Polynomial Identities And Combinatorial Methods

Author : Antonio Giambruno,Amitai Regev,Mikhail Zaicev
Publisher : CRC Press
Page : 442 pages
File Size : 53,7 Mb
Release : 2003-05-20
Category : Mathematics
ISBN : 0203911547

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Polynomial Identities And Combinatorial Methods by Antonio Giambruno,Amitai Regev,Mikhail Zaicev Pdf

Polynomial Identities and Combinatorial Methods presents a wide range of perspectives on topics ranging from ring theory and combinatorics to invariant theory and associative algebras. It covers recent breakthroughs and strategies impacting research on polynomial identities and identifies new concepts in algebraic combinatorics, invariant and representation theory, and Lie algebras and superalgebras for novel studies in the field. It presents intensive discussions on various methods and techniques relating the theory of polynomial identities to other branches of algebraic study and includes discussions on Hopf algebras and quantum polynomials, free algebras and Scheier varieties.

Author : Vladimir Shpilrain,Alexander Mikhalev,Jie-tai Yu
Publisher : Springer Science & Business Media
Page : 322 pages
File Size : 50,9 Mb
Release : 2012-11-12
Category : Mathematics
ISBN : 9780387217246

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Combinatorial Methods by Vladimir Shpilrain,Alexander Mikhalev,Jie-tai Yu Pdf

The main purpose of this book is to show how ideas from combinatorial group theory have spread to two other areas of mathematics: the theory of Lie algebras and affine algebraic geometry. Some of these ideas, in turn, came to combinatorial group theory from low-dimensional topology in the beginning of the 20th Century.

Author : A. Giambruno,Mikhail Zaicev
Publisher : American Mathematical Soc.
Page : 370 pages
File Size : 45,9 Mb
Release : 2005
Category : Mathematics
ISBN : 9780821838297

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Polynomial Identities and Asymptotic Methods by A. Giambruno,Mikhail Zaicev Pdf

This book gives a state of the art approach to the study of polynomial identities satisfied by a given algebra by combining methods of ring theory, combinatorics, and representation theory of groups with analysis. The idea of applying analytical methods to the theory of polynomial identities appeared in the early 1970s and this approach has become one of the most powerful tools of the theory. A PI-algebra is any algebra satisfying at least one nontrivial polynomial identity. This includes the polynomial rings in one or several variables, the Grassmann algebra, finite-dimensional algebras, and many other algebras occurring naturally in mathematics. The core of the book is the proof that the sequence of co-dimensions of any PI-algebra has integral exponential growth - the PI-exponent of the algebra. Later chapters further apply these results to subjects such as a characterization of varieties of algebras having polynomial growth and a classification of varieties that are minimal for a given exponent.

Author : Vesselin Drensky,Edward Formanek
Publisher : Birkhäuser
Page : 197 pages
File Size : 42,8 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9783034879347

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Polynomial Identity Rings by Vesselin Drensky,Edward Formanek Pdf

These lecture notes treat polynomial identity rings from both the combinatorial and structural points of view. The greater part of recent research in polynomial identity rings is about combinatorial questions, and the combinatorial part of the lecture notes gives an up-to-date account of recent research. On the other hand, the main structural results have been known for some time, and the emphasis there is on a presentation accessible to newcomers to the subject.

Author : Alexei Kanel-Belov,Yakov Karasik,Louis Halle Rowen
Publisher : CRC Press
Page : 418 pages
File Size : 40,6 Mb
Release : 2015-10-22
Category : Mathematics
ISBN : 9781498720090

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Computational Aspects of Polynomial Identities by Alexei Kanel-Belov,Yakov Karasik,Louis Halle Rowen Pdf

