Bücher der Reihe Encyclopedia of Mathematics and its Applications

Compound renewal processes (CRPs) are among the most ubiquitous models arising in applications of probability. At the same time, they are a natural generalization of random walks, the most well-studied classical objects in probability theory. This monograph, written for researchers and graduate students, presents the general asymptotic theory and generalizes many well-known results concerning random walks. The book contains the key limit theorems for CRPs, functional limit theorems, integro-local limit theorems, large and moderately large deviation principles for CRPs in the state space and in the space of trajectories, including large deviation principles in boundary crossing problems for CRPs, with an explicit form of the rate functionals, and an extension of the invariance principle for CRPs to the domain of moderately large and small deviations. Applications establish the key limit laws for Markov additive processes, including limit theorems in the domains of normal and large deviations.

Understanding the interaction of waves with obstacles is fundamental in many areas of science and engineering. This book synthesises mathematical methods for solving such scattering problems, with an emphasis on time-dependent problems. It will be a valuable resource for applied mathematicians, physicists and engineers.

Using Bishop's work on constructive analysis as a framework, this monograph gives a systematic, detailed and general constructive theory of probability theory and stochastic processes. It is the first extended account of this theory: almost all of the constructive existence and continuity theorems that permeate the book are original. It also contains results and methods hitherto unknown in the constructive and nonconstructive settings. The text features logic only in the common sense and, beyond a certain mathematical maturity, requires no prior training in either constructive mathematics or probability theory. It will thus be accessible and of interest, both to probabilists interested in the foundations of their speciality and to constructive mathematicians who wish to see Bishop's theory applied to a particular field.

The study of graph structure has advanced in recent years with great strides: finite graphs can be described algebraically, enabling them to be constructed out of more basic elements. Separately the properties of graphs can be studied in a logical language called monadic second-order logic. In this book, these two features of graph structure are brought together for the first time in a presentation that unifies and synthesizes research over the last 25 years. The authors not only provide a thorough description of the theory, but also detail its applications, on the one hand to the construction of graph algorithms, and, on the other to the extension of formal language theory to finite graphs. Consequently the book will be of interest to graduate students and researchers in graph theory, finite model theory, formal language theory, and complexity theory.

Internationally recognised researchers look at developing trends in combinatorics with applications in the study of words and in symbolic dynamics. They explain the important concepts, providing a clear exposition of some recent results, and emphasise the emerging connections between these different fields. Topics include combinatorics on words, pattern avoidance, graph theory, tilings and theory of computation, multidimensional subshifts, discrete dynamical systems, ergodic theory, numeration systems, dynamical arithmetics, automata theory and synchronised words, analytic combinatorics, continued fractions and probabilistic models. Each topic is presented in a way that links it to the main themes, but then they are also extended to repetitions in words, similarity relations, cellular automata, friezes and Dynkin diagrams. The book will appeal to graduate students, research mathematicians and computer scientists working in combinatorics, theory of computation, number theory, symbolic dynamics, tilings and stringology. It will also interest biologists using text algorithms.

The authors explain in this work a new approach to observing and controlling linear systems whose inputs and outputs are not fixed in advance. They cover a class of linear time-invariant state/signal system that is general enough to include most of the standard classes of linear time-invariant dynamical systems, but simple enough that it is easy to understand the fundamental principles. They begin by explaining the basic theory of finite-dimensional and bounded systems in a way suitable for graduate courses in systems theory and control. They then proceed to the more advanced infinite-dimensional setting, opening up new ways for researchers to study distributed parameter systems, including linear port-Hamiltonian systems and boundary triplets. They include the general non-passive part of the theory in continuous and discrete time, and provide a short introduction to the passive situation. Numerous examples from circuit theory are used to illustrate the theory.

Strongly regular graphs are found in statistical design, finite group theory, coding theory and quantum information theory. This detailed survey of the theory and examples gathers the major results for the first time. It is an invaluable reference for researchers in graph theory, algebraic combinatorics, information theory and group theory.

This book gathers all the main results on the spectral representation of the most important one-dimensional Markov processes, with examples and applications. Covering sixty years of research, it is an essential resource for graduate students and researchers interested in the connection between orthogonal polynomials and stochastic processes.

Written by acknowledged international experts in the field, this book presents fifteen expository chapters in the rapidly expanding area of algorithmic graph theory. It serves graduate students and researchers in graph theory, combinatorics and computer science, as well as mathematicians and computer scientists in related fields.

