Applied Mathematical Sciences – serie

The first edition of this book was originally published in 1985 under the ti­ tle "Probabilistic Properties of Deterministic Systems. " In the intervening years, interest in so-called "chaotic" systems has continued unabated but with a more thoughtful and sober eye toward applications, as befits a ma­ turing field. This interest in the serious usage of the concepts and techniques of nonlinear dynamics by applied scientists has probably been spurred more by the availability of inexpensive computers than by any other factor. Thus, computer experiments have been prominent, suggesting the wealth of phe­ nomena that may be resident in nonlinear systems. In particular, they allow one to observe the interdependence between the deterministic and probabilistic properties of these systems such as the existence of invariant measures and densities, statistical stability and periodicity, the influence of stochastic perturbations, the formation of attractors, and many others. The aim of the book, and especially of this second edition, is to present recent theoretical methods which allow one to study these effects. We have taken the opportunity in this second edition to not only correct the errors of the first edition, but also to add substantially new material in five sections and a new chapter.

A study of topology and geometry, beginning with a comprehensible account of the extraordinary and rather mysterious impact of mathematical physics, and especially gauge theory, on the study of the geometry and topology of manifolds. The focus of the book is the Yang-Mills-Higgs field and some considerable effort is expended to make clear its origin and significance in physics. Much of the mathematics developed here to study these fields is standard, but the treatment always keeps one eye on the physics and sacrifices generality in favor of clarity. This second edition has replaced a brief appendix in the first on the Seiberg-Witten equations with a much more detailed survey of Donaldson-Witten Theory and the Witten Conjecture regarding the relationship between Donaldson and Seiberg-Witten invariants.A large number of exercises are included to encourage active participation on the part of the reader.Review from first edition:"It is unusual to find a book so carefully tailored to the needs of this interdisciplinary area of mathematical physics...Naber combines a knowledge of his subject with an excellent informal writing style."SIAM REVIEW

The first of three volumes on partial differential equations, this one introduces basic examples arising in continuum mechanics, electromagnetism, complex analysis and other areas, and develops a number of tools for their solution, in particular Fourier analysis, distribution theory, and Sobolev spaces. These tools are then applied to the treatment of basic problems in linear PDE, including the Laplace equation, heat equation, and wave equation, as well as more general elliptic, parabolic, and hyperbolic equations. The book is targeted at graduate students in mathematics and at professional mathematicians with an interest in partial differential equations, mathematical physics, differential geometry, harmonic analysis, and complex analysis.The third edition further expands the material by incorporating new theorems and applications throughout the book, and by deepening connections and relating concepts across chapters.  In includes new sections on rigid body motion, on probabilistic results related to random walks, on aspects of operator theory related to quantum mechanics, on overdetermined systems, and on the Euler equation for incompressible fluids.  The appendices have also been updated with additional results, ranging from weak convergence of measures to the curvature of Kahler manifolds.Michael E. Taylor is a Professor of Mathematics at the University of North Carolina, Chapel Hill, NC.Review of first edition: “These volumes will be read by several generations of readers eager to learn the modern theory of partial differential equations of mathematical physics and the analysis in which this theory is rooted.”(Peter Lax, SIAM review, June 1998)

This book is a revised edition of my earlier book of the same title. The cur­ rent edition adopts the structure of the earlier version but is much changed. The introduction now contains definitions of stability. Chapters 2 to 4 ex­ plain stability and the energy method in more depth and new sections dealing with porous media are provided. Chapters 5 to 13 are revisions of those in the earlier edition. However, chapters 6 to 12 are substantially revised, brought completely up to date, and have much new material in. Throughout the book new results are provided which are not available elsewhere. Six new chapters, 14 - 19, are provided dealing with topics of current interest. These cover the topics of multi-component convection diffusion, convection in a compressible fluid, convection with temperature dependent viscosity and thermal conductivity, the subject of penetrative convection whereby part of the fluid layer can penetrate into another, nonlinear sta­ bility in the oceans, and finally in chapter 19 practical methods for solving numerically the eigenvalue problems which arise are presented. The book presents convection studies in a variety of fluid and porous media contexts. It should be accessible to a wide audience and begins at an elementary level. Many new references are provided.

