Alexander Rubinov
School of Information Technology and Mathematical Sciences, University of Ballarat, Vict., Australia

Abstract Convexity and Global Optimization

NONCONVEX OPTIMIZATION AND ITS APPLICATIONS Volume 44

This book consists of two parts. Firstly, the main notions of abstract convexity and their applications in the study of some classes of functions and sets are presented. Secondly, both theoretical and numerical aspects of global optimization based on abstract convexity are examined. Most of the book does not require knowledge of advanced mathematics.

Classical methods of nonconvex mathematical programming, being based on a local approximation, cannot be used to examine and solve many problems of global optimization, and so there is a clear need to develop special global tools for solving these problems. Some of these tools are based on abstract convexity, that is, on the representation of a function of a rather complicated nature as the upper envelope of a set of fairly simple functions.

Audience: The book will be of interest to specialists in global optimization, mathematical programming, and convex analysis, as well as engineers using mathematical tools and optimization techniques and specialists in
mathematical modelling.

Hardbound, ISBN 0-7923-6323-X
May 2000, 508 pp.

Norman E. Hurt
Zeta Associates, Fairfax, VA, USA

Mathematical Physics of Quantum Wires and Devices
From Spectral Resonances to Anderson Localization

MATHEMATICS AND ITS APPLICATIONS Volume 506

This is the first book to present a comprehensive treatment of the mathematical physics of quantum wires and devices. The focus is on the recent results in the area of the spectral theory of bent and deformed
quantum wires, simple quantum devices, Anderson localization, the quantum Hall effect and graphical models for quantum wire systems. The Selberg trace formula for finite volume graphical models is reviewed. Examples and relationships to recent work on acoustic and fluid flow, trapped states and spectral resonances, quantum chaos, random matrix theory, spectral statistics, point interactions, photonic crystals, Landau models, quantum transistors, edge states and metalnsulator transitions are developed. Problems related to modeling open quantum devices are discussed. The research of Exner and co-workers in quantum wires, Stollmann, Figotin, Bellissard et al. in the area of Anderson localization and the quantum Hall effect, and Bird, Ferry, Berggren and others in the area of quantum devices and their modeling is surveyed. The work on finite volume graphical models is interconnected to recent work on Ramanujan graphs and diagrams, the Phillipsarnak conjectures, L-functions and scattering theory.

Audience: This book will be of use to physicists, mathematicians and engineers interested in quantum wires, quantum devices and related mesoscopic systems.

Contents
1. Quantum Wires and Devices. 2. Acoustic and Fluid Flow. 3. Trapped States in Bent Quantum Wires. 4. Mathematical Modeling and Quantum Devices. 5. Problems in Open Quantum Devices. 6. Anderson Localization.
7. Landau Models and the Quantum Hall Effect. 8. Selberg Trace Formula. 9. STF and Finite Volume Graphs. References. Index.


Hardbound, ISBN 0-7923-6288-8
May 2000, 320 pp.

M.M. Rao
University of California, Riverside, USA

Stochastic Processes
Inference Theory

MATHEMATICS AND ITS APPLICATIONS Volume 508

This book presents a complete mathematical treatment of classical inference theory (Neyman earson, Fisher, and Wald) from the point of using it in stochastic processes, including some generalizations. It includes detailed
analysis of likelihood ratios for both Gaussian and several other classes (infinitely divisible, jump Markov, diffusion and additive). Both linear and nonlinear filtering (also for general nonquadratic criteria) are treated. The corresponding Kalmanucy filters for continuous parameter processes are presented. Consistency and limit distributions of estimations of biospectral densities of harmonizable processes are given.

Audience: Researchers and graduate students working in mathematics, statistics, and systems and communication engineering.

Contents
Preface. I. Introduction and Preliminaries. II. Some Principles of Hypothesis Testing. III. Parameter Estimation and Asymptotics. IV. Inferences for Classes of Processes. V. Likelihood Ratios for Processes. VI. Sampling
Methods for Processes. VII. More on Stochastic Inference. VIII. Prediction and Filtering of Processes. IX. Nonparametric Estimation for Processes. Bibliography. Notation index. Author index. Subject index.


Hardbound, ISBN 0-7923-6324-8
May 2000, 664 pp.

B.E. Brodsky / State University, Higher School of Economics, Moscow, Russia
B.S. Darkhovsky / Institute for Systems Analysis, Russian Academy of Sciences, Moscow, Russia

Non-Parametric Statistical Diagnosis
Problems and Methods

MATHEMATICS AND ITS APPLICATIONS Volume 509

This volume gives a systematic account of different problems of statistical diagnostics, i.e. the detection of changes in probabilistic characteristics of random processes and fields. Methods of solving such problems are
proposed, based upon a unified nonparametric approach. Two general formalisations of the problems of statistical diagnostics are considered. Firstly, the detection of changes in arbitrary probabilistic distributions of random processes and fields, `glued' from different stationary pieces: in other words, the detection of moments or areas of such `glueing'; and secondly, the detection of statistical `contaminations' in data (realisations of random fields or processes), or `abnormal' observations with deviating statistical
characteristics. A general approach to solving such problems is proposed, which is based upon the principle of reduction to certain standard situations and which does not use a priori data about probabilistic distributions. Much attention is paid to applications in such diverse areas as biology (EECs) and economics.

