Carlos Berenstein, University of Maryland, College Park, MD, Der-Chen Chang, Georgetown University, Washington, DC, and Jingzhi Tie, University of Georgia, Athens, GA

Laguerre Calculus and Its Applications on the Heisenberg Group

Expected publication date is May 6, 2001

Description

For nearly two centuries, the relation between analytic functions of one complex variable, their boundary values, harmonic functions, and the theory of Fourier series has been one of the central topics of study in mathematics. The topic stands on its own, yet also provides very useful mathematical applications.

This text provides a self-contained introduction to the corresponding questions in several complex variables: namely, analysis on the Heisenberg group and the study of the solutions of the boundary Cauchy-Riemann equations. In studying this material, readers are exposed to analysis in non-commutative compact and Lie groups, specifically the rotation group and the Heisenberg groups--both fundamental in the theory of group representations and physics.

Introduced in a concrete setting are the main ideas of the Calder?n-Zygmund-Stein school of harmonic analysis. Also considered in the book are some less conventional problems of harmonic and complex analysis, in particular, the Morera and Pompeiu problems for the Heisenberg group, which relates to questions in optics, tomography, and engineering.

The book was borne of graduate courses and seminars held at the University of Maryland (College Park), the University of Toronto (ON), Georgetown University (Washington, DC), and the University of Georgia (Athens). Readers should have an advanced undergraduate understanding of Fourier analysis and complex analysis in one variable.

Titles in this series are copublished with International Press, Cambridge, MA.

Contents

The Laguerre calculus
Estimates for powers of the sub-Laplacian
Estimates for the spectrum projection operators of the sub-Laplacian
The inverse of the operator $\square_{\alpha} = {\sum}^n_{j=1}({X^2_j} - {X^2_{j+n}}) - 2i{\alpha}$T
The explicit solution of the $\bar{\partial}$-Neumann problem in a non-isotropic Siegel domain
Injectivity of the Pompeiu transform in the isotropic H$_n$
Morera-type theorems for holomorphic $\mathcal H^p$ spaces in H$_n$ (I)
Morera-type theorems for holomorphic $\mathcal H^p$ spaces in H$_n$ (II)

Details:

Series: AMS/IP Studies in Advanced Mathematics, Volume: 22
Publication Year: 2001
ISBN: 0-8218-2761-8
Paging: approximately 328 pp.
Binding: Hardcover

Shigeyuki Morita, Tokyo Institute of Technology, Japan

Geometry of Characteristic Classes

Iwanami Series in Modern Mathematics
Expected publication date is May 24, 2001

Description

Characteristic classes are central to the modern study of the topology and geometry of manifolds. They were first introduced in topology, where, for instance, they could be used to define obstructions to the existence of certain fiber bundles. Characteristic classes were later defined (via the Chern-Weil theory) using connections on vector bundles, thus revealing their geometric side.

In the late 1960s new theories arose that described still finer structures. Examples of the so-called secondary characteristic classes came from Chern-Simons invariants, Gelfand-Fuks cohomology, and the characteristic classes of flat bundles. The new techniques are particularly useful for the study of fiber bundles whose structure groups are not finite dimensional.

The theory of characteristic classes of surface bundles is perhaps the most developed. Here the special geometry of surfaces allows one to connect this theory to the theory of moduli space of Riemann surfaces, i.e., Teichm?ller theory. In this book Morita presents an introduction to the modern theories of characteristic classes.

Contents

De Rham homotopy theory
Characteristic classes of flat bundles
Characteristic classes of foliations
Characteristic classes of surface bundles
Directions and problems for future research
Bibliography
Index

Details:

Series: Translations of Mathematical Monographs, Volume: 199
Publication Year: 2001
ISBN: 0-8218-2139-3
Paging: 180 pp.
Binding: Softcover

Edited by: Nathan Morrison

Foundations of the Theory of Probability

Description

Indispensable for anyone who wishes a thorough understanding of modern statistics, this basic tract develops probability theory on a postulational basis. The translation is edited by N. Morrison, with a Bibliography and notes by A. T. Bharucha-Reid.

Contents

Elementary theory of probability: 1.1 Axioms; 1.2 The relation to experimental data; 1.3 Notes on terminology; 1.4 Immediate corollaries of the axioms; conditional probabilities; Theorem of Bayes; 1.5 Independence; 1.6 Conditional probabilities as random variables; Markov chains Infinite probability fields: 2.1 Axiom of continuity; 2.2 Borel fields of probability; 2.3 Examples of infinite fields of probability Random variables: 3.1 Probability functions; 3.2 Definition of random variables and of distribution functions; 3.3 Multi-dimensional distribution functions; 3.4 Probabilities in infinite-dimensional spaces; 3.5 Equivalent random variables; various kinds of convergence Mathematical expectations: 4.1 Abstract Lebesgue integrals; 4.2 Absolute and conditional mathematical expectations; 4.3 The Tchebycheff inequality; 4.4 Some criteria for convergence; 4.5 Differentiation and integration of mathematical expectations with respect to a parameter Conditional probabilities and mathematical expectations: 5.1 Conditional probabilities; 5.2 Explanation of a Borel paradox; 5.3 Conditional probabilities with respect to a random variable; 5.4 Conditional mathematical expectations Independence; The law of large numbers: 6.1 Independence; 6.2 Independent random variables; 6.3 The law of large numbers; 6.4 Notes on the concept of mathematical expectation; 6.5 The strong law of large numbers; Convergence of a series Appendix--Zero-or-one law in the theory of probability
Bibliography
Notes to supplementary bibliography
Supplementary bibliography

Details:

Publisher: AMS Chelsea Publishing
Publication Year: 1950
Reprint/Revision History: reprinted 1956; first AMS printing 2000
ISBN: 0-8218-2648-4
Paging: 84 pp.
Binding: Hardcover

Steven G. Krantz, Washington University, St. Louis, MO

Function Theory of Several Complex Variables: Second Edition

Description

This work departs from earlier treatments of the subject by emphasizing integral formulas, the geometric theory of pseudoconvexity, estimates, partial differential equations, approximation theory, the boundary behavior of holomorphic functions, inner functions, invariant metrics, and mapping theory. While due homage is paid to the more traditional algebraic theory (sheaves, Cousin problems, etc.), the student with a background in real and complex variable theory, harmonic analysis, and differential equations will be most comfortable with this treatment.

It is currently the only book on the subject with exercises and a large number of examples.

Contents

An introduction to the subject
Some integral formulas
Subharmonicity and its applications
Convexity
H?rmander's solution of the $\bar\partial$ equation
Solution of the Levi problem and other applications of $\bar\partial$ techniques
Cousin problems, cohomology, and sheaves
The zero set of a holomorphic function
Some harmonic analysis
Constructive methods
Integral formulas for solutions to the $\bar\partial$ problem and norm estimates
Holomorphic mappings and invariant metrics
Manifolds
Area measures
Exterior algebra
Vectors, covectors, and differential forms
List of notation
Bibliography
Index

Details:

Series: AMS Chelsea Publishing
Publication Year: 2001
ISBN: 0-8218-2724-3
Paging: 564 pp.
Binding: Hardcover