Size function: Difference between revisions

Revision as of 00:29, 16 July 2017

Revision as of 23:49, 21 November 2017

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Assume that <math>M</math> is a compact locally connected Hausdorff space. The following statements hold:

¤* every size function <math>\ell_{(M,\varphi)}(x,y)</math> is a [[non-decreasing function]] in the variable <math>x</math> and a [[Nonincreasing function|non-increasing function]] in the variable <math>y</math>.

¤* every size function <math>\ell_{(M,\varphi)}(x,y)</math> is locally right-constant in both its variables.

¤* for every <math>x<y</math>, <math>\ell_{(M,\varphi)}(x,y)</math> is finite.

¤* for every <math>x<\min \varphi</math> and every <math>y>x</math>, <math>\ell_{(M,\varphi)}(x,y)=0</math>.

¤* for every <math>y\ge\max \varphi</math> and every <math>x<y</math>, <math>\ell_{(M,\varphi)}(x,y)</math> equals the number of connected components of <math>M</math> on which the minimum value of <math>\varphi</math> is smaller than or equal to <math>x</math>.

If we also assume that <math>M</math> is a smooth [[closed manifold]] and <math>\varphi</math> is a <math>C^1</math>-function, the following useful property holds:

¤* in order that <math>(x,y)</math> is a discontinuity point for <math>\ell_{(M,\varphi)}</math> it is necessary that either <math>x</math> or <math>y</math> or both are critical values for <math>\varphi</math>

.<ref name="Fro96">Patrizio Frosini, ''Connections between size functions and critical points'', Mathematical Methods In The Applied Sciences, 19:555–569, 1996.</ref>

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<ref name="DoFro04">Pietro Donatini and Patrizio Frosini, ''Lower bounds for natural pseudodistances via size functions'', Archives of Inequalities and Applications, 2(1):1–12, 2004.</ref>

¤* if <math>\ell_{(N,\psi)}(\bar x,\bar y)>\ell_{(M,\varphi)}(\tilde x,\tilde y)</math> then <math>d((M,\varphi),(N,\psi))\ge \min\{\tilde x-\bar x,\bar y-\tilde y\}</math>.

The previous result gives an easy way to get lower bounds for the [[natural pseudodistance]] and is one of the main motivation to introduce the concept of size function.