Notices of the American Mathematical Society

We deal in this work with quantitative results in the pointwise approximation of func­ tions by positive linear functionals and operators. One of the main objectives is to obtain estimates for the degree of approximation in terms of various types of second order moduli of continuity. In the category of sec­ ond order moduli we include both classical and newly introduced moduli. Particular attention is paid to optimizing the constants appearing in such estimates. In the last decades, the study of linear positive operators with the aid of second order moduli was intensive, thanks to their refinements in characterization of the smoothness of functions. As promoters of this direction of research we mention Yu. Brudnyi, G. Freud, and J. Petree. Our approach is more akin to the approach taken by H. Gonska, who obtained the first general estimates for second order moduli with precise constants and with free parameters. Two new methods will be presented. The first one, based on decomposition of functionals and the use of moments, can be applied to diverse types of moduli and leads to simple estimates. The second method gives sufficient conditions for obtaining absolute optimal constants. The benefits of these more direct methods, compared with the known method based on K-functionals, consist in the improvement and even the optimization of the constants, and in the generalization of the framework.

This is the first monograph to exclusively treat Kac-Moody (K-M) groups, a standard tool in mathematics and mathematical physics. K-M Lie algebras were introduced in the mid-sixties independently by V. Kac and R. Moody, generalizing finite-dimensional semisimple Lie algebras. K-M theory has since undergone tremendous developments in various directions and has profound connections with a number of diverse areas, including number theory, combinatorics, topology, singularities, quantum groups, completely integrable systems, and mathematical physics.
This comprehensive, well-written text moves from K-M Lie algebras to the broader K-M Lie group setting, and focuses on the study of K-M groups and their flag varieties. In developing K-M theory from scratch, the author systematically leads readers to the forefront of the subject, treating the algebro-geometric, topological, and representation-theoretic aspects of the theory. Most of the material presented here is not available anywhere in the book literature.
{\it Kac—Moody Groups, their Flag Varieties and Representation Theory} is suitable for an advanced graduate course in representation theory, and contains a number of examples, exercises, challenging open problems, comprehensive bibliography, and index. Research mathematicians at the crossroads of representation theory, geometry, and topology will learn a great deal from this text; although the book is devoted to the general K-M case, those primarily interested in the finite-dimensional case will also benefit. No prior knowledge of K-M Lie algebras or of (finite-dimensional) algebraic groups is required, but some basic knowledge would certainly be helpful. For the reader's convenience some of the basic results needed from other areas, including ind-varieties, pro-algebraic groups and pro-Lie algebras, Tits systems, local cohomology, equivariant cohomology, and homological algebra are included.

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 Work Treats Quantitative Aspects Of The Approximation Of Functions Using Positive Linear Operators. The Theory Of These Operators Has Been An Important Area Of Research In The Last Few Decades, Particularly As It Affects Computer-aided Geometric Design. In This Book, The Crucial Role Of The Second Order Moduli Of Continuity In The Study Of Such Operators Is Emphasized. New And Efficient Methods, Applicable To General Operators And To Diverse Concrete Moduli, Are Presented. The Advantages Of These Methods Consist In Obtaining Improved And Even Optimal Estimates, As Well As In Broadening The Applicability Of The Results. Additional Topics And Features: * Examination Of The Multivariate Approximation Case * Special Focus On The Bernstein Operators, Including Applications, And On Two New Classes Of Bernstein-type Operators * Many General Estimates, Leaving Room For Future Applications (e.g. The B-spline Case) * Extensions To Approximation Operators Acting On Spaces Of Vector Functions * Historical Perspective In The Form Of Previous Significant Results This Monograph Will Be Of Interest To Those Working In The Field Of Approximation Or Functional Analysis. Requiring Only Familiarity With The Basics Of Approximation Theory, The Book May Serve As A Good Supplementary Text For Courses In Approximation Theory, Or As A Reference Text On The Subject. Introduction -- Estimates With Second Order Moduli -- Absolute Optimal Constants -- Estimates For The Bernstein Operators -- Two Classes Of Bernstein Type Operators -- Approximation Operators For Vector Valued Function -- References -- Index. By Radu Păltănea.

Complex variables is a precise, elegant, and captivating subject. Presented from the point of view of modern work in the field, this new book addresses advanced topics in complex analysis that verge on current areas of research, including invariant geometry, the Bergman metric, the automorphism groups of domains, harmonic measure, boundary regularity of conformal maps, the Poisson kernel, the Hilbert transform, the boundary behavior of harmonic and holomorphic functions, the inhomogeneous Cauchy–Riemann equations, and the corona problem.

