Compact,convex sets in R n and a new banach lattice. II.- Application: An approximation problem.

We develop a calculus structure in the Banach lattice introduced in a preceding paper, having in mind an approximation problem appearing in non-smooth optimization. We show that the essential results depend as much on the order structure as on the analytical one.     

Richman extensions of an integral domain

We introduce the notion of a Richman extension of an integral domain, and show its basic properties. We further show that a flat extension need not be a Richman extension, and a Richman extension need not be an LCM-stable extension.     

A distance on curves modulo rigid transformations

We propose a geometric method for quantifying the difference between parametrized curves in Euclidean space by introducing a distance function on the space of parametrized curves up to rigid transformations (rotations and translations). Given two curves, the distance between them is defined as the infimum of an energy functional which, roughly speaking, measures the extent to which the jet field of the first curve needs to be rotated to match up with the jet field of the second curve. We show that this energy functional attains a global minimum on the appropriate function space, and we derive a set of first-order ODEs for the minimizer.     

Numerical solution for exterior problems

Summary We solve the Helmholtz equation in an exterior domain in the plane. A perfect absorption condition is introduced on a circle which contains the obstacle. This boundary condition is given explicitly by Bessel functions. We use a finite element method in the bounded domain. An explicit formula is used to compute the solution out of the circle. We give an error estimate and we present relevant numerical results.     

Kufarev's method for determining the Schwarz-Christoffel parameters

Summary We present an analysis of a method suggested by Kufarev for the numerical determination of the parameters involved in the Schwarz-Christoffel integral. We test the approach for certain polygonal domains including two with re-entrant corners and one with a cut. Finally, we consider the Motz problem from this point of view.     

A partition relation for partially ordered sets

We introduce a partition relation which is an alternate for measuring how badly the ordinary partition relation fails, we develop its corresponding partition calculus and we determine its status for various typical partially ordered sets.     

On nilpotent ideals in the cohomology ring of a finite group

In this paper we find upper bounds for the nilpotency degree of some ideals in the cohomology ring of a finite group by studying fixed point free actions of the group on suitable spaces. The ideals we study are the kernels of restriction maps to certain collections of proper subgroups. We recover the Quillen-Venkov lemma and the Quillen F-injectivity theorem as corollaries, and discuss some generalizations and further applications.We then consider the essential cohomology conjecture, and show that it is related to group actions on connected graphs. We discuss an obstruction for constructing a fixed point free action of a group on a connected graph with zero “k-invariant” and study the class related to this obstruction. It turns out that this class is a “universal essential class” for the group and controls many questions about the groups essential cohomology and transfers from proper subgroups.     

Double coset decompositions and computational harmonic analysis on groups

In this paper we introduce new techniques for the efficient computation of a Fourier transform on a finite group. We use the decomposition of a group into double cosets and a graph theoretic indexing scheme to derive algorithms that generalize the Cooley-Tukey FFT to arbitrary finite group. We apply our general results to special linear groups and low rank symmetric groups, and obtain new efficient algorithms for harmonic analysis on these classes of groups, as well as the two-sphere.     

Classes of 2-step nilpotent lie algebras

We give a characterization of the Lie algebras of H-type independent of the inner product used in the definition. We classify the real 2-step nilpotent Lie algebras with 2-dimensional center. Using these results we give examples of regular Lie algebras that are not H-type.     

Clones on three elements preserving a binary relation

We describe the clones on 3 elements that can be expressed as Pol ρ for ρ a binary relation. We present the poset of these clones ordered by inclusion. This article is a shortened version of the author’s thesis, to give an idea of the whole work. Presented by R. P?schel. Received March 24, 2005; accepted in final form December 20, 2005.     

Comparison of splittings used with the conjugate gradient algorithm

We consider several possible criteria for comparing splittings used with the conjugate gradient algorithm. We present sharp upper bounds for the error at each step of the algorithm and compare several widely used splittings with respect to their effect on these sharp upper bounds. We then consider a more stringent comparison test, and present necessary and sufficient conditions for the error at each step with one splitting to be smaller than that with another, for all pairs of corresponding initial guesses.     

