(* Content-type: application/mathematica *) (*** Wolfram Notebook File ***) (* http://www.wolfram.com/nb *) (* CreatedBy='Mathematica 7.0' *) (*CacheID: 234*) (* Internal cache information: NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 145, 7] NotebookDataLength[ 54982, 1488] NotebookOptionsPosition[ 50567, 1343] NotebookOutlinePosition[ 51057, 1362] CellTagsIndexPosition[ 51014, 1359] WindowFrame->Normal*) (* Beginning of Notebook Content *) Notebook[{ Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"2", "-", "2"}]], "Input", CellChangeTimes->{{3.5541326395864162`*^9, 3.554132639860406*^9}}], Cell[BoxData["0"], "Output", CellChangeTimes->{3.554132640905608*^9, 3.568746737547873*^9}] }, Open ]], Cell[BoxData["\[IndentingNewLine]"], "Input", CellChangeTimes->{3.554132645874547*^9}], Cell[CellGroupData[{ Cell["Ray's Argument", "Title", CellChangeTimes->{{3.554132651007717*^9, 3.554132654989085*^9}}], Cell["\<\ Nicholas Wheeler 16 August 2012\ \>", "Text", CellChangeTimes->{{3.5541326597673063`*^9, 3.554132672998567*^9}}, FontSize->10], Cell[BoxData["\[IndentingNewLine]"], "Input", CellChangeTimes->{3.5541326824047194`*^9}], Cell[CellGroupData[{ Cell["Introduction", "Subsection", CellChangeTimes->{{3.554132688887019*^9, 3.554132691142321*^9}}], Cell[TextData[{ "\"Partitons of unity\" (notebook dated 12 August) begins with discussion of \ random ", StyleBox["bisections", FontSlant->"Italic"], " of unity, and exposes an unanticipated problem: I found that two equally \ \"natural\" approaches to the production of random bisections led to \ populations that were statistically distinct. 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Look first to trisections. \ Write\ \>", "Text", CellChangeTimes->{{3.5542154218882847`*^9, 3.554215461658319*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"{", RowBox[{ FractionBox["x", RowBox[{"x", "+", "y", "+", "z"}]], ",", FractionBox["y", RowBox[{"x", "+", "y", "+", "z"}]], ",", FractionBox["z", RowBox[{"x", "+", "y", "+", "z"}]]}], "}"}], "=", RowBox[{"{", RowBox[{"\[Alpha]", ",", "\[Beta]", ",", RowBox[{"\[Gamma]", "=", RowBox[{"1", "-", "\[Alpha]", "-", "\[Beta]"}]}]}], "}"}]}]], "Output", CellChangeTimes->{{3.5542154696277647`*^9, 3.554215552883943*^9}}], Cell["\<\ The points {x, y, z} that contribute to a specified value of \[Alpha] lie on \ a plane (actually: on the part of a plane that lies interior to the unit \ cube):\ \>", "Text", CellChangeTimes->{{3.554215616051462*^9, 3.554215698946542*^9}, { 3.554215965289443*^9, 3.5542159701826067`*^9}}], Cell[BoxData[ RowBox[{ RowBox[{ RowBox[{ RowBox[{"(", RowBox[{"\[Alpha]", "-", "1"}], ")"}], "x"}], "+", RowBox[{"\[Alpha]", " ", "y"}], "+", RowBox[{"\[Alpha]", " ", "z"}]}], "\[Equal]", "0"}]], "Output"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Manipulate", "[", RowBox[{ RowBox[{"ContourPlot3D", "[", RowBox[{ RowBox[{ RowBox[{ RowBox[{ RowBox[{"(", RowBox[{"\[Alpha]", "-", "1"}], ")"}], "x"}], "+", RowBox[{"\[Alpha]", " ", "y"}], "+", RowBox[{"\[Alpha]", " ", "z"}]}], "\[Equal]", "0"}], ",", RowBox[{"{", RowBox[{"x", ",", "0", ",", "1"}], "}"}], ",", RowBox[{"{", RowBox[{"y", ",", "0", ",", "1"}], "}"}], ",", RowBox[{"{", RowBox[{"z", ",", "0", ",", "1"}], "}"}]}], "]"}], ",", RowBox[{"{", RowBox[{"\[Alpha]", ",", "0.1", ",", " ", "0.9"}], "}"}]}], "]"}]], "Input", CellChangeTimes->{{3.554215739163143*^9, 3.554215747385857*^9}}], Cell[BoxData[ TagBox[ StyleBox[ DynamicModuleBox[{$CellContext`\[Alpha]$$ = 0.4, Typeset`show$$ = True, Typeset`bookmarkList$$ = {}, Typeset`bookmarkMode$$ = "Menu", Typeset`animator$$, Typeset`animvar$$ = 1, Typeset`name$$ = "\"untitled\"", Typeset`specs$$ = {{ Hold[$CellContext`\[Alpha]$$], 0.1, 0.9}}, Typeset`size$$ = { 540., {296., 306.}}, Typeset`update$$ = 0, Typeset`initDone$$, Typeset`skipInitDone$$ = True, $CellContext`\[Alpha]$894$$ = 0}, DynamicBox[Manipulate`ManipulateBoxes[ 1, StandardForm, "Variables" :> {$CellContext`\[Alpha]$$ = 0.1}, "ControllerVariables" :> { Hold[$CellContext`\[Alpha]$$, $CellContext`\[Alpha]$894$$, 0]}, "OtherVariables" :> { Typeset`show$$, Typeset`bookmarkList$$, Typeset`bookmarkMode$$, Typeset`animator$$, Typeset`animvar$$, Typeset`name$$, Typeset`specs$$, Typeset`size$$, Typeset`update$$, Typeset`initDone$$, Typeset`skipInitDone$$}, "Body" :> ContourPlot3D[($CellContext`\[Alpha]$$ - 1) $CellContext`x + $CellContext`\[Alpha]$$ $CellContext`y + \ $CellContext`\[Alpha]$$ $CellContext`z == 0, {$CellContext`x, 0, 1}, {$CellContext`y, 0, 1}, {$CellContext`z, 0, 1}], "Specifications" :> {{$CellContext`\[Alpha]$$, 0.1, 0.9}}, "Options" :> {}, "DefaultOptions" :> {}], ImageSizeCache->{602., {380.03125, 387.96875}}, SingleEvaluation->True], Deinitialization:>None, DynamicModuleValues:>{}, SynchronousInitialization->True, UnsavedVariables:>{Typeset`initDone$$}, UntrackedVariables:>{Typeset`size$$}], "Manipulate", Deployed->True, StripOnInput->False], Manipulate`InterpretManipulate[1]]], "Output", CellChangeTimes->{ 3.5542157864717712`*^9, {3.568747204262323*^9, 3.568747221703288*^9}}] }, Open ]], Cell[TextData[{ "Ditto the points that contribute to a specified value of \[Beta]. The \ points that contribute to {\[Alpha], \[Beta]}\[LongDash]which together assign \ value to \[Gamma] (note that the requirement 0 < \[Gamma] < 1 serves to \ constrain the values that can be assigned to \[Alpha] and \[Beta]) lie \ therefore on the ", StyleBox["intersection", FontWeight->"Bold"], " of two planes: " }], "Text", CellChangeTimes->{{3.5542159032226343`*^9, 3.554215952172039*^9}, { 3.5542159860655317`*^9, 3.554216244337576*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Manipulate", "[", RowBox[{ RowBox[{"ContourPlot3D", "[", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{ RowBox[{ RowBox[{ RowBox[{"(", RowBox[{"\[Alpha]", "-", "1"}], ")"}], "x"}], "+", RowBox[{"\[Alpha]", " ", "y"}], "+", RowBox[{"\[Alpha]", " ", "z"}]}], "\[Equal]", "0"}], ",", RowBox[{ RowBox[{ RowBox[{"\[Beta]", " ", "x"}], "+", RowBox[{ RowBox[{"(", RowBox[{"\[Beta]", "-", "1"}], ")"}], " ", "y"}], "+", RowBox[{"\[Beta]", " ", "z"}]}], "\[Equal]", "0"}]}], "}"}], ",", RowBox[{"{", RowBox[{"x", ",", "0", ",", "1"}], "}"}], ",", RowBox[{"{", RowBox[{"y", ",", "0", ",", "1"}], "}"}], ",", RowBox[{"{", RowBox[{"z", ",", "0", ",", "1"}], "}"}]}], "]"}], ",", RowBox[{"{", RowBox[{"\[Alpha]", ",", "0.1", ",", "0.9"}], "}"}], ",", RowBox[{"{", RowBox[{"\[Beta]", ",", "0.1", ",", "0.9"}], "}"}]}], "]"}]], "Input", CellChangeTimes->{{3.554166905001853*^9, 3.554167013422689*^9}, { 3.554216451939024*^9, 3.5542164660667543`*^9}}], Cell[BoxData[ TagBox[ StyleBox[ DynamicModuleBox[{$CellContext`\[Alpha]$$ = 0.423, $CellContext`\[Beta]$$ = 0.308, Typeset`show$$ = True, Typeset`bookmarkList$$ = {}, Typeset`bookmarkMode$$ = "Menu", Typeset`animator$$, Typeset`animvar$$ = 1, Typeset`name$$ = "\"untitled\"", Typeset`specs$$ = {{ Hold[$CellContext`\[Alpha]$$], 0.1, 0.9}, { Hold[$CellContext`\[Beta]$$], 0.1, 0.9}}, Typeset`size$$ = { 540., {296., 306.}}, Typeset`update$$ = 0, Typeset`initDone$$, Typeset`skipInitDone$$ = True, $CellContext`\[Alpha]$1172$$ = 0, $CellContext`\[Beta]$1173$$ = 0}, DynamicBox[Manipulate`ManipulateBoxes[ 1, StandardForm, "Variables" :> {$CellContext`\[Alpha]$$ = 0.1, $CellContext`\[Beta]$$ = 0.1}, "ControllerVariables" :> { Hold[$CellContext`\[Alpha]$$, $CellContext`\[Alpha]$1172$$, 0], Hold[$CellContext`\[Beta]$$, $CellContext`\[Beta]$1173$$, 0]}, "OtherVariables" :> { Typeset`show$$, Typeset`bookmarkList$$, Typeset`bookmarkMode$$, Typeset`animator$$, Typeset`animvar$$, Typeset`name$$, Typeset`specs$$, Typeset`size$$, Typeset`update$$, Typeset`initDone$$, Typeset`skipInitDone$$}, "Body" :> ContourPlot3D[{($CellContext`\[Alpha]$$ - 1) $CellContext`x + $CellContext`\[Alpha]$$ $CellContext`y + \ $CellContext`\[Alpha]$$ $CellContext`z == 0, $CellContext`\[Beta]$$ $CellContext`x + ($CellContext`\[Beta]$$ - 1) $CellContext`y + $CellContext`\[Beta]$$ $CellContext`z == 0}, {$CellContext`x, 0, 1}, {$CellContext`y, 0, 1}, {$CellContext`z, 0, 1}], "Specifications" :> {{$CellContext`\[Alpha]$$, 0.1, 0.9}, {$CellContext`\[Beta]$$, 0.1, 0.9}}, "Options" :> {}, "DefaultOptions" :> {}], ImageSizeCache->{602., {381.03125, 388.96875}}, SingleEvaluation->True], Deinitialization:>None, DynamicModuleValues:>{}, SynchronousInitialization->True, UnsavedVariables:>{Typeset`initDone$$}, UntrackedVariables:>{Typeset`size$$}], "Manipulate", Deployed->True, StripOnInput->False], Manipulate`InterpretManipulate[1]]], "Output", CellChangeTimes->{3.5541670214827642`*^9, 3.554216471387617*^9, 3.568747255480564*^9}] }, Open ]], Cell[TextData[{ "Evidently \n\nprob[\[Alpha] < a, given \[Beta]] = ", StyleBox["area swept out", FontWeight->"Bold"], " by the line of intersection as \[Alpha] advances from 0 to a\n\nprob[\ \[Beta] < b, given \[Alpha]] = ", StyleBox["area swept out", FontWeight->"Bold"], " by the line of intersection as \[Beta] advances from 0 to b\n\nconditional \ prob density[a givenb] = conditional prob density[b given a] = ", StyleBox["length of the line of