On the measure of noncompactness of linear operators in spaces of strongly α-summable and bounded sequences

In this paper, we characterize classes of matrix transformations from BK spaces into spaces of bounded sequences and their subclasses of infinite matrices that define compact operators. Furthermore, using these results and the solvability of certain infinite linear systems we give necessary and sufficient conditions for A to be a compact operator on spaces that are strongly α-bounded or summable. Research is supported by the German DAAD foundation (German Academic Exchange Service), the University of Le Havre and the research project #1232 of the Serbian Ministry of Science, Technology and Development.     

Representation of measures with simultaneous polynomial denseness inL p (R,dμ), 1≤p<∞

We give characterisations of certain positive finite Borel measures with unbounded support on the real axis so that the algebraic polynomials are dense in all spacesL _p (R,dμ),p≥1. These conditions apply, in particular, to the measures satisfying the classical Carleman conditions. This work was completed while A. Bakan was visiting Würzburg University, supported by the German Academic Exchange Service (DAAD). S. Ruscheweyh received partial support from the German-Israeli Foundation (grant G-643-117.6/1999) and from INTAS (Project 99-00089).     

Yet more on the linear search problem

The linear search problem has been discussed previously by one of the present authors. In this paper, the probability distribution of the point sought in the real line is not known to the searcher. Since there is noa priori choice of distribution which recommends itself above all others, we treat the situation as a game and obtain minimax type solutions. Different minimaxima apply depending on the factors which one wishes to minimize (resp. maximize). Certain criteria are developed which help the reader judge whether the results obtained can be considered “good advice” in the solution of real problems analogous to this one. The research of this paper has been supported in part by the following agencies: National Science Foundation, Wisconsin Alumni Research Foundation, German Academic Exchange Service (DAAD), and Air Force Office of Scientific Research.     

Spectral conditioning and pseudospectral growth

Using the language of pseudospectra, we study the behavior of matrix eigenvalues under two scales of matrix perturbation. First, we relate Lidskii’s analysis of small perturbations to a recent result of Karow on the growth rate of pseudospectra. Then, considering larger perturbations, we follow recent work of Alam and Bora in characterizing the distance from a given matrix to the set of matrices with multiple eigenvalues in terms of the number of connected components of pseudospectra. J. V. Burke’s research was supported in part by National Science Foundation Grant DMS-0505712. A. S. Lewis’s research was supported in part by National Science Foundation Grant DMS-0504032. M. L. Overton’s research was supported in part by National Science Foundation Grant DMS-0412049.     

A posteriori error analysis for time-dependent Ginzburg-Landau type equations

Summary. This work presents an a posteriori error analysis for the finite element approximation of time-dependent Ginzburg-Landau type equations in two and three space dimensions. The solution of an elliptic, self-adjoint eigenvalue problem as a post-processing procedure in each time step of a finite element simulation leads to a fully computable upper bound for the error. Theoretical results for the stability of degree one vortices in Ginzburg-Landau equations and of generic interfaces in Allen-Cahn equations indicate that the error estimate only depends on the inverse of a small parameter in a low order polynomial. The actual dependence of the error estimate upon this parameter is explicitly determined by the computed eigenvalues and can therefore be monitored within an approximation scheme. The error bound allows for the introduction of local refinement indicators which may be used for adaptive mesh and time step size refinement and coarsening. Numerical experiments underline the reliability of this approach.Mathematics Subject Classification(2000): 65M15, 65M60, 65M50.AcknowledgmentS.B. is thankful to G. Dolzmann and R.H. Nochetto for stimulating discussions. This work was supported by a fellowship within the Postdoc-Programme of the German Academic Exchange Service (DAAD).     

Zeros of Asai-Eisenstein series

We generalize a result of Max Deuring on the zeros of Zeta-function of quadratic forms to Asai's non-holomophic Eisenstein series E(z,s) of the Hilbert modular group. We prove that inside the rectangular and −1≤Re(s) ≤ 2 the function E(z,s) has only simple zeros on the line Re(s)=1/2 and two simple real zeros, if |N(y)| is large. The research was supported by a fellowship within the Post-doc-Program of the DAAD (German Academic Exchange Service)     

Multi-Implicit Peer Two-Step W-Methods for Parallel Time Integration

Peer two-step W-methods are designed for integration of stiff initial value problems with parallelism across the method. The essential feature is that in each time step s ‘peer’ approximations are employed having similar properties. In fact, no primary solution variable is distinguished. Parallel implementation of these stages is easy since information from one previous time step is used only and the different linear systems may be solved simultaneously. This paper introduces a subclass having order s−1 where optimal damping for stiff problems is obtained by using different system parameters in different stages. Favourable properties of this subclass are uniform stability for realistic stepsize sequences and a superconvergence property which is proved using a polynomial collocation formulation. Numerical tests on a shared memory computer of a matrix-free implementation with Krylov methods are included. AMS subject classification (2000) 65L06, 65Y05.Received June 2004. Revised January 2005. Communicated by Timo Eirola.Helmut Podhaisky: The work of this author was supported by the German Academic Exchange Service, DAAD.     

