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Difference between revisions of "User:Boris Tsirelson/sandbox2"

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−
the method is as follows:
+
==Experiments==
−
\begin{equation}
 
−
\begin{gathered}
 
−
k_1 = h f(x_0,y_0), \quad y_0 = y(x_0),\\
 
−
k_2 = h f \big( x_0 + \tfrac13 h, y_0 + \tfrac13 k_1 \big) \, , \\
 
−
k_3 = h f \big( x_0 + \tfrac13 h, y_0 + \tfrac16 k_1 + \tfrac16 k_2 \big) \, , \\
 
−
k_4 = h f \big( x_0 + \tfrac12 h, y_0 + \tfrac18 k_1 + \tfrac38 k_3 \big) \, , \\
 
  
−
\end{gathered}
+
Note a fine distinction from [http://ada00.math.uni-bielefeld.de/MW1236/index.php/User:Boris_Tsirelson/sandbox#Experiments Ada]:
−
\end{equation}
 
  
 +
<center><asy>
 +
fill( box((-1,-1),(1,1)), white );
 +
draw( (-1.2,-0.5)--(1.2,-0.5) );
 +
label("Just a text",(0,0));
 +
filldraw( box((-0.7,-1),(0.7,1)), white, opacity(0) );
 +
shipout(scale(15)*currentpicture);
 +
</asy></center>
  
−
The number
+
I guess, the reason is that there Asy generates pdf file (converted into png afterwards), and here something else (probably ps).
  
 +
No, it seems, it generates eps, both here and there. Then, what could be the reason?
  
 +
More.
  
 +
<center><asy>
 +
label("Just a text",(0,0));
 +
fill( box((-2,-1),(2,1)), white );
 +
//draw( box((-2,-1),(2,1)), green );
 +
shipout(scale(15)*currentpicture);
 +
</asy></center>
  
  
−
the method is as follows:
+
<center><asy>
 +
label("Just a text",(0,0));
 +
fill( box((-2,-1),(2,1)), white );
 +
draw( box((-2,-1),(2,1)), green );
 +
shipout(scale(15)*currentpicture);
 +
</asy></center>
  
−
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056060/k0560602.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
+
Mysterious.
  
−
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056060/k0560603.png" /></td> </tr></table>
+
==Three dimensions==
  
−
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056060/k0560604.png" /></td> </tr></table>
+
<center><asy>
 +
settings.render = 0;
  
−
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056060/k0560605.png" /></td> </tr></table>
+
unitsize(100);
  
−
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056060/k0560606.png" /></td> </tr></table>
+
import three;
 +
import tube;
  
−
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056060/k0560607.png" /></td> </tr></table>
+
import graph;
 +
path unitCircle = Circle((0,0),1,35);
  
−
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056060/k0560608.png" /></td> </tr></table>
+
currentprojection = perspective((900,-350,-650));
 +
currentlight=light(gray(0.4),specularfactor=3,viewport=false,(-0.5,-0.5,-0.75),(0,-0.5,0.5),(0.5,0.5,0.75));
 +
// currentlight=light(gray(0.4),specularfactor=3,viewport=false,(-0.5,-0.5,-0.75),(0.5,-0.5,0.5),(0.5,0.5,0.75));
  
−
The number
+
triple horn_start=(0,-1,0.6);
 +
triple horn_end=(0,0.4,0.2);
 +
real horn_radius=0.2;
  
−
*******************************************
+
real ratio=horn_end.z/(-horn_start.y);    // fractal levels ratio
  
−
[http://www.mediawiki.org/wiki/Extension_talk:MathJax#.5BWORKAROUND.5D_MathJax_works_in_preview.2C_but_not_in_page_display here]
+
transform3 implode_right = shift(horn_end) * scale3(ratio) * rotate(-90,X) * shift(-horn_start.y*Y);
 +
transform3 left_right = reflect(O,X,Z)*rotate(90,Y);
  
−
$$\text{ $K$ compact}$$
+
path[] cover_with_holes = scale(horn_radius/ratio)*unitCircle^^
 +
  shift((horn_start.z,0))*scale(0.9horn_radius)*reverse(unitCircle)^^
 +
  shift((-horn_start.z,0))*scale(0.9horn_radius)*reverse(unitCircle);
 +
surface cover = surface(cover_with_holes,ZXplane);
 +
surface cover_left = shift((horn_start.x,horn_start.y,0))*cover;
 +
surface two_covers = surface(cover_left,left_right*cover_left);
  
