(* Content-type: application/mathematica *) (*** Wolfram Notebook File ***) (* http://www.wolfram.com/nb *) (* CreatedBy='Mathematica 7.0' *) (*CacheID: 234*) (* Internal cache information: NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 145, 7] NotebookDataLength[ 830231, 13856] NotebookOptionsPosition[ 828204, 13787] NotebookOutlinePosition[ 828568, 13803] CellTagsIndexPosition[ 828525, 13800] WindowFrame->Normal*) (* Beginning of Notebook Content *) Notebook[{ Cell["Next we consider the criterion S2", "Text", CellChangeTimes->{{3.489916436201599*^9, 3.4899164383266125`*^9}, { 3.4899165082489357`*^9, 3.4899165116395817`*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"s2", "=", RowBox[{ FractionBox[ RowBox[{"(", RowBox[{ RowBox[{ SuperscriptBox["k", "2"], "r"}], "+", "q"}], ")"}], RowBox[{"2", RowBox[{"(", RowBox[{"1", "+", "k"}], ")"}]}]], "-", SuperscriptBox[ RowBox[{"(", RowBox[{ SuperscriptBox["r", "2"], "+", " ", RowBox[{"r", " ", "q"}]}], ")"}], 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PlotRangePadding->{ Scaled[0.02], Scaled[0.02], Scaled[0.02]}]}], "}"}]], "Output", CellChangeTimes->{{3.4899222182510557`*^9, 3.4899222305480094`*^9}, 3.4899227725514784`*^9, 3.4910571820189047`*^9, 3.491057660318155*^9}] }, Open ]], Cell["\<\ The maximum seems to happen at r = 0 or r = 1. Lets look at each case \ separatly. For the first\ \>", "Text", CellChangeTimes->{{3.491057441852005*^9, 3.491057452235305*^9}, { 3.4910579364587803`*^9, 3.491057972786905*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"s2", "/.", RowBox[{"{", RowBox[{"r", "\[Rule]", "0"}], "}"}]}]], "Input", CellChangeTimes->{{3.4910572511439047`*^9, 3.4910572548314047`*^9}}], Cell[BoxData[ FractionBox["q", RowBox[{"2", " ", RowBox[{"(", RowBox[{"1", "+", "k"}], ")"}]}]]], "Output", CellChangeTimes->{3.4910572553939047`*^9}] }, Open ]], Cell["For the second", "Text", CellChangeTimes->{{3.491057975661905*^9, 3.491057977005655*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"s2At1", "=", RowBox[{"s2", "/.", RowBox[{"{", RowBox[{"r", "\[Rule]", "1"}], "}"}]}]}]], "Input", CellChangeTimes->{{3.4910572748470297`*^9, 3.4910572778782797`*^9}, { 3.4910576927712803`*^9, 3.491057694974405*^9}}], Cell[BoxData[ RowBox[{"1", "-", SqrtBox[ RowBox[{"1", "+", "q"}]], "+", FractionBox[ RowBox[{ SuperscriptBox["k", "2"], "+", "q"}], RowBox[{"2", " 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