(************** 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).

NOTE: If you modify the data for this notebook not in a Mathematica-
compatible application, you must delete the line below containing
the word CacheID, otherwise Mathematica-compatible applications may
try to use invalid cache data.

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[     17745,        581]*)
(*NotebookOutlinePosition[     18391,        603]*)
(*  CellTagsIndexPosition[     18347,        599]*)
(*WindowFrame->Normal*)



Notebook[{

Cell[CellGroupData[{
Cell["\<\


Gessel-Viennot Long Term #7\
\>", "Subsection"],

Cell["\<\

\"Actual value\" code\
\>", "Text"],

Cell[BoxData[
    \(\(\(\[IndentingNewLine]\)\(\(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]
    \(Actual[alpha_, \ N_, \ d_, \ paths_] := \(1\/N\) 
          Log[\ Det[\ \((AllConfigs[paths, \ N, d, \ 
                  alpha\ ]\ )\)]];\)\)\)\)], "Input"],

Cell["\<\
\"Approximate\" code
\
\>", "Text"],

Cell[BoxData[
    \(\(\(\[IndentingNewLine]\)\(Result[alpha_, \ N_, \ d_, \ 
          paths_]\  = \ \(1\/N\) 
          Log[\[Product]\+\(p = 1\)\%paths\((\[Sum]\+\(i = 1\)\%p\((\(\((\(-1\
\))\)\^\(i\  - \ 1\)\/Binomial[N, \ alpha\ N]\) \((Binomial[N, \ 
                            alpha\ N + \((i - 1)\)\ d]\ Binomial[N, \ 
                            alpha\ N - \((i - 
                                    1)\)\ d])\))\))\)];\)\)\)], "Input"],

Cell[CellGroupData[{

Cell[BoxData[{
    \(Actual[0.5, \ 15, \ 3, 14]\), "\[IndentingNewLine]", 
    \(Result[0.5, 15, \ 3, 14]\)}], "Input"],

Cell[BoxData[
    \(8.113043296714888`\)], "Output"],

Cell[BoxData[
    \(8.122742794902962`\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[{
    \(Actual[0.5, \ 30, \ 3, 29]\), "\[IndentingNewLine]", 
    \(Result[0.5, 30, \ 3, 29]\)}], "Input"],

Cell[BoxData[
    \(17.74494130771226`\)], "Output"],

Cell[BoxData[
    \(17.89449655414103`\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[{
    \(Actual[0.5, \ 100, \ 3, 99]\), "\[IndentingNewLine]", 
    \(Result[0.5, 100, \ 3, 99]\)}], "Input"],

Cell[BoxData[
    \(62.991991004617099462734657782`19.75387510561904\)], "Output"],

Cell[BoxData[
    \(65.43765457853309453611771755`18.65647414281004\)], "Output"]
}, Open  ]],

Cell["\<\
The conclusion here is that I may not let N and P go to infinity together.  \
The approximation becomes progressively less accurate.
\
\>", "Text"],

Cell[CellGroupData[{

Cell[BoxData[{
    \(Actual[0.5, \ 10\^2, \ 3, 10]\), "\[IndentingNewLine]", 
    \(Result[0.5, 10\^2, \ 3, 10]\)}], "Input"],

Cell[BoxData[
    \(6.448821400132036`\)], "Output"],

Cell[BoxData[
    \(6.611983257121648`\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[{
    \(Actual[0.5, \ 50\^2, \ 3, 50]\), "\[IndentingNewLine]", 
    \(Result[0.5, 50\^2, \ 3, 50]\)}], "Input"],

Cell[BoxData[
    \(34.1863271635388078998011002621`15.733325999602368\)], "Output"],

Cell[BoxData[
    \(34.558890028818634176865616861`14.163565811379105\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[{
    \(Actual[0.5, \ 150\^2, \ 3, 150]\), "\[IndentingNewLine]", 
    \(Result[0.5, 150\^2, \ 3, 150]\)}], "Input"],

Cell[BoxData[
    \(103.7928944932677371405742431207`15.73670060643549 + 
      0.0001396263401595463661537155`9.865500337640604\ \[ImaginaryI]\)], \
"Output"],

Cell[BoxData[
    \(103.9317578556414545213141100224`13.679277875240931\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(\(\(\[IndentingNewLine]\)\(Actual[0.5, \ 200\^2, \ 3, 
      200]\[IndentingNewLine]
    Result[0.5, 200\^2, \ 3, 200]\)\)\)], "Input"],