−
\[\text{ $K$ compact}\]
+
path3 horn_axis = horn_start..horn_start+(0,0.01,0)..(0,0,0.7)..(0,0.2,0.6)..horn_end+(0,0,0.01)..horn_end;
  
−
\begin{equation}
+
surface horn = tube( horn_axis, scale(horn_radius)*unitCircle );
−
\mu (B)= \sup \{\mu(K): K\subset B, \text{ $K$ compact}\}\,  
+
surface two_horns = surface(horn,reflect(O,X,Y)*horn);
−
\end{equation}
+
surface two_horns = surface(horn,reflect(O,X,Y)*horn);
 +
surface four_horns = surface(two_horns,left_right*two_horns,two_covers);
  
−
and having the following property:
+
surface four_small_horns = implode_right*four_horns;
−
\begin{equation}\label{e:tight}
+
surface eight_small_horns = surface(four_small_horns,left_right*four_small_horns);
−
\mu (B)= \sup \{\mu(K): K\subset B, \mbox{ $K$ compact}\}\,
 
−
\end{equation}
 
−
(see {{Cite|Sc}}).
 
  
 +
surface big_surface = surface(four_horns,eight_small_horns);
  
−
The total variation measure of a $\mathbb C$-valued measure is defined on $\mathcal{B}$ as:
+
real R = horn_radius/ratio;
−
\[
 
−
\abs{\mu}(B) :=\sup\left\{ \sum \abs{\mu(B_i)}: \text{$\{B_i\}\subset\mathcal{B}$ is a countable partition of $B$}\right\}.
 
−
\]
 
−
In the real-valued case the above definition simplifies as
 
  
−
-----------------------
+
draw ( circle((0,1,0), 1.005R, Y ), currentpen+2 );
 +
draw ( circle((horn_start.z,1.01,horn_start.x), horn_radius, Y ), currentpen+2 );
 +
draw ( circle((-horn_start.z,1.01,horn_start.x), horn_radius, Y ), currentpen+2 );
  
−
and the following identity  holds:
+
draw (big_surface, yellow);
−
\begin{equation}\label{e:area_formula}
 
−
\int_A J f (y) \, dy =  \int_{\mathbb R^m} \mathcal{H}^0 (A\cap f^{-1} (\{z\}))\, d\mathcal{H}^n (z)\, .
 
−
\end{equation}
 
  
−
Cp. with 3.2.2 of {{Cite|EG}}. From \eqref{e:area_formula} it is not  difficult to conclude the following generalization (which also goes  often under the same name):
+
pen blackpen = currentpen+1.5;
  
 +
draw ( circle((0,-1,0), 1.005R, Y ), blackpen );
 +
draw ( circle(horn_start, 0.98horn_radius, Y ), blackpen );
 +
draw ( circle((horn_start.x,horn_start.y,-horn_start.z), 0.98horn_radius, Y ), blackpen );
  
 +
real phi=0.9;  // adjust to the projection
 +
triple u = (cos(phi),0,sin(phi));
 +
draw( R*u-Y -- R*u+Y, blackpen );
 +
draw( -R*u-Y -- -R*u+Y, blackpen );
  
 +
</asy></center>
  
−
\begin{equation}\label{ab}
+
 
−
E=mc^2
+
<center><asy>
−
\end{equation}
+
settings.render = 0;
−
By \eqref{ab}, it is possible. But see \eqref{ba} below:
+
 
−
\begin{equation}\label{ba}
+
size(200);
−
E\ne mc^3,
+
import graph3;
−
\end{equation}
+
 
−
which is a pity.
+
currentprojection=perspective((2,2,5));
 +
 
 +
real R=1;
 +
real a=1;
 +
 
 +
real co=0.6;
 +
real colo=0.3;
 +
 
 +
triple f(pair t) {
 +
  return ((R+a*cos(t.y))*cos(t.x),(R+a*cos(t.y))*sin(t.x),a*sin(t.y));
 +
}
 +
 
 +
surface s=surface(f,(0,0),(2pi,2pi),20,20,Spline);
 +
 
 +
draw(s,rgb(co,co,co),meshpen=rgb(colo,colo,colo));
 +
 
 +
</asy></center>
 +
 
 +
==Sinusoid==
 +
 
 +
<center><asy>
 +
import graph;
 +
size(450);
 +
real f(real x) {return sin(x);};
 +
 
 +
real f1(real x) {return cos(x);};
 +
draw(graph(f1,-2*pi,2*pi),blue+1,"$\cos(x)$");
 +
draw(graph(f,-2*pi,2*pi),red+1,"$\sin(x)$");
 +
xaxis("$x$",Arrow);
 +
yaxis();
 +
 
