Semi-Lipschitz Functions and Best Approximation in Quasi-Metric Spaces

We show that the set of semi-Lipschitz functions, defined on a quasi-metric space (X, d), that vanish at a fixed point x₀X can be endowed with the structure of a quasi-normed semilinear space. This provides an appropriate setting in which to characterize both the points of best approximation and the semi-Chebyshev subsets of quasi-metric spaces. We also show that this space is bicomplete.     

A Characterization of Generalized Monotone Normed Cones

Let C be a cone and consider a quasi-norm p defined on it. We study the structure of the couple （C, p） as a topological space in the case where the function p is also monotone. We characterize when the topology of a quasi-normed cone can be defined by means of a monotone norm. We also define and study the dual cone of a monotone normed cone and the monotone quotient of a general cone. We provide a decomposition theorem which allows us to write a cone as a direct sum of a monotone subcone that is isomorphic to the monotone quotient and other particular subcone.     

The distributional dimension of fractals

In the book [1] H.Triebel introduces the distributional dimension of fractals in an analytical form and proves that： for Г as a non-empty set in R^n with Lebesgue measure ｜Г｜ = 0, one has dimH Г = dimD Г, where dimD Г and dimH Г are the Hausdorff dimension and distributional dimension, respectively. Thus we might say that the distributional dimension is an analytical definition for Hausdorff dimension. Therefore we can study Hausdorff dimension through the distributional dimension analytically. By discussing the distributional dimension, this paper intends to set up a criterion for estimating the upper and lower bounds of Hausdorff dimension analytically. Examples illustrating the criterion are included in the end.     

A posteriori FE error control for p-Laplacian by gradient recovery in quasi-norm

A posteriori error estimators based on quasi-norm gradient recovery are established for the finite element approximation of the p-Laplacian on unstructured meshes. The new a posteriori error estimators provide both upper and lower bounds in the quasi-norm for the discretization error. The main tools for the proofs of reliability are approximation error estimates for a local approximation operator in the quasi-norm.

     

A posteriori analysis of finite element discretizations of stochastic partial differential delay equations

In this paper, we study a posteriori error estimates for finite element approximation of stochastic partial differential delay equations containing a noise. We derive an energy norm a posteriori bounds for an Euler time-stepping method combined with a standard Galerkin schemes for the problems. For accessibility, we first address the spatially semidiscrete case and then move to the fully discrete scheme.     

On paratopological vector spaces

We show that each first countable paratopological vector space X has a compatible translation invariant quasi-metric such that the open balls are convex whenever X is a pseudoconvex vector space. We introduce the notions of a right-bounded subset and of a right-precompact subset of a paratopological vector space X and prove that X is quasi-normable if and only if the origin has a convex and right-bounded neighborhood. Duality in this context is also discussed. Furthermore, it is shown that the bicompletion of any paratopological vector space (respectively, of any quasi-metric vector space) admits the structure of a paratopological vector space (respectively, of a quasi-metric vector space). Finally, paratopological vector spaces of finite dimension are considered. This revised version was published online in June 2006 with corrections to the Cover Date.