Series: Advanced Courses in Mathematics - CRM Barcelona
2009, Approx. 220 p., Softcover
ISBN: 978-3-7643-8939-0
Due: December 2008
The review on central schemes, on error estimates for discontinuous Galerkin methods and on the use of wavelets in scientific computing form excellent teaching material for graduate students
This volume contains three sets of revised and augmented notes prepared by the authors for the Advanced School on Numerical Solutions of Partial Differential Equations held at the Centre de Recerca Matematica in November 2007.
The book is aimed at master students, PhD students, recent PhD doctorates, and researchers in general willing to update their research skills with recent developments on numerical techniques that play an important role in the international scene.
The text offers to researchers the opportunity to catch up with important developments in the field of numerical analysis and scientific computing and/or to get in touch with state-of-art numerical techniques that are not covered in usual books.
I. Wavelets and PDE’s.- Wavelets - The Fundamental Property of Wavelets -Wavelets for PDEs.- II. High Order Shock Capturing Schemes for Balance Laws - The Numerical Flux Function - Nonlinear Reconstruction and High Order Schemes - Central Schemes - Systems with Stiff Source.- III. Discontinuous Galerkin Methods: General Approach and Stability - Time Discretization - Discontinuous Galerkin Method for Conservation Laws - Discontinuous Galerkin Method for Convection-Diffusion Equations - Discontinuous Galerkin Method for PDEs.
2009, XVIII, 635 p. 212 illus., Softcover
ISBN: 978-0-387-87362-6
Due: December 2008
This commemorative book contains the 28 major articles that appeared in
the 2008 Twentieth Anniversary Issue of the journal Discrete & Computational
Geometry, and presents a comprehensive picture of the current state of
the field. Formed during the past few decades by the merger of the classical
discipline of combinatorial and discrete geometry with the new field of
computational geometry that sprang up in the 1970s, discrete and computational
geometry now claims the allegiance of a sizeable number of mathematicians
and computer scientists all over the world, whose most important work has
been appearing since 1986 in the pages of the journal.
The articles in this volume, a number of which solve long-outstanding problems in the field, were chosen by the editors of DCG for the importance of their results, for the breadth of their scope, and to show the intimate connections that have arisen between discrete and computational geometry and other areas of both computer science and mathematics. Apart from the articles, the editors present an expanded preface, along with a set of photographs of groups and individuals who have played a major role in the history of the field during the past twenty years.
Preface.- There are Not too Many Magic Configurations.- Computing the Detour and Spanning Ratio of Paths, Trees, and Cycles in 2D and 3D.- Robus Shape Fitting via Peeling and Grating Coresets.- Siegel's LEmaa and Sum-Distinct Sets.- Slicing Convex Sets and Measures by a Hyperplane.- A Centrally Symmetric Version of the Cyclic Polytope.- On Projections of Semi-Algebraic Sets Defined by Few Quadratic Inequalities.- Enumeration in Convex Geometries and Associated Polytopal Subdivisions of Spheres.- Isotopic Implicit Surface Meshing.- Line Transversals to Disjoint Balls.- Generating all Vertices of a Polyhedron is Hard.- Norm Bounds for Ehrhart Polynominal Roots.- Helly-Type Theorems for Line Transversals to Disjoint Unit Balls.- Grid Vertex-Unfolding Orthogonal Polyhedra.- Empty Convex Hexagons in Planar Point Sets.- Affinely Regular Polygons are Extremals of Area Functionals.- Improved Output-Sensitive Snap Rounding.- Pure Point Diffractive Substitution Delone Sets Have the Meyer Property.- Metric Combinatorics of Convex Polyhedra: Cut Loci and Nonoverlapping Unfoldings.- Empty Simplices of Polytopes and Graded Betti Numbers.- Rigidity and the Lower Bound Theorem for Doubly Cohen?Macaulay Complexes.- Finding the Homology of Submanifolds with HIgh Confidence from Random Samples.- Odd Crossing Number and Crossing Number are Not the Same.- Visibility Graphs of Point Sets in the Plane.- Decomposability of Polytopes.- An Inscribing Model for Random Polytopes.- An Optimal-Time Algorithm for Shortest Paths on a Convex Polytope in Three Dimensions.- Genral-Dimensional Constrained Delaunay nad Constrained Regular Triangulation, I: Combinatorial Properties.
Series: Probability and its Applications
2009, Approx. 280 p., Hardcover
ISBN: 978-3-540-85635-1
Due: December 4, 2008
Self-normalized processes are of common occurrence in probabilistic and statistical studies. A prototypical example is Student's t-statistic introduced in 1908 by Gosset, whose portrait is on the front cover. Due to the highly non-linear nature of these processes, the theory experienced a long period of slow development. In recent years there have been a number of important advances in the theory and applications of self-normalized processes. Some of these developments are closely linked to the study of central limit theorems, which imply that self-normalized processes are approximate pivots for statistical inference.
