SQLite format 3@  \ -m 1#*O @11+tableMapleNetPropertiesMapleNetPropertiesCREATE TABLE `MapleNetProperties` (pkid INTEGER PRIMARY KEY, `elisiondigitsbefore` INTEGER, `labelling` INTEGER, `indentamount` INTEGER, `elisiontermsthreshold` TEXT, `ansi` INTEGER, `errorbreak` INTEGER, `useclientjvm` INTEGER, `echo` INTEGER, `imaginaryunit` TEXT, `labelwidth` INTEGER, `plotdriver` TEXT, `elisiondigitsafter` INTEGER, `plotoutput` TEXT, `rtablesize` INTEGER, `elisiontermsbefore` INTEGER, `elisiondigitsthreshold` TEXT, `typesetting` TEXT, `plotdevice` TEXT, `verboseproc` INTEGER, `showassumed` INTEGER, `errorcursor` INTEGER, `longdelim` INTEGER, `plotoptions` TEXT, `quiet` INTEGER, `elisiontermsafter` INTEGER, `screenwidth` INTEGER, `preplot` TEXT, `prettyprint` INTEGER, `displayprecision` INTEGER, `warnlevel` INTEGER, `screenheight` INTEGER, `latexwidth` TEXT, `postplot` TEXT, `prompt` TEXT, `showlabels` INTEGER, `contextmenusize` TEXT, `helpbrowser` TEXT, `screenpixelheight` INTEGER, `unitattributes` TEXT, `worksheetdir` TEXT, `termelisionthreshold` INTEGER, `elisionthreshold` INTEGER, `proxyserver` TEXT)PtableHideHideCREATE TABLE `Hide` (pkid INTEGER PRIMARY KEY, `name` TEXT)))GtableViewPropertiesViewPropertiesCREATE TABLE `ViewProperties` (pkid INTEGER PRIMARY KEY, `presentation` INTEGER, `autoexpanding_sections` INTEGER, `assignment` INTEGER, `userprofilename` TEXT, `numericformat_applyinteger` INTEGER, `numericformat_applyrational` INTEGER, `numericformat_applyexponent` INTEGER, `numericformat_outputmask` TEXT, `numericformat_defaultunitformat` TEXT)x##7tableLabelSchemeLabelSchemeCREATE TABLE `LabelScheme` (pkid INTEGER PRIMARY KEY, `value` INTEGER, `prefix` TEXT)}QtableVersionVersionCREATE TABLE `Version` (pkid INTEGER PRIMARY KEY, `major` INTEGER, `minor` INTEGER, `mpldoc` TEXT) 559tableWorkbookExplorerTreeWorkbookExplorerTreeCREATE TABLE `WorkbookExplorerTree` (`id` INTEGER PRIMARY KEY AUTOINCREMENT, `parentid` INTEGER NOT NULL, `name` TEXT NOT NULL DEFAULT "", `expanded` INTEGER NOT NULL DEFAULT 0, `hidden` INTEGER NOT NULL DEFAULT 0)P ++Ytablesqlite_sequencesqlite_sequenceCREATE TABLE sqlite_sequence(name,seq) ''QtableWorkbookModelWorkbookModel CREATE TABLE `WorkbookModel` (`id` INTEGER PRIMARY KEY AUTOINCREMENT, `model_type` INTEGER NOT NULL, `parentid` INTEGER DEFAULT -1, `workbookid` INTEGER DEFAULT 0, `documentid` INTEGER DEFAULT 0, `position` INTEGER DEFAULT -1): CC{tableMapleLibraryObjectReferenceMapleLibraryObjectReference CREATE TABLE `MapleLibraryObjectReference` (`id` INTEGER PRIMARY KEY, `timestamp` INTEGER, `size` INTEGER, `dotm` TEXT)! 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`password` INTEGER, `tooltip` TEXT, `fillcolor` TEXT, `fontcolor` TEXT)sZ5tableECSpeakerECSpeakerbCREATE TABLE `ECSpeaker` (pkid INTEGER PRIMARY KEY, `id` TEXT, `enabled` INTEGER, `visible` INTEGER, `samplerate` INTEGER, `stereo` INTEGER, `volume` INTEGER, `mute` INTEGER, `tooltip` TEXT, `speakerdevice` TEXT)Y##etableECSliderIntECSliderIntaCREATE TABLE `ECSliderInt` (pkid INTEGER PRIMARY KEY, `id` TEXT, `lower_bound` INTEGER, `upper_bound` INTEGER, `control_position` INTEGER, `major_tick_spacing` INTEGER, `minor_ticks` INTEGER, `inputenabled` INTEGER, `visible` INTEGER, `filled` INTEGER, `orientation` INTEGER, `show_labels` INTEGER, `show_ticks` INTEGER, `snap_to_ticks` INTEGER, `continuous_update` INTEGER, `tooltip` TEXT, `enabled` INTEGER, `pixel_width` INTEGER, `pixel_height` INTEGER, `fillcolor` TEXT, `fontcolor` TEXT)X''KtableECSliderFloatECSliderFloat_CREATE TABLE `ECSliderFloat` (pkid INTEGER PRIMARY KEY, `id` TEXT, `lower_bound` REAL, `upper_bound` REAL, `control_position` REAL, `major_tick_spacing` REAL, `minor_ticks` REAL, `inputenabled` INTEGER, `visible` INTEGER, `filled` INTEGER, `orientation` INTEGER, `show_labels` INTEGER, `show_ticks` INTEGER, `snap_to_ticks` INTEGER, `continuous_update` INTEGER, `tooltip` TEXT, `enabled` INTEGER, `pixel_width` INTEGER, `pixel_height` INTEGER, `fillcolor` TEXT, `fontcolor` TEXT)eW!!tableECShortcutECShortcut^CREATE TABLE `ECShortcut` (pkid INTEGER PRIMARY KEY, `id` TEXT, `caption` TEXT, `tooltip` TEXT, `enabled` INTEGER, `visible` INTEGER, `visiblecharacterwidth` INTEGER, `pixel_width` INTEGER, `pixel_height` INTEGER, `user_size` INTEGER, `user_width` INTEGER, `linktarget` TEXT, `image_reference` INTEGER, `imagefilename` TEXT)           "  BOOLEANreadonly0library"  INTEGERpriority0library* ! INTEGERcreated1478892986library readonlylibraryprioritylibrary createdlibrary Ba F 5 'BLf'kindexidx2WorkbookModeloCREATE INDEX idx2 ON WorkbookModel (model_type)Je'gindexidx0WorkbookModelnCREATE INDEX idx0 ON WorkbookModel (parentId)Id]7indexsqlite_autoindex_WorkbookModelProperty_1WorkbookModelPropertylc77EtableWorkbookModelPropertyWorkbookModelPropertykCREATE TABLE `WorkbookModelProperty` (`id` INTEGER PRIMARY KEY, `modelid` INTEGER DEFAULT 0, `type` INTEGER DEFAULT 0, `name` TEXT DEFAULT '', `value` TEXT DEFAULT '', `category` TEXT DEFAULT '', `editable` INTEGER DEFAULT 0, `removable` INTEGER DEFAULT 0, `visible` INTEGER DEFAULT 0, `position` INTEGER DEFAULT 0, UNIQUE(modelId, name, category))vb77 tableTableOfContentsFolderTableOfContentsFolderjCREATE TABLE `TableOfContentsFolder` (pkid INTEGER PRIMARY KEY)a55?tableTableOfContentsEntryTableOfContentsEntryiCREATE TABLE `TableOfContentsEntry` (pkid INTEGER PRIMARY KEY, `reference` INTEGER, `topic` TEXT, `browser` TEXT, `aliases` TEXT, `category` TEXT, `active` INTEGER, `priority` INTEGER, `language` TEXT, `product` TEXT)q`1tableMapleCodeMapleCodehCREATE TABLE `MapleCode` (pkid INTEGER PRIMARY KEY, `action` TEXT, `time` INTEGER)%_--}tableECVolumeGaugeIntECVolumeGaugeIntgCREATE TABLE `ECVolumeGaugeInt` (pkid INTEGER PRIMARY KEY, `id` TEXT, `lower_bound` INTEGER, `upper_bound` INTEGER, `control_position` INTEGER, `major_tick_spacing` INTEGER, `minor_ticks` INTEGER, `inputenabled` INTEGER, `visible` INTEGER, `show_labels` INTEGER, `show_ticks` INTEGER, `continuous_update` INTEGER, `pixel_width` INTEGER, `pixel_height` INTEGER, `tooltip` TEXT)^11ctableECVolumeGaugeFloatECVolumeGaugeFloatfCREATE TABLE `ECVolumeGaugeFloat` (pkid INTEGER PRIMARY KEY, `id` TEXT, `lower_bound` REAL, `upper_bound` REAL, `control_position` REAL, `major_tick_spacing` REAL, `minor_ticks` REAL, `inputenabled` INTEGER, `visible` INTEGER, `show_labels` INTEGER, `show_ticks` INTEGER, `continuous_update` INTEGER, `pixel_width` INTEGER, `pixel_height` INTEGER, `tooltip` TEXT)ztnhb\VPJD>82,& ~xrlf`ZTNHC=6/(!  } u m e ] U M E = 5 - %    } u m e ] U M E = 5 - %    } u m e ] U M E = 5 - %    } u m e ] U M E = 5 - %    } u m e ] U M E = 5 - %   }ume]UME=5-% }ume]UME=5-% }ume]UME=5-% }ume]UME=5-% }ume]UME=5-% }ume"! YUQMIEA>:73/+#         ~|}{|yzxywxuvtursqrpqnomnkljkijghfgeeeeeee~e{ewetepemeiefcdbc`a_`]^\][\YZXYVWUVTURSPRPQOPN_N[NXNTNOLMKLIJGHFIFGEFBC@?>=<;:9876543210/.-,+*)('&%$#"!          ` ]       e b N K E D B A     ZW[Xroihfe        /0#$yrkd]VOHA:3,% }vohaZSLE>70)"  { t m f _ X Q J C < 5 . '     x q j c \ U N G @ 9 2 + $      | u n g ` Y R K D = 6 / ( !    y r k d ] V O H A : 3 , %     } v o h a Z S L E > 7 0 ) "   zsle^WPIB;4-& ~wpib[TMF?81*#{tmf_XQJC<5.'  xqjc\UNG@92+$|ung`YRKD=6/(! mieaXTPLHD@<840,($  |xoi}yuqmiea]YUQMIEA=951-)%!  |lh]YUQMIEA>:73/+$  ~{wtpmib_[XTKEhDgCfBdcba`@?>=<_^]\[;:987ZYXWVUTSRQPONMLKJIHGFEDCB6543210/.-,+*)('&%$#"! A@?>=<;:9876543210/.-,+*)('&%$#"!                    