Content-Type: multipart/mixed; boundary="-------------0011071730765" This is a multi-part message in MIME format. ---------------0011071730765 Content-Type: text/plain; name="00-435.comments" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="00-435.comments" AMS-Code: 46L55, 47C05, 81S05, 81T05. Accepted for publication in Pacific J. Math. ---------------0011071730765 Content-Type: text/plain; name="00-435.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="00-435.keywords" unbounded operators, Hilbert space, quantum deformation, positive definite forms ---------------0011071730765 Content-Type: application/x-tex; name="mwickpjm.tex" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="mwickpjm.tex" %%%%%%%%%%%%Pacific Journal of Mathematics %%%%%%%%% \documentclass[11pt]{pjm} \usepackage{amsthm} \usepackage{amsfonts} \usepackage{amsmath} \usepackage{eucal} \renewcommand{\labelenumi}{\textup{\theenumi.}} \newcommand{\tenz}{\ensuremath{T_{ij}^{kl}}} \newcommand{\otenz}{\ensuremath{T_{ik}^{lj}}} \newcommand{\lm}{\ensuremath{\lambda_{ij}}} \newcommand{\lf}{\ensuremath{\alpha_{ij}}} \newcommand{\qu}{\ensuremath{q_{ij}}} \newcommand{\hb}{\ensuremath{\mathcal{H}}} \newcommand{\ub}{\ensuremath{\mathcal{U}}} \newcommand{\tpr}{\ensuremath{e_{i}\otimes e_{j}}} \newcommand{\otpr}{\ensuremath{e_{j}\otimes e_{i}}} \newcommand{\alp}{\alpha^{\pi}(i,j)} \newcommand{\alg}{\ensuremath{\mathcal{U}(A,\Lambda)}} \newcommand{\img}{\operatorname{ran}} \newcommand{\capker}{\ker} \newcommand{\id}{\operatorname{id}} \newlength{\qedskip} \newlength{\qedadjust} \title[Braided operator commutators] {The kernel of Fock representations of Wick algebras with braided operator of coefficients} \author[P. E. T. J{\o}rgensen]{Palle E. T. J{\o}rgensen} \address{ \textup{(P. J.)} Department of Mathematics\\The University of Iowa\\ Iowa City, Iowa 52242-1419 U.S.A.} \email{jorgen@math.uiowa.edu} %\urladdr{Put the URL for your home page here if you have one} \thanks{P. J. was partially supported by the NSF} \author[D. P. Proskurin]{Daniil P. Proskurin} \address{ \textup{(D. P. and Yu. S.)} Institute of Mathematics\\Ukrain\-i\-an National Academy of Sciences\\ Tereshchinkivs'ka, 3, Kiev, 252601, Ukraine} \email{prosk@imath.kiev.ua} %\urladdr{Put the URL for your home page here if you have one} \author[Yu. S. Samo\u\i{}lenko]{Yuri\u\i{}~S.~Samo\u\i{}lenko} %\address{ %Institute of Mathematics\\Ukrain\-i\-an National Academy of %Sciences\\ %Tereshchinkivs'ka, 3, Kiev, 252601, Ukraine} \email{Yurii\_Sam@imath.kiev.ua} %\urladdr{Put the URL for your home page here if you have one} \thanks{Yu. S. was partially supported by CRDF grant no. UM1-311} \subjclass{46L55, 47C05, 81S05, 81T05} \keywords{unbounded operators, Hilbert space, quantum deformation, positive definite forms} \theoremstyle{plain} \newtheorem{definition}{Definition} \newtheorem{proposition}{Proposition} \newtheorem{theorem}{Theorem} \newtheorem{corollary}{Corollary} \theoremstyle{remark} \newtheorem{remark}{\bf Remark} \newtheorem{example}{\bf Example} \date{Received date / Revised version date} \begin {document} \begin{abstract} It is shown that the kernel of the Fock representation of a certain Wick algebra with braided operator of coefficients $T$, $||T\,||\le 1$, coincides with the largest quadratic Wick ideal. Improved conditions on the operator $T$ for the Fock inner product to be strictly positive are given. \end{abstract} \maketitle \section{Introduction}\label{intro} The problem of positivity of the Fock space inner product is central in the study of the Fock representation of Wick algebras (see \cite{bib:2}, \cite{bib:7}, \cite{bib:8}, \cite{bib:1}). The paper \cite{bib:1} presents several conditions on the coefficients of the Wick algebra for the Fock inner product to be positive. If the operator of coefficients of the Wick algebra $T$ satisfies the braid condition and the norm restriction $\Vert T\Vert \le 1$, then, as proved in \cite{bib:2}, the Fock inner product is positive. Moreover if $-1< T <1$, it was shown in \cite{bib:2} that the Fock inner product is strictly positive. In this article we prove that, for braided $T$ with $\Vert T\Vert\le 1$, the kernel of the Fock inner product coincides with the largest quadratic Wick ideal. In particular this implies that, for $-1< T\le 1$, the Fock inner product is strictly positive definite, and the Fock representation is faithful. This article is organized as follows. In sec.~\ref{S1} we present definitions of Wick algebras and the Fock representation and show that, in the braided case, the kernel of the Fock representation is generated by the kernel of the Fock inner product. In sec.~\ref{S2} we prove that if the operator $T$ is braided and $\Vert T\Vert\le 1$, then the kernel of the Fock inner product coincides with the two-sided ideal generated by $\ker (1+T)$. In sec.~\ref{S3} we combine results obtained in sec.~\ref{S1} and sec.~\ref{S2} to examine the $C^*$-representability of certain Wick algebras or their quotients. All results are illustrated by examples of different kinds of $q_{ij}$-CCR. \section{Preliminaries}\label{S1} For more detailed information about Wick algebras and the Fock representation we refer the reader to \cite{bib:1}. In this section we present only the basic definitions and properties. \medskip \noindent {\bf 1.} The notion of a $*$-algebra allowing Wick ordering (Wick algebra) was presented in the paper \cite{bib:1} as a generalization of a wide class of $*$-algebras, including the twisted CCR and CAR algebras (see \cite{bib:3}), the $q$-CCR (see \cite{bib:5}) algebra, etc. \begin {definition} Let $ \mathbb{J}=\mathbb{J}_{d} = \{1,2,\dots ,d\}$, $ \tenz\in C$, $i, j, k, l\in\mathbb{J}$, be such that $\tenz=\overline{T_{ji}^{lk}}$. The Wick algebra with the set of coefficients $\{T_{ij}^{kl}\}$ is denoted $W(T)$, and is a $*$-algebra, defined by generators $a_{i}$, $a_{i}^{*}$, $i\in\mathbb{J}$, which satisfy the basic relations: \[ a_i^*a_j=\delta_{ij}1+\sum_{k,l=1}^{d} \tenz a_{l}a_{k}^{*} \] \end {definition} \begin{definition} Monomials of the form $a_{i_1}a_{i_2}\cdots a_{i_m}a_{j_1}^*a_{j_2}^* \cdots a_{j_k}^*$ are called Wick ordered monomials. \end{definition} It was proved in \cite{bib:1} that the Wick ordered monomials form a basis for $W(T)$. Let $\hb= \left\langle e_{1}, \dots ,e_{d} \right\rangle $. Consider the full tensor algebra over $\hb,\, \hb^*$, denoted by $\mathcal{T}(\hb,\hb^{*})$. Then \[ W(T) \simeq \mathcal{T}(\hb,\hb^{*}) \biggm/ \left\langle e_{i}^{*}\otimes e_{j} - \delta_{ij}1 -\sum \tenz e_{l}\otimes e_{k}^{*} \right\rangle . \] To study the structure of Wick algebras, and the structure of the Fock representation, it is useful to introduce the following operators on $\mathcal{H}^{\otimes\,n}:= \smash{\underbrace{\mathcal{H}\otimes \cdots\otimes \mathcal{H}}_{n}}$ (see \cite{bib:1}): \begin{align*} T\colon & \mathcal{H} \otimes \mathcal{H}\mapsto \mathcal{H} \otimes\mathcal{H}, \quad T e_{k}\otimes e_{l} = \sum_{i,j} T_{ik}^{lj}e_{i}\otimes e_{j},\ T=T^*, \\ T_{i}\colon & \mathcal{H}^{\otimes\,n}\mapsto\mathcal{H}^{\otimes\,n}, \quad T_{i}=\underbrace{1\otimes \cdots\otimes 1}_{i-1}\otimes T \otimes\underbrace{1\otimes\cdots\otimes 1}_{n-i-1}\,, \\ R_{n}\colon & \mathcal{H}^{\otimes\,n}\mapsto \mathcal{H}^{\otimes\,n}, \quad R_{n}=1+T_1+T_1 T_2+\cdots + T_1 T_2\cdots T_{n-1}, \\ P_n\colon & \mathcal{H}^{\otimes\,n}\mapsto \mathcal{H}^{\otimes\,n}, \quad P_2=R_2,\ P_{n+1}=(1\otimes P_n)R_{n+1}. \end{align*} In this article we suppose that the operator $T$ is contractive, i.e., $\Vert T\Vert\le 1$, and satisfies the {\it braid condition}, i.e., on $\hb^{\otimes\,3}$ the equality $T_1 T_2 T_1 = T_2 T_1 T_2$ holds. It follows from the definition of $T_i$ that then $T_iT_j=T_jT_i$ if $\left| i-j\right|\ge 2$, and for the braided $T$ one has $T_iT_{i+1}T_i=T_{i+1}T_iT_{i+1}$. \begin{remark} These conditions hold for such well-known algebras as $q_{ij}$-CCR, $\mu$- CCR, $\mu$-CAR (see \cite{bib:1}). \end{remark} The Fock representation of a Wick $*$-algebra is determined by a vector $\Omega$ such that $a_i^*\Omega=0$ for all $i=1,\dots,d$ (see \cite{bib:1}). \begin{definition} [The Fock representation] The representation $\lambda_0$, acting on the space $\mathcal{T}(\hb)$ by formulas \begin{align*} \lambda_0(a_i)e_{i_1}\otimes\cdots\otimes e_{i_n}& =e_i\otimes e_{i_1}\otimes\cdots\otimes e_{i_n},\quad n\in\mathbb{N}\cup\{0\},\\ \lambda_0(a_i^*)1&= 0, \end{align*} where the action of $\lambda_0(a_i^*)$ on the monomials of degree $n\ge 1$ is determined inductively using the basic relations, is called the Fock representation. \end{definition} Note that the Fock representation is not a $*$-representation with respect to the standard inner product on $\mathcal{T}(\hb)$. However, it was proved in \cite{bib:1} that there exists a unique Hermitian sesquilinear form $\bigl<\,\cdot \,,\,\cdot \,\bigr>_0$ on $\mathcal{T}(\hb)$ such that $\lambda_0$ is a $*$-representation on $(\mathcal{T}(\hb),\bigl<\,\cdot \,,\,\cdot \,\bigr>_0)$. This form is called the Fock inner product on $\mathcal{T}(\hb)$. The subspaces $\hb^{\otimes\,n}$, $\hb^{\otimes\,m}$, $n\ne m$, are orthogonal with respect to $\bigl<\,\cdot \,,\,\cdot \,\bigr>_0$, and on $\hb^{\otimes\,n}$ we have the following formula (see \cite{bib:1}): \[ \bigl_0 =\bigl,\ n\ge 2. \] So, the positivity of the Fock inner product is equivalent to the positivity of operators $P_n$, $n\ge 2$, and $\mathcal{I}=\bigoplus_{n\ge 2\vphantom{\ge_{\mathstrut}}}\ker P_n$ determines the kernel of the Fock inner product. It was noted in \cite{bib:1} that the Fock representation is the GNS representation associated with the linear functional $f$ on a Wick algebra such that $f(1)=1$ and, for any Wick ordered monomial, $f(a_{i_1}\cdots a_{i_n}a_{j_1}^*\cdots a_{j_m}^*)=0$. Then for any $X,Y\in\mathcal{T}(\hb)$ we have (see \cite{bib:1}): \[ \bigl< X,Y \bigr>_0=f(X^*Y). \] \noindent {\bf 2.} In the following proposition we describe the kernel of the Fock representation of a Wick algebra with braided operator $T$ in terms of the Fock inner product. \begin{proposition} Let $W(T)$ be the Wick algebra with braided operator $T$, and let the Fock representation $\lambda_0$ be positive \textup{(}i.e., the Fock inner product is positive definite\textup{).} Then $\ker \lambda_0 = \mathcal{I}\otimes\mathcal{T}(\hb^*) + \mathcal{T}(\hb)\otimes\mathcal{I}^*$. \end{proposition} \begin{proof} First, we show that $X\in\ker P_m$ implies $X\in\ker\lambda_0$. Indeed, let $Y\in\hb^{\otimes\,n}$; then \[ \lambda_0(X)Y=X\otimes Y. \] Note that for braided $T$ we have the following decomposition (see \cite{bib:2} and sec.~\ref{S2} for more details): \[ P_{n+m}=P(D_m)(P_m\otimes\mathbf{1}_n), \] where \begin{align*} & P(D_m)=\widetilde{R}_{n+m}\widetilde{R}_{n+m-1}\cdots \widetilde{R}_{m+1},\\ &\widetilde{R}_k =1+T_{k-1}+T_{k-2}T_{k-1}+\cdots+ T_1T_2\cdots T_{k-1},\quad k\ge 2. \end{align*} Then \[ P_{n+m}(\lambda_0(X)Y)=P_{n+m}(X\otimes Y)=P(D_m)(P_m X\otimes Y)=0, \] and $\lambda_0(X)=0$ on $\left(\mathcal{T}(\hb)\, ,\, \bigl<\,\cdot \,,\,\cdot \,\bigr>_0\right)$. Therefore $\mathcal{I}\subset\ker\lambda_0$, and since $\ker\lambda_0$ is a $*$-ideal, \begin{equation}\label{incl} \mathcal{I}\otimes\mathcal{T}(\hb^*)+\mathcal{T}(\hb) \otimes\mathcal{I}^*\subset\ker\lambda_0. \end{equation} To prove the converse inclusion, we need a formula determining the action of $\lambda_0(X^*)$ on $\mathcal{T}(\hb)$ for any $X\in\hb^{\otimes\,k}$, $k\in\mathbb{N}$. For $k=1$, $X=e_i$, $i=1,\dots,d$, it was proved in \cite{bib:1} that: \[ \lambda_0(e_i^*) Y=\mu(e_i^*)R_n Y,\quad \forall\, Y\in\hb^{\otimes\,n}, \] where $\mu(e_i^*)\colon\mathcal{T}(\hb)\mapsto\mathcal{T}(\hb)$ is the annihilation operator: \[ \mu(e_i^*)e_{i_1}\otimes e_{i_2}\otimes\cdots\otimes e_{i_n}= \delta_{ii_1} e_{i_2}\otimes\cdots\otimes e_{i_n} . \] Then, using the definition of $P_n$, it is easy to see that, for $X\in\hb^{\otimes\,n}$ and $Y\in\hb^{\otimes\,n}$, \[ \lambda_0(X^*)Y=\bigl< X,P_n Y\bigr>=\bigl< X, Y\bigr>_0 . \] Let now \[ Z=\sum_{i=1}^n Y_i X_i^*+\sum_{j=n+1}^l Y_jX_j^*\in\ker\lambda_0, \] where $Y_i\in\mathcal{T}(\hb)$, $i=1,\dots,l$, \[ X_i\in\hb^{\otimes\,{m}},\; i=1,\dots,n\, ,\quad X_j\in\hb^{\otimes\,{n_j}},\; n_j>m\,,\quad j=n+1,\dots,l. \] Now (\ref{incl}) implies that we can suppose that the elements $X_i$ are linearly independent modulo $\mathcal{I}$. Denote by $\{\widehat{X}_i,\ i=1,\dots,n\}\subset\hb^{\otimes\,m}$ a family dual to the $\{X_i,\ i=1,\dots,n\}$ with respect to $\bigl<\, \cdot\, ,\, \cdot\, \bigr>_0$, i.e., such that \[ \bigl= \bigl< X_i\, ,\, \widehat{X}_j\bigr>_0=\delta_{ij}, \quad i,j=1,\dots,n. \] Since, for any $j=n+1,\dots,l$ and $i=1,\dots,n$, \[ \lambda_0(X_j^*)\widehat{X}_i=0, \] we have, in $\left(\mathcal{T}(\hb)\, ,\, \bigl<\,\cdot \,,\,\cdot \,\bigr>_0\right)$, \[ 0=\lambda_0(Z)\widehat{X}_i = Y_i,\quad i=1,\dots,n, \] which implies $Y_i\in\mathcal{I}$, $i=1,\dots,n$. The proof can be completed by evident induction. \end{proof} \begin{remark} In particular, we have shown, for braided $T$, and for any $X\in\hb^{\otimes\,n}$ and $Y\in\ker P_m$, that \[ X\otimes Y\in\ker P_{n+m}. \] By similar arguments, $Y\otimes X\in\ker P_{n+m}$, i.e., $\mathcal{I}=\smash{ \left\langle\bigotimes_{n\ge 2}\ker P_n \right\rangle}$ is a two-sided ideal in $\mathcal{T}(\mathcal{H})$. \end{remark} The two-sided ideal $\mathcal{J}\subset\mathcal{T}(\hb)$ is called a Wick ideal (see \cite{bib:1}) if it satisfies the following condition: \begin{equation}\label{eq:wick} \mathcal{T}\left(\hb^*\right)\otimes\mathcal{J} \subset\mathcal{J}\otimes\mathcal{T}\left(\hb^*\right). \end{equation} If $\mathcal{J}$ is generated by some subspace of $\hb^{\otimes\,n}$, then $\mathcal{J}$ is called a homogeneous Wick ideal of degree $n$. We show that for Wick algebras with braided operator of coefficients, $\mathcal{I}$ is a Wick ideal . \begin{proposition} Let $T$ satisfy the braid condition, and $\mathcal{I}=\smash{ \left\langle\bigoplus_{n\ge 2}\ker P_n\right\rangle}$; then \begin{equation}\label{eq:sufwick} \hb^*\otimes\mathcal{I}\subset\mathcal{I}+\mathcal{I}\otimes\hb^*. \end{equation} \end{proposition} \begin{proof} Note that conditions (\ref{eq:wick}) and (\ref{eq:sufwick}) are equivalent (see \cite{bib:1}). To prove the proposition, it is sufficient to show that, if $X\in\ker P_n$ for some $n\ge 2$, then for any $i=1,\dots,d$, \[ e_i^*\otimes X\in\ker P_{n-1}+\ker P_n\otimes\hb^*. \] Indeed, for any $X\in\hb^{\otimes\,n}$, we have the following formula (see \cite{bib:6}): \[ e_i^*\otimes X=\mu(e_i^*)R_n X+\mu(e_i^*)\sum_{k=1}^d T_1T_2\cdots T_n (X\otimes e_k)\otimes e_k^*. \] Then for $X\in\ker P_n$, we have \[ P_{n-1}\mu(e_i^*)R_n X=\mu(e_i^*)(1\otimes P_{n-1})R_nX= \mu(e_i^*)P_n X=0. \] Note that, for braided $T$, for any $k=2,\dots,n$, \[ T_k (T_1T_2\cdots T_n)=(T_1T_2\cdots T_n)T_{k-1}, \] which implies that \[ (1\otimes P_n)(T_1T_2\cdots T_n)=(T_1T_2\cdots T_n) (P_n\otimes 1). \] Then for any $k=1,\dots,d$, \begin{align*} P_n\mu(e_i^*)T_1T_2\cdots T_n (X\otimes e_k) & =\mu(e_i^*)(1\otimes P_n)T_1T_2\cdots T_n (X\otimes e_k)\\ & =\mu(e_i^*)T_1T_2\cdots T_n (P_n\otimes 1)(X\otimes e_k)=0. \settowidth{\qedskip}{$\displaystyle P_n\mu(e_i^*)T_1T_2\cdots T_n (X\otimes e_k) =\mu(e_i^*)T_1T_2\cdots T_n (P_n\otimes 1)(X\otimes e_k)=0.$} \addtolength{\qedskip}{-\textwidth} \rlap{\hskip-0.5\qedskip\llap{\qedsymbol}} \end{align*} \renewcommand{\qed}{\relax} \end{proof} For Wick algebras with braided $T$, the largest homogeneous ideal of degree $n$ is generated by $\ker R_n$ (see \cite{bib:1} and \cite{bib:6}), i.e., the condition $\ker R_n\ne\{0\}$ is necessary and sufficient for the existence of homogeneous Wick ideals. In the following proposition we show that the same is true for arbitrary Wick ideals. \begin{theorem}\label{prop:idew} If $\mathcal{J}\subset\mathcal{T}(\hb)$ is a non-trivial Wick ideal, then there exists $n\ge 2$ such that $\ker R_n\ne\{0\}$. \end{theorem} \begin{proof} For any $X\in\mathcal{T}(\hb)$, by $\deg X$ we denote the highest degree of its homogeneous components. Let $Y\in\mathcal{J}$ be of minimal degree. \[ Y=Y_1+Y_2+\cdots +Y_k,\quad Y_i\in\hb^{\otimes\,{n_i}},\ i=1,\dots,k ,\ n_i\in\mathbb{N}\cup\{0\}. \] Suppose that $\deg Y\ge 2$: then for any $i=1,\dots,d$, we have \[ e_i^*\otimes Y=\sum_{j=1}^k \mu(e_i^*)R_{n_j}Y_j + \mu(e_i^*)\sum_{j=1}^k\sum_{l=1}^d \widetilde{T}_{n_j}(Y_j\otimes e_l) \otimes e_l^*, \] where we put $R_0=1$, $R_1=1$, and \[ \widetilde{T}_k= \begin{cases} T_1T_2\cdots T_{k},&\quad k\ge 2,\\ T ,&\quad k=1,\\ 1,&\quad k=0. \end{cases} \] Then condition (\ref{eq:sufwick}) implies that for any $i=1,\dots,d$, \[ \sum_{j=1}^k \mu(e_i^*)R_{n_j}Y_j \in\mathcal{J}. \] Since the degrees of these elements are less than the degree of $Y$, we conclude that \[ \sum_{j=1}^k \mu(e_i^*)R_{n_j}Y_j =0,\quad i=1,\dots,d, \] and the independence of the Wick ordered monomials then implies \[ \mu(e_i^*)R_{n_j}Y_j =0,\quad i=1,\dots,d. \] Let $Y_k$ be the highest homogeneous component of $Y$; then, by our assumption, $\deg Y_k\ge 2$, and $\sum_{i=1}^d e_i\mu(e_i^*)R_{n_k}Y_k=R_{n_k}Y_k=0$, i.e., $Y_k\in\ker R_{n_k}$. To complete the proof, note that if $X=\beta+\sum_{i=1}^d \alpha_i e_i\in\mathcal{J}$, then for any $j$, we have \[ e_j^*\otimes X=\alpha_j +\beta e_j^*+\sum_{i=1}^d\alpha_i\sum_{k,l=1}^d T_{ji}^{kl}e_l\otimes e_k^*, \] and (\ref{eq:sufwick}) implies $\alpha_j=0$, $j=1,\dots,d$, $\beta =0$. \end{proof} \section{The structure of $\ker P_n$}\label{S2} In this section, we show that for Wick algebras with braided $T$ satisfying the condition $-10$, $n\ge 2$, i.e., the Fock inner product is strictly positive, and the Fock representation acts in the whole space $\mathcal{T}(\hb)$. \end{proposition} \begin{proof} Recall that if $T$ is braided and $\Vert T\Vert\le 1$ then $P_n\ge 0$ (see \cite{bib:2}). It remains only to show that $\ker P_n=\{0\}$ for $-1. \] Moreover, the Fock representation of this algebra is bounded. 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\or{9}{11}\or{10}{12}\or{\@xipt}{13}% \or{\@xiipt}{14}\or{\@xivpt}{17}}% \normalsize \linespacing=\baselineskip } \DeclareOption{9pt}{\def\@mainsize{9}\def\@ptsize{9}% \def\@typesizes{% \or{5}{6}\or{5}{6}\or{6}{7}\or{7}{8}\or{8}{10}% \or{9}{11}% normalsize \or{10}{12}\or{\@xipt}{13}\or{\@xiipt}{14}% \or{\@xivpt}{17}\or{\@xviipt}{20}}% \normalsize \linespacing=\baselineskip } \def\ps@empty{\let\@mkboth\@gobbletwo \def\@oddhead{\hyper@anchor{page.\thepage}{}} \let\@evenhead\@oddhead \let\@oddfoot\@empty \let\@evenfoot\@empty \global\topskip\normaltopskip} \def\ps@plain{\ps@empty \def\@oddfoot{\runhdfnt \hfil\thepage\hfil}% removed \scriptsize \let\@evenfoot\@oddfoot} \def\ps@headings{\ps@empty \def\@evenhead{\runhdfnt \rlap{\hyper@anchor{page.\thepage}\thepage}\hfil \leftmark{}{}\hfil}% \def\@oddhead{\runhdfnt \hfil \rightmark{}{}\hfil \llap{\hyper@anchor{page.\thepage}\thepage}}% \let\@mkboth\markboth } \let\sectionname\@empty \let\subsectionname\@empty \let\subsubsectionname\@empty \let\paragraphname\@empty \let\subparagraphname\@empty \def\leftmark{\expandafter\@firstoftwo\topmark{}{}} \def\rightmark{\expandafter\@secondoftwo\botmark{}{}} \def\ps@firstpage{\ps@plain} \long\def\@nilgobble#1\@nil{} \def\markboth#1#2{% \begingroup \@temptokena{{#1}{#2}}\xdef\@themark{\the\@temptokena}% \mark{\the\@temptokena}% \endgroup \if@nobreak\ifvmode\nobreak\fi\fi} \def\ps@myheadings{\ps@headings \let\@mkboth\@gobbletwo} \newskip\normaltopskip \normaltopskip=10pt \relax \let\sectionmark\@gobble \let\subsectionmark\@gobble \let\subsubsectionmark\@gobble \let\paragraphmark\@gobble \DeclareOption{makeidx}{} \input{amsgen.sty} \ExecuteOptions{leqno,centertags,letterpaper,portrait,% 10pt,twoside,onecolumn,final} \ProcessOptions\relax \if@compatibility \def\@tempa{\RequirePackage{amstex}\relax} \else \@ifclasswith{\@classname}{nomath}{% \let\@tempa\relax }{% \def\@tempa{\RequirePackage{amsmath}\relax}% }% \fi \@tempa % load amstex.sty or amsmath.sty \providecommand\numberwithin[2]{% \@ifundefined{c@#1}{\@nocounterr{#1}}{% \@ifundefined{c@#2}{\@nocounterr{#2}}{% \@addtoreset{#1}{#2}% \toks@\@xp\@xp\@xp{\csname the#1\endcsname}% \@xp\xdef\csname the#1\endcsname {\@xp\@nx\csname the#2\endcsname .