by Enrico Giusti (Universit di Firenze, Italy)

DIRECT METHODS IN THE CALCULUS OF VARIATIONS

This book provides a comprehensive discussion on the existence and regularity of minima of regular integrals in the calculus of variations and of solutions to elliptic partial differential equations and systems of the second order. While direct methods for the existence of solutions are well known and have been widely used in the last century, the regularity of the minima was always obtained by means of the Euler equation as a part of the general theory of partial differential equations. In this book, using the notion of the quasi-minimum introduced by Giaquinta and the author, the direct methods are extended to the regularity of the minima of functionals in the calculus of variations, and of solutions to partial differential equations. This unified treatment offers a substantial economy in the assumptions, and permits a deeper understanding of the nature of the regularity and singularities of the solutions. The book is essentially self-contained, and requires only a general knowledge of the elements of Lebesgue integration theory.

Contents:

・Semi-Classical Theory
・Integrable Functions
・Sobolev Spaces
・Semicontinuity
・Quasi-Convex Functionals
・Quasi-Minima
・Regularity of Quasi-Minima
・First Derivatives
・Partial Regularity
・Higher Derivatives

Readership: Graduate students, academics and researchers in the field of analysis and differential equations.

370pp (approx.) Pub. date: Scheduled Fall 2002
ISBN 981-238-043-4

by Samuel Kotz (George Washington University, USA), Yan Lumelskii (Statistics Laboratory, Technion, Israel) & Marianna Pensky (University of Central Florida, USA)

THE STRESS-STRENGTH MODEL AND ITS GENERALIZATIONS
Theory and Applications

This important book presents developments in a remarkable field of inquiry in statistical/probability theory ・the stress穆trength model.
Many papers in the field include the enigmatic "words" P(X<Y) ・or something similar ・in the title. This reflects the long-established concept of ordering of distributions. The basic impetus for the study carried out by the authors of this book is the general concept of stress穆trength as an interpretation of the P(X<Y) relationships, which leads to applications in reliability engineering, economics and modern medicine.

The Stress亡trength Model and Its Generalizations collects and digests theoretical and practical results on the theory and applications of the stress穆trength relationships in industrial and economic systems ・results that have been scattered in the literature during the last 40-odd years ・and augments and presents them for the first time in a unified manner suitable for practitioners as well as probabilists and theoretical and applied statisticians.

Contents:

・Stress亡trength Models: History, Mathematical Tools and Survey of Applications
・Theory and General Estimation Procedures
・Parametric Point Estimation
・Parametric Statistical Inference
・Nonparametric Methods
・Special Cases and Generalizations
・Examples and Details on Applications

Readership: Applied probabilists, statisticians (theoretical and consultant), and reliability engineers.

220pp (approx.) Pub. date: Scheduled Fall 2002
ISBN 981-238-057-4

by Tsoy-Wo Ma (University of Western Australia)

BANACH-HILBERT SPACES, VECTOR MEASURES AND GROUP REPRESENTATIONS

This book provides an elementary introduction to classical analysis on normed spaces, with special attention paid to fixed points, calculus, and ordinary differential equations. It contains a full treatment of vector measures on delta rings without assuming any scalar measure theory and hence should fit well into existing courses. The relation between group representations and almost periodic functions is presented. The mean values offer an infinite-dimensional analogue of measure theory on finite-dimensional Euclidean spaces. This book is ideal for beginners who want to get through the basic material as soon as possible and then do their own research immediately.

Contents:

・Metric Spaces
・Complete, Compact and Connected Sets
・Banach Spaces
・Simplicial Complexes
・Topological Fixed Points
・Foundation of Functional Analysis
・Natural Constructions
・Complex Analysis
・Differentiation in Banach Spaces
・Polynomials and Higher Derivatives
・Ordinary Differential Equations
・Compact Linear Operators
・Operators on Hilbert Spaces
・Spectral Properties of Hilbert Spaces
・Tensor Products
・Complex Vector Lattices
・Vector Measures on Semirings
・Extensions of Positive Measures
・Measurable Objects
・Integrals of Upper Functions
・Vector Integrals
・Finite Products of Measures
・Measures on Finite Dimensional Spaces
・Indefinite Integrals
・Differentiation of Measures
・Spectral Measures
・Locally Compact Spaces
・Almost Periodic Functions on Groups
・Group Representations
・Saturated Closed Invariant Ideals
・Mean Spaces

Readership: Upper level undergraduates, graduate students, academics and researchers in the field of analysis.

