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Difference between revisions of "User:Boris Tsirelson/sandbox1"

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<ref> [http://hea-www.harvard.edu/AstroStat http://hea-www.harvard.edu/AstroStat]; <nowiki> http://www.incagroup.org </nowiki>; <nowiki> http://astrostatistics.psu.edu </nowiki> </ref>
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====Notes====
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<references />
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-------------------------------------------
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{|
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| A || B || C
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|-
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| X || Y || Z
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|}
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-----------------------------------------
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-----------------------------------------
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$\newcommand*{\longhookrightarrow}{\lhook\joinrel\relbar\joinrel\rightarrow}$
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<asy>
 
<asy>
−
size(80,40,keepAspect=false);
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size(100,100);
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pen p2=red+1.2;
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label(scale(1.7)*'$T(\\Sigma)\hookrightarrow T(\\Sigma,X)$',(0,0));
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draw(((-4,0){right}..(-3,0.004)..(-2,0.054)..(-1,0.242)..(0,0.399)..(1,0.242)..(2,0.054)..(3,0.004)..{right}(4,0)),p=p2);
 
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defaultpen(linewidth(0.8));
 
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draw((-4,0)--(4,0));
 
 
</asy>
 
</asy>
  
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--------------------------------------
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<asy>
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size(220,220);
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import math;
  
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if for every $\varepsilon$ there is a $\delta > 0$ such that,
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int kmax=40;
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for any $a_1<b_1<a_2<b_2<\ldots < a_n<b_n \in I$ with $\sum_i |a_i -b_i| <\delta$, we have
 
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\[
 
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\sum_i d (f (b_i), f(a_i)) <\varepsilon\, .
 
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\]
 
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The  absolute continuity guarantees the uniform continuity. As for real  valued functions, there is a characterization through an appropriate  notion of derivative.
 
  
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'''Theorem 1'''
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guide g;
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A continuous function $f$ is absolutely continuous if and only if there is a function $g\in L^1_{loc} (I, \mathbb R)$ such that
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for (int k=-kmax; k<=kmax; ++k) {
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\begin{equation}\label{e:metric}
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  real phi = 0.2*k*pi;
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d (f(b), f(a))\leq \int_a^b g(t)\, dt \qquad \forall a<b\in I\,
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  real rho = 1;
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\end{equation}
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  if (k!=0) {
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(cp. with ). This theorem motivates the following
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    rho = sin(phi)/phi;
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  }
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  pair z=rho*expi(phi);
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  g=g..z;
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}
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draw (g);
  
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'''Definition 2'''
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defaultpen(0.75);
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If  $f:I\to X$ is a absolutely continuous and $I$ is compact, the metric  derivative of $f$ is the function $g\in L^1$ with the smalles $L^1$ norm  such that \ref{e:metric} holds (cp. with )
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draw ( (0,0)--(1.3,0), dotted, Arrow(SimpleHead,5) );
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dot ( (1,0) );
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label ( "$a$", (1,0), NE );
  
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--------------------------------------------
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</asy>

Latest revision as of 07:12, 13 March 2016

[1]

Notes

  1. ↑ http://hea-www.harvard.edu/AstroStat; http://www.incagroup.org ; http://astrostatistics.psu.edu


A B C
X Y Z


$\newcommand*{\longhookrightarrow}{\lhook\joinrel\relbar\joinrel\rightarrow}$

How to Cite This Entry:
Boris Tsirelson/sandbox1. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Boris_Tsirelson/sandbox1&oldid=27474
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