The origins of the finite element method can be traced back to the 1950s when engineers started to solve numerically structural mechanics problems in aeronautics. Since then, the field of applications has widened steadily and nowadays encompasses nonlinear solid mechanics, fluid/structure interactions, flows in industrial or geophysical settings, multicomponent reactive turbulent flows, mass transfer in porous media, viscoelastic flows in medical sciences, electromagnetism, wave scattering problems, and option pricing (to cite a few examples). Numerous commercial and academic codes based on the finite element method have been developed over the years. The method has been so successful to solve Partial Differential Equations (PDEs) that the term "Finite Element Method" nowadays refers not only to the mere interpolation technique it is, but also to a fuzzy set of PDEs and approximation techniques. The efficiency of the finite element method relies on two distinct ingredi­ ents: the interpolation capability of finite elements (referred to as the approx­ imability property in this book) and the ability of the user to approximate his model (mostly a set of PDEs) in a proper mathematical setting (thus guar­ anteeing continuity, stability, and consistency properties). Experience shows that failure to produce an approximate solution with an acceptable accuracy is almost invariably linked to departure from the mathematical foundations. Typical examples include non-physical oscillations, spurious modes, and lock­ ing effects. In most cases, a remedy can be designed if the mathematical framework is properly set up.

During the nine years since the publication of the ?rst edition of this book, therehasbeensubstantialprogressonthetreatmentofwell-setpr- lemsofnonlinearsolidmechanics. Themainpurposesofthissecondedition are to update the ?rst edition by giving a coherent account of some of the new developments, to correct errors, and to re?ne the exposition. Much of the text has been rewritten, reorganized, and extended. The philosophy underlying my approach is exactly that given in the following (slightly modi?ed) Preface to the First Edition. In particular, I continue to adhere to my policy of eschewing discussions relying on tech- cal aspects of theories of nonlinear partial di?erential equations (although I give extensive references to pertinent work employing such methods). Thus I intend that this edition, like the ?rst, be accessible to a wide circle of readers having the traditional prerequisites given in Sec. 1.2. I welcome corrections and comments, which can be sent to my electronic mail address: ssa@math.umd.edu. In due time, corrections will be placed on my web page: http://www.ipst.umd.edu/Faculty/antman.htm.

This book is aimed at postgraduate students in applied mathematics as well as at engineering and physics students with a ?rm background in mathem- ics. The ?rst four chapters can be used as the material for a ?rst course on inverse problems with a focus on computational and statistical aspects. On the other hand, Chapters 3 and 4, which discuss statistical and nonstati- ary inversion methods, can be used by students already having knowldege of classical inversion methods. There is rich literature, including numerous textbooks, on the classical aspects of inverse problems. From the numerical point of view, these books concentrate on problems in which the measurement errors are either very small or in which the error properties are known exactly. In real-world pr- lems, however, the errors are seldom very small and their properties in the deterministic sensearenot wellknown.For example,inclassicalliteraturethe errornorm is usuallyassumed to be a known realnumber. In reality,the error norm is a random variable whose mean might be known.

Since their emergence in the early 1950s, ?nite element methods have become one of the most versatile and powerful methodologies for the approximate numerical solution of partial differential equations. At the time of their inception, ?nite e- ment methods were viewed primarily as a tool for solving problems in structural analysis. However, it did not take long to discover that ?nite element methods could be applied with equal success to problems in other engineering and scienti?c ?elds. Today, ?nite element methods are also in common use, and indeed are often the method of choice, for incompressible ?uid ?ow, heat transfer, electromagnetics, and advection-diffusion-reaction problems, just to name a few. Given the early conn- tion between ?nite element methods and problems engendered by energy minimi- tion principles, it is not surprising that the ?rst mathematical analyses of ?nite e- ment methods were given in the environment of the classical Rayleigh–Ritz setting. Yet again, using the fertile soil provided by functional analysis in Hilbert spaces, it did not take long for the rigorous analysis of ?nite element methods to be extended to many other settings. Today, ?nite element methods are unsurpassed with respect to their level of theoretical maturity.

