Series: Progress in Mathematics, Vol. 289
1st Edition., 2011, VI, 222 p., Hardcover
ISBN: 978-3-0348-0109-6
Due: April 30, 2011
The book contains recent results concerning a functional calulus for n-tuples of not necessarily commuting linear operators and for quaternionic linear operators. These calculi are based on a new theory of hyperholomorphicity for functions with values in a Clifford algebra the so-called slice monogenic functions which are carefully described in the book and, in particular, for functions with values in the algebra of quaternions the so-called slice regular functions.
All the results in the book are new (except for an Appendix on the Riesz-Dunford functional calculus and for a short introduction to Clifford algebras) and appear for the first time organized in a monograph. All the material presented is self contained.
The intended audience consists of researcher in in operator theory, spectral theory, hypercomplex analysis, and mathematical physics as well as graduate and postgraduate students.
Content Level : Research
Keywords : Riesz-Dunford functional calculus - spectral theory - theory of slice hyperholomorphic functions
Related subjects : Analysis
1 Introduction.- 2 Slice monogenic functions.- 3 Functional calculus for n-tuples of operators.- 4 Quaternionic Functional Calculus.- 5 Appendix: The Riesz-Dunford functional calculus.- Bibliography.- Index.
Series: Progress in Nonlinear Differential Equations and Their Applications, Vol. 81
1st Edition., 2011, VIII, 242 p., Hardcover
ISBN: 978-0-8176-8116-6
Due: May 2, 2011
With the goal of establishing a version for partial differential equations (PDEs) of the Aubry?Mather theory of monotone twist maps, Moser and then Bangert studied solutions of their model equations that possessed certain minimality and monotonicity properties. This monograph presents extensions of the Moser?Bangert approach that include solutions of a family of nonlinear elliptic PDEs on R^n and an Allen?Cahn PDE model of phase transitions.
After recalling the relevant Moser?Bangert results, Extensions of Moser?Bangert Theory pursues the rich structure of the set of solutions of a simpler model case, expanding upon the studies of Moser and Bangert to include solutions that merely have local minimality properties. Subsequent chapters build upon the introductory results, making the monograph self contained.
Part I introduces a variational approach involving a renormalized functional to characterize the basic heteroclinic solutions obtained by Bangert. Following that, Parts II and III employ these basic solutions together with constrained minimization methods to construct multitransition heteroclinic and homoclinic solutions on R x T^{n-1} and R^2 x T^{n-2}, respectively, as local minima of the renormalized functional. The work is intended for mathematicians who specialize in partial differential equations and may also be used as a text for a graduate topics course in PDEs.
1 Introduction.- Part I: Basic Solutions.- 2 Function Spaces and the First Renormalized Functional.- 3 The Simplest Heteroclinics.- 4 Heteroclinics in x1 and x2.- 5 More Basic Solutions.- Part II: Shadowing Results.- 6 The Simplest Cases.- 7 The Proof of Theorem 6.8.- 8 k-Transition Solutions for k > 2.- 9 Monotone 2-Transition Solutions.- 10 Monotone Multitransition Solutions.- 11 A Mixed Case.- Part III: Solutions of (PDE) Defined on R^2 x T^{n-2}.- 12 A Class of Strictly 1-Monotone Infinite Transition Solutions of (PDE).- 13 Solutions of (PDE) with Two Transitions in x1 and Heteroclinic Behavior in x2
Series: International Series of Numerical Mathematics, Vol. 159
1st Edition., 2011, 250 p., Hardcover
ISBN: 978-3-0348-0106-5
Due: June 25, 2011
The book provides a comprehensive introduction to compact finite difference methods for solving boundary value ODEs with high accuracy. The corresponding theory is based on exact difference schemes (EDS) from which the implementable truncated difference schemes (TDS) are derived. The TDS are now competitive in terms of efficiency and accuracy with the well-studied numerical algorithms for the solution of initial value ODEs.
Moreover, various a posteriori error estimators are presented which can be used in adaptive algorithms as important building blocks.
The new class of EDS and TDS treated in this book can be considered as further developments of the results presented in the highly respected books of the Russian mathematician A. A. Samarskii. It is shown that the new Samarskii-like techniques open the horizon for the numerical treatment of more complicated problems.
The book contains exercises and the corresponding solutions enabling use as a course text or for self-study.
Researchers and students from numerical methods, engineering and other sciences will find this book provides an accessible and self-contained introduction to numerical methods for solving boundary value ODEs.
Content Level : Research
Related subjects : Dynamical Systems & Differential Equations
Series: Pseudo-Differential Operators, Preliminary entry 6
1st Edition., 2011, Approx. 150 p., Softcover
ISBN: 978-3-0348-0115-7
Due: July 2011
The mathematical notions are presented both on a basic level, making thus the content accessible to a wide audience, and later on, on a more sophisticated level, which can bring students more effectively to the frontier of research<br>Pseudo-differential operators are presented in the perspectives of signal analysis<br>Exercises are included to enhance the use of the book as a textbook<br>Includes introduction to wavelets and to pseudo-differential operators
This is a mathematical book on Fourier analysis. The aim is to present the basic notions and techniques of Fourier analysis in discrete settings, thus opening up this area of mathematics to a wide readership. The book consists of two parts. The first thirteen chapters contain topics related to the finite Fourier transform that can be completely understood by undergraduate students with basic knowledge of linear algebra and calculus. The last ten chapters are built on Hilbert spaces and Fourier series. Notwithstanding the use of the language of measure theory in the second part of the book, much of the contents are accessible to students familiar with an undergraduate course in real analysis. An average graduate student in mathematics should be able to benefit from the entire book and hence equipped to do research in this subject.
Content Level : Graduate
Related subjects : Analysis - Dynamical Systems & Differential Equations - Numerical and Computational Mathematics