The aim of this book is to extend the understanding of the fundamental role of generalizations of Lie and related non-commutative and non-associative structures in Mathematics and Physics. This is a thematic volume devoted to the interplay between several rapidly exp- ding research ?elds in contemporary Mathematics and Physics, such as generali- tions of the main structures of Lie theory aimed at quantization and discrete and non-commutative extensions of differential calculus and geometry, non-associative structures, actions of groups and semi-groups, non-commutative dynamics, n- commutative geometry and applications in Physics and beyond. The speci?c ?elds covered by this volume include: * Applications of Lie, non-associative and non-commutative associative structures to generalizations of classical and quantum mechanics and non-linear integrable systems, operadic and group theoretical methods; * Generalizations and quasi-deformations of Lie algebras such as color and super Lie algebras, quasi-Lie algebras, Hom-Lie algebras, in?nite-dimensional Lie algebras of vector ?elds associated to Riemann surfaces, quasi-Lie algebras of Witt type and their central extensions and deformations important for in- grable systems, for conformal ? eld theory and for string theory; * Non-commutative deformation theory, moduli spaces and interplay with n- commutativegeometry,algebraicgeometryandcommutativealgebra,q-deformed differential calculi and extensions of homological methods and structures; * Crossed product algebras and actions of groups and semi-groups, graded rings and algebras, quantum algebras, twisted generalizations of coalgebras and Hopf algebra structures such as Hom-coalgebras, Hom-Hopf algebras, and super Hopf algebras and their applications to bosonisation, parastatistics, parabosonic and parafermionic algebras, orthoalgebas and root systems in quantum mechanics;

Algorithms in algebraic geometry go hand in hand with software packages that implement them. Together they have established the modern field of computational algebraic geometry which has come to play a major role in both theoretical advances and applications. Over the past fifteen years, several excellent general purpose packages for computations in algebraic geometry have been developed, such as, CoCoA, Singular and Macaulay 2. While these packages evolve continuously, incorporating new mathematical advances, they both motivate and demand the creation of new mathematics and smarter algorithms.

This volume reflects the workshop a oeSoftware for Algebraic Geometrya held in the week from 23 to 27 October 2006, as the second workshop in the thematic year on Applications of Algebraic Geometry at the IMA. The papers in this volume describe the software packages Bertini, PHClab, Gfan, DEMiCs, SYNAPS, TrIm, Gambit, ApaTools, and the application of Risa/Asir to a conjecture on multiple zeta values. They offer the reader a broad view of current trends in computational algebraic geometry through software development and applications.

This 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.

There exists a vast literature on numerical methods of linear algebra. In our bibliography list, which is by far not complete, we included some monographs on the subject [46], [15], [32], [39], [11], [21]. The present book is devoted to the theory of algorithms for a single problem of linear algebra, namely, for the problem of solving systems of linear equations with non-full-rank matrix of coefficients. The solution of this problem splits into many steps, the detailed discussion of which are interest ing problems on their own (bidiagonalization of matrices, computation of singular values and eigenvalues, procedures of deflation of singular values, etc. ). Moreover, the theory of algorithms for solutions of the symmetric eigenvalues problem is closely related to the theory of solv ing linear systems (Householder's algorithms of bidiagonalization and tridiagonalization, eigenvalues and singular values, etc. ). It should be stressed that in this book we discuss algorithms which to computer programs having the virtue that the accuracy of com lead putations is guaranteed. As far as the final program product is con cerned, this means that the user always finds an unambiguous solution of his problem. This solution might be of two kinds: 1. Solution of the problem with an estimate of errors, where abso lutely all errors of input data and machine round-offs are taken into account. 2.

Introduction In the last few years a few monographs dedicated to the theory of topolog ical rings have appeared [Warn27], [Warn26], [Wies 19], [Wies 20], [ArnGM]. Ring theory can be viewed as a particular case of Z-algebras. Many general results true for rings can be extended to algebras over commutative rings. In topological algebra the structure theory for two classes of topological algebras is well developed: Banach algebras; and locally compact rings. The theory of Banach algebras uses results of Banach spaces, and the theory of locally compact rings uses the theory of LCA groups. As far as the author knows, the first papers on the theory of locally compact rings were [Pontr1]' [J1], [J2], [JT], [An], lOt], [K1]' [K2]' [K3], [K4], [K5], [K6]. Later two papers, [GS1,GS2]appeared, which contain many results concerning locally compact rings. This book can be used in two w.ays. It contains all necessary elementary results from the theory of topological groups and rings. In order to read these parts of the book the reader needs to know only elementary facts from the theories of groups, rings, modules, topology. The book consists of two parts.

