Introduction to the Representation Theory of Algebras
|
|
- Adele Benson
- 5 years ago
- Views:
Transcription
1 Introduction to the Representation Theory of Algebras
2
3 Michael Barot Introduction to the Representation Theory of Algebras 123
4 Michael Barot Instituto de Matemáticas Universidad Nacional Autónoma de México Mexico City Mexico ISBN ISBN (ebook) DOI / Springer Cham Heidelberg New York Dordrecht London Library of Congress Control Number: Mathematics Subject Classification (2010): 16G20 Springer International Publishing Switzerland 2015 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. Exempted from this legal reservation are brief excerpts in connection with reviews or scholarly analysis or material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Duplication of this publication or parts thereof is permitted only under the provisions of the Copyright Law of the Publisher s location, in its current version, and permission for use must always be obtained from Springer. Permissions for use may be obtained through RightsLink at the Copyright Clearance Center. Violations are liable to prosecution under the respective Copyright Law. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. While the advice and information in this book are believed to be true and accurate at the date of publication, neither the authors nor the editors nor the publisher can accept any legal responsibility for any errors or omissions that may be made. The publisher makes no warranty, express or implied, with respect to the material contained herein. Printed on acid-free paper Springer is part of Springer Science+Business Media (
5 To Angélica
6
7 Preface The aim of these notes is to give a brief and elementary introduction to the representation theory of finite-dimensional algebras. The notes originated from an undergraduate course I gave in two occasions at Universidad Nacional Autonóma de México. The plan of the course was to try to cope with two competing demands: to expect as little as possible and to reach as much as possible: to expect only linear algebra as background and yet to make way to substantial and central ideas and results during its progress. Therefore some crucial decisions were necessary. We opted for the model case rather than the most general situation, for the most illustrating example rather than the most extravagant one. We sought a guideline through this vast field which conducts to as many important notions, techniques and questions as possible in the limited space of a one-semester course. So, it is a book written from a specific point of view and the title should really be Introduction to the theory of algebras, which are finite dimensional over some algebraically closed field. The book starts with the most difficult chapter: matrix problems. Conceptually there is little to understand in that chapter, but it requires a considerable effort from the reader to follow the argumentation within. However, this chapter is central: it prepares all the main examples which later will guide through the rest. In the following two chapters we consider the main languages of representation theory. Since there are several competing languages in representation theory, a considerable amount of our effort is directed towards mastering and combining all of them. As you will see each of these languages has its own advantages and it therefore not only enables the reader to consult the majority of all research articles in the field, but also enriches the way we may think about the notions themselves. The rest of the book is devoted to gain structural insight into the categories of modules of a given algebra. In the chapter about module categories some older results are proved, whereas in the next four chapters more recent developments are discussed. The last chapter is more of an open-minded collection concerning indecomposable modules, the building bricks of the module categories, with respect to certain invariants, called dimension vectors. vii