intersection", FontWeight->"Bold"], " at {a, b}\n\nOne must, however, be careful to ", StyleBox["avoid impossible situations", FontVariations->{"Underline"->True}], " (a + b > 1), which are indicated by the disappearance of the 3rd plane \ from the following figure:" }], "Text", CellChangeTimes->{{3.554217142594677*^9, 3.554217197671523*^9}, { 3.5542172487568913`*^9, 3.554217343409107*^9}, {3.5542173827503653`*^9, 3.554217625859096*^9}, 3.554217689096929*^9}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Manipulate", "[", RowBox[{ RowBox[{"ContourPlot3D", "[", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{ RowBox[{ RowBox[{ RowBox[{"(", RowBox[{"\[Alpha]", "-", "1"}], ")"}], "x"}], "+", RowBox[{"\[Alpha]", " ", "y"}], "+", RowBox[{"\[Alpha]", " ", "z"}]}], "\[Equal]", "0"}], ",", RowBox[{ RowBox[{ RowBox[{"\[Beta]", " ", "x"}], "+", RowBox[{ RowBox[{"(", RowBox[{"\[Beta]", "-", "1"}], ")"}], " ", "y"}], "+", RowBox[{"\[Beta]", " ", "z"}]}], "\[Equal]", "0"}], ",", RowBox[{ RowBox[{ RowBox[{ RowBox[{"(", RowBox[{"1", "-", "\[Alpha]", "-", "\[Beta]"}], ")"}], " ", "x"}], "+", RowBox[{ RowBox[{"(", RowBox[{ RowBox[{"(", RowBox[{"1", "-", "\[Alpha]", "-", "\[Beta]"}], ")"}], "-", "1"}], ")"}], " ", "y"}], "+", RowBox[{ RowBox[{"(", RowBox[{"1", "-", "\[Alpha]", "-", "\[Beta]"}], ")"}], " ", "z"}]}], "\[Equal]", "0"}]}], "}"}], ",", RowBox[{"{", RowBox[{"x", ",", "0", ",", "1"}], "}"}], ",", RowBox[{"{", RowBox[{"y", ",", "0", ",", "1"}], "}"}], ",", RowBox[{"{", RowBox[{"z", ",", "0", ",", "1"}], "}"}]}], "]"}], ",", RowBox[{"{", RowBox[{"\[Alpha]", ",", "0.1", ",", "0.9"}], "}"}], ",", RowBox[{"{", RowBox[{"\[Beta]", ",", "0.1", ",", "0.9"}], "}"}]}], "]"}]], "Input", CellChangeTimes->{{3.554216360728382*^9, 3.554216443980692*^9}, { 3.5542164826782923`*^9, 3.554216485563203*^9}}], Cell[BoxData[ TagBox[ StyleBox[ DynamicModuleBox[{$CellContext`\[Alpha]$$ = 0.429, $CellContext`\[Beta]$$ = 0.524, Typeset`show$$ = True, Typeset`bookmarkList$$ = {}, Typeset`bookmarkMode$$ = "Menu", Typeset`animator$$, Typeset`animvar$$ = 1, Typeset`name$$ = "\"untitled\"", Typeset`specs$$ = {{ Hold[$CellContext`\[Alpha]$$], 0.1, 0.9}, { Hold[$CellContext`\[Beta]$$], 0.1, 0.9}}, Typeset`size$$ = { 540., {296., 306.}}, Typeset`update$$ = 0, Typeset`initDone$$, Typeset`skipInitDone$$ = True, $CellContext`\[Alpha]$7867$$ = 0, $CellContext`\[Beta]$7868$$ = 0}, DynamicBox[Manipulate`ManipulateBoxes[ 1, StandardForm, "Variables" :> {$CellContext`\[Alpha]$$ = 0.1, $CellContext`\[Beta]$$ = 0.1}, "ControllerVariables" :> { Hold[$CellContext`\[Alpha]$$, $CellContext`\[Alpha]$7867$$, 0], Hold[$CellContext`\[Beta]$$, $CellContext`\[Beta]$7868$$, 0]}, "OtherVariables" :> { Typeset`show$$, Typeset`bookmarkList$$, Typeset`bookmarkMode$$, Typeset`animator$$, Typeset`animvar$$, Typeset`name$$, Typeset`specs$$, Typeset`size$$, Typeset`update$$, Typeset`initDone$$, Typeset`skipInitDone$$}, "Body" :> ContourPlot3D[{($CellContext`\[Alpha]$$ - 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