(*^
::[	Information =

	"This is a Mathematica Notebook file.  It contains ASCII text, and can be
	transferred by email, ftp, or other text-file transfer utility.  It should
	be read or edited using a copy of Mathematica or MathReader.  If you 
	received this as email, use your mail application or copy/paste to save 
	everything from the line containing (*^ down to the line containing ^*)
	into a plain text file.  On some systems you may have to give the file a 
	name ending with ".ma" to allow Mathematica to recognize it as a Notebook.
	The line below identifies what version of Mathematica created this file,
	but it can be opened using any other version as well.";

	FrontEndVersion = "Macintosh Mathematica Notebook Front End Version 2.2";

	MacintoshStandardFontEncoding; 
	
	fontset = title, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeTitle, center, M7, bold, L3, e8,  24, "New York"; 
	fontset = subtitle, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeTitle, center, M7, bold, L2, e6,  18, "New York"; 
	fontset = subsubtitle, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeTitle, center, M7, bold, L2, e6,  14, "New York"; 
	fontset = section, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeSection, grayBox, M22, bold, L2, a20,  14, "New York"; 
	fontset = subsection, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeSection, blackBox, M19, bold, L2, a15,  12, "New York"; 
	fontset = subsubsection, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeSection, whiteBox, M18, bold, L2, a12,  10, "New York"; 
	fontset = text, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L2,  12, "New York"; 
	fontset = smalltext, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L2,  10, "New York"; 
	fontset = input, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeInput, M42, N23, bold, L2,  12, "Courier"; 
	fontset = output, output, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, L2,  12, "Courier"; 
	fontset = message, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, R65535, L2,  12, "Courier"; 
	fontset = print, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, L2,  12, "Courier"; 
	fontset = info, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, L2,  12, "Courier"; 
	fontset = postscript, PostScript, formatAsPostScript, output, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeGraphics, M7, l34, w282, h287, L2,  12, "Courier"; 
	fontset = name, inactive, nowordwrap, nohscroll, noKeepOnOnePage, preserveAspect, M7, italic, B65535, L2,  10, "Geneva"; 
	fontset = header, inactive, noKeepOnOnePage, preserveAspect, M7, L2,  10, "Times"; 
	fontset = leftheader, inactive, L2,  10, "Times"; 
	fontset = footer, inactive, noKeepOnOnePage, preserveAspect, center, M7, L2,  12, "Times"; 
	fontset = leftfooter, inactive, center, L2,  12, "Times"; 
	fontset = help, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L2,  10, "Geneva"; 
	fontset = clipboard, inactive, noKeepOnOnePage, preserveAspect, M7, L2,  12, "New York"; 
	fontset = completions, inactive, nowordwrap, noKeepOnOnePage, preserveAspect, M7, L2,  12, "New York"; 
	fontset = special1, inactive, nowordwrap, noKeepOnOnePage, preserveAspect, M7, L2,  12, "New York"; 
	fontset = special2, inactive, nowordwrap, noKeepOnOnePage, preserveAspect, center, M7, L2,  12, "New York"; 
	fontset = special3, inactive, nowordwrap, noKeepOnOnePage, preserveAspect, right, M7, L2,  12, "New York"; 
	fontset = special4, inactive, nowordwrap, noKeepOnOnePage, preserveAspect, M7, L2,  12, "New York"; 
	fontset = special5, inactive, nowordwrap, noKeepOnOnePage, preserveAspect, M7, L2,  12, "New York"; 
	paletteColors = 128; currentKernel; 
]
:[font = title; inactive; preserveAspect; fontSize = 19; fontName = "Times"; startGroup]
Linear Methods of Applied Mathematics
Calculations of some Fourier series
:[font = subtitle; inactive; dontPreserveAspect]
Solving the Wave Equation
:[font = text; inactive; preserveAspect; plain; bold; fontName = "Times"]
(c) Copyright 1994-1997 by Evans M. Harrell II and James V. Herod.     All rights reserved.
:[font = text; inactive; Cclosed; preserveAspect; plain; bold; fontSize = 13; fontName = "Times"; startGroup]
Notes for the instructor.
:[font = text; inactive; preserveAspect; fontSize = 13; fontName = "Times"; endGroup]
This contains calculations and examples which correlate with chapter 6 of the WWW text by Harrell and Herod.  Students can be encouraged to cut and paste from this notebook to do homework.
:[font = subsubsection; inactive; Cclosed; preserveAspect; fontSize = 11; fontName = "Times"; startGroup]
Instructions
:[font = text; inactive; preserveAspect; fontSize = 13; fontName = "Times"; endGroup]
This notebook uses Mathematica  to perform calculations for Harrell and Herod's hypertext book, Linear Methods of Applied Mathematics.   The student needs only a basic knowledge of Mathematica  to use the notebook, which is designed both to show how to work problems in the text and to provide a template for using Mathematica  to work other problems of the student's own design.   Calculations will be performed when the reader presses enter in a given calculation cell (bold print).  It is best to activate the cells in order, so that Mathematica  will be able to call on operators and functions defined in earlier cells.  Red color coding is used to warn the reader when a given calculation relies on an earlier one.
;[s]
11:0,0;19,1;30,0;96,1;133,0;181,1;192,0;315,1;326,0;537,1;548,0;720,-1;
2:6,15,10,Times,0,13,0,0,0;5,15,10,Times,2,13,0,0,0;
:[font = text; inactive; preserveAspect; fontSize = 14; fontName = "Times"]
A useful substitution, which we shall often make, is:
:[font = input; preserveAspect]
TrigId = {Cos[Pi n_] -> (-1)^n, Sin[Pi n_] -> 0};
:[font = text; inactive; dontPreserveAspect]
In Mathematica  notation, the wave equation could be written 
					
					D[u, {t,2}] = c^2 D[u, {x,2}] 
					
We classify this PDE as second order, linear, and homogeneous.  
;[s]
3:0,0;3,1;14,0;175,-1;
2:2,17,12,New York,0,12,0,0,0;1,17,12,New York,2,12,0,0,0;
:[font = subsubsection; inactive; Cclosed; preserveAspect; startGroup]
Definitions
:[font = text; inactive; preserveAspect; endGroup]
"Second order" means that derivatives up to second order occur in the equation, but no higher-order derivatives.  Linear means that the differential equation is formed from a linear combination of the solution u and its derivatives, and that u comes into the equation only in this linear combination.  Homogeneous  

This will be the case for the great majority of PDE's in this class.  
:[font = section; inactive; Cclosed; preserveAspect; startGroup]
Discussion of the wave operator and the method of separation of variables
:[font = text; inactive; preserveAspect]
We shall solve the wave equation in two entirely different ways.  The method in chapter VI is known as separation of variables, and later we shall solve the same equation with d'Alembert's method of characteristics.  The first method is often particularly good for linear equations.  "Linear" refers to the superposition property, i.e., if two functions u1 and u2 are solutions, then so is any linear combination, a1 u1 + a2 u2. 

In fact, (WE) is a statement that the function u is in the kernel of a linear operator. 

A convenient way to set up a linear differential operator in Mathematica is as follows - the operator acts on functions, so identify a function f with an understroke to show that it is the independent variable in the 
wave operator:
;[s]
15:0,0;355,1;356,0;362,1;363,0;414,2;415,1;416,0;418,1;419,0;422,2;423,1;424,0;426,1;427,0;754,-1;
3:7,17,12,New York,0,12,0,0,0;6,25,16,New York,64,12,0,0,0;2,19,13,Symbol,0,12,0,0,0;
:[font = input; preserveAspect]
WaveOp[f_,c_] := (1/c^2) D[f, {t,2}] -  D[f, {x,2}]
:[font = text; inactive; preserveAspect]
We'll often set c = 1 for convenience , so let's also define
:[font = input; dontPreserveAspect]
 WaveOp[f_] := WaveOp[f,1]
:[font = text; inactive; preserveAspect]
(Equivalently, we could choose a clock with time variable t' = c t.)  Let's
test out the wave operator
:[font = input; dontPreserveAspect; startGroup]
WaveOp[t Sin[x]^2]
:[font = output; output; inactive; preserveAspect; endGroup]
-2*t*Cos[x]^2 + 2*t*Sin[x]^2
;[o]
           2             2
-2 t Cos[x]  + 2 t Sin[x]
:[font = input; preserveAspect; startGroup]
 WaveOp[Sin[t - x]]
:[font = output; output; inactive; dontPreserveAspect; endGroup]
0
;[o]
0
:[font = text; inactive; dontPreserveAspect]
We next try the ansatz that u[t,x] is of the special form T[t] X[x], known as a product solution.
:[font = input; preserveAspect]
u[t_,x_] := T[t] X[x]
:[font = input; preserveAspect; startGroup]
WaveOp[u[t,x]]
:[font = output; output; inactive; dontPreserveAspect; endGroup]
X[x]*Derivative[2][T][t] - T[t]*Derivative[2][X][x]
;[o]
X[x] T''[t] - T[t] X''[x]
:[font = text; inactive; dontPreserveAspect]

This simplifies if we divide through by T[t] X[x]:
:[font = input; preserveAspect; startGroup]
%/(T[t] X[x])
:[font = output; output; inactive; preserveAspect; endGroup]
(X[x]*Derivative[2][T][t] - T[t]*Derivative[2][X][x])/
 
  (T[t]*X[x])
;[o]
X[x] T''[t] - T[t] X''[x]
-------------------------
        T[t] X[x]
:[font = input; dontPreserveAspect; startGroup]
Simplify[%]
:[font = output; output; inactive; dontPreserveAspect; endGroup]
Derivative[2][T][t]/T[t] - Derivative[2][X][x]/X[x]
;[o]
T''[t]   X''[x]
------ - ------
 T[t]     X[x]
:[font = text; inactive; preserveAspect]
If u solves the wave equation, this expression must be zero. (We could quibble about what happens when T or X = 0, but that would either mean that u is the uninteresting "trivial" solution or else the problem occurs only at isolated values of x and t.)  Thus 
                        T''/T = X''/X.
Since the left side is independent of x and the right side is independent of t, both sides must in fact equal a constant. We don't know its value yet, so let us call it - mu. (It turns out to be negative.)
Thus X''[x] = - mu X[x]:
:[font = input; preserveAspect; startGroup]
 DSolve[- X''[x]== mu X[x],X[x],x]
:[font = output; output; inactive; dontPreserveAspect; endGroup]
{{X[x] -> E^(-I*mu^(1/2)*x)*C[1] + E^(I*mu^(1/2)*x)*C[2]}}
;[o]
           -I Sqrt[mu] x         I Sqrt[mu] x
{{X[x] -> E              C[1] + E             C[2]}}
:[font = text; inactive; dontPreserveAspect; startGroup]

We now remind Mathematica about Euler's interesting formula for complex exponentials:
:[font = input; preserveAspect; startGroup]
 % /. {E^(I Sqrt[mu] x) -> Cos[Sqrt[mu] x] + 
               I Sin[Sqrt[mu] x],
         E^(- I Sqrt[mu] x) -> Cos[Sqrt[mu] x] -
                  I Sin[Sqrt[mu] x]}
:[font = output; output; inactive; preserveAspect; endGroup; endGroup]
{{X[x] -> C[1]*(Cos[mu^(1/2)*x] - I*Sin[mu^(1/2)*x]) + 
 
     C[2]*(Cos[mu^(1/2)*x] + I*Sin[mu^(1/2)*x])}}
;[o]
{{X[x] -> C[1] (Cos[Sqrt[mu] x] - I Sin[Sqrt[mu] x]) + 
 
     C[2] (Cos[Sqrt[mu] x] + I Sin[Sqrt[mu] x])}}
:[font = text; inactive; preserveAspect]

Evidently, we could reexpress this as 
:[font = input; preserveAspect]
 Clear[X,D1,D2]
:[font = input; preserveAspect]
X[x_] := D1 Cos[Sqrt[mu] x] + 
        D1 Sin[Sqrt[mu] x]
:[font = input; preserveAspect; startGroup]
X[0]
:[font = output; output; inactive; preserveAspect; endGroup]
D1
;[o]
D1
:[font = text; inactive; preserveAspect]

This means that in order to satisfy the BC at x=0, we must have:
:[font = input; preserveAspect]
X[x_] := Sin[Sqrt[mu] x]
:[font = input; preserveAspect; startGroup]
Solve[X[L] == 0, mu]
:[font = message; inactive; preserveAspect]
Solve::ifun: 
   Warning: Inverse functions are being used by Solve, so
     some solutions may not be found.
:[font = output; output; inactive; preserveAspect; endGroup]
{{mu -> 0}}
;[o]
{{mu -> 0}}
:[font = input; preserveAspect]
Clear[X]
:[font = input; preserveAspect]
 X[x_,n_Integer] := Sin[n Pi x/L]

:[font = text; inactive; preserveAspect]

Since the second derivative of this is - (n Pi/L)^2 Sin[n Pi x/L], we see that
:[font = input; preserveAspect]
mu[n_Integer] := -(n Pi/L)^2
:[font = text; inactive; preserveAspect]
Now let's look at the other part of the PDE which we "separated off":
T''[t] = - mu T[t]

As we saw, there are many possible values of \mu which are consistent with the boundary condition for the function X, and the same possibilities occur in equation (6.3) (multiply \mu by c^2 here if c is not set to 1). Therefore
         T[t_,n_] = A[n] Cos[n \pi t] + B[n] Sin[n \pi t ]. 
         
We cannot go further with the function T without being given the initial condition, that is, information about what happens at t=0.

The general solution we come up with is a linear combination of the particular solutions we get by separating the equation. It is not yet obvious that this is a completely general solution, but it is. The most general linear combination is of the form:
:[font = input; preserveAspect; startGroup]
u[t_,x_] = Sum[(A[n] Cos[n Pi t ] + B[n] Sin[n Pi t ] ) \
          Sin[n Pi x],  {n,1,infinity}] 
:[font = message; inactive; preserveAspect]
General::spell1: 
   Possible spelling error: new symbol name "infinity"
     is similar to existing symbol "Infinity".
:[font = output; output; inactive; preserveAspect; endGroup]
Sum[(A[n]*Cos[n*Pi*t] + B[n]*Sin[n*Pi*t])*Sin[n*Pi*x], 
 
  {n, 1, infinity}]
;[o]
Sum[(A[n] Cos[n Pi t] + B[n] Sin[n Pi t]) Sin[n Pi x], 
 
  {n, 1, infinity}]
:[font = text; inactive; preserveAspect]
The coefficients A and B have to be determined from the initial conditions.

Suppose that 

										u[0,x] = f[x]

and 

                    ut[0,x] =   g[x].
                    
Then we get:
;[s]
3:0,0;144,1;145,0;195,-1;
2:2,17,12,New York,0,12,0,0,0;1,25,16,New York,64,12,0,0,0;
:[font = input; preserveAspect; endGroup]
A[n_,f_] := 2 Integrate[f[x] Sin[n Pi x], {x,0,1}]
B[n_,g_] := (2/(n Pi)) Integrate[g[x] Sin[n Pi x], {x,0,1}]
:[font = section; inactive; Cclosed; preserveAspect; startGroup]
Illustration of normal mode solutions.  (Thanks to Alfred D. Andrew for this section.)
;[s]
2:0,0;40,1;87,-1;
2:1,19,14,New York,1,14,0,0,0;1,14,10,New York,1,10,0,0,0;
:[font = text; inactive; preserveAspect]

The normal modes are the solutions which have a single frequency (pure frequency) in time.  For the one dimensional wave equation, for each frequency there is basically a single product solution  T[n,t] X[n,t] which is a normal mode for that frequency.  All others are simply phase-shifted versions of this one.  You can sketch a few of them by executing the following cells, and the results can be animated to show how the string vibrates.

:[font = input; preserveAspect; startGroup]
T[n_,t_] = Cos[ n Pi t]
:[font = output; output; inactive; preserveAspect; endGroup]
Cos[n*Pi*t]
;[o]
Cos[n Pi t]
:[font = input; preserveAspect; startGroup]
X[n_,x_] = Sin[n Pi x]
:[font = output; output; inactive; preserveAspect; endGroup]
Sin[n*Pi*x]
;[o]
Sin[n Pi x]
:[font = input; preserveAspect; startGroup]
U[n_,x_,t_] = T[n,t] X[n,x]
:[font = output; output; inactive; preserveAspect; endGroup]
Cos[n*Pi*t]*Sin[n*Pi*x]
;[o]
Cos[n Pi t] Sin[n Pi x]
:[font = input; preserveAspect]
Table[Plot[U[1,x,t],{x,0,1},
	PlotRange->{-1,1},
	PlotStyle->
		Thickness[.007]],
	{t,0,.9,.1}]
:[font = input; preserveAspect; endGroup]
Table[Plot[U[3,x,t],{x,0,1},
	PlotRange->{-1,1},
	PlotStyle->
		Thickness[.007]],
	{t,0,.3,.1/3}]
:[font = section; inactive; Cclosed; preserveAspect; startGroup]
Model Problem.  
Solve the problem of the vibrating string for 0 < x < 1, c=1, 
         u[t,0] = u[t,1] = 0
using Fourier sine series, and initial conditions 
              u(x,0) = x - x^3, 
              u_t(x,0) = 0.
;[s]
3:0,0;17,2;220,1;221,-1;
3:1,19,14,New York,1,14,0,0,0;1,23,17,New York,0,18,0,0,0;1,17,12,New York,0,12,0,0,0;
:[font = input; preserveAspect]
TrigId = {Cos[Pi n_] -> (-1)^n, Sin[Pi n_] -> 0}
:[font = input; preserveAspect]
f[x_] := x - x^3
:[font = input; preserveAspect; startGroup]
A[n,f]
:[font = input; preserveAspect; endGroup]
(2*(-6*n*Pi*Cos[n*Pi] + 6*Sin[n*Pi] - 2*n^2*Pi^2*Sin[n*Pi]))/
 
