(************** 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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For more information on notebooks and Mathematica-compatible applications, contact Wolfram Research: web: http://www.wolfram.com email: info@wolfram.com phone: +1-217-398-0700 (U.S.) Notebook reader applications are available free of charge from Wolfram Research. *******************************************************************) (*CacheID: 232*) (*NotebookFileLineBreakTest NotebookFileLineBreakTest*) (*NotebookOptionsPosition[ 388827, 15343]*) (*NotebookOutlinePosition[ 390355, 15389]*) (* CellTagsIndexPosition[ 390264, 15383]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell["Babbage", "Title", FontSize->18], Cell["\<\ A brief explanation of the method of differences, used in Charles Babbage's \ Difference Engine #2\ \>", "Subtitle", FontSize->16], Cell["\<\ Original Mathematica code by Ed Theren, rewritten by Scott Centoni.\ \>", "Subtitle", FontSize->16], Cell["\<\ The Difference Engines were designs for large machines to automatically print \ out tables of mathematical functions by extrapolation from a small number of \ points, using the method of differences.\ \>", "Text"], Cell[CellGroupData[{ Cell["initialization", "Section"], Cell[BoxData[ \(CleanSlate[]\)], "Input"], Cell[CellGroupData[{ Cell["imports", "Subsection"], Cell[BoxData[ \(Needs["\"]\)], "Input"] }, Open ]], Cell[CellGroupData[{ Cell["prologs", "Subsection"], Cell["Automatically display matrices in MatrixForm.", "Text"], Cell[BoxData[ \(\($PrePrint = If[MatrixQ[#], MatrixForm[#], #] &;\)\)], "Input"], Cell["Improve the appearance of text in graphs.", "Text"], Cell[BoxData[ \(\(\(\($TextStyle = {FontFamily \[Rule] "\", FontSize \[Rule] 12}\) \)\(;\)\(\ \)\(\($FormatType = TraditionalForm\) \)\(;\)\(\ \)\)\)], "Input"] }, Open ]], Cell[CellGroupData[{ Cell["misc", "Subsection"], Cell["\<\ Print non-matrix 2D tables in a format that keeps the items centered on each \ line.\ \>", "Text"], Cell[BoxData[ \(CheckerInterleave[x_] := Most@Flatten[\({#, "\<\>"} &\) /@ x, 1]\)], "Input"], Cell[BoxData[ \(\(CheckerPad[n_]\)[x_] := Module[{\[IndentingNewLine]padding = Table["\<\>", {k, n - Length[x]}]\[IndentingNewLine]}, \[IndentingNewLine]Join[ padding, CheckerInterleave[x], padding\ ]\[IndentingNewLine]]\)], "Input"], Cell[BoxData[ \(CheckerForm[x_] := Module[{\[IndentingNewLine]width = Max[Length /@ x]\[IndentingNewLine] (*\(,\)\(strlen = Max@Map[Length@ToString, x, {2}]\)*) \[IndentingNewLine]}, \ \[IndentingNewLine]TableForm[\[IndentingNewLine]CheckerPad[width] /@ x, \[IndentingNewLine]TableSpacing \[Rule] {4, 1}\[IndentingNewLine]]\[IndentingNewLine]]\)], "Input"], Cell["Pascal's triangle", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Table[Binomial[n, k], {n, 0, 8}, {k, 0, n}]\)], "Input"], Cell[BoxData[ \({{1}, {1, 1}, {1, 2, 1}, {1, 3, 3, 1}, {1, 4, 6, 4, 1}, {1, 5, 10, 10, 5, 1}, {1, 6, 15, 20, 15, 6, 1}, {1, 7, 21, 35, 35, 21, 7, 1}, {1, 8, 28, 56, 70, 56, 28, 8, 1}}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(CheckerForm[Table[Binomial[n, k], {n, 0, 8}, {k, 0, n}]]\)], "Input"], Cell[BoxData[ TagBox[GridBox[{ {"\<\"\"\>", "\<\"\"\>", "\<\"\"\>", "\<\"\"\>", "\<\"\"\>", \ "\<\"\"\>", "\<\"\"\>", "\<\"\"\>", "1", "\<\"\"\>", "\<\"\"\>", "\<\"\"\>", "\<\"\"\>", "\<\"\"\>", \ "\<\"\"\>", "\<\"\"\>", "\<\"\"\>"}, {"\<\"\"\>", "\<\"\"\>", "\<\"\"\>", "\<\"\"\>", "\<\"\"\>", \ "\<\"\"\>", "\<\"\"\>", "1", "\<\"\"\>", "1", "\<\"\"\>", "\<\"\"\>", "\<\"\"\>", "\<\"\"\>", "\<\"\"\>", \ "\<\"\"\>", "\<\"\"\>"}, {"\<\"\"\>", "\<\"\"\>", "\<\"\"\>", "\<\"\"\>", "\<\"\"\>", \ "\<\"\"\>", "1", "\<\"\"\>", "2", "\<\"\"\>", "1", "\<\"\"\>", "\<\"\"\>", "\<\"\"\>", "\<\"\"\>", "\<\"\"\>", \ "\<\"\"\>"}, {"\<\"\"\>", "\<\"\"\>", "\<\"\"\>", "\<\"\"\>", "\<\"\"\>", "1", "\<\"\"\>", "3", "\<\"\"\>", "3", "\<\"\"\>", "1", "\<\"\"\>", "\<\"\"\>", "\<\"\"\>", "\<\"\"\>", "\<\"\"\>"}, {"\<\"\"\>", "\<\"\"\>", "\<\"\"\>", "\<\"\"\>", "1", "\<\"\"\>", "4", "\<\"\"\>", "6", "\<\"\"\>", "4", "\<\"\"\>", "1", "\<\"\"\>", "\<\"\"\>", "\<\"\"\>", "\<\"\"\>"}, {"\<\"\"\>", "\<\"\"\>", "\<\"\"\>", "1", "\<\"\"\>", "5", "\<\"\"\>", "10", "\<\"\"\>", "10", "\<\"\"\>", "5", "\<\"\"\>", "1", "\<\"\"\>", "\<\"\"\>", "\<\"\"\>"}, {"\<\"\"\>", "\<\"\"\>", "1", "\<\"\"\>", "6", "\<\"\"\>", "15", "\<\"\"\>", "20", "\<\"\"\>", "15", "\<\"\"\>", "6", "\<\"\"\>", "1", "\<\"\"\>", "\<\"\"\>"}, {"\<\"\"\>", "1", "\<\"\"\>", "7", "\<\"\"\>", "21", "\<\"\"\>", "35", "\<\"\"\>", "35", "\<\"\"\>", "21", "\<\"\"\>", "7", "\<\"\"\>", "1", "\<\"\"\>"}, {"1", "\<\"\"\>", "8", "\<\"\"\>", "28", "\<\"\"\>", "56", "\<\"\"\>", "70", "\<\"\"\>", "56", "\<\"\"\>", "28", "\<\"\"\>", "8", "\<\"\"\>", "1"} }, RowSpacings->4, ColumnSpacings->1, RowAlignments->Baseline, ColumnAlignments->{Left}], Function[ BoxForm`e$, TableForm[ BoxForm`e$, TableSpacing -> {4, 1}]]]], "Output"] }, Open ]] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["method of differences", "Section"], Cell["\<\ Perhaps the best way of thinking of the Difference Engine is as a machine for \ extrapolating tables of numbers. For instance, if we have a sequence of \ numbers 0,1,8,27,64, we recognize this as a list of the first few cubes. But \ if we did not make use of this information, how could we predict the next \ number? One way to approach this is to look at the successive differences \ from one item to the next:\ \>", "Text"], Cell[BoxData[ \(Differences[x_] := Rest[x] - Most[x]\)], "Input"], Cell[BoxData[ \(\(seq = {0, 1, 8, 27, 64, 125, 216};\)\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(Differences@%\)], "Input"], Cell[BoxData[ \({1, 7, 19, 37, 61, 91}\)], "Output"] }, Open ]], Cell["\<\ Hmm, not quite obvious what the pattern is, so let's repeat the process\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Differences@%\)], "Input"], Cell[BoxData[ \({6, 12, 18, 24, 30}\)], "Output"] }, Open ]], Cell["\<\ Aha...well, we can predict the next item in this sequence, but let's do it \ again anyway\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Differences@%\)], "Input"], Cell[BoxData[ \({6, 6, 6, 6}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Differences@%\)], "Input"], Cell[BoxData[ \({0, 0, 0}\)], "Output"] }, Open ]], Cell["\<\ We can see the pattern more clearly by showing all the differences \ together.