Algebra Books

Book SynopsisThis book is a revised version of the first edition and is intended as a Linear Algebra sequel and companion volume to the fourth edition of (Graduate Texts in Mathematics 23). In this new version of Multilinear Algebra, Chapters 1-5 remain essen­ tially unchanged from the previous edition.Table of Contents1 Tensor Products.- 2 Tensor Products of Vector Spaces with Additional Structure.- 3 Tensor Algebra.- 4 Skew-Symmetry and Symmetry in the Tensor Algebra.- 5 Exterior Algebra.- 6 Mixed Exterior Algebra.- 7 Applications to Linear Transformations.- 8 Skew and Skew-Hermitian Transformations.- 9 Symmetric Tensor Algebra.- 10 Clifford Algebras.- 11 Representations of Clifford Algebras.

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Book SynopsisThe nature of $C^*$-algebras is such that one cannot study perturbation without also studying the theory of lifting and the theory of extensions. This book focuses on the approximation questions involving representations of relations in matrices and $C^*$-algebras.Table of ContentsIntroduction Part I. Rings and $C^*$-algebras: $\sigma$-unital $C*$-algebras Order and factoring Generators and relations Basic perturbation Push-outs and pull-backs Matrix algebras Multipliers Part II. Lifting: Easy lifting Multiplier algebras Projectivity Properties of projective $C^*$-algebras Heavy lifting Part III. Perturbing: Inductive limits Stable relations Applications Extensions Part IV. Almost Normal: Normals, limits Almost normal elements Almost normal matrices Bibliography Index.

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Book SynopsisLarge sparse linear systems of equations are ubiquitous in science, engineering and beyond. This open access monograph focuses on factorization algorithms for solving such systems. It presents classical techniques for complete factorizations that are used in sparse direct methods and discusses the computation of approximate direct and inverse factorizations that are key to constructing general-purpose algebraic preconditioners for iterative solvers. A unified framework is used that emphasizes the underlying sparsity structures and highlights the importance of understanding sparse direct methods when developing algebraic preconditioners. Theoretical results are complemented by sparse matrix algorithm outlines.This monograph is aimed at students of applied mathematics and scientific computing, as well as computational scientists and software developers who are interested in understanding the theory and algorithms needed to tackle sparse systems. It is assumed that the reader has completed a basic course in linear algebra and numerical mathematics. Table of ContentsAn introduction to sparse matrices.- Sparse matrices and their graphs.- Introduction to matrix factorizations.- Sparse Cholesky sovler: The symbolic phase.- Sparse Cholesky solver: The factorization phase.- Sparse LU factorizations.- Stability, ill-conditioning and symmetric indefinite factorizations.- Sparse matrix ordering algorithms.- Algebraic preconditioning and approximate factorizations.- Incomplete factorizations.- Sparse approximate inverse preconditioners.

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Book SynopsisThis volume contains refereed papers related to the lectures and talks given at a conference held in Siena (Italy) in June 2004. Also included are research papers that grew out of discussions among the participants and their collaborators. All the papers are research papers, but some of them also contain expository sections which aim to update the state of the art on the classical subject of special projective varieties and their applications and new trends like phylogenetic algebraic geometry. The topic of secant varieties and the classification of defective varieties is central and ubiquitous in this volume. Besides the intrinsic interest of the subject, it turns out that it is also relevant in other fields of mathematics like expressions of polynomials as sums of powers, polynomial interpolation, rank tensor computations, Bayesian networks, algebraic statistics and number theory.

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Book SynopsisThis volume is the proceedings of a conference on Finite Geometries, Groups, and Computation that took place on September 4-9, 2004, at Pingree Park, Colorado (a campus of Colorado State University). Not accidentally, the conference coincided with the 60th birthday of William Kantor, and the topics relate to his major research areas. Participants were encouraged to explore the deeper interplay between these fields. The survey papers by Kantor, O'Brien, and Penttila should serve to introduce both students and the broader mathematical community to these important topics and some of their connections while the volume as a whole gives an overview of current developments in these fields.

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Book SynopsisThis is a memorial volume dedicated to A. L. S. Corner, previously Professor in Oxford, who published important results on algebra, especially on the connections of modules with endomorphism algebras. The volume contains refereed contributions which are related to the work of Corner. It contains also an unpublished extended paper of Corner himself. A memorial volume with important contributions related to algebra.

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Book SynopsisThis monograph presents recent developments of the theory of algebraic dynamical systems and their applications to computer sciences, cryptography, cognitive sciences, psychology, image analysis, and numerical simulations. The most important mathematical results presented in this book are in the fields of ergodicity, p-adic numbers, and noncommutative groups. For students and researchers working on the theory of dynamical systems, algebra, number theory, measure theory, computer sciences, cryptography, and image analysis.

