\magnification=\magstep1 \input amstex \input psfig \documentstyle{amsppt} \vsize=22 truecm \hsize=16 truecm \TagsOnRight \NoRunningHeads %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \def\today {\ifcase\month\or January \or February \or March \or April \or May \or June \or July \or August \or September \or October \or November \or December \fi \number\day~\number\year} %=============postscript======= %figure number psfile caption % (will be centered) \def\figure #1 #2 #3\cr { \bigskip \centerline{ \psfig {figure=#2} } \vbox{\eightpoint\noindent {\bf Figure #1} #3} \medskip } %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \def \real{{\Bbb R}} \def \complex{{\Bbb C}} \def \integer{{\Bbb Z}} \def \rational{{\Bbb Q}} \redefine \natural{{\Bbb N}} % \def\AA{{\Cal A}} \def\BB{{\Cal B}} \def\CC{{\Cal C}} \def\DD{{\Cal D}} \def\EE{{\Cal E}} \def\FF{{\Cal F}} \def\HH{{\Cal H}} \def\II{{\Cal I}} \def\JJ{{\Cal J}} \def\KK{{\Cal K}} \def\LL{{\Cal L}} \def\MM{{\Cal M}} \def\NN{{\Cal N}} \def\OO{{\Cal O}} \def\PP{{\Cal P}} \def\QQ{{\Cal Q}} \def\RR{{\Cal R}} \def\SS{{\Cal S}} \def\TT{{\Cal T}} \def\UU{{\Cal U}} \def\VV{{\Cal V}} \def\XX{{\Cal X}} \def\ZZ{{\Cal Z}} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \topmatter \title Lyapunov exponents for non-classical multidimensional continued fraction algorithms \endtitle \author V. Baladi and A. Nogueira \endauthor \address V. Baladi: Math\'ematiques, Universit\'e de Gen\`eve, CH-1211 Geneva 24, SWITZERLAND\newline \phantom{vb} (on leave from CNRS, UMR 128, ENS Lyon, France) \endaddress \email baladi\@sc2a.unige.ch \endemail \address A. Nogueira: Instituto de Matem\'atica, Universidade Federal do Rio de Janeiro \newline \phantom{vb} Caixa Postal 68.530, 21.945 Rio de Janeiro RJ, BRAZIL \endaddress \email nogueira\@vms1.nce.ufrj.br \endemail \date{November 1995} \enddate \subjclass 11J70, 11K50, 28D05 \endsubjclass \abstract We introduce a simple geometrical two-dimensional continued fraction algorithm inspired from dynamical renormalization. We prove that the algorithm is weakly convergent, and that the associated transformation admits an ergodic absolutely continuous invariant probability measure. Following Lagarias, its Lyapunov exponents are related to the approximation exponents which measure the diophantine quality of the continued fraction. The Lyapunov exponents for our algorithm and related ones, also introduced in this article, are studied numerically. \endabstract \endtopmatter %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \document \head 1. Introduction \endhead Most one-dimensional or multidimensional continued fraction algorithms, for example the Jacobi-Perron algorithm, can be given a simple geometrical definition as follows. Fixing the dimension $d \ge 1$, our aim is to obtain successive rational approximations of vectors $\theta$ in $[0,1]^d$. We view such a vector as the direction $\Theta=(\theta_1, \ldots, \theta_d,1)$ of a half-line in the positive quadrant of $\real^{d+1}$ and the algorithm produces a sequence of unimodular bases $(\ell_{1j}, \ell_{2j}, \ldots, \ell_{(d+1) j})$ of the lattice $\integer^{d+1}$ such that the direction $\Theta$ lies in the positive quadrant of all bases and the directions of the basis vectors converge to $\Theta$ (an algorithm with this last property for all $\theta$ --- sometimes only almost all $\theta$ --- is called {\it weakly convergent}). The approximations for $\theta$ are given by the vectors $(\ell_{1j}/\ell_{(d+1) j},\ldots, \ell_{d j} /\ell_{(d+1) j} )$. See e.g. [Br2] for references about this geometrical description, and also [B] or [L] for other definitions of convergence and a list of classical continued fraction algorithms. Inspired by the renormalization techniques described in [BRTT] for torus rotations, we introduce in this paper a simple two-dimensional continued fraction algorithm, called algorithm $\AA$, which has a rather natural geometric definition. This algorithm can be viewed as a path in a tree of possible algorithms, and its weak convergence can be proved by a very economical application of finite-dimensional projective metrics. Other ergodic properties are obtained in a similar way, allowing us to apply a theorem of Lagarias [L] on the existence of Lyapunov exponents for the algorithm and their relationship with the approximation exponents (see Section 2 for definitions). A numerical study of this algorithm and of the ones corresponding to some other paths (including randomly chosen ones) in our tree indicates that they can be very favorably compared with commonly used methods such as the Jacobi-Perron algorithm. We also define (and obtain partial results on) adaptations of the algorithm to higher-dimensions. The paper is organized as follows: In Section 2 we give the geometrical and analytical definitions of algorithm $\AA$, show how it can be viewed as a path in a tree of algorithms related to the renormalizations introduced in [BRTT], and state our main result (Theorem 1). In Section 3, we show that our algorithm is (a little bit better than) weakly convergent (Lemma 2). In Section 4, we prove that the transformation associated with the algorithm admits an ergodic absolutely continuous invariant probability measure with good properties (Lemma 3) which yields Theorem 1 by using [L]. Finally, we present our numerical results in Section 5. \smallskip We are very grateful to E. Ghys and K. Khanin for useful conversations. V.B. is also specially thankful to S.~Oliffson Kamphorst and C.~Tresser. This article was initiated during a visit of A.N. to E.N.S. Lyon and completed during visits of V.B. to U.F.~ Rio de Janeiro and I.M.P.A., and of A.N. to E.T.H. Z\"urich. The hospitality and financial support of these institutions is gratefully acknowledged. V.B. is also thankful to the C.C.C.I. (France) and the Soci\'et\'e Acad\'emique de Gen\`eve (Switzerland) for financial support. \head 2. Definition of the algorithms -- Statement of the main result \endhead We shall describe below in each dimension $d \ge 1$ a tree of algorithms related to the renormalizations studied in [BRTT]. We concentrate now on one specific two-dimensional algorithm (that we call algorithm $\AA$) which will occur as a particular path in the tree, and which turns out to be the easiest to study rigorously. As we shall see in the course of this article, some of our proofs apply to other paths, whereas we only have conjectures and/or numerical evidence in the general case. \smallskip \subhead{2.1 Geometric and analytical description of the algorithm} \endsubhead Let $\theta=(\theta_1, \theta_2)$ be the vector to be approximated (we may assume $0 < \theta_1 \le \theta_2 \le 1+\theta_1$) and $\Theta=(\theta_1, \theta_2,1)$ be the direction we wish to approach. Representing $\Theta$ by the first two coordinates $\alpha(\theta)=(\alpha_1, \alpha_2) =(\theta_1/(\theta_1+\theta_2+1), \theta_2/(\theta_1+\theta_2+1))$ of the intersection of the ray $\lambda \Theta$, $\lambda > 0$, with the two-dimensional simplex $\bar \Delta= \{ (x_1, x_2, x_3)\in \real_+^3 \mid x_1+x_2+x_3 =1\}$, we observe that $0 < \alpha_1 \le\alpha_2\le 1/2$. (See Figure 1.) We denote by $$ \Delta=\Delta_2= \{ (\alpha_1, \alpha_2, \alpha_3) \in \real^3_+ \mid \alpha_1 + \alpha_2 + \alpha_3 =1 \, , 0 \le \alpha_1 \le\alpha_2\le 1/2\} $$ the subset of the simplex $\bar \Delta$ thus obtained. \figure 1 fig1.eps Intersection of the ray $\lambda\Theta$ with $\Delta=\Delta_2$ \cr Writing points in the coordinates given by the basis $e_1, e_2, e_3$ constructed at the $n-1$-th step, the ray $\lambda \Theta$ intersects each of the three two-dimensional planes $P_i=\{ x\in \real^3\mid x_i =1\}$ ($i=1,2,3$) at a point $\lambda_{j(i)} \Theta$ for some $0 < \lambda_1 \le \lambda_2\le \lambda_3$. Observe that if the three intersection points coincide, then we may (but will not) stop iterating the algorithm because our ray has positive integer coordinates in the current basis. Let us first assume that the three intersection points are distinct. Then the {\it last intersection point} $p=\lambda_3 \Theta$ lies in the interior of a square $\{ (u,v) \mid 0 \le u, v \le 1\}$ of the lattice $\integer^2$ of the corresponding hyperplane $P_{i(3)}$. Dividing this square in four subsquares according to whether $u \ge 1/2$ or $u \le 1/2$ and $v \ge 1/2$ or $v \le 1/2$, and assuming first that $u(p)$ and $v(p)$ are both different from $1/2$, the point $p$ lies in a well-defined subsquare. (See Figure 2.a.) We shall now define the new simplex $\bar \Delta'$ and the new basis $e'_1$, $e'_2$, $e'_3$. We will write $(\alpha'_1$, $\alpha'_2)$ for the ($e'_j$) coordinates of the intersection of the ray $\lambda \Theta$ and the new simplex $\bar \Delta'$. We choose the three new basis elements by setting $e_3'$ to be the common vertex of the square and the selected subsquare, and $e'_1$, $e'_2$ two vertices joined to $e'_3$ by an edge of the square, ordered in such a way as to guarantee the property $\alpha'_1 \le \alpha'_2$. Then $\bar \Delta'$ is the simplex generated by the $e_i'$ and we may define a subsimplex $\Delta' \subset \bar \Delta'$ containing $p$ as above. (See Figure 2.b.) Observe that we could have used exactly the same procedure by choosing the second or first intersected plane $P_i$ instead of the last one. \figure 2 fig2.eps a) Intersection of the ray $\Theta$ with the planes $\PP_i$ b) Determination of the new basis \cr If there is any ambiguity in the above choices ($\alpha'_1=\alpha'_2$, or $u(p)$ or $v(p)$ equal to $1/2$, or coincidence of intersection points of the ray and the hyperplanes) we may either use a predefined rule or pick arbitrarily among the several possibilities: this will not influence the discussion in the present work. Clearly, by construction, the ray to be approximated lies in the positive quadrant of each basis and the property $0 \le \alpha_1 \le \alpha_2 \le 1/2$ is preserved. However, the property that $\alpha_1 > 0$ is not necessarily preserved. If at some iterate $\alpha_1=0$ but $\alpha_2 > 0$, the ray $\lambda \Theta$ will only intersect the hyperplanes $P_2$ and $P_3$. In this case we could (but will not) decide to switch to a one-dimensional continued fraction algorithm, instead, we will just continue applying algorithm $\AA$ taking at each step the last (i.e. second) intersected hyperplane (note that the property $\alpha_1=0$ will remain true for further iterations). If at some step $\alpha_1=\alpha_2=0$ then the ray only intersects the hyperplane $P_3$. We could then decide to stop iterating our algorithm (the continued fraction having been completely obtained) but will instead continue iterating using the only available hyperplane. We shall call the cases where $\alpha_1=0$ {\it degenerate.} \medskip We now turn to the analytic description of the algorithm $\AA$. We describe the algorithm with an auxiliary map $T: \Delta \to \Delta$ and partial quotient matrix $A:\Delta \to GL (3, \integer)$ (both defined Lebesgue almost everywhere), associating to $\alpha \in \Delta$ a sequence of matrices in $GL(3, \integer)$ by setting $L^{(0)}\equiv Id$ and $$ L^{(n)} (\alpha) = A(\alpha) A(T(\alpha)) A(T^2(\alpha)) \cdots A(T^{n-1} (\alpha)) \, .\tag{2.1} $$ The three columns $(\ell_{1j}, \ell_{2j}, \ell_{3j})\in \integer^3$, $j=1, 2, 3$ of the matrix $L^{(n)}(\alpha)$ are the elements of the $n$-th basis and the three rational vectors $(\ell_{1j}/\ell_{3 j}, \ell_{2 j}/\ell_{3 j})$, $j=1, 2, 3$ are the successive approximations of $(\theta_1, \theta_2)$ if $\alpha=\alpha(\theta)$. An algorithm defined by two maps $T$, $A$ as above (up to trivial changes of coordinates such as viewing $T$ as a function of $\theta$ instead of $\alpha$ and transposing matrices) is called a {\it Markovian multidimensional continued fraction algorithm} (see e.g. [L]). We now give the definitions for $T$ and $L$, using $\{x\}$ and $[x]$ do denote the fractional and integer parts of a real number $x$. First consider the case where the last intersected hyperplane is $P_1$: this is equivalent to the condition $$ \theta_2/\theta_1=\alpha_2/\alpha_1\ge 1 \text{ (always satisfied) } \quad \text{ and } 1/\theta_1= (1-\alpha_1-\alpha_2)/\alpha_1\ge 1 \, . \tag{2.2} $$ Then we set $$ \cases \beta_1(\alpha_1, \alpha_2) &= \min (\{ {\alpha_2 \over \alpha_1} \}, 1- \{ {\alpha_2 \over \alpha_1}\} )\, , \\ \beta_2 (\alpha_1, \alpha_2)&= \min(\{ {1-\alpha_1-\alpha_2 \over \alpha_1} \}, 1- \{ {1-\alpha_1-\alpha_2 \over \alpha_1}\} ) \, , \endcases \tag{2.3} $$ and $$ T(\alpha_1, \alpha_2)= (\min (\beta_1, \beta_2), \max(\beta_1, \beta_2)) \, .\tag{2.4} $$ Setting $$ \cases k&=[\theta_2/\theta_1]=[\alpha_2/\alpha_1] \cr j&=[1/\theta_1] = [(1-\alpha_1-\alpha_2)/\alpha_1] \, , \endcases \tag{2.5} $$ we have $k\ge 1$ and $j \ge 1$ except if we are in a degenerate case. We also have, for example, $$ A(\alpha_1, \alpha_2)=\cases \left ( \matrix 1& 1& 1\\ k+1 &k&k \\ j& j+1&j\\ \endmatrix \right ) \, , & \text{if $\{ {\alpha_2 \over \alpha_1} \}< \{ {1-\alpha_1-\alpha_2 \over \alpha_1} \}< 1/2$,}\cr \left ( \matrix 1& 1& 1\\ k &k+1&k \\ j+1& j&j\\ \endmatrix \right ) \, , & \text{if $\{ {1-\alpha_1-\alpha_2 \over \alpha_1} \} <\{ {\alpha_2 \over \alpha_1} \} < 1/2$.