Peter Mörters, Roger Moser, Mathew Penrose, Hartmut Schwetlick, and Johannes Zimmer (eds)
- Published in print:
- 2008
- Published Online:
- September 2008
- ISBN:
- 9780199239252
- eISBN:
- 9780191716911
- Item type:
- book
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199239252.001.0001
- Subject:
- Mathematics, Probability / Statistics, Analysis
There has been a significant increase recently in activities on the interface between applied analysis and probability theory. With the potential of a combined approach to the study of various ...
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There has been a significant increase recently in activities on the interface between applied analysis and probability theory. With the potential of a combined approach to the study of various physical systems in view, this book is a collection of topical survey articles by leading researchers in both fields, working on the mathematical description of growth phenomena in the broadest sense. The main aim of the book is to foster interaction between researchers in probability and analysis, and to inspire joint efforts to attack important physical problems. Mathematical methods discussed in the book comprise large deviation theory, lace expansion, harmonic analysis, multi-scale techniques, and homogenization of partial differential equations. Models based on the physics of individual particles are discussed alongside models based on the continuum description of large collections of particles, and the mathematical theories are used to describe physical phenomena such as droplet formation, Bose–Einstein condensation, Anderson localization, Ostwald ripening, or the formation of the early universe.Less
There has been a significant increase recently in activities on the interface between applied analysis and probability theory. With the potential of a combined approach to the study of various physical systems in view, this book is a collection of topical survey articles by leading researchers in both fields, working on the mathematical description of growth phenomena in the broadest sense. The main aim of the book is to foster interaction between researchers in probability and analysis, and to inspire joint efforts to attack important physical problems. Mathematical methods discussed in the book comprise large deviation theory, lace expansion, harmonic analysis, multi-scale techniques, and homogenization of partial differential equations. Models based on the physics of individual particles are discussed alongside models based on the continuum description of large collections of particles, and the mathematical theories are used to describe physical phenomena such as droplet formation, Bose–Einstein condensation, Anderson localization, Ostwald ripening, or the formation of the early universe.
Ali Taheri
- Published in print:
- 2015
- Published Online:
- September 2015
- ISBN:
- 9780198733133
- eISBN:
- 9780191797712
- Item type:
- book
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198733133.001.0001
- Subject:
- Mathematics, Analysis
This book, which is the first volume of two, presents a comprehensive treatment of aspects of classical and modern analysis relating to theory of ‘partial differential equations’ and the associated ...
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This book, which is the first volume of two, presents a comprehensive treatment of aspects of classical and modern analysis relating to theory of ‘partial differential equations’ and the associated ‘function spaces’. It begins with a quick review of basic properties of harmonic functions and Poisson integrals and then moves into a detailed study of Hardy spaces. The classical Dirichlet problem is considered and a variety of methods for its resolution ranging from potential theoretic (Perron’s method of sub-harmonic functions and Wiener’s criterion, Green’s functions and Poisson integrals, the method of layered potentials or integral equations) to variational (Dirichlet principle) are presented. Parallel to this is the development of the necessary function spaces: Lorentz and Marcinkiewicz spaces, Sobolev spaces (integer as well as fractional order), Hardy spaces, the John-Nirenberg space BMO, Morrey and Campanato spaces, Besov spaces and Triebel-Lizorkin spaces. Harmonic analysis is deeply intertwined with the topics covered and the subjects of summability methods, Tauberian theorems, convolution algebras, Calderon-Zygmund theory of singular integrals and Littlewood-Paley theory that on the one hand connect to various PDE estimates (Calderon-Zygmund inequality, Strichartz estimates, Mihlin-Hormander multipliers, etc.) and on the other lead to a unified characterisation of various function spaces are discussed in great depth. The book ends by a discussion of regularity theory for second order elliptic equations in divergence form— first with continuous and next with measurable coefficients—and covers, in particular, De Giorgi’s theorem, Moser iteration, Harnack inequality and local boundedness of solutions. (The case of elliptic systems and related topics is discussed in the exercises.)Less
