series: De Gruyter Studies in Mathematics

Published by De Gruyter

De Gruyter Studies in Mathematics

Edited by: Guozhen Lu, Carsten Carstensen, Gavril Farkas, Nicola Fusco, Fritz Gesztesy, Zenghu Li, Karl-Hermann Neeb, René L. Schilling and Volkmar Welker

ISSN: 0179-0986

Overview
Volumes

The series De Gruyter Studies in Mathematics was founded in 1982 by the late Professor Heinz Bauer and Professor Peter Gabriel.

The series is devoted to the publication of monographs and high-level textbooks in mathematics, mathematical methods and their applications. Apart from covering important areas of current interest, a major aim is to make topics of an interdisciplinary nature accessible to the non-specialist.

The works in this series are addressed to advanced students and researchers in mathematics and theoretical physics. In addition, it can serve as a guide for lectures and seminars on a graduate level.

Each volume undergoes peer review using a double-blind reviewing process. Reviewers are asked to judge the validity, significance, and originality of the work. Series editor(s) assess comments and make necessary recommendations to the author.

The series De Gruyter Studies in Mathematics is indexed in MathSciNet (Mathematical Reviews) and Scopus.

Editor-in-Chief

Guozhen Lu, University of Connecticut, USA

Editorial Board

Carstensen Carsten, Humboldt-Universitat zu Berlin, Germany

Gavril Farkas, Humboldt-Universitat zu Berlin, Germany

Nicola Fusco, Università di Napoli "Federico II", Italy

Fritz Gesztesy, Baylor University, USA

Zenghu Li, Beijing Normal University, China

Karl-Hermann Neeb, Friedrich-Alexander-Universität Erlangen-Nürnberg, Germany

René L. Schilling, Technische Universität Dresden, Germany

Volkmar Welker, Philipps-Universität Marburg, Germany

Please submit book proposals to Guozhen Lu

Supplementary Materials

Publication Ethics Statement

Topics

book: Boundary Value Problems for Second-Order Finite Difference Equations and Systems

Book 2023

Boundary Value Problems for Second-Order Finite Difference Equations and Systems

Johnny Henderson, Rodica Luca

Volume 91 in this series

This is an indispensable reference for those mathematicians that conduct research activity in applications of fixed-point theory to boundary value problems for nonlinear difference equations. Coverage includes second-order finite difference equations and systems of second-order finite difference equations subject to diverse multi-point boundary conditions, and various methods to study the existence of positive solutions for these problems.

DOI:: https://doi.org/10.1515/9783111040370
ISBN:: 9783111040370
ISBN:: 9783111040455
ISBN:: 9783111039312
Subject:: Mathematics
Subject:: Differential Equations and Dynamical Systems
Publisher:: De Gruyter

book: Maximal Subellipticity

Book 2023

Maximal Subellipticity

Brian Street

Volume 93 in this series

Maximally subelliptic partial differential equations (PDEs) are a far-reaching generalization of elliptic PDEs. Elliptic PDEs hold a special place: sharp results are known for general linear and even fully nonlinear elliptic PDEs. Over the past half-century, important results for elliptic PDEs have been generalized to maximally subelliptic PDEs. This text presents this theory and generalizes the sharp, interior regularity theory for general linear and fully nonlinear elliptic PDEs to the maximally subelliptic setting.

DOI:: https://doi.org/10.1515/9783111085647
ISBN:: 9783111085647
ISBN:: 9783111085944
ISBN:: 9783111085173
Subject:: Mathematics
Subject:: Analysis
Subject:: Differential Equations and Dynamical Systems
Subject:: Geometry and Topology
Publisher:: De Gruyter

book: Minimal Surfaces through Nevanlinna Theory

Book 2023

Minimal Surfaces through Nevanlinna Theory

Min Ru

Volume 92 in this series

The study of minimal surfaces is an important subject in differential geometry, and Nevanlinna theory is an important subject in complex analysis and complex geometry. This book discusses the interaction between these two subjects. In particular, it describes the study of the value distribution properties of the Gauss map of minimal surfaces through Nevanlinna theory, a project initiated by the prominent differential geometers Shiing-Shen Chern and Robert Osserman.

DOI:: https://doi.org/10.1515/9783110989557
ISBN:: 9783110989557
ISBN:: 9783110999822
ISBN:: 9783110989960
Subject:: Mathematics
Subject:: Analysis
Subject:: Applied Mathematics
Publisher:: De Gruyter

book: Polynomial Sequences

Book 2024

Polynomial Sequences

Basic Methods, Special Classes, and Computational Applications

Francesco Aldo Costabile, Maria Italia Gualtieri, Anna Napoli

Volume 96 in this series

Polynomials are useful mathematical tools. They are simply defined and can be calculated quickly on computer systems. They can be differentiated and integrated easily and can be pieced together to form spline curves. After Weierstrass approximation Theorem, polynomial sequences have acquired considerable importance not only in the various
branches of Mathematics, but also in Physics, Chemistry and Engineering disciplines. There is a wide literature on specific polynomial sequences. But there is no literature that attempts a systematic exposition of the main basic methods for the study of a generic polynomial sequence and, at the same time, gives an overview of the main polynomial classes and related applications, at least in numerical analysis. In this book, through an elementary matrix calculus-based approach, an attempt is made to fill this gap by exposing dated and very recent results, both theoretical and applied.

DOI:: https://doi.org/10.1515/9783110757248
ISBN:: 9783110757248
ISBN:: 9783110757231
ISBN:: 9783110757323
Subject:: Mathematics
Subject:: Analysis
Subject:: Differential Equations and Dynamical Systems
Publisher:: De Gruyter

book: Representation Theory and Geometry of the Flag Variety

Book 2023

Representation Theory and Geometry of the Flag Variety

William M. McGovern

Volume 90 in this series

This comprehensive reference begins with a review of the basics followed by a presentation of flag varieties and finite- and infinite-dimensional representations in classical types and subvarieties of flag varieties and their singularities. Associated varieties and characteristic cycles are covered as well and Kazhdan-Lusztig polynomials are treated. The coverage concludes with a discussion of pattern avoidance and singularities and some recent results on Springer fibers.

DOI:: https://doi.org/10.1515/9783110766943
ISBN:: 9783110766943
ISBN:: 9783110766967
ISBN:: 9783110766905
Subject:: Mathematics
Subject:: Algebra and Number Theory
Subject:: Geometry and Topology
Subject:: General Mathematics
Publisher:: De Gruyter

book: Holomorphic Dynamics on Hyperbolic Riemann Surfaces

Book 2023

Holomorphic Dynamics on Hyperbolic Riemann Surfaces

Marco Abate

Volume 89 in this series

This completely revised and updated edition of the one variable part of the author's classic older book "Iteration Theory of Holomorphic Maps on Taut Manifolds" presents the theory of holomorphic dynamical systems on hyperbolic Riemann surfaces from the very beginning of the subject up to the most recent developments. It is intended both as a reference book for the experts and as an accessible gateway to this beautiful theory for Master and Ph.D. students. It also contains extensive historical notes and references for further readings.

DOI:: https://doi.org/10.1515/9783110601978
ISBN:: 9783110601978
ISBN:: 9783110601053
ISBN:: 9783110598742
Subject:: Mathematics
Subject:: Analysis
Publisher:: De Gruyter

book: Topics in Complex Analysis

Book 2022

Topics in Complex Analysis

Joel L. Schiff

Volume 88 in this series

Complex analysis is found in many areas of applied mathematics, from fluid mechanics, thermodynamics, signal processing, control theory, mechanical and electrical engineering to quantum mechanics, among others. And of course, it is a fundamental branch of pure mathematics. The coverage in this text includes advanced topics that are not always considered in more elementary texts. These topics include, a detailed treatment of univalent functions, harmonic functions, subharmonic and superharmonic functions, Nevanlinna theory, normal families, hyperbolic geometry, iteration of rational functions, and analytic number theory. As well, the text includes in depth discussions of the Dirichlet Problem, Green’s function, Riemann Hypothesis, and the Laplace transform. Some beautiful color illustrations supplement the text of this most elegant subject.

DOI:: https://doi.org/10.1515/9783110757828
ISBN:: 9783110757828
ISBN:: 9783110757873
ISBN:: 9783110757699
Subject:: Mathematics
Subject:: Analysis
Subject:: Applied Mathematics
Publisher:: De Gruyter

book: Fourier Meets Hilbert and Riesz

Book 2023

Fourier Meets Hilbert and Riesz

An Introduction to the Corresponding Transforms

René Erlin Castillo

Volume 87 in this series

This book provides an introduction into the modern theory of classical harmonic analysis, dealing with Fourier analysis and the most elementary singular integral operators, the Hilbert transform and Riesz transforms. Ideal for self-study or a one semester course in Fourier analysis, included are detailed examples and exercises.

DOI:: https://doi.org/10.1515/9783110784091
ISBN:: 9783110784091
ISBN:: 9783110784121
ISBN:: 9783110784053
Subject:: Mathematics
Subject:: Analysis
Subject:: Differential Equations and Dynamical Systems
Subject:: Numerical and Computational Mathematics
Subject:: Applied Mathematics
Publisher:: De Gruyter

book: Measure Theory and Nonlinear Evolution Equations

Book 2022

Measure Theory and Nonlinear Evolution Equations

Flavia Smarrazzo, Alberto Tesei

Volume 86 in this series

This text on measure theory with applications to partial differential equations covers general measure theory, Lebesgue spaces of real-valued and vector-valued functions, different notions of measurability for the latter, weak convergence of functions and measures, Radon and Young measures, capacity. A comprehensive discussion of applications to quasilinear parabolic and hyperbolic problems is provided.

DOI:: https://doi.org/10.1515/9783110556902
ISBN:: 9783110556902
ISBN:: 9783110556001
ISBN:: 9783110556049
Subject:: Mathematics
Subject:: Analysis
Subject:: Differential Equations and Dynamical Systems
Publisher:: De Gruyter

book: Pointwise Variable Anisotropic Function Spaces on ℝⁿ

Book 2022

Pointwise Variable Anisotropic Function Spaces on ℝⁿ

Shai Dekel

Volume 85 in this series

Spaces of homogeneous type were introduced as a generalization to the Euclidean space and serve as a suffi cient setting in which one can generalize the classical isotropic Harmonic analysis and function space theory. This setting is sometimes too general, and the theory is limited. Here, we present a set of fl exible ellipsoid covers of ℝn that replace the Euclidean balls and support a generalization of the theory with fewer limitations.

DOI:: https://doi.org/10.1515/9783110761795
ISBN:: 9783110761795
ISBN:: 9783110761870
ISBN:: 9783110761764
Subject:: Mathematics
Subject:: Analysis
Subject:: Probability and Statistics
Subject:: Applied Mathematics
Subject:: Physics
Subject:: Theoretical and Mathematical Physics
Publisher:: De Gruyter

book: Selected Topics in Almost Periodicity

Book 2022

Selected Topics in Almost Periodicity

Marko Kostić

Volume 84 in this series

Covers uniformly recurrent solutions and c-almost periodic solutions of abstract Volterra integro-differential equations as well as various generalizations of almost periodic functions in Lebesgue spaces with variable coefficients. Treats multi-dimensional almost periodic type functions and their generalizations in adequate detail.

DOI:: https://doi.org/10.1515/9783110763522
ISBN:: 9783110763522
ISBN:: 9783110763607
ISBN:: 9783110763225
Subject:: Mathematics
Subject:: Analysis
Subject:: Differential Equations and Dynamical Systems
Publisher:: De Gruyter

book: Topological Groups and the Pontryagin-van Kampen Duality

Book 2022

Topological Groups and the Pontryagin-van Kampen Duality

An Introduction

Lydia Außenhofer, Dikran Dikranjan, Anna Giordano Bruno

Volume 83 in this series

This book provides an introduction to topological groups and the structure theory of locally compact abelian groups, with a special emphasis on Pontryagin-van Kampen duality, including a completely self-contained elementary proof of the duality theorem. Further related topics and applications are treated in separate chapters and in the appendix.

DOI:: https://doi.org/10.1515/9783110654936
ISBN:: 9783110654936
ISBN:: 9783110653496
ISBN:: 9783110653557
Subject:: Mathematics
Subject:: Algebra and Number Theory
Subject:: Geometry and Topology
Publisher:: De Gruyter

book: Crossed Modules

Book 2021

Crossed Modules

Friedrich Wagemann

Volume 82 in this series

This book presents material in two parts. Part one provides an introduction to crossed modules of groups, Lie algebras and associative algebras with fully written out proofs and is suitable for graduate students interested in homological algebra. In part two, more advanced and less standard topics such as crossed modules of Hopf algebra, Lie groups, and racks are discussed as well as recent developments and research on crossed modules.

DOI:: https://doi.org/10.1515/9783110750959
ISBN:: 9783110750959
ISBN:: 9783110750997
ISBN:: 9783110750768
Subject:: Mathematics
Subject:: Algebra and Number Theory
Subject:: Geometry and Topology
Publisher:: De Gruyter

book: USCO and Quasicontinuous Mappings

Book 2021

USCO and Quasicontinuous Mappings

L’ubica Holá, Dušan Holý, Warren Moors

Volume 81 in this series

This book presents two natural generalizations of continuous mappings, namely usco and quasicontinuous mappings. The first class considers set-valued mappings, the second class relaxes the definition of continuity. Both these topological concepts stem naturally from basic mathematical considerations and have numerous applications that are covered in detail.

DOI:: https://doi.org/10.1515/9783110750188
ISBN:: 9783110750188
ISBN:: 9783110750225
ISBN:: 9783110750157
Subject:: Engineering
Subject:: Mechanical Engineering
Subject:: Process Engineering and Industrial Engineering
Subject:: Mathematics
Subject:: Analysis
Subject:: Geometry and Topology
Subject:: General Mathematics
Publisher:: De Gruyter

book: Formal Analysis

Book 2021

Formal Analysis

An Introduction

Xiao-Xiong Gan

Volume 80 in this series

Formal analysis is the study of formal power series, formal Laurent series, formal root series, and other formal series or formal functionals. This book is the first comprehensive presentation of the topic that systematically introduces formal analysis, including its algebraic, analytic, and topological structure, along with various applications.

DOI:: https://doi.org/10.1515/9783110599459
ISBN:: 9783110599459
ISBN:: 9783110597851
ISBN:: 9783110598605
Subject:: Mathematics
Subject:: Analysis
Publisher:: De Gruyter

book: Affine Space Fibrations

Book 2021

Affine Space Fibrations

Rajendra V. Gurjar, Kayo Masuda, Masayoshi Miyanishi

Volume 79 in this series

Affine algebraic geometry has progressed remarkably in the last half a century, and its central topics are affine spaces and affine space fibrations. This authoritative book is aimed at graduate students and researchers alike, and studies the geometry and topology of morphisms of algebraic varieties whose general fibers are isomorphic to the affine space while describing structures of algebraic varieties with such affine space fibrations.

DOI:: https://doi.org/10.1515/9783110577563
ISBN:: 9783110577563
ISBN:: 9783110577365
ISBN:: 9783110577426
Subject:: Mathematics
Subject:: Algebra and Number Theory
Subject:: Geometry and Topology
Publisher:: De Gruyter

book: Complex Delay-Differential Equations

Book 2021

Complex Delay-Differential Equations

Kai Liu, Ilpo Laine, Lianzhong Yang

Volume 78 in this series

This book presents developments and new results on complex differential-difference equations, an area with important and interesting applications, which also gathers increasing attention. Key problems, methods, and results related to complex differential-difference equations are collected to offer an up-to-date overview of the field.

