R.P. Bambah V.C. Dumir R.J. Hans Gill, (Eds.) all of Panjab University, Chandigarh, India
Number Theory
3-7643-6259-6 * 2000 * Hardcover * 536 pages
This book contains 23 papers on various branches of number theory contributed by leading mathematicians, giving an overview of the developments in their respective fields together with open problems. These will be of interest to mathematicians at various levels providing the reader with ready access to meaningful problems and results, which have attracted the attention of some leading number theorists.
Series: Trends in Mathematics
Contents:
Preface
A Centennial History of the Prime Number Theorem T.M. Apostol
Non-Homogeneous Problems: Conjectures of Minkowski and Watson R.P. Bambah, V.C. Dumir & R.J. Hans-Gill
On the Oscillation Theorems of Pringsheim and Landau P.T. Bateman & H.G. Diamond
Modular Equations in Ramanujan's Lost Notebook B.C. Berndt
The abc-Conjecture J. Browkin
On Values of Linear and Quadratic Forms at Integral Points S.G. Dani
Variants of the Second Borel-Cantelli Lemma and their Applications in Metric Number Theory G. Harman
Pythagorean Triples E. Hlawka
Integer Points in Plane Regions and Exponential Sums M.N. Huxley
Artin's Conjecture for Polynomials Over Finite Fields E. Jensen & M. Ram Murty
Continuous Homomorphisms as Arithmetical Functions, and Sets of Uniqueness I. Katai
Hamburger's Theorem on $\zeta (s)$ and the Abundance Principle for Dirichlet Series with Functional Equations M.I. Knopp
A Survey of Number Theory and Cryptography N. Koblitz
Recent Developments in the Mean Square Theory of the Riemann Zeta and Other Zeta-Functions
K. Matsumoto
Algebraic Curves Over Finite Fields with many Rational Points and their Applications H. Niederreiter & C. Xing
A Report on Artin's Holomorphy Conjecture D. Prasad & C.S. Yogananda
Siegel's Main Theorem for Quadratic Forms S. Raghavan
Pfister's Work on Sums of Squares A.R. Rajwade
Notes on the Prime Number Theorem-I K. Ramachandra
Sums of Squares: An Elementary Method R.A. Rankin
Solution of the Basic Problems of Discrete Geometry in the Plane S.S. Ryshkov, R.G. Barykinskii & Y.V. Kucherinenko
Exponential Diophantine Equations Involving Products of Consecutive Integers and Related Equations
T.N. Shorey
Algebraic Independence of Transcendental Numbers: A Survey M. Waldschmidt
Israel Gohberg, Tel Aviv University, Israel
Seymour Goldberg, Silver Spring, MD
Nahum Krupnik, Bar Ilan University, Israel
Traces and Determinants of Linear Operators
3-7643-6177-8 * 2000 * Hardcover * 272 pages
This book is dedicated to a theory of traces and determinants on embedded algebras of linear operators, where the trace and determinant are extended from finite rank operators by a limit process. All the important classical examples of traces and determinants suggested by Hill, von Koch, Fredholm, Poincaré, Ruston and Grothendieck are exhibited. In particular, the determinants which were first introduced by Hill and Poincaré in their investigations of infinite systems of linear equations stemming from problems in celestial mechanics are studied. Most of Fredholm's seminal results are presented in this book by the editors' own means. Formulas for traces and determinants in a Hilbert space setting are readily derived, again. Also investigated are
generalizations to Banach spaces. Furthermore, a large part of this book is devoted to generalizations of the regularized determinants introduced by Hilbert and Carleman, and regularized determinants of higher order in embedded algebras are introduced. Much attention is paid to integral operators with semi-separable kernels, and explicit formulas of traces and determinants are given.
One of the conclusions of this book (based on results of Ben-Artziand Perelson) is that the trace and determinant, which are considered here, essentially depend not only on the operator but also on the algebra containing this operator. In fact, it turns out that by considering the same operator in different algebras, the trace and determinant of non nuclear operators can be almost any complex number. However, an operator is invertible if and only if each determinant is different from zero. Also each of the determinants can be used in the inversion formula.
