Algebra Books

Book SynopsisTable of ContentsPreface.- Study of ?X.- Completions, primary decomposition and length.- Depth and dimension.- Duality theorems.- Flat morphisms.- Étale morphisms.- Smooth morphisms.- Curves.

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Table of ContentsLocal Theory.- Global Theory.

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Table of ContentsAspects of non-commutative order.- Correspondences between von neumann algebras and discrete automorphism groups.- The construction and decomposition of quantum fields using operator theory, probability and fiber bundles.- On KMS states of a C* dynamical system.- Recent developments in the theory of unbounded derivations in C*-algebras.- Quasi-expectations and injective operator algebras.- General short exact sequence theorem for toeplitz operators of uniform algebras.- AW*-algebras with monotone convergence property and type III, non W*, AW*-factors.- A non-W*, AW*-factor.- Fixed points and commutation theorems.- Algebraic features of equilibrium states.- Minimal dilations of CP-flows.- Resistance inequalities for the isotropic heisenberg model.- Homogeneity of the state space of factors of type III1.- Product isometries and automorphisms of the car algebra.- Construction of ITPFI with non-trivial uncountable T-set.- On the algebraic reduction theory for countable direct summand C*-algebras of separable C*-algebras.- C*-algebras and applications to physics.

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Table of ContentsPreparatory remarks.- Formal constructions.- Simple Lie superalgebras.- A survey of some further developments.

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Table of ContentsPreliminaries.- Extensions of structure.- The category G..- Examples.

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Book SynopsisLet us first state exactly what this book is and what it is not. It is a compendium of equations for the physicist and the engineer working with electrostatics, magne­ tostatics, electric currents, electromagnetic fields, heat flow, gravitation, diffusion, optics, or acoustics. It tabulates the properties of 40 coordinate systems, states the Laplace and Helmholtz equations in each coordinate system, and gives the separation equations and their solutions. But it is not a textbook and it does not cover relativistic and quantum phenomena. The history of classical physics may be regarded as an interplay between two ideas, the concept of action-at-a-distance and the concept of a field. Newton's equation of universal gravitation, for instance, implies action-at-a-distance. The same form of equation was employed by COULOMB to express the force between charged particles. AMPERE and GAUSS extended this idea to the phenomenological action between currents. In 1867, LUDVIG LORENZ formulated electrodynamics as retarded action-at-a-distance. At almost the same time, MAXWELL presented the alternative formulation in terms of fields. In most cases, the field approach has shown itself to be the more powerful.Table of ContentsI. Eleven coordinate systems.- II. Transformations in the complex plane.- III. Cylindrical systems.- IV. Rotational systems.- V. The vector Helmholtz equation.- VI. Differential equations.- VII. Functions.- Appendix. Symbols.- Author Index.

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Book SynopsisAn invaluable summary of research work done in the period from 1978 to the presentTrade ReviewFrom the reviews: "It is a full-fledged advanced course on themes in higher algebra suited for a specialized graduate seminar, a research seminar, and of course, self-study by an aspiring researcher. … Serre’s Problem on Projective Modules, is very clear and well written … and quickly gets the reader properly air-borne. … the pay-off is huge: this is fantastic stuff. … is a superb book. It’s highly recommended." (Michael Berg, MathDL, March, 2007) "The book starts with the basics of projective modules and the K0 and K1 groups, and then gives the classical, partial results about Serre’s conjecture. … This well-written book is the definitive treatment of ‘Serre’s conjecture’ – its history, solution, and generalizations – and will be of interest to both beginning graduate students and advanced researchers in this field." (David F. Anderson, Zentralblatt MATH, Vol. 1101 (3), 2007) "Lam has done a magnificent job of organizing the mated al and presenting complete proofs of all the results directly connected with Sen-e's problem. ... The references are complete and make the book a very valuable reference even for experts in the field.... It will be very useful to students wishing to learn about projective modules ... . This is definitely a book that anyone ... interested in projective modules should have on his or her shelf!" (Richard G. Swan, Bulletin of the American Mathematical Society, Vol. 45 (3), July, 2008)Table of Contentsto Serre’s Conjecture: 1955–1976.- Foundations.- The “Classical” Results on Serre’s Conjecture.- The Basic Calculus of Unimodular Rows.- Horrocks’ Theorem.- Quillen’s Methods.- K1-Analogue of Serre’s Conjecture.- The Quadratic Analogue of Serre’s Conjecture.- References for Chapters I–VII.- Appendix: Complete Intersections and Serre’s Conjecture.- New Developments (since 1977).- References for Chapter VIII.

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Book SynopsisDas Buch richtet sich an eine Leserschaft, die bereits Grundkenntnisse in der Computergrafik hat. Vorwiegend ist hierbei an Studenten der Informatik gedacht, die bereits eine Computeranimationsvorlesung belegt haben oder die ein vertieftes Interesse an diesem Gebiet besitzen. Neben einem Überblick über die relevanten Themen der Computeranimation wurde ein besonderes Schwergewicht auf die physikalisch-basierten Animationsmethoden gelegt. Zum einfacheren Verständnis, speziell der physikalisch-basierten Methoden, sind allerdings Grundkenntnisse in der Physik sowie in der Analysis sehr hilfreich. Das Buch zeichnet sich im Besonderen dadurch aus, dass es auch exemplarisch wichtige Details einiger Animationsmethoden behandelt, die deren Implementierungen erleichtern.Table of ContentsEinführung.- Globale Bewegungen.- Deformationen.- Warping und Morphing.- Gesichtsanimation.- Prozedurale Animationstechniken.- Motion Capturing und Motion Editing.- Modellierung und Animation von Naturerscheinungen.- Modellierung und Animation von Stoffen.- Animationen mit neuronalen Netzen.- Physikalisch-basierte Animation mechanischer Systeme.

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Book SynopsisCategories and sheaves appear almost frequently in contemporary advanced mathematics. This book covers categories, homological algebra and sheaves in a systematic manner starting from scratch and continuing with full proofs to the most recent results in the literature, and sometimes beyond. The authors present the general theory of categories and functors, emphasizing inductive and projective limits, tensor categories, representable functors, ind-objects and localization.Trade ReviewFrom the reviews: "This book of Kashiwara and Schapira, recognized specialists in algebraic analysis, is a detailed full-scale exposition of categories, homological algebra and sheaves. These notions are presented from scratch up to the most recent (sometimes new) results … ." (Corrado Marastoni, Mathematical Reviews, Issue 2006 k)Table of ContentsThe Language of Categories.- Limits.- Filtrant Limits.- Tensor Categories.- Generators and Representability.- Indization of Categories.- Localization.- Additive and Abelian Categories.- ?-accessible Objects and F-injective Objects.- Triangulated Categories.- Complexes in Additive Categories.- Complexes in Abelian Categories.- Derived Categories.- Unbounded Derived Categories.- Indization and Derivation of Abelian Categories.- Grothendieck Topologies.- Sheaves on Grothendieck Topologies.- Abelian Sheaves.- Stacks and Twisted Sheaves.

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Book SynopsisThis book offers a comprehensive introduction to the general theory of C*-algebras and von Neumann algebras. Beginning with the basics, the theory is developed through such topics as tensor products, nuclearity and exactness, crossed products, K-theory, and quasidiagonality. The presentation carefully and precisely explains the main features of each part of the theory of operator algebras; most important arguments are at least outlined and many are presented in full detail.Trade ReviewFrom the reviews: "The volume under review is supposed to cover basics on operator algebras … . Blackadar’s book is very well written and pleasant to read. It is especially suited to readers who already know the basics of operator algebras but who need a reference for some result or who wish to have a unified approach to topics treated by them." (Paul Jolissaint, Mathematical Reviews, Issue 2006 k) "This volume is an important and useful contribution to the literature on C*-algebras and von Neumann algebras. … The book is extremely well written. It can be recommended as a reference to graduate students working in operator algebra theory and to other mathematicians who want to bring themselves up-to-date on the subject." (V. M. Manuilov, Zentralblatt MATH, Vol. 1092 (18), 2006)Table of ContentsOperators on Hilbert Space.- C*-Algebras.- Von Neumann Algebras.- Further Structure.- K-Theory and Finiteness.

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Book SynopsisCe troisième volume du Livre sur les Groupes et algèbres de Lie, neuvième Livre du traité, poursuit l’étude des algèbres de Lie et leurs représentations. Il comprend les chapitres: 7. Sous-algèbres de Cartan, éléments réguliers; 8. Algèbres de Lie semi-simples déployées.Trade ReviewFrom the reviews of the second edition:“The volume under review is the faithful reprint of Chapters 7 and 8 of Book 9 within Nicolas Bourbaki’s fundamental and sweeping collection ‘Éléments de Mathématique’ … . As usual and typical for Bourbaki’s books, each section comes with a wealth of complementing and further-leading exercises, for many of which detailed hints are given. No doubt, this volume was, is, and will remain one of the great source books in the general theory of Lie groups and Lie algebras.” (Werner Kleinert, Zentralblatt MATH, Vol. 1181, 2010)Table of ContentsSous-algèbres de Cartan éléments réguliers.- Algèbres de Lie semi-simples déployées.

