Polynomial Identities and Asymptotic Methods

Polynomial Identities and Asymptotic Methods
Title Polynomial Identities and Asymptotic Methods PDF eBook
Author A. Giambruno
Publisher American Mathematical Soc.
Pages 370
Release 2005
Genre Mathematics
ISBN 0821838296

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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.

Polynomial Identities in Algebras

Polynomial Identities in Algebras
Title Polynomial Identities in Algebras PDF eBook
Author Onofrio Mario Di Vincenzo
Publisher Springer Nature
Pages 421
Release 2021-03-22
Genre Mathematics
ISBN 3030631117

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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.

Identical Relations in Lie Algebras

Identical Relations in Lie Algebras
Title Identical Relations in Lie Algebras PDF eBook
Author Yuri Bahturin
Publisher Walter de Gruyter GmbH & Co KG
Pages 542
Release 2021-08-23
Genre Mathematics
ISBN 3110566656

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This updated edition of a classic title studies identical relations in Lie algebras and also in other classes of algebras, a theory with over 40 years of development in which new methods and connections with other areas of mathematics have arisen. New topics covered include graded identities, identities of algebras with actions and coactions of various Hopf algebras, and the representation theory of the symmetric and general linear group.

Geometric Approximation Algorithms

Geometric Approximation Algorithms
Title Geometric Approximation Algorithms PDF eBook
Author Sariel Har-Peled
Publisher American Mathematical Soc.
Pages 378
Release 2011
Genre Computers
ISBN 0821849115

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Exact algorithms for dealing with geometric objects are complicated, hard to implement in practice, and slow. Over the last 20 years a theory of geometric approximation algorithms has emerged. These algorithms tend to be simple, fast, and more robust than their exact counterparts. This book is the first to cover geometric approximation algorithms in detail. In addition, more traditional computational geometry techniques that are widely used in developing such algorithms, like sampling, linear programming, etc., are also surveyed. Other topics covered include approximate nearest-neighbor search, shape approximation, coresets, dimension reduction, and embeddings. The topics covered are relatively independent and are supplemented by exercises. Close to 200 color figures are included in the text to illustrate proofs and ideas.

The Ricci Flow: Techniques and Applications

The Ricci Flow: Techniques and Applications
Title The Ricci Flow: Techniques and Applications PDF eBook
Author Bennett Chow
Publisher American Mathematical Soc.
Pages 542
Release 2010-04-21
Genre Mathematics
ISBN 0821846612

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The Ricci flow uses methods from analysis to study the geometry and topology of manifolds. With the third part of their volume on techniques and applications of the theory, the authors give a presentation of Hamilton's Ricci flow for graduate students and mathematicians interested in working in the subject, with an emphasis on the geometric and analytic aspects. The topics include Perelman's entropy functional, point picking methods, aspects of Perelman's theory of $\kappa$-solutions including the $\kappa$-gap theorem, compactness theorem and derivative estimates, Perelman's pseudolocality theorem, and aspects of the heat equation with respect to static and evolving metrics related to Ricci flow. In the appendices, we review metric and Riemannian geometry including the space of points at infinity and Sharafutdinov retraction for complete noncompact manifolds with nonnegative sectional curvature. As in the previous volumes, the authors have endeavored, as much as possible, to make the chapters independent of each other. The book makes advanced material accessible to graduate students and nonexperts. It includes a rigorous introduction to some of Perelman's work and explains some technical aspects of Ricci flow useful for singularity analysis. The authors give the appropriate references so that the reader may further pursue the statements and proofs of the various results.

Polynomial Identity Rings

Polynomial Identity Rings
Title Polynomial Identity Rings PDF eBook
Author Vesselin Drensky
Publisher Birkhäuser
Pages 197
Release 2012-12-06
Genre Mathematics
ISBN 3034879342

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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.

Groups, Rings and Group Rings

Groups, Rings and Group Rings
Title Groups, Rings and Group Rings PDF eBook
Author A. Giambruno
Publisher American Mathematical Soc.
Pages 283
Release 2009
Genre Mathematics
ISBN 0821847716

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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.