(************** 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[ 111917, 3482]*) (*NotebookOutlinePosition[ 112566, 3504]*) (* CellTagsIndexPosition[ 112522, 3500]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell[BoxData[ \(A\ = \ Plot[d\ Log[2]\ - \ \(\[Pi]\^2\/24\) d\^3, \ {d, \ 0, \ 1}]\)], "Input"], Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart /Mabs { Mgmatrix idtransform Mtmatrix dtransform } bind def /Mabsadd { Mabs 3 -1 roll add 3 1 roll add exch } bind def %% Graphics %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10 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of the G-V matrix for N~P\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\(\(\[IndentingNewLine]\)\(AllConfigs[3, \ 12, \ 4, 1\/2]\)\)\)], "Input"], Cell[BoxData[ \({{924, 66, 66}, {66, 924, 0}, {66, 0, 924}}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[{ \(\(AC\ = \ AllConfigs[3, \ 12, \ 4, 1\/2];\)\), "\[IndentingNewLine]", \(\(AC[\([1]\)]\)[\([1]\)]\ \ \(AC[\([2]\)]\)[\([2]\)]\ \ \(AC[\([3]\)]\)[\([3]\)]\ \), \ "\[IndentingNewLine]", \(\(-\(AC[\([1]\)]\)[\([1]\)]\)\ \ \(AC[\([2]\)]\)[\([3]\)]\ \ \(AC[\([3]\)]\)[\([2]\)]\ \), \ "\[IndentingNewLine]", \(\(AC[\([1]\)]\)[\([2]\)]\ \ \(AC[\([2]\)]\)[\([3]\)]\ \ \(AC[\([3]\)]\)[\([1]\)]\ \), \ "\[IndentingNewLine]", \(\(-\(AC[\([1]\)]\)[\([2]\)]\)\ \ \(AC[\([2]\)]\)[\([1]\)]\ \ \(AC[\([3]\)]\)[\([3]\)]\ \), \ "\[IndentingNewLine]", \(\(AC[\([1]\)]\)[\([3]\)]\ \ \(AC[\([2]\)]\)[\([1]\)]\ \ \(AC[\([3]\)]\)[\([2]\)]\ \), \ "\[IndentingNewLine]", \(\(-\(AC[\([1]\)]\)[\([3]\)]\)\ \ \(AC[\([2]\)]\)[\([2]\)]\ \ \(AC[\([3]\)]\)[\([1]\)]\ \), \ "\[IndentingNewLine]", \(Det[AC]\)}], "Input"], Cell[BoxData[ \(788889024\)], "Output"], Cell[BoxData[ \(0\)], "Output"], Cell[BoxData[ \(0\)], "Output"], Cell[BoxData[ \(\(-4024944\)\)], "Output"], Cell[BoxData[ \(0\)], "Output"], Cell[BoxData[ \(\(-4024944\)\)], "Output"], Cell[BoxData[ \(780839136\)], "Output"] }, Open ]], Cell["\<\ My \"approximate\" behavior\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\(\(\[IndentingNewLine]\)\(APX1\ = \(AC[\([1]\)]\)[\([1]\)]\ \ \(AC[\([1]\)]\)[\([1]\)]\ \(AC[\([1]\)]\)[\([1]\)]\[IndentingNewLine] APX2\ = N[\(-\ \(AC[\([1]\)]\)[\([1]\)]\)\ \(AC[\([1]\)]\)[\([1]\)]\ \ \(\(AC[\([1]\)]\)[\([2]\)]\ \(AC[\([1]\)]\)[\([2]\)]\)\/\(AC[\([1]\)]\)[\([1]\ \)]]\[IndentingNewLine] APX3\ = N[\(AC[\([1]\)]\)[\([1]\)]\ \(\(AC[\([1]\)]\)[\([2]\)]\ \ \(AC[\([1]\)]\)[\([2]\)]\)\/\(AC[\([1]\)]\)[\([1]\)]\ \(\(AC[\([1]\)]\)[\([2]\ \)]\ \(AC[\([1]\)]\)[\([2]\)]\)\/\(AC[\([1]\)]\)[\([1]\)]]\[IndentingNewLine] APX4\ = \(-\ \(AC[\([1]\)]\)[\([1]\)]\)\ \(AC[\([1]\)]\)[\([1]\)]\ \ \(\(AC[\([1]\)]\)[\([2]\)]\ \(AC[\([1]\)]\)[\([2]\)]\)\/\(AC[\([1]\)]\)[\([1]\ \)]\[IndentingNewLine] APX5\ = \ \(AC[\([1]\)]\)[\([1]\)]\ \(AC[\([1]\)]\)[\([1]\)]\ \ \(\(AC[\([2]\)]\)[\([3]\)]\ \(AC[\([2]\)]\)[\([3]\)]\)\/\(AC[\([1]\)]\)[\([1]\ \)]\[IndentingNewLine] APX6\ = \(-\ \(AC[\([1]\)]\)[\([1]\)]\)\ \ \(\(\(AC[\([1]\)]\)[\([2]\)]\ \ \(AC[\([1]\)]\)[\([2]\)]\)\/\(AC[\([1]\)]\)[\([1]\)]\) \ \(\(AC[\([2]\)]\)[\([3]\)]\ \(AC[\([2]\)]\)[\([3]\)]\)\/\(AC[\([1]\)]\)[\([1]\ \)]\[IndentingNewLine] N[APX1\ + APX2\ + APX3\ + APX4\ + APX5\ + APX6]\)\)\)], "Input"], Cell[BoxData[ \(788889024\)], "Output"], Cell[BoxData[ \(\(-4.024944`*^6\)\)], "Output"], Cell[BoxData[ \(20535.428571428572`\)], "Output"], Cell[BoxData[ \(\(-4024944\)\)], "Output"], Cell[BoxData[ \(0\)], "Output"], Cell[BoxData[ \(0\)], "Output"], Cell[BoxData[ \(7.808596714285715`*^8\)], "Output"] }, Open ]], Cell["\<\ So at this point my approximation is a little low, but it's really not clear \ why it should definitely disagree substantially from the prediction\ \>", "Text"], Cell["\<\ Trying now to produce a nearest-neighbor approximation.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[{ \(Clear[R, \ RR]\), "\[IndentingNewLine]", \(\(\(GSP[index_, \ coord_]\ := \ If[coord\ \[Equal] \ 1, \ \((index\ - \ 1)\), \ \(-\ \((index\ - \ 1)\)\)];\);\)\), "\[IndentingNewLine]", \(\(MiniPathCount[i_, \ j_, \ d_, \ param_]\ := R[Abs[GSP[i, \ 1, d] - GSP[j, \ 1, d]]\ , \ d, \ param];\)\), "\[IndentingNewLine]", \(Matrix[p_, \ d_, \ param_]\ := \ Assuming[{Abs[d] == \ d}, Table[MiniPathCount[i, \ j\ , d, \ param]\ \ , \ {i, \ 1, \ p}, \ {j, \ 1, \ p}]]\[IndentingNewLine]\), "\[IndentingNewLine]", \(R[x_, d_, \ param_]\ = \ If[x\ < \ 2, \ RR[x, d, param], \ 0]\[IndentingNewLine]\), "\[IndentingNewLine]", \(Det[Matrix[3, 1, 1]]\ \), "\[IndentingNewLine]", \(Det[Matrix[5, 1, 1]]\), "\[IndentingNewLine]", \(Det[Matrix[7, 1, 1]]\)}], "Input"], Cell[BoxData[ \(If[x < 2, RR[x, d, param], 0]\)], "Output"], Cell[BoxData[ \(RR[0, 1, 1]\^3 - 2\ RR[0, 1, 1]\ RR[1, 1, 1]\^2\)], "Output"], Cell[BoxData[ \(RR[0, 1, 1]\^5 - 4\ RR[0, 1, 1]\^3\ RR[1, 1, 1]\^2 + 3\ RR[0, 1, 1]\ RR[1, 1, 1]\^4\)], "Output"], Cell[BoxData[ \(RR[0, 1, 1]\^7 - 6\ RR[0, 1, 1]\^5\ RR[1, 1, 1]\^2 + 10\ RR[0, 1, 1]\^3\ RR[1, 1, 1]\^4 - 4\ RR[0, 1, 1]\ RR[1, 1, 1]\^6\)], "Output"] }, Open ]], Cell["\<\ Before I get too advanced in attempting to produce the desired behavior here, \ let me see if this alone is sufficient to see the behavior I desire. \ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[{ \(Clear[RR]\), "\[IndentingNewLine]", \(\(RR[x_, d_, \ N_]\ = \ Binomial[N, \(1\/2\) N\ + \ x\ d];\)\), "\[IndentingNewLine]", \(DataApproximateNearest = N[\ Table[{1\/d, \ \(1\/\((41\ d)\)\^2\) Log[Det[Matrix[41, \ d, \ 41\ d]]]}, \ {d, 2, \ 22, 2}]]\)}], "Input"], Cell[BoxData[ \({{0.5`, 0.33174310441894506`}, {0.25`, 0.16905318393975013`}, {0.16666666666666666`, 0.11350591669361236`}, 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