Aspects of operator theory in hereditarily indecomposable Banach spaces

We examine certain special features exhibited by various classes of linear operators acting in a hereditarily indecomposable Banach space. For instance, we show that the family of all Riesz operators in a H.I. space forms a closed, 2-sided ideal. We also give further characterizations of the class of scalar-type spectral operators (to those already given in [16]). The final section discusses some properties of the spectral maximal spaces of (necessarily decomposable) linear operators in such spaces. Conferenza tenuta il 16 settembre 1997 The support of the German Academic Exchange Scheme (DAAD) is gratefully acknowledged     

On Faith's correspondence theorem

A simple proof of an extension of Faith's correspondence theorem for projective modules is given for a Morita context (R, V, W, S) in whichVWV=V andWVW=W.Research supported by N.S.E.R.C. (Canada), Grant No. A 8075.     

Integral inequalities for the fundamental solutions of diffusions on manifolds with divergence-free drift

The main purpose of this article is to present uniform integral inequalities for the fundamental solutions of diffusions on compact manifolds with divergence free drift vector fields. The method relies on the fact that the heat flow depends on the isoperimetric function. The isoperimetric function is used to construct a suitable comparison manifold. The heat kernel of this comparison manifold gives uniform bounds for the fundamental solutions of the original diffusion problem. The results presented here can be used to solve some open problem of Bhattacharya and Götze about diffusions with periodic, divergence free drift vector fields (see [Bhagöt1] and [Bhagöt2]). Mathematics Subject Classification (2000): 58J35, 47D07This research was supported by the German Exchange Service DAAD     

Extensions of group-valued set functions

The problem of common extension ofcharges (finitely additive measures) is generalised to include group-valued functions defined on a system of sets (u-systems). To eachu-systemU an Abelian groupH(U) is attached. Every Abelian group is isomorphic to one of the formH(U). The groupH(U) is an indicator for extendability of charges fromU to the Boolean algebra generated byU. AllG-valued measures extend if and only if Ext(H(U),G)=0, for instance. Supported as van Vleck visiting professor at Wesleyan University, Connecticut in 1993. Partially supported by the Graduierten KollogTheoretische und experimentelle Methoden der reinen Mathematik of Essen University, a project No. G-0294-081.06/93 of the German-Israeli Foundation for Scientific Research & Development and by the German Academic Exchange, DAAD 1994.     

2–3 brother trees

2–3 brother trees form a dense class of search trees havingO(logN) insertion and deletion algorithms. In this paper we provide anO(logN) insertion algorithm and show that these trees have much better density and storage utilization than 2–3 trees. Thus we demonstrate that the brother property which has so far been used only for binary trees can be usefully applied to 2–3 trees.Work supported partially by a Natural Sciences and Engineering Research Council of Canada Grant No. A-7700 and partially by the German Academic Exchange Service under Nato Research Grant No. 430/402/584/8.     

Classes of plus matrices in finite dimensional indefinite scalar product spaces

We describe in terms of canonical forms various classes of plus matrices in real and complex finite dimensional spaces with indefinite scalar product and study their topological structures (such as closure and interior).Dedicated to M.G. Kreîn, one of the founding fathers of the theory of plus-operators.The work of this author was performed under the auspices of C.N.R.-G.N.F.M. and partially supported by the research project, Nonlinear problems in analysis and its physical, chemical and biological applications: Analytical, modelling and computational aspects, of the Italian Ministry of Higher Education and Research (M.U.R.S.T.)The work of this author partially supported by the NSF grant DMS 9500924.The work of all authors is partially supported by the NATO grant CGR 960700.     

