By A. D. Alexandrov, Yu. G. Reshetnyak (auth.)
Read or Download General Theory of Irregular Curves PDF
Similar general books
Bioelectrochemistry: General Introduction
Quantity 1 of this sequence is meant to provide the reader a primary realizing of the most important parts deemed necessary to the learn of bioelec trochemistry. a radical clutch of the idea and technique of those simple issues is essential to manage effectively with the advanced phenomena that presently face investigators in so much bioelectrochemical laboratories.
Geometric Aspects of General Topology
This ebook is designed for graduate scholars to procure wisdom of measurement idea, ANR thought (theory of retracts), and similar themes. those theories are attached with a variety of fields in geometric topology and usually topology in addition. for this reason, for college students who desire to learn matters normally and geometric topology, figuring out those theories might be helpful.
Growth Hormone and Somatomedins during Lifespan
Many of the congresses on development hormone (GH) that have been held in Milan on the grounds that 1967, the Milan Congresses, have witnessed over 25 years the super growth of a study box that was once established in the beginning upon the scarce wisdom of the organic houses of a protein. GH, whose chemical constitution had simply been pointed out and a radioimmunoassay built for its dimension in blood, turned within the following years a massive sector of organic examine.
- Physics: a general course. Volume II : Electricity and Magnetism Waves Optics
- General Concepts in Integrated Pest and Disease Management (Integrated Management of Plant Pests and Diseases, Volume 1)
- Ray Optics, Fermat's Principle, and Applns to General Relativity
- Compilation of Input-Output Tables: Proceedings of a Session of the 17th General Conference of the International Association for Research in Income and Wealth, Gouvieux, France, August 16 – 22, 1981
- Introduccion general a la crítica de la economía politica / 1857
Extra info for General Theory of Irregular Curves
Sample text
Let us add to it the points A, C and B as new vertices. Then we get a new chain C for which, obviously, s(C') ~ s(C). With the point C the chain C' is subdivided into two chains ~ and TJ, where ~ is formed by the vertices of the chain C' lying on the arc [AC] (the point C included), and TJ consists of those vertices of the chain l;' which lie on [CB] (the point C included again). We have: s(C) ~ s(C') = s(~) + s(TJ) ~ s(AC) From this, due to the arbitrariness of s(K) = sup s(C) ~ s(AC) I,; + r;, + s(CB).
Let I: [a, b] - M and g: [c, d] - M be two parametrized curves. Let us assume that there exists a system of chains Z which obeys each of the above parametrized curves. Let us prove that in this case these curves are equivalent. Let us take c > 0 and find the chain ~ E Z, such that A(~) < c. Let Xi' i = 1, 2, ... , m, be a set of this chain. In line with the definition, Xi coincides with the totality of all x = I(t), where t;-I ~ t ~ tj and with the totality of all y = g(u), where ui_1 ~ U ~ u;.
In this case, for any rectifiable curve [{ lying in U, the curve cp(K) in IR n is rectifiable. Proof. Let [{ be a rectifiable curve lying in the domain of definition of an admissible chart cp: U __ IRH. According to the definition, for any point X of the curve K there exists an admissible chart Bx:Vx __ IR n, such that the image of a certain closed neighbourhood of the point X on the curve K is a rectifiable curve in IRn. Applying the Borel theorem we get a certain sequence of the points Xo = A < Xl < ...