Courses on mathematical programming are now part of standard teaching programs of universities and institutes. The aim of this book is to introduce students of mathematics, economics, technology and other related subjects to the qualitative theory of mathematical programming in paired vector spaces. Prerequisite for the study of this book is only a basic knowledge of analysis, of elements of functional analysis and linear algebra. The application of elementary ideas of functional analysis is convenient for a more rigorous construction of proofs and for some generalizations of the finite dimensional theory on infinite dimensional Banach-spaces. An important feature of this book is the use of a principle of duality to formulate the theoretical basis of many different concrete programming problems. The main idea of the book is to present relations of duality and to construct a general theoretical basis for different special programming problems.

This volume focuses on group theory and model theory with a particular emphasis on the interplay of the two areas. The survey papers provide an overview of the developments across group, module, and model theory while the research papers present the most recent study in those same areas. With introductory sections that make the topics easily accessible to students, the papers in this volume will appeal to beginning graduate students and experienced researchers alike. As a whole, this book offers a cross-section view of the areas in group, module, and model theory, covering topics such as DP-minimal groups, Abelian groups, countable 1-transitive trees, and module approximations. The papers in this book are the proceedings of the conference "New Pathways between Group Theory and Model Theory," which took place February 1-4, 2016, in Mulheim an der Ruhr, Germany, in honor of the editors' colleague Rudiger Goebel. This publication is dedicated to Professor Goebel, who passed away in 2014. He was one of the leading experts in Abelian group theory.

This is a self-contained introduction to algebraic curves over finite fields and geometric Goppa codes. There are four main divisions in the book. The first is a brief exposition of basic concepts and facts of the theory of error-correcting codes (Part I). The second is a complete presentation of the theory of algebraic curves, especially the curves defined over finite fields (Part II). The third is a detailed description of the theory of classical modular curves and their reduction modulo a prime number (Part III). The fourth (and basic) is the construction of geometric Goppa codes and the production of asymptotically good linear codes coming from algebraic curves over finite fields (Part IV). The theory of geometric Goppa codes is a fascinating topic where two extremes meet: the highly abstract and deep theory of algebraic (specifically modular) curves over finite fields and the very concrete problems in the engineering of information transmission. At the present time there are two essentially different ways to produce asymptotically good codes coming from algebraic curves over a finite field with an extremely large number of rational points. The first way, developed by M. A. Tsfasman, S. G. Vladut and Th. Zink [210], is rather difficult and assumes a serious acquaintance with the theory of modular curves and their reduction modulo a prime number. The second way, proposed recently by A.

The series is aimed specifically at publishing peer reviewed reviews and contributions presented at workshops and conferences. Each volume is associated with a particular conference, symposium or workshop. These events cover various topics within pure and applied mathematics and provide up-to-date coverage of new developments, methods and applications.

Graph models are extremely useful for a large number of applications as they play an important role as structuring tools. They allow to model net structures - like roads, computers, telephones, social networks - instances of abstract data structures - like lists, stacks, trees - and functional or object oriented programming. The focus of this highly self-contained book is on homomorphisms and endomorphisms, matrices and eigenvalues.

This book explores how different social psychology theories and concepts can be applied to practice. Considering theories from attribution theory to coercion theory, social identity theories to ostracism, the authors offer a greater understanding and appreciation of the ways in which social psychology can contribute to forensic practice. The book argues that social psychology is useful for carrying out assessments (including risk assessments), formulations, and interventions with clients in forensic settings, as well as for psychological consultation, training, and the development of services. These theories are also important when understanding multi-disciplinary and multi-agency working, staff-client relationships, and peer-to-peer relationships. Through illustrative composite case examples, taken from the authors' experiences in forensic settings, the chapters demonstrate effective ways to pursue a theoretically informed practice. Exploring a broad range of theories and a timely topic, Social Psychology in Forensic Practice will interest a wide readership including graduate and undergraduate students and researchers in criminology, sociology, and forensic, social and clinical psychology. It will also be of practical use to health professionals and non-health professionals working in forensic settings as well as policy makers and others commissioning forensic services.

Clifford analysis, a branch of mathematics that has been developed since about 1970, has important theoretical value and several applications. In this book, the authors introduce many properties of regular functions and generalized regular functions in real Clifford analysis, as well as harmonic functions in complex Clifford analysis. It covers important developments in handling the incommutativity of multiplication in Clifford algebra, the definitions and computations of high-order singular integrals, boundary value problems, and so on. In addition, the book considers harmonic analysis and boundary value problems in four kinds of characteristic fields proposed by Luogeng Hua for complex analysis of several variables. The great majority of the contents originate in the authors' investigations, and this new monograph will be interesting for researchers studying the theory of functions.

