Content-Type: multipart/mixed; boundary="-------------9810091121965" This is a multi-part message in MIME format. ---------------9810091121965 Content-Type: text/plain; name="98-636.comments" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="98-636.comments" PACS: 02.30.Jr, 42.15.-i, 42.60.Da, 42.65.Yj ---------------9810091121965 Content-Type: text/plain; name="98-636.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="98-636.keywords" oscillating cavity, wave equation, circle maps, devil's staircase ---------------9810091121965 Content-Type: application/x-tex; name="paper_figs.tex" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="paper_figs.tex" % ****** Start of file apssamp.tex ****** % % This file is part of the APS files in the REVTeX 3.1 distribution. % Version 3.1 of REVTeX, September 1996. % % Copyright (c) 1992 The American Physical Society. % % See the REVTeX 3.1 README file for restrictions and more information. % % % \documentstyle[preprint,eqsecnum,aps,epsfig]{revtex} % \documentstyle[eqsecnum,aps]{revtex} \def\btt#1{{\tt$\backslash$#1}} \def\BibTeX{\rm B{\sc ib}\TeX} \newtheorem{lem}{Lemma}[section] \newtheorem{preremark}{Remark}[section] \newenvironment{rem}{\begin{preremark}\rm}{\end{preremark}} \newtheorem{defi}{Definition}[section] \newtheorem{theorem}{Theorem}[section] \newtheorem{prop}{Proposition}[section] \newtheorem{cor}{Corollary}[section] % % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % New commands %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \newcommand{\bequ}[1]{\begin{equation}\label{#1}} \newcommand{\eequ}{\end{equation}} \newcommand{\barr}[1]{\begin{eqnarray}\label{#1}} \newcommand{\earr}{\end{eqnarray}} \newcommand{\barrz}{\begin{eqnarray*}} \newcommand{\earrz}{\end{eqnarray*}} \newcommand{\Rf}[1]{(\ref{#1})} \newcommand{\SSS}{{S^{{\rm 1}}}} \newcommand{\RR}{{\bf {R}}} \newcommand{\ZZ}{{\bf {Z}}} \newcommand{\QQ}{{\bf {Q}}} \newcommand{\RRR}{ {\bf R}^2} \newcommand{\CC}{{\bf {C}}} \newcommand{\NN}{{\bf {N}}} \newcommand{\OPH}{orientation preserving homeomorphism} \newcommand{\OPD}{orientation preserving diffeomorphism} \newcommand{\degree}{\mathop{\rm deg}\nolimits} \newcommand{\pomal}{\prec} \newcommand{\mod}{\mathop{\rm mod}\nolimits} \newcommand{\id}{\mathop{\rm Id}\nolimits} \newcommand{\der}{{\rm d}} \newcommand{\differ}[1]{{{#1}'}} \newcommand{\fal}{r_{\alpha}} \newcommand{\Fal}{R_{\alpha}} \newcommand{\calO}{{\cal O}} \newcommand{\A}{{\cal A}} \newcommand{\tim}{t} \newcommand{\len}{a} \newcommand{\ave}{A} \newcommand{\ampl}{B} \newcommand{\freq}{\Omega} \newcommand{\mov}{\theta} \newcommand{\fun}{a} \newcommand{\al}{\alpha} \newcommand{\be}{\beta} \newcommand{\coord}{x} \newcommand{\integer}{\bf Z} \newcommand{\torus}{\bf T} \newcommand{\real}{\bf R} \newcommand{\Gtil}{{\widetilde{G}}} \newcommand{\atil}{{\tilde{a}}} \newcommand{\Heps}{{H_\varepsilon}} \newcommand{\heps}{{h_\varepsilon}} \newcommand{\htil}{{\eta}} \newcommand{\calN}[2]{{{N}_{#1}(#2)}} \newcommand{\addition}{{b}} \newcommand{\psiminus}{{\Psi^{-}}} \newcommand{\psiplus}{{\Psi^{+}}} \newcommand{\psipm}{{\Psi^{\pm}}} \newcommand{\tempnote}[1]{\marginpar{\scriptsize{#1}}} \newcommand{\opensquare}{{\bf Q.E.D.}} \newcommand{\etal}{{\bf et al}} \newcommand{\proofend}{\vspace{3mm}} \renewcommand{\baselinestretch}{1.1} \begin{document} \draft \title{Theory of Circle Maps and the Problem of One-Dimensional Optical Resonator with a Periodically Moving Wall} \author{Rafael de la Llave\cite{email_rl}} \address{Department of Mathematics \\ University of Texas at Austin \\ Austin, TX 78712, USA} \author{Nikola P. Petrov\cite{email_np}} \address{Department of Physics \\ University of Texas at Austin \\ Austin, TX 78712, USA} \date{October 7, 1998} \maketitle \begin{abstract} We consider the electromagnetic field in a cavity with a periodically oscillating perfectly reflecting boundary and show that the mathematical theory of circle maps leads to several physical predictions. Notably, well-known results in the theory of circle maps (which we review briefly) imply that there are intervals of parameters where the waves in the cavity get concentrated in wave packets whose energy grows exponentially. Even if these intervals are dense for typical motions of the reflecting boundary, in the complement there is a positive measure set of parameters where the energy remains bounded. \end{abstract} \pacs{02.30.Jr, 42.15.-i, 42.60.Da, 42.65.Yj} \narrowtext %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{INTRODUCTION} In this paper, we consider the behavior of the electromagnetic field in a resonator one of whose walls is at rest and the other moving periodically. The main point we want to make is that several results in the mathematical literature of circle maps immediately yield physically important conclusions. The problem at hand is mathematically equivalent to the study of the motion of vibrating strings with a periodically moving boundary \cite{Balazs1961,Dittrichetal1997}, or the classical electromagnetic field in a periodically pulsating cavity \cite{Cooper1993b,ColeSchieve1995}. It is connected with the vacuum quantum effects in such region \cite{Moore1970,Law1994a}. The problem is also of practical importance, e.g., for the formation of short laser pulses \cite{HennebergerSchulte1966}. The goal of this paper is to show that the problem of a classical wave with a periodically moving boundary can be easily reformulated in terms of the study of long term behavior of circle maps and, therefore, that many well known results in this theory lead to physical predictions. In particular, we give proofs of several results obtained numerically by Cole and Schieve \cite{ColeSchieve1995} and others. Extensions of this approach, that will be discussed elsewhere, allow to reach conclusions for some quasi-periodic motions of small amplitude or possibly for non-homogeneous media. In the case of more than one spatial dimension, the analogous problem \cite{CooperStrauss1976} is much more complicated, so the predictions are not as clear as in the one-dimensional case and we will not discuss them further. We emphasize that the mathematical theory presented is completely rigorous and, hence, the physical predictions made are general for the assumptions stated. There are other intriguing relations for which we have no conceptual explanation. We observe that a calculation of Fulling and Davies \cite{FullingDavies1976} leads to the conclusion that the energy density radiated by a moving mirror is equal to the Schwarzian derivative of the motion of the mirror (for details see Sec.~\ref{sec:schwarz}). This Schwarzian derivative is a differential operator frequently used in the theory of one dimensional dynamical systems and particularly in the theory of circle maps. The plan of the exposition is the following. In Sec.~\ref{phys_set} we show how the physical problem can be formulated in terms of circle maps. Sec.~\ref{sec:maps-cir} contains a brief exposition of the necessary facts from the theory of circle maps, in Sec.~\ref{sec:appl-th} these facts are applied to the problem at hand and illustrated numerically, and in the conclusion we discuss the advantages of our approach. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Physical setting} \label{phys_set} \subsection{Description of the system} \label{sec:descr_system} We consider a one-dimensional optical resonator consisting of two parallel perfectly reflecting mirrors. For simplicity of notation, we will consider only the situation in which one of them is at rest at the origin of the $x$ axis while the other one is moving periodically with period $T$. The case where the two mirrors are moving periodically with a common period can be treated in a similar manner. We assume that the resonator is empty, so that the speed of the electromagnetic waves in it is equal to the speed of light, $c$. The speed of the moving mirror cannot exceed~$c$. We note that the experimental situation does not necessarily require that there is a physically moving mirror. One experimental possibility -- among others -- would be to have a material that is a good conductor or not depending on whether a magnetic field of sufficient intensity is applied to it, and then have a magnetic field applied to it in a changing region. This induces reflecting boundaries that are moving with time. Note that the boundaries of this region could move even faster than~$c$, hence the study of mirrors moving at a speed comparable to $c$ is not unphysical (even if, presumably, one would also have to discuss corrections to the boundary conditions depending on the details of the experimental realizations). We shall use dimensionless time $t$ and length $\ell$ connected with the physical (i.e., dimensional) time $t_{\rm phys}$ and length $\ell_{\rm phys}$ by $t := t_{\rm phys}/T$, $\ell := \ell_{\rm phys}/(cT)$. Let the coordinate of the moving mirror be $x=a(t)$, where $a$ is a $C^k$ function ($k=1,\,\ldots,\,\infty,\,\omega$) satisfying the conditions \begin{equation} \label{properties_a} a(t)>0 \,, \qquad |\differ{a}(t)| < 1 \,, \qquad a(t+1)=a(t) \,. \end{equation} The meaning of the first condition is that the cavity does not collapse, the second one means that the speed of the moving mirror cannot exceed the speed of light, and the third one is that the mirror's motion is periodic of period~1. An example which we will use for numerical illustrations is \begin{equation} \label{equ:a1} %\fl a(t) = \frac{\alpha}{2} + {\beta} \sin{2\pi t} \qquad \left( |\beta| < \frac{1}{2\pi}\,, \quad 0 < |\beta| < \frac{\alpha}{2} \right) \,. \end{equation} Since there are no charges and no currents, we impose the gauge conditions $A_0=0$, $\nabla\cdot{\bf A}=0$ on the 4-potential $A_\mu=(A_0, {\bf A})$ and obtain that ${\bf A}$ satisfies the homogeneous wave equation. We consider plane waves traveling in $x$-direction, so that without loss of generality, we assume that ${\bf A}(t,x) = A(t,x)\,{\bf e}_y$, and obtain that $A(t,x)$ must satisfy the homogeneous $(1+1)$-dimensional wave equation, \begin{equation} \label{eq:eq} A_{tt}(t,x) - A_{xx}(t,x) = 0 \,, \end{equation} in the domain $\Sigma := \{(t,x)\in\RRR \,|\, t_0 < t, \,\, 0\nu+2$, then the coefficients $\widehat{\psi}_k$ define a smooth function (for more details see, e.g., \cite[Sec.\ XIII.4]{Herman1979}). Of course, once we know $a\circ h^{-1}$, then, since $h^{-1}$ depends only on $G$ and is therefore determined, we can obtain~$a$. In summary, there are maps $G$ that do not come from any $a$ at all, come from infinitely many $a$'s, or come from one and only one $a$. The maps $F$ can always be obtained from one and only one~$a$. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Maps of the circle} \label{sec:maps-cir} In this section we recall briefly some facts from the theory of the dynamics of the orientation preserving homeomorphisms (OPHs) and diffeomorphisms (OPDs) of the circle $\SSS$, following closely the book of Katok and Hasselblatt \cite[Ch.\ 11, 12]{KatokHasselblatt1995}; see also \cite{deMelovanStrien1993} and \cite{Herman1979}. This is a very rich theory and we will only recall the facts that we will need in the physical application. We shall identify $\SSS$ with the quotient $\RR/\ZZ$ and use the universal covering projection $$ \pi : \RR \to \SSS \equiv \RR/\ZZ : x \mapsto \pi(x):= x \,(\mod 1) \,. $$ Another way of thinking about $\SSS$ is identifying it with the unit circle in $\CC$, and using the universal covering projection $x\mapsto e^{2\pi i x}$. Let $f:\SSS\to\SSS$ be an OPH and $F:\RR\to\RR$ be its {\em lift\/} to $\RR$, i.e., a map satisfying $f \circ \pi = \pi \circ F$. The fact that $f$ is an OPH implies that $F(x+1) = F(x) + 1$ for each $x\in \RR$, which is equivalent to saying that $F-\id$ is 1-periodic. The lift $F$ of $f$ is unique up to an additive integer constant. If a point $x\in\SSS$ is $q$-periodic, i.e., $f^q(x)=x$, then $F^q(x)=x+p$ for some $p\in\NN$. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsection{Rotation number} \label{sec:rot-num} A very important number to associate to a map of the circle is its rotation number, introduced by Poincar\'{e}. It is a measure of the average amount of rotation of a point along an orbit. \begin{defi} Let $f:\SSS\to\SSS$ be an \OPH\ and $F:\RR\to\RR$ a lift of $f$. Define \begin{equation} \label{eq:rotnum_def} \tau_0(F) := \lim_{n\to\infty} \frac{F^n(x)-x}{n} \,, \qquad \tau(f) := \tau_0(F) \,(\mod 1) \end{equation} and call $\tau(f)$ a {\em rotation number} of $f$. \end{defi} It was proven by Poincar\'{e} that the limit in \Rf{eq:rotnum_def} exists and is independent of $x$. Hence, $\tau(f)$ is well defined. The rotation number is a very important tool in classifying the possible types of behavior of the iterates of the OPHs of $\SSS$. The simplest example of an OPH of $\SSS$ is the {\em rotation\/} by $\alpha$ on $\SSS\equiv\RR/\ZZ$, $\fal:x\mapsto x+\alpha \,(\mod 1)$ (corresponding to a rotation by $2\pi\alpha$ radians on $\SSS$ thought of as the unit circle in $\CC$). The map $\Fal:x\mapsto x+\alpha$ is a lift of $\fal$, and $\tau(\fal)=\alpha \,(\mod 1)$. In the case of $r_\alpha$ there are two possibilities: \begin{itemize} % \item[(a)] If $\tau(\fal) = p/q \in \QQ$, then $R_{p/q}^q(x) = x + p$ for each $x\in \RR$, so every point in $\SSS$ is $q$-periodic for $r_{p/q}$. If $p$ and $q$ are relatively prime, $q$ is the minimal period. % \item[(b)] If $\tau(\fal) \notin \QQ$, then $\fal$ has no periodic points; every point in $\SSS$ has a dense orbit. Thus, the $\alpha$- and $\omega$-limit sets of any point $x\in\SSS$ are the whole $\SSS$, which is usually described as saying that $\SSS$ is a {\it minimal set\/} for $\fal$. [Recall that $\alpha(x)$ is the set of the points at which the orbit of $x$ accumulates in the past, and $\omega(x)$ those points where it accumulates in the future.] \end{itemize} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsection{Types of orbits of OPHs of the circle} \label{sec:typ-orb} To classify the possible orbits of OPHs of the circle, we need the following definition (for the particular case $f:\SSS\to\SSS$). \begin{defi} \label{def:hom_hetero} \begin{itemize} \item[(a)] On orbit ${\cal O}$ of $f$ is called {\em homoclinic\/} to an invariant set $T\in\SSS\setminus{\cal O}$ if $\alpha(x)=\omega(x)=T$ for any $x\in{\cal O}$. \item[(b)] An orbit ${\cal O}$ of $f$ is said to be {\em heteroclinic\/} to two disjoint invariant sets $T_1$ and $T_2$ if ${\cal O}$ is disjoint from each of them and $\alpha(x)=T_1$, $\omega(x)=T_2$ for any $x\in{\cal O}$. \end{itemize} \end{defi} With this definition, the possible types of orbits of circle OPHs were classified by Poincar\'{e} \cite{Poincare1885} as follows (for a modern pedagogical treatment see, e.g., \cite[Sec.