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    • Handbook of the Geometry of Banach Spaces

      • 1st Edition
      • Volume 1
      • August 15, 2001
      • English
      • eBook
        9 7 8 0 0 8 0 5 3 2 8 0 6
      The Handbook presents an overview of most aspects of modernBanach space theory and its applications. The up-to-date surveys, authored by leading research workers in the area, are written to be accessible to a wide audience. In addition to presenting the state of the art of Banach space theory, the surveys discuss the relation of the subject with such areas as harmonic analysis, complex analysis, classical convexity, probability theory, operator theory, combinatorics, logic, geometric measure theory, and partial differential equations.The Handbook begins with a chapter on basic concepts in Banachspace theory which contains all the background needed for reading any other chapter in the Handbook. Each of the twenty one articles in this volume after the basic concepts chapter is devoted to one specific direction of Banach space theory or its applications. Each article contains a motivated introduction as well as an exposition of the main results, methods, and open problems in its specific direction. Most have an extensive bibliography. Many articles contain new proofs of known results as well as expositions of proofs which are hard to locate in the literature or are only outlined in the original research papers.As well as being valuable to experienced researchers in Banach space theory, the Handbook should be an outstanding source for inspiration and information to graduate students and beginning researchers. The Handbook will be useful for mathematicians who want to get an idea of the various developments in Banach space theory.

    • Hilbert Spaces

      • 1st Edition
      • Volume 4
      • July 11, 2001
      • English
      • Paperback
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      • Hardback
        9 7 8 0 4 4 4 5 0 7 5 2 5
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      This book has evolved from the lecture course on Functional Analysis I had given several times at the ETH. The text has a strict logical order, in the style of “Definition – Theorem – Proof - Example - Exercises”. The proofs are rather thorough and there many examples. The first part of the book(the first three chapters, resp. the first two volumes) is devoted to the theory of Banach spaces in the most general sense of the term. The purpose of the first chapter (resp. first volume) is to introduce those results on Banach spaces which are used later or which are closely connected with the book. It therefore only contains a small part of the theory, and several results are stated (and proved) in a diluted form. The second chapter (which together with Chapter 3 makes the second volume) deals with Banach algebras (and involutive Banach algebras), which constitute the main topic of the first part of the book. The third chapter deals with compact operators on Banach spaces and linear (ordinary and partial) differential equations - applications of the, theory of Banach algebras.

    • General Theory of C*-Algebras

      • 1st Edition
      • Volume 3
      • July 11, 2001
      • English
      • Paperback
        9 7 8 0 4 4 4 5 4 2 1 9 9
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    • Partial Differential Equations

