Part of Encyclopedia of Mathematics and its Applications
Not yet published - available from June 2021
FORMAT: Hardback ISBN: 9781108835435
Using Bishop's work on constructive analysis as a framework, this monograph gives a systematic, detailed and general constructive theory of probability theory and stochastic processes. It is the first extended account of this theory: almost all of the constructive existence and continuity theorems that permeate the book are original. It also contains results and methods hitherto unknown in the constructive and nonconstructive settings. The text features logic only in the common sense and, beyond a certain mathematical maturity, requires no prior training in either constructive mathematics or probability theory. It will thus be accessible and of interest, both to probabilists interested in the foundations of their speciality and to constructive mathematicians who wish to see Bishop's theory applied to a particular field.
The first systematic account of probability theory in the framework of Bishop's constructive analysis
Contains many original existence and continuity results in constructive probability theory
Suitable for beginning graduate students without prior training in probability or constructive mathematics
Part I. Introduction and Preliminaries:
1. Introduction
2. Preliminaries
3. Partition of unity
Part II. Probability Theory:
4. Integration and measure
5. Probability space
Part III. Stochastic Process:
6. Random field and stochastic process
7. Measurable random field
8. Martingale
9. a.u. continuous process
10. a.u. cadlag process
11. Markov process
Appendix A. Inverse function theorem
Appendix B. Change of integration variables
Appendix C. Taylor's theorem
References
Index.
Not yet published - available from July 2021
FORMAT: Hardback ISBN: 9781733146654
Linear algebra is something all mathematics undergraduates and many other students, in subjects ranging from engineering to economics, have to learn. The
fifth edition of this hugely successful textbook retains the quality of earlier editions while at the same time seeing numerous minor improvements and major
additions. The latter include: a new chapter on singular values and singular vectors, including ways to analyze a matrix of data; a revised chapter on
computing in linear algebra, with professional-level algorithms and code that can be downloaded for a variety of languages; a new section on linear algebra
and cryptography; and a new chapter on linear algebra in probability and statistics. A dedicated and active website also offers solutions to exercises as well
as new exercises from many different sources (e.g. practice problems, exams, development of textbook examples), plus codes in MATLAB, Julia, and Python.
This fifth edition contains numerous minor improvements and major additions
Provides a new chapter on singular values and singular vectors, as well as a revised chapter on computing in linear algebra
A dedicated and active website offers solutions to exercises, new exercises from several sources, and codes in MATLAB, Julia, and Python
1. Introduction to vectors
2. Solving linear equations
3. Vector spaces and subspaces
4. Orthogonality
5. Determinants
6. Eigenvalues and eigenvectors
7. The singular value decomposition (SVD)
8. Linear transformations
9. Complex vectors and matrices
10. Applications
11. Numerical linear algebra
12. Linear algebra in probability and statistics
Matrix factorizations Index
Six great theorems/linear algebra in a nutshell.
Not yet published - available from September 2021
FORMAT: Hardback ISBN: 9781316518939
This classic work has been a unique resource for thousands of mathematicians, scientists and engineers since its first appearance in 1902. Never out of
print, its continuing value lies in its thorough and exhaustive treatment of special functions of mathematical physics and the analysis of differential
equations from which they emerge. The book also is of historical value as it was the first book in English to introduce the then modern methods of complex
analysis. This fifth edition preserves the style and content of the original, but it has been supplemented with more recent results and references where
appropriate. All the formulas have been checked and many corrections made. A complete bibliographical search has been conducted to present the
references in modern form for ease of use. A new foreword by Professor S.J. Patterson sketches the circumstances of the book's genesis and explains
the reasons for its longevity. A welcome addition to any mathematician's bookshelf, this will allow a whole new generation to experience the beauty
contained in this text.
The all-time classic textbook, now with references presented in the modern form
Never out of print since it first appeared in 1902
A unique reference for special functions and the analysis of equations of mathematical physics from which they emerge
Introduction
Part I. The Process of Analysis:
1. Complex numbers
2. The theory of convergence
3. Continuous functions and uniform convergence
4. The theory of Riemann integration
5. The fundamental properties of analytic functions
Taylor's, Laurent's and Liouville's theorems
6. The theory of residues
application to the evaluation of definite integrals
7. The expansion of functions in infinite series
8. Asymptotic expansions and summable series
9. Fourier series and trigonometric series
10. Linear differential equations
11. Integral equations
Part II. The Transcendental Functions:
12. The Gamma-function
13. The zeta-function of Riemann
14. The hypergeometric function
15. Legendre functions
16. The confluent hypergeometric function
17. Bessel functions
18. The equations of mathematical physics
19. Mathieu functions
20. Elliptic functions. General theorems and the Weierstrassian functions
21. The theta-functions
22. The Jacobian elliptic functions
23. Ellipsoidal harmonics and Lame's equation
Appendix. The elementary transcendental functions
References
Author index
Subject index
Part of Cambridge Tracts in Mathematics
Not yet published - available from September 2021
FORMAT: Hardback ISBN: 9781108842471
This book provides a general framework for doing geometric group theory for many non-locally-compact topological transformation groups that arise in
mathematical practice, including homeomorphism and diffeomorphism groups of manifolds, isometry groups of separable metric spaces and automorphism
groups of countable structures. Using Roe's framework of coarse structures and spaces, the author defines a natural coarse geometric structure on all
topological groups. This structure is accessible to investigation, especially in the case of Polish groups, and often has an explicit description, generalising
well-known structures in familiar cases including finitely generated discrete groups, compactly generated locally compact groups and Banach spaces. In
most cases, the coarse geometric structure is metrisable and may even be refined to a canonical quasimetric structure on the group. The book contains
many worked examples and sufficient introductory material to be accessible to beginning graduate students. An appendix outlines several open problems in
this young and rich theory.
