Content-Type: multipart/mixed; boundary="-------------0312161327389" This is a multi-part message in MIME format. ---------------0312161327389 Content-Type: text/plain; name="03-545.comments" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="03-545.comments" AMS-Code: 37E20, 37E35, 37E45, 70H08 PACS-Code: 05.45.-a, 45.20.Jj E-mail: gaidash@math.toronto.edu koch@math.utexas.edu WWW: http://www.math.toronto.edu/gaidash/ http://www.ma.utexas.edu/users/koch/ For additional data and possible updates, see ftp://ftp.ma.utexas.edu/pub/papers/koch/shearless/ ---------------0312161327389 Content-Type: text/plain; name="03-545.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="03-545.keywords" hamiltonian flow, quasi periodic, irrational, golden mean, invariant torus, critical, shearless, non-twist, breakup, renormalization, universal ---------------0312161327389 Content-Type: application/x-tex; name="shearless6.tex" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="shearless6.tex" %PARAM.2 \magnification=\magstep1 \def\firstpage{1} \pageno=\firstpage %fonts.5 \font\fiverm=cmr5 \font\sevenrm=cmr7 \font\sevenbf=cmbx7 \font\eightrm=cmr8 \font\eightbf=cmbx8 \font\ninerm=cmr9 \font\ninebf=cmbx9 \font\tenbf=cmbx10 \font\magtenbf=cmbx10 scaled\magstep1 % % from amssym.def \font\tenmsb=msbm10 \font\sevenmsb=msbm7 \font\fivemsb=msbm5 \newfam\msbfam \textfont\msbfam=\tenmsb \scriptfont\msbfam=\sevenmsb \scriptscriptfont\msbfam=\fivemsb \def\Bbb#1{{\fam\msbfam\relax#1}} %titles.5 % requires fonts.5 or higher \count5=0 \count6=1 \count7=1 \count8=1 \count9=1 \def\proof{\medskip\noindent{\bf Proof.\ }} \def\qed{\hfill{\sevenbf QED}\par\medskip} \def\references{\bigskip\noindent\hbox{\bf References}\medskip} \def\remark{\medskip\noindent{\bf Remark.\ }} \def\nextremark{\smallskip\noindent$\circ$\hskip1.5em} \def\firstremark{\bigskip\noindent{\bf Remarks.}\nextremark} \def\abstract#1\par{{\baselineskip=10pt \eightrm\narrower\noindent{\eightbf Abstract.} #1\par}} \def\equ(#1){\hskip-0.03em\csname e#1\endcsname} \def\clm(#1){\csname c#1\endcsname} \def\equation(#1){\eqno\tag(#1)} %\def\equation(#1){\eqno\tag(#1) {\rm #1}} \def\tag(#1){(\number\count5. \number\count6) \expandafter\xdef\csname e#1\endcsname{ (\number\count5.\number\count6)} \global\advance\count6 by 1} \def\claim #1(#2) #3\par{ \vskip.1in\medbreak\noindent {\bf #1\ \number\count5.\number\count7.\ }{\sl #3}\par \expandafter\xdef\csname c#2\endcsname{#1~\number\count5.\number\count7} \global\advance\count7 by 1 \ifdim\lastskip<\medskipamount \removelastskip\penalty55\medskip\fi} \def\section#1\par{\vskip0pt plus.3\vsize\penalty-75 \vskip0pt plus -.3\vsize\bigskip\bigskip \global\advance\count5 by 1 \message{#1}\leftline {\magtenbf \number\count5.\ #1} \count6=1 \count7=1 \count8=1 \nobreak\smallskip\noindent} \def\subsection#1\par{\vskip0pt plus.2\vsize\penalty-75 \vskip0pt plus 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\def\id{{\rm I}} \def\modulo{{\rm mod~}} \def\std{{\rm std}} \def\Re{{\rm Re}} \def\Im{{\rm Im}} \def\defeq{\mathrel{\mathop=^{\rm def}}} % \def\mapright#1{\smash{\mathop{\longrightarrow}\limits^{#1}}} % \def\half{{1\over 2}} \def\third{{1\over 3}} \def\quarter{{1\over 4}} % \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\GG{{\cal G}} \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\WW{{\cal W}} \def\XX{{\cal X}} \def\YY{{\cal Y}} \def\ZZ{{\cal Z}} %smallfonts.tex % \newskip\ttglue % \font\fiverm=cmr5 \font\fivei=cmmi5 \font\fivesy=cmsy5 \font\fivebf=cmbx5 \font\sixrm=cmr6 \font\sixi=cmmi6 \font\sixsy=cmsy6 \font\sixbf=cmbx6 \font\eightrm=cmr8 \font\eighti=cmmi8 \font\eightsy=cmsy8 \font\eightit=cmti8 \font\eightsl=cmsl8 \font\eighttt=cmtt8 \font\eightbf=cmbx8 \font\ninerm=cmr9 \font\ninei=cmmi9 \font\ninesy=cmsy9 \font\nineit=cmti9 \font\ninesl=cmsl9 \font\ninett=cmtt9 \font\ninebf=cmbx9 %% EIGHT POINT FONT FAMILY \def\eightpoint{\def\rm{\fam0\eightrm} \textfont0=\eightrm \scriptfont0=\sixrm \scriptscriptfont0=\fiverm \textfont1=\eighti \scriptfont1=\sixi \scriptscriptfont1=\fivei \textfont2=\eightsy \scriptfont2=\sixsy \scriptscriptfont2=\fivesy \textfont3=\tenex \scriptfont3=\tenex \scriptscriptfont3=\tenex \textfont\itfam=\eightit \def\it{\fam\itfam\eightit} \textfont\slfam=\eightsl \def\sl{\fam\slfam\eightsl} \textfont\ttfam=\eighttt \def\tt{\fam\ttfam\eighttt} \textfont\bffam=\eightbf \scriptfont\bffam=\sixbf \scriptscriptfont\bffam=\fivebf \def\bf{\fam\bffam\eightbf} \tt \ttglue=.5em plus.25em minus.15em \normalbaselineskip=9pt \setbox\strutbox=\hbox{\vrule height7pt depth2pt width0pt} \let\sc=\sixrm \let\big=\eightbig \normalbaselines\rm} \def\eightbig#1{{\hbox{$\textfont0=\ninerm\textfont2=\ninesy \left#1\vbox to6.5pt{}\right.$}}} %% NINE POINT FONT FAMILY \def\ninepoint{\def\rm{\fam0\ninerm} \textfont0=\ninerm \scriptfont0=\sixrm \scriptscriptfont0=\fiverm \textfont1=\ninei \scriptfont1=\sixi \scriptscriptfont1=\fivei \textfont2=\ninesy \scriptfont2=\sixsy \scriptscriptfont2=\fivesy \textfont3=\tenex \scriptfont3=\tenex \scriptscriptfont3=\tenex \textfont\itfam=\nineit \def\it{\fam\itfam\nineit} \textfont\slfam=\ninesl \def\sl{\fam\slfam\ninesl} \textfont\ttfam=\ninett \def\tt{\fam\ttfam\ninett} \textfont\bffam=\ninebf \scriptfont\bffam=\sixbf \scriptscriptfont\bffam=\fivebf \def\bf{\fam\bffam\ninebf} \tt \ttglue=.5em plus.25em minus.15em \normalbaselineskip=11pt \setbox\strutbox=\hbox{\vrule height8pt depth3pt width0pt} \let\sc=\sevenrm \let\big=\ninebig \normalbaselines\rm} \def\ninebig#1{{\hbox{$\textfont0=\tenrm\textfont2=\tensy \left#1\vbox to7.25pt{}\right.$}}} \global\let\figures\relax % % ??/epsf.tex (written by Radical Eye Software and copied below) % defines the macro \epsfbox with one argument, % the encapsulated PostScript file to include. % Invoking it causes a \vbox with the natural size of the drawing % to be inserted at the point of invocation. % Usually figures are meant to be centered and set off, and possibly % to have a title and/or a figure number. The macros below do that. % You should assign values that please you to the variables % \abovefigskip, \belowfigskip, figtitleskip, figtitlefont,... % AFTER \inputting the present file: \input /u2/kbi/tex/epsfig % and BEFORE the first invocation of any of the macros below. % RESET AT YOUR PLEASURE THE VARIABLES AT THE VERY BOTTOM! % % For convenience we make a dimension for figures: \newdimen\FigSize \FigSize=.9\hsize % alter at your convenience % % For a SCALED HORIZONTALLY CENTERED FIGURE use \epsfig. % First argument is the horizontal width of the figure, given % in any way TeX can understand. % The second argument is an encapsulated PostScript file name (filnam.eps). % Note the mandatory semicolons between arguments in this example!: % \epsfig .8\hsize; example.ps; % will put a centered scaled \vbox of width .8\hsize suitably offset % at the point of invocation \newskip\abovefigskip \newskip\belowfigskip \gdef\epsfig#1;#2;{\par\vskip\abovefigskip\penalty -500 {\everypar={}\epsfxsize=#1\noindent \centerline{\epsfbox{#2}}}% \vskip\belowfigskip}% % % SCALED TITLED EPSFIG HORIZONTALLY CENTERED: \tepsfig. % First argument is the horizontal width of the figure, % second an encapsulated PostScript file name, % third a title for the figure. % Note the mandatory semicolons between arguments! % example: \tepsfig5truein; example.ps;{This is a figure} \newskip\figtitleskip \gdef\tepsfig#1;#2;#3{\par\vskip\abovefigskip\penalty -500 {\everypar={}\epsfxsize=#1\noindent \vbox {\centerline{\epsfbox{#2}}\vskip\figtitleskip \centerline{\figtitlefont#3}}}% \vskip\belowfigskip}% % % SCALED NUMBERED TITLED EPSFIG HORIZONTALLY CENTERED: \nepsfig % The figure number is automatically increased for every % invocation of \nepsfig or \nipsfig. \newcount\FigNr \global\FigNr=0 \gdef\nepsfig#1;#2;#3{\global\advance\FigNr by 1 \tepsfig#1;#2;{Figure\space\the\FigNr.\space#3}}% % % % Often you would rather have TeX decide where to put the figure % by using \midinsert. Here are macros that do that % (mnemonics ``ipsfig'' is for ``midInsert PS FIGure'') % % TeX-PLACED SCALED EPSFIG HORIZONTALLY CENTERED: \ipsfig \gdef\ipsfig#1;#2;{%\goodbreak ?? \midinsert{\everypar={}\epsfxsize=#1\noindent \centerline{\epsfbox{#2}}}% \endinsert}% % % TeX-PLACED SCALED TITLED EPSFIG HORIZONTALLY CENTERED: \tipsfig \gdef\tipsfig#1;#2;#3{\midinsert {\everypar={}\epsfxsize=#1\noindent \vbox{\centerline{\epsfbox{#2}}% \vskip\figtitleskip \centerline{\figtitlefont#3}}}\endinsert}% % % TeX-PLACED SCALED NUMBERED TITLED EPSFIG HORIZONTALLY CENTERED: \nipsfig % example: \nipsfigd.9\hsize;example.ps;{This is an example figure} \gdef\nipsfig#1;#2;#3{\global\advance\FigNr by1% \tipsfig#1;#2;{Figure\space\the\FigNr.\space#3}}% % % ================================================================ % old: % \ifx\Input\undefined\let\next\input\else\let\next\Input\fi % \next /usr/local/lib/tex/inputs/epsf.tex %On Suns %% \next /usr/lib/tex/inputs/epsf.tex %On NexTs % % to make sure no new version of epsf.tex corrupts the above, % I copy here that file in toto: % % EPSF.TEX macro file: % Written by Tomas Rokicki of Radical Eye Software, 29 Mar 1989. % Revised by Don Knuth, 3 Jan 1990. % Revised by Tomas Rokicki to accept bounding boxes with no % space after the colon, 18 Jul 1990. % % TeX macros to include an Encapsulated PostScript graphic. % Works by finding the bounding box comment, % calculating the correct scale values, and inserting a vbox % of the appropriate size at the current position in the TeX document. % % To use with the center environment of LaTeX, preface the \epsffile % call with a \leavevmode. (LaTeX should probably supply this itself % for the center environment.) % % To use, simply say % \input epsf % somewhere early on in your TeX file % \epsfbox{filename.ps} % where you want to insert a vbox for a figure % % Alternatively, you can type % % \epsfbox[0 0 30 50]{filename.ps} % to supply your own BB % % which will not read in the file, and will instead use the bounding % box you specify. % % The effect will be to typeset the figure as a TeX box, at the % point of your \epsfbox command. By default, the graphic will have its % `natural' width (namely the width of its bounding box, as described % in filename.ps). The TeX box will have depth zero. % % You can enlarge or reduce the figure by saying % \epsfxsize= \epsfbox{filename.ps} % (or % \epsfysize= \epsfbox{filename.ps}) % instead. Then the width of the TeX box will be \epsfxsize and its % height will be scaled proportionately (or the height will be % \epsfysize and its width will be scaled proportiontally). The % width (and height) is restored to zero after each use. % % A more general facility for sizing is available by defining the % \epsfsize macro. Normally you can redefine this macro % to do almost anything. The first parameter is the natural x size of % the PostScript graphic, the second parameter is the natural y size % of the PostScript graphic. It must return the xsize to use, or 0 if % natural scaling is to be used. Common uses include: % % \epsfxsize % just leave the old value alone % 0pt % use the natural sizes % #1 % use the natural sizes % \hsize % scale to full width % 0.5#1 % scale to 50% of natural size % \ifnum#1>\hsize\hsize\else#1\fi % smaller of natural, hsize % % If you want TeX to report the size of the figure (as a message % on your terminal when it processes each figure), say `\epsfverbosetrue'. % \newread\epsffilein % file to \read \newif\ifepsffileok % continue looking for the bounding box? \newif\ifepsfbbfound % success? \newif\ifepsfverbose % report what you're making? \newdimen\epsfxsize % horizontal size after scaling \newdimen\epsfysize % vertical size after scaling \newdimen\epsftsize % horizontal size before scaling \newdimen\epsfrsize % vertical size before scaling \newdimen\epsftmp % register for arithmetic manipulation \newdimen\pspoints % conversion factor % \pspoints=1bp % Adobe points are `big' \epsfxsize=0pt % Default value, means `use natural size' \epsfysize=0pt % ditto % \def\epsfbox#1{\global\def\epsfllx{72}\global\def\epsflly{72}% \global\def\epsfurx{540}\global\def\epsfury{720}% \def\lbracket{[}\def\testit{#1}\ifx\testit\lbracket \let\next=\epsfgetlitbb\else\let\next=\epsfnormal\fi\next{#1}}% % \def\epsfgetlitbb#1#2 #3 #4 #5]#6{\epsfgrab #2 #3 #4 #5 .