Bøger i Encyclopedia of Mathematics and its Applications serien

Written by prominent experts in the field, this monograph provides the first comprehensive, unified presentation of the structural, algorithmic and applied aspects of the theory of Boolean functions. The book focuses on algebraic representations of Boolean functions and offers a unique in-depth treatment that provides emphasis on algorithms and applications.

This book provides a lucid and comprehensive introduction to the differential geometric study of partial differential equations. It was the first book to present substantial results on local solvability of general and, in particular, nonlinear PDE systems without using power series techniques.

Like differentiability, convexity is a natural and powerful property of functions that plays a significant role in many areas of mathematics, both pure and applied. It ties together notions from topology, algebra, geometry and analysis, and is an important tool in optimization, mathematical programming and game theory. This book, which is the product of a collaboration of over 15 years, is unique in that it focuses on convex functions themselves, rather than on convex analysis. The authors explore the various classes and their characteristics and applications, treating convex functions in both Euclidean and Banach spaces. The book can either be read sequentially for a graduate course, or dipped into by researchers and practitioners. Each chapter contains a variety of specific examples, and over 600 exercises are included, ranging in difficulty from early graduate to research level.

This major revision of Berstel and Perrin's classic Theory of Codes has been rewritten with a more modern focus and a much broader coverage of the subject. The concept of unambiguous automata, which is intimately linked with that of codes, now plays a significant role throughout the book, reflecting developments of the last 20 years. This is complemented by a discussion of the connection between codes and automata, and new material from the field of symbolic dynamics. The authors have also explored links with more practical applications, including data compression and cryptography. The treatment remains self-contained: there is background material on discrete mathematics, algebra and theoretical computer science. The wealth of exercises and examples make it ideal for self-study or courses. In sum this is a comprehensive reference on the theory of variable-length codes and their relation to automata.

Ordered sets are ubiquitous in mathematics and have significant applications in computer science, statistics, biology and the social sciences. As the first book to deal exclusively with finite ordered sets, this book will be welcomed by graduate students and researchers in all of these areas. Beginning with definitions of key concepts and fundamental results (Dilworth's and Sperner's theorem, interval and semiorders, Galois connection, duality with distributive lattices, coding and dimension theory), the authors then present applications of these structures in fields such as preference modelling and aggregation, operational research and management, cluster and concept analysis, and data mining. Exercises are included at the end of each chapter with helpful hints provided for some of the most difficult examples. The authors also point to further topics of ongoing research.

This largely self-contained treatment surveys, unites and extends some 20 years of research on direct and inverse problems for canonical systems of integral and differential equations and related systems. Five basic inverse problems are studied in which the main part of the given data is either a monodromy matrix; an input scattering matrix; an input impedance matrix; a matrix valued spectral function; or an asymptotic scattering matrix. The corresponding direct problems are also treated. The book incorporates introductions to the theory of matrix valued entire functions, reproducing kernel Hilbert spaces of vector valued entire functions (with special attention to two important spaces introduced by L. de Branges), the theory of J-inner matrix valued functions and their application to bitangential interpolation and extension problems, which can be used independently for courses and seminars in analysis or for self-study. A number of examples are presented to illustrate the theory.

This third volume of four describes all the most important techniques, mainly based on Groebner bases. It covers the 'standard' solutions (Gianni-Kalkbrener, Auzinger-Stetter, Cardinal-Mourrain) as well as the more innovative (Lazard-Rouillier, Giusti-Heintz-Pardo). The author also explores the historical background, from Bezout to Macaulay.

This first volume in a series provides a graduate-level introduction to the many facets of aperiodic order. Special attention is given to methods from algebra, discrete geometry and harmonic analysis, while the main focus is on topics motivated by physics and crystallography. Numerous illustrations and examples are included.

Now in its second edition, this classic text has been expanded to reflect significant developments in Brunn-Minkowski theory over the past two decades. It gives a complete presentation from basics to the exposition of current research, with full proofs, pointers to other fields and a fully updated reference list.

An extensive synthesis of recent work in the study of endomorphism rings and their modules, this book considers direct sum decompositions of modules, the class number of an algebraic number field, point set topological spaces, and classical noncommutative localization.

The first book in the literature devoted to ellipsoidal harmonics includes topics drawn from geometry, physics, biosciences and inverse problems. It serves as a reference for applied mathematicians and for anyone else who needs to know the current state of the art in this fascinating subject.

For over a century lattice sums have been studied by mathematicians and scientists in diverse areas of science, in some cases unwittingly duplicating previous work. Here, at last, is a comprehensive overview of the substantial body of knowledge that now exists on lattice sums and their applications.

In recent years the traditional subject of continuum mechanics has grown rapidly and many new techniques have emerged. This text provides a rigorous, yet accessible introduction to the basic concepts of the network approximation method and provides a unified approach for solving a wide variety of applied problems. As a unifying theme, the authors discuss in detail the transport problem in a system of bodies. They solve the problem of closely placed bodies using the new method of network approximation for PDE with discontinuous coefficients, developed in the 2000s by applied mathematicians in the USA and Russia. Intended for graduate students in applied mathematics and related fields such as physics, chemistry and engineering, the book is also a useful overview of the topic for researchers in these areas.

