An elementary proof of an equivalence theorem relevant in the theory of optimization

The authors give an elementary proof of an equivalence theorem of analysis which is often used in optimization theory. The theorem asserts that certain conditions are equivalent to weak convergence inL ₁. One is the Dunford-Pettis condition concerning absolute integrability. Two others are expressed in terms of Nagumo functions, and can be thought of as growth properties. The original proofs of the various parts of the theorem are scattered in different and specialized mathematical publications. The authors feel it useful to present here a straightforward proof of the various parts in terms of standard Lebesgue integration theory.     

Maximal functions,Hardy spaces and Fourier multiplier theorems on unbounded Vilenkin groups

We consider relations between various existing concepts of Hardy spaces on Vilenkin groups. The problem of an appropriate Fourier multiplier theorem for unbounded Vilenkin groups is revisited and the conditions made more precise under which such a theorem is established. Theorem 3.1 from Avdispahi? and Memi? (2010) [1] is rectified.     

The small index property for homogeneous models in AEC’s

We prove a version of a small index property theorem for strong amalgamation classes. Our result builds on an earlier theorem by Lascar and Shelah (in their case, for saturated models of uncountable first-order theories). We then study versions of the small index property for various non-elementary classes. In particular, we obtain the small index property for quasiminimal pregeometry structures.     

Global solutions of initial value problems for nonlinear second-order impulsive integro-differential equations of mixed type in Banach spaces

In this paper, we develop a theorem for the existence of global solutions of an initial value problems for a class of second-order impulsive integro-differential equations of mixed type in a real Banach space. The results obtained are the generalizations and improvements of various recent results. As an application of the theorem, the existence of global solutions of two mixed boundary value problems for two classes of fourth-order impulsive differential equations are established.     

Local Approximation Properties of Modified Baskakov Operators

In this paper, we introduce a general modification of the classical Baskakov operators which do not need to preserve the test function x ². Then, we study an approximation theorem, a Voronovskaya theorem, and various local approximation results for our modified Baskakov operators.     

New Braided Crossed Categories and Drinfel'd Quantum Double for Weak Hopf Group Coalgebras

A simple and nice structure theorem for orthogroups was given by Petrich in 1987. In this paper, we consider a generalized orthogroup, that is, a quasi-completely regular semigroup with a band of idempotents in which its set of regular elements, namely, RegS, forms an ideal of S. A method of construction of such semigroups is provided and as a result, the Petrich structure theorem of orthogroups becomes an immediate corollary of our theorem on generalized orthogroups. An example of such generalized orthogroup is also constructed. This example provides some useful information for the construction of various kinds of quasi-completely regular semigroups.     

关于求解弹性力学平面问题的功的互等定理法

本文推广功的互等定理法于求解具有复杂边界条件矩形板的弹性力学平面问题.首先,我们给出了作为基本系统的四边固定矩形板平面问题的基本解,然后基于在具有复杂边界条件的实际系统与基本系统之间应用功的互等定理,从而求得实际系统的位移表达式.当只存在位移边界条件时,用功的互等定理法求得的位移表达式就是真实的.但是在另一些情况下,当有静力边界条件或混合边界条件时,所求得的位移是容许的.为求得真实位移.必须应用最小势能原理.一些计算表明.对于求解具有复杂边界条件的矩形板弹性力学平面问题,功的互等定理法是一简便通用的方法.显然,这是一个新方法.     

Abelian groups with layered tiles and the sumset phenomenon

We prove a generalization of the main theorem in Jin, The sumset phenomenon, about the sumset phenomenon in the setting of an abelian group with layered tiles of cell measures. Then we give some applications of the theorem for multi-dimensional cases of the sumset phenomenon. Several examples are given in order to show that the applications obtained are not vacuous and cannot be improved in various directions. We also give a new proof of Shnirel'man's theorem to illustrate a different approach (which uses the sumset phenomenon) to some combinatorial problems.

     

A Note on Nonconvex Minimax Theorem with Separable Homogeneous Polynomials

The minimax theorem for a convex-concave bifunction is a fundamental theorem in optimization and convex analysis, and has a lot of applications in economics. In the last two decades, a nonconvex extension of this minimax theorem has been well studied under various generalized convexity assumptions. In this note, by exploiting the hidden convexity (joint range convexity) of separable homogeneous polynomials, we establish a nonconvex minimax theorem involving separable homogeneous polynomials. Our result complements the existing study of nonconvex minimax theorem by obtaining easily verifiable conditions for the nonconvex minimax theorem to hold.     

Autour du théorème de Stein-Rosenberg

Summary The aim of this note is to present a synthesis of recent applications or extensions of the Stein-Rosenberg theorem on nonnegative matrices: convergence ofnon linear iterative process; truncated, or chaotic, or boolean Stein-Rosenberg theorems.The conclusion is the following: As it works through various mathematical contexts, the Stein-Rosenberg theorem expersses a rather deap idea of an essentially algorithmic nature.     