This book focuses on variational methods in imaging science. Many numerical examples accompany the theory throughout the text. This systematic presentation includes additional material and images available on the website. It is geared towards graduate students and researchers in applied mathematics. Researchers in the area of imaging science will also find this book appealing. It can serve as a main text in courses in image processing or as a supplemental text for courses on regularization and inverse problems at the graduate level.

Partial differential equations (PDEs) and variational methods were introduced into image processing about fifteen years ago. Since then, intensive research has been carried out. The goals of this book are to present a variety of image analysis applications, the precise mathematics involved and how to discretize them.Thus, this book is intended for two audiences. The first is the mathematical community by showing the contribution of mathematics to this domain. It is also the occasion to highlight some unsolved theoretical questions. The second is the computer vision community by presenting a clear, self-contained and global overview of the mathematics involved in image procesing problems. This work will serve as a useful source of reference and inspiration for fellow researchers in Applied Mathematics and Computer Vision, as well as being a basis for advanced courses within these fields.During the four years since the publication of the first edition, there has been substantial progress in the range of image processing applications covered by the PDE framework. The main goals of the second edition are to update the first edition by giving a coherent account of some of the recent challenging applications, and to update the existing material. In addition, this book provides the reader with the opportunity to make his own simulations with a minimal effort. To this end, programming tools are made available, which will allow the reader to implement and test easily some classical approaches.

This second edition is the successor to "Direct methods in the calculus of variations" which was published in the Applied Mathematical Sciences series and is currently out of print. Although the core and the structure of the present book is similar to the first edition, it is much more than a revised version. Fifteen years have passed since the publication of the "Direct methods in the calculus of variations" book and since the subject is a very active one, almost half of the book presently consists of new material. The perspective has also slightly changed, indeed, a new subject, "quasiconvex analysis" has now been developed. The present edition, which is essentially a reference book on the subject of quasiconvex analysis can be used, as was the earlier book, for an advanced course on the calculus of variations.

Homology is a powerful tool used by mathematicians to study the properties of spaces and maps that are insensitive to small perturbations. This book uses a computer to develop a combinatorial computational approach to the subject. The core of the book deals with homology theory and its computation. Following this is a section containing extensions to further developments in algebraic topology, applications to computational dynamics, and applications to image processing. Included are exercises and software that can be used to compute homology groups and maps. The book will appeal to researchers and graduate students in mathematics, computer science, engineering, and nonlinear dynamics.

Perturbation theory and in particular normal form theory has shown strong growth during the last decades. So it is not surprising that the authors have presented an extensive revision of the first edition of the Averaging Methods in Nonlinear Dynamical Systems book. There are many changes, corrections and updates in chapters on Basic Material and Asymptotics, Averaging, and Attraction. Chapters on Periodic Averaging and Hyperbolicity, Classical (first level) Normal Form Theory, Nilpotent (classical) Normal Form, and Higher Level Normal Form Theory are entirely new and represent new insights in averaging, in particular its relation with dynamical systems and the theory of normal forms. Also new are surveys on invariant manifolds in Appendix C and averaging for PDEs in Appendix E. Since the first edition, the book has expanded in length and the third author, James Murdock has been added.Review of First Edition"One of the most striking features of the book is the nice collection of examples, which range from the very simple to some that are elaborate, realistic, and of considerable practical importance. Most of them are presented in careful detail and are illustrated with profuse, illuminating diagrams." - Mathematical Reviews

This book presents important recent developments in mathematical and computational methods used in impedance imaging and the theory of composite materials. The exposition brings simplicity to the advanced material by augmenting the theory with interesting practical examples and numerical illustrations. An introductory chapter covers the necessary basics. An extensive bibliography and open problems at the end of each chapter enhance the text. Graduate students and researchers in mathematics will benefit from this book. Researchers in engineering and physics might also find it helpful.

