By M. Hazewinkel (auth.), M. Hazewinkel (eds.)

ISBN-10: 9048182387

ISBN-13: 9789048182381

ISBN-10: 9401512337

ISBN-13: 9789401512336

This ENCYCLOPAEDIA OF arithmetic goals to be a reference paintings for all components of mathe­ matics. it's a translation with updates and editorial reviews of the Soviet Mathematical Encyclopaedia released via 'Soviet Encyclopaedia Publishing apartment' in 5 volumes in 1977-1985. The annotated translation comprises ten volumes together with a unique index quantity. There are 3 varieties of articles during this ENCYCLOPAEDIA. firstly there are survey-type articles facing many of the major instructions in arithmetic (where a slightly fme subdivi­ sion has been used). the most requirement for those articles has been that they need to supply a fairly whole updated account of the present situation in those parts and they may be maximally available. in most cases, those articles can be comprehensible to arithmetic scholars of their first specialization years, to graduates from different mathematical parts and, reckoning on the explicit topic, to experts in different domain names of technological know-how, en­ gineers and lecturers of arithmetic. those articles deal with their fabric at a reasonably common point and target to provide an idea of the type of difficulties, strategies and ideas excited by the realm in query. additionally they include historical past and motivation instead of distinct statements of special theorems with designated definitions and technical information on how you can perform proofs and buildings. the second one type of article, of medium size, comprises extra designated concrete difficulties, effects and techniques.

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Editorial comments. References [A1] SZEP,1. : Introduction to the theory of games, Reidel, 1985. [A2] NEUMANN, J. : Theory of games and economic behavior, Princeton Univ. Press, 1953. AMS 1980 Subject Classification: 90D99 STRATIFICATION - A decomposition of a (possibly infinite-dimensional) manifold into connected submanifolds of strictly-diminishing dimensions. I. Vo'itsekhovski'i Editorial comments. Usually a 'stratification' of a space means more than just some decomposition into connected pieces with diminishing dimensions, Let (P, <) be a partially ordered set.

F. Cantelli [2] stated sufficient conditions for the strong law of large numbers for independent random variables Xn in terms of the second and fourth central moments of the summands (these conditions are fulfilled for the Bernoulli scheme). Putting B n •k = "± E I Xj-EXj I k, j=l the Cantelli condition can be written in the form ~ B n. 2 ~ 11=1 n 2 < 00.

References [A 1] ZYGMUND, A: 'On the differentiability of multiple integrals', Fund. Math. 23 (1934), 143-149. AMS 1980 Subject Classification: 28A 15 STRONG ERGODICITY of a topological dynamical system in the narrow sense (a flow or a cascade) - A property examined in ergodic theory. : Ergodic theory, Cambridge Univ. Press, 1983. AMS 1980 Subject Classification: 28DXX 58F11, 54H20, STRONG EXTREMUM - A minimal or maximal value J(j) taken by a functional J(y) at a curve y(x), for which one of the inequalities Xl ~X~X2' J(j) ~ l(y) or ley) ~ l(y) holds for all comparison curves y(x) in an {neighbourhood of y(x).

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Encyclopaedia of Mathematics: Stochastic Approximation — Zygmund Class of Functions by M. Hazewinkel (auth.), M. Hazewinkel (eds.)

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