Adelmann, C., Technical University, Braunschweig, Germany

The Decomposition of Primes in Torsion Point Fields

2001. VI, 142 pp. Softcover
3-540-42035-5

We investigate the decomposition of prime ideals in non-abelian extensions of number fields. These fields are generated by the coordinates of torsion points of elliptic curves without complex multiplications. We explain the necessary prerequisites from the theory of elliptic curves, modular forms, algebraic number theory, and invariant theory. Due to the complexity of the problem, complete results are restricted to torsion points of low order. These results are complemented by computational data which also cover some unsolved cases.

Keywords: number fields, non-abelian ; decomposition of primes MSC : 11-02, 11R21, 11R32, 11R09, 11G05, 11F11, 13A50, 12Y05

Contents: Introduction.- Decomposition laws.- Elliptic curves.- Elliptic modular curves.- Torsion point fields.- Invariants and resolvent polynomials.- Appendix: Invariants of elliptic modular curves; L-series coefficients a p; Fully decomposed prime numbers; Resolvent polynomials; Free resolution of the invariant algebra.

Series: Lecture Notes in Mathematics.VOL. 1761

Alevras, D., IBM Corporation, West Chester, PA, USA
Padberg, M.W., New York University, NY, USA

Linear Optimization and Extensions
Problems and Solutions

2001. X, 450 pp. 67 figs., 30 tabs. Softcover
3-540-41744-3

This book offers a comprehensive treatment of the exercises and case studies as well as summaries of the chapters of the book "Linear Optimization and Extensions" by Manfred Padberg. It covers the areas of linear programming and the optimization of linear functions over polyhedra in finite dimensional Euclidean vector spaces.
Here are the main topics treated in the book: Simplex algorithms and their derivatives including the duality theory of linear programming. Polyhedral theory, pointwise and linear descriptions of polyhedra, double description algorithms, Gaussian elimination with and without division, the complexity of simplex steps. Projective algorithms, the geometry of projective algorithms, Newtonian barrier methods. Ellipsoids algorithms in perfect and in finite precision arithmetic, the equivalence of linear optimization and polyhedral separation. The foundations of mixed-integer programming and combinatorial optimization.

Keywords: Linear programming, linear optimization, mixed-integer programming

Contents: Preface.- 1. Introduction.- 2. The Linear Programming Problem.- 3. Basic Concepts.- 4. Five Preliminaries.- 5. Simple Algorithms.- 6. Primal-Dual Pairs.- 7. Analytical Geometry.- 8. Projective Algorithms.- 9. Ellipsoid Algorithms.- 10. Combinatorial Optimization: An Introduction.- A. Short-Term Management.- B. Operations Management in a Refinery.- C. Automatized Production: PCBs and Ulysses'Problem.- Bibliography.- Index.

Series: Universitext.

Fleuriot, J., University of Edinburgh, UK

A Combination of Geometry Theorem Proving and Nonstandard Analysis
with Application to Newton's Principia

2001. Approx. 160 pp. 30 figs. Hardcover
1-85233-466-5

Sir Isaac Newton's philosophi Naturalis Principia Mathematica'(the Principia) contains a prose-style mixture of geometric and limit reasoning that has often been viewed as logically vague.
In A Combination of Geometry Theorem Proving and Nonstandard Analysis , Jacques Fleuriot presents a formalization of Lemmas and Propositions from the Principia using a combination of methods from geometry and nonstandard analysis. The mechanization of the procedures, which respects much of Newton's original reasoning, is developed within the theorem prover Isabelle. The application of this framework to the mechanization of elementary real analysis using nonstandard techniques is also discussed.

Contents: Introduction .- A Brief History of the Infinitesimal.- The Principia and its Methods.- On Nonstandard Analysis.- Objectives.- Achieving our Goals.- Organisation of this book.- Geometry Theorem Proving.- Historical Background.- Algebraic Techniques.- Coordinate-Free Techniques.- Formalizing Geometry in Isabelle.- Concluding remarks.- Constructing the Hy perreals.- Isabelle/HOL.- Properties of an Infinitesimal Calculus.- Internal Set Theory.- Constructions Leading to the Reals.- Filters and Ultrafilters.- Ultrapower Construction of the Hyperreals.- Structure of the Hyperreal Number Line.- The Hypernatural Numbers.- An Alternative Construction for the Reals.- Related Work.- Concluding Remarks.- Infinitesimal and Analytic Geometry .- Non-Archimedean Geometry.- New Definitions and Relations.- Infinitesimal Geometry Proofs.- Verifying the Axioms of Geometry.- Concluding Remarks.- Mechanising Newton's Principia.- Formalizing Newton's Properties.- Mechanized Propositions and Lemmas.- Ratios of Infinitesimals.- Case Study: Propositio Kepleriana.- Expanding Newton's Proof.- Conclusions.- Nonstandard Real Analysis .- Extending a Relation to the Hyperreals.- Towards an Intuitive Calculus.- Real Sequences and Series.- Some Elementary Topology of the Reals.- Limits and Continuity.- Differentiation.- On the Transfer Principle.- Related Work and Conclusions.- Conclusions.- Geometry, Newton and the Principia.- Hyperreal Analysis.- Further Work.- Concluding Remarks.-

Series: Distinguished Dissertations.