This book presents an account of theories of ?ow in porous media which have proved tractable to analysis and computation. In particular, the t- ories of Darcy, Brinkman, and Forchheimer are presented and analysed in detail. In addition, we study the theory of voids in an elastic material due to J. Nunziato and S. Cowin. The range of validity of each theory is outlined and the mathematical properties are considered. The questions of structural stability, where the stability of the model itself is under cons- eration, and spatial stability are investigated. We believe this is the ?rst such account of these topics in book form. Throughout, we include several new results not published elsewhere. Temporal stability studies of a variety of problems are included, indic- ingpracticalapplicationsofeach.Bothlinearinstabilityanalysisandglobal nonlinear stability thresholds are presented where possible. The mundane, importantproblemofstabilityof?owinasituationwhereaporousmedium adjoins a clear ?uid is also investigated in some detail. In particular, the chapter dealing with this problem contains some new material only p- lished here. Since stability properties inevitably end up requiring to solve a multi-parameter eigenvalue problem by computational means, a separate chapter is devoted to this topic. Contemporary methods for solving such eigenvalue problems are presented in some detail.

This book gives a systematic investigation of convection in systems comprised of liquid layers with deformatable interfaces.This new edition includes completely updated and new material on flows in ultra thin films and brings up to date progress made in the technology on micro and nano scales. Also, this revised edition will reflect progress in the dynamics of complex fluids.

These notes originate from a one semester course which forms part of the "Math Methods" cycle at Brown. In the hope that these notes might prove useful for reference purposes several additional sections have been included and also a table of contents and index. Although asymptotic analysis is now enjoying a period of great vitality, these notes do not reflect a research oriented course. The course is aimed toward people in applied mathematics, physics, engineering, etc., who have a need for asymptotic analysis in their work. The choice of subjects has been largely dictated by the likelihood of application. Also abstraction and generality have not been pursued. Technique and computation are given equal prominence with theory. Both rigorous and formal theory is presented --very often in tandem. In practice, the means for a rigorous analysis are not always available. For this reason a goal has been the cultivation of mature formal reasoning. Therefore, during the course of lectures formal presentations gradually eclipse rigorous presentations. When this occurs, rigorous proofs are given as exercises or in the case of lengthy proofs, reference is made to the Reading List at the end.

It is not a large overstatement to claim that mathematics has traditionally arisen from attempts to understand quite concrete events in the physical world. The accelerated sophistication of the mathematical community has perhaps obscured this fact, especially during the present century, with the abstract becoming the hallmark of much of respectable mathematics. As a result of the inaccessibility of such work, practicing scientists have often been compelled to fashion their own mathematical tools, blissfully unaware of their prior existence in far too elegant and far too general form. But the mathematical sophistication of scientists has grown rapidly too, as has the scientific sophistication of many mathematicians, and the real worl- suitably defined - is once more serving its traditional role. One of the fields most enriched by this infusion has been that of combinatorics. This book has been written in a way as a tribute to those natural scientists whose breadth of vision has inparted a new vitality to a dormant giant. The present text arose out of a course in Combinatorial Methods given by the writer at the Courant Institute during 1967-68. Its structure has been determined by an attempt to reach an informed but heterogeneous group of students in mathematics, physics, and chemistry. Its lucidity has been enhanced immeasurably by the need to satisfy a very resolute critic, Professor Ora E. Percus, who is responsible for the original lecture notes as well as for their major modifications.