8 viii Preface My thanks go to Jan Schröer, for it was him who suggested to write jointly a book on representation theory but unfortunately gave up on it, to Juraj Hromkovič who invited me to ETH Zürich for a sabbatical in 2010 and made thus possible to finish the manuscript, to Manuela Tschabold who tirelessly read the whole manuscript carefully eliminating thus many errors and misprints, to Karin Baur who took it as a base for a course in 2011 suggesting many improvements, to Christof Geiss who was able to improve the manuscript still further, to Mario Aigner from Springer who tirelessly fought on my side for a good layout and to Angélica Herrera Loyo for her patience. I apologize to the reader for all the errors which still remain and for the poor English in which it is written. Mexico City, Mexico January 2012 Michael Barot
9 Contents 1 Matrix Problems Introduction The Two Subspace Problem Decomposition into Indecomposables The KroneckerProblem The ThreeKroneckerProblem Comments Representations of Quivers Quivers Representations Categories andfunctors The Path Category Equivalenceof Categories A New Example Algebras Definition and Generalities About Algebras The Path Algebra Quotients by Ideals Idempotents Morita Equivalence The Radical The Quiverof an Algebra Relations Summary Module Categories The CategoricalPoint of View Duality Kernels and Cokernels Unique Decomposition ix
10 x Contents 4.5 Projectives andinjectives ProjectiveCovers andinjectivehulls Simple Modules Elements of Homological Algebra Short Exact Sequences The Baer Sum Resolutions LongExact Sequences The Auslander-Reiten Theory The Auslander-ReitenQuiverandthe InfiniteRadical Example:The KroneckerAlgebra The Auslander-ReitenTranslate Auslander-ReitenSequences Knitting Sourceand Sink Maps The Knitting Technique Knitting with Dimension Vectors Limits of Knitting Preprojective Components Combinatorial Invariants GrothendieckGroup The Homological Bilinear Form The CoxeterTransformation QuadraticForms Roots Indecomposables and Dimensions The Brauer-ThrallConjectures Gabriel s Theorem Reflection Functors Kac s Theorem Tame and Wild Module Varieties References Index of Symbols Index of Notions
Mathematics, Computer Science and Logic - A Never Ending Story
Mathematics, Computer Science and Logic - A Never Ending Story Peter Paule Editor Mathematics, Computer Science and Logic - A Never Ending Story The Bruno Buchberger Festschrift 123 Editor Peter Paule
More informationPaul M. Gauthier. Lectures on Several Complex
Paul M. Gauthier Lectures on Several Complex Variables Paul M. Gauthier Départment de Mathématiques et de Statistique Université de Montréal Montreal, QC, Canada ISBN 978-3-319-11510-8 ISBN 978-3-319-11511-5
More informationProtecting Chips Against Hold Time Violations Due to Variability
Protecting Chips Against Hold Time Violations Due to Variability Gustavo Neuberger Ricardo Reis Gilson Wirth Protecting Chips Against Hold Time Violations Due to Variability Gustavo Neuberger Instituto
More informationCalculation of Demographic Parameters in Tropical Livestock Herds
Matthieu Lesnoff Renaud Lancelot Charles-Henri Moulin Samir Messad Xavier Juanès Christian Sahut Calculation of Demographic Parameters in Tropical Livestock Herds A discrete time approach with LASER animal-based
More informationZdravko Cvetkovski. Inequalities. Theorems, Techniques and Selected Problems
Inequalities Zdravko Cvetkovski Inequalities Theorems, Techniques and Selected Problems Dipl. Math. Zdravko Cvetkovski Informatics Department European University-Republic of Macedonia Skopje, Macedonia
More informationCollected Papers VI. Literary Reality and Relationships
Collected Papers VI. Literary Reality and Relationships PHAENOMENOLOGICA SERIES FOUNDED BY H.L. VAN BREDA AND PUBLISHED UNDER THE AUSPICES OF THE HUSSERL-ARCHIVES 206 ALFRED SCHUTZ COLLECTED PAPERS VI.
More informationRacial Profiling and the NYPD
Racial Profiling and the NYPD Jay L. Newberry Racial Profiling and the NYPD The Who, What, When, and Why of Stop and Frisk Jay L. Newberry Department of Geography Binghamton University Binghamton, NY USA
More informationHow to Write Technical Reports
How to Write Technical Reports Lutz Hering Heike Hering How to Write Technical Reports Understandable Structure, Good Design, Convincing Presentation 123 Dr. Lutz Hering Am Ricklinger Holze 14 30966 Hemmingen