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piecewise(And(x>=0, x<=1), c2*x, 0);%ZOplot([f(x), 2*x], x=-0.2 .. 1.2);"VICDF(X, t^2) assuming t>0, t<1;+R[Probability(X < t^2) assuming t>0, t<1;0NePDF(Triangular(0,1,1), t) assuming t>0, t<1;*JYPDF(sqrt(X), t) assuming t > 0, t < 1;&FQX := RandomVariable(Uniform(0,1));B?plot([f,g], -0.2 .. 1.2); > better)<WLet's make it more efficient: choose -5_plots:-display(hh, r1, transparency=0.5);,1]hh := Histogram(samplePoints[accepted]);=-}r1 := plots:-display( myPointPlot(samplePoints[rejected], yCoordinates[rejected], red, "rejected"), myPointPlot(samplePoints[accepted], yCoordinates[accepted], green, "accepted"));)-myPointPlot := proc( xvalues :: {list, rtable}, yvalues :: {list, rtable}, color, label :: {name, string}, $) Statistics:-PointPlot(yvalues, ':-xcoords' = convert(xvalues, 'list'), ':-color' = color, ':-symbolsize' = 1, ':-legend' = label); end proc;"'Accept/Reject  PDF? custom%OHow do I generate samples from a ,]p1 := plot(f, -0.2 .. 1.2, thickness=3);W 1### Student exercise: determine c so that f is a probability distribution function f(x);;{f := x -> piecewise(And(x>=0, x<=1), c*x^2*(4/3-x), 0);Cz plot([PDF(H, t), PDF(H2, t)], t=150..190, legend=['H', 'H2']);8suH2 := 153 + RandomVariable(BetaDistribution(5,2))*31(lUH := RandomVariable(Normal(175, 5));)hWHow do I generate samples from a PDF?b^Gheights2 := [176, 176, 175, 178, 180, 167, 177, 173, 185, 175, 173, 172, 182, 176, 187, 182, 185, 179, 174, 173, 176, 176, 177, 177, 169, 185, 179, 181, 178, 185, 176, 178, 176, 174, 174, 174, 167, 169, 176, 182, 178, 181, 176, 172, 182, 181, 176, 175, 177, 169, 177, 169, 178, 184, 185, 182, 168, 180, 183, 170, 173, 173, 171, 178, 179, 185, 182, 170, 173, 164, 177, 183, 174, 174, 179, 182, 173, 182, 182, 172, 177, 161, 182, 171, 172, 177, 173, 173, 170, 184, 176, 172, 181, 174, 180, 175, 175, 173, 175, 170, 187, 172, 170, 183, 182, 180, 175, 183, 173, 177, 175, 180, 177, 176, 180, 177, 169, 188, 180, 176, 174, 180, 174, 173, 172, 183, 185, 165, 179, 176, 176, 183, 179, 178, 178, 176, 166, 173, 176, 180, 173, 177, 177, 172, 180, 182, 171, 168, 178, 175, 181, 176, 173, 175, 178, 174, 180, 178, 189, 182, 167, 180, 175, 180, 178, 163, 168, 182, 167, 179, 182, 180, 185, 177, 168, 180, 182, 181, 179, 175, 174, 174, 173, 175, 173, 168, 168, 175, 171, 186, 177, 176, 172, 179, 172, 172, 180, 165, 181, 175, 169, 169, 176, 174, 179, 180, 177, 168, 165, 175, 184, 173, 171, 179, 179, 166, 173, 179, 171, 172, 183, 171, 164, 179, 179, 182, 174, 168, 178, 177, 175, 174, 170, 171, 180, 175, 171, 172, 176, 176, 174, 164, 169, 178, 171, 175, 173, 173, 171, 177, 180, 185, 164, 174, 174, 171, 170, 173, 179, 178, 179, 179, 173, 177, 174, 177, 176, 172, 180, 171, 173, 171, 180, 176, 181, 172, 170, 174, 182, 162, 176, 169, 163, 172, 184, 179, 175, 175, 183, 179, 178, 177, 174, 190, 179, 171, 175, 173, 184, 176, 174, 169, 180, 174, 179, 169, 170, 174, 177, 184, 175, 172, 180, 179, 176, 170, 170, 171, 172, 173, 171, 179, 177, 177, 174, 175, 177, 171, 176, 176, 178, 173, 176, 171, 180, 179, 176, 169, 171, 174, 182, 171, 173, 180, 179, 176, 170, 185, 172, 181, 177, 170, 165, 179, 182, 185, 175, 164, 177, 171, 181, 176, 184, 183, 175, 174, 175, 174, 171, 170, 177, 173, 174, 176, 173, 171, 173, 180, 182, 178, 168, 166, 175, 175, 182, 172, 181, 176, 181, 177, 180, 184, 176, 170, 174, 180, 171, 172, 169, 176];tWkheights := [170, 173, 172, 176, 175, 184, 167, 183, 176, 180, 176, 168, 175, 175, 175, 174, 176, 175, 175, 174] NkE p P D  o m A 5 " x 3wT6fEiE wX@$Z0Y`cLIcolors := [seq(ColorTools:-Color("HSV", [(i+2)*(1/8), .9, .9]), i = 1 .. nops(boundaries)-1)];HCplot([f,g,h], -0.2 .. 1.2);SD)h := unapply(piecewise( x <= boundaries[1], 0, x <= boundaries[2], tangents[1](x), x <= boundaries[3], tangents[2](x), x <= boundaries[4], chords[3](x), x <= boundaries[5], chords[4](x), 0), x);S@)g := unapply(piecewise( x <= boundaries[1], 0, x <= boundaries[2], chords[1](x), x <= boundaries[3], chords[2](x), x <= boundaries[4], tangents[2](x), x <= boundaries[5], tangents[3](x), 0), x);'<Splot([f, op(chords)], -0.2 .. 1.2);E8 chords := zip(makechord, boundaries[ .. -2], boundaries[2 .. ]);-4_boundaries[1 .. -2], boundaries[2 .. -1];@0boundaries := sort([op(tangentpoints), op(intersections)]);g,Qintersections := [solve(tangents[1](x) = tangents[2](x)), solve(tangents[2](x) = tangents[3](x))];)(Wplot([f, op(tangents)], -0.2 .. 1.2);6$qtangents := map2(maketangent, f01, tangentpoints);! Gtangentpoints := [0, 4/9, 1];7solve(fdouble(x), x);7fdouble := D(fprime);/fprime := D(f01);=f01 := unapply(f01x, x);# Kf01x := f(x) assuming x>0, x<1;!upperbound!lowerbound yconcaveq!lowerboundk!upperbound econvex ]tangent Vchord,A]plot([-exp(x),ln(x), sqrt(x)], x=-2..2);!=Gplot([x^2, exp(x)], x=-2..2);)9W if its second derivative is negative 8concave/6c if its second derivative is positive and 5convex-3_A sufficiently "nice" function is called 0/eplot([f, maketangent(f, 1/2)], -0.2 .. 1.2);+Aplot([f, %], -0.2 .. 1.2);'3maketangent(f, 1/5)F#maketangent := (f, x0) -> unapply(D(f)(x0) * (x - x0) + f(x0), x)' is the line  at / to the graph of tangentThe 9plots:-display(%, fp);;plot(ccc, 1/10 .. 4/5); Eccc := makechord(1/10, 4/5);n_makechord := (x0, x1) -> unapply(CurveFitting:-PolynomialInterpolation([[x0, f(x0)], [x1, f(x1)]], x), x)Hplots:-display(fp, plottools:-line([1/10, f(1/10)], [4/5, f(4/5)]))Bplots:-display(fp, plottools:-line([0, f(0)], [1/2, f(1/2)])) z and /xc is the line segment connecting the points v and t! between r+ of a graph of qchordoThe kCfp := plot(f, -0.2 .. 1.2);g% even better e and )cWLet's make it more efficient: choose \-nops(probation);X+nops(accepted); Tplots:-display( myPointPlot(samplePoints[accepted], yCoordinates[accepted], green, "accepted"), myPointPlot(samplePoints[probation], yCoordinates[probation], blue, "probation"), myPointPlot(samplePoints[rejected], yCoordinates[rejected], red, "rejected"));_PAprobation, rejected := selectremove(i -> yCoordinates[i] < f(samplePoints[i]), probation):aLEaccepted, probation := selectremove(i -> yCoordinates[i] < h(samplePoints[i]), [seq(1..N)]):HCplot([f,g,h], -0.2 .. 1.2); Dh(x);d=K### Student exercise: determine c3 so that h is tangent to f somewhere strictly between 0 and 1=6h := x -> piecewise(And(x>=0, x<=1), 12/7 + c3*(x-1), 0); 2f(1);.?plot([f,g], -0.2 .. 1.2);;){ to pre-screen for values that are certainly less than '7Use a lowerbound for x!s that is expensive: you may not have an explicit formula for it, just some program that computes a value for given #KIn many cases, it's evaluating 3kLet's make it more efficient: add pre-screening&QHistogram(samplePoints[accepted]); +nops(accepted);7qplots:-display( myPointPlot(samplePoints[accepted], yCoordinates[accepted], green, "accepted"), myPointPlot(samplePoints[rejected], yCoordinates[rejected], red, "rejected"));bGaccepted, rejected := selectremove(i -> yCoordinates[i] < f(samplePoints[i]), [seq(1 .. N)]): #[3 : / R3>_SEeQ.UV-acceptedX, acceptedY := op(map[3](map, `?