(*CacheID: 234*) (* Internal cache information: NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 145, 0] NotebookDataLength[ 74846, 2428] NotebookOptionsPosition[ 65779, 2113] NotebookOutlinePosition[ 66785, 2148] CellTagsIndexPosition[ 66573, 2140] WindowFrame->Normal ContainsDynamic->True *) (* Beginning of Notebook Content *) Notebook[{ Cell[CellGroupData[{ Cell[TextData[{ StyleBox["Solutions: ", FontColor->RGBColor[0, 0, 1]], StyleBox["Mathematica", FontSlant->"Italic", FontColor->RGBColor[0, 0, 1]], StyleBox[" 6 Lab 3", FontColor->RGBColor[0, 0, 1]] }], "Subtitle", CellChangeTimes->{{3.396207261561866*^9, 3.396207292942472*^9}, 3.396210579961336*^9, 3.396217977491781*^9, 3.396230358125164*^9, 3.3963075557269373`*^9}, FontWeight->"Bold"], Cell[RawData[""], "Subsubtitle"], Cell[CellGroupData[{ Cell[TextData[{ StyleBox["Problem 1", FontColor->RGBColor[1, 0, 0]], ":" }], "Subsubtitle", CellChangeTimes->{{3.396483280580566*^9, 3.396483283893361*^9}}], Cell[CellGroupData[{ Cell[BoxData[{ RowBox[{"TrigExpand", "[", RowBox[{"Tan", "[", RowBox[{"2", "\[Theta]"}], "]"}], "]"}], "\[IndentingNewLine]", RowBox[{"TrigExpand", "[", RowBox[{"Cos", "[", RowBox[{"6", "\[Theta]"}], "]"}], "]"}]}], "Input", CellChangeTimes->{{3.396229244026065*^9, 3.396229244527893*^9}}], Cell[BoxData[ FractionBox[ RowBox[{"2", " ", RowBox[{"Cos", "[", "\[Theta]", "]"}], " ", RowBox[{"Sin", "[", "\[Theta]", "]"}]}], RowBox[{ SuperscriptBox[ RowBox[{"Cos", "[", "\[Theta]", "]"}], "2"], "-", SuperscriptBox[ RowBox[{"Sin", "[", "\[Theta]", "]"}], "2"]}]]], "Output", CellChangeTimes->{3.396229252616992*^9, 3.396230488540641*^9}], Cell[BoxData[ RowBox[{ SuperscriptBox[ RowBox[{"Cos", "[", "\[Theta]", "]"}], "6"], "-", RowBox[{"15", " ", SuperscriptBox[ RowBox[{"Cos", "[", "\[Theta]", "]"}], "4"], " ", SuperscriptBox[ RowBox[{"Sin", "[", "\[Theta]", "]"}], "2"]}], "+", RowBox[{"15", " ", SuperscriptBox[ RowBox[{"Cos", "[", "\[Theta]", "]"}], "2"], " ", SuperscriptBox[ RowBox[{"Sin", "[", "\[Theta]", "]"}], "4"]}], "-", SuperscriptBox[ RowBox[{"Sin", "[", "\[Theta]", "]"}], "6"]}]], "Output", CellChangeTimes->{3.396229252616992*^9, 3.396230488663989*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[{ RowBox[{"Simplify", "[", FractionBox[ RowBox[{"2", " ", RowBox[{"Cos", "[", "\[Theta]", "]"}], " ", RowBox[{"Sin", "[", "\[Theta]", "]"}]}], RowBox[{ SuperscriptBox[ RowBox[{"Cos", "[", "\[Theta]", "]"}], "2"], "-", SuperscriptBox[ RowBox[{"Sin", "[", "\[Theta]", "]"}], "2"]}]], "]"}], "\[IndentingNewLine]", RowBox[{"Simplify", "[", RowBox[{ SuperscriptBox[ RowBox[{"Cos", "[", "\[Theta]", "]"}], "6"], "-", RowBox[{"15", " ", SuperscriptBox[ RowBox[{"Cos", "[", "\[Theta]", "]"}], "4"], " ", SuperscriptBox[ RowBox[{"Sin", "[", "\[Theta]", "]"}], "2"]}], "+", RowBox[{"15", " ", SuperscriptBox[ RowBox[{"Cos", "[", "\[Theta]", "]"}], "2"], " ", SuperscriptBox[ RowBox[{"Sin", "[", "\[Theta]", "]"}], "4"]}], "-", SuperscriptBox[ RowBox[{"Sin", "[", "\[Theta]", "]"}], "6"]}], "]"}]}], "Input", CellChangeTimes->{{3.396229272970181*^9, 3.396229297074024*^9}, { 3.3964835277572527`*^9, 3.396483537083359*^9}}], Cell[BoxData[ RowBox[{"Tan", "[", RowBox[{"2", " ", "\[Theta]"}], "]"}]], "Output", CellChangeTimes->{3.3962292988875923`*^9, 3.3962304917106943`*^9, 3.3964835406521683`*^9}], Cell[BoxData[ RowBox[{"Cos", "[", RowBox[{"6", " ", "\[Theta]"}], "]"}]], "Output", CellChangeTimes->{3.3962292988875923`*^9, 3.3962304917106943`*^9, 3.396483540879691*^9}] }, Open ]], Cell["\<\ \ \>", "Text", CellChangeTimes->{3.396229325771659*^9}] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ StyleBox["Problem 2", FontColor->RGBColor[1, 0, 0]], ":" }], "Subsubtitle", CellChangeTimes->{{3.3962293322724953`*^9, 3.3962293390217133`*^9}}], Cell[CellGroupData[{ Cell[BoxData[{ RowBox[{"TrigReduce", "[", SuperscriptBox[ RowBox[{"Sin", "[", "\[Alpha]", "]"}], "2"], "]"}], "\[IndentingNewLine]", RowBox[{"TrigReduce", "[", SuperscriptBox[ RowBox[{"Tan", "[", FractionBox["\[Theta]", "2"], "]"}], "2"], "]"}]}], "Input", CellChangeTimes->{{3.396229372747871*^9, 3.396229373462263*^9}}], Cell[BoxData[ RowBox[{ FractionBox["1", "2"], " ", RowBox[{"(", RowBox[{"1", "-", RowBox[{"Cos", "[", RowBox[{"2", " ", "\[Alpha]"}], "]"}]}], ")"}]}]], "Output", CellChangeTimes->{3.396229376283249*^9, 3.396230494553771*^9}], Cell[BoxData[ FractionBox[ RowBox[{"1", "-", RowBox[{"Cos", "[", "\[Theta]", "]"}]}], RowBox[{"1", "+", RowBox[{"Cos", "[", "\[Theta]", "]"}]}]]], "Output", CellChangeTimes->{3.396229376283249*^9, 3.396230494569388*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[{ RowBox[{"Simplify", "[", RowBox[{ FractionBox["1", "2"], " ", RowBox[{"(", RowBox[{"1", "-", RowBox[{"Cos", "[", RowBox[{"2", " ", "\[Alpha]"}], "]"}]}], ")"}]}], "]"}], "\[IndentingNewLine]", RowBox[{"Simplify", "[", FractionBox[ RowBox[{"1", "-", RowBox[{"Cos", "[", "\[Theta]", "]"}]}], RowBox[{"1", "+", RowBox[{"Cos", "[", "\[Theta]", "]"}]}]], "]"}]}], "Input", CellChangeTimes->{{3.39648341537768*^9, 3.3964834310412617`*^9}, { 3.39648346960257*^9, 3.396483486345656*^9}}], Cell[BoxData[ SuperscriptBox[ RowBox[{"Sin", "[", "\[Alpha]", "]"}], "2"]], "Output", CellChangeTimes->{{3.39648343685616*^9, 3.396483457463565*^9}, 3.3964834883041477`*^9}], Cell[BoxData[ SuperscriptBox[ RowBox[{"Tan", "[", FractionBox["\[Theta]", "2"], "]"}], "2"]], "Output", CellChangeTimes->{{3.39648343685616*^9, 3.396483457463565*^9}, 3.396483488583991*^9}] }, Open ]], Cell["\<\ \ \>", "Text", CellChangeTimes->{3.396229543645007*^9}] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ StyleBox["Problem 3", FontColor->RGBColor[1, 0, 0]], ":" }], "Subsubtitle", CellChangeTimes->{{3.3962295489439793`*^9, 3.396229555481319*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Hyperlink", "[", RowBox[{ "\"\\"", ",", "\"\\""}], "]"}]], "Input", CellChangeTimes->{{3.396229574751835*^9, 3.396229668482975*^9}}], Cell[BoxData[ TagBox[ ButtonBox[ PaneSelectorBox[{False->"\<\"MagicSquare\"\>", True-> StyleBox["\<\"MagicSquare\"\>", "HyperlinkActive"]}, Dynamic[ CurrentValue["MouseOver"]], BaselinePosition->Baseline, FrameMargins->0, ImageSize->Automatic], BaseStyle->"Hyperlink", ButtonData->{ URL["http://mathworld.wolfram.com/DuerersMagicSquare.html"], None}, ButtonNote->"http://mathworld.wolfram.com/DuerersMagicSquare.html"], Annotation[#, "http://mathworld.wolfram.com/DuerersMagicSquare.html", "Hyperlink"]& ]], "Output", CellChangeTimes->{3.396229678599649*^9, 3.396230499971781*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"M", "=", RowBox[{"(", GridBox[{ {"16", "3", "2", "13"}, {"5", "10", "11", "8"}, {"9", "6", "7", "12"}, {"4", "15", "14", "1"} }], ")"}]}]], "Input"], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{"16", ",", "3", ",", "2", ",", "13"}], "}"}], ",", RowBox[{"{", RowBox[{"5", ",", "10", ",", "11", ",", "8"}], "}"}], ",", RowBox[{"{", RowBox[{"9", ",", "6", ",", "7", ",", "12"}], "}"}], ",", RowBox[{"{", RowBox[{"4", ",", "15", ",", "14", ",", "1"}], "}"}]}], "}"}]], "Output", CellChangeTimes->{3.396229775937223*^9, 3.396230504464281*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Tr", "[", "M", "]"}]], "Input"], Cell[BoxData["34"], "Output", CellChangeTimes->{3.396229835059676*^9, 3.396230507028364*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Det", "[", "M", "]"}]], "Input"], Cell[BoxData["0"], "Output", CellChangeTimes->{3.396229838270314*^9, 3.396230511865601*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Eigenvalues", "[", "M", "]"}]], "Input"], Cell[BoxData[ RowBox[{"{", RowBox[{"34", ",", RowBox[{"-", "8"}], ",", "8", ",", "0"}], "}"}]], "Output", CellChangeTimes->{3.396229841569015*^9, 3.396230513766077*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Eigenvectors", "[", "M", "]"}]], "Input"], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{"1", ",", "1", ",", "1", ",", "1"}], "}"}], ",", RowBox[{"{", RowBox[{ RowBox[{"-", "1"}], ",", "0", ",", RowBox[{"-", "1"}], ",", "2"}], "}"}], ",", RowBox[{"{", RowBox[{ RowBox[{"-", "2"}], ",", "1", ",", "0", ",", "1"}], "}"}], ",", RowBox[{"{", RowBox[{ RowBox[{"-", "1"}], ",", "3", ",", RowBox[{"-", "3"}], ",", "1"}], "}"}]}], "}"}]], "Output", CellChangeTimes->{3.396229845909422*^9, 