Series: Springer Series in Computational Mathematics, Vol. 18
2017, XIV, 455 p. 55 illus., 15 illus. in color.
Hardcover
ISBN 978-3-662-54960-5
* Provides a detailed analysis of both the continuous boundary value
problems and the discretisation methods
* Includes numerous exercises for readers to test their understanding
of the text
* Discusses in detail the topics regularity of the solution of a boundary
value problem and eigenvalue problems
This book simultaneously presents the theory and the numerical treatment of elliptic
boundary value problems, since an understanding of the theory is necessary for the
numerical analysis of the discretisation. It first discusses the Laplace equation and its
finite difference discretisation before addressing the general linear differential equation
of second order. The variational formulation together with the necessary background
from functional analysis provides the basis for the Galerkin and finite-element methods,
which are explored in detail. A more advanced chapter leads the reader to the theory of
regularity. Individual chapters are devoted to singularly perturbed as well as to elliptic
eigenvalue problems. The book also presents the Stokes problem and its discretisation as
an example of a saddle-point problem taking into account its relevance to applications in
fluid dynamics.
Series: Universitext
1st ed. 2017, XIV, 508 p. 1 illus.
Softcover
ISBN 978-3-319-58645-8
* Covers material for about 40 hours of lectures for everybody working
in the area of stochastic analysis, from beginning graduate students
to experts in the field
* Presents exercise material to fill potential exams and homework
assignments, and gives hints that can lead to solutions.
* Provides all the necessary background from functional analysis and
the theory of PDEs
* Gives all necessary tools to understand the proofs of existing
theorems about SPDEs (from other sources) and even to formulate and prove a few new ones
Taking readers with a basic knowledge of probability and real analysis to the frontiers of a
very active research discipline, this textbook provides all the necessary background from
functional analysis and the theory of PDEs. It covers the main types of equations (elliptic,
hyperbolic and parabolic) and discusses different types of random forcing. The objective is
to give the reader the necessary tools to understand the proofs of existing theorems about
SPDEs (from other sources) and perhaps even to formulate and prove a few new ones.
Most of the material could be covered in about 40 hours of lectures, as long as not too
much time is spent on the general discussion of stochastic analysis in infinite dimensions.
As the subject of SPDEs is currently making the transition from the research level to that of
a graduate or even undergraduate course, the book attempts to present enough exercise
material to fill potential exams and homework assignments. Exercises appear throughout
and are usually directly connected to the material discussed at a particular place in the
text. The questions usually ask to verify something, so that the reader already knows the
answer and, if pressed for time, can move on. Accordingly, no solutions are provided, but
there are often hints on how to proceed.
The book will be of interest to everybody working in the area of stochastic analysis, from
beginning graduate students to experts in the field.
1st ed. 2017, XV, 122 p.
Softcover
ISBN 978-3-319-57912-2
* Features a step-by-step presentation of results of Goldbach's
conjecture
* Accessible to those that have mastered classical number theory and
fundamental notions of mathematical analysis
* Equips readers with an extensive bibliography
Important results surrounding the proof of Goldbach's ternary conjecture are presented
in this book. Beginning with an historical perspective along with an overview of essential
lemmas and theorems, this monograph moves on to a detailed proof of Vinogradov's
theorem. The principles of the Hardy-Littlewood circle method are outlined and applied
to Goldbach's ternary conjecture. New results due to H. Maier and the author on
Vinogradov's theorem are proved under the assumption of the Riemann hypothesis. The
final chapter discusses an approach to Goldbach's conjecture through theorems by L. G.
Schnirelmann. This book concludes with an Appendix featuring a sketch of H. Helfgott's
proof of Goldbach's ternary conjecture. The Appendix also presents some biographical
remarks of mathematicians whose research has played a seminal role on the Goldbach
ternary problem.
The author's step-by-step approach makes this book accessible to those that have
mastered classical number theory and fundamental notions of mathematical analysis.