[]`, generator:-accepted, [[1], [2]]));/Rcto N do generator:-generateValue(); end do: NEgenerator:-setup(f01, 0, 1);J generator := module() local dispatcher, f, fprime, uniform01, createInterval, splitInterval, generateChords, generateTangents, resetDispatcher; export intervals, accepted, probation, rejected, setup, generateValue; uses ST = Statistics; generateChords := proc(xleft, xmiddle, xright, yleft, ymiddle, yright) local x; return unapply(piecewise( x < xmiddle, ((x - xleft) * ymiddle + (xmiddle - x) * yleft) / (xmiddle - xleft), ((x - xmiddle) * yright + (xright - x) * ymiddle) / (xright - xmiddle)), x); end proc; generateTangents := proc(xleft, xmiddle, xright, yleft, yright, derivativeLeft, derivativeRight) vgFQhttp://www.mapleprimes.com/posts/96015-Generating-Samples-From-Custom-Probability-Distributions-IVsCiThis is certainly too complicated to explain in this session. The program that does it is shown here, though: ? online = and 4;mLet's make it more efficient: automate updating 44mnops(accepted), nops(probation), nops(rejected); 0plots:-display( myPointPlot(samplePoints[accepted], yCoordinates[accepted], green, "accepted"), myPointPlot(samplePoints[probation], yCoordinates[probation], blue, "probation"), myPointPlot(samplePoints[rejected], yCoordinates[rejected], red, "rejected"));_,Aprobation, rejected := selectremove(i -> yCoordinates[i] < f(samplePoints[i]), probation):a(Eaccepted, probation := selectremove(i -> yCoordinates[i] < h(samplePoints[i]), [seq(1..N)]):7$syCoordinates := gvalues *~ Sample(Uniform(0,1), N);$ Mgvalues := map(g, samplePoints);-_plots:-display(%, plot(g/areasum, 0..1));=Histogram(sampleValues);BsampleValues := ArrayTools:-Concatenate(2, op(sampleValues));U-sampleValues := zip((t, var) -> Sample(var, rhs(t)), tally, compositeVariables); =compositeVariables := [seq('`if`'(RandomVariable(Bernoulli(rectangleProbabilities[i])) < 1/2, triangleVariables[i], rectangleVariables[i]), i=1..nb-1)];eMrectangleVariables := [seq(RandomVariable(Uniform(boundaries[i], boundaries[i+1])), i=1..nb-1)];._triangleVariables := [seq(RandomVariable(Triangular(boundaries[i], boundaries[i+1], `if`(minvalues[i] = g(boundaries[i]), boundaries[i+1], boundaries[i]))), i=1..nb-1)];6qrectangleProbabilities := rectangleAreas /~ areas;I|rectangleAreas := minvalues *~ (boundaries[2..] - boundaries[..-2]);Vx/plots:-display( seq( plottools:-polygon([ [boundaries[i], minvalues[i]], [boundaries[i+1], minvalues[i]], [boundaries[i+1], g(boundaries[i+1])], [boundaries[i], g(boundaries[i])]], color=colors[i]), i = 1 .. nb - 1), seq( plottools:-polygon([ [boundaries[i], minvalues[i]], [boundaries[i+1], minvalues[i]], [boundaries[i+1], 0], [boundaries[i], 0]], color=darker[i]), i = 1 .. nb - 1)); .tadarker := map(ColorTools:-Darken, colors);Xp3minvalues := zip((b1, b2) -> min(g(b1), g(b2)), boundaries[..-2], boundaries[2..]);*lYtally := sort(Tally(pieces), key=lhs);&hQpieces := Sample(areaVariable, N);Ed areaVariable := RandomVariable(ProbabilityTable(probabilities));%`Oprobabilities := areas / areasum;"\Iareasum := add(a, a in areas);rXgareas := [seq((boundaries[i+1] - boundaries[i]) * (g(boundaries[i+1]) + g(boundaries[i]))/2, i = 1 .. nb-1)];T;nb := nops(boundaries);iPUplots:-display(seq( plottools:-polygon([ [boundaries[i], g(boundaries[i])], [boundaries[i+1], g(boundaries[i+1])], [boundaries[i+1], 0], [boundaries[i], 0]], color = colors[i]), i = 1 .. nops(boundaries)-1))w local x; return unapply(piecewise( x < xmiddle, yleft + derivativeLeft * (x - xleft), yright + derivativeRight * (x - xright)), x); end proc; setup := proc(ff :: appliable, a :: numeric, b :: numeric, $) local areas, fdoubleprime, i, lastpoint, derivativeLeft, pointTable, points, derivativeRight, xleft, xright, yleft, yright; if not a < b then error "invalid input, %1 expects its second argument, a, to be " "less than its third argument, but received %2 and %3", procname, a, b; end if; f := ff; fprime := D(f); fdoubleprime := D(fprime); uniform01 := ST:-Sample(ST:-RandomVariable(':-Uniform'(0, 1))); accepted, probation, rejected := Array(1..0), Array(1..0), Array(1..0); # Find all inflection points. We set them as keys in a table, # then gather them at the end. pointTable := table([b = 1]); lastpoint := a; while lastpoint <> FAIL and lastpoint < b do pointTable[lastpoint] := 1; lastpoint := RootFinding:-NextZero(fdoubleprime, lastpoint); end do; points := sort([indices(pointTable, ':-nolist')]); # Now set up the intervals in between these points. An # interval will be a record with these keys: # * xleft; the left boundary of the interval. # * xright; the right boundary of the interval. # * yleft, yright; the corresponding values of f. # * derivativeLeft, derivativeRight; the values of f' at xleft # and xright. # * lowerbound; a procedure that returns the lower bound for # points in this interval. # * upperbound; a procedure that returns the upper bound for # points in this interval. # * sample; a procedure that generates one sample value from # this interval. # * area; the total area under the upper curve defined for # this interval. intervals := Array(1 .. nops(points) - 1); xleft := points[1]; yleft := f(xleft); derivativeLeft := fprime(xleft); for i to nops(points) - 1 do xright := points[i+1]; yright := f(xright); derivativeRight := fprime(xright); intervals[i] := createInterval(xleft, xright, yleft, yright, derivativeLeft, derivativeRight); xleft := xright; yleft := yright; derivativeLeft := derivativeRight; end do; resetDispatcher(); return NULL; end proc; createInterval := proc(xleft, xright, yleft, yright, derivativeLeft, derivativeRight) local area, areas, generators, leftRectangleHeight, leftTriangleHeight, lowerbound, rightRectangleHeight, rightTriangleHeight, sample, selector, upperbound, xmiddle, ymiddle; # The intersection of the two tangents: xmiddle := (yright - yleft + derivativeLeft * xleft - derivativeRight * xright) / (derivativeLeft - derivativeRight); ymiddle := f(xmiddle); if ymiddle < ((xmiddle - xleft) * yright + (xright - xmiddle) * yleft) / (xright - xleft) then # Middle point is lower than chord: convex, so chords # define upper curve, tangents lower curve. lowerbound := generateTangents(xleft, xmiddle, xright, yleft, yright, derivativeLeft, derivativeRight); upperbound := generateChords(xleft, xmiddle, xright, yleft, ymiddle, yright); else # Middle point is higher than chord: concave, so # tangents define upper curve, chords lower curve. lowerbound := generateChords(xleft, xmiddle, xright, yleft, ymiddle, yright); upperbound := generateTangents(xleft, xmiddle, xright, yleft, yright, derivativeLeft, derivativeRight); # Instead of f at xmiddle, we use the intersection of # the tangents for ymiddle in the remainder of this # iteration: ymiddle := upperbound(xmiddle); end if; leftRectangleHeight := min(yleft, ymiddle); leftTriangleHeight := abs(yleft - ymiddle); rightRectangleHeight := min(ymiddle, yright); rightTriangleHeight := abs(yright - ymiddle); areas := [leftRectangleHeight * (xmiddle - xleft), leftTriangleHeight * (xmiddle - xleft)/2, rightRectangleHeight * (xright - xmiddle), rightTriangleHeight * (xright - xmiddle)/2]; generators := map(ST:-Sample @ ST:-RandomVariable, [ ':-Uniform'(xleft, xmiddle), ':-Triangular'( xleft, xmiddle, `if`(yleft > ymiddle, xleft, xmiddle)), ':-Uniform'(xmiddle, xright), ':-Triangular'( xmiddle, xright, `if`(ymiddle > yright, xmiddle, xright))]); selector := ST:-Sample(ST:-RandomVariable( ':-ProbabilityTable'(areas / add(x, x in areas)))); sample := proc() local i; i := trunc(selector(1)[1]); return generators[i](1)[1]; end proc; area := add(x, x in areas); return