Series: Applied Mathematical Sciences, Vol. 185
2013, X, 367 p. 50 illus. in color.
Hardcover
ISBN 978-1-4614-6994-0
Due: May 31, 2013
.This book fills an important gap in the literature, bridging PDE and dynamical systems approach to stability
Presents a unified treatment of the dynamical systems and functional analysis background of nonlinear stability
Includes illustrative examples and a variety of exercises
This book unifies the dynamical systems and functional analysis approaches to the linear and nonlinear stability of waves. It synthesizes fundamental ideas of the past 20+ years of research, carefully balancing theory and application. The book isolates and methodically develops key ideas by working through illustrative examples that are subsequently synthesized into general principles.
Many of the seminal examples of stability theory, including orbital stability of the KdV solitary wave, and asymptotic stability of viscous shocks for scalar conservation laws, are treated in a textbook fashion for the first time. It presents spectral theory from a dynamical systems and functional analytic point of view, including essential and absolute spectra, and develops general nonlinear stability results for dissipative and Hamiltonian systems. The structure of the linear eigenvalue problem for Hamiltonian systems is carefully developed, including the Krein signature and related stability indices. The Evans function for the detection of point spectra is carefully developed through a series of frameworks of increasing complexity. Applications of the Evans function to the Orientation index, edge bifurcations, and large domain limits are developed through illustrative examples. The book is intended for first or second year graduate students in mathematics, or those with equivalent mathematical maturity. It is highly illustrated and there are many exercises scattered throughout the text that highlight and emphasize the key concepts. Upon completion of the book, the reader will be in an excellent position to understand and contribute to current research in nonlinear stability.
Introduction.- Background material and notation.- Essential and absolute spectra.- Dynamical implications of spectra: dissipative systems.- Dynamical implications of spectra: Hamiltonian systems.- Dynamical implications of spectra: Hamiltonian systems.- Point spectrum: reduction to finite-rank eigenvalue problems.- Point spectrum: linear Hamiltonian systems.- The Evans function for boundary value problems.- The Evans function for Sturm-Liouville operators on the real line.- The Evans function for nth-order operators on the real line.- Index.- References.
Series: Lecture Notes in Mathematics, Vol. 2076
2013, X, 171 p.
Softcover
ISBN 978-3-642-37069-4
Due: May 31, 2013
May serve as the convenient introduction into intensively developing and interesting branches of contemporary nonlinear analysis, theory of differential equations and inclusions and control theory
The presentation is self-contained and directed to a non-specialist Contains interesting applications of the theory in control theory, theory of bifurcations and physics
This book offers a self-contained introduction to the theory of guiding functions methods, which can be used to study the existence of periodic solutions and their bifurcations in ordinary differential equations, differential inclusions and in control theory. It starts with the basic concepts of nonlinear and multivalued analysis, describes the classical aspects of the method of guiding functions, and then presents recent findings only available in the research literature. It describes essential applications in control theory, the theory of bifurcations, and physics, making it a valuable resource not only for gpureh mathematicians, but also for students and researchers working in applied mathematics, the engineering sciences and physics.
1 Background.- 2 MGF in Finite-Dimensional Spaces.- 3 Guiding Functions in Hilbert Spaces.- 4 Second-Order Differential Inclusions.- 5 Nonlinear Fredholm Inclusions.
Series: Lecture Notes in Mathematics, Vol. 2077
2013, VIII, 264 p. 17 illus.
Softcover
ISBN 978-3-319-00356-6
Due: May 31, 2013
Systematic step-by-step approach to asymptotic algorithms that enables the reader to develop an insight to compound asymptotic approximations Presents a novel, well-explained method of meso-scale approximations for bodies with non-periodic multiple perforations Contains illustrations and numerical examples for a range of physically realisable configurations
There are a wide range of applications in physics and structural mechanics involving domains with singular perturbations of the boundary. Examples include perforated domains and bodies with defects of different types. The accurate direct numerical treatment of such problems remains a challenge. Asymptotic approximations offer an alternative, efficient solution.
Greenfs function is considered here as the main object of study rather than a tool for generating solutions of specific boundary value problems. The uniformity of the asymptotic approximations is the principal point of attention. We also show substantial links between Greenfs functions and solutions of boundary value problems for meso-scale structures. Such systems involve a large number of small inclusions, so that a small parameter, the relative size of an inclusion, may compete with a large parameter, represented as an overall number of inclusions.
The main focus of the present text is on two topics: (a) asymptotics of Greenfs kernels in domains with singularly perturbed boundaries and (b) meso-scale asymptotic approximations of physical fields in non-periodic domains with many inclusions. The novel feature of these asymptotic approximations is their uniformity with respect to the independent variables.
