acute{e}t}}(K(X),M) \quad \text{ is bijective.} \] \end{prop-tweak} \begin{proof} It is clear that (i) is a particular case of (ii). Let us show that (ii) is a particular case of (iii). Let $A\to B$ be a connected finite \'etale Galois covering that splits $N$. Take $X'\colonequals X\times_AB$. As $A$ is normal and $X \to \Spec A$ is smooth, $X$ is also normal. Then, since $X\to \Spec A$ has geometrically integral generic fiber, the natural map $\pi_1^{\mathrm{\acute{e}t}}(X)\to \pi_1^{\mathrm{\acute{e}t}}(\Spec A)$ is surjective. This implies that any subcovering $X''\to X$ of $X'\to X$ is of the form $X''=X\times_AC$ for some subcovering $A\to C$ of $A\to B$. By assumption, $\text{Pic}\,X_{\Frac (C)}=0$, so we may apply \Cref{non-Noeth-pullback line bundle} and the morphism $X\times_AC \to \Spec C$ to deduce that \[ \text{the pullback map \quad $\Pic C \rightarrow \Pic (X\times_A C)$\quad is surjective. } \] Since $C$ is semilocal, we conclude that $\Pic C=0= \Pic (X\times_AC)$. It is thus enough to prove all assertions only for (iii). Assume first that $M=T$ is an $X$-torus. Take \[ \text{a flasque resolution\quad\quad $1\to F \to P \to T \to 1$,\quad\quad} \] where $F$ is a flasque $X$-torus and $P$ is a quasitrivial $X$-torus. This yields a commutative diagram \[ \begin{tikzcd} {H^1_{\mathrm{\acute{e}t}}(X,P)} \arrow[r] & {H^1_{\mathrm{\acute{e}t}}(X,T)} \arrow[r] \arrow[d, "\rho_1"] & {H^2_{\mathrm{\acute{e}t}}(X,F)} \arrow[d, "\rho_2"] \\ & {H^1_{\mathrm{\acute{e}t}}(K(X),T)} \arrow[r] & {H^2_{\mathrm{\acute{e}t}}(K(X),F)} \end{tikzcd} \] with exact rows. Now the quasitrivial torus $P$ is isomorphic to a finite direct product of tori $\mathrm{Res}_{X^{\prime\prime}/X}\mathbb{G}_{m,X^{\prime\prime}}$ for finite \'etale subcoverings $X''\to X$ of $X'\to X$. Hence, assumption (iii) implies that $H^1_{\mathrm{\acute{e}t}}(X,P)=0$, and so the injectivity of $\rho_1$ reduces to that of $\rho_2$. To prove that $\rho_2$ is injective we pick $a\in H^2_{\mathrm{\acute{e}t}}(X,F)$ for which $a\|_{K(X)}=0$. By spreading out, we may assume that $X$ is a localization of an irreducible, smooth, affine $R$-scheme $\widetilde{X}$, $F=\widetilde{F}_X$ for a flasque $\widetilde{X}$-torus $\widetilde{F}$, and $a=\widetilde{a}\|_X$ for some class $\widetilde{a} \in H^2(\widetilde{X},\widetilde{F})$. Since $\widetilde{a}\|_{K(X)}=0$, for a proper closed subset $Z\subset \widetilde{X}$, $$ \widetilde{a}\|_{\widetilde{X}\backslash Z}=0 \in H^2_{\mathrm{\acute{e}t}}(\widetilde{X}\backslash Z,\widetilde{F}). $$ By \Cref{purity for gp of mult type}\ref{gl-H1-H2-flasque}, $\widetilde{a}=0$, so $a=\widetilde{a}\|_X=0$. This proves the injectivity of $\rho_2$ and hence also of $\rho_1$. Now let $M$ be an arbitrary $X$-group of multiplicative type, then there is an $X$-subtorus $T\subset M$ such that $\mu\colonequals M/T$ is $X$-finite. Consequently, for any generically trivial $M$-torsor $\mathcal{P}$, the $\mu$-torsor $\mathcal{P}/T$ is finite over $X$; as $X$ is normal, this implies $(\mathcal{P}/T)(X)=(\mathcal{P}/T)(K(X))$. Therefore, $\mathcal{P}/T\to X$ has a section that lifts to a generic section of $\mathcal{P}\to X$, that is, $\mathcal{P}$ reduces to a generically trivial $T$-torsor $\mathcal{P}_T$. By the injectivity of $\rho_1$, $\mathcal{P}_T$ and hence also $\mathcal{P}$ is trivial. This proves the injectivity of $H^1_{\mathrm{\acute{e}t}}(X,M) \to H^1_{\mathrm{\acute{e}t}}(K(X),M)$. On the other hand, there is a short exact sequence \[ 1\to M\to F \to P \to 1 \] of $X$-groups of multiplicative type, where $F$ is flasque and $P$ is quasitrivial, both split after base change by $X'\to X$. This yields the following commutative diagram with exact rows \[ \begin{tikzcd} {H^1_{\mathrm{fppf}}(X,P)} \arrow[r] & {H^2_{\mathrm{fppf}}(X,M)} \arrow[r] \arrow[d,"\rho_3"] & {H^2_{\mathrm{fppf}}(X,F)} \arrow[d,"\rho_2"] \\ & {H^2_{\mathrm{fppf}}(K(X),M)} \arrow[r] & {H^2_{\mathrm{fppf}}(K(X),F)} \end{tikzcd} \] Since we have already shown that $H^1_{\mathrm{fppf}}(X,P)=0$ and $\rho_2$ is injective, the injectivity of $\rho_3$ follows. Finally, if $M$ is a flasque $X$-torus, the bijectivity of $H^1_{\mathrm{fppf}}(X,M) \to H^1_{\mathrm{fppf}}(K(X),M)$ will follow if one proves its surjectivity, but the latter follows from \Cref{purity for gp of mult type}\ref{gl-H1-H2-flasque} via a limit argument. \end{proof} \section{Geometric lemmata} \label{sect-geom lem} \begin{pp}[Geometric presentation lemma in the style of Gabber--Quillen] In both of the works of Fedorov and $\check{\mathrm{C}}$esnavi$\check{\mathrm{c}}$ius on mixed charateristic Grothendieck--Serre, a certain type geometric results in the style of Gabber--Quillen play a prominent role, see \cite{Fed22b}{Proposition~3.18} and \cite{Ces22a}{Variant~3.7}, respectively. We begin with the following analog of \emph{loc.~cit}. \end{pp} \begin{lemma-tweak} \label{Ces's Variant 3.7} For a semilocal Pr\"{u}fer ring $R$, a projective, flat $R$-scheme $\overline{X}$ with fibers of pure dimension $d>0$, an $R$-smooth open subscheme $X$, a finite subset $\textbf{x}\subset X$, and a closed subscheme $Y\subset X$ that is $R$-fiberwise of codimension $\ge 1$ such that $\overline{Y}\backslash Y$ is $R$-fiberwise of codimension $\ge 2$ in $\overline{X}$, there are affine opens $S\subset \mathbb{A}_R^{d-1}$ and $\textbf{x}\subset U\subset X$ and a smooth morphism $\pi:U\to S$ of relative dimension 1 such that $Y \cap U$ is $S$-finite. \end{lemma-tweak} \begin{proof} This can be proved similarly as \cite{Ces22a}{Variant~3.7}. \end{proof} \begin{pp}[A variant of Lindel's lemma] According to a lemma of Lindel \cite{Lin81}{Proposition~1 \emph{et seq} Lemma}, an \'etale extension of local rings $A\to B$ with trivial extension of residue fields induces isomorphisms $$A/r^nA\isoto B/r^nB, \quad \text{ where } \quad n\ge1,$$ for a well-chosen non-unit $r\in A$. In our context in which the prescribed $B$ is essentially smooth over a valuation ring, we will prove the following variant of \emph{loc. cit.} that allows us to fix the $r\in B$ in advance, at the cost of that $A$ is a carefully-chosen local ring of an affine space over that valuation ring. This result will be the key geometric input for dealing with torsors under a reductive group scheme that descends to the Pr\"{u}fer base ring, and, as the cited work of Lindel on the Bass--Quillen conjecture for vector bundles, it reduces us to studying torsors on opens of affine spaces. \end{pp} \begin{prop-tweak} \label{variant of Lindel's lem} Let $\Lambda$ be a semilocal Pr\"{u}fer domain, $X$ an irreducible, $\Lambda$-smooth affine scheme of pure relative dimension $d>0$, $Y\subset X$ a finitely presented closed subscheme that avoids all the maximal points of the $\Lambda$-fibers of $X$, and $\textbf{x} \subset X$ a finite subset. Assume that for every maximal ideal $\mathfrak{m}\subset \Lambda$ with finite residue field, there are at most $\max(\#\,\kappa(\mathfrak{m}),d)-1$ points of $\textbf{x}$ lying over $\mathfrak{m}$. There are an affine open neighbourhood $W\subset X$ of $\textbf{x}$, an affine open $ U\subset \mathbb{A}_{\Lambda}^d$, and an \'etale surjective $\Lambda$-morphism $f:W\to U$ such that the restriction $f\|_{W\cap Y}:W\cap Y \to U$ is a closed immersion and $f$ induces a Cartesian square: \begin{equation} \begin{tikzcd} W\cap Y \arrow[r, hook] \arrow[d, equal] & W \arrow[d, "f"] \\ W\cap Y \arrow[r, hook] & U. \end{tikzcd} \end{equation} \end{prop-tweak} \begin{rem-tweak}\label{rem on Lindel's lem} The assumption on the cardinality of $\textbf{x}$ holds, for instance, if either $\textbf{x}$ is a singleton or $d>\# \,\textbf{x}$. The latter case will be critical for us to settle the general semilocal case of \Cref{G-S for constant reductive gps}. On the other hand, the following finite field obstruction shows a certain assumption on $\# \textbf{x}$ is necessary: if $d=1$ and $\Lambda=k$ is a finite field, then the map $f$ delivered from \Cref{variant of Lindel's lem} gives a closed immersion $\textbf{x} \hookrightarrow \mathbb{A}_k^1$, which is impossible as soon as $\# \, \textbf{x} > \# \,k$. \end{rem-tweak} To prove \Cref{variant of Lindel's lem}, we begin with the following reduction to considering closed points. \begin{lemma-tweak} \label{reduction to closed points} The proof of \Cref{variant of Lindel's lem} reduces to the case when $\textbf{x}$ consists of closed points of the closed $\Lambda$-fibers of $X$. \end{lemma-tweak} \begin{proof} First, by a standard limit argument involving \Cref{approxm semi-local Prufer ring}, we can reduce to the case when $\Lambda$ has finite Krull dimension. Next, if for each $x\in \textbf{x}$ the closure $\overline{\{x\}}$ contains a closed point $x'$ of the closed $\Lambda$-fibers of $X$ and if the new collection $\{x':x\in \textbf{x}\}$ satisfies the same cardinality assumption on $\textbf{x}$, we can simply replace each $x$ by $x'$ to complete the reduction process. However, it may happen that $\overline{\{x\}}$ does not contain any point of the closed $\Lambda$-fibers of $X$, and even if it does, the new collection $\{x':x\in \textbf{x}\}$ may not satisfy the cardinality assumption on $\textbf{x}$. To overcome this difficulty, we will use a trick by adding auxiliary primes to $\Spec \Lambda$ (and adding the corresponding fibers to $X$ and $Y$) so that $\overline{\{x\}}$ contains closed points of the closed $\Lambda$-fibers of $X$ for all $x\in \textbf{x}$. More precisely, we will show that there are a semilocal Pr\"{u}fer domain $\Lambda'$, an open embedding $\Spec \Lambda \subset \Spec \Lambda'$, an irreducible, affine, $\Lambda'$-smooth scheme $X'$ of pure relative dimension $d$, a closed $\Lambda'$-subscheme $Y'\subset X'$ that avoids all the maximal points of the $\Lambda'$-fibers of $X'$, and a $\Lambda$-isomorphism $X'_{\Lambda} \simeq X$ that identifies $Y'_{\Lambda}$ with $Y$ such that the assumptions of the second sentence of this paragraph hold for our new $X'$ and $Y'$. To construct the desired $\Lambda'$ (and $X',Y'$), we can first use the specialization technique to reduce to the case when all points of $\textbf{x}$ are closed in the \emph{corresponding} $\Lambda$-fibers of $X$, that is, if $x\in \textbf{x}$ lies over $\mathfrak{p}\subset \Lambda$, then $x$ is $\kappa(\mathfrak{p})$-finite. For the rest of proof we will assume, without lose of generality, that there is exactly one point of $\textbf{x}$, say $x$, that lies over some \emph{non-maximal} prime of $ \Lambda$, say $\mathfrak{p}$. Write $\Lambda_{\mathfrak{p}}=\bigcup A$ as a filtered union of its finitely generated $\textbf{Z}$-subalgebras $A$. By a standard limit argument (\SPD{0EY1}{0C0C}), for large enough $A$, \begin{itemize} \item [(a)] $X_{\Lambda_{\mathfrak{p}}}$ descends to an irreducible, affine, $A$-smooth scheme $\mathcal{X}$ of pure relative dimension $d$; \item [(b)] the finitely presented closed subscheme $Y_{\Lambda_{\mathfrak{p}}}\subset X_{\Lambda_{\mathfrak{p}}}$ descends to a closed $A$-subscheme $\mathcal{Y} \subset \mathcal{X}$ which, upon enlarging $A$, avoids all the maximal points of the $A$-fibers of $\mathcal{X}$: by \cite{EGAIV3}{Proposition~9.2.6.1}, the subset \[ \text{$\{s\in \Spec A: \dim \mathcal{Y}_s=d\} \subset \Spec A$\quad is constructible,} \] and its base change over $\Lambda_{\mathfrak{p}}=\lim_A A$ is empty, so we may assume that it is already empty; \item [(c)] the $\kappa(\mathfrak{p})$-finite point $x$ descends to a $A/\mathfrak{p}_A$-finite closed subscheme $\widetilde{x}\subset \mathcal{X}_{A/\mathfrak{p}_A}$, where $\mathfrak{p}_A\colonequals A\cap \mathfrak{p}$; \end{itemize} For any prime $\Lambda \supset \mathfrak{q} \supset \mathfrak{p}$ with $\mathrm{ht}(\mathfrak{q})=\text{ht}(\mathfrak{p})+1$, choose an element $a_{\mathfrak{q}} \in \mathfrak{q} \backslash \mathfrak{p}$. We assume that \begin{itemize} \item [(d)] $a_{\mathfrak{q}}^{-1} \in A$ for all such $\mathfrak{q}$. (This guarantees the equality $A\cdot \Lambda_{\mathfrak{m}}=\Lambda_{\mathfrak{p}}$ for every maximal ideal $\mathfrak{m}\subset \Lambda$ containing $\mathfrak{p}$.) \end{itemize} Since a maximal ideal $\mathfrak{m}\subset \Lambda$ containing $\mathfrak{p}$ gives rise to a non-trivial valuation ring $\Lambda_{\mathfrak{m}}/\mathfrak{p}\Lambda_{\mathfrak{m}}$ of $\kappa(\mathfrak{p})$, we see that the field $\kappa(\mathfrak{p})$ is not finite. As $\kappa(\mathfrak{p}) =\bigcup_A A/\mathfrak{p}_A$, by enlarging $A$ we may assume that $A/\mathfrak{p}_A$ is also not a finite, so we can find a nonzero prime $ \mathfrak{p}' \subset A/\mathfrak{p}_A$.\footnote{We use the following fact: a prime ideal of a finite type $\mathbf{Z}$-algebra is maximal if and only if its residue field is finite.} Choose a valuation ring of $\kappa(\mathfrak{p}_A)$ with center $\mathfrak{p}'$ in $A/\mathfrak{p}_A$, and then extend it to a valuation ring $V_{\mathfrak{p}'}$ of $\kappa(\mathfrak{p})$. Let $V$ be the composite of $\Lambda_{\mathfrak{p}}$ and $V_{\mathfrak{p}'}$; explicitly, $V$ is the preimage of $V_{\mathfrak{p}'}$ under the reduction map $\Lambda_{\mathfrak{p}} \twoheadrightarrow \kappa(\mathfrak{p})$. Then $V$ is a valuation ring of $\Frac (\Lambda)$, and, by the above assumption (d), the equality $V\cdot \Lambda_{\mathfrak{m}}=\Lambda_{\mathfrak{p}}$ holds for any maximal ideal $\mathfrak{m}\subset \Lambda$ containing $\mathfrak{p}$. Therefore, by \cite{Bou98}{VI, \S7, Propositions~1--2}, \[ \Lambda^{\prime} \colonequals \Lambda \cap V \] is a semilocal Pr\"{u}fer domain whose spectrum is obtained by glueing $\Spec \Lambda$ with $\Spec V$ along their common open $\Spec \Lambda_{\mathfrak{p}}$. Consequently, we may glue $X$ with $\mathcal{X}_{V}$ along $X_{\Lambda_{\mathfrak{p}}}$ to extend $X$ to an irreducible, affine, $\Lambda'$-smooth scheme $X'$ of pure relative dimension $d$, with a closed $\Lambda'$-subscheme $Y'\subset X'$ obtained by glueing $Y$ with $\mathcal{Y}_{V}$ along $Y_{\Lambda_{\mathfrak{p}}}$; by construction, $Y'$ avoids all the maximal points of the $\Lambda'$-fibers of $X'$. Since the closed subscheme $\widetilde{x}_V \subset \mathcal{X}_V$ is $V$-finite, we may specialize $x$ to a point of $\widetilde{x}_V\subset X'$ that lies over the closed point of $\Spec V$. Hence, by replacing $\Lambda$ by $\Lambda'$, $X$ by $X'$ and $Y$ by $Y'$, we can reduce to the already treated case when all points of $\textbf{x}$ specialize to closed points of the closed $\Lambda$-fibers of $X$. \end{proof} As the relative dimension of $X/\Lambda$ is $d>0$, the closed subset $\textbf{x}\cup Y$ avoids all maximal points of the $R$-fibers of $X$. By prime avoidance, there is an $a\in \Gamma(X,\mathscr{O}_X)$ that vanishes on $\textbf{x}\cup Y$ but does not vanish at any maximal points of $\Lambda$-fibers of $X$. Replacing $Y$ by $V(a)$, we may assume the following in the sequel: \begin{itemize} \item for some $a\in \Gamma(X,\mathscr{O}_X)$, $Y=V(a)$ avoids all the maximal points of $\Lambda$-fibers of $X$; and \item $ \textbf{x}$ consists of closed points of the closed $\Lambda$-fibers of $X$ and is contained in $Y$. \end{itemize} \begin{lemma-tweak} \label{lem on h:X to A1} For $k$ a finite product of fields, a finite type affine $k$-scheme $X'$, a closed subscheme $Y'\subset X'$ of pure dimension $e>0$, a finite subset of closed points $\textbf{x}\subset Y'\cap X'^{\mathrm{sm}}$, and an element $(t(x))\in \prod_{x\in \textbf{x}}\kappa(x)$, there is a $k$-morphism $h:X'\to \mathbb{A}_k^1$ that is smooth at $\textbf{x}$ such that \[ \text{$h\|_{Y'}$ has fiber dimension $e-1$\quad\quad and \quad\quad $h(x)=t(x)$\quad for every $x\in \textbf{x}$.} \] \end{lemma-tweak} \begin{proof} Assume that $k$ is a field. Pick a finite subset of closed points $\textbf{y}\subset Y'$ that is disjoint from $\textbf{x}$ and meets each irreducible component of $Y'$ in exactly one point. For every integer $n>0$ denote by $\textbf{x}^{(n)}$ (resp., $\textbf{y}^{(n)}$) the $n^{\text{th}}$ infinitesimal neighbourhood of $\textbf{x}$ (resp., $\textbf{y}$) in $X'$. Pick $h_{\textbf{x}}\in H^0(\textbf{x}^{(1)}, \mathscr{O}_{\textbf{x}^{(1)}})$ such that \begin{equation}\label{value and differential at x} \text{$h_{\textbf{x}}(x)= t(x) $ \quad and \quad $\text{d}h_{\textbf{x}}(x)\neq 0\in \mathfrak{m}_{X',x}/\mathfrak{m}_{X',x}^2$ \quad for every \quad $x\in \textbf{x}$.} \end{equation} Pick an $h_{\textbf{y}} \in H^0(X',\mathscr{O}_{X'})$ such that its restriction to every irreducible component of $Y'_{\text{red}}$ is not identically zero. By the faithfully flatness of the completion map \[ \textstyle \mathscr{O}_{Y'_{\text{red}},\textbf{y}}=\prod_{y\in \textbf{y}} \mathscr{O}_{Y'_{\text{red}},y} \to \prod_{y\in \textbf{y}} \widehat{\mathscr{O}_{Y'_{\text{red}},y}}=\lim_{n\rightarrow \infty} H^0(\textbf{y}^{(n)}\cap Y'_{\text{red}},\mathscr{O}_{\textbf{y}^{(n)}\cap Y'_{\text{red}}} ), \] we see that for large $n$, the restriction of $h_{\textbf{y}}$ to every component of $\textbf{y}^{(n)}\cap Y'_{\text{red}}$ is nonzero. Let $h\in H^0(X',\mathscr{O}_{X'})$ be an element whose restriction to $\textbf{x}^{(1)}$ is $h_{\textbf{x}}$ and whose restriction to $\textbf{y}^{(n)}$ is congruent to $h_{\textbf{y}}$. As $X'$ is $k$-smooth at $\textbf{x}$, (\ref{value and differential at x}) implies that the morphism $h:X'\to \mathbb{A}_k^1$ (obtained by sending the standard coordinate of $\mathbb{A}_k^1$ to $h$) is smooth at $\textbf{x}$ and $h(x)=t(x)$ for every $x\in \textbf{x}$. For large $n$, as the restriction of $h$ to each irreducible component of $\textbf{y}^{(n)}\cap Y'_{\text{red}}$ and hence also to $Y'_{\text{red}}$ is nonzero, the morphism $h$ is non-constant on each irreducible component of $Y'$, so $h\|_{Y'}$ has fiber dimension $e-1$. \end{proof} \begin{lemma-tweak} \label{lem on g} With the setup in \Cref{reduction to closed points}, there exists a $\Lambda$-morphism $g:X\to \mathbb{A}_{\Lambda}^{d-1}$ such that \begin{enumerate}[label={{\upshape(\roman)}}] \item it smooth of relative dimension 1 at $\textbf{x}$; \item the restriction $g\|_Y$ is quasi-finite at $\textbf{x}$; and \item for every maximal ideal $\mathfrak{m}\subset \Lambda$ and every $x\in \textbf{x}$ lying over $\mathfrak{m}$, one has $\kappa(\mathfrak{m})=\kappa(g(x))$. \end{enumerate} In addition, if $d>\#(\textbf{x} \cap X_{\kappa(\mathfrak{m})})$ for every maximal ideal $\mathfrak{m}\subset \Lambda$ with finite residue field, then we may find such a morphism $g$ under which all points of $\textbf{x}$ have pairwise distinct images. \end{lemma-tweak} \begin{proof} We first reduce the lemma to the case when $\Lambda=k$ is a field. Assume that for every maximal ideal $\mathfrak{m}\subset \Lambda$ there exists a $\kappa(\mathfrak{m})$-morphism $g_{\mathfrak{m}}:X_{\kappa(\mathfrak{m})}\to \mathbb{A}_{\kappa(\mathfrak{m})}^{d-1}$ that is smooth at $\textbf{x}\cap X_{\kappa(\mathfrak{m})}$ such that the restriction $g_{\mathfrak{m}}\|_{Y_{\kappa(\mathfrak{m})}}$ is quasi-finite at $\textbf{x}\cap X_{\kappa(\mathfrak{m})}$. We then use Chinese remainder theorem to lift the maps $\{g_{\mathfrak{m}}\}_{\mathfrak{m}}$ simultaneously to obtain a $\Lambda$-morphism $g:X\to \mathbb{A}_{\Lambda}^{d-1}$ which would verify the first assertion of the lemma: only the flatness of $g$ at $\textbf{x}$ need to be checked, but this follows from the fibral criterion of flatness \cite{EGAIV3}{Théorème~11.3.10}. In addition, if all points of $\textbf{x} \cap X_{\kappa(\mathfrak{m})} $ have pairwise distinct images under $g_{\mathfrak{m}}$, then the resulting morphism $g$ verifies the second assertion of the lemma. Assume now that $\Lambda=k$ is a field. Given a collection of maps $t_1,\cdots,t_{d-1}:\textbf{x} \to k$, taking products yields maps $(t_1,\cdots,t_i):\textbf{x}\to \mathbb{A}_k^i(k)=k^{i}$ for $1\le i \le d-1$. We now apply \Cref{lem on h:X to A1} inductively to show: \begin{cl-tweak} \label{inductively construct g_i} For $1\le i\le d-1$, there exists a $k$-morphism $g_i:X\to \mathbb{A}_k^i$ such that \begin{itemize} \item $g_i$ is smooth at $\textbf{x}$ with $g_i\|_{\textbf{x}}=(t_1,\cdots,t_i)$; and \item every irreducible component of $(g_i\|_Y)^{-1}(g_i(\textbf{x}))$ intersecting $\textbf{x}$ has dimension $d-1-i$. \end{itemize} \end{cl-tweak} \begin{proof} [Proof of the claim] Set $g_0:X\to \Spec k$, the structural morphism. Assume the morphism $g_{i-1}$ has been constructed. We apply \Cref{lem on h:X to A1}, with $k$ being the ring $k'$ of global sections of $g_{i-1}(\textbf{x})$ here, $X'$ being $g_{i-1}^{-1}(g_{i-1}(\textbf{x}))$, $Y'$ being the union of all the irreducible components of $(g_{i-1}\|_Y)^{-1}(g_{i-1}(\textbf{x}))$ meeting $\textbf{x}$, and $t$ being $t_i\|_{k'}$, to obtain a $k^{\prime} $-morphism $h\colon g_{i-1}^{-1}(g_{i-1}(\textbf{x}))\to \mathbb{A}_{k'}^1$ that is smooth at $\textbf{x}$ such that $h\|_{Y'}$ has fiber dimension $d-1-i$ and such that $h\|_{\textbf{x}}=t_i\|_{k^{\prime}}$, where $t_i\|_{k'}:\textbf{x} \xrightarrow{t_i} k\to k'$. It remains to take $g_i\colonequals (g_{i-1},\widetilde{h}):X\to \mathbb{A}_k^i=\mathbb{A}_k^{i-1}\times_k\mathbb{A}_k^1$ for any lifting $\widetilde{h}\in H^0(X,\mathscr{O}_X)$ of \[ h \in H^0\bigl(g_{i-1}^{-1}(g_{i-1}(\textbf{x})),\mathscr{O}_{g_{i-1}^{-1}(g_{i-1}(\textbf{x}))}\bigr). \qedhere \] \end{proof} Starting from any map $(t_1,\cdots,t_{d-1})\colon \textbf{x}\to k^{d-1}$, the morphism $g\colonequals g_{d-1}$ from \Cref{inductively construct g_i} immediately settles the first assertion of the lemma. For the second assertion, it suffices to note that, under the stated assumption, there always exists an injection $\textbf{x} \hookrightarrow k^{d-1}$: for an infinite field $k$, the cardinality of $k^{d-1}$ is infinite, and, for a finite field $k$, we have $\# (k^{d-1}) \ge d-1$. \end{proof} Consider the morphism $(g,a)\colon X\to \mathbb{A}_{\Lambda}^d=\mathbb{A}_{\Lambda}^{d-1}\times_{\Lambda} \mathbb{A}_{\Lambda}^1$. By construction, it is quasi-finite at $\textbf{x}$, and, by the openness of the quasi-finite locus of a finite type morphism, up to shrinking $X$, we may assume that it is already quasi-finite; since the generic $\Lambda$-fibers of its domain and codomain are irreducible varieties of the same dimension $d$, it is also dominant. Consequently, by Zariski's main theorem \cite{EGAIV4}{Corollaire~18.12.13}, the morphism $(g,a)\colon X\rightarrow \mathbb{A}^d_{\Lambda}$ factors as $$X \xrightarrow{j} \overline{X} \xrightarrow {h_1} \mathbb{A}_{\Lambda}^d,$$ where $\overline{X}$ is an integral affine scheme, $j$ is an open immersion, and $h_1$ is finite, dominant. Denote $\overline{g}\colonequals \mathrm{pr}_1 \circ h_1$, where $\mathrm{pr}_1\colon \mathbb{A}_{\Lambda}^d \to \mathbb{A}_{\Lambda}^{d-1}$ is the projection onto the first $(d-1)$-coordinates, and let $\overline{a}\in \Gamma(\overline{X},\mathscr{O}_{\overline{X}})$ be the pullback of the last standard coordinate of $\mathbb{A}_{\Lambda}^d$. Then $h_1=(\overline{g},\overline{a})$, and $\overline{g}$ (resp., $\overline{a}$) restricts to $g$ (resp., $a$) on $X$. In what follows, we shall identify $j(x)$ with $x$ for every $x\in \textbf{x}$. Write $ S\subset \Spec \Lambda$ for the union of the closed points of $\Spec \Lambda$ (with the reduced structure). \begin{lemma-tweak} \label{lem on b} There exists an element $b\in \Gamma(\overline{X}, \mathscr{O}_{\overline{X}})$ such that the morphism $$h_2\colonequals (\overline{g}, b):\overline{X}\to \mathbb{A}_{\Lambda}^d=\mathbb{A}_{\Lambda}^{d-1}\times_{\Lambda} \mathbb{A}_{\Lambda}^1$$ has the following properties: \begin{enumerate}[label={{\upshape(\roman)}}] \item \label{lem on b 1} set-theoretically we have $h_1^{-1}(h_1(\textbf{x})) \cap h_2^{-1}(h_2(\textbf{x}))= \textbf{x}$; \item \label{lem on b 2} $h_2$ is \'etale around $\textbf{x}$ and induces a bijection $\textbf{x} \xrightarrow{\sim} h_2(\textbf{x})$; and \item \label{lem on b 3} $h_2$ induces an isomorphism of residue fields $\kappa(h_2(x)) \xrightarrow{\sim} \kappa(x)$ for every $x \in \textbf{x}$. \end{enumerate} \end{lemma-tweak} \begin{proof} Since $h_1$ is finite, surjective, we see that $\overline{g}^{-1}(g(\textbf{x}))$ is an $S$-curve that contains $g^{-1}(g(\textbf{x}))$ as an open subcurve, so it is $S$-smooth around $\textbf{x}$. For a point $x\in \textbf{x}$ lying over a maximal ideal $\mathfrak{m}\subset \Lambda$, its first infinitesimal neighbourhood in $\overline{g}^{-1}(g(\textbf{x}))$ is isomorphic to $\Spec (\kappa(x)[u_x]/(u_x^2))$, where $u_x$ is a uniformizer of $\overline{g}^{-1}(g(\textbf{x}))$ at $x$. Recall that the residue field of a point on a smooth curve over a field is a simple extension of that field, see \cite{Ces22a}{Lemma~6.3}. It follows that, for $x\in \textbf{x} $ lying over $\mathfrak{m}$, there exists a closed $\kappa(\mathfrak{m})$-immersion $x^{(1)} \hookrightarrow \mathbb{A}_{\kappa(\mathfrak{m})}^1=\mathbb{A}_{g(x)}^1$. For a maximal ideal $\mathfrak{m} \subset \Lambda$ with finite residue field, under the assumption that $\#(\textbf{x}\cap X_{\kappa(\mathfrak{m})}) < \max (\#\,\kappa(\mathfrak{m}),d)$, either $\textbf{x}$ contains at most $\#\,\kappa(\mathfrak{m})-1$ points lying over $\mathfrak{m}$ or every fiber of $g_{\kappa(\mathfrak{m})}$ contains at most 1 point of $\textbf{x}$ (\Cref{lem on g}). Consequently, we may arrange the above immersions so that they jointly give a closed immersion over $\mathbb{A}_{\Lambda}^{d-1}$: \begin{equation} \label{embeds 1st neighb of x} \textstyle \bigsqcup_{x\in \textbf{x}} x^{(1)} \hookrightarrow \mathbb{A}_{g(\textbf{x})}^1\subset \mathbb{A}_{\mathbb{A}_{\Lambda}^{d-1}}^1=\mathbb{A}_{\Lambda}^d, \end{equation} where we regard $g(\textbf{x})\subset \mathbb{A}_{\Lambda}^{d-1}$ as a closed subscheme. Note that the complement of the image of the morphism (\ref{embeds 1st neighb of x}) in $\mathbb{A}_{\Lambda}^d$ has at least 1 rational point fiberwisely over $\mathbb{A}_{\Lambda}^{d-1}$. Thus, by sending any $y\in (h_1^{-1}(h_1(\textbf{x})) \backslash \textbf{x}) $ to a suitable rational point of $\mathbb{A}_{g(y)}^1$, we may further extend (\ref{embeds 1st neighb of x}) to a $\mathbb{A}_{\Lambda}^{d-1}$-morphism \[ \textstyle u\colon Z\colonequals \left(\bigsqcup_{x\in \textbf{x}} x^{(1)}\right) \bigsqcup \left(\bigsqcup_{y\in h_1^{-1}(h_1(\textbf{x})) \backslash \textbf{x}}y\right) \to \mathbb{A}_{\Lambda}^d \] such that $u(\textbf{x})\cap u(h_1^{-1}(h_1(\textbf{x})) \backslash \textbf{x}) =\emptyset$, or, equivalently, \begin{equation}\label{intersection equality} h_1^{-1}(h_1(\textbf{x})) \cap u^{-1}(u(\textbf{x}))= \textbf{x}. \end{equation} Let $b\in \Gamma(\overline{X},\mathscr{O}_{\overline{X}})$ be a lifting of $u^(t)\in \Gamma(Z,\mathscr{O}_Z)$, where $t$ is the standard coordinate on $\mathbb{A}_{\Lambda}^1$. Consider the morphism $h_2\colonequals (\overline{g}, b):\overline{X}\to \mathbb{A}_{\Lambda}^d=\mathbb{A}_{\Lambda}^{d-1}\times_{\Lambda} \mathbb{A}_{\Lambda}^1$. Viewing $\overline{X}$ as a $\mathbb{A}_{\Lambda}^{d-1}$-scheme via $\overline{g}$, the base change of $h_2$ to $g(\mathbf{x})\subset \mathbb{A}_{\Lambda}^{d-1}$ restricts to $u$ on $Z$, so $h_2$ is unramified at $\textbf{x}$. Now \ref{lem on b 1} follows from (\ref{intersection equality}) and \ref{lem on b 3} is a consequence of our choice of the morphism (\ref{embeds 1st neighb of x}). For \ref{lem on b 2}, it suffices to argue that $h_2$ is flat at $\textbf{x}$; however, since the domain and the codomain of $h_2$ are $\Lambda$-flat of finite presentation, the fibral criterion of flatness \cite{EGAIV3}{Théorème~11.3.10} reduces us to checking the flatness of the $\Lambda$-fibers of $h_2$ at $\textbf{x}$, while the latter follows from the flatness criterion \cite{EGA IV2}{Proposition~6.1.5}. \end{proof} Let $\Lambda[h_1^{}(t_1),\cdots, h_1^(t_{d-1}), a,b]\subset \Gamma(\overline{X},\mathscr{O}_{\overline{X}})$ be the $\Lambda$-subalgebra generated by $a,b$ and $h_2^{}(t_i)(=h_1^(t_i)=\overline{g}^(t_i))$ for $1\le i \le d-1$. We introduce the following notations. \begin{itemize} \item Let $V\colonequals \Spec (\Lambda[h_1^{}(t_1),\cdots, h_1^(t_{d-1}), a,b])$, and let $h_3:\overline{X}\to V$ be the morphism induced by the inclusion $\Lambda[h_1^{}(t_1),\cdots, h_1^(t_{d-1}), a,b]\subset \Gamma(\overline{X},\mathscr{O}_{\overline{X}})$. \item Let $v_1:V\to \mathbb{A}_{\Lambda}^d$ be the morphism such that $v_1^(t_i)=h_1^(t_i)$ for $1\le i \le d-1$ and $v_1^(t_d)=a$. \item Let $v_2:V\to \mathbb{A}_{\Lambda}^d$ be the morphism such that $v_2^(t_i)=h_1^(t_i)$ for $1\le i \le d-1$ and $v_2^(t_d)=b$. Note that there is a natural surjection \[ \Lambda[h_1^{}(t_1),\cdots, h_1^(t_{d-1}),b] \twoheadrightarrow \Lambda[h_1^{}(t_1),\cdots, h_1^(t_{d-1}), a,b]/(a)= \Gamma(V,\mathscr{O}_{V})/(a); \] this implies that $v_2$ induces a closed immersion \[ \overline{v}_2:\Spec (\Gamma(V,\mathscr{O}_{V})/(a)) \hookrightarrow V \xrightarrow{v_2} \mathbb{A}_{\Lambda}^d. \] \end{itemize} We have the following commutative diagram of morphisms of affine schemes: \begin{equation} \begin{tikzcd} X \arrow[r, hook, "j"] & \overline{X} \arrow[drr, bend left, "h_2"] \arrow[ddr, bend right, "h_1"] \arrow[dr, "h_3"] & & \\ & & V \arrow[r, "v_2"] \arrow[d, "v_1"] & \mathbb{A}_{\Lambda}^d \\ & & \mathbb{A}_{\Lambda}^d & \end{tikzcd}. \end{equation} \begin{lemma-tweak} \label{isom of loc rings} The morphism $h_3$ induces a bijection $\textbf{x} \xrightarrow{\sim} h_3(\textbf{x})$ and $h_3^{-1}(h_3(\textbf{x}))=\textbf{x}$. Further, $h_3$ induces an isomorphism of semilocal rings $$\mathscr{O}_{V,h_3(\textbf{x})} \simeq \mathscr{O}_{\overline{X},\textbf{x}}\,=\mathscr{O}_{X,\textbf{x}}. $$ \end{lemma-tweak} \begin{proof} By \Cref{lem on b}~\ref{lem on b 2}--\ref{lem on b 3}, we see that $h_3$ induces a bijection $\textbf{x} \xrightarrow{\sim} h_3(\textbf{x})$ and an isomorphism of residue fields $\kappa(h_3(x)) \xrightarrow{\sim} \kappa(x)$ for every $x\in \textbf{x}$. Chasing the above diagram we see that \[ h_3^{-1}(h_3(\textbf{x}))\subset h_1^{-1}(h_1(\textbf{x})) \cap h_2^{-1}(h_2(\textbf{x}))=\textbf{x}, \] where the last equality is \Cref{lem on b}~\ref{lem on b 1}. As $h_3$ is finite, surjective, we see that $h_3^{-1}(h_3(\textbf{x}))=\textbf{x}$. By \Cref{lem on b}~\ref{lem on b 2}, $h_3$ is unramified at $\textbf{x}$. It follows that the base change of $h_3$ to $\Spec \,\mathscr{O}_{V,h_3(\textbf{x})}$ is $$ \Spec \,\mathscr{O}_{\overline{X},\textbf{x}}\to \Spec \,\mathscr{O}_{V,h_3(\textbf{x})}, $$ and it is actually an isomorphism: letting $J$ be the Jacobson radical of the semilocal ring $\mathscr{O}_{V,h_3(\textbf{x})}$, since the natural map \[ \textstyle \prod_{x\in \textbf{x}}\kappa(h_3(x)) \simeq \mathscr{O}_{V,h_3(\textbf{x})}/J \xrightarrow{h_3^} \mathscr{O}_{\overline{X},\textbf{x}}/J\mathscr{O}_{\overline{X},\textbf{x}} \simeq \prod_{x\in \textbf{x}} \kappa(x) \] is an isomorphism (in particular, surjective), an application of Nakayama lemma shows \[ \textstyle h_3^: \mathscr{O}_{V,h_3(\textbf{x})} \simeq \mathscr{O}_{\overline{X},\textbf{x}}=\mathscr{O}_{X,\textbf{x}}. \qedhere \] \end{proof} \begin{pp} [Proof of \Cref{variant of Lindel's lem}] Define $f\colonequals h_2\circ j:X\to \mathbb{A}_{\Lambda}^d$, which we may assume to be \'etale upon replacing $X$ by an affine open neighbourhood of $\textbf{x}$. By \Cref{isom of loc rings}, there exists an affine open neighbourhood $W_0'\subset V$ of $h_3(\textbf{x})$ such that $W_0\colonequals h_3^{-1}(W_0) \subset j(X)$ and $h_3\|_{W_0}:W_0\xrightarrow{\sim} W_0'$. We shall identify $W_0$ as an open subscheme of $X$ via $j$. As noted above, $v_2$ induces a closed immersion \[ \overline{v}_2:Y'\colonequals \Spec (\Gamma(V,\mathscr{O}_{V})/(a))\hookrightarrow \mathbb{A}_{\Lambda}^d. \] In particular, the topology of $Y'$ is induced from that of $\mathbb{A}_{\Lambda}^d$ via $\overline{v}_2$. Note also that, since $a$ vanishes on $\textbf{x}$, $h_3(\textbf{x})\subset Y' \subset V$. Consequently, there exists an affine open neighbourhood $U\subset \mathbb{A}_{\Lambda}^d$ of $f(\textbf{x})=v_2(h_3(\textbf{x}))$ such that $\overline{v}_2^{-1}(U)\subset W_0'$. Therefore, $f$ induces a closed immersion of affine schemes \[ Y_U:=f^{-1}(U)\cap Y = (h_3\circ j)^{-1}(v_2^{-1}(U) \cap Y')= (h_3\circ j)^{-1}(\overline{v}_2^{-1}(U))\simeq \overline{v}_2^{-1}(U) \hookrightarrow U. \] Since $f$ is separated and \'etale, any section of $f\times_{\mathbb{A}_{\Lambda}^d,f}Y_U$, such as the one $s$ induced by the above inclusion $Y_U\hookrightarrow U$, is an inclusion of a clopen, so \[ X\times_{\mathbb{A}_{\Lambda}^d,f}Y_U=\widetilde{Y}_1 \sqcup \widetilde{Y}_2 \quad\quad \text{ with } \quad\quad \widetilde{Y}_1=\text{im}(s)\xrightarrow{\sim} Y_U. \] Let $W\subset f^{-1}(U)$ be an affine open whose preimage in $X\times_{\mathbb{A}_{\Lambda}^d,f}Y_U$ is $\widetilde{Y}_1$. Then $f\|_W:W\to U$ is \'etale such that $f\|_{W\cap Y}:W\cap Y \hookrightarrow U$ is a closed immersion and such that $W\times_{U,f}(W\cap Y) \xrightarrow{\sim} W\cap Y$. As \'etale maps are open, we may shrink $U$ around $f(\textbf{x})$ to ensure that $f\|_W:W\to U$ is also surjective. \QED \end{pp} \section{ Torsors on a smooth affine relative curve} \label{section-torsors on sm aff curves} In this section we prove the following result concerning triviality of torsors on a smooth affine relative curve. A similar result can also be found in the recent preprint \cite{Ces22b}{Theorem~4.4}. \begin{thm-tweak}[Section theorem]\label{triviality on sm rel. affine curves} For a semilocal domain $R$ with geometrically unibranch\footnote{According to \SP{0BPZ}, a local ring $A$ is \emph{geometrically unibranch} if its reduction $A_{\text{red}}\colonequals A/\sqrt{(0)}$ is a domain, and if the integral closure of $A_{\text{red}}$ in its fraction field is a local ring whose residue field is purely inseparable over that of $A$. By \SP{06DM}, $A$ is geometrically unibranch if and only if its strict Henselization $A^{\text{sh}}$ has a unique minimal prime.} local rings, a smooth affine $R$-curve $C$ with a section $s\in C(R)$, a reductive $C$-group scheme $G$, an $R$-algebra $A$, and a $G$-torsor $\mathcal{P}$ over $C_A\colonequals C\times_RA$ that trivializes over $C_A\backslash Z_A$ for an $R$-finite closed subscheme $Z\subset C$, if \begin{enumerate}[label={{\upshape(\roman)}}] \item \label{sec-thm-semilocal} either $A$ is semilocal, \item \label{sec-thm-totally isotropic} or $s_A^(G)$ is totally isotropic, \end{enumerate} then the pullback $s_A^(\mathcal{P})$ is a trivial $s_A^(G)$-torsor, where $s_A$ denotes the image of $s$ in $C_A(A)$. \end{thm-tweak} To prove Theorem \ref{triviality on sm rel. affine curves}, we first use \Cref{equating red gps} to reduce to the case when $G$ is the base change of a reductive $R$-group scheme, and then to the case when $C=\mathbb{A}^1_R$, see \Cref{1st reduction of sec thm}. As for the latter, one can approach it via the geometry of affine Grassmannians. \begin{pp}[Reduction to the case when $C=\mathbb{A}_R^1$ and $G$ is constant] We start with the following result for equating reductive group schemes, which was already known to experts, see also \cite{Ces22b}{Lemma~3.5}. \end{pp} \begin{lemma-tweak}[Equating reductive group schemes]\label{equating red gps} For a semilocal ring $B$ with geometrically unibranch local rings, reductive $B$-group schemes $G_1$ and $G_2$ whose geometric $B$-fibers have the same type, maximal $B$-tori $T_1\subset G_1$ and $T_2\subset G_2$, and an ideal $I\subset B$, if there is an isomorphism of $B/I$-group schemes \[ \text{$\iota:(G_1)_{B/I} \isoto (G_2)_{B/I}$ \quad such that \quad $\iota((T_1)_{B/I})=(T_2)_{B/I}$.} \] then, there are a faithfully flat, finite, \'etale $B$-algebra $B'$, a section $s:B'\twoheadrightarrow B/I$, and an isomorphism of $B'$-group schemes $\iota': (G_1)_{B'} \simeq (G_2)_{B'}$ such that $\iota((T_1)_{B'})=(T_2)_{B'}$ and whose $s$-pullback is $\iota$. \end{lemma-tweak} \begin{proof} By \cite{SGA3IIInew}{Exposé~XXIV, Corollaire~2.2}, the condition on geometric $B$-fibers ensures that \[ \text{the functor \quad $X\colonequals \underline{\text{Isom}}_B((G_1,T_1),(G_2,T_2))$} \] parameterizing the isomorphisms of the pairs $(G_1,T_1)$ and $(G_2,T_2)$ is representable by a $B$-scheme and is an $H\colonequals \underline{\text{Aut}}_B((G_1,T_1))$-torsor. We need to show that, for any $\iota\in X(B/I)$, there are a faithfully flat, finite, \'etale $B$-algebra $B'$, an $\iota'\in X(B')$, and a section $s:B'\twoheadrightarrow B/I$ such that $s(\iota')=\iota\in X(B/I)$. By \emph{loc.~cit.}, $H$ is an extension of an \'etale locally constant $B$-group scheme by $T_1^{\text{ad}}$, the quotient of $T_1$ by the center of $G_1$. According to \cite{SGA3IIInew}{Exposé~XXIV, Proposition~2.6}, $T_1^{\text{ad}}$ acts freely on $X$ and \[ \text{the fppf quotient \quad $\overline{X}\colonequals X/T_1^{\text{ad}}$ } \] is representable by a faithfully flat $B$-scheme that is \'etale locally constant on $B$. As all local rings of $B$ are geometrically unibranch, by \cite{SGA3IIInew}{Exposé~X, Corollaire~5.14}, every connected component of $\overline{X}$ is finite \'etale over $B$. As the image of $\iota:\text{Spec}(B/I)\to X \to \overline{X}$ intersects only finitely many connected components of $\overline{X}$, the union of these components is the spectrum of a finite \'etale $B$-algebra $A$, and there are an $\overline{\iota}\in \overline{X}(A)$ and a section $t: A\twoheadrightarrow B/I$ such that $t(\overline{\iota})=\iota$. By adding more connected components of $\overline{X}$ into $\Spec A$ if needed, we may assume that $A$ is faithfully flat over $B$. Consider the fiber product \[ Y\colonequals X\times_{\overline{X},\overline{\iota}}\Spec A; \] it is a $T^{\mathrm{ad}}_A$-torsor equipped with a point $\iota\in Y(A/J)\subset X(A/J)$, where $J\colonequals \ker \,(A\twoheadrightarrow B/I)$. By \cite{Ces22}{Corollary~6.3.2}, there are a faithfully flat, finite, \'etale $A$-algebra $B'$, a section $$s':B'\twoheadrightarrow A/J\simeq B/I,$$ and an $\iota'\in Y(B')\subset X(B')$ such that $s'(\iota')=\iota$. \end{proof} \begin{lemma-tweak}\label{1st reduction of sec thm} The proof of the \Cref{triviality on sm rel. affine curves} reduces to the case when $C=\mathbb{A}_R^1$ and $G$ is the base change of a reductive $R$-group scheme. \end{lemma-tweak} \begin{proof} Let $B$ be the semilocal ring of $C$ at the closed points of $\text{im}(s) \cup Z$; its local rings are geometrically unibranch. By abuse of notation, we may view $s:B \twoheadrightarrow R$ as a section of the $R$-algebra $B$. As $B$ is semilocal, by \cite{SGA3II}{Exposé~XIV, Corollaire~3.20}, $G_B$ admits a maximal $B$-torus $T_B$. Since the pullbacks of the pairs $(G_B,T)$ and $((s^G)_B,(s^T)_B)$ along $s$ are the same, by \Cref{equating red gps}, there are a faithfully flat, finite, \'etale $B$-algebra $B'$, a section $s':B'\twoheadrightarrow R$ that lifts $s$, and a $B'$-isomorphism $$ \iota:(G_{B'},T_{B'}) \isoto \left((s^(G))_{B'},(s^(T)_{B'}\right) $$ whose $s$-pullback is the identity. We spread out $\Spec B^{\prime} \to \Spec B$ to obtain a finite \'etale cover $C'\to U $ of a small affine open neighbourhood $U$ of $\text{im}(s) \cup Z$ in $C$. Shrinking $U$ if necessary, we may assume that the isomorphism $\iota$ is defined over $C'$. In both cases of \Cref{triviality on sm rel. affine curves} we may replace $C$ by $C'$, $Z$ by $C'\times_CZ$, $s$ by $s'$, and $\mathcal{P}$ by $\mathcal{P}\|_{C'_A}$ to reduce to the case when $G$ is the base change of the reductive $R$-group scheme $s^(G)$. Next, in order to apply glueing \cite{Ces22a}{Lemma~7.1} to achieve that $C=\mathbb{A}_R^1$, we need to modify $C$ so that $Z$ embeds into $\mathbb{A}_R^1$. For this, we first replace $Z$ by $Z\cup \text{im}(s)$ to assume that $s$ factors through $Z$. Then we apply Panin's `finite field tricks' \cite{Ces22a}{Proposition~7.4} to obtain a finite morphism $\widetilde{C}\to C$ that is \'etale at the points in $\widetilde{Z}\colonequals \widetilde{C}\times_CZ$ such that $s$ lifts to $\widetilde{s}\in \widetilde{C}(R)$, and there are no finite fields obstruction to embedding $\widetilde{Z}$ into $\mathbb{A}_R^1$ in the following sense: for every maximal ideal $\mathfrak{m}\subset R$, \[ \# \bigl\{ z\in \widetilde{Z}_{\kappa(\mathfrak{m})}:[\kappa(z):\kappa(\mathfrak{m})]=d \bigr\}< \# \bigl\{y\in \mathbb{A}_{\kappa(\mathfrak{m})}^1: [\kappa(y):\kappa(\mathfrak{m})]=d \bigr\} \quad \text{for every} \quad d\ge 1. \] Then, by \cite{Ces22a}{Lemma~6.3}, there are an affine open $C''\subset \widetilde{C}$ containing $\text{im}(\widetilde{s})$, a quasi-finite, flat $R$-map $C''\to \mathbb{A}_R^1$ that maps $Z$ isomorphically to a closed subscheme $Z'\subset \mathbb{A}_R^1$ with \[ Z\simeq Z' \times_{\mathbb{A}_R^1}C''. \] (Actually, $C''\to \mathbb{A}_R^1$ can be \'etale by shrinking $C''$ around $\text{im}(\widetilde{s})$). For both cases of \Cref{triviality on sm rel. affine curves}, since $\mathcal{P}\|_{C''_A}$ is a $G$-torsors that trivializes over $C''_A\backslash \widetilde{Z}_A$, we may use \cite{Ces22a}{Lemma~7.1} to glue $\mathcal{P}_{C''_A}$ with the trivial $G$-torsor over $\mathbb{A}_A^1$ to obtain a $G$-torsor $\mathcal{P}'$ over $\mathbb{A}_A^1$ that trivializes over $\mathbb{A}_A^1 \backslash Z'_A$. Let $s'\in \mathbb{A}_R^1(R)$ be the image of $\widetilde{s}$; then $s'^(\mathcal{P}')\simeq s^(\mathcal{P})$. It remains to replace $C$ by $\mathbb{A}_R^1$, $Z$ by $Z'$, $s$ by $s'$, and $\mathcal{P}$ by $\mathcal{P}'$. \end{proof} \begin{pp}[Studying torsors on $\mathbb{P}_R^1$ via affine Grassmannian] The analysis of torsors on $\mathbb{A}_R^1$ ultimately depends on the geometry of affine Grassmannians. A nice summary of and complement on the relevant techniques can be found in \cite{Ces22}{\S5.3}. In particular, we will use the following slight variant of \cite{Ces22}{Proposition~5.3.6}, which in turn is a mild generalization of \cite{Fed22b}{Theorem~6}. \end{pp} \begin{prop-tweak}\label{main tech for torsor on A1} For a semilocal ring $R$ with $\mathrm{connected}$ spectrum and a reductive $R$-group scheme $G$, let \[ \textstyle G^{\mathrm{ad}}\simeq \prod_{i} \mathrm{Res}_{R_i/R}(G_i) \] be the canonical decomposition of the adjoint quotient $G^{\mathrm{ad}}$ in \cite{SGA3IIInew}{Exposé~XXIV, Proposition~5.10}, where $G_i$ is a simple adjoint $R_i$-group scheme, and $R_i$ is a finite, \'etale $R$-algebra with $\mathrm{connected}$ spectrum\footnote{To ensure that the $R_i$'s have connected spectra, we decompose $R_i\simeq \prod_{j=1}^{n_i}R_{ij}$, where $R_{ij}$ have connected spectra, and then use the canonical isomorphism $\mathrm{Res}_{R_i/R}(G_i)\simeq \prod_{j=1}^{n_i}\mathrm{Res}_{R_{ij}/R}(G_{i,R_{ij}})$.}. Let $Y\subset \mathbb{A}_R^1$ be a $R$-finite, \'etale, closed subscheme with the following properties: \begin{enumerate}[label={{\upshape(\roman)}}] \item\label{main tech i} for every $i$, there is a clopen $Y_i\subset Y\times _R R_i$ such that $(G_i)_{Y_i}$ contains a copy of $\mathbb{G}_{\text{m},Y_i}$; \item\label{main tech ii} $\mathscr{O}_{\mathbb{P}_{\kappa(\mathfrak{m})}^1}(1)$ is trivial on $\mathbb{P}_{\kappa(\mathfrak{m})}^1 \backslash (Y_i)_{\kappa(\mathfrak{m})}$ for each maximal ideal $\mathfrak{m} \subset R_i$ such that $(G_i)_{\kappa(\mathfrak{m})}$ is isotropic; \item\label{main tech iii} $\mathscr{O}_{\mathbb{P}_R^1}(1)$ is trivial on $\mathbb{P}_R^1 \backslash Y$. \end{enumerate} Let $\mathcal{P}$ be a $G$-torsor over $\mathbb{P}_R^1$ that trivializes over $\mathbb{P}_R^1\backslash Z$ for some $R$-finite closed subscheme $Z\subset \mathbb{A}_R^1\backslash Y$. Assume that for every maximal ideal $\mathfrak{m} \subset R$ the $G^{\mathrm{ad}}$-torsor over $\mathbb{P}_{\kappa(\mathfrak{m})}^1$ induced by $\mathcal{P}$ lifts to a $\mathrm{generically}$ $\mathrm{trivial}$ $(G^{\mathrm{ad}})^{\mathrm{sc}}$-torsor over $\mathbb{P}_{\kappa(\mathfrak{m})}^1$. Then the restriction $\mathcal{P}\|_{\mathbb{P}_R^1 \backslash Y}$ is trivial. \end{prop-tweak} By \cite{SGA3IIInew}{Exposé~XXVI, Corollaire~6.12}, (i) is equivalent to that the base change of $(G_i)_{Y_i}$ to every connected component of $Y_i$ contains proper parabolics. For instance, if $G$ is quasi-split, we can take $Y_i=Y\times_RR_i$ to ensure (i). In practice, we achieve (i) by guaranteeing base changes of $(G_i)_{Y_i}$ to connected components of $Y_i$ contain proper parabolics. For (ii), we can take $Y_i$ so that $Y_i(\kappa(\mathfrak{m}))\neq \emptyset$ for every maximal ideal $\mathfrak{m}\subset R_i$ with $(G_i)_{\kappa(\mathfrak{m})}$ isotropic. For (iii), we just choose $Y$ so that it contains finite \'etale $R$-schemes of degrees $d$ and $d+1$ for some $d\ge 1$, then $\mathscr{O}(d)$ and $\mathscr{O}(n+1)$ are both trivial on $\mathbb{P}_R^1\backslash Y$, and so is $\mathscr{O}(1)$. \begin{proof} We will deduce \Cref{main tech for torsor on A1} from (the proof of) a particular case of \cite{Ces22}{Proposition~5.3.6}. (We remind that the assumption (ii) of \emph{loc.~cit.} should read as `$(G_i)_{Y_i}$ contains a copy of $\mathbb{G}_{m,Y_i}$', as its proof shows.) The $R$-finite \'etale $Y$ is the vanishing locus of a monic polynomial $t$ in the standard coordinate of $\mathbb{A}_R^1$; namely, $t$ is the characteristic polynomial of this standard coordinate acting on $\widetilde{R}:=\Gamma(Y,\mathscr{O}_Y)$. The formal completion of $\mathbb{P}_R^1$ along $Y$ has coordinate ring $\widetilde{R}\fps{t}$. Recall that, by formal glueing, a $G$-torsor over $\mathbb{P}_R^1$ can be viewed as the glueing of its restriction to $\mathbb{P}_R^1\backslash Y$ and to $\widetilde{R}\fps{t}$ along the `intersection' $\widetilde{R}\lps{t}$; since our torsor $\mathcal{P}$ is trivial over an open neighbourhood $U\subset \mathbb{P}_R^1$ of $Y$, both of the restriction $\mathcal{P}\|_{U\backslash Y}$ and $\mathcal{P}\|_{\widetilde{R}\fps{t}}$ are trivial, and once a trivialization of the former was chosen, all such glueings are parameterized by elements of $G(\widetilde{R}\lps{t})/G(\widetilde{R}\fps{t})$. In particular, since $G(\widetilde{R}\lps{t})$ acts on $G(\widetilde{R}\lps{t})/G(\widetilde{R}\fps{t})$ (via left multiplication), an element of $G(\widetilde{R}\lps{t})$ yields a modification of $\mathcal{P}$ along $Y$: it is the $G$-torsor over $\mathbb{P}_R^1$ whose restriction to $\mathbb{P}_R^1\backslash Y$ and to $\widetilde{R}\fps{t}$ are the same as $\mathcal{P}$, but their corresponding glueings, viewed as elements of $G(\widetilde{R}\lps{t})/G(\widetilde{R}\fps{t})$, differ by a left translation by the element of $G(\widetilde{R}\lps{t})$ we choose. Denote by $\mathcal{P}^{\text{ad}}$ the $G^{\text{ad}}$-torsor over $\mathbb{P}_R^1$ induced by $\mathcal{P}$. Since the formation of $H^1(\mathbb{P}_R^1,-)$ commutes with taking products, $\mathcal{P}^{\text{ad}}$ corresponds to a collection $(\mathcal{P}^{\text{ad}}_i)$, where $\mathcal{P}^{\text{ad}}_i$ is a $\mathrm{Res}_{R_i/R}(G_i)$-torsor over $\mathbb{P}_R^1$ satisfying the analogous assumptions \ref{main tech i}--\ref{main tech iii} of the \Cref{main tech for torsor on A1}. Since $R\to R_i$ is finite \'etale and $G_i$ is $R_i$-smooth, we have $R^1f_G_i=1$ for the map $f\colon \Spec R_i\to \Spec R$ induced by $R\to R_i$. We have the following exact sequence of nonabelian pointed sets from \cite{Gir71}{Chapitre~V, Proposition~3.1.3} \[ 1\to H^1(\mathbb{P}_R^1,\mathrm{Res}_{R_i/R}(G_i))\to H^1(\mathbb{P}_{R_i}^1, G_i) \to H^1(\mathbb{P}_R^1,R^1f_G_i). \] Thus $\mathcal{Q} \mapsto \mathrm{Res}_{R_i/R}(\mathcal{Q})$ defines a bijection of pointed sets $H^1(\mathbb{P}_{R_i}^1, G_i) \isoto H^1(\mathbb{P}_R^1,\mathrm{Res}_{R_i/R}(G_i)) $. In particular, each $\mathcal{P}^{\text{ad}}_i$ corresponds to a $G_i$-torsor $\mathcal{Q}_i$ over $\mathbb{P}_{R_i}^1$. As one can see now, the assumptions \ref{main tech i}--\ref{main tech iii} of \Cref{main tech for torsor on A1} for the $\mathrm{Res}_{R_i/R}(G_i)$-torsor $\mathcal{P}^{\text{ad}}_i$ translate into the assumptions \cite{Ces22}{Proposition~5.3.6} (i)--(iv) for the $G_i$-torsor $\mathcal{Q}_i$ over $\mathbb{P}_{R_i}^1$. By the proof of \emph{loc.~cit.} (using the condition that over closed fibers of $\mathbb{P}^1_R$, the simply-connected lifting of the torsor induced by $\mathcal{P}$ is generically trivial), for some \[ \alpha_i\in \text{im}\bigl(G_i^{\text{sc}} ((\widetilde{R}\otimes_RR_i)\lps{t})\to G_i((\widetilde{R}\otimes_RR_i)\lps{t})\bigr), \] the corresponding modification of $\mathcal{Q}_i$ along $Y\times_RR_i$ is trivial. We can view the element \[ \alpha\colonequals (\alpha_i) \in \text{im} \bigl((G^{\mathrm{ad}})^{\text{sc}}(\widetilde{R}\lps{t}) \to G^{\mathrm{ad}}(\widetilde{R}\lps{t}) \bigr); \] as $(G^{\mathrm{ad}})^{\text{sc}} \to G^{\mathrm{ad}}$ factors through $(G^{\mathrm{ad}})^{\text{sc}} \to G$, $\alpha$ lifts to $\widetilde{\alpha}\in G(\widetilde{R}\lps{t})$. Denote by $\mathcal{Q}$ the modification of $\mathcal{P}$ along $Y$ using $\widetilde{\alpha}$. By construction, the $G^{\text{ad}}$-torsor $\mathcal{Q}^{\text{ad}}$ over $\mathbb{P}_R^1$ induced by $\mathcal{Q}$ corresponds to the collection of modifications of the $\mathcal{P}^{\text{ad}}_i=\text{Res}_{R_i/R}(\mathcal{Q}_i)$ along $Y$ using $\alpha_i \in G_i((\widetilde{R}\otimes_RR_i)\lps{t})= \text{Res}_{R_i/R}G_i(\widetilde{R}\lps{t})$, which are trivial, so that $\mathcal{Q}^{\text{ad}}$ is trivial, to the effect that $\mathcal{Q}$ reduces to a torsor over $\mathbb{P}_R^1$ under the center $Z_G$ of $G$. Now, as the last paragraph of the proof of \cite{Ces22}{Proposition~5.3.6} showed, any $Z_G$-torsor over $\mathbb{P}_R^1$ is isomorphic to the sum of a constant torsor (i.e., the pullback of a $Z_G$-torsor over $R$) and $\lambda_(\mathscr{O}(1))$ for a unique cocharacter $\lambda $ of $Z_G$. Therefore, by assumption \ref{main tech iii}, $\mathcal{Q}\|_{\mathbb{P}_R^1 \backslash Y}$ is a constant torsor, and, by checking along the infinity section, it is even trivial, so is $\mathcal{P}\|_{\mathbb{P}_R^1 \backslash Y}=\mathcal{Q}\|_{\mathbb{P}_R^1 \backslash Y}$, as desired. \end{proof} The following helps us to construct the desired $R$-finite, \'etale schemes $Y_i$ and $Y$ in \Cref{main tech for torsor on A1}. \begin{lemma-tweak}\label{lem on isotropicity} Let $R\rightarrow R_1$ be a finite \'etale ring map of semilocal rings with connected spectra, let $W \subset \mathbb{A}_R^1$ be an $R$-finite closed scheme, and let $G_1$ be a simple $R_1$-group scheme. There is an $R_1$-finite \'etale scheme $Y_1$ and a closed immersion $Y_1 \subset \mathbb{A}_R^1\backslash W$ over $R$ such that $(G_1)_{Y_1}$ contains a copy of $\mathbb{G}_{\text{m},Y_1}$, and, for every maximal ideal $\mathfrak{m} \subset R_1$ with $(G_1)_{\kappa(\mathfrak{m})}$ isotropic, the line bundle $\mathscr{O}_{\mathbb{P}_{\kappa(\mathfrak{m})}^1}(1)$ is trivial over $\mathbb{P}_{\kappa(\mathfrak{m})}^1 \backslash (Y_1)_{\kappa(\mathfrak{m})}$.\footnote{Note that $Y_1$ is a clopen of $Y_1\times_RR_1$, thus naturally embeds into $\mathbb{A}_{R_1}^1$.} In addition, there is an $R_1$-finite, \'etale scheme $\widetilde{Y}$ and a closed immersion $\widetilde{Y} \hookrightarrow \mathbb{A}_R^1\backslash W$ over $R$ such that the line bundle $\mathscr{O}_{\mathbb{P}_R^1}(1)$ is trivial over $\mathbb{P}_R^1 \backslash \widetilde{Y}$. \end{lemma-tweak} \begin{proof} Let $\text{Par}^{\prime} \to \Spec R_1$ be the scheme parameterizing \emph{proper} parabolic subgroup schemes of the reductive $R_1$-group scheme $G_1$; it is smooth projective over $R_1$ (\cite{SGA3IIInew}{Exposé~XXVI, Corollaire~3.5}). Fix an embedding $\text{Par}' \hookrightarrow \mathbb{P}_{R_1}^N$ over $R_1$. Write $\text{Par}'=\bigsqcup_{i=1}^t P_t$ as a disjoint union of its connected components; every $P_t$ has a constant relative dimension $d_t$ over $R_1$. For every maximal ideal $\mathfrak{m} \subset R_1$ with $(G_1)_{\kappa(\mathfrak{m})}$ isotropic, a proper parabolic subgroup of $(G_1)_{\kappa(\mathfrak{m})}$ gives a point $b_{\mathfrak{m}}\in \text{Par}'(\kappa(\mathfrak{m}))$. Fix an $i=1,\cdots, t$. For every maximal ideal $\mathfrak{m} \subset R_1$, by Bertini theorems (including Poonen's version \cite{Poo04}{Theorem~1.2} over finite fields), one can find a hypersurface in $\mathbb{P}_{\kappa(\mathfrak{m})}^N$ of large enough degree such that it passes through all points $b_{\mathfrak{m}}$ that lies in $P_i$ and has smooth intersection with $(P_i)_{\kappa(\mathfrak{m})}$. We may assume that the above hypersurfaces have the same degree for all $\mathfrak{m}$. By the Chinese Remainder theorem, one can lift these simultaneously to get a hypersurfaces $H\subset \mathbb{P}_{R_1}^N$. Then $H\cap P_i$ is a smooth projective $R_1$-scheme of pure relative dimension $d_i-1$, and $b_{\mathfrak{m}}\in H\cap P_i$ whenever $b_{\mathfrak{m}}\in P_i$. The same argument can be applied to the hypersurface section $H\cap P_i$. Continuing in this way, we finally arrive at an $R_1$-finite, \'etale, closed subscheme $Y_i \subset P_i$ such that $b_{\mathfrak{m}}\in Y_i$ whenever $b_{\mathfrak{m}}\in P_i$. Denote $Y_1^{\prime} \colonequals \bigsqcup_{i=1}^t Y_i$. Unfortunately, $Y_1^{\prime}$ may not embed into $\mathbb{A}_R^1\backslash W$. So we first modify $Y_1'$ by using Panin's `finite field tricks'. For large enough integers $d>0$, we can choose a monic polynomial $h_{\mathfrak{n}' }\in \kappa(\mathfrak{n}')[u]$ of degree $2d+1$ for each maximal ideal $\mathfrak{n}' \subset \Gamma(Y_1',\mathscr{O}_{Y_1'})$ such that: \begin{itemize} \item [(1)] if $\kappa(\mathfrak{n}')$ is finite, $h_{\mathfrak{n}'}$ is a product of two irreducible polynomials of degrees $d$ and $d+1$\footnote{This follows from the following fact: for a finite field $k$, there are asymptotically $(\# \,k)^d/d$ points on $\mathbb{A}_k^1$ of exact degree $d$.}, respectively; \item [(2)] if $\kappa(\mathfrak{n}')$ is infinite, $h_{\mathfrak{n}'}$ is a separable polynomial and has at least one root in $\kappa(\mathfrak{n}')$. \end{itemize} Let $h\in \Gamma(Y_1',\mathscr{O}_{Y_1'})[u] $ be a common monic lifting of $h_{\mathfrak{n}' }$ for all $\mathfrak{n}' \subset \Gamma(Y_1',\mathscr{O}_{Y_1'})$. Define \[ \textstyle Y_1\colonequals \Spec \bigl(\Gamma(Y_1',\mathscr{O}_{Y_1'})[u]/(h)\bigr); \] it is finite, \'etale over $Y_1'$, and hence also over $R_1$. By (1), for each maximal ideal $\mathfrak{n}\subset R$ with finite residue field, $(Y_1)_{\kappa(\mathfrak{n})}$ has at most $2\,\text{deg}(Y_1'/R)$ points, each of degree $\ge d$ over $\mathfrak{n}$, so, there is no finite fields obstruction to embedding $Y_1$ into $\mathbb{A}_R^1\backslash W$ over $R$ in the following sense: for every maximal ideal $\mathfrak{n}\subset R$, \[ \# \left\{ z\in (Y_1)_{\kappa(\mathfrak{n})}:[\kappa(z):\kappa(\mathfrak{n})]=e \right\}\le \# \left\{z\in \mathbb{A}_{\kappa(\mathfrak{n})}^1\backslash W: [\kappa(z):\kappa(\mathfrak{n})]=e \right\} \quad \text{for every} \quad e\ge 1, \] because the left hand side is zero for $e<d$ and is uniformly bounded for $e\ge d$, but the right hand side is asymptotically $(\# \,\kappa(\mathfrak{n}))^e/e$, which tends to infinity as $e$ grows. Consequently, for such $d$, there exists \[ \textstyle \text{a closed immersion \quad $\bigsqcup_{\mathfrak{n} \subset R} (Y_1)_{\kappa(\mathfrak{n})} \hookrightarrow \mathbb{A}_{R}^1\backslash W $ \quad over $R$; } \] by Nakayama lemma, any of its lifting $Y_1 \hookrightarrow \mathbb{A}_{R}^1\backslash W$ over $R$ (which exists since $Y_1$ is affine) is also a closed immersion. By construction, the restriction of $(G_1)_{Y_1'}$ to every connected component of $Y_1'$ contains a proper parabolic subgroup scheme, so, by \cite{SGA3IIInew}{Exposé~XXVI, Corollaire~6.12}, $(G_1)_{Y_1^{\prime}}$ contains $\mathbb{G}_{m,Y_1'}$, and also $(G_1)_{Y_1}$ contains $\mathbb{G}_{m,Y_1}$. For every maximal ideal $\mathfrak{m}\subset R_1$ with $(G_1)_{\kappa(\mathfrak{m})}$ isotropic, by construction, there exists a point $\mathfrak{n}^{\prime} \colonequals b_{\mathfrak{m}}\in Y_1'$, that is, $\kappa(\mathfrak{m})=\kappa(\mathfrak{n}^{\prime})$. Thus, in case (1) (resp., in case (2)), $(Y_1)_{\kappa(\mathfrak{m})}$ contains points of both degrees $d$ and $d+1$ (resp., a point of exact degree $1$) over $\mathfrak{m}$, so the line bundles $\mathscr{O}_{\mathbb{P}_{\kappa(\mathfrak{m})}^1}(d)$ and $\mathscr{O}_{\mathbb{P}_{\kappa(\mathfrak{m})}^1}(d+1)$ are trivial over $\mathbb{P}_{\kappa(\mathfrak{m})}^1 \backslash (Y_1)_{\kappa(\mathfrak{m})}$, hence so is $\mathscr{O}_{\mathbb{P}_{\kappa(\mathfrak{m})}^1}(1)$. To construct $\widetilde{Y}$, it suffices to produce, for a large $d$, $R$-finite, \'etale, closed subschemes $\widetilde{Y}_1, \widetilde{Y}_2\subset \mathbb{A}_R^1$ of $R$-degrees $d$ and $d+1$ which are disjoint from $W$, and then take $\widetilde{Y}\colonequals \widetilde{Y}_1\sqcup \widetilde{Y}_2$. To achieve this, one just need to imitate the above procedure for constructing $Y_1$ from $Y_1'$. Details omitted. \end{proof} \begin{pp}[Proof of \Cref{triviality on sm rel. affine curves}] \Cref{1st reduction of sec thm} reduces us to the case when $C=\mathbb{A}_R^1$ and $G$ is a reductive $R$-group scheme. Up to shifting we may assume that $s=0_R\in \mathbb{A}_R^1(R)$ is the zero section, and base changing to $A$ reduces us further to the case $A=R$ at the cost of that $R$ need not be a domain or geometrically unibranch. Thus, in case \ref{sec-thm-semilocal}, our $R$ is semilocal, and, in case \ref{sec-thm-totally isotropic}, our $G$ is totally isotropic. For both cases \ref{sec-thm-semilocal}--\ref{sec-thm-totally isotropic}, by glueing $\mathcal{P}$ with the trivial $G$-torsor over $\mathbb{P}_R^1\backslash Z$ we extend $\mathcal{P}$ to a $G$-torsor $\mathcal{Q}$ over $\mathbb{P}_R^1$. By \cite{Fed22b}{Proposition~2.3} or \cite{Ces22}{Lemma~5.3.5}, up to replacing $\mathcal{Q}$ and $Z$ by their pullbacks by $\mathbb{P}_R^1\to \mathbb{P}_R^1, t\mapsto t^d$, where $d$ is divisible by the $R$-fibral degrees of the simply-connected central cover $(G^{\text{ad}})^{\text{sc}}\to G^{\text{ad}}$, we may assume that for every maximal ideal $\mathfrak{m} \subset R$ the $G^{\text{ad}}$-torsor over $\mathbb{P}_{\kappa(\mathfrak{m})}^1$ induced by $\mathcal{Q}$ lifts to a generically trivial $(G^{\text{ad}})^{\text{sc}}$-torsor over $\mathbb{P}_{\kappa(\mathfrak{m})}^1$. \end{pp} \begin{cl-tweak} \label{lem on trivialize ouside Y} In both cases \ref{sec-thm-semilocal}--\ref{sec-thm-totally isotropic}, assume that $R$ is semilocal. For any $R$-finite closed subscheme $W_0\subset \mathbb{A}_R^1$, there exists an $R$-finite, \'etale, closed subscheme $Y\subset \mathbb{A}_R^1 \backslash W_0$ such that $\mathcal{Q}\|_{\mathbb{P}_R^1 \backslash Y}$ is trivial. \end{cl-tweak} \begin{proof} [Proof of the claim] Restricting ourselves on each connected component, we may assume that $\Spec R$ is connected. Write the canonical decomposition of $G^{\text{ad}}$ as in \Cref{main tech for torsor on A1}. Replace $W_0$ by $W_0\cup Z$ to assume that $Z\subset W_0$. Applying Lemma \ref{lem on isotropicity} separately to each simple $R_i$-group scheme $G_i$ (with appropriate choices of $W$'s), we get $R_i$-finite, \'etale schemes $Y_i$ such that $(G_i)_{Y_i}$ are totally isotropic, and a closed immersion $\bigsqcup_i Y_i \hookrightarrow \mathbb{A}_R^1 \backslash W_0$ over $R$ such that, for every maximal ideal $\mathfrak{m} \subset R_i$ with $(G_i)_{\kappa(\mathfrak{m})}$ isotropic, the line bundle $\mathscr{O}_{\mathbb{P}_{\kappa(\mathfrak{m})}^1}(1)$ is trivial over $\mathbb{P}_{\kappa(\mathfrak{m})}^1 \backslash (Y_i)_{\kappa(\mathfrak{m})}$. Applying the second part of Lemma \ref{lem on isotropicity} to $W\colonequals (\sqcup_i Y_i) \bigsqcup W_0$, we obtain an $R$-finite, \'etale, closed subscheme $$ \textstyle Y'\subset \mathbb{A}_R^1\backslash \left((\sqcup_i Y_i) \bigsqcup W_0\right) $$ such that $\mathscr{O}_{\mathbb{P}_R^1}(1)$ is trivial over $\mathbb{P}_R^1 \backslash Y'$. Denote $Y\colonequals Y^{\prime} \bigsqcup (\sqcup_i Y_i) $. Then all the assumptions \ref{main tech i}--\ref{main tech iii} of \Cref{main tech for torsor on A1} are verified, so we conclude that $\mathcal{Q}\|_{\mathbb{P}_R^1 \backslash Y}$ is trivial. \end{proof} For \ref{sec-thm-semilocal}, we take $W_0=Z\cup 0_R$, then \Cref{lem on trivialize ouside Y} gives a $R$-finite \'etale closed $Y\subset \mathbb{A}_R^1 \backslash W_0$ such that $\mathcal{Q}\|_{\mathbb{P}_R^1 \backslash Y}$ is trivial. Since $Y\cap 0_R =\emptyset$, we deduce that the pullback of $\mathcal{Q}$ along $s=0_R$ is also trivial, as wanted. For \ref{sec-thm-totally isotropic}, we will follow \cite{Ces22b}{Lemma~4.3} to show that both $\mathcal{P}=\mathcal{Q}\|_{\mathbb{A}_R^1}$ and $\mathcal{Q}\|_{\mathbb{P}_R^1 \backslash 0_R}$ descend to $G$-torsors over $R$, and then we are done: both of these descendants agree with the restriction of $\mathcal{Q}$ along $1_R\in \mathbb{A}_R^1(R)$, so they agree with the restriction of $\mathcal{Q}$ along $\infty_R$, which is trivial, and hence they must be trivial. By Quillen patching \cite{Ces22}{Corollary~5.1.5~(b)}, for the descent claim we may replace $R$ by its localizations at maximal ideals to assume that $R$ is local. Now, since $R$ is local, we apply \Cref{lem on trivialize ouside Y} to $W_0=0_R$ to get a $R$-finite \'etale closed $Z'\subset \mathbb{A}_R^1 \backslash 0_R$ such that $\mathcal{Q}\|_{\mathbb{P}_R^1\backslash Z'}$ is trivial. It remains to apply \Cref{main tech for torsor on A1} twice using that $G$ is totally isotropic, with $Y=0_R$ (resp., $Y=\infty_R$) and $Y_i=Y\times_{R}R_i$, to show that both $\mathcal{Q}\|_{\mathbb{P}_R^1\backslash 0_R}$ and $\mathcal{Q}\|_{\mathbb{P}_R^1 \backslash \infty_R}$ are trivial. \QED \section{ Torsors under a reductive group scheme over a smooth projective base} \label{sect-torsor on sm proj base} The main result of this section is the following: \begin{thm-tweak}\label{torsors-Sm proj base} For a semilocal Pr\"{u}fer domain $R$, an $r\in R\backslash \{0\}$, an irreducible, smooth, projective $R$-scheme $X$, a finite subset $\textbf{x} \subset X$ with semilocal ring $A\colonequals \mathscr{O}_{X,\textbf{x}}$, and a reductive $X$-group scheme $G$, \begin{enumerate}[label={{\upshape(\roman)}}] \item \label{loc-gen-trivial-sm-proj} any generically trivial $G$-torsor over $A$ is trivial, that is, \[ \ker\,(H^1(A,G)\rightarrow H^1(\Frac A, G))=\{\ast\}; \] \item \label{Nis-sm-proj}if $G_{A[\f{1}{r}]}$ is totally isotropic, then any generically trivial $G$-torsor over $A[\f{1}{r}]$ is trivial, that is, \[ \textstyle \ker\,(H^1(A[\f{1}{r}],G)\rightarrow H^1(\Frac A,G))=\{\ast\}. \] \end{enumerate} \end{thm-tweak} The case (i) is a version of the Grothendieck--Serre conjecture in the case the relevant reductive group scheme $G_A$ has a reductive model over some smooth projective compactification of $\Spec A$. The case (ii) provides a version of Nisnevich conjecture for such `nice' reductive groups satisfying the total isotropicity assumption: if $R$ is a discrete valuation ring with uniformizer $r$ and if $R\to A$ is a local homomorphism of local rings, then $r\in \mathfrak{m}_A\backslash \mathfrak{m}_A^2$, and (ii) says that any generically trivial $G$-torsor over $A[\f{1}{r}]$ is trivial (the isotropicity assumption on $G_A$ is essential, see, for instance, \cite{Fed21}). \begin{rem-tweak} An inspection of the proof below shows that \Cref{torsors-Sm proj base} still holds provided that $X$ is only a flat projective $R$-scheme such that $X\backslash X^{\text{sm}}$ is $R$-fiberwise of codimension $\ge 2$ in $X$, $\textbf{x}\subset X^{\text{sm}}$, and $G$ is a reductive $X^{\text{sm}}$-group scheme, where $X^{\text{sm}}$ denotes the smooth locus of $X\to \Spec R$. \end{rem-tweak} To prove \Cref{torsors-Sm proj base}, we first derive from \Cref{extend generically trivial torsors} and \Cref{Ces's Variant 3.7} the following key result, which reduces the proof of \Cref{torsors-Sm proj base} to studying torsors on a smooth affine relative curve. \begin{lemma-tweak}\label{nicely spread out lem} For a semilocal Pr\"{u}fer domain $R$ of finite Krull dimension, an irreducible, smooth, projective $R$-scheme $X$ of pure relative dimension $d> 0$, a finite subset $\textbf{x} \subset X$, and a reductive $X$-group scheme $G$, the following assertions hold. \begin{enumerate}[label={{\upshape(\roman)}}] \item \label{GS-nicely spread out} Given a generically trivial $G$-torsor $\mathcal{P}$ over $A\colonequals \mathscr{O}_{X,\textbf{x}}$, there are \begin{itemize} \item [-] a smooth, affine $A$-curve $C$, an $A$-finite closed subscheme $Z\subset C$, and a section $s\in C(A)$; \item [-] a reductive $C$-group scheme $\mathscr{G}$ satisfying $s^{\ast}\mathscr{G}\simeq G_A$ and a $\mathscr{G}$-torsor $\mathcal{F}$ such that $\mathcal{F}\|_{C\backslash Z}$ is trivial and $s^{\ast}\mathcal{F} \simeq \mathcal{P}$. \end{itemize} \item \label{Nis-nicely spread out} Given an $r\in R\backslash \{0\}$ and a generically trivial $G$-torsor $\widetilde{\mathcal{P}}$ over $A[\f{1}{r}]$, there are \begin{itemize} \item [-] a smooth, affine $A$-curve $C$, an $A$-finite closed subscheme $Z\subset C$, and a section $s\in C(A)$; \item [-] a reductive $C$-group scheme $\mathscr{G}$ such that $s^{\ast}\mathscr{G}\simeq G_A$, a $\mathscr{G}$-torsor $\widetilde{\mathcal{F}}$ over $C[\f{1}{r}]\colonequals C\times_AA[\f{1}{r}]$ such that $\widetilde{\mathcal{F}}\|_{C[\f{1}{r}]\backslash Z[\f{1}{r}]}$ is trivial and $(s\|_{A[\f{1}{r}]})^{\ast}(\widetilde{\mathcal{F}}) \simeq \widetilde{\mathcal{P}}$. \end{itemize} \end{enumerate} \end{lemma-tweak} \begin{proof} By \Cref{extend generically trivial torsors}, $\mathcal{P}$ (resp., $\widetilde{\mathcal{P}}$) extends to a $G$-torsor $\mathcal{P}_0$ (resp., $\widetilde{\mathcal{P}_0}$) over an open neighbourhood $W\subset X$ of $\textbf{x}$ (resp., an open neighbourhood $\widetilde{W}\subset X$ of $\Spec (A[\f{1}{r}])$) such that \[ \text{$\codim((X\backslash W)_K, X_K)\geq 3$ \quad and\quad $\codim((X\backslash W)_s,X_s)\geq 2$ for all $s\in \Spec (R)$}; \] and \[ \text{$\codim((X\backslash \widetilde{W})_K, X_K)\geq 3$ \quad and\quad $\codim((X\backslash \widetilde{W})_s,X_s)\geq 2$ for all $s\in \Spec (R)$.