DOI:: https://doi.org/10.1515/9783110560565
ISBN:: 9783110560565
ISBN:: 9783110560169
ISBN:: 9783110560404
Subject:: Mathematics
Subject:: Analysis
Subject:: Differential Equations and Dynamical Systems
Publisher:: De Gruyter

book: Ulrich Bundles

Book 2021

Ulrich Bundles

From Commutative Algebra to Algebraic Geometry

Laura Costa, Rosa María Miró-Roig, Joan Pons-Llopis

Volume 77 in this series

The goal of this book is to cover the active developments of arithmetically Cohen-Macaulay and Ulrich bundles and related topics in the last 30 years, and to present relevant techniques and multiple applications of the theory of Ulrich bundles to a wide range of problems in algebraic geometry as well as in commutative algebra.

DOI:: https://doi.org/10.1515/9783110647686
ISBN:: 9783110647686
ISBN:: 9783110645408
ISBN:: 9783110645804
Subject:: Mathematics
Subject:: Algebra and Number Theory
Subject:: Geometry and Topology
Publisher:: De Gruyter

book: Recent Developments in Sturm-Liouville Theory

Book 2021

Recent Developments in Sturm-Liouville Theory

Anton Zettl

Volume 76 in this series

This book provides a modern survey of some basic properties of Sturm-Liouville problems and to bring the reader to the forefront of knowledge of some areas of the theory. For example, some special Sturm-Liouville eigenvalue problems are equivalent to certain Jacobi and cyclic Jacobi matrix eigenvalue problems. A new approach to problems with periodic conditions is developed.

DOI:: https://doi.org/10.1515/9783110719000
ISBN:: 9783110719000
ISBN:: 9783110719383
ISBN:: 9783110718843
Subject:: Mathematics
Subject:: Analysis
Subject:: Differential Equations and Dynamical Systems
Subject:: General Mathematics
Publisher:: De Gruyter

book: Time-Frequency Analysis of Operators

Book 2020

Time-Frequency Analysis of Operators

Elena Cordero, Luigi Rodino

Volume 75 in this series

This authoritative text studies pseudodifferential and Fourier integral operators in the framework of time-frequency analysis, providing an elementary approach, along with applications to almost diagonalization of such operators and to the sparsity of their Gabor representations. Moreover, Gabor frames and modulation spaces are employed to study dispersive equations such as the Schrödinger, wave, and heat equations and related Strichartz problems.　　　　

The first part of the book is addressed to non-experts, presenting the basics of time-frequency analysis: short time Fourier transform, Wigner distribution and other representations, function spaces and frames theory, and it can be read independently as a short text-book on this topic from graduate and under-graduate students, or scholars in other disciplines.

DOI:: https://doi.org/10.1515/9783110532456
ISBN:: 9783110532456
ISBN:: 9783110530353
ISBN:: 9783110530605
Subject:: Mathematics
Subject:: Analysis
Publisher:: De Gruyter

book: Local Density of Solutions to Fractional Equations

Book 2019

Local Density of Solutions to Fractional Equations

Alessandro Carbotti, Serena Dipierro, Enrico Valdinoci

Volume 74 in this series

This book presents in a detailed and self-contained way a new and important density result in the analysis of fractional partial differential equations, while also covering several fundamental facts about space- and time-fractional equations.

DOI:: https://doi.org/10.1515/9783110664355
ISBN:: 9783110664355
ISBN:: 9783110660692
ISBN:: 9783110661316
Subject:: Mathematics
Subject:: Analysis
Subject:: Differential Equations and Dynamical Systems
Publisher:: De Gruyter

book: Abelian Groups

Book 2019

Abelian Groups

Structures and Classifications

Carol Jacoby, Peter Loth

Volume 73 in this series

This monograph covers in a comprehensive manner the current state of classification theory with respect to infinite abelian groups. A wide variety of ways to characterise different classes of abelian groups by invariants, isomorphisms and duality principles are discussed.

DOI:: https://doi.org/10.1515/9783110427684
ISBN:: 9783110427684
ISBN:: 9783110432114
ISBN:: 9783110427868
Subject:: Mathematics
Subject:: Algebra and Number Theory
Publisher:: De Gruyter

book: Modern Umbral Calculus

Book 2019

Modern Umbral Calculus

An Elementary Introduction with Applications to Linear Interpolation and Operator Approximation Theory

Francesco Aldo Costabile

Volume 72 in this series

This book presents a novel approach to umbral calculus, which uses only elementary linear algebra (matrix calculus) based on the observation that there is an isomorphism between Sheffer polynomials and Riordan matrices, and that Sheffer polynomials can be expressed in terms of determinants. Additionally, applications to linear interpolation and operator approximation theory are presented in many settings related to various families of polynomials.

DOI:: https://doi.org/10.1515/9783110652925
ISBN:: 9783110652925
ISBN:: 9783110649963
ISBN:: 9783110650099
Subject:: Mathematics
Subject:: Analysis
Subject:: Numerical and Computational Mathematics
Publisher:: De Gruyter

book: Periodic Locally Compact Groups

Book 2019

Periodic Locally Compact Groups

A Study of a Class of Totally Disconnected Topological Groups

Wolfgang Herfort, Karl H. Hofmann, Francesco G. Russo

Volume 71 in this series

This authoritative book on periodic locally compact groups is divided into three parts: The first part covers the necessary background material on locally compact groups including the Chabauty topology on the space of closed subgroups of a locally compact group, its Sylow theory, and the introduction, classifi cation and use of inductively monothetic groups.

The second part develops a general structure theory of locally compact near abelian groups, pointing out some of its connections with number theory and graph theory and illustrating it by a large exhibit of examples.

Finally, the third part uses this theory for a complete, enlarged and novel presentation of Mukhin’s pioneering work generalizing to locally compact groups Iwasawa’s early investigations of the lattice of subgroups of abstract groups.

Contents
Part I: Background information on locally compact groups
Locally compact spaces and groups
Periodic locally compact groups and their Sylow theory
Abelian periodic groups
Scalar automorphisms and the mastergraph
Inductively monothetic groups
Part II: Near abelian groups
The definition of near abelian groups
Important consequences of the definitions
Trivial near abelian groups
The class of near abelian groups
The Sylow structure of periodic nontrivial near abelian groups and their prime graphs
A list of examples
Part III: Applications
Classifying topologically quasihamiltonian groups
Locally compact groups with a modular subgroup lattice
Strongly topologically quasihamiltonian groups

DOI:: https://doi.org/10.1515/9783110599190
ISBN:: 9783110599190
ISBN:: 9783110599084
ISBN:: 9783110598476
Subject:: Mathematics
Subject:: Algebra and Number Theory
Publisher:: De Gruyter

book: Spectral Theory of Canonical Systems

Book 2018

Spectral Theory of Canonical Systems

Christian Remling

Volume 70 in this series

Canonical systems occupy a central position in the spectral theory of second order differential operators. They may be used to realize arbitrary spectral data, and the classical operators such as Schrödinger, Jacobi, Dirac, and Sturm-Liouville equations can be written in this form. ‘Spectral Theory of Canonical Systems’ offers a selfcontained and detailed introduction to this theory. Techniques to construct self-adjoint realizations in suitable Hilbert spaces, a modern treatment of de Branges spaces, and direct and inverse spectral problems are discussed.

Contents
Basic definitions
Symmetric and self-adjoint relations
Spectral representation
Transfer matrices and de Branges spaces
Inverse spectral theory
Some applications
The absolutely continuous spectrum

DOI:: https://doi.org/10.1515/9783110563238
ISBN:: 9783110563238
ISBN:: 9783110562286
ISBN:: 9783110562026
Subject:: Mathematics
Subject:: Analysis
Subject:: Differential Equations and Dynamical Systems
Subject:: Physics
Subject:: Theoretical and Mathematical Physics
Publisher:: De Gruyter

book: Univalent Functions

Book 2018

Univalent Functions

Derek K. Thomas, Nikola Tuneski, Allu Vasudevarao

Volume 69 in this series

The study of univalent functions dates back to the early years of the 20th century, and is one of the most popular research areas in complex analysis. This book is directed at introducing and bringing up to date current research in the area of univalent functions, with an emphasis on the important subclasses, thus providing an accessible resource suitable for both beginning and experienced researchers.

Contents
Univalent Functions – the Elementary Theory
Definitions of Major Subclasses
Fundamental Lemmas
Starlike and Convex Functions
Starlike and Convex Functions of Order α
Strongly Starlike and Convex Functions
Alpha-Convex Functions
Gamma-Starlike Functions
Close-to-Convex Functions
Bazilevič Functions
B1(α) Bazilevič Functions
The Class U(λ)
Convolutions
Meromorphic Univalent Functions
Loewner Theory
Other Topics
Open Problems

DOI:: https://doi.org/10.1515/9783110560961
ISBN:: 9783110560961
ISBN:: 9783110560121
ISBN:: 9783110560091
Subject:: Mathematics
Subject:: Analysis
Publisher:: De Gruyter

book: Volume 2 Fundamentals of Functions and Measure Theory

Book 2018

Volume 2 Fundamentals of Functions and Measure Theory

Valeriy K. Zakharov, Timofey V. Rodionov, Alexander V. Mikhalev

Part of the multi-volume work Sets, Functions, Measures

Volume 68/2 in this series

This comprehensive two-volume work is devoted to the most general beginnings of mathematics. It goes back to Hausdorff’s classic Set Theory (2nd ed., 1927), where set theory and the theory of functions were expounded as the fundamental parts of mathematics in such a way that there was no need for references to other sources. Along the lines of Hausdorff’s initial work (1st ed., 1914), measure and integration theory is also included here as the third fundamental part of contemporary mathematics. The material about sets and numbers is placed in Volume 1 and the material about functions and measures is placed in Volume 2.

Contents
Historical foreword on the centenary after Felix Hausdorff’s classic Set Theory
Fundamentals of the theory of functions
Fundamentals of the measure theory
Historical notes on the Riesz – Radon – Frechet problem of characterization of Radon integrals as linear functionals

DOI:: https://doi.org/10.1515/9783110550962
ISBN:: 9783110550962
ISBN:: 9783110550092
ISBN:: 9783110550221
Subject:: Mathematics
Subject:: Logic and Set Theory
Subject:: Analysis
Publisher:: De Gruyter

book: Volume 1 Fundamentals of Set and Number Theory

Book 2018

Volume 1 Fundamentals of Set and Number Theory

Valeriy K. Zakharov, Timofey V. Rodionov

Part of the multi-volume work Sets, Functions, Measures

Volume 68/1 in this series

This comprehensive two-volume work is devoted to the most general beginnings of mathematics. It goes back to Hausdorff’s classic Set Theory (2nd ed., 1927), where set theory and the theory of functions were expounded as the fundamental parts of mathematics in such a way that there was no need for references to other sources. Along the lines of Hausdorff’s initial work (1st ed., 1914), measure and integration theory is also included here as the third fundamental part of contemporary mathematics.The material about sets and numbers is placed in Volume 1 and the material about functions and measures is placed in Volume 2.

　

Contents
Fundamentals of the theory of classes, sets, and numbers
Characterization of all natural models of Neumann – Bernays – Godel and Zermelo – Fraenkel set theories
Local theory of sets as a foundation for category theory and its connection with the Zermelo – Fraenkel set theory
Compactness theorem for generalized second-order language

DOI:: https://doi.org/10.1515/9783110550948
ISBN:: 9783110550948
ISBN:: 9783110550085
ISBN:: 9783110550214
Subject:: Mathematics
Subject:: Logic and Set Theory
Subject:: Analysis
Publisher:: De Gruyter

book: Ergodic Behavior of Markov Processes

Book 2018

Ergodic Behavior of Markov Processes

With Applications to Limit Theorems

Alexei Kulik

Volume 67 in this series

The general topic of this book is the ergodic behavior of Markov processes. A detailed introduction to methods for proving ergodicity and upper bounds for ergodic rates is presented in the first part of the book, with the focus put on weak ergodic rates, typical for Markov systems with complicated structure. The second part is devoted to the application of these methods to limit theorems for functionals of Markov processes. The book is aimed at a wide audience with a background in probability and measure theory. Some knowledge of stochastic processes and stochastic differential equations helps in a deeper understanding of specific examples.

Contents　
Part I: Ergodic Rates for Markov Chains and Processes
Markov Chains with Discrete State Spaces
General Markov Chains: Ergodicity in Total Variation
MarkovProcesseswithContinuousTime
Weak Ergodic Rates

Part II: Limit Theorems
The Law of Large Numbers and the Central Limit Theorem
Functional Limit Theorems

DOI:: https://doi.org/10.1515/9783110458930
ISBN:: 9783110458930
ISBN:: 9783110458718
ISBN:: 9783110458701
Subject:: Mathematics
Subject:: Differential Equations and Dynamical Systems
Subject:: Probability and Statistics
Publisher:: De Gruyter

book: Noncommutative Geometry

Book 2022

Noncommutative Geometry

A Functorial Approach

Igor V. Nikolaev

Volume 66 in this series

Noncommutative geometry studies an interplay between spatial forms and algebras with non-commutative multiplication. This book covers the key concepts of noncommutative geometry and its applications in topology, algebraic geometry, and number theory. Our presentation is accessible to the graduate students as well as nonexperts in the field. The second edition includes two new chapters on arithmetic topology and quantum arithmetic.

DOI:: https://doi.org/10.1515/9783110788709
ISBN:: 9783110788709
ISBN:: 9783110788815
ISBN:: 9783110788600
Subject:: Mathematics
Subject:: Algebra and Number Theory
Subject:: Geometry and Topology
Publisher:: De Gruyter

book: Noncommutative Geometry

Book 2018

Noncommutative Geometry

A Functorial Approach

Igor V. Nikolaev

Volume 66 in this series

This book covers the basics of noncommutative geometry (NCG) and its applications in topology, algebraic geometry, and number theory. The author takes up the practical side of NCG and its value for other areas of mathematics. A brief survey of the main parts of NCG with historical remarks, bibliography, and a list of exercises is included. The presentation is intended for graduate students and researchers with interests in NCG, but will also serve nonexperts in the field.

Contents
Part I: Basics
Model examples
Categories and functors
C∗-algebras

Part II: Noncommutative invariants
Topology
Algebraic geometry
Number theory

Part III: Brief survey of NCG
Finite geometries
Continuous geometries
Connes geometries
Index theory
Jones polynomials
Quantum groups
Noncommutative algebraic geometry
Trends in noncommutative geometry

DOI:: https://doi.org/10.1515/9783110545258
ISBN:: 9783110545258
ISBN:: 9783110543483
Subject:: Mathematics
Subject:: Algebra and Number Theory
Subject:: Geometry and Topology
Publisher:: De Gruyter

book: Interval Analysis

Book 2017

Interval Analysis

and Automatic Result Verification

Günter Mayer

Volume 65 in this series

This self-contained text is a step-by-step introduction and a complete overview of interval computation and result verification, a subject whose importance has steadily increased over the past many years. The author, an expert in the field, gently presents the theory of interval analysis through many examples and exercises, and guides the reader from the basics of the theory to current research topics in the mathematics of computation.

