Part of Mathematical Sciences Research Institute Publications
Date Published: November 2015
format: Hardback
isbn: 9781107149724
In the 2012?13 academic year, the Mathematical Sciences Research Institute, Berkeley, hosted programs in Commutative Algebra (Fall 2012 and Spring 2013) and Noncommutative Algebraic Geometry and Representation Theory (Spring 2013). There have been many significant developments in these fields in recent years; what is more, the boundary between them has become increasingly blurred. This was apparent during the MSRI program, where there were a number of joint seminars on subjects of common interest: birational geometry, D-modules, invariant theory, matrix factorizations, noncommutative resolutions, singularity categories, support varieties, and tilting theory, to name a few. These volumes reflect the lively interaction between the subjects witnessed at MSRI. The Introductory Workshops and Connections for Women Workshops for the two programs included lecture series by experts in the field. The volumes include a number of survey articles based on these lectures, along with expository articles and research papers by participants of the programs.
Part of Encyclopedia of Mathematics and its Applications
Publication planned for: March 2016
This book provides researchers and graduate students with a thorough introduction to the variational analysis of nonlinear problems described by nonlocal operators. The authors give a systematic treatment of the basic mathematical theory and constructive methods for these classes of nonlinear equations, plus their application to various processes arising in the applied sciences. The equations are examined from several viewpoints, with the calculus of variations as the unifying theme. Part I begins the book with some basic facts about fractional Sobolev spaces. Part II is dedicated to the analysis of fractional elliptic problems involving subcritical nonlinearities, via classical variational methods and other novel approaches. Finally, Part III contains a selection of recent results on critical fractional equations. A careful balance is struck between rigorous mathematics and physical applications, allowing readers to see how these diverse topics relate to other important areas, including topology, functional analysis, mathematical physics, and potential theory.
Part of Cambridge Studies in Advanced Mathematics
Publication planned for: June 2016
format: Hardback
isbn: 9781107106987
The subject of special functions is often presented as a collection of disparate results, rarely organized in a coherent way. This book emphasizes general principles that unify and demarcate the subjects of study. The authors' main goals are to provide clear motivation, efficient proofs, and original references for all of the principal results. The book covers standard material, but also much more. It is shown how much of the subject can be traced back to two equations ? the hypergeometric equation and confluent hypergeometric equation ? and it details the ways in which these equations are canonical and special. There is extended coverage of orthogonal polynomials, including connections to approximation theory, continued fractions, and the moment problem, as well as an introduction to new asymptotic methods. There are chapters on Meijer G-functions and elliptic functions. The final chapter introduces Painleve transcendents, which have been termed the 'special functions of the twenty-first century'.
1. Orientation
2. Gamma, beta, zeta
3. Second-order differential equations
4. Orthogonal polynomials on an interval
5. The classical orthogonal polynomials
6. Semiclassical orthogonal polynomials
7. Asymptotics of orthogonal polynomials: two methods
8. Confluent hypergeometric functions
9. Cylinder functions
10. Hypergeometric functions
11. Spherical functions
12. Generalized hypergeometric functions
G-functions
13. Asymptotics
14. Elliptic functions
15. Painleve transcendents
appendix A. Complex analysis
appendix B. Fourier analysis
References
Index.
Part of London Mathematical Society Lecture Note Series
Publication planned for: July 2016
format: Paperback
isbn: 9781107541481
This comprehensive monograph is ideal for established researchers in the field and also graduate students who wish to learn more about the subject. The text is made accessible to a broad audience as it does not require any knowledge of Lie groups and only a limited knowledge of differential geometry. The author's primary emphasis is on potential theory on the hyperbolic ball, but many other relevant results for the hyperbolic upper half-space are included both in the text and in the end-of-chapter exercises. These exercises expand on the topics covered in the chapter and involve routine computations and inequalities not included in the text. The book also includes some open problems, which may be a source for potential research projects.
Preface
1. Mobius transformations
2. Mobius self-maps of the unit ball
3. Invariant Laplacian, gradient and measure
4. H-harmonic and H-subharmonic functions
5. The Poisson kernel
6. Spherical harmonic expansions
7. Hardy-type spaces
8. Boundary behavior of Poisson integrals
9. The Riesz decomposition theorem
10. Bergman and Dirichlet spaces
References
Index of symbols
Index.
Part of Cambridge Studies in Advanced Mathematics
Publication planned for: May 2016
format: Hardback
isbn: 9781107113886
The synthesis of symplectic geometry, the calculus of variations and control theory offered in this book provides a crucial foundation for the understanding of many problems in applied mathematics. Focusing on the theory of integrable systems, this book introduces a class of optimal control problems on Lie groups, whose Hamiltonians, obtained through the Maximum Principle of optimality, shed new light on the theory of integrable systems. These Hamiltonians provide an original and unified account of the existing theory of integrable systems. The book particularly explains much of the mystery surrounding the Kepler problem, the Jacobi problem and the Kovalevskaya Top. It also reveals the ubiquitous presence of elastic curves in integrable systems up to the soliton solutions of the non-linear Schroedinger's equation. Containing a useful blend of theory and applications, this is an indispensable guide for graduates and researchers in many fields, from mathematical physics to space control.
1. The orbit theorem and Lie determined systems
2. Control systems. Accessibility and controllability
3. Lie groups and homogeneous spaces
4. Symplectic manifolds. Hamiltonian vector fields
5. Poisson manifolds, Lie algebras and coadjoint orbits
6. Hamiltonians and optimality: the Maximum Principle
7. Hamiltonian view of classic geometry
8. Symmetric spaces and sub-Riemannian problems
9. Affine problems on symmetric spaces
10. Cotangent bundles as coadjoint orbits
11. Elliptic geodesic problem on the sphere
12. Rigid body and its generalizations
13. Affine Hamiltonians on space forms
14. Kowalewski?Lyapunov criteria
15. Kirchhoff?Kowalewski equation
16. Elastic problems on symmetric spaces: Delauney?Dubins problem
17. Non-linear Schroedinger's equation and Heisenberg's magnetic equation. Solitons.