Approximation theory and numerical analysis are central to the creation of accurate computer simulations and mathematical models. Research in these areas can influence the computational techniques used in a variety of mathematical and computational sciences.

This collection of contributed chapters, dedicated to renowned mathematician Gradimir V. Milovanovi, represent the recent work of experts in the fields of approximation theory and numerical analysis. These invited contributions describe new trends in these important areas of research including theoretic developments, new computational algorithms, and multidisciplinary applications.

Special features of this volume:

- Presents results and approximation methods in various computational settings including: polynomial and orthogonal systems, analytic functions, and differential equations.

- Provides a historical overview of approximation theory and many of its subdisciplines;

- Contains new results from diverse areas of research spanning mathematics, engineering, and the computational sciences.

"Approximation and Computation" is intended for mathematicians and researchers focusing on approximation theory and numerical analysis, but can also be a valuable resource to students and researchers in the computational and applied sciences."

This book deals with the numerical analysis and efficient numerical treatment of high-dimensional integrals using sparse grids and other dimension-wise integration techniques with applications to finance and insurance. The book focuses on providing insights into the interplay between coordinate transformations, effective dimensions and the convergence behaviour of sparse grid methods. The techniques, derivations and algorithms are illustrated by many examples, figures and code segments. Numerical experiments with applications from finance and insurance show that the approaches presented in this book can be faster and more accurate than (quasi-) Monte Carlo methods, even for integrands with hundreds of dimensions.

The book contains a selection of high quality papers, chosen among the best presentations during the International Conference on Spectral and High-Order Methods (2009), and provides an overview of the depth and breadth of the activities within this important research area. The carefully reviewed selection of the papers will provide the reader with a snapshot of state-of-the-art and help initiate new research directions through the extensive bibliography.

Readers of this book will learn how to solve a wide range of optimal investment problems arising in finance and economics.
Starting from the fundamental Merton problem, many variants are presented and solved, often using numerical techniques
that the book also covers. The final chapter assesses the relevance of many of the models in common use when applied to data.

In the framework of the Diderot Mathematical Forum (DMF) of the European Mathematical Society (EMS), December 19-20, 1997, a Videoconference was held linking three teams of specialists in Amsterdam, Madrid and Venice respectively. The general subject of this videoconference, the second one of the DMF series, was Mathematics and Environment and more specifically, Problems related to Water.

This volume contains the texts of the Madrid site contributions with important, new and unpublished, examples on the modeling, mathematical and numerical analysis and treatment of the associated control problems of relevant questions arising in Oceanography and Environment.

This book is devoted to the deformation and failure in metallic materials, summarizing the results of a research programme financed by the "Deutsche Forschungsgemeinschaft." It presents the recent engineering as well as mathematical key aspects of this field for a broad community. Its main focus is on the constitutive behaviour as well as the damage and fracture of metallic materials, covering their mathematical foundation, modelling and numerics, but also relevant experiments and their verification.

In recent years there has been an increasing interest in problems involving closed form evaluations of (and representations of the Riemann Zeta function at positive integer arguments as) various families of series associated with the Riemann Zeta function ((s), the Hurwitz Zeta function ((s, a), and their such extensions and generalizations as (for example) Lerch's transcendent (or the Hurwitz-Lerch Zeta function) iI>(z, s, a). Some of these developments have apparently stemmed from an over two-century-old theorem of Christian Goldbach (1690-1764), which was stated in a letter dated 1729 from Goldbach to Daniel Bernoulli (1700-1782), from recent rediscoveries of a fairly rapidly convergent series representation for ((3), which is actually contained in a 1772 paper by Leonhard Euler (1707-1783), and from another known series representation for ((3), which was used by Roger Apery (1916-1994) in 1978 in his celebrated proof of the irrationality of ((3). This book is motivated essentially by the fact that the theories and applications of the various methods and techniques used in dealing with many different families of series associated with the Riemann Zeta function and its aforementioned relatives are to be found so far only"in widely scattered journal articles. Thus our systematic (and unified) presentation of these results on the evaluation and representation of the Zeta and related functions is expected to fill a conspicuous gap in the existing books dealing exclusively with these Zeta functions."

