(* Content-type: application/vnd.wolfram.mathematica *) (*** Wolfram Notebook File ***) (* http://www.wolfram.com/nb *) (* CreatedBy='Mathematica 11.3' *) (*CacheID: 234*) (* Internal cache information: NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 158, 7] NotebookDataLength[ 40434, 1119] NotebookOptionsPosition[ 38199, 1074] NotebookOutlinePosition[ 38534, 1089] CellTagsIndexPosition[ 38491, 1086] WindowFrame->Normal*) (* Beginning of Notebook Content *) Notebook[{ Cell[BoxData[ RowBox[{ RowBox[{ RowBox[{"H", "[", "\[Alpha]_", "]"}], "[", "M_", "]"}], ":=", RowBox[{ RowBox[{"-", RowBox[{"Sqrt", "[", RowBox[{"Length", "[", "M", "]"}], "]"}]}], RowBox[{"Total", "[", RowBox[{"Flatten", "@", SuperscriptBox[ RowBox[{"Abs", "[", "M", "]"}], "\[Alpha]"]}], "]"}]}]}]], "Input", CellChangeTimes->{{3.7342716312099323`*^9, 3.734271669554571*^9}, { 3.734271757134116*^9, 3.734271786252825*^9}, 3.7342719075284567`*^9, { 3.734272098689764*^9, 3.7342721527262774`*^9}, {3.73427470566724*^9, 3.7342747105637836`*^9}}, CellLabel->"In[64]:=",ExpressionUUID->"cac6654a-3c27-42b6-bbd5-5239c5df5c3b"], Cell[BoxData[ RowBox[{ RowBox[{ RowBox[{"metroStep", "[", RowBox[{"\[Beta]_", ",", "H_", ",", "\[Delta]_"}], "]"}], "[", "M_", "]"}], ":=", RowBox[{ RowBox[{ RowBox[{"If", "[", RowBox[{ RowBox[{ RowBox[{"RandomReal", "[", "]"}], "<", RowBox[{"Exp", "[", RowBox[{ RowBox[{"-", "\[Beta]"}], RowBox[{"(", RowBox[{ RowBox[{"H", "[", "#", "]"}], "-", RowBox[{"H", "[", "M", "]"}]}], ")"}]}], "]"}]}], ",", "#", ",", "M"}], "]"}], "&"}], "@", RowBox[{"(", RowBox[{ RowBox[{"RotationMatrix", "[", RowBox[{"\[Delta]", ",", RowBox[{"{", RowBox[{ RowBox[{"RandomVariate", "[", RowBox[{ RowBox[{"NormalDistribution", "[", "]"}], ",", RowBox[{"Length", "[", "M", "]"}]}], "]"}], ",", RowBox[{"RandomVariate", "[", RowBox[{ RowBox[{"NormalDistribution", "[", RowBox[{"0", ",", "\[Delta]"}], "]"}], ",", RowBox[{"Length", "[", "M", "]"}]}], "]"}]}], "}"}]}], "]"}], ".", "M"}], ")"}]}]}]], "Input", CellChangeTimes->{{3.734272867895825*^9, 3.734272943942663*^9}, { 3.7342730534804497`*^9, 3.734273127233576*^9}, 3.734273158298628*^9, { 3.734273189842609*^9, 3.7342731907711067`*^9}, {3.73427336373703*^9, 3.734273367014591*^9}, {3.734273414425611*^9, 3.734273418183498*^9}, 3.7342734529306*^9, {3.734274581545105*^9, 3.734274621274729*^9}, { 3.73427466158144*^9, 3.734274677227866*^9}, {3.734274801271907*^9, 3.734274818100624*^9}}, CellLabel->"In[80]:=",ExpressionUUID->"19d4accf-8194-401d-8675-3877ed3ddff3"], Cell[BoxData[ RowBox[{ RowBox[{ RowBox[{ RowBox[{"metroRun", "[", "n_", "]"}], "[", RowBox[{"\[Beta]_", ",", "H_", ",", "\[Delta]_"}], "]"}], "[", "M_", "]"}], ":=", RowBox[{"Nest", "[", RowBox[{ RowBox[{"metroStep", "[", RowBox[{"\[Beta]", ",", "H", ",", "\[Delta]"}], "]"}], ",", "M", ",", "n"}], "]"}]}]], "Input", CellChangeTimes->{{3.734273210708605*^9, 3.73427327014806*^9}, { 3.734285605976254*^9, 3.734285612166654*^9}}, CellLabel->"In[93]:=",ExpressionUUID->"d2362d48-1e47-4024-84eb-f7df8b2144b8"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"data", "=", RowBox[{"{", RowBox[{ RowBox[{"metroStep", "[", RowBox[{"1", ",", RowBox[{"H", "[", "1", "]"}], ",", "0.2"}], "]"}], "[", RowBox[{"IdentityMatrix", "[", "4", "]"}], "]"}], "}"}]}]], "Input", CellChangeTimes->{{3.734274626852656*^9, 3.734274641458256*^9}, { 3.734274805719406*^9, 3.734274808884956*^9}, {3.734287938167347*^9, 3.7342879424533854`*^9}}, CellLabel-> "In[101]:=",ExpressionUUID->"c9d601ae-0194-4ee1-8bb9-1d47a0e08241"], Cell[BoxData[ RowBox[{"{", RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{"0.9894335592089142`", ",", "0.11383926607460329`", ",", RowBox[{"-", "0.054638913910017725`"}], ",", RowBox[{"-", "0.07124915787228168`"}]}], "}"}], ",", RowBox[{"{", RowBox[{ RowBox[{"-", "0.11177863228621562`"}], ",", "0.9877751298731109`", ",", "0.09719473382689221`", ",", RowBox[{"-", "0.04856968071442663`"}]}], "}"}], ",", RowBox[{"{", RowBox[{"0.07058800047857862`", ",", RowBox[{"-", "0.0852908485761985`"}], ",", "0.9902464183860918`", ",", "0.08458626490811345`"}], "}"}], ",", RowBox[{"{", RowBox[{"0.05953237324212077`", ",", "0.06376821663576948`", ",", RowBox[{"-", "0.08354521935198095`"}], ",", "0.9926780482143663`"}], "}"}]}], "}"}], "}"}]], "Output", CellChangeTimes->{ 3.734274641697309*^9, 3.734274681366783*^9, 3.734274715235586*^9, { 3.7342748095363417`*^9, 3.7342748192959623`*^9}, 3.7342879427917843`*^9}, CellLabel-> "Out[101]=",ExpressionUUID->"002a6413-df1b-4458-aa32-43b079bd8dee"] }, Open ]], Cell[BoxData[ RowBox[{ RowBox[{"data", "=", RowBox[{ RowBox[{"NestList", "[", RowBox[{ RowBox[{ RowBox[{"metroRun", "[", "40", "]"}], "[", RowBox[{"10", ",", RowBox[{"H", "[", "2", "]"}], ",", "0.5"}], "]"}], ",", RowBox[{"data", "[", RowBox[{"[", RowBox[{"-", "1"}], "]"}], "]"}], ",", "5000"}], "]"}], "[", RowBox[{"[", RowBox[{"5", ";;"}], "]"}], "]"}]}], ";"}]], "Input", CellChangeTimes->{{3.7342732719025373`*^9, 3.734273305620448*^9}, { 3.734273534615774*^9, 3.7342735363594418`*^9}, {3.734273707702042*^9, 3.734273744859934*^9}, {3.7342737891908092`*^9, 3.734273882134091*^9}, { 3.7342740010182257`*^9, 3.734274001240144*^9}, {3.7342741839813137`*^9, 3.734274193498411*^9}, {3.7342743177916317`*^9, 3.734274318053464*^9}, { 3.734274440321534*^9, 3.734274477154084*^9}, {3.7342745106187153`*^9, 3.734274525458152*^9}, {3.734274753280533*^9, 3.7342747572515373`*^9}, { 3.734274873496624*^9, 3.7342748749184723`*^9}, {3.7342749167040253`*^9, 3.734274916789953*^9}, {3.7342855859334106`*^9, 3.734285618831058*^9}, { 3.734288985024776*^9, 3.734288990982616*^9}}, CellLabel-> "In[111]:=",ExpressionUUID->"b09286f6-c846-44a0-80a1-7b04a057b628"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Histogram", "[", RowBox[{ RowBox[{ RowBox[{"H", "[", "2", "]"}], "/@", "data"}], "/", SuperscriptBox["4", "2"]}], "]"}]], "Input", CellChangeTimes->{{3.734273752373975*^9, 3.73427376143603*^9}, { 3.734273843840104*^9, 3.7342738442696733`*^9}, {3.7342747278700113`*^9, 3.734274728899703*^9}, {3.7342878512782087`*^9, 3.7342879137088537`*^9}, { 3.734288766802822*^9, 3.734288772339603*^9}, 3.734288950921043*^9, { 3.734289062722124*^9, 3.7342890628556347`*^9}}, CellLabel-> "In[113]:=",ExpressionUUID->"78504373-95bf-4da0-82a1-e5822cb71c01"], Cell[BoxData[ GraphicsBox[{ {RGBColor[0.987148, 0.8073604000000001, 0.49470040000000004`], EdgeForm[{ Opacity[0.574], Thickness[Small]}], {}, {RGBColor[0.987148, 0.8073604000000001, 0.49470040000000004`], EdgeForm[{ Opacity[0.574], Thickness[Small]}], TagBox[ TooltipBox[ TagBox[ DynamicBox[{ FEPrivate`If[ CurrentValue["MouseOver"], EdgeForm[{ GrayLevel[0.5], AbsoluteThickness[1.5], Opacity[0.66]}], {}, {}], RectangleBox[{-0.500000000019, 0}, {-0.500000000018, 101.}, "RoundingRadius" -> 0]}, ImageSizeCache->{{30.17578125, 50.76171875}, {34.95624234768917, 90.50980111011306}}], StatusArea[#, 101]& , TagBoxNote->"101"], StyleBox["101", {}, StripOnInput -> False]], Annotation[#, Style[101, {}], "Tooltip"]& ], TagBox[ TooltipBox[ TagBox[ DynamicBox[{ FEPrivate`If[ CurrentValue["MouseOver"], EdgeForm[{ GrayLevel[0.5], AbsoluteThickness[1.5], Opacity[0.66]}], {}, {}], RectangleBox[{-0.500000000018, 0}, {-0.500000000017, 346.}, "RoundingRadius" -> 0]}, ImageSizeCache->{{50.26171875, 70.849609375}, {-98.5895190067054, 90.50980111011306}}], StatusArea[#, 346]& , TagBoxNote->"346"], StyleBox["346", {}, StripOnInput -> False]], Annotation[#, Style[346, {}], "Tooltip"]& ], TagBox[ TooltipBox[ TagBox[ DynamicBox[{ FEPrivate`If[ CurrentValue["MouseOver"], EdgeForm[{ GrayLevel[0.5], AbsoluteThickness[1.5], Opacity[0.66]}], {}, {}], RectangleBox[{-0.500000000017, 0}, {-0.500000000016, 353.}, "RoundingRadius" -> 0]}, ImageSizeCache->{{70.349609375, 90.935546875}, {-102.40511218825954`, 90.50980111011306}}], StatusArea[#, 353]& , TagBoxNote->"353"], StyleBox["353", {}, StripOnInput -> False]], Annotation[#, Style[353, {}], "Tooltip"]& ], TagBox[ TooltipBox[ TagBox[ DynamicBox[{ FEPrivate`If[ CurrentValue["MouseOver"], EdgeForm[{ GrayLevel[0.5], AbsoluteThickness[1.5], Opacity[0.66]}], {}, {}], RectangleBox[{-0.500000000016, 0}, {-0.500000000015, 343.