This book, which is the first volume of two, presents a comprehensive treatment of aspects of classical and modern analysis relating to theory of ‘partial differential equations’ and the associated ‘function spaces’. It begins with a quick review of basic properties of harmonic functions and Poisson integrals and then moves into a detailed study of Hardy spaces. The classical Dirichlet problem is considered and a variety of methods for its resolution ranging from potential theoretic (Perron’s method of sub-harmonic functions and Wiener’s criterion, Green’s functions and Poisson integrals, the method of layered potentials or integral equations) to variational (Dirichlet principle) are presented. Parallel to this is the development of the necessary function spaces: Lorentz and Marcinkiewicz spaces, Sobolev spaces (integer as well as fractional order), Hardy spaces, the John-Nirenberg space BMO, Morrey and Campanato spaces, Besov spaces and Triebel-Lizorkin spaces. Harmonic analysis is deeply intertwined with the topics covered and the subjects of summability methods, Tauberian theorems, convolution algebras, Calderon-Zygmund theory of singular integrals and Littlewood-Paley theory that on the one hand connect to various PDE estimates (Calderon-Zygmund inequality, Strichartz estimates, Mihlin-Hormander multipliers, etc.) and on the other lead to a unified characterisation of various function spaces are discussed in great depth. The book ends by a discussion of regularity theory for second order elliptic equations in divergence form— first with continuous and next with measurable coefficients—and covers, in particular, De Giorgi’s theorem, Moser iteration, Harnack inequality and local boundedness of solutions. (The case of elliptic systems and related topics is discussed in the exercises.)
Ali Taheri
- Published in print:
- 2015
- Published Online:
- September 2015
- ISBN:
- 9780198733157
- eISBN:
- 9780191797729
- Item type:
- book
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198733157.001.0001
- Subject:
- Mathematics, Analysis
This book presents a comprehensive treatment of aspects of classical and modern analysis relating to theory of ‘partial differential equations’ and the associated ‘function spaces’. It begins with a ...
More
This book presents a comprehensive treatment of aspects of classical and modern analysis relating to theory of ‘partial differential equations’ and the associated ‘function spaces’. It begins with a quick review of basic properties of harmonic functions and Poisson integrals and then moves into a detailed study of Hardy spaces. The classical Dirichlet problem is considered and a variety of methods for its resolution ranging from potential theoretic (Perron’s method of sub-harmonic functions and Wiener’s criterion, Green’s functions and Poisson integrals, the method of layered potentials or integral equations) to variational (Dirichlet principle) are presented. Parallel to this is the development of the necessary function spaces: Lorentz and Marcinkiewicz spaces, Sobolev spaces (integer as well as fractional order), Hardy spaces, the John-Nirenberg space BMO, Morrey and Campanato spaces, Besov spaces and Triebel-Lizorkin spaces. Harmonic analysis is deeply intertwined with the topics covered and the subjects of summability methods, Tauberian theorems, convolution algebras, Calderon-Zygmund theory of singular integrals and Littlewood-Paley theory that on the one hand connect to various PDE estimates (Calderon-Zygmund inequality, Strichartz estimates, Mihlin-Hormander multipliers, etc.) and on the other lead to a unified characterisation of various function spaces are discussed in great depth. The book ends by a discussion of regularity theory for second order elliptic equations in divergence form— first with continuous and next with measurable coefficients—and covers, in particular, De Giorgi’s theorem, Moser iteration, Harnack inequality and local boundedness of solutions. (The case of elliptic systems and related topics is discussed in the exercises.)Less
This book presents a comprehensive treatment of aspects of classical and modern analysis relating to theory of ‘partial differential equations’ and the associated ‘function spaces’. It begins with a quick review of basic properties of harmonic functions and Poisson integrals and then moves into a detailed study of Hardy spaces. The classical Dirichlet problem is considered and a variety of methods for its resolution ranging from potential theoretic (Perron’s method of sub-harmonic functions and Wiener’s criterion, Green’s functions and Poisson integrals, the method of layered potentials or integral equations) to variational (Dirichlet principle) are presented. Parallel to this is the development of the necessary function spaces: Lorentz and Marcinkiewicz spaces, Sobolev spaces (integer as well as fractional order), Hardy spaces, the John-Nirenberg space BMO, Morrey and Campanato spaces, Besov spaces and Triebel-Lizorkin spaces. Harmonic analysis is deeply intertwined with the topics covered and the subjects of summability methods, Tauberian theorems, convolution algebras, Calderon-Zygmund theory of singular integrals and Littlewood-Paley theory that on the one hand connect to various PDE estimates (Calderon-Zygmund inequality, Strichartz estimates, Mihlin-Hormander multipliers, etc.) and on the other lead to a unified characterisation of various function spaces are discussed in great depth. The book ends by a discussion of regularity theory for second order elliptic equations in divergence form— first with continuous and next with measurable coefficients—and covers, in particular, De Giorgi’s theorem, Moser iteration, Harnack inequality and local boundedness of solutions. (The case of elliptic systems and related topics is discussed in the exercises.)
