(************** Content-type: application/mathematica ************** CreatedBy='Mathematica 5.1' 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[ 610089, 12635]*) (*NotebookOutlinePosition[ 610778, 12659]*) (* CellTagsIndexPosition[ 610734, 12655]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell[TextData[{ StyleBox["Mathematica", FontSlant->"Italic"], " Lesson III" }], "Title"], Cell["Physics 80301 Fall 2005", "Subtitle"], Cell[CellGroupData[{ Cell["Functions in Mathematica", "Section"], Cell["\<\ Notice the use of the underscore on the left hand side of this function \ definition. 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CellLabel->"From In[7]:="], Cell[BoxData[ \(1\/4\)], "Print", CellLabel->"From In[7]:="] }, Closed]], Cell["\<\ Here is a more complicated For-loop (an you guess what it does?)\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(For[i = 1; \ t = \ 1/x, \ i^2 < 10, \(i++\), t = t/x; Print["\", i + 1, "\<]=\>", Integrate[t, {x, 1, Infinity}]]]\)], "Input", CellLabel->"In[8]:="], Cell[BoxData[ InterpretationBox[\("Integral(1--infinity)[1/x^"\[InvisibleSpace]2\ \[InvisibleSpace]"]="\[InvisibleSpace]1\), SequenceForm[ "Integral(1--infinity)[1/x^", 2, "]=", 1], Editable->False]], "Print", CellLabel->"From In[8]:="], Cell[BoxData[ InterpretationBox[\("Integral(1--infinity)[1/x^"\[InvisibleSpace]3\ \[InvisibleSpace]"]="\[InvisibleSpace]1\/2\), SequenceForm[ "Integral(1--infinity)[1/x^", 3, "]=", Rational[ 1, 2]], Editable->False]], "Print", CellLabel->"From In[8]:="], Cell[BoxData[ InterpretationBox[\("Integral(1--infinity)[1/x^"\[InvisibleSpace]4\ \[InvisibleSpace]"]="\[InvisibleSpace]1\/3\), 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As an example," }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Clear[nn]; nn = 10; While[\((nn = nn - 1)\) > 5, Print[nn]]\)], "Input",\ CellLabel->"In[9]:="], Cell[BoxData[ \(9\)], "Print", CellLabel->"From In[9]:="], Cell[BoxData[ \(8\)], "Print", CellLabel->"From In[9]:="], Cell[BoxData[ \(7\)], "Print", CellLabel->"From In[9]:="], Cell[BoxData[ \(6\)], "Print", CellLabel->"From In[9]:="] }, Closed]] }, Open ]], Cell[" FixedPoint and NestWhile (see help browser)", "Subsection"] }, Closed]], Cell[CellGroupData[{ Cell["Nonlinear Curve Fitting", "Section"], Cell[TextData[{ "In this section, we consider the problem of fitting experimental data to a \ ", StyleBox["nonlinear", FontSlant->"Italic"], " function. We have in mind the problem of fitting the intensity profile of \ a laser beam. Such a problem was assigned this semester in the Laser Physics \ course. The form of the intensity profile is assumed to be F(x) = a", Cell[BoxData[ \(TraditionalForm\`\[ExponentialE]\^\(\(-\((x - c)\)\^2\)/b\^2\)\)]], ". We want to choose parameters (a,b,c) to fit experimental data to this \ functional form using the nonlinear fitting routine from the \"Statistics\" \ package. The package is loaded by the command:" }], "Text"], Cell[BoxData[ \(<< \ Statistics`NonlinearFit`\)], "Input", CellLabel->"In[10]:="], Cell["\<\ First, let's construct some psuedo data for the intensity:\ \>", "Text"], Cell[BoxData[ \(\(intens\ = \ Table[\((4 + 0.4 \((Random[] - 0.5)\))\)\ Exp[\(-\((2\ + 0.2 \((Random[] - 0.5)\))\)\) \((x - \((2\ + 0.2 \((Random[] - 0.5)\))\))\)^2], {x, 0, 4, 0.05}];\)\)], "Input", CellLabel->"In[11]:="], Cell["\<\ Save the distance coordinates used to generate the intensity