Series: Classics in Mathematics
1st Edition., 1991, XII, 480 p. 1 illus.
Softcover, ISBN 978-3-642-20211-7
Due: June 15, 2011
.Isoperimetric, measure concentration and random process techniques appear at the basis of the modern understanding of Probability in Banach spaces. Based on these tools, the book presents a complete treatment of the main aspects of Probability in Banach spaces (integrability and limit theorems for vector valued random variables, boundedness and continuity of random processes) and of some of their links to Geometry of Banach spaces (via the type and cotype properties). Its purpose is to present some of the main aspects of this theory, from the foundations to the most important achievements. The main features of the investigation are the systematic use of isoperimetry and concentration of measure and abstract random process techniques (entropy and majorizing measures). Examples of these probabilistic tools and ideas to classical Banach space theory are further developed.
Content Level ≫ Research
Related subjects ≫ Analysis - Applications - Mathematics - Probability Theory and Stochastic Processes
Introduction.- Notation.- Part 0. Isoperimetric Background and Generalities.- Chapter 1. Isoperimetric Inequalities and the Concentration of Measure Phenomenon.- Chapter 2. Generalities on Banach Space Valued Random Variables and Random Processes.- Part I. Banach Space Valued Random Variables and Their Strong Limiting Properties.- Chapter 3. Gaussian Random Variables.- Chapter 4. Rademacher Averages.- Chapter 5. Stable Random Variables.- Chapter 6. Sums of Independent Random Variables.- Chapter 7. The Strong Law of Large Numbers.- Chapter 8. The Law of the Iterated Logarithm.- Part II. Tightness of Vector Valued Random Variables and Regularity of Random Processes.- Chapter 9. Type and Cotype of Banach Spaces.- Chapter 10. The Central Limit Theorem.- Chapter 11. Regularity of Random Processes.- Chapter 12. Regularity of Gaussian and Stable Processes.- Chapter 13. Stationary Processes and Random Fourier Series.- Chapter 14. Empirical Process Methods in Probability in Banach Spaces.- Chapter 15. Applications to Banach Space Theory.
Series: Universitext
1st Edition., 2011, XIII, 216 p. 62 illus.
Softcover, ISBN 978-0-85729-602-3
Due: May 2011
Bringing together two fundamental texts from Frederic Pham’s research on singular integrals, the first part of this book focuses on topological and geometrical aspects while the second explains the analytic approach. Using notions developed by J. Leray in the calculus of residues in several variables and R. Thom’s isotopy theorems, Frederic Pham’s foundational study of the singularities of integrals lies at the interface between analysis and algebraic geometry, culminating in the Picard-Lefschetz formulae. These mathematical structures, enriched by the work of Nilsson, are then approached using methods from the theory of differential equations and generalized from the point of view of hyperfunction theory and microlocal analysis.
Content Level ≫ Graduate
Keywords ≫ Homology - Landau variety - Leray residue - Picard-Lefschetz formulae - boundary value - current - family of supports - functions in the Nilsson class - hyperfunctions - hyperfunctions in the Nilsson class - microsupport - residue theorem - singularity of integrals depending on parameters
Related subjects ≫ Algebra - Analysis
Differentiable manifolds.- Homology and cohomology of manifolds.- Leray’s theory of residues.- Thom’s isotopy theorem.- Ramification around Landau varieties.- Analyticity of an integral depending on a parameter.- Ramification of an integral whose integrand is itself ramified.- Functions of a complex variable in the Nilsson class.- Functions in the Nilsson class on a complex analytic manifold.- Analyticity of integrals depending on parameters.- Sketch of a proof of Nilsson’s theorem.- Examples: how to analyze integrals with singular integrands.- Hyperfunctions in one variable, hyperfunctions in the Nilsson class.- Introduction to Sato’s microlocal analysis.
Series: Universitext
1st Edition, 2011, XI, 523 p. 80 illus.
Softcover, ISBN 978-3-642-20553-8
Due: June 2011
The idea of this book is to give an extensive description of the classical complex analysis, here ''classical'' means roughly that sheaf theoretical and cohomological methods are omitted.
