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Line 27: Line 27:
 
draw(Label("$A\cap B$",0),g,Arrow);
 
draw(Label("$A\cap B$",0),g,Arrow);
 
</asy>
 
</asy>
====Case 2====
 
<asy>
 
int M=30;
 
real a = 0.07;
 
real a0 = 0.15;
 
real b = 0.02;
 
real c = 0.6;
 
real d = -0.2;
 
real x0 = -0.22;
 
real u = 0.2;
 
real v = 0.32;
 
  
draw ((-0.7,0)--(0.3,0),Arrow);
 
draw ((x0,-0.02)--(x0,1.2),Arrow);
 
 
label("$x$",(0.3,0),E);
 
label(rotate(90)*"$y$",(x0,1.2),N);
 
label("$x_0$",(x0,-0.02),S);
 
 
guide g1; guide g2; guide g3; guide g4; guide g5;
 
for (int k=floor(-0.7M); k<floor(0.3M); ++k) {
 
  real x = k/M;
 
  real z = 1+3*x^2;
 
  real y1 = 1/(z-2a-a0)+2b*(1+c*x)+d;
 
  real y2 = 1/(z-a-a0)+b*(1+c*x)+d;
 
  real y3 = 1/(z-a0)+d;
 
  real y4 = 1/(z+a-a0)-b*(1+c*x)+d;
 
  real y5 = 1/(z+2a-a0)-2b*(1+c*x)+d;
 
  g1=g1..(x,y1);
 
  g2=g2..(x,y2);
 
  g3=g3..(x,y3);
 
  g4=g4..(x,y4);
 
  g5=g5..(x,y5);
 
}
 
draw(g1,defaultpen+1);
 
draw(g2,defaultpen+1);
 
draw(g3,defaultpen+1);
 
draw(g4,defaultpen+1);
 
draw(g5,defaultpen+1);
 
 
real x = x0;
 
real z = 1+3*x^2;
 
real y1 = 1/(z-2a-a0)+2b*(1+c*x)+d;
 
real y2 = 1/(z-a-a0)+b*(1+c*x)+d;
 
real y3 = 1/(z-a0)+d;
 
real y4 = 1/(z+a-a0)-b*(1+c*x)+d;
 
real y5 = 1/(z+2a-a0)-2b*(1+c*x)+d;
 
path g = (x,y1)..(x-u,y2)..(x-v,y3)..(x-u,y4)..(x,y5);
 
draw( g );
 
 
pair w = (0.1,-0.6);
 
 
pair p = point(g,0.5);
 
dot ( p );
 
draw( p--p-0.5w, dashed );
 
label(rotate(90)*"$\Psi_{x_0}(y)$",p-0.5w,N);
 
 
draw( (x,y2)--(x-u,y2) );
 
draw( (x,y3)--(x-v,y3) );
 
draw( (x,y4)--(x-u,y4) );
 
 
draw( (x-u,y2+0.05)--(x-u,y4-0.05) );
 
draw( (x-v,y2+0.1)--(x-v,y4-0.1) );
 
 
real x = -0.15;
 
real z = 1+3*x^2;
 
real y4 = 1/(z+a-a0)-b*(1+c*x)+d;
 
dot( (x,y4) );
 
draw( (x,y4)--(x,y4)+w, dashed );
 
label("$\scriptstyle \underline f_\alpha(x)$",(x,y4)+w,SE);
 
 
real x = -0.05;
 
real z = 1+3*x^2;
 
real y3 = 1/(z-a0)+d;
 
dot( (x,y3) );
 
draw( (x,y3)--(x,y3)+w, dashed );
 
label("$\scriptstyle \underline f_1(x)=\overline f_1(x)$",(x,y3)+w,SE);
 
 
real x = 0.05;
 
real z = 1+3*x^2;
 
real y2 = 1/(z-a-a0)+b*(1+c*x)+d;
 
dot( (x,y2) );
 
draw( (x,y2)--(x,y2)+w, dashed );
 
label("$\scriptstyle \overline f_\alpha(x)$",(x,y2)+w,SE);
 
 
label("\small Fig. a4: Non-precise function",(x0,-0.2));
 
 
shipout(scale(250,120)*currentpicture);
 
</asy>
 
 
===Tests November 4th===
 
===Tests November 4th===
 
====Case 1====
 
====Case 1====

Revision as of 17:35, 17 November 2014

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[asy] pair A,B,C,X,Y,Z; A = (0,0); B = (1,0); C = (0.3,0.8); draw(A--B--C--A); X = (B+C)/2; Y = (A+C)/2; Z = (A+B)/2; draw(A--X, red); draw(B--Y,red); draw(C--Z,red); [/asy]

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\begin{align} \dot{x} & = \sigma(y-x) \\ \dot{y} & = \rho x - y - xz \\ \dot{z} & = -\beta z + xy \end{align}


\[ \frac{1}{(\sqrt{\phi \sqrt{5}}-\phi) e^{\frac25 \pi}} = 1+\frac{e^{-2\pi}} {1+\frac{e^{-4\pi}} {1+\frac{e^{-6\pi}} {1+\frac{e^{-8\pi}} {1+\ldots} } } } \]


Some Text \( \frac{1}{(\sqrt{\phi \sqrt{5}}-\phi) e^{\frac25 \pi}} = 1+\frac{e^{-2\pi}} {1+\frac{e^{-4\pi}} {1+\frac{e^{-6\pi}} {1+\frac{e^{-8\pi}} {1+\ldots} } } } \)


Some Text \[ \frac{1}{(\sqrt{\phi \sqrt{5}}-\phi) e^{\frac25 \pi}} = 1+\frac{e^{-2\pi}} {1+\frac{e^{-4\pi}} {1+\frac{e^{-6\pi}} {1+\frac{e^{-8\pi}} {1+\ldots} } } } \]

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How to Cite This Entry:
WikiSysop. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=WikiSysop&oldid=34564

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