\\% \epsfsetgraph{#6}}% % \def\epsfnormal#1{\epsfgetbb{#1}\epsfsetgraph{#1}}% % \def\epsfgetbb#1{% % % The first thing we need to do is to open the % PostScript file, if possible. % \openin\epsffilein=#1 \ifeof\epsffilein\errmessage{I couldn't open #1, will ignore it}\else % % Okay, we got it. Now we'll scan lines until we find one that doesn't % start with %. We're looking for the bounding box comment. % {\epsffileoktrue \chardef\other=12 \def\do##1{\catcode`##1=\other}\dospecials \catcode`\ =10 \loop \read\epsffilein to \epsffileline \ifeof\epsffilein\epsffileokfalse\else % % We check to see if the first character is a % sign; % if not, we stop reading (unless the line was entirely blank); % if so, we look further and stop only if the line begins with % `%%BoundingBox:'. % \expandafter\epsfaux\epsffileline:. \\% \fi \ifepsffileok\repeat \ifepsfbbfound\else \ifepsfverbose\message{No bounding box comment in #1; using defaults}\fi\fi }\closein\epsffilein\fi}% % % Now we have to calculate the scale and offset values to use. % First we compute the natural sizes. % \def\epsfsetgraph#1{% \epsfrsize=\epsfury\pspoints \advance\epsfrsize by-\epsflly\pspoints \epsftsize=\epsfurx\pspoints \advance\epsftsize by-\epsfllx\pspoints % % If `epsfxsize' is 0, we default to the natural size of the picture. % Otherwise we scale the graph to be \epsfxsize wide. % \epsfxsize\epsfsize\epsftsize\epsfrsize \ifnum\epsfxsize=0 \ifnum\epsfysize=0 \epsfxsize=\epsftsize \epsfysize=\epsfrsize % % We have a sticky problem here: TeX doesn't do floating point arithmetic! % Our goal is to compute y = rx/t. The following loop does this reasonably % fast, with an error of at most about 16 sp (about 1/4000 pt). % \else\epsftmp=\epsftsize \divide\epsftmp\epsfrsize \epsfxsize=\epsfysize \multiply\epsfxsize\epsftmp \multiply\epsftmp\epsfrsize \advance\epsftsize-\epsftmp \epsftmp=\epsfysize \loop \advance\epsftsize\epsftsize \divide\epsftmp 2 \ifnum\epsftmp>0 \ifnum\epsftsize<\epsfrsize\else \advance\epsftsize-\epsfrsize \advance\epsfxsize\epsftmp \fi \repeat \fi \else\epsftmp=\epsfrsize \divide\epsftmp\epsftsize \epsfysize=\epsfxsize \multiply\epsfysize\epsftmp \multiply\epsftmp\epsftsize \advance\epsfrsize-\epsftmp \epsftmp=\epsfxsize \loop \advance\epsfrsize\epsfrsize \divide\epsftmp 2 \ifnum\epsftmp>0 \ifnum\epsfrsize<\epsftsize\else \advance\epsfrsize-\epsftsize \advance\epsfysize\epsftmp \fi \repeat \fi % % Finally, we make the vbox and stick in a \special that dvips can parse. % \ifepsfverbose\message{#1: width=\the\epsfxsize, height=\the\epsfysize}\fi \epsftmp=10\epsfxsize \divide\epsftmp\pspoints \vbox to\epsfysize{\vfil\hbox to\epsfxsize{% \special{PSfile=#1 llx=\epsfllx\space lly=\epsflly\space urx=\epsfurx\space ury=\epsfury\space rwi=\number\epsftmp}% \hfil}}% \epsfxsize=0pt\epsfysize=0pt}% % % We still need to define the tricky \epsfaux macro. This requires % a couple of magic constants for comparison purposes. % {\catcode`\%=12 \global\let\epsfpercent=%\global\def\epsfbblit{%BoundingBox}}% % % So we're ready to check for `%BoundingBox:' and to grab the % values if they are found. % \long\def\epsfaux#1#2:#3\\{\ifx#1\epsfpercent \def\testit{#2}\ifx\testit\epsfbblit \epsfgrab #3 . . . \\% \epsffileokfalse \global\epsfbbfoundtrue \fi\else\ifx#1\par\else\epsffileokfalse\fi\fi}% % % Here we grab the values and stuff them in the appropriate definitions. % \def\epsfgrab #1 #2 #3 #4 #5\\{% \global\def\epsfllx{#1}\ifx\epsfllx\empty \epsfgrab #2 #3 #4 #5 .\\\else \global\def\epsflly{#2}% \global\def\epsfurx{#3}\global\def\epsfury{#4}\fi}% % % We default the epsfsize macro. % \def\epsfsize#1#2{\epsfxsize}% % % Finally, another definition for compatibility with older macros. % \let\epsffile=\epsfbox % ================================================================ % execution: why not set \epsfverbosetrue % reset at your pleasure \abovefigskip=\baselineskip % reset at your pleasure \belowfigskip=\baselineskip % reset at your pleasure \global\let\figtitlefont\bf % reset at your pleasure \global\figtitleskip=.5\baselineskip % reset at your pleasure %DRAFT6.TEX %\input param.2 %\input fonts.5 %\input titles.5 %\input macros.18 %\input smallfonts.tex %\input epsfig.tex % \baselineskip=12.09pt % \def\iminus{I^{^-}\!} \def\iplus{I^{^+}\!} \def\iplusminus{I^{^\pm}\!} \def\Iminus{{\Bbb I}^{^-}\!} \def\Iplus{{\Bbb I}^{^+}\!} \def\Iplusminus{{\Bbb I}^{^\pm}\!} \def\ssH{{\scriptscriptstyle H}} \def\ssHp{{\scriptscriptstyle H'}} \def\ssHpp{{\scriptscriptstyle H''}} \def\O{\phantom{0}} % \def\rKy{1} \def\rCM{2} \def\rC{3} \def\rBa{4} \def\rHPKSMC{5} \def\rCFU{6} \def\rCGMi{7} \def\rCGMii{8} \def\rAWMi{9} \def\rAWMii{10} \def\rVVKP{11} \def\rVV{12} \def\rSi{13} \def\rDMS{14} \def\rDL{15} \def\rP{16} \def\rFG{17} \def\rG{18} \def\rED{19} \def\rSKad{20} \def\rMcKi{21} \def\rMcKii{22} \def\rMEsc{23} \def\rCGJii{24} \def\rCJ{25} \def\rKi{26} \def\rAKW{27} \def\rAK{28} \def\rKii{29} \def\rKiii{30} \def\rGK{31} % \def\rightheadline{\sevenrm\hfil RENORMALIZATION AND SHEARLESS TORI\hfil} \def\leftheadline{\sevenrm\hfil DENIS GAIDASHEV AND HANS KOCH\hfil} % \cl{{\magtenbf Renormalization and Shearless Invariant Tori:}} \cl{{\magtenbf Numerical Results}} % \bigskip \cl{Denis Gaidashev} \cl{Department of Mathematics, University of Toronto} \cl{Toronto, Ontario, Canada M5S 3G4} \bigskip \cl{Hans Koch} \cl{Department of Mathematics, University of Texas at Austin} \cl{Austin, TX 78712} % \footnote{} {{\sevenrm This work was supported in part by the National Science Foundation under Grant No. DMS-0088935.}} % \bigskip \abstract We present some numerical evidence for universality associated with the breakup of shearless invariant tori, by studying a renormalization group transformation acting on an appropriate space of Hamiltonians. \section Introduction Shearless tori appear in many physical systems, where they play an important role as transport barriers. This includes models of the atmosphere, toroidal plasma devices, fluid dynamics, and others [\rKy--\rCFU]. In the context of this paper, an invariant torus of a Hamiltonian system with two degrees of freedom is called {\it shearless}, if its rotation number is a local maximum or minimum. Such tori are observed to be surprisingly stable. Some bifurcations involving shearless tori have been described in [\rVVKP--\rG]. At the other end of the spectrum are systems that satisfy a twist condition, which guarantees that the rotation number varies monotonely from one torus to the next. In both types of system, some of the most interesting questions relate to the breakup of invariant tori and the resulting loss of stability. The twist case has been studied extensively, and for quadratic irrational rotation numbers such as the golden mean $\vartheta=\half+\half\sqrt{5}$, all of the evidence indicates that the breakup of invariant tori involves self-similar structures and universal scaling ratios. There has been considerable success in describing the underlying mechanisms with renormalization group methods [\rED--\rKiii]. But shearless tori are different in many respects. Visually, a shearless torus near breakup separates two totally chaotic looking regions, while in the twist case, elliptic islands dominate a non-trivial fraction of phase space. It seems surprising that the breakup of such tori should also be governed by universality and scaling. Nevertheless, this appears to be what happens. In the case of the golden mean and related rotation numbers, numerical investigations of specific two-parameter families [\rCGMi--\rAWMii] reveal self-similarity phenomena, with asymptotic scaling ratios (both in parameter space and phase space) that seem to be independent of the family considered. Furthermore, the self-similarity transformation involves as $12$-step shift in the sequence of continued fraction approximants for $\vartheta$, as opposed to the ordinary $1$-step shift describing the similarity of periodic orbits in the twist case. Our goal is to provide some evidence of universality for these phenomena, by relating the observed scaling ratios to properties of a renormalization group (RG) transformation, near a fixed point of its $12$-th iterate. We will use a RG transformation $\RR$ of the type introduced in [\rKi]. These transformations have been studied mostly in connection with systems that satisfy a twist condition [\rAKW--\rKiii]. But as was shown in [\rG], they apply equally well to the study of isoenergetically degenerate Hamiltonians and shearless invariant tori. $\RR$ will be chosen in such a way that it has a ``trivial'' period two, consisting of Hamiltonians of the form $$ H_\gamma^0(q,p)=\omega\cdot p+\gamma(\Omega\cdot p)^3\,, \qquad q\in\torus^2\,, \quad p\in\real^2\,, \equation(HZeroGamma) $$ where $\omega=(\vartheta^{-1},1)$ and $\Omega=(1,-\vartheta^{-1})$. We recall that the flow associated with a Hamiltonian $H(q,p)$ is given by $\dot q=\nabla_2 H$ and $\dot p=-\nabla_1 H$, where $\nabla_nH$ denotes the partial gradient of $H$ with respect to its $n$-th argument. In the case of $H_\gamma^0\,$, the torus $p=p_0$ is invariant for every value of $p_0\,$, and the angular velocity $\dot q=\omega+3\gamma(\Omega\cdot p)^2\Omega$ is constant on this torus. Thus, the rotation number is simply $\dot q_1/\dot q_2\,$. The torus for $p=0$ is shearless: Its rotation number $\vartheta^{-1}$ is maximal if $\gamma\le 0$, or minimal if $\gamma\ge 0$, among the rotation numbers of nearby orbits. In what follows, we consider Hamiltonians on $\torus^2\times\real^2$ of the form $$ H(q,p)=\omega\cdot p +\!\sum_{(\nu,k)\in I}H_{\nu,k}e^{i\nu\cdot q}z^k\,,\qquad z=\Omega\cdot p\,, \equation(HDef1) $$ where $I=\integer^2\times\natural$. We assume that $H$ takes real values for real arguments. The renormalization of $H$ involves the composition with a ``scaling'' of phase space $T_\mu(q,p)=\bigl(Tq,\mu T^{-1}p\bigr)$, where $\mu$ is some nonzero real number, and $T=\left[{0~1\atop 1~1}\right]$ is a matrix related to the golden mean: Its eigenvalues are $\vartheta$ and $-\vartheta^{-1}$, and the corresponding eigenvectors are $\omega$ and $\Omega$, respectively. Our RG transformation $\RR$ is of the form $$ \RR(H)={\vartheta\over\mu}H'\circ\UU_\ssHp\,,\quad H'=H\circ T_\mu\,. \equation(RGDef) $$ where $\UU_\ssHp$ is a canonical transformation described below, and where $\mu$ is a normalization constant depending on $H$, which is chosen in such a way that the coefficient of $(\Omega\cdot p)^3$ in the renormalized Hamiltonian $\RR(H)$ is equal to $\pm\half$, if possible. The change of variables $H'\mapsto H'\circ\UU_\ssHp$ can also be regarded as a normalization procedure, but one that involves infinitely many normalization conditions. This feature of $\RR$ arises from the need of working with periodic variables. To be more precise, we note that the scaling $T_\mu$ has an eigenvalue $\vartheta>1$. (The other eigenvalues are of modulus $<1$, for the values of $\mu$ that appear in our analysis.) As a result, the rescaled Hamiltonian $H\circ T_\mu$ is less regular than $H$ in the direction of the corresponding eigenvector, which is $(\omega,0)$. It needs to be re-normalized, in order for $\RR$ to define a dynamical system on some fixed space of Hamiltonians. The modes $e^{i\nu\cdot q}z^k$ that are negatively affected by the scaling $T_\mu$ are those for which $n$ is relatively small and $\nu$ has a significant component in the expanding direction $\omega$ of $T$. These modes will be referred to as ``non-resonant''. To be more precise, let $$ \iplus=\bigl\{(\nu,k)\in I: \ |\omega\cdot\nu|\le\sigma|\Omega\cdot\nu| {\rm ~or~ } |\omega\cdot\nu|\le\kappa k\bigr\}\,, \equation(iplusdef) $$ where $\sigma$ and $\kappa$ are fixed positive real numbers that will be specified later. The resonant part $\Iplus H$ of a Hamiltonian $H$ is defined by restricting the sum in equation \equ(HDef1) to the index set $\iplus$, and the non-resonant part is defined as $\Iminus H=H-\Iplus H$. In order to compensate for the abovementioned loss of regularity in the scaling $H\mapsto H'=H\circ T_\mu\,$, we will determine $\UU_\ssHp$ in such a way that $$ \Iminus\bigl(H'\circ\UU_\ssHp\bigr)=0\,, \equation(Elim) $$ if possible. In other words, the change of variables $H'\mapsto H'\circ\UU_\ssHp$ is designed to eliminate the non-resonant modes of $H'$. The solution of equation \equ(Elim) is in general not unique. In our computations, we choose for $\UU_\ssHp$ a canonical transformation of the form $$ U_\phi(q,p) =\Bigl(q+\partial_2\varphi\bigl(q,z+Z(q,z)\bigr)\Omega\,\hbox{\bf ,}\, p-\nabla_1\varphi\bigl(q,z+Z(q,z)\bigr)\Bigr)\,, \equation(Uphi) $$ associated with a non-resonant generating function $\phi(q,p)=\varphi(q,z)$, where $z=\Omega\cdot p$, and where $Z$ is defined implicitly by the equation $$ Z(q,z)=-\Omega\cdot\nabla_1\varphi\bigl(q,z+Z(q,z)\bigr)\,. \equation(Zequ) $$ This makes \equ(Elim) an equation for $\phi$. Our approach is to solve this equation by Newton's method, starting with $0$ as an initial guess. We note that if $H'$ is purely resonant, then $\UU_\ssHp$ is the identity. Examples of such Hamiltonians are functions $H(q,p)$ that only depend on the action variable $p$. Thus, it straightforward to verify e.g. that the two Hamiltonians $H_\gamma^0\,$, with $\gamma=\pm\half$, form an orbit of period $2$ for $\RR$. This period is hyperbolic, with one unstable direction (not counting constant Hamiltonians) represented by the function $(q,p)\mapsto\Omega\cdot p$, and expanding eigenvalue $-\vartheta^2$. \section Results The RG transformation $\RR$ can be defined for a more general class of Hamiltonians than the ones considered here; see e.g. [\rKi]. However, the subclass of Hamiltonians of the form \equ(HDef1) is invariant under $\RR$, and numerical experiments suggest that families of such Hamiltonians exhibit the phenomena that we are interested in. For the same reason, we now restrict our analysis to Hamiltonians $H$ satisfying $\JJ H=-H$, where $\JJ H=H\circ J$ and $J(q,p)=(-q,-p)$. Formally, this symmetry also ensures that an invariant torus passing through the origin is shearless. By requiring that our generating functions satisfy $\JJ\phi=\phi$, we find that $U_\phi$ and $\RR$ commute with $J$ and $\JJ$, respectively. In our numerical implementation of $\RR$, we approximate functions like $H$ and $\phi$ by finite linear combinations of all modes $\cos(\nu\cdot q)z^k$ and $\sin(\nu\cdot q)z^k$ below a certain