Part of Cambridge Studies in Advanced Mathematics
Not yet published - available from November 2022
FORMAT: HardbackI SBN: 9781009218467
The analysis of eigenvalues of Laplace and Schrodinger operators is an important and classical topic in mathematical physics with many applications. This book presents a thorough introduction to the area, suitable for masters and graduate students, and includes an ample amount of background material on the spectral theory of linear operators in Hilbert spaces and on Sobolev space theory. Of particular interest is a family of inequalities by Lieb and Thirring on eigenvalues of Schrodinger operators, which they used in their proof of stability of matter. The final part of this book is devoted to the active research on sharp constants in these inequalities and contains state-of-the-art results, serving as a reference for experts and as a starting point for further research.
Contains complete proofs of all assertions to facilitate self-study and remain accessible for students with no previous exposure to the material
Detailed presentation of examples, familiarizing readers with concepts and techniques before they are discussed abstractly
Includes recent research results (published only in 2021), providing jumping-off points for future research in this active area
'In 1975, Lieb and Thirring proved a remarkable bound of the sum of the negative eigenvalues of a Schrodinger operator in three dimensions in terms of the L^{5/2}-norm of the potential and used it in their proof of the stability of matter. Shortly thereafter, they realized it was a case of a lovely set of inequalities which generalize Sobolev inequalities and have come to be called Lieb-Thirring bounds. This has spawned an industry with literally hundreds of papers on extensions, generalizations and optimal constants. It is wonderful to have the literature presented and synthesized by three experts who begin by giving the background necessary for this book to be useful not only to specialists but to the novice wishing to understand a deep chapter in mathematical analysis.' Barry Simon, California Institute of Technology
'In a difficult 1968 paper Dyson and Lenard succeeded in proving the 'Stability of Matter' in quantum mechanics. In 1975 a much simpler proof was developed by Thirring and me with a new, multi-function, Sobolev like inequality, as well as a bound on the negative spectrum of Schrodinger operators. These and other bounds have become an important and useful branch of functional analysis and differential equations generally and quantum mechanics in particular. This book, written by three of the leading contributors to the area, carefully lays out the entire subject in a highly readable, yet complete description of these inequalities. They also give gently, yet thoroughly, all the necessary spectral theory and Sobolev theory background that a beginning student might need.' Elliott Lieb, Princeton University
Overview
Part I. Background Material:
1. Elements of operator theory
2. Elements of Sobolev space theory
Part II. The Laplace and Schrodinger Operators:
3. The Laplacian on a domain
4. The Schrodinger operator
Part III. Sharp Constants in Lieb?Thirring Inequalities:5. Sharp Lieb?Thirring inequalities
6. Sharp Lieb?Thirring inequalities in higher dimensions
7. More on sharp Lieb?Thirring inequalities
8. More on the Lieb?Thirring constants
References
Index.
Not yet published - available from November 2022
FORMAT: Hardback ISBN: 9781009187053
Continuous Groups for Physicists is written for graduate students as well as researchers working in the field of theoretical physics. The text has been designed uniquely and it balances coverage of advanced and non-standard topics with an equal focus on the basic concepts for a thorough understanding. The book describes the general theory of Lie groups and Lie algebras, the passage between them, and their unitary/ Hermitian representations in the quantum mechanical setting. The four infinite classical families of compact simple Lie groups and their representations are covered in detail. Readers will benefit from the discussions on topics like spinor representations of real orthogonal groups, the Schwinger representation of a group, induced representations, systems of coherent states, real symplectic groups important in quantum mechanics, Wigner's theorem on symmetry operations in quantum mechanics, ray representations of Lie groups, and groups associated with non-relativistic and relativistic space-time.
