Notices of the American Mathematical Society

'An important topic, which is on the boundary between numerical analysis and computer science .... I found the book well written and containing much interesting material, most of the time disseminated in specialized papers published in specialized journals difficult to find. Moreover, there are very few books on these topics and they are not recent.'Numerical Algorithms (review of the first edition) This unique book provides concepts and background necessary to understand and build algorithms for computing the elementary functions. The author presents and structures the algorithms (hardware-oriented as well as software-oriented), and also discusses issues related to accurate floating-point implementation. The purpose is not to give'cookbook recipes'that allow one to implement a given function, but rather to provide the reader with tools necessary to build, or adapt, algorithms to their specific computing environment. The expanded second edition contains a number of revisions and additions, which incorporate numerous new results obtained during the last few years. New algorithms invented since 1997 – such as Matula's bipartite method, another table-based method due to Ercegovac, Lang, Tisserand, and Muller – as well as a new chapter on multiple-precision arithmetic have been added. In addition, the section on correct rounding of elementary functions has been fully reworked, also in the context of new results. Finally, the introductory presentation of floating-point arithmetic has been expanded, with more emphasis given to the use of the fused multiply-accumulate instruction. The book is an up-to-date presentation of information needed to understand and accurately use mathematical functions and algorithms in computational work and design. Graduates, professionals, and researchers in scientific computing, software engineering, and computer engineering will find the book a useful reference and resource.

"An important topic, which is on the boundary between numerical analysis and computer science.... I found the book well written and containing much interesting material, most of the time disseminated in specialized papers published in specialized journals difficult to find. Moreover, there are very few books on these topics and they are not recent."

–Numerical Algorithms (review of the first edition)

This unique book provides concepts and background necessary to understand and build algorithms for computing the elementary functions—sine, cosine, tangent, exponentials, and logarithms. The author presents and structures the algorithms, hardware-oriented as well as software-oriented, and also discusses issues related to accurate floating-point implementation. The purpose is not to give "cookbook recipes" that allow one to implement a given function, but rather to provide the reader with tools necessary to build or adapt algorithms for their specific computing environment.

This expanded second edition contains a number of revisions and additions, which incorporate numerous new results obtained during the last few years. New algorithms invented since 1997—such as Matula’s bipartite method, another table-based method due to Ercegovac, Lang, Tisserand, and Muller—as well as new chapters on multiple-precision arithmetic and examples of implementation have been added. In addition, the section on correct rounding of elementary functions has been fully reworked, also in the context of new results. Finally, the introductory presentation of floating-point arithmetic has been expanded, with more emphasis given to the use of the fused multiply-accumulate instruction.

The book is an up-to-date presentation of information needed to understand and accurately use mathematical functions and algorithms in computational work and design. Graduate and advanced undergraduate students, professionals, and researchers in scientific computing, numerical analysis, software engineering, and computer engineering will find the book a useful reference and resource.

Basic Algebra and Advanced Algebra systematically develop concepts and tools in algebra that are vital to every mathematician, whether pure or applied, aspiring or established. Together, the two books give the reader a global view of algebra and its role in mathematics as a whole.

Key topics and features of Basic Algebra:

*Linear algebra and group theory build on each other continually

*Chapters on modern algebra treat groups, rings, fields, modules, and Galois groups, with emphasis on methods of computation throughout

*Three prominent themes recur and blend together at times: the analogy between integers and polynomials in one variable over a field, the interplay between linear algebra and group theory, and the relationship between number theory and geometry

*Many examples and hundreds of problems are included, along with a separate 90-page section giving hints or complete solutions for most of the problems

*The exposition proceeds from the particular to the general, often providing examples well before a theory that incorporates them; includes blocks of problems that introduce additional topics and applications for further study

*Applications to science and engineering (e.g., the fast Fourier transform, the theory of error-correcting codes, the use of the Jordan canonical form in solving linear systems of ordinary differential equations, and constructions of interest in mathematical physics) appear in sequences of problems

Basic Algebra presents the subject matter in a forward-looking way that takes into account its historical development. It is suitable as a text in a two-semester advanced undergraduate or first-year graduate sequence in algebra, possibly supplemented by some material from Advanced Algebra at the graduate level. It requires of the reader only familiarity with matrix algebra, an understanding of the geometry and reduction of linear equations, and an acquaintance with proofs.