In the summer of 1941 Brown University undertook a Program of Advanced Instruction and Research in Mechanics. This in fact was the precursor to the present day Division of Applied Mathematics. Certainly an outstanding feature of this program must have been the lectures in Fluid Dynamics by Professor Friedrichs and the late Professor von Mises. Their notes were prepared in mimeograph form and given a wide distribution at that time. Since their appearance these lectures have had a strong influence on teaching and research in the subject. As the reader soon learns the notes have lost none of their vitality over the years. Indeed in certain instances only in the last few years has the -field caught up with the ideas developed in the course of these lectures. Many ideas of value are still to be found in these notes and the Editors are most happy to be able to include this volume in the series. The corrections which have accumulated over the years have been incorporated, and in addition an index has been added. With these exceptions all desire to revise has been resisted. Also in this connection we are very grateful to Dr. T. H. Chong for carefully overseeing the preparation of the present manuscript.

This book arose out of a number of different contexts, and numerous persons have contributed to its conception and development. It had its origin in a project initiated jointly with the IBM Cambridge Scien­ tific Center, particularly with Dr. Rhett Tsao, then of that Center. We are grateful to Mr. Norman Rasmussen, Manager of the IBM Scientific Center Complex, for his initial support. The work is being carried on at Brown University with generous support from the Office of Computing Activities of the National Science Foundation (grants GJ-174 and GJ-7l0); we are grateful to Dr. John Lehmann of this Office for his interest and encouragement. Professors Donald McClure and Richard Vitale of the Division of Applied Mathematics at Brown University contributed greatly to the project and taught courses in its spirit. We are indebted to them and to Dr. Tore Dalenius of the University of Stockholm for helpful criticisms of the manuscript. The final stimulus to the book's completion came from an invLtation to teach a course at the IBM European Systems Research Institute at Geneva. We are grateful to Dr. J.F. Blackburn, Director of the Institute, for his invitation, and to him and his wife Beverley for their hospitality. We are greatly indebted to Mrs. Katrina Avery for her splendid secretarial and editorial work on the manuscript.

This volume is intended to provide a comprehensive treatment of recent developments in methods of perturbation for nonlinear systems of ordinary differ­ ential equations. In this respect, it appears to be a unique work. The main goal is to describe perturbation techniques, discuss their ad­ vantages and limitations and give some examples. The approach is founded on analytical and numerical methods of nonlinear mechanics. Attention has been given to the extension of methods to high orders of approximation, required now by the increased accuracy of measurements in all fields of science and technology. The main theorems relevant to each perturbation technique are outlined, but they only provide a foundation and are not the objective of these notes. Each chapter concludes with a detailed survey of the pertinent literature, supplemental information and more examples to complement the text, when necessary, for better comprehension. The references are intended to provide a guide for background information and for the reader who wishes to analyze any particular point in more detail. The main sources referenced are in the fields of differential equations, nonlinear oscillations and celestial mechanics. Thanks are due to Katherine MacDougall and Sandra Spinacci for their patience and competence in typing these notes. Partial support from the Mathematics Program of the Office of Naval Research is gratefully acknowledged.

The present lectures intend to provide an introduction to the spectral analysis of self-adjoint operators within the framework of Hilbert space theory. The guiding notion in this approach is that of spectral representation. At the same time the notion of function of an operator is emphasized. The formal aspects of these concepts are explained in the first two chapters. Only then is the notion of Hilbert space introduced. The following three chapters concern bounded, completely continuous, and non-bounded operators. Next, simple differential operators are treated as operators in Hilbert space, and the final chapter deals with the perturbation of discrete and continuous spectra. The preparation of the original version of these lecture notes was greatly helped by the assistance of P. Rejto. Various valuable suggestions made by him and by R. Lewis have been incorporated. The present version of the notes contains extensive modifica­ tions, in particular in the chapters on bounded and unbounded operators. February, 1973 K.O.F. PREFACE TO THE SECOND PRINTING The second printing (1980) is a basically unchanged reprint in which a number of minor errors were corrected. The author wishes to thank Klaus Schmidt (Lausanne) and John Sylvester (New York) for their lists of errors. v TABLE OF CONTENTS I. Spectral Representation 1 1. Three typical problems 1 12 2. Linear space and functional representation.