Cell[BoxData[
    \(138.4953841361072577621606468691`15.737384952739786 + 
      0.000078539816339744830961465`9.491039534912154\ \[ImaginaryI]\)], \
"Output"],

Cell[BoxData[
    \(138.597739143604802468318207059`13.573580908315517\)], "Output"]
}, Open  ]],

Cell["\<\

So it looks like N should be at least the square of P, in order to ensure \
convergence.  \
\>", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(Actual[0.5, \ 1000, 3, 1]\)], "Input"],

Cell[BoxData[
    \(0.6894672615678509`\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(N[Log[2]]\)], "Input"],

Cell[BoxData[
    \(0.6931471805599453`\)], "Output"]
}, Open  ]],

Cell["\<\

Series of the 2F2 function, where I have let D = 1\
\>", "Text"],

Cell[CellGroupData[{

Cell[BoxData[{
    \(Clear[a]\), "\[IndentingNewLine]", 
    \(FullSimplify[
      Series[\(1\/\(Gamma[a\ N]\ Gamma[\((1\  - \ a)\) N]\)\) 
          HypergeometricPFQRegularized[{1\  - \ \((1\  - a)\)\ N, \ 
              1\  - \ a\ N}, \ {a\ N, \ \((1\  - \ a)\)\ N}, \ \(-\ 
              z\)], \ {z, \ 0, \ 3}]]\)}], "Input"],

Cell[BoxData[
    InterpretationBox[
      RowBox[{\(1\/\(Gamma[a\ N]\^2\ Gamma[N - a\ N]\^2\)\), 
        "-", \(z\/\(Gamma[\(-1\) + N - a\ N]\ Gamma[
                1 + N - a\ N]\ Gamma[\(-1\) + a\ N]\ Gamma[1 + a\ N]\)\), 
        "+", \(z\^2\/\(2\ Gamma[\(-2\) + N - a\ N]\ Gamma[
                2 + N - a\ N]\ Gamma[\(-2\) + a\ N]\ Gamma[2 + a\ N]\)\), 
        "-", \(z\^3\/\(6\ \((Gamma[\(-3\) + N - a\ N]\ Gamma[
                    3 + N - a\ N]\ Gamma[\(-3\) + a\ N]\ Gamma[
                    3 + a\ N])\)\)\), "+", 
        InterpretationBox[\(O[z]\^4\),
          SeriesData[ z, 0, {}, 0, 4, 1],
          Editable->False]}],
      SeriesData[ z, 0, {
        Times[ 
          Power[ 
            Gamma[ 
              Times[ a, N]], -2], 
          Power[ 
            Gamma[ 
              Plus[ N, 
                Times[ -1, a, N]]], -2]], 
        Times[ -1, 
          Power[ 
            Gamma[ 
              Plus[ -1, N, 
                Times[ -1, a, N]]], -1], 
          Power[ 
            Gamma[ 
              Plus[ 1, N, 
                Times[ -1, a, N]]], -1], 
          Power[ 
            Gamma[ 
              Plus[ -1, 
                Times[ a, N]]], -1], 
          Power[ 
            Gamma[ 
              Plus[ 1, 
                Times[ a, N]]], -1]], 
        Times[ 
          Rational[ 1, 2], 
          Power[ 
            Gamma[ 
              Plus[ -2, N, 
                Times[ -1, a, N]]], -1], 
          Power[ 
            Gamma[ 
              Plus[ 2, N, 
                Times[ -1, a, N]]], -1], 
          Power[ 
            Gamma[ 
              Plus[ -2, 
                Times[ a, N]]], -1], 
          Power[ 
            Gamma[ 
              Plus[ 2, 
                Times[ a, N]]], -1]], 
        Times[ 
          Rational[ -1, 6], 
          Power[ 
            Gamma[ 
              Plus[ -3, N, 
                Times[ -1, a, N]]], -1], 
          Power[ 
            Gamma[ 
              Plus[ 3, N, 
                Times[ -1, a, N]]], -1], 
          Power[ 
            Gamma[ 
              Plus[ -3, 
                Times[ a, N]]], -1], 
          Power[ 
            Gamma[ 
              Plus[ 3, 
                Times[ a, N]]], -1]]}, 0, 4, 1],
      Editable->False]], "Output"]
}, Open  ]],

