Format: Hardback, height x width: 235x155 mm, Approx. 300 p.
Pub. Date: 04-Mar-2024
ISBN-13: 9789819986910
This book focuses on the following three topics in the theory of boundary value problems of nonlinear second order elliptic partial differential equations and systems: (i) eigenvalue problem, (ii) upper and lower solutions method, (iii) topological degree method, and deals with the existence of solutions, more specifically non-constant positive solutions, as well as the uniqueness, stability and asymptotic behavior of such solutions.
While not all-encompassing, these topics represent major approaches to the theory of partial differential equations and systems, and should be of significant interest to graduate students and researchers. Two appendices have been included to provide a good gauge of the prerequisites for this book and make it reasonably self-contained.
A notable strength of the book is that it contains a large number of substantial examples. Exercises for the reader are also included. Therefore, this book is suitable as a textbook for graduate students who have already had an introductory course on PDE and some familiarity with functional analysis and nonlinear functional analysis, and as a reference for researchers.
Preface.- Preliminaries.- Eigenvalue problems of second order linear
elliptic operators.- Upper and lower solutions method for single
equations.- Upper and lower solutions method for systems.- Theory of
topological degree in cones and applications.- Systems with homogeneous
Neumann boundary conditions.- P-Laplace equations and systems.- Appendix A:
Basic results of Sobolev spaces and nonlinear functional analysis.- Appendix
B: Basic theory of elliptic equations.- References.- Index.
Format: Hardback, height x width: 235x155 mm, Approx. 200 p.
Pub. Date: 16-Feb-2024
ISBN-13: 9789819986675
This book explains the basic theory of Hilbert C*-module in detail, covering a wide range of applications from generalized index to module framework. At the center of the book, the Beurling-Deny criterion is characterized between operator valued Dirichlet forms and quantum Markov semigroups, hence opening a new field of quantum probability research. The general scope of the book includes: basic theory of Hilbert C*-modules; generalized indices and module frames; operator valued Dirichlet forms; and quantum Markov semigroups. This book will be of value to scholars and graduate students in the fields of operator algebra, quantum probability and quantum information.
Chapter
1. Basic theory of Hilbert C*-modules1.1 Hilbert C^*-modules and
bounded module mappings1.2 Polar decomposition and Wold
decomposition1.3 Tensor products of Hilbert C*-modules1.4 KSGNS
constructionChapter
2. Kasproves stabilization and Fredholm generalized
index theory2.1 Kasproves stabilization theorem2.2 Morita equivalence and
C*-correspondence2.3 Generalized index theory of Fredholm module
operators2.4 Introduction to module frames theoryChapter
3. Quantum Markov
semigroups and operator-valued Dirichlet forms3.1 Module operator
semigroups3.2 Abstract Cauchy problems based on Hilbert A-modules3.3 Quantum
Stone theorem and its application3.4 Operator-valued Dirichlet forms and
quantum Markov semigroups3.5 Application in operator-valued free probability
Format: Hardback, 260 pages, height x width: 235x155 mm, XIII, 260 p
Series: Texts and Readings in Mathematics 48
Pub. Date: 19-Feb-2024
ISBN-13: 9789819982769
This book serves as an advanced text for a graduate course on stochastic algorithms for the students of probability and statistics, engineering, economics and machine learning. This second edition gives a comprehensive treatment of stochastic approximation algorithms based on the ordinary differential equation (ODE) approach which analyses the algorithm in terms of a limiting ODE. It has a streamlined treatment of the classical convergence analysis and includes several recent developments such as concentration bounds, avoidance of traps, stability tests, distributed and asynchronous schemes, multiple time scales, general noise models, etc., and a category-wise exposition of many important applications. It is also a useful reference for researchers and practitioners in the field.
1. Introduction.- 2. Convergence Analysis.-
3. Finite Time Bounds and
Traps.-
4. Stability Criteria.- 5. Stochastic Recursive Inclusions.-
6.
Asynchronous Schemes.- 7. A Limit Theorem for Fluctuations.- 8. Multiple
Timescales.- 9. Constant Stepsize Algorithms.- 10. General Noise Models.-
11.
Stochastic Gradient Schemes.- 12. Liapunov and Related Systems.- 13. Appendix
A: Topics in Analysis.- 14. Appendix B: Ordinary Differential Equations.-
15. Appendix C: Topics in Probability.- Bibliography.- Index.
Format: Hardback, 490 pages, height x width: 235x155 mm, 35 Illustrations, black and white; XIII, 490 p. 35 illus.
Series: Texts in Applied Mathematics 77
Pub. Date: 20-Feb-2024
ISBN-13: 9783031493058
This book provides the elements of probability and stochastic processes of direct interest to the applied sciences where probabilistic models play an important role, most notably in the information and communications sciences, computer sciences, operations research, and electrical engineering, but also in fields like epidemiology, biology, ecology, physics, and the earth sciences.
