Graduate Studies in Mathematics, Volume: 116
2010; 330 pp; hardcover
ISBN-13: 978-0-8218-5115-9
Expected publication date is July 16, 2010.
Functional analysis studies the algebraic, geometric, and topological structures of spaces and operators that underlie many classical problems. Individual functions satisfying specific equations are replaced by classes of functions and transforms that are determined by the particular problems at hand.
This book presents the basic facts of linear functional analysis as related to fundamental aspects of mathematical analysis and their applications. The exposition avoids unnecessary terminology and generality and focuses on showing how the knowledge of these structures clarifies what is essential in analytic problems.
The material in the first part of the book can be used for an introductory course on functional analysis, with an emphasis on the role of duality. The second part introduces distributions and Sobolev spaces and their applications. Convolution and the Fourier transform are shown to be useful tools for the study of partial differential equations. Fundamental solutions and Green's functions are considered and the theory is illustrated with several applications. In the last chapters, the Gelfand transform for Banach algebras is used to present the spectral theory of bounded and unbounded operators, which is then used in an introduction to the basic axioms of quantum mechanics.
The presentation is intended to be accessible to readers whose backgrounds include basic linear algebra, integration theory, and general topology. Almost 240 exercises will help the reader in better understanding the concepts employed.
Graduate students interested in functional analysis, PDEs, analysis.
Introduction
Normed spaces and operators
Frechet spaces and Banach theorems
Duality
Weak topologies
Distributions
Fourier transform and Sobolev spaces
Banach algebras
Unbounded operators in a Hilbert space
Hints to exercises
Bibliography
Index
Mathematical Surveys and Monographs, Volume: 164
2010; approx. 501 pp; hardcover
ISBN-13: 978-0-8218-4993-4
Expected publication date is August 2, 2010.
This book provides the reader with the principal concepts and results related to differential properties of measures on infinite dimensional spaces. In the finite dimensional case such properties are described in terms of densities of measures with respect to Lebesgue measure. In the infinite dimensional case new phenomena arise. For the first time a detailed account is given of the theory of differentiable measures, initiated by S. V. Fomin in the 1960s; since then the method has found many various important applications. Differentiable properties are described for diverse concrete classes of measures arising in applications, for example, Gaussian, convex, stable, Gibbsian, and for distributions of random processes. Sobolev classes for measures on finite and infinite dimensional spaces are discussed in detail. Finally, we present the main ideas and results of the Malliavin calculus--a powerful method to study smoothness properties of the distributions of nonlinear functionals on infinite dimensional spaces with measures.
The target readership includes mathematicians and physicists whose research is related to measures on infinite dimensional spaces, distributions of random processes, and differential equations in infinite dimensional spaces. The book includes an extensive bibliography on the subject.
Graduate students and research mathematicians interested in measure theory and random processes.
Background material
Sobolev spaces on mathbb{R}^n
Differentiable measures on linear spaces
Some classes of differentiable measures
Subspaces of differentiability of measures
Integration by parts and logarithmic derivatives
Logarithmic gradients
Sobolev classes on infinite dimensional spaces
The Malliavin calculus
Infinite dimensional transformations
Measures on manifolds
Applications
References
Subject index
Translations of Mathematical Monographs, Volume: 238
2010; approx. 270 pp; hardcover
ISBN-13: 978-0-8218-4909-5
Expected publication date is July 9, 2010.
This book provides exposition of the basic theory of quantum bounded symmetric domains. The area became active in the late 1990s at a junction of noncommutative complex analysis and extensively developing theory of quantum groups. It is well known that the classical bounded symmetric domains involve a large number of nice constructions and results of the theory of C^*-algebras, theory of functions and functional analysis, representation theory of real reductive Lie groups, harmonic analysis, and special functions. In a surprising advance of the theory of quantum bounded symmetric domains, it turned out that many classical problems admit elegant quantum analogs. Some of those are expounded in the book. Anyone with an interest in the subject will welcome this unique treatment of quantum groups.
Graduate students and research mathematicians interested in representation theory and noncommutative geometry.
Quantum disc
Basic quantum theory of bounded symmetric domains
Conclusion
Bibliography
Index
Contemporary Mathematics, Volume: 515
2010; approx. 152 pp; softcover
ISBN-13: 978-0-8218-4673-5
Expected publication date is July 10, 2010.
This volume contains survey papers on mathematical finance based on some courses given at the "Lluis Santalo" Summer School of the Real Sociedad Matematica Espanola, held in July 2007 at the Universidad Internacional Menendez Pelayo, Santander Spain). The primary topics are pathwise approximations of stochastic differential equations, hedge funds, and credit derivatives.
The paper by L. Seco and F. Chen provides a systematic survey of hedge funds from a rigorous mathematical point of view. The related paper by M. Escobar, S. Kramer, F. Scheibl, L. Seco and R. Zagst, introduces a new theoretical framework for the pricing of hedge funds' equity, inspired by the framework of Black and Cox for the valuation of company equity as a call option.