Cell["\<\

Verify that the 3F2 function works well enough\
\>", "Text"],

Cell[CellGroupData[{

Cell[BoxData[{
    \(Clear[a, \ P]\), "\[IndentingNewLine]", 
    \(\(a\  = \ 1\/3;\)\), "\[IndentingNewLine]", 
    \(\(P = 4;\)\), "\[IndentingNewLine]", 
    \(N[\(1\/\(Gamma[a\ P]\ Gamma[\((1\  - \ a)\) P]\)\) 
        HypergeometricPFQRegularized[{1, 1\  - \ \((1\  - a)\)\ P, \ 
            1\  - \ a\ P}, \ {a\ P, \ \((1\  - \ a)\)\ P}, \ \(-\ 
            1\)]]\), "\[IndentingNewLine]", 
    \(N[\[Sum]\+\(i = 0\)\%1\((\(-1\))\)\^i\/\(\(Gamma[a\ P + \ i]\) \
\(Gamma[a\ P\  - \ i]\)\(\ \)\(Gamma[\((1\  - \ a)\)\ P\  - \ i]\)\(\ \
\)\(Gamma[\((1\  - \ a)\) P\  + \ i]\)\(\ \)\)]\), "\[IndentingNewLine]", 
    \(N[\[Sum]\+\(i = 0\)\%3\((\(-1\))\)\^i\/\(\(Gamma[a\ P + \ i]\) \
\(Gamma[a\ P\  - \ i]\)\(\ \)\(Gamma[\((1\  - \ a)\)\ P\  - \ i]\)\(\ \
\)\(Gamma[\((1\  - \ a)\) P\  + \ i]\)\(\ \)\)]\), "\[IndentingNewLine]", 
    \(N[\[Sum]\+\(i = 0\)\%100\((\(-1\))\)\^i\/\(\(Gamma[a\ P + \ i]\) \
\(Gamma[a\ P\  - \ i]\)\(\ \)\(Gamma[\((1\  - \ a)\)\ P\  - \ i]\)\(\ \
\)\(Gamma[\((1\  - \ a)\) P\  + \ i]\)\(\ \)\)]\)}], "Input"],

Cell[BoxData[
    \(0.4630602879870958`\)], "Output"],

Cell[BoxData[
    \(0.46741520100753164`\)], "Output"],

Cell[BoxData[
    \(0.46307925294623725`\)], "Output"],

Cell[BoxData[
    \(0.4630602879870588`\)], "Output"]
}, Open  ]],

Cell["Does it converge to my other result?  Looks likely!", "Text"],

Cell[CellGroupData[{

Cell[BoxData[{
    \(Clear[a, P]\), "\[IndentingNewLine]", 
    \(\(a\  = \ 1\/2;\)\), "\[IndentingNewLine]", 
    \(\(P\  = 50;\)\), "\[IndentingNewLine]", 
    \(N[\(1\/P\^2\) \((P\ Log[
                Gamma[P\^2] 
                  HypergeometricPFQRegularized[{1, 
                      1\  - \ \((1\  - a)\)\ P\^2, \ 
                      1\  - \ a\ P\^2}, \ {a\ P\^2, \ \((1\  - \ 
                            a)\)\ P\^2}, \ \(-\ 1\)]])\)] - 
      Result[0.5, \ 50\^2, \ 1, 50]\), "\[IndentingNewLine]", 
    \(\(a\  = \ 1\/2;\)\), "\[IndentingNewLine]", 
    \(\(P\  = 10;\)\), "\[IndentingNewLine]", 
    \(N[\(1\/P\^2\) \((P\ Log[
                Gamma[P\^2] 
                  HypergeometricPFQRegularized[{1, 
                      1\  - \ \((1\  - a)\)\ P\^2, \ 
                      1\  - \ a\ P\^2}, \ {a\ P\^2, \ \((1\  - \ 
                            a)\)\ P\^2}, \ \(-\ 1\)]])\)] - 
      Result[0.5, \ 10\^2, \ 1, 10]\), "\[IndentingNewLine]", 
    \(\)}], "Input"],

Cell[BoxData[
    \(0.13585776809456718`\)], "Output"],

Cell[BoxData[
    \(0.34326548699911097`\)], "Output"]
}, Open  ]],

Cell["\<\
Compare this, my regular result, and the actual value for P going to infinity \
like N.