Contents

Preliminaries
Real intervals
Interval vectors, interval matrices
Expressions, P-contraction, ε-inflation
Linear systems of equations
Nonlinear systems of equations
Eigenvalue problems
Automatic differentiation
Complex intervals

DOI:: https://doi.org/10.1515/9783110499469
ISBN:: 9783110499469
ISBN:: 9783110500639
ISBN:: 9783110498059
Subject:: Computer Sciences
Subject:: Algorithms
Subject:: Mathematics
Subject:: Analysis
Subject:: Numerical and Computational Mathematics
Publisher:: De Gruyter

book: The Hodge-Laplacian

Book 2016

The Hodge-Laplacian

Boundary Value Problems on Riemannian Manifolds

Dorina Mitrea, Irina Mitrea, Marius Mitrea, Michael Taylor

Volume 64 in this series

The core of this monograph is the development of tools to derive well-posedness results in very general geometric settings for elliptic differential operators. A new generation of Calderón-Zygmund theory is developed for variable coefficient singular integral operators, which turns out to be particularly versatile in dealing with boundary value problems for the Hodge-Laplacian on uniformly rectifiable subdomains of Riemannian manifolds via boundary layer methods. In addition to absolute and relative boundary conditions for differential forms, this monograph treats the Hodge-Laplacian equipped with classical Dirichlet, Neumann, Transmission, Poincaré, and Robin boundary conditions in regular Semmes-Kenig-Toro domains.
Lying at the intersection of partial differential equations, harmonic analysis, and differential geometry, this text is suitable for a wide range of PhD students, researchers, and professionals.

Contents:
Preface
Introduction and Statement of Main Results
Geometric Concepts and Tools
Harmonic Layer Potentials Associated with the Hodge-de Rham Formalism on UR Domains
Harmonic Layer Potentials Associated with the Levi-Civita Connection on UR Domains
Dirichlet and Neumann Boundary Value Problems for the Hodge-Laplacian on Regular SKT Domains
Fatou Theorems and Integral Representations for the Hodge-Laplacian on Regular SKT Domains
Solvability of Boundary Problems for the Hodge-Laplacian in the Hodge-de Rham Formalism
Additional Results and Applications
Further Tools from Differential Geometry, Harmonic Analysis, Geometric Measure Theory, Functional Analysis, Partial Differential Equations, and Clifford Analysis
Bibliography
Index

DOI:: https://doi.org/10.1515/9783110484380
ISBN:: 9783110484380
ISBN:: 9783110483390
ISBN:: 9783110482669
Subject:: Mathematics
Subject:: Differential Equations and Dynamical Systems
Subject:: Geometry and Topology
Publisher:: De Gruyter

book: Orthogonal Polynomials

Book Ahead of Publication 2025

Orthogonal Polynomials

Evguenii A. Rakhmanov

Volume 63 in this series

The origins of the theory of orthogonal polynomials go back at least to the 18th century when they were studied in terms of continued fractions. The theory is now large and complex: a crossroad of several important domains of analysis such as analytic function theory, analytic theory of differential equations, Fourier and harmonic analysis, spectral theory of Sturm-Liouville operators, and approximation and interpolation, among others.

DOI:: https://doi.org/10.0000/9783110313864
ISBN:: 9783110313864
ISBN:: 9783110383256
ISBN:: 9783110313857
Subject:: Mathematics
Subject:: Analysis
Subject:: Differential Equations and Dynamical Systems
Subject:: Physics
Subject:: Theoretical and Mathematical Physics
Publisher:: De Gruyter

book: Positive Dynamical Systems in Discrete Time

Book 2015

Positive Dynamical Systems in Discrete Time

Theory, Models, and Applications

Ulrich Krause

Volume 62 in this series

This book provides a systematic, rigorous and self-contained treatment of positive dynamical systems. A dynamical system is positive when all relevant variables of a system are nonnegative in a natural way. This is in biology, demography or economics, where the levels of populations or prices of goods are positive. The principle also finds application in electrical engineering, physics and computer sciences.

"The author has greatly expanded the field of positive systems in surprising ways." - Prof. Dr. David G. Luenberger, Stanford University(USA)

DOI:: https://doi.org/10.1515/9783110365696
ISBN:: 978-3-11-036975-5
ISBN:: 978-3-11-036569-6
ISBN:: 9783110391343
Subject:: Mathematics
Subject:: Analysis
Publisher:: De Gruyter

book: Markov Operators, Positive Semigroups and Approximation Processes

Book 2015

Markov Operators, Positive Semigroups and Approximation Processes

Francesco Altomare, Mirella Cappelletti, Vita Leonessa, Ioan Rasa

Volume 61 in this series

This research monograph gives a detailed account of a theory which is mainly concerned with certain classes of degenerate differential operators, Markov semigroups and approximation processes. These mathematical objects are generated by arbitrary Markov operators acting on spaces of continuous functions defined on compact convex sets; the study of the interrelations between them constitutes one of the distinguishing features of the book.

Among other things, this theory provides useful tools for studying large classes of initial-boundary value evolution problems, the main aim being to obtain a constructive approximation to the associated positive C₀-semigroups by means of iterates of suitable positive approximating operators. As a consequence, a qualitative analysis of the solutions to the evolution problems can be efficiently developed.

The book is mainly addressed to research mathematicians interested in modern approximation theory by positive linear operators and/or in the theory of positive C₀-semigroups of operators and evolution equations. It could also serve as a textbook for a graduate level course.

DOI:: https://doi.org/10.1515/9783110366976
ISBN:: 978-3-11-037274-8
ISBN:: 978-3-11-036697-6
ISBN:: 9783110386417
Subject:: Mathematics
Subject:: Analysis
Subject:: Probability and Statistics
Publisher:: De Gruyter

book: Hörmander Spaces, Interpolation, and Elliptic Problems

Book 2014

Hörmander Spaces, Interpolation, and Elliptic Problems

Vladimir A. Mikhailets, Aleksandr A. Murach, Peter V. Malyshev

Volume 60 in this series

The monograph gives a detailed exposition of the theory of general elliptic operators (scalar and matrix) and elliptic boundary value problems in Hilbert scales of Hörmander function spaces. This theory was constructed by the authors in a number of papers published in 2005–2009. It is distinguished by a systematic use of the method of interpolation with a functional parameter of abstract Hilbert spaces and Sobolev inner product spaces. This method, the theory and their applications are expounded for the first time in the monographic literature.

The monograph is written in detail and in a reader-friendly style. The complete proofs of theorems are given. This monograph is intended for a wide range of mathematicians whose research interests concern with mathematical analysis and differential equations.

DOI:: https://doi.org/10.1515/9783110296891
ISBN:: 978-3-11-029685-3
ISBN:: 978-3-11-029689-1
ISBN:: 9783110369069
Subject:: Mathematics
Subject:: Analysis
Publisher:: De Gruyter

book: Topological Dynamical Systems

Book 2014

Topological Dynamical Systems

An Introduction to the Dynamics of Continuous Mappings

Jan de Vries

Volume 59 in this series

There is no recent elementary introduction to the theory of discrete dynamical systems that stresses the topological background of the topic. This book fills this gap: it deals with this theory as 'applied general topology'. We treat all important concepts needed to understand recent literature. The book is addressed primarily to graduate students. The prerequisites for understanding this book are modest: a certain mathematical maturity and course in General Topology are sufficient.

DOI:: https://doi.org/10.1515/9783110342406
ISBN:: 978-3-11-034073-0
ISBN:: 978-3-11-034240-6
ISBN:: 9783110374599
Subject:: Mathematics
Subject:: Analysis
Subject:: Geometry and Topology
Publisher:: De Gruyter

book: Stochastic Ferromagnetism

Book 2014

Stochastic Ferromagnetism

Analysis and Numerics

Lubomir Banas, Zdzislaw Brzezniak, Mikhail Neklyudov, Andreas Prohl

Volume 58 in this series

This monograph examines magnetization dynamics at elevated temperatures which can be described by the stochastic Landau-Lifshitz-Gilbert equation (SLLG). The first part of the book studies the role of noise in finite ensembles of nanomagnetic particles: we show geometric ergodicity of a unique invariant measure of Gibbs type and study related properties of approximations of the SLLG, including time discretization and Ginzburg-Landau type penalization. In the second part we propose an implementable space-time discretization using random walks to construct a weak martingale solution of the corresponding stochastic partial differential equation which describes the magnetization process of infinite spin ensembles. The last part of the book is concerned with a macroscopic deterministic equation which describes temperature effects on macro-spins, i.e. expectations of the solutions to the SLLG. Furthermore, comparative computational studies with the stochastic model are included.

We use constructive tools such as e.g. finite element methods to derive the theoretical results, which are then used for computational studies. The numerical experiments motivate an interesting interplay between inherent geometric and stochastic effects of the SLLG which still lack a rigorous analytical understanding: the role of space-time white noise, possible finite time blow-up behavior of solutions, long-time asymptotics, and effective dynamics.

DOI:: https://doi.org/10.1515/9783110307108
ISBN:: 978-3-11-030699-6
ISBN:: 978-3-11-030710-8
Subject:: Mathematics
Subject:: Probability and Statistics
Publisher:: De Gruyter

book: Volume 2 American-Type Options

Book 2015

Volume 2 American-Type Options

Stochastic Approximation Methods, Volume 2

Dmitrii S. Silvestrov

Part of the multi-volume work American-Type Options

Volume 57 in this series

The book gives a systematical presentation of stochastic approximation methods for discrete time Markov price processes. Advanced methods combining backward recurrence algorithms for computing of option rewards and general results on convergence of stochastic space skeleton and tree approximations for option rewards are applied to a variety of models of multivariate modulated Markov price processes. The principal novelty of presented results is based on consideration of multivariate modulated Markov price processes and general pay-off functions, which can depend not only on price but also an additional stochastic modulating index component, and use of minimal conditions of smoothness for transition probabilities and pay-off functions, compactness conditions for log-price processes and rate of growth conditions for pay-off functions. The volume presents results on structural studies of optimal stopping domains, Monte Carlo based approximation reward algorithms, and convergence of American-type options for autoregressive and continuous time models, as well as results of the corresponding experimental studies.

DOI:: https://doi.org/10.1515/9783110329841
ISBN:: 978-3-11-032968-1
ISBN:: 978-3-11-032984-1
ISBN:: 9783110389906
Subject:: Business and Economics
Subject:: Business Management
Subject:: Management Accounting, Financial Controlling, Cost Calculation, Investment
Subject:: Mathematics
Subject:: Probability and Statistics
Publisher:: De Gruyter

book: Volume 1 American-Type Options

Book 2014

Volume 1 American-Type Options

Stochastic Approximation Methods, Volume 1

Dmitrii S. Silvestrov

Part of the multi-volume work American-Type Options

Volume 56 in this series

The book gives a systematical presentation of stochastic approximation methods for models of American-type options with general pay-off functions for discrete time Markov price processes. Advanced methods combining backward recurrence algorithms for computing of option rewards and general results on convergence of stochastic space skeleton and tree approximations for option rewards are applied to a variety of models of multivariate modulated Markov price processes. The principal novelty of presented results is based on consideration of multivariate modulated Markov price processes and general pay-off functions, which can depend not only on price but also an additional stochastic modulating index component, and use of minimal conditions of smoothness for transition probabilities and pay-off functions, compactness conditions for log-price processes and rate of growth conditions for pay-off functions. The book also contains an extended bibliography of works in the area.

This book is the first volume of the comprehensive two volumes monograph. The second volume will present results on structural studies of optimal stopping domains, Monte Carlo based approximation reward algorithms, and convergence of American-type options for autoregressive and continuous time models, as well as results of the corresponding experimental studies.

DOI:: https://doi.org/10.1515/9783110329827
ISBN:: 978-3-11-032967-4
ISBN:: 978-3-11-032982-7
Subject:: Business and Economics
Subject:: Mathematics and Statistics for Economists
Subject:: Mathematics
Subject:: Mathematics
Subject:: Probability and Statistics
Publisher:: De Gruyter

book: Elliptic Partial Differential Equations

Book 2013

Elliptic Partial Differential Equations

Existence and Regularity of Distributional Solutions

Lucio Boccardo, Gisella Croce

Volume 55 in this series

Elliptic partial differential equations is one of the main and most active areas in mathematics. This book is devoted to the study of linear and nonlinear elliptic problems in divergence form, with the aim of providing classical results, as well as more recent developments about distributional solutions. For this reason this monograph is addressed to master's students, PhD students and anyone who wants to begin research in this mathematical field.

DOI:: https://doi.org/10.1515/9783110315424
ISBN:: 978-3-11-031540-0
ISBN:: 978-3-11-031542-4
Subject:: Mathematics
Subject:: Analysis
Publisher:: De Gruyter

book: Stochastic Calculus of Variations

Book 2023

Stochastic Calculus of Variations

For Jump Processes

Yasushi Ishikawa

Volume 54 in this series

This book is a concise introduction to the stochastic calculus of variations for processes with jumps. The author provides many results on this topic in a self-contained way for e.g., stochastic differential equations (SDEs) with jumps. The book also contains some applications of the stochastic calculus for processes with jumps to the control theory, mathematical finance and so. This third and entirely revised edition of the work is updated to reflect the latest developments in the theory and some applications with graphics.

DOI:: https://doi.org/10.1515/9783110675290
ISBN:: 9783110675290
ISBN:: 9783110675320
ISBN:: 9783110675283
Subject:: Mathematics
Subject:: Differential Equations and Dynamical Systems
Subject:: Probability and Statistics
Publisher:: De Gruyter

book: Stochastic Calculus of Variations

Book 2016

Stochastic Calculus of Variations

For Jump Processes

Yasushi Ishikawa

Volume 54 in this series

This monograph is a concise introduction to the stochastic calculus of variations (also known as Malliavin calculus) for processes with jumps. It is written for researchers and graduate students who are interested in Malliavin calculus for jump processes. In this book "processes with jumps" includes both pure jump processes and jump-diffusions. The author provides many results on this topic in a self-contained way; this also applies to stochastic differential equations (SDEs) "with jumps".
The book also contains some applications of the stochastic calculus for processes with jumps to the control theory and mathematical finance. Namely, asymptotic expansions functionals related with financial assets of jump-diffusion are provided based on the theory of asymptotic expansion on the Wiener–Poisson space. Solving the Hamilton–Jacobi–Bellman (HJB) equation of integro-differential type is related with solving the classical Merton problem and the Ramsey theory.
The field of jump processes is nowadays quite wide-ranging, from the Lévy processes to SDEs with jumps. Recent developments in stochastic analysis have enabled us to express various results in a compact form. Up to now, these topics were rarely discussed in a monograph.

Contents:
Preface
Preface to the second edition
Introduction
Lévy processes and Itô calculus
Perturbations and properties of the probability law
Analysis of Wiener–Poisson functionals
Applications
Appendix
Bibliography
List of symbols
Index

DOI:: https://doi.org/10.1515/9783110378078
ISBN:: 9783110378078
Subject:: Mathematics
Subject:: Differential Equations and Dynamical Systems
Subject:: Probability and Statistics
Publisher:: De Gruyter

book: Stochastic Calculus of Variations for Jump Processes

Book 2013

Stochastic Calculus of Variations for Jump Processes

Yasushi Ishikawa

Volume 54 in this series

This monograph is a concise introduction to the stochastic calculus of variations (also known as Malliavin calculus) for processes with jumps. It is written for researchers and graduate students who are interested in Malliavin calculus for jump processes. In this book processes "with jumps" includes both pure jump processes and jump-diffusions. The author provides many results on this topic in a self-contained way; this also applies to stochastic differential equations (SDEs) "with jumps".