cut-off, and represent them as arrays of Fourier--Taylor coefficients. (Roughly half of them are zero, due to the abovementioned symmetry.) The cut-off used is of the form $$ |\nu_1|+|\nu_2|\le N_1\,,\quad k\le N_2\,,\quad |\nu_1|+|\nu_2|+k\le N_3\,,\qquad N\in\natural^3\,. \equation(Cut) $$ We have written procedures (in the programming language Ada95) for the the addition, differentiation, multiplication, and composition of such functions. Implicit equations like \equ(Zequ) and \equ(Elim) are solved numerically by using Newton's method. Our numerical implementation of $\RR$ is obtained by combining these steps, according to the definition of $\RR$. In what follows, we will not always distinguish between $\RR$ (or other objects) and its numerical implementation (or approximation). As mentioned earlier, $\RR^2$ has one unstable direction at the trivial fixed points $H_\gamma^0\,$. (Here, and in what follows, $\gamma=\pm\half$.) This unstable direction corresponds to a change of rotation number for the orbit passing through the origin. At the nontrivial fixed points of $\RR^{12}$, we expect one additional unstable direction, corresponding to a change of stability of this orbit. In order to find these fixed points, it is desirable to use modified RG transformations that have no (or fewer) unstable directions. Our construction of such transformations is based on the following principle. Consider a map $\FF$ on $\real^n$, that has a hyperbolic fixed point at $x_0\,$, with $d>0$ unstable directions. Let $p$ be a $\real^d$-valued function on the domain of $\FF$, such that the equation $p(\FF(x))=p(x)$ defines a manifold $\Sigma$ of codimension $d$ near $x_0\,$. Let $\PP$ be a projection in $\real^n$ onto $\Sigma$, leaving points on $\Sigma$ fixed, and define $\MM=\FF\circ\PP$. If $p$ is chosen in such a way that $\Sigma$ coincides with the local stable manifold of $F$ at $x_0\,$, then $x_0\,$ is clearly a stable fixed point for $\MM$. But since the property of having a stable fixed point is preserved under small perturbations of the map, we can expect the same to hold for other ``reasonable'' choices of $p$, and compatible projections $\PP$. We have used a procedure of this type, with $d=1$, to define a map $\MM_1=\RR^{12}\circ\PP$ for which $H_\gamma^0$ is a stable fixed point. This map was then used in a bisection procedure, along lines of Hamiltonians passing through $H_\gamma^0\,$, to search for points on the boundary of the basin of attraction for $H_\gamma^0\,$. Ideally, this boundary coincides in some regions with the codimension one stable manifold of a non-trivial fixed point for $\MM_1$ that is also a fixed point of $\RR^{12}$. This task turned out to be extremely tedious. For the vast majority of Hamiltonians considered, the numerical procedure broke down before reaching the $12$-th iterate of $\RR$. And for most of the others, the same happened during subsequent iterations of $\MM_1\,$, which should bring a ``critical'' Hamiltonian closer to the expected fixed point. (We also used Hamiltonians whose criticality was determined by investigating their dynamics.) But for some of the better Hamiltonians, $\RR^{n+6}(H)$ was noticeably similar to $\RR^n(H)$, modulo a rotation $\VV_\alpha\,$, where $$ \VV_\alpha H=H\circ V_\alpha\,,\qquad V_\alpha(q,p)=(q+\pi\alpha,p)\,, $$ with $\alpha$ a vector from $A=\{(1,0), (0,1), (1,1)\}$. Thus, we continued our search with $\NN=\VV_\alpha\circ\RR^6$ in place of $\RR^{12}$. It should be noted that $\NN^2=\RR^{12}$. This follows from an explicit computation [\rKi], which shows that for any $\alpha$, $$ \RR\circ\VV_\alpha=\VV_{T^{-1}\alpha}\circ\RR\,. $$ Our next goal was to get closer to the expected fixed point $H_0$ of $\NN$, by iterating another map $\MM_2$ of the form $\NN\circ\PP$, this time with $d=2$, designed to have $H_0$ as a stable fixed point. As starting points for this iteration, we used Hamiltonians obtained with the abovementioned bisection procedure. This task turned out to be much more difficult than expected. (Some possible reasons will be mentioned later.) All computations, including the bisection procedure, were repeated for various choices of the constants $\sigma$ and $\kappa$ defining the projections $\Iplusminus$. The values that produced the best convergence, at the degrees considered, are $\sigma=1.1$ and $\kappa=\sigma/1.5$. Thus, we will consider these values fixed from now on. One of the intermediate results of this iteration was that certain Fourier--Taylor coefficients of our Hamiltonians approached zero, indicating that each of the Hamiltonians $H_n=\RR^n(H_0)$ in the expected period $12$ for $\RR$ is invariant under a reflection $\JJ\circ\JJ_\alpha\,$, $$ \JJ_\alpha H=H\circ J_\alpha\,,\qquad J_\alpha(q,p)=(q+\pi\alpha,-p)\,, $$ for some $\alpha=\alpha(n)$ in $A$. One of the other transformations of this type maps $H_n$ to $H_{n+6}\,$. Interestingly, if we assume the existence of such a periodic orbit for $\RR$, and rotate $H_n$ by $\half\pi\alpha(n)$, for each $n$, we obtain another orbit $\{H^n\}$ of period $12$ for $\RR$. Each of these new Hamiltonians is of the form $$ H(q,p)=\omega\cdot p\,\pm\third z^3 +\!\!\sum_{(\nu,k)\in\iplus}h_{\nu,k}\cos(\nu\cdot q)z^k\,. \equation(HDef2) $$ In addition to being an even function of $q$, the Hamiltonian $H^n$ is invariant under $\JJ_\alpha\,$, with $\alpha=\alpha(n)$. In terms of its Fourier--Taylor coefficients, this means that $h^n_{\nu,k}=0$ whenever $\nu\cdot\alpha+k$ is even. We note that the symmetries of a periodic RG orbit like $\{H^n\}$ reflects an asymptotic symmetry of every Hamiltonian in the corresponding universality class. Thus, when studying the breakup of shearless golden tori, such symmetries can be imposed without loss of generality, which greatly simplifies the analysis. The bisection procedure was performed mostly at degrees $N=(7,7,9)$, that is, with a cutoff \equ(Cut) corresponding to this vector $N$. The subsequent iterations of $\MM_2$ were carried out first at the same degree, and later for successively higher degrees, using as input the approximate fixed point from previous computations. Once a certain accuracy was reached, it became possible to replace $\MM_2$ by a computationally much more efficient (but less stable) transformation: a Newton-like map, with a fixed linear approximation for the derivative $D\NN(H)$. In addition, we have also computed the largest two eigenvalues $\delta_1$ and $\delta_2$ of the derivative $D\NN(H_0)$, by following the orbit of a two dimensional subspace under a difference quotient approximation to $D\NN$. Table 1 lists the sixth roots of these eigenvalues, together with the average scaling $\bar\mu$, defined by $\bar\mu^6=\mu(H_0)\mu(H_1)\cdots\mu(H_5)$. If the renormalization group picture is correct, then the eigenvalues $\delta_1$ and $\delta_2$ should describe the accumulation of bifurcation points in two-parameter families of Hamiltonians, near a ``critical point'' at the end of a line of Hamiltonians possessing a shearless invariant torus, with rotation number $\vartheta$. Such a bifurcation analysis was carried out in [\rCGMii] and [\rAWMi]. For the ratios corresponding to $\bigl(\delta_1^{1/6},\,\delta_2^{1/6}\bigr)$, the values obtained in these two references $(2.683,\,1.511)$ and $(2.678,\,1.583)$, respectively. \bigskip $$ \vbox{\tabskip=0pt\offinterlineskip\halign to178.6pt{ \strut#&\vrule#\tabskip=0.5em&\hfil#&\vrule#&\hfil#&\vrule#&\hfil# &\vrule#&\hfil#&\vrule#\tabskip=0pt\cr \noalign{\hrule} & &$N$\hfil & &$\bar\mu$\hfil & &$\delta_1^{1/6}$\hfil & &$\delta_2^{1/6}$\hfil&\cr \noalign{\hrule} & &(\O7, 7, \O9)& &$0.36546$& &$2.6620$& &$1.5850$&\cr \noalign{\hrule} & &(12, 8, 14)& &$0.36594$& &$2.6613$& &$1.5850$&\cr \noalign{\hrule} & &(16, 8, 18)& &$0.36589$& &$2.6612$& &$1.5850$&\cr \noalign{\hrule}}} $$ \vskip-0.5em \cl{{\sevenrm Table 1. Scaling $\scriptstyle\bar\mu$, and eigenvalues $\scriptstyle\delta_j$ of $\scriptstyle D\NN(H_0)$}} \bigskip\bigskip One notable fact about all these numbers is that they are not nearly as accurate as the corresponding data obtained in the twist case [\rMcKii,\rAKW]. In experiments on two-parameter families, this is largely due to the fact that $12$ bifurcations are involved at each level of self-similarity, which limits the number of levels that can be studied at a given numerical precision. An analogous complication in our analysis is that the RG orbits pass through $6$ very different regions of function space. In order to renormalize Hamiltonians accurately, it is necessary to adapt the procedure to the region considered. It cannot be expected that the same domains and normal forms (determined by the choice of $\Iplusminus$) are appropriate along the entire orbit of a critical Hamiltonian. But in the analysis presented here, which is only a first step, the projections $\Iplusminus$ have been kept fixed. This makes the condition \equ(Elim) effectively much more restrictive in some regions than in others. In two of the six regions (depending on the choice of $\sigma$ and $\tau$), we obtain canonical transformations $\UU_\ssHp$ that are clearly incompatible with the domains of the Hamiltonians $H'$, as determined by the decay rate of the Fourier--Taylor coefficients. Furthermore, the determinant of their derivative is around $1.006$, instead of $1$, independently of the degrees considered. Thus, we estimate that there is a systematic error in our data of the order of $1$ to $2$ percent. A measure of such an error can be seen in the fourth column of Table 2. We believe that working with fixed projections $\Iplusminus$ is the main source of these errors, but there may be others. Some of the difficulties may also be due to the fact that the RG transformation considered here has two non-trivial unstable directions, as opposed to one in the twist case. Among other things, this increases the number of near-resonances between eigenvalues, which is a well known source of numerical problems. There are additional universal quantities, associated with $\RR$ and its orbit $\{H_n\}$ of period $12$, that can be observed in experiments with two-parameter families. (We assume of course that such an orbit exists, close to our approximate orbit and with the same symmetries.) {}From the fixed point equation for $\RR^{12}$ we have $(\vartheta/\bar\mu)^{12}H_n\circ\Lambda_n=H_n\,$, where $\Lambda_n$ is a composition of $12$ maps of the form $T_\mu\circ\UU_\ssHp\,$. Notice that the scaling transformations $\Lambda_n$ are all conjugate to each other. Under the action of $\Lambda_n\,$, an orbit of $H_n$ with frequency vector $w$ is mapped to an orbit of $H_n$ whose frequency vector is a constant multiple of $T^{-12}w$. Thus, assuming that $H_n$ has a unique invariant torus (at some fixed energy) with frequency vector parallel to $\omega$, this torus has to pass through the fixed points of $\Lambda_n\,$. In addition, the set of periodic orbits whose rotation numbers are the diophantine approximants for $\vartheta^{-1}$, should accumulate at any fixed point of $\Lambda_n\,$, at rates determined by the eigenvalues of $D\Lambda_n$ at that fixed point. Some fixed points of $D\Lambda_n$ can be determined by symmetries alone and thus will be referred to as ``symmetry points''. One of them is the origin, since all of our canonical transformations commute with the reflection $J: x\mapsto -x$ and leave $\omega\cdot q$ invariant. We have computed one of the the matrices $D\Lambda_n(0)$ and its four eigenvalues $\lambda_j\,$. Table 2 lists our numerical value for the $12$-th root of $\lambda_2\,$, and for three quantities that should be equal to zero, due to known relations among the eigenvalues. Two of these relations come from the fact that $\Lambda_n$ is a the composition of canonical transformations and a scaling $(q,p)\mapsto(q,\bar\mu^{12}p)$. As a consequence, the eigenvalues of $D\Lambda_n(0)$ come in pairs whose product is equal to $\bar\mu^{12}$. The identity $\lambda_1=\vartheta^{12}$ expresses the fact that an invariant torus is smooth in the direction of the flow. The relations involving $\lambda_1$ are satisfied accurately in our approximations, due to the special form \equ(Uphi) of the transformations $\UU_\ssHp\,$. But as commented on earlier, we find a $1.7\%$ deviation from the identity $(\lambda_2\lambda_3)^{1/12}=\bar\mu^{12}$, which is essentially independent of $N$ (we checked this for higher degrees as well). \bigskip \def\foo{\phantom{000)}} $$ \vbox{\tabskip=0pt\offinterlineskip\halign to312.2pt{ \strut#&\vrule#\tabskip=0.5em&\hfil#&\vrule#&\hfil#&\vrule#&\hfil# &\vrule#&\hfil#&\vrule#&\hfil#&\vrule#\tabskip=0pt\cr \noalign{\hrule} & &$N$\hfil & &$\lambda_1^{1/12}\!/\vartheta-1$\hfil & &$\lambda_2$\hfil & &$(\lambda_2\lambda_3)^{1/12}\!/\bar\mu-1$\hfil & &$\vartheta\lambda_4^{1/12}\!