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Generally, in any human field, a Smarandache Structure on a set A means a weak structure W on A such that there exists a proper subset B in A which is embedded with a stronger structure S.

These types of structures occur in our everyday's life, that's why we study them in this book.

Thus, as a particular case:

A Non-associative ring is a non-empty set R together with two binary operations '+' and '.' such that (R, +) is an additive abelian group and (R, .) is a groupoid. For all a, b, c in R we have (a + b) . c = a . c + b . c and c . (a + b) = c . a + c . b.

A Smarandache non-associative ring is a non-associative ring (R, +, .) which has a proper subset P in R, that is an associative ring (with respect to the same binary operations on R). ... Read more

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The present volume contains the written version of threeinvited lectures and sixteen papers presented in the InternationalConference on Near-Rings and Near-Fields held in Stellenbosch, SouthAfrica. These articles reflect contemporary research activities on thealgebraic structure theory of near-rings and the interaction they havewith group theory, geometry and combinatorics.

Audience: This book will be of value to graduate students ofmathematics and algebraists interested in all aspects of the near-ringtheory. ... Read more

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Focusing mainly on locally compact and compact rings, the text begins by illustrating results from the theory of topological groups, some of which are very recent results published for the first time. Constructions and examples of topological groups, relations between quasi-components and components in topological groups, minimal topological groups and properties of the finest precompact topology on an Abelian group are all covered. The text then goes on to examine various aspects of the theory of topological rings, covering constructions and examples of compact right topological rings, the local structure of locally compact rings, structure of compact rings, the local nilpotence of nilrings satisfying compactness conditions, groups of units of compact rings, bounded locally compact rings, topologically locally finite and locally topologically nilpotent radicals in the class of locally compact rings, rings of endomorphisms of a locally compact module and tensor products of compact rings. Results on totally bounded topological rings are also given and developed e.g. the structure theory for countably compact rings. This book will be invaluable to researchers and mathematicians with a basic grounding in the theory of rings and general topology. ... Read more

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The book lays algebraic foundations for real geometry through a systematic investigation of partially ordered rings of semi-algebraic functions. Real spectra serve as primary geometric objects, the maps between them are determined by rings of functions associated with the spectra. The many different possible choices for these rings of functions are studied via reflections of partially ordered rings. Readers should feel comfortable using basic algebraic and categorical concepts. As motivational background some familiarity with real geometry will be helpful. The book aims at researchers and graduate students with an interest in real algebra and geometry, ordered algebraic structures, topology and rings of continuous functions. ... Read more

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The theory of automorphisms and derivations of associative rings is a direct descendant of the development of classical Galois theory and the theory of invariants. This volume presents a comprehensive overview of the methods and results of that theory, which has been greatly enriched during the last twenty years. Some of the material included appears for the first time. Among the problems discussed in this book are the following: construction of a Galois theory for prime and semiprime rings and its application to domains and free algebras; investigation of the problems of the algebraic dependence of automorphisms and derivations; studies of the fixed rings for finite groups and rings of constants for differential Lie algebras acting on the rings; non-commutative invariants of linear groups; theorems of finite groups acting on modular lattices; actions of Hopf algebras. The monograph is meant for specialists in algebra, but it can also be useful for a wider range of mathematicians. The inclusions in the book of the latest achievements on the structural theory of rings with generalized identities makes it desirable reading for graduate students as well. ... Read more

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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.

