A pólya interpretation of the schur function

When the Schur function is written as a linear combination of products of symmetric power sums, it takes the form of a character-weighted cycle index polynomial. A Pólya-like interpretation is given to this formula and a purely combinatorial proof is given. Some observations concerning general group actions are made.     

On schur rings over cyclic groups

In this paper, we introduce the notion of wedge product of Schur rings. We show that for any nontrivial Schur ringS over a cyclic groupG, if there is a subgroupH such that Σ _g ε _H g Σ _g∈H g ∉S, thenS is either a dot product or wedge product for some Schur rings over smaller cyclic groups.     

On minimal non- -groups

A finite group G is called an J N J-group if every proper subgroup H of G is either subnormal in G or self-normalizing. We determinate the structure of non-J N J-groups in which all proper subgroups are J N J- groups.     

On schur complements in an ep matrix

It is established that under certain conditions a Schur complement in an EP matrix is as well an EP matrix. As an application a decomposition of a partitioned matrix into a sum of EP matrices is given.     

On the multiplier rules

We establish new results of first-order necessary conditions of optimality for finite-dimensional problems with inequality constraints and for problems with equality and inequality constraints, in the form of John’s theorem and in the form of Karush–Kuhn–Tucker’s theorem. In comparison with existing results, we weaken assumptions of continuity and of differentiability.     

On the structure of CDN[]-groups

We describe nilpotent non-Dedekind CDN[]-groups. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 9, pp. 1250–1261, September, 1998.     

M p -groups not containing groups of quaternions

This paper continues the study of the class of Mp-groups introduced by V. P. Shunkov. We obtain a criterion of nonsimplicity of an infinite group which contains anMp-group and contains no group of quaternions. We also obtain a criterion for an infinite group to be an Mp-group.     

On the multiplier conjecture

The Multiplier Theorem is a celebrated theorem in the Design theory. The conditionp>λ is crucial to all known proofs of the multiplier theorem. However in all known examples of difference sets μ _p . is a multiplier for every primep with (p, v)=1 andp‖n. Thus there is the multiplier conjecture: “The multiplier theorem holds without the assumption thatp>λ”. The general form of the multiplier theorem may be viewed as an attempt to partially resolve the multiplier conjecture, where the assumption “p>λ” is replaced by “n ₁>λ”. Since then Newman (1963), Turyn (1964), and McFarland (1970) attempted to partially resolve the multiplier conjecture (see [7], [8], [9]). This paper will prove the following result using the representation theory of finite groups and the algebraic number theory: LetG be an abelian group of orderv,v ₀ be the exponent ofG, andD be a (v, k, λ)-difference set inG. Ifn=2_{n 1}, then the general form of the multiplier theorem holds without the assumption thatn ₁>λ in any of the following cases:

Block-matrix generalizations of infinite-dimensional schur products and schur multipliers

We consider an extention of the familiar Schur product to a bilinear product on the space of matrices whose entries are either bounded operators on a fixed Hilbert space or bounded "square" operator matrices. We show that this is a "natural" non-commutative extention of the Schur product, which retains many of its properties. The work is done mainly in infinite dimensions, where we concentrate on the maps induced on the space of bounded operator matrices via left or right "Schur block-multiplication" by a fixed "Schur block-multiplier". Our main goal is to study the distinctions between left and right multipliers, as well as the behaviour of ideals of operators under action of maps induced bu such.     

On inertia and schur complement in optimization

Partitioned symmetric matrices, in particular the Hessian of the Lagrangian, play a fundamental role in nonlinear optimization. For this type of matrices S.-P. Han and O. Fujiwara recently presented an inertia theorem under a certain regularity assumption. We prove that this theorem is true without any regularity assumption. Then we consider matrix extensions preserving the sign of the determinant. Such extensions are shown to be related with the positive definiteness of some Schur complement. Under a regularity assumption this shows, from the viewpoint of linear algebra, the equivalence of strong stability in the sense of M. Kojima and strong regularity in the sense of S. M. Robinson. Finally, we discuss the inertia of a typical one-parameter family of symmetric matrices, occurring in various places in optimization (augmented Lagrangians, focal-point theory, etc.).     

内PN*-群的结构(英文)

若有限群G的每个极小子群及4阶循环子群在G中正规,则称G为PN~*-群.本文给出了有限群的每个真子群都是PN~*-群但其本身不是PN~*-群的完全分类.