DISCRETE UNIFORM STRUCTURES AND QUASITOPOI

It is known that no non-trivial subcategory of the category of topological spares is a quasitopos in the sense of Penon. The purpose of this note is to establish the existence of a proper class of sub-categories of the category of uniform spaces which are quasitopoi. These subcategories are generated by certain proximally discrete uniform spaces which correspond to each infinite regular cardinal.     

FUZZY SYNTOPOGENEOUS STRUCTURES

Abstract

The author gives a detailed analysis of the relation between the theories of realcompactifications and compactifications in the category of ditopological texture spaces and in the categories of bitopological spaces and topological spaces.     

SOME OF MY FAVORITE SOLVED AND UNSOLVED PROBLEMS IN GRAPH THEORY

Abstract

During my long life I published many papers with related titles. To keep this paper short I will not give proofs and will restrict myself to problems in graph theory, but I will try to give references and make these as complete as possible.

I will start with Turk type problems in extremal graph theory.     

A CHARACTERIZATION OF LINEAR OPERATORS IN CERTAIN PARANORMED SEQUENCE SPACES

Abstract

We define a new class B of sets of strongly summable and bounded sequences, determine their first and second α- and β-duals, and characterize linear operators on the spaces in B.     

Smoothing properties of transition semigroups in hilbert spaces

The paper gives conditions under which the transition semigroup corresponding to a large class of semilinear equations on a Hilbert space transforms Borel functions onto Frechet differentia hies ones.     

Multiresolution schemes for conservation laws

Summary. In recent years a variety of high–order schemes for the numerical solution of conservation laws has been developed. In general, these numerical methods involve expensive flux evaluations in order to resolve discontinuities accurately. But in large parts of the flow domain the solution is smooth. Hence in these regions an unexpensive finite difference scheme suffices. In order to reduce the number of expensive flux evaluations we employ a multiresolution strategy which is similar in spirit to an approach that has been proposed by A. Harten several years ago. Concrete ingredients of this methodology have been described so far essentially for problems in a single space dimension. In order to realize such concepts for problems with several spatial dimensions and boundary fitted meshes essential deviations from previous investigations appear to be necessary though. This concerns handling the more complex interrelations of fluxes across cell interfaces, the derivation of appropriate evolution equations for multiscale representations of cell averages, stability and convergence, quantifying the compression effects by suitable adapted multiscale transformations and last but not least laying grounds for ultimately avoiding the storage of data corresponding to a full global mesh for the highest level of resolution. The objective of this paper is to develop such ingredients for any spatial dimension and block structured meshes obtained as parametric images of Cartesian grids. We conclude with some numerical results for the two–dimensional Euler equations modeling hypersonic flow around a blunt body. Received June 24, 1998 / Revised version received February 21, 2000 / Published online November 8, 2000     

The sdfem on Shishkin meshes for linear convection-diffusion problems

Summary. We consider singularly perturbed linear elliptic problems in two dimensions. The solutions of such problems typically exhibit layers and are difficult to solve numerically. The streamline diffusion finite element method (SDFEM) has been proved to produce accurate solutions away from any layers on uniform meshes, but fails to compute the boundary layers precisely. Our modified SDFEM is implemented with piecewise linear functions on a Shishkin mesh that resolves boundary layers, and we prove that it yields an accurate approximation of the solution both inside and outside these layers. The analysis is complicated by the severe nonuniformity of the mesh. We give local error estimates that hold true uniformly in the perturbation parameter , provided only that , where mesh points are used. Numerical experiments support these theoretical results. Received February 19, 1999 / Revised version received January 27, 2000 / Published online August 2, 2000     

Ryser's Conjecture for Tripartite 3-Graphs

We prove that in a tripartite 3-graph . Received March 17, 1999     

Rank reduction, factorization and conjugation

We investigate the rank reduction procedure, related factorizations and conjugation algorithms. An exact characterization of the full rank factorization produced by the rank reduction algorithm is given. This result is then used to derive matrix decompositions and conjugation procedures.