~11.2]{KatokHasselblatt1995}): \begin{itemize} \item[(1)] For $\tau(f)=p/q\in\QQ$, all orbits of $f$ are of the following types: \begin{itemize} \item[(a)] a periodic orbit with the same period as the rotation $r_{p/q}$ and ordered in the same way as an orbit of $r_{p/q}$; \item[(b)] an orbit homoclinic to the periodic orbit if there is only one periodic orbit; \item[(c)] an orbit heteroclinic to two different periodic orbits if there are two or more periodic orbits. \end{itemize} \item[(2)] When $\tau(f)\notin\QQ$, the possible types of orbits are: \begin{itemize} \item[(a)] an orbit dense in $\SSS$ that is ordered in the same way as an orbit of $r_{\tau(f)}$ (as are the two following cases); \item[(b)] an orbit dense in a Cantor set; \item[(c)] an orbit homoclinic to a Cantor set. \end{itemize} \end{itemize} We also note that in cases 2(b) and 2(c), the Cantor set that has a dense orbit is unique and can be obtained as the set of accumulation points of any orbit. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsection{Poincar\'{e} and Denjoy theorems} \label{sec:PoincareDenjoy} Because of the simplicity of the rotations it is natural to ask whether a particular OPH of $\SSS$ is equivalent in some sense to a rotation. To state the results, we give a precise definition of ``equivalence'' and the important concept of topological transitivity. \begin{defi} \label{def:conju} \begin{itemize} \item[(a)] Let $f:M\to M$ and $g:N\to N$ be $C^m$ maps, $m\geq0$. The maps $f$ and $g$ are {\em topologically conjugate} if there exists a homeomorphism $h:M\to N$ such that $f=h^{-1}\circ g\circ h$. \item[(b)] The map $g$ is a {\em topological factor} of $f$ (or $f$ is {\em semiconjugate} to $g$) if there exists a surjective continuous map $h:M\to N$ such that $h\circ f=g\circ h$; the map $h$ is called a {\em semiconjugacy}. \item[(c)] A map $f:M\to M$ is {\em topologically transitive} provided the orbit, $\{f^k(x)\}_{k\in\ZZ}$, of some point $x$ is dense in $M$. \end{itemize} \end{defi} The meaning of the conjugacy is that $g$ becomes $f$ under a change of variables, so that from the point of coordinate independent physical quantities, $f$ and $g$ are equivalent. The meaning of the semiconjugacy is that, embedded in the dynamics of~$f$, we can find the dynamics of~$g$. The following theorem of Poincar\'{e} \cite{Poincare1885} was chronologically the first theorem classifying circle maps. \begin{theorem} [Poincar\'{e} Classification Theorem] \label{poincTh} Let $f:\SSS\to\SSS$ be an OPH with irrational rotation number. Then: \begin{itemize} \item[(a)] if $f$ is topologically transitive, then $f$ is topologically conjugate to the rotation $r_{\tau(f)}$; \item[(b)] if $f$ is not topologically transitive, then there exists a non-invertible continuous monotone map $h:\SSS\to\SSS$ such that $h \circ f = r_{\tau(f)} \circ h$; in other words, $f$ is semiconjugate to the rotation $r_{\tau(f)}$. \end{itemize} \end{theorem} If we restrict ourselves to considering not OPHs, but OPDs of the circle, we can say more about the conjugacy problem. An important result in this direction was the theorem of Denjoy~\cite{Denjoy1932}. \begin{theorem}[Denjoy Theorem] \label{th:denjoy} A $C^1$ OPD of $S^1$ with irrational rotation number and derivative of bounded variation is topologically transitive and hence (according to Poincar\'{e} theorem) topologically conjugate to a rotation. In particular, every $C^2$ OPD $f:\SSS\to\SSS$ is topologically conjugate to $r_{\tau(f)}$. \end{theorem} We note that this condition is extremely sharp. For every $\varepsilon>0$ there are $C^{2-\varepsilon}$ maps with irrational rotation number and semiconjugate but not conjugate to a rotation (see~\cite[Sec.\ X.3.19]{Herman1979}). %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsection{Smoothness of the conjugacy} \label{sec:smoothness} So far we have discussed only the conditions for existence of a conjugacy $h$ to a rotation, requiring $h$ to be only a homeomorphism. Can anything more be said about the differentiability properties of $h$ in the case of smooth or analytic maps of the circle? As we will see later, this is a physically important question since physical quantities such as energy density depend on the smoothness of the conjugacy. To answer this question precisely, we need two definitions. \begin{defi} \label{def:Dioph_cond} A number $\rho$ is called {\em Diophantine} of type $(K,\nu)$ (or simply of type~$\nu$) for $K>0$ and $\nu\geq1$, if $\left| \rho - \frac pq \right| > K \, |q|^{-1-\nu}$ for all $\frac pq \in \QQ$. The number $\rho$ is called {\em Diophantine} if it is Diophantine for some $K>0$ and $\nu\geq1$. A number which is not Diophantine is called a {\em Liouville number}. \end{defi} It can be proved that for $K\to0$, the set of all Diophantine numbers of type $(K,\nu)$ has Lebesgue measure as close to full as desired. \begin{defi} A function $f$ is said to be $C^{m-\delta}$ where $m\geq1$ is an integer and $\delta\in(0,1)$, if it is $C^{m-1}$ and its $(m-1)$st derivative is $(1-\delta)$-H\"{o}lder continuous, i.e., $$ \left|D^{m-1}f(x) - D^{m-1}f(y)\right| < {\rm const}\, \left|x-y\right|^{1-\delta} \,. $$ \end{defi} The first theorem answering the question about the smoothness of the conjugacy was the theorem of Arnold \cite{Arnold1961}. He proved that if the analytic map $f:\SSS\to\SSS$ is sufficiently close (in the sup-norm) to a rotation and $\tau(f)$ is Diophantine of type $\nu\geq1$, then $f$ is analytically conjugate to the rotation $r_{\tau(f)}$, i.e., there exists an analytic function $h:\SSS\to\SSS$ such that $h\circ f=r_{\tau(f)}\circ h$. The iterative technique applied by Arnold was fruitfully used later in the proof of the celebrated Kol\-mo\-go\-rov-Ar\-nold-Mo\-ser (KAM) theorem -- see, e.g., \cite{Wayne1996}. Arnold's result was extended to the case of finite differentiability by Moser~\cite{Moser1966}. In such a case, the Diophantine exponent $\nu$ has to be related to the number of derivatives one assumes for the map. Arnold's theorem is local, i.e., it is important that $f$ is close to a rotation. Arnold conjectured that any analytic map with a rotation number in a set of full measure is analytically conjugate to a rotation. Herman~\cite{Herman1979} proved that there exists a set ${\cal A}\subset[0,1]$ of full Lebesgue measure such that if $f\in C^k$ for $3\leq k\leq\omega$ and $\tau(f)\in {\cal A}$, then the conjugacy is $C^{k-2-\varepsilon}$ for any $\varepsilon>0$. The set ${\cal A}$ is characterized in terms of the growth of the partial quotients of the continued fraction expansions of its members; all numbers in ${\cal A}$ are Diophantine of order $\nu$ for any $\nu\geq1$. His result was improved by Yoccoz~\cite{Yoccoz1984} who showed that if $f\in C^k$, $3\leq k\leq\omega$, $\tau(f)$ is a Diophantine number of type $\nu\geq1$, and $k>2\nu-1$, then there exists a $C^{k-\nu-\varepsilon}$ conjugacy $h$ between $f$ and $r_{\tau(f)}$ for any $\varepsilon>0$, and by several others. The best result on smooth conjugacy we know of, is the following version of Herman's theorem as extended by Katznelson and Ornstein~\cite{KatznelsonOrnstein1989}. \begin{theorem}[Herman, Katznelson and Ornstein] Assume that $f$ is a $C^k$ circle OPD whose rotation number is Diophantine of order $\nu$, and $k>\nu+1$. Then the homeomorphism $h$ which conjugates $f$ with the rotation $r_{\tau(f)}$ is of class $C^{k-\nu-\varepsilon}$ for any $\varepsilon>0$. \end{theorem} There are examples of $C^{2-\varepsilon}$ maps with a Diophantine rotation number arbitrarily close to a rotation and not conjugated by an absolutely continuous function to a rotation -- see, e.g.,~\cite{HawkinsSchmidt1982}. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsection{Devil's staircase, frequency locking, Arnold's tongues} Let $\{f_\alpha\}_{\alpha\in A}$ be a one-parameter family of circle OPHs such that $f_\alpha(x)$ is increasing in $\alpha$ for every $x$. Then the function $\alpha\mapsto\tau(f_\alpha)$ is non-decreasing. (Since the maps are only defined modulo an integer and so is the rotation number, some care needs to be taken to define increasing and non-decreasing when some of the objects we are considering change integer parts. What is meant precisely is that if one takes the numbers with their integer parts, they can be made increasing or non-decreasing. This is done in detail in \cite[Sec.~11.1]{KatokHasselblatt1995}, and we will dispense with making it explicit since it does not lead to confusion.) For such a family the following fact holds: if $\tau(f_\alpha)\notin\QQ$, then $\alpha\mapsto\tau(f_\alpha)$ is strictly increasing locally at $\alpha$; on the other hand, if $f_\alpha$ has rational rotation number and the periodic point is attracting or repelling (i.e., there is a neighborhood of the point that gets mapped into itself by forwards or backwards iteration), then $\alpha\mapsto\tau(f_\alpha)$ is locally constant at this particular value of~$\alpha$, i.e., for all $\alpha'$ sufficiently close to~$\alpha$, $\tau(f_{\alpha'})=\tau(f_\alpha)$. The local constancy of the function $\alpha\mapsto\tau(f_\alpha)$ is known as {\em frequency (phase, mode) locking\/}. Note that, since the rotation number is continuous, when it indeed changes, it has to go through rational numbers. The described phenomenon suggests the following definition. \begin{defi} A monotone continuous function $\psi:[0,1]\to\RR$ is called a {\em devil's staircase} if there exists a family $\{I_\xi\}_{\xi\in\Xi}$ of disjoint open subintervals of $[0,1]$ with dense union such that $\psi$ takes constant values on these subintervals. (We call attention to the fact that the complement of the intervals in which the function is constant can be of positive measure.) The devil's staircase is said to be {\em complete\/} if the union of all intervals $I_\xi$ has a full Lebesgue measure. \end{defi} A very common way of phase locking for differentiable mappings arises when the map we consider has a periodic point and that the derivative of the return map at the periodic point is not equal to~$1$. By the implicit function theorem, such a periodic orbit persists, and the existence of a periodic orbit implies that the rotation number is locally constant. At the end of the phase locking interval the map has derivative one and experiences a saddle-node (tangent) bifurcation. We note that, unless certain combinations of derivatives vanish (see, e.g., \cite{Ruelle1989}), the saddle-node bifurcation happens in a universal way. That is, there are analytic changes of variables sending one into another. This leads to quantitative predictions. For example, the Lyapunov exponents of a periodic orbit should behave as a square root of the distance of the parameter to the edge of the phase locking interval. Of course, other things can happen in special cases: the fixed point may be attractive but only neutrally so, there may be an interval of fixed points, the family may be such that there are no frequency locking intervals (e.g., the rotation). Nevertheless, all these conditions are exceptional and can be excluded in concrete examples by explicit calculations. (For example, if the family of maps is analytic but not a root of the identity, it is impossible to have an interval of fixed points.) In the example we will consider, we will not perform a complete proof that a devil's staircase occurs, but rather we will present numerical evidence. In particular, the square root behavior of the Lyapunov exponent with the distance to the edge of the phase locking interval seems to be verified. Let us now consider two-parameter families of OPDs of the circle, $\{\phi_{\alpha,\beta}\}$, depending smoothly on $\alpha$ and $\beta$. Assume that when $\beta=0$, the maps of the family are rotations by $\alpha$, i.e., $\phi_{\alpha,\,0}=r_\alpha$. We will call $\beta$ the {\em nonlinearity parameter}. Assume also that $\partial\phi_{\alpha,\beta}/\partial\alpha>0$. An example of this type is the family studied by Arnold~\cite{Arnold1961}, \bequ{eq:Arnmap} \eta_{\alpha,\beta} : \SSS \to \SSS : x \mapsto \eta_{\alpha,\beta} (x) := x + \alpha + \beta\sin 2\pi x \,(\mod 1) \,, \eequ where $\alpha\in [0,1)$, $\beta\in(0,1/2\pi)$. The rotation number $\tau$ is a continuous map in the uniform topology, and $\phi_{\alpha,\beta}$ is a continuous function of $\alpha$ and $\beta$, so the function $(\alpha,\beta)\mapsto\tau(\phi_{\alpha,\beta}) =:\tau_\beta(\alpha)$ depends continuously on $\alpha$ and $\beta$. The map $\tau_\beta$ is non-decreasing; for $\beta>0$, $\tau_\beta$ is locally constant at each $\alpha$ for which $\tau_\beta(\alpha)$ is rational and strictly increasing if $\tau_\beta(\alpha)$ is irrational. Thus, $\tau_\beta$ is a devil's staircase. Since $\tau_\beta$ is strictly increasing for irrational values of $\tau_\beta(\alpha)$, the set $I_\nu := \{(\alpha,\beta)\,|\,\tau_\beta(\alpha) = \nu\}$ for an irrational $\nu\in[0,1]$ is a graph of a continuous function. For a rational $\nu$, $I_\nu$ has a non-empty interior and is bounded by two continuous curves. The wedges between these two curves are often referred to as {\em Arnold's tongues}. The fact that $\tau(\phi_{\alpha,0})=\tau(r_\alpha)=\alpha$ implies that for $\beta=0$, the set of $\alpha$'s for which there is frequency locking coincides with the rational numbers between~0 and~1, so its Lebesgue measure is zero. When $\beta>0$, its Lebesgue measure is positive. The width of the Arnold's tongues for small $\beta$ for the Arnold's map \Rf{eq:Arnmap} is investigated, e.g., in \cite{Davie1996}. Much of this analysis carries out for more general functions such as the ones we encounter in the problem of the periodically pulsating resonator. The total Lebesgue measure of the frequency locking intervals, $m( \{\tau^{-1}_\beta (\nu) \,| \, \nu\in \QQ\cap[0,1] \} )$, becomes equal to~$1$ when the family of circle maps consists of maps with a horizontal point (so that the map, even if having a continuous inverse, fails to have a differentiable one) -- see \cite{Jensenetal1984,Lanford1985} for numerical results and \cite{Swiatek1988} for analytical proof. With the Arnold's map $\eta_{\alpha,\beta}$ this happens when $\beta=1/2\pi$. In our case this happens when the mirror goes at one instant at the speed of light. We note also that the numerical papers \cite{Shenker1982,Jensenetal1984,% Lanford1985,Cvitanovicetal1985} contain not only conjectures about the measure of the phase locking intervals but, perhaps more importantly, conjectures about scaling relations that hold ``universally''. In particular, the dimension of the set of parameters not covered by the phase locking intervals should be the same for all non-degenerate families. These universality conjectures are supported not only by numerical evidence but also by a renormalization group picture -- see, e.g., \cite{Lanford1986} and the references therein. These universality predictions have been verified in several physical contexts. Notably in turbulence by Glazier and Libchaber ~\cite{GlazierLibchaber1988}. As we will see in Sec.~\ref{sec:small_amplitude}, we do not expect that the families obtained in (\ref{eq:Fexp}) for mirrors oscillating with different amplitudes belong to the same universality class as typical mappings, but they should have universality properties that are easy to figure out from those of the above references. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsection{Distribution of orbits} \label{sec:distr} For the physical problem at hand it is also important to know how the iterates of the circle map $x\mapsto g(x):=G(x)(\mod 1)$ are distributed. As we shall see in lemma~\ref{cor:measure}, if the iterates of $g$ are well distributed (in an appropriate sense), the energy of the field in the resonator does not build up. The distribution of an orbit is conveniently formalized by using the concept of invariant measures. We recall that a measure $\mu$ on $X$ is {\em invariant\/} under the measurable map $f:X\to X$ if $\mu(f^{-1}(A)) = \mu(A)$ for each measurable set $A$. Given a point $x\in\SSS$, the frequency of visit of the orbit of $x$ to $I\subset\SSS$ can be defined by \begin{equation} \label{eq:inv_measure} \mu_x(I) := \lim_{n\to\infty} \frac{\# \{ i\,|\,0\leq i\leq n \mbox{ and } f^i(x)\in I \} }{n} \,. \end{equation} It is easy to check that if for every interval $I$, the limit \Rf{eq:inv_measure} exists, it defines an invariant measure describing the frequency of visit of the orbit of $x$. Therefore, if there are orbits which have asymptotic frequencies of visit, we can find invariant measures. A trivial example of the existence of such measures is when $x$ is periodic. In such a case, the measure $\mu_x$ is a sum of Dirac delta functions concentrated on the periodic orbit. The measure of an interval is proportional to the number of points in the orbit it contains. We also note that it is easy to construct two-dimensional systems [e.g., $(r,\theta) \mapsto (1 + 0.1 (r-1), \, \theta + (r-1)^2 \sin(\theta)^2)$] for which the limits like the one in \Rf{eq:inv_measure} do not exist except for measures concentrated on the fixed points, so that even the existence of such equidistributed orbits is not obvious. There are also relations going in the opposite direction -- if invariant measures exist, they imply the existence of well distributed orbits. We recall that the Krylov-Bogolyubov theorem \cite[Thm.~4.1.1]{KatokHasselblatt1995} asserts that any continuous map on a compact metrizable space has an invariant probability measure. Moreover, the Birkhoff ergodic theorem \cite[Thm.~4.1.2]{KatokHasselblatt1995} implies that given any invariant measure $\mu$, the set of points for which $\mu_x$ as in \Rf{eq:inv_measure} does not exist has measure zero. Certain measures have the property that $\mu_x = \mu$ for $\mu$-almost all points. These measures are called {\em ergodic}. (There are several equivalent definitions of ergodicity and this is one of them.) From the physical point of view, we note that a measure is ergodic if all the points in the measure are distributed according to it. For maps of the circle, there are several criteria that allow to conclude that a map is ergodic. For rotations of the circle with an irrational rotation number we recall the classical Kronecker-Weyl equidistribution theorem \cite[Thm.~4.2.1]{KatokHasselblatt1995} which shows that any irrational rotation is uniquely ergodic, i.e., has only one invariant measure -- the Lebesgue measure~$m$. (Such uniquely ergodic maps are, obviously, ergodic because, by Birkhoff ergodic theorem, the limiting distribution has to exist almost everywhere, but, since there is only one invariant measure, all these invariant distributions have to agree with the original measure.) Thus, the iterates of any $x\in\SSS$ under an irrational rotation are uniformly distributed on the circle. For general non-linear circle OPDs the situation may be quite different. As an example, consider Arnold's map $\eta_{\alpha,\beta}$~\Rf{eq:Arnmap}. If it is conjugate to an irrational rotation by $h$, i.e., $\eta_{\alpha,\beta} = h^{-1} \circ r_{\tau(\eta_{\alpha,\beta})} \circ h$, then there is a unique invariant probability measure $\mu$ defined for each measurable set $A$ by $\mu(A) := m(h(A))$. This implies that if $I$ is an interval in $\SSS$, then the frequency with which a point $x$ visits $I$ is equal to $\mu(I)$. On the other hand, if $\tau(\eta_{\alpha,\beta}) = p/q \in \QQ$, then all orbits are periodic or asymptotic to periodic. Thus, the only possible invariant measure is concentrated at the periodic points and therefore singular, if the periodic points are isolated. Let us now assume that $\alpha$ is very close to $\tau_\beta^{-1}(p/q)$, but does not belong to it. Then $\eta_{\alpha,\beta}$ has no periodic orbits, but still there exists a point $x$ which is ``almost periodic'', i.e., the orbits linger for an extremely long time near the points $x,\, \eta_{\alpha,\beta}(x),\, \cdots ,\, \eta_{\alpha,\beta}^{q-1}(x)$. So that, even if the invariant measure is absolutely continuous, one expects that it is nevertheless quite peaked around the periodic orbit -- see Fig.~\ref{fig:singular_measures}. The behavior of such maps is described quantitatively by the ``intermittency theory'' \cite{PomeauManneville1980}. The continuity properties of the measures of the circle are not so easy to ascertain. Nevertheless, there are certain results that are easy to establish. Of course, in the case that we have a rational rotation number and isolated periodic orbits, some of them attracting and some of them repelling, the only possible invariant measures are measures concentrated in the periodic orbits. For the irrational rotation number case, the Kronecker-Weyl theorem implies that all the maps with an irrational rotation number -- since they are semi-conjugate to a rotation by Poincar\'{e} theorem -- are uniquely ergodic. In the situations where Herman's theorem applies, this measure will have a smooth density since it is the push-forward of Lebesgue measure by a smooth diffeomorphism. We also recall that by Banach-Alaoglu theorem and the Riesz representation theorem, the set of Borel probability measures is compact when we give it the topology of $\mu_n \rightarrow \mu \iff \mu_n(A) \rightarrow \mu(A) $ for all Borel measurable sets~$A$. (This convergence is called weak-* convergence by functional analysts and convergence in probability by probabilists.) \begin{lem} \label{continuity} If $\lambda^*$ is a parameter value for which $f_{\lambda^*} $ admits only one invariant measure $\mu_{\lambda^*}$, given $\mu_{\lambda_i} $ invariant measures for $f_{\lambda_i}$, with $\lambda_i \rightarrow \lambda^*$, then $\mu_i$ converges in the weak-* sense to $\mu_{\lambda^*}$. \end{lem} Note that we are not assuming that $f_{\lambda_i}$ are uniquely ergodic. In particular, the lemma says that in the set of uniquely ergodic maps, the map that a parameter associates the invariant measure is continuous if we give the measures the topology of weak-* convergence. \proofend \noindent {\bf Proof.} Let $\mu_{\lambda_{i_k}}$ be a convergent subsequence. The limit should be an invariant measure for $f_{\lambda^*}$. Hence, it should be $\mu_{\lambda^*}$. It is an easy point set topology lemma that for functions taking values in a compact metrizable space, if all subsequences converge to the same point, then this point is a limit. The space of measures with weak-* topology is metrizable because by Riesz representation theorem is the dual of the space of continuous functions with sup-norm, which is metrizable. \proofend We also point out that as a corollary of KAM theory \cite{Arnold1961} we can obtain that for non-degenerate families, if we consider the parameter values for which the rotation number is Diophantine with uniform constants, the measures are differentiable jointly on $x$ and in the parameter. (For the differentiability in the parameter, we need to use Whitney differentiability or, equivalently, declare that there is a family of densities differentiable both in $x$ and in $\lambda$ that agrees with the densities for these values of $\lambda$.) On the other hand, we point out that there are situations where the invariant measure is not unique (e.g., a rational rotation or a map with more than one periodic orbit). In such cases, it is not difficult to approximate them by maps in such a way that the invariant measure is discontinuous in the weak-* topology as a function of the parameter. The discontinuity of the measures with respect to parameters, as we shall see, has the physical interpretation that, by changing the oscillation parameters by arbitrarily small amounts, we can go from unbounded growth in the energy to the energy remaining bounded. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Application to the resonator problem} \label{sec:appl-th} Now we return to the problem of a one-dimensional optical resonator with a periodically moving wall to discuss the physical implications of circle maps theory, and illustrate with numerical results in an example. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsection{Circle maps in the resonator problem} If we take $a(t)$ to depend on two parameters, $\alpha$ and $\beta$, as in \Rf{equ:a1}, then, as we saw in Sec.~\ref{sec:sol_problem}, the time between the consecutive reflections at the mirrors can be described in terms of the functions $F_{\alpha,\beta}$ and $G_{\alpha,\beta}$ defined by \Rf{eq:Fexp}. These maps are lifts of circle maps that we will denote by $f_{\alpha,\beta}$ and $g_{\alpha,\beta}$. The restriction on the range of $\beta$ in \Rf{equ:a1} implies that $f_{\alpha,\beta}$ and $g_{\alpha,\beta}$ are analytic circle OPDs. Therefore, we can apply the results about the types of orbits of OPHs of $\SSS$, Poincar\'{e} and Denjoy theorems, as well as the smooth conjugacy results and the facts about the distribution of orbits. In an application where the motion of the mirror [i.e., $a(t)$] is given, one needs to compute $F_{\alpha,\beta}$ and $G_{\alpha,\beta}$~\Rf{eq:Fexp}, which cannot be expressed explicitly from $a(t)$ but they require only to solve one variable implicit equation. In the numerical computations we used the subroutine {\sc zeroin} \cite{Forsytheetal1977} to solve implicit equations. If $y=F_{\alpha,\beta}(t)$ and $z=G_{\alpha,\beta}(t)$, then for $a(t)$ given by \Rf{equ:a1}, $y$ and $z$ are given implicitly by \barrz &- y + t + \alpha + 2\beta \sin[\pi(y+t)] = 0 \\ &- z + t + \alpha + \beta \left[\sin(2\pi t) + \sin(2\pi z) \right] = 0 \,. \earrz Given $t$, we can find $y$, $z$ applying {\sc zeroin}. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsection{Rotation number, phase locking} \label{sec:rot_num} In this section, our goal is to translate the mathematical predictions from the theory of circle maps into physical predictions for the resonator problem. The theory of circle maps guarantees that the measure of the frequency locking intervals for $g_{\alpha,\beta}$ is small when $\beta$ is small and becomes $1$ when $\beta=1/2\pi$. The theory also guarantees for analytic maps that, unless a power of the map is the identity, the frequency locking intervals are non-trivial. For the example that we have at hand, it is very easy to verify that this does not happen and, therefore, we can predict that there will be frequency locking intervals and that as the amplitude of the oscillations of the moving mirror increases so that the maximum speed of the moving mirror reaches the speed of light, the devil's staircase becomes complete. Fig.~\ref{fig:rotation_numbers} % %\begin{figure} % \centering % \includegraphics[width=4in,height=3in] % {fig_devil_iop.eps} % \caption{A part of the graph of % $\tau(g_{\alpha,1/2\pi})$ % vs.\ $\alpha$.} % \label{fig:rotation_numbers} %\end{figure} % shows a part of the complete devil's staircase -- the situation which happens when the maps $g_{\alpha,\beta}$ and $f_{\alpha,\beta}$ lose their invertibility, i.e., for $\beta=1/2\pi$. We also recall that the theory of circle maps makes predictions about what happens for non-degenerate phase locking intervals. Namely, for parameters inside the phase locking interval the map has a periodic fixed point and the Lyapunov exponent is smaller than~$0$, while at the edges of the phase locking interval the map experiences a non-degenerate saddle-node bifurcation -- provided that certain combinations of the derivatives do not vanish~\cite{Ruelle1989}. We note that for parameters for which the map is in non-degenerate frequency locking, i.e., $\tau(g_{\alpha,\beta}) = p/q$ and the attractive periodic point of period $q$ has a negative Lyapunov exponent, $\{G^{nq}_{\alpha,\beta}(x)\}_{n=0}^\infty$ will converge exponentially to the fixed point for all $x$ in a certain interval, according to the results about the types of orbits of circle maps (Sec.~\ref{sec:typ-orb}). The whole circle can be divided into such intervals and a finite number of periodic points. Therefore, the graph of $G^{nq}_{\alpha,\beta}$, and hence of $g^{nq}_{\alpha,\beta}$, will look -- up to errors exponentially small in~$n$ -- like a piecewise-constant function % %\begin{figure} % \centering % \includegraphics[width=4in,height=3in] % {iter_staircase.eps} % \caption{Development of the % piecewise-constant structure % of $g^{6n}_{0.2545,\,0.1}$ % (the rotation number of % $g_{0.2545,\,0.1}$ % is~$1/6$). % Graphs of $g^{6n}_{0.2545,\,0.1}$ % are plotted for % $n=1$ ($\dotted$), %% dotted % $n=5$ ($\dashed$), %% dashed % $n=10$ ($\broken$), %% long dashed % $n=100$ ($\full$).} %% solid % \label{fig:staircase_for_g} %\end{figure} % with values (up to integers) in the fixed points of $g^{q}_{\alpha,\beta}$ -- see Fig.~\ref{fig:staircase_for_g}. The fact that certain functions tend to piecewise-constant functions for large values of the argument (which follows from what we found about $G^{nq}_{\alpha,\beta}$ for large $n$) was observed numerically for particular motions of the mirror in \cite{Law1994a,ColeSchieve1995}. In physical terms, this means that the rays will be getting closer and closer together, so with the time the wave packets will become narrower and narrower and more and more sharply peaked. The number of wave packets is equal to $q$. The number of reflections from the moving mirror per unit time will tend to the inverse of the rotation number. In the next section we discuss how this yields an increase of the field energy which happens exponentially fast on time. The fact that for $\tau(g_{\alpha,\beta}) \in\QQ$ the rays approach periodic orbits, is also interesting from a quantum mechanical point of view due to the relation between the periodic orbits in a classical system and the energy levels of the corresponding quantum system, given by the Gutzwiller's trace formula (see, e.g., \cite{Gutzwiller1990}). We also note that we expect that slightly away from the edges of a phase locking interval, the invariant density will be sharply peaked around the points in which it was concentrated in the phase locking intervals. This is described by the ``intermittency theory'' ~\cite{PomeauManneville1980}. To observe numerically in our example what happens when $\alpha$ enters or leaves a frequency locking interval, we set $\calN{\beta}{\nu} := \{\alpha\in[0,1) \, | \, \tau(g_{\alpha,\beta}) = \nu \}$. Fig.~\ref{fig:singular_measures} represents the Radon-Nikodym derivative ${\der\mu}/{\der m}$ of the invariant probability measure $\mu$ with respect to the Lebesgue measure $m$ [i.e., of the density of the invariant measures, which, as we saw in \Rf{eq:inv_measure}, is the frequency of visit of the iterates]. % %\begin{figure} % \centering % \includegraphics[width=4in,height=3in] % {fig_singular_measure.eps} % \caption{Density of the invariant measures %for $\beta=0.1$ and %$\alpha=0.253$ ($\broken$), $\alpha=0.2539$ ($\full$), and %$\alpha=0.253975$ ($\dotted$).} % \label{fig:singular_measures} %\end{figure} % The figure shows ${\der\mu}/{\der m}$ for $\alpha$ close to the left end of $\calN{0.1}{1/6}$. When $\alpha$ approaches (from the left) the left end of $\calN{0.1}{1/6}$, ${\der\mu}/{\der m}$ becomes sharply peaked at some points, and when $\alpha$ enters the frequency locking interval, the invariant measure becomes singular ($g_{\alpha,\,0.1}$ undergoes tangent bifurcation at $\alpha = 0.253977\ldots$). All seems to be consistent with the conjecture that all the frequency-locking intervals in the family (away of $\beta = 0$) are non-degenerate, i.e., that at the boundaries of the phase locking intervals the map satisfies the hypothesis of the saddle-node bifurcation theorem. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsection{Doppler shift} \label{sec:Dop} One of the most interesting parts of the applications of circle map theory is the ease with which we can describe the effect on the energy after repeated reflections. Recall that in Sec.~\ref{sec:Dop_shift}, we found the time dependence of the field energy under the assumption that at time $t$ all rays are going to the right. This assumption is not very restrictive in the case of a rational rotation number since, as we found in Sec.~\ref{sec:rot_num}, the field develops wave packets that become narrower with time, so \Rf{eq:locDopfac} and \Rf{eq:energy_time} hold for the asymptotic behavior of the energy. Note that \Rf{eq:locDopfac} expresses the Doppler shift factor in terms of the derivatives of the map~$G$. This gives a very close relation between the dynamics and the behavior of the wave packets. \begin{prop} Let $\alpha$ and $\beta$ be such that $\tau(g_{\alpha,\beta})=p/q$, and that the map $G:=G_{\alpha,\beta}$ has a stable periodic orbit $\Theta_q=\{\theta_1,\,\ldots,\,\theta_q\}$ such that $(G^q)'(\theta_1)<1$. Assume that the initial electromagnetic field in the cavity is not zero at some space-time point for which the phase of the first reflection from the moving mirror is in the basin of attraction of~$\Theta_q$. Then the energy of the field in the resonator will be asymptotically increasing at an exponential rate: \begin{equation} \label{eq:energy_exp} E(t) \sim \exp \left\{\frac{\ln D(\Theta_q)}{p}\,t\right\} \,. \end{equation} \end{prop} \noindent {\bf Proof.} First notice that the number of reflections from the moving mirror per unit time reaches a well defined limit (one and the same for all rays) -- the inverse of the rotation number. Secondly, as was discussed in Sec.~\ref{phys_set}, at reflection from the moving mirror at phase~$\theta$, a wave packet becomes narrower by a factor of $D(\theta)$~(\ref{Doppler_factor}), which leads to a $D(\theta)$~times increase in its energy. Asymptotically, the phases at reflection will approach the stable periodic orbit $\Theta_q=\{\theta_1,\,\ldots,\,\theta_q\}$ of $g_{\alpha,\beta}$. The Doppler factors at reflection will tend correspondingly to $\{D(\theta_1),\,\ldots,\,D(\theta_q)\}$ ~(\ref{Doppler_factor}). Hence, in time~$p$ each ray will undergo $q$ reflections from the moving mirror, the total Doppler shift factor along the periodic orbit $\Theta_q$ being $$ D(\Theta_q) := \prod_{i=1}^{q} D(\theta_i) = \prod_{i=1}^{q} \displaystyle{ \frac{1-a'(\theta_i)}{1+a'(\theta_i)}} \,. $$ On the other hand, the definition of the map $G$ as the advance in the time between successive reflections from the moving mirror yields $\theta_i=G^{i-1}(\theta_1)$. The chain rule applied to the explicit expression \Rf{eq:Fexp} for $G$ yields $$ (G^{q-1})' (\theta_1) = \prod_{j=1}^{q-1} G'(\theta_j) = \prod_{j=1}^{q-1} \displaystyle{ \frac{1+a'(\theta_j)}{1-a'(\theta_{j+1})}} \,, $$ which gives the following expression for $D(\Theta_q)$ [cf.~\Rf{eq:locDopfac}]: \begin{equation} \label{eq:Dop_orbit} D(\Theta_q) := \displaystyle{ \frac{1-a'(\theta_1)}{1+a'(\theta_q)} \left[ (G^{q-1})' (\theta_1) \right]^{-1} = \frac{1-a'(\theta_1)}{1+a'(\theta_q)} (G^{1-q})' (\theta_q) } \,. \end{equation} Hence, the energy density grows by a factor of $D(\Theta_q)^2$. Since after $q$ reflections the wave packet is concentrated in a length $D(\Theta_q)$ times smaller, the total energy grows by a factor of $D(\Theta_q)$ in $p$ units of time, which implies~\Rf{eq:energy_exp}. \proofend The quantities $(G^n)'(\theta)$ that appear in \Rf{eq:Dop_orbit} have been studied intensively in dynamical systems theory since they control the growth of infinitesimal perturbations of trajectories. Similarly, they are factors that multiply the invariant densities when they get transported, as we will see in~\Rf{eq:inv_density}. We found numerically the total Doppler factors $D(\Theta_q)$ for some particular choices of the parameters. %\begin{figure} % \centering % \includegraphics[width=4in,height=3in] % {dop_tau_1_6_portrait.eps} % \caption{$\log_{10}D(\Theta_6)$ vs.\ % $\alpha\in\calN{\beta}{1/6}$ % for different values of $\beta$.} % \label{fig:log_doppler} %\end{figure} % In Fig.~\ref{fig:log_doppler}, $\log_{10} D(\Theta_6)$ is shown for different values of $\beta$ and for $\alpha\in\calN{\beta}{1/6}$. Obviously, the maximum value of $D(\Theta_6)$ depends strongly on~$\beta$, becoming infinite for $\beta=1/{2\pi}$ and some $\alpha\in N_{1/2\pi}(1/6)$. For smaller values of~$\beta$, the Doppler factor is much smaller. Moreover, the width of the frequency locking intervals for small~$\beta$ is small, so the probability of hitting a frequency locking interval with arbitrarily chosen $\alpha$ and $\beta$ is small. [The likelihood of frequency locking for the Arnold's map~(\ref{eq:Arnmap}) is studied numerically in~\cite{Lanford1985}.] In the case when Herman's theorem apply, the derivatives of $G^n$ are bounded independently of $n$, which causes the energy of the system to be bounded for all times, which is proved in the following proposition. \begin{prop} If $G_{\alpha,\beta}$ is such that it satisfies the hypothesis of Herman's theorem, then the energy density remains bounded for all times. \end{prop} \proofend \noindent {\bf Proof.} In such a case $G_{\alpha,\beta}=h^{-1}\circ R\circ h$ with $h$ differentiable and $R$ a rotation by $\tau(g_{\alpha,\beta})$. Therefore $G_{\alpha,\beta}^n = h^{-1} \circ R^n \circ h$ and $$ (G_{\alpha,\beta}^n)' (\theta) = (h^{-1})' (R^n\circ h(\theta)) \, (R^n)' (h(\theta)) \, h'(\theta) = (h^{-1})' (R^n\circ h(\theta)) \, h'(\theta) $$ because $(R^n)'=1$. The two factors in the right-hand side of the above equation are bounded uniformly in $\theta$ and $n$. Thus, the ``local Doppler factors'' \Rf{eq:locDopfac} will be bounded, which implies the boundedness of the energy \Rf{eq:energy_time}. \proofend There is an interesting connection between the invariant densities of the system and the growth of the electromagnetic energy density. Recall that if a density $\mu$ is invariant, $\mu(G(\theta)) = \mu(\theta) / G'(\theta)$. Hence, if the density $\mu$ never vanishes, $G'(\theta) = \mu(\theta) / \mu(G(\theta))$ and, therefore, $(G^i)'(\theta) = \mu(\theta) / \mu(G^i(\theta))$. Let us assume that there is only one characteristic passing through the space-time point $(t,x)$, and this characteristic is going to the right. Then, using the notations of Sec.~\ref{sec:sol_problem}, we can write the energy density at $(t,x)$ as [cf.