The author adroitly weaves these varied topics to reveal a number of delightful interactions. Perhaps more importantly, the topics are presented with an understanding and explanation of their interrelations with other important parts of mathematics: harmonic analysis, differential geometry, partial differential equations, potential theory, abstract algebra, and invariant theory. Although the book examines complex analysis from many different points of view, it uses geometric analysis as its unifying theme.

This methodically designed book contains a rich collection of exercises, examples, and illustrations within each individual chapter, concluding with an extensive bibliography of monographs, research papers, and a thorough index. Seeking to capture the imagination of advanced undergraduate and graduate students with a basic background in complex analysis—and also to spark the interest of seasoned workers in the field—the book imparts a solid education both in complex analysis and in how modern mathematics works.

Complex variables is a precise, elegant, and captivating subject. Presented from a geometric analytical viewpoint, this work addresses advanced topics in complex analysis that verge on modern areas of research, including: invariant geometry, the Bergman metric, the automorphism groups of domains, extremal length, harmonic measure, boundary regularity of conformal maps, the inhomogeneous Cauchy-Riemann equations, and the corona problem. The author adroitly weaves these varied topics to reveal a number of delightful interactions. Perhaps more importantly, they are presented with an understanding and explanation of their interrelations with other important parts of mathematics: harmonic analysis, differential geometry, partial differential equations, potential theory, abstract algebra, and invariant theory. Containing an extensive bibliography of both monographs and research papers and a thorough index, the book is methodically designed with individualchapters containing a rich collection of exercises, examples, and illustrations.Seeking to capture the imagination of both advancedundergraduate and graduate students with a basic background in complex analysis,the book impartsa solid educationboth in complex analysis and in how modern mathematics works. TOC:Preface * Invariant Geometry * The Bergman Metric * Automorphism Groups of Domains * Extremal Length * Harmonic Measure * Boundary Regularity of Conformal Maps * The InhomogeneousCauchy-Riemann Equations * The Corona Problem * Bibliography * Index This volume! aims at introducing some basic ideas for studying approxima­ tion processes and, more generally, discrete processes. The study of discrete processes, which has grown together with the study of infinitesimal calcu­ lus, has become more and more relevant with the use of computers. The volume is suitably divided in two parts. In the first part we illustrate the numerical systems of reals, of integers as a subset of the reals, and of complex numbers. In this context we intro­ duce, in Chapter 2, the notion of sequence which invites also a rethinking of the notions of limit and continuity2 in terms of discrete processes; then, in Chapter 3, we discuss some elements of combinatorial calculus and the mathematical notion of infinity. In Chapter 4 we introduce complex num­ bers and illustrate some of their applications to elementary geometry; in Chapter 5 we prove the fundamental theorem of algebra and present some of the elementary properties of polynomials and rational functions, and of finite sums of harmonic motions. In the second part we deal with discrete processes, first with the process of infinite summation, in the numerical case, i.e., in the case of numerical series in Chapter 6, and in the case of power series in Chapter 7. The last chapter provides an introduction to discrete dynamical systems; it should be regarded as an invitation to further study. Complex variables is a precise, elegant, and captivating subject. Presented from the point of view of modern work in the field, this new book addresses advanced topics in complex analysis that verge on current areas of research. The author adroitly weaves these varied topics to reveal a number of delightful interactions. Perhaps more importantly, the topics are presented with an understanding and explanation of their interrelations with other important parts of mathematics: harmonic analysis, differential geometry, partial differential equations, potential theory, abstract algebra, and invariant theory. Although the book examines complex analysis from many different points of view, it uses geometric analysis as its unifying theme. This methodically designed book contains a rich collection of exercises, examples, and illustrations within each individual chapter, concluding with an extensive bibliography of monographs, research papers, and a thorough index. Seeking to capture the imagination of advanced undergraduate and graduate students with a basic background in complex analysis –and also to spark the interest of seasoned workers in the field – the book imparts a solid education both in complex analysis and in how modern mathematics works. 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 Over Two Hundred Novel And Innovative Computer Algebra Worksheets Or Recipes Will Enable Readers In Engineering, Physics, And Mathematics To Easily And Rapidly Solve And Explore Most Problems They Encounter In Their Mathematical Physics Studies. While The Aim Of This Text Is To Illustrate Applications, A Brief Synopsis Of The Fundamentals For Each Topic Is Presented, The Topics Being Organized To Correlate With Those Found In Traditional Mathematical Physics Texts. The Recipes Are Presented In The Form Of Stories And Anecdotes, A Pedagogical Approach That Makes A Mathematically Challenging Subject Easier And More Fun To Learn.