American Mathematical Society

Advances in information recording.

Developed from a working group and workshop held at Rutgers U. in March 2004, these articles examine interconnections amongst coding theory, theoretical computer science, information theory and related areas of computer science and mathematics. The articles focus on the challenges of the next 50 years, particularly those in wireless communications, magnetic or optical storage, signal processing in networks, network information theory and the creation of a quantum information theory. Specific topics include modulation coding for a two-dimensional optical storage channel, characterization of heat-assisted magnetic recording channels, Cramer-Rao bound for timing recovery on channels with inter-symbol interference, the faithful communication of genetic information, an information theoretic approach to data storage and processing in cells, coding for optical communications, and macro-molecular data storage with such features as highly parallel read and write operations and three-dimensional storage capacity. (Annotation ©2008 Book News Inc. Portland, OR)

Banach spaces of analytic functions; proceedings.

Eleven papers from the April 2006 special session study functions mapping the open unit disk in the Bloch space, Hardy and Bergman spaces, a vanishing mean oscillation space, Cauchy transforms, and Fock spaces. Other topics include the inverse of an analytic mapping, pluripolarity of manifolds, a class of square-summable sequences, multipliers and composition operators, indestructible Blaschke products, and order bounded weighted composition operators. No index is provided. (Annotation ©2008 Book News Inc. Portland, OR)

Basic quadratic forms.

This rich branch of number theory is useful to group theory, topology, cryptography and coding theory. Gerstein (mathematics, U. of California at Santa Barbara) works from his graduate-level courses to develop his material from the theoretical and basic up to topics of current research. Using mainly concrete constructions, Gerstein gives a brief introduction to classical forms, then moves to quadratic spaces and lattices, valuations, local fields, p-adic numbers, quadratic spaces over Qp and over Q, lattices over principal ideal domains, initial integral results, the local-global approach to lattices, and applications to cryptography. He provides exercises for each concept along with a full list of further reading. Readers should have a basic background in linear and abstract algebra. (Annotation ©2008 Book News Inc. Portland, OR)

The Beltrami equation.

This book gives an account of the measurable Riemann mapping theorem, explains how it relates to the existence and uniqueness theory of the planar Beltrami equations, and explores the various properties of the solutions to this equation. After background on the classical theory, the theory is developed in a more general framework of mappings of finite distortion and the associated degenerate elliptic equations, drawing on recent advances in non-linear harmonic analysis, Sobolev theory, and geometric theory. Emphasis is on the concept of a principal solution and its fundamental role in understanding the natural domain of definition of a given Beltrami operator. The results shed new light on the theory of planar quasiconformal mappings and have the potential for wide applications, some of which are discussed. Author information is not given, and there is no subject index. (Annotation ©2008 Book News Inc. Portland, OR)

C*-algebras and finite-dimensional approximations.

Built from lectures and other work on approximation theory in the context of operator algebras, this text explains most of the numerous types of approximation properties important to recent and current research and provides proofs of fundamental results only previously available scattered across the literature. After an introduction of fundamental facts (including material on von Neumann algebras and Aarveson's extension theorem), this covers basic theory, such as nuclear and exact C*-algebras, tensor products, constructions, exact groups, amenable traces and Kirchberg's factorization property, quasidiagonal C&-algebras, AF imbeddability, and local reflexivity. Special topics include simple C*-algebras, approximation properties for groups, the weak expectation property and local lifting property, weakly exact von Neumann algebras, and such applications as Herrero's approximation problem and classification of von Neumann algebras. (Annotation ©2008 Book News Inc. Portland, OR)

Chinese mathematicians; proceedings; 2v.

With dozens of lectures, expository and survey articles, these proceedings reflect the international interest in both pure and applied mathematics. General topics include algebra, number theory, cryptography, algebraic geometry, algebraic topology, geometric analysis, complex analysis, complex geometry, harmonic analysis, functional analysis, applied mathematics, dynamical systems, fractals, wavelets, numerical analysis, partial differential equations, probability, statistics, combinatorics, numerical analysis and scientific computing, applications of mathematics in the sciences, financial mathematics, control theory of optimization, ordinary differential equations, and education. Specific lecture topics include shock waves and cosmology, number theory, and analysis. Each entry contains its own references and there are no general appendices or indices. (Annotation ©2008 Book News Inc. Portland, OR)

Classifying spaces of sporadic groups.

