Courant Lecture Notes, Volume: 18
2009; 217 pp; softcover
ISBN-13: 978-0-8218-4737-4
Expected publication date is July 19, 2009.
This book features a unified derivation of the mathematical theory of the three classical types of invariant random matrix ensembles--orthogonal, unitary, and symplectic. The authors follow the approach of Tracy and Widom, but the exposition here contains a substantial amount of additional material, in particular, facts from functional analysis and the theory of Pfaffians. The main result in the book is a proof of universality for orthogonal and symplectic ensembles corresponding to generalized Gaussian type weights following the authors' prior work. New, quantitative error estimates are derived.
The book is based in part on a graduate course given by the first author at the Courant Institute in fall 2005. Subsequently, the second author gave a modified version of this course at the University of Rochester in spring 2007. Anyone with some background in complex analysis, probability theory, and linear algebra and an interest in the mathematical foundations of random matrix theory will benefit from studying this valuable reference.
Titles in this series are co-published with the Courant Institute of Mathematical Sciences at New York University.
Graduate students and research mathematicians interested in mathematical foundations of random matrix theory.
Invariant random matrix ensembles: unified derivation of eigenvalue cluster and correlation functions
Introduction and examples
Three classes of invariant ensembles
Auxiliary facts from functional analysis, Pfaffians, and three integral identities
Eigenvalue statistics for the three types of ensembles
Universality for orthogonal and symplectic ensembles
Widom's formulae for the beta=1 and 4 correlation kernels
Large N eigenvalue statistics for the beta=1,4 ensembles with monomial
potentials: universality
Bibliography
Index
AMS/IP Studies in Advanced Mathematics, Volume: 45
2009; 491 pp; hardcover
ISBN-13: 978-0-8218-4823-4
Expected publication date is July 18, 2009.
This book consists of two independent works: Part I is "Solutions of the Einstein Vacuum Equations", by Lydia Bieri. Part II is "Solutions of the Einstein-Maxwell Equations", by Nina Zipser.
A famous result of Christodoulou and Klainerman is the global nonlinear
stability of Minkowski spacetime. In this book, Bieri and Zipser provide
two extensions to this result. In the first part, Bieri solves the Cauchy
problem for the Einstein vacuum equations with more general, asymptotically
flat initial data, and describes precisely the asymptotic behavior. In
particular, she assumes less decay in the power of r and one less derivative
than in the Christodoulou-Klainerman result. She proves that in this case,
too, the initial data, being globally close to the trivial data, yields
a solution which is a complete spacetime, tending to the Minkowski spacetime
at infinity along any geodesic. In contrast to the original situation,
certain estimates in this proof are borderline in view of decay, indicating
that the conditions in the main theorem on the decay at infinity on the
initial data are sharp.
In the second part, Zipser proves the existence of smooth, global solutions
to the Einstein-Maxwell equations. A nontrivial solution of these equations
is a curved spacetime with an electromagnetic field. To prove the existence
of solutions to the Einstein-Maxwell equations, Zipser follows the argument
and methodology introduced by Christodoulou and Klainerman. To generalize
the original results, she needs to contend with the additional curvature
terms that arise due to the presence of the electromagnetic field F; in
her case the Ricci curvature of the spacetime is not identically zero but
rather represented by a quadratic in the components of F. In particular
the Ricci curvature is a constant multiple of the stress-energy tensor
for F. Furthermore, the traceless part of the Riemann curvature tensor
no longer satisfies the homogeneous Bianchi equations but rather inhomogeneous
equations including components of the spacetime Ricci curvature. Therefore,
the second part of this book focuses primarily on the derivation of estimates
for the new terms that arise due to the presence of the electromagnetic
field.
Graduate students and research mathematicians interested in general relativity.
Solutions of the Einstein vacuum equations, by Lydia Bieri
Introduction
Preliminary tools
Main theorem
Comparison
Error estimates
Second fundamental form k: estimates for the components of k
Second fundamental form {mathbf{chi}}: estimating {mathbf{chi}} and {mathbf{zeta}}
Uniformization theorem
boldsymbol{chi} on the surfaces boldsymbol{S}-changes in boldsymbol{r}
and boldsymbol{s}
The last slice
Curvature tensor-components
Uniformation theorem: standard situation, cases 1 and 2
Bibliography
Index
Solutions of the Einstein-Maxwell equations, by Nina Zipser
Introduction
Norms and notation
Existence theorem
The electromagnetic field
Error estimates for F
Interior estimates for F
Comparison theorem for the Weyl tensor
Error estimates for W
Second fundamental form
The lapse function
Optical function
Conclusion
Bibliography
Index
CRM Proceedings & Lecture Notes, Volume: 47
2009; 366 pp; softcover
ISBN-13: 978-0-8218-4481-6
Expected publication date is August 8, 2009.
This volume contains papers presented at a conference held in April 2007 at the CRM in Montreal honouring the remarkable contributions of John McKay over four decades of research. Papers by invitees who were unable to attend the conference are also included.
The papers cover a wide range of topics, including group theory, symmetries, modular functions, and geometry, with particular focus on two areas in which John McKay has made pioneering contributions: "Monstrous Moonshine" and the "McKay Correspondence". This book will be a valuable reference for graduate students and researchers interested in these and related areas and serve as a stimulus for new ideas.
Graduate students and research mathematicians interested in group theory, symmetries, modular functions, and geometry.
