series: De Gruyter Expositions in Mathematics

Veröffentlicht von De Gruyter

De Gruyter Expositions in Mathematics

Herausgegeben von: Lev Birbrair, Victor P. Maslov, Walter D. Neumann, Markus J. Pflaum, Dierk Schleicher und Katrin Wendland

ISSN: 0938-6572

Übersicht
Bände

The aim of the Expositions is to present new and important developments in pure and applied mathematics. Well established in the community over more than two decades, the series offers a large library of mathematical works, including several important classics.

The volumes supply thorough and detailed expositions of the methods and ideas essential to the topics in question. In addition, they convey their relationships to other parts of mathematics. The series is addressed to advanced readers interested in a thorough study of the subject.

Editorial Board

Lev Birbrair, Universidade Federal do Ceará, Fortaleza, Brasil
Walter D. Neumann, Columbia University, New York, USA
Markus J. Pflaum, University of Colorado, Boulder, USA
Dierk Schleicher, Aix-Marseille Université, France
Katrin Wendland, Trinity College Dublin, Dublin, Ireland

Honorary Editor

Victor P. Maslov, Russian Academy of Sciences, Moscow, Russia

Titles in planning include

Yuri A. Bahturin, Identical Relations in Lie Algebras (2019)
Yakov G. Berkovich, Lev G. Kazarin, and Emmanuel M. Zhmud', Characters of Finite Groups, Volume 2 (2019)
Jorge Herbert Soares de Lira, Variational Problems for Hypersurfaces in Riemannian Manifolds (2019)
Volker Mayer, Mariusz Urbański, and Anna Zdunik, Random and Conformal Dynamical Systems (2021)
Ioannis Diamantis, Boštjan Gabrovšek, Sofia Lambropoulou, and Maciej Mroczkowski, Knot Theory of Lens Spaces (2021)

Fachgebiete

book: Derived Algebraic Geometry

Buch 2024

Derived Algebraic Geometry

An Elemental Approach

Renaud Gauthier

Band 75 in dieser Reihe

The second edition presents schemes, simplicial sets, higher categories, model categories, derived algebraic geometry, and spectral algebraic geometry in a self-contained manner. It discusses Motives, Goodwillie Calculus, Higher Galois, Supersymmetry, and topics in physical mathematics. A new chapter on Derived Motivic Spectrais now included as is an extended introduction to Infinity Category as well as a revised chapter on Stacks.

DOI:: https://doi.org/10.1515/9783111334073
ISBN:: 9783111334073
ISBN:: 9783111333663
ISBN:: 9783111334219
Fachgebiet:: Mathematik
Fachgebiet:: Algebra und Zahlentheorie
Fachgebiet:: Geometrie und Topologie
Verlag:: De Gruyter

book: Volume 1-3 [Set Non-Invertible Dynamical Systems, Vol 1-3]

Buch Nur in gedruckter Form 2023

Volume 1-3 [Set Non-Invertible Dynamical Systems, Vol 1-3]

Teil des mehrbändigen Werks Non-Invertible Dynamical Systems

The book contains a detailed treatment of thermodynamic formalism on general compact metrizable spaces. Topological pressure, topological entropy, variational principle, and equilibrium states are presented in detail. Abstract ergodic theory is also given a significant attention. Ergodic theorems, ergodicity, and Kolmogorov-Sinai metric entropy are fully explored. Furthermore, the book gives the reader an opportunity to find rigorous presentation of thermodynamic formalism for distance expanding maps and, in particular, subshifts of finite type over a finite alphabet. It also provides a fairly complete treatment of subshifts of finite type over a countable alphabet. Transfer operators, Gibbs states and equilibrium states are, in this context, introduced and dealt with. Their relations are explored. All of this is applied to fractal geometry centered around various versions of Bowen’s formula in the context of expanding conformal repellors, limit sets of conformal iterated function systems and conformal graph directed Markov systems. A unique introduction to iteration of rational functions is given with emphasize on various phenomena caused by rationally indifferent periodic points. Also, a fairly full account of the classicaltheory of Shub’s expanding endomorphisms is given; it does not have a book presentation in English language mathematical literature.

ISBN:: 9783110706796
Fachgebiet:: Mathematik
Fachgebiet:: Analysis
Fachgebiet:: Differentialgleichungen und dynamische Systeme
Fachgebiet:: Geometrie und Topologie
Fachgebiet:: Wahrscheinlichkeitstheorie und Statistik
Verlag:: De Gruyter

book: Integral Representation

Buch 2023

Integral Representation

Choquet Theory for Linear Operators on Function Spaces

Walter Roth

Band 74 in dieser Reihe

This book presents a wide-ranging approach to operator-valued measures and integrals of both vector-valued and set-valued functions. It covers convergence theorems and an integral representation for linear operators on spaces of continuous vector-valued functions on a locally compact space. These are used to extend Choquet theory, which was originally formulated for linear functionals on spaces of real-valued functions, to operators of this type.

DOI:: https://doi.org/10.1515/9783111315478
ISBN:: 9783111315478
ISBN:: 9783111314501
ISBN:: 9783111315584
Fachgebiet:: Mathematik
Fachgebiet:: Analysis
Verlag:: De Gruyter

book: Stereotype Spaces and Algebras

Buch 2022

Stereotype Spaces and Algebras

Sergei S. Akbarov

Band 73 in dieser Reihe

The term “stereotype space” was introduced in 1995 and denotes a category of locally convex spaces with surprisingly elegant properties. Its study gives an unexpected point of view on functional analysis that brings this fi eld closer to other main branches of mathematics, namely, to algebra and geometry. This volume contains the foundations of the theory of stereotype spaces, with accurate definitions, formulations, proofs, and numerous examples illustrating the interaction of this discipline with the category theory, the theory of Hopf algebras, and the four big geometric disciplines: topology, differential geometry, complex geometry, and algebraic geometry.

DOI:: https://doi.org/10.1515/9783110780918
ISBN:: 9783110780918
ISBN:: 9783110780963
ISBN:: 9783110780864
Fachgebiet:: Mathematik
Fachgebiet:: Algebra und Zahlentheorie
Fachgebiet:: Analysis
Fachgebiet:: Geometrie und Topologie
Verlag:: De Gruyter

book: Deformation Theory of Discontinuous Groups

Buch 2022

Deformation Theory of Discontinuous Groups

Ali Baklouti

Band 72 in dieser Reihe

This book contains the latest developments of the theory of discontinuous groups acting on homogenous spaces, from basic concepts to a comprehensive exposition. It develops the newest approaches and methods in the deformation theory of topological modules and unitary representations and focuses on the geometry of discontinuous groups of solvable Lie groups and their compact extensions. It also presents proofs of recent results, computes fundamental examples, and serves as an introduction and reference for students and experienced researchers in Lie theory, discontinuous groups, and deformation (and moduli) spaces.

DOI:: https://doi.org/10.1515/9783110765304
ISBN:: 9783110765304
ISBN:: 9783110765298
ISBN:: 9783110765397
Fachgebiet:: Mathematik
Fachgebiet:: Algebra und Zahlentheorie
Fachgebiet:: Analysis
Fachgebiet:: Geometrie und Topologie
Fachgebiet:: Mathematik, Allgemeines
Verlag:: De Gruyter

book: The Canonical Operator in Many-Particle Problems and Quantum Field Theory

Buch 2022

The Canonical Operator in Many-Particle Problems and Quantum Field Theory

Victor P. Maslov, Oleg Yu. Shvedov

Band 71 in dieser Reihe

In this monograph we study the problem of construction of asymptotic solutions of equations for functions whose number of arguments tends to infinity as the small parameter tends to zero. Such equations arise in statistical physics and in quantum theory of a large number of fi elds. We consider the problem of renormalization of quantum field theory in the Hamiltonian formalism, which encounters additional difficulties related to the Stückelberg divergences and the Haag theorem. Asymptotic methods for solving pseudodifferential equations with small parameter multiplying the derivatives, as well as the asymptotic methods developed in the present monograph for solving problems in statistical physics and quantum field theory, can be considered from a unified viewpoint if one introduces the notion of abstract canonical operator. The book can be of interest for researchers – specialists in asymptotic methods, statistical physics, and quantum fi eld theory as well as for graduate and undergraduate students of these specialities.

DOI:: https://doi.org/10.1515/9783110762709
ISBN:: 9783110762709
ISBN:: 9783110762389
ISBN:: 9783110762747
Fachgebiet:: Mathematik
Fachgebiet:: Analysis
Fachgebiet:: Angewandte Mathematik
Fachgebiet:: Physik
Fachgebiet:: Theoretische und mathematische Physik
Fachgebiet:: Computational Physics
Verlag:: De Gruyter

book: Graded Algebras in Algebraic Geometry

Buch 2022

Graded Algebras in Algebraic Geometry

Aron Simis, Zaqueu Ramos

Band 70 in dieser Reihe

The objective of this book is to look at certain commutative graded algebras that appear frequently in algebraic geometry. By studying classical constructions from geometry from the point of view of modern commutative algebra, this carefully-written book is a valuable source of information, offering a careful algebraic systematization and treatment of the problems at hand, and contributing to the study of the original geometric questions. In greater detail, the material covers aspects of rational maps (graph, degree, birationality, specialization, combinatorics), Cremona transformations, polar maps, Gauss maps, the geometry of Fitting ideals, tangent varieties, joins and secants, Aluffi algebras. The book includes sections of exercises to help put in practice the theoretic material instead of the mere complementary additions to the theory.

DOI:: https://doi.org/10.1515/9783110640694
ISBN:: 9783110640694
ISBN:: 9783110637540
ISBN:: 9783110637755
Fachgebiet:: Mathematik
Fachgebiet:: Algebra und Zahlentheorie
Fachgebiet:: Geometrie und Topologie
Verlag:: De Gruyter

book: Volume 3 Analytic Endomorphisms of the Riemann Sphere

Buch 2023

Volume 3 Analytic Endomorphisms of the Riemann Sphere

Mariusz Urbański, Mario Roy, Sara Munday

Teil des mehrbändigen Werks Non-Invertible Dynamical Systems

Band 69/3 in dieser Reihe

Complex dynamics is one of the most fascinating subjects of study and research in mathematics. This third volume in the series entitled Non-Invertible Dynamical Systems not only examines topological and analytical properties of the iteration of rational functions on the Riemann sphere (in particular, the Fatou and Julia sets) but also focuses on thermodynamic, ergodic and fractal properties of these functions (notably, equilibrium states, Bowen's formula and Sullivan’s conformal measures). This volume builds on the first two volumes in the series while simultaneously developing some methods and techniques specific to rational functions.