A general framework for deriving high order, stable and tractable path-wise approximations of Stratonovich stochastic differential equations as applied to finance is the subject of the paper by L.G. Gyurko and T. Lyons.
The paper by R. Zagst and M. Scherer is a short course on the different approaches used for pricing, hedging and risk management of credit derivatives.
Researchers and practitioners in mathematical finance will find in this book a collection of excellent, up-to-date and mathematically rigorous presentations of some of the most advanced techniques for pricing and risk management.
Graduate students and research mathematicians interested in mathematical finance.
M. Escobar, S. Kramer, F. Scheibl, L.A. Seco, and R. Zagst -- Hedge funds as knock-out options
L. G. Gyurko and T. Lyons -- Rough paths based numerical algorithms in computational finance
L. A. Seco and F. Chen -- Hedge funds
R. Zagst and M. Scherer -- Modeling and pricing credit derivatives
Contemporary Mathematics, Volume: 516
2010; 345 pp; softcover
ISBN-13: 978-0-8218-4891-3
Expected publication date is July 11, 2010.
This volume is based on lectures presented at the AMS Special Session on Algebraic Methods in Statistics and Probability--held March 27-29, 2009, at the University of Illinois at Urbana-Champaign--and on contributed articles solicited for this volume.
A decade after the publication of Contemporary Mathematics Vol. 287, the present volume demonstrates the consolidation of important areas, such as algebraic statistics, computational commutative algebra, and deeper aspects of graphical models.
In statistics, this volume includes, among others, new results and applications in cubic regression models for mixture experiments, multidimensional Fourier regression experiments, polynomial characterizations of weakly invariant designs, toric and mixture models for the diagonal-effect in two-way contingency tables, topological methods for multivariate statistics, structural results for the Dirichlet distributions, inequalities for partial regression coefficients, graphical models for binary random variables, conditional independence and its relation to sub-determinants covariance matrices, connectivity of binary tables, kernel smoothing methods for partially ranked data, Fourier analysis over the dihedral groups, properties of square non-symmetric matrices, and Wishart distributions over symmetric cones.
In probability, this volume includes new results related to discrete-time semi Markov processes, weak convergence of convolution products in semigroups, Markov bases for directed random graph models, functional analysis in Hardy spaces, and the Hewitt-Savage zero-one law.
Graduate students and research mathematicians interested in applied mathematics, statistics, probability, and algebra.
S. A. Andersson and T. Klein -- Kiefer-complete classes of designs for cubic mixture models
V. S. Barbu and N. Limnios -- Some algebraic methods in semi-Markov chains
R. A. Bates, H. Maruri-Aguilar, E. Riccomagno, R. Schwabe, and H. P. Wynn -- Self-avoiding generating sequences for Fourier lattice designs
F. Bertrand -- Weakly invariant designs, rotatable designs and polynomial designs
C. Bocci, E. Carlini, and F. Rapallo -- Geometry of diagonal-effect models for contingency tables
P. Bubenik, G. Carlsson, P. T. Kim, and Z.-M. Luo -- Statistical topology via Morse theory persistence and nonparametric estimation
G. Budzban and G. Hognas -- Convolution products of probability measures on a compact semigroup with applications to random measures
S. Chakraborty and A. Mukherjea -- Completely simple semigroups of real dtimes d matrices and recurrent random walks
W.-Y. Chang, R. D. Gupta, and D. S. P. Richards -- Structural properties of the generalized Dirichlet distributions
S. Chaudhuri and G. L. Tan -- On qualitative comparison of partial regression coefficients for Gaussian graphical Markov models
M. A. Cueto, J. Morton, and B. Sturmfels -- Geometry of the restricted Boltzmann machine
M. Drton and H. Xiao -- Smoothness of Gaussian conditional independence models
W. Ehm -- Projections on invariant subspaces
S. M. Evans -- A zero-one law for linear transformations of Levy noise
H. Hara and A. Takemura -- Connecting tables with zero-one entries by a subset of a Markov basis
K. Khare and B. Rajaratnam -- Covariance trees and Wishart distributions on cones
P. Kidwell and G. Lebanon -- A kernel smoothing approach to censored preference data
M. S. Massa and S. L. Lauritzen -- Combining statistical models
S. Petrovi, A. Rinaldo, and S. E. Fienberg -- Algebraic statistics for
a directed random graph model with reciprocation
G. Pistone and M. P. Rogantin -- Regular fractions and indicator polynomials
M. A. G. Viana -- Dihedral Fourier analysis
T. von Rosen and D. Von Rosen -- On a class of singular nonsymmetric matrices with nonnegative integer spectra
A. S. Yasamin -- Some hypothesis tests for Wishart models on symmetric cones