      • 1st Edition
      • Volume 7
      • July 10, 2001
      • D. Sloan + 2 more
      • English
      • Paperback
        9 7 8 0 4 4 4 5 0 6 1 6 0
      • eBook
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      /homepage/sac/cam/na... Set now available at special set price !Over the second half of the 20th century the subject area loosely referred to as numerical analysis of partial differential equations (PDEs) has undergone unprecedented development. At its practical end, the vigorous growth and steady diversification of the field were stimulated by the demand for accurate and reliable tools for computational modelling in physical sciences and engineering, and by the rapid development of computer hardware and architecture. At the more theoretical end, the analytical insight into the underlying stability and accuracy properties of computational algorithms for PDEs was deepened by building upon recent progress in mathematical analysis and in the theory of PDEs.To embark on a comprehensive review of the field of numerical analysis of partial differential equations within a single volume of this journal would have been an impossible task. Indeed, the 16 contributions included here, by some of the foremost world authorities in the subject, represent only a small sample of the major developments. We hope that these articles will, nevertheless, provide the reader with a stimulating glimpse into this diverse, exciting and important field.The opening paper by Thomée reviews the history of numerical analysis of PDEs, starting with the 1928 paper by Courant, Friedrichs and Lewy on the solution of problems of mathematical physics by means of finite differences. This excellent survey takes the reader through the development of finite differences for elliptic problems from the 1930s, and the intense study of finite differences for general initial value problems during the 1950s and 1960s. The formulation of the concept of stability is explored in the Lax equivalence theorem and the Kreiss matrix lemmas. Reference is made to the introduction of the finite element method by structural engineers, and a description is given of the subsequent development and mathematical analysis of the finite element method with piecewise polynomial approximating functions. The penultimate section of Thomée's survey deals with `other classes of approximation methods', and this covers methods such as collocation methods, spectral methods, finite volume methods and boundary integral methods. The final section is devoted to numerical linear algebra for elliptic problems.The next three papers, by Bialecki and Fairweather, Hesthaven and Gottlieb and Dahmen, describe, respectively, spline collocation methods, spectral methods and wavelet methods. The work by Bialecki and Fairweather is a comprehensive overview of orthogonal spline collocation from its first appearance to the latest mathematical developments and applications. The emphasis throughout is on problems in two space dimensions. The paper by Hesthaven and Gottlieb presents a review of Fourier and Chebyshev pseudospectral methods for the solution of hyperbolic PDEs. Particular emphasis is placed on the treatment of boundaries, stability of time discretisations, treatment of non-smooth solutions and multidomain techniques. The paper gives a clear view of the advances that have been made over the last decade in solving hyperbolic problems by means of spectral methods, but it shows that many critical issues remain open. The paper by Dahmen reviews the recent rapid growth in the use of wavelet methods for PDEs. The author focuses on the use of adaptivity, where significant successes have recently been achieved. He describes the potential weaknesses of wavelet methods as well as the perceived strengths, thus giving a balanced view that should encourage the study of wavelet methods.Aspects of finite element methods and adaptivity are dealt with in the three papers by Cockburn, Rannacher and Suri. The paper by Cockburn is concerned with the development and analysis of discontinuous Galerkin (DG) finite element methods for hyperbolic problems. It reviews the key properties of DG methods for nonlinear hyperbolic conservation laws from a novel viewpoint that stems from the observation that hyperbolic conservation laws are normally arrived at via model reduction, by elimination of dissipation terms. Rannacher's paper is a first-rate survey of duality-based a posteriori error estimation and mesh adaptivity for Galerkin finite element approximations of PDEs. The approach is illustrated for simple examples of linear and nonlinear PDEs, including also an optimal control problem. Several open questions are identified such as the efficient determination of the dual solution, especially in the presence of oscillatory solutions. The paper by Suri is a lucid overview of the relative merits of the hp and p versions of the finite element method over the h version. The work is presented in a non-technical manner by focusing on a class of problems concerned with linear elasticity posed on thin domains. This type of problem is of considerable practical interest and it generates a number of significant theoretical problems.Iterative methods and multigrid techniques are reviewed in a paper by Silvester, Elman, Kay and Wathen, and in three papers by Stüben, Wesseling and Oosterlee and Xu. The paper by Silvester et al. outlines a new class of robust and efficient methods for solving linear algebraic systems that arise in the linearisation and operator splitting of the Navier-Stokes equations. A general preconditioning strategy is described that uses a multigrid V-cycle for the scalar convection-diffusion operator and a multigrid V-cycle for a pressure Poisson operator. This two-stage approach gives rise to a solver that is robust with respect to time-step-variation and for which the convergence rate is independent of the grid. The paper by Stüben gives a detailed overview of algebraic multigrid. This is a hierarchical and matrix-based approach to the solution of large, sparse, unstructured linear systems of equations. It may be applied to yield efficient solvers for elliptic PDEs discretised on unstructured grids. The author shows why this is likely to be an active and exciting area of research for several years in the new millennium. The paper by Wesseling and Oosterlee reviews geometric multigrid methods, with emphasis on applications in computational fluid dynamics (CFD). The paper is not an introduction to multigrid: it is more appropriately described as a refresher paper for practitioners who have some basic knowledge of multigrid methods and CFD. The authors point out that textbook multigrid efficiency cannot yet be achieved for all CFD problems and that the demands of engineering applications are focusing research in interesting new directions. Semi-coarsening, adaptivity and generalisation to unstructured grids are becoming more important. The paper by Xu presents an overview of methods for solving linear algebraic systems based on subspace corrections. The method is motivated by a discussion of the local behaviour of high-frequency components in the solution of an elliptic problem. Of novel interest is the demonstration that the method of subspace corrections is closely related to von Neumann's method of alternating projections. This raises the question as to whether certain error estimates for alternating directions that are available in the literature may be used to derive convergence estimates for multigrid and/or domain decomposition methods.Moving finite element methods and moving mesh methods are presented, respectively, in the papers by Baines and Huang and Russell. The paper by Baines reviews recent advances in Galerkin and least-squares methods for solving first- and second-order PDEs with moving nodes in multidimensions. The methods use unstructured meshes and they minimise the norm of the residual of the PDE over both the computed solution and the nodal positions. The relationship between the moving finite element method and L2 least-squares methods is discussed. The paper also describes moving finite volume and discrete l2 least-squares methods. Huang and Russell review a class of moving mesh algorithms based upon a moving mesh partial differential equation (MMPDE). The authors are leading players in this research area, and the paper is largely a review of their own work in developing viable MMPDEs and efficient solution strategies.The remaining three papers in this special issue are by Budd and Piggott, Ewing and Wang and van der Houwen and Sommeijer. The paper by Budd and Piggott on geometric integration is a survey of adaptive methods and scaling invariance for discretisations of ordinary and partial differential equations. The authors have succeeded in presenting a readable account of material that combines abstract concepts and practical scientific computing. Geometric integration is a new and rapidly growing area which deals with the derivation of numerical methods for differential equations that incorporate qualitative information in their structure. Qualitative features that may be present in PDEs might include symmetries, asymptotics, invariants or orderings and the objective is to take these properties into account in deriving discretisations. The paper by Ewing and Wang gives a brief summary of numerical methods for advection-dominated PDEs. Models arising in porous medium fluid flow are presented to motivate the study of the advection-dominated flows. The numerical methods reviewed are applicable not only to porous medium flow problems but second-order PDEs with dominant hyperbolic behaviour in general. The paper by van der Houwen and Sommeijer deals with approximate factorisation for time-dependent PDEs. The paper begins with some historical notes and it proceeds to present various approximate factorisation techniques. The objective is to show that the linear system arising from linearisation and discretisation of the PDE may be solved more efficiently if the coefficient matrix is replaced by an approximate factorisation based on splitting. The paper presents a number of new stability results obtained by the group at CWI Amsterdam for the resulting time integration methods.

