Translations of Mathematical Monographs, Volume: 78
2009; 339 pp; softcover
ISBN-13: 978-0-8218-4686-5
Expected publication date is January 30, 2009.
Written by one of the foremost Soviet experts in the field, this book is intended for specialists in the theory of random processes and its applications. The author's 1982 monograph on stochastic differential equations, written with Iosif Il'ich Gikhman, did not include a number of topics important to applications. The present work begins to fill this gap by investigating the asymptotic behavior of stochastic differential equations. The main topics are ergodic theory for Markov processes and for solutions of stochastic differential equations, stochastic differential equations containing a small parameter, and stability theory for solutions of systems of stochastic differential equations.
Ergodic theorems
General ergodic theorems
Densities for transition probabilities and resolvents for Markov solutions of stochastic differential equations
Ergodic theorems for one-dimensional stochastic equations
Ergodic theorems for solutions of stochastic equations in $R^d$
Asymptotic behavior of systems of stochastic equations containing a small parameter
Equations with a small right-hand side
Processes with rapid switching
Averaging over variables for systems of stochastic differential equations
Stability. Linear systems
Stability of sample paths of homogeneous Markov processes
Linear equations in $R^d$ and the stochastic semigroups connected with them. Stability
Stability of solutions of stochastic differential equations
Linear stochastic equations in Hilbert space. Stochastic semigroups. Stability
Linear equations with bounded coefficients
Strong stochastic semigroups with second moments
Stability
Bibliography
Graduate Studies in Mathematics, Volume: 100
2009; 516 pp; hardcover
ISBN-13: 978-0-8218-4799-2
Expected publication date is January 29, 2009.
This book, based on a first-year graduate course the author taught at the University of Wisconsin, contains more than enough material for a two-semester graduate-level abstract algebra course, including groups, rings and modules, fields and Galois theory, an introduction to algebraic number theory, and the rudiments of algebraic geometry. In addition, there are some more specialized topics not usually covered in such a course. These include transfer and character theory of finite groups, modules over artinian rings, modules over Dedekind domains, and transcendental field extensions.
This book could be used for self study as well as for a course text, and so full details of almost all proofs are included, with nothing being relegated to the chapter-end problems. There are, however, hundreds of problems, many being far from trivial. The book attempts to capture some of the informality of the classroom, as well as the excitement the author felt when taking the corresponding course as a student.
Graduate students and research mathematicians interested in algebra.
Part One, Noncommutative Algebra
Definitions and examples of groups
Subgroups and cosets
Homomorphisms
Group actions
The Sylow theorems and $p$-groups
Permutation groups
New groups from old
Solvable and nilpotent groups
Transfer
Operator groups and unique decompositions
Module theory without rings
Rings, ideals, and modules
Simple modules and primitive rings
Artinian rings and projective modules
An introduction to character theory
Part Two, Commutative Algebra
Polynomial rings, PIDs, and UFDs
Field extensions
Galois theory
Separability and inseparability
Cyclotomy and geometric constructions
Finite fields
Roots, radicals, and real numbers
Norms, traces, and discriminants
Transcendental extensions
The Artin-Schreier theorem
Ideal theory
Noetherian rings
Integrality
Dedekind domains
Algebraic sets and the nullstellensatz
Index
Graduate Studies in Mathematics, Volume: 101
2009; 359 pp; hardcover
ISBN-13: 978-0-8218-4798-5
Expected publication date is March 26, 2009.
This textbook is designed for graduate students in mathematics, physics, engineering, and computer science. Its purpose is to guide the reader in exploring contemporary approximation theory. The emphasis is on multi-variable approximation theory, i.e., the approximation of functions in several variables, as opposed to the classical theory of functions in one variable.
Most of the topics in the book, heretofore accessible only through research papers, are treated here from the basics to the currently active research, often motivated by practical problems arising in diverse applications such as science, engineering, geophysics, and business and economics. Among these topics are projections, interpolation paradigms, positive definite functions, interpolation theorems of Schoenberg and Micchelli, tomography, artificial neural networks, wavelets, thin-plate splines, box splines, ridge functions, and convolutions.
An important and valuable feature of the book is the bibliography of almost 600 items directing the reader to important books and research papers. There are 438 problems and exercises scattered through the book allowing the student reader to get a better understanding of the subject.
