(*CacheID: 234*) (* Internal cache information: NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 0, 0] NotebookDataLength[ 364506, 9069] NotebookOptionsPosition[ 343411, 8430] NotebookOutlinePosition[ 344376, 8463] CellTagsIndexPosition[ 344203, 8456] WindowFrame->Normal*) (* Beginning of Notebook Content *) Notebook[{ Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"2", "-", "2"}]], "Input", CellChangeTimes->{{3.536086476742803*^9, 3.536086477222335*^9}}], Cell[BoxData["0"], "Output", CellChangeTimes->{3.5360864907840843`*^9, 3.536502917863607*^9, 3.536502956314784*^9}] }, Open ]], Cell[BoxData["\[IndentingNewLine]"], "Input", CellChangeTimes->{3.536086501118999*^9}], Cell[CellGroupData[{ Cell["Hexagonal Coordinate System", "Title", CellChangeTimes->{{3.536086506975305*^9, 3.536086514168624*^9}}], Cell["\<\ Nicholas Wheeler 20 January 2012\ \>", "Text", CellChangeTimes->{{3.536086522888753*^9, 3.536086534552115*^9}}, FontSize->10], Cell[BoxData["\[IndentingNewLine]"], "Input", CellChangeTimes->{3.5360865442335567`*^9}], Cell[CellGroupData[{ Cell["Pre-introduction", "Subsection", CellChangeTimes->{{3.536594438974654*^9, 3.536594445619626*^9}}], Cell["\<\ While this work began as an attempt simply to discover an efficient way to \ coordinatize/identify/distinguish the nodes of a hexagonal grid, it proceeds \ to a sketch\[LongDash]made possible by that accomplishment\[LongDash]of the \ Markovian approach to random walks on such a grid. I am led to the conclusion \ that such an approach would require the resources of a supercomputer.\ \>", "Text", CellChangeTimes->{{3.536594454387376*^9, 3.536594588579398*^9}, { 3.536594635648868*^9, 3.536594637248518*^9}, {3.5365946769742527`*^9, 3.5365946805497723`*^9}, {3.536594800767836*^9, 3.536594817405842*^9}, { 3.536594877309581*^9, 3.536595030816996*^9}, {3.536595075606574*^9, 3.5365951272198353`*^9}}] }, Open ]], Cell[CellGroupData[{ Cell["Introduction", "Subsection", CellChangeTimes->{{3.536086567653121*^9, 3.5360865699077663`*^9}}], Cell["\<\ Here is the a portion of the unbounded hexagonal lattice on which, in recent \ work, I have constructed weighted random walks: \ \>", "Text", CellChangeTimes->{{3.536086624990773*^9, 3.53608668381006*^9}}], Cell[BoxData[ GraphicsBox[{LineBox[{{0, 0}, {0, 1}}], LineBox[NCache[{{0, 0}, { Rational[1, 2] 3^Rational[1, 2], Rational[-1, 2]}}, {{0, 0}, { 0.8660254037844386, -0.5}}]], LineBox[NCache[{{0, 0}, { Rational[-1, 2] 3^Rational[1, 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To \ that end, I construct\ \>", "Text", CellChangeTimes->{{3.536513486566249*^9, 3.53651354880392*^9}, { 3.536515423788518*^9, 3.5365154759459763`*^9}, 3.536515602502203*^9}], Cell[CellGroupData[{ Cell[BoxData[{ RowBox[{ RowBox[{"\[DoubleStruckCapitalZ]", "=", RowBox[{"Table", "[", RowBox[{"0", ",", RowBox[{"{", RowBox[{"k", ",", "1", ",", "81"}], "}"}]}], "]"}]}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"\[DoubleStruckCapitalO]", "=", RowBox[{"DiagonalMatrix", "[", "\[DoubleStruckCapitalZ]", "]"}]}], ";"}], "\[IndentingNewLine]"}], "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"\[DoubleStruckCapitalT]", "=", RowBox[{"Transpose", "[", "\[DoubleStruckCapitalS]", "]"}]}], ";"}], "\[IndentingNewLine]"}], "\[IndentingNewLine]", RowBox[{"Dimensions", "[", "\[DoubleStruckCapitalO]", "]"}], "\[IndentingNewLine]", RowBox[{"Dimensions", "[", "\[DoubleStruckCapitalS]", "]"}], "\[IndentingNewLine]", RowBox[{"Dimensions", "[", "\[DoubleStruckCapitalT]", "]"}], "\[IndentingNewLine]"}], "Input", CellChangeTimes->{{3.536513988159583*^9, 