Mathematica is a computer program (software) for doing symbolic, numeric and graphical analysis of mathematical problems. In the hands of economists, financial analysts and other professionals in econometrics and the quantitative sector of economic and financial modeling, it can be an invaluable tool for modeling and simulation on a large number of issues and problems, besides easily grinding out numbers, doing statistical estimations and rendering graphical plots and visuals. Mathematica enables these individuals to do all of this in a unified environment. This book's main use is that of an applications handbook. Modeling in Economics and Finance with Mathematica is a compilation of contributed papers prepared by experienced, "hands on" users of the Mathematica program. They come from

Originally published in 1999, "Wavelets Made Easy"offers a lucid and concise explanation of mathematical wavelets.Written at the level of a first course in calculus and linear algebra, its accessible presentation is designed for undergraduates in a variety of disciplines computer science, engineering, mathematics, mathematical sciences as well as for practicing professionals in these areas.

The presentsoftcover reprintretainsthecorrections fromthesecond printing (2001) andmakesthis uniquetext available to a wider audience. The first chapter startswith a description of the key features and applications of wavelets, focusing on Haar's wavelets but using only high-school mathematics. The next two chapters introduce one-, two-, and three-dimensional wavelets, with only the occasional use of matrix algebra.

The second part of this book provides the foundations of least-squares approximation, the discrete Fourier transform, and Fourier series. The third part explains the Fourier transform and then demonstrates how to apply basic Fourier analysis to designing and analyzing mathematical wavelets. Particular attention is paid to Daubechies wavelets.

Numerous exercises, a bibliography, and a comprehensive index combine to make this book an excellent text for the classroom as well as a valuable resource for self-study.

"

Arguably, many industrial optimization problems are of the multiobjective type. The present work, after providing a survey of the state of the art in multiobjective optimization, gives new insight into this important mathematical field by consequently taking up the viewpoint of differential geometry. This approach, unprecedented in the literature, very naturally results in a generalized homotopy method for multiobjective optimization which is theoretically well-founded and numerically efficient. The power of the new method is demonstrated by solving two real-life problems of industrial optimization.
The book presents recent results obtained by the author and is aimed at mathematicians, scientists, students and practitioners interested in optimization and numerical homotopy methods.

The present collection contains the results reported in 1970 at the Seminar on Approximate Com putations held by the Leningrad Section of the Mathematical Institute. Two trends are represented in the collection: automatic programming and numerical methods of analysis. V. N. Faddeeva CONTENTS On the Main Concepts of Parallel Sequencing. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 T. A. Tushkina and K. V. Shakhbazyan The Solution of Certain Parallel Sequencing Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 T. A. Tushkina and K. V. Shakhbazyan Choice of Enumeration in Parallel Sequencing Problems. . . . . . . . . . . . . . . . . . . . . . . . . . 13 K. V. Shakhbaz yan The PRORAB-Computer III (P, v) M-20 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 T. N. Smirnova, A. A. Aleksandrova, Yu. V. Rybakova, and N. A. Solov'eva Application of the PRORAB-Computer III (P, v) M-20 to the Solving of Linear Programming Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 T. N. Smirnova On a Matrix Inversion Method. . * . . . . . . . * . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 V. D. Vulichevich The Solution of a Particular Eigenvalue Problem for Certain Matrices of Special Form. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 V. D. Vulichevich and V. N. Kublanovskaya Solution of a Particular Eigenvalue Problem for a Polynomial Matrix. . . . . . . . . . . . . . . . . 65 M. I. Mavlyanova On a Method for Constructing the Matrix Solution for a Polynomial Matrix. . . . . . . . . . . . . . . 71 M. I. Mavlyanova On One Approach to the Solution of the Inverse Eigenvalue Problem. . . . . . . . . . . . . . . . . . . 80 V. N. Kublanovskaya Convergence of the Method of Lines when Solving Nonlinear Parabolic Boundary Value Problems with Discontinuous Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 A. P. Kubanskaya Some Applications of the Five-Point Scheme of the Method of Lines . . . . . . . . . . . . . . . . . . 93 A. P. Kubanskaya On Expansions into Nonminimal Sequences. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 L. N.

