A.G. Kusraev - Böcker

"Nonstandard Methods of Analysis" is concerned with the main trends in this field: infinitesimal analysis and Boolean-valued analysis. The methods that have been developed in the last 25 years are explained. Attention is paid to general principles and fundamentals of formalisms for infinitesimals, as well as to the technique of descents and ascents in a Boolean-valued universe. The book also includes various applications of non-standard methods to ordered algebraic systems, vector lattices, subdifferentials and convex programming that have been developed in recent years. The text is designed for undergraduate students, postgraduates and all researchers interested in applying non-standard methods in their work.

This text presents the main results since the 1990s on dominated operators, demonstrating a well-developed theory with a range of applications. The exposition focuses on the fundamental properties of dominated operators with special emphasis on their particular classes: integral and pseudointegral operators, disjointness preserving and decomposable operators, and summing and cyclically compact operators. This volume should be of interest to postgraduate students and researchers whose work involves geometric functional analysis, operator theory, vector lattices, measure and integration theory, and mathematical logic and foundations.

Infinitesimal analysis, once a synonym for calculus, is a technique for studying the properties of an arbitrary mathematical object by discriminating between its standard and nonstandard constituents. Resurrected by A. Robinson in the early 1960s with the epithet "nonstandard", infinitesimal analysis not only has revived the methods of infinitely small and infinitely large quantities, which go back to the very beginning of calculus, but also has suggested many powerful tools for research in every branch of modern mathematics. This book sets forth the basics of the theory, as well as up-to-date applications in, for example, functional analysis, optimization, and harmonic analysis.

This book presents the main results of the last fifteen years on dominated operators, demonstrating a well-developed theory with a wide range of applications. The exposition focuses on the fundamental properties of dominated operators with special emphasis on their particular classes: integral and pseudointegral operators, disjointness preserving and decomposable operators, summing and cyclically compact operators, etc. Audience: This volume will be of interest to postgraduate students and researchers whose work involves geometric functional analysis, operator theory, vector lattices, measure and integration theory, and mathematical logic and foundations.

Infinitesimal analysis, once a synonym for calculus, is now viewed as a technique for studying the properties of an arbitrary mathematical object by discriminating between its standard and nonstandard constituents. Resurrected by A. Robinson in the early 1960's with the epithet 'nonstandard', infinitesimal analysis not only has revived the methods of infinitely small and infinitely large quantities, which go back to the very beginning of calculus, but also has suggested many powerful tools for research in every branch of modern mathematics.The book sets forth the basics of the theory, as well as the most recent applications in, for example, functional analysis, optimization, and harmonic analysis. The concentric style of exposition enables this work to serve as an elementary introduction to one of the most promising mathematical technologies, while revealing up-to-date methods of monadology and hyperapproximation.This is a companion volume to the earlier works on nonstandard methods of analysis by A.G. Kusraev and S.S. Kutateladze (1999), ISBN 0-7923-5921-6 and Nonstandard Analysis and Vector Lattices edited by S.S. Kutateladze (2000), ISBN 0-7923-6619-0

The subject of the present book is sub differential calculus. The main source of this branch of functional analysis is the theory of extremal problems. For a start, we explicate the origin and statement of the principal problems of sub differential calculus. To this end, consider an abstract minimization problem formulated as follows: x E X, f(x) --+ inf. Here X is a vector space and f : X --+ iR is a numeric function taking possibly infinite values. In these circumstances, we are usually interested in the quantity inf f( x), the value of the problem, and in a solution or an optimum plan of the problem (i. e. , such an x that f(x) = inf f(X», if the latter exists. It is a rare occurrence to solve an arbitrary problem explicitly, i. e. to exhibit the value of the problem and one of its solutions. In this respect it becomes necessary to simplify the initial problem by reducing it to somewhat more manageable modifications formulated with the details of the structure of the objective function taken in due account. The conventional hypothesis presumed in attempts at theoretically approaching the reduction sought is as follows. Introducing an auxiliary function 1, one considers the next problem: x EX, f(x) -l(x) --+ inf. Furthermore, the new problem is assumed to be as complicated as the initial prob­ lem provided that 1 is a linear functional over X, i. e.

Nonstandard Methods of Analysis is concerned with the main trends in this field; infinitesimal analysis and Boolean-valued analysis. The methods that have been developed in the last twenty-five years are explained in detail, and are collected in book form for the first time. Special attention is paid to general principles and fundamentals of formalisms for infinitesimals as well as to the technique of descents and ascents in a Boolean-valued universe. The book also includes various novel applications of nonstandard methods to ordered algebraic systems, vector lattices, subdifferentials, convex programming etc. that have been developed in recent years. For graduate students, postgraduates and all researchers interested in applying nonstandard methods in their work.