Analytical solutions to the orbital motion of celestial objects have been nowadays mostly replaced by numerical solutions, but they are still irreplaceable whenever speed is to be preferred to accuracy, or to simplify a dynamical model. In this book, the most common orbital perturbations problems are discussed according to the Lie transforms method, which is the de facto standard in analytical orbital motion calculations.

This is an updated English translation of "Cohomologie Galoisienne", published more than 30 years ago as one of the very first Lecture Notes in Mathematics. It includes a reproduction of an influential paper of R. Steinberg, together with some new material and an expanded bibliography.

The book contains seven refereed research papers on locally compact quantum groups and groupoids by leading experts in the respective fields. These contributions are based on talks presented on the occasion of the meeting between mathematicians and theoretical physicists held in Strasbourg from February 21 to February 23, 2002. Topics covered are: various constructions of locally compact quantum groups and their multiplicative unitaries; duality theory for locally compact quantum groups; combinatorial quantization of flat connections associated with SL(2,c); quantum groupoids, especially coming from Depth 2 Extensions of von Neumann algebras, C*-algebras and Rings. Many mathematical results are motivated by problems in theoretical physics. Historical remarks set the results presented in perspective. Directed at research mathematicians and theoretical physicists as well as graduate students, the volume will give an overview of a field of research in which great progress has been achieved in the last few years, with new ties to many other areas of mathematics and physics.

This 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.

This 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.

This monograph provides a comprehensive introduction to the Kazhdan-Lusztig theory of cells in the broader context of the unequal parameter case. Serving as a useful reference, the present volume offers a synthesis of significant advances made since Lusztig's seminal work on the subject was published in 2002. The focus lies on the combinatorics of the partition into cells for general Coxeter groups, with special attention given to induction methods, cellular maps and the role of Lusztig's conjectures. Using only algebraic and combinatorial methods, the author carefully develops proofs, discusses open conjectures, and presents recent research, including a chapter on the action of the cactus group. Kazhdan-Lusztig Cells with Unequal Parameters will appeal to graduate students and researchers interested in related subject areas, such as Lie theory, representation theory, and combinatorics of Coxeter groups. Useful examples and various exercises make this book suitable for self-study and use alongside lecture courses. Information for readers: The character {\mathbb{Z}} has been corrupted in the print edition of this book and appears incorrectly with a diagonal line running through the symbol.

The volume is the outcome of the conference "Lie superalgebras," which was held at the Istituto Nazionale di Alta Matematica, Rome, in 2012. The conference gathered virtually all the main specialists in the subject, and the talks held provided comprehensive insights into the newest trends in research on Lie superalgebras (and related topics like vertex algebras, representation theory and supergeometry). The book includes both extended abstracts of the conference papers and new original works related to the theme of the conference.

This book presents the proceedings of the international conference Analytic Aspects in Convexity, which was held in Rome in October 2016. It offers a collection of selected articles, written by some of the world's leading experts in the field of Convex Geometry, on recent developments in this area: theory of valuations; geometric inequalities; affine geometry; and curvature measures. The book will be of interest to a broad readership, from those involved in Convex Geometry, to those focusing on Functional Analysis, Harmonic Analysis, Differential Geometry, or PDEs. The book is a addressed to PhD students and researchers, interested in Convex Geometry and its links to analysis.

Based on lectures given at Claremont McKenna College, this text constitutes a substantial, abstract introduction to linear algebra. The presentation emphasizes the structural elements over the computational - for example by connecting matrices to linear transformations from the outset - and prepares the student for further study of abstract mathematics. Uniquely among algebra texts at this level, it introduces group theory early in the discussion, as an example of the rigorous development of informal axiomatic systems.

