Series: Springer Monographs in Mathematics
1st ed. 2017, XXI, 495 p. 11 illus., 2 illus. incolor.
Hardcover
ISBN 978-3-319-64276-5
* Presents a detailed study of regularity properties of mappings in metric spaces
* Covers mappings with specific structures in Banach and finite dimensional spaces
* Offers new and previously unrecorded applications of regularity theory
* Emphasizes the quantitative character of regularity theory
This monograph offers the first systematic account of (metric) regularity theory in
variational analysis. It presents new developments alongside classical results and
demonstrates the power of the theory through applications to various problems in
analysis and optimization theory.
The origins of metric regularity theory can be traced back to a series of fundamental ideas
and results of nonlinear functional analysis and global analysis centered around problems
of existence and stability of solutions of nonlinear equations. In variational analysis,
regularity theory goes far beyond the classical setting and is also concerned with nondifferentiable
and multi-valued operators. The present volume explores all basic aspects
of the theory, from the most general problems for mappings between metric spaces to
those connected with fairly concrete and important classes of operators acting in Banach
and finite dimensional spaces. Written by a leading expert in the field, the book covers
new and powerful techniques, which have proven to be highly efficient even in classical
settings, and outlines the theoryfs predominantly quantitative character, leading to a
variety of new and unexpected applications.
Variational Analysis of Regular Mappings is aimed at graduate students and researchers in
nonlinear and functional analysis, especially those working in areas close to optimization
and optimal control, and will be suitable to anyone interested in applying new concepts
and ideas to operations research, control engineering and numerical analysis.
Series: SpringerBriefs in Statistics
1st ed. 2017, IX, 113 p. 27 illus. in color.
Softcover
ISBN 978-3-319-64670-1
* Presents and explains the theory of the recursive Bayesian estimation
algorithms for dynamic mixture models
* Develops a unified scheme for constructing the estimation algorithm
of dynamic mixtures with reproducible statistics
* Includes open source programs that can be easily modified or extended by readers
This book provides a general theoretical background for constructing the recursive
Bayesian estimation algorithms for mixture models. It collects the recursive algorithms
for estimating dynamic mixtures of various distributions and brings them in the unified
form, providing a scheme for constructing the estimation algorithm for a mixture of
components modeled by distributions with reproducible statistics. It offers the recursive
estimation of dynamic mixtures, which are free of iterative processes and close to
analytical solutions as much as possible. In addition, these methods can be used online
and simultaneously perform learning, which improves their efficiency during estimation.
The book includes detailed program codes for solving the presented theoretical tasks.
Codes are implemented in the open source platform for engineering computations. The
program codes given serve to illustrate the theory and demonstrate the work of the
included algorithms.
Series: Universitext
1st ed. 2017, XII, 190 p.
Softcover
ISBN 978-3-319-65905-3
* Designed for non-mathematicians, physics students as well
for example, who want to learn about this important area of mathematics
* Well organized and touches upon the main subjects, which offer a
deeper understanding of the orbit structure of an algebraic group
* Painless presentation places the subject within reasonable reach for
mathematics and physics student at the graduate level
Geometric Invariant Theory (GIT) is developed in this text within the context of algebraic
geometry over the real and complex numbers. This sophisticated topic is elegantly
presented with enough background theory included to make the text accessible to
advanced graduate students in mathematics and physics with diverse backgrounds in
algebraic and differential geometry. Throughout the book, examples are emphasized.
Exercises add to the readerfs understanding of the material; most are enhanced with hints.
The exposition is divided into two parts.
The first part, eBackground Theoryf, is organized
as a reference for the rest of the book. It contains two chapters developing material in
complex and real algebraic geometry and algebraic groups that are difficult to find in the
literature. Chapter 1 emphasizes the relationship between the Zariski topology and the
canonical Hausdorff topology of an algebraic variety over the complex numbers. Chapter
2 develops the interaction between Lie groups and algebraic groups.