~\Rf{eq:locDopfac}] \begin{equation} \label{eq:inv_density} {\cal T}^{00} (t,x) = \displaystyle{ \left[ \frac{1-\differ{a}(\theta^{-}_{-n_{-}})} {1+\differ{a}(\theta^{-}_0)} \, \frac{\mu(G^{n_{-}}(\theta^{-}_{-n_-}))} {\mu(\theta^{-}_{-n_-})} \right]^2 {\cal T}^{00}(t_0,x_0^-)} \,. \end{equation} In the general case [with two characteristics through $(x,t)$], one can use \Rf{eq:vect_pot} and \Rf{eq:energy} to prove the following result: \begin{lem} \label{cor:measure} If a system has an invariant density $\mu$ which is bounded away from zero, then the electromagnetic energy density of a $C^1$ initial data is smaller than $C\mu^2$ for all times. \end{lem} In the cases that Herman's theorem applies, there is an invariant density bounded away from zero (and also bounded). Hence, we conclude that there are values of the amplitude of mirror's oscillations for which the energy density of the field remains bounded. This set is typically a Cantor set interspersed with values for which the energy increases exponentially. Some other results about the behavior of the energy with respect to time and parameters are obtained in~\cite{Dittrichetal1997}. We call attention to the fact that \cite{Arnold1961} contains examples of analytic maps whose rotation numbers are very closely approximated by rationals and that are arbitrarily close to a rotation such that they preserve no invariant density and, therefore, are not smoothly conjugate to a rotation. It is also known that for all rotation numbers one can construct $C^{2-\varepsilon}$ maps arbitrarily close to rotations with this rotation number and such that they do not preserve any invariant measures~\cite{HawkinsSchmidt1982}. It is a testament to the ubiquity of these maps that these questions were motivated and found applications in the theory of classification of $C^*$~algebras. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsection{The behavior for small amplitude and universality} \label{sec:small_amplitude} We note that, even if all the motions of the mirror lead to a circle map as in (\ref{eq:Fper}), it does not seem clear to us that all the maps of the circle can appear as $F$, $G$ for a certain~$a$. This makes it impossible to conclude that the theory of generic circle maps applies directly to obtain conclusions for a generic motion of the mirror. Of course, all the conclusions of the general theory that apply to all maps of the circle apply to our case. Those conclusions that require non-degeneracy assumptions will need that we verify the assumptions. Nevertheless, the very developed mathematical theory of generic or universal circle maps cannot be applied without caution to maps that appear as the result of generic or universal oscillations of the mirror. One aspect that we have found makes a big difference with the generic theory is the situation where the mirror oscillates with small amplitude, i.e., $a_\varepsilon(t) = \bar a + \varepsilon \addition(t)$ with $\addition$ a periodic function of zero average and period $1$, and $\varepsilon\ll 1$. The first parameter, $\bar{a}$, is the average length of the resonator, while $\varepsilon=0$ is called the ``nonlinearity parameter'' for obvious reasons. If we denote by $F_{\bar{a},\varepsilon}$ and $G_{\bar{a},\varepsilon}$ the corresponding 2-parametric families of maps of the circle constructed according to (\ref{eq:Fexp}), then we have, for three times differentiable families, \begin{eqnarray} \label{eq:small_amp} \FL F_{\bar{a},\varepsilon}(t) &=& t + 2 \bar a + 2 \varepsilon \addition (t + \bar a) + 2 \varepsilon^2 \differ{\addition} (t + \bar a) \addition (t + \bar a) + O(\varepsilon^3) \,, \nonumber \\ [-2mm] \FL \\ [-2mm] \FL G_{\bar{a},\varepsilon}(t) &=& t + 2 \bar a + \varepsilon [ \addition (t) + \addition (t + 2 \bar a) ] + \varepsilon^2 \differ{\addition} (t + 2 \bar a) [ \addition (t) + \addition (t + 2 \bar a) ] + O(\varepsilon^3) \,. \nonumber \end{eqnarray} Note that the term of order $\varepsilon$ has vanishing average. As we will immediately show, this property causes that some well known generic properties of families of circle mappings do not hold for families of maps constructed as in \Rf{eq:Fexp}. Indeed, if we consider the expressions for small amplitude developed in \Rf{eq:small_amp}, we can write the maps as $$ H_\varepsilon (t) = t + 2 \bar{a} + \varepsilon H_1(t) + \varepsilon^2 H_2(t) + O(\varepsilon^3) \,. $$ Since the conclusions of the theory of circle maps are independent of the coordinate system chosen, it is natural to try to choose a coordinate system where these expressions are as simple as possible. Hence, we choose $\heps(t):=t+\varepsilon\htil(t)$, a perturbation of the identity, and consider $h_\varepsilon^{-1}\circ\Heps\circ\heps$, which is just $\Heps$ in another system of coordinates, related to the original one by $\heps$. Then, up to terms of order $\varepsilon^3$, we have \begin{eqnarray} \label{eq:transformed} \FL h_\varepsilon^{-1}\circ\Heps\circ\heps(t) &=& t + 2 \bar{a} + \varepsilon \left[ \htil(t) - \htil(t+2\bar{a}) + H_1(t) \right] \nonumber\\[-2mm] \FL \\[-2mm] \FL &&\hspace{-15mm}+\varepsilon^2 \left\{ \htil'(t+2\bar{a})\htil(t+2\bar{a}) - \htil'(t+2\bar{a}) \left[\htil(t)+H_1(t)\right] + H'_1(t) \htil(t) + H_2(t) \right\} \,. \nonumber \end{eqnarray} We would like to choose $\htil$ in such a way that the $\varepsilon$ term is not present. Note that since $\int \htil(t+2\bar{a}) \,\der t = \int \htil(t) \,\der t$, this is impossible unless $\int H_1(t) \,\der t = 0$. When $\int H_1(t) \,\der t = 0$, $H_1$ is smooth and $2\bar{a}$ is Diophantine, a well-known result (see, e.g., \cite[Sec.\ XIII.4]{Herman1979}) shows that in such a case we can obtain one $\htil$ satisfying \begin{equation} \label{eq:linearized} \htil(t) - \htil(t+2\bar{a}) + H_1(t) = 0 \end{equation} and $\overline{\htil}=0$. [Such $\htil$ is conventionally obtained by using Fourier coefficients. Note that in Fourier coefficients, \Rf{eq:linearized} amounts to $\widehat{\htil}_k (e^{2\pi i k 2 \bar{a}}-1) = \widehat{(H_1)}_k$. If $H_1$ is smooth, the Fourier coefficients decrease fast and if $2\bar{a}$ is Diophantine, then $(e^{2\pi i k 2 \bar{a}}-1)^{-1}$ does not grow too fast. For more details we refer to the reference above.] Since for the functions $F_{\bar{a},\varepsilon}$ and $G_{\bar{a},\varepsilon}$ the term of order $\varepsilon$ has a zero average, we can transform these functions into lifts of rotations plus $O(\varepsilon^2)$. This implies, in particular, that their rotation number is $\tau(F_{\bar{a},\varepsilon}) = \tau(G_{\bar{a},\varepsilon}) = 2 \bar{a} + O(\varepsilon^2)$. One could wonder if it would be possible to continue the process and eliminate also to order~$\varepsilon^2$. If we look at the $\varepsilon^2$ terms in \Rf{eq:transformed}, we see that $\overline{\htil'(t) \htil(t)} = 0$, and, when $\htil$ is chosen as in \Rf{eq:linearized}, $$ \htil'(t+2\bar{a}) [ \htil(t) + H_1(t) ] = \htil'(t+2\bar{a}) \htil(t+2\bar{a}) \,, $$ which also has average zero. Therefore, a necessary condition for the $\varepsilon^2$ term in $h_\varepsilon^{-1}\circ\Heps\circ\heps(t)$ to be zero is $\overline{H'_1(t)\htil(t)} + \overline{H_2(t)} = 0$. For the $F_{\bar{a},\varepsilon}$ in \Rf{eq:small_amp} we see that $F_2$ has zero average. Nevertheless, the term $F_1'(t) \htil(t)$ does not in general have average zero as can be seen in examples. Hence, we see that the rotation number indeed changes by an order which is $O(\varepsilon^2)$ and not higher in general. This property is not generic for families of circle maps starting with a rotation $2\bar{a}$ and it puts them outside of the universality classes considered in \cite{Shenker1982,Lanford1986}, etc., since the correspondence between rotation numbers and parameters is not the same. According to the geometric picture of renormalization developed in \cite{Lanford1986}, the space of circle maps is divided into slices of rational rotation numbers, which are -- in appropriate sense -- parallel. In that language -- in which we think of families of circle maps as curves in the space of mappings -- the families of advance maps $F_{\bar{a},\varepsilon}$ and $G_{\bar{a},\varepsilon}$ (for fixed $\bar{a}$) have second order tangency to the foliation of rational rotation numbers rather than being transversal. Hence, the scaling predicted by universality theory should be true for $\varepsilon^2$ in place of $\varepsilon$. We have not verified this prediction, but we expect to come back to it soon. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsection{Schwarzian derivative in the problem of moving mirrors} \label{sec:schwarz} Fulling and Davies \cite{FullingDavies1976} calculated the energy-momentum tensor in the two-dimensional quantum field theory of a massless scalar field influenced by the motion of a perfectly reflecting mirror (see also \cite{MostepanenkoTrunov1997}). They obtained that the ``renormalized'' vacuum expectation value of the energy density radiated by a moving mirror into initially empty space is $$ {\cal T}^{00}(u) = - \frac1{24\pi} \left[ \frac{F'''(u)}{F'(u)} - \frac32 \left(\frac{F''(u)}{F'(u)}\right)^2 \right] \,, $$ where $u=t-x$, and $F$ is related to the law of the motion of the mirror, $x=a(t)$, by~\Rf{eq:Fexp}. The right-hand side of this equation is nothing but (up to a constant factor) the Schwarzian derivative of $F$ -- a differential operator which naturally appears in complex analysis, e.g., it is invariant under a fractional linear transformation; vanishing Schwarzian derivative of a function is the necessary and sufficient condition that the function is fractional linear transformation, etc. More interestingly, the Schwarzian derivative has been used as an important tool in the proof of several important theorems in the theory of circle maps -- see, e.g.,~\cite{Yoccoz1984,Herman1985}. In the light of the connection between the solutions of the wave equation in a periodically pulsating domain and the theory of circle maps it is not impossible that this is not just a coincidence. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Conclusion} Using the method of characteristics for solving the wave equation, we reformulated the problem of studying the electromagnetic field in a resonator with a periodically oscillating wall into the language of circle maps. Then we used some results of the theory of circle maps in order to make predictions about the long time behavior of the field. We found that many results in the theory of circle maps have a directly observable physical meaning. Notably, for a typical family of mirror motions we expect that the electromagnetic energy grows exponentially fast in a dense set of intervals in the parameters. Nevertheless, it remains bounded for all times for a Cantor set of parameters that has positive measure. There are several advantages of the approach presented here. % First, it allows us to understand better the time evolution of the electromagnetic field in the resonator and the mechanism of the change in the field energy. % Second, the predictions are based on the general theory of circle maps so they are valid for any periodic motion of the mirror; let us also emphasize that our method is non-perturbative. % Last, but not least, for a given motion of the mirror, one can easily make certain predictions about the behavior of the field by simply calculating the rotation number of the corresponding circle map, and without solving any partial differential equations. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \acknowledgments This research was partially supported by N.S.F. grants. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{references} \bibitem[*]{email_rl} E-mail address: \verb!llave@math.utexas.edu!. \bibitem[\dagger]{email_np} E-mail address: \verb!npetrov@math.utexas.edu!. \bibitem{Balazs1961} N.~Balazs, {\rm J. 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Phys.\/} {\bf 176}, 227 (1996). \end{references} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{figure} \centerline{\epsfig{file=charact_landscape.eps, height= 2 in, angle=-90}} \bigskip \caption{Finding $A(t,x)$ by the method of characteristics.} \label{fig:charact_dop} \end{figure} \pagebreak \begin{figure} \centerline{\epsfig{file=reflection_landscape.eps, height= 5 in, angle=-90}} \bigskip \caption{Reflection by the moving mirror.} \label{fig:reflection} \end{figure} \pagebreak \begin{figure} \centerline{\epsfig{file=devil_landscape.eps, height= 5 in, angle=-90}} \bigskip \caption{A part of the graph of $\tau(g_{\alpha,1/2\pi})$ vs.\ $\alpha$.} \label{fig:rotation_numbers} \end{figure} \pagebreak \begin{figure} \centerline{\epsfig{file=constant_landscape.eps, height= 5 in, angle=-90}} \bigskip \caption{Development of the piecewise-constant structure of $g^{6n}_{0.2545,\,0.1}$ (the rotation number of $g_{0.2545,\,0.1}$ is~$1/6$). Graphs of $g^{6n}_{0.2545,\,0.1}$ are plotted for $n=1$ (dotted line), %% ($\dotted$) $n=5$ (dashed line), %% ($\dashed$) $n=10$ (long dashed line), %% ($\broken$) $n=100$ (solid line).} %% ($\full$) \label{fig:staircase_for_g} \end{figure} \pagebreak \begin{figure} \centerline{\epsfig{file=measures_landscape.eps, height= 5 in, angle=-90}} \bigskip \caption{Density of the invariant measures for $\beta=0.1$ and $\alpha=0.253$ (dashed line), %% ($\broken$) $\alpha=0.2539$ (solid line), %% ($\full$) and $\alpha=0.253975$ (dotted line).} %% ($\dotted$) \label{fig:singular_measures} \end{figure} \pagebreak \begin{figure} \centerline{\epsfig{file=doppler_landscape.eps, height= 5 in, angle=-90}} \bigskip \caption{A log-linear graph of the total Doppler factor for $g_{\alpha,\beta}$ in the phase locking interval of rotation number $1/6$ for different $\beta$. The insert [linear-linear graph of $D(\Theta_6)$ vs.\ $\alpha-\alpha_{\rm c}$] calls attention to the square-root behavior at edges; $\alpha_{\rm c}$ is the value of $\alpha$ at the left end of $\calN{0.14}{1/6}$.} % $\log_{10}D(\Theta_6)$ vs.\ % $\alpha\in\calN{\beta}{1/6}$ % for different values of $\beta$.