It doesn't even look like my regular result works well at N = P\
\>", "Text"],

Cell[CellGroupData[{

Cell[BoxData[{
    \(Clear[a, P]\), "\[IndentingNewLine]", 
    \(\(a\  = \ 1\/2;\)\), "\[IndentingNewLine]", 
    \(\(P\  = 100;\)\), "\[IndentingNewLine]", 
    \(N[\(1\/P\) \((P\ Log[
              Gamma[P] 
                HypergeometricPFQRegularized[{1, 1\  - \ \((1\  - a)\)\ P, \ 
                    1\  - \ a\ P}, \ {a\ P, \ \((1\  - \ a)\)\ P}, \ \(-\ 
                    1\)]])\)]\), "\[IndentingNewLine]", 
    \(N[Result[a, \ P, \ 1, P]]\), "\[IndentingNewLine]", 
    \(N[Actual[a, \ P, \ 1, \ P]]\), "\[IndentingNewLine]", 
    \(\(a\  = \ 1\/2;\)\), "\[IndentingNewLine]", 
    \(\(P\  = 50;\)\), "\[IndentingNewLine]", 
    \(N[\(1\/P\) \((P\ Log[
              Gamma[P] 
                HypergeometricPFQRegularized[{1, 1\  - \ \((1\  - a)\)\ P, \ 
                    1\  - \ a\ P}, \ {a\ P, \ \((1\  - \ a)\)\ P}, \ \(-\ 
                    1\)]])\)]\), "\[IndentingNewLine]", 
    \(N[Result[a, \ P, \ 1, P]]\), "\[IndentingNewLine]", 
    \(N[Actual[a, \ P, \ 1, \ P]]\), "\[IndentingNewLine]", 
    \(\(a\  = \ 1\/2;\)\), "\[IndentingNewLine]", 
    \(\(P\  = 10;\)\), "\[IndentingNewLine]", 
    \(N[\(1\/P\) \((P\ Log[
              Gamma[P] 
                HypergeometricPFQRegularized[{1, 1\  - \ \((1\  - a)\)\ P, \ 
                    1\  - \ a\ P}, \ {a\ P, \ \((1\  - \ a)\)\ P}, \ \(-\ 
                    1\)]])\)]\), "\[IndentingNewLine]", 
    \(N[Result[a, \ P, \ 1, P]]\), "\[IndentingNewLine]", 
    \(N[Actual[a, \ P, \ 1, \ P]]\)}], "Input"],

Cell[BoxData[
    \(69.30957029632569`\)], "Output"],

Cell[BoxData[
    \(66.06939607331923`\)], "Output"],

Cell[BoxData[
    \(30.066868787557272`\)], "Output"],

Cell[BoxData[
    \(34.30313796956033`\)], "Output"],

Cell[BoxData[
    \(31.755861557860403`\)], "Output"],

Cell[BoxData[
    \(15.028231726001675`\)], "Output"],

Cell[BoxData[
    \(5.766757273817589`\)], "Output"],

Cell[BoxData[
    \(4.865075694338127`\)], "Output"],

Cell[BoxData[
    \(2.9838840166269276`\)], "Output"]
}, Open  ]],

Cell[TextData[{
  "\nMy regular result does, however, work well at P = ",
  Cell[BoxData[
      \(TraditionalForm\`N\^2\)]]
}], "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(\(\(\[IndentingNewLine]\)\(Actual[0.5, \ 200\^2, \ 1, 
      200]\[IndentingNewLine]
    Result[0.5, 200\^2, \ 1, 200]\)\)\)], "Input"],

Cell[BoxData[
    \(138.4933181513226036074978218772`15.737378474166588\)], "Output"],

Cell[BoxData[
    \(138.5963882471040197330559918848`12.873795693102304\)], "Output"]
}, Open  ]],

Cell["\<\
My result looks even worse for P > N.  The only conclusion to draw is that \
while my approximation works fine for finite matrices, it does not extend to \
the infinite case.
In fact, no function I have here works when the number of paths becomes \
larger than their length because the binomial truncates as soon as a matrix \
element registers such an impossible event:  e.g., many paths become \
impossible.  Let me rewrite my \"Actual\" portion of the code to be exact, \
rather than t alowe negative binomials,\
\>", "Text"],