    • Operator Theory and Numerical Methods

      • 1st Edition
      • Volume 30
      • July 3, 2001
      • H. Fujita + 2 more
      • English
      • Paperback
        9 7 8 0 4 4 4 5 4 6 7 3 9
      • Hardback
        9 7 8 0 4 4 4 5 0 4 7 4 6
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      In accordance with the developments in computation, theoretical studies on numerical schemes are now fruitful and highly needed. In 1991 an article on the finite element method applied to evolutionary problems was published. Following the method, basically this book studies various schemes from operator theoretical points of view. Many parts are devoted to the finite element method, but other schemes and problems (charge simulation method, domain decomposition method, nonlinear problems, and so forth) are also discussed, motivated by the observation that practically useful schemes have fine mathematical structures and the converses are also true.

    • Neuro-informatics and Neural Modelling

      • 1st Edition
      • Volume 4
      • June 26, 2001
      • F. Moss + 1 more
      • English
      • eBook
        9 7 8 0 0 8 0 5 3 7 4 2 9
      How do sensory neurons transmit information about environmental stimuli to the central nervous system? How do networks of neurons in the CNS decode that information, thus leading to perception and consciousness? These questions are among the oldest in neuroscience. Quite recently, new approaches to exploration of these questions have arisen, often from interdisciplinary approaches combining traditional computational neuroscience with dynamical systems theory, including nonlinear dynamics and stochastic processes. In this volume in two sections a selection of contributions about these topics from a collection of well-known authors is presented. One section focuses on computational aspects from single neurons to networks with a major emphasis on the latter. The second section highlights some insights that have recently developed out of the nonlinear systems approach.