Computational Aspects of Polynomial Identities: Volume l, Kemer’s Theorems, 2nd Edition presents the underlying ideas in recent polynomial identity (PI)-theory and demonstrates the validity of the proofs of PI-theorems. This edition gives all the details involved in Kemer’s proof of Specht’s conjecture for affine PI-algebras in characteristic 0. The book first discusses the theory needed for Kemer’s proof, including the featured role of Grassmann algebra and the translation to superalgebras. The authors develop Kemer polynomials for arbitrary varieties as tools for proving diverse theorems. They also lay the groundwork for analogous theorems that have recently been proved for Lie algebras and alternative algebras. They then describe counterexamples to Specht’s conjecture in characteristic p as well as the underlying theory. The book also covers Noetherian PI-algebras, Poincaré–Hilbert series, Gelfand–Kirillov dimension, the combinatoric theory of affine PI-algebras, and homogeneous identities in terms of the representation theory of the general linear group GL. Through the theory of Kemer polynomials, this edition shows that the techniques of finite dimensional algebras are available for all affine PI-algebras. It also emphasizes the Grassmann algebra as a recurring theme, including in Rosset’s proof of the Amitsur–Levitzki theorem, a simple example of a finitely based T-ideal, the link between algebras and superalgebras, and a test algebra for counterexamples in characteristic p.

Author : Eli Aljadeff,Antonio Giambruno,Claudio Procesi,Amitai Regev
Publisher : American Mathematical Soc.
Page : 630 pages
File Size : 52,5 Mb
Release : 2020-12-14
Category : Education
ISBN : 9781470451745

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Rings with Polynomial Identities and Finite Dimensional Representations of Algebras by Eli Aljadeff,Antonio Giambruno,Claudio Procesi,Amitai Regev Pdf

A polynomial identity for an algebra (or a ring) A A is a polynomial in noncommutative variables that vanishes under any evaluation in A A. An algebra satisfying a nontrivial polynomial identity is called a PI algebra, and this is the main object of study in this book, which can be used by graduate students and researchers alike. The book is divided into four parts. Part 1 contains foundational material on representation theory and noncommutative algebra. In addition to setting the stage for the rest of the book, this part can be used for an introductory course in noncommutative algebra. An expert reader may use Part 1 as reference and start with the main topics in the remaining parts. Part 2 discusses the combinatorial aspects of the theory, the growth theorem, and Shirshov's bases. Here methods of representation theory of the symmetric group play a major role. Part 3 contains the main body of structure theorems for PI algebras, theorems of Kaplansky and Posner, the theory of central polynomials, M. Artin's theorem on Azumaya algebras, and the geometric part on the variety of semisimple representations, including the foundations of the theory of Cayley–Hamilton algebras. Part 4 is devoted first to the proof of the theorem of Razmyslov, Kemer, and Braun on the nilpotency of the nil radical for finitely generated PI algebras over Noetherian rings, then to the theory of Kemer and the Specht problem. Finally, the authors discuss PI exponent and codimension growth. This part uses some nontrivial analytic tools coming from probability theory. The appendix presents the counterexamples of Golod and Shafarevich to the Burnside problem.

Author : Onofrio Mario Di Vincenzo,Antonio Giambruno
Publisher : Springer Nature
Page : 421 pages
File Size : 51,7 Mb
Release : 2021-03-22
Category : Mathematics
ISBN : 9783030631116

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Polynomial Identities in Algebras by Onofrio Mario Di Vincenzo,Antonio Giambruno Pdf

This volume contains the talks given at the INDAM workshop entitled "Polynomial identites in algebras", held in Rome in September 2019. The purpose of the book is to present the current state of the art in the theory of PI-algebras. The review of the classical results in the last few years has pointed out new perspectives for the development of the theory. In particular, the contributions emphasize on the computational and combinatorial aspects of the theory, its connection with invariant theory, representation theory, growth problems. It is addressed to researchers in the field.