}, "RoundingRadius" -> 0]}, ImageSizeCache->{{90.435546875, 111.021484375}, {-96.95426478603936, 90.50980111011306}}], StatusArea[#, 343]& , TagBoxNote->"343"], StyleBox["343", {}, StripOnInput -> False]], Annotation[#, Style[343, {}], "Tooltip"]& ], TagBox[ TooltipBox[ TagBox[ DynamicBox[{ FEPrivate`If[ CurrentValue["MouseOver"], EdgeForm[{ GrayLevel[0.5], AbsoluteThickness[1.5], Opacity[0.66]}], {}, {}], RectangleBox[{-0.500000000015, 0}, {-0.500000000014, 346.}, "RoundingRadius" -> 0]}, ImageSizeCache->{{110.521484375, 131.107421875}, {-98.5895190067054, 90.50980111011306}}], StatusArea[#, 346]& , TagBoxNote->"346"], StyleBox["346", {}, StripOnInput -> False]], Annotation[#, Style[346, {}], "Tooltip"]& ], TagBox[ TooltipBox[ TagBox[ DynamicBox[{ FEPrivate`If[ CurrentValue["MouseOver"], EdgeForm[{ GrayLevel[0.5], AbsoluteThickness[1.5], Opacity[0.66]}], {}, {}], RectangleBox[{-0.500000000014, 0}, {-0.500000000013, 344.}, "RoundingRadius" -> 0]}, ImageSizeCache->{{130.607421875, 151.193359375}, {-97.49934952626138, 90.50980111011306}}], StatusArea[#, 344]& , TagBoxNote->"344"], StyleBox["344", {}, StripOnInput -> False]], Annotation[#, Style[344, {}], "Tooltip"]& ], TagBox[ TooltipBox[ TagBox[ DynamicBox[{ FEPrivate`If[ CurrentValue["MouseOver"], EdgeForm[{ GrayLevel[0.5], AbsoluteThickness[1.5], Opacity[0.66]}], {}, {}], RectangleBox[{-0.500000000013, 0}, {-0.500000000012, 342.}, "RoundingRadius" -> 0]}, ImageSizeCache->{{150.693359375, 171.28125}, {-96.40918004581734, 90.50980111011306}}], StatusArea[#, 342]& , TagBoxNote->"342"], StyleBox["342", {}, StripOnInput -> False]], Annotation[#, Style[342, {}], "Tooltip"]& ], TagBox[ TooltipBox[ TagBox[ DynamicBox[{ FEPrivate`If[ CurrentValue["MouseOver"], EdgeForm[{ GrayLevel[0.5], AbsoluteThickness[1.5], Opacity[0.66]}], {}, {}], RectangleBox[{-0.500000000012, 0}, {-0.500000000011, 345.}, "RoundingRadius" -> 0]}, ImageSizeCache->{{170.78125, 191.3671875}, {-98.04443426648338, 90.50980111011306}}], StatusArea[#, 345]& , TagBoxNote->"345"], StyleBox["345", {}, StripOnInput -> False]], Annotation[#, Style[345, {}], "Tooltip"]& ], TagBox[ TooltipBox[ TagBox[ DynamicBox[{ FEPrivate`If[ CurrentValue["MouseOver"], EdgeForm[{ GrayLevel[0.5], AbsoluteThickness[1.5], Opacity[0.66]}], {}, {}], RectangleBox[{-0.500000000011, 0}, {-0.50000000001, 353.}, "RoundingRadius" -> 0]}, ImageSizeCache->{{190.8671875, 211.453125}, {-102.40511218825954`, 90.50980111011306}}], StatusArea[#, 353]& , TagBoxNote->"353"], StyleBox["353", {}, StripOnInput -> False]], Annotation[#, Style[353, {}], "Tooltip"]& ], TagBox[ TooltipBox[ TagBox[ DynamicBox[{ FEPrivate`If[ CurrentValue["MouseOver"], EdgeForm[{ GrayLevel[0.5], AbsoluteThickness[1.5], Opacity[0.66]}], {}, {}], RectangleBox[{-0.50000000001, 0}, {-0.500000000009, 347.}, "RoundingRadius" -> 0]}, ImageSizeCache->{{210.953125, 231.5390625}, {-99.13460374692743, 90.50980111011306}}], StatusArea[#, 347]& , TagBoxNote->"347"], StyleBox["347", {}, StripOnInput -> False]], Annotation[#, Style[347, {}], "Tooltip"]& ], TagBox[ TooltipBox[ TagBox[ DynamicBox[{ FEPrivate`If[ CurrentValue["MouseOver"], EdgeForm[{ GrayLevel[0.5], AbsoluteThickness[1.5], Opacity[0.66]}], {}, {}], RectangleBox[{-0.500000000009, 0}, {-0.500000000008, 348.}, "RoundingRadius" -> 0]}, ImageSizeCache->{{231.0390625, 251.625}, {-99.67968848714945, 90.50980111011306}}], StatusArea[#, 348]& , TagBoxNote->"348"], StyleBox["348", {}, StripOnInput -> False]], Annotation[#, Style[348, {}], "Tooltip"]& ], TagBox[ TooltipBox[ TagBox[ DynamicBox[{ FEPrivate`If[ CurrentValue["MouseOver"], EdgeForm[{ GrayLevel[0.5], AbsoluteThickness[1.5], Opacity[0.66]}], {}, {}], RectangleBox[{-0.500000000008, 0}, {-0.500000000007, 347.