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This is perhaps the last remaining field comprehended and used by mathematicians in all areas of the discipline. New inequalities are discovered every year, some for their intrinsic interest whilst others flow from results obtained in various branches of mathematics. The study of inequalities reflects the many and various aspects of mathematics. There are numerous applications in a wide variety of fields, from mathematical physics to biology and economics. This volume contains the proceedings of the General Inequalities 7 meeting held at Oberwolfach in November 1995. It is conceived in the spirit of the preceding volumes in the sense that it not only contains the latest results, but it is also a useful reference for lecturers and researchers. ... Read more

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Riemannian Topology and Geometric Structures on Manifolds results from a similarly entitled conference held at the University of New Mexico in Albuquerque. The various contributions to this volume discuss recent advances in the areas of positive sectional curvature, Kähler and Sasaki geometry, and their interrelation to mathematical physics, notably M and superstring theory. Focusing on these fundamental ideas, this collection presents articles with original results, and plausible problems of interest for future research.

Contributors: C.P. Boyer, J. Cheeger, X. Dai, K. Galicki, P. Gauduchon, N. Hitchin, L. Katzarkov, J. Kollár, C. LeBrun, P. Rukimbira, S.R. Simanca, J. Sparks, C. van Coevering, and W. Ziller.

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This digital document is a journal article from Mathematical Social Sciences, published by Elsevier in . The article is delivered in HTML format and is available in your Amazon.com Media Library immediately after purchase. You can view it with any web browser.

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In this paper we introduce a new axiom for choice functions, equivalent to the path independence axiom. We call it the congruence condition. This axiom allows to construct directly the anti-exchange closure operators and the convex geometries associated to path independent choice functions. ... Read more

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Convex bodies are at once simple and amazingly rich in structure.While the classical results go back many decades, during the past ten years the integral geometry of convex bodies has undergone a dramatic revitalization, brought about by the introduction of methods, results and, most importantly, new viewpoints, from probability theory, harmonic analysis and the geometry of finite-dimensional normed spaces. This collection arises from an MSRI program held in the Spring of 1996, involving researchers in classical convex geometry, geometric functional analysis, computational geometry, and related areas of harmonic analysis.It is representative of the best research in a very active field that brings together ideas from several major strands in mathematics. ... Read more

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This text puts together "folklore results" touching weak Asplund spaces. It presents a thorough examination of weak Asplund cases. Nonseparable Banach spaces, renorming, and differentiability are stressed throughout the text and all subclasses, including inferences and counterexamples are discussed. It also covers Stegall's classes, fragmentability, and "long sequences" of linear projections. Notes, remarks and questions end each chapter. ... Read more

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Santaló (Spain 1911 - Argentina 2001) contributed to several branches of Geometry, perhaps his most outstanding achievement being his having laid the mathematical foundations of Stereology and its applications. Considerable power of abstraction, a brilliant geometric intuition and his outstanding gifts as a disseminator of science were among his virtues. The present volume contains a selection of his best papers edited by Antonio M. Naveira and Augustí Reventós in collaboration with Graciela S. Birman and Ximo Gual-Arnau.

Part I consists of a short biography and some photographs along with a complete list of his publications, classified into research papers, books, and articles on education and the popularization of mathematics, as well as a comprehensive analysis of his contribution to science.

Part II, the main part of the book, includes the selected papers, arranged in five sections, according to the nature of their contents: Differential Geometry, Integral Geometry, Convex Geometry, Affine Geometry, and Statistics and Stereology. Each section is preceded by a comment by a renowned specialist: Teufel, Langevin, Schneider, Leichtweiss, and Cruz-Orive respectively.

Finally, Part III emphasizes the influence of his work. It contains comments by several specialists about modern results based on, or closely related to, those of Santaló, some book reviews written by Santaló, as well as some reviews of his books. As a curious addendum, a ranking of his own articles, given by Santaló himself, is included.

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This volume provides a systematic examination of Lagrange-typefunctions and augmented Lagrangians. Weak duality, zero duality gapproperty and the existence of an exact penalty parameter are examined.Weak duality allows one to estimate a global minimum. The zero dualitygap property allows one to reduce the constrained optimization problemto a sequence of unconstrained problems, and the existence of an exactpenalty parameter allows one to solve only one unconstrained problem.

By applying Lagrange-type functions, a zero duality gap property fornonconvex constrained optimization problems is established under acoercive condition. It is shown that the zero duality gap property isequivalent to the lower semi-continuity of a perturbation function.

