INVARIANT SUBSPACES AND EIGEVALUES OF ELEMENTS IN C-ALGEBRAS

Let A be a C~*-algebra and x an element in A. the following invariant subspace problem is considered: Does there exist an irreducible representation π of A such that π(x) has a non-trivial invarint subspace? And a positive solution of the problem for finite separable matroid C~*-algebras is given. Also the eigenvalues Of elements in C~*-algebras is considered. Some versions of Fredholm Alternatives are given.     

Coloring by two-way independent sets

The paper stems from an attempt to investigate a somewhat mysterious phenomenon: conditions which suffice for the existence of a “large” set satisfying certain conditions (e.g., a large independent set in a graph) often suffice (or at least are conjectured to suffice) for the existence of a covering of the ground set by few sets satisfying these conditions (in the example of independent sets in a graph this means that the graph has small chromatic number). We consider two conjectures of this type, on coloring by sets which are “two-way independent”, in the sense of belonging to a matroid and at the same time being independent in a graph sharing its ground set with the matroid. We prove these conjectures for matroids of rank 2. We also consider dual conjectures, on packing bases of a matroid, which are independent in a given graph.     

Lifting Property of Direct Sums of Hollow Modules

We give necessary and sufficient conditions for a module M with a decomposition that complements maximal direct summands to be a lifting module. We apply our results to get a necessary and sufficient condition for a direct sum of modules to be a quasi-discrete (discrete) module. We also give some sufficient conditions for a lifting module to be a direct sum of indecomposables.     

Infinite Matroidal Version of Hall's Matching Theorem

Hall's theorem for bipartite graphs gives a necessary and sufficientcondition for the existence of a matching in a given bipartitegraph. Aharoni and Ziv have generalized the notion of matchabilityto a pair of possibly infinite matroids on the same set andgiven a condition that is sufficient for the matchability ofa given pair (M, W) of finitary matroids, where the matroidM is SCF (a sum of countably many matroids of finite rank).The condition of Aharoni and Ziv is not necessary for matchability.The paper gives a condition that is necessary for the existenceof a matching for any pair of matroids (not necessarily finitary)and is sufficient for any pair (M, W) of finitary matroids,where the matroid M is SCF.     

关于全不变扩张环和模(英文)

In this paper, we discuss FI-extending property of rings and modules. The main results are the following： a characterization of von Neumann regular rings which are two-sided FI-extending is given; sufficient conditions for direct summands of FI-extending modules to be FI-extending are obtained; and at last, a necessary and sufficient condition for nonsingular modules over nonsingular rings to be FI-extending is given.     

On critical exponents of Dowling matroids

The critical problem in matroid theory is the problem to determine the critical exponent of a given representable matroid over a finite field. In this paper, we study the critical exponents of a class of representable matroids over finite fields, called Dowling matroids. Then the critical problem for a Dowling matroid is corresponding to the classical problem in coding theory to determine the maximum dimension k such that there exists an \([n,k,d]_q\) code for given n, d and q. We give a necessary and sufficient condition on the critical exponents of Dowling matroids by using a coding theoretical approach.     

广义正交分解定理与Tseng度量广义逆

本文将Banach空间中广义正交分解定理从线性子空间拓广至非线性集—太阳集,分别给出了一算子为度量投影算子和一度量投影算子为有界线性算子的充要条件;得到了判别Banach空间中子空间广义正交可补的充要条件;建立了王玉文和季大琴(2000年)新近引入的Banach空间中的线性算子的Tseng度量广义逆存在的特征刻划条件;这些工作本质地把王玉文等人的新近结果从自反空间拓广至非自反空间的情形.     

Uniformly resolvable pairwise balanced designs with blocksizes two and three

A uniformly resolvable pairwise balanced design is a pairwise balanced design whose blocks can be resolved into parallel classes in such a way that all blocks in a given parallel class have the same size. We are concerned here with designs in which each block has size two or three, and we prove that the obvious necessary conditions on the existence of such designs are also sufficient, with two exceptions, corresponding to the non-existence of Nearly Kirkman Triple Systems of orders 6 and 12.     

On the Best Approximation in Non-Archimedean Normed Linear Spaces

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New Light on Bergman Complexes by Decomposing Matroid Types

We give a shorter proof of the fact, that Bergman complexes of matroids can be subdivided to realizations of the nested set complexes of the lattice of flats. Then, we present a direct sum decomposition into connected summands of the matroid types of faces of Bergman complexes.     

The Metapositive Definite Self－Conjugate Solution of the Matrix Equation AXB＝C over a Skew Field

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Complexity of Boolean algebras and their Scott rank

We deal with problems associated with Scott ranks of Boolean algebras. The Scott rank can be treated as some measure of complexity of an algebraic system. Our aim is to propound and justify the procedure which, given any countable Boolean algebra, will allow us to construct a Boolean algebra of a small Scott rank that has the same natural algebraic complexity as has the initial algebra. In particular, we show that the Scott rank does not always serve as a good measure of complexity for the class of Boolean algebras. We also study into the question as to whether or not a Boolean algebra of a big Scott rank can be decomposed into direct summands with intermediate ranks. Examples are furnished in which Boolean algebras have an arbitrarily big Scott rank such that direct summands in them either have a same rank or a fixed small one, and summands of intermediate ranks are altogether missing. This series of examples indicates, in particular, that there may be no nontrivial mutual evaluations for the Scott and Frechet ranks on a class of countable Boolean algebras. Supported by RFFR grant No. 99-01-00485, by a grant for Young Scientists from SO RAN, 1997, and by the Federal Research Program (FRP) “Integration”. Translated fromAlgebra i Logika, Vol. 38, No. 6, pp. 643–666, November–December, 1999.     