    • Handbook of Automated Reasoning

      • 1st Edition
      • Volume II
      • June 21, 2001
      • Alan J.A. Robinson + 1 more
      • English
      • Hardback
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    • Ordinary Differential Equations and Integral Equations

      • 1st Edition
      • Volume 6
      • June 20, 2001
      • C.T.H. Baker + 2 more
      • J.D. Pryce
      • English
      • Paperback
        9 7 8 0 4 4 4 5 0 6 0 0 9
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      /homepage/sac/cam/na... Set now available at special set price !This volume contains contributions in the area of differential equations and integral equations. Many numerical methods have arisen in response to the need to solve "real-life" problems in applied mathematics, in particular problems that do not have a closed-form solution. Contributions on both initial-value problems and boundary-value problems in ordinary differential equations appear in this volume. Numerical methods for initial-value problems in ordinary differential equations fall naturally into two classes: those which use one starting value at each step (one-step methods) and those which are based on several values of the solution (multistep methods).John Butcher has supplied an expert's perspective of the development of numerical methods for ordinary differential equations in the 20th century. Rob Corless and Lawrence Shampine talk about established technology, namely software for initial-value problems using Runge-Kutta and Rosenbrock methods, with interpolants to fill in the solution between mesh-points, but the 'slant' is new - based on the question, "How should such software integrate into the current generation of Problem Solving Environments?"Natali... Borovykh and Marc Spijker study the problem of establishing upper bounds for the norm of the nth power of square matrices.The dynamical system viewpoint has been of great benefit to ODE theory and numerical methods. Related is the study of chaotic behaviour.Willy Govaerts discusses the numerical methods for the computation and continuation of equilibria and bifurcation points of equilibria of dynamical systems.Arieh Iserles and Antonella Zanna survey the construction of Runge-Kutta methods which preserve algebraic invariant functions.Valeria Antohe and Ian Gladwell present numerical experiments on solving a Hamiltonian system of Hénon and Heiles with a symplectic and a nonsymplectic method with a variety of precisions and initial conditions.Stiff differential equations first became recognized as special during the 1950s. In 1963 two seminal publications laid to the foundations for later development: Dahlquist's paper on A-stable multistep methods and Butcher's first paper on implicit Runge-Kutta methods.Ernst Hairer and Gerhard Wanner deliver a survey which retraces the discovery of the order stars as well as the principal achievements obtained by that theory.Guido Vanden Berghe, Hans De Meyer, Marnix Van Daele and Tanja Van Hecke construct exponentially fitted Runge-Kutta methods with s stages.Differential-... equations arise in control, in modelling of mechanical systems and in many other fields.Jeff Cash describes a fairly recent class of formulae for the numerical solution of initial-value problems for stiff and differential-algebra... systems.Shengtai Li and Linda Petzold describe methods and software for sensitivity analysis of solutions of DAE initial-value problems.Again in the area of differential-algebra... systems, Neil Biehn, John Betts, Stephen Campbell and William Huffman present current work on mesh adaptation for DAE two-point boundary-value problems.Contrasting approaches to the question of how good an approximation is as a solution of a given equation involve (i) attempting to estimate the actual error (i.e., the difference between the true and the approximate solutions) and (ii) attempting to estimate the defect - the amount by which the approximation fails to satisfy the given equation and any side-conditions.The paper by Wayne Enright on defect control relates to carefully analyzed techniques that have been proposed both for ordinary differential equations and for delay differential equations in which an attempt is made to control an estimate of the size of the defect.Many phenomena incorporate noise, and the numerical solution of stochastic differential equations has developed as a relatively new item of study in the area.Keven Burrage, Pamela Burrage and Taketomo Mitsui review the way numerical methods for solving stochastic differential equations (SDE's) are constructed.One of the more recent areas to attract scrutiny has been the area of differential equations with after-effect (retarded, delay, or neutral delay differential equations) and in this volume we include a number of papers on evolutionary problems in this area.The paper of Genna Bocharov and Fathalla Rihan conveys the importance in mathematical biology of models using retarded differential equations.The contribution by Christopher Baker is intended to convey much of the background necessary for the application of numerical methods and includes some original results on stability and on the solution of approximating equations.Alfredo Bellen, Nicola Guglielmi and Marino Zennaro contribute to the analysis of stability of numerical solutions of nonlinear neutral differential equations.Koen Engelborghs, Tatyana Luzyanina, Dirk Roose, Neville Ford and Volker Wulf consider the numerics of bifurcation in delay differential equations.Evelyn Buckwar contributes a paper indicating the construction and analysis of a numerical strategy for stochastic delay differential equations (SDDEs).This volume contains contributions on both Volterra and Fredholm-type integral equations.Christophe... Baker responded to a late challenge to craft a review of the theory of the basic numerics of Volterra integral and integro-differential equations.Simon Shaw and John Whiteman discuss Galerkin methods for a type of Volterra integral equation that arises in modelling viscoelasticity.A subclass of boundary-value problems for ordinary differential equation comprises eigenvalue problems such as Sturm-Liouville problems (SLP) and Schrödinger equations.Liviu Ixaru describes the advances made over the last three decades in the field of piecewise perturbation methods for the numerical solution of Sturm-Liouville problems in general and systems of Schrödinger equations in particular.Alan Andrew surveys the asymptotic correction method for regular Sturm-Liouville problems.Leon Greenberg and Marco Marletta survey methods for higher-order Sturm-Liouville problems.R. Moore in the 1960s first showed the feasibility of validated solutions of differential equations, that is, of computing guaranteed enclosures of solutions.Boundary integral equations. Numerical solution of integral equations associated with boundary-value problems has experienced continuing interest.Peter Junghanns and Bernd Silbermann present a selection of modern results concerning the numerical analysis of one-dimensional Cauchy singular integral equations, in particular the stability of operator sequences associated with different projection methods.Johannes Elschner and Ivan Graham summarize the most important results achieved in the last years about the numerical solution of one-dimensional integral equations of Mellin type of means of projection methods and, in particular, by collocation methods.A survey of results on quadrature methods for solving boundary integral equations is presented by Andreas Rathsfeld.Wolfgang Hackbusch and Boris Khoromski present a novel approach for a very efficient treatment of integral operators.Ernst Stephan examines multilevel methods for the h-, p- and hp- versions of the boundary element method, including pre-conditioning techniques.George Hsiao, Olaf Steinbach and Wolfgang Wendland analyze various boundary element methods employed in local discretization schemes.