\the\toks@}}}} \if@compatibility \else \@ifclasswith{\@classname}{noamsfonts}{% % amsfonts package is not wanted }{% % amsfonts package IS wanted; test whether a recent enough version % seems to be installed \begingroup \fontencoding{U}\fontfamily{msa}\try@load@fontshape\endgroup \global\@xp\let\csname U+msa\endcsname\relax % reset \@ifundefined{U/msa/m/n}{% \ClassError{\@classname}{% Package `amsfonts' not installed, or version too old?\MessageBreak Unable to get font info for the `msam' fonts in the expected form% }{% The amsfonts package will not be loaded, to avoid probable\MessageBreak incompatibility problems. You can (a) use the `noamsfonts' documentclass\MessageBreak option next time, or (b) check that the amsfonts package is installed\MessageBreak correctly, and is not too old to be compatible.% }% }{% \RequirePackage{amsfonts}[1995/01/01]\relax }% } \fi % end yesamsfonts branch \ifhyp %%%%%%%%%%%%%%%%%%%%%% If using hypertex macros \newcommand{\hyper@quote}{\string"} \edef\hyper@hash{\string#} \edef\hyperhash{\string#} \newcommand{\hyper@link}[2]{% \special{html:}#2\special{html:}} \newcommand{\hyper@anchor}[1]{% \special{html:}\special{html:}} \newcommand{\anchor@ref}[2]{\hyper@link{\hyper@hash#1}{\mbox{#2}}} \newcommand{\anchor@longref}[2]{\hyper@link{\hyper@hash#1}{#2}} \else %%%%%%%%%%%%%%%%%%%%%%% If not using hypertex macros \newcommand{\hyper@link}[2]{#2} \edef\hyperhash{} \newcommand{\hyper@anchor}[1]{} \newcommand{\anchor@ref}[2]{\mbox{#2}} \newcommand{\anchor@longref}[2]{#2} \fi %%%%%%%%%%%%%%%%%%%%%%% End of conditional for for hypertex \ifyandy \def\bluetext#1{\special{color push}\special{color rgb 0 0 1}#1\special{color pop}} \newcount\height \newcount\width \long\def\anchor@ref#1#2{\leavevmode\setbox0\hbox{#2}\height=\ht0\width=\wd0% \special{button: \the\width\space \the\height\space #1}\bluetext{\box0}} \long\def\anchor@longref#1#2{\leavevmode\setbox0\hbox{#2}\height=\ht0\width=\wd0% \special{button: \the\width\space \the\height\space #1}\bluetext{\unhbox0}} \def\hyper@anchor#1{\special{mark: #1}} \def\hyper@link#1#2{\leavevmode\setbox0\hbox{#2}\height=\ht0\width=\wd0% \special{button: \the\width\space \the\height\space #1}\bluetext{\unhbox0}} \fi %%%%%%%%%%%%%%%%%%%% End of conditional for yandy \newcommand{\bib@link}[2]{\anchor@ref{bibcite#1}{#2}} \newcommand{\bib@anchor}[1]{\hyper@anchor{bibcite#1}} \newcommand{\hyperlink}[2]{\hyper@link{#1}{#2}} \newcommand{\hlabel}[1]{\hyper@anchor{#1}} \newcommand{\shlabel}[1]{\hyper@anchor{#1}{#1}} \newcommand{\href}[2]{\anchor@ref{#1}{#2}} \newcommand{\shref}[1]{\anchor@ref{#1}{#1}} \newcommand{\shyperlink}[1]{\hyper@link{#1}{#1}} %%@ Here are hypertex versions of various macros defined in the format \def\@citex[#1]#2{% \let\@citea\@empty \@cite{\@for\@citeb:=#2\do {\@citea\def\@citea{,\penalty\@m\ }% \edef\@citeb{\expandafter\@iden\@citeb}% \if@filesw\immediate\write\@auxout{\string\citation{\@citeb}}\fi \@ifundefined{b@\@citeb}{{\reset@font\bfseries ?}% \G@refundefinedtrue\@latex@warning {Citation `\@citeb' on page \thepage \space undefined}}% {\hbox{\bib@link{\@citeb}{\bf\csname b@\@citeb\endcsname}}}}}{#1}} \def\@bibitem#1{\item\if@filesw \immediate\write\@auxout {\string\bibcite{#1}{\the\value{\@listctr}}}\fi% \hyper@anchor{bibcite#1}\ignorespaces} \def\@lbibitem[#1]#2{\item[\@biblabel{\bib@anchor{#2}#1}\hfill]\if@filesw {\let\protect\noexpand \immediate \write\@auxout{\string\bibcite{#2}{#1}}}\fi\ignorespaces} %\def\@setref#1#2#3{\ifx#1\relax % \protect\@noref{#3}% % \else % \anchor@ref{#3}{\protect\textup{\expandafter#2#1\hbox{}}}% % \fi} \newcommand{\@noref}[1]{% \G@refundefinedtrue \nfss@text{\reset@font\bfseries ??}% \@latex@warning{Reference `#1' on page \thepage\space undefined}% } \def\old@setref#1#2#3{\ifx#1\relax \protect\@noref{#3}% \else \protect\textup{\expandafter#2#1\hbox{}}% \fi} \def\page@setref#1#2#3{\ifx#1\relax \protect\@noref{#3}% \else \anchor@ref{page\expandafter#2#1}{\protect\textup{\expandafter#2#1\hbox{}}}% \fi} \def\sectionlabel#1{\@bsphack \protected@write\@auxout{}% {\string\newlabel{#1}{{\@currentlabel}{\thepage}}}% \protected@write\@auxout{}% {\string\newlabel{sectionlabel@#1}{sectunit\thetoclink}}% \@esphack\hyper@anchor{#1}} \def\@setref#1#2#3{\ifx#1\relax \protect\@noref{#3}% \else \@ifundefined{r@sectionlabel@#3}{\anchor@ref{#3}}{\anchor@ref{\csname r@sectionlabel@#3\endcsname}}% {\protect\textup{\expandafter#2#1\hbox{}}}% \fi} \def\label#1{\@bsphack \protected@write\@auxout{}% {\string\newlabel{#1}{{\@currentlabel}{\thepage}}}% \@esphack\hyper@anchor{#1}} \def\oldpageref#1{\expandafter\old@setref\csname r@#1\endcsname \@secondoftwo{#1}} \def\pageref#1{\expandafter\page@setref\csname r@#1\endcsname \@secondoftwo{#1}} \newcounter{totfootns} \def\footnote{\stepcounter{totfootns}\@ifnextchar[\@xfootnote{\stepcounter\@mpfn \protected@xdef\@thefnmark{\thempfn}% \h@footnotemark\h@footnotetext}} \def\@xfootnote[#1]{% \begingroup \csname c@\@mpfn\endcsname #1\relax \unrestored@protected@xdef\@thefnmark{\thempfn}% \endgroup \h@footnotemark\@hfootnotetext} \long\def\h@footnotetext#1{\insert\footins{% \normalfont\footnotesize \interlinepenalty\interfootnotelinepenalty \splittopskip\footnotesep \splitmaxdepth \dp\strutbox \floatingpenalty\@MM \hsize\columnwidth \@parboxrestore \parindent\normalparindent \sloppy \edef\@currentlabel{\p@footnote\@thefnmark}% \@makefntext{\rule\z@\footnotesep\ignorespaces% \hyper@anchor{footnote\thetotfootns}#1\unskip\strut\par}}} \def\h@footnotemark{% \leavevmode \ifhmode\edef\@x@sf{\the\spacefactor}\nobreak\fi \anchor@ref{footnote\thetotfootns}{\@makefnmark} \ifhmode\spacefactor\@x@sf\fi \relax} %% End of redefinitions for hypertex \newcommand{\uppercasenonmath}[1]{\toks@\@emptytoks \@xp\@skipmath\@xp\@empty#1$$% \edef#1{\@nx\@upprep\the\toks@}% } \newcommand{\@upprep}{% \spaceskip1.3\fontdimen2\font plus1.3\fontdimen3\font \upchars@} \newcommand{\upchars@}{% \def\ss{SS}\def\i{I}\def\j{J}\def\ae{\AE}\def\oe{\OE}% \def\o{\O}\def\aa{\AA}\def\l{\L}\def\Mc{M{\scshape c}}} \newcommand{\@skipmath}{} \long\def\@skipmath#1$#2${% \@xskipmath#1\(\)% \@ifnotempty{#2}{\toks@\@xp{\the\toks@$#2$}\@skipmath\@empty}}% \newcommand{\@xskipmath}{} \long\def\@xskipmath#1\(#2\){% \uppercase{\toks@\@xp\@xp\@xp{\@xp\the\@xp\toks@#1}}% \@ifnotempty{#2}{\toks@\@xp{\the\toks@\(#2\)}\@xskipmath\@empty}}% \newcommand{\today}{% \relax\ifcase\month\or January\or February\or March\or April\or May\or June\or July\or August\or September\or October\or November\or December\fi \space\number\day, \number\year} \DeclareOldFontCommand{\rm}{\normalfont\rmfamily}{\mathrm} \DeclareOldFontCommand{\sf}{\normalfont\sffamily}{\mathsf} \DeclareOldFontCommand{\tt}{\normalfont\ttfamily}{\mathtt} \DeclareOldFontCommand{\bf}{\normalfont\bfseries}{\mathbf} \DeclareOldFontCommand{\it}{\normalfont\itshape}{\mathit} \DeclareOldFontCommand{\sl}{\normalfont\slshape}{\@nomath\sl} \DeclareOldFontCommand{\sc}{\normalfont\scshape}{\@nomath\sc} \renewcommand*{\title}[2][]{\gdef\shorttitle{#1}\gdef\@title{#2}} \edef\title{\@nx\@dblarg \@xp\@nx\csname\string\title\endcsname} \renewcommand{\author}[2][]{% \ifx\@empty\authors \gdef\shortauthors{#1}\gdef\authors{#2}% \else \g@addto@macro\shortauthors{\and#1}% \g@addto@macro\authors{\and#2}% \g@addto@macro\addresses{\author{}}% \fi } \edef\author{\@nx\@dblarg \@xp\@nx\csname\string\author\endcsname} \let\shortauthors\@empty \let\authors\@empty \let\addresses\@empty \let\thankses\@empty \let\finalcomment\@empty \newcommand{\address}[2][]{\g@addto@macro\addresses{\address{#1}{#2}}} \newcommand{\curraddr}[2][]{\g@addto@macro\addresses{\curraddr{#1}{#2}}} \newcommand{\email}[2][]{\g@addto@macro\addresses{\email{#1}{#2}}} \newcommand{\urladdr}[2][]{\g@addto@macro\addresses{\urladdr{#1}{#2}}} \renewcommand{\thanks}[1]{\g@addto@macro\thankses{\thanks{#1}}} \def\@setaddresses{\par \begingroup \footnotesize \def\author##1{\nobreak\addvspace\bigskipamount}% %\def\\{\unskip, \ignorespaces}% \interlinepenalty\@M \def\address##1##2{\begingroup \par\addvspace\bigskipamount\noindent \@ifnotempty{##1}{(\ignorespaces##1\unskip) }% {\scshape\ignorespaces##2}\par\endgroup}% \def\curraddr##1##2{\noindent\begingroup \@ifnotempty{##2}{{\itshape Current address}% \@ifnotempty{##1}{, \ignorespaces##1\unskip}\/:\newline ##2\par\endgroup}}% \def\email##1##2{\noindent\begingroup\normalfont\footnotesize \@ifnotempty{##2}{\nobreak{\itshape E-mail address}: %\@ifnotempty{##1}{##1:} ##2\par}\endgroup}% \def\urladdr##1##2{\noindent\begingroup\normalfont\footnotesize \@ifnotempty{##2}{\nobreak \@ifnotempty{##1}{##1:\space}% \shyperlink{##2}\par}\endgroup}% \addresses \endgroup } \let\@date\@empty \newcommand\recdate[1]{\gdef\@recdate{#1}} \let\@recdate\@empty \def\dedicatory#1{\def\@dedicatory{#1}} \let\@dedicatory=\@empty \def\keywords#1{\def\@keywords{#1}} \let\@keywords=\@empty \def\subjclass#1{\def\@subjclass{#1}} \let\@subjclass=\@empty \def\commby#1{\def\@commby{(Communicated by #1)}} \let\@commby=\@empty \def\translator#1{% \ifx\@empty\@translators \def\@translators{#1}% \else\g@addto@macro\@translators{\and#1}\fi} \let\@translators=\@empty \def\@settranslators{\par\begingroup \addvspace{6\p@\@plus9\p@}% \hbox to\columnwidth{\hss\normalfont\normalsize Translated by % \andify\@translators %\uppercasenonmath\@translators \@translators} \endgroup } \newcommand{\xandlist}[4]{\@andlista{{#1}{#2}{#3}}#4\and\and} \def\@andlista#1#2\and#3\and{\@andlistc{#2}\@ifnotempty{#3}{% \@andlistb#1{#3}}} \def\@andlistb#1#2#3#4#5\and{% \@ifempty{#5}{% \@andlistc{#2#4}% }{% \@andlistc{#1#4}\@andlistb{#1}{#3}{#3}{#5}% }} \let\@andlistc\@iden \newcommand{\nxandlist}[4]{% \def\@andlistc##1{\toks@\@xp{\the\toks@##1}}% \toks@{\toks@\@emptytoks \@andlista{{#1}{#2}{#3}}}% \the\@xp\toks@#4\and\and \edef#4{\the\toks@}% \let\@andlistc\@iden} \newcommand{\andify}{% \nxandlist{\unskip, }{\unskip{} and~}{\unskip, and~}} \def\and{\unskip{ }and \ignorespaces} \def\maketitle{\par \@topnum\z@ % this prevents figures from falling at the top of page 1 \uppercasenonmath\shorttitle \ifx\@empty\shortauthors \let\shortauthors\shorttitle \else \andify\shortauthors \uppercasenonmath\shortauthors\fi \@maketitle@hook \begingroup \@maketitle \toks@\@xp{\shortauthors}\@temptokena\@xp{\shorttitle}% \edef\@tempa{\@nx\markboth{\the\toks@}{\the\@temptokena}}\@tempa \endgroup \thispagestyle{firstpage}% this sets first page specifications \c@footnote\z@ \def\do##1{\let##1\relax}% \do\maketitle \do\@maketitle \do\title \do\@xtitle \do\@title \do\author \do\@xauthor \do\address \do\@xaddress \do\email \do\@xemail \do\curraddr \do\@xcurraddr \do\commby \do\@commby \do\dedicatory \do\@dedicatory \do\thanks \do\thankses \do\keywords \do\@keywords \do\subjclass \do\@subjclass } \def\@maketitle@hook{\global\let\@maketitle@hook\@empty} \def\@setrecdate{\ifx\@empty\@recdate% \@footnotetext{\bfseries Please supply date!} \else\@footnotetext{Received \@recdate.}\fi} \newbox\thanksbox \newcommand{\pubyr}[1]{\def\pubyear{#1}} \newcommand{\vol}[1]{\def\volumenum{#1}} \newcommand{\iss}[1]{\def\issuenum{#1}} \iffin\RequirePackage{pjmlogo}\fi \def\@maketitle{% \normalfont\normalsize \let\@makefnmark\relax \let\@thefnmark\relax \ifx\@empty\@subjclass\else \@footnotetext{\@setsubjclass}\fi \ifx\@empty\@keywords\else \@footnotetext{\@setkeywords}\fi \ifx\@empty\thankses\else %\@footnotetext{% % \def\par{\let\par\@par}\@setthanks}\fi \global\setbox\thanksbox=\vtop \bgroup\noindent\normalfont\footnotesize \parindent=\z@ \def\par{\let\par\@par}\@setthanks \egroup\fi \@mkboth{\@nx\shortauthors}{\@nx\shorttitle}% \iffin\pjmlogobox\else\vbox to 44pt{\vss}\fi % \global\topskip42\p@\relax % 5.5pc " " " " " \@settitle \ifx\@empty\authors \else \@setauthors \fi \ifx\@empty\@dedicatory \else \baselineskip18\p@ \vtop{\centering{\footnotesize\itshape\@dedicatory\@@par}% \global\dimen@i\prevdepth}\prevdepth\dimen@i \fi \@setabstract \normalsize \if@titlepage \newpage \else \dimen@12\p@ \advance\dimen@-\baselineskip \vskip\dimen@ plus 14\p@\relax \fi } % end \@maketitle \@ifundefined{ISSN}{\def\ISSN{0000-0000}}{} \newcommand\PII[1]{\def\@PII{#1}} \PII{S \ISSN(XX)0000-0} \newinsert\copyins \skip\copyins=1.5pc \count\copyins=1000 % magnification factor, 1000 = 100% \dimen\copyins=.5\textheight % maximum allowed per page \def\@combinefloats{% \ifx \@toplist\@empty \else \@cflt \fi \ifx \@botlist\@empty \else \@cflb \fi \ifvoid\copyins \else \@cflci \fi } \newcommand{\abstractname}{Abstract} \newcommand{\keywordsname}{Key words and phrases} \newcommand{\subjclassname}{Mathematics Subject Classification} \def\@tempb{pjm} \ifx\@classname\@tempb \newcommand{\datename}{Received} \else \newcommand{\datename}{Received} \fi \def\@settitle{\begin{center}\bfseries\mathversion{bold}\uppercasenonmath\@title\@title\end{center}} \def\@setauthors{% \begingroup \trivlist \centering \@topsep24\p@\relax \advance\@topsep by -\baselineskip \item\relax \andify\authors \scshape \leavevmode\authors \endtrivlist \endgroup } \def\@setdate{\noindent\footnotesize\datename\ \@date} \def\@setfinalcomment{\noindent\footnotesize\scshape\finalcomment} \def\@setsubjclass{% {\itshape\subjclassname.}\enspace\@subjclass\@addpunct.} \def\@setkeywords{% {\itshape \keywordsname.}\enspace \@keywords\@addpunct.} \def\@setthanks{\def\thanks##1{\par##1\@addpunct.