 +
xtick("$\frac{\pi}{6}$",pi/6,N);
 +
xtick("$\frac{\pi}{4}$",pi/4,N);
 +
xtick("$\frac{\pi}{3}$",pi/3,N);
 +
xtick("$\frac{\pi}{2}$",pi/2,N);
 +
xtick("$\frac{3\pi}{2}$",3*pi/2,N);
 +
xtick("$\pi$",pi,N);
 +
xtick("$2\pi$",2*pi,N);
 +
xtick("$-\frac{\pi}{2}$",-pi/2,N);
 +
xtick("$-\frac{3\pi}{2}$",-3*pi/2,N);
 +
xtick("$-\pi$",-pi,N);
 +
xtick("$-2\pi$",-2*pi,N);
 +
 
 +
ytick("$1/2$",0.5,1,fontsize(8pt));
 +
ytick("$\sqrt{2}/2$",sqrt(2)/2,1,fontsize(8pt));
 +
ytick("$\sqrt{3}/2$",sqrt(3)/2,1,fontsize(8pt));
 +
ytick("$1$",1,1,fontsize(8pt));
 +
ytick("$-1/2$",-0.5,-1,fontsize(8pt));
 +
ytick("$-\sqrt{2}/2$",-sqrt(2)/2,-1,fontsize(8pt));
 +
ytick("$-\sqrt{3}/2$",-sqrt(3)/2,-1,fontsize(8pt));
 +
ytick("$-1$",-1,-1,fontsize(8pt));
 +
 
 +
attach(legend(),truepoint(E),10E,UnFill);
 +
</asy></center>
 +
 
 +
==Sinusoidal spiral==
 +
 
 +
<center><asy>
 +
import graph;
 +
size (200);
 +
 
 +
real r = 2.3;
 +
real m = 4;
 +
 
 +
real eps=10.^(-10);
 +
for  (int k=0; k<m; ++k) {
 +
  draw ( polargraph(  new real(real x) {return cos(m*x)^(1/m);}, -(pi/2m)+eps+k*2pi/m, (pi/2m)-eps+k*2pi/m ),
 +
defaultpen+1.5 );
 +
  draw ( -r*expi(-pi/2m+k*2pi/m)..r*expi(-pi/2m+k*2pi/m), dashed );
 +
  draw ( -r*expi(pi/2m+k*2pi/m)..r*expi(pi/2m+k*2pi/m), dashed );
 +
}
 +
label( "$m=4$", (0.58,0.02), fontsize(7pt) );
 +
 
 +
real eps=10.^(-2);
 +
for  (int k=0; k<m; ++k) {
 +
  draw ( polargraph(  new real(real x) {return cos(m*x)^(-1/m);}, -(pi/2m)+eps+k*2pi/m, (pi/2m)-eps+k*2pi/m ),
 +
defaultpen+1.5 );
 +
}
 +
label( "$m=-4$", (1.55,0.02), fontsize(7pt) );
 +
 
 +
label( "sinusoidal spiral: $a=1$", (0,2.3) );
 +
draw ( unitcircle, dashed );
 +
</asy></center>
 +
 
 +
==Power function==
 +
 
 +
<center><asy>
 +
import graph;
 +
picture whole;
 +
 
 +
real sc=0.8;
 +
 
 +
draw ( graph( new real(real x) {return x;}, -2, 2), red+1.2, "$y=x$" );
 +
draw ( graph( new real(real x) {return 2x;}, -1, 1), blue+1.2, "$y=2x$" );
 +
draw ( graph( new real(real x) {return x/2;}, -2, 2), green+1.2, "$y=x/2$" );
 +
 
 +
xaxis(-2.1,2.1, LeftTicks(Label(fontsize(8pt)),Step=1,step=0.2,Size=2,size=1,NoZero));
 +
yaxis(-2,2, RightTicks(Label(fontsize(8pt)),Step=0.5,step=0.1,Size=2,size=1,NoZero));
 +
labelx("$x$",(2.3,0.25));
 +
labely("$y$",(0.15,2.3));
 +
 
