S. R. S. Varadhan, New York University - Courant Institute of Mathematical Sciences, NY
Probability Theory
Description
This volume presents topics in probability theory covered during a first-year graduate course given at the Courant Institute of Mathematical Sciences. The necessary background material in measure theory is developed, including the standard topics, such as extension theorem, construction of measures, integration, product spaces, Radon-Nikodym theorem, and conditional expectation.
In the first part of the book, characteristic functions are introduced, followed by the study of weak convergence of probability distributions. Then both the weak and strong limit theorems for sums of independent random variables are proved, including the weak and strong laws of large numbers, central limit theorems, laws of the iterated logarithm, and the Kolmogorov three series theorem. The first part concludes with infinitely divisible distributions and limit theorems for sums of uniformly infinitesimal independent random variables.
The second part of the book mainly deals with dependent random variables, particularly martingales and Markov chains. Topics include standard results regarding discrete parameter martingales and Doob's inequalities. The standard topics in Markov chains are treated, i.e., transience, and null and positive recurrence. A varied collection of examples is given to demonstrate the connection between martingales and Markov chains.
Additional topics covered in the book include stationary Gaussian processes, ergodic theorems, dynamic programming, optimal stopping, and filtering. A large number of examples and exercises is included. The book is a suitable text for a first-year graduate course in probability.
Contents
Measure theory
Weak convergence
Independent sums
Dependent random variables
Martingales
Stationary stochastic processes
Dynamic programming and filtering
Bibliography
Index
Details:
Publisher: American Mathematical Society
Distributor: American Mathematical Society
Series: Courant Lecture Notes, Volume: 7
Publication Year: 2001
ISBN: 0-8218-2852-5
Paging: 167 pp.
Binding: Softcover
Larry C. Grove, University of Arizona, Tucson, AZ
Classical Groups and Geometric Algebra
Expected publication date is October 31, 2001
Description
"Classical groups", named so by Hermann Weyl, are groups of matrices or quotients of matrix groups by small normal subgroups.
Thus the story begins, as Weyl suggested, with "Her All-embracing Majesty", the general linear group $GL_n(V)$ of all invertible linear transformations of a vector space $V$ over a field $F$. All further groups discussed are either subgroups of $GL_n(V)$ or closely related quotient groups.
Most of the classical groups consist of invertible linear transformations that respect a bilinear form having some geometric significance, e.g., a quadratic form, a symplectic form, etc. Accordingly, the author develops the required geometric notions, albeit from an algebraic point of view, as the end results should apply to vector spaces over more-or-less arbitrary fields, finite or infinite.
The classical groups have proved to be important in a wide variety of venues, ranging from physics to geometry and far beyond. In recent years, they have played a prominent role in the classification of the finite simple groups.
This text provides a single source for the basic facts about the classical groups and also includes the required geometrical background information from the first principles. It is intended for graduate students who have completed standard courses in linear algebra and abstract algebra. The author, L. C. Grove, is a well-known expert who has published extensively in the subject area.
Contents
Permutation actions
The basic linear groups
Bilinear forms
Symplectic groups
Symmetric forms and quadratic forms
Orthogonal geometry (char $F\not= 2$)
Orthogonal groups (char $F \not= 2$), I
$O(V)$, $V$ Euclidean
Clifford algebras (char $F \not = 2$)
Orthogonal groups (char $F \not = 2$), II
Hermitian forms and unitary spaces
Unitary groups
Orthogonal geometry (char $F = 2$)
Clifford algebras (char $F = 2$)
Orthogonal groups (char $F = 2$)
Further developments
Bibliography
List of notation
Index
Details:
Publisher: American Mathematical Society
Distributor: American Mathematical Society
Series: Graduate Studies in Mathematics Volume: 39
Publication Year: 2002
ISBN: 0-8218-2019-2
Paging: 169 pp.
Binding: Hardcover
Kenichi Ohshika, Osaka University, Japan
Discrete Groups
Expected publication date is November 18, 2001
Description
This book deals with geometric and topological aspects of discrete groups. The main topics are hyperbolic groups due to Gromov, automatic group theory, invented and developed by Epstein, whose subjects are groups that can be manipulated by computers, and Kleinian group theory, which enjoys the longest tradition and the richest contents within the theory of discrete subgroups of Lie groups.