Structural graph theory uses ideas of 'connectivity' to explore various aspects of graph theory and vice versa. Written by acknowledged international experts in the field, this reference for researchers and graduate students in graph theory and network flows also serves as a quick introduction for mathematicians in other fields.

This unique book covers most of the known results on reducibility of polynomials over arbitrary fields, algebraically closed fields and finitely generated fields. Also included are results based on recent work of E. Bombieri and U. Zannier.

Information content and programming semantics are just two of the applications of the mathematical concepts of order, continuity and domains. The authors develop the mathematical foundations of partially ordered sets with completeness properties of various degrees, in particular directed complete ordered sets and complete lattices. Uniquely, they focus on partially ordered sets that have an extra order relation, modelling the notion that one element 'finitely approximates' another, something closely related to intrinsic topologies linking order and topology. Extensive use is made of topological ideas, both by defining useful topologies on the structures themselves and by developing close connections with numerous aspects of topology. The theory so developed not only has applications to computer science but also within mathematics to such areas as analysis, the spectral theory of algebras and the theory of computability. This authoritative, comprehensive account of the subject will be essential for all those working in the area.

Sub-Riemannian manifolds are manifolds with the Heisenberg principle built in. This comprehensive text and reference introduces the theory and applications of sub-Riemannian geometry for graduate students and researchers in pure and applied mathematics, theoretical physics, control theory, and thermodynamics. Potential applications include quantum mechanics and quantum field theory.

Now in its second edition, this book gives a systematic and self-contained presentation of basic results on stochastic evolution equations in infinite dimensional, typically Hilbert and Banach, spaces. Thoroughly updated, it also includes two brand new chapters surveying recent developments in the area.

Monoidal Topology will appeal to a broad range of mathematicians and theoretical computer scientists who work with ordered, metric and topological structures. It presents frontline research in a number of fields and provides all the necessary pre-requisites in order and category theory.

Serving both as an introduction to the subject and as a reference, this book covers the general theory and emphasizes the classical types of orthogonal polynomials, or those of Gaussian type. Containing 25% brand new material, this revised edition reflects progress made in the field over the past decade.

Written by acknowledged international experts in the field, this book provides a broad survey of chromatic graph theory. It serves as a valuable reference for researchers and graduate students in graph theory and combinatorics and as a useful introduction to the topic for mathematicians in related fields.

In this fourth and final volume the author covers extensions of Buchberger's Algorithm, including a discussion of the most promising recent alternatives to Groebner bases: Gerdt's involutive bases and Faugere's F4 and F5 algorithms. This completes the author's comprehensive treatise, which is a fundamental reference for any mathematical library.

This book provides researchers and graduate students with a thorough introduction to the variational analysis of nonlinear problems described by nonlocal operators. The authors give a systematic treatment of the basic mathematical theory and constructive methods for these classes of nonlinear equations, plus their application to various processes arising in the applied sciences. The equations are examined from several viewpoints, with the calculus of variations as the unifying theme. Part I begins the book with some basic facts about fractional Sobolev spaces. Part II is dedicated to the analysis of fractional elliptic problems involving subcritical nonlinearities, via classical variational methods and other novel approaches. Finally, Part III contains a selection of recent results on critical fractional equations. A careful balance is struck between rigorous mathematics and physical applications, allowing readers to see how these diverse topics relate to other important areas, including topology, functional analysis, mathematical physics, and potential theory.

This two-volume work presents the main known equivalents to the Riemann hypothesis, perhaps the most important problem in mathematics. Volume 1 presents classical and modern arithmetic equivalents, with some analytic methods. Accompanying software is freely available online.

This two-volume work presents the main known equivalents to the Riemann hypothesis, perhaps the most important problem in mathematics. Volume 2 covers equivalents with a strong analytic orientation and is supported by an extensive set of appendices.

The second volume in this series exploring the mathematics of Aperiodic Order begins to develop the theory in more depth. A collection of leading experts cover various aspects of crystallography with a strong focus on almost periodicity and its close links to Fourier analysis.

Presents the state of the art of ultrametric pseudodifferential equations, relevant not only in mathematics but also in fields such as engineering, geophysics, and physics. Results previously scattered across many diverse journals are usefully consolidated here alongside novel ideas and applications.

Proof complexity is a rich subject drawing on methods from logic, combinatorics, algebra and computer science. This self-contained book presents the basic concepts, classical results, current state of the art and possible future directions in the field. Suitable for doctoral students and researchers in mathematics and theoretical computer science.

This self-contained book is the first to be dedicated entirely to Drinfeld's quasi-Hopf algebras, from the basics to the state of the art. It includes a detailed introduction to (braided) monoidal categories, the main tool used to study quasi-Hopf algebras. It is ideal for graduate students and researchers in mathematics and mathematical physics.

The Representation Theory of the Symmetric Group provides an account of both the ordinary and modular representation theory of the symmetric groups. The range of applications of this theory is vast, varying from theoretical physics, through combinatories to the study of polynomial identity algebras; and new uses are still being found.