(************** Content-type: application/mathematica ************** CreatedBy='Mathematica 5.2' 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[ 10932, 397]*) (*NotebookOutlinePosition[ 43585, 1557]*) (* CellTagsIndexPosition[ 43541, 1553]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell[TextData[StyleBox["Introduction to GoodDyson", FontColor->RGBColor[1, 0, 0]]], "Title"], Cell["Andrew Sills", "Subtitle"], Cell["\<\ Department of Mathematics, Rutgers University Hill Center, Busch Campus Piscataway, NJ 08854-8019 USA\ \>", "Subsubtitle"], Cell["\<\ 14 March 2005, updated 6 January 2006.\ \>", "Subsubtitle"], Cell[CellGroupData[{ Cell["Introduction", "Section", Background->RGBColor[1, 1, 0]], Cell[TextData[{ "This notebook accompanies the ", StyleBox["Mathematica", FontSlant->"Italic"], " package GoodDyson, which accompanies the following paper." }], "Text"], Cell[TextData[{ "A. Sills and D. Zeilberger, ``Disturbing the Dyson conjecture (in a GOOD \ way),'' ", StyleBox["Experimental Math", FontSlant->"Italic"], "., to appear." }], "Text", CellFrame->True, CellMargins->{{50, 50}, {Inherited, Inherited}}, Background->GrayLevel[0.666667]], Cell["\<\ This paper, as well as the latest version of GoodDyson, is \ available also at my home page\ \>", "Text"], Cell[TextData[ButtonBox["http://www.math.rutgers.edu/~asills/papers.html", ButtonData:>{ URL[ "http://www.math.rutgers.edu/~asills/papers.html"], None}, Active->True, ButtonStyle->"Hyperlink"]], "Text", CellFrame->True, CellMargins->{{50, 50}, {Inherited, Inherited}}, TextAlignment->Center, Background->GrayLevel[0.900008]], Cell["\<\ Please send comments, questions, bug reports, or other feedback \ to\ \>", "Text"], Cell[TextData[ButtonBox["asills@math.rutgers.edu", ButtonData:>{ URL[ "mailto:asills@math.rutgers.edu"], None}, Active->True, ButtonStyle->"Hyperlink"]], "Text", CellFrame->True, CellMargins->{{50, 50}, {Inherited, Inherited}}, TextAlignment->Center, Background->GrayLevel[0.900008]] }, Closed]], Cell[CellGroupData[{ Cell["Some Examples", "Section", Background->RGBColor[1, 1, 0]], Cell[TextData[{ "As mentioned in the introduction, this ", StyleBox["Mathematica", FontSlant->"Italic"], " package accompanies the paper ``Disturbing the Dyson conjecture (in a \ GOOD way\", and accordingly the paper should be read before (or at least \ concurrently with) this demonstration package." }], "Text"], Cell[CellGroupData[{ Cell["Loading the Package", "Subsection", Background->RGBColor[0.610178, 1, 0.928832]], Cell[TextData[{ "First, load the package. Of course, you will need to specify the correct \ path to where you have stored the file ``GoodDyson.m\" if you have not \ launched ", StyleBox["Mathematica", FontSlant->"Italic"], " from the command line while your current working directory is the one \ where the file ``GoodDyson.m\" is located." }], "Text"], Cell[BoxData[ \(<< GoodDyson.m\)], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["GuessDysonCoeff", "Subsection"], Cell[TextData[{ "To have ", StyleBox["Mathematica", FontSlant->"Italic"], " conjecture the co\[EDoubleDot]fficient of ", Cell[BoxData[ \(TraditionalForm\`x\^2\)]], Cell[BoxData[ \(TraditionalForm\`y\^\(-1\)\)]], Cell[BoxData[ \(TraditionalForm\`z\^\(-1\)\)]], " in the expansion of \n[(1-", Cell[BoxData[ \(TraditionalForm\`y\/x\)]], ")(1-", Cell[BoxData[ \(TraditionalForm\`z\/x\)]], ")]", Cell[BoxData[ \(TraditionalForm\`\^a\)]], "[(1-", Cell[BoxData[ \(TraditionalForm\`x\/y\)]], ")(1-", Cell[BoxData[ \(TraditionalForm\`z\/y\)]], ")]", Cell[BoxData[ \(TraditionalForm\`\^b\)]], "[(1-", Cell[BoxData[ \(TraditionalForm\`x\/z\)]], ")(1-", Cell[BoxData[ \(TraditionalForm\`y\/z\)]], ")]", Cell[BoxData[ \(TraditionalForm\`\^c\)]], " , enter" }], "Text"], Cell[BoxData[ \(GuessDysonCoeff[{2, \(-1\), \(-1\)}, {a, b, c}]\)], "Input"], Cell[TextData[{ "The output represents the rational function multiple of the multinomial co\ \[EDoubleDot]fficient. Thus, in this case the requested \ co\[EDoubleDot]fficient is the output times ", Cell[BoxData[ \(TraditionalForm\`\(\((a\ + \ b\ + \ c)\)!