J. McKay -- Introduction and background
H. Airault -- Symmetric sums associated to the factorization of Grunsky coefficients
D. Allcock -- A monstrous proposal
P. Boalch -- Quivers and difference Painleve equations
D. X. Charles, E. Z. Goren, and K. E. Lauter -- Families of Ramanujan graphs and quaternion algebras
A. Clingher, C. F. Doran, J. Lewis, and U. Whitcher -- Normal forms, K3 surface moduli, and modular parametrizations
H. Cohn -- Spontaneous generation of Hilbert modular functions
C. J. Cummins and N. S. Haghighi -- On a class of congruence subgroups
I. V. Dolgachev -- McKay's correspondenxe for cocompact discrete subgroups of SU(1, 1)
J. F. Duncan -- Arithmetic groups and the affine mathrm{E}_8 Dynkin diagram
T. Gannon -- The Galois action on character tables
N. Ganter -- Hecke operators in equivariant elliptic cohomology and generalized moonshine
G. W. Hart -- Sculptural presentation of the icosahedral rotation group
N. Hitchin -- Spherical harmonics and the iscoahedron
A. Klimyk and J. Patera -- Alternating group and multivariate exponential functions
J. Morava -- Moonshine elements in elliptic cohomology
M. R. Murty and K. L. Petersen -- The generalized Artin conjecture and arithmetic orbifolds
I. Nakamura -- McKay correspondence
V. V. Nikulin -- On ground fields of arithmetic hyperbolic reflection groups
S. P. Norton -- Moonshine-type functions and the CRM correspondence
Y. Ohyama -- Monodromy evolving deformations and Halphen's equation
D. Zagier -- Integral solutions of Apery-like recurrence equations
Mathematical Surveys and Monographs, Volume: 155
2009; 502 pp; hardcover
ISBN-13: 978-0-8218-4813-5
Expected publication date is July 25, 2009.
Descriptive Set Theory is the study of sets in separable, complete metric spaces that can be defined (or constructed), and so can be expected to have special properties not enjoyed by arbitrary pointsets. This subject was started by the French analysts at the turn of the 20th century, most prominently Lebesgue, and, initially, was concerned primarily with establishing regularity properties of Borel and Lebesgue measurable functions, and analytic, coanalytic, and projective sets. Its rapid development came to a halt in the late 1930s, primarily because it bumped against problems which were independent of classical axiomatic set theory. The field became very active again in the 1960s, with the introduction of strong set-theoretic hypotheses and methods from logic (especially recursion theory), which revolutionized it.
This monograph develops Descriptive Set Theory systematically, from its classical roots to the modern "effective" theory and the consequences of strong (especially determinacy) hypotheses. The book emphasizes the foundations of the subject, and it sets the stage for the dramatic results (established since the 1980s) relating large cardinals and determinacy or allowing applications of Descriptive Set Theory to classical mathematics.
The book includes all the necessary background from (advanced) set theory, logic and recursion theory.
Graduate students and research mathematicians interested in set theory. especially descriptive set theory.
Introduction
The basic classical notions
kappa-Suslin and lambda-Borel
Basic notions of the effective theory
Structure theory for pointclasses
The constructible universe
The playful universe
The recursion theorem
Metamathematics
The axiomatics of pointclasses
References
Index
2009; approx. 439 pp; softcover
ISBN-13: 978-0-8218-4807-4
Expected publication date is August 13, 2009.
Who would have thought that listing the positive integers along with their most remarkable properties could end up being such an engaging and stimulating adventure? The author uses this approach to explore elementary and advanced topics in classical number theory. A large variety of numbers are contemplated: Fermat numbers, Mersenne primes, powerful numbers, sublime numbers, Wieferich primes, insolite numbers, Sastry numbers, voracious numbers, to name only a few. The author also presents short proofs of miscellaneous results and constantly challenges the reader with a variety of old and new number theory conjectures.
This book becomes a platform for exploring new concepts such as the index of composition and the index of isolation of an integer. In addition, the book displays several tables of particular families of numbers, including the list of all 88 narcissistic numbers and the list of the eight known numbers which are not prime powers but which can be written as the sum of the cubes of their prime factors, and in each case with the algorithm used to create them.
Undergraduates, graduate students, and research mathematicians interested in number theory.
Those fascinating numbers
Appendix: The prime numbers <10 000
Bibliography
Index
Student Mathematical LibraryAVolume: 50
2009; 252 pp; softcover
ISBN-13: 978-0-8218-4772-5
Expected publication date is August 19, 2009.
The calculus of variations is used to find functions that optimize quantities expressed in terms of integrals. Optimal control theory seeks to find functions that minimize cost integrals for systems described by differential equations.
This book is an introduction to both the classical theory of the calculus of variations and the more modern developments of optimal control theory from the perspective of an applied mathematician. It focuses on understanding concepts and how to apply them. The range of potential applications is broad: the calculus of variations and optimal control theory have been widely used in numerous ways in biology, criminology, economics, engineering, finance, management science, and physics. Applications described in this book include cancer chemotherapy, navigational control, and renewable resource harvesting.
The prerequisites for the book are modest: the standard calculus sequence, a first course on ordinary differential equations, and some facility with the use of mathematical software. It is suitable for an undergraduate or beginning graduate course, or for self study. It provides excellent preparation for more advanced books and courses on the calculus of variations and optimal control theory.
Undergraduate and graduate students interested in the calculus of variations and optimal control.
The Brachistochrone
The fundamental problem. Extremals
The insufficiency of extremality
Important first integrals
The du Bois-Reymond equation
The corner conditions
Legendre's necessary condition
Jacobi's necessary condition
Weak versus strong variations
Weierstrass's necessary condition
The transversality conditions
Hilbert's invariant integral
The fundamental sufficient condition
Jacobi's condition revisited
Isoperimetrical problems
Optimal control problems
Necessary conditions for optimality
Time-optional control
A singular control problem
A biological control problem
Optimal control to a general target
Navigational control problems
State variable restrictions
Optimal harvesting
Afterword
Solutions or hints for selected exercises
Bibliography
Index