This is a textbook for a one semester course on numerical analysis for senior undergraduate or beginning graduate students with no previous knowledge of the subject. The prerequisites are calculus, some knowledge of ordinary differential equations, and knowledge of computer programming using Fortran. Normally this should be half of a two semester course, the other semester covering numerical solution of linear systems, inversion of matrices and roots of polynomials. Neither semester should be a prerequisite for the other. This would prepare the student for advanced topics on numerical analysis such as partial differential equations. We are philosophically opposed to a one semester surveyor "numerical methods" course which covers all of the above mentioned topics, plus perhaps others, in one semester. We believe the student in such a course does not learn enough about anyone topic to develop an appreciation for it. For reference Chapter I contains statements of results from other branches of mathematics needed for the numerical analysis. The instructor may have to review some of these results. Chapter 2 contains basic results about interpolation. We spend only about one week of a semester on interpolation and divide the remainder of the semester between quadrature and differential equations. Most of the sections not marked with an * can be covered in one semester. The sections marked with an * are included as a guide for further study.

This text was developed over a three year period of time (1971- 1973) from a variety of notes and references used in the presentation of a senior/first year graduate level course in the Division of En­ gineering at Brown University titled Linear System Theory. The in­ tent of the course was not only to introduce students to the more modern, state-space approach to multivariable control system analysis and design, as opposed to the classical, frequency domain approach, but also to draw analogies between the two approaches whenever and wherever possible. It is therefore felt that the material presented will have broader appeal to practicing engineers than a text devoted exclusively to the state-space approach. It was assumed that students taking the course had also taken, as a prerequisite, an undergraduate course in classical control theory and also were familiar with certain standard linear algebraic notions as well as the theory of ordinary differential equations, although a substantial effort was expended to make the material as self-contained as possible. In particular, Chapter 2 is employed to familiarize the reader with a good deal of the mathematical material employed through­ out the remainder of the text. Chapters 3 through 5 were drawn, in part, from a number of contemporary state-space and matrix algebraic references, as well as some recent research of the author, especially those portions which deal with polynomial matrices and the differential operator approach.

This book is an introduction to the mathematical theory of optimal control of processes governed by ordinary differential eq- tions. It is intended for students and professionals in mathematics and in areas of application who want a broad, yet relatively deep, concise and coherent introduction to the subject and to its relati- ship with applications. In order to accommodate a range of mathema- cal interests and backgrounds among readers, the material is arranged so that the more advanced mathematical sections can be omitted wi- out loss of continuity. For readers primarily interested in appli- tions a recommended minimum course consists of Chapter I, the sections of Chapters II, III, and IV so recommended in the introductory sec­ tions of those chapters, and all of Chapter V. The introductory sec­ tion of each chapter should further guide the individual reader toward material that is of interest to him. A reader who has had a good course in advanced calculus should be able to understand the defini­ tions and statements of the theorems and should be able to follow a substantial portion of the mathematical development. The entire book can be read by someone familiar with the basic aspects of Lebesque integration and functional analysis. For the reader who wishes to find out more about applications we recommend references [2], [13], [33], [35], and [50], of the Bibliography at the end of the book.