} \] Here, $K$ is the fraction field of $R$. Let $\textbf{z}\subset X$ be the set of maximal points of the $R$-fibers of $X$; the above codimension bounds implies $\textbf{z}\subset W$ (resp., $\textbf{z}\subset \widetilde{W}$). By \Cref{geom}\ref{geo-iii}, the semilocal ring $\mathscr{O}_{X,\textbf{z}}$, and hence also $\mathscr{O}_{X,\textbf{z}}[\f{1}{r}]$, is a Pr\"{u}fer domain. By the Grothendieck--Serre on semilocal Pr\"{u}fer schemes (\Cref{G-S over semi-local prufer}), the generically trivial $G$-torsor $(\mathcal{P}_0)\|_{\mathscr{O}_{X,\textbf{z}}}$ (resp., $(\widetilde{\mathcal{P}_0})\|_{\mathscr{O}_{X,\textbf{z}}[\f{1}{r}]}$) is actually trivial. Thus there exists a closed subscheme $Y\subset X$ (resp., $\widetilde{Y}\subset X$) that avoids all the maximal points of $R$-fibers of $X$ such that the restriction $(\mathcal{P}_0)\|_{X\backslash Y}$ (resp., $(\widetilde{\mathcal{P}_0})\|_{(X\backslash \widetilde{Y})[\f{1}{r}]}$) is trivial; such a $Y$ (resp., $\widetilde{Y}$) is $R$-fiberwise of codimension $>0$ in $X$. Now, we treat the two cases \ref{GS-nicely spread out}--\ref{Nis-nicely spread out} separately. For \ref{GS-nicely spread out}, by the above, $X\backslash W$ is $R$-fiberwise of codimension $\ge 2$ in $X$; \emph{a fortiori,} the same codimension bound holds for $Y\backslash W $ in $X$. Consequently, we can apply \Cref{Ces's Variant 3.7} to obtain an affine open $S\subset \mathbb{A}_{R}^{d-1}$, an affine open neighbourhood $ U\subset W$ of $\textbf{x}$, and a smooth morphism $\pi\colon U\to S$ of pure relative dimension 1 such that $U\cap Y$ is $S$-finite. Let $\tau:C\colonequals U\times_S\Spec A\to \Spec A$ be the base change of $\pi$ to $\Spec A$. Let $Z$ and $\mathcal{F}$ be the pullbacks of $U\cap Y$ and $(\mathcal{P}_0)\|_U$ under $\text{pr}_1:C\to U$, respectively. Then, via $\tau,$ $C$ is a smooth affine $A$-curve, $Z\subset C$ is a $A$-finite closed subscheme, and $\mathcal{F}$ is a $\mathscr{G}\colonequals \text{pr}_1^(G_U)$-torsor that trivializes over $C\backslash Z$. Finally, the diagonal in $C$ induces a section $s\in C(A)$ with $s^{\ast}\mathcal{F}\simeq \mathcal{P}$ (as $s^{\ast}\mathscr{G}=G_A$-torsors). For \ref{Nis-nicely spread out}, since $\Spec (A[\f{1}{r}])$ consists of points of $X[\f{1}{r}]\colonequals X\times_RR[\f{1}{r}]$ that specializes to some point of $\textbf{x}$, we deduce from the inclusion $\Spec A[\f{1}{r}] \subset \widetilde{W}$ that no points of $(X\backslash \widetilde{W})[\f{1}{r}] =X[\f{1}{r}]\backslash \widetilde{W}[\f{1}{r}]$ specializes to any points of $\textbf{x}$. Hence, the closure $\overline{(X\backslash \widetilde{W})[\f{1}{r}]}$ (in $X$) is disjoint from $\textbf{x}$, so $\widetilde{W}'\colonequals X\backslash \overline{(X\backslash \widetilde{W})[\f{1}{r}]}$ is an open neighbourhood of $\textbf{x}$. Notice that, since $\Spec R$ has finite Krull dimension, $X$ is topological Noetherian. Since $(X\backslash \widetilde{W})[\f{1}{r}]$ is $R[\f{1}{r}]$-fiberwise of codimension $\ge 2$ in $X[\f{1}{r}]$, by \Cref{geom}\ref{geo-i} applied to the closures of the (finitely many) maximal points of $(X\backslash \widetilde{W})[\f{1}{r}]$, the closure $\overline{(X\backslash \widetilde{W})[\f{1}{r}]}=X \backslash \widetilde{W}'$ is $R$-fiberwise of codimension $\ge 2$ in $X$; \emph{a fortiori}, the same holds for $\widetilde{Y}\backslash \widetilde{W}'$ in $X$. Consequently, we can apply \Cref{Ces's Variant 3.7} to obtain an affine open $\widetilde{S}\subset \mathbb{A}_{R}^{d-1}$, an affine open neighbourhood $ \widetilde{U}\subset \widetilde{W}'$ of $\textbf{x}$, and a smooth morphism $\widetilde{\pi}:\widetilde{U}\to \widetilde{S}$ of pure relative dimension 1 such that $\widetilde{U}\cap \widetilde{Y}$ is $\widetilde{S}$-finite. Notice that $\widetilde{U}[\f{1}{r}]\subset \widetilde{W}'[\f{1}{r}] =\widetilde{W}[\f{1}{r}]$, so we have the restriction $(\widetilde{\mathcal{P}_0})\|_{\widetilde{U}[\f{1}{r}]}$. Let $\tau\colon C\colonequals \widetilde{U}\times_{\widetilde{S}}\Spec A\to \Spec A$ be the base change of $\widetilde{\pi}$ to $\Spec A$. Let $Z$ be the pullback of $\widetilde{U}\cap \widetilde{Y}$ under $\mathrm{pr}_1\colon C\to \widetilde{U}$. Let $\widetilde{\mathcal{F}}$ be the pullback of $(\widetilde{\mathcal{P}_0})\|_{\widetilde{U}[\f{1}{r}]}$ under $\mathrm{pr}_1\colon C[\f{1}{r}]\to \widetilde{U}[\f{1}{r}]$. Then, via $\tau,$ $C$ is a smooth affine $A$-curve, $Z\subset C$ is a $A$-finite closed subscheme, and $\widetilde{\mathcal{F}}$ is a $\mathscr{G}\colonequals \text{pr}_1^{\ast}(G_{\widetilde{U}})$-torsor over $C[\f{1}{r}]$ that trivializes over $C[\f{1}{r}]\backslash Z[\f{1}{r}]$. Finally, the diagonal in $C$ induces a section $s\in C(A)$ such that $s^{\ast}\mathscr{G}=G_A$ and $s_{A[\f{1}{r}]}^{\ast}(\widetilde{\mathcal{F}})\simeq \widetilde{\mathcal{P}}$.\qedhere \end{proof} \begin{pp}[Proof of \Cref{torsors-Sm proj base}] By a standard limit argument involving \Cref{approxm semi-local Prufer ring}, one easily reduces to the case when $R$ has finite Krull dimension. Now, let $\mathcal{P}$ (resp., $\widetilde{\mathcal{P}}$) be a generically trivial $G$-torsor over $A\colonequals \mathscr{O}_{X,\textbf{x}}$ (resp., over $A[\f{1}{r}]$) that we want to trivialize. Let $d$ be the relative dimension of $X$ over $R$. If $d=0$, then $A$ and $A[\f{1}{r}]$ are semilocal Pr\"{u}fer domains, so, by the Grothendieck--Serre on semilocal Pr\"{u}fer schemes (\Cref{G-S over semi-local prufer}), the torsors $\mathcal{P}$ and $\widetilde{\mathcal{P}}$ are trivial. Hence we may assume that $d>0$. Then, by \Cref{nicely spread out lem}, there are a smooth, affine $A$-curve $C$, an $A$-finite closed subscheme $Z\subset C$, a section $s\in C(A)$, a reductive $C$-group scheme $\mathscr{G}$ with $s^{\ast}\mathscr{G}\simeq G_A$, \begin{itemize} \item [-] a $\mathscr{G}$-torsor $\mathcal{F}$ over $C$ that trivializes over $C\backslash Z$ such that $s^{\ast}\mathcal{F} \simeq \mathcal{P}$, and \item [-] a $\mathscr{G}$-torsor $\widetilde{\mathcal{F}}$ over $C[\f{1}{r}]$ that trivializes over $C[\f{1}{r}]\backslash Z[\f{1}{r}]$ such that $(s\|_{A[\f{1}{r}]})^{\ast}(\widetilde{\mathcal{F}}) \simeq \widetilde{\mathcal{P}}$. \end{itemize} By \Cref{triviality on sm rel. affine curves} \ref{sec-thm-semilocal}, the $G$-torsor $s^{\ast}\mathcal{F} \simeq \mathcal{P}$ is trivial. By \Cref{triviality on sm rel. affine curves} \ref{sec-thm-totally isotropic}, in case $(s\|_{A[\f{1}{r}]})^{\ast}(\mathscr{G})\simeq G_{A[\f{1}{r}]}$ is totally isotropic, the $G_{A[\f{1}{r}]}$-torsor $(s\|_{A[\f{1}{r}]})^{\ast}(\widetilde{\mathcal{F}}) \simeq \widetilde{\mathcal{P}}$ is trivial. \QED \end{pp} \section{ Torsors under a constant reductive group scheme} \label{sect-torsor under constant redu} In this section we prove the first main result of this paper, namely, the Grothendieck--Serre conjecture and a version of Nisnevich conjecture, for `constant' reductive group schemes. The proof uses a variant of Lindel's Lemma (\Cref{variant of Lindel's lem}) and glueing techniques to reduce to the resolved case \Cref{torsors-Sm proj base}. \begin{thm-tweak}\label{G-S for constant reductive gps} For a semilocal Pr\"ufer domain $R$, an $r\in R\backslash \{0\}$, an irreducible affine $R$-smooth scheme $X$, a finite subset $\textbf{x}\subset X$, and a reductive $R$-group scheme $G$, \begin{enumerate}[label={{\upshape(\roman)}}] \item\label{G-S for constant reductive gps i} any generically trivial $G$-torsor over $A\colonequals \mathscr{O}_{X,\textbf{x}}$ is trivial, that is, \[ \mathrm{ker}\left(H^1(A,G)\to H^1(\Frac A,G)\right)=\{\}; \] \item\label{G-S for constant reductive gps ii} if $G_{R[\f{1}{r}]}$ is totally isotropic, then any generically trivial $G$-torsor over $A[\f{1}{r}]$ is trivial, that is, \[ \textstyle \mathrm{ker}\left(H^1(A[\f{1}{r}],G)\to H^1(\Frac A,G)\right)=\{\}. \] \end{enumerate} \end{thm-tweak} \begin{proof} Let $\mathcal{P}$ (resp., $\widetilde{\mathcal{P}}$) be a generically trivial $G$-torsor over $A$ (resp., over $A[\f{1}{r}]$). By shrinking $X$ around $\textbf{x}$, we may assume that $\mathcal{P}$ is defined on the whole $X$ (resp., $\widetilde{\mathcal{P}}$ is defined on the whole $X[\f{1}{r}]\colonequals X\times_RR[\f{1}{r}]$). Let $d$ be the relative dimension of $X$ over $R$. As pointed out by $\check{\mathrm{C}}$esnavi$\check{\mathrm{c}}$ius, since it suffices to argue that $\mathcal{P}$ (resp., $\widetilde{\mathcal{P}}$) is trivial Zariski-semilocally on $X$, we may replace $X$ by $X\times_R \mathbb{A}_R^N$ for large $N$ to assume that $d>\#\, \textbf{x}$: by pulling back along the zero section $X\to X\times_R \mathbb{A}_R^N$, the Zariski-semilocal triviality of $\mathcal{P}_{X\times_R \mathbb{A}_R^N}$ (resp., $\widetilde{\mathcal{P}}_{X[\f{1}{r}]\times_R \mathbb{A}_R^N}$) on $X\times_R \mathbb{A}_R^N$ implies that of $\mathcal{P}$ (resp., $\widetilde{\mathcal{P}}$) on $X$. Our goal is to show that $\mathcal{P}\|_A$ (resp., $\widetilde{\mathcal{P}}\|_{A[\f{1}{r}]}$) is trivial. A limit argument involving \Cref{approxm semi-local Prufer ring} reduces us to the case when $R$ has finite Krull dimension. By specialization, we can assume that each point of $\textbf{x}$ is closed in its \emph{corresponding} $R$-fiber of $X$ (but not necessarily lies in the closed $R$-fibers of $X$). If $d=0$, then $A$ (resp., $A[\f{1}{r}]$) is a semilocal Pr\"{u}fer domain, so, by \Cref{G-S over semi-local prufer}, the torsor $\mathcal{P}\|_A$ (resp., $\widetilde{\mathcal{P}}\|_{A[\f{1}{r}]}$) is trivial. Thus we may assume that $d>0$ for what follows. Let $\textbf{y}$ be the set of maximal points of the $R$-fibers of $X$. \begin{cl-tweak} No points of $\textbf{x}$ specializes to any point of $\textbf{y}$, that is, $\overline{\textbf{x}} \cap \textbf{y} =\emptyset$. \end{cl-tweak} \begin{proof} [Proof of the claim] Let $\pi:X\to S\colonequals \Spec R$ be the structural morphism. If not, say $x\in \textbf{x}$ specializes to $y\in \textbf{y}$, then, by \Cref{geom}\ref{geo-i}, \[ \dim \overline{\{x\}}_{\pi(x)} = \dim \overline{\{x\}}_{\pi(y)}, \] which is $\ge \dim \overline{\{y\}}_{\pi(y)}=d$ (because $y$ is a maximal point in the fiber $ \pi^{-1}(\pi(y))$ which has pure dimension $d$). Since $\dim \pi^{-1}(\pi(x))=d>0$, $x$ can not be a closed point of the fiber $ \pi^{-1}(\pi(x))$, a contradiction. \end{proof} By \Cref{geom}~\ref{geo-iii} again, the semilocal ring $\mathscr{O}_{X,\textbf{y}}$, and hence also $\mathscr{O}_{X,\textbf{y}}[\f{1}{r}]$, are Pr\"{u}fer domains, so, by \Cref{G-S over semi-local prufer}, the $G$-torsor $\mathcal{P}\|_{\mathscr{O}_{X,\textbf{y}}}$ (resp., $\widetilde{\mathcal{P}}\|_{\mathscr{O}_{X,\textbf{y}}[\f{1}{r}]}$) is trivial. Therefore, using the above claim and prime avoidance, we can find an element $a\in \Gamma(X,\mathscr{O}_X)$ such that, if denote $ Y\colonequals V(a)\subset X$, then $\textbf{x}\subset Y$, $\textbf{y}\cap Y=\emptyset$, and the restriction $\mathcal{P}\|_{X\backslash Y}$ (resp., $\widetilde{\mathcal{P}}\|_{(X\backslash Y)[\f{1}{r}]}$) is trivial. (We just take $a=a_1a_2$, where $a_1$ is such that $\textbf{y}\cap V(a_1)=\emptyset$ and $\mathcal{P}\|_{X\backslash V(a_1)}$ (resp., $\widetilde{\mathcal{P}}\|_{(X\backslash V(a_1))[\f{1}{r}]}$) is trivial, and $a_2$ is delivered from prime avoidance utilizing the fact $\overline{\textbf{x}} \cap \textbf{y} =\emptyset$, so that $\textbf{x}\subset V(a_2)$ and $\textbf{y} \cap V(a_2)=\emptyset$.) Since $d>\#\, \textbf{x}$, we may apply \Cref{variant of Lindel's lem} to obtain an affine open neighbourhood $W\subset X$ of $\textbf{x}$, an affine open subscheme $ U\subset \mathbb{A}_R^d$, and an \'etale surjective $R$-morphism $f:W\to U$ such that the restriction $f\|_{W\cap Y}$ is a closed immersion and $f$ induces a Cartesian square \begin{equation} \begin{tikzcd} W\cap Y \arrow[r, hook] \arrow[d, equal] & W \arrow[d, "f"] \\ W\cap Y \arrow[r, hook] & U. \end{tikzcd} \end{equation} Applying $(-)\times_RR[\f{1}{r}]$ yields a similar Cartesian square. By glueing \cite{Ces22a}{Lemma~7.1}, \begin{enumerate}[label={{\upshape(\roman)}}] \item we may (non-canonically) glue $\mathcal{P}\|_{W}$ and the trivial $G$-torsor over $U\backslash f(W\cap Y)$ to descend $\mathcal{P}\|_{W}$ to a $G$-torsor $\mathcal{Q}$ over $U$ that trivializes over $U\backslash f(W\cap Y)$. Since $U$ has a smooth, projective compactification $\mathbb{P}_R^d$, we may apply \Cref{torsors-Sm proj base}~\ref{loc-gen-trivial-sm-proj} to deduce that $\mathcal{Q}\|_{\mathscr{O}_{U,f(\textbf{x})}}$ is trivial, so $\mathcal{P}\|_A=\mathcal{P}\|_{\mathscr{O}_{W,\textbf{x}}}$ is trivial, as desired. \item we may (non-canonically) glue $\widetilde{\mathcal{P}}\|_{W[\f{1}{r}]}$ and the trivial $G$-torsor over $(U\backslash f(W\cap Y))[\f{1}{r}]$ to descend $\widetilde{\mathcal{P}}\|_{W[\f{1}{r}]}$ to a $G$-torsor $\widetilde{\mathcal{Q}}$ over $U[\f{1}{r}]$ that trivializes over $U[\f{1}{r}]\backslash f(W\cap Y)[\f{1}{r}]$. Since $U$ has a smooth, projective compactification $\mathbb{P}_R^d$, we may apply \Cref{torsors-Sm proj base}~\ref{Nis-sm-proj} to conclude that $\widetilde{\mathcal{Q}}\|_{\mathscr{O}_{U,f(\textbf{x})}[\f{1}{r}]}$ is trivial, so $\widetilde{\mathcal{P}}\|_{A[\f{1}{r}]}=\widetilde{\mathcal{P}}\|_{\mathscr{O}_{W,\textbf{x}}[\f{1}{r}]}$ is trivial, as desired. \qedhere \end{enumerate} \end{proof} \section{The Bass--Quillen conjecture for torsors} \label{sect-Bass-Quillen} In this section, we prove the following variant of Bass--Quillen conjecture for torsors: \begin{thm-tweak} \label{B-Q over val rings} For a ring $A$ that is smooth over a Pr\"{u}fer ring $R$, a totally isotropic reductive $R$-group scheme $G$, then, via pullback, \[ \text{$H^1_{\mathrm{Nis}}(A,G)\isoto H^1_{\mathrm{Nis}}(\mathbb{A}^N_A,G)$\quad or equivalently, \quad $H^1_{\mathrm{Zar}}(A,G)\isoto H^1_{\mathrm{Zar}}(\mathbb{A}^N_A,G)$}. \] \end{thm-tweak} We notice that very special instances of \Cref{B-Q over val rings} for $G=\GL_n$ (that is, for vector bundles) was already known: Simis--Vasconcelos in \cite{SV71} considered the case when $A$ is a valuation ring and $N=1$, and Lequain--Simis in \cite{LS80} treated the case when $A$ is a Pr\"{u}fer ring. Apart from that, we are not aware of any instance of \Cref{B-Q over val rings} even for $G=\GL_n$ in our non-Noetherian context. \begin{rem-tweak} By \Cref{G-S for constant reductive gps}, for a reductive $R$-group scheme $G$, a $G$-torsor on $\mathbb{A}_R^N$ is Nisnevich-locally trivial, if and only if it is generically trivial, if and only if it is Zariski-locally trivial. This implies the equivalence of the formulation of \Cref{B-Q over val rings} for the topologies `Nis' and `Zar'. \end{rem-tweak} \begin{pp}[The `inverse' to Quillen patching] Compared to Quillen patching \cite{Ces22}{Corollary~5.1.5}, the following `inverse' to Quillen patching is more elementary but is also useful. Its case when $G=\text{GL}_n$ and $A=R[t_1,\cdots,t_N]$ is due to Roitman \cite{Roi79}{Proposition 2}. \end{pp} \begin{lemma-tweak}\label{inverse patching} Let $R$ be a ring, let $G$ be a quasi-affine, flat, finitely presented $R$-group scheme, let $A=\bigoplus_{i_1,\cdots,i_N\ge 0}A_{i_1,\cdots,i_N}$ be a $\mathbb{Z}_{\ge 0}^{\oplus N}$-graded $R$-algebra (resp., a $\mathbb{Z}_{\ge 0}^{\oplus N}$-graded domain over $R$) such that $R\xrightarrow{\sim}A_{0.\cdots,0}$, and suppose that every $G$-torsor on $A$ (resp., every generically trivial $G$-torsor on $A$) descends to a $G$-torsor on $R$. Then, for any multiplicative subset $S\subset R$, every $G$-torsor on $A_S$ (resp., every generically trivial $G$-torsor on $A_S$) whose restriction to each local ring of $(A_{0.\cdots,0})_S\simeq R_S$ extends to a $G$-torsor on $R$ descends to a $G$-torsor on $R_S$. \end{lemma-tweak} (The relevant case for us is when $A=R[t_1,\cdots,t_N]$.) \begin{proof} We focus on the part on generically trivial torsors, since the other is \cite[Proposition 5.1.10]{Ces22}. Let $X$ be a generically trivial $G$-torsor on $A_S$ whose restriction to each local ring of $(A_{0.\cdots,0})_S\simeq R_S$ extends to a $G$-torsor on $R$. By Quillen patching \cite{Ces22}{Corollary~5.1.5}, we may enlarge $S$ to reduce to the case when $R_S$ is local. Then, by assumption, the restriction of $X$ to $(A_{0.\cdots,0})_S\simeq R_S$ extends to a $G$-torsor $X_0$ on $R$. A limit argument reduces us further to the case when $S=\{r\}$ is a singleton at the cost of $R_S$ no longer being local. Notice that the projection onto the $(0,\cdots,0)$-th component \[ \textstyle R\oplus \bigl( \bigoplus_{(i_1,\cdots,i_N)\neq (0,\cdots, 0)} A_{i_1,\cdots,i_N}[\frac{1}{r}] \bigr)\simeq A[\frac{1}{r}] \times_{R[\frac{1}{r}]}R \twoheadrightarrow R \] induces an isomorphism both modulo $r^n$ and on $r^n$-torsion for every $n>0$. So, by \cite[Proposition 4.2.2]{Ces22}, we can glue the $G$-torsor $X$ on $A[\frac{1}{r}]$ with the $G$-torsor $X_0$ on $R$ to obtain a generically trivial $G$-torsor $\widetilde{X}$ on $A[\frac{1}{r}] \times_{R[\frac{1}{r}]}R$. However, since \[ \textstyle A[\frac{1}{r}] \times_{R[\frac{1}{r}]}R \simeq \underset{i\in \mathbb{N}}{\text{colim}} \,A, \] where the transition maps $A\to A$ are given by multiplications by $r^{i_1+\cdots+i_N}$ on the degree $(i_1,\cdots,i_N)$-part $A_{i_1,\cdots,i_N}$, so, by a standard limit argument, $\widetilde{X}$ descends to a generically trivial $G$-torsor on some copy of $A$ in the direct colimit. Therefore, by assumption, it descends further to a $G$-torsor on $R$. The base change to $R_S$ of this final descendant gives a desired descendant of $X$ to a $G$-torsor on $R_S$. \end{proof} \begin{pp}[Torsors on $\mathbb{A}_R^N$ under reductive $R$-group schemes] The following was conjectured in \cite{Ces22}{Conjecture 3.5.1} and settled recently in \cite{Ces22b}{Theorem 2.1(a)}. \end{pp} \begin{thm-tweak}\label{triviality over relative affine space for totally isotropic} For a ring $R$ and a $\mathrm{totally}$ $\mathrm{isotropic}$ reductive $R$-group scheme $G$, any $G$-torsor on $\mathbb{A}_R^N$ that is trivial away from some $R$-finite closed subscheme of $\mathbb{A}_R^N$ is trivial. \end{thm-tweak} \begin{lemma-tweak} \label{B-Q for mult type} For a normal domain $A$, an $A$-group $M$ of multiplicative type, the pullback \[ \text{$H^1_{\mathrm{fppf}}(A,M)\isoto H^1_{\mathrm{fppf}}(\mathbb{A}_A^n,M)$ \quad is bijective. } \] \end{lemma-tweak} \begin{proof} When $A$ is Noetherian, this is \cite{CTS87}{Lemma 2.4}. For a general normal domain $A$, we write it as a filtered union of its finitely generated $\mathbf{Z}$-subalgebras $A_i$, and, by replacing $A_i$ with its normalizations (which is again of finite type over $\mathbf{Z}$), we may assume that each $A_i$ is normal, so we may conclude from the Noetherian case via a limit argument, because $M$ is finitely presented over $A$. \end{proof} For any commutative unital ring $A$, denote by $A(t)$ the localization of $A[t]$ with respect to the multiplicative system of \emph{monic} polynomials. \begin{lemma-tweak}\label{triviality over R(t)} Let $R$ be a semilocal Pr\"ufer domain and $G$ a reductive $R$-group scheme. \[ \text{Every $G$-torsor over $R(t)$ is trivial \quad if and only if\quad it is generically trivial.} \] \end{lemma-tweak} \begin{proof} Let $\mathcal{E}$ be a generically trivial $G$-torsor over $R(t)$. Denote by $\mathfrak{m}$ the maximal ideal of $R$. We observe that $R(t)$ is the local ring of the projective $t$-line $\mathbb{P}_R^1$ over $R$ at $\infty \in \mathbb{P}_{V/\mathfrak{m}}^1$, with $s\colonequals \frac{1}{t}$ inverted: \[ \textstyle R(t)=\left(R[s]\right)_{(\mathfrak{m},s)}\left[ \frac{1}{s}\right]. \] Hence, $\mathcal{E}$ spreads out to a $G$-torsor $\mathcal{E}_{U^{\circ}}$ over $U^{\circ}\colonequals U\backslash \{s=0\} $ for an affine open $U\subset \mathbb{P}_R^1$ containing $\infty$. The fiber of $\{s=0\}$ over $\Frac R$ is the singleton $\{\xi\}$, where $\xi$ denotes the generic point of $\{s=0\}$, and \[ W\,\cap \,\text{Spec}\,\mathscr{O}_{\mathbb{P}_R^1,\xi} = \{\text{the generic point of } \mathbb{P}_R^1\}. \] Since $\mathcal{E}_{U^{\circ}}$ is generically trivial and $U^{\circ}$ is affine hence quasi-compact, there is an open neighbourhood $U_{\xi} \subset U$ of $\xi$ such that $\mathcal{E}_{U^{\circ}}\|_{U^{\circ}\cap U_{\xi}}$ is trivial. Consequently, we may (non-canonically) glue $\mathcal{E}_{U^{\circ}}$ with the trivial $G$-torsor over $U_{\xi}$ to obtain a $G$-torsor $\widetilde{\mathcal{E}}$ over the open subset $U^{\circ} \cup U_{\xi}$ of $U$. By construction, $\widetilde{\mathcal{E}}\|_{U^{\circ}}\cong \mathcal{E}_{U^{\circ}}$. Now the complementary closed $Z\colonequals U\backslash (U^{\circ} \cup U_{\xi})$ is contained in $\{s=0\}\backslash \{\xi\}$. In particular, \[ \text{$Z\times_R \Frac R= \emptyset$ \quad and\quad $\codim(Z_s,U_s)\ge 1$ for all $s\in \text{Spec}\,R$.} \] By purity \Cref{purity for rel. dim 1} of reductive torsors on smooth relative curves, $\widetilde{\mathcal{E}}$ extends to a $G$-torsor, which is also denoted by $\widetilde{\mathcal{E}}$, over entire $U$. As $\widetilde{\mathcal{E}}$ is generically trivial, by \Cref{G-S for constant reductive gps} \ref{G-S for constant reductive gps i}, its restriction to $\mathscr{O}_{\mathbb{P}_R^1,\infty}=\left(R[s]\right)_{(\mathfrak{m},s)}$, and its further restriction to $R(t)$, is trivial, that is, $\mathcal{E}\|_{R(t)}\cong \widetilde{\mathcal{E}}\|_{R(t)}$ is trivial. \end{proof} \begin{pp}[Proof of \Cref{B-Q over val rings}] Since any section of $\mathbb{A}_A^N\to \Spec A$ induces sections to the pullback maps in \Cref{B-Q over val rings}, these pullback maps are injective, so it suffices to show that they are surjective. By Quillen patching \cite{Ces22}{Corollary~5.1.5}, we may replace $R$ by its localizations to assume that $R$ is a valuation ring. By a limit argument involving \Cref{approxm semi-local Prufer ring}, we reduce to the case of a finite-rank $R$. \textbf{Step 1:} \emph{$A$ is a polynomial ring over $R$.} It suffices to show that every generically trivial $G$-torsor $\mathcal{E}$ over $R[t_1,\cdots,t_N]$ is trivial; \emph{a fortiori, } it descends to a $G$-torsor over $R$. To prove this we will argue by double induction on the pair $(N,\text{rank}(R))$. If $N=0$, then by convention $R[t_1,\cdots,t_N]=R$, we know from \cite{Guo20b} that a generically trivial $G$-torsor over $R$ is trivial. Now assume $N\ge 1$ and set $A':=R(t_N)[t_1,\cdots,t_{N-1}]$. \end{pp} \begin{cl-tweak} The $G_{A'}$-torsor $\mathcal{E}_{A'}$ descends to a $G_{R(t_N)}$-torsor $\mathcal{E}_0$. \end{cl-tweak} \begin{proof}[Proof of the claim] Consider the natural projection $\pi:\text{Spec}\,R(t_N) \to \Spec R$. Since by definition, $R(t_N)$ is the localization of $R[t_N]$ with respect to the multiplicative system of monic polynomials, the closed fiber of $\pi$ consists of the singleton $\{\mathfrak{p}_0\}$; further, the local ring $R(t_N)_{\mathfrak{p}_0}$ is a valuation ring of $\text{Frac}(R)(t_N)$ whose valuation restricts to the Gauss valuation associated to $R$ on $R[t_N]$: $$ \textstyle R[t_N]\to \Gamma_R , \quad \sum_{i\ge 0} a_i t_N^i \mapsto \min_i v(a_i), $$ where $v:R\to \Gamma_R$ is the (additive) valuation on $R$. In particular, $R$ and $R(t_N)_{\mathfrak{p}_0}$ have the same value group. To apply the Quillen patching \cite{Ces22}{Corollary~5.1.5} and conclude, it suffices to show that the base change of $\mathcal{E}_{A'}$ to $R(t_N)_{\mathfrak{p}}[t_1,\cdots,t_{N-1}]$ is trivial for every prime ideal $\mathfrak{p} \subset R(t_N)$. Indeed, if $\mathfrak{p}=\mathfrak{p}_0$, then the above discussion implies $\text{rank}(R_{\pi(\mathfrak{p})})=\text{rank}(R)$, so the desired triviality of the base change follows from induction hypothesis. If $\mathfrak{p}\neq \mathfrak{p}_0$, then $\pi(\mathfrak{p}) \in \Spec R$ is not the closed point and $\text{rank}(R_{\pi(\mathfrak{p})})<\text{rank}(R)$, so, by induction hypothesis, $\mathcal{E}_{R_{\pi(\mathfrak{p})}[t_1,\cdots,t_N]}$ and hence also its further base change along $R_{\pi(\mathfrak{p})}[t_1,\cdots,t_N]\to R(t_N)_{\mathfrak{p}}[t_1,\cdots,t_{N-1}]$ is trivial. \end{proof} Since $\mathcal{E}$ is generically trivial, by considering the pullback of $\mathcal{E}_{A'}$ along a general section $s\in \mathbb{A}_{R(t_N)}^{N-1}(R(t_N))$, we see that $\mathcal{E}_0$ is also generically trivial. By Lemma \ref{triviality over R(t)}, $\mathcal{E}_0$, and hence also $\mathcal{E}_{A'}$, is trivial. Consequently, $\mathcal{E}$ is trivial away from the $R$-finite closed subset $\{f(t_N)=0\} \subset \mathbb{A}_R^N$ for some monic polynomial $f\in R [t_N]$. By \Cref{triviality over relative affine space for totally isotropic}, $\mathcal{E}$ is trivial. This completes the induction process. \textbf{Step 2:} \emph{$A$ is the localization of a polynomial ring $\widetilde{R}:=R[u_1,\cdots,u_d]$ with respect to some multiplicative subset $S\subset \widetilde{R}$.} We wish to apply the `inverse' to Quillen patching \Cref{inverse patching}, with $R$ being $\widetilde{R}$ here and $A$ being $\widetilde{R}[t_1,\cdots,t_N]$. It remains to check the assumptions of \emph{loc.~cit}. Firstly, by Step 1, a generically trivial $G$-torsor over $\widetilde{R}[t_1,\cdots,t_N]$ descends to a $G$-torsor over $\widetilde{R}$. Secondly, for any multiplicative subset $S\subset \widetilde{R}$ and any generically trivial $G$-torsor $\mathcal{E}$ over $\widetilde{R}_S[t_1,\cdots,t_N]$, the restriction of $\mathcal{E}$ to each local ring of the closed subscheme $$ \Spec \widetilde{R}_S:=\left\{t_1=\cdots=t_N=0\right\}\subset \mathbb{A}_{\widetilde{R}_S}^N $$ is trivial (so trivially extends to a $G$-torsor over $\widetilde{R}$): by Bass--Quillen in the field case, the restriction of $\mathcal{E}$ to $\text{Frac}(\widetilde{R}_S)[t_1,\cdots,t_N]$ is trivial. Thus the restriction $\mathcal{E}\|_{\widetilde{R}_S}$ is generically trivial and hence, by \Cref{G-S for constant reductive gps}\ref{G-S for constant reductive gps i}, is Zariski locally trivial. All assumptions of \Cref{inverse patching} are thus verified. \textbf{Step 3:} \emph{$A$ is an arbitrary smooth $R$-algebra.} Let $\mathcal{E}$ be a generically trivial $G$-torsor over $\mathbb{A}_A^N$. We need to show that $\mathcal{E}$ descends to a $G$-torsor over $A$. By Quillen patching \cite{Ces22}{Corollary~5.1.5}, we may assume that $A$ is an essentially smooth local algebra over a valuation ring $R$. By localizing $R$, we assume that the homomorphism $R \to A$ is local. We will argue by double induction on the pair $(\dim R, \dim A-\dim R)$ to show that $\mathcal{E}$ is even trivial. If $\dim R=0$, then $R$ is a field and we come back to the classical already settled case. If $\dim A=\dim R$, by \Cref{geom}~\ref{geo-iii}, then $A$ is a valuation ring, and we come back to the case already settled in Step 1. Assume now $\dim R>0$ and $\dim A-\dim R>0$. By \Cref{enlarge valuation rings}, up to enlarging $R$ (without changing $\dim R$) we may assume that $A=\mathscr{O}_{X,x}$, where $X$ is an irreducible affine $R$-smooth scheme of pure relative dimension $d>0$, and $x\in X$ is a \emph{closed} point in the \emph{closed} $R$-fiber. Then necessarily we have $d=\dim A-\dim R$. By \Cref{variant of Lindel's lem}, up to shrinking $X$, there are an \'etale $R$-morphism $f:X\to \mathbb{A}_R^d$ and an element $r_0\in A_0:=\mathscr{O}_{\mathbb{A}_R^d,f(x)}$ such that \[ \text{$f$ \quad induces \quad \quad $A_0/r_0A_0 \xrightarrow{\sim} A/r_0A$.} \] On the other hand, our induction hypothesis implies that $\mathcal{E}_{\mathbb{A}_{A_{\mathfrak{p}}}^N}$ is trivial for any $\mathfrak{p}\in \text{Spec}\,A\backslash \{\mathfrak{m}_A\}$ (thus descends to the trivial $G$-torsor over $A_{\mathfrak{p}}$): \begin{itemize} \item either $\mathfrak{p}$ lies over a non-maximal ideal $ \mathfrak{q}\subset R$, in which case $R_{\mathfrak{q}} \to A_{\mathfrak{p}}$ is a local homomorphism, \[ \dim R_{\mathfrak{q}} <\dim R, \quad \text{ and } \quad \dim A_{\mathfrak{p}}-\dim R_{\mathfrak{q}}\le d=\dim A-\dim R; \] \item or $R \to A_{\mathfrak{p}}$ is a local homomorphism, in which case $$ \dim A_{\mathfrak{p}}-\dim R<\dim A-\dim R. $$ \end{itemize} By Quillen patching, we deduce that $\mathcal{E}_{\mathbb{A}_{A[\frac{1}{r_0}]}^N}$ descends to a $G$-torsor $\mathcal{F}$ over $A[\frac{1}{r_0}]$. However, $\mathcal{F}$ must be trivial, because it extends to a generically trivial $G$-torsor over $A$, such as the restriction of $\mathcal{E}$ along any section $s\in \mathbb{A}_{A}^N(A)$, and, by \Cref{G-S for constant reductive gps}\ref{G-S for constant reductive gps i}, any such an extension is trivial. Now by the \cite{Ces22a}{Lemma~7.1} applied to the Cartesian square \begin{equation} \begin{tikzcd} \mathbb{A}_{A/r_0A}^N \arrow[hookrightarrow]{r} \arrow[d, "\sim"] & \mathbb{A}_A^N \arrow{d} \\ \mathbb{A}_{A_0/r_0A_0}^N \arrow[hookrightarrow]{r} & \mathbb{A}_{A_0}^N, \end{tikzcd} \end{equation} we may glue $\mathcal{E}$ with the trivial $G$-torsor over $\mathbb{A}_{A_0[\frac{1}{r_0}]}^N $ to obtain a $G$-torsor $\mathcal{E}_0$ over $\mathbb{A}_{A_0}^N$ that trivializes over $\mathbb{A}_{A_0[\frac{1}{r_0}]}^N$. By Step 2, $\mathcal{E}_0$ is trivial, so $\mathcal{E}$ is trivial as well. \QED \begin{appendix} \section{Grothendieck--Serre on a semilocal Pr\"{u}fer domain} \label{section-G-S on semilocal prufer} The main result of this section is the following generalization of the main results of \cite{Guo22} and \cite{Guo20b}. \begin{thm} \label{G-S over semi-local prufer} For a semilocal Pr\"{u}fer domain $R$ and a reductive $R$-group scheme $G$, we have \[ \text{$\mathrm{ker}\left(\text{H}_{\mathrm{\acute{e}t}}^1(R,G)\to \text{H}_{\mathrm{\acute{e}t}}^1(\Frac R,G)\right)=\{\}$}. \] \end{thm} \begin{pp-t}[Setup]\label{pd-setup} We fix the following notations. For a semilocal Pr\"ufer domain $R$ of finite Krull dimension, all the maximal ideals $(\mathfrak{m}_i)_{i=1}^r$ of $R$, the local rings $\mathcal{O}_i\colonequals R_{\mathfrak{m}_i}$, an element $a\in R$ such that $V(a)=\{\mathfrak{m}_i\}_{i=1}^r$, let $\widehat{R}$ (resp., $\widehat{\mathcal{O}}_i$) denote the $a$-adic completion of $R$ (resp., of $\mathcal{O}_i$). Then $\widehat{\mathcal{O}}_i$ is an $a$-adic complete valuation ring of rank 1, and we have $\widehat{R}\simeq \prod_{i=1}^r\widehat{\mathcal{O}}_i$, compatibly with the topologizes. Denote $\widehat{K}_i\colonequals \Frac \widehat{\mathcal{O}}_i=\widehat{\mathcal{O}}_i[\f{1}{a}]$. Topologize $R[\f{1}{a}]$ by declaring $\{\mathrm{im}(a^nR\rightarrow R[\f{1}{a}])\}_{n\ge 1}$ to be a fundamental system of open neighbourhood of $0$; the associated completion is \[ \textstyle \text{$R[\f{1}{a}]\to \widehat{R}[\f{1}{a}]\simeq \prod_{i=1}^r \widehat{\mathcal{O}}_i[\f{1}{a}]=\prod_{i=1}^r \widehat{K}_i,$} \] where each $\widehat{K}_i$ is a complete valued field, with pseudo-uniformizer (the image of) $a$. In particular, for an $R$-scheme $X$, we have a map \[ \textstyle \text{$\Phi_X\colon X(R[\f{1}{a}])\rightarrow \prod_{i=1}^rX(\widehat{K}_i)$.} \] If $X$ is locally of finite type over $R$, we endow the right hand side with the product topology where each $X(\widehat{K}_i)$, by, for example, Conrad, has a natural topology induced from that of $\widehat{K}_i$, which we will call the $a$-adic topology. If moreover $X$ is affine, we can canonically topologize $X(R[\f{1}{a}])$ by choosing a closed embedding $X\hookrightarrow \mathbb{A}_R^N$ and endowing $X(R[\f{1}{a}]) \hookrightarrow R[\f{1}{a}]^N$ with the subspace topology (this is independent of the choices of the embeddings), then $\Phi_X$ is a continuous map. \end{pp-t} \csub[Lifting maximal tori of reductive group schemes over semilocal rings] Without any difficulty, one can generalize the lifting of maximal tori \cite{Guo20b} to semilocal case. \begin{lemma}\label{lift-tor} For a semilocal scheme $S$ and an $S$-smooth finitely presented group scheme $G$ whose $S$-fibers are connected and affine. Assume that \begin{enumerate}[label={{\upshape(\roman)}}] \item for each residue field $\kappa(s)$ of $S$ at a closed point $s$, the fiber $G_{\kappa(s)}$ is a $\kappa(s)$-reductive group; and \item $\# \kappa(s)\geq \dim (G_{\kappa(s)}/Z_{\kappa(s)})$ for the center $Z_{\kappa(s)}\subset G_{\kappa(s)}$, \end{enumerate} then the following natural map is surjective \[ \textstyle \un{\mathrm{Tor}}(G)(S)\twoheadrightarrow \prod_{s\in S\mathrm{\, closed}}\un{\mathrm{Tor}}(G)(\kappa(s)). \] \end{lemma} \begin{lemma}\label{tor-dense} For a semilocal Pr\"ufer domain $R$ of finite Krull dimension, we use the notations in the setup \S\ref{pd-setup}. For a reductive $R$-group scheme $G$, the scheme $\un{\mathrm{Tor}}(G)$ of maximal tori of $G$, and the $a$-adic topology on $\un{\mathrm{Tor}}(G)(\widehat{K}_i)$, the image of the following map is dense: \[ \textstyle \un{\mathrm{Tor}}(G)(R[\f{1}{a}])\rightarrow \prod_{i=1}^r\un{\mathrm{Tor}}(G)(\widehat{K}_i). \] \end{lemma} \csub[Harder's weak approximation] \vskip -0.5cm \begin{lemma}\label{tor-open-tor} For a semilocal Pr\"ufer domain $R$ of finite Krull dimension, we use the setup \S\ref{pd-setup}. For a $R[\f{1}{a}]$-torus $T$, let $L_i/\widehat{K}_i$ be minimal Galois field extensions splitting $T_{\widehat{K}_i}$ and consider the norm map \[ N_i\colon T(L_i)\rightarrow T(\widehat{K}_i). \] Then, the image $U$ of $\prod_{i=1}^r N_i$ is open and is contained in $\overline{\mathrm{im}(T(R[\f{1}{a}])\rightarrow \prod_{i=1}^rT(\widehat{K}_i))}$. \end{lemma} \begin{proof} The proof proceeds as the following steps. \vskip 0.1cm \textbf{Step 1.} The image $U$ is open. For each $i$, there is a short exact sequence of tori \[ 1\rightarrow \mathcal{T}_i\rightarrow \Res_{L_i/\widehat{K}_i}T_{L_i}\rightarrow T_{\widehat{K}_i}\rightarrow 1 \] and the norm map $N_i\colon (\Res_{L_i/\widehat{K}_i}T_{L_i})(\widehat{K}_i)\rightarrow (\Res_{L_i/\widehat{K}_i}T_{L_i}/\mathcal{T}_i)(\widehat{K}_i)$, which by \cite{Ces15d}{Proposition~4.3~(a) and \S2.8~(2)} is $a$-adically open. As a product of open subsets, $U$ is open in $\prod_{i=1}^r T(\widehat{K}_i)$. \vskip 0.1cm \textbf{Step 2.} We prove that $U$ is contained in the closure of $\mathrm{im}(T(\ria))$. Equivalently, we show that every $u\in U$ and every open neighbourhood $B_u\subset U$ satisfy that $B_u\cap \mathrm{im}(T(\ria))\neq \emptyset$. Let $\widetilde{R}/\ria$ be a minimal Galois cover splitting $T$. Consider the following commutative diagram \[ \begin{tikzcd} T(\tilde{R}) \arrow[d, "N_{\widetilde{R}/\ria}"'] \arrow[r] & \prod_{i=1}^rT(L_i) \arrow[d, "\prod_{i=1}^rN_i"] \\ T(\ria) \arrow[r] & \prod_{i=1}^rT(\widehat{K}_i) . \end{tikzcd} \] Take a preimage $v\in (\prod_{i=1}^rN_i)^{-1}(u) \subset \prod_{i=1}^rT(L_i)$ and let $B_v\subset \prod_{i=1}^r T(L_i)$ be the preimage of $B_u$. Since $T_{\widetilde{R}}$ splits, the image of $T(\widetilde{R})$ in $\prod_{i=1}^rT(L_i)$ is dense, hence $T(\widetilde{R})\times_{\prod_{i=1}^rT(L_i)}B_v\neq \emptyset$, namely, there is $r\in T(\widetilde{R})$ whose image is in $B_v$. Let $s\colonequals N_{\widetilde{R}/\ria}(r)\in T(\ria)$, then the image of $s$ under the map $T(\ria)\rightarrow \prod_{i=1}^rT(\widehat{K}_i)$ is contained in $B_u$, so the assertion follows. \end{proof} \begin{lemma}\label{tor-i-open} For a semilocal Pr\"ufer domain $R$ of finite Krull dimension, we use the setup \S\ref{pd-setup}. For a reductive $R$-group scheme $G$ and for each $i$ a fixed maximal torus $T_i\subset G_{\widehat{K}_i}$ with minimal Galois field extension $L_i/\widehat{K}_i$ splitting $T_i$, consider the following norm map \[ N_i\colon T_i(L_i)\rightarrow T_i(\widehat{K}_i). \] Then the image $U$ of the map $\prod_{i=1}^{r}N_i$ is an open subgroup of $\prod_{i=1}^r T_i(\widehat{K}_i)$ and is contained in $\overline{\mathrm{im}(G(\ria))}$, the closure of $\mathrm{im}(G(\ria)\rightarrow \prod_{i=1}^rG(\widehat{K}_i))$. \end{lemma} \begin{proof} By the same arguement in \Cref{tor-open-tor}, the image $U$ is open in $\prod_{i=1}T_i(\widehat{K}_i)$. It remains to show that $U\subset \overline{\mathrm{im}(G(\ria))}$, which proceeds as the following steps. \vskip 0.1cm \textbf{Step 1.} The map $\phi_i\colon G(\widehat{K}_i)\rightarrow \un{\mathrm{Tor}}(G)(\widehat{K}_i)$ defined by $g\mapsto gTg^{-1}$ is $a$-adically open for each $i$. Since the image of $\un{\mathrm{Tor}}(G)(\ria)\rightarrow \prod_{i=1}^r \un{\mathrm{Tor}}(G)(\widehat{K}_i)$ is dense, for every open neighbourhood $W\subset \prod_{i=1}^rG(\widehat{K}_i)$ of $\mathrm{id}$, we have $((\prod_{i=1}^r\phi_i)(W))\cap \mathrm{im}(\un{\mathrm{Tor}}(G)(\ria)\rightarrow \prod_{i=1}^r\un{\mathrm{Tor}}(G)(\widehat{K}_i))\neq \emptyset$. Therefore, there exist a torus $T^{\prime}\in \un{\mathrm{Tor}}(G)(\ria)$ and a $(g_i)_{i=1}^r\in W$ such that $g_iT_ig_i^{-1}=T^{\prime}_{\widehat{K}_i}$ for all $i$. \vskip 0.1cm \textbf{Step 2.} For any $u\in U$, consider the map $\prod_{i=1}^rG(\widehat{K}_i)\rightarrow \prod_{i=1}^rG(\widehat{K}_i)$ defined by $g\mapsto g^{-1}ug$. Then, we apply the Step 1 to the preimage $W$ of $U$ under this map: there is a $\gamma=(\gamma_i)_{i=1}^r\in W$ and a torus $T^{\prime}\in \un{\mathrm{Tor}}(G)(\ria)$ such that $\gamma_iT_i\gamma_i^{-1}=T^{\prime}_{\widehat{K}_i}$ for each $i$. Then, $u\in \gamma U\gamma^{-1}=\gamma (\prod_{i=1}^rN_i (T_i(L_i)))\gamma^{-1}$, which by transport of structure, is $\prod_{i=1}^rN_i(T^{\prime}_{\widehat{K}_i}(L_i))$. By \Cref{tor-open-tor}, the last term is contained in the closure of $\mathrm{im}(T^{\prime}(\ria)\rightarrow \prod_{i=1}^rT^{\prime}(\widehat{K}_i))$, so is contained in $\overline{\mathrm{im}(G(\ria))}$. \end{proof} \begin{prop}\label{open-normal} For a semilocal Pr\"ufer domain $R$ of finite Krull dimension, we use the setup \S\ref{pd-setup}. For a reductive $R$-group scheme $G$, the closure $\overline{\mathrm{im}(G(\ria))}$ of the image of $G(\ria)\rightarrow \prod_{i=1}^rG(\widehat{K}_i)$, \[ \textstyle \text{$\overline{\mathrm{im}(G(\ria))}$\quad contains an open normal subgroup $N$ of $\prod_{i=1}^rG(\widehat{K}_i)$.} \] \end{prop} \begin{proof} The proof proceeds in the following steps. \begin{enumerate}[label={{\upshape(\roman)}}] \item For each $i$, we fix a maximal torus $T_i\subset G_{\widehat{K}_i}$. Then \Cref{tor-i-open} provides the open subgroup $U\subset \prod_{i=1}^rT_i(\widehat{K}_i)$. Since each component of the norm map defining $U$ is the image of the $\widehat{K}_i$-points of $\Res_{L_i/\widehat{K}_i}(T_{i,L_i})\rightarrow T_i$, and $\Res_{L_i/\widehat{K}_i}(T_{i,L_i})$ is a Zariski dense open subset of an affine space over $\widehat{K}_i$, we have $U\cap \prod_{i=1}^rT_i^{\mathrm{reg}}(\widehat{K}_i)\neq \emptyset$. \item For any $\tau=(\tau_i)_{i=1}^r\in U\cap \prod_{i=1}^rT^{\mathrm{reg}}_i(\widehat{K}_i)$, by \cite{SGA3II}{Exposé~XIII, Corollaire~2.2}, for each $i$, \[ \text{ $f_i\colon G_{\widehat{K}_i}\times T_i\rightarrow G_{\widehat{K}_i},\quad (g,t)\mapsto gtg^{-1}$\quad is smooth at $(\mathrm{id}, \tau_i)$.} \] Hence, there is a Zariski open neighbourhood $B\subset \prod_{i=1}^r ( G_{\widehat{K}_i}\times T_i)$ of $(\mathrm{id},\tau)$ such that \[ \textstyle \text{$(\prod_{i=1}^rf_i)\|_B\colon B\rightarrow \prod_{i=1}^r G_{\widehat{K}_i}$ \quad is smooth.} \] By \cite{GGMB14}{Proposition~3.1.4}, the map $B(\prod_{i=1}^r\widehat{K}_i)\rightarrow \prod_{i=1}^rG(\widehat{K}_i)$ is open. In particular, since $(\prod_{i=1}^rG(\widehat{K}_i))\times U$ is an open neighbourhood of $(\mathrm{id},\tau)$, $E:=(\prod_{i=1}^{r}f_i)((\prod_{i=1}^rG(\widehat{K}_i))\times U)$ contains a non-empty open subset of $\prod_{i=1}^rG(\widehat{K}_i)$. Define $N$ as the subgroup of $\prod_{i=1}^rG(\widehat{K}_i)$ generated by $E$; it is open. By construction, $E$ is stable under conjugations by $\prod_{i=1}^rG(\widehat{K}_i)$, thus $N$ is normal. \item We prove that $N$ is in the closure of $\mathrm{im}(G(\ria)\rightarrow \prod_{i=1}^rG(\widehat{K}_i))$. As $E$ is the union of conjugates of $U$, which are contained in $\overline{G(\ria)}$ by \Cref{tor-i-open}, so $E$ is in this closure, and so is $N$. \qedhere \end{enumerate} \end{proof} \begin{cor}\label{lift-r-torus} For a semilocal Pr\"ufer domain $R$ of finite Krull dimension, we use the setup \S\ref{pd-setup}. For a reductive group scheme $G$ over $R$, a maximal torus $T_i\subset G_{\widehat{\mathcal{O}}_i}$ for each $i$, and any open neighbourhood $W$ of $\mathrm{id}\in \prod_{i=1}^rG(\widehat{K}_i)$ such that $W\subset \overline{\mathrm{im}(G(\ria))}\cap \prod_{i=1}^r G(\widehat{\mathcal{O}}_i)$, there exist $g=(g_i)_i\in W$ and a maximal torus $T\in \un{\mathrm{Tor}}(G)(R)$ such that for every $i$, we have \[ T_{\widehat{K}_i}=g_iT_{i,\widehat{K}_i}g_i^{-1}. \] \end{cor} \begin{proof} By \Cref{open-normal}, $\overline{\mathrm{im}(G(\ria))}\cap \prod_{i=1}^rG(\widehat{\mathcal{O}}_i)$ is an $a$-adically open neighbourhood of $\mathrm{id}\in \prod_{i=1}^rG(\widehat{K}_i)$, so it makes sense to take its subset $W$ such that $W$ is a neighbourhood of $\mathrm{id}$. Now consider the $a$-adically open map $\phi\colon \prod_{i=1}^rG(\widehat{K}_i)\rightarrow \prod_{i=1}^r\un{\mathrm{Tor}}(G)(\widehat{K}_i)$ defined by $g_i\mapsto g_iT_{i,\widehat{K}_i}g_i^{-1}$. Then $\phi(W)$ is an $a$-adically open neighbourhood of $(T_i)_i\in \prod_{i=1}^{r}\un{\mathrm{Tor}}(G)(\widehat{K}_i)$. Since $\prod_{i=1}^r\un{\mathrm{Tor}}(G)(\widehat{\mathcal{O}}_i)\subset \prod_{i=1}^r\un{\mathrm{Tor}}(G)(\widehat{K}_i)$ is also an $a$-adically open neighbourhood of $(T_i)_i$, we have an open intersection $\phi(W)\cap \prod_{i=1}^r\un{\mathrm{Tor}}(G)({\widehat{\mathcal{O}}_i})\neq \emptyset$. Then the density of the image of $\un{\mathrm{Tor}}(G)(\ria)\rightarrow \prod_{i=1}^r\un{\mathrm{Tor}}(G)(\widehat{K}_i)$ provided by \Cref{tor-dense} yields \[ \textstyle T\in \un{\mathrm{Tor}}(G)(R)\isoto \un{\mathrm{Tor}}(G)(\ria)\times_{\prod_{i=1}^r\un{\mathrm{Tor}}(G)(\widehat{K}_i)}\prod_{i=1}^r\un{\mathrm{Tor}}(G)(\widehat{\mathcal{O}}_i), \] thanks to the identification $R\isoto \ria \times_{\prod_{i=1}^r\widehat{K}_i}\prod_{i=1}^r\widehat{\mathcal{O}}_i$ and the affineness of $\un{\mathrm{Tor}}(G)$ over $R$. Then $T$ is a maximal $R$-torus of $G$ satisfying the conditions. \end{proof} \begin{lemma} \label{replace im by it closure} With the notations in \Cref{open-normal}, we have \[ \textstyle \overline{\mathrm{im}(G(\ria))}\cdot \prod_{i=1}^rG(\widehat{\mathcal{O}}_i)=\mathrm{im}(G(\ria)\rightarrow \prod_{i=1}^r G(\widehat{K}_i))\cdot \prod_{i=1}^r G(\widehat{\mathcal{O}}_i). \] \end{lemma} \csub[Product formula over semilocal Pr\"ufer domains] The goal of this section is to prove the following product formula for reductive group schemes: \begin{prop}\label{decomp-gp} For a semilocal Pr\"ufer domain $R$ of finite Krull dimension, we use the notations in the setup \S\ref{pd-setup}. For a reductive $R$-group scheme $G$, we have \[ \textstyle \prod_{i=1}^r G(\widehat{K}_i)=\mathrm{im}\bigl(G(R[\f{1}{a}])\rightarrow \prod_{i=1}^r G(\widehat{K}_i)\bigr)\cdot \prod_{i=1}^rG(\widehat{\mathcal{O}}_i). \] \end{prop} Before proceeding, we recall the following consequence of the Beauville-Laszlo type glueing of torsors: \begin{lemma} \label{double cosets} The $G$-torsors on $R$ that trivialize both on $R[\f{1}{a}]$ and on $\widehat{R}\simeq \prod_{i=1}^r\widehat{\mathcal{O}}_i$ are in bijection with the following double cosets \[ \textstyle \mathrm{im}\bigl(G(R[\f{1}{a}])\rightarrow \prod_{i=1}^r G(\widehat{K}_i)\bigr)\backslash \prod_{i=1}^r G(\widehat{K}_i)/ \prod_{i=1}^rG(\widehat{\mathcal{O}}_i). \] \end{lemma} \begin{cor} \label{cor to double cosets} \begin{enumerate}[label={{\upshape(\roman)}}] \item \label{pd-tori} \Cref{decomp-gp} holds when $G$ is an $R$-group scheme of multiplicative type. \item \label{decomp-gp implies G-S on semi pruf} \Cref{decomp-gp} implies \Cref{G-S over semi-local prufer}. \end{enumerate} \end{cor} \begin{proof} By \Cref{double cosets}, \ref{pd-tori} follows from \Cref{G-S type results for mult type}~\ref{G-S for mult type gp}, since then no nontrivial $G$-torsor on $R$ trivializes on $R[\f{1}{a}]$. For \ref{decomp-gp implies G-S on semi pruf}, by the approximation \Cref{approxm semi-local Prufer ring}, we may assume that the ring $R$ in \Cref{G-S over semi-local prufer} has finite Krull dimension. The case $\dim R=0$ is trivial, so we assume $\dim R>0$ for what follows. Since $R[\f{1}{a}]$ is also a semilocal Pr\"ufer domain and $\dim R[\f{1}{a}]<\dim R$, by induction on $\dim R$, we may assume that every generically trivial $G$-torsor on $R$ trivializes on $R[\f{1}{a}]$ and, by \cite{Guo20b}, also on each $\widehat{\mathcal{O}}_i$ (since $\widehat{\mathcal{O}}_i$ is a rank-one valuation ring). Therefore, once the product formula in \Cref{decomp-gp} holds for $G$, \Cref{double cosets} would imply that every generically trivial $G$-torsor on $R$ is itself trivial. \end{proof} \begin{proof}[Proof of \Cref{decomp-gp}] We will proceed verbatim as in \cite{Guo20b}{\S4}. We choose a minimal parabolic $\widehat{\mathcal{O}}_i$-subgroup $P_i$ for each $G_i\colonequals G\times_{R}\widehat{\mathcal{O}}_i$. Denote $U_i\colonequals \mathrm{rad}^u(P_i)$. \begin{enumerate}[label={{\upshape(\roman)}}] \item For the maximal split torus $T_i\subset P_i$, we have $\prod_{i=1}^rT_i(\widehat{K}_i)\subset \mathrm{im}(G(R[\f{1}{a}])\rightarrow \prod_{i=1}^rG(\widehat{K}_i))\cdot \prod_{i=1}^rG(\widehat{\mathcal{O}}_i)$. By \cite{SGA3IIInew}{Exposé~XXVI, Corollaire~6.11}, there is a maximal torus $\widetilde{T}_i$ of $G_i$ containing $T_i$. In particular, $\widetilde{T}_{i,\widehat{K}_i}$ is a maximal torus of $G_{\widehat{K}_i}$. Then we apply \Cref{lift-r-torus} to all $\widetilde{T}_i$: there are a $g=(g_i)_i\in \overline{\mathrm{im}(G(\ria))}\cap \prod_{i=1}^rG(\widehat{\mathcal{O}}_i)$ and a maximal torus $T_0\subset G$ such that $T_{0,\widehat{K}_i}=g_i\widetilde{T}_{i,\widehat{K}_i}g_i^{-1}$ for every $i$, which combined with \Cref{cor to double cosets}~\ref{pd-tori} for $T_0$ yields \[ \textstyle \prod_{i=1}^r\widetilde{T}(\widehat{K}_i)=g^{-1}\left(\prod_{i=1}^rT_0(\widehat{K}_i)\right)g\subset g^{-1}\left(\overline{\mathrm{im}(G(\ria))}\cdot \prod_{i=1}^rG(\widehat{\mathcal{O}}_i)\right)g. \] Since $g\in \overline{\mathrm{im}(G(\ria))}\cap \prod_{i=1}^rG(\widehat{\mathcal{O}}_i)$, the inclusion displayed above implies that $\prod_{i=1}^r\widetilde{T}_i(\widehat{K}_i)\subset \overline{\mathrm{im}(G(\ria))}\cdot \prod_{i=1}^rG(\widehat{\mathcal{O}}_i)$. Therefore, by \Cref{replace im by it closure}, we obtain the following inclusion \[ \textstyle \prod_{i=1}^rT_i(\widehat{K}_i)\subset \prod_{i=1}^r\widetilde{T}_i(\widehat{K}_i)\subset \mathrm{im}(G(\ria))\cdot \prod_{i=1}^rG(\widehat{\mathcal{O}}_i). \] \item We prove $\prod_{i=1}^rU_i(\widehat{K}_i)\subset \overline{\mathrm{im}(G(\ria))}$. Consider the $T_i$-action on $G_i$ defined by \[ \textstyle T_i\times G_i\rightarrow G_i,\quad (t,g)\mapsto tgt^{-1}. \] Recall the open normal subgroup $N\subset \prod_{i=1}^rG(\widehat{K}_i)$ constructed in \Cref{open-normal}, then each $N\cap U_i(\widehat{K}_i)$ is open in $U_i(\widehat{K}_i)$. The dynamic argument in \cite{Guo20b} shows that $U_i(\widehat{K}_i)=N\cap U_i(\widehat{K}_i)$, hence $U_i(\widehat{K}_i)\subset N$ for each $i$. Therefore, we have $\prod_{i=1}^rU_i(\widehat{K}_i)\subset \overline{\mathrm{im}(G(\ria))}$. \item We prove $\prod_{i=1}^rP_i(\widehat{K}_i)\subset \mathrm{im}(G(\ria)\rightarrow \prod_{i=1}^rG(\widehat{K}_i))\cdot \prod_{i=1}^rG(\widehat{\mathcal{O}}_i)$. The quotient $H_i\colonequals L_i/T_i$ is anisotropic, therefore we have $H_i(\widehat{K}_i)=H_i(\widehat{\mathcal{O}}_i)$ for every $i$. Consider the commutative diagram \[ \begin{tikzcd} 0 \arrow{r} & T_i(\widehat{\mathcal{O}}_i) \arrow{d} \arrow{r} & L_i(\widehat{\mathcal{O}}_i) \arrow{d} \arrow{r} & H_i(\widehat{\mathcal{O}}_i) \ar[equal]{d} \arrow{r} & H^1(\widehat{\mathcal{O}}_i, T_i)=0 \arrow{d} \\ 0 \arrow{r} & T_i(\widehat{K}_i) \arrow{r} & L_i(\widehat{K}_i) \arrow{r} & H_i(\widehat{K}_i) \arrow{r} & H^1(\widehat{K}_i,T_i)=0 \end{tikzcd} \] with exact rows. By diagram chase, we have $L_i(\widehat{K}_i)=T_i(\widehat{K}_i)\cdot L_i(\widehat{\mathcal{O}}_i)$ for every $i$. Subsequently, the combination of (i) and (ii) yields the inclusion \[ \textstyle \prod_{i=1}^rP_i(\widehat{K}_i)\subset \mathrm{im}(G(\ria)\rightarrow \prod_{i=1}^rG(\widehat{K}_i))\cdot \prod_{i=1}^rG(\widehat{\mathcal{O}}_i). \] \item Recall \cite{SGA3IIInew}{Exposé~XXVI, Théorème~4.3.2 and Corollaire~5.2} that for each $P_i$, there is a parabolic subgroup $Q_i$ of $G_i$ such that $P_i\cap Q_i=L_i$ fitting into the following surjection \[ \mathrm{rad}^u(P_i)(\widehat{K}_i)\cdot \mathrm{rad}^u(Q_i)(\widehat{K}_i)\twoheadrightarrow G(\widehat{K}_i)/P_i(\widehat{K}_i). \] This surjection, combined with the result of (ii) gives an inclusion \[ \textstyle \prod_{i=1}^rG(\widehat{K}_i)\subset \overline{\mathrm{im}(G(\ria)\rightarrow \prod_{i=1}^rG(\widehat{K}_i))}\cdot \prod_{i=1}^rP_i(\widehat{K}_i). \] Combined with (iii) and \Cref{replace im by it closure}, this yields the following desired product formula \[ \textstyle \prod_{i=1}^r G(\widehat{K}_i)=\mathrm{im}\p{G(R[\f{1}{a}])\rightarrow \prod_{i=1}^r G(\widehat{K}_i)}\cdot \prod_{i=1}^rG(\widehat{\mathcal{O}}_i).\qedhere \] \end{enumerate} \end{proof} \end{appendix} \begin{bibdiv} \begin{biblist} \bibselect{bibliography} \end{biblist} \end{bibdiv} \end{document}	arXiv
Averages Formulas Concept of Average: Assume 5 friends went to a movie. If the total money with the friends is equal to 1000 then we say that average money with each person is Rs.200/- But it is not necessary that each person has Rs.200. Some may have more money than the others, but the total money is equal to Rs.1000. Let us say Person E is rich and he bought Rs.500 to the movie. So the money with the remaining friends is equal to Rs.500. If E has not come to the movie, the average money with the other friends comes down to $\dfrac{{500}}{4}$ = Rs.125. If E has bought Rs.125 to the movie the average stands at Rs.125. But he has Rs.375 more than the average required. And this extra amount is distributed among all the friends equally, so that each person gets $\dfrac{{375}}{5} = 75$ extra. That is why final average = 125 + 75 = Rs.200. Formula 1: Average or mean: The Mean (Average) of a group of numbers is the sum of the numbers divided by the number of numbers: Average or Mean = Sum of the observations / Number of observations If the average of 'm ' quantities is 'x ' and the average age of ' n ' other quantities is 'y ' then the average of all of them put together is = $\displaystyle\frac{{mx + ny}}{{m + n}}$ If the average age of 'm ' quantities is 'x ' and the average age of 'n ' quantities out of them (m quantities) is ' y ' then the average of the rest of the quantities is = .$\displaystyle\frac{{mx - ny}}{{m - n}}$ If the average of ' n ' numbers is ' x ' and if ' k ' is added to or subtracted from each given number the average of ' n ' numbers becomes (x+k) or (x-k) respectively. In the other words average value will be increased or decreased by ' k '. If the average of ' n ' numbers is ' x ' and if each given number is multiplied to or divided by ' k ' then the average of n numbers becomes kx or $\displaystyle\frac{x}{k}$ respectively. If a person travels a distance at a speed of x km/hr and the same distance at a speed of y km/hr then the average speed during the whole journey is given by $\displaystyle\frac{{2xy}}{{x + y}}$ km/hr. If a person covers A km at x km/hr and B km at y km/hr and C km at z km/hr, then the average speed in covering the whole distance is $\displaystyle\frac{{A + B + C}}{{\displaystyle\frac{A}{x} + \frac{B}{y} + \frac{C}{z}}}$km/hr. The average of n (where n is an odd number) consecutive numbers is always the middle number E.g. 1, 3, 5, 7, 9. The Average = Middle number = 5 The average ' n ' (where n is even number) consectuive numbers is the average of the two middle numbers. E.g. Average of (2, 4, 6, 8, 10, 12) = $\displaystyle\frac{{6 + 8}}{2}$ = 7 Averages: Formulas Averages: Exercise	CommonCrawl
\begin{definition}[Definition:Underlying Module of Algebra] Let $R$ be a commutative ring. Let $(A, *)$ be an algebra over $R$. Its '''underlying module''' is the $R$-module $A$. \end{definition}	ProofWiki
Alan saved 500 dollars in a bank account that compounds 3 percent annually. Assuming there are no other transactions, after 10 years, how much is in Alan's bank account? (Give your answer to the nearest dollar.) After ten years, at a three percent annual interest rate, the bank account will have grown to $500 \cdot 1.03^{10} = \boxed{672}$, to the nearest dollar.	Math Dataset
Is the electron actually a standing wave? I studied in my physics class that de Broglie proposed that electrons are actually standing waves and that is the reason why their energy levels are quantised. But I studied that the wave function of an electron is what we call the atomic orbital and wave functions of electrons come in various shapes depending on the energy levels (i.e. the eigenfunctions for the corresponding eigenvalues). For instance, some wave functions are spherical in shape (s-orbitals), some are dumbbell-shaped (p-orbitals), etc. But if the wave functions of the electrons are of these shapes then how can the electron be a standing wave? For e.g., if the wave function of an electron is spherically symmetric (s-orbital) that means that there is 99% probability of finding the electron in that spherical region. But if the electron can be anywhere in that 3-dimensional space then how can it behave like a standing wave as proposed by de Broglie? This is because if the electron were a standing wave it would be a standing wave in its orbit and it will be a 2-dimensional 'thing'. But on the other hand we are also saying that the wave function of the electron is spherically symmetric and thus can be anywhere in the 3-dimensional space. How can an electron be both a standing wave and have its wave function spherically symmetric? Is the electron even a standing wave? I am so confused. Can someone please provide the explanation? electrons atomic-physics wave-particle-duality Pranita Baruah 1Pranita Baruah 1 $\begingroup$ Beware that there are already related questions & answers. If you won't get a response dig more in this site. Possibile starting point physics.stackexchange.com/q/638189 $\endgroup$ – Alchimista $\begingroup$ For instance, some wave functions are spherical in shape (s-orbitals), some are dumbbell-shaped (p-orbitals), etc. But if the wave functions of the electrons are of these shapes then how can the electron be a standing wave? Standing waves can take all kind of shapes. See e.g. the solutions of the Classic wave equation $\psi_t=c^2\nabla^2 \psi$ (plus boundary conditions). $\endgroup$ – Gert $\begingroup$ Related/possible duplicates: physics.stackexchange.com/q/137207/50583, physics.stackexchange.com/q/196002/50583 and their linked questions $\endgroup$ – ACuriousMind ♦ $\begingroup$ My understanding is that a "standing wave" doesn't have to be 2D. The functions involved describing the bounded electron are spherical harmonics, which are 3D standing waves. The 3D model is a more accurate model than the 2D model, because it handles all three dimensions and we live in a 3D world. The 2D model (developed by Bohr, de Broglie, etc) is just a crude approximation that was used when people didn't have a full understanding of quantum mechanics. $\endgroup$ – Maximal Ideal The Bohr model is a primitive first-exploration of quantization model, . The de Broglie standing wave interpretation of the Bohr model is not the wave function calculated with the Schrodinger equation and the Coulomb potential. The wave in the wavefunction is a probability wave. A probability means that many measurements should be taken of the electron to define its position in (x,y,z) at time t, i.e.to define the orbitals, here for hydrogen. The electron itself is a point particle according to the standard model, not a wave. Depending on the boundary conditions a wave function can be calculated, as happens with hydrogen. If you look at the hydrogen orbitals (calculated locations from the wavefunctions where electrons can be bound) you will see that they are not all spherically symmetric. anna vanna v $\begingroup$ What I've never quite understood is that if you take the particle in a $1D$ box with zero potential (e.g.) the particle forms standing waves. The unit of measurement of the wave function $\psi$ is $\mathrm{m^{-1/2}}$ but the UoM of a wave would be $\mathrm{m}$? $\endgroup$ $\begingroup$ @Gert The wavefunction in any quantum mechanical solution is $Ψ$, no units, and $Ψ^Ψ$ is the probability for the reaction to happen, which is a number from zero to 1, so I do not understand where you find the units you are talking about. Are you mixing the de Broglie wave , a handwaving wave, with the solution of the QM equation, the wavefunction? $\endgroup$ – anna v $\begingroup$ Wave function: $\psi_n(x) = \sqrt{\dfrac{2}{L}}\sin{\dfrac{n\pi}{L}}x$ from chem.libretexts.org/Bookshelves/… with $L$ in $\mathrm{m}$ $\endgroup$ $\begingroup$ @annav Your comment about units is incorrect. For a one-dimensional wave function $\psi(x)$, the normalized total probability must be $1 = \int_{-\infty}^{+\infty} \psi^\psi\,\mathrm dx$. This integral is dimensionally inconsistent unless $\psi$ has dimension $(\text{length})^{-1/2}$. A probability density over an interval that's not a one-dimensional length will have some different unit, but in general they're not dimensionless. $\endgroup$ – rob ♦ $\begingroup$ @rob thanks, one tends to see the wavefunction without any units, as in the hydrogen link I give, But I can see that one has to define the "length" on which the measurement of the probability is taken $\endgroup$ We often think of a standing wave as a one-dimensional function along a line - for example, a vibrating string or a sound wave in an organ pipe - or around a circle. But there are also two-dimensional standing wave functions that live on the surface of a sphere - they are called spherical harmonics. The shapes shown for atomic orbitals for a hydrogen atom, for example (which are actually depicting the spatial distribution of the amplitude of an electron's wave function) are based on these spherical harmonic functions. gandalf61gandalf61 $\begingroup$ The 3D electron standing waves are not restricted to the surface of a sphere, but their resonant states might still be referred to as spherical harmonics. $\endgroup$ – R.W. Bird $\begingroup$ I think this is a simple and clear answer but perhaps you could stress more that the standing wave of concern is the orbital intended as solution of the Schrödinger eq and not the orbital intended as the volume in which the probability of finding e is somewhat close to 1. It seems to me a point which wasn't clear to OP when s/he mixes the two saying about "99%.....".Anyway plus 1. $\endgroup$ An electron is a (3D) standing wave only when is is bounded (as by the electric potential well of a nucleus). There is no rule that says a 3D standing wave must be spherical. R.W. BirdR.W. Bird 10.4k22 gold badges66 silver badges1818 bronze badges You are confused and I understand, because there is this wiki article that definitely says yes: The electrons do not orbit the nucleus in the manner of a planet orbiting the sun, but instead exist as standing waves. https://en.wikipedia.org/wiki/Atomic_orbital#Electron_properties Though, as you can see from other answers, the electron cannot always be represented as a standing wave because: Quantum objects are not waves. Quantum obejcts are not classical point-like particles. They are quantum objects, which may show wave-like and particle-like properties. You may represent a quantum state by its "probability wave" or wavefunction, whose square gives the probability density to find the object "as a particle" at certain locations. It is not a wave in the classical sense that anything physical would be oscillating here, and the Schrödinger equation does not always look like a wave equation. Electrons may be in more than one orbital at once, due to the general possibility of superposition of quantum states. But since the orbital are the solutions to the time-independent Schrödinger equation, being in one - and only one - orbital is the only stable state for an electron, while all other states will be changed by time evolution. The orbitals don't "interfere" because, well, they aren't actual waves. Electron as a standing wave and its stability an electron is a quantum object, it shows in certain experiments wave like properties, and particle like properties in others, but it is neither, in reality it is best described by QM. the electron is not a wave in the classical sense, nothing is oscillating, and the Schrodinger equation cannot always be expressed as a wave equation. the electron might be in superposition (more then one orbital), but being in only one is a stable state and all other states will be changed by time evolution, and the orbitals do not interfere, because they aren't waves. So the answer to your question is that although for simplicity it might be advantageous to depict the electron in certain cases as standing waves, in reality the universe is fundamentally quantum mechanical and the atoms (and electrons) are modeled using QM resources (including the Schrodinger equation) which cannot always be expressed as wave equations. Árpád SzendreiÁrpád Szendrei Not the answer you're looking for? Browse other questions tagged electrons atomic-physics wave-particle-duality or ask your own question. Dual behaviour of matter Trouble understanding the Bohr model of the atom Why do we rule out orbits with non-constructive interference for the atom? Is it that electron of an atom can be found anywhere in the space? Can I steal your electron? Atomic orbitals How do electrons remain as standing waves? de Broglie's Hypothesis and the Stability of Electron Orbits How are line spectra explained after rejecting/improving Bohr's theory? A few questions about momentum and energy of electrons in the quantum mechanical model of the electron	CommonCrawl