 +
add(scale(0.72sc,1.2sc)*legend(),(0.5,-0.75));
 +
 
 +
real mrg=1.3;
 +
draw( scale(mrg)*box((-2,-2),(2,2)), white );
 +
 
 +
add (whole,shift(-sc*230,0)*currentpicture.fit(sc*mrg*6.5cm));
 +
erase();
 +
 
 +
 
 +
draw ( graph( new real(real x) {return 1/x;}, -4, -0.25), red+1.2, "$y=1/x$" );
 +
draw ( graph( new real(real x) {return 1/x;}, 0.25, 4), red+1.2 );
 +
draw ( graph( new real(real x) {return 2/x;}, -4, -0.5), blue+1.2, "$y=2/x$" );
 +
draw ( graph( new real(real x) {return 2/x;}, 0.5, 4), blue+1.2 );
 +
draw ( graph( new real(real x) {return 1/(2x);}, -4, -0.125), green+1.2, "$y=1/(2x)$" );
 +
draw ( graph( new real(real x) {return 1/(2x);}, 0.125, 4), green+1.2 );
 +
 
 +
xaxis(-4.2,4.2, LeftTicks(Label(fontsize(8pt)),Step=2,step=0.5,Size=2,size=1,NoZero));
 +
yaxis(-4,4, RightTicks(Label(fontsize(8pt)),Step=1,step=0.2,Size=2,size=1,NoZero));
 +
labelx("$x$",(4.6,0.5));
 +
labely("$y$",(0.3,4.6));
 +
 
 +
add(scale(0.75sc,0.75sc)*legend(),(0.95,-1.2));
 +
 
 +
real mrg=1.3;
 +
draw( scale(mrg)*box((-4,-4),(4,4)), white );
 +
 
 +
add (whole,shift(0,0)*currentpicture.fit(sc*mrg*6.5cm,mrg*6.5cm,false));
 +
erase();
 +
 
 +
 
 +
draw ( graph( new real(real x) {return x^3;}, -4^(1/3), 4^(1/3)), red+1.2, "$y=x^3$" );
 +
draw ( graph( new real(real x) {return x^2;}, -2, 2), blue+1.2, "$y=x^2$" );
 +
draw ( graph( new real(real x) {return sqrt(x);}, 0, 4), green+1.2, "$y=x^{1/2}$" );
 +
draw ( graph( new real(real x) {return -sqrt(x);}, 0, 4), green+1.2 );
 +
 
 +
xaxis(-4.2,4.2, LeftTicks(Label(fontsize(8pt)),Step=2,step=0.5,Size=2,size=1,NoZero));
 +
yaxis(-4,4, RightTicks(Label(fontsize(8pt)),Step=1,step=0.2,Size=2,size=1,NoZero));
 +
labelx("$x$",(4.6,0.5));
 +
labely("$y$",(0.3,4.6));
 +
 
 +
add(scale(0.5sc,0.75sc)*legend(),(0.6,-2.5));
 +
 
 +
real mrg=1.3;
 +
draw( scale(mrg)*box((-4,-4),(4,4)), white );
 +
 
 +
add (whole,shift(sc*230,0)*currentpicture.fit(sc*mrg*6.5cm,mrg*6.5cm,false));
 +
erase();
 +
 
 +
shipout(whole);
 +
</asy></center>
 +
 
 +
==Kolmogorov test==
 +
 
 +
<center><asy>
 +
 
 +
srand(2014011);
 +
 
 +
import stats;
 +
 
 +
int size = 13;
 +
real [] sample = new real[size+1];
 +
real lambda = 1.3/size;
 +
real width = 2.0;
 +
 
 +
for (int k=0; k<size; ++k) {
 +
  sample[k] = Gaussrand();
 +
}
 +
sample[size] = 10;
 +
 
 +
sample = sort(sample);
 +
 
 +
// for (real x : sample ) {
 +
//  write(x);
 +
// }
 +
 
 +
real x0 = -10;
 +
int k = 0;
 +
for (real x : sample ) {
 +
  filldraw( box( (x0,k/size-lambda), (x,k/size+lambda) ), rgb(0.8,0.8,0.8) );
 +
  draw( (x0,k/size-lambda)..(x,k/size-lambda), currentpen+1.5 );
 +
  draw( (x0,k/size)..(x,k/size), currentpen+1.5 );
 +
  draw( (x0,k/size+lambda)..(x,k/size+lambda), currentpen+1.5 );
 +
  k += 1;
 +
  x0 = x;
 +
  draw( (x,(k-1)/size-lambda)..(x,k/size+lambda) );
 +
}
 +
 
 +
clip( box((-width,-0.005),(width,1.005)) );
 +
 
 +
draw ((-width,0)--(width,0),Arrow);
 +
draw ((0,-0.1)--(0,1.3),Arrow);
 +
draw ((-width,1)--(width,1));
 +
 
 +
draw ((sample[2],0)..(sample[2],2/size));
 +
draw ((sample[size-1],0)..(sample[size-1],0.48), dashed);
 +
draw ((sample[size-1],0.7)..(sample[size-1],1-1/size), dashed);
 +
 