Cell[BoxData[{
    \(\(PathCountCorrect[i_, \ j_, \ length_, \ d_, \ slope_]\  := \ 
        If[And[GetEndPoint[j, \ 1, length, \ d, \ slope] - 
                GetStartPoint[i, \ 1, d]\  \[GreaterEqual] \ 0, 
            GetEndPoint[j, 2, length, d, slope] - 
                GetStartPoint[i, \ 2, d] \[GreaterEqual] 0\ ], 
          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]\ ], 0];\)\), "\[IndentingNewLine]", 
    \(\(AllConfigsCorrect[count_, length_, \ d_, \ slope_]\  := \ 
        Table[\ PathCountCorrect[i, \ j, \ length, \ d, \ slope], \ {i, \ 
            1, \ count}, \ {j, \ 1, \ count}];\)\), "\[IndentingNewLine]", 
    \(\(ActualCorrect[alpha_, \ N_, \ d_, \ paths_] := \(1\/N\) 
          Log[\ Det[\ \((AllConfigsCorrect[paths, \ N, d, \ 
                  alpha\ ]\ )\)]];\)\)}], "Input"],

Cell[CellGroupData[{

Cell[BoxData[
    \(ActualCorrect[0.5, \ 10, \ 1, 200]\)], "Input"],

Cell[BoxData[
    \(9.734567291221065`\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(\(\(\[IndentingNewLine]\)\(Actual[0.5, 10, 1, 200]\)\)\)], "Input"],

Cell[BoxData[
    \(9.734567291221065`\)], "Output"]
}, Open  ]],

Cell["\<\

Shockingly, the \"correct\" actual value, for a very great path count versus \
length does not differ much from the value for N > P without truncation.\
\>", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(ActualCorrect[0.5, \ 100, \ 1, 1000]\)], "Input"],

Cell[BoxData[
    \(469.7604394659684727519814725132`20.626466210378336 + 
      0.0314159265358979323846264338`16.451739642885137\ \[ImaginaryI]\)], \
"Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(Actual[0.5, 100, 1, 1000]\)], "Input"],

Cell[BoxData[
    \(469.7604394659684727519814725132`20.626466210378336 + 
      0.0314159265358979323846264338`16.451739642885137\ \[ImaginaryI]\)], \
"Output"]
}, Open  ]],

Cell["\<\

Of course, the approximation doesn't give a result close to the actual one \
here either, but if the actual doesn't go to what I expect, then making the \
approximation match it is moot.\
\>", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(\(\(\[IndentingNewLine]\)\(Result[0.5, 100, 1, 1000]\)\)\)], "Input"],

Cell[BoxData[
    \(660.8856463164364187859265216286`18.677806076774274\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(Result[0.5, 10, 1, 200]\)], "Input"],

Cell[BoxData[
    \(96.8296795495220861740930469774`18.639568269118026\)], "Output"]
}, Open  ]]
}, Open  ]]
},
FrontEndVersion->"5.0 for Microsoft Windows",
ScreenRectangle->{{0, 1280}, {0, 955}},
WindowSize->{1072, 734},
WindowMargins->{{Automatic, 96}, {Automatic, 92}}
]

(*******************************************************************
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.  The cache data will then be recreated when
you save this file from within Mathematica.
*******************************************************************)

(*CellTagsOutline
CellTagsIndex->{}
*)

(*CellTagsIndex
CellTagsIndex->{}
*)

(*NotebookFileOutline
Notebook[{

Cell[CellGroupData[{
Cell[1776, 53, 59, 4, 74, "Subsection"],
Cell[1838, 59, 46, 3, 52, "Text"],
Cell[1887, 64, 1191, 21, 183, "Input"],
Cell[3081, 87, 45, 3, 52, "Text"],
Cell[3129, 92, 446, 7, 88, "Input"],

Cell[CellGroupData[{
Cell[3600, 103, 119, 2, 50, "Input"],
Cell[3722, 107, 52, 1, 29, "Output"],
Cell[3777, 110, 52, 1, 29, "Output"]
}, Open  ]],

Cell[CellGroupData[{
Cell[3866, 116, 119, 2, 50, "Input"],
Cell[3988, 120, 52, 1, 29, "Output"],
Cell[4043, 123, 52, 1, 29, "Output"]
}, Open  ]],

Cell[CellGroupData[{
Cell[4132, 129, 121, 2, 50, "Input"],
Cell[4256, 133, 82, 1, 29, "Output"],
Cell[4341, 136, 81, 1, 29, "Output"]
}, Open  ]],
Cell[4437, 140, 157, 4, 52, "Text"],

Cell[CellGroupData[{
Cell[4619, 148, 125, 2, 52, "Input"],
Cell[4747, 152, 52, 1, 29, "Output"],
Cell[4802, 155, 52, 1, 29, "Output"]
}, Open  ]],

Cell[CellGroupData[{
Cell[4891, 161, 125, 2, 52, "Input"],
Cell[5019, 165, 84, 1, 29, "Output"],
Cell[5106, 168, 83, 1, 29, "Output"]
}, Open  ]],