Lucid content representation
Dedicated chapter on the structures and unitary representations of the Euclidean, Galilei, Lorentz and Poincare groups related to space-time
Exercises at the end of each chapter to test the understanding of readers
1. Basic Group Theory and Representation Theory
2. The Symmetric Group
3. Rotations in 2 and 3 Dimensions, SU(2)
4. General Theory of Lie Groups and Lie Algebras
5. Compact Simple Lie Algebras ? Classification and Irreducible Representations
6. Spinor Representations of the Orthogonal Groups
7. Properties of Some Reducible Group Representations, and Systems of Generalised Coherent States
8. Structure and some Properties and Applications of the Groups Sp(2n, R)
9. Wigner's Theorem, Ray Representations and Neutral Elements
10. Groups Related to Space-Time.
Not yet published - available from December 2022
FORMAT: Hardback ISBN: 9781108837705
Numerical Analysis is a broad field, and coming to grips with all of it may seem like a daunting task. This text provides a thorough and comprehensive exposition of all the topics contained in a classical graduate sequence in numerical analysis. With an emphasis on theory and connections with linear algebra and analysis, the book shows all the rigor of numerical analysis. Its high level and exhaustive coverage will prepare students for research in the field and become a valuable reference as they continue their career. Students will appreciate the simple notation, clear assumptions and arguments, as well as the many examples and classroom-tested exercises ranging from simple verification to qualifying exam-level problems. In addition to the many examples with hand calculations, readers will also be able to translate theory into practical computational codes by running sample MATLAB codes as they try out new concepts.
'This impressive volume covers an unusually broad range of topics in the field of numerical analysis, including numerical linear algebra, polynomial and trigonometric interpolation, best approximation, numerical quadrature, the approximate solution of nonlinear equations and convex optimization, and the numerical solution of ordinary and partial differential equations by finite difference, spectral and finite element methods. A particularly appealing feature of the text is the way in which it integrates a mathematically rigorous exposition with a wealth of illustrative examples, including numerical simulations, sample codes, and exercises. I warmly recommend the book to students and lecturers as an advanced undergraduate or introductory graduate level text.' Endre Suli, University of Oxford
'This long-awaited graduate text covers every topic of a traditional, one-year Numerical Analysis course in an accessible, rigorous, and comprehensive way. It's an indispensable assistant for anyone offering the class and a precious source of knowledge for a junior researcher in the field.' Maxim Olshanskii, University of Houston
'This is the book I have been waiting for: a textbook of numerical analysis fit for the Twenty-First Century. It sketches a path from the mathematical foundations of the subject to the wide range of its modern methods and algorithms, compromising on neither rigour nor clarity.' Arieh Iserles, University of Cambridge
'The book is the only textbook I know that covers the current topics for beginning graduate students in numerical analysis. The chosen topics in the book match exactly what one wishes to cover in a two-semester course sequence in computational mathematics, as the selection of the numerical methods is in align with the modern treatment of the subjects. Many instructors in the field have struggled to find two or more textbooks for the same coverage, but you can have all of them in this book.' Xiaofan Li, Illinois Institute of Technology
Part of London Mathematical Society Lecture Note Series
Not yet published - available from February 2023
FORMAT: Paperback ISBN: 9781009183291
Since their introduction by Gromov in the 1980s, the study of bounded cohomology and simplicial volume has developed into an active field connected to geometry and group theory. This monograph, arising from a learning seminar for young researchers working in the area, provides a collection of different perspectives on the subject, both classical and recent. The book's introduction presents the main definitions of the theories of bounded cohomology and simplicial volume, outlines their history, and explains their principal motivations and applications. Individual chapters then present different aspects of the theory, with a focus on examples. Detailed references to foundational papers and the latest research are given for readers wishing to dig deeper. The prerequisites are only basic knowledge of classical algebraic topology and of group theory, and the presentations are gentle and informal in order to be accessible to beginning graduate students wanting to enter this lively and topical field.
