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

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−
$\newcommand{\Om}{\Omega}
+
==Experiments==
−
\newcommand{\A}{\mathcal A}
 
−
\newcommand{\B}{\mathcal B}
 
−
\newcommand{\M}{\mathcal M} $
 
−
A '''measure space''' is a triple $(X,\A,\mu)$ where $X$ is a set, $\A$ a [[Algebra of sets|σ-algebra]] of its subsets, and $\mu:\A\to[0,+\infty]$ a [[measure]]. Thus, a measure space consists of a [[measurable space]] and a measure. The notation $(X,\A,\mu)$ is often shortened to $(X,\mu)$ and one says that $\mu$  is a measure on $X$; sometimes the notation is shortened to $X$.
 
  
−
====Basic notions and constructions====
+
Note a fine distinction from [http://ada00.math.uni-bielefeld.de/MW1236/index.php/User:Boris_Tsirelson/sandbox#Experiments Ada]:
  
−
''Inner measure'' $\mu_*$ and ''outer measure'' $\mu^*$ are defined for all subsets $A\subset X$ by
+
<center><asy>
−
: $ \mu_*(A) = \max\{\mu(B):B\in\A,B\subset A\}\,,\quad
+
fill( box((-1,-1),(1,1)), white );
−
\mu^*(A) = \min\{\mu(B):B\in\A,B\supset A\}\,;$
+
draw( (-1.2,-0.5)--(1.2,-0.5) );
−
{{Anchor|null}}{{Anchor|full}}{{Anchor|almost}}
+
label("Just a text",(0,0));
−
$A$ is called a ''null'' (or ''negligible'') set if $\mu^*(A)=0$; in this case the complement $X\setminus A$ is called a set of ''full measure'' (or ''conegligible''), and one says that $x\notin A$ for ''almost all'' $x$ (in other words, ''almost everywhere''). Two sets $A,B\subset X$ are ''almost equal'' (or ''equal mod 0'') if $(x\in A)\iff(x\in B)$ for almost all $x$ (in other words, $A\setminus B$ and $B\setminus A$ are negligible). Two functions $f,g:X\to Y$ are ''almost equal'' (or ''equal mod 0'', or ''equivalent'') if they are equal almost everywhere.
+
filldraw( box((-0.7,-1),(0.7,1)), white, opacity(0) );
 +
shipout(scale(15)*currentpicture);
 +
</asy></center>
  
−
A subset $A\subset X$ is called ''measurable'' (or $\mu$-measurable) if it is almost equal to some $B\in\A$. In this case $\mu_*(A)=\mu^*(A)=\mu(B)$. If $\mu_*(A)=\mu^*(A)<\infty$ then $A$ is $\mu$-measurable. All $\mu$-measurable sets are a σ-algebra $\A_\mu$ containing $\A$.
+
I guess, the reason is that there Asy generates pdf file (converted into png afterwards), and here something else (probably ps).
  
−
The ''[[Measure#product|product]]'' of two (or finitely many) measure spaces is a well-defined measure space.
+
No, it seems, it generates eps, both here and there. Then, what could be the reason?
  
−
A ''[[probability space]]'' is a measure space $(X,\A,\mu)$ satisfying $\mu(X)=1$. The product of infinitely many probability spaces is a well-defined probability space. (See {{Cite|D|Sect. 8.2}}, {{Cite|B|Sect. 3.5}}, {{Cite|P}}.)
+
More.
  
−
A ''strict isomorphism'' (or ''point isomorphism'', or ''[[metric isomorphism]]'') between two measure spaces $(X_1,\A_1,\mu_1)$ and $(X_2,\A_2,\mu_2)$ is a bijection $f:X_1\to X_2$ such that, first, the conditions $A_1\in\A_1$ and $A_2\in\A_2$ are equivalent whenever $A_1\subset X_1$, $A_2\subset X_2$, $A_2=f(A_1)$, and second, $\mu_1(A_1)=\mu_2(A_2)$ under these conditions.
+
<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>
  
−
A ''mod 0 isomorphism'' (or ''almost isomorphism'') between two complete measure spaces $(X_1,\A_1,\mu_1)$ and $(X_2,\A_2,\mu_2)$ is a strict isomorphism between some full measure sets $Y_1\subset X_1$ and $Y_2\subset X_2$ treated as measurable subspaces.
 
