American Mathematical Society

Advances in non-Archimedean analysis; proceedings.

This collection contains 19 research articles based on 30-minute talks delivered during the July 2010 conference on functional analysis over non-Archimedean valued complete fields. A professor from Pierre and Marie Curie University presents an algorithm for computing the radius of convergence function for first order differential equations, and a professor from the University of North Texas proves the existence of greatest common divisors and factorization in rings of non-Archimedean entire functions. Other topics include primitives of p-adic meromorphic functions, the Lipschitz condition for rational functions on ultrametric valued functions, the geometry of p-adic fractal strings, identities and congruencies for Genocchi numbers, and ultrametric calculus in field K. No index is provided. (Annotation ©2011 Book News Inc. Portland, OR)

Affine algebraic geometry; the Russell festschrift.

In June 2009, mathematicians from 10 countries gathered at McGill University in Canada to celebrate Peter Russell's retirement from there and his 40 years of contributions to the relatively new field of algebraic geometry. Of the 25 talks delivered, 19 are presented here. Their topics include Newton trees at infinity of algebraic curves, group actions on affine cones, holomorphic curves on irregular varieties of general type starting from surfaces, equivariant derivations and additive group actions, and embeddings of hyperbolas. They are not indexed. (Annotation ©2011 Book News Inc. Portland, OR)

Algebra.

Sepanski offers an introductory undergraduate algebra text beginning with integers and modular arithmetic and following with group theory, ring theory, and field theory. The final section of the text deals with Sylow's theorems and Galois theory. Included are more than 750 exercises in a range of skill levels including some complex problems rarely seen in first course texts. (Annotation ©2011 Book News Inc. Portland, OR)

Arithmetic of L-functions.

The 40 lectures gathered in this collection are from the 2009 iteration of the annual three-week graduate summer institute, each focusing on a different aspect of mathematics. They cover Stark's conjecture, the Birch and Swinnerton-Dyer conjecture, and analytic and cohomological methods. Among specific topics are Harold Stark revealing where his conjectures came from, special values of L-functions at negative integers, introduction to elliptic curves, complex multiplication, root numbers, and Euler systems and Kolyvagin systems. No index is provided. (Annotation ©2011 Book News Inc. Portland, OR)

Asymptotic analysis for periodic structures. (reprint 1978)

Since this treatise was first published, in 1978 by North-Holland Publishing Company, many books have appeared on homogenization, or the theory of partial differential equations with rapidly oscillating coefficients. Nevertheless, Bensoussan and G. Papanicolaou decided to reprint it with minor corrections and updated bibliography partly in memory of the third author J.-L. Lions (1925-2007), but also because it can still be useful to those learning homogenization. (Annotation ©2011 Book News Inc. Portland, OR)

Commutative algebra and its connections to geometry; proceedings.

The conference shared findings from the well developed Brazilian study in commutative algebra with young mathematicians in the outlying parts of Brazil and elsewhere in Latin American. The 15 papers included consider such topics as the Plücker-Clebsch formula in higher dimension, polynomial vector fields with algebraic trajectories, uniform bounds for Hilbert coefficients of parameters, a property of the Frobenius map of a polynomial ring, and some homological properties of almost Gorensein rings. They are not indexed. (Annotation ©2011 Book News Inc. Portland, OR)

Complex analysis and dynamical systems IV, pt. 2: General relativity, geometry, and PDE; proceedings.

The 18 papers contained in the second volume from the May 2009 conference investigate differentiable dynamical systems and functions of complex variables. The first two chapters survey recent results on multiplicity-free representations of compact Lie groups and the existence and properties of marginally outer trapped surfaces. Other topics include the stationary n-body problem in general relativity, asymptotic gluing of asymptotically hyperbolic vacuum initial data sets, the global geometry of spacetimes with toroidal or hyperbolic symmetry, and rates of decay for structural damped models with coefficients strictly increasing in time. No index is provided. (Annotation ©2011 Book News Inc. Portland, OR)

Complex analysis and dynamical systems IV; proceedings.

The first of two volumes from the May 2009 conference presents new results in function theory and optimization. The 21 papers propose an algorithm for continuous piecewise affine maps of compact support, investigate the stability of cycles in gene networks with variable feedbacks, and describe polynomial maps of the affine space. Other topics include optimal control of a dynamical biological system, algebraic and analytic properties of quasimetric space with dilations, the Schwarz kernel in Clifford analysis, and global holomorphic approximations of Cauchy-Riemann functions. No index is provided. (Annotation ©2011 Book News Inc. Portland, OR)

Complex variables.

Taylor (mathematics, U. of Utah) offers a textbook based on lecture notes from years of teaching a junior undergraduate one-semester course on complex variables. Students should have completed three semesters of calculus, had some linear algebra, and taken at least one semester of the foundations of analysis. The course is intended to link freshman and sophomore calculus, linear algebra, and differential equations, with the much more sophisticated senior level of mathematics. The book contains enough material for a full-year course. (Annotation ©2011 Book News Inc. Portland, OR)

Differential forms on Wasserstein space and infinite-dimensional Hamiltonian systems.

Gangbo (Georgia Institute of Technology, Atlanta), H.K. Kim (New York U.), and T. Pacini (Scuola normale Superiiore, Pisa, Italy) present the foundation for a new framework for defining and studying Hamiltonian partial differential equations. Over the past decade, they explain, it has become clear that the Wasserstein space provides a very useful space of weak solutions for those equations in which total mass is preserved. The theory of Wasserstein spaces is founded on optimal transport and calculus of variation, which provide a toolbox that can be expected to be uniformly useful throughout the theory, thus offering a unified theory for studying the equations. There is no index. (Annotation ©2011 Book News Inc. Portland, OR)

Entropy and the quantum II; proceedings.

