The aim of the Expositions is to present new and important developments in pure and applied mathematics. Well established in the community over more than two decades, the series offers a large library of mathematical works, including several important classics. The volumes supply thorough and detailed expositions of the methods and ideas essential to the topics in question. In addition, they convey their relationships to other parts of mathematics. The series is addressed to advanced readers interested in a thorough study of the subject. Editorial Board Lev Birbrair, Universidade Federal do Ceara, Fortaleza, BrasilWalter D. Neumann, Columbia University, New York, USAMarkus J. Pflaum, University of Colorado, Boulder, USADierk Schleicher, Jacobs University, Bremen, GermanyKatrin Wendland, University of Freiburg, Germany Honorary Editor Victor P. Maslov, Russian Academy of Sciences, Moscow, Russia Titles in planning include Yuri A. Bahturin, Identical Relations in Lie Algebras (2019)Yakov G. Berkovich, Lev G. Kazarin, and Emmanuel M. Zhmud', Characters of Finite Groups, Volume 2 (2019)Jorge Herbert Soares de Lira, Variational Problems for Hypersurfaces in Riemannian Manifolds (2019)Volker Mayer, Mariusz Urbanski, and Anna Zdunik, Random and Conformal Dynamical Systems (2021)Ioannis Diamantis, Bostjan Gabrovsek, Sofia Lambropoulou, and Maciej Mroczkowski, Knot Theory of Lens Spaces (2021)

This work is based on the lecture notes of the course M742: Topics in Partial Dif- ferential Equations, which I taught in the Spring semester of 1997 at Indiana Univer- sity. My main intention in this course was to give a concise introduction to solving two-dimensional compressibleEuler equations with Riemann data, which are special Cauchy data. This book covers new theoretical developments in the field over the past decade or so. Necessary knowledge of one-dimensional Riemann problems is reviewed and some popularnumerical schemes are presented. Multi-dimensional conservation laws are more physical and the time has come to study them. The theory onbasicone-dimensional conservation laws isfairly complete providing solid foundation for multi-dimensional problems. The rich theory on ellip- tic and parabolic partial differential equations has great potential in applications to multi-dimensional conservation laws. And faster computers make itpossible to reveal numerically more details for theoretical pursuitin multi-dimensional problems. Overview and highlights Chapter 1is an overview ofthe issues that concern us inthisbook. It lists theEulersystemandrelatedmodelssuch as theunsteady transonic small disturbance, pressure-gradient, and pressureless systems. Itdescribes Mach re- flection and the von Neumann paradox. In Chapters 2-4, which form Part I of the book, we briefly present the theory of one-dimensional conservation laws, which in- cludes solutions to the Riemann problems for the Euler system and general strictly hyperbolic and genuinely nonlinearsystems, Glimm's scheme, and large-time asymp- toties.

The series is devoted to the publication of high-level monographs which cover the whole spectrum of current nonlinear analysis and applications in various fields, such as optimization, control theory, systems theory, mechanics, engineering, and other sciences. One of its main objectives is to make available to the professional community expositions of results and foundations of methods that play an important role in both the theory and applications of nonlinear analysis. Contributions which are on the borderline of nonlinear analysis and related fields and which stimulate further research at the crossroads of these areas are particularly welcome. Editor-in-ChiefJurgen Appell, Wurzburg, Germany Honorary and Advisory EditorsCatherine Bandle, Basel, SwitzerlandAlain Bensoussan, Richardson, Texas, USAAvner Friedman, Columbus, Ohio, USAUmberto Mosco, Worcester, Massachusetts, USALouis Nirenberg, New York, USAAlfonso Vignoli, Rome, Italy Editorial BoardManuel del Pino, Bath, UK, and Santiago, ChileMikio Kato, Nagano, JapanWojciech Kryszewski, Torun, PolandVicentiu D. Radulescu, Krakow, PolandSimeon Reich, Haifa, Israel Please submit book proposals to Jurgen Appell. Titles in planning include Lucio Damascelli and Filomena Pacella, Morse Index of Solutions of Nonlinear Elliptic Equations (2019)Tomasz W. Dlotko and Yejuan Wang, Critical Parabolic-Type Problems (2019)Rafael Ortega, Periodic Differential Equations in the Plane: A Topological Perspective (2019)Ireneo Peral Alonso and Fernando Soria, Elliptic and Parabolic Equations Involving the Hardy-Leray Potential (2020)Cyril Tintarev, Profile Decompositions and Cocompactness: Functional-Analytic Theory of Concentration Compactness (2020)Takashi Suzuki, Semilinear Elliptic Equations: Classical and Modern Theories (2021)