Numerous exercises including hints for solutions and many figures make this an attractive, indispensable book for students who would like to have a sound introduction to classical complex analysis.
Content Level ≫ Graduate
Keywords ≫ Abelian Functions - Analytic Functions - Modular Forms - Riemannian Surfaces
Related subjects ≫ Analysis
Chapter I. Riemann Surfaces.- Chapter II. Harmonic Functions on Riemann Surfaces.- Chapter III. Uniformization.- Chapter IV. Compact Riemann Surfaces.- Appendices to Chapter IV.- Chapter V. Analytic Functions of Several Complex Variables.- Chapter V. Analytic Functions of Several Complex Variable.- Chapter VI. Abelian Functions.- Chapter VII. Modular Forms of Several Variables.- Chapter VIII. Appendix: Algebraic Tools.- References.- Index.
Series: Springer Optimization and Its Applications, Vol. 51
1st Edition., 2011, X, 116 p. 24 illus.
Hardcover, ISBN 978-1-4419-9806-4
Due: July 29, 2011
This volume presents a self-contained introduction to the theory of minisum hyperspheres. The minisum hypersphere problem is a generalization of the famous Fermat-Torricelli problem. The problem asks for a hypersphere minimizing the weighted sum of distances to a given point set. In the general framework of finite dimensional real Banach spaces, the minisum hypersphere problem involves defining a hypersphere and calculating the distance between points and hyperspheres. The theory of minisum hyperspheres is full of interesting open problems which impinge upon the larger field of geometric optimization.
This work provides an overview of the history of minisum hyperspheres as well as describes the best techniques for analyzing and solving minisum hypersphere problems. Related areas of geometric and nonlinear optimization are also discussed.
Content Level ≫ Research
Keywords ≫ Convex Optimization - Geometric Optimization - Minisum Hypersphere Problem
Related subjects ≫ Geometry & Topology - Mathematics
-Preface.- 1. Basic Concepts (Circles and Hyperspheres, Minisum Hyperspheres,
Mathematical Preliminaries, Finite Dominating Sets).- 2. Euclidean Minisum
Hyperspheres (Basic Assumptions, Distance, Degenerated Solutions, Existence
of Optimal Solutions, Incidence Properties, Solution Approaches for the
Planar Case, Concluding Remarks).- 3. Minisum Hyperspheres in Normed Spaces
(Basic Assumptions, Distance, Degenerated Solutions, Existence of Minisum
Hyperspheres, Incidence Properties, Polyhedral Norms in the Plane, Concluding
Remarks).- 4. Minisum Circle Problem with Unequal Norms (Basic Assumptions,
Distance, Properties of Minisum Circles, Polyhedral Norms, Concluding Remarks).-
5. Minisum Rectangles in a Manhattan Plane (Basic Assumptions, Notations,
Point-Rectangle Distance, Restricted Problems, Unrestricted Problem, Concluding
Remarks).- 6. Extensions.- Bibliiography.- Index.
Series: Modern Birkhauser Classics
Corrected reprint of the 1988 Edition, 2011, 244 p.
Softcover, ISBN 978-3-0348-0150-8
Due: May 13, 2011
.Concrete and elementary introduction to classification of stable vector bundles
Readable by students with basic knowledge of analytic - or algebraic geometry
Many examples, historical remarks, unsolved problems
This expository treatment is based on a survey given by one of the authors at the Seminaire Bourbaki in November 1978 and on a subsequent course held at the University of Gottingen. It is intended to serve as an introduction to the topical question of classification of holomorphic vector bundles on complex projective spaces, and can easily be read by students with a basic knowledge of analytic of algebraic geometry. Short supplementary sections describe more advanced topics, further results, and unsolved problems.
Content Level ≫ Research
Keywords ≫ Complex Geometry - Moduli Spaces - Monads - Stability
Related subjects ≫ Mathematics
Introduction.- Chapter 1. Holomorphic vector bundles and the geometry of Pn.- Chapter 2. Stability and moduli spaces.- Bibliography.- Supplemental Bibliography.- Index.- Appendix A. Sheaves on Pn and problems in linear algebra.- Bibliography for Appendix A.