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Book SynopsisCe deuxième volume du Livre sur les Groupes et algèbres de Lie, neuvième Livre du traité, comprend les chapitres: 2. Algèbres de Lie libres; 3. Groupes de Lie. Le chapitre 2 poursuit la présentation des notions fondamentales des algèbres de Lie avec l’introduction des algèbres de Lie libres et de la série de Hausdorff. Le chapitre 3 est consacré aux concepts de base pour les groupes de Lies sur un corps archimédien ou ultramétrique.Table of ContentsAlgèbres de Lie libres.- Groupes de Lie.

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Book SynopsisLes Éléments de mathématique de Nicolas Bourbaki ont pour objet une présentation rigoureuse, systématique et sans prérequis des mathématiques depuis leurs fondements. Ce volume du Livre d’Algèbre commutative, septième Livre du traité, est la continuation des chapitres antérieurs. Il introduit notamment les notions de profondeur et de lissité, fondamentales en géometrie algébrique. Il se termine par l’introduction des modules dualisants et de la dualité de Grothendieck. Ce volume est paru en 1998.Trade ReviewFrom the reviews: "The book under review is the faithful and unabridged reprint of the French original of Chapter 10 of N. Bourbaki’s ‘Commutative Algebra’ … . a highly important and valuable source book for seasoned mathematicians working in the fields of commutative algebra and algebraic geometry … ." (Werner Kleinert, Zentralblatt MATH, Vol. 1107 (9), 2007) "The book provides the basic theory and interesting examples, and further developments are proposed as exercises. With only 187 pages, exercises included, it gives a fairly good account of the current state of knowledge of … part of commutative algebra which is so important in algebraic geometry. … also systematically explores the behaviour of the notions considered under some base change. It is thus a useful reference. … is divided into ten sections." (Anne-Marie Simon, Mathematical Reviews, Issue 2008 h)Table of ContentsProfondeur, régularité, dualité.

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Book SynopsisCe huitième chapitre du Livre d'Algèbre, deuxième Livre des Éléments de mathématique, est consacré à l'étude de certaines classes d'anneaux et des modules sur ces anneaux.Il couvre les notions de module et d'anneau noethérien et artinien, ainsi que celle de radical. Ce chapitre décrit également la structure des anneaux semi-simples. Nous y donnons aussi la définition de divers groupes de Grothendieck qui jouent un rôle universel pour les invariants de modules et plusieurs descriptions du groupe de Brauer qui intervient dans la classification des anneaux simples.Une note historique en fin de volume, reprise de l'édition précédente, retrace l'émergence d'une grande partie des notions développées.Ce volume est une deuxième édition entièrement refondue de l'édition de 1958.Trade ReviewFrom the reviews of the second edition:“This book is intended as a comprehensive exposition of the theory of semi-simple rings and modules, with special emphasis on the Noetherian and Artinian cases. … Each section ends with a large collection of related exercises in the typical Bourbaki-style … . Certainly, it has been both a splendid idea and a great undertaking to rewrite N. Bourbaki’s classic Chapter 8 of Book II of the ‘Elements of Mathematics’ in such excellent a manner, very much so to the benefit of further generations of mathematicians.” (Werner Kleinert, Zentralblatt MATH, Vol. 1245, 2012)Table of ContentsIntroduction.- Chapitre VIII. Modules et anneaux semi-simples.- 1. Modules artiniens et modules noethériens.- 2. Structure des modules de longueur finie.- 3. Modules simples.- 4. Modules semi-simples.- 5. Commutation.- 6. Équivalence de Morita des modules et des algèbres.- 7. Anneaux simples.- 8. Anneaux semi-simples.- 9. Radical.- 10. Modules sur un anneau artinien.- 11. Groupes de Grothendieck.- 12. Produit tensoriel de modules semi-simples.- 13. Algèbres absolument semi-simples.- 14. Algèbres centrales et simples.- 15. Groupes de Brauer.- 16. Autres descriptions du groupe de Brauer.- 17. Normes et traces réduites.- 18. Algèbres simples sur un corps fini.- 20. Représentations linéaires des algèbres.- 21. Représentations linéaires des groupes finis.- Appendice 1. Algèbres sans élément unité.- Appendice 2. Déterminants sur un corps non commutatif.- Appendice 3. Le théorème des zéros de Hilbert.- Appendice 4. Trace d’un endomorphisme de rang fini.- Note Historique.- Bibliographie.- Index des notations.- Index terminologique

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Book Synopsisto the Encyclopaedia Subseries on Operator Algebras and Non-Commutative Geometry The theory of von Neumann algebras was initiated in a series of papers by Murray and von Neumann in the 1930's and 1940's. A von Neumann algebra is a self-adjoint unital subalgebra M of the algebra of bounded operators of a Hilbert space which is closed in the weak operator topology. According to von Neumann's bicommutant theorem, M is closed in the weak operator topology if and only if it is equal to the commutant of its commutant. Afactor is a von Neumann algebra with trivial centre and the work of Murray and von Neumann contained a reduction of all von Neumann algebras to factors and a classification of factors into types I, II and III. C* -algebras are self-adjoint operator algebras on Hilbert space which are closed in the norm topology. Their study was begun in the work of Gelfand and Naimark who showed that such algebras can be characterized abstractly as involutive Banach algebras, satisfying an algebraic relation connecting the norm and the involution. They also obtained the fundamental result that a commutative unital C* -algebra is isomorphic to the algebra of complex valued continuous functions on a compact space - its spectrum. Since then the subject of operator algebras has evolved into a huge mathematical endeavour interacting with almost every branch of mathematics and several areas of theoretical physics.Trade ReviewFrom the reviews:"... These notes [by E.Stormer] describe the main approaches to noncommutative entropy, together with several ramifications and variants. The notion of generator and variational principle are used to give applications to subfactors and C*-algebra formalism of quantum statistical mechanics. The author considers the most frequently studied examples, including Bernoulli shifts, Bogolyubov automorphisms, dual automorphisms on crossed products, shifts on infinite free products, and binary shifts on the CAR-algebra. The mathematical techniques and ideas are beautifully exposed, and the whole paper is a rich resource on the subject, either for the expert or the beginner. ..."V.Deaconu, Mathematical Reviews 2004"... the author gives a clear presentation of the dramatic developments in the classification theory for simple C*-algebras that have taken place over the past 25 years or so. ... As there is such a large amount of literature on the subject, this monograph article is particularly useful to the relative novice who wants to know the fundamental results in the theory without wading through a massive amount of detail. ...This monograph-length article is extremely well-written, filled with concrete examples, and has an exhaustive bibliography. I recommend it as an excellent introduction to graduate students and other mathematicians who want to bring themselves up-to-date on the subject. .."J.A.Packer, Mathematical Reviews 2004“Both contributions to this volume are high-end, excellently written research reviews, reflecting very thoroughly the current status in the respectively treated subbranches of the quickly evolving complex field of C* algebra theory. They both give a beautiful lay-out of the vast research program in the field which has been going on for decades … as well as to the standard works. … an excellent, very thorough, concise and needed overview for the researcher who is active in this field.” (Mark Sioen, Bulletin of the Belgian Mathematical Society, 2007) Table of ContentsI. Classification of Nuclear, Simple C*-algebras.- II. A Survey of Noncommutative Dynamical Entropy.

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Book SynopsisHere, the eminent algebraist, Nathan Jacobsen, concentrates on those algebras that have an involution. Although they appear in many contexts, these algebras first arose in the study of the so-called "multiplication algebras of Riemann matrices". Of particular interest are the Jordan algebras determined by such algebras, and thus their structure is discussed in detail. Two important concepts also dealt with are the universal enveloping algebras and the reduced norm. However, the largest part of the book is the fifth chapter, which focuses on involutorial simple algebras of finite dimension over a field.Trade Review"...the author takes us on a tour of division algebras, pointing out the salient facts, often with little-known proofs, but never going on so long as to bore the reader. This makes the book a pleasure to read" Bulletin of the London Mathematical SocietyTable of ContentsSkew Polynomials and Division Algebras.- Brauer Factor Sets and Noether Factor Sets.- Galois Descent and Generic Splitting Fields.- p-Algebras.- Simple Algebras with Involution.

