(* Content-type: application/mathematica *) (*** Wolfram Notebook File ***) (* http://www.wolfram.com/nb *) (* CreatedBy='%s' *) (*CacheID: 234*) (* Internal cache information: NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 131, 6] NotebookDataLength[ 220737, 11879] NotebookOptionsPosition[ 216682, 11742] NotebookOutlinePosition[ 217292, 11768] CellTagsIndexPosition[ 217249, 11765] WindowFrame->Normal ContainsDynamic->False*) (* Beginning of Notebook Content *) Notebook[{ Cell[CellGroupData[{ Cell["Mixmaster Cosmology", "Title", ImageRegion->{{0., 1.}, {0., 1.}}], Cell[CellGroupData[{ Cell["BKL Map", "Section", ImageRegion->{{0., 1.}, {0., 1.}}], Cell[CellGroupData[{ Cell["The BKL parameter u and pj[u]", "Subsection", ImageRegion->{{0., 1.}, {0., 1.}}], Cell["\<\ Let U[u] be the function that takes an old value of u into a new value\ \>", "Text", ImageRegion->{{0., 1.}, {0., 1.}}], Cell["U[u_] := If[u>2, u-1, 1/(u-1)]", "Input", ImageRegion->{{0., 1.}, {0., 1.}}], Cell["\<\ Let p1[u], p2[u], p3[u] be the Kasner exponents in ascending order for u>1:\ \>", "Text", ImageRegion->{{0., 1.}, {0., 1.}}], Cell["\<\ p1[u_] := -u/(1+u+u^2); p2[u_] := (1+u)/(1+u+u^2); p3[u_] := u(1+u)/(1+u+u^2);\ \>", "Input", ImageRegion->{{0., 1.}, {0., 1.}}], Cell["Check that Kasner equations are satisfied:", "Text", ImageRegion->{{0., 1.}, {0., 1.}}], Cell[CellGroupData[{ Cell["Together[p1[u] + p2[u] + p3[u]]", "Input", ImageRegion->{{0., 1.}, {0., 1.}}], Cell[OutputFormData["\<\ 1\ \>", "\<\ 1\ \>"], "Output", ImageRegion->{{0., 1.}, {0., 1.}}] }, Open ]], Cell[CellGroupData[{ Cell["Together[p1[u]^2 + p2[u]^2 + p3[u]^2]", "Input", ImageRegion->{{0., 1.}, {0., 1.}}], Cell[OutputFormData["\<\ 1\ \>", "\<\ 1\ \>"], "Output", ImageRegion->{{0., 1.}, {0., 1.}}] }, Open ]], Cell["Define a vectorial p[u]:", "Text", ImageRegion->{{0., 1.}, {0., 1.}}], Cell["p[u_] := {p1[u], p2[u], p3[u]}", "Input", ImageRegion->{{0., 1.}, {0., 1.}}], Cell["Check it:", "Text", ImageRegion->{{0., 1.}, {0., 1.}}], Cell[CellGroupData[{ Cell["p[u]", "Input", ImageRegion->{{0., 1.}, {0., 1.}}], Cell[OutputFormData["\<\ {-(u/(1 + u + u^2)), (1 + u)/(1 + u + u^2), (u*(1 + u))/(1 + u + u^2)}\ \>", "\<\ u 1 + u u (1 + u) {-(----------), ----------, ----------} 2 2 2 1 + u + u 1 + u + u 1 + u + u\ \>"], "Output", ImageRegion->{{0., 1.}, {0., 1.}}] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["The map of pl, pm, pn between epochs and eras", "Subsection", ImageRegion->{{0., 1.}, {0., 1.}}], Cell["\<\ Suppose that at some epoch, when u has some value, pm [or pl or pn] is equal \ to pN, with N=1 or 2 or 3. We want to know what pm will be at the next \ epoch. Its value pM, with M=1, or 2, or 3 will be given by the following \ map; call the map S for shift. S depends on the u at the past epoch: For u>2 (so the transition is a change of epoch not era), S:{1,2,3} -> \ {2,1,3} For u<2 (so the transition is a change of era) S:{1,2,3} -> \ {1,3,2}\ \>", "Text", ImageRegion->{{0., 1.}, {0., 1.}}], Cell["Clear[S];", "Input", ImageRegion->{{0., 1.}, {0., 1.}}], Cell["\<\ S[u_,v_] := If[u>2, If[v>2.9, v, v+1 - 2 Floor[v/2]] , If[v<1.1, v, v+1 - 2 Floor[v/3]] ] \ \>", "Input", ImageRegion->{{0., 1.}, {0., 1.}}], Cell["Check of this map:", "Text", ImageRegion->{{0., 1.}, {0., 1.}}], Cell[CellGroupData[{ Cell["{S[2.1,1], S[2.1,2], S[2.1,3] }", "Input", ImageRegion->{{0., 1.}, {0., 1.}}], Cell[OutputFormData["\<\ {2, 1, 3}\ \>", "\<\ {2, 1, 3}\ \>"], "Output", ImageRegion->{{0., 1.}, {0., 1.}}] }, Open ]], Cell[CellGroupData[{ Cell["{S[1.9,1], S[1.9,2], S[1.9,3] }", "Input", ImageRegion->{{0., 1.}, {0., 1.}}], Cell[OutputFormData["\<\ {1, 3, 2}\ \>", "\<\ {1, 3, 2}\ \>"], "Output", ImageRegion->{{0., 1.}, {0., 1.}}] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Table of Sequence of Epochs and Eras", "Subsection", ImageRegion->{{0., 1.}, {0., 1.}}], Cell["\<\ We will generate a table, which we later shall plot, containing the BKL \ sequence {u,l,pl,m,pm,n,pn}. Here if l=j, then pl=pj and similarly for m and \ n. Begin with the following initial values:. Note that we have carefully \ chosen uo to be an irrational number.\ \>", "Text", ImageRegion->{{0., 1.}, {0., 1.}}], Cell["lo=1; mo=2; no=3; uo=1/N[Pi-E];", "Input", ImageRegion->{{0., 1.}, {0., 1.