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(dat = NestList[Differences, \ seq, \ 5]\)], "Input"], Cell[BoxData[ \({{0, 1, 8, 27, 64, 125, 216}, {1, 7, 19, 37, 61, 91}, {6, 12, 18, 24, 30}, {6, 6, 6, 6}, {0, 0, 0}, {0, 0}}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(CheckerForm[dat]\)], "Input"], Cell[BoxData[ TagBox[GridBox[{ {"0", "\<\"\"\>", "1", "\<\"\"\>", "8", "\<\"\"\>", "27", "\<\"\"\>", "64", "\<\"\"\>", "125", "\<\"\"\>", "216"}, {"\<\"\"\>", "1", "\<\"\"\>", "7", "\<\"\"\>", "19", "\<\"\"\>", "37", "\<\"\"\>", "61", "\<\"\"\>", "91", "\<\"\"\>"}, {"\<\"\"\>", "\<\"\"\>", "6", "\<\"\"\>", "12", "\<\"\"\>", "18", "\<\"\"\>", "24", "\<\"\"\>", "30", "\<\"\"\>", "\<\"\"\>"}, {"\<\"\"\>", "\<\"\"\>", "\<\"\"\>", "6", "\<\"\"\>", "6", "\<\"\"\>", "6", "\<\"\"\>", "6", "\<\"\"\>", "\<\"\"\>", "\<\"\"\>"}, {"\<\"\"\>", "\<\"\"\>", "\<\"\"\>", "\<\"\"\>", "0", "\<\"\"\>", "0", "\<\"\"\>", "0", "\<\"\"\>", "\<\"\"\>", "\<\"\"\>", "\<\"\"\>"}, {"\<\"\"\>", "\<\"\"\>", "\<\"\"\>", "\<\"\"\>", "\<\"\"\>", "0", "\<\"\"\>", "0", "\<\"\"\>", "\<\"\"\>", "\<\"\"\>", "\<\"\"\>", "\<\"\"\>"} }, RowSpacings->4, ColumnSpacings->1, RowAlignments->Baseline, ColumnAlignments->{Left}], Function[ BoxForm`e$, TableForm[ BoxForm`e$, TableSpacing -> {4, 1}]]]], "Output"] }, Open ]], Cell["\<\ Of course, all the higher-order differences are exactly zero, so that taking \ the first item in each list of differences forms a new sequence, 0, 1, 6, 6, \ 0, ...\ \>", "Text"], Cell["\<\ Now, with a bit of thought, it should be evident that this process can be \ reversed: if we have the sequence of coefficients, we can generate the \ sequence of values. That is precisely what the Difference Engine does: One \ inputs the difference coefficients, and cranks the machine to repeatedly add \ them up to obtain the values. Matrix multiplication is a useful way to \ express this process (though it was not developed until after the Difference \ Engine was designed).\ \>", "Text"], Cell[BoxData[ \(crm[x_] := \(crm[x] = Table[If[0 \[LessEqual] j - i < 2, 1, 0], {i, x}, {j, x}]\)\)], "Input"], Cell[BoxData[ \(cri[x_] := \(cri[x] = Table[If[i \[LessEqual] j, \((\(-1\))\)\^\(i + j\), 0], {i, x}, {j, x}]\)\)], "Input"], Cell["\<\ This is the matrix representing cranking through a Difference Engine with 6 \ wheels:\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(crm[6]\)], "Input"], Cell[BoxData[ TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {"1", "1", "0", "0", "0", "0"}, {"0", "1", "1", "0", "0", "0"}, {"0", "0", "1", "1", "0", "0"}, {"0", "0", "0", "1", "1", "0"}, {"0", "0", "0", "0", "1", "1"}, {"0", "0", "0", "0", "0", "1"} }], "\[NoBreak]", ")"}], Function[ BoxForm`e$, MatrixForm[ BoxForm`e$]]]], "Output"] }, Open ]], Cell["\<\ This is the matrix that represents cranking that machine backward:\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(cri[6]\)], "Input"], Cell[BoxData[ TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {"1", \(-1\), "1", \(-1\), "1", \(-1\)}, {"0", "1", \(-1\), "1", \(-1\), "1"}, {"0", "0", "1", \(-1\), "1", \(-1\)}, {"0", "0", "0", "1", \(-1\), "1"}, {"0", "0", "0", "0", "1", \(-1\)}, {"0", "0", "0", "0", "0", "1"} }], "\[NoBreak]", ")"}], Function[ BoxForm`e$, MatrixForm[ BoxForm`e$]]]], "Output"] }, Open ]], Cell["\<\ If we do first one, then the other, then we get the identity matrix\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(crm[6] . cri[6]\)], "Input"], Cell[BoxData[ TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {"1", "0", "0", "0", "0", "0"}, {"0", "1", "0", "0", "0", "0"}, {"0", "0", "1", "0", "0", "0"}, {"0", "0", "0", "1", "0", "0"}, {"0", "0", "0", "0", "1", "0"}, {"0", "0", "0", "0", "0", "1"} }], "\[NoBreak]", ")"}], Function[ BoxForm`e$, MatrixForm[ BoxForm`e$]]]], "Output"] }, Open ]], Cell[BoxData[ \(Crank[x_] := crm[Length[x]] x\)], "Input"], Cell[BoxData[ \(ReverseCrank[x_] := cri[Length[x]] . x\)], "Input"], Cell["This basically cranks the machine once:", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Crank[{0, 1, 6, 6}]\)], "Input"], Cell[BoxData[ \({1, 7, 12, 6}\)], "Output"] }, Open ]], Cell["That is, 0+1=1, 1+6=7, 6+6=12.", "Text"], Cell["Let's crank the machine again:", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Crank@%\)], "Input"], Cell[BoxData[ \({8, 19, 18, 6}\)], "Output"] }, Open ]], Cell["So 1+7=8, 7+12=19, 12+6=18.", "Text"], Cell["Again:", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Crank@%\)], "Input"], Cell[BoxData[ \({27, 37, 24, 6}\)], "Output"] }, Open ]], Cell["Or, better yet, several steps shown in sequence:", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(NestList[Crank, \ {0, 1, 6, 6, 0}, \ 7]\)], "Input"], Cell[BoxData[ TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {"0", "1", "6", "6", "0"}, {"1", "7", "12", "6", "0"}, {"8", "19", "18", "6", "0"}, {"27", "37", "24", "6", "0"}, {"64", "61", "30", "6", "0"}, {"125", "91", "36", "6", "0"}, {"216", "127", "42", "6", "0"}, {"343", "169", "48", "6", "0"} }], "\[NoBreak]", ")"}], Function[ BoxForm`e$, MatrixForm[ BoxForm`e$]]]], "Output"] }, Open ]], Cell["\<\ And we see that the first column has reproduced our table of function values, \ plus one more--and we could keep on cranking the machine as long as we \ wanted. In most cases, we're only concerned with the values taken on \ sequentially by the register shown in the first column.\ \>", "Text"], Cell[BoxData[ \(CrankTimes[regs_, n_] := First /@ NestList[Crank, regs, n]\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(CrankTimes[{0, 1, 6, 6, 0}, 20]\)], "Input"], Cell[BoxData[ \({0, 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000, 1331, 1728, 2197, 2744, 3375, 4096, 4913, 5832, 6859, 8000}\)], "Output"] }, Open ]], Cell[BoxData[ \(CrankThrough[regs_] := CrankTimes[regs, Length[regs] - 1]\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(CrankThrough[{0, 1, 6, 6, 0}]\)], "Input"], Cell[BoxData[ \({0, 1, 8, 27, 64}\)], "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["using the machine to crank out its own coefficients", "Section"], Cell["\<\ There is a trick to use the machine to generate its own coefficients from a \ table of regularly-spaced values. If you reverse the sign of every other item \ in the sequence before and after cranking through the list, it just churns \ out the coefficients.\ \>", "Text"], Cell[BoxData[ \(Alternate[x_] := x \((\(-1\))\)\^Range[0, Length[x] - 1]\)], "Input"], Cell[BoxData[ \(DeCrankThrough[regs_] := Alternate@\(CrankThrough@\(Alternate@regs\)\)\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(DeCrankThrough[{0, 1, 8, 27, 64}]\)], "Input"], Cell[BoxData[ \({0, 1, 6, 6, 0}\)], "Output"] }, Open ]], Cell[TextData[{ "If we input ", Cell[BoxData[ \(TraditionalForm\`n\)]], " numbers as function values, we can output ", Cell[BoxData[ \(TraditionalForm\`n\)]], " numbers as coefficients without making any assumptions about higher-order \ terms. If we output more than ", Cell[BoxData[ \(TraditionalForm\`n\)]], " numbers, we assume that the higher-order differences are all zero." }], "Text"], Cell["Let's look at what happens in the general case:", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Array[a, \ 5]\)], "Input"], Cell[BoxData[ \({a[1], a[2], a[3], a[4], a[5]}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(DeCrankThrough[Array[a, \ 5]] // TableForm\)], "Input"], Cell[BoxData[ InterpretationBox[GridBox[{ {\(a[1]\)}, {\(\(-a[1]\) + a[2]\)}, {\(a[1] - 2\ a[2] + a[3]\)}, {\(\(-a[1]\) + 3\ a[2] - 3\ a[3] + a[4]\)}, {\(a[1] - 4\ a[2] + 6\ a[3] - 