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Book SynopsisA. Lineare Algebra I.- 1. Vektorräume.- 2. Matrizen.- 3. Determinant en.- B. Analytische Geometrie.- 4. Elementar-Geometrie in der Ebene.- 5. Euklidische Vektorräume.- 6. Der ?aun als Euklidischer Vektorraum.- 7. Geometrie im dreidimensionalen Raum.- C. Lineare Algebra II.- 8. Polynome und Matrizen.- 9. Homomorphismen von Vektorräumen.- Literatur.- Namenverzeichnis.Table of ContentsA. Lineare Algebra I.- 1. Vektorräume.- § 1. Der Begriff eines Vektorraumes.- 1. Vorbemerkung.- 2. Vektorräume.- 3. Unterräume.- 4. Geraden.- 5. Das Standardbeispiel Kn.- 6. Geometrische Deutung.- 7. Anfänge einer Geometrie im ?2.- § 2*. Über den Ursprung der Vektorräume.- 1. Die Grassmannsche Ausdehnungslehre.- 2. Grassmann: Übersicht über die allgemeine Formenlehre.- 3. Extensive Größen als Elemente eines Vektorraumes.- 4. Reaktion der Mathematiker.- 5. Der moderne Vektorraumbegriff.- § 3. Beispiele von Vektorräumen.- 1. Einleitung.- 2. Reelle Folgen.- 3. Vektorräume von Abbildungen.- 4. Stetige Funktionen.- 5. Reelle Polynome.- 6*. Reell-analytische Funktionen.- 7* Lineare Differentialgleichungen n-ter Ordnung mit konstanten Koeffizienten.- 8. Die Vektorräume Abb[M, K].- § 4. Elementare Theorie der Vektorräume.- 1. Vorbemerkung.- 2. Homogene Gleichungen.- 3. Erzeugung von Unterräumen.- 4. Lineare Abhängigkeit.- 5. Der Begriff einer Basis.- 6. Die Dimension eines Vektorraums.- 7. Der Dimensions-Satz.- 8*. Der Basis-Satz für beliebige Vektorraume.- 9*. Ein Glasperlen-Spiel.- § 5. Anwendungen.- 1. Die reellen Zahlen als Vektorraum über Q.- 2. Beispiele.- 3. Der Rang einer Teilmenge.- 4. Anwendung auf lineare Gleichungssysteme.- § 6. Homomorphismen von Vektorräumen.- 1. Einleitung.- 2. Definition und einfachste Eigenschaften.- 3. Kern und Bild.- 4. Die Dimensionsformel für Homomorphismen.- 5. Äquivalenz-Satz fÄr Homomorphismen.- 6. Der Rang eines Homomorphismus.- 7. Anwendung auf homogene lineare Gleichungen.- 8. Beispiele.- 9*. Die Funktionalgleichung f(x + y) = f(x) + f(y).- § 7*. Linearformen und der duale Raum.- 1. Vorbemerkungen.- 2. Definition und Beispiele.- 3. Existenz von Linearformen.- 4. Der Dual-Raum.- 5. Linearformen des Vektorraums der stetigen Funktionen.- § 8*. Direkte Summen und Komplemente.- 1. Summe und direkte Summe.- 2. Komplemente.- 3. Die Dimensionsformel für Summen.- 4. Die Bild-Kern-Zerlegung.- 2. Matrizen.- § 1. Erste Eigenschaften.- 1. Der Begriff einer Matrix.- 2. Über den Vorteil von Doppelindizes.- 3. Mat(m, n; K) als K-Vektorraum.- 4. Das Transponierte einer Matrix.- 5. Spalten- und Zeilenrang.- 6. Elementare Umformungen.- 7. Die Ranggleichung.- 8. Kästchenschreibweise und Rangberechnung.- 9. Zur Geschichte des Rang-Begriffes.- § 2. Matrizenrechnung.- 1. Arthur Cayley oder die Erfindung der Matrizenrechnung.- 2. Produkte von Matrizen.- 3. Produkte von Vektoren.- 4. Homomorphismen zwischen Standard-Raumen.- 5. Erntezeit.- 6. Das Skalarprodukt.- 7*. Rang A ? r.- 8. Kästchenrechnung.- § 3. Algebren.- 1. Einleitung.- 2. Der Begriff einer Algebra.- 3. Invertierbare Elemente.- 4. Ringe.- 5. Beispiele.- § 4. Der Begriff einer Gruppe.- 1. Halbgruppen.- 2. Gruppen.- 3. Untergruppen.- 4. Kommutative Gruppen.- 5. Homomorphismen.- 6. Normalteiler.- 7. Historische Bemerkungen.- § 5. Matrix-Algebren.- 1. Mat(n; K) und GL(n; K).- 2. Der Äquivalenz-Satz für invertierbare Matrizen.- 3. Die Invarianz des Ranges.- 4. Spezielle invertierbare Matrizen.- 5*. Zentralisator und Zentrum.- 6. Die Spur einer Matrix.- 7. Die Algebra Mat(2; K).- § 6. Der Normalformen-Satz.- 1. Elementar-Matrizen.- 2. Zusammenhang mit elementaren Umformungen.- 3. Anwendungen.- 4*. Die Weyr-Frobenius-Ungleichungen.- 5. Aufgaben zum Normalformen-Satz.- 6. Zur Geschichte des Normalformen-Satzes.- § 7. Gleichungssysteme.- 1. Erinnerung an lineare Gleichungen.- 2. Wiederholung von Problemen und Ergebnissen.- 3. Der Fall m = n.- 4. Anwendung des Normalformen-Satzes.- 5. Lösungsverfahren.- 6. Basiswechsel in Vektorräumen.- § 8*. Pseudo-Inverse.- 1. Motivation.- 2. Der Begriff des Pseudo-Inversen.- 3. Ein Kriterium für Gleichungssysteme.- 4. Zerlegung in eine direkte Summe.- 3. Determinant en.- § 1. Erste Ergebnisse über Determinanten.- 1. Eine Motivation.- 2. Determinanten-Funktionen.- 3. Existenz.- 4. Eigenschaften.- 5. Anwendungen auf die Gruppe GL(n; K).- 6. Die Cramerche Regel.- § 2. Das Inverse einer Matrix.- 1. Vorbemerkung.- 2. Die Entwicklungs-Sätze.- 3. Die komplementäre Matrix.- 4. Beschreibung des Inversen.- § 3. Existenzbeweise.- 1. Durch Induktion.- 2. Permutationen.- 3. Die Leibnizsche Formel.- 4. Permutationsmatrizen.- 5. Ein weiterer Existenzbeweis.- § 4. Erste Anwendungen.- 1. Lineare Gleichungssysteme.- 2. Zweidimensionale Geometrie.- 3. Lineare Abhängigkeit.- 4. Rangberechnung.- 5. Die Determinanten-Rekursionsformel.- 6. Das charakteristische Polynom.- 7*. Mehrfache Nullstellen von Polynomen.- 8*. Eine Funktionalgleichung.- 9. Orientierung von Vektorräumen.- § 5. Symmetrische Matrizen.- 1. Einleitung.- 2. Der Vektorraum der symmetrischen Matrizen.- 3. Quadratische Ergänzung.- 4. Die Jacobische Normalform.- 5. Normalformen-Satz.- 6*. Trägheits-Satz.- § 6. Spezielle Matrizen.- 1. Schiefsymmetrische Matrizen.- 2. Die Vandermondesche Determinante.- 3. Bandmatrizen.- 4. Aufgaben.- § 7. Zur Geschichte der Determinanten.- 1. Gottfried Wilhelm LEIBNIZ.- 2. BALTZER’S Lehrbuch.- 3. Die weitere Entwicklung.- B. Analytische Geometrie.- 4. Elementar-Geometrie in der Ebene.- § 1. Grundlagen.- 1. Skalarprodukt, Abstand und Winkel.- 2. Die Abbildung x ? x? 3..- 3. Geraden.- 4. Schnittpunkt zwischen zwei Geraden.- 5. Abstand zwischen Punkt und Gerade.- 6. Fläche eines Dreiecks.- 7. Der Höhenschnittpunkt.- § 2. Die Gruppe O(2).- 1. Drehungen und Spiegelungen.- 2. Orthogonale Matrizen.- 3. Bewegungen.- 4. Ein Beispiel.- 5. Die Hauptachsentransformation fur 2 Matrizen.- 6. Fix-Geraden.- 7. Die beiden Orientierungen der Ebene.- § 3. Geometrische Sätze.- 1. Der Kreis.- 2. Tangente.- 3. Die beiden Sehnensätze.- 4. Der Umkreis eines Dreiecks.- 5. Die Euler-Gerade.- 6. Der Feuerbach-Kreis.- 7. Das Mittendreieck.- 5. Euklidische Vektorräume.- § 1. Positiv definite Bilinearformen.- 1. Symmetrische Bilinearformen.- 2. Beispiele.- 3. Positiv definite Bilinearformen.- 4. Positiv definite Matrizen.- 5. Die Cauchy-Schwrzsche Ungleichung.- 6. Normierte Vektorraume.- § 2. Das Skalarprodukt.- 1. Der Begriff eines euklidischen Vektorraumes.- 2. Winkelmessung.- 3. Orthonormalbasen.- 4. Basisdarstellung.- 5. Orthogonales Komplement und orthogonale Summe.- 6. Linearformen.- § 3. Erste Anwendungen.- 1. Positiv definite Matrizen.- 2. Die adjungierte Abbildung.- 3. Systeme linearer Gleichungen.- 4. Ein Kriterium für gleiche Orientierung.- 5*. Legendre-Polynome.- §4. Geometrie in euklidischen Vektorräumen.- 1. Geraden.- 2. Hyperebenen.- 3. Schnittpunkt von Gerade und Hyperebene.- 4. Abstand von einer Hyperebene.- 5*. Orthogonale Projektion.- 6*. Abstand zweier Unterräume.- 7*. Volumenberechnung.- 8*. Duale Basen.- § 5. Die orthogonale Gruppe.- 1. Bewegungen.- 2. Spiegelungen.- 3. Die Transitivitat von O(V,?) auf Sphären.- 4*. Die Erzeugung von O(V,?) durch Spiegelungen.- 5*. Winkeltreue Abbildungen.- 6. Der ?aun als Euklidischer Vektorraum.- § 1. Der ?n und die orthogonale Gruppe O(n).- 1. Der euklidische Vektorraum ?n.- 2. Orthogonale Matrizen.- 3. Die Gruppe O(n).- 4. Spiegelungen.- 5. Erzeugung von O(n) durch Spiegelungen.- 6*. Drehungen.- 7. Anwendung der Determinanten-Theorie.- 8*. Eine Parameterdarstellung.- 9. Euler, Cauchy, Jacobi Und Cayley.- § 2. Die Hauptachsentransformation.- 1. Problemstellung.- 2. Der Vektorraum der symmetrischen Matrizen.- 3. Positiv semi-definite Matrizen.- 4. Das Minimum einer quadratischen Form.- 5. Satz uber die Hauptachsentransformation.- 6. Eigenwerte.- 7. Eigenräume.- § 3. Anwendungen.- 1. Vorbemerkung.- 2. Positiv definite Matrizen.- 3. Hyperflächen.- 2. Grades.- 4*. Der Quadratwurzel-Satz.- 5*. Polar-Zerlegung.- 6*. Orthogonale Normalform.- 7*. Das Moorw-Penrose-Inverse.- § 4*. Topologische Eigenschaften.- 1. Zusammenhang.- 2. Kompaktheit.- 3. Hauptachsentransformation.- 7. Geometrie im dreidimensionalen Raum.- § 1. Das Vektorprodukt.- 1. Definition und erste Eigenschaften.- 2. Zusammenhang mit Determinanten.- 3. Geometrische Deutung.- 4. Ebenen.- 5. Parallelotope.- 6. Vektorrechnung im Anschauungsraum.- § 2*. Sphärische Geometrie.- 1. Über den Ursprung der Sphärik.- 2. Das sphärische Dreieck.- 3. Das Polardreieck.- 4. Entfernung auf der Erde.- § 3. Die Gruppe O(3).- 1. Beschreibung durch das Vektorprodukt.- 2. Erzeugung durch Drehungen.- 3. Spiegelungen.- 4. Fix-Geraden.- 5. Die Normalform.- 6. Die Drehachse.- 7*. Die Eulersche Formel.- 8*. Drehungen um eine Achse.- § 4. Bewegungen.- 1. Fixpunkte.- 2. Bewegungen mit Fixpunkt.- 3. Schraubungen.- C. Lineare Algebra II.- 8. Polynome und Matrizen.- § 1. Polynome.- 1. Der Vektorraum Pol K.- 2. Pol K als Ring.- 3. Zerfallende Polynome.- 4. Pol K als Hauptidealring.- 5*. Unbestimmte.- § 2. Die komplexen Zahlen.- 1. Der Körper C der komplexen Zahlen.- 2. Konjugation und Betrag.- 3. Der Fundamentalsatz der Algebra.- § 3. Struktursatz für zerfallende Matrizen.- 1. Der Begriff der Diagonalisierbarkeit.- 2. Das charakteristische Polynom.- 3. Äquivalenz-Satz für Eigenwerte.- 4. Nilpotente Matrizen.- 5. Idempotente Matrizen.- 6. Zerfallende Matrizen.- 7. Diagonalisierbarkeits-Kriterium.- 8*. Ein Beispiel zum Struktur-Satz.- 9*. Elementarsymmetrische Funktionen und Potenzsummen.- §4. Die Algebra K[A].- 1. Eine Warnung.- 2. Matrix-Polynome.- 3. Das Minimalpolynom.- 4. Eigenwerte.- 5. Das Rechnen mit Kästchen-Diagonalmatrizen.- 6. Satz von Cayley.- 7. Äquivalenz-Satz für Diagonalisierbarkeit.- 8. Spektralscharen.- 9. Eigenräume.- §5. Die Jordan-Chevalley-Zerlegung.- 1. Existenz-Satz.- 2. Summen von diagonalisierbaren Matrizen.- 3. Die Eindeutigkeit.- 4. Anwendungen.- § 6. Normalformen reeller und komplexer Matrizen.- 1. Normalformen komplexer Matrizen.- 2. Reelle und komplexe Matrizen.- 3*. Hermitesche Matrizen.- 4. Invariante Unterräume.- 5. Die Stufenform.- 6. Der Satz über die Stufenform.- 7. Orthogonale Matrizen.- 8. Schiefsymmetrische Matrizen.- 9*. Normale Matrizen.- § 7*. Der höhere Standpunkt.- 1. Einfache und halbeinfache Algebren.- 2. Kommutative Algebren.- 3. Die Struktursätze.- 4. Die weitere Entwicklung.- 5. Der generische Standpunkt.- 9. Homomorphismen von Vektorräumen.- § 1. Der Vektorraum Hom(V, V?).- 1. Der Vektorraum Abb(M, V?).- 2. Hom(V, V?) als Unterraum von Abb(V, V?).- 3. Mat(m, n; K) als Beispiel.- 4. Verknüpfungen von Hom(V, V?) und Hom(V?, V?).- § 2. Beschreibung der Homomorphismen im endlich-dimensionalen Fall.- 1. Isomorphic mit Standard-Räumen.- 2. Darstellung der Homomorphismen.- 3. Basiswechsel.- 4. Die Algebra End V.- 5. Diagonalisierbarkeit.- 6. Die Linksmultiplikation in Mat(n; K).- 7. Polynome.- § 3. Euklische Vektorräume.- 1. Der Satz über die Hauptachsentransformation.- 2. Spiegelungen.- 3*. Unitäre Vektorräume.- § 4. Der Quotientenraum.- 1. Einleitung.- 2. Nebenklassen.- 3. Der Satz über den Quotientenraum.- 4. Der Satz über den kanonischen Epimorphismus.- 5. Kanonische Faktorisierung.- 6. Anwendungen.- 7. Beispiele.- § 5*. Nilpotente Endomorphismen.- 1. Problemstellung.- 2. Zyklische Unterräume.- 3. Der Struktur-Satz.- 4. Nilzyklische Matrizen.- 5. Die Normalform.- 6. Satz von der JoRDANSchen Normalform.- 7. Anwendungen auf Differentialgleichungen.- Literatur.- Namenverzeichnis.

