Series: Lecture Notes in Mathematics , Vol. 1975
2009, Approx. 200 p., Softcover
ISBN: 978-3-642-00638-8
Due: May 8, 2009
The main goal of this book is the construction of families of Calabi-Yau 3-manifolds with dense sets of complex multiplication fibers. The new families are determined by combining and generalizing two methods.
Firstly, the method of E. Viehweg and K. Zuo, who have constructed a deformation of the Fermat quintic with a dense set of CM fibers by a tower of cyclic coverings. Using this method, new families of K3 surfaces with dense sets of CM fibers and involutions are obtained.
Secondly, the construction method of the Borcea-Voisin mirror family, which in the case of the author's examples yields families of Calabi-Yau 3-manifolds with dense sets of CM fibers, is also utilized. Moreover fibers with complex multiplication of these new families are also determined.
This book was written for young mathematicians, physicists and also for experts who are interested in complex multiplication and varieties with complex multiplication. The reader is introduced to generic Mumford-Tate groups and Shimura data, which are among the main tools used here. The generic Mumford-Tate groups of families of cyclic covers of the projective line are computed for a broad range of examples.
1 An introduction to Hodge structures and Shimura varieties.- 2 Cyclic covers of the projective line.- 3 Some preliminaries for families of cyclic covers.- 4 The Galois group decomposition of the Hodge structure.- 5 The computation of the Hodge group.- 6 Examples of families with dense sets of complex multiplication fibers.- 7 The construction of Calabi-Yau manifolds with complex multiplication.- 8 The degree 3 case.- 9 Other examples and variations.- 10 Examples of CMCY families of 3-manifolds and their invariants.- 11 Maximal families of CMCY type.
Series: Springer Monographs in Mathematics
2009, Approx. 845 p. 16 illus., Hardcover
ISBN: 978-0-387-89491-1
Due: June 2009
Functional Equations and Inequalities with Applications presents a comprehensive, nearly encyclopedic, study of the classical topic of functional equations. Nowadays, the field of functional equations is an ever-growing branch of mathematics with far-reaching applications; it is increasingly used to investigate problems in mathematical analysis, combinatorics, biology, information theory, statistics, physics, the behavioral sciences, and engineering.
This self-contained monograph explores all aspects of functional equations and their applications to related topics, such as differential equations, integral equations, the Laplace transformation, the calculus of finite differences, and many other basic tools in analysis. Each chapter examines a particular family of equations and gives an in-depth study of its applications; examples and exercises to support the material.
The book is intended as a reference tool for any student, professional (researcher), or mathematician studying in a field where functional equations can be applied. It can also be used as a primary text in a classroom setting or for self-study. Finally, it could be an inspiring entree into an active area of mathematical exploration for engineers and other scientists who would benefit from this careful, rigorous exposition.
The book is intended as a reference tool for any student, professional (researcher), or mathematician studying in a field where functional equations can be applied. It can also be used as a primary text in a classroom setting or for self-study. Finally, it could be an inspiring entree into an active area of mathematical exploration for engineers and other scientists who would benefit from this careful, rigorous exposition.
- Preface.- 1. Basic Equations. Cauchy and Pexider Equations.- 2. Matrix Equations.- 3. Trigonometric Functional Equations.- 4. Quadratic Functional Equations.- 5. Characterization of Inner Product Spaces.- 6. Stability.- 7. Characterization of Polynomials.- 8. Nondierentiable Functions.- 9. Characterization of Groups, Loops and Closure Conditions.- 10. Functional Equations from Information Theory.- 11. Abel Equations and Generalizations.- 12. Regularity Conditions|Christensen Measurability.- 13. Dierence Equations.- 14. Characterization of Special Functions.- 15. Miscellaneous Equations.- 16. General Inequalities.- 17. Applications.- Symbols.- Bibliography.- Author Index.- Subject Index.
Series: Abel Symposia , Vol. 5
2009, Approx. 400 p., Hardcover
ISBN: 978-3-642-00872-6
Due: June 18, 2009
The Abel Symposium 2008 focused on the modern theory of differential equations and their applications in geometry, mechanics, and mathematical physics. Following the tradition of Monge, Abel and Lie, the scientific program emphasized the role of algebro-geometric methods, which nowadays permeate all mathematical models in natural and engineering sciences. The ideas of invariance and symmetry are of fundamental importance in the geometric approach to differential equations, with a serious impact coming from the area of integrable systems and field theories.
This volume consists of original contributions and broad overview lectures of the participants of the Symposium. The papers in this volume present the modern approach to this classical subject.
I.Anderson, M.Fels: Internal Equivalences for Darboux Integrable.- Ph.Delanoe: Differential Geometric Heuristics for Riemannian Optimal Mass Transportation.- V.V.Goldberg, V.V. Lychagin : On Rank Problems for Planar Webs and Projective Structures.- H.L.Huru: The Polynomial Algebra and Quantizations of Electromagnetic Fields.- N.H.Ibragimov: A Bridge Between Lie Symmetries and Galois Groups.- N.Kamran: Focal Systems for Pfaffian Systems with Characteristics.- P.Kersten, I.S.Krasil: Hamiltonian Structures for General PDE.- B.Kruglikov: Point Classification of 2nd Order ODEs: Tresse Classification Revisited and Beyond.- A.G.Kushner: Classification of Monge-Ampere Equations.- A.Marshakov: On Nonabelian Theories and Abelian Differentials.- R.Moeckel: Shooting for the Eight - A Topological Existence Proof for a Figure-Eight Orbit of the Three-Body Problem.- R.J.Alonso, S.Jimenez, J. Rodriguez: Some Canonical Structures of Cartan Planes in Jet Spaces and Applications.- V.Roubtsov , T.Skrypnyk: Compatible Poisson Brackets, Quadratic Poisson Algebras and Classical r-matrices.- M.Modugno, C.Tejero Prieto: Geometric Aspects of the Quantization of a Rigid Body.- K.Yamaguchi: Contact Geometry of Second Order I
Series: Springer Monographs in Mathematics
2009, Approx. 675 p., Hardcover
ISBN: 978-1-84882-532-1
Due: July 2009
This book presents the first systematic and unified treatment of the theory of mean periodic functions on homogeneous spaces. This area has its classical roots in the beginning of the twentieth century and is now a very active research area, having close connections to harmonic analysis, complex analysis, integral geometry, and analysis on symmetric spaces.
