(************** Content-type: application/mathematica ************** CreatedBy='Mathematica 5.0' Mathematica-Compatible Notebook This notebook can be used with any Mathematica-compatible application, such as Mathematica, MathReader or Publicon. The data for the notebook starts with the line containing stars above. To get the notebook into a Mathematica-compatible application, do one of the following: * Save the data starting with the line of stars above into a file with a name ending in .nb, then open the file inside the application; * Copy the data starting with the line of stars above to the clipboard, then use the Paste menu command inside the application. Data for notebooks contains only printable 7-bit ASCII and can be sent directly in email or through ftp in text mode. Newlines can be CR, LF or CRLF (Unix, Macintosh or MS-DOS style). 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It \ does not seem to have useful output for m > 6.\ \>", "Section"], Cell[BoxData[ \(Clear[DTL]; DTL[m_, \ s_]\ := \ \[IndentingNewLine]Module[{TL, styles, ge, \ go, \ G, plt1, \ plt2, \ a, \ b, lste, \ lsto, \ lt}, \[IndentingNewLine]TL[x_, \ c_]\ := \ \(G'\)[c]\ \((x - c)\)\ + \ G[c]; \[IndentingNewLine]\[IndentingNewLine]styles[k_]\ := Join[\ {{Thickness[0.01]}}, \ Table[{Thickness[0.005]}, \ {i, \ 1, \ k}\ ]]; \[IndentingNewLine]ge[ x_]\ := \ \[Product]\+\(k = 1\)\%\(m\/2\)\((x\^2\ - \ \((2 \ k)\)\^2)\)\^2\ ; \ go[x_]\ := x\^2\ \(\[Product]\+\(k = 1\)\%\(\(m - 1\)\/2\)\((x\^2\ - \ \((2 \ k)\)\^2)\)\^2\)\ ; \[IndentingNewLine]If[OddQ[m], \ a\ = \ \(-m\)\ + 0.5; \ b\ = \ m\ - \ 0.5, \ \ a\ = \ \(-m\)\ - 0.5; \ b\ = \ m\ + \ 0.5]; If[OddQ[m], g[x_]\ := s + \(\(\ \)\(go[x]\)\)\/go[1], \ g[x_]\ := \ s + \ ge[x]\/ge[1]]; \[IndentingNewLine]G[x_]\ = Expand[\[Integral]g[ x] \[DifferentialD]x]; \[IndentingNewLine]\ \[IndentingNewLine]lste[k_]\ := \ Join[\ {G[x]}, \ Table[TL[x, i], \ {i, \ 2, \ k, 2}\ ], \ Table[TL[x, \(-i\)], \ {i, \ 2, \ k, 2}\ ]]; \[IndentingNewLine]lsto[k_]\ := \ Join[\ {G[x], \ TL[x, 0]}, \ Table[TL[x, i], \ {i, \ 2, \ k - 1, 2}\ ], \ Table[TL[x, \(-i\)], \ {i, \ 2, \ k - 1, 2}\ ]]; \[IndentingNewLine]lt[q_] := \ If[OddQ[q], lsto[q], \ lste[q]]; \[IndentingNewLine]\[IndentingNewLine]plt1 = Plot[Evaluate[lt[m]], \ {x, \ a, \ b}, \ \[IndentingNewLine]PlotStyle \[Rule] styles[m], ImageSize\ \[Rule] \ 3.2\ 72, \ DisplayFunction\ -> \ Identity]; plt2 = Plot[{\ g[x]}, \ {x, \ a, \ b}, AxesOrigin\ \[Rule] {0, \ 0}, \ \[IndentingNewLine]PlotStyle \[Rule] styles[m], ImageSize\ \[Rule] \ 3.2\ 72, \ DisplayFunction\ -> \ Identity]; \[IndentingNewLine]Show[ GraphicsArray[{plt1, \ plt2}], \ \[IndentingNewLine]DisplayFunction\ -> \ \ $DisplayFunction, \ ImageSize\ \[Rule] \ 7.8\ 72];\[IndentingNewLine]]\)], "Input", InitializationCell->True], Cell["Try it here. ", "Subsection"], Cell[BoxData[ RowBox[{"DTL", "[", RowBox[{ StyleBox["6", FontColor->RGBColor[1, 0, 0]], ",", StyleBox["1", FontColor->RGBColor[1, 0, 0]]}], "]"}]], "Input"], Cell["Scratch work below. You may wish to play with this.", "Section"], Cell[BoxData[ \(Clear[g, \ G, \ TL, \ styles]; g[x_]\ := \ \((x\^2\ - \ 2\^2)\)\^2 + \ 4; G[x_]\ = Expand[\[Integral]g[x] \[DifferentialD]x]; \ TL[x_, \ c_]\ := \ \(G'\)[c]\ \((x - c)\)\ + \ G[c]; \ styles := {{Thickness[0.008]}, \ {Thickness[0.005]}, \ {Thickness[ 0.005]}}\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(Clear[curve];\)\), "\n", \(\(curve\ = \ Plot[{\ G[x], \ TL[x, \ 2], \ TL[x, \ \(-2\)]}, \ {x, \ \(-3\), \ 3}, \ \[IndentingNewLine]PlotStyle \[Rule] styles, ImageSize\ \[Rule] \ 3.2\ 72, \n\t\tDisplayFunction\ -> \ Identity];\)\), "\n", \(\t\(pt1\ = \ Graphics[{PointSize[ .025], Point[{2, G[2]}]}];\)\), "\n", \(\t\(pt2\ = \ Graphics[{PointSize[ .025], Point[{\(-2\), \ G[\(-2\)]}]}];\)\), "\n", \(\t\(pt3 = \ Graphics[{PointSize[ .025], Point[{0, \ G[0]}]}];\)\), "\[IndentingNewLine]", \(\(Show[curve, \ \ pt1, \ pt2, pt3, \ ImageSize\ \[Rule] \ 3.2\ 72, \n DisplayFunction\ -> \ $DisplayFunction];\)\ \ \)}], "Input"], 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