¿The author describes this marvelous book as designed for beginning graduate students in mathematics¿-in particular for those who intend to specialize in applied mathematics, and for graduate students in other disciplines such as engineering, physics and computer science. The first six chapters contain enough material for a year course, and the final two chapters contain related material¿ Those who are familiar with the author¿s earlier books will not be surprised by its excellence. It is businesslike and will be found to be demanding, but it is user-friendly. It is the reviewer¿s opinion that it will be extremely useful and popular as a text; institutions that do not already require their students to take such a course no longer have an excuse, and should immediately organize one based on this book.¿ ¿Mathematical Reviews

Gian-Carlo Rota was born in Vigevano, Italy, in 1932. He died in Cambridge, Mas sachusetts, in 1999. He had several careers, most notably as a mathematician, but also as a philosopher and a consultant to the United States government. His mathe matical career was equally varied. His early mathematical studies were at Princeton (1950 to 1953) and Yale (1953 to 1956). In 1956, he completed his doctoral thesis under the direction of Jacob T. Schwartz. This thesis was published as the pa per "Extension theory of differential operators I", the first paper reprinted in this volume. Rota's early work was in analysis, more specifically, in operator theory, differ ential equations, ergodic theory, and probability theory. In the 1960's, Rota was motivated by problems in fluctuation theory to study some operator identities of Glen Baxter (see [7]). Together with other problems in probability theory, this led Rota to study combinatorics. His series of papers, "On the foundations of combi natorial theory", led to a fundamental re-evaluation of the subject. Later, in the 1990's, Rota returned to some of the problems in analysis and probability theory which motivated his work in combinatorics. This was his intention all along, and his early death robbed mathematics of his unique perspective on linkages between the discrete and the continuous. Glimpses of his new research programs can be found in [2,3,6,9,10].

The articles in this volume are an outgrowth of an international conference entitled Variational and Topological Methods in the Study of Nonlinear Phe- nomena, held in Pisa in January-February 2000. Under the framework of the research project Differential Equations and the Calculus of Variations, the conference was organized to celebrate the 60th birthday of Antonio Marino, one of the leaders of the research group and a significant contrib- utor to the mathematical activity in this area of nonlinear analysis. The volume highlights recent advances in the field of nonlinear functional analysis and its applications to nonlinear partial and ordinary differential equations, with particular emphasis on variational and topological meth- ods. A broad range of topics is covered, including: concentration phenomena in PDEs, variational methods with applications to PDEs and physics, pe- riodic solutions of ODEs, computational aspects in topological methods, and mathematical models in biology. Though well-differentiated, the topics covered are unified through a com- mon perspective and approach. Unique to the work are several chapters on computational aspects and applications to biology, not usually found with such basic studies on PDEs and ODEs. The volume is an excellent reference text for researchers and graduate students in the above mentioned fields. Contributors are M. Clapp, M.J. Esteban, P. Felmer, A. Ioffe, W. Marzan- towicz, M. Mrozek, M. Musso, R. Ortega, P. Pilarczyk, M. del Pino, E. Sere, E. Schwartzman, P. Sintzoff, R. Turner, and I\f. Willem.

In the modern theory of boundary value problems the following ap proach to investigation is agreed upon (we call it the functional approach): some functional spaces are chosen; the statements of boundary value prob the basis of these spaces; and the solvability of lems are formulated on the problems, properties of solutions, and their dependence on the original data of the problems are analyzed. These stages are put on the basis of the correct statement of different problems of mathematical physics (or of the definition of ill-posed problems). For example, if the solvability of a prob lem in the functional spaces chosen cannot be established then, probably, the reason is in their unsatisfactory choice. Then the analysis should be repeated employing other functional spaces. Elliptical problems can serve as an example of classical problems which are analyzed by this approach. Their investigations brought a number of new notions and results in the theory of Sobolev spaces W;(D) which, in turn, enabled us to create a sufficiently complete theory of solvability of elliptical equations. Nowadays the mathematical theory of radiative transfer problems and kinetic equations is an extensive area of modern mathematical physics. It has various applications in astrophysics, the theory of nuclear reactors, geophysics, the theory of chemical processes, semiconductor theory, fluid mechanics, etc. 25,29,31,39,40, 47, 52, 78, 83, 94, 98, 120, 124, 125, 135, 146]."