  (n^4*Pi^4)
;[o]
                                        2   2
2 (-6 n Pi Cos[n Pi] + 6 Sin[n Pi] - 2 n  Pi  Sin[n Pi])
--------------------------------------------------------
                          4   4
                         n  Pi
:[font = input; preserveAspect; startGroup]
% /. TrigId
:[font = input; preserveAspect; endGroup]
(-12*(-1)^n)/(n^3*Pi^3)
;[o]
        n
-12 (-1)
---------
  3   3
 n  Pi
:[font = text; inactive; preserveAspect]
The B's are clearly zero, even without the aid of software.  Thus the solution for times t>0 is:
:[font = input; preserveAspect]
  U[t_,x_,infinity_] := \
       - Sum[Cos[n Pi t] Sin[n Pi x] 12 (-1)^n /(n^3 Pi^3), \
           {n,1,infinity}]
:[font = input; preserveAspect; startGroup]
  Table[Plot[U[t/4,x,7], {x, 0, 1}, 
        PlotRange -> {-.5,.5}], {t,0,8}]
:[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 282; pictureHeight = 174]
%!
%%Creator: Mathematica
%%AspectRatio: .61803 
MathPictureStart
%% Graphics
/Courier findfont 10  scalefont  setfont
% Scaling calculations
0.0238095 0.952381 0.309017 0.618034 [
[(0.2)] .21429 .30902 0 2 Msboxa
[(0.4)] .40476 .30902 0 2 Msboxa
[(0.6)] .59524 .30902 0 2 Msboxa
[(0.8)] .78571 .30902 0 2 Msboxa
[(1)] .97619 .30902 0 2 Msboxa
[(-0.4)] .01131 .0618 1 0 Msboxa
[(-0.2)] .01131 .18541 1 0 Msboxa
[(0.2)] .01131 .43262 1 0 Msboxa
[(0.4)] .01131 .55623 1 0 Msboxa
[ -0.001 -0.001 0 0 ]
[ 1.001 .61903 0 0 ]
] MathScale
% Start of Graphics
1 setlinecap
1 setlinejoin
newpath
[ ] 0 setdash
0 g
p
p
.002 w
.21429 .30902 m
.21429 .31527 L
s
P
[(0.2)] .21429 .30902 0 2 Mshowa
p
.002 w
.40476 .30902 m
.40476 .31527 L
s
P
[(0.4)] .40476 .30902 0 2 Mshowa
p
.002 w
.59524 .30902 m
.59524 .31527 L
s
P
[(0.6)] .59524 .30902 0 2 Mshowa
p
.002 w
.78571 .30902 m
.78571 .31527 L
s
P
[(0.8)] .78571 .30902 0 2 Mshowa
p
.002 w
.97619 .30902 m
.97619 .31527 L
s
P
[(1)] .97619 .30902 0 2 Mshowa
p
.001 w
.0619 .30902 m
.0619 .31277 L
s
P
p
.001 w
.1 .30902 m
.1 .31277 L
s
P
p
.001 w
.1381 .30902 m
.1381 .31277 L
s
P
p
.001 w
.17619 .30902 m
.17619 .31277 L
s
P
p
.001 w
.25238 .30902 m
.25238 .31277 L
s
P
p
.001 w
.29048 .30902 m
.29048 .31277 L
s
P
p
.001 w
.32857 .30902 m
.32857 .31277 L
s
P
p
.001 w
.36667 .30902 m
.36667 .31277 L
s
P
p
.001 w
.44286 .30902 m
.44286 .31277 L
s
P
p
.001 w
.48095 .30902 m
.48095 .31277 L
s
P
p
.001 w
.51905 .30902 m
.51905 .31277 L
s
P
p
.001 w
.55714 .30902 m
.55714 .31277 L
s
P
p
.001 w
.63333 .30902 m
.63333 .31277 L
s
P
p
.001 w
.67143 .30902 m
.67143 .31277 L
s
P
p
.001 w
.70952 .30902 m
.70952 .31277 L
s
P
p
.001 w
.74762 .30902 m
.74762 .31277 L
s
P
p
.001 w
.82381 .30902 m
.82381 .31277 L
s
P
p
.001 w
.8619 .30902 m
.8619 .31277 L
s
P
p
.001 w
.9 .30902 m
.9 .31277 L
s
P
p
.001 w
.9381 .30902 m
.9381 .31277 L
s
P
p
.002 w
0 .30902 m
1 .30902 L
s
P
p
.002 w
.02381 .0618 m
.03006 .0618 L
s
P
[(-0.4)] .01131 .0618 1 0 Mshowa
p
.002 w
.02381 .18541 m
.03006 .18541 L
s
P
[(-0.2)] .01131 .18541 1 0 Mshowa
p
.002 w
.02381 .43262 m
.03006 .43262 L
s
P
[(0.2)] .01131 .43262 1 0 Mshowa
p
.002 w
.02381 .55623 m
.03006 .55623 L
s
P
[(0.4)] .01131 .55623 1 0 Mshowa
p
.001 w
.02381 .08652 m
.02756 .08652 L
s
P
p
.001 w
.02381 .11125 m
.02756 .11125 L
s
P
p
.001 w
.02381 .13597 m
.02756 .13597 L
s
P
p
.001 w
.02381 .16069 m
.02756 .16069 L
s
P
p
.001 w
.02381 .21013 m
.02756 .21013 L
s
P
p
.001 w
.02381 .23485 m
.02756 .23485 L
s
P
p
.001 w
.02381 .25957 m
.02756 .25957 L
s
P
p
.001 w
.02381 .2843 m
.02756 .2843 L
s
P
p
.001 w
.02381 .33374 m
.02756 .33374 L
s
P
p
.001 w
.02381 .35846 m
.02756 .35846 L
s
P
p
.001 w
.02381 .38318 m
.02756 .38318 L
s
P
p
.001 w
.02381 .4079 m
.02756 .4079 L
s
P
p
.001 w
.02381 .45735 m
.02756 .45735 L
s
P
p
.001 w
.02381 .48207 m
.02756 .48207 L
s
P
p
.001 w
.02381 .50679 m
.02756 .50679 L
s
P
p
.001 w
.02381 .53151 m
.02756 .53151 L
s
P
p
.001 w
.02381 .03708 m
.02756 .03708 L
s
P
p
.001 w
.02381 .01236 m
.02756 .01236 L
s
P
p
.001 w
.02381 .58095 m
.02756 .58095 L
s
P
p
.001 w
.02381 .60567 m
.02756 .60567 L
s
P
p
.002 w
.02381 0 m
.02381 .61803 L
s
P
P
0 0 m
1 0 L
1 .61803 L
0 .61803 L
closepath
clip
newpath
p
p
.004 w
.02381 .30902 m
.06349 .33496 L
.10317 .36042 L
.14286 .38511 L
.18254 .40895 L
.22222 .4319 L
.2619 .45378 L
.30159 .47414 L
.34127 .49245 L
.38095 .50833 L
.42063 .52165 L
.46032 .53242 L
.5 .54056 L
.51984 .54355 L
.52976 .54474 L
.53968 .54572 L
.54464 .54613 L
.5496 .54648 L
.55456 .54677 L
.55952 .54699 L
.562 .54709 L
.56448 .54716 L
.56696 .54722 L
.56944 .54727 L
.57068 .54729 L
.57192 .5473 L
.57316 .54731 L
.5744 .54731 L
.57564 .54731 L
.57688 .54731 L
.57813 .5473 L
.57937 .54729 L
.58061 .54727 L
.58185 .54725 L
.58433 .5472 L
.58681 .54713 L
.58929 .54705 L
.59425 .54683 L
.59921 .54654 L
.60913 .54576 L
.61905 .54471 L
.63889 .54178 L
.65873 .53777 L
.69841 .52659 L
.7381 .51136 L
.77778 .49184 L
.81746 .46721 L
.85714 .43643 L
.89683 .399 L
.93651 .35574 L
Mistroke
.97619 .30902 L
Mfstroke
P
P
% End of Graphics
MathPictureEnd

:[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 282; pictureHeight = 174]
%!
%%Creator: Mathematica
%%AspectRatio: .61803 
MathPictureStart
%% Graphics
/Courier findfont 10  scalefont  setfont
% Scaling calculations
0.0238095 0.952381 0.309017 0.618034 [
[(0.2)] .21429 .30902 0 2 Msboxa
[(0.4)] .40476 .30902 0 2 Msboxa
[(0.6)] .59524 .30902 0 2 Msboxa
[(0.8)] .78571 .30902 0 2 Msboxa
[(1)] .97619 .30902 0 2 Msboxa
[(-0.4)] .01131 .0618 1 0 Msboxa
[(-0.2)] .01131 .18541 1 0 Msboxa
[(0.2)] .01131 .43262 1 0 Msboxa
[(0.4)] .01131 .55623 1 0 Msboxa
[ -0.001 -0.001 0 0 ]
[ 1.001 .61903 0 0 ]
] MathScale
% Start of Graphics
1 setlinecap
1 setlinejoin
newpath
[ ] 0 setdash
0 g
p
p
.002 w
.21429 .30902 m
.21429 .31527 L
s
P
[(0.2)] .21429 .30902 0 2 Mshowa
p
.002 w
.40476 .30902 m
.40476 .31527 L
s
P
[(0.4)] .40476 .30902 0 2 Mshowa
p
.002 w
.59524 .30902 m
.59524 .31527 L
s
P
[(0.6)] .59524 .30902 0 2 Mshowa
p
.002 w
.78571 .30902 m
.78571 .31527 L
s
P
[(0.8)] .78571 .30902 0 2 Mshowa
p
.002 w
.97619 .30902 m
.97619 .31527 L
s
P
[(1)] .97619 .30902 0 2 Mshowa
p
.001 w
.0619 .30902 m
.0619 .31277 L
s
P
p
.001 w
.1 .30902 m
.1 .31277 L
s
P
p
.001 w
.1381 .30902 m
.1381 .31277 L
s
P
p
.001 w
.17619 .30902 m
.17619 .31277 L
s
P
p
.001 w
.25238 .30902 m
.25238 .31277 L
s
P
p
.001 w
.29048 .30902 m
.29048 .31277 L
s
P
p
.001 w
.32857 .30902 m
.32857 .31277 L
s
P
p
.001 w
.36667 .30902 m
.36667 .31277 L
s
P
p
.001 w
.44286 .30902 m
.44286 .31277 L
s
P
p
.001 w
.48095 .30902 m
.48095 .31277 L
s
P
p
.001 w
.51905 .30902 m
.51905 .31277 L
s
P
p
.001 w
.55714 .30902 m
.55714 .31277 L
s
P
p
.001 w
.63333 .30902 m
.63333 .31277 L
s
P
p
.001 w
.67143 .30902 m
.67143 .31277 L
s
P
p
.001 w
.70952 .30902 m
.70952 .31277 L
s
P
p
.001 w
.74762 .30902 m
.74762 .31277 L
s
P
p
.001 w
.82381 .30902 m
.82381 .31277 L
s
P
p
.001 w
.8619 .30902 m
.8619 .31277 L
s
P
p
.001 w
.9 .30902 m
.9 .31277 L
s
P
p
.001 w
.9381 .30902 m
.9381 .31277 L
s
P
p
.002 w
0 .30902 m
1 .30902 L
s
P
p
.002 w
.02381 .0618 m
.03006 .0618 L
s
P
[(-0.4)] .01131 .0618 1 0 Mshowa
p
.002 w
.02381 .18541 m
.03006 .18541 L
s
P
[(-0.2)] .01131 .18541 1 0 Mshowa
p
.002 w
.02381 .43262 m
.03006 .43262 L
s
P
[(0.2)] .01131 .43262 1 0 Mshowa
p
.002 w
.02381 .55623 m
.03006 .55623 L
s
P
[(0.4)] .01131 .55623 1 0 Mshowa
p
.001 w
.02381 .08652 m
.02756 .08652 L
s
P
p
.001 w
.02381 .11125 m
.02756 .11125 L
s
P
p
.001 w
.02381 .13597 m
.02756 .13597 L
s
P
p
.001 w
.02381 .16069 m
.02756 .16069 L
s
P
p
.001 w
.02381 .21013 m
.02756 .21013 L
s
P
p
.001 w
.02381 .23485 m
.02756 .23485 L
s
P
p
.001 w
.02381 .25957 m
.02756 .25957 L
s
P
p
.001 w
.02381 .2843 m
.02756 .2843 L
s
P
p
.001 w
.02381 .33374 m
.02756 .33374 L
s
P
p
.001 w
.02381 .35846 m
.02756 .35846 L
s
P
p
.001 w
.02381 .38318 m
.02756 .38318 L
s
P
p
.001 w
.02381 .4079 m
.02756 .4079 L
s
P
p
.001 w
.02381 .45735 m
.02756 .45735 L
s
P
p
.001 w
.02381 .48207 m
.02756 .48207 L
s
P
p
.001 w
.02381 .50679 m
.02756 .50679 L
s
P
p
.001 w
.02381 .53151 m
.02756 .53151 L
s
P
p
.001 w
.02381 .03708 m
.02756 .03708 L
s
P
p
.001 w
.02381 .01236 m
.02756 .01236 L
s
P
p
.001 w
.02381 .58095 m
.02756 .58095 L
s
P
p
.001 w
.02381 .60567 m
.02756 .60567 L
s
P
p
.002 w
.02381 0 m
.02381 .61803 L
s
P
P
0 0 m
1 0 L
1 .61803 L
0 .61803 L
closepath
clip
newpath
p
p
.004 w
.02381 .30902 m
.06349 .33013 L
.10317 .35077 L
.14286 .37063 L
.18254 .38963 L
.22222 .40775 L
.2619 .42479 L
.30159 .44034 L
.34127 .45385 L
.38095 .46491 L
.40079 .46946 L
.42063 .47336 L
.44048 .47662 L
.46032 .47925 L
.47024 .48033 L
.48016 .48124 L
.49008 .48199 L
.5 .48257 L
.50496 .48279 L
.50992 .48297 L
.5124 .48304 L
.51488 .4831 L
.51736 .48314 L
.51984 .48318 L
.52108 .48319 L
.52232 .4832 L
.52356 .4832 L
.5248 .48321 L
.52604 .48321 L
.52728 .4832 L
.52852 .4832 L
.52976 .48319 L
.531 .48317 L
.53224 .48316 L
.53472 .48311 L
.5372 .48306 L
.53968 .48299 L
.54464 .4828 L
.5496 .48256 L
.55952 .4819 L
.56944 .481 L
.57937 .47983 L
.59921 .47668 L
.61905 .47238 L
.65873 .46033 L
.69841 .44408 L
.7381 .42479 L
.77778 .40401 L
.81746 .38316 L
.85714 .36316 L
Mistroke
.89683 .3443 L
.93651 .32639 L
.97619 .30902 L
Mfstroke
P
P
% End of Graphics
MathPictureEnd

:[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 282; pictureHeight = 174]
%!
%%Creator: Mathematica
%%AspectRatio: .61803 
MathPictureStart
%% Graphics
/Courier findfont 10  scalefont  setfont
% Scaling calculations
0.0238095 0.952381 0.309017 0.618034 [
[(0.2)] .21429 .30902 0 2 Msboxa
[(0.4)] .40476 .30902 0 2 Msboxa
[(0.6)] .59524 .30902 0 2 Msboxa
[(0.8)] .78571 .30902 0 2 Msboxa
[(1)] .97619 .30902 0 2 Msboxa
[(-0.4)] .01131 .0618 1 0 Msboxa
[(-0.2)] .01131 .18541 1 0 Msboxa
[(0.2)] .01131 .43262 1 0 Msboxa
[(0.4)] .01131 .55623 1 0 Msboxa
[ -0.001 -0.001 0 0 ]
[ 1.001 .61903 0 0 ]
] MathScale
% Start of Graphics
1 setlinecap
1 setlinejoin
newpath
[ ] 0 setdash
0 g
p
p
.002 w
.21429 .30902 m
.21429 .31527 L
s
P
[(0.2)] .21429 .30902 0 2 Mshowa
p
.002 w
.40476 .30902 m
.40476 .31527 L
s
P
[(0.4)] .40476 .30902 0 2 Mshowa
p
.002 w
.59524 .30902 m
.59524 .31527 L
s
P
[(0.6)] .59524 .30902 0 2 Mshowa
p
.002 w
.78571 .30902 m
.78571 .31527 L
s
P
[(0.8)] .78571 .30902 0 2 Mshowa
p
.002 w
.97619 .30902 m
.97619 .31527 L
s
P
[(1)] .97619 .30902 0 2 Mshowa
p
.001 w
.0619 .30902 m
.0619 .31277 L
s
P
p
.001 w
.1 .30902 m
.1 .31277 L
s
P
p
.001 w
.1381 .30902 m
.1381 .31277 L
s
P
p
.001 w
.17619 .30902 m
.17619 .31277 L
s
P
p
.001 w
.25238 .30902 m
.25238 .31277 L
s
P
p
.001 w
.29048 .30902 m
.29048 .31277 L
s
P
p
.001 w
.32857 .30902 m
.32857 .31277 L
s
P
p
.001 w
.36667 .30902 m
.36667 .31277 L
s
P
p
.001 w
.44286 .30902 m
.44286 .31277 L
s
P
p
.001 w
.48095 .30902 m
.48095 .31277 L
s
P
p
.001 w
.51905 .30902 m
.51905 .31277 L
s
P
p
.001 w
.55714 .30902 m
.55714 .31277 L
s
P
p
.001 w
.63333 .30902 m
.63333 .31277 L
s
P
p
.001 w
.67143 .30902 m
.67143 .31277 L
s
P
p
.001 w
.70952 .30902 m
.70952 .31277 L
s
P
p
.001 w
.74762 .30902 m
.74762 .31277 L
s
P
p
.001 w
.82381 .30902 m
.82381 .31277 L
s
P
p
.001 w
.8619 .30902 m
.8619 .31277 L
s
P
p
.001 w
.9 .30902 m
.9 .31277 L
s
P
p
.001 w
.9381 .30902 m
.9381 .31277 L
s
P
p
.002 w
0 .30902 m
1 .30902 L
s
P
p
.002 w
.02381 .0618 m
.03006 .0618 L
s
P
[(-0.4)] .01131 .0618 1 0 Mshowa
p
.002 w
.02381 .18541 m
.03006 .18541 L
s
P
[(-0.2)] .01131 .18541 1 0 Mshowa
p
.002 w
.02381 .43262 m
.03006 .43262 L
s
P
[(0.2)] .01131 .43262 1 0 Mshowa
p
.002 w
.02381 .55623 m
.03006 .55623 L
s
P
[(0.4)] .01131 .55623 1 0 Mshowa
p
.001 w
.02381 .08652 m
.02756 .08652 L
s
P
p
.001 w
.02381 .11125 m
.02756 .11125 L
s
P
p
.001 w
.02381 .13597 m
.02756 .13597 L
s
P
p
.001 w
.02381 .16069 m
.02756 .16069 L
s
P
p
.001 w
.02381 .21013 m
.02756 .21013 L
s
P
p
.001 w
.02381 .23485 m
.02756 .23485 L
s
P
p
.001 w
.02381 .25957 m
.02756 .25957 L
s
P
p
.001 w
.02381 .2843 m
.02756 .2843 L
s
P
p
.001 w
.02381 .33374 m
.02756 .33374 L
s
P
p
.001 w
.02381 .35846 m
.02756 .35846 L
s
P
p
.001 w
.02381 .38318 m
.02756 .38318 L
s
P
p
.001 w
.02381 .4079 m
.02756 .4079 L
s
P
p
.001 w
.02381 .45735 m
.02756 .45735 L
s
P
p
.001 w
.02381 .48207 m
.02756 .48207 L
s
P
p
.001 w
.02381 .50679 m
.02756 .50679 L
s
P
p
.001 w
.02381 .53151 m
.02756 .53151 L
s
P
p
.001 w
.02381 .03708 m
.02756 .03708 L
s
P
p
.001 w
.02381 .01236 m
.02756 .01236 L
s
P
p
.001 w
.02381 .58095 m
.02756 .58095 L
s
P
p
.001 w
.02381 .60567 m
.02756 .60567 L
s
P
p
.002 w
.02381 0 m
.02381 .61803 L
s
P
P
0 0 m
1 0 L
1 .61803 L
0 .61803 L
closepath
clip
newpath
p
p
.004 w
.02381 .30902 m
.06349 .31567 L
.10317 .32184 L
.14286 .3272 L
.18254 .33167 L
.22222 .33525 L
.24206 .33667 L
.25198 .33728 L
.2619 .33781 L
.27183 .33825 L
.28175 .3386 L
.28671 .33874 L
.29167 .33885 L
.29663 .33893 L
.29911 .33896 L
.30035 .33897 L
.30159 .33898 L
.30283 .33899 L
.30407 .339 L
.30531 .339 L
.30655 .339 L
.30779 .339 L
.30903 .339 L
.31027 .339 L
.31151 .33899 L
.31275 .33898 L
.31399 .33897 L
.31647 .33894 L
.31895 .3389 L
.32143 .33886 L
.32639 .33874 L
.33135 .33858 L
.34127 .33815 L
.35119 .33755 L
.36111 .33677 L
.38095 .33468 L
.40079 .33185 L
.42063 .32831 L
.46032 .31941 L
.5 .30902 L
.53968 .29863 L
.57937 .28972 L
.59921 .28618 L
.61905 .28335 L
.62897 .28222 L
.63889 .28126 L
.64881 .28049 L
.65873 .27989 L
.66369 .27965 L
.66865 .27945 L
Mistroke
.67361 .27929 L
.67857 .27918 L
.68105 .27913 L
.68353 .27909 L
.68601 .27906 L
.68725 .27905 L
.68849 .27904 L
.68973 .27904 L
.69097 .27903 L
.69221 .27903 L
.69345 .27903 L
.69469 .27903 L
.69593 .27904 L
.69717 .27904 L
.69841 .27905 L
.70089 .27907 L
.70337 .2791 L
.70833 .27918 L
.71329 .27929 L
.71825 .27943 L
.72817 .27978 L
.7381 .28023 L
.75794 .28136 L
.77778 .28279 L
.81746 .28636 L
.85714 .29083 L
.89683 .2962 L
.93651 .30236 L
.97619 .30902 L
Mfstroke
P
P
% End of Graphics
MathPictureEnd