What is common among these three classes of groups is that when seen as geometric objects, they have the properties of a negatively curved space rather than a positively curved space. As Kleinian groups are groups acting on a hyperbolic space of constant negative curvature, the technique employed to study them is that of hyperbolic manifolds, typical examples of negatively curved manifolds. Although hyperbolic groups in the sense of Gromov are much more general objects than Kleinian groups, one can apply for them arguments and techniques that are quite similar to those used for Kleinian groups. Automatic groups are further general objects, including groups having properties of spaces of curvature 0. Still, relationships between automatic groups and hyperbolic groups are examined here using ideas inspired by the study of hyperbolic manifolds. In all of these three topics, there is a "soul" of negative curvature upholding the theory. The volume would make a fine textbook for a graduate-level course in discrete groups.
Contents
Basic notions for infinite group
Hyperbolic groups
Automatic groups
Kleinian groups
Prospects
Bibliography
Index
Details:
Publisher: American Mathematical Society
Distributor: American Mathematical Society
Series: Translations of Mathematical Monographs,
Subseries: Iwanami Series in Modern Mathematics
Publication Year: 2002
ISBN: 0-8218-2080-X
Paging: approximately 207 pp.
Binding: Softcover
A. M. Vinogradov, University of Salerno, Baronossi (SA), Italy
Cohomological Analysis of Partial Differential Equations
and Secondary Calculus
Expected publication date is November 9, 2001
Description
This book is dedicated to fundamentals of a new theory, which is an analog of affine algebraic geometry for (nonlinear) partial differential equations. This theory grew up from the classical geometry of PDE's originated by S. Lie and his followers by incorporating some nonclassical ideas from the theory of integrable systems, the formal theory of PDE's in its modern cohomological form given by D. Spencer and H. Goldschmidt and differential calculus over commutative algebras (Primary Calculus). The main result of this synthesis is Secondary Calculus on diffieties, new geometrical objects which are analogs of algebraic varieties in the context of (nonlinear) PDE's.
Secondary Calculus surprisingly reveals a deep cohomological nature of the general theory of PDE's and indicates new directions of its further progress. Recent developments in quantum field theory showed Secondary Calculus to be its natural language, promising a nonperturbative formulation of the theory.
In addition to PDE's themselves, the author describes existing and potential applications of Secondary Calculus ranging from algebraic geometry to field theory, classical and quantum, including areas such as characteristic classes, differential invariants, theory of geometric structures, variational calculus, control theory, etc. This book, focused mainly on theoretical aspects, forms a natural dipole with Symmetries and Conservation Laws for Differential Equations of Mathematical Physics, Volume 182 in this same series, Translations of Mathematical Monographs, and shows the theory "in action".
Contents
From symmetries of partial differential equations to Secondary Calculus
Elements of differential calculus in commutative algebras
Geometry of finite-order contact structures and the classical theory of symmetries of partial differential equations
Geometry of infinitely prolonged differential equations and higher symmetries
$\mathcal{C}$-spectral sequence and some applications
Introduction to Secondary Calculus
Bibliography
Index
Details:
Publisher: American Mathematical Society
Distributor: American Mathematical Society
Series: Translations of Mathematical Monographs, Volume: 204
Publication Year: 2001
ISBN: 0-8218-2922-X
Paging: approximately 264 pp.
Binding: Hardcover
Edited by: Gaston M. N'Guerekata and Asamoah Nkwanta,
Morgan State University, Baltimore, MD
Council for African American Researchers in the Mathematical Sciences: Volume IV
Expected publication date is October 25, 2001
Description
This volume contains selected papers from the Sixth Conference for African American Researchers in the Mathematical Sciences (CAARMS), held at Morgan State University in Baltimore (MD). The CAARMS organizes this annual conference showcasing the current research primarily, but not exclusively, of African Americans in the mathematical sciences. Since the first conference in 1995, significant numbers of researchers have presented their current work in technical talks, and graduate students have presented their work in organized poster sessions.
Research topics include mathematics (number theory, analysis, topology, differential equations, algebra, combinatorics, etc.), mathematical physics, mathematical biology, operations research, probability and statistics, and computer science. In addition to the invited talks, tutorials and group discussions on various topics are organized to stimulate, nurture, and encourage increased participation by African Americans and other underrepresented groups in the mathematical sciences. These events create an ideal forum for mentoring and networking where attendees can meet researchers and graduate students who are interested in the same fields.
For volumes based on previous CAARMS proceedings, see African Americans in Mathematics, Volume 34, in the AMS Series in Discrete Mathematics and Theoretical Computer Science, African Americans in Mathematics II, Volume 252, and Council for African American Researchers in the Mathematical Sciences: Volume III, Volume 275, in the AMS series, Contemporary Mathematics.