Dusa McDuff and Dietmar Salamon
- Published in print:
- 2017
- Published Online:
- June 2017
- ISBN:
- 9780198794899
- eISBN:
- 9780191836411
- Item type:
- book
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198794899.001.0001
- Subject:
- Mathematics, Analysis, Geometry / Topology
Over the past number of years powerful new methods in analysis and topology have led to the development of the modern global theory of symplectic topology, including several striking and important ...
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Over the past number of years powerful new methods in analysis and topology have led to the development of the modern global theory of symplectic topology, including several striking and important results. The first edition of Introduction to Symplectic Topology was published in 1995. The book was the first comprehensive introduction to the subject and became a key text in the area. In 1998, a significantly revised second edition contained new sections and updates. This third edition includes both further updates and new material on this fast-developing area. All chapters have been revised to improve the exposition, new material has been added in many places, and various proofs have been tightened up. Copious new references to key papers have been added to the bibliography. In particular, the material on contact geometry has been significantly expanded, many more details on linear complex structures and on the symplectic blowup and blowdown have been added, the section on J-holomorphic curves in Chapter 4 has been thoroughly revised, there are new sections on GIT and on the topology of symplectomorphism groups, and the section on Floer homology has been revised and updated. Chapter 13 has been completely rewritten and has a new title (Questions of Existence and Uniqueness). It now contains an introduction to existence and uniqueness problems in symplectic topology, a section describing various examples, an overview of Taubes–Seiberg–Witten theory and its applications to symplectic topology, and a section on symplectic 4-manifolds. Chapter 14 on open problems has been added.Less
Over the past number of years powerful new methods in analysis and topology have led to the development of the modern global theory of symplectic topology, including several striking and important results. The first edition of Introduction to Symplectic Topology was published in 1995. The book was the first comprehensive introduction to the subject and became a key text in the area. In 1998, a significantly revised second edition contained new sections and updates. This third edition includes both further updates and new material on this fast-developing area. All chapters have been revised to improve the exposition, new material has been added in many places, and various proofs have been tightened up. Copious new references to key papers have been added to the bibliography. In particular, the material on contact geometry has been significantly expanded, many more details on linear complex structures and on the symplectic blowup and blowdown have been added, the section on J-holomorphic curves in Chapter 4 has been thoroughly revised, there are new sections on GIT and on the topology of symplectomorphism groups, and the section on Floer homology has been revised and updated. Chapter 13 has been completely rewritten and has a new title (Questions of Existence and Uniqueness). It now contains an introduction to existence and uniqueness problems in symplectic topology, a section describing various examples, an overview of Taubes–Seiberg–Witten theory and its applications to symplectic topology, and a section on symplectic 4-manifolds. Chapter 14 on open problems has been added.
Peter Grindrod
- Published in print:
- 2014
- Published Online:
- March 2015
- ISBN:
- 9780198725091
- eISBN:
- 9780191792526
- Item type:
- book
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198725091.001.0001
- Subject:
- Mathematics, Analysis, Probability / Statistics
This book presents analytics within a framework of mathematical theory and concepts, building upon firm theory and foundations of probability theory, graphs, and networks, random matrices, linear ...