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Book SynopsisThe book timely surveys new research results and related developments in Diophantine approximation, a division of number theory which deals with the approximation of real numbers by rational numbers. The book is appended with a list of challenging open problems and a comprehensive list of references. From the contents: Field extensions • Algebraic numbers • Algebraic geometry • Height functions • The abc-conjecture • Roth's theorem • Subspace theorems • Vojta's conjectures • L-functions.

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Book SynopsisThis volume contains selected refereed papers based on lectures presented at the ‘Fifth International Fez Conference on Commutative Algebra and Applications’ that was held in Fez, Morocco in June 2008. The volume represents new trends and areas of classical research within the field, with contributions from many different countries. In addition, the volume has as a special focus the research and influence of Alain Bouvier on commutative algebra over the past thirty years.

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Book SynopsisThis is the second revised edition of an introduction to contemporary relative homological algebra. It supplies important material essential to understand topics in algebra, algebraic geometry and algebraic topology. Each section comes with exercises providing practice problems for students as well as additional important results for specialists. In this new edition the authors have added well-known additional material in the first three chapters, and added new material that was not available at the time the original edition was published. In particular, the major changes are the following: Chapter 1: Section 1.2 has been rewritten to clarify basic notions for the beginner, and this has necessitated a new Section 1.3. Chapter 3: The classic work of D. G. Northcott on injective envelopes and inverse polynomials is finally included. This provides additional examples for the reader. Chapter 11: Section 11.9 on Kaplansky classes makes volume one more up to date. The material in this section was not available at the time the first edition was published. The authors also have clarified some text throughout the book and updated the bibliography by adding new references. The book is also suitable for an introductory course in commutative and ordinary homological algebra.

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Book SynopsisThis second volume deals with the relative homological algebra of complexes of modules and their applications. It is a concrete and easy introduction to the kind of homological algebra which has been developed in the last 50 years. The book serves as a bridge between the traditional texts on homological algebra and more advanced topics such as triangulated and derived categories or model category structures. It addresses to readers who have had a course in classical homological algebra, as well as to researchers.Table of ContentsDedication Preface Chapter I: Complexes of Modules 1. Definitions and basic constructions 2. Complexes formed from Modules 3. Free Complexes 4. Projective and Injective Complexes Chapter II: Short Exact Sequences of Complexe 1. The groups Extn(C, D) 2. The Group Ext1(C, D) 3. The Snake Lemma for Complexes 4. Mapping Cones Chapter III: The Category K(R-Mod) 1. Homotopies 2. The category K(R-Mod) 3. Split short exact sequences 4. The complexes Hom(C, D) 5. The Koszul Complex Chapter IV: Cotorsion Pairs and Triplets in C(R-Mod) 1. Cotorsion Pairs 2. Cotorsion triplets 3. The Dold triplet 4. More on cotorsion pairs and triplets Chapter V: Adjoint Functors 1. Adjoint functors Chapter VI: Model Structures 1. Model Structures on C(R-Mod) Chapter VII: Creating Cotorsion Pairs 1. Creating Cotorsion pairs in C(R-Mod) in a Termwise Manner 2. The Hill lemma 3. More cotorsion pairs 4. More Hovey pairs Chapter VIII: Minimal Complexes 1. Minimal resolutions 2. Decomposing a complex Chapter IX: Cartan and Eilenberg Resolutions 1. Cartan-Eilenberg Projective Complexes 2. Cartan and Eilenberg Projective resolutions 3. C - E injective complexes and resolutions 4. Cartan and Eilenberg Balance Bibliographical Notes References Index

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Book SynopsisThis second, revised and substantially extended edition of Approximations and Endomorphism Algebras of Modules reflects both the depth and the width of recent developments in the area since the first edition appeared in 2006. The new division of the monograph into two volumes roughly corresponds to its two central topics, approximation theory (Volume 1) and realization theorems for modules (Volume 2). It is a widely accepted fact that the category of all modules over a general associative ring is too complex to admit classification. Unless the ring is of finite representation type we must limit attempts at classification to some restricted subcategories of modules. The wild character of the category of all modules, or of one of its subcategories C, is often indicated by the presence of a realization theorem, that is, by the fact that any reasonable algebra is isomorphic to the endomorphism algebra of a module from C. This results in the existence of pathological direct sum decompositions, and these are generally viewed as obstacles to classification. In order to overcome this problem, the approximation theory of modules has been developed. The idea here is to select suitable subcategories C whose modules can be classified, and then to approximate arbitrary modules by those from C. These approximations are neither unique nor functorial in general, but there is a rich supply available appropriate to the requirements of various particular applications. The authors bring the two theories together. The first volume, Approximations, sets the scene in Part I by introducing the main classes of modules relevant here: the S-complete, pure-injective, Mittag-Leffler, and slender modules. Parts II and III of the first volume develop the key methods of approximation theory. Some of the recent applications to the structure of modules are also presented here, notably for tilting, cotilting, Baer, and Mittag-Leffler modules. In the second volume, Predictions, further basic instruments are introduced: the prediction principles, and their applications to proving realization theorems. Moreover, tools are developed there for answering problems motivated in algebraic topology. The authors concentrate on the impossibility of classification for modules over general rings. The wild character of many categories C of modules is documented here by the realization theorems that represent critical R-algebras over commutative rings R as endomorphism algebras of modules from C. The monograph starts from basic facts and gradually develops the theory towards its present frontiers. It is suitable both for graduate students interested in algebra and for experts in module and representation theory.Trade ReviewI strongly recommend the monograph to anyone who is interested in the modern theory of modules. (pruz), EMS Newsletter 9/2007 All in all, I highly recommend the book to everyone interested in cotorsion pairs, approximation theory, realization of algebras or application of set theory to algebra. Gabor Braun, Zentralblatt MATH 1121/2007 The monograph starts from basic facts and gradually develops the theory towards its present frontiers. It is suitable both for graduate students interested in algebra and for experts in module and representation theory. L'Enseignement Mathematique 3-4/2006

