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$CellContext`genPtJProbs[$CellContext`M, K, $CellContext`rsd]; $CellContext`gMatNum = \ $CellContext`genGamE[$CellContext`jProbsNum, $CellContext`lOrd, \ $CellContext`rsd]; $CellContext`pts = N[$CellContext`gMatNum, 3]; $CellContext`lOrd = \ $CellContext`lOrdFun[$CellContext`gMatNum]; Dynamic[$CellContext`lOrd]; Grid[{{Null}, { Grid[{{ Style["Structural Functions ", Bold, Black, FontSize -> 14], Style[" Notation ", Bold, Black, FontSize -> 14]}, { Null, Null}, { DynamicModule[{$CellContext`loc = \ $CellContext`matToLoc[$CellContext`gMatNum]}, LocatorPane[ Dynamic[$CellContext`loc, Function[{$CellContext`value, $CellContext`var}, \ $CellContext`var = $CellContext`locFun[$CellContext`value, $CellContext`M, K]; $CellContext`pts = \ $CellContext`locToMat[$CellContext`value, $CellContext`M, K]; $CellContext`pts = \ $CellContext`matMonotone[$CellContext`pts]; $CellContext`lOrd = \ $CellContext`lOrdBackDef[ $CellContext`lOrdFun[$CellContext`pts]], HoldAll]], Graphics[{ PointSize[Large], 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TableHeadings -> { Array[#& , $CellContext`M], Array[Subscript[$CellContext`x, #]& , K]}]]}}]}, {Null}, { Null}, { Style[ " Inequalities Defining the Identified Set \[ScriptCapitalE](z) \ ", Black, Bold, FontSize -> 14]}, {Null}, { Dynamic[ TableForm[ $CellContext`eSetMat[ $CellContext`jProbsDef[$CellContext`lOrd, $CellContext`M, K, $CellContext`type], $CellContext`lOrd], TableHeadings -> { Array[ Subscript[$CellContext`\[Gamma], # - 1]& , ($CellContext`M - 1) K + 2], Array[ Subscript[$CellContext`\[Gamma], # - 1]& , ($CellContext`M - 1) K + 2]}]]}}, Spacings -> {1, 1}]], $CellContext`lOrd = {1, 2, 3, 4, 5, 6}, $CellContext`jProbsDef[ Pattern[$CellContext`lOrd, Blank[]], Pattern[$CellContext`M, Blank[]], Pattern[K, Blank[]], Optional[ Pattern[$CellContext`type, Blank[]], "Arg"]] := Module[{$CellContext`Q, $CellContext`delta, $CellContext`rhoVec, \ $CellContext`jProbs, $CellContext`tmp}, $CellContext`Q = ($CellContext`M - 1) K; $CellContext`delta = Array[Subscript[$CellContext`\[Delta], 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Blank[]], Pattern[$CellContext`lOrd, Blank[]], Pattern[$CellContext`M, Blank[]], Pattern[K, Blank[]]] := Module[{$CellContext`mNow, $CellContext`ordNow}, $CellContext`ordNow = Part[$CellContext`lOrd, $CellContext`which]; $CellContext`mNow = Ceiling[$CellContext`ordNow/K]; Return[$CellContext`mNow]], $CellContext`kappa[ Pattern[$CellContext`which, Blank[]], Pattern[$CellContext`lOrd, Blank[]], Pattern[$CellContext`M, Blank[]], Pattern[K, Blank[]]] := Module[{$CellContext`kNow, $CellContext`mNow, $CellContext`ordNow}, \ $CellContext`ordNow = Part[$CellContext`lOrd, $CellContext`which]; $CellContext`mNow = \ $CellContext`mu[$CellContext`which, $CellContext`lOrd, $CellContext`M, K]; $CellContext`kNow = $CellContext`ordNow - ($CellContext`mNow - 1) K; Return[$CellContext`kNow]], $CellContext`gMatDef[ Pattern[$CellContext`lOrd, Blank[]], Pattern[$CellContext`M, Blank[]], Pattern[K, Blank[]], Optional[ Pattern[$CellContext`type, Blank[]], "Arg"]] := Module[{$CellContext`Q, $CellContext`gVec, $CellContext`gMat, \ $CellContext`tmp}, $CellContext`Q = ($CellContext`M - 1) K; $CellContext`gVec = Array[Subscript[$CellContext`\[Gamma], #]& , $CellContext`Q]; \ $CellContext`gMat = Array[0& , {$CellContext`M - 1, K}]; Do[Part[$CellContext`gMat, $CellContext`mu[$CellContext`q, $CellContext`lOrd, \ $CellContext`M, K], $CellContext`kappa[$CellContext`q, $CellContext`lOrd, \ $CellContext`M, K]] = Part[$CellContext`gVec, $CellContext`q], {$CellContext`q, 1, $CellContext`Q}]; If[$CellContext`type != "Arg", $CellContext`gMat = Array[Subscript[$CellContext`\[Gamma], #, #2]& , {$CellContext`M - 1, K}], $CellContext`gMat]; Return[$CellContext`gMat]], $CellContext`genPtJProbs[ Pattern[$CellContext`M, Blank[]], Pattern[K, Blank[]], Optional[ Pattern[$CellContext`seed, Blank[]], Null], Optional[ Pattern[$CellContext`valMax, Blank[]], 10000]] := Module[{$CellContext`alphaMat, $CellContext`delta, \ $CellContext`deltaMat, $CellContext`res}, If[ IntegerQ[$CellContext`seed], SeedRandom[$CellContext`seed]]; $CellContext`alphaMat = \ $CellContext`genPtAlpha[$CellContext`M, K, $CellContext`seed, $CellContext`valMax]; $CellContext`delta = \ $CellContext`genDelta[ K, $CellContext`seed, $CellContext`valMax]; $CellContext`deltaMat = \ {}; Do[$CellContext`deltaMat = Append[$CellContext`deltaMat, $CellContext`delta], {$CellContext`i, 1, $CellContext`M}]; $CellContext`res = $CellContext`alphaMat \ $CellContext`deltaMat; Return[$CellContext`res]], $CellContext`genPtAlpha[ Pattern[$CellContext`M, Blank[]], Pattern[K, Blank[]], Optional[ Pattern[$CellContext`seed, Blank[]], Null], Optional[ Pattern[$CellContext`valMax, Blank[]], 10000]] := Module[{$CellContext`Q, $CellContext`tmpVec, $CellContext`tmpMat, \ $CellContext`tmpSm, $CellContext`sm}, If[ IntegerQ[$CellContext`seed], SeedRandom[$CellContext`seed]]; $CellContext`tmpMat = {}; Do[$CellContext`tmpVec = RandomInteger[{ 1, $CellContext`valMax}, $CellContext`M]; $CellContext`tmpSm = Apply[Plus, $CellContext`tmpVec]; $CellContext`tmpMat = Append[$CellContext`tmpMat, \ $CellContext`tmpVec/$CellContext`tmpSm], {$CellContext`k, 1, K}]; Return[ Transpose[$CellContext`tmpMat]]; Null], $CellContext`genDelta[ Pattern[K, Blank[]], Optional[ Pattern[$CellContext`seed, Blank[]], Null], Optional[ Pattern[$CellContext`valMax, Blank[]], 10000]] := Module[{$CellContext`res, $CellContext`tmp, $CellContext`sm}, If[ IntegerQ[$CellContext`seed], SeedRandom[$CellContext`seed]]; $CellContext`tmp = RandomInteger[{1, $CellContext`valMax}, K]; $CellContext`sm = Apply[Plus, $CellContext`tmp]; $CellContext`res = \ $CellContext`tmp/$CellContext`sm; Return[$CellContext`res]], $CellContext`genGamE[ Pattern[$CellContext`jProbs, Blank[]], Pattern[$CellContext`lOrd, Blank[]], Optional[ Pattern[$CellContext`seed, Blank[]], Null]] := Module[{K, $CellContext`M, $CellContext`eSet, $CellContext`gVec, \ $CellContext`gMat}, If[ IntegerQ[$CellContext`seed], SeedRandom[$CellContext`seed], SeedRandom[]]; $CellContext`M = Part[ Dimensions[$CellContext`jProbs], 1]; K = Part[ Dimensions[$CellContext`jProbs], 2]; $CellContext`eSet = $CellContext`eSetMat[$CellContext`jProbs, \ $CellContext`lOrd]; $CellContext`gVec = \ $CellContext`genGamFromSet[$CellContext`eSet, $CellContext`seed]; \ $CellContext`gVec = Part[$CellContext`gVec, $CellContext`lOrdBackDef[$CellContext`lOrd]]; $CellContext`gMat = \ {}; Do[$CellContext`gMat = Append[$CellContext`gMat, Part[$CellContext`gVec, Span[($CellContext`m - 1) K + 1, $CellContext`m K]]], {$CellContext`m, 1, $CellContext`M - 1}]; Return[$CellContext`gMat]], $CellContext`eSetMat[ Pattern[$CellContext`jProbs, Blank[]], Pattern[$CellContext`lOrd, Blank[]]] := Module[{$CellContext`M, K, $CellContext`Q, $CellContext`eSet}, $CellContext`M = Part[ Dimensions[$CellContext`jProbs], 1]; K = Part[ Dimensions[$CellContext`jProbs], 2]; $CellContext`Q = ($CellContext`M - 1) K; $CellContext`eSet = $CellContext`cSetMat[$CellContext`jProbs, \ $CellContext`lOrd]; Part[$CellContext`eSet, Span[2, $CellContext`Q + 1], Span[ 2, $CellContext`Q + 1]] = $CellContext`eSubSetMat[$CellContext`jProbs, \ $CellContext`lOrd]; Return[$CellContext`eSet]], $CellContext`cSetMat[ Pattern[$CellContext`jProbs, Blank[]], Pattern[$CellContext`lOrd, Blank[]]] := Module[{$CellContext`nRow, $CellContext`nCol, $CellContext`lSeq, \ $CellContext`jProbsFlat, $CellContext`cSetMat, $CellContext`lbVec, \ $CellContext`ubVec}, $CellContext`nRow = Part[ Dimensions[$CellContext`jProbs], 1]; $CellContext`nCol = Part[ Dimensions[$CellContext`jProbs], 2]; $CellContext`jProbsFlat = Flatten[$CellContext`jProbs]; $CellContext`lbVec = Part[$CellContext`jProbsFlat, $CellContext`lOrd]; Do[Part[$CellContext`lbVec, $CellContext`i] = Part[$CellContext`lbVec, $CellContext`i] + Part[$CellContext`lbVec, $CellContext`i - 1], {$CellContext`i, 2, Part[ Dimensions[$CellContext`lbVec], 1]}]; $CellContext`ubVec = Join[{ Apply[Plus, Part[$CellContext`jProbs, 1, All]]}, Part[$CellContext`jProbsFlat, $CellContext`lOrd + \ $CellContext`nCol]]; $CellContext`ubVec = Part[$CellContext`ubVec, Span[1, Part[ Dimensions[$CellContext`ubVec], 1] - 1]]; Do[Part[$CellContext`ubVec, $CellContext`i] = Part[$CellContext`ubVec, $CellContext`i] + Part[$CellContext`ubVec, $CellContext`i - 1], {$CellContext`i, 2, Part[ Dimensions[$CellContext`ubVec], 1]}]; $CellContext`lSeq = ($CellContext`nRow - 1) $CellContext`nCol + 2; $CellContext`cSetMat = Array[0& , {$CellContext`lSeq, $CellContext`lSeq}]; Part[$CellContext`cSetMat, 1] = Join[{0}, $CellContext`lbVec, {1}]; Part[$CellContext`cSetMat, All, $CellContext`lSeq] = Join[{1}, 1 - $CellContext`ubVec, {0}]; Return[$CellContext`cSetMat]], $CellContext`eSubSetMat[ Pattern[$CellContext`jProbs, Blank[]], Pattern[$CellContext`lOrd, Blank[]]] := Module[{$CellContext`M, K, $CellContext`Q, $CellContext`dIdx, $CellContext`lIdx, \ $CellContext`pIdx, $CellContext`lMat, $CellContext`pMat, \ $CellContext`eSetSub}, $CellContext`M = Part[ Dimensions[$CellContext`jProbs], 1]; K = Part[ Dimensions[$CellContext`jProbs], 2]; $CellContext`lIdx = \ $CellContext`lambdaMatIdxDef[$CellContext`lOrd, $CellContext`M, K]; $CellContext`pIdx = \ $CellContext`piMatIdxDef[$CellContext`lOrd, $CellContext`M, K]; $CellContext`lMat = \ $CellContext`lambdaMatDef[$CellContext`jProbs, $CellContext`lOrd, \ $CellContext`M, K]; $CellContext`pMat = \ $CellContext`piMatDef[$CellContext`jProbs, $CellContext`lOrd, $CellContext`M, K]; $CellContext`Q = ($CellContext`M - 1) K; $CellContext`eSetSub = Array[0& , {$CellContext`Q, $CellContext`Q}]; Do[$CellContext`dIdx = Part[$CellContext`lIdx, $CellContext`p] - Part[$CellContext`pIdx, $CellContext`s]; If[Part[$CellContext`dIdx, $CellContext`k] > 0, Part[$CellContext`eSetSub, $CellContext`s, $CellContext`p] = Part[$CellContext`eSetSub, $CellContext`s, $CellContext`p] + Part[$CellContext`lMat, $CellContext`p, $CellContext`k] - Part[$CellContext`pMat, $CellContext`s, $CellContext`k]]; Null, {$CellContext`k, 1, K}, {$CellContext`s, 1, $CellContext`Q}, {$CellContext`p, $CellContext`s, \ $CellContext`Q}]; Return[$CellContext`eSetSub]], $CellContext`lambdaMatIdxDef[ Pattern[$CellContext`lOrd, Blank[]], Pattern[$CellContext`M, Blank[]], Pattern[K, Blank[]]] := Module[{$CellContext`Q, $CellContext`lMat, $CellContext`rowTmp}, \ $CellContext`Q = ($CellContext`M - 1) K; $CellContext`rowTmp = Array[0& , K]; $CellContext`lMat = {}; Do[Part[$CellContext`rowTmp, $CellContext`kappa[$CellContext`q, $CellContext`lOrd, \ $CellContext`M, K]] = $CellContext`q; $CellContext`lMat = Append[$CellContext`lMat, $CellContext`rowTmp]; Null, {$CellContext`q, 1, $CellContext`Q}]; Return[$CellContext`lMat]], $CellContext`piMatIdxDef[ Pattern[$CellContext`lOrd, Blank[]], Pattern[$CellContext`M, Blank[]], Pattern[K, Blank[]]] := Module[{$CellContext`Q, $CellContext`lMat, $CellContext`rowTmp}, \ $CellContext`Q = ($CellContext`M - 1) K; $CellContext`rowTmp = Array[$CellContext`Q + #& , K]; $CellContext`lMat = {}; Do[Part[$CellContext`rowTmp, $CellContext`kappa[$CellContext`Q + 1 - $CellContext`q, $CellContext`lOrd, $CellContext`M, K]] = $CellContext`Q + 1 - $CellContext`q; $CellContext`lMat = Prepend[$CellContext`lMat, $CellContext`rowTmp]; Null, {$CellContext`q, 1, $CellContext`Q}]; Return[$CellContext`lMat]], $CellContext`lambdaMatDef[ Pattern[$CellContext`jProbs, Blank[]], Pattern[$CellContext`lOrd, Blank[]], Pattern[$CellContext`M, Blank[]], Pattern[K, Blank[]]] := Module[{$CellContext`Q, $CellContext`lMat, $CellContext`rowTmp, \ $CellContext`jProbsCm}, $CellContext`jProbsCm = Accumulate[$CellContext`jProbs]; $CellContext`Q = ($CellContext`M - 1) K; $CellContext`rowTmp = Array[0& , K]; $CellContext`lMat = {}; Do[Part[$CellContext`rowTmp, $CellContext`kappa[$CellContext`q, $CellContext`lOrd, \ $CellContext`M, K]] = Part[$CellContext`jProbsCm, $CellContext`mu[$CellContext`q, $CellContext`lOrd, \ $CellContext`M, K], $CellContext`kappa[$CellContext`q, $CellContext`lOrd, \ $CellContext`M, K]]; $CellContext`lMat = Append[$CellContext`lMat, $CellContext`rowTmp]; Null, {$CellContext`q, 1, $CellContext`Q}]; Return[$CellContext`lMat]], $CellContext`piMatDef[ Pattern[$CellContext`jProbs, Blank[]], Pattern[$CellContext`lOrd, Blank[]], Pattern[$CellContext`M, Blank[]], Pattern[K, Blank[]]] := Module[{$CellContext`Q, $CellContext`lMat, $CellContext`rowTmp, \ $CellContext`jProbsCm}, $CellContext`jProbsCm = Accumulate[$CellContext`jProbs]; $CellContext`Q = ($CellContext`M - 1) K; $CellContext`rowTmp = Part[$CellContext`jProbsCm, $CellContext`M, All]; $CellContext`lMat = {}; Do[Part[$CellContext`rowTmp, $CellContext`kappa[$CellContext`Q + 1 - $CellContext`q, $CellContext`lOrd, $CellContext`M, K]] = Part[$CellContext`jProbsCm, $CellContext`mu[$CellContext`Q + 1 - $CellContext`q, $CellContext`lOrd, $CellContext`M, K], $CellContext`kappa[$CellContext`Q + 1 - $CellContext`q, $CellContext`lOrd, $CellContext`M, K]]; $CellContext`lMat = Prepend[$CellContext`lMat, $CellContext`rowTmp]; Null, {$CellContext`q, 1, $CellContext`Q}]; Return[$CellContext`lMat]], $CellContext`genGamFromSet[ Pattern[$CellContext`set, Blank[]], Optional[ Pattern[$CellContext`seed, Blank[]], Null]] := Module[{$CellContext`len, $CellContext`gVec, $CellContext`ubVec, \ $CellContext`lbVec, $CellContext`lb, $CellContext`ub, $CellContext`range, \ $CellContext`den, $CellContext`lOrdBack}, If[ IntegerQ[$CellContext`seed], SeedRandom[$CellContext`seed]]; $CellContext`len = Part[ Dimensions[$CellContext`set], 1]; $CellContext`gVec = Array[0& , $CellContext`len]; $CellContext`ubVec = Part[$CellContext`set, All, $CellContext`len]; Do[$CellContext`lbVec = Part[$CellContext`set, All, $CellContext`i + 1]; $CellContext`lb = Max[$CellContext`gVec + $CellContext`lbVec]; $CellContext`ub = 1 - Part[$CellContext`ubVec, $CellContext`i + 1]; $CellContext`den = Denominator[$CellContext`lb] Denominator[$CellContext`ub]; $CellContext`range = { Numerator[$CellContext`lb] Denominator[$CellContext`ub], Numerator[$CellContext`ub] Denominator[$CellContext`lb]}; Part[$CellContext`gVec, $CellContext`i + 1] = RandomInteger[$CellContext`range]/$CellContext`den; Null, {$CellContext`i, 1, $CellContext`len - 