More informationQuantum Theory and Local Causality
SPRINGER BRIEFS IN PHILOSOPHY Gábor Hofer-Szabó Péter Vecsernyés Quantum Theory and Local Causality SpringerBriefs in Philosophy SpringerBriefs present concise summaries of cutting-edge research and practical
More informationInnovations Lead to Economic Crises
Innovations Lead to Economic Crises Jon-Arild Johannessen Innovations Lead to Economic Crises Explaining the Bubble Economy Jon-Arild Johannessen Kristiania University College and Nord University Oslo/Bodø,
More informationGuide to Computing for Expressive Music Performance
Guide to Computing for Expressive Music Performance Alexis Kirke Eduardo R. Miranda Editors Guide to Computing for Expressive Music Performance Editors Alexis Kirke Interdisciplinary Centre for Computer
More informationProblem Books in Mathematics
Problem Books in Mathematics Series Editor: Peter Winkler Department of Mathematics Dartmouth College Hanover, NH 03755 USA More information about this series at http://www.springer.com/series/714 Hayk
More informationMATLAB Ò and Design Recipes for Earth Sciences
MATLAB Ò and Design Recipes for Earth Sciences Martin H. Trauth Elisabeth Sillmann MATLAB Ò and Design Recipes for Earth Sciences How to Collect, Process and Present Geoscientific Information 123 Martin
More informationTrends in Mathematics
Trends in Mathematics Trends in Mathematics is a series devoted to the publication of volumes arising from conferences and lecture series focusing on a particular topic from any area of mathematics. Its
More informationFoundations of Mathematics
Erwin Engeler Foundations of Mathematics Questions of Analysis, Geometry & Algorithmics Translated by Charles B. Thomas With 29 Figures Springer-Verlag Berlin Heidelberg New York London Paris Tokyo Hong
More informationUrbanization and the Migrant in British Cinema
Urbanization and the Migrant in British Cinema Gareth Millington Urbanization and the Migrant in British Cinema Spectres of the City Gareth Millington University of York York, UK ISBN 978-1-137-47398-1
More informationCompanion to European Heritage Revivals / edited by Linde Egberts and Koos Bosma
Companion to European Heritage Revivals / edited by Linde Egberts and Koos Bosma Companion to European Heritage Revivals / edited by Linde Egberts and Koos Bosma Linde Egberts and Koos Bosma (eds.) CLUE
More informationAn Introduction to Well Control Calculations for Drilling Operations
An Introduction to Well Control Calculations for Drilling Operations Dave Cormack An Introduction to Well Control Calculations for Drilling Operations 123 Dave Cormack Consultant, Auriga Training Ltd.
More informationBeing Agile. Your Roadmap to Successful Adoption of Agile. Mario E. Moreira
Being Agile Your Roadmap to Successful Adoption of Agile Mario E. Moreira Being Agile: Your Roadmap to Successful Adoption of Agile Copyright 2013 by Mario E. Moreira This work is subject to copyright.
More informationFreshwater Invertebrates in Central Europe
Freshwater Invertebrates in Central Europe György Kriska Freshwater Invertebrates in Central Europe A Field Guide György Kriska Group for Methodology in Biology Teaching, Biological Institute Eötvös Loránd
More informationThe Discourse of Peer Review
The Discourse of Peer Review Brian Paltridge The Discourse of Peer Review Reviewing Submissions to Academic Journals Brian Paltridge Sydney School of Education & Social Work University of Sydney Sydney,
More informationEdible Medicinal and Non-Medicinal Plants
Edible Medicinal and Non-Medicinal Plants T.K. Lim Edible Medicinal and Non-Medicinal Plants Volume 5, Fruits ISBN 978-94-007-5652-6 ISBN 978-94-007-5653-3 (ebook) DOI 10.1007/978-94-007-5653-3 Springer
More informationDamage Mechanics with Finite Elements
Damage Mechanics with Finite Elements Springer-Verlag Berlin Heidelberg GmbH P. 1. Kattan. G. Z. Voyiadjis Damage Mechanics with Finite Elements Practical Applications with Computer Tools With 28 Figures
More informationLecture Notes in Mathematics 2164
Lecture Notes in Mathematics 2164 Editors-in-Chief: Jean-Michel Morel, Cachan Bernard Teissier, Paris Advisory Board: Michel Brion, Grenoble Camillo De Lellis, Zurich Alessio Figalli, Zurich Davar Khoshnevisan,