  
−
Thus we have two equivalence relations between complete measure spaces: ''"strictly isomorphic"'' and ''"almost isomorphic"''. (See {{Cite|I|Sect. 2.4}}, {{Cite|B|Sect. 9.2}}.)
+
<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>
  
−
====Some classes of measure spaces====
+
Mysterious.
  
−
Let $(X,\A,\mu)$ be a measure space.
+
==Three dimensions==
  
−
Both $(X,\A,\mu)$ and $\mu$ are called ''complete'' if $\A_\mu=\A$ or, equivalently, if $\A$ contains all null sets. The ''completion'' of $(X,\A,\mu)$ is the complete measure space $(X,\A_\mu,\tilde\mu)$ where $\tilde\mu(A)=\mu(B)$ whenever $A\in\A_\mu$ is almost equal to $B\in\A$.
+
<center><asy>
 +
settings.render = 0;
  
−
If $X$ is a set of finite measure, that is, $\mu(X)<\infty$, then $\mu$, and sometimes also $(X,\A,\mu)$, is called ''finite.''
+
unitsize(100);
  
−
Both $(X,\A,\mu)$ and $\mu$ are called ''σ-finite'' if $X$ can be split into countably many sets of finite measure, that is, $X=A_1\cup A_2\cup\dots$ for some $A_n\in\A$ such that $\forall n \;\; \mu(A_n)<\infty$. (Finite measures are also σ-finite.)
+
import three;
 +
import tube;
  
−
Let $\mu(X)<\infty$. Both $(X,\A,\mu)$ and $\mu$ are called [[Perfect measure|''perfect'']] if for every $\mu$-measurable (or equivalently, for every $\A$-measurable) function $f:X\to\R$ the image $f(X)$ contains a Borel (or equivalently, σ-compact) subset $B$ whose preimage $f^{-1}(B)$ is of full measure. (See {{Cite|B|Sect. 7.5}}.)
+
import graph;
 +
path unitCircle = Circle((0,0),1,35);
  
−
For ''[[standard probability space]]s'' see the separate article. Standard measure spaces are defined similarly. They are perfect, and admit a complete classification (unlike perfect measure spaces in general).
+
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));
  
−
''Examples.'' The real line with Lebesgue measure on Borel σ-algebra is an incomplete σ-finite measure space. The real line with Lebesgue measure on Lebesgue σ-algebra is a complete σ-finite measure space. The unit interval $(0,1)$ with Lebesgue measure on Lebesgue σ-algebra is a standard probability space. The product of countably many copies of this space is standard; for uncountably many factors the product is perfect but nonstandard. The one-dimensional [[Hausdorff measure]] on the plane is not σ-finite.
+
triple horn_start=(0,-1,0.6);
 +
triple horn_end=(0,0.4,0.2);
 +
real horn_radius=0.2;
  
−
Let $\mu(X)<\infty$. An ''atom'' of $(X,\A,\mu)$ (and of $\mu$) is a non-negligible measurable set $A\subset X$ such that every measurable subset of $A$ is either negligible or almost equal to $A$. Both $(X,\A,\mu)$ and $\mu$ are called ''atomless'' or ''nonatomic'' if they have no atoms; on the other hand, they are called ''purely atomic'' if there exists a partition of $X$ into atoms. (See {{Cite|D|Sect. 3.5}}, {{Cite|B|Sect. 1.12(iii)}}, {{Cite|M|Sect. 6.4.1}}.)
+
real ratio=horn_end.z/(-horn_start.y);   // fractal levels ratio
  
−
If $x\in X$ is such that the single-point set $\{x\}$ is a non-negligible measurable set then clearly $\{x\}$ is an atom. If $(X,\A,\mu)$ is standard then every atom is almost equal to some $\{x\}$, but in general it is not.
+
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);
  