3.396230515370919*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[{ RowBox[{"a", "=", RowBox[{"{", RowBox[{"1", ",", "1", ",", "1", ",", "1"}], "}"}]}], "\[IndentingNewLine]", RowBox[{"b", "=", RowBox[{"{", RowBox[{ RowBox[{"-", "1"}], ",", "0", ",", RowBox[{"-", "1"}], ",", "2"}], "}"}]}], "\[IndentingNewLine]", RowBox[{"c", "=", RowBox[{"{", RowBox[{ RowBox[{"-", "2"}], ",", "1", ",", "0", ",", "1"}], "}"}]}], "\[IndentingNewLine]", RowBox[{"d", "=", RowBox[{"{", RowBox[{ RowBox[{"-", "1"}], ",", "3", ",", RowBox[{"-", "3"}], ",", "1"}], "}"}]}]}], "Input"], Cell[BoxData[ RowBox[{"{", RowBox[{"1", ",", "1", ",", "1", ",", "1"}], "}"}]], "Output", CellChangeTimes->{3.396229896877222*^9, 3.396230517993023*^9, 3.396483957468405*^9}], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{"-", "1"}], ",", "0", ",", RowBox[{"-", "1"}], ",", "2"}], "}"}]], "Output", CellChangeTimes->{3.396229896877222*^9, 3.396230517993023*^9, 3.396483957498549*^9}], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{"-", "2"}], ",", "1", ",", "0", ",", "1"}], "}"}]], "Output", CellChangeTimes->{3.396229896877222*^9, 3.396230517993023*^9, 3.3964839575300007`*^9}], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{"-", "1"}], ",", "3", ",", RowBox[{"-", "3"}], ",", "1"}], "}"}]], "Output", CellChangeTimes->{3.396229896877222*^9, 3.396230517993023*^9, 3.396483957584741*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[{ RowBox[{"a", ".", "a"}], "\[IndentingNewLine]", RowBox[{"a", ".", "b"}], "\[IndentingNewLine]", RowBox[{"a", ".", "c"}], "\[IndentingNewLine]", RowBox[{"a", ".", "d"}], "\[IndentingNewLine]", RowBox[{"b", ".", "b"}], "\[IndentingNewLine]", RowBox[{"b", ".", "c"}], "\[IndentingNewLine]", RowBox[{"b", ".", "d"}], "\[IndentingNewLine]", RowBox[{"c", ".", "c"}], "\[IndentingNewLine]", RowBox[{"c", ".", "d"}], "\[IndentingNewLine]", RowBox[{"d", ".", "d"}]}], "Input"], Cell[BoxData["4"], "Output", CellChangeTimes->{3.3962299005476627`*^9, 3.3962305211462708`*^9, 3.396483962597546*^9}], Cell[BoxData["0"], "Output", CellChangeTimes->{3.3962299005476627`*^9, 3.3962305211462708`*^9, 3.396483962629788*^9}], Cell[BoxData["0"], "Output", CellChangeTimes->{3.3962299005476627`*^9, 3.3962305211462708`*^9, 3.396483962644285*^9}], Cell[BoxData["0"], "Output", CellChangeTimes->{3.3962299005476627`*^9, 3.3962305211462708`*^9, 3.3964839627509212`*^9}], Cell[BoxData["6"], "Output", CellChangeTimes->{3.3962299005476627`*^9, 3.3962305211462708`*^9, 3.396483962969584*^9}], Cell[BoxData["4"], "Output", CellChangeTimes->{3.3962299005476627`*^9, 3.3962305211462708`*^9, 3.39648396306879*^9}], Cell[BoxData["6"], "Output", CellChangeTimes->{3.3962299005476627`*^9, 3.3962305211462708`*^9, 3.39648396317758*^9}], Cell[BoxData["6"], "Output", CellChangeTimes->{3.3962299005476627`*^9, 3.3962305211462708`*^9, 3.396483963242888*^9}], Cell[BoxData["6"], "Output", CellChangeTimes->{3.3962299005476627`*^9, 3.3962305211462708`*^9, 3.396483963301984*^9}], Cell[BoxData["20"], "Output", CellChangeTimes->{3.3962299005476627`*^9, 3.3962305211462708`*^9, 3.396483963369001*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[{ RowBox[{ RowBox[{"M", ".", "a"}], "\[Equal]", RowBox[{"34", "a"}]}], "\[IndentingNewLine]", RowBox[{ RowBox[{"M", ".", "b"}], "\[Equal]", RowBox[{ RowBox[{"-", "8"}], " ", "b"}]}], "\[IndentingNewLine]", RowBox[{ RowBox[{"M", ".", "c"}], "\[Equal]", RowBox[{"8", "c"}]}], "\[IndentingNewLine]", RowBox[{ RowBox[{"M", ".", "d"}], "\[Equal]", RowBox[{"0", "d"}]}]}], "Input"], Cell[BoxData["True"], "Output", CellChangeTimes->{3.396229904031138*^9, 3.396230523857772*^9}], Cell[BoxData["True"], "Output", CellChangeTimes->{3.396229904031138*^9, 3.3962305239399023`*^9}], Cell[BoxData["True"], "Output", CellChangeTimes->{3.396229904031138*^9, 3.396230524027459*^9}], Cell[BoxData["True"], "Output", CellChangeTimes->{3.396229904031138*^9, 3.396230524097513*^9}] }, Open ]], Cell["\<\ \ \>", "Text", CellChangeTimes->{3.396229916465781*^9}] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ StyleBox["Problem 4", FontColor->RGBColor[1, 0, 0]], ":" }], "Subsubtitle", CellChangeTimes->{{3.396229923209708*^9, 3.396229928663159*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Hyperlink", "[", RowBox[{ "\"\\"", ",", "\"\\""}], "]"}]], "Input", CellChangeTimes->{{3.396229574751835*^9, 3.396229668482975*^9}, { 3.396230136734187*^9, 3.396230187445108*^9}}], Cell[BoxData[ TagBox[ ButtonBox[ PaneSelectorBox[{False->"\<\"LatinSquare\"\>", True-> StyleBox["\<\"LatinSquare\"\>", "HyperlinkActive"]}, Dynamic[ CurrentValue["MouseOver"]], BaselinePosition->Baseline, FrameMargins->0, ImageSize->Automatic], BaseStyle->"Hyperlink", ButtonData->{ URL["http://en.wikipedia.org/wiki/Latin_square"], None}, ButtonNote->"http://en.wikipedia.org/wiki/Latin_square"], Annotation[#, "http://en.wikipedia.org/wiki/Latin_square", "Hyperlink"]& ]], "Output", CellChangeTimes->{3.396230195558777*^9, 3.3962305301040373`*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"latin1", "=", RowBox[{"(", GridBox[{ {"1", "2", "3", "4"}, {"2", "1", "4", "3"}, {"4", "3", "1", "2"}, {"3", "4", "2", "1"} }], ")"}]}]], "Input"], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{"1", ",", "2", ",", "3", ",", "4"}], "}"}], ",", RowBox[{"{", RowBox[{"2", ",", "1", ",", "4", ",", "3"}], "}"}], ",", RowBox[{"{", RowBox[{"4", ",", "3", ",", "1", ",", "2"}], "}"}], ",", RowBox[{"{", RowBox[{"3", ",", "4", ",", "2", ",", "1"}], "}"}]}], "}"}]], "Output", CellChangeTimes->{3.396298214480112*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Det", "[", "latin1", "]"}]], "Input"], Cell[BoxData[ RowBox[{"-", "80"}]], "Output", CellChangeTimes->{3.396298233694605*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Eigenvalues", "[", "latin1", "]"}]], "Input"], Cell[BoxData[ RowBox[{"{", RowBox[{"10", ",", RowBox[{"-", "4"}], ",", RowBox[{ RowBox[{"-", "1"}], "+", "\[ImaginaryI]"}], ",", RowBox[{ RowBox[{"-", "1"}], "-", "\[ImaginaryI]"}]}], "}"}]], "Output", CellChangeTimes->{3.396298290164815*^9}] }, Open ]], Cell["\<\ \ \>", "Text", CellChangeTimes->{3.396298339373116*^9}] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ StyleBox["Problem 5", FontColor->RGBColor[1, 0, 0]], ":" }], "Subsubtitle", CellChangeTimes->{{3.396298344597847*^9, 3.396298349526299*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"latin2", "=", RowBox[{"(", GridBox[{ {"1", "4", "2", "5", "3"}, {"4", "2", "5", "3", "1"}, {"2", "5", "3", "1", "4"}, {"5", "3", "1", "4", "2"}, {"3", "1", "4", "2", "5"} }], ")"}]}]], "Input"], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{"1", ",", "4", ",", "2", ",", "5", ",", "3"}], "}"}], ",", RowBox[{"{", RowBox[{"4", ",", "2", ",", "5", ",", "3", ",", "1"}], "}"}], ",", RowBox[{"{", RowBox[{"2", ",", "5", ",", "3", ",", "1", ",", "4"}], "}"}], ",", RowBox[{"{", RowBox[{"5", ",", "3", ",", "1", ",", "4", ",", "2"}], "}"}], ",", RowBox[{"{", RowBox[{"3", ",", "1", ",", "4", ",", "2", ",", "5"}], "}"}]}], "}"}]], "Output", CellChangeTimes->{3.396298449904168*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Det", "[", "latin2", "]"}]], "Input"], Cell[BoxData["1875"], "Output", CellChangeTimes->{3.396298464101677*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Eigenvalues", "[", "latin2", "]"}]], "Input"], Cell[BoxData[ RowBox[{"{", RowBox[{"15", ",", RowBox[{"-", SqrtBox[ RowBox[{ FractionBox["25", "2"], "+", FractionBox[ RowBox[{"5", " ", SqrtBox["5"]}], "2"]}]]}], ",", SqrtBox[ RowBox[{ FractionBox["25", "2"], "+", FractionBox[ RowBox[{"5", " ", SqrtBox["5"]}], "2"]}]], ",", RowBox[{"-", SqrtBox[ RowBox[{ FractionBox["5", "2"], " ", RowBox[{"(", RowBox[{"5", "-", SqrtBox["5"]}], ")"}]}]]}], ",", SqrtBox[ RowBox[{ FractionBox["5", "2"], " ", RowBox[{"(", RowBox[{"5", "-", SqrtBox["5"]}], ")"}]}]]}], "}"}]], "Output", CellChangeTimes->{3.396298477994639*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"N", "[", "%", "]"}]], "Input"], Cell[BoxData[ RowBox[{"{", RowBox[{"15.