New Edition available hereGalois' Theory of Algebraic Equations gives a detailed account of the development of the theory of algebraic equations, from its origins in ancient times to its completion by Galois in the nineteenth century. The main emphasis is placed on equations of at least the third degree, i.e. on the developments during the period from the sixteenth to the nineteenth century. The appropriate parts of works by Cardano, Lagrange, Vandermonde, Gauss, Abel and Galois are reviewed and placed in their historical perspective, with the aim of conveying to the reader a sense of the way in which the theory of algebraic equations has evolved and has led to such basic mathematical notions as “group” and “field”. A brief discussion on the fundamental theorems of modern Galois theory is included. Complete proofs of the quoted results are provided, but the material has been organized in such a way that the most technical details can be skipped by readers who are interested primarily in a broad survey of the theory.This book will appeal to both undergraduate and graduate students in mathematics and the history of science, and also to teachers and mathematicians who wish to obtain a historical perspective of the field. The text has been designed to be self-contained, but some familiarity with basic mathematical structures and with some elementary notions of linear algebra is desirable for a good understanding of the technical discussions in the later chapters.

This concise, fast-paced text introduces the concepts and applications behind plane networks. It presents fundamental material from linear algebra and differential equations, and offers several different applications of the continuous theory. Practical problems, supported by MATLAB files, underscore the theory; additional material can be downloaded from the author's website.

With applications in quantum field theory, elementary particle physics and general relativity, this two-volume work studies invariance of differential operators under Lie algebras, quantum groups, superalgebras including infinite-dimensional cases, Schroedinger algebras, applications to holography. This first volume covers the general aspects of Lie algebras and group theory supplemented by many concrete examples for a great variety of noncompact semisimple Lie algebras and groups. Contents: Introduction Lie Algebras and Groups Real Semisimple Lie Algebras Invariant Differential Operators Case of the Anti-de Sitter Group Conformal Case in 4D Kazhdan-Lusztig Polynomials, Subsingular Vectors, and Conditionally Invariant Equations Invariant Differential Operators for Noncompact Lie Algebras Parabolically Related to Conformal Lie Algebras Multilinear Invariant Differential Operators from New Generalized Verma Modules Bibliography Author Index Subject Index

Elliptic cohomology is an extremely beautiful theory with both geometric and arithmetic aspects. The former is explained by the fact that the theory is a quotient of oriented cobordism localised away from 2, the latter by the fact that the coefficients coincide with a ring of modular forms. The aim of the book is to construct this cohomology theory, and evaluate it on classifying spaces BG of finite groups G. This class of spaces is important, since (using ideas borrowed from Monstrous Moonshine') it is possible to give a bundle-theoretic definition of EU-(BG). Concluding chapters also discuss variants, generalisations and potential applications.

Intended for graduate courses or for independent study, this book presents the basic theory of fields. The first part begins with a discussion of polynomials over a ring, the division algorithm, irreducibility, field extensions, and embeddings. The second part is devoted to Galois theory. The third part of the book treats the theory of binomials. The book concludes with a chapter on families of binomials - the Kummer theory. This new edition has been completely rewritten in order to improve the pedagogy and to make the text more accessible to graduate students. The exercises have also been improved and a new chapter on ordered fields has been included. About the first edition: ... the author has gotten across many important ideas and results. This book should not only work well as a textbook for a beginning graduate course in field theory, but also for a student who wishes to take a field theory course as independent study. - J. N. Mordeson, Zentralblatt. The book is written in a clear and explanatory style. It contains over 235 exercises which provide a challenge to the reader. The book is recommended for a graduate course in field theory as well as for independent study. - T.