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Book SynopsisThis is the first of two volumes of a state-of-the-art survey article collection which originates from three commutative algebra sessions at the 2009 Fall Southeastern American Mathematical Society Meeting at Florida Atlantic University. The articles reach into diverse areas of commutative algebra and build a bridge between Noetherian and non-Noetherian commutative algebra. These volumes present current trends in two of the most active areas of commutative algebra: non-noetherian rings (factorization, ideal theory, integrality), and noetherian rings (the local theory, graded situation, and interactions with combinatorics and geometry). This volume contains combinatorial and homological surveys. The combinatorial papers document some of the increasing focus in commutative algebra recently on the interaction between algebra and combinatorics. Specifically, one can use combinatorial techniques to investigate resolutions and other algebraic structures as with the papers of Fløystad on Boij-Söderburg theory, of Geramita, Harbourne and Migliore, and of Cooper on Hilbert functions, of Clark on minimal poset resolutions and of Mermin on simplicial resolutions. One can also utilize algebraic invariants to understand combinatorial structures like graphs, hypergraphs, and simplicial complexes such as in the paper of Morey and Villarreal on edge ideals. Homological techniques have become indispensable tools for the study of noetherian rings. These ideas have yielded amazing levels of interaction with other fields like algebraic topology (via differential graded techniques as well as the foundations of homological algebra), analysis (via the study of D-modules), and combinatorics (as described in the previous paragraph). The homological articles the editors have included in this volume relate mostly to how homological techniques help us better understand rings and singularities both noetherian and non-noetherian such as in the papers by Roberts, Yao, Hummel and Leuschke.Table of ContentsPreface 1 Boij-Soederberg Theory: Introduction and SurveyGunnar Fløystad1.1 Introduction1.2 The Boij-Söderberg Conjectures1.2.1 Resolutions and Betti Diagrams1.2.2 The Positive Cone of Betti Diagrams1.2.3 Herzog-Kühl Equations1.2.4 Pure Resolutions1.2.5 Linear Combinations of Pure Diagrams1.2.6 The Boij-Söderberg Conjectures1.2.7 Algorithmic Interpretation1.2.8 Geometric Interpretation1.3 The Exterior Facets of the Boij-Söderberg Fan and their Supporting Hyperplanes1.3.1 The Exterior Facets1.3.2 The Supporting Hyperplanes1.3.3 Pairings of vector Bundles and Resolutions1.4 The Existence of Pure Free Resolutions and of Vector Bundles with Supernatural Cohomology1.4.1 The Equivariant Pure Free Resolution1.4.2 Equivariant Supernatural Bundles1.4.3 Characteristic Free Supernatural Bundles1.4.4 The Characteristic Free Pure Resolution1.4.5 Pure Resolutions Constructed from Generic Matrices1.5 Cohomology of Vector Bundles on Projective Spaces1.5.1 Cohomology Tables1.5.2 The Fan of Cohomology Tables of Vector Bundles1.5.3 Facet Equations1.6 Extensions to Non-Cohen-Macaulay Modules and to Coherent Sheaves1.6.1 Betti Diagrams of Graded Modules in General1.6.2 Cohomology of Coherent Sheaves1.7 Further Topics1.7.1 The Semigroup of Betti Diagrams of Modules1.7.2 Variants on the Grading1.7.3 Poset Structures1.7.4 Computer Packages1.7.5 Three Basic Problems 2 Hilbert Functions of Fat Point Subschemes of the Plane: the Two-fold WayAnthony V. Geramita, Brian Harbourne, and Juan Migliore2.1 Introduction2.2 Approach I: Nine Double Points2.3 Approach I: Points on Cubics2.4 Approach II: Points on Cubics 3 Edge Ideals: Algebraic and Combinatorial PropertiesSusan Morey and Rafael H. Villarreal3.1 Introduction3.2 Algebraic and Combinatorial Properties of Edge Ideals3.3 Invariants of Edge Ideals: Regularity, Projective Dimension, Depth3.4 Stability of Associated Primes 4 Three Simplicial ResolutionsJeff Mermin4.1 Introduction4.2 Background and Notation4.2.1 Algebra4.2.2 Combinatorics4.3 The Taylor Resolution4.4 Simplicial Resolutions4.5 The Scarf Complex4.6 The Lyubeznik Resolutions4.7 Intersections4.8 Questions 5 A Minimal Poset Resolution of Stable IdealsTimothy B. P. Clark5.1 Introduction5.2 Poset Resolutions and Stable Ideals5.3 The Shallability of PN5.4 The Topology of PN and Properties of D(PN)5.5 Proof of Theorem 2.45.6 A Minimal Cellular Resolution of R/N 6 Subsets of Complete Intersections and the EGH ConjectureSusan M. Cooper6.1 Introduction6.2 Preliminary Definitions and Results6.2.1 The Eisenbud-Green-Harris Conjecture and Complete Intersections6.3 Rectangular Complete Intersections6.4 Some Key Tools6.4.1 Pairs of Hilbert Functions and Maximal Growth6.4.2 Ideals Containing Regular Sequences6.5 Subsets of Complete Intersections in P26.6 Subsets of C.I.(2, d2, d3) with d2 = d36.7 Subsets of C.I.(3, d2, d3) with d3 = d26.8 An Application: The Cayley-Bacharach Property 7 The Homological ConjecturesPaul C. Roberts7.1 Introduction7.2 The Serre Multiplicity Conjectures7.2.1 The Vanishing Conjecture7.2.2 Gabber's Proof of the Nonnegativity Conjecture7.2.3 The Positivity Conjecture7.3 The Peskine-Szpiro Intersection Conjecture7.3.1 Hochester's Metatheorem7.4 Generalizations of the Multiplicity Conjectures7.4.1 The Graded Case7.4.2 The Generalized Rigidity Conjecture7.5 The Monomial, Direct Summand, and Canonical Element Conjectures7.6 Cohen-Macaulay Modules and Algebras7.6.1 Weakly Functorial Big Cohen-Macaulay Algebras7.7 The Syzygy Conjecture and the Improved New Intersection Conjecture7.8 Tight Closure Theory7.9 The Strong Direct Summand Conjecture7.10 Almost Cohen-Macaulay Algebras7.11 A Summary of Open Questions7.11.1 The Serre Positivity Conjecture7.11.2 Partial Euler Characteristics7.11.3 Strong Multiplicity Conjectures7.11.4 Cohen-Macaulay Modules and Related Conjectures7.11.5 Almost Cohen-Macaulay Algebras 8 The Compatibility, Independence, and Linear Growth PropertiesYongwei Yao8.1 Introduction8.2 Primary Decomposition8.3 Compatibility of Primary Components8.4 Maximal Primary Components, Independence8.5 Linear Growth of Primary Components8.6 Linear growth of {?Tor?_c^R (?(M/(I^m M )),?(N/(J^n N)))}8.7 Secondary Representation8.8 Compatibility of Secondary Components8.9 Applying a Result of Sharp on Artinian Modules8.10 Independence8.11 Minimal Secondary Components8.12 Linear Growth of Secondary Components 9 Recent Progress in Coherent Rings: A Homological PerspectiveLivia Hummel9.1 Introduction9.2 Coherent Rings and Grade9.2.1 Coherent Rings and ?(FP)?_?^R Modules9.2.2 Non-Noetherian Grade9.3 Cohen-Macaulay Rings9.4 Gorenstein Dimensions and the Auslander-Bridger Property9.4.1 Gorenstein Dimenstions9.4.2 The Auslander-Bridger Formula9.5 Gorenstein Rings and Injective Dimensions9.6 Foundations for Coherent Complete Intersections 10 Non-commutative Crepant Resolutions: Scenes from Categorical GeometryGraham J. Leuschke10.1 Introduction10.2 Morita Equivalence10.3 (Quasi)coherent Sheaves10.4 Derived Categories of Modules10.5 Derived Categories of Sheaves10.6 Example: Tilting on Projective Space10.7 The Non-existence of Non-commutative Spaces10.8 Resolutions of Singularities10.9 The Minimal Model Program10.10. Categorical Desingularizations10.11 Example: the McKay Correspondence10.12 Non-commutative Crepant Resolutions10.13 Example: Normalization10.14 MCM Endomorphism Rings10.15 Global Dimension of Endomorphism Rings10.16 Rational Singularities10.17 Examples: Finite Representation Type10.18 Example: the Generic Determinant10.19 Non-commutative Blowups10.20 Omissions and Open Questions