`", ",", RowBox[{"-", "4.2532540417602`"}], ",", "4.2532540417602`", ",", RowBox[{"-", "2.628655560595668`"}], ",", "2.628655560595668`"}], "}"}]], "Output", CellChangeTimes->{3.396298495646412*^9}] }, Open ]], Cell["\<\ \ \>", "Text", CellChangeTimes->{3.396298547468658*^9}] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ StyleBox["Problem 6", FontColor->RGBColor[1, 0, 0]], ":" }], "Subsubtitle", CellChangeTimes->{{3.3962985527075768`*^9, 3.396298558857376*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"latin3", "=", RowBox[{"(", GridBox[{ {"1", "4", "2", "5", "3"}, {"4", StyleBox["3", FontColor->RGBColor[1, 0, 0]], "5", "3", "1"}, {"2", "5", "3", "1", "4"}, {"5", "3", "1", "4", "2"}, {"3", "1", "4", "2", "5"} }], ")"}]}]], "Input"], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{"1", ",", "4", ",", "2", ",", "5", ",", "3"}], "}"}], ",", RowBox[{"{", RowBox[{"4", ",", "3", ",", "5", ",", "3", ",", "1"}], "}"}], ",", RowBox[{"{", RowBox[{"2", ",", "5", ",", "3", ",", "1", ",", "4"}], "}"}], ",", RowBox[{"{", RowBox[{"5", ",", "3", ",", "1", ",", "4", ",", "2"}], "}"}], ",", RowBox[{"{", RowBox[{"3", ",", "1", ",", "4", ",", "2", ",", "5"}], "}"}]}], "}"}]], "Output", CellChangeTimes->{3.396298615862154*^9, 3.396484547836939*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Det", "[", "latin3", "]"}]], "Input"], Cell[BoxData["1900"], "Output", CellChangeTimes->{3.396298633831863*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Eigenvalues", "[", "latin3", "]"}]], "Input"], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{"Root", "[", RowBox[{ RowBox[{ RowBox[{ RowBox[{"-", "1900"}], "-", RowBox[{"25", " ", "#1"}], "+", RowBox[{"375", " ", SuperscriptBox["#1", "2"]}], "-", RowBox[{"12", " ", SuperscriptBox["#1", "3"]}], "-", RowBox[{"16", " ", SuperscriptBox["#1", "4"]}], "+", SuperscriptBox["#1", "5"]}], "&"}], ",", "5"}], "]"}], ",", RowBox[{"Root", "[", RowBox[{ RowBox[{ RowBox[{ RowBox[{"-", "1900"}], "-", RowBox[{"25", " ", "#1"}], "+", RowBox[{"375", " ", SuperscriptBox["#1", "2"]}], "-", RowBox[{"12", " ", SuperscriptBox["#1", "3"]}], "-", RowBox[{"16", " ", SuperscriptBox["#1", "4"]}], "+", SuperscriptBox["#1", "5"]}], "&"}], ",", "4"}], "]"}], ",", RowBox[{"Root", "[", RowBox[{ RowBox[{ RowBox[{ RowBox[{"-", "1900"}], "-", RowBox[{"25", " ", "#1"}], "+", RowBox[{"375", " ", SuperscriptBox["#1", "2"]}], "-", RowBox[{"12", " ", SuperscriptBox["#1", "3"]}], "-", RowBox[{"16", " ", SuperscriptBox["#1", "4"]}], "+", SuperscriptBox["#1", "5"]}], "&"}], ",", "1"}], "]"}], ",", RowBox[{"Root", "[", RowBox[{ RowBox[{ RowBox[{ RowBox[{"-", "1900"}], "-", RowBox[{"25", " ", "#1"}], "+", RowBox[{"375", " ", SuperscriptBox["#1", "2"]}], "-", RowBox[{"12", " ", SuperscriptBox["#1", "3"]}], "-", RowBox[{"16", " ", SuperscriptBox["#1", "4"]}], "+", SuperscriptBox["#1", "5"]}], "&"}], ",", "3"}], "]"}], ",", RowBox[{"Root", "[", RowBox[{ RowBox[{ RowBox[{ RowBox[{"-", "1900"}], "-", RowBox[{"25", " ", "#1"}], "+", RowBox[{"375", " ", SuperscriptBox["#1", "2"]}], "-", RowBox[{"12", " ", SuperscriptBox["#1", "3"]}], "-", RowBox[{"16", " ", SuperscriptBox["#1", "4"]}], "+", SuperscriptBox["#1", "5"]}], "&"}], ",", "2"}], "]"}]}], "}"}]], "Output", CellChangeTimes->{3.3962986463150387`*^9, 3.3964845575138607`*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"N", "[", "%", "]"}]], "Input", CellChangeTimes->{{3.396484561160734*^9, 3.3964845632355556`*^9}}], Cell[BoxData[ RowBox[{"{", RowBox[{"15.21070409105077`", ",", "4.266017568849558`", ",", RowBox[{"-", "3.906367208312248`"}], ",", "2.961054077460109`", ",", RowBox[{"-", "2.5314085290481896`"}]}], "}"}]], "Output", CellChangeTimes->{3.396484564759507*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"CharacteristicPolynomial", "[", RowBox[{"latin3", ",", "x"}], "]"}]], "Input"], Cell[BoxData[ RowBox[{"1900", "+", RowBox[{"25", " ", "x"}], "-", RowBox[{"375", " ", SuperscriptBox["x", "2"]}], "+", RowBox[{"12", " ", SuperscriptBox["x", "3"]}], "+", RowBox[{"16", " ", SuperscriptBox["x", "4"]}], "-", SuperscriptBox["x", "5"]}]], "Output", CellChangeTimes->{3.396298670899951*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"spectrum", "=", RowBox[{"x", "/.", RowBox[{"NSolve", "[", RowBox[{ RowBox[{"%", "\[Equal]", "0"}], ",", "x"}], "]"}]}]}]], "Input"], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{"-", "3.906367208312248`"}], ",", RowBox[{"-", "2.5314085290481896`"}], ",", "2.961054077460109`", ",", "4.266017568849558`", ",", "15.210704091050768`"}], "}"}]], "Output", CellChangeTimes->{3.3962986860549*^9}] }, Open ]], Cell["\<\ \ \>", "Text", CellChangeTimes->{3.3962987448138647`*^9}] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ StyleBox["Problem 7", FontColor->RGBColor[1, 0, 0]], ":" }], "Subsubtitle", CellChangeTimes->{{3.396298752205175*^9, 3.396298757924367*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"S", "=", RowBox[{"(", GridBox[{ {"2", RowBox[{"-", "1"}], "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0"}, { RowBox[{"-", "1"}], "2", RowBox[{"-", "1"}], "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0"}, {"0", RowBox[{"-", "1"}], "2", RowBox[{"-", "1"}], "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0"}, {"0", "0", RowBox[{"-", "1"}], "2", RowBox[{"-", "1"}], "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0"}, {"0", "0", "0", RowBox[{"-", "1"}], "2", RowBox[{"-", "1"}], "0", "0", "0", "0", "0", "0", "0", "0", "0", "0"}, {"0", "0", "0", "0", RowBox[{"-", "1"}], "2", RowBox[{"-", "1"}], "0", "0", "0", "0", "0", "0", "0", "0", "0"}, {"0", "0", "0", "0", "0", RowBox[{"-", "1"}], "2", RowBox[{"-", "1"}], "0", "0", "0", "0", "0", "0", "0", "0"}, {"0", "0", "0", "0", "0", "0", RowBox[{"-", "1"}], "2", RowBox[{"-", "1"}], "0", "0", "0", "0", "0", "0", "0"}, {"0", "0", "0", "0", "0", "0", "0", RowBox[{"-", "1"}], "2", RowBox[{"-", "1"}], "0", "0", "0", "0", "0", "0"}, {"0", "0", "0", "0", "0", "0", "0", "0", RowBox[{"-", "1"}], "2", RowBox[{"-", "1"}], "0", "0", "0", "0", "0"}, {"0", "0", "0", "0", "0", "0", "0", "0", "0", RowBox[{"-", "1"}], "2", RowBox[{"-", "1"}], "0", "0", "0", "0"}, {"0", "0", "0", "0", "0", "0", "0", "0", "0", "0", RowBox[{"-", "1"}], "2", RowBox[{"-", "1"}], "0", "0", "0"}, {"0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", RowBox[{"-", "1"}], "2", RowBox[{"-", "1"}], "0", "0"}, {"0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", RowBox[{"-", "1"}], "2", RowBox[{"-", "1"}], "0"}, {"0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", RowBox[{"-", "1"}], "2", RowBox[{"-", "1"}]}, {"0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", RowBox[{"-", "1"}], "2"} }], ")"}]}]], "Input", CellChangeTimes->{{3.3964850350146303`*^9, 3.396485035460663*^9}}], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{"2", ",", RowBox[{"-", "1"}], ",", "0", ",", "0", ",", "0", ",", "0", ",", "0", ",", "0", ",", "0", ",", "0", ",", "0", ",", "0", ",", "0", ",", "0", ",", "0", ",", "0"}], "}"}], ",", RowBox[{"{", RowBox[{ RowBox[{"-", "1"}], ",", "2", ",", RowBox[{"-", "1"}], ",", "0", ",", "0", ",", "0", ",", "0", ",", "0", ",", "0", ",", "0", ",", "0", ",", "0", ",", "0", ",", "0", ",", "0", ",", "0"}], "}"}], ",", RowBox[{"{", RowBox[{"0", ",", RowBox[{"-", "1"}], ",", "2", ",", RowBox[{"-", "1"}], ",", "0", ",", "0", ",", "0", ",", "0", ",", "0", ",", "0", ",", "0", ",", "0", ",", "0", ",", "0", ",", "0", ",", "0"}], "}"}], ",", RowBox[{"{", RowBox[{"0", ",", "0", ",", RowBox[{"-", "1"}], ",", "2", ",", RowBox[{"-", "1"}], ",", "0", ",", "0", ",", "0", ",", "0", ",", "0", ",", "0", ",", "0", ",", "0", ",", "0", ",", "0", ",", "0"}], "}"}], ",", RowBox[{"{", RowBox[{"0", ",", "0", ",", "0", ",", RowBox[{"-", "1"}], ",", "2", ",", RowBox[{"-", "1"}], ",", "0", ",", "0", ",", "0", ",", "0", ",", "0", ",", "0", ",", "0", ",", "0", ",", "0", ",", "0"}], "}"}], ",", RowBox[{"{", RowBox[{"0", ",", "0", ",", "0", ",", "0", ",", RowBox[{"-", "1"}], ",", "2", ",", RowBox[{"-", "1"}], ",", "0", ",", "0", ",", "0", ",", "0", ",", "0", ",", "0", ",", "0", ",", "0", ",", "0"}], "}"}], ",", RowBox[{"{", RowBox[{"0", ",", "0", ",", "0", ",", "0", ",", "0", ",", RowBox[{"-", "1"}], ",", "2", ",", RowBox[{"-", "1"}], ",", "0", ",", "0", ",", "0", ",", "0", ",", "0", ",", "0", ",", "0", ",", "0"}], "}"}], ",", RowBox[{"{", RowBox[{"0", ",", "0", ",", "0", ",", "0", ",", "0", ",", "0", ",", RowBox[{"-", "1"}], ",", "2", ",", RowBox[{"-", "1"}], ",", "0", ",", "0", ",", "0", ",", "0", ",", "0", ",", "0", ",", "0"}], "}"}], ",", RowBox[{"{", RowBox[{ "0", ",", "0", ",", "0", ",", "0", ",", "0", ",", "0", ",", "0", ",", RowBox[{"-", "1"}], ",", "2", ",", RowBox[{"-", "1"}], ",", "0", ",", "0", ",", "0", ",", "0", ",", "0", ",", "0"}], "}"}], ",", RowBox[{"{", RowBox[{ "0", ",", "0", ",", "0", ",", "0", ",", "0", ",", "0", ",", "0", ",", "0", ",", RowBox[{"-", "1"}], ",", "2", ",", RowBox[{"-", "1"}], ",", "0", ",", "0", ",", "0", ",", "0", ",", "0"}], "}"}], ",", RowBox[{"{", RowBox[{ "0", ",", "0", ",", "0", ",", "0", ",", "0", ",", "0", ",", "0", ",", "0", ",", "0", ",", RowBox[{"-", "1"}], ",", "2", ",", RowBox[{"-", "1"}], ",", "0", ",", "0", ",", "0", ",", "0"}], "}"}], ",", RowBox[{"{", RowBox[{ "0", ",", "0", ",", "0", ",", "0", ",", "0", ",", "0", ",", "0", ",", "0", ",", "0", ",", "0", ",", RowBox[{"-", "1"}], ",", "2", ",", RowBox[{"-", "1"}], ",", "0", ",", "0", ",", "0"}], "}"}], ",", RowBox[{"{", RowBox[{ "0", ",", "0", ",", "0", ",", "0", ",", "0", ",", "0", ",", "0", ",", "0", ",", "0", ",", "0", ",", "0", ",", RowBox[{"-", "1"}], ",", "2", ",", RowBox[{"-", "1"}], ",", "0", ",", "0"}], "}"}], ",", RowBox[{"{", RowBox[{ "0", ",", "0", ",", "0", ",", "0", ",", "0", ",", "0", ",", "0", ",", "0", ",", "0", ",", "0", ",", "0", ",", "0", ",", RowBox[{"-", "1"}], ",", "2", ",", RowBox[{"-", "1"}], ",", "0"}], "}"}], ",", RowBox[{"{", RowBox[{ "0", ",", "0", ",", "0", ",", "0", ",", "0", ",", "0", ",", "0", ",", "0", ",", "0", ",", "0", ",", "0", ",", "0", ",", "0", ",", RowBox[{"-", "1"}], ",", "2", ",", RowBox[{"-", "1"}]}], "}"}], ",", RowBox[{"{", RowBox[{ "0", ",", "0", ",", "0", ",", "0", ",", "0", ",", "0", ",", "0", ",", "0", ",", "0", ",", "0", ",", "0", ",", "0", ",", "0", ",", "0", ",", RowBox[{"-", "1"}], ",", "2"}], "}"}]}], "}"}]], "Output", CellChangeTimes->{3.3962988086385517`*^9, 3.396485040455741*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Eigenvalues", "[", "S", "]"}]], "Input", CellChangeTimes->{{3.396485045561075*^9, 3.396485050178421*^9}}], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{"2", "+", RowBox[{"\[Sqrt]", RowBox[{"(", RowBox[{"4", "+", RowBox[{"Root", "[", RowBox[{ RowBox[{ RowBox[{"17", "+", RowBox[{"204", " ", "#1"}], "+", RowBox[{"714", " ", SuperscriptBox["#1", "2"]}], "+", RowBox[{"1122", " ", SuperscriptBox["#1", "3"]}], "+", RowBox[{"935", " ", SuperscriptBox["#1", "4"]}], "+", RowBox[{"442", " ", SuperscriptBox["#1", "5"]}], "+", RowBox[{"119", " ", SuperscriptBox["#1", "6"]}], "+", RowBox[{"17", " ", SuperscriptBox["#1", "7"]}], "+", SuperscriptBox["#1", "8"]}], "&"}], ",", "8"}], "]"}]}], ")"}]}]}], ",", RowBox[{"2", "+", RowBox[{"\[Sqrt]", RowBox[{"(", RowBox[{"4", "+", RowBox[{"Root", "[", RowBox[{ RowBox[{ RowBox[{"17", "+", RowBox[{"204", " ", "#1"}], "+", RowBox[{"714", " ", SuperscriptBox["#1", "2"]}], "+", RowBox[{"1122", " ", SuperscriptBox["#1", "3"]}], "+", RowBox[{"935", " ", SuperscriptBox["#1", "4"]}], "+", RowBox[{"442", " ", SuperscriptBox["#1", "5"]}], "+", RowBox[{"119", " ", SuperscriptBox["#1", "6"]}], "+", RowBox[{"17", " ", SuperscriptBox["#1", "7"]}], "+", SuperscriptBox["#1", "8"]}], "&"}], ",", "7"}], "]"}]}], ")"}]}]}], ",", RowBox[{"2", "+", RowBox[{"\[Sqrt]", RowBox[{"(", RowBox[{"4", "+", RowBox[{"Root", "[", RowBox[{ RowBox[{ RowBox[{"17", "+", RowBox[{"204", " ", "#1"}], "+", RowBox[{"714", " ", SuperscriptBox["#1", "2"]}], "+", RowBox[{"1122", " ", SuperscriptBox["#1", "3"]}], "+", RowBox[{"935", " ", SuperscriptBox["#1", "4"]}], "+", RowBox[{"442", " ", SuperscriptBox["#1", "5"]}], "+", RowBox[{"119", " ", SuperscriptBox["#1", "6"]}], "+", RowBox[{"17", " ", SuperscriptBox["#1", "7"]}], "+", SuperscriptBox["#1", "8"]}], "&"}], ",", "6"}], "]"}]}], ")"}]}]}], ",", RowBox[{"2", "+", RowBox[{"\[Sqrt]", RowBox[{"(", RowBox[{"4", "+", RowBox[{"Root", "[", RowBox[{ RowBox[{ RowBox[{"17", "+", RowBox[{"204", " ", "#1"}], "+", RowBox[{"714", " ", SuperscriptBox["#1", "2"]}], "+", RowBox[{"1122", " ", SuperscriptBox["#1", "3"]}], "+", RowBox[{"935", " ", SuperscriptBox["#1", "4"]}], "+", RowBox[{"442", " ", SuperscriptBox["#1", "5"]}], "+", RowBox[{"119", " ", SuperscriptBox["#1", "6"]}], "+", RowBox[{"17", " ", SuperscriptBox["#1", "7"]}], "+", SuperscriptBox["#1", "8"]}], "&"}], ",", "5"}], "]"}]}], ")"}]}]}], ",", RowBox[{"2", "+", RowBox[{"\[Sqrt]", RowBox[{"(", RowBox[{"4", "+", RowBox[{"Root", "[", RowBox[{ RowBox[{ RowBox[{"17", "+", RowBox[{"204", " ", "#1"}], "+", RowBox[{"714", " ", SuperscriptBox["#1", "2"]}], "+", RowBox[{"1122", " ", SuperscriptBox["#1", "3"]}], "+", RowBox[{"935", " ", SuperscriptBox["#1", "4"]}], "+", RowBox[{"442", " ", SuperscriptBox["#1", "5"]}], "+", RowBox[{"119", " ", SuperscriptBox["#1", "6"]}], "+", RowBox[{"17", " ", SuperscriptBox["#1", "7"]}], "+", SuperscriptBox["#1", "8"]}], "&"}], ",", "4"}], "]"}]}], ")"}]}]}], ",", RowBox[{"2", "+", RowBox[{"\[Sqrt]", RowBox[{"(", RowBox[{"4", "+", RowBox[{"Root", "[", RowBox[{ RowBox[{ RowBox[{"17", "+", RowBox[{"204", " ", "#1"}], "+", RowBox[{"714", " ", SuperscriptBox["#1", "2"]}], "+", RowBox[{"1122", " ", SuperscriptBox["#1", "3"]}], "+", RowBox[{"935", " ", SuperscriptBox["#1", "4"]}], "+", RowBox[{"442", " ", SuperscriptBox["#1", "5"]}], "+", RowBox[{"119", " ", SuperscriptBox["#1", "6"]}], "+", RowBox[{"17", " ", SuperscriptBox["#1", "7"]}], "+", SuperscriptBox["#1", "8"]}], "&"}], ",", "3"}], "]"}]}], ")"}]}]}], ",", RowBox[{"2", "+", RowBox[{"\[Sqrt]", RowBox[{"(", RowBox[{"4", "+", RowBox[{"Root", "[", RowBox[{ RowBox[{ RowBox[{"17", "+", RowBox[{"204", " ", "#1"}], "+", RowBox[{"714", " ", SuperscriptBox["#1", "2"]}], "+", RowBox[{"1122", " ", SuperscriptBox["#1", "3"]}], "+", RowBox[{"935", " ", SuperscriptBox["#1", "4"]}], "+", RowBox[{"442", " ", SuperscriptBox["#1", "5"]}], "+", RowBox[{"119", " ", SuperscriptBox["#1", "6"]}], "+", RowBox[{"17", " ", SuperscriptBox["#1", "7"]}], "+", SuperscriptBox["#1", "8"]}], "&"}], ",", "2"}], "]"}]}], ")"}]}]}], ",", RowBox[{"2", "+", RowBox[{"\[Sqrt]", RowBox[{"(", RowBox[{"4", "+", RowBox[{"Root", "[", RowBox[{ RowBox[{ RowBox[{"17", "+", RowBox[{"204", " ", "#1"}], "+", RowBox[{"714", " ", SuperscriptBox["#1", "2"]}], "+", RowBox[{"1122", " ", SuperscriptBox["#1", "3"]}], "+", RowBox[{"935", " ", SuperscriptBox["#1", "4"]}], "+", RowBox[{"442", " ", SuperscriptBox["#1", "5"]}], "+", RowBox[{"119", " ", SuperscriptBox["#1", "6"]}], "+", RowBox[{"17", " ", SuperscriptBox["#1", "7"]}], "+", SuperscriptBox["#1", "8"]}], "&"}], ",", "1"}], "]"}]}], ")"}]}]}], ",", RowBox[{"2", "-", RowBox[{"\[Sqrt]", RowBox[{"(", RowBox[{"4", "+", RowBox[{"Root", "[", RowBox[{ RowBox[{ RowBox[{"17", "+", RowBox[{"204", " ", "#1"}], "+", RowBox[{"714", " ", SuperscriptBox["#1", "2"]}], "+", RowBox[{"1122", " ", SuperscriptBox["#1", "3"]}], "+", RowBox[{"935", " ", SuperscriptBox["#1", "4"]}], "+", RowBox[{"442", " ", SuperscriptBox["#1", "5"]}], "+", RowBox[{"119", " ", SuperscriptBox["#1", "6"]}], "+", RowBox[{"17", " ", SuperscriptBox["#1", "7"]}], "+", SuperscriptBox["#1", "8"]}], "&"}], ",", "1"}], "]"}]}], ")"}]}]}], ",", RowBox[{"2", "-", RowBox[{"\[Sqrt]", RowBox[{"(", RowBox[{"4", "+", RowBox[{"Root", "[", RowBox[{ RowBox[{ RowBox[{"17", "+", RowBox[{"204", " ", "#1"}], "+", RowBox[{"714", " ", SuperscriptBox["#1", "2"]}], "+", RowBox[{"1122", " ", SuperscriptBox["#1", "3"]}], "+", RowBox[{"935", " ", SuperscriptBox["#1", "4"]}], "+", RowBox[{"442", " ", SuperscriptBox["#1", "5"]}], "+", RowBox[{"119", " ", SuperscriptBox["#1", "6"]}], "+", RowBox[{"17", " ", SuperscriptBox["#1", "7"]}], "+", SuperscriptBox["#1", "8"]}], "&"}], ",", "2"}], "]"}]}], ")"}]}]}], ",", RowBox[{"2", "-", RowBox[{"\[Sqrt]", RowBox[{"(", RowBox[{"4", "+", RowBox[{"Root", "[", RowBox[{ RowBox[{ RowBox[{"17", "+", RowBox[{"204", " ", "#1"}], "+", RowBox[{"714", " ", SuperscriptBox["#1", "2"]}], "+", RowBox[{"1122", " ", SuperscriptBox["#1", "3"]}], "+", RowBox[{"935", " ", SuperscriptBox["#1", "4"]}], "+", RowBox[{"442", " ", SuperscriptBox["#1", "5"]}], "+", RowBox[{"119", " ", SuperscriptBox["#1", "6"]}], "+", RowBox[{"17", " ", SuperscriptBox["#1", "7"]}], "+", SuperscriptBox["#1", "8"]}], "&"}], ",", "3"}], "]"}]}], ")"}]}]}], ",", RowBox[{"2", "-", RowBox[{"\[Sqrt]", RowBox[{"(", RowBox[{"4", "+", RowBox[{"Root", "[", RowBox[{ RowBox[{ RowBox[{"17", "+", RowBox[{"204", " ", "#1"}], "+", RowBox[{"714", " ", SuperscriptBox["#1", "2"]}], "+", RowBox[{"1122", " ", SuperscriptBox["#1", "3"]}], "+", RowBox[{"935", " ", SuperscriptBox["#1", "4"]}], "+", RowBox[{"442", " ", SuperscriptBox["#1", "5"]}], "+", RowBox[{"119", " ", SuperscriptBox["#1", "6"]}], "+", RowBox[{"17", " ", SuperscriptBox["#1", "7"]}], "+", SuperscriptBox["#1", "8"]}], "&"}], ",", "4"}], "]"}]}], ")"}]}]}], ",", RowBox[{"2", "-", RowBox[{"\[Sqrt]", RowBox[{"(", RowBox[{"4", "+", RowBox[{"Root", "[", RowBox[{ RowBox[{ RowBox[{"17", "+", RowBox[{"204", " ", "#1"}], "+", RowBox[{"714", " ", SuperscriptBox["#1", "2"]}], "+", RowBox[{"1122", " ", SuperscriptBox["#1", "3"]}], "+", RowBox[{"935", " ", SuperscriptBox["#1", "4"]}], "+", RowBox[{"442", " ", SuperscriptBox["#1", "5"]}], "+", RowBox[{"119", " ", SuperscriptBox["#1", "6"]}], "+", RowBox[{"17", " ", SuperscriptBox["#1", "7"]}], "+", SuperscriptBox["#1", "8"]}], "&"}], ",", "5"}], "]"}]}], ")"}]}]}], ",", RowBox[{"2", "-", RowBox[{"\[Sqrt]", RowBox[{"(", RowBox[{"4", "+", RowBox[{"Root", "[", RowBox[{ RowBox[{ RowBox[{"17", "+", RowBox[{"204", " ", "#1"}], "+", RowBox[{"714", " ", SuperscriptBox["#1", "2"]}], "+", RowBox[{"1122", " ", SuperscriptBox["#1", "3"]}], "+", RowBox[{"935", " ", SuperscriptBox["#1", "4"]}], "+", RowBox[{"442", " ", SuperscriptBox["#1", "5"]}], "+", RowBox[{"119", " ", SuperscriptBox["#1", "6"]}], "+", RowBox[{"17", " ", SuperscriptBox["#1", "7"]}], "+", SuperscriptBox["#1", "8"]}], "&"}], ",", "6"}], "]"}]}], ")"}]}]}], ",", RowBox[{"2", "-", RowBox[{"\[Sqrt]", RowBox[{"(", RowBox[{"4", "+", RowBox[{"Root", "[", RowBox[{ RowBox[{ RowBox[{"17", "+", RowBox[{"204", " ", "#1"}], "+", RowBox[{"714", " ", SuperscriptBox["#1", "2"]}], "+", RowBox[{"1122", " ", SuperscriptBox["#1", "3"]}], "+", RowBox[{"935", " ", SuperscriptBox["#1", "4"]}], "+", RowBox[{"442", " ", SuperscriptBox["#1", "5"]}], "+", RowBox[{"119", " ", SuperscriptBox["#1", "6"]}], "+", RowBox[{"17", " ", SuperscriptBox["#1", "7"]}], "+", SuperscriptBox["#1", "8"]}], "&"}], ",", "7"}], "]"}]}], ")"}]}]}], ",", RowBox[{"2", "-", RowBox[{"\[Sqrt]", RowBox[{"(", RowBox[{"4", "+", RowBox[{"Root", "[", RowBox[{ RowBox[{ RowBox[{"17", "+", RowBox[{"204", " ", "#1"}], "+", RowBox[{"714", " ", SuperscriptBox["#1", "2"]}], "+", RowBox[{"1122", " ", SuperscriptBox["#1", "3"]}], "+", RowBox[{"935", " ", SuperscriptBox["#1", "4"]}], "+", RowBox[{"442", " ", SuperscriptBox["#1", "5"]}], "+", RowBox[{"119", " ", SuperscriptBox["#1", "6"]}], "+", RowBox[{"17", " ", SuperscriptBox["#1", "7"]}], "+", SuperscriptBox["#1", "8"]}], "&"}], ",", "8"}], "]"}]}], ")"}]}]}]}], "}"}]], "Output", CellChangeTimes->{3.3964850531484957`*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"N", "[", "%", "]"}]], "Input", CellChangeTimes->{{3.396485070472897*^9, 3.396485072488637*^9}}], Cell[BoxData[ RowBox[{"{", RowBox[{ "3.965946199367804`", ",", "3.8649444588087114`", ",", "3.7004342714592284`", ",", "3.478017834441318`", ",", "3.205269272758513`", ",", "2.891476711553077`", ",", "2.5473259801441657`", ",", "2.1845367189266045`", ",", "1.8154632810733955`", ",", "1.4526740198558343`", ",", "1.1085232884469234`", ",", "0.7947307272414872`", ",", "0.5219821655586818`", ",", "0.29956572854077157`", ",", "0.13505554119128838`", ",", "0.03405380063219643`"}], "}"}]], "Output", CellChangeTimes->{3.396485074018005*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"CharacteristicPolynomial", "[", RowBox[{"%", ",", "x"}], "]"}]], "Input"], Cell[BoxData[ RowBox[{"17", "-", RowBox[{"816", " ", "x"}], "+", RowBox[{"11628", " ", SuperscriptBox["x", "2"]}], "-", RowBox[{"77520", " ", SuperscriptBox["x", "3"]}], "+", RowBox[{"293930", " ", SuperscriptBox["x", "4"]}], "-", RowBox[{"705432", " ", SuperscriptBox["x", "5"]}], "+", RowBox[{"1144066", " ", SuperscriptBox["x", "6"]}], "-", RowBox[{"1307504", " ", SuperscriptBox["x", "7"]}], "+", RowBox[{"1081575", " ", SuperscriptBox["x", "8"]}], "-", RowBox[{"657800", " ", SuperscriptBox["x", "9"]}], "+", RowBox[{"296010", " ", SuperscriptBox["x", "10"]}], "-", RowBox[{"98280", " ", SuperscriptBox["x", "11"]}], "+", RowBox[{"23751", " ", SuperscriptBox["x", "12"]}], "-", RowBox[{"4060", " ", SuperscriptBox["x", "13"]}], "+", RowBox[{"465", " ", SuperscriptBox["x", "14"]}], "-", RowBox[{"32", " ", SuperscriptBox["x", "15"]}], "+", SuperscriptBox["x", "16"]}]], "Output", CellChangeTimes->{3.396298821597732*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"spectrum", "=", RowBox[{"x", "/.", RowBox[{"NSolve", "[", RowBox[{ RowBox[{"%", "\[Equal]", "0"}], ",", "x"}], "]"}]}]}]], "Input"], Cell[BoxData[ RowBox[{"{", RowBox[{ "0.034053800632196436`", ",", "0.13505554119128863`", ",", "0.29956572854077523`", ",", "0.5219821655584377`", ",", "0.7947307272477455`", ",", "1.108523288365079`", ",", "1.4526740204721977`", ",", "1.8154632784564735`", ",", "2.1845367232427644`", ",", "2.5473259956948255`", ",", "2.8914765890721483`", ",", "3.2052696703493697`", ",", "3.478017038454895`", ",", "3.7004353079524717`", ",", "3.8649436359731726`", ",", "3.9659464887961584`"}], "}"}]], "Output", CellChangeTimes->{3.3962988363369007`*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"ListPlot", "[", RowBox[{"spectrum", ",", RowBox[{"PlotStyle", "\[Rule]", RowBox[{"PointSize", "[", "Large", "]"}]}]}], "]"}]], "Input", CellChangeTimes->{{3.396298856889975*^9, 3.396298914135538*^9}}], Cell[BoxData[ GraphicsBox[ {Hue[0.67, 0.6, 0.6], PointSize[Large], PointBox[{{1., 0.034053800632196436`}, {2., 0.13505554119128863`}, {3., 0.29956572854077523`}, {4., 0.5219821655584377}, {5., 0.7947307272477455}, {6., 1.108523288365079}, {7., 1.4526740204721977`}, { 8., 1.8154632784564735`}, {9., 2.1845367232427644`}, {10., 2.5473259956948255`}, {11., 2.8914765890721483`}, {12., 3.2052696703493697`}, {13., 3.478017038454895}, {14., 3.7004353079524717`}, {15., 3.8649436359731726`}, {16., 3.9659464887961584`}}]}, AspectRatio->NCache[GoldenRatio^(-1), 0.6180339887498948], Axes->True, PlotRange->Automatic, PlotRangeClipping->True]], "Output", CellChangeTimes->{{3.396298862256336*^9, 3.396298915683179*^9}}] }, Open ]], Cell[BoxData["\[IndentingNewLine]"], "Input", CellChangeTimes->{3.396485411037814*^9}], Cell[BoxData[ RowBox[{ RowBox[{ SubscriptBox["S", "ring"], "=", RowBox[{"(", GridBox[{ {"2", RowBox[{"-", "1"}], "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", StyleBox[ RowBox[{"-", "1"}], FontColor->RGBColor[1, 0, 0]]}, { RowBox[{"-", "1"}], "2", RowBox[{"-", "1"}], "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0"}, {"0", RowBox[{"-", "1"}], "2", RowBox[{"-", "1"}], "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0"}, {"0", "0", RowBox[{"-", "1"}], "2", RowBox[{"-", "1"}], "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0"}, {"0", "0", "0", RowBox[{"-", "1"}], "2", RowBox[{"-", "1"}], "0", "0", "0", "0", "0", "0", "0", "0", "0", "0"}, {"0", "0", "0", "0", RowBox[{"-", "1"}], "2", RowBox[{"-", "1"}], "0", "0", "0", "0", "0", "0", "0", "0", "0"}, {"0", "0", "0", "0", "0", RowBox[{"-", "1"}], "2", RowBox[{"-", "1"}], "0", "0", "0", "0", "0", "0", "0", "0"}, {"0", "0", "0", "0", "0", "0", RowBox[{"-", "1"}], "2", RowBox[{"-", "1"}], "0", "0", "0", "0", "0", "0", "0"}, {"0", "0", "0", "0", "0", "0", "0", RowBox[{"-", "1"}], "2", RowBox[{"-", "1"}], "0", "0", "0", "0", "0", "0"}, {"0", "0", "0", "0", "0", "0", "0", "0", RowBox[{"-", "1"}], "2", RowBox[{"-", "1"}], "0", "0", "0", "0", "0"}, {"0", "0", "0", "0", "0", "0", "0", "0", "0", RowBox[{"-", "1"}], "2", RowBox[{"-", "1"}], "0", "0", "0", "0"}, {"0", "0", "0", "0", "0", "0", "0", "0", "0", "0", RowBox[{"-", "1"}], "2", RowBox[{"-", "1"}], "0", "0", "0"}, {"0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", RowBox[{"-", "1"}], "2", RowBox[{"-", "1"}], "0", "0"}, {"0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", RowBox[{"-", "1"}], "2", RowBox[{"-", "1"}], "0"}, {"0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", RowBox[{"-", "1"}], "2", RowBox[{"-", "1"}]}, { StyleBox[ RowBox[{"-", "1"}], FontColor->RGBColor[1, 0, 