This book introduces mathematicians, physicists, and philosophers to a new, coherent approach to theory and interpretation of quantum physics, in which classical and quantum thinking live peacefully side by side and jointly fertilize the intuition. The formal, mathematical core of quantum physics is cleanly separated from the interpretation issues. The book demonstrates that the universe can be rationally and objectively understood from the smallest to the largest levels of modeling. The thermal interpretation featured in this book succeeds without any change in the theory. It involves one radical step, the reinterpretation of an assumption that was virtually never questioned before - the traditional eigenvalue link between theory and observation is replaced by a q-expectation link: Objective properties are given by q-expectations of products of quantum fields and what is computable from these. Averaging over macroscopic spacetime regions produces macroscopic quantities with negligible uncertainty, and leads to classical physics. - Reflects the actual practice of quantum physics. - Models the quantum-classical interface through coherent spaces. - Interprets both quantum mechanics and quantum field theory. - Eliminates probability and measurement from the foundations. - Proposes a novel solution of the measurement problem.

Equivariant cohomology on smooth manifolds is the subject of this book which is part of a collection of volumes edited by J. Bruning and V.W. Guillemin. The point of departure are two relatively short but very remarkable papers be Henry Cartan, published in 1950 in the Proceedings of the "Colloque de Topologie." These papers are reproduced here, together with a modern introduction to the subject, written by two of the leading experts in the field. This "introduction" comes as a textbook of its own, though, presenting the first full treatment of equivariant cohomology in the de Rahm setting. The well known topological approach is linked with the differential form aspect through the equivariant de Rahm theorem. The systematic use of supersymmetry simplifies considerably the ensuing development of the basic technical tools which are then applied to a variety of subjects, leading up to the localization theorems and other very recent results."

The Influence Line Approach to the Analysis of Rigid Frames offers a simple method of analysis of indeterminate structures. It is original and independent of other methods. The author derived these equations by applying an algebraic rather than an arithmetical method of distribution of fixed-end moments. His method is fully explained and illustrated by worked examples.

The equations listed in the Tables in The Influence Line Approach to the Analysis of Rigid Frames offer a simple approach to the analysis of rigid frames, including building frames, rendering them statically determinate for any system of loading, static or moving and including the self weight of a structure.

Particularly useful aspects to the reader are:
-The equations are of an elementary nature consisting only of distribution factors and the co-efficient of a span length and to which values from zero to unity are given.
-The equations can be used to analyze frames the members of which can be either of constant or variable cross-section, and in both cases distributions of fixed-end moments are not required.
-In addition, the evaluation of fixed-end moments is not required when the frame consists of members of constant cross-section.
-The equations are independent of other methods of analysis requiring neither the use of model analysis nor the application of linear equations.
-The equations offer a good indication of structural behavior.
-The Tables lend themselves to expansion catering for different degrees of end fixation. The Influence Line Approach to the Analysis of Rigid Frames can be taught not only to university undergraduate students, but also to those pursuing middle-levelcourses in Civil Engineering, Structural Engineering and Building. In addition, practicing assistant structural designers will find it a useful reference work.

Field Arithmetic explores Diophantine fields through their absolute Galois groups. This largely self-contained treatment starts with techniques from algebraic geometry, number theory, and profinite groups. Graduate students can effectively learn generalizations of finite field ideas. We use Haar measure on the absolute Galois group to replace counting arguments. New Chebotarev density variants interpret diophantine properties. Here we have the only complete treatment of Galois stratifications, used by Denef and Loeser, et al, to study Chow motives of Diophantine statements.

Progress from the first edition starts by characterizing the finite-field like P(seudo)A(lgebraically)C(losed) fields. We once believed PAC fields were rare. Now we know they include valuable Galois extensions of the rationals that present its absolute Galois group through known groups. PAC fields have projective absolute Galois group. Those that are Hilbertian are characterized by this group being pro-free. These last decade results are tools for studying fields by their relation to those with projective absolute group. There are still mysterious problems to guide a new generation: Is the solvable closure of the rationals PAC; and do projective Hilbertian fields have pro-free absolute Galois group (includes Shafarevich's conjecture)?

The third edition improves the second edition in two ways: First it removes many typos and mathematical inaccuracies that occur in the second edition (in particular in the references). Secondly, the third edition reports on five open problems (out of thirtyfour open problems of the second edition) that have been partially or fully solved since that edition appeared in 2005.