An attractive feature of this book is that it contains the charming classical theory of determinants together with its most recent concrete andabstract developments and applications. The general presentation of thebook is based on the editors' work. The material is self-contained and may be used for advanced courses and seminars.
Series: Operator Theory: Advances and Applications, Vol. 116
Table of Contents:
Preface
Introduction
1. Finite Rank Operators
2. Continuous Extension of Trace and Determinant
3. First Examples
4. Trace Class and Hilbert-Schmidt Operators in Hilbert Space
5. Nuclear Operators in Banach Spaces
6. The Fredholm Determinant
7. Possible Values of Traces and Determinants, Perelson Algebras
8. Inversion Formulas
9. Regularized Determinants
10. Hilbert-Carleman Determinants
11. Regularized Determinants of Higher Order
12. Inversion Formulas via Generalized Determinants
13. Determinants of Integral Operators with Semi-separable Kernels
14. Algebras without the Approximation Theory
Bibliography
Index
List of Symbols
Herwig Hauser, Universität Innsbruck, Austria /Joseph Lipman, Purdue University, West Lafayette, IN
Frans Oort, Universiteit Utrecht, The Netherlands /Adolfo Quirós, Universidad Autónoma de Madrid, Spain (Eds.)
Resolution of Singularities
A Research Textbook in Tribute to Oscar Zariski
3-7643-6178-6 * 2000 * Hardcover * 624 pages
In September 1997, the Working Week on Resolution of Singularities was held at Obergurgi in the Tyrolean Alps. Its objective was to manifest the state of the art in the field and to formulate major questions for future research. The four courses given during this week were written up by the speakers and make up Part I of this volume. They are complemented in Part II by fifteen selected contributions on specific topics and resolution theories.
The volume is intended to provide a broad and accessible introduction to resolution of singularities leading the reader directly to concrete research problems.
Series: Progress in Mathematics, Vol. 181
Table of Contents
Preface
Program of Working Week on Resolution of Singularities
Participants
Oscar Zariski (1899-1986) J. Lipman
Resolution of singularities, 1860-1999 H. Hauser
1. Classes of the Working Week
Alterations and resolution of singularities D. Abramovich and F. Oort
Reduction of singularities for differential equations J.-M. Aroca
Puiseux solutions of singular differential equations J.-M. Aroca
A course on constructive desingularization and equivariance S. Encinas and O. Villamayor
Valuations, deformations and toric geometry B. Teissier
2. Contributions
A computer program for the resolution of singularities G. Bodnár and J. Schicho
Uniformisation et désingularisation des surfaces d'aprés Zariski V. Cossart
Toric varieties and toric resolution D. Cox
A compactification of a fine moduli space of curves B. van Geemen and F. Oort
T. GeisserApplications of de Jong's theorem on alterations
Resolving singularities of plane analytic branches with one toric morphism R. Goldin and B. Teissier
Excellent surfaces over fields and their taut resolution H. Hauser
An application of alterations to Dieudonné modules A.J. de Jong
Valuation theoretic and model theoretic aspects of local uniformization F.-V. Kuhlmann
Les singuarités Sandwich D.T. Lê
Equisingularity and simultaneous resolution of singularities J. Lipman
Resolution of weighted homogeneous surface singularities G. Müller
Alterations and birational anabelian geometry F. Pop
The turbulent fifties in resolution of singularities H. ReitbergerValuations M. Vaquié
Geir Ellingsrud, University of Oslo, Norway
William Fulton, University of Michigan, Ann Arbor
Angelo Vistoli, Università di Bologna, Italy (Eds.)
Recent Progress in Intersection Theory
0-8176-4122-X * 2000 * Hardcover * 400 pages
This collection of papers focuses on new concepts and results in intersection theory, enumerative geometry, and related topics; it is an outgrowth of a conference in intersection theory held in Bologna, Italy, in December 1997.