/\bar\mu-1$\hfil &\cr \noalign{\hrule} & &(\O7, 7, \O9)& &$5.2\times 10^{-12}$& &$0.65643$& & $0.016\foo$& &$-4.1\times 10^{-11}$&\cr \noalign{\hrule} & &(12, 8, 14)& &$5.2\times 10^{-12}$& &$0.65704$& & $0.017\foo$& &$-2.2\times 10^{-11}$&\cr \noalign{\hrule} & &(16, 8, 18)& &$5.2\times 10^{-12}$& &$0.65694$& & $0.017\foo$& &$ 5.6\times 10^{-11}$&\cr \noalign{\hrule}}} $$ \hskip-0.5em \cl{\sevenrm Table 2. Scaling parameters $\scriptstyle\lambda_j$ at a symmetry point} \bigskip\bigskip We note that $J$ has three other fixed points on $\torus^2\times\real$ besides the origin: the vectors $(\pi\alpha,0)=J_\alpha(0)$ with $\alpha\in A$. They are all fixed points of $\Lambda_n$ as well. The eigenvalues of $D\Lambda_n$ at these points agree, since $J_\alpha^{-1}\circ\Lambda_n\circ J_\alpha$ is either $\Lambda_n$ or $\Lambda_{n+6}\,$, due to a symmetry mentioned earlier. In addition, $\Lambda_n$ has most likely one additional fixed point (and its image under $J$) that depends on $n$. This case is described best in the coordinates in which $H_n$ is represented by a Hamiltonian $H^n$ of the form \equ(HDef2). The scaling $\Lambda_n$ in these coordinates will be denoted by $S_n\,$. Since $H^n(q,p)$ is invariant under the reflection $q\mapsto -q$, a closed orbit for $H^n$ whose rotation number is a ratio of odd integers, if unique, will have to intersect the symmetry plane $q=0$. But if such an orbit exists close to the invariant $\omega$-torus, then there exists an infinite sequence of them, with rotation numbers approaching $1/\vartheta$. Their intersection with the plane $q=0$ should accumulate at a point $(0,p_n)$ on the invariant $\omega$-torus, and if $S_n$ is analytic as expected, then $(0,p_n)$ has to be a fixed point of $S_n\,$. Here, we have also used that $S_n$ leaves the plane $q=0$ invariant, which follows from the fact that Hamiltonians of the form \equ(HDef2) lead to generating functions $\phi(q,p)$ that are odd in $q$. This situation is in fact similar to the one encountered in the twist case in [\rAKW]. It would be interesting to determine the point $(0,p_n)$ and the eigenvalues of $DS(0,p_n)$ from our numerical data, but we have not yet done so. The analogue of $\lambda_2$ is expected to be the accumulation rate of orbits along a ``symmetry plane''. The numerical value determined in [\rAWMi] for this rate is $0.6035$. Our approximations for the Hamiltonians $H_n$ can be found in [\rGK]. For a more qualitative impression, we have visualized some orbits of two Hamiltonians. The first Hamiltonian, $H$, is our numerical approximation to the fixed point $H_0$ of $\RR^{12}$. Figure 1 shows orbits for the return map for $H$ to the cylinder $q_2=0$ at energy zero. Only orbits in the lower ``half'' of this cylinder are shown. The topmost (twistless) orbit passes thorough the origin and has rotation number $0.604\ldots$, instead of $\vartheta^{-1}=0.618\ldots$, as expected for $H_0\,$. Also shown is the golden curve with rotation number $\vartheta$, and adjacent elliptic periods $5/8$ and $8/13$. The orbits in the lower part of Figure 1 appear to be chaotic. Clearly, $H$ is still quite far from having a critical golden torus with rotation number $\vartheta$. For comparison, Figure 2 shows the orbits for such a critical Hamiltonian. The Hamiltonian used is of the form $H+f$, where $f$ is a linear combination of the two expanding eigenvectors of $D\RR^{12}(H_0)$, determined in such a way that $H+f$ has a critical golden torus with rotation number $\vartheta$. The necessary correction $f$ is quite large: roughly a quarter of the size of $H$, measured in some appropriate norm. \section Conclusion The scaling phenomena observed in [\rCGMi--\rAWMi] suggest that the breakup of shearless invariant tori is governed by a periodic RG orbit of length $12$. Our goal was to lend support to this hypothesis, by constructing such a periodic orbit and analyzing its properties. To this end, we chose an RG transformation $\RR$ of the type introduced in [\rKi], which acts on Hamiltonians with two degrees of freedom. By extending techniques used in the case of shear flows [\rAKW], it was indeed possible to find an approximate period $12$ for $\RR$. The analysis of the shearless case turned out to be much more difficult, and the numerical results are correspondingly less accurate. Still, our data are comparable in accuracy, and consistent, with the ones obtained in [\rCGMii,\rAWMi]. Thus, our results represent the first real evidence that the breakup of shearless golden tori is universal, in the sense that the observed asymptotic ratios are independent of the system considered. They also indicate that the period $12$ for $\RR$ is invariant under a certain group of symmetries. This symmetry should be universal as well, and thus reflected in the structure of shearless golden tori and nearby orbits, at criticality. 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World Scientific, London (1993). \item{[\rMcKii]} R.S.~MacKay, {\it Renormalisation Approach to Invariant Circles in Area--Preserving Maps,} Physica D {\bf 7}, 283--300 (1983). \item{[\rMEsc]} A.~Mehr and D.F.~Escande, {\it Destruction of KAM Tori in Hamiltonian Systems: Link with the Destabilization of Nearby Cycles and Calculation of Residues,} Physica D {\bf 13}, 302--338 (1984). \item{[\rCGJii]} M.~Govin, C.~Chandre, and H.R.~Jauslin, {\it KAM--Renormalization Group Approach to the Breakup of Invariant Tori in Hamiltonian Systems.} Phys. Rev. E {\bf 57}, 1536--1543 (1998). \item{[\rCJ]} C.~Chandre and H.R.~Jauslin, {\it Renormalization-group analysis for the transition to chaos in Hamiltonian systems.} Physics Reports {\bf 365}, 1--64 (2002). %%%%%%%% RENORMALIZATION, SAME RG \item{[\rKi]} H.~Koch, {\it A Renormalization Group for Hamiltonians, with Applications to KAM Tori.} Erg.~Theor.~Dyn.~Syst.~{\bf 19}, 1--47 (1999). \item{[\rAKW]} J.J.~Abad, H.~Koch, and P.~Wittwer, {\it A Renormalization Group for Hamiltonians: Numerical Results.} Nonlinearity {\bf 11}, 1185--1194 (1998). \item{[\rAK]} J.J.~Abad and H.~Koch, {\it Renormalization and Periodic Orbits for Hamiltonian Flows,} Commun.~Math.~Phys. {\bf 212}, 371--394 (2000). \item{[\rKii]} H.~Koch, {\it On the renormalization of Hamiltonian flows, and critical invariant tori.} Discrete Contin.~Dynam.~Systems A, {\bf 8}, 633--646 (2002). \item{[\rKiii]} H.~Koch, {\it A Renormalization Group Fixed Point Associated with the Breakup of Golden Invariant Tori.} preprint {\tt mp\_arc 02-175} (2002), to appear in Discrete Contin. Dynam. Systems (special volume on Hamiltonian systems). \item{[\rGK]} Degree $(7,7,9)$ Hamiltonians are included with the preprint {\tt mp\_arc 03-545} of this paper; additional data can be found at {\tt ftp://ftp.ma.utexas.edu/pub/papers/koch/shearless/} } \vfil\eject %FIGURES \ipsfig 8truein; fig1.ps; \vfill\eject \ipsfig 8truein; fig2.ps; \bye ---------------0312161327389 Content-Type: application/postscript; name="fig1.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="fig1.ps" %!PS-Adobe-2.0 %%Title: Two paramter fixed nontwist family %%Creator: dstool %%CreationDate: Wed Jun 4 16:53:18 2003 %%For: Hans Koch,dept,fac,000000 %%Pages: 1 %%DocumentFonts: /Times-Roman %%BoundingBox: 0 0 612 792 %%EndComments % begin the prolog... % Prolog.ps % Define plotting symbols. % Symbols are defined in two parts: a "glyph" which is an absolute symbol, and the command % for plotting the glyph, which scales the glyph into user coordinates. % To make symbols have different linewidths, uncomment the commands "linewidth setlinewidth" % and replace it with something like "0.1 setlinewidth". % Also, boxes and triangles can be filled in by adding the command % "fill" instead of the "stroke" command. % First, we define the glyphs... /Boxglyph {newpath dup scale %linewidth setlinewidth 0.5 0.5 moveto -0.5 0.5 lineto -0.5 -0.5 lineto 0.5 -0.5 lineto 0.5 0.5 lineto closepath stroke} def /Triangleglyph {newpath dup scale %linewidth setlinewidth 0 1 moveto 0.866 -0.5 lineto -0.866 -0.5 lineto closepath stroke} def /Circleglyph {newpath dup scale %linewidth setlinewidth 0 0 0.5 0 360 arc fill} def /Crossglyph {newpath dup scale %linewidth setlinewidth 0.75 0 moveto -0.75 0 lineto 0 0.75 moveto 0 -0.75 lineto stroke} def % Now define the symbol commands... % procedure to draw a box. Called as: x y scalefactor Box -- /Box{3 1 roll moveto currentpoint gsave translate xscalei yscalei scale Boxglyph grestore }def % procedure to draw a triangle. Called as: x y scalefactor Triangle -- /Triangle{3 1 roll moveto currentpoint gsave translate xscalei yscalei scale Triangleglyph grestore }def % procedure to draw a cross. Called as: x y scalefactor Cross -- /Cross{3 1 roll moveto currentpoint gsave translate xscalei yscalei scale Crossglyph grestore }def % procedure to draw a dot. Called as: x y scalefactor Dot -- /Dot{3 1 roll moveto currentpoint gsave translate xscalei yscalei scale Circleglyph grestore }def % procedure to draw a line to a symbol. Called as: x y Line x y /Line{linewidth 8 div setlinewidth lineto currentpoint stroke} def % draws ticks on the bounding box. Called as: num_x_ticks num_y_ticks Ticks -- % For thicker tick marks, uncomment the end of the first line /Ticks{ gsave % 1 setlinewidth 1 add yBBoxSide exch div dup 0 0 moveto dup yBBoxSide{0 exch moveto TickSize 0 rlineto}for %left xBBoxSide 0 moveto dup yBBoxSide{xBBoxSide exch moveto TickSize neg 0 rlineto}for %rt 1 add xBBoxSide exch div dup 0 0 moveto dup xBBoxSide{0 moveto 0 TickSize rlineto}for %bottom 0 yBBoxSide moveto dup xBBoxSide{yBBoxSide moveto 0 TickSize neg rlineto}for %top stroke grestore }def % draws grid on the bounding box. Called as: num_x_ticks num_y_ticks Grid -- % For thicker tick marks, uncomment the end of the first line /Grid{ gsave % 1 setlinewidth 1 add yBBoxSide exch div 0 0 moveto dup yBBoxSide{0 exch moveto xBBoxSide 0 rlineto}for % left to right 1 add xBBoxSide exch div 0 0 moveto dup xBBoxSide{0 moveto 0 yBBoxSide rlineto}for % bottom to top stroke grestore }def % draws box of sidelength (xBBoxSide, yBBoxSide) beginning at origin /BoundingBox{ gsave 1 setlinewidth 0 0 moveto xBBoxSide 0 rlineto 0 yBBoxSide rlineto xBBoxSide neg 0 rlineto closepath stroke grestore }def % macros for string positioning and manipulation /Display{0 Yp moveto show /Yp Yp HeadPtSize sub def }def % display header info % centers arbitrary string at current point. Called as: (str) CenterString (str) /CenterString { dup stringwidth pop 2 div currentpoint 3 -2 roll exch sub exch moveto }def % vertically centers string at current point. Called as:fontheight (str) VCenterString (str) /VCenterString{dup length .5 add 2 div 3 -1 roll mul currentpoint 3 -1 roll add moveto }def % prints figure titles centered at given pt % requires: CenterString; TitlePtSize; TitleFont % called as: x y (title) (caption) FigureTitle -- /FigureTitle { gsave TitleFont setfont 4 2 roll moveto gsave exch CenterString show grestore currentpoint TitlePtSize sub moveto CenterString show grestore }def % prints x-axis label % requires: CenterString; LabelPtSize; LabelFont % called as: (label) XLabel -- /XLabel { gsave LabelFont setfont xBBoxSide 2 div LabelPtSize -3 mul moveto CenterString show grestore }def % prints y-axis label horizontally % requires: CenterString; LabelFont % called as: (label) HorYLabel -- /HorYLabel { gsave LabelFont setfont dup stringwidth pop -2 div HorSpace 2 mul sub yBBoxSide 2 div moveto CenterString show grestore }def % prints list of n strings along x axis at evenly spaced intervals % requires: LabelFont; LabelPtSize; CenterString % called as: (s_1) (s_2) ... (s_n) n PlaceXLabels -- /PlaceXLabels { gsave LabelFont setfont dup 1 le { % if n==1, center string pop xBBoxSide 2 div LabelPtSize -1.5 mul moveto CenterString show } { % else place labels at intervals of length xBBoxSide/(n-1) 1 sub xBBoxSide exch div % set length -1 mul xBBoxSide exch 0{ % for i=xBBoxside to 0 step (-length) LabelPtSize -1.5 mul moveto CenterString show }for } ifelse grestore}def % prints list of n strings along y axis at evenly spaced intervals % requires: LabelFont; LabelPtSize;CenterString % called as: (s_1) (s_2) ... (s_n) n PlaceYLabels -- /PlaceYLabels { gsave LabelFont setfont dup 1 le { % if n==1, center string pop dup stringwidth pop -2 div HorSpace sub yBBoxSide 2 div moveto CenterString show } { % else place labels at intervals of length yBBoxSide/(n-1) 1 sub yBBoxSide exch div % set length -1 mul yBBoxSide exch 0 { % for i=yBBoxside to 0 step (-length) exch dup stringwidth pop -2 div HorSpace sub 3 -1 roll moveto CenterString show }for } ifelse grestore}def % % The following macros are not explicitly used by dstool but may be useful % % prints y-axis label vertically % requires: VCenterString; LabelPtSize; LabelFont % called as: (label) VerYLabel -- /VerYLabel { gsave LabelFont setfont LabelPtSize -1.5 mul yBBoxSide 2 div moveto LabelPtSize exch VCenterString LabelPtSize exch vshow grestore }def % The following proc is from PostScript Tutorial and Cookbook by Adobe Systems Inc., p. 165 /vshowdict 4 dict def % called as: vert_space (string) vshow -- /vshow{ vshowdict begin /thestring exch def /lineskip exch def thestring { /charcode exch def /thechar ( ) dup 0 charcode put def 0 lineskip neg rmoveto gsave thechar stringwidth pop 2 div neg 0 rmoveto thechar show grestore } forall end }def % prints annotations on graph. Called as: (label) xpos ypos Annotation -- % requires: CenterString; LabelPtSize; LabelFont % xpos and ypos are between 0 and 1 /Annotation{ gsave LabelFont setfont yBBoxSide mul exch xBBoxSide mul exch moveto CenterString show grestore }def % declare global constants (edit with care). 