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This volume is devoted to the theory of nilpotent Lie algebrasand their applications. Nilpotent Lie algebras have played animportant role over the last years both in the domain of algebra,considering its role in the classification problems of Lie algebras,and in the domain of differential geometry. Among the topics discussedhere are the following: cohomology theory of Lie algebras,deformations and contractions, the algebraic variety of the laws ofLie algebras, the variety of nilpotent laws, and characteristicallynilpotent Lie algebras in nilmanifolds.
Audience: This book is intended for graduate studentsspecialising in algebra, differential geometry and in theoreticalphysics and for researchers in mathematics and in theoretical physics. ... Read more

Customer Reviews (1)

Interesting title, quite bad book.
The title of this book does not properly correspond to its contents. It is supposed to develop the general theory of nilpotent Lie algebras, but the most part of the book deals with a quite special class of nilpotent algebras, called filiform. Only the first two chapters present some general material, along with a list of nilpotent algebras in low dimensions which is full of mistakes. On the other hand, the authors use many different notations for the same objects, which confuses the reader. The quantity of wrongly ennounced results (due to misprints, etc) is discouraging, and shows how careless the final text has been prepared by the editors. This suffices to give it up.
It is a pity, since there is no book dealing with nilpotent algebras only, but certainly this is not the way to write it. ... Read more

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This new proceedings is a collection of papers presenting the recent developments and research in various fields of algebra, especially Lie Algebras, Rings and their related topics undertaken in the U.S.A., Russia, North Asia and Israel. Contributors include E. Zelmanov, the 1994 Fields Medalist, L. Bokut, Günter F. Pilz, Koichiro Harada, Alexander Kemer, V.K. Kharchenko, L. Makar-Limanov and Louis H. Rowen. ... Read more

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The book is devoted to the geometrical construction of the representations of Lusztig's small quantum groups at roots of unity. These representations are realized as some spaces of vanishing cycles of perverse sheaves over configuration spaces. As an application, the bundles of conformal blocks over the moduli spaces of curves are studied. The book is intended for specialists in group representations and algebraic geometry. ... Read more

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This is the first of three volumes which present, in anoriginal way, some of the most important tools of applied mathematics,in areas such as probability theory, operator calculus, representationtheory, and special functions, used in solving problems inmathematics, physics and computer science.
Volume I - Representations and Probability Theory- deals with probability theory in connection with grouprepresentations. It presents an introduction to Lie algebras and Liegroups which emphasises the connections with probability theory andrepresentation theory. The book contains an introduction and sevenchapters which treat, respectively, noncommutative algebra,hypergeometric functions, probability and Fock spaces, moment systems,Bernoulli processes/systems, and matrix elements.
Each chapter contains exercises which range in difficulty from easy toadvanced. The text is written so as to be suitable for self-study forboth beginning graduate students and researchers.
For students, teachers and researchers with an interest in algebraicstructures and operator calculus. ... Read more

Customer Reviews (1)

first volume of three: operatrors as Heissenberg-Weyl algebras
This set of three volumes appear to be an attempt at codification of
operator theory using an Heissenberg-Weyl algebra and hypergeometric functions.
Since the traditional mathematical analysis approach is called Banach algebra
and although generalized as Hardy spaces is mostly based on the complex plane,
this more recent quantum mechanics/ group theory based approach is kind of new.
The authors have a real favorite in Bernoulli systems( which are mostly boson related)
and pretty much ignores Euler numbers and Eulerian functional system.
The Umbral calculus is not mentioned at all that I can find, just like Banach algebra.
In spite of these limitations this book is well written
and has very important information presented in a pretty universally understandable manner in contrast to books
like Banach Algebras and Several Complex Variables (Graduate Texts in Mathematics, Vol 35).
Later volumes promise to include Lie groups and Appell systems,
but Bell polynomials are replaced by a Krawtchouk approach?
I find this book both interesting and useful. ... Read more

Product Description
The geometrical view of mechanics is based on the study ofcertain exterior systems, the most classical of which are Pfaffiansystems. In this book, we present the classification theorems(Frobenius, Darboux) and the local classification of Pfaffian systemsof five variables, following Cartan.
We also present a new class of exterior systems, calledk-symplectic systems, generalizing the notion of symplecticform. These systems permit us to write in the language of exteriorforms the equations proposed by Nambu for a model of statisticalmechanics.
Audience: This book is aimed at graduate students and atresearch workers in the field of mathematics, differential geometry,statistical mechanics, mathematics of physics and Lie algebras. ... Read more