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Book SynopsisThis is the softcover reprint of the English translation of 1972 (available from Springer since 1989) of the first 7 chapters of Bourbaki's 'Algèbre commutative'. It provides a very complete treatment of commutative algebra, enabling the reader to go further and study algebraic or arithmetic geometry. The first 3 chapters treat in succession the concepts of flatness, localization and completions (in the general setting of graduations and filtrations). Chapter 4 studies associated prime ideals and the primary decomposition. Chapter 5 deals with integers, integral closures and finitely generated algebras over a field (including the Nullstellensatz). Chapter 6 studies valuation (of any rank), and the last chapter focuses on divisors (Krull, Dedekind, or factorial domains) with a final section on modules over integrally closed Noetherian domains, not usually found in textbooks. Useful exercises appear at the ends of the chapters.Table of ContentsFlat Modules; Localization; Graduations, Filtrations and Topologies; Associated Prime Ideals and Primary Decomposition, Integers, Valuations, Divisors, Exercises

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Book SynopsisThe 1963 Göttingen notes of T. A. Springer are well known in the field but have been unavailable for some time. This book is a translation of those notes, completely updated and revised. The part of the book dealing with the algebraic structures is on a fairly elementary level, presupposing basic results from algebra. Trade ReviewFrom the reviews of the first edition: "This book is an updated and revised English version of the German notes on octaves, Jordan algebras and exceptional groups which appeared as mimeographed lecture notes of Göttingen University in 1963. It is still an excellent reference on the subject … ." (Huberta Lausch, Zentralblatt MATH, Vol. 1087, 2006)Table of Contents1. Composition Algebras.- 2. The Automorphism Group of an Octonion Algebra.- 3. Triality.- 4. Twisted Composition Algebras.- 5. J-algebras and Albert Algebras.- 6. Proper J-algebras and Twisted Composition Algebras.- 7. Exceptional Groups.- 8. Cohomological Invariants.