Many of the papers included here have a strongly expository content, yet they lead to the forefront of our knowledge. For this reason the work will be very useful to experts in intersection theory, as well as to graduate students and specialists in other areas of mathematics and physics.
The broad range of topics covered include:
algebraic stacks, moduli theory and Gromov-Witten invariants
recent applications of homotopy theory to intersection theory
an introduction to the formalism of motives, and to the Bloch-Beilinson filtration
an introduction to the excess intersection algorithm of Stückrad and Vogel and its geometric aspects
This volume will be an excellent resource for researchers and graduate students in algebraic geometry, and to other mathematicians interested in intersection theory and algebraic geometry.
Series: Trends in Mathematics
Contents
Preface
Complete Moduli for Fibered Surfaces D. Abramovich and A. Vistoli
Normal Differential Operators and Deformation Theory P. Burchard and H. Clemens
Notes on the Construction of the Moduli Space of Curves D. Edidin
The Excess Intersection Formula and Gravitational Correlators L. Ernström
Join Varieties and Intersection Theory H. Flenner
Intersection Products for Spaces of Algebraic Cycles E.M. Friedlander
Schur Q-Functions and Degeneracy Locus Fromulas for Morphisms with Symmetries A. Lascoux and P. Pragacz
Algebraic Cycles and Motives: An Introduction R. Laterveer
Equimultiplicity and Equidimensionality of Normal Cones H. Flenner and M. Manaresi
Zero Estimates on Abelian Varieties M. Nakamaye
Generalized Plücker Formulas A. Thorup
T. Andreescu, American Mathematics Competition, Univ. of NE, Lincoln
R. Gelca, University of Michigan, Ann Arbor, MI
Mathematical Olympiad Challenges
0-8176-4190-4 * 2000 * Hardcover * 300 pages * 75 Illustrations
Also available in Softcover.
Mathematical Olympiad Challenges is a rich collection of problems put together by two experienced and well-known professors and coaches of the U.S. International Mathematical Olympiad Team. Hundreds of beautiful, challenging, and instructive problems from algebra, geometry, trigonometry, combinatorics, and number theory were selected from numerous mathematical competitions and journals.
The problems are clustered by topic into self-contained sections with solutions provided separately. All sections start with an essay discussing basic facts and one or two representative examples. A list of carefully chosen problems follows and the reader is invited to take them on. Additionally, historical insights and asides are presented to stimulate further inquiry. The emphasis throughout is on encouraging readers to move away from routine exercises and memorized algorithms toward creative solutions to open-ended problems.
Key features of this work include:
Problems clustered in three self-contained sections Complete solutions provided separately Background material, representative examples, and beautiful diagrams to supplement each problem set Material successfully tested in classrooms as well as in national and international competitions Historical insights and asides presented to stimulate further inquiry
Aimed at motivated high school and beginning college students and instructors, this work can be used as a text for advanced problem-solving courses, for self-study, or as a resource for teachers and students training for mathematical competitions and for teacher professional development, seminars, and workshops.