}, "RoundingRadius" -> 0]}, ImageSizeCache->{{251.125, 271.712890625}, {-99.13460374692743, 90.50980111011306}}], StatusArea[#, 347]& , TagBoxNote->"347"], StyleBox["347", {}, StripOnInput -> False]], Annotation[#, Style[347, {}], "Tooltip"]& ], TagBox[ TooltipBox[ TagBox[ DynamicBox[{ FEPrivate`If[ CurrentValue["MouseOver"], EdgeForm[{ GrayLevel[0.5], AbsoluteThickness[1.5], Opacity[0.66]}], {}, {}], RectangleBox[{-0.500000000007, 0}, {-0.500000000006, 343.}, "RoundingRadius" -> 0]}, ImageSizeCache->{{271.212890625, 291.796875}, {-96.95426478603936, 90.50980111011306}}], StatusArea[#, 343]& , TagBoxNote->"343"], StyleBox["343", {}, StripOnInput -> False]], Annotation[#, Style[343, {}], "Tooltip"]& ], TagBox[ TooltipBox[ TagBox[ DynamicBox[{ FEPrivate`If[ CurrentValue["MouseOver"], EdgeForm[{ GrayLevel[0.5], AbsoluteThickness[1.5], Opacity[0.66]}], {}, {}], RectangleBox[{-0.500000000006, 0}, {-0.500000000005, 344.}, "RoundingRadius" -> 0]}, ImageSizeCache->{{291.296875, 311.8828125}, {-97.49934952626138, 90.50980111011306}}], StatusArea[#, 344]& , TagBoxNote->"344"], StyleBox["344", {}, StripOnInput -> False]], Annotation[#, Style[344, {}], "Tooltip"]& ], TagBox[ TooltipBox[ TagBox[ DynamicBox[{ FEPrivate`If[ CurrentValue["MouseOver"], EdgeForm[{ GrayLevel[0.5], AbsoluteThickness[1.5], Opacity[0.66]}], {}, {}], RectangleBox[{-0.500000000005, 0}, {-0.500000000004, 346.}, "RoundingRadius" -> 0]}, ImageSizeCache->{{311.3828125, 331.96875}, {-98.5895190067054, 90.50980111011306}}], StatusArea[#, 346]& , TagBoxNote->"346"], StyleBox["346", {}, StripOnInput -> False]], Annotation[#, Style[346, {}], "Tooltip"]& ], TagBox[ TooltipBox[ TagBox[ DynamicBox[{ FEPrivate`If[ CurrentValue["MouseOver"], EdgeForm[{ GrayLevel[0.5], AbsoluteThickness[1.5], Opacity[0.66]}], {}, {}], RectangleBox[{-0.500000000004, 0}, {-0.500000000003, 49.}, "RoundingRadius" -> 0]}, ImageSizeCache->{{331.46875, 352.0546875}, {63.30064883923414, 90.50980111011306}}], StatusArea[#, 49]& , TagBoxNote->"49"], StyleBox["49", {}, StripOnInput -> False]], Annotation[#, Style[49, {}], "Tooltip"]& ]}, {}, {}}, {{}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, \ {}, {}, {}, {}, {}}}, AspectRatio->NCache[GoldenRatio^(-1), 0.6180339887498948], Axes->{True, True}, AxesLabel->{None, None}, AxesOrigin->{-0.50000000001932, 0}, FrameLabel->{{None, None}, {None, None}}, FrameTicks->{{Automatic, Automatic}, {Automatic, Automatic}}, GridLines->{None, None}, GridLinesStyle->Directive[ GrayLevel[0.5, 0.4]], PlotRange->{{-0.500000000019, -0.500000000003}, {All, All}}, PlotRangePadding->{{ Scaled[0.02], Scaled[0.02]}, { Scaled[0.02], Scaled[0.05]}}, Ticks->{Automatic, Automatic}]], "Output", CellChangeTimes->{{3.734273761777768*^9, 3.734273884833589*^9}, 3.734274004165744*^9, {3.7342741873550997`*^9, 3.734274200692301*^9}, 3.734274323949729*^9, {3.7342744516925364`*^9, 3.7342745452507067`*^9}, 3.734274687792594*^9, {3.7342747227550697`*^9, 3.7342747292049713`*^9}, 3.7342747824257727`*^9, 3.734274842614546*^9, 3.7342748973582373`*^9, 3.734274958534041*^9, 3.734287658450778*^9, {3.734287856214024*^9, 3.734287884218719*^9}, 3.734287914640992*^9, 3.734288250350131*^9, { 3.734288767453093*^9, 3.7342887727803373`*^9}, 3.734288951631302*^9, 3.734289022685177*^9, 3.734289063156498*^9}, CellLabel-> "Out[113]=",ExpressionUUID->"46945c0b-e240-4c95-a92c-0c4d9ab027bd"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"1", "+", RowBox[{ RowBox[{"Mean", "[", RowBox[{ RowBox[{"H", "[", "1", "]"}], "/@", "data"}], "]"}], "/", SuperscriptBox["4", "2"]}]}]], "Input", CellChangeTimes->{{3.734273895200355*^9, 3.734273899086282*^9}, { 3.734274175722883*^9, 3.734274176138234*^9}, {3.734274734703082*^9, 3.734274740756043*^9}, {3.734274962847492*^9, 3.734274963568054*^9}, 3.734288955633256*^9}, CellLabel-> "In[110]:=",ExpressionUUID->"97f199bc-0af4-44b6-aa4a-fcb6cd45e868"], Cell[BoxData["0.018554016903542525`"], "Output", CellChangeTimes->{ 3.7342738992890987`*^9, 3.734274005102474*^9, {3.7342741763677053`*^9, 