1st ed. 2017, XIV, 456 p. 10 illus.
Hardcover
ISBN 978-3-319-51717-9
* Provides new and striking material in group theory and model theory
* Presents a balanced mix of survey papers that will appeal to a wide
audience ranging from beginning graduate students to experienced
researchers
* Includes research papers that reflect the most current research in the
field
This volume focuses on group theory and model theory with a particular emphasis on the
interplay of the two areas. The survey papers provide an overview of the developments
across group, module, and model theory while the research papers present the most
recent study in those same areas. With introductory sections that make the topics easily
accessible to students, the papers in this volume will appeal to beginning graduate
students and experienced researchers alike. As a whole, this book offers a cross-section
view of the areas in group, module, and model theory, covering topics such as DP-minimal
groups, Abelian groups, countable 1-transitive trees, and module approximations.
The papers in this book are the proceedings of the conference gNew Pathways between
Group Theory and Model Theory,h which took place February 1-4, 2016, in Mulheim an
der Ruhr, Germany, in honor of the editorsf colleague Rudiger Gobel. This publication
is dedicated to Professor Gobel, who passed away in 2014. He was one of the leading
experts in Abelian group theory.
Series: Springer Monographs in Mathematics
1st ed. 2017, VIII, 435 p.
Hardcover
ISBN 978-3-319-53042-0
* Focuses on a coherent representation of the main approaches to
analyze the dynamics of cellular automata
* Emphasises on classification strategies and methods that allow
a rigorous analysis and precise understanding of their long term
dynamics
* Provides basic information to make the book accessible also
to physicists, chemists and biologists
This book focuses on a coherent representation of the main approaches to analyze the
dynamics of cellular automata. Cellular automata are an inevitable tool in mathematical
modeling. In contrast to classical modeling approaches as partial differential equations,
cellular automata are straightforward to simulate but hard to analyze. In this book we
present a review of approaches and theories that allow the reader to understand the
behavior of cellular automata beyond simulations.
The first part consists of an introduction of cellular automata on Cayley graphs, and their
characterization via the fundamental Cutis-Hedlund-Lyndon theorems in the context of
different topological concepts (Cantor, Besicovitch and Weyl topology). The second part
focuses on classification results: What classification follows from topological concepts
(Hurley classification), Lyapunov stability (Gilman classification), and the theory of
formal languages and grammars (K*rka classification). These classifications suggest to
cluster cellular automata, similar to the classification of partial differential equations
in hyperbolic, parabolic and elliptic equations. This part of the book culminates in
the question, whether properties of cellular automata are decidable. Surjectivity, and
injectivity are examined, and the seminal Garden of Eden theorems are discussed.
The third part focuses on the analysis of cellular automata that inherit distinct properties,
often based on mathematical modeling of biological, physical or chemical systems.
Linearity is a concept that allows to define self-similar limit sets. Models for particle
motion show how to bridge the gap between cellular automata and partial differential
equations (HPP model and ultradiscrete limit). Pattern formation is related to linear
cellular automata, to the Bar-Yam model for Turing pattern, and Greenberg-Hastings
automata for excitable media. Also models for sandpiles, the dynamics of infectious
diseases and evolution of predator-prey systems are discussed. Mathematicians find
an overview about theory and tools for the analysis of cellular automata. The book
contains an appendix introducing basic mathematical techniques and notations, such
that also physicists, chemists and biologists interested in cellular automata beyond pure
simulations will benefit.
Series: Fields Institute Monographs, Vol. 35
1st ed. 2017, VIII, 352 p. 37 illus.
Hardcover
ISBN 978-1-4939-6941-8
* Covers a variety of different facets of free probability, giving a flavor
of the breadth of the subject
* Features exercises scattered throughout the text
* Showcases basic ideas and results in order to focus on their relation
This volume opens the world of free probability to a wide variety of readers. From its roots
in the theory of operator algebras, free probability has intertwined with non-crossing
partitions, random matrices, applications in wireless communications, representation
theory of large groups, quantum groups, the invariant subspace problem, large
deviations, subfactors, and beyond. This book puts a special emphasis on the relation
of free probability to random matrices, but also touches upon the operator algebraic,
combinatorial, and analytic aspects of the theory.
The book serves as a combination textbook/research monograph, with self-contained
chapters, exercises scattered throughout the text, and coverage of important ongoing
progress of the theory. It will appeal to graduate students and all mathematicians
interested in random matrices and free probability from the point of view of operator
algebras, combinatorics, analytic functions, or applications in engineering and statistical
physics.
Series: Lecture Notes in Mathematics, Vol. 2183
1st ed. 2017, Approx. 200 p.
Softcover
ISBN 978-3-319-54207-2
* Contains a gentle introduction to Monge-Ampere equationsOffers a
starting point to learn the theory of viscosity solutions (see appendix
of part 2)Provides up-to-date research directions in the fields of
Hamilton-Jacobi and linearized Monge-Ampere equations
Consisting of two parts, the first part of this volume is an essentially self-contained
exposition of the geometric aspects of local and global regularity theory for the Monge*
Ampere and linearized Monge*Ampere equations. As an application, we solve the second
boundary value problem of the prescribed affine mean curvature equation, which can be
viewed as a coupling of the latter two equations. Of interest in its own right, the linearized
Monge*Ampere equation also has deep connections and applications in analysis, fluid
mechanics and geometry, including the semi-geostrophic equations in atmospheric flows,
the affine maximal surface equation in affine geometry and the problem of finding Kahler
metrics of constant scalar curvature in complex geometry.
Among other topics, the second part provides a thorough exposition of the large time
behavior and discounted approximation of Hamilton*Jacobi equations, which have
received much attention in the last two decades, and a new approach to the subject, the
nonlinear adjoint method, is introduced. The appendix offers a short introduction to the
theory of viscosity solutions of first-order Hamilton*Jacobi equations.
Series: Frontiers in Mathematics
1st ed. 2017, X, 134 p. 43 illus.
Softcover
ISBN 978-3-319-51743-8
* Provides a lucid exposition of modern complex theory of differential
equations, which has its origins in the works of Jean Leray
* Includes more than 40 figures and numerous examples and exercises
* Defines all the required terms along the way, making the book selfcontained
This book discusses the complex theory of differential equations or more precisely, the
theory of differential equations on complex-analytic manifolds.
Although the theory of differential equations on real manifolds is well known * it is
described in thousands of papers and its usefulness requires no comments or explanations
* to date specialists on differential equations have not focused on the complex theory
of partial differential equations. However, as well as being remarkably beautiful, this
theory can be used to solve a number of problems in real theory, for instance, the Poincare
balayage problem and the mother body problem in geophysics.
The monograph does not require readers to be familiar with advanced notions in complex
analysis, differential equations, or topology. With its numerous examples and exercises,
it appeals to advanced undergraduate and graduate students, and also to researchers
wanting to familiarize themselves with the subject.