Provides a coherent framework for doing geometric group theory in the context of all topological groups, the first complete account of this subject
Contains sufficient introductory material to make it accessible to beginning graduate students
An appendix outlines several open problems in this new area
1. Introduction
2. Coarse structure and metrisability
3. Structure theory
4. Sections, cocycles and group extensions
5. Polish groups of bounded geometry
6. Automorphism groups of countable structures
7. Zappa-Szep products
Appendix A. Open problems.
Part of Cambridge Tracts in Mathematics
Not yet published - available from October 2021
FORMAT: Hardback ISBN: 9781316516911
This monograph is the first to present the theory of global attractors of Hamiltonian partial differential equations. A particular focus is placed on the results
obtained in the last three decades, with chapters on the global attraction to stationary states, to solitons, and to stationary orbits. The text includes many
physically relevant examples and will be of interest to graduate students and researchers in both mathematics and physics. The proofs involve novel
applications of methods of harmonic analysis, including Tauberian theorems, Titchmarsh's convolution theorem, and the theory of quasimeasures. As well as
the underlying theory, the authors discuss the results of numerical simulations and formulate open problems to prompt further research.
The first monograph on the theory of global attractors of Hamiltonian partial differential equations
Covers a range of applications in mathematical physics
Formulates many open problems to prompt research
Introduction
1. Global attraction to stationary states
2. Global attraction to solitons
3. Global attraction to stationary orbits
4. Asymptotic stability of stationary orbits and solitons
5. Adiabatic effective dynamics of solitons
6. Numerical simulation of solitons
7. Dispersive decay
8. Attractors and quantum mechanics
References
Index.
Part of Cambridge Monographs on Mechanics
Not yet published - available from January 2022
FORMAT: HardbackI SBN: 9781108481526
FORMAT: PaperbackISBN: 9781108722902
Covering both theory and experiment, this text describes the behaviour of homogeneous and density-stratified fluids over and around topography. Its
presentation is suitable for advanced undergraduate and graduate students in fluid mechanics, as well as for practising scientists, engineers, and
researchers. Using laboratory experiments and illustrations to further understanding, the author explores topics ranging from the classical hydraulics of
single-layer flow to more complex situations involving stratified flows over two- and three-dimensional topography, including complex terrain. A particular
focus is placed on applications to the atmosphere and ocean, including discussions of downslope windstorms, and of oceanic flow over continental shelves
and slopes. This new edition has been restructured to make it more digestible, and updated to cover significant developments in areas such as exchange
flows, gravity currents, waves in stratified fluids, stability, and applications to the atmosphere and ocean.
Highly illustrated with photographs and line diagrams
Second edition includes new chapters on exchange flows, gravity currents, waves in stratified fluids, stability, and application to the atmosphere and ocean
Discusses a wide range of applications, providing the basis for understanding complex phenomena in the atmosphere and ocean
1. Background
2. Nonlinear single-layer flow: classical hydraulics
3. Nonlinear single-layer flow past obstacles ? jumps, bores and wave dispersion
4. Two-layer flow with jumps and topography
5. Two-layer and stratified flow through contractions
6. Exchange flows
7. Gravity currents, downslope and anabatic flows, and stratified hydraulic jumps
8. Waves in stratified fluids
9. The stability of stratified flows
10. Stratified flow over two-dimensional obstacles ? linear and near-linear theory
11. Stratified flow over two-dimensional obstacles ? nonlinear hydraulic models with applications
12. Stratified flow over three-dimensional topography ? linear theory
13. Three-dimensional stratified flow over finite obstacles
14. Flow over complex and realistic terrain in the atmosphere and ocean
15. Applications to practical modelling of flow over complex terrain
References
Index.
Part of Cambridge Studies in Advanced Mathematics
available from January 2022
FORMAT: HardbackISBN: 9781108837989
The language of ∞-categories provides an insightful new way of expressing many results in higher-dimensional mathematics but can be challenging for the
uninitiated. To explain what exactly an ∞-category is requires various technical models, raising the question of how they might be compared. To overcome
this, a model-independent approach is desired, so that theorems proven with any model would apply to them all. This text develops the theory of ∞-
categories from first principles in a model-independent fashion using the axiomatic framework of an ∞-cosmos, the universe in which ∞-categories live as
objects. An ∞-cosmos is a fertile setting for the formal category theory of ∞-categories, and in this way the foundational proofs in ∞-category theory
closely resemble the classical foundations of ordinary category theory. Equipped with exercises and appendices with background material, this first
introduction is meant for students and researchers who have a strong foundation in classical 1-category theory.
Presents a first introduction to infinity categories for students and researchers with a strong foundation in classical 1-category theory
Takes a model-independent approach, meaning that theorems proven with any model would apply to all models
Provides exercises at the end of each section, allowing the reader to test their understanding of the material
Includes four expository appendices, which provide background material that might not be familiar to all readers.
Part I. Basic ∞-Category Theory:
1. ∞-Cosmoi and their homotopy 2-categories
2. Adjunctions, limits, and colimits I
3. Comma ∞-categories
4. Adjunctions, limits, and colimits II
5. Fibrations and Yoneda's lemma
6. Exotic ∞-cosmoi
Part II. The Calculus of Modules:
7. Two-sided fibrations and modules
8. The calculus of modules
9. Formal category theory in a virtual equipment
Part III. Model Independence:
10. Change-of-model functors
11. Model independence
12. Applications of model independence.