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Book SynopsisThis work deals with the matrix methods of continuous signal and image processing according to which strip-transformation is used. The authors suggest ways to solve a problem of evaluating potential noise immunity and synthesis of an optimal filter for the case of pulse noises, of applying the two-dimensional strip-transformation for storage and noise immune transmission of images. The strip-transformation of images is illustrated by examples and classes of images invariant relative to symmetrical orthogonal transformations. The monograph is intended for scientists and specialists whose activities are connected with computer signals and images processing, instrumentation and metrology. It can also be used by undergraduates, as well as by post-graduates for studying computer methods of signal and image processing.Table of ContentsPreface Introduction CHAPTER 1. THE STRIP-METHOD OF TRANSFORMATION OF SIGNALS 1.1. The strip-method linear encoding, and problems solved by it 1.2. Conservation of a "smoothness" of the transformed signal 1.3. Alignment of a variance of a non-stationary signal 1.4. Alignment comprehension a non-stationary signal 1.5. Spectrum narrowing of an encoding signal CHAPTER 2. OPTIMAL CHEBYSHEV ENCODING AND THE FILTRATION 2.1. Preliminary remarks 2.2. Problem statement 2.3. An estimation of a potential noise-immune in case of single pulse noise 2.4. An estimation of a potential noise-immune in case of multiple pulse noise 2.5. Synthesis optimal Chebyshev filter 2.6. Quasi-optimal encoding 2.7. Redundancy introduction in a strip-method linear encoding CHAPTER 3. THE STRIP-METHOD OF TRANSFORMATION OF IMAGES 3.1. A two-dimensional strip-transformation 3.2. A choice of optimal matrices of transformation 3.3. Examples of strip-transformation of images 3.4. Definition of critical multiplicity of pulse noise 3.5. Root images of bilateral strip-transformation CHAPTER 4. HARDWARE REALIZATION OF THE STRIP-METHOD 4.1. Realization of a strip-method with use of magnetic recording-reproduction 4.2. Realization of a strip-method with a cyclic matrix 4.3. The device of alignment of a signal variance 4.4. Devices for introduction of information redundancy Conclusion Appendix. Hadamard matrices and matrices close to them Bibliography