\)\/\(\(a!\)\ \ \(b!\)\ \ \ \(c!\)\)\)]], ".\n" }], "Text"] }, Closed]], Cell[CellGroupData[{ Cell["GetDysonCoeff", "Subsection"], Cell[TextData[{ "`GetDysonCoeff' is similar to `GuessDysonCoeff` except that it first \ checks to see if the requested co\[EDoubleDot]fficient has been previously \ calculated and stored. If so, the earlier result is retrieved; if not, the \ requested co\[EDoubleDot]fficient is calculated. Note also that with \ `GetDysonCoeff`, the \"exponent variables\" from the Dyson product are \ automatically specified as ", Cell[BoxData[ \(TraditionalForm\`a\_1, \ a\_2, \ \(\(.\)\(\ \)\(.\)\(\ \)\(.\)\)\)]] }], "Text"], Cell[BoxData[ \(GetDysonCoeff[{2, \(-1\), \(-1\)}]\)], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["WritePaper (3 variable case)", "Subsection"], Cell[TextData[{ "To have ", StyleBox["Mathematica", FontSlant->"Italic"], " write a paper on the variation of Dyson's conjecture where we seek the \ co\[EDoubleDot]fficient of ", Cell[BoxData[ \(TraditionalForm\`\(\(x\_1\^2\)\(/\)\)\)]], Cell[BoxData[ \(TraditionalForm\`\(x\_2\) x\_3\)]], " in the expansion of \n[(1- ", Cell[BoxData[ \(TraditionalForm\`x\_2/x\_1\)]], ") (1 - ", Cell[BoxData[ \(TraditionalForm\`x\_3/x\_1\)]], ")]", Cell[BoxData[ \(TraditionalForm\`\^a\_1\)]], "[(1- ", Cell[BoxData[ \(TraditionalForm\`x\_1/x\_2\)]], ") (1 - ", Cell[BoxData[ \(TraditionalForm\`x\_3/x\_2\)]], ")]", Cell[BoxData[ \(TraditionalForm\`\^a\_2\)]], "[(1- ", Cell[BoxData[ \(TraditionalForm\`x\_1/x\_3\)]], ") (1 - ", Cell[BoxData[ \(TraditionalForm\`x\_2/x\_3\)]], ")]", Cell[BoxData[ \(TraditionalForm\`\^a\_3\)]], ",\nincluding an automatically generated analogue of I.J. Good's proof, \ enter " }], "Text"], Cell[BoxData[ \(WritePaper[{2, \(-1\), \(-1\)}]\)], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["WritePaper (2 variable case)", "Subsection"], Cell[TextData[{ "For ", StyleBox["n", FontSlant->"Italic"], "=2 variables, the `WritePaper` function is considerably more terse, since \ the requested result follows easily from the binomial theorem." }], "Text"], Cell[BoxData[ \(WritePaper[{3, \(-3\)}]\)], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["WritePaper (4 or more variables)", "Subsection"], Cell[TextData[{ "For ", StyleBox["n", FontSlant->"Italic"], ">3 variables, the `WritePaper` function will generate much more output \ than in the ", StyleBox["n", FontSlant->"Italic"], "=3 case, because the recursions depend on expressions in ", StyleBox["n", FontSlant->"Italic"], "-1 variables, which must also be verified. Thus the requested result is \ labeled as a theorem, then the function recuresively calls itself to verify \ the formulas requred for the theorem, labeling all subresults as 'lemma'. In \ general, the calculation of a 4-variable result takes considerably longer \ than a three variable result. For purposes of this demo, I have included the \ simplest possible four-variable result, which should execute quite quickly." }], "Text"], Cell[BoxData[ \(WritePaper[{0, 0, 0, 0}]\)], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["TurboDyson", "Subsection"], Cell[TextData[{ "Let ", StyleBox["b", FontWeight->"Bold"], " = {", Cell[BoxData[ \(TraditionalForm\`b\_1, \ b\_2, \ \[Ellipsis]\ , \ b\_n\)]], "}", " be an ", StyleBox["n", FontSlant->"Italic"], "-dimensional vector whose components sum to zero. The vector ", StyleBox["b", FontWeight->"Bold"], " is said to be in ", StyleBox["canonical form", FontSlant->"Italic"], " if its components are in nondecreasing order. Define the complexity of \ ", StyleBox["b", FontWeight->"Bold"], " to be the some of its positive components. `TurboDyson (n, C)` exploits \ symmetries and algebraic relations to find (and store) the \ co\[EDoubleDot]fficients of ", Cell[BoxData[ \(TraditionalForm\`x\_1\^b\_1\)]], Cell[BoxData[ \(TraditionalForm\`x\_2\^b\_2\)]], "\[CenterEllipsis] ", Cell[BoxData[ \(TraditionalForm\`x\_n\^b\_n\)]], " for all vectors ", StyleBox["b", FontWeight->"Bold"], " of complexity \[LessEqual] C. 