3.734274201476615*^9}, 3.73427432516159*^9, {3.7342744529386673`*^9, 3.734274515900497*^9}, 3.734274546217537*^9, 3.734274689505424*^9, { 3.734274723753941*^9, 3.734274741243452*^9}, 3.734274783331256*^9, 3.734274846195489*^9, 3.734274910563336*^9, 3.734274964016556*^9, 3.734287660285326*^9, 3.734288731274395*^9, 3.734288956152451*^9}, CellLabel-> "Out[110]=",ExpressionUUID->"0d8d0d0e-edfa-449a-bb6b-bdeff666bd45"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{ RowBox[{ RowBox[{"H", "[", "1", "]"}], "[", RowBox[{"IdentityMatrix", "[", "12", "]"}], "]"}], "/", SuperscriptBox["12", "2"]}], "//", "N"}]], "Input", CellChangeTimes->{{3.734288925327345*^9, 3.734288938277343*^9}}, CellLabel-> "In[108]:=",ExpressionUUID->"174a3e7f-be6f-47e2-8dab-4d7035aaccc1"], Cell[BoxData[ RowBox[{"-", "0.2886751345948129`"}]], "Output", CellChangeTimes->{{3.7342889337709303`*^9, 3.7342889384536867`*^9}}, CellLabel-> "Out[108]=",ExpressionUUID->"11e30345-b1c9-47bd-90e5-22985ea861c8"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"t1", "=", RowBox[{"FullSimplify", "[", RowBox[{ RowBox[{"Total", "[", RowBox[{"Flatten", "[", SuperscriptBox[ RowBox[{"Abs", "[", RowBox[{"RotationMatrix", "[", "\[Theta]", "]"}], "]"}], RowBox[{"2", "-", "\[Epsilon]"}]], "]"}], "]"}], ",", RowBox[{"{", RowBox[{ RowBox[{"\[Theta]", "\[Element]", "Reals"}], ",", RowBox[{"\[Epsilon]", "\[Element]", "Reals"}]}], "}"}]}], "]"}]}]], "Input", CellChangeTimes->{{3.734289182979594*^9, 3.734289291611445*^9}}, CellLabel-> "In[124]:=",ExpressionUUID->"1b4380fb-27f4-452b-bb0b-712dad575f2f"], Cell[BoxData[ RowBox[{"2", " ", RowBox[{"(", RowBox[{ SuperscriptBox[ RowBox[{"Abs", "[", RowBox[{"Cos", "[", "\[Theta]", "]"}], "]"}], RowBox[{"2", "-", "\[Epsilon]"}]], "+", SuperscriptBox[ RowBox[{"Abs", "[", RowBox[{"Sin", "[", "\[Theta]", "]"}], "]"}], RowBox[{"2", "-", "\[Epsilon]"}]]}], ")"}]}]], "Output", CellChangeTimes->{{3.7342891874014263`*^9, 3.734289292101226*^9}}, CellLabel-> "Out[124]=",ExpressionUUID->"49bc1d2c-058f-4279-988d-355c973860cb"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"FullSimplify", "[", RowBox[{ RowBox[{"Series", "[", RowBox[{"t1", ",", RowBox[{"{", RowBox[{"\[Epsilon]", ",", "0", ",", "2"}], "}"}]}], "]"}], ",", RowBox[{"{", RowBox[{"\[Theta]", "\[Element]", "Reals"}], "}"}]}], "]"}]], "Input", CellChangeTimes->{{3.73428929267728*^9, 3.734289312380084*^9}, { 3.734289446985054*^9, 3.734289447589985*^9}}, CellLabel-> "In[127]:=",ExpressionUUID->"2a832002-adf6-4c60-bb56-92e9cf4c32f6"], Cell[BoxData[ InterpretationBox[ RowBox[{"2", "-", RowBox[{"2", " ", RowBox[{"(", RowBox[{ RowBox[{ SuperscriptBox[ RowBox[{"Cos", "[", "\[Theta]", "]"}], "2"], " ", RowBox[{"Log", "[", RowBox[{"Abs", "[", RowBox[{"Cos", "[", "\[Theta]", "]"}], "]"}], "]"}]}], "+", RowBox[{ RowBox[{"Log", "[", RowBox[{"Abs", "[", RowBox[{"Sin", "[", "\[Theta]", "]"}], "]"}], "]"}], " ", SuperscriptBox[ RowBox[{"Sin", "[", "\[Theta]", "]"}], "2"]}]}], ")"}], " ", "\[Epsilon]"}], "+", RowBox[{ RowBox[{"(", RowBox[{ RowBox[{ SuperscriptBox[ RowBox[{"Cos", "[", "\[Theta]", "]"}], "2"], " ", SuperscriptBox[ RowBox[{"Log", "[", RowBox[{"Abs", "[", RowBox[{"Cos", "[", "\[Theta]", "]"}], "]"}], "]"}], "2"]}], "+", RowBox[{ SuperscriptBox[ RowBox[{"Log", "[", RowBox[{"Abs", "[", RowBox[{"Sin", "[", "\[Theta]", "]"}], "]"}], "]"}], "2"], " ", SuperscriptBox[ RowBox[{"Sin", "[", "\[Theta]", "]"}], "2"]}]}], ")"}], " ", SuperscriptBox["\[Epsilon]", "2"]}], "+", InterpretationBox[ SuperscriptBox[ RowBox[{"O", "[", "\[Epsilon]", "]"}], "3"], SeriesData[$CellContext`\[Epsilon], 0, {}, 0, 3, 1], Editable->False]}], SeriesData[$CellContext`\[Epsilon], 0, { 2, (-2) (Cos[$CellContext`\[Theta]]^2 Log[ Abs[ Cos[$CellContext`\[Theta]]]] + Log[ Abs[ Sin[$CellContext`\[Theta]]]] Sin[$CellContext`\[Theta]]^2), Cos[$CellContext`\[Theta]]^2 Log[ Abs[ Cos[$CellContext`\[Theta]]]]^2 + Log[ Abs[ Sin[$CellContext`\[Theta]]]]^2 Sin[$CellContext`\[Theta]]^2}, 