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Book SynopsisThis text provides a lively introduction to pure mathematics. It begins with sets, functions and relations, proof by induction and contradiction, complex numbers, vectors and matrices, and provides a brief introduction to group theory. It moves onto analysis, providing a gentle introduction to epsilon-delta technology and finishes with continuity and functions. The book features numerous exercises of varying difficulty throughout the text.Table of Contents1. Sets, Functions and Relations.- 1.1 Sets.- 1.2 Subsets.- 1.3 Well-known Sets.- 1.4 Rationals, Reals and Pictures.- 1.5 Set Operations.- 1.6 Sets of Sets.- 1.7 Paradox.- 1.8 Set-theoretic Constructions.- 1.9 Notation.- 1.10 Venn Diagrams.- 1.11 Quantifiers and Negation.- 1.12 Informal Description of Maps.- 1.13 Injective, Surjective and Bijective Maps.- 1.14 Composition of Maps.- 1.15 Graphs and Respectability Reclaimed.- 1.16 Characterizing Bijections.- 1.17 Sets of Maps.- 1.18 Relations.- 1.19 Intervals.- 2. Proof.- 2.1 Induction.- 2.2 Complete Induction.- 2.3 Counter-examples and Contradictions.- 2.4 Method of Descent.- 2.5 Style.- 2.6 Implication.- 2.7 Double Implication.- 2.8 The Master Plan.- 3. Complex Numbers and Related Functions.- 3.1 Motivation.- 3.2 Creating the Complex Numbers.- 3.3 A Geometric Interpretation.- 3.4 Sine, Cosine and Polar Form.- 3.5 e.- 3.6 Hyperbolic Sine and Hyperbolic Cosine.- 3.7 Integration Tricks.- 3.8 Extracting Roots and Raising to Powers.- 3.9 Logarithm.- 3.10 Power Series.- 4. Vectors and Matrices.- 4.1 Row Vectors.- 4.2 Higher Dimensions.- 4.3 Vector Laws.- 4.4 Lengths and Angles.- 4.5 Position Vectors.- 4.6 Matrix Operations.- 4.7 Laws of Matrix Algebra.- 4.8 Identity Matrices and Inverses.- 4.9 Determinants.- 4.10 Geometry of Determinants.- 4.11 Linear Independence.- 4.12 Vector Spaces.- 4.13 Transposition.- 5. Group Theory.- 5.1 Permutations.- 5.2 Inverse Permutations.- 5.3 The Algebra of Permutations.- 5.4 The Order of a Permutation.- 5.5 Permutation Groups.- 5.6 Abstract Groups.- 5.7 Subgroups.- 5.8 Cosets.- 5.9 Cyclic Groups.- 5.10 Isomorphism.- 5.11 Homomorphism.- 6. Sequences and Series.- 6.1 Denary and Decimal Sequences.- 6.2 The Real Numbers.- 6.3 Notation for Sequences.- 6.4 Limits of Sequences.- 6.5 The Completeness Axiom.- 6.6 Limits of Sequences Revisited.- 6.7 Series.- 7. Mathematical Analysis.- 7.1 Continuity.- 7.2 Limits.- 8. Creating the Real Numbers.- 8.1 Dedekind’s Construction.- 8.2 Construction via Cauchy Sequences.- 8.3 A Sting in the Tail: p-adic numbers.- Further Reading.- Solutions.

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Book SynopsisThis is comprehensive basic monograph on mixed Hodge structures. Building up from basic Hodge theory the book explains Delingne's mixed Hodge theory in a detailed fashion. Then both Hain's and Morgan's approaches to mixed Hodge theory related to homotopy theory are sketched. Next comes the relative theory, and then the all encompassing theory of mixed Hodge modules. The book is interlaced with chapters containing applications. Three large appendices complete the book.Trade ReviewFrom the reviews: "This book is dealing with Hodge Theory ... which generalizes in a functorial way the variations of MHS. ... The clarity of the presentation and the wealth of information are both remarkable. This book ... is a masterpiece that anyone working in Algebraic Geometry, Singularities or Analytic/Complex Geometry would like to have in his own library." (Alexandru Dimca, Zentralblatt MATH, Vol. 1138 (16), 2008) "The book under review … focuses mainly on the ‘pure’ story just summarized, is aimed at graduate students and researchers … . The book begins with a brief historical survey; each chapter is headed by a good summary of its contents and concluded by historical remarks (with references). … this work is a thoroughly readable and very up-to-date account of mixed Hodge theory, written by masters of the subject, and will undoubtedly serve as a basic reference for years to come." (Matt Kerr, Mathematical Reviews, Issue 2009 C) “This book has been awaited for many years. … the book which is now available will certainly rapidly become one of the standard references on the topic. Hodge theory assigns to a complex variety data which come from linear algebra. … I heartily recommend the book.” (Helene Esnault, Jahresbericht der Deutsche Mathematiker Vereinigung, Vol. 112 (1), 2010)Table of ContentsBasic Hodge Theory.- Compact Kähler Manifolds.- Pure Hodge Structures.- Abstract Aspects of Mixed Hodge Structures.- Mixed Hodge Structures on Cohomology Groups.- Smooth Varieties.- Singular Varieties.- Singular Varieties: Complementary Results.- Applications to Algebraic Cycles and to Singularities.- Mixed Hodge Structures on Homotopy Groups.- Hodge Theory and Iterated Integrals.- Hodge Theory and Minimal Models.- Hodge Structures and Local Systems.- Variations of Hodge Structure.- Degenerations of Hodge Structures.- Applications of Asymptotic Hodge Theory.- Perverse Sheaves and D-Modules.- Mixed Hodge Modules.

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Book SynopsisA matroid is an abstract mathematical structure that captures combinatorial properties of matrices. This book offers a unique introduction to matroid theory, emphasizing motivations from matrix theory and applications to systems analysis. This book serves also as a comprehensive presentation of the theory and application of mixed matrices, developed primarily by the present author in the 1990's. A mixed matrix is a convenient mathematical tool for systems analysis, compatible with the physical observation that "fixed constants" and "system parameters" are to be distinguished in the description of engineering systems. This book will be extremely useful to graduate students and researchers in engineering, mathematics and computer science. From the reviews: "…The book has been prepared very carefully, contains a lot of interesting results and is highly recommended for graduate and postgraduate students." András Recski, Mathematical Reviews Clippings 2000m:93006Table of ContentsPreface I. Introduction to Structural Approach --- Overview of the Book 1 Structural Approach to Index of DAE 1.1 Index of differential-algebraic equations 1.2 Graph-theoretic structural approach 1.3 An embarrassing phenomenon 2 What Is Combinatorial Structure? 2.1 Two kinds of numbers 2.2 Descriptor form rather than standard form 2.3 Dimensional analysis 3 Mathematics on Mixed Polynomial Matrices 3.1 Formal definitions 3.2 Resolution of the index problem 3.3 Block-triangular decomposition II. Matrix, Graph and Matroid 4 Matrix 4.1 Polynomial and algebraic independence 4.2 Determinant 4.3 Rank, term-rank and generic-rank 4.4 Block-triangular forms 5 Graph 5.1 Directed graph and bipartite graph 5.2 Jordan-Holder-type theorem for submodular functions 5.3 Dulmage-Mendelsohn decomposition 5.4 Maximum flow and Menger-type linking 5.5 Minimum cost flow and weighted matching 6 Matroid 6.1 From matrix to matroid 6.2 Basic concepts 6.3 Examples 6.4 Basis exchange properties 6.5 Independent matching problem 6.6 Union 6.7 Bimatroid (linking system) III. Physical Observations for Mixed Matrix Formulation 7 Mixed Matrix for Modeling Two Kinds of Numbers 7.1 Two kinds of numbers 7.2 Mixed matrix and mixed polynomial matrix 8 Algebraic Implications of Dimensional Consistency 8.1 Introductory comments 8.2 Dimensioned matrix 8.3 Total unimodularity of dimensioned matrices 9 Physical Matrix 9.1 Physical matrix 9.2 Physical matrices in a dynamical system IV. Theory and Application of Mixed Matrices 10 Mixed Matrix and Layered Mixed Matrix 11 Rank of Mixed Matrices 11.1 Rank identities for LM-matrices 11.2 Rank identities for mixed matrices 11.3 Reduction to independent matching problems 11.4 Algorithms for the rank 11.4.1 Algorithm for LM-matrices 11.4.2 Algorithm for mixed matrices 12 Structural Solvability of Systems of Equations 12.1 Formulation of structural solvability 12.2 Graphical conditions for structural solvability 12.3 Matroidal conditions for structural solvability 13. Combinatorial Canonical Form of LM-matrices 13.1 LM-equivalence 13.2 Theorem of CCF 13.3 Construction of CCF 13.4 Algorithm for CCF 13.5 Decomposition of systems of equations by CCF 13.6 Application of CCF 13.7 CCF over rings 14 Irreducibility of LM-matrices 14.1 Theorems on LM-irreducibility 14.2 Proof of the irreducibility of determinant 15 Decomposition of Mixed Matrices 15.1 LU-decomposition of invertible mixed matrices 15.2 Block-triangularization of general mixed matrices 16 Related Decompositions 16.1 Partition as a matroid union 16.2 Multilayered matrix 16.3 Electrical network with admittance expression 17 Partitioned Matrix 17.1 Definitions 17.2 Existence of proper block-triangularization 17.3 Partial order among blocks 17.4 Generic partitioned matrix 18 Principal Structures of LM-matrices 18.1 Motivations 18.2 Principal structure of submodular systems 18.3 Principal structure of generic matrices 18.4 Vertical principal structure of LM-matrices 18.5 Horizontal principal structure of LM-matrices V. Polynomial Matrix and Valuated Matroid 19 Polynomial/Rational Matrix 19.1 Polynomial matrix and Smith form 19.2 Rational matrix and Smith-McMillan form at infinity 19.3 Matrix pencil and Kronecker form 20 Valuated Matroid 20.1 Introduction 20.2 Examples 20.3 Basic operations 20.4 Greedy algorithms 20.5 Valuated bimatroid 20.6 Induction through bipartite graphs 20.7 Characterizations 20.8 Further exchange properties 20.9 Valuated independent assignment problem 20.10 Optimality criteria 20.10.1 Potential criterion 20.10.2 Negative-cycle criterion 20.10.3 Proof of the optimality criteria 20.10.4 Extension to VIAP(k) 20.11 Application to triple matrix product 20.12 Cycle-canceling algorithms 20.12.1 Algorithms 20.12.2 Validity of the minimum-ratio cycle algorithm 20.13 Augmenting algorithms 20.13.1 Algorithms 20.13.2 Validity of the augmenting algorithm VI. Theory and Application of Mixed Polynomial Matrices 21 Descriptions of Dynamical Systems 21.1 Mixed polynomial mat

