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"Times"; fontset = special5, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 12, "Times"; paletteColors = 128; automaticGrouping; currentKernel; ] :[font = input; preserveAspect] (* In this notebook we investigate the dynamics of the logistic function: x(t+1) = lambda x(t) (1-x(t)). *) :[font = input; preserveAspect] Logistic[x_,n_,lambda_] := If[n==0,{}, With[{y=lambda x(1-x)}, Prepend[Logistic[y,n-1,lambda],x]]] :[font = input; preserveAspect] (* Consider two special cases of the logistic function for lambda = 1 and lambda = 4. *) :[font = input; preserveAspect] FixedPointLogistic[] := Logistic[.9,100,1] :[font = input; preserveAspect] ChaoticLogistic[] := Logistic[.9,100,4] :[font = input; preserveAspect] Exercise1[timeSeries_,flag_] := ListPlot[Table[{timeSeries[[i]], timeSeries[[i+1]]}, {i,1,Length[timeSeries]-1}], PlotJoined -> flag] :[font = input; preserveAspect] (* Examine the behavior of the systems in phase space. *) :[font = input; preserveAspect; startGroup] 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.57981 L s P p .002 w .02381 0 m .02381 .61803 L s P P 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath .004 w .88101 .22663 m .36669 .55721 L .90158 .18484 L .29908 .49854 L .80666 .35932 L .58139 .58617 L .94845 .08143 L .13176 .25133 L .40667 .58073 L .93964 .10174 L .16461 .31134 L .50377 .60332 L .97619 .01486 L .02404 .01529 L .02475 .01703 L .02756 .02395 L .03875 .05106 L .08262 .15112 L .24452 .4339 L .70206 .49741 L .80482 .36223 L .5861 .58413 L .94514 .08911 L .14418 .27469 L .44445 .59534 L .96328 .04635 L .075 .13445 L .21755 .39624 L .64113 .55168 L .89263 .20328 L .32892 .52736 L .85328 .27946 L .45217 .59741 L .96663 .03825 L .06189 .1051 L .17006 .32074 L .51897 .60242 L .97474 .01843 L .02983 .0295 L .04773 .07235 L .11707 .22269 L .36032 .55269 L .89428 .19992 L .32347 .52244 L .84533 .29387 L .4755 .60179 L .97372 .02095 L .03389 .03938 L .06372 .10924 L .17675 .33209 L Mistroke .53733 .59974 L .97041 .02907 L .04704 .07074 L .11446 .21748 L .35189 .5464 L .88409 .2205 L .35678 .55009 L .89007 .20849 L .33734 .53466 L .86509 .25744 L .41655 .58527 L .94698 .08485 L .13729 .26183 L .42364 .58821 L .95175 .07373 L .1193 .22712 L .36749 .55776 L .90247 .18297 L .29605 .49536 L .80151 .36743 L .59452 .58018 L .93875 .10377 L .1679 .31704 L .51298 .60292 L .97555 .01645 L .02662 .02165 L .03503 .04213 L .06816 .11926 L .19297 .35862 L .58026 .58665 L .94921 .07966 L .12889 .24582 L .39775 .5762 L .93231 .11832 L .19145 .35621 L .57635 .58824 L .95178 .07364 L .11915 .22682 L .36701 .55743 L .90194 .18409 L .29787 .49727 L .8046 .36257 L .58666 .58388 L .94473 .09005 L .1457 .27749 L .44898 .59659 L .96531 .04146 L .06708 .11683 L .18903 .35231 L Mfstroke % End of Graphics MathPictureEnd :[font = input; preserveAspect; animationSpeed = 30] Exercise2[min_,max_,incr_] := Do[ListPlot[Logistic[.7,100,x], PlotJoined -> True, PlotRange -> {{0,100},{0,1.0}}, AxesOrigin -> {0,0}], {x,min,max,incr}] :[font = input; preserveAspect] (* Consider the behavior as we vary lambda from 1 to 4. Animate the resulting graphics and slow the animation. *) :[font = input; preserveAspect; startGroup] Exercise2[1.0,4.0,0.2] :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 282; pictureHeight = 174; startGroup] %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0 0.01 0 0.618034 [ [(20)] .2 0 0 2 Msboxa [(40)] .4 0 0 2 Msboxa [(60)] .6 0 0 2 Msboxa [(80)] .8 0 0 2 Msboxa [(100)] 1 0 0 2 Msboxa [(0)] -0.0125 0 1 0 Msboxa [(0.2)] -0.0125 .12361 1 0 Msboxa [(0.4)] -0.0125 .24721 1 0 Msboxa [(0.6)] -0.0125 .37082 1 0 Msboxa [(0.8)] -0.0125 .49443 1 0 Msboxa [(1)] -0.0125 .61803 1 0 Msboxa [ -0.001 -0.001 0 0 ] [ 1.001 .61903 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath [ ] 0 setdash 0 g p p .002 w .2 0 m .2 .00625 L s P [(20)] .2 0 0 2 Mshowa p .002 w .4 0 m .4 .00625 L s P [(40)] .4 0 0 2 Mshowa p .002 w .6 0 m .6 .00625 L s P [(60)] .6 0 0 2 Mshowa p .002 w .8 0 m .8 .00625 L s P [(80)] .8 0 0 2 Mshowa p .002 w 1 0 m 1 .00625 L s P [(100)] 1 0 0 2 Mshowa p .001 w .04 0 m .04 .00375 L s P p .001 w .08 0 m .08 .00375 L s P p .001 w .12 0 m .12 .00375 L s P p .001 w .16 0 m .16 .00375 L s P p .001 w .24 0 m .24 .00375 L s P p .001 w .28 0 m .28 .00375 L s P p .001 w .32 0 m .32 .00375 L s P p .001 w .36 0 m .36 .00375 L s P p .001 w .44 0 m .44 .00375 L s P p .001 w .48 0 m .48 .00375 L s P p .001 w .52 0 m .52 .00375 L s P p .001 w .56 0 m .56 .00375 L s P p .001 w .64 0 m .64 .00375 L s P p .001 w .68 0 m .68 .00375 L s P p .001 w .72 0 m .72 .00375 L s P p .001 w .76 0 m .76 .00375 L s P p .001 w .84 0 m .84 .00375 L s P p .001 w .88 0 m .88 .00375 L s P p .001 w .92 0 m .92 .00375 L s P p .001 w .96 0 m .96 .00375 L s P p .002 w 0 0 m 1 0 L s P p .002 w 0 0 m .00625 0 L s P [(0)] -0.0125 0 1 0 Mshowa p .002 w 0 .12361 m .00625 .12361 L s P [(0.2)] -0.0125 .12361 1 0 Mshowa p .002 w 0 .24721 m .00625 .24721 L s P [(0.4)] -0.0125 .24721 1 0 Mshowa p .002 w 0 .37082 m .00625 .37082 L s P [(0.6)] -0.0125 .37082 1 0 Mshowa p .002 w 0 .49443 m .00625 .49443 L s P [(0.8)] -0.0125 .49443 1 0 Mshowa p .002 w 0 .61803 m .00625 .61803 L s P [(1)] -0.0125 .61803 1 0 Mshowa p .001 w 0 .02472 m .00375 .02472 L s P p .001 w 0 .04944 m .00375 .04944 L s P p .001 w 0 .07416 m .00375 .07416 L s P p .001 w 0 .09889 m .00375 .09889 L s P p .001 w 0 .14833 m .00375 .14833 L s P p .001 w 0 .17305 m .00375 .17305 L s P p .001 w 0 .19777 m .00375 .19777 L s P p .001 w 0 .22249 m .00375 .22249 L s P p .001 w 0 .27193 m .00375 .27193 L s P p .001 w 0 .29666 m .00375 .29666 L s P p .001 w 0 .32138 m .00375 .32138 L s P p .001 w 0 .3461 m .00375 .3461 L s P p .001 w 0 .39554 m .00375 .39554 L s P p .001 w 0 .42026 m .00375 .42026 L s P p .001 w 0 .44498 m .00375 .44498 L s P p .001 w 0 .46971 m .00375 .46971 L s P p .001 w 0 .51915 m .00375 .51915 L s P p .001 w 0 .54387 m .00375 .54387 L s P p .001 w 0 .56859 m .00375 .56859 L s P p .001 w 0 .59331 m .00375 .59331 L s P p .002 w 0 0 m 0 .61803 L s P P 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath .004 w .01 .43262 m .02 .12979 L .03 .10253 L .04 .08552 L .05 .07369 L .06 .0649 L .07 .05809 L .08 .05263 L .09 .04815 L .1 .0444 L .11 .04121 L .12 .03846 L .13 .03607 L .14 .03396 L .15 .03209 L .16 .03043 L .17 .02893 L .18 .02758 L .19 .02635 L .2 .02522 L .21 .02419 L .22 .02325 L .23 .02237 L .24 .02156 L .25 .02081 L .26 .02011 L .27 .01945 L .28 .01884 L .29 .01827 L .3 .01773 L .31 .01722 L .32 .01674 L .33 .01629 L .34 .01586 L .35 .01545 L .36 .01506 L .37 .0147 L .38 .01435 L .39 .01401 L .4 .0137 L .41 .01339 L .42 .0131 L .43 .01282 L .44 .01256 L .45 .0123 L .46 .01206 L .47 .01182 L .48 .0116 L .49 .01138 L .5 .01117 L Mistroke .51 .01097 L .52 .01077 L .53 .01059 L .54 .0104 L .55 .01023 L .56 .01006 L .57 .0099 L .58 .00974 L .59 .00958 L .6 .00944 L .61 .00929 L .62 .00915 L .63 .00902 L .64 .00888 L .65 .00876 L .66 .00863 L .67 .00851 L .68 .0084 L .69 .00828 L .7 .00817 L .71 .00806 L .72 .00796 L .73 .00785 L .74 .00775 L .75 .00766 L .76 .00756 L .77 .00747 L .78 .00738 L .79 .00729 L .8 .00721 L .81 .00712 L .82 .00704 