}\cr \endcases \tag{2.6} $$ In the cases left undefined in \thetag{2.6}, the matrix $A$ takes one of the following forms (up to exchanging the first two columns): $$ \left ( \matrix 1& 1& 1\\ k &k+1&k+1 \\ j+1& j&j+1\\ \endmatrix \right )\, , \left ( \matrix 1& 1& 1\\ k &k+1&k \\ j& j+1&j+1\\ \endmatrix \right )\, , \left ( \matrix 1& 1& 1\\ k &k+1&k+1 \\ j& j+1&j\\ \endmatrix \right )\, . \tag{2.7} $$ The maps $T$ and $A$ corresponding to the last intersection with $P_2$ or $P_3$ are similar. More precisely, one just needs to exchange $\alpha_2$ and $\alpha_1$ in \thetag{2.3--2.6} to get the formula for the intersection with $P_2$ (this case only occurs on the zero-measure set defined by $\alpha_1=\alpha_2$), and one should take $$ \cases \beta_1 (\alpha_1, \alpha_2)&= \min (\{ {\alpha_1 \over 1-\alpha_1-\alpha_2 } \}, 1- \{ {\alpha_1 \over 1-\alpha_1-\alpha_2 } \} )\, , \cr \beta_2 (\alpha_1, \alpha_2)&= \min (\{ {\alpha_2 \over 1-\alpha_1-\alpha_2 } \}, 1- \{ {\alpha_2 \over 1-\alpha_1-\alpha_2 } \} ) \, , \endcases \tag{2.8} $$ and $$ \cases k&=[\alpha_1/(1-\alpha_1-\alpha_2)] \cr j& = [\alpha_2/(1-\alpha_1-\alpha_2)] \, , \endcases \tag{2.9} $$ when considering the intersection with $P_3$ (again, $k\ge 1$ and $j \ge 1$ except in the degenerate cases). Note that the formulas for $T$ and $A$ that we would obtain by considering the first (respectively the second) intersected hyperplane instead of the last one would be exactly the same: the only difference is that $k$ and/or $j$ in the expressions for $A(\alpha_1, \alpha_2)$ would vanish even in non degenerate cases. Observe also that the geometrical description of algorithm $\AA$ implies that it is of the {\it linear simplex-splitting} type (see e.g. [L]), i.e., the matrix $\widetilde A := A^{-1}$ defines a piecewise linear map $\widetilde T$ on the homogeneous cone $\{ y \in \real^3 \mid y_i \ge 0 \, , y_3=\max_i y_i \}$, for which there is a countable partition into homogeneous cones on each of which $\widetilde T$ is homogeneous linear with $\Tilde T(y)=\tilde A(y)$. Finally, the map $\widetilde T$ is constant on rays, and a suitable choice of representative produces the map $T$. (See the proof of Lemma 3 in Section 4 for more about this.) \smallskip \subhead{2.2 Approximation exponents} \endsubhead \smallskip In order to state our main result, we need some definitions valid for general $d \ge 1$ (we follow the terminology in [L]). The {\it Diophantine approximation exponent} $\eta(\theta)$ of $\theta \in [0,1]^d$ is defined by $$ \eta(\theta) = \limsup_{q \to \infty} \sup \Sb w=(p_1, \ldots, p_d, q) \in \integer^{d+1} \\ 0 \le p_i \le q \endSb \eta(w,\theta)\, , \tag{2.10} $$ where the {\it Roth exponent} $\eta(w,\theta)$ for $w=(p_1, \ldots, p_d, q)$ is set to be $$ \eta(w, \theta) = {-\log \|\theta-\tilde w\|\over \log q } \, , \tag{2.11} $$ where $\tilde w = (p_1/q, \ldots, p_d/q) \in \rational ^d$ and $\|\cdot \|$ denotes euclidean norm. If an algorithm $\DD$ produces positive matrices $L^{(n)}\in GL(d+1, \integer)$ whose columns $w_i^{(n)}$ describe the successive approximations of $\Theta(\theta)$, we define the {\it best approximation exponent for $\theta$ using the algorithm $\DD$} by $$ \eta_\DD(\theta) := \limsup_{n \to \infty} \bigr \{ \max_{1 \le i \le d+1} \eta(w_i^{(n)}(\theta),\theta) \bigr \} \, , \tag{2.12} $$ and the {\it uniform approximation exponent for $\theta$ using $\DD$} by $$ \eta^*_\DD(\theta) := \liminf_{n \to \infty} \bigr \{ \min_{1 \le i \le d+1} \eta(w_i^{(n)}(\theta),\theta) \bigr \} \le \eta_\DD(\theta) \, . \tag{2.13} $$ It follows from Dirichlet's theorem (see e.g. [L]) that $\eta^*_\DD(\theta) \le 1+(1/d)$ for any algorithm $\DD$ and almost all $\theta$. An algorithm $\DD$ which realizes the Dirichlet bound $\eta^*_\DD(\theta) = 1+(1/d)$ for almost every $\theta$ is called an {\it optimal} algorithm. The ordinary one-dimensional continued fraction agorithm is optimal but the existence of optimal algorithms for any $d \ge 2$ is a basic open question. \smallskip The relevance of Oseledec's theorem in the study of these approximation algorithms was first pointed out by Kosygin ([K]) and then applied by several authors (in particular [Ba1, Ba2], [L], and [Br2] who gives a clear overview). Referring the reader e.g. to Section 3 in [L] for a statement of Oseledec's theorem and a definition of the Lyapunov exponents $\lambda_i$, we now state our main theoretical result: \proclaim{Theorem 1 (Approximation exponents and Lyapunov exponents)} Let $\AA$ be the two-dimensional continued fraction algorithm defined by the maps $T$ and $A$ in Section 2.1. There is a unique absolutely continuous $T$-invariant probability measure $\mu$ on $\Delta_2$. This measure is ergodic and Oseledec's theorem holds for $(T,A,\mu)$, yielding Lyapunov exponents $\lambda_1 > \lambda_2\ge \lambda_3$. The maximal exponent $\lambda_1$ is strictly positive and the corresponding space has dimension one. There are constants $c(\AA), c^*(\AA)$ such that for Lebesgue-almost every $\theta \in [0,1]^2$ $$ \eta_{\AA} (\theta) =c(\AA) \ge 1-{ \lambda_2 \over \lambda_1} \, , \quad \eta^*_\AA (\theta) = c^*(\AA)= 1-{ \lambda_2 \over \lambda_1} \, . \tag{2.14} $$ \endproclaim The numerical experiments described in Section 5 seem to indicate that $\lambda_1(\AA) \sim 2.329$, $\lambda_2(\AA) \sim -0.722$ ($\lambda_3 = -\lambda_1-\lambda_2$ since all matrices $A$ have determinant $1$), so that $c^*_\AA\sim 1.31$. It would be interesting to prove rigorously that $\lambda_2 (\AA)< 0$. To prove Theorem 1, we shall use a slight modification of results of Lagarias [L, Theorem 4.1, Theorem 4.2] which gives conditions under which Oseledec's theorem can be applied to control convergence exponents. Since some of these conditions are interesting in themselves, we state them now: \proclaim{Lemma 2 (Semi-weak convergence of $\AA$)} Let $w_j^{(n)}(\theta)$ be the columns of the matrices $L^{(n)}(\alpha(\theta))$ produced by $\AA$. There are $c_0 > 1$ and $n_0 \ge 1$ so that for all $\theta \in [0,1]^2$ we have $$ \max_{1 \le j \le 3} \| \theta -\tilde w^{(n)}_j (\theta)\| \le c_0^{-n}\, , \quad \forall n \ge n_0\, . \tag{2.15} $$ \endproclaim The semi-weak convergence property \thetag{2.15} is defined in [L] and is stronger than the weak convergence property in our introduction. If it holds for an algorithm $\DD$ and a set of values of $\theta$ of Lebesgue measure one, we say that algorithm $\DD$ is {\it semi-weakly convergent.} Recall also that an algorithm described by matrices $L^{(n)}$ is called {\it strongly convergent} if all vectors given by the columns of $L^{(n)}(\alpha(\theta))$ become arbitrarily close to the ray $\lambda \Theta$ as $n \to \infty$. The strong convergence of algorithm $\AA$ (maybe only for a set of $\theta$ values of Lebesgue measure one) is an open problem. \proclaim{Lemma 3 (Absolutely continuous ergodic probability for $T$)} The auxiliary map $T : \Delta \to \Delta$ associated with algorithm $\AA$ admits an ergodic absolutely continuous invariant probability measure $\mu$ with a continuous density which is bounded away from zero. \endproclaim The proof of Lemma 3 uses results of Mayer [M] (which yield in fact stronger properties for $\mu$, in particular mixing) and is an application of transfer operator techniques for locally expanding maps, using an infinite-dimensional version of the projective methods applied in the proof of Lemma 2. \smallskip \subhead {2.3 Generalizations and variants: a tree of algorithms} \endsubhead \smallskip Before we mention higher-dimensional generalizations, we would like to point out that instead of assuming that $\alpha_i \le 1/2$ and considering partitions of the squares into four subsquares, it would also be possible to consider the full simplex $\bar \Delta$ of points $0 \le \alpha_1 \le \alpha_2 \le 1$ and divide the squares into two simplices along the anti-diagonal. The numerical experiments described in Section 5 (cf. Table 2) seem to indicate that the subsquare procedure is preferable. At first sight the generalization of algorithm $\AA$ to arbitrary dimension $d$ seems quite straightforward: consider the $\lambda \Theta$ ray in $\real^{d+1}$ and the $d$-dimensional hyperplanes $P_i$, $i=1, \ldots, d+1$, and divide the $d$-dimensional hypercube into $2^d$ sub-hypercubes. However, there is a small complication as pointed out in [BRTT]. Indeed, if $d\ge 3$, the union of all the $d$-dimensional simplices generated by $d$ edges of a $d$-dimensional cube having a common vertex does not cover the cube. (Points in the cube whose co-ordinates $(x_1, \ldots x_d)$ in any basis formed by $d$ such edges satisfy $\sum_i x_i > 1$ --- such as $(1/2, 1/2 , \ldots, 1/2)$ --- will never be covered.) One way out is to perform a basis change as described in Remark 2.1 of [BRTT]: this corresponds to selecting a subdivision of the $d$-dimensional cube into simplices whose vertices are vertices of the cube, and then choosing the new basis by taking vectors whose endpoints are given by the vertices of the simplex containing the intersection point of our ray with the hyperplane. We call this the modification {\it of the first type} and note that it yields again a Markovian linear simplex-splitting algorithm (see Figure 3). Another solution, described in Remark 2.5 of [BRTT], is to consider, for each hyperplane $P_i$, the first intersection of the positive ray $\lambda \Theta$ which {\it does} belong to a suitable simplex (such an intersection always exist by Kronecker's theorem). This algorithm (which we will say is obtained by a {\it modification of the second type}) does not seem to be very tractable analytically, and has the annoying property that the corresponding matrices in general do not have determinant equal to one. Observe finally that for both types of modification, one may (or may not) decide to use the subcube selection algorithm. \figure 3 fig3.eps Decomposition of the $3$-dimensional hypercube into $6$ simplices (two of which are not visible) \cr We now explain how algorithm $\AA$ can be viewed as a path in a $d+1$-ary tree of algorithms in any dimension $d$. Recall that a generic ray intersects the $d+1$ hyperplanes successively in $d+1$ distinct points (where we use perhaps the modification of the second type of the notion of first intersection with a hyperplane) and that we choose to construct the $n$-th approximation basis by considering the last such intersection. At each step of the algorithm, we could have used some other selection rule to choose among the $d+1$ hyperplanes. (This idea is similar, but not identical, to the branching algorithm studied by Pipping, see [B] for references.) In this paper, we restrict ourselves to Markovian choices, namely we use a selection process (which may depend on the current value of the point in $\Delta_d$) which is {\it the same at each iteration} (i.e., does not depend on the ``past''). We will also consider (Markovian) {\it random} algorithms, where we establish a list of $N$ selection rules and choose at each iteration the rule to apply using a probability vector $p_i > 0$, with $\sum_{i=1}^N p_i=1$. Note that the one-dimensional reduction of our algorithm $\AA$ is essentially the classical Gauss (division) continued fraction algorithm, whereas the one-dimensional path corresponding to choosing the first intersection at each step would be related to the Farey (subtraction) algorithm. Observe also that in dimension $d=2$, the algorithm corresponding to selecting the first intersected hyperplane is similar, but not at all identical with the Poincar\'e algorithm described e.g. in [N]. Indeed, the Poincar\'e algorithm always uses the same partition of the two-dimensional cube into six simplices. This leads to convergence problems and undesirable ergodic properties thoroughly analysed in [N]. The numerical results in Section 5 indicate that our simplex selection procedure (see Figure 2) suppresses this difficulty. \smallskip \subhead{2.4 Renormalization} \endsubhead \smallskip It remains to discuss the relationship between the tree of algorithms described above and the higher-dimensional renormalizations introduced in [BRTT]. The renormalizations there simply correspond to the map $T(\alpha_1,\alpha_2)$ described in \thetag{2.3-2.4, 2.8}. More precisely, [BRTT] associated to a torus rotation of angle $(\alpha_1, \alpha_2)$ a partition of the torus in subdomains on which the first return map was again a torus rotation. The precise claim is that the choice of a subdomain there corresponds to the choice of a hyperplane $P_i$ here, and the new rotation angle is given by the associated map $T(\alpha_1,\alpha_2)$. This analogy also holds in higher dimensions. Since [BRTT] were not concerned by approximations, and in particular did not need to preserve the property that the original vector lied in the positive quadrant of each basis, the fact that the $d$-dimensional simplices described above do not cover the $d$-dimensional cube when $d \ge 3$ was not as problematic there as it is here. We refer to [KO] for a discussion of a related continued fractions algorithm also connected with dynamical renormalization. \head 3. Proof of the semi-weak convergence of algorithm $\AA$ \endhead In this section, we use projective metric techniques in finite dimension to show Lemma 2 for Algorithm $\AA$. We also explain how to adapt the argument to make it work for some variants of algorithm $\AA$ described in Section 2. The version of the projective results for linear transformations of positive cones that we shall use may be found in [F]. We first recall that the {\it projective distance} $\Gamma(x,y)=\Gamma_V(x,y)$ between two vectors $x, y$ in the positive cone $V=V_{d+1}= \{ (v_1, \ldots , v_d) \mid v_i \ge 0 \}$ of $\real^{d+1}$ ($d \ge 1$) can be defined by the formula $$ \Gamma(x,y) = \cases 0&\text{if $x_i y_i =0$ and $x_i + y_i>0$ for some $i$,}\cr \inf_{x_i, y_j \ne 0} ( x_j y_i )/ (x_i y_j)&\text{otherwise.} \endcases \tag{3.1} $$ Recall also that the {\it $V$-projective bound} $\delta(A)=\delta_V(A)$ of a $(d+1) \times (d+1)$ matrix $A$ with nonnegative real coefficients $a_{ij}$ (which is a measure of the projective diameter of the image of the positive cone, as we shall see) is given by $$ \delta(A) = \cases 0&\text{if $a_{ir} a_{js} =0$ and $a_{ir} + a_{js}>0$,} \cr & \qquad \text{ for some $i=j$ and $ r, s $,}\cr &\qquad \text{ or $r=s$ and $i,j$,}\cr \min_{i,j,r,s} (a_{ir} a_{js}) / (a_{is} a_{jr}) &\text{otherwise.} \endcases\tag{3.2} $$ (In other words, $\delta(A) = \inf_{x,y \in V_{d+1}} \Gamma (Ax, Ay)$.) We shall consider the projective space $\PP=\PP_V$ obtained by identifying two points $x, y \in V$ such that $x =\lambda y$ for some $\lambda > 0$ (we then write $x \sim y$), denoting $\bar x \in \PP$ for the equivalence class of $x \in V$. \proclaim{Lemma 4 (Projective metrics [F, Lemmas 15.1, 15.2])} \roster \item The function $D_V (\bar x,\bar y) = -\log \Gamma(x,y)$ where $x$ and $y$ are arbitrary representatives of $\bar x, \bar y \in \PP$ is a metric on $\PP_V$. \item The action induced on $\PP_V$ by a $(d+1) \times (d+1)$ matrix $A$ with nonnegative coefficients is a contraction for the metric $D_V$. More precisely, we have $$ \Gamma_V(Ax, Ay) \ge { \Gamma_V(x,y) + \delta_V(A) \over \delta_V(A) \Gamma_V(x,y) +1} \ge \Gamma_V(x,y) \, . \tag{3.3} $$ \endroster \endproclaim (The more general formulation given in [F] allows replacing $V$ by another cone $W$ -- see the proof of Lemma 3.) \proclaim{Corollary 5} The transformation of $\PP$ induced by a nonnegative matrix $A$ with projective bound $\delta(A) >0$ is a {\it strict} contraction for the metric $D$ by a factor $$ \epsilon\le (1-\delta(A))/(1+\delta(A))< 1\, . $$ \endproclaim \demo{Proof of Corollary 5} It suffices to check that for any $\epsilon < 1$ with $\epsilon > (1-\delta)/(1+\delta)$ we have $\Gamma (A x, Ay) \ge \Gamma(x,y) ^{\epsilon}$. This follows from from \thetag{3.3} by a concavity argument using the auxiliary functions $f(\Gamma) = (\Gamma+\delta)/(\delta \Gamma+1)$ and $g(\Gamma)=\Gamma^\epsilon$ on $[0,1]$. \qed \enddemo In order to prove Lemma 3, we need a comparison lemma between the projective distance $D$ and the distance $D_1(u,v) = \sum_{i=1}^{d}|u_i-v_i|$ on the simplex $\bar \Delta_d=\{ u \in \real^{d+1} \mid u_i \ge 0 \, , \sum_{i=1}^{d+1}u_i =1\}$: \proclaim{Lemma 6 (Comparison between distances)} For any $d \ge 1$ and $u, v \in \bar \Delta_d$: $$ D_1(u,v) \le d \cdot D(\bar u, \bar v) \, .