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Book SynopsisAn invaluable summary of research work done in the period from 1978 to the presentTrade ReviewFrom the reviews: "It is a full-fledged advanced course on themes in higher algebra suited for a specialized graduate seminar, a research seminar, and of course, self-study by an aspiring researcher. … Serre’s Problem on Projective Modules, is very clear and well written … and quickly gets the reader properly air-borne. … the pay-off is huge: this is fantastic stuff. … is a superb book. It’s highly recommended." (Michael Berg, MathDL, March, 2007) "The book starts with the basics of projective modules and the K0 and K1 groups, and then gives the classical, partial results about Serre’s conjecture. … This well-written book is the definitive treatment of ‘Serre’s conjecture’ – its history, solution, and generalizations – and will be of interest to both beginning graduate students and advanced researchers in this field." (David F. Anderson, Zentralblatt MATH, Vol. 1101 (3), 2007) "Lam has done a magnificent job of organizing the mated al and presenting complete proofs of all the results directly connected with Sen-e's problem. ... The references are complete and make the book a very valuable reference even for experts in the field.... It will be very useful to students wishing to learn about projective modules ... . This is definitely a book that anyone ... interested in projective modules should have on his or her shelf!" (Richard G. Swan, Bulletin of the American Mathematical Society, Vol. 45 (3), July, 2008)Table of Contentsto Serre’s Conjecture: 1955–1976.- Foundations.- The “Classical” Results on Serre’s Conjecture.- The Basic Calculus of Unimodular Rows.- Horrocks’ Theorem.- Quillen’s Methods.- K1-Analogue of Serre’s Conjecture.- The Quadratic Analogue of Serre’s Conjecture.- References for Chapters I–VII.- Appendix: Complete Intersections and Serre’s Conjecture.- New Developments (since 1977).- References for Chapter VIII.

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Book SynopsisThis textbook introduces students of economics to the fundamental notions and instruments in linear algebra. Linearity is used as a first approximation to many problems that are studied in different branches of science, including economics and other social sciences. Linear algebra is also the most suitable to teach students what proofs are and how to prove a statement. The proofs that are given in the text are relatively easy to understand and also endow the student with different ways of thinking in making proofs. Theorems for which no proofs are given in the book are illustrated via figures and examples. All notions are illustrated appealing to geometric intuition. The book provides a variety of economic examples using linear algebraic tools. It mainly addresses students in economics who need to build up skills in understanding mathematical reasoning. Students in mathematics and informatics may also be interested in learning about the use of mathematics in economics.Table of ContentsSome Basic Concepts.- Vectors and Matrices.- Square Matrices and Determinants.- Inverse Matrix.- Systems of Linear Equations.- Linear Spaces.- Euclidean Spaces.- Linear Transformations.- Eigenvectors and Eigenvalues.- Linear Model of Production in a Classical Setting.- Linear Programming.- Natural Numbers and Induction.- Methods of Evaluating Determinants.- Complex Numbers.- Pseudoinverse.- Answers and Solutions.

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Book SynopsisTable of ContentsInhaltsübersicht: Abhandlungen zur Zahlentheorie und Algebra.- Abhandlungen zur Funktionentheorie.- Abhandlungen zur Mengenlehre.- Abhandlungen zur Geschichte der Mathematik und zur Philosophie des Unendlichen.- Anhang: Aus dem Briefwechsel zwischen Cantor und Dedekind.- Das Leben Georg Cantors.- Bibliographie weiterer Arbeiten von Georg Cantor.

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Book SynopsisThis monograph provides an introduction to the theory of Clifford algebras, with an emphasis on its connections with the theory of Lie groups and Lie algebras. The book starts with a detailed presentation of the main results on symmetric bilinear forms and Clifford algebras. It develops the spin groups and the spin representation, culminating in Cartan’s famous triality automorphism for the group Spin(8). The discussion of enveloping algebras includes a presentation of Petracci’s proof of the Poincaré–Birkhoff–Witt theorem.This is followed by discussions of Weil algebras, Chern--Weil theory, the quantum Weil algebra, and the cubic Dirac operator. The applications to Lie theory include Duflo’s theorem for the case of quadratic Lie algebras, multiplets of representations, and Dirac induction. The last part of the book is an account of Kostant’s structure theory of the Clifford algebra over a semisimple Lie algebra. It describes his “Clifford algebra analogue” of the Hopf–Koszul–Samelson theorem, and explains his fascinating conjecture relating the Harish-Chandra projection for Clifford algebras to the principal sl(2) subalgebra.Aside from these beautiful applications, the book will serve as a convenient and up-to-date reference for background material from Clifford theory, relevant for students and researchers in mathematics and physics.Table of ContentsPreface.- Conventions.- List of Symbols.- 1 Symmetric bilinear forms.- 2 Clifford algebras.- 3 The spin representation.- 4 Covariant and contravariant spinors.- 5 Enveloping algebras.- 6 Weil algebras.- 7 Quantum Weil algebras.- 8 Applications to reductive Lie algebras.- 9 D(g; k) as a geometric Dirac operator.- 10 The Hopf–Koszul–Samelson Theorem.- 11 The Clifford algebra of a reductive Lie algebra.- A Graded and filtered super spaces.- B Reductive Lie algebras.- C Background on Lie groups.- References.- Index.

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Book SynopsisTable of Contentsto Volume II.- V. Ring Theory.- 17. Modules of Finite Length and their Endomorphism Rings.- 18. Semilocal Rings and the Jacobson Radical.- 19. Quasinjective Modules and Selfinjective Rings.- 20. Direct Sum Representations of Rings and Modules.- 21. Azumaya Diagrams.- 22. Projective Covers and Perfect Rings.- 23. Morita Duality.- 24. Quasi-Frobenius Rings.- 25. Sigma Cyclic and Serial Rings.- 26. Semiprimitive Rings, Semiprime Rings, and the Nil Radical.- Register of Names.