L .83 .00696 L .84 .00688 L .85 .0068 L .86 .00673 L .87 .00666 L .88 .00658 L .89 .00651 L .9 .00645 L .91 .00638 L .92 .00631 L .93 .00625 L .94 .00618 L .95 .00612 L .96 .00606 L .97 .006 L .98 .00594 L .99 .00589 L 1 .00583 L Mistroke Mfstroke % End of Graphics MathPictureEnd :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 282; pictureHeight = 174] %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0 0.01 0 0.618034 [ [(20)] .2 0 0 2 Msboxa [(40)] .4 0 0 2 Msboxa [(60)] .6 0 0 2 Msboxa [(80)] .8 0 0 2 Msboxa [(100)] 1 0 0 2 Msboxa [(0)] -0.0125 0 1 0 Msboxa [(0.2)] -0.0125 .12361 1 0 Msboxa [(0.4)] -0.0125 .24721 1 0 Msboxa [(0.6)] -0.0125 .37082 1 0 Msboxa [(0.8)] -0.0125 .49443 1 0 Msboxa [(1)] -0.0125 .61803 1 0 Msboxa [ -0.001 -0.001 0 0 ] [ 1.001 .61903 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath [ ] 0 setdash 0 g p p .002 w .2 0 m .2 .00625 L s P [(20)] .2 0 0 2 Mshowa p .002 w .4 0 m .4 .00625 L s P [(40)] .4 0 0 2 Mshowa p .002 w .6 0 m .6 .00625 L s P [(60)] .6 0 0 2 Mshowa p .002 w .8 0 m .8 .00625 L s P [(80)] .8 0 0 2 Mshowa p .002 w 1 0 m 1 .00625 L s P [(100)] 1 0 0 2 Mshowa p .001 w .04 0 m .04 .00375 L s P p .001 w .08 0 m .08 .00375 L s P p .001 w .12 0 m .12 .00375 L s P p .001 w .16 0 m .16 .00375 L s P p .001 w .24 0 m .24 .00375 L s P p .001 w .28 0 m .28 .00375 L s P p .001 w .32 0 m .32 .00375 L s P p .001 w .36 0 m .36 .00375 L s P p .001 w .44 0 m .44 .00375 L s P p .001 w .48 0 m .48 .00375 L s P p .001 w .52 0 m .52 .00375 L s P p .001 w .56 0 m .56 .00375 L s P p .001 w .64 0 m .64 .00375 L s P p .001 w .68 0 m .68 .00375 L s P p .001 w .72 0 m .72 .00375 L s P p .001 w .76 0 m .76 .00375 L s P p .001 w .84 0 m .84 .00375 L s P p .001 w .88 0 m .88 .00375 L s P p .001 w .92 0 m .92 .00375 L s P p .001 w .96 0 m .96 .00375 L s P p .002 w 0 0 m 1 0 L s P p .002 w 0 0 m .00625 0 L s P [(0)] -0.0125 0 1 0 Mshowa p .002 w 0 .12361 m .00625 .12361 L s P [(0.2)] -0.0125 .12361 1 0 Mshowa p .002 w 0 .24721 m .00625 .24721 L s P [(0.4)] -0.0125 .24721 1 0 Mshowa p .002 w 0 .37082 m .00625 .37082 L s P [(0.6)] -0.0125 .37082 1 0 Mshowa p .002 w 0 .49443 m .00625 .49443 L s P [(0.8)] -0.0125 .49443 1 0 Mshowa p .002 w 0 .61803 m .00625 .61803 L s P [(1)] -0.0125 .61803 1 0 Mshowa p .001 w 0 .02472 m .00375 .02472 L s P p .001 w 0 .04944 m .00375 .04944 L s P p .001 w 0 .07416 m .00375 .07416 L s P p .001 w 0 .09889 m .00375 .09889 L s P p .001 w 0 .14833 m .00375 .14833 L s P p .001 w 0 .17305 m .00375 .17305 L s P p .001 w 0 .19777 m .00375 .19777 L s P p .001 w 0 .22249 m .00375 .22249 L s P p .001 w 0 .27193 m .00375 .27193 L s P p .001 w 0 .29666 m .00375 .29666 L s P p .001 w 0 .32138 m .00375 .32138 L s P p .001 w 0 .3461 m .00375 .3461 L s P p .001 w 0 .39554 m .00375 .39554 L s P p .001 w 0 .42026 m .00375 .42026 L s P p .001 w 0 .44498 m .00375 .44498 L s P p .001 w 0 .46971 m .00375 .46971 L s P p .001 w 0 .51915 m .00375 .51915 L s P p .001 w 0 .54387 m .00375 .54387 L s P p .001 w 0 .56859 m .00375 .56859 L s P p .001 w 0 .59331 m .00375 .59331 L s P p .002 w 0 0 m 0 .61803 L s P P 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath .004 w .01 .43262 m .02 .15574 L .03 .1398 L .04 .12981 L .05 .12305 L .06 .11826 L .07 .11476 L .08 .11214 L .09 .11015 L .1 .10862 L .11 .10744 L .12 .10651 L .13 .10579 L .14 .10522 L .15 .10477 L .16 .10441 L .17 .10412 L .18 .1039 L .19 .10372 L .2 .10357 L .21 .10346 L .22 .10337 L .23 .1033 L .24 .10324 L .25 .10319 L .26 .10315 L .27 .10312 L .28 .1031 L .29 .10308 L .3 .10307 L .31 .10305 L .32 .10304 L .33 .10304 L .34 .10303 L .35 .10303 L .36 .10302 L .37 .10302 L .38 .10302 L .39 .10301 L .4 .10301 L .41 .10301 L .42 .10301 L .43 .10301 L .44 .10301 L .45 .10301 L .46 .10301 L .47 .10301 L .48 .10301 L .49 .10301 L .5 .10301 L Mistroke .51 .10301 L .52 .10301 L .53 .10301 L .54 .10301 L .55 .10301 L .56 .10301 L .57 .10301 L .58 .10301 L .59 .10301 L .6 .10301 L .61 .10301 L .62 .10301 L .63 .10301 L .64 .10301 L .65 .10301 L .66 .10301 L .67 .10301 L .68 .10301 L .69 .10301 L .7 .10301 L .71 .10301 L .72 .10301 L .73 .10301 L .74 .10301 L .75 .10301 L .76 .10301 L .77 .10301 L .78 .10301 L .79 .10301 L .8 .10301 L .81 .10301 L .82 .10301 L .83 .10301 L .84 .10301 L .85 .10301 L .86 .10301 L .87 .10301 L .88 .10301 L .89 .10301 L .9 .10301 L .91 .10301 L .92 .10301 L .93 .10301 L .94 .10301 L .95 .10301 L .96 .10301 L .97 .10301 L .98 .10301 L .99 .10301 L 1 .10301 L Mistroke Mfstroke % End of Graphics MathPictureEnd :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 282; pictureHeight = 174] %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0 0.01 0 0.618034 [ [(20)] .2 0 0 2 Msboxa [(40)] .4 0 0 2 Msboxa [(60)] .6 0 0 2 Msboxa [(80)] .8 0 0 2 Msboxa [(100)] 1 0 0 2 Msboxa [(0)] -0.0125 0 1 0 Msboxa [(0.2)] -0.0125 .12361 1 0 Msboxa [(0.4)] -0.0125 .24721 1 0 Msboxa [(0.6)] -0.0125 .37082 1 0 Msboxa [(0.8)] -0.0125 .49443 1 0 Msboxa [(1)] -0.0125 .61803 1 0 Msboxa [ -0.001 -0.001 0 0 ] [ 1.001 .61903 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath [ ] 0 setdash 0 g p p .002 w .2 0 m .2 .00625 L s P [(20)] .2 0 0 2 Mshowa p .002 w .4 0 m .4 .00625 L s P [(40)] .4 0 0 2 Mshowa p .002 w .6 0 m .6 .00625 L s P [(60)] .6 0 0 2 Mshowa p .002 w .8 0 m .8 .00625 L s P [(80)] .8 0 0 2 Mshowa p .002 w 1 0 m 1 .00625 L s P [(100)] 1 0 0 2 Mshowa p .001 w .04 0 m .04 .00375 L s P p .001 w .08 0 m .08 .00375 L s P p .001 w .12 0 m .12 .00375 L s P p .001 w .16 0 m .16 .00375 L s P p .001 w .24 0 m .24 .00375 L s P p .001 w .28 0 m .28 .00375 L s P p .001 w .32 0 m .32 .00375 L s P p .001 w .36 0 m .36 .00375 L s P p .001 w .44 0 m .44 .00375 L s P p .001 w .48 0 m .48 .00375 L s P p .001 w .52 0 m .52 .00375 L s P p .001 w .56 0 m .56 .00375 L s P p .001 w .64 0 m .64 .00375 L s P p .001 w .68 0 m .68 .00375 L s P p .001 w .72 0 m .72 .00375 L s P p .001 w .76 0 m .76 .00375 L s P p .001 w .84 0 m .84 .00375 L s P p .001 w .88 0 m .88 .00375 L s P p .001 w .92 0 m .92 .00375 L s P p .001 w .96 0 m .96 .00375 L s P p .002 w 0 0 m 1 0 L s P p .002 w 0 0 m .00625 0 L s P [(0)] -0.0125 0 1 0 Mshowa p .002 w 0 .12361 m .00625 .12361 L s P [(0.2)] -0.0125 .12361 1 0 Mshowa p .002 w 0 .24721 m .00625 .24721 L s P [(0.4)] -0.0125 .24721 1 0 Mshowa p .002 w 0 .37082 m .00625 .37082 L s P [(0.6)] -0.0125 .37082 1 0 Mshowa p .002 w 0 .49443 m .00625 .49443 L s P [(0.8)] -0.0125 .49443 1 0 Mshowa p .002 w 0 .61803 m .00625 .61803 L s P [(1)] -0.0125 .61803 1 0 Mshowa p .001 w 0 .02472 m .00375 .02472 L s P p .001 w 0 .04944 m .00375 .04944 L s P p .001 w 0 .07416 m .00375 .07416 L s P p .001 w 0 .09889 m .00375 .09889 L s P p .001 w 0 .14833 m .00375 .14833 L s P p .001 w 0 .17305 m .00375 .17305 L s P p .001 w 0 .19777 m .00375 .19777 L s P p .001 w 0 .22249 m .00375 .22249 L s P p .001 w 0 .27193 m .00375 .27193 L s P p .001 w 0 .29666 m .00375 .29666 L s P p .001 w 0 .32138 m .00375 .32138 L s P p .001 w 0 .3461 m .00375 .3461 L s P p .001 w 0 .39554 m .00375 .39554 L s P p .001 w 0 .42026 m .00375 .42026 L s P p .001 w 0 .44498 m .00375 .44498 L s P p .001 w 0 .46971 m .00375 .46971 L s P p .001 w 0 .51915 m .00375 .51915 L s P p .001 w 0 .54387 m .00375 .54387 