Analysis of the generalized total least squares problemAX≈B when some columns ofA are free of error

Summary This paper studies the algebraic properties and perturbation theory of the generalized total least squares problem (GTLS)AXB in whichA =(A ₁,A ₂),A ₁ is free of error, and the error contained in (A ₂,B) is of the formEC withC a given nonsingular matrix. The problem was proposed by Van Huffel and Vandewalle in [15]. The solvability conditions, formulas for the GTLS solutions, their residuals, and the minimum norm correction matrices are obtained, and a perturbation theory for the GTLS problem is given.This author was supported by NSERC of Canada Grant No. A9236This author was supported by the National Natural Sciences Foundation, P.R. China. This work was carried out when this author was visiting the School of Computer Science. McGill University, Montreal, Quebec, Canada     

Minimum residual methods for augmented systems

For large systems of linear equations, iterative methods provide attractive solution techniques. We describe the applicability and convergence of iterative methods of Krylov subspace type for an important class of symmetric and indefinite matrix problems, namely augmented (or KKT) systems. Specifically, we consider preconditioned minimum residual methods and discuss indefinite versus positive definite preconditioning. For a natural choice of starting vector we prove that when the definite and indenfinite preconditioners are related in the obvious way, MINRES (which is applicable in the case of positive definite preconditioning) and full GMRES (which is applicable in the case of indefinite preconditioning) give residual vectors with identical Euclidean norm at each iteration. Moreover, we show that the convergence of both methods is related to a system of normal equations for which the LSQR algorithm can be employed. As a side result, we give a rare example of a non-trivial normal(1) matrix where the corresponding inner product is explicitly known: a conjugate gradient method therefore exists and can be employed in this case. This work was supported by British Council/German Academic Exchange Service Research Collaboration Project 465 and NATO Collaborative Research Grant CRG 960782     

A note on matrix equivalence

Let R be a principal ideal ringR_n the ring of n × n it matrices over R. It is shown that if A, B, X, Y are elements of R* such that A = XB, B = YA, then A and B are left equivalent. Some consequences are given.     

On group strategy-proof mechanisms for a many-to-one matching model

For the many-to-one matching model in which firms have substitutable and quota q–separable preferences over subsets of workers we show that the workers-optimal stable mechanism is group strategy-proof for the workers. Therefore, in centralized markets like entry-level professional labor markets if the proposed matching is the workers-optimal stable matching then, no group of workers can never benefit by reporting untruthfully their preference relations. We exhibit an example showing that this property fails if the preferences of firms are substitutable but not quota q–separable.Revised August 2004We are grateful to Flip Klijn, Howard Petith, William Thomson, a referee and an associate editor of this journal for helpful comments. The work of R. Martínez, A. Neme, and J. Oviedo is partially supported by the Universidad Nacional de San Luis through Grant 319502, by the Consejo Nacional de Investigaciones Científicas y Técnicas CONICET, through Grant PICT-02114, and by the Agencia Nacional de Promoción Científica y Técnica, through Grant 03-10814. The work of J. Massó is partially supported by the Spanish Ministry of Science and Technology, through Grant BEC2002–2130, and by the Generalitat de Catalunya, through Grant 2001SGR-00162 and the Barcelona Economics Program (CREA). All authors acknowledge financial support from the Grant PCI España-Iberoamérica 2003 (Programa de Cooperación Interuniversitaria de la Agencia Española de Cooperación Internacional-AECI).     

A new proof of <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript> estimates for the parabolic polyharmonic equations

In this paper we obtain local L ^p estimates for the parabolic polyharmonic equations by a straightforward approach. Yao was supported by the Innovation Foundation of Shanghai University (Grant No. A10-0101-08-905), Shanghai Leading Academic Discipline Project (Grant No. J50101) and Key Disciplines of Shanghai Municipality (Grant No. S30104). Zhou was supported by the National Basic Research Program of China (Grant No. 2006CB705700), National Natural Science Foundation of China (Grant No. 60532080), and the Key Project of Chinese Ministry of Education (Grant No. 306017)     

The numerically stable reconstruction of Jacobi matrices from spectral data

Summary We present an exposé of the elementary theory of Jacobi matrices and, in particular, their reconstruction from the Gaussian weights and abscissas. Many recent works propose use of the diagonal Hermitian Lanczos process for this purpose. We show that this process is numerically unstable. We recall Rutishauser's elegant and stable algorithm of 1963, based on plane rotations, implement it efficiently, and discuss our numerical experience. We also apply Rutishauser's algorithm to reconstruct a persymmetric Jacobi matrix from its spectrum in an efficient and stable manner.Dedicated to Professor F.L. Bauer on the occasion of his 60th birthdayThis work was supported by the National Science Foundation under grant MCS-81-02344, and by the Mathematics Research Institute of the Swiss Federal Institute of Technology, Zürich