This book is intended to provide a reasonably self-contained account of a major portion of the general theory of rings and modules suitable as a text for introductory and more advanced graduate courses. We assume the famil iarity with rings usually acquired in standard undergraduate algebra courses. Our general approach is categorical rather than arithmetical. The continuing theme of the text is the study of the relationship between the one-sided ideal structure that a ring may possess and the behavior of its categories of modules. Following a brief outline of set-theoretic and categorical foundations, the text begins with the basic definitions and properties of rings, modules and homomorphisms and ranges through comprehensive treatments of direct sums, finiteness conditions, the Wedderburn-Artin Theorem, the Jacobson radical, the hom and tensor functions, Morita equivalence and duality, de composition theory of injective and projective modules, and semi perfect and perfect rings. In this second edition we have included a chapter containing many of the classical results on artinian rings that have hdped to form the foundation for much of the contemporary research on the representation theory of artinian rings and finite dimensional algebras. Both to illustrate the text and to extend it we have included a substantial number of exercises covering a wide spectrum of difficulty. There are, of course" many important areas of ring and module theory that the text does not touch upon."

This book is the first monograph on the theory of endomorphism rings of Abelian groups. The theory is a rapidly developing area of algebra and has its origin in the theory of operators of vector spaves. The text contains additional information on groups themselves, introducing new concepts, methods, and classes of groups. All the main fields of the theory of endomorphism rings of Abelian groups from early results to the most recent are covered. Neighbouring results on endomorphism rings of modules are also mentioned.
This text has many pedagogical features:

-all the necessary definitions and formulations of assertions on Abelian groups, rings, and modules are gathered in the first two sections;
-each chapter begins with a brief summary of results;
-there are exercises of varying difficulty in each section;
-lesser known facts on rings and modules are presented with proofs;
-there are comments at the end of each chapter together with a brief historical review as well as a look at the future direction of modern research;
-an extensive bibliography is provided. This book will be invaluable as a background text for introductory as well as advanced graduate courses. Professional algebraists might find it useful as a first systematic presentation of results previously only to be found scattered throughout various journals.

This two-volume work introduces the theory and applications of Schur-convex functions. The first volume introduces concepts and properties of Schur-convex functions, including Schur-geometrically convex functions, Schur-harmonically convex functions, Schur-power convex functions, etc. and also discusses applications of Schur-convex functions in symmetric function inequalities.

This book is concerned with discontinuous groups of motions of the unique connected and simply connected Riemannian 3-manifold of constant curva ture -1, which is traditionally called hyperbolic 3-space. This space is the 3-dimensional instance of an analogous Riemannian manifold which exists uniquely in every dimension n:::: 2. The hyperbolic spaces appeared first in the work of Lobachevski in the first half of the 19th century. Very early in the last century the group of isometries of these spaces was studied by Steiner, when he looked at the group generated by the inversions in spheres. The ge ometries underlying the hyperbolic spaces were of fundamental importance since Lobachevski, Bolyai and Gauss had observed that they do not satisfy the axiom of parallels. Already in the classical works several concrete coordinate models of hy perbolic 3-space have appeared. They make explicit computations possible and also give identifications of the full group of motions or isometries with well-known matrix groups. One such model, due to H. Poincare, is the upper 3 half-space IH in JR . The group of isometries is then identified with an exten sion of index 2 of the group PSL(2,"

Algebraic K-Theory plays an important role in many areas of modern mathematics: most notably algebraic topology, number theory, and algebraic geometry, but even including operator theory. The broad range of these topics has tended to give the subject an aura of inapproachability. This book, based on a course at the University of Maryland in the fall of 1990, is intended to enable graduate students or mathematicians working in other areas not only to learn the basics of algebraic K-Theory, but also to get a feel for its many applications. The required prerequisites are only the standard one-year graduate algebra course and the standard introductory graduate course on algebraic and geometric topology. Many topics from algebraic topology, homological algebra, and algebraic number theory are developed as needed. The final chapter gives a concise introduction to cyclic homology and its interrelationship with K-Theory.

The analysis of orthogonal polynomials associated with general weights has been a major theme in classical analysis this century. The use of potential theory since the early 1980¿s had a dramatic influence on the development of orthogonal polynomials associated with weights on the real line. For many applications of orthogonal polynomials, for example in approximation theory and numerical analysis, it is not asymptotics but certain bounds that are most important. In this monograph, the authors define and discuss their classes of weights, state several of their results on Christoffel functions, Bernstein inequalities, restricted range inequalities, and record their bounds on the orthogonal polynomials as well as their asymptotic results. This book will be of interest to researchers in approximation theory and potential theory, as well as in some branches of engineering.