\begin{document} \maketitle \begin{abstract} Suppose that $f$ is a homomorphism from the mapping class group $\mathcal{M}(N_{g,n})$ of a nonorientable surface of genus $g$ with $n$ boundary components, to $\mathrm{GL}(m,\mathbb{C})$. We prove that if $g\ge 5$, $n\le 1$ and $m\le g-2$, then $f$ factors through the abelianization of $\mathcal{M}(N_{g,n})$, which is $\mathbb{Z}_2\times\mathbb{Z}_2$ for $g\in\{5,6\}$ and $\mathbb{Z}_2$ for $g\ge 7$. If $g\ge 7$, $n=0$ and $m=g-1$, then either $f$ has finite image (of order at most two if $g\ne 8$), or it is conjugate to one of four ``homological representations''. As an application we prove that for $g\ge 5$ and $h<g$, every homomorphism $\mathcal{M}(N_{g,0})\to\mathcal{M}(N_{h,0})$ factors through the abelianization of $\mathcal{M}(N_{g,0})$. \end{abstract} \section{Introduction} For a compact surface $F$, its {\it mapping class group} $\mathcal{M}(F)$ is the group of isotopy classes of all, orientation preserving if $F$ is orientable, homeomorphisms $F\to F$ equal to the identity on the boundary of $F$. A compact surface of genus $g$ with $n$ boundary components will be denoted by $S_{g,n}$ if it is orientable, or by $N_{g,n}$ if it is nonorientable. If $n=0$ then we drop it in the notation and write simply $S_g$ or $N_g$. The first integral homology group of $F$ will be denoted by $H_1(F)$. After fixing a basis of $H_1(S_g)$, the action of $\mathcal{M}(S_g)$ on $H_1(S_g)$ gives rise to a homomorphism $\mathcal{M}(S_g)\to\mathrm{Sp}(2g,\mathbb{Z})$, which is well known to be surjective, and whose kernel is known as the Torelli group. Gluing a disc along each boundary component of $S_{g,n}$ induces an epimorphism $\mathcal{M}(S_{g,n})\to\mathcal{M}(S_g)$, and by composing it with $\mathcal{M}(S_g)\to\mathrm{Sp}(2g,\mathbb{Z})$, and then with the inclusion $\mathrm{Sp}(2g,\mathbb{Z})\hookrightarrow\mathrm{GL}(2g,\mathbb{C})$ we obtain the map $\Phi\colon\mathcal{M}(S_{g,n})\to\mathrm{GL}(2g,\mathbb{C})$. Recently, the following two results were proved by J. Franks, M. Handel and M. Korkmaz. \begin{theorem}[\cite{FH,KorkRep}]\label{FHK} Let $g\ge 2$, $m\le 2g-1$ and let $f\colon\mathcal{M}(S_{g,n})\to\mathrm{GL}(m,\mathbb{C})$ be a homomorphism. Then $f$ is trivial if $g\ge 3$, and $\mathrm{Im}(f)$ is a quotient of $\mathbb{Z}_{10}$ if $g=2$. \end{theorem} We say that two homomorphism $f_1$, $f_2$ from a group $G$ to a group $H$ are {\it conjugate} if there exits $h\in H$ such that $f_2(x)=hf_1(x)h^{-1}$ for $x\in G$. \begin{theorem}[\cite{KorkSymp}]\label{KorU} For $g\ge 3$, every nontrivial homomorphism $f\colon\mathcal{M}(S_{g,n})\to\mathrm{GL}(2g,\mathbb{C})$ is conjugate to the map $\Phi$. \end{theorem} In this paper we prove analogous results for $\mathcal{M}(N_g)$. Fix $g\ge 3$. Let $R_g$ denote the quotient of $H_1(N_g)$ by its torsion. Hence, $R_g$ is a free $\mathbb{Z}$-module of rank $g-1$. There is covering $P\colon S_{g-1}\to N_g$ of degree two. By a theorem of Birman and Chillingworth \cite{BC}, $\mathcal{M}(N_g)$ is isomorphic to the subgroup of $\mathcal{M}(S_{g-1})$ consisting of the isotopy classes of orientation preserving lifts of homeomorphisms of $N_g$, which gives an action of $N_g$ on $H_1(S_{g-1})$. Let $K_g\subset H_1(S_{g-1})$ be the kernel of the composition of the induced map $P_\ast\colon H_1(S_{g-1})\to H_1(N_g)$ with the canonical projection $H_1(N_g)\to R_g$. Then $K_g$ is $\mathcal{M}(N_g)$-invariant subgroup of rank $g-1$ and we have two homomorphisms \[\Psi_1\colon\mathcal{M}(N_g)\to\mathrm{GL}(K_g)\quad\textrm{and}\quad \Psi_2\colon\mathcal{M}(N_g)\to\mathrm{GL}(H_1(S_{g-1})/K_g),\] which after fixing bases will be treated as representations of $\mathcal{M}(N_g)$ in $\mathrm{GL}(g-1,\mathbb{C})$. We will see that these representations are not conjugate, although $\ker\Psi_1=\ker\Psi_2$. Our first result is the following. \begin{theorem}\label{MNtoGLfact} Suppose that $n\le 1$, $g\ge 5$, $m\le g-2$ and $f\colon\mathcal{M}(N_{g,n})\to\mathrm{GL}(m,\mathbb{C})$ is a nontrivial homomorphism. Then $\mathrm{Im}(f)$ is ether $\mathbb{Z}_2$ or $\mathbb{Z}_2\times\mathbb{Z}_2$, the latter case being possible only for $g=5$ or $6$. \end{theorem} Theorem \ref{MNtoGLfact} was proved in \cite{KorkRep}, in a more general setting of punctured surfaces, under additional assumption that $m\le g-3$ if $g$ is even. Therefore, the only novelty of our result is that it also covers the case $m=g-2$ for even $g$. As an application of Theorem \ref{MNtoGLfact} we prove the following result, which solves Problem 3.3 in \cite{KorkProb}. \begin{theorem}\label{MNgtoMNh} Suppose that $g\ge 5$, $h<g$ and $f\colon\mathcal{M}(N_g)\to\mathcal{M}(N_h)$ is a nontrivial homomorphism. Then $\mathrm{Im}(f)$ is as in Theorem \ref{MNtoGLfact}. \end{theorem} Analogous theorem for mapping class groups of orientable surfaces was proved in \cite{HK}, see also \cite{AS}. We will prove that both Theorem \ref{MNtoGLfact} and Theorem \ref{MNgtoMNh} fail for $g=4$, by showing that there is a homomorphism from $\mathcal{M}(N_4)$ to $\mathcal{M}(N_3)\cong\mathrm{GL}(2,\mathbb{Z})$, whose image is isomorphic to the infinite dihedral group. Suppose that $g\ge 7$. Then the abelianization of $\mathcal{M}(N_g)$ is $\mathbb{Z}_2$ and we denote by $\mathrm{ab}\colon\mathcal{M}(N_g)\to\mathbb{Z}_2$ the canonical projection. For $i=1,2$ we set $\Psi'_i=(-1)^\mathrm{ab}\Psi_i$. Our next result is the following. \begin{theorem}\label{MNtoGLg-1} Suppose that $g\ge 7$, $g\ne 8$ and $f\colon\mathcal{M}(N_g)\to\mathrm{GL}(g-1,\mathbb{C})$ is a nontrivial homomorphism. Then either $\mathrm{Im}(f)\cong\mathbb{Z}_2$, or $f$ is conjugate to one of $\Psi_1$, $\Psi'_1$, $\Psi_2$, $\Psi'_2$. \end{theorem} For $g=8$ other representations of $\mathcal{M}(N_8)$ in $\mathrm{GL}(7,\mathbb{C})$ occur, related to the fact that there is an epimorphism $\epsilon\colon\mathcal{M}(N_8)\to\mathrm{Sp}(6,\mathbb{Z}_2)$ and the last group admits irreducible representations in $\mathrm{GL}(7,\mathbb{C})$ (see \cite{Atlas}). We prove the following result. \begin{theorem}\label{MN8toGL7} Suppose that $f\colon\mathcal{M}(N_8)\to\mathrm{GL}(7,\mathbb{C})$ is a homomorphism. Then one of the following holds. \begin{itemize} \item[(1)] $\mathrm{Im}(f)\cong\mathbb{Z}_2$. \item[(2)] $f$ or $(-1)^\mathrm{ab} f$ factors through $\epsilon\colon\mathcal{M}(N_8)\to\mathrm{Sp}(6,\mathbb{Z}_2)$. \item[(3)] $f$ is conjugate to one of $\Psi_1$, $\Psi'_1$, $\Psi_2$, $\Psi'_2$. \end{itemize} \end{theorem} To prove our theorems we use the ideas and results from \cite{FH,KorkRep,KorkSymp} with necessary modifications. While the case of odd genus is relatively easy, the case of even genus requires much more effort. This phenomenon is typical for the mapping group of a nonorientable surface. Throughout this paper we will often have to solve an equation of the form $L=R$, where $L$ and $R$ are products of matrices from $\mathrm{GL}(m,\mathbb{C})$ with some unknown coefficients. Although the dimension $m$ is variable, the calculations of $L$ and $R$ always reduce to multiplication of blocks of size at most $7\times 7$. With some patience, such calculations could be done by hand, but it is definitely easier to use a computer. We used GAP, but of course, any program that performs symbolic operations on matrices, could be used as well. \section{Notation and algebraic preliminaries}\label{nota} Suppose that $m\ge 2$ is fixed. We denote by $I_m$ the identity matrix of dimension $m$. We will sometimes write simply $I$, if $m$ is clear from the context. We denote by $E_{ij}$ the elementary matrix with $1$ on the position $(i,j)$ and $0$ elsewhere. Suppose that $M_1,\dots,M_k$ are nonsingular square matrices of dimensions $m_1,\dots,m_k$, where $m_1+\cdots+m_k=m$. Then we denote by $\mathrm{diag}\left(M_1,\dots,M_k\right)$ the $m\times m$ matrix with $M_1,\dots,M_k$ on the main diagonal and zeros elsewhere. Set \[ V=\begin{pmatrix}1&1\\0&1\end{pmatrix},\quad \widehat{V}=\begin{pmatrix}1&0\\-1&1\end{pmatrix},\quad W=\begin{pmatrix}1&1&0&-1\\0&1&0&0\\0&-1&1&1\\0&0&0&1\end{pmatrix}\] For $2\le 2i\le m$ we define \[A_i=\mathrm{diag}\left(I_{2i-2},V,I_{m-2i}\right),\quad B_i=\mathrm{diag}\left(I_{2i-2},\widehat{V},I_{m-2i}\right),\] and for $2\le 2j\le m-2$, \[C_j=\mathrm{diag}\left(I_{2j-2},W,I_{m-2-2j}\right).\] The proof of the following lemma is straightforward and we leave it as an exercise (c.f. \cite[Lemma 2.2]{KorkSymp}). \begin{lemma}\label{diag} Suppose that $1\le k\le l\le m/2$ and $M\in\mathrm{GL}(m,\mathbb{C})$ satisfies $A_iM=MA_i$, $B_iM=MB_i$ and $C_jM=MC_j$ for all $i,j$ such that $k\le i\le l$, $k\le j\le l-1$. Then $M$ has the form \begin{equation}\label{stars} \begin{pmatrix}\ast&0&\ast\\ 0&\lambda I_{2(l-k+1)}&0\\ \ast&0&\ast\end{pmatrix}, \end{equation} for some $\lambda\in\mathbb{C}^\ast$, where the top-left $\lambda$ of the block $\lambda I_{2(l-k+1)}$ is at the position $(2k-1,2k-1)$. \end{lemma} Suppose that $L\in\mathrm{GL}(m,\mathbb{C})$ and $\lambda$ is an eigenvalue of $L$. Then we denote by $\#\lambda$ the multiplicity of $\lambda$. For $k\ge 1$ we denote by $E^k(L,\lambda)$ the space $\ker(E-\lambda I)^k$. Thus $E^1(L,\lambda)$ is the eigenspace of $L$ with respect to $\lambda$, and it will be also denoted by $E(L,\lambda)$. Note that if $L'\in\mathrm{GL}(m,\mathbb{C})$ commutes with $L$, then the spaces $E^k(L,\lambda)$ are $L'$-invariant for $k\ge 1$. For $k\ge 2$ we denote by $\mathfrak{S}_k$ the full symmetric group of the set $\{1,\dots,k\}$. It is generated by the transpositions $\sigma_i=(i,i+1)$ for $1\le i\le k-1$. We will need the following result from the representation theory of the symmetric group, see for example \cite[Exercise 4.14]{FulHar}. \begin{lemma}\label{repsym} For $k\ge 5$, $\mathfrak{S}_k$ has no irreducible representation (over $\mathbb{C}$) of dimension $1<m<k-1$. If $k\ge 7$, then $\mathfrak{S}_k$ has two irreducible representations of dimension $k-1$: the standard one and the tensor product of the standard and sign representations. \end{lemma} \section{Mapping class group of a nonorientable surface} Let $n\in\{0,1\}$ and $g\ge 2$. Let us represent $N_{g,n}$ as a sphere (if $n=0$) or a disc (if $n=1$) with $g$ crosscaps. This means that interiors of $g$ small pairwise disjoint discs should be removed from the sphere/disc, and then antipodal points in each of the resulting boundary components should be identified. Let us arrange the crosscaps as shown on Figure \ref{xiI} and number them from $1$ to $g$. For each nonempty subset $I\subseteq\{1,\dots,g\}$ let $\xi_I$ be the simple closed curve shown on Figure \ref{xiI}. Note that $\xi_I$ is two-sided if and only if $I$ has even number of elements. In such case $t_{\xi_I}$ will be the Dehn twist about $\gamma_I$ in the direction indicated by arrows on Figure \ref{xiI}. \begin{figure} \caption{ The surface $N_{g,n}$ and the curve $\xi_I$ for $I=\{i_1,i_2,\dots,i_k\}$.} \label{xiI} \end{figure} We will write $\xi_i$ instead of $\xi_{\{i\}}$. The following curves will play a special role and so we give them different names. \begin{itemize} \item $\delta_i=\xi_{\{i,i+1\}}$ for $1\le i\le g-1$, \item $\varepsilon_j=\xi_{\{1,2,\dots,2j\}}$ for $2\le 2j\le g$. \end{itemize} Note that $\varepsilon_1=\delta_1$. For $1\le i\le g-1$ we define the {\it crosscap transposition} $u_i$ to be the isotopy class of the homeomorphism interchanging the $i$'th and the $(i+1)$'st crosscaps as shown on Figure \ref{U}, and equal to the identity outside a disc containing these crosscaps. \begin{figure} \caption{The crosscap transposition $u_i$.} \label{U} \end{figure} The groups $\mathcal{M}(N_{1,n})$ are trivial for $n\le 1$ by \cite[Theorem 3.4]{E}, we have $\mathcal{M}(N_2)\cong\mathbb{Z}_2\times\mathbb{Z}_2$ by \cite{Lick}, and it follows from \cite{BC} that $\mathcal{M}(N_3)\cong\mathrm{GL}(2,\mathbb{Z})$. For $g\ge 3$, a finite generating set for $\mathcal{M}(N_{g,n})$ was given in \cite{Chill} for $n=0$ and \cite{Stu_bdr} for $n>0$. For $n\le 1$ this set can be reduced to the one given in the following theorem, which can be deduced form the main result of \cite{PSz}. \begin{theorem}\label{gener} For $g\ge 4$ and $n\in\{0,1\}$, $\mathcal{M}(N_{g,n})$ is generated by $u_{g-1}$, $t_{\varepsilon_2}$ and $t_{\delta_i}$ for $1\le i\le g-1$. \end{theorem} If $n>1$, then we consider $N_{g,n}$ as the result of gluing $S_{0,n+1}$ to $N_{g,1}$ along the boundary component. We will need the following relations, satisfied in $\mathcal{M}(N_{g,n})$. Those between Dehn twists are the well know disjointness and braid relations. \begin{itemize} \item[(R1)] $t_{\delta_i}t_{\delta_j}=t_{\delta_j}t_{\delta_i}\quad$ for $\|i-j\|>1$, \item[(R2)] $t_{\varepsilon_i}t_{\varepsilon_j}=t_{\varepsilon_j}t_{\varepsilon_i}\quad$ for all $i,j$, \item[(R3)] $t_{\varepsilon_i}t_{\delta_j}=t_{\delta_j}t_{\varepsilon_i}\quad$ for $j\ne 2i$, \item[(R4)] $t_{\delta_i}t_{\delta_{i+1}}t_{\delta_i}=t_{\delta_{i+1}}t_{\delta_i}t_{\delta_{i+1}}\quad$ for $1\le i\le g-2$, \item[(R5)] $t_{\varepsilon_i}t_{\delta_{2i}}t_{\varepsilon_i}=t_{\delta_{2i}}t_{\varepsilon_i}t_{\delta_{2i}}$ for $2i<g$; \end{itemize} The relations involving crosscap transpositions are not so well known and we refer the reader to \cite{PSz} and \cite{SzepB} for their proofs. \begin{itemize} \item[(R6)] $t_{\delta_i}u_j=u_jt_{\delta_i}\quad$ for $\|i-j\|>1$, \item[(R7)] $u_iu_j=u_ju_i\quad$ for $\|i-j\|>1$, \item[(R8)] $t_{\varepsilon_i}u_j=u_jt_{\varepsilon_i}\quad$ for $j>2i$, \item[(R9)] $u_iu_{i+1}u_i=u_{i+1}u_iu_{i+1}\quad$ for $1\le i\le g-2$, \item[(R10)] $t_{\delta_i}u_{i+1}u_i=u_{i+1}u_it_{\delta_{i+1}}\quad$ for $1\le i\le g-2$, \item[(R11)] $u_{i+1}t_{\delta_i}t_{\delta_{i+1}}u_i=t_{\delta_i}t_{\delta_{i+1}}\quad$ for $1\le i\le g-2$; \item[(R12)] $t_{\delta_i}u_it_{\delta_i}=u_i\quad$ for $1\le i\le g-1$. \end{itemize} If follows from (R4) that all $t_{\delta_i}$ are conjugate for $1\le i\le g-1$, by (R5) $t_{\varepsilon_j}$ is conjugate to $t_{\delta_{2j}}$ for $2j<g$, and by (R12) $t_{\delta_i}$ is conjugate to $t_{\delta_i}^{-1}$. Similarly, by (R9) all $u_i$ are conjugate for $1\le i\le g-1$, and by (R11) $u_i$ is conjugate to $u_i^{-1}$. For a group $G$ we denote the abelianization $G/[G,G]$ by $G^\mathrm{ab}$. The following theorem is proved in \cite{KorkH1} for $n=0$ and generalised to $n>0$ in \cite{Stu_bdr}. \begin{theorem}\label{abNg} For $n\le 1$ and $g\ge 3$, $\mathcal{M}(N_{g,n})^\mathrm{ab}$ has the following presentation as a $\mathbb{Z}$-module. \begin{align} &\lr{[t_{\delta_1}], [t_{\varepsilon_2}], [u_1]\,\|\,2[t_{\delta_1}]= 2[t_{\varepsilon_2}]=2[u_1]=0}\quad\textrm{if }g=4,\\ &\lr{[t_{\delta_1}], [u_1]\,\|\,2[t_{\delta_1}]=2[u_1]=0}\quad\textrm{if }g\in\{3,5,6\},\\ &\lr{[u_1]\,\|\,2[u_1]=0}\quad\textrm{if }g\ge 7. \end{align} In particular, for $g\ge 7$ we have $[t_{\delta_1}]=0$. \end{theorem} \begin{lemma}\label{com_norm} For $g\ge 5$ and $n\le 1$ let $\alpha$, $\beta$ be two-sided curves on $N_{g,n}$, intersecting transversally in one point. If $f\colon\mathcal{M}(N_{g,n})\to G$ is a homomorphism, such that $f(t_\alpha)$ commutes with $f(t_\beta)$, then $\mathrm{Im}(f)$ is abelian. \end{lemma} \begin{proof} Let $N=N_{g,n}$ and $\mathcal{M}=\mathcal{M}(N_{g,n})$. Fix a regular neighbourhood $A$ of $\alpha\cup\beta$. Note that $A$ is homeomorphic to $S_{1,1}$ and $N\backslash A$ is homeomorphic to $N_{g-2,1}$. It follows that for each $i\le g-2$ there is a homeomorphism $h\colon N\to N$ such that $h(\alpha)=\delta_i$ and $h(\beta)=\delta_{i+1}$. It follows that $ht_\alpha h^{-1}=t^{\varepsilon_1}_{\delta_i}$ and $ht_\beta h^{-1}=t^{\varepsilon_2}_{\delta_{i+1}}$, where $\varepsilon_j\in\{-1,1\}$ for $j=1,2$. Hence $f(t_{\delta_i})$ commutes with $f(t_{\delta_{i+1}})$ and by the braid relation (R4) $f(t_{\delta_i})=f(t_{\delta_{i+1}})$. Analogously, $f(t_{\varepsilon_2})=f(t_{\delta_4})$. By Theorem \ref{gener}, $\mathrm{Im}(f)$ is generated by $f(t_{\delta_1})$ and $f(u_{g-1})$, and since $u_{g-1}$ commutes with $t_{\delta_1}$, thus $\mathrm{Im}(f)$ is abelian. \end{proof} \begin{lemma}\label{tsq} Suppose that $g\ge 4$ and $f\colon\mathcal{M}(N_{g,n})\to G$ is a homomorphism. If $f(t_{\varepsilon_i})=f(t_{\delta_{j}})$ for some $2i+1\le j\le g-1$, then $f(t^2_{\delta_1})=1$. \end{lemma} \begin{proof} Set $x=f(t_{\varepsilon_i})=f(t_{\delta_j})$ and $y=f(u_j)$. By the relation (R8) we have $xy=yx$, and by (R12) $xyx=y$. Hence $x^2=1$ which finishes the proof, because $t_{\delta_j}$ is conjugate to $t_{\delta_1}$. \end{proof} \begin{figure} \caption{The surface $S_{g-1}$ for $g=2r+1$ (top) and $g=2r+2$ (bottom).} \label{S2r} \end{figure} Let $g=2r+s$, where $r\ge 1$, $s\in\{1,2\}$ and $S=S_{g-1}$. Consider $S$ as being embedded in $\mathbb{R}^3$ in such a way that it is invariant under the reflections about the $xy$, $xz$ and $yz$ planes, as shown on Figure \ref{S2r}. We define a homeomorphism $j\colon S\to S$ as $j(x,y,z)=(-x,-y,-z)$. The quotient space $S/j$ is a nonorientable surface of genus $g$ and the projection $p\colon S\to S/j$ is a covering map of degree $2$. Let $S'$ be the subsurface of $S$ consisting of points $(x,y,z)\in S$ with $x\le-\varepsilon$, where $\varepsilon$ is a positive constant, so small that $S'$ is homeomorphic to $S_{r,s}$. If $g$ is even, then one of the boundary components of $S'$ is isotopic to $\alpha_{r+1}$. In this paper we identify isotopic curves, and therefore we will treat $\alpha_{r+1}$ as a curve on $S'$. Note that the restriction of $p$ to $S'$ is an embedding. For odd $g$ we define $\gamma'$ to be the arc of $\gamma_r$ consisting of points with $x\le 0$. For even $g$ we define $\beta'$ to be the arc of $\beta_{r+1}$ consisting of points with $x\le 0$. Note that $p(\gamma')$ and $p(\beta')$ are one-sided simple closed curves on $S/j$. \begin{prop}\label{HomeoCov} There is a homeomorphism $\varphi\colon S_{g-1}/j\to N_{g}$ such that, for $P=\varphi\circ p$, up to isotopy \begin{itemize} \item[(1)] $P(\beta_i)=\delta_{2i}$ for $1\le i\le r$, \item[(2)] $P(\alpha_i)=\varepsilon_i$ for $2\le 2i\le g$, \item[(3)] $P(\gamma_i)=\delta_{2i+1}$ for $2\le 2i\le g-2$, \item[(4)] $P(\gamma')=\xi_g$ if $g$ is odd, \item[(5)] $P(\beta')=\xi_g$ if $g$ is even. \end{itemize} \end{prop} \begin{proof} Observe that the curves $\delta_i$ for $1\le i\le g-1$ form a chain of two-sided curves, which means that $\delta_i$ and $\delta_j$ intersect at one point if $\|i-j\|=1$, and they are disjoint otherwise. It follows that a regular neighbourhood of the union of $\delta_i$ for $1\le i\le g-1$ is homeomorphic to $S_{r,s}$. Let $\Sigma$ be such a neighbourhood, which may be taken to contain the curves $\varepsilon_i$ for $2\le 2i\le g$ (if $g$ is even, then one of the boundary components of $\Sigma$ is isotopic to $\varepsilon_{r+1}$). Note that $\varepsilon_i$, $\varepsilon_{i+1}$ and $\delta_{2i+1}$ bound a pair of pants for $2\le 2i\le g-2$. It follows that there exists a homeomorphism $\varphi\colon S_{g-1}/j\to N_{g}$ such that, for $P=\varphi\circ p$, we have $P(S')=\Sigma$ and the conditions (1, 2, 3) are satisfied. Observe that $N_g\backslash\Sigma$ is a M\"obius strip (if $g$ is odd) or an annulus (if $g$ is even), whose core (isotopic to $\xi_{\{1,\dots,g\}}$) intersects $\xi_g$ once. By looking at the intersection of $\xi_g$ with the curves $\delta_i$, $\varepsilon_j$ it is easy to see that $\varphi$ can be taken to satisfy also the condition (4) or (5). \end{proof} \begin{cor}\label{HomSN} There is a homomorphism $\iota\colon\mathcal{M}(S')\to\mathcal{M}(N_{g,n})$ such that \begin{itemize} \item $\iota(t_{\beta_i})=t_{\delta_{2i}}$ for $1\le i\le r$, \item $\iota(t_{\alpha_i})=t_{\varepsilon_i}$ for $2\le 2i\le g$, \item $\iota(t_{\gamma_i})=t_{\delta_{2i+1}}$ for $2\le 2i\le g-2$, \end{itemize} where the Dehn twists about the curves on $S'$ are right with respect to the standard orientation. \end{cor} \begin{proof} By the proof of Proposition \ref{HomeoCov}, the restriction of $P$ to $S'$ is a homeomorphism onto $\Sigma$ satisfying the conditions (1,2,3). There is an induced isomorphism $\mathcal{M}(S')\to\mathcal{M}(\Sigma)$, which may be composed with the homomorphism $\mathcal{M}(\Sigma)\to\mathcal{M}(N_{g,n})$ induced by the inclusion $\Sigma\hookrightarrow N_{g,n}$, for any $n\ge 0$, to obtain $\iota$. \end{proof} For any homeomorphism $h\colon N_g\to N_g$ there is a unique orientation preserving lift $\widetilde{h}\colon S_{g-1}\to S_{g-1}$ such that $h\circ P=P\circ\widetilde{h}$. By \cite{BC}, the mapping $h\mapsto\widetilde{h}$ induces a monomorphism $\theta\colon\mathcal{M}(N_g)\to\mathcal{M}(S_{g-1})$. The following proposition follows from \cite{BC} and \cite[Theorem 10]{SzepB}, where the lift of a crosscap transposition is determined. \begin{prop}\label{lifts} There is a monomorphism $\theta\colon\mathcal{M}(N_g)\to\mathcal{M}(S_{g-1})$ such that \[\theta(t_{\varepsilon_i})=t_{\alpha_{i}}t^{-1}_{\alpha_{g-i}},\quad \theta(t_{\delta_{2i}})=t_{\beta_{i}}t^{-1}_{\beta_{g-i}},\quad \theta(t_{\delta_{2j+1}})=t_{\gamma_{j}}t^{-1}_{\gamma_{g-1-j}},\] for $1\le i\le r$, $2\le 2j\le g-2$ and \[\theta(u_{g-1})=\begin{cases} t^{-1}_{\beta_r}t_{\beta_{r+1}}(t_{\gamma_r}t_{\beta_r}t_{\beta_{r+1}})^2t^{-1}_{\epsilon}&\textrm{if\ }g=2r+1,\\ t^{-1}_{\gamma_r}t_{\gamma_{r+1}}(t_{\beta_{r+1}}t_{\gamma_r}t_{\gamma_{r+1}})^2t^{-1}_{\phi}&\textrm{if\ }g=2r+2. \end{cases}\] \end{prop} \section{Homological representations} Fix $g\ge 3$ and let $S=S_{g-1}$, $N=N_g$ and $P\colon S\to N$ be as in the previous section. The group $H_1(S)$ is a free $\mathbb{Z}$-module of rank $2(g-1)$ and the homology classes $a_i=[\alpha_i]$, $b_i=[\beta_i]$ for $1\le i\le g-1$ form its basis, which is a symplectic basis with respect to the algebraic intersection form: \[\lr{a_i,a_j}=0,\quad \lr{b_i,b_j}=0,\quad \lr{a_i,b_j}=\delta_{ij}.\] Let $\Phi\colon\mathcal{M}(S)\to\mathrm{Sp}(H_1(S))$ be the homomorphism induced by the action of $\mathcal{M}(S)$ on $H_1(S)$. If $\gamma$ is an oriented simple closed curve on $S$, $[\gamma]\in H_1(S)$ is its homology class, and $t_\gamma$ is the right Dehn twist, then $\Phi(t_\gamma)$ is the transvection \begin{equation}\label{transv} \Phi(t_\gamma)(h)=h+\lr{[\gamma],h}[\gamma],\quad\textrm{for\ }h\in H_1(S). \end{equation} From this formula we immediately obtain that, with respect to the basis $(a_1, b_1,\dots,a_{g-1}, b_{g-1})$, we have \[\Phi(t_{\alpha_i})=A_i,\quad \Phi(t_{\beta_i})=B_i,\quad \Phi(t_{\gamma_j})=C_j,\] for $1\le i\le g-1$, $1\le j\le g-2$, where $A_i$, $B_i$ and $C_j$ are the matrices defined in Section \ref{nota}. The group $H_1(N)$ has the following presentation, as a $\mathbb{Z}$-module: \[H_1(N)=\lr{x_1,\dots,x_g\,\|\,2(x_1+\cdots+x_g)=0},\] where $x_i=[\xi_i]$. Set $k=x_1+\dots+x_g$ and $R=H_1(N)/\lr{k}$. Observe that $k$ is the unique element of order two in $H_1(N)$ and $R$ is a free $\mathbb{Z}$-module of rank $g-1$. The map $P\colon S\to N$ induces $P_\ast\colon H_1(S)\to H_1(N)$, such that, for $1\le i\le r$ \begin{align} &P_\ast(a_i)=x_1+\cdots+x_{2i}=-P_\ast(a_{g-i}),\\ &P_\ast(b_i)=x_{2i}+x_{2i+1}=P_\ast(b_{g-i}), \end{align} and if $g=2r+2$ then \[P_\ast(a_{r+1})=x_1+\cdots+x_g=k,\quad P_\ast(b_{r+1})=2x_g.\] Let $q\colon H_1(S)\to R$ be the composition of $P_\ast$ with the canonical projection $H_1(N)\to R$, and set $K=\ker q$. It is easy to verify that $K$ has rank $g-1$ and the following elements form its basis: \begin{align} &e_i=a_i+a_{g-i},\quad e_{r+i}=b_i-b_{g-i}\quad\textrm{for\ }1\le i\le r,\\ &e_{2r+1}=a_{r+1}\quad\textrm{\ for\ }g=2r+2. \end{align} We also set \begin{align} &f_i=b_i,\quad f_{r+i}=a_{g-i}\quad\textrm{for\ }1\le i\le r,\\ &f_{2r+1}=b_{r+1}\quad\textrm{\ for\ }g=2r+2. \end{align} Observe that the elements $e_i$, $f_i$ for $1\le i\le g-1$ form a symplectic basis of $H_1(S)$. It follows that $H_1(S)/K$ is a free $\mathbb{Z}$-module of rank $g-1$, which is canonically isomorphic to $R$ if $g$ is odd, or to an index-two subgroup of $R$ if $g$ is even. The group $\mathcal{M}(N)$ acts on $H_1(S)$ by the composition $\Phi\circ\theta\colon\mathcal{M}(N)\to\mathrm{Sp}(H_1(S))$. Observe that $K$ is $M(N)$-invariant and hence we have two $(g-1)$-dimensional representations \[\psi_1\colon\mathcal{M}(N)\to\mathrm{GL}(K),\quad \psi_2\colon\mathcal{M}(N)\to\mathrm{GL}(H_1(S)/K).\] \begin{lemma} $\ker\Psi_1=\ker\Psi_2$ and $\theta(\ker\Psi_1)\subset\ker\Phi$. \end{lemma} \begin{proof} Fix the basis $(e_1,\dots,e_{g-1},f_1,\dots,f_{g-1})$ of $H_1(S)$. For any $x\in\mathcal{M}(N)$ let $X$ be the matrix of $\Phi(\theta(x))$. We have $X=\begin{pmatrix}X_1&Y\\0&X_2\end{pmatrix}$, where $X_1, X_2, Y$ are $(g-1)\times(g-1)$ matrices. The matrix of the algebraic intersection form is $\Omega=\begin{pmatrix}0&I_{g-1}\\-I_{g-1}&0\end{pmatrix}$ and since $X$ is symplectic, we have $X^t\Omega X=\Omega$, which gives $X_1^tX_2=I$. Therefore $X_1=I\Leftrightarrow X_2=I$, which proves $\ker\Psi_1=\ker\Psi_2$. To prove the second part of the lemma, assume $X_1=X_2=I$. Let $j_\ast\colon\mathcal{M}(S)\to\mathcal{M}(S)$ be the map induced by the covering involution $j$. It is easy to check that the matrix of $j_\ast$ has the form $J=\begin{pmatrix}-I_{g-1}&T\\0&I_{g-1}\end{pmatrix}$ for some $T$. We have $XJ=JX$, which implies $Y=0$. \end{proof} Note that $\ker\Phi$ is the Torelli group, which is well known to be torsion free, and since $\theta$ is a monomorphism, we immediately obtain the following. \begin{cor}\label{kerTF} $\ker\Psi_1$ is torsion free. {$\Box$} \end{cor} \begin{rem} Let $H$ denote the subgroup of $\mathcal{M}(N)$ consisting of the elements inducing the identity on $H_1(N)$. It was proved in \cite{Gas} that $\theta(H)\subset\ker\Phi$. We leave it as an exercise to check that if $g$ is odd, then $H=\ker\Psi_2$, whereas if $g$ is even, then $H$ is an index-two subgroup of $\ker\Psi_2$. In the latter case, if $g=2r+2$, then we have $\ker\Psi_2=H\cup t_{\varepsilon_{r+1}}H$. \end{rem} \begin{rem} There is a nontrivial action of $\pi_1(N)$ on $\mathbb{Z}$ defined as follows: $\gamma\in\pi_1(N)$ acts by multiplication by $1$ or $-1$ according to whether $\gamma$ preserves or reverses local orientations of $N$. This action gives rise to homology groups with local coefficients $H_\ast(N,\widetilde{\mathbb{Z}})$, where $\widetilde{\mathbb{Z}}$ is $\mathbb{Z}$ with the nontrivial $\mathbb{Z}[\pi_1(N)]$-module structure. By \cite[Example 3H.3]{Hat}, we have the exact sequence \[H_2(N)\to H_1(N,\widetilde{\mathbb{Z}})\to H_1(S)\stackrel{P_\ast}{\longrightarrow}H_1(N),\] which is a part of a long exact sequence of homology groups. Since $H_2(N)=0$, we have a $\mathcal{M}(N)$-equivariant isomorphism $H_1(N,\widetilde{\mathbb{Z}})\cong\ker P_\ast$. If $g$ is odd, then $\ker P_\ast=K$, whereas if $g$ is even, then $\ker P_\ast$ is an index-two subgroup of $K$. Therefore the representations $\Psi_1$ and $\Psi_2$ may be seen as coming from the actions of $\mathcal{M}(N)$ on $H_1(N,\widetilde{\mathbb{Z}})$ and $H_1(N)$ respectively. \end{rem} For $K$ we fix the basis \begin{align} &(e_1,e_{r+1},\dots,e_r,e_{2r})\quad\textrm{if\ }g=2r+1,\\ &(e_1,e_{r+1},\dots,e_r,e_{2r},e_{2r+1})\quad\textrm{if\ }g=2r+2. \end{align} For $H_1(S)/K$ we fix the basis \begin{align} &(a_1+K,b_1+K,\dots,a_r+K,b_r+K)\quad\textrm{if\ }g=2r+1,\\ &(a_1+K,b_1+K,\dots,a_r+K,b_r+K,b_{r+1}+K)\quad\textrm{if\ }g=2r+2. \end{align} Having fixed bases for $K$ and $H_1(S)/K$ we can now compute, for $\Psi_1$ and $\Psi_2$, the images of the generators of $\mathcal{M}(N)$. This is done by a straightforward calculation, using Proposition \ref{lifts} and the formula (\ref{transv}). For $k=1,2$ and $1\le i\le r$, $1\le j\le r-1$ we have \[\Psi_k(t_{\varepsilon_i})=A_i,\quad \Psi_k(t_{\delta_{2i}})=B_i,\quad \Psi_k(t_{\delta_{2j+1}})=C_j.\] If $g=2r+1$ then \[\Psi_1(u_{g-1})=\begin{pmatrix}I_{g-3}&0&0\\0&1&0\\0&1&-1\end{pmatrix},\quad \Psi_2(u_{g-1})=\begin{pmatrix}I_{g-3}&0&0\\0&-1&0\\0&-1&1\end{pmatrix}. \] If $g=2r+2$ then \[\Psi_1(t_{\delta_{g-1}})=\begin{pmatrix}I_{g-4}&0&0&0\\0&1&1&0\\0&0&1&0\\0&0&-2&1\end{pmatrix},\quad \Psi_2(t_{\delta_{g-1}})=\begin{pmatrix}I_{g-4}&0&0&0\\0&1&1&-2\\0&0&1&0\\0&0&0&1\end{pmatrix}, \] \[\Psi_1(u_{g-1})=\begin{pmatrix}I_{g-4}&0&0&0\\0&1&-1&1\\0&0&1&0\\0&0&2&-1\end{pmatrix},\quad \Psi_2(u_{g-1})=\begin{pmatrix}I_{g-4}&0&0&0\\0&1&1&-2\\0&0&1&0\\0&0&1&-1\end{pmatrix}. \] Now it is easy to see that $\Psi_1$ and $\Psi_2$ are not conjugate as homomorphism to $\mathrm{GL}(g-1,\mathbb{C})$. For suppose that there is $M\in\mathrm{GL}(g-1,\mathbb{C})$, such that $\Psi_1(x)=M\Psi_2(x)M^{-1}$ for all $x\in\mathcal{M}(N)$. Then $M$ commutes with $A_i$, $B_i$, $C_j$ for $1\le i\le r$, $1\le j\le r-1$, and by Lemma \ref{diag}, $M=\alpha I_{2r}$ if $g=2r+1$, or $M=\mathrm{diag}(\alpha I_{2r},\beta)$ if $g=2r+2$, for $\alpha,\beta\in\mathbb{C}$. In either case it is impossible that $\Psi_1(u_{g-1})=M\Psi_2(u_{g-1})M^{-1}$. \section{Homomorphisms from $\mathcal{M}(N_{g,n})$ to $\mathrm{GL}(m,\mathbb{C})$ for $m<g-1$} The aim of this section is to prove Theorem \ref{MNtoGLfact}. The proof is divided in two parts. \begin{proof}[Proof of Theorem \ref{MNtoGLfact} for $(g,m)\ne(6,4)$.] Suppose that $n\in\{0,1\}$, $g=2r+s$ for $r\ge 2$, $s\in\{1,2\}$, $m\le g-2$ and $f\colon\mathcal{M}(N_{g,n})\to\mathrm{GL}(m,\mathbb{C})$ is a homomorphism. By Theorem \ref{abNg}, it suffices to prove that $\mathrm{Im}(f)$ is abelian. Let $S'=S_{r,s}$ and $\iota\colon\mathcal{M}(S')\to\mathcal{M}(N_{g,n})$ be the homomorphism from Corollary \ref{HomSN}. Set $f'=f\circ\iota$ and observe that if $\mathrm{Im}(f')$ is abelian, then so is $\mathrm{Im}(f)$, by Lemma \ref{com_norm}. Suppose that $m\le 2r-1$. Then $\mathrm{Im}(f')$ is either trivial or cyclic by Theorem \ref{FHK} and we are done. This finishes the proof for odd $g$. Suppose that $g=2r+2$ for $r\ge 3$ and $m=2r$. By Theorem \ref{KorU}, $f'$ is either trivial or conjugate to the homological representation $\Phi$. In the former case we are done. In the latter case, by the definition of $\Phi$ we have $\Phi(t_{\gamma_r})=\Phi(t_{\alpha_r})$ because the curves $\gamma_r$ and $\alpha_r$ become isotopic after gluing discs to the boundary of $S'$. It follows that $f(t_{\delta_{2r+1}})=f(t_{\varepsilon_r})$ and by Lemma \ref{tsq} $f(t_{\delta_1}^2)=1$. This is a contradiction because $\Phi(t_{\alpha_1})$ has infinite order. \end{proof} In order to prove Theorem \ref{MNtoGLfact} for $(g,m)=(6,4)$, we first prove some lemmas. \begin{lemma}\label{MN4toGL2} Suppose that $f\colon\mathcal{M}(N_{4,n})\to\mathrm{GL}(2,\mathbb{C})$ is a homomorphism. Then, with respect to some basis one of the following cases holds. \begin{itemize} \item[(1)] $f(t_{\delta_1})=f(t_{\delta_2})=f(t_{\delta_3})=\lambda I$, $\lambda\in\{-1,1\}$ \item[(2)] $f(t_{\delta_1})=f(t_{\delta_2})=f(t_{\delta_3})=\begin{pmatrix}1&0\\0&-1\end{pmatrix}$ \item[(3)] $f(t_{\delta_1})=f(t_{\delta_3})=\begin{pmatrix}1&1\\0&-1\end{pmatrix}$, $f(t_{\delta_2})=\begin{pmatrix}-1&0\\1&1\end{pmatrix}$. \end{itemize} In particular $f(t_{\delta_1}^2)=1$. \end{lemma} \begin{proof} For $i=1,2,3$ let $L_i=f(t_{\delta_i})$ and $U=f(u_{3})$. Suppose that $L_1$ has only one eigenvalue $\lambda$. Since $L_1$ is conjugate to $L_1^{-1}$ (by (R12)), we have $\lambda\in\{-1,1\}$. If $\dim E(L_1,\lambda)=2$, then we have the case (1). Suppose that $\dim E(L_1,\lambda)=1$. If $E(L_1,\lambda)\ne E(L_2,\lambda)$, then with respect to some basis we have $L_1=\begin{pmatrix}\lambda&1\\0&\lambda\end{pmatrix}$, $L_2=\begin{pmatrix}\lambda&0\\x&\lambda\end{pmatrix}$ for some $x$, and from the braid relation $L_1L_2L_1=L_2L_1L_2$ we have $x=-1$. Since $L_3$ commutes with $L_1$ we have $L_3=\begin{pmatrix}\lambda&y\\0&\lambda\end{pmatrix}$ for some $y$, and from $L_2L_3L_2=L_3L_2L_3$ we obtain $y=1$, hence $L_1=L_3$. Since $\delta_1=\varepsilon_1$, we have $L_1^2=I$ by Lemma \ref{tsq} (for $i=1$, $j=3$), which is a contradiction. If $E(L_1,\lambda)=E(L_2,\lambda)$, then with respect to some basis we have $L_1=\begin{pmatrix}\lambda&1\\0&\lambda\end{pmatrix}$, $L_2=\begin{pmatrix}\lambda&x\\0&\lambda\end{pmatrix}$, and it is easy to obtain a contradiction as above, by showing that $L_1=L_2=L_3$. Suppose that $L_1$ has two eigenvalues $\lambda, \mu$. Then with respect to some basis we have $L_1=\begin{pmatrix}\lambda&0\\0&\mu\end{pmatrix}$, and since $L_3$ and $U$ commute with $L_1$, they are also diagonal. In particular we have $UL_3=L_3U$ and $L_3UL_3=U$ (R12) gives $L_3^2=1$, which implies $\{\lambda, \mu\}=\{-1,1\}$. We have $L_3=L_1$ or $L_3=-L_1$. In the latter case the braid relations $L_3L_2L_3=L_2L_3L_2$ and $L_1L_2L_1=L_2L_1L_2$ imply $L_2L_1L_2=0$, a contradiction, hence $L_1=L_3$. If $E(L_1,1)\ne E(L_2,1)$, then with respect to some basis we have $L_1=\begin{pmatrix}1&1\\0&-1\end{pmatrix}$, $L_2=\begin{pmatrix}-1&0\\x&1\end{pmatrix}$. From $L_1L_2L_1=L_2L_1L_2$ we have $x=1$ and we are in the case (3). Analogously, if $E(L_1,-1)\ne E(L_2,-1)$, then with respect to some basis we have $L_1=\begin{pmatrix}-1&1\\0&1\end{pmatrix}$, $L_2=\begin{pmatrix}1&0\\1&-1\end{pmatrix}$, and since $E(L_1,1)\ne E(L_1,1)$, we are in the case (3) again. Finally, if $E(L_1,1)=E(L_2,1)$ and $E(L_1,-1)=E(L_2,-1)$, then with respect to some basis we have $L_1=L_2=\begin{pmatrix}1&0\\0&-1\end{pmatrix}$ and we are in the case (2). \end{proof} \begin{lemma}\label{MN6toGL4a^2} Suppose that $n\le 1$ and $f\colon\mathcal{M}(N_{6,n})\to\mathrm{GL}(4,\mathbb{C})$ is a homomorphism such that $f(t_{\delta_1}^2)=1$. Then $\mathrm{Im}(f)$ is abelian. \end{lemma} \begin{proof} Let $H$ be the normal closure of $t_{\delta_1}^2$ in $\mathcal{M}(N_{6,n})$ and set $G=\mathcal{M}(N_{6,n})/H$. We have an induced homomorphism $f'\colon G\to\mathrm{GL}(4,\mathbb{C})$ such that $f=f'\circ\pi$, where $\pi\colon\mathcal{M}(N_{6,n})\to G$ is the canonical projection. By the relations (R1, R4), the mapping $\rho(\sigma_i)=\pi(t_{\delta_i})$, where $\sigma_i$ is the transposition $(i,i+1)$ for $1\le i\le 5$, defines a homomorphism $\rho\colon\mathfrak{S}_6\to G$. Let $\phi\colon\mathfrak{S}_6\to\mathrm{GL}(4,\mathbb{C})$ be the composition $f'\circ \rho$. By Lemma \ref{repsym}, $\phi$ is the direct sum of one-dimensional representations. In particular the image of $\phi$ is abelian, and so is $\mathrm{Im}(f)$ by Lemma \ref{com_norm}. \end{proof} Let $R$ be the subsurface obtained by removing from $N_{6,n}$ a regular neighbourhood of $\delta_1\cup\delta_2$. Note that $R$ is homeomorphic to $N_{4,n+1}$. The homomorphism $\mathcal{M}(R)\to\mathcal{M}(N_{6,n})$ induced by the inclusion of $R$ in $N_{6,n}$ is injective, and we will treat $\mathcal{M}(R)$ as a subgroup of $\mathcal{M}(N_{6,n})$. \begin{lemma}\label{MN6toGL4splitting} Suppose that $n\le 1$, $f\colon\mathcal{M}(N_{6,n})\to\mathrm{GL}(4,\mathbb{C})$ is a homomorphism and there exists a splitting $\mathbb{C}^4=V_1\oplus V_2$ such that $V_i$ is a $2$-dimensional $\mathcal{M}(R)$-invariant subspace for $i=1,2$. Then $\mathrm{Im}(f)$ is abelian.\end{lemma} \begin{proof} Let $f'$ be the restriction of $f$ to $\mathcal{M}(R)$. With respect to the splitting $\mathbb{C}^4=V_1\oplus V_2$ we have $f'=f_1\oplus f_2$ for some $f_i\colon\mathcal{M}(R)\to\mathrm{GL}(2,\mathbb{C})$, $i=1,2$. By Lemma \ref{MN4toGL2} we have $f_i(t_{\delta_4}^2)=1$ for $i=1,2$, hence $f(t_{\delta_4}^2)=1$ and we are done by Lemma \ref{MN6toGL4a^2}. \end{proof} \begin{lemma}\label{MN6toGL4inv} Suppose that $n\le 1$, $f\colon\mathcal{M}(N_{6,n})\to\mathrm{GL}(4,\mathbb{C})$ is a homomorphism, $f(t_{\delta_1})$ has only one eigenvalue and there exists a $2$-dimensional $\mathcal{M}(R)$-invariant subspace. Then $\mathrm{Im} f$ is abelian. \end{lemma} \begin{proof} Let $\lambda$ be the eigenvalue of $f(t_{\delta_1})$. Fix a basis of $\mathbb{C}^4$ whose first two vectors span the $\mathcal{M}(R)$-invariant subspace. By the case (1) of Lemma \ref{MN4toGL2}, with respect to such basis we have $f(t_{\delta_4})=\begin{pmatrix}\lambda I&X\\0&\lambda I\end{pmatrix}$, $f(t_{\delta_5})=\begin{pmatrix}\lambda I&Y\\0&\lambda I\end{pmatrix}$, for some $2\times 2$ matrices $X, Y$. In particular $f(t_{\delta_4})$ and $f(t_{\delta_5})$ commute and we are done by Lemma \ref{com_norm}. \end{proof} \begin{proof}[Proof of Theorem \ref{MNtoGLfact} for $g=6$, $m=4$.] Suppose that $n\in\{0,1\}$ and $f\colon\mathcal{M}(N_{6,n})\to\mathrm{GL}(4,\mathbb{C})$ is a homomorphism. For $1\le i\le 5$ we set $L_i=f(t_{\delta_i})$ and $M=f(t_{\varepsilon_2})$, $U_5=f(u_5)$. We consider the following cases. \begin{itemize} \item[(1)] $L_1$ has $4$ eigenvalues. \item[(2)] $L_1$ has $3$ eigenvalues. \item[(3)] $L_1$ has $2$ eigenvalues with equal multiplicities. \item[(4)] $L_1$ has $2$ eigenvalues with different multiplicities. \item[(5)] $L_1$ has $1$ eigenvalue. \end{itemize} In the cases (1, 2, 3) it is easy to find a splitting $\mathbb{C}^4=V_1\oplus V_2$ such that $V_i$ is a $2$-dimensional $\mathcal{M}(R)$-invariant subspace for $i=1,2$. For example, suppose that $L_1$ has three eigenvalues $\lambda_1, \lambda_2, \lambda_3$ such that $\#\lambda_1=\#\lambda_2=1$ and $\#\lambda_3=2$. Then we take $V_1=E(L_1,\lambda_1)\oplus E(L_1,\lambda_2)$ and $V_2=E(L_1,\lambda_3)$ if $\dim E(L_1,\lambda_3)=2$ or $V_2=E^2(L_1,\lambda_3)$ if $\dim E(L_1,\lambda_3)=1$. Therefore in the cases (1, 2, 3) we are done by Lemma \ref{MN6toGL4splitting}. Assume (5). Let $\lambda$ be the unique eigenvalue of $L_1$ and $k=\dim E(L_1,\lambda)$. If $k=4$ then $L_1=\lambda I$ and the image of $f$ is cyclic. If $k=2$ or $k=1$ then respectively $E(L_1,\lambda)$ or $E^2(L_1,\lambda)$ is a $2$-dimensional $\mathcal{M}(R)$-invariant subspace, and we are done by Lemma \ref{MN6toGL4inv}. Suppose that $k=3$. If $E(L_1,\lambda)\ne E(L_2,\lambda)$ then $E(L_1,\lambda)\cap E(L_2,\lambda)$ is a $2$-dimensional $\mathcal{M}(R)$-invariant subspace, and we are done by Lemma \ref{MN6toGL4inv}. If $E(L_1,\lambda)=E(L_2,\lambda)$ then with respect to some basis we have \[L_1=\begin{pmatrix}\lambda&0&0&0\\0&\lambda&0&0\\0&0&\lambda&1\\0&0&0&\lambda\end{pmatrix},\quad L_2=\begin{pmatrix}\lambda&0&0&x\\0&\lambda&0&y\\0&0&\lambda&z\\0&0&0&\lambda\end{pmatrix}.\] In particular $L_1$ and $L_2$ commute and we are done by Lemma \ref{com_norm}. It remains to consider the case (4). Suppose that $L_1$ has eigenvalues $\mu$, $\lambda$, with $\#\mu=1$ and $\#\lambda=3$. Since $L_1$ is conjugate to $L_1^{-1}$, we have $\{\mu,\lambda\}=\{-1,1\}$. It follows from Theorem \ref{abNg} that there is a homomorphism $\tau(\mathcal{M}(N_6))\to\{-1,1\}$ such that $\tau(a_1)=-1$. By multiplying $f$ by $\tau$ if necessary, we may assume $\mu=-1$, $\lambda=1$. The Jordan form of $L_1$ is one of the following three matrices. \[(i)\ \begin{pmatrix}-1&0&0&0\\0&1&0&0\\0&0&1&0\\0&0&0&1\end{pmatrix}\quad (ii)\ \begin{pmatrix}-1&0&0&0\\0&1&1&0\\0&0&1&1\\0&0&0&1\end{pmatrix}\quad (iii)\ \begin{pmatrix}-1&0&0&0\\0&1&0&0\\0&0&1&1\\0&0&0&1\end{pmatrix} \] In the case (i) we have $L_1^2=I$ and we are done by Lemma \ref{MN6toGL4a^2}. In the case (ii) the following subspaces are $\mathcal{M}(R)$-invariant: $E(L_1,-1)$, $E(L_1,1)$, $E^2(L_1,1)$, $E^3(L_1,1)$. It follows that \[M=\begin{pmatrix}x_1&0&0&0\\0&x_2&v_1&v_2\\0&0&x_3&v_3\\0&0&0&x_4\end{pmatrix},\quad L_4=\begin{pmatrix}y_1&0&0&0\\0&y_2&w_1&w_2\\0&0&y_3&w_3\\0&0&0&y_4\end{pmatrix}.\] The braid relation $ML_4M=L_4ML_4$ (R5) implies $x_i=y_i$ for $1\le i\le 4$. Since the first two vectors of the basis are eigenvectors of $M$, they have to correspond to different eigenvalues of $M$. Therefore $x_2=-x_1$, $x_3=x_4=1$ and $x_1=1$ or $x_1=-1$. In either case it is not difficult to check that $ML_4M=L_4ML_4$ holds if and only if $M=L_4$. We are done by Lemma \ref{com_norm}. In the case (iii) the following subspaces are $\mathcal{M}(R)$-invariant: $E(L_1,-1)$, $E(L_1,1)$, $E^2(L_1,1)$. We have $\dim E(L_1,1)=2$ and by applying Lemma \ref{MN4toGL2} to the action of $\mathcal{M}(R)$ on this subspace, we obtain three sub-cases. Sub-case (iiia). If the action of $\mathcal{M}(R)$ on $E(L_1,1)$ is trivial, then we have \[M=\begin{pmatrix}-1&0&0&0\\0&1&0&x_1\\0&0&1&x_2\\0&0&0&1\end{pmatrix},\quad L_4=\begin{pmatrix}-1&0&0&0\\0&1&0&y_1\\0&0&1&y_2\\0&0&0&1\end{pmatrix}\] As in the case (ii), the braid relation implies $M=L_4$ and we are done by Lemma \ref{com_norm}. Sub-case (iiib). By changing the basis of $E(L_1,1)$ we may assume that \[M=\begin{pmatrix}1&0&0&0\\0&-1&0&x_1\\0&0&1&x_2\\0&0&0&1\end{pmatrix},\quad L_4=\begin{pmatrix}1&0&0&0\\0&-1&0&y_1\\0&0&1&y_2\\0&0&0&1\end{pmatrix}\] As in the case (ii), the braid relation implies $M=L_4$ and we are done by Lemma \ref{com_norm}. Sub-case (iiic). By changing the basis of $E(L_1,1)$ we may assume that \[M=\begin{pmatrix}1&0&0&0\\0&1&1&x_1\\0&0&-1&x_2\\0&0&0&1\end{pmatrix}, L_4=\begin{pmatrix}1&0&0&0\\0&-1&0&y_1\\0&1&1&y_2\\0&0&0&1\end{pmatrix}, L_5=\begin{pmatrix}1&0&0&0\\0&1&1&z_1\\0&0&-1&z_2\\0&0&0&1\end{pmatrix}.\] By solving the equations $ML_4M=L_4ML_4$ and $L_5L_4L_5=L_4L_5L_4$ we obtain $x_2=-(2x_1+y_1+2y_2)$, $z_2=-(2z_1+y_1+2y_2)$, and from $ML_5=L_5M$ we obtain $x_2=z_2$. Thus $M=L_5$ and by Lemma \ref{tsq} $L_1^2=1$. We are done by Lemma \ref{MN6toGL4a^2}. \end{proof} \section{Homomorphisms between mapping class groups} The aim of this section is to prove Theorem \ref{MNgtoMNh}. Fix $g\ge 5$ and set $\mathcal{M}=\mathcal{M}(N_g)$. We are going to use the fact that $s=t_{\delta_1}\cdots t_{\delta_{g-1}}$ has finite order in $\mathcal{M}$ (equal to $g$ if it is even, or $2g$ otherwise, see \cite{PSz}). By the relations (R1,R4) we have \begin{equation}\label{srel} t_{\delta_{i+1}}s=st_{\delta_i}\quad\textrm{for\ }1\le i\le g-2. \end{equation} By Theorem \ref{abNg} we have $s\in[\mathcal{M},\mathcal{M}]$ for $g\ge 7$ and $g=5$, $s^2\in[\mathcal{M},\mathcal{M}]$ for $g=6$. \begin{proof}[Proof of Theorem \ref{MNgtoMNh}] Suppose that $g\ge 5$, $h<g$ and $f\colon\mathcal{M}(N_g)\to\mathcal{M}(N_h)$ is a homomorphism. Since $M(N_h)$ is abelian for $h\le 2$, we are assuming $h\ge 3$. Let $f'\colon\mathcal{M}(N_{g})\to\mathrm{GL}(h-1,\mathbb{C})$ be the composition $\Psi_1\circ f$ and $K=\ker \Psi_1$. By Theorem \ref{MNtoGLfact}, $\mathrm{Im}(f')$ is abelian, hence $f([\mathcal{M}(N_g),\mathcal{M}(N_g)])\subseteq K$. Suppose that $g\ge 7$ or $g=5$. Then $f(s)\in K$, and since $K$ is torsion free by Lemma \ref{kerTF}, thus $f(s)=1$. This gives, by (\ref{srel}), $f(t_{\delta_1})=f(t_{\delta_2})$ and we are done by Lemma \ref{com_norm}. If $g=6$ then $f(s^2)\in K$, which gives $f(s^2)=1$ and $f(t_{\delta_2})=f(t_{\delta_4})$. Since $t_{\delta_1}$ commutes with $t_{\delta_4}$, thus $f(t_{\delta_1})$ commutes with $f(t_{\delta_2})$ and we are done by Lemma \ref{com_norm}. \end{proof} Note that Theorems \ref{MNtoGLfact} and \ref{MNgtoMNh} are trivially true for $g\le 3$ because $\mathrm{GL}(1,\mathbb{C})=\mathbb{C}^\ast$, $\mathcal{M}(N_2)\cong\mathbb{Z}_2\times\mathbb{Z}_2$, $M(N_1)=1$ are abelian groups. On the other hand, Corollary \ref{g4cex} below shows that both theorems are false for $g=4$ (recall that $\mathcal{M}(N_3)\cong\mathrm{GL}(2,\mathbb{Z})$). Let $D_\infty$ denote the infinite dihedral group, defined by the presentation \[D_\infty=\lr{x,y\,\|\,x^2=y^2=1}.\] \begin{lemma}\label{MN4onD} There is an epimorphism $\phi\colon\mathcal{M}(N_4)\to D_\infty$. \end{lemma} \begin{proof} According to the main result of \cite{Szepg4} simplified in \cite{PSz}, $\mathcal{M}(N_4)$ admits a presentation with generators $t_{\varepsilon_2}$, $t_{\delta_i}$, $u_i$ for $i=1,2,3$ and relations (R1, R3, R4, R6, R7, R9, R10, R11, R12) and \begin{align} &t_{\delta_{i+1}}u_1u_{i+1}=u_iu_{i+1}t_{\delta_i}\quad\mathrm{for\ }i=1,2\\ &(t_{\varepsilon_2}u_3)^2=1,\quad t_{\delta_1}(t_{\delta_2}t_{\delta_3}u_3u_2)t_{\delta_1}=(t_{\delta_2}t_{\delta_3}u_3u_2). \end{align} It is easy to check that the mapping $\phi(t_{\varepsilon_2})=xy$, $\phi(t_{\delta_i})=1$, $\phi(u_i)=y$ for $i=1,2,3$, respects the defining relations of $\mathcal{M}(N_4)$, hence it defines a homomorphism onto $D_\infty$. \end{proof} \begin{cor}\label{g4cex} For $h\ge 3$ there is a homomorphism $f\colon\mathcal{M}(N_4)\to\mathcal{M}(N_h)$, such that $\mathrm{Im}(f)$ is isomorphic to $D_\infty$. \end{cor} \begin{proof} Fix $h\ge 3$. By the proof of \cite[Theorem 3]{SzepCR}, $t_{\delta_1}$ can be written in $\mathcal{M}(N_h)$ as a product of two involutions $\sigma, \tau$. Since $t_{\delta_1}$ has infinite order in $\mathcal{M}(N_h)$, the mapping $x\mapsto\sigma$, $y\mapsto\tau$ defines an embedding $D_\infty\to\mathcal{M}(N_h)$. By pre-composing this embedding with the epimorphism $\phi$ from Lemma \ref{MN4onD}, we obtain $f$. \end{proof} The following two theorems can be proved by the same method as Theorem \ref{MNgtoMNh}. We leave the details to the reader. \begin{theorem} Suppose that $g\ge 5$, $g\ge 2h+2$ and $f\colon\mathcal{M}(N_g)\to\mathcal{M}(S_h)$ is a homomorphism. Then $\mathrm{Im}(f)$ is abelian. \end{theorem} \begin{theorem} Suppose that $g\ge 3$ and $h\le 2g$. Then the only homomorphism from $\mathcal{M}(S_g)$ to $\mathcal{M}(N_h)$ is the trivial one. \end{theorem} \section{Homomorphisms from $\mathcal{M}(N_g)$ to $\mathrm{GL}(g-1,\mathbb{C})$.} The aim of this section is to prove Theorem \ref{MNtoGLg-1}. The prove it divided in two cases, according to the parity of the genus. Let $g=2r+s$, $s\in\{1,2\}$, $S'=S_{r,s}$ and $\iota\colon\mathcal{M}(S')\to\mathcal{M}(N_{g,n})$ be the homomorphism from Corollary \ref{HomSN}. If $f\colon\mathcal{M}(N_{g,n})\to\mathrm{GL}(m,\mathbb{C})$ is a homomorphism, then we set $f'=f\circ\iota$. \begin{proof}[Proof of Theorem \ref{MNtoGLg-1} for odd $g$.] Suppose that $N=N_{2r+1}$, $r\ge 3$ and $f\colon\mathcal{M}(N)\to\mathrm{GL}(2r,\mathbb{C})$ is a homomorphism, such that $\mathrm{Im}(f)$ is not abelian. By Theorem \ref{KorU}, $f'$ is conjugate to the homological representation $\Phi$, and thus the exists a basis, such that $f(t_{\varepsilon_i})=f'(t_{\alpha_i})=A_i$, $f(t_{\delta_{2i}})=f'(t_{\beta_i})=B_i$ for $1\le i\le r$ and $f(t_{\delta_{2j+1}})=f'(t_{\gamma_j})=C_j$ for $1\le j\le r-1$. Set $U_k=f(u_k)$ for $1\le k\le 2r$. Since $U_{2r}$ commutes with $A_i$ and $B_i$ for $1\le i\le r$, and with $C_j$ for $j=1,\dots,r-2$ (R6,R8) thus, by Lemma \ref{diag}, \[U_{2r}=\begin{pmatrix}\lambda I_{2r-2}&0\\0&X\end{pmatrix},\] for some $2\times 2$ matrix $X$. Since $U_{2r}$ is conjugate to $U_{2r}^{-1}$ we have $\lambda\in\{-1,1\}$ and by multiplying $f$ by $(-1)^\mathrm{ab}$ if necessary, we may assume $\lambda=1$. The relation $B_rU_{2r}B_r=U_{2r}$ (R12) implies $X=\begin{pmatrix}x&0\\y&-x\end{pmatrix}$. From (R11) and (R7) we have \begin{align} &U_{2r-2}=(C_{r-1}B_rB_{r-1}C_{r-1})^{-1}U_{2r}(C_{r-1}B_rB_{r-1}C_{r-1}),\\ &U_{2r}U_{2r-2}-U_{2r-2}U_{2r}=0, \end{align} and since the left hand side of the last equation is equal to \[(1-x^2)(E_{2r,2r-3}+E_{2r-2,2r-1}),\] thus $x^2=1$. We have $U_{2r}^{-1}=U_{2r}$, and from (R11) and (R9) \begin{align} &U_{2r-1}=(C_{r-1}B_r)^{-1}U_{2r}(C_{r-1}B_r),\\ &U_{2r}U_{2r-1}U_{2r}-U_{2r-1}U_{2r}U_{2r-1}=0. \end{align} By considering the cases $x=1$ and $x=-1$ separately, we find that the left hand side of the last equation is of the form $(y-x)^2Z$, where $Z\ne 0$. Hence $x=y$ and $U_{2r}=\Psi_1(u_{2r})$ if $x=1$, or $U_{2r}=\Psi_2(u_{2r})$ if $x=-1$. By Theorem \ref{gener}, $f$ is equal to $\Psi_1$ or $\Psi_2$ on generators of $\mathcal{M}(N)$. \end{proof} Now we will borrow some arguments from \cite{KorkSymp} to prove Lemma \ref{AB} below, which will be a starting point for the proof of Theorem \ref{MNtoGLg-1} for even genus. \begin{lemma}\label{flag} Suppose that $n\le 1$, $g\ge 5$ and $f\colon\mathcal{M}(N_{g,n})\to\mathrm{GL}(m,\mathbb{C})$ is a homomorphism. If there is a flag $0=W_0\subset W_1\subset\cdots\subset W_k=\mathbb{C}^m$ of $\mathcal{M}(N_{g,n})$-invariant subspaces such that $\dim(W_i/W_{i-1})<g-1$ for $i=1,\dots,k$, then $\mathrm{Im}(f)$ is abelian. \end{lemma} \begin{proof} The same argument as in the proof of \cite[Lemma 4.8]{KorkSymp} can be applied, using Theorem \ref{MNtoGLfact}, to show that with respect to some basis $f[\mathcal{M}(N_{g,n}),\mathcal{M}(N_{g,n})]$ is contained in the subgroup of upper triangular matrices with $1$ on the diagonal. Since this subgroup is nilpotent and $[\mathcal{M}(S'),\mathcal{M}(S')]$ is perfect, it follows that $f'[\mathcal{M}(S'),\mathcal{M}(S')]$ is trivial, which means that $\mathrm{Im}(f')$ is abelian, and so is $\mathrm{Im}(f)$. \end{proof} \begin{lemma}\label{dim_eigen} Suppose that $N=N_{2r+2}$, $r\ge 3$ and $f\colon\mathcal{M}(N)\to\mathrm{GL}(2r+1,\mathbb{C})$ is a homomorphism, such that $\mathrm{Im}(f)$ is not abelian. Then $L_1=f(t_{\delta_1})$ has an eigenvalue $\lambda$ such that $\dim E(L_1,\lambda)=2r$. \end{lemma} \begin{proof} By \cite[Corollary 4.6]{KorkSymp} applied to $f'$, $L_1$ has at most two eigenvalues. It follows that there is an eigenvalue $\lambda$ with $\#\lambda\ge r+1\ge 4$. Set $m=\dim E(L_1,\lambda)$. Since $\mathrm{Im}(f)$ is not abelian, thus $m\le 2r$. We are going to show that $m=2r$. Let $R$ be the subsurface obtained by removing from $N$ a regular neighbourhood of $\delta_1\cup\delta_2$. We have $R\approx N_{2r,1}$. We treat $\mathcal{M}(R)$ as a subgroup of $\mathcal{M}(N)$. Suppose $m\le 2r-2$. Let $W=E^k(L_1,\lambda)$, where $k=\max\{4-m, 1\}$. Observe that $W$ is a $\mathcal{M}(R)$-invariant subspace with $3\le\dim W\le 2r-2$. By Lemma \ref{flag}, $f(\mathcal{M}(R))$ is abelian, which means $f(t_{\delta_4})=f(t_{\delta_5})$. By Lemma \ref{com_norm}, $\mathrm{Im}(f)$ is abelian, a contradiction. Suppose that $m=2r-1$ and set $L_2=f(t_{\delta_2})$. If $E(L_1,\lambda)\ne E(L_2,\lambda)$ then $E(L_1,\lambda)\cap E(L_2,\lambda)$ is a $\mathcal{M}(R)$-invariant subspace of dimension $2r-3$ or $2r-2$ and we can use the same argument as above to obtain a contradiction. If $E(L_1,\lambda)=E(L_2,\lambda)$, then by \cite[Lemma 4.3]{KorkSymp} applied to $f'$, $E(L_1,\lambda)$ is a $\mathcal{M}(S')$-invariant subspace of dimension $2r-1$, and by \cite[Lemma 4.8]{KorkSymp} $f'$ is trivial. It follows that $\mathrm{Im} f$ is abelian, a contradiction. \end{proof} \begin{lemma}\label{AB} Suppose that $N=N_{2r+2}$, $r\ge 3$ and $f\colon\mathcal{M}(N)\to\mathrm{GL}(2r+1,\mathbb{C})$ is a homomorphism. If $r=3$ then assume that $1$ is the unique eigenvalue of $f(t_{\delta_1})$. Then either $\mathrm{Im}(f)$ is abelian, or with respect to some basis $f(t_{\varepsilon_i})=A_i$, $f(t_{\delta_{2i}})=B_i$ for $i=1,\dots,r$. \end{lemma} \begin{proof} Suppose that $\mathrm{Im}(f)$ is not abelian. By Lemma \ref{dim_eigen}, $L_1=f(t_{\delta_1})$ has an eigenvalue $\lambda$ with $\dim E(L_1,\lambda)=2r$. If $r=3$ then $\lambda=1$ by assumption, and for $r\ge 4$, $\lambda=1$ by the proof of \cite[Lemma 5.2]{KorkSymp}. Since $\mathcal{M}(S')$ is perfect, thus $\det L_1=1$ and $\lambda=1$ is the unique eigenvalue. Set $L_2=f(t_{\delta_2})$. We claim that $E(L_1,1)\ne E(L_2,1)$. For otherwise it is easy to prove that $L_1$ and $L_2$ commute (see the proof of Theorem \ref{MNtoGLfact} for $(g,m)=(6,4)$, case (5)), and $\mathrm{Im}(f)$ is abelian by Lemma \ref{com_norm}, a contradiction. Now we can apply \cite[Lemma 4.7]{KorkSymp} to $f'$ to conclude that with respect to some basis we have $f(t_{\varepsilon_i})=f'(t_{\alpha_i})=A_i$, $f(t_{\delta_{2i}})=f'(t_{\beta_i})=B_i$ for $i=1,\dots,r$. \end{proof} \begin{proof}[Proof of Theorem \ref{MNtoGLg-1} for even $g$.] Suppose that $N=N_{2r+2}$, $r\ge 4$ and $f\colon\mathcal{M}(N)\to\mathrm{GL}(2r+1,\mathbb{C})$ is a homomorphism, such that $\mathrm{Im}(f)$ is not abelian. By Lemma \ref{AB} there is a basis such that $f(t_{\varepsilon_i})=A_i$ and $f(t_{\delta_{2i}})=B_i$ for $1\le i\le r$. Set $D_i=f(t_{\delta_{2i+1}})$ for $1\le i\le r$ and $U_j=f(u_j)$ for $1\le j\le 2r+1$. Fix $i\in\{1,\dots,r-1\}$. Since $D_i$ is conjugate to $A_1$, it has one eigenvalue $\lambda=1$. For $j\notin\{i,i+1\}$ the relations $D_iA_j=A_jD_i$ and $D_iB_j=B_jD_i$ imply, by Lemma \ref{diag}, that $D_i$ has the form \[D_i=\begin{pmatrix} I_{2(i-1)}&0&0&0&0\\ 0&F_{11}&F_{12}&0&X_1\\ 0&F_{21}&F_{22}&0&X_2\\ 0&0&0&I_{2(g-i-1)}&0\\ 0&Y_1&Y_2&0&z \end{pmatrix},\] where $F_{kl}$ are $2\times 2$ matrices, $X_k$ are $2\times 1$ vectors, $Y_l$ are $1\times 2$ vectors and $z$ is a complex number. The relations $D_iA_i=A_iD_i$ and $D_iA_{i+1}=A_{i+1}D_i$ imply, for $k,l\in\{1,2\}$, $VF_{kl}=F_{kl}V$, $VF_{kl}=F_{kl}$ for $k\ne l$, $VX_k=X_k$, $Y_lV=Y_l$, hence \begin{align} &F_{11}=\begin{pmatrix}s_1&t_1\\0&s_1\end{pmatrix}, F_{12}=\begin{pmatrix}0&v_1\\0&0\end{pmatrix}, X_1=\begin{pmatrix}x_1\\0\end{pmatrix},\\ &F_{21}=\begin{pmatrix}0&v_2\\0&0\end{pmatrix}, F_{22}=\begin{pmatrix}s_2&t_2\\0&s_2\end{pmatrix}, X_2=\begin{pmatrix}x_2\\0\end{pmatrix},\\ &Y_1=\begin{pmatrix}0&y_1\end{pmatrix}, Y_2=\begin{pmatrix}0&y_2\end{pmatrix}. \end{align} Since $s_1$, $s_2$ are eigenvalues, we have $s_1=s_2=1$ and $\det D_i=z$, which gives $z=1$. Now, by solving the equations $B_iD_iB_i-D_iB_iD_i=0$ and $B_{i+1}D_iB_{i+1}-D_iB_{i+1}D_i=0$ we obtain $t_1=t_2=1$, $v_1v_2=1$, $y_2=y_1v_1$, $x_2=x_1v_2$, $x_1y_1=0$. Thus, for $i=1,\dots,r-1$ we have \[D_i=\begin{pmatrix} I_{2(i-1)}&0&0&0&0&0&0\\ 0&1&1&0&\alpha_i&0&\alpha_ix_i\\ 0&0&1&0&0&0&0\\ 0&0&\alpha_i^{-1}&1&1&0&x_i\\ 0&0&0&0&1&0&0\\ 0&0&0&0&0&I_{2(g-i-1)}&0\\ 0&0&y_i&0&\alpha_iy_i&0&1 \end{pmatrix},\quad x_iy_i=0.\] Similarly, using the relations between $D_r$ and $A_i$, $B_i$ it can be shown that \[D_r=\begin{pmatrix}I_{2g-2}&0&0&0\\ 0&1&1&x_r\\ 0&0&1&0\\ 0&0&y_r&1\end{pmatrix},\quad x_ry_r=0.\] It is not possible that $x_r=y_r=0$, because then $D_r=A_r$ and Lemma \ref{tsq} would give a contradiction. For $1\le i\le r-1$, by solving the equation $D_iD_r-D_rD_i=0$ we obtain $x_iy_r=0$ and $x_ry_i=0$. It follows that either $x_i=0$ for all $i=1,\dots,r$, or $y_i=0$ for all $i=1,\dots,r$. We are going to show that it is possible to change the basis so that $\alpha_i=-1$ for $i=1,\dots,r-1$ and $x_r+y_r=-2$. Suppose that the old basis is $\beta_1=(v_1,w_1,\dots,v_r,w_r,v_{r+1})$. We consider two cases. {\bf Case 1:} $x_r=0$. Then $y_r\ne 0$ and the new basis is: \begin{align} &v'_{i}=(-1)^{r-i}\alpha_i\cdots\alpha_{r-1}v_i,\ w'_{i}=(-1)^{r-i}\alpha_i\cdots\alpha_{r-1}w_i,\ i=1,\dots,r-1,\\ &v'_{r}=v_r,\ w'_{r}=w_r,\quad v'_{r+1}=-\frac{y_r}{2}v_{r+1}. \end{align} In the new basis we have: \[ D_r=\Psi_1(t_{\delta_{2r+1}}),\quad D_i=C_i+x_i'\left(E_{2r+1,2i}-E_{2r+1,2i+2}\right),\] for $i=1,\dots,r-1$. {\bf Case 2:} $y_r=0$. Then $x_r\ne 0$ and the new basis is: \begin{align} &v'_{i}=(-1)^{r-i+1}\alpha_i\cdots\alpha_{r-1}\frac{x_r}{2}v_i,\ w'_{i}=(-1)^{r-i+1}\alpha_i\cdots\alpha_{r-1}\frac{x_r}{2}w_i,\\ &i=1,\dots,r-1,\, v'_{r}=-\frac{x_r}{2}v_r,\ w'_{r}=-\frac{x_r}{2}w_r,\quad v'_{r+1}=v_{r+1}. \end{align} In the new basis we have: \[ D_r=\Psi_2(t_{\delta_{2r+1}}),\quad D_i=C_i+x_i'\left(E_{2i-1,2r+1}-E_{2i+1,2r+1}\right),\] for $i=1,\dots,r-1$. Since $U_{2r+1}$ commutes with $A_i$ and $B_i$ for $1\le i\le r-1$, thus, by Lemma \ref{diag}, \[U_{2r+1}=\mathrm{diag}\left(\lambda_1I_2,\lambda_2I_2,\dots,\lambda_{r-1}I_2,X\right),\] for some $3\times 3$ matrix $X$. The relations $A_rU_{2r+1}=U_{2r+1}A_r$ (R8) and $D_rU_{2r+1}D_r=U_{2r+1}$ (R12) imply that $X$ has the form \[X=\begin{pmatrix}\lambda_r&\alpha&\lambda_r\\0&\lambda_r&0\\0&\beta&-\lambda_r\end{pmatrix} \quad\mathrm{or}\quad X=\begin{pmatrix}\lambda_r&\alpha&\beta\\0&\lambda_r&0\\0&\lambda_r&-\lambda_r\end{pmatrix}\] respectively in case 1 and case 2. For $1\le i\le r-1$, by the relation (R6) we have $D_iU_{2r+1}-U_{2r+1}D_i=0$. By solving this equation we obtain $\lambda_i=\lambda_{i+1}$ and $x'_i=0$, hence $D_i=C_i$. We also see that $U_{2r+1}$ has two eigenvalues $\lambda_r$,$-\lambda_r$ with $\#\lambda_r=2r$. Since $U_{2r+1}$ is conjugate to $U_{2r+1}^{-1}$ we have $\lambda_r\in\{-1,1\}$ and by multiplying $f$ by $(-1)^\mathrm{ab}$ if necessary, we may assume $\lambda_r=1$. By the relation (R11) we have \begin{align} &U_{2r}=(B_rC_r)^{-1}U_{2r+1}^{-1}(B_rC_r),\\ &U_{2r-1}=(B_rC_rC_{r-1}B_r)^{-1}U_{2r+1}(B_rC_rC_{r-1}B_r). \end{align} Similarly as in the proof for odd $g$, by solving $U_{2r+1}U_{2r-1}-U_{2r-1}U_{2r+1}=0$ we obtain $\beta=-2\alpha$, and then by solving $U_{2r+1}U_{2r}U_{2r+1}-U_{2r}U_{2r+1}U_{2r}=0$ we obtain $\alpha=-1$ in the case 1, or $\alpha=1$ in the case 2. Hence $U_{2r+1}=\Psi_1(u_{2r+1})$ in the case 1, or $U_{2r+1}=\Psi_2(u_{2r+1})$ in the case 2. By Theorem \ref{gener}, $f$ is equal to $\Psi_1$ in the case 1, and equal to $\Psi_2$ in the case 2, on generators of $\mathcal{M}(N)$. \end{proof} \section{Homomorphisms from $\mathcal{M}(N_{8})$ to $\mathrm{GL}(7,\mathbb{C})$} The aim of this section is to prove Theorem \ref{MN8toGL7}. First we have to define the epimorphism $\epsilon\colon\mathcal{M}(N_{2r+2})\to\mathrm{Sp}(2r,\mathbb{Z}_2)$. Fix $r\ge 1$ and set $V=H_1(N_{2r+2},\mathbb{Z}_2)$. $V$ is a vector space over $\mathbb{Z}_2$ of dimension $2r+2$ with basis $\overline{x_i}=[\xi_i]_2$ for $1\le i\le 2r+2$, where $[\xi_i]_2$ denotes the mod 2 homology class of the curve $\xi_i$. The mod 2 intersection pairing is the symmetric bilinear form on $V$ satisfying $\lr{\overline{x_i},\overline{x_j}}_2=\delta_{ij}$. We define another basis for $V$. For $1\le i\le r$ we set \begin{align} &v_i=[\varepsilon_i]_2=\overline{x_1}+\cdots+\overline{x_{2i}}, \quad w_i=[\delta_{2i}]_2=\overline{x_{2i}}+\overline{x_{2i+1}},\\ &c=\overline{x_{2r+2}},\quad d=\overline{x_1}+\cdots+\overline{x_{2r+2}}. \end{align} Let $\mathrm{Iso}(V)$ denote the group of automorphisms of $V$ preserving $\lr{\cdot,\cdot}_2$. \begin{lemma}\label{semidir} The group $\mathrm{Iso}(V)$ is isomorphic to a semi-direct product $\mathrm{Sp}(2r,\mathbb{Z}_2)\ltimes\mathbb{Z}_2^{2r+1}$. \end{lemma} \begin{proof} It is easy to check that $d$ is the unique vector of $V$ satisfying $\lr{x,d}_2=\lr{x,x}_2$ for all $x\in V$, which implies that $d$ is fixed by all elements of $\mathrm{Iso}(V)$. Let $W=\mathrm{span}\{v_i,w_i\,\|\,i=1,\dots,r\}$ and observe that the restriction of $\lr{\cdot,\cdot}_2$ to $W$ is nondegenerate and $\lr{x,x}_2=0$ for $x\in W$, hence it is a symplectic form on $W$. For $R\in\mathrm{Sp}(W)$ we define $A_R\in\mathrm{Iso}(V)$ as \[A_R(d)=d,\quad A_R(c)=c,\quad A_R(x)=R(x)\quad\textrm{for\ }x\in W.\] It is easy to check that $W=\{x\in V\,\|\,\lr{x,d}_2=\lr{x,c}_2=0\}$. It follows that if $L\in\mathrm{Iso}(V)$ fixes $c$, then since $L(d)=d$, $L$ preserves $W$, and hence $L=A_R$ for some $R\in\mathrm{Sp}(W)$. Thus the mapping $R\mapsto A_R$ defines an isomorphism $\mathrm{Sp}(W)\to\mathrm{Stab}_{\mathrm{Iso}(V)}(c)$. For $x\in\mathbb{Z}_2$ and $z\in W$ we define $B_{x,z}\in\mathrm{Iso}(V)$ as \[B_{x,z}(d)=d,\quad B_{x,z}(c)=c+xd+z,\quad B_{x,z}(w)=w+\lr{w,z}_2d\quad\textrm{for\ }w\in W.\] Let \[N=\{B_{x,z}\,\|\,x\in\mathbb{Z}_2, z\in W\}.\] This is a subgroup of $\mathrm{Iso}(V)$ with the group law \[B_{x_1,z_1}B_{x_2,z_2}=B_{x_1+x_2+\lr{z_1,z_2}_2,z_1+z_2}.\] It follows that $N$ is abelian and $B_{x,z}^2=1$ for all $x, z$. Thus $N$ is isomorphic to $\mathbb{Z}_2^{2r+1}$. Let $L\in\mathrm{Iso}(V)$ be arbitrary. Since $\lr{L(c),d}=\lr{L(c),L(d)}=\lr{c,d}=1$, thus $L(c)=c+xd+z$ for some $x\in\mathbb{Z}_2$, $z\in W$. It follows that $B_{x,z}^{-1}L\in\mathrm{Stab}_{\mathrm{Iso}(V)}(c)$ and hence $L=B_{x,z}A_R$ for some $R\in\mathrm{Sp}(W)$. This decomposition is clearly unique, and since $A_RB_{x,z}A_R^{-1}=B_{x,R(z)},$ thus $N$ is normal in $\mathrm{Iso}(V)$ and $\mathrm{Iso}(V)=N\rtimes\mathrm{Stab}_{\mathrm{Iso}(V)}(c)$. \end{proof} \begin{lemma}\label{epionSp} For $r\ge 2$ there is an epimorphism \[\epsilon\colon\mathcal{M}(N_{2r+2})\to\mathrm{Sp}(2r,\mathbb{Z}_2),\] whose kernel is normally generated by $t_{\delta_{2r+1}}u_{2r+1}$ and $t_{\delta_{2r+1}}t_{\varepsilon_r}^{-1}$. \end{lemma} \begin{proof} Let $\mathcal{M}=\mathcal{M}(N_{2r+2})$. The action of $\mathcal{M}$ on $V=H_1(N_{2r+2},\mathbb{Z}_2)$ induces a homomorphism $\rho\colon\mathcal{M}\to\mathrm{Iso}(V)$, which was proved to be surjective in \cite{GP} and \cite{McCP}, and whose kernel is the normal closure of $t_{\delta_{2r+1}}u_{2r+1}$ by \cite{SzepGD}. By Lemma \ref{semidir}, there exists a normal subgroup $N$ of $\mathrm{Iso}(V)$, such that $\mathrm{Iso}(V)/N$ is isomorphic to $\mathrm{Sp}(2r,\mathbb{Z}_2)$. We define $\epsilon$ to be the composition of $\rho$ with the canonical projection $\mathrm{Iso}(V)\to\mathrm{Iso}(V)/N$. Let $K$ be the normal closure of $t_{\delta_{2r+1}}u_{2r+1}$ and $t_{\delta_{2r+1}}t_{\varepsilon_r}^{-1}$ in $\mathcal{M}$. We claim that $K\subseteq\ker\epsilon$. We have $t_{\delta_{2r+1}}u_{2r+1}\in\ker\rho\subset\ker\epsilon$. For $x\in V$ we have $\rho(t_{\varepsilon_r})(x)=x+\lr{v_r,x}_2v_r$ and $\rho(t_{\delta_{2r+1}})(x)=x+\lr{[\delta_{2r+1}]_2,x}[\delta_{2r+1}]_2$. Since $[\delta_{2r+1}]_2=v_r+d$, it is not difficult to check that $\rho(t_{\delta_{2r+1}})=B_{1,v_r}\circ\rho(t_{\varepsilon_r})$, which gives $\rho(t_{\delta_{2r+1}}t_{\varepsilon_r}^{-1})\in N$ and $t_{\delta_{2r+1}}t_{\varepsilon_r}^{-1}\in\ker\epsilon$. It follows that there is an induced epimorphism \[\epsilon'\colon\mathcal{M}/K\to\mathrm{Iso}(V)/N\cong\mathrm{Sp}(2r,\mathbb{Z}_2).\] To prove that $\epsilon'$ is an isomorphism, it suffices to show $[\mathcal{M}:K]\le\|\mathrm{Sp}(2r,\mathbb{Z}_2)\|$. We are going to prove the last inequality by exhibiting an epimorphism $\mathrm{Sp}(2r,\mathbb{Z}_2)\to\mathcal{M}/K$. Observe that the map $\eta\colon\mathcal{M}(S')\to\mathcal{M}/K$ defined to be the composition of $\iota\colon\mathcal{M}(S')\to\mathcal{M}$ from Corollary \ref{HomSN} with the canonical projection $\pi\colon\mathcal{M}\to\mathcal{M}/K$ is surjective, because $\mathcal{M}$ is generated by twists about curves on $P(S')$ and $t_{\delta_{2r+1}}u_{2r+1}$ by Theorem \ref{gener}. Gluing a disc along the boundary component of $S'$ bounding a pair of pants with $\alpha_r$ and $\gamma_r$ induces an epimomorphism $\mathcal{M}(S')\to\mathcal{M}(S_{r,1})$ whose kernel is normally generated by $t_{\gamma_r}t_{\alpha_r}^{-1}$ (see \cite[Proposition 3.8]{KorkSymp}). Since $\iota(t_{\gamma_r}t_{\alpha_r}^{-1})=t_{\delta_{2r+1}}t_{\varepsilon_r}^{-1}\in K$, it follows that we have an induced epimorphism $\eta'\colon\mathcal{M}(S_{r,1})\to\mathcal{M}/K$. There is an epimorphism $\mathcal{M}(S_{r,1})\to\mathrm{Sp}(2r,\mathbb{Z}_2)$ induced by the action of $\mathcal{M}(S_{r,1})$ on $H_1(S_{r,1},\mathbb{Z}_2)$, whose kernel is normally generated by $t_{\alpha_1}^2$ (see \cite[Theorem 5.7]{BGP}, here we are using the assumption $r\ge 2$). By applying Lemma \ref{tsq} (with $i=r$, $j=2r+1$) to $\pi\colon\mathcal{M}\to\mathcal{M}/K$, we have $\eta'(t_{\alpha_1}^2)=\pi(t^2_{\delta_1})=1$. It follows that there is an induced epimorphism $\eta''\colon\mathrm{Sp}(2r,\mathbb{Z}_2)\to\mathcal{M}/K$. \begin{displaymath} \xymatrix{ \mathcal{M}(S') \ar[r]^\iota \ar[d] & \mathcal{M} \ar[r]^\pi & \mathcal{M}/K\\ \mathcal{M}(S_{r,1}) \ar[d] \ar[urr]^{\eta'} & & \\ \mathrm{Sp}(2r,\mathbb{Z}_2) \ar[uurr]^{\eta''} & & } \end{displaymath} The existence of $\eta''$ proves that $\epsilon'$ is an isomorphism and $K=\ker\epsilon$.\end{proof} \begin{lemma}\label{MN8factSp} Suppose that $f\colon\mathcal{M}(N_8)\to\mathrm{GL}(7,\mathbb{C})$ is a homomorphism, such that $f(t_{\delta_1})$ has order $2$. Then $f$ or $(-1)^\mathrm{ab} f$ factors through the epimorphism $\epsilon\colon\mathcal{M}(N_8)\to\mathrm{Sp}(6,\mathbb{Z}_2)$. \end{lemma} \begin{proof} Let $H$ be the normal closure of $t_{\delta_1}^2$ in $\mathcal{M}=\mathcal{M}(N_8)$ and $G=\mathcal{M}/H$. Since $H\subseteq\ker f$, we have a homomorphism $f'\colon G\to\mathrm{GL}(7,\mathbb{C})$ such that $f=f'\circ\pi$, where $\pi\colon\mathcal{M}\to G$ is the canonical projection. There is a homomorphism $\rho\colon\mathfrak{S}_8\to G$, defined as $\rho(\sigma_i)=\pi(t_{\delta_i})$, where $\sigma_i=(i,i+1)$, for $1\le i\le 7$. Let $\phi\colon\mathfrak{S}_8\to\mathrm{GL}(7,\mathbb{C})$ be the composition $\phi=f'\circ \rho$. If $\phi$ is reducible, then $\mathrm{Im}(\phi)$ is abelian by Lemma \ref{repsym}, $f(t_{\delta_1})=\phi(\sigma_1)=\phi(\sigma_2)=f(t_{\delta_2})$, and $\mathrm{Im}(f)$ is also abelian by Lemma \ref{com_norm}, which implies $f(t_{\delta_1})=1$ by Theorem \ref{abNg}, a contradiction. Hence $\phi$ is irreducible and since $\det f(t_{\delta_1})=1$ (by Theorem \ref{abNg}), $\phi$ is the tensor product of the standard and sign representations (by Lemma \ref{repsym}). For $1\le i\le 7$ set $L_i=f(t_{\delta_i})=\phi(\sigma_i)$. With respect to some basis $(v_1,\dots,v_7)$ we have \begin{align} L_1=\mathrm{diag}\left(A,-I_5\right),\quad L_7=\mathrm{diag}\left(-I_5,B\right),\quad L_i=\mathrm{diag}\left(-I_{i-2},C,-I_{6-i}\right) \end{align} for $2\le i\le 6$, where \[A=\begin{pmatrix}1&-1\\0&-1\end{pmatrix},\quad B=\begin{pmatrix}-1&0\\-1&1\end{pmatrix},\quad C=\begin{pmatrix}-1&0&0\\-1&1&-1\\0&0&-1\end{pmatrix}.\quad \] Let $M$ be the matrix of $f(\varepsilon_3)$. Since $M$ commutes with $L_i$ for $i\ne 6$ (R5), it preserves $E(L_i,1)=\mathrm{span}\{v_i\}$. Hence $M(v_i)=x_iv_i$ for $i\ne 6$ and $M(v_6)=y_1v_1+\cdots+y_7v_7$, for some complex numbers $x_i, y_j$. By solving the equations $ML_i=L_iM$ for $1\le i\le 5$ and $i=7$ we obtain \[x_i=x_1,\ y_i=iy_1\ \textrm{\ for\ }1\le i\le 5,\quad y_6=x_1+6y_1,\ x_7=y_6-2y_7.\] Since $M$ and $L_i$ are conjugate, they have the same eigenvalues, which gives $x_1=-1$ and $y_6=-x_7$. If $y_6=1$, then $y_1=1/3$, $y_7=1$, which contradicts the braid relation $ML_6M=L_6ML_6$ (R5). Hence $y_6=-1$, $y_1=0$, $y_7=-1$, which means $M=L_7$. For $i=1,\dots 7$ let $U_i$ be the matrix of $f(u_i)$. Since $U_7$ commutes with $L_j$ for $1\le j\le 5$ (R6) and with $M=L_7$ (R8), we obtain, as above, that \begin{align} &U_7(v_i)=xv_i\quad\textrm{for\ }1\le i\le 5,\\ &U_7(v_6)=y(v_1+2v_2+3v_3+4v_4+5v_5)+(x+6y)v_6+zv_7\\ &U_7(v_7)=(x+6y-2z)v_7 \end{align} for some complex numbers $x,y,z$. Since $U_7$ is conjugate to its inverse, and $x$ is an eigenvalue of multiplicity at least $5$, thus $x=\pm 1$, and by multiplying $f$ by $(-1)^\mathrm{ab}$ if necessary, we may assume $x=-1$. By (R11) we have $U_5=(L_6L_7L_5L_6)^{-1}U_7(L_6L_7L_5L_6)$ and by solving $U_5U_7=U_7U_5$ we obtain $y=0$. Since $\det U_7=\pm 1$, either $-1-2z=1$ or $-1-2z=-1$. In the latter case we have $U_7=-I$, and since $U_6$ is conjugate to $U_7$, thus $U_6=-I$, and the relation $L_6U_7U_6=U_7U_6L_7$ (R10) gives $L_6=L_7$, a contradiction. Hence $z=-1$ and $U_7=L_7$. We have $M=U_7=L_7$ and since $L_7^2=I$, thus $\{t_{\delta_7}t_{\varepsilon_3}^{-1}, t_{\delta_7}u_7\}\subset\ker f$, which implies, by Lemma \ref{epionSp}, that $f$ factors through $\epsilon$. \end{proof} \begin{proof}[Proof of Theorem \ref{MN8toGL7}] Suppose that $f\colon\mathcal{M}(N_8)\to\mathrm{GL}(7,\mathbb{C})$ is a homomorphism, such that $\mathrm{Im}(f)$ is not abelian. By Lemma \ref{dim_eigen}, $L=f(t_{\delta_1})$. has an eigenvalue $\lambda$ such that $\dim E(L,\lambda)=6$. Since $L$ is conjugate to $L^{-1}$ we have $\lambda^2=1$. Suppose that $\lambda=-1$. Then since $\det L=1$ we have $\#\lambda=6$, and there is another eigenvalue $\mu=1$. It follows that $L$ has order $2$ and the case (2) holds by Lemma \ref{MN8factSp}. If $\lambda=1$ then it must be the unique eigenvalue, and the case (3) holds by Lemma \ref{AB} and the proof of Theorem \ref{MNtoGLg-1} for even $g$. \end{proof} \begin{rem} Suppose that $G$ is a finite quotient of $\mathcal{M}(N_g)$ for $g\ge 7$, $g\ne 8$, and $f\colon G\to\mathrm{GL}(g-1,\mathbb{C})$ is a homomorphism. Then, by Theorem \ref{MNtoGLg-1}, $\mathrm{Im}(f)$ is abelian, and if $G$ is perfect, then $f$ must be trivial. For example, by Lemma \ref{epionSp}, for $r\ge 4$, the only homomorphism from $\mathrm{Sp}(2r,\mathbb{Z}_2)$ to $\mathrm{GL}(2r+1,\mathbb{C})$ is the trivial one. \end{rem} \end{document}	arXiv