0]], "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", RowBox[{"-", "1"}], "2"} }], ")"}]}], ";"}]], "Input", CellChangeTimes->{{3.396485422603156*^9, 3.396485440113867*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"CharacteristicPolynomial", "[", RowBox[{ SubscriptBox["S", "ring"], ",", "x"}], "]"}]], "Input", CellChangeTimes->{3.396485458299617*^9}], Cell[BoxData[ RowBox[{ RowBox[{ RowBox[{"-", "256"}], " ", "x"}], "+", RowBox[{"5440", " ", SuperscriptBox["x", "2"]}], "-", RowBox[{"45696", " ", SuperscriptBox["x", "3"]}], "+", RowBox[{"201552", " ", SuperscriptBox["x", "4"]}], "-", RowBox[{"537472", " ", SuperscriptBox["x", "5"]}], "+", RowBox[{"940576", " ", SuperscriptBox["x", "6"]}], "-", RowBox[{"1136960", " ", SuperscriptBox["x", "7"]}], "+", RowBox[{"980628", " ", SuperscriptBox["x", "8"]}], "-", RowBox[{"615296", " ", SuperscriptBox["x", "9"]}], "+", RowBox[{"283360", " ", SuperscriptBox["x", "10"]}], "-", RowBox[{"95680", " ", SuperscriptBox["x", "11"]}], "+", RowBox[{"23400", " ", SuperscriptBox["x", "12"]}], "-", RowBox[{"4032", " ", SuperscriptBox["x", "13"]}], "+", RowBox[{"464", " ", SuperscriptBox["x", "14"]}], "-", RowBox[{"32", " ", SuperscriptBox["x", "15"]}], "+", SuperscriptBox["x", "16"]}]], "Output", CellChangeTimes->{3.3962990316431227`*^9, 3.39648546498241*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"spectrum", "=", RowBox[{"x", "/.", RowBox[{"NSolve", "[", RowBox[{ RowBox[{"%", "\[Equal]", "0"}], ",", "x"}], "]"}]}]}]], "Input"], Cell[BoxData[ RowBox[{"{", RowBox[{ "0.`", ",", "0.1522409331604236`", ",", "0.15224093679442957`", ",", "0.5857863718757075`", ",", "0.5857865033782763`", ",", "1.2346328959887243`", ",", "1.2346333745542706`", ",", "1.9999880821239244`", ",", "2.0000119185784064`", ",", "2.765313420271791`", ",", "2.7654203191178803`", ",", "3.41408123397623`", ",", "3.414345940684123`", ",", "3.8476064438616526`", ",", "3.847911692081391`", ",", "3.9999999335527696`"}], "}"}]], "Output", CellChangeTimes->{3.396299047187426*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"ListPlot", "[", RowBox[{"spectrum", ",", RowBox[{"PlotStyle", "\[Rule]", RowBox[{"PointSize", "[", "0.02", "]"}]}]}], "]"}]], "Input", CellChangeTimes->{3.396299070854514*^9}], Cell[BoxData[ GraphicsBox[ {Hue[0.67, 0.6, 0.6], PointSize[0.02], PointBox[{{1., 0.}, {2., 0.1522409331604236}, {3., 0.15224093679442957`}, { 4., 0.5857863718757075}, {5., 0.5857865033782763}, {6., 1.2346328959887243`}, {7., 1.2346333745542706`}, {8., 1.9999880821239244`}, {9., 2.0000119185784064`}, {10., 2.765313420271791}, {11., 2.7654203191178803`}, {12., 3.41408123397623}, { 13., 3.414345940684123}, {14., 3.8476064438616526`}, {15., 3.847911692081391}, {16., 3.9999999335527696`}}]}, AspectRatio->NCache[GoldenRatio^(-1), 0.6180339887498948], Axes->True, PlotRange->Automatic, PlotRangeClipping->True]], "Output", CellChangeTimes->{3.3962990744786673`*^9}] }, Open ]], Cell[BoxData["\[IndentingNewLine]"], "Input", CellChangeTimes->{3.396485477309347*^9}], Cell[BoxData[ RowBox[{ RowBox[{ SubscriptBox["S", "impure"], "=", RowBox[{"(", GridBox[{ {"2", RowBox[{"-", "1"}], "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0"}, { RowBox[{"-", "1"}], "2", RowBox[{"-", "1"}], "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0"}, {"0", RowBox[{"-", "1"}], "2", RowBox[{"-", "1"}], "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0"}, {"0", "0", RowBox[{"-", "1"}], "2", StyleBox[ RowBox[{"-", "2"}], FontColor->RGBColor[1, 0, 0]], "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0"}, {"0", "0", "0", StyleBox[ RowBox[{"-", "2"}], FontColor->RGBColor[1, 0, 0]], StyleBox["3", FontColor->RGBColor[1, 0, 0]], StyleBox[ RowBox[{"-", "2"}], FontColor->RGBColor[1, 0, 0]], "0", "0", "0", "0", "0", "0", "0", "0", "0", "0"}, {"0", "0", "0", "0", StyleBox[ RowBox[{"-", "2"}], FontColor->RGBColor[1, 0, 0]], "2", RowBox[{"-", "1"}], "0", "0", "0", "0", "0", "0", "0", "0", "0"}, {"0", "0", "0", "0", "0", RowBox[{"-", "1"}], "2", RowBox[{"-", "1"}], "0", "0", "0", "0", "0", "0", "0", "0"}, {"0", "0", "0", "0", "0", "0", RowBox[{"-", "1"}], "2", RowBox[{"-", "1"}], "0", "0", "0", "0", "0", "0", "0"}, {"0", "0", "0", "0", "0", "0", "0", RowBox[{"-", "1"}], "2", RowBox[{"-", "1"}], "0", "0", "0", "0", "0", "0"}, {"0", "0", "0", "0", "0", "0", "0", "0", RowBox[{"-", "1"}], "2", RowBox[{"-", "1"}], "0", "0", "0", "0", "0"}, {"0", "0", "0", "0", "0", "0", "0", "0", "0", RowBox[{"-", "1"}], "2", RowBox[{"-", "1"}], "0", "0", "0", "0"}, {"0", "0", "0", "0", "0", "0", "0", "0", "0", "0", RowBox[{"-", "1"}], "2", RowBox[{"-", "1"}], "0", "0", "0"}, {"0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", RowBox[{"-", "1"}], "2", RowBox[{"-", "1"}], "0", "0"}, {"0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", RowBox[{"-", "1"}], "2", RowBox[{"-", "1"}], "0"}, {"0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", RowBox[{"-", "1"}], "2", RowBox[{"-", "1"}]}, {"0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", RowBox[{"-", "1"}], "2"} }], ")"}]}], ";"}]], "Input", CellChangeTimes->{{3.396485511859725*^9, 3.3964855260666857`*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"CharacteristicPolynomial", "[", RowBox[{ SubscriptBox["S", "impure"], ",", "x"}], "]"}]], "Input", CellChangeTimes->{3.3964855401964912`*^9}], Cell[BoxData[ RowBox[{ RowBox[{"-", "232"}], "+", RowBox[{"5266", " ", "x"}], "-", RowBox[{"38408", " ", SuperscriptBox["x", "2"]}], "+", RowBox[{"129451", " ", SuperscriptBox["x", "3"]}], "-", RowBox[{"210441", " ", SuperscriptBox["x", "4"]}], "+", RowBox[{"79583", " ", SuperscriptBox["x", "5"]}], "+", RowBox[{"329503", " ", SuperscriptBox["x", "6"]}], "-", RowBox[{"734815", " ", SuperscriptBox["x", "7"]}], "+", RowBox[{"812547", " ", SuperscriptBox["x", "8"]}], "-", RowBox[{"579582", " ", SuperscriptBox["x", "9"]}], "+", RowBox[{"286412", " ", SuperscriptBox["x", "10"]}], "-", RowBox[{"100387", " ", SuperscriptBox["x", "11"]}], "+", RowBox[{"24941", " ", SuperscriptBox["x", "12"]}], "-", RowBox[{"4299", " ", SuperscriptBox["x", "13"]}], "+", RowBox[{"489", " ", SuperscriptBox["x", "14"]}], "-", RowBox[{"33", " ", SuperscriptBox["x", "15"]}], "+", SuperscriptBox["x", "16"]}]], "Output", CellChangeTimes->{3.396299133322596*^9, 3.396485543185549*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"spectrum", "=", RowBox[{"x", "/.", RowBox[{"NSolve", "[", RowBox[{ RowBox[{"%", "\[Equal]", "0"}], ",", "x"}], "]"}]}]}]], "Input"], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{"-", "0.6477896162157661`"}], ",", "0.07879718867410519`", ",", "0.28773973525395835`", ",", "0.4877723171946824`", ",", "0.715140267874472`", ",", "1.0877066873465076`", ",", "1.4520266081428557`", ",", "1.8265264075737209`", ",", "2.2261867218078724`", ",", "2.550090550700265`", ",", "2.9308473473899315`", ",", "3.3266958401105335`", ",", "3.5322400678245267`", ",", "3.7156863691022304`", ",", "3.9245810351867156`", ",", "5.505752472033391`"}], "}"}]], "Output", CellChangeTimes->{3.396299144427267*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"ListPlot", "[", RowBox[{"spectrum", ",", RowBox[{"PlotStyle", "\[Rule]", RowBox[{"PointSize", "[", "0.02", "]"}]}]}], "]"}]], "Input", CellChangeTimes->{3.3962991592644444`*^9}], Cell[BoxData[ GraphicsBox[ {Hue[0.67, 0.6, 0.6], PointSize[0.02], PointBox[{{1., -0.6477896162157661}, {2., 0.07879718867410519}, {3., 0.28773973525395835`}, {4., 0.4877723171946824}, {5., 0.715140267874472}, {6., 1.0877066873465076`}, {7., 1.4520266081428557`}, {8., 1.8265264075737209`}, {9., 2.2261867218078724`}, {10., 2.550090550700265}, {11., 2.9308473473899315`}, {12., 3.3266958401105335`}, {13., 3.5322400678245267`}, {14., 3.7156863691022304`}, {15., 3.9245810351867156`}, {16., 5.505752472033391}}]}, AspectRatio->NCache[GoldenRatio^(-1), 0.6180339887498948], Axes->True, PlotRange->Automatic, PlotRangeClipping->True]], "Output", CellChangeTimes->{3.396299163404332*^9}] }, Open ]], Cell[BoxData["\[IndentingNewLine]"], "Input", CellChangeTimes->{3.399389788352503*^9}] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ StyleBox["Problem 8", FontColor->RGBColor[1, 0, 0]], ":" }], "Subsubtitle", CellChangeTimes->{{3.3962293322724953`*^9, 3.3962293390217133`*^9}, 3.399389797483961*^9, 3.3993932657678843`*^9}], Cell[BoxData[ RowBox[{"Needs", "[", "\"\\"", "]"}]], "Input", CellChangeTimes->{{3.396222985351863*^9, 3.3962230018936653`*^9}, { 3.396223140034974*^9, 3.396223143561356*^9}}], Cell[BoxData[ RowBox[{"Needs", "[", "\"\\"", "]"}]], "Input", CellChangeTimes->{{3.3962232252775497`*^9, 3.396223226473337*^9}}], Cell[CellGroupData[{ Cell[BoxData[{ RowBox[{"?", "SolarMass"}], "\[IndentingNewLine]", RowBox[{"?", "SolarRadius"}], "\[IndentingNewLine]", RowBox[{"?", "ProtonMass"}], "\[IndentingNewLine]", RowBox[{"?", "PlanckConstant"}]}], "Input", CellChangeTimes->{{3.396223296481978*^9, 3.396223323946487*^9}, { 3.396223355818775*^9, 3.39622339546568*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ StyleBox["\<\"SolarMass is a unit of mass.