\tag{3.4} $$ \endproclaim \noindent The proof is probably standard, we include it for completeness: \demo{Proof of Lemma 6} If $D(\bar u, \bar v) = \infty$, there is nothing to prove. Otherwise there are indices $k(u,v)$ and $\ell(u,v)$ so that $$ \eqalign { D(\bar u,\bar v) &= \log v_\ell - \log u_\ell + \log u_k - \log v_k\cr &= |\log v_\ell - \log u_\ell| + |\log u_k - \log v_k|\cr &\ge \max_{1\le i\le d+1} |\log u_i - \log v_i|\cr &\ge \max_{1\le i\le d} |\log u_i - \log v_i|\, .\cr }\tag{3.5} $$ Therefore, since $t-s \le \log t -\log s$ for any $0 < s \le t \le 1$, we have $$ \eqalign { D_1(u,v)&\le \sum_{i=1}^d |\log u_i - \log v_i| \le d\cdot D(\bar u,\bar v) \, . \qed } \tag{3.6} $$ \enddemo \medskip We now prove Lemma 2 from Section 2: \demo{Proof of Lemma 2} The basic remark is that, except in the degenerate cases, all matrices $A(\alpha_1, \alpha_2)$ appearing in algorithm $\AA$ have a projective bound satisfying $$ \delta(A(\alpha_1, \alpha_2)) \ge 1/4 \, . \tag{3.7} $$ (The degenerate cases, which only concern a set of Lebesgue measure zero of values of $\theta$, may be dealt with by hand.) Indeed, it is obvious from the definitions that $$ \delta(A(\alpha_1, \alpha_2)) = { j \over j+1} \cdot { k \over k+1} \ge {1\over 4}\, . \tag{3.8} $$ (We used that $k \ge 1$ and $j \ge 1$.) Corollary 5 thus implies that the transformation induced by each $A(\alpha_1, \alpha_2)$ is a contraction by a factor $\epsilon\le 3/5$. Therefore, writing $e_1$, $e_2$, $e_3$ for the standard basis in $\real^3$, and using $\bar u$ to denote also the representative of $u \in V$ in $\Delta_d$, we get by applying Corollary 5 and Lemma 6 that for any $n \ge 1$, and $\theta$ $$ \eqalign { {1 \over \sqrt 2} \max_{1\le j \le 3} \| \theta - \tilde w_j^{(n)}(\theta) \| &\le{ 1 \over \sqrt 2} \max_{1\le j \le 3} \| \theta - \tilde w_j^{(n)}(\theta)\|_1\cr &\le \max_{1\le j \le 3} D_1(\overline{ \Theta(\theta)}, \overline{L^{(n)}(\alpha(\theta)) e_j})\cr &\le \max_{1\le i < j \le 3} D_1(\overline{ L^{(n)}(\alpha(\theta)) e_i}, \overline{L^{(n)}(\alpha(\theta)) e_j})\cr &\le 2 \max_{1\le i < j \le 3} D(\overline{L^{(n)} (\alpha(\theta))e_i }, \overline{L^{(n)} (\alpha(\theta))e_j})\cr &\le 2 \, (3/5)^{n-1} \max_{1\le i < j \le 3\, , \alpha \in \Delta_2} D( \overline{A(\alpha) e_i}, \overline{A(\alpha) e_j})\cr &\le {1\over 2} \, (3/5)^{n-1}\, . } \tag{3.9} $$ (We have used that the ray $\lambda \Theta$ is always in the positive quadrant of the basis $L^{(n)}(\alpha)(e_i)$, $i=1,2,3$, and also the fact that the maximum distance between any vertex of a simplex and an interior point of the simplex is bounded by the maximum distance between two vertices.) \qed \enddemo \remark{Remark 3.1} Exactly the same arguments as those used in the proof of Lemma 2 show that this lemma holds for the higher dimensional version of algorithm $\AA$ obtained by the modifications of the first and second type. In particular, these algorithms have the semi-weak convergence property. (Indeed, the corresponding matrices for nondegenerate points are all projectively bounded by $\delta \ge 1/4$ in any dimension.) \endremark \remark{Remark 3.2} In dimension $d=2$ or higher, one may consider other paths in the tree of algorithm described in Section 2. Since the cases $k=0$ and/or $j=0$ can then occur, it is necessary to work with a cone strictly larger than the positive quadrant $V_{d+1}$ (and use the corresponding generalizations of $\Gamma$, $\delta$, and Lemmas 3 and 4 which are to be found in Lemma 15.1, 15.2 of [F]) in order to get strictly positive projective bounds $\delta(A)$ for the matrices $A$ which appear. We now explain what can be done. In dimension $d=2$, we have verified by constructing explicitly such suitable cones that for {\it any} $\Theta$ the transformations induced by the matrices $A$ are strict contractions with a uniform coefficient $\delta(A) \ge \delta > 0$, for all paths (i.e. selection rules) which avoid the choice ``intersection with $P_3$'' whenever this intersection is the first one on our ray. Let us call this choice the {\it least efficient} one. Therefore all selection rules which can be proved to avoid the least efficient choice infinitely many often (for all or almost all $\theta$) yield a weakly convergent algorithm, whereas all paths which avoid this choice for a given (uniform) proportion of iterations (for all or almost all $\theta$) yield a semi-weakly convergent algorithm. In particular, any random path which chooses the first intersected plane with probability strictly below $1$ or which avoids the intersection with $P_3$ with probability $1$ will be semi-weakly convergent. (A random path is called semi-weakly convergent if \thetag{2.15} for the random approximation $(w_j^{(n)})_{\omega}$ associated to a given sequence of selection rules $(\omega_0, \omega_1, \ldots)$ holds with probability $1$.) We {\it conjecture} (see the experimental evidence presented in Section 5) that {\it any} deterministic or random Markov path in the tree of two-dimensional algorithms is semi-weakly convergent (for almost each vector $\theta$). We also expect the (almost everywhere) semi-weak convergence to hold in any dimension $d \ge 3$ for both modifications of our tree of algorithms. \endremark \head 4. Ergodic properties and Lyapunov exponents \endhead The main goal of this section is to prove Lemma 3 for the map $T$ arising in the two-dimensional algorithm $\AA$. Using also Lemma 2 which was proved in the previous section, we shall then apply the result from [L] on Lyapunov exponents. As in Section 3, we also mention several more or less straightforward generalizations to other paths in dimensions two and higher. We shall use the following result of Mayer: \proclaim{Theorem 7 (Main Theorem in [M])} Let $I^d$ be the $d$-dimensional unit cube $\{ x \in \real^d \mid 0\le x_j \le 1 \, , 1\le j \le d \}$ endowed with Lebesgue measure, and let $T : I^d \to I^d$ be an almost everywhere defined map such that there is a finite or countable (measure-theoretical) partition $I^d = \cup_{i\in \II} O_i$ into disjoint open sets with the measure of $\cup_i \partial O_i$ equal to zero and such that: \roster \item $T$ maps $O_i$ bijectively to the interior of $I^d$ for all $i$; each inverse branch $\psi_i : I^d \to O_i$ is real analytic and may be extended to a holomorphic open domain $I^d \subset \Omega$ of $\complex^d$ such that every $\psi_i (\overline \Omega)$ is a strict subset of $\Omega$. Moreover, on some open domain $\Omega'$ with $\overline \Omega \subset \Omega'$ we have $\text{det} \, \psi_i'(z) \ne 0$ and $\sum_{i \in \II} |\text{det}\, \psi_i'(z)| < \infty$. \item The set $\text{Per}\, (T) = \{ x \in I^d \mid \exists \, n \, , T^n x=x\}$ is dense in $I^d$. \endroster Then there is a $T$-invariant absolutely continuous probability measure $\mu$ with an analytic density $f$. The measure $\mu$ is ergodic (in fact, mixing) for $T$ and has full support. \endproclaim Mayer in fact proves much more in [M] and requires weaker assumptions, in particular, he applies his result to the Jacobi-Perron algorithm (see also [Br1]). \demo{Proof of Lemma 3} One may obviously replace $I^d$ by a simplex of the form $\Delta_d=\{ x \in \real^d \mid 0\le x_1 \le x_2 \ldots \le x_d \le 1/2 \}$ in the statement of Theorem 7. Therefore, we only need to verify that assumptions \therosteritem{1-2} in that theorem are satisfied by our map $T :\Delta_2 \to \Delta_2$. Clearly, there is a countable partition of $\Delta_2$ into open two-dimensional simplices $O_i$, for $i \in \II \subset\{ (j,k,\ell,m) \mid j \ge 1 \, , k \ge 1 \, , 1\le \ell \le 3\, , 1 \le m \le 4\}$ with the property that $T$ maps $O_i$ bijectively to the interior of $\Delta_2$. (The index $\ell$ corresponds to the number of the last intersected hyperplane, $m$ tells us which subsquare is selected, and $k$ and $j$ are as in \thetag{2.5, 2.9}. Note that the degenerate cases correspond to an edge of $\Delta_2$ and may thus be neglected.) Also, it follows from the formulas in Section 2 that $T$ and its local inverses $\psi_i$ are projective transformations. In particular, the $\psi_i$ are analytic maps on $\Delta_2$ which admit a holomorphic extension to the entire complex plane. The key remark now is that the map $\psi_i$ associated to a quadruple $i=(j,k,l,m)$ is nothing but the projective map induced by the corresponding matrix $A$. To check that these two maps coincide, observe that it suffices to verify that they send the vertices of the simplex $\Delta_2$ to the same three points by projectivity. (Note that this is the linear simplex-splitting property.) Since these matrices are uniformly projectively bounded by \thetag{3.7}, it will not be difficult to see that there is a domain $\Omega$ as in assumption \therosteritem{1} by applying Corollary 5. Indeed, we only need to show that the euclidean or $D_1$ metric on $\Delta_2$ is equivalent to a suitable projective metric. We may use Lemma 6 to bound the metric on the simplex by the projective metric $D_V$. However, since the projective metric $D_V$ diverges on the boundary of the positive cone $V$ it cannot be controlled by a bounded metric such as $D_1$. To obtain the reverse bound, we thus need to find a strictly larger cone $ V \subset V'$ with the property that the associated projective bound $\delta_{V'}(A)$ is bounded away from zero for all matrices $A$ appearing in our algorithm. (We are using here the projective distance $\Gamma_W(x,y)$ between two vectors $x, y$ in a cone $W$, given by $\Gamma_W(x,y) = K_W(x,y) K_W(y,x)$ where $K_W(x,y)=\sup \{ \lambda\in \real \mid x-\lambda y \in W\}$, to define the projective bound $\delta_W(A)$ of a matrix $A$ such that that $A W \subset W$ by setting $\delta_W(A)=\inf_{x,y \in W} \Gamma_W (A x, Ay)$.) We then wish to apply the general version of Lemma 4 (Lemmas 15.1, 15.2 from [F]). To prove that a cone $V'$ with the above property exists, we first use the fact that $\Gamma_W(r,s) \ge \Gamma_V(r,s)$ for any $r, s \in V \subset W$ (in particular $r=Ax$ and $s =Ay$ with $x,y \in V$), and then apply continuity in $r,s$ of $\Gamma_W(r,s)$. Since the restriction to $V$ of the projective distance on $V'$ (or any cone $W$ whose interior contains the closure of $V$: just note that the continuous function $\Gamma_W(x,y)$ does not vanish on the boundary of $V$) is bounded above by a constant multiple of the distance $D_1$ on $\Delta_2$, we get the desired reverse bound. (Note that the bound from Lemma 6 still holds for $V'$.) To check the summability condition on the determinants we need to quantify the contraction rate for areas. One way to do this is to note that the inverse branch $\psi_{j,k,l,m} (\alpha_1,\alpha_2)= (\alpha'_1,\alpha'_2)$ has the following form when $\ell=1$ (i.e., when the last intersection is with $P_1$) and $m=1$ (i.e., we are in the subsquare which does not require any change of coordinates) $$ \cases \alpha'_1 &= {1 \over \alpha_1 + \alpha_2 +j+k+1}\cr \alpha'_2 &= { \alpha_1 + k \over \alpha_1 + \alpha_2 +j+k+1}\, ,\cr \endcases \tag4.1 $$ (the formula for $\ell=2$ and $m=1$ is symmetric) and the following form when $\ell=3$ and $m=1$ $$ \cases \alpha'_1 &= {\alpha_1+k \over \alpha_1 + \alpha_2 +j+k+1}\cr \alpha'_2 &= { \alpha_2 + j\over \alpha_1 + \alpha_2 +j+k+1}\, .\cr \endcases \tag4.2 $$ Explicit computation of the partial derivatives yields a constant $C$ such that $$ |\det \psi'_{j,k,\ell,1}(\alpha_1, \alpha_2)| \le C {1 \over( k+j)^3} \tag{4.3} $$ for $\ell=1$ or $\ell=3$, all $j$, $k$, $\alpha_1$, and $\alpha_2$. By symmetry, \thetag{4.3} also holds for $\ell=2$, and since the additional changes of coordinates associated with the cases $m=2,3,4$ are isometries (they are suitable compositions of the maps $(x,y) \mapsto (y,x)$ and $(x,y) \mapsto (1-x,y)$), the same bound holds also there. The condition $\sup_{\alpha_1, \alpha_2} \sum_{j,k,l,m} |\det \psi'_{j,k,l,m}(\alpha_1, \alpha_2)| < \infty$ then follows from \thetag{4.3}. It remains to prove that periodic points of $T$ are dense in $\Delta_2$. But this is an obvious consequence of the fact that each injective branch of $T$ uniformly expands a simplex to the full domain of definition $\Delta_2$ so that each branch of $T^n$ expands a simplex of exponentially shrinking diameter to $\Delta_2$ (then apply the Brouwer fixed point theorem to $T^n$). \qed \enddemo \demo{Proof of Theorem 1} In order to apply Theorem 4.1 and Theorem 4.2 in [L] we only need to check that properties (H1)-(H5) from [L, Section 4] are satisfied. (Note that the formulation there uses maps on the unit square but that one may again translate this into our simplicial setting in a straightforward manner.) Property (H1) says that $T$ must have an ergodic absolutely continuous invariant measure $d\mu$ with support $\Omega$ a $T$-invariant Borel subset of positive Lebesgue measure. This is clearly a consequence of our Lemma 3. Since $\Omega$ is in our case the full simplex $\Delta_2$, property (H2) in [L] reduces to the requirement that $T$ is piecewise continuous with almost everywhere nonvanishing Jacobian, and this is true too. Property (H3) is the semi-weak convergence of the algorithm which we proved in Lemma 2, and (H5) is a ``partial quotient mixing'' assumption which trivially holds in our case because all of our matrices are strictly positive. It remains to check the boundedness condition (H4) which reads $$ E\bigl [ \log (\max \| A\|, 1)\bigr ] = \int_\Delta \log (\max(1, \|A(\alpha_1, \alpha_2)\|)\, d\mu(\alpha_1, \alpha_2) < \infty \, .\tag{4.4} $$ Since $\mu$ has a continuous density we may replace it by Lebesgue measure in \thetag{4.3}. To obtain the bound, we may use any matrix norm for example the $l^1$ norm for which we have $$ \| A(\alpha_1, \alpha_2)\|_1 \le \max(j(\alpha_1, \alpha_2),k(\alpha_1,\alpha_2)) +1 \, .