Bøger udgivet af Birkhauser Boston

Algebraic, differential, and integral equations are used in the applied sciences, en- gineering, economics, and the social sciences to characterize the current state of a physical, economic, or social system and forecast its evolution in time. Generally, the coefficients of and/or the input to these equations are not precisely known be- cause of insufficient information, limited understanding of some underlying phe- nomena, and inherent randonmess. For example, the orientation of the atomic lattice in the grains of a polycrystal varies randomly from grain to grain, the spa- tial distribution of a phase of a composite material is not known precisely for a particular specimen, bone properties needed to develop reliable artificial joints vary significantly with individual and age, forces acting on a plane from takeoff to landing depend in a complex manner on the environmental conditions and flight pattern, and stock prices and their evolution in time depend on a large number of factors that cannot be described by deterministic models. Problems that can be defined by algebraic, differential, and integral equations with random coefficients and/or input are referred to as stochastic problems. The main objective of this book is the solution of stochastic problems, that is, the determination of the probability law, moments, and/or other probabilistic properties of the state of a physical, economic, or social system. It is assumed that the operators and inputs defining a stochastic problem are specified.

Overview Since the early 1990s, fuzzy modeling and identification from process data have been and continue to be an evolving subject of interest. Although the application of fuzzy models proved to be effective for the approxima- tion of uncertain nonlinear processes, the data-driven identification offuzzy models alone sometimes yields complex and unrealistic models. Typically, this is due to the over-parameterization of the model and insufficient in- formation content of the identification data set. These difficulties stem from a lack of initial a priori knowledge or information about the system to be modeled. To solve the problem of limited knowledge, in the area of modeling and identification, there is a tendency to blend information of different natures to employ as much knowledge for model building as possible. Hence, the incorporation of different types of a priori knowledge into the data-driven fuzzy model generation is a challenging and important task. Motivated by our research into this topic, our book presents new ap- proaches to the construction of fuzzy models for model-based control. New model structures and identification algorithms are described for the effec- tive use of heterogenous information in the form of numerical data, qualita- tive knowledge and first-principle models. By exploiting the mathematical properties of the proposed model structures, such as invertibility and local linearity, new control algorithms will be presented.

Structured matrices serve as a natural bridge between the areas of algebraic computations with polynomials and numerical matrix computations, allowing cross-fertilization of both fields. This book covers most fundamental numerical and algebraic computations with Toeplitz, Hankel, Vandermonde, Cauchy, and other popular structured matrices. Throughout the computations, the matrices are represented by their compressed images, called displacements, enabling both a unified treatment of various matrix structures and dramatic saving of computer time and memory. The resulting superfast algorithms allow further dramatic parallel acceleration using FFT and fast sine and cosine transforms. Included are specific applications to other fields, in particular, superfast solutions to: various fundamental problems of computer algebra; the tangential Nevanlinna--Pick and matrix Nehari problems The primary intended readership for this work includes researchers, algorithm designers, and advanced graduate students in the fields of computations with structured matrices, computer algebra, and numerical rational interpolation. The book goes beyond research frontiers and, apart from very recent research articles, includes yet unpublished results. To serve a wider audience, the presentation unfolds systematically and is written in a user-friendly engaging style. Only some preliminary knowledge of the fundamentals of linear algebra is required. This makes the material accessible to graduate students and new researchers who wish to study the rapidly exploding area of computations with structured matrices and polynomials. Examples, tables, figures, exercises, extensive bibliography, and index lend this text to classroom use or self-study.

On becoming familiar with difference equations and their close re- lation to differential equations, I was in hopes that the theory of difference equations could be brought completely abreast with that for ordinary differential equations. [HUGH L. TURRITTIN, My Mathematical Expectations, Springer Lecture Notes 312 (page 10), 1973] A major task of mathematics today is to harmonize the continuous and the discrete, to include them in one comprehensive mathematics, and to eliminate obscurity from both. [E. T. BELL, Men of Mathematics, Simon and Schuster, New York (page 13/14), 1937] The theory of time scales, which has recently received a lot of attention, was introduced by Stefan Hilger in his PhD thesis [159] in 1988 (supervised by Bernd Aulbach) in order to unify continuous and discrete analysis. This book is an intro- duction to the study of dynamic equations on time scales. Many results concerning differential equations carryover quite easily to corresponding results for difference equations, while other results seem to be completely different in nature from their continuous counterparts. The study of dynamic equations on time scales reveals such discrepancies, and helps avoid proving results twice, once for differential equa- tions and once for difference equations. The general idea is to prove a result for a dynamic equation where the domain of the unknown function is a so-called time scale, which is an arbitrary nonempty closed subset of the reals.