 +
label("$x$",(width,0),S);
 +
label("$y$",(0,1.3),W);
 +
label("$0$",(0,0),SW);
 +
label("$1$",(0,1),NE);
 +
 
 +
label("$X_{(1)}$",(sample[0],0),S);
 +
label("$X_{(2)}$",(sample[1],0),S);
 +
label("$X_{(3)}$",(sample[2],0),S);
 +
label("$X_{(n)}$",(sample[size-1],0),S);
 +
 
 +
label("$F_n(x)+\lambda_n(\alpha)$",(-1.55,0.35));
 +
draw ((-1.35,0.25)..(-1.2,1/size+lambda));
 +
dot((-1.2,1/size+lambda));
 +
 
 +
label("$F_n(x)$",(0.4,0.3));
 +
draw ((0.4,0.4)..(0.3,8/size));
 +
dot((0.3,8/size));
 +
 
 +
label("$F_n(x)-\lambda_n(\alpha)$",(1.5,0.6));
 +
draw ((1.6,0.7)..(1.7,1-lambda));
 +
dot((1.7,1-lambda));
 +
 
 +
shipout(scale(100,100)*currentpicture);
 +
</asy></center>
 +
 
 +
==Golden ratio==
 +
 
 +
Strangely, the figure in EoM is erroneous! ED=EB, not BD=EB.
 +
 
 +
<center><asy>
 +
 
 +
pair A=(-1,0);
 +
pair B=(0,0);
 +
pair E=(0,0.5);
 +
pair C=A+(0.5*(sqrt(5)-1),0);
 +
pair D=(-1/sqrt(5), 0.5*(1-1/sqrt(5)));
 +
 
 +
draw( A--B--E--cycle,currentpen+1.5 );
 +
dot(A,currentpen+3.5); dot(B,currentpen+3.5); dot(E,currentpen+3.5); dot(C,currentpen+3.5); dot(D,currentpen+3.5);
 +
 
 +
draw( shift(E)*scale(0.5)*unitcircle,currentpen+1 );
 +
draw( shift(A)*scale(0.5*(sqrt(5)-1))*unitcircle,currentpen+1 );
 +
 
 +
draw( shift(B)*scale(0.5)*unitcircle, dashed+red );
 +
 
 +
clip(A+(-0.15,-0.15)--B+(0.15,-0.15)--E+(0.15,0.15)--A+(-0.15,0.15)--cycle);
 +
 
 +
label("$A$",A,S); label("$B$",B,S); label("$C$",C,S);
 +
label("$E$",E,N); label("$D$",D,N);
 +
 
 +
label( "\small Golden Ratio construction", (-0.5,0.8) );
 +
 
 +
shipout(scale(100)*currentpicture);
 +
</asy></center>
 +
 
 +
 
 +
 
 +
 
 +
 
 +
 
 +
 
 +
 
 +
[Calculus: ] the art of numbering and measuring exactly a thing whose existence cannot be conceived. (Voltaire, [http://www.fordham.edu/halsall/mod/1778voltaire-newton.asp Letter XVII: On Infinites In Geometry, And Sir Isaac Newton's Chronology])
 +
 
 +
And what are these fluxions? The velocities of evanescent increments? They are neither finite quantities, nor quantities infinitely small, nor yet nothing. May we not call them ghosts of departed quantities?  (Berkeley, [http://www-history.mcs.st-and.ac.uk/Quotations/Berkeley.html The Analyst])
 +
 
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Latest revision as of 20:14, 12 December 2014

Experiments

Note a fine distinction from Ada:

I guess, the reason is that there Asy generates pdf file (converted into png afterwards), and here something else (probably ps).

No, it seems, it generates eps, both here and there. Then, what could be the reason?

More.


Mysterious.

Three dimensions


Sinusoid

Sinusoidal spiral

Power function

Kolmogorov test

Golden ratio

Strangely, the figure in EoM is erroneous! ED=EB, not BD=EB.





[Calculus: ] the art of numbering and measuring exactly a thing whose existence cannot be conceived. (Voltaire, Letter XVII: On Infinites In Geometry, And Sir Isaac Newton's Chronology)

And what are these fluxions? The velocities of evanescent increments? They are neither finite quantities, nor quantities infinitely small, nor yet nothing. May we not call them ghosts of departed quantities? (Berkeley, The Analyst)


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How to Cite This Entry:
Boris Tsirelson/sandbox2. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Boris_Tsirelson/sandbox2&oldid=32517
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