data\ \>", "Text"], Cell[BoxData[ \(\(dist = Table[x, {x, 0, 4, 0.05}];\)\)], "Input", CellLabel->"In[12]:="], Cell["Make up a data table with two columns [dist, intens]", "Text"], Cell[BoxData[ \(\(data\ = \ Transpose[{dist, intens}];\)\)], "Input", CellLabel->"In[13]:="], Cell["Plot the data table, but hold back the display:", "Text"], Cell[BoxData[ \(\(plot1\ = ListPlot[data, PlotStyle \[Rule] {PointSize[0.02], RGBColor[1, 0, 0]}, AxesLabel \[Rule] {"\<\[Rho]\>", "\"}, DisplayFunction\ -> Identity];\)\)], "Input", CellLabel->"In[14]:="], Cell["Do the nonlinear fit:", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(f\ = NonlinearFit[data, a\ Exp[\(-\((x - c)\)^2\)/b^2], x, {a, b, c}]\)], "Input", CellLabel->"In[15]:="], Cell[BoxData[ \(4.0153347056182485`\ \[ExponentialE]\^\(\(-2.0058337968038487`\)\ \ \((\(-1.99366544923536`\) + x)\)\^2\)\)], "Output", CellLabel->"Out[15]="] }, Open ]], Cell[TextData[{ "Examine the fit parameters (a,b,c) :. [We expect approximately (a=4,b=1/", Cell[BoxData[ \(TraditionalForm\`\@2\)]], ",c=2)]" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(NonlinearRegress[data, a\ Exp[\(-\((x - c)\)^2\)/b^2], x, {a, b, c}, \[IndentingNewLine]RegressionReport \[Rule] ParameterCITable]\)], "Input", CellLabel->"In[16]:="], Cell[BoxData[ RowBox[{"{", RowBox[{"ParameterCITable", "\[Rule]", TagBox[GridBox[{ {"\<\"\"\>", "\<\"Estimate\"\>", "\<\"Asymptotic SE\"\>", \ "\<\"CI\"\>"}, {"a", "4.0153347056182485`", "0.05145189436747197`", \({3.912901852531958`, 4.117767558704539`}\)}, {"b", "0.7060777524992228`", "0.010447258743904885`", \({0.6852788580518024`, 0.7268766469466431`}\)}, {"c", "1.99366544923536`", "0.007387308496733765`", \({1.9789584477682298`, 2.00837245070249`}\)} }, RowSpacings->1, ColumnSpacings->3, RowAlignments->Baseline, ColumnAlignments->{Left}], Function[ BoxForm`e$, TableForm[ BoxForm`e$, TableDepth -> 2, TableHeadings -> {{a, b, c}, { "Estimate", "Asymptotic SE", "CI"}}]]]}], "}"}]], "Output", CellLabel->"Out[16]="] }, Closed]], Cell["Plot of fit function and hold it back the display.", "Text"], Cell[BoxData[ \(\(plot2 = Plot[f, {x, 0, 4}, PlotStyle \[Rule] Thickness[0.01], DisplayFunction \[Rule] Identity];\)\)], "Input", CellLabel->"In[17]:="], Cell["Now display the two graphs together:", "Text"] }, Closed]], Cell[CellGroupData[{ Cell["Modules in Mathematica", "Section"], Cell["\<\ Here we give an example of a module. This is Mathematica's version of a Maple \ procedure or a Fortran subroution. In this example, we solve a Laplace equation \ boundary-value problem in 2 dimensions and plot the results. a) In this example \"imax\" designates number of x intervals, \"jmax\" \ designates number of y intervals amd \"maxit\" designates the maximum number of iterations permitted. These \ three quantities are input parameters. b) The terms {v,v0,del=1./jmax,err} are local variables, known only inside \ the module. c) The statements enclosed in (* .... *) are comments and disregarded in the \ execution block d) Notice the \"do loop\" used to initialize the arraya v and v0: Do[v[i,j]=0;v0[i,j]=0,{i,0,imax},{j,0,jmax}]; e) Notice the \"for loop\" used in the iteration: the notation m++ has \ the same meaning as in C f) Notice how Show[] is used to display the table as a surface. \ \ \>", "Text"], Cell[BoxData[ \(laplace[imax_, jmax_, maxit_]\ := \ Module[{v, v0, del = 1. /jmax, err}, \[IndentingNewLine] (*\ clear\ arrays\ v\ and\ v0\ to\ start\ *) \[IndentingNewLine]\ \ \ \ \ \ \ Do[v[i, j] = 0; v0[i, j] = 0, {i, 0, imax}, {j, 0, jmax}]; \[IndentingNewLine] (*\ impose\ boundary\ condition\ V = \(1\ at\ x = \ \(0\[IndentingNewLine] \ --\)\ all\ other\ boundary\ values\ are\ 0\)\ \ \ *) \[IndentingNewLine]\ \ \ \ \ \ \ Do[v[0, j] = 1; v0[0, j] = 1, {j, 0, jmax}]; \[IndentingNewLine] (*\ Set\ up\ an\ iteration\ loop\ *) \[IndentingNewLine]For[ m = 1, \[IndentingNewLine]\ m < maxit\ , \[IndentingNewLine]\(m++\), \[IndentingNewLine]\ \ \ \ \ \ Do[Do[v[i, j]\ = \ 0.25\ \((v0[i - 1, j] + v0[i + 1, j] + v0[i, j - 1] + v0[i, j + 1])\), {i, 1, imax - 1}], {j, 1, jmax - 1}]; \[IndentingNewLine]\ \ \ \ \ err\ = \ Max[Table[ Abs[v[i, j] - v0[i, j]], {i, 1, imax - 1}, {j, 1, jmax - 1}]]; \[IndentingNewLine]\ \ \ \ \ \ Do[ Do[v0[i, j]\ = \ v[i, j], {i, 1, imax - 1}], {j, 1, jmax - 1}];\ \[IndentingNewLine]]; \[IndentingNewLine]\ \ \ Print["\< Number of iterations = \>", maxit, "\< Max Error = \>", err]; \[IndentingNewLine]Show[ SurfaceGraphics[\[IndentingNewLine]Table[ v[i, j], {j, 0, jmax}, {i, 0, imax}], \[IndentingNewLine]MeshRange \[Rule] {{0, jmax + del}, {0, imax + del}}], \[IndentingNewLine]BoxRatios\ \[Rule] \ {1, 1, 1}, \ PlotRange \[Rule] {0, 1}, \[IndentingNewLine]Boxed \[Rule] False, \ Axes \[Rule] True, \[IndentingNewLine]AxesLabel \[Rule] 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.69803 .43235 .68765 .42012 Metetra .723 .773 .893 r .68765 .42012 .69803 .43235 .72478 .41996 .71454 .40697 Metetra .707 .759 .892 r .71454 .40697 .72478 .41996 .75181 .40848 .74172 .39492 Metetra .696 .748 .891 r .74172 .39492 .75181 .40848 .77914 .39763 .7692 .38362 Metetra .689 .741 .891 r .7692 .38362 .77914 .39763 .80676 .38719 .79698 .37283 Metetra .683 .736 .89 r .79698 .37283 .80676 .38719 .83468 .37699 .82506 .36234 Metetra .68 .732 .89 r .82506 .36234 .83468 .37699 .8629 .36689 .85345 .352 Metetra .679 .73 .889 r .85345 .352 .8629 .36689 .89142 .35678 .88213 .34166 Metetra .913 .742 .657 r .31568 1.02589 .33039 1.03625 .36271 .9326 .34816 .92995 Metetra .94 .784 .661 r .34816 .92995 .36271 .9326 .39338 .83975 .37919 .8419 Metetra .959 .825 .668 r .37919 .8419 .39338 .83975 .42257 .75999 .40888 .76387 Metetra .967 .859 .681 r .40888 .76387 .42257 .75999 .45058 .69296 .43742 .6963 Metetra .964 .883 .703 r .43742 .6963 .45058 .69296 .47773 .63716 .46506 .63865 Metetra .953 .896 .733 r .46506 .63865 .47773 .63716 .50429 .59081 .49206 .5899 Metetra .933 .899 .769 r .49206 .5899 .50429 .59081 .53048 .55232 .51862 .5489 Metetra .907 .891 .803 r .51862 .5489 .53048 .55232 .55649 .52023 .54494 .5145 Metetra .875 .874 .832 r .54494 .5145 .55649 .52023 .58246 .49338 .57116 .48561 Metetra .84 .852 .854 r .57116 .48561 .58246 .49338 .60849 .47073 .5974 .46127 Metetra .806 .829 .868 r .5974 .46127 .60849 .47073 .63466 .45144 .62375 .44061 Metetra .775 .806 .877 r .62375 .44061 .63466 .45144 .66103 .43477 .65029 .42286 Metetra .749 .786 .883 r .65029 .42286 .66103 .43477 .68765 .42012 .67707 .40736 Metetra .727 .769 .886 r .67707 .40736 .68765 .42012 .71454 .40697 .70412 .39356 Metetra .711 .757 .888 r .70412 .39356 .71454 .40697 .74172 .39492 .73145 .381 Metetra .699 .747 .889 r .73145 .381 .74172 .39492 .7692 .38362 .75909 .36931 Metetra .691 .74 .889 r .75909 .36931 .7692 .38362 .79698 .37283 .78703 .3582 Metetra .685 .736 .889 r .78703 .3582 .79698 .37283 .82506 .36234 .81528 .34744 Metetra .681 .732 .889 r .81528 .34744 .82506 .36234 .85345 .352 .84384 .33685 Metetra .68 .731 .889 r .84384 .33685 .85345 .352 .88213 .34166 .8727 .32629 Metetra .911 .745 .663 r .30071 1.01536 .31568 1.02589 .34816 .92995 .33347 .92399 Metetra .928 .771 .667 r .33347 .92399 .34816 .92995 .37919 .8419 .3649 .83904 Metetra .94 .798 .676 r .3649 .83904 .37919 .8419 .40888 .76387 .39504 .76236 Metetra .946 .821 .689 r .39504 .76236 .40888 .76387 .43742 .6963 .42407 .69467 Metetra .945 .84 .708 r .42407 .69467 .43742 .6963 .46506 .63865 .45217 .63595 Metetra .936 .851 .733 r .45217 .63595 .46506 .63865 .49206 .5899 .47959 .58562 Metetra .92 .856 .761 r .47959 .58562 .49206 .5899 .51862 .5489 .50651 .54287 Metetra .898 .853 .789 r .50651 .54287 .51862 .5489 .54494 .5145 .53314 .50676 Metetra .87 .843 .816 r .53314 .50676 .54494 .5145 .57116 .48561 .55962 .47634 Metetra .839 .829 .838 r .55962 .47634 .57116 .48561 .5974 .46127 .58608 .45068 Metetra .808 .812 .855 r .58608 .45068 .5974 .46127 .62375 .44061 .61264 .42893 Metetra .778 .795 .867 r .61264 .42893 .62375 .44061 .65029 .42286 .63936 .41029 Metetra .752 .779 .875 r .63936 .41029 .65029 .42286 .67707 .40736 .6663 .39408 Metetra .731 .766 .881 r .6663 .39408 .67707 .40736 .70412 .39356 .69351 .37973 Metetra .714 .754 .884 r .69351 .37973 .70412 .39356 .73145 .381 .72101 .36673 Metetra .701 .746 .886 r .72101 .36673 .73145 .381 .75909 .36931 .74881 .35468 Metetra .692 .74 .888 r .74881 .35468 .75909 .36931 .78703 .3582 .77692 .34328 Metetra .686 .735 .888 r .77692 .34328 .78703 .3582 .81528 .34744 .80534 .33226 Metetra .682 .732 .889 r .80534 .33226 .81528 .34744 .84384 .33685 .83407 .32145 Metetra .68 .731 .888 r .83407 .32145 .84384 .33685 .8727 .32629 .86311 .31066 Metetra .91 .746 .667 r .28548 1.00464 .30071 1.01536 .33347 .92399 .31861 .91596 Metetra .92 .764 .671 r .31861 .91596 .33347 .92399 .3649 .83904 .35045 .8329 Metetra .927 .781 .679 r .35045 .8329 .3649 .83904 .39504 .76236 .38105 .7571 Metetra .931 .797 .691 r .38105 .7571 .39504 .76236 .42407 .69467 .41053 .6894 Metetra .929 .811 .708 r .41053 .6894 .42407 .69467 .45217 .63595 .43908 .63 Metetra .921 .82 .729 r .43908 .63 .45217 .63595 .47959 .58562 .4669 .57861 Metetra .908 .825 .753 r .4669 .57861 .47959 .58562 .50651 .54287 .49418 .53463 Metetra .888 .825 .779 r .49418 .53463 .50651 .54287 .53314 .50676 .52111 .49729 Metetra .864 .82 .804 r .52111 .49729 .53314 .50676 .55962 .47634 .54785 .46571 Metetra .837 .811 .826 r .54785 .46571 .55962 .47634 .58608 .45068 .57455 .43905 Metetra .808 .799 .844 r .57455 .43905 .58608 .45068 .61264 .42893 .60131 .41643 Metetra .78 .786 .858 r .60131 .41643 .61264 .42893 .63936 .41029 .62822 .39708 Metetra .754 .773 .869 r .62822 .39708 .63936 .41029 .6663 .39408 .65534 .3803 Metetra .733 .762 .876 r .65534 .3803 .6663 .39408 .69351 .37973 .68272 .36548 Metetra .716 .752 .881 r .68272 .36548 .69351 .37973 .72101 .36673 .71039 .3521 Metetra .702 .744 .884 r .71039 .3521 .72101 .36673 .74881 .35468 .73835 .33974 Metetra .693 .739 .886 r .73835 .33974 .74881 .35468 .77692 .34328 .76664 .32806 Metetra .687 .735 .887 r .76664 .32806 .77692 .34328 .80534 .33226 .79523 .31682 Metetra .683 .732 .888 r .79523 .31682 .80534 .33226 .83407 .32145 .82415 .30578 Metetra .681 .731 .888 r .82415 .30578 .83407 .32145 .86311 .31066 .85337 .29479 Metetra .909 .748 .669 r .26999 .99374 .28548 1.00464 .31861 .91596 .30354 .90653 Metetra .915 .759 .673 r .30354 .90653 .31861 .91596 .35045 .8329 .33583 .82452 Metetra .919 .77 .68 r .33583 .82452 .35045 .8329 .38105 .7571 .36689 .7492 Metetra .92 .781 .691 r .36689 .7492 .38105 .7571 .41053 .6894 .39682 .68144 Metetra .917 .791 .707 r .39682 .68144 .41053 .6894 .43908 .63 .42579 .62156 Metetra .91 .799 .725 r .42579 .62156 .43908 .63 .4669 .57861 .454 .56942 Metetra .898 .804 .747 r .454 .56942 .4669 .57861 .49418 .53463 .48162 .52457 Metetra .88 .805 .771 r .48162 .52457 .49418 .53463 .52111 .49729 .50885 .48632 Metetra .859 .803 .795 r .50885 .48632 .52111 .49729 .54785 .46571 .53587 .4539 Metetra .834 .797 .817 r .53587 .4539 .54785 .46571 .57455 .43905 .5628 .42647 Metetra .807 .789 .836 r .5628 .42647 .57455 .43905 .60131 .41643 .58977 .40319 Metetra .78 .779 .852 r .58977 .40319 .60131 .41643 .62822 .39708 .61688 .38328 Metetra .755 .768 .863 r .61688 .38328 .62822 .39708 .65534 .3803 .64419 .36604 Metetra .734 .758 .872 r .64419 .36604 .65534 .3803 .68272 .36548 .67174 .35082 Metetra .717 .75 .878 r .67174 .35082 .68272 .36548 .71039 .3521 .69958 .33712 Metetra .703 .743 .882 r .69958 .33712 .71039 .3521 .73835 .33974 .72773 .32448 Metetra .694 .738 .885 r .72773 .32448 .73835 .33974 .76664 .32806 .75618 .31256 Metetra .687 .734 .887 r .75618 .31256 .76664 .32806 .79523 .31682 .78496 .30109 Metetra .683 .732 .888 r .78496 .30109 .79523 .31682 .82415 .30578 .81405 .28984 Metetra .681 .731 .888 r .81405 .28984 .82415 .30578 .85337 .29479 .84346 .27865 Metetra .909 .748 .671 r .25422 .98264 .26999 .99374 .30354 .90653 .28825 .89611 Metetra .912 .755 .674 r .28825 .89611 .30354 .90653 .33583 .82452 .32102 .81454 Metetra .913 .763 .681 r .32102 .81454 .33583 .82452 .36689 .7492 .35254 .73937 Metetra .912 .771 .691 r .35254 .73937 .36689 .7492 .39682 .68144 .38292 .67146 Metetra .909 .778 .705 r .38292 .67146 .39682 .68144 .42579 .62156 .41231 .61119 Metetra .901 .784 .722 r .41231 .61119 .42579 .62156 .454 .56942 .44089 .5585 Metetra .89 .789 .743 r .44089 .5585 .454 .56942 .48162 .52457 .46886 .513 Metetra .874 .791 .765 r .46886 .513 .48162 .52457 .50885 .48632 .49639 .4741 Metetra .854 .79 .788 r .49639 .4741 .50885 .48632 .53587 .4539 .52367 .44105 Metetra .83 .786 .81 r .52367 .44105 .53587 .4539 .5628 .42647 .55084 .41306 Metetra .805 .78 .83 r .55084 .41306 .5628 .42647 .58977 .40319 .57803 .38927 Metetra .78 .772 .846 r .57803 .38927 .58977 .40319 .61688 .38328 .60534 .36893 Metetra .756 .764 .859 r .60534 .36893 .61688 .38328 .64419 .36604 .63284 .35131 Metetra .735 .755 .869 r .63284 .35131 .64419 .36604 .67174 .35082 .66058 .33578 Metetra .717 .748 .876 r .66058 .33578 .67174 .35082 .69958 .33712 .68859 .32179 Metetra .704 .742 .881 r .68859 .32179 .69958 .33712 .72773 .32448 .71692 .3089 Metetra .694 .737 .884 r .71692 .3089 .72773 .32448 .75618 .31256 .74555 .29676 Metetra .687 .734 .886 r .74555 .29676 .75618 .31256 .78496 .30109 .77451 .28508 Metetra .683 .732 .887 r .77451 .28508 .78496 .30109 .81405 .28984 .80379 .27363 Metetra .681 .731 .888 r .80379 .27363 .81405 .28984 .84346 .27865 .83339 .26224 Metetra .908 .749 .672 r .23817 .97135 .25422 .98264 .28825 .89611 .27273 .88491 Metetra .909 .753 .675 r .27273 .88491 .28825 .89611 .32102 .81454 .306 .80335 Metetra .909 .758 .681 r .306 .80335 .32102 .81454 .35254 .73937 .338 .72806 Metetra .907 .763 .691 r .338 .72806 .35254 .73937 .38292 .67146 .36883 .65989 Metetra .902 .769 .704 r .36883 .65989 .38292 .67146 .41231 .61119 .39863 .59926 Metetra .895 .773 .72 r .39863 .59926 .41231 .61119 .44089 .5585 .42759 .54612 Metetra .883 .777 .74 r .42759 .54612 .44089 .5585 .46886 .513 .45589 .50016 Metetra .868 .78 .761 r .45589 .50016 .46886 .513 .49639 .4741 .48372 .46079 Metetra .849 .78 .783 r .48372 .46079 .49639 .4741 .52367 .44105 .51127 .4273 Metetra .827 .778 .805 r .51127 .4273 .52367 .44105 .55084 .41306 .53868 .39888 Metetra .803 .773 .825 r .53868 .39888 .55084 .41306 .57803 .38927 .56609 .37473 Metetra .779 .767 .842 r .56609 .37473 .57803 .38927 .60534 .36893 .5936 .35406 Metetra .755 .76 .856 r .5936 .35406 .60534 .36893 .63284 .35131 .62129 .33614 Metetra .735 .753 .867 r .62129 .33614 .63284 .35131 .66058 .33578 .64922 .32034 Metetra .718 .746 .874 r .64922 .32034 .66058 .33578 .68859 .32179 .67742 .30612 Metetra .704 .741 .88 r .67742 .30612 .68859 .32179 .71692 .3089 .70593 .29301 Metetra .694 .737 .883 r .70593 .29301 .71692 .3089 .74555 .29676 .73475 .28066 Metetra .688 .734 .886 r .73475 .28066 .74555 .29676 .77451 .28508 .76389 .26878 Metetra .683 .732 .887 r .76389 .26878 .77451 .28508 .80379 .27363 .79336 .25714 Metetra .681 .731 .887 r .79336 .25714 .80379 .27363 .83339 .26224 .82315 .24555 Metetra .908 .749 .672 r .22183 .95985 .23817 .97135 .27273 .88491 .25696 .87305 Metetra .908 .751 .675 r .25696 .87305 .27273 .88491 .306 .80335 .29076 .79114 Metetra .906 .754 .681 r .29076 .79114 .306 .80335 .338 .72806 .32326 .71549 Metetra .903 .757 .69 r .32326 .71549 .338 .72806 .36883 .65989 .35455 .64698 Metetra .897 .761 .703 r .35455 .64698 .36883 .65989 .39863 .59926 .38476 .58598 Metetra .889 .765 .718 r .38476 .58598 .39863 .59926 .42759 .54612 .41409 .53249 Metetra .878 .768 .737 r .41409 .53249 .42759 .54612 .45589 .50016 .44272 .48618 Metetra .863 .771 .758 r .44272 .48618 .45589 .50016 .48372 .46079 .47085 .44649 Metetra .845 .771 .78 r .47085 .44649 .48372 .46079 .51127 .4273 .49866 .41269 Metetra .824 .77 .802 r .49866 .41269 .51127 .4273 .53868 .39888 .52632 .384 Metetra .801 .767 .822 r .52632 .384 .53868 .39888 .56609 .37473 .55395 .35959 Metetra .777 .762 .839 r .55395 .35959 .56609 .37473 .5936 .35406 .58166 .33868 Metetra .755 .756 .854 r .58166 .33868 .5936 .35406 .62129 .33614 .60955 .32054 Metetra .734 .75 .865 r .60955 .32054 .62129 .33614 .64922 .32034 .63767 .30453 Metetra .718 .745 .873 r .63767 .30453 .64922 .32034 .67742 .30612 .66606 .2901 Metetra .704 .74 .879 r .66606 .2901 .67742 .30612 .70593 .29301 .69475 .2768 Metetra .694 .736 .883 r .69475 .2768 .70593 .29301 .73475 .28066 .72376 .26426 Metetra .688 .733 .885 r .72376 .26426 .73475 .28066 .76389 .26878 .75309 .2522 Metetra .683 .731 .887 r .75309 .2522 .76389 .26878 .79336 .25714 .78275 .24037 Metetra .681 .73 .887 r .78275 .24037 .79336 .25714 .82315 .24555 .81274 .22858 Metetra .908 .749 .672 r .20519 .94814 .22183 .95985 .25696 .87305 .24094 .86056 Metetra .907 .749 .675 r .24094 .86056 .25696 .87305 .29076 .79114 .27532 .77797 Metetra .904 .75 .68 r .27532 .77797 .29076 .79114 .32326 .71549 .30833 .70175 Metetra .899 .752 .689 r .30833 .70175 .32326 .71549 .35455 .64698 .34008 .63277 Metetra .893 .754 .701 r .34008 .63277 .35455 .64698 .38476 .58598 .37071 .57141 Metetra .884 .757 .717 r .37071 .57141 .38476 .58598 .41409 .53249 .4004 .51764 Metetra .873 .76 .735 r .4004 .51764 .41409 .53249 .44272 .48618 .42935 .47111 Metetra .859 .763 .755 r .42935 .47111 .44272 .48618 .47085 .44649 .45777 .43122 Metetra .841 .764 .777 r .45777 .43122 .47085 .44649 .49866 .41269 .48585 .39727 Metetra .821 .763 .799 r .48585 .39727 .49866 .41269 .52632 .384 .51375 .36841 Metetra .798 .761 .819 r .51375 .36841 .52632 .384 .55395 .35959 .5416 .34385 Metetra .775 .758 .837 r .5416 .34385 .55395 .35959 .58166 .33868 .56952 .32279 Metetra .753 .753 .852 r .56952 .32279 .58166 .33868 .60955 .32054 .59761 .3045 Metetra .733 .748 .863 r .59761 .3045 .60955 .32054 .63767 .30453 .62592 .28833 Metetra .717 .743 .872 r .62592 .28833 .63767 .30453 .66606 .2901 .6545 .27374 Metetra .704 .738 .878 r .6545 .27374 .66606 .2901 .69475 .2768 .68338 .26027 Metetra .694 .735 .882 r .68338 .26027 .69475 .2768 .72376 .26426 .71258 .24756 Metetra .688 .733 .885 r .71258 .24756 .72376 .26426 .75309 .2522 .7421 .23531 Metetra .683 .731 .886 r .7421 .23531 .75309 .2522 .78275 .24037 .77196 .2233 Metetra .681 .73 .887 r .77196 .2233 .78275 .24037 .81274 .22858 .80215 .21133 Metetra .909 .748 .671 r .18825 .93622 .20519 .94814 .24094 .86056 .22467 .84738 Metetra .905 .746 .674 r .22467 .84738 .24094 .86056 .27532 .77797 .25966 .76374 Metetra .901 .745 .679 r .25966 .76374 .27532 .77797 .30833 .70175 .29322 .68671 Metetra .895 .746 .688 r .29322 .68671 .30833 .70175 .34008 .63277 .32544 .61718 Metetra .888 .747 .7 r .32544 .61718 .34008 .63277 .37071 .57141 .35648 .55546 Metetra .879 .749 .715 r .35648 .55546 .37071 .57141 .4004 .51764 .38653 .50149 Metetra .868 .752 .733 r .38653 .50149 .4004 .51764 .42935 .47111 .4158 .45486 Metetra .853 .754 .753 r .4158 .45486 .42935 .47111 .45777 .43122 .44451 .41495 Metetra .836 .756 .775 r .44451 .41495 .45777 .43122 .48585 .39727 .47284 .38096 Metetra .816 .756 .796 r .47284 .38096 .48585 .39727 .51375 .36841 .50097 .3521 Metetra .795 .755 .817 r .50097 .3521 .51375 .36841 .5416 .34385 .52905 .3275 Metetra .772 .753 .835 r .52905 .3275 .5416 .34385 .56952 .32279 .55718 .30638 Metetra .751 .749 .85 r .55718 .30638 .56952 .32279 .59761 .3045 .58547 .28801 Metetra .732 .745 .862 r .58547 .28801 .59761 .3045 .62592 .28833 .61397 .27173 Metetra .716 .741 .871 r .61397 .27173 .62592 .28833 .6545 .27374 .64274 .25702 Metetra .703 .737 .877 r .64274 .25702 .6545 .27374 .68338 .26027 .67182 .2434 Metetra .694 .734 .882 r .67182 .2434 .68338 .26027 .71258 .24756 .70121 .23053 Metetra .687 .732 .885 r .70121 .23053 .71258 .24756 .7421 .23531 .73093 .21811 Metetra .683 .731 .886 r .73093 .21811 .7421 .23531 .77196 .2233 .76098 .20592 Metetra .681 .73 .887 r .76098 .20592 .77196 .2233 .80215 .21133 .79137 .19377 Metetra .909 .748 .669 r .17099 .92407 .18825 .93622 .22467 .84738 .20818 .83338 Metetra .904 .743 .672 r .20818 .83338 .22467 .84738 .25966 .76374 .24383 .74822 Metetra .898 .74 .677 r .24383 .74822 .25966 .76374 .29322 .68671 .27796 .67011 Metetra .891 .739 .686 r .27796 .67011 .29322 .68671 .32544 .61718 .31066 .5999 Metetra .883 .739 .698 r .31066 .5999 .32544 .61718 .35648 .55546 .3421 .53787 Metetra .873 .74 .713 r .3421 .53787 .35648 .55546 .38653 .50149 .3725 .48383 Metetra .861 .742 .731 r .3725 .48383 .38653 .50149 .4158 .45486 .40208 .43729 Metetra .847 .745 .751 r .40208 .43729 .4158 .45486 .44451 .41495 .43106 .39751 Metetra .83 .747 .772 r .43106 .39751 .44451 .41495 .47284 .38096 .45964 .36371 Metetra .811 .748 .794 r .45964 .36371 .47284 .38096 .50097 .3521 .488 .33498 Metetra .79 .748 .815 r .488 .33498 .50097 .3521 .52905 .3275 .51629 .31049 Metetra .769 .747 .833 r .51629 .31049 .52905 .3275 .55718 .30638 .54463 .28942 Metetra .748 .745 .849 r .54463 .28942 .55718 .30638 .58547 .28801 .57312 .27104 Metetra .73 .742 .861 r .57312 .27104 .58547 .28801 .61397 .27173 .60182 .25472 Metetra .714 .738 .87 r .60182 .25472 .61397 .27173 .64274 .25702 .63079 .23992 Metetra .702 .736 .877 r .63079 .23992 .64274 .25702 .67182 .2434 .66005 .22619 Metetra .693 .733 .881 r .66005 .22619 .67182 .2434 .70121 .23053 .68964 .21318 Metetra .687 .731 .884 r .68964 .21318 .70121 .23053 .73093 .21811 .71956 .2006 Metetra .683 .73 .886 r .71956 .2006 .73093 .21811 .76098 .20592 .74982 .18824 Metetra .681 .73 .887 r .74982 .18824 .76098 .20592 .79137 .19377 .78041 .17591 Metetra .91 .746 .667 r .15341 .9117 .17099 .92407 .20818 .83338 .19147 .81827 Metetra .902 .739 .669 r .19147 .81827 .20818 .83338 .24383 .74822 .22787 .73097 Metetra .894 .733 .675 r .22787 .73097 .24383 .74822 .27796 .67011 .26261 .65143 Metetra .885 .729 .684 r .26261 .65143 .27796 .67011 .31066 .5999 .29579 .58047 Metetra .875 .728 .695 r .29579 .58047 .31066 .5999 .3421 .53787 .32763 .51821 Metetra .864 .728 .71 r .32763 .51821 .3421 .53787 .3725 .48383 .35835 .46431 Metetra .852 .73 .728 r .35835 .46431 .3725 .48383 .40208 .43729 .38821 .41809 Metetra .838 .733 .748 r .38821 .41809 .40208 .43729 .43106 .39751 .41744 .37874 Metetra .822 .736 .77 r .41744 .37874 .43106 .39751 .45964 .36371 .44625 .34533 Metetra .804 .739 .792 r .44625 .34533 .45964 .36371 .488 .33498 .47483 .31696 Metetra .784 .741 .813 r .47483 .31696 .488 .33498 .51629 .31049 .50333 .29273 Metetra .764 .741 .832 r .50333 .29273 .51629 .31049 .54463 .28942 .53187 .27185 Metetra .744 .74 .848 r .53187 .27185 .54463 .28942 .57312 .27104 .56056 .25357 Metetra .727 .738 .86 r .56056 .25357 .57312 .27104 .60182 .25472 .58946 .23728 Metetra .712 .736 .87 r .58946 .23728 .60182 .25472 .63079 .23992 .61863 .22243 Metetra .701 .734 .876 r .61863 .22243 .63079 .23992 .66005 .22619 .64809 .20862 Metetra .692 .732 .881 r .64809 .20862 .66005 .22619 .68964 .21318 .67787 .19549 Metetra .686 .731 .884 r .67787 .19549 .68964 .21318 .71956 .2006 .70799 .18277 Metetra .682 .73 .886 r 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Metetra .781 .689 .762 r .37584 .31153 .38979 .33606 .41898 .30452 .40514 .28167 Metetra .768 .699 .786 r .40514 .28167 .41898 .30452 .44793 .2777 .43423 .2562 Metetra .754 .707 .808 r .43423 .2562 .44793 .2777 .47681 .25468 .46326 .2342 Metetra .739 .714 .828 r .46326 .2342 .47681 .25468 .50574 .23465 .49237 .21492 Metetra .725 .719 .845 r .49237 .21492 .50574 .23465 .53482 .21693 .52162 .1977 Metetra .712 .723 .858 r .52162 .1977 .53482 .21693 .56411 .20095 .55111 .18202 Metetra .701 .725 .868 r .55111 .18202 .56411 .20095 .59367 .18621 .58086 .16744 Metetra .693 .726 .876 r .58086 .16744 .59367 .18621 .62353 .17235 .61092 .15361 Metetra .687 .727 .881 r .61092 .15361 .62353 .17235 .65371 .15905 .64131 .14027 Metetra .682 .727 .884 r .64131 .14027 .65371 .15905 .68424 .14608 .67205 .1272 Metetra .68 .728 .886 r .67205 .1272 .68424 .14608 .71512 .13325 .70314 .11424 Metetra .679 .728 .888 r .70314 .11424 .71512 .13325 .74635 .12041 .73459 .10125 Metetra .919 .734 .636 r .07969 .85982 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