Davenport, H./ Montgomery, H.L.,
University of Michigan, Ann Arbor, MI, USA
Multiplicative Number Theory
3rd ed. 2000. Approx. 190 pp.
0-387-95097-4
This book thoroughly examines the distribution of prime numbers in arithmetic progressions. It covers
many classical results, including the Dirichlet theorem on the existence of prime numbers in arithmetical
progressions, the theorem of Siegel, and functional equations of the L-functions and their consequences
for the distribution of prime numbers. In addition, a simplified, improved version of the large sieve method is
presented. The 3rd edition includes a large number of revisions and corrections as well as a new section
with references to more recent work in the field.
Contents: From the contents: Primes in Arithmetic Progression.- Gauss' Sum.- Cyclotomy.- Primes in
Arithmetic Progression: The General Modulus.- Primitive Characters.- Dirichlet's Class Number Formula.-
The Distribution of the Primes.- Riemann's Memoir.- The Functional Equation of the L Function.- Properties
of the Gamma Function.- Integral Functions of Order 1.- The Infinite Products for xi(s) and xi(s,Zero-Free
Region for zeta(s).- Zero-Free Regions for L(s, chi).- The Number N(T).- The Number N(T, chi).- The explicit
Formula for psi(x).- The Prime Number Theorem.- The Explicit Formula for psi(x,chi).- The Prime Number
Theorem for Arithmetic Progressions (I).- Siegel's Theorem.- The Prime Number Theorem for Arithmetic
Progressions (II).- The Polya-Vinogradov Inequality.- Further Prime Number Sums.
Series: Graduate Texts in Mathematics.VOL. 74
Szasz, D., Technical University, Budapest, Hungary
(Ed.)
Hard Ball Systems and the Lorentz Gas
MSC 2000: 37-XX, 82-XX. With contributions by numerous experts
2000. Approx. 400 pp. 75 figs.
3-540-67620-1
Hard Ball Systems and the Lorentz Gas are fundamental models arising in the theory of Hamiltonian
dynamical systems. Moreover, in these models, some key laws of statistical physics can also be tested or
even established by mathematically rigorous tools. The mathematical methods are most beautiful but
sometimes quite involved. This collection of surveys written by leading researchers of the fields -
mathematicians, physicists or mathematical physicists - treat both mathematically rigourous results, and
evolving physical theories where the methods are analytic or computational. Some basic topics:
hyperbolicity and ergodicity, correlation decay, Lyapunov exponents, Kolmogorov-Sinai entropy, entropy
production, irreversibility. This collection is a unique introduction into the subject for graduate students,
postdocs or researchers - in both mathematics and physics - who want to start working in the field.
Keywords: hard ball systems, Lorentz gas, ergodic theory of hyperbolic dynamical systems, nonequilibrium
stationary states
Contents: Part I. Mathematics: 1. D. Burago, S. Ferleger, A. Kononenko: A Geometric Approach to
Semi-Dispersing Billiards.- 2. T. J. Murphy, E. G. D. Cohen: On the Sequences of Collisions Among Hard
Spheres in Infinite Spacel- 3. N. Simanyi: Hard Ball Systems and Semi-Dispersive Billiards: Hyperbolicity
and Ergodicity.- 4. N. Chernov, L.-S. Young: Decay of Correlations for Lorentz Gases and Hard Balls.- 5. N.
Chernov: Entropy Values and Entropy Bounds.- 6. L. A. Bunimovich: Existence of Transport Coefficients.-
7. C. Liverani: Interacting Particles.- 8. J. L. Lebowitz, J. Piasecki and Ya. G. Sinai: Scaling Dynamics of a
Massive Piston in an Ideal Gas .- Part II. Physics: 1. H. van Beijeren, R. van Zon, J. R. Dorfman: Kinetic
Theory Estimates for the Kolmogorov-Sinai Entropy, and the Largest Lyapunov Exponents for Dilute,
Hard-Ball Gases and for Dilute, Random Lorentz Gases.- 2. H. A. Posch and R. Hirschl: Simulation of
Billiards and of Hard-Body Fluids.- 3. C. P. Dettmann: The Lorentz Gas: a Paradigm for Nonequilibrium
Stationary States.- 4. T. Tl, J. Vollmer: Entropy Balance, Multibaker Maps, and the Dynamics of the Lorentz
Gas.- Appendix: 1. D. Szasz: Boltzmann Ergodic Hypothesis, a Conjecture for Centuries?
Series: Encyclopaedia of Mathematical Sciences.VOL. 101
Borel, A., Institute for Advanced Study, Princeton, NJ, USA
Collected Papers IV
2000. Approx. 645 pp.
3-540-67640-6
This volume contains the papers published by A. Borel from 1983 to 1999. About half of them are research
papers, written singly or in collaboration, on various topics pertaining mainly to algebraic or Lie groups,
homogeneous spaces, arithmetic groups (L2-spectrum, automorphic forms, cohomology and covolumes),
L2-cohomology of symmetric or locally symmetric spaces, and to the Oppenheim conjecture. Other
publications include some surveys, some personal recollections (of D. Montgomery, Harish-Chandra, A.
Weil), some considerations on mathematics in general and several articles of a historical nature : on the
School of Mathematics at the Institute for Advanced Study, on N. Bourbaki and on parts of the works of H.
Weyl, C. Chevalley, E. Kolchin, J. Leray, A. Weil. The volume concludes with an essay on H. PoincarEand
special relativity. Some comments on, and corrections to, a number of papers have been added.
Keywords: Algebra, algebraic number theory, Lie groups, algebraic topology, history of mathematics
Contents: Papers by Armand Borel published from 1983 to 1999.
Heinonen, J.M., University of Michigan, Ann Arbor, MI, USA
Lectures on Analysis on Metric Spaces
2000. Approx. 135 pp.
0-387-95104-0
Analysis in spaces with no a priori smooth structure has progressed to include concepts from the first order
calculus. In particular, there have been important advances in understanding the infinitesimal versus global
behavior of Lipschitz functions and quasiconformal mappings in rather general settings; abstract Sobolev
space theories have been instrumental in this development. The purpose of this book is to communicate
some of the recent work in the area while preparing the reader to study more substantial, related articles.
The material can be roughly divided into three different types: classical, standard but sometimes with a new
twist, and recent. The author first studies basic covering theorems and their applications to analysis in
metric measure spaces. This is followed by a discussion on Sobolev spaces emphasizing principles that are
valid in larger contexts. The last few sections of the book present a basic theory of quasisymmetric maps
between metric spaces. Much of the material is relatively recent and appears for the first time in book
format. There are plenty of exercises. The book is well suited for self-study, or as a text in a graduate course
or seminar. The material is relevant to anyone who is interested in analysis and geometry in nonsmooth
settings.
Contents: Covering theorems.- Maximal functions.- Sobolev spaces.- PoincarE inequality.- Sobolev spaces
on metric spaces.- Lipschitz functions.- Modulus of a curve family, capacity, and upper gradients.- Loewner
spaces.- Loewner spaces and PoincarE inequalities.- Quasisymmetric maps. Basic theory I.- Quasisymmetric
maps. Basic theory II.- Quasisymmetric embeddings of metric spaces in Euclidean space.- Existence of
doubling measures.- Doubling measures and quasisymmetric maps.- Conformal gauges.
Series: Universitext.