3.5365140342092657`*^9}, { 3.536515512949889*^9, 3.536515579224594*^9}}], Cell[BoxData[ RowBox[{"{", RowBox[{"81", ",", "81"}], "}"}]], "Output", CellChangeTimes->{3.536515585660243*^9}], Cell[BoxData[ RowBox[{"{", RowBox[{"81", ",", "81"}], "}"}]], "Output", CellChangeTimes->{3.536515585667755*^9}], Cell[BoxData[ RowBox[{"{", RowBox[{"81", ",", "81"}], "}"}]], "Output", CellChangeTimes->{3.536515585675871*^9}] }, Open ]], Cell[TextData[{ "and borrow from \"Linear Algebra of Supersymmetric Matrices\" (4-11 August \ 2011)the ", StyleBox["ArrayFlatten", "Input"], " command, which works this way:" }], "Text", CellChangeTimes->{{3.5365156113178387`*^9, 3.536515660838216*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"ArrayFlatten", "[", RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"(", "\[NoBreak]", GridBox[{ {"0", "0"}, {"0", "0"} }], "\[NoBreak]", ")"}], ",", RowBox[{"(", "\[NoBreak]", GridBox[{ {"1", "2"}, {"3", "4"} }], "\[NoBreak]", ")"}]}], "}"}], ",", RowBox[{"{", RowBox[{ RowBox[{"(", "\[NoBreak]", GridBox[{ {"5", "6"}, {"7", "8"} }], "\[NoBreak]", ")"}], ",", RowBox[{"(", "\[NoBreak]", GridBox[{ {"0", "0"}, {"0", "0"} }], "\[NoBreak]", ")"}]}], "}"}]}], "}"}], "]"}], "//", "MatrixForm"}]], "Input", CellChangeTimes->{{3.5220123393706923`*^9, 3.522012462817273*^9}, { 3.536513563498403*^9, 3.536513588064837*^9}}], Cell[BoxData[ TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {"0", "0", "1", "2"}, {"0", "0", "3", "4"}, {"5", "6", "0", "0"}, {"7", "8", "0", "0"} }, GridBoxAlignment->{ "Columns" -> {{Center}}, "ColumnsIndexed" -> {}, "Rows" -> {{Baseline}}, "RowsIndexed" -> {}}, GridBoxSpacings->{"Columns" -> { Offset[0.27999999999999997`], { Offset[0.7]}, Offset[0.27999999999999997`]}, "ColumnsIndexed" -> {}, "Rows" -> { Offset[0.2], { Offset[0.4]}, Offset[0.2]}, "RowsIndexed" -> {}}], "\[NoBreak]", ")"}], Function[BoxForm`e$, MatrixForm[BoxForm`e$]]]], "Output", CellChangeTimes->{3.536513593062993*^9}] }, Open ]], Cell["We arrive thus at", "Text", CellChangeTimes->{{3.53651572121093*^9, 3.536515725466209*^9}}], Cell[CellGroupData[{ Cell[BoxData[{ RowBox[{ RowBox[{ RowBox[{"\[DoubleStruckCapitalA]", "=", RowBox[{"ArrayFlatten", "[", RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{"\[DoubleStruckCapitalO]", ",", "\[DoubleStruckCapitalS]"}], "}"}], ",", RowBox[{"{", RowBox[{"\[DoubleStruckCapitalT]", ",", "\[DoubleStruckCapitalO]"}], "}"}]}], "}"}], "]"}]}], ";"}], "\[IndentingNewLine]"}], "\[IndentingNewLine]", RowBox[{"Dimensions", "[", "\[DoubleStruckCapitalA]", "]"}]}], "Input", CellChangeTimes->{{3.5365140702914753`*^9, 3.536514164778919*^9}, { 3.5365143596279297`*^9, 3.53651438846871*^9}, {3.536515712554488*^9, 3.5365157130877943`*^9}}], Cell[BoxData[ RowBox[{"{", RowBox[{"162", ",", "162"}], "}"}]], "Output", CellChangeTimes->{3.5365157340394983`*^9}] }, Open ]], Cell["But this\[LongDash]by", "Text", CellChangeTimes->{{3.5365167045929832`*^9, 3.536516709567811*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Det", "[", "\[DoubleStruckCapitalA]", "]"}]], "Input", CellChangeTimes->{{3.5365144266036453`*^9, 3.536514431041049*^9}}], Cell[BoxData["0"], "Output", CellChangeTimes->{3.536514432369315*^9}] }, Open ]], Cell["\<\ \[LongDash]cannot be correct. 