}}], Cell["\<\ Make a table with 1000 entries to hold the sequence. Call it Ev for \ \"Evolution\":\ \>", "Text", ImageRegion->{{0., 1.}, {0., 1.}}], Cell["Clear[Ev]", "Input", ImageRegion->{{0., 1.}, {0., 1.}}], Cell["Ev = Table[0,{1000},{7}];", "Input", ImageRegion->{{0., 1.}, {0., 1.}}], Cell[CellGroupData[{ Cell["\<\ Ev[[1]] = {uo, lo, p[uo][[lo]], mo, p[uo][[mo]], no, p[uo][[no]] }\ \>", "Input", ImageRegion->{{0., 1.}, {0., 1.}}], Cell[OutputFormData["\<\ {2.362330327109234, 1, -0.2641560464356995, 2, 0.3759761604156717, 3, 0.888179886020028}\ \>", "\<\ {2.36233, 1, -0.264156, 2, 0.375976, 3, 0.88818}\ \>"], "Output", ImageRegion->{{0., 1.}, {0., 1.}}] }, Open ]], Cell["\<\ For[j=2, j<1001, j++, Ev[[j,1]] = U[Ev[[j-1,1]] ]; Ev[[j,2]] = S[Ev[[j-1,1]], Ev[[j-1,2]] ]; Ev[[j,3]] = p[ Ev[[j,1]] ][[ Ev[[j,2]] ]]; Ev[[j,4]] = S[Ev[[j-1,1]], Ev[[j-1,4]] ]; Ev[[j,5]] = p[ Ev[[j,1]] ][[ Ev[[j,4]] ]]; Ev[[j,6]] = S[Ev[[j-1,1]], Ev[[j-1,6]] ]; Ev[[j,7]] = p[ Ev[[j,1]] ][[ Ev[[j,6]] ]] ]\ \>", "Input", ImageRegion->{{0., 1.}, {0., 1.}}] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Overview of Sequence of 1000 Epochs", "Section", ImageRegion->{{0., 1.}, {0., 1.}}], Cell["\<\ We plot out pl, pm, and pn as a function of epoch number for the full 1000 \ epochs\ \>", "Text", ImageRegion->{{0., 1.}, {0., 1.}}], Cell[CellGroupData[{ Cell["\<\ PL = ListPlot[ Ev[[Range[1,1000],3]], PlotJoined->True, PlotStyle->{ Thickness[0.002]} ]\ \>", "Input", ImageRegion->{{0., 1.}, {0., 1.}}], Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0.02381 0.000952 0.161925 0.441632 [ [(200)] .21429 .16192 0 2 Msboxa [(400)] .40476 .16192 0 2 Msboxa [(600)] .59524 .16192 0 2 Msboxa [(800)] .78571 .16192 0 2 Msboxa [(1000)] .97619 .16192 0 2 Msboxa [(-0.2)] .01131 .0736 1 0 Msboxa [(0.2)] .01131 .25025 1 0 Msboxa [(0.4)] .01131 .33858 1 0 Msboxa [(0.6)] .01131 .4269 1 0 Msboxa [(0.8)] .01131 .51523 1 0 Msboxa [(1)] .01131 .60356 1 0 Msboxa 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.58857 .15526 L .58952 .16868 L .59048 .15506 L .59143 .16889 L .59238 .15484 L .59333 .16912 L .59429 .15461 L .59524 .16937 L Mistroke .59619 .15436 L .59714 .16963 L .5981 .1541 L .59905 .16992 L .6 .15382 L .60095 .17022 L .6019 .15351 L .60286 .17055 L .60381 .15318 L .60476 .17091 L .60571 .15283 L .60667 .17129 L .60762 .15244 L .60857 .17171 L .60952 .15202 L .61048 .17217 L .61143 .15156 L .61238 .17268 L .61333 .15106 L .61429 .17323 L .61524 .1505 L .61619 .17385 L .61714 .14989 L .6181 .17453 L .61905 .1492 L .62 .17531 L .62095 .14843 L .6219 .17618 L .62286 .14756 L .62381 .17717 L .62476 .14657 L .62571 .1783 L .62667 .14543 L .62762 .17962 L .62857 .14411 L .62952 .18118 L .63048 .14256 L .63143 .18302 L .63238 .14072 L .63333 .18526 L .63429 .13848 L .63524 .18802 L .63619 .13572 L .63714 .19153 L .6381 .13223 L .63905 .1961 L .64 .12766 L .64095 .20234 L .6419 .12145 L .64286 .21134 L Mistroke .64381 .11254 L .64476 .22539 L .64571 .09874 L .64667 .25027 L .64762 .07503 L .64857 .30504 L .64952 .0309 L .65048 .01658 L .65143 .08614 L .65238 .27715 L .65333 .05138 L .65429 .3831 L .65524 .51996 L .65619 .38603 L .65714 .52545 L .6581 .40564 L .65905 .55926 L .66 .51376 L .66095 .36345 L .6619 .48135 L .66286 .24486 L .66381 .08008 L .66476 .2918 L .66571 .03994 L .66667 .43426 L .66762 .59411 L .66857 .59077 L .66952 .58525 L .67048 .57519 L .67143 .55405 L .67238 .49894 L .67333 .30874 L .67429 .02864 L .67524 .01913 L .67619 .0561 L .67714 .36496 L .6781 .4844 L .67905 .2557 L .68 .07004 L .68095 .3191 L .6819 .02305 L .68286 .03326 L .68381 .01516 L .68476 .122 L .68571 .21052 L .68667 .11335 L .68762 .22405 L .68857 .10005 L .68952 .24774 L .69048 .07738 L Mistroke .69143 .29875 L .69238 .03502 L .69333 .01475 L .69429 .15018 L .69524 .17421 L .69619 .14952 L .69714 .17494 L .6981 .14879 L .69905 .17576 L .7 .14797 L .70095 .1767 L .7019 .14704 L .70286 .17776 L .70381 .14597 L .70476 .17899 L .70571 .14474 L .70667 .18043 L .70762 .14331 L .70857 .18213 L .70952 .1416 L .71048 .18418 L .71143 .13956 L .71238 .18668 L .71333 .13706 L .71429 .18981 L .71524 .13394 L .71619 .19384 L .71714 .12992 L .7181 .19922 L .71905 .12456 L .72 .20677 L .72095 .11706 L .7219 .21808 L .72286 .1059 L .72381 .23685 L .72476 .08767 L .72571 .27363 L .72667 .0543 L .72762 .3717 L .72857 .49786 L .72952 .30477 L .73048 .03107 L .73143 .01645 L .73238 .08856 L .73333 .27162 L .73429 .056 L .73524 .36532 L .73619 .48513 L .73714 .25833 L .7381 .06766 L Mistroke .73905 .32624 L .74 .01991 L .74095 .05008 L .7419 .38835 L .74286 .52974 L .74381 .42069 L .74476 .58027 L .74571 .56517 L .74667 .52964 L .74762 .42032 L .74857 .57982 L .74952 .56422 L .75048 .52716 L .75143 .41165 L .75238 .56826 L .75333 .53752 L .75429 .44715 L .75524 .6017 L .75619 .60149 L .75714 .60123 L .7581 .60091 L .75905 .6005 L .76 .59999 L .76095 .59933 L .7619 .59845 L .76286 .59725 L .76381 .59553 L .76476 .59299 L .76571 .58897 L .76667 .5821 L .76762 .56894 L .76857 .53921 L .76952 .45275 L .77048 .60309 L .77143 .60308 L .77238 .60306 L .77333 .60305 L .77429 .60304 L .77524 .60302 L .77619 .60301 L .77714 .60299 L .7781 .60297 L .77905 .60295 L .78 .60292 L .78095 .6029 L .7819 .60287 L .78286 .60284 L .78381 .6028 L .78476 .60277 L .78571 .60273 L Mistroke .78667 .60268 L .78762 .60263 L .78857 .60257 L .78952 .6025 L .79048 .60242 L .79143 .60234 L .79238 .60224 L .79333 .60212 L .79429 .60198 L .79524 .60182 L .79619 .60163 L .79714 .6014 L .7981 .60111 L .79905 .60077 L .8 .60033 L .80095 .59977 L .8019 .59903 L .80286 .59804 L .80381 .59668 L .80476 .5947 L .80571 .5917 L .80667 .58684 L .80762 .57821 L .80857 .56076 L .80952 .51791 L .81048 .37858 L .81143 .51133 L .81238 .35455 L .81333 .46328 L .81429 .18387 L .81524 .13987 L .81619 .1863 L .81714 .13745 L .8181 .18932 L .81905 .13442 L .82 .19321 L .82095 .13055 L .8219 .19836 L .82286 .12541 L .82381 .20553 L .82476 .11829 L .82571 .21617 L .82667 .10778 L .82762 .23351 L .82857 .09087 L .82952 .26652 L .83048 .06039 L .83143 .34969 L .83238 .01473 L .83333 .15357 L Mistroke .83429 .17049 L .83524 .15324 L .83619 .17084 L .83714 .15289 L .8381 .17122 L .83905 .15251 L .84 .17164 L .84095 .1521 L .8419 .17209 L .84286 .15164 L .84381 .17259 L .84476 .15115 L .84571 .17313 L .84667 .1506 L .84762 .17374 L .84857 .14999 L .84952 .17441 L .85048 .14932 L .85143 .17517 L .85238 .14856 L .85333 .17602 L .85429 .14771 L .85524 .17699 L .85619 .14674 L .85714 .1781 L .8581 .14563 L .85905 .17939 L .86 .14435 L .86095 .1809 L .8619 .14284 L .86286 .18269 L .86381 .14105 L .86476 .18485 L .86571 .13889 L .86667 .18752 L .86762 .13623 L .86857 .19088 L .86952 .13287 L .87048 .19524 L .87143 .12852 L .87238 .20115 L .87333 .12264 L .87429 .20957 L .87524 .11429 L .87619 .22252 L .87714 .10154 L .8781 .24488 L .87905 .08006 L .88 .29184 L .88095 .03991 L Mistroke .8819 .43443 L .88286 .59424 L .88381 .59098 L .88476 .58562 L .88571 .5759 L .88667 .55567 L .88762 .50362 L .88857 .32605 L .88952 .01999 L .89048 .04956 L .89143 .39052 L .89238 .53369 L .89333 .43427 L .89429 .59412 L .89524 .59078 L .89619 .58528 L .89714 .57525 L .8981 .55418 L .89905 .49931 L .9 .31013 L .90095 .02782 L .9019 .0204 L .90286 .04676 L .90381 .40241 L .90476 .55411 L .90571 .49912 L .90667 .3094 L .90762 .02825 L .90857 .01971 L .90952 .05153 L .91048 .3825 L .91143 .51882 L .91238 .38191 L .91333 .51771 L .91429 .37788 L .91524 .50997 L .91619 .34952 L .91714 .01474 L .9181 .15255 L .91905 .1716 L .92 .15214 L .92095 .17205 L .9219 .15169 L .92286 .17254 L .92381 .1512 L .92476 .17308 L .92571 .15065 L .92667 .17368 L .92762 .15005 L .92857 .17435 L Mistroke .92952 .14939 L .93048 .1751 L .93143 .14864 L .93238 .17594 L .93333 .1478 L .93429 .1769 L .93524 .14684 L .93619 .17799 L .93714 .14574 L .9381 .17926 L .93905 .14448 L .94 .18075 L .94095 .14299 L .9419 .18251 L .94286 .14123 L .94381 .18463 L .94476 .13911 L .94571 .18724 L .94667 .1365 L .94762 .19053 L .94857 .13322 L .94952 .19478 L .95048 .12898 L .95143 .20051 L .95238 .12328 L .95333 .20863 L .95429 .11521 L .95524 .22102 L .95619 .10301 L .95714 .24212 L .9581 .08265 L .95905 .28542 L .96 .04476 L .96095 .41132 L .9619 .56779 L .96286 .53634 L .96381 .44321 L .96476 .60001 L .96571 .59936 L .96667 .59848 L .96762 .59729 L .96857 .5956 L .96952 .59309 L .97048 .58914 L .97143 .58239 L .97238 .56955 L .97333 .54071 L .97429 .45765 L .97524 .16634 L .97619 .15739 L Mistroke Mfstroke P p [ .05 .01 ] 0 setdash .002 w .02476 .32785 m .02571 .01929 L .02667 .05478 L .02762 .36988 L .02857 .49426 L .02952 .29151 L .03048 .04015 L .03143 .4332 L .03238 .59323 L .03333 .58936 L .03429 .58279 L .03524 .57035 L .03619 .54268 L .03714 .464 L .0381 .18615 L .03905 .13759 L .04 .18914 L .04095 .1346 L .0419 .19297 L .04286 .13078 L .04381 .19804 L .04476 .12573 L .04571 .20508 L .04667 .11874 L .04762 .21547 L .04857 .10846 L .04952 .23231 L .05048 .09203 L .05143 .26402 L .05238 .06258 L .05333 .34233 L .05429 .01541 L .05524 .11304 L .05619 .22456 L .05714 .09955 L .0581 .24869 L .05905 .07649 L .06 .3011 L .06095 .03344 L .0619 .0151 L .06286 .12474 L .06381 .2065 L .06476 .11733 L .06571 .21766 L .06667 .10631 L .06762 .23611 L .06857 .08838 L .06952 .27203 L .07048 .05565 L .07143 .3666 L Mistroke .07238 .4877 L .07333 .26758 L .07429 .05947 L .07524 .35288 L .07619 .45988 L .07714 .17319 L .0781 .15055 L .07905 .1738 L .08 .14994 L .08095 .17448 L .0819 .14926 L .08286 .17524 L .08381 .14849 L .08476 .1761 L .08571 .14763 L .08667 .17708 L .08762 .14665 L .08857 .17821 L .08952 .14553 L .09048 .17951 L .09143 .14422 L .09238 .18105 L .09333 .14269 L .09429 .18287 L .09524 .14087 L .09619 .18507 L .09714 .13867 L .0981 .18779 L .09905 .13596 L .1 .19122 L .10095 .13253 L .1019 .1957 L .10286 .12807 L .10381 .20178 L .10476 .12201 L .10571 .2105 L .10667 .11337 L .10762 .22403 L .10857 .10007 L .10952 .24769 L .11048 .07743 L .11143 .29862 L .11238 .0351 L .11333 .01474 L .11429 .15153 L .11524 .17271 L .11619 .15102 L .11714 .17327 L .1181 .15046 L .11905 .17389 L Mistroke .12 .14984 L .12095 .17459 L .1219 .14915 L .12286 .17536 L .12381 .14837 L .12476 .17624 L .12571 .14749 L .12667 .17724 L .12762 .14649 L .12857 .17839 L .12952 .14534 L .13048 .17973 L .13143 .14401 L .13238 .1813 L .13333 .14244 L .13429 .18317 L .13524 .14057 L .13619 .18544 L .13714 .1383 L .1381 .18825 L .13905 .1355 L .14 .19181 L .14095 .13194 L .1419 .19649 L .14286 .12728 L .14381 .20288 L .14476 .12092 L .14571 .21214 L .14667 .11175 L .14762 .22671 L .14857 .09746 L .14952 .25281 L .15048 .07269 L .15143 .31151 L .15238 .02704 L .15333 .02184 L .15429 .03865 L .15524 .4408 L .15619 .5987 L .15714 .59759 L .1581 .59602 L .15905 .59373 L .16 .59017 L .16095 .58421 L .1619 .57317 L .16286 .5494 L .16381 .48504 L .16476 .25798 L .16571 .06797 L .16667 .32527 L Mistroke .16762 .0203 L .16857 .04745 L .16952 .39944 L .17048 .54923 L .17143 .48452 L .17238 .25614 L .17333 .06964 L .17429 .32029 L .17524 .02248 L .17619 .03563 L .17714 .01472 L .1781 .16014 L .17905 .1636 L .18 .16013 L .18095 .16362 L .1819 .16012 L .18286 .16363 L .18381 .1601 L .18476 .16365 L .18571 .16009 L .18667 .16366 L .18762 .16007 L .18857 .16368 L .18952 .16006 L .19048 .16369 L .19143 .16004 L .19238 .16371 L .19333 .16003 L .19429 .16372 L .19524 .16001 L .19619 .16374 L .19714 .16 L .1981 .16375 L .19905 .15998 L .2 .16377 L .20095 .15997 L .2019 .16379 L .20286 .15995 L .20381 .1638 L .20476 .15993 L .20571 .16382 L .20667 .15991 L .20762 .16384 L .20857 .1599 L .20952 .16385 L .21048 .15988 L .21143 .16387 L .21238 .15986 L .21333 .16389 L .21429 .15984 L Mistroke .21524 .16391 L .21619 .15982 L .21714 .16393 L .2181 .15981 L .21905 .16395 L .22 .15979 L .22095 .16397 L .2219 .15977 L .22286 .16399 L .22381 .15975 L .22476 .16401 L .22571 .15973 L .22667 .16403 L .22762 .1597 L .22857 .16405 L .22952 .15968 L .23048 .16407 L .23143 .15966 L .23238 .1641 L .23333 .15964 L .23429 .16412 L .23524 .15962 L .23619 .16414 L .23714 .15959 L .2381 .16416 L .23905 .15957 L .24 .16419 L .24095 .15955 L .2419 .16421 L .24286 .15952 L .24381 .16424 L .24476 .1595 L .24571 .16426 L .24667 .15947 L .24762 .16429 L .24857 .15944 L .24952 .16432 L .25048 .15942 L .25143 .16435 L .25238 .15939 L .25333 .16437 L .25429 .15936 L .25524 .1644 L .25619 .15933 L .25714 .16443 L .2581 .1593 L .25905 .16446 L .26 .15927 L .26095 .16449 L .2619 .15924 L Mistroke .26286 .16452 L .26381 .15921 L .26476 .16456 L .26571 .15918 L .26667 .16459 L .26762 .15914 L .26857 .16462 L .26952 .15911 L .27048 .16466 L .27143 .15908 L .27238 .16469 L .27333 .15904 L .27429 .16473 L .27524 .159 L .27619 .16477 L .27714 .15897 L .2781 .16481 L .27905 .15893 L .28 .16485 L .28095 .15889 L .2819 .16489 L .28286 .15885 L .28381 .16493 L .28476 .1588 L .28571 .16497 L .28667 .15876 L .28762 .16502 L .28857 .15872 L .28952 .16506 L .29048 .15867 L .29143 .16511 L .29238 .15862 L .29333 .16516 L .29429 .15858 L .29524 .16521 L .29619 .15853 L .29714 .16526 L .2981 .15847 L .29905 .16531 L .3 .15842 L .30095 .16537 L .3019 .15837 L .30286 .16542 L .30381 .15831 L .30476 .16548 L .30571 .15825 L .30667 .16554 L .30762 .15819 L .30857 .1656 L .30952 .15813 L Mistroke .31048 .16567 L .31143 .15807 L .31238 .16574 L .31333 .158 L .31429 .1658 L .31524 .15793 L .31619 .16588 L .31714 .15786 L .3181 .16595 L .31905 .15778 L .32 .16603 L .32095 .15771 L .3219 .16611 L .32286 .15763 L .32381 .16619 L .32476 .15754 L .32571 .16628 L .32667 .15746 L .32762 .16637 L .32857 .15737 L .32952 .16646 L .33048 .15728 L .33143 .16656 L .33238 .15718 L .33333 .16666 L .33429 .15708 L .33524 .16676 L .33619 .15697 L .33714 .16688 L .3381 .15686 L .33905 .16699 L .34 .15674 L .34095 .16711 L .3419 .15662 L .34286 .16724 L .34381 .15649 L .34476 .16738 L .34571 .15636 L .34667 .16752 L .34762 .15622 L .34857 .16767 L .34952 .15607 L .35048 .16782 L .35143 .15591 L .35238 .16799 L .35333 .15575 L .35429 .16816 L .35524 .15557 L .35619 .16835 L .35714 .15539 L Mistroke .3581 .16854 L .35905 .15519 L .36 .16875 L .36095 .15498 L .3619 .16897 L .36286 .15476 L .36381 .16921 L .36476 .15453 L .36571 .16946 L .36667 .15427 L .36762 .16973 L .36857 .154 L .36952 .17002 L .37048 .15371 L .37143 .17033 L .37238 .1534 L .37333 .17067 L .37429 .15306 L .37524 .17104 L .37619 .1527 L .37714 .17143 L .3781 .1523 L .37905 .17187 L .38 .15187 L .38095 .17234 L .3819 .15139 L .38286 .17286 L .38381 .15087 L .38476 .17344 L .38571 .1503 L .38667 .17408 L .38762 .14966 L .38857 .17479 L .38952 .14894 L .39048 .1756 L .39143 .14814 L .39238 .17651 L .39333 .14723 L .39429 .17754 L .39524 .14619 L .39619 .17874 L .39714 .14499 L .3981 .18014 L .39905 .1436 L .4 .18178 L .40095 .14196 L .4019 .18375 L .40286 .13999 L .40381 .18615 L .40476 .13759 L Mistroke .40571 .18915 L .40667 .1346 L .40762 .19298 L .40857 .13078 L .40952 .19805 L .41048 .12572 L .41143 .20509 L .41238 .11873 L .41333 .21548 L .41429 .10845 L .41524 .23234 L .41619 .09201 L .41714 .26407 L .4181 .06253 L .41905 .34248 L .42 .01538 L .42095 .11377 L .4219 .22336 L .42286 .10072 L .42381 .24644 L .42476 .0786 L .42571 .29559 L .42667 .03721 L .42762 .44826 L .42857 .60207 L .42952 .60193 L .43048 .60176 L .43143 .60155 L .43238 .6013 L .43333 .601 L .43429 .60062 L .43524 .60015 L .43619 .59953 L .43714 .59872 L .4381 .59762 L .43905 .59607 L .44 .59379 L .44095 .59027 L .4419 .58439 L .44286 .57353 L .44381 .55023 L .44476 .48756 L .44571 .26709 L .44667 .05989 L .44762 .3514 L .44857 .45687 L .44952 .16395 L .45048 .15978 L .45143 .16397 L .45238 .15976 L Mistroke .45333 .16399 L .45429 .15974 L .45524 .16401 L .45619 .15972 L .45714 .16403 L .4581 .1597 L .45905 .16405 L .46 .15968 L .46095 .16408 L .4619 .15966 L .46286 .1641 L .46381 .15964 L .46476 .16412 L .46571 .15961 L .46667 .16414 L .46762 .15959 L .46857 .16417 L .46952 .15957 L .47048 .16419 L .47143 .15954 L .47238 .16422 L .47333 .15952 L .47429 .16424 L .47524 .15949 L .47619 .16427 L .47714 .15947 L .4781 .1643 L .47905 .15944 L .48 .16432 L .48095 .15941 L .4819 .16435 L .48286 .15938 L .48381 .16438 L .48476 .15936 L .48571 .16441 L .48667 .15933 L .48762 .16444 L .48857 .1593 L .48952 .16447 L .49048 .15927 L .49143 .1645 L .49238 .15924 L .49333 .16453 L .49429 .15921 L .49524 .16456 L .49619 .15917 L .49714 .1646 L .4981 .15914 L .49905 .16463 L .5 .15911 L Mistroke .50095 .16466 L .5019 .15907 L .50286 .1647 L .50381 .15903 L .50476 .16474 L .50571 .159 L .50667 .16477 L .50762 .15896 L .50857 .16481 L .50952 .15892 L .51048 .16485 L .51143 .15888 L .51238 .16489 L .51333 .15884 L .51429 .16494 L .51524 .1588 L .51619 .16498 L .51714 .15875 L .5181 .16502 L .51905 .15871 L .52 .16507 L .52095 .15866 L .5219 .16512 L .52286 .15862 L .52381 .16517 L .52476 .15857 L .52571 .16522 L .52667 .15852 L .52762 .16527 L .52857 .15847 L .52952 .16532 L .53048 .15841 L .53143 .16538 L .53238 .15836 L .53333 .16543 L .53429 .1583 L .53524 .16549 L .53619 .15824 L .53714 .16555 L .5381 .15818 L .53905 .16561 L .54 .15812 L .54095 .16568 L .5419 .15806 L .54286 .16575 L .54381 .15799 L .54476 .16581 L .54571 .15792 L .54667 .16589 L .54762 .15785 L Mistroke .54857 .16596 L .54952 .15777 L .55048 .16604 L .55143 .1577 L .55238 .16612 L .55333 .15762 L .55429 .1662 L .55524 .15753 L .55619 .16629 L .55714 .15745 L .5581 .16638 L .55905 .15735 L .56 .16647 L .56095 .15726 L .5619 .16657 L .56286 .15716 L .56381 .16667 L .56476 .15706 L .56571 .16678 L .56667 .15695 L .56762 .16689 L .56857 .15684 L .56952 .16701 L .57048 .15672 L .57143 .16713 L .57238 .1566 L .57333 .16726 L .57429 .15647 L .57524 .1674 L .57619 .15634 L .57714 .16754 L .5781 .1562 L .57905 .16769 L .58 .15605 L .58095 .16785 L .5819 .15589 L .58286 .16801 L .58381 .15572 L .58476 .16819 L .58571 .15555 L .58667 .16837 L .58762 .15536 L .58857 .16857 L .58952 .15516 L .59048 .16878 L .59143 .15495 L .59238 .16901 L .59333 .15473 L .59429 .16924 L 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.02531 L .65524 .02582 L .65619 .02452 L .65714 .02809 L .6581 .01996 L .65905 .04974 L .66 .38975 L .66095 .53229 L .6619 .42946 L .66286 .58985 L .66381 .58365 L .66476 .57207 L .66571 .5468 L .66667 .47705 L .66762 .22977 L .66857 .09449 L .66952 .25886 L .67048 .06718 L .67143 .32768 L .67238 .01936 L .67333 .05428 L .67429 .37175 L .67524 .49797 L .67619 .30516 L .67714 .03082 L .6781 .01665 L .67905 .08502 L .68 .27975 L .68095 .04925 L .6819 .39178 L .68286 .53595 L .68381 .44191 L .68476 .59933 L .68571 .59845 L .68667 .59724 L .68762 .59553 L .68857 .59298 L .68952 .58895 L .69048 .58206 L Mistroke .69143 .56888 L .69238 .53905 L .69333 .4522 L .69429 .60301 L .69524 .60299 L .69619 .60297 L .69714 .60295 L .6981 .60293 L .69905 .6029 L .7 .60287 L .70095 .60284 L .7019 .60281 L .70286 .60277 L .70381 .60273 L .70476 .60269 L .70571 .60263 L .70667 .60258 L .70762 .60251 L .70857 .60243 L .70952 .60235 L .71048 .60225 L .71143 .60214 L .71238 .602 L .71333 .60184 L .71429 .60166 L .71524 .60143 L .71619 .60116 L .71714 .60082 L .7181 .60039 L .71905 .59985 L .72 .59914 L .72095 .5982 L .7219 .59689 L .72286 .59502 L .72381 .5922 L .72476 .58767 L .72571 .57974 L .72667 .56405 L .72762 .52671 L .72857 .41009 L .72952 .566 L .73048 .53177 L .73143 .4277 L .73238 .5881 L .73333 .58053 L .73429 .56571 L .73524 .53104 L .73619 .42517 L .73714 .58543 L .7381 .57554 L Mistroke .73905 .55485 L .74 .50127 L .74095 .31735 L .7419 .02391 L .74286 .03007 L .74381 .01737 L .74476 .07451 L .74571 .30646 L .74667 .03002 L .74762 .01742 L .74857 .07381 L .74952 .30838 L .75048 .02885 L .75143 .01883 L .75238 .05875 L .75333 .3554 L .75429 .46501 L .75524 .1894 L .75619 .13435 L .75714 .1933 L .7581 .13046 L .75905 .19849 L .76 .12529 L .76095 .20571 L .7619 .11811 L .76286 .21645 L .76381 .1075 L .76476 .234 L .76571 .09041 L .76667 .26753 L .76762 .05951 L .76857 .35273 L .76952 .45957 L .77048 .17222 L .77143 .15151 L .77238 .17273 L .77333 .151 L .77429 .1733 L .77524 .15044 L .77619 .17392 L .77714 .14982 L .7781 .17462 L .77905 .14912 L .78 .1754 L .78095 .14834 L .7819 .17628 L .78286 .14746 L .78381 .17728 L .78476 .14645 L .78571 .17844 L Mistroke .78667 .1453 L .78762 .17978 L .78857 .14395 L .78952 .18136 L .79048 .14238 L .79143 .18325 L .79238 .14049 L .79333 .18553 L .79429 .13821 L .79524 .18837 L .79619 .13538 L .79714 .19197 L .7981 .13179 L .79905 .19669 L .8 .12707 L .80095 .20317 L .8019 .12063 L .80286 .21257 L .80381 .11132 L .80476 .22743 L .80571 .09676 L .80667 .25421 L .80762 .0714 L .80857 .31517 L .80952 .02503 L .81048 .02658 L .81143 .02276 L .81238 .03445 L .81333 .01483 L .81429 .14091 L .81524 .18502 L .81619 .13872 L .81714 .18772 L .8181 .13602 L .81905 .19114 L .82 .13261 L .82095 .19559 L .8219 .12817 L .82286 .20163 L .82381 .12216 L .82476 .21028 L .82571 .11359 L .82667 .22366 L .82762 .10043 L .82857 .247 L .82952 .07807 L .83048 .29695 L .83143 .03625 L .83238 .45337 L .83333 .60317 L Mistroke .83429 .60316 L .83524 .60315 L .83619 .60315 L .83714 .60314 L .8381 .60313 L .83905 .60312 L .84 .60311 L .84095 .6031 L .8419 .60309 L .84286 .60308 L .84381 .60307 L .84476 .60306 L .84571 .60305 L .84667 .60303 L .84762 .60301 L .84857 .603 L .84952 .60298 L .85048 .60296 L .85143 .60294 L .85238 .60291 L .85333 .60289 L .85429 .60286 L .85524 .60282 L .85619 .60279 L .85714 .60275 L .8581 .6027 L .85905 .60266 L .86 .6026 L .86095 .60254 L .8619 .60247 L .86286 .60239 L .86381 .60229 L .86476 .60219 L .86571 .60206 L .86667 .60191 L .86762 .60174 L .86857 .60153 L .86952 .60128 L .87048 .60097 L .87143 .60058 L .87238 .60009 L .87333 .59946 L .87429 .59863 L .87524 .59749 L .87619 .59589 L .87714 .59353 L .8781 .58984 L .87905 .58364 L .88 .57205 L .88095 .54675 L Mistroke .8819 .4769 L .88286 .22924 L .88381 .095 L .88476 .25779 L .88571 .06814 L .88667 .32475 L .88762 .02051 L .88857 .04606 L .88952 .40551 L .89048 .55905 L .89143 .51318 L .89238 .36131 L .89333 .47704 L .89429 .22973 L .89524 .09453 L .89619 .25877 L .89714 .06726 L .8981 .32745 L .89905 .01944 L .9 .05359 L .90095 .3744 L .9019 .50319 L .90286 .32449 L .90381 .02062 L .90476 .04538 L .90571 .40855 L .90667 .56371 L .90762 .52581 L .90857 .4069 L .90952 .5612 L .91048 .51909 L .91143 .38286 L .91238 .51951 L .91333 .38439 L .91429 .52239 L .91524 .39476 L .91619 .54122 L .91714 .4593 L .9181 .17139 L .91905 .15235 L .92 .17182 L .92095 .15192 L .9219 .17229 L .92286 .15145 L .92381 .1728 L .92476 .15093 L .92571 .17337 L .92667 .15036 L .92762 .17401 L .92857 .14973 L Mistroke .92952 .17471 L .93048 .14902 L .93143 .1755 L .93238 .14823 L .93333 .1764 L .93429 .14733 L .93524 .17742 L .93619 .14631 L .93714 .1786 L .9381 .14513 L .93905 .17997 L .94 .14376 L .94095 .18159 L .9419 .14215 L .94286 .18352 L .94381 .14022 L .94476 .18587 L .94571 .13788 L .94667 .18878 L .94762 .13496 L .94857 .19251 L .94952 .13125 L .95048 .19742 L .95143 .12635 L .95238 .20419 L .95333 .11962 L .95429 .21411 L .95524 .1098 L .95619 .23 L .95714 .09426 L .9581 .25932 L .95905 .06677 L .96 .32896 L .96095 .01889 L .9619 .05823 L .96286 .35724 L .96381 .46876 L .96476 .20166 L .96571 .12213 L .96667 .21033 L .96762 .11354 L .96857 .22374 L .96952 .10035 L .97048 .24715 L .97143 .07793 L .97238 .2973 L .97333 .03601 L .97429 .45469 L .97524 .60328 L .97619 .60328 L Mistroke Mfstroke P p [ .01 .01 ] 0 setdash .002 w .02476 .55396 m .02571 .49867 L .02667 .30776 L .02762 .02922 L .02857 .01834 L .02952 .06335 L .03048 .3398 L .03143 .01585 L .03238 .10073 L .03333 .24643 L .03429 .0786 L .03524 .29557 L .03619 .03722 L .03714 .4482 L .0381 .60206 L .03905 .60191 L .04 .60173 L .04095 .60152 L .0419 .60127 L .04286 .60096 L .04381 .60057 L .04476 .60008 L .04571 .59944 L .04667 .5986 L .04762 .59745 L .04857 .59583 L .04952 .59344 L .05048 .5897 L .05143 .5834 L .05238 .57157 L .05333 .5456 L .05429 .47329 L .05524 .21686 L .05619 .10709 L .05714 .23472 L .0581 .08972 L .05905 .26905 L .06 .0582 L .06095 .35735 L .0619 .469 L .06286 .20243 L .06381 .12137 L .06476 .21146 L .06571 .11242 L .06667 .2256 L .06762 .09854 L .06857 .25067 L .06952 .07466 L .07048 .30603 L .07143 .03028 L Mistroke .07238 .01715 L .07333 .0774 L .07429 .29869 L .07524 .03506 L .07619 .01475 L .07714 .15083 L .0781 .17348 L .07905 .15025 L .08 .17413 L .08095 .14961 L .0819 .17485 L .08286 .14889 L .08381 .17566 L .08476 .14808 L .08571 .17658 L .08667 .14716 L .08762 .17763 L .08857 .14611 L .08952 .17884 L .09048 .1449 L .09143 .18025 L .09238 .14349 L .09333 .18192 L .09429 .14182 L .09524 .18391 L .09619 .13983 L .09714 .18635 L .0981 .13739 L .09905 .1894 L .1 .13435 L .10095 .1933 L .1019 .13045 L .10286 .19849 L .10381 .12529 L .10476 .20572 L .10571 .11811 L .10667 .21645 L .10762 .1075 L .10857 .234 L .10952 .0904 L .11048 .26754 L .11143 .0595 L .11238 .35277 L .11333 .45965 L .11429 .17245 L .11524 .15128 L .11619 .17299 L .11714 .15075 L .1181 .17358 L .11905 .15016 L Mistroke .12 .17423 L .12095 .1495 L .1219 .17496 L .12286 .14877 L .12381 .17579 L .12476 .14795 L .12571 .17673 L .12667 .14701 L .12762 .1778 L .12857 .14594 L .12952 .17903 L .13048 .1447 L .13143 .18048 L .13238 .14326 L .13333 .18219 L .13429 .14155 L .13524 .18425 L .13619 .13949 L .13714 .18676 L .1381 .13698 L .13905 .18992 L .14 .13383 L .14095 .19398 L .1419 .12978 L .14286 .19942 L .14381 .12437 L .14476 .20704 L .14571 .11679 L .14667 .21851 L .14762 .10547 L .14857 .23761 L .14952 .08694 L .15048 .27529 L .15143 .05291 L .15238 .37704 L .15333 .50834 L .15429 .34353 L .15524 .01523 L .15619 .11915 L .15714 .21483 L .1581 .10909 L .15905 .23122 L .16 .09308 L .16095 .26179 L .1619 .06456 L .16286 .33589 L .16381 .01674 L .16476 .08353 L .16571 .28331 L .16667 .04641 L Mistroke .16762 .40397 L .16857 .55662 L .16952 .50636 L .17048 .33618 L .17143 .01667 L .17238 .08473 L .17333 .28044 L .17429 .0487 L .17524 .39411 L .17619 .54008 L .17714 .4556 L .1781 .60332 L .17905 .60332 L .18 .60332 L 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.24095 .60331 L .2419 .60331 L .24286 .60331 L .24381 .60331 L .24476 .60331 L .24571 .60331 L .24667 .60331 L .24762 .60331 L .24857 .60331 L .24952 .60331 L .25048 .60331 L .25143 .60331 L .25238 .60331 L .25333 .60331 L .25429 .60331 L .25524 .60331 L .25619 .60331 L .25714 .60331 L .2581 .60331 L .25905 .60331 L .26 .60331 L .26095 .60331 L .2619 .60331 L Mistroke .26286 .60331 L .26381 .60331 L .26476 .60331 L .26571 .60331 L .26667 .60331 L .26762 .60331 L .26857 .60331 L .26952 .60331 L .27048 .60331 L .27143 .60331 L .27238 .60331 L .27333 .60331 L .27429 .60331 L .27524 .60331 L .27619 .60331 L .27714 .60331 L .2781 .60331 L .27905 .60331 L .28 .60331 L .28095 .60331 L .2819 .60331 L .28286 .6033 L .28381 .6033 L .28476 .6033 L .28571 .6033 L .28667 .6033 L .28762 .6033 L .28857 .6033 L .28952 .6033 L .29048 .6033 L .29143 .6033 L .29238 .6033 L .29333 .6033 L .29429 .6033 L .29524 .6033 L .29619 .6033 L .29714 .6033 L .2981 .6033 L .29905 .6033 L .3 .6033 L .30095 .6033 L .3019 .6033 L .30286 .6033 L .30381 .6033 L .30476 .6033 L .30571 .6033 L .30667 .6033 L .30762 .60329 L .30857 .60329 L .30952 .60329 L Mistroke .31048 .60329 L .31143 .60329 L .31238 .60329 L .31333 .60329 L .31429 .60329 L .31524 .60329 L .31619 .60329 L .31714 .60329 L .3181 .60329 L .31905 .60329 L .32 .60329 L .32095 .60329 L .3219 .60329 L .32286 .60328 L .32381 .60328 L .32476 .60328 L .32571 .60328 L .32667 .60328 L .32762 .60328 L .32857 .60328 L .32952 .60328 L .33048 .60328 L .33143 .60328 L .33238 .60328 L .33333 .60327 L .33429 .60327 L .33524 .60327 L .33619 .60327 L .33714 .60327 L .3381 .60327 L .33905 .60327 L .34 .60327 L .34095 .60326 L .3419 .60326 L .34286 .60326 L .34381 .60326 L .34476 .60326 L .34571 .60326 L .34667 .60325 L .34762 .60325 L .34857 .60325 L .34952 .60325 L .35048 .60325 L .35143 .60324 L .35238 .60324 L .35333 .60324 L .35429 .60324 L .35524 .60323 L .35619 .60323 L .35714 .60323 L Mistroke .3581 .60323 L .35905 .60322 L .36 .60322 L .36095 .60322 L .3619 .60321 L .36286 .60321 L .36381 .60321 L .36476 .6032 L .36571 .6032 L .36667 .60319 L .36762 .60319 L .36857 .60318 L .36952 .60318 L .37048 .60317 L .37143 .60317 L .37238 .60316 L .37333 .60315 L .37429 .60315 L .37524 .60314 L .37619 .60313 L .37714 .60312 L .3781 .60311 L .37905 .6031 L .38 .60309 L .38095 .60308 L .3819 .60307 L .38286 .60306 L .38381 .60304 L .38476 .60303 L .38571 .60301 L .38667 .603 L .38762 .60298 L .38857 .60296 L .38952 .60294 L .39048 .60291 L .39143 .60288 L .39238 .60286 L .39333 .60282 L .39429 .60279 L .39524 .60275 L .39619 .6027 L .39714 .60265 L .3981 .6026 L .39905 .60254 L .4 .60246 L .40095 .60238 L .4019 .60229 L .40286 .60218 L .40381 .60206 L .40476 .60191 L Mistroke .40571 .60173 L .40667 .60152 L .40762 .60127 L .40857 .60096 L .40952 .60057 L .41048 .60008 L .41143 .59944 L .41238 .5986 L .41333 .59745 L .41429 .59583 L .41524 .59343 L .41619 .58969 L .41714 .58338 L .4181 .57153 L .41905 .54551 L .42 .47302 L .42095 .21593 L .4219 .10801 L .42286 .23311 L .42381 .09126 L .42476 .26568 L .42571 .06113 L .42667 .34719 L .42762 .01485 L .42857 .139 L .42952 .18738 L .43048 .13637 L .43143 .1907 L .43238 .13305 L .43333 .19501 L .43429 .12875 L .43524 .20082 L .43619 .12296 L .43714 .20909 L .4381 .11476 L .43905 .22175 L .44 .10229 L .44095 .24346 L .4419 .08139 L .44286 .28852 L .44381 .04238 L .44476 .42236 L .44571 .58226 L .44667 .56928 L .44762 .54005 L .44857 .45548 L .44952 .60332 L .45048 .60332 L .45143 .60332 L .45238 .60332 L Mistroke .45333 .60332 L .45429 .60332 L .45524 .60332 L .45619 .60332 L .45714 .60332 L .4581 .60331 L .45905 .60331 L .46 .60331 L .46095 .60331 L .4619 .60331 L .46286 .60331 L .46381 .60331 L .46476 .60331 L .46571 .60331 L .46667 .60331 L .46762 .60331 L .46857 .60331 L .46952 .60331 L .47048 .60331 L .47143 .60331 L .47238 .60331 L .47333 .60331 L .47429 .60331 L .47524 .60331 L .47619 .60331 L .47714 .60331 L .4781 .60331 L .47905 .60331 L .48 .60331 L .48095 .60331 L 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