72 points = 1 inch /x0 0 def /xf 1 def /y0 0 def /yf 1 def % dstool scales output into [0,1] x [0,1] /TickSize 5 def % size (in pts) of tick marks /linewidth 0.005 def % width of line for plotting (smallest resolution=0) /HeadPtSize 6 def % print the header in this font size /HorSpace 8 def % pts left of BBox for hor labels /Sm 1 def % typical sizes of symbols /Md 2 def /Lm 3 def /Lg 4 def /XL 6 def % The below are model-specific definitions. Written by dstool. /SetFonts { /LabelPtSize 10 def /TitlePtSize 14 def /LabelFont /Times-Roman findfont LabelPtSize scalefont def /TitleFont /Times-Roman findfont TitlePtSize scalefont def } def /xBBoxSide 450 def % length (in pts) of bounding box /yBBoxSide 450 def % height (in pts) of bounding box /xorigin 80 def % displacement (in pts) from left side of page /yorigin 145 def % displacement (in pts) from bottom of page % The below are not model specific but depend on the above definitions /Yp yBBoxSide 60 add 6 HeadPtSize mul add def % print 6 header lines at this height /xscale xBBoxSide xf x0 sub div def % horizontal scaling factor /yscale yBBoxSide yf y0 sub div def % vertical scaling factor /xscalei 1 xscale div def % horizontal scaling factor inverse /yscalei 1 yscale div def % vertical scaling factor inverse %%%%EndProlog gsave xorigin yorigin translate linewidth setlinewidth SetFonts xBBoxSide 2 div yBBoxSide TitlePtSize 2 mul add (Figure 1) () FigureTitle (-3.14159) (3.14159) 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dstool %%CreationDate: Wed Jun 4 17:47:42 2003 %%For: Hans Koch,dept,fac,000000 %%Pages: 1 %%DocumentFonts: /Times-Roman %%BoundingBox: 0 0 612 792 %%EndComments % begin the prolog... % Prolog.ps % Define plotting symbols. % Symbols are defined in two parts: a "glyph" which is an absolute symbol, and the command % for plotting the glyph, which scales the glyph into user coordinates. % To make symbols have different linewidths, uncomment the commands "linewidth setlinewidth" % and replace it with something like "0.1 setlinewidth". % Also, boxes and triangles can be filled in by adding the command % "fill" instead of the "stroke" command. % First, we define the glyphs... /Boxglyph {newpath dup scale %linewidth setlinewidth 0.5 0.5 moveto -0.5 0.5 lineto -0.5 -0.5 lineto 0.5 -0.5 lineto 0.5 0.5 lineto closepath stroke} def /Triangleglyph {newpath dup scale %linewidth setlinewidth 0 1 moveto 0.866 -0.5 lineto -0.866 -0.5 lineto closepath stroke} def /Circleglyph {newpath dup scale %linewidth setlinewidth 0 0 0.5 0 360 arc fill} def /Crossglyph {newpath dup scale %linewidth setlinewidth 0.75 0 moveto -0.75 0 lineto 0 0.75 moveto 0 -0.75 lineto stroke} def % Now define the symbol commands... % procedure to draw a box. Called as: x y scalefactor Box -- /Box{3 1 roll moveto currentpoint gsave translate xscalei yscalei scale Boxglyph grestore }def % procedure to draw a triangle. Called as: x y scalefactor Triangle -- /Triangle{3 1 roll moveto currentpoint gsave translate xscalei yscalei scale Triangleglyph grestore }def % procedure to draw a cross. Called as: x y scalefactor Cross -- /Cross{3 1 roll moveto currentpoint gsave translate xscalei yscalei scale Crossglyph grestore }def % procedure to draw a dot. Called as: x y scalefactor Dot -- /Dot{3 1 roll moveto currentpoint gsave translate xscalei yscalei scale Circleglyph grestore }def % procedure to draw a line to a symbol. Called as: x y Line x y /Line{linewidth 8 div setlinewidth lineto currentpoint stroke} def % draws ticks on the bounding box. Called as: num_x_ticks num_y_ticks Ticks -- % For thicker tick marks, uncomment the end of the first line /Ticks{ gsave % 1 setlinewidth 1 add yBBoxSide exch div dup 0 0 moveto dup yBBoxSide{0 exch moveto TickSize 0 rlineto}for %left xBBoxSide 0 moveto dup yBBoxSide{xBBoxSide exch moveto TickSize neg 0 rlineto}for %rt 1 add xBBoxSide exch div dup 0 0 moveto dup xBBoxSide{0 moveto 0 TickSize rlineto}for %bottom 0 yBBoxSide moveto dup xBBoxSide{yBBoxSide moveto 0 TickSize neg rlineto}for %top stroke grestore }def % draws grid on the bounding box. Called as: num_x_ticks num_y_ticks Grid -- % For thicker tick marks, uncomment the end of the first line /Grid{ gsave % 1 setlinewidth 1 add yBBoxSide exch div 0 0 moveto dup yBBoxSide{0 exch moveto xBBoxSide 0 rlineto}for % left to right 1 add xBBoxSide exch div 0 0 moveto dup xBBoxSide{0 moveto 0 yBBoxSide rlineto}for % bottom to top stroke grestore }def % draws box of sidelength (xBBoxSide, yBBoxSide) beginning at origin /BoundingBox{ gsave 1 setlinewidth 0 0 moveto xBBoxSide 0 rlineto 0 yBBoxSide rlineto xBBoxSide neg 0 rlineto closepath stroke grestore }def % macros for string positioning and manipulation /Display{0 Yp moveto show /Yp Yp HeadPtSize sub def }def % display header info % centers arbitrary string at current point. Called as: (str) CenterString (str) /CenterString { dup stringwidth pop 2 div currentpoint 3 -2 roll exch sub exch moveto }def % vertically centers string at current point. Called as:fontheight (str) VCenterString (str) /VCenterString{dup length .5 add 2 div 3 -1 roll mul currentpoint 3 -1 roll add moveto }def % prints figure titles centered at given pt % requires: CenterString; TitlePtSize; TitleFont % called as: x y (title) (caption) FigureTitle -- /FigureTitle { gsave TitleFont setfont 4 2 roll moveto gsave exch CenterString show grestore currentpoint TitlePtSize sub moveto CenterString show grestore }def % prints x-axis label % requires: CenterString; LabelPtSize; LabelFont % called as: (label) XLabel -- /XLabel { gsave LabelFont setfont xBBoxSide 2 div LabelPtSize -3 mul moveto CenterString show grestore }def % prints y-axis label horizontally % requires: CenterString; LabelFont % called as: (label) HorYLabel -- /HorYLabel { gsave LabelFont setfont dup stringwidth pop -2 div HorSpace 2 mul sub yBBoxSide 2 div moveto CenterString show grestore }def % prints list of n strings along x axis at evenly spaced intervals % requires: LabelFont; LabelPtSize; CenterString % called as: (s_1) (s_2) ... (s_n) n PlaceXLabels -- /PlaceXLabels { gsave LabelFont setfont dup 1 le { % if n==1, center string pop xBBoxSide 2 div LabelPtSize -1.5 mul moveto CenterString show } { % else place labels at intervals of length xBBoxSide/(n-1) 1 sub xBBoxSide exch div % set length -1 mul xBBoxSide exch 0{ % for i=xBBoxside to 0 step (-length) LabelPtSize -1.5 mul moveto CenterString show }for } ifelse grestore}def % prints list of n strings along y axis at evenly spaced intervals % requires: LabelFont; LabelPtSize;CenterString % called as: (s_1) (s_2) ... (s_n) n PlaceYLabels -- /PlaceYLabels { gsave LabelFont setfont dup 1 le { % if n==1, center string pop dup stringwidth pop -2 div HorSpace sub yBBoxSide 2 div moveto CenterString show } { % else place labels at intervals of length yBBoxSide/(n-1) 1 sub yBBoxSide exch div % set length -1 mul yBBoxSide exch 0 { % for i=yBBoxside to 0 step (-length) exch dup stringwidth pop -2 div HorSpace sub 3 -1 roll moveto CenterString show }for } ifelse grestore}def % % The following macros are not explicitly used by dstool but may be useful % % prints y-axis label vertically % requires: VCenterString; LabelPtSize; LabelFont % called as: (label) VerYLabel -- /VerYLabel { gsave LabelFont setfont LabelPtSize -1.5 mul yBBoxSide 2 div moveto LabelPtSize exch VCenterString LabelPtSize exch vshow grestore }def % The following proc is from PostScript Tutorial and Cookbook by Adobe Systems Inc., p. 165 /vshowdict 4 dict def % called as: vert_space (string) vshow -- /vshow{ vshowdict begin /thestring exch def /lineskip exch def thestring { /charcode exch def /thechar ( ) dup 0 charcode put def 0 lineskip neg rmoveto gsave thechar stringwidth pop 2 div neg 0 rmoveto thechar show grestore } forall end }def % prints annotations on graph. Called as: (label) xpos ypos Annotation -- % requires: CenterString; LabelPtSize; LabelFont % xpos and ypos are between 0 and 1 /Annotation{ gsave LabelFont setfont yBBoxSide mul exch xBBoxSide mul exch moveto CenterString show grestore }def % declare global constants (edit with care). 72 points = 1 inch /x0 0 def /xf 1 def /y0 0 def /yf 1 def % dstool scales output into [0,1] x [0,1] /TickSize 5 def % size (in pts) of tick marks /linewidth 0.005 def % width of line for plotting (smallest resolution=0) /HeadPtSize 6 def % print the header in this font size /HorSpace 8 def % pts left of BBox for hor labels /Sm 1 def % typical sizes of symbols /Md 2 def /Lg 4 def /XL 6 def % The below are model-specific definitions. Written by dstool. /SetFonts { /LabelPtSize 10 def /TitlePtSize 14 def /LabelFont /Times-Roman findfont LabelPtSize scalefont def /TitleFont /Times-Roman findfont TitlePtSize scalefont def } def /xBBoxSide 450 def % length (in pts) of bounding box /yBBoxSide 450 def % height (in pts) of bounding box /xorigin 80 def % displacement (in pts) from left side of page /yorigin 145 def % displacement (in pts) from bottom of page % The below are not model specific but depend on the above definitions /Yp yBBoxSide 60 add 6 HeadPtSize mul add def % print 6 header lines at this height /xscale xBBoxSide xf x0 sub div def % horizontal scaling factor /yscale yBBoxSide yf y0 sub div def % vertical scaling factor /xscalei 1 xscale div def % horizontal scaling factor inverse /yscalei 1 yscale div def % vertical scaling factor inverse %%%%EndProlog gsave xorigin yorigin translate linewidth setlinewidth SetFonts xBBoxSide 2 div yBBoxSide TitlePtSize 2 mul add (Figure 2) () FigureTitle (-3.14159) (3.14159) 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0.845846 0.744695 Md Dot 0.478331 0.686943 Md Dot 0.144861 0.570482 Md Dot 0.841631 0.463713 Md Dot 0.264953 0.859966 Md Dot 0.917783 0.731012 Md Dot 0.617817 0.574698 Md Dot 0.122584 0.747279 Md Dot 0.877956 0.590265 Md Dot 0.386280 0.753662 Md Dot 0.089081 0.595025 Md Dot 0.740558 0.468915 Md Dot 0.162516 0.871092 Md Dot 0.856513 0.764775 Md Dot 0.527378 0.642016 Md Dot 0.152994 0.595801 Md Dot 0.871280 0.479992 Md Dot 0.307808 0.815216 Md Dot 0.987950 0.669462 Md Dot 0.661146 0.527827 Md Dot 0.128600 0.796920 Md Dot 0.872441 0.646704 Md Dot 0.426633 0.715484 Md Dot 0.135360 0.548548 Md Dot 0.807975 0.449376 Md Dot 0.222596 0.878515 Md Dot 0.881776 0.766445 Md Dot 0.588628 0.602475 Md Dot 0.126085 0.705876 Md Dot 0.876162 0.552113 Md Dot 0.355614 0.785675 Md Dot 0.042307 0.638123 Md Dot 0.707759 0.498524 Md Dot 0.139560 0.858435 Md Dot 0.850506 0.744862 Md Dot 0.488799 0.673799 Md Dot 0.153070 0.563533 Md Dot 0.858629 0.463074 Md Dot 0.285426 0.834568 Md Dot 0.954624 0.693594 Md Dot 0.636806 0.548798 Md Dot 0.127806 0.762598 Md Dot 0.881553 0.606137 Md Dot 0.404705 0.729627 Md Dot 0.120705 0.557720 Md Dot 0.776909 0.450442 Md Dot 0.192005 0.878995 Md Dot 0.866919 0.773599 Md Dot 0.563740 0.618662 Md Dot 0.137290 0.659122 Md Dot 0.873212 0.516859 Md Dot 0.329420 0.807715 Md Dot 0.006101 0.664025 Md Dot 0.683123 0.519729 Md Dot 0.127916 0.836765 Md Dot 0.853891 0.706750 Md Dot 0.452968 0.704995 Md Dot 0.138085 0.563339 Md Dot 0.822004 0.457735 Md Dot 0.239713 0.877102 Md Dot 0.890912 0.761941 Md Dot 0.602971 0.594633 Md Dot 0.119150 0.735072 Md Dot 0.873977 0.578215 Md Dot 0.371239 0.774283 Md Dot 0.060249 0.625662 Md Dot 0.722269 0.488710 Md Dot 0.147740 0.870222 Md Dot 0.849868 0.767589 Md Dot 0.515326 0.654464 Md Dot 0.151339 0.588507 Md Dot 0.864518 0.475480 Md Dot 0.297514 0.827548 Md Dot 0.968554 0.685447 Md Dot 0.650089 0.540414 Md Dot 0.125608 0.786102 Md Dot 0.873918 0.633259 Md Dot 0.416597 0.725032 Md Dot 0.125087 0.559070 Md Dot 0.788284 0.451974 Md Dot 0.202340 0.883604 Md Dot 0.869426 0.781264 Md Dot 0.579992 0.612955 Md Dot 0.125262 0.697815 Md Dot 0.872601 0.545292 Md Dot 0.346194 0.798482 Md Dot 0.023231 0.655488 Md Dot 0.699062 0.511416 Md Dot 0.132503 0.858377 Md Dot 0.846944 0.746757 Md Dot 0.483155 0.681646 Md Dot 0.147830 0.568807 Md Dot 0.847953 0.463860 Md Dot 0.272737 0.851319 Md Dot 0.930232 0.717564 Md Dot 0.624113 0.565777 Md Dot 0.124477 0.751903 Md Dot 0.879619 0.594958 Md Dot 0.392494 0.745022 Md Dot 0.100736 0.581495 Md Dot 0.751030 0.461314 Md Dot 0.170647 0.873384 Md Dot 0.859413 0.767309 Md Dot 0.537541 0.634500 Md Dot 0.150143 0.610252 Md Dot 0.871920 0.487525 Md Dot 0.312151 0.814249 Md Dot 0.991080 0.669055 Md Dot 0.665703 0.526513 Md Dot 0.127712 0.806201 Md Dot 0.867945 0.659294 Md Dot 0.430959 0.715303 Md Dot 0.134203 0.553814 Md Dot 0.807640 0.451642 Md Dot 0.222433 0.880665 Md Dot 0.880340 0.770964 Md Dot 0.591441 0.603241 Md Dot 0.122578 0.715363 Md Dot 0.874460 0.560146 Md Dot 0.358832 0.785356 Md Dot 0.043443 0.639038 Md Dot 0.710669 0.498761 Md Dot 0.140182 0.863758 Md Dot 0.848669 0.755836 Md Dot 0.497902 0.667922 Md Dot 0.152208 0.572441 Md Dot 0.859843 0.467246 Md Dot 0.288458 0.833868 Md Dot 0.956923 0.693080 Md Dot 0.640182 0.547544 Md Dot 0.126468 0.769953 Md Dot 0.878957 0.614267 Md Dot 0.407723 0.729291 Md Dot 0.120836 0.559750 Md Dot 0.778568 0.451506 Md Dot 0.193348 0.880799 Md Dot 0.866669 0.777249 Md Dot 0.568058 0.617639 Md Dot 0.133523 0.670372 Md Dot 0.872768 0.524557 Md Dot 0.333377 0.806524 Md Dot 0.008989 0.663493 Md Dot 0.686964 0.518707 Md Dot 0.128266 0.843524 Md Dot 0.850981 0.718787 Md Dot 0.460068 0.700260 Md Dot 0.140207 0.564894 Md Dot 0.827888 0.459233 Md Dot 0.247041 0.872981 Md Dot 0.897561 0.753673 Md Dot 0.606321 0.589592 Md Dot 0.120368 0.736944 Md Dot 0.875410 0.580041 Md Dot 0.374670 0.768941 Md Dot 0.067787 0.617678 Md Dot 0.725968 0.483310 Md Dot 0.151069 0.869177 Md Dot 0.851821 0.764186 Md Dot 0.515593 0.652767 Md Dot 0.153086 0.585455 