L s P p .001 w 0 .56859 m .00375 .56859 L s P p .001 w 0 .59331 m .00375 .59331 L s P p .002 w 0 0 m 0 .61803 L s P P 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath .004 w .01 .43262 m .02 .1817 L .03 .17959 L .04 .17837 L .05 .17765 L .06 .17722 L .07 .17696 L .08 .17681 L .09 .17672 L .1 .17666 L .11 .17663 L .12 .17661 L .13 .1766 L .14 .17659 L .15 .17659 L .16 .17658 L .17 .17658 L .18 .17658 L .19 .17658 L .2 .17658 L .21 .17658 L .22 .17658 L .23 .17658 L .24 .17658 L .25 .17658 L .26 .17658 L .27 .17658 L .28 .17658 L .29 .17658 L .3 .17658 L .31 .17658 L .32 .17658 L .33 .17658 L .34 .17658 L .35 .17658 L .36 .17658 L .37 .17658 L .38 .17658 L .39 .17658 L .4 .17658 L .41 .17658 L .42 .17658 L .43 .17658 L .44 .17658 L .45 .17658 L .46 .17658 L .47 .17658 L .48 .17658 L .49 .17658 L .5 .17658 L Mistroke .51 .17658 L .52 .17658 L .53 .17658 L .54 .17658 L .55 .17658 L .56 .17658 L .57 .17658 L .58 .17658 L .59 .17658 L .6 .17658 L .61 .17658 L .62 .17658 L .63 .17658 L .64 .17658 L .65 .17658 L .66 .17658 L .67 .17658 L .68 .17658 L .69 .17658 L .7 .17658 L .71 .17658 L .72 .17658 L .73 .17658 L .74 .17658 L .75 .17658 L .76 .17658 L .77 .17658 L .78 .17658 L .79 .17658 L .8 .17658 L .81 .17658 L .82 .17658 L .83 .17658 L .84 .17658 L .85 .17658 L .86 .17658 L .87 .17658 L .88 .17658 L .89 .17658 L .9 .17658 L .91 .17658 L .92 .17658 L .93 .17658 L .94 .17658 L .95 .17658 L .96 .17658 L .97 .17658 L .98 .17658 L .99 .17658 L 1 .17658 L Mistroke Mfstroke % End of Graphics MathPictureEnd :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 282; pictureHeight = 174] %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0 0.01 0 0.618034 [ [(20)] .2 0 0 2 Msboxa [(40)] .4 0 0 2 Msboxa [(60)] .6 0 0 2 Msboxa [(80)] .8 0 0 2 Msboxa [(100)] 1 0 0 2 Msboxa [(0)] -0.0125 0 1 0 Msboxa [(0.2)] -0.0125 .12361 1 0 Msboxa [(0.4)] -0.0125 .24721 1 0 Msboxa [(0.6)] -0.0125 .37082 1 0 Msboxa [(0.8)] -0.0125 .49443 1 0 Msboxa [(1)] -0.0125 .61803 1 0 Msboxa [ -0.001 -0.001 0 0 ] [ 1.001 .61903 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath [ ] 0 setdash 0 g p p .002 w .2 0 m .2 .00625 L s P [(20)] .2 0 0 2 Mshowa p .002 w .4 0 m .4 .00625 L s P [(40)] .4 0 0 2 Mshowa p .002 w .6 0 m .6 .00625 L s P [(60)] .6 0 0 2 Mshowa p .002 w .8 0 m .8 .00625 L s P [(80)] .8 0 0 2 Mshowa p .002 w 1 0 m 1 .00625 L s P [(100)] 1 0 0 2 Mshowa p .001 w .04 0 m .04 .00375 L s P p .001 w .08 0 m .08 .00375 L s P p .001 w .12 0 m .12 .00375 L s P p .001 w .16 0 m .16 .00375 L s P p .001 w .24 0 m .24 .00375 L s P p .001 w .28 0 m .28 .00375 L s P p .001 w .32 0 m .32 .00375 L s P p .001 w .36 0 m .36 .00375 L s P p .001 w .44 0 m .44 .00375 L s P p .001 w .48 0 m .48 .00375 L s P p .001 w .52 0 m .52 .00375 L s P p .001 w .56 0 m .56 .00375 L s P p .001 w .64 0 m .64 .00375 L s P p .001 w .68 0 m .68 .00375 L s P p .001 w .72 0 m .72 .00375 L s P p .001 w .76 0 m .76 .00375 L s P p .001 w .84 0 m .84 .00375 L s P p .001 w .88 0 m .88 .00375 L s P p .001 w .92 0 m .92 .00375 L s P p .001 w .96 0 m .96 .00375 L s P p .002 w 0 0 m 1 0 L s P p .002 w 0 0 m .00625 0 L s P [(0)] -0.0125 0 1 0 Mshowa p .002 w 0 .12361 m .00625 .12361 L s P [(0.2)] -0.0125 .12361 1 0 Mshowa p .002 w 0 .24721 m .00625 .24721 L s P [(0.4)] -0.0125 .24721 1 0 Mshowa p .002 w 0 .37082 m .00625 .37082 L s P [(0.6)] -0.0125 .37082 1 0 Mshowa p .002 w 0 .49443 m .00625 .49443 L s P [(0.8)] -0.0125 .49443 1 0 Mshowa p .002 w 0 .61803 m .00625 .61803 L s P [(1)] -0.0125 .61803 1 0 Mshowa p .001 w 0 .02472 m .00375 .02472 L s P p .001 w 0 .04944 m .00375 .04944 L s P p .001 w 