More informationThe New Middle Ages. Series Editor Bonnie Wheeler English & Medieval Studies Southern Methodist University Dallas, Texas, USA
The New Middle Ages Series Editor Bonnie Wheeler English & Medieval Studies Southern Methodist University Dallas, Texas, USA The New Middle Ages is a series dedicated to pluridisciplinary studies of medieval
More informationTheory of Digital Automata
Theory of Digital Automata International Series on INTELLIGENT SYSTEMS, CONTROL AND AUTOMATION: SCIENCE AND ENGINEERING VOLUME 63 Editor Professor S. G. Tzafestas, National Technical University of Athens,
More informationPropaganda and Hogarth s Line of Beauty in the First World War
Propaganda and Hogarth s Line of Beauty in the First World War Georgina Williams Propaganda and Hogarth s Line of Beauty in the First World War Georgina Williams ISBN 978-1-137-57193-9 ISBN 978-1-137-57194-6
More informationSpringerBriefs in Electrical and Computer Engineering
SpringerBriefs in Electrical and Computer Engineering More information about this series at http://www.springer.com/series/10059 Fatima Hussain Internet of Things Building Blocks and Business Models 123
More informationBenedetto Cotrugli The Book of the Art of Trade
Benedetto Cotrugli The Book of the Art of Trade Carlo Carraro Giovanni Favero Editors Benedetto Cotrugli The Book of the Art of Trade With Scholarly Essays from Niall Ferguson, Giovanni Favero, Mario Infelise,
More informationIrish Women Writers and the Modern Short Story
Irish Women Writers and the Modern Short Story Elke D hoker Irish Women Writers and the Modern Short Story Elke D hoker University of Leuven Leuven, Belgium ISBN 978-3-319-30287-4 ISBN 978-3-319-30288-1
More informationMulticriteria Optimization
Multicriteria Optimization Matthias Ehrgott Multicriteria Optimization Second edition With 88 Figures and 12 Tables 12 Dr. habil. Matthias Ehrgott, Associate Professor The University of Auckland Department
More informationMarxism and Education. Series Editor Anthony Green Institute of Education University of London London, United Kingdom
Marxism and Education Series Editor Anthony Green Institute of Education University of London London, United Kingdom This series assumes the ongoing relevance of Marx s contributions to critical social
More informationThe Sound of Silence
The Sound of Silence Burkhard Vogel The Sound of Silence Lowest-Noise RIAA Phono-Amps: Designer s Guide 123 Dipl.-Ing. Burkhard Vogel BUVOCON-Burkhard Vogel Consult GmbH 70180 Stuttgart, Germany ISBN 978-3-540-76883-8
More informationFormal Concept Analysis
Formal Concept Analysis Springer Berlin Heidelberg New York Barcelona Hong Kong London Milan Paris Singapore Tokyo Bernhard Ganter Rudolf Wille Formal Concept Analysis Mathematical Foundations With 105
More informationThe Grotesque in Contemporary Anglophone Drama
The Grotesque in Contemporary Anglophone Drama wwwwwwwwwwwwwwwww Ondřej Pilný The Grotesque in Contemporary Anglophone Drama Ondřej Pilný Department of Anglophone Literatures and Cultures Faculty of Arts
More informationRhetoric, Politics and Society
Rhetoric, Politics and Society Series Editors Alan Finlayson University of East Anglia United Kingdom James Martin Goldsmiths, University of London United Kingdom Kendall Phillips University of Syracuse
More informationThe Language of Cosmetics Advertising
The Language of Cosmetics Advertising aheuhiuea Helen Ringrow The Language of Cosmetics Advertising Helen Ringrow School of Languages and Area Studies University of Portsmouth Portsmouth, United Kingdom
More informationTraining for Model Citizenship
Training for Model Citizenship Molly Sundberg Training for Model Citizenship An Ethnography of Civic Education and State- Making in Rwanda Molly Sundberg Uppsala University Uppsala Sweden ISBN 978-1-137-58421-2
More informationPerforming Age in Modern Drama
Performing Age in Modern Drama Valerie Barnes Lipscomb Performing Age in Modern Drama Valerie Barnes Lipscomb Liberal Arts University of South Florida Sarasota-Manatee Sarasota, Florida, USA ISBN 978-1-137-51251-2
More informationJane Dowson. Carol Ann Duffy. Poet for Our Times