# Introduction to Functional Analysis Part III, Autumn 2004 T. W. Körner October 21, 2004 Small print This is just a first draft for the course. The content of the course will be what I say, not what these notes say. Experience shows that skeleton notes (at least when I write them) are very error prone so use these notes with care. I should very much appreciate being told of any corrections or possible improvements and might even part with a small reward to the first finder of particular errors. ## The Rivlin-Shapiro formula ## Some notes on prerequisites Many years ago it was more or less clear what could and what could not be assumed in an introductory functional analysis course. Since then, however, many of the concepts have drifted into courses at lower levels. I shall therefore assume that you know what is a normed space, and what is a a linear map and that you can do the following exercise. Exercise 1. Let $\left(X,\\|\\|_{X}\right)$ and $\left(Y,\\|\\|_{Y}\right)$ be normed spaces. (i) If $T: X \rightarrow Y$ is linear, then $T$ is continuous if and only if there exists a constant $K$ such that $$ \\|T x\\|_{Y} \leq K\\|x\\|_{X} $$ for all $x \in X$. (ii) If $T: X \rightarrow Y$ is linear and $x_{0} \in X$, then $T$ is continuous at $x_{0}$ if and only if there exists a constant $K$ such that $$ \\|T x\\|_{Y} \leq K\\|x\\|_{X} $$ for all $x \in X$. (iii) If we write $\mathcal{L}(X, Y)$ for the space of continuous linear maps from $X$ to $Y$ and write $$ \\|T\\|=\sup \left\{\\|T x\\|_{Y}:\\|x\\|_{X}=1, x \in X\right\} $$ then $(\mathcal{L}(X, Y),\\|\\|)$ is a normed space. I also assume familiarity with the concept of a metric space and a complete metric space. You should be able to do at least parts (i) and (ii) of the following exercise (part (iii) is a little harder). Exercise 2. Let $\left(X,\\|\\|_{X}\right)$ and $\left(Y,\\|\\|_{Y}\right)$ be normed spaces. (i) If $\left(Y,\\|\\|_{Y}\right)$ is complete then $(\mathcal{L}(X, Y),\\|\\|)$ is. (ii) Consider the set $s$ of sequences $x=\left(x_{1}, x_{2}, \ldots\right)$ in which only finitely many of the $x_{j}$ are non-zero. Explain briefly how $s$ may be considered as a vector space. If we write $$ \\|x\\|=\sup _{j}\left\|x_{j}\right\| $$ show that $(s,\\|\\|)$ is a normed vector space which is not complete. (iii) If $\left(X,\\|\\|_{X}\right)$ is complete does it follow that $(\mathcal{L}(X, Y),\\|\\|)$ is? Give a proof or a counter-example. The reader will notice that I have not distinguished between vector spaces over $\mathbb{R}$ and those over $\mathbb{C}$. I shall try to make the distinction when it matters but, if the two cases are treated in the same way, I shall often proceed as above. Although I shall stick with metric spaces as much as possible, there will be points where we shall need the notions of a topological space, a compact topological space and a Hausdorff topological space. I would be happy, if requested, to give a supplementary lecture introducing these notions. (Even where I use them, no great depth of understanding is required.) I shall also use, without proof, the famous Stone-Weierstrass theorem. Theorem 3. (A) Let $X$ be a compact space and $C(X)$ the space of real valued continuous functions on $X$. Suppose $A$ is a subalgebra of $C(X)$ (that is a subspace which is algebraicly closed under multiplication) and (i) $1 \in A$, (ii) Given any two distinct points $x$ and $y$ in $X$ there is an $f \in A$ with $f(x) \neq f(y)$. Then $A$ is uniformly dense in $C(X)$. (B) Let $X$ be a compact space and $C(X)$ the space of complex valued continuous functions on $X$. Suppose $A$ is a subalgebra of $C(X)$ and (i) $1 \in A$, (ii) Given any two distinct points $x$ and $y$ in $X$ there is an $f \in A$ with $f(x) \neq f(y)$. (iii) If $f \in A$ then $f^{} \in A$. Then $A$ is uniformly dense in $C(X)$. The proof will not be examinable, but if you have not met it, you may wish to request a supplementary lecture on the topic. Functional analysis goes hand in hand with measure theory. Towards the end of the course I will need to refer Borel measures on the line. However, I will not use any theorems from measure theory proper and I will make my treatment independent of previous knowledge. Elsewhere I may make a few remarks involving measure theory. These are for interest only and will not be examinable ${ }^{1}$. I intend the course to be fully accessible without measure theory. ${ }^{1}$ In this course, as in other Part III courses you should assume that everything in the lectures and nothing outside them is examinable unless you are explicitly to the contrary. If you are in any doubt, ask the lecturer. ## Baire category If $(X, d)$ is a metric space we say that a set $E$ in $X$ has dense complement ${ }^{2}$ if, given $x \in E$ and $\delta>0$, we can find a $y \notin E$ such that $d(x, y)<\delta$. Exercise 4. Consider the space $M_{n}$ of $n \times n$ complex matrices with an appropriate norm. Show that the set of matrices which do not have $n$ distinct eigenvalues is a closed set with dense complement. Theorem 5 (Baire's theorem). If $(X, d)$ is a complete metric space and $E_{1}, E_{2}, \ldots$ are closed sets with dense complement then $X \neq \bigcup_{j=1}^{\infty} E_{j}$. Exercise 6. (If you are happy with general topology.) Show that a result along the same lines holds true for compact Hausdorff spaces. We call the countable union of closed sets with dense complement a set of first category. The following observations are trivial but useful. Lemma 7. (i) The countable union of first category sets is itself of first category. (ii) If $(X, d)$ is a complete metric space, then Baire's theorem asserts that $X$ is not of first category. Exercise 8. If $(X, d)$ is a complete metric space and $X$ is countable show that there is an $x \in X$ and $a \delta>0$ such that the ball $B(x, \delta)$ with centre $x$ and radius $\delta$ consists of one point. The following exercise is a standard application of Baire's theorem. Exercise 9. Consider the space $C([0,1])$ of continuous functions under the uniform norm \\|\\|. Let $$ \begin{aligned} E_{m}=\{f \in C([0,1]): & \text { there exists an } x \in[0,1] \text { with } \\ & \|f(x+h)-f(x)\| \leq m\|h\| \text { for all } x+h \in[0,1]\} . \end{aligned} $$ (i) Show that $E_{m}$ is closed in $\left(C\left([0,1], \\|_{\infty}\right)\right.$. (ii) If $f \in C([0,1])$ and $\epsilon>0$ explain why we can find an infinitely differentiable function $g$ such that $\\|f-g\\|_{\infty}<\epsilon / 2$. By considering the function $h$ given by $$ h(x)=g(x)+\frac{\epsilon}{2} \sin N x $$ with $N$ large show that $E_{m}$ has dense complement. (iii) Using Baire's theorem show that there exist continuous nowhere differentiable functions. ${ }^{2}$ If the lecturer uses the words 'nowhere dense' correct him for using an old fashioned and confusing terminology Exercise 10. (This is quite long and not very central.) (i) Consider the space $\mathcal{F}$ of non-empty closed sets in $[0,1]$. Show that if we write $$ d_{0}(x, E)=\inf _{e \in E}\|x-e\| $$ when $x \in[0,1]$ and $E \in \mathcal{F}$ and write $$ d(E, F)=\sup _{f \in F} d_{0}(f, E)+\sup _{e \in E} d_{0}(e, F) $$ then $d$ is a metric on $\mathcal{F}$. (ii) Suppose $E_{n}$ is a Cauchy sequence in $(\mathcal{F}, d)$. By considering $$ E=\left\{x: \text { there exist } e_{n} \in E_{n} \text { such that } e_{n} \rightarrow x\right\} $$ or otherwise, show that $E_{n}$ converges. Thus $(\mathcal{F}, d)$ is complete. (iii) Show that the set $$ \mathcal{A}_{n}=\{E \in \mathcal{F}: \text { there exists an } x \in E \text { with }(x-1 / n, x+1 / n) \cap E=\{x\}\} $$ is closed with dense complement in $(\mathcal{F}, d)$. Deduce that the set of elements of $\mathcal{F}$ with isolated points is of first category. (A set $E$ has an isolated point $e$ if we can find $a \delta>0$ such that $(e-\delta, e+\delta) \cap E=\{e\}$.) (iv) Let $I=[r / n,(r+1) / n]$ with $0 \leq r \leq n-1$ and $r$ and $n$ integers. Show that the set $$ \mathcal{B}_{r, n}=\{E \in \mathcal{F}: E \supseteq I\} $$ is closed with dense complement in $(\mathcal{F}, d)$. Deduce that the set of elements of $\mathcal{F}$ containing an open interval is of first category. (v) Deduce the existence of non-empty closed sets which have no isolated points and contain no intervals. ## Non-existence of functions of several vari- ables This course is very much a penny plain rather than tuppence coloured ${ }^{3}$. One exception is the theorem proved in this section. Theorem 11. Let $\lambda$ be irrational We can find increasing continuous functions $\phi_{j}:[0,1] \rightarrow \mathbb{R}[1 \leq j \leq 5]$ with the following property. Given any ${ }^{3}$ And thus suitable for those 'who want from books plain cooking made still plainer by plain cooks'. continuous function $f:[0,1]^{2} \rightarrow \mathbb{R}$ we can find a function $g: \mathbb{R} \rightarrow \mathbb{R}$ such that $$ f(x, y)=\sum_{j=1}^{5} g\left(\phi_{j}(x)+\lambda \phi_{j}(y)\right) $$ The main point of Theorem 11 may be expressed as follows. Theorem 12. Any continuous function of two variables can be written in terms of continuous functions of one variable and addition. That is, there are no true functions of two variables! (We shall explain why this statement is slightly less shocking than it seems at the end of this section.) For the moment we merely observe that the result is due in successively more exact forms to Kolmogorov, Arnol'd and a succession of mathematicians ending with Kahane whose proof we use here. It is, of course, much easier to prove a specific result like Theorem 11 than one like Theorem 12. Our first step is to observe that Theorem 11 follows from the apparently simpler result that follows. Lemma 13. Let $\lambda$ be irrational We can find increasing continuous functions $\phi_{j}:[0,1] \rightarrow \mathbb{R}[1 \leq j \leq 5]$ with the following property. Given any continuous function $F:[0,1]^{2} \rightarrow \mathbb{R}$ we can find a function $G: \mathbb{R} \rightarrow \mathbb{R}$ such that $\\|G\\|_{\infty} \leq\\|F\\|_{\infty}$ and $$ \sup _{(x, y) \in[0,1]^{2}}\left\|F(x, y)-\sum_{j=1}^{5} G\left(\phi_{j}(x)+\lambda \phi_{j}(y)\right)\right\| \leq \frac{999}{1000}\\|F\\|_{\infty} . $$ Next we make the following observation. Lemma 14. We can find a sequence of functions $f_{n}:[0,1]^{2} \rightarrow \mathbb{R}$ which are uniformly dense in $C([0,1])^{2}$. This enables us to obtain Lemma 13 from a much more specific result. Lemma 15. Let $\lambda$ be irrational and let the $f_{n}$ be as in Lemma 14. We can find increasing continuous functions $\phi_{j}:[0,1] \rightarrow \mathbb{R}[1 \leq j \leq 5]$ with the following property. We can find functions $g_{n}: \mathbb{R} \rightarrow \mathbb{R}$ such that $\left\\|g_{n}\right\\|_{\infty} \leq$ $\left\\|f_{n}\right\\|_{\infty}$ and $$ \sup _{(x, y) \in[0,1]^{2}}\left\|f_{n}(x, y)-\sum_{j=1}^{5} g_{n}\left(\phi_{j}(x)+\lambda \phi_{j}(y)\right)\right\| \leq \frac{998}{1000}\left\\|f_{n}\right\\|_{\infty} . $$ Now that we have reduced the matter to satisfying a countable set of conditions, we can use a Baire category argument. We need to use the correct metric space. Lemma 16. The space $Y$ of continuous functions $\phi:[0,1] \rightarrow \mathbb{R}^{5}$ with norm $$ \\|\phi\\|_{\infty}=\sup _{t \in[0,1]}\\|\phi(t)\\| $$ is complete. The subset $X$ of $Y$ consisting of those $\phi$ such that each $\phi_{j}$ is increasing is a closed subset of $Y$. Thus if $d$ is the metric on $X$ obtained by restricting the metric on $Y$ derived from \\|\\|$_{\infty}$ we have $(X, d)$ complete. Exercise 17. Prove Lemma 16 Lemma 18. Let $f:[0,1]^{2} \rightarrow \mathbb{R}$ be continuous and let $\lambda$ be irrational. Consider the set $E$ of $\phi \in X$ such that there exists a $g: \mathbb{R} \rightarrow \mathbb{R}$ such that $\\|g\\|_{\infty} \leq\\|f\\|_{\infty}$ $$ \sup _{(x, y) \in[0,1]^{2}}\left\|f(x, y)-\sum_{j=1}^{5} g\left(\phi_{j}(x)+\lambda \phi_{j}(y)\right)\right\|<\frac{998}{1000}\\|f\\|_{\infty} . $$ The $X \backslash E$ is a closed set with dense complement in $(X, d)$. (Notice that it is important to take ' $<$ ' rather than ' $\leq$ ' in the displayed formula of Lemma 18.) Lemma 18 is the heart of the proof and once it is proved we can easily retrace our steps and obtain Theorem 11. By using appropriate notions of information Vitushkin was able to show that we can not replace continuous by continuously differentiable in Theorem 12. Thus Theorem 11 is an 'exotic' rather than a 'central' result. ## The principle of uniform boundedness We start with a result which is sometimes useful by itself but which, for us, is merely a stepping stone to Theorem 22 . Lemma 19 (Principle of uniform boundedness). Suppose that $(X, d)$ is a complete metric space and we have a collection $\mathcal{F}$ of continuous functions $f: X \rightarrow \mathbb{R}$ which are pointwise bounded, that is, given any $x \in X$ we can find $a K(x)>0$ such that $$ \|f(x)\| \leq K(x) \text { for all } f \in \mathcal{F} . $$ Then we can find a ball $B\left(x_{0}, \delta\right)$ and a $K$ such that $$ \|f(x)\| \leq K \text { for all } f \in \mathcal{F} \text { and all } x \in B\left(x_{0}, \delta\right) . $$ Exercise 20. (i) Suppose that $(X, d)$ is a complete metric space and we have a sequence of continuous functions $f_{n}: X \rightarrow \mathbb{R}$ and a function $f: X \rightarrow \mathbb{R}$ such that $f_{n}$ converges pointwise that is $$ f_{n}(x) \rightarrow f(x) \text { for all } f \in \mathcal{F} . $$ Then we can find a ball $B\left(x_{0}, \delta\right)$ and a $K$ such that $$ \left\|f_{n}(x)\right\| \leq K \text { for all } n \text { and all } x \in B\left(x_{0}, \delta\right) . $$ (ii) (This is elementary but acts as a hint for (iii).) Suppose $y \in[0,1]$. Show that we can find a sequence of continuous function $f_{n}:[0,1] \rightarrow \mathbb{R}$ such that $1 \geq f_{n}(x) \geq 0$ for all $x$ and $n, f_{n}$ converges pointwise to 0 everywhere, $f_{n}$ converges uniformly on $[0,1] \backslash(y-\delta, y+\delta)$ and fails to converge uniformly on $[0,1] \cap(y-\delta, y+\delta)$ for all $\delta>0$. (iii) State with reasons whether the following statement is true or false. Under the conditions of (i) we can obtain the stronger conclusion that we can find a ball $B\left(x_{0}, \delta\right)$ such that $$ f_{n}(x) \rightarrow f(x) \text { uniformly on } B\left(x_{0}, \delta\right) . $$ Exercise 21. Suppose that $(X, d)$ is a complete metric space and $Y$ is a subset of $X$ which is of first category in $X$. Suppose further that we have a collection $\mathcal{F}$ of continuous functions $f: X \rightarrow \mathbb{R}$ which are pointwise bounded on $X \backslash Y$, that is, given any $x \notin Y$, we can find a $K(x)>0$ such that $$ \|f(x)\| \leq K(x) \text { for all } f \in \mathcal{F} \text {. } $$ Show that we can find a ball $B\left(x_{0}, \delta\right)$ and a $K$ such that $$ \|f(x)\| \leq K \text { for all } f \in \mathcal{F} \text { and all } x \in B\left(x_{0}, \delta\right) \text {. } $$ We now use the principle of uniform boundedness to prove the BanachSteinhaus theorem ${ }^{4}$. Theorem 22. (Banach-Steinhauss theorem) Let $\left(U,\\|\\|_{U}\right)$ and $\left(V,\\|\\|_{V}\right)$ be normed spaces and suppose \\|\\|$_{U}$ is complete. If we have a collection $\mathcal{F}$ of continuous linear maps from $U$ to $V$ which are pointwise bounded then we can find a $K$ such that $\\|T\\| \leq K$ for all $T \in \mathcal{F}$. Here is a typical use of the Banach-Steinhauss theorem. ${ }^{4}$ You should be warned that a lot of people, including the present writer, tend to confuse the names of these two theorems. My research supervisor took the simpler course of referring to all the theorems of functional analysis as 'Banach's theorem'. Theorem 23. There exists a continuous $2 \pi$ periodic function $f: \mathbb{R} \rightarrow \mathbb{R}$ whose Fourier series fails to converge at a given point. The next exercise contains results that most of you will have already met. Exercise 24. (i) Show that the set $l^{\infty}$ of bounded sequences over $\mathbb{F}$ (with $\mathbb{F}=\mathbb{R}$ or $\mathbf{F}=\mathbb{C}$ ) $$ \mathbf{a}=\left(a_{1}, a_{2}, \ldots\right) $$ can be made into vector space in a natural manner. Show that $\\|\mathbf{a}\\|_{\infty}=$ $\sup _{j \geq 1}\left\|a_{j}\right\|$ defines a complete norm on $l^{\infty}$. (ii) Show that s, the set of convergent sequences and $s_{0}$ the set of sequences convergent to 0 are both closed subspaces of $\left(l^{\infty},\\|\\|_{\infty}\right)$. (iii) Show that the set $l^{1}$ of sequences $$ \mathbf{a}=\left(a_{1}, a_{2}, \ldots\right) \text { such that } \sum_{j=1}^{\infty}\left\|a_{j}\right\| \text { converges } $$ can be made into vector space in a natural manner. Show that $\\|\mathbf{a}\\|_{1}=$ $\sum_{j=1}^{\infty}\left\|a_{j}\right\|$ defines a complete norm on $l^{1}$. (iv) Show that, if $\mathbf{a} \in l^{1}$, then $$ T_{\mathbf{a}}(\mathbf{b})=\sum_{j=1}^{\infty} a_{j} b_{j} $$ defines a continuous linear map from $l^{\infty}$ to $\mathbb{F}$ and that $\left\\|T_{\mathbf{a}}\right\\|=\\|\mathbf{a}\\|_{1}$. Here is another use of the Banach-Steinhaus theorem. Lemma 25. Let $a_{i j} \in \mathbb{R}[i, j \geq 1]$. We say that the $a_{i j}$ constitute a summation method if whenever $c_{j} \rightarrow c$ we have $\sum_{j=1}^{\infty} a_{i j} c_{j}$ convergent for each $i$ and $$ \sum_{j=1}^{\infty} a_{i j} c_{j} \rightarrow c $$ as $i \rightarrow \infty$. The following conditions are necessary and sufficient for the $a_{i j}$ to constitute a summation method:- (i) There exists a $K$ such that $$ \begin{aligned} & \qquad \sum_{j=1}^{\infty}\left\|a_{i j}\right\| \leq K \text { for all } i . \\ & \text { (ii) } \sum_{j=1}^{\infty} a_{i j} \rightarrow 1 \text { as } i \rightarrow \infty \\ & \text { (iii) } a_{i j} \rightarrow 0 \text { as } i \rightarrow \infty \text { for each } j . \end{aligned} $$ Exercise 26. Cesàro's summation method takes a sequence $c_{0}, c_{1}, c_{2}, \ldots$ and replaces it with a new sequence whose $n$th term $$ b_{n}=\frac{c_{1}+c_{2}+\cdots+c_{n}}{n} $$ is the average of the first $n$ terms of the old sequence. (i) By rewriting the statement above along the lines of Lemma 25 show that if the old sequence converges to $c$ so does the new one. (ii) Examine what happens when $c_{j}=(-1)^{j}$. Examine what happens if $c_{j}=(-1)^{k}$ when $2^{k} \leq j<2^{k+1}$. (iii) Show that, in the notation of Lemma 25, taking $a_{n, 2 n}=1, a_{n, m}=0$, otherwise, gives a summation method. Show that taking $a_{n, 2 n+1}=1, a_{n, m}=$ 0 , otherwise, also gives a summation method. Show that the two methods disagree when presented with the sequence $1,-1,1,-1, \ldots$ Another important consequence of the Baire category theorem is the open mapping theorem. (Recall that a complete normed space is called a Banach space.) Theorem 27 (Open mapping theorem). Let $E$ and $F$ be Banach spaces and $T: E \rightarrow F$ be a continuous linear surjection. Then $T$ is an open map (that is to say, if $U$ is open in $E$ we have $T U$ open in $F$.) This has an immediate corollary. Theorem 28 (Inverse mapping theorem). Let $E$ and $F$ be Banach spaces and let $T: E \rightarrow F$ be a continuous linear bijection. Then $T^{-1}$ is continuous. The next exercise is simple, and if you can not do it this reveals a gap in your knowledge (which can be remedied by asking the lecturer) rather that in intelligence. Exercise 29. Let $(X, d)$ and $(Y, \rho)$ be metric spaces with associated topologies $\tau$ and $\sigma$. Then the product topology induced on $X \times Y$ by $\tau$ and $\sigma$ is the same as the topology given by the metric $$ \triangle\left(\left(x_{1}, y_{1}\right),\left(x_{2}, y_{2}\right)\right)=d\left(x_{1}, x_{2}\right)+\rho\left(y_{1}, y_{2}\right) . $$ The inverse mapping theorem has the following useful consequence. Theorem 30 (Closed graph theorem). Let $E$ and $F$ be Banach spaces and let $T: E \rightarrow F$ be linear. Then $T$ is continuous if and only the graph $$ \{(x, T x): x \in E\} $$ is closed in $E \times F$ with the product topology. ## Zorn's lemma and Tychonov's theorem Let $A$ be a non empty set and, for each $\alpha \in A$, let $X_{\alpha}$ be a non-empty set. Is $\prod_{\alpha \in A} X_{\alpha}$ non-empty (or, equivalently, does there exist a function $f: A \rightarrow \bigcup_{\alpha \in A} X_{\alpha}$ with $\left.f(\alpha) \in X_{\alpha}\right)$ ? It is known that the standard axioms of set theory do not suffice to answer this question in general. (In particular cases they do suffice. If $X_{\alpha}=A$ for all $\alpha \in A$ then $f(\alpha)=\alpha$ will do.) Specifically, if there exists any model for standard set theory, then there exist models for set theory obeying the standard axioms in which the answer to our question is always yes (such systems are said to obey the axiom of choice) and there exist models in which the answer is sometimes no. Most mathematicians are happy to add the axiom of choice to the standard axioms and this is what we shall do. Note that if we prove something using the standard axioms and the axiom of choice then we will be unable to find a counter-example using only the standard axioms. Note also that, when dealing with specific systems we may be able to prove the result for that system without using the axiom of choice. The axiom of choice is not very easy to use in the form that we have stated it and it is usually more convenient to use an equivalent formulation called Zorn's lemma. Definition 31. Suppose $X$ is a non-empty set. We say that $\succeq$ is partial order on $X$, that is to say, that $\succeq$ is a relation on $X$ with (i) $x \succeq y, y \succeq z$ implies $x \succeq z$, (ii) $x \succeq y$ and $y \succeq x$ implies $x=y$, (iii) $x \succeq x$ for all $x, y, z$. We say that a subset $C$ of $X$ is a chain if, for every $x, y \in C$ at least one of the statements $x \succeq y, y \succeq x$ is true. If $Y$ is a non-empty subset of $X$ we say that $z \in X$ is an upper bound for $Y$ if $z \succeq y$ for all $y \in Y$. We say that $m$ is a maximal element for $(X, \succeq)$ if $x \succeq m$ implies $x=m$. You must be able to do the following exercise. Exercise 32. (i) Give an example of a partially ordered set which is not a chain. (ii) Give an example of a partially ordered set and a chain $C$ such that (a) the chain has an upper bound lying in $C$, (b) the chain has an upper bound but no upper bound within $C$, (c) the chain has no upper bound. (iii) If a chain $C$ has an upper bound lying in $C$, show that it is unique. Give an example to show that, even in this case $C$ may have infinitely many upper bounds (not lying in $C$ ). (iv) Give examples of partially ordered sets which have (a) no maximal elements, (b) exactly one maximal element, (b) infinitely many maximal elements. (v) how should a minimal element be defined? Give examples of partially ordered sets which have (a) no maximal or minimal elements, (b) exactly one maximal element and no minimal element, (c) infinitely many maximal elements and infinitely many minimal elements. Axiom 33 (Zorn's lemma). Let $(X, \succeq)$ be a partially ordered set. If every chain in $X$ has an upper bound then $X$ contains a maximal element. Zorn's lemma is associated with a proof routine which we illustrate in Lemmas 34 and 36 Lemma 34. Zorn's lemma implies the axiom of choice. The converse result is less important to us but we prove it for completeness. Lemma 35. The axiom of choice implies Zorn's lemma. Proof. (Since the proof we use is non-standard, I give it in detail.) Let $X$ be a non-empty set with a partial order $\succeq$ having no maximal elements. We show that the assumption that every chain has a upper bound leads to a contradiction. We write $x \succ y$ if $x \succeq y$ and $x \neq y$. If $C$ is a chain we write $$ C_{x}=\{c \in C: x \succ c\} . $$ Observe that, if $C$ is a chain in $X$, we can find an $x \in X$ such that $x \succ c$ for all $c \in C$. (By assumption, $C$ has an upper bound, $x^{\prime}$, say. Since $X$ has no maximal elements, we can find an $x \in X$ such that $x \succ x^{\prime}$.) We shall take $\emptyset$ to be a well ordered chain. We shall look at well ordered chains, that is to say, chains for which every non-empty subset has a minimum. (Formally, if $S \subseteq C$ is non-empty we can find an $s_{0} \in C$ such that $s \succeq s_{0}$ for all $s \in S$. We write $\min C=s_{0}$.) By the previous paragraph $$ A_{C}=\{x: x \succ c \text { for all } c \in C\} \neq \emptyset . $$ Thus, if we write $\mathcal{W}$ for the set of all well ordered chains, the axiom of choice, tells us that there is a function $\kappa: \mathcal{W} \rightarrow X$ such that $\kappa(C) \succ c$ for all $c \in C$. We now consider 'special chains' defined to be well ordered chains $C$ such that $$ \kappa\left(C_{x}\right)=x \text { for all } x \in C $$ (Note that 'well ordering' is an important general idea, but 'special chains' are an ad hoc notion for this particular proof. Note also that if $C$ is a special chain and $x \in C$ then $C_{x}$ is a special chain.) The key point is that, if $K$ and $L$ are special chains, then either $K=L$ or $K=L_{x}$ for some $x \in L$ or $L=K_{x}$ for some $x \in K$. Subproof If $K=L$, we are done. If not, at least one of $K \backslash L$ and $L \backslash K$ is non-empty. Suppose, without loss of generality, that $K \backslash L \neq \emptyset$. Since $K$ is well ordered, $x=\min K \backslash L$ exists. We observe that $K_{x} \subseteq L$. If $K_{x}=L$, we are done. We show that the remaining possibility $K_{x} \neq L$ leads to contradiction. In this case, $L \backslash K_{x} \neq \emptyset$ so $y=\min L \backslash K_{x}$ exists. By definition of $y$ and the fact that $K_{x} \subseteq L$, we have $K_{y}=L_{y}$. But $K$ and $L$ are special chains so $y=\kappa\left(K_{y}\right) \in K$ contradicting the definition of $y$. End subproof We now take $S$ to be the union of all special chains. Using the key observation, it is routine to see that: (i) $S$ is a chain. (If $a, b \in S$, then $a \in L$ and $b \in K$ for some special chains. By our key observation, either $L \supseteq K$ of $K \supseteq L$. Without loss of generality, $K \supseteq L$ so $a, b \in K$ and $a \succeq b$ or $b \succeq a$.) (ii) If $a \in S$, then $S_{a}$ is a special chain. (We must have $a \in K$ for some special chain $K$. Since $K \subseteq S$, we have $K_{a} \subseteq S_{a}$. On the other hand, if $b \in S_{a}$ then $b \in L$ for some special chain $L$ and each of the three possible relationships given in our key observation imply $b \in K_{a}$. Thus $S_{a} \subseteq K_{a}$, so $S_{a}=K_{a}$ and $S_{a}$ is a special chain.) (iii) $S$ is well ordered. (If $E$ is a non empty subset of $S$, pick an $x \in E$. If $S_{x} \cap E=\emptyset$, then $x$ is a minimum for $E$. If not, then $S_{x} \cap E$ is a non-empty subset of the special, so well ordered chain $S_{x}$, so $\min S_{x} \cap E$ exists and is a minimum for $E$.) (iv) $S$ is a special chain. (If $x \in S$, we can find a special chain $K$ such that $x \in K$. Let $y=\kappa(K)$. Then $L=K \cup\{y\}$ is a special chain. As in (ii), $S_{y}=L_{y}$, so $S_{x}=L_{x}$ and $\kappa\left(S_{x}\right)=\kappa\left(L_{x}\right)=x$.) We can now swiftly obtain a contradiction. Since $S$ is well ordered $\kappa(S)$ exists and does not lie in $S$. But $S$ is special, so $S \cup \kappa(S)$ is, so $S \cup \kappa(S) \subseteq S$, so $\kappa(s)$ lies in $S$. The required result follows by reductio ad absurdum ${ }^{5}$. Lemma 36 (Hammel basis theorem). (i) Every vector space has a basis. (ii) If $U$ is an infinite dimensional normed space over $\mathbb{F}$ (with $\mathbb{F}=\mathbb{R}$ or $\mathbb{F}=\mathbb{C})$ then we can find a discontinuous linear map $T: U \rightarrow \mathbb{F}$. ${ }^{5}$ To the best of my knowledge, this particular proof is due to Jonathon Letwin (American Mathematical Monthly, Volume 98, 1991, pp. 353-4). If you know about transfinite induction, there are more direct proofs. Exercise 37. (i) Show that if $f: \mathbb{R} \rightarrow \mathbb{R}$ is continuous and satisfies the equation $$ f(x+y)=f(x)+f(y) $$ for all $x, y \in \mathbb{R}$ then there exists a $c$ such that $f(x)=c x$ for all $x \in \mathbb{R}$. (ii) Show that there exists a discontinuous function $f: \mathbb{R} \rightarrow \mathbb{R}$ and satisfying the equation $$ f(x+y)=f(x)+f(y) $$ for all $x, y \in \mathbb{R}$. [Hint. Consider $\mathbb{R}$ as a vector space over $\mathbb{Q}$.] The rest of this section is devoted to a proof of Tychonov's theorem. Theorem 38 (Tychonov). The product of compact spaces is itself compact. We follow the presentation in [1]. (The method of proof is due to Bourbaki.) The following result should be familiar to almost all of my readers. Lemma 39 (Finite intersection property). (i) If a topological space is compact then, whenever a non-empty collection of closed sets $\mathcal{F}$ has the property that $\bigcap_{j=1}^{n} F_{j} \neq \emptyset$, for any $F_{1}, F_{2}, \ldots, F_{n} \in \mathcal{F}$ it follows that $\bigcap_{F \in \mathcal{F}} F \neq \emptyset$. (ii) A topological space is compact if whenever a non-empty collection of sets $\mathcal{A}$ has the property that $\bigcap_{j=1}^{n} A_{j} \neq \emptyset$ for any $A_{1}, A_{2}, \ldots, A_{n} \in \mathcal{A}$ it follows that $\bigcap_{A \in \mathcal{A}} \bar{A} \neq \emptyset$. Definition 40. A system $\mathcal{F}$ of subsets of a given set $S$ is said to be of finite character if whenever every finite subset of a set $A \subseteq S$ belongs to $\mathcal{F}$ it follows that $A \in \mathcal{F}$. Lemma 41 (Tukey's lemma). If a system $\mathcal{F}$ of subsets of a given set $S$ has finite character and $F \in \mathcal{F}$ then $\mathcal{F}$ has a maximal (with respect to inclusion) element containing $F$. We now prove Tychonov's theorem. The reason why Tychonov's theorem demands the axiom of choice is made clear by the final result of this section. Lemma 42. Tychonov's theorem implies the axiom of choice. ## The Hahn-Banach theorem A good example of the use of Zorn's lemma occurs when we ask if given a Banach space $(U,\\|\\|)$ (over $\mathbb{C}$, say) there exist any non-trivial continuous linear maps $T: U \rightarrow \mathbb{C}$. For any space that we can think of, the answer is obviously yes, but to show that the result is always yes we need Zorn's lemma ${ }^{6}$. Our proof uses the theorem of Hahn-Banach. One form of this theorem is the following. Theorem 43. (Hahn-Banach) Let $U$ be a real vector space. Suppose $p$ : $U \rightarrow \mathbb{R}$ is such that $$ p(u+v) \leq p(u)+p(v) \text { and } p(a u)=a p(u) $$ for all $u, v \in U$ and all real and positive a. If $E$ is a subspace of $U$ and there exists a linear map $T: E \rightarrow \mathbb{R}$ with $T x \leq p(x)$ for all $x \in E$ then there exists a linear map $\tilde{T}: U \rightarrow \mathbb{R}$ with $T x \leq p(x)$ for all $x \in U$ and $\tilde{T}(x)=T x$ for all $x \in E$. [Note that we do not assume that the vector space $U$ is normed but we do assume that the vector space is real.] We have the following important corollary Theorem 44. Let $(U,\\|\\|)$ be a real normed vector space. If $E$ is a subspace of $U$ and there exists a continuous linear map $T: E \rightarrow \mathbb{R}$, then there exists a continuous linear map $\tilde{T}: U \rightarrow \mathbb{R}$ with $\\|\tilde{T}\\|=\\|T\\|$. The next result is famous as 'the result that Banach did not prove'. Theorem 45. Let $(U,\\|\\|)$ be a complex normed vector space. If $E$ is a subspace of $U$ and there exists a continuous linear map $T: E \rightarrow \mathbb{C}$ then there exists a continuous linear map $\tilde{T}: U \rightarrow \mathbb{C}$ with $\\|\tilde{T}\\|=\\|T\\|$. We can now answer the question posed in the first sentence of this section. Lemma 46. If $(U,\\|\\|)$ is normed space over the field $\mathbb{F}$ of real or complex numbers and $a \in U$ with $a \neq 0$, then we can find a continuous linear map $T: U \rightarrow \mathbb{F}$ with $T a \neq 0$ Here are a couple of results proved by Banach using his theorem. ${ }^{6}$ In fact the statement is marginally weaker than Zorn's lemma but you need to be logician either to know or care about this. Theorem 47 (Generalised limits). Consider the vector space $l^{\infty}$ of bounded real sequences. There exists a linear map $L: l^{\infty} \rightarrow \mathbb{R}$ such that (i) If $x_{n} \geq 0$ for all $n$ then $L \mathbf{x} \geq 0$. (ii) $L\left(\left(x_{1}, x_{2}, x_{3}, \ldots\right)\right)=L\left(\left(x_{0}, x_{1}, x_{2}, \ldots\right)\right)$. (iii) $L((1,1,1, \ldots))=1$. The theorem is illustrated by the following lemma. Lemma 48. Let $L$ be as in Theorem 47. Then $$ \limsup _{n \rightarrow \infty} x_{n} \geq L(\mathbf{x}) \geq \liminf _{n \rightarrow \infty} x_{n} . $$ In particular, if $x_{n} \rightarrow x$ then $L(\mathbf{x})=x$. Exercise 49. (i) Show that, even though the sequence $x_{n}=(-1)^{n}$ has no limit, $L(\mathbf{x})$ is uniquely defined. (ii) Find, with reasons, a sequence $\mathbf{x} \in l^{\infty}$ for which $L(\mathbf{x})$ is not uniquely defined. Banach used the same idea to prove the following odd result. Lemma 50. Let $\mathbb{T}=\mathbb{R} / \mathbb{Z}$ be the unit circle and let $B(\mathbb{T})$ be the vector space of real valued bounded functions. Then we can find a linear map $I: B(\mathbb{T}) \rightarrow$ $\mathbb{R}$ obeying the following conditions. (i) $I(1)=1$ (ii) If $\geq 0$ if $f$ is positive. (iii) If $f \in B(\mathbb{T}), a \in \mathbb{T}$ and we write $f_{a}(x)=f(x-a)$ then I $f_{a}=I f$. Exercise 51. Show that if $I$ is as in Lemma 50 and $f$ is Riemann integrable then $$ I f=\int_{\mathbb{T}} f(t) d t $$ However, Lemma 50 is put in context by the following. Lemma 52. Let $G$ be the group freely generated by two generators and $B(G)$ be the vector space of real valued bounded functions on $G$. If $f \in B(G)$ let us write $f_{c}(x)=f\left(x c^{-1}\right)$ for all $x, c \in G$. There exists a function $f \in B(G)$ and $c_{1}, c_{2}, c_{3}$ such that $f(x) \geq 0$ for all $x \in G$ and $$ f(x)+f_{c_{1}}(x)-f_{c_{2}}(x)-f_{c_{3}}(x) \leq-1 $$ for all $x \in G$. Exercise 53. If $G$ is as in Lemma 52 then there is no linear map $I: B(G) \rightarrow$ $\mathbb{R}$ obeying the following conditions. (i) $I(1)=1$. (ii) If $\geq 0$ if $f$ is positive. (iii) $I f_{c}=$ If for all $c \in G$. It can be shown that there is a finitely additive, congruence respecting integral for $\mathbb{R}$ and $\mathbb{R}^{2}$ but not $\mathbb{R}^{n}$ for $n \geq 3$. ## Banach algebras Many of the objects studied in analysis turn out to be Banach algebras. Definition 54. An algebra $(B,+, ., \times)$ is a vector space $(B,+, ., \mathbb{C})$ equipped with a multiplication $\times$ such that (i) $x \times(y \times z)=(x \times y) \times z$, (ii) $(x+y) \times z=x \times z+y \times z$ and $z \times(x+y)=z \times x+z \times y$, (iii) $(\lambda x) \times y=x \times(\lambda y)=\lambda(x \times y)$ for all $x, y, z \in B$. [We shall write $x \times y=x y$.] Note that there is no assumption that multiplication is commutative. In principle, we could talk about real Banach algebras (in which $\mathbb{C}$ is replaced by $\mathbb{R}$ ) but, though some elementary results carry over, our treatment will only cover complex Banach algebras.) Definition 55. A Banach algebra $(B,+, ., \times,\\|\\|)$ is an algebra $(B,+, ., \times, \mathbb{C})$ such that $(B,+, ., \mathbb{C},\\|\\|)$ is a Banach space and such that the map $(x, y) \mapsto x y$ is continuous. Note that the two definitions above are not to be memorized; so far as this course is concerned the following definition is all that is required. Definition 56. A Banach algebra $(B,\\|\\|)$ is a Banach space equipped with a continuous multiplication which makes it an algebra. As usual there is a little amount of playing about with the definition. Lemma 57. The following statements about (B, \|\| \\|) a Banach space equipped with a multiplication are equivalent. (i) Multiplication is left and right continuous (that is, the map $x \mapsto x y$ is continuous for all $y$ and the map $y \mapsto x y$ is continuous for all $x$ ). (ii) There exists a $K$ such that $\\|x y\\| \leq K\\|x\\|\\|y\\|$ for all $x$ and $y$. (iii) $(B,\\|\\|)$ is a Banach algebra. Lemma 58. If $(B,\\|\\|)$ is a Banach algebra we can find a norm \\|\\|$_{B}$ on $B$ which is equivalent to \\|\\| (that is, there exists a $C>0$ such that $C^{-1}\\|x\\| \leq$ $\left.\\|x\\|_{B} \leq C\\|x\\|\right)$ such that $$ \\|x y\\|_{B} \leq\\|x\\|_{B}\\|y\\|_{B} $$ for all $x, y \in B$. Unless specifically indicated otherwise you may assume both in the rest of the notes and in the literature generally that the norm on a Banach algebra has been chosen to satisfy $$ \\|x y\\| \leq\\|x\\|\\|y\\| $$ for all $x$ and $y$. Definition 59. We say that a Banach algebra $B$ has a unit e if $x e=e x=x$ for all $x \in B$. The following remarks forms part of the course but are left as an exercise. Exercise 60. (i) If a Banach algebra has a unit that unit is unique. (ii) If $(B,\\|\\|)$ is a Banach algebra with unit $e$ we can find a norm \\|\\|$_{B}$ on $B$ which is equivalent to \\|\| \\| such that $$ \\|x y\\|_{B} \leq\\|x\\|_{B}\\|y\\|_{B} $$ for all $x, y \in B$ and $$ \\|e\\|_{B}=1 $$ Unless specifically indicated otherwise you may assume both in the rest of the notes and in the literature generally that the norm on a Banach algebra with unit $e$ has been chosen to satisfy $$ \\|e\\|=1 $$ for all $x$ and $y$. Example 61. (i) A Banach space (B,\\| \\|) becomes a commutative Banach algebra if we define $x y=0$ for all $x, y \in B$. If $B$ is non-trivial the resulting algebra has no unit. (ii) Consider the Banach space $l^{1}$ of sequences $\mathbf{a}=\left(a_{0}, a_{1}, \ldots\right)$. If we define $\mathbf{a} \mathbf{b}=\mathbf{c}$ with $$ c_{r}=\sum_{k+j=r, k \geq 0, j \geq 0} a_{j} b_{k} $$ then $$ is a well defined multiplication and $l^{1}$ is a Banach algebra with this multiplication. As a Banach algebra, $l^{1}$ is commutative with a unit. (ii) Consider the Banach space $l^{1}$ of sequences $\mathbf{a}=\left(a_{1}, a_{2}, \ldots\right)$. If we define $\mathbf{a} \mathbf{b}=\mathbf{c}$ with $$ c_{r}=\sum_{k+j=r, k \geq 1, j \geq 1} a_{j} b_{k} $$ then $$ is a well defined multiplication and $l^{1}$ is a Banach algebra with this multiplication. As a Banach algebra, $l^{1}$ is commutative but has no unit. (iii) Consider the Banach space $l^{1}$ of $i$ two sided sequences $\mathbf{a}=\left(\ldots, a-2, a_{-1}, a_{0}, a_{1}, \ldots\right)$. If we define $\mathbf{a} \mathbf{b}=\mathbf{c}$ with $$ c_{r}=\sum_{k+j=r} a_{j} b_{k} $$ then $$ is a well defined multiplication and $l^{1}$ is a Banach algebra with this multiplication. As a Banach algebra, $l^{1}$ is commutative and has a unit. Exercise 62. If you know measure theory you ought to work through this exercise. We work in $L^{1}$ the space of Lebesgue integrable functions $f: \mathbb{R} \rightarrow \mathbb{C}$. (i) Use Fubini's theorem to show that, if $f, g \in L^{1}$, then $$ f g(x)=\int_{-\infty}^{\infty} f(x-t) g(t) d x $$ is well defined almost everywhere and that $f * g \in L^{1}$ with $$ \\|f * g\\|_{1} \leq\\|f\\|_{1}\\|g\\|_{1} $$ (ii) Use Fubini's theorem to show that, if $f, g \in L^{1}$ then $$ \widehat{f * g}(\lambda)=\hat{f}(\lambda) \hat{g}(\lambda) $$ for all $\lambda \in \mathbb{R}$. (iii) If $e_{a}(t)=e^{-i a t}$ compute $\hat{e_{a} f}$ for $f \in L^{1}$. Show that if $e \in L^{1}$ is a unit $\hat{e}=1$. (iv) Show that if $f \in L^{1}$ then $\sup _{\lambda \in \mathbb{R}}\|\hat{f}(\lambda)\| \leq \mid f \\|_{1}$. Show that if $f$ is once continuously differentiable with $f, f^{\prime} \in L^{1}$ and $f(t), f^{\prime}(t) \rightarrow 0$ as $\|t\| \rightarrow \infty$ then $\hat{f}(\lambda) \rightarrow 0$ as $\|\lambda\| \rightarrow \infty$. Use a density argument to show that $\hat{g}(\lambda) \rightarrow 0$ as $\|\lambda\| \rightarrow \infty$ whenever $g \in L^{1}$ (this is the Lebesgue-Riemann lemma). (v) Use (iii) and (iv) to show that $\left(L^{1}, \right)$ has no unit. Lemma 63. If $B$ is a Banach algebra without unit we can find $\tilde{B}$ a Banach algebra with a unit $e$ such that (i) $B$ is a sub Banach algebra of $\tilde{B}$, (ii) $B$ is closed in $\tilde{B}$, (iii) $\tilde{B}=\operatorname{span}(B, e)$ in the algebraic sense. Exercise 64. (i) Suppose we apply the construction of Lemma 63 to a Banach algebra $B$ with unit $u$. Is $u$ a unit of the extended algebra $\tilde{B}$ ? Does $\tilde{B}$ have a unit? (ii) (Needs measure theory.) Can you find a natural identification for the unit of $\widetilde{L^{1}}$ where $L^{1}$ is the Banach algebra of Exercise 62. Thus any Banach algebra $B$ without a unit can be studied by 'adjoining a unit and then removing it'. This is our excuse for only studying Banach algebras with a unit. The following result is easy but fundamental. Lemma 65. Let $B$ be a Banach algebra with unit e. (i) If $\\|e-a\\|<1$ then a is invertible (that is has a multiplicative inverse). (ii) If $E$ is the set of invertible elements in $B$ then $E$ is open. Lemma 65 (i) can be improved in a useful way. Theorem 66. (i) If $B$ is a Banach algebra and $b \in B$ then, writing $\rho(b)=$ $\inf _{n}\left\\|b^{n}\right\\|^{1 / n}$ we have $$ \left\\|b^{n}\right\\|^{1 / n} \rightarrow \rho(b) $$ as $n \rightarrow \infty$. (ii) If $B$ is a Banach algebra with unit $e$ and $\rho(e-a)<1$ then $a$ is invertible. We call $\rho(a)$ the spectral radius of $a$. Exercise 67. Consider the space $M_{n}$ of $n \times n$ matrices over $\mathbb{C}$ with the operator norm. (i) Show that $M_{n}$ is a Banach algebra with unit. For which values of $n$ is it commutative? (ii) Give an example of an $A \in M_{2}$ with $A \neq 0$ but $\rho(A)=0$. (iii) If $A$ is diagonalisable show that $$ \rho(A)=\max \{\|\lambda\|: \lambda \text { an eigenvalue of } A\} \text {. } $$ (iv) (Harder and not essential.) Show that the formula of (iii) holds in general. ## Maximal ideals We now embark on a line of reasoning which will eventually lead to a characterization of a large class of commutative Banach algebras. Initially we continue to deal with Banach algebras which are not necessarily commutative. The generality is more apparent than real as the next exercise reveals. Exercise 68. Let $B$ be a Banach algebra with unit e. Let $A$ be the closed Banach algebra generated by e and some $a \in B$. (Formally, $A$ is the smallest closed sub Banach algebra containing e and a.) Then $A$ is commutative. Definition 69. Let $B$ be a Banach space with unit e. If $x \in B$ the resolvent $R(x)$ of $x$ is defined by $$ R(x)=\{\lambda \in \mathbb{C}: x-\lambda e \text { is invertible }\} . $$ Lemma 70. We use the notation of Definition 69. (i) $\mathbb{C} \backslash R(x)$ is bounded. (ii) $R(x)$ is open. (iii) If $\mu \in R(x)$ we can find $a \delta>0$ and $a_{0}, a_{1}, \ldots \in B$ such that $\sum_{j=0}^{\infty} a_{j} z^{j}$ converges for all $\|z\|<\delta$ and $$ (x-\lambda e)^{-1}=\sum_{j=0}^{\infty} a_{j}(\lambda-\mu)^{j} $$ for $\lambda \in \mathbb{C}$ and $\|\lambda-\mu\|<\delta$. (iv) $R(x) \neq \mathbb{C}$. Lemma 70 gives us our first substantial result on the nature of commutative Banach algebras. Theorem 71 (Gelfand-Mazur). Any Banach algebra which is also a field is isomorphic as a Banach algebra to $\mathbb{C}$. ## Analytic functions In order to extract more information on the resolvent we take a detour through a little (easy) integration theory and complex variable theory. Theorem 72. Let $U$ be a Banach space, $[a, b]$ a closed bounded interval in $\mathbb{R}$ Then we can define an integral $\int_{a}^{b} F(t) d t$ for every $F:[a, b] \rightarrow U a$ continuous function having the following properties (here $F, G:[a, b] \rightarrow U$ are continuous and $\lambda, \mu \in \mathbb{C}$ ). (i) $\int_{a}^{b} \lambda F(t)+\mu G(t) d t=\lambda \int_{a}^{b} F(t) d t+\mu \int_{a}^{b} G(t) d t$. (ii) If $a<c<b$ then $$ \int_{a}^{b} F(t) d t=\int_{a}^{c} F(t) d t+\int_{c}^{b} F(t) d t . $$ (iii) $\left\\|\int_{a}^{b} F(t) d t\right\\| \leq \int_{a}^{b}\\|F(t)\\| d t$. (iv) If $T: U \rightarrow \mathbb{C}$ is a continuous linear functional $$ \int_{a}^{b} T(F(t)) d t=T \int_{a}^{b} F(t) d t . $$ Using the integral just defined we can define contour integrals as we did in the complex variable course. Definition 73. If $\gamma:[a, b] \rightarrow \mathbb{C}$ is continuously differentiable with $\gamma(a)=$ $\gamma(b)$ and $F:[a, b] \rightarrow U$ a continuous function we define $$ \int_{\gamma} F(z) d z=\int_{a}^{b} F(\gamma(t)) \gamma^{\prime}(t) d t . $$ (We shall talk about the 'closed contour' $\gamma$.) We can now introduce the notion of an analytic Banach algebra valued function. Definition 74. Let $B$ be a Banach algebra and $\Omega$ a simply connected ${ }^{7}$ open set in $\mathbb{C}$. A function $f: \Omega \rightarrow B$ is said to be analytic on $\Omega$ if there exists an $f^{\prime}: \Omega \rightarrow B$ such that, for all $z \in \Omega$ $$ \left\\|\frac{f(z+h)-f(z)}{h}-f^{\prime}(z)\right\\| \rightarrow 0 $$ as $h \rightarrow 0$ through values of $h$ such that $z+h \in \Omega$. Theorem 75. Let $B$ be a Banach algebra, $\Omega$ an open simply connected set in $\mathbb{C}$, and $\gamma$ a closed contour in $\Omega$. Then $$ \int_{\gamma} f(z) d z=0 $$ ${ }^{7}$ Informally 'with no holes'. We can follow a first undergraduate complex variable course and show. Lemma 76. Let $B$ be a Banach algebra with a unit $e, \Omega$ an open set in $\mathbb{C}$ containing a disc $D\left(z_{0}, R\right)$, and $\gamma$ a contour describing a circle centre $z_{0}$ radius $0<r<R$. If $\left\|z_{0}-z\right\|<r$ then $$ f(z)=\frac{1}{2 \pi i} \int_{\gamma} \frac{f(w)}{z-w} d w . $$ Lemma 77. Let $B$ be a Banach algebra with a unit $e$ and $\Omega$ an open set in $\mathbb{C}$ containing a disc $D\left(z_{0}, R\right)$. There exist unique $a_{0}, a_{1}, a_{2}, \ldots \in B$ such that $\sum_{j=0}^{\infty} a_{r}\left(z-z_{0}\right)^{r}$ converges and $$ f(z)=\sum_{j=0}^{\infty} a_{r}\left(z-z_{0}\right)^{r} $$ for all $\left\|z-z_{0}\right\|<R$. Theorem 78. If $B$ is a Banach algebra with unit $$ \sup \{\|\lambda\|: \lambda \notin R(x)\}=\rho(x) $$ ## Maximal ideals One way of exploiting the Gelfand-Mazur theorem is to introduce the notion of maximal ideals. (From now on all our Banach algebras will be commutative.) Lemma 79. Every proper ideal in a commutative algebra with unit is contained in a maximal ideal. (Recall that an ideal $I$ in a commutative algebra $B$ is a vector subspace of $B$ such that if $a \in B$ and $b \in I$ then $a b \in I$. An ideal $J$ is maximal if $J \neq B$ but whenever an ideal $K$ satisfies $J \subseteq K \subseteq B$ either $K=J$ or $K=B$.) Lemma 80. Every maximal ideal $M$ in a commutative Banach algebra with unit is closed. Lemma 81. If $M$ is a maximal ideal in a commutative Banach algebra with unit then the quotient $B / M$ is isomorphic to $\mathbb{C}$ as a Banach algebra. The notion of a maximal ideal is closely linked to that of a multiplicative linear functional. Definition 82. A multiplicative linear functional on a Banach algebra is a non-trivial (i.e not the zero map) linear map $\chi: B \rightarrow \mathbb{C}$ such that $\chi(x y)=$ $\chi(x) \chi(y)$ for all $x, y \in B$. Lemma 83. If $B$ is commutative Banach algebra with identity and $\chi$ is a multiplicative linear functional then the following results hold. (i) $\operatorname{ker} \chi$ is a maximal ideal. (ii) The map $x+\operatorname{ker} \chi \mapsto \chi(x)$ is an algebraic isomorphism of $B / \operatorname{ker} \chi$ with $\mathbb{C}$. (iii) $\chi$ is continuous and $\\|\chi\\|=1$. Theorem 84. If $B$ is commutative Banach algebra with identity then the mapping $\chi \mapsto \operatorname{ker} \chi$ is a bijection between the set of multiplicative linear functionals on $B$ and its maximal ideals. We now have the following useful corollary. Lemma 85. If $B$ is commutative Banach algebra with identity then an element $x \in B$ is invertible if and only $\chi(x) \neq 0$ for all multiplicative linear functionals $\chi$. The Banach algebra proof Theorem 87 was the first result to convince classical analysts of the utility of these ideas. The lemma that precedes it places the result in context. Lemma 86. If $f \in C(\mathbb{T})$ has an absolutely convergent Fourier series (that is to say, $\left.\sum_{-\infty}^{\infty}\|\hat{f}(n)\|<\infty\right)$ then $$ f(t)=\sum_{-\infty}^{\infty} \hat{f}(n) \exp (i n t) $$ Theorem 87 (Wiener's theorem). Suppose $f \in C(\mathbb{T})$ has an absolutely convergent Fourier series. Then, if $f(t) \neq 0$ for all $t \in \mathbb{T}, 1 / f$ also has ian absolutely convergent Fourier series. Exercise 88. Let $B$ be any Banach space. Make it into a Banach algebra by defining $x y=0$ for all $x, y \in B$. Now add an identity in the usual manner. Identify all the multiplicative linear functionals. ## The Gelfand representation Throughout this section $B$ will be a commutative Banach algebra with a unit $e$ and $\mathcal{M}$ will be the space of maximal ideals. If $x \in B$ and $M \in \mathcal{M}$ we know by Theorem 84 that there is a unique multiplicative linear functional $\chi_{M}$ with kernel $M$ so we may write $M(x)=\chi_{M}(x)$ the space. We give $\mathcal{M}$ the weak star topology, that is to say, the smallest topology containing sets of the form $$ \left\{M \in \mathcal{M}:\left\|M(x)-M_{0}(x)\right\|<\epsilon\right\} $$ with $M_{0} \in \mathcal{M}$ and $x \in B$. Lemma 89. Under the weak topology $\mathcal{M}$ is a compact Hausdorff space. If $x \in B$ and $M \in \mathcal{M}$ we now write $\hat{x}(M)=M(x)$. Lemma 90. Let $B$ be a commutative Banach algebra with unit. The mapping $x \mapsto \hat{x}$ is an algebraic homomorphism of $B$ into $C(\mathcal{M})$. As linear map from $(B,\\|\\|)$ to $C\left(\mathcal{M},\\|\\|_{\infty}\right)$ it is continuous with operator norm exactly 1. We know that the homomorphism $x \mapsto \hat{x}$ need not be injective Exercise 91. Justify this statement by considering the Banach algebra of Exercise 88. The following simple observation is the key to the question of when we have isomorphism. Lemma 92. Suppose $x$ is an element of a commutative Banach algebra with unit. Then the complement of the resolvent $R(x)$ is the range of $\hat{x}$. That is to say, $$ \{\hat{x}(M): M \in \mathcal{M}\}=\{\lambda \in \mathbb{C}:(x-\lambda e) \text { is not invertible }\} . $$ There are two immediate corollaries. Lemma 93. If $x$ is an element of a commutative Banach algebra with unit, then $\\|\hat{x}\\|_{\infty}=\rho(x)$. Lemma 94. If $x$ is an element of a commutative Banach algebra with unit, then $\rho(x)=0$ if and only if $x$ is contained in every maximal ideal. We make the following definitions. Definition 95. If $B$ is a commutative Banach algebra with unit we define the radical of $B$ to be the set of all elements contained in every maximal ideal. Thus $x \in \operatorname{radical}(B)$ if and only if $\rho(x)=0$. Definition 96. We say that a commutative Banach algebra with unit is semi-simple if and only if its radical consists of 0 alone. Theorem 97. Let $B$ be a commutative Banach algebra with unit. The mapping $x \mapsto \hat{x}$ is injective if and only if $B$ is semi-simple. Exercise 98. Consider the Banach algebra $X$ of continuous linear maps $T: l^{\infty} \rightarrow l^{\infty}$. Let $S$ be the map given by $$ S\left(a_{1}, a_{2}, \ldots\right)=\left(0, c_{1} a_{1}, c_{2} a_{2}, \ldots\right), $$ (with the sequence $c_{j}$ bounded. Explain why the closed Banach subalgebra generated by $I$ and $S$ is a commutative Banach algebra. Show that with an appropriate choice of $c_{j}$ we can have $S^{n} \neq 0$ for all $n$ but $\rho(S)=0$. Theorem 99. Let $B$ be a commutative Banach algebra with unit. If there exists a $K>0$ such that $\\|x\\|^{2} \leq K\left\\|x^{2}\right\\|$ for all $x \in B$, then $\rho$ is a norm equivalent to the original norm on $B$. ## Finding the Gelfand representation Suppose we are given a commutative Banach algebra $B$ and we wish to find its Gelfand representation. It is not enough to find its maximal ideals (or, equivalently its multiplicative linear functionals). We must also find the correct topology on the space of maximal ideals. The following simple remarks resolve the problem in all the cases that we shall consider. Exercise 100. Write out the proof that if $(X, \tau)$ and $(Y, \sigma)$ are topological spaces with $(X, \tau)$ compact and $(Y, \sigma)$ Hausdorff then, if $f:(X, \tau) \rightarrow(Y, \sigma)$ is a continuous bijection, $f$ is a homeomorphism. Lemma 101. Suppose $\tau$ is a compact topology on the space $\mathcal{M}$ of maximal ideals of commutative Banach space $B$ with identity. If the maps $\hat{x}$ : $(\mathcal{M}, \tau) \rightarrow \mathbb{C}$ are continuous for each $x \in B$ then $\tau$ is the weak star topology on $\mathcal{M}$. Our first identification was adumbrated in our proof of Wiener's theorem. Example 102. Consider the space $A(\mathbb{T})$ of continuous functions $f: \mathbb{T} \rightarrow \mathbb{C}$ with absolutely convergent Fourier series (that is to say, $\sum_{-\infty}^{\infty}\|\tilde{f}(n)\|<\infty$ where $\tilde{f}(n)$ is the nth Fourier coefficient). If we set $$ \\|f\\|_{A}=\sum_{-\infty}^{\infty}\|\tilde{f}(n)\|, $$ then $\left(A(\mathbb{T}),\\|\\|_{A}\right)$ is a commutative Banach algebra with unit 1 under pointwise multiplication. $\left(A(\mathbb{T}),\\|\\|_{A}\right)$ has maximal ideal space (identified with) $\mathbb{T}$ under its usual topology. We have $\hat{f}(t)=f(t)$. Example 103. The 'transform' nature of the Gelfand transform is clearer if we seek the maximal ideal space and transform associated with the Banach algebra $l^{1}(\mathbb{Z})$ with standard norm and addition and multiplication given by convolution (that is $\mathbf{a} \mathbf{b}=\mathbf{c}$ where $c_{m}=\sum_{r=-\infty}^{\infty} a_{m-r} b_{r}$ ). Here is a variation on the theme. Lemma 104. Let $D=\{z \in \mathbb{C}:\|z\|<1\}$ and $\bar{D}=\{z \in \mathbb{C}:\|z\| \leq 1\}$. Consider $A(D)$ the set of continuous functions $f: \bar{D} \rightarrow \mathbb{C}$ such that $f$ is analytic in $D$. If $f_{1}, f_{2}, \ldots, f_{n} \in A(D)$ are such that $\sum_{j=1}^{n}\left\|f_{j}(z)\right\|>0$ for all $z \in \mathbb{C}$ (that is to say that the $f_{j}$ do not vanish simultaneously) show that we can find $g_{1}, g_{2}, \ldots, g_{n} \in A(D)$ such that $\sum_{j}^{n} f_{j=1}(z) g_{j}(z)=1$ for all $z \in \bar{D}$ The next example is a key one in understanding the kind of problem we face. Example 105. Consider the sub Banach algebra $A_{+}(\mathbb{T})$ of $A(\mathbb{T})$ consisting of elements $f$ of $A(\mathbb{T})$ with $\tilde{f}(n)=0$ for $n<0$. Show that $A_{+}(\mathbb{T})$ has maximal ideal space (identified with) $\mathbb{D}$ the closed unit disc. We have $\hat{f}(z)=$ $\sum_{n=0}^{\infty} \tilde{f}(n) z^{n}$. Exercise 106. Consider the sub Banach algebra $A_{-}(\mathbb{T})$ of $A(\mathbb{T})$ consisting of elements $f$ of $A(\mathbb{T})$ with $\tilde{f}(n)=0$ for $n>0$. Find the maximal ideal space and associated Gelfand transform. Exercise 107. Consider the space $B(\mathbb{T})$ of continuous functions $f: \mathbb{T} \rightarrow \mathbb{C}$ with $\sum_{-\infty}^{\infty}\|n \tilde{f}(n)\|<\infty$. Show that if we set $$ \\|f\\|_{B}=\sum_{-\infty}^{\infty}(\|n\|+1)\|\tilde{f}(n)\| $$ then $\left(B(\mathbb{T}),\\|\\|_{B}\right)$ is a commutative Banach algebra with unit 1 under pointwise multiplication. Find the maximal ideal space and associated Gelfand transform. Exercise 108. Consider the sub Banach algebra $B_{+}(\mathbb{T})$ of $B(\mathbb{T})$ consisting of elements $f$ of $B(\mathbb{T})$ with $\tilde{f}(n)=0$ for $n<0$. Find the maximal ideal space and associated Gelfand transform. Our next example is fundamental. Example 109. Let $(X, \tau)$ be a compact Hausdorff space. The space $C(X)$ of continuous functions $f: X \rightarrow \mathbb{C}$ with the uniform norm is a commutative Banach algebra with unit 1 under pointwise operations. $C(X)$ has maximal ideal space (identified with) $X$ under its usual topology. We have $\hat{f}(t)=f(t)$. One way of expressing many of our results is in terms of function algebras. Definition 110. Let $(X, \tau)$ be a compact Hausdorff space. If we consider $C(X)$ as a Banach algebra in the usual way then any subalgebra $A$ with a norm which makes it a Banach algebra is called a function algebra. Lemma 111. With the notation of Definition 110, if $A$ is a Banach algebra with norm \\|\\| containing 1 , then $\\|f\\| \geq\\|f\\|_{\infty}$ for all $f \in A$. Lemma 112. We use the notation of Definition 110. (i) If $A$ separates points (that is, given $x, y \in X$ with $x \neq y$, we can find an $f \in A$ such that $f(x) \neq f(y)$ ) and $f \in A$ implies $f^{} \in A$ then $A$ has maximal ideal space (identified with) $X$ under its usual topology. We have $\hat{f}(t)=f(t)$. (ii) If $A$ satisfies (i) and, in addition, there exists a $K$ such that $\\|f\\|^{2} \leq$ $K\left\\|f^{2}\right\\|$ then $A=C(X)$ and there exists a $\kappa$ such that $$ \kappa\\|f\\|_{\infty} \geq\\|f\\| \geq\\|f\\|_{\infty} $$ for all $f \in A$ (so the norms \\|\\| and \\|\\|$_{\infty}$ are Lipschitz equivalent) Exercise 113. Show that the space $B$ of continuous functions $f:[0,1] \cup$ $[2,3] \rightarrow \mathbb{C}$ such that $f(2+t)=f(t)$ for $t \in[0,1]$ equipped with the uniform norm is function algebra. Find the maximal ideal space and associated Gelfand transform. Exercise 114. Show that the space $C^{1}([0,1])$ of once continuously differentiable functions equipped with norm $$ \\|f\\|=\\|f\\|_{\infty}+\left\\|f^{\prime}\right\\|_{\infty} $$ is function algebra. Find the maximal ideal space and associated Gelfand transform. ## Three more uses of Hahn-Banach The following exercise provides background for our first discussion but is not examinable. For the moment $C([a, b])$ will be the set of real valued continuous functions. Exercise 115. We say that a function $G:[a, b] \rightarrow \mathbb{R}$ is of bounded variation if there exists a $K$ such that whenever we have a dissection $$ \mathcal{D}=\left\{x_{0}, x_{1}, x_{2}, \ldots, x_{n}\right\} $$ $a=x_{0}<x_{1}<x_{2}<\cdots<x_{n}=b$ we have $$ \sum_{j=1}^{n}\left\|G\left(x_{j}\right)-G\left(x_{j-1}\right)\right\| \leq K . $$ We write $$ \\|G\\|_{B V}=\sup _{\mathcal{D}} \sum_{j=1}^{n}\left\|G\left(x_{j}\right)-G\left(x_{j-1}\right)\right\| $$ where the supremum is taken over all possible dissections. Suppose $f:[a, b] \rightarrow \mathbb{R}$ is continuous. Let us write $$ S(\mathcal{D}, f, G)=\sum_{j=1}^{n} f\left(x_{j}\right)\left(G\left(x_{j}\right)-G\left(x_{j-1}\right) .