Customer Reviews (1)

An extremely weak book
This book intends to give new insights of Nambu mechanics, but up to some calculations in differential forms (in sense of Cartan, but were prolongations and other fundamental topics are not treated), authors do not provide any sustainable physical result. It seems obvious that they are far from having expertise on physics. There is no mention to multibrackets in this book, which is the well known technique to attack the problem. ... Read more

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The theory of Kac lagebras and their duality, elaboratedindependently in the seventies by Kac and Vainermann and bythe authors of this book, has nowreached a state ofmaturity which justifies the publication of acomprehensiveand authoritative account in bookform. Further, the topic of"quantum groups" has recently become very fashionable andattractedthe attention of more and more mathematicians andtheoretical physicists. However a good characterization ofquantum groups among Hopf algebras inanalogy to thecharacterization of Lie groups among locally compact groupsis still missing. It is thus very valuable to develop thegeneraltheory as does this book, with emphasis on theanalytical aspects of thesubject instead of the purelyalgebraic ones.While in the Pontrjagin duality theory of locally compactabelian groups a perfect symmetry existsbetween a group andits dual, this is no longer true in the various dualitytheorems of Tannaka, Krein, Stinespring and others dealingwith non-abelian locally compact groups. Kac (1961) andTakesaki (1972)formulated the objective of finding a goodcategory of Hopf algebras, containing the category oflocally compact groups and fulfilling a perfectduality.The category of Kac algebras developed in this bookfullyanswers the original duality problem, while not yetsufficientlynon-unimodular to include quantum groups.This self-contained account of thetheory will be ofinterest to all researchers working in quantumgroups,particularly those interested in the approach by Lie groupsand Lie algebras or by non-commutative geometry, and moregenerally also to those working in C* algebras ortheoretical physics. ... Read more

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Finite-Dimensional Division Algebras over fields determine, by the Wedderburn Theorem, the semi-simple finite-dimensional algebras over a field. They lead to the definition of the Brauer group and to certain geometric objects, the Brauer-Severi varieties. The book concentrates on those algebras that have an involution. Algebras with involution appear in many contexts; they arose first in the study of the so-called "multiplication algebras of Riemann matrices". The largest part of the book is the fifth chapter, dealing with involutorial simple algebras of finite dimension over a field. Of particular interest are the Jordan algebras determined by these algebras with involution;their structure is discussed. Two important concepts of these algebras with involution are the universal enveloping algebras and the reduced norm.

Corrections of the 1^st edition (1996) carried out on behalf of N. Jacobson (deceased) by Prof. P.M. Cohn (UC London, UK).

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This book is a detailed exposition of algebraic and geometricalaspects related to the theory of symmetries and recursion operatorsfor nonlinear partial differential equations (PDE), both in classicaland in super, or graded, versions. It contains an original theory ofFrolicher--Nijenhuis brackets which is the basis for aspecial cohomological theory naturally related to the equationstructure. This theory gives rise to infinitesimal deformations ofPDE, recursion operators being a particular case of such deformations. ... Read more

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This volume contains a re-edition of Max Koecher's famous Minnesota Notes. The main objects are homogeneous, but not necessarily convex, cones. They are described in terms of Jordan algebras. The central point is a correspondence between semisimple real Jordan algebras and so-called omega-domains. This leads to a construction of half-spaces which give an essential part of all bounded symmetric domains. The theory is presented in a concise manner, with only elementary prerequisites. The editors have added notes on each chapter containing an account of the relevant developments of the theory since these notes were first written. ... Read more

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This book is the first to systematically explore the classification and function theory of complex homogeneous bounded domains. The Siegel domains are discussed in detail, and proofs are presented. Using the normal Siegel domains to realize the homogeneous bounded domains, we can obtain more property of the geometry and the function theory on homogeneous bounded domains.