A p-version finite element method for nonlinear elliptic variational inequalities in 2D

This article introduces and analyzes a p-version FEM for variational inequalities resulting from obstacle problems for some quasi-linear elliptic partial differential operators. We approximate the solution by controlling the obstacle condition in images of the Gauss–Lobatto points. We show existence and uniqueness for the discrete solution u _p from the p-version for the obstacle problem. We prove the convergence of u _p towards the solution with respect to the energy norm, and assuming some additional regularity for the solution we derive an a priori error estimate. In numerical experiments the p-version turns out to be superior to the h-version concerning the convergence rate and the number of unknowns needed to achieve a certain exactness of the approximation.     

Buchsbaum homogeneous algebras with minimal multiplicity

In this paper we first give a lower bound on multiplicities for Buchsbaum homogeneous k-algebras A in terms of the dimension d, the codimension c, the initial degree q, and the length of the local cohomology modules of A. Next, we introduce the notion of Buchsbaum k-algebras with minimal multiplicity of degree q, and give several characterizations for those rings. In particular, we will show that those algebras have linear free resolutions. Further, we will give many examples of those algebras.     

Metric marginal problems for set-valued or non-measurable variables

Summary In a separable metric space, if two Borel probability measures (laws) are nearby in a suitable metric, then there exist random variables with those laws which are nearby in probability. Specifically, by a well-known theorem of Strassen, the Prohorov distance between two laws is the infimum of Ky Fan distances of random variables with those laws. The present paper considers possible extensions of Strassen's theorem to two random elements one of which may be (compact) set-valued and/or non-measurable. There are positive results in finite-dimensional spaces, but with factors depending on the dimension. Examples show that such factors cannot entirely be avoided, so that the extension of Strassen's theorem to the present situation fails in infinite dimensions.This research was partially supported by a Guggenheim Fellowship, by National Science Foundation grant DMS 8505550 at MSRI-Berkeley, and other NSF grants     

Asymptotic Distribution of Nodes for Near-Optimal Polynomial Interpolation on Certain Curves in R 2

Let E\subset \Bbb R ^s be compact and let d _n ^E denote the dimension of the space of polynomials of degree at most n in s variables restricted to E . We introduce the notion of an asymptotic interpolation measure (AIM). Such a measure, if it exists , describes the asymptotic behavior of any scheme τ _n ={ \bf x _k,n } _k=1 ^dnE , n=1,2,\ldots , of nodes for multivariate polynomial interpolation for which the norms of the corresponding interpolation operators do not grow geometrically large with n . We demonstrate the existence of AIMs for the finite union of compact subsets of certain algebraic curves in R ² . It turns out that the theory of logarithmic potentials with external fields plays a useful role in the investigation. Furthermore, for the sets mentioned above, we give a computationally simple construction for ``good' interpolation schemes. November 9, 2000. Date revised: August 4, 2001. Date accepted: September 14, 2001.     

A supraconvergent scheme for the Korteweg-de Vries equation

Summary A finite-difference method for the integration of the Korteweg-de Vries equation on irregular grids is analyzed. Under periodic boundary conditions, the method is shown to be supraconvergent in the sense that, though being inconsistent, it is second order convergent. However, such a convergence only takes place on grids with an odd number of points per period. When a grid with an even number of points is used, the inconsistency of the method leads to divergence. Numerical results backing the analysis are presented.     

Some characterisations of supported algebras

We give several equivalent characterisations of left (and hence, by duality, also of right) supported algebras. These characterisations are in terms of properties of the left and the right parts of the module category, or in terms of the classes L₀ and R₀ which consist respectively of the predecessors of the projective modules, and of the successors of the injective modules.     

On bimeasurings II

We answer some questions posed in [L. Grunenfelder, M. Mastnak, On bimeasurings, J. Pure Appl. Algebra 204 (2006) 258-269] concerning the universal bimeasuring bialgebra via a construction of suitable subcoalgebras of the universal measuring coalgebra.     

Motivic Serre invariants of curves

In this article, we generalize the theory of motivic integration on formal schemes topologically of finite type and the notion of motivic Serre invariant, to a relative point of view. We compute the relative motivic Serre invariant for curves defined over the field of fractions of a complete discrete valuation ring R of equicharacteristic zero. One aim of this study is to understand the behavior of motivic Serre invariants under ramified extension of the ring R. Thanks to our constructions, we obtain, in particular, an expression for the generating power series, whose coefficients are the motivic Serre invariant associated to a curve, computed on a tower of ramified extensions of R. We give an interpretation of this series in terms of the motivic zeta function of Denef and Loeser.