:[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 282; pictureHeight = 174]
%!
%%Creator: Mathematica
%%AspectRatio: .61803 
MathPictureStart
%% Graphics
/Courier findfont 10  scalefont  setfont
% Scaling calculations
0.0238095 0.952381 0.309017 0.618034 [
[(0.2)] .21429 .30902 0 2 Msboxa
[(0.4)] .40476 .30902 0 2 Msboxa
[(0.6)] .59524 .30902 0 2 Msboxa
[(0.8)] .78571 .30902 0 2 Msboxa
[(1)] .97619 .30902 0 2 Msboxa
[(-0.4)] .01131 .0618 1 0 Msboxa
[(-0.2)] .01131 .18541 1 0 Msboxa
[(0.2)] .01131 .43262 1 0 Msboxa
[(0.4)] .01131 .55623 1 0 Msboxa
[ -0.001 -0.001 0 0 ]
[ 1.001 .61903 0 0 ]
] MathScale
% Start of Graphics
1 setlinecap
1 setlinejoin
newpath
[ ] 0 setdash
0 g
p
p
.002 w
.21429 .30902 m
.21429 .31527 L
s
P
[(0.2)] .21429 .30902 0 2 Mshowa
p
.002 w
.40476 .30902 m
.40476 .31527 L
s
P
[(0.4)] .40476 .30902 0 2 Mshowa
p
.002 w
.59524 .30902 m
.59524 .31527 L
s
P
[(0.6)] .59524 .30902 0 2 Mshowa
p
.002 w
.78571 .30902 m
.78571 .31527 L
s
P
[(0.8)] .78571 .30902 0 2 Mshowa
p
.002 w
.97619 .30902 m
.97619 .31527 L
s
P
[(1)] .97619 .30902 0 2 Mshowa
p
.001 w
.0619 .30902 m
.0619 .31277 L
s
P
p
.001 w
.1 .30902 m
.1 .31277 L
s
P
p
.001 w
.1381 .30902 m
.1381 .31277 L
s
P
p
.001 w
.17619 .30902 m
.17619 .31277 L
s
P
p
.001 w
.25238 .30902 m
.25238 .31277 L
s
P
p
.001 w
.29048 .30902 m
.29048 .31277 L
s
P
p
.001 w
.32857 .30902 m
.32857 .31277 L
s
P
p
.001 w
.36667 .30902 m
.36667 .31277 L
s
P
p
.001 w
.44286 .30902 m
.44286 .31277 L
s
P
p
.001 w
.48095 .30902 m
.48095 .31277 L
s
P
p
.001 w
.51905 .30902 m
.51905 .31277 L
s
P
p
.001 w
.55714 .30902 m
.55714 .31277 L
s
P
p
.001 w
.63333 .30902 m
.63333 .31277 L
s
P
p
.001 w
.67143 .30902 m
.67143 .31277 L
s
P
p
.001 w
.70952 .30902 m
.70952 .31277 L
s
P
p
.001 w
.74762 .30902 m
.74762 .31277 L
s
P
p
.001 w
.82381 .30902 m
.82381 .31277 L
s
P
p
.001 w
.8619 .30902 m
.8619 .31277 L
s
P
p
.001 w
.9 .30902 m
.9 .31277 L
s
P
p
.001 w
.9381 .30902 m
.9381 .31277 L
s
P
p
.002 w
0 .30902 m
1 .30902 L
s
P
p
.002 w
.02381 .0618 m
.03006 .0618 L
s
P
[(-0.4)] .01131 .0618 1 0 Mshowa
p
.002 w
.02381 .18541 m
.03006 .18541 L
s
P
[(-0.2)] .01131 .18541 1 0 Mshowa
p
.002 w
.02381 .43262 m
.03006 .43262 L
s
P
[(0.2)] .01131 .43262 1 0 Mshowa
p
.002 w
.02381 .55623 m
.03006 .55623 L
s
P
[(0.4)] .01131 .55623 1 0 Mshowa
p
.001 w
.02381 .08652 m
.02756 .08652 L
s
P
p
.001 w
.02381 .11125 m
.02756 .11125 L
s
P
p
.001 w
.02381 .13597 m
.02756 .13597 L
s
P
p
.001 w
.02381 .16069 m
.02756 .16069 L
s
P
p
.001 w
.02381 .21013 m
.02756 .21013 L
s
P
p
.001 w
.02381 .23485 m
.02756 .23485 L
s
P
p
.001 w
.02381 .25957 m
.02756 .25957 L
s
P
p
.001 w
.02381 .2843 m
.02756 .2843 L
s
P
p
.001 w
.02381 .33374 m
.02756 .33374 L
s
P
p
.001 w
.02381 .35846 m
.02756 .35846 L
s
P
p
.001 w
.02381 .38318 m
.02756 .38318 L
s
P
p
.001 w
.02381 .4079 m
.02756 .4079 L
s
P
p
.001 w
.02381 .45735 m
.02756 .45735 L
s
P
p
.001 w
.02381 .48207 m
.02756 .48207 L
s
P
p
.001 w
.02381 .50679 m
.02756 .50679 L
s
P
p
.001 w
.02381 .53151 m
.02756 .53151 L
s
P
p
.001 w
.02381 .03708 m
.02756 .03708 L
s
P
p
.001 w
.02381 .01236 m
.02756 .01236 L
s
P
p
.001 w
.02381 .58095 m
.02756 .58095 L
s
P
p
.001 w
.02381 .60567 m
.02756 .60567 L
s
P
p
.002 w
.02381 0 m
.02381 .61803 L
s
P
P
0 0 m
1 0 L
1 .61803 L
0 .61803 L
closepath
clip
newpath
p
p
.004 w
.02381 .30902 m
.06349 .29164 L
.10317 .27374 L
.14286 .25488 L
.18254 .23487 L
.22222 .21402 L
.2619 .19324 L
.30159 .17396 L
.34127 .15771 L
.36111 .15111 L
.38095 .14565 L
.40079 .14136 L
.41071 .13964 L
.42063 .1382 L
.43056 .13704 L
.44048 .13613 L
.44544 .13577 L
.4504 .13547 L
.45536 .13523 L
.45784 .13513 L
.46032 .13505 L
.4628 .13498 L
.46528 .13492 L
.46652 .1349 L
.46776 .13488 L
.469 .13486 L
.47024 .13485 L
.47148 .13484 L
.47272 .13483 L
.47396 .13483 L
.4752 .13483 L
.47644 .13483 L
.47768 .13484 L
.47892 .13485 L
.48016 .13486 L
.48264 .13489 L
.48512 .13494 L
.4876 .135 L
.49008 .13507 L
.5 .13547 L
.50496 .13573 L
.50992 .13604 L
.51984 .13679 L
.53968 .13879 L
.55952 .14142 L
.57937 .14468 L
.61905 .15313 L
.65873 .16418 L
.69841 .17769 L
.7381 .19324 L
Mistroke
.77778 .21029 L
.81746 .2284 L
.85714 .2474 L
.89683 .26727 L
.93651 .2879 L
.97619 .30902 L
Mfstroke
P
P
% End of Graphics
MathPictureEnd

:[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 282; pictureHeight = 174]
%!
%%Creator: Mathematica
%%AspectRatio: .61803 
MathPictureStart
%% Graphics
/Courier findfont 10  scalefont  setfont
% Scaling calculations
0.0238095 0.952381 0.309017 0.618034 [
[(0.2)] .21429 .30902 0 2 Msboxa
[(0.4)] .40476 .30902 0 2 Msboxa
[(0.6)] .59524 .30902 0 2 Msboxa
[(0.8)] .78571 .30902 0 2 Msboxa
[(1)] .97619 .30902 0 2 Msboxa
[(-0.4)] .01131 .0618 1 0 Msboxa
[(-0.2)] .01131 .18541 1 0 Msboxa
[(0.2)] .01131 .43262 1 0 Msboxa
[(0.4)] .01131 .55623 1 0 Msboxa
[ -0.001 -0.001 0 0 ]
[ 1.001 .61903 0 0 ]
] MathScale
% Start of Graphics
1 setlinecap
1 setlinejoin
newpath
[ ] 0 setdash
0 g
p
p
.002 w
.21429 .30902 m
.21429 .31527 L
s
P
[(0.2)] .21429 .30902 0 2 Mshowa
p
.002 w
.40476 .30902 m
.40476 .31527 L
s
P
[(0.4)] .40476 .30902 0 2 Mshowa
p
.002 w
.59524 .30902 m
.59524 .31527 L
s
P
[(0.6)] .59524 .30902 0 2 Mshowa
p
.002 w
.78571 .30902 m
.78571 .31527 L
s
P
[(0.8)] .78571 .30902 0 2 Mshowa
p
.002 w
.97619 .30902 m
.97619 .31527 L
s
P
[(1)] .97619 .30902 0 2 Mshowa
p
.001 w
.0619 .30902 m
.0619 .31277 L
s
P
p
.001 w
.1 .30902 m
.1 .31277 L
s
P
p
.001 w
.1381 .30902 m
.1381 .31277 L
s
P
p
.001 w
.17619 .30902 m
.17619 .31277 L
s
P
p
.001 w
.25238 .30902 m
.25238 .31277 L
s
P
p
.001 w
.29048 .30902 m
.29048 .31277 L
s
P
p
.001 w
.32857 .30902 m
.32857 .31277 L
s
P
p
.001 w
.36667 .30902 m
.36667 .31277 L
s
P
p
.001 w
.44286 .30902 m
.44286 .31277 L
s
P
p
.001 w
.48095 .30902 m
.48095 .31277 L
s
P
p
.001 w
.51905 .30902 m
.51905 .31277 L
s
P
p
.001 w
.55714 .30902 m
.55714 .31277 L
s
P
p
.001 w
.63333 .30902 m
.63333 .31277 L
s
P
p
.001 w
.67143 .30902 m
.67143 .31277 L
s
P
p
.001 w
.70952 .30902 m
.70952 .31277 L
s
P
p
.001 w
.74762 .30902 m
.74762 .31277 L
s
P
p
.001 w
.82381 .30902 m
.82381 .31277 L
s
P
p
.001 w
.8619 .30902 m
.8619 .31277 L
s
P
p
.001 w
.9 .30902 m
.9 .31277 L
s
P
p
.001 w
.9381 .30902 m
.9381 .31277 L
s
P
p
.002 w
0 .30902 m
1 .30902 L
s
P
p
.002 w
.02381 .0618 m
.03006 .0618 L
s
P
[(-0.4)] .01131 .0618 1 0 Mshowa
p
.002 w
.02381 .18541 m
.03006 .18541 L
s
P
[(-0.2)] .01131 .18541 1 0 Mshowa
p
.002 w
.02381 .43262 m
.03006 .43262 L
s
P
[(0.2)] .01131 .43262 1 0 Mshowa
p
.002 w
.02381 .55623 m
.03006 .55623 L
s
P
[(0.4)] .01131 .55623 1 0 Mshowa
p
.001 w
.02381 .08652 m
.02756 .08652 L
s
P
p
.001 w
.02381 .11125 m
.02756 .11125 L
s
P
p
.001 w
.02381 .13597 m
.02756 .13597 L
s
P
p
.001 w
.02381 .16069 m
.02756 .16069 L
s
P
p
.001 w
.02381 .21013 m
.02756 .21013 L
s
P
p
.001 w
.02381 .23485 m
.02756 .23485 L
s
P
p
.001 w
.02381 .25957 m
.02756 .25957 L
s
P
p
.001 w
.02381 .2843 m
.02756 .2843 L
s
P
p
.001 w
.02381 .33374 m
.02756 .33374 L
s
P
p
.001 w
.02381 .35846 m
.02756 .35846 L
s
P
p
.001 w
.02381 .38318 m
.02756 .38318 L
s
P
p
.001 w
.02381 .4079 m
.02756 .4079 L
s
P
p
.001 w
.02381 .45735 m
.02756 .45735 L
s
P
p
.001 w
.02381 .48207 m
.02756 .48207 L
s
P
p
.001 w
.02381 .50679 m
.02756 .50679 L
s
P
p
.001 w
.02381 .53151 m
.02756 .53151 L
s
P
p
.001 w
.02381 .03708 m
.02756 .03708 L
s
P
p
.001 w
.02381 .01236 m
.02756 .01236 L
s
P
p
.001 w
.02381 .58095 m
.02756 .58095 L
s
P
p
.001 w
.02381 .60567 m
.02756 .60567 L
s
P
p
.002 w
.02381 0 m
.02381 .61803 L
s
P
P
0 0 m
1 0 L
1 .61803 L
0 .61803 L
closepath
clip
newpath
p
p
.004 w
.02381 .30902 m
.06349 .2623 L
.10317 .21903 L
.14286 .1816 L
.18254 .15082 L
.22222 .1262 L
.2619 .10667 L
.30159 .09144 L
.32143 .08534 L
.34127 .08027 L
.36111 .07625 L
.37103 .07465 L
.38095 .07332 L
.39087 .07227 L
.39583 .07185 L
.40079 .07149 L
.40575 .07121 L
.40823 .07109 L
.41071 .07099 L
.41319 .0709 L
.41567 .07083 L
.41691 .0708 L
.41815 .07078 L
.41939 .07076 L
.42063 .07075 L
.42187 .07073 L
.42312 .07073 L
.42436 .07072 L
.4256 .07072 L
.42684 .07073 L
.42808 .07074 L
.42932 .07075 L
.43056 .07077 L
.43304 .07081 L
.43552 .07087 L
.44048 .07104 L
.44544 .07127 L
.4504 .07156 L
.46032 .07231 L
.47024 .07329 L
.48016 .07448 L
.5 .07747 L
.53968 .08562 L
.57937 .09639 L
.61905 .1097 L
.65873 .12558 L
.69841 .1439 L
.7381 .16426 L
.77778 .18613 L
.81746 .20908 L
Mistroke
.85714 .23293 L
.89683 .25762 L
.93651 .28308 L
.97619 .30902 L
Mfstroke
P
P
% End of Graphics
MathPictureEnd

:[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 282; pictureHeight = 174]
%!
%%Creator: Mathematica
%%AspectRatio: .61803 
MathPictureStart
%% Graphics
/Courier findfont 10  scalefont  setfont
% Scaling calculations
0.0238095 0.952381 0.309017 0.618034 [
[(0.2)] .21429 .30902 0 2 Msboxa
[(0.4)] .40476 .30902 0 2 Msboxa
[(0.6)] .59524 .30902 0 2 Msboxa
[(0.8)] .78571 .30902 0 2 Msboxa
[(1)] .97619 .30902 0 2 Msboxa
[(-0.4)] .01131 .0618 1 0 Msboxa
[(-0.2)] .01131 .18541 1 0 Msboxa
[(0.2)] .01131 .43262 1 0 Msboxa
[(0.4)] .01131 .55623 1 0 Msboxa
[ -0.001 -0.001 0 0 ]
[ 1.001 .61903 0 0 ]
] MathScale
% Start of Graphics
1 setlinecap
1 setlinejoin
newpath
[ ] 0 setdash
0 g
p
p
.002 w
.21429 .30902 m
.21429 .31527 L
s
P
[(0.2)] .21429 .30902 0 2 Mshowa
p
.002 w
.40476 .30902 m
.40476 .31527 L
s
P
[(0.4)] .40476 .30902 0 2 Mshowa
p
.002 w
.59524 .30902 m
.59524 .31527 L
s
P
[(0.6)] .59524 .30902 0 2 Mshowa
p
.002 w
.78571 .30902 m
.78571 .31527 L
s
P
[(0.8)] .78571 .30902 0 2 Mshowa
p
.002 w
.97619 .30902 m
.97619 .31527 L
s
P
[(1)] .97619 .30902 0 2 Mshowa
p
.001 w
.0619 .30902 m
.0619 .31277 L
s
P
p
.001 w
.1 .30902 m
.1 .31277 L
s
P
p
.001 w
.1381 .30902 m
.1381 .31277 L
s
P
p
.001 w
.17619 .30902 m
.17619 .31277 L
s
P
p
.001 w
.25238 .30902 m
.25238 .31277 L
s
P
p
.001 w
.29048 .30902 m
.29048 .31277 L
s
P
p
.001 w
.32857 .30902 m
.32857 .31277 L
s
P
p
.001 w
.36667 .30902 m
.36667 .31277 L
s
P
p
.001 w
.44286 .30902 m
.44286 .31277 L
s
P
p
.001 w
.48095 .30902 m
.48095 .31277 L
s
P
p
.001 w
.51905 .30902 m
.51905 .31277 L
s
P
p
.001 w
.55714 .30902 m
.55714 .31277 L
s
P
p
.001 w
.63333 .30902 m
.63333 .31277 L
s
P
p
.001 w
.67143 .30902 m
.67143 .31277 L
s
P
p
.001 w
.70952 .30902 m
.70952 .31277 L
s
P
p
.001 w
.74762 .30902 m
.74762 .31277 L
s
P
p
.001 w
.82381 .30902 m
.82381 .31277 L
s
P
p
.001 w
.8619 .30902 m
.8619 .31277 L
s
P
p
.001 w
.9 .30902 m
.9 .31277 L
s
P
p
.001 w
.9381 .30902 m
.9381 .31277 L
s
P
p
.002 w
0 .30902 m
1 .30902 L
s
P
p
.002 w
.02381 .0618 m
.03006 .0618 L
s
P
[(-0.4)] .01131 .0618 1 0 Mshowa
p
.002 w
.02381 .18541 m
.03006 .18541 L
s
P
[(-0.2)] .01131 .18541 1 0 Mshowa
p
.002 w
.02381 .43262 m
.03006 .43262 L
s
P
[(0.2)] .01131 .43262 1 0 Mshowa
p
.002 w
.02381 .55623 m
.03006 .55623 L
s
P
[(0.4)] .01131 .55623 1 0 Mshowa
p
.001 w
.02381 .08652 m
.02756 .08652 L
s
P
p
.001 w
.02381 .11125 m
.02756 .11125 L
s
P
p
.001 w
.02381 .13597 m
.02756 .13597 L
s
P
p
.001 w
.02381 .16069 m
.02756 .16069 L
s
P
p
.001 w
.02381 .21013 m
.02756 .21013 L
s
P
p
.001 w
.02381 .23485 m
.02756 .23485 L
s
P
p
.001 w
.02381 .25957 m
.02756 .25957 L
s
P
p
.001 w
.02381 .2843 m
.02756 .2843 L
s
P
p
.001 w
.02381 .33374 m
.02756 .33374 L
s
P
p
.001 w
.02381 .35846 m
.02756 .35846 L
s
P
p
.001 w
.02381 .38318 m
.02756 .38318 L
s
P
p
.001 w
.02381 .4079 m
.02756 .4079 L
s
P
p
.001 w
.02381 .45735 m
.02756 .45735 L
s
P
p
.001 w
.02381 .48207 m
.02756 .48207 L
s
P
p
.001 w
.02381 .50679 m
.02756 .50679 L
s
P
p
.001 w
.02381 .53151 m
.02756 .53151 L
s
P
p
.001 w
.02381 .03708 m
.02756 .03708 L
s
P
p
.001 w
.02381 .01236 m
.02756 .01236 L
s
P
p
.001 w
.02381 .58095 m
.02756 .58095 L
s
P
p
.001 w
.02381 .60567 m
.02756 .60567 L
s
P
p
.002 w
.02381 0 m
.02381 .61803 L
s
P
P
0 0 m
1 0 L
1 .61803 L
0 .61803 L
closepath
clip
newpath
p
p
.004 w
.02381 .30902 m
.06349 .29164 L
.10317 .27374 L
.14286 .25488 L
.18254 .23487 L
.22222 .21402 L
.2619 .19324 L
.30159 .17396 L
.34127 .15771 L
.36111 .15111 L
.38095 .14565 L
.40079 .14136 L
.41071 .13964 L
.42063 .1382 L
.43056 .13704 L
.44048 .13613 L
.44544 .13577 L
.4504 .13547 L
.45536 .13523 L
.45784 .13513 L
.46032 .13505 L
.4628 .13498 L
.46528 .13492 L
.46652 .1349 L
.46776 .13488 L
.469 .13486 L
.47024 .13485 L
.47148 .13484 L
.47272 .13483 L
.47396 .13483 L
.4752 .13483 L
.47644 .13483 L
.47768 .13484 L
.47892 .13485 L
.48016 .13486 L
.48264 .13489 L
.48512 .13494 L
.4876 .135 L
.49008 .13507 L
.5 .13547 L
.50496 .13573 L
.50992 .13604 L
.51984 .13679 L
.53968 .13879 L
.55952 .14142 L
.57937 .14468 L
.61905 .15313 L
.65873 .16418 L
.69841 .17769 L
.7381 .19324 L
Mistroke
.77778 .21029 L
.81746 .2284 L
.85714 .2474 L
.89683 .26727 L
.93651 .2879 L
.97619 .30902 L
Mfstroke
P
P
% End of Graphics
MathPictureEnd