Carol Ann Duffy Jane Dowson Carol Ann Duffy Poet for Our Times Jane Dowson De Montfort University Leicester, UK ISBN 978-1-137-41562-2 ISBN 978-1-137-41563-9 (ebook) DOI 10.1057/978-1-137-41563-9 Library
More informationStudies in German Idealism
Studies in German Idealism Volume 17 Series Editor Reinier W. Munk, VU University, Amsterdam, The Netherlands Advisory Editorial Board Frederick Beiser, Syracuse University, U.S.A. Daniel Dahlstrom, Boston
More informationA Algorithms and Combinatorics 13
A Algorithms and Combinatorics 13 Editorial Board R. L. Graham, Murray Hill B. Korte, Bonn L. Lovasz, Budapest A.Wigderson, Jerusalem G. M. Ziegler, Berlin Springer Berlin Heidelberg New York Barcelona
More informationShame and Modernity in Britain
Shame and Modernity in Britain Anne-Marie Kilday David S. Nash Shame and Modernity in Britain 1890 to the Present Anne-Marie Kilday Department of History, Philosophy and Religion Oxford Brookes University
More informationJ. Andrew Hubbell. Byron s Nature. A Romantic Vision of Cultural Ecology
Byron s Nature Given the important role that Lord Byron s short but massively influential poem Darkness played in the development of eco-criticism, the lack of a systematic examination of Lord Byron s
More informationPostdisciplinary Studies in Discourse
Postdisciplinary Studies in Discourse Series Editors Johannes Angermuller University of Warwick Coventry, United Kingdom Judith Baxter Aston University Birmingham, UK Aim of the series Postdisciplinary
More informationAncient West Asian Civilization
Ancient West Asian Civilization Akira Tsuneki Shigeo Yamada Ken-ichiro Hisada Editors Ancient West Asian Civilization Geoenvironment and Society in the Pre- Islamic Middle East Editors Akira Tsuneki Faculty
More informationInjectable Fillers in Aesthetic Medicine
Mauricio de Maio Berthold Rzany Injectable Fillers in Aesthetic Medicine Second Edition 123 Injectable Fillers in Aesthetic Medicine Mauricio de Maio Berthold Rzany Injectable Fillers in Aesthetic Medicine
More informationLandolt-Börnstein / New Series
Landolt-Börnstein / New Series Landolt-Börnstein Numerical Data and Functional Relationships in Science and Technology New Series Editor in Chief: W. Martienssen Units and Fundamental Constants in Physics
More informationReasonably Simple Economics
Reasonably Simple Economics Why the World Works the Way It Does Evan Osborne Reasonably Simple Economics: Why the World Works the Way It Does Copyright 2013 by Evan Osborne This work is subject to copyright.
More informationComplicite, Theatre and Aesthetics
Complicite, Theatre and Aesthetics Tomasz Wiśniewski Complicite, Theatre and Aesthetics From Scraps of Leather Tomasz Wiśniewski University of Gdańsk, Poland ISBN 978-3-319-33442-4 ISBN 978-3-319-33443-1
More informationTheatre and Residual Culture
Theatre and Residual Culture Christopher Collins Theatre and Residual Culture J.M. Synge and Pre-Christian Ireland Christopher Collins School of English University of Nottingham Nottingham, UK ISBN 978-1-349-94871-0
More informationCommunicating Science
Communicating Science Pierre Laszlo Communicating Science A Practical Guide 123 Prof. Pierre Laszlo Cloud s Rest Prades F-12320 Senergues France DOI 10.1007/75951 ISBN-10 3-540-31919-0 Springer Berlin
More informationDada and Existentialism
Dada and Existentialism Elizabeth Benjamin Dada and Existentialism The Authenticity of Ambiguity Elizabeth Benjamin University of Birmingham Birmingham, United Kingdom ISBN 978-1-137-56367-5 DOI 10.1057/978-1-137-56368-2
More informationErgebnisse der Mathematik und ihrer Grenzgebiete
Ergebnisse der Mathematik und ihrer Grenzgebiete 3. Folge. Band 26 A Series of Modem Surveys in Mathematics Editorial Board E. Bombieri, Princeton S. Feferman, Stanford M. Gromov, Bures-sur-Yvette H.W.
More informationVision, Illusion and Perception
Vision, Illusion and Perception Volume 1 Series editor Nicholas Wade, University of Dundee, Department of Psychology, Dundee, DD1 4HN, Scotland, UK e-mail: n.j.wade@dundee.ac.uk Editorial Board Benjamin
More informationLandolt-Börnstein Numerical Data and Functional Relationships in Science and Technology New Series / Editor in Chief: W.