Kac-Moody Lie algebras 9 were introduced in the mid-1960s independently by V. Kac and R. Moody, generalizing the finite-dimensional semisimple Lie alge- bras which we refer to as the finite case. The theory has undergone tremendous developments in various directions and connections with diverse areas abound, including mathematical physics, so much so that this theory has become a stan- dard tool in mathematics. A detailed treatment of the Lie algebra aspect of the theory can be found in V. Kac's book [Kac-90l This self-contained work treats the algebro-geometric and the topological aspects of Kac-Moody theory from scratch. The emphasis is on the study of the Kac-Moody groups 9 and their flag varieties XY, including their detailed construction, and their applications to the representation theory of g. In the finite case, 9 is nothing but a semisimple Y simply-connected algebraic group and X is the flag variety 9 /Py for a parabolic subgroup p y C g.

Equations with Involutive Operators demonstrates an important interplay between abstract and concrete operator theory. The focus is on the investigation of a number of equations, which, while seemingly different, are all unified by the same idea: they are all realizations of some operator equations in Banach spaces. One permeating theme in these equations involves the role of the Fredholm property. The text is carefully written, self-contained, and covers a broad range of topics and results. Key ideas are developed in a step-by step approach, beginning with required background and historical material, and culminating in the final chapters with state-of-the art topics. Experts in operator theory, integral equations, and function theory as well as students in these areas will find open problems for further investigations. The book will also be useful to engineers using operator theory and integral equation techniques. Good examples, bibliography and index make this text a valuable classroom or reference resource.

The engineering objective of high performance control using the tools of optimal control theory, robust control theory, and adaptive control theory is more achiev- able now than ever before, and the need has never been greater. Of course, when we use the term high peiformance control we are thinking of achieving this in the real world with all its complexity, uncertainty and variability. Since we do not expect to always achieve our desires, a more complete title for this book could be "e;Towards High Performance Control"e;. To illustrate our task, consider as an example a disk drive tracking system for a portable computer. The better the controller performance in the presence of eccen- tricity uncertainties and external disturbances, such as vibrations when operated in a moving vehicle, the more tracks can be used on the disk and the more memory it has. Many systems today are control system limited and the quest is for high performance in the real world.

A Levy process is a continuous-time analogue of a random walk, and as such, is at the cradle of modern theories of stochastic processes. The need for modeling jumps, bursts, extremes and other irregular behavior of phenomena in nature and society has led to a renaissance of the theory of general Levy processes. Researchers and practitioners in fields as diverse as physics, meteorology, statistics, insurance, and finance have rediscovered the simplicity of Levy processes and their enormous flexibility in modeling tails, dependence and path behavior. This volume, with an excellent introductory preface, describes the state-of-the-art of this rapidly evolving subject with special emphasis on the non-Brownian world. It collects articles written by leading experts that will appeal to the non-specialist. An extensive bibliography at the end of each article makes this an invaluable comprehensive reference text. For the researcher and graduate student, every article contains open problems and points out directions for future research. The accessible nature of the work makes this an ideal introductory text for graduate seminars in applied probability, stochastic processes, physics, finance, and telecommunications, and a unique guide to the world of Lvy processes.

The goal of this book is to present a unified mathematical treatment of diverse problems in mathematics, physics, computer science, and engineer- ing using geometric algebra. Geometric algebra was invented by William Kingdon Clifford in 1878 as a unification and generalization of the works of Grassmann and Hamilton, which came more than a quarter of a century before. Whereas the algebras of Clifford and Grassmann are well known in advanced mathematics and physics, they have never made an impact in elementary textbooks where the vector algebra of Gibbs-Heaviside still predominates. The approach to Clifford algebra adopted in most of the ar- ticles here was pioneered in the 1960s by David Hestenes. Later, together with Garret Sobczyk, he developed it into a unified language for math- ematics and physics. Sobczyk first learned about the power of geometric algebra in classes in electrodynamics and relativity taught by Hestenes at Arizona State University from 1966 to 1967. He still vividly remembers a feeling of disbelief that the fundamental geometric product of vectors could have been left out of his undergraduate mathematics education. Geometric algebra provides a rich, general mathematical framework for the develop- ment of multilinear algebra, projective and affine geometry, calculus on a manifold, the representation of Lie groups and Lie algebras, the use of the horosphere and many other areas. This book is addressed to a broad audience of applied mathematicians, physicists, computer scientists, and engineers.