This book will be particularly useful to graduate students and mathematicians in analytic
number theory, approximation theory as well as to researchers working on Goldbach's
problem.
Series: Universitext
1st ed. 2017, XVIII, 235 p. 8 illus.
Softcover
ISBN 978-3-319-59799-7
* Provides a self-contained introduction to Nevanlinna theory and
normal families
* Collects recent results on first-order and Painleve differential
equations, and 2D Hamiltonian systems
* Presents applications of Nevanlinna theory to the value distribution
of differential polynomials and uniqueness of meromorphic functions
This book offers a modern introduction to Nevanlinna theory and its intricate relation
to the theory of normal families, algebraic functions, asymptotic series, and algebraic
differential equations.
Following a comprehensive treatment of Nevanlinna’s theory of value distribution,
the author presents advances made since Hayman’s work on the value distribution of
differential polynomials and illustrates how value- and pair-sharing problems are linked
to algebraic curves and Briot*Bouquet differential equations. In addition to discussing
classical applications of Nevanlinna theory, the book outlines state-of-the-art research,
such as the effect of the Yosida and Zalcman*Pang method of re-scaling to algebraic
differential equations, and presents the Painleve*Yosida theorem, which relates Painleve
transcendents and solutions to selected 2D Hamiltonian systems to certain Yosida classes
of meromorphic functions.
Aimed at graduate students interested in recent developments in the field and
researchers working on related problems, Nevanlinna Theory, Normal Families, and
Algebraic Differential Equations will also be of interest to complex analysts looking for an
introduction to various topics in the subject area. With examples, exercises and proofs
seamlessly intertwined with the body of the text, this book is particularly suitable for the
more advanced reader.
Series: Springer INdAM Series, Vol. 19
1st ed. 2017, X, 456 p. 18 illus., 5 illus. in
color.
Hardcover
ISBN 978-3-319-58970-1
* Presents lecture notes prepared by leading experts in the field
* Fully refereed original papers
* Covers a broad range of subjects
Lie theory is a mathematical framework for encoding the concept of symmetries of
a problem, and was the central theme of an INdAM intensive research period at the
Centro de Giorgi in Pisa, Italy, in the academic year 2014-2015. This book gathers the key
outcomes of this period, addressing topics such as: structure and representation theory of
vertex algebras, Lie algebras and superalgebras, as well as hyperplane arrangements with
different approaches, ranging from geometry and topology to combinatorics.
Series: Compact Textbooks in Mathematics
1st ed. 2017, XIII, 177 p. 28 illus., 1 illus. in color.
Softcover
ISBN 978-3-319-59432-3
* Written from the ground up to provide a comprehensive introduction
to wavelet analysis
* Provides an accessible working knowledge of the analytical and
computational methods of wavelet analysis
* Concise, yet highly accessible to both pure and applied
mathematicians, as well as scientists and engineers
This book provides a systematic exposition of the basic ideas and results of wavelet
analysis suitable for mathematicians, scientists, and engineers alike. The primary goal
of this text is to show how different types of wavelets can be constructed, illustrate why
they are such powerful tools in mathematical analysis, and demonstrate their use in
applications. It also develops the required analytical knowledge and skills on the part of
the reader, rather than focus on the importance of more abstract formulation with full
mathematical rigor.
These notes differs from many textbooks with similar titles in that a major emphasis
is placed on the thorough development of the underlying theory before introducing
applications and modern topics such as fractional Fourier transforms, windowed canonical
transforms, fractional wavelet transforms, fast wavelet transforms, spline wavelets,
Daubechies wavelets, harmonic wavelets and non-uniform wavelets.
The selection, arrangement, and presentation of the material in these lecture notes
have carefully been made based on the authors’ teaching, research and professional
experience. Drafts of these lecture notes have been used successfully by the authors in
their own courses on wavelet transforms and their applications at the University of Texas
Pan-American and the University of Kashmir in India.