DOI:: https://doi.org/10.1515/9783110769876
ISBN:: 9783110769876
ISBN:: 9783110769845
ISBN:: 9783110769890
Fachgebiet:: Mathematik
Fachgebiet:: Analysis
Fachgebiet:: Differentialgleichungen und dynamische Systeme
Fachgebiet:: Geometrie und Topologie
Fachgebiet:: Wahrscheinlichkeitstheorie und Statistik
Verlag:: De Gruyter

book: Volume 2 Finer Thermodynamic Formalism – Distance Expanding Maps and Countable State Subshifts of Finite Type, Conformal GDMSs, Lasota-Yorke Maps and Fractal Geometry

Buch 2022

Volume 2 Finer Thermodynamic Formalism – Distance Expanding Maps and Countable State Subshifts of Finite Type, Conformal GDMSs, Lasota-Yorke Maps and Fractal Geometry

Mariusz Urbański, Mario Roy, Sara Munday

Teil des mehrbändigen Werks Non-Invertible Dynamical Systems

Band 69/2 in dieser Reihe

This book consists of three volumes. The first volume contains introductory accounts of topological dynamical systems, fi nite-state symbolic dynamics, distance expanding maps, and ergodic theory of metric dynamical systems acting on probability measure spaces, including metric entropy theory of Kolmogorov and Sinai. More advanced topics comprise infi nite ergodic theory, general thermodynamic formalism, topological entropy and pressure. Thermodynamic formalism of distance expanding maps and countable-alphabet subshifts of fi nite type, graph directed Markov systems, conformal expanding repellers, and Lasota-Yorke maps are treated in the second volume, which also contains a chapter on fractal geometry and its applications to conformal systems. Multifractal analysis and real analyticity of pressure are also covered. The third volume is devoted to the study of dynamics, ergodic theory, thermodynamic formalism and fractal geometry of rational functions of the Riemann sphere.

DOI:: https://doi.org/10.1515/9783110702699
ISBN:: 9783110702699
ISBN:: 9783110702736
ISBN:: 9783110700619
Fachgebiet:: Mathematik
Fachgebiet:: Analysis
Fachgebiet:: Differentialgleichungen und dynamische Systeme
Fachgebiet:: Geometrie und Topologie
Fachgebiet:: Wahrscheinlichkeitstheorie und Statistik
Verlag:: De Gruyter

book: Volume 1 Ergodic Theory – Finite and Infinite, Thermodynamic Formalism, Symbolic Dynamics and Distance Expanding Maps

Buch 2022

Volume 1 Ergodic Theory – Finite and Infinite, Thermodynamic Formalism, Symbolic Dynamics and Distance Expanding Maps

Mariusz Urbański, Mario Roy, Sara Munday

Teil des mehrbändigen Werks Non-Invertible Dynamical Systems

Band 69/1 in dieser Reihe

The book contains a detailed treatment of thermodynamic formalism on general compact metrizable spaces. Topological pressure, topological entropy, variational principle, and equilibrium states are presented in detail. Abstract ergodic theory is also given a significant attention. Ergodic theorems, ergodicity, and Kolmogorov-Sinai metric entropy are fully explored. Furthermore, the book gives the reader an opportunity to find rigorous presentation of thermodynamic formalism for distance expanding maps and, in particular, subshifts of finite type over a finite alphabet. It also provides a fairly complete treatment of subshifts of finite type over a countable alphabet. Transfer operators, Gibbs states and equilibrium states are, in this context, introduced and dealt with. Their relations are explored. All of this is applied to fractal geometry centered around various versions of Bowen’s formula in the context of expanding conformal repellors, limit sets of conformal iterated function systems and conformal graph directed Markov systems. A unique introduction to iteration of rational functions is given with emphasize on various phenomena caused by rationally indifferent periodic points. Also, a fairly full account of the classicaltheory of Shub’s expanding endomorphisms is given; it does not have a book presentation in English language mathematical literature.

DOI:: https://doi.org/10.1515/9783110702682
ISBN:: 9783110702682
ISBN:: 9783110702750
ISBN:: 9783110702644
Fachgebiet:: Mathematik
Fachgebiet:: Analysis
Fachgebiet:: Differentialgleichungen und dynamische Systeme
Fachgebiet:: Geometrie und Topologie
Fachgebiet:: Wahrscheinlichkeitstheorie und Statistik
Verlag:: De Gruyter

book: Identical Relations in Lie Algebras

Buch 2021

Identical Relations in Lie Algebras

Yuri Bahturin

Band 68 in dieser Reihe

This updated edition of a classic title studies identical relations in Lie algebras and also in other classes of algebras, a theory with over 40 years of development in which new methods and connections with other areas of mathematics have arisen. New topics covered include graded identities, identities of algebras with actions and coactions of various Hopf algebras, and the representation theory of the symmetric and general linear group.

DOI:: https://doi.org/10.1515/9783110566659
ISBN:: 9783110566659
ISBN:: 9783110565577
ISBN:: 9783110565706
Fachgebiet:: Mathematik
Fachgebiet:: Algebra und Zahlentheorie
Verlag:: De Gruyter

book: Complex Algebraic Foliations

Buch 2020

Complex Algebraic Foliations

Alcides Lins Neto, Bruno Scárdua

Band 67 in dieser Reihe

This book is a basic reference in the modern theory of holomorphic foliations, presenting the interplay between various aspects of the theory and utilizing methods from algebraic and complex geometry along with techniques from complex dynamics and several complex variables. The result is a solid introduction to the theory of foliations, covering basic concepts through modern results on the structure of foliations on complex projective spaces.

DOI:: https://doi.org/10.1515/9783110602050
ISBN:: 9783110602050
ISBN:: 9783110601077
ISBN:: 9783110594515
Fachgebiet:: Mathematik
Fachgebiet:: Algebra und Zahlentheorie
Fachgebiet:: Analysis
Fachgebiet:: Geometrie und Topologie
Verlag:: De Gruyter

book: Pre-Riesz Spaces

Buch 2019

Pre-Riesz Spaces

Anke Kalauch, Onno van Gaans

Band 66 in dieser Reihe

This monograph develops the theory of pre-Riesz spaces, which are the partially ordered vector spaces that embed order densely into Riesz spaces. Concepts from Riesz space theory such as disjointness, ideals, and bands are extended to pre-Riesz spaces. The analysis revolves around embedding techniques, including the Riesz completion and the functional representation. In the same spirit, norms and topologies on a pre-Riesz space and their extensions to the Riesz completion are examined. The generalized concepts are used to investigate disjointness preserving operators on pre-Riesz spaces and related notions.

The monograph presents recent results as well as being an accessible introduction to the theory of partially ordered vector spaces and positive operators.

Contents
A primer on ordered vector spaces
Embeddings, covers, and completions
Seminorms on pre-Riesz spaces
Disjointness, bands, and ideals in pre-Riesz spaces
Operators on pre-Riesz spaces

DOI:: https://doi.org/10.1515/9783110476293
ISBN:: 9783110476293
ISBN:: 9783110475395
ISBN:: 9783110475449
Fachgebiet:: Mathematik
Fachgebiet:: Analysis
Verlag:: De Gruyter

Buch 2018

Volume 6

Yakov G. Berkovich, Zvonimir Janko

Teil des mehrbändigen Werks Groups of Prime Power Order

Band 65 in dieser Reihe

This is the sixth volume of a comprehensive and elementary treatment of finite group theory. This volume contains many hundreds of original exercises (including solutions for the more difficult ones) and an extended list of about 1000 open problems. The current book is based on Volumes 1–5 and it is suitable for researchers and graduate students working in group theory.

DOI:: https://doi.org/10.1515/9783110533149
ISBN:: 9783110533149
ISBN:: 9783110530971
ISBN:: 9783110531008
Fachgebiet:: Mathematik
Fachgebiet:: Algebra und Zahlentheorie
Verlag:: De Gruyter

Buch 2019

Volume 2

Teil des mehrbändigen Werks Characters of Finite Groups

Band 64 in dieser Reihe

This updated edition of this classic book is devoted to ordinary representation theory and is addressed to finite group theorists intending to study and apply character theory. It contains many exercises and examples, and the list of problems contains a number of open questions.

DOI:: https://doi.org/10.1515/9783110224092
ISBN:: 9783110224092
ISBN:: 9783110384802
ISBN:: 9783110224085
Fachgebiet:: Mathematik
Fachgebiet:: Algebra und Zahlentheorie
Verlag:: De Gruyter

Buch 2018

Volume 1

Teil des mehrbändigen Werks Characters of Finite Groups

Band 63 in dieser Reihe

This updated edition of this classic book is devoted to ordinary representation theory and is addressed to finite group theorists intending to study and apply character theory. It contains many exercises and examples, and the list of problems contains a number of open questions.

DOI:: https://doi.org/10.1515/9783110224078
ISBN:: 9783110224078
ISBN:: 9783110390544
ISBN:: 9783110224061
Fachgebiet:: Mathematik
Fachgebiet:: Algebra und Zahlentheorie
Verlag:: De Gruyter

Buch 2016

Volume 5

Yakov G. Berkovich, Zvonimir Janko

Teil des mehrbändigen Werks Groups of Prime Power Order

Band 62 in dieser Reihe

This is the fifth volume of a comprehensive and elementary treatment of finite p-group theory. Topics covered in this volume include theory of linear algebras and Lie algebras.

The book contains many dozens of original exercises (with difficult exercises being solved) and a list of about 900 research problems and themes.

DOI:: https://doi.org/10.1515/9783110295351
ISBN:: 9783110295351
ISBN:: 9783110389043
ISBN:: 9783110295344
Fachgebiet:: Mathematik
Fachgebiet:: Algebra und Zahlentheorie
Verlag:: De Gruyter

Buch 2016

Volume 4

Yakov G. Berkovich, Zvonimir Janko

Teil des mehrbändigen Werks Groups of Prime Power Order

Band 61 in dieser Reihe

This is the fourth volume of a comprehensive and elementary treatment of finite p-group theory. As in the previous volumes, minimal nonabelian p-groups play an important role. Topics covered in this volume include:

subgroup structure of metacyclic p-groups
Ishikawa’s theorem on p-groups with two sizes of conjugate classes
p-central p-groups
theorem of Kegel on nilpotence of H p-groups
partitions of p-groups
characterizations of Dedekindian groups
norm of p-groups
p-groups with 2-uniserial subgroups of small order

The book also contains hundreds of original exercises and solutions and a comprehensive list of more than 500 open problems. This work is suitable for researchers and graduate students with a modest background in algebra.