This is the proceedings of the "8th IMACS Seminar on Monte Carlo Methods" held from August 29 to September 2, 2011 in Borovets, Bulgaria, and organized by the Institute of Information and Communication Technologies of the Bulgarian Academy of Sciences in cooperation with the International Association for Mathematics and Computers in Simulation (IMACS). Included are 24 papers which cover all topics presented in the sessions of the seminar: stochastic computation and complexity of high dimensional problems, sensitivity analysis, high-performance computations for Monte Carlo applications, stochastic metaheuristics for optimization problems, sequential Monte Carlo methods for large-scale problems, semiconductor devices and nanostructures. The history of the IMACS Seminar on Monte Carlo Methods goes back to April 1997 when the first MCM Seminar was organized in Brussels: 1st IMACS Seminar, 1997, Brussels, Belgium 2nd IMACS Seminar, 1999, Varna, Bulgaria 3rd IMACS Seminar, 2001, Salzburg, Austria 4th IMACS Seminar, 2003, Berlin, Germany 5th IMACS Seminar, 2005, Tallahassee, USA 6th IMACS Seminar, 2007, Reading, UK 7th IMACS Seminar, 2009, Brussels, Belgium 8th IMACS Seminar, 2011, Borovets, Bulgaria

While there are many excellent books available on fundamental and applied electromagnetics, most introduce operator concepts in an ad hoc manner, and few discuss the subject within the general framework of operator theory. This is in contrast to quantum theory, where the use of operators and concepts from functional analysis is common. However, casting electromagnetic problems in terms of operator theory produces useful insights into the mathematical properties and physical characteristics of solutions. For instance, the commonly used modal expansion of fields in waveguides are immediately justified upon identifying the differential operators as being of the appropriate Sturm-Liouville type. As another example, existence, uniqueness and solvability of integral formulations can often be settled by appealing to the theory of Fredholm operators. Many other examples that illustrate the value of abstracting problems to an operator level are provided. Although the book focuses on mathematical fundamentals, it is written from the perspective of engineers and applied scientists working in electromagnetics. The book begins with a review of electromagnetic theory, including a discussion of singular integral operators commonly encountered in applications. It then turns to a self-contained introduction to operator theory, including basic functional analysis, linear operators, Green¿s functions and Green¿s operators, spectral theory, and Sturm-Liouville operators. The discussion is at an introductory mathematical level, presenting definitions and theorems, as well as proofs of the theorems when these are particularly simple or enlightening. The tools developed in this first part of the book are then applied to problems in classical electromagnetic theory: boundary-value problems and potential theory, transmission lines, waves in layered media, scattering problems in waveguides, and electromagnetic cavities.

With contributions by specialists in optimization and practitioners in the fields of aerospace engineering, chemical engineering, and fluid and solid mechanics, the major themes include an assessment of the state of the art in optimization algorithms as well as challenging applications in design and control, in the areas of process engineering and systems with partial differential equation models.

The aim of this monograph is to give a unified account fo the classical topics in fixed point theory that lie on the border-line of topology and non-linear functional analysis, emphasizing the topological developments related to the Leray-Schauder theory. The first part of this book is based on "Fixed Point Theory I" which was published by PWN, Warsaw in 1982. The second part follows the outline conceived by Andrzej Granas and the late James Dugunji. The completionof this work has been awaited for many years by researchers in this area. "If the authors do equally well with the second volume they will have produced the best monograph in this particular field."Math Reviews

This new edition is intended for third and fourth year undergraduates in Engineering, Physics, Mathematics, and the Applied Sciences, and can serve as a springboard for further work in Continuum Mechanics or General Relativity. Starting from a basic knowledge of calculus and matrix algebra, together with fundamental ideas from mechanics and geometry, the text gradually develops the tools for formulating and manipulating the field equations of Continuum Mechanics. The mathematics of tensor analysis is introduced in well-separated stages: the concept of a tensor as an operator; the representation of a tensor in terms of its Cartesian components; the components of a tensor relative to a general basis, tensor notation, and finally, tensor calculus. The physical interpretation and application of vectors and tensors are stressed throughout. Though concise, the text is written in an informal, non-intimidating style enhanced by worked-out problems and a meaningful variety of exercises. The new edition includes more exercises, especially at the end of chapter IV. Furthermore, the author has appended a section on Differential Geometry, the essential mathematical tool in the study of the 2-dimensional structural shells and 4-dimensional general relativity.

This book presents some of the latest research in critical point theory, describing methods and presenting the newest applications. Coverage includes extrema, even valued functionals, weak and double linking, sign changing solutions, Morse inequalities, and cohomology groups. Applications described include Hamiltonian systems, Schrodinger equations and systems, jumping nonlinearities, elliptic equations and systems, superlinear problems and beam equations. "

With contributions by specialists in optimization and practitioners in the fields of aerospace engineering, chemical engineering, and fluid and solid mechanics, the major themes include an assessment of the state of the art in optimization algorithms as well as challenging applications in design and control, in the areas of process engineering and systems with partial differential equation models.