Author : Larry Guth
Publisher : American Mathematical Soc.
Page : 273 pages
File Size : 45,9 Mb
Release : 2016-06-10
Category : Combinatorial analysis
ISBN : 9781470428907

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Polynomial Methods in Combinatorics by Larry Guth Pdf

This book explains some recent applications of the theory of polynomials and algebraic geometry to combinatorics and other areas of mathematics. One of the first results in this story is a short elegant solution of the Kakeya problem for finite fields, which was considered a deep and difficult problem in combinatorial geometry. The author also discusses in detail various problems in incidence geometry associated to Paul Erdős's famous distinct distances problem in the plane from the 1940s. The proof techniques are also connected to error-correcting codes, Fourier analysis, number theory, and differential geometry. Although the mathematics discussed in the book is deep and far-reaching, it should be accessible to first- and second-year graduate students and advanced undergraduates. The book contains approximately 100 exercises that further the reader's understanding of the main themes of the book.

Author : Jocelyn Quaintance,H W Gould
Publisher : World Scientific
Page : 276 pages
File Size : 50,6 Mb
Release : 2015-10-27
Category : Mathematics
ISBN : 9789814725293

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Combinatorial Identities for Stirling Numbers by Jocelyn Quaintance,H W Gould Pdf

' This book is a unique work which provides an in-depth exploration into the mathematical expertise, philosophy, and knowledge of H W Gould. It is written in a style that is accessible to the reader with basic mathematical knowledge, and yet contains material that will be of interest to the specialist in enumerative combinatorics. This book begins with exposition on the combinatorial and algebraic techniques that Professor Gould uses for proving binomial identities. These techniques are then applied to develop formulas which relate Stirling numbers of the second kind to Stirling numbers of the first kind. Professor Gould''s techniques also provide connections between both types of Stirling numbers and Bernoulli numbers. Professor Gould believes his research success comes from his intuition on how to discover combinatorial identities. This book will appeal to a wide audience and may be used either as lecture notes for a beginning graduate level combinatorics class, or as a research supplement for the specialist in enumerative combinatorics. Contents:Basic Properties of SeriesThe Binomial TheoremIterative SeriesTwo of Professor Gould''s Favorite Algebraic TechniquesVandermonde ConvolutionThe nth Difference Operator and Euler''s Finite Difference TheoremMelzak''s FormulaGeneralized Derivative FormulasStirling Numbers of the Second Kind S(n; k)Eulerian NumbersWorpitzky NumbersStirling Numbers of the First Kind s(n; k)Explicit Formulas for s(n; n — k)Number Theoretic Definitions of Stirling NumbersBernoulli NumbersAppendix A: Newton-Gregory ExpansionsAppendix B: Generalized Bernoulli and Euler Polynomials Readership: Undergraduates, graduates and researchers interested in combinatorial and algebraic techniques. Key Features:Professor Gould is an acknowledged expert in the field of Stirling number identitiesFor the first time in print, this book collects Professor''s Gould''s vast knowledge on this subject in one accessible locationThis book contains Professor Gould''s unique approaches to discovering and proving binomial identitiesThis book contains many fully-worked detailed proofs of the identities found in H W Gould''s "Combinatorial Identities: A Standardized Set of Tables Listing 500 Binomial Coefficient Summations"Keywords:Stirling Numbers of the First Kind;Stirling Numbers of the Second Kind;Bernoulli Numbers;Generalized Bernoulli Polynomials;Worpitzky Numbers;Eulerian Numbers;Binomial Theorem;Vandermonde Convolution;Euler''s Finite Difference Theorem;Melzak''s Formula "This book is a unique work that could appeal to a wide audience: from graduate students to specialists in enumerative combinatorics, to enthusiasts of Gould''s work." CERN Courier '

Author : Ernesto Damiani,Ottavio D'Antona,Vincenzo Marra,Fabrizio Palombi
Publisher : Springer Science & Business Media
Page : 267 pages
File Size : 40,6 Mb
Release : 2009-07-24
Category : Computers
ISBN : 9780387887531

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From Combinatorics to Philosophy by Ernesto Damiani,Ottavio D'Antona,Vincenzo Marra,Fabrizio Palombi Pdf

From Combinatorics to Philosophy: The Legacy of G. -C. Rota provides an assessment of G. -C. Rota's legacy to current international research issues in mathematics, philosophy and computer science. This volume includes chapters by leading researchers, as well as a number of invited research papers. Rota’s legacy connects European and Italian research communities to the USA by providing inspiration to several generations of researchers in combinatorics, philosophy and computer science. From Combinatorics to Philosophy: The Legacy of G. -C. Rota is of valuable interest to research institutions and university libraries worldwide. This book is also designed for advanced-level students in mathematics, computer science, and philosophy.