−
Let $\{x\}$ be measurable for all $x\in X$. Both $(X,\A,\mu)$ and $\mu$ are called ''continuous'' if $\mu(\{x\})=0$ for all $x\in X$; on the other hand, they are called ''discrete'' if $X$ is almost equal to some finite or countable set. (See {{Cite|C|Sect. 1.2}}, {{Cite|K|Sect. 17.A}}.) A discrete space cannot be atomless (unless $\mu(X)=0$), but a purely atomic nonstandard space can be continuous. (See {{Cite|B|Sect. 7.14(v)}}.)
+
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);
  
−
====On terminology====
+
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;
  
−
The phrase "separable measure space" is quite ambiguous. Some authors call $(X,\A,\mu)$ separable when the Hilbert space $L_2(X,\A,\mu)$ is separable; equivalently, when $\A$ contains a countably generated sub-σ-algebra $\B$ such that every set of $\A$ is almost equal to some set of $\B$. (See {{Cite|B|Sect. 7.14(iv)}}, {{Cite|M|Sect. IV.6.0}}.) But in {{Cite|I|Sect. 3.1}} it is required instead that $\B$ separates points and $(X,\A,\mu)$ is complete, while in {{Cite|H}} all these conditions are imposed together.
+
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);
  
−
====References====
+
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);
−
|valign="top"|{{Ref|T}}|| Terence Tao, "An introduction to measure  theory", AMS (2011). &nbsp; {{MR|2827917}} &nbsp; {{ZBL|05952932}}
+
 
−
|-
+
real R = horn_radius/ratio;
−
|valign="top"|{{Ref|C}}|| Donald L.  Cohn, "Measure theory", Birkhäuser (1993). &nbsp;   {{MR|1454121}}  &nbsp;   {{ZBL|0860.28001}}
+
 
−
|-
+
draw ( circle((0,1,0), 1.005R, Y ), currentpen+2 );
−
|valign="top"|{{Ref|P}}|| David Pollard, "A user's guide to measure theoretic probability", Cambridge (2002).  &nbsp; {{MR|1873379}} &nbsp; {{ZBL|0992.60001}}
+
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 );
−
|valign="top"|{{Ref|B}}|| V.I. Bogachev, "Measure theory", Springer-Verlag (2007). &nbsp;  {{MR|2267655}}  &nbsp;{{ZBL|1120.28001}}
+
 
−
|-
+
draw (big_surface, yellow);
−
|valign="top"|{{Ref|I}}||  Kiyosi Itô, "Introduction to probability  theory", Cambridge (1984). &nbsp; {{MR|0777504}} &nbsp; {{ZBL|0545.60001}}
+
 
−
|-
+
pen blackpen = currentpen+1.5;
−
|valign="top"|{{Ref|D}}||  Richard M. Dudley, "Real analysis and probability", Wadsworth&Brooks/Cole (1989). &nbsp; {{MR|0982264}} &nbsp; {{ZBL|0686.60001}}
+
 
−
|-
+
draw ( circle((0,-1,0), 1.005R, Y ), blackpen );
−
|valign="top"|{{Ref|K}}|| Alexander  S. Kechris, "Classical    descriptive set theory", Springer-Verlag  (1995). &nbsp;   {{MR|1321597}} &nbsp; {{ZBL|0819.04002}}
+
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 );
−
|valign="top"|{{Ref|M}}||Paul Malliavin, "Integration and probability", Springer-Verlag (1995). &nbsp; {{MR|1335234}} &nbsp; {{ZBL|0874.28001}}
+
 
−
|-
+
real phi=0.9;  // adjust to the projection
−
|valign="top"|{{Ref|H}}|| Jean Haezendonck, "Abstract Lebesgue-Rohlin spaces", ''Bull. Soc. Math. de Belgique'' '''25'''  (1973), 243–258. &nbsp; {{MR|0335733}} &nbsp;  {{ZBL|0308.60006}}
+
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=21296
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