Series: Developments in Mathematics , Vol. 19
2010, XVI, 176 p., Hardcover
ISBN: 978-1-4419-0433-1
Due: November 2009
This book covers the necessary topics for learning the basic properties of complex manifolds and their submanifolds, offering an easy, friendly, and accessible introduction into the subject while aptly guiding the reader to topics of current research and to more advanced publications.
The book begins with an introduction to the geometry of complex manifolds and their submanifolds and describes the properties of hypersurfaces and CR submanifolds, with particular emphasis on CR submanifolds of maximal CR dimension. The second part contains results which are not new, but recently published in some mathematical journals. The final part contains several original results by the authors, with complete proofs.
Key features of "CR Submanifolds of Complex Projective Space":
- Presents recent developments and results in the study of submanifolds previously published only in research papers.
- Special topics explored include: the Kahler manifold, submersion and immersion, codimension reduction of a submanifold, tubes over submanifolds, geometry of hypersurfaces and CR submanifolds of maximal CR dimension.
- Provides relevant techniques, results and their applications, and presents insight into the motivations and ideas behind the theory.
- Presents the fundamental definitions and results necessary for reaching the frontiers of research in this field.
This text is largely self-contained. Prerequisites include basic knowledge of introductory manifold theory and of curvature properties of Riemannian geometry. Advanced undergraduates, graduate students and researchers in differential geometry will benefit from this concise approach to an important topic.
1. Complex manifold.- 2. Almost complex structure.- 3. Complex vector space complexification.- 4. Kahler manifold.- 5. Structure equations of a submanifold.- 6. Submanifolds of a Euclidean space.- 7. Submanifolds of a complex manifold.- 8. The Levi form.- 9. The principal circle bundle S^{2n+1}({\bf P}^n({\bf C}),S^1).- 10. Submersion and immersion.- 11. Hypersurfaces of a Riemannian manifold of constant curvature.- 12. Hypersurfaces of a sphere S^{n+1}(1/a).- 13. Hypersurfaces of a sphere with parallel shape operator.- 14. Codimension reduction of a submanifold.- 15. CR submanifolds of maximal CR dimension.- 16. Real hypersurfaces of a complex projective space.- 17. Tubes around submanifolds.- 18. Levi form of CR submanifolds of maximal CR dimension of a complex space form.- 19. Eigenvalues of the shape operator A of CR submanifolds of maximal CR dimension of a complex space form.- 20. CR submanifolds of maximal CR dimension satisfying the condition h(FX,Y)+h(X,FY)=0.- 21. Contact CR submanifolds of maximal CR dimension.- 22. Invariant submanifolds of real hypersurfaces of complex space forms.- 23. The scalar curvature of CR submanifolds of maximal CR dimension.
Series: Stochastic Modelling and Applied Probability , Vol. 63
2010, X, 390 p., Hardcover
ISBN: 978-1-4419-1104-9
Due: November 2009
This book presents a comprehensive study of hybrid switching diffusion processes and their applications. The motivations for studying such processes originate from emerging and existing applications in wireless communications, signal processing, queueing networks, production planning, biological systems, ecosystems, financial engineering, and modeling, analysis, and control and optimization of large-scale systems, under the influence of random environment. One of the distinct features of the processes under consideration is the coexistence of continuous dynamics and discrete events. This book is written for applied mathematicians, applied probabilists, systems engineers, control scientists, operations researchers, and financial analysts. Selected materials from the book may also be used in a graduate level course on stochastic processes and applications or a course on hybrid systems. A large part of the book is concerned with the discrete event process depending on the continuous dynamics. In addition to the existence and uniqueness of solutions of switching diffusion equations, regularity, Feller and strong Feller properties, continuous and smooth dependence on initial data, recurrence, ergodicity, invariant measures, and stability are dealt with. Numerical methods for solutions of switching diffusions are developed; algorithms for approximation to invariant measures are investigated. Two-time-scale models are also examined. The results presented in the book are useful to researchers and practitioners who need to use stochastic models to deal with hybrid stochastic systems, and to treat real-world problems when continuous dynamics and discrete events are intertwined, in which the traditional approach using stochastic differential equations alone are no longer adequate.
Preface.-Introduction and Motivation.-Switching Diffusion.- Recurrence.-Ergodicity.-Numerical Approximation.-Numerical Approximation to Invariant Measures.-Stability.-Stability of Switching ODE.-Invariance Principles.-Positive Recurrence: Multi-ergodic-class of Switching Processes.-Stochastic Volatility Using Regime-switching Diffusions.- Two-time-scale Switching Jump-diffusions.-Appendix.-References.
Series: Progress in Mathematics , Vol. 280
2010, Approx. 300 p., Hardcover
ISBN: 978-3-0346-0200-6
Due: November 2009
This volume is a homage to the memory of the Spanish mathematician Federico Gaeta (1923-2007). Apart from a historical presentation of his life and interaction with the classical Italian school of algebraic geometry, the volume presents surveys and original research papers on the mathematics he studied. Specifically, it is divided into three parts: linkage theory, Schottky problem and invariant theory. On this last topic a hitherto unpublished article by Federico Gaeta is also included.