Author : Robert A. Beeler
Publisher : Springer
Page : 368 pages
File Size : 52,5 Mb
Release : 2015-03-14
Category : Mathematics
ISBN : 9783319138442

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How to Count by Robert A. Beeler Pdf

Providing a self-contained resource for upper undergraduate courses in combinatorics, this text emphasizes computation, problem solving, and proof technique. In particular, the book places special emphasis the Principle of Inclusion and Exclusion and the Multiplication Principle. To this end, exercise sets are included at the end of every section, ranging from simple computations (evaluate a formula for a given set of values) to more advanced proofs. The exercises are designed to test students' understanding of new material, while reinforcing a working mastery of the key concepts previously developed in the book. Intuitive descriptions for many abstract techniques are included. Students often struggle with certain topics, such as generating functions, and this intuitive approach to the problem is helpful in their understanding. When possible, the book introduces concepts using combinatorial methods (as opposed to induction or algebra) to prove identities. Students are also asked to prove identities using combinatorial methods as part of their exercises. These methods have several advantages over induction or algebra.

Author : A. Giambruno,César Polcino Milies,Sudarshan K. Sehgal
Publisher : American Mathematical Soc.
Page : 283 pages
File Size : 40,6 Mb
Release : 2009
Category : Mathematics
ISBN : 9780821847718

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Groups, Rings and Group Rings by A. Giambruno,César Polcino Milies,Sudarshan K. Sehgal Pdf

Represents the proceedings of the conference on Groups, Rings and Group Rings, held July 28 - August 2, 2008, in Ubatuba, Brazil. This title contains results in active research areas in the theory of groups, group rings and algebras (including noncommutative rings), polynomial identities, Lie algebras and superalgebras.

Author : John Riordan
Publisher : Unknown
Page : 280 pages
File Size : 45,7 Mb
Release : 1979
Category : Mathematics
ISBN : STANFORD:36105031609568

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Combinatorial Identities by John Riordan Pdf

Author : Alexei Kanel-Belov,Louis Halle Rowen
Publisher : CRC Press
Page : 400 pages
File Size : 44,6 Mb
Release : 2005-02-22
Category : Mathematics
ISBN : 9781439863725

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Computational Aspects of Polynomial Identities by Alexei Kanel-Belov,Louis Halle Rowen Pdf

A comprehensive study of the main research done in polynomial identities over the last 25 years, including Kemer's solution to the Specht problem in characteristic O and examples in the characteristic p situation. The authors also cover codimension theory, starting with Regev's theorem and continuing through the Giambruno-Zaicev exponential rank. T

Author : Jan Felipe Van Diejen,Luc Vinet
Publisher : American Mathematical Soc.
Page : 290 pages
File Size : 52,5 Mb
Release : 1999
Category : Fonctions spéciales - Congrès
ISBN : 9780821820261

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Algebraic Methods and $q$-Special Functions by Jan Felipe Van Diejen,Luc Vinet Pdf

There has been revived interest in recent years in the study of special functions. Many of the latest advances in the field were inspired by the works of R. A. Askey and colleagues on basic hypergeometric series and I. G. Macdonald on orthogonal polynomials related to root systems. Significant progress was made by the use of algebraic techniques involving quantum groups, Hecke algebras, and combinatorial methods. The CRM organized a workshop for key researchers in the field to present an overview of current trends. This volume consists of the contributions to that workshop. Topics include basic hypergeometric functions, algebraic and representation-theoretic methods, combinatorics of symmetric functions, root systems, and the connections with integrable systems.