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Book SynopsisThis work provides the current theory and observations behind the cosmological phenomenon of dark energy. The approach is comprehensive with rigorous mathematical theory and relevant astronomical observations discussed in context. The book treats the background and history starting with the new-found importance of Einstein’s cosmological constant (proposed long ago) in dark energy formulation, as well as the frontiers of dark energy. The authors do not presuppose advanced knowledge of astronomy, and basic mathematical concepts used in modern cosmology are presented in a simple, but rigorous way. All this makes the book useful for both astronomers and physicists, and also for university students of physical sciences.

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Book SynopsisThe Dirac equation is of fundamental importance for relativistic quantum mechanics and quantum electrodynamics. In relativistic quantum mechanics, the Dirac equation is referred to as one-particle wave equation of motion for electron in an external electromagnetic field. In quantum electrodynamics, exact solutions of this equation are needed to treat the interaction between the electron and the external field exactly. In this monograph, all propagators of a particle, i.e., the various Green's functions, are constructed in a certain way by using exact solutions of the Dirac equation.Table of ContentsI. Relativistic wave equations II. Solutions in physically important fields III. Charge in Aharonov-Bohm field and its combinations with some additional fields IV. Solutions in more general fields V. Dirac-Pauli equation VI. Charged particles in quantized plane-wave field VII. Attendant equations and their solutions Subject Index Bibliography

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Book SynopsisKrichever and Novikov introduced certain classes of infinite dimensional Lie algebras to extend the Virasoro algebra and its related algebras to Riemann surfaces of higher genus. The author of this book generalized and extended them to a more general setting needed by the applications. Examples of applications are Conformal Field Theory, Wess-Zumino-Novikov-Witten models, moduli space problems, integrable systems, Lax operator algebras, and deformation theory of Lie algebra. Furthermore they constitute an important class of infinite dimensional Lie algebras which due to their geometric origin are still manageable. This book gives an introduction for the newcomer to this exciting field of ongoing research in mathematics and will be a valuable source of reference for the experienced researcher. Beside the basic constructions and results also applications are presented.

