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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).
  
−
Every $\A_\mu$-measurable function $X\to\R$ is almost equal to some $\A$-measurable function $X\to\R$. The same holds for arbitrary [[Measurable space#countably generated|countably generated measurable space]] in place of $\R$.
+
No, it seems, it generates eps, both here and there. Then, what could be the reason?
  
−
''Example.'' Let $X$ be the real line, $\A$ the Borel σ-algebra and $\mu$ Lebesgue measure on $\A$, then $\A_\mu$ is the Lebesgue σ-algebra.
+
More.
  
−
The ''[[Measure#product|product]]'' of two (or finitely many) measure spaces is a well-defined measure space.
+
<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 ''[[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}}.)
 
  
−
====Completion====
+
<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>
  
−
Let $(X,\A,\mu)$ be a measure space. 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 $B\in\A$.
+
Mysterious.
  
−
Let $(X,\A,\mu)$ be complete, and $\B\subset\A$ a sub-σ-algebra. Then $(X,\A,\mu)$ is the completion of $(X,\B,\mu|_\B)$ if and only if for every $A\in\A$ there exist $B,C\in\B$ such that $B\subset A\subset C$ and $\mu(C\setminus B)=0$.
+
==Three dimensions==
  
−
Surprisingly, the Borel σ-algebra can be "almost restored" from the Lebesgue σ-algebra in the following sense.
+
<center><asy>
 +
settings.render = 0;
  
−
Let $(X,\A,\mu)$ be complete, and $\B_1\subset\A$, $\B_2\subset\A$ two countably generated sub-σ-algebras such that $(X,\A,\mu)$ is the completion of $(X,\B_i,\mu|_{\B_i})$ for $i=1,2$. Then there exists a set $Y\in\B_1\cap\B_2$ of full measure such that $\B_1|_Y=\B_2|_Y$. (Here $\B_i|_Y=\{B\cap Y:B\in\B_i\}=\{B\in\B_i:B\subset Y\}$.)  
+
unitsize(100);
  
−
====Isomorphism====
+
import three;
 +
import tube;
  
−
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.
+
import graph;
 +
path unitCircle = Circle((0,0),1,35);
  
−
A ''mod  0 isomorphism'' (or ''almost isomorphism'') between two measure spaces  $(X_1,\A_1,\mu_1)$ and $(X_2,\A_2,\mu_2)$ is a strict isomorphism  between some full measure sets $Y_1\in\A_1$ and $Y_2\in\A_2$ treated as  measurable subspaces.
+
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));
  
−
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}}.)
+
triple horn_start=(0,-1,0.6);
 +
triple horn_end=(0,0.4,0.2);
 +
real horn_radius=0.2;
  
−
Almost isomorphisms are completion insensitive in the following sense.
+
real ratio=horn_end.z/(-horn_start.y);    // fractal levels ratio
  
−
Let  measure spaces $(X_1,\A_1,\mu_1)$, $(X_2,\A_2,\mu_2)$ be such that (a) their completions are almost isomorphic, and (b) measurable spaces  $(X_1,\A_1)$, $(X_2,\A_2)$ are countably generated. Then  $(X_1,\A_1,\mu_1)$, $(X_2,\A_2,\mu_2)$ are almost isomorphic (under the  same isomorphism, restricted to a smaller subset of full measure).
+
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);
  
−
====Some classes of measure spaces====
+
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);
  
−
Let $(X,\A,\mu)$ be a measure space. Both $(X,\A,\mu)$ and $\mu$ are called ''totally finite'' if $\mu(X)<\infty$, and ''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 $\forall n \;\; \mu(A_n)<\infty$. (Totally finite measures are also σ-finite.)
+
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;
  
−
Let $(X,\A,\mu)$ be a totally finite measure space. 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}}.)
+
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);
  
−
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).
+
surface four_small_horns = implode_right*four_horns;
 +
surface eight_small_horns = surface(four_small_horns,left_right*four_small_horns);
  
−
''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.
+
surface big_surface = surface(four_horns,eight_small_horns);
  
−
====Atoms etc====
+
real R = horn_radius/ratio;
  
−
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'' (or ''diffused'') 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}}.)
+
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 );
  
−
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.
+
draw (big_surface, yellow);
  
−
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)}}.)
+
pen blackpen = currentpen+1.5;
  
−
====On terminology====
+
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 );
  
−
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.
+
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 );
  
−
====References====
+
</asy></center>
  
−
{|
+
 
−
|valign="top"|{{Ref|T}}|| Terence Tao, "An introduction to measure  theory", AMS (2011). &nbsp; {{MR|2827917}} &nbsp; {{ZBL|05952932}}
+
<center><asy>
−
|-
+
settings.render = 0;
−
|valign="top"|{{Ref|C}}|| Donald L.  Cohn, "Measure theory", Birkhäuser (1993). &nbsp;   {{MR|1454121}}  &nbsp;   {{ZBL|0860.28001}}
+
 
−
|-
+
size(200);
−
|valign="top"|{{Ref|P}}|| David Pollard, "A user's guide to measure theoretic probability", Cambridge (2002).  &nbsp; {{MR|1873379}} &nbsp; {{ZBL|0992.60001}}
+
import graph3;
−
|-
+
 
−
|valign="top"|{{Ref|B}}|| V.I. Bogachev, "Measure theory", Springer-Verlag (2007). &nbsp;  {{MR|2267655}}  &nbsp;{{ZBL|1120.28001}}
+
currentprojection=perspective((2,2,5));
−
|-
+
 
−
|valign="top"|{{Ref|I}}||  Kiyosi Itô, "Introduction to probability  theory", Cambridge (1984). &nbsp; {{MR|0777504}} &nbsp; {{ZBL|0545.60001}}
+
real R=1;
−
|-
+
real a=1;
−
|valign="top"|{{Ref|D}}||  Richard M. Dudley, "Real analysis and probability", Wadsworth&Brooks/Cole (1989). &nbsp; {{MR|0982264}} &nbsp; {{ZBL|0686.60001}}
+
 
−
|-
+
real co=0.6;
−
|valign="top"|{{Ref|K}}|| Alexander  S. Kechris, "Classical    descriptive set theory", Springer-Verlag  (1995). &nbsp;   {{MR|1321597}} &nbsp; {{ZBL|0819.04002}}
+
real colo=0.3;
−
|-
+
 
−
|valign="top"|{{Ref|M}}||Paul Malliavin, "Integration and probability", Springer-Verlag (1995). &nbsp; {{MR|1335234}} &nbsp; {{ZBL|0874.28001}}
+
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));
−
|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}}
+
}
−
|-
+
 
−
|valign="top"|{{Ref|F}}|| D.H. Fremlin, "Measure theory", Torres Fremlin, Colchester. Vol. 1: 2004 &nbsp; {{MR|2462519}} &nbsp; {{ZBL|1162.28001}}; Vol. 2: 2003  &nbsp; {{MR|2462280}}  &nbsp; {{ZBL|1165.28001}}; Vol. 3: 2004 &nbsp;   {{MR|2459668}} &nbsp; {{ZBL|1165.28002}}; Vol. 4: 2006 &nbsp;   {{MR|2462372}} &nbsp; {{ZBL|1166.28001}}
+
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=21315
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