The first book to explicitly use Mathematica so as to allow researchers and students to more easily compute and solve almost any kind of differential equation using Lie's theory. Previously time-consuming and cumbersome calculations are now much more easily and quickly performed using the Mathematica computer algebra software. The material in this book, and on the accompanying CD-ROM, will be of interest to a broad group of scientists, mathematicians and engineers involved in dealing with symmetry analysis of differential equations. Each section of the book starts with a theoretical discussion of the material, then shows the application in connection with Mathematica. The cross-platform CD-ROM contains Mathematica (version 3.0) notebooks which allow users to directly interact with the code presented within the book. In addition, the author's proprietary "MathLie" software is included, so users can readily learn to use this powerful tool in regard to performing algebraic computations.

Discrete Event Systems: Diagnosis and Diagnosability addresses the problem of fault diagnosis of Discrete Event Systems (DESs). This book provides the basic techniques and approaches necessary for the design of an efficient fault diagnosis system for a wide range of modern engineering applications. This book classifies the different techniques and approaches according to several criteria such as: modeling tools (Automata, Petri nets, Templates) that is used to construct the model; the information (qualitative based on events occurrences and/or states outputs, quantitative based on signal processing, data analysis) that is needed to analyze and achieve the diagnosis; the decision structure (centralized, decentralized) that is required to achieve the diagnosis; as well as the complexity (polynomial, exponential) of the algorithm that is used to determine the set of faults that the proposed approach is able to diagnose as well as the delay time required for this diagnosis. The goal of this classification is to select the efficient method to achieve the fault diagnosis according to the application constraints. This book will include illustrated examples of the presented methods and techniques as well as a discussion on the application of these methods on several real-world problems.

This book illustrates how models of complex systems are built up and provides indispensable mathematical tools for studying their dynamics. This second edition includes more recent research results and many new and improved worked out examples and exercises.

This volume contains the Proceedings of the International Workshop Variational Methods For Discontinuous Structures, which was jointly organized by the Dipar timento di Matematica Francesco Brioschi of Milano Politecnico and the Interna tional School for Advanced Studies (SISSA) of Trieste. The Conference took place at Villa Erba Antica (Cernobbio) on the Lago di Como on July 4- 6, 2001. In past years the calculus of variations faced mainly the study of continuous structures, say particularly problems with smooth solutions. One of the deepest and more delicate problems was the regularity of weak solutions. More recently, new sophisticated tools have been introduced in order to study discontinuities: in many variational problems solutions develop singularities, and sometimes the most interesting part of a solution is the singularity itself. The conference intended to focus on recent developments in this direction. Some of the talks were devoted to differential or variational modelling of image segmentation, occlusion and textures synthesizing in image analysis, varia tional description of micro-magnetic materials, dimension reduction and structured deformations in elasticity and plasticity, phase transitions, irrigation and drainage, evolution of crystalline shapes; in most cases theoretical and numerical analysis of these models were provided. viii Preface Other talks were dedicated to specific problems of the calculus of variations: variational theory of weak or lower-dimensional structures, optimal transport prob lems with free Dirichlet regions, higher order variational problems, symmetrization in the BV framework."

This monograph presents a collection of results, observations, and examples related to dynamical systems described by linear and nonlinear ordinary differential and difference equations. In particular, dynamical systems that are susceptible to analysis by the Liapunov approach are considered. The naive observation that certain "diagonal-type" Liapunov functions are ubiquitous in the literature attracted the attention of the authors and led to some natural questions. Why does this happen so often? What are the spe cial virtues of these functions in this context? Do they occur so frequently merely because they belong to the simplest class of Liapunov functions and are thus more convenient, or are there any more specific reasons? This monograph constitutes the authors' synthesis of the work on this subject that has been jointly developed by them, among others, producing and compiling results, properties, and examples for many years, aiming to answer these questions and also to formalize some of the folklore or "cul ture" that has grown around diagonal stability and diagonal-type Liapunov functions. A natural answer to these questions would be that the use of diagonal type Liapunov functions is frequent because of their simplicity within the class of all possible Liapunov functions. This monograph shows that, although this obvious interpretation is often adequate, there are many in stances in which the Liapunov approach is best taken advantage of using diagonal-type Liapunov functions. In fact, they yield necessary and suffi cient stability conditions for some classes of nonlinear dynamical systems."

This book contains a broad spectrum of plasma physics areas, from magnetic confinement (tokamaks) to spectroscopy in plasmas. The invited papers of the LAWPP present mini-courses for graduate students and review papers in each area, also updating the new ideas in the field.