Basic Algebra and Advanced Algebra systematically develop concepts and tools in algebra that are vital to every mathematician, whether pure or applied, aspiring or established. Together, the two books give the reader a global view of algebra and its role in mathematics as a whole. Key topics and features of Basic Algebra: *Linear algebra and group theory build on each other continually *Chapters on modern algebra treat groups, rings, fields, modules, and Galois groups, with emphasis on methods of computation throughout *Three prominent themes recur and blend together at times: the analogy between integers and polynomials in one variable over a field, the interplay between linear algebra and group theory, and the relationship between number theory and geometry *Many examples and hundreds of problems are included, along with a separate 90-page section giving hints or complete solutions for most of the problems *The exposition proceeds from the particular to the general, often providing examples well before a theory that incorporates them; includes blocks of problems that introduce additional topics and applications for further study *Applications to science and engineering (e.g., the fast Fourier transform, the theory of error-correcting codes, the use of the Jordan canonical form in solving linear systems of ordinary differential equations, and constructions of interest in mathematical physics) appear in sequences of problems Basic Algebra presents the subject matter in a forward-looking way that takes into account its historical development. It is suitable as a text in a two-semester advanced undergraduate or first-year graduate sequence in algebra, possibly supplemented by some material from Advanced Algebra at the graduate level. It requires of the reader only familiarity with matrix algebra, an understanding of the geometry and reduction of linear equations, and an acquaintance with proofs Recollection Of Tony Joseph / Jacques Dixmier -- Work Of Anthony Joseph In Classical Representation Theory / W. Mcgovern -- Quantized Representation Theory Following Joseph / Daniel R. Farkas And Gail Letzter -- Operateurs Differentiels Invariants Et Probleme De Noether / Jacques Alev And Francois Dumas -- Langlands Parameters For Heisenberg Modules / A. Beilinson -- Instanton Counting Via Affine Lie Algebras Ii : From Whittaker Vectors To The Seiberg-witten Prepotential / A. Braverman And P. Etingof -- Irreducibility Of Perfect Representations Of Double Affine Hecke Algebras / Ivan Cherednik -- Algebraic Groups Over A 2-dimensional Local Field : Some Further Constructions / Dennis Gaitsgory And David Kazhdan -- Modules With A Demazure Flag / Anthony Joseph -- Microlocalization Of Ind-sheaves / M. Kashiwara, P. Schapira, F. Ivorra And I. Waschkies -- Endoscopic Decomposition Of Certain Depth Zero Representations / David Kazhdan And Yakov Varshavsky -- Odd Family Algebras / A. A. Kirillov And L. G. Rybnikov -- Gelfand-zeitlin Theory From The Perspective Of Classical Mechanics : I / Bertram Kostant And Nolan Wallach -- Extensions Of Algebraic Groups / Shrawan Kumar And Karl-hermann Neeb -- Differential Operators And Cohomology Groups On The Basic Affine Space / Thierry Levasseur And J. T. Stafford -- Q-analogue Of An Identity Of N. Wallach / G. Lusztig -- Centralizers In The Quantum Plane Algebra / L. Makar-limanov -- Centralizer Construction Of The Yangian Of The Queer Lie Superalgebra / Maxim Nazarov And Alexander Sergeev -- Definito Nova Algebroidis Verticiani / Vadim Schechtman. Joseph Bernstein, Vladimir Hinich, Anna Melnikov, Editors. Includes Bibliographical References. Publications Of Anthony Joseph: P. [xiii]-xix. "This book provides concepts and background necessary to understand and build algorithms for computing the elementary functions - sine, cosine, tangent, exponentials, and logarithms. The author presents and structures the algorithms, hardware-oriented as well as software-oriented, and also discusses issues related to accurate floating-point implementation. The purpose is not to give "cookbook recipes" that allow one to implement a given function, but rather to provide the reader with tools necessary to build or adapt algorithms for their specific computing environment." "This expanded second edition contains a number of revisions and additions, which incorporate numerous new results obtained during the last few years. New algorithms invented since 1997 - such as Matula's bipartite method, another table-based method due to Ercegovac, Lang, Tisserand, and Muller - as well as new chapters on multiple-precision arithmetic and examples of implementation have been added. In addition, the section on correct rounding of elementary functions has been fully reworked, also in the context of new results. Finally, the introductory presentation of floating-point arithmetic has been expanded, with more emphasis given to the use of the fused multiply-accumulate instruction." "The book is an up-to-date presentation of information needed to understand and accurately use mathematical functions and algorithms in computational work and design. Graduate and advanced undergraduate students, professionals, and researchers in scientific computing, numerical analysis, software engineering, and computer engineering will find the book a useful reference and resource."