\right. $$ If $\mathcal{D}=\left\{x_{0}, x_{1}, x_{2}, \ldots, x_{n}\right\}$ and $\mathcal{D}^{\prime}=\left\{x_{0}^{\prime}, x_{1}^{\prime}, x_{2}^{\prime}, \ldots, x_{n^{\prime}}^{\prime}\right\}$ are such that $\|f(t)-f(s)\|<\epsilon$ for all $t, s \in\left[x_{j-1}, x_{j}\right][1 \leq j \leq n]$ and for all $t, s \in\left[x_{j-1}^{\prime}, x_{j}^{\prime}\right]$ $\left[1 \leq j \leq n^{\prime}\right]$ show by considering $\mathcal{D} \cup \mathcal{D}^{\prime}$, or otherwise that $$ \left\|S(\mathcal{D}, f, G)-S\left(\mathcal{D}^{\prime}, f, G\right)\right\| \leq 2 K \epsilon . $$ Hence, or otherwise, show that there exists a unique $I(f, G)$ such that, given any $\epsilon>0$ we can find a $\delta>0$ such that, given any $$ \mathcal{D}=\left\{x_{0}, x_{1}, x_{2}, \ldots, x_{n}\right\} $$ with $\left\|x_{j-1}-x_{j}\right\|<\delta[1 \leq j \leq n]$ we have $$ \|S(\mathcal{D}, f, G)-I(f, G)\|<\epsilon . $$ We write $$ I(f, G)=\int_{a}^{b} f(t) d G(t) $$ (i) Let $[a, b]=[0,1]$. Find elementary expressions for $\int_{a}^{b} f(t) d G(t)$ in the three cases when $G(t)=t$, when $G(t)=-t$ and when $G(t)=0$ for $t<1 / 2$, $G(t)=1$ for $t \geq 1 / 2$. (ii) Show that the map $T:\left(C([a, b]),\\|\\|_{\infty}\right) \rightarrow \mathbb{R}$ given by $$ T f=\int_{a}^{b} f(t) d G(t) $$ is linear and continuous with $\\|T\\|=\\|G\\|_{B V}$. Theorem 116. If $T: C([a, b]) \rightarrow \mathbb{R}$ is a continuous linear function then we can find a function $G:[a, b] \rightarrow \mathbb{R}$ of bounded variation such that $$ T f=\int_{a}^{b} f(t) d G(t) $$ for all $f \in C([a, b])$ If you know a little measure theory you can restate the theorem in more modern language. Theorem 117. (The Riesz representation theorem.) The dual of $C([a, b])$ is the space of Borel measures on $[a, b]$. The method used can easily be extended to all compact spaces. Our second result is more abstract. We require Aloaoglu's theorem. Theorem 118. The unit ball of the dual of a normed space $X$ is compact in the weak star topology. Our proof of the Riesz representation theorem used the Hahn-Banach theorem as a convenience. Our proof of the next result uses it as basic ingredient. Theorem 119. Every Banach space is isometrically isomorphic to some subspace of $C(K)$ for some compact space $K$. (In my opinion this result looks more interesting than it is.) Our third result requires us to recast the Hahn Banach theorem in a geometric form. Lemma 120. If $V$ is a real normed spaced and $E$ is a convex subset of $V$ containing $B(\mathbf{0}, \epsilon)$ for some $\epsilon>0$, then, given any $\mathbf{x} \notin E$ we can find a continuous linear map $T: V \rightarrow \mathbb{R}$ such that $T \mathbf{x} \geq T \mathbf{e}$ for all $\mathbf{e} \in E$. Theorem 121. If $V$ is a real normed spaced and $K$ is a compact convex subset of $V$, then, given any $\mathbf{x} \notin E$ we can find a continuous linear map $T: V \rightarrow \mathbb{R}$ and a real $\alpha$ such that $T \mathbf{x}>\alpha>T \mathbf{k}$ for all $\mathbf{k} \in K$. Definition 122. Let $V$ be a real or complex vector space. If $K$ is a nonempty subset of $V$ we say that $E \subseteq K$ is an extreme set of $K$ if, whenever $u, v \in K, 1>\lambda>0$ and $\lambda u+(1-\lambda) v \in E$, it follows that $u, v \in E$. If $\{e\}$ is an extreme set we call e an extreme point. Exercise 123. Define an extreme point directly. Exercise 124. We work in $\mathbb{R}^{2}$. Find the extreme points, if any, of the following sets and prove your statements. (i) $E_{1}=\{\mathbf{x}:\\|\mathbf{x}\\|<1\}$. (ii) $E_{2}=\{\mathbf{x}:\\|\mathbf{x}\\| \leq 1\}$. (iii) $E_{3}=\{(x, 0): x \in \mathbb{R}\}$. (iv) $E_{4}=\{(x, y):\|x\|,\|y\| \leq 1\}$. Theorem 125. (Krein-Milman). A non-empty compact convex subset $K$ of a normed vector space has at least one extreme point. Theorem 126. A non-empty compact convex subset $K$ of a normed vector space is the closed convex hull of its extreme points (that is, is the smallest closed convex set containing its extreme points). Our hypotheses in our version of the Krein-Milman theorem are so strong as to make the conclusion practically useless. However the hypotheses can be much weakened as is indicated by the following version. Theorem 127. (Krein-Milman). Let $E$ be the dual space of a normed vector space. A non-empty convex subset $K$ which is compact in the weak star topology has at least one extreme point. Theorem 128. Let $E$ be the dual space of a normed vector space. A nonempty convex subset $K$ which is compact in the weak star topology is the weak star closed convex hull of its extreme points. Lemma 129. The extreme points of the closed unit ball of the dual of $C([0,1])$ are the delta masses $\delta_{a}$ and $-\delta_{a}$ with $a \in[0,1]$. ## The Rivlin-Shapiro formula In this section we give an elegant use of extreme points due to Rivlin and Shapiro. Lemma 130. Carathéodory We work in $\mathbb{R}^{n}$. Suppose that $\mathbf{x} \in \mathbb{R}^{n}$ and we are given a finite set of points $\mathbf{e}_{1}, \mathbf{e}_{2}, \ldots, \mathbf{e}_{N}$ and positive real numbers $\lambda_{1}$, $\lambda_{2}, \ldots, \lambda_{N}$ such that $$ \sum_{j=1}^{N} \lambda_{j}=1, \sum_{j=1}^{N} \lambda_{j} \mathbf{e}_{j}=\mathbf{x} $$ Then after renumbering the $\mathbf{e}_{j}$ we can find positive real numbers $\lambda_{1}^{\prime}, \lambda_{2}^{\prime}$, $\ldots, \lambda_{m}^{\prime}$ with $m \leq n+1$ such that $$ \sum_{j=1}^{m} \lambda_{j}^{\prime}=1, \sum_{j=1}^{m} \lambda_{j}^{\prime} \mathbf{e}_{j}=\mathbf{x} . $$ Lemma 131. Consider $\mathcal{P}_{n}$, the subspace of $C([-1,1])$ consisting of real polynomials of degree $n$ or less. If $S: \mathcal{P}_{n} \rightarrow \mathbb{R}$ is linear then we can find an $N \leq n+1$ and distinct points $x_{0}, x_{1}, \ldots, x_{N} \in[-1,1]$ and non-zero real numbers $\lambda_{0}, \lambda_{1}, \ldots, \lambda_{N}$ such that $$ \sum_{j=0}^{N}\left\|\lambda_{j}\right\|=1,\\|S\\| \sum_{j=0}^{N} \lambda_{j} P\left(x_{j}\right)=S P . $$ for all $P \in \mathcal{P}_{n}$. Lemma 132. We continue with the hypotheses and notation of Lemma 131 There exists a $P_{} \in \mathcal{P}_{n}$ such that $$ P_{}\left(x_{j}\right)=\left\\|P_{}\right\\|_{\infty} \operatorname{sgn} \lambda_{j} $$ for all $j$ with $0 \leq j \leq N$. Further, if $P \in \mathcal{P}_{n}$ satisfies $$ P\left(x_{j}\right)=\\|P\\|_{\infty} \operatorname{sgn} \lambda_{j} $$ then $\\|P\\|_{\infty}\\|S\\|=S P$. The following results are of considerable interest in view of Lemma 132. Lemma 133. We have $\cos n \theta=T_{n}(\cos \theta)$ where $T_{n}$ is a real polynomial of degree n. Further (i) $\left\|T_{n}(x)\right\| \leq 1$ for all $x \in[-1,1]$. (ii) There exist $n+1$ distinct points $x_{1}, x_{2}, \ldots, x_{n+1} \in[-1,1]$ such that $\left\|T_{n}\left(x_{j}\right)\right\|=1$ for all $1 \leq j \leq n+1$. Lemma 134. If $P$ is a real polynomial of degree $n$ or less such that (i) $\|P(x)\| \leq 1$ for all $x \in[-1,1]$ and (ii) There exist $n+1$ distinct points $x_{1}, x_{2}, \ldots, x_{n+1} \in[-1,1]$ such that $\left\|P\left(x_{j}\right)\right\|=1$ for all $1 \leq j \leq n+1$, Then $P= \pm T_{n}$. Theorem 135. If $P$ is a real polynomial of degree at most $n$ and $t \notin$ $[-1,1]$ then $$ \|P(t)\| \leq \sup _{\|x\| \leq 1}\|P(x)\|\left\|T_{n}(t)\right\| $$ Exercise 136. If $P$ is a real polynomial of degree at most $n$ then $$ \left\|P^{(r)}(t)\right\| \leq\left\|T^{(r)}(t)\right\| \sup _{\|x\| \leq 1}\|P(x)\| . $$ Exercise 137. (This exercise is part of the course.) (i) Show that if $n \geq 1$ the coefficient of $t^{n}$ in $T_{n}(t)$ is $2^{n-1}$. (ii) Show that if $n \geq 1$ and $P$ is a real polynomial of degree $n$ or less with $\|P(t)\| \leq 1$ then the coefficient of $t^{n}$ in $P(t)$ has absolute value at most $2^{n-1}$. (iii) Find, with proof, a polynomial $P$ of degree at most $n-1$ which minimises $$ \sup _{t \in[-1,1]}\left\|t^{n}-P(t)\right\| $$ Show that $P$ is unique. (Tchebychev introduced his polynomials $T_{n}$ in this context.) ## References [1] B. Bollobás Linear Analysis : an Introductory Course (CUP 1991) [2] C. Gofman and G. Pedrick A First Course in Functional Analysis (Prentice Hall 1965, available as a Chelsea reprint from the AMS) [3] J. D. Pryce Basic Methods of Linear Functional Analysis (Hutchinson 1973) [4] W. Rudin Real and Complex Analysis (McGraw Hill, 2nd Edition, 1974) [5] W. Rudin Functional Analysis (McGraw Hill 1973)	Textbooks
Pontryagin class In mathematics, the Pontryagin classes, named after Lev Pontryagin, are certain characteristic classes of real vector bundles. The Pontryagin classes lie in cohomology groups with degrees a multiple of four. Definition Given a real vector bundle E over M, its k-th Pontryagin class $p_{k}(E)$ is defined as $p_{k}(E)=p_{k}(E,\mathbb {Z} )=(-1)^{k}c_{2k}(E\otimes \mathbb {C} )\in H^{4k}(M,\mathbb {Z} ),$ where: • $c_{2k}(E\otimes \mathbb {C} )$ denotes the $2k$-th Chern class of the complexification $E\otimes \mathbb {C} =E\oplus iE$ of E, • $H^{4k}(M,\mathbb {Z} )$ is the $4k$-cohomology group of M with integer coefficients. The rational Pontryagin class $p_{k}(E,\mathbb {Q} )$ is defined to be the image of $p_{k}(E)$ in $H^{4k}(M,\mathbb {Q} )$, the $4k$-cohomology group of M with rational coefficients. Properties The total Pontryagin class $p(E)=1+p_{1}(E)+p_{2}(E)+\cdots \in H^{}(M,\mathbb {Z} ),$ is (modulo 2-torsion) multiplicative with respect to Whitney sum of vector bundles, i.e., $2p(E\oplus F)=2p(E)\smile p(F)$ for two vector bundles E and F over M. In terms of the individual Pontryagin classes pk, $2p_{1}(E\oplus F)=2p_{1}(E)+2p_{1}(F),$ $2p_{2}(E\oplus F)=2p_{2}(E)+2p_{1}(E)\smile p_{1}(F)+2p_{2}(F)$ and so on. The vanishing of the Pontryagin classes and Stiefel–Whitney classes of a vector bundle does not guarantee that the vector bundle is trivial. For example, up to vector bundle isomorphism, there is a unique nontrivial rank 10 vector bundle $E_{10}$ over the 9-sphere. (The clutching function for $E_{10}$ arises from the homotopy group $\pi _{8}(\mathrm {O} (10))=\mathbb {Z} /2\mathbb {Z} $.) The Pontryagin classes and Stiefel-Whitney classes all vanish: the Pontryagin classes don't exist in degree 9, and the Stiefel–Whitney class w9 of E10 vanishes by the Wu formula w9 = w1w8 + Sq1(w8). Moreover, this vector bundle is stably nontrivial, i.e. the Whitney sum of E10 with any trivial bundle remains nontrivial. (Hatcher 2009, p. 76) Given a 2k-dimensional vector bundle E we have $p_{k}(E)=e(E)\smile e(E),$ where e(E) denotes the Euler class of E, and $\smile $ denotes the cup product of cohomology classes. Pontryagin classes and curvature As was shown by Shiing-Shen Chern and André Weil around 1948, the rational Pontryagin classes $p_{k}(E,\mathbf {Q} )\in H^{4k}(M,\mathbf {Q} )$ can be presented as differential forms which depend polynomially on the curvature form of a vector bundle. This Chern–Weil theory revealed a major connection between algebraic topology and global differential geometry. For a vector bundle E over a n-dimensional differentiable manifold M equipped with a connection, the total Pontryagin class is expressed as $p=\left[1-{\frac {{\rm {Tr}}(\Omega ^{2})}{8\pi ^{2}}}+{\frac {{\rm {Tr}}(\Omega ^{2})^{2}-2{\rm {Tr}}(\Omega ^{4})}{128\pi ^{4}}}-{\frac {{\rm {Tr}}(\Omega ^{2})^{3}-6{\rm {Tr}}(\Omega ^{2}){\rm {Tr}}(\Omega ^{4})+8{\rm {Tr}}(\Omega ^{6})}{3072\pi ^{6}}}+\cdots \right]\in H_{dR}^{}(M),$ where Ω denotes the curvature form, and H*dR(M) denotes the de Rham cohomology groups.[1] Pontryagin classes of a manifold The Pontryagin classes of a smooth manifold are defined to be the Pontryagin classes of its tangent bundle. Novikov proved in 1966 that if two compact, oriented, smooth manifolds are homeomorphic then their rational Pontryagin classes pk(M, Q) in H4k(M, Q) are the same. If the dimension is at least five, there are at most finitely many different smooth manifolds with given homotopy type and Pontryagin classes. Pontryagin classes from Chern classes The Pontryagin classes of a real vector bundle $\pi :E\to X$ can be completely determined by the Chern classes of its complexification. This follows from the fact that $E\otimes _{\mathbb {R} }\mathbb {C} \cong E\oplus {\bar {E}}$, the Whitney sum formula, and properties of Chern classes of its complex conjugate bundle. That is, $c_{i}({\bar {E}})=(-1)^{i}c_{i}(E)$ and $c(E\oplus {\bar {E}})=c(E)c({\bar {E}})$. Then, this given the relation $1-p_{1}(E)+p_{2}(E)-\cdots +(-1)^{n}p_{n}(E)=(1+c_{1}(E)+\cdots +c_{n}(E))\cdot (1-c_{1}(E)+c_{2}(E)-\cdots +(-1)^{n}c_{n}(E))$[2] for example, we can apply this formula to find the Pontryagin classes of a vector bundle on a curve and a surface. For a curve, we have $(1-c_{1}(E))(1+c_{1}(E))=1+c_{1}(E)^{2}$ so all of the Pontryagin classes of complex vector bundles are trivial. On a surface, we have $(1-c_{1}(E)+c_{2}(E))(1+c_{1}(E)+c_{2}(E))=1-c_{1}(E)^{2}+2c_{2}(E)$ showing $p_{1}(E)=c_{1}(E)^{2}-2c_{2}(E)$. On line bundles this simplifies further since $c_{2}(L)=0$ by dimension reasons. Pontryagin classes on a Quartic K3 Surface Recall that a quartic polynomial whose vanishing locus in $\mathbb {CP} ^{3}$ is a smooth subvariety is a K3 surface. If we use the normal sequence $0\to {\mathcal {T}}_{X}\to {\mathcal {T}}_{\mathbb {CP} ^{3}}\|_{X}\to {\mathcal {O}}(4)\to 0$ we can find ${\begin{aligned}c({\mathcal {T}}_{X})&={\frac {c({\mathcal {T}}_{\mathbb {CP} ^{3}}\|_{X})}{c({\mathcal {O}}(4))}}\\&={\frac {(1+[H])^{4}}{(1+4[H])}}\\&=(1+4[H]+6[H]^{2})\cdot (1-4[H]+16[H]^{2})\\&=1+6[H]^{2}\end{aligned}}$ showing $c_{1}(X)=0$ and $c_{2}(X)=6[H]^{2}$. Since $[H]^{2}$ corresponds to four points, due to Bezout's lemma, we have the second chern number as $24$. Since $p_{1}(X)=-2c_{2}(X)$ in this case, we have $p_{1}(X)=-48$. This number can be used to compute the third stable homotopy group of spheres.[3] Pontryagin numbers Pontryagin numbers are certain topological invariants of a smooth manifold. Each Pontryagin number of a manifold M vanishes if the dimension of M is not divisible by 4. It is defined in terms of the Pontryagin classes of the manifold M as follows: Given a smooth $4n$-dimensional manifold M and a collection of natural numbers $k_{1},k_{2},\ldots ,k_{m}$ such that $k_{1}+k_{2}+\cdots +k_{m}=n$, the Pontryagin number $P_{k_{1},k_{2},\dots ,k_{m}}$ is defined by $P_{k_{1},k_{2},\dots ,k_{m}}=p_{k_{1}}\smile p_{k_{2}}\smile \cdots \smile p_{k_{m}}([M])$ where $p_{k}$ denotes the k-th Pontryagin class and [M] the fundamental class of M. Properties 1. Pontryagin numbers are oriented cobordism invariant; and together with Stiefel-Whitney numbers they determine an oriented manifold's oriented cobordism class. 2. Pontryagin numbers of closed Riemannian manifolds (as well as Pontryagin classes) can be calculated as integrals of certain polynomials from the curvature tensor of a Riemannian manifold. 3. Invariants such as signature and ${\hat {A}}$-genus can be expressed through Pontryagin numbers. For the theorem describing the linear combination of Pontryagin numbers giving the signature see Hirzebruch signature theorem. Generalizations There is also a quaternionic Pontryagin class, for vector bundles with quaternion structure. See also • Chern–Simons form • Hirzebruch signature theorem References 1. "De Rham Cohomology - an overview \| ScienceDirect Topics". www.sciencedirect.com. Retrieved 2022-02-02. 2. Mclean, Mark. "Pontryagin Classes" (PDF). Archived (PDF) from the original on 2016-11-08. 3. "A Survey of Computations of Homotopy Groups of Spheres and Cobordisms" (PDF). p. 16. Archived (PDF) from the original on 2016-01-22. • Milnor John W.; Stasheff, James D. (1974). Characteristic classes. ISBN 0-691-08122-0. {{cite book}}: \|work= ignored (help) • Hatcher, Allen (2009). "Vector Bundles & K-Theory" (2.1 ed.). {{cite journal}}: Cite journal requires \|journal= (help) External links • "Pontryagin class", Encyclopedia of Mathematics, EMS Press, 2001 [1994]	Wikipedia
\begin{document} \title{Classification of Picard lattices of K3 surfaces \footnote{With support by Russian Sientific Fund N 14-50-00005.}} \date{5 September 2017} \author{Viacheslav V. Nikulin} \maketitle \begin{abstract} Using results of our papers \cite{Nik9}, \cite{Nik10} and \cite{Nik11} about classification of degenerations of K\"ahlerian K3 surfaces with finite symplectic automorphism groups, we classify Picard lattices of K\"ahlerian K3 surfaces. By classification we understand classification depending on their possible finite symplectic automorphism groups and their non-singular rational curves if a Picard lattice is negative definite. \end{abstract} \centerline{Dedicated to the memory of Igor Rostislavovich Shafarevich} \section{Introduction} \label{sec1:introduction} Let $X$ be a K\"ahlerian K3 surface with negative definite Picard lattice $S_X$. Then the full symplectic automorphism group $G=Aut\ (X)_0$ of $X$ is finite and and its coinvariant sublattice $S_G=((S_X)^G)^\perp_{S_X}$ is negative definite. We denote by $P(X)\subset S_X$ the set of classes of all irreducible non-singular rational curves of $X$. They have the square $(-2)$. The primitive sublattice $S=MS_X=[S_G,P(X)]_{pr}\subset S_X$ of $S_X$ generated by $S_G$ and $P(X)$ gives the {\it main part} of the Picard lattice $S_X$. The remaining part $(S)^\perp_{S_X}$ can be considered as {\it exotic part} of the Picard lattice $S_X$. If a K3 surface $X$ is algebraic, instead of $S_X$, one can take the orthogonal complement $(h)^\perp_{S_X}$ to a nef element $h\in S_X$ with $h^2>0$. We classify the main parts $S$ of Picard lattices of K3 if the group $G$ is large enough, it has order $\|G\|\ge 8$, equivalently, it is different from $D_6$, $C_4$, $(C_2)^2$, $C_3$, $C_2$ and $C_1$ where $C_m$ is a cyclic group of order $m$ and $D_m$ is a dihedral group of order $m$. It was known (for all $G$) if the set $P(X)$ is empty. See \cite{Hash}, \cite{Kon}, \cite{Muk}, \cite{Nik0}, \cite{Xiao}. The case, when $P(X)$ is non-empty, can be considered as a degeneration of K\"ahlerian K3 surfaces with the finite symplectic automorphism group $G$. Degenerations of K\"ahlerian K3 surfaces with finite symplectic automorphism groups were studied in our papers \cite{Nik9}, \cite{Nik10} and \cite{Nik11}. In Section \ref{sec2:degen}, we remind to a reader our results of \cite{Nik9}, \cite{Nik10} and \cite{Nik11} about classification of degenerations of K\"ahlerian K3 surfaces with finite symplectic automorphism groups. In Section \ref{sec3:piclatt}, we present their applications to classification of Picard lattices of K3 surfaces. In Section \ref{sec4:tables}, we give Tables 1---4 which contain our classification of Picard lattices of K3 surfaces: all possible cases are given in lines of the Tables 1---4 which are not marked by $o$. Lines which are marked by $o$ are also important. They give classification of cases when a degeneration of K3 with finite symplectic automorphism group $G$ has larger symplectic automorphism group $\tilde{G}$ which has $\|\tilde{G}\|>\|G\|$. The same happens for Kummer surfaces. They can be considered as a degeneration with the Dynkin diagram $16\mathbb A_1$ of K3 with trivial symplectic automorphism group. But, their symplectic automorphism group is $(C_2)^4$. It was shown in our paper \cite{Nik-1}. We hope to consider similar classification for remaining groups $D_6$, $C_4$, $(C_2)^2$, $C_3$, $C_2$ and $C_1$, and more details in further variants of the paper and further publications. \section{On classification of degenerations of\\ K\"ahlerian K3 surfaces with finite symplectic\\ automorphism groups. A reminder.} \label{sec2:degen} Here we remind to a reader our results from \cite{Nik9}, \cite{Nik10} and \cite{Nik11} about classification of degenerations of K\"ahlerian K3 surfaces with finite symplectic automorphism groups. We refer to these papers for more details. Let $X$ be a K\"ahlerian K3 surface (e. g. see \cite{Sh}, \cite{PS}, \cite{BR}, \cite{Siu}, \cite{Tod} about such surfaces). That is $X$ is a non-singular compact complex surface with the trivial canonical class $K_X$, and its irregularity $q(X)$ is equal to 0. Then $H^2(X,\mathbb Z)$ with the intersection pairing is an even unimodular lattice $L_{K3}$ of the signature $(3,19)$. For a non-zero holomorphic $2$-form $\omega_X\in \Omega^2[X]$, we have $H^{2,0}(X)=\Omega^2[X]=\mathbb C\omega_X$. The primitive sublattice $$ S_X=H^2(X,\mathbb Z)\cap H^{1,1}(X)=\{x\in H^2(X,\mathbb Z)\ \|\ x\cdot H^{2,0}(X)=0 \ \}\subset H^2(X,\mathbb Z) $$ is the {\it Picard lattice} of $X$ generated by first Chern classes of all line bundles over $X$. We remind that a primitive sublattice means that $H^2(X,\mathbb Z)/S_X$ has no torsion. Let $G$ be a finite symplectic automorphism group of $X$. Here symplectic means that for any $g\in G$, for a non-zero holomorphic $2$-form $\omega_X\in H^{2,0}(X)=\Omega^2[X]=\mathbb C\omega_X$, one has $g^\ast(\omega_X)=\omega_X$. For an $G$-invariant sublattice $M\subset H^2(X,\mathbb Z)$, we denote by $M^G=\{x\in M\ \|\ G(x)=x\}$ the {\it fixed sublattice of $M$,} and by $M_G=(M^G)^\perp_M$ the {\it coinvariant sublattice of $M$.} By \cite{Nik-1/2}, \cite{Nik0}, the coinvariant lattice $S_G=H^2(X,\mathbb Z)_G=(S_X)_G$ is {\it Leech type lattice:} i. e. it is negative definite, it has no elements with square $(-2)$, $G$ acts trivially on the discriminant group $A_{S_G}=(S_G)^\ast/S_G$, and $(S_G)^G=\{0\}$. For a general pair $(X,G)$, the $S_G=S_X$, and non-general $(X,G)$ can be considered as K\"ahlerian K3 surfaces with the condition $S_G\subset S_X$ on the Picard lattice (in terminology of \cite{Nik0}). The dimension of their moduli is equal to $20-\rk S_G$. Let $E\subset X$ be a non-singular irreducible rational curve (that is $E\cong \mathbb P^1$). It is equivalent to: $\alpha=cl(E)\in S_X$, $\alpha^2=-2$, $\alpha$ is effective and $\alpha$ is numerically effective: $\alpha\cdot D\ge 0$ for every irreducible curve $D$ on $X$ such that $cl(D)\not=\alpha$. Let us consider the primitive sublattice $S=[S_G, G(\alpha)]_{pr}\subset S_X$ of $S_X$ generated by the coinvariant sublattice $S_G$ and all classes of the orbit $G(E)$. Since $S_G$ has no elements with square $(-2)$, it follows that $\rk S=\rk S_G+1$ and $S=[S_G,\alpha]_{pr}\subset S_X$. Let us {\it assume that the lattice $S=[S_G,\alpha]_{pr}$ is negative definite.} Then the elements $G(\alpha)$ define the basis of the root system $\Delta(S)$ of all elements with square $(-2)$ of $S$. All curves $G(E)$ of $X$ can be contracted to Du Val singularities of types of connected components of the Dynkin diagram of the basis. The group $G$ will act on the corresponding singular K3 surface $\overline{X}$ with these Du Val singularities. For a general such triplet $(X,G,G(E))$, the Picard lattice $S_X=S$, and such triplets can be considered as {\it a degeneration of codimension $1$} of K\"ahlerian K3 surfaces $(X,G)$ with the finite symplectic automorphism group $G$. Really, the dimension of moduli of K\"ahlerian K3 surfaces with the condition $S\subset S_X$ on the Picard lattice is equal to $20-\rk S=20-\rk S_G-1$. By Global Torelli Theorem for K3 surfaces \cite{PS}, \cite{BR}, the main invariants of the degeneration is the {\it type of the abstract group $G$} which is equivalent to the isomorphism class of the coinvariant lattice $S_G$, and the type of the degeneration which is equivalent to the Dynkin diagram of the basis $G(\alpha)$ or the Dynkin diagram of the rational curves $G(E)$. We can consider only the maximal finite symplectic automorphism group $G$ with the same coinvariant lattice $S_G$, that is $G=Clos(G)$. By Global Torelly Theorem for K3 surfaces, this is equivalent to $$ G\|S_G=\{ g\in O(S_G)\ \|\ g\ is\ identity\ on\ A_{S_G}=(S_G)^\ast/S_G \}. $$ Indeed, $G$ and $Clos(G)$ have the same lattice $S_G$, the same orbits $G(E)$ and $Clos(G)(E)$, and the same sublattice $S\subset S_X$. In \cite{Nik9}, all types of $G=Clos(G)$ and types of degenerations (that is Dynkin diagrams of the orbits $G(E)$) are described. They are described in Table 1 of Section \ref{sec4:tables} which is the same as \cite[Table 1]{Nik10} where ${\bf n}$ gives types of possible $G=Clos(G)$, and we show all possible types of degenerations at the corresponding rows by their Dynkin diagrams. In the Table 1 of Section \ref{sec4:tables}, the type of $G=Clos(G) $ and the isomorphism class of the lattice $S_G$ is marked by ${\bf n}$. We also give the genus of $S_G$ which is defined by the discriminant quadratic form $q_{S_G}$. They were calculated in papers \cite{Nik0}, \cite{Muk}, \cite{Xiao}, \cite{Hash}. In \cite{Nik10}, with two exceptions which are marked by $I$ and $II$ for ${\bf n}=10$ (group $D_8$) and the degeneration $2\mathbb A_1$, and for ${\bf n}=34$ (group $\SSS_4$) and the degeneration $6\mathbb A_1$, it was shown that the lattice $S=[S_G,\alpha]_{pr}$ is unique, up to isomorphisms, and its genus (equivalent to $\rk S$ and the discriminant quadratic form $q_S$) is shown in Table 1. In \cite[Table 2]{Nik10}, we described possible markings of $S$, $G$ and $\alpha$ by Niemeier lattices which give exact lattices descriptions of $S$, and the action of $G$ on $S$ and $G(\alpha)$. For degenerations of arbitrary codimension $t\ge 1$ which were considered in \cite{Nik11}, instead of one orbit $G(E)$ of a non-singular rational curve, we should consider $t\ge 1$ different orbits $G(E_1),\dots, G(E_t)$ of non-singular rational curves on $X$, and their classes $G(\alpha_1), \dots, G(\alpha_t)$ in $S_X$, but we also assume that the sublattice $S=[S_G,G(\alpha_1),\dots,G(\alpha_t)]_{pr}\subset S_X$ is negative definite. Then the codimension of the degeneration is equal to $t=\rk S-\rk S_G$, and $S_X=S=[S_G,\alpha_1,\dots, \alpha_t]_{pr}$ in general. We remark that $\rk S=\rk S_G+t\le 19$ since $H^2(X,\mathbb Z)$ has the signature $(3,19)$. Thus, $t\le 19-\rk S_G$. The type of the degeneration is given by the Dynkin diagrams and subdiagrams $$ (Dyn(G(\alpha_1)),\dots, Dyn(G(\alpha_t)))\subset Dyn(G(\alpha_1)\cup\dots \cup G(\alpha_t)) $$ and their types. In difficult cases, we also consider the matrix of subdiagrams which is defined by $$ (Dyn (G(\alpha_i)),\, Dyn ( G(\alpha_j)))\subset Dyn(G(\alpha_i)\cup G(\alpha_j)) $$ and their types for $1\le i<j\le t$. In Table 2 of Section \ref{sec4:tables} which is similar to \cite[Table 2]{Nik11}, we give classification of types of degenerations of arbitrary codimension $\ge 2$ for ${\bf n}\ge 12$ (then the codimension is less or equal to $4$). Equivalently, either $\|G\|>8$ or $G\cong Q_8$ of order $8$ ($G$ is big enough). We calculate genuses of the lattices $S$ by $\rk S$ and $q_S$. By $\ast$, we mark cases when we prove that the lattice $S$ is unique up to isomorphisms for the given type. We mark by $o$ some cases (it will be important for classification of Picard lattices, and we shall discuss this later). The description of their markings by Niemeier lattices is given in \cite[Table 3]{Nik11}. For remaining groups $D_8$ (equivalently, ${\bf n}=10$) and $(C_2)^3$ (equivalently, ${\bf n}=9$) of order $8$, similar results are given in Tables 3 and 4 of Section \ref{sec4:tables} which are similar to tables of \cite{Nik11}. Their markings by Niemeier lattices are given in \cite[Table 5]{Nik11} and \cite[Table 7]{Nik11} respectively. For remaining finite symplectic automorphism groups $G=Clos(G)$ of order $\|G\|<8$, which are $D_6$ (${\bf n}=6$), $C_4$ (${\bf n}=4)$, $(C_2)^2$ (${\bf n}=3$), $C_3$ (${\bf n}=2$), $C_2$ (${\bf n}=1$) and $C_1$, similar classification results are unknown yet. \section{Classification of Picard lattices of K3 surfaces} \label{sec3:piclatt} Classification of degenerations in Section \ref{sec2:degen} and Tables 1---4 of Section \ref{sec4:tables} contain important classication of Picard lattices of K3 surfaces which is the main subject of this paper. Let $S_X$ be the Picard lattice of K\"ahlerian K3 surface $X$ and $S_X<0$ is negative definite. For algebraic K3 surfaces, instead of $S_X$, one can take $(h)^\perp_{S_X}$ where $h\in S_X$ is a nef element with $h^2>0$. Let $G=Aut(X)_0$ be the full symplectic automorphism subgroup of $X$ (it is finite) and let $E_1,\dots E_k$ are all non-singular rational curves of $X$. Then $$ S=[(S)_G=(S_X)_G, cl(E_1),\dots cl(E_k)]_{pr}\subset S_X $$ is the most important part of the Picard lattice $S_X$ {(\it the main part of $S_X$ or $MS_X$)}. The larger part $S\subset S_X$ of $S_X$ is generated by the main part $S$ and by the {\it exotic} part $ES_X=(S)^\perp_{S_X}$ of $S_X$. Classification of possible $S=MS_X$ is the most important. The main part $S=MS_X$ can be described purely lattice-theoretically for negative definite $S_X$; see \cite[Remark 1.14.7]{Nik1}. Let $H(S_X)\subset O(S_X)$ be the kernel of the action of $O(S_X)$ on the discriminant group $(S_X)^\ast/S_X$ (equivalently, on the discriminant quadratic form $q_{S_X}$). Then the exotic part $ES_X=(S_X)^{H(S_X)}$ is the fixed part of $H(S_X)$ on $S_X$, and the main part $MS_X=(S_X)_{H(S_X)}$ is the coinvariant part of $H(S_X)$. The group $H(S_X)=W^{(2)}(S_X)\rtimes Aut(X)_0$ where $W^{(2)}(S_X)$ is generated by reflections in all elements $\delta\in S_X$ with $\delta^2=-2$. K3 surfaces with the Picard lattice $S$ give the degeneration of K3 surfaces with the symplectic automorphism group $G=Aut (X)_0$. {\it Thus, the lattice $S=MS_X$ is one of lattices of Tables 1 --- 4 (if $G>D_6$ and $k\ge 1$).} The only difference is that {\it the group $G$ must be the maximal finite symplectic automorphism group of K3 surfaces with the Picard lattice $S$.} From the point of view of abstract lattices, $e_1=cl(E_1),\dots , e_k=cl(E_k)$ give a basis of the $(-2)$ root system of $S$ and $$ G\|S=\{g\in O(S)\ \|\ g(\{e_1,...,e_k\})=\{e_1,...,e_k\})\ and\ g\|(S^\ast/S)=id\}. $$ For some two cases $(S_1,G_1)$ and $(S,G)$ of the Tables 1 --- 4, the lattices $S_1\cong S$ can be isomorphic, but their symplectic automorphism groups $G_1\subset G$ are only subgroups of one another with different orders, $\|G_1\|<\|G\|$, where $G$ is the maximal symplectic automorphism group of K3 surfaces with the Picard lattice $S$, but $G_1\subset G$ is only a proper subgroup of the maximal symplectic automorphism group $G$ of K3 surfaces with the Picard lattice $S$. Thus, to get from Tables 1 --- 4 classification of Picards lattices $S$ of K3 surfaces, we should find all such pairs of degenerations $(S_1,G_1)$ and $(S,G)$ of Tables 1---4. Classification of such pairs is given in the List 1 below as pairs $$ (S_1,G_1)\Longleftarrow (S,G). $$ {\it In Tables 1---4, we mark the case $(S_1, G_1)$ by $o$ (old). For the classification of Picard lattices $S$ of K3 surfaces, we can remove this case $(S_1,G_1)$.} But, this case is important and interesting as itself: If a K3 surface has a symplectic automorphism group $G_1$ (not necessarily maximal) and the degeneration $S_1$, then the full symplectic automorphism group of $X$ is larger, it is $G$, with $\|G_1\|<\|G\|$. For $G$, we have less orbits and less codimension of the degeneration. The same happens for the classical case of Kummer surfaces: $\|G_1\|=1$, $G=(C_2)^4$, $S=[16\mathbb A_1]_{pr}$: Kummer surfaces give degeneration of the type $16\mathbb A_1$ (of codim. $16$) of K3 surfaces with trivial symplectic automorphism group, but their full symplectic automorphism group increases to $(C_2)^4$. It was shown in \cite{Nik-1}, that any K3 surface with 16 (sixteen) $\mathbb P^1$ and the Dynkin diagram $16\mathbb A_1$ is Kummer. Thus, we have the following result. \begin{theorem} Classification of the main parts $S=MS_X$ of Picard lattices $S_X<0$ of K3 surfaces with full finite symplectic automorphism group $G$ of order $\|G\|\ge 8$ (equivalently, ${\bf n}\ge 9$, or $G$ is different from $D_6$, $C_4$, $(C_2)^2$, $C_3$, $C_2$ and $C_1$) and with at least one non-singular rational curve is given in Tables 1---4 of Section \ref{sec4:tables} in lines which are not marked by $o$. Lines $(G_1,S_1)$ of Tables 1---4 of Section \ref{sec4:tables} which are marked by $o$ give classification of degenerations $X$ of K\"ahlerian K3 surfaces with finite symplectic automorphism groups $G_1$ such that the full finite symplectic automorphism group $G$ of $X$ is larger than $G_1$, it contains $G_1$ as a proper subroup. Such cases are shown in the List 1 below as $$ (G_1,S_1)\Longleftarrow (G,S) $$ where $(G,S)$ gives the corresponding line of Tables 1---4 with maximal finite symplectic automorphism group $G$ and $S\cong S_1$. \label{theorem1} \end{theorem} \vskip2cm {\bf List 1:} {\it The list or cases, when a degeneration of K3 surfaces with symplectic automorphism group $G_1$ has, actually, the maximal finite symplectic automorphism group $G$ which contains $G_1$ and $\|G_1\|<\|G\|$. The group $G$ has less orbits and less codimension of the degeneration than $G_1$.} $({\bf n}=9,\ ((2\mathbb A_1,2\mathbb A_1)\subset 4\mathbb A_1)_{II})\ \Longleftarrow ({\bf n}=21,4\mathbb A_1)$; $({\bf n}=9,\ (8\mathbb A_1,8\mathbb A_1)\subset 16\mathbb A_1)\ \Longleftarrow ({\bf n}=21,16\mathbb A_1)$; $({\bf n}=9,\ ((2\mathbb A_1,2\mathbb A_1)_{II},4\mathbb A_1)\subset 8\mathbb A_1)\ \Longleftarrow\ ({\bf n}=21,(4\mathbb A_1,4\mathbb A_1)\subset 8\mathbb A_1)$; ({\bf n}=9,\ $ \left(\begin{array}{cccc} 2\mathbb A_1 & (4\mathbb A_1)_I & (4\mathbb A_1)_I & (4\mathbb A_1)_I \\ & 2\mathbb A_1 & (4\mathbb A_1)_I & (4\mathbb A_1)_I \\ & & 2\mathbb A_1 & (4\mathbb A_1)_I \\ & & & 2\mathbb A_1 \end{array}\right) \subset 8\mathbb A_1) \Longleftarrow ({\bf n}=40,\ 8\mathbb A_1)$; $({\bf n}=9,\ ((2\mathbb A_1,2\mathbb A_1)_{II},4\mathbb A_1,4\mathbb A_1) \subset 12\mathbb A_1)\Longleftarrow ({\bf n}=49,\ 12\mathbb A_1)$; ({\bf n}=9,\ $ \left(\begin{array}{cccc} 2\mathbb A_1 & 2\mathbb A_3 & (4\mathbb A_1)_I & 6\mathbb A_1 \\ & 4\mathbb A_1 & 6\mathbb A_1 & 8\mathbb A_1 \\ & & 2\mathbb A_1 & 2\mathbb A_3 \\ & & & 4\mathbb A_1 \end{array}\right) \subset 4\mathbb A_3)\ \Longleftarrow\ ({\bf n}=22,(4\mathbb A_1,8\mathbb A_1)\subset 4\mathbb A_3)$; $({\bf n}=9,\ (4\mathbb A_1,4\mathbb A_1,4\mathbb A_1,4\mathbb A_1) \subset 16\mathbb A_1)\Longleftarrow ({\bf n}=39,\ 16\mathbb A_1)$; ({\bf n}=9,\ $ \left(\begin{array}{ccccc} 2\mathbb A_1 & (4\mathbb A_1)_I & (4\mathbb A_1)_I & (4\mathbb A_1)_I & 10\mathbb A_1 \\ & 2\mathbb A_1 & (4\mathbb A_1)_I & (4\mathbb A_1)_I & 10\mathbb A_1 \\ & & 2\mathbb A_1 & (4\mathbb A_1)_I & 10\mathbb A_1 \\ & & & 2\mathbb A_1 & 10\mathbb A_1\\ & & & & 8\mathbb A_1 \end{array}\right) \subset 16\mathbb A_1 ) \Longleftarrow ({\bf n}=56,\ 16\mathbb A_1)$; ({\bf n}=9,\ $ \left(\begin{array}{ccccc} 2\mathbb A_1 & (4\mathbb A_1)_I & 6\mathbb A_1 & (4\mathbb A_1)_I & 6\mathbb A_1 \\ & 2\mathbb A_1 & 2\mathbb A_3 & (4\mathbb A_1)_I & 6\mathbb A_1 \\ & & 4\mathbb A_1 & 6\mathbb A_1 & 8\mathbb A_1 \\ & & & 2\mathbb A_1 & 2\mathbb A_3\\ & & & & 4\mathbb A_1 \end{array}\right) \subset 2\mathbb A_1\amalg 4\mathbb A_3) $ $\Longleftarrow\ ({\bf n}=22,\ \left(\begin{array}{rrr} 2\mathbb A_1 & 6\mathbb A_1 & 10\mathbb A_1 \\ & 4\mathbb A_1 & 4\mathbb A_3 \\ & & 8\mathbb A_1 \end{array}\right)\subset 2\mathbb A_1\amalg 4\mathbb A_3)$; $({\bf n}=9,\ ((2\mathbb A_1,2\mathbb A_1)_{II},4\mathbb A_1,4\mathbb A_1,4\mathbb A_1) \subset 16\mathbb A_1)\Longleftarrow ({\bf n}=75,\ 16\mathbb A_1)$; $({\bf n}=9, \left(\begin{array}{ccccc} 2\mathbb A_1 & 2\mathbb A_3 & (4\mathbb A_1)_I & 6\mathbb A_1 & 6\mathbb A_1 \\ & 4\mathbb A_1 & 6\mathbb A_1 & 8\mathbb A_1 & 8\mathbb A_1 \\ & & 2\mathbb A_1 & 2\mathbb A_3 & 6\mathbb A_1 \\ & & & 4\mathbb A_1 & 8\mathbb A_1\\ & & & & 4\mathbb A_1 \end{array}\right) \subset 4\mathbb A_3\amalg 4\mathbb A_1) $ $\Longleftarrow ({\bf n}=22,\ \left(\begin{array}{rrr} 4\mathbb A_1 & (8\mathbb A_1)_{II} & 12\mathbb A_1 \\ & 4\mathbb A_1 & 4\mathbb A_3 \\ & & 8\mathbb A_1 \end{array}\right)\subset 4\mathbb A_1\amalg 4\mathbb A_3)$; ({\bf n}=10, $(\mathbb A_1,\mathbb A_1)\subset 2\mathbb A_1)\ \Longleftarrow\ ({\bf n}=22,2\mathbb A_1)$; ({\bf n}=10, $((4\mathbb A_1,4\mathbb A_1)\subset 8\mathbb A_1)_{II}) \Longleftarrow\ ({\bf n}=22,8\mathbb A_1)$; ({\bf n}=10, $(\mathbb A_1,\mathbb A_1,(2\mathbb A_1)_I)\subset 4\mathbb A_1)\ \Longleftarrow ({\bf n}=39,\ 4\mathbb A_1)$; ({\bf n}=10, $(\mathbb A_1,\mathbb A_1,4\mathbb A_1)\subset 6\mathbb A_1)\ \Longleftarrow\ ({\bf n}=22,(2\mathbb A_1,4\mathbb A_1)\subset 6\mathbb A_1)$; ({\bf n}=10, $(\mathbb A_1,\mathbb A_1,8\mathbb A_1)\subset 10\mathbb A_1)\ \Longleftarrow\ ({\bf n}=22,(2\mathbb A_1,8\mathbb A_1)\subset 10\mathbb A_1)$; ({\bf n}=10, $ \left(\begin{array}{ccc} (2\mathbb A_1)_{I} & (6\mathbb A_1)_I & (6\mathbb A_1)_I \\ & 4\mathbb A_1 & (8\mathbb A_1)_{II} \\ & & 4\mathbb A_1 \end{array}\right) \subset 10\mathbb A_1)\ \Longleftarrow\ ({\bf n}=22,(2\mathbb A_1,8\mathbb A_1)\subset 10\mathbb A_1)$; ({\bf n}=10, $ \left(\begin{array}{ccc} (2\mathbb A_1)_{II} & 6\mathbb A_1 & 6\mathbb A_1 \\ & 4\mathbb A_1 & (8\mathbb A_1)_{II} \\ & & 4\mathbb A_1 \end{array}\right) \subset 10\mathbb A_1)\ \Longleftarrow\ ({\bf n}=22,(2\mathbb A_1,8\mathbb A_1)\subset 10\mathbb A_1)$; ({\bf n}=10, $ \left(\begin{array}{ccc} 4\mathbb A_1 & (8\mathbb A_1)_I & (8\mathbb A_1)_{II} \\ & 4\mathbb A_1 & (8\mathbb A_1)_{I} \\ & & 4\mathbb A_1 \end{array}\right) \subset 12\mathbb A_1)\ \Longleftarrow\ ({\bf n}=22,(4\mathbb A_1,8\mathbb A_1)\subset 12\mathbb A_1)$; ({\bf n}=10, $ \left(\begin{array}{ccc} 4\mathbb A_1 & 4\mathbb A_2 & (8\mathbb A_1)_{II} \\ & 4\mathbb A_1 & 4\mathbb A_2 \\ & & 4\mathbb A_1 \end{array}\right) \subset 4\mathbb A_3) \Longleftarrow\ ({\bf n}=22,(4\mathbb A_1,8\mathbb A_1)\subset 4\mathbb A_3)$; ({\bf n}=10,\ $ \left(\begin{array}{ccc} 4\mathbb A_1 & (8\mathbb A_1)_{II} & 12\mathbb A_1 \\ & 4\mathbb A_1 & 12\mathbb A_1 \\ & & 8\mathbb A_1 \end{array}\right) \subset 16\mathbb A_1) \Longleftarrow ({\bf n}=39,\ 16\mathbb A_1)$; $({\bf n}=10,\ (4\mathbb A_1,4\mathbb A_1,2\mathbb A_2)\subset 6\mathbb A_2) \Longleftarrow ({\bf n}=34,\ 6\mathbb A_2)$; ({\bf n}=10,\ $ \left(\begin{array}{cccc} \mathbb A_1 & 2\mathbb A_1 & 3\mathbb A_1 & 5\mathbb A_1 \\ & \mathbb A_1 & 3\mathbb A_1 & 5\mathbb A_1 \\ & & (2\mathbb A_1)_I & (6\mathbb A_1)_I \\ & & & 4\mathbb A_1 \end{array}\right) \subset 8\mathbb A_1) \Longleftarrow ({\bf n}=56,\ 8\mathbb A_1)$; $({\bf n}=10,\ (\mathbb A_1,\mathbb A_1,(2\mathbb A_1)_{I},8\mathbb A_1) \subset 12\mathbb A_1)\Longleftarrow ({\bf n}=65,\ 12\mathbb A_1)$; ({\bf n}=10,\ $ \left(\begin{array}{cccc} \mathbb A_1 & 2\mathbb A_1 & 5\mathbb A_1 & 5\mathbb A_1 \\ & \mathbb A_1 & 5\mathbb A_1 & 5\mathbb A_1 \\ & & 4\mathbb A_1 & (8\mathbb A_1)_I \\ & & & 4\mathbb A_1 \end{array}\right) \subset 10\mathbb A_1)\ \Longleftarrow ({\bf n}=22,\ (2\mathbb A_1, (4\mathbb A_1,4\mathbb A_1)_{II})\subset 10\mathbb A_1)$; $({\bf n}=10,\ (\mathbb A_1,\,\mathbb A_1,\,4\mathbb A_1,\,8\mathbb A_1)\subset 14\mathbb A_1)\ \Longleftarrow\ ({\bf n}=22,\ (2\mathbb A_1,4\mathbb A_1,8\mathbb A_1)\subset 14\mathbb A_1)$; $({\bf n}=10,\ (\mathbb A_1,\,\mathbb A_1,\,4\mathbb A_1,\, 8\mathbb A_1)\subset 2\mathbb A_1\amalg 4\mathbb A_3)$ $\Longleftarrow\ ({\bf n}=22,\ \left(\begin{array}{rrr} 2\mathbb A_1 & 6\mathbb A_1 & 10\mathbb A_1 \\ & 4\mathbb A_1 & 4\mathbb A_3 \\ & & 8\mathbb A_1 \end{array}\right)\subset 2\mathbb A_1\amalg 4\mathbb A_3)$; ({\bf n}=10,\ $ \left(\begin{array}{cccc} \mathbb A_1 & \mathbb A_3 & 3\mathbb A_1 & 5\mathbb A_1 \\ & (2\mathbb A_1)_{II} & 4\mathbb A_1 & 6\mathbb A_1 \\ & & (2\mathbb A_1)_I & 2\mathbb A_3 \\ & & & 4\mathbb A_1 \end{array}\right) \subset 3\mathbb A_3)\ \Longleftarrow ({\bf n}=34,\ (3\mathbb A_1,(6\mathbb A_1)_{II})\subset 3\mathbb A_3)$; ({\bf n}=10,\ $ \left(\begin{array}{cccc} \mathbb A_1 & 3\mathbb A_1 & 5\mathbb A_1 & 9\mathbb A_1 \\ & (2\mathbb A_1)_{I} & (6\mathbb A_1)_I & 10\mathbb A_1 \\ & & 4\mathbb A_1 & 12\mathbb A_1 \\ & & & 8\mathbb A_1 \end{array}\right) \subset 15\mathbb A_1)\ \Longleftarrow ({\bf n}=34,\ (3\mathbb A_1,12\mathbb A_1))\subset 15\mathbb A_1))$; ({\bf n}=10,\ $(\mathbb A_1,\,4\mathbb A_1,\,4\mathbb A_1,\,2\mathbb A_2)\subset \mathbb A_1\amalg 6\mathbb A_2) \Longleftarrow ({\bf n}=34,\ (\mathbb A_1,6\mathbb A_2)\subset \mathbb A_1\amalg 6\mathbb A_2))$; ({\bf n}=10,\ $ \left(\begin{array}{cccc} (2\mathbb A_1)_I & 4\mathbb A_1 & (6\mathbb A_1)_I & (6\mathbb A_1)_{I} \\ & (2\mathbb A_1)_{II} & 6\mathbb A_1 & 6\mathbb A_1 \\ & & 4\mathbb A_1 & (8\mathbb A_1)_{II} \\ & & & 4\mathbb A_1 \end{array}\right) \subset 12\mathbb A_1)\ \Longleftarrow ({\bf n}=65,\ 12\mathbb A_1)$; ({\bf n}=10,\ $ \left(\begin{array}{cccc} (2\mathbb A_1)_I & (6\mathbb A_1)_{II} & (6\mathbb A_1)_I & (6\mathbb A_1)_I \\ & 4\mathbb A_1 & (8\mathbb A_1)_{I} & (8\mathbb A_1)_I \\ & & 4\mathbb A_1 & (8\mathbb A_1)_{II} \\ & & & 4\mathbb A_1 \end{array}\right) \subset 14\mathbb A_1)$ $\Longleftarrow\ ({\bf n}=22,\ (2\mathbb A_1,4\mathbb A_1,8\mathbb A_1)\subset 14\mathbb A_1)$; ({\bf n}=10,\ $ \left(\begin{array}{cccc} (2\mathbb A_1)_{II} & 6\mathbb A_1 & 6\mathbb A_1 & 6\mathbb A_1 \\ & 4\mathbb A_1 &(8\mathbb A_1)_I & (8\mathbb A_1)_I \\ & & 4\mathbb A_1 & (8\mathbb A_1)_{II} \\ & & & 4\mathbb A_1 \end{array}\right) \subset 14\mathbb A_1) $ $\Longleftarrow\ ({\bf n}=22,\ (2\mathbb A_1,4\mathbb A_1,8\mathbb A_1)\subset 14\mathbb A_1)$; ({\bf n}=10,\ $ \left(\begin{array}{cccc} (2\mathbb A_1)_{I} & (6\mathbb A_1)_I & (6\mathbb A_1)_I & (6\mathbb A_1)_I \\ & 4\mathbb A_1 & 4\mathbb A_2 & (8\mathbb A_1)_{II} \\ & & 4\mathbb A_1 & 4\mathbb A_2 \\ & & & 4\mathbb A_1 \end{array}\right) \subset 2\mathbb A_1\amalg 4\mathbb A_3) $,\ ({\bf n}=10,\ $ \left(\begin{array}{cccc} (2\mathbb A_1)_{II} & 6\mathbb A_1 & 6\mathbb A_1 & 6\mathbb A_1 \\ & 4\mathbb A_1 & 4\mathbb A_2 & (8\mathbb A_1)_{II} \\ & & 4\mathbb A_1 & 4\mathbb A_2 \\ & & & 4\mathbb A_1 \end{array}\right) \subset 2\mathbb A_1\amalg 4\mathbb A_3) $ $\Longleftarrow\ ({\bf n}=22,\ \left(\begin{array}{rrr} 2\mathbb A_1 & 6\mathbb A_1 & 10\mathbb A_1 \\ & 4\mathbb A_1 & 4\mathbb A_3 \\ & & 8\mathbb A_1 \end{array}\right)\subset 2\mathbb A_1\amalg 4\mathbb A_3)$; ({\bf n}=10,\ $ \left(\begin{array}{cccc} (2\mathbb A_1)_I & 2\mathbb A_3 & (6\mathbb A_1)_I & 10\mathbb A_1 \\ & 4\mathbb A_1 & (8\mathbb A_1)_{I} & 12\mathbb A_1 \\ & & 4\mathbb A_1 & 4\mathbb A_3 \\ & & & 8\mathbb A_1 \end{array}\right) \subset 6\mathbb A_3) \ \Longleftarrow ({\bf n}=34,\ ((6\mathbb A_1)_I,12\mathbb A_1)\subset 6\mathbb A_3)$; ({\bf n}=10,\ $ \left(\begin{array}{cccc} 4\mathbb A_1 & (8\mathbb A_1)_{II} & (8\mathbb A_1)_I & (8\mathbb A_1)_I \\ & 4\mathbb A_1 & (8\mathbb A_1)_I & (8\mathbb A_1)_I \\ & & 4\mathbb A_1 & (8\mathbb A_1)_{II}\\ & & & 4\mathbb A_1 \end{array}\right) \subset 16\mathbb A_1) \Longleftarrow ({\bf n}=56,\ 16\mathbb A_1)$; ({\bf n}=10,\ $ \left(\begin{array}{cccc} 4\mathbb A_1 & (8\mathbb A_1)_I & (8\mathbb A_1)_I & (8\mathbb A_1)_I \\ & 4\mathbb A_1 & 4\mathbb A_2 & (8\mathbb A_1)_{II}\\ & & 4\mathbb A_1 & 4\mathbb A_2 \\ & & & 4\mathbb A_1 \end{array}\right) \subset 4\mathbb A_1\amalg 4\mathbb A_3) $ $\Longleftarrow ({\bf n}=22,\ \left(\begin{array}{rrr} 4\mathbb A_1 & (8\mathbb A_1)_{II} & 12\mathbb A_1 \\ & 4\mathbb A_1 & 4\mathbb A_3 \\ & & 8\mathbb A_1 \end{array}\right)\subset 4\mathbb A_1\amalg 4\mathbb A_3)$; ({\bf n}=10,\ $ \left(\begin{array}{cccc} 4\mathbb A_1 & (8\mathbb A_1)_{I} & (8\mathbb A_1)_I & 4\mathbb A_1\amalg 2\mathbb A_2 \\ & 4\mathbb A_1 & 4\mathbb A_2 & 4\mathbb A_1\amalg 2\mathbb A_2 \\ & & 4\mathbb A_1 & 4\mathbb A_1\amalg 2\mathbb A_2 \\ & & & 2\mathbb A_2 \end{array}\right) \subset 4\mathbb A_1\amalg 6\mathbb A_2)$\ $\Longleftarrow ({\bf n}=34,\ (4\mathbb A_1,6\mathbb A_2)\subset 4\mathbb A_1\amalg 6\mathbb A_2)$; $({\bf n}=12,\ (8\mathbb A_1,8\mathbb A_1)\subset 16\mathbb A_1)\Longleftarrow ({\bf n}=75,\ 16\mathbb A_1)$; $({\bf n}=12,\ (\mathbb A_2,\mathbb A_2)\subset 2\mathbb A_2)\Longleftarrow ({\bf n}=26,\ 2\mathbb A_2)$; $({\bf n}=16,\ (5\mathbb A_1,5\mathbb A_1,5\mathbb A_1)\subset 15\mathbb A_1) \Longleftarrow ({\bf n}=55,\ 15\mathbb A_1)$; $({\bf n}=17,\ (\mathbb A_1,\mathbb A_1) \subset 2\mathbb A_1)\Longleftarrow ({\bf n}=34,\ 2\mathbb A_1)$; $({\bf n}=17,\ (\mathbb A_1,3\mathbb A_1) \subset 4\mathbb A_1)\Longleftarrow ({\bf n}=49,\ 4\mathbb A_1)$; $({\bf n}=17,\ (4\mathbb A_1,4\mathbb A_1) \subset 8\mathbb A_1)\Longleftarrow ({\bf n}=34,\ 8\mathbb A_1)$; $({\bf n}=17,\ (4\mathbb A_1,12\mathbb A_1) \subset 16\mathbb A_1)\Longleftarrow ({\bf n}=49,\ 16\mathbb A_1)$; $({\bf n}=17,\ (6\mathbb A_1,6\mathbb A_1) \subset 12\mathbb A_1)\Longleftarrow ({\bf n}=49,\ 12\mathbb A_1)$; $({\bf n}=17,\ (6\mathbb A_1,6\mathbb A_1) \subset 6\mathbb A_2)\Longleftarrow ({\bf n}=34,\ 6\mathbb A_2)$; $({\bf n}=17,\ (\mathbb A_1,\mathbb A_1,\mathbb A_1) \subset 3\mathbb A_1)\Longleftarrow ({\bf n}=61,\ 3\mathbb A_1)$; $({\bf n}=17,\ (\mathbb A_1,\mathbb A_1,4\mathbb A_1) \subset 6\mathbb A_1)\Longleftarrow ({\bf n}=34,\ (2\mathbb A_1,4\mathbb A_1)\subset 6\mathbb A_1)$; $({\bf n}=17,\ (\mathbb A_1,\mathbb A_1,6\mathbb A_1) \subset 8\mathbb A_1)\Longleftarrow ({\bf n}=34,\ (2\mathbb A_1,(6\mathbb A_1)_{II})\subset 8\mathbb A_1)$; $({\bf n}=17,\ (\mathbb A_1,\mathbb A_1,12\mathbb A_1)\subset 14\mathbb A_1) \Longleftarrow ({\bf n}=34,\ (2\mathbb A_1,12\mathbb A_1)\subset 14\mathbb A_1)$; $({\bf n}=17,\ (\mathbb A_1,3\mathbb A_1,4\mathbb A_1) \subset 8\mathbb A_1)\Longleftarrow ({\bf n}=65,\ 8\mathbb A_1)$; $({\bf n}=17,\ (\mathbb A_1,3\mathbb A_1,12\mathbb A_1) \subset 16\mathbb A_1)\Longleftarrow ({\bf n}=75,\ 16\mathbb A_1)$; $({\bf n}=17,\ (\mathbb A_1,4\mathbb A_1,4\mathbb A_1) \subset 9\mathbb A_1)\Longleftarrow ({\bf n}=34,\ (\mathbb A_1,8\mathbb A_1)\subset 9\mathbb A_1)$; $({\bf n}=17,\ (\mathbb A_1,6\mathbb A_1,6\mathbb A_1) \subset \mathbb A_1\amalg 6\mathbb A_2) \Longleftarrow ({\bf n}=34,\ (\mathbb A_1,6\mathbb A_2)\subset \mathbb A_1\amalg 6\mathbb A_2)$; $({\bf n}=17,\ (3\mathbb A_1,4\mathbb A_1,4\mathbb A_1)\subset 11\mathbb A_1) \Longleftarrow ({\bf n}=34,\ (3\mathbb A_1,8\mathbb A_1)\subset 11\mathbb A_1)$; $({\bf n}=17,\ (4\mathbb A_1,4\mathbb A_1,4\mathbb A_1)\subset 12\mathbb A_1) \Longleftarrow ({\bf n}=61,\ 12\mathbb A_1)$; ({\bf n}=17,\ $ \left(\begin{array}{rrr} 4\mathbb A_1 & 4\mathbb A_2 & 8\mathbb A_1 \\ & 4\mathbb A_1 & 4\mathbb A_2 \\ & & 4\mathbb A_1 \end{array}\right) \subset 4\mathbb A_3)\ \Longleftarrow ({\bf n}=34,\ (4\mathbb A_1,8\mathbb A_1)\subset 4\mathbb A_3)$; $({\bf n}=17,\ (4\mathbb A_1,4\mathbb A_1,6\mathbb A_1)\subset 14\mathbb A_1)\ \Longleftarrow ({\bf n}=34,\ ((6\mathbb A_1)_I,8\mathbb A_1)\subset 14\mathbb A_1)$; $({\bf n}=17,\ (4\mathbb A_1,6\mathbb A_1,6\mathbb A_1) \subset 16\mathbb A_1)\Longleftarrow ({\bf n}=75,\ 16\mathbb A_1)$; $({\bf n}=17,\ (4\mathbb A_1,6\mathbb A_1,6\mathbb A_1)\subset 4\mathbb A_1\amalg 6\mathbb A_2) \Longleftarrow ({\bf n}=34,\ (4\mathbb A_1,6\mathbb A_2)\subset 4\mathbb A_1\amalg 6\mathbb A_2)$; $({\bf n}=21,\ (4\mathbb A_1, 4\mathbb A_1,4\mathbb A_1) \subset 12\mathbb A_1)\Longleftarrow ({\bf n}=49,\ 12\mathbb A_1)$; $({\bf n}=21,\ (4\mathbb A_1, 4\mathbb A_1,4\mathbb A_1,4\mathbb A_1) \subset 16\mathbb A_1)\Longleftarrow ({\bf n}=75,\ 16\mathbb A_1)$; $({\bf n}=22,\ (2\mathbb A_1,2\mathbb A_1) \subset 4\mathbb A_1)\Longleftarrow ({\bf n}=39,\ 4\mathbb A_1)$; $({\bf n}=22,\ ((4\mathbb A_1,4\mathbb A_1) \subset 8\mathbb A_1)_I)\Longleftarrow ({\bf n}=40,\ 8\mathbb A_1)$; $({\bf n}=22,\ (8\mathbb A_1,8\mathbb A_1) \subset 16\mathbb A_1)\ \Longleftarrow ({\bf n}=39,\ 16\mathbb A_1)$; $({\bf n}=22,\ (2\mathbb A_1, 2\mathbb A_1,4\mathbb A_1) \subset 8\mathbb A_1)\Longleftarrow ({\bf n}=56,\ 8\mathbb A_1)$; $({\bf n}=22,\ (2\mathbb A_1, 2\mathbb A_1,8\mathbb A_1) \subset 12\mathbb A_1)\Longleftarrow ({\bf n}=65,\ 12\mathbb A_1)$; $({\bf n}=22,\ ((4\mathbb A_1,4\mathbb A_1)_I,8\mathbb A_1) \subset 16\mathbb A_1)\Longleftarrow ({\bf n}=56,\ 16\mathbb A_1)$; $({\bf n}=34,\ (\mathbb A_1,\mathbb A_1)\subset 2\mathbb A_1)\Longleftarrow ({\bf n}=51,\ 2\mathbb A_1)$; $({\bf n}=34,\ (\mathbb A_1,2\mathbb A_1)\subset 3\mathbb A_1)\Longleftarrow ({\bf n}=61,\ 3\mathbb A_1)$; $({\bf n}=34,\ (\mathbb A_1,3\mathbb A_1)\subset 4\mathbb A_1)\Longleftarrow ({\bf n}=65,\ 4\mathbb A_1)$; $({\bf n}=34,\ (4\mathbb A_1,4\mathbb A_1)\subset 8\mathbb A_1)\Longleftarrow ({\bf n}=51,\ 8\mathbb A_1)$; $({\bf n}=34,\ (4\mathbb A_1,8\mathbb A_1)\subset 12\mathbb A_1)\Longleftarrow ({\bf n}=61,\ 12\mathbb A_1)$; $({\bf n}=34,\ (4\mathbb A_1,12\mathbb A_1)\subset 16\mathbb A_1)\Longleftarrow ({\bf n}=65,\ 16\mathbb A_1)$; $({\bf n}=34,\ ((6\mathbb A_1)_{I},(6\mathbb A_1)_{II})\subset 12\mathbb A_1)\Longleftarrow ({\bf n}=65,\ 12\mathbb A_1)$; $({\bf n}=39,\ (4\mathbb A_1,4\mathbb A_1)\subset 8\mathbb A_1)\Longleftarrow ({\bf n}=56,\ 8\mathbb A_1)$; $({\bf n}=39,\ (4\mathbb A_1,8\mathbb A_1)\subset 12\mathbb A_1)\Longleftarrow ({\bf n}=65,\ 12\mathbb A_1)$; $({\bf n}=39,\ (8\mathbb A_1,8\mathbb A_1)\subset 16\mathbb A_1)\Longleftarrow ({\bf n}=75,\ 16\mathbb A_1)$; $({\bf n}=40,\ (8\mathbb A_1,8\mathbb A_1)\subset 16\mathbb A_1)\Longleftarrow ({\bf n}=56,\ 16\mathbb A_1)$; $({\bf n}=49,\ (4\mathbb A_1,4\mathbb A_1)\subset 8\mathbb A_1)\Longleftarrow ({\bf n}=65,\ 8\mathbb A_1)$; $({\bf n}=49,\ (4\mathbb A_1,12\mathbb A_1)\subset 16\mathbb A_1)\Longleftarrow ({\bf n}=75,\ 16\mathbb A_1)$. \begin{proof} Assume that for two degenerations given by two lines $({\bf n}, Deg, S)$ and $({\bf n}_1, Deg_1, S_1)$ of Tables 1---4 we have \begin{equation} ({\bf n_1}, Deg_1, S_1)\Longleftarrow ({\bf n}, Deg, S). \label{pairslist1} \end{equation} Then the group $G$ (marked by ${\bf n}$) is maximal symplectic automorphism group for $S$, or shortly \begin{equation} the\ degeneration\ ({\bf n},Deg,S)\ is\ maximal; \label{A} \end{equation} group $G_1$ (marked by ${\bf n}_1$) is isomorphic to a proper subgroup of $G$ (marked by ${\bf n}$), or shortly, \begin{equation} {\bf n}_1\subset {\bf n}, \label{B} \end{equation} full Dynkin diagrams of $Deg$ and $Deg_1$ are isomorphic: \begin{equation} Dyn(Deg_1)\cong Dyn(Deg); \label{C} \end{equation} genuses of lattices $S_1$ and $S$ are equal, equivalently, \begin{equation} \rk S_1=\rk S\ and\ q_{S_1}\cong q_S; \label{D} \end{equation} finally, each marking by Niemeier lattices of the degeneration $({\bf n}_1, Deg_1, S_1)$ is isomorphic to a restriction on an appropriate subgroup $G_1\subset G$ of a marking of the degeneration $({\bf n}, Deg, S)$ by Niemeier lattices, or shortly, \begin{equation} Niemeier\ markings\ of\ ({\bf n}_1, Deg_1, S_1)\ and\ ({\bf n}, Deg, S)\ agree. \label{E} \end{equation} Using calculations by Hashimoto in \cite{Hash} (which can be easily done by the Program GAP, see \cite{GAP}), we can find all possible pairs ${\bf n}$, ${\bf n_1}$ from Table 1 such that ${\bf n}\supset {\bf n}_1$. They are given in the list of subgroups \eqref{listsubgroups} below. \begin{equation} {\bf List\ of\ subgroups} \label{listsubgroups} \end{equation} $$ {\bf n}=75\supset {\bf n}_1=1,2,3,4,9,10,12,17,21,22,39,49; $$ $$ {\bf n}=65\supset {\bf n}_1=1,2,3,4,6,9,10,17,21,22,34,39,49; $$ $$ {\bf n}=61\supset {\bf n}_1=1,2,3,4,6,10,17,18,30,34; $$ $$ {\bf n}=56 \supset {\bf n}_1=1,3,4,9,10,21,22,39,40; $$ $$ {\bf n}=55 \supset {\bf n}_1=1,2,3,6,16,17; $$ $$ {\bf n}=51 \supset {\bf n}_1=1,2,3,4,6,9,10,17,18,22,34; $$ $$ {\bf n}=49 \supset {\bf n}_1=1,2,3,9,17,21; $$ $$ {\bf n}=48 \supset {\bf n}_1=1,2,3,6,18,30; $$ $$ {\bf n}=46 \supset {\bf n}_1=1,2,4,6,30; $$ $$ {\bf n}=40 \supset {\bf n}_1=1,3,4,9,10,22; $$ $$ {\bf n}=39 \supset {\bf n}_1=1,3,4,9,10,21,22; $$ $$ {\bf n}=34 \supset {\bf n}_1=1,2,3,4,6,10,17; $$ $$ {\bf n}=33 \supset {\bf n}_1=2; $$ $$ {\bf n}=32 \supset {\bf n}_1=1,4,16; $$ $$ {\bf n}=30 \supset {\bf n}_1=1,2,6; $$ $$ {\bf n}=26 \supset {\bf n}_1=1,3,4,10,12; $$ $$ {\bf n}=22 \supset {\bf n}_1=1,3,4,9,10; $$ $$ {\bf n}=21 \supset {\bf n}_1=1,3,9; $$ $$ {\bf n}=18 \supset {\bf n}_1=1,2,3,6; $$ $$ {\bf n}=17 \supset {\bf n}_1=1,2,3; $$ $$ {\bf n}=16 \supset {\bf n}_1=1; $$ $$ {\bf n}=12 \supset {\bf n}_1=1,4; $$ $$ {\bf n}=10 \supset {\bf n}_1=1,3,4; $$ $$ {\bf n}=9 \supset {\bf n}_1=1,3; $$ $$ {\bf n}=6 \supset {\bf n}_1=1,2; $$ $$ {\bf n}=4 \supset {\bf n}_1=1; $$ $$ {\bf n}=3 \supset {\bf n}_1=1. $$ All degenerations $({\bf n}, Deg, S)$ with one orbit of $(-2)$-curves (equivalently, of the codimension $1$) are, obviously, maximal. They are classified in Table 1. Using the list of subgroups \eqref{listsubgroups}, Tables 1---4 and our description of markings by Niemeier lattices in \cite{Nik10} and \cite{Nik11}, firstly, we find all pairs $({\bf n_1}, Deg_1, S_1)\Longleftarrow ({\bf n}, Deg, S)$ where the degeneration $({\bf n}, Deg, S)$ has one orbit. They give pairs of List 1 where $({\bf n}, Deg, S)$ belongs to Table 1. The corresponding cases $({\bf n_1}, Deg_1, S_1)$ we mark by $o$ in Tables 1---4. Then all remaining non-marked by $o$ degenerations (lines of Tables 2---4) with two orbits (equivalently, of the codimension $2$) are maximal. For them, ${\bf n}\le 34$. They are given in Tables 2---4 as degenerations with two orbits which are not marked by $o$. Let us denote the set of them by $D_2$. Next, in 3rd step, using the list of subgroups \eqref{listsubgroups}, Tables 2---4 and our description of markings by Niemeier lattices in \cite{Nik10} and \cite{Nik11}, we find all pairs $({\bf n_1}, Deg_1, S_1)\Longleftarrow ({\bf n}, Deg, S)$ where the degeneration $({\bf n}, Deg, S)$ belongs to the set $D_2$. They are given in List 1 as pairs $({\bf n_1}, Deg_1, S_1)\Longleftarrow ({\bf n}, Deg, S)$ where the degeneration $({\bf n}, Deg, S)$ has two orbits. The corresponding degenerations $({\bf n_1}, Deg_1, S_1)$ we mark by $o$. Let us denote the set of remaining (not marked by $o$) degenerations with $3$ orbits of Tables 2---4 as $D_3$. For them, ${\bf n}\le 30$. Next, in 4th step, using the list of subgroups \eqref{listsubgroups}, Tables 2---4 and our description of markings by Niemeier lattices in \cite{Nik10} and \cite{Nik11}, we find all pairs $({\bf n_1}, Deg_1, S_1)\Longleftarrow ({\bf n}, Deg, S)$ where the degeneration $({\bf n}, Deg, S)$ belongs to $D_3$. They are given in List 1 as pairs $({\bf n_1}, Deg_1, S_1)\Longleftarrow ({\bf n}, Deg, S)$ where the degeneration $({\bf n}, Deg, S)$ has three orbits (equivalently, it has the codimension $3$). The corresponding degenerations $({\bf n_1}, Deg_1, S_1)$ of Tables 2---4 we mark by $o$. Let us denote the set of remaining (nor marked by $o$) degenerations with $4$ orbits (equivalently, of the codimension $4$) of Tables 2---4 as $D_4$. For them, ${\bf n}\le 10$. Next, in 5th step, using the list of subgroups \eqref{listsubgroups}, Tables 2---4, we find all pairs \linebreak $({\bf n_1}, Deg_1, S_1)\Longleftarrow ({\bf n}, Deg, S)$ where the degeneration $({\bf n}, Deg, S)$ belongs to $D_4$ and ${\bf n}_1\ge 8$. We see that there are no such pairs satisfying conditions \eqref{A}, \eqref{B}. This gives the List 1 of the Theorem. Below, for all cases of the List 1, we give details of the check of the most complicated condition \eqref{E}. They are also important as itself because they describe exact isomorphisms between lattices $S$ and $S_1$ of the List 1. Case 1: $({\bf n}=9,\ ((2\mathbb A_1,2\mathbb A_1)\subset 4\mathbb A_1)_{II})\ \Longleftarrow ({\bf n}=21,4\mathbb A_1)$. For ${\bf n}=21$, the group $G\cong (C_2)^4$. By \cite{Nik10}, the case $({\bf n}=21,4\mathbb A_1)$ is marked by the Niemeier lattice $N_{23}$ with $G=H_{21,1}\subset A_{23}$ and by any of $4$-elements orbits $\{\alpha_1,\alpha_{16},\alpha_{14},\alpha_{18}\}$, $\{\alpha_2,\alpha_{20},\alpha_{19},\alpha_{24}\}$, $\{\alpha_3,\alpha_{10},\alpha_{5},\alpha_6\}$, $\{\alpha_8,\alpha_{11},\alpha_{9},\alpha_{21}\}$, $\{\alpha_{12},\alpha_{22},\alpha_{23},\alpha_{17}\}$ of $H_{21,1}$ in the set of simple $(-2)$-roots $\alpha_1, \dots \alpha_{24}$ of $N_{23}$. All other orbits of $H_{21,1}$ have one element. Exactly (see \cite{Nik10}), the group $G=H_{21,1}$ is equal to $$ G=H_{21,1}= $$ $$ [(\alpha_2\alpha_{20})(\alpha_{3}\alpha_{10})(\alpha_{5}\alpha_{6})(\alpha_{8}\alpha_{11}) (\alpha_{9}\alpha_{21})(\alpha_{12}\alpha_{22})(\alpha_{17}\alpha_{23})(\alpha_{19}\alpha_{24}), $$ $$ (\alpha_{2}\alpha_{19})(\alpha_{3}\alpha_{5})(\alpha_{6}\alpha_{10})(\alpha_{8}\alpha_{9}) (\alpha_{11}\alpha_{21})(\alpha_{12}\alpha_{23})(\alpha_{17}\alpha_{22})(\alpha_{20}\alpha_{24}), $$ $$ (\alpha_{1}\alpha_{16})(\alpha_{2}\alpha_{20})(\alpha_{3}\alpha_{6})(\alpha_{5}\alpha_{10}) (\alpha_{12}\alpha_{23})(\alpha_{14}\alpha_{18})(\alpha_{17}\alpha_{22})(\alpha_{19}\alpha_{24}), $$ $$ (\alpha_{1}\alpha_{14})(\alpha_{2}\alpha_{24})(\alpha_{3}\alpha_{5})(\alpha_{6}\alpha_{10}) (\alpha_{12}\alpha_{22})(\alpha_{16}\alpha_{18})(\alpha_{17}\alpha_{23})(\alpha_{19}\alpha_{20})]. $$ Here, we denote the corresponding subgroup of the symmetric group $\SSS_{24}$ on $\alpha_{1},\dots, \alpha_{24}$ by $G$ too. For ${\bf n}=9$, the group $G_1\cong (C_2)^3$ has the order $8$ and the identificator $i=5$ (see Hashimoto \cite{Hash}) for the program GAP (see \cite{GAP}). The group $G_1$ is isomorphic to ($G1:={\rm SmallGroup}(8,5);$) for the program GAP. Using the program GAP, we find all conjugacy classes \newline (${\rm F}:={\rm IsomorphicSubgroups}(G,G1)$;) of $G1$ in $G$. There are $15$ conjugacy classes. For the first of them $F[1]$ (for other conjugacy classes, the result is similar), we find the corresponding subgroup ($G_1:={\rm Image}(F[1]);)\subset G=H_{21,1}$ and its orbits (${\rm ORB}:={\rm Orbits}(G_1);$). These orbits of order $>1$ are $\{\alpha_1,\alpha_{14}\}$, $\{\alpha_{16},\alpha_{18}\}$, $\{\alpha_2,\alpha_{20},\alpha_{19},\alpha_{24}\}$, $\{\alpha_3,\alpha_{10},\alpha_{5},\alpha_6\}$, $\{\alpha_8,\alpha_{11},\alpha_{9},\alpha_{21}\}$, $\{\alpha_{12},\alpha_{22},\alpha_{23},\alpha_{17}\}$. Thus, for $({\bf n}=21,4\mathbb A_1)$ marked by $H_{21,1}$ with the orbit $\{\alpha_1,\alpha_{16},\alpha_{14},\alpha_{18}\}$, for the subgroup $G_1\subset