0, 3, 1], Editable->False]], "Output", CellChangeTimes->{{3.734289299725992*^9, 3.7342893131931143`*^9}, 3.7342894487288523`*^9}, CellLabel-> "Out[127]=",ExpressionUUID->"a27ce88a-34bd-4f78-ad90-ce5d10de1040"] }, Open ]], Cell[BoxData[ RowBox[{ RowBox[{"T", "[", "n_", "]"}], ":=", RowBox[{"Total", "[", RowBox[{ RowBox[{ RowBox[{"ArrayPad", "[", RowBox[{ RowBox[{"RotationMatrix", "[", RowBox[{"\[Theta]", "[", "#", "]"}], "]"}], ",", RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"2", RowBox[{"(", RowBox[{"#", "-", "1"}], ")"}]}], ",", RowBox[{"2", RowBox[{"(", RowBox[{"n", "-", "#", "-", "1"}], ")"}]}]}], "}"}], ",", RowBox[{"{", RowBox[{ RowBox[{"2", RowBox[{"(", RowBox[{"#", "-", "1"}], ")"}]}], ",", RowBox[{"2", RowBox[{"(", RowBox[{"n", "-", "#", "-", "1"}], ")"}]}]}], "}"}]}], "}"}]}], "]"}], "&"}], "/@", RowBox[{"Range", "[", "n", "]"}]}], "]"}]}]], "Input", CellChangeTimes->{{3.734358435933256*^9, 3.7343584643533373`*^9}, { 3.734358523955289*^9, 3.7343586601426973`*^9}}, CellLabel-> "In[128]:=",ExpressionUUID->"180ef9a5-3287-46d1-ae4b-ae9e85480093"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ SuperscriptBox[ RowBox[{"Abs", "[", RowBox[{"(", RowBox[{ RowBox[{"Transpose", "[", RowBox[{"Array", "[", RowBox[{"p", ",", RowBox[{"{", RowBox[{"10", ",", "10"}], "}"}]}], "]"}], "]"}], ".", RowBox[{"T", "[", "6", "]"}], ".", RowBox[{"Array", "[", RowBox[{"p", ",", RowBox[{"{", RowBox[{"10", ",", "10"}], "}"}]}], "]"}]}], ")"}], "]"}], RowBox[{"2", "-", "\[Epsilon]"}]], "//", "Total"}]], "Input", CellChangeTimes->{{3.734358661015789*^9, 3.734358807664146*^9}}, CellLabel-> "In[141]:=",ExpressionUUID->"e2107056-243c-408e-8546-20a935f61e1d"], Cell[BoxData[ InterpretationBox[ TagBox[ FrameBox[GridBox[{ { ItemBox[ TagBox[ RowBox[{"{", RowBox[{ RowBox[{ SuperscriptBox[ RowBox[{"Abs", "[", RowBox[{ RowBox[{ RowBox[{"p", "[", RowBox[{"2", ",", "1"}], "]"}], " ", RowBox[{"(", RowBox[{ RowBox[{ RowBox[{"Cos", "[", RowBox[{"\[Theta]", "[", "1", "]"}], "]"}], " ", RowBox[{"p", "[", RowBox[{"2", ",", "1"}], "]"}]}], "-", RowBox[{ RowBox[{"p", "[", RowBox[{"1", ",", "1"}], "]"}], " ", RowBox[{"Sin", "[", RowBox[{"\[Theta]", "[", "1", "]"}], "]"}]}]}], ")"}]}], "+", TemplateBox[{"8"}, "OutputSizeLimit`Skeleton"], "+", RowBox[{ RowBox[{"p", "[", RowBox[{"9", ",", "1"}], "]"}], " ", RowBox[{"(", RowBox[{ RowBox[{ RowBox[{"Cos", "[", RowBox[{"\[Theta]", "[", "5", "]"}], "]"}], " ", RowBox[{"p", "[", RowBox[{"9", ",", "1"}], "]"}]}], "+", RowBox[{ RowBox[{"p", "[", RowBox[{"10", ",", "1"}], "]"}], " ", RowBox[{"Sin", "[", RowBox[{"\[Theta]", "[", "5", "]"}], "]"}]}]}], ")"}]}]}], "]"}], RowBox[{"2", "-", "\[Epsilon]"}]], "+", SuperscriptBox[ RowBox[{"Abs", "[", RowBox[{ RowBox[{ RowBox[{"p", "[", RowBox[{"2", ",", "1"}], "]"}], " ", RowBox[{"(", TemplateBox[{"1"}, "OutputSizeLimit`Skeleton"], ")"}]}], "+", TemplateBox[{"8"}, "OutputSizeLimit`Skeleton"], "+", RowBox[{ RowBox[{"p", "[", RowBox[{"9", ",", "1"}], "]"}], " ", RowBox[{"(", TemplateBox[{"1"}, "OutputSizeLimit`Skeleton"], ")"}]}]}], "]"}], RowBox[{"2", "-", "\[Epsilon]"}]], "+", SuperscriptBox[ RowBox[{"Abs", "[", TemplateBox[{"1"}, "OutputSizeLimit`Skeleton"], "]"}], RowBox[{"2", "-", "\[Epsilon]"}]], "+", TemplateBox[{"4"}, "OutputSizeLimit`Skeleton"], "+", SuperscriptBox[ RowBox[{"Abs", "[", TemplateBox[{"1"}, "OutputSizeLimit`Skeleton"], "]"}], RowBox[{"2", "-", "\[Epsilon]"}]], "+", SuperscriptBox[ RowBox[{"Abs", "[", TemplateBox[{"1"}, "OutputSizeLimit`Skeleton"], "]"}], RowBox[{"2", "-", "\[Epsilon]"}]], "+", SuperscriptBox[ RowBox[{"Abs", "[", RowBox[{ RowBox[{ RowBox[{"p", "[", RowBox[{"2", ",", "1"}], "]"}], " ", RowBox[{"(", RowBox[{ RowBox[{ RowBox[{"Cos", "[", RowBox[{"\[Theta]", "[", "1", "]"}], "]"}], " ", RowBox[{"p", "[", RowBox[{"2", ",", "10"}], "]"}]}], "-", RowBox[{ RowBox[{"p", "[", RowBox[{"1", ",", "10"}], "]"}], " ", RowBox[{"Sin", "[", RowBox[{"\[Theta]", "[", "1", "]"}], "]"}]}]}], ")"}]}], "+", TemplateBox[{"8"}, "OutputSizeLimit`Skeleton"], "+", RowBox[{ RowBox[{"p", "[", RowBox[{"9", ",", "1"}], "]"}], " ", RowBox[{"(", TemplateBox[{"1"}, "OutputSizeLimit`Skeleton"], ")"}]}]}], "]"}], RowBox[{"2", "-", "\[Epsilon]"}]]}], ",", TemplateBox[{"8"}, "OutputSizeLimit`Skeleton"], ",", TemplateBox[{"1"}, "OutputSizeLimit`Skeleton"]}], "}"}], Short[#, 5]& ], BaseStyle->{Deployed -> False}, StripOnInput->False]}, {GridBox[{ { TagBox[ TooltipBox[ StyleBox[ StyleBox[ DynamicBox[ToBoxes[ FEPrivate`FrontEndResource[ "FEStrings", "sizeBriefExplanation"], StandardForm], ImageSizeCache->{61., {2., 8.}}], StripOnInput->False, DynamicUpdating->True], "OSLText", StripOnInput->False], StyleBox[ DynamicBox[ ToBoxes[ FEPrivate`FrontEndResource["FEStrings", "sizeExplanation"], StandardForm]], DynamicUpdating -> True, StripOnInput -> False]], Annotation[#, Style[ Dynamic[ FEPrivate`FrontEndResource["FEStrings", "sizeExplanation"]], DynamicUpdating -> True], "Tooltip"]& ], ButtonBox[ PaneSelectorBox[{False-> StyleBox[ StyleBox[ DynamicBox[ToBoxes[ FEPrivate`FrontEndResource["FEStrings", "sizeShowLess"], StandardForm], ImageSizeCache->{54., {0., 8.}}], StripOnInput->False, DynamicUpdating->True], "OSLControl", StripOnInput->False], True-> StyleBox[ StyleBox[ DynamicBox[ToBoxes[ FEPrivate`FrontEndResource["FEStrings", "sizeShowLess"], StandardForm]], StripOnInput->False, DynamicUpdating->True], "OSLControlActive", StripOnInput->False]}, Dynamic[ CurrentValue["MouseOver"]], Alignment->Center, FrameMargins->0, ImageSize->{Automatic, 25}], Appearance->None, BaselinePosition->Baseline, ButtonFunction:>OutputSizeLimit`ButtonFunction[ OutputSizeLimit`Defer, 141, 30935294013293723379, 5/2], Enabled->True, Evaluator->Automatic, Method->"Queued"], ButtonBox[ PaneSelectorBox[{False-> StyleBox[ StyleBox[ DynamicBox[ToBoxes[ FEPrivate`FrontEndResource["FEStrings", "sizeShowMore"], StandardForm], ImageSizeCache->{61., {0., 8.}}], StripOnInput->False, DynamicUpdating->True], "OSLControl", StripOnInput->False], True-> StyleBox[ StyleBox[ DynamicBox[ToBoxes[ FEPrivate`FrontEndResource["FEStrings", "sizeShowMore"], StandardForm]], StripOnInput->False, DynamicUpdating->True], "OSLControlActive", StripOnInput->False]}, Dynamic[ CurrentValue["MouseOver"]], Alignment->Center, FrameMargins->0, ImageSize->{Automatic, 25}], Appearance->None, BaselinePosition->Baseline, ButtonFunction:>OutputSizeLimit`ButtonFunction[ OutputSizeLimit`Defer, 141, 30935294013293723379, 5 2], Enabled->True, Evaluator->Automatic, Method->"Queued"], ButtonBox[ PaneSelectorBox[{False-> StyleBox[ StyleBox[ DynamicBox[ToBoxes[ FEPrivate`FrontEndResource["FEStrings", "sizeShowAll"], StandardForm], ImageSizeCache->{46., {0., 8.}}], StripOnInput->False, DynamicUpdating->True], "OSLControl", StripOnInput->False], True-> StyleBox[ StyleBox[ DynamicBox[ToBoxes[ FEPrivate`FrontEndResource["FEStrings", "sizeShowAll"], StandardForm]], StripOnInput->False, DynamicUpdating->True], "OSLControlActive", StripOnInput->False]}, Dynamic[ CurrentValue["MouseOver"]], Alignment->Center, FrameMargins->0, ImageSize->{Automatic, 25}], Appearance->None, BaselinePosition->Baseline, ButtonFunction:>OutputSizeLimit`ButtonFunction[ OutputSizeLimit`Defer, 141, 30935294013293723379, Infinity], Enabled->True, Evaluator->Automatic, Method->"Queued"], ButtonBox[ PaneSelectorBox[{False-> StyleBox[ StyleBox[ DynamicBox[ToBoxes[ FEPrivate`FrontEndResource["FEStrings", "sizeChangeLimit"], StandardForm], ImageSizeCache->{83., {0., 8.