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Book SynopsisThis monograph is devoted to various types of algebras of functions with n variables. It is the first complete monograph (in English) on this topic, covering mainly the Russian literature. It is addressed to all algebraists working in the area of universal algebras, semigroup theory, etc. It is also a useful source of information for graduate and PhD students who are starting their research in this area. The book is the first monograph in the English mathematical literature which provides readers with a very systematical study of the notion of Menger algebras, and its generalizations and applications. The results presented here were originally published mostly in the Russian literature: In 2006, the first version of this book was edited in Russian and it is now presented in an extended version, where two new and very important chapters are added. The monograph is a broad survey of unknown or little-known Russian literature on algebras of multiplace functions and presents to the mathematical community a beautiful and strongly developing theory.Table of ContentsDesignations Introduction 1 Main concepts 1.1 Elements of the theory of relations 1.2 Functions and operations 1.3 Algebraic systems 1.4 Closure operations 1.5 Notes on Chapter 1 2 Menger algebras of functions 2.1 Definitions and fundamental notions 2.2 Menger semigroups 2.3 v-regular Menger algebras 2.4 i-solvable Menger algebras 2.5 Group-like Menger algebras 2.6 Antisymmetric Menger algebras 2.7 Representations of Menger algebras 2.8 Notes on Chapter 2 3 Ordered Menger algebras 3.1 Menger algebras of relations 3.2 F.o. and p.q-o. Menger algebras 3.3 Algebras of reversive functions 3.4 (f)-, (g)-, (f,g)-Menger algebras 3.5 Subtraction Menger algebras 3.6 Restrictive Menger algebras 3.7 Functional Menger systems 3.8 Notes on Chapter 3 4 Relations between functions 4.1 Stabilizers of Menger algebras 4.2 Stabilizers of functional Menger systems 4.3 Stationary subsets 4.4 Semi-compatibility relation 4.5 Co-definability relation 4.6 Connectivity relation 4.7 Projection equivalence relation 4.8 Semiadjacency relation 4.9 Notes on Chapter 4 5 (2, n)-semigroups of functions 5.1 (2, n)-semigroups and their representations 5.2 Menger (2, n)-semigroups 5.3 Projection relations on (2, n)-semigroups 5.4 Notes on Chapter 5 6 Systems of multiplace functions 6.1 Menger systems 6.2 Menger T-systems 6.3 Positional algebras 6.4 Mal'cev-Post iterative algebras 6.5 Semigroups of functions 6.6 Central semigroups of operations 6.7 Algebras of vector-valued functions 6.8 Notes on Chapter 6 7 Open problems 7.1 Closure operations 7.2 Menger algebras of functions 7.3 Menger algebras of relations 7.4 (2, n)-semigroups 7.5 Positional algebras Bibliography Index

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Book SynopsisThis monograph is devoted to the creation of a comprehensive formalism for quantitative description of polarized modes’ linear interaction in modern single-mode optic fibers. The theory of random connections between polarized modes, developed in the monograph, allows calculations of the zero shift deviations for a fiber ring interferometer. The monograph addresses also the Sagnac effect and the Thomas precession. Devices such as gyroscopes, used in navigation and flight control, work based on this technology. Given the ever increasing market for navigation and air traffic, researchers and practitioners in research and industry need a fundamental and sound understanding of the principles. This work presents the underlying physical foundations.Table of Contents1. Introduction2. Fiber ring interferometry based on the Sagnac effect 3. Development of the theory of a polarization modes linear interaction (random coupling) in a singlemode fiber lightguides4. Experimental investigation of a polarization modes random coupling in a single-mode fiber lightguides5. Fiber ring interferometers with a minimum configuration6. Fiber ring interferometers with an original configuration7. Geometrical phases in optics - Application of the Poincare sphere method for determination of a zero shift presence in a fiber ring interferometers8. Unstable, nonlinear and magnetic effects and methods of removal of its influence on a zero shift in a fiber ring interferometers9. Relativistic effects in optical and nonoptical ring interferometers 10. Conclusion

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Book SynopsisThis is the second of two volumes of a state-of-the-art survey article collection which originates from three commutative algebra sessions at the 2009 Fall Southeastern American Mathematical Society Meeting at Florida Atlantic University. The articles reach into diverse areas of commutative algebra and build a bridge between Noetherian and non-Noetherian commutative algebra. These volumes present current trends in two of the most active areas of commutative algebra: non-noetherian rings (factorization, ideal theory, integrality), and noetherian rings (the local theory, graded situation, and interactions with combinatorics and geometry). This volume contains surveys on aspects of closure operations, finiteness conditions and factorization. Closure operations on ideals and modules are a bridge between noetherian and nonnoetherian commutative algebra. It contains a nice guide to closure operations by Epstein, but also contains an article on test ideals by Schwede and Tucker and one by Enescu which discusses the action of the Frobenius on finite dimensional vector spaces both of which are related to tight closure. Finiteness properties of rings and modules or the lack of them come up in all aspects of commutative algebra. However, in the study of non-noetherian rings it is much easier to find a ring having a finite number of prime ideals. The editors have included papers by Boynton and Sather-Wagstaff and by Watkins that discuss the relationship of rings with finite Krull dimension and their finite extensions. Finiteness properties in commutative group rings are discussed in Glaz and Schwarz's paper. And Olberding's selection presents us with constructions that produce rings whose integral closure in their field of fractions is not finitely generated. The final three papers in this volume investigate factorization in a broad sense. The first paper by Celikbas and Eubanks-Turner discusses the partially ordered set of prime ideals of the projective line over the integers. The editors have also included a paper on zero divisor graphs by Coykendall, Sather-Wagstaff, Sheppardson and Spiroff. The final paper, by Chapman and Krause, concerns non-unique factorization.