:[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 282; pictureHeight = 174]
%!
%%Creator: Mathematica
%%AspectRatio: .61803 
MathPictureStart
%% Graphics
/Courier findfont 10  scalefont  setfont
% Scaling calculations
0.0238095 0.952381 0.309017 0.618034 [
[(0.2)] .21429 .30902 0 2 Msboxa
[(0.4)] .40476 .30902 0 2 Msboxa
[(0.6)] .59524 .30902 0 2 Msboxa
[(0.8)] .78571 .30902 0 2 Msboxa
[(1)] .97619 .30902 0 2 Msboxa
[(-0.4)] .01131 .0618 1 0 Msboxa
[(-0.2)] .01131 .18541 1 0 Msboxa
[(0.2)] .01131 .43262 1 0 Msboxa
[(0.4)] .01131 .55623 1 0 Msboxa
[ -0.001 -0.001 0 0 ]
[ 1.001 .61903 0 0 ]
] MathScale
% Start of Graphics
1 setlinecap
1 setlinejoin
newpath
[ ] 0 setdash
0 g
p
p
.002 w
.21429 .30902 m
.21429 .31527 L
s
P
[(0.2)] .21429 .30902 0 2 Mshowa
p
.002 w
.40476 .30902 m
.40476 .31527 L
s
P
[(0.4)] .40476 .30902 0 2 Mshowa
p
.002 w
.59524 .30902 m
.59524 .31527 L
s
P
[(0.6)] .59524 .30902 0 2 Mshowa
p
.002 w
.78571 .30902 m
.78571 .31527 L
s
P
[(0.8)] .78571 .30902 0 2 Mshowa
p
.002 w
.97619 .30902 m
.97619 .31527 L
s
P
[(1)] .97619 .30902 0 2 Mshowa
p
.001 w
.0619 .30902 m
.0619 .31277 L
s
P
p
.001 w
.1 .30902 m
.1 .31277 L
s
P
p
.001 w
.1381 .30902 m
.1381 .31277 L
s
P
p
.001 w
.17619 .30902 m
.17619 .31277 L
s
P
p
.001 w
.25238 .30902 m
.25238 .31277 L
s
P
p
.001 w
.29048 .30902 m
.29048 .31277 L
s
P
p
.001 w
.32857 .30902 m
.32857 .31277 L
s
P
p
.001 w
.36667 .30902 m
.36667 .31277 L
s
P
p
.001 w
.44286 .30902 m
.44286 .31277 L
s
P
p
.001 w
.48095 .30902 m
.48095 .31277 L
s
P
p
.001 w
.51905 .30902 m
.51905 .31277 L
s
P
p
.001 w
.55714 .30902 m
.55714 .31277 L
s
P
p
.001 w
.63333 .30902 m
.63333 .31277 L
s
P
p
.001 w
.67143 .30902 m
.67143 .31277 L
s
P
p
.001 w
.70952 .30902 m
.70952 .31277 L
s
P
p
.001 w
.74762 .30902 m
.74762 .31277 L
s
P
p
.001 w
.82381 .30902 m
.82381 .31277 L
s
P
p
.001 w
.8619 .30902 m
.8619 .31277 L
s
P
p
.001 w
.9 .30902 m
.9 .31277 L
s
P
p
.001 w
.9381 .30902 m
.9381 .31277 L
s
P
p
.002 w
0 .30902 m
1 .30902 L
s
P
p
.002 w
.02381 .0618 m
.03006 .0618 L
s
P
[(-0.4)] .01131 .0618 1 0 Mshowa
p
.002 w
.02381 .18541 m
.03006 .18541 L
s
P
[(-0.2)] .01131 .18541 1 0 Mshowa
p
.002 w
.02381 .43262 m
.03006 .43262 L
s
P
[(0.2)] .01131 .43262 1 0 Mshowa
p
.002 w
.02381 .55623 m
.03006 .55623 L
s
P
[(0.4)] .01131 .55623 1 0 Mshowa
p
.001 w
.02381 .08652 m
.02756 .08652 L
s
P
p
.001 w
.02381 .11125 m
.02756 .11125 L
s
P
p
.001 w
.02381 .13597 m
.02756 .13597 L
s
P
p
.001 w
.02381 .16069 m
.02756 .16069 L
s
P
p
.001 w
.02381 .21013 m
.02756 .21013 L
s
P
p
.001 w
.02381 .23485 m
.02756 .23485 L
s
P
p
.001 w
.02381 .25957 m
.02756 .25957 L
s
P
p
.001 w
.02381 .2843 m
.02756 .2843 L
s
P
p
.001 w
.02381 .33374 m
.02756 .33374 L
s
P
p
.001 w
.02381 .35846 m
.02756 .35846 L
s
P
p
.001 w
.02381 .38318 m
.02756 .38318 L
s
P
p
.001 w
.02381 .4079 m
.02756 .4079 L
s
P
p
.001 w
.02381 .45735 m
.02756 .45735 L
s
P
p
.001 w
.02381 .48207 m
.02756 .48207 L
s
P
p
.001 w
.02381 .50679 m
.02756 .50679 L
s
P
p
.001 w
.02381 .53151 m
.02756 .53151 L
s
P
p
.001 w
.02381 .03708 m
.02756 .03708 L
s
P
p
.001 w
.02381 .01236 m
.02756 .01236 L
s
P
p
.001 w
.02381 .58095 m
.02756 .58095 L
s
P
p
.001 w
.02381 .60567 m
.02756 .60567 L
s
P
p
.002 w
.02381 0 m
.02381 .61803 L
s
P
P
0 0 m
1 0 L
1 .61803 L
0 .61803 L
closepath
clip
newpath
p
p
.004 w
.02381 .30902 m
.06349 .31567 L
.10317 .32184 L
.14286 .3272 L
.18254 .33167 L
.22222 .33525 L
.24206 .33667 L
.25198 .33728 L
.2619 .33781 L
.27183 .33825 L
.28175 .3386 L
.28671 .33874 L
.29167 .33885 L
.29663 .33893 L
.29911 .33896 L
.30035 .33897 L
.30159 .33898 L
.30283 .33899 L
.30407 .339 L
.30531 .339 L
.30655 .339 L
.30779 .339 L
.30903 .339 L
.31027 .339 L
.31151 .33899 L
.31275 .33898 L
.31399 .33897 L
.31647 .33894 L
.31895 .3389 L
.32143 .33886 L
.32639 .33874 L
.33135 .33858 L
.34127 .33815 L
.35119 .33755 L
.36111 .33677 L
.38095 .33468 L
.40079 .33185 L
.42063 .32831 L
.46032 .31941 L
.5 .30902 L
.53968 .29863 L
.57937 .28972 L
.59921 .28618 L
.61905 .28335 L
.62897 .28222 L
.63889 .28126 L
.64881 .28049 L
.65873 .27989 L
.66369 .27965 L
.66865 .27945 L
Mistroke
.67361 .27929 L
.67857 .27918 L
.68105 .27913 L
.68353 .27909 L
.68601 .27906 L
.68725 .27905 L
.68849 .27904 L
.68973 .27904 L
.69097 .27903 L
.69221 .27903 L
.69345 .27903 L
.69469 .27903 L
.69593 .27904 L
.69717 .27904 L
.69841 .27905 L
.70089 .27907 L
.70337 .2791 L
.70833 .27918 L
.71329 .27929 L
.71825 .27943 L
.72817 .27978 L
.7381 .28023 L
.75794 .28136 L
.77778 .28279 L
.81746 .28636 L
.85714 .29083 L
.89683 .2962 L
.93651 .30236 L
.97619 .30902 L
Mfstroke
P
P
% End of Graphics
MathPictureEnd

:[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 282; pictureHeight = 174]
%!
%%Creator: Mathematica
%%AspectRatio: .61803 
MathPictureStart
%% Graphics
/Courier findfont 10  scalefont  setfont
% Scaling calculations
0.0238095 0.952381 0.309017 0.618034 [
[(0.2)] .21429 .30902 0 2 Msboxa
[(0.4)] .40476 .30902 0 2 Msboxa
[(0.6)] .59524 .30902 0 2 Msboxa
[(0.8)] .78571 .30902 0 2 Msboxa
[(1)] .97619 .30902 0 2 Msboxa
[(-0.4)] .01131 .0618 1 0 Msboxa
[(-0.2)] .01131 .18541 1 0 Msboxa
[(0.2)] .01131 .43262 1 0 Msboxa
[(0.4)] .01131 .55623 1 0 Msboxa
[ -0.001 -0.001 0 0 ]
[ 1.001 .61903 0 0 ]
] MathScale
% Start of Graphics
1 setlinecap
1 setlinejoin
newpath
[ ] 0 setdash
0 g
p
p
.002 w
.21429 .30902 m
.21429 .31527 L
s
P
[(0.2)] .21429 .30902 0 2 Mshowa
p
.002 w
.40476 .30902 m
.40476 .31527 L
s
P
[(0.4)] .40476 .30902 0 2 Mshowa
p
.002 w
.59524 .30902 m
.59524 .31527 L
s
P
[(0.6)] .59524 .30902 0 2 Mshowa
p
.002 w
.78571 .30902 m
.78571 .31527 L
s
P
[(0.8)] .78571 .30902 0 2 Mshowa
p
.002 w
.97619 .30902 m
.97619 .31527 L
s
P
[(1)] .97619 .30902 0 2 Mshowa
p
.001 w
.0619 .30902 m
.0619 .31277 L
s
P
p
.001 w
.1 .30902 m
.1 .31277 L
s
P
p
.001 w
.1381 .30902 m
.1381 .31277 L
s
P
p
.001 w
.17619 .30902 m
.17619 .31277 L
s
P
p
.001 w
.25238 .30902 m
.25238 .31277 L
s
P
p
.001 w
.29048 .30902 m
.29048 .31277 L
s
P
p
.001 w
.32857 .30902 m
.32857 .31277 L
s
P
p
.001 w
.36667 .30902 m
.36667 .31277 L
s
P
p
.001 w
.44286 .30902 m
.44286 .31277 L
s
P
p
.001 w
.48095 .30902 m
.48095 .31277 L
s
P
p
.001 w
.51905 .30902 m
.51905 .31277 L
s
P
p
.001 w
.55714 .30902 m
.55714 .31277 L
s
P
p
.001 w
.63333 .30902 m
.63333 .31277 L
s
P
p
.001 w
.67143 .30902 m
.67143 .31277 L
s
P
p
.001 w
.70952 .30902 m
.70952 .31277 L
s
P
p
.001 w
.74762 .30902 m
.74762 .31277 L
s
P
p
.001 w
.82381 .30902 m
.82381 .31277 L
s
P
p
.001 w
.8619 .30902 m
.8619 .31277 L
s
P
p
.001 w
.9 .30902 m
.9 .31277 L
s
P
p
.001 w
.9381 .30902 m
.9381 .31277 L
s
P
p
.002 w
0 .30902 m
1 .30902 L
s
P
p
.002 w
.02381 .0618 m
.03006 .0618 L
s
P
[(-0.4)] .01131 .0618 1 0 Mshowa
p
.002 w
.02381 .18541 m
.03006 .18541 L
s
P
[(-0.2)] .01131 .18541 1 0 Mshowa
p
.002 w
.02381 .43262 m
.03006 .43262 L
s
P
[(0.2)] .01131 .43262 1 0 Mshowa
p
.002 w
.02381 .55623 m
.03006 .55623 L
s
P
[(0.4)] .01131 .55623 1 0 Mshowa
p
.001 w
.02381 .08652 m
.02756 .08652 L
s
P
p
.001 w
.02381 .11125 m
.02756 .11125 L
s
P
p
.001 w
.02381 .13597 m
.02756 .13597 L
s
P
p
.001 w
.02381 .16069 m
.02756 .16069 L
s
P
p
.001 w
.02381 .21013 m
.02756 .21013 L
s
P
p
.001 w
.02381 .23485 m
.02756 .23485 L
s
P
p
.001 w
.02381 .25957 m
.02756 .25957 L
s
P
p
.001 w
.02381 .2843 m
.02756 .2843 L
s
P
p
.001 w
.02381 .33374 m
.02756 .33374 L
s
P
p
.001 w
.02381 .35846 m
.02756 .35846 L
s
P
p
.001 w
.02381 .38318 m
.02756 .38318 L
s
P
p
.001 w
.02381 .4079 m
.02756 .4079 L
s
P
p
.001 w
.02381 .45735 m
.02756 .45735 L
s
P
p
.001 w
.02381 .48207 m
.02756 .48207 L
s
P
p
.001 w
.02381 .50679 m
.02756 .50679 L
s
P
p
.001 w
.02381 .53151 m
.02756 .53151 L
s
P
p
.001 w
.02381 .03708 m
.02756 .03708 L
s
P
p
.001 w
.02381 .01236 m
.02756 .01236 L
s
P
p
.001 w
.02381 .58095 m
.02756 .58095 L
s
P
p
.001 w
.02381 .60567 m
.02756 .60567 L
s
P
p
.002 w
.02381 0 m
.02381 .61803 L
s
P
P
0 0 m
1 0 L
1 .61803 L
0 .61803 L
closepath
clip
newpath
p
p
.004 w
.02381 .30902 m
.06349 .33013 L
.10317 .35077 L
.14286 .37063 L
.18254 .38963 L
.22222 .40775 L
.2619 .42479 L
.30159 .44034 L
.34127 .45385 L
.38095 .46491 L
.40079 .46946 L
.42063 .47336 L
.44048 .47662 L
.46032 .47925 L
.47024 .48033 L
.48016 .48124 L
.49008 .48199 L
.5 .48257 L
.50496 .48279 L
.50992 .48297 L
.5124 .48304 L
.51488 .4831 L
.51736 .48314 L
.51984 .48318 L
.52108 .48319 L
.52232 .4832 L
.52356 .4832 L
.5248 .48321 L
.52604 .48321 L
.52728 .4832 L
.52852 .4832 L
.52976 .48319 L
.531 .48317 L
.53224 .48316 L
.53472 .48311 L
.5372 .48306 L
.53968 .48299 L
.54464 .4828 L
.5496 .48256 L
.55952 .4819 L
.56944 .481 L
.57937 .47983 L
.59921 .47668 L
.61905 .47238 L
.65873 .46033 L
.69841 .44408 L
.7381 .42479 L
.77778 .40401 L
.81746 .38316 L
.85714 .36316 L
Mistroke
.89683 .3443 L
.93651 .32639 L
.97619 .30902 L
Mfstroke
P
P
% End of Graphics
MathPictureEnd

:[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 282; pictureHeight = 174]
%!
%%Creator: Mathematica
%%AspectRatio: .61803 
MathPictureStart
%% Graphics
/Courier findfont 10  scalefont  setfont
% Scaling calculations
0.0238095 0.952381 0.309017 0.618034 [
[(0.2)] .21429 .30902 0 2 Msboxa
[(0.4)] .40476 .30902 0 2 Msboxa
[(0.6)] .59524 .30902 0 2 Msboxa
[(0.8)] .78571 .30902 0 2 Msboxa
[(1)] .97619 .30902 0 2 Msboxa
[(-0.4)] .01131 .0618 1 0 Msboxa
[(-0.2)] .01131 .18541 1 0 Msboxa
[(0.2)] .01131 .43262 1 0 Msboxa
[(0.4)] .01131 .55623 1 0 Msboxa
[ -0.001 -0.001 0 0 ]
[ 1.001 .61903 0 0 ]
] MathScale
% Start of Graphics
1 setlinecap
1 setlinejoin
newpath
[ ] 0 setdash
0 g
p
p
.002 w
.21429 .30902 m
.21429 .31527 L
s
P
[(0.2)] .21429 .30902 0 2 Mshowa
p
.002 w
.40476 .30902 m
.40476 .31527 L
s
P
[(0.4)] .40476 .30902 0 2 Mshowa
p
.002 w
.59524 .30902 m
.59524 .31527 L
s
P
[(0.6)] .59524 .30902 0 2 Mshowa
p
.002 w
.78571 .30902 m
.78571 .31527 L
s
P
[(0.8)] .78571 .30902 0 2 Mshowa
p
.002 w
.97619 .30902 m
.97619 .31527 L
s
P
[(1)] .97619 .30902 0 2 Mshowa
p
.001 w
.0619 .30902 m
.0619 .31277 L
s
P
p
.001 w
.1 .30902 m
.1 .31277 L
s
P
p
.001 w
.1381 .30902 m
.1381 .31277 L
s
P
p
.001 w
.17619 .30902 m
.17619 .31277 L
s
P
p
.001 w
.25238 .30902 m
.25238 .31277 L
s
P
p
.001 w
.29048 .30902 m
.29048 .31277 L
s
P
p
.001 w
.32857 .30902 m
.32857 .31277 L
s
P
p
.001 w
.36667 .30902 m
.36667 .31277 L
s
P
p
.001 w
.44286 .30902 m
.44286 .31277 L
s
P
p
.001 w
.48095 .30902 m
.48095 .31277 L
s
P
p
.001 w
.51905 .30902 m
.51905 .31277 L
s
P
p
.001 w
.55714 .30902 m
.55714 .31277 L
s
P
p
.001 w
.63333 .30902 m
.63333 .31277 L
s
P
p
.001 w
.67143 .30902 m
.67143 .31277 L
s
P
p
.001 w
.70952 .30902 m
.70952 .31277 L
s
P
p
.001 w
.74762 .30902 m
.74762 .31277 L
s
P
p
.001 w
.82381 .30902 m
.82381 .31277 L
s
P
p
.001 w
.8619 .30902 m
.8619 .31277 L
s
P
p
.001 w
.9 .30902 m
.9 .31277 L
s
P
p
.001 w
.9381 .30902 m
.9381 .31277 L
s
P
p
.002 w
0 .30902 m
1 .30902 L
s
P
p
.002 w
.02381 .0618 m
.03006 .0618 L
s
P
[(-0.4)] .01131 .0618 1 0 Mshowa
p
.002 w
.02381 .18541 m
.03006 .18541 L
s
P
[(-0.2)] .01131 .18541 1 0 Mshowa
p
.002 w
.02381 .43262 m
.03006 .43262 L
s
P
[(0.2)] .01131 .43262 1 0 Mshowa
p
.002 w
.02381 .55623 m
.03006 .55623 L
s
P
[(0.4)] .01131 .55623 1 0 Mshowa
p
.001 w
.02381 .08652 m
.02756 .08652 L
s
P
p
.001 w
.02381 .11125 m
.02756 .11125 L
s
P
p
.001 w
.02381 .13597 m
.02756 .13597 L
s
P
p
.001 w
.02381 .16069 m
.02756 .16069 L
s
P
p
.001 w
.02381 .21013 m
.02756 .21013 L
s
P
p
.001 w
.02381 .23485 m
.02756 .23485 L
s
P
p
.001 w
.02381 .25957 m
.02756 .25957 L
s
P
p
.001 w
.02381 .2843 m
.02756 .2843 L
s
P
p
.001 w
.02381 .33374 m
.02756 .33374 L
s
P
p
.001 w
.02381 .35846 m
.02756 .35846 L
s
P
p
.001 w
.02381 .38318 m
.02756 .38318 L
s
P
p
.001 w
.02381 .4079 m
.02756 .4079 L
s
P
p
.001 w
.02381 .45735 m
.02756 .45735 L
s
P
p
.001 w
.02381 .48207 m
.02756 .48207 L
s
P
p
.001 w
.02381 .50679 m
.02756 .50679 L
s
P
p
.001 w
.02381 .53151 m
.02756 .53151 L
s
P
p
.001 w
.02381 .03708 m
.02756 .03708 L
s
P
p
.001 w
.02381 .01236 m
.02756 .01236 L
s
P
p
.001 w
.02381 .58095 m
.02756 .58095 L
s
P
p
.001 w
.02381 .60567 m
.02756 .60567 L
s
P
p
.002 w
.02381 0 m
.02381 .61803 L
s
P
P
0 0 m
1 0 L
1 .61803 L
0 .61803 L
closepath
clip
newpath
p
p
.004 w
.02381 .30902 m
.06349 .33496 L
.10317 .36042 L
.14286 .38511 L
.18254 .40895 L
.22222 .4319 L
.2619 .45378 L
.30159 .47414 L
.34127 .49245 L
.38095 .50833 L
.42063 .52165 L
.46032 .53242 L
.5 .54056 L
.51984 .54355 L
.52976 .54474 L
.53968 .54572 L
.54464 .54613 L
.5496 .54648 L
.55456 .54677 L
.55952 .54699 L
.562 .54709 L
.56448 .54716 L
.56696 .54722 L
.56944 .54727 L
.57068 .54729 L
.57192 .5473 L
.57316 .54731 L
.5744 .54731 L
.57564 .54731 L
.57688 .54731 L
.57813 .5473 L
.57937 .54729 L
.58061 .54727 L
.58185 .54725 L
.58433 .5472 L
.58681 .54713 L
.58929 .54705 L
.59425 .54683 L
.59921 .54654 L
.60913 .54576 L
.61905 .54471 L
.63889 .54178 L
.65873 .53777 L
.69841 .52659 L
.7381 .51136 L
.77778 .49184 L
.81746 .46721 L
.85714 .43643 L
.89683 .399 L
.93651 .35574 L
Mistroke
.97619 .30902 L
Mfstroke
P
P
% End of Graphics
MathPictureEnd