} \label{fig:log_doppler} \end{figure} \pagebreak \end{document} ---------------9810091121965 Content-Type: application/postscript; name="charact_landscape.eps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="charact_landscape.eps" %!PS-Adobe-2.0 EPSF-1.2 %%BoundingBox: 104 221 535 372 %%Title: charact_landscape.eps %%Creator: npetrov@linux54 with xmgr %%CreationDate: Wed Oct 7 14:57:14 1998 %%EndComments 80 dict begin /languagelevel where {pop /gs_languagelevel languagelevel def} {/gs_languagelevel 1 def} ifelse gs_languagelevel 1 gt { <} image end } >> matrix makepattern /Pat0 exch def <} image end } >> matrix makepattern /Pat1 exch def <} image end } >> 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Content-Type: application/postscript; name="constant_landscape.eps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="constant_landscape.eps" %!PS-Adobe-2.0 EPSF-1.2 %%BoundingBox: 97 67 554 608 %%Title: constant_landscape.eps %%Creator: npetrov@linux54 with xmgr %%CreationDate: Tue Oct 6 10:00:55 1998 %%EndComments 80 dict begin /languagelevel where {pop /gs_languagelevel languagelevel def} {/gs_languagelevel 1 def} ifelse gs_languagelevel 1 gt { <} image end } >> matrix makepattern /Pat0 exch def <} image end } >> matrix makepattern /Pat1 exch def <} image end } >> matrix makepattern /Pat2 exch def <} image end } >> matrix makepattern /Pat3 exch def <} image end } >> matrix makepattern /Pat4 exch def <} image end } >> matrix makepattern /Pat5 exch def <} image end } >> matrix makepattern /Pat6 exch def <} image end } >> matrix makepattern /Pat7 exch def <} image end } >> matrix makepattern /Pat8 exch def <} image end } >> matrix makepattern /Pat9 exch def <} image end } >> matrix makepattern /Pat10 exch def <} image end } >> matrix makepattern /Pat11 exch def <} image end } >> matrix makepattern /Pat12 exch def <} image end } >> matrix makepattern /Pat13 exch def <} image end } >> matrix makepattern /Pat14 exch def <} image end } >> matrix makepattern /Pat15 exch def }{ /Pat0 { 0.000000 setgray } def /Pat1 { 0.062500 setgray } def /Pat2 { 0.125000 setgray } def /Pat3 { 0.187500 setgray } def /Pat4 { 0.250000 setgray } def /Pat5 { 0.312500 setgray } def /Pat6 { 0.375000 setgray } def /Pat7 { 0.437500 setgray } def /Pat8 { 0.500000 setgray } def /Pat9 { 0.562500 setgray } def /Pat10 { 0.625000 setgray } def /Pat11 { 0.687500 setgray } def /Pat12 { 0.750000 setgray } def /Pat13 { 0.812500 setgray } def /Pat14 { 0.875000 setgray } def /Pat15 { 0.937500 setgray } def /setpattern { } def } ifelse /m {moveto} bind def /l {lineto} bind def /s {stroke} bind def % Symbol fill /f { gsave fill grestore stroke } bind def % Opaque symbol /o { gsave 1.000000 1.000000 1.000000 setrgbcolor fill grestore stroke } bind def % Circle symbol /a { 3 -1 roll 0 360 arc } bind def /da { a s } bind def /fa { a f } bind def /oa { a o } bind def % Square symbol /sq { moveto dup dup rmoveto 2 mul dup neg 0 rlineto dup neg 0 exch rlineto 0 rlineto closepath } bind def /dsq { sq s } bind def /fsq { sq f } bind def /osq { sq o } bind def % Triangle symbols /t1 { moveto dup 0 exch rmoveto dup neg dup 2 mul rlineto 2 mul 0 rlineto closepath } bind def /dt1 { t1 s } bind def /ft1 { t1 f } bind def /ot1 { t1 o } bind def /t2 { moveto dup neg 0 rmoveto dup dup 2 mul exch neg rlineto 2 mul 0 exch rlineto closepath } bind def /dt2 { t2 s } bind def /ft2 { t2 f } bind def /ot2 { t2 o } bind def /t3 { moveto dup neg 0 exch rmoveto dup dup 2 mul rlineto neg 2 mul 0 rlineto closepath } bind def /dt3 { t3 s } bind def /ft3 { t3 f } bind def /ot3 { t3 o } bind def /t4 { moveto dup 0 rmoveto dup dup -2 mul exch rlineto -2 mul 0 exch rlineto closepath } bind def /dt4 { t4 s } bind def /ft4 { t4 f } bind def /ot4 { t4 o } bind def % Diamond symbol /di { moveto dup 0 exch rmoveto dup neg dup rlineto dup dup neg rlineto dup dup rlineto closepath } bind def /ddi { di s } bind def /fdi { di f } bind def /odi { di o } bind def % Plus symbol /pl { dup 0 rmoveto dup -2 mul 0 rlineto dup dup rmoveto -2 mul 0 exch rlineto } bind def /dpl { m pl s } bind def % x symbol /x { dup dup rmoveto dup -2 mul dup rlineto 2 mul dup 0 rmoveto dup neg exch rlineto } bind def /dx { m x s } bind def % Splat symbol /dsp { m dup pl dup 0 exch rmoveto 0.707 mul x s } bind def /RJ { stringwidth neg exch neg exch rmoveto } bind def /CS { stringwidth 2 div neg exch 2 div neg exch rmoveto } bind def 0.24 0.24 scale 1 setlinecap mark /ISOLatin1Encoding 8#000 1 8#054 {StandardEncoding exch get} for /minus 8#056 1 8#217 {StandardEncoding exch get} for /dotlessi 8#301 1 8#317 {StandardEncoding exch get} for /space /exclamdown /cent /sterling /currency /yen /brokenbar /section /dieresis /copyright /ordfeminine /guillemotleft /logicalnot /hyphen /registered /macron /degree /plusminus /twosuperior /threesuperior /acute /mu /paragraph /periodcentered /cedilla /onesuperior /ordmasculine /guillemotright /onequarter /onehalf /threequarters /questiondown /Agrave /Aacute /Acircumflex /Atilde /Adieresis /Aring /AE /Ccedilla /Egrave /Eacute /Ecircumflex /Edieresis /Igrave /Iacute /Icircumflex /Idieresis /Eth /Ntilde /Ograve /Oacute /Ocircumflex /Otilde /Odieresis /multiply /Oslash /Ugrave /Uacute /Ucircumflex /Udieresis /Yacute /Thorn /germandbls /agrave /aacute /acircumflex /atilde /adieresis /aring /ae /ccedilla /egrave /eacute /ecircumflex /edieresis /igrave /iacute /icircumflex /idieresis /eth /ntilde /ograve /oacute /ocircumflex /otilde /odieresis /divide /oslash /ugrave /uacute /ucircumflex /udieresis /yacute /thorn /ydieresis /ISOLatin1Encoding where not {256 array astore def} if cleartomark /makeISOEncoded { findfont /curfont exch def /newfont curfont 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736 l 2505 736 l 2509 736 l 2513 736 l 2517 736 l s 506 434 m 506 2114 l 2521 2114 l 2521 434 l 506 434 l s end showpage %%Trailer ---------------9810091121965 Content-Type: application/postscript; name="devil_landscape.eps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="devil_landscape.eps" %!PS-Adobe-2.0 EPSF-1.2 %%BoundingBox: 104 67 554 605 %%Title: devil_landscape.eps %%Creator: npetrov@linux54 with xmgr %%CreationDate: Tue Oct 6 09:46:26 1998 %%EndComments 80 dict begin /languagelevel where {pop /gs_languagelevel languagelevel def} {/gs_languagelevel 1 def} ifelse gs_languagelevel 1 gt { <} image end } >> matrix makepattern /Pat0 exch def <} image end } >> matrix makepattern /Pat1 exch def <} image end } >> matrix makepattern /Pat2 exch def <} image end } >> matrix makepattern /Pat3 exch def <} image end } >> matrix makepattern /Pat4 exch def <} image end } >> matrix makepattern /Pat5 exch def <} image end } >> matrix makepattern /Pat6 exch def <} image end } >> matrix makepattern /Pat7 exch def <} image end } >> matrix makepattern /Pat8 exch def <} image end } >> matrix makepattern /Pat9 exch def <} image end } >> matrix makepattern /Pat10 exch def <} image end } >> matrix makepattern /Pat11 exch def <} image end } >> matrix makepattern /Pat12 exch def <} image end } >> matrix makepattern /Pat13 exch def <} image end } >> matrix makepattern /Pat14 exch def <} image end } >> matrix makepattern /Pat15 exch def }{ /Pat0 { 0.000000 setgray } def /Pat1 { 0.062500 setgray } def /Pat2 { 0.125000 setgray } def /Pat3 { 0.187500 setgray } def /Pat4 { 0.250000 setgray } def /Pat5 { 0.312500 setgray } def /Pat6 { 0.375000 setgray } def /Pat7 { 0.437500 setgray } def /Pat8 { 0.500000 setgray } def /Pat9 { 0.562500 setgray } def /Pat10 { 0.625000 setgray } def /Pat11 { 0.687500 setgray } def /Pat12 { 0.750000 setgray } def /Pat13 { 0.812500 setgray } def /Pat14 { 0.875000 setgray } def /Pat15 { 0.937500 setgray } def /setpattern { } def } ifelse /m {moveto} bind def /l {lineto} bind def /s {stroke} bind def % Symbol fill /f { gsave fill grestore stroke } bind def % Opaque symbol /o { gsave 1.000000 1.000000 1.000000 setrgbcolor fill grestore stroke } bind def % Circle symbol /a { 3 -1 roll 0 360 arc } bind def /da { a s } bind def /fa { a f } bind def /oa { a o } bind def % Square symbol /sq { moveto dup dup rmoveto 2 mul dup neg 0 rlineto dup neg 0 exch rlineto 0 rlineto closepath } bind def /dsq { sq s } bind def /fsq { sq f } bind def /osq { sq o } bind def % Triangle symbols /t1 { moveto dup 0 exch rmoveto dup neg dup 2 mul rlineto 2 mul 0 rlineto closepath } bind def /dt1 { t1 s } bind def /ft1 { t1 f } bind def /ot1 { t1 o } bind def /t2 { moveto dup neg 0 rmoveto dup dup 2 mul exch neg rlineto 2 mul 0 exch rlineto closepath } bind def /dt2 { t2 s } bind def /ft2 { t2 f } bind def /ot2 { t2 o } bind def /t3 { moveto dup neg 0 exch rmoveto dup dup 2 mul rlineto neg 2 mul 0 rlineto closepath } bind def /dt3 { t3 s } bind def /ft3 { t3 f } bind def /ot3 { t3 o } bind def /t4 { moveto dup 0 rmoveto dup dup -2 mul exch rlineto -2 mul 0 exch rlineto closepath } bind def /dt4 { t4 s } bind def /ft4 { t4 f } bind def /ot4 { t4 o } bind def % Diamond symbol /di { moveto dup 0 exch rmoveto dup neg dup rlineto dup dup neg rlineto dup dup rlineto closepath } bind def /ddi { di s } bind def /fdi { di f } bind def /odi { di o } bind def % Plus symbol /pl { dup 0 rmoveto dup -2 mul 0 rlineto dup dup rmoveto -2 mul 0 exch rlineto } bind def /dpl { m pl s } bind def % x symbol /x { dup dup rmoveto dup -2 mul dup rlineto 2 mul dup 0 rmoveto dup neg exch rlineto } bind def /dx { m x s } bind def % Splat symbol /dsp { m dup pl dup 0 exch rmoveto 0.707 mul x s } bind def /RJ { stringwidth neg exch neg exch rmoveto } bind def /CS { stringwidth 2 div neg exch 2 div neg exch rmoveto } bind def 0.24 0.24 scale 1 setlinecap mark /ISOLatin1Encoding 8#000 1 8#054 {StandardEncoding exch get} for /minus 8#056 1 8#217 {StandardEncoding exch get} for /dotlessi 8#301 1 8#317 {StandardEncoding exch get} for /space /exclamdown /cent /sterling /currency /yen /brokenbar /section /dieresis /copyright /ordfeminine /guillemotleft /logicalnot /hyphen /registered /macron /degree /plusminus /twosuperior /threesuperior /acute /mu /paragraph /periodcentered /cedilla /onesuperior /ordmasculine /guillemotright /onequarter /onehalf /threequarters /questiondown /Agrave /Aacute /Acircumflex /Atilde /Adieresis /Aring /AE /Ccedilla /Egrave /Eacute /Ecircumflex /Edieresis /Igrave /Iacute /Icircumflex /Idieresis /Eth /Ntilde /Ograve /Oacute /Ocircumflex /Otilde /Odieresis /multiply /Oslash /Ugrave /Uacute /Ucircumflex /Udieresis /Yacute /Thorn /germandbls /agrave /aacute /acircumflex /atilde /adieresis /aring /ae /ccedilla /egrave /eacute /ecircumflex /edieresis /igrave /iacute /icircumflex /idieresis /eth /ntilde /ograve /oacute /ocircumflex /otilde /odieresis /divide /oslash /ugrave /uacute /ucircumflex /udieresis /yacute /thorn /ydieresis /ISOLatin1Encoding where not {256 array astore def} if cleartomark /makeISOEncoded { findfont /curfont exch def /newfont curfont maxlength dict def /ISOLatin1 (-ISOLatin1) def /curfontname curfont /FontName get dup length string cvs def /newfontname curfontname length ISOLatin1 length add string dup 0 curfontname putinterval dup curfontname length ISOLatin1 putinterval def curfont { exch dup /FID ne { dup /Encoding eq { exch pop ISOLatin1Encoding exch } if dup /FontName eq { exch pop newfontname exch } if exch newfont 3 1 roll put } { pop pop } ifelse } forall newfontname newfont definefont } def /Times-Roman makeISOEncoded pop /Times-Bold makeISOEncoded pop /Times-Italic makeISOEncoded pop /Times-BoldItalic makeISOEncoded pop /Helvetica makeISOEncoded pop /Helvetica-Bold makeISOEncoded pop /Helvetica-Oblique makeISOEncoded pop /Helvetica-BoldOblique makeISOEncoded pop 2550 0 translate 90 rotate s 0.000000 0.000000 0.000000 setrgbcolor 1 setlinewidth 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(0.36) show grestore newpath 1402 385 m gsave 1402 385 translate 0 rotate 0 -20 m (0.38) CS (0.38) show grestore newpath 1849 385 m gsave 1849 385 translate 0 rotate 0 -20 m (0.4) CS (0.4) show grestore newpath 2297 385 m gsave 2297 385 translate 0 rotate 0 -20 m (0.42) CS (0.42) show grestore newpath /Times-Roman-ISOLatin1 findfont 60 scalefont setfont 1513 285 m gsave 1513 285 translate 0 rotate 0 0 m (a) CS /Symbol findfont 60 scalefont setfont (a) show grestore newpath 506 435 m 516 435 l 506 816 m 516 816 l 506 1198 m 516 1198 l 506 1580 m 516 1580 l 506 1962 m 516 1962 l 2521 435 m 2511 435 l 2521 816 m 2511 816 l 2521 1198 m 2511 1198 l 2521 1580 m 2511 1580 l 2521 1962 m 2511 1962 l s 506 435 m 526 435 l 506 1198 m 526 1198 l 506 1962 m 526 1962 l 2521 435 m 2501 435 l 2521 1198 m 2501 1198 l 2521 1962 m 2501 1962 l /Symbol findfont 60 scalefont setfont /Times-Roman-ISOLatin1 findfont 60 scalefont setfont s 466 435 m gsave 466 435 translate 0 rotate 0 -20 m (0.13) RJ (0.13) show grestore newpath 466 1198 m gsave 466 1198 translate 0 rotate 0 -20 m (0.23) RJ (0.23) show grestore newpath 466 1962 m gsave 466 1962 translate 0 rotate 0 -20 m (0.33) RJ (0.33) show grestore newpath /Times-Roman-ISOLatin1 findfont 60 scalefont setfont /Symbol findfont 60 scalefont setfont 311 1274 m gsave 311 1274 translate 0 rotate 0 0 m (t) RJ (t) show grestore newpath /Symbol findfont 61 scalefont setfont 506 483 m 506 483 l 508 483 m 508 483 l 510 483 m 510 483 l 512 488 m 512 488 l 514 495 m 514 495 l 516 533 m 516 533 l 518 533 m 518 533 l 520 533 m 520 533 l 522 533 m 522 533 l 524 533 m 524 533 l 526 533 m 526 533 l 528 533 m 528 533 l 530 533 m 530 533 l 532 533 m 532 533 l 534 533 m 