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Book SynopsisThis volume contains selected refereed papers based on lectures presented at the "Integers Conference 2011", an international conference in combinatorial number theory that was held in Carrollton, Georgia, United States in October 2011. This was the fifth Integers Conference, held bi-annually since 2003. It featured plenary lectures presented by Ken Ono, Carla Savage, Laszlo Szekely, Frank Thorne, and Julia Wolf, along with sixty other research talks. This volume consists of ten refereed articles, which are expanded and revised versions of talks presented at the conference. They represent a broad range of topics in the areas of number theory and combinatorics including multiplicative number theory, additive number theory, game theory, Ramsey theory, enumerative combinatorics, elementary number theory, the theory of partitions, and integer sequences.

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Book SynopsisThis is the fourth volume of a comprehensive and elementary treatment of finite p-group theory. As in the previous volumes, minimal nonabelian p-groups play an important role. Topics covered in this volume include: subgroup structure of metacyclic p-groups Ishikawa’s theorem on p-groups with two sizes of conjugate classes p-central p-groups theorem of Kegel on nilpotence of H p-groups partitions of p-groups characterizations of Dedekindian groups norm of p-groups p-groups with 2-uniserial subgroups of small order The book also contains hundreds of original exercises and solutions and a comprehensive list of more than 500 open problems. This work is suitable for researchers and graduate students with a modest background in algebra.

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Book SynopsisThis is the fifth volume of a comprehensive and elementary treatment of finite p-group theory. Topics covered in this volume include theory of linear algebras and Lie algebras. The book contains many dozens of original exercises (with difficult exercises being solved) and a list of about 900 research problems and themes.

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Book SynopsisThis book deals with analytic treatments of Markov processes. Symmetric Dirichlet forms and their associated Markov processes are important and powerful tools in the theory of Markov processes and their applications. The theory is well studied and used in various fields. In this monograph, we intend to generalize the theory to non-symmetric and time dependent semi-Dirichlet forms. By this generalization, we can cover the wide class of Markov processes and analytic theory which do not possess the dual Markov processes. In particular, under the semi-Dirichlet form setting, the stochastic calculus is not well established yet. In this monograph, we intend to give an introduction to such calculus. Furthermore, basic examples different from the symmetric cases are given. The text is written for graduate students, but also researchers.Table of ContentsChapter 1 Dirichlet forms 1.1 Semi-Dirichlet forms and resolvents 1.2 Closability and regular Dirichlet forms 1.3 Transience and recurrence of Dirichlet forms 1.4 An auxiliary bilinear forms 1.5 Examples Chapter 2 Some analytic properties of Dirichlet forms 2.1 Capacity 2.2 Qasi-continuity 2.3 Potential of measures 2.4 An orthogonal decomposition of Dirichlet forms Chapter 3 Markov processes 3.1 Hunt processes 3.2 Excessive functions and negligible sets 3.3 Hunt processes associated with Dirichlet forms 3.4 Negligible sets of Hunt processes 3.5 Decomposition of Dirichlet forms Chapter 4 Additive functionals and smooth measures 4.1 Positive continuous additive functionals 4.2 Dual PCAFs and duality measures 4.3 Time changes and killings by PCAFs Chapter 5 Martingale AFs and AFs of zero energy 5.1 Decomposition of AFs 5.2 Beurling-Deny type decompositions 5.3 CAFs of zero energy 5.4 Martingale AFs of local Dirichlet forms 5.5 Transformations by multiplicative functionals 5.6 Conservativeness and recurrence of Dirichlet forms Chapter 6 Time dependent Dirichlet forms 6.1 Time dependent Dirichlet forms and associated resolvents 6.2 Some parabolic potential theory 6.3 Associated space-time processes 6.4 Additive functionals and associated measures 6.5 Some stochastic calculus

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Book SynopsisThe book presents integral formulations for partial differential equations, with the focus on spherical and plane integral operators. The integral relations are obtained for different elliptic and parabolic equations, and both direct and inverse mean value relations are studied. The derived integral equations are used to construct new numerical methods for solving relevant boundary value problems, both deterministic and stochastic based on probabilistic interpretation of the spherical and plane integral operators.

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Book SynopsisThis monograph aspires to lay the foundations of a new scientific discipline, demoeconomics, representing the synthesis of demography and spatial economics. This synthesis is performed in terms of interaction between population and its economic activity. The monograph appears a unique research work having no analogs in scientific literature. Demoeconomic systems are studied involving the macrosystems approach which combines the generalized entropy maximization principle and the local equilibria principle. Demoeconomic systems operate in an uncertain environment; thus and so, the monograph develops the methodology and technique of probabilistic modeling and forecasting of their evolution.

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Book SynopsisThere is no recent elementary introduction to the theory of discrete dynamical systems that stresses the topological background of the topic. This book fills this gap: it deals with this theory as 'applied general topology'. We treat all important concepts needed to understand recent literature. The book is addressed primarily to graduate students. The prerequisites for understanding this book are modest: a certain mathematical maturity and course in General Topology are sufficient.