Approximation für Okasche Paare von Garben homogener Räume

This article brings an approximation theorem for sections of Oka pairs of sheaves of homogeneous spaces (such sheaves were introduced in [15]). Among the applications are: an Oka principle for the approximation of a generating system for a coherent analytic sheaf and an Oka principle for the approximation of a holomorphic fibre bundle with homogeneous fibre (without a structure group). Together with a homotopy theorem [15] for Oka pairs of sheaves of homogeneous spaces, a theoretical unification is thus given for various known results on the Oka principle [12,20,6,7].     

A generalized Kantorovich theorem for nonlinear equations based on function splitting

The Kantorovich theorem is a fundamental tool in nonlinear analysis for proving the existence and uniqueness of solutions of nonlinear equations arising in various fields. In the present paper we formulate and prove a generalized Kantorovich theorem that contains as special cases the Kantorovich theorem and a weak Kantorovich theorem recently proved by Uko and Argyros. An illustrative example is given to show that the new theorem is applicable in some situations in which the other two theorems are not applicable.     

Ideal version of Ramsey’s theorem

We consider various forms of Ramsey’s theorem, the monotone subsequence theorem and the Bolzano-Weierstrass theorem which are connected with ideals of subsets of natural numbers. We characterize ideals with properties considered. We show that, in a sense, Ramsey’s theorem, the monotone subsequence theorem and the Bolzano-Weierstrass theorem characterize the same class of ideals. We use our results to show some versions of density Ramsey’s theorem (these are similar to generalizations shown in [P. Frankl, R. L. Graham, and V. Rödl: Iterated combinatorial density theorems.     

A robust von Neumann minimax theorem for zero-sum games under bounded payoff uncertainty

The celebrated von Neumann minimax theorem is a fundamental theorem in two-person zero-sum games. In this paper, we present a generalization of the von Neumann minimax theorem, called robust von Neumann minimax theorem, in the face of data uncertainty in the payoff matrix via robust optimization approach. We establish that the robust von Neumann minimax theorem is guaranteed for various classes of bounded uncertainties, including the matrix 1-norm uncertainty, the rank-1 uncertainty and the columnwise affine parameter uncertainty.     

Mercer theorem for RKHS on noncompact sets

Reproducing kernel Hilbert spaces are an important family of function spaces and play useful roles in various branches of analysis and applications including the kernel machine learning. When the domain of definition is compact, they can be characterized as the image of the square root of an integral operator, by means of the Mercer theorem. The purpose of this paper is to extend the Mercer theorem to noncompact domains, and to establish a functional analysis characterization of the reproducing kernel Hilbert spaces on general domains.     

Ramsey's theorem with sums or unions

Using Hindman's theorem as a strong pigeonhole principle, we prove strengthened versions of Ramsey's theorem and of various generalizations of Ramsey's theorem due to Nash-Williams, Galvin and Prikry, and Silver.     

Strong Connectivity of Polyhedral Complexes

A classical theorem of Robbins states that the edges of a graph may be oriented, in such a way that an oriented path exists between any source and destination, if and only if the graph is both connected and two-connected (it cannot be disconnected by the removal of an edge). In this paper, an algebraic version of Robbins' result becomes a lemma on Hilbert bases for free abelian groups, which is then applied to generalize his theorem to higher dimensional complexes. An application to cycle bases for graphs is given, and various examples are presented.     

Bounded turning and quasiconformal maps

We considern-dimensional quasiconformal maps of an arbitrary domain onto a domain with specific properties and prove various results related to the Hayman-Wu theorem.     

模糊量及模糊数的表现特性

模糊集的表现定理是模糊数学的最基本理论.在表现定理的基础上,对各种模糊量:包括凸模糊量、正规模糊量、正规凸模糊量、凸有界模糊量、模糊数、有限模糊数、对称模糊数的表现定理进行了深入的研究,从而建立了不同类型模糊量与普通集合之间的联系.     

论Whittaker-Shannon抽样定理及其一些推广

The aim of this paper is to present a survey of results concerning the Whittaker-Kotel'nikov-Raabe-Shannon-Someya sampling theorem and its various extensions obtained at Aachen since 1977. This theorem, basic in communication engineering, is often called the cardinal interpolation series theorem in mathematical circles. The interconnections of the sampling theorem (in the setting of Paley-Wiener space) with the theory of Fourier series and integrals are examined. Emphasis is placed upon error analysis, including the aliasing, round-off (or quantization), and time jitter errors. Some new error estimates are given, others are improved; many of the proofs are reduced to a common structure. Both deterministic and probabilistic methods are employed. whereas these results are worked out in detail, the paper also contains a brief discussion of some of the various generalizations.