}\thankses} \newbox\abstractbox \newenvironment{abstract}{% \ifx\maketitle\relax \ClassWarning{\@classname}{Abstract should precede \protect\maketitle; reported}% \fi \global\setbox\abstractbox=\vtop \bgroup \normalfont\small \small\bfseries\mathversion{bold} \list{}{\labelwidth\z@ \leftmargin2.2pc \rightmargin\leftmargin \listparindent\normalparindent \itemindent\z@ \parsep\z@ \@plus\p@ \let\fullwidthdisplay\relax }% \item[\indent]% }{% \endlist\egroup \ifx\@setabstract\relax \@setabstracta \fi } \def\@setabstract{\@setabstracta \global\let\@setabstract\relax} \def\@setabstracta{% \ifvoid\abstractbox \else \skip@16\p@ plus 10\p@ \advance\skip@-\lastskip \advance\skip@-\baselineskip \vskip\skip@ \box\abstractbox \prevdepth\z@ % because \abstractbox is a vtop \fi } \def\titlepage{% \clearpage \thispagestyle{empty}\setcounter{page}{0}} \def\endtitlepage{\newpage} \def\labelenumi{\theenumi)} \def\theenumi{\@arabic\c@enumi} \def\labelenumii{\theenumii)} \def\theenumii{\@alph\c@enumii} \def\p@enumii{\theenumi} \def\labelenumiii{(\theenumiii)} \def\theenumiii{\@roman\c@enumiii} \def\p@enumiii{\theenumi(\theenumii)} \def\labelenumiv{(\theenumiv)} \def\theenumiv{\@Alph\c@enumiv} \def\p@enumiv{\p@enumiii\theenumiii} \def\labelitemi{$\m@th\bullet$} \def\labelitemii{\bfseries --}% \upshape already done by \itemize \def\labelitemiii{$\m@th\ast$} \def\labelitemiv{$\m@th\cdot$} \newenvironment{verse}{\let\\\@centercr \list{}{\itemsep\z@ \itemindent -1.5em\listparindent\itemindent \rightmargin\leftmargin \advance\leftmargin 1.5em}\item[]% }{% \endlist } \let\endverse=\endlist % for efficiency \newenvironment{quotation}{\list{}{% \leftmargin3pc \listparindent\normalparindent \itemindent\z@ \rightmargin\leftmargin \parsep\z@ \@plus\p@}% \item[]% }{% \endlist } \let\endquotation=\endlist % for efficiency \newenvironment{quote}{% \list{}{\rightmargin\leftmargin}\item[]% }{% \endlist } \let\endquote=\endlist % for efficiency \def\trivlist{\parsep\parskip\@nmbrlistfalse \@trivlist \labelwidth\z@ \leftmargin\z@ \itemindent\z@ \let\@itemlabel\@empty \def\makelabel##1{\upshape##1}} \renewenvironment{enumerate}{% \ifnum \@enumdepth >3 \@toodeep\else \advance\@enumdepth \@ne \edef\@enumctr{enum\romannumeral\the\@enumdepth}\list {\csname label\@enumctr\endcsname}{\usecounter {\@enumctr}\def\makelabel##1{\hss\llap{\upshape##1}}}\fi }{% \endlist } \let\endenumerate=\endlist % for efficiency \renewenvironment{itemize}{% \ifnum\@itemdepth>3 \@toodeep \else \advance\@itemdepth\@ne \edef\@itemitem{labelitem\romannumeral\the\@itemdepth}% \list{\csname\@itemitem\endcsname}% {\def\makelabel##1{\hss\llap{\upshape##1}}}% \fi }{% \endlist } \let\enditemize=\endlist % for efficiency \newcommand{\descriptionlabel}[1]{\hspace\labelsep \upshape\bfseries #1:} \newenvironment{description}{\list{}{% \advance\leftmargini6\p@ \itemindent-12\p@ \labelwidth\z@ \let\makelabel\descriptionlabel}% }{ \endlist } \let\enddescription=\endlist % for efficiency \let\upn=\textup \AtBeginDocument{% \settowidth\leftmargini{\labelenumi\hskip\labelsep}% \advance\leftmargini by \normalparindent \settowidth\leftmarginii{\labelenumii\hskip\labelsep}% \advance\leftmarginii by 6pt \settowidth\leftmarginiii{\labelenumiii\hskip\labelsep}% \advance\leftmarginiii by 6pt \settowidth\leftmarginiv{\labelenumiv\hskip\labelsep}% \advance\leftmarginiv by 10pt \leftmarginv=10pt \leftmarginvi=10pt \leftmargin=\leftmargini \labelsep=5pt \labelwidth=\leftmargini \advance\labelwidth-\labelsep \@listi} \newskip\listisep \listisep\smallskipamount \def\@listI{\leftmargin\leftmargini \parsep\z@skip \topsep\listisep \itemsep\z@skip \listparindent\normalparindent} \let\@listi\@listI \def\@listii{\leftmargin\leftmarginii \labelwidth\leftmarginii \advance\labelwidth-\labelsep \topsep\z@skip \parsep\z@skip \partopsep\z@skip \itemsep\z@skip} \def\@listiii{\leftmargin\leftmarginiii \labelwidth\leftmarginiii \advance\labelwidth-\labelsep} \def\@listiv{\leftmargin\leftmarginiv \labelwidth\leftmarginiv \advance\labelwidth-\labelsep} \def\@listv{\leftmargin\leftmarginv \labelwidth\leftmarginv \advance\labelwidth-\labelsep} \def\@listvi{\leftmargin\leftmarginvi \labelwidth\leftmarginvi \advance\labelwidth-\labelsep} \def\@startsection#1#2#3#4#5#6{% \if@noskipsec \leavevmode \fi \par \@tempskipa #4\relax \@afterindenttrue \ifdim \@tempskipa <\z@ \@tempskipa -\@tempskipa \@afterindentfalse\fi \if@nobreak \everypar{}\else \addpenalty\@secpenalty\addvspace\@tempskipa\fi \@ifstar{\@dblarg{\@sect{#1}{\@m}{#3}{#4}{#5}{#6}}}% {\@dblarg{\@sect{#1}{#2}{#3}{#4}{#5}{#6}}}% } \def\@secnumfont{\bfseries} \setcounter{tocdepth}{2} \newcounter{toclink}\setcounter{toclink}{\m@ne} \def\@sect#1#2#3#4#5#6[#7]#8{\stepcounter{toclink}% \edef\@toclevel{\ifnum#2=\@m 0\else\number#2\fi}% \ifnum #2>\c@secnumdepth \let\@secnumber\@empty \else \@xp\let\@xp\@secnumber\csname the#1\endcsname\fi \ifnum #2>\c@secnumdepth \let\@svsec\@empty \else \refstepcounter{#1}% \edef\@svsec{\ifnum#2<\@m \@ifundefined{#1name}{}{% \ignorespaces\csname #1name\endcsname\space}\fi \@nx\textup{% \@nx\@secnumfont \csname the#1\endcsname.}\enspace }% \fi \@tempskipa #5\relax \ifdim \@tempskipa>\z@ % then this is not a run-in section heading \begingroup #6\relax\hyper@anchor{sectunit\thetoclink}% \@hangfrom{\hskip #3\relax\@svsec}{\interlinepenalty\@M #8\par}% \endgroup \ifnum#2>\@m \else \@tocwrite{#1}{\protect\anchor@longref{sectunit\thetoclink}{#7}}\fi \else \def\@svsechd{\hyper@anchor{sectunit\thetoclink}#6\hskip #3\@svsec \@ifnotempty{#8}{\ignorespaces#8\unskip \@addpunct.}% \ifnum#2>\c@tocdepth \else \@tocwrite{#1}{\protect\anchor@longref{sectunit\thetoclink}{#7}}\fi }% \fi \global\@nobreaktrue \@xsect{#5}} \let\@ssect\relax \newcounter{part} \newcounter{section} \newcounter{subsection}[section] \newcounter{subsubsection}[subsection] \newcounter{paragraph}[subsubsection] \renewcommand\thepart {\Roman{part}} \renewcommand\thesection {\arabic{section}} \renewcommand\thesubsection {\thesection.\arabic{subsection}} \renewcommand\thesubsubsection {\thesubsection .\arabic{subsubsection}} \renewcommand\theparagraph {\thesubsubsection.\arabic{paragraph}} \setcounter{secnumdepth}{3} \def\partname{Part} \def\part{\@startsection{part}{0}% \z@{\linespacing\@plus\linespacing}{.5\linespacing}% {\normalfont\Large\bfseries\centering}} \def\specialsection{\@startsection{section}{1}% \z@{\linespacing\@plus\linespacing}{.5\linespacing}% {\normalfont\Large\bfseries\raggedright}} \def\section{\@startsection{section}{1}% \z@{-12.9pt plus -2.5pt minus -2pt}{8pt}% {\normalfont\bfseries\centering}} \def\subsection{\@startsection{subsection}{2}% \z@{-9pt plus -2.5pt minus -12pt}{-3pt}% {\normalfont\bfseries}} \def\subsubsection{\@startsection{subsubsection}{3}% \z@{-7pt plus -2.5pt minus -10pt}{-3pt}% {\normalfont\bfseries}} \def\paragraph{\@startsection{paragraph}{4}% \z@\z@{-\fontdimen2\font}% \normalfont\itshape} \def\subparagraph{\@startsection{subparagraph}{5}% \z@\z@{-\fontdimen2\font}% \normalfont\itshape} \def\appendix{\par\c@section\z@ \c@subsection\z@ \let\sectionname\appendixname \def\thesection{\@Alph\c@section}} \def\appendixname{Appendix} \def\@Roman#1{\@xp\@slowromancap \romannumeral#1@}% \def\@slowromancap#1{\ifx @#1% then terminate \else \if i#1I\else\if v#1V\else\if x#1X\else\if l#1L\else\if c#1C\else\if m#1M\else#1\fi\fi\fi\fi\fi\fi \@xp\@slowromancap \fi } \newcommand{\@pnumwidth}{1.6em} \newcommand{\@tocrmarg}{2.6em} \setcounter{tocdepth}{2} \def\@starttoc#1#2{\begingroup \par\removelastskip\vskip\z@skip \@startsection{}\@M\z@{\linespacing\@plus\linespacing}% {.5\linespacing}{\centering\scshape}{#2}% \ifx\contentsname#2% \else \addcontentsline{toc}{section}{#2}\fi \makeatletter \@input{\jobname.