Display variation

A variation of a mapping is a numerical characteristic of a mapping associated with its differential properties.

The concept of “display variation” was defined by S. Banach ^[1] .

Two-dimensional case

Consider the definition of map variation for the two-dimensional case.

Let a mapping be given

\alpha \colon x=f(u,\;v),\;y=\varphi (u,\;v),

{\ displaystyle \ alpha \ colon x = f (u, \; v), \; y = \ varphi (u, \; v),}

Where $f(u,\;v)$ ${\ displaystyle f (u, \; v)}$ and $\varphi (u,\;v)$ ${\ displaystyle \ varphi (u, \; v)}$ - continuous squared $D_{0}=[0,\;1]\times [0,\;1]$ ${\ displaystyle D_ {0} = [0, \; 1] \ times [0, \; 1]}$ functions. They say that mapping $\alpha$ ${\ displaystyle \ alpha}$ has limited variation if there is a number $M>0$ ${\ displaystyle M> 0}$ such that for any sequence of non-overlapping squares $D^{i}\subset D_{0}\;(i=1,\;2,\;\ldots )$ ${\ displaystyle D ^ {i} \ subset D_ {0} \; (i = 1, \; 2, \; \ ldots)}$ with sides parallel to the coordinate axes $u,\;v$ ${\ displaystyle u, \; v}$ , the inequality holds

\sum _{i}\mathrm {mes} \,D_{xy}^{i}\leqslant M,

{\ displaystyle \ sum _ {i} \ mathrm {mes} \, D_ {xy} ^ {i} \ leqslant M,}

Where $D_{xy}$ ${\ displaystyle D_ {xy}}$ - image of the set $D^{i}\subset D_{0}$ ${\ displaystyle D ^ {i} \ subset D_ {0}}$ when displaying $\alpha$ ${\ displaystyle \ alpha}$ ,

$\mathrm {mes} \,D$ ${\ displaystyle \ mathrm {mes} \, D}$ Is a flat Lebesgue measure of the set $D$ ${\ displaystyle D}$ .

Numerical value of display variation $V(\alpha )$ ${\ displaystyle V (\ alpha)}$ can be defined in various ways. For example, if the mapping $\alpha$ ${\ displaystyle \ alpha}$ has a limited variation, then its variation $V(\alpha )$ ${\ displaystyle V (\ alpha)}$ can be determined by the formula:

V(\alpha )=\iint \limits _{-\infty }^{\quad +\infty }N(s,\;t)\,ds\,dt,

{\ displaystyle V (\ alpha) = \ iint \ limits _ {- \ infty} ^ {\ quad + \ infty} N (s, \; t) \, ds \, dt,}

Where $N(s,\;t)$ ${\ displaystyle N (s, \; t)}$ - number of system solutions $f(u,\;v)=s,\;\varphi (u,\;v)=t$ ${\ displaystyle f (u, \; v) = s, \; \ varphi (u, \; v) = t}$ , or the so-called Banach indicatrix display $\alpha$ ${\ displaystyle \ alpha}$ .

It was shown ^[2] that if the map $\alpha$ ${\ displaystyle \ alpha}$ has a limited variation, then almost everywhere on $D_{0}$ ${\ displaystyle D_ {0}}$ there is a generalized Jacobian $J(P)$ ${\ displaystyle J (P)}$ where $P\subset D_{0}$ ${\ displaystyle P \ subset D_ {0}}$ which we integrate on $D_{0}$ ${\ displaystyle D_ {0}}$ . Wherein

J(P)\,{\stackrel {\mathrm {def} }{=}}\lim _{\mathrm {mes} \,K\to 0}{\frac {\mathrm {mes} \,K_{xy}}{\mathrm {mes} \,K}},

{\ displaystyle J (P) \, {\ stackrel {\ mathrm {def}} {=}} \ lim _ {\ mathrm {mes} \, K \ to 0} {\ frac {\ mathrm {mes} \, K_ {xy}} {\ mathrm {mes} \, K}},}

Where $K\subset D_{0}$ ${\ displaystyle K \ subset D_ {0}}$ - a square containing a point $P\subset D_{0}$ ${\ displaystyle P \ subset D_ {0}}$ whose sides are parallel to the axes $u,\;v$ ${\ displaystyle u, \; v}$ ;

$K_{xy}$ ${\ displaystyle K_ {xy}}$ - image of the set $K$ ${\ displaystyle K}$ ;

$\mathrm {mes} \,K$ ${\ displaystyle \ mathrm {mes} \, K}$ Is a flat Lebesgue measure of the set $K$ ${\ displaystyle K}$ .

Literature

Lavrentiev, M.A., Shabat, B.V. Methods of the theory of functions of a complex variable. - M .: Nauka, 1987 .-- 688 p.
Griffiths, F. External differential systems and the calculus of variations. - M .: Mir, 1986 .-- 360 p.
Kolmogorov, A.N., Fomin, S.V. Elements of the theory of functions and functional analysis. - 7th ed. - M .: FIZMATLIT, 2004 .-- 572 p. - ISBN 5-9221-0266-4 . .

Notes

↑ Banach S. Fundamenta Mathematicae. - 1925. - t. 7. - p. 225-236.
↑ Kudryavtsev L. D. Metric questions in the theory of functions and mappings. - at. 1. - K., 1969. - p. 34-108

[1] Banach S. Fundamenta Mathematicae. - 1925. - t. 7. - p. 225-236.

[2] Kudryavtsev L. D. Metric questions in the theory of functions and mappings. - at. 1. - K., 1969. - p. 34-108

Display variation

Two-dimensional case

Literature

Notes

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