Md Dot 0.867100 0.474363 Md Dot 0.300199 0.822565 Md Dot 0.975654 0.678644 Md Dot 0.653005 0.535544 Md Dot 0.127341 0.786945 Md Dot 0.875058 0.634168 Md Dot 0.419292 0.720829 Md Dot 0.130137 0.552303 Md Dot 0.796694 0.449729 Md Dot 0.210729 0.881457 Md Dot 0.874399 0.774668 Md Dot 0.582753 0.608786 Md Dot 0.126304 0.698964 Md Dot 0.874306 0.546324 Md Dot 0.349191 0.793608 Md Dot 0.030361 0.648729 Md Dot 0.701824 0.506447 Md Dot 0.135085 0.857446 Md Dot 0.848624 0.744102 Md Dot 0.483741 0.679718 Md Dot 0.149823 0.565583 Md Dot 0.851447 0.462919 Md Dot 0.276677 0.845775 Md Dot 0.937996 0.709313 Md Dot 0.627699 0.560265 Md Dot 0.125909 0.753920 Md Dot 0.880852 0.597019 Md Dot 0.396006 0.739793 Md Dot 0.107661 0.572962 Md Dot 0.758434 0.456977 Md Dot 0.176538 0.874973 Md Dot 0.861508 0.769046 Md Dot 0.544807 0.629588 Md Dot 0.147294 0.622285 Md Dot 0.872174 0.494138 Md Dot 0.315749 0.813567 Md Dot 0.993469 0.668952 Md Dot 0.669424 0.525632 Md Dot 0.127090 0.813814 Md Dot 0.864131 0.670232 Md Dot 0.434890 0.714778 Md Dot 0.133473 0.558025 Md Dot 0.808031 0.453570 Md Dot 0.223050 0.882207 Md Dot 0.879686 0.774151 Md Dot 0.594228 0.603489 Md Dot 0.119652 0.723804 Md Dot 0.872921 0.567634 Md Dot 0.361990 0.784656 Md Dot 0.044996 0.639346 Md Dot 0.713505 0.498626 Md Dot 0.141071 0.868342 Md Dot 0.847430 0.765583 Md Dot 0.507081 0.662037 Md Dot 0.150595 0.582981 Md Dot 0.860453 0.472251 Md Dot 0.291175 0.834166 Md Dot 0.957913 0.694035 Md Dot 0.643200 0.547282 Md Dot 0.124789 0.777599 Md Dot 0.875693 0.623112 Md Dot 0.410353 0.730033 Md Dot 0.119489 0.563994 Md Dot 0.778289 0.453490 Md Dot 0.192927 0.882847 Md Dot 0.865530 0.781986 Md Dot 0.571724 0.617567 Md Dot 0.129597 0.681251 Md Dot 0.871955 0.532390 Md Dot 0.337081 0.805732 Md Dot 0.011210 0.663568 Md Dot 0.690512 0.518176 Md Dot 0.128587 0.850074 Md Dot 0.848289 0.731117 Md Dot 0.468105 0.694583 Md Dot 0.142424 0.567015 Md Dot 0.834224 0.461077 Md Dot 0.255151 0.867551 Md Dot 0.906029 0.743695 Md Dot 0.610848 0.583266 Md Dot 0.121599 0.740289 Md Dot 0.876878 0.583312 Md Dot 0.379280 0.762328 Md Dot 0.077065 0.607761 Md Dot 0.731405 0.476816 Md Dot 0.155568 0.868961 Md Dot 0.854061 0.762290 Md Dot 0.518426 0.649216 Md Dot 0.154309 0.585560 Md Dot 0.869883 0.474799 Md Dot 0.303777 0.817373 Md Dot 0.983488 0.671756 Md Dot 0.656858 0.530315 Md Dot 0.128888 0.789612 Md Dot 0.875399 0.637354 Md Dot 0.422798 0.716617 Md Dot 0.134921 0.546477 Md Dot 0.805871 0.448224 Md Dot 0.220246 0.878189 Md Dot 0.880858 0.766101 Md Dot 0.586075 0.603487 Md Dot 0.127775 0.699976 Md Dot 0.876486 0.547274 Md Dot 0.352778 0.787519 Md Dot 0.039210 0.640038 Md Dot 0.705145 0.500152 Md Dot 0.138349 0.855778 Md Dot 0.851020 0.739815 Md Dot 0.483963 0.677512 Md Dot 0.152469 0.560837 Md Dot 0.856206 0.461530 Md Dot 0.281953 0.837903 Md Dot 0.949171 0.698034 Md Dot 0.633024 0.552431 Md Dot 0.127812 0.757408 Md Dot 0.882304 0.600618 Md Dot 0.401164 0.732566 Md Dot 0.117061 0.561243 Md Dot 0.770695 0.451709 Md Dot 0.186675 0.877404 Md Dot 0.865181 0.771519 Md Dot 0.556828 0.622169 Md Dot 0.141487 0.644555 Md Dot 0.872893 0.507413 Md Dot 0.323677 0.810493 Md Dot 0.000478 0.666516 Md Dot 0.677450 0.522424 Md Dot 0.127147 0.827888 Md Dot 0.857608 0.691948 Md Dot 0.444707 0.710358 Md Dot 0.135177 0.562326 Md Dot 0.814462 0.456304 Md Dot 0.230665 0.881268 Md Dot 0.883798 0.771188 Md Dot 0.599558 0.600091 Md Dot 0.117614 0.733727 Md Dot 0.872222 0.576891 Md Dot 0.367739 0.780107 Md Dot 0.052002 0.634290 Md Dot 0.718756 0.494689 Md Dot 0.144319 0.872022 Md Dot 0.847697 0.772818 Md Dot 0.516841 0.655170 Md Dot 0.148918 0.594320 Md Dot 0.861893 0.477988 Md Dot 0.295342 0.832819 Md Dot 0.961434 0.692853 Md Dot 0.647748 0.545422 Md Dot 0.123496 0.786882 Md Dot 0.871955 0.634373 Md Dot 0.414402 0.729605 Md Dot 0.119378 0.567187 Md Dot 0.780264 0.455159 Md Dot 0.194597 0.885279 Md Dot 0.865212 0.787119 Md Dot 0.577875 0.616487 Md Dot 0.124119 0.697560 Md Dot 0.871065 0.544990 Md Dot 0.343904 0.802539 Md Dot 0.017347 0.661114 Md Dot 0.696933 0.515543 Md Dot 0.130399 0.859379 Md Dot 0.845516 0.749463 Md Dot 0.483169 0.682835 Md Dot 0.146291 0.571651 Md Dot 0.845534 0.464810 Md Dot 0.270090 0.855185 Md Dot 0.924997 0.723535 Md Dot 0.622019 0.569548 Md Dot 0.123218 0.751532 Md Dot 0.878421 0.594592 Md Dot 0.390426 0.748685 Md Dot 0.095838 0.587676 Md Dot 0.746714 0.464672 Md Dot 0.167163 0.872956 Md Dot 0.857936 0.767428 Md Dot 0.534198 0.637282 Md Dot 0.150793 0.606008 Md Dot 0.871255 0.485203 Md Dot 0.310212 0.815555 Md Dot 0.988639 0.670529 Md Dot 0.663679 0.527911 Md Dot 0.127573 0.803191 Md Dot 0.869119 0.655169 Md Dot 0.429022 0.716253 Md Dot 0.133576 0.553483 Md Dot 0.805906 0.451288 Md Dot 0.220511 0.880972 Md Dot 0.879251 0.771901 Md Dot 0.590217 0.604179 Md Dot 0.122945 0.713310 Md Dot 0.874393 0.558354 Md Dot 0.357488 0.786635 Md Dot 0.041474 0.640614 Md Dot 0.709433 0.499968 Md Dot 0.139382 0.863109 Md Dot 0.848527 0.754690 Md Dot 0.496183 0.669390 Md Dot 0.151977 0.571484 Md Dot 0.858922 0.466676 Md Dot 0.287121 0.835246 Md Dot 0.954712 0.694904 Md Dot 0.638726 0.549002 Md Dot 0.126374 0.768087 Md Dot 0.879256 0.612190 Md Dot 0.406384 0.730416 Md Dot 0.119488 0.561042 Md Dot 0.776253 0.451952 Md Dot 0.191283 0.880384 Md Dot 0.865913 0.776874 Md Dot 0.565755 0.618829 Md Dot 0.134877 0.665560 Md Dot 0.872564 0.521176 Md Dot 0.331097 0.808042 Md Dot 0.006285 0.665085 Md Dot 0.684757 0.520145 Md Dot 0.127611 0.840730 Md Dot 0.851875 0.713843 Md Dot 0.456524 0.703115 Md Dot 0.138555 0.565179 Md Dot 0.824064 0.458847 Md Dot 0.242383 0.876189 Md Dot 0.892852 0.760006 Md Dot 0.604623 0.593147 Md Dot 0.118915 0.737502 Md Dot 0.873948 0.580562 Md Dot 0.372960 0.772593 Md Dot 0.062721 0.623515 Md Dot 0.723947 0.487178 Md Dot 0.148937 0.870882 Md Dot 0.850187 0.768672 Md Dot 0.517457 0.652838 Md Dot 0.151044 0.590877 Md Dot 0.864912 0.476693 Md Dot 0.298518 0.827028 Md Dot 0.969635 0.684887 Md Dot 0.651180 0.539803 Md Dot 0.125453 0.788063 Md Dot 0.873157 0.635693 Md Dot 0.417571 0.724764 Md Dot 0.125240 0.559529 Md Dot 0.788937 0.452245 Md Dot 0.202972 0.883949 Md Dot 0.869544 0.781887 Md Dot 0.581109 0.612674 Md Dot 0.124368 0.700625 Md Dot 0.872429 0.547565 Md Dot 0.347459 0.797784 Md Dot 0.024477 0.654872 Md Dot 0.700231 0.510831 Md Dot 0.132946 0.859844 Md Dot 0.846602 0.749696 Md Dot 0.485861 0.679556 Md Dot 0.148254 0.570157 Md Dot 0.849534 0.464704 Md Dot 0.274955 0.849364 Md Dot 0.933358 0.714727 Md Dot 0.626257 0.563616 Md Dot 0.124470 0.754749 Md Dot 0.879398 0.597903 Md Dot 0.394570 0.743066 Md Dot 0.103330 0.578897 Md Dot 0.754168 0.459962 Md Dot 0.172995 0.874760 Md Dot 0.859920 0.769626 Md Dot 0.541801 0.632115 Md Dot 0.147954 0.618261 Md Dot 0.871465 0.491811 Md Dot 0.313691 0.815147 Md Dot 0.990664 0.670786 Md Dot 0.667301 0.527266 Md Dot 0.126749 0.810973 Md Dot 0.865164 0.666153 Md Dot 0.432730 0.716140 Md Dot 0.132448 0.558226 Md Dot 0.805631 0.453379 Md Dot 0.220388 0.882943 Md Dot 0.878023 0.776147 Md Dot 0.593103 0.604827 Md Dot 0.119548 0.722685 Md Dot 0.872593 0.566604 Md Dot 0.360791 0.786223 Md Dot 0.042697 0.641475 Md Dot 0.712398 0.500181 Md Dot 0.140144 0.868383 Md Dot 0.846987 0.765979 Md Dot 0.506541 0.662777 Md Dot 0.150226 0.583221 Md Dot 0.859726 0.472263 Md Dot 0.290285 0.835438 Md Dot 0.956044 0.695775 Md Dot 0.642246 0.548545 Md Dot 0.124484 0.776828 Md Dot 0.875629 0.622227 Md Dot 0.409471 0.731082 Md Dot 0.118211 0.565498 Md Dot 0.776384 0.454083 Md Dot 0.191188 0.882766 Md Dot 0.864783 0.782284 Md Dot 0.570343 0.618456 Md Dot 0.130217 0.678699 Md Dot 0.871727 0.530480 Md Dot 0.335597 0.806998 Md Dot 0.009120 0.665020 Md Dot 0.689094 0.519378 Md Dot 0.127963 0.848702 Md Dot 0.848535 0.728615 Md Dot 0.465829 0.696621 Md Dot 0.141294 0.567317 Md Dot 0.831672 0.460860 Md Dot 0.251998 0.870206 Md Dot 0.902224 0.748475 Md Dot 0.609277 0.586100 Md Dot 0.120648 0.740077 Md Dot 0.875895 0.583096 Md Dot 0.377695 0.765226 Md Dot 0.073059 0.612396 Md Dot 0.729297 0.479777 Md Dot 0.153667 0.869641 Md Dot 0.852875 0.764375 Md Dot 0.518307 0.650211 Md Dot 0.153258 0.587443 Md Dot 0.868307 0.475486 Md Dot 0.302158 0.820467 Md Dot 0.979058 0.675968 Md Dot 0.655122 0.533365 Md Dot 0.127732 0.789360 Md Dot 0.874556 0.637131 Md Dot 0.421197 0.719261 Md Dot 0.131784 0.550829 Md Dot 0.800037 0.449440 Md Dot 0.214164 0.880742 Md Dot 0.876411 0.772475 Md Dot 0.584463 0.607026 Md Dot 0.126171 0.701081 Md Dot 0.874827 0.548078 Md Dot 0.351067 0.791355 Md Dot 0.033779 0.645769 Md Dot 0.703552 0.504210 Md Dot 0.136342 0.857870 Md Dot 0.849092 0.744617 Md Dot 0.485381 0.677893 Md Dot 0.150788 0.565085 Md Dot 0.853603 0.463015 Md Dot 0.279328 0.842530 Md Dot 0.942802 0.704674 Md Dot 0.630365 0.556924 Md Dot 0.126421 0.756505 Md Dot 0.881087 0.599701 Md Dot 0.398576 0.736762 Md Dot 0.111612 0.568396 Md Dot 0.763525 0.454843 Md Dot 0.180619 0.876421 Md Dot 0.862779 0.771049 Md Dot 0.550491 0.626295 Md Dot 0.144355 0.633123 Md Dot 0.872161 0.500387 Md Dot 0.318962 0.813096 Md Dot 0.995385 0.669112 Md Dot 0.672704 0.525054 Md Dot 0.126598 0.820612 Md Dot 0.860669 0.680519 Md Dot 0.438824 0.713844 Md Dot 0.133142 0.561536 Md Dot 0.809126 0.455293 Md Dot 0.224460 0.883224 Md Dot 0.879724 0.776129 Md Dot 0.596962 0.603282 Md Dot 0.117257 0.731275 Md Dot 0.871533 0.574542 Md Dot 0.365024 0.783692 Md Dot 0.046809 0.639203 Md Dot 0.716214 0.498225 Md Dot 0.142138 0.872250 Md Dot 0.846652 0.774094 Md Dot 0.515980 0.656543 Md Dot 0.148179 0.594962 Md Dot 0.860617 0.478137 Md Dot 0.293818 0.835169 Md Dot 0.958028 0.696071 Md Dot 0.646118 0.547722 Md Dot 0.122826 0.785802 Md Dot 0.871745 0.633088 Md Dot 0.412901 0.731522 Md Dot 0.117075 0.569954 Md Dot 0.777006 0.456284 Md Dot 0.191547 0.885387 Md Dot 0.863835 0.788244 Md Dot 0.576094 0.617866 Md Dot 0.124660 0.694687 Md Dot 0.870715 0.542662 Md Dot 0.341915 0.804512 Md Dot 0.014237 0.663475 Md Dot 0.695072 0.517417 Md Dot 0.129328 0.858099 Md Dot 0.845460 0.747090 Md Dot 0.480268 0.685496 Md Dot 0.145208 0.571360 Md Dot 0.842851 0.464292 Md Dot 0.266629 0.858631 Md Dot 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0.859170 0.751367 Md Dot 0.516367 0.646545 Md Dot 0.159683 0.573593 Md Dot 0.877977 0.470353 Md Dot 0.311554 0.801146 Md Dot 0.006577 0.649375 Md Dot 0.664835 0.514099 Md Dot 0.135689 0.787491 Md Dot 0.881717 0.634206 Md Dot 0.430269 0.702082 Md Dot 0.152166 0.520734 Md Dot 0.845423 0.443565 Md Dot 0.263885 0.843982 Md Dot 0.933178 0.705045 Md Dot 0.612767 0.563090 Md Dot 0.136413 0.714950 Md Dot 0.890581 0.560406 Md Dot 0.381148 0.742716 Md Dot 0.103045 0.569240 Md Dot 0.745197 0.454990 Md Dot 0.168981 0.859561 Md Dot 0.865594 0.739241 Md Dot 0.515583 0.642039 Md Dot 0.165904 0.560957 Md Dot 0.889891 0.466420 Md Dot 0.323777 0.776928 Md Dot 0.041495 0.614292 Md Dot 0.676952 0.488706 Md Dot 0.146732 0.781980 Md Dot 0.893139 0.626855 Md Dot 0.441914 0.678954 Md Dot 0.178610 0.479368 Md Dot 0.935819 0.441101 Md Dot 0.364146 0.680515 Md Dot 0.173650 0.429398 Md Dot 0.951563 0.426633 Md Dot 0.375951 0.646681 Md Dot 0.222268 0.352062 Md Dot 0.263334 0.202334 Md Dot 0.188206 0.873016 Md Dot 0.868176 0.761691 Md Dot 0.550753 0.622165 Md Dot 0.148453 0.625713 Md Dot 0.875989 0.496526 Md Dot 0.321753 0.805216 Md Dot 0.006242 0.657908 Md Dot 0.675477 0.517182 Md Dot 0.130421 0.816751 Md Dot 0.865085 0.674107 Md Dot 0.440486 0.707728 Md Dot 0.140645 0.549398 Md Dot 0.821906 0.451564 Md Dot 0.238470 0.872558 Md Dot 0.893726 0.753117 Md Dot 0.597275 0.592774 Md Dot 0.126113 0.715708 Md Dot 0.878638 0.560619 Md Dot 0.364947 0.773964 Md Dot 0.059646 0.621802 Md Dot 0.716813 0.486791 Md Dot 0.146550 0.859357 Md Dot 0.853713 0.744766 Md Dot 0.495822 0.665458 Md Dot 0.157240 0.561299 Md Dot 0.868244 0.463521 Md Dot 0.297118 0.818243 Md Dot 0.979191 0.671693 Md Dot 0.649626 0.532172 Md Dot 0.131407 0.773584 Md Dot 0.882469 0.618130 Md Dot 0.416623 0.715781 Md Dot 0.137456 0.537366 Md Dot 0.808591 0.444714 Md Dot 0.222898 0.874054 Md Dot 0.884909 0.757352 Md Dot 0.583438 0.600871 Md Dot 0.132971 0.687633 Md Dot 0.879054 0.537723 Md Dot 0.349908 0.785850 Md Dot 0.040615 0.636041 Md Dot 0.702521 0.497877 Md Dot 0.138922 0.848158 Md Dot 0.854845 0.724891 Md Dot 0.474897 0.682399 Md Dot 0.153419 0.552002 Md Dot 0.855772 0.457673 Md Dot 0.280107 0.837065 Md Dot 0.949147 0.696486 Md Dot 0.630886 0.552213 Md Dot 0.129553 0.751153 Md Dot 0.884631 0.594126 Md Dot 0.399215 0.731914 Md Dot 0.118017 0.558170 Md Dot 0.770802 0.450264 Md Dot 0.186967 0.875829 Md Dot 0.866099 0.768008 Md Dot 0.554303 0.622381 Md Dot 0.144144 0.637175 Md Dot 0.873794 0.502987 Md Dot 0.322419 0.809199 Md Dot 0.001589 0.664171 Md Dot 0.676186 0.521135 Md Dot 0.128069 0.823439 Md Dot 0.860317 0.684684 Md Dot 0.442384 0.710287 Md Dot 0.136360 0.558489 Md Dot 0.815355 0.454682 Md Dot 0.231448 0.879427 Md Dot 0.885378 0.767285 Md Dot 0.597873 0.598982 Md Dot 0.120134 0.727210 Md Dot 0.874032 0.570800 Md Dot 0.365845 0.779484 Md Dot 0.052525 0.632183 Md Dot 0.717138 0.493472 Md Dot 0.144164 0.868013 Md Dot 0.848991 0.763844 Md Dot 0.508549 0.659757 Md Dot 0.151905 0.581754 Md Dot 0.862896 0.472012 Md Dot 0.294074 0.829791 Md Dot 0.964313 0.688073 Md Dot 0.646338 0.542949 Md Dot 0.125942 0.779861 Md Dot 0.876062 0.625710 Md Dot 0.413251 0.726418 Md Dot 0.123864 0.558686 Md Dot 0.785108 0.451521 Md Dot 0.199318 0.882644 Md Dot 0.868520 0.779847 Md Dot 0.575945 0.614406 Md Dot 0.128140 0.688317 Md Dot 0.872846 