--Jacket The Study Of Cr Manifolds Lies At The Intersection Of Three Main Mathematical Disciplines: Partial Differential Equations, Complex Analysis In Several Complex Variables, And Differential Geometry. While The Pde And Complex Analytic Aspects Have Been Intensely Studied In The Last Fifty Years, Much Effort Has Recently Been Made To Understand The Differential Geometric Side Of The Subject. This Monograph Provides A Unified Presentation Of Several Differential Geometric Aspects In The Theory Of Cr Manifolds And Tangential Cauchy-riemann Equations. It Presents The Major Differential Geometric Acheivements In The Theory Of Cr Manifolds, Such As The Tanaka-webster Connection, Fefferman's Metric, Pseudo-einstein Structures And The Lee Conjecture, Cr Immersions, Subelliptic Harmonic Maps As A Local Manifestation Of Pseudoharmonic Maps From A Cr Manifold, Yang-mills Fields On Cr Manifolds, To Name A Few. It Also Aims At Explaining How Certain Results From Analysis Are Employed In Cr Geometry. Motivated By Clear Exposition, Many Examples, Explicitly Worked-out Geometric Results, And Stimulating Unproved Statements And Comments Referring To The Most Recent Aspects Of The Theory, This Monograph Is Suitable For Researchers And Graduate Students In Differential Geometry, Complex Analysis, And Pdes.--publisher's Website. Introduction -- Cr Manifolds -- Fefferman Metric -- Cr Yamabe Problem -- Pseudoharmonic Maps -- Pseudo-einsteinian Manifolds -- Pseudo-hermitian Immersions -- Quasiconfromal Mappings -- Yang-mills Fields On Cr Manifolds -- Spectral Geometry Sorin Dragomir, Giuseppe Tomassini. Includes Bibliographical References (p. [463]-482) And Index. Basic Algebra and Advanced Algebra systematically develop concepts and tools in algebra that are vital to every mathematician, whether pure or applied, aspiring or established. Together, the two books give the reader a global view of algebra and its role in mathematics as a whole. The exposition proceeds from the particular to the general, often providing examples well before a theory that incorporates them. The presentation includes blocks of problems that introduce additional topics and applications to science and engineering to guide further study. Many examples and hundreds of problems are included, along with a separate 90-page section giving hints or complete solutions for most of the problems. Basic Algebra presents the subject matter in a forward-looking way that takes into account its historical development. It is suitable as a text in a two-semester advanced undergraduate or first-year graduate sequence in algebra, possibly supplemented by some material from Advanced Algebra at the graduate level. It requires of the reader only familiarity with matrix algebra, an understanding of the geometry and reduction of linear equations, and an acquaintance with proofs. Dedicated to Anthony Joseph, this volume contains surveys and invited articles by leading specialists in representation theory. The focus here is on semisimple Lie algebras and quantum groups, where the impact of Joseph's work has been seminal and has changed the face of the subject. Two introductory biographical overviews of Joseph's contributions in classical representation theory (the theory of primitive ideals in semisimple Lie algebras) and quantized representation theory (the study of the quantized enveloping algebra) are followed by 16 research articles covering a number of varied and interesting topics in representation theory. Contributors: J. Alev; A. Beilinson; A. Braverman; I. Cherednik; J. Dixmier; F. Dumas; P. Etingof; D. Farkas; D. Gaitsgory; F. Ivorra; A. Joseph; D. Joseph; M. Kashiwara; D. Kazhdan; A.A. Kirillov; B. Kostant; S. Kumar; G. Letzter; T. Levasseur; G. Lusztig; L. Makar-Limanov; W. McGovern; M. Nazarov; K-H. Neeb; L.G. Rybnikov; P. Schapira; V. Schechtman; A. Sergeev; J.T. Stafford; Ya. Varshavsky; N. Wallach; and I. Waschkies This monograph presents a comprehensive treatment of important new ideas on Dirac operators and Dirac cohomology. Dirac operators are widely used in physics, differential geometry, and group-theoretic settings (particularly, the geometric construction of discrete series representations). The related concept of Dirac cohomology, which is defined using Dirac operators, is a far-reaching generalization that connects index theory in differential geometry to representation theory. Using Dirac operators as a unifying theme, the authors demonstrate how some of the most important results in representation theory fit together when viewed from this perspective. An excellent contribution to the mathematical literature of representation theory, this self-contained exposition offers a systematic examination and panoramic view of the subject. The material will be of interest to researchers and graduate students in representation theory, differential geometry, and physics. Cover Table of Contents Feature Articles Pictures of Hyperbolic