The book also contains some applications of the stochastic calculus for processes with jumps to the control theory and mathematical finance. Namely, asymptotic expansions functionals related with financial assets of jump-diffusion are provided based on the theory of asymptotic expansion on the Wiener–Poisson space. Solving the Hamilton–Jacobi–Bellman (HJB) equation of integro-differential type is related with solving the classical Merton problem and the Ramsey theory.

The field of jump processes is nowadays quite wide-ranging, from the Lévy processes to SDEs with jumps. Recent developments in stochastic analysis have enabled us to express various results in a compact form. Up to now, these topics were rarely discussed in a monograph.

DOI:: https://doi.org/10.1515/9783110282009
ISBN:: 978-3-11-028180-4
ISBN:: 978-3-11-028200-9
Subject:: Mathematics
Subject:: Probability and Statistics
Publisher:: De Gruyter

book: Krichever–Novikov Type Algebras

Book 2014

Krichever–Novikov Type Algebras

Theory and Applications

Martin Schlichenmaier

Volume 53 in this series

Krichever and Novikov introduced certain classes of infinite dimensional Lie algebras to extend the Virasoro algebra and its related algebras to Riemann surfaces of higher genus. The author of this book generalized and extended them to a more general setting needed by the applications. Examples of applications are Conformal Field Theory, Wess-Zumino-Novikov-Witten models, moduli space problems, integrable systems, Lax operator algebras, and deformation theory of Lie algebra. Furthermore they constitute an important class of infinite dimensional Lie algebras which due to their geometric origin are still manageable.

This book gives an introduction for the newcomer to this exciting field of ongoing research in mathematics and will be a valuable source of reference for the experienced researcher. Beside the basic constructions and results also applications are presented.

DOI:: https://doi.org/10.1515/9783110279641
ISBN:: 978-3-11-026517-0
ISBN:: 978-3-11-027964-1
ISBN:: 9783110381474
Subject:: Mathematics
Subject:: Algebra and Number Theory
Subject:: Analysis
Publisher:: De Gruyter

book: Function Classes on the Unit Disc

Book 2019

Function Classes on the Unit Disc

An Introduction

Miroslav Pavlović

Volume 52 in this series

This revised and extended edition of a well-established monograph in function theory contains a study on various function classes on the disc, a number of new results and new or easy proofs of old but interesting theorems (for example, the Fefferman–Stein theorem on subharmonic behavior or the theorem on conjugate functions in Bergman spaces) and a full discussion on g-functions.

DOI:: https://doi.org/10.1515/9783110630855
ISBN:: 9783110630855
ISBN:: 9783110628654
ISBN:: 9783110628449
Subject:: Mathematics
Subject:: Analysis
Publisher:: De Gruyter

book: Function Classes on the Unit Disc

Book 2014

Function Classes on the Unit Disc

An Introduction

Miroslav Pavlović

Volume 52 in this series

This monograph contains a study on various function classes, a number of new results and new or easy proofs of old results (Fefferman-Stein theorem on subharmonic behavior, theorems on conjugate functions and fractional integration on Bergman spaces, Fefferman's duality theorem), which are interesting for specialists; applications of the Hardy-Littlewood inequalities on Taylor coefficients to (C, α)-maximal theorems and (C, α)-convergence; a study of BMOA, due to Knese, based only on Green's formula; the problem of membership of singular inner functions in Besov and Hardy-Sobolev spaces; a full discussion of g-function (all p > 0) and Calderón's area theorem; a new proof, due to Astala and Koskela, of the Littlewood-Paley inequality for univalent functions; and new results and proofs on Lipschitz spaces, coefficient multipliers and duality, including compact multipliers and multipliers on spaces with non-normal weights.

It also contains a discussion of analytic functions and lacunary series with values in quasi-Banach spaces with applications to function spaces and composition operators. Sixteen open questions are posed.

The reader is assumed to have a good foundation in Lebesgue integration, complex analysis, functional analysis, and Fourier series.

Further information can be found at the author's website at http://poincare.matf.bg.ac.rs/~pavlovic.

DOI:: https://doi.org/10.1515/9783110281903
ISBN:: 978-3-11-028123-1
ISBN:: 978-3-11-028190-3
Subject:: Mathematics
Subject:: Analysis
Publisher:: De Gruyter

book: Multivariate Characteristic and Correlation Functions

Book 2013

Multivariate Characteristic and Correlation Functions

Zoltán Sasvári

Volume 50 in this series

In a certain sense characteristic functions and correlation functions are the same, the common underlying concept is positive definiteness. Many results in probability theory, mathematical statistics and stochastic processes can be derived by using these functions. While there are books on characteristic functions of one variable, books devoting some sections to the multivariate case, and books treating the general case of locally compact groups, interestingly there is no book devoted entirely to the multidimensional case which is extremely important for applications. This book is intended to fill this gap at least partially. It makes the basic concepts and results on multivariate characteristic and correlation functions easily accessible to both students and researchers in a comprehensive manner.

The first chapter presents basic results and should be read carefully since it is essential for the understanding of the subsequent chapters. The second chapter is devoted to correlation functions, their applications to stationary processes and some connections to harmonic analysis. In Chapter 3 we deal with several special properties, Chapter 4 is devoted to the extension problem while Chapter 5 contains a few applications. A relatively large appendix comprises topics like infinite products, functional equations, special functions or compact operators.

DOI:: https://doi.org/10.1515/9783110223996
ISBN:: 978-3-11-022398-9
ISBN:: 978-3-11-022399-6
Subject:: Mathematics
Subject:: Probability and Statistics
Publisher:: De Gruyter

book: Metric Embeddings

Book 2013

Metric Embeddings

Bilipschitz and Coarse Embeddings into Banach Spaces

Mikhail I. Ostrovskii

Volume 49 in this series

Embeddings of discrete metric spaces into Banach spaces recently became an important tool in computer science and topology. The purpose of the book is to present some of the most important techniques and results, mostly on bilipschitz and coarse embeddings.

The topics include: (1) Embeddability of locally finite metric spaces into Banach spaces is finitely determined; (2) Constructions of embeddings; (3) Distortion in terms of Poincaré inequalities; (4) Constructions of families of expanders and of families of graphs with unbounded girth and lower bounds on average degrees; (5) Banach spaces which do not admit coarse embeddings of expanders; (6) Structure of metric spaces which are not coarsely embeddable into a Hilbert space; (7) Applications of Markov chains to embeddability problems; (8) Metric characterizations of properties of Banach spaces; (9) Lipschitz free spaces. Substantial part of the book is devoted to a detailed presentation of relevant results of Banach space theory and graph theory. The final chapter contains a list of open problems. Extensive bibliography is also included. Each chapter, except the open problems chapter, contains exercises and a notes and remarks section containing references, discussion of related results, and suggestions for further reading.

The book will help readers to enter and to work in a very rapidly developing area having many important connections with different parts of mathematics and computer science.

DOI:: https://doi.org/10.1515/9783110264012
ISBN:: 978-3-11-026340-4
ISBN:: 978-3-11-026401-2
Subject:: Computer Sciences
Subject:: Fundamentals of Computer Sciences
Subject:: Mathematics
Subject:: Analysis
Subject:: Geometry and Topology
Subject:: Discrete Mathematics
Publisher:: De Gruyter

book: Semi-Dirichlet Forms and Markov Processes

Book 2013

Semi-Dirichlet Forms and Markov Processes

Yoichi Oshima

Volume 48 in this series

This book deals with analytic treatments of Markov processes. Symmetric Dirichlet forms and their associated Markov processes are important and powerful tools in the theory of Markov processes and their applications. The theory is well studied and used in various fields. In this monograph, we intend to generalize the theory to non-symmetric and time dependent semi-Dirichlet forms. By this generalization, we can cover the wide class of Markov processes and analytic theory which do not possess the dual Markov processes. In particular, under the semi-Dirichlet form setting, the stochastic calculus is not well established yet. In this monograph, we intend to give an introduction to such calculus. Furthermore, basic examples different from the symmetric cases are given. The text is written for graduate students, but also researchers.

DOI:: https://doi.org/10.1515/9783110302066
ISBN:: 978-3-11-030200-4
ISBN:: 978-3-11-030206-6
Subject:: Mathematics
Subject:: Probability and Statistics
Publisher:: De Gruyter

book: Inverse Problems and Nonlinear Evolution Equations

Book 2013

Inverse Problems and Nonlinear Evolution Equations

Solutions, Darboux Matrices and Weyl–Titchmarsh Functions

Alexander L. Sakhnovich, Lev A. Sakhnovich, Inna Ya. Roitberg

Volume 47 in this series

This book is based on the method of operator identities and related theory of S-nodes, both developed by Lev Sakhnovich. The notion of the transfer matrix function generated by the S-node plays an essential role.

The authors present fundamental solutions of various important systems of differential equations using the transfer matrix function, that is, either directly in the form of the transfer matrix function or via the representation in this form of the corresponding Darboux matrix, when Bäcklund–Darboux transformations and explicit solutions are considered. The transfer matrix function representation of the fundamental solution yields solution of an inverse problem, namely, the problem to recover system from its Weyl function. Weyl theories of selfadjoint and skew-selfadjoint Dirac systems, related canonical systems, discrete Dirac systems, system auxiliary to the N-wave equation and a system rationally depending on the spectral parameter are obtained in this way.

The results on direct and inverse problems are applied in turn to the study of the initial-boundary value problems for integrable (nonlinear) wave equations via inverse spectral transformation method. Evolution of the Weyl function and solution of the initial-boundary value problem in a semi-strip are derived for many important nonlinear equations. Some uniqueness and global existence results are also proved in detail using evolution formulas.

The reading of the book requires only some basic knowledge of linear algebra, calculus and operator theory from the standard university courses.

DOI:: https://doi.org/10.1515/9783110258615
ISBN:: 978-3-11-025860-8
ISBN:: 978-3-11-025861-5
Subject:: Mathematics
Subject:: Analysis
Subject:: Physics
Subject:: Theoretical and Mathematical Physics
Publisher:: De Gruyter

book: [Set Singular Traces, Volume 1+2]

Book Print Only 2023

[Set Singular Traces, Volume 1+2]

Part of the multi-volume work Singular Traces

Volume 46 in this series

This book is the second edition of the first complete study and monograph dedicated to singular traces. The first volume offers, due to the contributions of Albrecht Pietsch and Nigel Kalton, a complete theory of traces and their spectral properties on ideals of compact operators, updated in this second edition on the fundamental new approach for compact operators, the Pietsch correspondence. For mathematical physicists and other users of Connes’ noncommutative geometry the text offers a complete reference to Dixmier traces and associated formulas involving residues of spectral zeta functions and asymptotics of partition functions. An operator based theory of pseudodifferential operators is used to detail the deep association between the noncommutative residue in differential geometry and singular traces. The second volume introduces noncommutative integration theory on semifinite von Neumann algebras and the theory of singular traces for symmetric operator spaces. Deeper aspects of the association between measurability, poles and residues of spectral zeta functions, and asymptotics of heat traces are studied. Applications in Connes’ noncommutative geometry that are detailed include integration of quantum differentials, measures on fractals, and Connes’ character formula concerning the Hochschild class of the Chern character.

ISBN:: 9783110703184
Subject:: Mathematics
Subject:: Analysis
Subject:: Numerical and Computational Mathematics
Publisher:: De Gruyter

book: Volume 2 Trace Formulas

Book 2023

Volume 2 Trace Formulas

Steven Lord, Edward McDonald, Fedor Sukochev, Dmitriy Zanin

Part of the multi-volume work Singular Traces

Volume 46/2 in this series

This volume introduces noncommutative integration theory on semifinite von Neumann algebras and the theory of singular traces for symmetric operator spaces. Deeper aspects of the association between measurability, poles and residues of spectral zeta functions, and asymptotics of heat traces are studied. Applications in Connes’ noncommutative geometry that are detailed include integration of quantum differentials, measures on fractals, and Connes’ character formula concerning the Hochschild class of the Chern character.

DOI:: https://doi.org/10.1515/9783110700176
ISBN:: 9783110700176
ISBN:: 9783110700244
ISBN:: 9783110700008
Subject:: Mathematics
Subject:: Analysis
Subject:: Numerical and Computational Mathematics
Subject:: General Mathematics
Publisher:: De Gruyter

book: Volume 1 Theory

Book 2021

Volume 1 Theory

Steven Lord, Fedor Sukochev, Dmitriy Zanin

Part of the multi-volume work Singular Traces

Volume 46/1 in this series

This book is the second edition of the first complete study and monograph dedicated to singular traces. The text offers, due to the contributions of Albrecht Pietsch and Nigel Kalton, a complete theory of traces and their spectral properties on ideals of compact operators on a separable Hilbert space. The second edition has been updated on the fundamental approach provided by Albrecht Pietsch. For mathematical physicists and other users of Connes’ noncommutative geometry the text offers a complete reference to traces on weak trace class operators, including Dixmier traces and associated formulas involving residues of spectral zeta functions and asymptotics of partition functions.

DOI:: https://doi.org/10.1515/9783110378054
ISBN:: 9783110378054
ISBN:: 9783110392319
ISBN:: 9783110377798
Subject:: Mathematics
Subject:: Analysis
Subject:: Numerical and Computational Mathematics
Publisher:: De Gruyter

book: Singular Traces

Book 2012

Singular Traces

Theory and Applications

Steven Lord, Fedor Sukochev, Dmitriy Zanin

Volume 46 in this series

This book is the first complete study and monograph dedicated to singular traces. The text mathematically formalises the study of traces in a self contained theory of functional analysis. Extensive notes will treat the historical development. The final section will contain the most complete and concise treatment known of the integration half of Connes' quantum calculus.

Singular traces are traces on ideals of compact operators that vanish on the subideal of finite rank operators. Singular traces feature in A. Connes' interpretation of noncommutative residues. Particularly the Dixmier trace, which generalises the restricted Adler-Manin-Wodzicki residue of pseudo-differential operators and plays the role of the residue for a new catalogue of 'geometric' spaces, including Connes-Chamseddine standard models, Yang-Mills action for quantum differential forms, fractals, isospectral deformations, foliations and noncommutative index theory.

The theory of singular traces has been studied after Connes' application to non-commutative geometry and physics by various authors. Recent work by Nigel Kalton and the authors has advanced the theory of singular traces. Singular traces can be equated to symmetric functionals of symmetric
sequence or function spaces, residues of zeta functions and heat kernel asymptotics, and characterised by Lidksii and Fredholm formulas. The traces and formulas used in noncommutative geometry are now completely understood in this theory, with surprising new mathematical and physical consequences.

For mathematical readers the text offers fundamental functional analysis results and, due to Nigel Kalton's contribution, a now complete theory of traces on compact operators. For mathematical physicists and other users of Connes' noncommutative geometry the text offers a complete reference to Dixmier traces and access to the deeper mathematical features of traces on ideals associated to the harmonic sequence. These features, not known and not discussed in general texts on noncommutative geometry, are undoubtably physical and probe to the fascinating heart of classical limits and quantization.