0.537814 Md Dot 0.341685 0.801327 Md Dot 0.018271 0.658295 Md Dot 0.694862 0.513941 Md Dot 0.130856 0.853450 Md Dot 0.848078 0.737180 Md Dot 0.474527 0.688578 Md Dot 0.145650 0.566084 Md Dot 0.841549 0.461698 Md Dot 0.264238 0.859051 Md Dot 0.918311 0.729397 Md Dot 0.616701 0.574244 Md Dot 0.123870 0.743457 Md Dot 0.879266 0.586444 Md Dot 0.385173 0.753190 Md Dot 0.089651 0.593329 Md Dot 0.739971 0.467976 Md Dot 0.162470 0.869262 Md Dot 0.857291 0.760825 Md Dot 0.524103 0.643291 Md Dot 0.155122 0.588914 Md Dot 0.873037 0.476842 Md Dot 0.308517 0.811489 Md Dot 0.992743 0.664101 Md Dot 0.661867 0.524256 Md Dot 0.130542 0.793911 Md Dot 0.875130 0.642641 Md Dot 0.427317 0.711971 Md Dot 0.139866 0.541003 Md Dot 0.816460 0.447266 Md Dot 0.231687 0.872665 Md Dot 0.890285 0.753755 Md Dot 0.590574 0.595924 Md Dot 0.130033 0.700897 Md Dot 0.879606 0.548199 Md Dot 0.357616 0.778981 Md Dot 0.051517 0.627384 Md Dot 0.709824 0.491206 Md Dot 0.143023 0.853034 Md Dot 0.854777 0.733161 Md Dot 0.484321 0.674051 Md Dot 0.156608 0.553357 Md Dot 0.864385 0.459552 Md Dot 0.291019 0.823841 Md Dot 0.969614 0.678746 Md Dot 0.642905 0.538339 Md Dot 0.131223 0.764335 Md Dot 0.884401 0.607896 Md Dot 0.410566 0.720131 Md Dot 0.132669 0.541350 Md Dot 0.797610 0.445064 Md Dot 0.211594 0.876768 Md Dot 0.877556 0.764544 Md Dot 0.576407 0.607599 Md Dot 0.134188 0.677861 Md Dot 0.876957 0.530280 Md Dot 0.342394 0.794329 Md Dot 0.027478 0.647261 Md Dot 0.695488 0.506424 Md Dot 0.134361 0.845378 Md Dot 0.853590 0.720801 Md Dot 0.467662 0.690346 Md Dot 0.148305 0.555926 Md Dot 0.844010 0.457571 Md Dot 0.265989 0.853719 Md Dot 0.924416 0.720697 Md Dot 0.617056 0.569692 Md Dot 0.127198 0.737682 Md Dot 0.882568 0.580834 Md Dot 0.385536 0.748673 Md Dot 0.095653 0.584308 Md Dot 0.743164 0.462904 Md Dot 0.165620 0.866897 Md Dot 0.859964 0.754898 Md Dot 0.522705 0.642054 Md Dot 0.158102 0.582033 Md Dot 0.877132 0.474121 Md Dot 0.312225 0.803162 Md Dot 0.004487 0.652523 Md Dot 0.665607 0.515931 Md Dot 0.134228 0.791873 Md Dot 0.878858 0.639740 Md Dot 0.430804 0.704426 Md Dot 0.148850 0.527252 Md Dot 0.836997 0.444734 Md Dot 0.254492 0.854167 Md Dot 0.917995 0.720585 Md Dot 0.604122 0.574430 Md Dot 0.135243 0.706833 Md Dot 0.887618 0.553465 Md Dot 0.372075 0.755080 Md Dot 0.085627 0.589588 Md Dot 0.728370 0.466439 Md Dot 0.157903 0.851913 Md Dot 0.863675 0.727355 Md Dot 0.496652 0.656726 Md Dot 0.167399 0.542548 Md Dot 0.890940 0.458980 Md Dot 0.321713 0.774249 Md Dot 0.043586 0.609741 Md Dot 0.674836 0.486360 Md Dot 0.148969 0.773608 Md Dot 0.898157 0.617224 Md Dot 0.440600 0.675277 Md Dot 0.184183 0.467737 Md Dot 0.961176 0.437785 Md Dot 0.391781 0.627643 Md Dot 0.258582 0.309884 Md Dot 0.175827 0.090989 Md Dot 0.106650 0.523810 Md Dot 0.719277 0.434406 Md Dot 0.170221 0.794622 Md Dot 0.907164 0.640119 Md Dot 0.479523 0.633642 Md Dot 0.213860 0.439452 Md Dot 0.088069 0.383054 Md Dot 0.639274 0.392658 Md Dot 0.248072 0.508296 Md Dot 0.130388 0.368081 Md Dot 0.832777 0.394687 Md Dot 0.245071 0.813936 Md Dot 0.951643 0.660988 Md Dot 0.586559 0.551610 Md Dot 0.175352 0.608639 Md Dot 0.922637 0.491910 Md Dot 0.375327 0.701517 Md Dot 0.153474 0.476376 Md Dot 0.851488 0.428997 Md Dot 0.267388 0.827343 Md Dot 0.952598 0.680678 Md Dot 0.615515 0.548811 Md Dot 0.145531 0.700662 Md Dot 0.899250 0.548982 Md Dot 0.384349 0.728371 Md Dot 0.121953 0.540162 Md Dot 0.769116 0.442021 Md Dot 0.187048 0.865279 Md Dot 0.872033 0.746220 Md Dot 0.539171 0.624215 Md Dot 0.159707 0.593522 Md Dot 0.884860 0.480629 Md Dot 0.324544 0.787421 Md Dot 0.029269 0.631053 Md Dot 0.677842 0.499172 Md Dot 0.140191 0.798374 Md Dot 0.880868 0.647474 Md Dot 0.441765 0.691060 Md Dot 0.162410 0.510111 Md Dot 0.875637 0.444504 Md Dot 0.298697 0.799650 Md Dot 0.001543 0.645055 Md Dot 0.650833 0.515222 Md Dot 0.142348 0.752776 Md Dot 0.897081 0.595530 Md Dot 0.418849 0.697623 Md Dot 0.161166 0.495450 Md Dot 0.873271 0.438997 Md Dot 0.294221 0.801560 Md Dot 0.996804 0.647221 Md Dot 0.645912 0.518106 Md Dot 0.143588 0.743421 Md Dot 0.899514 0.586326 Md Dot 0.414408 0.699333 Md Dot 0.159670 0.494943 Md Dot 0.868537 0.438059 Md Dot 0.288627 0.808775 Md Dot 0.985422 0.656765 Md Dot 0.639866 0.525969 Md Dot 0.142256 0.737990 Md Dot 0.898480 0.581184 Md Dot 0.408555 0.706270 Md Dot 0.151217 0.506288 Md Dot 0.841310 0.437561 Md Dot 0.257883 0.844706 Md Dot 0.929132 0.705689 Md Dot 0.605828 0.566172 Md Dot 0.140590 0.698709 Md Dot 0.892449 0.547126 Md Dot 0.374011 0.746722 Md Dot 0.096774 0.573796 Md Dot 0.735709 0.457527 Md Dot 0.163826 0.850974 Md Dot 0.867634 0.724277 Md Dot 0.501462 0.649972 Md Dot 0.171070 0.539305 Md Dot 0.900744 0.458969 Md Dot 0.332860 0.754288 Md Dot 0.072657 0.577717 Md Dot 0.690349 0.464221 Md Dot 0.157261 0.778628 Md Dot 0.903739 0.622341 Md Dot 0.454706 0.657023 Md Dot 0.201500 0.444928 Md Dot 0.039622 0.413712 Md Dot 0.501586 0.472300 Md Dot 0.610287 0.209497 Md Dot 0.749695 0.247305 Md Dot 0.286219 0.521175 Md Dot 0.222828 0.275251 Md Dot 0.946889 0.207977 Md Dot 0.406410 0.451054 Md Dot 0.164102 0.119348 Md Dot 0.127980 0.515873 Md Dot 0.773939 0.432292 Md Dot 0.192343 0.856666 Md Dot 0.880592 0.728939 Md Dot 0.535693 0.619417 Md Dot 0.169170 0.571980 Md Dot 0.900271 0.472495 Md Dot 0.338660 0.755048 Md Dot 0.074674 0.579584 Md Dot 0.696346 0.464285 Md Dot 0.155955 0.791767 Md Dot 0.896726 0.637795 Md Dot 0.459012 0.659730 Md Dot 0.194961 0.460500 Md Dot 0.003449 0.430344 Md Dot 0.444134 0.542558 Md Dot 0.510405 0.204003 Md Dot 0.940520 0.163590 Md Dot 0.452093 0.390702 Md Dot 0.509030 0.088556 Md Dot 0.549561 0.365079 Md Dot 0.714273 0.226998 Md Dot 0.328078 0.420163 Md Dot 0.065639 0.186055 Md Dot 0.864012 0.359310 Md Dot 0.273517 0.752762 Md Dot 0.030519 0.581664 Md Dot 0.616662 0.489056 Md Dot 0.194019 0.601668 Md Dot 0.961152 0.489129 Md Dot 0.420182 0.619626 Md Dot 0.276116 0.305386 Md Dot 0.189460 0.741270 Md Dot 0.949494 0.582664 Md Dot 0.485164 0.592811 Md Dot 0.267912 0.353187 Md Dot 0.926322 0.212329 Md Dot 0.376623 0.482130 Md Dot 0.942807 0.229843 Md Dot 0.383007 0.488980 Md Dot 0.919118 0.229695 Md Dot 0.356610 0.517807 Md Dot 0.564026 0.177853 Md Dot 0.164890 0.151924 Md Dot 0.255973 0.059647 Md Dot 0.151819 0.660317 Md Dot 0.896473 0.519703 Md Dot 0.360753 0.751401 Md Dot 0.087661 0.577720 Md Dot 0.720474 0.460470 Md Dot 0.158176 0.832018 Md Dot 0.875596 0.693170 Md Dot 0.481042 0.659298 Md Dot 0.178443 0.508916 Md Dot 0.924972 0.450248 Md Dot 0.356088 0.703437 Md Dot 0.142925 0.479837 Md Dot 0.815148 0.424166 Md Dot 0.228643 0.852090 Md Dot 0.905412 0.717104 Md Dot 0.573409 0.587765 Md Dot 0.156857 0.631978 Md Dot 0.894139 0.502078 Md Dot 0.347189 0.763326 Md Dot 0.068272 0.595942 Md Dot 0.702280 0.472783 Md Dot 0.150871 0.815401 Md Dot 0.880300 0.669334 Md Dot 0.463461 0.670595 Md Dot 0.177310 0.497805 Md Dot 0.924417 0.446381 Md Dot 0.353751 0.704546 Md Dot 0.140393 0.482453 Md Dot 0.806532 0.423779 Md Dot 0.220413 0.853925 Md Dot 0.899051 0.720662 Md Dot 0.564772 0.594644 Md Dot 0.160400 0.616404 Md Dot 0.894825 0.493500 Md Dot 0.343189 0.763933 Md Dot 0.065972 0.596060 Md Dot 0.698176 0.473469 Md Dot 0.150749 0.808108 Md Dot 0.884210 0.659234 Md Dot 0.459753 0.670478 Md Dot 0.179976 0.489901 Md Dot 0.935780 0.444517 Md Dot 0.365598 0.680683 Md Dot 0.174244 0.429499 Md Dot 0.953745 0.426663 Md Dot 0.378465 0.642184 Md Dot 0.229880 0.341745 Md Dot 0.366587 0.170800 Md Dot 0.174404 0.174603 Md Dot 0.132455 0.117711 Md Dot 0.927632 0.198415 Md Dot 0.390443 0.458338 Md Dot 0.113155 0.160022 Md Dot 0.178863 0.233715 Md Dot 0.440465 0.150733 Md Dot 0.583438 0.261905 Md Dot 0.797855 0.249381 Md Dot 0.279893 0.580963 Md Dot 0.153344 0.374266 Md Dot 0.927610 0.410995 Md Dot 0.344076 0.691863 Md Dot 0.148083 0.459583 Md Dot 0.837378 0.421208 Md Dot 0.251002 0.835869 Md Dot 0.933972 0.691929 Md Dot 0.596257 0.563267 Md Dot 0.153285 0.663258 Md Dot 0.899806 0.521895 Md Dot 0.366339 0.742385 Md Dot 0.100545 0.561804 Md Dot 0.733272 0.451500 Md Dot 0.165137 0.839943 Md Dot 0.875190 0.704815 Md Dot 0.493872 0.649460 Md Dot 0.179255 0.517297 Md Dot 0.925700 0.453277 Md Dot 0.358290 0.701848 Md Dot 0.145900 0.476255 Md Dot 0.825648 0.424623 Md Dot 0.239201 0.847411 Md Dot 0.915799 0.709298 Md Dot 0.584214 0.578061 Md Dot 0.153446 0.649861 Md Dot 0.895200 0.512903 Md Dot 0.354924 0.757232 Md Dot 0.078552 0.586955 Md Dot 0.711849 0.466385 Md Dot 0.154303 0.824923 Md Dot 0.877242 0.682809 Md Dot 0.472510 0.665176 Md Dot 0.177791 0.503680 Md Dot 0.924256 0.448373 Md Dot 0.354450 0.704930 Md Dot 0.140344 0.483149 Md Dot 0.806351 0.423992 Md Dot 0.220238 0.854176 Md Dot 0.898741 0.721102 Md Dot 0.564706 0.594890 Md Dot 0.160231 0.616668 Md Dot 0.894593 0.493617 Md Dot 0.342973 0.764431 Md Dot 0.065275 0.596888 Md Dot 0.697847 0.474011 Md Dot 0.150492 0.808167 Md Dot 0.883984 0.659342 Md Dot 0.459462 0.670953 Md Dot 0.179532 0.490539 Md Dot 0.934086 0.444634 Md Dot 0.363753 0.684238 Md Dot 0.169092 0.437614 Md Dot 0.927434 0.427913 Md Dot 0.349603 0.696409 Md Dot 0.146956 0.466699 Md Dot 0.831257 0.422429 Md Dot 0.244781 0.841713 Md Dot 0.924496 0.700492 Md Dot 0.589575 0.570930 Md Dot 0.153891 0.654749 Md Dot 0.897765 0.516205 Md Dot 0.360157 0.749814 Md Dot 0.089470 0.574453 Md Dot 0.720988 0.458679 Md Dot 0.159048 0.830522 Md Dot 0.877094 0.690675 Md Dot 0.481139 0.657952 Md Dot 0.180110 0.505748 Md Dot 0.930918 0.449579 Md Dot 0.362432 0.690900 Md Dot 0.160658 0.452321 Md Dot 0.886338 0.427238 Md Dot 0.304965 0.774672 Md Dot 0.033366 0.609141 Md Dot 0.656507 0.490892 Md Dot 0.155648 0.732488 Md Dot 0.913456 0.576075 Md Dot 0.425555 0.673458 Md Dot 0.192068 0.441827 Md Dot 0.008930 0.423473 Md Dot 0.449758 0.531029 Md Dot 0.563670 0.199393 Md Dot 0.125755 0.185817 Md Dot 0.173716 0.243530 Md Dot 0.300304 0.171714 Md Dot 0.018821 0.631746 Md Dot 0.661560 0.503670 Md Dot 0.143968 0.764711 Md Dot 0.896186 0.607811 Md Dot 0.428589 0.688910 Md Dot 0.170336 0.485342 Md Dot 0.905748 0.440302 Md Dot 0.330656 0.742177 Md Dot 0.085120 0.556658 Md Dot 0.694480 0.452296 Md Dot 0.163189 0.770775 Md Dot 0.912294 0.613365 Md Dot 0.459336 0.645094 Md Dot 0.214989 0.423018 Md Dot 0.110653 0.359601 Md Dot 0.752048 0.380695 Md Dot 0.199836 0.765951 Md Dot 0.949807 0.606052 Md Dot 0.512058 0.577679 Md Dot 0.251497 0.395724 Md Dot 0.319720 0.188081 Md Dot 0.899624 0.734921 Md Dot 0.585388 0.589712 Md Dot 0.141250 0.674406 Md Dot 0.885711 0.528475 Md Dot 0.352657 0.773630 Md Dot 0.057116 0.615972 Md Dot 0.705693 0.484484 Md Dot 0.145336 0.837153 Md Dot 0.864351 0.703903 Md Dot 0.470747 0.677986 Md Dot 0.162315 0.532055 Md Dot 0.875845 0.452698 Md Dot 0.301867 0.801819 Md Dot 0.000795 0.648654 Md Dot 0.654434 0.516434 Md Dot 0.139337 0.764166 Md Dot 0.891974 0.607414 Md Dot 0.421742 0.700215 Md Dot 0.157086 0.505244 Md Dot 0.859457 0.440168 Md Dot 0.279001 0.823478 Md Dot 0.963516 0.676416 Md Dot 0.629152 0.541325 Md Dot 0.139186 0.730070 Md Dot 0.895198 0.573852 Md Dot 0.397902 0.720659 Md Dot 0.132832 0.531533 Md Dot 0.793703 0.440525 Md Dot 0.207874 0.872184 Md Dot 0.878589 0.755826 Md Dot 0.567012 0.609083 Md Dot 0.143234 0.651439 Md Dot 0.879063 0.512395 Md Dot 0.334363 0.795326 Md Dot 0.023502 0.646150 Md Dot 0.687803 0.507009 Md Dot 0.134107 0.830548 Md Dot 0.860744 0.695048 Md Dot 0.453973 0.697604 Md Dot 0.147377 0.547253 Md Dot 0.838546 0.453058 Md Dot 0.258240 0.857851 Md Dot 0.916440 0.726925 Md Dot 0.609602 0.575475 Md Dot 0.129125 0.724837 Md Dot 0.883611 0.568889 Md Dot 0.377877 0.754998 Md Dot 0.086602 0.592502 Md Dot 0.733232 0.467801 Md Dot 0.159307 0.860439 Md Dot 0.859902 0.743441 Md Dot 0.508070 0.651464 Md Dot 0.162485 0.561285 Md Dot 0.881369 0.465402 Md Dot 0.313398 0.793931 Md Dot 0.016089 0.638905 Md Dot 0.666500 0.506942 Md Dot 0.139496 0.781805 Md Dot 0.886973 0.627134 Md Dot 0.432141 0.694802 Md Dot 0.161273 0.505115 Md Dot 0.872471 0.442229 Md Dot 0.294358 0.804179 Md Dot 0.993912 0.650975 Md Dot 0.646190 0.520361 Md Dot 0.141788 0.747498 Md Dot 0.897235 0.590304 Md Dot 0.414511 0.701773 Md Dot 0.156424 0.501087 Md Dot 0.857580 0.438389 Md Dot 0.276393 0.825187 Md Dot 0.960159 0.678543 Md Dot 0.626145 0.543606 Md Dot 0.139961 0.724704 Md Dot 0.895800 0.569039 Md Dot 0.394921 0.722847 Md Dot 0.129934 0.534380 Md Dot 0.787109 0.441016 Md Dot 0.201794 0.871849 Md Dot 0.875703 0.756272 Md Dot 0.560982 0.612987 Md Dot 0.146211 0.639841 Md Dot 0.878660 0.505124 Md Dot 0.329737 0.797967 Md Dot 0.018450 0.648931 Md Dot 0.683319 0.509696 Md Dot 0.133148 0.824447 Md Dot 0.863182 0.685437 Md Dot 0.448545 0.700972 Md Dot 0.145840 0.545993 Md Dot 0.834083 0.451865 Md Dot 0.252693 0.861951 Md Dot 0.909797 0.733672 Md Dot 0.605158 0.580465 Md Dot 0.129104 0.719514 Md Dot 0.882809 0.564111 Md Dot 0.373225 0.760541 Md Dot 0.078689 0.600842 Md Dot 0.726953 0.473006 Md Dot 0.155016 0.858022 Md Dot 0.858830 0.739782 Md Dot 0.357591 0.712698 Md Dot 0.133063 0.499056 Md Dot 0.784347 0.426870 Md Dot 0.201047 0.855115 Md Dot 0.886768 0.724813 Md Dot 0.543804 0.611126 Md Dot 0.168599 0.580415 Md Dot 0.900871 0.476357 Md Dot 0.341281 0.753397 Md Dot 0.077856 0.576977 Md Dot 0.699872 0.462320 Md Dot 0.156532 0.796396 Md Dot 0.894902 0.643444 Md Dot 0.461839 0.658901 Md Dot 0.194132 0.464152 Md Dot 0.997845 0.432646 Md Dot 0.436930 0.553650 Md Dot 0.467502 0.210412 Md Dot 0.337516 0.782540 Md Dot 0.040730 0.626350 Md Dot 0.690757 0.493511 Md Dot 0.140878 0.819324 Md Dot 0.871058 0.676419 Md Dot 0.453877 