0 .07416 m .00375 .07416 L s P p .001 w 0 .09889 m .00375 .09889 L s P p .001 w 0 .14833 m .00375 .14833 L s P p .001 w 0 .17305 m .00375 .17305 L s P p .001 w 0 .19777 m .00375 .19777 L s P p .001 w 0 .22249 m .00375 .22249 L s P p .001 w 0 .27193 m .00375 .27193 L s P p .001 w 0 .29666 m .00375 .29666 L s P p .001 w 0 .32138 m .00375 .32138 L s P p .001 w 0 .3461 m .00375 .3461 L s P p .001 w 0 .39554 m .00375 .39554 L s P p .001 w 0 .42026 m .00375 .42026 L s P p .001 w 0 .44498 m .00375 .44498 L s P p .001 w 0 .46971 m .00375 .46971 L s P p .001 w 0 .51915 m .00375 .51915 L s P p .001 w 0 .54387 m .00375 .54387 L s P p .001 w 0 .56859 m .00375 .56859 L s P p .001 w 0 .59331 m .00375 .59331 L s P p .002 w 0 0 m 0 .61803 L s P P 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath .004 w .01 .43262 m .02 .20766 L .03 .22062 L .04 .22698 L .05 .22979 L .06 .23096 L .07 .23144 L .08 .23163 L .09 .23171 L .1 .23174 L .11 .23175 L .12 .23176 L .13 .23176 L .14 .23176 L .15 .23176 L .16 .23176 L .17 .23176 L .18 .23176 L .19 .23176 L .2 .23176 L .21 .23176 L .22 .23176 L .23 .23176 L .24 .23176 L .25 .23176 L .26 .23176 L .27 .23176 L .28 .23176 L .29 .23176 L .3 .23176 L .31 .23176 L .32 .23176 L .33 .23176 L .34 .23176 L .35 .23176 L .36 .23176 L .37 .23176 L .38 .23176 L .39 .23176 L .4 .23176 L .41 .23176 L .42 .23176 L .43 .23176 L .44 .23176 L .45 .23176 L .46 .23176 L .47 .23176 L .48 .23176 L .49 .23176 L .5 .23176 L Mistroke .51 .23176 L .52 .23176 L .53 .23176 L .54 .23176 L .55 .23176 L .56 .23176 L .57 .23176 L .58 .23176 L .59 .23176 L .6 .23176 L .61 .23176 L .62 .23176 L .63 .23176 L .64 .23176 L .65 .23176 L .66 .23176 L .67 .23176 L .68 .23176 L .69 .23176 L .7 .23176 L .71 .23176 L .72 .23176 L .73 .23176 L .74 .23176 L .75 .23176 L .76 .23176 L .77 .23176 L .78 .23176 L .79 .23176 L .8 .23176 L .81 .23176 L .82 .23176 L .83 .23176 L .84 .23176 L .85 .23176 L .86 .23176 L .87 .23176 L .88 .23176 L .89 .23176 L .9 .23176 L .91 .23176 L .92 .23176 L .93 .23176 L .94 .23176 L .95 .23176 L .96 .23176 L .97 .23176 L .98 .23176 L .99 .23176 L 1 .23176 L Mistroke Mfstroke % End of Graphics MathPictureEnd :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 282; pictureHeight = 174] %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0 0.01 0 0.618034 [ [(20)] .2 0 0 2 Msboxa [(40)] .4 0 0 2 Msboxa [(60)] .6 0 0 2 Msboxa [(80)] .8 0 0 2 Msboxa [(100)] 1 0 0 2 Msboxa [(0)] -0.0125 0 1 0 Msboxa [(0.2)] -0.0125 .12361 1 0 Msboxa [(0.4)] -0.0125 .24721 1 0 Msboxa [(0.6)] -0.0125 .37082 1 0 Msboxa [(0.8)] -0.0125 .49443 1 0 Msboxa [(1)] -0.0125 .61803 1 0 Msboxa [ -0.001 -0.001 0 0 ] [ 1.001 .61903 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath [ ] 0 setdash 0 g p p .002 w .2 0 m .2 .00625 L s P [(20)] .2 0 0 2 Mshowa p .002 w .4 0 m .4 .00625 L s P [(40)] .4 0 0 2 Mshowa p .002 w .6 0 m .6 .00625 L s P [(60)] .6 0 0 2 Mshowa p .002 w .8 0 m .8 .00625 L s P [(80)] .8 0 0 2 Mshowa p .002 w 1 0 m 1 .00625 L s P [(100)] 1 0 0 2 Mshowa p .001 w .04 0 m .04 .00375 L s P p .001 w .08 0 m .08 .00375 L s P p .001 w .12 0 m .12 .00375 L s P p .001 w .16 0 m .16 .00375 L s P p .001 w .24 0 m .24 .00375 L s P p .001 w .28 0 m .28 .00375 L s P p .001 w .32 0 m .32 .00375 L s P p .001 w .36 0 m .36 .00375 L s P p .001 w .44 0 m .44 .00375 L s P p .001 w .48 0 m .48 .00375 L s P p .001 w .52 0 m .52 .00375 L s P p .001 w .56 0 m .56 .00375 L s P p .001 w .64 0 m .64 .00375 L s P p .001 w .68 0 m .68 .00375 L s P p .001 w .72 0 m .72 .00375 L s P p .001 w .76 0 m .76 .00375 L s P p .001 w .84 0 m .84 .00375 L s P p .001 w .88 0 