Статьи журнала - Владикавказский математический журнал
Все статьи: 967
Unbounded convergence in the convergence vector lattices: a survey
Статья научная
Various convergences in vector lattices were historically a subject of deep investigation which stems from the begining of the 20th century in works of Riesz, Kantorovich, Nakano, Vulikh, Zanen, and many other mathematicians. The study of the unbounded order convergence had been initiated by Nakano in late 40th in connection with Birkhoff's ergodic theorem. The idea of Nakano was to define the almost everywhere convergence in terms of lattice operations without the direct use of measure theory. Many years later it was recognised that the unbounded order convergence is also rathe useful in probability theory. Since then, the idea of investigating of convergences by using their unbounded versions, have been exploited in several papers. For instance, unbounded convergences in vector lattices have attracted attention of many researchers in order to find new approaches to various problems of functional analysis, operator theory, variational calculus, theory of risk measures in mathematical finance, stochastic processes, etc. Some of those unbounded convergences, like unbounded norm convergence, unbounded multi-norm convergence, unbounded τ-convergence are topological. Others are not topological in general, for example: the unbounded order convergence, the unbounded relative uniform convergence, various unbounded convergences in lattice-normed lattices, etc. Topological convergences are, as usual, more flexible for an investigation due to the compactness arguments, etc. The non-topological convergences are more complicated in genelal, as it can be seen on an example of the a.e-convergence. In the present paper we present recent developments in convergence vector lattices with emphasis on related unbounded convergences. Special attention is paid to the case of convergence in lattice multi pseudo normed vector lattices that generalizes most of cases which were discussed in the literature in the last 5 years.
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Unbounded order convergence and the Gordon theorem
Статья научная
The celebrated Gordon's theorem is a natural tool for dealing with universal completions of Archimedean vector lattices. Gordon's theorem allows us to clarify some recent results on unbounded order convergence. Applying the Gordon theorem, we demonstrate several facts on order convergence of sequences in Archimedean vector lattices. We present an elementary Boolean-Valued proof of the Gao-Grobler-Troitsky-Xanthos theorem saying that a sequence xn in an Archimedean vector lattice X is uo- (uo-Cauchy) in X if and only if xn is o- (o-convergent) in Xu. We also give elementary proof of the theorem, which is a result of contributions of several authors, saying that an Archimedean vector lattice is sequentially uo-complete if and only if it is σ-universally complete. Furthermore, we provide a comprehensive solution to Azouzi's problem on characterization of an Archimedean vector lattice in which every uo-Cauchy net is o-convergent in its universal completion.
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Unicity on entire functions concerning their difference operators and derivatives
Статья научная
In this paper we study the uniqueness of entire functions concerning their difference operator and derivatives. The idea of entire and meromorphic functions relies heavily on this direction. Rubel and Yang considered the uniqueness of entire function and its derivative and proved that if f(z) and f′(z) share two values a,b counting multilicities then f(z)≡f′(z). Later, Li Ping and Yang improved the result given by Rubel and Yang and proved that if f(z) is a non-constant entire function and a,b are two finite distinct complex values and if f(z) and f(k)(z) share a counting multiplicities and b ignoring multiplicities then f(z)≡f(k)(z). In recent years, the value distribution of meromorphic functions of finite order with respect to difference analogue has become a subject of interest. By replacing finite distinct complex values by polynomials, we prove the following result: Let Δf(z) be trancendental entire functions of finite order, k≥0 be integer and P1 and P2 be two polynomials. If Δf(z) and f(k) share P1 CM and share P2 IM, then Δf≡f(k). A non-trivial proof of this result uses Nevanlinna's value distribution theory.