Johnson, D.L., University of Nottingham, UK

Symmetries

2001. Approx. 210 pp. 60 figs. Softcover
1-85233-270-0

Written by the author of Sets, Logic and Categories, the main object of study for this book is geometry, with group theory providing an appropriate language in which to express geometrical ideas. Key features include:
- An overview ofthe preliminaries from group theory and geometry
- Coverage of the discrete subgroups of the Euclidean group
- A clear and complete derivation and classification of the 17 plane crystallographic groups
- Tessellations of various spaces (they are constructed, described and classified)
- A brief introduction to hyperbolic geometry
Each chapter contains a number of exercises, most with solutions, and suggestions for background, alternative and further reading. The author's accessible and down-to-earth approach make this an ideal introduction for readers in the second or third year of a mathematics undergraduate course. It is also recommended for mechanical engineers, architects, physicists and crystallographers needing an understanding of 3-dimensional geometry, symmetry and trigonometry.

Series: Springer Undergraduate Mathematics Series.

Eggermont, P., University of Delaware, Newark, DE, USA
LaRiccia, V., University of Delaware, Newark, DE, USA

Maximum Penalized Likelihood Estimation
Volume I: Density Estimation

2001. Approx. 515 pp. 30 figs. Hardcover
0-387-95268-3

This book is intended for graduate students in statistics and industrial mathematics, as well as researchers and practitioners in the field. We cover both theory and practice of nonparametric estimation. The text is novel in its use of maximum penalized likelihood estimation, and the theory of convex minimization problems (fully developed in the text) to obtain convergence rates. We also use (and develop from an elementary view point) discrete parameter submartingales and exponential inequalities. A substantial effort has been made to discuss computational details, and to include simulation studies and analyses of some classical data sets using fully automatic (data driven) procedures. Some theoretical topics that appear in textbook form for the first time are definitive treatments of I.J. Good's roughness penalization, monotone and unimodal density estimation, asymptotic optimality of generalized cross validation for spline smoothing and analogous methods for ill-posed least squares problems, and convergence proofs of EM algorithms for random sampling problems.

Series: Springer Series in Statistics.

Glaz, J., University of Connecticut, Storrs, CT, USA
Naus, J., Rutgers University, Piscataway, NJ, USA
Wallenstein, S., New York, NY, USA

Scan Statistics

2001. Approx. 380 pp. 6 figs. Hardcover
0-387-98819-X

In many statistical applications the scientists have to analyze the occurrence of observed clusters of events in time or space. The scientists are especially interested to determine whether an observed cluster of events has occurred by chance if it is assumed that the events are distributed independently and uniformly over time or space. Applications of scan statistics have been recorded in many areas of science and technology including: geology, geography, medicine, minefield detection, molecular biology, photography, quality control and reliability theory and radio-optics.

Contents: Introduction.- Conditional Continuous Uniform Case: Exact Distribution.- Conditional Continuous Uniform Case: Bounds and Approximations.- Unconditional Continuous Case: Exact Distribution, Bounds and Approximations.- Conditional Discrete Case: Exact Distribution, Bounds and Approximations.- Unconditional Discrete Case: Sequences of iid Discrete Random Variables.- Scan Statistics on the Circle.- Power of Scan Statistics: Pulse Alternatives.- Scan Statistics in Two Dimensions.- Two Sequence Scan Statistics.- Generalized Scan-Type Statistics.- Coverage Problems: Unusually Small Clusters.

Series: Springer Series in Statistics.

Homer, S., University of Boston, MA, USA
Selman, A.L., State University of New York at Buffalo, NY, USA

Computability and Complexity Theory

2001. Approx. 215 pp. 40 figs. Hardcover
0-387-95055-9

Intended for use in an introductory graduate course in theoretical computer science, this text contains material that should be core knowledge in the theory of computation for all graduates in computer science. It is self-contained and is best suited for a one semester course. The text starts with classical computability theory which forms the basis for complexity theory. This has the pedagogical advantage that students learn a qualitative subject before advancing to a quantitative one. Since this is a graduate course, students should have some knowledge of such topics as automata theory, formal languages, computability theory, or complexity theory.

Contents: 1. Preliminaries.- 2. Introduction to Computability.- 3. Undecidability.- 4. Introduction to Complexity Theory.- 5. Basic Results.- 6. Nondeterminism and NP-Completeness.- 7. Relative Computability.

Series: Texts in Computer Science.