G=H_{21,1}$, the orbit splits in two suborbits $\{\alpha_1,\alpha_{14}\}$, $\{\alpha_{16},\alpha_{18}\}$ of the order two, and it gives $({\bf n}=9,\ ((2\mathbb A_1,2\mathbb A_1)\subset 4\mathbb A_1)_{II})$ with the same lattice. More exactly (by the program GAP), the corresponding subgroup $G_1\subset G$ is equal to $$ G=H_{21,2}\supset G_1= $$ $$ [(\alpha_{2}\alpha_{19})(\alpha_{3}\alpha_{5})(\alpha_{6}\alpha_{10})(\alpha_{8}\alpha_{9}) (\alpha_{11}\alpha_{21})(\alpha_{12}\alpha_{23})(\alpha_{17}\alpha_{22})(\alpha_{20}\alpha_{24}), $$ $$ (\alpha_{2}\alpha_{20})(\alpha_{3}\alpha_{10})(\alpha_{5}\alpha_{6})(\alpha_{8}\alpha_{11}) (\alpha_{9}\alpha_{21})(\alpha_{12}\alpha_{22})(\alpha_{17}\alpha_{23})(\alpha_{19}\alpha_{24}), $$ $$ (\alpha_{1}\alpha_{14})(\alpha_{3}\alpha_{10})(\alpha_{5}\alpha_{6})(\alpha_{8}\alpha_{21}) (\alpha_{9}\alpha_{11})(\alpha_{12}\alpha_{23})(\alpha_{16}\alpha_{18})(\alpha_{17}\alpha_{22})]. $$ Since both cases $({\bf n}=9,\ ((2\mathbb A_1,2\mathbb A_1)\subset 4\mathbb A_1)_{II})$ and $({\bf n}=21,4\mathbb A_1)$ have a unique lattices $S_1$ and $S$, up to isomorphism (they are marked by $\ast$ in Tables 1---4; for Table 1, all lattices are marked by $\ast$ by definition)), they have isomorphic lattices $S_1\cong S$ at any case: it is enough to find just one case when they are isomorphic. Case 2: $({\bf n}=9,\ (8\mathbb A_1,8\mathbb A_1)\subset 16\mathbb A_1)\ \Longleftarrow ({\bf n}=21,16\mathbb A_1)$. Similar to Case 1. By \cite{Nik10}, the $G\cong (C_2)^4$ is marked by $N_{23}$ and $$ G=H_{21,2}= $$ $$ [(\alpha_{1}\alpha_{3})(\alpha_{2}\alpha_{23}) (\alpha_{5}\alpha_{14})(\alpha_{6}\alpha_{16}) (\alpha_{10}\alpha_{18})(\alpha_{12}\alpha_{20}) (\alpha_{17}\alpha_{24})(\alpha_{19}\alpha_{22}), $$ $$ (\alpha_{1}\alpha_{2})(\alpha_{3}\alpha_{23}) (\alpha_{5}\alpha_{17})(\alpha_{6}\alpha_{12}) (\alpha_{10}\alpha_{22})(\alpha_{14}\alpha_{24}) (\alpha_{16}\alpha_{20})(\alpha_{18}\alpha_{19}), $$ $$ (\alpha_{1}\alpha_{16})(\alpha_{2}\alpha_{20}) (\alpha_{3}\alpha_{6})(\alpha_{5}\alpha_{10}) (\alpha_{12}\alpha_{23})(\alpha_{14}\alpha_{18}) (\alpha_{17}\alpha_{22})(\alpha_{19}\alpha_{24}), $$ $$ (\alpha_{1}\alpha_{14})(\alpha_{2}\alpha_{24}) (\alpha_{3}\alpha_{5})(\alpha_{6}\alpha_{10}) (\alpha_{12}\alpha_{22})(\alpha_{16}\alpha_{18}) (\alpha_{17}\alpha_{23})(\alpha_{19}\alpha_{20})] $$ with the orbit $ \{\alpha_{1},\alpha_{3},\alpha_{2},\alpha_{16},\alpha_{14}, \alpha_{23},\alpha_{6},\alpha_{5}, \alpha_{20},\alpha_{24},\alpha_{18},\alpha_{12},\alpha_{17}, \alpha_{10},\alpha_{19},\alpha_{22}\} $. The $G_1\cong (C_2)^3$ is marked by $$ G=H_{21,2}\supset G_1= $$ $$ [(\alpha_{1}\alpha_{3})(\alpha_{2}\alpha_{23}) (\alpha_{5}\alpha_{14})(\alpha_{6}\alpha_{16}) (\alpha_{10}\alpha_{18})(\alpha_{12}\alpha_{20}) (\alpha_{17}\alpha_{24})(\alpha_{19}\alpha_{22}), $$ $$ (\alpha_{1}\alpha_{2})(\alpha_{3}\alpha_{23}) (\alpha_{5}\alpha_{17})(\alpha_{6}\alpha_{12}) (\alpha_{10}\alpha_{22})(\alpha_{14}\alpha_{24}) (\alpha_{16}\alpha_{20})(\alpha_{18}\alpha_{19}), $$ $$ (\alpha_{1}\alpha_{16})(\alpha_{2}\alpha_{20}) (\alpha_{3}\alpha_{6})(\alpha_{5}\alpha_{10}) (\alpha_{12}\alpha_{23})(\alpha_{14}\alpha_{18}) (\alpha_{17}\alpha_{22})(\alpha_{19}\alpha_{24})] $$ with suborbits $\{\alpha_{1},\alpha_{3},\alpha_{2},\alpha_{16},\alpha_{23},\alpha_{6},\alpha_{20},\alpha_{12}\}$, $\{\alpha_{5},\alpha_{14},\alpha_{17},\alpha_{10},\alpha_{24},\alpha_{18},\alpha_{22},\alpha_{19}\}$ (of the orbit of $G$ above). Both $G$ and $G_1$ are marked by $\ast$ in Tables 1---4. Case 3: $({\bf n}=9,\ ((2\mathbb A_1,2\mathbb A_1)_{II},4\mathbb A_1)\subset 8\mathbb A_1)\ \Longleftarrow\ ({\bf n}=21,(4\mathbb A_1,4\mathbb A_1)\subset 8\mathbb A_1)$. Similar to Case 1. By \cite{Nik11}, the $G\cong (C_2)^4$ is marked by $N_{23}$ and $G=H_{21,2}$ of Case 1 with orbits $\{\alpha_1,\alpha_{16},\alpha_{14},\alpha_{18}\}$, $\{\alpha_2,\alpha_{20},\alpha_{19},\alpha_{24}\}$. The $G_1\cong (C_2)^3$ is marked by $G_1\subset H_{21,2}$ of Case 1 with suborbits $\{\alpha_1,\alpha_{14}\}$, $\{\alpha_{16},\alpha_{18}\}$, $\{\alpha_2,\alpha_{20},\alpha_{19},\alpha_{24}\}$. Both $G$ and $G_1$ are marked by $\ast$ in the tables 1---4. Case 4: ({\bf n}=9,\ $ \left(\begin{array}{cccc} 2\mathbb A_1 & (4\mathbb A_1)_I & (4\mathbb A_1)_I & (4\mathbb A_1)_I \\ & 2\mathbb A_1 & (4\mathbb A_1)_I & (4\mathbb A_1)_I \\ & & 2\mathbb A_1 & (4\mathbb A_1)_I \\ & & & 2\mathbb A_1 \end{array}\right) \subset 8\mathbb A_1) \Longleftarrow ({\bf n}=40,\ 8\mathbb A_1)$. Similar to Case 1. By \cite{Nik10}, the $G\cong Q_8\ast Q_8$ is marked by $N_{23}$ and $$ G=H_{40,1}= $$ $$ [(\alpha_{1}\alpha_{16})(\alpha_{2}\alpha_{18}) (\alpha_{4}\alpha_{9})(\alpha_{5}\alpha_{23}) (\alpha_{10}\alpha_{12})(\alpha_{14}\alpha_{20}) (\alpha_{15}\alpha_{21})(\alpha_{19}\alpha_{24}), $$ $$ (\alpha_{3}\alpha_{5})(\alpha_{6}\alpha_{12}) (\alpha_{8}\alpha_{13})(\alpha_{10}\alpha_{22}) (\alpha_{15}\alpha_{21})(\alpha_{16}\alpha_{19}) (\alpha_{17}\alpha_{23})(\alpha_{18}\alpha_{20}), $$ $$ (\alpha_{1}\alpha_{16})(\alpha_{2}\alpha_{20}) (\alpha_{3}\alpha_{6})(\alpha_{5}\alpha_{10}) (\alpha_{12}\alpha_{23})(\alpha_{14}\alpha_{18}) (\alpha_{17}\alpha_{22})(\alpha_{19}\alpha_{24}), $$ $$ (\alpha_{1}\alpha_{14})(\alpha_{2}\alpha_{24}) (\alpha_{3}\alpha_{5})(\alpha_{6}\alpha_{10}) (\alpha_{12}\alpha_{22})(\alpha_{16}\alpha_{18}) (\alpha_{17}\alpha_{23})(\alpha_{19}\alpha_{20})] $$ with the orbit $\{\alpha_{1},\alpha_{16},\alpha_{14},\alpha_{19},\alpha_{18},\alpha_{20}, \alpha_{24},\alpha_{2}\}$. The $G_1\cong (C_2)^3$ is marked by $$ G=H_{40,1}\supset G_1= $$ $$ [(\alpha_{1}\alpha_{24})(\alpha_{2}\alpha_{14})(\alpha_{3}\alpha_{23})(\alpha_{5}\alpha_{17}) (\alpha_{6}\alpha_{10})(\alpha_{8}\alpha_{13})(\alpha_{12}\alpha_{22})(\alpha_{15}\alpha_{21}), $$ $$ (\alpha_{3}\alpha_{5})(\alpha_{6}\alpha_{12})(\alpha_{8}\alpha_{13})(\alpha_{10}\alpha_{22}) (\alpha_{15}\alpha_{21})(\alpha_{16}\alpha_{19})(\alpha_{17}\alpha_{23})(\alpha_{18}\alpha_{20}), $$ $$ (\alpha_{2}\alpha_{14})(\alpha_{3}\alpha_{6})(\alpha_{4}\alpha_{9})(\alpha_{5}\alpha_{12}) (\alpha_{10}\alpha_{23})(\alpha_{15}\alpha_{21})(\alpha_{17}\alpha_{22})(\alpha_{18}\alpha_{20})] $$ with suborbits $\{\alpha_{1},\alpha_{24}\}$, $\{\alpha_{2},\alpha_{14}\}$, $\{\alpha_{16},\alpha_{19}\}$, $\{\alpha_{18},\alpha_{20}\}$. Both $G$ and $G_1$ are marked by $\ast$ in Tables 1---4. Case 5: $({\bf n}=9,\ ((2\mathbb A_1,2\mathbb A_1)_{II},4\mathbb A_1,4\mathbb A_1) \subset 12\mathbb A_1)\Longleftarrow ({\bf n}=49,\ 12\mathbb A_1)$. Similar to Case 1. By \cite{Nik10}, the $G\cong 2^4C_3$ is marked by $N_{23}$ and $$ G=H_{49,1}= $$ $$ [(\alpha_{1}\alpha_{22}\alpha_{19}) (\alpha_{3}\alpha_{16}\alpha_{17}) (\alpha_{4}\alpha_{20}\alpha_{9}) (\alpha_{7}\alpha_{10}\alpha_{8}) (\alpha_{12}\alpha_{13}\alpha_{23}) (\alpha_{14}\alpha_{18}\alpha_{21}), $$ $$ (\alpha_{2}\alpha_{12})(\alpha_{3}\alpha_{8}) (\alpha_{4}\alpha_{20})(\alpha_{7}\alpha_{16}) (\alpha_{9}\alpha_{11})(\alpha_{13}\alpha_{23}) (\alpha_{14}\alpha_{22})(\alpha_{18}\alpha_{19}), $$ $$ (\alpha_{2}\alpha_{13})(\alpha_{3}\alpha_{22}) (\alpha_{4}\alpha_{9})(\alpha_{7}\alpha_{18}) (\alpha_{8}\alpha_{14})(\alpha_{11}\alpha_{20}) (\alpha_{12}\alpha_{23})(\alpha_{16}\alpha_{19})] $$ with the orbit $\{\alpha_{1},\alpha_{22},\alpha_{21},\alpha_{10},\alpha_{19},\alpha_{14}, \alpha_{8},\alpha_{17},\alpha_{18},\alpha_{7},\alpha_{3},\alpha_{16}\}$. The $G_1\cong (C_2)^3$ is marked by $$ G=H_{49,1}\supset G_1= $$ $$ [(\alpha_{1}\alpha_{21})(\alpha_{2}\alpha_{23})(\alpha_{3}\alpha_{8})(\alpha_{4}\alpha_{9}) (\alpha_{10}\alpha_{17})(\alpha_{11}\alpha_{20})(\alpha_{12}\alpha_{13})(\alpha_{14}\alpha_{22}), $$ $$ (\alpha_{2}\alpha_{12})(\alpha_{3}\alpha_{8})(\alpha_{4}\alpha_{20})(\alpha_{7}\alpha_{16}) (\alpha_{9}\alpha_{11})(\alpha_{13}\alpha_{23})(\alpha_{14}\alpha_{22})(\alpha_{18}\alpha_{19}), $$ $$ (\alpha_{2}\alpha_{13})(\alpha_{3}\alpha_{22})(\alpha_{4}\alpha_{9})(\alpha_{7}\alpha_{18}) (\alpha_{8}\alpha_{14})(\alpha_{11}\alpha_{20})(\alpha_{12}\alpha_{23})(\alpha_{16}\alpha_{19})] $$ with suborbits $\{\alpha_{1},\alpha_{21}\}$, $\{\alpha_{10},\alpha_{17}\}$, $\{\alpha_{3},\alpha_{8},\alpha_{22},\alpha_{14}\}$, $\{\alpha_{7},\alpha_{16},\alpha_{18},\alpha_{19}\}$. Both $G$ and $G_1$ are marked by $\ast$ in Tables 1---4. Case 6: ({\bf n}=9,\ $ \left(\begin{array}{cccc} 2\mathbb A_1 & 2\mathbb A_3 & (4\mathbb A_1)_I & 6\mathbb A_1 \\ & 4\mathbb A_1 & 6\mathbb A_1 & 8\mathbb A_1 \\ & & 2\mathbb A_1 & 2\mathbb A_3 \\ & & & 4\mathbb A_1 \end{array}\right) \subset 4\mathbb A_3)\ \Longleftarrow\ ({\bf n}=22,(4\mathbb A_1,8\mathbb A_1)\subset 4\mathbb A_3)$. Similar to Case 1. By \cite{Nik11}, the $G\cong C_2\times D_8$ is marked by $N_{21}$ and $$ G=H_{22,1}= $$ $$ [(\alpha_{1,3}\alpha_{3,3})(\alpha_{1,4}\alpha_{3,5}) (\alpha_{2,4}\alpha_{2,5})(\alpha_{3,4}\alpha_{1,5}) (\alpha_{1,6}\alpha_{1,8}) (\alpha_{2,6}\alpha_{2,8}) (\alpha_{3,6}\alpha_{3,8})(\alpha_{1,7}\alpha_{3,7}), $$ $$ (\alpha_{1,2}\alpha_{3,2})(\alpha_{1,3}\alpha_{3,6}) (\alpha_{2,3}\alpha_{2,6})(\alpha_{3,3}\alpha_{1,6}) (\alpha_{1,4}\alpha_{3,4})(\alpha_{1,7}\alpha_{1,8}) (\alpha_{2,7}\alpha_{2,8})(\alpha_{3,7}\alpha_{3,8}), $$ $$ (\alpha_{1,1}\alpha_{3,1})(\alpha_{1,2}\alpha_{3,2}) (\alpha_{1,3}\alpha_{3,3})(\alpha_{1,4}\alpha_{3,4}) (\alpha_{1,5}\alpha_{3,5})(\alpha_{1,6}\alpha_{3,6}) (\alpha_{1,7}\alpha_{3,7})(\alpha_{1,8}\alpha_{3,8}) ] $$ with orbits $\{\alpha_{2,3},\alpha_{2,6},\alpha_{2,8},\alpha_{2,7}\}$, $\{\alpha_{1,3},\alpha_{3,3},\alpha_{3,6},\alpha_{1,6},\alpha_{3,8}, \alpha_{1,8},\alpha_{3,7},\alpha_{1,7}\}$. The $G_1\cong (C_2)^3$ is marked by $$ G=H_{22,1}\supset G_1= $$ $$ [(\alpha_{1,3}\alpha_{3,7})(\alpha_{2,3}\alpha_{2,7})(\alpha_{3,3}\alpha_{1,7})(\alpha_{1,4}\alpha_{1,5}) (\alpha_{2,4}\alpha_{2,5})(\alpha_{3,4}\alpha_{3,5})(\alpha_{1,6}\alpha_{3,6})(\alpha_{1,8}\alpha_{3,8}), $$ $$ (\alpha_{1,3}\alpha_{3,3})(\alpha_{1,4}\alpha_{3,5})(\alpha_{2,4}\alpha_{2,5})(\alpha_{3,4}\alpha_{1,5}) (\alpha_{1,6}\alpha_{1,8})(\alpha_{2,6}\alpha_{2,8})(\alpha_{3,6}\alpha_{3,8})(\alpha_{1,7}\alpha_{3,7}), $$ $$ (\alpha_{1,1}\alpha_{3,1})(\alpha_{1,2}\alpha_{3,2})(\alpha_{1,4}\alpha_{1,5})(\alpha_{2,4}\alpha_{2,5}) (\alpha_{3,4}\alpha_{3,5})(\alpha_{1,6}\alpha_{3,8})(\alpha_{2,6}\alpha_{2,8})(\alpha_{3,6}\alpha_{1,8})] $$ with suborbits $\{\alpha_{2,3},\alpha_{2,7}\}$, $\{\alpha_{1,3},\alpha_{3,7},\alpha_{3,3},\alpha_{1,7}\}$, $\{\alpha_{2,6},\alpha_{2,8}\}$, $\{\alpha_{1,6},\alpha_{3,6},\alpha_{1,8},\alpha_{3,8}\}$. Both $G$ and $G_1$ are marked by $\ast$ in Tables 1---4. Case 7: $({\bf n}=9,\ (4\mathbb A_1,4\mathbb A_1,4\mathbb A_1,4\mathbb A_1) \subset 16\mathbb A_1)\Longleftarrow ({\bf n}=39,\ 16\mathbb A_1)$. Similar to Case 1. By \cite{Nik10}, the $G\cong 2^4C_2$ is marked by $N_{23}$ and $$ G=H_{39,1}= $$ $$ [(\alpha_{5}\alpha_{10})(\alpha_{7}\alpha_{13}) (\alpha_{8}\alpha_{12})(\alpha_{9}\alpha_{22}) (\alpha_{11}\alpha_{23})(\alpha_{14}\alpha_{16}) (\alpha_{17}\alpha_{21})(\alpha_{19}\alpha_{20}), $$ $$ (\alpha_{2}\alpha_{12})(\alpha_{3}\alpha_{8}) (\alpha_{4}\alpha_{20})(\alpha_{7}\alpha_{16}) (\alpha_{9}\alpha_{11})(\alpha_{13}\alpha_{23}) (\alpha_{14}\alpha_{22})(\alpha_{18}\alpha_{19}), $$ $$ (\alpha_{2}\alpha_{13})(\alpha_{3}\alpha_{22}) (\alpha_{4}\alpha_{9})(\alpha_{7}\alpha_{18}) (\alpha_{8}\alpha_{14})(\alpha_{11}\alpha_{20}) (\alpha_{12}\alpha_{23})(\alpha_{16}\alpha_{19})] $$ with the orbit $\{\alpha_{2},\alpha_{12},\alpha_{13},\alpha_{3},\alpha_{18},\alpha_{8}, \alpha_{23},\alpha_{19},\alpha_{7},\alpha_{22},\alpha_{4}, \alpha_{14},\alpha_{20},\alpha_{11},\alpha_{16},\alpha_{9}\}$. The $G_1\cong (C_2)^3$ is marked by $$ G=H_{39,1}\supset G_1= $$ $$ [(\alpha_{2}\alpha_{3})(\alpha_{4}\alpha_{18})(\alpha_{5}\alpha_{10})(\alpha_{7}\alpha_{22}) (\alpha_{9}\alpha_{13})(\alpha_{11}\alpha_{14})(\alpha_{16}\alpha_{23})(\alpha_{17}\alpha_{21}), $$ $$ (\alpha_{2}\alpha_{18})(\alpha_{3}\alpha_{4})(\alpha_{5}\alpha_{10})(\alpha_{8}\alpha_{19}) (\alpha_{11}\alpha_{16})(\alpha_{12}\alpha_{20})(\alpha_{14}\alpha_{23})(\alpha_{17}\alpha_{21}), $$ $$ (\alpha_{5}\alpha_{10})(\alpha_{7}\alpha_{13})(\alpha_{8}\alpha_{12})(\alpha_{9}\alpha_{22}) (\alpha_{11}\alpha_{23})(\alpha_{14}\alpha_{16})(\alpha_{17}\alpha_{21})(\alpha_{19}\alpha_{20})] $$ with suborbits $\{\alpha_{2},\alpha_{3},\alpha_{18},\alpha_{4}\}$, $\{\alpha_{7},\alpha_{22},\alpha_{13},\alpha_{9}\}$, $\{\alpha_{8},\alpha_{19},\alpha_{12},\alpha_{20}\}$, $\{\alpha_{11},\alpha_{14},\alpha_{16},\alpha_{23}\}$. \newline Both $G$ and $G_1$ are marked by $\ast$ in Tables 1---4. Case 8: ({\bf n}=9,\ $ \left(\begin{array}{ccccc} 2\mathbb A_1 & (4\mathbb A_1)_I & (4\mathbb A_1)_I & (4\mathbb A_1)_I & 10\mathbb A_1 \\ & 2\mathbb A_1 & (4\mathbb A_1)_I & (4\mathbb A_1)_I & 10\mathbb A_1 \\ & & 2\mathbb A_1 & (4\mathbb A_1)_I & 10\mathbb A_1 \\ & & & 2\mathbb A_1 & 10\mathbb A_1\\ & & & & 8\mathbb A_1 \end{array}\right) \subset 16\mathbb A_1 ) \Longleftarrow ({\bf n}=56,\ 16\mathbb A_1)$. Similar to Case 1. By \cite{Nik10}, the $G\cong \Gamma_{25}a_1$ is marked by $N_{23}$ and $$ G=H_{56,2}= $$ $$ [(\alpha_{2}\alpha_{3})(\alpha_{5}\alpha_{6})(\alpha_{7}\alpha_{18}) (\alpha_{8}\alpha_{23})(\alpha_{10}\alpha_{20})(\alpha_{11}\alpha_{17}) (\alpha_{15}\alpha_{16})(\alpha_{19}\alpha_{24}), $$ $$ (\alpha_{1}\alpha_{14})(\alpha_{2}\alpha_{23})(\alpha_{3}\alpha_{5}) (\alpha_{6}\alpha_{18})(\alpha_{7}\alpha_{8})(\alpha_{9}\alpha_{13}) (\alpha_{10}\alpha_{16})(\alpha_{17}\alpha_{24}), $$ $$ (\alpha_{1}\alpha_{14})(\alpha_{2}\alpha_{24})(\alpha_{3}\alpha_{5}) (\alpha_{6}\alpha_{10})(\alpha_{12}\alpha_{22})(\alpha_{16}\alpha_{18}) (\alpha_{17}\alpha_{23})(\alpha_{19}\alpha_{20})] $$ with the orbit $\{\alpha_{2}, \alpha_{3}, \alpha_{23}, \alpha_{24}, \alpha_{5},\alpha_{8}, \alpha_{17}, \alpha_{19}, \alpha_{6}, \alpha_{7}, \alpha_{11}, \alpha_{20}, \alpha_{18}, \alpha_{10},\alpha_{16},\alpha_{15}\}$. The $G_1\cong (C_2)^3$ is marked by $$ G=H_{56,2}\supset G_1= $$ $$ [(\alpha_{2}\alpha_{6})(\alpha_{3}\alpha_{15})(\alpha_{5}\alpha_{11})(\alpha_{7}\alpha_{19}) (\alpha_{8}\alpha_{20})(\alpha_{9}\alpha_{13})(\alpha_{12}\alpha_{22})(\alpha_{16}\alpha_{17}), $$ $$ (\alpha_{3}\alpha_{11})(\alpha_{5}\alpha_{15})(\alpha_{7}\alpha_{20})(\alpha_{8}\alpha_{19}) (\alpha_{9}\alpha_{13})(\alpha_{10}\alpha_{24})(\alpha_{12}\alpha_{22})(\alpha_{18}\alpha_{23}), $$ $$ (\alpha_{1}\alpha_{14})(\alpha_{3}\alpha_{7})(\alpha_{5}\alpha_{8})(\alpha_{9}\alpha_{13}) (\alpha_{10}\alpha_{24})(\alpha_{11}\alpha_{20})(\alpha_{15}\alpha_{19})(\alpha_{16}\alpha_{17})] $$ with suborbits $\{\alpha_{2},\alpha_{6}\}$, $\{\alpha_{10},\alpha_{24}\}$, $\{\alpha_{16},\alpha_{17}\}$, $\{\alpha_{18},\alpha_{23}\}$, $\{\alpha_{3},\alpha_{15},\alpha_{11},\alpha_{7},\alpha_{5},\alpha_{19},\alpha_{20},\alpha_{8}\}$. Both $G$ and $G_1$ are marked by $\ast$ Tables 1---4. Case 9: ({\bf n}=9,\ $ \left(\begin{array}{ccccc} 2\mathbb A_1 & (4\mathbb A_1)_I & 6\mathbb A_1 & (4\mathbb A_1)_I & 6\mathbb A_1 \\ & 2\mathbb A_1 & 2\mathbb A_3 & (4\mathbb A_1)_I & 6\mathbb A_1 \\ & & 4\mathbb A_1 & 6\mathbb A_1 & 8\mathbb A_1 \\ & & & 2\mathbb A_1 & 2\mathbb A_3\\ & & & & 4\mathbb A_1 \end{array}\right) \subset 2\mathbb A_1\amalg 4\mathbb A_3) $ $\Longleftarrow\ ({\bf n}=22,\ \left(\begin{array}{rrr} 2\mathbb A_1 & 6\mathbb A_1 & 10\mathbb A_1 \\ & 4\mathbb A_1 & 4\mathbb A_3 \\ & & 8\mathbb A_1 \end{array}\right)\subset 2\mathbb A_1\amalg 4\mathbb A_3)$. Similar to Cases 1 and 6. By \cite{Nik11}, the $G\cong C_2\times D_8$ is marked by $N_{21}$ and $G=H_{22,1}$ of Case 6 with orbits $\{\alpha_{1,1},\alpha_{3,1}\}$, $\{\alpha_{2,3},\alpha_{2,6},\alpha_{2,8},\alpha_{2,7}\}$, $\{\alpha_{1,3},\alpha_{3,3},\alpha_{3,6},\alpha_{1,6}, \alpha_{3,8}, \alpha_{1,8},\alpha_{3,7},\alpha_{1,7}\}$. The $G_1\cong (C_2)^3$ is marked by $$ G=H_{22,1}\supset G_1= $$ $$ [(\alpha_{1,3}\alpha_{3,7})(\alpha_{2,3}\alpha_{2,7})(\alpha_{3,3}\alpha_{1,7})(\alpha_{1,4}\alpha_{1,5}) (\alpha_{2,4}\alpha_{2,5})(\alpha_{3,4}\alpha_{3,5})(\alpha_{1,6}\alpha_{3,6})(\alpha_{1,8}\alpha_{3,8}), $$ $$ (\alpha_{1,3}\alpha_{3,3})(\alpha_{1,4}\alpha_{3,5})(\alpha_{2,4}\alpha_{2,5})(\alpha_{3,4}\alpha_{1,5}) (\alpha_{1,6}\alpha_{1,8})(\alpha_{2,6}\alpha_{2,8})(\alpha_{3,6}\alpha_{3,8})(\alpha_{1,7}\alpha_{3,7}), $$ $$ (\alpha_{1,1}\alpha_{3,1})(\alpha_{1,2}\alpha_{3,2})(\alpha_{1,4}\alpha_{1,5})(\alpha_{2,4}\alpha_{2,5}) (\alpha_{3,4}\alpha_{3,5})(\alpha_{1,6}\alpha_{3,8})(\alpha_{2,6}\alpha_{2,8})(\alpha_{3,6}\alpha_{1,8})] $$ with suborbits $\{\alpha_{1,1},\alpha_{3,1}\}$, $\{\alpha_{2,3},\alpha_{2,7}\}$, $\{\alpha_{1,3},\alpha_{3,3}\alpha_{1,7},\alpha_{3,7}\}$, $\{\alpha_{2,6},\alpha_{2,8}\}$, $\{\alpha_{1,6},\alpha_{3,6},\alpha_{1,8},\alpha_{3,8}\}$. Both $G$ and $G_1$ are marked by $\ast$ in Tables 1---4. Case 10: $({\bf n}=9,\ ((2\mathbb A_1,2\mathbb A_1)_{II},4\mathbb A_1,4\mathbb A_1,4\mathbb A_1) \subset 16\mathbb A_1)\Longleftarrow ({\bf n}=75,\ 16\mathbb A_1)$. Similar to Case 1. By \cite{Nik10}, the $G\cong 4^2\AAA_4$ is marked by $N_{23}$ and $$ G=H_{75,1}= $$ $$ [(\alpha_{3}\alpha_{16})(\alpha_{4}\alpha_{5}) (\alpha_{6}\alpha_{21})(\alpha_{10}\alpha_{20}) (\alpha_{11}\alpha_{12})(\alpha_{13}\alpha_{17}) (\alpha_{14}\alpha_{22})(\alpha_{23}\alpha_{24}),\ $$ $$ (\alpha_{1}\alpha_{9}\alpha_{7}) (\alpha_{3}\alpha_{21}\alpha_{22}) (\alpha_{4}\alpha_{14}\alpha_{24}) (\alpha_{5}\alpha_{10}\alpha_{16}) (\alpha_{6}\alpha_{20}\alpha_{23}) (\alpha_{12}\alpha_{17}\alpha_{13}),\ $$ $$ (\alpha_{1}\alpha_{6}\alpha_{22}) (\alpha_{3}\alpha_{24}\alpha_{16}) (\alpha_{4}\alpha_{20}\alpha_{9}) (\alpha_{5}\alpha_{10}\alpha_{7}) (\alpha_{11}\alpha_{13}\alpha_{12}) (\alpha_{14}\alpha_{18}\alpha_{21})] $$ with the orbit $\{\alpha_{1},\alpha_{9},\alpha_{6},\alpha_{7},\alpha_{4}, \alpha_{21},\alpha_{20}, \alpha_{22},\alpha_{5},\alpha_{14},\alpha_{10},\alpha_{23}, \alpha_{3},\alpha_{24},\alpha_{18},\alpha_{16}\}$. The $G_1\cong (C_2)^3$ is marked by $$ G=H_{75,1}\supset G_1= $$ $$ [(\alpha_{1}\alpha_{7})(\alpha_{3}\alpha_{24})(\alpha_{4}\alpha_{21})(\alpha_{5}\alpha_{6}) (\alpha_{9}\alpha_{18})(\alpha_{11}\alpha_{12})(\alpha_{13}\alpha_{17})(\alpha_{16}\alpha_{23}), $$ $$ (\alpha_{3}\alpha_{24})(\alpha_{4}\alpha_{6})(\alpha_{5}\alpha_{21})(\alpha_{10}\alpha_{14}) (\alpha_{11}\alpha_{17})(\alpha_{12}\alpha_{13})(\alpha_{16}\alpha_{23})(\alpha_{20}\alpha_{22}), $$ $$ (\alpha_{3}\alpha_{16})(\alpha_{4}\alpha_{5})(\alpha_{6}\alpha_{21})(\alpha_{10}\alpha_{20}) (\alpha_{11}\alpha_{12})(\alpha_{13}\alpha_{17})(\alpha_{14}\alpha_{22})(\alpha_{23}\alpha_{24})] $$ with suborbits $\{\alpha_{1},\alpha_{7}\}$, $\{\alpha_{9},\alpha_{18}\}$, $\{\alpha_{3},\alpha_{24},\alpha_{16},\alpha_{23}\}$, $\{\alpha_{4},\alpha_{21},\alpha_{6},\alpha_{5}\}$, $\{\alpha_{10},\alpha_{14},\alpha_{20},\alpha_{22}\}$. Both $G$ and $G_1$ are marked by $\ast$ in Tables 1---4. Case 11: $({\bf n}=9, \left(\begin{array}{ccccc} 2\mathbb A_1 & 2\mathbb A_3 & (4\mathbb A_1)_I & 6\mathbb A_1 & 6\mathbb A_1 \\ & 4\mathbb A_1 & 6\mathbb A_1 & 8\mathbb A_1 & 8\mathbb A_1 \\ & & 2\mathbb A_1 & 2\mathbb A_3 & 6\mathbb A_1 \\ & & & 4\mathbb A_1 & 8\mathbb A_1\\ & & & & 4\mathbb A_1 \end{array}\right) \subset 4\mathbb A_3\amalg 4\mathbb A_1) $ $\Longleftarrow ({\bf n}=22,\ \left(\begin{array}{rrr} 4\mathbb A_1 & (8\mathbb A_1)_{II} & 12\mathbb A_1 \\ & 4\mathbb A_1 & 4\mathbb A_3 \\ & & 8\mathbb A_1 \end{array}\right)\subset 4\mathbb A_1\amalg 4\mathbb A_3)$. Similar to Cases 1 and 6. By \cite{Nik11}, the $G\cong C_2\times D_8$ is marked by $N_{21}$ and $G=H_{22,1}$ from Case 6 with orbits $\{\alpha_{1,4},\alpha_{3,5},\alpha_{3,4},\alpha_{1,5}\}$, $\{\alpha_{2,3},\alpha_{2,6},\alpha_{2,8},\alpha_{2,7}\}$, $\{\alpha_{1,3},\alpha_{3,3},\alpha_{3,6},\alpha_{1,6},\alpha_{3,8}, \linebreak \alpha_{1,8},\alpha_{3,7},\alpha_{1,7}\}$. The $G_1\cong (C_2)^3$ is marked by $G=H_{22,1}\supset G_1$ of Case 6 with suborbits $\{\alpha_{2,3},\alpha_{2,7}\}$, $\{\alpha_{1,3},\alpha_{3,7},\alpha_{3,3},\alpha_{1,7}\}$, $\{\alpha_{2,6},\alpha_{2,8}\}$, $\{\alpha_{1,6},\alpha_{3,6},\alpha_{1,8},\alpha_{3,8}\}$, $\{\alpha_{1,4},\alpha_{1,5},\alpha_{3,5},\alpha_{3,4}\}$. Both $G$ and $G_1$ are marked by $\ast$ in Tables 1---4. Case 12: ({\bf n}=10, $(\mathbb A_1,\mathbb A_1)\subset 2\mathbb A_1)\ \Longleftarrow\ ({\bf n}=22,2\mathbb A_1)$. Similar to Case 1. By \cite{Nik10}, the $G\cong C_2\times D_8$ is marked by $N_{23}$ and $$ G=H_{22,2}= $$ $$ [ (\alpha_{3}\alpha_{4})(\alpha_{6}\alpha_{15}) (\alpha_{8}\alpha_{11})(\alpha_{9}\alpha_{22}) (\alpha_{12}\alpha_{23})(\alpha_{14}\alpha_{20}) (\alpha_{16}\alpha_{19})(\alpha_{17}\alpha_{21}), $$ $$ (\alpha_{2}\alpha_{22})(\alpha_{3}\alpha_{13}) (\alpha_{4}\alpha_{18})(\alpha_{7}\alpha_{9}) (\alpha_{10}\alpha_{24})(\alpha_{11}\alpha_{20}) (\alpha_{15}\alpha_{17})(\alpha_{16}\alpha_{19}), $$ $$ (\alpha_{2}\alpha_{13})(\alpha_{3}\alpha_{22}) (\alpha_{4}\alpha_{9})(\alpha_{7}\alpha_{18}) (\alpha_{8}\alpha_{14})(\alpha_{11}\alpha_{20}) (\alpha_{12}\alpha_{23})(\alpha_{16}\alpha_{19})] $$ with the orbit $\{\alpha_{12},\alpha_{23}\}$. The $G_1\cong D_8$ is marked by $$ G=H_{22,2}\supset G_1= $$ $$ [(\alpha_{2}\alpha_{9})(\alpha_{3}\alpha_{18})(\alpha_{4}\alpha_{13})(\alpha_{6}\alpha_{21}) (\alpha_{7}\alpha_{22})(\alpha_{8}\alpha_{14})(\alpha_{10}\alpha_{24})(\alpha_{16}\alpha_{19}), $$ $$ (\alpha_{2}\alpha_{13})(\alpha_{3}\alpha_{9})(\alpha_{4}\alpha_{22})(\alpha_{6}\alpha_{15}) (\alpha_{7}\alpha_{18})(\alpha_{8}\alpha_{20})(\alpha_{11}\alpha_{14})(\alpha_{17}\alpha_{21})] $$ with suborbits $\{\alpha_{12}\}$, $\{\alpha_{23}\}$. Both $G$ and $G_1$ are marked by $\ast$ in Tables 1---4. Case 13: ({\bf n}=10, $((4\mathbb A_1,4\mathbb A_1)\subset 8\mathbb A_1)_{II}) \Longleftarrow\ ({\bf n}=22,8\mathbb A_1)$. Similar to Cases 1 and 12. By \cite{Nik10}, the $G\cong C_2\times D_8$ is marked by $N_{23}$ and $G=H_{22,2}$ from Case 12 with the orbit $\{\alpha_{2},\alpha_{22},\alpha_{13},\alpha_{7},\alpha_{9}, \alpha_{3},\alpha_{18},\alpha_{4}\}$. The $G_1\cong D_8$ is marked by $$ G=H_{22,2}\supset G_1= $$ $$ [(\alpha_{3}\alpha_{4})(\alpha_{6}\alpha_{15})(\alpha_{8}\alpha_{11})(\alpha_{9}\alpha_{22}) (\alpha_{12}\alpha_{23})(\alpha_{14}\alpha_{20})(\alpha_{16}\alpha_{19})(\alpha_{17}\alpha_{21}), $$ $$ (\alpha_{2}\alpha_{3})(\alpha_{4}\alpha_{7})(\alpha_{8}\alpha_{14})(\alpha_{9}\alpha_{18}) (\alpha_{10}\alpha_{24})(\alpha_{12}\alpha_{23})(\alpha_{13}\alpha_{22})(\alpha_{15}\alpha_{17})] $$ with suborbits $\{\alpha_{2},\alpha_{3},\alpha_{4},\alpha_{7}\}$, $\{\alpha_{9},\alpha_{22},\alpha_{18},\alpha_{13}\}$. Both $G$ and $G_1$ are marked by $\ast$ in Tables 1---4. Case 14: ({\bf n}=10, $(\mathbb A_1,\mathbb A_1,(2\mathbb A_1)_I)\subset 4\mathbb A_1)\ \Longleftarrow ({\bf n}=39,\ 4\mathbb A_1)$. Similar to Case 1. By \cite{Nik10}, the $G\cong 2^4C_2$ is marked by $N_{23}$ and $$ G=H_{39,2}= $$ $$ [(\alpha_{5}\alpha_{18})(\alpha_{6}\alpha_{7}) (\alpha_{10}\alpha_{17})(\alpha_{11}\alpha_{20}) (\alpha_{12}\alpha_{13})(\alpha_{14}\alpha_{22}) (\alpha_{15}\alpha_{16})(\alpha_{19}\alpha_{24}), $$ $$ (\alpha_{2}\alpha_{12})(\alpha_{3}\alpha_{8}) (\alpha_{4}\alpha_{20})(\alpha_{7}\alpha_{16}) (\alpha_{9}\alpha_{11})(\alpha_{13}\alpha_{23}) (\alpha_{14}\alpha_{22})(\alpha_{18}\alpha_{19}), $$ $$ (\alpha_{2}\alpha_{13})(\alpha_{3}\alpha_{22}) (\alpha_{4}\alpha_{9})(\alpha_{7}\alpha_{18}) (\alpha_{8}\alpha_{14})(\alpha_{11}\alpha_{20}) (\alpha_{12}\alpha_{23})(\alpha_{16}\alpha_{19})] $$ with the orbit $\{\alpha_{2},\alpha_{12},\alpha_{13},\alpha_{23}\}$. The $G_1\cong D_8$ is marked by $$ G=H_{39,2}\supset G_1= $$ $$ [(\alpha_{3}\alpha_{14}\alpha_{8}\alpha_{22})(\alpha_{4}\alpha_{20}\alpha_{9}\alpha_{11}) (\alpha_{5}\alpha_{7}\alpha_{15}\alpha_{19})(\alpha_{6}\alpha_{18}\alpha_{24}\alpha_{16}) (\alpha_{10}\alpha_{17})(\alpha_{12}\alpha_{13}), $$ $$ (\alpha_{5}\alpha_{18})(\alpha_{6}\alpha_{7})(\alpha_{10}\alpha_{17})(\alpha_{11}\alpha_{20}) (\alpha_{12}\alpha_{13})(\alpha_{14}\alpha_{22})(\alpha_{15}\alpha_{16})(\alpha_{19}\alpha_{24})] $$ with suborbits $\{\alpha_{2}\}$, $\{\alpha_{23}\}$, $\{\alpha_{12},\alpha_{13}\}$. Both $G$ and $G_1$ are marked by $\ast$ in Tables 1---4. Case 15: ({\bf n}=10, $(\mathbb A_1,\mathbb A_1,4\mathbb A_1)\subset 6\mathbb A_1)\ \Longleftarrow\ ({\bf n}=22,(2\mathbb A_1,4\mathbb A_1)\subset 6\mathbb A_1)$. Similar to Cases 1 and 12. By \cite{Nik11}, the $G\cong C_2\times D_8$ is marked by $N_{23}$ and $G=H_{22,2}$ of Case 12 with orbits $\{\alpha_{12},\alpha_{23}\}$, $\{\alpha_{6},\alpha_{15},\alpha_{21},\alpha_{17}\}$. The $G_1\cong D_8$ is marked by $$ G=H_{22,2}\supset G_1= $$ $$ [(\alpha_{2}\alpha_{9})(\alpha_{3}\alpha_{18})(\alpha_{4}\alpha_{13})(\alpha_{6}\alpha_{21}) (\alpha_{7}\alpha_{22})(\alpha_{8}\alpha_{14})(\alpha_{10}\alpha_{24})(\alpha_{16}\alpha_{19}), $$ $$ (\alpha_{2}\alpha_{13})(\alpha_{3}\alpha_{9})(\alpha_{4}\alpha_{22})(\alpha_{6}\alpha_{15}) (\alpha_{7}\alpha_{18})(\alpha_{8}\alpha_{20})(\alpha_{11}\alpha_{14})(\alpha_{17}\alpha_{21})] $$ with suborbits $\{\alpha_{12}\}$, $\{\alpha_{23}\}$, $\{\alpha_{6},\alpha_{15},\alpha_{21},\alpha_{17}\}$. Both $G$ and $G_1$ are marked by $\ast$ in Tables 1---4. Case 16: ({\bf n}=10, $(\mathbb A_1,\mathbb A_1,8\mathbb A_1)\subset 10\mathbb A_1)\ \Longleftarrow\ ({\bf n}=22,(2\mathbb A_1,8\mathbb A_1)\subset 10\mathbb A_1)$. By \cite{Nik11} which describes markings of these cases by Niemeier lattices, we should consider two cases, when $({\bf n}=22,(2\mathbb A_1,8\mathbb A_1)\subset 10\mathbb A_1)$ is marked by $N_{23}$ and by $N_{21}$. If $({\bf n}=22,(2\mathbb A_1,8\mathbb A_1)\subset 10\mathbb A_1)$ is marked by $N_{23}$, then $G$ is marked by $G=H_{22,2}$ from Case 12 with orbits $\{\alpha_{12},\alpha_{23}\}$, $\{\alpha_{2},\alpha_{22},\alpha_{13},\alpha_{7},\alpha_{9}, \alpha_{3},\alpha_{18},\alpha_{4}\}$, and $G_1\cong D_8$ is marked by $G_1\subset G=H_{22,2}$ from Case 15 with suborbits $\{\alpha_{12}\}$, $\{\alpha_{23}\}$, $\{\alpha_{2},\alpha_{22},\alpha_{13},\alpha_{7},\alpha_{9}, \alpha_{3},\alpha_{18},\alpha_{4}\}$. If $({\bf n}=22,(2\mathbb A_1,8\mathbb A_1)\subset 10\mathbb A_1)$ is marked by $N_{21}$, then $G$ is marked by $G=H_{22,1}$ from Case 6 with orbits $\{\alpha_{1,1},\alpha_{3,1}\}$, $\{\alpha_{1,3},\alpha_{3,3},\alpha_{3,6},\alpha_{1,6},\alpha_{3,8}, \alpha_{1,8},\alpha_{3,7},\alpha_{1,7}\}$, and $G_1\cong D_8$ is marked by $$ G=H_{22,1}\supset G_1= $$ $$ [(\alpha_{1,3}\alpha_{3,3})(\alpha_{1,4}\alpha_{3,5})(\alpha_{2,4}\alpha_{2,5})(\alpha_{3,4}\alpha_{1,5}) (\alpha_{1,6}\alpha_{1,8})(\alpha_{2,6}\alpha_{2,8})(\alpha_{3,6}\alpha_{3,8})(\alpha_{1,7}\alpha_{3,7}), $$ $$ (\alpha_{1,2}\alpha_{3,2})(\alpha_{1,3}\alpha_{3,6})(\alpha_{2,3}\alpha_{2,6})(\alpha_{3,3}\alpha_{1,6}) (\alpha_{1,4}\alpha_{3,4})(\alpha_{1,7}\alpha_{1,8})(\alpha_{2,7}\alpha_{2,8})(\alpha_{3,7}\alpha_{3,8})] $$ with suborbits $\{\alpha_{1,1}\}$, $\{\alpha_{3,1}\}$, $\{\alpha_{1,3},\alpha_{3,3},\alpha_{3,6},\alpha_{1,6},\alpha_{3,8}, \alpha_{1,8},\alpha_{3,7},\alpha_{1,7}\}$. Case 17: ({\bf n}=10, $ \left(\begin{array}{ccc} (2\mathbb A_1)_{I} & (6\mathbb A_1)_I & (6\mathbb A_1)_I \\ & 4\mathbb A_1 & (8\mathbb A_1)_{II} \\ & & 4\mathbb A_1 \end{array}\right) \subset 10\mathbb A_1)\ \Longleftarrow\ ({\bf n}=22,(2\mathbb A_1,8\mathbb A_1)\subset 10\mathbb A_1)$. Similar to Case 16. If $({\bf n}=22,(2\mathbb A_1,8\mathbb A_1)\subset 10\mathbb A_1)$ is marked by $N_{23}$, then $G=H_{22,2}$ from Case 12 with orbits $\{\alpha_{12},\alpha_{23}\}$, $\{\alpha_{2},\alpha_{22},\alpha_{13},\alpha_{7},\alpha_{9},\alpha_{3},\alpha_{18},\alpha_{4}\}$, and $G_1\cong D_8$ is marked by $$ G=H_{22,2}\supset G_1= $$ $$ [(\alpha_{3}\alpha_{4})(\alpha_{6}\alpha_{15})(\alpha_{8}\alpha_{11})(\alpha_{9}\alpha_{22}) (\alpha_{12}\alpha_{23})(\alpha_{14}\alpha_{20})(\alpha_{16}\alpha_{19})(\alpha_{17}\alpha_{21}), $$ $$ (\alpha_{2}\alpha_{9})(\alpha_{3}\alpha_{18})(\alpha_{4}\alpha_{13})(\alpha_{6}\alpha_{21}) (\alpha_{7}\alpha_{22})(\alpha_{8}\alpha_{14})(\alpha_{10}\alpha_{24})(\alpha_{16}\alpha_{19})] $$ with suborbits $\{\alpha_{12},\alpha_{23}\}$, $\{\alpha_{2},\alpha_{9},\alpha_{22},\alpha_{7}\}$, $\{\alpha_{3},\alpha_{4},\alpha_{18},\alpha_{13}\}$. If $({\bf n}=22,(2\mathbb A_1,8\mathbb A_1)\subset 10\mathbb A_1)$ is marked by $N_{21}$, then $G=H_{22,1}$ from Case 6 with orbits $\{\alpha_{1,1},\alpha_{3,1}\}$, $\{\alpha_{1,3},\alpha_{3,3},\alpha_{3,6},\alpha_{1,6},\alpha_{3,8}, \alpha_{1,8},\alpha_{3,7},\alpha_{1,7}\}$, and $G_1\cong D_8$ is marked by $$ G=H_{22,1}\supset G_1= $$ $$ [(\alpha_{1,2}\alpha_{3,2})(\alpha_{1,3}\alpha_{3,6})(\alpha_{2,3}\alpha_{2,6})(\alpha_{3,3}\alpha_{1,6}) (\alpha_{1,4}\alpha_{3,4})(\alpha_{1,7}\alpha_{1,8})(\alpha_{2,7}\alpha_{2,8})(\alpha_{3,7}\alpha_{3,8}), $$ $$ (\alpha_{1,1}\alpha_{3,1})(\alpha_{1,2}\alpha_{3,2})(\alpha_{1,4}\alpha_{1,5})(\alpha_{2,4}\alpha_{2,5}) (\alpha_{3,4}\alpha_{3,5})(\alpha_{1,6}\alpha_{3,8})(\alpha_{2,6}\alpha_{2,8})(\alpha_{3,6}\alpha_{1,8})] $$ with suborbits $\{\alpha_{1,1},\alpha_{3,1}\}$, $\{\alpha_{1,3},\alpha_{3,6},\alpha_{1,8},\alpha_{1,7}\}$, $\{\alpha_{3,3},\alpha_{1,6},\alpha_{3,8},\alpha_{3,7}\}$. Case 18: ({\bf n}=10, $ \left(\begin{array}{ccc} (2\mathbb A_1)_{II} & 6\mathbb A_1 & 6\mathbb A_1 \\ & 4\mathbb A_1 & (8\mathbb A_1)_{II} \\ & & 4\mathbb A_1 \end{array}\right) \subset 10\mathbb A_1)\ \Longleftarrow\ ({\bf n}=22,(2\mathbb A_1,8\mathbb A_1)\subset 10\mathbb A_1)$. Similar to Case 17. If $({\bf n}=22,(2\mathbb A_1,8\mathbb A_1)\subset 10\mathbb A_1)$ is marked by $N_{23}$, then $G=H_{22,2}$ from Case 12 with orbits $\{\alpha_{12},\alpha_{23}\}$, $\{\alpha_{2},\alpha_{22},\alpha_{13},\alpha_{7},\alpha_{9},\alpha_{3},\alpha_{18},\alpha_{4}\}$, and $D_8\cong G_1$ is marked by $$ G=H_{22,2}\supset G_1= $$ $$ [(\alpha_{3}\alpha_{4})(\alpha_{6}\alpha_{15})(\alpha_{8}\alpha_{11})(\alpha_{9}\alpha_{22}) (\alpha_{12}\alpha_{23})(\alpha_{14}\alpha_{20})(\alpha_{16}\alpha_{19})(\alpha_{17}\alpha_{21}), $$ $$ (\alpha_{2}\alpha_{3})(\alpha_{4}\alpha_{7})(\alpha_{8}\alpha_{14})(\alpha_{9}\alpha_{18}) (\alpha_{10}\alpha_{24})(\alpha_{12}\alpha_{23})(\alpha_{13}\alpha_{22})(\alpha_{15}\alpha_{17})] $$ with suborbits $\{\alpha_{12},\alpha_{23}\}$, $\{\alpha_{2},\alpha_{3},\alpha_{4},\alpha_{7}\}$, $\{\alpha_{9},\alpha_{22},\alpha_{18},\alpha_{13}\}$. If $({\bf n}=22,(2\mathbb A_1,8\mathbb A_1)\subset 10\mathbb A_1)$ is marked by $N_{21}$, then $G=H_{22,1}$ from Case 6 with orbits $\{\alpha_{1,1},\alpha_{3,1}\}$, $\{\alpha_{1,3},\alpha_{3,3},\alpha_{3,6},\alpha_{1,6},\alpha_{3,8}, \alpha_{1,8},\alpha_{3,7},\alpha_{1,7}\}$, and $G_1\cong D_8$ is marked by $$ G=H_{22,1}\supset G_1= $$ $$ [(\alpha_{1,1}\alpha_{3,1})(\alpha_{1,3}\alpha_{1,6})(\alpha_{2,3}\alpha_{2,6})(\alpha_{3,3}\alpha_{3,6}) (\alpha_{1,5}\alpha_{3,5})(\alpha_{1,7}\alpha_{3,8})(\alpha_{2,7}\alpha_{2,8})(\alpha_{3,7}\alpha_{1,8}), $$ $$ (\alpha_{1,1}\alpha_{3,1})(\alpha_{1,2}\alpha_{3,2})(\alpha_{1,4}\alpha_{1,5})(\alpha_{2,4}\alpha_{2,5}) (\alpha_{3,4}\alpha_{3,5})(\alpha_{1,6}\alpha_{3,8})(\alpha_{2,6}\alpha_{2,8})(\alpha_{3,6}\alpha_{1,8})] $$ with suborbits $\{\alpha_{1,1},\alpha_{3,1}\}$, $\{\alpha_{1,3},\alpha_{1,6},\alpha_{3,8},\alpha_{1,7}\}$, $\{\alpha_{3,3},\alpha_{3,6},\alpha_{1,8},\alpha_{3,7}\}$. Case 19: ({\bf n}=10, $ \left(\begin{array}{ccc} 4\mathbb A_1 & (8\mathbb A_1)_I & (8\mathbb A_1)_{II} \\ & 4\mathbb A_1 & (8\mathbb A_1)_{I} \\ & & 4\mathbb A_1 \end{array}\right) \subset 12\mathbb A_1)\ \Longleftarrow\ ({\bf n}=22,(4\mathbb A_1,8\mathbb A_1)\subset 12\mathbb A_1)$. Similar to Cases 1, 12 and 18. By \cite{Nik11}, the $G\cong C_2\times D_8$ is marked by $N_{23}$ and $G=H_{22,2}$ from Case 12 with orbits $\{\alpha_{6},\alpha_{15},\alpha_{17},\alpha_{21}\}$, $\{\alpha_{2},\alpha_{22},\alpha_{13},\alpha_{9},\alpha_{3},\alpha_{7},\alpha_{4},\alpha_{8}\}$. The $G_1\cong D_8$ is marked by $G_1\subset G=H_{22,2}$ of Case 18 with suborbits $\{\alpha_{6}, \alpha_{15},\alpha_{17},\alpha_{21}\}$ $\{\alpha_{2},\alpha_{3},\alpha_{4},\alpha_{7}\}$, $\{\alpha_{9},\alpha_{22},\alpha_{18},\alpha_{13}\}$. Both $G$ and $G_1$ are marked by $\ast$ in Tables 1---4. Case 20: ({\bf n}=10, $ \left(\begin{array}{ccc} 4\mathbb A_1 & 4\mathbb A_2 & (8\mathbb A_1)_{II} \\ & 4\mathbb A_1 & 4\mathbb A_2 \\ & & 4\mathbb A_1 \end{array}\right) \subset 4\mathbb A_3) \Longleftarrow\ ({\bf n}=22,(4\mathbb A_1,8\mathbb A_1)\subset 4\mathbb A_3)$. Similar to Cases 1 and 6. By \cite{Nik11}, the $G\cong C_2\times D_8$ is marked by $N_{21}$ and $G=H_{22,1}$ from Case 6 with orbits $\{\alpha_{2,3},\alpha_{2,6},\alpha_{2,8},\alpha_{2,7}\}$, $\{\alpha_{1,3},\alpha_{3,3},\alpha_{3,6},\alpha_{1,6},\alpha_{3,8}, \alpha_{1,8},\alpha_{3,7},\alpha_{1,7}\}$. The $G_1\cong D_8$ is marked by $$ G=H_{22,1}\supset G_1= $$ $$ [(\alpha_{1,2}\alpha_{3,2})(\alpha_{1,3}\alpha_{3,6})(\alpha_{2,3}\alpha_{2,6})(\alpha_{3,3}\alpha_{1,6}) (\alpha_{1,4}\alpha_{3,4})(\alpha_{1,7}\alpha_{1,8})(\alpha_{2,7}\alpha_{2,8})(\alpha_{3,7}\alpha_{3,8}), $$ $$ (\alpha_{1,1}\alpha_{3,1}) (\alpha_{1,3}\alpha_{3,6}\alpha_{1,7}\alpha_{1,8}) (\alpha_{2,3}\alpha_{2,6}\alpha_{2,7}\alpha_{2,8}) (\alpha_{3,3}\alpha_{1,6}\alpha_{3,7}\alpha_{3,8}) (\alpha_{1,4}\alpha_{1,5}\alpha_{3,4}\alpha_{3,5}) (\alpha_{2,4}\alpha_{2,5})] $$ with suborbits $\{ \alpha_{2,3},\alpha_{2,6},\alpha_{2,7},\alpha_{2,8} \}$, $\{ \alpha_{1,3}, \alpha_{3,6},\alpha_{1,7}, \alpha_{1,8}\}$, $\{ \alpha_{3,3}, \alpha_{1,6},\alpha_{3,7}, \alpha_{3,8} \}$. Both $G$ and $G_1$ are marked by $\ast$ in Tables 1---4. Case 21: ({\bf n}=10,\ $ \left(\begin{array}{ccc} 4\mathbb A_1 & (8\mathbb A_1)_{II} & 12\mathbb A_1 \\ & 4\mathbb A_1 & 12\mathbb A_1 \\ & & 8\mathbb A_1 \end{array}\right) \subset 16\mathbb A_1) \Longleftarrow ({\bf n}=39,\ 16\mathbb A_1)$. Similar to Cases 1 and 7. By \cite{Nik10}, the $G\cong 2^4C_2$ is marked by $N_{23}$ and $G=H_{39,1}$ from Case 7 with the orbit $\{\alpha_{2},\alpha_{12},\alpha_{13},\alpha_{3},\alpha_{18},\alpha_{8}, \alpha_{23},\alpha_{19},\alpha_{7},\alpha_{22},\alpha_{4}, \alpha_{14},\alpha_{20},\alpha_{11},\alpha_{16},\alpha_{9}\}$. The $G_1\cong D_8$ is marked by $$ G=H_{39,1}\supset G_1= $$ $$ [(\alpha_{2}\alpha_{7}\alpha_{18}\alpha_{13})(\alpha_{3}\alpha_{9}\alpha_{4}\alpha_{22}) (\alpha_{5}\alpha_{10}) (\alpha_{8}\alpha_{16}\alpha_{20}\alpha_{23})(\alpha_{11}\alpha_{19}\alpha_{14}\alpha_{12}) (\alpha_{17}\alpha_{21}), $$ $$ (\alpha_{5}\alpha_{10})(\alpha_{7}\alpha_{13})(\alpha_{8}\alpha_{12})(\alpha_{9}\alpha_{22}) (\alpha_{11}\alpha_{23})(\alpha_{14}\alpha_{16})(\alpha_{17}\alpha_{21})(\alpha_{19}\alpha_{20})] $$ with suborbits $\{\alpha_{2},\alpha_{7},\alpha_{18},\alpha_{13}\}$, $\{\alpha_{3},\alpha_{9},\alpha_{4},\alpha_{22}\}$, $\{\alpha_{8},\alpha_{16},\alpha_{12},\alpha_{20},\alpha_{14},\alpha_{11},\alpha_{23},\alpha_{19}\}$. Both $G$ and $G_1$ are marked by $\ast$ in Tables 1---4. Case 22: $({\bf n}=10,\ (4\mathbb A_1,4\mathbb A_1,2\mathbb A_2)\subset 6\mathbb A_2) \Longleftarrow ({\bf n}=34,\ 6\mathbb A_2)$. Similar to Case 1. By \cite{Nik10}, the $G\cong \SSS_4$ is marked by $N_{22}$ and $$ G=H_{34,1}= $$ $$ [(\alpha_{1,2}\alpha_{2,5})(\alpha_{2,2}\alpha_{1,5}) (\alpha_{1,4}\alpha_{2,8})(\alpha_{2,4}\alpha_{1,8}) (\alpha_{1,7}\alpha_{2,9})(\alpha_{2,7}\alpha_{1,9}) (\alpha_{1,11}\alpha_{1,12})(\alpha_{2,11}\alpha_{2,12}), $$ $$ (\alpha_{1,1}\alpha_{1,3}\alpha_{2,4}) (\alpha_{2,1}\alpha_{2,3}\alpha_{1,4}) (\alpha_{1,2}\alpha_{1,5}\alpha_{2,11}) (\alpha_{2,2}\alpha_{2,5}\alpha_{1,11}) (\alpha_{1,7}\alpha_{1,9}\alpha_{2,12}) (\alpha_{2,7}\alpha_{2,9}\alpha_{1,12})] $$ with the orbit $\{\alpha_{1,2},\alpha_{2,5}, \alpha_{1,5},\alpha_{1,11},\alpha_{2,2},\alpha_{2,11}, \alpha_{1,12},\alpha_{2,12},\alpha_{2,7},\alpha_{1,7},\alpha_{1,9},\alpha_{2,9}\}$. The $G_1\cong D_8$ is marked by $$ G=H_{34,1}\supset G_1= $$ $$ [(\alpha_{1,2}\alpha_{1,11})(\alpha_{2,2}\alpha_{2,11})(\alpha_{1,3}\alpha_{1,8})(\alpha_{2,3}\alpha_{2,8}) (\alpha_{1,5}\alpha_{1,9})(\alpha_{2,5}\alpha_{2,9})(\alpha_{1,7}\alpha_{1,12})(\alpha_{2,7}\alpha_{2,12}), $$ $$ (\alpha_{1,1}\alpha_{1,3})(\alpha_{2,1}\alpha_{2,3})(\alpha_{1,2}\alpha_{2,7})(\alpha_{2,2}\alpha_{1,7}) (\alpha_{1,4}\alpha_{2,8})(\alpha_{2,4}\alpha_{1,8})(\alpha_{1,5}\alpha_{2,9})(\alpha_{2,5}\alpha_{1,9})] $$ with suborbits $\{\alpha_{1,2},\alpha_{1,11},\alpha_{2,7},\alpha_{2,12}\}$, $\{\alpha_{2,2},\alpha_{2,11},\alpha_{1,7},\alpha_{1,12}\}$, $\{\alpha_{1,5},\alpha_{1,9},\alpha_{2,9},\alpha_{2,5}\}$. Both $G$ and $G_1$ are marked by $\ast$ in Tables 1---4. Case 23: ({\bf n}=10,\ $ \left(\begin{array}{cccc} \mathbb A_1 & 2\mathbb A_1 & 3\mathbb A_1 & 5\mathbb A_1 \\ & \mathbb A_1 & 3\mathbb A_1 & 5\mathbb A_1 \\ & & (2\mathbb A_1)_I & (6\mathbb A_1)_I \\ & & & 4\mathbb A_1 \end{array}\right) \subset 8\mathbb A_1) \Longleftarrow ({\bf n}=56,\ 8\mathbb A_1)$. Similar to Case 1. By \cite{Nik10}, the $G\cong \Gamma_{15}a_1$ is marked by $N_{23}$ and $$ G=H_{56,1}= $$ $$ [(\alpha_{5}\alpha_{23})(\alpha_{6}\alpha_{10})(\alpha_{7}\alpha_{9}) (\alpha_{8}\alpha_{16})(\alpha_{13}\alpha_{18})(\alpha_{14}\alpha_{24}) (\alpha_{15}\alpha_{20})(\alpha_{19}\alpha_{21}), $$ $$ (\alpha_{1}\alpha_{14})(\alpha_{2}\alpha_{23})(\alpha_{3}\alpha_{5}) (\alpha_{6}\alpha_{18})(\alpha_{7},\alpha_{8})(\alpha_{9}\alpha_{13}) (\alpha_{10}\alpha_{16})(\alpha_{17}\alpha_{24}), $$ $$ (\alpha_{1}\alpha_{14})(\alpha_{2}\alpha_{24})(\alpha_{3}\alpha_{5}) (\alpha_{6}\alpha_{10})(\alpha_{12}\alpha_{22})(\alpha_{16}\alpha_{18}) (\alpha_{17}\alpha_{23})(\alpha_{19}\alpha_{20})] $$ with the orbit $\{\alpha_{1}, \alpha_{14}, \alpha_{24}, \alpha_{17}, \alpha_{2},\alpha_{23}, \alpha_{5}, \alpha_{3} \}$. The $G_1\cong D_8$ is marked by $$ G=H_{56,1}\supset G_1= $$ $$ [(\alpha_{5}\alpha_{23})(\alpha_{6}\alpha_{10})(\alpha_{7}\alpha_{9})(\alpha_{8}\alpha_{16}) (\alpha_{13}\alpha_{18})(\alpha_{14}\alpha_{24})(\alpha_{15}\alpha_{20})(\alpha_{19}\alpha_{21}), $$ $$ (\alpha_{2}\alpha_{17})(\alpha_{6}\alpha_{16})(\alpha_{7}\alpha_{8})(\alpha_{9}\alpha_{13}) (\alpha_{10}\alpha_{18})(\alpha_{12}\alpha_{22})(\alpha_{19}\alpha_{20})(\alpha_{23}\alpha_{24})] $$ with suborbits $\{\alpha_{1}\}$, $\{\alpha_{3}\}$, $\{\alpha_{2},\alpha_{17}\}$, $\{\alpha_{5},\alpha_{23},\alpha_{24},\alpha_{14}\}$. Both $G$ and $G_1$ are marked by $\ast$ in Tables 1---4. Case 24: $({\bf n}=10,\ (\mathbb A_1,\mathbb A_1,(2\mathbb A_1)_{I},8\mathbb A_1) \subset 12\mathbb A_1)\Longleftarrow ({\bf n}=65,\ 12\mathbb A_1)$. Similar to Case 1. By \cite{Nik10}, the $G\cong 2^4D_6$ is marked by $N_{23}$ and $$ G=H_{65,3}= $$ $$ [(\alpha_{1}\alpha_{7}\alpha_{17})(\alpha_{3}\alpha_{22}\alpha_{5}) (\alpha_{4}\alpha_{23}\alpha_{20})(\alpha_{6}\alpha_{14}\alpha_{8}) (\alpha_{9}\alpha_{13}\alpha_{11})(\alpha_{10}\alpha_{19}\alpha_{21}), $$ $$ (\alpha_{1}\alpha_{11}\alpha_{23}\alpha_{8}) (\alpha_{4}\alpha_{6}\alpha_{13}\alpha_{20}) (\alpha_{5}\alpha_{16}) (\alpha_{7}\alpha_{17}\alpha_{9}\alpha_{14}) (\alpha_{10}\alpha_{21}\alpha_{19}\alpha_{15}) (\alpha_{12}\alpha_{24})]; $$ with the orbit $\{\alpha_{1},\alpha_{7},\alpha_{23},\alpha_{8},\alpha_{11},\alpha_{17},\alpha_{20},\alpha_{9}, \alpha_{6},\alpha_{4},\alpha_{14},\alpha_{13}\}$. The $G_1\cong D_8$ is marked by $$ G=H_{65,3}\supset G_1= $$ $$ [(\alpha_{4}\alpha_{7})(\alpha_{5}\alpha_{16})(\alpha_{6}\alpha_{14})(\alpha_{8}\alpha_{11}) (\alpha_{9}\alpha_{13})(\alpha_{12}\alpha_{24})(\alpha_{15}\alpha_{21})(\alpha_{17}\alpha_{20}), $$ $$ (\alpha_{3}\alpha_{5})(\alpha_{4}\alpha_{13})(\alpha_{6}\alpha_{7})(\alpha_{9}\alpha_{20}) (\alpha_{10}\alpha_{15})(\alpha_{14}\alpha_{17})(\alpha_{16}\alpha_{22})(\alpha_{19}\alpha_{21})] $$ with suborbits $\{\alpha_{1}\}$, $\{\alpha_{23}\}$, $\{\alpha_{8},\alpha_{11}\}$, $\{\alpha_{4},\alpha_{7},\alpha_{13},\alpha_{6},\alpha_{9},\alpha_{14},\alpha_{20},\alpha_{17}\}$. Both $G$ and $G_1$ are marked by $\ast$ in Tables 1---4. Case 25: ({\bf n}=10,\ $ \left(\begin{array}{cccc} \mathbb A_1 & 2\mathbb A_1 & 5\mathbb A_1 & 5\mathbb A_1 \\ & \mathbb A_1 & 5\mathbb A_1 & 5\mathbb A_1 \\ & & 4\mathbb A_1 & (8\mathbb A_1)_I \\ & & & 4\mathbb A_1 \end{array}\right) \subset 10\mathbb A_1)\ \Longleftarrow ({\bf n}=22,\ (2\mathbb A_1, (4\mathbb A_1,4\mathbb A_1)_{II})\subset 10\mathbb A_1)$. Similar to Case 1. By \cite{Nik11}, the $G\cong C_2\times D_8$ is marked by $N_{23}$ and $$ G=H_{22,3}= $$ $$ [(\alpha_{1}\alpha_{10})(\alpha_{3}\alpha_{22}) (\alpha_{4}\alpha_{9})(\alpha_{5}\alpha_{24}) (\alpha_{6}\alpha_{15})(\alpha_{8}\alpha_{14}) (\alpha_{11}\alpha_{20})(\alpha_{17}\alpha_{21}), $$ $$ (\alpha_{2}\alpha_{22})(\alpha_{3}\alpha_{13}) (\alpha_{4}\alpha_{18})(\alpha_{7}\alpha_{9}) (\alpha_{10}\alpha_{24})(\alpha_{11}\alpha_{20}) (\alpha_{15}\alpha_{17})(\alpha_{16}\alpha_{19}), $$ $$ (\alpha_{2}\alpha_{13})(\alpha_{3}\alpha_{22}) (\alpha_{4}\alpha_{9})(\alpha_{7}\alpha_{18}) (\alpha_{8}\alpha_{14})(\alpha_{11}\alpha_{20}) (\alpha_{12}\alpha_{23})(\alpha_{16}\alpha_{19})] $$ with orbits $\{\alpha_{8},\alpha_{14}\}$, $\{\alpha_{1},\alpha_{10},\alpha_{5},\alpha_{24}\}$, $\{\alpha_{2},\alpha_{22},\alpha_{13},\alpha_{3}\}$. The $G_1\cong D_8$ is marked by $$ G=H_{22,3}\supset G_1= $$ $$ [(\alpha_{2}\alpha_{22})(\alpha_{3}\alpha_{13})(\alpha_{4}\alpha_{18})(\alpha_{7}\alpha_{9}) (\alpha_{10}\alpha_{24})(\alpha_{11}\alpha_{20})(\alpha_{15}\alpha_{17})(\alpha_{16}\alpha_{19}), $$ $$ (\alpha_{1}\alpha_{10})(\alpha_{2}\alpha_{13})(\alpha_{5}\alpha_{24})(\alpha_{6}\alpha_{15}) (\alpha_{7}\alpha_{18})(\alpha_{12}\alpha_{23})(\alpha_{16}\alpha_{19})(\alpha_{17}\alpha_{21})] $$ with suborbits $\{\alpha_{8}\}$, $\{\alpha_{14}\}$, $\{\alpha_{1},\alpha_{10},\alpha_{24},\alpha_{5}\}$, $\{\alpha_{2},\alpha_{22},\alpha_{13},\alpha_{3}\}$. Both $G$ and $G_1$ are marked by $\ast$ in Tables 1---4. Case 26: $({\bf n}=10,\ (\mathbb A_1,\,\mathbb A_1,\,4\mathbb A_1,\,8\mathbb A_1)\subset 14\mathbb A_1)\ \Longleftarrow\ ({\bf n}=22,\ (2\mathbb A_1,4\mathbb A_1,8\mathbb A_1)\subset 14\mathbb A_1)$. Similar to Case 16. By \cite{Nik11}, we should consider two cases, when \newline $({\bf n}=22,(2\mathbb A_1,4\mathbb A_1,8\mathbb A_1)\subset 14\mathbb A_1)$ is marked by $N_{23}$ and by $N_{21}$. If $({\bf n}=22,(2\mathbb A_1,4\mathbb A_1,8\mathbb A_1)\subset 14\mathbb A_1)$ is marked by $N_{23}$, then $G$ is marked by $G=H_{22,2}$ from Case 12 with orbits $\{\alpha_{12},\alpha_{23}\}$, $\{\alpha_{6},\alpha_{15},\alpha_{21},\alpha_{17}\}$, $\{\alpha_{2},\alpha_{22},\alpha_{13},\alpha_{7},\alpha_{9}, \alpha_{3},\alpha_{18},\alpha_{4}\}$. The $G_1\cong D_8$ is marked by $G_1\subset G=H_{22,2}$ from Case 15 with suborbits $\{\alpha_{12}\}$, $\{\alpha_{23}\}$, $\{\alpha_{6},\alpha_{15},\alpha_{21},\alpha_{17}\}$, $\{\alpha_{2},\alpha_{22},\alpha_{13},\alpha_{7},\alpha_{9}, \alpha_{3},\alpha_{18},\alpha_{4}\}$. If $({\bf n}=22,(2\mathbb A_1,4\mathbb A_1,8\mathbb A_1)\subset 14\mathbb A_1)$ is marked by $N_{21}$, then $G$ is marked by $G=H_{22,1}$ from Case 6 with orbits $\{\alpha_{1,1},\alpha_{3,1}\}$, $\{\alpha_{1,4},\alpha_{3,5},\alpha_{3,4},\alpha_{1,5}$\}, $\{\alpha_{1,3},\alpha_{3,3},\alpha_{3,6},\alpha_{1,6},\alpha_{3,8}, \alpha_{1,8},\alpha_{3,7},\alpha_{1,7}\}$. The $G_1\cong D_8$ is marked by $G_1\subset G=H_{22,1}$ of Case 16 with suborbits $\{\alpha_{1,1}\}$, $\{\alpha_{3,1}\}$, $\{\alpha_{1,4},\alpha_{3,5}, \alpha_{3,4}, \alpha_{1,5}\}$ , $\{\alpha_{1,3},\alpha_{3,3},\alpha_{3,6},\alpha_{1,6},\alpha_{3,8}, \alpha_{1,8},\alpha_{3,7},\alpha_{1,7}\}$. Case 27: $({\bf n}=10,\ (\mathbb A_1,\,\mathbb A_1,\,4\mathbb A_1,\, 8\mathbb A_1)\subset 2\mathbb A_1\amalg 4\mathbb A_3)$ $\Longleftarrow\ ({\bf n}=22,\ \left(\begin{array}{rrr} 2\mathbb A_1 & 6\mathbb A_1 & 10\mathbb A_1 \\ & 4\mathbb A_1 & 4\mathbb A_3 \\ & & 8\mathbb A_1 \end{array}\right)\subset 2\mathbb A_1\amalg 4\mathbb A_3)$. Similar to Cases 1 and 9. The $G\cong C_2\times D_8$ is marked by $N_{21}$ and $G=H_{22,1}$ from Case 9 with orbits $\{\alpha_{1,1},\alpha_{3,1}\}$, $\{\alpha_{2,3},\alpha_{2,6},\alpha_{2,8},\alpha_{2,7}\}$, $\{\alpha_{1,3},\alpha_{3,3},\alpha_{3,6},\alpha_{1,6}, \alpha_{3,8}, \alpha_{1,8},\alpha_{3,7},\alpha_{1,7}\}$. The $G_1\cong D_8$ is marked by $$ G=H_{22,1}\supset G_1= $$ $$ [(\alpha_{1,3}\alpha_{3,3})(\alpha_{1,4}\alpha_{3,5})(\alpha_{2,4}\alpha_{2,5})(\alpha_{3,4}\alpha_{1,5}) (\alpha_{1,6}\alpha_{1,8})(\alpha_{2,6}\alpha_{2,8})(\alpha_{3,6}\alpha_{3,8})(\alpha_{1,7}\alpha_{3,7}), $$ $$ (\alpha_{1,2}\alpha_{3,2})(\alpha_{1,3}\alpha_{1,6}\alpha_{1,7}\alpha_{3,8}) (\alpha_{2,3}\alpha_{2,6}\alpha_{2,7}\alpha_{2,8}) (\alpha_{3,3}\alpha_{3,6}\alpha_{3,7}\alpha_{1,8}) (\alpha_{1,4}\alpha_{3,5}\alpha_{3,4}\alpha_{1,5})(\alpha_{2,4}\alpha_{2,5})] $$ with suborbits $\{\alpha_{1,1}\}$, $\{\alpha_{1,3}\}$, $\{\alpha_{2,3},\alpha_{2,7},\alpha_{2,6},\alpha_{2,8}\}$, $\{\alpha_{1,3},\alpha_{1,7},\alpha_{3,6},\alpha_{1,6},\alpha_{1,8},\alpha_{3,8},\alpha_{3,3},\alpha_{3,7}\}$. Both $G$ and $G_1$ are marked by $\ast$ in Tables 1---4. Case 28: ({\bf n}=10,\ $ \left(\begin{array}{cccc} \mathbb A_1 & \mathbb A_3 & 3\mathbb A_1 & 5\mathbb A_1 \\ & (2\mathbb A_1)_{II} & 4\mathbb A_1 & 6\mathbb A_1 \\ & & (2\mathbb A_1)_I & 2\mathbb A_3 \\ & & & 4\mathbb A_1 \end{array}\right) \subset 3\mathbb A_3)\ \Longleftarrow ({\bf n}=34,\ (3\mathbb A_1,(6\mathbb A_1)_{II})\subset 3\mathbb A_3)$. Similar to Case 1. By \cite{Nik11}, the $G\cong \SSS_4$ is marked by $N_{21}$ and $$ G=H_{34,2}= $$ $$ [(\alpha_{1,2}\alpha_{1,3}\alpha_{3,7})(\alpha_{2,2}\alpha_{2,3}\alpha_{2,7}) (\alpha_{3,2}\alpha_{3,3}\alpha_{1,7})(\alpha_{1,4}\alpha_{3,8}\alpha_{3,5}) (\alpha_{2,4}\alpha_{2,8}\alpha_{2,5}) (\alpha_{3,4}\alpha_{1,8}\alpha_{1,5}), $$ $$ (\alpha_{1,1}\alpha_{3,1}) (\alpha_{1,3}\alpha_{3,7}\alpha_{3,3}\alpha_{1,7}) (\alpha_{2,3}\alpha_{2,7}) (\alpha_{1,4}\alpha_{3,6}\alpha_{3,5}\alpha_{3,8}) (\alpha_{2,4}\alpha_{2,6}\alpha_{2,5}\alpha_{2,8}) (\alpha_{3,4}\alpha_{1,6}\alpha_{1,5}\alpha_{1,8})] $$ with orbits $\{\alpha_{2,2},\alpha_{2,3},\alpha_{2,7}\}$, $\{\alpha_{1,2},\alpha_{1,3},\alpha_{3,7},\alpha_{3,3},\alpha_{1,7},\alpha_{3,2}\}$. The $G_1\cong D_8$ is marked by $$ G=H_{34,2}\supset G_1= $$ $$ [(\alpha_{1,2}\alpha_{3,2})(\alpha_{1,3}\alpha_{3,3})(\alpha_{1,4}\alpha_{3,6})(\alpha_{2,4}\alpha_{2,6}) (\alpha_{3,4}\alpha_{1,6})(\alpha_{1,5}\alpha_{1,8})(\alpha_{2,5}\alpha_{2,8})(\alpha_{3,5}\alpha_{3,8}), $$ $$ (\alpha_{1,1}\alpha_{3,1})(\alpha_{1,2}\alpha_{3,2})(\alpha_{1,3}\alpha_{1,7})(\alpha_{2,3}\alpha_{2,7}) (\alpha_{3,3}\alpha_{3,7})(\alpha_{1,4}\alpha_{3,5})(\alpha_{2,4}\alpha_{2,5})(\alpha_{3,4}\alpha_{1,5})] $$ with suborbits $\{\alpha_{2,2}\}$, $\{\alpha_{1,2},\alpha_{3,2}\}$, $\{\alpha_{2,3},\alpha_{2,7}\}$, $\{\alpha_{1,3},\alpha_{3,3},\alpha_{1,7},\alpha_{3,7}\}$. Both $G$ and $G_1$ are marked by $\ast$ in Tables 1---4. Case 29: ({\bf n}=10,\ $ \left(\begin{array}{cccc} \mathbb A_1 & 3\mathbb A_1 & 5\mathbb A_1 & 9\mathbb A_1 \\ & (2\mathbb A_1)_{I} & (6\mathbb A_1)_I & 10\mathbb A_1 \\ & & 4\mathbb A_1 & 12\mathbb A_1 \\ & & & 8\mathbb A_1 \end{array}\right) \subset 15\mathbb A_1)\ \Longleftarrow ({\bf n}=34,\ (3\mathbb A_1,12\mathbb A_1)\subset 15\mathbb A_1)$. Similar to Case 1. By \cite{Nik11}, the $G\cong \SSS_4$ is marked by $N_{23}$ and $$ G=H_{34,2}= $$ $$ [(\alpha_{1}\alpha_{16}\alpha_{19}\alpha_{15}) (\alpha_{3}\alpha_{5}\alpha_{12}\alpha_{14}) (\alpha_{4}\alpha_{9}) (\alpha_{7}\alpha_{20}\alpha_{23}\alpha_{22}) (\alpha_{8}\alpha_{11}\alpha_{17}\alpha_{18}) (\alpha_{10}\alpha_{13}), $$ $$ (\alpha_{2}\alpha_{13})(\alpha_{3}\alpha_{22}) (\alpha_{4}\alpha_{9})(\alpha_{7}\alpha_{18}) (\alpha_{8}\alpha_{14})(\alpha_{11}\alpha_{20}) (\alpha_{12}\alpha_{23})(\alpha_{16}\alpha_{19})] $$ with orbits $\{\alpha_{2},\alpha_{10},\alpha_{13}\}$,\ $\{\alpha_{3},\alpha_{22},\alpha_{12},\alpha_{18},\alpha_{17}, \alpha_{20},\alpha_{7},\alpha_{11},\alpha_{23},\alpha_{8}, \alpha_{5},\alpha_{14}\}$. The $G_1\cong D_8$ is marked by $$ G=H_{34,2}\supset G_1= $$ $$ [(\alpha_{3}\alpha_{8})(\alpha_{4}\alpha_{9})(\alpha_{5}\alpha_{18})(\alpha_{10}\alpha_{13}) (\alpha_{11}\alpha_{14})(\alpha_{12}\alpha_{17})(\alpha_{15}\alpha_{16})(\alpha_{20}\alpha_{22}), $$ $$ (\alpha_{1}\alpha_{15})(\alpha_{3}\alpha_{18})(\alpha_{5}\alpha_{17})(\alpha_{7}\alpha_{22}) (\alpha_{8}\alpha_{14})(\alpha_{11}\alpha_{12})(\alpha_{16}\alpha_{19})(\alpha_{20}\alpha_{23})] $$ with suborbits $\{\alpha_{2}\}$, $\{\alpha_{10},\alpha_{13}\}$, $\{\alpha_{7},\alpha_{22},\alpha_{20},\alpha_{23}\}$, $\{\alpha_{3},\alpha_{8},\alpha_{18},\alpha_{14},\alpha_{5},\alpha_{11},\alpha_{17},\alpha_{12}\}$. Both $G$ and $G_1$ are marked by $\ast$ in Tables 1---4. Case 30: ({\bf n}=10,\ $(\mathbb A_1,\,4\mathbb A_1,\,4\mathbb A_1,\,2\mathbb A_2)\subset \mathbb A_1\amalg 6\mathbb A_2) \Longleftarrow ({\bf n}=34,\ (\mathbb A_1,6\mathbb A_2)\subset \mathbb A_1\amalg 6\mathbb A_2))$. Similar to Cases 1 and 22. By \cite{Nik11}, the $G\cong \SSS_4$ is marked by by $N_{22}$ and $G=H_{34,1}$ of Case 22 with orbits $\{\alpha_{1,6}\}$, $\{\alpha_{1,2},\alpha_{2,5}, \alpha_{1,5},\alpha_{1,11},\alpha_{2,2},\alpha_{2,11}, \alpha_{1,12},\alpha_{2,12},\alpha_{2,7},\alpha_{1,7},\alpha_{1,9},\alpha_{2,9}\}$. The $G_1\cong D_8$ is marked by $G_1\subset G=H_{34,1}$ of Case 22 with suborbits $\{\alpha_{1,6}\}$, $\{\alpha_{1,2},\alpha_{1,11},\alpha_{2,7},\linebreak \alpha_{2,12}\}$, $\{\alpha_{2,2},\alpha_{2,11},\alpha_{1,7},\alpha_{1,12}\}$, $\{\alpha_{1,5},\alpha_{1,9},\alpha_{2,9},\alpha_{2,5}\}$. Both $G$ and $G_1$ are marked by $\ast$ in Tables 1---4. Case 31: ({\bf n}=10,\ $ \left(\begin{array}{cccc} (2\mathbb A_1)_I & 4\mathbb A_1 & (6\mathbb A_1)_I & (6\mathbb A_1)_{I} \\ & (2\mathbb A_1)_{II} & 6\mathbb A_1 & 6\mathbb A_1 \\ & & 4\mathbb A_1 & (8\mathbb A_1)_{II} \\ & & & 4\mathbb A_1 \end{array}\right) \subset 12\mathbb A_1)\ \Longleftarrow ({\bf n}=65,\ 12\mathbb A_1)$. Similar to Cases 1 and 24. By \cite{Nik10}, the $G\cong 2^4D_6$ is marked by $N_{23}$ and $G=H_{65,3}$ of Case 24 with the orbit $\{\alpha_{1},\alpha_{7},\alpha_{23},\alpha_{8},\alpha_{11},\alpha_{17},\alpha_{20}, \alpha_{9}, \alpha_{6},\alpha_{4},\alpha_{14},\alpha_{13}\}$. The $G_1\cong D_8$ is marked by $$ G=H_{65,3}\supset G_1= $$ $$ [(\alpha_{1}\alpha_{13})(\alpha_{4}\alpha_{11})(\alpha_{5}\alpha_{22})(\alpha_{7}\alpha_{20}) (\alpha_{8}\alpha_{17})(\alpha_{10}\alpha_{15})(\alpha_{12}\alpha_{24})(\alpha_{14}\alpha_{23}), $$ $$ (\alpha_{3}\alpha_{22})(\alpha_{4}\alpha_{17})(\alpha_{5}\alpha_{16})(\alpha_{6}\alpha_{9}) (\alpha_{7}\alpha_{20})(\alpha_{10}\alpha_{19})(\alpha_{13}\alpha_{14})(\alpha_{15}\alpha_{21})] $$ with suborbits $\{\alpha_{6},\alpha_{9}\}$, $\{\alpha_{7},\alpha_{20}\}$, $\{\alpha_{1},\alpha_{13},\alpha_{14},\alpha_{23}\}$, $\{\alpha_{4},\alpha_{11},\alpha_{17},\alpha_{8}\}$ . The orbit $\{\alpha_{6},\alpha_{9}\}$ has the type $(2\mathbb A_1)_I$, and the orbit $\{\alpha_{7},\alpha_{20}\}$ has the type $(2\mathbb A_1)_{II}$ for $G_1$. Both $G$ and $G_1$ are marked by $\ast$ in Tables 1---4. Case 32: ({\bf n}=10,\ $ \left(\begin{array}{cccc} (2\mathbb A_1)_I & (6\mathbb A_1)_{II} & (6\mathbb A_1)_I & (6\mathbb A_1)_I \\ & 4\mathbb A_1 & (8\mathbb A_1)_{I} & (8\mathbb A_1)_I \\ & & 4\mathbb A_1 & (8\mathbb A_1)_{II} \\ & & & 4\mathbb A_1 \end{array}\right) \subset 14\mathbb A_1)$ $\Longleftarrow\ ({\bf n}=22,\ (2\mathbb A_1,4\mathbb A_1,8\mathbb A_1)\subset 14\mathbb A_1)$. Similar to Case 16. By \cite{Nik11}, we should consider two cases, when \newline $({\bf n}=22,(2\mathbb A_1,4\mathbb A_1,8\mathbb A_1)\subset 14\mathbb A_1)$ is marked by $N_{23}$ and by $N_{21}$. If $({\bf n}=22,(2\mathbb A_1,4\mathbb A_1,8\mathbb A_1)\subset 14\mathbb A_1)$ is marked by $N_{23}$, then $G$ is marked by $G=H_{22,2}$ from Case 12 with orbits $\{\alpha_{12},\alpha_{23}\}$, $\{\alpha_{6},\alpha_{15},\alpha_{21},\alpha_{17}\}$, $\{\alpha_{2},\alpha_{22},\alpha_{13},\alpha_{7},\alpha_{9}, \alpha_{3},\alpha_{18},\alpha_{4}\}$, and $G_1\cong D_8$ is marked by $$ G=H_{22,2}\supset G_1 $$ $$ [(\alpha_{3}\alpha_{4})(\alpha_{6}\alpha_{15})(\alpha_{8}\alpha_{11})(\alpha_{9}\alpha_{22}) (\alpha_{12}\alpha_{23})(\alpha_{14}\alpha_{20})(\alpha_{16}\alpha_{19})(\alpha_{17}\alpha_{21}), $$ $$ (\alpha_{2}\alpha_{9})(\alpha_{3}\alpha_{18})(\alpha_{4}\alpha_{13})(\alpha_{6}\alpha_{21}) (\alpha_{7}\alpha_{22})(\alpha_{8}\alpha_{14})(\alpha_{10}\alpha_{24})(\alpha_{16}\alpha_{19})] $$ with suborbits $\{\alpha_{12},\alpha_{23},\}$, $\{\alpha_{6},\alpha_{15},\alpha_{21},\alpha_{17}\}$, $\{\alpha_{2},\alpha_{9},\alpha_{22},\alpha_{7}\}$, $\{\alpha_{3},\alpha_{4},\alpha_{18},\alpha_{13}\}$. If $({\bf n}=22,(2\mathbb A_1,8\mathbb A_1)\subset 10\mathbb A_1)$ is marked by $N_{21}$, then $G$ is marked by $G=H_{22,1}$ from Case 6 with orbits $\{\alpha_{1,1},\alpha_{3,1}\}$, $\{\alpha_{1,4},\alpha_{3,5},\alpha_{3,4},\alpha_{1,5}$\}, $\{\alpha_{1,3},\alpha_{3,3},\alpha_{3,6},\alpha_{1,6},\alpha_{3,8}, \alpha_{1,8},\alpha_{3,7},\alpha_{1,7}\}$, and $G_1\cong D_8$ is marked by $$ G=H_{22,1}\supset G_1= $$ $$ [(\alpha_{1,2}\alpha_{3,2})(\alpha_{1,3}\alpha_{3,6})(\alpha_{2,3}\alpha_{2,6})(\alpha_{3,3}\alpha_{1,6}) (\alpha_{1,4}\alpha_{3,4})(\alpha_{1,7}\alpha_{1,8})(\alpha_{2,7}\alpha_{2,8})(\alpha_{3,7}\alpha_{3,8}), $$ $$ (\alpha_{1,1}\alpha_{3,1})(\alpha_{1,2}\alpha_{3,2})(\alpha_{1,4}\alpha_{1,5})(\alpha_{2,4}\alpha_{2,5}) (\alpha_{3,4}\alpha_{3,5})(\alpha_{1,6}\alpha_{3,8})(\alpha_{2,6}\alpha_{2,8})(\alpha_{3,6}\alpha_{1,8})] $$ with suborbits $\{\alpha_{1,1}\}$,$\{\alpha_{3,1}\}$,$\{\alpha_{1,4},\alpha_{3,5}, \alpha_{3,4}, \alpha_{1,5}\}$,$\{\alpha_{1,3},\alpha_{3,6},\alpha_{1,8},\alpha_{1,7}\}$,$\{\alpha_{3,3},\alpha_{1,6}, \alpha_{3,8},\alpha_{3,7}\}$. Case 33: ({\bf n}=10,\ $ \left(\begin{array}{cccc} (2\mathbb A_1)_{II} & 6\mathbb A_1 & 6\mathbb A_1 & 6\mathbb A_1 \\ & 4\mathbb A_1 &(8\mathbb A_1)_I & (8\mathbb A_1)_I \\ & & 4\mathbb A_1 & (8\mathbb A_1)_{II} \\ & & & 4\mathbb A_1 \end{array}\right) \subset 14\mathbb A_1) $ $\Longleftarrow\ ({\bf n}=22,\ (2\mathbb A_1,4\mathbb A_1,8\mathbb A_1)\subset 14\mathbb A_1)$. Similar to Case 16. By \cite{Nik11}, we should consider two cases, when \newline $({\bf n}=22,(2\mathbb A_1,4\mathbb A_1,8\mathbb A_1)\subset 14\mathbb A_1)$ is marked by $N_{23}$ and by $N_{21}$. If $({\bf n}=22,(2\mathbb A_1,4\mathbb A_1,8\mathbb A_1)\subset 14\mathbb A_1)$ is marked by $N_{23}$, then $G$ is marked by $G=H_{22,2}$ from Case 12 with orbits $\{\alpha_{12},\alpha_{23}\}$, $\{\alpha_{6},\alpha_{15},\alpha_{21},\alpha_{17}\}$, $\{\alpha_{2},\alpha_{22},\alpha_{13},\alpha_{7},\alpha_{9}, \alpha_{3},\alpha_{18},\alpha_{4}\}$, and $G_1\cong D_8$ is marked by $$ G=H_{22,2}\supset G_1= $$ $$ [(\alpha_{3}\alpha_{4})(\alpha_{6}\alpha_{15})(\alpha_{8}\alpha_{11})(\alpha_{9}\alpha_{22}) (\alpha_{12}\alpha_{23})(\alpha_{14}\alpha_{20})(\alpha_{16}\alpha_{19})(\alpha_{17}\alpha_{21}), $$ $$ (\alpha_{2}\alpha_{3})(\alpha_{4}\alpha_{7})(\alpha_{8}\alpha_{14})(\alpha_{9}\alpha_{18}) (\alpha_{10}\alpha_{24})(\alpha_{12}\alpha_{23})(\alpha_{13}\alpha_{22})(\alpha_{15}\alpha_{17})] $$ with suborbits $\{\alpha_{12},\alpha_{23},\}$,$\{\alpha_{6},\alpha_{15},\alpha_{17},\alpha_{21}\}$, $\{\alpha_{2},\alpha_{3},\alpha_{4},\alpha_{7}\}$, $\{\alpha_{9},\alpha_{22}, \alpha_{18},\alpha_{13}\}$. If $({\bf n}=22,(2\mathbb A_1,8\mathbb A_1)\subset 10\mathbb A_1)$ is marked by $N_{21}$, then $G$ is marked by $G=H_{22,1}$ from Case 6 with orbits $\{\alpha_{1,1},\alpha_{3,1}\}$, $\{\alpha_{1,4},\alpha_{3,5},\alpha_{3,4},\alpha_{1,5}$\}, $\{\alpha_{1,3},\alpha_{3,3},\alpha_{3,6},\alpha_{1,6},\alpha_{3,8}, \alpha_{1,8},\alpha_{3,7},\alpha_{1,7}\}$, and $G_1\cong D_8$ is marked by $$ G=H_{22,1}\supset G_1= $$ $$ [(\alpha_{1,1}\alpha_{3,1})(\alpha_{1,3}\alpha_{1,6})(\alpha_{2,3}\alpha_{2,6})(\alpha_{3,3}\alpha_{3,6}) (\alpha_{1,5}\alpha_{3,5})(\alpha_{1,7}\alpha_{3,8})(\alpha_{2,7}\alpha_{2,8})(\alpha_{3,7}\alpha_{1,8}), $$ $$ (\alpha_{1,1}\alpha_{3,1})(\alpha_{1,2}\alpha_{3,2})(\alpha_{1,4}\alpha_{1,5})(\alpha_{2,4}\alpha_{2,5}) (\alpha_{3,4}\alpha_{3,5})(\alpha_{1,6}\alpha_{3,8})(\alpha_{2,6}\alpha_{2,8})(\alpha_{3,6}\alpha_{1,8})] $$ with suborbits $\{\alpha_{1,1}\}$, $\{\alpha_{3,1}\}$,$\{\alpha_{1,4},\alpha_{3,5}, \alpha_{3,4}, \alpha_{1,5}\}$,$\{\alpha_{1,3},\alpha_{1,6},\alpha_{3,8},\alpha_{1,7}\}$,$\{\alpha_{3,3},\alpha_{3,6}, \alpha_{1,8},\alpha_{3,7}\}$. Case 34: ({\bf n}=10,\ $ \left(\begin{array}{cccc} (2\mathbb A_1)_{I} & (6\mathbb A_1)_I & (6\mathbb A_1)_I & (6\mathbb A_1)_I \\ & 4\mathbb A_1 & 4\mathbb A_2 & (8\mathbb A_1)_{II} \\ & & 4\mathbb A_1 & 4\mathbb A_2 \\ & & & 4\mathbb A_1 \end{array}\right) \subset 2\mathbb A_1\amalg 4\mathbb A_3) $,\ ({\bf n}=10,\ $ \left(\begin{array}{cccc} (2\mathbb A_1)_{II} & 6\mathbb A_1 & 6\mathbb A_1 & 6\mathbb A_1 \\ & 4\mathbb A_1 & 4\mathbb A_2 & (8\mathbb A_1)_{II} \\ & & 4\mathbb A_1 & 4\mathbb A_2 \\ & & & 4\mathbb A_1 \end{array}\right) \subset 2\mathbb A_1\amalg 4\mathbb A_3) $ $\Longleftarrow\ ({\bf n}=22,\ \left(\begin{array}{rrr} 2\mathbb A_1 & 6\mathbb A_1 & 10\mathbb A_1 \\ & 4\mathbb A_1 & 4\mathbb A_3 \\ & & 8\mathbb A_1 \end{array}\right)\subset 2\mathbb A_1\amalg 4\mathbb A_3)$. Similar to Cases 1 and 9. By \cite{Nik11}, the $G$ is marked by $N_{21}$ and $G=H_{22,1}$ from Case 9 with orbits $\{\alpha_{1,1},\alpha_{3,1}\}$, $\{\alpha_{2,3},\alpha_{2,6},\alpha_{2,8},\alpha_{2,7}\}$, $\{\alpha_{1,3},\alpha_{3,3},\alpha_{3,6},\alpha_{1,6}, \alpha_{3,8}, \alpha_{1,8},\alpha_{3,7},\alpha_{1,7}\}$. The $G_1\cong D_8$ is marked by $$ G=H_{22,1}\supset G_1= $$ $$ [(\alpha_{1,2}\alpha_{3,2})(\alpha_{1,3}\alpha_{3,6})(\alpha_{2,3}\alpha_{2,6})(\alpha_{3,3}\alpha_{1,6}) (\alpha_{1,4}\alpha_{3,4})(\alpha_{1,7}\alpha_{1,8})(\alpha_{2,7}\alpha_{2,8})(\alpha_{3,7}\alpha_{3,8}), $$ $$ (\alpha_{1,1}\alpha_{3,1}) (\alpha_{1,3}\alpha_{3,6}\alpha_{1,7}\alpha_{1,8})(\alpha_{2,3}\alpha_{2,6}\alpha_{2,7}\alpha_{2,8}) (\alpha_{3,3}\alpha_{1,6}\alpha_{3,7}\alpha_{3,8})(\alpha_{1,4}\alpha_{1,5}\alpha_{3,4}\alpha_{3,5}) (\alpha_{2,4}\alpha_{2,5})] $$ with suborbits $\{\alpha_{1,1},\alpha_{3,1}\}$, $\{\alpha_{2,3}\alpha_{2,6},\alpha_{2,7},\alpha_{2,8}\}$, $\{\alpha_{1,3}\alpha_{3,6},\alpha_{1,7},\alpha_{1,8}\}$, $\{\alpha_{3,3}\alpha_{1,6},\alpha_{3,7},\alpha_{3,8}\}$ for the first case, and by $$ G=H_{22,1}\supset G_1= $$ $$ [(\alpha_{1,1}\alpha_{3,1})(\alpha_{1,3}\alpha_{1,6})(\alpha_{2,3}\alpha_{2,6})(\alpha_{3,3}\alpha_{3,6}) (\alpha_{1,5}\alpha_{3,5})(\alpha_{1,7}\alpha_{3,8})(\alpha_{2,7}\alpha_{2,8})(\alpha_{3,7}\alpha_{1,8}), $$ $$ (\alpha_{1,2}\alpha_{3,2})(\alpha_{1,3}\alpha_{1,6}\alpha_{1,7}\alpha_{3,8}) (\alpha_{2,3}\alpha_{2,6}\alpha_{2,7}\alpha_{2,8}) (\alpha_{3,3}\alpha_{3,6}\alpha_{3,7}\alpha_{1,8}) (\alpha_{1,4}\alpha_{3,5}\alpha_{3,4}\alpha_{1,5})(\alpha_{2,4}\alpha_{2,5})] $$ with suborbits $\{\alpha_{1,1},\alpha_{3,1}\}$, $\{\alpha_{2,3},\alpha_{2,6},\alpha_{2,7},\alpha_{2,8}\}$, $\{\alpha_{1,3},\alpha_{1,6},\alpha_{1,7},\alpha_{3,8}\}$, $\{\alpha_{3,3},\alpha_{3,6},\alpha_{3,7},\alpha_{1,8}\}$ for the second case. All these $G$ and $G_1$ are marked by $\ast$ in Tables 1---4. Case 35: ({\bf n}=10,\ $ \left(\begin{array}{cccc} (2\mathbb A_1)_I & 2\mathbb A_3 & (6\mathbb A_1)_I & 10\mathbb A_1 \\ & 4\mathbb A_1 & (8\mathbb A_1)_{I} & 12\mathbb A_1 \\ & & 4\mathbb A_1 & 4\mathbb A_3 \\ & & & 8\mathbb A_1 \end{array}\right) \subset 6\mathbb A_3) \ \Longleftarrow ({\bf n}=34,\ ((6\mathbb A_1)_I,12\mathbb A_1)\subset 6\mathbb A_3)$. Similar to Case 1. By \cite{Nik11}, the $G\cong \SSS_4$ is marked by $N_{21}$ and $$ G=H_{34,1}= $$ $$ [(\alpha_{1,3}\alpha_{3,4}\alpha_{1,8}) (\alpha_{2,3}\alpha_{2,4}\alpha_{2,8}) (\alpha_{3,3}\alpha_{1,4}\alpha_{3,8}) (\alpha_{1,5}\alpha_{3,6}\alpha_{3,7}) (\alpha_{2,5}\alpha_{2,6}\alpha_{2,7}) (\alpha_{3,5}\alpha_{1,6}\alpha_{1,7}), $$ $$ (\alpha_{1,1}\alpha_{3,1}) (\alpha_{1,3}\alpha_{3,7}\alpha_{3,3}\alpha_{1,7}) (\alpha_{2,3}\alpha_{2,7}) (\alpha_{1,4}\alpha_{3,6}\alpha_{3,5}\alpha_{3,8}) (\alpha_{2,4}\alpha_{2,6}\alpha_{2,5}\alpha_{2,8}) (\alpha_{3,4}\alpha_{1,6}\alpha_{1,5}\alpha_{1,8})] $$ with orbits $\{\alpha_{2,3},\alpha_{2,4},\alpha_{2,7},\alpha_{2,8},\alpha_{2,6},\alpha_{2,5} \}$, $\{\alpha_{1,3}, \alpha_{3,4},\alpha_{3,7}, \alpha_{1,8}, \alpha_{1,6}, \alpha_{1,5}, \alpha_{3,3}, \alpha_{1,7}, \alpha_{3,6}, \alpha_{1,4}, \linebreak \alpha_{3,5}, \alpha_{3,8}\}$. The $G_1\cong D_8$ is marked by $$ G=H_{34,1}\supset G_1= $$ $$ [(\alpha_{1,3}\alpha_{1,7})(\alpha_{2,3}\alpha_{2,7})(\alpha_{3,3}\alpha_{3,7})(\alpha_{1,4}\alpha_{3,4}) (\alpha_{1,5}\alpha_{3,5})(\alpha_{1,6}\alpha_{3,8})(\alpha_{2,6}\alpha_{2,8})(\alpha_{3,6}\alpha_{1,8}), $$ $$ (\alpha_{1,1}\alpha_{3,1})(\alpha_{1,4}\alpha_{1,6})(\alpha_{2,4}\alpha_{2,6})(\alpha_{3,4}\alpha_{3,6}) (\alpha_{1,5}\alpha_{3,8})(\alpha_{2,5}\alpha_{2,8})(\alpha_{3,5}\alpha_{1,8})(\alpha_{1,7}\alpha_{3,7})] $$ with suborbits $\{\alpha_{2,3},\alpha_{2,7}\}$, $\{\alpha_{2,4},\alpha_{2,6},\alpha_{2,8},\alpha_{2,5}\}$, $\{\alpha_{1,3},\alpha_{1,7},\alpha_{3,7},\alpha_{3,3}\}$, $\{\alpha_{1,4},\alpha_{3,4},\alpha_{1,6},\alpha_{3,6}, \linebreak \alpha_{3,8},\alpha_{1,8},\alpha_{1,5},\alpha_{3,5}\}$. Both $G$ and $G_1$ are marked by $\ast$ in Tables 1---4. Case 36: ({\bf n}=10,\ $ \left(\begin{array}{cccc} 4\mathbb A_1 & (8\mathbb A_1)_{II} & (8\mathbb A_1)_I & (8\mathbb A_1)_I \\ & 4\mathbb A_1 & (8\mathbb A_1)_I & (8\mathbb A_1)_I \\ & & 4\mathbb A_1 & (8\mathbb A_1)_{II}\\ & & & 4\mathbb A_1 \end{array}\right) \subset 16\mathbb A_1) \Longleftarrow ({\bf n}=56,\ 16\mathbb A_1)$. Similar to Cases 1 and 8. By \cite{Nik10}, the $G\cong \Gamma_{25}a_1$ is marked by $N_{23}$ and $G=H_{56,2}$ from Case 8 with the orbit $\{\alpha_{2}, \alpha_{3}, \alpha_{23}, \alpha_{24}, \alpha_{5},\alpha_{8}, \alpha_{17}, \alpha_{19}, \alpha_{6}, \alpha_{7}, \alpha_{11}, \alpha_{20}, \alpha_{18}, \alpha_{10},\alpha_{16},\alpha_{15}\}$. The $G_1\cong D_8$ is marked by $$ G=H_{56,2}\supset G_1= $$ $$ [(\alpha_{2}\alpha_{17})(\alpha_{6}\alpha_{16})(\alpha_{7}\alpha_{8})(\alpha_{9}\alpha_{13}) (\alpha_{10}\alpha_{18})(\alpha_{12}\alpha_{22})(\alpha_{19}\alpha_{20})(\alpha_{23}\alpha_{24}), $$ $$ (\alpha_{1}\alpha_{14})(\alpha_{3}\alpha_{7})(\alpha_{5}\alpha_{8})(\alpha_{9}\alpha_{13}) (\alpha_{10}\alpha_{24})(\alpha_{11}\alpha_{20})(\alpha_{15}\alpha_{19})(\alpha_{16}\alpha_{17})] $$ with suborbits $\{\alpha_{2},\alpha_{17},\alpha_{16},\alpha_{6}\}$, $\{\alpha_{3},\alpha_{7},\alpha_{8},\alpha_{5}\}$, $\{\alpha_{10},\alpha_{18},\alpha_{24},\alpha_{23}\}$, $\{\alpha_{11},\alpha_{20},\alpha_{19},\alpha_{15}\}$. \linebreak Both $G$ and $G_1$ are marked by $\ast$ in Tables 1---4. Case 37: ({\bf n}=10,\ $ \left(\begin{array}{cccc} 4\mathbb A_1 & (8\mathbb A_1)_I & (8\mathbb A_1)_I & (8\mathbb A_1)_I \\ & 4\mathbb A_1 & 4\mathbb A_2 & (8\mathbb A_1)_{II}\\ & & 4\mathbb A_1 & 4\mathbb A_2 \\ & & & 4\mathbb A_1 \end{array}\right) \subset 4\mathbb A_1\amalg 4\mathbb A_3) $ $\Longleftarrow ({\bf n}=22,\ \left(\begin{array}{rrr} 4\mathbb A_1 & (8\mathbb A_1)_{II} & 12\mathbb A_1 \\ & 4\mathbb A_1 & 4\mathbb A_3 \\ & & 8\mathbb