The theory of approximation of functions is one of the central branches in mathematical analysis and has been developed over a number of decades. This monograph deals with a series of problems related to one of the directions of the theory, namely, the approximation of periodic functions by trigonometric polynomials generated by linear methods of summation of Fourier series. More specific, the following linear methods are investigated: classical methods of Fourier, Fejir, Riesz, and Roginski. For these methods the so-called Kolmogorov-Nikol'skii problem is considered, which consists of finding exact and asymptotically exact qualities for the upper bounds of deviations of polynomials generated by given linear methods on given classes of 2?-periodic functions. Much attention is also given to the multidimensional case. The material presented in this monograph did not lose its importance since the publication of the Russian edition (1981). Moreover, new material has been added and several corrections were made. In this field of mathematics numerous deep results were obtained, many important and complicated problems were solved, and new methods were developed, which can be extremely useful for many mathematicians. All principle problems considered in this monograph are given in the final form, i.e. in the form of exact asymptotic equalities, and, therefore, retain their importance and interest for a long time.

Many special functions occuring in physics and partial differential equations can be represented by integral transformatIons: the fundamental solutions of many PDE's, Newton-Coulomb potentials, hypergeometric functions, Feynman integrals, initial data of (inverse) tomography problems, etc. The general picture of such transfor- mations is as follows. There is an analytic fibre bundle E --+ T, a differential form w on E, whose restrictions on the fibres are closed, and a family of cycles in these fibres, parametrized by the points of T and depending continuously on these points. Then the integral of the form w along these cycles is a function on the base. The analytic properties of such functions depend on the monodromy action, i.e., on the natural action of the fundamental group of the base in the homology of the fibre: this action on the integration cycles defines the ramification of the analytic continuation of our function. The study of this action (which is a purely topological problem) can answer questions about the analytic behaviour of the integral function, for instance, is this function single-valued or at least algebraic, what are the singular points of this function, and what is its asymptotics close to these points. In this book, we study such analytic properties of three famous classes of func- tions: the volume functions, which appear in the Archimedes-Newton problem on in- tegrable bodies; the Newton-Coulomb potentials, and the Green functions of hyperbolic equations (studied, in particular, in the Hada- mard-Petrovskii-Atiyah-Bott-Garding lacuna theory).

Inverse problems and optimal design have come of age as a consequence of the availability of better, more accurate, and more efficient simulation packages. Many of these simulators, which can run on small workstations, can capture the complicated behavior of the physical systems they are modeling, and have become commonplace tools in engineering and science. There is a great desire to use them as part of a process by which measured field data are analyzed or by which design of a product is automated. A major obstacle in doing precisely this is that one is ultimately confronted with a large-scale optimization problem. This volume contains expository articles on both inverse problems and design problems formulated as optimization. Each paper describes the physical problem in some detail and is meant to be accessible to researchers in optimization as well as those who work in applied areas where optimization is a key tool. What emerges in the presentations is that there are features about the problem that must be taken into account in posing the objective function, and in choosing an optimization strategy. In particular there are certain structures peculiar to the problems that deserve special treatment, and there is ample opportunity for parallel computation. THIS IS BACK COVER TEXT Inverse problems and optimal design have come of age as a consequence of the availability of better, more accurate, and more efficient, simulation packages. The problem of determining the parameters of a physical system from

This corrected third printing retains the authors'main emphasis on ordinary differential equations. It is most appropriate for upper level undergraduate and graduate students in the fields of mathematics, engineering, and applied mathematics, as well as the life sciences, physics and economics. The authors have taken the view that a differential equations theory defines functions; the object of the theory is to understand the behaviour of these functions. The tools the authors use include qualitative and numerical methods besides the traditional analytic methods, and the companion software, MacMath, is designed to bring these notions to life.

In this book the author sets out to answer two important questions: 1. Which numerical methods may be combined together? 2. How can different numerical methods be matched together? In doing so the author presents a number of useful combinations, for instance, the combination of various FEMs, the combinations of FEM-FDM, REM-FEM, RGM-FDM, etc. The combined methods have many advantages over single methods: high accuracy of solutions, less CPU time, less computer storage, easy coupling with singularities as well as the complicated boundary conditions. Since coupling techniques are essential to combinations, various matching strategies among different methods are carefully discussed. The author provides the matching rules so that optimal convergence, even superconvergence, and optimal stability can be achieved, and also warns of the matching pitfalls to avoid. Audience: The book is intended for both mathematicians and engineers and may be used as text for advanced students.