Record(':-xleft' = xleft, ':-xright' = xright, ':-yleft' = yleft, ':-yright' = yright, ':-derivativeLeft' = derivativeLeft, ':-derivativeRight' = derivativeRight, ':-lowerbound' = lowerbound, ':-upperbound' = upperbound, ':-sample' = sample, ':-area' = area); end proc; resetDispatcher := proc() local areas, i, sample; areas := [seq(i:-area, i in intervals)]; sample := ST:-Sample(ST:-RandomVariable(':-ProbabilityTable'( areas / add(x, x in areas)))); dispatcher := proc() return trunc(sample(1)[1]); end proc; end proc; generateValue := proc() local fx, i, interval, lowerbound, x, y; while true do i := dispatcher(); interval := intervals[i]; x := interval:-sample(); y := uniform01(1)[1] * interval:-upperbound(x); lowerbound := interval:-lowerbound(x); if y < lowerbound then accepted(rtable_num_elems(accepted) + 1) := [x, y]; return x; end if; fx := f(x); if y > fx or lowerbound < 0.9 * fx then splitInterval(i, x, fx); end if; if y < fx then probation(rtable_num_elems(probation) + 1) := [x, y]; return x; end if; rejected(rtable_num_elems(rejected) + 1) := [x, y]; end do; end proc; splitInterval := proc(i, x, fx) local xDerivative, oldInterval; xDerivative := fprime(x); oldInterval := intervals[i]; intervals[i] := createInterval( oldInterval:-xleft, x, oldInterval:-yleft, fx, oldInterval:-derivativeLeft, xDerivative); intervals(rtable_num_elems(intervals) + 1) := createInterval( x, oldInterval:-xright, fx, oldInterval:-yright, xDerivative, oldInterval:-derivativeRight); resetDispatcher(); end proc; end module: Mp #>Yta~E-I%mrowG6#/I+modulenameG6"I,TypesettingGI(_syslibGF'6'-I#miGF$6'Q"fF'/%'italicGQ%trueF'/%,mathvariantGQ'italicF'/%*referenceGQ%2404F'/%2display_referenceGQ%2403F'/%+executableGQ&falseF'/F3Q'normalF'/F6Q%2418F'/F9Q%2417F'}}}-I%mrowG6#/I+modulenameG6"I,TypesettingGI(_syslibGF'6)-I#miGF$6)Q"hF'/%%boldGQ&falseF'/%'italicGQ%trueF'/%+executableGF4/%,mathvariantGQ'italicF'/%*referenceGQ%2414F'/%2display_referenceGQ%2413F'/%%sizeGQ#12F'F/F5/F8Q'normalF'/F;Q%2416F'/F>Q%2415F'}|}-I%mrowG6#/I+modulenameG6"I,TypesettingGI(_syslibGF'6)-I#miGF$6)Q"hF'/%%boldGQ&falseF'/%'italicGQ%trueF'/%+executableGF4/%,mathvariantGQ'italicF'/%*referenceGQ%2414F'/%2display_referenceGQ%2413F'/%%sizeGQ#12F'F/F5/F8Q'normalF'/F;Q%2416F'/F>Q%2415F'k{Y-I%mrowG6#/I+modulenameG6"I,TypesettingGI(_syslibGF'6)-I#miGF$6)Q"gF'/%%boldGQ&falseF'/%'italicGQ%trueF'/%+executableGF4/%,mathvariantGQ'italicF'/%*referenceGQ%2414F'/%2display_referenceGQ%2413F'/%%sizeGQ#12F'F/F5/F8Q'normalF'F:F=kzY-I%mrowG6#/I+modulenameG6"I,TypesettingGI(_syslibGF'6)-I#miGF$6)Q"gF'/%%boldGQ&falseF'/%'italicGQ%trueF'/%+executableGF4/%,mathvariantGQ'italicF'/%*referenceGQ%2414F'/%2display_referenceGQ%2413F'/%%sizeGQ#12F'F/F5/F8Q'normalF'F:F=ayE-I%mrowG6#/I+modulenameG6"I,TypesettingGI(_syslibGF'6'-I#miGF$6'Q"fF'/%'italicGQ%trueF'/%,mathvariantGQ'italicF'/%*referenceGQ%2404F'/%2display_referenceGQ%2403F'/%+executableGQ&falseF'/F3Q'normalF'/F6Q%2412F'/F9Q%2411F'axE-I%mrowG6#/I+modulenameG6"I,TypesettingGI(_syslibGF'6'-I#miGF$6'Q"fF'/%'italicGQ%trueF'/%,mathvariantGQ'italicF'/%*referenceGQ%2404F'/%2display_referenceGQ%2403F'/%+executableGQ&falseF'/F3Q'normalF'/F6Q%2412F'/F9Q%2411F'awE-I%mrowG6#/I+modulenameG6"I,TypesettingGI(_syslibGF'6'-I#miGF$6'Q"fF'/%'italicGQ%trueF'/%,mathvariantGQ'italicF'/%*referenceGQ%2404F'/%2display_referenceGQ%2403F'/%+executableGQ&falseF'/F3Q'normalF'/F6Q%2410F'/F9Q%2409F'avE-I%mrowG6#/I+modulenameG6"I,TypesettingGI(_syslibGF'6'-I#miGF$6'Q"fF'/%'italicGQ%trueF'/%,mathvariantGQ'italicF'/%*referenceGQ%2404F'/%2display_referenceGQ%2403F'/%+executableGQ&falseF'/F3Q'normalF'/F6Q%2410F'/F9Q%2409F'auE-I%mrowG6#/I+modulenameG6"I,TypesettingGI(_syslibGF'6'-I#miGF$6'Q"xF'/%'italicGQ%trueF'/%,mathvariantGQ'italicF'/%*referenceGQ%2404F'/%2display_referenceGQ%2403F'/%+executableGQ&falseF'/F3Q'normalF'/F6Q%2408F'/F9Q%2407F'atE-I%mrowG6#/I+modulenameG6"I,TypesettingGI(_syslibGF'6'-I#miGF$6'Q"xF'/%'italicGQ%trueF'/%,mathvariantGQ'italicF'/%*referenceGQ%2404F'/%2display_referenceGQ%2403F'/%+executableGQ&falseF'/F3Q'normalF'/F6Q%2408F'/F9Q%2407F'asE-I%mrowG6#/I+modulenameG6"I,TypesettingGI(_syslibGF'6'-I#miGF$6'Q"fF'/%'italicGQ%trueF'/%,mathvariantGQ'italicF'/%*referenceGQ%2404F'/%2display_referenceGQ%2403F'/%+executableGQ&falseF'/F3Q'normalF'/F6Q%2406F'/F9Q%2405F'arE-I%mrowG6#/I+modulenameG6"I,TypesettingGI(_syslibGF'6'-I#miGF$6'Q"fF'/%'italicGQ%trueF'/%,mathvariantGQ'italicF'/%*referenceGQ%2404F'/%2display_referenceGQ%2403F'/%+executableGQ&falseF'/F3Q'normalF'/F6Q%2406F'/F9Q%2405F'fqO-I%mrowG6#/I+modulenameG6"I,TypesettingGI(_syslibGF'6'-I#miGF$6(Q"gF'/%%sizeGQ#18F'/%'italicGQ%trueF'/%,mathvariantGQ'italicF'/%*referenceGQ%2404F'/%2display_referenceGQ%2403F'/%0font_style_nameGQ*Heading~1F'/F6Q'normalF'F8F;fpO-I%mrowG6#/I+modulenameG6"I,TypesettingGI(_syslibGF'6'-I#miGF$6(Q"gF'/%%sizeGQ#18F'/%'italicGQ%trueF'/%,mathvariantGQ'italicF'/%*referenceGQ%2404F'/%2display_referenceGQ%2403F'/%0font_style_nameGQ*Heading~1F'/F6Q'normalF'F8F;f7generator:-intervals;=b}plots:-display( myPointPlot(acceptedX, acceptedY, green, "accepted"), myPointPlot(probationX, probationY, blue, "probation"), myPointPlot(rejectedX, rejectedY, red, "rejected"));U^-rejectedX, rejectedY := op(map[3](map, `?[]`, generator:-rejected, [[1], [2]]));XZ3probationX, probationY := op(map[3](map, `?[]`, generator:-probation, [[1], [2]]));  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-I%mrowG6#/I+modulenameG6"I,TypesettingGI(_syslibGF'64-I#miGF$6'Q"yF'/%'italicGQ%trueF'/%,mathvariantGQ'italicF'/%*referenceGQ%2404F'/%2display_referenceGQ%2403F'-I#moGF$6/Q"~F'/F3Q'normalF'/F6Q%2424F'/F9Q%2423F'/%&fenceGQ&falseF'/%*separatorGFG/%)stretchyGFG/%*symmetricGFG/%(largeopGFG/%.movablelimitsGFG/%'accentGFG/%'lspaceGQ&0.0emF'/%'rspaceGFV-F<6/Q"=F'F?FAFCFEFHFJFLFNFPFR/FUQ,0.2777778emF'/FXFgnF;-F,6'Q"DF'/F0FGF?FAFC-I(mfencedGF$6&-F#6'-F,6'Q"fF'F/F2F5F8/%+executableGFGF?/F6Q%2432F'/F9Q%2431F'F?FAFC-F^o6&-F#6'-I%msubGF$6&-F,6'Q"xF'F/F2F5F8-F,6'Q"0F'F/F2F5F8/%/subscriptshiftGFgp/I+msemanticsGF$Q'atomicF'FeoF?FgoFioF?FAFC-F<6/Q'⋅F'F?FAFCFEFHFJFLFNFPFRFTFW-F^o6&-F#6+FbpF;-F<6/Q*&uminus0;F'F?FAFCFEFHFJFLFNFPFR/FUQ,0.2222222emF'/FXFhqF;F_pFeoF?FgoFioF?FAFCF;-F<6/Q"+F'F?FAFCFEFHFJFLFNFPFRFgqFiqF;FboF[pFeoF?FgoFio- ]-I%mrowG6#/I+modulenameG6"I,TypesettingGI(_syslibGF'6'-I%msubGF$6&-I#miGF$6'Q"xF'/%'italicGQ%trueF'/%,mathvariantGQ'italicF'/%*referenceGQ%2404F'/%2display_referenceGQ%2403F'-F/6'Q"0F'F2F5F8F;/%/subscriptshiftGF@/I+msemanticsGF$Q'atomicF'/%+executableGQ&falseF'/F6Q'normalF'/F9Q%2430F'/FX|^C&oZf,Y4o?3-7_yCoordinates := Sample(Uniform(0, 2), N);,3]samplePoints := Sample(Uniform(0,1), N);/%N := 3*10^4:+=plot([f,g], -0.2 ..1.2); 'g(x);-#_g := x -> piecewise(And(x>0, x<1), 2, 0);'Accept/Reject  PDF? custom%OHow do I generate samples from a %Oplots:-display(p1, Histogram(%));-Sample(X, 10^6); Mean(X);AX := RandomVariable(dist);"Idist := Distribution(PDF = f):,]p1 := plot(f, -0.2 .. 1.2, thickness=3);Wx1### Student exercise: determine c so that f is a probability distribution function qf(x);;m{f := x -> piecewise(And(x>=0, x<=1), c*x^2*(4/3-x), 0);7isplots:-display(%, DensityPlot(H2, range=154..186));eCHistogram(s, bincount=100);a9s := Sample(H2, 10^6);]%Skewness(H);Y9Skewness(H2, numeric);#UKStandardDeviation(H2, numeric);Q1Mean(H2, numeric);CM plot([PDF(H, t), PDF(H2, t)], t=150..190, legend=['H', 'H2']);I)Sample(H2, 5);8EuH2 := 153 + RandomVariable(BetaDistribution(5,2))*31A'Sample(H, 5);=!PDF(H, x);(9UH := RandomVariable(Normal(175, 5));)5WHow do I generate samples from a PDF?