1st Edition., 2011, XV, 301 p. 27 illus., Hardcover
ISBN: 978-1-4419-6054-2
Presents material accessible to advanced undergraduates
Contains new content on lie group theory that is accessible to anyone familiar with manifolds
Features self-contained chapters with bibliographical notes and practice exercises
In this text, integral geometry deals with Radonfs problem of representing a function on a manifold in terms of its integrals over certain submanifolds?hence the term the Radon transform. Examples and far-reaching generalizations lead to fundamental problems such as: (i) injectivity, (ii) inversion formulas, (iii) support questions, (iv) applications (e.g., to tomography, partial differential equations and group representations). For the case of the plane, the inversion theorem and the support theorem have had major applications in medicine through tomography and CAT scanning. While containing some recent research, the book is aimed at beginning graduate students for classroom use or self-study. A number of exercises point to further results with documentation. From the reviews: gIntegral Geometry is a fascinating area, where numerous branches of mathematics meet together. The contents of the book is concentrated around the duality and double fibration, which is realized through the masterful treatment of a variety of examples. The book is written by an expert, who has made fundamental contributions to the area.h ?Boris Rubin, Louisiana State University
Content Level ā Graduate
Keywords : Homogeneous spaces in duality - Manifolds and lie groups - Radio astronomy - Radon transform - Spaces of constant curvature - Topology of spaces - X-ray tranform on symmetric spaces
Related subjects : Algebra - Analysis - Geometry & Topology
Series: Graduate Texts in Mathematics, Vol. 202
2011, XVII, 433 p. 131 illus., Hardcover
ISBN: 978-1-4419-7939-1
This book is an introduction to manifolds at the beginning graduate level. It contains the essential topological ideas that are needed for the further study of manifolds, particularly in the context of differential geometry, algebraic topology, and related fields. Its guiding philosophy is to develop these ideas rigorously but economically, with minimal prerequisites and plenty of geometric intuition.
Although this second edition has the same basic structure as the first edition, it has been extensively revised and clarified; not a single page has been left untouched. The major changes include a new introduction to CW complexes (replacing most of the material on simplicial complexes in Chapter 5); expanded treatments of manifolds with boundary, local compactness, group actions, and proper maps; and a new section on paracompactness.
This text is designed to be used for an introductory graduate course on the geometry and topology of manifolds. It should be accessible to any student who has completed a solid undergraduate degree in mathematics. The authorfs book Introduction to Smooth Manifolds is meant to act as a sequel to this book.
Content Level : Graduate
Keywords : Cell complexes - Covering spaces - Homology - Surfaces - The fundamental group - Topological spaces - Topology
Related subjects : Geometry & Topology
Preface.- 1 Introduction.- 2 Topological Spaces.- 3 New Spaces from Old.- 4 Connectedness and Compactness.- 5 Cell Complexes.- 6 Compact Surfaces.- 7 Homotopy and the Fundamental Group.- 8 The Circle.- 9 Some Group Theory.- 10 The Seifert-Van Kampen Theorem.- 11 Covering Maps.- 12 Group Actions and Covering Maps.- 13 Homology.- Appendix A: Review of Set Theory.- Appendix B: Review of Metric Spaces.- Appendix C: Review of Group Theory.- References.- Notation Index.- Subject Index.
Series: Universitext
1st Edition., 2011, XVI, 386 p., Softcover
ISBN: 978-3-642-18323-2
Due: March 29, 2011
The theory of Markov Decision Processes focuses on controlled Markov chains in discrete time. The authors establish the theory for general state and action spaces and at the same time show its application by means of numerous examples, mostly taken from the fields of finance and operations research. By using a structural approach many technicalities (concerning measure theory) are avoided. They cover problems with finite and infinite horizons, as well as Partially Observable Markov Decision Processes, Piecewise Deterministic Markov Decision Processes and stopping problems.