DOI:: https://doi.org/10.1515/9783110281477
ISBN:: 9783110281477
ISBN:: 9783110381559
ISBN:: 9783110281453
Fachgebiet:: Mathematik
Fachgebiet:: Algebra und Zahlentheorie
Verlag:: De Gruyter

book: The Elementary Theory of Groups

Buch 2014

The Elementary Theory of Groups

A Guide through the Proofs of the Tarski Conjectures

Benjamin Fine, Anthony Gaglione, Alexei Myasnikov, Gerhard Rosenberger, Dennis Spellman

Band 60 in dieser Reihe

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

DOI:: https://doi.org/10.1515/9783110342031
ISBN:: 978-3-11-034199-7
ISBN:: 978-3-11-034203-1
ISBN:: 9783110382570
Fachgebiet:: Mathematik
Fachgebiet:: Algebra und Zahlentheorie
Verlag:: De Gruyter

book: The d-bar Neumann Problem and Schrödinger Operators

Buch 2023

The d-bar Neumann Problem and Schrödinger Operators

Friedrich Haslinger

Band 59 in dieser Reihe

This book's subject lies in the nexus of partial differential equations, operator theory, and complex analysis. The spectral analysis of the complex Laplacian and the compactness of the d-bar-Neumann operator are primary topics.The revised 2nd edition explores updates to Schrödinger operators with magnetic fields and connections to the Segal Bargmann space (Fock space), to quantum mechanics, and the uncertainty principle.

DOI:: https://doi.org/10.1515/9783111182926
ISBN:: 9783111182926
ISBN:: 9783111182902
ISBN:: 9783111184678
Fachgebiet:: Mathematik
Fachgebiet:: Analysis
Fachgebiet:: Differentialgleichungen und dynamische Systeme
Verlag:: De Gruyter

book: Current Algebras on Riemann Surfaces

Buch 2012

Current Algebras on Riemann Surfaces

New Results and Applications

Oleg K. Sheinman

Band 58 in dieser Reihe

This monograph is an introduction into a new and fast developing field on the crossroads of infinite-dimensional Lie algebra theory and contemporary mathematical physics. It contains a self-consistent presentation of the theory of Krichever-Novikov algebras, Lax operator algebras, their interaction, representation theory, relations to moduli spaces of Riemann surfaces and holomorphic vector bundles on them, to Lax integrable systems, and conformal field theory.

For beginners, the book provides a short way to join in the investigations in these fields. For experts, it sums up the recent advances in the theory of almost graded infinite-dimensional Lie algebras and their applications.
The book may serve as a base for semester lecture courses on finite-dimensional integrable systems, conformal field theory, almost graded Lie algebras. Majority of results are presented for the first time in the form of monograph.

DOI:: https://doi.org/10.1515/9783110264524
ISBN:: 978-3-11-026396-1
ISBN:: 978-3-11-026452-4
Fachgebiet:: Mathematik
Fachgebiet:: Algebra und Zahlentheorie
Fachgebiet:: Physik
Fachgebiet:: Theoretische und mathematische Physik
Verlag:: De Gruyter

book: Volume III Completion of the Classification

Buch 2013

Volume III Completion of the Classification

Helmut Strade

Teil des mehrbändigen Werks Simple Lie Algebras over Fields of Positive Characteristic

Teil des mehrbändigen Werks Simple Lie Algebras over Fields of Positive Characteristic

Band 57 in dieser Reihe

The problem of classifying the finite-dimensional simple Lie algebras over fields of characteristic p > 0 is a long-standing one. Work on this question during the last 45 years has been directed by the Kostrikin–Shafarevich Conjecture of 1966, which states that over an algebraically closed field of characteristic p > 5 a finite-dimensional restricted simple Lie algebra is classical or of Cartan type. This conjecture was proved for p > 7 by Block and Wilson in 1988. The generalization of the Kostrikin–Shafarevich Conjecture for the general case of not necessarily restricted Lie algebras and p > 7 was announced in 1991 by Strade and Wilson and eventually proved by Strade in 1998. The final Block–Wilson–Strade–Premet Classification Theorem is a landmark result of modern mathematics and can be formulated as follows: Every finite-dimensional simple Lie algebra over an algebraically closed field of characteristic p > 3 is of classical, Cartan, or Melikian type.

In the three-volume book, the author is assembling the proof of the Classification Theorem with explanations and references. The goal is a state-of-the-art account on the structure and classification theory of Lie algebras over fields of positive characteristic leading to the forefront of current research in this field.

This is the last of three volumes. In this monograph the proof of the Classification Theorem presented in the first volume is concluded. It collects all the important results on the topic which can be found only in scattered scientific literature so far.

DOI:: https://doi.org/10.1515/9783110263015
ISBN:: 978-3-11-026298-8
ISBN:: 978-3-11-026301-5
Fachgebiet:: Mathematik
Fachgebiet:: Algebra und Zahlentheorie
Verlag:: De Gruyter

Buch 2011

Volume 3

Yakov Berkovich, Zvonimir Janko

Teil des mehrbändigen Werks Groups of Prime Power Order

Band 56 in dieser Reihe

This is the third volume of a comprehensive and elementary treatment of finite p-group theory. Topics covered in this volume:

impact of minimal nonabelian subgroups on the structure of p-groups,
classification of groups all of whose nonnormal subgroups have the same order,
degrees of irreducible characters of p-groups associated with finite algebras,
groups covered by few proper subgroups,
p-groups of element breadth 2 and subgroup breadth 1,
exact number of subgroups of given order in a metacyclic p-group,
soft subgroups,
p-groups with a maximal elementary abelian subgroup of order p²,
p-groups generated by certain minimal nonabelian subgroups,
p-groups in which certain nonabelian subgroups are 2-generator.

The book contains many dozens of original exercises (with difficult exercises being solved) and a list of about 900 research problems and themes.

DOI:: https://doi.org/10.1515/9783110254488
ISBN:: 9783110254488
ISBN:: 9783110207170
Fachgebiet:: Mathematik
Fachgebiet:: Algebra und Zahlentheorie
Verlag:: De Gruyter

book: Partial Differential Equations

Buch 2011

Partial Differential Equations

A unified Hilbert Space Approach

Rainer Picard, Des McGhee

Band 55 in dieser Reihe

This book presents a systematic approach to a solution theory for linear partial differential equations developed in a Hilbert space setting based on a Sobolev lattice structure, a simple extension of the well-established notion of a chain (or scale) of Hilbert spaces.

The focus on a Hilbert space setting (rather than on an apparently more general Banach space) is not a severe constraint, but rather a highly adaptable and suitable approach providing a more transparent framework for presenting the main issues in the development of a solution theory for partial differential equations.

In contrast to other texts on partial differential equations, which consider either specific equation types or apply a collection of tools for solving a variety of equations, this book takes a more global point of view by focusing on the issues involved in determining the appropriate functional analytic setting in which a solution theory can be naturally developed. Applications to many areas of mathematical physics are also presented.

The book aims to be largely self-contained. Full proofs to all but the most straightforward results are provided, keeping to a minimum references to other literature for essential material. It is therefore highly suitable as a resource for graduate courses and also for researchers, who will find new results for particular evolutionary systems from mathematical physics.

DOI:: https://doi.org/10.1515/9783110250275
ISBN:: 978-3-11-025026-8
ISBN:: 978-3-11-025027-5
Fachgebiet:: Mathematik
Fachgebiet:: Analysis
Verlag:: De Gruyter

book: Volume 2 Relative Homological Algebra

Buch 2011

Volume 2 Relative Homological Algebra

Teil des mehrbändigen Werks Relative Homological Algebra

Band 54 in dieser Reihe

This second volume deals with the relative homological algebra of complexes of modules and their applications. It is a concrete and easy introduction to the kind of homological algebra which has been developed in the last 50 years. The book serves as a bridge between the traditional texts on homological algebra and more advanced topics such as triangulated and derived categories or model category structures. It addresses to readers who have had a course in classical homological algebra, as well as to researchers.

DOI:: https://doi.org/10.1515/9783110215236
ISBN:: 978-3-11-021522-9
ISBN:: 978-3-11-021523-6
Fachgebiet:: Mathematik
Fachgebiet:: Algebra und Zahlentheorie
Verlag:: De Gruyter

book: Products of Finite Groups

Buch 2010

Products of Finite Groups

Adolfo Ballester-Bolinches, Ramon Esteban-Romero, Mohamed Asaad

Band 53 in dieser Reihe

The study of finite groups factorised as a product of two or more subgroups has become a subject of great interest during the last years with applications not only in group theory, but also in other areas like cryptography and coding theory. It has experienced a big impulse with the introduction of some permutability conditions. The aim of this book is to gather, order, and examine part of this material, including the latest advances made, give some new approach to some topics, and present some new subjects of research in the theory of finite factorised groups.

Some of the topics covered by this book include groups whose subnormal subgroups are normal, permutable, or Sylow-permutable, products of nilpotent groups, and an exhaustive structural study of totally and mutually permutable products of finite groups and their relation with classes of groups.

This monograph is mainly addressed to graduate students and senior researchers interested in the study of products and permutability of finite groups. A background in finite group theory and a basic knowledge of representation theory and classes of groups is recommended to follow it.

DOI:: https://doi.org/10.1515/9783110220612
ISBN:: 978-3-11-020417-9
ISBN:: 978-3-11-022061-2
Fachgebiet:: Mathematik
Fachgebiet:: Algebra und Zahlentheorie
Verlag:: De Gruyter

book: Stability Analysis of Impulsive Functional Differential Equations

Buch 2009

Stability Analysis of Impulsive Functional Differential Equations

Ivanka Stamova

Band 52 in dieser Reihe

This book is devoted to impulsive functional differential equations which are a natural generalization of impulsive ordinary differential equations (without delay) and of functional differential equations (without impulses). At the present time the qualitative theory of such equations is under rapid development.