A_1 \end{array}\right)\subset 4\mathbb A_1\amalg 4\mathbb A_3)$. Similar to Cases 1 and 11. By \cite{Nik11}, the $G\cong C_2\times D_8$ is marked by $N_{21}$ and $G=H_{22,1}$ from Case 11 with orbits $\{\alpha_{1,4},\alpha_{3,5},\alpha_{3,4},\alpha_{1,5}\}$, $\{\alpha_{2,3},\alpha_{2,6},\alpha_{2,8},\alpha_{2,7}\}$, $\{\alpha_{1,3},\alpha_{3,3},\alpha_{3,6},\alpha_{1,6}, \linebreak \alpha_{3,8}, \alpha_{1,8},\alpha_{3,7},\alpha_{1,7}\}$. The $G_1\cong D_8$ is marked by $$ G=H_{22,1}\supset G_1= $$ $$ [(\alpha_{1,2}\alpha_{3,2})(\alpha_{1,3}\alpha_{3,6})(\alpha_{2,3}\alpha_{2,6})(\alpha_{3,3}\alpha_{1,6}) (\alpha_{1,4}\alpha_{3,4})(\alpha_{1,7}\alpha_{1,8})(\alpha_{2,7}\alpha_{2,8})(\alpha_{3,7}\alpha_{3,8}), $$ $$ (\alpha_{1,1}\alpha_{3,1}) (\alpha_{1,3}\alpha_{3,6}\alpha_{1,7}\alpha_{1,8})(\alpha_{2,3}\alpha_{2,6}\alpha_{2,7}\alpha_{2,8}) (\alpha_{3,3}\alpha_{1,6}\alpha_{3,7}\alpha_{3,8})(\alpha_{1,4}\alpha_{1,5}\alpha_{3,4}\alpha_{3,5}) (\alpha_{2,4}\alpha_{2,5})] $$ with suborbits $\{\alpha_{1,4},\alpha_{3,4},\alpha_{1,5},\alpha_{3,5}\}$, $\{\alpha_{1,3},\alpha_{3,6},\alpha_{1,7},\alpha_{1,8}\}$, $\{\alpha_{2,3},\alpha_{2,6},\alpha_{2,7},\alpha_{2,8}\}$, $\{\alpha_{3,3},\alpha_{1,6}, \linebreak \alpha_{3,7},\alpha_{3,8}\}$. Both $G$ and $G_1$ are marked by $\ast$ in Tables 1---4. Case 38: ({\bf n}=10,\ $ \left(\begin{array}{cccc} 4\mathbb A_1 & (8\mathbb A_1)_{I} & (8\mathbb A_1)_I & 4\mathbb A_1\amalg 2\mathbb A_2 \\ & 4\mathbb A_1 & 4\mathbb A_2 & 4\mathbb A_1\amalg 2\mathbb A_2 \\ & & 4\mathbb A_1 & 4\mathbb A_1\amalg 2\mathbb A_2 \\ & & & 2\mathbb A_2 \end{array}\right) \subset 4\mathbb A_1\amalg 6\mathbb A_2)$\ $\Longleftarrow ({\bf n}=34,\ (4\mathbb A_1,6\mathbb A_2)\subset 4\mathbb A_1\amalg 6\mathbb A_2)$. Similar to Cases 1 and 22. By \cite{Nik11}, the $G\cong \SSS_4$ is marked by $N_{22}$ and $G=H_{34,1}$ of Case 22 with orbits $\{\alpha_{1,1}, \alpha_{1,3}, \alpha_{2,4}, \alpha_{1,8}\}$, $\{\alpha_{1,2},\alpha_{2,5}, \alpha_{1,5},\alpha_{1,11},\alpha_{2,2},\alpha_{2,11}, \alpha_{1,12},\alpha_{2,12},\alpha_{2,7},\alpha_{1,7},\alpha_{1,9}, \linebreak \alpha_{2,9}\}$. The $G_1\cong D_8$ is marked by $G_1\subset G=H_{34,1}$ of Case 22 with suborbits $\{\alpha_{1,1}, \alpha_{1,3}, \alpha_{2,4}, \linebreak \alpha_{1,8}\}$, $\{\alpha_{1,2},\alpha_{1,11},\alpha_{2,7},\alpha_{2,12}\}$, $\{\alpha_{2,2},\alpha_{2,11},\alpha_{1,7},\alpha_{1,12}\}$, $\{\alpha_{1,5},\alpha_{1,9},\alpha_{2,9},\alpha_{2,5}\}$. Both $G$ and $G_1$ are marked by $\ast$ in Tables 1---4. Case 39: $({\bf n}=12,\ (8\mathbb A_1,8\mathbb A_1)\subset 16\mathbb A_1)\Longleftarrow ({\bf n}=75,\ 16\mathbb A_1)$. Similar to Cases 1 and 10. By \cite{Nik10}, the $G\cong 4^2\AAA_4$ is marked by $N_{23}$ and $G=H_{75,1}$ from Case 10 with the orbit $ \{\alpha_{1},\alpha_{9},\alpha_{6},\alpha_{7},\alpha_{4}, \alpha_{21},\alpha_{20}, \alpha_{22},\alpha_{5},\alpha_{14},\alpha_{10},\alpha_{23}, \alpha_{3},\alpha_{24},\alpha_{18},\alpha_{16}\}$. The $G_1\cong Q_8$ is marked by $$ G=H_{75,1}\supset G_1= $$ $$ [(\alpha_{1}\alpha_{4}\alpha_{9}\alpha_{5})(\alpha_{3}\alpha_{20}\alpha_{24}\alpha_{14}) (\alpha_{6}\alpha_{18}\alpha_{21}\alpha_{7})(\alpha_{10}\alpha_{16}\alpha_{22}\alpha_{23}) (\alpha_{11}\alpha_{12})(\alpha_{13}\alpha_{17}), $$ $$ (\alpha_{1}\alpha_{3}\alpha_{9}\alpha_{24})(\alpha_{4}\alpha_{14}\alpha_{5}\alpha_{20}) (\alpha_{6}\alpha_{22}\alpha_{21}\alpha_{10})(\alpha_{7}\alpha_{23}\alpha_{18}\alpha_{16}) (\alpha_{11}\alpha_{17})(\alpha_{12}\alpha_{13})] $$ with suborbits $\{\alpha_{1},\alpha_{4},\alpha_{3},\alpha_{9},\alpha_{14},\alpha_{20},\alpha_{5},\alpha_{24}\}$, $\{\alpha_{6},\alpha_{18},\alpha_{22},\alpha_{21},\alpha_{16},\alpha_{23},\alpha_{7},\alpha_{10}\}$. Both $G$ and $G_1$ are marked by $\ast$ in Tables 1---4. Case 40: $({\bf n}=12,\ (\mathbb A_2,\mathbb A_2)\subset 2\mathbb A_2)\Longleftarrow ({\bf n}=26,\ 2\mathbb A_2)$. By \cite{Nik10}, the $G\cong SD_{16}$ is marked by $N_{22}$ and $$ G=H_{26,1}= $$ $$ [(\alpha_{1,2}\alpha_{2,2})(\alpha_{1,4}\alpha_{2,4}) (\alpha_{1,5}\alpha_{2,6}\alpha_{2,11}\alpha_{2,9}) (\alpha_{2,5}\alpha_{1,6}\alpha_{1,11}\alpha_{1,9}) (\alpha_{1,7}\alpha_{2,10}\alpha_{2,8}\alpha_{1,12}) (\alpha_{2,7}\alpha_{1,10}\alpha_{1,8}\alpha_{2,12}), $$ $$ (\alpha_{1,3}\alpha_{2,4})(\alpha_{2,3}\alpha_{1,4})(\alpha_{1,6}\alpha_{2,12})(\alpha_{2,6}\alpha_{1,12}) (\alpha_{1,7}\alpha_{2,8})(\alpha_{2,7}\alpha_{1,8})(\alpha_{1,9}\alpha_{1,10})(\alpha_{2,9}\alpha_{2,10})] $$ with the orbit $\{\alpha_{1,3},\alpha_{2,4},\alpha_{1,4},\alpha_{2,3}\}$. The $G_1\cong Q_8$ is marked by $$ G=H_{26,1}\supset G_1 $$ $$ [(\alpha_{1,3}\alpha_{2,3})(\alpha_{1,4}\alpha_{2,4})(\alpha_{1,5}\alpha_{1,7}\alpha_{2,11}\alpha_{2,8}) (\alpha_{2,5}\alpha_{2,7}\alpha_{1,11}\alpha_{1,8}) (\alpha_{1,6}\alpha_{2,12}\alpha_{1,9}\alpha_{1,10})(\alpha_{2,6}\alpha_{1,12}\alpha_{2,9}\alpha_{2,10}), $$ $$ (\alpha_{1,2}\alpha_{2,2})(\alpha_{1,4}\alpha_{2,4})(\alpha_{1,5}\alpha_{2,6}\alpha_{2,11}\alpha_{2,9}) (\alpha_{2,5}\alpha_{1,6}\alpha_{1,11}\alpha_{1,9}) (\alpha_{1,7}\alpha_{2,10}\alpha_{2,8}\alpha_{1,12})(\alpha_{2,7}\alpha_{1,10}\alpha_{1,8}\alpha_{2,12})] $$ with suborbits $\{\alpha_{1,3},\alpha_{2,3}\}$, $\{\alpha_{1,4},\alpha_{2,4}\}$. Both $G$ and $G_1$ are marked by $\ast$ in Tables 1---4. Case 41: $({\bf n}=16,\ (5\mathbb A_1,5\mathbb A_1,5\mathbb A_1)\subset 15\mathbb A_1) \Longleftarrow ({\bf n}=55,\ 15\mathbb A_1)$. Similar to Case 1. By \cite{Nik10}, the $G\cong \AAA_5$ is marked by $N_{23}$ and $$ G=H_{55,2}= $$ $$[(\alpha_{2}\alpha_{8}\alpha_{7}\alpha_{17}\alpha_{11}) (\alpha_{3}\alpha_{14}\alpha_{10}\alpha_{13}\alpha_{12}) (\alpha_{4}\alpha_{19}\alpha_{16}\alpha_{15}\alpha_{9}) (\alpha_{6}\alpha_{23}\alpha_{20}\alpha_{18}\alpha_{22}), $$ $$ (\alpha_{2}\alpha_{13})(\alpha_{3}\alpha_{22}) (\alpha_{4}\alpha_{9})(\alpha_{7}\alpha_{18}) (\alpha_{8}\alpha_{14})(\alpha_{11}\alpha_{20}) (\alpha_{12}\alpha_{23})(\alpha_{16}\alpha_{19})] $$ with the orbit $\{\alpha_{2},\alpha_{8},\alpha_{13},\alpha_{7},\alpha_{14}, \alpha_{12},\alpha_{17},\alpha_{18},\alpha_{10},\alpha_{3}, \alpha_{23},\alpha_{11},\alpha_{22},\alpha_{20},\alpha_{6}\}$. The $G_1\cong D_{10}$ is marked by $$ G=H_{55,2}\supset G_1= $$ $$ [(\alpha_{2}\alpha_{6})(\alpha_{4}\alpha_{15})(\alpha_{7}\alpha_{23})(\alpha_{9}\alpha_{16}) (\alpha_{10}\alpha_{20})(\alpha_{11}\alpha_{14})(\alpha_{12}\alpha_{17})(\alpha_{13}\alpha_{22}), $$ $$ (\alpha_{3}\alpha_{7})(\alpha_{6}\alpha_{11})(\alpha_{8}\alpha_{22})(\alpha_{9}\alpha_{15}) (\alpha_{10}\alpha_{23})(\alpha_{13}\alpha_{17})(\alpha_{14}\alpha_{18})(\alpha_{16}\alpha_{19})] $$ with suborbits $\{\alpha_{2},\alpha_{6},\alpha_{11},\alpha_{14},\alpha_{18}\}$, $\{\alpha_{3},\alpha_{7},\alpha_{23},\alpha_{10},\alpha_{20}\}$, $\{\alpha_{8},\alpha_{22},\alpha_{13},\alpha_{17},\alpha_{12}\}$. Both $G$ and $G_1$ are marked by $\ast$ in Tables 1---4. Case 42: $({\bf n}=17,\ (\mathbb A_1,\mathbb A_1) \subset 2\mathbb A_1)\Longleftarrow ({\bf n}=34,\ 2\mathbb A_1)$. Similar to Cases 1 and 29. By \cite{Nik10}, the $G\cong \SSS_4$ is marked by $N_{23}$ and $G=H_{34,2}$ of Case 29 with the orbit $\{\alpha_{4},\alpha_{9}\}$. The $G_1\cong \AAA_{4}$ is marked by $$ G=H_{34,2}\supset G_1= $$ $$ [(\alpha_{2}\alpha_{10}\alpha_{13})(\alpha_{3}\alpha_{20}\alpha_{14})(\alpha_{5}\alpha_{18}\alpha_{7}) (\alpha_{8}\alpha_{11}\alpha_{22})(\alpha_{12}\alpha_{23}\alpha_{17})(\alpha_{15}\alpha_{16}\alpha_{19}), $$ $$ (\alpha_{1}\alpha_{15})(\alpha_{3}\alpha_{18})(\alpha_{5}\alpha_{17})(\alpha_{7}\alpha_{22}) (\alpha_{8}\alpha_{14})(\alpha_{11}\alpha_{12})(\alpha_{16}\alpha_{19})(\alpha_{20}\alpha_{23})] $$ with suborbits $\{\alpha_{4}\}$, $\{\alpha_{9}\}$. Both $G$ and $G_1$ are marked by $\ast$ in Tables 1---4. Case 43: $({\bf n}=17,\ (\mathbb A_1,3\mathbb A_1) \subset 4\mathbb A_1)\Longleftarrow ({\bf n}=49,\ 4\mathbb A_1)$. Similar to Cases 1 and 5. By \cite{Nik10}, the $G\cong 2^4C_3$ is marked by $N_{23}$ and $G=H_{49,1}$ from Case 5 with the orbit $\{\alpha_{2},\alpha_{12},\alpha_{13},\alpha_{23}\}$. The $G_1\cong \AAA_{4}$ is marked by $$ G=H_{49,1}\supset G_1= $$ $$ [(\alpha_{1}\alpha_{3}\alpha_{18})(\alpha_{7}\alpha_{17}\alpha_{22})(\alpha_{8}\alpha_{19}\alpha_{21}) (\alpha_{9}\alpha_{20}\alpha_{11})(\alpha_{10}\alpha_{14}\alpha_{16})(\alpha_{12}\alpha_{13}\alpha_{23}), $$ $$ (\alpha_{1}\alpha_{10})(\alpha_{3}\alpha_{8})(\alpha_{4}\alpha_{11})(\alpha_{7}\alpha_{18}) (\alpha_{9}\alpha_{20})(\alpha_{14}\alpha_{22})(\alpha_{16}\alpha_{19})(\alpha_{17}\alpha_{21})] $$ with suborbits $\{\alpha_{2}\}$, $\{\alpha_{12},\alpha_{13},\alpha_{23}\}$. Both $G$ and $G_1$ are marked by $\ast$ in Tables 1---4. Case 44: $({\bf n}=17,\ (4\mathbb A_1,4\mathbb A_1) \subset 8\mathbb A_1)\Longleftarrow ({\bf n}=34,\ 8\mathbb A_1)$. Similar to Case 1. By \cite{Nik10}, the $G\cong \SSS_4$ is marked by $N_{23}$ and $$ G=H_{34,1}= $$ $$ [(\alpha_{1}\alpha_{18}\alpha_{15}\alpha_{22}) (\alpha_{2}\alpha_{4}\alpha_{20}\alpha_{11}) (\alpha_{3}\alpha_{7}) (\alpha_{8}\alpha_{10}\alpha_{16}\alpha_{24}) (\alpha_{9}\alpha_{12}\alpha_{23}\alpha_{13}) (\alpha_{14}\alpha_{19}), $$ $$ (\alpha_{2}\alpha_{13})(\alpha_{3}\alpha_{22}) (\alpha_{4}\alpha_{9})(\alpha_{7}\alpha_{18}) (\alpha_{8}\alpha_{14})(\alpha_{11}\alpha_{20}) (\alpha_{12}\alpha_{23})(\alpha_{16}\alpha_{19})] $$ with the orbit $\{\alpha_{2},\alpha_{13},\alpha_{20},\alpha_{9},\alpha_{12},\alpha_{4},\alpha_{23},\alpha_{11}\}$. The $G_1\cong \AAA_{4}$ is marked by $$ G=H_{34,1}\supset G_1= $$ $$ [(\alpha_{1}\alpha_{3}\alpha_{22})(\alpha_{4}\alpha_{12}\alpha_{11})(\alpha_{7}\alpha_{18}\alpha_{15}) (\alpha_{8}\alpha_{10}\alpha_{14})(\alpha_{9}\alpha_{20}\alpha_{23})(\alpha_{16}\alpha_{24}\alpha_{19}), $$ $$ (\alpha_{2}\alpha_{9})(\alpha_{3}\alpha_{7})(\alpha_{4}\alpha_{13})(\alpha_{8}\alpha_{16}) (\alpha_{11}\alpha_{12})(\alpha_{14}\alpha_{19})(\alpha_{18}\alpha_{22})(\alpha_{20}\alpha_{23})] $$ with suborbits $\{\alpha_{2},\alpha_{9},\alpha_{20},\alpha_{23}\}$, $\{\alpha_{4},\alpha_{12},\alpha_{13},\alpha_{11}\}$. Both $G$ and $G_1$ are marked by $\ast$ in Tables 1---4. Case 45: $({\bf n}=17,\ (4\mathbb A_1,12\mathbb A_1) \subset 16\mathbb A_1)\Longleftarrow ({\bf n}=49,\ 16\mathbb A_1)$. Similar to Case 1. By \cite{Nik10}, the $G\cong 2^4C_3$ is marked by $N_{23}$ and $$ G=H_{49,2}= $$ $$ [(\alpha_{1}\alpha_{21}\alpha_{6}) (\alpha_{2}\alpha_{14}\alpha_{4}) (\alpha_{3}\alpha_{8}\alpha_{7}) (\alpha_{9}\alpha_{22}\alpha_{23}) (\alpha_{11}\alpha_{19}\alpha_{20}) (\alpha_{12}\alpha_{18}\alpha_{13}), $$ $$ (\alpha_{2}\alpha_{12})(\alpha_{3}\alpha_{8}) (\alpha_{4}\alpha_{20})(\alpha_{7}\alpha_{16}) (\alpha_{9}\alpha_{11})(\alpha_{13}\alpha_{23}) (\alpha_{14}\alpha_{22})(\alpha_{18}\alpha_{19}), $$ $$ (\alpha_{2}\alpha_{13})(\alpha_{3}\alpha_{22}) (\alpha_{4}\alpha_{9})(\alpha_{7}\alpha_{18}) (\alpha_{8}\alpha_{14})(\alpha_{11}\alpha_{20}) (\alpha_{12}\alpha_{23})(\alpha_{16}\alpha_{19})] $$ with the orbit $\{\alpha_{2},\alpha_{14},\alpha_{12},\alpha_{11},\alpha_{23}, \alpha_{19},\alpha_{4},\alpha_{22},\alpha_{18}, \alpha_{3}, \alpha_{9}, \alpha_{13},\alpha_{7}, \alpha_{20},\alpha_{8},\alpha_{16}\}$. The $G_1\cong \AAA_{4}$ is marked by $$ G=H_{49,2}\supset G_1= $$ $$ [(\alpha_{1}\alpha_{6}\alpha_{21})(\alpha_{3}\alpha_{22}\alpha_{13})(\alpha_{4}\alpha_{16}\alpha_{14}) (\alpha_{7}\alpha_{19}\alpha_{23})(\alpha_{8}\alpha_{18}\alpha_{20})(\alpha_{9}\alpha_{11}\alpha_{12}), $$ $$ (\alpha_{2}\alpha_{3})(\alpha_{4}\alpha_{18})(\alpha_{7}\alpha_{9})(\alpha_{8}\alpha_{12}) (\alpha_{11}\alpha_{16})(\alpha_{13}\alpha_{22})(\alpha_{14}\alpha_{23})(\alpha_{19}\alpha_{20})] $$ with suborbits $\{\alpha_{2},\alpha_{3},\alpha_{22},\alpha_{13}\}$, $\{\alpha_{4},\alpha_{16},\alpha_{18},\alpha_{14},\alpha_{11},\alpha_{20},\alpha_{23}, \alpha_{12},\alpha_{8},\alpha_{19},\alpha_{7},\alpha_{9}\}$. Both $G$ and $G_1$ are marked by $\ast$ in Tables 1---4. Case 46: $({\bf n}=17,\ (6\mathbb A_1,6\mathbb A_1) \subset 12\mathbb A_1)\Longleftarrow ({\bf n}=49,\ 12\mathbb A_1)$. Similar to Cases 1 and 5. By \cite{Nik10}, the $G\cong 2^4C_3$ is marked by $N_{23}$ and $G=H_{49,1}$ from Case 5 with the orbit $\{\alpha_{1},\alpha_{22},\alpha_{21},\alpha_{10},\alpha_{19},\alpha_{14}, \alpha_{8},\alpha_{17},\alpha_{18},\alpha_{7},\alpha_{3},\alpha_{16}\}$. The $G_1\cong \AAA_4$ is marked by $$ G=H_{49,1}\supset G_1= $$ $$ [(\alpha_{1}\alpha_{3}\alpha_{18})(\alpha_{7}\alpha_{17}\alpha_{22})(\alpha_{8}\alpha_{19}\alpha_{21}) (\alpha_{9}\alpha_{20}\alpha_{11})(\alpha_{10}\alpha_{14}\alpha_{16})(\alpha_{12}\alpha_{13}\alpha_{23}), $$ $$ (\alpha_{2}\alpha_{12})(\alpha_{3}\alpha_{8})(\alpha_{4}\alpha_{20})(\alpha_{7}\alpha_{16}) (\alpha_{9}\alpha_{11})(\alpha_{13}\alpha_{23})(\alpha_{14}\alpha_{22})(\alpha_{18}\alpha_{19})] $$ with suborbits $\{\alpha_{1},\alpha_{3},\alpha_{18},\alpha_{8},\alpha_{19},\alpha_{21}\}$, $\{\alpha_{7},\alpha_{17},\alpha_{16},\alpha_{22},\alpha_{10},\alpha_{14}\}$. Both $G$ and $G_1$ are marked by $\ast$ in Tables 1---4. Case 47: $({\bf n}=17,\ (6\mathbb A_1,6\mathbb A_1) \subset 6\mathbb A_2)\Longleftarrow ({\bf n}=34,\ 6\mathbb A_2)$. Similar to Cases 1 and 22. By \cite{Nik10}, the $G\cong \SSS_4$ is marked by $N_{22}$ and $G=H_{34,1}$ of Case 22 with the orbit $\{\alpha_{1,2},\alpha_{2,5},\alpha_{1,5},\alpha_{1,11},\alpha_{2,2}, \alpha_{2,11},\alpha_{1,12},\alpha_{2,12},\alpha_{2,7},\alpha_{1,7},\alpha_{1,9},\alpha_{2,9}\}$. The $G_1\cong \AAA_4$ is marked by $$ G=H_{34,1}\supset G_1= $$ $$ [(\alpha_{1,2}\alpha_{2,9}\alpha_{1,12})(\alpha_{2,2}\alpha_{1,9}\alpha_{2,12})(\alpha_{1,3}\alpha_{1,8}\alpha_{2,4}) (\alpha_{2,3}\alpha_{2,8}\alpha_{1,4})(\alpha_{1,5}\alpha_{2,11}\alpha_{2,7})(\alpha_{2,5}\alpha_{1,11}\alpha_{1,7}), $$ $$ (\alpha_{1,1}\alpha_{1,3})(\alpha_{2,1}\alpha_{2,3})(\alpha_{1,2}\alpha_{2,7})(\alpha_{2,2}\alpha_{1,7}) (\alpha_{1,4}\alpha_{2,8})(\alpha_{2,4}\alpha_{1,8})(\alpha_{1,5}\alpha_{2,9})(\alpha_{2,5}\alpha_{1,9})] $$ with suborbits $\{\alpha_{1,2},\alpha_{2,9},\alpha_{2,7},\alpha_{1,12},\alpha_{1,5},\alpha_{2,11}\}$, $\{\alpha_{2,2},\alpha_{1,9},\alpha_{1,7},\alpha_{2,12},\alpha_{2,5},\alpha_{1,11}\}$. Both $G$ and $G_1$ are marked by $\ast$ in Tables 1---4. Case 48: $({\bf n}=17,\ (\mathbb A_1,\mathbb A_1,\mathbb A_1) \subset 3\mathbb A_1)\Longleftarrow ({\bf n}=61,\ 3\mathbb A_1)$. Similar to Case 1. By \cite{Nik10}, the $G\cong \AAA_{4,3}$ is marked by $N_{23}$ and $$ G=H_{61,1}= $$ $$ [(\alpha_{1}\alpha_{10}\alpha_{23})(\alpha_{2}\alpha_{4}\alpha_{11}) (\alpha_{3}\alpha_{21}\alpha_{20})(\alpha_{8}\alpha_{9}\alpha_{17}) (\alpha_{13}\alpha_{22}\alpha_{14})(\alpha_{16}\alpha_{19}\alpha_{24}), $$ $$ (\alpha_{2}\alpha_{14})(\alpha_{3}\alpha_{17})(\alpha_{4}\alpha_{8})(\alpha_{7}\alpha_{15}) (\alpha_{9}\alpha_{13})(\alpha_{10}\alpha_{12})(\alpha_{11}\alpha_{21})(\alpha_{16}\alpha_{19}), $$ $$ (\alpha_{2}\alpha_{13})(\alpha_{3}\alpha_{22})(\alpha_{4}\alpha_{9})(\alpha_{7}\alpha_{18}) (\alpha_{8}\alpha_{14})(\alpha_{11}\alpha_{20})(\alpha_{12}\alpha_{23})(\alpha_{16}\alpha_{19})] $$ with the orbit $\{\alpha_{7},\alpha_{18},\alpha_{15}\}$. The $G_1\cong \AAA_4$ is marked by $$ G=H_{61,1}\supset G_1= $$ $$ [(\alpha_{2}\alpha_{14}\alpha_{20})(\alpha_{3}\alpha_{22}\alpha_{17})(\alpha_{4}\alpha_{21}\alpha_{9}) (\alpha_{8}\alpha_{13}\alpha_{11})(\alpha_{10}\alpha_{23}\alpha_{12})(\alpha_{16}\alpha_{24}\alpha_{19}), $$ $$ (\alpha_{1}\alpha_{10})(\alpha_{2}\alpha_{13})(\alpha_{3}\alpha_{14})(\alpha_{4}\alpha_{20}) (\alpha_{8}\alpha_{22})(\alpha_{9}\alpha_{11})(\alpha_{12}\alpha_{23})(\alpha_{17}\alpha_{21})] $$ with suborbits $\{\alpha_{7}\}$, $\{\alpha_{18}\}$, $\{\alpha_{15}\}$. Both $G$ and $G_1$ are marked by $\ast$ in Tables 1---4. Case 49: $({\bf n}=17,\ (\mathbb A_1,\mathbb A_1,4\mathbb A_1) \subset 6\mathbb A_1)\Longleftarrow ({\bf n}=34,\ (2\mathbb A_1,4\mathbb A_1)\subset 6\mathbb A_1)$. Similar to Cases 1, 29 and 42. By \cite{Nik11}, the $G\cong \SSS_4$ is marked by $N_{23}$ and $G=H_{34,2}$ of Case 29 with orbits $\{\alpha_{4},\alpha_{9}\}$, $\{\alpha_{1},\alpha_{16},\alpha_{19},\alpha_{15}\}$. The $G_1\cong \AAA_{4}$ is marked by $G_1\subset G=H_{34,2}$ of Case 42 with suborbits $\{\alpha_{4}\}$, $\{\alpha_{9}\}$, $\{\alpha_{1},\alpha_{16},\alpha_{19},\alpha_{15}\}$. Both $G$ and $G_1$ are marked by $\ast$ in Tables 1---4. Case 50: $({\bf n}=17,\ (\mathbb A_1,\mathbb A_1,6\mathbb A_1) \subset 8\mathbb A_1)\Longleftarrow ({\bf n}=34,\ (2\mathbb A_1,(6\mathbb A_1)_{II})\subset 8\mathbb A_1)$. Similar to Case 1. By \cite{Nik11}, the $G\cong \SSS_4$ is marked by $N_{23}$ and $$ G=H_{34,3}= $$ $$ [(\alpha_{1}\alpha_{6}\alpha_{18}\alpha_{7}) (\alpha_{3}\alpha_{24}\alpha_{20}\alpha_{17}) (\alpha_{4}\alpha_{9}) (\alpha_{8}\alpha_{13}\alpha_{14}\alpha_{19}) (\alpha_{10}\alpha_{15}\alpha_{12}\alpha_{23}) (\alpha_{11}\alpha_{22}), $$ $$ (\alpha_{2}\alpha_{13})(\alpha_{3}\alpha_{22}) (\alpha_{4}\alpha_{9})(\alpha_{7}\alpha_{18}) (\alpha_{8}\alpha_{14})(\alpha_{11}\alpha_{20}) (\alpha_{12}\alpha_{23})(\alpha_{16}\alpha_{19})] $$ with orbits $\{\alpha_{4},\alpha_{9}\}$, $\{\alpha_{2},\alpha_{8},\alpha_{14},\alpha_{16},\alpha_{13},\alpha_{19}\}$. The $G_1\cong \AAA_{4}$ is marked by $$ G=H_{34,3}\supset G_1= $$ $$ [(\alpha_{2}\alpha_{14}\alpha_{19})(\alpha_{3}\alpha_{22}\alpha_{24})(\alpha_{6}\alpha_{7}\alpha_{18}) (\alpha_{8}\alpha_{13}\alpha_{16})(\alpha_{11}\alpha_{17}\alpha_{20})(\alpha_{12}\alpha_{15}\alpha_{23}), $$ $$ (\alpha_{1}\alpha_{6})(\alpha_{2}\alpha_{16})(\alpha_{3}\alpha_{20})(\alpha_{7}\alpha_{18}) (\alpha_{10}\alpha_{15})(\alpha_{11}\alpha_{22})(\alpha_{12}\alpha_{23})(\alpha_{13}\alpha_{19})] $$ with suborbits $\{\alpha_{4}\}$, $\{\alpha_{9}\}$, $\{\alpha_{2},\alpha_{8},\alpha_{14},\alpha_{16},\alpha_{13},\alpha_{19}\}$. Both $G$ and $G_1$ are marked by $\ast$ in Tables 1---4. Case 51: $({\bf n}=17,\ (\mathbb A_1,\mathbb A_1,12\mathbb A_1)\subset 14\mathbb A_1) \Longleftarrow ({\bf n}=34,\ (2\mathbb A_1,12\mathbb A_1)\subset 14\mathbb A_1)$. Similar to Cases 1, 29 and 42. By \cite{Nik11}, the $G\cong \SSS_4$ is marked by $N_{23}$ and $G=H_{34,2}$ of Case 29 with orbits $\{\alpha_{4},\alpha_{9}\}$, $\{\alpha_{3},\alpha_{22},\alpha_{12},\alpha_{18},\alpha_{17}, \alpha_{20},\alpha_{7},\alpha_{11},\alpha_{23},\alpha_{8},\alpha_{5},\alpha_{14}\}$. The $G_1\cong \AAA_{4}$ is marked by $G_1\subset G=H_{34,2}$ of Case 42 with suborbits $\{\alpha_{4}\}$, $\{\alpha_{9}\}$, $\{\alpha_{3},\alpha_{22},\alpha_{12},\alpha_{18},\alpha_{17}, \linebreak \alpha_{20},\alpha_{7}, \alpha_{11},\alpha_{23},\alpha_{8},\alpha_{5},\alpha_{14}\}$. Both $G$ and $G_1$ are marked by $\ast$ in Tables 1---4. Case 52: $({\bf n}=17,\ (\mathbb A_1,3\mathbb A_1,4\mathbb A_1) \subset 8\mathbb A_1)\Longleftarrow ({\bf n}=65,\ 8\mathbb A_1)$. Similar to Case 1. By \cite{Nik10}, the $G\cong 2^4D_6$ is marked by $N_{23}$ and $$ G=H_{65,4}= $$ $$[(\alpha_{1}\alpha_{21}\alpha_{23})(\alpha_{3}\alpha_{22}\alpha_{12}) (\alpha_{4}\alpha_{20}\alpha_{13}) (\alpha_{5}\alpha_{18}\alpha_{16}) (\alpha_{8}\alpha_{11}\alpha_{10})(\alpha_{9}\alpha_{14}\alpha_{17}), $$ $$ (\alpha_{1}\alpha_{11}\alpha_{23}\alpha_{8}) (\alpha_{4}\alpha_{6}\alpha_{13}\alpha_{20}) (\alpha_{5}\alpha_{16}) (\alpha_{7}\alpha_{17}\alpha_{9}\alpha_{14}) (\alpha_{10}\alpha_{21}\alpha_{19}\alpha_{15}) (\alpha_{12}\alpha_{24})] $$ with the orbit $\{\alpha_{1},\alpha_{8},\alpha_{21},\alpha_{23},\alpha_{15}, \alpha_{11},\alpha_{10},\alpha_{19}\}$. The $G_1\cong \AAA_{4}$ is marked by $$ G=H_{65,4}\supset G_1= $$ $$ [(\alpha_{4}\alpha_{13}\alpha_{20})(\alpha_{5}\alpha_{16}\alpha_{18})(\alpha_{7}\alpha_{17}\alpha_{9}) (\alpha_{8}\alpha_{11}\alpha_{19})(\alpha_{12}\alpha_{24}\alpha_{22})(\alpha_{15}\alpha_{21}\alpha_{23}), $$ $$ (\alpha_{3}\alpha_{12})(\alpha_{4}\alpha_{13})(\alpha_{6}\alpha_{20})(\alpha_{7}\alpha_{14}) (\alpha_{8}\alpha_{10})(\alpha_{9}\alpha_{17})(\alpha_{11}\alpha_{19})(\alpha_{22}\alpha_{24})] $$ with suborbits $\{\alpha_{1}\}$, $\{\alpha_{15},\alpha_{21},\alpha_{23}\}$, $\{\alpha_{8},\alpha_{11},\alpha_{10},\alpha_{19}\}$. Both $G$ and $G_1$ are marked by $\ast$ in Tables 1---4. Case 53: $({\bf n}=17,\ (\mathbb A_1,3\mathbb A_1,12\mathbb A_1) \subset 16\mathbb A_1)\Longleftarrow ({\bf n}=75,\ 16\mathbb A_1)$. Similar to Cases 1 and 10. By \cite{Nik10}, the $G\cong 4^2\AAA_4$ is marked by $N_{23}$ and $G=H_{75,1}$ from Case 10 with the orbit $\{\alpha_{1},\alpha_{9},\alpha_{6},\alpha_{7},\alpha_{4}, \alpha_{21},\alpha_{20}, \alpha_{22},\alpha_{5},\alpha_{14},\alpha_{10},\alpha_{23}, \alpha_{3},\alpha_{24},\alpha_{18},\alpha_{16}\}$. The $G_1\cong \AAA_4$ is marked by $$ G=H_{75,1}\supset G_1= $$ $$ [(\alpha_{3}\alpha_{4}\alpha_{22})(\alpha_{5}\alpha_{20}\alpha_{23})(\alpha_{6}\alpha_{10}\alpha_{16}) (\alpha_{7}\alpha_{9}\alpha_{18})(\alpha_{11}\alpha_{17}\alpha_{12})(\alpha_{14}\alpha_{24}\alpha_{21}), $$ $$ (\alpha_{3}\alpha_{16})(\alpha_{4}\alpha_{5})(\alpha_{6}\alpha_{21})(\alpha_{10}\alpha_{20}) (\alpha_{11}\alpha_{12})(\alpha_{13}\alpha_{17})(\alpha_{14}\alpha_{22})(\alpha_{23}\alpha_{24})] $$ with suborbits $\{\alpha_{1}\}$, $\{\alpha_{7},\alpha_{9},\alpha_{18}\}$, $\{\alpha_{3},\alpha_{4},\alpha_{16},\alpha_{22},\alpha_{5},\alpha_{6}, \alpha_{14},\alpha_{20},\alpha_{10},\alpha_{21},\alpha_{24},\alpha_{23}\}$. Both $G$ and $G_1$ are marked by $\ast$ in Tables 1---4. Case 54: $({\bf n}=17,\ (\mathbb A_1,4\mathbb A_1,4\mathbb A_1) \subset 9\mathbb A_1)\Longleftarrow ({\bf n}=34,\ (\mathbb A_1,8\mathbb A_1)\subset 9\mathbb A_1)$. This is similar to Cases 1 and 44. By \cite{Nik11}, the $G\cong \SSS_4$ is marked by $N_{23}$ and $G=H_{34,1}$ of Case 44 with orbits $\{\alpha_5\}$, $\{\alpha_{2},\alpha_{13},\alpha_{20},\alpha_{9},\alpha_{12},\alpha_{4},\alpha_{23},\alpha_{11}\}$. The $G_1\cong \AAA_{4}$ is marked by $G_1\subset G=H_{34,1}$ of Case 44 with suborbits $\{\alpha_5\}$, $\{\alpha_{2},\alpha_{9},\alpha_{20},\alpha_{23}\}$, $\{\alpha_{4},\alpha_{12},\alpha_{13},\alpha_{11}\}$. Both $G$ and $G_1$ are marked by $\ast$ in Tables 1---4. Case 55: $({\bf n}=17,\ (\mathbb A_1,6\mathbb A_1,6\mathbb A_1) \subset \mathbb A_1\amalg 6\mathbb A_2) \Longleftarrow ({\bf n}=34,\ (\mathbb A_1,6\mathbb A_2)\subset \mathbb A_1\amalg 6\mathbb A_2)$. Similar to Cases 1 and 22. By \cite{Nik11}, the $G\cong \SSS_4$ is marked by $N_{22}$ and $G=H_{34,1}$ of Case 22 with orbits $\{\alpha_{1,6}\}$, $\{\alpha_{1,2},\alpha_{2,5}, \alpha_{1,5},\alpha_{1,11},\alpha_{2,2},\alpha_{2,11}, \alpha_{1,12},\alpha_{2,12},\alpha_{2,7},\alpha_{1,7},\alpha_{1,9},\alpha_{2,9}\}$. The $G_1\cong \AAA_4$ is marked by $$ G=H_{34,1}\supset G_1= $$ $$ [(\alpha_{1,2}\alpha_{2,9}\alpha_{1,12})(\alpha_{4 }\alpha_{1,9}\alpha_{2,12})(\alpha_{1,3}\alpha_{1,8}\alpha_{2,4}) (\alpha_{2,3}\alpha_{2,8}\alpha_{1,4})(\alpha_{1,5}\alpha_{2,11}\alpha_{2,7})(\alpha_{2,5}\alpha_{1,11}\alpha_{1,7}), $$ $$ (\alpha_{1,1}\alpha_{1,3})(\alpha_{2,1}\alpha_{2,3})(\alpha_{1,2}\alpha_{2,7})(\alpha_{2,2}\alpha_{1,7}) (\alpha_{1,4}\alpha_{2,8})(\alpha_{2,4}\alpha_{1,8})(\alpha_{1,5}\alpha_{2,6})(\alpha_{2,5}\alpha_{1,9})] $$ with suborbits $\{\alpha_{1,6}\}$, $\{\alpha_{1,2},\alpha_{2,9},\alpha_{2,7},\alpha_{1,12},\alpha_{1,5},\alpha_{2,11}\}$, $\{\alpha_{2,2},\alpha_{1,9},\alpha_{1,7},\alpha_{2,12},\alpha_{2,5},\alpha_{1,11}\}$. \newline Both $G$ and $G_1$ are marked by $\ast$ in Tables 1---4. Case 56: $({\bf n}=17,\ (3\mathbb A_1,4\mathbb A_1,4\mathbb A_1)\subset 11\mathbb A_1) \Longleftarrow ({\bf n}=34,\ (3\mathbb A_1,8\mathbb A_1)\subset 11\mathbb A_1)$. Similar to Case 1. By \cite{Nik11}, the $G\cong \SSS_4$ is marked by $N_{23}$ and $$ G=H_{34,4}= $$ $$ [(\alpha_{1}\alpha_{6}\alpha_{16}\alpha_{19}) (\alpha_{4}\alpha_{14}\alpha_{23}\alpha_{9}) (\alpha_{5}\alpha_{11}\alpha_{20}\alpha_{21}) (\alpha_{7}\alpha_{8}\alpha_{12}\alpha_{18}) (\alpha_{13}\alpha_{15})(\alpha_{17}\alpha_{22}), $$ $$ (\alpha_{2}\alpha_{13})(\alpha_{3}\alpha_{22}) (\alpha_{4}\alpha_{9})(\alpha_{7}\alpha_{18}) (\alpha_{8}\alpha_{14})(\alpha_{11}\alpha_{20}) (\alpha_{12}\alpha_{23})(\alpha_{16}\alpha_{19})] $$ with orbits $\{\alpha_{2},\alpha_{15},\alpha_{13}\}$, $\{\alpha_{4},\alpha_{7},\alpha_{18},\alpha_{23}, \alpha_{9},\alpha_{12},\alpha_{8},\alpha_{14}\}$. The $G_1\cong \AAA_{4}$ is marked by $$ G=H_{34,4}\supset G_1= $$ $$ [(\alpha_{2}\alpha_{13}\alpha_{15})(\alpha_{3}\alpha_{22}\alpha_{17})(\alpha_{4}\alpha_{23}\alpha_{18}) (\alpha_{5}\alpha_{11}\alpha_{20})(\alpha_{6}\alpha_{16}\alpha_{19})(\alpha_{7}\alpha_{12}\alpha_{9}), $$ $$ (\alpha_{1}\alpha_{6})(\alpha_{4}\alpha_{18})(\alpha_{5}\alpha_{21})(\alpha_{7}\alpha_{9}) (\alpha_{8}\alpha_{23})(\alpha_{11}\alpha_{20})(\alpha_{12}\alpha_{14})(\alpha_{16}\alpha_{19})] $$ with suborbits $\{\alpha_{2},\alpha_{15},\alpha_{13}\}$, $\{\alpha_{4},\alpha_{23},\alpha_{18},\alpha_{8}\}$, $\{\alpha_{7},\alpha_{12},\alpha_{9},\alpha_{14}\}$. Both $G$ and $G_1$ are marked by $\ast$ in Tables 1---4. Case 57: $({\bf n}=17,\ (4\mathbb A_1,4\mathbb A_1,4\mathbb A_1)\subset 12\mathbb A_1) \Longleftarrow ({\bf n}=61,\ 12\mathbb A_1)$. Similar to Cases 1 and 48. By \cite{Nik10}, the $G$ is marked by by $N_{23}$ and $G=H_{61,1}$ of Case 48 with the orbit $\{\alpha_{2},\alpha_{4},\alpha_{21},\alpha_{13},\alpha_{11},\alpha_{8}, \alpha_{20},\alpha_{17},\alpha_{22},\alpha_{14},\alpha_{9},\alpha_{3}\}$. The $G_1\cong \AAA_{4}$ is marked by $$ G=H_{61,1}\supset G_1= $$ $$ [(\alpha_{2}\alpha_{13}\alpha_{21})(\alpha_{4}\alpha_{20}\alpha_{8})(\alpha_{7}\alpha_{15}\alpha_{18}) (\alpha_{9}\alpha_{14}\alpha_{11})(\alpha_{10}\alpha_{12}\alpha_{23})(\alpha_{16}\alpha_{24}\alpha_{19}), $$ $$ (\alpha_{1}\alpha_{10})(\alpha_{2}\alpha_{13})(\alpha_{3}\alpha_{14})(\alpha_{4}\alpha_{20}) (\alpha_{8}\alpha_{22})(\alpha_{9}\alpha_{11})(\alpha_{12}\alpha_{23})(\alpha_{17}\alpha_{21})] $$ with suborbits $\{\alpha_{2},\alpha_{13},\alpha_{21},\alpha_{17}\}$, $\{\alpha_{3},\alpha_{14},\alpha_{11},\alpha_{9}\}$, $\{\alpha_{4},\alpha_{20},\alpha_{8},\alpha_{22}\}$. Both $G$ and $G_1$ are marked by $\ast$ in Tables 1---4. Case 58: ({\bf n}=17,\ $ \left(\begin{array}{rrr} 4\mathbb A_1 & 4\mathbb A_2 & 8\mathbb A_1 \\ & 4\mathbb A_1 & 4\mathbb A_2 \\ & & 4\mathbb A_1 \end{array}\right) \subset 4\mathbb A_3)\ \Longleftarrow ({\bf n}=34,\ (4\mathbb A_1,8\mathbb A_1)\subset 4\mathbb A_3)$. Similar to Case 1. By \cite{Nik11}, the $G\cong \SSS_4$ is marked by $N_{21}$ and $$ G=H_{34,3}= $$ $$ [(\alpha_{1,1}\alpha_{1,3}\alpha_{1,7}) (\alpha_{2,1}\alpha_{2,3}\alpha_{2,7}) (\alpha_{3,1}\alpha_{3,3}\alpha_{3,7}) (\alpha_{1,4}\alpha_{1,8}\alpha_{1,6}) $$ $$ (\alpha_{2,4}\alpha_{2,8}\alpha_{2,6}) (\alpha_{3,4}\alpha_{3,8}\alpha_{3,6}), $$ $$ (\alpha_{1,1}\alpha_{3,1}) (\alpha_{1,3}\alpha_{3,7}\alpha_{3,3}\alpha_{1,7}) (\alpha_{2,3}\alpha_{2,7}) (\alpha_{1,4}\alpha_{3,6}\alpha_{3,5}\alpha_{3,8}) $$ $$ (\alpha_{2,4}\alpha_{2,6}\alpha_{2,5}\alpha_{2,8}) (\alpha_{3,4}\alpha_{1,6}\alpha_{1,5}\alpha_{1,8}) ] $$ with orbits $\{\alpha_{2,4},\alpha_{2,8},\alpha_{2,6},\alpha_{2,5}\}$, $\{\alpha_{1,4}, \alpha_{1,8},\alpha_{3,6},\alpha_{1,6}, \alpha_{3,4},\alpha_{3,5},\alpha_{1,5},\alpha_{3,8}\}$. The $G_1\cong \AAA_{4}$ is marked by $$ G=H_{34,3}\supset G_1= $$ $$ [(\alpha_{1,1}\alpha_{1,3}\alpha_{1,7})(\alpha_{2,1}\alpha_{2,3}\alpha_{2,7})(\alpha_{3,1}\alpha_{3,3}\alpha_{3,7}) (\alpha_{1,4}\alpha_{1,8}\alpha_{1,6})(\alpha_{2,4}\alpha_{2,8}\alpha_{2,6})(\alpha_{3,4}\alpha_{3,8}\alpha_{3,6}), $$ $$ (\alpha_{1,3}\alpha_{3,3})(\alpha_{1,4}\alpha_{3,5})(\alpha_{2,4}\alpha_{2,5})(\alpha_{3,4}\alpha_{1,5}) (\alpha_{1,6}\alpha_{1,8})(\alpha_{2,6}\alpha_{2,8})(\alpha_{3,6}\alpha_{3,8})(\alpha_{1,7}\alpha_{3,7})] $$ with suborbits $\{\alpha_{1,4},\alpha_{1,8},\alpha_{3,5},\alpha_{1,6}\}$, $\{\alpha_{2,4},\alpha_{2,8},\alpha_{2,5},\alpha_{2,6}\}$, $\{\alpha_{3,4},\alpha_{3,8},\alpha_{1,5},\alpha_{3,6}\}$. Both $G$ and $G_1$ are marked by $\ast$ in Tables 1---4. Case 59: $({\bf n}=17,\ (4\mathbb A_1,4\mathbb A_1,6\mathbb A_1)\subset 14\mathbb A_1)\ \Longleftarrow ({\bf n}=34,\ ((6\mathbb A_1)_I,8\mathbb A_1)\subset 14\mathbb A_1)$. Similar to Cases 1 and 44. By \cite{Nik10}, the $G\cong \SSS_4$ is marked by $N_{23}$ and $G=H_{34,1}$ of Case 44 with orbits $\{\alpha_{1},\alpha_{3},\alpha_{22},\alpha_{15},\alpha_{7},\alpha_{18}\}$, $\{\alpha_{2},\alpha_{13},\alpha_{20},\alpha_{9},\alpha_{12},\alpha_{4},\alpha_{23},\alpha_{11}\}$. The $G_1\cong \AAA_{4}$ is marked by $G_1\subset G=H_{34,1}$ of Case 44 with suborbits $\{\alpha_{2},\alpha_{9},\alpha_{20},\alpha_{23}\}$, $\{\alpha_{4},\alpha_{12},\alpha_{13},\alpha_{11}\}$, $\{\alpha_{1},\alpha_{3},\alpha_{22},\alpha_{15},\alpha_{7},\alpha_{18}\}$. Both $G$ and $G_1$ are marked by $\ast$ in Tables 1---4. Case 60: $({\bf n}=17,\ (4\mathbb A_1,6\mathbb A_1,6\mathbb A_1) \subset 16\mathbb A_1)\Longleftarrow ({\bf n}=75,\ 16\mathbb A_1)$. Similar to Cases 1 and 10. By \cite{Nik10}, the $G\cong 4^2\AAA_4$ is marked by $N_{23}$ and $G=H_{75,1}$ from Case 10 with the orbit $\{\alpha_{1},\alpha_{9},\alpha_{6},\alpha_{7},\alpha_{4}, \alpha_{21},\alpha_{20}, \alpha_{22},\alpha_{5},\alpha_{14},\alpha_{10},\alpha_{23}, \alpha_{3},\alpha_{24},\alpha_{18},\alpha_{16}\}$. The $G_1\cong \AAA_4$ is marked by $$ G=H_{75,1}\supset G_1= $$ $$ [(\alpha_{1}\alpha_{3}\alpha_{10})(\alpha_{4}\alpha_{6}\alpha_{21})(\alpha_{7}\alpha_{24}\alpha_{14}) (\alpha_{9}\alpha_{16}\alpha_{20})(\alpha_{11}\alpha_{13}\alpha_{17})(\alpha_{18}\alpha_{23}\alpha_{22}), $$ $$ (\alpha_{3}\alpha_{16})(\alpha_{4}\alpha_{5})(\alpha_{6}\alpha_{21})(\alpha_{10}\alpha_{20}) (\alpha_{11}\alpha_{12})(\alpha_{13}\alpha_{17})(\alpha_{14}\alpha_{22})(\alpha_{23}\alpha_{24})] $$ with suborbits $\{\alpha_{4},\alpha_{6},\alpha_{5},\alpha_{21}\}$, $\{\alpha_{1},\alpha_{3},\alpha_{10},\alpha_{16},\alpha_{20},\alpha_{9}\}$, $\{\alpha_{7},\alpha_{24},\alpha_{14},\alpha_{23},\alpha_{22},\alpha_{18}\}$. Both $G$ and $G_1$ are marked by $\ast$ in Tables 1---4. Case 61: $({\bf n}=17,\ (4\mathbb A_1,6\mathbb A_1,6\mathbb A_1)\subset 4\mathbb A_1\amalg 6\mathbb A_2) \Longleftarrow ({\bf n}=34,\ (4\mathbb A_1,6\mathbb A_2)\subset 4\mathbb A_1\amalg 6\mathbb A_2)$. Similar to Cases 1, 22 and 55. By \cite{Nik11}, the $G\cong \SSS_4$ is marked by $N_{22}$ and $G=H_{34,1}$ of Case 22 with orbits $\{\alpha_{1,1}, \alpha_{1,3}, \alpha_{2,4}, \alpha_{1,8}\}$, $\{\alpha_{1,2},\alpha_{2,5}, \alpha_{1,5},\alpha_{1,11},\alpha_{2,2},\alpha_{2,11}, \alpha_{1,12},\alpha_{2,12},\alpha_{2,7},\linebreak \alpha_{1,7}, \alpha_{1,9},\alpha_{2,9}\}$. The $G_1\cong \AAA_4$ is marked by $G_1\subset G=H_{34,1}$ of Case 55 with suborbits $\{\alpha_{1,1}, \alpha_{1,3}, \alpha_{2,4}, \alpha_{1,8}\}$, $\{\alpha_{1,2},\alpha_{2,9},\alpha_{2,7},\alpha_{1,12},\alpha_{1,5},\alpha_{2,11}\}$, $\{\alpha_{2,2},\alpha_{1,9},\alpha_{1,7},\alpha_{2,12},\alpha_{2,5},\alpha_{1,11}\}$. Both $G$ and $G_1$ are marked by $\ast$ in Tables 1---4. Case 62: $({\bf n}=21,\ (4\mathbb A_1, 4\mathbb A_1,4\mathbb A_1) \subset 12\mathbb A_1)\Longleftarrow ({\bf n}=49,\ 12\mathbb A_1)$. Similar to Cases 1 and 5. By \cite{Nik10}, the $G\cong 2^4C_3$ is marked by $N_{23}$ and $G=H_{49,1}$ of Case 5 with the orbit $\{\alpha_{1},\alpha_{22},\alpha_{21},\alpha_{10},\alpha_{19},\alpha_{14}, \alpha_{8},\alpha_{17},\alpha_{18},\alpha_{7},\alpha_{3},\alpha_{16}\}$. The $G_1\cong (C_2)^4$ is marked by $$ G=H_{49,1}\supset G_1= $$ $$ [(\alpha_{1}\alpha_{21})(\alpha_{2}\alpha_{23})(\alpha_{3}\alpha_{8})(\alpha_{4}\alpha_{9}) (\alpha_{10}\alpha_{17})(\alpha_{11}\alpha_{20})(\alpha_{12}\alpha_{13})(\alpha_{14}\alpha_{22}), $$ $$ (\alpha_{2}\alpha_{12})(\alpha_{3}\alpha_{8})(\alpha_{4}\alpha_{20})(\alpha_{7}\alpha_{16}) (\alpha_{9}\alpha_{11})(\alpha_{13}\alpha_{23})(\alpha_{14}\alpha_{22})(\alpha_{18}\alpha_{19}), $$ $$ (\alpha_{1}\alpha_{17})(\alpha_{2}\alpha_{12})(\alpha_{3}\alpha_{22})(\alpha_{4}\alpha_{11}) (\alpha_{8}\alpha_{14})(\alpha_{9}\alpha_{20})(\alpha_{10}\alpha_{21})(\alpha_{13}\alpha_{23}), $$ $$ (\alpha_{2}\alpha_{13})(\alpha_{3}\alpha_{22})(\alpha_{4}\alpha_{9})(\alpha_{7}\alpha_{18}) (\alpha_{8}\alpha_{14})(\alpha_{11}\alpha_{20})(\alpha_{12}\alpha_{23})(\alpha_{16}\alpha_{19})] $$ with suborbits $\{\alpha_{1},\alpha_{21},\alpha_{17},\alpha_{10} \}$, $\{\alpha_{3},\alpha_{8},\alpha_{22},\alpha_{14} \}$, $\{\alpha_{7},\alpha_{16},\alpha_{18},\alpha_{19}\}$. Both $G$ and $G_1$ are marked by $\ast$ in Tables 1---4. Case 63: $({\bf n}=21,\ (4\mathbb A_1, 4\mathbb A_1,4\mathbb A_1,4\mathbb A_1) \subset 16\mathbb A_1)\Longleftarrow ({\bf n}=75,\ 16\mathbb A_1)$. Similar to Cases 1 and 10. By \cite{Nik10}, the $G\cong 4^2\AAA_4$ is marked by $N_{23}$ and $G=H_{75,1}$ of Case 10 with the orbit $\{\alpha_{1},\alpha_{9},\alpha_{6},\alpha_{7},\alpha_{4}, \alpha_{21},\alpha_{20}, \alpha_{22},\alpha_{5},\alpha_{14},\alpha_{10},\alpha_{23}, \alpha_{3},\alpha_{24},\alpha_{18},\alpha_{16}\}$. The $G_1\cong (C_2)^4$ is marked by $$ G=H_{75,1}\supset G_1= $$ $$ [(\alpha_{1}\alpha_{7})(\alpha_{3}\alpha_{16})(\alpha_{9}\alpha_{18})(\alpha_{10}\alpha_{22}) (\alpha_{11}\alpha_{17})(\alpha_{12}\alpha_{13})(\alpha_{14}\alpha_{20})(\alpha_{23}\alpha_{24}), $$ $$ (\alpha_{1}\alpha_{9})(\alpha_{3}\alpha_{23})(\alpha_{7}\alpha_{18})(\alpha_{10}\alpha_{14}) (\alpha_{11}\alpha_{12})(\alpha_{13}\alpha_{17})(\alpha_{16}\alpha_{24})(\alpha_{20}\alpha_{22}), $$ $$ (\alpha_{3}\alpha_{24})(\alpha_{4}\alpha_{6})(\alpha_{5}\alpha_{21})(\alpha_{10}\alpha_{14}) (\alpha_{11}\alpha_{17})(\alpha_{12}\alpha_{13})(\alpha_{16}\alpha_{23})(\alpha_{20}\alpha_{22}), $$ $$ (\alpha_{3}\alpha_{16})(\alpha_{4}\alpha_{5})(\alpha_{6}\alpha_{21})(\alpha_{10}\alpha_{20}) (\alpha_{11}\alpha_{12})(\alpha_{13}\alpha_{17})(\alpha_{14}\alpha_{22})(\alpha_{23}\alpha_{24})] $$ with suborbits $\{\alpha_{1},\alpha_{7},\alpha_{9},\alpha_{18}\}$, $\{\alpha_{3},\alpha_{16},\alpha_{23},\alpha_{24}\}$, $\{\alpha_{4},\alpha_{6},\alpha_{5},\alpha_{21}\}$, $\{\alpha_{10},\alpha_{22},\alpha_{14},\alpha_{20}\}$. \newline Both $G$ and $G_1$ are marked by $\ast$ in Tables 1---4. Case 64: $({\bf n}=22,\ (2\mathbb A_1,2\mathbb A_1) \subset 4\mathbb A_1)\Longleftarrow ({\bf n}=39,\ 4\mathbb A_1)$. Similar to Cases 1 and 14. By \cite{Nik10}, the $G\cong 2^4C_2$ is marked by $N_{23}$ and $G=H_{39,2}$ of Case 14 with the orbit $\{\alpha_{2},\alpha_{12},\alpha_{13},\alpha_{23}\}$. The $G_1\cong C_2\times D_8$ is marked by $$ G=H_{39,2}\supset G_1= $$ $$ [(\alpha_{3}\alpha_{14}\alpha_{8}\alpha_{22})(\alpha_{4}\alpha_{20}\alpha_{9}\alpha_{11})(\alpha_{5}\alpha_{7}\alpha_{15}\alpha_{19}) (\alpha_{6}\alpha_{18}\alpha_{24}\alpha_{16})(\alpha_{10}\alpha_{17})(\alpha_{12}\alpha_{13}), $$ $$ (\alpha_{5}\alpha_{18})(\alpha_{6}\alpha_{7})(\alpha_{10}\alpha_{17})(\alpha_{11}\alpha_{20}) (\alpha_{12}\alpha_{13})(\alpha_{14}\alpha_{22})(\alpha_{15}\alpha_{16})(\alpha_{19}\alpha_{24}), $$ $$ (\alpha_{2}\alpha_{23})(\alpha_{4}\alpha_{9})(\alpha_{5}\alpha_{24})(\alpha_{6}\alpha_{15}) (\alpha_{7}\alpha_{16})(\alpha_{11}\alpha_{20})(\alpha_{12}\alpha_{13})(\alpha_{18}\alpha_{19})] $$ with suborbits $\{\alpha_{2},\alpha_{23}\}$, $\{\alpha_{12},\alpha_{13}\}$. Both $G$ and $G_1$ are marked by $\ast$ in Tables 1---4. Case 65: $({\bf n}=22,\ ((4\mathbb A_1,4\mathbb A_1) \subset 8\mathbb A_1)_I)\Longleftarrow ({\bf n}=40,\ 8\mathbb A_1)$. Similar to cases 1 and 4. By \cite{Nik10}, the $G\cong Q_8\ast Q_8$ is marked by $N_{23}$ and $G=H_{40,1}$ of Case 4 with the orbit $\{\alpha_{1},\alpha_{16},\alpha_{14},\alpha_{19},\alpha_{18},\alpha_{20}, \alpha_{24},\alpha_{2}\}$. The $G_1\cong C_2\times D_8$ is marked by $$ G=H_{40,1}\supset G_1= $$ $$ [(\alpha_{3}\alpha_{5})(\alpha_{6}\alpha_{12})(\alpha_{8}\alpha_{13})(\alpha_{10}\alpha_{22}) (\alpha_{15}\alpha_{21})(\alpha_{16}\alpha_{19})(\alpha_{17}\alpha_{23})(\alpha_{18}\alpha_{20}), $$ $$ (\alpha_{1}\alpha_{16})(\alpha_{2}\alpha_{18})(\alpha_{4}\alpha_{9})(\alpha_{5}\alpha_{23}) (\alpha_{10}\alpha_{12})(\alpha_{14}\alpha_{20})(\alpha_{15}\alpha_{21})(\alpha_{19}\alpha_{24}), $$ $$ (\alpha_{2}\alpha_{14})(\alpha_{3}\alpha_{6})(\alpha_{4}\alpha_{9})(\alpha_{5}\alpha_{12}) (\alpha_{10}\alpha_{23})(\alpha_{15}\alpha_{21})(\alpha_{17}\alpha_{22})(\alpha_{18}\alpha_{20})] $$ with suborbits $\{\alpha_{1},\alpha_{16},\alpha_{19},\alpha_{24}\}$, $\{\alpha_{2},\alpha_{18},\alpha_{14},\alpha_{20}\}$. Both $G$ and $G_1$ are marked by $\ast$ in Tables 1---4. Case 66: $({\bf n}=22,\ (8\mathbb A_1,8\mathbb A_1) \subset 16\mathbb A_1)\ \Longleftarrow ({\bf n}=39,\ 16\mathbb A_1)$. Similar to Cases 1 and 7. By \cite{Nik10}, the $G\cong 2^4C_2$ is marked by $N_{23}$ and $G=H_{39,1}$ of Case 7 with the orbit $\{\alpha_{2},\alpha_{12},\alpha_{13},\alpha_{3},\alpha_{18},\alpha_{8}, \alpha_{23},\alpha_{19},\alpha_{7},\alpha_{22},\alpha_{4}, \alpha_{14},\alpha_{20},\alpha_{11},\alpha_{16},\alpha_{9}\}$. The $G_1\cong C_2\times D_8$ is marked by $$ G=H_{39,1}\supset G_1= $$ $$ [(\alpha_{5}\alpha_{10})(\alpha_{7}\alpha_{13})(\alpha_{8}\alpha_{12})(\alpha_{9}\alpha_{22}) (\alpha_{11}\alpha_{23})(\alpha_{14}\alpha_{16})(\alpha_{17}\alpha_{21})(\alpha_{19}\alpha_{20}), $$ $$ (\alpha_{2}\alpha_{7})(\alpha_{3}\alpha_{9})(\alpha_{4}\alpha_{22})(\alpha_{8}\alpha_{11}) (\alpha_{12}\alpha_{16})(\alpha_{13}\alpha_{18})(\alpha_{14}\alpha_{20})(\alpha_{19}\alpha_{23}), $$ $$ (\alpha_{2}\alpha_{3})(\alpha_{4}\alpha_{18})(\alpha_{7}\alpha_{9})(\alpha_{8}\alpha_{12}) (\alpha_{11}\alpha_{16})(\alpha_{13}\alpha_{22})(\alpha_{14}\alpha_{23})(\alpha_{19}\alpha_{20})] $$ with suborbits $\{\alpha_{2},\alpha_{7},\alpha_{3},\alpha_{13},\alpha_{9},\alpha_{18},\alpha_{22},\alpha_{4}\}$, $\{\alpha_{8},\alpha_{12},\alpha_{11},\alpha_{16},\alpha_{23},\alpha_{14},\alpha_{19},\alpha_{20}\}$. Both $G$ and $G_1$ are marked by $\ast$ in Tables 1---4. Case 67: $({\bf n}=22,\ (2\mathbb A_1, 2\mathbb A_1,4\mathbb A_1) \subset 8\mathbb A_1)\Longleftarrow ({\bf n}=56,\ 8\mathbb A_1)$. Similar to Cases 1 and 23. By \cite{Nik10}, the $G\cong \Gamma_{15}a_1$ is marked by $N_{23}$ and $G=H_{56,1}$ of Case 23 with the orbit $\{\alpha_{1}, \alpha_{14}, \alpha_{24}, \alpha_{17}, \alpha_{2},\alpha_{23}, \alpha_{5}, \alpha_{3} \}$. The $G_1\cong C_2\times D_8$ is marked by $$ G=H_{56,1}\supset G_1= $$ $$ [(\alpha_{1}\alpha_{17})(\alpha_{2}\alpha_{3})(\alpha_{6}\alpha_{9})(\alpha_{7}\alpha_{10}) (\alpha_{8}\alpha_{13})(\alpha_{15}\alpha_{20})(\alpha_{16}\alpha_{18})(\alpha_{19}\alpha_{21}), $$ $$ (\alpha_{2}\alpha_{17})(\alpha_{6}\alpha_{16})(\alpha_{7}\alpha_{8})(\alpha_{9}\alpha_{13}) (\alpha_{10}\alpha_{18})(\alpha_{12}\alpha_{22})(\alpha_{19}\alpha_{20})(\alpha_{23}\alpha_{24}), $$ $$ (\alpha_{5}\alpha_{14})(\alpha_{6}\alpha_{9})(\alpha_{7}\alpha_{10})(\alpha_{8}\alpha_{18}) (\alpha_{13}\alpha_{16})(\alpha_{15}\alpha_{21})(\alpha_{19}\alpha_{20})(\alpha_{23}\alpha_{24})] $$ with suborbits $\{\alpha_{5},\alpha_{14}\}$, $\{\alpha_{23},\alpha_{24}\}$, $\{\alpha_{1},\alpha_{17},\alpha_{2},\alpha_{3}\}$. Both $G$ and $G_1$ are marked by $\ast$ in Tables 1---4. Case 68: $({\bf n}=22,\ (2\mathbb A_1, 2\mathbb A_1,8\mathbb A_1) \subset 12\mathbb A_1)\Longleftarrow ({\bf n}=65,\ 12\mathbb A_1)$. Similar to Cases 1 and 24. By \cite{Nik10}, the $G\cong 2^4D_6$ is marked by $N_{23}$ and $G=H_{65,3}$ of Case 24 with the orbit $\{\alpha_{1},\alpha_{7},\alpha_{23},\alpha_{8},\alpha_{11},\alpha_{17},\alpha_{20}, \alpha_{9}, \alpha_{6},\alpha_{4},\alpha_{14},\alpha_{13}\}$. The $G_1\cong C_2\times D_8$ is marked by $$ G=H_{65,3}\supset G_1= $$ $$ [(\alpha_{1}\alpha_{17})(\alpha_{4}\alpha_{23})(\alpha_{5}\alpha_{22})(\alpha_{6}\alpha_{9}) (\alpha_{8}\alpha_{13})(\alpha_{11}\alpha_{14})(\alpha_{12}\alpha_{24})(\alpha_{19}\alpha_{21}), $$ $$ (\alpha_{1}\alpha_{8})(\alpha_{3}\alpha_{22})(\alpha_{4}\alpha_{13})(\alpha_{5}\alpha_{16}) (\alpha_{10}\alpha_{21})(\alpha_{11}\alpha_{23})(\alpha_{14}\alpha_{17})(\alpha_{15}\alpha_{19}), $$ $$ (\alpha_{1}\alpha_{8})(\alpha_{4}\alpha_{14})(\alpha_{6}\alpha_{9})(\alpha_{7}\alpha_{20}) (\alpha_{10}\alpha_{15})(\alpha_{11}\alpha_{23})(\alpha_{13}\alpha_{17})(\alpha_{19}\alpha_{21})] $$ with suborbits $\{\alpha_{6},\alpha_{9}\}$, $\{\alpha_{7},\alpha_{20}\}$, $\{\alpha_{1},\alpha_{17},\alpha_{8},\alpha_{14},\alpha_{13},\alpha_{11},\alpha_{4},\alpha_{23}\}$. Both $G$ and $G_1$ are marked by $\ast$ in Tables 1---4. Case 69: $({\bf n}=22,\ ((4\mathbb A_1,4\mathbb A_1)_I,8\mathbb A_1) \subset 16\mathbb A_1)\Longleftarrow ({\bf n}=56,\ 16\mathbb A_1)$. Similar to cases 1 and 8. By \cite{Nik10}, the $G\cong \Gamma_{25}a_1$ is marked by $N_{23}$ and $G=H_{56,2}$ of Case 8 with the orbit $\{\alpha_{2}, \alpha_{3}, \alpha_{23}, \alpha_{24}, \alpha_{5},\alpha_{8}, \alpha_{17}, \alpha_{19}, \alpha_{6}, \alpha_{7}, \alpha_{11}, \alpha_{20}, \alpha_{18}, \alpha_{10},\alpha_{16},\alpha_{15}\}$. The $G_1\cong C_2\times D_8$ is marked by $$ G=H_{56,2}\supset G_1= $$ $$ (\alpha_{3}\alpha_{11})(\alpha_{5}\alpha_{15})(\alpha_{7}\alpha_{20})(\alpha_{8}\alpha_{19}) (\alpha_{9}\alpha_{13})(\alpha_{10}\alpha_{24})(\alpha_{12}\alpha_{22})(\alpha_{18}\alpha_{23}), $$ $$ (\alpha_{2}\alpha_{10})(\alpha_{3}\alpha_{20})(\alpha_{5}\alpha_{19})(\alpha_{6}\alpha_{24}) (\alpha_{7}\alpha_{15})(\alpha_{8}\alpha_{11})(\alpha_{16}\alpha_{18})(\alpha_{17}\alpha_{23}), $$ $$ (\alpha_{1}\alpha_{14})(\alpha_{3}\alpha_{20})(\alpha_{5}\alpha_{19})(\alpha_{7}\alpha_{11}) (\alpha_{8}\alpha_{15})(\alpha_{12}\alpha_{22})(\alpha_{16}\alpha_{17})(\alpha_{18}\alpha_{23})] $$ with suborbits $\{\alpha_{2},\alpha_{10},\alpha_{24},\alpha_{6}\}$, $\{\alpha_{16},\alpha_{18},\alpha_{17},\alpha_{23}\}$, $\{\alpha_{3},\alpha_{11},\alpha_{20},\alpha_{8},\alpha_{7},\alpha_{19},\alpha_{15},\alpha_{5}\}$. Both $G$ and $G_1$ are marked by $\ast$ in Tables 1---4. Case 70: $({\bf n}=34,\ (\mathbb A_1,\mathbb A_1)\subset 2\mathbb A_1)\Longleftarrow ({\bf n}=51,\ 2\mathbb A_1)$. Similar to Case 1. By \cite{Nik10}, the $G\cong C_2\times \SSS_4$ is marked by $N_{23}$ and $$ G=H_{51,3}= $$ $$ [(\alpha_{1}\alpha_{15}\alpha_{18}\alpha_{20}) (\alpha_{3}\alpha_{6}\alpha_{23}\alpha_{17}) (\alpha_{4}\alpha_{19}) (\alpha_{7}\alpha_{21}\alpha_{24}\alpha_{11}) (\alpha_{8}\alpha_{9}\alpha_{14}\alpha_{16}) (\alpha_{12}\alpha_{22}), $$ $$ (\alpha_{2}\alpha_{13})(\alpha_{3}\alpha_{22}) (\alpha_{4}\alpha_{9})(\alpha_{7}\alpha_{18}) (\alpha_{8}\alpha_{14})(\alpha_{11}\alpha_{20}) (\alpha_{12}\alpha_{23})(\alpha_{16}\alpha_{19})] $$ with the orbit $\{\alpha_{2},\alpha_{13}\}$. The $G_1\cong \SSS_4$ is marked by $$ G=H_{51,3}\supset G_1= $$ $$ [(\alpha_{3}\alpha_{6}\alpha_{22})(\alpha_{4}\alpha_{16}\alpha_{8})(\alpha_{7}\alpha_{24}\alpha_{18}) (\alpha_{9}\alpha_{14}\alpha_{19})(\alpha_{11}\alpha_{20}\alpha_{15})(\alpha_{12}\alpha_{23}\alpha_{17}), $$ $$ (\alpha_{1}\alpha_{11}\alpha_{24}\alpha_{20})(\alpha_{3}\alpha_{22}\alpha_{23}\alpha_{12}) (\alpha_{4}\alpha_{9}\alpha_{19}\alpha_{16})(\alpha_{6}\alpha_{17}) (\alpha_{7}\alpha_{21}\alpha_{18}\alpha_{15})(\alpha_{8}\alpha_{14})] $$ with suborbits $\{\alpha_{2}\}$, $\{\alpha_{13}\}$. Both $G$ and $G_1$ are marked by $\ast$ in Tables 1---4. Case 71: $({\bf n}=34,\ (\mathbb A_1,2\mathbb A_1)\subset 3\mathbb A_1)\Longleftarrow ({\bf n}=61,\ 3\mathbb A_1)$. Similar to Cases 1 and 48. By \cite{Nik10}, $G\cong \AAA_{4,3}$ is marked by by $N_{23}$ and $G=H_{61,1}$ of Case 48 with the orbit $\{\alpha_{7},\alpha_{15},\alpha_{18}\}$. The $G_1\cong \SSS_4$ is marked by $$ G=H_{61,1}\supset G_1= $$ $$ [(\alpha_{2}\alpha_{20}\alpha_{14})(\alpha_{3}\alpha_{17}\alpha_{22})(\alpha_{4}\alpha_{9}\alpha_{21}) (\alpha_{8}\alpha_{11}\alpha_{13})(\alpha_{10}\alpha_{12}\alpha_{23})(\alpha_{16}\alpha_{19}\alpha_{24}), $$ $$ (\alpha_{1}\alpha_{10}\alpha_{23}\alpha_{12})(\alpha_{2}\alpha_{3}\alpha_{21}\alpha_{9})(\alpha_{4}\alpha_{22}\alpha_{8}\alpha_{20}) (\alpha_{7}\alpha_{15})(\alpha_{11}\alpha_{17}\alpha_{14}\alpha_{13})(\alpha_{16}\alpha_{19})] $$ with suborbits $\{\alpha_{18}\}$, $\{\alpha_{7},\alpha_{15}\}$. Both $G$ and $G_1$ are marked by $\ast$ in Tables 1---4. Case 72: $({\bf n}=34,\ (\mathbb A_1,3\mathbb A_1)\subset 4\mathbb A_1)\Longleftarrow ({\bf n}=65,\ 4\mathbb A_1)$. Similar to Cases 1 and 52. By \cite{Nik10}, the $G\cong 2^4D_6$ is marked by $N_{23}$ and $G=H_{65,4}$ of Case 52 with the orbit $\{\alpha_{3},\alpha_{22},\alpha_{24},\alpha_{12}\}$. The $G_1\cong \SSS_4$ is marked by $$ G=H_{65,4}\supset G_1= $$ $$ [(\alpha_{4}\alpha_{20}\alpha_{13})(\alpha_{5}\alpha_{18}\alpha_{16})(\alpha_{7}\alpha_{9}\alpha_{17}) (\alpha_{8}\alpha_{19}\alpha_{11})(\alpha_{12}\alpha_{22}\alpha_{24})(\alpha_{15}\alpha_{23}\alpha_{21}), $$ $$ (\alpha_{1}\alpha_{8}\alpha_{21}\alpha_{19})(\alpha_{4}\alpha_{13}\alpha_{20}\alpha_{6})(\alpha_{5}\alpha_{18}) (\alpha_{7}\alpha_{17}\alpha_{14}\alpha_{9})(\alpha_{10}\alpha_{15}\alpha_{11}\alpha_{23})(\alpha_{12}\alpha_{22})] $$ with suborbits $\{\alpha_{3}\}$, $\{\alpha_{22},\alpha_{24},\alpha_{12}\}$. Both $G$ and $G_1$ are marked by $\ast$ in Tables 1---4. Case 73: $({\bf n}=34,\ (4\mathbb A_1,4\mathbb A_1)\subset 8\mathbb A_1)\Longleftarrow ({\bf n}=51,\ 8\mathbb A_1)$. Similar to Cases 1 and 70. By \cite{Nik10}, the $G\cong C_2\times \SSS_4$ is marked by $N_{23}$ and $G=H_{51,3}$ of Case 70 with the orbit $\{\alpha_{1},\alpha_{15},\alpha_{21},\alpha_{24}, \alpha_{18},\alpha_{7},\alpha_{20},\alpha_{11}\}$. The $G_1\cong \SSS_4$ is marked by $$ G=H_{51,3}\supset G_1= $$ $$ [(\alpha_{3}\alpha_{22}\alpha_{6})(\alpha_{4}\alpha_{8}\alpha_{16})(\alpha_{7}\alpha_{18}\alpha_{24}) (\alpha_{9}\alpha_{19}\alpha_{14})(\alpha_{11}\alpha_{15}\alpha_{20})(\alpha_{12}\alpha_{17}\alpha_{23}), $$ $$ (\alpha_{1}\alpha_{7}\alpha_{24}\alpha_{18})(\alpha_{2}\alpha_{13}) (\alpha_{3}\alpha_{12}\alpha_{23}\alpha_{22})(\alpha_{4}\alpha_{9}\alpha_{19}\alpha_{16}) (\alpha_{6}\alpha_{17})(\alpha_{11}\alpha_{21}\alpha_{20}\alpha_{15})] $$ with suborbits $\{\alpha_1,\alpha_7,\alpha_{18},\alpha_{24}\}$, $\{\alpha_{11},\alpha_{15},\alpha_{21},\alpha_{20}\}$. Both $G$ and $G_1$ are marked by $\ast$ in Tables 1---4. Case 74: $({\bf n}=34,\ (4\mathbb A_1,8\mathbb A_1)\subset 12\mathbb A_1)\Longleftarrow ({\bf n}=61,\ 12\mathbb A_1)$. Similar to Cases 1 and 48. By \cite{Nik10}, the $G\cong \AAA_{4,3}$ is marked by by $N_{23}$ and $G=H_{61,1}$ of Case 48 with the orbit $\{\alpha_{2},\alpha_{4},\alpha_{21},\alpha_{13},\alpha_{11},\alpha_{8}, \alpha_{20},\alpha_{17},\alpha_{22},\alpha_{14},\alpha_{9},\alpha_{3}\}$. The $G_1\cong \SSS_4$ is marked by $$ G=H_{61,1}\supset G_1= $$ $$ [(\alpha_{2}\alpha_{13}\alpha_{21})(\alpha_{4}\alpha_{20}\alpha_{8})(\alpha_{7}\alpha_{15}\alpha_{18}) (\alpha_{9}\alpha_{14}\alpha_{11})(\alpha_{10}\alpha_{12}\alpha_{23})(\alpha_{16}\alpha_{24}\alpha_{19}), $$ $$ (\alpha_{1}\alpha_{10}\alpha_{23}\alpha_{12})(\alpha_{2}\alpha_{3}\alpha_{21}\alpha_{9}) (\alpha_{4}\alpha_{22}\alpha_{8}\alpha_{20})(\alpha_{7}\alpha_{15}) (\alpha_{11}\alpha_{17}\alpha_{14}\alpha_{13})(\alpha_{16}\alpha_{19})] $$ with suborbits $\{\alpha_{4},\alpha_{20},\alpha_{22},\alpha_{8}\}$, $\{\alpha_{2},\alpha_{13},\alpha_{3},\alpha_{21},\alpha_{11},\alpha_{9},\alpha_{17},\alpha_{14}\}$. Both $G$ and $G_1$ are marked by $\ast$ in Tables 1---4. Case 75: $({\bf n}=34,\ (4\mathbb A_1,12\mathbb A_1)\subset 16\mathbb A_1)\Longleftarrow ({\bf n}=65,\ 16\mathbb A_1)$. Similar to Case 1. By \cite{Nik10}, the $G\cong 2^4D_6$ is marked by $N_{23}$ and $$ G=H_{65,1}= $$ $$ [(\alpha_{1}\alpha_{6}\alpha_{21})(\alpha_{4}\alpha_{17}\alpha_{14}) (\alpha_{7}\alpha_{15}\alpha_{8})(\alpha_{9}\alpha_{10}\alpha_{23}) (\alpha_{11}\alpha_{20}\alpha_{19})(\alpha_{12}\alpha_{18}\alpha_{24}), $$ $$ (\alpha_{1}\alpha_{11}\alpha_{23}\alpha_{8}) (\alpha_{4}\alpha_{6}\alpha_{13}\alpha_{20}) (\alpha_{5}\alpha_{16}) (\alpha_{7}\alpha_{17}\alpha_{9}\alpha_{14}) (\alpha_{10}\alpha_{21}\alpha_{19}\alpha_{15}) (\alpha_{12}\alpha_{24})] $$ with the orbit $\{\alpha_{1},\alpha_{19},\alpha_{6},\alpha_{7},\alpha_{4}, \alpha_{17},\alpha_{20},\alpha_{11},\alpha_{14}, \alpha_{21},\alpha_{15},\alpha_{23},\alpha_{8},\alpha_{10}, \alpha_{9},\alpha_{13}\}$. The $G_1\cong \SSS_4$ is marked by $$ G=H_{65,1}\supset G_1= $$ $$ [(\alpha_{4}\alpha_{19}\alpha_{7})(\alpha_{6}\alpha_{13}\alpha_{21})(\alpha_{8}\alpha_{11}\alpha_{23}) (\alpha_{9}\alpha_{14}\alpha_{15})(\alpha_{10}\alpha_{20}\alpha_{17})(\alpha_{12}\alpha_{18}\alpha_{24}), $$ $$ (\alpha_{1}\alpha_{4}\alpha_{7}\alpha_{19})(\alpha_{5}\alpha_{16}) (\alpha_{6}\alpha_{10}\alpha_{11}\alpha_{13})(\alpha_{8}\alpha_{14}\alpha_{20}\alpha_{21}) (\alpha_{9}\alpha_{15}\alpha_{23}\alpha_{17})(\alpha_{12}\alpha_{28})] $$ with suborbits $\{\alpha_{1},\alpha_{4},\alpha_{19},\alpha_{7}\}$, $\{\alpha_{6},\alpha_{13},\alpha_{10},\alpha_{21},\alpha_{20},\alpha_{11},\alpha_{8},\alpha_{17}, \alpha_{23},\alpha_{14},\alpha_{9},\alpha_{15}\}$. Both $G$ and $G_1$ are marked by $\ast$ in Tables 1---4. Case 76: $({\bf n}=34,\ ((6\mathbb A_1)_{I},(6\mathbb A_1)_{II})\subset 12\mathbb A_1)\Longleftarrow ({\bf n}=65,\ 12\mathbb A_1)$. Similar to Cases 1 and 24. By \cite{Nik10}, the $G\cong 2^4D_6$ is marked by $N_{23}$ and $G=H_{65,3}$ of Case 24 with the orbit $\{\alpha_{1},\alpha_{7},\alpha_{23},\alpha_{8},\alpha_{11},\alpha_{17},\alpha_{20}, \alpha_{9}, \alpha_{6},\alpha_{4},\alpha_{14},\alpha_{13}\}$. The $G_1\cong \SSS_4$ is marked by $$ G=H_{65,3}\supset G_1= $$ $$ [(\alpha_{1}\alpha_{4}\alpha_{6})(\alpha_{5}\alpha_{16}\alpha_{22})(\alpha_{7}\alpha_{8}\alpha_{13}) (\alpha_{9}\alpha_{23}\alpha_{17})(\alpha_{10}\alpha_{21}\alpha_{19})(\alpha_{11}\alpha_{14}\alpha_{20}), $$ $$ (\alpha_{3}\alpha_{5}\alpha_{22}\alpha_{16})(\alpha_{4}\alpha_{6}\alpha_{17}\alpha_{9}) (\alpha_{7}\alpha_{13}\alpha_{20}\alpha_{14})(\alpha_{8}\alpha_{11}) (\alpha_{10}\alpha_{15}\alpha_{19}\alpha_{21})(\alpha_{12}\alpha_{24})] $$ with suborbits $\{\alpha_{7},\alpha_{8},\alpha_{13},\alpha_{11},\alpha_{20},\alpha_{14}\}$ of the type $I$, and $\{\alpha_{1},\alpha_{4},\alpha_{6},\alpha_{17},\alpha_{9},\alpha_{23}\}$ of the type $II$. Both $G$ and $G_1$ are marked by $\ast$ in Tables 1---4. Case 77: $({\bf n}=39,\ (4\mathbb A_1,4\mathbb A_1)\subset 8\mathbb A_1)\Longleftarrow ({\bf n}=56,\ 8\mathbb A_1)$. Similar to Cases 1 and 23. By \cite{Nik10}, the $G\cong \Gamma_{15}a_1$ is marked by $N_{23}$ and $G=H_{56,1}$ of Case 23 with the orbit $\{\alpha_{1}, \alpha_{14}, \alpha_{24}, \alpha_{17}, \alpha_{2},\alpha_{23}, \alpha_{5}, \alpha_{3}\}$. The $G_1\cong 2^4C_2$ is marked by $$ G=H_{56,1}\supset G_1= $$ $$ [(\alpha_{2}\alpha_{17})(\alpha_{6}\alpha_{16})(\alpha_{7}\alpha_{8})(\alpha_{9}\alpha_{13}) (\alpha_{10}\alpha_{18})(\alpha_{12}\alpha_{22})(\alpha_{19}\alpha_{20})(\alpha_{23}\alpha_{24}), $$ $$ (\alpha_{1}\alpha_{17})(\alpha_{2}\alpha_{3})(\alpha_{6}\alpha_{9})(\alpha_{7}\alpha_{10}) (\alpha_{8}\alpha_{13})(\alpha_{15}\alpha_{20})(\alpha_{16}\alpha_{18})(\alpha_{19}\alpha_{21}), $$ $$ (\alpha_{5}\alpha_{23})(\alpha_{6}\alpha_{10})(\alpha_{7}\alpha_{9})(\alpha_{8}\alpha_{16}) (\alpha_{13}\alpha_{18})(\alpha_{14}\alpha_{24})(\alpha_{15}\alpha_{20})(\alpha_{19}\alpha_{21})] $$ with suborbits $\{\alpha_{1},\alpha_{17},\alpha_{2},\alpha_{3}\}$, $\{\alpha_{5},\alpha_{23},\alpha_{24},\alpha_{14}\}$. Both $G$ and $G_1$ are marked by $\ast$ in Tables 1---4. Case 78: $({\bf n}=39,\ (4\mathbb A_1,8\mathbb A_1)\subset 12\mathbb A_1)\Longleftarrow ({\bf n}=65,\ 12\mathbb A_1)$. Similar to Cases 1 and 24. By \cite{Nik10}, the $G\cong 2^4D_6$ is marked by by $N_{23}$ and $G=H_{65,3}$ of Case 24 with the orbit $\{\alpha_{1},\alpha_{7},\alpha_{23},\alpha_{8},\alpha_{11},\alpha_{17},\alpha_{20}, \alpha_{9}, \alpha_{6},\alpha_{4},\alpha_{14},\alpha_{13}\}$. The $G_1\cong 2^4C_2$ is marked by $$ G=H_{65,3}\supset G_1= $$ $$ [(\alpha_{4}\alpha_{7})(\alpha_{5}\alpha_{16})(\alpha_{6}\alpha_{14})(\alpha_{8}\alpha_{11}) (\alpha_{9}\alpha_{13})(\alpha_{12}\alpha_{24})(\alpha_{15}\alpha_{21})(\alpha_{17}\alpha_{20}), $$ $$ (\alpha_{1}\alpha_{8})(\alpha_{3}\alpha_{16})(\alpha_{5}\alpha_{22})(\alpha_{6}\alpha_{7}) (\alpha_{9}\alpha_{20})(\alpha_{10}\alpha_{19})(\alpha_{11}\alpha_{23})(\alpha_{15}\alpha_{21}), $$ $$ (\alpha_{3}\alpha_{5})(\alpha_{4}\alpha_{13})(\alpha_{6}\alpha_{7})(\alpha_{9}\alpha_{20}) (\alpha_{10}\alpha_{15})(\alpha_{14}\alpha_{17})(\alpha_{16}\alpha_{22})(\alpha_{19}\alpha_{21})] $$ with suborbits $\{\alpha_{1},\alpha_{8},\alpha_{11},\alpha_{23}\}$, $\{\alpha_{4},\alpha_{7},\alpha_{13},\alpha_{6},\alpha_{9},\alpha_{14},\alpha_{20},\alpha_{17}\}$. Both $G$ and $G_1$ are mar\-ked by $\ast$ in Tables 1---4. Case 79: $({\bf n}=39,\ (8\mathbb A_1,8\mathbb A_1)\subset 16\mathbb A_1)\Longleftarrow ({\bf n}=75,\ 16\mathbb A_1)$. Similar to Cases 1 and 10. By \cite{Nik10}, the $G\cong 4^2\AAA_4$ is marked by $N_{23}$ and $G=H_{75,1}$ of Case 10 with the orbit $\{\alpha_{1},\alpha_{9},\alpha_{6},\alpha_{7},\alpha_{4}, \alpha_{21},\alpha_{20}, \alpha_{22},\alpha_{5},\alpha_{14},\alpha_{10},\alpha_{23}, \alpha_{3},\alpha_{24},\alpha_{18},\alpha_{16}\}$. The $G_1\cong 2^4C_2$ is marked by $$ G=H_{75,1}\supset G_1= $$ $$ [(\alpha_{1}\alpha_{3})(\alpha_{4}\alpha_{22})(\alpha_{5}\alpha_{10})(\alpha_{6}\alpha_{14}) (\alpha_{7}\alpha_{23})(\alpha_{9}\alpha_{24})(\alpha_{16}\alpha_{18})(\alpha_{20}\alpha_{21}), $$ $$ (\alpha_{3}\alpha_{24})(\alpha_{4}\alpha_{6})(\alpha_{5}\alpha_{21})(\alpha_{10}\alpha_{14}) (\alpha_{11}\alpha_{17})(\alpha_{12}\alpha_{13})(\alpha_{16}\alpha_{23})(\alpha_{20}\alpha_{22}), $$ $$ (\alpha_{3}\alpha_{16})(\alpha_{4}\alpha_{5})(\alpha_{6}\alpha_{21})(\alpha_{10}\alpha_{20}) (\alpha_{11}\alpha_{12})(\alpha_{13}\alpha_{17})(\alpha_{14}\alpha_{22})(\alpha_{23}\alpha_{24})] $$ with suborbits $\{\alpha_{1},\alpha_{3},\alpha_{24},\alpha_{16},\alpha_{9},\alpha_{23},\alpha_{18},\alpha_{7}\}$, $\{\alpha_{4},\alpha_{22},\alpha_{6},\alpha_{5},\alpha_{20},\alpha_{14},\alpha_{21},\alpha_{10}\}$. Both $G$ and $G_1$ are marked by $\ast$ in Tables 1---4. Case 80: $({\bf n}=40,\ (8\mathbb A_1,8\mathbb A_1)\subset 16\mathbb A_1)\Longleftarrow ({\bf n}=56,\ 16\mathbb A_1)$. Similar to Cases 1 and 8. By \cite{Nik10}, the $G\cong \Gamma_{25}a_1$ is marked by $N_{23}$ and $G=H_{56,2}$ of Case 8 with the orbit $\{\alpha_{2}, \alpha_{3}, \alpha_{23}, \alpha_{24}, \alpha_{5},\alpha_{8}, \alpha_{17}, \alpha_{19}, \alpha_{6}, \alpha_{7}, \alpha_{11}, \alpha_{20}, \alpha_{18}, \alpha_{10},\alpha_{16},\alpha_{15}\}$. The $G_1\cong Q_8\ast Q_8$ is marked by $$ G=H_{56,2}\supset G_1= $$ $$ [(\alpha_{2}\alpha_{16})(\alpha_{3}\alpha_{5})(\alpha_{6}\alpha_{17})(\alpha_{9}\alpha_{13}) (\alpha_{10}\alpha_{23})(\alpha_{11}\alpha_{15})(\alpha_{12}\alpha_{22})(\alpha_{18}\alpha_{24}), $$ $$ (\alpha_{2}\alpha_{10})(\alpha_{3}\alpha_{20})(\alpha_{5}\alpha_{19})(\alpha_{6}\alpha_{24}) (\alpha_{7}\alpha_{15})(\alpha_{8}\alpha_{11})(\alpha_{16}\alpha_{18})(\alpha_{17}\alpha_{23}), $$ $$ (\alpha_{3}\alpha_{11})(\alpha_{5}\alpha_{15})(\alpha_{7}\alpha_{20})(\alpha_{8}\alpha_{19}) (\alpha_{9}\alpha_{13})(\alpha_{10}\alpha_{24})(\alpha_{12}\alpha_{22})(\alpha_{18}\alpha_{23}), $$ $$ (\alpha_{1}\alpha_{14})(\alpha_{3}\alpha_{20})(\alpha_{5}\alpha_{19})(\alpha_{7}\alpha_{11}) (\alpha_{8}\alpha_{15})(\alpha_{12}\alpha_{22})(\alpha_{16}\alpha_{17})(\alpha_{18}\alpha_{23})] $$ with suborbits $\{\alpha_{2},\alpha_{16},\alpha_{10}\alpha_{18},\alpha_{17},\alpha_{23},\alpha_{24},\alpha_{6}\}$, $\{\alpha_{3},\alpha_{5},\alpha_{20}\alpha_{11},\alpha_{19},\alpha_{15},\alpha_{7},\alpha_{8}\}$. Both $G$ and $G_1$ are marked by $\ast$ in Tables 1---4. Case 81: $({\bf n}=49,\ (4\mathbb A_1,4\mathbb A_1)\subset 8\mathbb A_1)\Longleftarrow ({\bf n}=65,\ 8\mathbb A_1)$. Similar to Cases 1 and 52. By \cite{Nik10}, the $G\cong 2^4D_6$ is marked by $N_{23}$ and $G=H_{65,4}$ from Case 52 with the orbit $\{\alpha_{1},\alpha_{8},\alpha_{21},\alpha_{23},\alpha_{15}, \alpha_{11},\alpha_{10},\alpha_{19}\}$. The $G_1\cong 2^4C_3$ is marked by $$ G=H_{65,4}\supset G_1= $$ $$ [(\alpha_{4}\alpha_{13}\alpha_{20})(\alpha_{5}\alpha_{16}\alpha_{18})(\alpha_{7}\alpha_{17}\alpha_{9}) (\alpha_{8}\alpha_{11}\alpha_{19})(\alpha_{12}\alpha_{24}\alpha_{22})(\alpha_{15}\alpha_{21}\alpha_{23}), $$ $$ (\alpha_{1}\alpha_{15})(\alpha_{3}\alpha_{12})(\alpha_{4}\alpha_{20})(\alpha_{6}\alpha_{13}) (\alpha_{7}\alpha_{9})(\alpha_{14}\alpha_{17})(\alpha_{21}\alpha_{23})(\alpha_{22}\alpha_{24}), $$ $$ (\alpha_{3}\alpha_{12})(\alpha_{4}\alpha_{13})(\alpha_{6}\alpha_{20})(\alpha_{7}\alpha_{14}) (\alpha_{8}\alpha_{10})(\alpha_{9}\alpha_{17})(\alpha_{11}\alpha_{19})(\alpha_{22}\alpha_{24})] $$ with suborbits $\{\alpha_{1},\alpha_{15},\alpha_{21},\alpha_{23}\}$, $\{\alpha_{8},\alpha_{11},\alpha_{10},\alpha_{19}\}$. Both $G$ and $G_1$ are marked by $\ast$ in Tables 1---4. Case 82: $({\bf n}=49,\ (4\mathbb A_1,12\mathbb A_1)\subset 16\mathbb A_1)\Longleftarrow ({\bf n}=75,\ 16\mathbb A_1)$. Similar to Cases 1 and 10. By \cite{Nik10}, the $G\cong 4^2\AAA_4$ is marked by $N_{23}$ and $G=H_{75,1}$ of Case 10 with the orbit $\{\alpha_{1},\alpha_{9},\alpha_{6},\alpha_{7},\alpha_{4}, \alpha_{21},\alpha_{20}, \alpha_{22},\alpha_{5},\alpha_{14},\alpha_{10},\alpha_{23}, \alpha_{3},\alpha_{24},\alpha_{18},\alpha_{16}\}$. The $G_1\cong 2^4C_3$ is marked by $$ G=H_{75,1}\supset G_1= $$ $$ [(\alpha_{1}\alpha_{3}\alpha_{10})(\alpha_{4}\alpha_{6}\alpha_{21})(\alpha_{7}\alpha_{24}\alpha_{14}) (\alpha_{9}\alpha_{16}\alpha_{20})(\alpha_{11}\alpha_{13}\alpha_{17})(\alpha_{18}\alpha_{23}\alpha_{22}), $$ $$ (\alpha_{3}\alpha_{24})(\alpha_{4}\alpha_{6})(\alpha_{5}\alpha_{21})(\alpha_{10}\alpha_{14}) (\alpha_{11}\alpha_{17})(\alpha_{12}\alpha_{13})(\alpha_{16}\alpha_{23})(\alpha_{20}\alpha_{22}), $$ $$ (\alpha_{3}\alpha_{16})(\alpha_{4}\alpha_{5})(\alpha_{6}\alpha_{21})(\alpha_{10}\alpha_{20}) (\alpha_{11}\alpha_{12})(\alpha_{13}\alpha_{17})(\alpha_{14}\alpha_{22})(\alpha_{23}\alpha_{24})] $$ with suborbits $\{\alpha_{4},\alpha_{6},\alpha_{5},\alpha_{21}\}$, $\{\alpha_{1},\alpha_{3},\alpha_{10},\alpha_{24},\alpha_{16},\alpha_{14},\alpha_{20},\alpha_{23}, \alpha_{7},\alpha_{22},\alpha_{9},\alpha_{18}\}$. Both $G$ and $G_1$ are marked by $\ast$ in Tables 1---4. This completes the proof of the Theorem. \end{proof} Thus, we have finally obtained \vskip1cm \begin{theorem} Classification of Picard lattices $S=MS_X$ of K3 surfaces $X$ with finite symplectic automorphic groups which are big enough (larger than $D_6$, $C_4$, $(C_2)^2$, $C_3$, $C_2$ and $C_1$) and with at least one $-2$ curve is given in the Tables 1 --- 4 of Section \ref{sec4:tables} below in lines which are not marked by $o$. \end{theorem} We hope to consider remaining symplectic groups $D_6$, $C_4$, $(C_2)^2$, $C_3$, $C_2$, $C_1$ later as well. Now, we obtain for these groups \begin{theorem} If the Picard lattice $S=MS_X$ of a K3 surface $X$ with at least one $-2$ curve is different from all lattices of lines of Tables 1 --- 4 of Section \ref{sec4:tables} which are not marked by $o$ (for example, if the genus is different), then the symplectic automorphism group of $X$ is small, it is one of groups: $D_6$, $C_4$, $(C_2)^2$, $C_3$, $C_2$ or $C_1$. \end{theorem} \section{Tables.} \label{sec4:tables} Here we present tables of lattices $S$ of degenerations of K\"ahlerian K3 surfaces with finite symplectic automorphism groups and classification of Picard lattices of K3 surfaces which were discussed in previous sections. \begin{table} \label{table1} \caption{Types and lattices $S$ of degenerations of codimension $1$ of K\"ahlerian K3 surfaces with finite symplectic automorphism groups $G=Clos(G)$. All lines are marked by $\ast$, by definition.