Table of Contents
Preface
Part I: Problems
1. Geometry and Trigonometry
A property of equilateral triangles / Dissections of polygonal surfaces / Regular polygons
Cyclic quadrilaterals / Power of a point Geometric constructions and transformations
Problems with physical flavor / Tetrahedra inscribed in parallelepipeds
Telescopic sums and products in trigonometry / Trigonometric substitutions
2. Algebra and Analysis
No square is negative / Look at the endpoints / Systems of equations / The Abel summation formula
On an algebraic identity / Telescopic sums and products in algebra / Periodicity
Chebyshev polynomials / Matrices / The mean value theorem
3. Number Theory and Combinatorics
Arrange in order / Squares and cubes / Invariants / Repunits / Digits of numbers / Residues
Equations with unknowns as exponents / Numerical functions / Pell equations
Prime numbers and binomial coefficients
Part II: Solutions
To All Problems Posed in Part I
1. Geometry and Trigonometry
2. Algebra and Analysis
3. Number Theory and Combinatorics
Index
Glossary of Items
Bibliography
Albrecht Böttcher, University of Chemnitz, Germany
Sergei M. Grudsky, Rostov State University, Rostov-on-Don, Russia
Toeplitz Matrices, Asymptotic Linear Algebra, and Functional Analysis
3-7643-6290-1 * 2000 * Softcover * 104 pages
This text is a self-contained introduction to some problems for Toeplitz matrices that are placed in the borderland between linear algebra and functional analysis. The text looks at Toeplitz matrices with rational symbols, and focuses attention on the asymptotic behavior of the singular values, which includes the behavior of the norms, the norms of the inverses, and the condition numbers as special cases.
The text illustrates that the asymptotics of several linear algebra characteristics depend in a fascinating way on functional analytic properties of infinite matrices. Many convergence results can very comfortably be obtained by working with appropriate C*-algebras, while refinements of these results, for example, estimates of the convergence speed, nevertheless require hard analysis.
The book presents a course given by the author in New Delhi in January 2000. It is a self-contained introduction to some problems for Toeplitz matrices. The text illustrates that the asymptotics of several linear algebra characteristics depend in a fascinating way on functional analytic properties of
infinite matrices.
Contents:
Preface
1. Infinite Toeplitz Matrices
2.C*-Algebras in Action
3. Instability
4. Condition Numbers
5. Singular Values
Bibliography
Finn Aaserud, / Helge Kragh, / Erik Rüdinger, / Roger H. Stuewer, (Ed.)
Controversy and Consensus
Nuclear Beta Decay 1911-1934 Carsten Jensen (1948-1990)
3-7643-5313-9 * 2000 * Hardcover * 240 pages * 52 Illustrations
The book describes in detail the considerable efforts by theoretical and experimental physicists to understand the beta spectra of atomic nuclei. After a brief prehistory, the main narrative spans the period from 1911, when Rutherford and his collaborators in Manchester established that the atom had an extremely massive nucleus, until 1934, when the question of beta decay was settled theoretically by Fermi and others. The book discusses extensively the intense controversy over several years between Lise Meitner in Germany and C.D. Ellis in England about the origin of beta rays. Aside from providing a rich story in its own right, the narrative is uniquely suited to illuminate currently debated issues such as the relationship between theory and experiment, scientific controversy, the role and resolution of anomalies, and national differences in the approach to science.
Series: Science Networks: Historical Studies, Vol. 24
Contents:
Editor's Acknowledgments
Editor's Preface'
In Carsten Jensen's Memory
Author's Preface
Abbreviations
The Main Decay Chains
1. Prelude: Beta-Spectrum Research in the Pre-Nuclear Years, 1900-1911
2. The Origin of Beta Rays, and the Growing Complexity of Their Spectrum: The Rutherford Era, 1911-1919
3. The Rise of a Controversy: Ellis, Meitner and Smekal Advance Different Beta-Spectrum Theories, 1920-1922
4. Secondary Effects and Order of Emission: Two Main Questions in the Controversy, 1923-1925
5. The End of the Beginning: The Controversy Enters the Decisive Phase, 1925-1929
6. From Anomaly to Explanation: The Continuous Beta Spectrum, 1929-1934
7. Towards a Theory of Internal Conversion: The Beta Line-Spectrum, 1927-1934
Summary and Conclusion
Name Index
Alain Darte, Labor. de l'Informatique du Parallelisme, Lyon, France
Yves Robert, Labor. de l'Informatique du Parallelisme, Lyon, France
Frédéric Vivien, Université Louis Pasteur, Illkich, France
Scheduling and Automatic Parallelization
0-8176-4149-1 * 2000 * Hardcover * 280 pages
Program restructuring methods, particularly loop transformations, are important optimization techniques used in parallelizing compilers. New developments in loop transformations have emerged recently based on unimodular transformations and general affine transformations. The purpose of this new book is to offer an explanation of how to incorporate these transformations in algorithms, which transformations apply, and how to optimize them.