Matroids,generalized networks,and electric network synthesis

Matroid theory has been applied to solve problems in generalized assignment, operations research, control theory, network theory, flow theory, generalized flow theory or linear programming, coding theory, and telecommunication network design. The operations of matroid union, matroid partitioning, matroid intersection, and the theorem on the greedy algorithm, Rado's theorem, and Brualdi's symmetric version of Rado's theorem have been important for some of these applications. In this paper we consider the application of matroids to solve problems in network synthesis. Previously Bruno and Weinberg defined a generalized network, which is a network based on a matroid rather than a graph; for a generalized network the duality principle holds whereas it does not hold for a network based on a graph. We use the concept of the generalized network to formulate a solution to the following problem: What are the necessary and sufficient conditions for a singular matrix of real numbers, of order p and rank s, to be realizable as the open-circuit resistance matrix of a resistance p-port network. A simple algorithm is given for carriyng out the synthesis. We then present a number of unsolved problems, included among which is what could be called the four-color problem of network synthesis, namely, the resistance n-port problem.     

Existence results for a class of hemivariational inequality problems on Hadamard manifolds

In this paper, a class of hemivariational inequality problems are introduced and studied on Hadamard manifolds. Using the properties of Clarke’s generalized directional derivative and Fan-KKM lemma, an existence theorem of solution in connection with the hemivariational inequality problem is obtained when the constraint set is bounded. By employing some coercivity conditions and the properties of Clarke’s generalized directional derivative, an existence result and the boundedness of the set of solutions for the underlying problem are investigated when the constraint set is unbounded. Moreover, a sufficient and necessary condition for ensuring the nonemptiness of the set of solutions concerned with the hemivariational inequality problem is also given.     

On the (co)girth of a connected matroid

This article studies the girth and cogirth problems for a connected matroid. The problem of finding the cogirth of a graphic matroid has been intensively studied, but studies on the equivalent problem for a vector matroid or a general matroid have been rarely reported. Based on the duality and connectivity of a matroid, we prove properties associated with the girth and cogirth of a matroid whose contraction or restriction is disconnected. Then, we devise algorithms that find the cogirth of a matroid M from the matroids associated with the direct sum components of the restriction of M. As a result, the problem of finding the (co)girth of a matroid can be decomposed into a set of smaller sub-problems, which helps alleviate the computation. Finally, we implement and demonstrate the application of our algorithms to vector matroids.     

Face numbers of Engström representations of matroids

A classic problem in the theory of matroids is to find a subspace arrangement, such as a hyperplane or pseudosphere arrangement, whose intersection poset is isomorphic to a prescribed geometric lattice. Engström gave an explicit construction for an infinite family of such arrangements, indexed by the set of finite regular CW complexes. In this paper, we compute the face numbers of these topological representations in terms of the face numbers of the indexing complexes and give upper bounds on the total number of faces in these objects. Moreover, for a fixed rank, we show that the total number of faces in the Engström representation corresponding to a codimension one homotopy sphere arrangement is bounded above by a polynomial in the number of elements of the matroid, whose degree is one less than the matroid’s rank.     

Local sidon sets and uniform convergence of fourier series

For compact groups several necessary and sufficient conditions for a set to be local Sidon are given; these conditions are expressed in functional-analytic terms. An immediate corollary is the existence of a large class of compact groupsG, including all the connected non-Abelian Lie groups, which support functions not inA(G) but having Fourier series that are both unformly and absolutely convergent. This has been shown for the special case ofSU(2) by Mayer. Also several necessary and sufficient conditions for a set to be local Λ (p) are given.     

Decompositions of a Krein space in regular subspaces invariant under a uniformly bounded C0-semigroup of bi-contractions

We give necessary and sufficient conditions under which a C₀-semigroup of bi-contractions on a Krein space is similar to a semigroup of contractions on a Hilbert space. Under these and additional conditions we obtain direct sum decompositions of the Krein space into invariant regular subspaces and we describe the behavior of the semigroup on each of these summands. In the last section we give sufficient conditions for the co-generator of the semigroup to be power bounded.     

两大部类扩大再生产的充分必要条件与求解

两大部类扩大再生产的充分必要条件也就是马克思扩大再生产公式有解的充分必要条件.直到目前,由于还没有严谨地提出对于扩大再生产公式的一般求解方法,因而也没有严格地确定扩大再生产的充分必要条件.本文基于已经有研究获得的扩大再生产的一个必要条件,建立一种求解扩大再生产问题的一般方法,从而证明这个必要条件能够成为充分条件,由此确定它是充分必要条件.进而使用变量替换法,给出了通过直接求解两个部类的剩余价值积累率而求解扩大再生产问题的另一种方法.最后引用《资本论》中的两个实例,对所给出的两种求解扩大再生产问题的一般方法做了计算验证.     

Tags on subsets

A new definition of a tag on a subset of a finite set is given. Tags were recently defined in a joint paper of the author and J.S. Chahal. The new definition considerably simplifies the concepts further. Relationship with lexicographic ordering is much more visible. Applications to a general (t,k) existence problem which includes the existence conjecture for t-designs or characterizing degree sequences of a k-uniform hypergraphs as particular cases is discussed. Some new necessary inequalities, as well as some sufficient conditions for such existence questions are derived.