Gives an overview of the basics of the field as well as many current research directions, including references to classical papers and the latest research
Accessible to graduate students, with few prerequisites, an informal style, clear definitions, and many concrete examples
Written and edited by young researchers at the cutting edge of the field
Introduction Caterina Campagnolo, Francesco Fournier-Facio, Nicolaus Heuer, Marco Moraschini
Part I. Simplicial Volume:
1. Gromov's mapping theorem via multicomplexes Marco Moraschini
2. The proportionality principle via hyperbolic geometry Filippo Sarti
3. Positivity of simplicial volume via barycentric techniques Shi Wang
4. Gromov's systolic inequality via smoothing Lizhi Chen
Integral foliated simplicial volume Caterina Campagnolo
6. l2-Betti numbers Holger Kammeyer
Part II. Bounded Cohomology:
7. Stable commutator length Nicolaus Heuer
8. Quasimorphisms on negatively curved groups Biao Ma
9. Extension of quasicocycles from hyperbolically embedded subgroups Francesco Fournier-Facio
10. Lie groups and Symmetric spaces Anton Hase
11. Continuous bounded cohomology, representations and multiplicative constants Alessio Savini
12. The proportionality principle via bounded Filippo Baroni
References
Index.
Copyright Year 2023
ISBN 9781032350981
October 11, 2022 Forthcoming by Chapman and Hall/CRC
240 Pages 10 B/W Illustrations
This book is devoted to multiplicative analytic geometry. The book reflects recent investigations in the topic. The reader can use the main formulae for investigations of multiplicative differential equations, multiplicative integral equations, and multiplicative geometry.
The authors summarize the most recent contributions in this area. The goal of the authors is to bring the most recent research on the topic to capable senior undergraduate students, beginning graduate students of engineering and science, and researchers in a form to advance further study. The book contains seven chapters. The chapters in the book are pedagogically organized. Each chapter concludes with a section with practical problems.
Two operations, differentiation and integration, are basic in calculus and analysis. In fact, they are the infinitesimal versions of the subtraction and addition operations on numbers, respectively. In the period from 1967 till 1970 Michael Grossman and Robert Katz gave definitions of a new kind of derivative and integral, moving the roles of subtraction and addition to division and multiplication, and thus established a new calculus, called multiplicative calculus. Multiplicative calculus can especially be useful as a mathematical tool for economics and finance.
Multiplicative Analytic Geometry builds upon multiplicative calculus and advances the theory to the topics of analytic and differential geometry.