Dynamical Systems You Could Have Invented Spectral Sequences Mathematicians and Mathematics Textbooks for Prospective Elementary Teachers Communications WHAT IS...a Quantum Group? Aumann Awarded Nobel Prize MathSciNet Matters Commentary Letter from the Editor: Deaths and Didactics Letters to the Editor Probability Theory: The Logic of Science-A Book Review Departments Mathematics People Mathematics Opportunities For Your Information Inside the AMS Reference and Book List Mathematics Calendar New Publications Offered by the AMS Classified Advertisements AMS Standard Cover Sheet AAAS Conference in St. Louis, MO General Information Regarding Meetings & Conferences of the AMS Meetings & Conferences of the AMS Presenters of Papers, San Antonio Meeting Program of the Sessions, San Antonio Meeting Meetings and Conferences Table of Contents "This monograph examines and develops the Global Smoothness Preservation Property (GSPP) and the Shape Preservation Property (SPP) in the field of interpolation of functions. The study is developed for the univariate and bivariate cases using well-known classical interpolation operators of Lagrange, Grunwald, Hermite-Fejer and Shepard type. One of the first books on the subject, it presents interesting new results along with an excellent survey of past research." "This text is best suited to graduate students and researchers in mathematical analysis, interpolation of functions, pure and applied mathematicians in numerical analysis, approximation theory, data fitting, computer-aided geometric design, fluid mechanics, and engineering researchers."--Jacket "This monograph presents a comprehensive treatment of important new ideas on Dirac operators and Dirac cohomology. Dirac operators are widely used in physics, differential geometry, and group-theoretic settings (particularly, the geometric construction of discrete series representations). The related concept of Dirac cohomology, which is defined using Dirac operators, is a far-reaching generalization that connects index theory in differential geometry to representation theory. Using Dirac operators as a unifying theme, the authors demonstrate how some of the most important results in representation theory fit together when viewed from this perspective."--Publisher's website "This monograph provides a unified presentation of several differential geometric aspects in the theory of CR manifolds and tangential Cauchy-Riemann equations. It presents the major differential geometric acheivements in the theory of CR manifolds, such as the Tanaka-Webster connection, Fefferman's metric, pseudo-Einstein structures and the Lee conjecture, CR immersions, subelliptic harmonic maps as a local manifestation of pseudoharmonic maps from a CR manifold, Yang-Mills fields on CR manifolds, to name a few. It also aims at explaining how certain results from analysis are employed in CR geometry."--Jacket This work deals with Numerical Algorithms. This unique book provides concepts and background necessary to understand and build algorithms for computing the elementary functions - sine, cosine, tangent, exponentials, and logarithms. The author presents and structures the algorithms, hardware-oriented as well as software-oriented, and also discusses issues related to accurate floating-point implementation. The purpose is not to give ""cookbook recipes"" that allow one to implement a given function, but rather to provide the reader with tools necessary to build or adapt algorithms for their speci Presents many major differential geometric acheivements in the theory of CR manifolds for the first time in book form Explains how certain results from analysis are employed in CR geometry Many examples and explicitly worked-out proofs of main geometric results in the first section of the book making it suitable as a graduate main course or seminar textbook Provides unproved statements and comments inspiring further study "The book is an up-to-date presentation of information needed to understand and accurately use mathematical functions and algorithms in computational work and design. Graduate and advanced undergraduate students, professionals, and researchers in scientific computing, numerical analysis, software engineering, and computer engineering will find the book a useful reference and resource."--Résumé de l'éditeur Second Edition of successful, well-reviewed Birkhauser book, which sold 866 copies in North America Provides an up-to-date presentation by including new results, examples, and problems throughout the text The second edition adds a chapter on multiple-precision arithmetic, and new algorithms invented since 1997 This book presents a comprehensive treatment of important new ideas on Dirac operators and Dirac cohomology. Using Dirac operators as a unifying theme, the authors demonstrate how some of the most important results in representation theory fit together when viewed from this perspective. Contains new results on different aspects of Lie theory, including Lie superalgebras, quantum groups, crystal bases, representations of reductive groups in finite characteristic, and the geometric Langlands program