:[font = output; output; inactive; preserveAspect; endGroup]
{Graphics["<<>>"], Graphics["<<>>"], Graphics["<<>>"], 
 
  Graphics["<<>>"], Graphics["<<>>"], Graphics["<<>>"], 
 
  Graphics["<<>>"], Graphics["<<>>"], Graphics["<<>>"]}
;[o]
{-Graphics-, -Graphics-, -Graphics-, -Graphics-, 
 
  -Graphics-, -Graphics-, -Graphics-, -Graphics-, 
 
  -Graphics-}
:[font = text; inactive; preserveAspect; endGroup]
 If you are reading this notebook with <I>Mathematica</I> or MathReader you may
select the <BR>
graphics just calculated and animate them to see the string vibrate.
:[font = section; inactive; Cclosed; preserveAspect; startGroup]
Model Problem VI.1.  
Solve the problem of the vibrating string for 0 < x < 1, c=1, 
         u[t,0] = u[t,1] = 0
using Fourier sine series, and initial conditions 
              u(x,0) = 0, 
              u_t(x,0) = 1
;[s]
2:0,0;22,1;219,-1;
2:1,19,14,New York,1,14,0,0,0;1,17,12,New York,0,12,0,0,0;
:[font = input; preserveAspect]
TrigId = {Cos[Pi n_] -> (-1)^n, Sin[Pi n_] -> 0}
:[font = text; inactive; preserveAspect]
The A coefficients are clearly zero this time, even without the aid of software.  As for the B coefficients, 
:[font = input; preserveAspect]
g[x_] := 1
:[font = input; preserveAspect; startGroup]
B[n,g]
:[font = output; output; inactive; preserveAspect; endGroup]
(2*(1/(n*Pi) - Cos[n*Pi]/(n*Pi)))/(n*Pi)
;[o]
    1     Cos[n Pi]
2 (---- - ---------)
   n Pi     n Pi
--------------------
        n Pi
:[font = input; preserveAspect; startGroup]
% /. TrigId
:[font = output; output; inactive; preserveAspect; endGroup]
(2*(1/(n*Pi) - (-1)^n/(n*Pi)))/(n*Pi)
;[o]
              n
    1     (-1)
2 (---- - -----)
   n Pi   n Pi
----------------
      n Pi
:[font = input; preserveAspect]
  U[t_,x_,infinity_] := \
        Sum[Sin[n Pi t] Sin[n Pi x] 4 /(n^2 Pi^2), \
           {n,1,infinity, 2}]
:[font = text; inactive; preserveAspect]
The final 2 here ensures that only odd terms are included.  The even coefficients are 0
:[font = input; preserveAspect; startGroup]
  Table[Plot[U[t/4,x,7], {x, 0, 1}, 
        PlotRange -> {-.5,.5}], {t,0,8}]
:[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 282; pictureHeight = 174]
%!
%%Creator: Mathematica
%%AspectRatio: .61803 
MathPictureStart
%% Graphics
/Courier findfont 10  scalefont  setfont
% Scaling calculations
0.0238095 0.952381 0.309017 0.618034 [
[(0.2)] .21429 .30902 0 2 Msboxa
[(0.4)] .40476 .30902 0 2 Msboxa
[(0.6)] .59524 .30902 0 2 Msboxa
[(0.8)] .78571 .30902 0 2 Msboxa
[(1)] .97619 .30902 0 2 Msboxa
[(-0.4)] .01131 .0618 1 0 Msboxa
[(-0.2)] .01131 .18541 1 0 Msboxa
[(0.2)] .01131 .43262 1 0 Msboxa
[(0.4)] .01131 .55623 1 0 Msboxa
[ -0.001 -0.001 0 0 ]
[ 1.001 .61903 0 0 ]
] MathScale
% Start of Graphics
1 setlinecap
1 setlinejoin
newpath
[ ] 0 setdash
0 g
p
p
.002 w
.21429 .30902 m
.21429 .31527 L
s
P
[(0.2)] .21429 .30902 0 2 Mshowa
p
.002 w
.40476 .30902 m
.40476 .31527 L
s
P
[(0.4)] .40476 .30902 0 2 Mshowa
p
.002 w
.59524 .30902 m
.59524 .31527 L
s
P
[(0.6)] .59524 .30902 0 2 Mshowa
p
.002 w
.78571 .30902 m
.78571 .31527 L
s
P
[(0.8)] .78571 .30902 0 2 Mshowa
p
.002 w
.97619 .30902 m
.97619 .31527 L
s
P
[(1)] .97619 .30902 0 2 Mshowa
p
.001 w
.0619 .30902 m
.0619 .31277 L
s
P
p
.001 w
.1 .30902 m
.1 .31277 L
s
P
p
.001 w
.1381 .30902 m
.1381 .31277 L
s
P
p
.001 w
.17619 .30902 m
.17619 .31277 L
s
P
p
.001 w
.25238 .30902 m
.25238 .31277 L
s
P
p
.001 w
.29048 .30902 m
.29048 .31277 L
s
P
p
.001 w
.32857 .30902 m
.32857 .31277 L
s
P
p
.001 w
.36667 .30902 m
.36667 .31277 L
s
P
p
.001 w
.44286 .30902 m
.44286 .31277 L
s
P
p
.001 w
.48095 .30902 m
.48095 .31277 L
s
P
p
.001 w
.51905 .30902 m
.51905 .31277 L
s
P
p
.001 w
.55714 .30902 m
.55714 .31277 L
s
P
p
.001 w
.63333 .30902 m
.63333 .31277 L
s
P
p
.001 w
.67143 .30902 m
.67143 .31277 L
s
P
p
.001 w
.70952 .30902 m
.70952 .31277 L
s
P
p
.001 w
.74762 .30902 m
.74762 .31277 L
s
P
p
.001 w
.82381 .30902 m
.82381 .31277 L
s
P
p
.001 w
.8619 .30902 m
.8619 .31277 L
s
P
p
.001 w
.9 .30902 m
.9 .31277 L
s
P
p
.001 w
.9381 .30902 m
.9381 .31277 L
s
P
p
.002 w
0 .30902 m
1 .30902 L
s
P
p
.002 w
.02381 .0618 m
.03006 .0618 L
s
P
[(-0.4)] .01131 .0618 1 0 Mshowa
p
.002 w
.02381 .18541 m
.03006 .18541 L
s
P
[(-0.2)] .01131 .18541 1 0 Mshowa
p
.002 w
.02381 .43262 m
.03006 .43262 L
s
P
[(0.2)] .01131 .43262 1 0 Mshowa
p
.002 w
.02381 .55623 m
.03006 .55623 L
s
P
[(0.4)] .01131 .55623 1 0 Mshowa
p
.001 w
.02381 .08652 m
.02756 .08652 L
s
P
p
.001 w
.02381 .11125 m
.02756 .11125 L
s
P
p
.001 w
.02381 .13597 m
.02756 .13597 L
s
P
p
.001 w
.02381 .16069 m
.02756 .16069 L
s
P
p
.001 w
.02381 .21013 m
.02756 .21013 L
s
P
p
.001 w
.02381 .23485 m
.02756 .23485 L
s
P
p
.001 w
.02381 .25957 m
.02756 .25957 L
s
P
p
.001 w
.02381 .2843 m
.02756 .2843 L
s
P
p
.001 w
.02381 .33374 m
.02756 .33374 L
s
P
p
.001 w
.02381 .35846 m
.02756 .35846 L
s
P
p
.001 w
.02381 .38318 m
.02756 .38318 L
s
P
p
.001 w
.02381 .4079 m
.02756 .4079 L
s
P
p
.001 w
.02381 .45735 m
.02756 .45735 L
s
P
p
.001 w
.02381 .48207 m
.02756 .48207 L
s
P
p
.001 w
.02381 .50679 m
.02756 .50679 L
s
P
p
.001 w
.02381 .53151 m
.02756 .53151 L
s
P
p
.001 w
.02381 .03708 m
.02756 .03708 L
s
P
p
.001 w
.02381 .01236 m
.02756 .01236 L
s
P
p
.001 w
.02381 .58095 m
.02756 .58095 L
s
P
p
.001 w
.02381 .60567 m
.02756 .60567 L
s
P
p
.002 w
.02381 0 m
.02381 .61803 L
s
P
P
0 0 m
1 0 L
1 .61803 L
0 .61803 L
closepath
clip
newpath
p
p
.004 w
.02381 .30902 m
.06349 .30902 L
.10317 .30902 L
.14286 .30902 L
.18254 .30902 L
.22222 .30902 L
.2619 .30902 L
.30159 .30902 L
.34127 .30902 L
.38095 .30902 L
.42063 .30902 L
.46032 .30902 L
.5 .30902 L
.53968 .30902 L
.57937 .30902 L
.61905 .30902 L
.65873 .30902 L
.69841 .30902 L
.7381 .30902 L
.77778 .30902 L
.81746 .30902 L
.85714 .30902 L
.89683 .30902 L
.93651 .30902 L
.97619 .30902 L
s
P
P
% End of Graphics
MathPictureEnd

:[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 282; pictureHeight = 174]
%!
%%Creator: Mathematica
%%AspectRatio: .61803 
MathPictureStart
%% Graphics
/Courier findfont 10  scalefont  setfont
% Scaling calculations
0.0238095 0.952381 0.309017 0.618034 [
[(0.2)] .21429 .30902 0 2 Msboxa
[(0.4)] .40476 .30902 0 2 Msboxa
[(0.6)] .59524 .30902 0 2 Msboxa
[(0.8)] .78571 .30902 0 2 Msboxa
[(1)] .97619 .30902 0 2 Msboxa
[(-0.4)] .01131 .0618 1 0 Msboxa
[(-0.2)] .01131 .18541 1 0 Msboxa
[(0.2)] .01131 .43262 1 0 Msboxa
[(0.4)] .01131 .55623 1 0 Msboxa
[ -0.001 -0.001 0 0 ]
[ 1.001 .61903 0 0 ]
] MathScale
% Start of Graphics
1 setlinecap
1 setlinejoin
newpath
[ ] 0 setdash
0 g
p
p
.002 w
.21429 .30902 m
.21429 .31527 L
s
P
[(0.2)] .21429 .30902 0 2 Mshowa
p
.002 w
.40476 .30902 m
.40476 .31527 L
s
P
[(0.4)] .40476 .30902 0 2 Mshowa
p
.002 w
.59524 .30902 m
.59524 .31527 L
s
P
[(0.6)] .59524 .30902 0 2 Mshowa
p
.002 w
.78571 .30902 m
.78571 .31527 L
s
P
[(0.8)] .78571 .30902 0 2 Mshowa
p
.002 w
.97619 .30902 m
.97619 .31527 L
s
P
[(1)] .97619 .30902 0 2 Mshowa
p
.001 w
.0619 .30902 m
.0619 .31277 L
s
P
p
.001 w
.1 .30902 m
.1 .31277 L
s
P
p
.001 w
.1381 .30902 m
.1381 .31277 L
s
P
p
.001 w
.17619 .30902 m
.17619 .31277 L
s
P
p
.001 w
.25238 .30902 m
.25238 .31277 L
s
P
p
.001 w
.29048 .30902 m
.29048 .31277 L
s
P
p
.001 w
.32857 .30902 m
.32857 .31277 L
s
P
p
.001 w
.36667 .30902 m
.36667 .31277 L
s
P
p
.001 w
.44286 .30902 m
.44286 .31277 L
s
P
p
.001 w
.48095 .30902 m
.48095 .31277 L
s
P
p
.001 w
.51905 .30902 m
.51905 .31277 L
s
P
p
.001 w
.55714 .30902 m
.55714 .31277 L
s
P
p
.001 w
.63333 .30902 m
.63333 .31277 L
s
P
p
.001 w
.67143 .30902 m
.67143 .31277 L
s
P
p
.001 w
.70952 .30902 m
.70952 .31277 L
s
P
p
.001 w
.74762 .30902 m
.74762 .31277 L
s
P
p
.001 w
.82381 .30902 m
.82381 .31277 L
s
P
p
.001 w
.8619 .30902 m
.8619 .31277 L
s
P
p
.001 w
.9 .30902 m
.9 .31277 L
s
P
p
.001 w
.9381 .30902 m
.9381 .31277 L
s
P
p
.002 w
0 .30902 m
1 .30902 L
s
P
p
.002 w
.02381 .0618 m
.03006 .0618 L
s
P
[(-0.4)] .01131 .0618 1 0 Mshowa
p
.002 w
.02381 .18541 m
.03006 .18541 L
s
P
[(-0.2)] .01131 .18541 1 0 Mshowa
p
.002 w
.02381 .43262 m
.03006 .43262 L
s
P
[(0.2)] .01131 .43262 1 0 Mshowa
p
.002 w
.02381 .55623 m
.03006 .55623 L
s
P
[(0.4)] .01131 .55623 1 0 Mshowa
p
.001 w
.02381 .08652 m
.02756 .08652 L
s
P
p
.001 w
.02381 .11125 m
.02756 .11125 L
s
P
p
.001 w
.02381 .13597 m
.02756 .13597 L
s
P
p
.001 w
.02381 .16069 m
.02756 .16069 L
s
P
p
.001 w
.02381 .21013 m
.02756 .21013 L
s
P
p
.001 w
.02381 .23485 m
.02756 .23485 L
s
P
p
.001 w
.02381 .25957 m
.02756 .25957 L
s
P
p
.001 w
.02381 .2843 m
.02756 .2843 L
s
P
p
.001 w
.02381 .33374 m
.02756 .33374 L
s
P
p
.001 w
.02381 .35846 m
.02756 .35846 L
s
P
p
.001 w
.02381 .38318 m
.02756 .38318 L
s
P
p
.001 w
.02381 .4079 m
.02756 .4079 L
s
P
p
.001 w
.02381 .45735 m
.02756 .45735 L
s
P
p
.001 w
.02381 .48207 m
.02756 .48207 L
s
P
p
.001 w
.02381 .50679 m
.02756 .50679 L
s
P
p
.001 w
.02381 .53151 m
.02756 .53151 L
s
P
p
.001 w
.02381 .03708 m
.02756 .03708 L
s
P
p
.001 w
.02381 .01236 m
.02756 .01236 L
s
P
p
.001 w
.02381 .58095 m
.02756 .58095 L
s
P
p
.001 w
.02381 .60567 m
.02756 .60567 L
s
P
p
.002 w
.02381 0 m
.02381 .61803 L
s
P
P
0 0 m
1 0 L
1 .61803 L
0 .61803 L
closepath
clip
newpath
p
p
.004 w
.02381 .30902 m
.06349 .33249 L
.10317 .35844 L
.14286 .38705 L
.18254 .41552 L
.20238 .42834 L
.22222 .43953 L
.24206 .44872 L
.25198 .4525 L
.2619 .45574 L
.27183 .45844 L
.28175 .46063 L
.29167 .46234 L
.29663 .46303 L
.30159 .46362 L
.31151 .46451 L
.31647 .46482 L
.32143 .46506 L
.32639 .46524 L
.32887 .4653 L
.33135 .46535 L
.33383 .46538 L
.33507 .46539 L
.33631 .4654 L
.33755 .46541 L
.33879 .46541 L
.34003 .46541 L
.34127 .46541 L
.34251 .4654 L
.34375 .4654 L
.34623 .46538 L
.34871 .46535 L
.35119 .46531 L
.36111 .46509 L
.38095 .46449 L
.40079 .46388 L
.41071 .46362 L
.42063 .46341 L
.43056 .46325 L
.44048 .46314 L
.44544 .46309 L
.4504 .46306 L
.45536 .46303 L
.46032 .46302 L
.4628 .46301 L
.46528 .463 L
.46776 .463 L
.47024 .46299 L
.47272 .46299 L
.4752 .46299 L
Mistroke
.47768 .46299 L
.47892 .46299 L
.48016 .46299 L
.4814 .46298 L
.48264 .46298 L
.48388 .46298 L
.48512 .46298 L
.48636 .46298 L
.4876 .46298 L
.48884 .46298 L
.49008 .46298 L
.49132 .46298 L
.49256 .46298 L
.4938 .46298 L
.49504 .46298 L
.49628 .46298 L
.49752 .46298 L
.49876 .46298 L
.5 .46298 L
.50124 .46298 L
.50248 .46298 L
.50372 .46298 L
.50496 .46298 L
.5062 .46298 L
.50744 .46298 L
.50868 .46298 L
.50992 .46298 L
.51116 .46298 L
.5124 .46298 L
.51364 .46298 L
.51488 .46298 L
.51612 .46298 L
.51736 .46298 L
.51984 .46299 L
.52108 .46299 L
.52232 .46299 L
.5248 .46299 L
.52728 .46299 L
.52976 .46299 L
.53224 .463 L
.53472 .463 L
.53968 .46302 L
.54464 .46303 L
.5496 .46306 L
.55456 .46309 L
.55952 .46314 L
.56944 .46325 L
.57937 .46341 L
.58929 .46362 L
.59921 .46388 L
Mistroke
.61905 .46449 L
.62897 .46481 L
.63889 .46509 L
.64385 .46521 L
.64881 .46531 L
.65129 .46535 L
.65377 .46538 L
.65501 .46539 L
.65625 .4654 L
.65749 .4654 L
.65873 .46541 L
.65997 .46541 L
.66121 .46541 L
.66245 .46541 L
.66369 .4654 L
.66493 .46539 L
.66617 .46538 L
.66865 .46535 L
.67113 .4653 L
.67361 .46524 L
.67857 .46506 L
.68353 .46482 L
.68849 .46451 L
.69345 .46411 L
.69841 .46362 L
.70833 .46234 L
.71825 .46063 L
.72817 .45844 L
.7381 .45574 L
.75794 .44872 L
.77778 .43953 L
.81746 .41552 L
.85714 .38705 L
.89683 .35844 L
.93651 .33249 L
.97619 .30902 L
Mfstroke
P
P
% End of Graphics
MathPictureEnd