Audience: This book will be of interest to researchers in statistics and random processes, as well as advanced and postgraduate students in the same disciplines, and to specialists in control theory, systems analysis,
biomedical engineering, and econometrics.

Contents
Preface. Part 1: Theory. 1. Preliminary considerations. 2. State of the art review. 3. Retrospective methods of statistical diagnosis for random sequences: change-point problems. 4. Retrospective methods of statistical
diagnosis for random processes: `Contamination' problems. 5. Sequential methods of statistical diagnosis. 6. Statistical diagnosis problems for random fields. Part 2: Applications. 7. Application of the change-point analysis to investigation of the brain electrical activity. 8. Methods of statistical diagnosis in economic and financial systems. Appendix. Bibliography. Author Index. Subject Index. Main Notations and Abbreviations.

Hardbound, ISBN 0-7923-6328-0
April 2000, 468 pp.

edited by
Peter A. Loeb / Dept. of Mathematics, University of Illinois at Urbana, USA
Manfred P.H. Wolff / Mathematical Institute, University of T・ingen, Germany

Nonstandard Analysis for the Working Mathematician

MATHEMATICS AND ITS APPLICATIONS Volume 510

This book is addressed to mathematicians working in analysis and its applications. The aim is to provide an understandable introduction to the basic theory of non-standard analysis and to illuminate some of its most striking applications. Problems are posed in all chapters.

The opening chapter of the book presents a simplified form of the general theory that is suitable for the results of calculus and basic real analysis. The presentation is intended to facilitate the acquisition of basic skills in the subject, so that a reader who begins with no background in mathematical logic should find it relatively easy to continue. The book then proceeds with the full theory.

Following Part I, each chapter takes up a different field for applications, beginning with a gentle introduction that even non-experts can read with profit. The remainder of each chapter is then addressed to experts, showing how to use non-standard analysis in the search for solutions of open problems and how to obtain rich new structures that produce deep insights into the field under consideration. The particular applications discussed here are in functional analysis including operator theory, probability theory including stochastic processes, and economics including game theory and financial mathematics. In working through this book the reader should gain many new and helpful insights into the enterprise of mathematics.

Audience: This work will be of interest to specialists whose work involves real functions, probability theory, stochastic processes, logic and foundations. Much of the book, in particular the introductory Part I, can be used in a graduate course.

Contents and Contributors
Preface. I: An Introduction to Nonstandard Analysis; P.A. Loeb. 1. A Simple Introduction to Nonstandard Analysis. 2. An Introduction to General Nonstandard Analysis. 3. Topology and Measure Theory. II: Functional
Analysis; M.P.H. Wolff. 4. Functional Analysis. III: Measure and Probability Theory and Applications; H. Osswald, Y. Sun. 5. Measure Theory and Integration. 6. Probability Theory. 7. Conventional Operations on Nonstandard Constructions. IV: Economics and Nonstandard Analysis; Y. Sun. 8. Nonstandard Analysis in Mathematical Economics. Index.

Hardbound, ISBN 0-7923-6340-X
June 2000, 336 pp.
Paperback, ISBN 0-7923-6341-8
June 2000, 336 pp.

Liu Yanpei
Institute of Applied Mathematics, Academia Sinica, Beijing, PR China

Enumerative Theory of Maps

MATHEMATICS AND ITS APPLICATIONS Volume 468

This monograph provides a unified theory of maps and their enumerations. The crucial idea is to suitably decompose the given set of maps for extracting a functional equation, in order to have advantages for solving or transforming it into those that can be employed to derive as simple a formula as possible. It is shown that the foundation of the theory is for rooted planar maps, since other kinds of maps including nonrooted (or symmetrical) ones and those on general surfaces have been found to have relationships with particular types
in planar cases. A number of functional equations and close formulae are discovered in an exact or asymptotic manner.

Audience: This book will be of interest to college teachers, graduate students working in mathematics, especially in combinatorics and graph theory, functional and approximate analysis and algebraic systems.


Kluwer Academic Publishers, Dordrecht
Co-publication with Science Press Beijing

Hardbound, ISBN 0-7923-5599-7
June 2000, 415 pp.

Jerzy Urbanowicz / Institute of Mathematics, Polish Academy of Sciences, Warsaw, Poland
Kenneth S. Williams / Centre for Research in Algebra and Number Theory, Carleton University, Ottawa, ON,

Congruences for L-Functions

MATHEMATICS AND ITS APPLICATIONS Volume 511

This book provides a comprehensive and up-to-date treatment of research carried out in the last twenty years on congruences involving the values of L-functions (attached to quadratic characters) at certain special values.

There is no other book on the market which deals with this subject. The book presents in a unified way congruences found by many authors over the years, from the classical ones of Gauss and Dirichlet to the recent ones of Gras, Vehara, and others.

Audience: This book is aimed at graduate students and researchers interested in (analytic) number theory, functions of a complex variable and special functions.

Contents
Preface. I. Short Character Sums. II. Class Number Congruences. III. Congruences Between the Orders of K2-Groups. IV. Congruences among the Values of 2-Adic L-Functions. V. Applications of Zagier's Formula (I).
VI. Applications of Zagier's Formula (II). Bibliography. Author Index. Subject Index. List of symbols.


Hardbound, ISBN 0-7923-6379-5
June 2000, 272 pp.