After a presentation of the fundamental theory of existence, uniqueness and continuability of solutions, a systematic development of stability theory for that class of problems is given which makes the book unique. It addresses to a wide audience such as mathematicians, applied researches and practitioners.

DOI:: https://doi.org/10.1515/9783110221824
ISBN:: 978-3-11-022181-7
ISBN:: 978-3-11-022182-4
Fachgebiet:: Mathematik
Fachgebiet:: Analysis
Fachgebiet:: Wahrscheinlichkeitstheorie und Statistik
Verlag:: De Gruyter

book: State Observers for Linear Systems with Uncertainty

Buch 2009

State Observers for Linear Systems with Uncertainty

Sergey K. Korovin, Vasily V. Fomichev

Band 51 in dieser Reihe

This book presents the basic concepts and recent developments of linear control problems with perturbations. The presentation concerns both continuous and discrete dynamical systems. It is self-contained and illustrated by numerous examples.

From the contents:

Notion of state observers
Observability
Observers of full-phase vectors for fully determined linear systems
Functional observers for fully determined linear systems
Asymptotic observers for linear systems with uncertainty
Observers for bilinear and discrete systems

DOI:: https://doi.org/10.1515/9783110218138
ISBN:: 978-3-11-021812-1
ISBN:: 978-3-11-021813-8
Fachgebiet:: Mathematik
Fachgebiet:: Analysis
Fachgebiet:: Wahrscheinlichkeitstheorie und Statistik
Verlag:: De Gruyter

book: Ultrafilters and Topologies on Groups

Buch 2011

Ultrafilters and Topologies on Groups

Yevhen Zelenyuk

Band 50 in dieser Reihe

This book presents the relationship between ultrafilters and topologies on groups. It shows how ultrafilters are used in constructing topologies on groups with extremal properties and how topologies on groups serve in deriving algebraic results about ultrafilters.

The contents of the book fall naturally into three parts. The first, comprising Chapters 1 through 5, introduces to topological groups and ultrafilters insofar as the semigroup operation on ultrafilters is not required. Constructions of some important topological groups are given. In particular, that of an extremally disconnected topological group based on a Ramsey ultrafilter. Also one shows that every infinite group admits a nondiscrete zero-dimensional topology in which all translations and the inversion are continuous.

In the second part, Chapters 6 through 9, the Stone-Cêch compactification βG of a discrete group G is studied. For this, a special technique based on the concepts of a local left group and a local homomorphism is developed. One proves that if G is a countable torsion free group, then βG contains no nontrivial finite groups. Also the ideal structure of βG is investigated. In particular, one shows that for every infinite Abelian group G, βG contains 2²|G| minimal right ideals.

In the third part, using the semigroup βG, almost maximal topological and left topological groups are constructed and their ultrafilter semigroups are examined. Projectives in the category of finite semigroups are characterized. Also one shows that every infinite Abelian group with finitely many elements of order 2 is absolutely ω-resolvable, and consequently, can be partitioned into ω subsets such that every coset modulo infinite subgroup meets each subset of the partition.

The book concludes with a list of open problems in the field. Some familiarity with set theory, algebra and topology is presupposed. But in general, the book is almost self-contained. It is aimed at graduate students and researchers working in topological algebra and adjacent areas.

DOI:: https://doi.org/10.1515/9783110213225
ISBN:: 978-3-11-020422-3
ISBN:: 978-3-11-021322-5
Fachgebiet:: Mathematik
Fachgebiet:: Algebra und Zahlentheorie
Fachgebiet:: Geometrie und Topologie
Verlag:: De Gruyter

book: Applied Algebraic Dynamics

Buch 2009

Applied Algebraic Dynamics

Vladimir Anashin, Andrei Khrennikov

Band 49 in dieser Reihe

This monograph presents recent developments of the theory of algebraic dynamical systems and their applications to computer sciences, cryptography, cognitive sciences, psychology, image analysis, and numerical simulations. The most important mathematical results presented in this book are in the fields of ergodicity, p-adic numbers, and noncommutative groups.

For students and researchers working on the theory of dynamical systems, algebra, number theory, measure theory, computer sciences, cryptography, and image analysis.

DOI:: https://doi.org/10.1515/9783110203011
ISBN:: 978-3-11-020300-4
ISBN:: 978-3-11-020301-1
Fachgebiet:: Mathematik
Fachgebiet:: Algebra und Zahlentheorie
Fachgebiet:: Analysis
Fachgebiet:: Angewandte Mathematik
Verlag:: De Gruyter

book: Stochastic Dynamics and Boltzmann Hierarchy

Buch 2009

Stochastic Dynamics and Boltzmann Hierarchy

Dmitri Ya. Petrina

Band 48 in dieser Reihe

The monograph is devoted to one of the most important trends in contemporary mathematical physics, the investigation of evolution equations of many-particle systems of statistical mechanics. The book systematizes rigorous results obtained in this field in recent years, and it presents contemporary methods for the investigation of evolution equations of infinite-particle systems.

The book is intended for experts in statistical physics, mathematical physics, and probability theory and for students of universities specialized in mathematics and physics.

DOI:: https://doi.org/10.1515/9783110213201
ISBN:: 978-3-11-020804-7
ISBN:: 978-3-11-021320-1
Fachgebiet:: Mathematik
Fachgebiet:: Analysis
Fachgebiet:: Wahrscheinlichkeitstheorie und Statistik
Verlag:: De Gruyter

Buch 2008

Volume 2

Yakov Berkovich, Zvonimir Janko

Teil des mehrbändigen Werks Groups of Prime Power Order

Band 47 in dieser Reihe

This is the second of three volumes devoted to elementary finite p-group theory. Similar to the first volume, hundreds of important results are analyzed and, in many cases, simplified. Important topics presented in this monograph include: (a) classification of p-groups all of whose cyclic subgroups of composite orders are normal, (b) classification of 2-groups with exactly three involutions, (c) two proofs of Ward's theorem on quaternion-free groups, (d) 2-groups with small centralizers of an involution, (e) classification of 2-groups with exactly four cyclic subgroups of order 2ⁿ > 2, (f) two new proofs of Blackburn's theorem on minimal nonmetacyclic groups, (g) classification of p-groups all of whose subgroups of index p² are abelian, (h) classification of 2-groups all of whose minimal nonabelian subgroups have order 8, (i) p-groups with cyclic subgroups of index p² are classified.

This volume contains hundreds of original exercises (with all difficult exercises being solved) and an extended list of about 700 open problems. The book is based on Volume 1, and it is suitable for researchers and graduate students of mathematics with a modest background on algebra.

DOI:: https://doi.org/10.1515/9783110208238
ISBN:: 978-3-11-020419-3
ISBN:: 978-3-11-020823-8
Fachgebiet:: Mathematik
Fachgebiet:: Algebra und Zahlentheorie
Verlag:: De Gruyter

Buch 2008

Volume 1

Yakov Berkovich

Teil des mehrbändigen Werks Groups of Prime Power Order

Band 46 in dieser Reihe

This is the first of three volumes of a comprehensive and elementary treatment of finite p-group theory. Topics covered in this monograph include: (a) counting of subgroups, with almost all main counting theorems being proved, (b) regular p-groups and regularity criteria, (c) p-groups of maximal class and their numerous characterizations, (d) characters of p-groups, (e) p-groups with large Schur multiplier and commutator subgroups, (f) (p‒1)-admissible Hall chains in normal subgroups, (g) powerful p-groups, (h) automorphisms of p-groups, (i) p-groups all of whose nonnormal subgroups are cyclic, (j) Alperin's problem on abelian subgroups of small index.

The book is suitable for researchers and graduate students of mathematics with a modest background on algebra. It also contains hundreds of original exercises (with difficult exercises being solved) and a comprehensive list of about 700 open problems.

DOI:: https://doi.org/10.1515/9783110208221
ISBN:: 978-3-11-020418-6
ISBN:: 978-3-11-020822-1
Fachgebiet:: Mathematik
Fachgebiet:: Algebra und Zahlentheorie
Verlag:: De Gruyter

book: Distribution Theory of Algebraic Numbers

Buch 2008

Distribution Theory of Algebraic Numbers

Pei-Chu Hu, Chung-Chun Yang

Band 45 in dieser Reihe

The book timely surveys new research results and related developments in Diophantine approximation, a division of number theory which deals with the approximation of real numbers by rational numbers. The book is appended with a list of challenging open problems and a comprehensive list of references.

From the contents: Field extensions • Algebraic numbers • Algebraic geometry • Height functions • The abc-conjecture • Roth's theorem • Subspace theorems • Vojta's conjectures • L-functions.

DOI:: https://doi.org/10.1515/9783110208269
ISBN:: 978-3-11-020536-7
ISBN:: 978-3-11-020826-9
Fachgebiet:: Mathematik
Fachgebiet:: Algebra und Zahlentheorie
Verlag:: De Gruyter

book: Quasi-Stationary Phenomena in Nonlinearly Perturbed Stochastic Systems

Buch 2008

Quasi-Stationary Phenomena in Nonlinearly Perturbed Stochastic Systems

Mats Gyllenberg, Dmitrii S. Silvestrov

Band 44 in dieser Reihe

The book is devoted to studies of quasi-stationary phenomena in nonlinearly perturbed stochastic systems. New methods of asymptotic analysis for nonlinearly perturbed stochastic processes based on new types of asymptotic expansions for perturbed renewal equation and recurrence algorithms for construction of asymptotic expansions for Markov type processes with absorption are presented. Asymptotic expansions are given in mixed ergodic (for processes) and large deviation theorems (for absorption times) for nonlinearly perturbed regenerative processes, semi-Markov processes, and Markov chains. Applications to analysis of quasi-stationary phenomena in nonlinearly perturbed queueing systems, population dynamics and epidemic models, and for risk processes are presented. The book also contains an extended bibliography of works in the area. It is an essential reference for theoretical and applied researchers in the field of stochastic processes and their applications and may be also useful for doctoral and advanced undergraduate students.

DOI:: https://doi.org/10.1515/9783110208252
ISBN:: 978-3-11-020437-7
ISBN:: 978-3-11-020825-2
Fachgebiet:: Mathematik
Fachgebiet:: Wahrscheinlichkeitstheorie und Statistik
Verlag:: De Gruyter

book: Modules over Discrete Valuation Rings

Buch 2018

Modules over Discrete Valuation Rings

Piotr A. Krylov, Askar A. Tuganbaev

Band 43 in dieser Reihe

This book provides the first systematic treatment of modules over discrete valuation domains, which play an important role in various areas of algebra, especially in commutative algebra. Many important results representing the state of the art are presented in the text along with interesting open problems. This updated edition presents new approaches on p-adic integers and modules, and on the determinability of a module by its automorphism group.