534 533 l 536 533 m 536 533 l 538 533 m 538 533 l 540 533 m 540 533 l 542 533 m 542 533 l 544 533 m 544 533 l 546 533 m 546 533 l 549 533 m 549 533 l 551 533 m 551 533 l 553 533 m 553 533 l 555 533 m 555 533 l 557 533 m 557 533 l 559 533 m 559 533 l 561 533 m 561 533 l 563 533 m 563 533 l 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1987 m 2231 1987 l 2233 1987 m 2233 1987 l 2235 1987 m 2235 1987 l 2237 1987 m 2237 1987 l 2239 1987 m 2239 1987 l 2241 1987 m 2241 1987 l 2243 1987 m 2243 1987 l 2245 1987 m 2245 1987 l 2247 1987 m 2247 1987 l 2249 1987 m 2249 1987 l 2251 1987 m 2251 1987 l 2253 1987 m 2253 1987 l 2255 1987 m 2255 1987 l 2257 1987 m 2257 1987 l 2259 1987 m 2259 1987 l 2261 1987 m 2261 1987 l 2263 1987 m 2263 1987 l 2265 1987 m 2265 1987 l 2267 1987 m 2267 1987 l 2269 1987 m 2269 1987 l 2271 1987 m 2271 1987 l 2273 1987 m 2273 1987 l 2275 1987 m 2275 1987 l 2277 1987 m 2277 1987 l 2279 1987 m 2279 1987 l 2281 1987 m 2281 1987 l 2283 1987 m 2283 1987 l 2285 1987 m 2285 1987 l 2287 1987 m 2287 1987 l 2289 1987 m 2289 1987 l 2291 1987 m 2291 1987 l 2293 1987 m 2293 1987 l 2295 1987 m 2295 1987 l 2297 1987 m 2297 1987 l 2299 1987 m 2299 1987 l 2301 1987 m 2301 1987 l 2303 1987 m 2303 1987 l 2305 1987 m 2305 1987 l 2307 1987 m 2307 1987 l 2309 1987 m 2309 1987 l 2311 1987 m 2311 1987 l 2313 1987 m 2313 1987 l 2315 1987 m 2315 1987 l 2317 1987 m 2317 1987 l 2319 1987 m 2319 1987 l 2321 1987 m 2321 1987 l 2323 1987 m 2323 1987 l 2325 1987 m 2325 1987 l 2327 1987 m 2327 1987 l 2329 1987 m 2329 1987 l 2331 1987 m 2331 1987 l 2333 1987 m 2333 1987 l 2335 1987 m 2335 1987 l 2337 1987 m 2337 1987 l 2339 1987 m 2339 1987 l 2342 1987 m 2342 1987 l 2344 1995 m 2344 1995 l 2346 2001 m 2346 2001 l 2348 2005 m 2348 2005 l 2350 2009 m 2350 2009 l 2352 2012 m 2352 2012 l 2354 2015 m 2354 2015 l 2356 2018 m 2356 2018 l 2358 2020 m 2358 2020 l 2360 2023 m 2360 2023 l 2362 2026 m 2362 2026 l 2364 2028 m 2364 2028 l 2366 2030 m 2366 2030 l 2368 2033 m 2368 2033 l 2370 2035 m 2370 2035 l 2372 2037 m 2372 2037 l 2374 2039 m 2374 2039 l 2376 2041 m 2376 2041 l 2378 2043 m 2378 2043 l 2380 2045 m 2380 2045 l 2382 2047 m 2382 2047 l 2384 2049 m 2384 2049 l 2386 2051 m 2386 2051 l 2388 2054 m 2388 2054 l 2390 2055 m 2390 2055 l 2392 2057 m 2392 2057 l 2394 2060 m 2394 2060 l 2396 2061 m 2396 2061 l 2398 2063 m 2398 2063 l 2400 2065 m 2400 2065 l 2402 2067 m 2402 2067 l 2404 2068 m 2404 2068 l 2406 2070 m 2406 2070 l 2408 2072 m 2408 2072 l 2410 2075 m 2410 2075 l 2412 2075 m 2412 2075 l 2414 2077 m 2414 2077 l 2416 2079 m 2416 2079 l 2418 2081 m 2418 2081 l 2420 2083 m 2420 2083 l 2422 2085 m 2422 2085 l 2424 2085 m 2424 2085 l 2426 2088 m 2426 2088 l 2428 2089 m 2428 2089 l 2430 2091 m 2430 2091 l 2432 2093 m 2432 2093 l 2434 2095 m 2434 2095 l 2436 2098 m 2436 2098 l 2438 2098 m 2438 2098 l 2440 2099 m 2440 2099 l 2442 2101 m 2442 2101 l 2444 2103 m 2444 2103 l 2446 2106 m 2446 2106 l 2448 2106 m 2448 2106 l 2450 2108 m 2450 2108 l 2452 2110 m 2452 2110 l 2454 2114 m 2454 2114 l 2456 2114 m 2456 2114 l 2458 2114 m 2458 2114 l 2460 2114 m 2460 2114 l /Symbol findfont 60 scalefont setfont s 506 435 m 506 2114 l 2521 2114 l 2521 435 l 506 435 l s 865 830 m 1067 830 l s 877 836 m 865 830 l 877 824 l closepath gsave eofill grestore s 1213 1138 m 1359 1138 l s 1225 1144 m 1213 1138 l 1225 1132 l closepath gsave eofill grestore s 1264 1201 m 1514 1201 l s 1276 1207 m 1264 1201 l 1276 1195 l closepath gsave eofill grestore s 1683 1522 m 1794 1522 l s 1695 1528 m 1683 1522 l 1695 1516 l closepath gsave eofill grestore s 1814 1620 m 1913 1620 l s 1826 1626 m 1814 1620 l 1826 1614 l closepath gsave eofill grestore s 1937 1731 m 2012 1731 l s 1949 1737 m 1937 1731 l 1949 1725 l closepath gsave eofill grestore s 2004 1790 m 2213 1790 l s 2016 1796 m 2004 1790 l 2016 1784 l closepath gsave eofill grestore s /Symbol findfont 60 scalefont setfont /Times-Roman-ISOLatin1 findfont 60 scalefont setfont 577 496 m gsave 577 496 translate 0 rotate 0 -20 m (1/7) show grestore newpath /Times-Roman-ISOLatin1 findfont 60 scalefont setfont /Symbol findfont 60 scalefont setfont /Symbol findfont 60 scalefont setfont /Times-Roman-ISOLatin1 findfont 60 scalefont setfont 699 658 m gsave 699 658 translate 0 rotate 0 -20 m (1/6) show grestore newpath /Times-Roman-ISOLatin1 findfont 60 scalefont setfont /Symbol findfont 60 scalefont setfont /Symbol findfont 60 scalefont setfont /Times-Roman-ISOLatin1 findfont 60 scalefont setfont 932 915 m gsave 932 915 translate 0 rotate 0 -20 m (1/5) show grestore newpath /Times-Roman-ISOLatin1 findfont 60 scalefont setfont /Symbol findfont 60 scalefont setfont /Symbol findfont 60 scalefont setfont /Times-Roman-ISOLatin1 findfont 60 scalefont setfont 1347 1294 m gsave 1347 1294 translate 0 rotate 0 -20 m (1/4) show grestore newpath /Times-Roman-ISOLatin1 findfont 60 scalefont setfont /Symbol findfont 60 scalefont setfont /Symbol findfont 60 scalefont setfont /Times-Roman-ISOLatin1 findfont 60 scalefont setfont 2158 1931 m gsave 2158 1931 translate 0 rotate 0 -20 m (1/3) show grestore newpath /Times-Roman-ISOLatin1 findfont 60 scalefont setfont /Symbol findfont 60 scalefont setfont /Symbol findfont 60 scalefont setfont /Times-Roman-ISOLatin1 findfont 60 scalefont setfont 1090 822 m gsave 1090 822 translate 0 rotate 0 -20 m (2/11) show grestore newpath /Times-Roman-ISOLatin1 findfont 60 scalefont setfont /Symbol findfont 60 scalefont setfont /Symbol findfont 60 scalefont setfont /Times-Roman-ISOLatin1 findfont 60 scalefont setfont 1383 1134 m gsave 1383 1134 translate 0 rotate 0 -20 m (2/9) show grestore newpath /Times-Roman-ISOLatin1 findfont 60 scalefont setfont /Symbol findfont 60 scalefont setfont /Symbol findfont 60 scalefont setfont /Times-Roman-ISOLatin1 findfont 60 scalefont setfont 1545 1197 m gsave 1545 1197 translate 0 rotate 0 -20 m (3/13) show grestore newpath /Times-Roman-ISOLatin1 findfont 60 scalefont setfont /Symbol findfont 60 scalefont setfont /Symbol findfont 60 scalefont setfont /Times-Roman-ISOLatin1 findfont 60 scalefont setfont 1814 1518 m gsave 1814 1518 translate 0 rotate 0 -20 m (3/11) show grestore newpath /Times-Roman-ISOLatin1 findfont 60 scalefont setfont /Symbol findfont 60 scalefont setfont /Symbol findfont 60 scalefont setfont /Times-Roman-ISOLatin1 findfont 60 scalefont setfont 1929 1620 m gsave 1929 1620 translate 0 rotate 0 -20 m (2/7) show grestore newpath /Times-Roman-ISOLatin1 findfont 60 scalefont setfont /Symbol findfont 60 scalefont setfont /Symbol findfont 60 scalefont setfont /Times-Roman-ISOLatin1 findfont 60 scalefont setfont 2035 1727 m gsave 2035 1727 translate 0 rotate 0 -20 m (3/10) show grestore newpath /Times-Roman-ISOLatin1 findfont 60 scalefont setfont /Symbol findfont 60 scalefont setfont /Symbol findfont 60 scalefont setfont /Times-Roman-ISOLatin1 findfont 60 scalefont setfont 2241 1787 m gsave 2241 1787 translate 0 rotate 0 -20 m (4/13) show grestore newpath /Times-Roman-ISOLatin1 findfont 60 scalefont setfont /Symbol findfont 60 scalefont setfont end showpage %%Trailer ---------------9810091121965 Content-Type: application/postscript; name="doppler_landscape.eps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="doppler_landscape.eps" %!PS-Adobe-2.0 EPSF-1.2 %%BoundingBox: 95 65 554 605 %%Title: doppler_landscape.eps %%Creator: npetrov@linux54 with xmgr %%CreationDate: Wed Oct 7 11:00:47 1998 %%EndComments 80 dict begin /languagelevel where {pop /gs_languagelevel languagelevel def} {/gs_languagelevel 1 def} ifelse gs_languagelevel 1 gt { <} image end } >> matrix makepattern /Pat0 exch def <} image end } >> matrix makepattern /Pat1 exch def <} image end } >> matrix makepattern /Pat2 exch def <} image end } >> matrix makepattern /Pat3 exch def <} image end } >> matrix makepattern /Pat4 exch def <} image end } >> matrix makepattern /Pat5 exch def <} image end } >> matrix makepattern /Pat6 exch def <} image end } >> matrix makepattern /Pat7 exch def <} image end } >> matrix makepattern /Pat8 exch def <} image end } >> matrix makepattern /Pat9 exch def <} image end } >> matrix makepattern /Pat10 exch def <} image end } >> matrix makepattern /Pat11 exch def <} image end } >> matrix makepattern /Pat12 exch def <} image end } >> matrix makepattern /Pat13 exch def <} image end } >> matrix makepattern /Pat14 exch def <} image end } >> matrix makepattern /Pat15 exch def }{ /Pat0 { 0.000000 setgray } def /Pat1 { 0.062500 setgray } def /Pat2 { 0.125000 setgray } def /Pat3 { 0.187500 setgray } def /Pat4 { 0.250000 setgray } def /Pat5 { 0.312500 setgray } def /Pat6 { 0.375000 setgray } def /Pat7 { 0.437500 setgray } def /Pat8 { 0.500000 setgray } def /Pat9 { 0.562500 setgray } def /Pat10 { 0.625000 setgray } def /Pat11 { 0.687500 setgray } def /Pat12 { 0.750000 setgray } def /Pat13 { 0.812500 setgray } def /Pat14 { 0.875000 setgray } def /Pat15 { 0.937500 setgray } def /setpattern { } def } ifelse /m {moveto} bind def /l {lineto} bind def /s {stroke} bind def % Symbol fill /f { gsave fill grestore stroke } bind def % Opaque symbol /o { gsave 1.000000 1.000000 1.000000 setrgbcolor fill grestore stroke } bind def % Circle symbol /a { 3 -1 roll 0 360 arc } bind def /da { a s } bind def /fa { a f } bind def /oa { a o } bind def % Square symbol /sq { moveto dup dup rmoveto 2 mul dup neg 0 rlineto dup neg 0 exch rlineto 0 rlineto closepath } bind def /dsq { sq s } bind def /fsq { sq f } bind def /osq { sq o } bind def % Triangle symbols /t1 { moveto dup 0 exch rmoveto dup neg dup 2 mul rlineto 2 mul 0 rlineto closepath } bind def /dt1 { t1 s } bind def /ft1 { t1 f } bind def /ot1 { t1 o } bind def /t2 { moveto dup neg 0 rmoveto dup dup 2 mul exch neg rlineto 2 mul 0 exch rlineto closepath } bind def /dt2 { t2 s } bind def /ft2 { t2 f } bind def /ot2 { t2 o } bind def /t3 { moveto dup neg 0 exch rmoveto dup dup 2 mul rlineto neg 2 mul 0 rlineto closepath } bind def /dt3 { t3 s } bind def /ft3 { t3 f } bind def /ot3 { t3 o } bind def /t4 { moveto dup 0 rmoveto dup dup -2 mul exch rlineto -2 mul 0 exch rlineto closepath } bind def /dt4 { t4 s } bind def /ft4 { t4 f } bind def /ot4 { t4 o } bind def % Diamond symbol /di { moveto dup 0 exch rmoveto dup neg dup rlineto dup dup neg rlineto dup dup rlineto closepath } bind def /ddi { di s } bind def /fdi { di f } bind def /odi { di o } bind def % Plus symbol /pl { dup 0 rmoveto dup -2 mul 0 rlineto dup dup rmoveto -2 mul 0 exch rlineto } bind def /dpl { m pl s } bind def % x symbol /x { dup dup rmoveto dup -2 mul dup rlineto 2 mul dup 0 rmoveto dup neg exch rlineto } bind def /dx { m x s } bind def % Splat symbol /dsp { m dup pl dup 0 exch rmoveto 0.707 mul x s } bind def /RJ { stringwidth neg exch neg exch rmoveto } bind def /CS { stringwidth 2 div neg exch 2 div neg exch rmoveto } bind def 0.24 0.24 scale 1 setlinecap mark /ISOLatin1Encoding 8#000 1 8#054 {StandardEncoding exch get} for /minus 8#056 1 8#217 {StandardEncoding exch get} for /dotlessi 8#301 1 8#317 {StandardEncoding exch get} for /space /exclamdown /cent /sterling /currency /yen /brokenbar /section /dieresis /copyright /ordfeminine /guillemotleft /logicalnot /hyphen /registered /macron /degree /plusminus /twosuperior /threesuperior /acute /mu /paragraph /periodcentered /cedilla /onesuperior /ordmasculine /guillemotright /onequarter /onehalf /threequarters /questiondown /Agrave /Aacute /Acircumflex /Atilde /Adieresis /Aring /AE /Ccedilla /Egrave /Eacute /Ecircumflex /Edieresis /Igrave /Iacute /Icircumflex /Idieresis /Eth /Ntilde /Ograve /Oacute /Ocircumflex /Otilde /Odieresis /multiply /Oslash /Ugrave /Uacute /Ucircumflex /Udieresis /Yacute /Thorn /germandbls /agrave /aacute /acircumflex /atilde /adieresis /aring /ae /ccedilla /egrave /eacute /ecircumflex /edieresis /igrave /iacute /icircumflex /idieresis /eth /ntilde /ograve /oacute /ocircumflex /otilde /odieresis /divide /oslash /ugrave /uacute /ucircumflex /udieresis /yacute /thorn /ydieresis /ISOLatin1Encoding where not {256 array astore def} if cleartomark /makeISOEncoded { findfont /curfont exch def /newfont curfont maxlength dict def /ISOLatin1 (-ISOLatin1) def /curfontname curfont /FontName get dup length string cvs def /newfontname curfontname length ISOLatin1 length add string dup 0 curfontname putinterval dup curfontname length ISOLatin1 putinterval def curfont { exch dup /FID ne { dup /Encoding eq { exch pop ISOLatin1Encoding exch } if dup /FontName eq { exch pop newfontname exch } if exch newfont 3 1 roll put } { pop pop } ifelse } forall newfontname newfont definefont } def /Times-Roman makeISOEncoded pop /Times-Bold makeISOEncoded pop /Times-Italic makeISOEncoded pop /Times-BoldItalic makeISOEncoded pop /Helvetica makeISOEncoded pop /Helvetica-Bold makeISOEncoded pop /Helvetica-Oblique makeISOEncoded pop /Helvetica-BoldOblique makeISOEncoded pop 2550 0 translate 90 rotate s 0.000000 0.000000 0.000000 setrgbcolor 1 setlinewidth /Times-Italic-ISOLatin1 findfont 60 scalefont setfont [] 0 setdash 506 434 m 506 444 l 1346 434 m 1346 444 l 2185 434 m 2185 444 l 506 2115 m 506 2105 l 1346 2115 m 1346 2105 l 2185 2115 m 2185 2105 l s 506 434 m 506 454 l 1346 434 m 1346 454 l 2185 434 m 2185 454 l 506 2115 m 506 2095 l 1346 2115 m 1346 2095 l 2185 2115 m 2185 2095 l /Times-Italic-ISOLatin1 findfont 60 scalefont setfont /Times-Roman-ISOLatin1 findfont 60 scalefont setfont s 506 384 m gsave 506 384 translate 0 rotate 0 -20 m (0.24) CS (0.24) show grestore newpath 1346 384 m gsave 1346 384 translate 