A factorization theorem for matrices

It is shown that a nonscalar invertible square matrix can be written as a product of two square matrices with prescribed eigenvalues subject only to the obvious determinant condition. As corollaries, we give short proofs of some known results such as Ballantine's characterization of products of four or five positive definite matrices, the commutator theorem of Shoda-Thompson for fields with sufficiently many elements and other results.     

A factorization theorem for banded matrices

In this paper we prove a factorization theorem for strictly m-banded totally positive matrices. We show that such a matrix is a product of m one-banded matrices with positive entries.     

A unique factorization theorem for matroids

We study the combinatorial, algebraic and geometric properties of the free product operation on matroids. After giving cryptomorphic definitions of free product in terms of independent sets, bases, circuits, closure, flats and rank function, we show that free product, which is a noncommutative operation, is associative and respects matroid duality. The free product of matroids M and N is maximal with respect to the weak order among matroids having M as a submatroid, with complementary contraction equal to N. Any minor of the free product of M and N is a free product of a repeated truncation of the corresponding minor of M with a repeated Higgs lift of the corresponding minor of N. We characterize, in terms of their cyclic flats, matroids that are irreducible with respect to free product, and prove that the factorization of a matroid into a free product of irreducibles is unique up to isomorphism. We use these results to determine, for K a field of characteristic zero, the structure of the minor coalgebra of a family of matroids that is closed under formation of minors and free products: namely, is cofree, cogenerated by the set of irreducible matroids belonging to .     

A convergence theorem for spectral factorization

This paper presents a convergence theorem for an iterative method of spectral factorization in the context of multivariate prediction theory. It may be viewed as a constructive proof that the factorization exists, using only the analytic results of Hardy space theory.     

A characterization ofC 1-convex sets in Sobolev spaces

In this paper we prove that, ifK is a closed subset ofW ₀ ^1,p (Ω,R ^m ) with 1<p<+∞ andm≥1, thenK is stable under convex combinations withC ¹ coefficients if and only if there exists a closed and convex valued multifunction from Ω toR ^m such that The casem=1 has already been studied by using truncation arguments which rely on the order structure ofR (see [2]). In the casem>1 a different approach is needed, and new techniques involving suitable Lipschitz projections onto convex sets are developed. Our results are used to prove the stability, with respect to the convergence in the sense of Mosco, of the class of convex sets of the form (*); this property may be useful in the study of the limit behaviour of a sequence of variational problems of obstacle type. This article was processed by the author using the Springer-Verlag TEX mamath macro package 1990     

A theorem on the factorization of matrix polynomials

A similarity condition is developed for the factorization of monic matrix polynomials L(λ) into the forms L(λ) = L_k(λ) … L₁(λ), wihtout any restriction on the spectrum of factors L_j(λ).     

AnR danalogue of Valentine’s theorem on 3-convex sets

This paper deals with anR ^danalogue of a theorem of Valentine which states that a closed 3-convex setS in the plane is decomposable into 3 or fewer closed convex sets. In Valentine’s proof, the points of local nonconvexity ofS are treated as vertices of a polygonP contained in the kernel ofS, yielding a decomposition ofS into 2 or 3 convex sets, depending on whetherP has an even or odd number of edges. Thus the decomposition actually depends onc(P′), the chromatic number of the polytopeP′ dual toP. A natural analogue of this result is the following theorem: LetS be a closed subset ofR ^d, and letQ denote the set of points of local nonconvexity ofS. We require thatQ be contained in the kernel ofS and thatQ coincide with the set of points in the union of all the (d − 2)-dimensional faces of somed-dimensional polytopeP. ThenS is decomposable intoc(P′) closed convex sets.     

A general coincidence theorem on contractible spaces

We obtain a general coincidence theorem for multifunctions in very large classes defined on contractible spaces. Our theorem generalizes a recent result due to Tarafdar and Yuan (1994) and many other earlier works including the Fan-Browder fixed point theorem.

     

A theorem of Cohen on parapolar spaces

We give an elementary proof of a critical lemma of Arjeh Cohen used in his fundamental paper giving an axiomatic characterization of Grassmann spaces of finite singular rank.     

A mean ergodic theorem on vector spaces

A mean ergodic theorem for a matrix is first proved from which a mean ergodic theorem for affine operators on a vector space without any topological structure is obtained.     

A factorization theorem for comaps of geometric lattices

A function γ: K → L between two geometric lattices K and L is a normalized comap if it preserves the relations: x covers or equals y, meets of modular pairs, and the minimum. The theorem, a normalized comap can be factored into an injection followed by a retraction onto a modular flat, is proved.     

Embedding theorems and Kahlerity for 1-convex spaces

Unless the contrary is explicitly stated, allℂ-analytic spaces considered here are assumed to be non compact, countable at infinity and of bounded Zariski dimension. The author gratefully acknowledges the generous support from the N.S.F. Grant #MCS 81-02266. Also he would like to thank Prof. H. Hironaka for bringing this problem to his attention.