Contents
Preliminaries
Basic facts
Endomorphism rings of divisible and complete modules
Representation of rings by endomorphism rings
Torsion-free modules
Mixed modules
Determinity of modules by their endomorphism rings
Modules with many endomorphisms or automorphisms

DOI:: https://doi.org/10.1515/9783110611144
ISBN:: 9783110611144
ISBN:: 9783110609776
ISBN:: 9783110609851
Fachgebiet:: Mathematik
Fachgebiet:: Algebra und Zahlentheorie
Verlag:: De Gruyter

book: Modules over Discrete Valuation Domains

Buch 2008

Modules over Discrete Valuation Domains

Piotr A. Krylov, Askar A. Tuganbaev

Band 43 in dieser Reihe

This book provides the first systematic treatment of modules over discrete valuation domains which plays an important role in various areas of algebra, especially in commutative algebra. Many important results representing the state of the art are presented in the text which is supplemented by exercises and interesting open problems.

An important contribution to commutative algebra.

DOI:: https://doi.org/10.1515/9783110205787
ISBN:: 978-3-11-020053-9
ISBN:: 978-3-11-020578-7
Fachgebiet:: Mathematik
Fachgebiet:: Algebra und Zahlentheorie
Verlag:: De Gruyter

book: Volume 2 Classifying the Absolute Toral Rank Two Case

Buch 2017

Volume 2 Classifying the Absolute Toral Rank Two Case

Helmut Strade

Teil des mehrbändigen Werks Simple Lie Algebras over Fields of Positive Characteristic

Band 42 in dieser Reihe

The problem of classifying the finite dimensional simple Lie algebras over fields of characteristic p > 0 is a long standing one. Work on this question has been directed by the Kostrikin Shafarevich Conjecture of 1966, which states that over an algebraically closed field of characteristic p > 5 a finite dimensional restricted simple Lie algebra is classical or of Cartan type. This conjecture was proved for p > 7 by Block and Wilson in 1988. The generalization of the Kostrikin-Shafarevich Conjecture for the general case of not necessarily restricted Lie algebras and p > 7 was announced in 1991 by Strade and Wilson and eventually proved by Strade in 1998. The final Block-Wilson-Strade-Premet Classification Theorem is a landmark result of modern mathematics and can be formulated as follows: Every simple finite dimensional simple Lie algebra over an algebraically closed field of characteristic p > 3 is of classical, Cartan, or Melikian type.

This is the second part of a three-volume book about the classification of the simple Lie algebras over algebraically closed fields of characteristic > 3. The first volume contains the methods, examples and a first classification result. This second volume presents insight in the structure of tori of Hamiltonian and Melikian algebras. Based on sandwich element methods due to A. I. Kostrikin and A. A. Premet and the investigations of filtered and graded Lie algebras, a complete proof for the classification of absolute toral rank 2 simple Lie algebras over algebraically closed fields of characteristic > 3 is given.

Contents

Tori in Hamiltonian and Melikian algebras
1-sections
Sandwich elements and rigid tori
Towards graded algebras
The toral rank 2 case

DOI:: https://doi.org/10.1515/9783110517606
ISBN:: 9783110517606
ISBN:: 9783110516760
ISBN:: 9783110516890
Fachgebiet:: Mathematik
Fachgebiet:: Algebra und Zahlentheorie
Verlag:: De Gruyter

book: Volume II Classifying the Absolute Toral Rank Two Case

Buch 2009

Volume II Classifying the Absolute Toral Rank Two Case

Helmut Strade

Teil des mehrbändigen Werks Simple Lie Algebras over Fields of Positive Characteristic

Band 42 in dieser Reihe

The problem of classifying the finite-dimensional simple Lie algebras over fields of characteristic p > 0 is a long-standing one. Work on this question during the last 45 years has been directed by the Kostrikin–Shafarevich Conjecture of 1966, which states that over an algebraically closed field of characteristic p > 5 a finite-dimensional restricted simple Lie algebra is classical or of Cartan type. This conjecture was proved for p > 7 by Block and Wilson in 1988. The generalization of the Kostrikin–Shafarevich Conjecture for the general case of not necessarily restricted Lie algebras and p > 7 was announced in 1991 by Strade and Wilson and eventually proved by Strade in 1998. The final Block–Wilson–Strade–Premet Classification Theorem is a landmark result of modern mathematics and can be formulated as follows: Every finite-dimensional simple Lie algebra over an algebraically closed field of characteristic p > 3 is of classical, Cartan, or Melikian type.

In the three-volume book, the author is assembling the proof of the Classification Theorem with explanations and references. The goal is a state-of-the-art account on the structure and classification theory of Lie algebras over fields of positive characteristic leading to the forefront of current research in this field.

This is the second part of the three-volume book about the classification of the simple Lie algebras over algebraically closed fields of characteristics > 3. The first volume contains the methods, examples, and a first classification result. This second volume presents insight in the structure of tori of Hamiltonian and Melikian algebras. Based on sandwich element methods due to Aleksei. I. Kostrikin and Alexander A. Premet and the investigation of absolute toral rank 2 simple Lie algebras over algebraically closed fields of characteristics > 3 is given.

DOI:: https://doi.org/10.1515/9783110203059
ISBN:: 978-3-11-019701-3
ISBN:: 978-3-11-020305-9
Fachgebiet:: Mathematik
Fachgebiet:: Algebra und Zahlentheorie
Verlag:: De Gruyter

book: Approximations and Endomorphism Algebras of Modules

Buch 2012

Approximations and Endomorphism Algebras of Modules

Volume 1 – Approximations / Volume 2 – Predictions

Rüdiger Göbel, Jan Trlifaj

Band 41 in dieser Reihe

This second, revised and substantially extended edition of Approximations and Endomorphism Algebras of Modules reflects both the depth and the width of recent developments in the area since the first edition appeared in 2006. The new division of the monograph into two volumes roughly corresponds to its two central topics, approximation theory (Volume 1) and realization theorems for modules (Volume 2).

It is a widely accepted fact that the category of all modules over a general associative ring is too complex to admit classification. Unless the ring is of finite representation type we must limit attempts at classification to some restricted subcategories of modules. The wild character of the category of all modules, or of one of its subcategories C, is often indicated by the presence of a realization theorem, that is, by the fact that any reasonable algebra is isomorphic to the endomorphism algebra of a module from C. This results in the existence of pathological direct sum decompositions, and these are generally viewed as obstacles to classification. In order to overcome this problem, the approximation theory of modules has been developed. The idea here is to select suitable subcategories C whose modules can be classified, and then to approximate arbitrary modules by those from C. These approximations are neither unique nor functorial in general, but there is a rich supply available appropriate to the requirements of various particular applications.

The authors bring the two theories together. The first volume, Approximations, sets the scene in Part I by introducing the main classes of modules relevant here: the S-complete, pure-injective, Mittag-Leffler, and slender modules. Parts II and III of the first volume develop the key methods of approximation theory. Some of the recent applications to the structure of modules are also presented here, notably for tilting, cotilting, Baer, and Mittag-Leffler modules. In the second volume, Predictions, further basic instruments are introduced: the prediction principles, and their applications to proving realization theorems. Moreover, tools are developed there for answering problems motivated in algebraic topology. The authors concentrate on the impossibility of classification for modules over general rings. The wild character of many categories C of modules is documented here by the realization theorems that represent critical R-algebras over commutative rings R as endomorphism algebras of modules from C.

The monograph starts from basic facts and gradually develops the theory towards its present frontiers. It is suitable both for graduate students interested in algebra and for experts in module and representation theory.

DOI:: https://doi.org/10.1515/9783110218114
ISBN:: 978-3-11-021810-7
ISBN:: 978-3-11-021811-4
Fachgebiet:: Mathematik
Fachgebiet:: Algebra und Zahlentheorie
Verlag:: De Gruyter

book: Approximations and Endomorphism Algebras of Modules

Buch 2006

Approximations and Endomorphism Algebras of Modules

Rüdiger Göbel, Jan Trlifaj

Band 41 in dieser Reihe

The category of all modules over a general associative ring is too complex to admit any reasonable classification. Thus, unless the ring is of finite representation type, one must limit attempts at classification to some restricted subcategories of modules.

The wild character of the category of all modules, or of one of its subcategories C is often indicated by the presence of a realization theorem, that is, by the fact that any reasonable algebra is isomorphic to the endomorphism algebra of a module from C. This results in the existence of pathological direct sum decompositions and these are generally viewed as obstacles to the classification. Realization theorems have thus become important indicators of the non-classification theory of modules.

In order to overcome this problem, approximation theory of modules has been developed over the past few decades. The idea here is to select suitable subcategories C whose modules can be classified, and then to approximate arbitrary modules by ones from C. These approximations are neither unique nor functorial in general, but there is always a rich supply available appropriate to the requirements of various particular applications. Thus approximation theory has developed into an important part of the classification theory of modules.

In this monograph the two methods are brought together. First the approximation theory of modules is developed and some of its recent applications, notably to infinite dimensional tilting theory, are presented. Then some prediction principles from set theory are introduced and these become the principal tools in the establishment of appropriate realization theorems.

The monograph starts from basic facts and gradually develops the theory towards its present frontiers. It is suitable both for graduate students interested in algebra and for experts in module and representation theory.

DOI:: https://doi.org/10.1515/9783110199727
ISBN:: 978-3-11-011079-1
ISBN:: 978-3-11-019972-7
Fachgebiet:: Mathematik
Fachgebiet:: Algebra und Zahlentheorie
Verlag:: De Gruyter

book: Embedding Problems in Symplectic Geometry

Buch 2005

Embedding Problems in Symplectic Geometry

Felix Schlenk

Band 40 in dieser Reihe

Symplectic geometry is the geometry underlying Hamiltonian dynamics, and symplectic mappings arise as time-1-maps of Hamiltonian flows. The spectacular rigidity phenomena for symplectic mappings discovered in the last two decades show that certain things cannot be done by a symplectic mapping. For instance, Gromov's famous "non-squeezing'' theorem states that one cannot map a ball into a thinner cylinder by a symplectic embedding. The aim of this book is to show that certain other things can be done by symplectic mappings. This is achieved by various elementary and explicit symplectic embedding constructions, such as "folding", "wrapping'', and "lifting''. These constructions are carried out in detail and are used to solve some specific symplectic embedding problems.