ThesubjectofthisbookisSemi-In?niteAlgebra,ormorespeci?cally,Semi-In?nite Homological Algebra. The term "semi-in?nite" is loosely associated with objects that can be viewed as extending in both a "positive" and a "negative" direction, withsomenaturalpositioninbetween,perhapsde?nedupto a"?nite"movement. Geometrically, this would mean an in?nite-dimensional variety with a natural class of "semi-in?nite" cycles or subvarieties, having always a ?nite codimension in each other, but in?nite dimension and codimension in the whole variety [37]. (For further instances of semi-in?nite mathematics see, e. g. , [38] and [57], and references below. ) Examples of algebraic objects of the semi-in?nite type range from certain in?nite-dimensional Lie algebras to locally compact totally disconnected topolo- cal groups to ind-schemes of ind-in?nite type to discrete valuation ?elds. From an abstract point of view, these are ind-pro-objects in various categories, often - dowed with additional structures. One contribution we make in this monograph is the demonstration of another class of algebraic objects that should be thought of as "semi-in?nite", even though they do not at ?rst glance look quite similar to the ones in the above list. These are semialgebras over coalgebras, or more generally over corings - the associative algebraic structures of semi-in?nite nature. The subject lies on the border of Homological Algebra with Representation Theory, and the introduction of semialgebras into it provides an additional link with the theory of corings [23], as the semialgebrasare the natural objects dual to corings.

Science used to be experiments and theory, now it is experiments, theory and computations. The computational approach to understanding nature and technology is currently flowering in many fields such as physics, geophysics, astrophysics, chemistry, biology, and most engineering disciplines. This book is a gentle introduction to such computational methods where the techniques are explained through examples. It is our goal to teach principles and ideas that carry over from field to field. You will learn basic methods and how to implement them. In order to gain the most from this text, you will need prior knowledge of calculus, basic linear algebra and elementary programming.

OO It is a matter of general consensus that in the last decade the H _ optimization for robust control has dominated the research effort in control systems theory. Much attention has been paid equally to the mathematical instrumentation and the computational aspects. There are several excellent monographs that cover the standard topics in the area. Among the recent issues we have to cite here Linear Robust Control authored by Green and Limebeer (Prentice Hall 1995), Robust Controller Design Using Normalized Coprime Factor Plant Descriptions - by McFarlane and Glover (Springer Verlag 1989), Robust and Optimal Control - by Zhou, Doyle and Glover (Prentice Hall 1996). Thus, when the authors of the present monograph decided to start the work they were confronted with a very rich literature on the subject. However two reasons motivated their initiative. The first concerns the theory in which the whole development of the book was embedded. As is well known, there are several ways of approach oo ing H and robust control theory. Here we mention three relevant direc tions chronologically ordered: a) the first makes use of a generalization of the Beurling-Lax theorem to Krein spaces; b) the second makes use of a generalization of Nevanlinna-Pick interpolation theory and commutant lifting theorem; c) the third, and probably the most attractive from an el evate engineering viewpoint, is the two Riccati equations based approach which offers a complete solution in state space form."

This text provides the reader with a unique insight into the finite element method, along with symbolic programing that fundamentally changes the way applications can be developed. It is an essential tool for undergraduate or early postgraduate courses as well as an excellent reference book for engineers and scientists who want to quickly develop finite-element programs. The use of symbolic computation in Maple system delivers new benefits in the analysis and understanding of the finite element method.

Given a function x(t) E c{n) [a, bj, points a = al < a2 < ...< ar = b and subsets aj of {0,1,"',n -1} with L:j=lcard(aj) = n, the classical interpolation problem is to find a polynomial P - (t) of degree at most (n - 1) n l such that P~~l(aj) = x{i)(aj) for i E aj, j = 1,2," r. In the first four chapters of this monograph we shall consider respectively the cases: the Lidstone interpolation (a = 0, b = 1, n = 2m, r = 2, al = a2 = {a, 2", 2m - 2}), the Hermite interpolation (aj = {a, 1,' ", kj - I}), the Abel - Gontscharoff interpolation (r = n, ai ~ ai+l, aj = {j - I}), and the several particular cases of the Birkhoff interpolation. For each of these problems we shall offer: (1) explicit representations of the interpolating polynomial; (2) explicit representations of the associated error function e(t) = x(t) - Pn-l(t); and (3) explicit optimal/sharp constants Cn,k so that the inequalities k I e{k)(t) I < C k(b -at- max I x{n)(t) I, 0 n - 1 n -, a$t$b - are satisfied. In addition, for the Hermite interpolation we shall provide explicit opti- mal/sharp constants C(n,p, v) so that the inequality II e(t) lip:::; C(n,p, v) II x{n)(t) 1111, p, v ~ 1 holds.