The aim of this book is to provide a systematic and practical account of methods of integration of ordinary and partial differential equations based on invariance under continuous (Lie) groups of trans­ formations. The goal of these methods is the expression of a solution in terms of quadrature in the case of ordinary differential equations of first order and a reduction in order for higher order equations. For partial differential equations at least a reduction in the number of independent variables is sought and in favorable cases a reduction to ordinary differential equations with special solutions or quadrature. In the last century, approximately one hundred years ago, Sophus Lie tried to construct a general integration theory, in the above sense, for ordinary differential equations. Following Abel's approach for algebraic equations he studied the invariance of ordinary differential equations under transformations. In particular, Lie introduced the study of continuous groups of transformations of ordinary differential equations, based on the infinitesimal properties of the group. In a sense the theory was completely successful. It was shown how for a first-order differential equation the knowledge of a group leads immediately to quadrature, and for a higher order equation (or system) to a reduction in order. In another sense this theory is somewhat disappointing in that for a first-order differ­ ential equation essentially no systematic way can be given for finding the groups or showing that they do not exist for a first-order differential equation.

Since there are several excellent books on stability theory, the author selected some recent topics in stability theory which are related to existence theorems for periodic solutions and for almost periodic solutions. The author hopes that these notes will also serve as an introduction to stability theory. These notes contain stability theory by Liapunov's second method and somewhat extended discussion of stability properties in almost periodic systems, and the existence of a periodic solution in a periodic system is discussed in connection with the boundedness of solutions, and the existence of an almost periodic solution in an almost periodic system is considered in con­ nection with some stability property of a bounded solution. In the theory of almost periodic systems, one has to consider almost periodic functions depending on parameters, but most of text books on almost periodic functions do not contain this case. Therefore, as mathemati­ cal preliminaries, the first chapter is intended to provide a guide for some properties of almost periodic functions with parameters as well as for properties of asymptotically almost periodic functions. These notes originate from a seminar on stability theory given by the author at the Mathematics Department of Michigan State Univer­ sity during the academic year 1972-1973. The author is very grateful to Professor Pui-Kei Wong and members of the Department for their warm hospitality and many helpful conversations. The author wishes to thank Mrs.

This monograph is based, in part, upon lectures given in the Princeton School of Engineering and Applied Science. It presupposes mainly an elementary knowledge of linear algebra and of topology. In topology the limit is dimension two mainly in the latter chapters and questions of topological invariance are carefully avoided. From the technical viewpoint graphs is our only requirement. However, later, questions notably related to Kuratowski's classical theorem have demanded an easily provided treatment of 2-complexes and surfaces. January 1972 Solomon Lefschetz 4 INTRODUCTION The study of electrical networks rests upon preliminary theory of graphs. In the literature this theory has always been dealt with by special ad hoc methods. My purpose here is to show that actually this theory is nothing else than the first chapter of classical algebraic topology and may be very advantageously treated as such by the well known methods of that science. Part I of this volume covers the following ground: The first two chapters present, mainly in outline, the needed basic elements of linear algebra. In this part duality is dealt with somewhat more extensively. In Chapter III the merest elements of general topology are discussed. Graph theory proper is covered in Chapters IV and v, first structurally and then as algebra. Chapter VI discusses the applications to networks. In Chapters VII and VIII the elements of the theory of 2-dimensional complexes and surfaces are presented.

The German edition of this book, first published in 1966, has been quite popular; we did not, however, consider publishing an English edition because a number of excellent textbooks in this field already exist. In recent years, how­ ever, the wish was frequently expressed that, especially, the description of the relationships between optimization and other subfields of mathematics, which is not to be found in this form in other texts, might be made available to a wider readership; so it was with this in mind that, be­ latedly, a translation was undertaken after all. Since the appearance of the German edition, the field of optimization has continued to develop at an unabated rate. A completely current presentation would have required a total reworking of the book; unfortunately, this was not possible. For example, we had to ignore the extensive progress which has been made in the development of numerical methods which do not require convexity assumptions to find local maxima and minima of non-linear optimization problems. These methods are also applicable to boundary value, and other, problems. Many new results, both of a numerical and a theoretical na­ ture, which are especially relevant to applications, are to be found in the areas of optimal contol and integer optimiza­ tion.