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This book presents analytics within a framework of mathematical theory and concepts, building upon firm theory and foundations of probability theory, graphs, and networks, random matrices, linear algebra, optimization, forecasting, discrete dynamical systems, and more. Following on from the theoretical considerations, applications are given to data from commercially relevant interests: supermarket baskets; loyalty cards; mobile phone call records; smart meters; ‘omic‘ data; sales promotions; social media; and microblogging. Each chapter tackles a topic in analytics: social networks and digital marketing; forecasting; clustering and segmentation; inverse problems; Markov models of behavioural changes; multiple hypothesis testing and decision-making; and so on. Chapters start with background mathematical theory explained with a strong narrative and then give way to practical considerations and then to exemplar applications.Less
This book presents analytics within a framework of mathematical theory and concepts, building upon firm theory and foundations of probability theory, graphs, and networks, random matrices, linear algebra, optimization, forecasting, discrete dynamical systems, and more. Following on from the theoretical considerations, applications are given to data from commercially relevant interests: supermarket baskets; loyalty cards; mobile phone call records; smart meters; ‘omic‘ data; sales promotions; social media; and microblogging. Each chapter tackles a topic in analytics: social networks and digital marketing; forecasting; clustering and segmentation; inverse problems; Markov models of behavioural changes; multiple hypothesis testing and decision-making; and so on. Chapters start with background mathematical theory explained with a strong narrative and then give way to practical considerations and then to exemplar applications.
Fon-Che Liu
- Published in print:
- 2016
- Published Online:
- January 2017
- ISBN:
- 9780198790426
- eISBN:
- 9780191831676
- Item type:
- book
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198790426.001.0001
- Subject:
- Mathematics, Analysis
Real analysis in its modern aspect is presented concisely in this text for the beginning graduate student of mathematics and related disciplines to have a solid grounding in the general theory of ...
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Real analysis in its modern aspect is presented concisely in this text for the beginning graduate student of mathematics and related disciplines to have a solid grounding in the general theory of measure and to build helpful insights for effectively applying the general principles of real analysis to concrete problems. After an introductory chapter, a compact but precise treatment of general measure and integration is undertaken to provide the reader with an overall view of the general theory before delving into special measures. The universality of the method of outer measure in the construction of measures is emphasized, because it provides a unified way of looking for useful regularity properties of measures. The chapter on functions of real variables is the core of the book; it treats properties of functions that are not only basic for understanding the general features of functions but also relevant for the study of those function spaces which are important when application of functional analytical methods is in question. The chapter on basic principles of functional analysis and that on the Fourier integral reveal the intimate interplay between functional analysis and real analysis. Applications of many of the topics discussed are included; these contain explorations toward probability theory and partial differential equations.Less
Real analysis in its modern aspect is presented concisely in this text for the beginning graduate student of mathematics and related disciplines to have a solid grounding in the general theory of measure and to build helpful insights for effectively applying the general principles of real analysis to concrete problems. After an introductory chapter, a compact but precise treatment of general measure and integration is undertaken to provide the reader with an overall view of the general theory before delving into special measures. The universality of the method of outer measure in the construction of measures is emphasized, because it provides a unified way of looking for useful regularity properties of measures. The chapter on functions of real variables is the core of the book; it treats properties of functions that are not only basic for understanding the general features of functions but also relevant for the study of those function spaces which are important when application of functional analytical methods is in question. The chapter on basic principles of functional analysis and that on the Fourier integral reveal the intimate interplay between functional analysis and real analysis. Applications of many of the topics discussed are included; these contain explorations toward probability theory and partial differential equations.
Simon Donaldson
- Published in print:
- 2011
- Published Online:
- December 2013
- ISBN:
- 9780198526391
- eISBN:
- 9780191774874
- Item type:
- book
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198526391.001.0001
- Subject:
- Mathematics, Geometry / Topology, Analysis
The theory of Riemann surfaces occupies a very special place in mathematics. It is a culmination of much of traditional calculus, making surprising connections with geometry and arithmetic. It is an ...