\tag{4.5} $$ It is easy to check that the sets (corresponding to ``last intersection with $P_1$'') $$ \cases R^1_{k_0}&= \{ \alpha_1 \mid (\alpha_1, \alpha_2) \in \Delta_2 \, , k(\alpha_1, \alpha_2)=[\alpha_2/\alpha_1] = k_0 \} \, ,\cr S^1_{j_0}&= \{ \alpha_1 \mid (\alpha_1, \alpha_2) \in \Delta_2 \, , j(\alpha_1, \alpha_2) = [(1-\alpha_2)/\alpha_1]= j_0\} \endcases \tag{4.6} $$ for $j_0,k_0 \ge 1$ have Lebesgue measure bounded by $m(R^1_{k_0}) \le C/k_0^2$, $m(S^1_{j_0}) \le C /j_0^2$ where $C$ is some uniform constant. We obtain the same bounds for the sets $R^2_{k_0}$, $S^2_{j_0}$, $R^3_{k_0}$, $S^3_{j_0}$ associated with the intersection with $P_2$ or $P_3$ (recall \thetag{2.8}). Therefore, using \thetag{4.5}, we find $$ E\bigl [ \log (\max \| A\|, 1)\bigr ] \le \sum_{j=1}^\infty C {\log (j +1)\over j^2} + \sum_{k=1}^\infty C {\log (k +1)\over k^2} < \infty \tag{4.7} $$ as desired. \qed \enddemo \remark{Remark 4.1} The same arguments may be used to show that Lemma 3 and Theorem 1 hold for the higher-dimensional version of algorithm $\AA$ obtained by the modification of the first type. \endremark \remark{Remark 4.2} Consider the other paths in the tree of algorithms from Section 2. In dimension $d=2$, we may use Remark 3.2 to adapt the proofs of Lemma 3 and Theorem 1 for all paths in the tree which avoid the least efficient choice infinitely often (at least for almost all $\theta$) and which have the partial quotient mixing property (H5). It is also possible, although more cumbersome, to consider random paths. We sketch now how this can be done. The version of Lemma 3 relevant in the random case involves the invariant measure for the Markov chain associated to the random map. (See [K1, K2] for general information on random dynamical systems and their ergodic properties.) The definitions \thetag{2.12}-\thetag{2.13} can be modified to define random approximation exponents by taking the essential supremum (respectively infimum) over all random sequences of choices $(\omega_0, \omega_1, \ldots)$. Using the standard notion of Lyapunov exponents for random dynamical systems (see e.g. [ACE]), the results and arguments of Lagarias may be adapted to the random case in a straightforward manner. In particular \thetag{2.14} would have the same form where the Lyapunov exponents are replaced by the almost everywhere constant value of the random Lyapunov exponents. The final result is that the random analogue of Theorem 1 holds for any random path which chooses the first intersected plane with probability strictly below $1$, or which avoids the intersection with $P_3$ with strictly positive probability whenever the partial mixing condition (H5) from [L] holds. We {\it conjecture} (see also the numerical data in the next section) that {\it any} deterministic or random Markov path in the tree of two-dimensional algorithms satisfies the (if necessary, randomized) conclusions of Lemma 3 and Theorem 1. We also expect similar results to hold in any higher dimension $d \ge 3$ for both modifications of our tree of algorithms. \endremark \head 5. Numerical results \endhead In this section, we present our numerical study of the Lyapunov exponents of algorithm $\AA$ and three other two-dimensional algorithms constructed from our tree (Table 1). We include also the numerical results for the Jacobi-Perron and ordered Jacobi-Perron algorithms (they are consistent with Baldwin's results from [Ba2]), and we reproduce Baldwin's results on his GFCP algorithm for comparison. (See Broise [Br2] for rigorous results on the Lyapunov exponents of the Jacobi-Perron algorithm.) In Table 2 we present the data obtained by using the variant of algorithm $\AA$ and the three other algorithms when the subsquare selection is replaced by a triangular selection process as described in the first paragraph of Section 2.3. The three other two-dimensional algorithms correspond respectively to the choice of the second intersected hyperplane (algorithm $\BB$), the first intersected hyperplane (algorithm $\CC$) and a random algorithm where we choose the $j^{\text{th}}$ intersected hyperplane with probability $1/3$ for $1\le j \le 3$ at each step (algorithm $\RR$). Although we have written the proofs in detail only for algorithm $\AA$, Remarks 3.1--3.2, and 4.1--4.2 can be used to show that Theorem 1 also applies to algorithms $\BB$ and $\RR$. The case of algorithm $\CC$ is not at all clear from a mathematical point of view, but the numerical behaviour is quite satisfactory. Before discussing the values of the exponents in Table 1, we briefly explain how our programs work. (They have been written in the C language, using double precision, any request for the source code should be sent to the first named author.) We start from some ``arbitrary'' vector $(\alpha_1, \alpha_2)$, initialize $Q$ to be the $3\times 3$ identity matrix, and $\ell$ to be the vector $(0,0,0)$, and repeat a great number of times (one million iterates) the following procedure (inspired from the proof in [JPS] and suggested to us by S.~Oliffson Kamphorst): \smallskip \noindent {\tt - let $A$ be the matrix associated to $(\alpha_1,\alpha_2)$ by the algorithm;} \noindent {\tt - let $(\alpha_1,\alpha_2):= T(\alpha_1, \alpha_2)$;} \noindent {\tt - let $B := A Q$;} \noindent {\tt - decompose $B$ as $B=Q R$ where $Q$ is unitary and $R$ upper-triangular;} \noindent {\tt - let $\ell(i) = \ell(i)+ \log R(i,i)$ for $1 \le i \le 3$.} \smallskip At the end of the process, $\ell(i)$ divided by the number of iterations contains an approximation to the $i^{\text{th}}$ Lyapunov exponent. \medskip Recall that $\lambda_1> 0$ controls the exponential growth of denominators in the approximations (see [L, (4.22)]) and that $-(\lambda_1 - \lambda_2 )< 0$ controls the exponential decay of $ \|\theta- \tilde w_i^{(n)}(\theta)\|$ (see [L, (4.18), last formula of p. 317]). A high value for $\lambda_1 - \lambda_2$ seems therefore to be a desirable feature. Another goal is that the uniform approximation exponent $c^*(\AA) = 1-\lambda_2/\lambda_1 \le 1.5$ be as close as possible to $1.5$. Keeping this in mind, we see that algorithm $\AA$ compares very favorably to both versions of the Jacobi-Perron algorithm and to the GCFP algorithm of Baldwin in terms of control of the rate of convergence of $\|\theta- \tilde w_i^{(n)}(\theta)\|$. It is slightly better than the ordered Jacobi-Perron algorithm and slightly worse than the classical Jacobi-Perron algorithm as far as the uniform approximation exponent $c^*$ is considered. The GCFP algorithm produces a better convergence exponent than algorithm $\AA$. Considering now the other variants, we see that the best uniform approximation exponent of all algorithms (including the GCFP one) is obtained by algorithm $\BB$ (we find $c^*(\BB)\sim 1.396$), and the best convergence rate by far is given by algorithm $\AA$ ($\lambda_1-\lambda_2 \sim 3.05$). When both the approximation exponent and the convergence rate are taken into account, algorithm $\BB$ does quite well. \medskip \centerline{ \vbox{ \hrule \halign{ \hfil \bf#\hfil &\quad \hfil #\hfil &\quad\hfil #\hfil &\quad \hfil #\hfil &\quad\hfil #\hfil \cr \strut Algorithm & $\lambda_1$ & $\lambda_2$ & $c^*=1-\lambda_2/\lambda_1$ & $\lambda_1-\lambda_2$ \cr \noalign{\hrule}\cr $\AA$ & {\bf 2.3293} & $-0.7218$ & $1.310$ & {\bf 3.0511} \cr $\BB$ & $1.3091$ & $-0.5190$ & {\bf 1.396}& $1.8281$ \cr $\CC$ & $0.5091$ & $-0.1953$ & $1.384$ & $0.7044$ \cr $\RR$ & $1.4334$ & $-0.5080$ & $1.354$ & $1.9414$ \cr J-P & $1.2006$ & $-0.4486$ & $1.374$ & $1.6492$ \cr ordered J-P & $1.6421$ & $-0.4594$ & $1.280$ & $2.1015$ \cr \strut GCFP (from [Ba2]) & $0.6249$ & $-0.2418$ & $1.387$ & $0.8667$ \cr } \hrule}} \smallskip \centerline{\eightpoint {\bf Table 1} (Recall $\lambda_3=-\lambda_1-\lambda_2$)} \bigskip Table 2 shows that, although the process which cuts the squares along the antidiagonal (see Section 2.3) gives a very high value of $\lambda_1$ (and thus $\lambda_1 - \lambda_2$) for the modification $\AA'$ of algorithm $\AA$, the approximation exponents are on the whole not as good as those of the algorithms in Table 1. \medskip \centerline{ \vbox{ \hrule \halign{ \hfil \bf#\hfil &\quad \hfil #\hfil &\quad\hfil #\hfil &\quad \hfil #\hfil &\quad\hfil #\hfil \cr \strut Algorithm & $\lambda_1$ & $\lambda_2$ & $c^*=1-\lambda_2/\lambda_1$ & $\lambda_1-\lambda_2$ \cr \noalign{\hrule}\cr $\AA'$ & {\bf 2.5966} & $-0.5548$ & $1.214$ & {\bf 3.1514} \cr $\BB'$ & $1.3931$ & $-0.4318$ & $1.310$ & $1.8249$ \cr $\CC'$ & $0.4217$ & $-0.1627$ & {\bf 1.386} & $0.5844$ \cr \strut $\RR'$ & $1.5191$ & $-0.4080$ & $1.269$ & $1.9271$ \cr } \hrule}} \smallskip \centerline{\eightpoint {\bf Table 2} (Division along antidiagonal instead of subsquare selection)} \medskip \Refs \widestnumber\key{BRTT} \ref \key ACE \by L. Arnold, H. Crauel, and J.-P. Eckmann (eds.) \book Lyapunov exponents (Proceedings Oberwolfach 1990) \yr 1991 \publ Springer (Lecture Notes in Math. {\bf 1486}) \publaddr Berlin \endref \ref \key BRTT \by V. Baladi, D. Rockmore, N. Tongring, and C. Tresser \paper Renormalization on the $n$-dimensional torus \jour Nonlinearity \yr 1992 \vol 5 \pages 1111-1136 \endref \ref \key Ba1 \by P.R. Baldwin \paper A multidimensional continued fraction and some of its statistical properties \jour J. Statist. Phys. \vol 66 \pages 1463--1505 \yr 1992 \endref \ref \key Ba2 \by P.R. Baldwin \paper A convergence exponent for multidimensional continued fractions \jour J. Statist. Phys. \vol 66 \pages 1507--1526 \yr 1992 \endref \ref \key B \by A. J. Brentjes \book Multi-dimensional continued fraction algorithms (Math. Centre Tract. No 145) \publ Math. Centre \publaddr Amsterdam \yr 1981 \endref \ref \key Br1 \by A. Broise \paper Aspects stochastiques de certains syst\`emes dynamiques: transformations dilatantes de l'intervalle, fractions continues multidimensionnelles. Partie II: Fractions continues multidimensionnelles et lois stables \paperinfo PH.D. thesis, Universit\'e de Rennes, to appear Bull. Soc. Math. France \yr 1994 \endref \ref \key Br2 \by A. Broise \paper Aspects stochastiques de certains syst\`emes dynamiques: transformations dilatantes de l'intervalle, fractions continues multidimensionnelles. Partie III et Annexe: Fractions continues multidimensionnelles et approximations diophantiennes \paperinfo PH.D. thesis, Universit\'e de Rennes \yr 1994 \endref \ref \key F \by H. Furstenberg \book Stationary processes and prediction theory \yr 1960 \publ Princeton U. Press \publaddr Princeton, NJ \endref \ref \key JPS \by R.A. Johnson, K.J. Palmer, and G.R. Sell \paper Ergodic properties of linear dynamical systems \jour SIAM J. Math. Anal. \vol 18 \pages 1--33 \yr 1987 \endref \ref \key K1 \by Y. Kifer \book Ergodic Theory of Random Perturbations \publ Birkh\"auser \yr 1986 \publaddr Boston Basel \endref \ref \key K2 \by Y. Kifer \book Random Perturbations of Dynamical Systems \publ Birkh\"auser \yr 1988 \publaddr Boston Basel \endref \ref \key KO \by S. Kim and S. Ostlund \paper Simultaneous rational approximations in the study of dynamical systems \jour Phys. Rev. A \yr 1986 \pages 3426--3434 \vol 34 \endref \ref \key K \by D. Kosygin \paper Multidimensional KAM theory from the renormalization group viewpoint \jour Advances Sov. Math. \vol 3 \yr 1991 \pages 99--129 \endref \ref \key L \by J.C. Lagarias \paper The quality of the Diophantine approximations found by the Jacobi-Perron algorithm and related algorithms \jour Monatsh. Math. \vol 115 \pages 299-328 \yr 1993 \endref \ref \key M \by D. Mayer \paper Approach to equilibrium for locally expanding maps in $\real^k$ \jour Comm. Math. Phys. \yr 1984 \pages 1--15 \vol 95 \endref \ref \key N \by A. Nogueira \paper The $3$-dimensional Poincar\'e continued fraction algorithm \jour Israel J. Math. \yr 1995 \vol 90 \pages 373--401 \endref \endRefs \enddocument %!PS-Adobe-3.0 EPSF-3.0 %%Creator: Adobe Illustrator(TM) 5.5 %%For: () () %%Title: (fig1.eps) %%CreationDate: (11/5/95) ( 6:10 PM) %%BoundingBox: 182 461 477 717 %%HiResBoundingBox: 182 461.1308 476.5129 716.9161 %%DocumentProcessColors: Black %%DocumentFonts: Helvetica %%+ Symbol %%+ ZapfDingbats %%DocumentSuppliedResources: procset Adobe_level2_AI5 1.0 0 %%+ procset Adobe_typography_AI5 1.0 0 %%+ procset Adobe_IllustratorA_AI5 1.0 0 %AI5_FileFormat 3.0 %AI3_ColorUsage: Black&White %AI3_TemplateBox: 306 396 306 396 %AI3_TileBox: 30 31 582 761 %AI3_DocumentPreview: Macintosh_ColorPic %AI5_ArtSize: 612 792 %AI5_RulerUnits: 2 %AI5_ArtFlags: 1 0 0 1 0 0 1 1 0 %AI5_TargetResolution: 800 %AI5_NumLayers: 1 %AI5_OpenToView: 82 748 1.5 685 809 18 0 1 394 52 %AI5_OpenViewLayers: 7 %%EndComments %%BeginProlog %%BeginResource: procset Adobe_level2_AI5 1.0 0 %%Title: (Adobe Illustrator (R) Version 5.0 Level 2 Emulation) %%Version: 1.0 %%CreationDate: (04/10/93) () %%Copyright: ((C) 1987-1993 Adobe Systems Incorporated All Rights Reserved) userdict /Adobe_level2_AI5 21 dict dup begin put /packedarray where not { userdict begin /packedarray { array astore readonly } bind def /setpacking /pop load def /currentpacking false def end 0 } if pop userdict /defaultpacking currentpacking put true setpacking /initialize { Adobe_level2_AI5 begin } bind def /terminate { currentdict Adobe_level2_AI5 eq { end } if } bind def mark /setcustomcolor where not { /findcmykcustomcolor { 5 packedarray } bind def /setcustomcolor { exch aload pop pop 4 { 4 index mul 4 1 roll } repeat 5 -1 roll pop setcmykcolor } def } if /gt38? mark {version cvx exec} stopped {cleartomark true} {38 gt exch pop} ifelse def userdict /deviceDPI 72 0 matrix defaultmatrix dtransform dup mul exch dup mul add sqrt put userdict /level2? systemdict /languagelevel known dup { pop systemdict /languagelevel get 2 ge } if put level2? not { /setcmykcolor where not { /setcmykcolor { exch .11 mul add exch .59 mul add exch .3 mul add 1 exch sub setgray } def } if /currentcmykcolor where not { /currentcmykcolor { 0 0 0 1 currentgray sub } def } if /setoverprint where not { /setoverprint /pop load def } if /selectfont where not { /selectfont { exch findfont exch dup type /arraytype eq { makefont } { scalefont } ifelse setfont } bind def } if /cshow where not { /cshow { [ 0 0 5 -1 roll aload pop ] cvx bind forall } bind def } if } if cleartomark /anyColor? { add add add 0 ne } bind def /testColor { gsave setcmykcolor currentcmykcolor grestore } bind def /testCMYKColorThrough { testColor anyColor? } bind def userdict /composite? level2? { gsave 1 1 1 1 setcmykcolor currentcmykcolor grestore add add add 4 eq } { 1 0 0 0 testCMYKColorThrough 0 1 0 0 testCMYKColorThrough 0 0 1 0 testCMYKColorThrough 0 0 0 1 testCMYKColorThrough and and and } ifelse put composite? not { userdict begin gsave /cyan? 1 0 0 0 testCMYKColorThrough def /magenta? 0 1 0 0 testCMYKColorThrough def /yellow? 0 0 1 0 testCMYKColorThrough def /black? 