4\ a[4] + a[5]\)} }, RowSpacings->1, ColumnSpacings->3, RowAlignments->Baseline, ColumnAlignments->{Left}], TableForm[ { a[ 1], Plus[ Times[ -1, a[ 1]], a[ 2]], Plus[ a[ 1], Times[ -2, a[ 2]], a[ 3]], Plus[ Times[ -1, a[ 1]], Times[ 3, a[ 2]], Times[ -3, a[ 3]], a[ 4]], Plus[ a[ 1], Times[ -4, a[ 2]], Times[ 6, a[ 3]], Times[ -4, a[ 4]], a[ 5]]}]]], "Output"] }, Open ]], Cell["\<\ This is very similar to the definition of a derivative, but note that the \ derivatives of different orders are being approximated at different points. \ We do recover the original points by cranking through the machine in the \ normal way:\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(CrankThrough@%\)], "Input"], Cell[BoxData[ \({a[1], a[2], a[3], a[4], a[5]}\)], "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["scaling the step size", "Section"], Cell["\<\ The technique of using the Difference Engine to generate its own coefficients \ from a sequence of equally-spaced values is of limited practical benefit. \ It's more useful if we can adjust the coefficients afterward to produce a \ more finely-spaced sequence.\ \>", "Text"], Cell[TextData[{ "To begin with, we consider the coefficients that correspond to the simple \ integer powers, ", Cell[BoxData[ \(TraditionalForm\`1, x, x\^2, x\^3, ... \)]] }], "Text"], Cell[BoxData[ \(\(polycoeff[0] = {1};\)\)], "Input"], Cell[BoxData[ \(polycoeff[n_] := DeCrankThrough[Table[k\^n, {k, 0, n}]]\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(CheckerForm[Table[polycoeff[k], {k, 0, 8}]]\)], "Input"], Cell[BoxData[ TagBox[GridBox[{ {"\<\"\"\>", "\<\"\"\>", "\<\"\"\>", "\<\"\"\>", "\<\"\"\>", \ "\<\"\"\>", "\<\"\"\>", "\<\"\"\>", "1", "\<\"\"\>", "\<\"\"\>", "\<\"\"\>", "\<\"\"\>", "\<\"\"\>", \ "\<\"\"\>", "\<\"\"\>", "\<\"\"\>"}, {"\<\"\"\>", "\<\"\"\>", "\<\"\"\>", "\<\"\"\>", "\<\"\"\>", \ "\<\"\"\>", "\<\"\"\>", "0", "\<\"\"\>", "1", "\<\"\"\>", "\<\"\"\>", "\<\"\"\>", "\<\"\"\>", "\<\"\"\>", \ "\<\"\"\>", "\<\"\"\>"}, {"\<\"\"\>", "\<\"\"\>", "\<\"\"\>", "\<\"\"\>", "\<\"\"\>", \ "\<\"\"\>", "0", "\<\"\"\>", "1", "\<\"\"\>", "2", "\<\"\"\>", "\<\"\"\>", "\<\"\"\>", "\<\"\"\>", "\<\"\"\>", \ "\<\"\"\>"}, {"\<\"\"\>", "\<\"\"\>", "\<\"\"\>", "\<\"\"\>", "\<\"\"\>", "0", "\<\"\"\>", "1", "\<\"\"\>", "6", "\<\"\"\>", "6", "\<\"\"\>", "\<\"\"\>", "\<\"\"\>", "\<\"\"\>", "\<\"\"\>"}, {"\<\"\"\>", "\<\"\"\>", "\<\"\"\>", "\<\"\"\>", "0", "\<\"\"\>", "1", "\<\"\"\>", "14", "\<\"\"\>", "36", "\<\"\"\>", "24", "\<\"\"\>", "\<\"\"\>", "\<\"\"\>", "\<\"\"\>"}, {"\<\"\"\>", "\<\"\"\>", "\<\"\"\>", "0", "\<\"\"\>", "1", "\<\"\"\>", "30", "\<\"\"\>", "150", "\<\"\"\>", "240", "\<\"\"\>", "120", "\<\"\"\>", "\<\"\"\>", "\<\"\"\>"}, {"\<\"\"\>", "\<\"\"\>", "0", "\<\"\"\>", "1", "\<\"\"\>", "62", "\<\"\"\>", "540", "\<\"\"\>", "1560", "\<\"\"\>", "1800", "\<\"\"\>", "720", "\<\"\"\>", "\<\"\"\>"}, {"\<\"\"\>", "0", "\<\"\"\>", "1", "\<\"\"\>", "126", "\<\"\"\>", "1806", "\<\"\"\>", "8400", "\<\"\"\>", "16800", "\<\"\"\>", "15120", "\<\"\"\>", "5040", "\<\"\"\>"}, {"0", "\<\"\"\>", "1", "\<\"\"\>", "254", "\<\"\"\>", "5796", "\<\"\"\>", "40824", "\<\"\"\>", "126000", "\<\"\"\>", "191520", "\<\"\"\>", "141120", "\<\"\"\>", "40320"} }, RowSpacings->4, ColumnSpacings->1, RowAlignments->Baseline, ColumnAlignments->{Left}], Function[ BoxForm`e$, TableForm[ BoxForm`e$, TableSpacing -> {4, 1}]]]], "Output"] }, Open ]], Cell["\<\ We'll try the cube function again, but this time sample it with a step of two \ instead of one.\ \>", "Text"], Cell[BoxData[ \(\(func = x\^3;\)\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(Table[func, {x, 0, 14}]\)], "Input"], Cell[BoxData[ \({0, 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000, 1331, 1728, 2197, 2744}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(dat = Table[func, {x, 0, 14, 2}]\)], "Input"], Cell[BoxData[ \({0, 8, 64, 216, 512, 1000, 1728, 2744}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(coe = DeCrankThrough[dat]\)], "Input"], Cell[BoxData[ \({0, 8, 48, 48, 0, 0, 0, 0}\)], "Output"] }, Open ]], Cell["\<\ We see that instead of the coefficient list {0,1,6,6,0,...} we got before, we \ get larger values. But we can't just multiply the coefficients by a constant \ to change the step size.\ \>", "Text"], Cell["\<\ It turns out that we can multiply the coefficient list by a matrix that will \ scale the step. To scale a 7th-order coefficient list by one half, we \ multiply it on the left by\ \>", "Text"], Cell[BoxData[ RowBox[{ RowBox[{"halve", "=", TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {"1", "0", "0", "0", "0", "0", "0", "0"}, { "0", \(1\/2\), \(-\(1\/8\)\), \(1\/16\), \(-\(5\/128\)\), \ \(7\/256\), \(-\(21\/1024\)\), \(33\/2048\)}, {"0", "0", \(1\/4\), \(-\(1\/8\)\), \(5\/64\), \(-\(7\/128\)\), \ \(21\/512\), \(-\(33\/1024\)\)}, {"0", "0", "0", \(1\/8\), \(-\(3\/32\)\), \(9\/128\), \(-\(7\/128\)\), \ \(45\/1024\)}, {"0", "0", "0", "0", \(1\/16\), \(-\(1\/16\)\), \(7\/128\), \ \(-\(3\/64\)\)}, {"0", "0", "0", "0", "0", \(1\/32\), \(-\(5\/128\)\), \(5\/128\)}, {"0", "0", "0", "0", "0", "0", \(1\/64\), \(-\(3\/128\)\)}, {"0", "0", "0", "0", "0", "0", "0", \(1\/128\)} }], "\[NoBreak]", ")"}], Function[ BoxForm`e$, MatrixForm[ BoxForm`e$]]]}], ";"}]], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(halve coe\)], "Input"], Cell[BoxData[ \({0, 1, 6, 6, 0, 0, 0, 0}\)], "Output"] }, Open ]], Cell["\<\ This recovers the coefficient list we saw before, and the same technique \ works with any function.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(CrankThrough[coe]\)], "Input"], Cell[BoxData[ \({0, 8, 64, 216, 512, 1000, 1728, 2744}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(CrankThrough[halve . coe]\)], "Input"], Cell[BoxData[ \({0, 1, 8, 27, 64, 125, 216, 343}\)], "Output"] }, Open ]], Cell["We can repeat this to get a step size of one half:", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(halve . halve . coe\)], "Input"], Cell[BoxData[ \({0, 1\/8, 3\/4, 3\/4, 0, 0, 0, 0}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(CrankThrough[halve . halve . coe]\)], "Input"], Cell[BoxData[ \({0, 1\/8, 1, 27\/8, 8, 125\/8, 27, 343\/8}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(N@%\)], "Input"], Cell[BoxData[ \({0.`, 0.125`, 1.`, 3.375`, 8.`, 15.625`, 27.`, 42.875`}\)], "Output"] }, Open ]], Cell[TextData[{ "The general scaling matrix (of the appropriate order) is a function of the \ scaling factor ", Cell[BoxData[ \(TraditionalForm\`r\)]], "." }], "Text"], Cell[BoxData[ \(CoeffMat[n_] := Transpose@Table[PadRight[polycoeff[k], n + 1], {k, 0, n}]\)], "Input"], Cell[BoxData[ \(scalmat[r_, n_] := CoeffMat[n] . \((IdentityMatrix[n + 1] r\^Range[0, n])\) . 