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Book Synopsisby a more general quadratic algebra (possibly obtained by deformation) and then to derive Rq [G] by requiring it to possess the latter as a comodule. A third principle is to focus attention on the tensor structure of the cat­ egory of (!; modules. This means of course just defining an algebra structure on Rq[G]; but this is to be done in a very specific manner. Concretely the category is required to be braided and this forces (9.4.2) the existence of an "R-matrix" satisfying in particular the quantum Yang-Baxter equation and from which the algebra structure of Rq[G] can be written down (9.4.5). Finally there was a search for a perfectly self-dual model for Rq[G] which would then be isomorphic to Uq(g). Apparently this failed; but V. G. Drinfeld found that it could be essentially made to work for the "Borel part" of Uq(g) denoted U (b) and further found a general construction (the Drinfeld double) q mirroring a Lie bialgebra. This gives Uq(g) up to passage to a quotient. One of the most remarkable aspects of the above superficially different ap­ proaches is their extraordinary intercoherence. In particular they essentially all lead for G semisimple to the same and hence "canonical", objects Rq[G] and Uq(g), though this epithet may as yet be premature.Table of ContentsI. Hopf Algebras.- 1.1 Axioms of a Hopf Algebra.- 1.2 Group Algebras and Enveloping Algebras.- 1.3 Adjoint Action.- 1.4 The Hopf Dual.- 1.5 Comments and Complements.- 2. Excerpts from the Classical Theory.- 2.1 Lie Algebras.- 2.2 Algebraic Lie Algebras.- 2.3 Algebraic Groups.- 2.4 Lie Algebras of Algebraic Groups.- 2.5 Comments and Complements.- 3. Encoding the Cartan Matrix.- 3.1 Quantum Weyl Algebras.- 3.2 The Drinfeld Double.- 3.3 The Rosso Form and the Casimir Invariant.- 3.4 The Classical Limit and the Shapovalev Form.- 3.5 Comments and Complements.- 4. Highest Weight Modules.- 4.1 The Jantzen Filtration and Sum Formula.- 4.2 Kac-Moody Lie Algebras.- 4.3 Integrable Modules for Uq(gc).- 4.4 Demazure Modules and Product Formulae.- 4.5 Comments and Complements.- 5. The Crystal Basis.- 5.1 Operators in the Crystal Limit.- 5.2 Crystals.- 5.3 Ad-invariant Filtrations, Twisted Actions and the Crystal Basis for Uq(n-).- 5.4 The Grand Loop.- 5.5 Comments and Complements.- 6. The Global Bases.- 6.1 The ? Operation and the Embedding Theorem.- 6.2 Globalization.- 6.3 The Demazure Property.- 6.4 Littelmann’s Path Crystals.- 6.5 Comments and Complements.- 7. Structure Theorems for Uq(g).- 7.1 Local Finiteness for the Adjoint Action.- 7.2 Positivity of the Rosso Form.- 7.3 The Separation Theorem.- 7.4 Noetherianity.- 7.5 Comments and Complements.- 8. The Primitive Spectrum of Uq(g).- 8.1 The Poincaré Series of the Harmonic Space.- 8.2 Factorization of the Quantum PRV Determinants.- 8.3 Verma Module Annihilators.- 8.4 Equivalence of Categories.- 8.5 Comments and Complements.- 9. Structure Theorems for Rq[G].- 9.1 Commutativity Relations.- 9.2 Surjectivity and Injectivity Theorems.- 9.3 The Adjoint Action.- 9.4 The R-Matrix.- 9.5 Comments and Complements.- 10. The Prime Spectrum of Rq[G].- 10.1 Highest Weight Modules.- 10.2 The Quantum Weyl Group.- 10.3 Prime and Primitive Ideals of Rq[G].- 10.4 Hopf Algebra Automorphisms.- 10.5 Comments and Complements.- A.2 Excerpts from Ring Theory.- A.3 Combinatorial Identities and Dimension Theory.- A.4 Remarks on Constructions of Quantum Groups.- A.5 Comments and Complements.- Index of Notation.

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Table of ContentsI. Linear Spaces.- II. Normed Linear Spaces.- III. Completeness, Compactness, and Reflexivity.- IV. Unconditional Convergence and Bases.- V. Compact Convex Sets and Continuous Function Spaces.- VI. Norm and Order.- VII. Metric Geometry in Normed Spaces.- VIII. Reader’s Guide.- Index of Citations.- Index of Symbols.

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Book SynopsisDieser Buchtitel ist Teil des Digitalisierungsprojekts Springer Book Archives mit Publikationen, die seit den Anfängen des Verlags von 1842 erschienen sind. Der Verlag stellt mit diesem Archiv Quellen für die historische wie auch die disziplingeschichtliche Forschung zur Verfügung, die jeweils im historischen Kontext betrachtet werden müssen. Dieser Titel erschien in der Zeit vor 1945 und wird daher in seiner zeittypischen politisch-ideologischen Ausrichtung vom Verlag nicht beworben.

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Book SynopsisCe livre des Éléments de mathématique est consacré à la Topologie algébrique. Les quatre premiers chapitres présentent la théorie des revêtements d'un espace topologique et du groupe de Poincaré. On construit le revêtement universel d'un espace connexe pointé délaçable et on établit l'équivalence de catégories entre revêtements de cet espace et actions du groupe de Poincaré. On démontre une version générale du théorème de van Kampen exprimant le groupoïde de Poincaré d'un espace topologique comme un coégalisateur de diagrammes de groupoïdes. Dans de nombreuses situations géométriques, on en déduit une présentation explicite du groupe de Poincaré.Table of ContentsMode d'Emploi.- Introduction.- Chapitre I. Revêtements.- 1. Produits fibrés et carrés cartésiens.- 2. Applications étales.- 3. Faisceaux.- 4. Revêtements.- 5. Revêtements principaux.- 6. Espaces simplement connexes.- Exercices.- Chapitre II. Groupoïdes.- 1. Carquois.- 2. Graphes.- 3. Groupoïdes.- 4. Homotopies.- 5. Coégalisateur.- Exercices.- Chapitre III. Homotopie et Groupoïdes de Poincaré.- 1. Homotopies, homéotopies.- 2. Homotopie et chemins.- 3. Groupoïde de Poincaré.- 4. Homotopie et revêtements.- 5. Homotopie et revêtements (cas des espaces localement connexes par arcs).- Exercices.- Chapitre IV. Espaces Delaçables.- 1. Espaces délaçables.- 2. Groupes de Poincaré des espaces délaçables.- 3. Groupes de Poincaré des groupes topologiques.- 4. Théorie de la descente.- 5. Théorème de van Kampen.- 6. Espaces classifiants.- Exercices.- Index des notations.- Index terminologique.

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Book SynopsisDesigned to provide tools for independent study, this book contains student-tested mathematical exercises joined with MATLAB programming exercises.Most chapters open with a review followed by theoretical and programming exercises, with detailed solutions provided for all problems including programs. Many of the MATLAB exercises are presented as Russian dolls: each question improves and completes the previous program and results are provided to validate the intermediate programs.The book offers useful MATLAB commands, advice on tables, vectors, matrices and basic commands for plotting. It contains material on eigenvalues and eigenvectors and important norms of vectors and matrices including perturbation theory; iterative methods for solving nonlinear and linear equations; polynomial and piecewise polynomial interpolation; Bézier curves; approximations of functions and integrals and more. The last two chapters considers ordinary differential equations including two point boundary value problems, and deal with finite difference methods for some partial differential equations.The format is designed to assist students working alone, with concise Review paragraphs, Math Hint footnotes on the mathematical aspects of a problem and MATLAB Hint footnotes with tips on programming.Trade ReviewFrom the book reviews:“This is a very interesting and useful book for any advanced undergraduate and beginning graduate student on mathematics, statistics, computational physics, chemistry, and engineering, with a focus on numerical analysis and computational science. The main scope of this book is to provide students with the opportunity to apply numerical analysis and the well-known MATLAB to solve problems in their own specialties.” (T. E. Simos, Computing Reviews, January, 2015)“This is an interesting new kind of book in the area of numerical analysis. … It is widely accepted that solving exercises is essential to achieve a deeper understanding of a mathematical topic. Under this point of view the present book can be seen as an adequate vehicle to really get into the field of numerical analysis. … the book can also serve as a rich source of exercises for university courses.” (Rolf Dieter Grigorieff, zbMATH, Vol. 1304, 2015)Table of Contents1 An Introduction to MATLAB commands.- 2 Matrices and Linear Systems.- 3 Matrices, Eigenvalues and Eigenvectors.- 4 Matrices, Norms and Conditioning.- 5 Iterative Methods.- 6 Polynomial Interpolation.- 7 Bézier Curves and Bernstein Polynomials.- 8 Piecewise Polynomials, Interpolation and Applications.- 9 Approximation of Integrals.- 10 Linear Least Squares Methods.- 11 Continuous and Discrete Approximations.- 12 Ordinary Differential Equations, One Step Methods.- 13 Finite Differences for differential and partial differential equations.- References.- Index of Names.- Subject Index.- MATLAB Index.