Federico Gaeta, among the last classics (Ignacio Sols).- Federico Gaeta and his Italian heritage (Ciro Ciliberto).- Gaetas’s work on Liaison Theory: an appreciation (Rosa M. Miro-Roig).- Symmetric ladders and G-biliaison (Elisa Gorla).- Linkage invariants and the Hilbert scheme of codimension 2 subscheme in P{n+2} (Jan O. Kleppe).- Minimal links and a result of Gaeta (Juan Migliore and Uwe Nagel).- On the existence of maximal rank curves with prescribed Hartshorne?Rao module (Silvio Greco and Rosa Maria Miro-Roig).- Doubling rational normal curves (Roberto Notari, Ignacio Ojeda and Maria Luisa Spreafico).- Survey on the Schottky Problem (Esteban Gomez Gonzalez and Jose Maria Munoz Porras).- Abelian solutions on soliton equations and geometry of abelian varieties (I. Krichever and T. Shiota).- A special case of the \Gamma_{00} Conjecture (Samuel Grushevsky).- His ten last years (Maria Emilia Alonso).- Covariants vanishing on totally decomposable forms (Emmanuel Briand).- Symmetric functions and secant spaces of rational normal curves (Federico Gaeta, revised by Laureano Gonzalez Vega).- Articles published by Federico Gaeta.
Series: Grundlehren der mathematischen Wissenschaften , Vol. 325
2010, Approx. 755 p., Hardcover
ISBN: 978-3-642-04047-4
Due: November 19, 2009
This is a masterly exposition and an encyclopedic presentation of the theory of hyperbolic conservation laws. It illustrates the essential role of continuum thermodynamics in providing motivation and direction for the development of the mathematical theory while also serving as the principal source of applications. The reader is expected to have a certain mathematical sophistication and to be familiar with (at least) the rudiments of analysis and the qualitative theory of partial differential equations, whereas prior exposure to continuum physics is not required. The target group of readers would consist of
(a) experts in the mathematical theory of hyperbolic systems of conservation laws who wish to learn about the connection with classical physics;
(b) specialists in continuum mechanics who may need analytical tools;
(c) experts in numerical analysis who wish to learn the underlying mathematical theory; and
(d) analysts and graduate students who seek introduction to the theory of hyperbolic systems of conservation laws.
New to the 3rd edition is an account of the early history of the subject, spanning the period between 1800 to 1957. Also new is a chapter recounting the recent solution of open problems of long standing in classical aerodynamics. Furthermore, the presentation of a number of topics in the previous edition has been revised and brought up to date, and the collection of applications has been substantially enriched. The bibliography, also expanded and updated, now comprises over fifteen hundred titles.
Preface.- Acknowledgments.- Introduction.- A Sketch of the Early History of Hyperbolic Conservation Laws.- I.Balance Laws.- II.Introduction to Continuum Physics.- III.Hyperbolic Systems of Balance Laws.- IV.The Cauchy Problem.- V.Entropy and the Stability of Classical Solutions.- VI.The L1 Theory for Scalar Conservation Laws.- VII.Hyperbolic Systems of Balance Laws in One-Space Dimension.- VIII.Admissible Shocks.- IX.AdmissibleWave Fans and the Riemann Problem.- X.Generalized Characteristics.- XI.Genuinely Nonlinear Scalar Conservation Laws.- XII.Genuinely Nonlinear Systems of Two Conservation Laws.- XIII.The Random Choice Method.- XIV.The Front Tracking Method and Standard Riemann Semigroups.- XV.Construction of BV Solutions by the Vanishing Viscosity Method.- XVI.Compensated Compactness.-XVII.Conservation Laws in Two Space Dimensions.- Bibliography.- Author Index.- Subject Index.
Series: Trends in Mathematics
2010, Approx. 300 p., Hardcover
ISBN: 978-3-0346-0008-8
Due: March 2010
This volume represents the proceedings of a conference on Several Complex Variables, PDE's, geometry, and their interactions, held July 7-11, 2008 at the University of Fribourg, Switzerland, in honor of Linda Rothschild. The contributors are leading experts who were invited plenary speakers at the conference, or who were invited by the editors to contribute to this volume.
Preface.- A mathematical CV of Linda Rothschild: Her contributions to complex analysis.- Oblique polar lines of RX |f|2|g|2μ.- On involutive systems of first-order nonlinear PDEs.- Gevrey Hypoellipticity for an interesting variant of Kohn’s operator.- Subelliptic Estimates.- Invariant CR Mappings.- On the subellipticity of some hypoelliptic quasihomogeneous systems of complex vector fields.- Invariance of the parametric Oka property.- Positivity of the ∂-Neumann Laplacian.- Compactness estimates for the ∂-Neumann problem in weighted L2-spaces.- Remarks on the homogeneous complex Monge-Ampere equation.- A Rado theorem for locally solvable structures of co-rank one.- Applications of a parametric Oka principle for liftings.- Stability of the vanishing of the ∂b-cohomology under small horizontal perturbations of the CR structure in compact abstract q-concave CR manifolds.- coherent Sheaves and Cohesive Sheaves.- Characteristic classes of the boundary of a complex b-manifold.- Solvability of planar complex vector fields with applications to deformation of surfaces.- The Gauss map on complex hyperbolic space forms.- The large time asymptotics of the entropy.- The closed range property for ∂ on domains with pseudoconcave boundary.- New normal forms for Levi-nondegenerate hypersurfaces