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.01473 Mdot 32351 .01473 Mdot .32423 .01473 Mdot .32495 .01473 Mdot .32567 .01473 Mdot .32639 .01473 Mdot .3271 01473 Mdot .32782 .01473 Mdot .32854 .01473 Mdot 32926 .01473 Mdot .32998 .01473 Mdot .3307 .01473 Mdot .33142 .01473 Mdot .33213 .01473 Mdot .33285 01473 Mdot .33357 .01473 Mdot .33429 .01473 Mdot 33501 .01474 Mdot .33573 .01474 Mdot .33644 .01474 Mdot .33716 .01474 Mdot .33788 .01474 Mdot .3386 01474 Mdot .33932 .01474 Mdot .34003 .01474 Mdot 34075 .01474 Mdot .34147 .01474 Mdot .34219 .01474 Mdot .34291 .01474 Mdot .34362 .01474 Mdot .34434 01474 Mdot .34506 .01474 Mdot .34578 .01474 Mdot 3465 .01474 Mdot .34721 .01474 Mdot .34793 .01474 Mdot .34865 .01475 Mdot .34937 .01475 Mdot .35008 01475 Mdot .3508 .01475 Mdot .35152 .01475 Mdot 35224 .01475 Mdot .35295 .01475 Mdot .35367 .01475 Mdot .35439 .01475 Mdot .35511 .01475 Mdot .35582 01475 Mdot .35654 .01475 Mdot .35726 .01475 Mdot 35798 .01475 Mdot .35869 .01475 Mdot .35941 .01476 Mdot .36013 .01476 Mdot .36085 .01476 Mdot .36156 01476 Mdot .36228 .01476 Mdot .363 .01476 Mdot 36372 .01476 Mdot .36443 .01476 Mdot .36515 .01476 Mdot .36587 .01476 Mdot .36658 .01476 Mdot .3673 01476 Mdot .36802 .01477 Mdot .36873 .01477 Mdot 36945 .01477 Mdot .37017 .01477 Mdot .37089 .01477 Mdot .3716 .01477 Mdot .37232 .01477 Mdot .37304 01477 Mdot .37375 .01477 Mdot .37447 .01477 Mdot 37519 .01478 Mdot .3759 .01478 Mdot .37662 .01478 Mdot .37734 .01478 Mdot .37805 .01478 Mdot .37877 01478 Mdot .37949 .01478 Mdot .3802 .01478 Mdot 38092 .01479 Mdot .38163 .01479 Mdot .38235 .01479 Mdot .38307 .01479 Mdot .38378 .01479 Mdot .3845 01479 Mdot .38522 .01479 Mdot .38593 .0148 Mdot 38665 .0148 Mdot .38737 .0148 Mdot .38808 .0148 Mdot .3888 .0148 Mdot .38951 .0148 Mdot .39023 0148 Mdot .39095 .01481 Mdot .39166 .01481 Mdot 39238 .01481 Mdot .39309 .01481 Mdot .39381 .01481 Mdot .39453 .01481 Mdot .39524 .01482 Mdot .39596 01482 Mdot .39667 .01482 Mdot .39739 .01482 Mdot 3981 .01482 Mdot .39882 .01483 Mdot .39954 .01483 Mdot .40025 .01483 Mdot .40097 .01483 Mdot .40168 01483 Mdot .4024 .01483 Mdot .40311 .01484 Mdot 40383 .01484 Mdot .40454 .01484 Mdot .40526 .01484 Mdot .40597 .01485 Mdot .40669 .01485 Mdot .40741 01485 Mdot .40812 .01485 Mdot .40884 .01485 Mdot 40955 .01486 Mdot .41027 .01486 Mdot .41098 .01486 Mdot .4117 .01486 Mdot .41241 .01487 Mdot .41313 01487 Mdot .41384 .01487 Mdot .41456 .01487 Mdot 41527 .01488 Mdot .41599 .01488 Mdot .4167 .01488 Mdot .41742 .01489 Mdot .41813 .01489 Mdot .41885 01489 Mdot .41956 .01489 Mdot .42028 .0149 Mdot 42099 .0149 Mdot .4217 .0149 Mdot .42242 .01491 Mdot .42313 .01491 Mdot .42385 .01491 Mdot .42456 01492 Mdot .42528 .01492 Mdot .42599 .01492 Mdot 42671 .01493 Mdot .42742 .01493 Mdot .42813 .01493 Mdot .42885 .01494 Mdot .42956 .01494 Mdot .43028 01494 Mdot .43099 .01495 Mdot .43171 .01495 Mdot 43242 .01495 Mdot .43313 .01496 Mdot .43385 .01496 Mdot .43456 .01497 Mdot .43528 .01497 Mdot .43599 01497 Mdot .4367 .01498 Mdot .43742 .01498 Mdot 43813 .01499 Mdot .43885 .01499 Mdot .43956 .015 Mdot .44027 .015 Mdot .44099 .015 Mdot .4417 01501 Mdot .44241 .01501 Mdot .44313 .01502 Mdot 44384 .01502 Mdot .44455 .01503 Mdot .44527 .01503 Mdot .44598 .01504 Mdot .4467 .01504 Mdot .44741 01505 Mdot .44812 .01505 Mdot .44884 .01506 Mdot 44955 .01506 Mdot .45026 .01507 Mdot .45097 .01507 Mdot .45169 .01508 Mdot .4524 .01508 Mdot .45311 01509 Mdot .45383 .0151 Mdot .45454 .0151 Mdot 45525 .01511 Mdot .45597 .01511 Mdot .45668 .01512 Mdot .45739 .01513 Mdot .4581 .01513 Mdot .45882 01514 Mdot .45953 .01514 Mdot .46024 .01515 Mdot 46096 .01516 Mdot .46167 .01516 Mdot .46238 .01517 Mdot .46309 .01518 Mdot .46381 .01518 Mdot .46452 01519 Mdot .46523 .0152 Mdot .46594 .01521 Mdot 46666 .01521 Mdot .46737 .01522 Mdot .46808 .01523 Mdot .46879 .01524 Mdot .46951 .01524 Mdot .47022 01525 Mdot .47093 .01526 Mdot .47164 .01527 Mdot 47235 .01527 Mdot .47307 .01528 Mdot .47378 .01529 Mdot .47449 .0153 Mdot .4752 .01531 Mdot .47591 01532 Mdot .47663 .01532 Mdot .47734 .01533 Mdot 47805 .01534 Mdot .47876 .01535 Mdot .47947 .01536 Mdot .48018 .01537 Mdot .4809 .01538 Mdot .48161 01539 Mdot .48232 .0154 Mdot .48303 .01541 Mdot 48374 .01542 Mdot .48445 .01543 Mdot .48517 .01544 Mdot .48588 .01545 Mdot .48659 .01546 Mdot .4873 01547 Mdot .48801 .01548 Mdot .48872 .01549 Mdot 48943 .0155 Mdot .49014 .01551 Mdot .49085 .01552 Mdot .49157 .01553 Mdot .49228 .01555 Mdot .49299 01556 Mdot .4937 .01557 Mdot .49441 .01558 Mdot 49512 .01559 Mdot .49583 .01561 Mdot .49654 .01562 Mdot .49725 .01563 Mdot .49796 .01564 Mdot .49867 01566 Mdot .49938 .01567 Mdot .5001 .01568 Mdot 50081 .01569 Mdot .50152 .01571 Mdot .50223 .01572 Mdot .50294 .01573 Mdot .50365 .01575 Mdot .50436 01576 Mdot .50507 .01578 Mdot .50578 .01579 Mdot 50649 .01581 Mdot .5072 .01582 Mdot .50791 .01584 Mdot .50862 .01585 Mdot .50933 .01587 Mdot .51004 01588 Mdot .51075 .0159 Mdot .51146 .01591 Mdot 51217 .01593 Mdot .51288 .01594 Mdot .51359 .01596 Mdot .5143 .01598 Mdot .51501 .01599 Mdot .51572 01601 Mdot .51643 .01603 Mdot .51714 .01605 Mdot 51785 .01606 Mdot .51856 .01608 Mdot .51927 .0161 Mdot .51997 .01612 Mdot .52068 .01614 Mdot .52139 01615 Mdot .5221 .01617 Mdot .52281 .01619 Mdot 52352 .01621 Mdot .52423 .01623 Mdot .52494 .01625 Mdot .52565 .01627 Mdot .52636 .01629 Mdot .52707 01631 Mdot .52778 .01633 Mdot .52848 .01635 Mdot 52919 .01637 Mdot .5299 .0164 Mdot .53061 .01642 Mdot .53132 .01644 Mdot .53203 .01646 Mdot .53274 01648 Mdot .53345 .01651 Mdot .53415 .01653 Mdot 53486 .01655 Mdot .53557 .01658 Mdot .53628 .0166 Mdot .53699 .01662 Mdot .5377 .01665 Mdot .5384 01667 Mdot .53911 .0167 Mdot .53982 .01672 Mdot 54053 .01675 Mdot .54124 .01677 Mdot .54195 .0168 Mdot .54265 .01682 Mdot .54336 .01685 Mdot .54407 01688 Mdot .54478 .0169 Mdot .54549 .01693 Mdot 54619 .01696 Mdot .5469 .01699 Mdot .54761 .01702 Mdot .54832 .01704 Mdot .54902 .01707 Mdot .54973 0171 Mdot .55044 .01713 Mdot .55115 .01716 Mdot 55185 .01719 Mdot .55256 .01722 Mdot .55327 .01725 Mdot .55398 .01728 Mdot .55468 .01732 Mdot .55539 01735 Mdot .5561 .01738 Mdot .55681 .01741 Mdot 55751 .01744 Mdot .55822 .01748 Mdot .55893 .01751 Mdot .55963 .01755 Mdot .56034 .01758 Mdot .56105 01761 Mdot .56175 .01765 Mdot .56246 .01769 Mdot 56317 .01772 Mdot .56387 .01776 Mdot .56458 .01779 Mdot .56529 .01783 Mdot .56599 .01787 Mdot .5667 01791 Mdot .56741 .01794 Mdot .56811 .01798 Mdot 56882 .01802 Mdot .56953 .01806 Mdot .57023 .0181 Mdot .57094 .01814 Mdot .57165 .01818 Mdot .57235 01822 Mdot .57306 .01826 Mdot .57376 .01831 Mdot 57447 .01835 Mdot .57518 .01839 Mdot .57588 .01844 Mdot .57659 .01848 Mdot .57729 .01852 Mdot .578 01857 Mdot .5787 .01861 Mdot .57941 .01866 Mdot 58012 .0187 Mdot .58082 .01875 Mdot .58153 .0188 Mdot .58223 .01885 Mdot .58294 .01889 Mdot .58364 01894 Mdot .58435 .01899 Mdot .58505 .01904 Mdot 58576 .01909 Mdot .58646 .01914 Mdot .58717 .01919 Mdot .58787 .01924 Mdot .58858 .0193 Mdot .58928 01935 Mdot .58999 .0194 Mdot .59069 .01946 Mdot 5914 .01951 Mdot .5921 .01957 Mdot .59281 .01962 Mdot .59351 .01968 Mdot .59422 .01973 Mdot .59492 01979 Mdot .59563 .01985 Mdot .59633 .01991 Mdot 59704 .01996 Mdot .59774 .02002 Mdot .59845 .02008 Mdot .59915 .02014 Mdot .59985 .02021 Mdot .60056 02027 Mdot .60126 .02033 Mdot .60197 .02039 Mdot 60267 .02046 Mdot .60338 .02052 Mdot .60408 .02059 Mdot .60478 .02065 Mdot .60549 .02072 Mdot .60619 02079 Mdot .60689 .02085 Mdot .6076 .02092 Mdot 6083 .02099 Mdot .60901 .02106 Mdot .60971 .02113 Mdot .61041 .0212 Mdot .61112 .02127 Mdot .61182 02135 Mdot .61252 .02142 Mdot .61323 .02149 Mdot 61393 .02157 Mdot .61463 .02164 Mdot .61534 .02172 Mdot .61604 .02179 Mdot .61674 .02187 Mdot .61745 02195 Mdot .61815 .02203 Mdot .61885 .02211 Mdot 61955 .02219 Mdot .62026 .02227 Mdot .62096 .02235 Mdot .62166 .02244 Mdot .62237 .02252 Mdot .62307 0226 Mdot .62377 .02269 Mdot .62447 .02278 Mdot 62518 .02286 Mdot .62588 .02295 Mdot .62658 .02304 Mdot .62728 .02313 Mdot .62799 .02322 Mdot .62869 02331 Mdot .62939 .0234 Mdot .63009 .0235 Mdot 63079 .02359 Mdot .6315 .02368 Mdot .6322 .02378 Mdot .6329 .02388 Mdot .6336 .02397 Mdot .6343 02407 Mdot .63501 .02417 Mdot .63571 .02427 Mdot 63641 .02437 Mdot .63711 .02448 Mdot .63781 .02458 Mdot .63851 .02468 Mdot .63922 .02479 Mdot .63992 02489 Mdot .64062 .025 Mdot .64132 .02511 Mdot 64202 .02522 Mdot .64272 .02533 Mdot .64342 .02544 Mdot .64412 .02555 Mdot .64483 .02566 Mdot .64553 02578 Mdot .64623 .02589 Mdot .64693 .02601 Mdot 64763 .02613 Mdot .64833 .02624 Mdot .64903 .02636 Mdot .64973 .02649 Mdot .65043 .02661 Mdot .65113 02673 Mdot .65183 .02685 Mdot .65253 .02698 Mdot 65323 .02711 Mdot .65393 .02723 Mdot .65463 .02736 Mdot .65533 .02749 Mdot .65603 .02762 Mdot .65674 02775 Mdot .65744 .02789 Mdot .65814 .02802 Mdot 65884 .02816 Mdot .65954 .02829 Mdot .66024 .02843 Mdot .66093 .02857 Mdot .66163 .02871 Mdot .66233 02885 Mdot .66303 .029 Mdot .66373 .02914 Mdot 66443 .02929 Mdot .66513 .02944 Mdot .66583 .02958 Mdot .66653 .02973 Mdot .66723 .02989 Mdot .66793 03004 Mdot .66863 .03019 Mdot .66933 .03035 Mdot 67003 .0305 Mdot .67073 .03066 Mdot .67143 .03082 Mdot .67212 .03098 Mdot .67282 .03114 Mdot .67352 03131 Mdot .67422 .03147 Mdot .67492 .03164 Mdot 67562 .03181 Mdot .67632 .03198 Mdot .67702 .03215 Mdot .67771 .03232 Mdot .67841 .0325 Mdot .67911 03267 Mdot .67981 .03285 Mdot .68051 .03303 Mdot 68121 .03321 Mdot .6819 .03339 Mdot .6826 .03357 Mdot .6833 .03376 Mdot .684 .03394 Mdot .6847 03413 Mdot .68539 .03432 Mdot .68609 .03451 Mdot 68679 .03471 Mdot .68749 .0349 Mdot .68819 .0351 Mdot .68888 .0353 Mdot .68958 .0355 Mdot .69028 0357 Mdot .69098 .0359 Mdot .69167 .03611 Mdot 69237 .03631 Mdot .69307 .03652 Mdot .69377 .03673 Mdot .69446 .03695 Mdot .69516 .03716 Mdot .69586 03738 Mdot .69655 .03759 Mdot .69725 .03781 Mdot 69795 .03804 Mdot .69864 .03826 Mdot .69934 .03848 Mdot .70004 .03871 Mdot .70073 .03894 Mdot .70143 03917 Mdot .70213 .03941 Mdot .70282 .03964 Mdot 70352 .03988 Mdot .70422 .04012 Mdot .70491 .04036 Mdot .70561 .0406 Mdot .70631 .04085 Mdot .707 04109 Mdot .7077 .04134 Mdot .70839 .04159 Mdot 70909 .04185 Mdot .70979 .0421 Mdot .71048 .04236 Mdot .71118 .04262 Mdot .71187 .04288 Mdot .71257 04315 Mdot .71326 .04341 Mdot .71396 .04368 Mdot 71466 .04395 Mdot .71535 .04423 Mdot .71605 .0445 Mdot .71674 .04478 Mdot .71744 .04506 Mdot .71813 04534 Mdot .71883 .04563 Mdot .71952 .04591 Mdot 72022 .0462 Mdot .72091 .0465 Mdot .72161 .04679 Mdot .7223 .04709 Mdot .723 .04739 Mdot .72369 04769 Mdot .72439 .04799 Mdot .72508 .0483 Mdot 72578 .04861 Mdot .72647 .04892 Mdot .72716 .04923 Mdot .72786 .04955 Mdot .72855 .04987 Mdot .72925 05019 Mdot .72994 .05051 Mdot .73064 .05084 Mdot 73133 .05117 Mdot .73202 .0515 Mdot .73272 .05184 Mdot .73341 .05218 Mdot .73411 .05252 Mdot .7348 05286 Mdot .73549 .05321 Mdot .73619 .05355 Mdot 73688 .05391 Mdot .73757 .05426 Mdot .73827 .05462 Mdot .73896 .05498 Mdot .73965 .05534 Mdot .74035 05571 Mdot .74104 .05608 Mdot .74173 .05645 Mdot 74243 .05682 Mdot .74312 .0572 Mdot .74381 .05758 Mdot .74451 .05796 Mdot .7452 .05835 Mdot .74589 05874 Mdot .74658 .05913 Mdot .74728 .05953 Mdot 74797 .05993 Mdot .74866 .06033 Mdot .74936 .06074 Mdot .75005 .06115 Mdot .75074 .06156 Mdot .75143 06197 Mdot .75212 .06239 Mdot .75282 .06281 Mdot 75351 .06324 Mdot .7542 .06367 Mdot .75489 .0641 Mdot .75559 .06453 Mdot .75628 .06497 Mdot .75697 06542 Mdot .75766 .06586 Mdot .75835 .06631 Mdot 75904 .06676 Mdot .75974 .06722 Mdot .76043 .06768 Mdot .76112 .06814 Mdot .76181 .06861 Mdot .7625 06908 Mdot .76319 .06955 Mdot .76388 .07003 Mdot 76457 .07051 Mdot .76527 .071 Mdot .76596 .07149 Mdot .76665 .07198 Mdot .76734 .07247 Mdot .76803 07297 Mdot .76872 .07348 Mdot .76941 .07399 Mdot 7701 .0745 Mdot .77079 .07501 Mdot .77148 .07553 Mdot .77217 .07606 Mdot .77286 .07659 Mdot .77355 07712 Mdot .77424 .07765 Mdot .77493 .07819 Mdot 77562 .07874 Mdot .77631 .07929 Mdot .777 .07984 Mdot .77769 .08039 Mdot .77838 .08095 Mdot .77907 08152 Mdot .77976 .08209 Mdot .78045 .08266 Mdot 78114 .08324 Mdot .78183 .08382 Mdot .78252 .08441 Mdot .78321 .085 Mdot .7839 .0856 Mdot .78459 0862 Mdot .78528 .0868 Mdot .78597 .08741 Mdot 78666 .08802 Mdot .78735 .08864 Mdot .78804 .08926 Mdot .78873 .08989 Mdot .78941 .09052 Mdot .7901 09116 Mdot .79079 .0918 Mdot .79148 .09245 Mdot 79217 .0931 Mdot .79286 .09376 Mdot .79355 .09442 Mdot .79423 .09508 Mdot .79492 .09575 Mdot .79561 09643 Mdot .7963 .09711 Mdot .79699 .0978 Mdot 79768 .09849 Mdot .79836 .09918 Mdot .79905 .09988 Mdot .79974 .10059 Mdot .80043 .1013 Mdot .80111 10202 Mdot .8018 .10274 Mdot .80249 .10347 Mdot 80318 .1042 Mdot .80387 .10494 Mdot .80455 .10568 Mdot .80524 .10643 Mdot .80593 .10719 Mdot .80661 10795 Mdot .8073 .10871 Mdot .80799 .10948 Mdot 80868 .11026 Mdot .80936 .11104 Mdot .81005 .11183 Mdot .81074 .11262 Mdot .81142 .11342 Mdot .81211 11423 Mdot .8128 .11504 Mdot .81348 .11586 Mdot 81417 .11668 Mdot .81486 .11751 Mdot .81554 .11834 Mdot .81623 .11918 Mdot .81691 .12003 Mdot .8176 12088 Mdot .81829 .12174 Mdot .81897 .12261 Mdot 81966 .12348 Mdot .82034 .12436 Mdot .82103 .12524 Mdot .82172 .12613 Mdot .8224 .12703 Mdot .82309 12793 Mdot .82377 .12884 Mdot .82446 .12976 Mdot 82514 .13068 Mdot .82583 .13161 Mdot .82651 .13254 Mdot .8272 .13349 Mdot .82788 .13443 Mdot .82857 13539 Mdot .82926 .13635 Mdot .82994 .13732 Mdot 83062 .1383 Mdot .83131 .13928 Mdot .83199 .14027 Mdot .83268 .14127 Mdot .83336 .14227 Mdot .83405 14328 Mdot .83473 .1443 Mdot .83542 .14533 Mdot 8361 .14636 Mdot .83679 .1474 Mdot .83747 .14845 Mdot .83815 .1495 Mdot .83884 .15056 Mdot .83952 15163 Mdot .84021 .15271 Mdot .84089 .1538 Mdot 84157 .15489 Mdot .84226 .15599 Mdot .84294 .15709 Mdot .84362 .15821 Mdot .84431 .15933 Mdot .84499 16046 Mdot .84567 .1616 Mdot .84636 .16275 Mdot 84704 .1639 Mdot .84772 .16506 Mdot .84841 .16623 Mdot .84909 .16741 Mdot .84977 .1686 Mdot .85046 16979 Mdot .85114 .171 Mdot .85182 .17221 Mdot 8525 .17343 Mdot .85319 .17466 Mdot .85387 .17589 Mdot .85455 .17714 Mdot .85523 .17839 Mdot .85592 17965 Mdot .8566 .18092 Mdot .85728 .1822 Mdot 85796 .18349 Mdot .85864 .18479 Mdot .85933 .18609 Mdot .86001 .18741 Mdot .86069 .18873 Mdot .86137 19006 Mdot .86205 .1914 Mdot .86274 .19276 Mdot 86342 .19412 Mdot .8641 .19548 Mdot .86478 .19686 Mdot .86546 .19825 Mdot .86614 .19965 Mdot .86682 20105 Mdot .8675 .20247 Mdot .86819 .2039 Mdot 86887 .20533 Mdot .86955 .20678 Mdot .87023 .20823 Mdot .87091 .2097 Mdot .87159 .21117 Mdot .87227 21265 Mdot .87295 .21415 Mdot .87363 .21565 Mdot 87431 .21717 Mdot .87499 .21869 Mdot .87567 .22023 Mdot .87635 .22177 Mdot .87703 .22333 Mdot .87771 22489 Mdot .87839 .22647 Mdot .87907 .22805 Mdot 87975 .22965 Mdot .88043 .23126 Mdot .88111 .23288 Mdot .88179 .2345 Mdot .88247 .23614 Mdot .88315 2378 Mdot .88383 .23946 Mdot .88451 .24113 Mdot 88519 .24281 Mdot .88587 .24451 Mdot .88655 .24621 Mdot .88723 .24793 Mdot .8879 .24966 Mdot .88858 2514 Mdot .88926 .25315 Mdot .88994 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52888 Mdot .47734 .5292 Mdot .47805 .52952 Mdot 47876 .52983 Mdot .47947 .53014 Mdot .48018 .53044 Mdot .4809 .53073 Mdot .48161 .53101 Mdot .48232 53129 Mdot .48303 .53157 Mdot .48374 .53184 Mdot 48445 .5321 Mdot .48517 .53235 Mdot .48588 .5326 Mdot .48659 .53284 Mdot .4873 .53307 Mdot .48801 5333 Mdot .48872 .53352 Mdot .48943 .53373 Mdot 49014 .53394 Mdot .49085 .53414 Mdot .49157 .53433 Mdot .49228 .53452 Mdot .49299 .5347 Mdot .4937 53487 Mdot .49441 .53503 Mdot .49512 .53519 Mdot 49583 .53534 Mdot .49654 .53549 Mdot .49725 .53562 Mdot .49796 .53575 Mdot .49867 .53587 Mdot .49938 53598 Mdot .5001 .53609 Mdot .50081 .53619 Mdot 50152 .53628 Mdot .50223 .53636 Mdot .50294 .53644 Mdot .50365 .53651 Mdot .50436 .53657 Mdot .50507 53663 Mdot .50578 .53667 Mdot .50649 .53671 Mdot 5072 .53674 Mdot .50791 .53677 Mdot .50862 .53678 Mdot .50933 .53679 Mdot .51004 .53679 Mdot .51075 53679 Mdot .51146 .53677 Mdot .51217 .53675 Mdot 51288 .53672 Mdot .51359 .53668 Mdot .5143 .53664 Mdot .51501 .53659 Mdot .51572 .53653 Mdot .51643 53646 Mdot .51714 .53639 Mdot .51785 .5363 Mdot 51856 .53621 Mdot .51927 .53612 Mdot .51997 .53601 Mdot .52068 .5359 Mdot .52139 .53578 Mdot .5221 53565 Mdot .52281 .53552 Mdot .52352 .53538 Mdot 52423 .53523 Mdot .52494 .53507 Mdot .52565 .53491 Mdot .52636 .53474 Mdot .52707 .53456 Mdot .52778 53437 Mdot .52848 .53418 Mdot .52919 .53398 Mdot 5299 .53377 Mdot .53061 .53356 Mdot .53132 .53334 Mdot .53203 .53311 Mdot .53274 .53288 Mdot .53345 53264 Mdot .53415 .53239 Mdot .53486 .53213 Mdot 53557 .53187 Mdot .53628 .5316 Mdot .53699 .53132 Mdot .5377 .53104 Mdot .5384 .53075 Mdot .53911 53046 Mdot .53982 .53016 Mdot .54053 .52985 Mdot 54124 .52954 Mdot .54195 .52922 Mdot .54265 .52889 Mdot .54336 .52856 Mdot .54407 .52822 Mdot .54478 52787 Mdot .54549 .52752 Mdot .54619 .52716 Mdot 5469 .5268 Mdot .54761 .52643 Mdot .54832 .52606 Mdot .54902 .52568 Mdot .54973 .5253 Mdot .55044 52491 Mdot .55115 .52451 Mdot .55185 .52411 Mdot 55256 .5237 Mdot .55327 .52329 Mdot .55398 .52288 Mdot .55468 .52246 Mdot .55539 .52203 Mdot .5561 5216 Mdot .55681 .52116 Mdot .55751 .52072 Mdot 55822 .52028 Mdot .55893 .51983 Mdot .55963 .51938 Mdot .56034 .51892 Mdot .56105 .51846 Mdot .56175 51799 Mdot .56246 .51752 Mdot .56317 .51705 Mdot 56387 .51657 Mdot .56458 .51609 Mdot .56529 .51561 Mdot .56599 .51512 Mdot .5667 .51463 Mdot .56741 51414 Mdot .56811 .51364 Mdot .56882 .51314 Mdot 56953 .51264 Mdot .57023 .51213 Mdot .57094 .51162 Mdot .57165 .51111 Mdot .57235 .5106 Mdot .57306 51008 Mdot .57376 .50956 Mdot .57447 .50904 Mdot 57518 .50852 Mdot .57588 .50799 Mdot .57659 .50747 Mdot .57729 .50694 Mdot .578 .50641 Mdot .5787 50588 Mdot .57941 .50534 Mdot .58012 .50481 Mdot 58082 .50428 Mdot .58153 .50374 Mdot .58223 .5032 Mdot .58294 .50266 Mdot .58364 .50213 Mdot .58435 50159 Mdot .58505 .50105 Mdot .58576 .50051 Mdot 58646 .49997 Mdot .58717 .49943 Mdot .58787 .49889 Mdot .58858 .49835 Mdot .58928 .49781 Mdot .58999 49727 Mdot .59069 .49673 Mdot 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11645 .28985 Mdot .11717 .29377 Mdot .1179 .29769 Mdot .11862 .3016 Mdot .11934 .30551 Mdot .12007 30941 Mdot .12079 .31331 Mdot .12152 .3172 Mdot 12224 .32109 Mdot .12296 .32496 Mdot .12369 .32883 Mdot .12441 .33269 Mdot .12513 .33653 Mdot .12586 34036 Mdot .12658 .34418 Mdot .1273 .34799 Mdot 12803 .35178 Mdot .12875 .35555 Mdot .12947 .35931 Mdot .13019 .36305 Mdot .13092 .36677 Mdot .13164 37048 Mdot .13236 .37416 Mdot .13309 .37783 Mdot 13381 .38147 Mdot .13453 .38509 Mdot .13526 .38869 Mdot .13598 .39227 Mdot .1367 .39583 Mdot .13743 39936 Mdot .13815 .40287 Mdot .13887 .40635 Mdot 1396 .40981 Mdot .14032 .41324 Mdot .14104 .41665 Mdot .14177 .42002 Mdot .14249 .42338 Mdot .14321 4267 Mdot .14394 .42999 Mdot .14466 .43326 Mdot 14538 .4365 Mdot .1461 .43971 Mdot .14683 .44288 Mdot .14755 .44603 Mdot .14827 .44915 Mdot .149 45223 Mdot .14972 .45529 Mdot .15044 .45831 Mdot 15117 .46131 Mdot .15189 .46426 Mdot .15261 .46719 Mdot .15333 .47009 Mdot .15406 .47295 Mdot .15478 47578 Mdot .1555 .47857 Mdot .15623 .48133 Mdot 15695 .48406 Mdot .15767 .48675 Mdot .15839 .48941 Mdot .15912 .49204 Mdot .15984 .49463 Mdot .16056 49718 Mdot .16129 .4997 Mdot .16201 .50219 Mdot 16273 .50464 Mdot .16345 .50705 Mdot .16418 .50943 Mdot .1649 .51178 Mdot .16562 .51409 Mdot .16635 51636 Mdot .16707 .5186 Mdot .16779 .5208 Mdot 16851 .52297 Mdot .16924 .5251 Mdot .16996 .5272 Mdot .17068 .52925 Mdot .1714 .53128 Mdot .17213 53327 Mdot .17285 .53522 Mdot .17357 .53714 Mdot 1743 .53902 Mdot .17502 .54086 Mdot .17574 .54267 Mdot .17646 .54445 Mdot .17719 .54618 Mdot .17791 54789 Mdot .17863 .54955 Mdot .17935 .55119 Mdot 18008 .55278 Mdot .1808 .55435 Mdot .18152 .55587 Mdot .18224 .55737 Mdot .18297 .55882 Mdot .18369 56025 Mdot .18441 .56163 Mdot .18513 .56299 Mdot 18586 .56431 Mdot .18658 .56559 Mdot .1873 .56684 Mdot .18802 .56806 Mdot .18874 .56924 Mdot .18947 57039 Mdot .19019 .57151 Mdot .19091 .57259 Mdot 19163 .57364 Mdot .19236 .57466 Mdot .19308 .57565 Mdot .1938 .5766 Mdot .19452 .57752 Mdot .19525 57841 Mdot .19597 .57926 Mdot .19669 .58009 Mdot 19741 .58088 Mdot .19813 .58164 Mdot .19886 .58238 Mdot .19958 .58308 Mdot .2003 .58375 Mdot .20102 58438 Mdot .20174 .58499 Mdot .20247 .58557 Mdot 20319 .58612 Mdot .20391 .58665 Mdot .20463 .58714 Mdot .20535 .5876 Mdot .20608 .58803 Mdot .2068 58844 Mdot .20752 .58882 Mdot .20824 .58917 Mdot 20896 .58949 Mdot .20969 .58979 Mdot .21041 .59005 Mdot .21113 .5903 Mdot .21185 .59051 Mdot .21257 5907 Mdot .2133 .59086 Mdot .21402 .591 Mdot 21474 .59112 Mdot .21546 .5912 Mdot .21618 .59127 Mdot .2169 .5913 Mdot .21763 .59132 Mdot .21835 59131 Mdot .21907 .59128 Mdot .21979 .59122 Mdot 22051 .59114 Mdot .22124 .59104 Mdot .22196 .59091 Mdot .22268 .59077 Mdot .2234 .5906 Mdot .22412 59041 Mdot .22484 .5902 Mdot .22556 .58997 Mdot 22629 .58972 Mdot .22701 .58944 Mdot .22773 .58915 Mdot .22845 .58884 Mdot .22917 .58851 Mdot .22989 58816 Mdot .23062 .58779 Mdot .23134 .5874 Mdot 23206 .587 Mdot .23278 .58657 Mdot .2335 .58613 Mdot .23422 .58568 Mdot .23494 .5852 Mdot .23566 58471 