2}]; $CellContext`gVec = Part[$CellContext`gVec, Span[2, $CellContext`len - 1]]; Return[$CellContext`gVec]], $CellContext`lOrdBackDef[ Pattern[$CellContext`lOrd, Blank[]]] := Module[{$CellContext`lOrdBack}, $CellContext`lOrdBack = \ $CellContext`lOrd; Do[Part[$CellContext`lOrdBack, Part[$CellContext`lOrd, $CellContext`i]] = $CellContext`i, \ {$CellContext`i, 1, Length[$CellContext`lOrd]}]; Return[$CellContext`lOrdBack]; Null], $CellContext`lOrdFun[ Pattern[$CellContext`gMat, Blank[]]] := Module[{$CellContext`res, $CellContext`ord}, $CellContext`ord = Ordering[ Flatten[$CellContext`gMat]]; Return[ $CellContext`lOrdBackDef[$CellContext`ord]]], $CellContext`matToLoc[ Pattern[$CellContext`gMat, Blank[]]] := Module[{$CellContext`M, K, $CellContext`pts}, $CellContext`M = Part[ Dimensions[$CellContext`gMat], 1] + 1; K = Part[ Dimensions[$CellContext`gMat], 2]; $CellContext`pts = Transpose[{ $CellContext`supFun[$CellContext`M, K], Flatten[$CellContext`gMat]}]; Return[$CellContext`pts]], $CellContext`supFun[ Pattern[$CellContext`M, Blank[]], Pattern[K, Blank[]]] := Module[{$CellContext`sup}, $CellContext`sup = Flatten[ Array[Array[#/(K + 1)& , K]& , $CellContext`M - 1]]; Return[$CellContext`sup]], $CellContext`locFun[ Pattern[$CellContext`loc, Blank[]], Pattern[$CellContext`M, Blank[]], Pattern[K, Blank[]]] := Module[{$CellContext`ptVar, $CellContext`tmp}, $CellContext`tmp = Part[$CellContext`loc, All, 2]; $CellContext`ptVar = Transpose[{ $CellContext`supFun[$CellContext`M, K], $CellContext`tmp}]; Return[$CellContext`ptVar]], $CellContext`locToMat[ Pattern[$CellContext`loc, Blank[]], Pattern[$CellContext`M, Blank[]], Pattern[K, Blank[]]] := Module[{$CellContext`mat}, $CellContext`mat = Array[0& , {$CellContext`M - 1, K}]; Do[Part[$CellContext`mat, $CellContext`j, All] = Part[$CellContext`loc, Span[($CellContext`j - 1) K + 1, $CellContext`j K], 2], {$CellContext`j, 1, $CellContext`M - 1}]; Return[$CellContext`mat]], $CellContext`matMonotone[ Pattern[$CellContext`gMat, Blank[]]] := Module[{$CellContext`M, K, $CellContext`tmp, $CellContext`mat, $CellContext`ord}, \ $CellContext`M = Part[ Dimensions[$CellContext`gMat], 1]; K = Part[ Dimensions[$CellContext`gMat], 2]; $CellContext`mat = Array[0& , {$CellContext`M, K}]; Do[$CellContext`ord = Ordering[ Part[$CellContext`gMat, All, $CellContext`i]]; Part[$CellContext`mat, All, $CellContext`i] = Part[$CellContext`gMat, $CellContext`ord, $CellContext`i], \ {$CellContext`i, 1, K}]; Return[$CellContext`mat]], $CellContext`descriptionFun[ Pattern[$CellContext`type, Blank[]]] := Module[{}, If[$CellContext`type == "Arg", Grid[{{ Style[ " \[CenterDot] \!\(\*SubscriptBox[\(\[Gamma]\), \(i\)]\) \ signifies the object \!\(\*SubscriptBox[\(\[Gamma]\), \([i]\)]\) in the \ paper,", TextAlignment -> Left]}, { Style[ " \[CenterDot] 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