Part 2, eGeometric Invariant Theoryf
consists of three chapters (3*5). Chapter 3 centers on the Hilbert*
Mumford theorem and contains a complete development of the Kempf*Ness theorem
and Vindbergfs theory. Chapter 4 studies the orbit structure of a reductive algebraic
group on a projective variety emphasizing Kostantfs theory. The final chapter studies the
extension of classical invariant theory to products of classical groups emphasizing recent
applications of the theory to physics.
Series: Lecture Notes in Mathematics, Vol. 2185
1st ed. 2017, XXIII, 201 p. 6 illus.
Softcover
ISBN 978-3-319-63629-0
* Provides a clear and rigorous exposition of Ramanujan's theory of divergent series
* A special chapter is devoted to an algebraic formalism unifying the
most important summation processes
* Only little basic knowledge in analysis is required to read this monograph
The aim of this monograph is to give a detailed exposition of the summation method
that Ramanujan uses in Chapter VI of his second Notebook. This method, presented by
Ramanujan as an application of the Euler-MacLaurin formula, is here extended using
a difference equation in a space of analytic functions. This provides simple proofs of
theorems on the summation of some divergent series. Several examples and applications
are given. For numerical evaluation, a formula in terms of convergent series is provided
by the use of Newton interpolation. The relation with other summation processes such
as those of Borel and Euler is also studied. Finally, in the last chapter, a purely algebraic
theory is developed that unifies all these summation processes. This monograph is aimed
at graduate students and researchers who have a basic knowledge of analytic function
theory.
Series: Lecture Notes in Mathematics, Vol. 2193
1st ed. 2017, XIV, 216 p. 4 illus.
Softcover
ISBN 978-3-319-65183-5
* Provides a survey of current research and new approaches in the
theory of shadowing of approximate trajectories of dynamical systems
* Contains novelty approach for proving hyperbolicity by using the
sifting method of Liao, which is both powerful and self-contained
* A main feature is the direct, straightforward approach to the results,
well written, in an easy-to-read, comfortable pace
* Can be used right after a course of introduction to dynamical systems
or even hyperbolic dynamics
Focusing on the theory of shadowing of approximate trajectories (pseudotrajectories) of
dynamical systems, this book surveys recent progress in establishing relations between
shadowing and such basic notions from the classical theory of structural stability as
hyperbolicity and transversality.
Special attention is given to the study of "quantitative" shadowing properties, such as
Lipschitz shadowing (it is shown that this property is equivalent to structural stability both
for diffeomorphisms and smooth flows), and to the passage to robust shadowing (which
is also equivalent to structural stability in the case of diffeomorphisms, while the situation
becomes more complicated in the case of flows).
Relations between the shadowing property of diffeomorphisms on their chain transitive
sets and the hyperbolicity of such sets are also described.
The book will allow young researchers in the field of dynamical systems to gain a better
understanding of new ideas in the global qualitative theory. It will also be of interest to
specialists in dynamical systems and their applications.
Series: Ecole d'Ete de Probabilites de Saint-Flour, Vol. 2197
1st ed. 2017, XI, 164 p.
Softcover
ISBN 978-3-319-65815-5
* First book-length treatment of large deviations for random graphs,
plus a chapter on exponential random graphs
* Contains a summary of important results from graph limit theory
with complete proofs
* Written in a style for beginning graduate students, self-contained
with essentially no need for background knowledge other than some
amount of graduate probability and analysis
This book addresses the emerging body of literature on the study of rare events in random
graphs and networks. For example, what does a random graph look like if by chance it
has far more triangles than expected* Until recently, probability theory offered no tools
to help answer such questions. Important advances have been made in the last few years,
employing tools from the newly developed theory of graph limits. This work represents
the first book-length treatment of this area, while also exploring the related area of
exponential random graphs. All required results from analysis, combinatorics, graph
theory and classical large deviations theory are developed from scratch, making the text
self-contained and doing away with the need to look up external references. Further, the
book is written in a format and style that are accessible for beginning graduate students in
mathematics and statistics.