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Book SynopsisThis book provides a clear and easy-to-understand introduction to higher mathematics with numerous examples. The author shows how to solve typical problems in a recipe-like manner and divides the material into short, easily digestible learning units.Have you ever cooked a 3-course meal based on a recipe? That generally works quite well, even if you are not a great cook. What does this have to do with mathematics? Well, you can solve a lot of math problems recipe-wise: Need to solve a Riccati's differential equation or the singular value decomposition of a matrix? Look it up in this book, you'll find a recipe for it here. Recipes are available for problems from the· Calculus in one and more variables,· linear algebra,· Vector Analysis,· Theory on differential equations, ordinary and partial,· Theory of integral transformations,· Function theory.Other features of this book include:· The division of Higher Mathematics into approximately 100 chapters of roughly equal length. Each chapter covers approximately the material of a 90-minute lecture.· Many tasks, the solutions to which can be found in the accompanying workbook.· Many problems in higher mathematics can be solved with computers. We always indicate how it works with MATLAB®.For the present 3rd edition, the book has been completely revised and supplemented by a section on the solution of boundary value problems for ordinary differential equations, by the topic of residue estimates for Taylor expansions and by the characteristic method for partial differential equations of the 1st order, as well as by several additional problems.Table of ContentsPreface.- 1 Ways of speaking, symbols and quantities.- 2 The natural, whole and rational numbers.- 3 The real numbers.- 4 Machine numbers.- 5 Polynomials.- 6 Trigonometric functions.- 7 Complex numbers - Cartesian coordinates.- 8 Complex numbers - Polar coordinates.- 9 Systems of linear equations.- 10 Calculating with matrices.- 11 LR-decomposition of a matrix.- 12 The determinant.- 13 Vector spaces.- 14 Generating systems and linear (in)dependence.- 15 Bases of vector spaces.- 16 Orthogonality I.- 17 Orthogonality II.- 18 The linear balancing problem.- 14 The linear balancing problem. 14 Generating systems and linear (in)dependence.- 15 Bases of vector spaces.- 16 Orthogonality I.- 17 Orthogonality II.- 18 The linear compensation problem.- 19 The QR-decomposition of a matrix.- 20 Sequences.- 21 Computation of limit values of sequences.- 22 Series.- 23 Illustrations.- 24 Power series.- 25 Limit values and continuity.- 26 Differentiation.- 27 Applications of differential calculus I.- 28 Applications of differential calculus I.- 28 Applications of differential calculus II.- 28 Applications of differential calculus I.- 28 Applications of differential calculus II. 28 Applications of differential calculus II.- 29 Polynomial and spline interpolation.- 30 Integration I.- 31 Integration II.- 32 Improper integrals.- 33 Separable and linear differential equations of the 1st order.- 34 Linear differential equations with constant coefficients.- 35 Some special types of differential equations.- 36 Numerics of ordinary differential equations I.- 37 Linear mappings and representation matrices.- 38 Basic transformation.- 39 Diagonalization - Eigenvalues and eigenvectors.- 40 Numerical computation of eigenvalues and eigenvectors.- 41 Quadrics.- 42 Schurzdecomposition and singular value decomposition.- 43 Jordan normal form I.- 44 Jordan normal form II.- 45 Definiteness and matrix norms.- 46 Functions of several variables.- 47 Partial differentiation - gradient, Hessian matrix, Jacobian matrix.- 48 Applications of partial derivatives.- 49 Determination of extreme values.- 50 Determination of extreme values under constraints.- 51 Total differentiation, differential operators.- 52 Implicit functions.- 53 Coordinate transformations.- 54 Curves I.- 55 Curves II.- 56 Curve integrals.- 57 Gradient fields.- 58 Domain integrals.- 59 The transformation formula.- 60 Areas and area integrals.- 61 Integral theorems I.- 62 Integral theorems II.- 63 General about differential equations.- 64 The exact differential equation.- 65 Systems of linear differential equations I.- 66 Systems of linear differential equations II.- 67 Systems of linear differential equations II.- 68 Boundary value problems.- 69 Basic concepts of numerics.- 70 Fixed point iteration.- 71 Iterative methods for systems of linear equations.- 72 Optimization.- 73 Numerics of ordinary differential equations II.- 74 Fourier series - Calculation of Fourier coefficients.- 75 Fourier series - Background, theorems and application.- 76 Fourier transform I.- 77 Fourier transform II.- 78 Discrete Fourier transform.- 79 The Laplacian transform.- 80 Holomorphic functions.- 81 Complex integration.- 82 Laurent series.- 83 The residue calculus.- 84 Conformal mappings.- 85 Harmonic functions and Dirichlet's boundary value problem.- 86 Partial differential equations 1st order.- 87 Partial differential equations 2nd order - General.- 88 The Laplace or Poisson equation.- 89 The heat conduction equation.- 90 The wave equation.- 91 Solving pDGLs with Fourier and Laplace transforms.- Index.

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Book SynopsisThese lecture notes treat polynomial identity rings from both the combinatorial and structural points of view. The greater part of recent research in polynomial identity rings is about combinatorial questions, and the combinatorial part of the lecture notes gives an up-to-date account of recent research. On the other hand, the main structural results have been known for some time, and the emphasis there is on a presentation accessible to newcomers to the subject.Trade ReviewFrom the reviews: “The book under review consists of two excellent monographs on the PI-theory by two leading researchers, V. Drensky and E. Formanek … In summary, both expositions are very well written, and the book is recommended both for graduate students and researchers.” (MATHEMATICAL REVIEWS)Table of ContentsA Combinatorial Aspects in PI-Rings.- Vesselin Drensky.- 1 Basic Properties of PI-algebras.- 2 Quantitative Approach to PI-algebras.- 3 The Amitsur-Levitzki Theorem.- 4 Central Polynomials for Matrices.- 5 Invariant Theory of Matrices.- 6 The Nagata-Higman Theorem.- 7 The Shirshov Theorem for Finitely Generated PI-algebras.- 8 Growth of Codimensions of PI-algebras.- B Polynomial Identity Rings.- Edward Formanek.- 1 Polynomial Identities.- 2 The Amitsur-Levitzki Theorem.- 3 Central Polynomials.- 4 Kaplansky’s Theorem.- 5 Theorems of Amitsur and Levitzki on Radicals.- 6 Posner’s Theorem.- 7 Every PI-ring Satisfies a Power of the Standard Identity.- 8 Azumaya Algebras.- 9 Artin’s Theorem.- 10 Chain Conditions.- 11 Hilbert and Jacobson PI-Rings.- 12 The Ring of Generic Matrices.- 13 The Generic Division Ring of Two 2 x 2 Generic Matrices.- 14 The Center of the Generic Division Ring.- 15 Is the Center of the Generic Division Ring a Rational Function Field?.

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Book SynopsisIn July 2004, a conference on graph theory was held in Paris in memory of Claude Berge, one of the pioneers of the field. The event brought together many prominent specialists on topics such as perfect graphs and matching theory, upon which Claude Berge's work has had a major impact. This volume includes contributions to these and other topics from many of the participants.Table of ContentsClaude Berge — Sculptor of Graph Theory.- ?-path-connectivity and mk-generation: an Upper Bound on m.- Automated Results and Conjectures on Average Distance in Graphs.- Brambles, Prisms and Grids.- Dead Cell Analysis in Hex and the Shannon Game.- Ratios of Some Domination Parameters in Graphs and Claw-free Graphs.- Excessive Factorizations of Regular Graphs.- Odd Pairs of Cliques.- Recognition of Perfect Circular-arc Graphs.- On Edge-maps whose Inverse Preserves Flows or Tensions.- On the Extremal Number of Edges in 2-Factor Hamiltonian Graphs.- Generalized Colourings (Matrix Partitions) of Cographs.- A Note on [k, l]-sparse Graphs.- Even Pairs in Bull-reducible Graphs.- Kernels in Orientations of Pretransitive Orientable Graphs.- Nonrepetitive Graph Coloring.- A Characterization of the 1-well-covered Graphs with no 4-cycles.- A Graph-theoretical Generalization of Berge’s Analogue of the Erd?s-Ko-Rado Theorem.- Independence Polynomials and the Unimodality Conjecture for Very Well-covered, Quasi-regularizable, and Perfect Graphs.- Precoloring Extension on Chordal Graphs.- On the Enumeration of Bipartite Minimum Edge Colorings.- Kempe Equivalence of Colorings.- Acyclic 4-choosability of Planar Graphs with Girth at Least 5.- Automorphism Groups of Circulant Graphs — a Survey.- Hypo-matchings in Directed Graphs.- On Reed’s Conjecture about ?,? and ?.- On the Generalization of the Matroid Parity Problem.- Reconstruction of a Rank 3 Oriented Matroids from its Rank 2 Signed Circuits.- The Normal Graph Conjecture is True for Circulants.- Two-arc Transitive Near-polygonal Graphs.- Open Problems.

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