Graduate students and research mathematicians interested in approximation theory and applications.
Introductory discussion of interpolation
Linear interpolation operators
Optimization of the Lagrange operator
Multivariate polynomials
Moving the nodes
Projections
Tensor-product interpolation
The Boolean algebra of projections
The Newton paradigm for interpolation
The Lagrange paradigm for interpolation
Interpolation by translates of a single function
Positive definite functions
Strictly positive definite functions
Completely monotone functions
The Schoenberg interpolation theorem
The Micchelli interpolation theorem
Positive definite functions on spheres
Approximation by positive definite functions
Approximate reconstruction of functions and tomography
Approximation by convolution
The good kernels
Ridge functions
Ridge function approximation via convolutions
Density of ridge functions
Artificial neural networks
Chebyshev centers
Optimal reconstruction of functions
Algorithmic orthogonal projections
Cardinal B-splines and the sinc function
The Golomb-Weinberger theory
Hilbert function spaces and reproducing kernels
Spherical thin-plate splines
Box splines
Wavelets, I
Wavelets, II
Quasi-interpolation
Bibliography
Index
Index of symbols
Graduate Studies in Mathematics, Volume: 102
2009; 376 pp; hardcover
ISBN-13: 978-0-8218-4797-8
Expected publication date is February 5, 2009.
This book provides a concrete introduction to a number of topics in harmonic analysis, accessible at the early graduate level or, in some cases, at an upper undergraduate level. Necessary prerequisites to using the text are rudiments of the Lebesgue measure and integration on the real line. It begins with a thorough treatment of Fourier series on the circle and their applications to approximation theory, probability, and plane geometry (the isoperimetric theorem). Frequently, more than one proof is offered for a given theorem to illustrate the multiplicity of approaches.
The second chapter treats the Fourier transform on Euclidean spaces, especially the author's results in the three-dimensional piecewise smooth case, which is distinct from the classical Gibbs-Wilbraham phenomenon of one-dimensional Fourier analysis. The Poisson summation formula treated in Chapter 3 provides an elegant connection between Fourier series on the circle and Fourier transforms on the real line, culminating in Landau's asymptotic formulas for lattice points on a large sphere.
Much of modern harmonic analysis is concerned with the behavior of various linear operators on the Lebesgue spaces $L^p(\mathbb{R}^n)$. Chapter 4 gives a gentle introduction to these results, using the Riesz-Thorin theorem and the Marcinkiewicz interpolation formula. One of the long-time users of Fourier analysis is probability theory. In Chapter 5 the central limit theorem, iterated log theorem, and Berry-Esseen theorems are developed using the suitable Fourier-analytic tools.
The final chapter furnishes a gentle introduction to wavelet theory, depending only on the $L_2$ theory of the Fourier transform (the Plancherel theorem). The basic notions of scale and location parameters demonstrate the flexibility of the wavelet approach to harmonic analysis.
The text contains numerous examples and more than 200 exercises, each located in close proximity to the related theoretical material.
Undergraduate and graduate students interested in Fourier transform and harmonic analysis.
Fourier series on the circle
Fourier transforms on the line and space
Fourier analysis in $L^p$ spaces
Poisson summation formula and multiple Fourier series
Applications to probability theory
Introduction to wavelets
References
Notations
Index
2008; 192 pp; hardcover
ISBN-13: 978-0-8218-4426-7
This classic book, originally published in 1968, is based on notes of a year-long seminar the authors ran at Princeton University. The primary goal of the book was to give a rather complete presentation of algebraic aspects of global class field theory, and the authors accomplished this goal spectacularly: for more than 40 years since its first publication, the book has served as an ultimate source for many generations of mathematicians.
In this revised edition, two mathematical additions complementing the exposition in the original text are made. The new edition also contains several new footnotes, additional references, and historical comments.
Graduate students and research mathematicians interested in number theory.
Preliminaries
The first fundamental inequality
Second fundamental inequality
Reciprocity law
The existence theorem
Connected component of idele classes
The Grunwald-Wang theorem
Higher ramification theory
Explicit reciprocity laws
Group extensions
Abstract class field theory
Weil groups
Bibliography