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

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(Undo revision 35590 by Boris Tsirelson (talk))
 
(123 intermediate revisions by the same user not shown)
Line 1: Line 1:
−
==Negative results==
+
==Experiments==
  
−
$\newcommand{\M}{\mathscr M}$
+
Note a fine distinction from [http://ada00.math.uni-bielefeld.de/MW1236/index.php/User:Boris_Tsirelson/sandbox#Experiments Ada]:
−
As was noted, the normal  form of an object $M\in\M$ is a "selected representative"  from the equivalence class  $[M]$, usually possessing some nice properties. The set of all these "representatives" intersect each equivalence class exactly once; such set is called a ''transversal'' (for the given equivalence relation). Existence of a transversal is ensured by the [[axiom of choice]] for arbitrary equivalence relation on arbitrary set. However, a transversal in general is far from being nice. For example, consider the equivalence relation "$x-y$ is rational" for real numbers $x,y$. Its transversal (so-called [[Non-measurable set|Vitali set]]) cannot be Lebesgue measurable!
 
  
−
Typically, the set $\M$, endowed with its natural σ-algebra, is a [[standard Borel space]], and the set $\{(x,y)\in\M\times\M:x\sim y\}$ is a Borel subset of $\M\times\M$; this case is well-known as a "Borel equivalence relation". Still, existence of a ''Borel'' transversal is not guaranteed (for an example, use the Vitali set again).
+
<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>
  
−
Existence of Borel transversals and related properties of equivalence relations are investigated in [[descriptive set theory]]. According to {{Cite|K|Sect. 4}}, a lot of work in this area is philosophically motivated by problems of classification of objects up to some equivalence. A number of negative results are available. They show that in many cases, classification by a Borel transversal is impossible, and moreover, much weaker kinds of classification are also impossible. For example, a negative result is available for locally finite connected graphs (or trees) treated up to isomorphism, see {{Cite|K|Sect. 7, Item (B)(iii)}}.
+
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?
−
|valign="top"|{{Ref|K}}|| Alexander S.  Kechris,  "New directions in descriptive set theory", ''Bull. Symb. Logic''  '''5''' (1999), 161–174. {{MR|}}  {{ZBL|0933.03057}}
+
 
−
|}
+
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>
 +
 
 +
 
 +
<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>
 +
 
 +
Mysterious.
 +
 
 +
==Three dimensions==
 +
 
 +
<center><asy>
 +
settings.render = 0;
 +
 
 +
unitsize(100);
 +
 
 +
import three;
 +
import tube;
 +
 
 +
import graph;
 +
path unitCircle = Circle((0,0),1,35);
 +
 
 +
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));
 +
 
 +
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
 +
 
 +
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);
 +
 
 +
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);
 +
 
 +
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;
 +
 
 +
surface horn = tube( horn_axis, scale(horn_radius)*unitCircle );
 +
surface two_horns = surface(horn,reflect(O,X,Y)*horn);
 +
surface two_horns = surface(horn,reflect(O,X,Y)*horn);
 +
surface four_horns = surface(two_horns,left_right*two_horns,two_covers);
 +
 
 +
surface four_small_horns = implode_right*four_horns;
 +
surface eight_small_horns = surface(four_small_horns,left_right*four_small_horns);
 +
 
 +
surface big_surface = surface(four_horns,eight_small_horns);
 +
 
 +
real R = horn_radius/ratio;
 +
 
 +
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 );
 +
 
 +
draw (big_surface, yellow);
 +
 
 +
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>
 +
 
 +
 
 +
<center><asy>
 +
settings.render = 0;
 +
 
 +
size(200);
 +
import graph3;
 +
 
 +
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])
 +
 
 +
 
 +
WARNING: Asirra, the cat and dog CAPTCHA, is closing permanently on October 6, 2014. Please contact this site's administrator and ask them to switch to a different CAPTCHA. Thank you!

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)


WARNING: Asirra, the cat and dog CAPTCHA, is closing permanently on October 6, 2014. Please contact this site's administrator and ask them to switch to a different CAPTCHA. Thank you!

How to Cite This Entry:
Boris Tsirelson/sandbox2. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Boris_Tsirelson/sandbox2&oldid=25338
Retrieved from "https://encyclopediaofmath.org/index.php?title=User:Boris_Tsirelson/sandbox2&oldid=35591"