} \begin{tabular}{\|c\|\|c\|c\|c\|c\|c\|c\|c\|c\|} \hline {\bf n}& $\|G\|$& $i$& $G$ & $\rk S_G$ &$q_{S_G}$ &$Deg$& $\rk S$ &$q_S$ \\ \hline \hline $1$ &$2$ & $1$& $C_2$ & $8$ &$2_{II}^{+8}$ & $\mathbb A_1$ & $9$ & $2_7^{+9}$\\ \hline & & & & & & $2\mathbb A_1$ & $9$ & $2_{II}^{-6},4_3^{-1}$\\ \hline \hline $2$ & $3$& $1$&$C_3$ & $12$ & $3^{+6}$ & $\mathbb A_1$ & $13$ & $2_3^{-1},3^{+6}$\\ \hline & & & & & & $3\mathbb A_1$ & $13$ & $2_1^{+1},3^{-5}$\\ \hline \hline $3$ &$4$ & $2$&$C_2^2$& $12$ & $2_{II}^{-6},4_{II}^{-2}$& $\mathbb A_1$ & $13$ & $2_3^{+7},4_{II}^{+2}$\\ \hline & & & & & & $2\mathbb A_1$& $13$ & $2_{II}^{-4},4_7^{-3}$\\ \hline & & & & & & $4\mathbb A_1$ & $13$ & $2_{II}^{-6},8_3^{-1}$\\ \hline \hline $4$ &$4$& $1$ &$C_4$ & $14$ &$2_2^{+2},4_{II}^{+4}$ & $\mathbb A_1$ & $15$ & $2_5^{-3},4_{II}^{+4}$\\ \hline & & & & & & $2\mathbb A_1$ & $15$ & $4_1^{-5}$\\ \hline & & & & & & $4\mathbb A_1$ & $15$ & $2_2^{+2},4_{II}^{+2},8_7^{+1}$\\ \hline & & & & & & $\mathbb A_2$ & $15$ & $2_1^{+1},4_{II}^{-4}$\\ \hline \hline $6$ &$6$& $1$ & $D_6$ & $14$ &$2_{II}^{-2},3^{+5}$ & $\mathbb A_1$ & $15$ & $2_7^{-3},3^{+5}$\\ \hline & & & & & & $2\mathbb A_1$ & $15$ & $4_3^{-1},3^{+5}$\\ \hline & & & & & & $3\mathbb A_1$ & $15$ & $2_1^{-3},3^{-4}$\\ \hline & & & & & & $6\mathbb A_1$ & $15$ & $4_1^{+1},3^{+4}$\\ \hline \hline $9$ &$8$& $5$ &$C_2^3$& $14$ & $2_{II}^{+6},4_2^{+2}$ & $2\mathbb A_1$ & $15$ & $2_{II}^{-4},4_5^{-3}$\\ \hline & & & & & & $4\mathbb A_1$ & $15$ & $2_{II}^{+6},8_1^{+1}$\\ \hline & & & & & & $8\mathbb A_1$ & $15$ & $2_{II}^{+6},4_1^{+1}$\\ \hline \hline $10$&$8$& $3$ &$D_8$ & $15$ & $4_1^{+5}$ & $\mathbb A_1$ & $16$ & $2_1^{+1},4_7^{+5}$\\ \hline & & & & & &$(2\mathbb A_1)_I$& $16$ & $2_6^{-2},4_6^{-4}$\\ \hline & & & & & &$(2\mathbb A_1)_{II}$& $16$ & $2_{II}^{+2},4_{II}^{+4}$\\ \hline & & & & & & $4\mathbb A_1$ & $16$ & $4_7^{+3},8_1^{+1}$\\ \hline & & & & & & $8\mathbb A_1$ & $16$ & $4_0^{+4}$\\ \hline & & & & & & $2\mathbb A_2$ & $16$ & $4_{II}^{+4}$\\ \hline \hline $12$&$8$& $4$ & $Q_8$ & $17$ & $2_7^{-3},8_{II}^{-2}$ & $8\mathbb A_1$ & $18$ & $2_7^{-3},16_3^{-1}$\\ \hline & & & & & & $\mathbb A_2$ & $18$ & $2_6^{-2},8_{II}^{-2}$\\ \hline \hline $16$ &$10$&$1$ &$D_{10}$& $16$ & $5^{+4}$ & $\mathbb A_1$ & $17$ & $2_7^{+1},5^{+4}$\\ \hline & & & & & & $5\mathbb A_1$ &$17$ & $2_7^{+1},5^{-3}$\\ \hline \end{tabular} \end{table} \begin{table} \begin{tabular}{\|c\|\|c\|c\|c\|c\|c\|c\|c\|c\|} \hline {\bf n}& $\|G\|$& $i$& $G$ & $\rk S_G$ &$q_{S_G}$ &$Deg$& $\rk S$ &$q_S$ \\ \hline \hline $17$&$12$ & $3$&$\AAA_4$& $16$& $2_{II}^{-2},4_{II}^{-2},3^{+2}$ & $\mathbb A_1$ &$17$ & $2_7^{-3},4_{II}^{+2},3^{+2}$\\ \hline & & & & & &$3\mathbb A_1$ & $17$ & $2_1^{-3},4_{II}^{+2},3^{-1}$\\ \hline & & & & & &$4\mathbb A_1$ & $17$ & $2_{II}^{-2},8_3^{-1},3^{+2}$\\ \hline & & & & & &$6\mathbb A_1$ & $17$ & $4_1^{-3},3^{+1}$\\ \hline & & & & & &$12\mathbb A_1$& $17$ & $2_{II}^{-2},8_1^{+1},3^{-1}$\\ \hline \hline $18$&$12$ &$4$ &$D_{12}$& $16$&$2_{II}^{+4},3^{+4}$ & $\mathbb A_1$& $17$& $2_7^{+5},3^{+4}$\\ \hline & & & & & & $2\mathbb A_1$ & $17$ & $2_{II}^{+2},4_7^{+1},3^{+4}$\\ \hline & & & & & & $3\mathbb A_1$ & $17$ & $2_5^{-5},3^{-3}$\\ \hline & & & & & & $6\mathbb A_1$ & $17$ & $2_{II}^{-2},4_1^{+1},3^{+3}$\\ \hline \hline $21$&$16$ &$14$&$C_2^4$ & $15$ & $2_{II}^{+6},8_I^{+1}$ & $4\mathbb A_1$ & $16$ & $2_{II}^{+4},4_{II}^{+2}$\\ \hline & & & & & &$16\mathbb A_1$& $16$ & $2_{II}^{+6}$\\ \hline \hline $22$& $16$&$11$&$C_2\times D_8$& $16$& $2_{II}^{+2},4_0^{+4}$ &$2\mathbb A_1$&$17$ & $4_7^{+5}$\\ \hline & & & & & & $4\mathbb A_1$ & $17$ & $2_{II}^{+2},4_0^{+2},8_7^{+1}$\\ \hline & & & & & & $8\mathbb A_1$ & $17$ & $2_{II}^{+2},4_7^{+3}$\\ \hline \hline $26$& $16$& $8$&$SD_{16}$& $18$ &$2_7^{+1},4_7^{+1},8_{II}^{+2}$&$8\mathbb A_1$&$19$ & $2_1^{+1},4_1^{+1},16_3^{-1}$\\ \hline & & & & & & $2\mathbb A_2$ & $19$ & $2_5^{-1},8_{II}^{-2}$\\ \hline \hline $30$ & $18$& $4$&$\AAA_{3,3}$&$16$ &$3^{+4},9^{-1}$& $3\mathbb A_1$ & $17$ & $2_5^{-1},3^{-3},9^{-1}$\\ \hline & & & & & & $9\mathbb A_1$ & $17$ & $2_3^{-1},3^{+4}$\\ \hline \hline $32$& $20$& $3$&$Hol(C_5)$& $18$ &$2_6^{-2},5^{+3}$& $2\mathbb A_1$ &$19$ & $4_1^{+1},5^{+3}$\\ \hline & & & & & & $5\mathbb A_1$ &$19$& $2_1^{+3},5^{-2}$\\ \hline & & & & & & $10\mathbb A_1$ &$19$ & $4_5^{-1},5^{+2}$\\ \hline & & & & & & $5\mathbb A_2$ &$19$ & $2_5^{-1},5^{-2}$\\ \hline \hline $33$ & $21$&$1$ &$C_7\rtimes C_3$&$18$& $7^{+3}$ & $7\mathbb A_1$ & $19$ & $2_1^{+1},7^{+2}$\\ \hline \hline $34$& $24$&$12$&$\SSS_4$& $17$ & $4_3^{+3},3^{+2}$& $\mathbb A_1$& $18$ & $2_5^{-1},4_1^{+3},3^{+2}$\\ \hline & & & & & & $2\mathbb A_1$ & $18$ & $2_2^{+2},4_{II}^{+2},3^{+2}$\\ \hline & & & & & & $3\mathbb A_1$ & $18$ & $2_7^{+1},4_5^{-3},3^{-1}$ \\ \hline & & & & & & $4\mathbb A_1$ & $18$ & $4_3^{-1},8_3^{-1},3^{+2}$ \\ \hline & & & & & & $(6\mathbb A_1)_I$& $18$ & $2_4^{-2},4_0^{+2},3^{+1}$\\ \hline & & & & & & $(6\mathbb A_1)_{II}$&$18$ & $2_{II}^{+2},4_{II}^{-2},3^{+1}$ \\ \hline & & & & & & $8\mathbb A_1$ &$18$ & $4_2^{+2},3^{+2}$\\ \hline & & & & & & $12\mathbb A_1$ & $18$ & $4_5^{-1},8_7^{+1},3^{-1}$\\ \hline & & & & & & $6\mathbb A_2$ & $18$ & $4_{II}^{-2},3^{+1}$\\ \hline \end{tabular} \end{table} \begin{table} \begin{tabular}{\|c\|\|c\|c\|c\|c\|c\|c\|c\|c\|} \hline {\bf n}& $\|G\|$& $i$& $G$ & $\rk S_G$ &$q_{S_G}$ &$Deg$& $\rk S$ &$q_S$ \\ \hline \hline $39$&$32$ &$27$&$2^4C_2$& $17$ &$2_{II}^{+2},4_0^{+2},8_7^{+1}$ & $4\mathbb A_1$&$18$& $4_6^{+4}$\\ \hline & & & & & & $8\mathbb A_1$& $18$& $2_{II}^{+2},4_7^{+1},8_7^{+1}$\\ \hline & & & & & & $16\mathbb A_1$&$18$& $2_{II}^{+2},4_6^{+2}$\\ \hline \hline $40$ &$32$ &$49$&$Q_8Q_8$& $17$ & $4_7^{+5}$ & $8\mathbb A_1$ & $18$ & $4_6^{+4}$\\ \hline \hline $46$ &$36$ &$9$ &$3^2C_4$ & $18$ & $2_6^{-2},3^{+2},9^{-1}$& $6\mathbb A_1$ & $19$ & $4_7^{+1},3^{+1},9^{-1}$ \\ \hline & & & & & & $9\mathbb A_1$ & $19$ & $2_5^{-3},3^{+2}$ \\ \hline & & & & & & $9\mathbb A_2$ & $19$ & $2_5^{-1},3^{+2}$\\ \hline \hline $48$ &$36$ &$10$&$\SSS_{3,3}$&$18$ & $2_{II}^{-2},3^{+3},9^{-1}$ & $3\mathbb A_1$& $19$ & $2_5^{+3},3^{-2},9^{-1}$\\ \hline & & & & & & $6\mathbb A_1$& $19$ & $4_1^{+1},3^{+2},9^{-1}$\\ \hline & & & & & & $9\mathbb A_1$& $19$ & $2_7^{-3},3^{+3}$\\ \hline \hline $49$ & $48$&$50$&$2^4C_3$& $17$ & $2_{II}^{-4},8_1^{+1},3^{-1}$&$4\mathbb A_1$&$18$ & $2_{II}^{-2},4_{II}^{+2},3^{-1}$\\ \hline & & & & & &$12\mathbb A_1$&$18$ & $2_{II}^{-2},4_2^{-2}$\\ \hline & & & & & &$16\mathbb A_1$&$18$ & $2_{II}^{-4},3^{-1}$ \\ \hline \hline $51$ & $48$&$48$&$C_2\times \SSS_4$&$18$&$2_{II}^{+2},4_2^{+2},3^{+2}$ & $2\mathbb A_1$ & $19$ & $4_1^{+3},3^{+2}$\\ \hline & & & & & & $4\mathbb A_1$ & $19$ & $2_{II}^{+2},8_1^{+1},3^{+2}$\\ \hline & & & & & & $6\mathbb A_1$ & $19$ & $4_7^{-3},3^{+1}$\\ \hline & & & & & & $8\mathbb A_1$ & $19$ & $2_{II}^{-2},4_5^{-1},3^{+2}$\\ \hline & & & & & &$12\mathbb A_1$ & $19$ & $2_{II}^{-2},8_7^{+1},3^{-1}$\\ \hline \hline $55$ & $60$&$5$ &$\AAA_5$ & $18$ &$2_{II}^{-2},3^{+1},5^{-2}$& $\mathbb A_1$& $19$& $2_7^{-3},3^{+1},5^{-2}$ \\ \hline & & & & & &$5\mathbb A_1$& $19$& $2_3^{+3},3^{+1},5^{+1}$ \\ \hline & & & & & &$6\mathbb A_1$& $19$& $4_1^{+1},5^{-2}$\\ \hline & & & & & &$10\mathbb A_1$&$19$& $4_7^{+1},3^{+1},5^{-1}$\\ \hline & & & & & &$15\mathbb A_1$& $19$& $2_5^{+3},5^{-1}$\\ \hline \hline $56$ &$64$&$138$&$\Gamma_{25}a_1$&$18$& $4_5^{+3},8_1^{+1}$ & $8\mathbb A_1$ & $19$ & $4_4^{-2},8_5^{-1}$ \\ \hline & & & & & &$16\mathbb A_1$ & $19$ & $4_5^{+3}$ \\ \hline \hline $61$ &$72$ &$43$&$\AAA_{4,3}$& $18$ & $4_{II}^{-2},3^{-3}$&$3\mathbb A_1$&$19$& $2_5^{-1},4_{II}^{+2},3^{+2}$\\ \hline & & & & & &$12\mathbb A_1$&$19$ & $8_1^{+1},3^{+2}$\\ \hline \hline $65$ &$96$ &$227$&$2^4D_6$& $18$&$2_{II}^{-2},4_7^{+1},8_1^{+1},3^{-1}$&$4\mathbb A_1$&$19$& $4_3^{-3},3^{-1}$\\ \hline & & & & & &$8\mathbb A_1$&$19$& $2_{II}^{-2},8_7^{+1},3^{-1}$\\ \hline & & & & & & $12\mathbb A_1$&$19$& $4_5^{+3}$\\ \hline & & & & & & $16\mathbb A_1$&$19$& $2_{II}^{+2},4_3^{-1},3^{-1}$\\ \hline \hline $75$&$192$&$1023$&$4^2\AAA_4$&$18$&$2_{II}^{-2},8_6^{-2}$& $16\mathbb A_1$&$19$ &$2_{II}^{-2},8_5^{-1}$\\ \hline \end{tabular} \end{table} \begin{table} \label{table2} \caption{Types and lattices $S$ of degenerations of codimension $\ge 2$ of K\"ahlerian K3 surfaces with finite symplectic automorphism groups $G=Clos(G)$ for ${\bf n}\ge 12$.} \begin{tabular}{\|c\|\|c\|c\|c\|c\|c\|c\|c\|c\|} \hline {\bf n}& $\|G\|$& $i$& $G$ & $\rk S_G$ &$Deg$& $\rk S$ &$q_S$ \\ \hline \hline $12$&$8$& $4$ & $Q_8$ & $17$ & $(8\mathbb A_1,8\mathbb A_1)\subset 16\mathbb A_1$ $o$ & $19$ & $2_{II}^{-2},8_5^{-1}$ $\ast$\\ \hline & & & & & $(8\mathbb A_1,8\mathbb A_1)\subset 8\mathbb A_2$ & $19$ & $2_7^{-3},3^{-1}$ $\ast$\\ \hline & & & & & $(8\mathbb A_1,\mathbb A_2)\subset 8\mathbb A_1\amalg \mathbb A_2$ & $19$ & $2_2^{+2},16_3^{-1}$ $\ast$\\ \hline & & & & & $(\mathbb A_2,\mathbb A_2)\subset 2\mathbb A_2$ $o$ & $19$ & $2_5^{-1},8_{II}^{-2}$ $\ast$\\ \hline \hline $16$ &$10$&$1$ &$D_{10}$& $16$ & $(\mathbb A_1,\mathbb A_1)\subset 2\mathbb A_1$ & $18$ & $2_6^{+2},5^{+4}$\\ \hline & & & & & $(\mathbb A_1,5\mathbb A_1)\subset 6\mathbb A_1$ &$18$ & $2_6^{+2},5^{-3}$\\ \hline & & & & & $(5\mathbb A_1,5\mathbb A_1)\subset 10\mathbb A_1$ &$18$ & $2_6^{+2},5^{+2}$ $\ast$\\ \hline & & & & & $(5\mathbb A_1,5\mathbb A_1)\subset 5\mathbb A_2$ &$18$ & $3^{-1},5^{-2}$ $\ast$\\ \hline & & & & & $(\mathbb A_1,\mathbb A_1,5\mathbb A_1)\subset 7\mathbb A_1$ &$19$ & $2_5^{+3},5^{-3}$\\ \hline & & & & & $(\mathbb A_1,5\mathbb A_1,5\mathbb A_1)\subset 11\mathbb A_1$ &$19$ & $2_5^{+3},5^{+2}$\\ \hline & & & & & $(\mathbb A_1,5\mathbb A_1,5\mathbb A_1)\subset \mathbb A_1\amalg 5\mathbb A_2$ &$19$ & $2_7^{+1},3^{-1},5^{-2}$\\ \hline & & & & & $(5\mathbb A_1,5\mathbb A_1,5\mathbb A_1)\subset 15\mathbb A_1$ $o$ &$19$ & $2_1^{-3},5^{-1}$ $\ast$ \\ \hline & & & & & $(5\mathbb A_1,5\mathbb A_1,5\mathbb A_1)\subset 5\mathbb A_2\amalg 5\mathbb A_1$ &$19$ & $2_7^{+1},3^{-1},5^{+1}$ $\ast$\\ \hline & & & & & $ \left(\begin{array}{rrr} 5\mathbb A_1 & 5\mathbb A_2 & 10\mathbb A_1 \\ & 5\mathbb A_1 & 5\mathbb A_2 \\ & & 5\mathbb A_1 \end{array}\right) \subset 5\mathbb A_3 $ & $19$ & $4_1^{+1},5^{+1}$ $\ast$ \\ \hline \hline $17$&$12$ & $3$&$\AAA_4$& $16$ & $(\mathbb A_1,\mathbb A_1)\subset 2\mathbb A_1$\ $o$ & $18$ & $2_2^{+2},4_{II}^{+2},3^{+2}$ $\ast$\\ \hline & & & & & $(\mathbb A_1,3\mathbb A_1)\subset 4\mathbb A_1$\ $o$ & $18$ & $2_{II}^{-2},4_{II}^{+2},3^{-1}$ $\ast$ \\ \hline & & & & & $(\mathbb A_1,4\mathbb A_1)\subset 5\mathbb A_1$ & $18$ & $2_3^{+3},8_7^{+1},3^{+2}$ $\ast$\\ \hline & & & & & $(\mathbb A_1,6\mathbb A_1)\subset 7\mathbb A_1$ & $18$ & $2_1^{+1},4_7^{+3},3^{+1}$ $\ast$\\ \hline & & & & & $(\mathbb A_1,12\mathbb A_1)\subset 13\mathbb A_1$ & $18$ & $2_3^{+3},8_1^{+1},3^{-1}$ $\ast$ \\ \hline & & & & & $(3\mathbb A_1,4\mathbb A_1)\subset 7\mathbb A_1$ & $18$ & $2_5^{+3},8_7^{+1},3^{-1}$ $\ast$ \\ \hline & & & & & $(3\mathbb A_1,6\mathbb A_1)\subset 3\mathbb A_3$ & $18$ & $2_6^{+2},4_{II}^{-2}$ $\ast$ \\ \hline & & & & & $(3\mathbb A_1,12\mathbb A_1)\subset 15\mathbb A_1$ & $18$ & $2_1^{-3},8_1^{+1}$ $\ast$ \\ \hline & & & & & $(4\mathbb A_1,4\mathbb A_1)\subset 8\mathbb A_1$\ $o$ & $18$ & $4_2^{+2},3^{+2}$ $\ast$ \\ \hline & & & & & $(4\mathbb A_1,4\mathbb A_1)\subset 4\mathbb A_2$ & $18$ & $2_{II}^{-2},3^{+3}$ $\ast$ \\ \hline & & & & & $(4\mathbb A_1,6\mathbb A_1)\subset 10\mathbb A_1$ & $18$ & $4_1^{+1},8_3^{-1},3^{+1}$ $\ast$ \\ \hline & & & & & $(4\mathbb A_1,12\mathbb A_1)\subset 16\mathbb A_1$\ $o$ & $18$ & $2_{II}^{-4},3^{-1}$ $\ast$ \\ \hline & & & & & $(4\mathbb A_1,12\mathbb A_1)\subset 4\mathbb D_4$ & $18$ & $2_{II}^{-2},3^{-1}$ $\ast$ \\ \hline & & & & & $(6\mathbb A_1,6\mathbb A_1)\subset 12\mathbb A_1$ \ $o$& $18$ & $2_{II}^{-2},4_2^{-2}$ $\ast$ \\ \hline \end{tabular} \end{table} \begin{table} \begin{tabular}{\|c\|\|c\|c\|c\|c\|c\|c\|c\|c\|} \hline & & & & & $(6\mathbb A_1,6\mathbb A_1)\subset 6\mathbb A_2$\ $o$ & $18$ & $4_{II}^{-2},3^{+1}$ $\ast$ \\ \hline & & & & & $(6\mathbb A_1,12\mathbb A_1)\subset 6\mathbb A_3$ & $18$ & $4_6^{+2}$ $\ast$ \\ \hline & & & & & $(\mathbb A_1,\mathbb A_1,\mathbb A_1)\subset 3\mathbb A_1$ $o$ & $19$ & $2_{1}^{+1},4_{II}^{+2},3^{+2}$ $\ast$ \\ \hline & & & & & $(\mathbb A_1,\mathbb A_1,4\mathbb A_1)\subset 6\mathbb A_1$\ $o$ & $19$ & $2_{2}^{+2},8_{7}^{+1},3^{+2}$ $\ast$ \\ \hline & & & & & $(\mathbb A_1,\mathbb A_1,6\mathbb A_1)\subset 8\mathbb A_1$\ $o$ & $19$ & $4_{7}^{-3},3^{+1}$ $\ast$ \\ \hline & & & & & $(\mathbb A_1,\mathbb A_1,12\mathbb A_1)\subset 14\mathbb A_1$ \ $o$ & $19$ & $2_{2}^{+2},8_{5}^{-1},3^{-1}$ $\ast$ \\ \hline & & & & & $(\mathbb A_1,3\mathbb A_1,4\mathbb A_1)\subset 8\mathbb A_1$ $o$ & $19$ & $2_{II}^{-2},8_{7}^{+1},3^{-1}$ $\ast$ \\ \hline & & & & & $(\mathbb A_1,3\mathbb A_1,12\mathbb A_1)\subset 16\mathbb A_1$ $o$ & $19$ & $2_{II}^{-2},8_{5}^{-1}$ $\ast$ \\ \hline & & & & & $(\mathbb A_1,4\mathbb A_1,4\mathbb A_1)\subset 9\mathbb A_1$\ $o$ & $19$ & $2_{1}^{+1},4_{0}^{+2},3^{+2}$ $\ast$ \\ \hline & & & & & $(\mathbb A_1,4\mathbb A_1,4\mathbb A_1)\subset \mathbb A_1\amalg 4\mathbb A_2$ &$19$ & $2_{7}^{-3},3^{+3}$ $\ast$ \\ \hline & & & & & $(\mathbb A_1,4\mathbb A_1,6\mathbb A_1)\subset 11\mathbb A_1$ & $19$ & $2_{3}^{-1},4_{7}^{+1},8_{5}^{-1},3^{+1}$ \\ \hline & & & & & $(\mathbb A_1,4\mathbb A_1,12\mathbb A_1)\subset \mathbb A_1\amalg 4\mathbb D_4$ &$19$ & $2_{7}^{-3},3^{-1}$ $\ast$ \\ \hline & & & & & $(\mathbb A_1,6\mathbb A_1,6\mathbb A_1)\subset \mathbb A_1\amalg 6\mathbb A_2$\ $o$ &$19$ & $2_{3}^{-1},4_{II}^{-2},3^{+1}$ $\ast$ \\ \hline & & & & & $(\mathbb A_1,6\mathbb A_1,12\mathbb A_1)\subset \mathbb A_1\amalg 6\mathbb A_3$ & $19$ & $2_{7}^{+1},4_6^{+2}$ \\ \hline & & & & & $(3\mathbb A_1,4\mathbb A_1,4\mathbb A_1)\subset 11\mathbb A_1$ \ $o$ & $19$ & $2_{1}^{+1},4_{2}^{+2},3^{-1}$ $\ast$ \\ \hline & & & & & $(3\mathbb A_1,4\mathbb A_1,4\mathbb A_1)\subset 3\mathbb A_1\amalg 4\mathbb A_2$ & $19$ & $2_{5}^{+3},3^{-2}$ $\ast$ \\ \hline & & & & & $(3\mathbb A_1,4\mathbb A_1,12\mathbb A_1)\subset 3\mathbb A_1\amalg 4\mathbb D_4$ & $19$ & $2_{5}^{+3}$ $\ast$ \\ \hline & & & & & $(3\mathbb A_1,6\mathbb A_1,4\mathbb A_1)\subset 3\mathbb A_3\amalg 4\mathbb A_1$ & $19$ & $2_{6}^{+2},8_{3}^{-1}$ \\ \hline & & & & & $(4\mathbb A_1,4\mathbb A_1,4\mathbb A_1)\subset 12\mathbb A_1$ $o$ & $19$ & $8_{1}^{+1},3^{+2}$ $\ast$ \\ \hline & & & & & $ \left(\begin{array}{rrr} 4\mathbb A_1 & 4\mathbb A_2 & 8\mathbb A_1 \\ & 4\mathbb A_1 & 4\mathbb A_2 \\ & & 4\mathbb A_1 \end{array}\right) \subset 4\mathbb A_3 $\ $o$ & $19$ & $4_1^{+1},3^{+2}$ $\ast$ \\ \hline & & & & & $(4\mathbb A_1,4\mathbb A_1,6\mathbb A_1)\subset 14\mathbb A_1$ \ $o$ & $19$ & $2_{2}^{-2},4_{1}^{+1},3^{+1}$ $\ast$ \\ \hline & & & & & $(4\mathbb A_1,4\mathbb A_1,6\mathbb A_1)\subset 4\mathbb A_2\amalg 6\mathbb A_1$ & $19$ & $4_{1}^{+1},3^{+2}$ $\ast$ \\ \hline & & & & & $(4\mathbb A_1,6\mathbb A_1,6\mathbb A_1)\subset 16\mathbb A_1$ $o$ & $19$ & $2_{II}^{-2},8_{5}^{-1}$ $\ast$ \\ \hline & & & & & $(4\mathbb A_1,6\mathbb A_1,6\mathbb A_1)\subset 4\mathbb A_1\amalg 6\mathbb A_2$ \ $o$ & $19$ & $8_{3}^{-1},3^{+1}$ $\ast$ \\ \hline \end{tabular} \end{table} \begin{table} \begin{tabular}{\|c\|\|c\|c\|c\|c\|c\|c\|} \hline {\bf n}& $\|G\|$& $i$& $G$ &$Deg$& $\rk S$ & $q_S$ \\ \hline \hline $18$&$12$ &$4$ &$D_{12}$& $(\mathbb A_1,2\mathbb A_1)\subset 3\mathbb A_1$ & $18$ & $2_7^{+3},4_7^{+1},3^{+4}$\\ \hline & & & & $(\mathbb A_1,3\mathbb A_1)\subset 4\mathbb A_1$ & $18$ & $2_0^{+4},3^{-3}$ $\ast$\\ \hline & & & & $(\mathbb A_1,6\mathbb A_1)\subset 7\mathbb A_1$ & $18$ & $2_1^{-3},4_7^{+1},3^{+3}$ $\ast$\\ \hline & & & & $(2\mathbb A_1,3\mathbb A_1)\subset 5\mathbb A_1$ & $18$ & $2_1^{+3},4_7^{+1},3^{-3}$\\ \hline & & & & $\left((2\mathbb A_1,6\mathbb A_1)\subset 8\mathbb A_1\right)_I$ & $18$ & $4_4^{-2},3^{+3}$ $\ast$ \\ \hline & & & & $\left((2\mathbb A_1,6\mathbb A_1)\subset 8\mathbb A_1\right)_{II}$ & $18$ & $2_{II}^{-4},3^{+3}$ \\ \hline & & & & $(2\mathbb A_1,6\mathbb A_1)\subset 2\mathbb D_4$ & $18$ & $2_{II}^{-2},3^{+3}$ $\ast$\\ \hline & & & & $(3\mathbb A_1,6\mathbb A_1)\subset 9\mathbb A_1$ & $18$ & $2_7^{+3},4_7^{+1},3^{-2}$ $\ast$\\ \hline & & & & $(3\mathbb A_1,6\mathbb A_1)\subset 3\mathbb A_3$ & $18$ & $2_6^{-4},3^{+2}$ $\ast$\\ \hline & & & & $(6\mathbb A_1,6\mathbb A_1)\subset 12\mathbb A_1$ & $18$ & $4_2^{+2},3^{+2}$ $\ast$\\ \hline & & & & $(6\mathbb A_1,6\mathbb A_1)\subset 6\mathbb A_2$ & $18$ & $2_{II}^{-2},3^{-1},9^{-1}$ $\ast$\\ \hline & & & & $(\mathbb A_1,2\mathbb A_1,3\mathbb A_1)\subset 6\mathbb A_1$ & $19$ & $2_0^{+2},4_7^{+1},3^{-3}$ $\ast$ \\ \hline & & & & $ \left(\begin{array}{rrr} \mathbb A_1 & 3\mathbb A_1 & 7\mathbb A_1 \\ & 2\mathbb A_1 & (8\mathbb A_1)_I \\ & & 6\mathbb A_1 \end{array}\right) \subset 9\mathbb A_1 $ & $19$ & $2_1^{+1},4_2^{+2},3^{+3}$\\ \hline & & & & $ \left(\begin{array}{rrr} \mathbb A_1 & 3\mathbb A_1 & 7\mathbb A_1 \\ & 2\mathbb A_1 & 2\mathbb D_4 \\ & & 6\mathbb A_1 \end{array}\right) \subset \mathbb A_1\amalg 2\mathbb D_4 $ & $19$ & $2_7^{-3},3^{+3}$ $\ast$\\ \hline & & & & $(\mathbb A_1,3\mathbb A_1,6\mathbb A_1)\subset 10 \mathbb A_1$ & $19$ & $2_2^{-2},4_7^{+1},3^{-2}$ $\ast$\\ \hline & & & & $ \left(\begin{array}{rrr} \mathbb A_1 & 4\mathbb A_1 & 7\mathbb A_1 \\ & 3\mathbb A_1 & 3\mathbb A_3 \\ & & 6\mathbb A_1 \end{array}\right) \subset \mathbb A_1\amalg 3\mathbb A_3 $ & $19$ & $2_1^{+3},3^{+2}$ $\ast$ \\ \hline & & & & $(\mathbb A_1,6\mathbb A_1,6\mathbb A_1)\subset 13\mathbb A_1$ & $19$ & $2_7^{+1},4_2^{+2},3^{+2}$ $\ast$\\ \hline & & & & $(\mathbb A_1,6\mathbb A_1,6\mathbb A_1)\subset \mathbb A_1\amalg 6\mathbb A_2$ & $19$ & $2_3^{+3},3^{-1},9^{-1}$ $\ast$\\ \hline & & & & $ \left(\begin{array}{rrr} 2\mathbb A_1 & 5\mathbb A_1 & (8\mathbb A_1)_I \\ & 3\mathbb A_1 & 9\mathbb A_1 \\ & & 6\mathbb A_1 \end{array}\right) \subset 11\mathbb A_1 $ & $19$ & $2_7^{+1},4_6^{+2},3^{-2}$\\ \hline & & & & $ \left(\begin{array}{rrr} 2\mathbb A_1 & 5\mathbb A_1 & (8\mathbb A_1)_I \\ & 3\mathbb A_1 & 3\mathbb A_3 \\ & & 6\mathbb A_1 \end{array}\right) \subset 2\mathbb A_1\amalg 3\mathbb A_3$ & $19$ & $2_2^{+2},4_7^{+1},3^{+2}$ $\ast$\\ \hline \end{tabular} \end{table} \begin{table} \begin{tabular}{\|c\|\|c\|c\|c\|c\|c\|c\|c\|c\|} \hline & & & & $ \left(\begin{array}{rrr} 2\mathbb A_1 & 5\mathbb A_1 & (8\mathbb A_1)_{II} \\ & 3\mathbb A_1 & 3\mathbb A_3 \\ & & 6\mathbb A_1 \end{array}\right) \subset 2\mathbb A_1\amalg 3\mathbb A_3$ & $19$ & $2_{II}^{+2},4_1^{+1},3^{+2}$\\ \hline & & & & $ \left(\begin{array}{rrr} 2\mathbb A_1 & 5\mathbb A_1 & 2\mathbb D_4 \\ & 3\mathbb A_1 & 9\mathbb A_1 \\ & & 6\mathbb A_1 \end{array}\right) \subset 3\mathbb A_1\amalg 2\mathbb D_4$ & $19$ & $2_5^{+3},3^{-2}$ $\ast$\\ \hline & & & & $ \left(\begin{array}{rrr} 2\mathbb A_1 & (8\mathbb A_1)_I & (8\mathbb A_1)_{II} \\ & 6\mathbb A_1 & 12\mathbb A_1 \\ & & 6\mathbb A_1 \end{array}\right) \subset 14\mathbb A_1$ & $19$ & $2_{II}^{+2},4_1^{+1},3^{+2}$\\ \hline & & & & $ \left(\begin{array}{rrr} 2\mathbb A_1 & (8\mathbb A_1)_I & (8\mathbb A_1)_I \\ & 6\mathbb A_1 & 6\mathbb A_2 \\ & & 6\mathbb A_1 \end{array}\right) \subset 2\mathbb A_1\amalg 6\mathbb A_2$ & $19$ & $4_3^{-1},3^{-1},9^{-1}$ $\ast$\\ \hline & & & & $ \left(\begin{array}{rrr} 2\mathbb A_1 & 2\mathbb D_4 & (8\mathbb A_1)_I \\ & 6\mathbb A_1 & 12\mathbb A_1 \\ & & 6\mathbb A_1 \end{array}\right) \subset 2\mathbb D_4\amalg 6\mathbb A_1$ & $19$ & $4_1^{+1},3^{+2}$ $\ast$\\ \hline & & & & $(3\mathbb A_1,6\mathbb A_1,6\mathbb A_1)\subset 15 \mathbb A_1$ & $19$ & $2_3^{-1},4_4^{-2},3^{-1}$ $\ast$\\ \hline & & & & $(3\mathbb A_1,6\mathbb A_1,6\mathbb A_1)\subset 3\mathbb A_1\amalg 6\mathbb A_2 $ & $19$ & $2_1^{-3},9^{-1}$ $\ast$\\ \hline & & & & $ \left(\begin{array}{rrr} 3\mathbb A_1 & 3\mathbb A_3 & 9\mathbb A_1 \\ & 6\mathbb A_1 & 12\mathbb A_1 \\ & & 6\mathbb A_1 \end{array}\right) \subset 3\mathbb A_3\amalg 6\mathbb A_1$ & $19$ & $2_6^{-2},4_5^{-1},3^{+1}$ $\ast$\\ \hline & & & & $ \left(\begin{array}{rrr} 3\mathbb A_1 & 3\mathbb A_3 & 9\mathbb A_1 \\ & 6\mathbb A_1 & 6\mathbb A_2 \\ & & 6\mathbb A_1 \end{array}\right) \subset 3\mathbb A_5$ & $19$ & $2_3^{+3},3^{-1}$ $\ast$\\ \hline \hline {\bf n}& $\|G\|$& $i$& $G$ & $Deg$& $\rk S$ &$q_S$ \\ \hline \hline $21$&$16$ &$14$&$C_2^4$ & $(4\mathbb A_1,4\mathbb A_1)\subset 8\mathbb A_1$ & $17$ & $2_{II}^{+4},8_7^{+1}$ $\ast$\\ \hline & & & & $(4\mathbb A_1,4\mathbb A_1,4\mathbb A_1)\subset 12\mathbb A_1$\ $o$ & $18$ & $2_{II}^{-2},4_2^{-2}$ $\ast$ \\ \hline & & & & $(4\mathbb A_1,4\mathbb A_1,4\mathbb A_1,4\mathbb A_1)\subset 16\mathbb A_1$ $o$ & $19$ & $2_{II}^{-2},8_5^{-1}$ $\ast$ \\ \hline \end{tabular} \end{table} \begin{table} \begin{tabular}{\|c\|\|c\|c\|c\|c\|c\|c\|c\|c\|} \hline {\bf n}& $\|G\|$& $i$& $G$ & $Deg$& $\rk S$ &$q_S$ \\ \hline \hline $22$& $16$&$11$&$C_2\times D_8$& $(2\mathbb A_1,2\mathbb A_1)\subset 4\mathbb A_1$ \ $o$ & $18$ & $4_6^{+4}$ $\ast$\\ \hline & & & & $(2\mathbb A_1,4\mathbb A_1)\subset 6\mathbb A_1$ & $18$ & $4_7^{+3},8_7^{+1}$ $\ast$ \\ \hline & & & & $(2\mathbb A_1,8\mathbb A_1)\subset 10\mathbb A_1$ & $18$ & $4_6^{+4}$\\ \hline & & & & $((4\mathbb A_1,4\mathbb A_1)\subset 8\mathbb A_1)_I$\ $o$ & $18$ & $4_6^{+4}$ $\ast$\\ \hline & & & & $((4\mathbb A_1,4\mathbb A_1)\subset 8\mathbb A_1)_{II}$ & $18$ & $2_{II}^{-2},8_6^{-2}$ $\ast$\\ \hline & & & & $(4\mathbb A_1,8\mathbb A_1)\subset 12\mathbb A_1$ & $18$ & $2_{II}^{+2},4_7^{+1},8_7^{+1}$ $\ast$\\ \hline & & & & $(4\mathbb A_1,8\mathbb A_1)\subset 4\mathbb A_3$ & $18$ & $2_{II}^{-2},4_2^{-2}$ $\ast$\\ \hline & & & & $(8\mathbb A_1,8\mathbb A_1)\subset 16\mathbb A_1$ \ $o$ & $18$ & $2_{II}^{+2},4_6^{+2}$ $\ast$\\ \hline & & & & $(2\mathbb A_1,2\mathbb A_1,4\mathbb A_1)\subset 8\mathbb A_1$ $o$ & $19$ & $4_6^{+2},8_7^{+1}$ $\ast$\\ \hline & & & & $(2\mathbb A_1,2\mathbb A_1,8\mathbb A_1)\subset 12\mathbb A_1$ $o$ & $19$ & $4_5^{+3}$ $\ast$\\ \hline & & & & $(2\mathbb A_1,(4\mathbb A_1,4\mathbb A_1)_{II})\subset 10\mathbb A_1$ & $19$ & $4_5^{-1},8_4^{-2}$ $\ast$\\ \hline & & & & $(2\mathbb A_1,4\mathbb A_1,8\mathbb A_1)\subset 14\mathbb A_1$ & $19$ & $4_4^{-2},8_5^{-1}$\\ \hline & & & & $ \left(\begin{array}{rrr} 2\mathbb A_1 & 6\mathbb A_1 & 10\mathbb A_1 \\ & 4\mathbb A_1 & 4\mathbb A_3 \\ & & 8\mathbb A_1 \end{array}\right)\subset 2\mathbb A_1\amalg 4\mathbb A_3$ & $19$ & $4_5^{+3}$ $\ast$ \\ \hline & & & & $((4\mathbb A_1,\,4\mathbb A_1)_I,\,8\mathbb A_1)\subset 16\mathbb A_1$ $o$ & $19$ & $4_5^{+3}$ $\ast$\\ \hline & & & & $ \left(\begin{array}{rrr} 4\mathbb A_1 & (8\mathbb A_1)_{II} & 12\mathbb A_1 \\ & 4\mathbb A_1 & 4\mathbb A_3 \\ & & 8\mathbb A_1 \end{array}\right)\subset 4\mathbb A_1\amalg 4\mathbb A_3$ & $19$ & $2_{II}^{-2},8_5^{-1}$ $\ast$\\ \hline \hline $30$ & $18$& $4$&$\AAA_{3,3}$& $(3\mathbb A_1,\,3\mathbb A_1)\subset 6\mathbb A_1$ & $18$ & $2_2^{+2},3^{+2},9^{-1}$ $\ast$\\ \hline & & & & $(3\mathbb A_1,\,9\mathbb A_1)\subset 12\mathbb A_1$ & $18$ & $2_0^{+3},3^{-3}$ $\ast$\\ \hline & & & & $(3\mathbb A_1,\,9\mathbb A_1)\subset 3\mathbb D_4$ & $18$ & $3^{-3}$ $\ast$\\ \hline & & & & $(9\mathbb A_1,\,9\mathbb A_1)\subset 9\mathbb A_2$ & $18$ & $3^{-3}$ $\ast$\\ \hline & & & & $(3\mathbb A_1,\,3\mathbb A_1,\,3\mathbb A_1)\subset 9\mathbb A_1$& $19$& $2_7^{-3},3^{-1},9^{-1}$ $\ast$\\ \hline & & & &$(3\mathbb A_1,\,3\mathbb A_1,\,9\mathbb A_1)\subset 15\mathbb A_1$ &$19$ & $2_1^{+3},3^{+2}$ $\ast$\\ \hline & & & & $ \left(\begin{array}{rrr} 3\mathbb A_1 & 6\mathbb A_1 & 12\mathbb A_1 \\ & 3\mathbb A_1 & 3\mathbb D_4 \\ & & 9\mathbb A_1 \end{array}\right)\subset 3\mathbb A_1\amalg 3\mathbb D_4$ & $19$ & $2_1^{+1},3^{+2}$ $\ast$\\ \hline \end{tabular} \end{table} \begin{table} \begin{tabular}{\|c\|\|c\|c\|c\|c\|c\|c\|c\|c\|} \hline {\bf n}& $\|G\|$& $i$& $G$ & $\rk S_G$ & $Deg$& $\rk S$ &$q_S$ \\ \hline \hline $34$& $24$&$12$&$\SSS_4$ & $17$ & $(\mathbb A_1,\mathbb A_1)\subset 2\mathbb A_1$ $o$ & $19$ & $4_1^{+3},3^{+2}$ $\ast$\\ \hline & & & & & $(\mathbb A_1,2\mathbb A_1)\subset 3\mathbb A_1$ $o$ & $19$ & $2_1^{+1},4_{II}^{+2},3^{+2}$ $\ast$\\ \hline & & & & & $(\mathbb A_1,3\mathbb A_1)\subset 4\mathbb A_1$ $o$ & $19$ & $4_{3}^{-3},3^{-1}$ $\ast$\\ \hline & & & & & $(\mathbb A_1,4\mathbb A_1)\subset 5\mathbb A_1$& $19$ & $2_1^{+1},4_1^{+1},8_7^{+1},3^{+2}$ $\ast$\\ \hline & & & & & $(\mathbb A_1,(6\mathbb A_1)_{I})\subset 7\mathbb A_1$& $19$ & $2_1^{+1},4_{6}^{-2},3^{+1}$ $\ast$\\ \hline & & & & & $(\mathbb A_1,8\mathbb A_1)\subset 9\mathbb A_1$& $19$ & $2_1^{+1},4_{0}^{+2},3^{+2}$ $\ast$\\ \hline & & & & & $(\mathbb A_1,12\mathbb A_1)\subset 13\mathbb A_1$& $19$ & $2_7^{+1},4_{5}^{-1},8_7^{+1},3^{-1}$\\ \hline & & & & & $(\mathbb A_1,6\mathbb A_2)\subset \mathbb A_1\amalg 6\mathbb A_2$& $19$ & $2_7^{+1},4_{II}^{+2},3^{+1}$ $\ast$\\ \hline & & & & & $(2\mathbb A_1,4\mathbb A_1)\subset 6\mathbb A_1$ & $19$ & $2_2^{+2},8_{7}^{+1},3^{+2}$ $\ast$\\ \hline & & & & & $(2\mathbb A_1,(6\mathbb A_1)_{II})\subset 8\mathbb A_1$ & $19$ & $4_{7}^{-3},3^{+1}$ $\ast$\\ \hline & & & & & $(2\mathbb A_1,12\mathbb A_1)\subset 14\mathbb A_1$ & $19$ & $2_2^{+2},8_{1}^{+1},3^{-1}$ $\ast$ \\ \hline & & & & & $(3\mathbb A_1,4\mathbb A_1)\subset 7\mathbb A_1$ & $19$ & $2_1^{+1},4_1^{+1},8_{5}^{-1},3^{-1}$ $\ast$\\ \hline & & & & & $(3\mathbb A_1,(6\mathbb A_1)_{II}) \subset 3\mathbb A_3$ & $19$ & $4_5^{+3}$ $\ast$\\ \hline & & & & & $(3\mathbb A_1,8\mathbb A_1)\subset 11\mathbb A_1$ & $19$ & $2_1^{+1},4_2^{-2},3^{-1}$ $\ast$\\ \hline & & & & & $(3\mathbb A_1,12\mathbb A_1)\subset 15\mathbb A_1$ & $19$ & $2_7^{+1},4_7^{+1},8_{7}^{+1}$ $\ast$\\ \hline & & & & & $(4\mathbb A_1,4\mathbb A_1)\subset 8\mathbb A_1$ $o$ & $19$ & $2_{II}^{+2},4_1^{+1},3^{+2}$ $\ast$\\ \hline & & & & & $(4\mathbb A_1,(6\mathbb A_1)_I)\subset 10\mathbb A_1$ & $19$ & $2_{6}^{+2},8_1^{+1},3^{+1}$ $\ast$\\ \hline & & & & & $(4\mathbb A_1,(6\mathbb A_1)_{II})\subset 10\mathbb A_1$ & $19$ & $2_{II}^{+2},8_3^{-1},3^{+1}$\\ \hline & & & & & $(4\mathbb A_1,8\mathbb A_1)\subset 12\mathbb A_1$ $o$ & $19$ & $8_1^{+1},3^{+2}$ $\ast$\\ \hline & & & & & $(4\mathbb A_1,8\mathbb A_1)\subset 4\mathbb A_3$ & $19$ & $4_1^{+1},3^{+2}$ $\ast$\\ \hline & & & & & $(4\mathbb A_1,12\mathbb A_1)\subset 16\mathbb A_1$ $o$ & $19$ & $2_{II}^{+2},4_3^{-1},3^{-1}$ $\ast$\\ \hline & & & & & $(4\mathbb A_1,12\mathbb A_1)\subset 4\mathbb D_4$ & $19$ & $4_3^{-1},3^{-1}$ $\ast$\\ \hline & & & & & $(4\mathbb A_1,6\mathbb A_2)\subset 4\mathbb A_1\amalg 6\mathbb A_2$ & $19$ & $8_3^{-1},3^{+1}$ $\ast$\\ \hline & & & & & $((6\mathbb A_1)_I,(6\mathbb A_1)_{II})\subset 12\mathbb A_1$ $o$ & $19$ & $4_5^{+3}$ $\ast$\\ \hline & & & & & $((6\mathbb A_1)_I,8\mathbb A_1)\subset 14\mathbb A_1$& $19$ & $2_4^{-2},4_7^{+1},3^{+1}$ $\ast$\\ \hline & & & & & $((6\mathbb A_1)_I,12\mathbb A_1)\subset 6\mathbb A_3$& $19$ & $2_6^{-2},4_3^{-1}$ $\ast$\\ \hline \hline $39$&$32$ &$27$&$2^4C_2$& $17$ & $(4\mathbb A_1,4\mathbb A_1)\subset 8\mathbb A_1$ $o$ & $19$& $4_4^{-2},8_5^{-1}$ $\ast$\\ \hline & & & & & $(4\mathbb A_1,8\mathbb A_1)\subset 12\mathbb A_1$ $o$ & $19$& $4_5^{+3}$ $\ast$\\ \hline & & & & & $(8\mathbb A_1,8\mathbb A_1)\subset 16\mathbb A_1$ $o$ & $19$& $2_{II}^{-2},8_5^{-1}$ $\ast$\\ \hline \hline $40$ &$32$ &$49$&$Q_8Q_8$& $17$ & $(8\mathbb A_1,8\mathbb A_1)\subset 16\mathbb A_1$ $o$ & $19$ & $4_5^{+3}$ $\ast$\\ \hline \hline $49$ & $48$&$50$&$2^4C_3$& $17$ & $(4\mathbb A_1,4\mathbb A_1)\subset 8\mathbb A_1$ $o$ &$19$ & $2_{II}^{-2},8_7^{+1},3^{-1}$ $\ast$\\ \hline & & & & & $(4\mathbb A_1,12\mathbb A_1)\subset 16\mathbb A_1$ $o$ & $19$ & $2_{II}^{-2},8_5^{-1}$ $\ast$ \\ \hline \end{tabular} \end{table} \begin{table} \label{table3} \caption{Types and lattices $S$ of degenerations of codimension $\ge 2$ of K\"ahlerian K3 surfaces with symplectic automorphism group $D_8$.} \begin{tabular}{\|c\|\|c\|c\|c\|c\|c\|c\|c\|c\|} \hline {\bf n}& $\|G\|$& $i$& $G$ & $\rk S_G$ &$Deg$& $\rk S$ &$q_S$ \\ \hline \hline $10$&$8$& $3$ & $D_8$ & $15$ & $(\mathbb A_1,\mathbb A_1)\subset 2\mathbb A_1$\ $o$ & $17$ & $4_7^{+5}$ $\ast$\\ \hline & & & & & $(\mathbb A_1,(2\mathbb A_1)_I)\subset 3\mathbb A_1$ & $17$ & $2_7^{+1},4_0^{+4}$ \\ \hline & & & & & $(\mathbb A_1,(2\mathbb A_1)_{II})\subset \mathbb A_3$ & $17$ & $4_7^{+5}$ $\ast$\\ \hline & & & & & $(\mathbb A_1,4\mathbb A_1)\subset 5\mathbb A_1$ & $17$ & $2_7^{+1},4_1^{+3},8_7^{+1}$\\ \hline & & & & & $(\mathbb A_1,8\mathbb A_1)\subset 9\mathbb A_1$ & $17$ & $2_1^{+1},4_6^{+4}$\\ \hline & & & & & $(\mathbb A_1,2\mathbb A_2)\subset \mathbb A_1\amalg 2\mathbb A_2$ & $17$ & $2_7^{+1},4_{II}^{+4}$ $\ast$\\ \hline & & & & & $((2\mathbb A_1)_I,(2\mathbb A_1)_{II})\subset 4\mathbb A_1$ & $17$ & $4_7^{+5}$ $\ast$\\ \hline & & & & & $((2\mathbb A_1)_I,4\mathbb A_1)\subset 6\mathbb A_1)_I$ & $17$ & $2_6^{+2},4_0^{+2},8_1^{+1}$\\ \hline & & & & & $((2\mathbb A_1)_I,4\mathbb A_1)\subset 6\mathbb A_1)_{II}$ & $17$ & $2_6^{+2},4_{II}^{+2},8_1^{+1}$\\ \hline & & & & & $((2\mathbb A_1)_{II},4\mathbb A_1)\subset 6\mathbb A_1$ & $17$ & $2_{II}^{+2},4_{II}^{+2},8_7^{+1}$\\ \hline & & & & & $((2\mathbb A_1)_I,4\mathbb A_1)\subset 2\mathbb A_3$ & $17$ & $2_0^{+2},4_7^{+3}$ $\ast$\\ \hline & & & & & $((2\mathbb A_1)_I,8\mathbb A_1)\subset 10\mathbb A_1$ & $17$ & $2_2^{+2},4_5^{+3}$ $\ast$\\ \hline & & & & & $((4\mathbb A_1,4\mathbb A_1)\subset 8\mathbb A_1)_I$ & $17$ & $4_7^{+1},8_0^{+2}$ $\ast$\\ \hline & & & & & $((4\mathbb A_1,4\mathbb A_1)\subset 8\mathbb A_1)_{II}$\ $o$ & $17$ & $2_{II}^{+2},4_7^{+3}$ $\ast$\\ \hline & & & & & $(4\mathbb A_1,4\mathbb A_1)\subset 4\mathbb A_2$ & $17$ & $4_1^{-3},3^{+1}$ $\ast$ \\ \hline & & & & & $(4\mathbb A_1,8\mathbb A_1)\subset 12\mathbb A_1$ & $17$ & $4_6^{+2},8_1^{+1}$ $\ast$\\ \hline & & & & & $(4\mathbb A_1,8\mathbb A_1)\subset 4\mathbb A_3$ & $17$ & $4_7^{+3}$ $\ast$\\ \hline & & & & & $(4\mathbb A_1,2\mathbb A_2)\subset 4\mathbb A_1\amalg 2\mathbb A_2$ & $17$ & $4_{II}^{-2},8_3^{-1}$ $\ast$ \\ \hline \hline & & & & & $(\mathbb A_1,\mathbb A_1,(2\mathbb A_1)_I)\subset 4\mathbb A_1$\ $o$ & $18$ & $4_6^{+4}$ $\ast$ \\ \hline & & & & & $(\mathbb A_1,\mathbb A_1,4\mathbb A_1)\subset 6\mathbb A_1$\ $o$ & $18$ & $4_7^{+3},8_7^{+1}$ $\ast$ \\ \hline & & & & & $(\mathbb A_1,\mathbb A_1,8\mathbb A_1)\subset 10\mathbb A_1$\ $o$ & $18$ & $4_6^{+4}$ \\ \hline & & & & & $(\mathbb A_1,(2\mathbb A_1)_{II},(2\mathbb A_1)_I)\subset \mathbb A_3\amalg 2\mathbb A_1$ & $18$ & $4_6^{+4}$ $\ast$ \\ \hline & & & & & $ \left(\begin{array}{ccc} \mathbb A_1 & 3\mathbb A_1 & 5\mathbb A_1 \\ & (2\mathbb A_1)_I & (6\mathbb A_1)_I \\ & & 4\mathbb A_1 \end{array}\right) \subset 7\mathbb A_1 $ & $18$ & $2_7^{+1},4_6^{+2},8_5^{-1}$ \\ \hline & & & & & $(\mathbb A_1,(2\mathbb A_1)_{I},4\mathbb A_1)\subset \mathbb A_1\amalg 2\mathbb A_3$ & $18$ & $2_7^{+1},4_7^{+3}$ $\ast$ \\ \hline & & & & & $(\mathbb A_1,(2\mathbb A_1)_{II},4\mathbb A_1)\subset \mathbb A_3\amalg 4\mathbb A_1$ & $18$ & $4_5^{+3},8_1^{+1}$ \\ \hline & & & & & $(\mathbb A_1,(2\mathbb A_1)_{I},8\mathbb A_1)\subset 11\mathbb A_1$ & $18$ & $2_7^{+1},4_7^{+3}$ $\ast$ \\ \hline & & & & & $ \left(\begin{array}{ccc} \mathbb A_1 & 5\mathbb A_1 & 5\mathbb A_1 \\ & 4\mathbb A_1 & (8\mathbb A_1)_I \\ & & 4\mathbb A_1 \end{array}\right) \subset 9\mathbb A_1 $ & $18$ & $2_7^{+1},4_7^{+1},8_4^{-2}$ \\ \hline & & & & & $(\mathbb A_1,4\mathbb A_1,4\mathbb A_1)\subset \mathbb A_1\amalg 4\mathbb A_2$ & $18$ & $2_7^{+1},4_1^{+3},3^{+1}$ $\ast$\\ \hline \end{tabular} \end{table} \begin{table} \begin{tabular}{\|c\|\|c\|c\|c\|c\|c\|c\|c\|c\|} \hline {\bf n}& $G$ & $\rk S_G$ &$Deg$& $\rk S$ &$q_S$ \\ \hline \hline $10$ & $D_8$ & $15$ & $(\mathbb A_1,4\mathbb A_1,8\mathbb A_1)\subset 13\mathbb A_1$ & $18$ & $2_7^{+1},4_0^{+2},8_7^{+1}$ \\ \hline & & & $(\mathbb A_1,4\mathbb A_1,8\mathbb A_1)\subset \mathbb A_1\amalg 4\mathbb A_3$ & $18$ & $2_7^{+1},4_7^{+3}$ \\ \hline & & & $(\mathbb A_1,4\mathbb A_1,2\mathbb A_2)\subset 5\mathbb A_1\amalg 2\mathbb A_2$ & $18$ & $2_3^{-1},4_{II}^{+2},8_3^{-1}$ \\ \hline & & & $ \left(\begin{array}{ccc} (2\mathbb A_1)_{I} & 4\mathbb A_1 & (6\mathbb A_1)_I \\ & (2\mathbb A_1)_{II} & 6\mathbb A_1 \\ & & 4\mathbb A_1 \end{array}\right) \subset 8\mathbb A_1 $ & $18$ & $4_5^{+3},8_1^{+1}$ \\ \hline & & & $ \left(\begin{array}{ccc} (2\mathbb A_1)_{I} & 4\mathbb A_1 & (6\mathbb A_1)_{II} \\ & (2\mathbb A_1)_{II} & 6\mathbb A_1 \\ & & 4\mathbb A_1 \end{array}\right) \subset 8\mathbb A_1 $ & $18$ & $4_7^{+3},8_7^{+1}$ $\ast$ \\ \hline & & & $ \left(\begin{array}{ccc} (2\mathbb A_1)_{II} & 4\mathbb A_1 & 6\mathbb A_1 \\ & (2\mathbb A_1)_{I} & 2\mathbb A_3 \\ & & 4\mathbb A_1 \end{array}\right) \subset 2\mathbb A_1\amalg 2\mathbb A_3 $ & $18$ & $4_6^{+4}$ $\ast$ \\ \hline & & & $ \left(\begin{array}{ccc} (2\mathbb A_1)_{I} & (6\mathbb A_1)_I & (6\mathbb A_1)_{II} \\ & 4\mathbb A_1 & (8\mathbb A_1)_I \\ & & 4\mathbb A_1 \end{array}\right) \subset 10\mathbb A_1 $ & $18$ & $2_6^{+2},8_0^{+2}$ \\ \hline & & & $ \left(\begin{array}{ccc} (2\mathbb A_1)_{I} & (6\mathbb A_1)_I & (6\mathbb A_1)_I \\ & 4\mathbb A_1 & (8\mathbb A_1)_{II} \\ & & 4\mathbb A_1 \end{array}\right) \subset 10\mathbb A_1 $\ $o$ & $18$ & $4_6^{+4}$ \\ \hline & & & $ \left(\begin{array}{ccc} (2\mathbb A_1)_{II} & 6\mathbb A_1 & 6\mathbb A_1 \\ & 4\mathbb A_1 & (8\mathbb A_1)_I \\ & & 4\mathbb A_1 \end{array}\right) \subset 10\mathbb A_1 $ & $18$ & $2_{II}^{+2},8_6^{+2}$ \\ \hline & & & $ \left(\begin{array}{ccc} (2\mathbb A_1)_{II} & 6\mathbb A_1 & 6\mathbb A_1 \\ & 4\mathbb A_1 & (8\mathbb A_1)_{II} \\ & & 4\mathbb A_1 \end{array}\right) \subset 10\mathbb A_1 $\ $o$ & $18$ & $4_6^{+4}$ \\ \hline & & & $ \left(\begin{array}{ccc} (2\mathbb A_1)_{I} & (6\mathbb A_1)_I & (6\mathbb A_1)_I \\ & 4\mathbb A_1 & 4\mathbb A_2 \\ & & 4\mathbb A_1 \end{array}\right) \subset 2\mathbb A_1\amalg 4\mathbb A_2 $ & $18$ & $2_2^{+2},4_6^{+2},3^{+1}$ $\ast$ \\ \hline & & & $ \left(\begin{array}{ccc} (2\mathbb A_1)_{II} & 6\mathbb A_1 & 6\mathbb A_1 \\ & 4\mathbb A_1 & 4\mathbb A_2 \\ & & 4\mathbb A_1 \end{array}\right) \subset 2\mathbb A_1\amalg 4\mathbb A_2 $ & $18$ & $2_{II}^{+2},4_{II}^{-2},3^{+1}$ \\ \hline & & & $ \left(\begin{array}{ccc} (2\mathbb A_1)_{I} & 2\mathbb A_3 & (6\mathbb A_1)_I \\ & 4\mathbb A_1 & (8\mathbb A_1)_I \\ & & 4\mathbb A_1 \end{array}\right) \subset 2\mathbb A_3\amalg 4\mathbb A_1 $ & $18$ & $2_0^{+2},4_7^{+1},8_3^{-1}$ \\ \hline & & & $ \left(\begin{array}{ccc} (2\mathbb A_1)_{I} & 2\mathbb A_3 & (6\mathbb A_1)_{II} \\ & 4\mathbb A_1 & 4\mathbb A_2 \\ & & 4\mathbb A_1 \end{array}\right) \subset 2\mathbb A_5 $ & $18$ & $2_2^{-2},4_{II}^{+2}$ $\ast$ \\ \hline \end{tabular} \end{table} \begin{table} \begin{tabular}{\|c\|\|c\|c\|c\|c\|c\|c\|c\|c\|} \hline {\bf n}& $G$ & $\rk S_G$ &$Deg$& $\rk S$ &$q_S$ \\ \hline \hline $10$ & $D_8$ & $15$ & $ \left(\begin{array}{ccc} (2\mathbb A_1)_{I} & (6\mathbb A_1)_I & 10\mathbb A_1 \\ & 4\mathbb A_1 & 12\mathbb A_1 \\ & & 8\mathbb A_1 \end{array}\right) \subset 14\mathbb A_1 $ & $18$ & $2_2^{-2},4_1^{+1},8_3^{-1}$ $\ast$ \\ \hline & & & $ \left(\begin{array}{ccc} (2\mathbb A_1)_{I} & (6\mathbb A_1)_I & 10\mathbb A_1 \\ & 4\mathbb A_1 & 4\mathbb A_3 \\ & & 8\mathbb A_1 \end{array}\right) \subset 2\mathbb A_1\amalg 4\mathbb A_3 $ & $18$ & $2_2^{-2},4_0^{+2}$ \\ \hline & & & $ \left(\begin{array}{ccc} (2\mathbb A_1)_{I} & (6\mathbb A_1)_{II} & 10\mathbb A_1 \\ & 4\mathbb A_1 & 4\mathbb A_3 \\ & & 8\mathbb A_1 \end{array}\right) \subset 2\mathbb A_1\amalg 4\mathbb A_3 $ & $18$ & $2_2^{-2},4_{II}^{-2}$ $\ast$ \\ \hline & & & $((2\mathbb A_1)_{I},4\mathbb A_1,8\mathbb A_1)\subset 2\mathbb A_3\amalg 8\mathbb A_1$ & $18$ & $2_4^{-2},4_6^{-2}$ $\ast$ \\ \hline & & & $ \left(\begin{array}{ccc} 4\mathbb A_1 & (8\mathbb A_1)_I & (8\mathbb A_1)_{II} \\ & 4\mathbb A_1 & (8\mathbb A_1)_{I} \\ & & 4\mathbb A_1 \end{array}\right) \subset 12\mathbb A_1$\ $o$ & $18$ & $2_{II}^{-2},4_1^{+1},8_5^{-1}$ $\ast$ \\ \hline & & & $ \left(\begin{array}{ccc} 4\mathbb A_1 & (8\mathbb A_1)_I & (8\mathbb A_1)_{I} \\ & 4\mathbb A_1 & 4\mathbb A_2 \\ & & 4\mathbb A_1 \end{array}\right) \subset 4\mathbb A_1\amalg 4\mathbb A_2 $ & $18$ & $4_{7}^{+1},8_5^{-1},3^{+1}$ $\ast$ \\ \hline & & & $ \left(\begin{array}{ccc} 4\mathbb A_1 & 4\mathbb A_2 & (8\mathbb A_1)_{II} \\ & 4\mathbb A_1 & 4\mathbb A_2 \\ & & 4\mathbb A_1 \end{array}\right) \subset 4\mathbb A_3 $\ $o$ & $18$ & $2_{II}^{-2},4_2^{-2}$ $\ast$ \\ \hline & & & $ \left(\begin{array}{ccc} 4\mathbb A_1 & (8\mathbb A_1)_{II} & 12\mathbb A_1 \\ & 4\mathbb A_1 & 12\mathbb A_1 \\ & & 8\mathbb A_1 \end{array}\right) \subset 16\mathbb A_1 $\ $o$ & $18$ & $2_{II}^{+2},4_6^{+2}$ $\ast$ \\ \hline & & & $ \left(\begin{array}{ccc} 4\mathbb A_1 & (8\mathbb A_1)_I & 12\mathbb A_1 \\ & 4\mathbb A_1 & 4\mathbb A_3 \\ & & 8\mathbb A_1 \end{array}\right) \subset 4\mathbb A_1\amalg 4\mathbb A_3 $ & $18$ & $4_{5}^{-1},8_5^{-1}$ $\ast$ \\ \hline & & & $ \left(\begin{array}{ccc} 4\mathbb A_1 & 4\mathbb A_2 & 12\mathbb A_1 \\ & 4\mathbb A_1 & 4\mathbb A_3 \\ & & 8\mathbb A_1 \end{array}\right) \subset 4\mathbb D_4 $ & $18$ & $4_6^{+2}$ $\ast$ \\ \hline & & & $(4\mathbb A_1,4\mathbb A_1,2\mathbb A_2)\subset (8\mathbb A_1)_I\amalg 2\mathbb A_2$ & $18$ & $8_6^{+2}$ $\ast$ \\ \hline & & & $(4\mathbb A_1,4\mathbb A_1,2\mathbb A_2)\subset 6\mathbb A_2$ $o$ & $18$ & $4_{II}^{-2},3^{+1}$ $\ast$ \\ \hline \end{tabular} \end{table} \begin{table} \begin{tabular}{\|c\|\|c\|c\|c\|c\|c\|c\|c\|c\|} \hline {\bf n}& $G$ &$Deg$& $\rk S$ &$q_S$ \\ \hline \hline $10$ & $D_8$ & $ \left(\begin{array}{cccc} \mathbb A_1 & 2\mathbb A_1 & 3\mathbb A_1 & 5\mathbb A_1 \\ & \mathbb A_1 & 3\mathbb A_1 & 5\mathbb A_1 \\ & & (2\mathbb A_1)_I & (6\mathbb A_1)_I \\ & & & 4\mathbb A_1 \end{array}\right) \subset 8\mathbb A_1 $ $o$ & $19$ & $4_4^{-2},8_5^{-1}$ $\ast$ \\ \hline & & $ (\mathbb A_1,\,\mathbb A_1,\,(2\mathbb A_1)_I,\, 8\mathbb A_1)\subset 12\mathbb A_1$ $o$ & $19$ & $4_5^{+3}$ $\ast$ \\ \hline & & $ \left(\begin{array}{cccc} \mathbb A_1 & 2\mathbb A_1 & 5\mathbb A_1 & 5\mathbb A_1 \\ & \mathbb A_1 & 5\mathbb A_1 & 5\mathbb A_1 \\ & & 4\mathbb A_1 & (8\mathbb A_1)_I \\ & & & 4\mathbb A_1 \end{array}\right) \subset 10\mathbb A_1 $\ $o$ & $19$ & $4_5^{-1},8_4^{-2}$ $\ast$ \\ \hline & & $ (\mathbb A_1,\,\mathbb A_1,\,4\mathbb A_1,\,8\mathbb A_1)\subset 14\mathbb A_1$ $o$ & $19$ & $4_4^{-2},8_5^{-1}$ \\ \hline & & $ (\mathbb A_1,\,\mathbb A_1,\,4\mathbb A_1,\, 8\mathbb A_1)\subset 2\mathbb A_1\amalg 4\mathbb A_3$ \ $o$ & $19$ & $4_5^{+3}$ $\ast$ \\ \hline & & $ \left(\begin{array}{cccc} \mathbb A_1 & \mathbb A_3 & 3\mathbb A_1 & 5\mathbb A_1 \\ & (2\mathbb A_1)_{II} & 4\mathbb A_1 & 8\mathbb A_1 \\ & & (2\mathbb A_1)_I & (6\mathbb A_1)_I \\ & & & 4\mathbb A_1 \end{array}\right) \subset \mathbb A_3\amalg 6\mathbb A_1 $ & $19$ & $4_4^{-2},8_5^{-1}$ \\ \hline & & $ \left(\begin{array}{cccc} \mathbb A_1 & \mathbb A_3 & 3\mathbb A_1 & 5\mathbb A_1 \\ & (2\mathbb A_1)_{II} & 4\mathbb A_1 & 6\mathbb A_1 \\ & & (2\mathbb A_1)_I & 2\mathbb A_3 \\ & & & 4\mathbb A_1 \end{array}\right) \subset 3\mathbb A_3 $\ $o$ & $19$ & $4_5^{+3}$ $\ast$ \\ \hline & & $ \left(\begin{array}{cccc} \mathbb A_1 & 3\mathbb A_1 & 5\mathbb A_1 & 5\mathbb A_1 \\ & (2\mathbb A_1)_{I} & 2\mathbb A_3 & (6\mathbb A_1)_I \\ & & 4\mathbb A_1 & (8\mathbb A_1)_I \\ & & & 4\mathbb A_1 \end{array}\right) \subset \mathbb A_1\amalg 2\mathbb A_3\amalg 4\mathbb A_1 $ & $19$ & $2_3^{-1},4_5^{-1},8_1^{+1}$ $\ast$ \\ \hline & & $ \left(\begin{array}{cccc} \mathbb A_1 & \mathbb A_3 & 5\mathbb A_1 & 5\mathbb A_1 \\ & (2\mathbb A_1)_{II} & 6\mathbb A_1 & 6\mathbb A_1 \\ & & 4\mathbb A_1 & (8\mathbb A_1)_{I}\\ & & & 4\mathbb A_1 \end{array}\right) \subset \mathbb A_3\amalg 8\mathbb A_1 $ & $19$ & $4_7^{+1},8_6^{+2}$ \\ \hline & & $ \left(\begin{array}{cccc} \mathbb A_1 & 3\mathbb A_1 & 5\mathbb A_1 & 5\mathbb A_1 \\ & (2\mathbb A_1)_{I} & (6\mathbb A_1)_I & (6\mathbb A_1)_I \\ & & 4\mathbb A_1 & 4\mathbb A_2 \\ & & & 4\mathbb A_1 \end{array}\right) \subset 3\mathbb A_1\amalg 4\mathbb A_2 $ & $19$ & $2_3^{-1},4_0^{+2},3^{+1}$ $\ast$ \\ \hline & & $(\mathbb A_1,\,(2\mathbb A_1)_{II},\,4\mathbb A_1,\,4\mathbb A_1)\subset \mathbb A_3\amalg 4\mathbb A_2$ & $19$ & $4_7^{-3},3^{+1}$ $\ast$ \\ \hline & & $ \left(\begin{array}{cccc} \mathbb A_1 & 3\mathbb A_1 & 5\mathbb A_1 & 9\mathbb A_1 \\ & (2\mathbb A_1)_{I} & (6\mathbb A_1)_I & 10\mathbb A_1 \\ & & 4\mathbb A_1 & 12\mathbb A_1 \\ & & & 8\mathbb A_1 \end{array}\right) \subset 15\mathbb A_1 $ \ $o$ & $19$ & $2_1^{+1},4_5^{-1},8_3^{-1}$ $\ast$ \\ \hline \end{tabular} \end{table} \begin{table} \begin{tabular}{\|c\|\|c\|c\|c\|c\|c\|c\|c\|c\|} \hline {\bf n}& $G$ &$Deg$& $\rk S$ &$q_S$ \\ \hline \hline $10$ & $D_8$ & $ (\mathbb A_1,\,(2\mathbb A_1)_I,\,4\mathbb A_1,\, 8\mathbb A_1) \subset \mathbb A_1\amalg 2\mathbb A_3,\amalg 8\mathbb A_1$ & $19$ & $2_3^{-1},4_6^{-2}$ $\ast$ \\ \hline & & $ \left(\begin{array}{cccc} \mathbb A_1 & 3\mathbb A_1 & 5\mathbb A_1 & 9\mathbb A_1 \\ & (2\mathbb A_1)_I & (6\mathbb A_1)_{I} & 10\mathbb A_1 \\ & & 4\mathbb A_1 & 4\mathbb A_3 \\ & & & 8\mathbb A_1 \end{array}\right) \subset 3\mathbb A_1\amalg 4\mathbb A_3 $ & $19$ & $2_5^{-1},4_4^{-2}$ \\ \hline & & $ \left(\begin{array}{cccc} \mathbb A_1 & 5\mathbb A_1 & 5\mathbb A_1 & 5\mathbb A_1 \\ & 4\mathbb A_1 & (8\mathbb A_1)_I & (8\mathbb A_1)_I \\ & & 4\mathbb A_1 & 4\mathbb A_2 \\ & & & 4\mathbb A_1 \end{array}\right) \subset 5\mathbb A_1\amalg 4\mathbb A_2 $ & $19$ & $2_1^{+1},4_7^{+1},8_3^{-1},3^{+1}$ \\ \hline & & $ \left(\begin{array}{cccc} \mathbb A_1 & 5\mathbb A_1 & 5\mathbb A_1 & 9\mathbb A_1 \\ & 4\mathbb A_1 & (8\mathbb A_1)_{I} & 12\mathbb A_1 \\ & & 4\mathbb A_1 & 4\mathbb A_3 \\ & & & 8\mathbb A_1 \end{array}\right) \subset 5\mathbb A_1\amalg 4\mathbb A_3 $ & $19$ & $2_7^{+1},4_5^{-1},8_5^{-1}$ $\ast$ \\ \hline & & $ \left(\begin{array}{cccc} \mathbb A_1 & 5\mathbb A_1 & 5\mathbb A_1 & 9\mathbb A_1 \\ & 4\mathbb A_1 & 4\mathbb A_2 & 12\mathbb A_1 \\ & & 4\mathbb A_1 & 4\mathbb A_3 \\ & & & 8\mathbb A_1 \end{array}\right) \subset \mathbb A_1\amalg 4\mathbb D_4 $ & $19$ & $2_7^{+1},4_6^{+2}$ \\ \hline & & $ \left(\begin{array}{cccc} \mathbb A_1 & 5\mathbb A_1 & 5\mathbb A_1 & \mathbb A_1\amalg 2\mathbb A_2 \\ & 4\mathbb A_1 & (8\mathbb A_1)_I & 4\mathbb A_1\amalg 2\mathbb A_2 \\ & & 4\mathbb A_1 & 4\mathbb A_1\amalg 2\mathbb A_2 \\ & & & 2\mathbb A_2 \end{array}\right) \subset 9\mathbb A_1\amalg 2\mathbb A_2 $ & $19$ & $2_7^{+1},8_6^{+2}$ \\ \hline & & $(\mathbb A_1,\,4\mathbb A_1,\,4\mathbb A_1,\,2\mathbb A_2)\subset \mathbb A_1\amalg 6\mathbb A_2$\ $o$ & $19$ & $2_7^{+1},4_{II}^{+2},3^{+1}$ $\ast$ \\ \hline & & $ \left(\begin{array}{cccc} (2\mathbb A_1)_I & 4\mathbb A_1 & (6\mathbb A_1)_I & (6\mathbb A_1)_{II} \\ & (2\mathbb A_1)_{II} & 6\mathbb A_1 & 6\mathbb A_1 \\ & & 4\mathbb A_1 & (8\mathbb A_1)_I \\ & & & 4\mathbb A_1 \end{array}\right) \subset 12\mathbb A_1 $ & $19$ & $4_7^{+1},8_6^{+2}$ \\ \hline & & $ \left(\begin{array}{cccc} (2\mathbb A_1)_I & 4\mathbb A_1 & (6\mathbb A_1)_I & (6\mathbb A_1)_{I} \\ & (2\mathbb A_1)_{II} & 6\mathbb A_1 & 6\mathbb A_1 \\ & & 4\mathbb A_1 & (8\mathbb A_1)_{II} \\ & & & 4\mathbb A_1 \end{array}\right) \subset 12\mathbb A_1 $ $o$ & $19$ & $4_5^{+3}$\ $\ast$ \\ \hline & & $ \left(\begin{array}{cccc} (2\mathbb A_1)_I & 4\mathbb A_1 & (6\mathbb A_1)_I & (6\mathbb A_1)_{I} \\ & (2\mathbb A_1)_{II} & 6\mathbb A_1 & 6\mathbb A_1 \\ & & 4\mathbb A_1 & 4\mathbb A_2 \\ & & & 4\mathbb A_1 \end{array}\right) \subset 4\mathbb A_1\amalg 4\mathbb A_2 $ & $19$ & $4_7^{-3},3^{+1}$ $\ast$ \\ \hline \end{tabular} \end{table} \begin{table} \begin{tabular}{\|c\|\|c\|c\|c\|c\|c\|c\|c\|c\|} \hline {\bf n}& $G$ &$Deg$& $\rk S$ &$q_S$ \\ \hline \hline $10$ & $D_8$ & $ \left(\begin{array}{cccc} (2\mathbb A_1)_{II} & 6\mathbb A_1 & 4\mathbb A_1 & 6\mathbb A_1 \\ & 4\mathbb A_1 & (6\mathbb A_1)_{I} & (8\mathbb A_1)_I \\ & & (2\mathbb A_1)_I & 2\mathbb A_3 \\ & & & 4\mathbb A_1 \end{array}\right) \subset 6\mathbb A_1\amalg 2\mathbb A_3 $ & $19$ & $4_6^{+2},8_7^{+1}$ \\ \hline & & $ \left(\begin{array}{cccc} (2\mathbb A_1)_{II} & 4\mathbb A_1 & 6\mathbb A_1 & 6\mathbb A_1 \\ & (2\mathbb A_1)_{I} & 2\mathbb A_3 & (6\mathbb A_1)_{II} \\ & & 4\mathbb A_1 & 4\mathbb A_2 \\ & & & 4\mathbb A_1 \end{array}\right) \subset 2\mathbb A_1\amalg 2\mathbb A_5 $ & $19$ & $4_5^{+3}$ $\ast$ \\ \hline & & $ \left(\begin{array}{cccc} (2\mathbb A_1)_I & (6\mathbb A_1)_{II} & (6\mathbb A_1)_I & (6\mathbb A_1)_I \\ & 4\mathbb A_1 & (8\mathbb A_1)_{I} & (8\mathbb A_1)_I \\ & & 4\mathbb A_1 & (8\mathbb A_1)_{II} \\ & & & 4\mathbb A_1 \end{array}\right) \subset 14\mathbb A_1 $\ $o$ & $19$ & $4_6^{+2},8_7^{+1}$ \\ \hline & & $ \left(\begin{array}{cccc} (2\mathbb A_1)_{II} & 6\mathbb A_1 & 6\mathbb A_1 & 6\mathbb A_1 \\ & 4\mathbb A_1 &(8\mathbb A_1)_I & (8\mathbb A_1)_I \\ & & 4\mathbb A_1 & (8\mathbb A_1)_{II} \\ & & & 4\mathbb A_1 \end{array}\right) \subset 14\mathbb A_1 $\ $o$ & $19$ & $4_4^{-2},8_5^{-1}$ \\ \hline & & $ \left(\begin{array}{cccc} (2\mathbb A_1)_{I} & (6\mathbb A_1)_{II} & (6\mathbb A_1)_I & (6\mathbb A_1)_I \\ & 4\mathbb A_1 & (8\mathbb A_1)_I & (8\mathbb A_1)_I \\ & & 4\mathbb A_1 & 4\mathbb A_2 \\ & & & 4\mathbb A_1 \end{array}\right) \subset 6\mathbb A_1\amalg 4\mathbb A_2 $ & $19$ & $2_2^{-2},8_1^{+1},3^{+1}$ $\ast$ \\ \hline & & $ \left(\begin{array}{cccc} (2\mathbb A_1)_{II} & 6\mathbb A_1 & 6\mathbb A_1 & 6\mathbb A_1 \\ & 4\mathbb A_1 &(8\mathbb A_1)_I & (8\mathbb A_1)_I \\ & & 4\mathbb A_1 & 4\mathbb A_2 \\ & & & 4\mathbb A_1 \end{array}\right) \subset 6\mathbb A_1\amalg 4\mathbb A_2 $ & $19$ & $2_{II}^{+2},8_3^{-1},3^{+1}$ \\ \hline & & $ \left(\begin{array}{cccc} (2\mathbb A_1)_{I} & (6\mathbb A_1)_I & (6\mathbb A_1)_I & (6\mathbb A_1)_I \\ & 4\mathbb A_1 & 4\mathbb A_2 & (8\mathbb A_1)_{II} \\ & & 4\mathbb A_1 & 4\mathbb A_2 \\ & & & 4\mathbb A_1 \end{array}\right) \subset 2\mathbb A_1\amalg 4\mathbb A_3 $\ $o$ & $19$ & $4_5^{+3}$ $\ast$ \\ \hline & & $ \left(\begin{array}{cccc} (2\mathbb A_1)_{II} & 6\mathbb A_1 & 6\mathbb A_1 & 6\mathbb A_1 \\ & 4\mathbb A_1 & 4\mathbb A_2 & (8\mathbb A_1)_{II} \\ & & 4\mathbb A_1 & 4\mathbb A_2 \\ & & & 4\mathbb A_1 \end{array}\right) \subset 2\mathbb A_1\amalg 4\mathbb A_3 $\ $o$ & $19$ & $4_5^{+3}$ $\ast$ \\ \hline & & $ \left(\begin{array}{cccc} (2\mathbb A_1)_{I} & 2\mathbb A_3 & (6\mathbb A_1)_I & (6\mathbb A_1)_I \\ & 4\mathbb A_1 & (8\mathbb A_1)_{I} & (8\mathbb A_1)_{I} \\ & & 4\mathbb A_1 & 4\mathbb A_2 \\ & & & 4\mathbb A_1 \end{array}\right) \subset 2\mathbb A_3\amalg 4\mathbb A_2 $ & $19$ & $2_0^{+2},4_7^{+1},3^{+1}$ $\ast$ \\ \hline \end{tabular} \end{table} \begin{table} \begin{tabular}{\|c\|\|c\|c\|c\|c\|c\|c\|c\|c\|} \hline {\bf n}& $G$ &$Deg$& $\rk S$ &$q_S$ \\ \hline \hline $10$ & $D_8$ & $ \left(\begin{array}{cccc} 4\mathbb A_1 & (6\mathbb A_1)_I & (8\mathbb A_1)_I & (8\mathbb A_1)_I \\ & (2\mathbb A_1)_I & 2\mathbb A_3 & (6\mathbb A_1)_{II} \\ & & 4\mathbb A_1 & 4\mathbb A_2 \\ & & & 4\mathbb A_1 \end{array}\right) \subset 4\mathbb A_1\amalg 2\mathbb A_5 $ & $19$ & $2_6^{+2},8_7^{+1}$ \\ \hline & & $ \left(\begin{array}{cccc} (2\mathbb A_1)_{I} & (6\mathbb A_1)_I & (6\mathbb A_1)_{II} & 10\mathbb A_1 \\ & 4\mathbb A_1 & (8\mathbb A_1)_I & 12\mathbb A_1 \\ & & 4\mathbb A_1 & 4\mathbb A_3 \\ & & & 8\mathbb A_1 \end{array}\right) \subset 6\mathbb A_1\amalg 4\mathbb A_3 $ & $19$ & $2_{2}^{-2},8_3^{-1}$ $\ast$ \\ \hline & & $ \left(\begin{array}{cccc} (2\mathbb A_1)_I & 2\mathbb A_3 & (6\mathbb A_1)_I & 10\mathbb A_1 \\ & 4\mathbb A_1 & (8\mathbb A_1)_{I} & 12\mathbb A_1 \\ & & 4\mathbb A_1 & 4\mathbb A_3 \\ & & & 8\mathbb A_1 \end{array}\right) \subset 6\mathbb A_3 $ \ $o$ & $19$ & $2_4^{-2},4_5^{-1}$ $\ast$ \\ \hline & & $ \left(\begin{array}{cccc} (2\mathbb A_1)_{I} & (6\mathbb A_1)_I & (6\mathbb A_1)_I & 10\mathbb A_1 \\ & 4\mathbb A_1 & 4\mathbb A_2 & 12\mathbb A_1 \\ & & 4\mathbb A_1 & 4\mathbb A_3 \\ & & & 8\mathbb A_1 \end{array}\right) \subset 2\mathbb A_1\amalg 4\mathbb D_4 $ & $19$ & $2_6^{+2},4_7^{+1}$ $\ast$ \\ \hline & & $ \left(\begin{array}{cccc} 4\mathbb A_1 & (8\mathbb A_1)_{II} & (8\mathbb A_1)_I & (8\mathbb A_1)_I \\ & 4\mathbb A_1 & (8\mathbb A_1)_I & (8\mathbb A_1)_I \\ & & 4\mathbb A_1 & (8\mathbb A_1)_{II}\\ & & & 4\mathbb A_1 \end{array}\right) \subset 16\mathbb A_1 $ $o$ & $19$ & $4_5^{+3}$ $\ast$ \\ \hline & & $ \left(\begin{array}{cccc} 4\mathbb A_1 & (8\mathbb A_1)_I & (8\mathbb A_1)_I & (8\mathbb A_1)_I \\ & 4\mathbb A_1 & 4\mathbb A_2 & (8\mathbb A_1)_{II}\\ & & 4\mathbb A_1 & 4\mathbb A_2 \\ & & & 4\mathbb A_1 \end{array}\right) \subset 4\mathbb A_1\amalg 4\mathbb A_3 $\ $o$ & $19$ & $2_{II}^{-2},8_5^{-1}$ $\ast$ \\ \hline & & $ \left(\begin{array}{cccc} 4\mathbb A_1 & 4\mathbb A_2 & (8\mathbb A_1)_I & (8\mathbb A_1)_I \\ & 4\mathbb A_1 & (8\mathbb A_1)_I & (8\mathbb A_1)_{I} \\ & & 4\mathbb A_1 & 4\mathbb A_2 \\ & & & 4\mathbb A_1 \end{array}\right) \subset 8\mathbb A_2 $ & $19$ & $4_1^{+1},3^{+2}$ $\ast$ \\ \hline & & $ \left(\begin{array}{cccc} 4\mathbb A_1 & (8\mathbb A_1)_{I} & (8\mathbb A_1)_I & 4\mathbb A_1\amalg 2\mathbb A_2 \\ & 4\mathbb A_1 & 4\mathbb A_2 & 4\mathbb A_1\amalg 2\mathbb A_2 \\ & & 4\mathbb A_1 & 4\mathbb A_1\amalg 2\mathbb A_2 \\ & & & 2\mathbb A_2 \end{array}\right) \subset 4\mathbb A_1\amalg 6\mathbb A_2 $ \ $o$ & $19$ & $8_3^{-1},3^{+1}$ $\ast$ \\ \hline \end{tabular} \end{table} \begin{table} \label{table4} \caption{Types and lattices $S$ of degenerations of codimension $\ge 2$ of K\"ahlerian K3 surfaces with symplectic automorphism group $(C_2)^3$.} \begin{tabular}{\|c\|\|c\|c\|c\|c\|c\|c\|c\|c\|} \hline {\bf n}& $\|G\|$ & $G$ &$Deg$& $\rk S$ &$q_S$ \\ \hline \hline $9$&$8$& $(C_2)^3$ & $((2\mathbb A_1,2\mathbb A_1)\subset 4\mathbb A_1)_I$ & $16$ & $2_{II}^{+2},4_0^{+4}$ $\ast$\\ \hline & & & $((2\mathbb A_1,2\mathbb A_1)\subset 4\mathbb A_1)_{II}$\ $o$ & $16$ & $2_{II}^{+4},4_{II}^{+2}$ $\ast$ \\ \hline & & & $(2\mathbb A_1,4\mathbb A_1)\subset 6\mathbb A_1$ & $16$ & $2_{II}^{+4},4_7^{+1},8_1^{+1}$ $\ast$\\ \hline & & & $(2\mathbb A_1,4\mathbb A_1)\subset 2\mathbb A_3$ & $16$ & $2_{II}^{+4},4_{II}^{+2}$ $\ast$\\ \hline & & & $(2\mathbb A_1,8\mathbb A_1)\subset 10\mathbb A_1$ & $16$ & $2_{II}^{-4},4_4^{-2}$ $\ast$\\ \hline & & & $(4\mathbb A_1,4\mathbb A_1)\subset 8\mathbb A_1$ & $16$ & $2_{II}^{+4},4_{II}^{+2}$ \\ \hline & & & $(4\mathbb A_1,8\mathbb A_1)\subset 4\mathbb A_3$ & $16$ & $2_{II}^{+6}$ $\ast$\\ \hline & & & $(8\mathbb A_1,8\mathbb A_1)\subset 16\mathbb A_1$\ $o$ & $16$ & $2_{II}^{+6}$ $\ast$\\ \hline & & & $ \left(\begin{array}{ccc} 2\mathbb A_1 & (4\mathbb A_1)_I & (4\mathbb A_1)_I \\ & 2\mathbb A_1 & (4\mathbb A_1)_I \\ & & 2\mathbb A_1 \end{array}\right) \subset 6\mathbb A_1 $ & $17$ & $4_7^{+5}$ $\ast$ \\ \hline & & & $((2\mathbb A_1,2\mathbb A_1)_I,4\mathbb A_1)\subset 8\mathbb A_1$ & $17$ & $2_{II}^{+2},4_6^{+2},8_1^{+1}$ $\ast$\\ \hline & & & $((2\mathbb A_1,2\mathbb A_1)_{II},4\mathbb A_1)\subset 8\mathbb A_1$\ $o$ & $17$ & $2_{II}^{+4},8_7^{+1}$ $\ast$\\ \hline & & & $ \left(\begin{array}{ccc} 2\mathbb A_1 & (4\mathbb A_1)_I & 6\mathbb A_1 \\ & 2\mathbb A_1 & 2\mathbb A_3 \\ & & 4\mathbb A_1 \end{array}\right) \subset 2\mathbb A_1\amalg 2\mathbb A_3 $ & $17$ & $2_{II}^{+2},4_7^{+3}$ $\ast$ \\ \hline & & & $((2\mathbb A_1,2\mathbb A_1)_I,8\mathbb A_1)\subset 12\mathbb A_1$ & $17$ & $2_{II}^{+2},4_7^{+3}$ $\ast$\\ \hline & & & $(2\mathbb A_1,4\mathbb A_1,4\mathbb A_1)\subset 10\mathbb A_1$ & $17$ & $2_{II}^{+2},4_7^{+3}$ $\ast$ \\ \hline & & & $(2\mathbb A_1,4\mathbb A_1,4\mathbb A_1)\subset 2\mathbb A_3\amalg 4\mathbb A_1$ & $17$ & $2_{II}^{+4},8_7^{+1}$ $\ast$ \\ \hline & & & $(2\mathbb A_1,4\mathbb A_1,8\mathbb A_1)\subset 2\mathbb A_1\amalg 4\mathbb A_3$ & $17$ & $2_{II}^{+4},4_7^{+1}$ $\ast$ \\ \hline & & & $(4\mathbb A_1,4\mathbb A_1,4\mathbb A_1)\subset 12\mathbb A_1$ & $17$ & $2_{II}^{+4},8_7^{+1}$ \\ \hline & & & $ \left(\begin{array}{cccc} 2\mathbb A_1 & (4\mathbb A_1)_I & (4\mathbb A_1)_I & (4\mathbb A_1)_I \\ & 2\mathbb A_1 & (4\mathbb A_1)_I & (4\mathbb A_1)_I \\ & & 2\mathbb A_1 & (4\mathbb A_1)_I \\ & & & 2\mathbb A_1 \end{array}\right) \subset 8\mathbb A_1 $\ $o$ & $18$ & $4_6^{+4}$ $\ast$ \\ \hline & & & $ \left(\begin{array}{cccc} 2\mathbb A_1 & (4\mathbb A_1)_I & (4\mathbb A_1)_I & 6\mathbb A_1 \\ & 2\mathbb A_1 & (4\mathbb A_1)_I & 6\mathbb A_1 \\ & & 2\mathbb A_1 & 6\mathbb A_1 \\ & & & 4\mathbb A_1 \end{array}\right) \subset 10\mathbb A_1 $ & $18$ & $4_5^{+3},8_1^{+1}$ $\ast$ \\ \hline & & & $ \left(\begin{array}{cccc} 2\mathbb A_1 & (4\mathbb A_1)_I & (4\mathbb A_1)_I & 6\mathbb A_1 \\ & 2\mathbb A_1 & (4\mathbb A_1)_I & 6\mathbb A_1 \\ & & 2\mathbb A_1 & 2\mathbb A_3 \\ & & & 4\mathbb A_1 \end{array}\right) \subset 4\mathbb A_1\amalg 2\mathbb A_3 $ & $18$ & $4_6^{+4}$ $\ast$ \\ \hline \end{tabular} \end{table} \begin{table} \begin{tabular}{\|c\|\|c\|c\|c\|c\|c\|c\|c\|c\|} \hline {\bf n}& $G$ &$Deg$& $\rk S$ &$q_S$ \\ \hline \hline $9$& $(C_2)^3$ & $ \left(\begin{array}{cccc} 2\mathbb A_1 & (4\mathbb A_1)_I & (4\mathbb A_1)_I & 10\mathbb A_1 \\ & 2\mathbb A_1 & (4\mathbb A_1)_I & 10\mathbb A_1 \\ & & 2\mathbb A_1 & 10\mathbb A_1 \\ & & & 8\mathbb A_1 \end{array}\right) \subset 14\mathbb A_1 $ & $18$ & $4_6^{+4}$ \\ \hline & & $((2\mathbb A_1,2\mathbb A_1)_I,4\mathbb A_1,4\mathbb A_1)\subset 12\mathbb A_1$ & $18$ & $4_6^{+4}$ $\ast$ \\ \hline & & $((2\mathbb A_1,2\mathbb A_1)_{II},4\mathbb A_1,4\mathbb A_1)\subset 12\mathbb A_1$\ $o$ & $18$ & $2_{II}^{-2},4_2^{-2}$ $\ast$ \\ \hline & & $ \left(\begin{array}{cccc} 2\mathbb A_1 & 2\mathbb A_3 & (4\mathbb A_1)_I & 6\mathbb A_1 \\ & 4\mathbb A_1 & 6\mathbb A_1 & 8\mathbb A_1 \\ & & 2\mathbb A_1 & 6\mathbb A_1 \\ & & & 4\mathbb A_1 \end{array}\right) \subset 2\mathbb A_3\amalg 6\mathbb A_1 $ & $18$ & $2_{II}^{-2},4_1^{+1},8_5^{-1}$ $\ast$ \\ \hline & & $ \left(\begin{array}{cccc} 2\mathbb A_1 & 2\mathbb A_3 & (4\mathbb A_1)_I & 6\mathbb A_1 \\ & 4\mathbb A_1 & 6\mathbb A_1 & 8\mathbb A_1 \\ & & 2\mathbb A_1 & 2\mathbb A_3 \\ & & & 4\mathbb A_1 \end{array}\right) \subset 4\mathbb A_3 $\ $o$ & $18$ & $2_{II}^{-2},4_2^{-2}$ $\ast$ \\ \hline & & $((2\mathbb A_1,2\mathbb A_1)_I,4\mathbb A_1,8\mathbb A_1) \subset 4\mathbb A_1\amalg 4\mathbb A_3$ & $18$ & $2_{II}^{-2},4_2^{-2}$ $\ast$ \\ \hline & & $(2\mathbb A_1,4\mathbb A_1,4\mathbb A_1,4\mathbb A_1)\subset 14\mathbb A_1$ & $18$ & $2_{II}^{+2},4_5^{-1},8_5^{-1}$ $\ast$ \\ \hline & & $(2\mathbb A_1,4\mathbb A_1,4\mathbb A_1,4\mathbb A_1)\subset 2\mathbb A_3\amalg 8\mathbb A_1$ & $18$ & $2_{II}^{+2},4_6^{+2}$ $\ast$ \\ \hline & & $(4\mathbb A_1,4\mathbb A_1,4\mathbb A_1,4\mathbb A_1)\subset 16\mathbb A_1$\ $o$ & $18$ & $2_{II}^{+2},4_6^{+2}$ $\ast$ \\ \hline & & $ \left(\begin{array}{ccccc} 2\mathbb A_1 & (4\mathbb A_1)_I & (4\mathbb A_1)_I & (4\mathbb A_1)_I & 10\mathbb A_1 \\ & 2\mathbb A_1 & (4\mathbb A_1)_I & (4\mathbb A_1)_I & 10\mathbb A_1 \\ & & 2\mathbb A_1 & (4\mathbb A_1)_I & 10\mathbb A_1 \\ & & & 2\mathbb A_1 & 10\mathbb A_1\\ & & & & 8\mathbb A_1 \end{array}\right) \subset 16\mathbb A_1 $ $o$ & $19$ & $4_5^{+3}$ $\ast$ \\ \hline & & $ \left(\begin{array}{ccccc} 2\mathbb A_1 & (4\mathbb A_1)_I & 6\mathbb A_1 & 6\mathbb A_1 & 6\mathbb A_1 \\ & 2\mathbb A_1 & 6\mathbb A_1 & (4\mathbb A_1)_I & 6\mathbb A_1 \\ & & 4\mathbb A_1 & 6\mathbb A_1 & 8\mathbb A_1 \\ & & & 2\mathbb A_1 & 2\mathbb A_3\\ & & & & 4\mathbb A_1 \end{array}\right) \subset 8\mathbb A_1\amalg 2\mathbb A_3 $ & $19$ & $4_4^{-2},8_5^{-1}$ $\ast$ \\ \hline & & $ \left(\begin{array}{ccccc} 2\mathbb A_1 & (4\mathbb A_1)_I & 6\mathbb A_1 & (4\mathbb A_1)_I & 6\mathbb A_1 \\ & 2\mathbb A_1 & 2\mathbb A_3 & (4\mathbb A_1)_I & 6\mathbb A_1 \\ & & 4\mathbb A_1 & 6\mathbb A_1 & 8\mathbb A_1 \\ & & & 2\mathbb A_1 & 2\mathbb A_3\\ & & & & 4\mathbb A_1 \end{array}\right) \subset 2\mathbb A_1\amalg 4\mathbb A_3 $\ $o$& $19$ & $4_5^{+3}$ $\ast$ \\ \hline \end{tabular} \end{table} \begin{table} \begin{tabular}{\|c\|\|c\|c\|c\|c\|c\|c\|c\|c\|} \hline {\bf n}& $G$ &$Deg$& $\rk S$ &$q_S$ \\ \hline \hline $9$& $(C_2)^3$ & $ \left(\begin{array}{ccccc} 2\mathbb A_1 & (4\mathbb A_1)_I & (4\mathbb A_1)_I & 6\mathbb A_1 & 10\mathbb A_1 \\ & 2\mathbb A_1 & (4\mathbb A_1)_I & 6\mathbb A_1 & 10\mathbb A_1 \\ & & 2\mathbb A_1 & 6\mathbb A_1 & 10\mathbb A_1 \\ & & & 4\mathbb A_1 & 4\mathbb A_3\\ & & & & 8\mathbb A_1 \end{array}\right) \subset 6\mathbb A_1\amalg 4\mathbb A_3 $ & $19$ & $4_5^{+3}$ $\ast$ \\ \hline & & $((2\mathbb A_1,2\mathbb A_1)_{II},4\mathbb A_1,4\mathbb A_1,4\mathbb A_1)\subset 16\mathbb A_1$ $o$ & $19$ & $2_{II}^{-2},8_5^{-1}$ $\ast$ \\ \hline & & $ \left(\begin{array}{ccccc} 2\mathbb A_1 & 2\mathbb A_3 & (4\mathbb A_1)_I & 6\mathbb A_1 & 6\mathbb A_1 \\ & 4\mathbb A_1 & 6\mathbb A_1 & 8\mathbb A_1 & 8\mathbb A_1 \\ & & 2\mathbb A_1 & 2\mathbb A_3 & 6\mathbb A_1 \\ & & & 4\mathbb A_1 & 8\mathbb A_1\\ & & & & 4\mathbb A_1 \end{array}\right) \subset 4\mathbb A_3\amalg 4\mathbb A_1 $\ $o$ & $19$ & $2_{II}^{-2},8_5^{-1}$ $\ast$ \\ \hline \end{tabular} \end{table} V.V. Nikulin \par Steklov Mathematical Institute, \par ul. Gubkina 8, Moscow 117966, GSP-1, Russia; \vskip5pt Deptm. of Pure Mathem. The University of Liverpool, Liverpool\par L69 3BX, UK \par \vskip5pt [email protected]\, \ \ [email protected] \, \ \ [email protected] Personal page: http://vnikulin.com \end{document}	arXiv
Invariant decomposition The invariant decomposition is a decomposition of the elements of ${\text{Pin}}(p,q,r)$ groups into orthogonal commuting elements. It is also valid in their subgroups, e.g. orthogonal, pseudo-Euclidean, conformal, and classical groups. Because the elements of Pin groups are the composition of $k$ oriented reflections, the invariant decomposition theorem reads Every $k$-reflection can be decomposed into $ \lceil k/2\rceil $ commuting factors.[1] It is named the invariant decomposition because these factors are the invariants of the $k$-reflection $R\in {\text{Pin}}(p,q,r)$. A well known special case is the Chasles' theorem, which states that any rigid body motion in $ {\text{SE}}(3)$ can be decomposed into a rotation around, followed or preceded by a translation along, a single line. Both the rotation and the translation leave two lines invariant: the axis of rotation and the orthogonal axis of translation. Since both rotations and translations are bireflections, a more abstract statement of the theorem reads "Every quadreflection can be decomposed into commuting bireflections". In this form the statement is also valid for e.g. the spacetime algebra $ {\text{SO}}(3,1)$, where any Lorentz transformation can be decomposed into a commuting rotation and boost. Bivector decomposition Any bivector $F$ in the geometric algebra $\mathbb {R} _{p,q,r}$ of total dimension $n=p+q+r$ can be decomposed into $k=\lfloor n/2\rfloor $ orthogonal commuting simple bivectors that satisfy $F=F_{1}+F_{2}\ldots +F_{k}$ Defining $\lambda _{i}:=F_{i}^{2}\in \mathbb {C} $, their properties can be summarized as $F_{i}F_{j}=\delta _{ij}\lambda _{i}+F_{i}\wedge F_{j}$ (no sum). The $F_{i}$ are then found as solutions to the characteristic polynomial $0=(F_{1}-F_{i})(F_{2}-F_{i})\cdots (F_{k}-F_{i})$ Defining $W_{m}={\frac {1}{m!}}\langle F^{m}\rangle _{2m}={\frac {1}{m!}}\underbrace {F\wedge F\wedge \ldots \wedge F} _{m}$ and $r=\lfloor k/2\rfloor $, the solutions are given by $F_{i}={\begin{cases}{\frac {\lambda _{i}^{r}W_{0}+\lambda _{i}^{r-1}W_{2}+\ldots +W_{k}}{\lambda _{i}^{r-1}W_{1}+\lambda _{i}^{r-2}W_{3}+\ldots +W_{k-1}}}&k{\text{ even}}\\[1em]{\frac {\lambda _{i}^{r}W_{1}+\lambda _{i}^{r-1}W_{3}+\ldots +W_{k}}{\lambda _{i}^{r}W_{0}+\lambda _{i}^{r-1}W_{2}+\ldots +W_{k-1}}}&k{\text{ odd}}\end{cases}}$ The values of $\lambda _{i}$ are subsequently found by squaring this expression and rearranging, which yields the polynomial ${\begin{aligned}0&=\sum _{m=0}^{k}\langle W_{m}^{2}\rangle _{0}(-\lambda _{i})^{k-m}\\&=(F_{1}^{2}-\lambda _{i})(F_{2}^{2}-\lambda _{i})\cdots (F_{k}^{2}-\lambda _{i})\end{aligned}}$ By allowing complex values for $\lambda _{i}$, the counter example of Marcel Riesz can in fact be solved.[1] This closed form solution for the invariant decomposition is only valid for eigenvalues $\lambda _{i}$ with algebraic multiplicity of 1. For degenerate $\lambda _{i}$ the invariant decomposition still exists, but cannot be found using the closed form solution. Exponential map A $2k$-reflection $R\in {\text{Spin}}(p,q,r)$ can be written as $R=\exp(F)$ where $F\in {\mathfrak {spin}}(p,q,r)$ is a bivector, and thus permits a factorization $R=e^{F}=e^{F_{1}}e^{F_{2}}\cdots e^{F_{k}}$ The invariant decomposition therefore gives a closed form formula for exponentials, since each $F_{i}$ squares to a scalar and thus follows Euler's formula: $R_{i}=e^{F_{i}}=\cosh({\sqrt {\lambda _{i}}})+{\frac {\sinh({\sqrt {\lambda _{i}}})}{\sqrt {\lambda _{i}}}}F_{i}$ Carefully evaluating the limit $\lambda _{i}\to 0$ gives $R_{i}=e^{F_{i}}=1+F_{i}$ and thus translations are also included. Rotor factorization Given a $2k$-reflection $R\in {\text{Spin}}(p,q,r)$ we would like to find the factorization into $R_{i}=\exp(F_{i})$. Defining the simple bivector $t(F_{i}):={\frac {\tanh({\sqrt {\lambda _{i}}})}{\sqrt {\lambda _{i}}}}F_{i},$ where $\lambda _{i}=F_{i}^{2}$. These bivectors can be found directly using the above solution for bivectors by substituting[1] $W_{m}=\langle R\rangle _{2m}/\langle R\rangle _{0}$ where $\langle R\rangle _{2m}$ selects the grade $2m$ part of $R$. After the bivectors $t(F_{i})$ have been found, $R_{i}$ is found straightforwardly as $R_{i}={\frac {1+t(F_{i})}{\sqrt {1-t(F_{i})^{2}}}}$ Principal logarithm After the decomposition of $R\in {\text{Spin}}(p,q,r)$ into $R_{i}=\exp(F_{i})$ has been found, the principal logarithm of each simple rotor is given by $F_{i}={\text{Log}}(R_{i})={\begin{cases}{\frac {\langle R_{i}\rangle _{2}}{\sqrt {\langle R_{i}\rangle _{2}^{2}}}}\;{\text{arccosh}}(\langle R_{i}\rangle )&\lambda _{i}^{2}\neq 0\\\langle R_{i}\rangle _{2}&\lambda _{i}^{2}=0\end{cases}}$ and thus the logarithm of $R$ is given by ${\text{Log}}(R)=\sum _{i=1}^{k}{\text{Log}}(R_{i})$ General Pin group elements So far we have only considered elements of ${\text{Spin}}(p,q,r)$, which are $2k$-reflections. To extend the invariant decomposition to a $(2k+1)$-reflections $P\in {\text{Pin}}(p,q,r)$, we use that the vector part $r=\langle P\rangle _{1}$ is a reflection which already commutes with, and is orthogonal to, the $2k$-reflection $R=r^{-1}P=Pr^{-1}$. The problem then reduces to finding the decomposition of $R$ using the method described above. Invariant bivectors The bivectors $F_{i}$ are invariants of the corresponding $R\in {\text{Spin}}(p,q,r)$ since they commute with it, and thus under group conjugation $RF_{i}R^{-1}=F_{i}$ Going back to the example of Chasles' theorem as given in the introduction, the screw motion in 3D leaves invariant the two lines $F_{1}$ and $F_{2}$, which correspond to the axis of rotation and the orthogonal axis of translation on the horizon. While the entire space undergoes a screw motion, these two axes remain unchanged by it. History The invariant decomposition finds its roots in a statement made by Marcel Riesz about bivectors[2]: Can any bivector $F$ be decomposed into the direct sum of mutually orthogonal simple bivectors? Mathematically, this would mean that for a given bivector $F$ in an $n$ dimensional geometric algebra, it should be possible to find a maximum of $ k=\lfloor n/2\rfloor $ bivectors $F_{i}$, such that $ F=\sum _{i=1}^{\lfloor n/2\rfloor }F_{i}$, where the $F_{i}$ satisfy $F_{i}\cdot F_{j}=[F_{i},F_{j}]=0$ and should square to a scalar $\lambda _{i}:=F_{i}^{2}\in \mathbb {R} $. Marcel Riesz gave some examples which lead to this conjecture, but also one (seeming) counter example. A first more general solution to the conjecture in geometric algebras $\mathbb {R} _{n,0,0}$ was given by David Hestenes and Garret Sobczyck.[3] However, this solution was limited to purely Euclidean spaces. In 2011 the solution in $\mathbb {R} _{4,1,0}$ (3DCGA) was published by Leo Dorst and Robert Jan Valkenburg, and was the first solution in a Lorentzian signature.[4] Also in 2011, Charles Gunn was the first to give a solution in the degenerate metric $\mathbb {R} _{3,0,1}$.[5] This offered a first glimpse that the principle might be metric independent. Then, in 2021, the full metric and dimension independent closed form solution was given by Martin Roelfs in his PhD thesis.[6] And because bivectors in a geometric algebra $\mathbb {R} _{p,q,r}$ form the Lie algebra ${\mathfrak {spin}}(p,q,r)$, the thesis was also the first to use this to decompose elements of ${\text{Spin}}(p,q,r)$ groups into orthogonal commuting factors which each follow Euler's formula, and to present closed form exponential and logarithmic functions for these groups. Subsequently in a paper by Martin Roelfs and Steven De Keninck the invariant decomposition was extended to include elements of ${\text{Pin}}(p,q,r)$, not just ${\text{Spin}}(p,q,r)$, and the direct decomposition of elements of ${\text{Spin}}(p,q,r)$ without having to pass through ${\mathfrak {spin}}(p,q,r)$ was found.[1] References 1. Roelfs, Martin; De Keninck, Steven. "Graded Symmetry Groups: Plane and Simple".{{cite web}}: CS1 maint: url-status (link) 2. Riesz, Marcel (1993). Bolinder, E. Folke; Lounesto, Pertti (eds.). Clifford Numbers and Spinors. doi:10.1007/978-94-017-1047-3. ISBN 978-90-481-4279-8. 3. Hestenes, David (1984). Clifford algebra to geometric calculus: a unified language for mathematics and physics. Garret Sobczyk. Dordrecht: D. Reidel. ISBN 90-277-1673-0. OCLC 10726931. 4. Dorst, Leo; Valkenburg, Robert (2011), Dorst, Leo; Lasenby, Joan (eds.), "Square Root and Logarithm of Rotors in 3D Conformal Geometric Algebra Using Polar Decomposition", Guide to Geometric Algebra in Practice, London: Springer London, pp. 81–104, doi:10.1007/978-0-85729-811-9_5, ISBN 978-0-85729-810-2, retrieved 2021-11-13 5. Gunn, Charles (19 December 2011). Geometry, Kinematics, and Rigid Body Mechanics in Cayley-Klein Geometries (Thesis). Technische Universität Berlin. doi:10.14279/DEPOSITONCE-3058. 6. Roelfs, Martin (2021). Spectroscopic and Geometric Algebra Methods for Lattice Gauge Theory (Thesis). doi:10.13140/RG.2.2.23224.67848.	Wikipedia

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