DOI:: https://doi.org/10.1515/9783110262551
ISBN:: 978-3-11-026250-6
ISBN:: 978-3-11-026255-1
Subject:: Mathematics
Subject:: Algebra and Number Theory
Subject:: Analysis
Subject:: Physics
Subject:: Theoretical and Mathematical Physics
Publisher:: De Gruyter

book: Narrow Operators on Function Spaces and Vector Lattices

Book 2012

Narrow Operators on Function Spaces and Vector Lattices

Mikhail Popov, Beata Randrianantoanina

Volume 45 in this series

Most classes of operators that are not isomorphic embeddings are characterized by some kind of a “smallness” condition. Narrow operators are those operators defined on function spaces that are “small” at {-1,0,1}-valued functions, e.g. compact operators are narrow. The original motivation to consider such operators came from theory of embeddings of Banach spaces, but since then they were also applied to the study of the Daugavet property and to other geometrical problems of functional analysis. The question of when a sum of two narrow operators is narrow, has led to deep developments of the theory of narrow operators, including an extension of the notion to vector lattices and investigations of connections to regular operators.

Narrow operators were a subject of numerous investigations during the last 30 years. This monograph provides a comprehensive presentation putting them in context of modern theory. It gives an in depth systematic exposition of concepts related to and influenced by narrow operators, starting from basic results and building up to most recent developments. The authors include a complete bibliography and many attractive open problems.

DOI:: https://doi.org/10.1515/9783110263343
ISBN:: 978-3-11-026303-9
ISBN:: 978-3-11-026334-3
Subject:: Mathematics
Subject:: Analysis
Publisher:: De Gruyter

book: Topology of Algebraic Curves

Book 2012

Topology of Algebraic Curves

An Approach via Dessins d'Enfants

Alex Degtyarev

Volume 44 in this series

This monograph summarizes and extends a number of results on the topology of trigonal curves in geometrically ruled surfaces. An emphasis is given to various applications of the theory to a few related areas, most notably singular plane curves of small degree, elliptic surfaces, and Lefschetz fibrations (both complex and real), and Hurwitz equivalence of braid monodromy factorizations.

The approach relies on a close relation between trigonal curves/elliptic surfaces, a certain class of ribbon graphs, and subgroups of the modular group, which provides a combinatorial framework for the study of geometric objects. A brief summary of the necessary auxiliary results and techniques used and a background of the principal problems dealt with are included in the text.

The book is intended to researchers and graduate students in the field of topology of complex and real algebraic varieties.

DOI:: https://doi.org/10.1515/9783110258424
ISBN:: 978-3-11-025591-1
ISBN:: 978-3-11-025842-4
Subject:: Mathematics
Subject:: Algebra and Number Theory
Subject:: Geometry and Topology
Publisher:: De Gruyter

book: Stochastic Models for Fractional Calculus

Book 2019

Stochastic Models for Fractional Calculus

Mark M. Meerschaert, Alla Sikorskii

Volume 43 in this series

Fractional calculus is a rapidly growing field of research, at the interface between probability, differential equations, and mathematical physics. It is used to model anomalous diffusion, in which a cloud of particles spreads in a different manner than traditional diffusion. This monograph develops the basic theory of fractional calculus and anomalous diffusion, from the point of view of probability.

In this book, we will see how fractional calculus and anomalous diffusion can be understood at a deep and intuitive level, using ideas from probability. It covers basic limit theorems for random variables and random vectors with heavy tails. This includes regular variation, triangular arrays, infinitely divisible laws, random walks, and stochastic process convergence in the Skorokhod topology. The basic ideas of fractional calculus and anomalous diffusion are closely connected with heavy tail limit theorems. Heavy tails are applied in finance, insurance, physics, geophysics, cell biology, ecology, medicine, and computer engineering.

The goal of this book is to prepare graduate students in probability for research in the area of fractional calculus, anomalous diffusion, and heavy tails. Many interesting problems in this area remain open. This book will guide the motivated reader to understand the essential background needed to read and unerstand current research papers, and to gain the insights and techniques needed to begin making their own contributions to this rapidly growing field.

DOI:: https://doi.org/10.1515/9783110560244
ISBN:: 9783110560244
ISBN:: 9783110559149
ISBN:: 9783110559071
Subject:: Mathematics
Subject:: Probability and Statistics
Publisher:: De Gruyter

book: Stochastic Models for Fractional Calculus

Book 2012

Stochastic Models for Fractional Calculus

Mark M. Meerschaert, Alla Sikorskii

Volume 43 in this series

Fractional calculus is a rapidly growing field of research, at the interface between probability, differential equations, and mathematical physics. It is used to model anomalous diffusion, in which a cloud of particles spreads in a different manner than traditional diffusion. This monograph develops the basic theory of fractional calculus and anomalous diffusion, from the point of view of probability.

In this book, we will see how fractional calculus and anomalous diffusion can be understood at a deep and intuitive level, using ideas from probability. It covers basic limit theorems for random variables and random vectors with heavy tails. This includes regular variation, triangular arrays, infinitely divisible laws, random walks, and stochastic process convergence in the Skorokhod topology. The basic ideas of fractional calculus and anomalous diffusion are closely connected with heavy tail limit theorems. Heavy tails are applied in finance, insurance, physics, geophysics, cell biology, ecology, medicine, and computer engineering.

The goal of this book is to prepare graduate students in probability for research in the area of fractional calculus, anomalous diffusion, and heavy tails. Many interesting problems in this area remain open. This book will guide the motivated reader to understand the essential background needed to read and unerstand current research papers, and to gain the insights and techniques needed to begin making their own contributions to this rapidly growing field.

DOI:: https://doi.org/10.1515/9783110258165
ISBN:: 978-3-11-025869-1
ISBN:: 978-3-11-025816-5
Subject:: Mathematics
Subject:: Probability and Statistics
Publisher:: De Gruyter

book: Green's Functions

Book 2012

Green's Functions

Construction and Applications

Yuri A. Melnikov, Max Y. Melnikov

Volume 42 in this series

Green's functions represent one of the classical and widely used issues in the area of differential equations.

This monograph is looking at applied elliptic and parabolic type partial differential equations in two variables. The elliptic type includes the Laplace, static Klein-Gordon and biharmonic equation. The parabolic type is represented by the classical heat equation and the Black-Scholes equation which has emerged as a mathematical model in financial mathematics. The book is attractive for practical needs: It contains many easily computable or computer friendly representations of Green's functions, includes all the standard Green's functions and many novel ones, and provides innovative and new approaches that might lead to Green's functions.

The book is a useful source for everyone who is studying or working in the fields of science, finance, or engineering that involve practical solution of partial differential equations.

DOI:: https://doi.org/10.1515/9783110253399
ISBN:: 978-3-11-025302-3
ISBN:: 978-3-11-025339-9
Subject:: Business and Economics
Subject:: Mathematics and Statistics for Economists
Subject:: Mathematics
Subject:: Mathematics
Subject:: Analysis
Publisher:: De Gruyter

book: Algebraic Graph Theory

Book 2019

Algebraic Graph Theory

Morphisms, Monoids and Matrices

Ulrich Knauer, Kolja Knauer

Volume 41 in this series

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

DOI:: https://doi.org/10.1515/9783110617368
ISBN:: 9783110617368
ISBN:: 9783110616286
ISBN:: 9783110616125
Subject:: Mathematics
Subject:: Algebra and Number Theory
Subject:: Combinatorics and Graph Theory
Subject:: Discrete Mathematics
Publisher:: De Gruyter

book: Algebraic Graph Theory

Book 2011

Algebraic Graph Theory

Morphisms, Monoids and Matrices

Ulrich Knauer

Volume 41 in this series

Graph models are extremely useful for almost all applications and applicators as they play an important role as structuring tools. They allow to model net structures – like roads, computers, telephones – instances of abstract data structures – like lists, stacks, trees – and functional or object oriented programming. In turn, graphs are models for mathematical objects, like categories and functors.

This highly self-contained book about algebraic graph theory is written with a view to keep the lively and unconventional atmosphere of a spoken text to communicate the enthusiasm the author feels about this subject. The focus is on homomorphisms and endomorphisms, matrices and eigenvalues. It ends with a challenging chapter on the topological question of embeddability of Cayley graphs on surfaces.

DOI:: https://doi.org/10.1515/9783110255096
ISBN:: 978-3-11-025408-2
ISBN:: 978-3-11-025509-6
Subject:: Mathematics
Subject:: Algebra and Number Theory
Subject:: Discrete Mathematics
Publisher:: De Gruyter

book: Differential Equations with Impulse Effects

Book 2011

Differential Equations with Impulse Effects

Multivalued Right-hand Sides with Discontinuities

Nikolai A. Perestyuk, Viktor A. Plotnikov, Anatolii M. Samoilenko, Natalia V. Skripnik, Peter V. Malyshev

Volume 40 in this series

Significant interest in the investigation of systems with discontinuous trajectories is explained by the development of equipment in which significant role is played by impulsive control systems and impulsive computing systems. Impulsive systems are also encountered in numerous problems of natural sciences described by mathematical models with conditions reflecting the impulsive action of external forces with pulses whose duration can be neglected.

Differential equations with set-valued right-hand side arise in the investigation of evolution processes in the case of measurement errors, inaccuracy or incompleteness of information, action of bounded perturbations, violation of unique solvability conditions, etc. Differential inclusions also allow one to describe the dynamics of controlled processes and are widely used in the theory of optimal control.

This monograph is devoted to the investigation of impulsive differential equations with set-valued and discontinuous right-hand sides. It is intended for researchers, lecturers, postgraduate students, and students of higher schools specialized in the field of the theory of differential equations, the theory of optimal control, and their applications.

DOI:: https://doi.org/10.1515/9783110218176
ISBN:: 978-3-11-021816-9
ISBN:: 978-3-11-021817-6
Subject:: Mathematics
Subject:: Analysis
Subject:: Probability and Statistics
Publisher:: De Gruyter

book: Theory of Interacting Quantum Fields

Book 2012

Theory of Interacting Quantum Fields

Alexei L. Rebenko, Peter V. Malyshev

Volume 39 in this series

This monograph is devoted to the systematic presentation of foundations of the quantum field theory. Unlike numerous monographs devoted to this topic, a wide range of problems covered in this book are accompanied by their sufficiently clear interpretations and applications. An important significant feature of this monograph is the desire of the author to present mathematical problems of the quantum field theory with regard to new methods of the constructive and Euclidean field theory that appeared in the last thirty years of the 20th century and are based on the rigorous mathematical apparatus of functional analysis, the theory of operators, and the theory of generalized functions.

The monograph is useful for students, post-graduate students, and young scientists who desire to understand not only the formality of construction of the quantum field theory but also its essence and connection with the classical mechanics, relativistic classical field theory, quantum mechanics, group theory, and the theory of path integral formalism.

DOI:: https://doi.org/10.1515/9783110250633
ISBN:: 978-3-11-025062-6
ISBN:: 978-3-11-025063-3
Subject:: Mathematics
Subject:: Analysis
Subject:: Physics
Subject:: Theoretical and Mathematical Physics
Publisher:: De Gruyter

book: Markov Processes, Semigroups and Generators

Book 2011

Markov Processes, Semigroups and Generators

Vassili N. Kolokoltsov

Volume 38 in this series

Markov processes represent a universal model for a large variety of real life random evolutions. The wide flow of new ideas, tools, methods and applications constantly pours into the ever-growing stream of research on Markov processes that rapidly spreads over new fields of natural and social sciences, creating new streamlined logical paths to its turbulent boundary. Even if a given process is not Markov, it can be often inserted into a larger Markov one (Markovianization procedure) by including the key historic parameters into the state space.

This monograph gives a concise, but systematic and self-contained, exposition of the essentials of Markov processes, together with recent achievements, working from the "physical picture" - a formal pre-generator, and stressing the interplay between probabilistic (stochastic differential equations) and analytic (semigroups) tools.

The book will be useful to students and researchers. Part I can be used for a one-semester course on Brownian motion, Lévy and Markov processes, or on probabilistic methods for PDE. Part II mainly contains the author's research on Markov processes.

From the contents:

Tools from Probability and Analysis
Brownian motion
Markov processes and martingales
SDE, ψDE and martingale problems
Processes in Euclidean spaces
Processes in domains with a boundary
Heat kernels for stable-like processes
Continuous-time random walks and fractional dynamics
Complex chains and Feynman integral

DOI:: https://doi.org/10.1515/9783110250114
ISBN:: 978-3-11-025010-7
ISBN:: 978-3-11-025011-4
Subject:: Mathematics
Subject:: Algebra and Number Theory
Subject:: Probability and Statistics
Publisher:: De Gruyter

book: Bernstein Functions

Book 2010

Bernstein Functions

Theory and Applications

René L. Schilling, Renming Song, Zoran Vondracek

Volume 37 in this series

This text is a self-contained and unified approach to Bernstein functions and their subclasses, bringing together old and establishing new connections. Applications of Bernstein functions in different fields of mathematics are given, with special attention to interpretations in probability theory. An extensive list of complete Bernstein functions with their representations is provided.

A self-contained and unified approach to the topic
With applications to various fields of mathematics, such as probability theory, potential theory, operator theory, integral equations, functional calculi and complex analysis
With an extensive list of complete Bernstein functions.
Additional material and corrections can be found on the authors' website.

DOI:: https://doi.org/10.1515/9783110215311
ISBN:: 978-3-11-021530-4
ISBN:: 978-3-11-021531-1
Subject:: Mathematics
Subject:: Analysis
Subject:: Probability and Statistics
Publisher:: De Gruyter

book: Bernstein Functions

Book 2012

Bernstein Functions

Theory and Applications

René L. Schilling, Renming Song, Zoran Vondracek

Volume 37 in this series

Bernstein functions appear in various fields of mathematics, e.g. probability theory, potential theory, operator theory, functional analysis and complex analysis – often with different definitions and under different names. Among the synonyms are `Laplace exponent' instead of Bernstein function, and complete Bernstein functions are sometimes called `Pick functions', `Nevanlinna functions' or `operator monotone functions'.

This monograph – now in its second revised and extended edition – offers a self-contained and unified approach to Bernstein functions and closely related function classes, bringing together old and establishing new connections. For the second edition the authors added a substantial amount of new material. As in the first edition Chapters 1 to 11 contain general material which should be accessible to non-specialists, while the later Chapters 12 to 15 are devoted to more specialized topics. An extensive list of complete Bernstein functions with their representations is provided.

DOI:: https://doi.org/10.1515/9783110269338
ISBN:: 978-3-11-025229-3
ISBN:: 978-3-11-026933-8
Subject:: Mathematics
Subject:: Analysis
Subject:: Probability and Statistics
Publisher:: De Gruyter

book: Introduction to Harmonic Analysis and Generalized Gelfand Pairs

Book 2009

Introduction to Harmonic Analysis and Generalized Gelfand Pairs

Gerrit van Dijk

Volume 36 in this series

This book is intended as an introduction to harmonic analysis and generalized Gelfand pairs. Starting with the elementary theory of Fourier series and Fourier integrals, the author proceeds to abstract harmonic analysis on locally compact abelian groups and Gelfand pairs. Finally a more advanced theory of generalized Gelfand pairs is developed.