The main purpose of this book is the study of local aspects of spectral analysis and spectral synthesis on Euclidean spaces, Riemannian symmetric spaces of an arbitrary rank and Heisenberg groups. The subject can be viewed as arising from three classical topics: John's support theorem, Schwartz's fundamental principle, and Delsarte's two-radii theorem.
Highly topical, the book contains most of the significant recent results in this area with complete and detailed proofs. In order to make this book accessible to a wide audience, the authors have included an introductory section that develops analysis on symmetric spaces without the use of Lie theory. Challenging open problems are described and explained, and promising new research directions are indicated.
Designed for both experts and beginners in the field, the book is rich in methods for a wide variety of problems in many areas of mathematics.
Part 1; Symmetric Spaces. Harmonic Analysis on Spheres.- 1. General Considerations.- 2. Analogues of the Beltrami-Klein Model for Rank One Symmetric Spaces of Non-Compact Type.- 3. Realizations of Rank One Symmetric Spaces of Compact Type.- 4. Realizations of the Irreducible Components of the Quasi-Regular Representation of Groups Transitive on Spheres. Invariant Subspaces.- 5. Non-Euclidean Analogues of Plane Waves.- Comments, Further Results and Open Problems.- Part 2; Transformations with Generalized Transmutation Property Associated with Eigenfunctions Expansions.- 6. Preliminaries.- 7. Some Special Functions.- 8. Exponential Expansions.- 9. Multidimensional Euclidean Case.- 10. The Case of Symmetric Spaces X = G/K of Noncompact Type.- 11. The Case of Compact Symmetric Spaces.- 12. The Case of Phase Space.- Comments, Further Results and Open Problems.- Part 3; Mean periodicity.- 13. Mean Periodic Functions on Subsets of the Real Line.- 14. Mean Periodic Functions on Multidimensional Domains.- 15. Mean Periodic Functions on G/K.- 16. Mean Periodic Functions on Compact Symmetric Spaces of Rank One.- 17. Mean Periodicity on Phase Space and the Heisenberg Group.- Comments, Further Results and Open Problems.- Part 4. Local Aspects of Spectral Analysis and the Exponential Representation Problem.- 18. A New Look at the Schwartz Theory.- 19. Recent Developments in the Spectral Analysis Problem for Higher Dimensions.- 20. Spectral Analysis on Domains of Noncompact Symmetric Spaces of an Arbitrary Rank.- 21. Spherical Spectral Analysis on Subsets of Compact Symmetric Spaces.- Comments, Further Results and Open Problems.- Bibliography.
Series: Universitext
Original French edition published by Editions de l'Ecole Polytechnique, 2005. Second edition, 2006.
2009, Approx. 220 p. 24 illus., Hardcover
ISBN: 978-0-387-78865-4
Due: September 2009
- brisk review of the basic definitions and fundamental results of group theory, illustrated with examples;
- representation theory of finite groups (Schur’s Lemma and characters) and, using Haar measure, its generalization to compact groups;
- Lie algebras and linear Lie groups;
- detailed study of the group of rotations, the special unitary group in dimension 2 and their representations;
- spherical harmonics;
- representations of the special unitary group in dimension 3 (roots and weights) with quark theory as a consequence of the mathematical properties of the symmetry group.
This book is an introduction to both the theory of group representations and its applications in quantum mechanics. Unlike many other texts, it deals with finite groups, compact groups, linear Lie groups and their Lie algebras, concisely presented in one volume. With only linear algebra and calculus as prerequisites, it is accessible to advanced undergraduates in mathematics and physics, and will still be of interest to beginning graduate students. Exercises for each chapter and a collection of problems with complete solutions make this an ideal text for the classroom or for independent study.
Introduction.- General Facts about Groups.- Representations of Finite Groups.- Representations of Compact Groups.- Lie groups and Lie algebras.- Lie groups SU(2) and SO(3).- Representations of SU(2) and SO(3).- Spherical Harmonics.- Representations of SU(3) and Quarks.- Problems and Solutions.- Bibliography.- Index.
Series on Knots and Everything
This book is a unique summary of the results of a long research project undertaken by the authors on discreteness in modern physics. In contrast with the usual expectation that discreteness is the result of mathematical tools for insertion into a continuous theory, this more basic treatment builds up the world from the discrimination of discrete entities. This gives an algebraic structure in which certain fixed numbers arise. As such, one agrees with the measured value of the fine-structure constant to one part in 10,000,000 (107).
Combinatorial Space
The Story of the Particle Concept
Dimensionality
The Simple Bit-String Picture
The Fine Structure Constant Calculated
Process and Interaction
Pre-Space, High Energy Particles: Nonlocality
Space: Relativity and Classical Space
Perception: Current Quantum Physics ・The State-Observer Philosophy
Just Six Numbers
Quantum or Gravity?
Readership: Researchers in high energy physics, quantum physics and mathematical physics.
140pp (approx.) Pub. date: Scheduled Fall 2009
ISBN 978-981-4261-67-8