--jacket. I. The Appetizers -- 1. Linear Odes Of Physics -- 2. Applications Of Series -- 3. Vectors And Matrices -- Ii. The Entrees -- 4. Linear Pdes Of Physics -- 5. Complex Variables -- 6. Integral Transforms -- 7. Calculus Of Variations -- Iii. The Desserts -- 8. Nlodes And Pdes Of Physics -- 9. Numerical Methods. Richard H. Enns. Includes Bibliographical References (p. [375]-378) And Index. "This fairly self-contained work embraces a broad range of topics in analysis at the graduate level, requiring only a sound knowledge of calculus and the functions of one variable. A key feature of this lively yet rigorous and systematic exposition is the historical accounts of ideas and methods pertaining to the relevant topics. Most interesting and useful are the connections developed between analysis and other mathematical disciplines, in this case, numerical analysis and probability theory. The text is divided into two parts: The first examines the systems of real and complex numbers and deals with the notion of sequences in this context. After the presentation of natural numbers as a subset of the reals, elements of combinatorics and a discussion of the mathematical notion of the infinite are introduced. The second part is dedicated to discrete processes starting with a study of the processes of infinite summation both in the case of numerical series and of power series." -- Publisher Kac-Moody Lie algebras 9 were introduced in the mid-1960s independently by V. Kac and R. Moody, generalizing the finite-dimensional semisimple Lie alge­ bras which we refer to as the finite case. The theory has undergone tremendous developments in various directions and connections with diverse areas abound, including mathematical physics, so much so that this theory has become a stan­ dard tool in mathematics. A detailed treatment of the Lie algebra aspect of the theory can be found in V. Kac's book [Kac-90l This self-contained work treats the algebro-geometric and the topological aspects of Kac-Moody theory from scratch. The emphasis is on the study of the Kac-Moody groups 9 and their flag varieties XY, including their detailed construction, and their applications to the representation theory of g. In the finite case, 9 is nothing but a semisimple Y simply-connected algebraic group and X is the flag variety 9 /Py for a parabolic subgroup p y C g. Cover Table of Contents Feature Articles Discretizing Manifolds via Minimum Energy Points Comme Appelé du Néant—As if Summoned from the Void: The Life of Alexandre Grothendieck, Part II Communications WHAT IS...a Motive? The Elephant in the Internet Some of What Mathematicians Do Société Mathématique de France Commentary Opinion Letters to the Editor The Constants of Nature and Just Six Numbers —A Book Review Departments Mathematics People Mathematics Opportunities Inside the AMS Reference and Book List Mathematics Calendar New Publications Offered by the AMS Classified Advertisements 2005 Summer Research Institute—Algebraic Geometry Joint Summer Research Conferences 2005 Atlanta Meeting Registration Forms Meetings and Conferences Table of Contents From the AMS Secretary 2004 Spring Policy Committee Reports Reciprocity Agreements Over two hundred novel and innovative computer algebra worksheets or'recipes'will enable readers in engineering, physics, and mathematics to easily and rapidly solve and explore most problems they encounter in their mathematical physics studies. While the aim of this text is to illustrate applications, a brief synopsis of the fundamentals for each topic is presented, the topics being organized to correlate with those found in traditional mathematical physics texts. The recipes are presented in the form of stories and anecdotes, a pedagogical approach that makes a mathematically challenging subject easier and more fun to learn. This is a self-contained and standalone text using MAPLE that may be used in the classroom, for self-study, as a reference, or as a text for an online course. "This methodologically designed book contains a rich collection of exercises, examples, and illustrations within each individual chapter, concluding with an extensive bibliography of monographs, research papers, and a thorough index. Seeking to capture the imagination of advanced undergraduate and graduate students with a basic background in complex analysis -- and also to spark the interest of seasoned workers in the field -- the book imparts a solid education both in complex analysis and in how modern mathematics works"--Jacket * Uses a pedagogical approach that makes a mathematically challenging subject easier and more fun to learn * Self-contained and standalone text that may be used in the classroom, for an online course, for self-study, as a reference * Using MAPLE allows the reader to easily and quickly change the models and parameters "This monograph will be of interest to those working in the field of approximation or functional analysis. Requiring only familiarity with the basics of approximation theory, the book may serve as a good supplementary text for courses in approximation theory, or as a reference text on the subject."--Jacket Rudiments of mathematics, or even refined geometrical and algebraic rules appear in many ancient civilizations, as for instance the Babylonian, the Egyptian, the Hindu, the Chinese or some of the pre-Colombian civilizations. 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 The most constructive proofs of the Weierstrass theorem concerning the approximation of continuous functions on a compact interval by polynomials use some sequences of linear positive operators.