+9plots:-display(%, h3);'5plot(%, x=160..190);#;PDF(Normal(175, 5), x);-_plots:-display(h1, h3, transparency=0.5);:yh3 := Histogram(heights2, bincount=30, color=red): h3;-_plots:-display(h1, h2, transparency=0.5);<}h2 := Histogram(heights2, bincount=10, color=green): h2;bGheights2 := [176, 176, 175, 178, 180, 167, 177, 173, 185, 175, 173, 172, 182, 176, 187, 182, 185, 179, 174, 173, 176, 176, 177, 177, 169, 185, 179, 181, 178, 185, 176, 178, 176, 174, 174, 174, 167, 169, 176, 182, 178, 181, 176, 172, 182, 181, 176, 175, 177, 169, 177, 169, 178, 184, 185, 182, 168, 180, 183, 170, 173, 173, 171, 178, 179, 185, 182, 170, 173, 164, 177, 183, 174, 174, 179, 182, 173, 182, 182, 172, 177, 161, 182, 171, 172, 177, 173, 173, 170, 184, 176, 172, 181, 174, 180, 175, 175, 173, 175, 170, 187, 172, 170, 183, 182, 180, 175, 183, 173, 177, 175, 180, 177, 176, 180, 177, 169, 188, 180, 176, 174, 180, 174, 173, 172, 183, 185, 165, 179, 176, 176, 183, 179, 178, 178, 176, 166, 173, 176, 180, 173, 177, 177, 172, 180, 182, 171, 168, 178, 175, 181, 176, 173, 175, 178, 174, 180, 178, 189, 182, 167, 180, 175, 180, 178, 163, 168, 182, 167, 179, 182, 180, 185, 177, 168, 180, 182, 181, 179, 175, 174, 174, 173, 175, 173, 168, 168, 175, 171, 186, 177, 176, 172, 179, 172, 172, 180, 165, 181, 175, 169, 169, 176, 174, 179, 180, 177, 168, 165, 175, 184, 173, 171, 179, 179, 166, 173, 179, 171, 172, 183, 171, 164, 179, 179, 182, 174, 168, 178, 177, 175, 174, 170, 171, 180, 175, 171, 172, 176, 176, 174, 164, 169, 178, 171, 175, 173, 173, 171, 177, 180, 185, 164, 174, 174, 171, 170, 173, 179, 178, 179, 179, 173, 177, 174, 177, 176, 172, 180, 171, 173, 171, 180, 176, 181, 172, 170, 174, 182, 162, 176, 169, 163, 172, 184, 179, 175, 175, 183, 179, 178, 177, 174, 190, 179, 171, 175, 173, 184, 176, 174, 169, 180, 174, 179, 169, 170, 174, 177, 184, 175, 172, 180, 179, 176, 170, 170, 171, 172, 173, 171, 179, 177, 177, 174, 175, 177, 171, 176, 176, 178, 173, 176, 171, 180, 179, 176, 169, 171, 174, 182, 171, 173, 180, 179, 176, 170, 185, 172, 181, 177, 170, 165, 179, 182, 185, 175, 164, 177, 171, 181, 176, 184, 183, 175, 174, 175, 174, 171, 170, 177, 173, 174, 176, 173, 171, 173, 180, 182, 178, 168, 166, 175, 175, 182, 172, 181, 176, 181, 177, 180, 184, 176, 170, 174, 180, 171, 172, 169, 176];. ah1 := Histogram(heights, bincount=10): h1;5[min, max](heights);)Mean(heights);/with(Statistics):t{kheights := [170, 173, 172, 176, 175, 184, 167, 183, 176, 180, 176, 168, 175, 175, 175, 174, 176, 175, 175, 174]$wMThe probability density function$uMHow do you describe a data set? nErik Postma - epostma@maplesoft.com Manager of the mathematical software group / senior architect for mathematical software Jw`I0 o @  ` - }  } V .  qKvVJ usG;(~9}Z<lK/Xc if its second derivative is positive and Wconvex-U_A sufficiently "nice" function is called 0Qeplot([f, maketangent(f, 1/2)], -0.2 .. 1.2);MAplot([f, %], -0.2 .. 1.2);I3maketangent(f, 1/5)FEmaketangent := (f, x0) -> unapply(D(f)(x0) * (x - x0) + f(x0), x)@' is the line > at </ to the graph of ;tangent9The 59plots:-display(%, fp);1;plot(ccc, 1/10 .. 4/5); -Eccc := makechord(1/10, 4/5);n)_makechord := (x0, x1) -> unapply(CurveFitting:-PolynomialInterpolation([[x0, f(x0)], [x1, f(x1)]], x), x)H%plots:-display(fp, plottools:-line([1/10, f(1/10)], [4/5, f(4/5)]))B!plots:-display(fp, plottools:-line([0, f(0)], [1/2, f(1/2)]))  and /c is the line segment connecting the points  and ! between + of a graph of chordThe  Cfp := plot(f, -0.2 .. 1.2); % even better  and )WLet's make it more efficient: choose ~-nops(probation);z+nops(accepted); vplots:-display( myPointPlot(samplePoints[accepted], yCoordinates[accepted], green, "accepted"), myPointPlot(samplePoints[probation], yCoordinates[probation], blue, "probation"), myPointPlot(samplePoints[rejected], yCoordinates[rejected], red, "rejected"));_rAprobation, rejected := selectremove(i -> yCoordinates[i] < f(samplePoints[i]), probation):anEaccepted, probation := selectremove(i -> yCoordinates[i] < h(samplePoints[i]), [seq(1..N)]):jCplot([f,g,h], -0.2 .. 1.2); fh(x);d_K### Student exercise: determine c3 so that h is tangent to f somewhere strictly between 0 and 1=Xh := x -> piecewise(And(x>=0, x<=1), 12/7 + c3*(x-1), 0); Tf(1);P?plot([f,g], -0.2 .. 1.2);;K{ to pre-screen for values that are certainly less than I7Use a lowerbound for xCs that is expensive: you may not have an explicit formula for it, just some program that computes a value for given #AKIn many cases, it's evaluating 3<kLet's make it more efficient: add pre-screening&2QHistogram(samplePoints[accepted]);.+nops(accepted);7*qplots:-display( myPointPlot(samplePoints[accepted], yCoordinates[accepted], green, "accepted"), myPointPlot(samplePoints[rejected], yCoordinates[rejected], red, "rejected"));b&Gaccepted, rejected := selectremove(i -> yCoordinates[i] < f(samplePoints[i]), [seq(1 .. N)]):%"OyCoordinates := ysmpl *~ gvalues;%Oysmpl := Sample(Uniform(0,1), N);$Mgvalues := map(g, samplePoints);1gsamplePoints := Sample(Triangular(0,1,1), N);(Uplot([f , g ], -0.2 .. 1.2);(Uplot([f(x), g(x)], x = -0.2 .. 1.2);si### Student Exercise: Choose c2 such that the graphs of g and f are tangent somewhere strictly between 0 and 12ig := x -> piecewise(And(x>=0, x<=1), c2*x, 0);%|Oplot([f(x), 2*x], x=-0.2 .. 1.2);"xICDF(X, t^2) assuming t>0, t<1;+t[Probability(X < t^2) assuming t>0, t<1;0pePDF(Triangular(0,1,1), t) assuming t>0, t<1;*lYPDF(sqrt(X), t) assuming t > 0, t < 1;&hQX := RandomVariable(Uniform(0,1));d?plot([f,g], -0.2 .. 1.2); ` better)^WLet's make it more efficient: choose -W_plots:-display(hh, r1, transparency=0.5);,S]hh := Histogram(samplePoints[accepted]);=O}r1 := plots:-display( myPointPlot(samplePoints[rejected], yCoordinates[rejected], red, "rejected"), myPointPlot(samplePoints[accepted], yCoordinates[accepted], green, "accepted"));K-myPointPlot := proc( xvalues :: {list, rtable}, yvalues :: {list, rtable}, color, label :: {name, string}, $) Statistics:-PointPlot(yvalues, ':-xcoords' = convert(xvalues, 'list'), ':-color' = color, ':-symbolsize' = 1, ':-legend' = label); end proc;G-rejected[1..10];C-accepted[1..10];#?Knops(accepted), nops(rejected);`;Caccepted, rejected := selectremove(i -> yCoordinates[i] < f(samplePoints[i]), [seq(1..N)]): 53sgYL;* eA r / W j LB`:OFZ33_NAprobation, rejected := selectremove(i -> yCoordinates[i] < f(samplePoints[i]), probation):aJEaccepted, probation := selectremove(i -> yCoordinates[i] < h(samplePoints[i]), [seq(1..N)]):7FsyCoordinates := gvalues *~ Sample(Uniform(0,1), N);$BMgvalues := map(g, samplePoints);->_plots:-display(%, plot(g/areasum, 0..1));:=Histogram(sampleValues);B6sampleValues := ArrayTools:-Concatenate(2, op(sampleValues));U2-sampleValues := zip((t, var) -> Sample(var, rhs(t)), tally, compositeVariables);.=compositeVariables := [seq('`if`'(RandomVariable(Bernoulli(rectangleProbabilities[i])) < 1/2, triangleVariables[i], rectangleVariables[i]), i=1..nb-1)];e*MrectangleVariables := [seq(RandomVariable(Uniform(boundaries[i], boundaries[i+1])), i=1..nb-1)];.&_triangleVariables := [seq(RandomVariable(Triangular(boundaries[i], boundaries[i+1], `if`(minvalues[i] = g(boundaries[i]), boundaries[i+1], boundaries[i]))), i=1..nb-1)];6"qrectangleProbabilities := rectangleAreas /~ areas;IrectangleAreas := minvalues *~ (boundaries[2..] - boundaries[..