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Book SynopsisAfter being an open question for sixty years the Tarski conjecture was answered in the affirmative by Olga Kharlampovich and Alexei Myasnikov and independently by Zlil Sela. Both proofs involve long and complicated applications of algebraic geometry over free groups as well as an extension of methods to solve equations in free groups originally developed by Razborov. This book is an examination of the material on the general elementary theory of groups that is necessary to begin to understand the proofs. This material includes a complete exposition of the theory of fully residually free groups or limit groups as well a complete description of the algebraic geometry of free groups. Also included are introductory material on combinatorial and geometric group theory and first-order logic. There is then a short outline of the proof of the Tarski conjectures in the manner of Kharlampovich and Myasnikov.

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Book SynopsisThe problem of classifying the finite dimensional simple Lie algebras over fields of characteristic p > 0 is a long-standing one. Work on this question has been directed by the Kostrikin-Shafarevich Conjecture of 1966, which states that over an algebraically closed field of characteristic p > 5 a finite dimensional restricted simple Lie algebra is classical or of Cartan type. This conjecture was proved for p > 7 by Block and Wilson in 1988. The generalization of the Kostrikin-Shafarevich Conjecture for the general case of not necessarily restricted Lie algebras and p > 7 was announced in 1991 by Strade and Wilson and eventually proved by Strade in 1998. The final Block-Wilson-Strade-Premet Classification Theorem is a landmark result of modern mathematics and can be formulated as follows: Every simple finite dimensional simple Lie algebra over an algebraically closed field of characteristic p > 3 is of classical, Cartan, or Melikian type. In the three-volume book, the author is assembling the proof of the Classification Theorem with explanations and references. The goal is a state-of-the-art account on the structure and classification theory of Lie algebras over fields of positive characteristic. This first volume is devoted to preparing the ground for the classification work to be performed in the second and third volumes. The concise presentation of the general theory underlying the subject matter and the presentation of classification results on a subclass of the simple Lie algebras for all odd primes will make this volume an invaluable source and reference for all research mathematicians and advanced graduate students in algebra. The second edition is corrected. Contents Toral subalgebras in p-envelopesLie algebras of special derivationsDerivation simple algebras and modulesSimple Lie algebrasRecognition theoremsThe isomorphism problemStructure of simple Lie algebrasPairings of induced modulesToral rank 1 Lie algebras

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Book SynopsisThe problem of classifying the finite dimensional simple Lie algebras over fields of characteristic p > 0 is a long standing one. Work on this question has been directed by the Kostrikin Shafarevich Conjecture of 1966, which states that over an algebraically closed field of characteristic p > 5 a finite dimensional restricted simple Lie algebra is classical or of Cartan type. This conjecture was proved for p > 7 by Block and Wilson in 1988. The generalization of the Kostrikin-Shafarevich Conjecture for the general case of not necessarily restricted Lie algebras and p > 7 was announced in 1991 by Strade and Wilson and eventually proved by Strade in 1998. The final Block-Wilson-Strade-Premet Classification Theorem is a landmark result of modern mathematics and can be formulated as follows: Every simple finite dimensional simple Lie algebra over an algebraically closed field of characteristic p > 3 is of classical, Cartan, or Melikian type. This is the second part of a three-volume book about the classification of the simple Lie algebras over algebraically closed fields of characteristic > 3. The first volume contains the methods, examples and a first classification result. This second volume presents insight in the structure of tori of Hamiltonian and Melikian algebras. Based on sandwich element methods due to A. I. Kostrikin and A. A. Premet and the investigations of filtered and graded Lie algebras, a complete proof for the classification of absolute toral rank 2 simple Lie algebras over algebraically closed fields of characteristic > 3 is given. Contents Tori in Hamiltonian and Melikian algebras1-sectionsSandwich elements and rigid toriTowards graded algebrasThe toral rank 2 case

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Book SynopsisAs a first comprehensive overview on Farey sequences and subsequences, this monograph is intended as a reference for anyone looking for specific material or formulas related to the subject. Duality of subsequences and maps between them are discussed and explicit proofs are shown in detail. From the ContentBasic structural and enumerative properties of Farey sequences, Collective decision making, Committee methods in pattern recognition, Farey duality, Farey sequence, Fundamental Farey subsequences, Monotone bijections between Farey subsequences

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Book SynopsisThis book provides the first systematic treatment of modules over discrete valuation domains, which play an important role in various areas of algebra, especially in commutative algebra. Many important results representing the state of the art are presented in the text along with interesting open problems. This updated edition presents new approaches on p-adic integers and modules, and on the determinability of a module by its automorphism group. ContentsPreliminariesBasic factsEndomorphism rings of divisible and complete modulesRepresentation of rings by endomorphism ringsTorsion-free modulesMixed modulesDeterminity of modules by their endomorphism ringsModules with many endomorphisms or automorphisms

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