Series: Probability and Its Applications
2011, X, 340 p., Hardcover
ISBN: 978-3-642-15003-6
Due: November 2010
Measure-valued branching processes arise as high density limits of branching particle systems. The Dawson-Watanabe superprocess is a special class of those. The author constructs superprocesses with Borel right underlying motions and general branching mechanisms and shows the existence of their Borel right realizations. He then uses transformations to derive the existence and regularity of several different forms of the superprocesses. This treatment simplifies the constructions and gives useful perspectives. Martingale problems of superprocesses are discussed under Feller type assumptions. The most important feature of the book is the systematic treatment of immigration superprocesses and generalized Ornstein-Uhlenbeck processes based on skew convolution semigroups.
The volume addresses researchers in measure-valued processes, branching processes, stochastic analysis, biological and genetic models, and graduate students in probability theory and stochastic processes.
Preface.- 1. Random Measures on Metric Spaces.- 2. Measure-valued Branching Processes.- 3. One-dimensional Branching Processes.- 4. Branching Particle Systems.- 5. Basic Regularities of Superprocesses.- 6. Constructions by Transformations.- 7. Martingale Problems of Superprocesses.- 8. Entrance Laws and Excursion Laws.- 9. Structures of Independent Immigration.- 10. State-dependent Immigration Structures.- 11. Generalized Ornstein-Uhlenbeck Processes.- 12. Small Branching Fluctuation Limits.- 13. Appendix: Markov Processes.- Bibliography.- Index.
Series: Sources and Studies in the History of Mathematics and Physical Sciences
2011, 499 p. 129 illus., Hardcover
ISBN: 978-0-387-84825-9
Due: November 29, 2010
The "Almagest," by the Greek astronomer and mathematician Ptolemy, is the most important surviving treatise on early mathematical astronomy, offering historians valuable insight into the astronomy and mathematics of the ancient world.
Pedersen's 1974 publication, "A Survey of the Almagest," is the most recent in a long tradition of companions to the "Almagest." Part paraphrase and part commentary, Pedersen’s work has earned the universal praise of historians and serves as the definitive introductory text for students interested in studying the "Almagest."
In this revised edition, Alexander Jones, a distinguished authority on the history of early astronomy, provides supplementary information and commentary to the original text to account for scholarship that has appeared since 1974. This revision also incorporates various corrections to Pedersen's original text that have been identified since its publication.
This volume is intended to provide students of the history of astronomy with a self-contained introduction to the "Almagest," helping them to understand and appreciate Ptolemy’s great and classical work.
- Forward to the revised edition.- Preface.- The almagest through the ages.- Physics and philosophy in the almagest.- Ptolemy as a mathematician.- Spherical astronomy in the almagest.- The motion of the sun.- The theories of the moon.- Parallaxes and eclipses.- The fixed stars.- The superior planets.- The inferior planets.- Retrograde motions and maximum elongations.- Latitudes and visibility periods.- Epilogue-the other ptolemy.- Apendix A: dated observations.- Appendix B: numerical parameters.- Bibliography.- Index of names.- Index of subjects.- Supplementary notes.- Supplementary bibliography.
Series: Problem Books in Mathematics
2011, XII, 522 p., Hardcover
ISBN: 978-1-4419-7441-9
Due: November 29, 2010
The theory of function spaces endowed with the topology of pointwise convergence, or Cp-theory, exists at the intersection of three important areas of mathematics: topological algebra, functional analysis, and general topology. Cp-theory has an important role in the classification and unification of heterogeneous results from each of these areas of research. Through over 500 carefully selected problems and exercises, this volume provides a self-contained introduction to Cp-theory and general topology. By systematically introducing each of the major topics in Cp-theory, this volume is designed to bring a dedicated reader from basic topological principles to the frontiers of modern research. Key features include: - A unique problem-based introduction to the theory of function spaces. - Detailed solutions to each of the presented problems and exercises. - A comprehensive bibliography reflecting the state-of-the-art in modern Cp-theory. - Numerous open problems and directions for further research. This volume can be used as a textbook for courses in both Cp-theory and general topology as well as a reference guide for specialists studying Cp-theory and related topics. This book also provides numerous topics for PhD specialization as well as a large variety of material suitable for graduate research.
Detailed summary of exercise sections Preface Introduction 1.1. Basic notions of topology and function spaces 1.1.1. Topologies, axioms of separation and a glance at Cp(X) 1.1.2. Products, cardinal functions and convergence 1.1.3. Metrizability and completeness 1.1.4. Compactness type properties in function spaces 1.1.5. More on completeness. Realcompact spaces Bibliographic notes 1.2. Solutions of problems 1.001?1.500 1.3. Bonus results: some hidden statements 1.3.1. Standard spaces 1.3.2. Metrizable spaces and compact spaces 1.3.3. Properties of continuous maps 1.3.4. Covering properties, normality and open families 1.3.5. Product spaces and cardinal invariants 1.3.6. Raznoie (unclassified results) 1.4. Open problems 1.4.1. Local properties 1.4.2. Discreteness of X and completeness of Cp(X) 1.4.3. Dense subspaces 1.4.4. The Lindelof property in X and Cp(X) 1.4.5. Other covering properties 1.4.6. Mappings which involve Cp-spaces 1.4.7. Very general questions 1.4.8. Fuzzy questions 1.4.9. Naive questions 1.4.10. Raznoie (unclassified questions) 1.5. Bibliography 1.6. List of special symbols 1.7. Index
Series: Universitext
2011, XVIII, 412 p. 248 illus., 124 in color., Softcover
ISBN: 978-1-4419-7399-3
Due: October 29, 2010
Includes an extensive bibliography with added historical references
Provides hints and solutions for selected exercises making this book ideal for self-study
Improves upon an already successful first edition
Provides a comprehensive understanding of a large body of important mathematics in geometry and topology
Manifolds, the higher-dimensional analogues of smooth curves and surfaces, are fundamental objects in modern mathematics.