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The \"Hyperlink\" style is for links within the same Notebook, \ or between Notebooks.\ \>", "Text"], Cell[CellGroupData[{ Cell[StyleData["Hyperlink"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, FontColor->RGBColor[0, 0, 1], FontVariations->{"Underline"->True}, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`NotebookLocate[ #2]}]&), Active->True, ButtonNote->ButtonData}], Cell[StyleData["Hyperlink", "Presentation"], FontSize->16], Cell[StyleData["Hyperlink", "Condensed"], FontSize->11], Cell[StyleData["Hyperlink", "Printout"], FontSize->10, FontColor->GrayLevel[0], FontVariations->{"Underline"->False}] }, Closed]], Cell["\<\ The following styles are for linking automatically to the on-line \ help system.\ \>", "Text"], Cell[CellGroupData[{ Cell[StyleData["MainBookLink"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, FontColor->RGBColor[0, 0, 1], FontVariations->{"Underline"->True}, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { 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If a cell's FormatType matches the name of one of the styles \ defined below, then that style is applied between the cell's style and its \ own options. 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For example, \ \"UnmatchedBracket\" style defines how unmatched bracket, curly bracket, and \ parenthesis characters are displayed (typically by coloring them to make them \ stand out).\ \>", "Text"], Cell[StyleData["UnmatchedBracket"], StyleMenuListing->None, FontColor->RGBColor[0.760006, 0.330007, 0.8]] }, Closed]] }, Open ]] }] ] (******************************************************************* 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, 94, 1, 102, "Title"], Cell[1873, 56, 32, 0, 54, "Subtitle"], Cell[1908, 58, 132, 4, 89, "Subsubtitle"], Cell[2043, 64, 70, 3, 71, "Subsubtitle"], Cell[CellGroupData[{ Cell[2138, 71, 64, 1, 77, "Section"], Cell[2205, 74, 178, 5, 52, "Text"], Cell[2386, 81, 297, 9, 68, "Text"], Cell[2686, 92, 115, 3, 33, "Text"], Cell[2804, 97, 342, 8, 49, "Text"], Cell[3149, 107, 92, 3, 33, "Text"], Cell[3244, 112, 301, 8, 49, "Text"] }, Closed]], Cell[CellGroupData[{ Cell[3582, 125, 65, 1, 57, "Section"], Cell[3650, 128, 323, 7, 74, "Text"], Cell[CellGroupData[{ Cell[3998, 139, 88, 1, 62, "Subsection"], Cell[4089, 142, 362, 8, 74, "Text"], Cell[4454, 152, 47, 1, 27, "Input"] }, Closed]], Cell[CellGroupData[{ Cell[4538, 158, 37, 0, 46, "Subsection"], Cell[4578, 160, 882, 39, 58, "Text"], Cell[5463, 201, 80, 1, 27, "Input"], Cell[5546, 204, 329, 8, 76, "Text"] }, Closed]], Cell[CellGroupData[{ Cell[5912, 217, 35, 0, 46, "Subsection"], Cell[5950, 219, 524, 9, 94, "Text"], Cell[6477, 230, 67, 1, 27, "Input"] }, Closed]], Cell[CellGroupData[{ Cell[6581, 236, 50, 0, 46, "Subsection"], Cell[6634, 238, 1026, 39, 94, "Text"], Cell[7663, 279, 64, 1, 27, "Input"] }, Closed]], Cell[CellGroupData[{ Cell[7764, 285, 50, 0, 46, "Subsection"], Cell[7817, 287, 222, 6, 54, "Text"], Cell[8042, 295, 56, 1, 27, "Input"] }, Closed]], Cell[CellGroupData[{ Cell[8135, 301, 54, 0, 46, "Subsection"], Cell[8192, 303, 786, 17, 134, "Text"], Cell[8981, 322, 57, 1, 27, "Input"] }, Closed]], Cell[CellGroupData[{ Cell[9075, 328, 32, 0, 46, "Subsection"], Cell[9110, 330, 1137, 37, 134, "Text"], Cell[10250, 369, 49, 1, 27, "Input"], Cell[10302, 372, 200, 4, 54, "Text"], Cell[10505, 378, 49, 1, 27, "Input"], Cell[10557, 381, 248, 6, 54, "Text"], Cell[10808, 389, 64, 1, 27, "Input"] }, Closed]] }, Closed]] }, Open ]], Cell[10911, 395, 17, 0, 102, "Title"] } ] *) (******************************************************************* End of Mathematica Notebook file. *******************************************************************)