:[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 282; pictureHeight = 174]
%!
%%Creator: Mathematica
%%AspectRatio: .61803 
MathPictureStart
%% Graphics
/Courier findfont 10  scalefont  setfont
% Scaling calculations
0.0238095 0.952381 0.309017 0.618034 [
[(0.2)] .21429 .30902 0 2 Msboxa
[(0.4)] .40476 .30902 0 2 Msboxa
[(0.6)] .59524 .30902 0 2 Msboxa
[(0.8)] .78571 .30902 0 2 Msboxa
[(1)] .97619 .30902 0 2 Msboxa
[(-0.4)] .01131 .0618 1 0 Msboxa
[(-0.2)] .01131 .18541 1 0 Msboxa
[(0.2)] .01131 .43262 1 0 Msboxa
[(0.4)] .01131 .55623 1 0 Msboxa
[ -0.001 -0.001 0 0 ]
[ 1.001 .61903 0 0 ]
] MathScale
% Start of Graphics
1 setlinecap
1 setlinejoin
newpath
[ ] 0 setdash
0 g
p
p
.002 w
.21429 .30902 m
.21429 .31527 L
s
P
[(0.2)] .21429 .30902 0 2 Mshowa
p
.002 w
.40476 .30902 m
.40476 .31527 L
s
P
[(0.4)] .40476 .30902 0 2 Mshowa
p
.002 w
.59524 .30902 m
.59524 .31527 L
s
P
[(0.6)] .59524 .30902 0 2 Mshowa
p
.002 w
.78571 .30902 m
.78571 .31527 L
s
P
[(0.8)] .78571 .30902 0 2 Mshowa
p
.002 w
.97619 .30902 m
.97619 .31527 L
s
P
[(1)] .97619 .30902 0 2 Mshowa
p
.001 w
.0619 .30902 m
.0619 .31277 L
s
P
p
.001 w
.1 .30902 m
.1 .31277 L
s
P
p
.001 w
.1381 .30902 m
.1381 .31277 L
s
P
p
.001 w
.17619 .30902 m
.17619 .31277 L
s
P
p
.001 w
.25238 .30902 m
.25238 .31277 L
s
P
p
.001 w
.29048 .30902 m
.29048 .31277 L
s
P
p
.001 w
.32857 .30902 m
.32857 .31277 L
s
P
p
.001 w
.36667 .30902 m
.36667 .31277 L
s
P
p
.001 w
.44286 .30902 m
.44286 .31277 L
s
P
p
.001 w
.48095 .30902 m
.48095 .31277 L
s
P
p
.001 w
.51905 .30902 m
.51905 .31277 L
s
P
p
.001 w
.55714 .30902 m
.55714 .31277 L
s
P
p
.001 w
.63333 .30902 m
.63333 .31277 L
s
P
p
.001 w
.67143 .30902 m
.67143 .31277 L
s
P
p
.001 w
.70952 .30902 m
.70952 .31277 L
s
P
p
.001 w
.74762 .30902 m
.74762 .31277 L
s
P
p
.001 w
.82381 .30902 m
.82381 .31277 L
s
P
p
.001 w
.8619 .30902 m
.8619 .31277 L
s
P
p
.001 w
.9 .30902 m
.9 .31277 L
s
P
p
.001 w
.9381 .30902 m
.9381 .31277 L
s
P
p
.002 w
0 .30902 m
1 .30902 L
s
P
p
.002 w
.02381 .0618 m
.03006 .0618 L
s
P
[(-0.4)] .01131 .0618 1 0 Mshowa
p
.002 w
.02381 .18541 m
.03006 .18541 L
s
P
[(-0.2)] .01131 .18541 1 0 Mshowa
p
.002 w
.02381 .43262 m
.03006 .43262 L
s
P
[(0.2)] .01131 .43262 1 0 Mshowa
p
.002 w
.02381 .55623 m
.03006 .55623 L
s
P
[(0.4)] .01131 .55623 1 0 Mshowa
p
.001 w
.02381 .08652 m
.02756 .08652 L
s
P
p
.001 w
.02381 .11125 m
.02756 .11125 L
s
P
p
.001 w
.02381 .13597 m
.02756 .13597 L
s
P
p
.001 w
.02381 .16069 m
.02756 .16069 L
s
P
p
.001 w
.02381 .21013 m
.02756 .21013 L
s
P
p
.001 w
.02381 .23485 m
.02756 .23485 L
s
P
p
.001 w
.02381 .25957 m
.02756 .25957 L
s
P
p
.001 w
.02381 .2843 m
.02756 .2843 L
s
P
p
.001 w
.02381 .33374 m
.02756 .33374 L
s
P
p
.001 w
.02381 .35846 m
.02756 .35846 L
s
P
p
.001 w
.02381 .38318 m
.02756 .38318 L
s
P
p
.001 w
.02381 .4079 m
.02756 .4079 L
s
P
p
.001 w
.02381 .45735 m
.02756 .45735 L
s
P
p
.001 w
.02381 .48207 m
.02756 .48207 L
s
P
p
.001 w
.02381 .50679 m
.02756 .50679 L
s
P
p
.001 w
.02381 .53151 m
.02756 .53151 L
s
P
p
.001 w
.02381 .03708 m
.02756 .03708 L
s
P
p
.001 w
.02381 .01236 m
.02756 .01236 L
s
P
p
.001 w
.02381 .58095 m
.02756 .58095 L
s
P
p
.001 w
.02381 .60567 m
.02756 .60567 L
s
P
p
.002 w
.02381 0 m
.02381 .61803 L
s
P
P
0 0 m
1 0 L
1 .61803 L
0 .61803 L
closepath
clip
newpath
p
p
.004 w
.02381 .30902 m
.06349 .3331 L
.10317 .35891 L
.14286 .38646 L
.18254 .41399 L
.22222 .43955 L
.2619 .46298 L
.30159 .48648 L
.34127 .51283 L
.38095 .54252 L
.42063 .57191 L
.44048 .58439 L
.4504 .58965 L
.46032 .59413 L
.47024 .59771 L
.4752 .59914 L
.48016 .60033 L
.48512 .60126 L
.4876 .60162 L
.49008 .60192 L
.49256 .60216 L
.4938 .60225 L
.49504 .60232 L
.49628 .60238 L
.49752 .60243 L
.49876 .60245 L
.5 .60246 L
.50124 .60245 L
.50248 .60243 L
.50372 .60238 L
.50496 .60232 L
.5062 .60225 L
.50744 .60216 L
.50992 .60192 L
.5124 .60162 L
.51488 .60126 L
.51984 .60033 L
.52976 .59771 L
.53968 .59413 L
.55952 .58439 L
.57937 .57191 L
.61905 .54252 L
.65873 .51283 L
.69841 .48648 L
.7381 .46298 L
.77778 .43955 L
.81746 .41399 L
.85714 .38646 L
.89683 .35891 L
.93651 .3331 L
Mistroke
.97619 .30902 L
Mfstroke
P
P
% End of Graphics
MathPictureEnd

:[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 282; pictureHeight = 174]
%!
%%Creator: Mathematica
%%AspectRatio: .61803 
MathPictureStart
%% Graphics
/Courier findfont 10  scalefont  setfont
% Scaling calculations
0.0238095 0.952381 0.309017 0.618034 [
[(0.2)] .21429 .30902 0 2 Msboxa
[(0.4)] .40476 .30902 0 2 Msboxa
[(0.6)] .59524 .30902 0 2 Msboxa
[(0.8)] .78571 .30902 0 2 Msboxa
[(1)] .97619 .30902 0 2 Msboxa
[(-0.4)] .01131 .0618 1 0 Msboxa
[(-0.2)] .01131 .18541 1 0 Msboxa
[(0.2)] .01131 .43262 1 0 Msboxa
[(0.4)] .01131 .55623 1 0 Msboxa
[ -0.001 -0.001 0 0 ]
[ 1.001 .61903 0 0 ]
] MathScale
% Start of Graphics
1 setlinecap
1 setlinejoin
newpath
[ ] 0 setdash
0 g
p
p
.002 w
.21429 .30902 m
.21429 .31527 L
s
P
[(0.2)] .21429 .30902 0 2 Mshowa
p
.002 w
.40476 .30902 m
.40476 .31527 L
s
P
[(0.4)] .40476 .30902 0 2 Mshowa
p
.002 w
.59524 .30902 m
.59524 .31527 L
s
P
[(0.6)] .59524 .30902 0 2 Mshowa
p
.002 w
.78571 .30902 m
.78571 .31527 L
s
P
[(0.8)] .78571 .30902 0 2 Mshowa
p
.002 w
.97619 .30902 m
.97619 .31527 L
s
P
[(1)] .97619 .30902 0 2 Mshowa
p
.001 w
.0619 .30902 m
.0619 .31277 L
s
P
p
.001 w
.1 .30902 m
.1 .31277 L
s
P
p
.001 w
.1381 .30902 m
.1381 .31277 L
s
P
p
.001 w
.17619 .30902 m
.17619 .31277 L
s
P
p
.001 w
.25238 .30902 m
.25238 .31277 L
s
P
p
.001 w
.29048 .30902 m
.29048 .31277 L
s
P
p
.001 w
.32857 .30902 m
.32857 .31277 L
s
P
p
.001 w
.36667 .30902 m
.36667 .31277 L
s
P
p
.001 w
.44286 .30902 m
.44286 .31277 L
s
P
p
.001 w
.48095 .30902 m
.48095 .31277 L
s
P
p
.001 w
.51905 .30902 m
.51905 .31277 L
s
P
p
.001 w
.55714 .30902 m
.55714 .31277 L
s
P
p
.001 w
.63333 .30902 m
.63333 .31277 L
s
P
p
.001 w
.67143 .30902 m
.67143 .31277 L
s
P
p
.001 w
.70952 .30902 m
.70952 .31277 L
s
P
p
.001 w
.74762 .30902 m
.74762 .31277 L
s
P
p
.001 w
.82381 .30902 m
.82381 .31277 L
s
P
p
.001 w
.8619 .30902 m
.8619 .31277 L
s
P
p
.001 w
.9 .30902 m
.9 .31277 L
s
P
p
.001 w
.9381 .30902 m
.9381 .31277 L
s
P
p
.002 w
0 .30902 m
1 .30902 L
s
P
p
.002 w
.02381 .0618 m
.03006 .0618 L
s
P
[(-0.4)] .01131 .0618 1 0 Mshowa
p
.002 w
.02381 .18541 m
.03006 .18541 L
s
P
[(-0.2)] .01131 .18541 1 0 Mshowa
p
.002 w
.02381 .43262 m
.03006 .43262 L
s
P
[(0.2)] .01131 .43262 1 0 Mshowa
p
.002 w
.02381 .55623 m
.03006 .55623 L
s
P
[(0.4)] .01131 .55623 1 0 Mshowa
p
.001 w
.02381 .08652 m
.02756 .08652 L
s
P
p
.001 w
.02381 .11125 m
.02756 .11125 L
s
P
p
.001 w
.02381 .13597 m
.02756 .13597 L
s
P
p
.001 w
.02381 .16069 m
.02756 .16069 L
s
P
p
.001 w
.02381 .21013 m
.02756 .21013 L
s
P
p
.001 w
.02381 .23485 m
.02756 .23485 L
s
P
p
.001 w
.02381 .25957 m
.02756 .25957 L
s
P
p
.001 w
.02381 .2843 m
.02756 .2843 L
s
P
p
.001 w
.02381 .33374 m
.02756 .33374 L
s
P
p
.001 w
.02381 .35846 m
.02756 .35846 L
s
P
p
.001 w
.02381 .38318 m
.02756 .38318 L
s
P
p
.001 w
.02381 .4079 m
.02756 .4079 L
s
P
p
.001 w
.02381 .45735 m
.02756 .45735 L
s
P
p
.001 w
.02381 .48207 m
.02756 .48207 L
s
P
p
.001 w
.02381 .50679 m
.02756 .50679 L
s
P
p
.001 w
.02381 .53151 m
.02756 .53151 L
s
P
p
.001 w
.02381 .03708 m
.02756 .03708 L
s
P
p
.001 w
.02381 .01236 m
.02756 .01236 L
s
P
p
.001 w
.02381 .58095 m
.02756 .58095 L
s
P
p
.001 w
.02381 .60567 m
.02756 .60567 L
s
P
p
.002 w
.02381 0 m
.02381 .61803 L
s
P
P
0 0 m
1 0 L
1 .61803 L
0 .61803 L
closepath
clip
newpath
p
p
.004 w
.02381 .30902 m
.06349 .33249 L
.10317 .35844 L
.14286 .38705 L
.18254 .41552 L
.20238 .42834 L
.22222 .43953 L
.24206 .44872 L
.25198 .4525 L
.2619 .45574 L
.27183 .45844 L
.28175 .46063 L
.29167 .46234 L
.29663 .46303 L
.30159 .46362 L
.31151 .46451 L
.31647 .46482 L
.32143 .46506 L
.32639 .46524 L
.32887 .4653 L
.33135 .46535 L
.33383 .46538 L
.33507 .46539 L
.33631 .4654 L
.33755 .46541 L
.33879 .46541 L
.34003 .46541 L
.34127 .46541 L
.34251 .4654 L
.34375 .4654 L
.34623 .46538 L
.34871 .46535 L
.35119 .46531 L
.36111 .46509 L
.38095 .46449 L
.40079 .46388 L
.41071 .46362 L
.42063 .46341 L
.43056 .46325 L
.44048 .46314 L
.44544 .46309 L
.4504 .46306 L
.45536 .46303 L
.46032 .46302 L
.4628 .46301 L
.46528 .463 L
.46776 .463 L
.47024 .46299 L
.47272 .46299 L
.4752 .46299 L
Mistroke
.47768 .46299 L
.47892 .46299 L
.48016 .46299 L
.4814 .46298 L
.48264 .46298 L
.48388 .46298 L
.48512 .46298 L
.48636 .46298 L
.4876 .46298 L
.48884 .46298 L
.49008 .46298 L
.49132 .46298 L
.49256 .46298 L
.4938 .46298 L
.49504 .46298 L
.49628 .46298 L
.49752 .46298 L
.49876 .46298 L
.5 .46298 L
.50124 .46298 L
.50248 .46298 L
.50372 .46298 L
.50496 .46298 L
.5062 .46298 L
.50744 .46298 L
.50868 .46298 L
.50992 .46298 L
.51116 .46298 L
.5124 .46298 L
.51364 .46298 L
.51488 .46298 L
.51612 .46298 L
.51736 .46298 L
.51984 .46299 L
.52108 .46299 L
.52232 .46299 L
.5248 .46299 L
.52728 .46299 L
.52976 .46299 L
.53224 .463 L
.53472 .463 L
.53968 .46302 L
.54464 .46303 L
.5496 .46306 L
.55456 .46309 L
.55952 .46314 L
.56944 .46325 L
.57937 .46341 L
.58929 .46362 L
.59921 .46388 L
Mistroke
.61905 .46449 L
.62897 .46481 L
.63889 .46509 L
.64385 .46521 L
.64881 .46531 L
.65129 .46535 L
.65377 .46538 L
.65501 .46539 L
.65625 .4654 L
.65749 .4654 L
.65873 .46541 L
.65997 .46541 L
.66121 .46541 L
.66245 .46541 L
.66369 .4654 L
.66493 .46539 L
.66617 .46538 L
.66865 .46535 L
.67113 .4653 L
.67361 .46524 L
.67857 .46506 L
.68353 .46482 L
.68849 .46451 L
.69345 .46411 L
.69841 .46362 L
.70833 .46234 L
.71825 .46063 L
.72817 .45844 L
.7381 .45574 L
.75794 .44872 L
.77778 .43953 L
.81746 .41552 L
.85714 .38705 L
.89683 .35844 L
.93651 .33249 L
.97619 .30902 L
Mfstroke
P
P
% End of Graphics
MathPictureEnd

:[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 282; pictureHeight = 174]
%!
%%Creator: Mathematica
%%AspectRatio: .61803 
MathPictureStart
%% Graphics
/Courier findfont 10  scalefont  setfont
% Scaling calculations
0.0238095 0.952381 0.309017 0.618034 [
[(0.2)] .21429 .30902 0 2 Msboxa
[(0.4)] .40476 .30902 0 2 Msboxa
[(0.6)] .59524 .30902 0 2 Msboxa
[(0.8)] .78571 .30902 0 2 Msboxa
[(1)] .97619 .30902 0 2 Msboxa
[(-0.4)] .01131 .0618 1 0 Msboxa
[(-0.2)] .01131 .18541 1 0 Msboxa
[(0.2)] .01131 .43262 1 0 Msboxa
[(0.4)] .01131 .55623 1 0 Msboxa
[ -0.001 -0.001 0 0 ]
[ 1.001 .61903 0 0 ]
] MathScale
% Start of Graphics
1 setlinecap
1 setlinejoin
newpath
[ ] 0 setdash
0 g
p
p
.002 w
.21429 .30902 m
.21429 .31527 L
s
P
[(0.2)] .21429 .30902 0 2 Mshowa
p
.002 w
.40476 .30902 m
.40476 .31527 L
s
P
[(0.4)] .40476 .30902 0 2 Mshowa
p
.002 w
.59524 .30902 m
.59524 .31527 L
s
P
[(0.6)] .59524 .30902 0 2 Mshowa
p
.002 w
.78571 .30902 m
.78571 .31527 L
s
P
[(0.8)] .78571 .30902 0 2 Mshowa
p
.002 w
.97619 .30902 m
.97619 .31527 L
s
P
[(1)] .97619 .30902 0 2 Mshowa
p
.001 w
.0619 .30902 m
.0619 .31277 L
s
P
p
.001 w
.1 .30902 m
.1 .31277 L
s
P
p
.001 w
.1381 .30902 m
.1381 .31277 L
s
P
p
.001 w
.17619 .30902 m
.17619 .31277 L
s
P
p
.001 w
.25238 .30902 m
.25238 .31277 L
s
P
p
.001 w
.29048 .30902 m
.29048 .31277 L
s
P
p
.001 w
.32857 .30902 m
.32857 .31277 L
s
P
p
.001 w
.36667 .30902 m
.36667 .31277 L
s
P
p
.001 w
.44286 .30902 m
.44286 .31277 L
s
P
p
.001 w
.48095 .30902 m
.48095 .31277 L
s
P
p
.001 w
.51905 .30902 m
.51905 .31277 L
s
P
p
.001 w
.55714 .30902 m
.55714 .31277 L
s
P
p
.001 w
.63333 .30902 m
.63333 .31277 L
s
P
p
.001 w
.67143 .30902 m
.67143 .31277 L
s
P
p
.001 w
.70952 .30902 m
.70952 .31277 L
s
P
p
.001 w
.74762 .30902 m
.74762 .31277 L
s
P
p
.001 w
.82381 .30902 m
.82381 .31277 L
s
P
p
.001 w
.8619 .30902 m
.8619 .31277 L
s
P
p
.001 w
.9 .30902 m
.9 .31277 L
s
P
p
.001 w
.9381 .30902 m
.9381 .31277 L
s
P
p
.002 w
0 .30902 m
1 .30902 L
s
P
p
.002 w
.02381 .0618 m
.03006 .0618 L
s
P
[(-0.4)] .01131 .0618 1 0 Mshowa
p
.002 w
.02381 .18541 m
.03006 .18541 L
s
P
[(-0.2)] .01131 .18541 1 0 Mshowa
p
.002 w
.02381 .43262 m
.03006 .43262 L
s
P
[(0.2)] .01131 .43262 1 0 Mshowa
p
.002 w
.02381 .55623 m
.03006 .55623 L
s
P
[(0.4)] .01131 .55623 1 0 Mshowa
p
.001 w
.02381 .08652 m
.02756 .08652 L
s
P
p
.001 w
.02381 .11125 m
.02756 .11125 L
s
P
p
.001 w
.02381 .13597 m
.02756 .13597 L
s
P
p
.001 w
.02381 .16069 m
.02756 .16069 L
s
P
p
.001 w
.02381 .21013 m
.02756 .21013 L
s
P
p
.001 w
.02381 .23485 m
.02756 .23485 L
s
P
p
.001 w
.02381 .25957 m
.02756 .25957 L
s
P
p
.001 w
.02381 .2843 m
.02756 .2843 L
s
P
p
.001 w
.02381 .33374 m
.02756 .33374 L
s
P
p
.001 w
.02381 .35846 m
.02756 .35846 L
s
P
p
.001 w
.02381 .38318 m
.02756 .38318 L
s
P
p
.001 w
.02381 .4079 m
.02756 .4079 L
s
P
p
.001 w
.02381 .45735 m
.02756 .45735 L
s
P
p
.001 w
.02381 .48207 m
.02756 .48207 L
s
P
p
.001 w
.02381 .50679 m
.02756 .50679 L
s
P
p
.001 w
.02381 .53151 m
.02756 .53151 L
s
P
p
.001 w
.02381 .03708 m
.02756 .03708 L
s
P
p
.001 w
.02381 .01236 m
.02756 .01236 L
s
P
p
.001 w
.02381 .58095 m
.02756 .58095 L
s
P
p
.001 w
.02381 .60567 m
.02756 .60567 L
s
P
p
.002 w
.02381 0 m
.02381 .61803 L
s
P
P
0 0 m
1 0 L
1 .61803 L
0 .61803 L
closepath
clip
newpath
p
p
.004 w
.02381 .30902 m
.06349 .30902 L
.10317 .30902 L
.14286 .30902 L
.1627 .30902 L
.17262 .30902 L
.18254 .30902 L
.1875 .30902 L
.19246 .30902 L
.19742 .30902 L
.20238 .30902 L
.20486 .30902 L
.20734 .30902 L
.20982 .30902 L
.21106 .30902 L
.2123 .30902 L
.21354 .30902 L
.21478 .30902 L
.21602 .30902 L
.21726 .30902 L
.2185 .30902 L
.21974 .30902 L
.22098 .30902 L
.22222 .30902 L
.2247 .30902 L
.22718 .30902 L
.23214 .30902 L
.2371 .30902 L
.24206 .30902 L
.25198 .30902 L
.2619 .30902 L
.30159 .30902 L
.34127 .30902 L
.36111 .30902 L
.38095 .30902 L
.39087 .30902 L
.40079 .30902 L
.41071 .30902 L
.42063 .30902 L
.4256 .30902 L
.43056 .30902 L
.44048 .30902 L
.44544 .30902 L
.4504 .30902 L
.45536 .30902 L
.45784 .30902 L
.46032 .30902 L
.4628 .30902 L
.46528 .30902 L
.46776 .30902 L
Mistroke
.47024 .30902 L
.47272 .30902 L
.4752 .30902 L
.47768 .30902 L
.47892 .30902 L
.48016 .30902 L
.4814 .30902 L
.48264 .30902 L
.48388 .30902 L
.48512 .30902 L
.48636 .30902 L
.4876 .30902 L
.48884 .30902 L
.49008 .30902 L
.49132 .30902 L
.49256 .30902 L
.4938 .30902 L
.49504 .30902 L
.49628 .30902 L
.49752 .30902 L
.49876 .30902 L
.5 .30902 L
.50124 .30902 L
.50248 .30902 L
.50372 .30902 L
.50496 .30902 L
.5062 .30902 L
.50744 .30902 L
.50868 .30902 L
.50992 .30902 L
.51116 .30902 L
.5124 .30902 L
.51364 .30902 L
.51488 .30902 L
.51612 .30902 L
.51736 .30902 L
.51984 .30902 L
.52108 .30902 L
.52232 .30902 L
.5248 .30902 L
.52728 .30902 L
.52976 .30902 L
.53224 .30902 L
.53472 .30902 L
.5372 .30902 L
.53968 .30902 L
.54464 .30902 L
.5496 .30902 L
.55456 .30902 L
.55952 .30902 L
Mistroke
.56944 .30902 L
.5744 .30902 L
.57937 .30902 L
.58929 .30902 L
.59921 .30902 L
.61905 .30902 L
.63889 .30902 L
.65873 .30902 L
.69841 .30902 L
.71825 .30902 L
.72817 .30902 L
.7381 .30902 L
.74802 .30902 L
.75794 .30902 L
.7629 .30902 L
.76786 .30902 L
.77282 .30902 L
.7753 .30902 L
.77654 .30902 L
.77778 .30902 L
.77902 .30902 L
.78026 .30902 L
.7815 .30902 L
.78274 .30902 L
.78398 .30902 L
.78522 .30902 L
.78646 .30902 L
.7877 .30902 L
.79018 .30902 L
.79266 .30902 L
.79514 .30902 L
.79762 .30902 L
.80258 .30902 L
.80754 .30902 L
.81746 .30902 L
.82738 .30902 L
.8373 .30902 L
.85714 .30902 L
.89683 .30902 L
.93651 .30902 L
.97619 .30902 L
Mfstroke
P
P
% End of Graphics
MathPictureEnd