Cell[CellGroupData[{
Cell[5226, 174, 129, 2, 52, "Input"],
Cell[5358, 178, 159, 3, 29, "Output"],
Cell[5520, 183, 85, 1, 29, "Output"]
}, Open  ]],

Cell[CellGroupData[{
Cell[5642, 189, 155, 3, 72, "Input"],
Cell[5800, 194, 159, 3, 29, "Output"],
Cell[5962, 199, 84, 1, 29, "Output"]
}, Open  ]],
Cell[6061, 203, 115, 4, 52, "Text"],

Cell[CellGroupData[{
Cell[6201, 211, 58, 1, 30, "Input"],
Cell[6262, 214, 53, 1, 29, "Output"]
}, Open  ]],

Cell[CellGroupData[{
Cell[6352, 220, 42, 1, 30, "Input"],
Cell[6397, 223, 53, 1, 29, "Output"]
}, Open  ]],
Cell[6465, 227, 75, 3, 52, "Text"],

Cell[CellGroupData[{
Cell[6565, 234, 329, 6, 65, "Input"],
Cell[6897, 242, 2266, 75, 83, "Output"]
}, Open  ]],
Cell[9178, 320, 71, 3, 52, "Text"],

Cell[CellGroupData[{
Cell[9274, 327, 1039, 16, 290, "Input"],
Cell[10316, 345, 53, 1, 29, "Output"],
Cell[10372, 348, 54, 1, 29, "Output"],
Cell[10429, 351, 54, 1, 29, "Output"],
Cell[10486, 354, 53, 1, 29, "Output"]
}, Open  ]],
Cell[10554, 358, 67, 0, 33, "Text"],

Cell[CellGroupData[{
Cell[10646, 362, 988, 20, 247, "Input"],
Cell[11637, 384, 54, 1, 29, "Output"],
Cell[11694, 387, 54, 1, 29, "Output"]
}, Open  ]],
Cell[11763, 391, 176, 5, 71, "Text"],

Cell[CellGroupData[{
Cell[11964, 400, 1487, 28, 414, "Input"],
Cell[13454, 430, 52, 1, 29, "Output"],
Cell[13509, 433, 52, 1, 29, "Output"],
Cell[13564, 436, 53, 1, 29, "Output"],
Cell[13620, 439, 52, 1, 29, "Output"],
Cell[13675, 442, 53, 1, 29, "Output"],
Cell[13731, 445, 53, 1, 29, "Output"],
Cell[13787, 448, 52, 1, 29, "Output"],
Cell[13842, 451, 52, 1, 29, "Output"],
Cell[13897, 454, 53, 1, 29, "Output"]
}, Open  ]],
Cell[13965, 458, 135, 4, 52, "Text"],

Cell[CellGroupData[{
Cell[14125, 466, 155, 3, 72, "Input"],
Cell[14283, 471, 85, 1, 29, "Output"],
Cell[14371, 474, 85, 1, 29, "Output"]
}, Open  ]],
Cell[14471, 478, 538, 9, 71, "Text"],
Cell[15012, 489, 1063, 18, 143, "Input"],

Cell[CellGroupData[{
Cell[16100, 511, 67, 1, 30, "Input"],
Cell[16170, 514, 52, 1, 29, "Output"]
}, Open  ]],

Cell[CellGroupData[{
Cell[16259, 520, 87, 1, 50, "Input"],
Cell[16349, 523, 52, 1, 29, "Output"]
}, Open  ]],
Cell[16416, 527, 176, 4, 52, "Text"],

Cell[CellGroupData[{
Cell[16617, 535, 69, 1, 30, "Input"],
Cell[16689, 538, 161, 3, 29, "Output"]
}, Open  ]],

Cell[CellGroupData[{
Cell[16887, 546, 58, 1, 30, "Input"],
Cell[16948, 549, 161, 3, 29, "Output"]
}, Open  ]],
Cell[17124, 555, 211, 5, 52, "Text"],

Cell[CellGroupData[{
Cell[17360, 564, 89, 1, 50, "Input"],
Cell[17452, 567, 85, 1, 29, "Output"]
}, Open  ]],

Cell[CellGroupData[{
Cell[17574, 573, 56, 1, 30, "Input"],
Cell[17633, 576, 84, 1, 29, "Output"]
}, Open  ]]
}, Open  ]]
}
]
*)



(*******************************************************************
End of Mathematica Notebook file.
*******************************************************************)