Looking back at the years that have passed since the realization of the very first electronic, multi-purpose computers, one observes a tremendous growth in hardware and software performance. Today, researchers and engi neers have access to computing power and software that can solve numerical problems which are not fully understood in terms of existing mathemati cal theory. Thus, computational sciences must in many respects be viewed as experimental disciplines. As a consequence, there is a demand for high quality, flexible software that allows, and even encourages, experimentation with alternative numerical strategies and mathematical models. Extensibil ity is then a key issue; the software must provide an efficient environment for incorporation of new methods and models that will be required in fu ture problem scenarios. The development of such kind of flexible software is a challenging and expensive task. One way to achieve these goals is to in vest much work in the design and implementation of generic software tools which can be used in a wide range of application fields. In order to provide a forum where researchers could present and discuss their contributions to the described development, an International Work shop on Modern Software Tools for Scientific Computing was arranged in Oslo, Norway, September 16-18, 1996. This workshop, informally referred to as Sci Tools '96, was a collaboration between SINTEF Applied Mathe matics and the Departments of Informatics and Mathematics at the Uni versity of Oslo."

This monograph is a slightly revised version of my PhD thesis [86], com pleted in the Department of Computer Science at the University of Edin burgh in June 1988, with an additional chapter summarising more recent developments. Some of the material has appeared in the form of papers [50,88]. The underlying theme of the monograph is the study of two classical problems: counting the elements of a finite set of combinatorial structures, and generating them uniformly at random. In their exact form, these prob lems appear to be intractable for many important structures, so interest has focused on finding efficient randomised algorithms that solve them ap proxim~ly, with a small probability of error. For most natural structures the two problems are intimately connected at this level of approximation, so it is natural to study them together. At the heart of the monograph is a single algorithmic paradigm: sim ulate a Markov chain whose states are combinatorial structures and which converges to a known probability distribution over them. This technique has applications not only in combinatorial counting and generation, but also in several other areas such as statistical physics and combinatorial optimi sation. The efficiency of the technique in any application depends crucially on the rate of convergence of the Markov chain.

This book is devoted to the relation between two different concepts of integrability: the complete integrability of complex analytical Hamiltonian systems and the integrability of complex analytical linear differential equations. For linear differential equations, integrability is made precise within the framework of differential Galois theory. The connection of these two integrability notions is given by the variational equation (i.e. linearized equation) along a particular integral curve of the Hamiltonian system. The underlying heuristic idea, which motivated the main results presented in this monograph, is that a necessary condition for the integrability of a Hamiltonian system is the integrability of the variational equation along any of its particular integral curves. This idea led to the algebraic non-integrability criteria for Hamiltonian systems. These criteria can be considered as generalizations of classical non-integrability results by Poincare and Lyapunov, as well as more recent results by Ziglin and Yoshida. Thus, by means of the differential Galois theory it is not only possible to understand all these approaches in a unified way but also to improve them. Several important applications are also included: homogeneous potentials, Bianchi IX cosmological model, three-body problem, Henon-Heiles system, etc. The book is based on the original joint research of the author with J.M. Peris, J.P. Ramis and C. Simo, but an effort was made to present these achievements in their logical order rather than their historical one. The necessary background on differential Galois theory and Hamiltonian systems is included, and several new problems and conjectures which open new lines of research are proposed. - - - The book is an excellent introduction to non-integrability methods in Hamiltonian mechanics and brings the reader to the forefront of research in the area. The inclusion of a large number of worked-out examples, many of wide applied interest, is commendable. There are many historical references, and an extensive bibliography. (Mathematical Reviews) For readers already prepared in the two prerequisite subjects [differential Galois theory and Hamiltonian dynamical systems], the author has provided a logically accessible account of a remarkable interaction between differential algebra and dynamics. (Zentralblatt MATH)