Landolt-Börnstein Numerical Data and Functional Relationships in Science and Technology New Series / Editor in Chief: W. Martienssen Group III: Condensed Matter Volume 44 Semiconductors Subvolume C New
More informationCognitive Studies in Literature and Performance
Cognitive Studies in Literature and Performance Series Editors Bruce McConachie Department of Theatre Arts University of Pittsburgh Pittsburgh, PA, USA Blakey Vermeule Department of English Stanford University
More informationThe Marmoset Brain in Stereotaxic Coordinates
The Marmoset Brain in Stereotaxic Coordinates Xavier Palazzi Nicole Bordier The Marmoset Brain in Stereotaxic Coordinates 13 Xavier Palazzi Biomatech Chasse-sur-Rhoˆne x.palazzi@biomatech.fr Nicole Bordier
More informationCultural Sociology. Series Editors Jeffrey C. Alexander Center for Cultural Sociology Yale University New Haven, CT, USA
Cultural Sociology Series Editors Jeffrey C. Alexander Center for Cultural Sociology Yale University New Haven, CT, USA Ron Eyerman Center for Cultural Sociology Yale University New Haven, CT, USA David
More informationThe Language of Suspense in Crime Fiction
The Language of Suspense in Crime Fiction Reshmi Dutta-Flanders The Language of Suspense in Crime Fiction A Linguistic Stylistic Approach Reshmi Dutta-Flanders English Language and Linguistics School of
More informationJourney through Mathematics
Journey through Mathematics Enrique A. González-Velasco Journey through Mathematics Creative Episodes in Its History Enrique A. González-Velasco Department of Mathematical Sciences University of Massachusetts
More informationBirkhäuser Advanced Texts
Birkhäuser Advanced Texts Basler Lehrbücher Edited by Herbert Amann, University of Zürich Steven G. Krantz, Washington University, St. Louis Shrawan Kumar, University of North Carolina at Chapel Hili Steven
More informationSpringer Praxis Books
Springer Praxis Books More information about this series at http://www.springer.com/series/4097 J.B. Williams The Electronics Revolution Inventing the Future J.B. Williams Offord Darcy St Neots Cambridgeshire,
More informationPhase Equilibria, Crystallographic and Thermodynamic Data of Binary Alloys
Landolt-Börnstein Numerical Data and Functional Relationships in Science and Technology New Series / Editor in Chief: W. Martienssen Group IV: Physical Chemistry Volume 12 Phase Equilibria, Crystallographic
More informationTHEORY AND APPLICATIONS OF SPECIAL FUNCTIONS. A Volume Dedicated to Mizan Rahman
- - - - -- - THEORY AND APPLICATIONS OF SPECIAL FUNCTIONS A Volume Dedicated to Mizan Rahman Developments in Mathematics VOLUME 13 Series Editor: Krishnaswami Alladi, University of Florida, U.S.A. Aims
More informationLandolt-Börnstein Numerical Data and Functional Relationships in Science and Technology New Series / Editor in Chief: W.
Landolt-Börnstein Numerical Data and Functional Relationships in Science and Technology New Series / Editor in Chief: W. Martienssen Group III: Condensed Matter Volume 44 Semiconductors Subvolume D New
More informationEvolution of Broadcast Content Distribution
Evolution of Broadcast Content Distribution Roland Beutler Evolution of Broadcast Content Distribution 123 Roland Beutler Südwestrundfunk (SWR) Stuttgart, Germany ISBN 978-3-319-45972-1 ISBN 978-3-319-45973-8
More informationHuman Rights Violation in Turkey
Human Rights Violation in Turkey Human Rights Violation in Turkey Rethinking Sociological Perspectives David Straw University of Manchester, UK David Straw 2013 Softcover reprint of the hardcover 1st edition
More informationHANDBOOK OF RECORDING ENGINEERING FOURTH EDITION
HANDBOOK OF RECORDING ENGINEERING FOURTH EDITION HANDBOOK OF RECORDING ENGINEERING FOURTH EDITION by John Eargle JME Consulting Corporation Springe] John Eargle JME Consulting Corporation Los Angeles,
More informationEATCS Monographs on Theoretical Computer Science
EATCS Monographs on Theoretical Computer Science Volume 23 Editors: W. Brauer G. Rozenberg A. Salomaa Advisory Board: G.Ausiello M.Broy S.Even IHartmanis N.Jones T. Leighton M.Nivat C. Papadimitriou D.
