Series: Operator Theory: Advances and Applications, Vol. 225
2012, 2012, X, 240 p.
Hardcover, ISBN 978-3-0348-0355-7
Due: June 29, 2012
Investigation of the interconnection between probability problems and analysis problems Consideration of the statistical problems using the game theory ideas Construction of special examples instead of well-known existence theorems Generalization and investigation of the notion of integrable operators
In a number of famous works, M. Kac showed that various methods of probability theory can be fruitfully applied to important problems of analysis. The interconnection between probability and analysis also plays a central role in the present book. However, our approach is mainly based on the application of analysis methods (the method of operator identities, integral equations theory, dual systems, integrable equations) to probability theory (Levy processes, M. Kac's problems, the principle of imperceptibility of the boundary, signal theory). The essential part of the book is dedicated to problems of statistical physics (classical and quantum cases). We consider the corresponding statistical problems (Gibbs-type formulas, non-extensive statistical mechanics, Boltzmann equation) from the game point of view (the game between energy and entropy). One chapter is dedicated to the construction of special examples instead of existence theorems (D. Larson's theorem, Ringrose's hypothesis, the Kadison-Singer and Gohberg-Krein questions). We also investigate the Bezoutiant operator. In this context, we do not make the assumption that the Bezoutiant operator is normally solvable, allowing us to investigate the special classes of the entire functions.
Introduction.- 1 Levy processes.- 2 The principle of imperceptibility of the boundary.- 3 Approximation of positive functions.- 4 Optimal prediction and matched filtering.- 5 Effective construction of a class of non-factorable operators.- 6 Comparison of thermodynamic characteristics.- 7 Dual canonical systems and dual matrix string equations.- 8 Integrable operators and Canonical Differential Systems.- 9 The game between energy and entropy.- 10 Inhomogeneous Boltzmann equations.- 11 Operator Bezoutiant and concrete examples.- Comments.- Bibliography.- Glossary.- Index.
Series: Springer Optimization and Its Applications, Vol. 68
2012, 2012, XXX, 770 p. 25 illus., 12 in color.
Hardcover, ISBN 978-1-4614-3497-9
Due: June 30, 2012
Nonlinear Analysis: Stability, Approximation, and Inequalities presents some of the most recent results in the field of nonlinear analysis. Dedicated to Themistocles M. Rassias on the occasion of his 60th birthday, this volume contains 44 articles on various developments in the field, pertaining to subjects such as the stability of functional equations, variational systems, geometric analysis, analytic inequalities, approximation theory and optimization, as well as their applications. Many of the chapters are related to the seminal contributions of Th. M. Rassias and are based upon his initial findings.
This book is well suited to researchers working in nonlinear analysis and approximation theory, differential equations, variational analysis, optimization and their applications, and also to mathematically oriented engineers. It can be used as a valuable source of supplementary material for graduate research and course work.
Part I, Stability
1. Stability problems for functional equations,
2. Fixed point theorems,
3. Critical point theorems,
4. Well posed optimization problems,
5. Parametric dependence of solutions of variational systems,
6. Applications.
Part II, Approximation
7. Nonlinear approximation theory,
8. Interpolation problems,
9. Approximations in partial differential equations
10. Approximations in integral equations,
11. Approximations with polynomials
12. Special functions,
13. Applications to learning theory.
Part III, Inequalities,
14. Inequalities in analysis
15. Inequalities in approximation theory,
16. Variational inequalities and optimization
17. Multivalued variational inequalities and set-valued analysis
18. Applications.
Series: Trends in Mathematics
2012, 2012, Approx. 250 p. 12 illus. in color.
Hardcover, ISBN 978-3-0348-0416-5
Due: July 27, 2012
This book contains survey papers based on the lectures presented at the 3rd International Winter School gModern Problems of Mathematics and Mechanicsh held in January 2010 at the Belarusian State University, Minsk. These lectures are devoted to different problems of modern analysis and its applications. An extended presentation of modern problems of applied analysis will enable the reader to get familiar with new approaches of mostly interdisciplinary character. The results discussed are application oriented and present new insight into applied problems of growing importance such as applications to composite materials, anomalous diffusion, and fluid dynamics.
Introduction.- Kisil, Vladimir V.: Erlangen Program at Large: Brief Outline.- Laurincikas, A.: The Riemann zeta-function: approximation of analytic functions.- Luchko, Yury: Anomalous diffusion: models, their analysis, and interpretation.- Mityushev, Vladimir, V.: R-linear and Riemann-Hilbert problems for multiply connected domains.- Plaksa, S. A.: Commutative algebras associated with classic equations of mathematical physics.- Rogosin, Sergei V.: 2D Free Boundary Value Problems.
Series: Advanced Courses in Mathematics - CRM Barcelona
2013, 2013, 190 p.
Softcover, ISBN 978-3-0348-0407-3
Due: September 30, 2012
Contains two surveys of new results on linear and multilinear analysis
Offers a very nice presentation of the De Giorgi?Moser?Nash result
Contains elegant applications of harmonic analysis to human vision
This book contains an expanded version of lectures delivered by the authors at the CRM in the spring of 2009. It contains four series of lectures. The first one is an application of harmonic analysis and the Heisenberg group to understand human vision. The second and third series of lectures cover some of the main topics on linear and multilinear harmonic analysis. The last one is a nice introduction to a deep result of De Giorgi, Moser and Nash on regularity of elliptic partial differential equations in divergence form.
1 Models of the Visual Cortex in Lie Groups.- 2 Multilinear Calderon?Zygmund
Singular Integrals.- 3 Singular Integrals and Weights.- 4 De Giorgi-Nash-Moser
Theory.