Durrett, R., Cornell University, Ithaca, NY, USA

Probability Models for DNA Sequence Evolution

2002. Approx. 250 pp. Hardcover
0-387-95435-X

Our basic question is: Given a collection of DNA sequences, what underlying forces are responsible for the observed patterns of variability? To approach this question we introduce and analyze a number of probability models: the Wright-Fisher model, the coalescent, the infinite alleles model, and the infinite sites model. We study the complications that come from nonconstant population size, recombination, population subdivision, and three forms of natural selection: directional selection, balancing selection, and background selection. These theoretical results set the stage for the investigation of various statistical tests to detect departures from "neutral evolution." The final chapter studies the evolution of whole genomes by chromosomal inversions, reciprocal translocations, and genome duplication.
Throughout the book, the theory is developed in close connection with data from more than 60 experimental studies from the biology literature that illustrate the use of these results. This book is written for mathematicians and for biologists alike. We assume no previous knowledge of concepts from biology and only a basic knowledge of probability: a one semester undergraduate course and some familiarity with Markov chains and Poisson processes.
Rick Durrett received his Ph.D. in operations research from Stanford University in 1976. He taught in the UCLA mathematics department before coming to Cornell in 1985. He is the author of six books and 125 research papers, and is the academic father of more than 30 Ph.D. students. His current interests are the use of probability models in genetics and ecology, and decreasing the mean and variance of his golf.

Contents: Basic Models.- Neutral Complications.- Natural Selection.- Statistical Tests.- Genome Rearrangement.

Series: Probability and its Applications.

Derksen, H., University of Michigan, Ann Arbor, MI, USA;
Kemper, G., University of Heidelberg, Germany

Computational Invariant Theory

2002. X, 268 pp. Hardcover
3-540-43476-3

Throughout the history of invariant theory, computational methods have always been at the center of attention. This book, the first volume of the new subseries on "Invariant Theory and Algebraic Transformation Groups", provides a comprehensive and up-to-date overview of the algorithmic aspects of invariant theory. Special features are an introductory chapter on Grobner basis methods and a chapter on applications, covering fields as disparate as graph theory, coding theory, dynamical systems, and computer vision. Both authors have made significant contributions to the theory and practice of algorithmic invariant theory. Numerous illustrative examples and a careful selection of proofs make the book accessible to non-specialists.
The book will be very useful to postgraduate students as well as researchers in geometry, computer algebra, and, of course, invariant theory.

Keywords: Invariant theory, computational commutative algebra

Series: Encyclopaedia of Mathematical Sciences. VOL. 130

Holden, H., Norwegian University of Science and Technology, Trondheim, Norway; Risebro, N.H., University of Oslo, Norway

Front Tracking for Hyperbolic Conservation Laws

2002. XI, 380 pp. Hardcover
3-540-43289-2

Hyperbolic conservation laws are central in the theory of nonlinear partial differential equations, and in many applications in science and technology. In this book the reader is given a detailed, rigorous, and self-contained presentation of the theory of hyperbolic conservation laws from the basic theory up to the research front. The approach is constructive, and the mathematical approach using front tracking can be applied directly as a numerical method. After a short introduction on the fundamental properties of conservation laws, the theory of scalar conservation laws in one dimension is treated in detail, showing the stability of the Cauchy problem using front tracking. The extension to multidimensional scalar conservation laws is obtained using dimensional splitting. Inhomogeneous equations and equations with diffusive terms are included as well as a discussion of convergence rates. The classical theory of Kruzkov and Kuznetsov is covered. Systems of conservation laws in one dimension are treated in detail, starting with the solution of the Riemann problem. Solutions of the Cauchy problem are proved to exist in a constructive manner using front tracking, amenable to numerical computations. The book includes a detailed discussion of the very recent proof of wellposedness of the Cauchy problem for one-dimensional hyperbolic conservation laws. The book includes a chapter on traditional finite difference methods for hyperbolic conservation laws with error estimates and a section on measure valued solutions. Extensive examples are given, and many exercises are included with hints and answers. Additional background material not easily available elsewhere is given in appendices.

Keywords: front tracking, hyperbolic partial differential equations, conservation laws

Contents: 1. Introduction.- 2. Scalar Conservation Laws.- 3. A Short Course in Difference Methods.- 4. Multidimensional Scalar Conservation Laws.- 5. The Riemann Problem for Systems.- 6. Existence of Solutions of the Cauchy Problem.- 7. Wellposedness of the Cauchy Problem.- Appendix A: Total Variation, Compactedness, etc.- Appendix B: The Method of Vanishing Viscosity.- Appendix C: Answers and Hints.- References.- Index.