\"\>", "MSG"], " ", ButtonBox[ StyleBox["\[RightSkeleton]", "SR"], Active->True, BaseStyle->"Link", ButtonData->"paclet:Units/ref/SolarMass"]}]], "Print", "PrintUsage", CellChangeTimes->{3.3962234003630466`*^9}, CellTags->"Info3396198200-1999354"], Cell[BoxData[ RowBox[{ StyleBox["\<\"SolarRadius is a physical constant.\"\>", "MSG"], " ", ButtonBox[ StyleBox["\[RightSkeleton]", "SR"], Active->True, BaseStyle->"Link", ButtonData->"paclet:PhysicalConstants/ref/SolarRadius"]}]], "Print", \ "PrintUsage", CellChangeTimes->{3.396223400697917*^9}, CellTags->"Info3396198200-7824464"], Cell[BoxData[ RowBox[{ StyleBox["\<\"ProtonMass is the mass of a proton.\"\>", "MSG"], " ", ButtonBox[ StyleBox["\[RightSkeleton]", "SR"], Active->True, BaseStyle->"Link", ButtonData->"paclet:PhysicalConstants/ref/ProtonMass"]}]], "Print", \ "PrintUsage", CellChangeTimes->{3.39622340116225*^9}, CellTags->"Info3396198200-8319869"], Cell[BoxData[ RowBox[{ StyleBox["\<\"PlanckConstant is a universal constant of nature which \ relates the energy of a quantum of radiation to the frequency of the \ oscillator which emitted it.\"\>", "MSG"], " ", ButtonBox[ StyleBox["\[RightSkeleton]", "SR"], Active->True, BaseStyle->"Link", ButtonData->"paclet:PhysicalConstants/ref/PlanckConstant"]}]], "Print", \ "PrintUsage", CellChangeTimes->{3.396223401557703*^9}, CellTags->"Info3396198201-8621354"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[{"SolarMass", "\[IndentingNewLine]", "SolarRadius", "\ \[IndentingNewLine]", "ProtonMass", "\[IndentingNewLine]", \ "PlanckConstant"}], "Input", CellChangeTimes->{{3.3962234269283133`*^9, 3.396223443963512*^9}}], Cell[BoxData["SolarMass"], "Output", CellChangeTimes->{3.396223445334861*^9}], Cell[BoxData[ RowBox[{"6.9599`*^8", " ", "Meter"}]], "Output", CellChangeTimes->{3.3962234454177027`*^9}], Cell[BoxData[ RowBox[{"1.67262158`*^-27", " ", "Kilogram"}]], "Output", CellChangeTimes->{3.396223445517346*^9}], Cell[BoxData[ RowBox[{"6.62606876`*^-34", " ", "Joule", " ", "Second"}]], "Output", CellChangeTimes->{3.3962234456063137`*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Convert", "[", RowBox[{"SolarMass", ",", "Kilogram"}], "]"}]], "Input", CellChangeTimes->{{3.396223277959597*^9, 3.396223289774984*^9}}], Cell[BoxData[ RowBox[{"1.9891000000000002`*^30", " ", "Kilogram"}]], "Output", CellChangeTimes->{3.396223291449514*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"SolarMomentOfInertia", "=", RowBox[{ FractionBox["2", "5"], RowBox[{"(", "SolarMass", ")"}], SuperscriptBox[ RowBox[{"(", "SolarRadius", ")"}], "2"]}]}]], "Input", CellChangeTimes->{{3.396223492875244*^9, 3.396223600322043*^9}, { 3.396223677437488*^9, 3.3962236798187*^9}}], Cell[BoxData[ RowBox[{"1.9376083204`*^17", " ", SuperscriptBox["Meter", "2"], " ", "SolarMass"}]], "Output", CellChangeTimes->{3.396223602995929*^9, 3.396223688436614*^9, 3.396223832514407*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"SolarAngularMomentum", "=", RowBox[{"%", RowBox[{"(", FractionBox[ RowBox[{"2", "\[Pi]"}], RowBox[{"25.36", "Day"}]], ")"}]}]}]], "Input", CellChangeTimes->{{3.3962236119891644`*^9, 3.3962236635845413`*^9}}], Cell[BoxData[ FractionBox[ RowBox[{"4.800612038567112`*^16", " ", SuperscriptBox["Meter", "2"], " ", "SolarMass"}], "Day"]], "Output", CellChangeTimes->{3.396223695230422*^9, 3.39622384704294*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"SolarSpin", "=", RowBox[{"Convert", "[", RowBox[{"%", ",", RowBox[{"Kilogram", " ", RowBox[{ SuperscriptBox["Meter", "2"], "/", "Second"}]}]}], "]"}]}]], "Input", CellChangeTimes->{{3.396223740917708*^9, 3.396223771244103*^9}, { 3.3962238562160463`*^9, 3.3962238576342497`*^9}}], Cell[BoxData[ FractionBox[ RowBox[{"1.1051964590178056`*^42", " ", "Kilogram", " ", SuperscriptBox["Meter", "2"]}], "Second"]], "Output", CellChangeTimes->{3.396223801020873*^9, 3.396223859056595*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"ProtonNumber", "=", FractionBox["SolarMass", "ProtonMass"]}]], "Input", CellChangeTimes->{{3.396223913648678*^9, 3.396223984900373*^9}}], Cell[BoxData[ FractionBox[ RowBox[{"5.978638635046189`*^26", " ", "SolarMass"}], "Kilogram"]], "Output",\ CellChangeTimes->{3.3962239886957083`*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"ProtonNumber", "=", RowBox[{"Convert", "[", RowBox[{"%", ",", RowBox[{"Kilogram", "/", "Kilogram"}]}], "]"}]}]], "Input", CellChangeTimes->{{3.396223999202384*^9, 3.396224024179138*^9}}], Cell[BoxData["1.1892110108970376`*^57"], "Output", CellChangeTimes->{3.396224026308827*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"hbar", "=", FractionBox["PlanckConstant", RowBox[{"2", "\[Pi]"}]]}]], "Input", CellChangeTimes->{{3.396224107464921*^9, 3.396224141305971*^9}}], Cell[BoxData[ RowBox[{"1.0545715964207855`*^-34", " ", "Joule", " ", "Second"}]], "Output",\ CellChangeTimes->{3.396224142845358*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"TotalProtonSpin", "=", RowBox[{"ProtonNumber", " ", FractionBox["hbar", "2"]}]}]], "Input", CellChangeTimes->{{3.396224039154154*^9, 3.396224054963397*^9}, { 3.396224157438137*^9, 3.396224171890092*^9}, 3.396224635216371*^9}], Cell[BoxData[ RowBox[{"6.2705407712143256`*^22", " ", "Joule", " ", "Second"}]], "Output", CellChangeTimes->{3.396224181107518*^9, 3.396224639366406*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ FractionBox["TotalProtonSpin", "SolarSpin"]], "Input", CellChangeTimes->{{3.396224210021583*^9, 3.39622423106844*^9}}], Cell[BoxData[ FractionBox[ RowBox[{"5.673688799896256`*^-20", " ", "Joule", " ", SuperscriptBox["Second", "2"]}], RowBox[{"Kilogram", " ", SuperscriptBox["Meter", "2"]}]]], "Output", CellChangeTimes->{3.396224232695891*^9, 3.396224645234057*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Convert", "[", " ", RowBox[{ RowBox[{"Joule", " ", SuperscriptBox["Second", "2"]}], ",", RowBox[{"Kilogram", " ", SuperscriptBox["Meter", "2"]}]}], "]"}]], "Input", CellChangeTimes->{{3.396224257793273*^9, 3.3962242878254766`*^9}, { 3.396224373875567*^9, 3.3962243926794643`*^9}, {3.3962246649523983`*^9, 3.396224676350163*^9}}], Cell[BoxData[ RowBox[{"Kilogram", " ", SuperscriptBox["Meter", "2"]}]], "Output", CellChangeTimes->{3.396224292379342*^9, 3.396224398170986*^9, 3.396224681061569*^9}] }, Open ]] }, Open ]] }, Open ]] }, WindowToolbars->"EditBar", WindowSize->{532, 810}, WindowMargins->{{Automatic, 428}, {Automatic, 89}}, Magnification->1.25, FrontEndVersion->"6.0 for Mac OS X PowerPC (32-bit) (June 19, 2007)", StyleDefinitions->"Default.nb" ] (* End of Notebook Content *) (* Internal cache information *) (*CellTagsOutline CellTagsIndex->{ "Info3396198200-1999354"->{ Cell[59000, 1879, 332, 9, 50, "Print", CellTags->"Info3396198200-1999354"]}, "Info3396198200-7824464"->{ Cell[59335, 1890, 353, 10, 50, "Print", CellTags->"Info3396198200-7824464"]}, "Info3396198200-8319869"->{ Cell[59691, 1902, 351, 10, 50, "Print", CellTags->"Info3396198200-8319869"]}, "Info3396198201-8621354"->{ Cell[60045, 1914, 477, 12, 88, "Print", CellTags->"Info3396198201-8621354"]} } *) (*CellTagsIndex CellTagsIndex->{ {"Info3396198200-1999354", 66132, 2125}, {"Info3396198200-7824464", 66243, 2128}, {"Info3396198200-8319869", 66355, 2131}, {"Info3396198201-8621354", 66467, 2134} } *) (*NotebookFileOutline Notebook[{ Cell[CellGroupData[{ Cell[590, 16, 409, 12, 57, "Subtitle"], Cell[1002, 30, 32, 0, 25, "Subsubtitle"], Cell[CellGroupData[{ Cell[1059, 34, 161, 5, 32, "Subsubtitle"], Cell[CellGroupData[{ Cell[1245, 43, 307, 7, 53, "Input"], Cell[1555, 52, 372, 10, 59, "Output"], Cell[1930, 64, 582, 16, 38, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[2549, 85, 1039, 29, 115, "Input"], Cell[3591, 116, 182, 4, 33, "Output"], Cell[3776, 122, 180, 4, 33, "Output"] }, Open ]], Cell[3971, 129, 67, 4, 51, "Text"] }, Open ]], Cell[CellGroupData[{ Cell[4075, 138, 166, 5, 32, "Subsubtitle"], Cell[CellGroupData[{ Cell[4266, 147, 345, 9, 84, "Input"], Cell[4614, 158, 247, 7, 53, "Output"], Cell[4864, 167, 234, 6, 56, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[5135, 178, 551, 16, 101, "Input"], Cell[5689, 196, 181, 4, 38, "Output"], Cell[5873, 202, 201, 5, 54, "Output"] }, Open ]], Cell[6089, 210, 67, 4, 51, "Text"] }, Open ]], Cell[CellGroupData[{ Cell[6193, 219, 163, 5, 32, "Subsubtitle"], Cell[CellGroupData[{ Cell[6381, 228, 238, 6, 72, "Input"], Cell[6622, 236, 628, 15, 33, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[7287, 256, 199, 7, 88, "Input"], Cell[7489, 265, 440, 11, 53, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[7966, 281, 55, 1, 33, "Input"], Cell[8024, 284, 93, 1, 33, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[8154, 290, 56, 1, 33, "Input"], Cell[8213, 293, 92, 1, 33, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[8342, 299, 64, 1, 33, "Input"], Cell[8409, 302, 177, 4, 33, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[8623, 311, 65, 1, 33, "Input"], Cell[8691, 314, 538, 16, 53, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[9266, 335, 577, 19, 91, "Input"], Cell[9846, 356, 182, 4, 33, "Output"], Cell[10031, 362, 220, 6, 33, "Output"], Cell[10254, 370, 203, 5, 33, "Output"], Cell[10460, 377, 220, 6, 33, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[10717, 388, 501, 10, 205, "Input"], Cell[11221, 400, 121, 2, 33, "Output"], Cell[11345, 404, 121, 2, 33, "Output"], Cell[11469, 408, 121, 2, 33, "Output"], Cell[11593, 412, 123, 2, 33, "Output"], Cell[11719, 416, 121, 2, 33, "Output"], Cell[11843, 420, 120, 2, 33, "Output"], Cell[11966, 424, 120, 2, 33, "Output"], Cell[12089, 428, 121, 2, 33, "Output"], Cell[12213, 432, 121, 2, 33, "Output"], Cell[12337, 436, 122, 2, 33, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[12496, 443, 418, 13, 91, "Input"], Cell[12917, 458, 95, 1, 33, "Output"], Cell[13015, 461, 97, 1, 33, "Output"], Cell[13115, 464, 95, 1, 33, "Output"], Cell[13213, 467, 95, 1, 33, "Output"] }, Open ]], Cell[13323, 471, 67, 4, 51, "Text"] }, Open ]], Cell[CellGroupData[{ Cell[13427, 480, 161, 5, 32, "Subsubtitle"], Cell[CellGroupData[{ Cell[13613, 489, 273, 6, 72, "Input"], Cell[13889, 497, 597, 15, 33, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[14523, 517, 197, 7, 88, "Input"], Cell[14723, 526, 411, 11, 53, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[15171, 542, 61, 1, 33, "Input"], Cell[15235, 545, 88, 2, 33, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[15360, 552, 69, 1, 33, "Input"], Cell[15432, 555, 267, 8, 33, "Output"] }, Open ]], Cell[15714, 566, 67, 4, 51, "Text"] }, Open ]], Cell[CellGroupData[{ Cell[15818, 575, 161, 5, 32, "Subsubtitle"], Cell[CellGroupData[{ Cell[16004, 584, 249, 8, 106, "Input"], Cell[16256, 594, 543, 14, 72, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[16836, 613, 61, 1, 33, "Input"], Cell[16900, 616, 73, 1, 33, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[17010, 622, 69, 1, 33, "Input"], Cell[17082, 625, 722, 29, 120, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[17841, 659, 54, 1, 33, "Input"], Cell[17898, 662, 253, 6, 33, "Output"] }, Open ]], Cell[18166, 671, 67, 4, 51, "Text"] }, Open ]], Cell[CellGroupData[{ Cell[18270, 680, 163, 5, 32, "Subsubtitle"], Cell[CellGroupData[{ Cell[18458, 689, 303, 10, 106, "Input"], Cell[18764, 701, 565, 14, 72, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[19366, 720, 61, 1, 33, "Input"], Cell[19430, 723, 73, 1, 33, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[19540, 729, 69, 1, 33, "Input"], Cell[19612, 732, 2234, 69, 255, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[21883, 806, 122, 2, 33, "Input"], Cell[22008, 810, 269, 5, 53, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[22314, 820, 105, 2, 33, "Input"], Cell[22422, 824, 331, 10, 38, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[22790, 839, 172, 5, 33, "Input"], Cell[22965, 846, 272, 6, 53, "Output"] }, Open ]], Cell[23252, 855, 69, 4, 51, "Text"] }, Open ]], Cell[CellGroupData[{ Cell[23358, 864, 161, 5, 32, "Subsubtitle"], Cell[CellGroupData[{ Cell[23544, 873, 2232, 54, 644, "Input"], Cell[25779, 929, 4099, 95, 471, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[29915, 1029, 130, 2, 33, "Input"], Cell[30048, 1033, 12772, 371, 1199, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[42857, 1409, 120, 2, 33, "Input"], Cell[42980, 1413, 573, 11, 91, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[43590, 1429, 100, 2, 33, "Input"], Cell[43693, 1433, 1031, 32, 131, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[44761, 1470, 172, 5, 33, "Input"], Cell[44936, 1477, 578, 11, 91, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[45551, 1493, 237, 5, 53, "Input"], Cell[45791, 1500, 762, 15, 264, "Output"] }, Open ]], Cell[46568, 1518, 87, 1, 53, "Input"], Cell[46658, 1521, 2494, 63, 646, "Input"], Cell[CellGroupData[{ Cell[49177, 1588, 168, 4, 40, "Input"], Cell[49348, 1594, 1059, 33, 108, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[50444, 1632, 172, 5, 33, "Input"], Cell[50619, 1639, 550, 10, 91, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[51206, 1654, 212, 5, 53, "Input"], Cell[51421, 1661, 708, 14, 263, "Output"] }, Open ]], Cell[52144, 1678, 87, 1, 53, "Input"], Cell[52234, 1681, 2622, 67, 646, "Input"], Cell[CellGroupData[{ Cell[54881, 1752, 172, 4, 40, "Input"], Cell[55056, 1758, 1071, 33, 108, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[56164, 1796, 172, 5, 33, "Input"], Cell[56339, 1803, 589, 11, 91, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[56965, 1819, 214, 5, 53, "Input"], Cell[57182, 1826, 733, 15, 255, "Output"] }, Open ]], Cell[57930, 1844, 87, 1, 53, "Input"] }, Open ]], Cell[CellGroupData[{ Cell[58054, 1850, 216, 6, 32, "Subsubtitle"], Cell[58273, 1858, 200, 3, 33, "Input"], Cell[58476, 1863, 139, 2, 33, "Input"], Cell[CellGroupData[{ Cell[58640, 1869, 335, 6, 91, "Input"], Cell[CellGroupData[{ Cell[59000, 1879, 332, 9, 50, "Print", CellTags->"Info3396198200-1999354"], Cell[59335, 1890, 353, 10, 50, "Print", CellTags->"Info3396198200-7824464"], Cell[59691, 1902, 351, 10, 50, "Print", CellTags->"Info3396198200-8319869"], Cell[60045, 1914, 477, 12, 88, "Print", CellTags->"Info3396198201-8621354"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[60571, 1932, 225, 3, 91, "Input"], Cell[60799, 1937, 78, 1, 33, "Output"], Cell[60880, 1940, 107, 2, 38, "Output"], Cell[60990, 1944, 114, 2, 38, "Output"], Cell[61107, 1948, 128, 2, 38, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[61272, 1955, 164, 3, 33, "Input"], Cell[61439, 1960, 121, 2, 38, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[61597, 1967, 318, 8, 76, "Input"], Cell[61918, 1977, 201, 4, 38, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[62156, 1986, 255, 7, 58, "Input"], Cell[62414, 1995, 204, 4, 59, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[62655, 2004, 329, 8, 60, "Input"], Cell[62987, 2014, 208, 4, 57, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[63232, 2023, 164, 3, 56, "Input"], Cell[63399, 2028, 153, 4, 59, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[63589, 2037, 223, 5, 53, "Input"], Cell[63815, 2044, 92, 1, 37, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[63944, 2050, 175, 4, 56, "Input"], Cell[64122, 2056, 136, 3, 38, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[64295, 2064, 260, 5, 55, "Input"], Cell[64558, 2071, 155, 2, 38, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[64750, 2078, 134, 2, 59, "Input"], Cell[64887, 2082, 260, 6, 61, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[65184, 2093, 378, 9, 42, "Input"], Cell[65565, 2104, 174, 4, 38, "Output"] }, Open ]] }, Open ]] }, Open ]] } ] *)