0 0 0 1 testCMYKColorThrough def grestore /isCMYKSep? cyan? magenta? yellow? black? or or or def /customColor? isCMYKSep? not def end } if end defaultpacking setpacking %%EndResource %%BeginResource: procset Adobe_typography_AI5 1.0 1 %%Title: (Typography Operators) %%Version: 1.0 %%CreationDate:(03/26/93) () %%Copyright: ((C) 1987-1993 Adobe Systems Incorporated All Rights Reserved) currentpacking true setpacking userdict /Adobe_typography_AI5 54 dict dup begin put /initialize { begin begin Adobe_typography_AI5 begin Adobe_typography_AI5 { dup xcheck { bind } if pop pop } forall end end end Adobe_typography_AI5 begin } def /terminate { currentdict Adobe_typography_AI5 eq { end } if } def /modifyEncoding { /_tempEncode exch ddef /_pntr 0 ddef { counttomark -1 roll dup type dup /marktype eq { pop pop exit } { /nametype eq { _tempEncode /_pntr dup load dup 3 1 roll 1 add ddef 3 -1 roll put } { /_pntr exch ddef } ifelse } ifelse } loop _tempEncode } def /TE { StandardEncoding 256 array copy modifyEncoding /_nativeEncoding exch def } def % /TZ { dup type /arraytype eq { /_wv exch def } { /_wv 0 def } ifelse /_useNativeEncoding exch def pop pop findfont _wv type /arraytype eq { _wv makeblendedfont } if dup length 2 add dict begin mark exch { 1 index /FID ne { def } if cleartomark mark } forall pop /FontName exch def counttomark 0 eq { 1 _useNativeEncoding eq { /Encoding _nativeEncoding def } if cleartomark } { /Encoding load 256 array copy modifyEncoding /Encoding exch def } ifelse FontName currentdict end definefont pop } def /tr { _ax _ay 3 2 roll } def /trj { _cx _cy _sp _ax _ay 6 5 roll } def /a0 { /Tx { dup currentpoint 3 2 roll tr _psf newpath moveto tr _ctm _pss } ddef /Tj { dup currentpoint 3 2 roll trj _pjsf newpath moveto trj _ctm _pjss } ddef } def /a1 { /Tx { dup currentpoint 4 2 roll gsave dup currentpoint 3 2 roll tr _psf newpath moveto tr _ctm _pss grestore 3 1 roll moveto tr sp } ddef /Tj { dup currentpoint 4 2 roll gsave dup currentpoint 3 2 roll trj _pjsf newpath moveto trj _ctm _pjss grestore 3 1 roll moveto tr jsp } ddef } def /e0 { /Tx { tr _psf } ddef /Tj { trj _pjsf } ddef } def /e1 { /Tx { dup currentpoint 4 2 roll gsave tr _psf grestore 3 1 roll moveto tr sp } ddef /Tj { dup currentpoint 4 2 roll gsave trj _pjsf grestore 3 1 roll moveto tr jsp } ddef } def /i0 { /Tx { tr sp } ddef /Tj { trj jsp } ddef } def /i1 { W N } def /o0 { /Tx { tr sw rmoveto } ddef /Tj { trj swj rmoveto } ddef } def /r0 { /Tx { tr _ctm _pss } ddef /Tj { trj _ctm _pjss } ddef } def /r1 { /Tx { dup currentpoint 4 2 roll currentpoint gsave newpath moveto tr _ctm _pss grestore 3 1 roll moveto tr sp } ddef /Tj { dup currentpoint 4 2 roll currentpoint gsave newpath moveto trj _ctm _pjss grestore 3 1 roll moveto tr jsp } ddef } def /To { pop _ctm currentmatrix pop } def /TO { iTe _ctm setmatrix newpath } def /Tp { pop _tm astore pop _ctm setmatrix _tDict begin /W { } def /h { } def } def /TP { end iTm 0 0 moveto } def /Tr { _render 3 le { currentpoint newpath moveto } if dup 8 eq { pop 0 } { dup 9 eq { pop 1 } if } ifelse dup /_render exch ddef _renderStart exch get load exec } def /iTm { _ctm setmatrix _tm concat 0 _rise translate _hs 1 scale } def /Tm { _tm astore pop iTm 0 0 moveto } def /Td { _mtx translate _tm _tm concatmatrix pop iTm 0 0 moveto } def /iTe { _render -1 eq { } { _renderEnd _render get dup null ne { load exec } { pop } ifelse } ifelse /_render -1 ddef } def /Ta { pop } def /Tf { dup 1000 div /_fScl exch ddef % selectfont } def /Tl { pop 0 exch _leading astore pop } def /Tt { pop } def /TW { 3 npop } def /Tw { /_cx exch ddef } def /TC { 3 npop } def /Tc { /_ax exch ddef } def /Ts { /_rise exch ddef currentpoint iTm moveto } def /Ti { 3 npop } def /Tz { 100 div /_hs exch ddef iTm } def /TA { pop } def /Tq { pop } def /Th { pop pop pop pop pop } def /TX { pop } def /Tk { exch pop _fScl mul neg 0 rmoveto } def /TK { 2 npop } def /T* { _leading aload pop neg Td } def /T*- { _leading aload pop Td } def /T- { _hyphen Tx } def /T+ { } def /TR { _ctm currentmatrix pop _tm astore pop iTm 0 0 moveto } def /TS { currentfont 3 1 roll /_Symbol_ _fScl 1000 mul selectfont 0 eq { Tx } { Tj } ifelse setfont } def /Xb { pop pop } def /Tb /Xb load def /Xe { pop pop pop pop } def /Te /Xe load def /XB { } def /TB /XB load def currentdict readonly pop end setpacking %%EndResource %%BeginResource: procset Adobe_IllustratorA_AI5 1.1 0 %%Title: (Adobe Illustrator (R) Version 5.0 Abbreviated Prolog) %%Version: 1.1 %%CreationDate: (3/7/1994) () %%Copyright: ((C) 1987-1994 Adobe Systems Incorporated All Rights Reserved) currentpacking true setpacking userdict /Adobe_IllustratorA_AI5_vars 70 dict dup begin put /_lp /none def /_pf { } def /_ps { } def /_psf { } def /_pss { } def /_pjsf { } def /_pjss { } def /_pola 0 def /_doClip 0 def /cf currentflat def /_tm matrix def /_renderStart [ /e0 /r0 /a0 /o0 /e1 /r1 /a1 /i0 ] def /_renderEnd [ null null null null /i1 /i1 /i1 /i1 ] def /_render -1 def /_rise 0 def /_ax 0 def /_ay 0 def /_cx 0 def /_cy 0 def /_leading [ 0 0 ] def /_ctm matrix def /_mtx matrix def /_sp 16#020 def /_hyphen (-) def /_fScl 0 def /_cnt 0 def /_hs 1 def /_nativeEncoding 0 def /_useNativeEncoding 0 def /_tempEncode 0 def /_pntr 0 def /_tDict 2 dict def /_wv 0 def /Tx { } def /Tj { } def /CRender { } def /_AI3_savepage { } def /_gf null def /_cf 4 array def /_if null def /_of false def /_fc { } def /_gs null def /_cs 4 array def /_is null def /_os false def /_sc { } def /discardSave null def /buffer 256 string def /beginString null def /endString null def /endStringLength null def /layerCnt 1 def /layerCount 1 def /perCent (%) 0 get def /perCentSeen? false def /newBuff null def /newBuffButFirst null def /newBuffLast null def /clipForward? false def end userdict /Adobe_IllustratorA_AI5 74 dict dup begin put /initialize { Adobe_IllustratorA_AI5 dup begin Adobe_IllustratorA_AI5_vars begin discardDict { bind pop pop } forall dup /nc get begin { dup xcheck 1 index type /operatortype ne and { bind } if pop pop } forall end newpath } def /terminate { end end } def /_ null def /ddef { Adobe_IllustratorA_AI5_vars 3 1 roll put } def /xput { dup load dup length exch maxlength eq { dup dup load dup length 2 mul dict copy def } if load begin def end } def /npop { { pop } repeat } def /sw { dup length exch stringwidth exch 5 -1 roll 3 index mul add 4 1 roll 3 1 roll mul add } def /swj { dup 4 1 roll dup length exch stringwidth exch 5 -1 roll 3 index mul add 4 1 roll 3 1 roll mul add 6 2 roll /_cnt 0 ddef { 1 index eq { /_cnt _cnt 1 add ddef } if } forall pop exch _cnt mul exch _cnt mul 2 index add 4 1 roll 2 index add 4 1 roll pop pop } def /ss { 4 1 roll { 2 npop (0) exch 2 copy 0 exch put pop gsave false charpath currentpoint 4 index setmatrix stroke grestore moveto 2 copy rmoveto } exch cshow 3 npop } def /jss { 4 1 roll { 2 npop (0) exch 2 copy 0 exch put gsave _sp eq { exch 6 index 6 index 6 index 5 -1 roll 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pop cf } if setflat } def /j { setlinejoin } def /J { setlinecap } def /M { setmiterlimit } def /w { setlinewidth } def /H { } def /h { closepath } def /N { _pola 0 eq { _doClip 1 eq { clip /_doClip 0 ddef } if newpath } { /CRender { N } ddef } ifelse } def /n { N } def /F { _pola 0 eq { _doClip 1 eq { gsave _pf grestore clip newpath /_lp /none ddef _fc /_doClip 0 ddef } { _pf } ifelse } { /CRender { F } ddef } ifelse } def /f { closepath F } def /S { _pola 0 eq { _doClip 1 eq { gsave _ps grestore clip newpath /_lp /none ddef _sc /_doClip 0 ddef } { _ps } ifelse } { /CRender { S } ddef } ifelse } def /s { closepath S } def /B { _pola 0 eq { _doClip 1 eq gsave F grestore { gsave S grestore clip newpath /_lp /none ddef _sc /_doClip 0 ddef } { S } ifelse } { /CRender { B } ddef } ifelse } def /b { closepath B } def /W { /_doClip 1 ddef } def /* { count 0 ne { dup type /stringtype eq { pop } if } if newpath } def /u { } def /U { } def /q { _pola 0 eq { gsave } if } def /Q { _pola 0 eq { 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_lp /fill ne { _of setoverprint _cf aload pop setcmykcolor /_lp /fill ddef } if } ddef /_pf { _fc fill } ddef /_psf { _fc ashow } ddef /_pjsf { _fc awidthshow } ddef /_lp /none ddef } def /K { _cs astore pop /_sc { _lp /stroke ne { _os setoverprint _cs aload pop setcmykcolor /_lp /stroke ddef } if } ddef /_ps { _sc stroke } ddef /_pss { _sc ss } ddef /_pjss { _sc jss } ddef /_lp /none ddef } def /x { /_gf exch ddef findcmykcustomcolor /_if exch ddef /_fc { _lp /fill ne { _of setoverprint _if _gf 1 exch sub setcustomcolor /_lp /fill ddef } if } ddef /_pf { _fc fill } ddef /_psf { _fc ashow } ddef /_pjsf { _fc awidthshow } ddef /_lp /none ddef } def /X { /_gs exch ddef findcmykcustomcolor /_is exch ddef /_sc { _lp /stroke ne { _os setoverprint _is _gs 1 exch sub setcustomcolor /_lp /stroke ddef } if } ddef /_ps { _sc stroke } ddef /_pss { _sc ss } ddef /_pjss { _sc jss } ddef /_lp /none ddef } def /A { pop } def /annotatepage { userdict /annotatepage 2 copy known {get exec} {pop pop} ifelse } def /discard { save /discardSave exch store discardDict begin /endString exch store gt38? { 2 add } if load stopped pop end discardSave restore } bind def userdict /discardDict 7 dict dup begin put /pre38Initialize { /endStringLength endString length store /newBuff buffer 0 endStringLength getinterval store /newBuffButFirst newBuff 1 endStringLength 1 sub getinterval store /newBuffLast newBuff endStringLength 1 sub 1 getinterval store } def /shiftBuffer { newBuff 0 newBuffButFirst putinterval newBuffLast 0 currentfile read not { stop } if put } def 0 { pre38Initialize mark currentfile newBuff readstring exch pop { { newBuff endString eq { cleartomark stop } if shiftBuffer } loop } { stop } ifelse } def 1 { pre38Initialize /beginString exch store mark currentfile newBuff readstring exch pop { { newBuff beginString eq { /layerCount dup load 1 add store } { newBuff endString eq { /layerCount dup load 1 sub store layerCount 0 eq { cleartomark stop } if } if } ifelse shiftBuffer } loop } { stop } ifelse } def 2 { mark { currentfile buffer readline not { stop } if endString eq { cleartomark stop } if } loop } def 3 { /beginString exch store /layerCnt 1 store mark { currentfile buffer readline not { stop } if dup beginString eq { pop /layerCnt dup load 1 add store } { endString eq { layerCnt 1 eq { cleartomark stop } { /layerCnt dup load 1 sub store } ifelse } if } ifelse } loop } def end userdict /clipRenderOff 15 dict dup begin put { /n /N /s /S /f /F /b /B } { { _doClip 1 eq { /_doClip 0 ddef clip } if newpath } def } forall /Tr /pop load def /Bb {} def /BB /pop load def /Bg {12 npop} def /Bm {6 npop} def /Bc /Bm load def /Bh {4 npop} def end /Lb { 4 npop 6 1 roll pop 4 1 roll pop pop pop 0 eq { 0 eq { (%AI5_BeginLayer) 1 (%AI5_EndLayer--) discard } { /clipForward? true def /Tx /pop load def /Tj /pop load def currentdict end clipRenderOff begin begin } ifelse } { 0 eq { save /discardSave exch store } if } ifelse } bind def /LB { discardSave dup null ne { restore } { pop clipForward? { currentdict end end begin /clipForward? false ddef } if } ifelse } bind def /Pb { pop pop 0 (%AI5_EndPalette) discard } bind def /Np { 0 (%AI5_End_NonPrinting--) discard } bind def /Ln /pop load def /Ap /pop load def /Ar { 72 exch div 0 dtransform dup mul exch dup mul add sqrt dup 1 lt { pop 1 } if setflat } def /Mb { q } def /Md { } def /MB { Q } def /nc 3 dict def nc begin /setgray { pop } bind def /setcmykcolor { 4 npop } bind def /setcustomcolor { 2 npop } bind def currentdict readonly pop end currentdict readonly pop end setpacking %%EndResource %%EndProlog %%BeginSetup %%IncludeFont: Helvetica %%IncludeFont: Symbol %%IncludeFont: ZapfDingbats Adobe_level2_AI5 /initialize get exec Adobe_IllustratorA_AI5_vars Adobe_IllustratorA_AI5 Adobe_typography_AI5 /initialize get exec Adobe_IllustratorA_AI5 /initialize get exec [ 39/quotesingle 96/grave 128/Adieresis/Aring/Ccedilla/Eacute/Ntilde/Odieresis /Udieresis/aacute/agrave/acircumflex/adieresis/atilde/aring/ccedilla/eacute /egrave/ecircumflex/edieresis/iacute/igrave/icircumflex/idieresis/ntilde /oacute/ograve/ocircumflex/odieresis/otilde/uacute/ugrave/ucircumflex /udieresis/dagger/degree/cent/sterling/section/bullet/paragraph/germandbls /registered/copyright/trademark/acute/dieresis/.notdef/AE/Oslash /.notdef/plusminus/.notdef/.notdef/yen/mu/.notdef/.notdef /.notdef/.notdef/.notdef/ordfeminine/ordmasculine/.notdef/ae/oslash /questiondown/exclamdown/logicalnot/.notdef/florin/.notdef/.notdef /guillemotleft/guillemotright/ellipsis/.notdef/Agrave/Atilde/Otilde/OE/oe /endash/emdash/quotedblleft/quotedblright/quoteleft/quoteright/divide /.notdef/ydieresis/Ydieresis/fraction/currency/guilsinglleft/guilsinglright /fi/fl/daggerdbl/periodcentered/quotesinglbase/quotedblbase/perthousand /Acircumflex/Ecircumflex/Aacute/Edieresis/Egrave/Iacute/Icircumflex /Idieresis/Igrave/Oacute/Ocircumflex/.notdef/Ograve/Uacute/Ucircumflex /Ugrave/dotlessi/circumflex/tilde/macron/breve/dotaccent/ring/cedilla /hungarumlaut/ogonek/caron TE %AI3_BeginEncoding: _Helvetica Helvetica [39/quotesingle 96/grave 128/Adieresis/Aring/Ccedilla/Eacute/Ntilde/Odieresis /Udieresis/aacute/agrave/acircumflex/adieresis/atilde/aring/ccedilla/eacute /egrave/ecircumflex/edieresis/iacute/igrave/icircumflex/idieresis/ntilde /oacute/ograve/ocircumflex/odieresis/otilde/uacute/ugrave/ucircumflex /udieresis/dagger/degree/cent/sterling/section/bullet/paragraph/germandbls /registered/copyright/trademark/acute/dieresis/.notdef/AE/Oslash /.notdef/plusminus/.notdef/.notdef/yen/mu/.notdef/.notdef /.notdef/.notdef/.notdef/ordfeminine/ordmasculine/.notdef/ae/oslash /questiondown/exclamdown/logicalnot/.notdef/florin/.notdef/.notdef /guillemotleft/guillemotright/ellipsis/.notdef/Agrave/Atilde/Otilde/OE/oe /endash/emdash/quotedblleft/quotedblright/quoteleft/quoteright/divide /.notdef/ydieresis/Ydieresis/fraction/currency/guilsinglleft/guilsinglright /fi/fl/daggerdbl/periodcentered/quotesinglbase/quotedblbase/perthousand /Acircumflex/Ecircumflex/Aacute/Edieresis/Egrave/Iacute/Icircumflex /Idieresis/Igrave/Oacute/Ocircumflex/.notdef/Ograve/Uacute/Ucircumflex /Ugrave/dotlessi/circumflex/tilde/macron/breve/dotaccent/ring/cedilla /hungarumlaut/ogonek/caron /_Helvetica/Helvetica 0 0 1 TZ %AI3_EndEncoding AdobeType %AI3_BeginEncoding: _Symbol Symbol [/_Symbol/Symbol 0 0 0 TZ %AI3_EndEncoding AdobeType %AI3_BeginEncoding: _ZapfDingbats ZapfDingbats [/_ZapfDingbats/ZapfDingbats 0 0 0 TZ %AI3_EndEncoding AdobeType %AI5_Begin_NonPrinting Np %AI3_BeginPattern: (Yellow Stripe) (Yellow Stripe) 8.4499 4.6 80.4499 76.6 [ %AI3_Tile (0 O 0 R 0 0.4 1 0 k 0 0.4 1 0 K) @ ( 800 Ar 0 J 0 j 3.6 w 4 M []0 d %AI3_Note: 0 D 8.1999 8.1999 m 80.6999 8.1999 L S 8.1999 22.6 m 80.6999 22.6 L S 8.1999 37.0001 m 80.6999 37.0001 L S 8.1999 51.3999 m 80.6999 51.3999 L S 8.1999 65.8 m 80.6999 65.8 L S 8.1999 15.3999 m 80.6999 15.3999 L S 8.1999 29.8 m 80.6999 29.8 L S 8.1999 44.1999 m 80.6999 44.1999 L S 8.1999 58.6 m 80.6999 58.6 L S 8.1999 73.0001 m 80.6999 73.0001 L S ) & ] E %AI3_EndPattern %AI5_End_NonPrinting-- %AI5_Begin_NonPrinting Np 3 Bn %AI5_BeginGradient: (Black & White) (Black & White) 0 2 Bd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r [ 0 0 50 100 %_Bs 1 0 50 0 %_Bs BD %AI5_EndGradient %AI5_BeginGradient: (Red & Yellow) (Red & Yellow) 0 2 Bd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r [ 0 1 0.6 0 1 50 100 %_Bs 0 0 1 0 1 50 0 %_Bs BD %AI5_EndGradient %AI5_BeginGradient: (Yellow & Blue Radial) (Yellow & Blue Radial) 1 2 Bd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0 k Pc 0.75 0 0 0 k Pc 1 0 0 0 k Pc 0.25 0.25 0 0 k Pc 0.5 0.5 0 0 k Pc 0.75 0.75 0 0 k Pc 1 1 0 0 k Pc Bb 2 (Red & Yellow) -4014 4716 0 0 1 0 0 1 0 0 Bg 0 BB Pc 0 0.25 0 0 k Pc 0 0.5 0 0 k Pc 0 0.75 0 0 k Pc 0 1 0 0 k Pc 0 0.25 0.25 0 k Pc 0 0.5 0.5 0 k Pc 0 0.75 0.75 0 k Pc 0 1 1 0 k Pc Bb 0 0 0 0 Bh 2 (Yellow & Blue Radial) -4014 4716 0 0 1 0 0 1 0 0 Bg 0 BB Pc 0 0 0.25 0 k Pc 0 0 0.5 0 k Pc 0 0 0.75 0 k Pc 0 0 1 0 k Pc 0.25 0 0.25 0 k Pc 0.5 0 0.5 0 k Pc 0.75 0 0.75 0 k Pc 1 0 1 0 k Pc (Yellow Stripe) 0 0 1 1 0 0 0 0 0 [1 0 0 1 0 0] p Pc 0.25 0.125 0 0 k Pc 0.5 0.25 0 0 k Pc 0.75 0.375 0 0 k Pc 1 0.5 0 0 k Pc 0.125 0.25 0 0 k Pc 0.25 0.5 0 0 k Pc 0.375 0.75 0 0 k Pc 0.5 1 0 0 k Pc 0 0 0 0 k Pc 0 0.25 0.125 0 k Pc 0 0.5 0.25 0 k Pc 0 0.75 0.375 0 k Pc 0 1 0.5 0 k Pc 0 0.125 0.25 0 k Pc 0 0.25 0.5 0 k Pc 0 0.375 0.75 0 k Pc 0 0.5 1 0 k Pc 0 0 0 0 k Pc 0.125 0 0.25 0 k Pc 0.25 0 0.5 0 k Pc 0.375 0 0.75 0 k Pc 0.5 0 1 0 k Pc 0.25 0 0.125 0 k Pc 0.5 0 0.25 0 k Pc 0.75 0 0.375 0 k Pc 1 0 0.5 0 k Pc 0 0 0 0 k Pc 0.25 0.125 0.125 0 k Pc 0.5 0.25 0.25 0 k Pc 0.75 0.375 0.375 0 k Pc 1 0.5 0.5 0 k Pc 0.25 0.25 0.125 0 k Pc 0.5 0.5 0.25 0 k Pc 0.75 0.75 0.375 0 k Pc 1 1 0.5 0 k Pc 0 0 0 0 k Pc 0.125 0.25 0.125 0 k Pc 0.25 0.5 0.25 0 k Pc 0.375 0.75 0.375 0 k Pc 0.5 1 0.5 0 k Pc 0.125 0.25 0.25 0 k Pc 0.25 0.5 0.5 0 k Pc 0.375 0.75 0.75 0 k Pc 0.5 1 1 0 k Pc 0 0 0 0 k Pc 0.125 0.125 0.25 0 k Pc 0.25 0.25 0.5 0 k Pc 0.375 0.375 0.75 0 k Pc 0.5 0.5 1 0 k Pc 0.25 0.125 0.25 0 k Pc 0.5 0.25 0.5 0 k Pc 0.75 0.375 0.75 0 k Pc 1 0.5 1 0 k Pc PB %AI5_EndPalette %%EndSetup %AI5_BeginLayer 1 1 1 1 0 0 0 79 128 255 Lb (Layer 1) Ln 0 A 1 Ap 0 O 1 g 1 R 0 G 800 Ar 0 J 0 j 1 w 4 M []0 d %AI3_Note: 0 D 306.5256 468.391 m 306.5256 589.0644 L 186.1978 589.0644 L 186.1978 468.391 L 306.5256 468.391 L b 306.5256 468.391 m 306.5256 589.0644 L 186.1978 589.0644 L 186.1978 468.391 L 306.5256 468.391 L b 306.5256 468.391 m 306.5256 589.0644 L 186.1978 589.0644 L 186.1978 468.391 L 306.5256 468.391 L b 306.5256 468.391 m 306.5256 589.0644 L 186.1978 589.0644 L 186.1978 468.391 L 306.5256 468.391 L b 0 Ap 186.1978 589.0644 m B 186.1978 589.0644 m B 0 To 1 0 0 1 182 709 0 Tp TP 0 Tr 0 g /_ZapfDingbats 9.6539 Tf 0 Ts 99.7131 Tz 0 Tt 0 TA %_ 0 XL 28.8787 0 Xb XB 0 0 5 TC 100 100 200 TW 0 0 0 Ti 0 Ta 0 0 2 2 3 Th 0 Tq 0 0 Tl 0 Tc 0 Tw (s) Tx (\r) TX TO 0 R 0 G 306.5256 468.391 m 468.567 468.391 468.567 468.391 y B 0 To -0.8488 0.5288 -0.5266 -0.8501 475.7866 467.8547 0 Tp TP 0 Tr /_ZapfDingbats 9.6459 Tf 99.8714 Tz 28.9007 0 Xb XB (s) Tx (\r) TX TO 1 Ap 1 g 1 R 0 G 306.5256 468.391 m 306.5256 589.0644 L 186.1978 589.0644 L 186.1978 468.391 L 306.5256 468.391 L b 306.5256 468.391 m 306.5256 589.0644 L 186.1978 589.0644 L 186.1978 468.391 L 306.5256 468.391 L b 306.5256 468.391 m 306.5256 589.0644 L 186.1978 589.0644 L 186.1978 468.391 L 306.5256 468.391 L s 383.0005 552.326 m 383.0005 672.9994 L 262.6728 672.9994 L 262.6728 552.326 L 383.0005 552.326 L s 0 Ap 262.6728 672.9994 m 262.6728 672.9994 186.1978 589.0644 v S 306.5256 589.0644 m 382.1983 671.6586 l S 262.6728 552.326 m S 287.0058 577.2652 m 286.4709 577.2652 186.1978 468.391 v S 0 To 0.6085 -0.7935 0.7919 0.6107 284 579.3333 0 Tp TP 0 Tr 0 O 0 g /_ZapfDingbats 9.6365 Tf 100.0615 Tz 28.9276 0 Xb XB (s) Tx (\r) TX TO 1 R 0 G 305.7234 470 m 264.0098 552.5942 l 264.0098 552.5942 187.5348 588.528 v 306.5256 468.391 l 383.0005 552.326 l 306.5256 468.391 l S 0 O 0.5 g 0.5 G 231.9224 567.0751 m 281.6578 518.2693 l 186.4652 590.6734 l B 0 G 187 470 m 234.0615 553.1305 233.5267 553.1305 y S 0 To 1 0 0 1 214.8091 578.3379 0 Tp TP 0 Tr 0 O 0 g /_Symbol 14.4808 Tf 99.7131 Tz 28.8787 0 Xb XB (D) Tx (\r) TX TO 0 To 1 0 0 1 262.4054 485.5534 0 Tp TP 0 Tr /_Symbol 16.0898 Tf (D) Tx (\r) TX TO 0 To 1 0 0 1 261.8706 500.5706 0 Tp TP 0 Tr /_Helvetica 16.0898 Tf (_) Tx (\r) TX TO 0 To 0.8595 -0.5111 0.5089 0.8608 274.7344 638.8765 0 Tp TP 0 Tr /_ZapfDingbats 9.6468 Tf 99.8622 Tz 28.9007 0 Xb XB (5) Tx (\r) TX TO 0 To 1 0 0 1 285.3333 637.3333 0 Tp TP 0 Tr /_Symbol 9.6539 Tf 99.7131 Tz 28.8787 0 Xb XB (\( q , q , ) Tx /_Helvetica 9.6539 Tf (1 \)) Tx (\r) TX TO 0 To 0.8595 -0.5111 0.5089 0.8608 230.8271 558.6446 0 Tp TP 0 Tr /_ZapfDingbats 9.6468 Tf 99.8622 Tz 28.9007 0 Xb XB (5) Tx (\r) TX TO 1 R 0 G 235.6659 557.4212 m S 235.6659 557.4212 m 235.6659 493.5983 235.6659 493.5983 y S 0 To 1 0 0 1 227.1092 478.5812 0 Tp TP 0 Tr 0 O 0 g /_Symbol 6.4359 Tf 99.7131 Tz 28.8787 0 Xb XB (1) Tx (\r) TX TO 0 To 1 0 0 1 296 633.3333 0 Tp TP 0 Tr (1) Tx (\r) TX TO 0 To 1 0 0 1 242.0834 478.0449 0 Tp TP 0 Tr (2 ) Tx (\r) TX TO 0 To 1 0 0 1 312 634 0 Tp TP 0 Tr (2) Tx (\r) TX TO 0 To 1 0 0 1 214.2743 480.7265 0 Tp TP 0 Tr /_Symbol 9.6539 Tf (\( a , a \)) Tx (\r) TX TO 0 To 0.3573 0.934 -0.9333 0.3591 237.3333 488 0 Tp TP 0 Tr /_ZapfDingbats 9.6303 Tf 100.2122 Tz 28.9523 0 Xb XB (5) Tx (\r) TX TO 1 R 0.5 G 306.5256 468.391 m S 0 G 306.5256 468.391 m S 186 709 m 186 592 l S 236.6667 561.3333 m S 236.6667 561.3333 m 316.6667 712.6667 316.6667 712.6667 y S 0 To 1 0 0 1 340.6667 636 0 Tp TP 0 Tr 0 O 0 g /_Symbol 14.4808 Tf 99.7131 Tz 28.8787 0 Xb XB (Q) Tx (\r) TX TO 0 To 1 0 0 1 334 638 0 Tp TP 0 Tr /_Symbol 10 Tf 100 Tz 36 0 Xb XB (=) Tx (\r) TX TO 0 To 1 0 0 1 286.6667 701.3333 0 Tp TP 0 Tr (l) Tx /_Symbol 14 Tf (Q) Tx (\r) TX TO LB %AI5_EndLayer-- %%PageTrailer gsave annotatepage grestore showpage %%Trailer Adobe_IllustratorA_AI5 /terminate get exec Adobe_typography_AI5 /terminate get exec Adobe_level2_AI5 /terminate get exec %%EOF %!PS-Adobe-3.0 EPSF-3.0 %%Creator: Adobe Illustrator(TM) 5.5 %%For: () () %%Title: (fig2.eps) %%CreationDate: (11/5/95) ( 6:32 PM) %%BoundingBox: 95 265 512 724 %%HiResBoundingBox: 95.1916 265.8825 511.414 723.957 %%DocumentProcessColors: Black %%DocumentFonts: DorchesterScriptMT %%+ Helvetica %%+ Symbol %%+ ZapfDingbats %%DocumentSuppliedResources: procset Adobe_level2_AI5 1.0 0 %%+ procset Adobe_typography_AI5 1.0 0 %%+ procset Adobe_IllustratorA_AI5 1.0 0 %AI5_FileFormat 3.0 %AI3_ColorUsage: Black&White %AI3_TemplateBox: 306 396 306 396 %AI3_TileBox: 30 31 582 761 %AI3_DocumentPreview: Macintosh_ColorPic %AI5_ArtSize: 612 792 %AI5_RulerUnits: 2 %AI5_ArtFlags: 1 0 0 1 0 0 1 1 0 %AI5_TargetResolution: 800 %AI5_NumLayers: 1 %AI5_OpenToView: -30 804 1 685 809 18 0 1 394 52 %AI5_OpenViewLayers: 7 %%EndComments %%BeginProlog %%BeginResource: procset Adobe_level2_AI5 1.0 0 %%Title: (Adobe Illustrator (R) Version 5.0 Level 2 Emulation) %%Version: 1.0 %%CreationDate: (04/10/93) () %%Copyright: ((C) 1987-1993 Adobe Systems Incorporated All Rights Reserved) userdict /Adobe_level2_AI5 21 dict dup begin put /packedarray where not { userdict begin /packedarray { array astore readonly } bind def /setpacking /pop load def /currentpacking false def end 0 } if pop userdict /defaultpacking currentpacking put true setpacking /initialize { Adobe_level2_AI5 begin } bind def /terminate { currentdict Adobe_level2_AI5 eq { end } if } bind def mark /setcustomcolor where not { /findcmykcustomcolor { 5 packedarray } bind def /setcustomcolor { exch aload pop pop 4 { 4 index mul 4 1 roll } repeat 5 -1 roll pop setcmykcolor } def } if /gt38? mark {version cvx exec} stopped {cleartomark true} {38 gt exch pop} ifelse def userdict /deviceDPI 72 0 matrix defaultmatrix dtransform dup mul exch dup mul add sqrt put userdict /level2? systemdict /languagelevel known dup { pop systemdict /languagelevel get 2 ge } if put level2? not { /setcmykcolor where not { /setcmykcolor { exch .11 mul add exch .59 mul add exch .3 mul add 1 exch sub setgray } def } if /currentcmykcolor where not { /currentcmykcolor { 0 0 0 1 currentgray sub } def } if /setoverprint where not { /setoverprint /pop load def } if /selectfont where not { /selectfont { exch findfont exch dup type /arraytype eq { makefont } { scalefont } ifelse setfont } bind def } if /cshow where not { /cshow { [ 0 0 5 -1 roll aload pop ] cvx bind forall } bind def } if } if cleartomark /anyColor? { add add add 0 ne } bind def /testColor { gsave setcmykcolor currentcmykcolor grestore } bind def /testCMYKColorThrough { testColor anyColor? } bind def userdict /composite? level2? { gsave 1 1 1 1 setcmykcolor currentcmykcolor grestore add add add 4 eq } { 1 0 0 0 testCMYKColorThrough 0 1 0 0 testCMYKColorThrough 0 0 1 0 testCMYKColorThrough 0 0 0 1 testCMYKColorThrough and and and } ifelse put composite? not { userdict begin gsave /cyan? 1 0 0 0 testCMYKColorThrough def /magenta? 0 1 0 0 testCMYKColorThrough def /yellow? 0 0 1 0 testCMYKColorThrough def /black? 0 0 0 1 testCMYKColorThrough def grestore /isCMYKSep? cyan? magenta? yellow? black? or or or def /customColor? isCMYKSep? not def end } if end defaultpacking setpacking %%EndResource %%BeginResource: procset Adobe_typography_AI5 1.0 1 %%Title: (Typography Operators) %%Version: 1.0 %%CreationDate:(03/26/93) () %%Copyright: ((C) 1987-1993 Adobe Systems Incorporated All Rights Reserved) currentpacking true setpacking userdict /Adobe_typography_AI5 54 dict dup begin put /initialize { begin begin Adobe_typography_AI5 begin Adobe_typography_AI5 { dup xcheck { bind } if pop pop } forall end end end Adobe_typography_AI5 begin } def /terminate { currentdict Adobe_typography_AI5 eq { end } if } def /modifyEncoding { /_tempEncode exch ddef /_pntr 0 ddef { counttomark -1 roll dup type dup /marktype eq { pop pop exit } { /nametype eq { _tempEncode /_pntr dup load dup 3 1 roll 1 add ddef 3 -1 roll put } { /_pntr exch ddef } ifelse } ifelse } loop _tempEncode } def /TE { StandardEncoding 256 array copy modifyEncoding /_nativeEncoding exch def } def % /TZ { dup type /arraytype eq { /_wv exch def } { /_wv 0 def } ifelse /_useNativeEncoding exch def pop pop findfont _wv type /arraytype eq { _wv makeblendedfont } if dup length 2 add dict begin mark exch { 1 index /FID ne { def } if cleartomark mark } forall pop /FontName exch def counttomark 0 eq { 1 _useNativeEncoding eq { /Encoding _nativeEncoding def } if cleartomark } { /Encoding load 256 array copy modifyEncoding /Encoding exch def } ifelse FontName currentdict end definefont pop } def /tr { _ax _ay 3 2 roll } def /trj { _cx _cy _sp _ax _ay 6 5 roll } def /a0 { /Tx { dup currentpoint 3 2 roll tr _psf newpath moveto tr _ctm _pss } ddef /Tj { dup currentpoint 3 2 roll trj _pjsf newpath moveto trj _ctm _pjss } ddef } def /a1 { /Tx { dup currentpoint 4 2 roll gsave dup currentpoint 3 2 roll tr _psf newpath moveto tr _ctm _pss grestore 3 1 roll moveto tr sp } ddef /Tj { dup currentpoint 4 2 roll gsave dup currentpoint 3 2 roll trj _pjsf newpath moveto trj _ctm _pjss grestore 3 1 roll moveto tr jsp } ddef } def /e0 { /Tx { tr _psf } ddef /Tj { trj _pjsf } ddef } def /e1 { /Tx { dup currentpoint 4 2 roll gsave tr _psf grestore 3 1 roll moveto tr sp } ddef /Tj { dup currentpoint 4 2 roll gsave trj _pjsf grestore 3 1 roll moveto tr jsp } ddef } def /i0 { /Tx { tr sp } ddef /Tj { trj jsp } ddef } def /i1 { W N } def /o0 { /Tx { tr sw rmoveto } ddef /Tj { trj swj rmoveto } ddef } def /r0 { /Tx { tr _ctm _pss } ddef /Tj { trj _ctm _pjss } ddef } def /r1 { /Tx { dup currentpoint 4 2 roll currentpoint gsave newpath moveto tr _ctm _pss grestore 3 1 roll moveto tr sp } ddef /Tj { dup currentpoint 4 2 roll currentpoint gsave newpath moveto trj _ctm _pjss grestore 3 1 roll moveto tr jsp } ddef } def /To { pop _ctm currentmatrix pop } def /TO { iTe _ctm setmatrix newpath } def /Tp { pop _tm astore pop _ctm setmatrix _tDict begin /W { } def /h { } def } def /TP { end iTm 0 0 moveto } def /Tr { _render 3 le { currentpoint newpath moveto } if dup 8 eq { pop 0 } { dup 9 eq { pop 1 } if } ifelse dup /_render exch ddef _renderStart exch get load exec } def /iTm { _ctm setmatrix _tm concat 0 _rise translate _hs 1 scale } def /Tm { _tm astore pop iTm 0 0 moveto } def /Td { _mtx translate _tm _tm concatmatrix pop iTm 0 0 moveto } def /iTe { _render -1 eq { } { _renderEnd _render get dup null ne { load exec } { pop } ifelse } ifelse /_render -1 ddef } def /Ta { pop } def /Tf { dup 1000 div /_fScl exch ddef % selectfont } def /Tl { pop 0 exch _leading astore pop } def /Tt { pop } def /TW { 3 npop } def /Tw { /_cx exch ddef } def /TC { 3 npop } def /Tc { /_ax exch ddef } def /Ts { /_rise exch ddef currentpoint iTm moveto } def /Ti { 3 npop } def /Tz { 100 div /_hs exch ddef iTm } def /TA { pop } def /Tq { pop } def /Th { pop pop pop pop pop } def /TX { pop } def /Tk { exch pop _fScl mul neg 0 rmoveto } def /TK { 2 npop } def /T* { _leading aload pop neg Td } def /T*- { _leading aload pop Td } def /T- { _hyphen Tx } def /T+ { } def /TR { _ctm currentmatrix pop _tm astore pop iTm 0 0 moveto } def /TS { currentfont 3 1 roll /_Symbol_ _fScl 1000 mul selectfont 0 eq { Tx } { Tj } ifelse setfont } def /Xb { pop pop } def /Tb /Xb load def /Xe { pop pop pop pop } def /Te /Xe load def /XB { } def /TB /XB load def currentdict readonly pop end setpacking %%EndResource %%BeginResource: procset Adobe_IllustratorA_AI5 1.1 0 %%Title: (Adobe Illustrator (R) Version 5.0 Abbreviated Prolog) %%Version: 1.1 %%CreationDate: (3/7/1994) () %%Copyright: ((C) 1987-1994 Adobe Systems Incorporated All Rights Reserved) currentpacking true setpacking userdict /Adobe_IllustratorA_AI5_vars 70 dict dup begin put /_lp /none def /_pf { } def /_ps { } def /_psf { } def /_pss { } def /_pjsf { } def /_pjss { } def /_pola 0 def /_doClip 0 def /cf currentflat def /_tm matrix def /_renderStart [ /e0 /r0 /a0 /o0 /e1 /r1 /a1 /i0 ] def /_renderEnd [ null null null null /i1 /i1 /i1 /i1 ] def /_render -1 def /_rise 0 def /_ax 0 def /_ay 0 def /_cx 0 def /_cy 0 def /_leading [ 0 0 ] def /_ctm matrix def /_mtx matrix def /_sp 16#020 def /_hyphen (-) def /_fScl 0 def /_cnt 0 def /_hs 1 def /_nativeEncoding 0 def /_useNativeEncoding 0 def /_tempEncode 0 def /_pntr 0 def /_tDict 2 dict def /_wv 0 def /Tx { } def /Tj { } def /CRender { } def /_AI3_savepage { } def /_gf null def /_cf 4 array def /_if null def /_of false def /_fc { } def /_gs null def /_cs 4 array def /_is null def /_os false def /_sc { } def /discardSave null def /buffer 256 string def /beginString null def /endString null def /endStringLength null def /layerCnt 1 def /layerCount 1 def /perCent (%) 0 get def /perCentSeen? false def /newBuff null def /newBuffButFirst null def /newBuffLast null def /clipForward? false def end userdict /Adobe_IllustratorA_AI5 74 dict dup begin put /initialize { Adobe_IllustratorA_AI5 dup begin Adobe_IllustratorA_AI5_vars begin discardDict { bind pop pop } forall dup /nc get begin { dup xcheck 1 index type /operatortype ne and { bind } if pop pop } forall end newpath } def /terminate { end end } def /_ null def /ddef { Adobe_IllustratorA_AI5_vars 3 1 roll put } def /xput { dup load dup length exch maxlength eq { dup dup load dup length 2 mul dict copy def } if load begin def end } def /npop { { pop } repeat } def /sw { dup length exch stringwidth exch 5 -1 roll 3 index mul add 4 1 roll 3 1 roll mul add } def /swj { dup 4 1 roll dup length exch stringwidth exch 5 -1 roll 3 index mul add 4 1 roll 3 1 roll mul add 6 2 roll /_cnt 0 ddef { 1 index eq { /_cnt _cnt 1 add ddef } if } forall pop exch _cnt mul exch _cnt mul 2 index add 4 1 roll 2 index add 4 1 roll pop pop } def /ss { 4 1 roll { 2 npop (0) exch 2 copy 0 exch put pop gsave false charpath currentpoint 4 index setmatrix stroke grestore moveto 2 copy rmoveto } exch cshow 3 npop } def /jss { 4 1 roll { 2 npop (0) exch 2 copy 0 exch put gsave _sp eq { exch 6 index 6 index 6 index 5 -1 roll widthshow currentpoint } { false charpath currentpoint 4 index setmatrix stroke } ifelse grestore moveto 2 copy rmoveto } exch cshow 6 npop } def /sp { { 2 npop (0) exch 2 copy 0 exch put pop false charpath 2 copy rmoveto } exch cshow 2 npop } def /jsp { { 2 npop (0) exch 2 copy 0 exch put _sp eq { exch 5 index 5 index 5 index 5 -1 roll widthshow } { false charpath } ifelse 2 copy rmoveto } exch cshow 5 npop } def /pl { transform 0.25 sub round 0.25 add exch 0.25 sub round 0.25 add exch itransform } def /setstrokeadjust where { pop true setstrokeadjust /c { curveto } def /C /c load def /v { currentpoint 6 2 roll curveto } def /V /v load def /y { 2 copy curveto } def /Y /y load def /l { lineto } def /L /l load def /m { moveto } def } { /c { pl curveto } def /C /c load def /v { currentpoint 6 2 roll pl curveto } def /V /v load def /y { pl 2 copy curveto } def /Y /y load def /l { pl lineto } def /L /l load def /m { pl moveto } def } ifelse /d { setdash } def /cf { } def /i { dup 0 eq { pop cf } if setflat } def /j { setlinejoin } def /J { setlinecap } def /M { setmiterlimit } def /w { setlinewidth } def /H { } def /h { closepath } def /N { _pola 0 eq { _doClip 1 eq { clip /_doClip 0 ddef } if newpath } { /CRender { N } ddef } ifelse } def /n { N } def /F { _pola 0 eq { _doClip 1 eq { gsave _pf grestore clip newpath /_lp /none ddef _fc /_doClip 0 ddef } { _pf } ifelse } { /CRender { F } ddef } ifelse } def /f { closepath F } def /S { _pola 0 eq { _doClip 1 eq { gsave _ps grestore clip newpath /_lp /none ddef _sc /_doClip 0 ddef } { _ps } ifelse } { /CRender { S } ddef } ifelse } def /s { closepath S } def /B { _pola 0 eq { _doClip 1 eq gsave F grestore { gsave S grestore clip newpath /_lp /none ddef _sc /_doClip 0 ddef } { S } ifelse } { /CRender { B } ddef } ifelse } def /b { closepath B } def /W { /_doClip 1 ddef } def /* { count 0 ne { dup type /stringtype eq { pop } if } if newpath } def /u { } def /U { } def /q { _pola 0 eq { gsave } if } def /Q { _pola 0 eq { grestore } if } def /*u { _pola 1 add /_pola exch ddef } def /*U { _pola 1 sub /_pola exch ddef _pola 0 eq { CRender } if } def /D { pop } def /*w { } def /*W { } def /` { /_i save ddef clipForward? { nulldevice } if 6 1 roll 4 npop concat pop userdict begin /showpage { } def 0 setgray 0 setlinecap 1 setlinewidth 0 setlinejoin 10 setmiterlimit [] 0 setdash /setstrokeadjust where {pop false setstrokeadjust} if newpath 0 setgray false setoverprint } def /~ { end _i restore } def /O { 0 ne /_of exch ddef /_lp /none ddef } def /R { 0 ne /_os exch ddef /_lp /none ddef } def /g { /_gf exch ddef /_fc { _lp /fill ne { _of setoverprint _gf setgray /_lp /fill ddef } if } ddef /_pf { _fc fill } ddef /_psf { _fc ashow } ddef /_pjsf { _fc awidthshow } ddef /_lp /none ddef } def /G { /_gs exch ddef /_sc { _lp /stroke ne { _os setoverprint _gs setgray /_lp /stroke ddef } if } ddef /_ps { _sc stroke } ddef /_pss { _sc ss } ddef /_pjss { _sc jss } ddef /_lp /none ddef } def /k { _cf astore pop /_fc { _lp /fill ne { _of setoverprint _cf aload pop setcmykcolor /_lp /fill ddef } if } ddef /_pf { _fc fill } ddef /_psf { _fc ashow } ddef /_pjsf { _fc awidthshow } ddef /_lp /none ddef } def /K { _cs astore pop /_sc { _lp /stroke ne { _os setoverprint _cs aload pop setcmykcolor /_lp /stroke ddef } if } ddef /_ps { _sc stroke } ddef /_pss { _sc ss } ddef /_pjss { _sc jss } ddef /_lp /none ddef } def /x { /_gf exch ddef findcmykcustomcolor /_if exch ddef /_fc { _lp /fill ne { _of setoverprint _if _gf 1 exch sub setcustomcolor /_lp /fill ddef } if } ddef /_pf { _fc fill } ddef /_psf { _fc ashow } ddef /_pjsf { _fc awidthshow } ddef /_lp /none ddef } def /X { /_gs exch ddef findcmykcustomcolor /_is exch ddef /_sc { _lp /stroke ne { _os setoverprint _is _gs 1 exch sub setcustomcolor /_lp /stroke ddef } if } ddef /_ps { _sc stroke } ddef /_pss { _sc ss } ddef /_pjss { _sc jss } ddef /_lp /none ddef } def /A { pop } def /annotatepage { userdict /annotatepage 2 copy known {get exec} {pop pop} ifelse } def /discard { save /discardSave exch store discardDict begin /endString exch store gt38? { 2 add } if load stopped pop end discardSave restore } bind def userdict /discardDict 7 dict dup begin put /pre38Initialize { /endStringLength endString length store /newBuff buffer 0 endStringLength getinterval store /newBuffButFirst newBuff 1 endStringLength 1 sub getinterval store /newBuffLast newBuff endStringLength 1 sub 1 getinterval store } def /shiftBuffer { newBuff 0 newBuffButFirst putinterval newBuffLast 0 currentfile read not { stop } if put } def 0 { pre38Initialize mark currentfile newBuff readstring exch pop { { newBuff endString eq { cleartomark stop } if shiftBuffer } loop } { stop } ifelse } def 1 { pre38Initialize /beginString exch store mark currentfile newBuff readstring exch pop { { newBuff beginString eq { /layerCount dup load 1 add store } { newBuff endString eq { /layerCount dup load 1 sub store layerCount 0 eq { cleartomark stop } if } if } ifelse shiftBuffer } loop } { stop } ifelse } def 2 { mark { currentfile buffer readline not { stop } if endString eq { cleartomark stop } if } loop } def 3 { /beginString exch store /layerCnt 1 store mark { currentfile buffer readline not { stop } if dup beginString eq { pop /layerCnt dup load 1 add store } { endString eq { layerCnt 1 eq { cleartomark stop } { /layerCnt dup load 1 sub store } ifelse } if } ifelse } loop } def end userdict /clipRenderOff 15 dict dup begin put { /n /N /s /S /f /F /b /B } { { _doClip 1 eq { /_doClip 0 ddef clip } if newpath } def } forall /Tr /pop load def /Bb {} def /BB /pop load def /Bg {12 npop} def /Bm {6 npop} def /Bc /Bm load def /Bh {4 npop} def end /Lb { 4 npop 6 1 roll pop 4 1 roll pop pop pop 0 eq { 0 eq { (%AI5_BeginLayer) 1 (%AI5_EndLayer--) discard } { /clipForward? true def /Tx /pop load def /Tj /pop load def currentdict end clipRenderOff begin begin } ifelse } { 0 eq { save /discardSave exch store } if } ifelse } bind def /LB { discardSave dup null ne { restore } { pop clipForward? { currentdict end end begin /clipForward? false ddef } if } ifelse } bind def /Pb { pop pop 0 (%AI5_EndPalette) discard } bind def /Np { 0 (%AI5_End_NonPrinting--) discard } bind def /Ln /pop load def /Ap /pop load def /Ar { 72 exch div 0 dtransform dup mul exch dup mul add sqrt dup 1 lt { pop 1 } if setflat } def /Mb { q } def /Md { } def /MB { Q } def /nc 3 dict def nc begin /setgray { pop } bind def /setcmykcolor { 4 npop } bind def /setcustomcolor { 2 npop } bind def currentdict readonly pop end currentdict readonly pop end setpacking %%EndResource %%EndProlog %%BeginSetup %%IncludeFont: DorchesterScriptMT %%IncludeFont: Helvetica %%IncludeFont: Symbol %%IncludeFont: ZapfDingbats Adobe_level2_AI5 /initialize get exec Adobe_IllustratorA_AI5_vars Adobe_IllustratorA_AI5 Adobe_typography_AI5 /initialize get exec Adobe_IllustratorA_AI5 /initialize get exec [ 39/quotesingle 96/grave 128/Adieresis/Aring/Ccedilla/Eacute/Ntilde/Odieresis /Udieresis/aacute/agrave/acircumflex/adieresis/atilde/aring/ccedilla/eacute /egrave/ecircumflex/edieresis/iacute/igrave/icircumflex/idieresis/ntilde /oacute/ograve/ocircumflex/odieresis/otilde/uacute/ugrave/ucircumflex /udieresis/dagger/degree/cent/sterling/section/bullet/paragraph/germandbls /registered/copyright/trademark/acute/dieresis/.notdef/AE/Oslash /.notdef/plusminus/.notdef/.notdef/yen/mu/.notdef/.notdef /.notdef/.notdef/.notdef/ordfeminine/ordmasculine/.notdef/ae/oslash /questiondown/exclamdown/logicalnot/.notdef/florin/.notdef/.notdef /guillemotleft/guillemotright/ellipsis/.notdef/Agrave/Atilde/Otilde/OE/oe /endash/emdash/quotedblleft/quotedblright/quoteleft/quoteright/divide /.notdef/ydieresis/Ydieresis/fraction/currency/guilsinglleft/guilsinglright /fi/fl/daggerdbl/periodcentered/quotesinglbase/quotedblbase/perthousand /Acircumflex/Ecircumflex/Aacute/Edieresis/Egrave/Iacute/Icircumflex /Idieresis/Igrave/Oacute/Ocircumflex/.notdef/Ograve/Uacute/Ucircumflex /Ugrave/dotlessi/circumflex/tilde/macron/breve/dotaccent/ring/cedilla /hungarumlaut/ogonek/caron TE %AI3_BeginEncoding: _DorchesterScriptMT DorchesterScriptMT [39/quotesingle 96/grave 128/Adieresis/Aring/Ccedilla/Eacute/Ntilde/Odieresis /Udieresis/aacute/agrave/acircumflex/adieresis/atilde/aring/ccedilla/eacute /egrave/ecircumflex/edieresis/iacute/igrave/icircumflex/idieresis/ntilde /oacute/ograve/ocircumflex/odieresis/otilde/uacute/ugrave/ucircumflex /udieresis/dagger/degree/cent/sterling/section/bullet/paragraph/germandbls /registered/copyright/trademark/acute/dieresis/.notdef/AE/Oslash /.notdef/plusminus/.notdef/.notdef/yen/mu/.notdef/.notdef /.notdef/.notdef/.notdef/ordfeminine/ordmasculine/.notdef/ae/oslash /questiondown/exclamdown/logicalnot/.notdef/florin/.notdef/.notdef /guillemotleft/guillemotright/ellipsis/.notdef/Agrave/Atilde/Otilde/OE/oe /endash/emdash/quotedblleft/quotedblright/quoteleft/quoteright/divide /.notdef/ydieresis/Ydieresis/fraction/currency/guilsinglleft/guilsinglright /fi/fl/daggerdbl/periodcentered/quotesinglbase/quotedblbase/perthousand /Acircumflex/Ecircumflex/Aacute/Edieresis/Egrave/Iacute/Icircumflex /Idieresis/Igrave/Oacute/Ocircumflex/.notdef/Ograve/Uacute/Ucircumflex /Ugrave/dotlessi/circumflex/tilde/macron/breve/dotaccent/ring/cedilla /hungarumlaut/ogonek/caron /_DorchesterScriptMT/DorchesterScriptMT 0 0 1 TZ %AI3_EndEncoding AdobeType %AI3_BeginEncoding: _Helvetica Helvetica [39/quotesingle 96/grave 128/Adieresis/Aring/Ccedilla/Eacute/Ntilde/Odieresis /Udieresis/aacute/agrave/acircumflex/adieresis/atilde/aring/ccedilla/eacute /egrave/ecircumflex/edieresis/iacute/igrave/icircumflex/idieresis/ntilde /oacute/ograve/ocircumflex/odieresis/otilde/uacute/ugrave/ucircumflex /udieresis/dagger/degree/cent/sterling/section/bullet/paragraph/germandbls /registered/copyright/trademark/acute/dieresis/.notdef/AE/Oslash /.notdef/plusminus/.notdef/.notdef/yen/mu/.notdef/.notdef /.notdef/.notdef/.notdef/ordfeminine/ordmasculine/.notdef/ae/oslash /questiondown/exclamdown/logicalnot/.notdef/florin/.notdef/.notdef /guillemotleft/guillemotright/ellipsis/.notdef/Agrave/Atilde/Otilde/OE/oe /endash/emdash/quotedblleft/quotedblright/quoteleft/quoteright/divide /.notdef/ydieresis/Ydieresis/fraction/currency/guilsinglleft/guilsinglright /fi/fl/daggerdbl/periodcentered/quotesinglbase/quotedblbase/perthousand /Acircumflex/Ecircumflex/Aacute/Edieresis/Egrave/Iacute/Icircumflex 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/terminate get exec Adobe_level2_AI5 /terminate get exec %%EOF %!PS-Adobe-3.0 EPSF-3.0 %%Creator: Adobe Illustrator(TM) 5.5 %%For: () () %%Title: (fig3.eps) %%CreationDate: (11/5/95) ( 6:16 PM) %%BoundingBox: 178 504 415 735 %%HiResBoundingBox: 178.4354 504.3591 414.8647 734.6801 %%DocumentProcessColors: Cyan Magenta Yellow Black %%DocumentSuppliedResources: procset Adobe_level2_AI5 1.0 0 %%+ procset Adobe_IllustratorA_AI5 1.0 0 %AI5_FileFormat 3.0 %AI3_ColorUsage: Color %AI3_TemplateBox: 306 396 306 396 %AI3_TileBox: 30 31 582 761 %AI3_DocumentPreview: Macintosh_ColorPic %AI5_ArtSize: 612 792 %AI5_RulerUnits: 2 %AI5_ArtFlags: 1 0 0 1 0 0 1 1 0 %AI5_TargetResolution: 800 %AI5_NumLayers: 1 %AI5_OpenToView: -30 804 1 685 809 18 0 1 394 52 %AI5_OpenViewLayers: 7 %%EndComments %%BeginProlog %%BeginResource: procset Adobe_level2_AI5 1.0 0 %%Title: (Adobe Illustrator (R) Version 5.0 Level 2 Emulation) %%Version: 1.0 %%CreationDate: (04/10/93) () %%Copyright: ((C) 1987-1993 Adobe Systems Incorporated All Rights Reserved) userdict /Adobe_level2_AI5 21 dict dup begin put /packedarray where not { userdict begin /packedarray { array astore readonly } bind def /setpacking /pop load def /currentpacking false def end 0 } if pop userdict /defaultpacking currentpacking put true setpacking /initialize { Adobe_level2_AI5 begin } bind def /terminate { currentdict Adobe_level2_AI5 eq { end } if } bind def mark /setcustomcolor where not { /findcmykcustomcolor { 5 packedarray } bind def /setcustomcolor { exch aload pop pop 4 { 4 index mul 4 1 roll } repeat 5 -1 roll pop setcmykcolor } def } if /gt38? mark {version cvx exec} stopped {cleartomark true} {38 gt exch pop} ifelse def userdict /deviceDPI 72 0 matrix defaultmatrix dtransform dup mul exch dup mul add sqrt put userdict /level2? systemdict /languagelevel known dup { pop systemdict /languagelevel get 2 ge } if put level2? not { /setcmykcolor where not { /setcmykcolor { exch .11 mul add exch .59 mul add exch .3 mul add 1 exch sub setgray } def } if /currentcmykcolor where not { /currentcmykcolor { 0 0 0 1 currentgray sub } def } if /setoverprint where not { /setoverprint /pop load def } if /selectfont where not { /selectfont { exch findfont exch dup type /arraytype eq { makefont } { scalefont } ifelse setfont } bind def } if /cshow where not { /cshow { [ 0 0 5 -1 roll aload pop ] cvx bind forall } bind def } if } if cleartomark /anyColor? { add add add 0 ne } bind def /testColor { gsave setcmykcolor currentcmykcolor grestore } bind def /testCMYKColorThrough { testColor anyColor? } bind def userdict /composite? level2? { gsave 1 1 1 1 setcmykcolor currentcmykcolor grestore add add add 4 eq } { 1 0 0 0 testCMYKColorThrough 0 1 0 0 testCMYKColorThrough 0 0 1 0 testCMYKColorThrough 0 0 0 1 testCMYKColorThrough and and and } ifelse put composite? not { userdict begin gsave /cyan? 1 0 0 0 testCMYKColorThrough def /magenta? 