A baire approach to quantitative resonance principles

The main purpose of this paper is to give a proof of quantitative resonance and condensation principles via Baire category arguments in contrast to the gliding hump method used previously. Thereby a new nonquantitative resonance principle is established which is concerned with dominated convergence in Frechet spaces, by the way yielding more detailed information on the structure of the underlying spaces. On the other hand, Baire's approach is an easy tool to develop condensation principles by considering residual sets. Finally, some typical applications concerning trigonometric Fourier partial sums may illustrate the result received.     

A note on determinants,minors and a boundary map

We give an explicit formula for the first non-trivial boundary map of the Akin–Buchsbaum–Weyman resolution (Schur functors and Schur complexes, Adv. Math 1982;44:207-278). The role of the Laplace expansion formula is emphasized.     

Associated Prime Submodules of Finitely Generated Modules

As generalizations of annihilators and associated primes, we introduce the notions of weak annihilators and weak associated primes, respectively. We first study the properties of the weak annihilator of a subset X in a ring R. We next investigate how the weak associated primes of a ring R behave under passage to the skew monoid ring R*M. Let R be a semicommutative ring, and M an ordered monoid and φ: M → Aut(R) a compatible monoid homomorphism. Then we can describe all weak associated primes of the skew monoid ring R*M in terms of the weak associated primes of R in a very straightforward way.     

On preconditioned eigensolvers and Invert-Lanczos processes

This paper deals with the convergence analysis of various preconditioned iterations to compute the smallest eigenvalue of a discretized self-adjoint and elliptic partial differential operator. For these eigenproblems several preconditioned iterative solvers are known, but unfortunately, the convergence theory for some of these solvers is not very well understood.The aim is to show that preconditioned eigensolvers (like the preconditioned steepest descent iteration (PSD) and the locally optimal preconditioned conjugate gradient method (LOPCG)) can be interpreted as truncated approximate Krylov subspace iterations. In the limit of preconditioning with the exact inverse of the system matrix (such preconditioning can be approximated by multiple steps of a preconditioned linear solver) the iterations behave like Invert-Lanczos processes for which convergence estimates are derived.     

Isolated complex singularities and their CR links

In this paper, we discuss some recent studies on the complex structure of an isolated normal singularity by using the information from its link. We also give some open problems to be further pursued. Dedicated to Professor Sheng GONG on the occasion of his 75th birthday     

Studies on Jacobi–Davidson,Rayleigh quotient iteration,inverse iteration generalized Davidson and Newton updates

We study Davidson‐type subspace eigensolvers. Correction equations of Jacobi–Davidson and several other schemes are reviewed. New correction equations are derived. A general correction equation is constructed, existing correction equations may be considered as special cases of this general equation. The main theme of this study is to identify the essential common ingredient that leads to the efficiency of a diverse form of Davidson‐type methods. We emphasize the importance of the approximate Rayleigh‐quotient‐iteration direction. Copyright © 2006 John Wiley & Sons, Ltd.     

Decomposability of graph -algebras

We give conditions on an arbitrary directed graph for the associated Cuntz-Krieger algebra to be decomposable as a direct sum. We describe the direct summands as certain graph algebras.

     

Orderings and maximal ideals of rings of analytic functions

We prove that there is a natural injective correspondence between the maximal ideals of the ring of analytic functions on a real analytic set and those of its subring of bounded analytic functions. By describing the maximal ideals in terms of ultrafilters we see that this correspondence is surjective if and only if is compact. This approach is also useful for studying the orderings of the field of meromorphic functions on .

     

Local Łojasiewicz exponents,Milnor numbers and mixed multiplicities of ideals

Let be a finite analytic map. We give an expression for the local Łojasiewicz exponent and for the multiplicity of g when the component functions of g satisfy certain condition with respect to a set of n monomial ideals I ₁,..., I _n . We give an effective method to compute Łojasiewicz exponents based on the computation of mixed multiplicities. As a consequence of our study, we give a numerical characterization of a class of functions that includes semi-weighted homogenous functions and Newton non-degenerate functions. Work supported by DGICYT Grant MTM2006-06027.     

A Fibonacci quotient

In this article, we describe the outcome of a mathematical collaboration between a university lecturer and an undergraduate student. The resulting investigation concerned a particular divisibility property of the Fibonacci numbers, and indeed it seems that a new result was found in this regard. An interesting point to be made here is that, although the mathematical content was relatively straightforward, this joint exploration did, in a very modest sense, mirror certain key aspects of the research process.