(************** Content-type: application/mathematica ************** CreatedBy='Mathematica 5.0' Mathematica-Compatible Notebook This notebook can be used with any Mathematica-compatible application, such as Mathematica, MathReader or Publicon. The data for the notebook starts with the line containing stars above. To get the notebook into a Mathematica-compatible application, do one of the following: * Save the data starting with the line of stars above into a file with a name ending in .nb, then open the file inside the application; * Copy the data starting with the line of stars above to the clipboard, then use the Paste menu command inside the application. Data for notebooks contains only printable 7-bit ASCII and can be sent directly in email or through ftp in text mode. Newlines can be CR, LF or CRLF (Unix, Macintosh or MS-DOS style). 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For more information on notebooks and Mathematica-compatible applications, contact Wolfram Research: web: http://www.wolfram.com email: info@wolfram.com phone: +1-217-398-0700 (U.S.) Notebook reader applications are available free of charge from Wolfram Research. *******************************************************************) (*CacheID: 232*) (*NotebookFileLineBreakTest NotebookFileLineBreakTest*) (*NotebookOptionsPosition[ 226688, 6902]*) (*NotebookOutlinePosition[ 227332, 6924]*) (* CellTagsIndexPosition[ 227288, 6920]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell["The Gessel-Viennot Algorithm", "Subtitle"], Cell[CellGroupData[{ Cell["\<\ Notebook 5: Allow the Free Length, Separation to Grow \ \>", "Section"], Cell["\<\ Here I have created a plot of the results of the Gessel-Viennot algorithm for \ given lengths, separations, slope and path count. I let either length or \ separation vary.\ \>", "Text"] }, Open ]], Cell[CellGroupData[{ Cell["\<\ Generalized Case-n Paths \ \>", "Section"], Cell["\<\ Try 1: Classic example. 2 paths, 0.5 alpha, one separation This does appear to approach our limit, but rather slowly. Note the strange \"sawtooth\" effect. I wonder if this is just an artifact \ of incrementing the steps one at a time.\ \>", "Text"], Cell[BoxData[{ \(\(slopes\ = \ {0.5, \ 0.5};\)\), "\[IndentingNewLine]", \(\(startPoints\ = \ {{0, \ 0}, \ {1, \ \(-1\)}};\)\), "\[IndentingNewLine]", \(\(sizeMax\ = 5000;\)\)}], "Input"], Cell["\<\ This algorithm takes [paths] paths, and assigns each a long-term slope shown. \ It then finds how the entropy per unit changes with length. \ \>", "Text"], Cell[BoxData[{ \(\(GetStartPoint[index_, \ coord_, \ d_]\ := \ If[coord\ \[Equal] \ 1, \ d\ \((index\ - \ 1)\), \ \(-\ d\)\ \((index\ - \ 1)\)];\)\), "\[IndentingNewLine]", \(\(GetEndPoint[index_, \ coord_, length_, \ d_, \ slope_]\ := \ GetStartPoint[index, \ coord, \ d]\ + \ If[coord \[Equal] 1, \ length\ slope, \ length\ \((1\ - \ slope)\)];\)\), "\[IndentingNewLine]", \(\(PathCount[i_, \ j_, \ length_, \ d_, \ slope_]\ := Binomial[ GetEndPoint[j, \ 1, length, \ d, \ slope] - GetStartPoint[i, \ 1, d] + GetEndPoint[j, 2, length, d, slope] - GetStartPoint[i, \ 2, d]\ , GetEndPoint[j, \ 1, length, d, slope] - GetStartPoint[i, \ 1, d]\ ];\)\), "\[IndentingNewLine]", \(\(AllConfigs[count_, length_, \ d_, \ slope_]\ := \ Table[\ PathCount[i, \ j, \ length, \ d, \ slope], \ {i, \ 1, \ count}, \ {j, \ 1, \ count}];\)\), "\[IndentingNewLine]", \(\(dataGeneralLENGTH[sizeMax_, \ d_, \ slope_, \ count_] := \ Table[N[\(1\/sz\) Log[\ Det[\ \((AllConfigs[count, \ sz, d, \ slope\ ]\ )\)]]], {sz, \ 1, \ sizeMax}];\)\), "\[IndentingNewLine]", \(\(dataGeneralDIST[size_, \ dMax_, \ slope_, \ count_] := \ Table[N[\(1\/size\) Log[\ Det[\ \((AllConfigs[count, \ size, d, \ slope\ ]\ )\)]]], {d, \ 1, \ dMax}];\)\)}], "Input"], Cell["\<\ Two path, increasing length, separation 1. \ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(ListPlot[\(-1\)\ *\ dataGeneralLENGTH[500, \ 1, 0.5, \ 2], PlotJoined \[Rule] True]\)], "Input"], Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart /Mabs { Mgmatrix 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