(************** 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[ 142692, 4585]*) (*NotebookOutlinePosition[ 143339, 4607]*) (* CellTagsIndexPosition[ 143295, 4603]*) (*WindowFrame->Normal*) Notebook[{ Cell[BoxData[{ \(\(Array1[i_, \ D_, N_, \ \[Alpha]_]\ = \ \(D\ + \ i\ - \ 1\)\/D - \ N\/D + \ \(N\ \[Alpha]\)\/D;\)\), "\[IndentingNewLine]", \(\(Array2[i_, \ D_, N_, \ \[Alpha]_]\ = \ \(D\ + \ i\)\/D + \ \(N\ \ \[Alpha]\)\/D;\)\), "\[IndentingNewLine]", \(\(FActual[N_, \ \[Lambda]_, \ D_, \ \[Alpha]_]\ := \ Binomial[N, \ \[Alpha]\ N]\ - 4\ \(\((\(-1\))\)\^\(D\ - \ 1\)\) \((Sin[\[Pi]\ \((1\ + \ N\ - \ N\ \[Alpha])\)]\ 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Maybe integrating the \ log won't be so bad.\ \>", "Text"], Cell[BoxData[{ \(\(GetStartPoint[index_, \ coord_, \ d_]\ := If\ [Mod[index, \ 2] \[Equal] 1, \ If[coord\ \[Equal] \ 1, \ \ d\ \((\(index - 1\)\/2)\), \ \(-\ d\)\ \((\(index - 1\)\/2\ )\)], \ If[coord\ \[Equal] \ 1, \(-\ \ d\)\ \((index\/2\ )\), \ \ d\ \((index\/2\ )\)]\ ];\)\ \), "\[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]", \(\(Actual[alpha_, \ d_, \ p_] := \ \(1\/\(d\ p\)\) Log[Det[\ \((AllConfigs[p, \ d\ p, D, \ alpha\ ]\ )\)]];\)\)}], "Input"], Cell[BoxData[ \(\(\(\[IndentingNewLine]\)\(\(GetStartPoint[index_, \ coord_, \ d_]\ := If\ [Mod[index, \ 2] \[Equal] 1, \ If[coord\ \[Equal] \ 1, \ \ d\ \((\(index - 1\)\/2)\), \ \(-\ d\)\ \((\(index - 1\)\/2\ )\)], \ If[coord\ \[Equal] \ 1, \(-\ \ d\)\ \((index\/2\ )\), \ \ d\ \((index\/2\ )\)]\ ];\)\ \[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] \(Actual[alpha_, \ d_, \ p_] := \ \(1\/p\) Log[Det[\ \((AllConfigs[p, \ d\ p, d, \ alpha\ ]\ )\)]];\)\)\)\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(\(\(\[IndentingNewLine]\)\(\(Num\ = 26;\)\[IndentingNewLine] \(Dst\ = 2;\)\[IndentingNewLine] \(\[Alpha]\ = \ 1\/2;\)\[IndentingNewLine] NIntegrate[\(1\/\(2\ \[Pi]\)\) Log[\(\(2\ \@\(2\ \[Pi]\ Num\)\)\/Num\) \(\((\[Alpha]\ Num)\)\^\(\(-2\ \)\ \[Alpha]\ Num\)\) \(Num\^Num\) \(\[ExponentialE]\^\(\(-\ \[Lambda]\^2\)\/\ \(\(\ \)\(4 \((Dst\^2\/\(\[Alpha]\ Num\))\)\)\)\)\) \(\[Sum]\+\(x = \ \(-200\)\)\%200 \[ExponentialE]\^\(\(-\((Dst\^2\/\(\[Alpha]\ Num\))\)\) \((x \ - 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If your integrand is oscillatory try using the option \ Method->Oscillatory in NIntegrate. \ \\!\\(\\*ButtonBox[\\\"More\[Ellipsis]\\\", ButtonStyle->\\\"RefGuideLinkText\ \\\", ButtonFrame->None, ButtonData:>\\\"NIntegrate::slwcon\\\"]\\)\"\>"}]], \ "Message"], Cell[BoxData[ RowBox[{\(NIntegrate::"ncvb"\), \(\(:\)\(\ \)\), "\<\"NIntegrate failed \ to converge to prescribed accuracy after \\!\\(7\\) recursive bisections in \ \\!\\(\[Lambda]\\) near \\!\\(\[Lambda]\\) = \\!\\(-2.7243498792848984`\\). \ \\!\\(\\*ButtonBox[\\\"More\[Ellipsis]\\\", ButtonStyle->\\\"RefGuideLinkText\ \\\", ButtonFrame->None, ButtonData:>\\\"NIntegrate::ncvb\\\"]\\)\"\>"}]], \ "Message"], Cell[BoxData[ \(\(\(90.40732004642165`\)\(\[InvisibleSpace]\)\) + 0.005263724252184172`\ \[ImaginaryI]\)], "Output"], Cell[BoxData[ \(88.52708242337967`\)], "Output"], Cell[BoxData[ \(88.52708242337967`\)], "Output"], Cell[BoxData[ \(\(\(18.276093658504237`\)\(\[InvisibleSpace]\)\) + 0.`\ \[ImaginaryI]\)], "Output"], Cell[BoxData[ \(16.46344140234418`\)], "Output"] }, Open ]], Cell["\<\ Great! It looks like the approximation only differs by the same small amount \ even for large numbers--this indicates that once I apply the additional \ \"area-wise\" factor, these will converge to the same number at infinity. Now I just need to find a generating function for this series, since I do \ notice that you can't just drop the result past a certain point: note that I \ had to incease the range on the sum each time. \ \>", "Text"] }, FrontEndVersion->"5.0 for Microsoft Windows", ScreenRectangle->{{0, 1600}, {0, 1091}}, WindowSize->{899, 740}, WindowMargins->{{315, Automatic}, {Automatic, 91}} ] (******************************************************************* Cached data follows. If you edit this Notebook file directly, not using Mathematica, you must remove the line containing CacheID at the top of the file. 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