Mathematical models of bond markets are of interest to researchers working in applied mathematics, especially in mathematical finance. This book concerns bond market models in which random elements are represented by Levy processes. These are more flexible than classical models and are well suited to describing prices quoted in a discontinuous fashion. The book's key aims are to characterize bond markets that are free of arbitrage and to analyze their completeness. Nonlinear stochastic partial differential equations (SPDEs) are an important tool in the analysis. The authors begin with a relatively elementary analysis in discrete time, suitable for readers who are not familiar with finance or continuous time stochastic analysis. The book should be of interest to mathematicians, in particular to probabilists, who wish to learn the theory of the bond market and to be exposed to attractive open mathematical problems.

Suitable for graduate students and researchers in diverse areas of mathematics, this monograph offers a new perspective on the classical theory of connected Hopf algebras, and extends it to a new setting where a real hyperplane arrangement is a central feature.

The first systematic account of the basic theory of normed algebras, without assuming associativity, includes many new and unpublished results and is sure to become a central resource for researchers and graduate students in the field. This first volume of two focuses on C*-algebras and related structure.

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.

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.

This book, first published in 1991, is devoted to the exposition of combinatorial matrix theory. This subject concerns itself with the use of matrix theory and linear algebra in proving results in combinatorics (and vice versa), and with the intrinsic properties of matrices viewed as arrays of numbers rather than algebraic objects in themselves.

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.

Provided the first self-contained comprehensive exposition of the theory of dynamical systems as a core mathematical discipline closely intertwined with all main areas of mathematics. The authors introduce and rigorously develop the theory while providing researchers interested in applications with fundamental tools and paradigms.

Originally published in 1977, this volume is concerned with the relationship between symmetries of a linear second-order partial differential equation of mathematical physics, the coordinate systems in which the equation admits solutions via separation of variables, and the properties of the special functions that arise in this manner.

Originally published in 2000, this is the first volume of a comprehensive treatise on the mathematical theory of deterministic control systems modeled by multi-dimensional partial differential equations (distributed parameter systems). Volume 1 presents the abstract parabolic and hyperbolic settings, including many fascinating results.

This book looks at Volterra integral and functional equations, and shows that the theory of Volterra equations exhibits a rich variety of features not present in the theory of ordinary differential equations. The book is generally self-contained and assumes only a basic knowledge of analysis.

With applications in communication, transportation, and production, graph connectivity has thrived using the theory of complexity and algorithms in modern computer science. This is the first thorough book on this central notion in graph and network theory, and covers new concepts and algorithms for quicker and more efficient computing.

Geometric tomography overlaps with convex geometry and employs many tools from that area, including some formulas from integral geometry. This comprehensive study provides a rigorous treatment of the subject. Although meant primarily for researchers and graduate students in geometry and tomography, brief introductions, suitable for advanced undergraduates, are provided to the basic concepts.

A Handbook of Categorical Algebra is designed to give a detailed account of what should be known by everybody working in, or using, category theory. As such it will be a unique reference. The three volumes are written in sequence, with the first being essentially self-contained. Volume 1 is devoted to general concepts and can be used for advanced undergraduate courses on category theory.

This book presents the mathematical theory of turbulence to engineers and physicists, and the physical theory of turbulence to mathematicians. The mathematical technicalities are kept to a minimum within the book, enabling the language to be at a level understood by a broad audience.

This should be a useful reference for anybody with interests in quantum theory and related areas of function theory, functional analysis, differential geometry or topological invariant theory.

This is the first volume of a two volume set that provides a modern account of basic Banach algebra theory including all known results on general Banach *-algebras. This account emphasises the role of *-algebra structure and explores the algebraic results which underlie the theory of Banach algebras and *-algebras.

This treatise deals with modern theory of functional equations in several variables and their applications to mathematics, information theory, and the natural, behavioural and social sciences. The authors have chosen to emphasize applications, though not at the expense of theory, so they have kept the prerequisites to a minimum.

Based on the author's LMS lecture note volume Skew Field Constructions, the present work offers a comprehensive account of skew fields. Numerous exercises test the reader's understanding, presenting further aspects and open problems in a concise form. The notes and comments at the end of each chapter provide an historical background.

The notion of 'stopping times' is a useful one in probability theory; it can be applied to both classical problems and fresh ones. This book presents this technique in the context of the directed set, stochastic processes indexed by directed sets and many applications in probability, analysis and ergodic theory.

This comprehensive study of the analytic aspects of mathematical tomography contains elementary and graphical introductions to the Radon transform, tomography and CT scanners, to the development of the basic properties of the Radon transform, to Grassmann manifolds, and to the study of k-plane transform, plus coverage of more advanced topics.