Mdot .23639 .5842 Mdot .23711 .58368 Mdot 23783 .58314 Mdot .23855 .58259 Mdot .23927 .58202 Mdot .23999 .58144 Mdot .24071 .58084 Mdot .24143 58023 Mdot .24216 .5796 Mdot .24288 .57896 Mdot 2436 .57831 Mdot .24432 .57764 Mdot .24504 .57697 Mdot .24576 .57628 Mdot .24648 .57557 Mdot .2472 57486 Mdot .24792 .57414 Mdot .24865 .5734 Mdot 24937 .57265 Mdot .25009 .5719 Mdot .25081 .57113 Mdot .25153 .57035 Mdot .25225 .56957 Mdot .25297 56877 Mdot .25369 .56797 Mdot .25441 .56715 Mdot 25513 .56633 Mdot .25585 .5655 Mdot .25657 .56466 Mdot .2573 .56382 Mdot .25802 .56296 Mdot .25874 5621 Mdot .25946 .56124 Mdot .26018 .56037 Mdot 2609 .55949 Mdot .26162 .5586 Mdot .26234 .55771 Mdot .26306 .55682 Mdot .26378 .55592 Mdot .2645 55501 Mdot .26522 .5541 Mdot .26594 .55319 Mdot 26666 .55227 Mdot .26738 .55135 Mdot .2681 .55042 Mdot .26882 .54949 Mdot .26954 .54856 Mdot .27027 54763 Mdot .27099 .54669 Mdot .27171 .54575 Mdot 27243 .54481 Mdot .27315 .54387 Mdot .27387 .54292 Mdot .27459 .54198 Mdot .27531 .54103 Mdot .27603 54009 Mdot .27675 .53914 Mdot .27747 .53819 Mdot 27819 .53724 Mdot .27891 .5363 Mdot .27963 .53535 Mdot .28035 .5344 Mdot .28107 .53346 Mdot .28179 53251 Mdot .28251 .53157 Mdot .28323 .53063 Mdot 28395 .52969 Mdot .28467 .52875 Mdot .28539 .52781 Mdot .28611 .52688 Mdot .28683 .52595 Mdot .28755 52502 Mdot .28827 .52409 Mdot .28899 .52317 Mdot 28971 .52225 Mdot .29043 .52133 Mdot .29115 .52042 Mdot .29187 .51951 Mdot .29259 .51861 Mdot .29331 51771 Mdot .29403 .51682 Mdot .29475 .51592 Mdot 29547 .51504 Mdot .29618 .51416 Mdot .2969 .51328 Mdot .29762 .51241 Mdot .29834 .51154 Mdot .29906 51068 Mdot .29978 .50983 Mdot .3005 .50898 Mdot 30122 .50814 Mdot .30194 .5073 Mdot .30266 .50647 Mdot .30338 .50565 Mdot .3041 .50483 Mdot .30482 50402 Mdot .30554 .50322 Mdot .30626 .50243 Mdot 30698 .50164 Mdot .3077 .50086 Mdot .30841 .50008 Mdot .30913 .49932 Mdot .30985 .49856 Mdot .31057 49781 Mdot .31129 .49706 Mdot .31201 .49633 Mdot 31273 .4956 Mdot .31345 .49488 Mdot .31417 .49417 Mdot .31489 .49347 Mdot .3156 .49278 Mdot .31632 49209 Mdot .31704 .49142 Mdot .31776 .49075 Mdot 31848 .49009 Mdot .3192 .48944 Mdot .31992 .4888 Mdot .32064 .48817 Mdot .32136 .48755 Mdot .32207 48694 Mdot .32279 .48633 Mdot .32351 .48574 Mdot 32423 .48516 Mdot .32495 .48458 Mdot .32567 .48402 Mdot .32639 .48346 Mdot .3271 .48292 Mdot .32782 48238 Mdot .32854 .48186 Mdot .32926 .48134 Mdot 32998 .48084 Mdot .3307 .48034 Mdot .33142 .47986 Mdot .33213 .47938 Mdot .33285 .47892 Mdot .33357 47846 Mdot .33429 .47802 Mdot .33501 .47759 Mdot 33573 .47716 Mdot .33644 .47675 Mdot .33716 .47635 Mdot .33788 .47595 Mdot .3386 .47557 Mdot .33932 4752 Mdot .34003 .47484 Mdot .34075 .47449 Mdot 34147 .47415 Mdot .34219 .47382 Mdot .34291 .4735 Mdot .34362 .47319 Mdot .34434 .4729 Mdot .34506 47261 Mdot .34578 .47233 Mdot .3465 .47207 Mdot 34721 .47181 Mdot .34793 .47157 Mdot .34865 .47133 Mdot .34937 .47111 Mdot .35008 .47089 Mdot .3508 47069 Mdot .35152 .4705 Mdot .35224 .47032 Mdot 35295 .47015 Mdot .35367 .46998 Mdot .35439 .46983 Mdot .35511 .46969 Mdot .35582 .46956 Mdot .35654 46944 Mdot .35726 .46933 Mdot .35798 .46923 Mdot 35869 .46915 Mdot .35941 .46907 Mdot .36013 .469 Mdot .36085 .46894 Mdot .36156 .46889 Mdot .36228 46885 Mdot .363 .46882 Mdot .36372 .4688 Mdot 36443 .4688 Mdot .36515 .4688 Mdot .36587 .46881 Mdot .36658 .46883 Mdot .3673 .46886 Mdot .36802 4689 Mdot .36873 .46895 Mdot .36945 .469 Mdot 37017 .46907 Mdot .37089 .46915 Mdot .3716 .46924 Mdot .37232 .46933 Mdot .37304 .46944 Mdot .37375 46955 Mdot .37447 .46967 Mdot .37519 .4698 Mdot 3759 .46995 Mdot .37662 .47009 Mdot .37734 .47025 Mdot .37805 .47042 Mdot .37877 .47059 Mdot .37949 47078 Mdot .3802 .47097 Mdot .38092 .47117 Mdot 38163 .47138 Mdot .38235 .47159 Mdot .38307 .47182 Mdot .38378 .47205 Mdot .3845 .47229 Mdot .38522 47254 Mdot .38593 .47279 Mdot .38665 .47306 Mdot 38737 .47333 Mdot .38808 .47361 Mdot .3888 .47389 Mdot .38951 .47418 Mdot .39023 .47448 Mdot .39095 47479 Mdot .39166 .4751 Mdot .39238 .47542 Mdot 39309 .47575 Mdot .39381 .47608 Mdot .39453 .47642 Mdot .39524 .47677 Mdot .39596 .47712 Mdot .39667 47748 Mdot .39739 .47784 Mdot .3981 .47821 Mdot 39882 .47859 Mdot .39954 .47897 Mdot .40025 .47936 Mdot .40097 .47975 Mdot .40168 .48015 Mdot .4024 48056 Mdot .40311 .48096 Mdot .40383 .48138 Mdot 40454 .4818 Mdot .40526 .48222 Mdot .40597 .48265 Mdot .40669 .48308 Mdot .40741 .48352 Mdot .40812 48396 Mdot .40884 .48441 Mdot .40955 .48486 Mdot 41027 .48531 Mdot .41098 .48577 Mdot .4117 .48623 Mdot .41241 .4867 Mdot .41313 .48716 Mdot .41384 48764 Mdot .41456 .48811 Mdot .41527 .48859 Mdot 41599 .48907 Mdot .4167 .48956 Mdot .41742 .49005 Mdot .41813 .49054 Mdot .41885 .49103 Mdot .41956 49152 Mdot .42028 .49202 Mdot .42099 .49252 Mdot 4217 .49302 Mdot .42242 .49353 Mdot .42313 .49403 Mdot .42385 .49454 Mdot .42456 .49505 Mdot .42528 49556 Mdot .42599 .49607 Mdot .42671 .49659 Mdot 42742 .4971 Mdot .42813 .49762 Mdot .42885 .49813 Mdot .42956 .49865 Mdot .43028 .49917 Mdot .43099 49968 Mdot .43171 .5002 Mdot .43242 .50072 Mdot 43313 .50124 Mdot .43385 .50176 Mdot .43456 .50228 Mdot .43528 .5028 Mdot .43599 .50332 Mdot .4367 50384 Mdot .43742 .50435 Mdot .43813 .50487 Mdot 43885 .50539 Mdot .43956 .5059 Mdot .44027 .50642 Mdot .44099 .50693 Mdot .4417 .50744 Mdot .44241 50795 Mdot .44313 .50846 Mdot .44384 .50897 Mdot 44455 .50948 Mdot .44527 .50998 Mdot .44598 .51048 Mdot .4467 .51099 Mdot .44741 .51148 Mdot .44812 51198 Mdot .44884 .51247 Mdot .44955 .51297 Mdot 45026 .51345 Mdot .45097 .51394 Mdot .45169 .51442 Mdot .4524 .51491 Mdot .45311 .51538 Mdot .45383 51586 Mdot .45454 .51633 Mdot .45525 .5168 Mdot 45597 .51726 Mdot .45668 .51773 Mdot .45739 .51818 Mdot .4581 .51864 Mdot .45882 .51909 Mdot .45953 51954 Mdot .46024 .51998 Mdot .46096 .52042 Mdot 46167 .52085 Mdot .46238 .52128 Mdot .46309 .52171 Mdot .46381 .52213 Mdot .46452 .52255 Mdot .46523 52296 Mdot .46594 .52337 Mdot .46666 .52377 Mdot 46737 .52417 Mdot .46808 .52457 Mdot .46879 .52496 Mdot .46951 .52534 Mdot .47022 .52572 Mdot .47093 52609 Mdot .47164 .52646 Mdot .47235 .52682 Mdot 47307 .52718 Mdot .47378 .52753 Mdot .47449 .52788 Mdot .4752 .52822 Mdot .47591 .52855 Mdot .47663 52888 Mdot .47734 .5292 Mdot .47805 .52952 Mdot 47876 .52983 Mdot .47947 .53014 Mdot .48018 .53044 Mdot .4809 .53073 Mdot .48161 .53101 Mdot .48232 53129 Mdot .48303 .53157 Mdot .48374 .53184 Mdot 48445 .5321 Mdot .48517 .53235 Mdot .48588 .5326 Mdot .48659 .53284 Mdot .4873 .53307 Mdot .48801 5333 Mdot .48872 .53352 Mdot .48943 .53373 Mdot 49014 .53394 Mdot .49085 .53414 Mdot .49157 .53433 Mdot .49228 .53452 Mdot .49299 .5347 Mdot .4937 53487 Mdot .49441 .53503 Mdot .49512 .53519 Mdot 49583 .53534 Mdot .49654 .53549 Mdot .49725 .53562 Mdot .49796 .53575 Mdot .49867 .53587 Mdot .49938 53598 Mdot .5001 .53609 Mdot .50081 .53619 Mdot 50152 .53628 Mdot .50223 .53636 Mdot .50294 .53644 Mdot .50365 .53651 Mdot .50436 .53657 Mdot .50507 53663 Mdot .50578 .53667 Mdot .50649 .53671 Mdot 5072 .53674 Mdot .50791 .53677 Mdot .50862 .53678 Mdot .50933 .53679 Mdot .51004 .53679 Mdot .51075 53679 Mdot .51146 .53677 Mdot .51217 .53675 Mdot 51288 .53672 Mdot .51359 .53668 Mdot .5143 .53664 Mdot .51501 .53659 Mdot .51572 .53653 Mdot .51643 53646 Mdot .51714 .53639 Mdot .51785 .5363 Mdot 51856 .53621 Mdot .51927 .53612 Mdot .51997 .53601 Mdot .52068 .5359 Mdot .52139 .53578 Mdot .5221 53565 Mdot .52281 .53552 Mdot .52352 .53538 Mdot 52423 .53523 Mdot .52494 .53507 Mdot .52565 .53491 Mdot .52636 .53474 Mdot .52707 .53456 Mdot .52778 53437 Mdot .52848 .53418 Mdot .52919 .53398 Mdot 5299 .53377 Mdot .53061 .53356 Mdot .53132 .53334 Mdot .53203 .53311 Mdot .53274 .53288 Mdot .53345 53264 Mdot .53415 .53239 Mdot .53486 .53213 Mdot 53557 .53187 Mdot .53628 .5316 Mdot .53699 .53132 Mdot .5377 .53104 Mdot .5384 .53075 Mdot .53911 53046 Mdot .53982 .53016 Mdot .54053 .52985 Mdot 54124 .52954 Mdot .54195 .52922 Mdot .54265 .52889 Mdot .54336 .52856 Mdot .54407 .52822 Mdot .54478 52787 Mdot .54549 .52752 Mdot .54619 .52716 Mdot 5469 .5268 Mdot .54761 .52643 Mdot .54832 .52606 Mdot .54902 .52568 Mdot .54973 .5253 Mdot .55044 52491 Mdot .55115 .52451 Mdot .55185 .52411 Mdot 55256 .5237 Mdot .55327 .52329 Mdot .55398 .52288 Mdot .55468 .52246 Mdot .55539 .52203 Mdot .5561 5216 Mdot .55681 .52116 Mdot .55751 .52072 Mdot 55822 .52028 Mdot .55893 .51983 Mdot .55963 .51938 Mdot .56034 .51892 Mdot .56105 .51846 Mdot .56175 51799 Mdot .56246 .51752 Mdot .56317 .51705 Mdot 56387 .51657 Mdot .56458 .51609 Mdot .56529 .51561 Mdot .56599 .51512 Mdot .5667 .51463 Mdot .56741 51414 Mdot .56811 .51364 Mdot .56882 .51314 Mdot 56953 .51264 Mdot .57023 .51213 Mdot .57094 .51162 Mdot .57165 .51111 Mdot .57235 .5106 Mdot .57306 51008 Mdot .57376 .50956 Mdot .57447 .50904 Mdot 57518 .50852 Mdot .57588 .50799 Mdot .57659 .50747 Mdot .57729 .50694 Mdot .578 .50641 Mdot .5787 50588 Mdot .57941 .50534 Mdot .58012 .50481 Mdot 58082 .50428 Mdot .58153 .50374 Mdot .58223 .5032 Mdot .58294 .50266 Mdot .58364 .50213 Mdot .58435 50159 Mdot .58505 .50105 Mdot .58576 .50051 Mdot 58646 .49997 Mdot .58717 .49943 Mdot .58787 .49889 Mdot .58858 .49835 Mdot .58928 .49781 Mdot .58999 49727 Mdot .59069 .49673 Mdot .5914 .4962 Mdot 5921 .49566 Mdot .59281 .49512 Mdot .59351 .49459 Mdot .59422 .49406 Mdot .59492 .49352 Mdot .59563 49299 Mdot .59633 .49247 Mdot .59704 .49194 Mdot 59774 .49141 Mdot .59845 .49089 Mdot .59915 .49037 Mdot .59985 .48985 Mdot .60056 .48933 Mdot .60126 48882 Mdot .60197 .48831 Mdot .60267 .4878 Mdot 60338 .48729 Mdot .60408 .48679 Mdot .60478 .48629 Mdot .60549 .48579 Mdot .60619 .4853 Mdot .60689 48481 Mdot .6076 .48432 Mdot .6083 .48384 Mdot 60901 .48336 Mdot .60971 .48289 Mdot .61041 .48242 Mdot .61112 .48195 Mdot .61182 .48149 Mdot .61252 48103 Mdot .61323 .48058 Mdot .61393 .48013 Mdot 61463 .47969 Mdot .61534 .47925 Mdot .61604 .47882 Mdot .61674 .47839 Mdot .61745 .47797 Mdot .61815 47755 Mdot .61885 .47714 Mdot .61955 .47673 Mdot 62026 .47633 Mdot .62096 .47594 Mdot .62166 .47555 Mdot .62237 .47517 Mdot .62307 .4748 Mdot .62377 47443 Mdot .62447 .47406 Mdot .62518 .47371 Mdot 62588 .47336 Mdot .62658 .47302 Mdot .62728 .47269 Mdot .62799 .47236 Mdot .62869 .47204 Mdot .62939 47173 Mdot .63009 .47142 Mdot .63079 .47112 Mdot 6315 .47083 Mdot .6322 .47055 Mdot .6329 .47028 Mdot .6336 .47001 Mdot .6343 .46975 Mdot .63501 4695 Mdot .63571 .46926 Mdot .63641 .46903 Mdot 63711 .4688 Mdot .63781 .46859 Mdot .63851 .46838 Mdot .63922 .46818 Mdot .63992 .46799 Mdot .64062 46781 Mdot .64132 .46764 Mdot .64202 .46748 Mdot 64272 .46733 Mdot .64342 .46718 Mdot .64412 .46705 Mdot .64483 .46692 Mdot .64553 .46681 Mdot .64623 4667 Mdot .64693 .46661 Mdot .64763 .46652 Mdot 64833 .46645 Mdot .64903 .46638 Mdot .64973 .46633 Mdot .65043 .46628 Mdot .65113 .46625 Mdot .65183 46622 Mdot .65253 .46621 Mdot .65323 .46621 Mdot 65393 .46621 Mdot .65463 .46623 Mdot .65533 .46626 Mdot .65603 .4663 Mdot .65674 .46635 Mdot .65744 46641 Mdot .65814 .46648 Mdot .65884 .46656 Mdot 65954 .46665 Mdot .66024 .46676 Mdot .66093 .46687 Mdot .66163 .467 Mdot .66233 .46714 Mdot .66303 46729 Mdot .66373 .46745 Mdot .66443 .46762 Mdot 66513 .4678 Mdot .66583 .468 Mdot .66653 .4682 Mdot .66723 .46842 Mdot .66793 .46865 Mdot .66863 46889 Mdot .66933 .46914 Mdot .67003 .4694 Mdot 67073 .46967 Mdot .67143 .46996 Mdot .67212 .47026 Mdot .67282 .47057 Mdot .67352 .47089 Mdot .67422 47122 Mdot .67492 .47156 Mdot .67562 .47192 Mdot 67632 .47229 Mdot .67702 .47267 Mdot .67771 .47306 Mdot .67841 .47346 Mdot .67911 .47387 Mdot .67981 4743 Mdot .68051 .47473 Mdot .68121 .47518 Mdot 6819 .47564 Mdot .6826 .47611 Mdot .6833 .47659 Mdot .684 .47709 Mdot .6847 .47759 Mdot .68539 47811 Mdot .68609 .47864 Mdot .68679 .47918 Mdot 68749 .47973 Mdot .68819 .48029 Mdot .68888 .48086 Mdot .68958 .48145 Mdot .69028 .48204 Mdot .69098 48265 Mdot .69167 .48326 Mdot .69237 .48389 Mdot 69307 .48453 Mdot .69377 .48518 Mdot .69446 .48584 Mdot .69516 .48651 Mdot .69586 .48719 Mdot .69655 48788 Mdot .69725 .48858 Mdot .69795 .4893 Mdot 69864 .49002 Mdot .69934 .49075 Mdot .70004 .49149 Mdot .70073 .49225 Mdot .70143 .49301 Mdot .70213 49378 Mdot .70282 .49456 Mdot .70352 .49535 Mdot 70422 .49615 Mdot .70491 .49696 Mdot .70561 .49778 Mdot .70631 .49861 Mdot .707 .49945 Mdot .7077 50029 Mdot .70839 .50115 Mdot .70909 .50201 Mdot 70979 .50288 Mdot .71048 .50376 Mdot .71118 .50465 Mdot .71187 .50555 Mdot .71257 .50645 Mdot .71326 50736 Mdot .71396 .50828 Mdot .71466 .50921 Mdot 71535 .51014 Mdot .71605 .51108 Mdot .71674 .51203 Mdot .71744 .51299 Mdot .71813 .51395 Mdot .71883 51491 Mdot .71952 .51589 Mdot .72022 .51687 Mdot 72091 .51785 Mdot .72161 .51885 Mdot .7223 .51984 Mdot .723 .52085 Mdot .72369 .52185 Mdot .72439 52287 Mdot .72508 .52388 Mdot .72578 .5249 Mdot 72647 .52593 Mdot .72716 .52696 Mdot .72786 .52799 Mdot .72855 .52903 Mdot .72925 .53007 Mdot .72994 53112 Mdot .73064 .53217 Mdot .73133 .53322 Mdot 73202 .53427 Mdot .73272 .53532 Mdot .73341 .53638 Mdot .73411 .53744 Mdot .7348 .5385 Mdot .73549 53956 Mdot .73619 .54063 Mdot .73688 .54169 Mdot 73757 .54276 Mdot .73827 .54382 Mdot .73896 .54489 Mdot .73965 .54595 Mdot .74035 .54702 Mdot .74104 54808 Mdot .74173 .54914 Mdot .74243 .5502 Mdot 74312 .55127 Mdot .74381 .55232 Mdot .74451 .55338 Mdot .7452 .55443 Mdot .74589 .55549 Mdot .74658 55654 Mdot .74728 .55758 Mdot .74797 .55862 Mdot 74866 .55966 Mdot .74936 .5607 Mdot .75005 .56173 Mdot .75074 .56275 Mdot .75143 .56377 Mdot .75212 56479 Mdot .75282 .5658 Mdot .75351 .5668 Mdot 7542 .5678 Mdot .75489 .56879 Mdot .75559 .56977 Mdot .75628 .57075 Mdot .75697 .57172 Mdot .75766 57268 Mdot .75835 .57363 Mdot .75904 .57458 Mdot 75974 .57551 Mdot .76043 .57644 Mdot .76112 .57735 Mdot .76181 .57826 Mdot .7625 .57916 Mdot .76319 58005 Mdot .76388 .58092 Mdot .76457 .58179 Mdot 76527 .58264 Mdot .76596 .58348 Mdot .76665 .58431 Mdot 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