A maximality theorem concerning extreme points

title: Bemerkungen zum von Neumannschen Minimaxtheorem

title: Varianten zum Konvergenzsatz von Simons und Anwendungen in der Choquettheorie

title: Continuity of sublinear operators on F-spaces
abstract: In this note, some of the fundamental theorems concerning the automatic continuity of positive linear functionals and positive linear operators (see [9,14,21]) as well as certain uniform boundedness type theorems (see [5,13,15,19]) will be extended to derive various continuity properties of (even discontinuous) sublinear operators from some metrizable topological vector space to some ordered topological vector space.

title: Automatic Continuity of Linear Operators
abstract: The present survey is essentially based on some recent automatic continuity results which have been obtained by Ernst Albrecht and the author in the joint papers [1;2;3]. In contrast to the excellent account of automatic continuity theory in Sinclair's lecture notes [29] and in Dales' recent survey article , the emphasis of the present exposition lies on the automatic continuity problem for linear operators acting between spaces of functions and distributions. The continuity of such operators will be derived from certain conditions which naturally arise in the theory of linear systems, namely from time-invariance and causality on the one side and from dissipativity on the other. The automatic continuity of all time-invariant and causal linear operators turns out to be just one achievement of a far-reaching new theory on automatic continuity which also applies to the problem of continuity for algebrahomomorphisms, derivations, intertwining operators, and local operators. Some central features of this theory will be sketched; in particular, some extensions and variants of the classical principle of uniform boundedness will be discussed.

title: Beschränktheitsaussagen für sublineare und bisublineare Funktionale

title: Monotonic sequences and rates of convergence of asynchronized iterative methods
abstract: In a recent paper B. Vemmer and the authors investigated the effect of varying the number of processors on the rate of convergence of the asynchronized parallel block Jacobi method associated with monotone matrices. It was found that, under certain simplifying assumptions, increasing the number of processors in relation to the number of blocks (or, what comes to the same in more general settings, the number of iteration operators) slows down the convergence. One interpretation for these results given in that paper was that increasing the number of processors means that when the current global approximation is updated by a local approximation from one of the processors, that local approximation was computed from a “much” earlier global approximation received from the host node. Hence the slowdown in the rate of convergence. The principal purpose of this paper is to remove some of the simplifying assumptions that were made in the above-mentioned paper and to prove that many of the results there hold under much more general conditions. Our present assumptions do not yield a fixed iteration matrix which models the process as was the case previously. This means that different tools have to be developed to establish results comparing the rate of convergence of two assynchronized processes.

title: On the spectrum of the SOR operator for symmetric positive definite matrices
abstract: One attractive feature of the classical successive overrelaxation (SOR) iterative method is its algorithmic simplicity. Whereas for linear systems with symmetric positive definite matrices A, the theorem of Ostrowski guarantees convergence as long as the relaxation parameter ω∈(0,2), results on the optimal choice of ω can only be obtained under certain additional assumtions on A. Here, for positive definite systems with coefficient matrices A=I−L−LT in which L is not necessarily strictly lower triangular, we study the behavior of the spectrum of the SOR operator Lω as ω→0 and ω→2 and describe enclosure sets which may be used to estimate the spectral radius of Lω for 0<ω<2. Numerical experiments show why results on the optimal relaxation parameter are difficult to obtain in the general case of positive definite systems.

title: Subproper splitting for rectangular matrices
abstract: In this paper iterative schemes for approximating a solution to a rectangular but consistent linear system Ax = b are studied. Let A ϵ Cm × nr. The splitting A = M − N is called subproper if R(A) ⊆ R(M) and R(A ∗ ) ⊆R(M ∗ ) . Consider the iteration x i = M † Nx i −1 + M † b . We characterize the convergence of this scheme to a solution of the linear system. When A ϵ Rm×nr, monotonicity and the concept of subproper regular splitting are used to determine a necessary and a sufficient condition for the scheme to converge to a solution.

title: The university of South Carolina mini-conference on matrix theory and linear algebra

title: Boundary results for positively invariant cones and their reachability cones
abstract: We say that a closed, convex, and solid cone is positively invariant with respect to the linear system of differential equations . In a previous paper, , the authors obtained formulas for the closure of the reachability cone under certain additional assumptions on A. The purpose of the present paper is to characterize the set XA (K) ⋂ ∂ XA (K) which consists of those points on the boundary of XA (K) which actually reach A, in the case when K is a simplicial cone. We use this characterization to construct an algorithm for the computation of XA (K) from its closure. Furthermore, to obtain our characterization we develop a result which is of interest in itself, namely, that the maximal positively invariant subset of the boundary of a positively invariant simplicial cone k is, precisely, the union of its positively invariant faces.