Combining aspects of algebra, topology, and analysis, manifolds have also been applied to classical mechanics, general relativity, and quantum field theory. In this streamlined introduction to the subject, the theory of manifolds is presented with the aim of helping the reader achieve a rapid mastery of the essential topics. By the end of the book the reader should be able to compute, at least for simple spaces, one of the most basic topological invariants of a manifold, its de Rham cohomology. Along the way the reader acquires the knowledge and skills necessary for further study of geometry and topology. The second edition contains fifty pages of new material. Many passages have been rewritten, proofs simplified, and new examples and exercises added. This work may be used as a textbook for a one-semester graduate or advanced undergraduate course, as well as by students engaged in self-study. The requisite point-set topology is included in an appendix of twenty-five pages; other appendices review facts from real analysis and linear algebra. Hints and solutions are provided to many of the exercises and problems. Requiring only minimal undergraduate prerequisites, "An Introduction to Manifolds" is also an excellent foundation for the author's publication with Raoul Bott, "Differential Forms in Algebraic Topology."
Preface to the Second Edition.- Preface to the First Edition.-Chapter 1. Eudlidean Spaces. 1. Smooth Functions on a Euclidean Space.- 2. Tangent Vectors in R(N) as Derivativations.- 3. The Exterior Algebra of Multicovectors.- 4. Differential Forms on R(N).- Chapter 2. Manifolds.- 5. Manifolds.- 6. Smooth Maps on a Manifold.- 7. Quotients.- Chapter 3. The Tangent Space.- 8. The Tangent Space.- 9. Submanifolds.- 10. Categories and Functors.- 11. The Rank of a Smooth Map.- 12. The Tangent Bundle.- 13. Bump Functions and Partitions of Unity.- 14. Vector Fields.-Chapter 4. Lie Groups and Lie Algebras.- 15. Lie Groups.- 16. Lie Algebras.- Chapter 5. Differential Forms.- 17. Differential 1-Forms.- 18. Differential k-Forms.- 19. The Exterior Derivative.- 20. The Lie Derivative and Interior Multiplication.- Chapter 6. Integration.- 21. Orientations.- 22. Manifolds with Boundary.- 23. Integration on Manifolds.- Chapter 7. De Rham Theory.- 24. De Rham Cohomology.- 25. The Long Exact Sequence in Cohomology.- 26. The Mayer ?Vietoris Sequence.- 27. Homotopy Invariance.- 28. Computation of de Rham Cohomology.- 29. Proof of Homotopy Invariance.-Appendices.- A. Point-Set Topology.- B. The Inverse Function Theorem on R(N) and Related Results.- C. Existence of a Partition of Unity in General.- D. Linear Algebra.- E. Quaternions and the Symplectic Group.- Solutions to Selected Exercises.- Hints and Solutions to Selected End-of-Section Problems.- List of Symbols.- References.- Index.
Series: Grundlehren der mathematischen Wissenschaften, Vol. 342
Originally published under Vladimir G. Maz'ja in Springer Series of Soviet Mathematics
2011, X, 840 p., Hardcover
ISBN: 978-3-642-15563-5
Due: December 2010
Sobolev spaces play an outstanding role in modern analysis, in particular, in the theory of partial differential equations and its applications in mathematical physics. They form an indispensable tool in approximation theory, spectral theory, differential geometry etc. The theory of these spaces is of interest in itself being a beautiful domain of mathematics. The present volume includes basics on Sobolev spaces, approximation and extension theorems, embedding and compactness theorems, their relations with isoperimetric and isocapacitary inequalities, capacities with applications to spectral theory of elliptic differential operators as well as pointwise inequalities for derivatives. The selection of topics is mainly influenced by the author’s involvement in their study, a considerable part of the text is a report on his work in the field. Part of this volume ?rst appeared in German as three booklets of Teubner-Texte zur Mathematik (1979,1980). In the Springer volume “Sobolev Spaces”, published in English in 1985, the material was expanded and revised. The present 2nd edition is enhanced by many recent results and it includes new applications to linear and nonlinear partial differential equations. New historical comments, five new chapters and a signi?cantly augmented list of references aim to create a broader and modern view of the area.
Introduction.- 1 .Basic Properties of Sobolev Spaces.- 2 .Inequalities for Functions Vanishing at the Boundary.- 3.Conductor and Capacitary Inequalities with Applications to Sobolev-type Embeddings.- 4.Generalizations for Functions on Manifolds and Topological Spaces.- 5.Integrability of Functions in the Space L 1/1(Ω).- 6.Integrability of Functions in the Space L 1/p (Ω).- 7.Continuity and Boundedness of Functions in Sobolev Spaces.- 8.Localization Moduli of Sobolev Embeddings for General Domains.- 9.Space of Functions of Bounded Variation.- 10.Certain Function Spaces, Capacities and Potentials.- 11 Capacitary and Trace Inequalities for Functions in Rn with Derivatives of an Arbitrary Order.-12.Pointwise Interpolation Inequalities for Derivatives and Potentials.- 13.A Variant of Capacity.- 14.-Integral Inequality for Functions on a Cube.- 15.Embedding of the Space L l/p(Ω) into Other Function Spaces.- 16.Embedding L l/p(Ω) ⊂ W m/r(Ω).-17.Approximation in Weighted Sobolev Spaces.-18.Spectrum of the Schrodinger operator and the Dirichlet Laplacian.- References.- List of Symbols.- Subject Index.- Author Index.