:[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 282; pictureHeight = 174]
%!
%%Creator: Mathematica
%%AspectRatio: .61803 
MathPictureStart
%% Graphics
/Courier findfont 10  scalefont  setfont
% Scaling calculations
0.0238095 0.952381 0.309017 0.618034 [
[(0.2)] .21429 .30902 0 2 Msboxa
[(0.4)] .40476 .30902 0 2 Msboxa
[(0.6)] .59524 .30902 0 2 Msboxa
[(0.8)] .78571 .30902 0 2 Msboxa
[(1)] .97619 .30902 0 2 Msboxa
[(-0.4)] .01131 .0618 1 0 Msboxa
[(-0.2)] .01131 .18541 1 0 Msboxa
[(0.2)] .01131 .43262 1 0 Msboxa
[(0.4)] .01131 .55623 1 0 Msboxa
[ -0.001 -0.001 0 0 ]
[ 1.001 .61903 0 0 ]
] MathScale
% Start of Graphics
1 setlinecap
1 setlinejoin
newpath
[ ] 0 setdash
0 g
p
p
.002 w
.21429 .30902 m
.21429 .31527 L
s
P
[(0.2)] .21429 .30902 0 2 Mshowa
p
.002 w
.40476 .30902 m
.40476 .31527 L
s
P
[(0.4)] .40476 .30902 0 2 Mshowa
p
.002 w
.59524 .30902 m
.59524 .31527 L
s
P
[(0.6)] .59524 .30902 0 2 Mshowa
p
.002 w
.78571 .30902 m
.78571 .31527 L
s
P
[(0.8)] .78571 .30902 0 2 Mshowa
p
.002 w
.97619 .30902 m
.97619 .31527 L
s
P
[(1)] .97619 .30902 0 2 Mshowa
p
.001 w
.0619 .30902 m
.0619 .31277 L
s
P
p
.001 w
.1 .30902 m
.1 .31277 L
s
P
p
.001 w
.1381 .30902 m
.1381 .31277 L
s
P
p
.001 w
.17619 .30902 m
.17619 .31277 L
s
P
p
.001 w
.25238 .30902 m
.25238 .31277 L
s
P
p
.001 w
.29048 .30902 m
.29048 .31277 L
s
P
p
.001 w
.32857 .30902 m
.32857 .31277 L
s
P
p
.001 w
.36667 .30902 m
.36667 .31277 L
s
P
p
.001 w
.44286 .30902 m
.44286 .31277 L
s
P
p
.001 w
.48095 .30902 m
.48095 .31277 L
s
P
p
.001 w
.51905 .30902 m
.51905 .31277 L
s
P
p
.001 w
.55714 .30902 m
.55714 .31277 L
s
P
p
.001 w
.63333 .30902 m
.63333 .31277 L
s
P
p
.001 w
.67143 .30902 m
.67143 .31277 L
s
P
p
.001 w
.70952 .30902 m
.70952 .31277 L
s
P
p
.001 w
.74762 .30902 m
.74762 .31277 L
s
P
p
.001 w
.82381 .30902 m
.82381 .31277 L
s
P
p
.001 w
.8619 .30902 m
.8619 .31277 L
s
P
p
.001 w
.9 .30902 m
.9 .31277 L
s
P
p
.001 w
.9381 .30902 m
.9381 .31277 L
s
P
p
.002 w
0 .30902 m
1 .30902 L
s
P
p
.002 w
.02381 .0618 m
.03006 .0618 L
s
P
[(-0.4)] .01131 .0618 1 0 Mshowa
p
.002 w
.02381 .18541 m
.03006 .18541 L
s
P
[(-0.2)] .01131 .18541 1 0 Mshowa
p
.002 w
.02381 .43262 m
.03006 .43262 L
s
P
[(0.2)] .01131 .43262 1 0 Mshowa
p
.002 w
.02381 .55623 m
.03006 .55623 L
s
P
[(0.4)] .01131 .55623 1 0 Mshowa
p
.001 w
.02381 .08652 m
.02756 .08652 L
s
P
p
.001 w
.02381 .11125 m
.02756 .11125 L
s
P
p
.001 w
.02381 .13597 m
.02756 .13597 L
s
P
p
.001 w
.02381 .16069 m
.02756 .16069 L
s
P
p
.001 w
.02381 .21013 m
.02756 .21013 L
s
P
p
.001 w
.02381 .23485 m
.02756 .23485 L
s
P
p
.001 w
.02381 .25957 m
.02756 .25957 L
s
P
p
.001 w
.02381 .2843 m
.02756 .2843 L
s
P
p
.001 w
.02381 .33374 m
.02756 .33374 L
s
P
p
.001 w
.02381 .35846 m
.02756 .35846 L
s
P
p
.001 w
.02381 .38318 m
.02756 .38318 L
s
P
p
.001 w
.02381 .4079 m
.02756 .4079 L
s
P
p
.001 w
.02381 .45735 m
.02756 .45735 L
s
P
p
.001 w
.02381 .48207 m
.02756 .48207 L
s
P
p
.001 w
.02381 .50679 m
.02756 .50679 L
s
P
p
.001 w
.02381 .53151 m
.02756 .53151 L
s
P
p
.001 w
.02381 .03708 m
.02756 .03708 L
s
P
p
.001 w
.02381 .01236 m
.02756 .01236 L
s
P
p
.001 w
.02381 .58095 m
.02756 .58095 L
s
P
p
.001 w
.02381 .60567 m
.02756 .60567 L
s
P
p
.002 w
.02381 0 m
.02381 .61803 L
s
P
P
0 0 m
1 0 L
1 .61803 L
0 .61803 L
closepath
clip
newpath
p
p
.004 w
.02381 .30902 m
.06349 .28555 L
.10317 .2596 L
.14286 .23098 L
.18254 .20251 L
.20238 .18969 L
.22222 .17851 L
.24206 .16932 L
.25198 .16553 L
.2619 .1623 L
.27183 .15959 L
.28175 .1574 L
.29167 .15569 L
.29663 .155 L
.30159 .15442 L
.31151 .15353 L
.31647 .15321 L
.32143 .15297 L
.32639 .1528 L
.32887 .15274 L
.33135 .15269 L
.33383 .15265 L
.33507 .15264 L
.33631 .15263 L
.33755 .15263 L
.33879 .15262 L
.34003 .15262 L
.34127 .15262 L
.34251 .15263 L
.34375 .15264 L
.34623 .15266 L
.34871 .15269 L
.35119 .15273 L
.36111 .15294 L
.38095 .15354 L
.40079 .15415 L
.41071 .15441 L
.42063 .15462 L
.43056 .15478 L
.44048 .1549 L
.44544 .15494 L
.4504 .15497 L
.45536 .155 L
.46032 .15502 L
.4628 .15503 L
.46528 .15503 L
.46776 .15504 L
.47024 .15504 L
.47272 .15504 L
.4752 .15505 L
Mistroke
.47768 .15505 L
.47892 .15505 L
.48016 .15505 L
.4814 .15505 L
.48264 .15505 L
.48388 .15505 L
.48512 .15505 L
.48636 .15505 L
.4876 .15505 L
.48884 .15505 L
.49008 .15505 L
.49132 .15505 L
.49256 .15505 L
.4938 .15505 L
.49504 .15505 L
.49628 .15505 L
.49752 .15505 L
.49876 .15505 L
.5 .15505 L
.50124 .15505 L
.50248 .15505 L
.50372 .15505 L
.50496 .15505 L
.5062 .15505 L
.50744 .15505 L
.50868 .15505 L
.50992 .15505 L
.51116 .15505 L
.5124 .15505 L
.51364 .15505 L
.51488 .15505 L
.51612 .15505 L
.51736 .15505 L
.51984 .15505 L
.52108 .15505 L
.52232 .15505 L
.5248 .15505 L
.52728 .15504 L
.52976 .15504 L
.53224 .15504 L
.53472 .15503 L
.53968 .15502 L
.54464 .155 L
.5496 .15497 L
.55456 .15494 L
.55952 .1549 L
.56944 .15478 L
.57937 .15462 L
.58929 .15441 L
.59921 .15415 L
Mistroke
.61905 .15354 L
.62897 .15323 L
.63889 .15294 L
.64385 .15282 L
.64881 .15273 L
.65129 .15269 L
.65377 .15266 L
.65501 .15265 L
.65625 .15264 L
.65749 .15263 L
.65873 .15262 L
.65997 .15262 L
.66121 .15262 L
.66245 .15263 L
.66369 .15263 L
.66493 .15264 L
.66617 .15265 L
.66865 .15269 L
.67113 .15274 L
.67361 .1528 L
.67857 .15297 L
.68353 .15321 L
.68849 .15353 L
.69345 .15393 L
.69841 .15442 L
.70833 .15569 L
.71825 .1574 L
.72817 .15959 L
.7381 .1623 L
.75794 .16932 L
.77778 .17851 L
.81746 .20251 L
.85714 .23098 L
.89683 .2596 L
.93651 .28555 L
.97619 .30902 L
Mfstroke
P
P
% End of Graphics
MathPictureEnd

:[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 282; pictureHeight = 174]
%!
%%Creator: Mathematica
%%AspectRatio: .61803 
MathPictureStart
%% Graphics
/Courier findfont 10  scalefont  setfont
% Scaling calculations
0.0238095 0.952381 0.309017 0.618034 [
[(0.2)] .21429 .30902 0 2 Msboxa
[(0.4)] .40476 .30902 0 2 Msboxa
[(0.6)] .59524 .30902 0 2 Msboxa
[(0.8)] .78571 .30902 0 2 Msboxa
[(1)] .97619 .30902 0 2 Msboxa
[(-0.4)] .01131 .0618 1 0 Msboxa
[(-0.2)] .01131 .18541 1 0 Msboxa
[(0.2)] .01131 .43262 1 0 Msboxa
[(0.4)] .01131 .55623 1 0 Msboxa
[ -0.001 -0.001 0 0 ]
[ 1.001 .61903 0 0 ]
] MathScale
% Start of Graphics
1 setlinecap
1 setlinejoin
newpath
[ ] 0 setdash
0 g
p
p
.002 w
.21429 .30902 m
.21429 .31527 L
s
P
[(0.2)] .21429 .30902 0 2 Mshowa
p
.002 w
.40476 .30902 m
.40476 .31527 L
s
P
[(0.4)] .40476 .30902 0 2 Mshowa
p
.002 w
.59524 .30902 m
.59524 .31527 L
s
P
[(0.6)] .59524 .30902 0 2 Mshowa
p
.002 w
.78571 .30902 m
.78571 .31527 L
s
P
[(0.8)] .78571 .30902 0 2 Mshowa
p
.002 w
.97619 .30902 m
.97619 .31527 L
s
P
[(1)] .97619 .30902 0 2 Mshowa
p
.001 w
.0619 .30902 m
.0619 .31277 L
s
P
p
.001 w
.1 .30902 m
.1 .31277 L
s
P
p
.001 w
.1381 .30902 m
.1381 .31277 L
s
P
p
.001 w
.17619 .30902 m
.17619 .31277 L
s
P
p
.001 w
.25238 .30902 m
.25238 .31277 L
s
P
p
.001 w
.29048 .30902 m
.29048 .31277 L
s
P
p
.001 w
.32857 .30902 m
.32857 .31277 L
s
P
p
.001 w
.36667 .30902 m
.36667 .31277 L
s
P
p
.001 w
.44286 .30902 m
.44286 .31277 L
s
P
p
.001 w
.48095 .30902 m
.48095 .31277 L
s
P
p
.001 w
.51905 .30902 m
.51905 .31277 L
s
P
p
.001 w
.55714 .30902 m
.55714 .31277 L
s
P
p
.001 w
.63333 .30902 m
.63333 .31277 L
s
P
p
.001 w
.67143 .30902 m
.67143 .31277 L
s
P
p
.001 w
.70952 .30902 m
.70952 .31277 L
s
P
p
.001 w
.74762 .30902 m
.74762 .31277 L
s
P
p
.001 w
.82381 .30902 m
.82381 .31277 L
s
P
p
.001 w
.8619 .30902 m
.8619 .31277 L
s
P
p
.001 w
.9 .30902 m
.9 .31277 L
s
P
p
.001 w
.9381 .30902 m
.9381 .31277 L
s
P
p
.002 w
0 .30902 m
1 .30902 L
s
P
p
.002 w
.02381 .0618 m
.03006 .0618 L
s
P
[(-0.4)] .01131 .0618 1 0 Mshowa
p
.002 w
.02381 .18541 m
.03006 .18541 L
s
P
[(-0.2)] .01131 .18541 1 0 Mshowa
p
.002 w
.02381 .43262 m
.03006 .43262 L
s
P
[(0.2)] .01131 .43262 1 0 Mshowa
p
.002 w
.02381 .55623 m
.03006 .55623 L
s
P
[(0.4)] .01131 .55623 1 0 Mshowa
p
.001 w
.02381 .08652 m
.02756 .08652 L
s
P
p
.001 w
.02381 .11125 m
.02756 .11125 L
s
P
p
.001 w
.02381 .13597 m
.02756 .13597 L
s
P
p
.001 w
.02381 .16069 m
.02756 .16069 L
s
P
p
.001 w
.02381 .21013 m
.02756 .21013 L
s
P
p
.001 w
.02381 .23485 m
.02756 .23485 L
s
P
p
.001 w
.02381 .25957 m
.02756 .25957 L
s
P
p
.001 w
.02381 .2843 m
.02756 .2843 L
s
P
p
.001 w
.02381 .33374 m
.02756 .33374 L
s
P
p
.001 w
.02381 .35846 m
.02756 .35846 L
s
P
p
.001 w
.02381 .38318 m
.02756 .38318 L
s
P
p
.001 w
.02381 .4079 m
.02756 .4079 L
s
P
p
.001 w
.02381 .45735 m
.02756 .45735 L
s
P
p
.001 w
.02381 .48207 m
.02756 .48207 L
s
P
p
.001 w
.02381 .50679 m
.02756 .50679 L
s
P
p
.001 w
.02381 .53151 m
.02756 .53151 L
s
P
p
.001 w
.02381 .03708 m
.02756 .03708 L
s
P
p
.001 w
.02381 .01236 m
.02756 .01236 L
s
P
p
.001 w
.02381 .58095 m
.02756 .58095 L
s
P
p
.001 w
.02381 .60567 m
.02756 .60567 L
s
P
p
.002 w
.02381 0 m
.02381 .61803 L
s
P
P
0 0 m
1 0 L
1 .61803 L
0 .61803 L
closepath
clip
newpath
p
p
.004 w
.02381 .30902 m
.06349 .28493 L
.10317 .25913 L
.14286 .23157 L
.18254 .20404 L
.22222 .17849 L
.2619 .15505 L
.30159 .13155 L
.34127 .1052 L
.38095 .07551 L
.42063 .04612 L
.44048 .03365 L
.4504 .02838 L
.46032 .02391 L
.47024 .02032 L
.4752 .01889 L
.48016 .01771 L
.48512 .01678 L
.4876 .01641 L
.49008 .01611 L
.49256 .01588 L
.4938 .01578 L
.49504 .01571 L
.49628 .01565 L
.49752 .01561 L
.49876 .01558 L
.5 .01558 L
.50124 .01558 L
.50248 .01561 L
.50372 .01565 L
.50496 .01571 L
.5062 .01578 L
.50744 .01588 L
.50992 .01611 L
.5124 .01641 L
.51488 .01678 L
.51984 .01771 L
.52976 .02032 L
.53968 .02391 L
.55952 .03365 L
.57937 .04612 L
.61905 .07551 L
.65873 .1052 L
.69841 .13155 L
.7381 .15505 L
.77778 .17849 L
.81746 .20404 L
.85714 .23157 L
.89683 .25913 L
.93651 .28493 L
Mistroke
.97619 .30902 L
Mfstroke
P
P
% End of Graphics
MathPictureEnd