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Статья научная
In the fall of 1990 a small colloquium on nonstandard analysis was arranged at the request of a group of graduate and postgraduate students of Novosibirsk State University. At the meetings many unsolved problems were formulated stemming from various branches of analysis and seemingly deserving attention of the novices of nonstandard analysis. In 1994 some discussion took place on combining nonstandard methods at the international conference "Interaction Between Functional Analysis, Harmonic Analysis and Probability" (Missouri University, Columbia USA). The same topics were submitted to the international conference "Analysis and Logic" held in Belgium in 1997. In 1998 an INTAS research project was submitted. The problems raised in the framework of these projects are the core of this article. The list of the problems contains not only simple questions for drill but also topics for serious research intended mostly at the graduate and post graduate level. Some problems need creative thought to clarify and specify them.
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Upper semilattices of finite-dimensional gauges
Статья научная
This is a brief overview of some applications of the ideas of abstract convexity to the upper semilattices of gauges in finite dimensions.
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Персоналии
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Vector Lattice Powers: Continuous and Measurable Vector Functions
Статья научная
In the study of order properties of homogeneous polynomials in vector lattices two constructions are of fundamental importance: the symmetric positive tensor product and the vector lattice power. Both associate a canonical n-homogeneous polynomial with each Archimedean vector lattice, such that any other homogeneous polynomial of an appropriate class defined on the same vector lattice is the composition of the canonical polynomial with a linear operator. With this so called “linearization” in hand, various tools of the theory of positive linear operators can be used to study homogeneous polynomials. Thus, the problem of description of the Fremlin symmetric tensor products and the vector lattice powers for special vector lattices arises. The former enables one to study a large class of order bounded homogeneous polynomials, but has a very complicated structure; the latter has a much more transparent structure, but handles a narrower class of homogeneous polynomials, namely orthogonally additive ones. The purpose of this note is to describe the power of the vector lattice of continuous or Bochner measurable vector functions with values in a Banach lattice and to apply this result to the representation of homogeneous orthogonally additive polynomials.
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Weakly N1-separable quasi-complete Abelian p-groups are bounded
Статья научная
We prove that each weakly N1-separable quasi-complete abelian p-group is bounded, thus extending recent results of ours in (Vladikavkaz Math. J., 2007 and 2008).
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Weakly compact-friendly operators
Статья научная
We introduce weak compact-friendliness as an extension of compact-friendliness, and and prove that if a non-zero weakly compact-friendly operator B: E→ E on a Banach lattice is quasi-nilpotent at some non-zero positive vector, then B has a non-trivial closed invariant ideal. Relevant facts related to compact-friendliness are also discussed.
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When all separately band preserving bilinear operators are symmetric?
Статья научная
A purely algebraic characterization of universally complete vector lattices in which all separately band preserving bilinear operators are symmetric is obtained: this class consists of universally complete vector lattices with \sigma-distributive Boolean algebra of bands.
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When are the nonstandard hulls of normed lattices discrete or continuous?
Статья научная
This note is a nonstandard analysis version of the paper "When are ultrapowers of normed lattices discrete or continuous?" by W. Wnuk and B. Wiatrowski.
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ξ-лиевы дифференцирования на алгебрах локально измеримых операторов
Статья научная
Изучаются ξ-лиевы дифференцирования на алгебрах локально измеримых операторов LS(M), где M - алгебра фон Неймана, не содержащая прямых абелевых слагаемых
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Статья научная
В статье рассматривается вопрос о существовании абсолютно представляющих систем экспонент в весовом пространстве Фреше \tilde{A}(\varPhi) функций, аналитических в выпуклой области G из \mathbb C^p, p\ge 1. При некоторых довольно общих предположениях относительно последовательности весов \varPhi=\left\{f_n(z)\right\}_{n=1}^\infty доказывается обобщенная теорема Пэли - Винера - Шварца для \tilde A(\varPhi).
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Статья научная
Установлено, что справедливость аналога теоремы Уитни для пространств ультраджетов нормального типа эквивалентна существованию в этих пространствах абсолютно представляющих систем экспонент с мнимыми показателями.
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