Series: Lecture Notes in Mathematics, Vol. 2002
1st Edition., 2010, X, 398 p. 30 illus., 15 in color., Softcover
ISBN: 978-3-642-14257-4
Due: September 2010
Nonautonomous dynamical systems provide a mathematical framework for temporally changing phenomena, where the law of evolution varies in time due to seasonal, modulation, controlling or even random effects. Our goal is to provide an approach to the corresponding geometric theory of nonautonomous discrete dynamical systems in infinite-dimensional spaces by virtue of 2-parameter semigroups (processes). These dynamical systems are generated by implicit difference equations, which explicitly depend on time. Compactness and dissipativity conditions are provided for such problems in order to have attractors using the natural concept of pullback convergence. Concerning a necessary linear theory, our hyperbolicity concept is based on exponential dichotomies and splittings. This concept is in turn used to construct nonautonomous invariant manifolds, so-called fiber bundles, and deduce linearization theorems. The results are illustrated using temporal and full discretizations of evolutionary differential equations.
Content Level ≫ Research
Keywords ≫Exponential dichotomy - Invariant fiber bundles - Nonautonomous difference equations - Nonautonomous dynamical systems - Topological linearization
Related subjects ≫ Dynamical Systems & Diff. Equations
1 Nonautonomous dynamical systems.- 2 Nonautonomous difference equations.- 3 Linear difference equations.- 3 Linear difference equations.- 4 Invariant fiber bundles.- 5 Linearization.
Series: Undergraduate Texts in Mathematics
1st Edition., 2010, XXII, 182 p. 23 illus., Hardcover
ISBN: 978-1-4419-7022-0
Due: October 26, 2010
Presents fundamental mathematics, integers and real numbers, in a way that asks for student participation, while teaching how mathematics is done Provides students with methods and ideas they can use in future courses Primarily for: undergraduates who have studied calculus or linear algebra; mathematics teachers and teachers-in-training; scientists and social scientists who want to strengthen their command of mathematical methods Extra topics in appendices give instructor flexibility
The Art of Proof is designed for a one-semester or two-quarter course. A typical student will have studied calculus (perhaps also linear algebra) with reasonable success. With an artful mixture of chatty style and interesting examples, the student's previous intuitive knowledge is placed on solid intellectual ground. The topics covered include: integers, induction, algorithms, real numbers, rational numbers, modular arithmetic, limits, and uncountable sets. Methods, such as axiom, theorem and proof, are taught while discussing the mathematics rather than in abstract isolation. Some of the proofs are presented in detail, while others (some with hints) may be assigned to the student or presented by the instructor. The authors recommend that the two parts of the book -- Discrete and Continuous -- be given equal attention. The book ends with short essays on further topics suitable for seminar-style presentation by small teams of students, either in class or in a mathematics club setting. These include: continuity, cryptography, groups, complex numbers, ordinal number, and generating functions.
Preface.- Notes for the Student.- Notes for Instructors.- Part I: The Discrete.- 1 Integers.- 2 Natural Numbers and Induction.- 3 Some Points of Logic.- 4 Recursion.- 5 Underlying Notions in Set Theory.- 6 Equivalence Relations and Modular Arithmetic.- 7 Arithmetic in Base Ten.- Part II: The Continuous.- 8 Real Numbers.- 9 Embedding Z in R.- 10. Limits and Other Consequences of Completeness.- 11 Rational and Irrational Numbers.- 12 Decimal Expansions.- 13 Cardinality.- 14 Final Remarks.- Further Topics.- A Continuity and Uniform Continuity.- B Public-Key Cryptography.- C Complex Numbers.- D Groups and Graphs.- E Generating Functions.- F Cardinal Number and Ordinal Number.- G Remarks on Euclidean Geometry.- List of Symbols.- Index.
Series: Springer Monographs in Mathematics
1st Edition., 2010, XII, 672 p. 3 illus., Softcover
ISBN: 978-1-4419-6949-1
Due: October 29, 2010
Opens new possibilities to analysis of statistical functionals and gives an alternative approach to stochastic calculus
Contains new material about the existence and smoothnesss of several nonlinear operators acting between spaces of functions having bounded p-variation
Will appeal to graduate students and researchers working on various aspects of calculus of non-smooth functions
Concrete Functional Calculus focuses primarily on differentiability of some nonlinear operators on functions or pairs of functions, including composition of two functions, and the product integral, taking a matrix- or operator-valued coefficient function into a solution of a system of linear differential equations with the given coefficients. Key features and topics: * Extensive usage of p-variation of functions * Applications to stochastic processes. This work will serve as a thorough reference on its main topics for researchers and graduate students with a background in real analysis and, for Chapter 12, in probability.
Preface.- 1 Introduction and Overview.- 2 Definitions and Basic Properties of Extended Riemann-Stieltjes integrals.- 3 Phi-variation and p-variation; Inequalities for Integrals.- 4 Banach Algebras.- 5 Derivatives and Analyticity in Normed Spaces.- 6 Nemytskii Operators on Function Spaces.- 7 Nemytskii Oerators on Lp Spaces.- 8 Two-Function Composition.- 9 Product Integration.- 10 Nonlinear Differential and Integral Equations.- 11 Fourier Series.- 12 Stochastic Processes and Phi-Variation.- Appendix Nonatomic Measure Spaces.- References.- Subject Index.- Author Index.- Index of Notation.-
Series: Applied Mathematical Sciences, Vol. 172
1st Edition., 2011, X, 344 p., Hardcover
ISBN: 978-1-4419-6869-2
Due: November 2010
Written by a pioneer in dynamical systems
Bridges both abstract and applied areas of dynamical systems
Comprehensive in scope
Over the last four decades there has been extensive development in the theory of dynamical systems. This book starts from the phenomenological point of view reviewing examples. This book aims at a wide audience where the first four chapters have been used for an undergraduate course in Dynamical Systems. Material from the last two chapters and from the appendices has been used quite a lot for master and PhD courses. All chapters are concluded by an exercise section. The book is also directed towards researchers, where one of the challenges is to help applied researchers acquire background for a better understanding of the data that computer simulation or experiment may provide them with the development of the theory. Henk Broer and Floris Takens, professors at the Institute for Mathematics and Computer Science of the University of Groningen, are leaders in the field of dynamical systems. They have published a wealth of scientific papers and books in this area. Both are members of the Royal Netherlands Academy of Arts and Sciences (KNAW).
Content Level ≫ Graduate
Keywords ≫ Bifurcation theory - Chaotic Dynamics - Dynamical Systems - Time series analysis - periodic and quasi-periodic orbits
Related subjects ≫ Applications - Dynamical Systems & Diff. Equations