Series: Universitext
1st ed. 2016, X, 240 p.
Softcover
ISBN 978-3-319-34071-5
* Presents a unique collection of approximation methods in various
metrics
* Explains the essential aspects of each method in detail
* Includes many exercises with solutions as well as bibliographical
notes
This book presents a wide range of well-known and less common methods used for
estimating the accuracy of probabilistic approximations, including the Esseen type
inversion formulas, the Stein method as well as the methods of convolutions and triangle
function. Emphasising the correct usage of the methods presented, each step required
for the proofs is examined in detail. As a result, this textbook provides valuable tools for
proving approximation theorems.
While Approximation Methods in Probability Theory will appeal to everyone interested in
limit theorems of probability theory, the book is particularly aimed at graduate students
who have completed a standard intermediate course in probability theory. Furthermore,
experienced researchers wanting to enlarge their toolkit will also find this book useful.
Series: SpringerBriefs in Mathematics
1st ed. 2016, IX, 128 p. 1 illus. in color.
Softcover
ISBN 978-3-319-34218-4
* Provides an easy-to-read overview of entropy methods for diffusive
equations
* The first book to summarize entropy methods for cross-diffusion
systems
* The majority of the content should be accessible for advanced
undergraduate and graduate students
This book presents a range of entropy methods for diffusive PDEs devised by many
researchers in the course of the past few decades, which allow us to understand the
qualitative behavior of solutions to diffusive equations (and Markov diffusion processes).
Applications include the large-time asymptotics of solutions, the derivation of convex
Sobolev inequalities, the existence and uniqueness of weak solutions, and the analysis
of discrete and geometric structures of the PDEs. The purpose of the book is to provide
readers an introduction to selected entropy methods that can be found in the research
literature. In order to highlight the core concepts, the results are not stated in the widest
generality and most of the arguments are only formal (in the sense that the functional
setting is not specified or sufficient regularity is supposed). The text is also suitable for
advanced master and PhD students and could serve as a textbook for special courses and
seminars..
Series: Springer Series in Statistics
1st ed. 2016, XII, 488 p. 77 illus.
Hardcover
ISBN 978-1-4939-4018-9
* Presents a complete and detailed exposition on statistical analysis
of shapes that includes appendices, background material, and
exercises, making this text a self-contained reference
* Addresses and explores the next generation of shape analysis
* Focuses on providing a working knowledge of a broad range of
relevant material, foregoing in-depth technical details and elaborate
mathematical explanations
This textbook for courses on function data analysis and shape data analysis describes
how to define, compare, and mathematically represent shapes, with a focus on statistical
modeling and inference. It is aimed at graduate students in analysis in statistics,
engineering, applied mathematics, neuroscience, biology, bioinformatics, and other
related areas. The interdisciplinary nature of the broad range of ideas covered?from
introductory theory to algorithmic implementations and some statistical case studies
?is meant to familiarize graduate students with an array of tools that are relevant
in developing computational solutions for shape and related analyses. These tools,
gleaned from geometry, algebra, statistics, and computational science, are traditionally
scattered across different courses, departments, and disciplines; Functional and Shape
Data Analysis offers a unified, comprehensive solution by integrating the registration
problem into shape analysis, better preparing graduate students for handling future
scientific challenges.
Recently, a data-driven and application-oriented focus on shape analysis has been
trending. This text offers a self-contained treatment of this new generation of methods
in shape analysis of curves. Its main focus is shape analysis of functions and curves?in
one, two, and higher dimensions?both closed and open. It develops elegant Riemannian
frameworks that provide both quantification of shape differences and registration
of curves at the same time. Additionally, these methods are used for statistically
summarizing given curve data, performing dimension reduction, and modeling observed
variability. It is recommended that the reader have a background in calculus, linear
algebra, numerical analysis, and computation
Series: Progress in Mathematics, Vol. 318
2016, X, 446 p. 65 illus., 29 illus. in
color.
Hardcover
ISBN 978-3-319-33875-0
* Provides an accessible introduction to geometric scattering theory
and the theory of resonances
* Discusses important developments such as resonance counting,
analysis of the Selberg zeta function, and the Poisson formula
* New chapters cover resolvent estimates, wave propagation, and
Naudfs proof of a spectral gap for convex hyperbolic surfaces
* Makes use of new techniques for resonance plotting that more
clearly illustrate existing results of resonance distribution
This text introduces geometric spectral theory in the context of infinite-area Riemann
surfaces, providing a comprehensive account of the most recent developments in the
field. For the second edition the context has been extended to general surfaces with
hyperbolic ends, which provides a natural setting for development of the spectral theory
while still keeping technical difficulties to a minimum. All of the material from the first
edition is included and updated, and new sections have been added.
Topics covered include an introduction to the geometry of hyperbolic surfaces, analysis
of the resolvent of the Laplacian, scattering theory, resonances and scattering poles,
the Selberg zeta function, the Poisson formula, distribution of resonances, the inverse
scattering problem, Patterson-Sullivan theory, and the dynamical approach to the zeta
function. The new sections cover the latest developments in the field, including the
spectral gap, resonance asymptotics near the critical line, and sharp geometric constants
for resonance bounds. A new chapter introduces recently developed techniques for
resonance calculation that illuminate the existing results and conjectures on resonance
distribution.
The spectral theory of hyperbolic surfaces is a point of intersection for a great variety of
areas, including quantum physics, discrete groups, differential geometry, number theory,
complex analysis, and ergodic theory. This book will serve as a valuable resource for
graduate students and researchers from these and other related fields.
Review of the first edition:
"The exposition is very clear and thorough, and essentially self-contained; the proofs are
detailed...The book gathers together some material which is not always easily available in
the literature...To conclude, the book is certainly at a level accessible to graduate students
and researchers from a rather large range of fields. Clearly, the reader...would certainly
benefit greatly from it." (Colin Guillarmou, Mathematical Reviews, Issue 2008 h)
Series: SpringerBriefs in Mathematics
1st ed. 2016, XV, 110 p.
Printed book
Hardcover
ISBN 978-3-319-38932-5
* Details key elements of the regularity theory for mean-field games
* Presents a series of techniques for well-posedness
* Explores stationary and time-dependent MFGs through a series of apriori
estimates
Beginning with a concise introduction to the theory of mean-field games (MFGs), this
book presents the key elements of the regularity theory for MFGs. It then introduces
a series of techniques for well-posedness in the context of mean-field problems,
including stationary and time-dependent MFGs, subquadratic and superquadratic MFG
formulations, and distinct classes of mean-field couplings. It also explores stationary and
time-dependent MFGs through a series of a-priori estimates for solutions of the Hamilton-
Jacobi and Fokker-Planck equation. It shows sophisticated a-priori systems derived using a
range of analytical techniques, and builds on previous results to explain classical solutions.
The final chapter discusses the potential applications, models and natural extensions of
MFGs. As MFGs connect common problems in pure mathematics, engineering, economics
and data management, this book is a valuable resource for researchers and graduate
students in these fields.