The exposition is self-contained and addressed to students and researchers interested in geometry or dynamics.

DOI:: https://doi.org/10.1515/9783110199697
ISBN:: 978-3-11-017876-0
ISBN:: 978-3-11-019969-7
Fachgebiet:: Mathematik
Fachgebiet:: Geometrie und Topologie
Verlag:: De Gruyter

book: Trigonometric Sums in Number Theory and Analysis

Buch 2004

Trigonometric Sums in Number Theory and Analysis

Gennady I. Arkhipov, Vladimir N. Chubarikov, Anatoly A. Karatsuba, Maria Shishkova

Band 39 in dieser Reihe

Das Buch stellt die von den Autoren entwickelte Theorie der mehrfachen trigonometrischen Summen vor. Einem integrierten Ansatz folgend, erhalten die Autoren Abschätzungen für diese Summen, vergleichbar den klassischen Abschätzungen von I. M. Vinogradov, und verwenden diese, um verschiedene Probleme der analytischen Zahlentheorie zu lösen. Sie untersuchen trigonometrische Integrale, die häufig in der Physik, der mathematischen Statistik und der Analysis anzutreffen sind, und präsentieren rein arithmetische Ergebnisse, die die Lösbarkeit von Gleichungen in ganzen Zahlen betreffen.

DOI:: https://doi.org/10.1515/9783110197983
ISBN:: 978-3-11-016266-0
ISBN:: 978-3-11-019798-3
Fachgebiet:: Mathematik
Fachgebiet:: Algebra und Zahlentheorie
Fachgebiet:: Analysis
Verlag:: De Gruyter

book: Volume 1 Structure Theory

Buch 2017

Volume 1 Structure Theory

Helmut Strade

Teil des mehrbändigen Werks Simple Lie Algebras over Fields of Positive Characteristic

Band 38 in dieser Reihe

The problem of classifying the finite dimensional simple Lie algebras over fields of characteristic p > 0 is a long-standing one. Work on this question has been directed by the Kostrikin-Shafarevich Conjecture of 1966, which states that over an algebraically closed field of characteristic p > 5 a finite dimensional restricted simple Lie algebra is classical or of Cartan type. This conjecture was proved for p > 7 by Block and Wilson in 1988. The generalization of the Kostrikin-Shafarevich Conjecture for the general case of not necessarily restricted Lie algebras and p > 7 was announced in 1991 by Strade and Wilson and eventually proved by Strade in 1998. The final Block-Wilson-Strade-Premet Classification Theorem is a landmark result of modern mathematics and can be formulated as follows: Every simple finite dimensional simple Lie algebra over an algebraically closed field of characteristic p > 3 is of classical, Cartan, or Melikian type.

In the three-volume book, the author is assembling the proof of the Classification Theorem with explanations and references. The goal is a state-of-the-art account on the structure and classification theory of Lie algebras over fields of positive characteristic.

This first volume is devoted to preparing the ground for the classification work to be performed in the second and third volumes. The concise presentation of the general theory underlying the subject matter and the presentation of classification results on a subclass of the simple Lie algebras for all odd primes will make this volume an invaluable source and reference for all research mathematicians and advanced graduate students in algebra. The second edition is corrected.

Contents

Toral subalgebras in p-envelopes
Lie algebras of special derivations
Derivation simple algebras and modules
Simple Lie algebras
Recognition theorems
The isomorphism problem
Structure of simple Lie algebras
Pairings of induced modules
Toral rank 1 Lie algebras

DOI:: https://doi.org/10.1515/9783110515442
ISBN:: 9783110515442
ISBN:: 9783110515169
ISBN:: 9783110515237
Fachgebiet:: Mathematik
Fachgebiet:: Algebra und Zahlentheorie
Verlag:: De Gruyter

book: Volume I Structure Theory

Buch 2004

Volume I Structure Theory

Helmut Strade

Teil des mehrbändigen Werks Simple Lie Algebras over Fields of Positive Characteristic

Band 38 in dieser Reihe

The problem of classifying the finite-dimensional simple Lie algebras over fields of characteristic p > 0 is a long-standing one. Work on this question during the last 45 years has been directed by the Kostrikin–Shafarevich Conjecture of 1966, which states that over an algebraically closed field of characteristic p > 5 a finite-dimensional restricted simple Lie algebra is classical or of Cartan type. This conjecture was proved for p > 7 by Block and Wilson in 1988. The generalization of the Kostrikin–Shafarevich Conjecture for the general case of not necessarily restricted Lie algebras and p > 7 was announced in 1991 by Strade and Wilson and eventually proved by Strade in 1998. The final Block–Wilson–Strade–Premet Classification Theorem is a landmark result of modern mathematics and can be formulated as follows: Every finite-dimensional simple Lie algebra over an algebraically closed field of characteristic p > 3 is of classical, Cartan, or Melikian type.

In the three-volume book, the author is assembling the proof of the Classification Theorem with explanations and references. The goal is a state-of-the-art account on the structure and classification theory of Lie algebras over fields of positive characteristic leading to the forefront of current research in this field.

This first volume is devoted to preparing the ground for the classification work to be performed in the second and third volume. The concise presentation of the general theory underlying the subject matter and the presentation of classification results on a subclass of the simple Lie algebras for all odd primes make this volume an invaluable source and reference for all research mathematicians and advanced graduate students in albegra.

DOI:: https://doi.org/10.1515/9783110197945
ISBN:: 978-3-11-014211-2
ISBN:: 978-3-11-019794-5
Fachgebiet:: Mathematik
Fachgebiet:: Algebra und Zahlentheorie
Verlag:: De Gruyter

book: Error Calculus for Finance and Physics

Buch 2003

Error Calculus for Finance and Physics

The Language of Dirichlet Forms

Nicolas Bouleau

Band 37 in dieser Reihe

Many recent advances in modelling within the applied sciences and engineering have focused on the increasing importance of sensitivity analyses. For a given physical, financial or environmental model, increased emphasis is now placed on assessing the consequences of changes in model outputs that result from small changes or errors in both the hypotheses and parameters. The approach proposed in this book is entirely new and features two main characteristics. Even when extremely small, errors possess biases and variances. The methods presented here are able, thanks to a specific differential calculus, to provide information about the correlation between errors in different parameters of the model, as well as information about the biases introduced by non-linearity. The approach makes use of very powerful mathematical tools (Dirichlet forms), which allow one to deal with errors in infinite dimensional spaces, such as spaces of functions or stochastic processes. The method is therefore applicable to non-elementary models along the lines of those encountered in modern physics and finance. This text has been drawn from presentations of research done over the past ten years and that is still ongoing. The work was presented in conjunction with a course taught jointly at the Universities of Paris 1 and Paris 6. The book is intended for students, researchers and engineers with good knowledge in probability theory.

DOI:: https://doi.org/10.1515/9783110199291
ISBN:: 978-3-11-018036-7
ISBN:: 978-3-11-019929-1
Fachgebiet:: Wirtschaftswissenschaften
Fachgebiet:: Mathematik und Statistik für Ökonomen
Fachgebiet:: Mathematik
Fachgebiet:: Mathematik
Fachgebiet:: Wahrscheinlichkeitstheorie und Statistik
Fachgebiet:: Angewandte Mathematik
Fachgebiet:: Physik
Fachgebiet:: Theoretische und mathematische Physik
Verlag:: De Gruyter

book: Automatic Sequences

Buch 2003

Automatic Sequences

von Friedrich Haeseler

Band 36 in dieser Reihe

Automatic sequences are sequences which are produced by a finite automaton. Although they are not random they may look as being random. They are complicated, in the sense of not being not ultimately periodic, they may look rather complicated, in the sense that it may not be easy to name the rule by which the sequence is generated, however there exists a rule which generates the sequence. The concept automatic sequences has special applications in algebra, number theory, finite automata and formal languages, combinatorics on words.

The text deals with different aspects of automatic sequences, in particular:
· a general introduction to automatic sequences
· the basic (combinatorial) properties of automatic sequences
· the algebraic approach to automatic sequences
· geometric objects related to automatic sequences.

DOI:: https://doi.org/10.1515/9783110197969
ISBN:: 978-3-11-015629-4
ISBN:: 978-3-11-019796-9
Fachgebiet:: Mathematik
Fachgebiet:: Algebra und Zahlentheorie
Fachgebiet:: Diskrete Mathematik
Fachgebiet:: Angewandte Mathematik
Verlag:: De Gruyter

book: Loops in Group Theory and Lie Theory

Buch 2002

Loops in Group Theory and Lie Theory

Péter T. Nagy, Karl Strambach

Band 35 in dieser Reihe

In this book the theory of binary systems is considered as a part of group theory and, in particular, within the framework of Lie groups. The novelty is the consequent treatment of topological and differentiable loops as topological and differentiable sections in Lie groups. The interplay of methods and tools from group theory, differential geometry and topology, symmetric spaces, topological geometry, and the theory of foliations is what gives a special flavour to the results presented in this book. It is the first monograph devoted to the study of global loops. So far books on differentiable loops deal with local loops only. This theory can only be used partially for the theory of global loops since non-associative local structures have, in general, no global forms.

The text is addressed to researchers in non-associative algebra and foundations of geometry. It should prove enlightening to a broad range of readers, including mathematicians working in group theory, the theory of Lie groups, in differential and topological geometry, and in algebraic groups.

The authors have produced a text that is suitable not only for a graduate course, but also for selfstudy in the subjectby interested graduate students. Moreover, the material presented can be used for lectures and seminars in algebra, topological algebra and geometry.

DOI:: https://doi.org/10.1515/9783110900583
ISBN:: 978-3-11-017010-8
ISBN:: 978-3-11-090058-3
Fachgebiet:: Mathematik
Fachgebiet:: Algebra und Zahlentheorie
Fachgebiet:: Geometrie und Topologie
Verlag:: De Gruyter

book: Extension of Holomorphic Functions

Buch 2020

Extension of Holomorphic Functions

Marek Jarnicki, Peter Pflug

Band 34 in dieser Reihe

This second extended edition of the classic reference on the extension problem of holomorphic functions in pluricomplex analysis contains a wealth of additional material, organized under the original chapter structure, and covers in a self-contained way all new and recent developments and theorems that appeared since the publication of the first edition about twenty years ago.