0.688168 Md Dot 0.160005 0.523676 Md Dot 0.868554 0.448377 Md Dot 0.291937 0.813023 Md Dot 0.982477 0.663273 Md Dot 0.643765 0.528715 Md Dot 0.137483 0.752932 Md Dot 0.892241 0.595794 Md Dot 0.411883 0.710000 Md Dot 0.145920 0.518520 Md Dot 0.826798 0.439800 Md Dot 0.242121 0.859641 Md Dot 0.906333 0.729524 Md Dot 0.592295 0.583881 Md Dot 0.139106 0.686030 Md Dot 0.886823 0.537036 Md Dot 0.359682 0.766713 Md Dot 0.067911 0.606206 Md Dot 0.713261 0.477495 Md Dot 0.149498 0.841784 Md Dot 0.864360 0.710997 Md Dot 0.478811 0.670736 Md Dot 0.165478 0.532222 Md Dot 0.884775 0.454082 Md Dot 0.312642 0.785585 Md Dot 0.025514 0.626173 Md Dot 0.665333 0.499099 Md Dot 0.145058 0.767921 Md Dot 0.896342 0.611204 Md Dot 0.431991 0.685403 Md Dot 0.173933 0.481007 Md Dot 0.919380 0.440388 Md Dot 0.345655 0.714593 Md Dot 0.124528 0.503922 Md Dot 0.759667 0.426611 Md Dot 0.184874 0.838824 Md Dot 0.888699 0.699839 Md Dot 0.515617 0.624048 Md Dot 0.187447 0.518640 Md Dot 0.949775 0.454829 Md Dot 0.386459 0.650772 Md Dot 0.221806 0.360471 Md Dot 0.238957 0.219280 Md Dot 0.185696 0.317460 Md Dot 0.159496 0.284585 Md Dot 0.106431 0.329148 Md Dot 0.770494 0.374908 Md Dot 0.207432 0.777312 Md Dot 0.950416 0.617704 Md Dot 0.527217 0.570099 Md Dot 0.238469 0.429264 Md Dot 0.185408 0.284698 Md Dot 0.234332 0.208439 Md Dot 0.902133 0.300000 Md Dot 0.312957 0.640848 Md Dot 0.164228 0.397239 Md Dot 0.946287 0.417855 Md Dot 0.366750 0.656081 Md Dot 0.204419 0.375265 Md Dot 0.136730 0.321823 Md Dot 0.929655 0.398612 Md Dot 0.342784 0.683320 Md Dot 0.156117 0.443378 Md Dot 0.874493 0.422674 Md Dot 0.290898 0.791892 Md Dot 0.005539 0.633218 Md Dot 0.641507 0.510813 Md Dot 0.152174 0.719850 Md Dot 0.909717 0.565023 Md Dot 0.410919 0.691027 Md Dot 0.171072 0.470848 Md Dot 0.914105 0.436621 Md Dot 0.338380 0.724837 Md Dot 0.108878 0.524615 Md Dot 0.725639 0.434499 Md Dot 0.170656 0.805580 Md Dot 0.901082 0.653823 Md Dot 0.484235 0.635270 Md Dot 0.207096 0.455724 Md Dot 0.048982 0.412531 Md Dot 0.521640 0.456040 Md Dot 0.529289 0.221738 Md Dot 0.596821 0.105278 Md Dot 0.089408 0.225896 Md Dot 0.886707 0.372070 Md Dot 0.294680 0.738983 Md Dot 0.061613 0.558257 Md Dot 0.647157 0.463876 Md Dot 0.182536 0.661788 Md Dot 0.946175 0.522636 Md Dot 0.425323 0.636250 Md Dot 0.246227 0.347901 Md Dot 0.567435 0.155662 Md Dot 0.179399 0.141572 Md Dot 0.570891 0.515873 Md Dot 0.232313 0.476641 Md Dot 0.111536 0.375104 Md Dot 0.740711 0.384012 Md Dot 0.196725 0.755316 Md Dot 0.951885 0.595495 Md Dot 0.503113 0.580557 Md Dot 0.261005 0.374616 Md Dot 0.569143 0.158063 Md Dot 0.163508 0.154684 Md Dot 0.146614 0.105362 Md Dot 0.209536 0.817460 Md Dot 0.922557 0.666451 Md Dot 0.539764 0.586106 Md Dot 0.201489 0.510911 Md Dot 0.994353 0.448588 Md Dot 0.439523 0.561021 Md Dot 0.428964 0.221698 Md Dot 0.176913 0.869841 Md Dot 0.864199 0.757941 Md Dot 0.536495 0.631395 Md Dot 0.154802 0.600492 Md Dot 0.877109 0.483114 Md Dot 0.316396 0.803562 Md Dot 0.005945 0.654049 Md Dot 0.669968 0.515880 Md Dot 0.132686 0.802288 Md Dot 0.873346 0.653397 Md Dot 0.434629 0.705896 Md Dot 0.145503 0.536227 Md Dot 0.829991 0.447233 Md Dot 0.246969 0.862690 Md Dot 0.906196 0.734842 Md Dot 0.599241 0.583390 Md Dot 0.132362 0.706510 Md Dot 0.884005 0.553032 Md Dot 0.366897 0.764718 Md Dot 0.072123 0.605648 Md Dot 0.720198 0.476434 Md Dot 0.151538 0.851959 Md Dot 0.860037 0.728965 Md Dot 0.490723 0.664343 Md Dot 0.163360 0.545587 Md Dot 0.880450 0.458729 Md Dot 0.309535 0.794692 Md Dot 0.013204 0.639388 Md Dot 0.662424 0.508317 Md Dot 0.140483 0.773475 Md Dot 0.890557 0.617534 Md Dot 0.428830 0.694500 Md Dot 0.162775 0.499833 Md Dot 0.877764 0.441216 Md Dot 0.299979 0.794826 Md Dot 0.007782 0.638252 Md Dot 0.651996 0.510574 Md Dot 0.144844 0.749199 Md Dot 0.900126 0.591920 Md Dot 0.420177 0.692996 Md Dot 0.167050 0.485345 Md Dot 0.894834 0.438868 Md Dot 0.318120 0.763097 Md Dot 0.054326 0.591715 Md Dot 0.671744 0.476147 Md Dot 0.156856 0.751224 Md Dot 0.912008 0.593602 Md Dot 0.439402 0.662536 Md Dot 0.202197 0.432705 Md Dot 0.054295 0.402983 Md Dot 0.532909 0.443927 Md Dot 0.497136 0.231510 Md Dot 0.110414 0.561905 Md Dot 0.755764 0.451440 Md Dot 0.175986 0.866091 Md Dot 0.865716 0.750457 Md Dot 0.530238 0.633435 Md Dot 0.159635 0.585670 Md Dot 0.881897 0.476470 Md Dot 0.319007 0.793640 Md Dot 0.019172 0.639542 Md Dot 0.672353 0.505919 Md Dot 0.137882 0.794610 Md Dot 0.880678 0.642873 Md Dot 0.436895 0.696231 Md Dot 0.157569 0.515596 Md Dot 0.860954 0.444157 Md Dot 0.281863 0.823253 Md Dot 0.965371 0.676470 Md Dot 0.632473 0.540231 Md Dot 0.137331 0.738045 Md Dot 0.893238 0.581236 Md Dot 0.401103 0.719812 Md Dot 0.133890 0.532079 Md Dot 0.796815 0.441099 Md Dot 0.210831 0.872806 Md Dot 0.879687 0.756576 Md Dot 0.570537 0.607355 Md Dot 0.140947 0.659290 Md Dot 0.878985 0.517513 Md Dot 0.337258 0.794021 Md Dot 0.026181 0.644984 Md Dot 0.690585 0.505735 Md Dot 0.134616 0.834691 Md Dot 0.858979 0.701849 Md Dot 0.457789 0.695326 Md Dot 0.148160 0.548690 Md Dot 0.841077 0.454043 Md Dot 0.261495 0.855460 Md Dot 0.920409 0.723164 Md Dot 0.612471 0.572546 Md Dot 0.128922 0.728730 Md Dot 0.883820 0.572443 Md Dot 0.380850 0.751801 Md Dot 0.091138 0.587800 Md Dot 0.737441 0.464983 Md Dot 0.162110 0.862494 Md Dot 0.860340 0.746821 Md Dot 0.513293 0.647611 Md Dot 0.161686 0.567167 Md Dot 0.880907 0.467893 Md Dot 0.313938 0.795117 Md Dot 0.014948 0.640791 Md Dot 0.667119 0.507999 Md Dot 0.138570 0.784786 Md Dot 0.885115 0.630728 Md Dot 0.432539 0.696276 Md Dot 0.159158 0.509384 Md Dot 0.865722 0.442668 Md Dot 0.286850 0.815262 Md Dot 0.977109 0.665735 Md Dot 0.638035 0.532023 Md Dot 0.139173 0.741750 Md Dot 0.895033 0.584760 Md Dot 0.406594 0.712192 Md Dot 0.143594 0.518786 Md Dot 0.820054 0.438894 Md Dot 0.234648 0.863433 Md Dot 0.899091 0.736372 Md Dot 0.586217 0.589981 Md Dot 0.140093 0.677476 Md Dot 0.885196 0.530633 Md Dot 0.353374 0.773913 Md Dot 0.056960 0.616738 Md Dot 0.706331 0.484876 Md Dot 0.145223 0.838828 Md Dot 0.863367 0.706774 Md Dot 0.471984 0.677703 Md Dot 0.161760 0.534055 Md Dot 0.874494 0.453270 Md Dot 0.300522 0.804393 Md Dot 0.997083 0.652181 Md Dot 0.653060 0.519055 Md Dot 0.138351 0.764244 Md Dot 0.891035 0.607535 Md Dot 0.420409 0.702583 Md Dot 0.154240 0.509556 Md Dot 0.850691 0.440278 Md Dot 0.269084 0.835461 Md Dot 0.944952 0.692679 Md Dot 0.618008 0.554487 Md Dot 0.138559 0.717248 Md Dot 0.893445 0.562467 Md Dot 0.386602 0.733826 Md Dot 0.115167 0.553433 Md Dot 0.760699 0.447569 Md Dot 0.180070 0.865721 Md Dot 0.868066 0.748702 Md Dot 0.533348 0.630038 Md Dot 0.159939 0.587840 Md Dot 0.883341 0.477691 Md Dot 0.321309 0.790666 Md Dot 0.023844 0.635438 Md Dot 0.674623 0.502734 Md Dot 0.139049 0.795676 Md Dot 0.881155 0.644120 Md Dot 0.438929 0.693670 Md Dot 0.160159 0.512246 Md Dot 0.868712 0.444180 Md Dot 0.290730 0.811134 Md Dot 0.983958 0.660396 Md Dot 0.642347 0.527395 Md Dot 0.139414 0.747081 Md Dot 0.894862 0.589928 Md Dot 0.410712 0.708243 Md Dot 0.148390 0.513115 Md Dot 0.833277 0.438772 Md Dot 0.249141 0.853471 Md Dot 0.915767 0.719227 Md Dot 0.597750 0.576441 Md Dot 0.139981 0.690533 Md Dot 0.889426 0.540607 Md Dot 0.365496 0.758235 Md Dot 0.080216 0.592697 Md Dot 0.721156 0.468741 Md Dot 0.154859 0.843784 Md Dot 0.866486 0.713365 Md Dot 0.485961 0.662879 Md Dot 0.169511 0.530025 Md Dot 0.896142 0.454810 Md Dot 0.325849 0.763378 Md Dot 0.058420 0.592558 Md Dot 0.680227 0.474712 Md Dot 0.154125 0.770110 Md Dot 0.904076 0.613127 Md Dot 0.445999 0.664469 Md Dot 0.196308 0.448521 Md Dot 0.018064 0.422844 Md Dot 0.464278 0.513632 Md Dot 0.613793 0.200933 Md Dot 0.752639 0.248605 Md Dot 0.284173 0.528264 Md Dot 0.209109 0.291362 Md Dot 0.437305 0.156068 Md Dot 0.982434 0.625397 Md Dot 0.591842 0.528163 Md Dot 0.190379 0.583005 Md Dot 0.952055 0.480648 Md Dot 0.403542 0.641066 Md Dot 0.240381 0.342740 Md Dot 0.507028 0.155726 Md Dot 0.100212 0.174035 Md Dot 0.074665 0.330566 Md Dot 0.624259 0.379504 Md Dot 0.282681 0.430110 Md Dot 0.442326 0.170905 Md Dot 0.914680 0.461905 Md Dot 0.349972 0.724901 Md Dot 0.115270 0.523857 Md Dot 0.742235 0.434009 Md Dot 0.174768 0.829161 Md Dot 0.888694 0.686262 Md Dot 0.498888 0.634867 Md Dot 0.192654 0.494942 Md Dot 0.974365 0.445747 Md Dot 0.411601 0.600639 Md Dot 0.317348 0.263250 Md Dot 0.568381 0.577778 Md Dot 0.172895 0.595198 Md Dot 0.914196 0.484627 Md Dot 0.361196 0.722502 Md Dot 0.123126 0.519694 Md Dot 0.761735 0.432963 Md Dot 0.184245 0.848732 Md Dot 0.881280 0.716321 Md Dot 0.520889 0.626699 Md Dot 0.178181 0.541393 Md Dot 0.920299 0.461855 Md Dot 0.356299 0.712874 Md Dot 0.132235 0.499442 Md Dot 0.781867 0.426752 Md Dot 0.199156 0.854013 Md Dot 0.886468 0.723149 Md Dot 0.540952 0.612583 Md Dot 0.170460 0.574158 Md Dot 0.903835 0.473864 Md Dot 0.343460 0.747238 Md Dot 0.086153 0.566064 Md Dot 0.705820 0.455695 Md Dot 0.159525 0.799194 Md Dot 0.895865 0.646686 Md Dot 0.466993 0.653582 Md Dot 0.197533 0.461390 Md Dot 0.011597 0.428585 Md Dot 0.455923 0.526778 Md Dot 0.565999 0.200668 Md Dot 0.106454 0.203072 Md Dot 0.031849 0.359568 Md Dot 0.481907 0.462364 Md Dot 0.844980 0.235917 Md Dot 0.301696 0.571990 Md Dot 0.206297 0.312499 Md Dot 0.283794 0.186467 Md Dot 0.089084 0.444444 Md Dot 0.634832 0.415223 Md Dot 0.234220 0.533332 Md Dot 0.075648 0.423343 Md Dot 0.591923 0.421403 Md Dot 0.299552 0.380515 Md Dot 0.087687 0.166597 Md Dot 0.031453 0.352185 Md Dot 0.482160 0.458617 Md Dot 0.864921 0.239889 Md Dot 0.309180 0.575639 Md Dot 0.218320 0.301983 Md Dot 0.502094 0.152360 Md Dot 0.500627 0.387302 Md Dot 0.069256 0.214740 Md Dot 0.806671 0.354173 Md Dot 0.231612 0.767280 Md Dot 0.979873 0.604709 Md Dot 0.560147 0.535852 Md Dot 0.227728 0.476240 Md Dot 0.097732 0.386816 Md Dot 0.674434 0.388431 Md Dot 0.216195 0.616103 Md Dot 0.007100 0.490064 Md Dot 0.484294 0.533160 Md Dot 0.414523 0.241252 Md Dot 0.912171 0.401587 Md Dot 0.325162 0.717521 Md Dot 0.107439 0.514705 Md Dot 0.715844 0.430680 Md Dot 0.172146 0.783098 Md Dot 0.914975 0.626438 Md Dot 0.477680 0.628270 Md Dot 0.223083 0.421087 Md Dot 0.141654 0.328998 Md Dot 0.940969 0.400940 Md Dot 0.355597 0.662048 Md Dot 0.189318 0.393095 Md Dot 0.052559 0.393039 Md Dot 0.528158 0.442654 Md Dot 0.539458 0.222037 Md Dot 0.309673 0.097201 Md Dot 0.628607 0.515873 Md Dot 0.159480 0.688528 Md Dot 0.915596 0.540369 Md Dot 0.399372 0.694116 Md Dot 0.167291 0.469296 Md Dot 0.901680 0.434645 Md Dot 0.324092 0.748964 Md Dot 0.073632 0.568590 Md Dot 0.682466 0.460629 Md Dot 0.162019 0.755119 Md Dot 0.916307 0.597260 Md Dot 0.449433 0.649564 Md Dot 0.215217 0.415701 Md Dot 0.120508 0.348469 Md Dot 0.811478 0.385952 Md Dot 0.227889 0.810247 Md Dot 0.942438 0.656234 Md Dot 0.563262 0.563285 Md Dot 0.194269 0.546762 Md Dot 0.963475 0.465153 Md Dot 0.408323 0.620863 Md Dot 0.274631 0.300421 Md Dot 0.623588 0.546032 Md Dot 0.140286 0.720840 Md Dot 0.895861 0.565640 Md Dot 0.392349 0.725262 Md Dot 0.126691 0.538095 Md Dot 0.780315 0.441961 Md Dot 0.195823 0.870909 Md Dot 0.873225 0.755582 Md Dot 0.554339 0.617066 Md Dot 0.149688 0.626817 Md Dot 0.878788 0.497477 Md Dot 0.325743 0.798990 Md Dot 0.015586 0.649406 Md Dot 0.679370 0.510800 Md Dot 0.133185 0.817210 Md Dot 0.866780 0.674394 Md Dot 0.443878 0.702468 Md Dot 0.146152 0.541933 Md Dot 0.833537 0.450094 Md Dot 0.251657 0.861327 Md Dot 0.909832 0.732556 Md Dot 0.603717 0.580482 Md Dot 0.130469 0.715268 Md Dot 0.883634 0.560432 Md Dot 0.371679 0.760881 Md Dot 0.077999 0.600692 Md Dot 0.725516 0.473026 Md Dot 0.154498 0.855922 Md Dot 0.859620 0.735806 Md Dot 0.498047 0.658951 Md Dot 0.163267 0.551613 Md Dot 0.881265 0.461312 Md Dot 0.311543 0.793572 Md Dot 0.015568 0.638044 Md Dot 0.664516 0.506902 Md Dot 0.140405 0.776797 Md Dot 0.889452 0.621289 Md Dot 0.430596 0.693847 Md Dot 0.163079 0.500568 Md Dot 0.878626 0.441605 Md Dot 0.301077 0.793468 Md Dot 0.009962 0.636413 Md Dot 0.653137 0.509065 Md Dot 0.145170 0.750108 Md Dot 0.900322 0.592810 Md Dot 0.421255 0.691730 Md Dot 0.168506 0.483390 Md Dot 0.900310 0.438971 Md Dot 0.324178 0.752610 Md Dot 0.069657 0.574668 Md Dot 0.681105 0.464245 Md Dot 0.160294 0.757121 Md Dot 0.914025 0.599326 Md Dot 0.447961 0.653045 Md Dot 0.211140 0.422136 Md Dot 0.098044 0.370177 Md Dot 0.686672 0.382450 Md Dot 0.212138 0.639962 Md Dot 0.996920 0.504975 Md Dot 0.480605 0.547081 Md Dot 0.379467 0.255181 Md Dot 0.966123 0.661905 Md Dot 0.613739 0.538871 Md Dot 0.155528 0.678761 Md Dot 0.907833 0.533008 Md Dot 0.384013 0.717155 Md Dot 0.136175 0.514901 Md Dot 0.797414 0.434341 Md Dot 0.211580 0.865773 Md Dot 0.884901 0.742764 Md Dot 0.563692 0.605363 Md Dot 0.150828 0.633796 Md Dot 0.884021 0.502090 Md Dot 0.334752 0.786475 Md Dot 0.034753 0.632027 Md Dot 0.688047 0.497692 Md Dot 0.138960 0.819056 Md Dot 0.869863 0.676301 Md Dot 0.451533 0.691630 Md Dot 0.156715 0.528102 Md Dot 0.859523 0.448633 Md Dot 0.281586 0.827470 Md Dot 0.960528 0.682530 Md Dot 0.632250 0.543829 Md Dot 0.134798 0.742738 Md Dot 0.890441 0.585730 Md Dot 0.400769 0.723454 Md Dot 0.129119 0.540238 Md Dot 0.787771 0.443558 Md Dot 0.202219 0.874717 Md Dot 0.874180 0.762070 Md Dot 0.565260 0.612834 Md Dot 0.141463 0.652965 Md Dot 0.876904 0.513165 Md Dot 0.332050 0.800123 Md Dot 0.016613 0.653018 Md Dot 0.685628 0.511964 Md Dot 0.131717 0.832275 Md Dot 0.858376 0.698347 Md Dot 0.452868 0.700941 Md Dot 0.143551 0.553407 Md Dot 0.830993 0.454572 Md Dot 0.249671 0.866864 Md Dot 0.903930 0.742316 Md Dot 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