620pp (approx.) Pub. date: Scheduled Fall 2002
ISBN 981-238-038-8

by Jaroslav Kurzweil (Academy of Sciences, Czech Republic)

INTEGRATION BETWEEN THE LEBESGUE INTEGRAL
AND THE HENSTOCK-KURZWEIL INTEGRAL
Its Relation to Locally Convex Vector Spaces

The main topics of this book are convergence and topologization. Integration on a compact interval on the real line is treated with Riemannian sums for various integration bases. General results are specified to a spectrum of integrations, including Lebesgue integration, the Denjoy integration in the restricted sense, the integrations introduced by Pfeffer and by Bongiorno, and many others. Morever, some relations between integration and differentiation are made clear.
The book is self-contained. It is of interest to specialists in the field of real functions, and it can also be read by students, since only the basics of mathematical analysis and vector spaces are required.

Contents:

・Basic Concepts and Properties of y-Integration
・Convergence
・Convergence and Locally Convex Spaces
・An Auxiliary Locally Convex Space
・L-Integration
・M-Integration
・Noncompleteness
・S-Integration
・R-Integration
・An Extension of the Concept of y-Integration
・Differentiation and Integration

Readership: Researchers, academics and graduate students interested in real analysis.

130pp (approx.) Pub. date: Scheduled Fall 2002
ISBN 981-238-046-9

by Jian-Ke Lu, Shou-Guo Zhang (Wuhan University, China) & Shi-Giang Liu (Ningxia University, China)

INTRODUCTION TO THE THEORY OF COMPLEX FUNCTIONS

Series in Pure Mathematics - Vol. 25

This book is based on the teaching experience of the authors, and therefore some of the topics are presented in a new form. For instance, the multi-valued properties of the argument function are discussed in detail so that the beginner may readily grasp the elementary multi-valued analytic functions. The residue theorem is extended to the case where poles of analytic functions considered may occur on the boundary of a region ・which is very useful in applications but not seen in textbooks written in English.

Contents:

・Complex Numbers and Complex Functions
・Fundamentals of Analytic Functions
・Complex Integrals
・Theory of Series for Analytic Functions
・Theory of Residue
・Analytic Extension
・Conformal Mapping
・Harmonic Functions
・Analytic Functions Applied to Planar Flow

Readership: Upper level undergraduates and graduate students in mathematics, mechanics and physics.

300pp (approx.) Pub. date: Scheduled Fall 2002
ISBN 981-238-047-7

by Supriva Kar (Indian Institute of Technology, Kanpur)

NON-COMMUTATIVE GEOMETRY
A Perspective on String and Field Theories

This book provides a systematic, comprehensive and up-to-date account of the recent developments in non-commutative geometry, at a pedagogical level. It does not go into the details of rigorous (advanced level) mathematical formulation of non-commutative geometry; rather, it restricts itself to the domain of strings and quantum fields.
Since non-commutative geometry has recently aroused revived renewed interest in open string theory, the author motivates the text from the viewpoint of a string theory. He begins with an introduction to the subject, explaining what one means by non-commutative geometry and why it is relevant to study such geometry, and discussing its possible origin in a string theory.

The book comprises five chapters. Chapter 1 gives a mathematical introduction. In Chapter 2, non-commutativity in an open bosonic string theory is discussed with explicit calculations. Chapter 3 deals with non-commutative quantum fields, their dynamics and some of their interactions. In Chapter 4, some of the classical solutions of non-commutative string and field theories are discussed. Chapter 5 treats some applications of non-commutative theory.

Students will find this book useful as a bridge between string and field theories. In addition, it will prove invaluable for interdisciplinary areas of study.

Contents:

・Non-commutativity in Open String Theory
・Non-commutative Quantum Field Theory
・Solutions in a Non-commutative Theory
・Applications in a Non-commutative Space

Readership: Senior undergraduates, graduate students and researchers in theoretical, mathematical, high energy and condensed matter physics.

250pp (approx.) Pub. date: Scheduled Fall 2002
ISBN 981-238-052-3