The theoretical tools are presented gradually, not deterring the readers with a wall of technicalities before they have the opportunity to understand their relevance in simple situations. In particular, the use of the so-called modern integration theory (the Lebesgue integral) is postponed until the fifth chapter, where it is reviewed in sufficient detail for a rigorous treatment of the topics of interest in the various domains of application listed above.
The treatment, while mathematical, maintains a balance between depth and accessibility that is suitable for the efficient manipulation, based on solid theoretical foundations, of the four most important and ubiquitous categories of probabilistic models:
Markov chains, which are omnipresent and versatile models in applied probability
Poisson processes (on the line and in space), occurring in a range of applications from ecology to queuing and mobile communications networks
Brownian motion, which models fluctuations in the stock market and the "white noise" of physics
Wide-sense stationary processes, of special importance in signal analysis and design, as well as in the earth sciences.
This book can be used as a text in various ways and at different levels
of study. Essentially, it provides the material for a two-semester graduate
course on probability and stochastic processes in a department of applied
mathematics or for students in departments where stochastic models play
an essential role. The progressive introduction of concepts and tools,
along with the inclusion of numerous examples, also makes this book well-adapted
for self-study
Preface.- Basic Notions.- Discrete Random Variables.- Continuous Random
Vectors.- The Lebesgue Integral.- From Integral to Expectation.- Convergence
Almost Sure.- Convergence in Distribution.- Martingales.- Markov Chains.-
Poisson Processes.- Brownian Motion.- Wide-sense Stationary Processes.- A
Review of Hilbert Spaces.- Bibliography.- Index.
Format: Hardback, 241 pages, height x width: 235x155 mm, XI, 241 p.,
Series: Texts and Readings in Mathematics 84
Pub. Date: 19-Feb-2024
ISBN-13: 9789819983513
This book introduces the dilation theory of operators on Hilbert spaces and its relationship to complex geometry. Classical as well as very modern topics are covered in the book. On the one hand, it introduces the reader to the characteristic function, a classical object used by Sz.-Nagy and Foias and still a topic of current research. On the other hand, it describes the dilation theory of the symmetrized bidisc which has been developed mostly in the present century and is a very active topic of research. It also describes an abstract theory of dilation in the setting of set theory. This was developed very recently.
A good portion of the book discusses various geometrical objects like the bidisc, the Euclidean unit ball, and the symmetrized bidisc. It shows the similarities and differences between the dilation theory in these domains. While completely positive maps play a big role in the dilation theory of the Euclidean unit ball, this is not so in the symmetrized bidisc for example. There, the central role is played by an operator equation. Targeted to graduate students and researchers, the book introduces the reader to different techniques applicable in different domains.
Dilation for One Operator.- C*-Algebras and Completely Positive Maps.-
Dilation Theory in Two Variables - The Bidisc.- Dilation Theory in Several
Variables - the Euclidean Ball.- The Euclidean Ball - The Drury Arveson
Space.- Dilation Theory in Several Variables - The Symmetrized Bidisc.- An
Abstract Dilation Theory.
Format: Hardback, 580 pages, height x width: 235x155 mm, 20 Illustrations, black and white; X, 580 p. 20 illus.
Pub. Date: 29-May-2024
ISBN-13: 9784431544265
In modern mathematics, a Lie group means a group which is also a manifold. While Lie groups are geometric objects, Lie algebras are algebraic elements that characterize the infinitesimal structure of Lie groups. The theories of Lie groups and Lie algebras originated in the late nineteenth century and nowadays they comprise an indispensable part of modern mathematics connecting almost all of its sub-disciplines. This book, written by one of the leading experts in the field, aims to provide a pedagogical introduction to essential concepts and methods in the theories of Lie groups and their representations in a manner comprehensible to a wide audience in mathematics and mathematical physics. The book begins with an elementary introduction to the structure of topological groups, their representation theory and Fourier analysis, followed by a detailed account of the PeterWeyl theorem for compact groups. The second half of the book starts with the classification theory of finite dimensional irreducible representations and leads the reader to the CartanWeyl highest weight theory, including abundant enlightening examples of classical matrix groups. A unique feature of the book is the final chapters dedicated to the infinite dimensional unitary representation theory and the BorelWeil theory, for which no introductory textbook exists yet. The authors writing style is such that whenever an abstract concept is introduced, he tries to motivate the reader before giving a formal definition. Special efforts are made to bring those techniques and ideas to light that are often hidden behind theorems; every explanation is well thought out to minimize the background knowledge needed to understand it; exercises are attached at the end of each chapter with fully worked-out solutions to help readers test their understanding. These features jointly make this book a precious resource for both students and researchers engaging in this rapidly growing field of research.