Intended for non-experts, this begins with background material on the relevant constructions from algebraic topology and on local geometries from group theory, reviewing aspects of group cohomology, simplicial sets and their equivalence with topological space, Bousfield-Kan completions and homotopy colimits, decompositions and p-subgroups, and 2-local geometries for simple groups. Working on each of 26 sporadic finite simple groups, this also constructs a 2-completed classifying space using a homotopy decomposition with classifying spaces of suitable 2-local subgroups, with decompositions for individual sporadic Mathieu, Janki, Higman-Sims, McLaughlin. Suzuki, Conway, Fischer, Harada-Norton, Thompson, Baby Monster, Fischer-Griess, Held, Rudvalis, O'Nan and Lyons groups, and details of proofs for individual groups. The result is remarkably accessible at explaining sporadic groups and also is successful at working with the cohomology of the other simple groups. (Annotation ©2008 Book News Inc. Portland, OR)

Complex analysis and dynamical systems III; a conference in honor of the retirement of Professors Dov Aharonov, Lev Aizenberg, Samuel Krushkal, and Uri Srebro.

These proceedings of the January 2006 conference include 31 papers covering a wide range of topics in the geometric theory of functions of one and several complex variables, including univalent functions, conformal and quasi-conformal mappings, minimal surfaces and dynamics in infinite-dimensional spaces along with other related subjects. Specific topics include stable expansions in homogeneous polynomials, Schottky's theorem on comformal mappings between annuli, Univalent convex functions in the positive direction of the real axis, controlled approximation for some classes of homomorphic functions, connected complex orbits, the QC Reimann mapping theorem in space, analytic properties of Besov spaces via Bergman projections, analytic functions in algebras, quadratic forms in geometric function theory (including quasi-conformal extensions and Fredhom eigenvalues), the Cauchy problem of couple-stress elasticity, mappings associated with weighted Sobolev spaces, the parabolicity of minimal groups, and harmonic polynomial interpolation. Articles include references. (Annotation ©2008 Book News Inc. Portland, OR)

Comparison theorems in Riemannian geometry, 2d ed.

Cheeger and Ebin take a sensible approach to this new edition of their mid-1970s original: the enormous growth in this field has made attempting to upgrade their first edition unmanageable. Instead, they offer updated and expanded references and a sincere conviction readers can still use significant parts of their findings. They begin with basic concepts and results, such as the exponential map and normal coordinates, the Hopf-Rinow theorem, Jacobi fields, basic index lemmas, the Rauch comparison theorem and theorems to which Cartan was involved. They then move briskly through Toponogov's theorem, homogeneous spaces, Morse theory, closed deodesics and the cut locus, the sphere theorem and its generalizations, the differentiable space theorem, complete manifolds of non-negative curvature, and compact manifolds of non-positive curvature. (Annotation ©2008 Book News Inc. Portland, OR)

A course on the Web graph.

This rigorous mathematical study of web graphs and graph theory as it pertains to the Internet is the first of its kind, and introduces the world to the concept of Internet mathematics. Bonato (Dalhousie U.) developed this textbook from the groundbreaking graduate course he has taught for the last few years, and shows how mathematical models can be applied to such web-based entities as search engines, viruses and networks. This book is the latest entry in the American Mathematical Society's Graduate Studies in Mathematics series. (Annotation ©2008 Book News Inc. Portland, OR)

Data mining and mathematical programming.

Pardalos (U. of Florida) and Hansen (HEC Montreal) have edited this collection of mathematical programming concepts from contributors all over the world on the relatively new subject of data mining, or extracting useful and specific information from large databases. Based upon lectures presented at the "Data Mining and Mathematical Programming" workshop held in Montreal in October 2006, this technically dense resource is designed primarily for research scientists in the field of optimization, data analysis and applied mathematics, as well as advanced engineering students. (Annotation ©2008 Book News Inc. Portland, OR)

Elementary geometry.

Working from their one-semester lecture course for future teachers at Humboldt-U. in Berlin, Agricola and Friedrich give a solid background while also providing resources for teachers to use in the classroom and ideas for further research by students. They cover the basics of Euclidean space, elementary geometrical figures and their properties, symmetries of the plane and of space (including affine mappings and centroids, projections, central dilations and translations, plane transformations and discrete and finite subgroups), hyperbolic geometry (including the Poincaré and disc models) and spherical geometry. The authors include exercises for each chapter and a concise bibliography. Readers should have completed a first-year course in linear algebra and calculus. (Annotation ©2008 Book News Inc. Portland, OR)

Finsler geometry; proceedings.

These proceedings of the September 2005 symposium reflect the respect participants felt for their late colleague Makato Matsumoto and his work in Finsler geometry. Contributors address such topics as two curvature-driven problems in Riemann-Finsler geometry, curvature properties of certain metrics, a connectiveness principle in positively curved Finsler manifolds, Riemann-Finsler surfaces, Finsler geometry in the tangent bundle, and topics in Finsler-inspired differential geometry such as perturbations of constant connection Wagner spaces, path geometries of almost-Grassmann structures, Ehresmann connections in relation to metrics and good metric derivatives and dynamical systems of the Lagrangian and Hamiltonian mechanical systems. The collection closes with topics on complex Finsler geometry, including a survey and a paper on the Chern-Finsler connection with Finsler-Kähler manifolds. Contributors include a charming recount of the life and work of Professor Matsumoto. (Annotation ©2008 Book News Inc. Portland, OR)

Frames and operator theory in analysis and signal processing; proceedings.