#1}% \if@filesw \@xp\newwrite\csname tf@#1\endcsname \immediate\@xp\openout\csname tf@#1\endcsname \jobname.#1\relax \fi \global\@nobreakfalse \endgroup \addvspace{20\p@\@plus14\p@}% \let\tableofcontents\relax } \def\contentsname{Contents} \def\listfigurename{List of Figures} \def\listtablename{List of Tables} \def\tableofcontents{\@starttoc{toc}\contentsname} \def\listoffigures{\@starttoc{lof}\listfigurename} \def\listoftables{\@starttoc{lot}\listtablename} \AtBeginDocument{% \@for\@tempa:=-1,0,1,2,3\do{% \@ifundefined{r@tocindent\@tempa}{% \@xp\gdef\csname r@tocindent\@tempa\endcsname{0pt}}{}% }% } \def\@writetocindents{% \begingroup \@for\@tempa:=-1,0,1,2,3\do{% \immediate\write\@auxout{% \string\newlabel{tocindent\@tempa}{% \csname r@tocindent\@tempa\endcsname}}% }% \endgroup} \AtEndDocument{\@writetocindents} \let\indentlabel\@empty \def\@tochangmeasure#1{\sbox\z@{#1}% \ifdim\wd\z@>\csname r@tocindent\@toclevel\endcsname\relax \@xp\xdef\csname r@tocindent\@toclevel\endcsname{\the\wd\z@}% \fi } \def\@toclevel{0} \def\@tocline#1#2#3#4#5#6#7{\relax \ifnum #1>\c@tocdepth % then omit \else \par \addpenalty\@secpenalty\addvspace{#2}% \begingroup \hyphenpenalty\@M \@ifempty{#4}{% \@tempdima\csname r@tocindent\number#1\endcsname\relax }{% \@tempdima#4\relax }% \parindent\z@ \leftskip#3\relax \advance\leftskip\@tempdima\relax \rightskip\@pnumwidth plus1em \parfillskip-\@pnumwidth #5\leavevmode\hskip-\@tempdima #6\relax \hfil\hbox to\@pnumwidth{\@tocpagenum{#7}}\par \nobreak \endgroup \fi} \def\@tocpagenum#1{\hss{\mdseries #1}} \def\@tocwrite#1{\@xp\@tocwriteb\csname toc#1\endcsname{#1}} \def\@tocwriteb#1#2#3{% \begingroup \def\@tocline##1##2##3##4##5##6{% \ifnum##1>\c@tocdepth \else \sbox\z@{##5\let\indentlabel\@tochangmeasure##6}\fi}% \csname l@#2\endcsname{#1{\csname#2name\endcsname}{\@secnumber}{}}% \endgroup \addcontentsline{toc}{#2}% {\protect#1{\csname#2name\endcsname}{\@secnumber}{#3}}} \def\l@section{\@tocline{1}{2pt}{1pc}{}{}} \newcommand{\tocsection}[3]{% \indentlabel{\@ifnotempty{#2}{\ignorespaces#1 #2.\quad}}#3} \def\l@subsection{\@tocline{2}{2pt}{3pc}{5pc}{}} \let\tocsubsection\tocsection \def\l@subsubsection{\@tocline{3}{2pt}{5pc}{7pc}{}} \let\tocparagraph\tocsection \let\tocsubsubsection\tocsection \def\l@part{\@tocline{-1}{12pt plus2pt}{0pt}{}{\bfseries}} \let\tocpart\tocsection \def\l@chapter{\@tocline{0}{8pt plus1pt}{0pt}{}{}} \let\tocchapter\tocsection \let\tocappendix\tocchapter \def\l@figure{\@tocline{0}{3pt plus2pt}{0pt}{}{}} \let\l@table=\l@figure \def\refname{References} \def\bibname{Bibliography} \def\bibliographystyle#1{% \if@filesw\immediate\write\@auxout {\string\bibstyle{#1}}\fi \def\@tempa{#1}% \def\@tempb{amsplain}% \def\@tempc{}% \ifx\@tempa\@tempb \def\@biblabel##1{##1.}% \def\bibsetup{}% \else \def\bibsetup{\labelsep6\p@}% \ifx\@tempa\@tempc \def\@biblabel##1{}% \def\bibsetup{\labelwidth\z@ \leftmargin24\p@ \itemindent-24\p@ \labelsep\z@ }% \fi \fi} \newenvironment{thebibliography}[1]{% \@xp\section\@xp*\@xp{\refname}% \normalfont\footnotesize\labelsep .5em\relax \renewcommand\theenumiv{\arabic{enumiv}}\let\p@enumiv\@empty \list{\@biblabel{\theenumiv}}{\settowidth\labelwidth{\@biblabel{#1}}% \leftmargin\labelwidth \advance\leftmargin\labelsep \setlength{\itemsep}{3pt} \usecounter{enumiv}}% \sloppy \clubpenalty\@M \widowpenalty\clubpenalty \sfcode`\.=\@m }{% \def\@noitemerr{\@latex@warning{Empty `thebibliography' environment}}% \endlist } \def\bysame{\leavevmode\hbox to3em{\hrulefill}\thinspace} \def\newblock{} \newcommand\MR[1]{\relax\ifhmode\unskip\spacefactor3000 \space\fi \def\@tempa##1:##2:##3\@nil{% \ifx @##2\@empty##1\else\textbf{##1:}##2\fi}% \MRhref{#1}{MR \@tempa#1:@:\@nil}} \newcommand\URL{\begingroup \def\@sverb##1{% \def\@tempa####1##1{\@URL{####1}\egroup\endgroup}% \@tempa}% \verb} \let\URLhref\@gobble \def\@URL#1{\URLhref{#1}#1} \newif\if@restonecol \def\theindex{\@restonecoltrue\if@twocolumn\@restonecolfalse\fi \columnseprule\z@ \columnsep 35\p@ \twocolumn[\@xp\section\@xp*\@xp{\indexname}]% \thispagestyle{plain}% \let\item\@idxitem \parindent\z@ \parskip\z@\@plus.3\p@\relax \footnotesize} \def\indexname{Index} \def\@idxitem{\par\hangindent 2em} \def\subitem{\par\hangindent 2em\hspace*{1em}} \def\subsubitem{\par\hangindent 3em\hspace*{2em}} \def\endtheindex{\if@restonecol\onecolumn\else\clearpage\fi} \def\indexspace{\par\bigskip} \def\footnoterule{\kern-.4\p@ \hrule\@width 5pc\kern11\p@\kern-\footnotesep} \def\@makefnmark{\hbox{$\m@th^{\@thefnmark}$}} \def\@makefntext{\indent\@makefnmark} \long\def\@footnotetext#1{\insert\footins{% \normalfont\footnotesize \interlinepenalty\interfootnotelinepenalty \splittopskip\footnotesep \splitmaxdepth \dp\strutbox \floatingpenalty\@MM \hsize\columnwidth \@parboxrestore \parindent\normalparindent \sloppy \edef\@currentlabel{\p@footnote\@thefnmark}% \@makefntext{\rule\z@\footnotesep\ignorespaces#1\unskip\strut\par}}} \hfuzz=1pt \vfuzz=\hfuzz \def\sloppy{\tolerance9999 \emergencystretch 3em\relax} \setcounter{topnumber}{4} \setcounter{bottomnumber}{4} \setcounter{totalnumber}{4} \setcounter{dbltopnumber}{4} \renewcommand{\topfraction}{.97} \renewcommand{\bottomfraction}{.97} \renewcommand{\textfraction}{.03} \renewcommand{\floatpagefraction}{.9} \renewcommand{\dbltopfraction}{.97} \renewcommand{\dblfloatpagefraction}{.9} \setlength{\floatsep}{12pt plus 6pt minus 4pt} \setlength{\textfloatsep}{15pt plus 8pt minus 5pt} \setlength{\intextsep}{12pt plus 6pt minus 4pt} \setlength{\dblfloatsep}{12pt plus 6pt minus 4pt} \setlength{\dbltextfloatsep}{15pt plus 8pt minus 5pt} \setlength{\@fptop}{0pt}% removed ``plus 1fil'' \setlength{\@fpsep}{8pt}% removed ``plus 2fil'' \setlength{\@fpbot}{0pt plus 1fil} \setlength{\@dblfptop}{0pt}% removed ``plus 1fil'' \setlength{\@dblfpsep}{8pt}% removed ``plus 2fil'' \setlength{\@dblfpbot}{0pt plus 1fil} \newcommand{\fps@figure}{tbp} \newcommand{\fps@table}{tbp} \newcounter{figure} \def\@captionheadfont{\bfseries} \def\@captionfont{\normalfont} \def\ftype@figure{1} \def\ext@figure{lof} \def\fnum@figure{\figurename\ \thefigure} \def\figurename{Figure} \newenvironment{figure}{% \@float{figure}% }{% \end@float } \newcounter{table} \def\ftype@table{2} \def\ext@table{lot} \def\fnum@table{\tablename\ \thetable} \def\tablename{Table} \newenvironment{table}{% \@float{table}% }{% \end@float } \def\@floatboxreset{\global\@minipagefalse \centering} \long\def\@makecaption#1#2{% \setbox\@tempboxa\vbox{\color@setgroup \advance\hsize-2\captionindent\noindent \@captionfont\@captionheadfont#1\@xp\@ifnotempty\@xp {\@cdr#2\@nil}{.