    • Inherently Parallel Algorithms in Feasibility and Optimization and their Applications

      • 1st Edition
      • Volume 8
      • June 18, 2001
      • D. Butnariu + 2 more
      • English
      • Hardback
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      The Haifa 2000 Workshop on "Inherently Parallel Algorithms for Feasibility and Optimization and their Applications" brought together top scientists in this area. The objective of the Workshop was to discuss, analyze and compare the latest developments in this fast growing field of applied mathematics and to identify topics of research which are of special interest for industrial applications and for further theoretical study.Inherently parallel algorithms, that is, computational methods which are, by their mathematical nature, parallel, have been studied in various contexts for more than fifty years. However, it was only during the last decade that they have mostly proved their practical usefulness because new generations of computers made their implementation possible in order to solve complex feasibility and optimization problems involving huge amounts of data via parallel processing. These led to an accumulation of computational experience and theoretical information and opened new and challenging questions concerning the behavior of inherently parallel algorithms for feasibility and optimization, their convergence in new environments and in circumstances in which they were not considered before their stability and reliability. Several research groups all over the world focused on these questions and it was the general feeling among scientists involved in this effort that the time has come to survey the latest progress and convey a perspective for further development and concerted scientific investigations. Thus, the editors of this volume, with the support of the Israeli Academy for Sciences and Humanities, took the initiative of organizing a Workshop intended to bring together the leading scientists in the field. The current volume is the Proceedings of the Workshop representing the discussions, debates and communications that took place. Having all that information collected in a single book will provide mathematicians and engineers interested in the theoretical and practical aspects of the inherently parallel algorithms for feasibility and optimization with a tool for determining when, where and which algorithms in this class are fit for solving specific problems, how reliable they are, how they behave and how efficient they were in previous applications. Such a tool will allow software creators to choose ways of better implementing these methods by learning from existing experience.

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