This work contains articles based on talks presented at a January 2006 meeting held in San Antonio, Texas. One of the goals of the meeting was to integrate both the industrial and theoretical aspects of frames and operator theory, in particular their applications to image and signal processing. In addition, the meeting dealt with work related to sampling and numerical solutions of partial differential equations. Some specific topics examined include classes of finite equal norm parseval frames, short-time Fourier transform analysis of localization operators, and operator theory and modulation spaces. The readership for the book includes pure mathematicians working on the foundations of frame and operator theory, as well as applied mathematicians investigating applications. The book will also be of interest to physicists and engineers employing these designs. (Annotation ©2008 Book News Inc. Portland, OR)

Group representations, ergodic theory, and mathematical physics; a tribute to George W. Mackey; proceedings.

This book is based on lectures presented at an American Mathematical Society special session held in January 2007 in New Orleans, dedicated to the memory of George Mackey, who made contributions to representation theory, harmonic analysis, ergodic theory, and mathematical physics. The papers, written especially for this book by internationally known mathematicians and mathematical physicists, range from expository and historical surveys to original high-level research articles. Topics examined include recent results on induced representations, virtual groups, the Mackey Machine and cross products, representations of Baumslag-Solitar groups, the Radon transform and the heat equation, groupoids in the study of wavelets, and quantum theory. There is no subject index. (Annotation ©2008 Book News Inc. Portland, OR)

Hardy spaces and potential theory on C¹ domains in Riemannian manifolds.

This work studies Hardy spaces on C¹ and Lipschitz domains in Riemannian manifolds. It provides an equivalence theorem proving that the definition of Hardy spaces by conjugate harmonic functions is equivalent to the atomic definition of these spaces. The theorem is established in any dimension of the domain of C¹. The material required to develop an approach for C¹ domains on Riemannian manifolds draws on earlier work by Fabes, Jodeit, and Rivière and recent results by Mitrea and Taylor. The first part of the book is of interest in itself, as it considers the boundary value problems for the Laplace-Beltrami operator. Author information is not given, and there is no subject index. (Annotation ©2008 Book News Inc. Portland, OR)

Higher arithmetic; an algorithmic introduction to number theory.

Following Gauss in his first systematic treatise on it in 1801, Edwards (emeritus, mathematics, New York U.) prefers higher arithmetic to number theory as the name for the general study of specific relations among whole numbers. Theory is about thinking, he explains, and arithmetic is about doing, and he found in his courses that students would rather do calculations than listen to him talk about them. Therefore, he includes many exercises in this introduction for readers who do not necessarily have a deep background in mathematics. (Annotation ©2008 Book News Inc. Portland, OR)

Integer points in polyhedra — geometry, number theory, representation theory, algebra, optimization, statistics; proceedings.

Research and survey articles from a June 2006 conference highlight recent advances related to lattice-point questions. Some specific topics explored include lattice reformulation of integer programming problems, Ehrhart polynomial roots and Stanley's non-negativity theorem, enumeration of integer solutions to linear inequalities defined by digraphs, and perfect Delaunay polytopes and perfect quadratic functions on lattices. The book is suitable for researchers and graduate students interested in combinatorial aspects of commutative algebra, optimization, discrete geometry, statistics, mirror symmetry, and geometry of numbers. (Annotation ©2008 Book News Inc. Portland, OR)

Poisson geometry in mathematics and physics; proceedings.

Sixteen papers from the June 2006 conference present new results in normal forms of Poisson structures, deformation of Poisson structures, reduction of systems with symmetry, Kontsevich formality and its variants, and quantization of canonical transformations via their graphs. Topics include orbifold cohomology of abelian symplectic reductions, generalized Kähler and hyper-Kähler quotients, pure spinors and moment maps, and locally noncommutative space-times. No index is provided. (Annotation ©2008 Book News Inc. Portland, OR)

Positive polynomials and sums of squares.

Developed from lectures given by Marshall at the U. of Saskatchewan and elsewhere, this provides an elementary introduction and includes such subjects as the moment problem and applications to polynomial optimization. Marshall addresses Krivine's Positivstellensatz, the non-compact case, Archimedean T-modules, Schmüdgen's Positivstellensatz, Putinar's question, the weak isotropy of quadratic forms, Scheiderer's local-global principle, and semidefinite programming and operation. Appendices cover such background material as the Tarksi-Seidenberg theorem and algebraic sets. Designed for students at the beginning graduate level, this concentrates on concrete objects, such as polynomials in n variables with real coefficients, and Marshall includes plenty of examples and new, simple proofs. He also provides a very useful bibliography for further study. (Annotation ©2008 Book News Inc. Portland, OR)