m .88 .00375 L s P p .001 w .92 0 m .92 .00375 L s P p .001 w .96 0 m .96 .00375 L s P p .002 w 0 0 m 1 0 L s P p .002 w 0 0 m .00625 0 L s P [(0)] -0.0125 0 1 0 Mshowa p .002 w 0 .12361 m .00625 .12361 L s P [(0.2)] -0.0125 .12361 1 0 Mshowa p .002 w 0 .24721 m .00625 .24721 L s P [(0.4)] -0.0125 .24721 1 0 Mshowa p .002 w 0 .37082 m .00625 .37082 L s P [(0.6)] -0.0125 .37082 1 0 Mshowa p .002 w 0 .49443 m .00625 .49443 L s P [(0.8)] -0.0125 .49443 1 0 Mshowa p .002 w 0 .61803 m .00625 .61803 L s P [(1)] -0.0125 .61803 1 0 Mshowa p .001 w 0 .02472 m .00375 .02472 L s P p .001 w 0 .04944 m .00375 .04944 L s P p .001 w 0 .07416 m .00375 .07416 L s P p .001 w 0 .09889 m .00375 .09889 L s P p .001 w 0 .14833 m .00375 .14833 L s P p .001 w 0 .17305 m .00375 .17305 L s P p .001 w 0 .19777 m .00375 .19777 L s P p .001 w 0 .22249 m .00375 .22249 L s P p .001 w 0 .27193 m .00375 .27193 L s P p .001 w 0 .29666 m .00375 .29666 L s P p .001 w 0 .32138 m .00375 .32138 L s P p .001 w 0 .3461 m .00375 .3461 L s P p .001 w 0 .39554 m .00375 .39554 L s P p .001 w 0 .42026 m .00375 .42026 L s P p .001 w 0 .44498 m .00375 .44498 L s P p .001 w 0 .46971 m .00375 .46971 L s P p .001 w 0 .51915 m .00375 .51915 L s P p .001 w 0 .54387 m .00375 .54387 L s P p .001 w 0 .56859 m .00375 .56859 L s P p .001 w 0 .59331 m .00375 .59331 L s P p .002 w 0 0 m 0 .61803 L s P P 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath .004 w .01 .43262 m .02 .23362 L .03 .26156 L .04 .27156 L .05 .27403 L .06 .27455 L .07 .27466 L .08 .27468 L .09 .27468 L .1 .27468 L .11 .27468 L .12 .27468 L .13 .27468 L .14 .27468 L .15 .27468 L .16 .27468 L .17 .27468 L .18 .27468 L .19 .27468 L .2 .27468 L .21 .27468 L .22 .27468 L .23 .27468 L .24 .27468 L .25 .27468 L .26 .27468 L .27 .27468 L .28 .27468 L .29 .27468 L .3 .27468 L .31 .27468 L .32 .27468 L .33 .27468 L .34 .27468 L .35 .27468 L .36 .27468 L .37 .27468 L .38 .27468 L .39 .27468 L .4 .27468 L .41 .27468 L .42 .27468 L .43 .27468 L .44 .27468 L .45 .27468 L .46 .27468 L .47 .27468 L .48 .27468 L .49 .27468 L .5 .27468 L Mistroke .51 .27468 L .52 .27468 L .53 .27468 L .54 .27468 L .55 .27468 L .56 .27468 L .57 .27468 L .58 .27468 L .59 .27468 L .6 .27468 L .61 .27468 L .62 .27468 L .63 .27468 L .64 .27468 L .65 .27468 L .66 .27468 L .67 .27468 L .68 .27468 L .69 .27468 L .7 .27468 L .71 .27468 L .72 .27468 L .73 .27468 L .74 .27468 L .75 .27468 L .76 .27468 L .77 .27468 L .78 .27468 L .79 .27468 L .8 .27468 L .81 .27468 L .82 .27468 L .83 .27468 L .84 .27468 L .85 .27468 L .86 .27468 L .87 .27468 L .88 .27468 L .89 .27468 L .9 .27468 L .91 .27468 L .92 .27468 L .93 .27468 L .94 .27468 L .95 .27468 L .96 .27468 L .97 .27468 L .98 .27468 L .99 .27468 L 1 .27468 L Mistroke Mfstroke % End of Graphics MathPictureEnd :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 282; pictureHeight = 174] %! 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%%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0 0.01 0 0.618034 [ [(20)] .2 0 0 2 Msboxa [(40)] .4 0 0 2 Msboxa [(60)] .6 0 0 2 Msboxa [(80)] .8 0 0 2 Msboxa [(100)] 1 0 0 2 Msboxa [(0)] -0.0125 0 1 0 Msboxa [(0.2)] -0.0125 .12361 1 0 Msboxa [(0.4)] -0.0125 .24721 1 0 Msboxa [(0.6)] -0.0125 .37082 1 0 Msboxa [(0.8)] -0.0125 .49443 1 0 Msboxa [(1)] -0.0125 .61803 1 0 Msboxa [ -0.001 -0.001 0 0 ] [ 1.001 .61903 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath [ ] 0 setdash 0 g p p .002 w .2 0 m .2 .00625 L s P [(20)] .2 0 0 2 Mshowa p .002 w .4 0 m .4 .00625 L s P [(40)] .4 0 0 2 Mshowa p .002 w .6 0 m .6 .00625 L s P [(60)] .6 0 0 2 Mshowa p .002 w .8 0 m .8 .00625 L s P [(80)] .8 0 0 2 Mshowa p .002 w 1 0 m 1 .00625 L s P [(100)] 1 0 0 2 Mshowa p .001 w .04 0 m .04 .00375 L s P p .001 w .08 0 m .08 .00375 L s P p .001 w .12 0 m .12 .00375 L s P p .001 w .16 0 m .16 .00375 L s P p .001 w .24 0 m .24 .00375 L s P p .001 w .28 0 m .28 .00375 L s P p .001 w .32 0 m .32 .00375 L s P p .001 w .36 0 m .36 .00375 L s P p .001 w .44 0 m .44 .00375 L s P p .001 w .48 0 m .48 .00375 L s P p .001 w .52 0 m .52 .00375 L s P p .001 w .56 0 m .56 .00375 L s P p .001 w .64 0 m .64 .00375 L s P p .001 w .68 0 m .68 .00375 L s P p .001 w .72 0 m .72 .00375 L s P p .001 w .76 0 m .76 .00375 L s P p .001 w .84 0 m .84 .00375 L s P p .001 w .88 0 m .88 .00375 L s P p .001 w .92 0 m .92 .00375 L s P p .001 w .96 0 m .96 .00375 L s P p .002 w 0 0 m 1 0 L s P p .002 w 0 0 m .00625 0 L s P [(0)] -0.0125 0 1 0 Mshowa p .002 w 0 .12361 m .00625 .12361 L s P [(0.2)] -0.0125 .12361 1 0 Mshowa p .002 w 0 .24721 m .00625 .24721 L s P [(0.4)] -0.0125 .24721 1 0 Mshowa p .002 w 0 .37082 m .00625 .37082 L s P [(0.6)] -0.0125 .37082 1 0 Mshowa p .002 w 0 .49443 m .00625 .49443 L s P [(0.8)] -0.0125 .49443 1 0 Mshowa p .002 w 0 .61803 m .00625 .61803 L s P [(1)] -0.0125 .61803 1 0 Mshowa p .001 w 0 .02472 m .00375 .02472 L s P p .001 w 0 .04944 m .00375 .04944 L s P p .001 w 0 .07416 m .00375 .07416 L s P p .001 w 0 .09889 m .00375 .09889 L s P p .001 w 0 .14833 m .00375 .14833 L s P p .001 w 0 .17305 m .00375 .17305 L s P p .001 w 0 .19777 m .00375 .19777 L s P p .001 w 0 .22249 m .00375 .22249 L s P p .001 w 0 .27193 m .00375 .27193 L s P p .001 w 0 .29666 m .00375 .29666 L s P p .001 w 0 .32138 m .00375 .32138 L s P p .001 w 0 .3461 m .00375 .3461 L s P p .001 w 0 .39554 m .00375 .39554 L s P p .001 w 0 .42026 m .00375 .42026 L s P p .001 w 0 .44498 m .00375 .44498 L s P p .001 w 0 .46971 m .00375 .46971 L s P p .001 w 0 .51915 m .00375 .51915 L s P p .001 w 0 .54387 m .00375 .54387 L s P p .001 w 0 .56859 m .00375 .56859 L s P p .001 w 0 .59331 m .00375 .59331 L s P p .002 w 0 0 m 0 .61803 L s P P 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath .004 w .01 .43262 m .02 .28553 L .03 .33796 L .04 .33694 L .05 .33714 L .06 .3371 L .07 .33711 L .08 .33711 L .09 .33711 L .1 .33711 L .11 .33711 L .12 .33711 L .13 .33711 L .14 .33711 L .15 .33711 L .16 .33711 L .17 .33711 L .18 .33711 L .19 .33711 L .2 .33711 L .21 .33711 L .22 .33711 L .23 .33711 L .24 .33711 L .25 .33711 L .26 .33711 L .27 .33711 L .28 .33711 L .29 .33711 L .3 .33711 L .31 .33711 L .32 .33711 L .33 .33711 L .34 .33711 L .35 .33711 L .36 .33711 L .37 .33711 L .38 .33711 L .39 .33711 L .4 .33711 L .41 .33711 L .42 .33711 L .43 .33711 L .44 .33711 L .45 .33711 L .46 .33711 L .47 .33711 L .48 .33711 L .49 .33711 L .5 .33711 L Mistroke .51 .33711 L .52 .33711 L .53 .33711 L .54 .33711 L .55 .33711 L .56 .33711 L .57 .33711 L .58 .33711 L .59 .33711 L .6 .33711 L .61 .33711 L .62 .33711 L .63 .33711 L .64 .33711 L .65 .33711 L .66 .33711 L .67 .33711 L .68 .33711 L .69 .33711 L .7 .33711 L .71 .33711 L .72 .33711 L .73 .33711 L .74 .33711 L .75 .33711 L .76 .33711 L .77 .33711 L .78 .33711 L .79 .33711 L .8 .33711 L .81 .33711 L .82 .33711 L .83 .33711 L .84 .33711 L .85 .33711 L .86 .33711 L .87 .33711 L .88 .33711 L .89 .33711 L .9 .33711 L .91 .33711 L .92 .33711 L .93 .33711 L .94 .33711 L .95 .33711 L .96 .33711 L .97 .33711 L .98 .33711 L .99 .33711 L 1 .33711 L Mistroke Mfstroke % End of Graphics MathPictureEnd :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 282; pictureHeight = 174] %! 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