This book is aimed at advanced undergraduates or beginning graduate students. The scope of the book is limited, with the aim of enabling students to reach a level suitable for starting PhD research. The main prerequisites for the book are elementary real, complex and functional analysis. In the later chapters, familiarity with some more advanced functional analysis is assumed, in particular with the spectral theory of (unbounded) self-adjoint operators on a Hilbert space.

From the contents

Fourier series
Fourier integrals
Locally compact groups
Haar measures
Harmonic analysis on locally compact abelian groups
Theory and examples of Gelfand pairs
Theory and examples of generalized Gelfand pairs

DOI:: https://doi.org/10.1515/9783110220209
ISBN:: 978-3-11-022019-3
ISBN:: 978-3-11-022020-9
Subject:: Mathematics
Subject:: Algebra and Number Theory
Subject:: Analysis
Publisher:: De Gruyter

book: Integral Representation Theory

Book 2010

Integral Representation Theory

Applications to Convexity, Banach Spaces and Potential Theory

Jaroslav Lukeš, Jan Malý, Ivan Netuka, Jirí Spurný

Volume 35 in this series

This monograph presents the state of the art of convexity, with an emphasis to integral representation. The exposition is focused on Choquet’s theory of function spaces with a link to compact convex sets. An important feature of the book is an interplay between various mathematical subjects, such as functional analysis, measure theory, descriptive set theory, Banach spaces theory and potential theory. A substantial part of the material is of fairly recent origin and many results appear in the book form for the first time. The text is self-contained and covers a wide range of applications.

From the contents:

Geometry of convex sets
Choquet theory of function spaces
Affine functions on compact convex sets
Perfect classes of functions and representation of affine functions
Simplicial function spaces
Choquet's theory of function cones
Topologies on boundaries
Several results on function spaces and compact convex sets
Continuous and measurable selectors
Construction of function spaces
Function spaces in potential theory and Dirichlet problem
Applications

DOI:: https://doi.org/10.1515/9783110203219
ISBN:: 978-3-11-020320-2
ISBN:: 978-3-11-020321-9
Subject:: Mathematics
Subject:: Analysis
Publisher:: De Gruyter

book: Volume 2 Applications in Rigorous Quantum Field Theory

Book 2020

Volume 2 Applications in Rigorous Quantum Field Theory

Fumio Hiroshima, József Lörinczi

Part of the multi-volume work Feynman-Kac-Type Theorems and Gibbs Measures on Path Space

Volume 34/2 in this series

This is the second updated and extended edition of the successful book on Feynman-Kac theory. It offers a state-of-the-art mathematical account of functional integration methods in the context of self-adjoint operators and semigroups using the concepts and tools of modern stochastic analysis. In the second volume, these ideas are applied principally to a rigorous treatment of some fundamental models of quantum field theory.

DOI:: https://doi.org/10.1515/9783110403541
ISBN:: 9783110403541
ISBN:: 9783110403602
ISBN:: 9783110403503
Subject:: Mathematics
Subject:: Analysis
Subject:: Probability and Statistics
Publisher:: De Gruyter

book: Volume 1 Feynman-Kac-Type Formulae and Gibbs Measures

Book 2020

Volume 1 Feynman-Kac-Type Formulae and Gibbs Measures

József Lörinczi, Fumio Hiroshima, Volker Betz

Part of the multi-volume work Feynman-Kac-Type Theorems and Gibbs Measures on Path Space

Volume 34/1 in this series

This is the second updated and extended edition of the successful book on Feynman-Kac theory. It offers a state-of-the-art mathematical account of functional integration methods in the context of self-adjoint operators and semigroups using the concepts and tools of modern stochastic analysis. The first volume concentrates on Feynman-Kac-type formulae and Gibbs measures.

DOI:: https://doi.org/10.1515/9783110330397
ISBN:: 9783110330397
ISBN:: 9783110389937
ISBN:: 9783110330045
Subject:: Mathematics
Subject:: Analysis
Subject:: Probability and Statistics
Publisher:: De Gruyter

book: Feynman-Kac-Type Theorems and Gibbs Measures on Path Space

Book 2011

Feynman-Kac-Type Theorems and Gibbs Measures on Path Space

With Applications to Rigorous Quantum Field Theory

József Lörinczi, Fumio Hiroshima, Volker Betz

Volume 34 in this series

This monograph offers a state-of-the-art mathematical account of functional integration methods in the context of self-adjoint operators and semigroups using the concepts and tools of modern stochastic analysis. These ideas are then applied principally to a rigorous treatment of some fundamental models of quantum field theory.

In this self-contained presentation of the material both beginners and experts are addressed, while putting emphasis on the interdisciplinary character of the subject.

DOI:: https://doi.org/10.1515/9783110203738
ISBN:: 9783110201482
ISBN:: 9783110203738
Subject:: Mathematics
Subject:: Analysis
Subject:: Probability and Statistics
Publisher:: De Gruyter

book: Computer Arithmetic and Validity

Book 2012

Computer Arithmetic and Validity

Theory, Implementation, and Applications

Ulrich Kulisch

Volume 33 in this series

This is the revised and extended second edition of the successful basic book on computer arithmetic. It is consistent with the newest recent standard developments in the field. The book shows how the arithmetic and mathematical capability of the digital computer can be enhanced in a quite natural way. The work is motivated by the desire and the need to improve the accuracy of numerical computing and to control the quality of the computed results (validity). The accuracy requirements for the elementary floating-point operations are extended to the customary product spaces of computations including interval spaces. The mathematical properties of these models are extracted into an axiomatic approach which leads to a general theory of computer arithmetic. Detailed methods and circuits for the implementation of this advanced computer arithmetic on digital computers are developed in part two of the book. Part three then illustrates by a number of sample applications how this extended computer arithmetic can be used to compute highly accurate and mathematically verified results. The book can be used as a high-level undergraduate textbook but also as reference work for research in computer arithmetic and applied mathematics.

DOI:: https://doi.org/10.1515/9783110301793
ISBN:: 978-3-11-030173-1
ISBN:: 978-3-11-030179-3
Subject:: Computer Sciences
Subject:: Computer Architecture
Subject:: Mathematics
Subject:: Numerical and Computational Mathematics
Subject:: Applied Mathematics
Publisher:: De Gruyter

book: Computer Arithmetic and Validity

Book 2008

Computer Arithmetic and Validity

Theory, Implementation, and Applications

Ulrich Kulisch

Volume 33 in this series

The present book deals with the theory of computer arithmetic, its implementation on digital computers and applications in applied mathematics to compute highly accurate and mathematically verified results. The aim is to improve the accuracy of numerical computing (by implementing advanced computer arithmetic) and to control the quality of the computed results (validity). The book can be useful as high-level undergraduate textbook but also as reference work for scientists researching computer arithmetic and applied mathematics.

DOI:: https://doi.org/10.1515/9783110203196
ISBN:: 978-3-11-020318-9
ISBN:: 978-3-11-020319-6
Subject:: Computer Sciences
Subject:: Computer Architecture
Subject:: Mathematics
Subject:: Applied Mathematics
Publisher:: De Gruyter

book: Circle-valued Morse Theory

Book 2006

Circle-valued Morse Theory

Andrei V. Pajitnov

Volume 32 in this series

In the early 1920s M. Morse discovered that the number of critical points of a smooth function on a manifold is closely related to the topology of the manifold. This became a starting point of the Morse theory which is now one of the basic parts of differential topology.

Circle-valued Morse theory originated from a problem in hydrodynamics studied by S. P. Novikov in the early 1980s. Nowadays, it is a constantly growing field of contemporary mathematics with applications and connections to many geometrical problems such as Arnold's conjecture in the theory of Lagrangian intersections, fibrations of manifolds over the circle, dynamical zeta functions, and the theory of knots and links in the three-dimensional sphere.

The aim of the book is to give a systematic treatment of geometric foundations of the subject and recent research results. The book is accessible to first year graduate students specializing in geometry and topology.

DOI:: https://doi.org/10.1515/9783110197976
ISBN:: 978-3-11-015807-6
ISBN:: 978-3-11-019797-6
Subject:: Mathematics
Subject:: Analysis
Subject:: Geometry and Topology
Publisher:: De Gruyter

book: Elliptic Curves

Book 2003

Elliptic Curves

A Computational Approach

Susanne Schmitt, Horst G. Zimmer, Attila Pethö

Volume 31 in this series

The basics of the theory of elliptic curves should be known to everybody, be he (or she) a mathematician or a computer scientist. Especially everybody concerned with cryptography should know the elements of this theory.

The purpose of the present textbook is to give an elementary introduction to elliptic curves. Since this branch of number theory is particularly accessible to computer-assisted calculations, the authors make use of it by approaching the theory under a computational point of view. Specifically, the computer-algebra package SIMATH can be applied on several occasions. However, the book can be read also by those not interested in any computations. Of course, the theory of elliptic curves is very comprehensive and becomes correspondingly sophisticated. That is why the authors made a choice of the topics treated. Topics covered include the determination of torsion groups, computations regarding the Mordell-Weil group, height calculations, S-integral points. The contents is kept as elementary as possible. In this way it becomes obvious in which respect the book differs from the numerous textbooks on elliptic curves nowadays available.

DOI:: https://doi.org/10.1515/9783110198010
ISBN:: 978-3-11-016808-2
ISBN:: 978-3-11-019801-0
Subject:: Mathematics
Subject:: Algebra and Number Theory
Publisher:: De Gruyter

book: The Reidemeister Torsion of 3-Manifolds

Book 2003

The Reidemeister Torsion of 3-Manifolds

Liviu I. Nicolaescu

Volume 30 in this series

This is a state-of-the-art introduction to the work of Franz Reidemeister, Meng Taubes, Turaev, and the author on the concept of torsion and its generalizations. Torsion is the oldest topological (but not with respect to homotopy) invariant that in its almost eight decades of existence has been at the center of many important and surprising discoveries. During the past decade, in the work of Vladimir Turaev, new points of view have emerged, which turned out to be the "right ones" as far as gauge theory is concerned.

The book features mostly the new aspects of this venerable concept. The theoretical foundations of this subject are presented in a style accessible to those, who wish to learn and understand the main ideas of the theory. Particular emphasis is upon the many and rather diverse concrete examples and techniques which capture the subleties of the theory better than any abstract general result. Many of these examples and techniques never appeared in print before, and their choice is often justified by ongoing current research on the topology of surface singularities.

The text is addressed to mathematicians with geometric interests who want to become comfortable users of this versatile invariant.

DOI:: https://doi.org/10.1515/9783110198102
ISBN:: 978-3-11-017383-3
ISBN:: 978-3-11-019810-2
Subject:: Mathematics
Subject:: Geometry and Topology
Publisher:: De Gruyter

book: Discontinuous Groups of Isometries in the Hyperbolic Plane

Book 2002

Discontinuous Groups of Isometries in the Hyperbolic Plane

Werner Fenchel, Jakob Nielsen, Asmus L. Schmidt

Volume 29 in this series

This is an introductory textbook on isometry groups of the hyperbolic plane. Interest in such groups dates back more than 120 years. Examples appear in number theory (modular groups and triangle groups), the theory of elliptic functions, and the theory of linear differential equations in the complex domain (giving rise to the alternative name Fuchsian groups).

The current book is based on what became known as the famous Fenchel-Nielsen manuscript. Jakob Nielsen (1890-1959) started this project well before World War II, and his interest arose through his deep investigations on the topology of Riemann surfaces and from the fact that the fundamental group of a surface of genus greater than one is represented by such a discontinuous group. Werner Fenchel (1905-1988) joined the project later and overtook much of the preparation of the manuscript. The present book is special because of its very complete treatment of groups containing reversions and because it avoids the use of matrices to represent Moebius maps.

This text is intended for students and researchers in the many areas of mathematics that involve the use of discontinuous groups.

DOI:: https://doi.org/10.1515/9783110891355
ISBN:: 978-3-11-017526-4
ISBN:: 978-3-11-089135-5
Subject:: Mathematics
Subject:: Analysis
Subject:: Geometry and Topology
Publisher:: De Gruyter

book: Painlevé Differential Equations in the Complex Plane

Book 2002

Painlevé Differential Equations in the Complex Plane

Valerii I. Gromak, Ilpo Laine, Shun Shimomura

Volume 28 in this series

This book is the first comprehensive treatment of Painlevé differential equations in the complex plane. Starting with a rigorous presentation for the meromorphic nature of their solutions, the Nevanlinna theory will be applied to offer a detailed exposition of growth aspects and value distribution of Painlevé transcendents. The subsequent main part of the book is devoted to topics of classical background such as representations and expansions of solutions, solutions of special type like rational and special transcendental solutions, Bäcklund transformations and higher order analogues, treated separately for each of these six equations. The final chapter offers a short overview of applications of Painlevé equations, including an introduction to their discrete counterparts. Due to the present important role of Painlevé equations in physical applications, this monograph should be of interest to researchers in both mathematics and physics and to graduate students interested in mathematical physics and the theory of differential equations.

DOI:: https://doi.org/10.1515/9783110198096
ISBN:: 978-3-11-017379-6
ISBN:: 978-3-11-019809-6
Subject:: Mathematics
Subject:: Analysis
Publisher:: De Gruyter

book: Stochastic Finance

Book 2004

Stochastic Finance

An Introduction in Discrete Time

Hans Föllmer, Alexander Schied

Volume 27 in this series

This book is an introduction to financial mathematics.

The first part of the book studies a simple one-period model which serves as a building block for later developments. Topics include the characterization of arbitrage-free markets, preferences on asset profiles, an introduction to equilibrium analysis, and monetary measures of risk.

In the second part, the idea of dynamic hedging of contingent claims is developed in a multiperiod framework. Such models are typically incomplete: They involve intrinsic risks which cannot be hedged away completely. Topics include martingale measures, pricing formulas for derivatives, American options, superhedging, and hedging strategies with minimal shortfall risk.

In addition to many corrections and improvements, this second edition contains several new sections, including a systematic discussion of law-invariant risk measures and of the connections between American options, superhedging, and dynamic risk measures.

DOI:: https://doi.org/10.1515/9783110212075
ISBN:: 9783110212075
ISBN:: 9783110183467
Subject:: Mathematics
Subject:: Probability and Statistics
Subject:: Applied Mathematics
Publisher:: De Gruyter

book: Measure and Integration Theory

Book 2001

Measure and Integration Theory

Heinz Bauer, Robert B. Burckel

Volume 26 in this series

This book gives a straightforward introduction to the field as it is nowadays required in many branches of analysis and especially in probability theory. The first three chapters (Measure Theory, Integration Theory, Product Measures) basically follow the clear and approved exposition given in the author's earlier book on "Probability Theory and Measure Theory". Special emphasis is laid on a complete discussion of the transformation of measures and integration with respect to the product measure, convergence theorems, parameter depending integrals, as well as the Radon-Nikodym theorem.

The final chapter, essentially new and written in a clear and concise style, deals with the theory of Radon measures on Polish or locally compact spaces. With the main results being Luzin's theorem, the Riesz representation theorem, the Portmanteau theorem, and a characterization of locally compact spaces which are Polish, this chapter is a true invitation to study topological measure theory.

The text addresses graduate students, who wish to learn the fundamentals in measure and integration theory as needed in modern analysis and probability theory. It will also be an important source for anyone teaching such a course.