Preface
1 The Field R*
1.1 Definition
1.2 An Order in R?
1.3 Multiplicative Absolute Value
1.4 The Power Function
1.5 Multiplicative Trigonometric Functions
1.6 Multiplicative Inverse Trigonometric Functions
1.7 Multiplicative Hyperbolic Functions
1.8 Multiplicative Inverse Hyperbolic Functions
1.9 Multiplicative Matrices
1.10 Advanced Practical Problems
2 Multiplicative Plane Euclidean Geometry
2.1 The Multiplicative Vector Space R2 ?
2.2 The Multiplicative Inner Product Space R2 ?
2.3 The Multiplicative Euclidean Plane E?2
2.4 Multiplicative Lines
2.5 Multiplicative Orthonormal Pairs
2.6 Equations of a Multiplicative Line
2.7 Perpendicular Multiplicative Lines
2.8 Multiplicative Parallel and Intersecting Multiplicative Lines
2.9 Multiplicative Reflections
2.10 Multiplicative Congruence and Multiplicative Isometries
2.11 Multiplicative Translations
2.12 Multiplicative Rotations
2.13 Multiplicative Glide Reflections
2.14 Structure of the Multiplicative Isometry Group
2.15 Fixed Points and Fixed Multiplicative Lines
2.16 Advanced Practical Problems
3 Multiplicative Affine Transformations in the Multiplicative
Euclidean Plane
3.1 Multiplicative Affine Transformations
3.2 Fixed Multiplicative Lines
3.3 The Fundamental Theorem
3.4 Multiplicative Affine Reflections
3.5 Multiplicative Shears
3.6 Multiplicative Dilatations
3.7 Multiplicative Similarities
3.8 Multiplicative Affine Symmetries
3.9 Multiplicative Rays and Multiplicative Angles
3.10 Multiplicative Rectilinear Figures
3.11 The Multiplicative Centroid
3.12 Multiplicative Symmetries of a Multiplicative Segment
3.13 Multiplicative Symmetries of a Multiplicative Angle
3.14 Multiplicative Barycentric Coordinates
3.15 Multiplicative Addition of Multiplicative Angles
3.16 Multiplicative Triangles
3.17 Multiplicative Symmetries of a Multiplicative Triangle
3.18 Congruence of Multiplicative Angles
3.19 Congruence Theorems for Multiplicative Triangles
3.20 Multiplicative Angle Sum of Multiplicative Triangles
3.21 Advanced Practical Problems
4 Finite Groups of Multiplicative Isometries of E?2
4.1 Cyclic and Dihedral Groups
4.2 Conjugate Subgroups
4.3 Orbits and Stabilizers
4.4 Regular Multiplicative Polygons
4.5 Similar Regular Multiplicative Polygons
4.6 Advanced Practical Problems
5 Multiplicative Geometry on the Multiplicative Sphere
5.1 The Space E?3
5.2 The Multiplicative Cross Product
5.3 Multiplicative Orthonormal Bases
5.4 Multiplicative Planes
5.5 Incidence Multiplicative Geometry of the Multiplicative Sphere
5.6 The Multiplicative Distance
5.7 Multiplicative Motions on S?2
5.8 Multiplicative Orthogonal Transformations
5.9 The Euler Theorem
5.10 Multiplicative Isometries
5.11 Multiplicative Segments
5.12 Multiplicative Rays, Multiplicative Angles and Multiplicative Triangles
5.13 Multiplicative Spherical Trigonometry
5.14 A Multiplicative Congruence Theorem
5.15 Multiplicative Right Triangles
5.16 Advanced Practical Problems
6 The Projective Multiplicative Plane P?2
6.1 Definition. Incidence Properties of P?2
6.2 Multiplicative Homogeneous Coordinates
6.3 The Desargues Theorem. The Pappus Theorem
6.4 The Projective Multiplicative Group
6.5 The Fundamental Theorem of the Projective Multiplicative Geometry
6.6 Multiplicative Polarities
6.7 Multiplicative Cross Product
6.8 Advanced Practical Problems
7 The Multiplicative Distance Geometry on P?2
7.1 The Multiplicative Distance
7.2 Multiplicative Isometries
7.3 Multiplicative Motions
7.4 Elliptic Multiplicative Geometry
7.5 Advanced Practical Problems
8 The Hyperbolic Multiplicative Plane
8.1 Introduction
8.2 Definition of H2?
8.3 Multiplicative Perpendicular Lines
8.4 Multiplicative Distance of H?2
8.5 Multiplicative Isometries
8.6 Multiplicative Reflections of H?2
8.7 Multiplicative Motions
8.8 Multiplicative Reflections
8.9 Multiplicative Parallel Displacements
8.10 Multiplicative Translations
8.11 Multiplicative Glide Reflections
8.12 Multiplicative Angles, Multiplicative Rays and Multiplicative
Triangles
8.13 Advanced Practical Problems
References
Index
Copyright Year 2023
ISBN 9781032007960
October 25, 2022 Forthcoming by Chapman and Hall/CRC
501 Pages 98 Color Illustrations
Wavelet Transforms: Kith and Kin serves as an introduction to contemporary aspects of time-frequency analysis encompassing the theories of Fourier transforms, wavelet transforms and their respective offshoots.
This book is the first of its kind totally devoted to the treatment of continuous signals and it systematically encompasses the theory of Fourier transforms, wavelet transforms, geometrical wavelet transforms and their ramifications. The authors intend to motivate and stimulate interest among mathematicians, computer scientists, engineers and physical, chemical and biological scientists.