This book addresses the needs of mathematicians, physicists and engineers, as well as research students interested in asymptotic analysis and numerical computations for solutions to partial differential equations.
Part I: Greenfs functions in singularly perturbed domains: Uniform asymptotic formulae for Greenfs functions for the Laplacian in domains with small perforations.- Mixed and Neumann boundary conditions for domains with small holes and inclusions. Uniform asymptotics of Greenfs kernels.- Greenfs function for the Dirichlet boundary value problem in a domain with several inclusions.- Numerical simulations based on the asymptotic approximations.- Other examples of asymptotic approximations of Greenfs functions in singularly perturbed domains.- Part II: Greenfs tensors for vector elasticity in bodies with small defects: Greenfs tensor for the Dirichlet boundary value problem in a domain with a single inclusion.- Greenfs tensor in bodies with multiple rigid inclusions.- Greenfs tensor for the mixed boundary value problem in a domain with a small hole.- Part III Meso-scale approximations. Asymptotic treatment of perforated domains without homogenization: Meso-scale approximations for solutions of Dirichlet problems.- Mixed boundary value problems in multiply-perforated domains.
Series: Lecture Notes in Mathematics, Vol. 2079
2013, VIII, 270 p.
Softcover
ISBN 978-3-642-37616-0
Due: May 31, 2013
This work starts with the study of those limit theorems in probability theory for which classical methods do not work. In many cases some form of linearization can help to solve the problem, because the linearized version is simpler. But in order to apply such a method we have to show that the linearization causes a negligible error. The estimation of this error leads to some important large deviation type problems, and the main subject of this work is their investigation. We provide sharp estimates of the tail distribution of multiple integrals with respect to a normalized empirical measure and so-called degenerate U-statistics and also of the supremum of appropriate classes of such quantities. The proofs apply a number of useful techniques of modern probability that enable us to investigate the non-linear functionals of independent random variables.
This lecture note yields insights into these methods, and may also be useful for those who only want some new tools to help them prove limit theorems when standard methods are not a viable option.
1 Introduction.- 2 Motivation of the investigation. Discussion of some problems.- 3 Some estimates about sums of independent random variables.- 4 On the supremum of a nice class of partial sums.- 5 Vapnik? ?ervonenkis classes and L2-dense classes of functions .- 6 The proof of Theorems 4.1 and 4.2 on the supremum of random sums.- 7 The completion of the proof of Theorem 4.1.- 8 Formulation of the main results of this work.- 9 Some results about U-statistics.- 10 MultipleWiener?Ito integrals and their properties.- 11 The diagram formula for products of degenerate U-statistics.- 12 The proof of the diagram formula for U-statistics.- 13 The proof of Theorems 8.3, 8.5 and Example 8.7.- 14 Reduction of the main result in this work.- 15 The strategy of the proof for the main result of this work.- 16 A symmetrization argument.- 17 The proof of the main result.- 18 An overview of the results and a discussion of the literature.
Series: Springer Texts in Statistics, Vol. 103
2013, XII, 430 p. 142 illus., 138 in color.
Hardcover
ISBN 978-1-4614-7137-0
Due: June 28, 2013
.Provides tools for Statistical Learning that are essential for practitioners in science, industry and other fields
Analyses and methods are presented in R
Topics include linear regression, classification, resampling methods, shrinkage approaches, tree-based methods, support vector machines, and clustering
Extensive use of color graphics assist reader
An Introduction to Statistical Learning provides an accessible overview of the field of statistical learning, an essential toolset for making sense of the vast and complex data sets that have emerged in fields ranging from biology to finance to marketing to astrophysics in the past twenty years. This book presents some of the most important modeling and prediction techniques, along with relevant applications. Topics include linear regression, classification, resampling methods, shrinkage approaches, tree-based methods, support vector machines, clustering, and more. Color graphics and real-world examples are used to illustrate the methods presented. Since the goal of this textbook is to facilitate the use of these statistical learning techniques by practitioners in science, industry, and other fields, each chapter contains a tutorial on implementing the analyses and methods presented in R, an extremely popular open source statistical software platform.
Two of the authors co-wrote The Elements of Statistical Learning (Hastie, Tibshirani and Friedman, 2nd edition 2009), a popular reference book for statistics and machine learning researchers. An Introduction to Statistical Learning covers many of the same topics, but at a level accessible to a much broader audience. This book is targeted at statisticians and non-statisticians alike who wish to use cutting-edge statistical learning techniques to analyze their data. The text assumes only a previous course in linear regression and no knowledge of matrix algebra.
Introduction.- Statistical Learning.- Linear Regression.- Classification.- Resampling Methods.- Linear Model Selection and Regularization.- Moving Beyond Linearity.- Tree-Based Methods.- Support Vector Machines.- Unsupervised Learning.- Index.