The school is designed to introduce young researchers to current topics in mathematical physics that involve the analytical setting and a dose of physical intuition. Nine lectures consider such topics as kinetic theory and the Kac master equation, max-to-mean ration estimates for the fundamental eigenfunction of the Dirichlet Laplacian, random unitary models and their localization properties, the universality of correlations for random analytic functions, a Wegner estimate for Wigner matrices, and quantum Heisenberg models and their probabilistic representations. They are not indexed. (Annotation ©2011 Book News Inc. Portland, OR)

Evolutionary game dynamics; proceedings.

Evolutionary game theory combines the strategic viewpoint of classical game theory with population dynamics to provide a mathematical tool for reducing social phenomena to the level of individual actions. Six lectures from the short course introduce the theory and explore such aspects as deterministic evolutionary game dynamics, some global and unilateral adaptive dynamics, stochastic evolutionary game dynamics, and the evolution of cooperation in finite populations. (Annotation ©2011 Book News Inc. Portland, OR)

Foundations and applications of statistics; an introduction using R.

Intended for a two-semester sequence, this undergraduate textbook introduces the concepts of distributions, variability, hypothesis testing, confidence intervals, likelihood-based statistics, and linear models. Pruim (mathematics, Calvin College) integrates the free R software environment for statistical computing and graphics throughout the text to manipulate data and perform calculations for the examples. (Annotation ©2011 Book News Inc. Portland, OR)

Geometry; a guide for teachers.

Ten seminars encourage middle school geometry teachers to deepen their understanding of polygons and circles, tessellation, symmetry, dissection, lattice squares, and geometry in three dimensions. Judith Sally (emeritus, Northwestern U.) and her husband, who is director of undergraduate studies at the University of Chicago, explain the construction of the five regular polyhedra, the area of polygonal regions and disks, and the density of disk packing. (Annotation ©2011 Book News Inc. Portland, OR)

An introduction to complex analysis and geometry.

Based on a freshman mathematics course taught at the University of Illinois, this textbook explains the basic properties of complex numbers, analyzes the zero-sets of quadratic equations from the point of view of complex rather than real variables, and introduces convergent power series in one complex variable. The Cauchy integral formula is applied to ordinary integrals, the Fourier transform, and the Gamma function. (Annotation ©2011 Book News Inc. Portland, OR)

Introduction to differential equations.

Designed to be a concise, readable introductory textbook for undergraduates, this volume on differential equations provides clear instruction with copious examples for developing an understanding of this foundational area of advanced mathematics and engineering. The work is divided into four sections covering single differential equations, linear algebra, linear systems of differential equations and nonlinear systems of differential equations. Individual sections cover specific aspects of equations as well as theoretical and real world example problems. The volume includes numerous illustrations and sample equations and access to additional online resources is provided. Taylor is a professor of mathematics at the University of North Carolina. (Annotation ©2011 Book News Inc. Portland, OR)

Introduction to functional equations; theory and problem-solving strategies for mathematical competitions and beyond.

Efthimiou (U. of Central Florida) presents material from a lecture series he delivered as part of training a math team. Students with backgrounds ranging from function theory to differentiability should be able to follow the book, particularly high school students who participate in math competitions and are interested in the International Math Olympiads. Most of the questions he deals with, he warns, require ingenuity and insight rather than knowing a lot of techniques. After setting out the background, he covers basic equations, generalization, changing the rules, equations with no parameters, and getting additional experience. (Annotation ©2011 Book News Inc. Portland, OR)

An introduction to measure theory.

Tao (mathematics, U. of California-Los Angeles) presents a textbook based on his introductory one-quarter graduate course on real analysis, focusing particularly on the basics of measure and integration theory both in Euclidean spaces and in abstract measure spaces. It is meant to precede his 2010 An Epsilon of Room, Volume 1, which introduces the analysis of Hilbert and Banach spaces, point-set topology, and related topics. Together, the two volumes serve as a text for complete first-year graduate course in real analysis. The only prerequisite is undergraduate-level real analysis. (Annotation ©2011 Book News Inc. Portland, OR)

Iterated function systems, moments, and tranformations of infinite matrices.

Jorgensen (U. of Iowa), Keri A. Kornelson (U. of Oklahoma-Normal), and Karen L. Shuman (Grinnell College, Iowa) examine the moments of equilibrium measures for iterated function systems, and draw connections to operator theory. In order to encode the salient features of a given system into precise moment data, they establish an interdependence between the system's equilibrium measures, the encoding of the sequence of moments of these measures into operators, and a new correspondence between the system moment and this family of operators. Among their topics are the moment problem, the Kato-Friedrichs operator, the integral operator of a moment matrix, and boundedness and spectral properties. The book is not indexed. (Annotation ©2011 Book News Inc. Portland, OR)

Jumping numbers of a simple complete ideal in a two-dimensional regular local ring.

More specifically, Järvilehto obtains a formula for the jumping numbers of an analytically irreducible plane curve, then shows that the jumping numbers determine the equi-singularity class of the curve. His topics include preliminaries on complete ideals, the dual graph, multiplier ideals and jumping numbers, and jumping numbers of a simple ideal. He has not indexed the book. (Annotation ©2011 Book News Inc. Portland, OR)