Structured matrices serve as a natural bridge between the areas of algebraic computations with polynomials and numerical matrix computations, allowing cross-fertilization of both fields. This book covers most fundamental numerical and algebraic computations with Toeplitz, Hankel, Vandermonde, Cauchy, and other popular structured matrices. Throughout the computations, the matrices are represented by their compressed images, called displacements, enabling both a unified treatment of various matrix structures and dramatic saving of computer time and memory. The resulting superfast algorithms allow further dramatic parallel acceleration using FFT and fast sine and cosine transforms. Included are specific applications to other fields, in particular, superfast solutions to: various fundamental problems of computer algebra; the tangential Nevanlinna--Pick and matrix Nehari problems The primary intended readership for this work includes researchers, algorithm designers, and advanced graduate students in the fields of computations with structured matrices, computer algebra, and numerical rational interpolation. The book goes beyond research frontiers and, apart from very recent research articles, includes yet unpublished results. To serve a wider audience, the presentation unfolds systematically and is written in a user-friendly engaging style. Only some preliminary knowledge of the fundamentals of linear algebra is required. This makes the material accessible to graduate students and new researchers who wish to study the rapidly exploding area of computations with structured matrices and polynomials. Examples, tables, figures, exercises, extensive bibliography, and index lend this text toclassroom use or self-study.