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The theory of Riemann surfaces occupies a very special place in mathematics. It is a culmination of much of traditional calculus, making surprising connections with geometry and arithmetic. It is an extremely useful part of mathematics, knowledge of which is needed by specialists in many other fields. It provides a model for a large number of more recent developments in areas including manifold topology, global analysis, algebraic geometry, Riemannian geometry, and diverse topics in mathematical physics. This text on Riemann surface theory proves the fundamental analytical results on the existence of meromorphic functions and the Uniformisation Theorem. The approach taken emphasises PDE methods, applicable more generally in global analysis. The connection with geometric topology, and in particular the role of the mapping class group, is also explained. To this end, some more sophisticated topics have been included, compared with traditional texts at this level. While the treatment is novel, the roots of the subject in traditional calculus and complex analysis are kept well in mind. Part I sets up the interplay between complex analysis and topology, with the latter treated informally. Part II works as a rapid first course in Riemann surface theory, including elliptic curves. The core of the book is contained in Part III, where the fundamental analytical results are proved.Less
The theory of Riemann surfaces occupies a very special place in mathematics. It is a culmination of much of traditional calculus, making surprising connections with geometry and arithmetic. It is an extremely useful part of mathematics, knowledge of which is needed by specialists in many other fields. It provides a model for a large number of more recent developments in areas including manifold topology, global analysis, algebraic geometry, Riemannian geometry, and diverse topics in mathematical physics. This text on Riemann surface theory proves the fundamental analytical results on the existence of meromorphic functions and the Uniformisation Theorem. The approach taken emphasises PDE methods, applicable more generally in global analysis. The connection with geometric topology, and in particular the role of the mapping class group, is also explained. To this end, some more sophisticated topics have been included, compared with traditional texts at this level. While the treatment is novel, the roots of the subject in traditional calculus and complex analysis are kept well in mind. Part I sets up the interplay between complex analysis and topology, with the latter treated informally. Part II works as a rapid first course in Riemann surface theory, including elliptic curves. The core of the book is contained in Part III, where the fundamental analytical results are proved.
Simon Scott
- Published in print:
- 2010
- Published Online:
- January 2011
- ISBN:
- 9780198568360
- eISBN:
- 9780191594748
- Item type:
- book
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198568360.001.0001
- Subject:
- Mathematics, Analysis
This text provides a broad account of the theory of traces and determinants on geometric algebras of differential and pseudodifferential operators over compact manifolds. Trace and determinant ...
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This text provides a broad account of the theory of traces and determinants on geometric algebras of differential and pseudodifferential operators over compact manifolds. Trace and determinant functionals on geometric operator algebras provide a means of constructing refined invariants in analysis, topology, differential geometry, analytic number theory and QFT. The consequent interactions around such invariants have led to significant advances both in pure mathematics and theoretical physics. As the fundamental tools of trace theory have become well understood and clear general structures have emerged, so the need for specialist texts which explain the basic theoretical principles and the computational techniques has become increasingly exigent. This text is the first to deal with the general theory of traces and determinants of operators on manifolds in a broad context, encompassing a number of the principle applications and backed up by specific computations which set out in detail to newcomers the nuts-and-bolts of the basic theory. Both the microanalytic approach to traces and determinants via pseudodifferential operator theory and the more computational approach directed by applications in geometric analysis, are developed in a general framework that will be of interest to mathematicians and physicists in a number of different fields.Less
This text provides a broad account of the theory of traces and determinants on geometric algebras of differential and pseudodifferential operators over compact manifolds. Trace and determinant functionals on geometric operator algebras provide a means of constructing refined invariants in analysis, topology, differential geometry, analytic number theory and QFT. The consequent interactions around such invariants have led to significant advances both in pure mathematics and theoretical physics. As the fundamental tools of trace theory have become well understood and clear general structures have emerged, so the need for specialist texts which explain the basic theoretical principles and the computational techniques has become increasingly exigent. This text is the first to deal with the general theory of traces and determinants of operators on manifolds in a broad context, encompassing a number of the principle applications and backed up by specific computations which set out in detail to newcomers the nuts-and-bolts of the basic theory. Both the microanalytic approach to traces and determinants via pseudodifferential operator theory and the more computational approach directed by applications in geometric analysis, are developed in a general framework that will be of interest to mathematicians and physicists in a number of different fields.
Accounting for Oneself
Adjusting to a World in Motion
Advancing the Human Right to Health
Aid and Development
The AIDS Generation
Aliens and Strangers?
Altruism in Humans
American Arcadia
Analogies and Theories