Divided power algebras and distributive laws

We study divided power structures over a product of operads with distributive law. We give a systematic method to characterise the divided power algebras over such a product from the structures of divided power algebra coming from each of the factor operads. We characterise divided power algebras with operadic derivation, as well as divided power p-level algebras in characteristic p, and divided power Poisson algebras in characteristic 3.     

Isolated singularities of solutions to the Yamabe equation

In this paper we study the asymptotic behavior of local solutions to the Yamabe equation near an isolated singularity, when the metric is not necessarily conformally flat. We are able to prove, when the dimension is less than or equal to five, that any solution is asymptotic to a rotationally symmetric Fowler solution. We also prove refined asymptotics if deformed Fowler solutions are allowed in the expansion.     

Neat rings

A ring is called clean if every element is the sum of a unit and an idempotent. Throughout the last 30 years several characterizations of commutative clean rings have been given. We have compiled a thorough list, including some new equivalences, in hopes that in the future there will be a better understanding of this interesting class of rings. One of the fundamental properties of clean rings is that every homomorphic image of a clean ring is clean. We define a neat ring to be one for which every proper homomorphic image is clean. In particular, the ring of integers, Z, and any nonlocal PID are examples neat rings which are not clean. We characterize neat Bézout domains using the group of divisibility. In particular, it is shown that a neat Bézout domain has stranded primes, that is, for every nonzero prime ideal the set of primes either containing or contained in the given prime forms a chain under set-theoretic inclusion.     

Irreducible maps of commutative rings

We give necessary conditions for a map to be irreducible (in the category of finitely generated, torsion free modules) over a non-local, commutative ring and sufficient conditions when the ring is Bass. In particular, we show that an irreducible map of ZG, where G is a square free abelian group, must be a monomorphism with a simple cokernel. We also show that local endomorphism rings are necessary and sufficient for the existence of almost split sequences over a commutative Bass ring and we explicitly describe the modules and the maps in those sequences. The results in this paper enable us to describe the Auslander-Reiten quiver of a non-local Bass ring in [8].     

Jensen's Inequality for polynomials with concentration at low degrees

Summary We give a generalization of Jensen's Inequality, valid for polynomials having some concentration at low degrees. We investigate the constants involved, both from a theoretical and a numerical point of view.     

Spline approximant in Hilbertian subspaces ofC(Ω)

Summary In this paper we give a new approach of approximation by spline functions. We define and study approximant spline functions which can be easly calculated without solving a linear system. We investigate also the error in using approximant spline functions.     

Poincaré’s inequalities and Talagrand’s concentration phenomenon for the exponential distribution

Summary. We present a simple proof, based on modified logarithmic Sobolev inequalities, of Talagrand’s concentration inequality for the exponential distribution. We actually observe that every measure satisfying a Poincaré inequality shares the same concentration phenomenon. We also discuss exponential integrability under Poincaré inequalities and its consequence to sharp diameter upper bounds on spectral gaps. Received: 10 June 1996 / In revised form: 9 August 1996     

The critical exponent for continuous conventional powers of doubly nonnegative matrices

We prove that there exists an exponent beyond which all continuous conventional powers of n-by-n doubly nonnegative matrices are doubly nonnegative. We show that this critical exponent cannot be less than n-2 and we conjecture that it is always n-2 (as it is with Hadamard powering). We prove this conjecture when n<6 and in certain other special cases. We establish a quadratic bound for the critical exponent in general.     

Horizontal submanifolds of groups of Heisenberg type

We study maximal horizontal subgroups of Carnot groups of Heisenberg type. We classify those of dimension half of that of the canonical distribution (“lagrangians”) and illustrate some notable ones of small dimension. An infinitesimal classification of the arbitrary maximal horizontal submanifolds follows as a consequence. This work was supported by CONICET, Antorchas, FONCyT and Secyt (UNC).