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adjacency matrices. On finite honeycombs (however large) we expect to be \ led asymptotically to a distorted version of blilnking equilibration. But we \ cannot expect in such cases to guess the structure of the relevant \ eigenvectors (those associated with eigenvalues \[PlusMinus]1). And to obtain \ them by computation would\[LongDash]if the lattice is of physically \ interesting size\[LongDash]require a supercomputer.\n\nI am reminded that the \ computational requirements of lattice gauge theory" }], "Text", CellChangeTimes->{{3.536593223877356*^9, 3.5365932403801327`*^9}, { 3.536593285505941*^9, 3.536593328423416*^9}, {3.536593359253483*^9, 3.536593660418655*^9}, {3.536593925191403*^9, 3.536593944353746*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Hyperlink", "[", RowBox[{ "\"\\"", ",", "\"\\""}], "]"}]], "Input", CellChangeTimes->{{3.5365939499713097`*^9, 3.536593983095025*^9}}], Cell[BoxData[ TagBox[ ButtonBox[ PaneSelectorBox[{False->"\<\"Lattice Gauge Theory\"\>", True-> StyleBox["\<\"Lattice Gauge Theory\"\>", "HyperlinkActive"]}, Dynamic[ CurrentValue["MouseOver"]], BaseStyle->{"Hyperlink"}, BaselinePosition->Baseline, FrameMargins->0, ImageSize->Automatic], BaseStyle->"Hyperlink", ButtonData->{ URL["http://en.wikipedia.org/wiki/Lattice_gauge_theory"], None}, ButtonNote->"http://en.wikipedia.org/wiki/Lattice_gauge_theory"], Annotation[#, "http://en.wikipedia.org/wiki/Lattice_gauge_theory", "Hyperlink"]& ]], "Output", CellChangeTimes->{3.5365939882264214`*^9}] }, Open ]], Cell["\<\ served historically as a major stimulus toward the development of \ supercomputers. I have not attempted to introduce site-specific weighting, whether along \ lines suggested by naive corrugation models or otherwise. I think I see how \ to do so (by adapting techniques explored in an earlier notebook), but in \ view of the computational problem encountered in the preceding section I see \ no reason to introduce that added degree of complexity, since I would be \ computationally prevented from exploring its consequences.\ \>", "Text", CellChangeTimes->{{3.53659400298559*^9, 3.5365940410199413`*^9}, 3.536594097507393*^9, {3.536606361469027*^9, 3.536606361608086*^9}, { 3.536606540484291*^9, 3.536606669513494*^9}, {3.536606733307226*^9, 3.536606740591345*^9}, {3.5366068081283817`*^9, 3.536606844274941*^9}, { 3.5366068772742033`*^9, 3.536606945517622*^9}}] }, Open ]] }, Open ]] }, WindowToolbars->"EditBar", WindowSize->{1069, 628}, WindowMargins->{{19, Automatic}, {Automatic, 0}}, ShowSelection->True, Magnification:>FEPrivate`If[ FEPrivate`Equal[FEPrivate`$VersionNumber, 6.], 1.5, 1.5 Inherited], FrontEndVersion->"7.0 for Mac OS X PowerPC (32-bit) (November 11, 2008)", StyleDefinitions->"Default.nb" ] 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