In particular, for a type of kth power penalty functions, this bookobtains an analytic expression of the least exact penalty parameterand establishes that a fairly small exact penalty parameter can beachieved. As shown by numerical experiments, this property is veryimportant for some global methods of Lipschitz programming, otherwiseill conditioning may occur. Audience: The book is suitable for researchers in mathematicalprogramming and optimization and postgraduate students in appliedmathematics. ... Read more

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Vol. 1200 of the LNM series deals with the geometrical structure of finite dimensional normed spaces. One of the main topics is the estimation of the dimensions of euclidean and l^n p spaces which nicely embed into diverse finite-dimensional normed spaces. An essential method here is the concentration of measure phenomenon which is closely related to large deviation inequalities in Probability on the one hand, and to isoperimetric inequalities in Geometry on the other.
The book contains also an appendix, written by M. Gromov, which is an introduction to isoperimetric inequalities on riemannian manifolds. Only basic knowledge of Functional Analysis and Probability is expected of the reader. The book can be used (and was used by the authors) as a text for a first or second graduate course. The methods used here have been useful also in areas other than Functional Analysis (notably, Combinatorics). ... Read more

Customer Reviews (1)

very clear
I would suggest that anyone interested in "asymptotic geometry" start with this short book, which is written with love by two of the people who created the field (the results of which are not only interesting in and of themselves, but are used in many other areas). Yes, the price is outrageous (for a mimeographed paperback), but that's why God created libraries. ... Read more

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This book is the first systematic and comprehensive study ofthe theory of continuous selections of multivalued mappings. Thisinteresting branch of modern topology was introduced by E.A. Michaelin the 1950s and has since witnessed an intensive development withvarious applications outside topology, e.g. in geometry of Banachspaces, manifolds theory, convex sets, fixed points theory,differential inclusions, optimal control, approximation theory, andmathematical economics.
The work can be used in different ways: the first part is anexposition of the basic theory, with details. The second part is acomprehensive survey of the main results. Lastly, the third partcollects various kinds of applications of the theory.
Audience: This volume will be of interest to graduate studentsand research mathematicians whose work involves general topology,convex sets and related geometric topics, functional analysis, globalanalysis, analysis on manifolds, manifolds and cell complexes, andmathematical economics. ... Read more

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This book is the result of 18 years of research into Steiner'sproblem and its relatives in theory and application. Starting withinvestigations of shortest networks for VLSI layout and, on the otherhand, for certain facility location problems, the author has foundmany common properties for Steiner's problem in various spaces. Thepurpose of the book is to sum up and generalize many of these resultsfor arbitrary finite-dimensional Banach spaces. It shows that we cancreate a homogeneous and general theory when we consider twodimensions of such spaces, and that we can find many facts which arehelpful in attacking Steiner's problem in the higher-dimensionalcases. The author examines the underlying mathematical properties ofthis network design problem and demonstrates how it can be attacked byvarious methods of geometry, graph theory, calculus, optimization andtheoretical computer science.
Audience: All mathematicians and users of applied graph theory. ... Read more

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Metric fixed point theory encompasses the branch of fixed pointtheory which metric conditions on the underlying space and/or on themappings play a fundamental role. In some sense the theory is afar-reaching outgrowth of Banach's contraction mapping principle. Anatural extension of the study of contractions is the limiting casewhen the Lipschitz constant is allowed to equal one. Such mappings arecalled nonexpansive. Nonexpansive mappings arise in a variety ofnatural ways, for example in the study of holomorphic mappings andhyperconvex metric spaces.
Because most of the spaces studied in analysis share many algebraicand topological properties as well as metric properties, there is noclear line separating metric fixed point theory from the topologicalor set-theoretic branch of the theory. Also, because of its metricunderpinnings, metric fixed point theory has provided the motivationfor the study of many geometric properties of Banach spaces. Thecontents of this Handbook reflect all of these facts.
The purpose of the Handbook is to provide a primary resource foranyone interested in fixed point theory with a metric flavor. The goalis to provide information for those wishing to find results that mightapply to their own work and for those wishing to obtain a deeperunderstanding of the theory. The book should be of interest to a widerange of researchers in mathematical analysis as well as to thosewhose primary interest is the study of fixed point theory and theunderlying spaces. The level of exposition is directed to a wideaudience, including students and established researchers. ... Read more

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The notion of convex set, known according to its numerousapplications in linear spaces due to its connectivity which leads toseparation and support properties, does not imply, in fact,necessarily, the connectivity. This aspect of non-connectivity hiddenunder the convexity is discussed in this book. The property ofnon-preserving the connectivity leads to a huge extent of the domainof convexity. The book contains the classification of 100 notions ofconvexity, using a generalised convexity notion, which is theclassifier, ordering the domain of concepts of convex sets. Also, itopens the wide range of applications of convexity in non-connectedenvironment. Applications in pattern recognition, in discreteprogramming, with practical applications in pharmaco-economics arediscussed. Both the synthesis part and the applied part make the bookuseful for more levels of readers.Audience: Researchers dealing with convexity and related topics,young researchers at the beginning of their approach to convexity, PhDand master students. ... Read more