The present volume covers recent developments in the area, including self-normalized large and moderate deviations, and laws of the iterated logarithms for self-normalized martingales. This is the first book that systematically treats the theory and applications of self-normalization.
1. Introduction.- Part I Independent Random Variables.- 2. Classical Limit Theorems and Preliminary Tools.- 3. Self-Normalized Large Deviations.- 4. Weak Convergence of Self-Normalized Sums.- 5. Stein’s Method and Self-Normalized Berry?Esseen Inequality.- 6. Self-Normalized Moderate Deviations and Law of the Iterated Logarithm.- 7. Cramer-type Moderate Deviations for Self-Normalized Sums.- 8. Self-Normalized Empirical Processes and U-Statistics.- Part II Martingales and Dependent Random Vectors.- 9. Martingale Inequalities and Related Tools.- 10. A General Framework for Self-Normalization.- 11. Pseudo-Maximization via Method of Mixtures.- 12. Moment and Exponential Inequalities for Self-Normalized Processes.- 13. Laws of the Iterated Logarithm for Self-Normalized Processes and Martingales.- 14. Multivariate Matrix-Normalized Processes.- Part III Statistical Applications.- 15. The t-Statistic and Studentized Statistics.- 16. Self-Normalization and Approximate Pivots for Bootstrapping.- 17. Self-Normalized Martingales and Pseudo-Maximization in Likelihood or Bayesian Inference.- 18. Information Bounds and Boundary Crossing Probabilities for Self-Normalized Statistics in Sequential Analysis.- References.- Index.
Series: Probability and its Applications
2009, Approx. 460 p., Hardcover
ISBN: 978-3-540-89331-8
Due: December 4, 2008
Stochastic processes are mathematical models of random phenomena that evolve according to prescribed dynamics. Processes commonly used in applications are Markov chains in discrete and continuous time, renewal and regenerative processes, Poisson processes, and Brownian motion. This volume gives an in-depth description of the structure and basic properties of these stochastic processes. A main focus is on equilibrium distributions, strong laws of large numbers, and ordinary and functional central limit theorems for cost and performance parameters. Although these results differ for various processes, they have a common trait of being limit theorems for processes with regenerative increments. Extensive examples and exercises show how to formulate stochastic models of systems as functions of a system’s data and dynamics, and how to represent and analyze cost and performance measures. Topics include stochastic networks, spatial and space-time Poisson processes, queueing, reversible processes, simulation, Brownian approximations, and varied Markovian models.
The technical level of the volume is between that of introductory texts that focus on highlights of applied stochastic processes, and advanced texts that focus on theoretical aspects of processes. Intended readers are researchers and graduate students in mathematics, statistics, operations research, computer science, engineering, and business.
1.Markov Chains .- 2.Renewal and Regenerative Processes.- 3.Poisson Processes.- 4.Continuous-Time Markov Chains.- 5.Brownian Motion.- 6.Appendix.- References.- Notation.- Index.
Series: Universitext
2009, XIV, 212 p., Softcover
ISBN: 978-0-387-87574-3
Due: December 2008
The book discusses classical Galois theory in considerable generality, treating fields of characteristic zero and of positive characteristic with consideration of both separable and inseparable extensions, but with a particular emphasis on algebraic extensions of the field of rational numbers. While most of the book is concerned with finite extensions, it discusses algebraic closure and infinite Galois extensions, and concludes with a new chapter on transcendental extensions.
Key topics and features of this second edition:
- Approaches Galois theory from the linear algebra point of view, following Artin;
- Presents a number of applications of Galois theory, including symmetric functions, finite fields, cyclotomic fields, algebraic number fields, solvability of equations by radicals, and the impossibility of solution of the three geometric problems of Greek antiquity.
"The text offers the standard material of classical field theory and Galois theory, though in a remarkably original, unconventional and comprehensive manner … . the book under review must be seen as a highly welcome and valuable complement to existing textbook literature … . It comes with its own features and advantages … it surely is a perfect introduction to this evergreen subject. The numerous explaining remarks, hints, examples and applications are particularly commendable … just as the outstanding clarity and fullness of the text." (Zentralblatt MATH, Vol. 1089 (15), 2006)
Steven H. Weintraub is a Professor of Mathematics at Lehigh University and the author of seven books. This book grew out of a graduate course he taught at Lehigh. He is also the author of Algebra: An Approach via Module Theory (with W. A. Adkins).
Introduction to Galois Theory.- Field Theory and Galois Theory.- Development and Applications of Galois Theory.- Extensions of the Field of Rational Numbers.- Further Topics in Field Theory.- Transcendental Extensions.- A. Some Results from Group Theory.- B. A Lemma on Constructing Fields.- C. A Lemma from Elementary Number Theory.- References.- Index.