\@captionfont\upshape\enspace#2}% \unskip\kern-2\captionindent\par \global\setbox\@ne\lastbox\color@endgroup}% \ifhbox\@ne % the normal case \setbox\@ne\hbox{\unhbox\@ne\unskip\unskip\unpenalty\unkern}% \fi \ifdim\wd\@tempboxa=\z@ % this means caption will fit on one line \setbox\@ne\hbox to\columnwidth{\hss\kern-2\captionindent\box\@ne\hss}% \else % tempboxa contained more than one line \setbox\@ne\vbox{\unvbox\@tempboxa\parskip\z@skip \noindent\unhbox\@ne\advance\hsize-2\captionindent\par}% \fi \ifnum\@tempcnta<64 % if the float IS a figure... \addvspace\abovecaptionskip \moveright\captionindent\box\@ne \else % if the float IS NOT a figure... \moveright\captionindent\box\@ne \nobreak \vskip\belowcaptionskip \fi \relax } \newskip\abovecaptionskip \abovecaptionskip=12pt \relax \newskip\belowcaptionskip \belowcaptionskip=12pt \relax \newdimen\captionindent \captionindent=3pc \RequirePackage{amsthm}[1996/09/24] \def\@swapped#1#2{#2% \@ifnotempty{#1}{\@addpunct{.}\quad#1\unskip}} \def\thmhead@plain#1#2#3{% \thmname{#1}\thmnumber{\@ifnotempty{#1}{ }\@upn{#2}}% \thmnote{ \textmd{\upshape(#3)}}} \def\swappedhead@plain#1#2#3{% \thmnumber{\@upn{#2}}\thmname{\@ifnotempty{#2}{. }#1}% \thmnote{ \textmd{\upshape(#3)}}} \def\th@plain{% \let\thmhead\thmhead@plain \let\swappedhead\swappedhead@plain \thm@preskip.5\baselineskip\@plus.2\baselineskip \@minus.2\baselineskip \thm@postskip\thm@preskip \itshape } \def\th@definition{% \let\thmhead\thmhead@plain \let\swappedhead\swappedhead@plain \thm@preskip.5\baselineskip\@plus.2\baselineskip \@minus.2\baselineskip \thm@postskip\thm@preskip \upshape } \def\th@remark{% \thm@headfont{\scshape}% heading font bold \let\thmhead\thmhead@plain \let\swappedhead\swappedhead@plain \thm@preskip.5\baselineskip\@plus.2\baselineskip \@minus.2\baselineskip \thm@postskip\thm@preskip \upshape } \if@compatibility \let\@newpf\proof \let\proof\relax \let\endproof\relax \newenvironment{pf}{\@newpf[\proofname]}{\qed\endtrivlist} \newenvironment{pf*}[1]{\@newpf[#1]}{\qed\endtrivlist} \fi \def\nonbreakingspace{\unskip\nobreak\ \ignorespaces} \def~{\protect\nonbreakingspace} \def\@biblabel#1{\@ifnotempty{#1}{[#1]}} \def\@cite#1#2{{% \m@th\upshape\mdseries[{#1\if@tempswa, #2\fi}]}} \@ifundefined{cite }{% \expandafter\let\csname cite \endcsname\cite \edef\cite{\@nx\protect\@xp\@nx\csname cite \endcsname}% }{} \def\fullwidthdisplay{\displayindent\z@ \displaywidth\columnwidth} \edef\@tempa{\noexpand\fullwidthdisplay\the\everydisplay} \everydisplay\expandafter{\@tempa} \newcommand\seename{see also}% \newcommand\see[2]{{\em \seename\/} #1}% \newcommand\printindex{\@input{\jobname.ind}}% \DeclareRobustCommand\textprime{\leavevmode \raise.8ex\hbox{\check@mathfonts\the\scriptfont2 \char48 }} \hyphenation{acad-e-my acad-e-mies af-ter-thought anom-aly anom-alies an-ti-deriv-a-tive an-tin-o-my an-tin-o-mies apoth-e-o-ses apoth-e-o-sis ap-pen-dix ar-che-typ-al as-sign-a-ble as-sist-ant-ship as-ymp-tot-ic asyn-chro-nous at-trib-uted at-trib-ut-able bank-rupt bank-rupt-cy bi-dif-fer-en-tial blue-print busier busiest cat-a-stroph-ic cat-a-stroph-i-cally con-gress cross-hatched data-base de-fin-i-tive de-riv-a-tive dis-trib-ute dri-ver dri-vers eco-nom-ics econ-o-mist elit-ist equi-vari-ant ex-quis-ite ex-tra-or-di-nary flow-chart for-mi-da-ble forth-right friv-o-lous ge-o-des-ic ge-o-det-ic geo-met-ric griev-ance griev-ous griev-ous-ly hexa-dec-i-mal ho-lo-no-my ho-mo-thetic ideals idio-syn-crasy in-fin-ite-ly in-fin-i-tes-i-mal ir-rev-o-ca-ble key-stroke lam-en-ta-ble light-weight mal-a-prop-ism man-u-script mar-gin-al meta-bol-ic me-tab-o-lism meta-lan-guage me-trop-o-lis met-ro-pol-i-tan mi-nut-est mol-e-cule mono-chrome mono-pole mo-nop-oly mono-spline mo-not-o-nous mul-ti-fac-eted mul-ti-plic-able non-euclid-ean non-iso-mor-phic non-smooth par-a-digm par-a-bol-ic pa-rab-o-loid pa-ram-e-trize para-mount pen-ta-gon phe-nom-e-non post-script pre-am-ble pro-ce-dur-al pro-hib-i-tive pro-hib-i-tive-ly pseu-do-dif-fer-en-tial pseu-do-fi-nite pseu-do-nym qua-drat-ic quad-ra-ture qua-si-smooth qua-si-sta-tion-ary qua-si-tri-an-gu-lar quin-tes-sence quin-tes-sen-tial re-arrange-ment rec-tan-gle ret-ri-bu-tion retro-fit retro-fit-ted right-eous right-eous-ness ro-bot ro-bot-ics sched-ul-ing se-mes-ter semi-def-i-nite semi-ho-mo-thet-ic set-up se-vere-ly side-step sov-er-eign spe-cious spher-oid spher-oid-al star-tling star-tling-ly sta-tis-tics sto-chas-tic straight-est strange-ness strat-a-gem strong-hold sum-ma-ble symp-to-matic syn-chro-nous topo-graph-i-cal tra-vers-a-ble tra-ver-sal tra-ver-sals treach-ery turn-around un-at-tached un-err-ing-ly white-space wide-spread wing-spread wretch-ed wretch-ed-ly Eng-lish Euler-ian Feb-ru-ary Gauss-ian Hamil-ton-ian Her-mit-ian Jan-u-ary Japan-ese Kor-te-weg Le-gendre Mar-kov-ian Noe-ther-ian No-vem-ber Rie-mann-ian Sep-tem-ber} \def\calclayout{\advance\textheight -\headheight \advance\textheight -\headsep \oddsidemargin\paperwidth \advance\oddsidemargin -\textwidth \divide\oddsidemargin\tw@ \ifdim\oddsidemargin<.5truein \oddsidemargin.5truein \fi \advance\oddsidemargin -1truein \evensidemargin\oddsidemargin \topmargin\paperheight \advance\topmargin -\textheight \advance\topmargin -\headheight \advance\topmargin -\headsep \divide\topmargin\tw@ \ifdim\topmargin<.5truein \topmargin.5truein \fi \advance\topmargin -1.18truein\relax } \calclayout % initialize \newcommand{\authnames}[1]{\iffin\def\authornames{#1}\fi} \newcommand{\beginningpage}[1]{\iffin\setcounter{page}{#1}\def\beginpage{#1}\fi} \pagenumbering{arabic} \pagestyle{headings} \thispagestyle{plain} \renewenvironment{proof}[1][\proofname]{\par \normalfont \topsep6\p@\@plus6\p@ \trivlist \item[\hskip\labelsep\itshape #1\@addpunct{.}]\ignorespaces }{% \qed\endtrivlist } \RequirePackage{amssymb} \renewcommand{\openbox}{\leavevmode \hbox to .77778em{\hfil${\square}$\hfil}} \newcommand{\papnum}[1]{\def\papernum{#1}} \def\enddoc@text{\vspace{6pt}\par \ifx\@empty\@translators \else\@settranslators\fi \ifx\@empty\@date\else \@setdate\fi \ifx\@empty\addresses \else\noindent\@setaddresses\fi \ifvoid\thanksbox\else\vspace{6pt}\par \noindent\box\thanksbox\fi \ifx\@empty\finalcomment\else\vspace{6pt}\par\noindent\@setfinalcomment\fi \iffin\@setpaphomepage\fi \iffin\label{finalpage}\fi} \AtEndDocument{\enddoc@text} \if@compatibility \else\endinput\fi \def\tiny{\Tiny} \def\defaultfont{\normalfont} \def\rom{\textup} \endinput %% %% End of file `pjm.cls'. ---------------0011071730765--