Part of Cambridge Studies in Advanced Mathematics
availability: Not yet published - available from April 2016
format: Hardback
isbn: 9781107104099
Reproducing kernel Hilbert spaces have developed into an important tool in many areas, especially statistics and machine learning, and they play a valuable role in complex analysis, probability, group representation theory, and the theory of integral operators. This unique text offers a unified overview of the topic, providing detailed examples of applications, as well as covering the fundamental underlying theory, including chapters on interpolation and approximation, Cholesky and Schur operations on kernels, and vector-valued spaces. Self-contained and accessibly written, with exercises at the end of each chapter, this unrivalled treatment of the topic serves as an ideal introduction for graduate students across mathematics, computer science, and engineering, as well as a useful reference for researchers working in functional analysis or its applications.
Part I. General Theory:
1. Introduction
2. Fundamental results
3. Interpolation and approximation
4. Cholesky and Schur
5. Operations on kernels
6. Vector-valued spaces
Part II. Applications and Examples:
7. Power series on balls and pull-backs
8. Statistics and machine learning
9. Negative definite functions
10. Positive definite functions on groups
11. Applications of RKHS to integral operators
12. Stochastic processes.
Part of Encyclopedia of Mathematics and its Applications
Not yet published - available from May 2016
format: Hardback
isbn: 9781107042599
The Banach?Tarski Paradox is a most striking mathematical construction: it asserts that a solid ball can be taken apart into finitely many pieces that can be rearranged using rigid motions to form a ball twice as large. This volume explores the consequences of the paradox for measure theory and its connections with group theory, geometry, set theory, and logic. This new edition of a classic book unifies contemporary research on the paradox. It has been updated with many new proofs and results, and discussions of the many problems that remain unsolved. Among the new results presented are several unusual paradoxes in the hyperbolic plane, one of which involves the shapes of Escher's famous 'Angel and Devils' woodcut. A new chapter is devoted to a complete proof of the remarkable result that the circle can be squared using set theory, a problem that had been open for over sixty years.
Part I. Paradoxical Decompositions, or the Nonexistence of Finitely Additive Measures:
1. Introduction
2. The Hausdorff paradox
3. The Banach?Tarski paradox: duplicating spheres and balls
4. Hyperbolic paradoxes
5. Locally commutative actions: minimizing the number of pieces in a paradoxical decomposition
6. Higher dimensions
7. Free groups of large rank: getting a continuum of spheres from one
8. Paradoxes in low dimensions
9. Squaring the circle
10. The semigroup of equidecomposability types
Part II: Finitely Additive Measures, or the Nonexistence of Paradoxical Decompositions:
11. Transition
12. Measures in groups
13. Applications of amenability
14. Growth conditions in groups and supramenability
15. The role of the axiom of choice.
A publication of the Societe Mathematique de France.
Seminaires et Congres, Number: 29
2015; 119 pp; softcover
ISBN-13: 978-2-85629-817-6
This volume is a result of lectures given at the CIMPA School on Control and Stabilization of PDEs, held from May 9-19, 2011, in Monastir, Tunisia. Different control techniques for linear parabolic equations were presented and the deduction of the null controllability of such equations from local Carleman inequality was described. Overall, Carleman-type and Hardy type inequalities for the null controllability of degenerate parabolic equations were discussed.
Current issues in the control of conservation laws, such as the control of classical solutions in singular control limits and the control solutions with shock waves, were also highlighted during this school. Finally, different techniques and methods for the stability of evolution equations with and without delay, applicable to Navier-Stokes equations, were presented.
A publication of the Societe Mathematique de France, Marseilles (SMF), distributed by the AMS in the U.S., Canada, and Mexico. Orders from other countries should be sent to the SMF. Members of the SMF receive a 30% discount from list.
Graduate students and research mathematicians interested in PDEs.
V. Barbu -- Stabilization of the Navier-Stokes Equation
G. Lebeau -- Introduction aux inegalites de Carleman
S. Nicaise -- Stabilization of second order evolution equations with unbounded feedback delay
ISBN: 978-1-118-63219-2
304 pages
March 2016
Written by a team of international experts, Extremes and Recurrence in Dynamical Systems presents a unique point of view on the mathematical theory of extremes and how it relates to applications in the natural and social sciences. Featuring an interdisciplinary approach to new concepts in pure and applied mathematical research, the book skillfully combines the areas of statistical mechanics, probability theory, measure theory, dynamical systems, statistical inference, geophysics, and software application. Emphasizing statistical mechanics, the book introduces robust theoretical embedding for the application of extreme value theory and modeling through dynamical systems. Extremes and Recurrence in Dynamical Systems also features:
A careful examination of how a dynamical system can be taken as a generator of stochastic processes
Discussions on the applications of statistical inference in the theoretical and heuristic use of extremes
Several examples of analysis of extremes in a physical and geophysical context
A final summary of the main results presented with a discussion of forthcoming research guidelines
An appendix with software Matlab and Octave programming language to help readers to develop further understanding of the presented concepts
Extremes and Recurrence in Dynamical Systems is ideal for academics and practitioners in pure and applied mathematics, probability theory, statistics, chaos, theoretical and applied dynamical systems, statistical mechanics, geophysical fluid dynamics, geosciences and complexity science.
1 Introduction 3
1.1 A Transdisciplinary Research Area 3
1.2 Some Mathematical Ideas 6
1.3 Some Difficulties and Challenges in Studying Extremes 8
1.3.1 Finiteness of Data 8
and more
ISBN: 978-1-119-18947-3
352 pages
April 2016
Introducing the various classes of functional differential equations, Functional Differential Equations: Advances and Applications presents the needed tools and topics to study the various classes of functional differential equations and is primarily concerned with the existence, uniqueness, and estimates of solutions to specific problems. The book focuses on the general theory of functional differential equations, provides the requisite mathematical background, and details the qualitative behavior of solutions to functional differential equations.
The book addresses problems of stability, particularly for ordinary differential equations in which the theory can provide models for other classes of functional differential equations, and the stability of solutions is useful for the application of results within various fields of science, engineering, and economics. Functional Differential Equations: Advances and Applications also features:
Discussions on the classes of equations that cannot be solved to the highest order derivative, and in turn, addresses existence results and behavior types
Oscillatory motion and solutions that occur in many real-world phenomena as well as in man-made machines
Numerous examples and applications with a specific focus on ordinary differential equations and functional differential equations with finite delay
An appendix that introduces generalized Fourier series and Fourier analysis after periodicity and almost periodicity
An extensive bibliography with over 550 references that connects the presented concepts to further topical exploration
Functional Differential Equations: Advances and Applications is an ideal reference for academics and practitioners in applied mathematics, engineering, economics, and physics. The book is also an appropriate textbook for graduate- and PhD-level courses in applied mathematics, differential and difference equations, differential analysis, and dynamics processes.
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Preface v
1 Introduction 1
1.1 Classical and New Types of Functional Equations 2
1.2 Main Directions in the Study of FDE 5
1.3 Metric Spaces and Related Concepts 11
and more