The text is written from the ground up with target readers being senior undergraduate and first-year graduate students and it can serve as a reference for professionals in mathematics, engineering and applied sciences.
Flexibility in the book’s organization enables instructors to select chapters appropriate to courses of different lengths, emphasis and levels of difficulty
Self-contained, the text provides an impetus to the contemporary developments in the signal processing aspects of wavelet theory at the forefront of research
A large number of worked-out examples are included
Every major concept is presented with explanations, limitations and subsequent developments, with emphasis on applications in science and engineering
A wide range of exercises are incoporated in varying levels from elementary to challenging so readers may develop both manipulative skills in theory wavelets and deeper insight
Answers and hints for selected exercises appear at the end
The origin of the theory of wavelet transforms dates back to the 1980s as an outcome of the intriguing efforts of mathematicians, physicists and engineers. Owing to the lucid mathematical framework and versatile applicability, the theory of wavelet transforms is now a nucleus of shared aspirations and ideas.
List of Figures xix
1 The Fourier Transforms 1
1.1 Introduction
1.2 The Fourier Transform
1.2.1 Definition and Examples
1.2.2 Basic Properties of the Fourier Transform
1.2.3 Convolution and Correlation
1.2.4 Shannon’s Sampling Theorem
1.2.5 Uncertainty Principle for the Fourier Transform
1.3 The Fractional Fourier Transform
1.3.1 Definition and Basic Properties
1.3.2 Fractional Convolution and Correlation
1.3.3 Uncertainty Principle for the Fractional Fourier Transform
1.4 The Linear Canonical Transform
1.4.1 Definition and Basic Properties
1.4.2 Linear Canonical Convolution and Correlation
Copyright Year 2023
ISBN 9780367707057
November 10, 2022 Forthcoming by Chapman and Hall/CRC
180 Pages 32 B/W Illustrations
Introduction to Traveling Waves is an invitation to research focused on traveling waves for undergraduate and masters level students. Traveling waves are not typically covered in the undergraduate curriculum, and topics related to traveling waves are usually only covered in research papers, except for a few texts designed for students. This book includes techniques that are not covered in those texts.
Through their experience involving undergraduate and graduate students in a research topic related to traveling waves, the authors found that the main difficulty is to provide reading materials that contain the background information sufficient to start a research project without an expectation of an extensive list of prerequisites beyond regular undergraduate coursework. This book meets that need and serves as an entry point into research topics about the existence and stability of traveling waves.
Self-contained, step-by-step introduction to nonlinear waves written assuming minimal prerequisites, such as an undergraduate course on linear algebra and differential equations.
Suitable as a textbook for a special topics course, or as supplementary reading for courses on modeling.
Contains numerous examples to support the theoretical material.
Supplementary MATLAB codes available via GitHub.
1. Nonlinear Traveling Waves. 1.1. Traveling Waves. 1.2. Reaction-Diffusion Equations. 1.3. Traveling Waves as Solutions of Reaction-Diffusion Equations. 1.4. Planar Waves. 1.5. Examples of Reaction-Diffusion Equations. 1.6. Other Partial Differential Equations that Support Waves. 2. Systems of Reaction-Diffusion Equations posed on Infinite Domains. 2.1. Systems of Reaction-Diffusion Equations. 2.2. Examples of Reaction-Diffusion Systems. 3. Existence of Fronts, Pulses, and Wavetrains. 3.1. Traveling Waves as Orbits in the Associated Dynamical Systems. 3.2. Dynamical Systems Approach: Equilibrium Points. 3.3. Existence of Fronts in Fisher-KPP Equation: Trapping Region Technique. 3.4. Existence of Fronts in Solid Fuel Combustion Model. 3.5. Wavetrains. 4. Stability of Fronts and Pulses. 4.1. Stability: Introduction. 4.2. A Heuristic Presentation of Spectral Stability for Front and Pulse Traveling Wave Solutions. 4.3. Location of the Point Spectrum. 4.4. Beyond Spectral Stability.