Despite the fact that fluid dynamics and filtration through porous media and mathematics, there are classical research areas in engineering, physics, are still many industrial processes that require the study of new mathemat- ical models for flows of particular complexity, due to the peculiar properties of the systems involved. The aim of this book is to provide a number of examples showing how frequently such situations arise in various branches of industrial technology. The selection of the subjects was motivated not only by their industrial rel- evance and mathematical interest. What I had in mind was a collection of problems having a really distinctive character, thus bringing some fresh air into one of the oldest and most revered domains of applied mathematics. The incredible richness of nonstandard flow problems in industrial appli- cations has always been, and still is, a constant surprise to me. Therefore I tried to offer a very large spectrum of subjects, with special attention devoted to those problems in which the modeling phase is far from being obvious, and the mathematical content is absolutely nontrivial. With such a view to diversity, topics have been selected from a variety of sources (such as glass industry, polymers science, coffee brewing, fuels pipelining), and contributors from different backgrounds (mathematics, physics, chemical engineering) have been included. Consequently, the mathematical nature of the problems formulated spans over a large range, so that their theoret- ical investigation and numerical computation require a variety of different techniques.

Semiconcavity is a natural generalization of concavity that retains most of the good properties known in convex analysis, but arises in a wider range of applications. This text is the first comprehensive exposition of the theory of semiconcave functions, and of the role they play in optimal control and Hamilton-Jacobi equations.The first part covers the general theory, encompassing all key results and illustrating them with significant examples. The latter part is devoted to applications concerning the Bolza problem in the calculus of variations and optimal exit time problems for nonlinear control systems. The exposition is essentially self-contained since the book includes all prerequisites from convex analysis, nonsmooth analysis, and viscosity solutions.

A number of important topics in complex analysis and geometry are covered in this excellent introductory text. Written by experts in the subject, each chapter unfolds from the basics to the more complex. The exposition is rapid-paced and efficient, without compromising proofs and examples that enable the reader to grasp the essentials. The most basic type of domain examined is the bounded symmetric domain, originally described and classified by Cartan and Harish- Chandra. Two of the five parts of the text deal with these domains: one introduces the subject through the theory of semisimple Lie algebras (Koranyi), and the other through Jordan algebras and triple systems (Roos). Larger classes of domains and spaces are furnished by the pseudo-Hermitian symmetric spaces and related R-spaces. These classes are covered via a study of their geometry and a presentation and classification of their Lie algebraic theory (Kaneyuki). In the fourth part of the book, the heat kernels of the symmetric spaces belonging to the classical Lie groups are determined (Lu). Explicit computations are made for each case, giving precise results and complementing the more abstract and general methods presented. Also explored are recent developments in the field, in particular, the study of complex semigroups which generalize complex tube domains and function spaces on them (Faraut). This volume will be useful as a graduate text for students of Lie group theory with connections to complex analysis, or as a self-study resource for newcomers to the field. Readers will reach the frontiers of the subject in a considerably shorter time than with existing texts.

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 "e;diagonal-type"e; 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 "e;cul- ture"e; 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.

The goal of this is book to give a detailed presentation of multicomponent flow models and to investigate the mathematical structure and properties of the resulting system of partial differential equations. These developments are also illustrated by simulating numerically a typical laminar flame. Our aim in the chapters is to treat the general situation of multicomponent flows, taking into account complex chemistry and detailed transport phe- nomena. In this book, we have adopted an interdisciplinary approach that en- compasses a physical, mathematical, and numerical point of view. In par- ticular, the links between molecular models, macroscopic models, mathe- matical structure, and mathematical properties are emphasized. We also often mention flame models since combustion is an excellent prototype of multicomponent flow. This book still does not pretend to be a complete survey of existing models and related mathematical results. In particular, many subjects like multi phase-flows , turbulence modeling, specific applications, porous me- dia, biological models, or magneto-hydrodynamics are not covered. We rather emphasize the fundamental modeling of multicomponent gaseous flows and the qualitative properties of the resulting systems of partial dif- ferential equations. Part of this book was taught at the post-graduate level at the Uni- versity of Paris, the University of Versailles, and at Ecole Poly technique in 1998-1999 to students of applied mathematics.