Contents
Research articles
K. M. Lewis -- Hyponormality and a family of Toeplitz operators on the Bergman space
E. Goins -- A ternary algebra with applications to binary quadratic forms
I. Assani -- Spectral characterization of ergodic dynamical systems
C. Castillo-Chavez and A.-A. Yakubu -- Epidemics on attractors
K. F. Sellers -- A definition of vague coherent systems
M. C. Jackson -- Spatial data analysis for discrete data on a lattice
C. R. Handy -- New perspectives in moment-wavelet analysis from quantum operator theory: Scalets and local quantization
Historical articles
J. L. Houston -- Numbers that count-Persons who impact, mathematically!
Details:
Publisher: American Mathematical Society
Distributor: American Mathematical Society
Series: Contemporary Mathematics, Volume: 284
Publication Year: 2002
ISBN: 0-8218-2793-6
Paging: 135 pp.
Binding: Softcover
Edited by: Bruce C. Berndt, University of Illinois, Urbana-Champaign, IL,
and Robert A. Rankin, University of Glasgow, Scotland
Ramanujan: Essays and Surveys
Expected publication date is November 14, 2001
Description
This book contains essays on Ramanujan and his work that were written especially for this volume. It also includes important survey articles in areas influenced by Ramanujan's mathematics. Most of the articles in the book are nontechnical, but even those that are more technical contain substantial sections that will engage the general reader.
The book opens with the only four existing photographs of Ramanujan, presenting historical accounts of them and information about other people in the photos. This section includes an account of a cryptic family history written by his younger brother, S. Lakshmi Narasimhan. Following are articles on Ramanujan's illness by R. A. Rankin, the British physician D. A. B. Young, and Nobel laureate S. Chandrasekhar. They present a study of his symptoms, a convincing diagnosis of the cause of his death, and a thorough exposition of Ramanujan's life as a patient in English sanitariums and nursing homes.
Following this are biographies of S. Janaki (Mrs. Ramanujan) and S. Narayana Iyer, Chief Accountant of the Madras Port Trust Office, who first communicated Ramanujan's work to the Journal of the Indian Mathematical Society. The last half of the book begins with a section on "Ramanujan's Manuscripts and Notebooks". Included is an important article by G. E. Andrews on Ramanujan's lost notebook.
The final two sections feature both nontechnical articles, such as Jonathan and Peter Borwein's "Ramanujan and pi", and more technical articles by Freeman Dyson, Atle Selberg, Richard Askey, and G. N. Watson.
This volume complements the book Ramanujan: Letters and Commentary, Volume 9, in the AMS series, History of Mathematics. For more on Ramanujan, see these AMS publications Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work, Volume 136.H, and Collected Papers of Srinivasa Ramanujan, Volume 159.H, in the AMS Chelsea Publishing series.
Copublished with the London Mathematical Society. Members of the LMS may order directly from the AMS at the AMS member price. The LMS is registered with the Charity Commissioners.
Contents
・R. A. Rankin -- Commentary (by R. A. R.)
The life of Ramanujan
・The four photographs of Ramanujan
・The books studied by Ramanujan in India
・The influence of Carr's synopsis on Ramanujan
・The notebooks of Srinivasa Ramanujan
・A recently discovered letter giving Ramanujan's examination scores On Ramanujan
・The Ramanujan family record
Ramanujan's illness
・Ramanujan as a patient
・Ramanujan's illness
・An incident in the life of S. Ramanujan, F.R.S.: Conversations with G. H. Hardy, F.R.S. and J. E. Littlewood, F.R.S. and their sequel
S. Janaki
・S. Janaki Ammal (Mrs. Ramanujan)
・Conversation "I didn't understand his work, but I knew his worth"
S. Narayana Iyer
・A short biography of S. Narayana Iyer
・The distribution of primes
・Some theorems in summation
E. H. Neville
・Srinivasa Ramanujan
・University lectures in Madras
Ramanujan's manuscripts and notebooks
・Ramanujan's manuscripts and notebooks
・Ramanujan's manuscripts and notebooks, II
・An overview of Ramanujan's notebooks
・An introduction to Ramanujan's "lost" notebook
Nontechnical articles on Ramanujan's work
・Ramanujan and pi
・$\pi$ related developments since 1988
・Reflections around the Ramanujan centenary
・The problems submitted by Ramanujan to the Journal of the Indian Mathematical Society
Somewhat more technical articles on Ramanujan's work
・A walk through Ramanujan's garden
・Ramanujan and hypergeometric and basic hypergeometric series
・The final problem: An account of the mock theta functions
Details:
Publisher: American Mathematical Society, London Mathematical Society
Distributor: American Mathematical Society
Series: History of Mathematics
Publication Year: 2001
ISBN: 0-8218-2624-7
Paging: 347 pp.
Binding: Hardcover