The book contains the methods and bases of functional analysis that are directly adjacent to the problems of numerical mathematics and its applications; they are what one needs for the understand ing from a general viewpoint of ideas and methods of computational mathematics and of optimization problems for numerical algorithms. Functional analysis in mathematics is now just the small visible part of the iceberg. Its relief and summit were formed under the influence of this author's personal experience and tastes. This edition in English contains some additions and changes as compared to the second edition in Russian; discovered errors and misprints had been corrected again here; to the author's distress, they jump incomprehensibly from one edition to another as fleas. The list of literature is far from being complete; just a number of textbooks and monographs published in Russian have been included. The author is grateful to S. Gerasimova for her help and patience in the complex process of typing the mathematical manuscript while the author corrected, rearranged, supplemented, simplified, general ized, and improved as it seemed to him the book's contents. The author thanks G. Kontarev for the difficult job of translation and V. Klyachin for the excellent figures."

In some domains of mechanics, physics and control theory boundary value problems arise for nonlinear first order PDEs. A well-known classical result states a sufficiency condition for local existence and uniqueness of twice differentiable solution. This result is based on the method of characteristics (MC). Very often, and as a rule in control theory, the continuous nonsmooth (non-differentiable) functions have to be treated as a solutions to the PDE. At the points of smoothness such solutions satisfy the equation in classical sense. But if a function satisfies this condition only, with no requirements at the points of nonsmoothness, the PDE may have nonunique solutions. The uniqueness takes place if an appropriate matching principle for smooth solution branches defined in neighboring domains is applied or, in other words, the notion of generalized solution is considered. In each field an appropriate matching principle are used. In Optimal Control and Differential Games this principle is the optimality of the cost function. In physics and mechanics certain laws must be fulfilled for correct matching. A purely mathematical approach also can be used, when the generalized solution is introduced to obtain the existence and uniqueness of the solution, without being aimed to describe (to model) some particular physical phenomenon. Some formulations of the generalized solution may meet the modelling of a given phenomenon, the others may not.

Operational methods have been used for over a century to solve problems such as ordinary and partial differential equations. When solving such problems, in many cases it is fairly easy to obtain the Laplace transform, while it is very demanding to determine the inverse Laplace transform which is the solution of a given problem. Sometimes, after some difficult contour integration we may find that a series solution results, but this may be quite difficult to evaluate in order to get an answer at a particular time value. The advent of computers has given an impetus to developing numerical methods for the determination of the inverse Laplace transform. This book gives background material on the theory of Laplace transforms, together with a fairly comprehensive list of methods which are available at the current time. Computer programs are included for those methods which perform consistently well on a wide range of Laplace transforms.

An approach to complexity theory which offers a means of analysing algorithms in terms of their tractability. The authors consider the problem in terms of parameterized languages and taking "k-slices" of the language, thus introducing readers to new classes of algorithms which may be analysed more precisely than was the case until now. The book is as self-contained as possible and includes a great deal of background material. As a result, computer scientists, mathematicians, and graduate students interested in the design and analysis of algorithms will find much of interest.

This volume of the Proceedings of the congress ISAAC '97 collects the con tributions of the four sections 1. Function theoretic and functional analytic methods for pde, 2. Applications of function theory of several complex variables to pde, 3. Integral equations and boundary value problems, 4. Partial differential equations. Most but not all of the authors have participated in the congress. Unfortunately some from Eastern Europe and Asia have not managed to come because of lack of financial support. Nevertheless their manuscripts of the proposed talks are included in this volume. The majority of the papers deal with complex methods. Among them boundary value problems in particular the Riemann-Hilbert, the Riemann (Hilbert) and related problems are treated. Boundary behaviour of vector-valued functions are studied too. The Riemann-Hilbert problem is solved for elliptic complex equations, for mixed complex equations, and for several complex variables. It is considered in a general topological setting for mappings into q;n and related to Toeplitz operators. Convolution operators are investigated for nilpotent Lie groups leading to some consequences for the null space of the tangential Cauchy Riemann operator. Some boundary value problems for overdetermined systems in balls of q;n are solved explicitly. A survey is given for the Gauss-Manin connection associated with deformations of curve singularities. Several papers deal with generalizations of analytic functions with various applications to mathematical physics. Singular integrals in quaternionic anal ysis are studied which are applied to the time-harmonic Maxwell equations."

Contains well-chosen examples and exercises

A student-friendly introduction that follows a workbook type approach