}}], StripOnInput->False, DynamicUpdating->True], "OSLControl", StripOnInput->False], True-> StyleBox[ StyleBox[ DynamicBox[ToBoxes[ FEPrivate`FrontEndResource["FEStrings", "sizeChangeLimit"], StandardForm]], StripOnInput->False, DynamicUpdating->True], "OSLControlActive", StripOnInput->False]}, Dynamic[ CurrentValue["MouseOver"]], Alignment->Center, FrameMargins->0, ImageSize->{Automatic, 25}], Appearance->None, BaselinePosition->Baseline, ButtonFunction:>FrontEndExecute[{ FrontEnd`SetOptions[ FrontEnd`$FrontEnd, FrontEnd`PreferencesSettings -> {"Page" -> "Advanced"}], FrontEnd`FrontEndToken["PreferencesDialog"]}], Evaluator->None, Method->"Preemptive"]} }, AutoDelete->False, FrameStyle->GrayLevel[0.85], GridBoxDividers->{"Columns" -> {False, {True}}}, GridBoxItemSize->{"Columns" -> {{Automatic}}, "Rows" -> {{Automatic}}}, GridBoxSpacings->{"Columns" -> {{2}}}]} }, DefaultBaseStyle->"Column", GridBoxAlignment->{ "Columns" -> {{Left}}, "ColumnsIndexed" -> {}, "Rows" -> {{Baseline}}, "RowsIndexed" -> {}}, GridBoxDividers->{ "Columns" -> {{False}}, "ColumnsIndexed" -> {}, "Rows" -> {{False}}, "RowsIndexed" -> {}}, GridBoxItemSize->{ "Columns" -> {{Automatic}}, "ColumnsIndexed" -> {}, "Rows" -> {{1.}}, "RowsIndexed" -> {}}, GridBoxSpacings->{"Columns" -> { Offset[0.27999999999999997`], { Offset[0.5599999999999999]}, Offset[0.27999999999999997`]}, "ColumnsIndexed" -> {}, "Rows" -> { Offset[0.2], Offset[1.2], { Offset[0.4]}, Offset[0.2]}, "RowsIndexed" -> {}}], BaseStyle->"OutputSizeLimit", FrameMargins->{{12, 12}, {0, 15}}, FrameStyle->GrayLevel[0.85], RoundingRadius->5, StripOnInput->False], Deploy, DefaultBaseStyle->"Deploy"], If[30935294013293723379 === $SessionID, Out[141], Message[ MessageName[Syntax, "noinfoker"]]; Missing["NotAvailable"]; Null]]], "Output", CellChangeTimes->{{3.734358662284535*^9, 3.73435870049967*^9}, { 3.734358744797121*^9, 3.734358807935054*^9}}, CellLabel-> "Out[141]=",ExpressionUUID->"f7409e5e-82fc-4b61-b8cb-344a548f2b33"] }, Open ]] }, WindowSize->{932, 1044}, WindowMargins->{{18, Automatic}, {Automatic, 18}}, FrontEndVersion->"11.3 for Linux x86 (64-bit) (March 6, 2018)", StyleDefinitions->"Default.nb" ] (* End of Notebook Content *) (* Internal cache information *) (*CellTagsOutline CellTagsIndex->{} *) (*CellTagsIndex CellTagsIndex->{} *) (*NotebookFileOutline Notebook[{ Cell[558, 20, 678, 16, 34, "Input",ExpressionUUID->"cac6654a-3c27-42b6-bbd5-5239c5df5c3b"], Cell[1239, 38, 1684, 43, 78, "Input",ExpressionUUID->"19d4accf-8194-401d-8675-3877ed3ddff3"], Cell[2926, 83, 551, 14, 31, "Input",ExpressionUUID->"d2362d48-1e47-4024-84eb-f7df8b2144b8"], Cell[CellGroupData[{ Cell[3502, 101, 504, 12, 31, "Input",ExpressionUUID->"c9d601ae-0194-4ee1-8bb9-1d47a0e08241"], Cell[4009, 115, 1103, 25, 59, "Output",ExpressionUUID->"002a6413-df1b-4458-aa32-43b079bd8dee"] }, Open ]], Cell[5127, 143, 1253, 26, 31, "Input",ExpressionUUID->"b09286f6-c846-44a0-80a1-7b04a057b628"], Cell[CellGroupData[{ Cell[6405, 173, 594, 12, 39, "Input",ExpressionUUID->"78504373-95bf-4da0-82a1-e5822cb71c01"], Cell[7002, 187, 12109, 354, 238, "Output",ExpressionUUID->"46945c0b-e240-4c95-a92c-0c4d9ab027bd"] }, Open ]], Cell[CellGroupData[{ Cell[19148, 546, 510, 12, 36, "Input",ExpressionUUID->"97f199bc-0af4-44b6-aa4a-fcb6cd45e868"], Cell[19661, 560, 581, 9, 35, "Output",ExpressionUUID->"0d8d0d0e-edfa-449a-bb6b-bdeff666bd45"] }, Open ]], Cell[CellGroupData[{ Cell[20279, 574, 351, 9, 36, "Input",ExpressionUUID->"174a3e7f-be6f-47e2-8dab-4d7035aaccc1"], Cell[20633, 585, 216, 4, 35, "Output",ExpressionUUID->"11e30345-b1c9-47bd-90e5-22985ea861c8"] }, Open ]], Cell[CellGroupData[{ Cell[20886, 594, 638, 17, 39, "Input",ExpressionUUID->"1b4380fb-27f4-452b-bb0b-712dad575f2f"], Cell[21527, 613, 518, 14, 40, "Output",ExpressionUUID->"49bc1d2c-058f-4279-988d-355c973860cb"] }, Open ]], Cell[CellGroupData[{ Cell[22082, 632, 486, 12, 31, "Input",ExpressionUUID->"2a832002-adf6-4c60-bb56-92e9cf4c32f6"], Cell[22571, 646, 2012, 58, 67, "Output",ExpressionUUID->"a27ce88a-34bd-4f78-ad90-ce5d10de1040"] }, Open ]], Cell[24598, 707, 1126, 33, 55, "Input",ExpressionUUID->"180ef9a5-3287-46d1-ae4b-ae9e85480093"], Cell[CellGroupData[{ Cell[25749, 744, 678, 19, 33, "Input",ExpressionUUID->"e2107056-243c-408e-8546-20a935f61e1d"], Cell[26430, 765, 11753, 306, 219, "Output",ExpressionUUID->"f7409e5e-82fc-4b61-b8cb-344a548f2b33"] }, Open ]] } ] *) (* End of internal cache information *)