DOI:: https://doi.org/10.1515/9783110630275
ISBN:: 9783110630275
ISBN:: 9783110627664
ISBN:: 9783110627695
Fachgebiet:: Mathematik
Fachgebiet:: Analysis
Verlag:: De Gruyter

book: Extension of Holomorphic Functions

Buch 2000

Extension of Holomorphic Functions

Marek Jarnicki, Peter Pflug

Band 34 in dieser Reihe

DOI:: https://doi.org/10.1515/9783110809787
ISBN:: 978-3-11-015363-7
ISBN:: 978-3-11-080978-7
Fachgebiet:: Mathematik
Fachgebiet:: Analysis
Verlag:: De Gruyter

book: Compositions of Quadratic Forms

Buch 2000

Compositions of Quadratic Forms

Daniel B. Shapiro

Band 33 in dieser Reihe

DOI:: https://doi.org/10.1515/9783110824834
ISBN:: 978-3-11-012629-7
ISBN:: 978-3-11-082483-4
Fachgebiet:: Mathematik
Fachgebiet:: Analysis
Verlag:: De Gruyter

book: Conformal Geometry of Discrete Groups and Manifolds

Buch 2000

Conformal Geometry of Discrete Groups and Manifolds

Boris N. Apanasov

Band 32 in dieser Reihe

DOI:: https://doi.org/10.1515/9783110808056
ISBN:: 978-3-11-014404-8
ISBN:: 978-3-11-080805-6
Fachgebiet:: Mathematik
Fachgebiet:: Geometrie und Topologie
Verlag:: De Gruyter

book: Nonlinear Wave Equations Perturbed by Viscous Terms

Buch 2000

Nonlinear Wave Equations Perturbed by Viscous Terms

Petr P. Mosolov, Viktor P. Maslov, Maria A. Shishkova

Band 31 in dieser Reihe

DOI:: https://doi.org/10.1515/9783110811902
ISBN:: 978-3-11-015282-1
ISBN:: 978-3-11-081190-2
Fachgebiet:: Mathematik
Fachgebiet:: Analysis
Fachgebiet:: Angewandte Mathematik
Verlag:: De Gruyter

book: Relative Homological Algebra

Buch 2000

Relative Homological Algebra

Edgar E. Enochs, Overtoun M. G. Jenda

Band 30 in dieser Reihe

DOI:: https://doi.org/10.1515/9783110803662
ISBN:: 978-3-11-016633-0
ISBN:: 978-3-11-080366-2
Fachgebiet:: Mathematik
Fachgebiet:: Algebra und Zahlentheorie
Verlag:: De Gruyter

book: Volume 1 Relative Homological Algebra

Buch 2011

Volume 1 Relative Homological Algebra

Teil des mehrbändigen Werks Relative Homological Algebra

Band 30 in dieser Reihe

This is the second revised edition of an introduction to contemporary relative homological algebra. It supplies important material essential to understand topics in algebra, algebraic geometry and algebraic topology. Each section comes with exercises providing practice problems for students as well as additional important results for specialists.

In this new edition the authors have added well-known additional material in the first three chapters, and added new material that was not available at the time the original edition was published. In particular, the major changes are the following:

Chapter 1: Section 1.2 has been rewritten to clarify basic notions for the beginner, and this has necessitated a new Section 1.3.
Chapter 3: The classic work of D. G. Northcott on injective envelopes and inverse polynomials is finally included. This provides additional examples for the reader.
Chapter 11: Section 11.9 on Kaplansky classes makes volume one more up to date. The material in this section was not available at the time the first edition was published.

The authors also have clarified some text throughout the book and updated the bibliography by adding new references.

The book is also suitable for an introductory course in commutative and ordinary homological algebra.

DOI:: https://doi.org/10.1515/9783110215212
ISBN:: 978-3-11-021520-5
ISBN:: 978-3-11-021521-2
Fachgebiet:: Mathematik
Fachgebiet:: Algebra und Zahlentheorie
Verlag:: De Gruyter

book: Monoids, Acts and Categories

Buch 2000

Monoids, Acts and Categories

With Applications to Wreath Products and Graphs. A Handbook for Students and Researchers

Mati Kilp, Ulrich Knauer, Alexander V. Mikhalev

Band 29 in dieser Reihe

DOI:: https://doi.org/10.1515/9783110812909
ISBN:: 978-3-11-015248-7
ISBN:: 978-3-11-081290-9
Fachgebiet:: Mathematik
Fachgebiet:: Algebra und Zahlentheorie
Verlag:: De Gruyter

book: Holomorphy and Convexity in Lie Theory

Buch 2000

Holomorphy and Convexity in Lie Theory

Karl-Hermann Neeb

Band 28 in dieser Reihe

DOI:: https://doi.org/10.1515/9783110808148
ISBN:: 978-3-11-015669-0
ISBN:: 978-3-11-080814-8
Fachgebiet:: Mathematik
Fachgebiet:: Algebra und Zahlentheorie
Fachgebiet:: Analysis
Verlag:: De Gruyter

book: Algebra in the Stone-Cech Compactification

Buch 1998

Algebra in the Stone-Cech Compactification

Theory and Applications

Neil Hindman, Dona Strauss

Band 27 in dieser Reihe

DOI:: https://doi.org/10.1515/9783110809220
ISBN:: 9783110154207
ISBN:: 9783110809220
Fachgebiet:: Mathematik
Fachgebiet:: Algebra und Zahlentheorie
Verlag:: De Gruyter

book: Positivity in Lie Theory

Buch 1998

Positivity in Lie Theory

Joachim Hilgert, Jimmie D. Lawson, Karl-Hermann Neeb, Ernest B. Vinberg

Band 26 in dieser Reihe

DOI:: https://doi.org/10.1515/9783110811186
ISBN:: 978-3-11-016112-0
ISBN:: 978-3-11-081118-6
Fachgebiet:: Mathematik
Fachgebiet:: Algebra und Zahlentheorie
Fachgebiet:: Analysis
Verlag:: De Gruyter

book: Character Theory of Finite Groups

Buch 1998

Character Theory of Finite Groups

Bertram Huppert

Band 25 in dieser Reihe

DOI:: https://doi.org/10.1515/9783110809237
ISBN:: 9783110154214
ISBN:: 9783110809237
Fachgebiet:: Mathematik
Fachgebiet:: Algebra und Zahlentheorie
Verlag:: De Gruyter

book: Evolution Equations and Lagrangian Coordinates

Buch 1997

Evolution Equations and Lagrangian Coordinates

Anvarbek M. Meirmanov, Vladislav V. Pukhnachov, Sergei I. Shmarev

Band 24 in dieser Reihe

DOI:: https://doi.org/10.1515/9783110874440
ISBN:: 978-3-11-014875-6
ISBN:: 978-3-11-087444-0
Fachgebiet:: Mathematik
Fachgebiet:: Analysis
Verlag:: De Gruyter

book: Lectures in Real Geometry

Buch 1996

Lectures in Real Geometry

Fabrizio Broglia

Band 23 in dieser Reihe

DOI:: https://doi.org/10.1515/9783110811117
ISBN:: 978-3-11-015095-7
ISBN:: 978-3-11-081111-7
Fachgebiet:: Mathematik
Fachgebiet:: Algebra und Zahlentheorie
Fachgebiet:: Geometrie und Topologie
Verlag:: De Gruyter

book: An Introduction to Lorentz Surfaces

Buch 1996

An Introduction to Lorentz Surfaces

Tilla Weinstein

Band 22 in dieser Reihe

DOI:: https://doi.org/10.1515/9783110821635
ISBN:: 978-3-11-014333-1
ISBN:: 978-3-11-082163-5
Fachgebiet:: Mathematik
Fachgebiet:: Geometrie und Topologie
Verlag:: De Gruyter

book: Compact Projective Planes

Buch 1995

Compact Projective Planes

With an Introduction to Octonion Geometry

Helmut Salzmann, Dieter Betten, Theo Grundhöfer, Hermann Hähl, Rainer Löwen, Markus Stroppel

Band 21 in dieser Reihe

DOI:: https://doi.org/10.1515/9783110876833
ISBN:: 978-3-11-011480-5
ISBN:: 978-3-11-087683-3
Fachgebiet:: Mathematik
Fachgebiet:: Algebra und Zahlentheorie
Fachgebiet:: Geometrie und Topologie
Verlag:: De Gruyter

book: Semigroups in Algebra, Geometry and Analysis

Buch 1995

Semigroups in Algebra, Geometry and Analysis

Karl H. Hofmann, Jimmie D. Lawson, Ernest B. Vinberg

Band 20 in dieser Reihe

DOI:: https://doi.org/10.1515/9783110885583
ISBN:: 978-3-11-014319-5
ISBN:: 978-3-11-088558-3
Fachgebiet:: Mathematik
Fachgebiet:: Algebra und Zahlentheorie
Fachgebiet:: Geometrie und Topologie
Verlag:: De Gruyter

book: Blow-Up in Quasilinear Parabolic Equations

Buch 1995

Blow-Up in Quasilinear Parabolic Equations

A. A. Samarskii, Victor a. Galaktionov, Sergey p. Kurdyumov, A. P. Mikhailov, Michael Grinfeld

Band 19 in dieser Reihe

DOI:: https://doi.org/10.1515/9783110889864
ISBN:: 978-3-11-012754-6
ISBN:: 978-3-11-088986-4
Fachgebiet:: Mathematik
Fachgebiet:: Analysis
Fachgebiet:: Differentialgleichungen und dynamische Systeme
Verlag:: De Gruyter

book: Unitary Representation Theory of Exponential Lie Groups

Buch 1994

Unitary Representation Theory of Exponential Lie Groups

Horst Leptin, Jean Ludwig

Band 18 in dieser Reihe

DOI:: https://doi.org/10.1515/9783110874235
ISBN:: 978-3-11-013938-9
ISBN:: 978-3-11-087423-5
Fachgebiet:: Mathematik
Fachgebiet:: Algebra und Zahlentheorie
Fachgebiet:: Analysis
Verlag:: De Gruyter

book: The Restricted 3-Body Problem: Plane Periodic Orbits

Buch 1994

The Restricted 3-Body Problem: Plane Periodic Orbits

Alexander D. Bruno

Band 17 in dieser Reihe

DOI:: https://doi.org/10.1515/9783110901733
ISBN:: 9783110137033
ISBN:: 9783110901733
Fachgebiet:: Mathematik
Fachgebiet:: Analysis
Fachgebiet:: Angewandte Mathematik
Verlag:: De Gruyter