Series: Applied Mathematical Sciences. VOL. 152

Goldschmidt, D., IDA Center for Communications Research, Princeton, NJ, USA

Algebraic Functions and Projective Curves

2002. Approx. 195 pp. Hardcover
0-387-95432-5

This book provides a self-contained exposition of the theory of algebraic curves without requiring any of the prerequisites of modern algebraic geometry. The self-contained treatment makes this important and mathematically central subject accessible to non-specialists. At the same time, specialists in the field may be interested to discover several unusual topics. Among these are Tate theory of residues, higher derivatives and Weierstrass points in characteristic p, the Stohr--Voloch proof of the Riemann hypothesis, and a treatment of inseparable residue field extensions. Although the exposition is based on the theory of function fields in one variable, the book is unusual in that it also covers projective curves, including singularities and a section on plane curves.
David Goldschmidt has served as the Director of the Center for Communications Research since 1991. Prior to that he was Professor of Mathematics at the University of California, Berkeley.

Contents: Preface.- Introduction.- Background.- Function Fields.- Finite Extensions.- Projective Curves.- Zeta Functions.- Appendix: Field Extensions.- Bibliography.- Index.

Series: Graduate Texts in Mathematics. VOL. 215

Puig, L., Universit'e de Paris VII, France

Theory of Blocks of the Finite Groups
The Hyperfocal Subalgebra of a Block

2002. VI, 209 pp. Hardcover
3-540-43514-X

About 60 years ago, R. Brauer introduced "block theory"; his purpose was to study the group algebra kG of a finite group G over a field k of nonzero characteristic p: any indecomposable two-sided ideal that also is a direct summand of kG determines a G-block.
But the main discovery of Brauer is perhaps the existence of families of infinitely many nonisomorphic groups having a "common block"; i.e., blocks having mutually isomorphic "source algebras".
In this book, based on a course given by the author at Wuhan University in 1999, all the concepts mentioned are introduced, and all the proofs are developed completely. Its main purpose is the proof of the existence and the uniqueness of the "hyperfocal subalgebra" in the source algebra. This result seems fundamental in block theory; for instance, the structure of the source algebra of a nilpotent block, an important fact in block theory, can be obtained as a corollary.

Keywords: Group, block, source algebra, hyperfocal algebra MSC ( 2000 ): 20C11

Series: Springer Monographs in Mathematics.

Gelfand, S.I., American Mathematical Society, Providence, RI, USA;
Manin, Y.I., Max-Planck-Institut fur Mathematik, Bonn, Germany

Methods of Homological Algebra, 2nd ed.

2002. Approx. 370 pp. Hardcover
3-540-43583-2

Homological algebra first arose as a language for describing topological prospects of geometrical objects. As with every successful language it quickly expanded its coverage and semantics, and its contemporary applications are many and diverse. This modern approach to homological algebra, by two leading writers in the field, is based on the systematic use of the language and ideas of derived categories and derived functors. Relations with standard cohomology theory (sheaf cohomology, spectral sequences, etc.) are described. In most cases complete proofs are given. Basic concepts and results of homotopical algebra are also presented. The book addresses people who want to learn a modern approach to homological algebra and to use it in their work. For the second edition the authors have made numerous corrections.

Keywords: Homological algebra, category theory, homotopical algebra MSC ( 2000 ): 18-XX

Contents: I. Simplical Sets.- II. Main Notions of the Category Theory.- III. Derived Categories and Derived Functors.- IV. Triangulated Categories.- V. Introduction to Homotopic Algebra.

Series: Springer Monographs in Mathematics.

Jost, J., MPI fur Mathematik in den Naturwissenschaften, Leipzig, Germany

Partial Differential Equations

2002. Approx. 345 pp. 10 figs. Hardcover
0-387-95428-7

This book is intended for students who wish to get an introduction to the theory of partial differential equations. The author focuses on elliptic equations and systematically develops the relevant existence schemes, always with a view towards nonlinear problems. These are maximum principle methods (particularly important for numerical analysis schemes), parabolic equations, variational methods, and continuity methods. This book also develops the main methods for obtaining estimates for solutions of elliptic equations: Sobolev space theory, weak and strong solutions, Schauder estimates, and Moser iteration. Connections between elliptic, parabolic, and hyperbolic equations are explored, as well as the connection with Brownian motion and semigroups.
This book can be utilized for a one-year course on partial differential equations. Jurgen Jost is Director of the Max Planck Institute for Mathematics in the Sciences and Professor of Mathematics at the University of Leipzig. He is the author of a number of Springer books, including Postmodern Analysis (1998), Compact Riemann Surfaces (1997) and Riemannian Geometry and Geometric Analysis (1995). The present book is an expanded translation of the original German version, Partielle Differentialgleichungen (1998).

Keywords: PDE, Partial Differential Equations

Contents: Introduction.- The Laplace equation as the prototype of an elliptic partial differential equation of 2nd order.- The maximum principle.- Existence techniques I: methods based on the maximum principle.- Existence techniques II: Parabolic methods. The Head equation.- The wave equation and its connections with the Laplace and heat equation.- The heat equation, semigroups, and Brownian motion.- The Dirichlet principle. Variational methods for the solution of PDE (Existence techniques III).- Sobolev spaces and L2 regularity theory.- Strong solutions.- The regularity theory of Schauder and the continuity method (Existence techniques IV).- The Moser iteration method and the reqularity theorem of de Giorgi and Nash.- Banach and Hilbert spaces. The Lp-spaces.- Bibliography.

Series: Graduate Texts in Mathematics. VOL. 214