This volume contains the combined Proceedings of the Second International Meeting on Commutative Algebra and Related Areas (SIMCARA) held from July 22-26, 2019, at the Universidade de Sao Paulo, Sao Carlos, Brazil, and the AMS Special Session on Commutative Algebra, held from September 14-15, 2019, at the University of Wisconsin-Madison, Wisconsin. These two meetings celebrated the combined 150th birthday of Roger and Sylvia Wiegand. The Wiegands have been a fixture in the commutative algebra community, as well as the wider mathematical community, for over 40 years. Articles in this volume cover various areas of factorization theory, homological algebra, ideal theory, representation theory, homological rigidity, maximal Cohen-Macaulay modules, and the behavior of prime spectra under completion, as well as some topics in related fields. The volume itself bears evidence that the area of commutative algebra is a vibrant one and highlights the influence of the Wiegands on generations of researchers. It will be useful to researchers and graduate students.

The fusion of algebra, analysis and geometry, and their application to real world problems, have been dominant themes underlying mathematics for over a century. Geometric algebras, introduced and classified by Clifford in the late 19th century, have played a prominent role in this effort, as seen in the mathematical work of Cartan, Brauer, Weyl, Chevelley, Atiyah, and Bott, and in applications to physics in the work of Pauli, Dirac and others. One of the most important applications of geometric algebras to geometry is to the representation of groups of Euclidean and Minkowski rotations. This aspect and its direct relation to robotics and vision will be discussed in several chapters of this multi-authored textbook, which resulted from the ASI meeting.

Moreover, group theory, beginning with the work of Burnside, Frobenius and Schur, has been influenced by even more general problems. As a result, general group actions have provided the setting for powerful methods within group theory and for the use of groups in applications to physics, chemistry, molecular biology, and signal processing. These aspects, too, will be covered in detail.

With the rapidly growing importance of, and ever expanding conceptual and computational demands on signal and image processing in remote sensing, computer vision, medical image processing, and biological signal processing, and on neural and quantum computing, geometric algebras, and computational group harmonic analysis, the topics of the book have emerged as key tools. The list of authors includes many of the world's leading experts in the development of new algebraic modeling and signal representation methodologies, novel Fourier-based andgeometric transforms, and computational algorithms required for realizing the potential of these new application fields.

This book presents a unified mathematical treatment of diverse problems in the fields of cognitive systems using Clifford, or geometric, algebra. Geometric algebra provides a rich general mathematical framework for the development of the ideas of multilinear algebra, projective and affine geometry, calculus on manifolds, the representation of Lie groups and Lie algebras, and many other areas of applications. By treating a wide spectrum of problems in a common geometric language, the book offers both new insights and new solutions that should be useful to scientists and engineers working in different but related areas of artificial intelligence. It looks at building intelligence systems through the construction of Perception Action Cycles; critical to this concept is incorporating representation and learning in a flexible geometric system. Each chapter is written in accessible terms accompanied by numerous examples and figures that clarify the application of geometric algebra to problems in geometric computing, image processing, computer vision, robotics, neural computing and engineering. Topics and features: *Introduces a nonspecialist to Clifford, or geometric, algebra and it shows applications in artificial intelligence *Thorough discussion of several tasks of signal and image processing, computer vision, robotics, neurocomputing and engineering using the geometric algebra framework *Features the computing frameworks of the linear model n-dimensional affine plane and the nonlinear model of Euclidean space known as the horosphere, and addresses the relationship of these models to conformal, affine and projective geometries *Applications of geometric algebra to other related areas: aeronautics, mechatronics, graphics engineering, and speech processing *Exercises and hints for the development of future computer software packages for extensive calculations in geometric algebra The book is an essential resource for computer scientists, AI researchers, and electrical engineers and includes computer programs to clarify and demonstrate the importance of geometric computing for cognitive systems and artificial autonomous systems research.