:[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 282; pictureHeight = 174]
%!
%%Creator: Mathematica
%%AspectRatio: .61803 
MathPictureStart
%% Graphics
/Courier findfont 10  scalefont  setfont
% Scaling calculations
0.0238095 0.952381 0.309017 0.618034 [
[(0.2)] .21429 .30902 0 2 Msboxa
[(0.4)] .40476 .30902 0 2 Msboxa
[(0.6)] .59524 .30902 0 2 Msboxa
[(0.8)] .78571 .30902 0 2 Msboxa
[(1)] .97619 .30902 0 2 Msboxa
[(-0.4)] .01131 .0618 1 0 Msboxa
[(-0.2)] .01131 .18541 1 0 Msboxa
[(0.2)] .01131 .43262 1 0 Msboxa
[(0.4)] .01131 .55623 1 0 Msboxa
[ -0.001 -0.001 0 0 ]
[ 1.001 .61903 0 0 ]
] MathScale
% Start of Graphics
1 setlinecap
1 setlinejoin
newpath
[ ] 0 setdash
0 g
p
p
.002 w
.21429 .30902 m
.21429 .31527 L
s
P
[(0.2)] .21429 .30902 0 2 Mshowa
p
.002 w
.40476 .30902 m
.40476 .31527 L
s
P
[(0.4)] .40476 .30902 0 2 Mshowa
p
.002 w
.59524 .30902 m
.59524 .31527 L
s
P
[(0.6)] .59524 .30902 0 2 Mshowa
p
.002 w
.78571 .30902 m
.78571 .31527 L
s
P
[(0.8)] .78571 .30902 0 2 Mshowa
p
.002 w
.97619 .30902 m
.97619 .31527 L
s
P
[(1)] .97619 .30902 0 2 Mshowa
p
.001 w
.0619 .30902 m
.0619 .31277 L
s
P
p
.001 w
.1 .30902 m
.1 .31277 L
s
P
p
.001 w
.1381 .30902 m
.1381 .31277 L
s
P
p
.001 w
.17619 .30902 m
.17619 .31277 L
s
P
p
.001 w
.25238 .30902 m
.25238 .31277 L
s
P
p
.001 w
.29048 .30902 m
.29048 .31277 L
s
P
p
.001 w
.32857 .30902 m
.32857 .31277 L
s
P
p
.001 w
.36667 .30902 m
.36667 .31277 L
s
P
p
.001 w
.44286 .30902 m
.44286 .31277 L
s
P
p
.001 w
.48095 .30902 m
.48095 .31277 L
s
P
p
.001 w
.51905 .30902 m
.51905 .31277 L
s
P
p
.001 w
.55714 .30902 m
.55714 .31277 L
s
P
p
.001 w
.63333 .30902 m
.63333 .31277 L
s
P
p
.001 w
.67143 .30902 m
.67143 .31277 L
s
P
p
.001 w
.70952 .30902 m
.70952 .31277 L
s
P
p
.001 w
.74762 .30902 m
.74762 .31277 L
s
P
p
.001 w
.82381 .30902 m
.82381 .31277 L
s
P
p
.001 w
.8619 .30902 m
.8619 .31277 L
s
P
p
.001 w
.9 .30902 m
.9 .31277 L
s
P
p
.001 w
.9381 .30902 m
.9381 .31277 L
s
P
p
.002 w
0 .30902 m
1 .30902 L
s
P
p
.002 w
.02381 .0618 m
.03006 .0618 L
s
P
[(-0.4)] .01131 .0618 1 0 Mshowa
p
.002 w
.02381 .18541 m
.03006 .18541 L
s
P
[(-0.2)] .01131 .18541 1 0 Mshowa
p
.002 w
.02381 .43262 m
.03006 .43262 L
s
P
[(0.2)] .01131 .43262 1 0 Mshowa
p
.002 w
.02381 .55623 m
.03006 .55623 L
s
P
[(0.4)] .01131 .55623 1 0 Mshowa
p
.001 w
.02381 .08652 m
.02756 .08652 L
s
P
p
.001 w
.02381 .11125 m
.02756 .11125 L
s
P
p
.001 w
.02381 .13597 m
.02756 .13597 L
s
P
p
.001 w
.02381 .16069 m
.02756 .16069 L
s
P
p
.001 w
.02381 .21013 m
.02756 .21013 L
s
P
p
.001 w
.02381 .23485 m
.02756 .23485 L
s
P
p
.001 w
.02381 .25957 m
.02756 .25957 L
s
P
p
.001 w
.02381 .2843 m
.02756 .2843 L
s
P
p
.001 w
.02381 .33374 m
.02756 .33374 L
s
P
p
.001 w
.02381 .35846 m
.02756 .35846 L
s
P
p
.001 w
.02381 .38318 m
.02756 .38318 L
s
P
p
.001 w
.02381 .4079 m
.02756 .4079 L
s
P
p
.001 w
.02381 .45735 m
.02756 .45735 L
s
P
p
.001 w
.02381 .48207 m
.02756 .48207 L
s
P
p
.001 w
.02381 .50679 m
.02756 .50679 L
s
P
p
.001 w
.02381 .53151 m
.02756 .53151 L
s
P
p
.001 w
.02381 .03708 m
.02756 .03708 L
s
P
p
.001 w
.02381 .01236 m
.02756 .01236 L
s
P
p
.001 w
.02381 .58095 m
.02756 .58095 L
s
P
p
.001 w
.02381 .60567 m
.02756 .60567 L
s
P
p
.002 w
.02381 0 m
.02381 .61803 L
s
P
P
0 0 m
1 0 L
1 .61803 L
0 .61803 L
closepath
clip
newpath
p
p
.004 w
.02381 .30902 m
.06349 .28555 L
.10317 .2596 L
.14286 .23098 L
.18254 .20251 L
.20238 .18969 L
.22222 .17851 L
.24206 .16932 L
.25198 .16553 L
.2619 .1623 L
.27183 .15959 L
.28175 .1574 L
.29167 .15569 L
.29663 .155 L
.30159 .15442 L
.31151 .15353 L
.31647 .15321 L
.32143 .15297 L
.32639 .1528 L
.32887 .15274 L
.33135 .15269 L
.33383 .15265 L
.33507 .15264 L
.33631 .15263 L
.33755 .15263 L
.33879 .15262 L
.34003 .15262 L
.34127 .15262 L
.34251 .15263 L
.34375 .15264 L
.34623 .15266 L
.34871 .15269 L
.35119 .15273 L
.36111 .15294 L
.38095 .15354 L
.40079 .15415 L
.41071 .15441 L
.42063 .15462 L
.43056 .15478 L
.44048 .1549 L
.44544 .15494 L
.4504 .15497 L
.45536 .155 L
.46032 .15502 L
.4628 .15503 L
.46528 .15503 L
.46776 .15504 L
.47024 .15504 L
.47272 .15504 L
.4752 .15505 L
Mistroke
.47768 .15505 L
.47892 .15505 L
.48016 .15505 L
.4814 .15505 L
.48264 .15505 L
.48388 .15505 L
.48512 .15505 L
.48636 .15505 L
.4876 .15505 L
.48884 .15505 L
.49008 .15505 L
.49132 .15505 L
.49256 .15505 L
.4938 .15505 L
.49504 .15505 L
.49628 .15505 L
.49752 .15505 L
.49876 .15505 L
.5 .15505 L
.50124 .15505 L
.50248 .15505 L
.50372 .15505 L
.50496 .15505 L
.5062 .15505 L
.50744 .15505 L
.50868 .15505 L
.50992 .15505 L
.51116 .15505 L
.5124 .15505 L
.51364 .15505 L
.51488 .15505 L
.51612 .15505 L
.51736 .15505 L
.51984 .15505 L
.52108 .15505 L
.52232 .15505 L
.5248 .15505 L
.52728 .15504 L
.52976 .15504 L
.53224 .15504 L
.53472 .15503 L
.53968 .15502 L
.54464 .155 L
.5496 .15497 L
.55456 .15494 L
.55952 .1549 L
.56944 .15478 L
.57937 .15462 L
.58929 .15441 L
.59921 .15415 L
Mistroke
.61905 .15354 L
.62897 .15323 L
.63889 .15294 L
.64385 .15282 L
.64881 .15273 L
.65129 .15269 L
.65377 .15266 L
.65501 .15265 L
.65625 .15264 L
.65749 .15263 L
.65873 .15262 L
.65997 .15262 L
.66121 .15262 L
.66245 .15263 L
.66369 .15263 L
.66493 .15264 L
.66617 .15265 L
.66865 .15269 L
.67113 .15274 L
.67361 .1528 L
.67857 .15297 L
.68353 .15321 L
.68849 .15353 L
.69345 .15393 L
.69841 .15442 L
.70833 .15569 L
.71825 .1574 L
.72817 .15959 L
.7381 .1623 L
.75794 .16932 L
.77778 .17851 L
.81746 .20251 L
.85714 .23098 L
.89683 .2596 L
.93651 .28555 L
.97619 .30902 L
Mfstroke
P
P
% End of Graphics
MathPictureEnd

:[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 282; pictureHeight = 174]
%!
%%Creator: Mathematica
%%AspectRatio: .61803 
MathPictureStart
%% Graphics
/Courier findfont 10  scalefont  setfont
% Scaling calculations
0.0238095 0.952381 0.309017 0.618034 [
[(0.2)] .21429 .30902 0 2 Msboxa
[(0.4)] .40476 .30902 0 2 Msboxa
[(0.6)] .59524 .30902 0 2 Msboxa
[(0.8)] .78571 .30902 0 2 Msboxa
[(1)] .97619 .30902 0 2 Msboxa
[(-0.4)] .01131 .0618 1 0 Msboxa
[(-0.2)] .01131 .18541 1 0 Msboxa
[(0.2)] .01131 .43262 1 0 Msboxa
[(0.4)] .01131 .55623 1 0 Msboxa
[ -0.001 -0.001 0 0 ]
[ 1.001 .61903 0 0 ]
] MathScale
% Start of Graphics
1 setlinecap
1 setlinejoin
newpath
[ ] 0 setdash
0 g
p
p
.002 w
.21429 .30902 m
.21429 .31527 L
s
P
[(0.2)] .21429 .30902 0 2 Mshowa
p
.002 w
.40476 .30902 m
.40476 .31527 L
s
P
[(0.4)] .40476 .30902 0 2 Mshowa
p
.002 w
.59524 .30902 m
.59524 .31527 L
s
P
[(0.6)] .59524 .30902 0 2 Mshowa
p
.002 w
.78571 .30902 m
.78571 .31527 L
s
P
[(0.8)] .78571 .30902 0 2 Mshowa
p
.002 w
.97619 .30902 m
.97619 .31527 L
s
P
[(1)] .97619 .30902 0 2 Mshowa
p
.001 w
.0619 .30902 m
.0619 .31277 L
s
P
p
.001 w
.1 .30902 m
.1 .31277 L
s
P
p
.001 w
.1381 .30902 m
.1381 .31277 L
s
P
p
.001 w
.17619 .30902 m
.17619 .31277 L
s
P
p
.001 w
.25238 .30902 m
.25238 .31277 L
s
P
p
.001 w
.29048 .30902 m
.29048 .31277 L
s
P
p
.001 w
.32857 .30902 m
.32857 .31277 L
s
P
p
.001 w
.36667 .30902 m
.36667 .31277 L
s
P
p
.001 w
.44286 .30902 m
.44286 .31277 L
s
P
p
.001 w
.48095 .30902 m
.48095 .31277 L
s
P
p
.001 w
.51905 .30902 m
.51905 .31277 L
s
P
p
.001 w
.55714 .30902 m
.55714 .31277 L
s
P
p
.001 w
.63333 .30902 m
.63333 .31277 L
s
P
p
.001 w
.67143 .30902 m
.67143 .31277 L
s
P
p
.001 w
.70952 .30902 m
.70952 .31277 L
s
P
p
.001 w
.74762 .30902 m
.74762 .31277 L
s
P
p
.001 w
.82381 .30902 m
.82381 .31277 L
s
P
p
.001 w
.8619 .30902 m
.8619 .31277 L
s
P
p
.001 w
.9 .30902 m
.9 .31277 L
s
P
p
.001 w
.9381 .30902 m
.9381 .31277 L
s
P
p
.002 w
0 .30902 m
1 .30902 L
s
P
p
.002 w
.02381 .0618 m
.03006 .0618 L
s
P
[(-0.4)] .01131 .0618 1 0 Mshowa
p
.002 w
.02381 .18541 m
.03006 .18541 L
s
P
[(-0.2)] .01131 .18541 1 0 Mshowa
p
.002 w
.02381 .43262 m
.03006 .43262 L
s
P
[(0.2)] .01131 .43262 1 0 Mshowa
p
.002 w
.02381 .55623 m
.03006 .55623 L
s
P
[(0.4)] .01131 .55623 1 0 Mshowa
p
.001 w
.02381 .08652 m
.02756 .08652 L
s
P
p
.001 w
.02381 .11125 m
.02756 .11125 L
s
P
p
.001 w
.02381 .13597 m
.02756 .13597 L
s
P
p
.001 w
.02381 .16069 m
.02756 .16069 L
s
P
p
.001 w
.02381 .21013 m
.02756 .21013 L
s
P
p
.001 w
.02381 .23485 m
.02756 .23485 L
s
P
p
.001 w
.02381 .25957 m
.02756 .25957 L
s
P
p
.001 w
.02381 .2843 m
.02756 .2843 L
s
P
p
.001 w
.02381 .33374 m
.02756 .33374 L
s
P
p
.001 w
.02381 .35846 m
.02756 .35846 L
s
P
p
.001 w
.02381 .38318 m
.02756 .38318 L
s
P
p
.001 w
.02381 .4079 m
.02756 .4079 L
s
P
p
.001 w
.02381 .45735 m
.02756 .45735 L
s
P
p
.001 w
.02381 .48207 m
.02756 .48207 L
s
P
p
.001 w
.02381 .50679 m
.02756 .50679 L
s
P
p
.001 w
.02381 .53151 m
.02756 .53151 L
s
P
p
.001 w
.02381 .03708 m
.02756 .03708 L
s
P
p
.001 w
.02381 .01236 m
.02756 .01236 L
s
P
p
.001 w
.02381 .58095 m
.02756 .58095 L
s
P
p
.001 w
.02381 .60567 m
.02756 .60567 L
s
P
p
.002 w
.02381 0 m
.02381 .61803 L
s
P
P
0 0 m
1 0 L
1 .61803 L
0 .61803 L
closepath
clip
newpath
p
p
.004 w
.02381 .30902 m
.06349 .30902 L
.10317 .30902 L
.14286 .30902 L
.1627 .30902 L
.17262 .30902 L
.18254 .30902 L
.1875 .30902 L
.19246 .30902 L
.19742 .30902 L
.20238 .30902 L
.20486 .30902 L
.20734 .30902 L
.20982 .30902 L
.21106 .30902 L
.2123 .30902 L
.21354 .30902 L
.21478 .30902 L
.21602 .30902 L
.21726 .30902 L
.2185 .30902 L
.21974 .30902 L
.22098 .30902 L
.22222 .30902 L
.2247 .30902 L
.22718 .30902 L
.23214 .30902 L
.2371 .30902 L
.24206 .30902 L
.25198 .30902 L
.2619 .30902 L
.30159 .30902 L
.34127 .30902 L
.36111 .30902 L
.38095 .30902 L
.39087 .30902 L
.40079 .30902 L
.41071 .30902 L
.42063 .30902 L
.4256 .30902 L
.43056 .30902 L
.44048 .30902 L
.44544 .30902 L
.4504 .30902 L
.45536 .30902 L
.45784 .30902 L
.46032 .30902 L
.4628 .30902 L
.46528 .30902 L
.46776 .30902 L
Mistroke
.47024 .30902 L
.47272 .30902 L
.4752 .30902 L
.47768 .30902 L
.47892 .30902 L
.48016 .30902 L
.4814 .30902 L
.48264 .30902 L
.48388 .30902 L
.48512 .30902 L
.48636 .30902 L
.4876 .30902 L
.48884 .30902 L
.49008 .30902 L
.49132 .30902 L
.49256 .30902 L
.4938 .30902 L
.49504 .30902 L
.49628 .30902 L
.49752 .30902 L
.49876 .30902 L
.5 .30902 L
.50124 .30902 L
.50248 .30902 L
.50372 .30902 L
.50496 .30902 L
.5062 .30902 L
.50744 .30902 L
.50868 .30902 L
.50992 .30902 L
.51116 .30902 L
.5124 .30902 L
.51364 .30902 L
.51488 .30902 L
.51612 .30902 L
.51736 .30902 L
.51984 .30902 L
.52108 .30902 L
.52232 .30902 L
.5248 .30902 L
.52728 .30902 L
.52976 .30902 L
.53224 .30902 L
.53472 .30902 L
.5372 .30902 L
.53968 .30902 L
.54464 .30902 L
.5496 .30902 L
.55456 .30902 L
.55952 .30902 L
Mistroke
.56944 .30902 L
.5744 .30902 L
.57937 .30902 L
.58929 .30902 L
.59921 .30902 L
.61905 .30902 L
.63889 .30902 L
.65873 .30902 L
.69841 .30902 L
.71825 .30902 L
.72817 .30902 L
.7381 .30902 L
.74802 .30902 L
.75794 .30902 L
.7629 .30902 L
.76786 .30902 L
.77282 .30902 L
.7753 .30902 L
.77654 .30902 L
.77778 .30902 L
.77902 .30902 L
.78026 .30902 L
.7815 .30902 L
.78274 .30902 L
.78398 .30902 L
.78522 .30902 L
.78646 .30902 L
.7877 .30902 L
.79018 .30902 L
.79266 .30902 L
.79514 .30902 L
.79762 .30902 L
.80258 .30902 L
.80754 .30902 L
.81746 .30902 L
.82738 .30902 L
.8373 .30902 L
.85714 .30902 L
.89683 .30902 L
.93651 .30902 L
.97619 .30902 L
Mfstroke
P
P
% End of Graphics
MathPictureEnd

:[font = output; output; inactive; preserveAspect; endGroup]
{Graphics["<<>>"], Graphics["<<>>"], Graphics["<<>>"], 
 
  Graphics["<<>>"], Graphics["<<>>"], Graphics["<<>>"], 
 
  Graphics["<<>>"], Graphics["<<>>"], Graphics["<<>>"]}
;[o]
{-Graphics-, -Graphics-, -Graphics-, -Graphics-, 
 
  -Graphics-, -Graphics-, -Graphics-, -Graphics-, 
 
  -Graphics-}
:[font = text; inactive; preserveAspect; endGroup]
If you are reading this notebook with Mathematica or MathReader you may select the graphics just calculated and animate them to see the string vibrate.
;[s]
3:0,0;38,1;49,0;152,-1;
2:2,17,12,New York,0,12,0,0,0;1,17,12,New York,2,12,0,0,0;
:[font = section; inactive; Cclosed; preserveAspect; startGroup]
Model Problem VI.2.  
Set up the general solution of the vibrating string with Neumann BC., with the simplifications c = L = 1.  
;[s]
2:0,0;22,1;130,-1;
2:1,19,14,New York,1,14,0,0,0;1,17,12,New York,0,12,0,0,0;
:[font = input; preserveAspect]
X[x_,mu_] := D1 Cos[Sqrt[mu] x] + D1 Sin[Sqrt[mu] x]
:[font = input; preserveAspect; startGroup]
D[X[x,mu],x]
:[font = output; output; inactive; preserveAspect; endGroup]
D1*mu^(1/2)*Cos[mu^(1/2)*x] - D1*mu^(1/2)*Sin[mu^(1/2)*x]
;[o]
D1 Sqrt[mu] Cos[Sqrt[mu] x] - D1 Sqrt[mu] Sin[Sqrt[mu] x]
:[font = text; inactive; preserveAspect]
Possibility 1. mu=0, which means that the output just given is not really the solution to (6.1), which has become X''[x] = 0.  The solutions of this are of the form A + Bx, but B has to be 0 so that X'[0] = 0. Thus with this possibility, X[x] = 1 (times any constant). Eq. (6.3) now also becomes 
          T''(t) = 0, 
          which leads to the product solution of (WE):
	(A0 + B0 t) 1 = A0 + B0 t.

Since the boundary conditions do not apply to the t variable, we keep the
general linear combination.  A0 and B0 will be determined by the initial conditions.

Possibility 2. mu > 0. The boundary condition at x=0 now forces 
    X[x] =  Cos[Sqrt[mu] x] 
(again, up to a constant multiple), and the other boundary condition forces one of the 
values mun = (n Pi)^2.  The product solutions become
	(An Cos[n Pi t] + Bn Sin[n Pi t] ) Cos[n Pi t],
and the general solution will be
;[s]
7:0,0;508,1;509,0;515,1;516,0;755,1;756,0;881,-1;
2:4,17,12,New York,0,12,0,0,0;3,25,16,New York,64,12,0,0,0;
:[font = input; preserveAspect; startGroup]
  u[t_,x_] = A[0] + B[0] t + Sum[(A[n] Cos[n Pi t ] + B[n] Sin[n Pi t ] ) \
                 Cos[n Pi x], {n,1,infinity}]
:[font = output; output; inactive; preserveAspect; endGroup; endGroup; endGroup]
A[0] + t*B[0] + Sum[Cos[n*Pi*x]*
 
    (A[n]*Cos[n*Pi*t] + B[n]*Sin[n*Pi*t]), {n, 1, infinity}]
;[o]
A[0] + t B[0] + Sum[Cos[n Pi x] 
 
    (A[n] Cos[n Pi t] + B[n] Sin[n Pi t]), {n, 1, infinity}]
^*)