DOI:: https://doi.org/10.1515/9783110866209
ISBN:: 978-3-11-016719-1
ISBN:: 978-3-11-086620-9
Subject:: Mathematics
Subject:: Analysis
Publisher:: De Gruyter

book: The Structure of Compact Groups

Book 2020

The Structure of Compact Groups

A Primer for the Student – A Handbook for the Expert

Karl H. Hofmann, Sidney A. Morris

Volume 25 in this series

This book is designed both as a textbook for high-level graduate courses and as a reference for researchers who need to apply the structure and representation theory of compact groups. A gentle introduction to compact groups and their representation theory is followed by self-contained courses on linear and compact Lie groups, and on locally compact abelian groups.

This fourth edition was updated with the latest developments in the field.

DOI:: https://doi.org/10.1515/9783110695991
ISBN:: 9783110695991
Subject:: Mathematics
Subject:: Algebra and Number Theory
Subject:: Geometry and Topology
Publisher:: De Gruyter

book: The Structure of Compact Groups

Book 2013

The Structure of Compact Groups

A Primer for the Student - A Handbook for the Expert

Karl H. Hofmann, Sidney A. Morris

Volume 25 in this series

The subject matter of compact groups is frequently cited in fields like algebra, topology, functional analysis, and theoretical physics. This book serves the dual purpose of providing a textbook on it for upper level graduate courses or seminars, and of serving as a source book for research specialists who need to apply the structure and representation theory of compact groups.

After a gentle introduction to compact groups and their representation theory, the book presents self-contained courses on linear Lie groups, on compact Lie groups, and on locally compact abelian groups. Separate appended chapters contain the material for courses on abelian groups and on category theory. However, the thrust of the book points in the direction of the structure theory of not necessarily finite dimensional, nor necessarily commutative, compact groups, unfettered by weight restrictions or dimensional bounds. In the process it utilizes infinite dimensional Lie algebras and the exponential function of arbitrary compact groups.

The first edition of 1998 and the second edition of 2006 were well received by reviewers and have been frequently quoted in the areas of instruction and research. For the present new edition the text has been cleaned of typographical flaws and the content has been conceptually sharpened in some places and polished and improved in others. New material has been added to various sections taking into account the progress of research on compact groups both by the authors and other writers. Motivation was provided, among other things, by questions about the structure of compact groups put to the authors by readers through the years following the earlier editions. Accordingly, the authors wished to clarify some aspects of the book which they felt needed improvement. The list of references has increased as the authors included recent publications pertinent to the content of the book.

DOI:: https://doi.org/10.1515/9783110296792
ISBN:: 978-3-11-029655-6
ISBN:: 978-3-11-029679-2
Subject:: Mathematics
Subject:: Algebra and Number Theory
Subject:: Geometry and Topology
Publisher:: De Gruyter

book: The Structure of Compact Groups

Book 2006

The Structure of Compact Groups

A Primer for Students - A Handbook for the Expert

Karl H. Hofmann, Sidney A. Morris

Volume 25 in this series

Dealing with subject matter of compact groups that is frequently cited in fields like algebra, topology, functional analysis, and theoretical physics, this book has been conceived with the dual purpose of providing a text book for upper level graduate courses or seminars, and of serving as a source book for research specialists who need to apply the structure and representation theory of compact groups.

After a gentle introduction to compact groups and their representation theory, the book presents self-contained courses on linear Lie groups, on compact Lie groups, and on locally compact abelian groups. However, the thrust of book points in the direction of the structure theory of infinite dimensional, not necessarily commutative compact groups, unfettered by weight restrictions or dimensional bounds. In the process it utilizes infinite dimensional Lie algebras and the exponential function of arbitrary compact groups.

The first edition of 1998 was well received by reviewers and has been frequently quoted in the areas of instruction and research. For the present new edition the text has been cleaned of typographical flaws and some minor inaccuracies of content; it has been edited and improved in various sections. New material has been added in order to reflect ongoing research. In the process of revising the original edition, the integrity of the original section numbering was carefully respected so that citations of material from the first edition remains perfectly viable to the users of this edition.

DOI:: https://doi.org/10.1515/9783110199772
ISBN:: 978-3-11-019006-9
ISBN:: 978-3-11-019977-2
Subject:: Mathematics
Subject:: Algebra and Number Theory
Subject:: Geometry and Topology
Publisher:: De Gruyter

book: Variational Methods for Potential Operator Equations

Book 1997

Variational Methods for Potential Operator Equations

With Applications to Nonlinear Elliptic Equations

Jan H. Chabrowski

Volume 24 in this series

DOI:: https://doi.org/10.1515/9783110809374
ISBN:: 978-3-11-015269-2
ISBN:: 978-3-11-080937-4
Subject:: Mathematics
Subject:: Analysis
Publisher:: De Gruyter

book: Probability Theory

Book 1996

Probability Theory

Heinz Bauer, Robert B. Burckel

Volume 23 in this series

DOI:: https://doi.org/10.1515/9783110814668
ISBN:: 978-3-11-013935-8
ISBN:: 978-3-11-081466-8
Subject:: Mathematics
Subject:: Probability and Statistics
Publisher:: De Gruyter

book: Methods of Noncommutative Analysis

Book 1996

Methods of Noncommutative Analysis

Theory and Applications

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

Volume 22 in this series

DOI:: https://doi.org/10.1515/9783110813548
ISBN:: 978-3-11-014632-5
ISBN:: 978-3-11-081354-8
Subject:: Mathematics
Subject:: Analysis
Publisher:: De Gruyter

book: Potential Theory on Infinite-Dimensional Abelian Groups

Book 1995

Potential Theory on Infinite-Dimensional Abelian Groups

Alexander Bendikov, Carol Regher

Volume 21 in this series

DOI:: https://doi.org/10.1515/9783110876840
ISBN:: 978-3-11-014283-9
ISBN:: 978-3-11-087684-0
Subject:: Mathematics
Subject:: Algebra and Number Theory
Subject:: Analysis
Publisher:: De Gruyter

book: Harmonic Analysis of Probability Measures on Hypergroups

Book 1995

Harmonic Analysis of Probability Measures on Hypergroups

Walter R. Bloom, Herbert Heyer

Volume 20 in this series

DOI:: https://doi.org/10.1515/9783110877595
ISBN:: 9783110121056
ISBN:: 9783110877595
Subject:: Mathematics
Subject:: Algebra and Number Theory
Subject:: Analysis
Subject:: Probability and Statistics
Publisher:: De Gruyter

book: Dirichlet Forms and Symmetric Markov Processes

Book 1994

Dirichlet Forms and Symmetric Markov Processes

Masatoshi Fukushima, Yoichi Oshima, Masayoshi Takeda

Volume 19 in this series

DOI:: https://doi.org/10.1515/9783110889741
ISBN:: 978-3-11-011626-7
ISBN:: 978-3-11-088974-1
Subject:: Mathematics
Subject:: Probability and Statistics
Publisher:: De Gruyter

book: Dirichlet Forms and Symmetric Markov Processes

Book 2010

Dirichlet Forms and Symmetric Markov Processes

Masatoshi Fukushima, Yoichi Oshima, Masayoshi Takeda

Volume 19 in this series

This book contains an introductory and comprehensive account of the theory of (symmetric) Dirichlet forms. Moreover this analytic theory is unified with the probabilistic potential theory based on symmetric Markov processes and developed further in conjunction with the stochastic analysis based on additive functional.

Since the publication of the first edition in 1994, this book has attracted constant interests from readers and is by now regarded as a standard reference for the theory of Dirichlet forms. For the present second edition, the authors not only revised the existing text, but also added sections on capacities and Sobolev type inequalities, irreducible recurrence and ergodicity, recurrence and Poincaré type inequalities, the Donsker-Varadhan type large deviation principle, as well as several new exercises with solutions.

The book addresses to researchers and graduate students who wish to comprehend the area of Dirichlet forms and symmetric Markov processes.

DOI:: https://doi.org/10.1515/9783110218091
ISBN:: 978-3-11-021808-4
ISBN:: 978-3-11-021809-1
Subject:: Mathematics
Subject:: Probability and Statistics
Publisher:: De Gruyter

book: Quantum Invariants of Knots and 3-Manifolds

Book 2010

Quantum Invariants of Knots and 3-Manifolds

Vladimir G. Turaev

Volume 18 in this series

Due to the strong appeal and wide use of this monograph, it is now available in its second revised edition. The monograph gives a systematic treatment of 3-dimensional topological quantum field theories (TQFTs) based on the work of the author with N. Reshetikhin and O. Viro. This subject was inspired by the discovery of the Jones polynomial of knots and the Witten-Chern-Simons field theory. On the algebraic side, the study of 3-dimensional TQFTs has been influenced by the theory of braided categories and the theory of quantum groups.

The book is divided into three parts. Part I presents a construction of 3-dimensional TQFTs and 2-dimensional modular functors from so-called modular categories. This gives a vast class of knot invariants and 3-manifold invariants as well as a class of linear representations of the mapping class groups of surfaces. In Part II the technique of 6j-symbols is used to define state sum invariants of 3-manifolds. Their relation to the TQFTs constructed in Part I is established via the theory of shadows. Part III provides constructions of modular categories, based on quantum groups and skein modules of tangles in the 3-space.

This fundamental contribution to topological quantum field theory is accessible to graduate students in mathematics and physics with knowledge of basic algebra and topology. It is an indispensable source for everyone who wishes to enter the forefront of this fascinating area at the borderline of mathematics and physics.

From the contents:

Invariants of graphs in Euclidean 3-space and of closed 3-manifolds
Foundations of topological quantum field theory
Three-dimensional topological quantum field theory
Two-dimensional modular functors
6j-symbols
Simplicial state sums on 3-manifolds
Shadows of manifolds and state sums on shadows
Constructions of modular categories

DOI:: https://doi.org/10.1515/9783110221848
ISBN:: 978-3-11-022183-1
ISBN:: 978-3-11-022184-8
Subject:: Mathematics
Subject:: Algebra and Number Theory
Subject:: Geometry and Topology
Subject:: Physics
Subject:: Quantum Physics
Publisher:: De Gruyter

book: Quantum Invariants of Knots and 3-Manifolds

Book 1994

Quantum Invariants of Knots and 3-Manifolds

Vladimir G. Turaev

Volume 18 in this series

This monograph provides a systematic treatment of topological quantum field theories (TQFT's) in three dimensions, inspired by the discovery of the Jones polynomial of knots, the Witten-Chern-Simons field theory, and the theory of quantum groups. The author, one of the leading experts in the subject, gives a rigorous and self-contained exposition of new fundamental algebraic and topological concepts that emerged in this theory.

The book is divided into three parts. Part I presents a construction of 3-dimensional TQFT's and 2-dimensional modular functors from so-called modular categories. This gives new knot and 3-manifold invariants as well as linear representations of the mapping class groups of surfaces. In Part II the machinery of 6j-symbols is used to define state sum invariants of 3-manifolds. Their relation to the TQFT's constructed in Part I is established via the theory of shadows. Part III provides constructions of modular categories, based on quantum groups and Kauffman's skein modules.

This book is accessible to graduate students in mathematics and physics with a knowledge of basic algebra and topology. It will be an indispensable source for everyone who wishes to enter the forefront of this rapidly growing and fascinating area at the borderline of mathematics and physics. Most of the results and techniques presented here appear in book form for the first time.

DOI:: https://doi.org/10.1515/9783110883275
ISBN:: 9783110883275
ISBN:: 9783110137040
Subject:: Mathematics
Subject:: Algebra and Number Theory
Subject:: Geometry and Topology
Subject:: Physics
Subject:: Quantum Physics
Publisher:: De Gruyter

book: Quantum Invariants of Knots and 3-Manifolds

Book 2016

Quantum Invariants of Knots and 3-Manifolds

Vladimir G. Turaev

Volume 18 in this series

Due to the strong appeal and wide use of this monograph, it is now available in its third revised edition. The monograph gives a systematic treatment of 3-dimensional topological quantum field theories (TQFTs) based on the work of the author with N. Reshetikhin and O. Viro. This subject was inspired by the discovery of the Jones polynomial of knots and the Witten-Chern-Simons field theory. On the algebraic side, the study of 3-dimensional TQFTs has been influenced by the theory of braided categories and the theory of quantum groups.
The book is divided into three parts. Part I presents a construction of 3-dimensional TQFTs and 2-dimensional modular functors from so-called modular categories. This gives a vast class of knot invariants and 3-manifold invariants as well as a class of linear representations of the mapping class groups of surfaces. In Part II the technique of 6j-symbols is used to define state sum invariants of 3-manifolds. Their relation to the TQFTs constructed in Part I is established via the theory of shadows. Part III provides constructions of modular categories, based on quantum groups and skein modules of tangles in the 3-space.
This fundamental contribution to topological quantum field theory is accessible to graduate students in mathematics and physics with knowledge of basic algebra and topology. It is an indispensable source for everyone who wishes to enter the forefront of this fascinating area at the borderline of mathematics and physics.

Contents:
Invariants of graphs in Euclidean 3-space and of closed 3-manifolds
Foundations of topological quantum field theory
Three-dimensional topological quantum field theory
Two-dimensional modular functors
6j-symbols
Simplicial state sums on 3-manifolds
Shadows of manifolds and state sums on shadows
Constructions of modular categories

DOI:: https://doi.org/10.1515/9783110435221
ISBN:: 9783110435221
ISBN:: 9783110442663
ISBN:: 9783110434569
Subject:: Mathematics
Subject:: Algebra and Number Theory
Subject:: Geometry and Topology
Subject:: Physics
Subject:: Quantum Physics
Publisher:: De Gruyter

book: Korovkin-type Approximation Theory and Its Applications

Book 1994

Korovkin-type Approximation Theory and Its Applications

Francesco Altomare, Michele Campiti

Volume 17 in this series

DOI:: https://doi.org/10.1515/9783110884586
ISBN:: 9783110141788
ISBN:: 9783110884586
Subject:: Mathematics
Subject:: Analysis
Publisher:: De Gruyter

book: Rational Iteration

Book 1993

Rational Iteration

Complex Analytic Dynamical Systems

Norbert Steinmetz

Volume 16 in this series

DOI:: https://doi.org/10.1515/9783110889314
ISBN:: 9783110137651
ISBN:: 9783110889314
Subject:: Mathematics
Subject:: Analysis
Subject:: Differential Equations and Dynamical Systems
Publisher:: De Gruyter

book: Nevanlinna Theory and Complex Differential Equations

Book 1993

Nevanlinna Theory and Complex Differential Equations

Ilpo Laine

Volume 15 in this series

DOI:: https://doi.org/10.1515/9783110863147
ISBN:: 978-3-11-013422-3
ISBN:: 978-3-11-086314-7
Subject:: Mathematics
Subject:: Analysis
Publisher:: De Gruyter

book: Dirichlet Forms and Analysis on Wiener Space

Book 1991

Dirichlet Forms and Analysis on Wiener Space

Nicolas Bouleau, Francis Hirsch

Volume 14 in this series

The subject of this book is analysis on Wiener space by means of Dirichlet forms and Malliavin calculus. There are already several literature on this topic, but this book has some different viewpoints.