1 Grundlagen.- 1.1 Allgemeine Grundlagen.- 1.1.1 Ziele und Aufgaben.- 1.1.2 Methoden.- 1.1.3 Geschichte und Einordnung.- 1.1.3.1 Geschichte der Bauwerksvermessung.- 1.1.3.2 Geschichte des Vermessungswesens.- 1.1.3.3 Geschichte der Architekturphotogrammetrie.- 1.1.4 Rechtliche Grundlagen und Rahmenbedingungen.- 1.1.4.1 Internationale Vereinbarungen und Organisationen.- 1.1.4.2 Baugesetzbuch, Denkmalpflegegesetze, Vermessungsgesetze.- 1.2 Messgroessen und Masseinheiten.- 1.2.1 Strecken.- 1.2.2 Winkel.- 1.3 Bezugssysteme und Koordinaten.- 1.3.1 Bezugsflachen.- 1.3.2 Koordinaten.- 1.3.3 Koordinatensysteme.- 1.3.3.1 Polarkoordinaten.- 1.3.3.2 Lokale Koordinatensysteme.- 1.3.3.3 Regionale Koordinatensysteme.- 1.3.3.4 Globale Koordinatensysteme.- 1.3.3.5 Geographische Koordinaten.- 1.3.3.6 Geozentrische Koordinaten.- 1.3.4 Koordinatentransformationen.- 1.3.4.1 Translation (2D).- 1.3.4.2 Massstabslose Transformation (2D).- 1.3.4.3 AEhnlichkeitstransformation (2D).- 1.3.4.4 Vereinfachte AEhnlichkeitstransformation mit 2 Passpunkten (2D).- 1.3.4.5 Affintransformation (2D).- 1.3.4.6 Weitere ebene Koordinatentransformationen.- 1.3.4.7 Raumliche Koordinatentransformation (3D).- 1.3.5 Festpunktfelder.- 1.3.5.1 Netz trigonometrischer Punkte zur Lagedefinition.- 1.3.5.2 Hoehennetz.- 1.3.6 Vermessungsnetze fur die Bauwerksvermessung.- 1.3.6.1 Netzdesign.- 1.3.6.2 Vermarkung.- 1.3.6.3 Design und Fertigung von Punktsignalisierungen.- 1.3.6.4 Auswahl naturlicher Passpunkte.- 1.3.6.5 Schnurnetz zur temporaren Vermarkung.- 1.3.6.6 Punktubersichten und Einmessskizzen.- 1.4 Fehlerlehre und Statistik.- 1.4.1 Fehlerarten und ihre Wirkung.- 1.4.1.1 Zufallige Fehler.- 1.4.1.2 Systematische Fehler.- 1.4.1.3 Grobe Fehler.- 1.4.2 Fehlerfortpflanzung und Ausgleichsrechnung.- 1.4.3 Rechenscharfe und Rundung.- 1.4.4 Toleranzen im Bauwesen.- 2 Dokumentation von Gebauden und Ensembles.- 2.1 Amtliche Dokumentation.- 2.1.1 Katasterunterlagen.- 2.1.2 Amtliche Karten.- 2.1.3 Lageplan.- 2.1.4 Geoinformationssysteme (GIS).- 2.2 Plane.- 2.2.1 Grundriss.- 2.2.2 Schnitt.- 2.2.3 Ansicht.- 2.2.4 Detaildarstellungen.- 2.2.5 Massstabe und Detaillierungsgrad.- 2.2.6 Materialien und Aufbewahrung.- 2.3 3D-Beschreibungen.- 2.3.1 CAD-Modell.- 2.3.2 Animation.- 2.3.3 Virtual Reality.- 2.3.4 Augmented Reality.- 2.4 Fotografie.- 2.4.1 Analoge Fotografie.- 2.4.1.1 Fotografisches Material.- 2.4.1.2 Kameras.- 2.4.1.3 Objektive.- 2.4.1.4 Licht.- 2.4.1.5 Belichtung.- 2.4.1.6 Archivierungen von Fotomaterialien.- 2.4.2 Digitale Bilder.- 2.4.2.1 Flachensensoren.- 2.4.2.2 Zeilenkameras.- 2.4.2.3 Spezialkameras.- 2.4.3 Scannen analoger Fotovorlagen.- 2.4.4 Digitale Bildverarbeitung.- 2.5 Textliche und hybride Beschreibungen.- 2.5.1 Raumbuch.- 2.5.2 Hypertext Dokumente.- 2.5.3 Informationssystem.- 2.6 Archivierung digitaler Daten.- 2.6.1 Datentrager.- 2.6.2 Datenformate.- 2.6.2.1 Texte.- 2.6.2.2 Datenbanken.- 2.6.2.3 Vektordaten.- 2.6.2.4 Rasterdaten.- 2.6.2.5 Hypermedia.- 3 Erfassung von Messelementen.- 3.1 Messprinzipien.- 3.1.1 Vom-Grossen-ins-Kleine.- 3.1.2 UEberbestimmungen.- 3.1.3 Vermeidung von systematischen Fehlern.- 3.2 Gerate und Instrumente.- 3.2.1 Bauteile, Kleingerate und Zubehoer.- 3.2.1.1 Lote und Libellen.- 3.2.1.2 Fernrohr.- 3.2.1.3 Stative.- 3.2.1.4 Fluchtstab.- 3.2.1.5 Nivellierlatten und Kleingerat.- 3.2.1.6 Aufstellen eines Instruments.- 3.2.2 Winkelmessung.- 3.2.2.1 Bestimmung rechter Winkel.- 3.2.2.2 Theodolit.- 3.2.2.3 Satzmessung.- 3.2.2.4 Berechnung von Richtungswinkeln aus Koordinaten.- 3.2.3 Streckenmessung.- 3.2.3.1 Streckenmessung mit dem Messband.- 3.2.3.2 Optische Streckenmessung.- 3.2.3.3 Elektro-optische Entfernungsmessung (EDM).- 3.2.4 Hoehenmessung.- 3.2.4.1 Einfache Werkzeuge.- 3.2.4.2 Nivellement.- 3.2.4.3 Rotationslaser.- 3.3 Beschaffung einer Vermessungsausrustung.- 4 Messverfahren.- 4.1 Schrittskizze.- 4.2 Handaufmass.- 4.3 Punktbestimmung ohne Theodolit.- 4.3.1 Bogenschlag.- 4.3.2 Einbindeverfahren.- 4.3.3 Orthogonalverfahren.- 4.3.4

The aim of this book is to present a substantial part of matrix analysis that is functional analytic in spirit. Much of this will be of interest to graduate students and research workers in operator theory, operator algebras, mathematical physics and numerical analysis. The book can be used as a basic text for graduate courses on advanced linear algebra and matrix analysis. It can also be used as supplementary text for courses in operator theory and numerical analysis. Among topics covered are the theory of majorization, variational principles for eigenvalues, operator monotone and convex functions, perturbation of matrix functions and matrix inequalities. Much of this is presented for the first time in a unified way in a textbook. The reader will learn several powerful methods and techniques of wide applicability, and see connections with other areas of mathematics. A large selection of matrix inequalities will make this book a valuable reference for students and researchers who are working in numerical analysis, mathematical physics and operator theory.

This is an English translation of the now classic "Algèbre Locale - Multiplicités" originally published by Springer as LNM 11, in several editions since 1965. It gives a short account of the main theorems of commutative algebra, with emphasis on modules, homological methods and intersection multiplicities ("Tor-formula"). Many modifications to the original French text have been made by the author for this English edition: they make the text easier to read, without changing its intended informal character.