0 1 0 0 testCMYKColorThrough def /yellow? 0 0 1 0 testCMYKColorThrough def /black? 0 0 0 1 testCMYKColorThrough def grestore /isCMYKSep? cyan? magenta? yellow? black? or or or def /customColor? isCMYKSep? not def end } if end defaultpacking setpacking %%EndResource %%BeginResource: procset Adobe_IllustratorA_AI5 1.1 0 %%Title: (Adobe Illustrator (R) Version 5.0 Abbreviated Prolog) %%Version: 1.1 %%CreationDate: (3/7/1994) () %%Copyright: ((C) 1987-1994 Adobe Systems Incorporated All Rights Reserved) currentpacking true setpacking userdict /Adobe_IllustratorA_AI5_vars 70 dict dup begin put /_lp /none def /_pf { } def /_ps { } def /_psf { } def /_pss { } def /_pjsf { } def /_pjss { } def /_pola 0 def /_doClip 0 def /cf currentflat def /_tm matrix def /_renderStart [ /e0 /r0 /a0 /o0 /e1 /r1 /a1 /i0 ] def /_renderEnd [ null null null null /i1 /i1 /i1 /i1 ] def /_render -1 def /_rise 0 def /_ax 0 def /_ay 0 def /_cx 0 def /_cy 0 def /_leading [ 0 0 ] def /_ctm matrix def /_mtx matrix def /_sp 16#020 def /_hyphen (-) def /_fScl 0 def /_cnt 0 def /_hs 1 def /_nativeEncoding 0 def /_useNativeEncoding 0 def /_tempEncode 0 def /_pntr 0 def /_tDict 2 dict def /_wv 0 def /Tx { } def /Tj { } def /CRender { } def /_AI3_savepage { } def /_gf null def /_cf 4 array def /_if null def /_of false def /_fc { } def /_gs null def /_cs 4 array def /_is null def /_os false def /_sc { } def /discardSave null def /buffer 256 string def /beginString null def /endString null def /endStringLength null def /layerCnt 1 def /layerCount 1 def /perCent (%) 0 get def /perCentSeen? false def /newBuff null def /newBuffButFirst null def /newBuffLast null def /clipForward? false def end userdict /Adobe_IllustratorA_AI5 74 dict dup begin put /initialize { Adobe_IllustratorA_AI5 dup begin Adobe_IllustratorA_AI5_vars begin discardDict { bind pop pop } forall dup /nc get begin { dup xcheck 1 index type /operatortype ne and { bind } if pop pop } forall end newpath } def /terminate { end end } def /_ null def /ddef { Adobe_IllustratorA_AI5_vars 3 1 roll put } def /xput { dup load dup length exch maxlength eq { dup dup load dup length 2 mul dict copy def } if load begin def end } def /npop { { pop } repeat } def /sw { dup length exch stringwidth exch 5 -1 roll 3 index mul add 4 1 roll 3 1 roll mul add } def /swj { dup 4 1 roll dup length exch stringwidth exch 5 -1 roll 3 index mul add 4 1 roll 3 1 roll mul add 6 2 roll /_cnt 0 ddef { 1 index eq { /_cnt _cnt 1 add ddef } if } forall pop exch _cnt mul exch _cnt mul 2 index add 4 1 roll 2 index add 4 1 roll pop pop } def /ss { 4 1 roll { 2 npop (0) exch 2 copy 0 exch put pop gsave false charpath currentpoint 4 index setmatrix stroke grestore moveto 2 copy rmoveto } exch cshow 3 npop } def /jss { 4 1 roll { 2 npop (0) exch 2 copy 0 exch put gsave _sp eq { exch 6 index 6 index 6 index 5 -1 roll widthshow currentpoint } { false charpath currentpoint 4 index setmatrix stroke } ifelse grestore moveto 2 copy rmoveto } exch cshow 6 npop } def /sp { { 2 npop (0) exch 2 copy 0 exch put pop false charpath 2 copy rmoveto } exch cshow 2 npop } def /jsp { { 2 npop (0) exch 2 copy 0 exch put _sp eq { exch 5 index 5 index 5 index 5 -1 roll widthshow } { false charpath } ifelse 2 copy rmoveto } exch cshow 5 npop } def /pl { transform 0.25 sub round 0.25 add exch 0.25 sub round 0.25 add exch itransform } def /setstrokeadjust where { pop true setstrokeadjust /c { curveto } def /C /c load def /v { currentpoint 6 2 roll curveto } def /V /v load def /y { 2 copy curveto } def /Y /y load def /l { lineto } def /L /l load def /m { moveto } def } { /c { pl curveto } def /C /c load def /v { currentpoint 6 2 roll pl curveto } def /V /v load def /y { pl 2 copy curveto } def /Y /y load def /l { pl lineto } def /L /l load def /m { pl moveto } def } ifelse /d { setdash } def /cf { } def /i { dup 0 eq { pop cf } if setflat } def /j { setlinejoin } def /J { setlinecap } def /M { setmiterlimit } def /w { setlinewidth } def /H { } def /h { closepath } def /N { _pola 0 eq { _doClip 1 eq { clip /_doClip 0 ddef } if newpath } { /CRender { N } ddef } ifelse } def /n { N } def /F { _pola 0 eq { _doClip 1 eq { gsave _pf grestore clip newpath /_lp /none ddef _fc /_doClip 0 ddef } { _pf } ifelse } { /CRender { F } ddef } ifelse } def /f { closepath F } def /S { _pola 0 eq { _doClip 1 eq { gsave _ps grestore clip newpath /_lp /none ddef _sc /_doClip 0 ddef } { _ps } ifelse } { /CRender { S } ddef } ifelse } def /s { closepath S } def /B { _pola 0 eq { _doClip 1 eq gsave F grestore { gsave S grestore clip newpath /_lp /none ddef _sc /_doClip 0 ddef } { S } ifelse } { /CRender { B } ddef } ifelse } def /b { closepath B } def /W { /_doClip 1 ddef } def /* { count 0 ne { dup type /stringtype eq { pop } if } if newpath } def /u { } def /U { } def /q { _pola 0 eq { gsave } if } def /Q { _pola 0 eq { grestore } if } def /*u { _pola 1 add /_pola exch ddef } def /*U { _pola 1 sub /_pola exch ddef _pola 0 eq { CRender } if } def /D { pop } def /*w { } def /*W { } def /` { /_i save ddef clipForward? { nulldevice } if 6 1 roll 4 npop concat pop userdict begin /showpage { } def 0 setgray 0 setlinecap 1 setlinewidth 0 setlinejoin 10 setmiterlimit [] 0 setdash /setstrokeadjust where {pop false setstrokeadjust} if newpath 0 setgray false setoverprint } def /~ { end _i restore } def /O { 0 ne /_of exch ddef /_lp /none ddef } def /R { 0 ne /_os exch ddef /_lp /none ddef } def /g { /_gf exch ddef /_fc { _lp /fill ne { _of setoverprint _gf setgray /_lp /fill ddef } if } ddef /_pf { _fc fill } ddef /_psf { _fc ashow } ddef /_pjsf { _fc awidthshow } ddef /_lp /none ddef } def /G { /_gs exch ddef /_sc { _lp /stroke ne { _os setoverprint _gs setgray /_lp /stroke ddef } if } ddef /_ps { _sc stroke } ddef /_pss { _sc ss } ddef /_pjss { _sc jss } ddef /_lp /none ddef } def /k { _cf astore pop /_fc { _lp /fill ne { _of setoverprint _cf aload pop setcmykcolor /_lp /fill ddef } if } ddef /_pf { _fc fill } ddef /_psf { _fc ashow } ddef /_pjsf { _fc awidthshow } ddef /_lp /none ddef } def /K { _cs astore pop /_sc { _lp /stroke ne { _os setoverprint _cs aload pop setcmykcolor /_lp /stroke ddef } if } ddef /_ps { _sc stroke } ddef /_pss { _sc ss } ddef /_pjss { _sc jss } ddef /_lp /none ddef } def /x { /_gf exch ddef findcmykcustomcolor /_if exch ddef /_fc { _lp /fill ne { _of setoverprint _if _gf 1 exch sub setcustomcolor /_lp /fill ddef } if } ddef /_pf { _fc fill } ddef /_psf { _fc ashow } ddef /_pjsf { _fc awidthshow } ddef /_lp /none ddef } def /X { /_gs exch ddef findcmykcustomcolor /_is exch ddef /_sc { _lp /stroke ne { _os setoverprint _is _gs 1 exch sub setcustomcolor /_lp /stroke ddef } if } ddef /_ps { _sc stroke } ddef /_pss { _sc ss } ddef /_pjss { _sc jss } ddef /_lp /none ddef } def /A { pop } def /annotatepage { userdict /annotatepage 2 copy known {get exec} {pop pop} ifelse } def /discard { save /discardSave exch store discardDict begin /endString exch store gt38? { 2 add } if load stopped pop end discardSave restore } bind def userdict /discardDict 7 dict dup begin put /pre38Initialize { /endStringLength endString length store /newBuff buffer 0 endStringLength getinterval store /newBuffButFirst newBuff 1 endStringLength 1 sub getinterval store /newBuffLast newBuff endStringLength 1 sub 1 getinterval store } def /shiftBuffer { newBuff 0 newBuffButFirst putinterval newBuffLast 0 currentfile read not { stop } if put } def 0 { pre38Initialize mark currentfile newBuff readstring exch pop { { newBuff endString eq { cleartomark stop } if shiftBuffer } loop } { stop } ifelse } def 1 { pre38Initialize /beginString exch store mark currentfile newBuff readstring exch pop { { newBuff beginString eq { /layerCount dup load 1 add store } { newBuff endString eq { /layerCount dup load 1 sub store layerCount 0 eq { cleartomark stop } if } if } ifelse shiftBuffer } loop } { stop } ifelse } def 2 { mark { currentfile buffer readline not { stop } if endString eq { cleartomark stop } if } loop } def 3 { /beginString exch store /layerCnt 1 store mark { currentfile buffer readline not { stop } if dup beginString eq { pop /layerCnt dup load 1 add store } { endString eq { layerCnt 1 eq { cleartomark stop } { /layerCnt dup load 1 sub store } ifelse } if } ifelse } loop } def end userdict /clipRenderOff 15 dict dup begin put { /n /N /s /S /f /F /b /B } { { _doClip 1 eq { /_doClip 0 ddef clip } if newpath } def } forall /Tr /pop load def /Bb {} def /BB /pop load def /Bg {12 npop} def /Bm {6 npop} def /Bc /Bm load def /Bh {4 npop} def end /Lb { 4 npop 6 1 roll pop 4 1 roll pop pop pop 0 eq { 0 eq { (%AI5_BeginLayer) 1 (%AI5_EndLayer--) discard } { /clipForward? true def /Tx /pop load def /Tj /pop load def currentdict end clipRenderOff begin begin } ifelse } { 0 eq { save /discardSave exch store } if } ifelse } bind def /LB { discardSave dup null ne { restore } { pop clipForward? { currentdict end end begin /clipForward? false ddef } if } ifelse } bind def /Pb { pop pop 0 (%AI5_EndPalette) discard } bind def /Np { 0 (%AI5_End_NonPrinting--) discard } bind def /Ln /pop load def /Ap /pop load def /Ar { 72 exch div 0 dtransform dup mul exch dup mul add sqrt dup 1 lt { pop 1 } if setflat } def /Mb { q } def /Md { } def /MB { Q } def /nc 3 dict def nc begin /setgray { pop } bind def /setcmykcolor { 4 npop } bind def /setcustomcolor { 2 npop } bind def currentdict readonly pop end currentdict readonly pop end setpacking %%EndResource %%EndProlog %%BeginSetup Adobe_level2_AI5 /initialize get exec Adobe_IllustratorA_AI5 /initialize get exec %AI5_Begin_NonPrinting Np %AI3_BeginPattern: (Yellow Stripe) (Yellow Stripe) 8.4499 4.6 80.4499 76.6 [ %AI3_Tile (0 O 0 R 0 0.4 1 0 k 0 0.4 1 0 K) @ ( 800 Ar 0 J 0 j 3.6 w 4 M []0 d %AI3_Note: 0 D 8.1999 8.1999 m 80.6999 8.1999 L S 8.1999 22.6 m 80.6999 22.6 L S 8.1999 37.0001 m 80.6999 37.0001 L S 8.1999 51.3999 m 80.6999 51.3999 L S 8.1999 65.8 m 80.6999 65.8 L S 8.1999 15.3999 m 80.6999 15.3999 L S 8.1999 29.8 m 80.6999 29.8 L S 8.1999 44.1999 m 80.6999 44.1999 L S 8.1999 58.6 m 80.6999 58.6 L S 8.1999 73.0001 m 80.6999 73.0001 L S ) & ] E %AI3_EndPattern %AI5_End_NonPrinting-- %AI5_Begin_NonPrinting Np 3 Bn %AI5_BeginGradient: (Black & White) (Black & White) 0 2 Bd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r [ 0 0 50 100 %_Bs 1 0 50 0 %_Bs BD %AI5_EndGradient %AI5_BeginGradient: (Red & Yellow) (Red & Yellow) 0 2 Bd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r [ 0 1 0.6 0 1 50 100 %_Bs 0 0 1 0 1 50 0 %_Bs BD %AI5_EndGradient %AI5_BeginGradient: (Yellow & Blue Radial) (Yellow & Blue Radial) 1 2 Bd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0.75 0 0 k Pc 1 1 0 0 k Pc Bb 2 (Red & Yellow) -4014 4716 0 0 1 0 0 1 0 0 Bg 0 BB Pc 0 0.25 0 0 k Pc 0 0.5 0 0 k Pc 0 0.75 0 0 k Pc 0 1 0 0 k Pc 0 0.25 0.25 0 k Pc 0 0.5 0.5 0 k Pc 0 0.75 0.75 0 k Pc 0 1 1 0 k Pc Bb 0 0 0 0 Bh 2 (Yellow & Blue Radial) -4014 4716 0 0 1 0 0 1 0 0 Bg 0 BB Pc 0 0 0.25 0 k Pc 0 0 0.5 0 k Pc 0 0 0.75 0 k Pc 0 0 1 0 k Pc 0.25 0 0.25 0 k Pc 0.5 0 0.5 0 k Pc 0.75 0 0.75 0 k Pc 1 0 1 0 k Pc (Yellow Stripe) 0 0 1 1 0 0 0 0 0 [1 0 0 1 0 0] p Pc 0.25 0.125 0 0 k Pc 0.5 0.25 0 0 k Pc 0.75 0.375 0 0 k Pc 1 0.5 0 0 k Pc 0.125 0.25 0 0 k Pc 0.25 0.5 0 0 k Pc 0.375 0.75 0 0 k Pc 0.5 1 0 0 k Pc 0 0 0 0 k Pc 0 0.25 0.125 0 k Pc 0 0.5 0.25 0 k Pc 0 0.75 0.375 0 k Pc 0 1 0.5 0 k Pc 0 0.125 0.25 0 k Pc 0 0.25 0.5 0 k Pc 0 0.375 0.75 0 k Pc 0 0.5 1 0 k Pc 0 0 0 0 k Pc 0.125 0 0.25 0 k Pc 0.25 0 0.5 0 k Pc 0.375 0 0.75 0 k Pc 0.5 0 1 0 k Pc 0.25 0 0.125 0 k Pc 0.5 0 0.25 0 k Pc 0.75 0 0.375 0 k Pc 1 0 0.5 0 k Pc 0 0 0 0 k Pc 0.25 0.125 0.125 0 k Pc 0.5 0.25 0.25 0 k Pc 0.75 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209.1094 609.3594 c 209.1094 609.3594 l 207.25 611.0156 204.4062 610.8437 202.7656 608.9844 c 201.1094 607.125 201.2812 604.2812 203.1406 602.6406 c 203.1406 602.6406 l b 208.25 598 m 208.25 595.5156 210.2656 593.5 212.75 593.5 c 215.2344 593.5 217.25 595.5156 217.25 598 c 217.25 598 l 217.25 600.4844 215.2344 602.5 212.75 602.5 c 210.2656 602.5 208.25 600.4844 208.25 598 c 208.25 598 l b 181 621.75 m 181 619.2656 183.0156 617.25 185.5 617.25 c 187.9844 617.25 190 619.2656 190 621.75 c 190 621.75 l 190 624.2344 187.9844 626.25 185.5 626.25 c 183.0156 626.25 181 624.2344 181 621.75 c 181 621.75 l b 0 0.5 0.5 0 k 0 1 1 0 K 212.5974 651.2379 m B 212.5974 651.2379 m B 289.0724 614.4995 m B 262.0655 619.5947 m B 0.5 G 332.9252 530.5645 m S 0 G 332.9252 530.5645 m S 0 O 0 0.5 0.5 0 k 0 1 1 0 K 263.0663 623.5068 m B 287.0662 733.5068 m B 212.5974 651.2379 m 213.7329 532.8401 l 406.3996 734.1735 l 287.0662 733.5068 l 213.0662 653.5068 l B 1 g 0 G 343.3321 504.7033 m B 0 0.5 0.5 0 k 0 0.75 0.75 0 K 289.3371 734.2568 m 216.0037 533.5901 l B 0 1 1 0 K 215.0662 652.8401 m 406.3996 734.1735 l B 0.5 0 0.5 0 k 1 0 1 0 K 298.4043 507.7943 m B 1 0 1 0 k 0.25 0 0.25 0 K 333.8067 534.9568 m 215.4157 533.402 l 412.3162 730.5901 l 414.3611 611.2744 l 336.0643 535.4771 l B 413.8093 611.8117 m 214.8639 533.9393 l B 0.5 0 0.5 0 k 334.1024 535.7114 m 411.0662 728.8401 l B 0 0 0.25 0 k 0 0.5 0.5 0 K 182.2746 625.5276 m 374.9412 706.1943 l 298.9412 504.861 l 178.9412 505.5276 l 180.3218 624.1526 l 298.9412 504.861 l B 0 0.25 0.25 0 K 373.6079 703.5276 m 178.9412 505.5276 l B 0.75 0 0 0 k 0.75 0.75 0 0 K 202.2329 636.2776 m 320.8941 517.6459 l 400.9774 720.2985 l 202.8941 638.3126 l 324.2274 636.9792 l 320.8941 517.6459 l 400.9774 720.2985 l B 400.2329 719.0761 m 323.4829 635.7568 l B 0 0 0.25 0 k 0 0 0.25 0 K 195.25 623.25 m 195.25 620.7656 197.2656 618.75 199.75 618.75 c 202.2344 618.75 204.25 620.7656 204.25 623.25 c 204.25 623.25 l 204.25 625.7344 202.2344 627.75 199.75 627.75 c 197.2656 627.75 195.25 625.7344 195.25 623.25 c 195.25 623.25 l b 207.75 600 m 207.75 597.5156 209.7656 595.5 212.25 595.5 c 214.7344 595.5 216.75 597.5156 216.75 600 c 216.75 600 l 216.75 602.4844 214.7344 604.5 212.25 604.5 c 209.7656 604.5 207.75 602.4844 207.75 600 c 207.75 600 l b LB %AI5_EndLayer-- %%PageTrailer gsave annotatepage grestore showpage %%Trailer Adobe_IllustratorA_AI5 /terminate get exec Adobe_level2_AI5 /terminate get exec %%EOF