0 rotate 0 -20 m (0.29) CS (0.29) show grestore newpath 2185 384 m gsave 2185 384 translate 0 rotate 0 -20 m (0.34) CS (0.34) show grestore newpath /Times-Roman-ISOLatin1 findfont 60 scalefont setfont /Symbol findfont 60 scalefont setfont 1513 284 m gsave 1513 284 translate 0 rotate 0 0 m (a) CS (a) show grestore newpath 506 434 m 516 434 l 506 714 m 516 714 l 506 994 m 516 994 l 506 1275 m 516 1275 l 506 1555 m 516 1555 l 506 1835 m 516 1835 l 506 2115 m 516 2115 l 2521 434 m 2511 434 l 2521 714 m 2511 714 l 2521 994 m 2511 994 l 2521 1275 m 2511 1275 l 2521 1555 m 2511 1555 l 2521 1835 m 2511 1835 l 2521 2115 m 2511 2115 l s 506 434 m 526 434 l 506 994 m 526 994 l 506 1555 m 526 1555 l 506 2115 m 526 2115 l 2521 434 m 2501 434 l 2521 994 m 2501 994 l 2521 1555 m 2501 1555 l 2521 2115 m 2501 2115 l /Symbol findfont 60 scalefont setfont /Times-Roman-ISOLatin1 findfont 60 scalefont setfont s 466 434 m gsave 466 434 translate 0 rotate 0 -20 m (0) RJ (0) show grestore newpath 466 994 m gsave 466 994 translate 0 rotate 0 -20 m (1) RJ (1) show grestore newpath 466 1555 m gsave 466 1555 translate 0 rotate 0 -20 m (2) RJ (2) show grestore newpath 466 2115 m gsave 466 2115 translate 0 rotate 0 -20 m (3) RJ (3) show grestore newpath /Times-Roman-ISOLatin1 findfont 60 scalefont setfont /Helvetica-ISOLatin1 findfont 60 scalefont setfont 334 1275 m gsave 334 1275 translate 90 rotate 0 0 m (log10 D\(Q6\) for aNNb\(1/6\)) CS /Times-Roman-ISOLatin1 findfont 60 scalefont setfont (log) show /Times-Roman-ISOLatin1 findfont 35 scalefont setfont 0 -17 rmoveto (10) show /Times-Roman-ISOLatin1 findfont 60 scalefont setfont 0 17 rmoveto ( ) show /Times-Italic-ISOLatin1 findfont 60 scalefont setfont (D) show /Times-Roman-ISOLatin1 findfont 60 scalefont setfont (\() show /Symbol findfont 60 scalefont setfont (Q) show /Times-Roman-ISOLatin1 findfont 60 scalefont setfont /Times-Roman-ISOLatin1 findfont 35 scalefont setfont 0 -17 rmoveto (6) show /Times-Roman-ISOLatin1 findfont 60 scalefont setfont 0 17 rmoveto (\) for ) show /Symbol findfont 60 scalefont setfont (a) show /Times-Roman-ISOLatin1 findfont 60 scalefont setfont /Symbol findfont 60 scalefont setfont (\316) show /Times-Roman-ISOLatin1 findfont 60 scalefont setfont /Times-Italic-ISOLatin1 findfont 60 scalefont setfont (N) show /Times-Italic-ISOLatin1 findfont 35 scalefont setfont 0 -17 rmoveto /Symbol findfont 35 scalefont setfont (b) show /Symbol findfont 60 scalefont setfont 0 17 rmoveto /Times-Roman-ISOLatin1 findfont 60 scalefont setfont (\(1/6\)) show grestore newpath 741 434 m 741 438 l 741 439 l 741 440 l 741 441 l 741 442 l 741 443 l 741 444 l 741 445 l 741 446 l 741 447 l 741 448 l 741 449 l 741 450 l 741 451 l 741 452 l 742 452 l 742 453 l 742 454 l 742 455 l 742 456 l 742 457 l 742 458 l 742 459 l 742 460 l 742 461 l 743 461 l 743 462 l 743 463 l 743 464 l 743 465 l 743 466 l 744 466 l 744 467 l 744 468 l 744 469 l 744 470 l 745 470 l 745 471 l 745 472 l 746 472 l 746 473 l 747 473 l 748 473 l 749 473 l 749 472 l 749 471 l 750 471 l 750 470 l 750 469 l 751 469 l 751 468 l 751 467 l 751 466 l 751 465 l 752 465 l 752 464 l 752 463 l 752 462 l 752 461 l 752 460 l 753 460 l 753 459 l 753 458 l 753 457 l 753 456 l 753 455 l 753 454 l 753 453 l 753 452 l 753 451 l 754 451 l 754 450 l 754 449 l 754 448 l 754 447 l 754 446 l 754 445 l 754 444 l 754 443 l 754 442 l 754 441 l 754 440 l 754 439 l 754 438 l 754 437 l s 1246 434 m 1246 441 l 1246 443 l 1246 444 l 1246 446 l 1246 447 l 1246 448 l 1246 449 l 1246 450 l 1246 451 l 1246 452 l 1246 453 l 1246 454 l 1246 455 l 1246 456 l 1246 457 l 1246 458 l 1246 459 l 1246 460 l 1246 461 l 1246 462 l 1246 463 l 1246 464 l 1246 465 l 1246 466 l 1246 467 l 1246 468 l 1246 469 l 1246 470 l 1246 471 l 1246 472 l 1246 473 l 1246 474 l 1247 474 l 1247 475 l 1247 476 l 1247 477 l 1247 478 l 1247 479 l 1247 480 l 1247 481 l 1247 482 l 1247 483 l 1247 484 l 1247 485 l 1247 486 l 1247 487 l 1247 488 l 1247 489 l 1248 489 l 1248 490 l 1248 491 l 1248 492 l 1248 493 l 1248 494 l 1248 495 l 1248 496 l 1248 497 l 1248 498 l 1248 499 l 1249 500 l 1249 501 l 1249 502 l 1249 503 l 1249 504 l 1249 505 l 1249 506 l 1249 507 l 1250 507 l 1250 508 l 1250 509 l 1250 510 l 1250 511 l 1250 512 l 1250 513 l 1251 514 l 1251 515 l 1251 516 l 1251 517 l 1251 518 l 1252 518 l 1252 519 l 1252 520 l 1252 521 l 1252 522 l 1253 522 l 1253 523 l 1253 524 l 1253 525 l 1254 525 l 1254 526 l 1254 527 l 1255 527 l 1255 528 l 1256 528 l 1256 529 l 1257 529 l 1258 529 l 1259 529 l 1259 528 l 1260 528 l 1260 527 l 1261 527 l 1261 526 l 1261 525 l 1262 525 l 1262 524 l 1262 523 l 1263 523 l 1263 522 l 1263 521 l 1263 520 l 1264 520 l 1264 519 l 1264 518 l 1264 517 l 1265 517 l 1265 516 l 1265 515 l 1265 514 l 1265 513 l 1266 513 l 1266 512 l 1266 511 l 1266 510 l 1266 509 l 1266 508 l 1267 508 l 1267 507 l 1267 506 l 1267 505 l 1267 504 l 1267 503 l 1268 503 l 1268 502 l 1268 501 l 1268 500 l 1268 499 l 1268 498 l 1268 497 l 1269 497 l 1269 496 l 1269 495 l 1269 494 l 1269 493 l 1269 492 l 1269 491 l 1269 490 l 1269 489 l 1270 489 l 1270 488 l 1270 487 l 1270 486 l 1270 485 l 1270 484 l 1270 483 l 1270 482 l 1270 481 l 1270 480 l 1271 480 l 1271 479 l 1271 478 l 1271 477 l 1271 476 l 1271 475 l 1271 474 l 1271 473 l 1271 472 l 1271 471 l 1271 470 l 1271 469 l 1271 468 l 1272 468 l 1272 467 l 1272 466 l 1272 465 l 1272 464 l 1272 463 l 1272 462 l 1272 461 l 1272 460 l 1272 459 l 1272 458 l 1272 457 l 1272 456 l 1272 455 l 1272 454 l 1272 453 l 1272 452 l 1272 451 l 1272 450 l 1272 449 l 1272 448 l 1272 447 l 1272 446 l 1272 445 l 1273 444 l 1273 443 l 1273 442 l 1273 441 l 1273 439 l 1273 436 l s 1778 434 m 1778 450 l 1778 455 l 1778 458 l 1778 461 l 1778 464 l 1778 467 l 1778 469 l 1778 472 l 1778 474 l 1778 476 l 1778 478 l 1778 479 l 1778 481 l 1778 483 l 1778 484 l 1778 486 l 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1677 995 l 1677 1835 l 1006 1835 l s 506 434 m 506 2115 l 2521 2115 l 2521 434 l 506 434 l s 1004 996 m 1004 996 l s 1004 996 m 1004 1834 l s 5 setlinewidth 1739 466 m 1351 838 l 1718 473 m 1739 466 l 1730 487 l s 1 setlinewidth /Times-Roman-ISOLatin1 findfont 60 scalefont setfont 1007 936 m gsave 1007 936 translate 0 rotate 0 -20 m (0) CS (0) show grestore newpath /Times-Roman-ISOLatin1 findfont 60 scalefont setfont /Helvetica-ISOLatin1 findfont 60 scalefont setfont /Helvetica-ISOLatin1 findfont 60 scalefont setfont /Times-Roman-ISOLatin1 findfont 60 scalefont setfont 1680 932 m gsave 1680 932 translate 0 rotate 0 -20 m (6410-7) CS (6) show /Symbol findfont 60 scalefont setfont (\264) show /Times-Roman-ISOLatin1 findfont 60 scalefont setfont (10) show /Times-Roman-ISOLatin1 findfont 35 scalefont setfont 0 35 rmoveto (-7) show grestore newpath /Times-Roman-ISOLatin1 findfont 60 scalefont setfont /Helvetica-ISOLatin1 findfont 60 scalefont setfont /Helvetica-ISOLatin1 findfont 60 scalefont setfont /Times-Roman-ISOLatin1 findfont 60 scalefont setfont 1415 921 m gsave 1415 921 translate 0 rotate 0 -20 m (a-ac) RJ /Symbol findfont 60 scalefont setfont (a) show /Times-Roman-ISOLatin1 findfont 60 scalefont setfont (-) show /Symbol findfont 60 scalefont setfont (a) show /Times-Roman-ISOLatin1 findfont 60 scalefont setfont /Times-Roman-ISOLatin1 findfont 35 scalefont setfont 0 -17 rmoveto (c) show grestore newpath /Times-Roman-ISOLatin1 findfont 60 scalefont setfont /Helvetica-ISOLatin1 findfont 60 scalefont setfont /Helvetica-ISOLatin1 findfont 60 scalefont setfont /Times-Roman-ISOLatin1 findfont 60 scalefont setfont 968 1012 m gsave 968 1012 translate 0 rotate 0 -20 m (1.0000) RJ (1.0000) show grestore newpath /Times-Roman-ISOLatin1 findfont 60 scalefont setfont /Helvetica-ISOLatin1 findfont 60 scalefont setfont /Helvetica-ISOLatin1 findfont 60 scalefont setfont /Times-Roman-ISOLatin1 findfont 60 scalefont setfont 960 1830 m gsave 960 1830 translate 0 rotate 0 -20 m (1.0152) RJ (1.0152) show grestore newpath /Times-Roman-ISOLatin1 findfont 60 scalefont setfont /Helvetica-ISOLatin1 findfont 60 scalefont setfont /Helvetica-ISOLatin1 findfont 60 scalefont setfont /Times-Roman-ISOLatin1 findfont 60 scalefont setfont 751 541 m gsave 751 541 translate 0 rotate 0 -20 m (b=0.10) CS /Symbol findfont 60 scalefont setfont (b) show /Times-Roman-ISOLatin1 findfont 60 scalefont setfont (=0.10) show grestore newpath /Times-Roman-ISOLatin1 findfont 60 scalefont setfont /Helvetica-ISOLatin1 findfont 60 scalefont setfont /Helvetica-ISOLatin1 findfont 60 scalefont setfont /Times-Roman-ISOLatin1 findfont 60 scalefont setfont 1260 589 m gsave 1260 589 translate 0 rotate 0 -20 m (b=0.12) CS /Symbol findfont 60 scalefont setfont (b) show /Times-Roman-ISOLatin1 findfont 60 scalefont setfont (=0.12) show grestore newpath /Times-Roman-ISOLatin1 findfont 60 scalefont setfont /Helvetica-ISOLatin1 findfont 60 scalefont setfont /Helvetica-ISOLatin1 findfont 60 scalefont setfont /Times-Roman-ISOLatin1 findfont 60 scalefont setfont 1794 699 m gsave 1794 699 translate 0 rotate 0 -20 m (b=0.14) CS /Symbol findfont 60 scalefont setfont (b) show /Times-Roman-ISOLatin1 findfont 60 scalefont setfont (=0.14) show grestore newpath /Times-Roman-ISOLatin1 findfont 60 scalefont setfont /Helvetica-ISOLatin1 findfont 60 scalefont setfont /Helvetica-ISOLatin1 findfont 60 scalefont setfont /Times-Roman-ISOLatin1 findfont 60 scalefont setfont 2059 822 m gsave 2059 822 translate 0 rotate 0 -20 m (b=0.15) CS /Symbol findfont 60 scalefont setfont (b) show /Times-Roman-ISOLatin1 findfont 60 scalefont setfont (=0.15) show grestore newpath /Times-Roman-ISOLatin1 findfont 60 scalefont setfont /Helvetica-ISOLatin1 findfont 60 scalefont setfont /Helvetica-ISOLatin1 findfont 60 scalefont setfont /Times-Roman-ISOLatin1 findfont 60 scalefont setfont 2276 1494 m gsave 2276 1494 translate 0 rotate 0 -20 m () RJ /Symbol findfont 60 scalefont setfont () show grestore newpath /Symbol findfont 60 scalefont setfont /Helvetica-ISOLatin1 findfont 60 scalefont setfont /Helvetica-ISOLatin1 findfont 60 scalefont setfont /Times-Roman-ISOLatin1 findfont 60 scalefont setfont 2273 1407 m gsave 2273 1407 translate 0 rotate 0 -20 m (b=1/2p) RJ /Symbol findfont 60 scalefont setfont (b) show /Times-Roman-ISOLatin1 findfont 60 scalefont setfont (=1/2) show /Symbol findfont 60 scalefont setfont (p) show grestore newpath /Symbol findfont 60 scalefont setfont /Helvetica-ISOLatin1 findfont 60 scalefont setfont /Helvetica-ISOLatin1 findfont 60 scalefont setfont /Times-Roman-ISOLatin1 findfont 60 scalefont setfont 968 1423 m gsave 968 1423 translate 0 rotate 0 -20 m (D\(Q6\)) RJ /Times-Italic-ISOLatin1 findfont 60 scalefont setfont (D) show /Times-Roman-ISOLatin1 findfont 60 scalefont setfont (\() show /Symbol findfont 60 scalefont setfont (Q) show /Times-Roman-ISOLatin1 findfont 60 scalefont setfont /Times-Roman-ISOLatin1 findfont 35 scalefont setfont 0 -17 rmoveto (6) show 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putinterval def curfont { exch dup /FID ne { dup /Encoding eq { exch pop ISOLatin1Encoding exch } if dup /FontName eq { exch pop newfontname exch } if exch newfont 3 1 roll put } { pop pop } ifelse } forall newfontname newfont definefont } def /Times-Roman makeISOEncoded pop /Times-Bold makeISOEncoded pop /Times-Italic makeISOEncoded pop /Times-BoldItalic makeISOEncoded pop /Helvetica makeISOEncoded pop /Helvetica-Bold makeISOEncoded pop /Helvetica-Oblique makeISOEncoded pop /Helvetica-BoldOblique makeISOEncoded pop 2550 0 translate 90 rotate s 0.000000 0.000000 0.000000 setrgbcolor 1 setlinewidth /Times-Italic-ISOLatin1 findfont 60 scalefont setfont [] 0 setdash 3 setlinewidth 1510 434 m 1996 1692 l s 1 setlinewidth [40 20] 0 setdash 1992 1688 m 1992 1146 l s [] 0 setdash 1652 802 m 1213 434 l s 1794 1174 m 909 434 l s 1794 1174 m 1312 1577 l s 1652 802 m 707 1577 l s [20 20] 0 setdash 3 setlinewidth 901 1419 m 1498 1419 l s 1 setlinewidth [] 0 setdash [20 20] 0 setdash 3 setlinewidth 1007 509 m 1304 509 l s 1 setlinewidth [] 0 setdash [20 20] 0 setdash 3 setlinewidth 1296 509 m 1296 509 l s 1 setlinewidth [] 0 setdash /Times-Italic-ISOLatin1 findfont 60 scalefont setfont /Times-Roman-ISOLatin1 findfont 60 scalefont setfont 1126 1466 m gsave 1126 1466 translate 0 rotate 0 -20 m (D") CS /Symbol findfont 60 scalefont setfont (D) show /Symbol findfont 60 scalefont setfont (\242) show grestore newpath /Symbol findfont 60 scalefont setfont /Times-Italic-ISOLatin1 findfont 60 scalefont setfont /Times-Italic-ISOLatin1 findfont 60 scalefont setfont /Times-Roman-ISOLatin1 findfont 60 scalefont setfont 1166 561 m gsave 1166 561 translate 0 rotate 0 -20 m ( D) CS ( ) show /Symbol findfont 60 scalefont setfont (D) show grestore newpath /Symbol findfont 60 scalefont setfont /Times-Italic-ISOLatin1 findfont 60 scalefont setfont /Times-Italic-ISOLatin1 findfont 60 scalefont setfont /Times-Roman-ISOLatin1 findfont 60 scalefont setfont 1933 1391 m gsave 1933 1391 translate 0 rotate 0 -20 m (d) CS /Symbol findfont 60 scalefont setfont (d) show grestore newpath /Symbol findfont 60 scalefont setfont /Times-Italic-ISOLatin1 findfont 60 scalefont setfont /Times-Italic-ISOLatin1 findfont 60 scalefont setfont /Times-Roman-ISOLatin1 findfont 60 scalefont setfont 1608 573 m gsave 1608 573 translate 0 rotate 0 -20 m ( m) CS ( ) show /Times-Italic-ISOLatin1 findfont 60 scalefont setfont (m) show grestore newpath /Times-Italic-ISOLatin1 findfont 60 scalefont setfont end showpage %%Trailer ---------------9810091121965--