The book presents Markov Decision Processes in action and includes various state-of-the-art applications with a particular view towards finance. It is useful for upper-level undergraduates, Master students and researchers both in applied probability and finance and provides exercises (without solutions).
Content Level : Graduate
Keywords : - Markov Decision Processes - Partially Observable Markov Decision Processes - Portfolio optimization - Stochastic dynamic programming
Related subjects : Applications - Probability Theory and Stochastic Processes - Quantitative Finance
Preface.- 1.Introduction and First Examples.- Part I Finite Horizon Optimization Problems and Financial Markets.- 2.Theory of Finite Horizon Markov Decision Processes.- 3.The Financial Markets.- 4.Financial Optimization Problems.- Part II Partially Observable Markov Decision Problems.- 5.Partially Observable Markov Decision Processes.- 6.Partially Observable Markov Decision Problems in Finance.- Part III Infinite Horizon Optimization Problems.- 7.Theory of Infinite Horizon Markov Decision Processes.- 8.Piecewise Deterministic Markov Decision Processes.- 9.Optimization Problems in Finance and Insurance.- Part IV Stopping Problems.- 10.Theory of Optimal Stopping Problems.- 11.Stopping Problems in Finance.- Part V Appendix.- A.Tools from Analysis.- B.Tools from Probability.- C.Tools from Mathematical Finance.- References.- Index.
Series: Lecture Notes in Mathematics, Vol. 2015
Subseries: Ecole d'Ete de Probabilites de Saint-Flour
1st Edition., 2011, 140 p. 10 illus., Softcover
ISBN: 978-3-642-18230-3
Due: March 2011
This volume deals with the random perturbation of PDEs which lack well-posedness, mainly because of their non-uniqueness, in some cases because of blow-up. The aim is to show that noise may restore uniqueness or prevent blow-up. This is not a general or easy-to-apply rule, and the theory presented in the book is in fact a series of examples with a few unifying ideas. The role of additive and bilinear multiplicative noise is described and a variety of examples are included, from abstract parabolic evolution equations with non-Lipschitz nonlinearities to particular fluid dynamic models, like the dyadic model, linear transport equations and motion of point vortices.
Content Level : Research
Keywords : - Blow-up - Stochastic Fluid Dynamics - Stochastic Partial Differential Equations - Uniqueness
Related subjects : Probability Theory and Stochastic Processes
1. Introduction to Uniqueness and Blow-up.- 2. Regularization by Additive Noise.- 3. Dyadic Models.- 4. Transport Equation.- 5. Other Models. Uniqueness and Singularities
Series: Lecture Notes in Mathematics, Vol. 2016
1st Edition., 2011, CCXXIV, 12 p. 10 illus., Softcover
ISBN: 978-3-642-18267-9
Due: April 2011
.About this book.
The main theme of the book is the study, from the standpoint of s-numbers, of integral operators of Hardy type and related Sobolev embeddings. In the theory of s-numbers the idea is to attach to every bounded linear map between Banach spaces a monotone decreasing sequence of non-negative numbers with a view to the classification of operators according to the way in which these numbers approach a limit: approximation numbers provide an especially important example of such numbers. The asymptotic behavior of the s-numbers of Hardy operators acting between Lebesgue spaces is determined here in a wide variety of cases. The proof methods involve the geometry of Banach spaces and generalized trigonometric functions; there are connections with the theory of the p-Laplacian.
Content Level : Graduate
Keywords : - eigenvalues and eigenfunctions - generalized trigonometric functions - p-Laplacian - s-numbers - spectral theory on Banach spaces
Related subjects : Analysis - Dynamical Systems & Differential Equations - Mathematics Education
1 Basic material.- 2 Trigonometric generalisations.- 3 The Laplacian and some natural variants.- 4 Hardy operators.- 5 s-Numbers and generalised trigonometric functions.- 6 Estimates of s-numbers of weighted Hardy operators.- 7 More refined estimates.- 8 A non-linear integral system.- 9 Hardy operators on variable exponent spaces