Barreira, Luis, Valls, Claudia, Universidade de Lisboa, Lisbon, Portugal

Dynamical Systems by Example

Due 2019-06-19
1st ed. 2019, VIII, 214 p.
50 illus.
Printed book
Hardcover
ISBN 978-3-030-15914-6

Excellent companion to a dynamical systems textbook

Includes large variety of problems and accompanying solutions

Problems range from simple to complex

Emphasis on dynamical systems with discrete time

This book comprises an impressive collection of problems that cover a variety of carefully
selected topics on the core of the theory of dynamical systems. Aimed at the graduate/upper
undergraduate level, the emphasis is on dynamical systems with discrete time. In addition to
the basic theory, the topics include topological, low-dimensional, hyperbolic and symbolic
dynamics, as well as basic ergodic theory. As in other areas of mathematics, one can gain the
first working knowledge of a topic by solving selected problems. It is rare to find large
collections of problems in an advanced field of study much less to discover accompanying
detailed solutions. This text fills a gap and can be used as a strong companion to an
analogous dynamical systems textbook such as the authors’ own Dynamical Systems
(Universitext, Springer) or another text designed for a one- or two-semesteradvanced
undergraduate/graduate course. The book is also intended for independent study. Problems
often begin with specific cases and then move on to general results, following a natural path
of learning. They are also well-graded in terms of increasing the challenge to the reader.
Anyone who works through the theory and problems in Part I will have acquired the
background and techniques needed to do advanced studies in this area. Part II includes
complete solutions to every problem given in Part I with each conveniently restated. Beyond
basic prerequisites from linear algebra, differential and integral calculus, and complex analysis
and topology, in each chapter the authors recall the notions and results (without proofs) that
are necessary to treat the challenges set for that chapter, thus making the text self-contained

Lerner, Nicolas, Sorbonne Universite, Paris, France

Carleman Inequalities An Introduction and More

Due 2019-06-14
1st ed. 2019, XXI, 520 p. 22
illus., 11 illus. in color.
Printed book
Hardcover
ISBN 978-3-030-15992-4

Provides an accessible introduction to Carleman estimates

Includes recent results, examples, and applications

Written by an expert in the field

Over the past 25 years, Carleman estimates have become an essential tool in several areas
related to partial differential equations such as control theory, inverse problems, or fluid
mechanics. This book provides a detailed exposition of the basic techniques of Carleman
Inequalities, driven by applications to various questions of unique continuation. Beginning with
an elementary introduction to the topic, including examples accessible to readers without prior
knowledge of advanced mathematics, the book's first five chapters contain a thorough
exposition of the most classical results, such as Calderon's and Hormander's theorems. Later
chapters explore a selection of results of the last four decades around the themes of
continuation for elliptic equations, with the Jerison-Kenig estimates for strong unique
continuation, counterexamples to Cauchy uniqueness of Cohen and Alinhac & Baouendi,
operators with partially analytic coefficients with intermediate results between Holmgren's and
Hormander's uniqueness theorems, Wolff's modification of Carleman's method, conditional
pseudo-convexity, and more. With examples and special cases motivating the general theory, as
well as appendices on mathematical background, this monograph provides an accessible, selfcontained
basic reference on the subject, including a selection of the developments of the past
thirty years in unique continuation.

Armstrong, S., Kuusi, T., Mourrat, J.-C.,
Courant Institute of Mathematical Sciences, New York, NY, USA

Quantitative Stochastic Homogenization and LargeScale Regularity

Due 2019-06-29
1st ed. 2019, Approx. 540 p.
Printed book
Hardcover
ISBN 978-3-030-15544-5

First book focusing on stochastic (as opposed to periodic) homogenization,
presenting the quantitative theory, and exposing the renormalization
approach to stochastic homogenization

Collects the essential ideas and results of the theory of quantitative
stochastic homogenization, including the optimal error estimates and scaling
limit of the first-order correctors to a variant of the Gaussian free field

Proves for the first time important new results, including optimal estimates
for the first-order correctors in negative Sobolev spaces, optimal error
estimates for Dirichlet and Neumann problems and the optimal quantitative
description of the parabolic and elliptic Green functions

Contains an original con

The focus of this book is the large-scale statistical behavior of solutions of divergence-form
elliptic equations with random coefficients, which is closely related to the long-time
asymptotics of reversible diffusions in random media and other basic models of statistical
physics. Of particular interest is the quantification of the rate at which solutions converge to
those of the limiting, homogenized equation in the regime of large scale separation, and the
description of their fluctuations around this limit. This self-contained presentation gives a
complete account of the essential ideas and fundamental results of this new theory of
quantitative stochastic homogenization, including the latest research on the topic, and is
supplemented with many new results. The book serves as an introduction to the subject for
advanced graduate students and researchers working in partial differential equations, statistical
physics, probability and related fields, as well as a comprehensive reference for experts in
homogenization

Harumi Hattori (West Virginia University, USA)

Partial Differential Equations, 2nd Edition
Methods, Applications and Theories

ISBN: 978-981-12-0223-0 (hardcover)

Description

This is an introductory level textbook for partial differential equations (PDE's). It is suitable for a one-semester undergraduate level or two-semester graduate level course in PDE's or applied mathematics. This volume is application-oriented and rich in examples. Going through these examples, the reader is able to easily grasp the basics of PDE's.

Chapters One to Five are organized to aid understanding of the basic PDE's. They include the first-order equations and the three fundamental second-order equations, i.e. the heat, wave and Laplace equations. Through these equations, we learn the types of problems, how we pose the problems, and the methods of solutions such as the separation of variables and the method of characteristics. The modeling aspects are explained as well. The methods introduced in earlier chapters are developed further in Chapters Six to Twelve. They include the Fourier series, the Fourier and the Laplace transforms, and the Green's functions. Equations in higher dimensions are also discussed in detail. In this second edition, a new chapter is added and numerous improvements have been made including the reorganization of some chapters. Extensions of nonlinear equations treated in earlier chapters are also discussed.

Partial differential equations are becoming a core subject in Engineering and the Sciences. This textbook will greatly benefit those studying in these subjects by covering basic and advanced topics in PDE based on applications.