Commutative Ring Theory emerged as a distinct field of research in math ematics only at the beginning of the twentieth century. It is rooted in nine teenth century major works in Number Theory and Algebraic Geometry for which it provided a useful tool for proving results. From this humble origin, it flourished into a field of study in its own right of an astonishing richness and interest. Nowadays, one has to specialize in an area of this vast field in order to be able to master its wealth of results and come up with worthwhile contributions. One of the major areas of the field of Commutative Ring Theory is the study of non-Noetherian rings. The last ten years have seen a lively flurry of activity in this area, including: a large number of conferences and special sections at national and international meetings dedicated to presenting its results, an abundance of articles in scientific journals, and a substantial number of books capturing some of its topics. This rapid growth, and the occasion of the new Millennium, prompted us to embark on a project aimed at presenting an overview of the recent research in the area. With this in mind, we invited many of the most prominent researchers in Non-Noetherian Commutative Ring Theory to write expository articles representing the most recent topics of research in this area."

This self-contained work, focusing on the theory of state spaces of C*-algebras and von Neumann algebras, explains how the oriented state space geometrically determines the algebra. The theory of orientation of C*-algebra state spaces is presented with a new approach that does not depend on Jordan algebras, and the theory of orientation of normal state spaces of von Neumann algebras is presented with complete proofs for the first time. The theory of operator algebras was initially motivated by applications to physics, but has recently found unexpected new applications to fields of pure mathematics as diverse as foliations and knot theory. Key features include: * first and only work devoted to state spaces of operator algebras--contains much material not available in existing books * prerequisites are standard graduate courses in real and complex variables, measure theory, and functional analysis * complete proofs of basic results on operator algebras presented so that no previous knowledge in the field is needed * detailed introduction develops basic tools used throughout the text * numerous chapter remarks on advanced topics of independent interest with references to the literature, or discussion of applications to physics "State Spaces of Operator Algebras" is intended for specialists in operator algebras, as well as graduate students and mathematicians seeking an overview of the field. The introduction to C*-algebras and von Neumann algebras may also be of interest in it own right for those wanting a quick introduction to basic concepts in those fields.

In semigroup theory there are certain kinds of band decompositions, which are very useful in the study of the structure semigroups. There are a number of special semigroup classes in which these decompositions can be used very successfully. The book focuses attention on such classes of semigroups. Some of them are partially discussed in earlier books, but in the last thirty years new semigroup classes have appeared and a fairly large body of material has been published on them. The book provides a systematic review on this subject. The first chapter is an introduction. The remaining chapters are devoted to special semigroup classes. These are Putcha semigroups, commutative semigroups, weakly commutative semigroups, R-Commutative semigroups, conditionally commutative semigroups, RC-commutative semigroups, quasi commutative semigroups, medial semigroups, right commutative semigroups, externally commutative semigroups, E-m semigroups, WE-m semigroups, weakly exponential semigroups, (m, n)-commutative semigroups and n(2)-permutable semigroups. Audience: Students and researchers working in algebra and computer science.

Some Historical Background This book deals with the cohomology of groups, particularly finite ones. Historically, the subject has been one of significant interaction between algebra and topology and has directly led to the creation of such important areas of mathematics as homo logical algebra and algebraic K-theory. It arose primarily in the 1920's and 1930's independently in number theory and topology. In topology the main focus was on the work ofH. Hopf, but B. Eckmann, S. Eilenberg, and S. MacLane (among others) made significant contributions. The main thrust of the early work here was to try to understand the meanings of the low dimensional homology groups of a space X. For example, if the universal cover of X was three connected, it was known that H2(X; A. ) depends only on the fundamental group of X. Group cohomology initially appeared to explain this dependence. In number theory, group cohomology arose as a natural device for describing the main theorems of class field theory and, in particular, for describing and analyzing the Brauer group of a field. It also arose naturally in the study of group extensions, N"