-2]);V/plots:-display( seq( plottools:-polygon([ [boundaries[i], minvalues[i]], [boundaries[i+1], minvalues[i]], [boundaries[i+1], g(boundaries[i+1])], [boundaries[i], g(boundaries[i])]], color=colors[i]), i = 1 .. nb - 1), seq( plottools:-polygon([ [boundaries[i], minvalues[i]], [boundaries[i+1], minvalues[i]], [boundaries[i+1], 0], [boundaries[i], 0]], color=darker[i]), i = 1 .. nb - 1)); .adarker := map(ColorTools:-Darken, colors);X3minvalues := zip((b1, b2) -> min(g(b1), g(b2)), boundaries[..-2], boundaries[2..]);*Ytally := sort(Tally(pieces), key=lhs);& Qpieces := Sample(areaVariable, N);E areaVariable := RandomVariable(ProbabilityTable(probabilities));%Oprobabilities := areas / areasum;"~Iareasum := add(a, a in areas);rzgareas := [seq((boundaries[i+1] - boundaries[i]) * (g(boundaries[i+1]) + g(boundaries[i]))/2, i = 1 .. nb-1)];v;nb := nops(boundaries);irUplots:-display(seq( plottools:-polygon([ [boundaries[i], g(boundaries[i])], [boundaries[i+1], g(boundaries[i+1])], [boundaries[i+1], 0], [boundaries[i], 0]], color = colors[i]), i = 1 .. nops(boundaries)-1))cnIcolors := [seq(ColorTools:-Color("HSV", [(i+2)*(1/8), .9, .9]), i = 1 .. nops(boundaries)-1)];jCplot([f,g,h], -0.2 .. 1.2);Sf)h := unapply(piecewise( x <= boundaries[1], 0, x <= boundaries[2], tangents[1](x), x <= boundaries[3], tangents[2](x), x <= boundaries[4], chords[3](x), x <= boundaries[5], chords[4](x), 0), x);Sb)g := unapply(piecewise( x <= boundaries[1], 0, x <= boundaries[2], chords[1](x), x <= boundaries[3], chords[2](x), x <= boundaries[4], tangents[2](x), x <= boundaries[5], tangents[3](x), 0), x);'^Splot([f, op(chords)], -0.2 .. 1.2);EZ chords := zip(makechord, boundaries[ .. -2], boundaries[2 .. ]);-V_boundaries[1 .. -2], boundaries[2 .. -1];@Rboundaries := sort([op(tangentpoints), op(intersections)]);gNQintersections := [solve(tangents[1](x) = tangents[2](x)), solve(tangents[2](x) = tangents[3](x))];)JWplot([f, op(tangents)], -0.2 .. 1.2);6Fqtangents := map2(maketangent, f01, tangentpoints);!BGtangentpoints := [0, 4/9, 1];>7solve(fdouble(x), x);:7fdouble := D(fprime);6/fprime := D(f01);2=f01 := unapply(f01x, x);#.Kf01x := f(x) assuming x>0, x<1;'!upperbound!!lowerbound concave!lowerbound !upperbound convex tangent xchord,c]plot([-exp(x),ln(x), sqrt(x)], x=-2..2);!_Gplot([x^2, exp(x)], x=-2..2);)[W if its second derivative is negative Zconcave 8wi u R m T87generator:-intervals;=}plots:-display( myPointPlot(acceptedX, acceptedY, green, "accepted"), myPointPlot(probationX, probationY, blue, "probation"), myPointPlot(rejectedX, rejectedY, red, "rejected"));U-rejectedX, rejectedY := op(map[3](map, `?[]`, generator:-rejected, [[1], [2]]));X|3probationX, probationY := op(map[3](map, `?[]`, generator:-probation, [[1], [2]]));Ux-acceptedX, acceptedY := op(map[3](map, `?[]`, generator:-accepted, [[1], [2]]));/tcto N do generator:-generateValue(); end do: pEgenerator:-setup(f01, 0, 1);l generator := module() local dispatcher, f, fprime, uniform01, createInterval, splitInterval, generateChords, generateTangents, resetDispatcher; export intervals, accepted, probation, rejected, setup, generateValue; uses ST = Statistics; generateChords := proc(xleft, xmiddle, xright, yleft, ymiddle, yright) local x; return unapply(piecewise( x < xmiddle, ((x - xleft) * ymiddle + (xmiddle - x) * yleft) / (xmiddle - xleft), ((x - xmiddle) * yright + (xright - x) * ymiddle) / (xright - xmiddle)), x); end proc; generateTangents := proc(xleft, xmiddle, xright, yleft, yright, derivativeLeft, derivativeRight) ghQhttp://www.mapleprimes.com/posts/96015-Generating-Samples-From-Custom-Probability-Distributions-IVseiThis is certainly too complicated to explain in this session. The program that does it is shown here, though: a online _ and 4]mLet's make it more efficient: automate updating 4Vmnops(accepted), nops(probation), nops(rejected); Rplots:-display( myPointPlot(samplePoints[accepted], yCoordinates[accepted], green, "accepted"), myPointPlot(samplePoints[probation], yCoordinates[probation], blue, "probation"), myPointPlot(samplePoints[rejected], yCoordinates[rejected], red, "rejected")); local x; return unapply(piecewise( x < xmiddle, yleft + derivativeLeft * (x - xleft), yright + derivativeRight * (x - xright)), x); end proc; setup := proc(ff :: appliable, a :: numeric, b :: numeric, $) local areas, fdoubleprime, i, lastpoint, derivativeLeft, pointTable, points, derivativeRight, xleft, xright, yleft, yright; if not a < b then error "invalid input, %1 expects its second argument, a, to be " "less than its third argument, but received %2 and %3", procname, a, b; end if; f := ff; fprime := D(f); fdoubleprime := D(fprime); uniform01 := ST:-Sample(ST:-RandomVariable(':-Uniform'(0, 1))); accepted, probation, rejected := Array(1..0), Array(1..0), Array(1..0); # Find all inflection points. We set them as keys in a table, # then gather them at the end. pointTable := table([b = 1]); lastpoint := a; while lastpoint <> FAIL and lastpoint < b do pointTable[lastpoint] := 1; lastpoint := RootFinding:-NextZero(fdoubleprime, lastpoint); end do; points := sort([indices(pointTable, ':-nolist')]); # Now set up the intervals in between these points. An # interval will be a record with these keys: # * xleft; the left boundary of the interval. # * xright; the right boundary of the interval. # * yleft, yright; the corresponding values of f. # * derivativeLeft, derivativeRight; the values of f' at xleft # and xright. # * lowerbound; a procedure that returns the lower bound for # points in this interval. # * upperbound; a procedure that returns the upper bound for # points in this interval. # * sample; a procedure that generates one sample value from # this interval. # * area; the total area under the upper curve defined for # this interval. intervals := Array(1 .. nops(points) - 1); xleft := points[1]; yleft := f(xleft); derivativeLeft := fprime(xleft); for i to nops(points) - 1 do xright := points[i+1]; yright := f(xright); derivativeRight := fprime(xright); intervals[i] := createInterval(xleft, xright, yleft, yright, derivativeLeft, derivativeRight); xleft := xright; yleft := yright; derivativeLeft := derivativeRight; end do; resetDispatcher(); return NULL; end proc; createInterval := proc(xleft, xright, yleft, yright, derivativeLeft, derivativeRight) local area, areas, generators, leftRectangleHeight, leftTriangleHeight, lowerbound, rightRectangleHeight, rightTriangleHeight, sample, selector, upperbound, xmiddle, ymiddle; # The intersection of the two tangents: xmiddle := (yright - yleft + derivativeLeft * xleft - derivativeRight * xright) / (derivativeLeft - derivativeRight); ymiddle := f(xmiddle); if ymiddle < ((xmiddle - xleft) * yright + (xright - xmiddle) * yleft) / (xright - xleft) then # Middle point is lower than chord: convex, so chords # define upper curve, tangents lower curve. lowerbound := generateTangents(xleft, xmiddle, xright, yleft, yright, derivativeLeft, derivativeRight); upperbound := generateChords(xleft, xmiddle, xright, yleft, ymiddle, yright); else # Middle point is higher than chord: concave, so # tangents define upper curve, chords lower curve. lowerbound := generateChords(xleft, xmiddle, xright, yleft, ymiddle, yright); upperbound := generateTangents(xleft, xmiddle, xright, yleft, yright, derivativeLeft, derivativeRight); # Instead of f at xmiddle, we use the intersection of # the tangents for ymiddle in the remainder of this # iteration: ymiddle := upperbound(xmiddle); end if; leftRectangleHeight := min(yleft, ymiddle); leftTriangleHeight := abs(yleft - ymiddle); rightRectangleHeight := min(ymiddle, yright); rightTriangleHeight := abs(yright - ymiddle); areas := [leftRectangleHeight * (xmiddle - xleft), leftTriangleHeight * (xmiddle - xleft)/2, rightRectangleHeight * (xright - xmiddle), rightTriangleHeight * (xright - xmiddle)/2]; generators := map(ST:-Sample @ ST:-RandomVariable, [ ':-Uniform'(xleft, xmiddle), ':-Triangular'( xleft, xmiddle, `if`(yleft > ymiddle, xleft, xmiddle)), ':-Uniform'(xmiddle, xright), ':-Triangular'( xmiddle, xright, `if`(ymiddle > yright, xmiddle, xright))]); selector := ST:-Sample(ST:-RandomVariable( ':-ProbabilityTable'(areas / add(x, x in areas)))); sample := proc() local i; i := trunc(selector(1)[1]); return generators[i](1)[1]; end proc; area := add(x, x in areas); return Record(':-xleft' = xleft, ':-xright' = xright, ':-yleft' = yleft, ':-yright' = yright, ':-derivativeLeft' = derivativeLeft, ':-derivativeRight' = derivativeRight, ':-lowerbound' = lowerbound, ':-upperbound' = upperbound, ':-sample' = sample, ':-area' = area); end proc; resetDispatcher := proc() local areas, i, sample; areas := [seq(i:-area, i in intervals)]; sample := ST:-Sample(ST:-RandomVariable(':-ProbabilityTable'( areas / add(x, x in areas)))); dispatcher := proc() return trunc(sample(1)[1]); end proc; end proc; generateValue := proc() local fx, i, interval, lowerbound, x, y; while true do i := dispatcher(); interval := intervals[i]; x := interval:-sample(); y := uniform01(1)[1] * interval:-upperbound(x); lowerbound := interval:-lowerbound(x); if y < lowerbound then accepted(rtable_num_elems(accepted) + 1) := [x, y]; return x; end if; fx := f(x); if y > fx or lowerbound < 0.9 * fx then splitInterval(i, x, fx); end if; if y < fx then probation(rtable_num_elems(probation) + 1) := [x, y]; return x; end if; rejected(rtable_num_elems(rejected) + 1) := [x, y]; end do; end proc; splitInterval := proc(i, x, fx) local xDerivative, oldInterval; xDerivative := fprime(x); oldInterval := intervals[i]; intervals[i] := createInterval( oldInterval:-xleft, x, oldInterval:-yleft, fx, oldInterval:-derivativeLeft, xDerivative); intervals(rtable_num_elems(intervals) + 1) := createInterval( x, oldInterval:-xright, fx, oldInterval:-yright, xDerivative, oldInterval:-derivativeRight); resetDispatcher(); end proc; end module: }rg\QF;0%uj_TI:* { k [ K ; +  { k [ K ; +  { k [ K ; +  { k [ K ; +  s b Q @ /   vgWH9* sdTE6&~o`PA2#zk\M=.wgXI:* scTD5&~o_PA2" 9 %  ! 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G>  >  =  = ` A < e;  ; G:  :  9  9 ` ? 8 e7  7 G6  6  5  5 ` = 4 e3  3 G2  2  1  1 ` ; 0 e/  / G.  .  -  - ` 9 , e+  + G*  *  )  ) ` 7 ( e'  ' G&  &  %  % ` 5 $ e#  # G"  "  !  ! ` 3  e   G       ` 1  e   G       ` /  e   G       ` -  e   G       ` +  e   G       ` )  e   G       ` '  G       ` %  e   G              F   e  ~ G  }    |    {   z  F  y e  x !  w G  v    u    t   s F  r E  q e  p G  o    n    m   l F  k e  j G  i    h    g   f  F  e e  d !  c G  b    a    `   _ F  ^ E  ] e  \ !  [ G  Z    Y    X   W F  V e  U !  T G  S    R    Q   P  F  O #  N G  M    L    K   J F  I E  H D  G  D  F D  E C` # D G  C    B ` " A e  @ G  ?    > `  = e  < G  ;    : `  9 e  8 e  7 !  6 e  5 e  4  !  3 e  2 G  1    0 `  / e  . 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G- -  , ,  + e* * G) )  ( (  ' e& & G% %  $ $  # e" " G! !       e  !  e  e   !  e  G  &  %  G       e  G     0  e  G       .  e   G      ,  e  G     *  e  G     (  e ~ G }   | & { G z   y $ x e w G v   u " t G s   r  q e p G o   n  m e l G k   j  i e h G g   f  e e d G c   b  a e ` G _   ^  ] e \ G [   Z  Y e X G W   V  U e T G S   R  Q e P G O   N  M e L G K   J  I e H G G   F  E e D G C   B  A e @ G ?   >  = e < G ;   :  9 e 8 G 7   6  5 e 4 G 3 & 2 % 1 G 0   /  . G -   , r + e * G )   ( r ' e & G %   $ r # e " G !    r  e  G     r  e  G     r  e  G     r  e  G     r  e  G     r  qcUF8* qbTF8) ~ p b T F 8 *    q c U G 9 *  ~ p a S E 6 (  } o ` R D 6 '  | n ` R D 6 (  |m_QC5' tfXI;-teWI;,}oaSD6( }oaSD5' {l^PA2$|m_QC5'  Eg  e~ ~ !} } G| |  { {  z z y y Fk x ew w !v v Gu u  t t  s s r r  Fk q #p p Go o  n n  m m l l Fk k Eg j Dg i  Dg h Dg g C# f Ge e  d d " c eb b Ga a  ` `  _ e^ ^ G] ]  \ \  [ eT Z eY Y !T X eT W eV V  !T U eT T GS S  R R  Q eP P GO O  N N  M eL L GK K  J J  I eH H GG G  F F  E eD D GC C  B B  A #8 @ e8 ? #8 > e8 = #8 < e8 ; e: :  !8 9 e8 8 G7 7  6 6  5 e4 4 G3 3  2 2  1 e0 0 G/ /  . .  - e, , G+ +  * *  ) e( ( G' '  & &  % e$ $ G# #  " "  ! e   G       #  e  #  e  #  e  #  e  #  e  e   !  e  G       e   G         e  #  e   #  e  G  &  %)  G     ' ~ e } G |   {  z e y G x   w  v e u G t   s  r e q G p   o  n e m G l   k  j e i G h   g  f e e G d   c  b G a   `  _ e ^ G ]   \  [ G Z   Y  X e W G V   U  T e S G R   Q  P e O G N   M  L # K e J  # I e H G G   F   E  D # C e B  # A e @ G ?   >   =   < e ; G : & 9 %% 8 G 7   6 # 5 G 4   3 [, 2 e 1 G 0   / [* . e - G ,   + [( * e ) G (   ' [& & e % G $   # [$ " e ! G     ["  e  G     [  e  G     [  e  G     [  e  G     [  e  G     [  G     [  e tfXJ;-vhZL>0! w h Z L > / !   v h Z K = / !   v g Y K = .   u g Y J < .   u f X J < -   tfXI;-teWI;,seWH:,sdVH:+rdVG9+qcUG9+rcUG9* | e{ { Gz z  y y Z x ew w Gv v  u u Z t es s Gr r  q q Z p eo o Gn n  m m Z l ek k Gj j  i i Z h eg g !f f  $d e ed d Gc c  b b Z a e\ ` #\ _ e\ ^  #\ ] e\ \ G[ [ &Z Z %- Y GX X  W W + V eU U GT T  S S { R eQ Q GP P  O O y N eM M GL L  K K w J eI I GH H  G G u F eE E GD D  C C s B eA A G@ @  ? ? q > e= = G< <  ; ; o : e9 9 G8 8  7 7 m 6 e5 5 G4 4  3 3 k 2 e1 1 G0 0  / / i . e- - G, ,  + + g * e) ) G( (  ' ' e & e% % G$ $  # # c " e! ! G      a  e  G     _  e  G     ]  e  G     [  e  G     Y  e   G       W  e   G     U  e  G     S  e  G     Q ~ e } G |   { O z e y G x   w M v e u G t   s K r e q G p   o I n e m G l   k G j e i G h   g E f e e G d   c C b e a G `   _ A ^ e ] G \   [ ? Z e Y G X   W = V e U G T   S ; R e Q G P   O 9 N e M G L   K 7 J e I G H   G 5 F e E G D   C 3 B e A G @   ? 1 > e = G <   ; / : e 9 G 8   7 - 6 e 5 G 4   3 + 2 e 1 G 0   / ) . e - G ,   + ' * G )   ( % ' e & G %   $   #  " F ! e  G           F  e  !  G          F  Eg  e  G          F  e  G           F  e  !  G          F qcUF8*   G     /  G     Z  G     Z  e  G     Z  e  G     Z  e  G~ ~  } } Z yrkd]VOHA:3,% }vohaZSLE>70)"  { t m f _ X Q J C < 5 . '     x q j c \ U N G @ 9 2 + $      | u n g ` Y R K D = 6 / ( !    y r k d ] V O H A : 3 , %     } v o h a Z S L E > 7 0 ) "   zsle^WPIB;4-& ~wpib[TMF?81*#{tmf_XQJC<5.'  xqjc\UNG@92+$|ung`YRKD=6/(! mieaXTPLHD@<840,($  |xoi}yuqmiea]YUQMIEA=951-)%!  |lh]YUQMIEA>:73/+$  ~{wtpmib_[XTKEhDgCfBdcba`@?>=<_^]\[;:987ZYXWVUTSRQPONMLKJIHGFEDCB6543210/.-,+*)('&%$#"! A@?>=<;:9876543210/.-,+*)('&%$#"!                      {tmf_XQJC<5.'  xqjc\UNG@92+$ | u n g ` Y R K D = 6 / ( !    y r k d ] V O H A : 3 , %     } v o h a Z S L E > 7 0 ) "    z s l e ^ W P I B ; 4 - &     ~ w p i b [ T M F ? 8 1 * #     {tmf_XQJC<5.'  xqjc\UNG@92+$|ung`YRKD=6/(! yrkd]VOHA:3,% }vohaZSLE>70)"  P L H D @ < 8 4 0 , ( $                                        | u o e a ] S O K G C 7 3 / + ' #                                    ~ z v r n j f b Y U Q M I E A = 9 5 1 - ) % !                                         } y p j                           ~ z v r n j f b ^ Z V R N J F B > : 6 2 . * & "                                } m i ^ Z V R N J F B ? ; 8 4 0 , %                                        | x u q n j c ` \ Y U L Fzsm#}yuqmibWSOKGC?;73/+'# {tnd`\RNJFB62.*&" }yu T {tmf_XQJC<5.'  xqjc\UNG@92+$ | u n g ` Y R K D = 6 / ( !    y r k d ] V O H A : 3 , %     } v o h a Z S L E > 7 0 ) "    z s l e ^ W P I B ; 4 - &     ~ w p i b [ T M F ? 8 1 * #     {tmf_XQJC<5.'  xqjc\UNG@92+$|ung`YRKD=6/(! yrkd]VOHA:3,% }vohaZSLE>70)" G!GGGGG G GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG}GvGpGfGbG^GTGPGLGHGDG8G4G0G,G(G$G GG GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG{GwGsGoGkGgGcG]GZGVGRGNGJGFGBG>G:G6G2G.G*G&G"GGGGG GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG~GzGtGqGmGkGGGGGGGGGGGGGGGGGGGGGGGGGGGGG{GwGsGoGkGgGcG_G[GWGSGOGKGGGCG?G;G7G3G/G+G'G#GGGGGG GGGGGGGGGGGGGGGGGGGGGGGGG~GnGjGbG_G[GWGSGOGKGGGCG@G<G9G5G1G-G&GGGGGG GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG}GyGvGrGoGkGgGdGaG]GZGVGPGMGIGGFFFFFFFyFrFlFFFFFFFFFEEEkEEEDjDiDhDDDCgC)e)A&[&&&\&&&s&&a&&&&f&O%Z%%%[%%%r%%`%%%%e%N$f$#`#^#q#A#?#=############_#######{#y#w#u#s#f#d#*#(#"# #!g!!!~!w!Y!V!:!!!!v!!!!!!!!!p!!!R      ~ z v r n j cG% v{tmf_XQJC<5.'  xqjc\UNG@92+$ | u n g ` Y R K D = 6 / ( !    y r k d ] V O H A : 3 , %     } v o h a Z S L E > 7 0 ) "    z s l e ^ W P I B ; 4 - &     ~ w p i b [ T M F ? 8 1 * #     {tmf_XQJC<5.'  xqjc\UNG@92+$|ung`YRKD=6/(! eeee|exetepeleheeeae_e]eVeReNeJeFeBe>e:e6e2e.e*e&e"eeeeee eeeeeeeeeeeeeeeeeeeeeeeeeeeeeeexece_e[eZeXeWeUeQeMeIeEe@e>e<e;e9e5e1e-e)e%e!eeeeeeee e eeeeeeeeeeeeeeeeeeeeeeeeeeeeeee|exetepelehede`e^eWeSeOeKeGeCe?e;e7e3e/e+e'e#eeeeeee eeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeee{eweueneleeeeeeeeeeeeeeeeeeeeeeeeeeee|exetepelehede`e\eXeTePeLeHeDe@e<e8e4e0e,e(e$e eeeee eeeeeeeeeeeeeeeeeeeeeeeeeeeeeeezexevetereqeoekegeeece\eXeTePeLeHeDe=e6e2e.e)e'e!eeee eeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeezeselehe^eWeSeQeJeHGGGGGGG{GwGsGoGkGdG\GYGUGQGMGIGEGAG=G9G5G1G-G) eztnhb\VPJD>82,& ~xrlf`ZTNHC=6/(!  } u m e ] U M E = 5 - %    } u m e ] U M E = 5 - %    } u m e ] U M E = 5 - %    } u m e ] U M E = 5 - %    } u m e ] U M E = 5 - %   }ume]UME=5-% }ume]UME=5-% }ume]UME=5-% }ume]UME=5-% }ume]UME=5-% }ume"! YUQMIEA>:73/+#         ~|}{|yzxywxuvtursqrpqnomnkljkijghfgeeeeeee~e{ewetepemeiefcdbc`a_`]^\][\YZXYVWUVTURSPRPQOPN_N[NXNTNOLMKLIJGHFIFGEFBC@?>=<;:9876543210/.-,+*)('&%$#"!          ` ]       e b N K E D B A     ZW[Xroihfe          @xph`XPH@80( xph`XPH@80(  x p h ` X P H @ 8 0 (    x p h ` X P H @ 8 0 (    x p h ` X P H @ 8 0 (    x p h ` X P H @ 8 0 (    x p h ` X P H @ 8 0 (    xph`XPH@80( xph`XPH@80( xph`XPH@80( xph`XPH@80( xph`XPH@80( xph`XPH@~}~{|z{yzwxvwuvstrsqropnomnkljkijghfgefcdbcab_`^_]^[\Z[YZWXVWUVSTRSQROPNOMNKLJKIJGHFGEFCDBCAB?@>?=>;<:;9:786756342312/0./-.+,*+)*'(&'%&#$"#!"        ~}~|}pqn{nznynxnwnvnuntnsnrnpnomnlmjkijhibgbfbebdbcab``````````````}`y`u`q`m`i`e`a`]`Y`U`Q`M`I`E`A`=`9`5`1`-`)`%`!````` ` `````````````````|`l`h`a^_]^[\Z[YZWXVWUVSTRSQROPNOMNKLJKIJGHFGEFCDBCAB?@>?<=;<:;89785645341201/0-.,-+,&*&)&(&'%&$% @xph`XPH@80( xph`XPH@80(  x p h ` X P H @ 8 0 (    x p h ` X P H @ 8 0 (    x p h ` X P H @ 8 0 (    x p h ` X P H @ 8 0 (    x p h ` X P H @ 8 0 (    xph`XPH@80( xph`XPH@80( xph`XPH@80( xph`XPH@80( xph`XPH@80( xph`XPH@~}~{|z{yzwxvwuvstrsqropnomnkljkijghfgefcdbcab]`]_]^\][[[[[[[[[[[[[[[}[y[u[q[m[i[e[a[\YZXYVWUVTURSQRPQNOMNLMJKIJHIFGEFDEBCAB@A>?=><=:;9:89675645231201./-.,-*+)*()&'%&$%"#!" !TPLHD@<840,($          ~}~|}z{yzxyvwtvtustrrrrrrrrrrrrr|rxrspqopmnkljmjkijfgdcba`_^]\[ZYXWVUTSRQPONMLKJIHGFEDCBA@?>=<;:9876543210/.-,+*)('&%$#"!  @xph`XPH@80( xph`XPH@80(  x p h ` X P H @ 8 0 (    x p h ` X P H @ 8 0 (    x p h ` X P H @ 8 0 (    x p h ` X P H @ 8 0 (    x p h ` X P H @ 8 0 (    xph`XPH@80( xph`XPH@80( xph`XPH@80( xph`XPH@80( xph`XPH@80( xph`XPH@-.,-+,)*()'(%&$%#$!" !         ~}~|}{|z{yzwxvwuvtustrspqopnomnlmkykrklgggkgjgighefdebcab`a^_]^\]YZVWT[TYTXTVTUSTRSPQOPNOLMKLJKHIGHFGDECDBC:;8A8@8?8>8=8<8:89786745342301/0./,-+,*+()'(&'$%#$"# !        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Dictionary Hyperlink[255,255,255]Times New Roman[147,0,15] B -'#  HyperlinkWarning[255,255,255]Courier New[0,0,255] G +'+# Atomic Variable[255,255,255]Times New Roman[175,0,175] B)'##  HyperlinkError[255,255,255]Courier New[255,0,255] E/'+Header and Footer[255,255,255]Times New Roman[0,0,0] D-' +Annotation Title[255,255,255]Times New Roman[0,0,0]B)'+Ordered List 5[255,255,255]Times New Roman[0,0,0] B)'+Ordered List 4[255,255,255]Times New Roman[0,0,0] B)'+Ordered List 3[255,255,255]Times New Roman[0,0,0] B)'+Ordered List 2[255,255,255]Times New Roman[0,0,0] =#'# Text Output[255,255,255]Courier New[0,0,255] >#' #!Maple Input[255,255,255]Courier New[120,0,14] B)'+Ordered List 1[255,255,255]Times New Roman[0,0,0] >~!'## Diagnostic[255,255,255]Courier New[40,120,40] =}' + Heading 3[255,255,255]Times New Roman[0,0,0]I|+' +'2D Inert Output[255,255,255]Times New Roman[144,144,144] @{%'+Maple Output[255,255,255]Times New Roman[0,0,0] =z' +Heading 2[255,255,255]Times New Roman[0,0,0]Ey3'# Line Printed Output[255,255,255]Courier New[0,0,255] =x'+ Heading 4[255,255,255]Times New Roman[0,0,0] ?w'+ 2D Output[255,255,255]Times New Roman[0,0,255] 9v'# Warning[255,255,255]Courier New[0,0,255] =u' +Heading 1[255,255,255]Times New Roman[0,0,0]  s)X O  X  _  K @A%' +Caption Text[255,255,255]Times New Roman[0,0,0] A@'+# Hyperlink[255,255,255]Times New Roman[0,128,128] :?'+Author[255,255,255]Times New Roman[0,0,0] B>)' +Equation Label[255,255,255]Times New Roman[0,0,0] K=;' ## Maple Input Placeholder[255,255,255]Courier New[200,0,200] 6<'#Code[255,255,255]Courier New[255,0,0] E;/' +Caption Reference[255,255,255]Times New Roman[0,0,0] ::'+Normal[255,255,255]Times New Roman[0,0,0] 89'+Text[255,255,255]Times New Roman[0,0,0] 98' +Title[255,255,255]Times New Roman[0,0,0]97'## Error[255,255,255]Courier New[255,0,255] <6' +2D Input[255,255,255]Times New Roman[0,0,0] =5'+Dash Item[255,255,255]Times New Roman[0,0,0] =4'+List Item[255,255,255]Times New Roman[0,0,0] C3+'+Annotation Text[255,255,255]Times New Roman[0,0,0] >2!'+Maple Plot[255,255,255]Times New Roman[0,0,0] ?1#'+Bullet Item[255,255,255]Times New Roman[0,0,0] ;0' +2D Math[255,255,255]Times New Roman[0,0,0] K/5'+! Dictionary Hyperlink[255,255,255]Times New Roman[147,0,15] B.-'#  HyperlinkWarning[255,255,255]Courier New[0,0,255] G-+'+# Atomic Variable[255,255,255]Times New Roman[175,0,175] B,)'##  HyperlinkError[255,255,255]Courier New[255,0,255] E+/'+Header and Footer[255,255,255]Times New Roman[0,0,0] qaA"_> x Y 8  t S 2  l K * d C "  q D # ~]<vU4nM, fEd7 nM, fE$^=U  L863Q  L864M  L835I  L834E  L784A  L783=  L8009  L8055  L8011  L804-  L802)  L782%  L781!  L780  L803  L778  L777  L689   L688   L700 L702* % L735presentation*| % L736presentation*u % L743presentation*n % L742presentation*h % L741presentation*a % L740presentation*Y % L734presentation*R % L739presentation*L % L738presentationB L776>  L687:  L6860 L628,  L671(  L672$  L673   L644 L627  L647   L646  L660  L645  L626|  L625l L624h  L280] L279Y  L598U  L597Q  L596M  L595I  L587E  L586A  L585> L984: L9857 L9863  L581/  L278+  L362*$ % L573presentation* % L574presentation L277 L354   L353   L351  L350  L321}  L320y  L349u  L319q  L318m  L305i  L322f L983b  L295_ L982[  L294W  L293S  L292O  L291K  L284G  L283C  L282?  L2766 L72  L204.  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