More informationSpringer-Verlag Berlin Heidelberg GmbH
Reporting on Income Distribution and Poverty Springer-Verlag Berlin Heidelberg GmbH Richard Hauser Irene Becker Editors Reporting on Incotne Distribution and Poverty Perspectives from a German and a European
More informationNEUROANATOMY 3D-Stereoscopic Atlas of the Human Brain
Martin C. Hirsch Thomas Kramer NEUROANATOMY 3D-Stereoscopic Atlas of the Human Brain Springer-Verlag Berlin Heidelberg GmbH Martin C. Hirsch Thomas Kramer NEUROANATOMY 3D-Stereoscopic Atlas of the Human
More informationTranscultural Research Heidelberg Studies on Asia and Europe in a Global Context
Transcultural Research Heidelberg Studies on Asia and Europe in a Global Context Series Editors Madeleine Herren Axel Michaels Rudolf G. Wagner For further volumes: http://www.springer.com/series/8753
More informationRamanujan's Notebooks
Ramanujan's Notebooks Part II Springer New York Berlin Heidelberg Barcelona Hong Kong London Milan Paris Singapore Tokyo Bust of Ramanujan by Paul Granlund Bruce C. Berndt Ramanujan's Notebooks Part II
More informationRadiology for Undergraduate Finals and Foundation Years
MasterPass Radiology for Undergraduate Finals and Foundation Years key topics and question types Tristan Barrett, Nadeem Shaida and Ashley Shaw Foreword by Adrian K Dixon Radiology for Undergraduate Finals
More informationBurkhard Vogel. How to Gain Gain. A Reference Book on Triodes in Audio Pre-Amps
How to Gain Gain Burkhard Vogel How to Gain Gain A Reference Book on Triodes in Audio Pre-Amps 123 Dipl. -Ing. Burkhard Vogel BUVOCON GmbH 70180 Stuttgart Germany ISBN: 978-3-540-69502-8 e-isbn: 978-3-540-69505-9
More informationEncyclopedia of Marine Sciences
1. G. Baretta-Bekker. E. K. Duursma B. R. Kuipers (Eds.) Encyclopedia of Marine Sciences With 127 Figures Springer-Verlag Berlin Heidelberg New York London Paris Tokyo Hong Kong Barcelona Budapest Drs.
More informationTexts in Theoretical Computer Science An EATCS Series
Texts in Theoretical Computer Science An EATCS Series Editors: W. Brauer G. Rozenberg A. Salomaa Advisory Board: G. Ausiello M. Broy S. Even J. Hartmanis N. Jones T. Leighton M. Nivat C. Papadimitriou
More informationCognitive Studies in Literature and Performance
Cognitive Studies in Literature and Performance Series Editors Bruce McConachie Department of Theatre Arts University of Pittsburgh Pittsburgh, Pennsylvania, USA Blakey Vermeule Department of English Stanford
More informationWittgenstein and Interreligious Disagreement
Wittgenstein and Interreligious Disagreement Gorazd Andrejč Wittgenstein and Interreligious Disagreement A Philosophical and Theological Perspective Gorazd Andrejč St Edmunds College University of Cambridge
More informationJill Scott Esther Stoeckli Editors. Neuromedia. Art and Neuroscience Research
Neuromedia Jill Scott Esther Stoeckli Editors Neuromedia Art and Neuroscience Research Editors Jill Scott Institute for Cultural Studies in Art, Media and Design Zurich University of the Arts (ZHdk) Zurich,
More informationTrends in Mathematics
Trends in Mathematics Trends in Mathematics is a series devoted to the publication of volumes arising from conferences and lecture series focusing on a particular topic from any area of mathematics. Its
More informationPerforming Arts Medicine in Clinical Practice
Performing Arts Medicine in Clinical Practice Howard A. Bird Performing Arts Medicine in Clinical Practice Howard A. Bird University College London London UK ISBN 978-3-319-12426-1 DOI 10.1007/978-3-319-12427-8
More informationYorick Wilks. Machine Translation. Its Scope and Limits
Machine Translation Yorick Wilks Machine Translation Its Scope and Limits 123 Yorick Wilks Department of Computer Science The University of Sheffield Regent Court, 211 Portobello Street Sheffield, S1 4DP,
More informationTen Essays in the Development of Economic Thought. Ronald L. Meek Tyler Professor of Economics at the University of Leicester
SMITH, MARX, & AFTER SMITH, MARX, & AFTER Ten Essays in the Development of Economic Thought Ronald L. Meek Tyler Professor of Economics at the University of Leicester SPRINGER-SCIENCE+BUSINESS MEDIA. B.V.