First the authors review the theory of Dirichlet forms, but they observe only functional analytic, potential theoretical and algebraic properties. They do not mention the relation with Markov processes or stochastic calculus as discussed in usual books (e.g. Fukushima’s book). Even on analytic properties, instead of mentioning the Beuring-Deny formula, they discuss “carré du champ” operators introduced by Meyer and Bakry very carefully. Although they discuss when this “carré du champ” operator exists in general situation, the conditions they gave are rather hard to verify, and so they verify them in the case of Ornstein-Uhlenbeck operator in Wiener space later. (It should be noticed that one can easily show the existence of “carré du champ” operator in this case by using Shigekawa’s H-derivative.)

In the part on Malliavin calculus, the authors mainly discuss the absolute continuity of the probability law of Wiener functionals. The Dirichlet form corresponds to the first derivative only, and so it is not easy to consider higher order derivatives in this framework. This is the reason why they discuss only the first step of Malliavin calculus. On the other hand, they succeeded to deal with some delicate problems (the absolute continuity of the probability law of the solution to stochastic differential equations with Lipschitz continuous coefficients, the domain of stochastic integrals (Itô-Ramer-Skorokhod integrals), etc.).

This book focuses on the abstract structure of Dirichlet forms and Malliavin calculus rather than their applications. However, the authors give a lot of exercises and references and they may help the reader to study other topics which are not discussed in this book.

Zentralblatt Math, Reviewer: S.Kusuoka (Hongo)

DOI:: https://doi.org/10.1515/9783110858389
ISBN:: 978-3-11-012919-9
ISBN:: 978-3-11-085838-9
Subject:: Mathematics
Subject:: Analysis
Subject:: Probability and Statistics
Publisher:: De Gruyter

book: Ordinary Differential Equations

Book 1990

Ordinary Differential Equations

An Introduction to Nonlinear Analysis

Herbert Amann, Gerhard Metzen

Volume 13 in this series

DOI:: https://doi.org/10.1515/9783110853698
ISBN:: 9783110115154
ISBN:: 9783110853698
Subject:: Mathematics
Subject:: Analysis
Publisher:: De Gruyter

book: Transcendental Numbers

Book 1989

Transcendental Numbers

Andrei B. Shidlovskii, W. D. Brownawell, Neal Koblitz

Volume 12 in this series

DOI:: https://doi.org/10.1515/9783110889055
ISBN:: 978-3-11-011568-0
ISBN:: 978-3-11-088905-5
Subject:: Mathematics
Subject:: Algebra and Number Theory
Publisher:: De Gruyter

book: Elementary Geometry in Hyperbolic Space

Book 1989

Elementary Geometry in Hyperbolic Space

Werner Fenchel

Volume 11 in this series

DOI:: https://doi.org/10.1515/9783110849455
ISBN:: 978-3-11-011734-9
ISBN:: 978-3-11-084945-5
Subject:: Mathematics
Subject:: Geometry and Topology
Publisher:: De Gruyter

book: Analyticity in Infinite Dimensional Spaces

Book 1989

Analyticity in Infinite Dimensional Spaces

Michel Hervé

Volume 10 in this series

DOI:: https://doi.org/10.1515/9783110856941
ISBN:: 978-3-11-010995-5
ISBN:: 978-3-11-085694-1
Subject:: Mathematics
Subject:: Analysis
Publisher:: De Gruyter

book: Gibbs Measures and Phase Transitions

Book 2011

Gibbs Measures and Phase Transitions

Hans-Otto Georgii

Volume 9 in this series

DOI:: https://doi.org/10.1515/9783110250329
ISBN:: 978-3-11-025029-9
ISBN:: 978-3-11-025032-9
Subject:: Mathematics
Subject:: Probability and Statistics
Subject:: Physics
Subject:: Theoretical and Mathematical Physics
Publisher:: De Gruyter

book: Transformation Groups

Book 1987

Transformation Groups

Tammo tom Dieck

Volume 8 in this series

“This book is a jewel – it explains important, useful and deep topics in Algebraic Topology that you won’t find elsewhere, carefully and in detail.”
Prof. Günter M. Ziegler, TU Berlin

DOI:: https://doi.org/10.1515/9783110858372
ISBN:: 978-3-11-009745-0
ISBN:: 978-3-11-085837-2
Subject:: Mathematics
Subject:: Algebra and Number Theory
Subject:: Analysis
Subject:: Geometry and Topology
Publisher:: De Gruyter

book: Mathematical Theory of Statistics

Book 1985

Mathematical Theory of Statistics

Statistical Experiments and Asymptotic Decision Theory

Helmut Strasser

Volume 7 in this series

DOI:: https://doi.org/10.1515/9783110850826
ISBN:: 9783110102581
ISBN:: 9783110850826
Subject:: Mathematics
Subject:: Probability and Statistics
Publisher:: De Gruyter

book: Ergodic Theorems

Book 1985

Ergodic Theorems

Ulrich Krengel, Antoine Brunel

Volume 6 in this series

DOI:: https://doi.org/10.1515/9783110844641
ISBN:: 978-3-11-008478-8
ISBN:: 978-3-11-084464-1
Subject:: Mathematics
Subject:: Analysis
Subject:: Probability and Statistics
Publisher:: De Gruyter

Book 2013

Knots

Gerhard Burde, Heiner Zieschang, Michael Heusener

Volume 5 in this series

This 3. edition is an introduction to classical knot theory. It contains many figures and some tables of invariants of knots. This comprehensive account is an indispensable reference source for anyone interested in both classical and modern knot theory. Most of the topics considered in the book are developed in detail; only the main properties of fundamental groups and some basic results of combinatorial group theory are assumed to be known.

DOI:: https://doi.org/10.1515/9783110270785
ISBN:: 978-3-11-027074-7
ISBN:: 978-3-11-027078-5
Subject:: Mathematics
Subject:: Geometry and Topology
Publisher:: De Gruyter

Book 2002

Knots

Gerhard Burde, Heiner Zieschang

Volume 5 in this series

This book is an introduction to classical knot theory. Topics covered include: different constructions of knots, knot diagrams, knot groups, fibred knots, characterisation of torus knots, prime decomposition of knots, cyclic coverings and Alexander polynomials and modules together with the free differential calculus, braids, branched coverings and knots, Montesinos links, representations of knot groups, surgery of 3-manifolds and knots.

Knot theory has expanded enormously since the first edition of this book published in 1985. A special feature of this second completely revised and extended edition is the introduction to two new constructions of knot invariants, namely the Jones and homfly polynomials and the Vassiliev invariants.

The book contains many figures and some tables of invariants of knots. This comprehensive account is an indispensable reference source for anyone interested in both classical and modern knot theory.

Most of the topics considered in the book are developed in detail; only the main properties of fundamental groups and some basic results of combinatorial group theory are assumed to be known. The text is accessible to advanced undergraduate and graduate students in mathematics.

DOI:: https://doi.org/10.1515/9783110198034
ISBN:: 978-3-11-017005-4
ISBN:: 978-3-11-019803-4
Subject:: Mathematics
Subject:: Geometry and Topology
Publisher:: De Gruyter

book: Spaces of Measures

Book 1984

Spaces of Measures

Frank A. Chervenak

Volume 4 in this series

DOI:: https://doi.org/10.1515/9783110853995
ISBN:: 978-3-11-008784-0
ISBN:: 978-3-11-085399-5
Subject:: Mathematics
Subject:: Analysis
Subject:: Geometry and Topology
Publisher:: De Gruyter

book: Holomorphic Functions of Several Variables

Book 1983

Holomorphic Functions of Several Variables

An Introduction to the Fundamental Theory

Ludger Kaup, Burchard Kaup, Gottfried Barthel, Michael Bridgland

Volume 3 in this series

DOI:: https://doi.org/10.1515/9783110838350
ISBN:: 978-3-11-004150-7
ISBN:: 978-3-11-083835-0
Subject:: Mathematics
Subject:: Analysis
Publisher:: De Gruyter

book: Semimartingales

Book 1982

Semimartingales

A Course on Stochastic Processes

Michel Métivier

Volume 2 in this series

DOI:: https://doi.org/10.1515/9783110845563
ISBN:: 978-3-11-008674-4
ISBN:: 978-3-11-084556-3
Subject:: Mathematics
Subject:: Analysis
Subject:: Probability and Statistics
Publisher:: De Gruyter

book: Riemannian Geometry

Book 1995

Riemannian Geometry

Wilhelm P.A. Klingenberg

Volume 1 in this series

DOI:: https://doi.org/10.1515/9783110905120
ISBN:: 9783110145939
ISBN:: 9783110905120
Subject:: Mathematics
Subject:: Geometry and Topology
Publisher:: De Gruyter

book: Commutative Algebra Methods for Coding Theory

Book Ahead of Publication 2024

Commutative Algebra Methods for Coding Theory

Ştefan Ovidiu I. Tohăneanu

Volume 97 in this series

This book aims to be a comprehensive treatise on the interactions between Coding Theory and Commutative Algebra. With the help of a multitude of examples, it expands and systematizes the known and versatile commutative algebraic framework used, since the early 90’s, to study linear codes. The book provides the necessary background for the reader to advance with similar research on coding theory topics from commutative algebraic perspectives.

DOI:: https://doi.org/10.0000/9783111212920
ISBN:: 9783111212920
ISBN:: 9783111214795
ISBN:: 9783111215389
Subject:: Mathematics
Subject:: Algebra and Number Theory
Subject:: Geometry and Topology
Publisher:: De Gruyter

book: Weighted Morrey Spaces

Book Ahead of Publication 2024

Weighted Morrey Spaces

Calderón-Zygmund Theory and Boundary Problems

Marcus Laurel, Marius Mitrea

Volume 99 in this series

This monograph is a testament to the potency of the method of singular integrals of layer potential type in solving boundary value problems for weakly elliptic systems in the setting of Muckenhoupt-weighted Morrey spaces and their pre-duals.

A functional analytic framework for Muckenhoupt-weighted Morrey spaces in the rough setting of Ahlfors regular sets is built from the ground up and subsequently supports a Calderón-Zygmund theory on this brand of Morrey space in the optimal geometric environment of uniformly rectifiable sets. A thorough duality theory for such Morrey spaces is also developed and ushers in a never-before-seen Calderón-Zygmund theory for Muckenhoupt-weighted Block spaces. Both weighted Morrey and Block spaces are also considered through the lens of (generalized) Banach function spaces, and ultimately, a variety of boundary value problems are formulated and solved with boundary data arbitrarily prescribed from either scale of space.

The fairly self-contained nature of this monograph ensures that graduate students, researchers, and professionals in a variety of fields, e.g., function space theory, harmonic analysis, and PDE, will find this monograph a welcome and valuable addition to the mathematical literature.

DOI:: https://doi.org/10.0000/9783111458274
ISBN:: 9783111458274
ISBN:: 9783111461458
ISBN:: 9783111458168
Subject:: Mathematics
Subject:: Logic and Set Theory
Subject:: Analysis
Subject:: Differential Equations and Dynamical Systems
Subject:: Geometry and Topology
Publisher:: De Gruyter

book: Continuous-Parameter Time Series

Book Ahead of Publication 2024

Continuous-Parameter Time Series

Peter J. Brockwell, Alexander M. Lindner

Volume 98 in this series

This book provides a self-contained account of continuous-parameter time series, starting with second-order models. Integration with respect to orthogonal increment processes, spectral theory and linear prediction are treated in detail. Lévy-driven models are incorporated, extending coverage to allow for infinite variance, a variety of marginal distributions and sample paths having jumps. The necessary theory of Lévy processes and integration of deterministic functions with respect to these processes is developed at length. Special emphasis is given to the analysis of continuous-time ARMA processes.

DOI:: https://doi.org/10.0000/9783111325200
ISBN:: 9783111325200
ISBN:: 9783111324999
ISBN:: 9783111325033
Subject:: Mathematics
Subject:: Probability and Statistics
Subject:: Applied Mathematics
Publisher:: De Gruyter

book: The Structure of Compact Groups

Book 2023

The Structure of Compact Groups

A Primer for the Student – A Handbook for the Expert

Karl H. Hofmann, Sidney A. Morris

Volume 25 in this series

The subject matter of compact groups is frequently cited in fi elds like algebra, topology, functional analysis, and theoretical physics. This book serves the dual purpose of providing a text for upper level graduate students, and of being a source book for researchers who need the structure and representation theory of compact groups. After a gentle introduction to compact groups and their representation theory, the book presents self-contained courses on linear Lie groups and on locally compact abelian groups.Appended chapters contain the material for self-contained courses on abelian groups and on category theory.Using the Lie algebras and the exponential function of arbitrary compact groups, the book avoids unnecessary restrictions to finite dimensional or abelian compact groups. Earlier editions of 1998, 2006, 2013, and 2020 have been quoted for instruction and research. The present edition conceptually sharpens, polishes, and improves the earlier material. For instance, it includes a treatment of the Bohr compactifi cation of topological groups which fi ts perfectly into the general treatment of adjoint functors that the book treats in an appendix of its own, and which, in the abelian environment, connects neatly with the Pontryagin--van Kampen duality of compact abelian groups having been discussed in the book in great detail. The link between arbitrary compact groups and their weakly complete group algebras is as extensively discussed as is now the theory of weakly complete universal enveloping algebras of the Lie algebras of compact groups. All of this is based on the category of weakly complete real and complex vector spaces and its precise duality to the category of ordinary real, respectively, complex vector spaces, is treated in an appendix systematically.

DOI:: https://doi.org/10.1515/9783111172606
ISBN:: 9783111172606
ISBN:: 9783111174051
ISBN:: 9783111171630
Subject:: Mathematics
Subject:: Algebra and Number Theory
Subject:: Combinatorics and Graph Theory
Publisher:: De Gruyter

book: Spectral Flow

Book Open Access 2023

Spectral Flow

A Functional Analytic and Index-Theoretic Approach

Nora Doll, Hermann Schulz-Baldes, Nils Waterstraat

Volume 94 in this series

More Cite Download EPUB Download PDF

This is the first treatment entirely dedicated to an analytic study of spectral flow for paths of selfadjoint Fredholm operators, possibly unbounded or understood in a semifinite sense. The importance of spectral flow for homotopy and index theory is discussed in detail. Applications concern eta-invariants, the Bott-Maslov and Conley-Zehnder indices, Sturm-Liouville oscillation theory, the spectral localizer and bifurcation theory.

DOI:: https://doi.org/10.1515/9783111172477
ISBN:: 9783111172477
ISBN:: 9783111173085
ISBN:: 9783111169897
Subject:: Mathematics
Subject:: Differential Equations and Dynamical Systems
Subject:: Numerical and Computational Mathematics
Subject:: Applied Mathematics
Publisher:: De Gruyter

book: Metrical Almost Periodicity and Applications to Integro-Differential Equations

Book 2023

Metrical Almost Periodicity and Applications to Integro-Differential Equations

Marko Kostić

Volume 95 in this series

The theory of almost periodic functions is a very active field of research for scholars. This research monograph analyzes various classes of multi-dimensional metrically almost periodic type functions with values in complex Banach spaces. We provide many applications of our theoretical results to the abstract Volterra integro-differential inclusions in Banach spaces.

DOI:: https://doi.org/10.1515/9783111233871
ISBN:: 9783111233871
ISBN:: 9783111234175
ISBN:: 9783111233031
Subject:: Mathematics
Subject:: Analysis
Subject:: Differential Equations and Dynamical Systems
Publisher:: De Gruyter

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