Scheduling and Automatic Parallelization offers a detailed and self-contained presentation for studying loop transformations, the detection of parallel loops, and how to use them to detect parallelism in a specific program. It provides careful explanation and exposition for all parallel-loop algorithms that have been designed recently in a framework of scheduling algorithms on cyclic graphs, primarily task graph scheduling and loop nest scheduling perspectives.
Features and Topics:
Complete state-of-the-art coverage of task graph scheduling Self-contained presentations that use an optimization perspective End-of-chapter exercises for reinforcement and practice Up-to-date results on acyclic scheduling in models with and without communication costs Scheduling cyclic graphs with one-dimensional and multi-dimensional weights Numerous detailed work examples used in text exposition
Detailed coverage of parallel-loop detection algorithms Optimization perspective used for presentations
High-performance Fortran (HPF) code used for algorithms
The book is an essential text/reference for the latest developments in automatic parallelization methods used for scheduling, compilers, and program transformations. Professionals, researchers and graduates in computer science, software engineering, and computer engineering will find it an authoritative resource and reference. It is also suitable for self-study purposes by professionals and practitioners.
Contents
Preface
Introduction
I. Unidimensional Problems
1. Scheduling DAGs without Communications
2. Scheduling DAGs with Communications
3. Cyclic Scheduling
II. Multidimensional problems
4. Systems of Uniform Recurrence Equations
5. Parallelism Detection in Nested Loops
Bibliography
Index
R. Vinter, Imperial College of Science, Technology & Medicine, London
Optimal Control
0-8176-4075-4 * 2000 * Hardcover * 365 pages
Optimal control emerged as a distinct field of research only in recent decades. It provides a unified perspective of optimization problems, arising in scheduling and the control of engineering devices, that are beyond the reach of traditional analytical and computational techniques. In addition, the field has contributed significant advances to branches of applied mathematics and broad applications in process control, scheduling, robotics, resource economics, and other areas.
Optimal Control brings together many of the important advances in 'nonsmooth' optimal control over the last two decades concerning necessary conditions, minimizer regularity and global optimality conditions associated with the Hamilton-Jacobi equation. The book's development and analysis is largely self-contained and incorporates many simplifications and unifying features for subject.
Features and Topics:
A comprehensive overview is provided for specialists and non-specialists Authoritative, coherent and accessible coverage of the role of nonsmooth analysis in investigating minimizing curves for optimization
Chapter coverage of dynamic programming from a system-theoretic viewpoint Thorough coverage of necessary conditions, including nonsmooth maximum principles and versions of the Euler Lagrange condition for differential inclusion problems Accessible treatment of topics, such as minimizer regularity and free end-time necessary conditions for problems with data discontinuous in time
This new book is an essential resource for an authoritative and comprehensive presentation of the foundations and application of nonsmooth optimal control. Postgraduates, researchers and professionals in systems, control, optimization and applied mathematics will find that the book is accessible, provides a rich source of insights and offers clear exposition.
Series: Systems and Control: Foundations and Applications
Contents
Chapter 1. Overview
Chapter 2. Measurable Multifunctions and Differential Inclusions
Chapter 3. Variational Principles
Chapter 4. Nonsmooth Analysis
Chapter 5. Subdifferential Calculus
Chapter 6. The Maximum Principle
Chapter 7. The Extended Euler and Hamilton Conditions
Chapter 8. Necessary Conditions for Free Time Problems
Chapter 9. The Maximum Principle for State Constrained Problems
Chapter 10. The Extended Euler Lagrange Condition and Hamilton Inclusion for State Constrained Problems
Chapter 11. Regularity of Minimizers
Chapter 12. Dynamic Programming
Index