book: The Adjunction Theory of Complex Projective Varieties

Buch 1995

The Adjunction Theory of Complex Projective Varieties

Mauro C. Beltrametti, Andrew J. Sommese

Band 16 in dieser Reihe

DOI:: https://doi.org/10.1515/9783110871746
ISBN:: 9783110143553
ISBN:: 9783110871746
Fachgebiet:: Mathematik
Fachgebiet:: Algebra und Zahlentheorie
Fachgebiet:: Geometrie und Topologie
Verlag:: De Gruyter

book: Orthogonal Decompositions and Integral Lattices

Buch 1994

Orthogonal Decompositions and Integral Lattices

Alexei Kostrikin, Pham Huu Tiep

Band 15 in dieser Reihe

DOI:: https://doi.org/10.1515/9783110901757
ISBN:: 978-3-11-013783-5
ISBN:: 978-3-11-090175-7
Fachgebiet:: Mathematik
Fachgebiet:: Algebra und Zahlentheorie
Verlag:: De Gruyter

book: Subgroup Lattices of Groups

Buch 1994

Subgroup Lattices of Groups

Roland Schmidt

Band 14 in dieser Reihe

DOI:: https://doi.org/10.1515/9783110868647
ISBN:: 9783110112139
ISBN:: 9783110868647
Fachgebiet:: Mathematik
Fachgebiet:: Algebra und Zahlentheorie
Verlag:: De Gruyter

book: Elliptic Problems in Domains with Piecewise Smooth Boundaries

Buch 1994

Elliptic Problems in Domains with Piecewise Smooth Boundaries

Sergey Nazarov, Boris A. Plamenevsky

Band 13 in dieser Reihe

DOI:: https://doi.org/10.1515/9783110848915
ISBN:: 978-3-11-013522-0
ISBN:: 978-3-11-084891-5
Fachgebiet:: Mathematik
Fachgebiet:: Analysis
Verlag:: De Gruyter

book: Moduli Spaces of Abelian Surfaces

Buch 1993

Moduli Spaces of Abelian Surfaces

Compactification, Degenerations and Theta Functions

Klaus Hulek, Constantin Kahn, Steven H. Weintraub

Band 12 in dieser Reihe

DOI:: https://doi.org/10.1515/9783110891928
ISBN:: 9783110138511
ISBN:: 9783110891928
Fachgebiet:: Mathematik
Fachgebiet:: Algebra und Zahlentheorie
Fachgebiet:: Analysis
Fachgebiet:: Geometrie und Topologie
Verlag:: De Gruyter

book: Global Affine Differential Geometry of Hypersurfaces

Buch 2015

Global Affine Differential Geometry of Hypersurfaces

An-Min Li, Udo Simon, Guosong Zhao, Zejun Hu

Band 11 in dieser Reihe

This book draws a colorful and widespread picture of global affine hypersurface theory up to the most recent state. Moreover, the recent development revealed that affine differential geometry – as differential geometry in general – has an exciting intersection area with other fields of interest, like partial differential equations, global analysis, convex geometry and Riemann surfaces.
The second edition of this monograph leads the reader from introductory concepts to recent research. Since the publication of the first edition in 1993 there appeared important new contributions, like the solutions of two different affine Bernstein conjectures, due to Chern and Calabi, respectively. Moreover, a large subclass of hyperbolic affine spheres were classified in recent years, namely the locally strongly convex Blaschke hypersurfaces that have parallel cubic form with respect to the Levi-Civita connection of the Blaschke metric. The authors of this book present such results and new methods of proof.

DOI:: https://doi.org/10.1515/9783110268898
ISBN:: 9783110266672
ISBN:: 9783110268898
ISBN:: 9783110390902
Fachgebiet:: Mathematik
Fachgebiet:: Analysis
Fachgebiet:: Geometrie und Topologie
Verlag:: De Gruyter

book: The Link Invariants of the Chern-Simons Field Theory

Buch 1993

The Link Invariants of the Chern-Simons Field Theory

New Developments in Topological Quantum Field Theory

E. Guadagnini

Band 10 in dieser Reihe

DOI:: https://doi.org/10.1515/9783110879643
ISBN:: 978-3-11-014028-6
ISBN:: 978-3-11-087964-3
Fachgebiet:: Mathematik
Fachgebiet:: Geometrie und Topologie
Fachgebiet:: Angewandte Mathematik
Verlag:: De Gruyter

book: Invariant Distances and Metrics in Complex Analysis

Buch 2013

Invariant Distances and Metrics in Complex Analysis

Marek Jarnicki, Peter Pflug

Band 9 in dieser Reihe

As in the field of "Invariant Distances and Metrics in Complex Analysis" there was and is a continuous progress this is now the second extended edition of the corresponding monograph. This comprehensive book is about the study of invariant pseudodistances (non-negative functions on pairs of points) and pseudometrics (non-negative functions on the tangent bundle) in several complex variables. It is an overview over a highly active research area at the borderline between complex analysis, functional analysis and differential geometry. New chapters are covering the Wu, Bergman and several other metrics.

The book considers only domains in Cⁿ and assumes a basic knowledge of several complex variables. It is a valuable reference work for the expert but is also accessible to readers who are knowledgeable about several complex variables. Each chapter starts with a brief summary of its contents and continues with a short introduction. It ends with an "Exercises" and a "List of problems" section that gathers all the problems from the chapter. The authors have been highly successful in giving a rigorous but readable account of the main lines of development in this area.

DOI:: https://doi.org/10.1515/9783110253863
ISBN:: 978-3-11-025043-5
ISBN:: 978-3-11-025386-3
Fachgebiet:: Mathematik
Fachgebiet:: Analysis
Fachgebiet:: Geometrie und Topologie
Verlag:: De Gruyter

book: Invariant Distances and Metrics in Complex Analysis

Buch 1993

Invariant Distances and Metrics in Complex Analysis

Marek Jarnicki, Peter Pflug

Band 9 in dieser Reihe

DOI:: https://doi.org/10.1515/9783110870312
ISBN:: 978-3-11-013251-9
ISBN:: 978-3-11-087031-2
Fachgebiet:: Mathematik
Fachgebiet:: Analysis
Verlag:: De Gruyter

book: Nilpotent Groups and their Automorphisms

Buch 1993

Nilpotent Groups and their Automorphisms

Evgenii I. Khukhro

Band 8 in dieser Reihe

DOI:: https://doi.org/10.1515/9783110846218
ISBN:: 978-3-11-013672-2
ISBN:: 978-3-11-084621-8
Fachgebiet:: Mathematik
Fachgebiet:: Algebra und Zahlentheorie
Verlag:: De Gruyter

book: Infinite Dimensional Lie Superalgebras

Buch 1992

Infinite Dimensional Lie Superalgebras

Yuri Bahturin, Alexander V. Mikhalev, Viktor M. Petrogradsky, Mikhail V. Zaicev

Band 7 in dieser Reihe

DOI:: https://doi.org/10.1515/9783110851205
ISBN:: 978-3-11-012974-8
ISBN:: 978-3-11-085120-5
Fachgebiet:: Mathematik
Fachgebiet:: Algebra und Zahlentheorie
Verlag:: De Gruyter

book: Contact Geometry and Linear Differential Equations

Buch 1992

Contact Geometry and Linear Differential Equations

Vladimir E. Nazaikinskii, Victor E. Shatalov, Boris Yu. Sternin

Band 6 in dieser Reihe

DOI:: https://doi.org/10.1515/9783110873108
ISBN:: 978-3-11-013381-3
ISBN:: 978-3-11-087310-8
Fachgebiet:: Mathematik
Fachgebiet:: Differentialgleichungen und dynamische Systeme
Fachgebiet:: Geometrie und Topologie
Verlag:: De Gruyter

book: The Riemann Zeta-Function

Buch 1992

The Riemann Zeta-Function

Anatoly A. Karatsuba, S. M. Voronin, Neal Koblitz

Band 5 in dieser Reihe

DOI:: https://doi.org/10.1515/9783110886146
ISBN:: 978-3-11-013170-3
ISBN:: 978-3-11-088614-6
Fachgebiet:: Mathematik
Fachgebiet:: Algebra und Zahlentheorie
Verlag:: De Gruyter

book: Finite Soluble Groups

Buch 1992

Finite Soluble Groups

Klaus Doerk, Trevor O. Hawkes

Band 4 in dieser Reihe

DOI:: https://doi.org/10.1515/9783110870138
ISBN:: 978-3-11-012892-5
ISBN:: 978-3-11-087013-8
Fachgebiet:: Mathematik
Fachgebiet:: Algebra und Zahlentheorie
Verlag:: De Gruyter

book: The Stefan Problem

Buch 1992

The Stefan Problem

A.M. Meirmanov, Marek Niezgodka, Anna Crowley

Band 3 in dieser Reihe

DOI:: https://doi.org/10.1515/9783110846720
ISBN:: 978-3-11-011479-9
ISBN:: 978-3-11-084672-0
Fachgebiet:: Mathematik
Fachgebiet:: Analysis
Verlag:: De Gruyter

book: Combinatorial Homotopy and 4-Dimensional Complexes

Buch 1991

Combinatorial Homotopy and 4-Dimensional Complexes

Hans-Joachim Baues, Ronald Brown

Band 2 in dieser Reihe

DOI:: https://doi.org/10.1515/9783110854480
ISBN:: 978-3-11-012488-0
ISBN:: 978-3-11-085448-0
Fachgebiet:: Mathematik
Fachgebiet:: Algebra und Zahlentheorie
Fachgebiet:: Geometrie und Topologie
Verlag:: De Gruyter

book: The Analytical and Topological Theory of Semigroups

Buch 1990

The Analytical and Topological Theory of Semigroups

Trends and Developments

Karl H. Hofmann, Jimmie D. Lawson, John S. Pym

Band 1 in dieser Reihe

DOI:: https://doi.org/10.1515/9783110856040
ISBN:: 9783110124897
ISBN:: 9783110856040
Fachgebiet:: Mathematik
Fachgebiet:: Algebra und Zahlentheorie
Fachgebiet:: Analysis
Fachgebiet:: Geometrie und Topologie
Verlag:: De Gruyter

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