More informationProgress in Mathematics Volume 249. Series Editors Hyman Bass Joseph Oesterle Alan Weinstein
Progress in Mathematics Volume 249 Series Editors Hyman Bass Joseph Oesterle Alan Weinstein Joachim Kock Israel Vainsencher An Invitation to Quantum Cohomology Kontsevich's Formula for Rational Plane Curves
More informationNumerical Analysis. Ian Jacques and Colin Judd. London New York CHAPMAN AND HALL. Department of Mathematics Coventry Lanchester Polytechnic
Numerical Analysis Numerical Analysis Ian Jacques and Colin Judd Department of Mathematics Coventry Lanchester Polytechnic London New York CHAPMAN AND HALL First published in 1987 by Chapman and Hall Ltd
More informationMusic Recommendation and Discovery
Music Recommendation and Discovery Òscar Celma Music Recommendation and Discovery The Long Tail, Long Fail, and Long Play in the Digital Music Space 123 Òscar Celma BMAT Bruniquer 49 08024 Barcelona Spain
More informationMarx, Engels, and Marxisms
Marx, Engels, and Marxisms Series Editors Terrell Carver University of Bristol Bristol, United Kingdom Marcello Musto York University Toronto, Ontario, Canada The volumes of this series challenge the Marxist
More informationLOCALITY DOMAINS IN THE SPANISH DETERMINER PHRASE
LOCALITY DOMAINS IN THE SPANISH DETERMINER PHRASE Studies in Natural Language and Linguistic Theory VOLUME 79 Managing Editors Marcel den Dikken, City University of New York Liliane Haegeman, University
More informationA Glossary of Anesthesia and Related Terminology. Second Edition
A Glossary of Anesthesia and Related Terminology Second Edition Sanford L. Klein A Glossary of Anesthesia and Related Terminology Second Edition With 228 Illustrations Springer-Verlag New York Berlin Heidelberg
More informationPolymer Technology Dictionary
Polymer Technology Dictionary Polymer Technology Dictionary Tony Whelan MSC, Consultant Formerly Director London Polymer Consultants Ltd New Southgate, London, UK SPRINGER-SCIENCE+BUSINESS MEDIA, B.V First
More informationLecture Notes in Computer Science 7020
Lecture Notes in Computer Science 7020 Commenced Publication in 1973 Founding and Former Series Editors: Gerhard Goos, Juris Hartmanis, and Jan van Leeuwen Editorial Board David Hutchison Lancaster University,
More informationQuality Assurance in Seafood Processing: A Practical Guide
Quality Assurance in Seafood Processing: A Practical Guide Quality Assurance in Seafood Processing: A Practical Guide A. D. Bonnell SPRINGER-SCIENCE+BUSINESS MEDIA, B.V. 1994 Springer Science+Business
More informationEdited by: Wolfgang Dietrich UNESCO Chair for Peace Studies University of Innsbruck/Austria
Masters of Peace Masters of Peace is a book series edited by the University of Innsbruck s UNESCO Chair for Peace Studies. It has been founded to honour outstanding works of young academics in the field
More informationEnglish for Biomedical Scientists Ramón Ribes Palma Iannarelli Rafael F. Duarte
English for Biomedical Scientists Ramón Ribes Palma Iannarelli Rafael F. Duarte Ramón Ribes Palma Iannarelli Rafael F. Duarte English for Biomedical Scientists Ramón Ribes Palma Iannarelli Hospital Reina
More informationPostnarrativist Philosophy of Historiography
Postnarrativist Philosophy of Historiography This page intentionally left blank Postnarrativist Philosophy of Historiography Jouni-Matti Kuukkanen University of Oulu, Finland Jouni-Matti Kuukkanen 2015
More informationThe Second French Republic
The Second French Republic 1848 1852 Christopher Guyver The Second French Republic 1848 1852 A Political Reinterpretation Christopher Guyver Chertsey, Surrey, United Kingdom ISBN 978-1-137-59739-7 ISBN
More informationThe. Craft of. Editing
The Craft of Editing Springer Science+ Business Media, LLC The Craft of Editing A Guide for Managers, Scientists, and Engineers Michael Alley. ~. T Springer Michael Alley College of Engineering VIrginia
More informationPublications des Archives Henri-Poincaré Publications of the Henri Poincaré Archives
Publications des Archives Henri-Poincaré Publications of the Henri Poincaré Archives Textes et Travaux, Approches Philosophiques en Logique, Mathématiques et Physique autour de 1900 Texts, Studies and
More information